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Metamath Proof Explorer |
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| Ref | Description |
| idi 1 | (_Note_: This inference r... |
| a1ii 2 | (_Note_: This inference r... |
| mp2 9 | A double modus ponens infe... |
| mp2b 10 | A double modus ponens infe... |
| a1i 11 | Inference introducing an a... |
| 2a1i 12 | Inference introducing two ... |
| mp1i 13 | Inference detaching an ant... |
| a2i 14 | Inference distributing an ... |
| mpd 15 | A modus ponens deduction. ... |
| imim2i 16 | Inference adding common an... |
| syl 17 | An inference version of th... |
| 3syl 18 | Inference chaining two syl... |
| 4syl 19 | Inference chaining three s... |
| mpi 20 | A nested modus ponens infe... |
| mpisyl 21 | A syllogism combined with ... |
| id 22 | Principle of identity. Th... |
| idALT 23 | Alternate proof of ~ id . ... |
| idd 24 | Principle of identity ~ id... |
| a1d 25 | Deduction introducing an e... |
| 2a1d 26 | Deduction introducing two ... |
| a1i13 27 | Add two antecedents to a w... |
| 2a1 28 | A double form of ~ ax-1 . ... |
| a2d 29 | Deduction distributing an ... |
| sylcom 30 | Syllogism inference with c... |
| syl5com 31 | Syllogism inference with c... |
| com12 32 | Inference that swaps (comm... |
| syl11 33 | A syllogism inference. Co... |
| syl5 34 | A syllogism rule of infere... |
| syl6 35 | A syllogism rule of infere... |
| syl56 36 | Combine ~ syl5 and ~ syl6 ... |
| syl6com 37 | Syllogism inference with c... |
| mpcom 38 | Modus ponens inference wit... |
| syli 39 | Syllogism inference with c... |
| syl2im 40 | Replace two antecedents. ... |
| syl2imc 41 | A commuted version of ~ sy... |
| pm2.27 42 | This theorem, sometimes ca... |
| mpdd 43 | A nested modus ponens dedu... |
| mpid 44 | A nested modus ponens dedu... |
| mpdi 45 | A nested modus ponens dedu... |
| mpii 46 | A doubly nested modus pone... |
| syld 47 | Syllogism deduction. Dedu... |
| syldc 48 | Syllogism deduction. Comm... |
| mp2d 49 | A double modus ponens dedu... |
| a1dd 50 | Double deduction introduci... |
| 2a1dd 51 | Double deduction introduci... |
| pm2.43i 52 | Inference absorbing redund... |
| pm2.43d 53 | Deduction absorbing redund... |
| pm2.43a 54 | Inference absorbing redund... |
| pm2.43b 55 | Inference absorbing redund... |
| pm2.43 56 | Absorption of redundant an... |
| imim2d 57 | Deduction adding nested an... |
| imim2 58 | A closed form of syllogism... |
| embantd 59 | Deduction embedding an ant... |
| 3syld 60 | Triple syllogism deduction... |
| sylsyld 61 | A double syllogism inferen... |
| imim12i 62 | Inference joining two impl... |
| imim1i 63 | Inference adding common co... |
| imim3i 64 | Inference adding three nes... |
| sylc 65 | A syllogism inference comb... |
| syl3c 66 | A syllogism inference comb... |
| syl6mpi 67 | A syllogism inference. (C... |
| mpsyl 68 | Modus ponens combined with... |
| mpsylsyld 69 | Modus ponens combined with... |
| syl6c 70 | Inference combining ~ syl6... |
| syl6ci 71 | A syllogism inference comb... |
| syldd 72 | Nested syllogism deduction... |
| syl5d 73 | A nested syllogism deducti... |
| syl7 74 | A syllogism rule of infere... |
| syl6d 75 | A nested syllogism deducti... |
| syl8 76 | A syllogism rule of infere... |
| syl9 77 | A nested syllogism inferen... |
| syl9r 78 | A nested syllogism inferen... |
| syl10 79 | A nested syllogism inferen... |
| a1ddd 80 | Triple deduction introduci... |
| imim12d 81 | Deduction combining antece... |
| imim1d 82 | Deduction adding nested co... |
| imim1 83 | A closed form of syllogism... |
| pm2.83 84 | Theorem *2.83 of [Whitehea... |
| peirceroll 85 | Over minimal implicational... |
| com23 86 | Commutation of antecedents... |
| com3r 87 | Commutation of antecedents... |
| com13 88 | Commutation of antecedents... |
| com3l 89 | Commutation of antecedents... |
| pm2.04 90 | Swap antecedents. Theorem... |
| com34 91 | Commutation of antecedents... |
| com4l 92 | Commutation of antecedents... |
| com4t 93 | Commutation of antecedents... |
| com4r 94 | Commutation of antecedents... |
| com24 95 | Commutation of antecedents... |
| com14 96 | Commutation of antecedents... |
| com45 97 | Commutation of antecedents... |
| com35 98 | Commutation of antecedents... |
| com25 99 | Commutation of antecedents... |
| com5l 100 | Commutation of antecedents... |
| com15 101 | Commutation of antecedents... |
| com52l 102 | Commutation of antecedents... |
| com52r 103 | Commutation of antecedents... |
| com5r 104 | Commutation of antecedents... |
| imim12 105 | Closed form of ~ imim12i a... |
| jarr 106 | Elimination of a nested an... |
| jarri 107 | Inference associated with ... |
| pm2.86d 108 | Deduction associated with ... |
| pm2.86 109 | Converse of Axiom ~ ax-2 .... |
| pm2.86i 110 | Inference associated with ... |
| loolin 111 | The Linearity Axiom of the... |
| loowoz 112 | An alternate for the Linea... |
| con4 113 | Alias for ~ ax-3 to be use... |
| con4i 114 | Inference associated with ... |
| con4d 115 | Deduction associated with ... |
| mt4 116 | The rule of modus tollens.... |
| mt4d 117 | Modus tollens deduction. ... |
| mt4i 118 | Modus tollens inference. ... |
| pm2.21i 119 | A contradiction implies an... |
| pm2.24ii 120 | A contradiction implies an... |
| pm2.21d 121 | A contradiction implies an... |
| pm2.21ddALT 122 | Alternate proof of ~ pm2.2... |
| pm2.21 123 | From a wff and its negatio... |
| pm2.24 124 | Theorem *2.24 of [Whitehea... |
| jarl 125 | Elimination of a nested an... |
| jarli 126 | Inference associated with ... |
| pm2.18d 127 | Deduction form of the Clav... |
| pm2.18 128 | Clavius law, or "consequen... |
| pm2.18i 129 | Inference associated with ... |
| notnotr 130 | Double negation eliminatio... |
| notnotri 131 | Inference associated with ... |
| notnotriALT 132 | Alternate proof of ~ notno... |
| notnotrd 133 | Deduction associated with ... |
| con2d 134 | A contraposition deduction... |
| con2 135 | Contraposition. Theorem *... |
| mt2d 136 | Modus tollens deduction. ... |
| mt2i 137 | Modus tollens inference. ... |
| nsyl3 138 | A negated syllogism infere... |
| con2i 139 | A contraposition inference... |
| nsyl 140 | A negated syllogism infere... |
| nsyl2 141 | A negated syllogism infere... |
| notnot 142 | Double negation introducti... |
| notnoti 143 | Inference associated with ... |
| notnotd 144 | Deduction associated with ... |
| con1d 145 | A contraposition deduction... |
| con1 146 | Contraposition. Theorem *... |
| con1i 147 | A contraposition inference... |
| mt3d 148 | Modus tollens deduction. ... |
| mt3i 149 | Modus tollens inference. ... |
| pm2.24i 150 | Inference associated with ... |
| pm2.24d 151 | Deduction form of ~ pm2.24... |
| con3d 152 | A contraposition deduction... |
| con3 153 | Contraposition. Theorem *... |
| con3i 154 | A contraposition inference... |
| con3rr3 155 | Rotate through consequent ... |
| nsyld 156 | A negated syllogism deduct... |
| nsyli 157 | A negated syllogism infere... |
| nsyl4 158 | A negated syllogism infere... |
| nsyl5 159 | A negated syllogism infere... |
| pm3.2im 160 | Theorem *3.2 of [Whitehead... |
| jc 161 | Deduction joining the cons... |
| jcn 162 | Theorem joining the conseq... |
| jcnd 163 | Deduction joining the cons... |
| impi 164 | An importation inference. ... |
| expi 165 | An exportation inference. ... |
| simprim 166 | Simplification. Similar t... |
| simplim 167 | Simplification. Similar t... |
| pm2.5g 168 | General instance of Theore... |
| pm2.5 169 | Theorem *2.5 of [Whitehead... |
| conax1 170 | Contrapositive of ~ ax-1 .... |
| conax1k 171 | Weakening of ~ conax1 . G... |
| pm2.51 172 | Theorem *2.51 of [Whitehea... |
| pm2.52 173 | Theorem *2.52 of [Whitehea... |
| pm2.521g 174 | A general instance of Theo... |
| pm2.521g2 175 | A general instance of Theo... |
| pm2.521 176 | Theorem *2.521 of [Whitehe... |
| expt 177 | Exportation theorem ~ pm3.... |
| impt 178 | Importation theorem ~ pm3.... |
| pm2.61d 179 | Deduction eliminating an a... |
| pm2.61d1 180 | Inference eliminating an a... |
| pm2.61d2 181 | Inference eliminating an a... |
| pm2.61i 182 | Inference eliminating an a... |
| pm2.61ii 183 | Inference eliminating two ... |
| pm2.61nii 184 | Inference eliminating two ... |
| pm2.61iii 185 | Inference eliminating thre... |
| ja 186 | Inference joining the ante... |
| jad 187 | Deduction form of ~ ja . ... |
| pm2.01 188 | Weak Clavius law. If a fo... |
| pm2.01d 189 | Deduction based on reducti... |
| pm2.6 190 | Theorem *2.6 of [Whitehead... |
| pm2.61 191 | Theorem *2.61 of [Whitehea... |
| pm2.65 192 | Theorem *2.65 of [Whitehea... |
| pm2.65i 193 | Inference for proof by con... |
| pm2.21dd 194 | A contradiction implies an... |
| pm2.65d 195 | Deduction for proof by con... |
| mto 196 | The rule of modus tollens.... |
| mtod 197 | Modus tollens deduction. ... |
| mtoi 198 | Modus tollens inference. ... |
| mt2 199 | A rule similar to modus to... |
| mt3 200 | A rule similar to modus to... |
| peirce 201 | Peirce's axiom. A non-int... |
| looinv 202 | The Inversion Axiom of the... |
| bijust0 203 | A self-implication (see ~ ... |
| bijust 204 | Theorem used to justify th... |
| impbi 207 | Property of the biconditio... |
| impbii 208 | Infer an equivalence from ... |
| impbidd 209 | Deduce an equivalence from... |
| impbid21d 210 | Deduce an equivalence from... |
| impbid 211 | Deduce an equivalence from... |
| dfbi1 212 | Relate the biconditional c... |
| dfbi1ALT 213 | Alternate proof of ~ dfbi1... |
| biimp 214 | Property of the biconditio... |
| biimpi 215 | Infer an implication from ... |
| sylbi 216 | A mixed syllogism inferenc... |
| sylib 217 | A mixed syllogism inferenc... |
| sylbb 218 | A mixed syllogism inferenc... |
| biimpr 219 | Property of the biconditio... |
| bicom1 220 | Commutative law for the bi... |
| bicom 221 | Commutative law for the bi... |
| bicomd 222 | Commute two sides of a bic... |
| bicomi 223 | Inference from commutative... |
| impbid1 224 | Infer an equivalence from ... |
| impbid2 225 | Infer an equivalence from ... |
| impcon4bid 226 | A variation on ~ impbid wi... |
| biimpri 227 | Infer a converse implicati... |
| biimpd 228 | Deduce an implication from... |
| mpbi 229 | An inference from a bicond... |
| mpbir 230 | An inference from a bicond... |
| mpbid 231 | A deduction from a bicondi... |
| mpbii 232 | An inference from a nested... |
| sylibr 233 | A mixed syllogism inferenc... |
| sylbir 234 | A mixed syllogism inferenc... |
| sylbbr 235 | A mixed syllogism inferenc... |
| sylbb1 236 | A mixed syllogism inferenc... |
| sylbb2 237 | A mixed syllogism inferenc... |
| sylibd 238 | A syllogism deduction. (C... |
| sylbid 239 | A syllogism deduction. (C... |
| mpbidi 240 | A deduction from a bicondi... |
| biimtrid 241 | A mixed syllogism inferenc... |
| biimtrrid 242 | A mixed syllogism inferenc... |
| imbitrid 243 | A mixed syllogism inferenc... |
| syl5ibcom 244 | A mixed syllogism inferenc... |
| imbitrrid 245 | A mixed syllogism inferenc... |
| syl5ibrcom 246 | A mixed syllogism inferenc... |
| biimprd 247 | Deduce a converse implicat... |
| biimpcd 248 | Deduce a commuted implicat... |
| biimprcd 249 | Deduce a converse commuted... |
| imbitrdi 250 | A mixed syllogism inferenc... |
| imbitrrdi 251 | A mixed syllogism inferenc... |
| biimtrdi 252 | A mixed syllogism inferenc... |
| biimtrrdi 253 | A mixed syllogism inferenc... |
| syl7bi 254 | A mixed syllogism inferenc... |
| syl8ib 255 | A syllogism rule of infere... |
| mpbird 256 | A deduction from a bicondi... |
| mpbiri 257 | An inference from a nested... |
| sylibrd 258 | A syllogism deduction. (C... |
| sylbird 259 | A syllogism deduction. (C... |
| biid 260 | Principle of identity for ... |
| biidd 261 | Principle of identity with... |
| pm5.1im 262 | Two propositions are equiv... |
| 2th 263 | Two truths are equivalent.... |
| 2thd 264 | Two truths are equivalent.... |
| monothetic 265 | Two self-implications (see... |
| ibi 266 | Inference that converts a ... |
| ibir 267 | Inference that converts a ... |
| ibd 268 | Deduction that converts a ... |
| pm5.74 269 | Distribution of implicatio... |
| pm5.74i 270 | Distribution of implicatio... |
| pm5.74ri 271 | Distribution of implicatio... |
| pm5.74d 272 | Distribution of implicatio... |
| pm5.74rd 273 | Distribution of implicatio... |
| bitri 274 | An inference from transiti... |
| bitr2i 275 | An inference from transiti... |
| bitr3i 276 | An inference from transiti... |
| bitr4i 277 | An inference from transiti... |
| bitrd 278 | Deduction form of ~ bitri ... |
| bitr2d 279 | Deduction form of ~ bitr2i... |
| bitr3d 280 | Deduction form of ~ bitr3i... |
| bitr4d 281 | Deduction form of ~ bitr4i... |
| bitrid 282 | A syllogism inference from... |
| bitr2id 283 | A syllogism inference from... |
| bitr3id 284 | A syllogism inference from... |
| bitr3di 285 | A syllogism inference from... |
| bitrdi 286 | A syllogism inference from... |
| bitr2di 287 | A syllogism inference from... |
| bitr4di 288 | A syllogism inference from... |
| bitr4id 289 | A syllogism inference from... |
| 3imtr3i 290 | A mixed syllogism inferenc... |
| 3imtr4i 291 | A mixed syllogism inferenc... |
| 3imtr3d 292 | More general version of ~ ... |
| 3imtr4d 293 | More general version of ~ ... |
| 3imtr3g 294 | More general version of ~ ... |
| 3imtr4g 295 | More general version of ~ ... |
| 3bitri 296 | A chained inference from t... |
| 3bitrri 297 | A chained inference from t... |
| 3bitr2i 298 | A chained inference from t... |
| 3bitr2ri 299 | A chained inference from t... |
| 3bitr3i 300 | A chained inference from t... |
| 3bitr3ri 301 | A chained inference from t... |
| 3bitr4i 302 | A chained inference from t... |
| 3bitr4ri 303 | A chained inference from t... |
| 3bitrd 304 | Deduction from transitivit... |
| 3bitrrd 305 | Deduction from transitivit... |
| 3bitr2d 306 | Deduction from transitivit... |
| 3bitr2rd 307 | Deduction from transitivit... |
| 3bitr3d 308 | Deduction from transitivit... |
| 3bitr3rd 309 | Deduction from transitivit... |
| 3bitr4d 310 | Deduction from transitivit... |
| 3bitr4rd 311 | Deduction from transitivit... |
| 3bitr3g 312 | More general version of ~ ... |
| 3bitr4g 313 | More general version of ~ ... |
| notnotb 314 | Double negation. Theorem ... |
| con34b 315 | A biconditional form of co... |
| con4bid 316 | A contraposition deduction... |
| notbid 317 | Deduction negating both si... |
| notbi 318 | Contraposition. Theorem *... |
| notbii 319 | Negate both sides of a log... |
| con4bii 320 | A contraposition inference... |
| mtbi 321 | An inference from a bicond... |
| mtbir 322 | An inference from a bicond... |
| mtbid 323 | A deduction from a bicondi... |
| mtbird 324 | A deduction from a bicondi... |
| mtbii 325 | An inference from a bicond... |
| mtbiri 326 | An inference from a bicond... |
| sylnib 327 | A mixed syllogism inferenc... |
| sylnibr 328 | A mixed syllogism inferenc... |
| sylnbi 329 | A mixed syllogism inferenc... |
| sylnbir 330 | A mixed syllogism inferenc... |
| xchnxbi 331 | Replacement of a subexpres... |
| xchnxbir 332 | Replacement of a subexpres... |
| xchbinx 333 | Replacement of a subexpres... |
| xchbinxr 334 | Replacement of a subexpres... |
| imbi2i 335 | Introduce an antecedent to... |
| bibi2i 336 | Inference adding a bicondi... |
| bibi1i 337 | Inference adding a bicondi... |
| bibi12i 338 | The equivalence of two equ... |
| imbi2d 339 | Deduction adding an antece... |
| imbi1d 340 | Deduction adding a consequ... |
| bibi2d 341 | Deduction adding a bicondi... |
| bibi1d 342 | Deduction adding a bicondi... |
| imbi12d 343 | Deduction joining two equi... |
| bibi12d 344 | Deduction joining two equi... |
| imbi12 345 | Closed form of ~ imbi12i .... |
| imbi1 346 | Theorem *4.84 of [Whitehea... |
| imbi2 347 | Theorem *4.85 of [Whitehea... |
| imbi1i 348 | Introduce a consequent to ... |
| imbi12i 349 | Join two logical equivalen... |
| bibi1 350 | Theorem *4.86 of [Whitehea... |
| bitr3 351 | Closed nested implication ... |
| con2bi 352 | Contraposition. Theorem *... |
| con2bid 353 | A contraposition deduction... |
| con1bid 354 | A contraposition deduction... |
| con1bii 355 | A contraposition inference... |
| con2bii 356 | A contraposition inference... |
| con1b 357 | Contraposition. Bidirecti... |
| con2b 358 | Contraposition. Bidirecti... |
| biimt 359 | A wff is equivalent to its... |
| pm5.5 360 | Theorem *5.5 of [Whitehead... |
| a1bi 361 | Inference introducing a th... |
| mt2bi 362 | A false consequent falsifi... |
| mtt 363 | Modus-tollens-like theorem... |
| imnot 364 | If a proposition is false,... |
| pm5.501 365 | Theorem *5.501 of [Whitehe... |
| ibib 366 | Implication in terms of im... |
| ibibr 367 | Implication in terms of im... |
| tbt 368 | A wff is equivalent to its... |
| nbn2 369 | The negation of a wff is e... |
| bibif 370 | Transfer negation via an e... |
| nbn 371 | The negation of a wff is e... |
| nbn3 372 | Transfer falsehood via equ... |
| pm5.21im 373 | Two propositions are equiv... |
| 2false 374 | Two falsehoods are equival... |
| 2falsed 375 | Two falsehoods are equival... |
| pm5.21ni 376 | Two propositions implying ... |
| pm5.21nii 377 | Eliminate an antecedent im... |
| pm5.21ndd 378 | Eliminate an antecedent im... |
| bija 379 | Combine antecedents into a... |
| pm5.18 380 | Theorem *5.18 of [Whitehea... |
| xor3 381 | Two ways to express "exclu... |
| nbbn 382 | Move negation outside of b... |
| biass 383 | Associative law for the bi... |
| biluk 384 | Lukasiewicz's shortest axi... |
| pm5.19 385 | Theorem *5.19 of [Whitehea... |
| bi2.04 386 | Logical equivalence of com... |
| pm5.4 387 | Antecedent absorption impl... |
| imdi 388 | Distributive law for impli... |
| pm5.41 389 | Theorem *5.41 of [Whitehea... |
| imbibi 390 | The antecedent of one side... |
| pm4.8 391 | Theorem *4.8 of [Whitehead... |
| pm4.81 392 | A formula is equivalent to... |
| imim21b 393 | Simplify an implication be... |
| pm4.63 396 | Theorem *4.63 of [Whitehea... |
| pm4.67 397 | Theorem *4.67 of [Whitehea... |
| imnan 398 | Express an implication in ... |
| imnani 399 | Infer an implication from ... |
| iman 400 | Implication in terms of co... |
| pm3.24 401 | Law of noncontradiction. ... |
| annim 402 | Express a conjunction in t... |
| pm4.61 403 | Theorem *4.61 of [Whitehea... |
| pm4.65 404 | Theorem *4.65 of [Whitehea... |
| imp 405 | Importation inference. (C... |
| impcom 406 | Importation inference with... |
| con3dimp 407 | Variant of ~ con3d with im... |
| mpnanrd 408 | Eliminate the right side o... |
| impd 409 | Importation deduction. (C... |
| impcomd 410 | Importation deduction with... |
| ex 411 | Exportation inference. (T... |
| expcom 412 | Exportation inference with... |
| expdcom 413 | Commuted form of ~ expd . ... |
| expd 414 | Exportation deduction. (C... |
| expcomd 415 | Deduction form of ~ expcom... |
| imp31 416 | An importation inference. ... |
| imp32 417 | An importation inference. ... |
| exp31 418 | An exportation inference. ... |
| exp32 419 | An exportation inference. ... |
| imp4b 420 | An importation inference. ... |
| imp4a 421 | An importation inference. ... |
| imp4c 422 | An importation inference. ... |
| imp4d 423 | An importation inference. ... |
| imp41 424 | An importation inference. ... |
| imp42 425 | An importation inference. ... |
| imp43 426 | An importation inference. ... |
| imp44 427 | An importation inference. ... |
| imp45 428 | An importation inference. ... |
| exp4b 429 | An exportation inference. ... |
| exp4a 430 | An exportation inference. ... |
| exp4c 431 | An exportation inference. ... |
| exp4d 432 | An exportation inference. ... |
| exp41 433 | An exportation inference. ... |
| exp42 434 | An exportation inference. ... |
| exp43 435 | An exportation inference. ... |
| exp44 436 | An exportation inference. ... |
| exp45 437 | An exportation inference. ... |
| imp5d 438 | An importation inference. ... |
| imp5a 439 | An importation inference. ... |
| imp5g 440 | An importation inference. ... |
| imp55 441 | An importation inference. ... |
| imp511 442 | An importation inference. ... |
| exp5c 443 | An exportation inference. ... |
| exp5j 444 | An exportation inference. ... |
| exp5l 445 | An exportation inference. ... |
| exp53 446 | An exportation inference. ... |
| pm3.3 447 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 448 | Theorem *3.31 (Imp) of [Wh... |
| impexp 449 | Import-export theorem. Pa... |
| impancom 450 | Mixed importation/commutat... |
| expdimp 451 | A deduction version of exp... |
| expimpd 452 | Exportation followed by a ... |
| impr 453 | Import a wff into a right ... |
| impl 454 | Export a wff from a left c... |
| expr 455 | Export a wff from a right ... |
| expl 456 | Export a wff from a left c... |
| ancoms 457 | Inference commuting conjun... |
| pm3.22 458 | Theorem *3.22 of [Whitehea... |
| ancom 459 | Commutative law for conjun... |
| ancomd 460 | Commutation of conjuncts i... |
| biancomi 461 | Commuting conjunction in a... |
| biancomd 462 | Commuting conjunction in a... |
| ancomst 463 | Closed form of ~ ancoms . ... |
| ancomsd 464 | Deduction commuting conjun... |
| anasss 465 | Associative law for conjun... |
| anassrs 466 | Associative law for conjun... |
| anass 467 | Associative law for conjun... |
| pm3.2 468 | Join antecedents with conj... |
| pm3.2i 469 | Infer conjunction of premi... |
| pm3.21 470 | Join antecedents with conj... |
| pm3.43i 471 | Nested conjunction of ante... |
| pm3.43 472 | Theorem *3.43 (Comp) of [W... |
| dfbi2 473 | A theorem similar to the s... |
| dfbi 474 | Definition ~ df-bi rewritt... |
| biimpa 475 | Importation inference from... |
| biimpar 476 | Importation inference from... |
| biimpac 477 | Importation inference from... |
| biimparc 478 | Importation inference from... |
| adantr 479 | Inference adding a conjunc... |
| adantl 480 | Inference adding a conjunc... |
| simpl 481 | Elimination of a conjunct.... |
| simpli 482 | Inference eliminating a co... |
| simpr 483 | Elimination of a conjunct.... |
| simpri 484 | Inference eliminating a co... |
| intnan 485 | Introduction of conjunct i... |
| intnanr 486 | Introduction of conjunct i... |
| intnand 487 | Introduction of conjunct i... |
| intnanrd 488 | Introduction of conjunct i... |
| adantld 489 | Deduction adding a conjunc... |
| adantrd 490 | Deduction adding a conjunc... |
| pm3.41 491 | Theorem *3.41 of [Whitehea... |
| pm3.42 492 | Theorem *3.42 of [Whitehea... |
| simpld 493 | Deduction eliminating a co... |
| simprd 494 | Deduction eliminating a co... |
| simprbi 495 | Deduction eliminating a co... |
| simplbi 496 | Deduction eliminating a co... |
| simprbda 497 | Deduction eliminating a co... |
| simplbda 498 | Deduction eliminating a co... |
| simplbi2 499 | Deduction eliminating a co... |
| simplbi2comt 500 | Closed form of ~ simplbi2c... |
| simplbi2com 501 | A deduction eliminating a ... |
| simpl2im 502 | Implication from an elimin... |
| simplbiim 503 | Implication from an elimin... |
| impel 504 | An inference for implicati... |
| mpan9 505 | Modus ponens conjoining di... |
| sylan9 506 | Nested syllogism inference... |
| sylan9r 507 | Nested syllogism inference... |
| sylan9bb 508 | Nested syllogism inference... |
| sylan9bbr 509 | Nested syllogism inference... |
| jca 510 | Deduce conjunction of the ... |
| jcad 511 | Deduction conjoining the c... |
| jca2 512 | Inference conjoining the c... |
| jca31 513 | Join three consequents. (... |
| jca32 514 | Join three consequents. (... |
| jcai 515 | Deduction replacing implic... |
| jcab 516 | Distributive law for impli... |
| pm4.76 517 | Theorem *4.76 of [Whitehea... |
| jctil 518 | Inference conjoining a the... |
| jctir 519 | Inference conjoining a the... |
| jccir 520 | Inference conjoining a con... |
| jccil 521 | Inference conjoining a con... |
| jctl 522 | Inference conjoining a the... |
| jctr 523 | Inference conjoining a the... |
| jctild 524 | Deduction conjoining a the... |
| jctird 525 | Deduction conjoining a the... |
| iba 526 | Introduction of antecedent... |
| ibar 527 | Introduction of antecedent... |
| biantru 528 | A wff is equivalent to its... |
| biantrur 529 | A wff is equivalent to its... |
| biantrud 530 | A wff is equivalent to its... |
| biantrurd 531 | A wff is equivalent to its... |
| bianfi 532 | A wff conjoined with false... |
| bianfd 533 | A wff conjoined with false... |
| baib 534 | Move conjunction outside o... |
| baibr 535 | Move conjunction outside o... |
| rbaibr 536 | Move conjunction outside o... |
| rbaib 537 | Move conjunction outside o... |
| baibd 538 | Move conjunction outside o... |
| rbaibd 539 | Move conjunction outside o... |
| bianabs 540 | Absorb a hypothesis into t... |
| pm5.44 541 | Theorem *5.44 of [Whitehea... |
| pm5.42 542 | Theorem *5.42 of [Whitehea... |
| ancl 543 | Conjoin antecedent to left... |
| anclb 544 | Conjoin antecedent to left... |
| ancr 545 | Conjoin antecedent to righ... |
| ancrb 546 | Conjoin antecedent to righ... |
| ancli 547 | Deduction conjoining antec... |
| ancri 548 | Deduction conjoining antec... |
| ancld 549 | Deduction conjoining antec... |
| ancrd 550 | Deduction conjoining antec... |
| impac 551 | Importation with conjuncti... |
| anc2l 552 | Conjoin antecedent to left... |
| anc2r 553 | Conjoin antecedent to righ... |
| anc2li 554 | Deduction conjoining antec... |
| anc2ri 555 | Deduction conjoining antec... |
| pm4.71 556 | Implication in terms of bi... |
| pm4.71r 557 | Implication in terms of bi... |
| pm4.71i 558 | Inference converting an im... |
| pm4.71ri 559 | Inference converting an im... |
| pm4.71d 560 | Deduction converting an im... |
| pm4.71rd 561 | Deduction converting an im... |
| pm4.24 562 | Theorem *4.24 of [Whitehea... |
| anidm 563 | Idempotent law for conjunc... |
| anidmdbi 564 | Conjunction idempotence wi... |
| anidms 565 | Inference from idempotent ... |
| imdistan 566 | Distribution of implicatio... |
| imdistani 567 | Distribution of implicatio... |
| imdistanri 568 | Distribution of implicatio... |
| imdistand 569 | Distribution of implicatio... |
| imdistanda 570 | Distribution of implicatio... |
| pm5.3 571 | Theorem *5.3 of [Whitehead... |
| pm5.32 572 | Distribution of implicatio... |
| pm5.32i 573 | Distribution of implicatio... |
| pm5.32ri 574 | Distribution of implicatio... |
| pm5.32d 575 | Distribution of implicatio... |
| pm5.32rd 576 | Distribution of implicatio... |
| pm5.32da 577 | Distribution of implicatio... |
| sylan 578 | A syllogism inference. (C... |
| sylanb 579 | A syllogism inference. (C... |
| sylanbr 580 | A syllogism inference. (C... |
| sylanbrc 581 | Syllogism inference. (Con... |
| syl2anc 582 | Syllogism inference combin... |
| syl2anc2 583 | Double syllogism inference... |
| sylancl 584 | Syllogism inference combin... |
| sylancr 585 | Syllogism inference combin... |
| sylancom 586 | Syllogism inference with c... |
| sylanblc 587 | Syllogism inference combin... |
| sylanblrc 588 | Syllogism inference combin... |
| syldan 589 | A syllogism deduction with... |
| sylbida 590 | A syllogism deduction. (C... |
| sylan2 591 | A syllogism inference. (C... |
| sylan2b 592 | A syllogism inference. (C... |
| sylan2br 593 | A syllogism inference. (C... |
| syl2an 594 | A double syllogism inferen... |
| syl2anr 595 | A double syllogism inferen... |
| syl2anb 596 | A double syllogism inferen... |
| syl2anbr 597 | A double syllogism inferen... |
| sylancb 598 | A syllogism inference comb... |
| sylancbr 599 | A syllogism inference comb... |
| syldanl 600 | A syllogism deduction with... |
| syland 601 | A syllogism deduction. (C... |
| sylani 602 | A syllogism inference. (C... |
| sylan2d 603 | A syllogism deduction. (C... |
| sylan2i 604 | A syllogism inference. (C... |
| syl2ani 605 | A syllogism inference. (C... |
| syl2and 606 | A syllogism deduction. (C... |
| anim12d 607 | Conjoin antecedents and co... |
| anim12d1 608 | Variant of ~ anim12d where... |
| anim1d 609 | Add a conjunct to right of... |
| anim2d 610 | Add a conjunct to left of ... |
| anim12i 611 | Conjoin antecedents and co... |
| anim12ci 612 | Variant of ~ anim12i with ... |
| anim1i 613 | Introduce conjunct to both... |
| anim1ci 614 | Introduce conjunct to both... |
| anim2i 615 | Introduce conjunct to both... |
| anim12ii 616 | Conjoin antecedents and co... |
| anim12dan 617 | Conjoin antecedents and co... |
| im2anan9 618 | Deduction joining nested i... |
| im2anan9r 619 | Deduction joining nested i... |
| pm3.45 620 | Theorem *3.45 (Fact) of [W... |
| anbi2i 621 | Introduce a left conjunct ... |
| anbi1i 622 | Introduce a right conjunct... |
| anbi2ci 623 | Variant of ~ anbi2i with c... |
| anbi1ci 624 | Variant of ~ anbi1i with c... |
| bianbi 625 | Exchanging conjunction in ... |
| anbi12i 626 | Conjoin both sides of two ... |
| anbi12ci 627 | Variant of ~ anbi12i with ... |
| anbi2d 628 | Deduction adding a left co... |
| anbi1d 629 | Deduction adding a right c... |
| anbi12d 630 | Deduction joining two equi... |
| anbi1 631 | Introduce a right conjunct... |
| anbi2 632 | Introduce a left conjunct ... |
| anbi1cd 633 | Introduce a proposition as... |
| an2anr 634 | Double commutation in conj... |
| pm4.38 635 | Theorem *4.38 of [Whitehea... |
| bi2anan9 636 | Deduction joining two equi... |
| bi2anan9r 637 | Deduction joining two equi... |
| bi2bian9 638 | Deduction joining two bico... |
| anbiim 639 | Adding biconditional when ... |
| 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 ... |
| biadanid 821 | Deduction associated with ... |
| pm5.1 822 | Two propositions are equiv... |
| pm5.21 823 | Two propositions are equiv... |
| pm5.35 824 | Theorem *5.35 of [Whitehea... |
| abai 825 | Introduce one conjunct as ... |
| pm4.45im 826 | Conjunction with implicati... |
| impimprbi 827 | An implication and its rev... |
| nan 828 | Theorem to move a conjunct... |
| pm5.31 829 | Theorem *5.31 of [Whitehea... |
| pm5.31r 830 | Variant of ~ pm5.31 . (Co... |
| pm4.15 831 | Theorem *4.15 of [Whitehea... |
| pm5.36 832 | Theorem *5.36 of [Whitehea... |
| annotanannot 833 | A conjunction with a negat... |
| pm5.33 834 | Theorem *5.33 of [Whitehea... |
| syl12anc 835 | Syllogism combined with co... |
| syl21anc 836 | Syllogism combined with co... |
| syl22anc 837 | Syllogism combined with co... |
| syl1111anc 838 | Four-hypothesis eliminatio... |
| syldbl2 839 | Stacked hypotheseis implie... |
| mpsyl4anc 840 | An elimination deduction. ... |
| pm4.87 841 | Theorem *4.87 of [Whitehea... |
| bimsc1 842 | Removal of conjunct from o... |
| a2and 843 | Deduction distributing a c... |
| animpimp2impd 844 | Deduction deriving nested ... |
| pm4.64 847 | Theorem *4.64 of [Whitehea... |
| pm4.66 848 | Theorem *4.66 of [Whitehea... |
| pm2.53 849 | Theorem *2.53 of [Whitehea... |
| pm2.54 850 | Theorem *2.54 of [Whitehea... |
| imor 851 | Implication in terms of di... |
| imori 852 | Infer disjunction from imp... |
| imorri 853 | Infer implication from dis... |
| pm4.62 854 | Theorem *4.62 of [Whitehea... |
| jaoi 855 | Inference disjoining the a... |
| jao1i 856 | Add a disjunct in the ante... |
| jaod 857 | Deduction disjoining the a... |
| mpjaod 858 | Eliminate a disjunction in... |
| ori 859 | Infer implication from dis... |
| orri 860 | Infer disjunction from imp... |
| orrd 861 | Deduce disjunction from im... |
| ord 862 | Deduce implication from di... |
| orci 863 | Deduction introducing a di... |
| olci 864 | Deduction introducing a di... |
| orc 865 | Introduction of a disjunct... |
| olc 866 | Introduction of a disjunct... |
| pm1.4 867 | Axiom *1.4 of [WhiteheadRu... |
| orcom 868 | Commutative law for disjun... |
| orcomd 869 | Commutation of disjuncts i... |
| orcoms 870 | Commutation of disjuncts i... |
| orcd 871 | Deduction introducing a di... |
| olcd 872 | Deduction introducing a di... |
| orcs 873 | Deduction eliminating disj... |
| olcs 874 | Deduction eliminating disj... |
| olcnd 875 | A lemma for Conjunctive No... |
| orcnd 876 | A lemma for Conjunctive No... |
| mtord 877 | A modus tollens deduction ... |
| pm3.2ni 878 | Infer negated disjunction ... |
| pm2.45 879 | Theorem *2.45 of [Whitehea... |
| pm2.46 880 | Theorem *2.46 of [Whitehea... |
| pm2.47 881 | Theorem *2.47 of [Whitehea... |
| pm2.48 882 | Theorem *2.48 of [Whitehea... |
| pm2.49 883 | Theorem *2.49 of [Whitehea... |
| norbi 884 | If neither of two proposit... |
| nbior 885 | If two propositions are no... |
| orel1 886 | Elimination of disjunction... |
| pm2.25 887 | Theorem *2.25 of [Whitehea... |
| orel2 888 | Elimination of disjunction... |
| pm2.67-2 889 | Slight generalization of T... |
| pm2.67 890 | Theorem *2.67 of [Whitehea... |
| curryax 891 | A non-intuitionistic posit... |
| exmid 892 | Law of excluded middle, al... |
| exmidd 893 | Law of excluded middle in ... |
| pm2.1 894 | Theorem *2.1 of [Whitehead... |
| pm2.13 895 | Theorem *2.13 of [Whitehea... |
| pm2.621 896 | Theorem *2.621 of [Whitehe... |
| pm2.62 897 | Theorem *2.62 of [Whitehea... |
| pm2.68 898 | Theorem *2.68 of [Whitehea... |
| dfor2 899 | Logical 'or' expressed in ... |
| pm2.07 900 | Theorem *2.07 of [Whitehea... |
| pm1.2 901 | Axiom *1.2 of [WhiteheadRu... |
| oridm 902 | Idempotent law for disjunc... |
| pm4.25 903 | Theorem *4.25 of [Whitehea... |
| pm2.4 904 | Theorem *2.4 of [Whitehead... |
| pm2.41 905 | Theorem *2.41 of [Whitehea... |
| orim12i 906 | Disjoin antecedents and co... |
| orim1i 907 | Introduce disjunct to both... |
| orim2i 908 | Introduce disjunct to both... |
| orim12dALT 909 | Alternate proof of ~ orim1... |
| orbi2i 910 | Inference adding a left di... |
| orbi1i 911 | Inference adding a right d... |
| orbi12i 912 | Infer the disjunction of t... |
| orbi2d 913 | Deduction adding a left di... |
| orbi1d 914 | Deduction adding a right d... |
| orbi1 915 | Theorem *4.37 of [Whitehea... |
| orbi12d 916 | Deduction joining two equi... |
| pm1.5 917 | Axiom *1.5 (Assoc) of [Whi... |
| or12 918 | Swap two disjuncts. (Cont... |
| orass 919 | Associative law for disjun... |
| pm2.31 920 | Theorem *2.31 of [Whitehea... |
| pm2.32 921 | Theorem *2.32 of [Whitehea... |
| pm2.3 922 | Theorem *2.3 of [Whitehead... |
| or32 923 | A rearrangement of disjunc... |
| or4 924 | Rearrangement of 4 disjunc... |
| or42 925 | Rearrangement of 4 disjunc... |
| orordi 926 | Distribution of disjunctio... |
| orordir 927 | Distribution of disjunctio... |
| orimdi 928 | Disjunction distributes ov... |
| pm2.76 929 | Theorem *2.76 of [Whitehea... |
| pm2.85 930 | Theorem *2.85 of [Whitehea... |
| pm2.75 931 | Theorem *2.75 of [Whitehea... |
| pm4.78 932 | Implication distributes ov... |
| biort 933 | A disjunction with a true ... |
| biorf 934 | A wff is equivalent to its... |
| biortn 935 | A wff is equivalent to its... |
| biorfi 936 | A wff is equivalent to its... |
| pm2.26 937 | Theorem *2.26 of [Whitehea... |
| pm2.63 938 | Theorem *2.63 of [Whitehea... |
| pm2.64 939 | Theorem *2.64 of [Whitehea... |
| pm2.42 940 | Theorem *2.42 of [Whitehea... |
| pm5.11g 941 | A general instance of Theo... |
| pm5.11 942 | Theorem *5.11 of [Whitehea... |
| pm5.12 943 | Theorem *5.12 of [Whitehea... |
| pm5.14 944 | Theorem *5.14 of [Whitehea... |
| pm5.13 945 | Theorem *5.13 of [Whitehea... |
| pm5.55 946 | Theorem *5.55 of [Whitehea... |
| pm4.72 947 | Implication in terms of bi... |
| imimorb 948 | Simplify an implication be... |
| oibabs 949 | Absorption of disjunction ... |
| orbidi 950 | Disjunction distributes ov... |
| pm5.7 951 | Disjunction distributes ov... |
| jaao 952 | Inference conjoining and d... |
| jaoa 953 | Inference disjoining and c... |
| jaoian 954 | Inference disjoining the a... |
| jaodan 955 | Deduction disjoining the a... |
| mpjaodan 956 | Eliminate a disjunction in... |
| pm3.44 957 | Theorem *3.44 of [Whitehea... |
| jao 958 | Disjunction of antecedents... |
| jaob 959 | Disjunction of antecedents... |
| pm4.77 960 | Theorem *4.77 of [Whitehea... |
| pm3.48 961 | Theorem *3.48 of [Whitehea... |
| orim12d 962 | Disjoin antecedents and co... |
| orim1d 963 | Disjoin antecedents and co... |
| orim2d 964 | Disjoin antecedents and co... |
| orim2 965 | Axiom *1.6 (Sum) of [White... |
| pm2.38 966 | Theorem *2.38 of [Whitehea... |
| pm2.36 967 | Theorem *2.36 of [Whitehea... |
| pm2.37 968 | Theorem *2.37 of [Whitehea... |
| pm2.81 969 | Theorem *2.81 of [Whitehea... |
| pm2.8 970 | Theorem *2.8 of [Whitehead... |
| pm2.73 971 | Theorem *2.73 of [Whitehea... |
| pm2.74 972 | Theorem *2.74 of [Whitehea... |
| pm2.82 973 | Theorem *2.82 of [Whitehea... |
| pm4.39 974 | Theorem *4.39 of [Whitehea... |
| animorl 975 | Conjunction implies disjun... |
| animorr 976 | Conjunction implies disjun... |
| animorlr 977 | Conjunction implies disjun... |
| animorrl 978 | Conjunction implies disjun... |
| ianor 979 | Negated conjunction in ter... |
| anor 980 | Conjunction in terms of di... |
| ioran 981 | Negated disjunction in ter... |
| pm4.52 982 | Theorem *4.52 of [Whitehea... |
| pm4.53 983 | Theorem *4.53 of [Whitehea... |
| pm4.54 984 | Theorem *4.54 of [Whitehea... |
| pm4.55 985 | Theorem *4.55 of [Whitehea... |
| pm4.56 986 | Theorem *4.56 of [Whitehea... |
| oran 987 | Disjunction in terms of co... |
| pm4.57 988 | Theorem *4.57 of [Whitehea... |
| pm3.1 989 | Theorem *3.1 of [Whitehead... |
| pm3.11 990 | Theorem *3.11 of [Whitehea... |
| pm3.12 991 | Theorem *3.12 of [Whitehea... |
| pm3.13 992 | Theorem *3.13 of [Whitehea... |
| pm3.14 993 | Theorem *3.14 of [Whitehea... |
| pm4.44 994 | Theorem *4.44 of [Whitehea... |
| pm4.45 995 | Theorem *4.45 of [Whitehea... |
| orabs 996 | Absorption of redundant in... |
| oranabs 997 | Absorb a disjunct into a c... |
| pm5.61 998 | Theorem *5.61 of [Whitehea... |
| pm5.6 999 | Conjunction in antecedent ... |
| orcanai 1000 | Change disjunction in cons... |
| pm4.79 1001 | Theorem *4.79 of [Whitehea... |
| pm5.53 1002 | Theorem *5.53 of [Whitehea... |
| ordi 1003 | Distributive law for disju... |
| ordir 1004 | Distributive law for disju... |
| andi 1005 | Distributive law for conju... |
| andir 1006 | Distributive law for conju... |
| orddi 1007 | Double distributive law fo... |
| anddi 1008 | Double distributive law fo... |
| pm5.17 1009 | Theorem *5.17 of [Whitehea... |
| pm5.15 1010 | Theorem *5.15 of [Whitehea... |
| pm5.16 1011 | Theorem *5.16 of [Whitehea... |
| xor 1012 | Two ways to express exclus... |
| nbi2 1013 | Two ways to express "exclu... |
| xordi 1014 | Conjunction distributes ov... |
| pm5.54 1015 | Theorem *5.54 of [Whitehea... |
| pm5.62 1016 | Theorem *5.62 of [Whitehea... |
| pm5.63 1017 | Theorem *5.63 of [Whitehea... |
| niabn 1018 | Miscellaneous inference re... |
| ninba 1019 | Miscellaneous inference re... |
| pm4.43 1020 | Theorem *4.43 of [Whitehea... |
| pm4.82 1021 | Theorem *4.82 of [Whitehea... |
| pm4.83 1022 | Theorem *4.83 of [Whitehea... |
| pclem6 1023 | Negation inferred from emb... |
| bigolden 1024 | Dijkstra-Scholten's Golden... |
| pm5.71 1025 | Theorem *5.71 of [Whitehea... |
| pm5.75 1026 | Theorem *5.75 of [Whitehea... |
| ecase2d 1027 | Deduction for elimination ... |
| ecase2dOLD 1028 | Obsolete version of ~ ecas... |
| ecase3 1029 | Inference for elimination ... |
| ecase 1030 | Inference for elimination ... |
| ecase3d 1031 | Deduction for elimination ... |
| ecased 1032 | Deduction for elimination ... |
| ecase3ad 1033 | Deduction for elimination ... |
| ecase3adOLD 1034 | Obsolete version of ~ ecas... |
| ccase 1035 | Inference for combining ca... |
| ccased 1036 | Deduction for combining ca... |
| ccase2 1037 | Inference for combining ca... |
| 4cases 1038 | Inference eliminating two ... |
| 4casesdan 1039 | Deduction eliminating two ... |
| cases 1040 | Case disjunction according... |
| dedlem0a 1041 | Lemma for an alternate ver... |
| dedlem0b 1042 | Lemma for an alternate ver... |
| dedlema 1043 | Lemma for weak deduction t... |
| dedlemb 1044 | Lemma for weak deduction t... |
| cases2 1045 | Case disjunction according... |
| cases2ALT 1046 | Alternate proof of ~ cases... |
| dfbi3 1047 | An alternate definition of... |
| pm5.24 1048 | Theorem *5.24 of [Whitehea... |
| 4exmid 1049 | The disjunction of the fou... |
| consensus 1050 | The consensus theorem. Th... |
| pm4.42 1051 | Theorem *4.42 of [Whitehea... |
| prlem1 1052 | A specialized lemma for se... |
| prlem2 1053 | A specialized lemma for se... |
| oplem1 1054 | A specialized lemma for se... |
| dn1 1055 | A single axiom for Boolean... |
| bianir 1056 | A closed form of ~ mpbir ,... |
| jaoi2 1057 | Inference removing a negat... |
| jaoi3 1058 | Inference separating a dis... |
| ornld 1059 | Selecting one statement fr... |
| dfifp2 1062 | Alternate definition of th... |
| dfifp3 1063 | Alternate definition of th... |
| dfifp4 1064 | Alternate definition of th... |
| dfifp5 1065 | Alternate definition of th... |
| dfifp6 1066 | Alternate definition of th... |
| dfifp7 1067 | Alternate definition of th... |
| ifpdfbi 1068 | Define the biconditional a... |
| anifp 1069 | The conditional operator i... |
| ifpor 1070 | The conditional operator i... |
| ifpn 1071 | Conditional operator for t... |
| ifptru 1072 | Value of the conditional o... |
| ifpfal 1073 | Value of the conditional o... |
| ifpid 1074 | Value of the conditional o... |
| casesifp 1075 | Version of ~ cases express... |
| ifpbi123d 1076 | Equivalence deduction for ... |
| ifpbi23d 1077 | Equivalence deduction for ... |
| ifpimpda 1078 | Separation of the values o... |
| 1fpid3 1079 | The value of the condition... |
| elimh 1080 | Hypothesis builder for the... |
| dedt 1081 | The weak deduction theorem... |
| con3ALT 1082 | Proof of ~ con3 from its a... |
| 3orass 1087 | Associative law for triple... |
| 3orel1 1088 | Partial elimination of a t... |
| 3orrot 1089 | Rotation law for triple di... |
| 3orcoma 1090 | Commutation law for triple... |
| 3orcomb 1091 | Commutation law for triple... |
| 3anass 1092 | Associative law for triple... |
| 3anan12 1093 | Convert triple conjunction... |
| 3anan32 1094 | Convert triple conjunction... |
| 3ancoma 1095 | Commutation law for triple... |
| 3ancomb 1096 | Commutation law for triple... |
| 3anrot 1097 | Rotation law for triple co... |
| 3anrev 1098 | Reversal law for triple co... |
| anandi3 1099 | Distribution of triple con... |
| anandi3r 1100 | Distribution of triple con... |
| 3anidm 1101 | Idempotent law for conjunc... |
| 3an4anass 1102 | Associative law for four c... |
| 3ioran 1103 | Negated triple disjunction... |
| 3ianor 1104 | Negated triple conjunction... |
| 3anor 1105 | Triple conjunction express... |
| 3oran 1106 | Triple disjunction in term... |
| 3impa 1107 | Importation from double to... |
| 3imp 1108 | Importation inference. (C... |
| 3imp31 1109 | The importation inference ... |
| 3imp231 1110 | Importation inference. (C... |
| 3imp21 1111 | The importation inference ... |
| 3impb 1112 | Importation from double to... |
| 3impib 1113 | Importation to triple conj... |
| 3impia 1114 | Importation to triple conj... |
| 3expa 1115 | Exportation from triple to... |
| 3exp 1116 | Exportation inference. (C... |
| 3expb 1117 | Exportation from triple to... |
| 3expia 1118 | Exportation from triple co... |
| 3expib 1119 | Exportation from triple co... |
| 3com12 1120 | Commutation in antecedent.... |
| 3com13 1121 | Commutation in antecedent.... |
| 3comr 1122 | Commutation in antecedent.... |
| 3com23 1123 | Commutation in antecedent.... |
| 3coml 1124 | Commutation in antecedent.... |
| 3jca 1125 | Join consequents with conj... |
| 3jcad 1126 | Deduction conjoining the c... |
| 3adant1 1127 | Deduction adding a conjunc... |
| 3adant2 1128 | Deduction adding a conjunc... |
| 3adant3 1129 | Deduction adding a conjunc... |
| 3ad2ant1 1130 | Deduction adding conjuncts... |
| 3ad2ant2 1131 | Deduction adding conjuncts... |
| 3ad2ant3 1132 | Deduction adding conjuncts... |
| simp1 1133 | Simplification of triple c... |
| simp2 1134 | Simplification of triple c... |
| simp3 1135 | Simplification of triple c... |
| simp1i 1136 | Infer a conjunct from a tr... |
| simp2i 1137 | Infer a conjunct from a tr... |
| simp3i 1138 | Infer a conjunct from a tr... |
| simp1d 1139 | Deduce a conjunct from a t... |
| simp2d 1140 | Deduce a conjunct from a t... |
| simp3d 1141 | Deduce a conjunct from a t... |
| simp1bi 1142 | Deduce a conjunct from a t... |
| simp2bi 1143 | Deduce a conjunct from a t... |
| simp3bi 1144 | Deduce a conjunct from a t... |
| 3simpa 1145 | Simplification of triple c... |
| 3simpb 1146 | Simplification of triple c... |
| 3simpc 1147 | Simplification of triple c... |
| 3anim123i 1148 | Join antecedents and conse... |
| 3anim1i 1149 | Add two conjuncts to antec... |
| 3anim2i 1150 | Add two conjuncts to antec... |
| 3anim3i 1151 | Add two conjuncts to antec... |
| 3anbi123i 1152 | Join 3 biconditionals with... |
| 3orbi123i 1153 | Join 3 biconditionals with... |
| 3anbi1i 1154 | Inference adding two conju... |
| 3anbi2i 1155 | Inference adding two conju... |
| 3anbi3i 1156 | Inference adding two conju... |
| syl3an 1157 | A triple syllogism inferen... |
| syl3anb 1158 | A triple syllogism inferen... |
| syl3anbr 1159 | A triple syllogism inferen... |
| syl3an1 1160 | A syllogism inference. (C... |
| syl3an2 1161 | A syllogism inference. (C... |
| syl3an3 1162 | A syllogism inference. (C... |
| 3adantl1 1163 | Deduction adding a conjunc... |
| 3adantl2 1164 | Deduction adding a conjunc... |
| 3adantl3 1165 | Deduction adding a conjunc... |
| 3adantr1 1166 | Deduction adding a conjunc... |
| 3adantr2 1167 | Deduction adding a conjunc... |
| 3adantr3 1168 | Deduction adding a conjunc... |
| ad4ant123 1169 | Deduction adding conjuncts... |
| ad4ant124 1170 | Deduction adding conjuncts... |
| ad4ant134 1171 | Deduction adding conjuncts... |
| ad4ant234 1172 | Deduction adding conjuncts... |
| 3adant1l 1173 | Deduction adding a conjunc... |
| 3adant1r 1174 | Deduction adding a conjunc... |
| 3adant2l 1175 | Deduction adding a conjunc... |
| 3adant2r 1176 | Deduction adding a conjunc... |
| 3adant3l 1177 | Deduction adding a conjunc... |
| 3adant3r 1178 | Deduction adding a conjunc... |
| 3adant3r1 1179 | Deduction adding a conjunc... |
| 3adant3r2 1180 | Deduction adding a conjunc... |
| 3adant3r3 1181 | Deduction adding a conjunc... |
| 3ad2antl1 1182 | Deduction adding conjuncts... |
| 3ad2antl2 1183 | Deduction adding conjuncts... |
| 3ad2antl3 1184 | Deduction adding conjuncts... |
| 3ad2antr1 1185 | Deduction adding conjuncts... |
| 3ad2antr2 1186 | Deduction adding conjuncts... |
| 3ad2antr3 1187 | Deduction adding conjuncts... |
| simpl1 1188 | Simplification of conjunct... |
| simpl2 1189 | Simplification of conjunct... |
| simpl3 1190 | Simplification of conjunct... |
| simpr1 1191 | Simplification of conjunct... |
| simpr2 1192 | Simplification of conjunct... |
| simpr3 1193 | Simplification of conjunct... |
| simp1l 1194 | Simplification of triple c... |
| simp1r 1195 | Simplification of triple c... |
| simp2l 1196 | Simplification of triple c... |
| simp2r 1197 | Simplification of triple c... |
| simp3l 1198 | Simplification of triple c... |
| simp3r 1199 | Simplification of triple c... |
| simp11 1200 | Simplification of doubly t... |
| simp12 1201 | Simplification of doubly t... |
| simp13 1202 | Simplification of doubly t... |
| simp21 1203 | Simplification of doubly t... |
| simp22 1204 | Simplification of doubly t... |
| simp23 1205 | Simplification of doubly t... |
| simp31 1206 | Simplification of doubly t... |
| simp32 1207 | Simplification of doubly t... |
| simp33 1208 | Simplification of doubly t... |
| simpll1 1209 | Simplification of conjunct... |
| simpll2 1210 | Simplification of conjunct... |
| simpll3 1211 | Simplification of conjunct... |
| simplr1 1212 | Simplification of conjunct... |
| simplr2 1213 | Simplification of conjunct... |
| simplr3 1214 | Simplification of conjunct... |
| simprl1 1215 | Simplification of conjunct... |
| simprl2 1216 | Simplification of conjunct... |
| simprl3 1217 | Simplification of conjunct... |
| simprr1 1218 | Simplification of conjunct... |
| simprr2 1219 | Simplification of conjunct... |
| simprr3 1220 | Simplification of conjunct... |
| simpl1l 1221 | Simplification of conjunct... |
| simpl1r 1222 | Simplification of conjunct... |
| simpl2l 1223 | Simplification of conjunct... |
| simpl2r 1224 | Simplification of conjunct... |
| simpl3l 1225 | Simplification of conjunct... |
| simpl3r 1226 | Simplification of conjunct... |
| simpr1l 1227 | Simplification of conjunct... |
| simpr1r 1228 | Simplification of conjunct... |
| simpr2l 1229 | Simplification of conjunct... |
| simpr2r 1230 | Simplification of conjunct... |
| simpr3l 1231 | Simplification of conjunct... |
| simpr3r 1232 | Simplification of conjunct... |
| simp1ll 1233 | Simplification of conjunct... |
| simp1lr 1234 | Simplification of conjunct... |
| simp1rl 1235 | Simplification of conjunct... |
| simp1rr 1236 | Simplification of conjunct... |
| simp2ll 1237 | Simplification of conjunct... |
| simp2lr 1238 | Simplification of conjunct... |
| simp2rl 1239 | Simplification of conjunct... |
| simp2rr 1240 | Simplification of conjunct... |
| simp3ll 1241 | Simplification of conjunct... |
| simp3lr 1242 | Simplification of conjunct... |
| simp3rl 1243 | Simplification of conjunct... |
| simp3rr 1244 | Simplification of conjunct... |
| simpl11 1245 | Simplification of conjunct... |
| simpl12 1246 | Simplification of conjunct... |
| simpl13 1247 | Simplification of conjunct... |
| simpl21 1248 | Simplification of conjunct... |
| simpl22 1249 | Simplification of conjunct... |
| simpl23 1250 | Simplification of conjunct... |
| simpl31 1251 | Simplification of conjunct... |
| simpl32 1252 | Simplification of conjunct... |
| simpl33 1253 | Simplification of conjunct... |
| simpr11 1254 | Simplification of conjunct... |
| simpr12 1255 | Simplification of conjunct... |
| simpr13 1256 | Simplification of conjunct... |
| simpr21 1257 | Simplification of conjunct... |
| simpr22 1258 | Simplification of conjunct... |
| simpr23 1259 | Simplification of conjunct... |
| simpr31 1260 | Simplification of conjunct... |
| simpr32 1261 | Simplification of conjunct... |
| simpr33 1262 | Simplification of conjunct... |
| simp1l1 1263 | Simplification of conjunct... |
| simp1l2 1264 | Simplification of conjunct... |
| simp1l3 1265 | Simplification of conjunct... |
| simp1r1 1266 | Simplification of conjunct... |
| simp1r2 1267 | Simplification of conjunct... |
| simp1r3 1268 | Simplification of conjunct... |
| simp2l1 1269 | Simplification of conjunct... |
| simp2l2 1270 | Simplification of conjunct... |
| simp2l3 1271 | Simplification of conjunct... |
| simp2r1 1272 | Simplification of conjunct... |
| simp2r2 1273 | Simplification of conjunct... |
| simp2r3 1274 | Simplification of conjunct... |
| simp3l1 1275 | Simplification of conjunct... |
| simp3l2 1276 | Simplification of conjunct... |
| simp3l3 1277 | Simplification of conjunct... |
| simp3r1 1278 | Simplification of conjunct... |
| simp3r2 1279 | Simplification of conjunct... |
| simp3r3 1280 | Simplification of conjunct... |
| simp11l 1281 | Simplification of conjunct... |
| simp11r 1282 | Simplification of conjunct... |
| simp12l 1283 | Simplification of conjunct... |
| simp12r 1284 | Simplification of conjunct... |
| simp13l 1285 | Simplification of conjunct... |
| simp13r 1286 | Simplification of conjunct... |
| simp21l 1287 | Simplification of conjunct... |
| simp21r 1288 | Simplification of conjunct... |
| simp22l 1289 | Simplification of conjunct... |
| simp22r 1290 | Simplification of conjunct... |
| simp23l 1291 | Simplification of conjunct... |
| simp23r 1292 | Simplification of conjunct... |
| simp31l 1293 | Simplification of conjunct... |
| simp31r 1294 | Simplification of conjunct... |
| simp32l 1295 | Simplification of conjunct... |
| simp32r 1296 | Simplification of conjunct... |
| simp33l 1297 | Simplification of conjunct... |
| simp33r 1298 | Simplification of conjunct... |
| simp111 1299 | Simplification of conjunct... |
| simp112 1300 | Simplification of conjunct... |
| simp113 1301 | Simplification of conjunct... |
| simp121 1302 | Simplification of conjunct... |
| simp122 1303 | Simplification of conjunct... |
| simp123 1304 | Simplification of conjunct... |
| simp131 1305 | Simplification of conjunct... |
| simp132 1306 | Simplification of conjunct... |
| simp133 1307 | Simplification of conjunct... |
| simp211 1308 | Simplification of conjunct... |
| simp212 1309 | Simplification of conjunct... |
| simp213 1310 | Simplification of conjunct... |
| simp221 1311 | Simplification of conjunct... |
| simp222 1312 | Simplification of conjunct... |
| simp223 1313 | Simplification of conjunct... |
| simp231 1314 | Simplification of conjunct... |
| simp232 1315 | Simplification of conjunct... |
| simp233 1316 | Simplification of conjunct... |
| simp311 1317 | Simplification of conjunct... |
| simp312 1318 | Simplification of conjunct... |
| simp313 1319 | Simplification of conjunct... |
| simp321 1320 | Simplification of conjunct... |
| simp322 1321 | Simplification of conjunct... |
| simp323 1322 | Simplification of conjunct... |
| simp331 1323 | Simplification of conjunct... |
| simp332 1324 | Simplification of conjunct... |
| simp333 1325 | Simplification of conjunct... |
| 3anibar 1326 | Remove a hypothesis from t... |
| 3mix1 1327 | Introduction in triple dis... |
| 3mix2 1328 | Introduction in triple dis... |
| 3mix3 1329 | Introduction in triple dis... |
| 3mix1i 1330 | Introduction in triple dis... |
| 3mix2i 1331 | Introduction in triple dis... |
| 3mix3i 1332 | Introduction in triple dis... |
| 3mix1d 1333 | Deduction introducing trip... |
| 3mix2d 1334 | Deduction introducing trip... |
| 3mix3d 1335 | Deduction introducing trip... |
| 3pm3.2i 1336 | Infer conjunction of premi... |
| pm3.2an3 1337 | Version of ~ pm3.2 for a t... |
| mpbir3an 1338 | Detach a conjunction of tr... |
| mpbir3and 1339 | Detach a conjunction of tr... |
| syl3anbrc 1340 | Syllogism inference. (Con... |
| syl21anbrc 1341 | Syllogism inference. (Con... |
| 3imp3i2an 1342 | An elimination deduction. ... |
| ex3 1343 | Apply ~ ex to a hypothesis... |
| 3imp1 1344 | Importation to left triple... |
| 3impd 1345 | Importation deduction for ... |
| 3imp2 1346 | Importation to right tripl... |
| 3impdi 1347 | Importation inference (und... |
| 3impdir 1348 | Importation inference (und... |
| 3exp1 1349 | Exportation from left trip... |
| 3expd 1350 | Exportation deduction for ... |
| 3exp2 1351 | Exportation from right tri... |
| exp5o 1352 | A triple exportation infer... |
| exp516 1353 | A triple exportation infer... |
| exp520 1354 | A triple exportation infer... |
| 3impexp 1355 | Version of ~ impexp for a ... |
| 3an1rs 1356 | Swap conjuncts. (Contribu... |
| 3anassrs 1357 | Associative law for conjun... |
| ad5ant245 1358 | Deduction adding conjuncts... |
| ad5ant234 1359 | Deduction adding conjuncts... |
| ad5ant235 1360 | Deduction adding conjuncts... |
| ad5ant123 1361 | Deduction adding conjuncts... |
| ad5ant124 1362 | Deduction adding conjuncts... |
| ad5ant125 1363 | Deduction adding conjuncts... |
| ad5ant134 1364 | Deduction adding conjuncts... |
| ad5ant135 1365 | Deduction adding conjuncts... |
| ad5ant145 1366 | Deduction adding conjuncts... |
| ad5ant2345 1367 | Deduction adding conjuncts... |
| syl3anc 1368 | Syllogism combined with co... |
| syl13anc 1369 | Syllogism combined with co... |
| syl31anc 1370 | Syllogism combined with co... |
| syl112anc 1371 | Syllogism combined with co... |
| syl121anc 1372 | Syllogism combined with co... |
| syl211anc 1373 | Syllogism combined with co... |
| syl23anc 1374 | Syllogism combined with co... |
| syl32anc 1375 | Syllogism combined with co... |
| syl122anc 1376 | Syllogism combined with co... |
| syl212anc 1377 | Syllogism combined with co... |
| syl221anc 1378 | Syllogism combined with co... |
| syl113anc 1379 | Syllogism combined with co... |
| syl131anc 1380 | Syllogism combined with co... |
| syl311anc 1381 | Syllogism combined with co... |
| syl33anc 1382 | Syllogism combined with co... |
| syl222anc 1383 | Syllogism combined with co... |
| syl123anc 1384 | Syllogism combined with co... |
| syl132anc 1385 | Syllogism combined with co... |
| syl213anc 1386 | Syllogism combined with co... |
| syl231anc 1387 | Syllogism combined with co... |
| syl312anc 1388 | Syllogism combined with co... |
| syl321anc 1389 | Syllogism combined with co... |
| syl133anc 1390 | Syllogism combined with co... |
| syl313anc 1391 | Syllogism combined with co... |
| syl331anc 1392 | Syllogism combined with co... |
| syl223anc 1393 | Syllogism combined with co... |
| syl232anc 1394 | Syllogism combined with co... |
| syl322anc 1395 | Syllogism combined with co... |
| syl233anc 1396 | Syllogism combined with co... |
| syl323anc 1397 | Syllogism combined with co... |
| syl332anc 1398 | Syllogism combined with co... |
| syl333anc 1399 | A syllogism inference comb... |
| syl3an1b 1400 | A syllogism inference. (C... |
| syl3an2b 1401 | A syllogism inference. (C... |
| syl3an3b 1402 | A syllogism inference. (C... |
| syl3an1br 1403 | A syllogism inference. (C... |
| syl3an2br 1404 | A syllogism inference. (C... |
| syl3an3br 1405 | A syllogism inference. (C... |
| syld3an3 1406 | A syllogism inference. (C... |
| syld3an1 1407 | A syllogism inference. (C... |
| syld3an2 1408 | A syllogism inference. (C... |
| syl3anl1 1409 | A syllogism inference. (C... |
| syl3anl2 1410 | A syllogism inference. (C... |
| syl3anl3 1411 | A syllogism inference. (C... |
| syl3anl 1412 | A triple syllogism inferen... |
| syl3anr1 1413 | A syllogism inference. (C... |
| syl3anr2 1414 | A syllogism inference. (C... |
| syl3anr3 1415 | A syllogism inference. (C... |
| 3anidm12 1416 | Inference from idempotent ... |
| 3anidm13 1417 | Inference from idempotent ... |
| 3anidm23 1418 | Inference from idempotent ... |
| syl2an3an 1419 | ~ syl3an with antecedents ... |
| syl2an23an 1420 | Deduction related to ~ syl... |
| 3ori 1421 | Infer implication from tri... |
| 3jao 1422 | Disjunction of three antec... |
| 3jaob 1423 | Disjunction of three antec... |
| 3jaoi 1424 | Disjunction of three antec... |
| 3jaod 1425 | Disjunction of three antec... |
| 3jaoian 1426 | Disjunction of three antec... |
| 3jaodan 1427 | Disjunction of three antec... |
| mpjao3dan 1428 | Eliminate a three-way disj... |
| 3jaao 1429 | Inference conjoining and d... |
| syl3an9b 1430 | Nested syllogism inference... |
| 3orbi123d 1431 | Deduction joining 3 equiva... |
| 3anbi123d 1432 | Deduction joining 3 equiva... |
| 3anbi12d 1433 | Deduction conjoining and a... |
| 3anbi13d 1434 | Deduction conjoining and a... |
| 3anbi23d 1435 | Deduction conjoining and a... |
| 3anbi1d 1436 | Deduction adding conjuncts... |
| 3anbi2d 1437 | Deduction adding conjuncts... |
| 3anbi3d 1438 | Deduction adding conjuncts... |
| 3anim123d 1439 | Deduction joining 3 implic... |
| 3orim123d 1440 | Deduction joining 3 implic... |
| an6 1441 | Rearrangement of 6 conjunc... |
| 3an6 1442 | Analogue of ~ an4 for trip... |
| 3or6 1443 | Analogue of ~ or4 for trip... |
| mp3an1 1444 | An inference based on modu... |
| mp3an2 1445 | An inference based on modu... |
| mp3an3 1446 | An inference based on modu... |
| mp3an12 1447 | An inference based on modu... |
| mp3an13 1448 | An inference based on modu... |
| mp3an23 1449 | An inference based on modu... |
| mp3an1i 1450 | An inference based on modu... |
| mp3anl1 1451 | An inference based on modu... |
| mp3anl2 1452 | An inference based on modu... |
| mp3anl3 1453 | An inference based on modu... |
| mp3anr1 1454 | An inference based on modu... |
| mp3anr2 1455 | An inference based on modu... |
| mp3anr3 1456 | An inference based on modu... |
| mp3an 1457 | An inference based on modu... |
| mpd3an3 1458 | An inference based on modu... |
| mpd3an23 1459 | An inference based on modu... |
| mp3and 1460 | A deduction based on modus... |
| mp3an12i 1461 | ~ mp3an with antecedents i... |
| mp3an2i 1462 | ~ mp3an with antecedents i... |
| mp3an3an 1463 | ~ mp3an with antecedents i... |
| mp3an2ani 1464 | An elimination deduction. ... |
| biimp3a 1465 | Infer implication from a l... |
| biimp3ar 1466 | Infer implication from a l... |
| 3anandis 1467 | Inference that undistribut... |
| 3anandirs 1468 | Inference that undistribut... |
| ecase23d 1469 | Deduction for elimination ... |
| 3ecase 1470 | Inference for elimination ... |
| 3bior1fd 1471 | A disjunction is equivalen... |
| 3bior1fand 1472 | A disjunction is equivalen... |
| 3bior2fd 1473 | A wff is equivalent to its... |
| 3biant1d 1474 | A conjunction is equivalen... |
| intn3an1d 1475 | Introduction of a triple c... |
| intn3an2d 1476 | Introduction of a triple c... |
| intn3an3d 1477 | Introduction of a triple c... |
| an3andi 1478 | Distribution of conjunctio... |
| an33rean 1479 | Rearrange a 9-fold conjunc... |
| 3orel2 1480 | Partial elimination of a t... |
| 3orel3 1481 | Partial elimination of a t... |
| 3orel13 1482 | Elimination of two disjunc... |
| 3pm3.2ni 1483 | Triple negated disjunction... |
| nanan 1486 | Conjunction in terms of al... |
| dfnan2 1487 | Alternative denial in term... |
| nanor 1488 | Alternative denial in term... |
| nancom 1489 | Alternative denial is comm... |
| nannan 1490 | Nested alternative denials... |
| nanim 1491 | Implication in terms of al... |
| nannot 1492 | Negation in terms of alter... |
| nanbi 1493 | Biconditional in terms of ... |
| nanbi1 1494 | Introduce a right anti-con... |
| nanbi2 1495 | Introduce a left anti-conj... |
| nanbi12 1496 | Join two logical equivalen... |
| nanbi1i 1497 | Introduce a right anti-con... |
| nanbi2i 1498 | Introduce a left anti-conj... |
| nanbi12i 1499 | Join two logical equivalen... |
| nanbi1d 1500 | Introduce a right anti-con... |
| nanbi2d 1501 | Introduce a left anti-conj... |
| nanbi12d 1502 | Join two logical equivalen... |
| nanass 1503 | A characterization of when... |
| xnor 1506 | Two ways to write XNOR (ex... |
| xorcom 1507 | The connector ` \/_ ` is c... |
| xorass 1508 | The connector ` \/_ ` is a... |
| excxor 1509 | This tautology shows that ... |
| xor2 1510 | Two ways to express "exclu... |
| xoror 1511 | Exclusive disjunction impl... |
| xornan 1512 | Exclusive disjunction impl... |
| xornan2 1513 | XOR implies NAND (written ... |
| xorneg2 1514 | The connector ` \/_ ` is n... |
| xorneg1 1515 | The connector ` \/_ ` is n... |
| xorneg 1516 | The connector ` \/_ ` is u... |
| xorbi12i 1517 | Equality property for excl... |
| xorbi12d 1518 | Equality property for excl... |
| anxordi 1519 | Conjunction distributes ov... |
| xorexmid 1520 | Exclusive-or variant of th... |
| norcom 1523 | The connector ` -\/ ` is c... |
| nornot 1524 | ` -. ` is expressible via ... |
| noran 1525 | ` /\ ` is expressible via ... |
| noror 1526 | ` \/ ` is expressible via ... |
| norasslem1 1527 | This lemma shows the equiv... |
| norasslem2 1528 | This lemma specializes ~ b... |
| norasslem3 1529 | This lemma specializes ~ b... |
| norass 1530 | A characterization of when... |
| trujust 1535 | Soundness justification th... |
| tru 1537 | The truth value ` T. ` is ... |
| dftru2 1538 | An alternate definition of... |
| trut 1539 | A proposition is equivalen... |
| mptru 1540 | Eliminate ` T. ` as an ant... |
| tbtru 1541 | A proposition is equivalen... |
| bitru 1542 | A theorem is equivalent to... |
| trud 1543 | Anything implies ` T. ` . ... |
| truan 1544 | True can be removed from a... |
| fal 1547 | The truth value ` F. ` is ... |
| nbfal 1548 | The negation of a proposit... |
| bifal 1549 | A contradiction is equival... |
| falim 1550 | The truth value ` F. ` imp... |
| falimd 1551 | The truth value ` F. ` imp... |
| dfnot 1552 | Given falsum ` F. ` , we c... |
| inegd 1553 | Negation introduction rule... |
| efald 1554 | Deduction based on reducti... |
| pm2.21fal 1555 | If a wff and its negation ... |
| truimtru 1556 | A ` -> ` identity. (Contr... |
| truimfal 1557 | A ` -> ` identity. (Contr... |
| falimtru 1558 | A ` -> ` identity. (Contr... |
| falimfal 1559 | A ` -> ` identity. (Contr... |
| nottru 1560 | A ` -. ` identity. (Contr... |
| notfal 1561 | A ` -. ` identity. (Contr... |
| trubitru 1562 | A ` <-> ` identity. (Cont... |
| falbitru 1563 | A ` <-> ` identity. (Cont... |
| trubifal 1564 | A ` <-> ` identity. (Cont... |
| falbifal 1565 | A ` <-> ` identity. (Cont... |
| truantru 1566 | A ` /\ ` identity. (Contr... |
| truanfal 1567 | A ` /\ ` identity. (Contr... |
| falantru 1568 | A ` /\ ` identity. (Contr... |
| falanfal 1569 | A ` /\ ` identity. (Contr... |
| truortru 1570 | A ` \/ ` identity. (Contr... |
| truorfal 1571 | A ` \/ ` identity. (Contr... |
| falortru 1572 | A ` \/ ` identity. (Contr... |
| falorfal 1573 | A ` \/ ` identity. (Contr... |
| trunantru 1574 | A ` -/\ ` identity. (Cont... |
| trunanfal 1575 | A ` -/\ ` identity. (Cont... |
| falnantru 1576 | A ` -/\ ` identity. (Cont... |
| falnanfal 1577 | A ` -/\ ` identity. (Cont... |
| truxortru 1578 | A ` \/_ ` identity. (Cont... |
| truxorfal 1579 | A ` \/_ ` identity. (Cont... |
| falxortru 1580 | A ` \/_ ` identity. (Cont... |
| falxorfal 1581 | A ` \/_ ` identity. (Cont... |
| trunortru 1582 | A ` -\/ ` identity. (Cont... |
| trunorfal 1583 | A ` -\/ ` identity. (Cont... |
| falnortru 1584 | A ` -\/ ` identity. (Cont... |
| falnorfal 1585 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1588 | Equality theorem for the a... |
| hadbi123i 1589 | Equality theorem for the a... |
| hadass 1590 | Associative law for the ad... |
| hadbi 1591 | The adder sum is the same ... |
| hadcoma 1592 | Commutative law for the ad... |
| hadcomb 1593 | Commutative law for the ad... |
| hadrot 1594 | Rotation law for the adder... |
| hadnot 1595 | The adder sum distributes ... |
| had1 1596 | If the first input is true... |
| had0 1597 | If the first input is fals... |
| hadifp 1598 | The value of the adder sum... |
| cador 1601 | The adder carry in disjunc... |
| cadan 1602 | The adder carry in conjunc... |
| cadbi123d 1603 | Equality theorem for the a... |
| cadbi123i 1604 | Equality theorem for the a... |
| cadcoma 1605 | Commutative law for the ad... |
| cadcomb 1606 | Commutative law for the ad... |
| cadrot 1607 | Rotation law for the adder... |
| cadnot 1608 | The adder carry distribute... |
| cad11 1609 | If (at least) two inputs a... |
| cad1 1610 | If one input is true, then... |
| cad0 1611 | If one input is false, the... |
| cad0OLD 1612 | Obsolete version of ~ cad0... |
| cadifp 1613 | The value of the carry is,... |
| cadtru 1614 | The adder carry is true as... |
| minimp 1615 | A single axiom for minimal... |
| minimp-syllsimp 1616 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1617 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1618 | Derivation of a commuted f... |
| minimp-ax2 1619 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1620 | Derivation of ~ pm2.43 (al... |
| impsingle 1621 | The shortest single axiom ... |
| impsingle-step4 1622 | Derivation of impsingle-st... |
| impsingle-step8 1623 | Derivation of impsingle-st... |
| impsingle-ax1 1624 | Derivation of impsingle-ax... |
| impsingle-step15 1625 | Derivation of impsingle-st... |
| impsingle-step18 1626 | Derivation of impsingle-st... |
| impsingle-step19 1627 | Derivation of impsingle-st... |
| impsingle-step20 1628 | Derivation of impsingle-st... |
| impsingle-step21 1629 | Derivation of impsingle-st... |
| impsingle-step22 1630 | Derivation of impsingle-st... |
| impsingle-step25 1631 | Derivation of impsingle-st... |
| impsingle-imim1 1632 | Derivation of impsingle-im... |
| impsingle-peirce 1633 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1634 | Derivation of ~ ax-2 from ... |
| meredith 1635 | Carew Meredith's sole axio... |
| merlem1 1636 | Step 3 of Meredith's proof... |
| merlem2 1637 | Step 4 of Meredith's proof... |
| merlem3 1638 | Step 7 of Meredith's proof... |
| merlem4 1639 | Step 8 of Meredith's proof... |
| merlem5 1640 | Step 11 of Meredith's proo... |
| merlem6 1641 | Step 12 of Meredith's proo... |
| merlem7 1642 | Between steps 14 and 15 of... |
| merlem8 1643 | Step 15 of Meredith's proo... |
| merlem9 1644 | Step 18 of Meredith's proo... |
| merlem10 1645 | Step 19 of Meredith's proo... |
| merlem11 1646 | Step 20 of Meredith's proo... |
| merlem12 1647 | Step 28 of Meredith's proo... |
| merlem13 1648 | Step 35 of Meredith's proo... |
| luk-1 1649 | 1 of 3 axioms for proposit... |
| luk-2 1650 | 2 of 3 axioms for proposit... |
| luk-3 1651 | 3 of 3 axioms for proposit... |
| luklem1 1652 | Used to rederive standard ... |
| luklem2 1653 | Used to rederive standard ... |
| luklem3 1654 | Used to rederive standard ... |
| luklem4 1655 | Used to rederive standard ... |
| luklem5 1656 | Used to rederive standard ... |
| luklem6 1657 | Used to rederive standard ... |
| luklem7 1658 | Used to rederive standard ... |
| luklem8 1659 | Used to rederive standard ... |
| ax1 1660 | Standard propositional axi... |
| ax2 1661 | Standard propositional axi... |
| ax3 1662 | Standard propositional axi... |
| nic-dfim 1663 | This theorem "defines" imp... |
| nic-dfneg 1664 | This theorem "defines" neg... |
| nic-mp 1665 | Derive Nicod's rule of mod... |
| nic-mpALT 1666 | A direct proof of ~ nic-mp... |
| nic-ax 1667 | Nicod's axiom derived from... |
| nic-axALT 1668 | A direct proof of ~ nic-ax... |
| nic-imp 1669 | Inference for ~ nic-mp usi... |
| nic-idlem1 1670 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1671 | Lemma for ~ nic-id . Infe... |
| nic-id 1672 | Theorem ~ id expressed wit... |
| nic-swap 1673 | The connector ` -/\ ` is s... |
| nic-isw1 1674 | Inference version of ~ nic... |
| nic-isw2 1675 | Inference for swapping nes... |
| nic-iimp1 1676 | Inference version of ~ nic... |
| nic-iimp2 1677 | Inference version of ~ nic... |
| nic-idel 1678 | Inference to remove the tr... |
| nic-ich 1679 | Chained inference. (Contr... |
| nic-idbl 1680 | Double the terms. Since d... |
| nic-bijust 1681 | Biconditional justificatio... |
| nic-bi1 1682 | Inference to extract one s... |
| nic-bi2 1683 | Inference to extract the o... |
| nic-stdmp 1684 | Derive the standard modus ... |
| nic-luk1 1685 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1686 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1687 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1688 | This alternative axiom for... |
| lukshefth1 1689 | Lemma for ~ renicax . (Co... |
| lukshefth2 1690 | Lemma for ~ renicax . (Co... |
| renicax 1691 | A rederivation of ~ nic-ax... |
| tbw-bijust 1692 | Justification for ~ tbw-ne... |
| tbw-negdf 1693 | The definition of negation... |
| tbw-ax1 1694 | The first of four axioms i... |
| tbw-ax2 1695 | The second of four axioms ... |
| tbw-ax3 1696 | The third of four axioms i... |
| tbw-ax4 1697 | The fourth of four axioms ... |
| tbwsyl 1698 | Used to rederive the Lukas... |
| tbwlem1 1699 | Used to rederive the Lukas... |
| tbwlem2 1700 | Used to rederive the Lukas... |
| tbwlem3 1701 | Used to rederive the Lukas... |
| tbwlem4 1702 | Used to rederive the Lukas... |
| tbwlem5 1703 | Used to rederive the Lukas... |
| re1luk1 1704 | ~ luk-1 derived from the T... |
| re1luk2 1705 | ~ luk-2 derived from the T... |
| re1luk3 1706 | ~ luk-3 derived from the T... |
| merco1 1707 | A single axiom for proposi... |
| merco1lem1 1708 | Used to rederive the Tarsk... |
| retbwax4 1709 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1710 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1711 | Used to rederive the Tarsk... |
| merco1lem3 1712 | Used to rederive the Tarsk... |
| merco1lem4 1713 | Used to rederive the Tarsk... |
| merco1lem5 1714 | Used to rederive the Tarsk... |
| merco1lem6 1715 | Used to rederive the Tarsk... |
| merco1lem7 1716 | Used to rederive the Tarsk... |
| retbwax3 1717 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1718 | Used to rederive the Tarsk... |
| merco1lem9 1719 | Used to rederive the Tarsk... |
| merco1lem10 1720 | Used to rederive the Tarsk... |
| merco1lem11 1721 | Used to rederive the Tarsk... |
| merco1lem12 1722 | Used to rederive the Tarsk... |
| merco1lem13 1723 | Used to rederive the Tarsk... |
| merco1lem14 1724 | Used to rederive the Tarsk... |
| merco1lem15 1725 | Used to rederive the Tarsk... |
| merco1lem16 1726 | Used to rederive the Tarsk... |
| merco1lem17 1727 | Used to rederive the Tarsk... |
| merco1lem18 1728 | Used to rederive the Tarsk... |
| retbwax1 1729 | ~ tbw-ax1 rederived from ~... |
| merco2 1730 | A single axiom for proposi... |
| mercolem1 1731 | Used to rederive the Tarsk... |
| mercolem2 1732 | Used to rederive the Tarsk... |
| mercolem3 1733 | Used to rederive the Tarsk... |
| mercolem4 1734 | Used to rederive the Tarsk... |
| mercolem5 1735 | Used to rederive the Tarsk... |
| mercolem6 1736 | Used to rederive the Tarsk... |
| mercolem7 1737 | Used to rederive the Tarsk... |
| mercolem8 1738 | Used to rederive the Tarsk... |
| re1tbw1 1739 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1740 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1741 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1742 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1743 | Justification for ~ rb-imd... |
| rb-imdf 1744 | The definition of implicat... |
| anmp 1745 | Modus ponens for ` { \/ , ... |
| rb-ax1 1746 | The first of four axioms i... |
| rb-ax2 1747 | The second of four axioms ... |
| rb-ax3 1748 | The third of four axioms i... |
| rb-ax4 1749 | The fourth of four axioms ... |
| rbsyl 1750 | Used to rederive the Lukas... |
| rblem1 1751 | Used to rederive the Lukas... |
| rblem2 1752 | Used to rederive the Lukas... |
| rblem3 1753 | Used to rederive the Lukas... |
| rblem4 1754 | Used to rederive the Lukas... |
| rblem5 1755 | Used to rederive the Lukas... |
| rblem6 1756 | Used to rederive the Lukas... |
| rblem7 1757 | Used to rederive the Lukas... |
| re1axmp 1758 | ~ ax-mp derived from Russe... |
| re2luk1 1759 | ~ luk-1 derived from Russe... |
| re2luk2 1760 | ~ luk-2 derived from Russe... |
| re2luk3 1761 | ~ luk-3 derived from Russe... |
| mptnan 1762 | Modus ponendo tollens 1, o... |
| mptxor 1763 | Modus ponendo tollens 2, o... |
| mtpor 1764 | Modus tollendo ponens (inc... |
| mtpxor 1765 | Modus tollendo ponens (ori... |
| stoic1a 1766 | Stoic logic Thema 1 (part ... |
| stoic1b 1767 | Stoic logic Thema 1 (part ... |
| stoic2a 1768 | Stoic logic Thema 2 versio... |
| stoic2b 1769 | Stoic logic Thema 2 versio... |
| stoic3 1770 | Stoic logic Thema 3. Stat... |
| stoic4a 1771 | Stoic logic Thema 4 versio... |
| stoic4b 1772 | Stoic logic Thema 4 versio... |
| alnex 1775 | Universal quantification o... |
| eximal 1776 | An equivalence between an ... |
| nf2 1779 | Alternate definition of no... |
| nf3 1780 | Alternate definition of no... |
| nf4 1781 | Alternate definition of no... |
| nfi 1782 | Deduce that ` x ` is not f... |
| nfri 1783 | Consequence of the definit... |
| nfd 1784 | Deduce that ` x ` is not f... |
| nfrd 1785 | Consequence of the definit... |
| nftht 1786 | Closed form of ~ nfth . (... |
| nfntht 1787 | Closed form of ~ nfnth . ... |
| nfntht2 1788 | Closed form of ~ nfnth . ... |
| gen2 1790 | Generalization applied twi... |
| mpg 1791 | Modus ponens combined with... |
| mpgbi 1792 | Modus ponens on biconditio... |
| mpgbir 1793 | Modus ponens on biconditio... |
| nex 1794 | Generalization rule for ne... |
| nfth 1795 | No variable is (effectivel... |
| nfnth 1796 | No variable is (effectivel... |
| hbth 1797 | No variable is (effectivel... |
| nftru 1798 | The true constant has no f... |
| nffal 1799 | The false constant has no ... |
| sptruw 1800 | Version of ~ sp when ` ph ... |
| altru 1801 | For all sets, ` T. ` is tr... |
| alfal 1802 | For all sets, ` -. F. ` is... |
| alim 1804 | Restatement of Axiom ~ ax-... |
| alimi 1805 | Inference quantifying both... |
| 2alimi 1806 | Inference doubly quantifyi... |
| ala1 1807 | Add an antecedent in a uni... |
| al2im 1808 | Closed form of ~ al2imi . ... |
| al2imi 1809 | Inference quantifying ante... |
| alanimi 1810 | Variant of ~ al2imi with c... |
| alimdh 1811 | Deduction form of Theorem ... |
| albi 1812 | Theorem 19.15 of [Margaris... |
| albii 1813 | Inference adding universal... |
| 2albii 1814 | Inference adding two unive... |
| 3albii 1815 | Inference adding three uni... |
| sylgt 1816 | Closed form of ~ sylg . (... |
| sylg 1817 | A syllogism combined with ... |
| alrimih 1818 | Inference form of Theorem ... |
| hbxfrbi 1819 | A utility lemma to transfe... |
| alex 1820 | Universal quantifier in te... |
| exnal 1821 | Existential quantification... |
| 2nalexn 1822 | Part of theorem *11.5 in [... |
| 2exnaln 1823 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1824 | Theorem *11.25 in [Whitehe... |
| alimex 1825 | An equivalence between an ... |
| aleximi 1826 | A variant of ~ al2imi : in... |
| alexbii 1827 | Biconditional form of ~ al... |
| exim 1828 | Theorem 19.22 of [Margaris... |
| eximi 1829 | Inference adding existenti... |
| 2eximi 1830 | Inference adding two exist... |
| eximii 1831 | Inference associated with ... |
| exa1 1832 | Add an antecedent in an ex... |
| 19.38 1833 | Theorem 19.38 of [Margaris... |
| 19.38a 1834 | Under a nonfreeness hypoth... |
| 19.38b 1835 | Under a nonfreeness hypoth... |
| imnang 1836 | Quantified implication in ... |
| alinexa 1837 | A transformation of quanti... |
| exnalimn 1838 | Existential quantification... |
| alexn 1839 | A relationship between two... |
| 2exnexn 1840 | Theorem *11.51 in [Whitehe... |
| exbi 1841 | Theorem 19.18 of [Margaris... |
| exbii 1842 | Inference adding existenti... |
| 2exbii 1843 | Inference adding two exist... |
| 3exbii 1844 | Inference adding three exi... |
| nfbiit 1845 | Equivalence theorem for th... |
| nfbii 1846 | Equality theorem for the n... |
| nfxfr 1847 | A utility lemma to transfe... |
| nfxfrd 1848 | A utility lemma to transfe... |
| nfnbi 1849 | A variable is nonfree in a... |
| nfnbiOLD 1850 | Obsolete version of ~ nfnb... |
| nfnt 1851 | If a variable is nonfree i... |
| nfn 1852 | Inference associated with ... |
| nfnd 1853 | Deduction associated with ... |
| exanali 1854 | A transformation of quanti... |
| 2exanali 1855 | Theorem *11.521 in [Whiteh... |
| exancom 1856 | Commutation of conjunction... |
| exan 1857 | Place a conjunct in the sc... |
| alrimdh 1858 | Deduction form of Theorem ... |
| eximdh 1859 | Deduction from Theorem 19.... |
| nexdh 1860 | Deduction for generalizati... |
| albidh 1861 | Formula-building rule for ... |
| exbidh 1862 | Formula-building rule for ... |
| exsimpl 1863 | Simplification of an exist... |
| exsimpr 1864 | Simplification of an exist... |
| 19.26 1865 | Theorem 19.26 of [Margaris... |
| 19.26-2 1866 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1867 | Theorem ~ 19.26 with tripl... |
| 19.29 1868 | Theorem 19.29 of [Margaris... |
| 19.29r 1869 | Variation of ~ 19.29 . (C... |
| 19.29r2 1870 | Variation of ~ 19.29r with... |
| 19.29x 1871 | Variation of ~ 19.29 with ... |
| 19.35 1872 | Theorem 19.35 of [Margaris... |
| 19.35i 1873 | Inference associated with ... |
| 19.35ri 1874 | Inference associated with ... |
| 19.25 1875 | Theorem 19.25 of [Margaris... |
| 19.30 1876 | Theorem 19.30 of [Margaris... |
| 19.43 1877 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1878 | Obsolete proof of ~ 19.43 ... |
| 19.33 1879 | Theorem 19.33 of [Margaris... |
| 19.33b 1880 | The antecedent provides a ... |
| 19.40 1881 | Theorem 19.40 of [Margaris... |
| 19.40-2 1882 | Theorem *11.42 in [Whitehe... |
| 19.40b 1883 | The antecedent provides a ... |
| albiim 1884 | Split a biconditional and ... |
| 2albiim 1885 | Split a biconditional and ... |
| exintrbi 1886 | Add/remove a conjunct in t... |
| exintr 1887 | Introduce a conjunct in th... |
| alsyl 1888 | Universally quantified and... |
| nfimd 1889 | If in a context ` x ` is n... |
| nfimt 1890 | Closed form of ~ nfim and ... |
| nfim 1891 | If ` x ` is not free in ` ... |
| nfand 1892 | If in a context ` x ` is n... |
| nf3and 1893 | Deduction form of bound-va... |
| nfan 1894 | If ` x ` is not free in ` ... |
| nfnan 1895 | If ` x ` is not free in ` ... |
| nf3an 1896 | If ` x ` is not free in ` ... |
| nfbid 1897 | If in a context ` x ` is n... |
| nfbi 1898 | If ` x ` is not free in ` ... |
| nfor 1899 | If ` x ` is not free in ` ... |
| nf3or 1900 | If ` x ` is not free in ` ... |
| empty 1901 | Two characterizations of t... |
| emptyex 1902 | On the empty domain, any e... |
| emptyal 1903 | On the empty domain, any u... |
| emptynf 1904 | On the empty domain, any v... |
| ax5d 1906 | Version of ~ ax-5 with ant... |
| ax5e 1907 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1908 | If a formula holds for som... |
| nfv 1909 | If ` x ` is not present in... |
| nfvd 1910 | ~ nfv with antecedent. Us... |
| alimdv 1911 | Deduction form of Theorem ... |
| eximdv 1912 | Deduction form of Theorem ... |
| 2alimdv 1913 | Deduction form of Theorem ... |
| 2eximdv 1914 | Deduction form of Theorem ... |
| albidv 1915 | Formula-building rule for ... |
| exbidv 1916 | Formula-building rule for ... |
| nfbidv 1917 | An equality theorem for no... |
| 2albidv 1918 | Formula-building rule for ... |
| 2exbidv 1919 | Formula-building rule for ... |
| 3exbidv 1920 | Formula-building rule for ... |
| 4exbidv 1921 | Formula-building rule for ... |
| alrimiv 1922 | Inference form of Theorem ... |
| alrimivv 1923 | Inference form of Theorem ... |
| alrimdv 1924 | Deduction form of Theorem ... |
| exlimiv 1925 | Inference form of Theorem ... |
| exlimiiv 1926 | Inference (Rule C) associa... |
| exlimivv 1927 | Inference form of Theorem ... |
| exlimdv 1928 | Deduction form of Theorem ... |
| exlimdvv 1929 | Deduction form of Theorem ... |
| exlimddv 1930 | Existential elimination ru... |
| nexdv 1931 | Deduction for generalizati... |
| 2ax5 1932 | Quantification of two vari... |
| stdpc5v 1933 | Version of ~ stdpc5 with a... |
| 19.21v 1934 | Version of ~ 19.21 with a ... |
| 19.32v 1935 | Version of ~ 19.32 with a ... |
| 19.31v 1936 | Version of ~ 19.31 with a ... |
| 19.23v 1937 | Version of ~ 19.23 with a ... |
| 19.23vv 1938 | Theorem ~ 19.23v extended ... |
| pm11.53v 1939 | Version of ~ pm11.53 with ... |
| 19.36imv 1940 | One direction of ~ 19.36v ... |
| 19.36imvOLD 1941 | Obsolete version of ~ 19.3... |
| 19.36iv 1942 | Inference associated with ... |
| 19.37imv 1943 | One direction of ~ 19.37v ... |
| 19.37iv 1944 | Inference associated with ... |
| 19.41v 1945 | Version of ~ 19.41 with a ... |
| 19.41vv 1946 | Version of ~ 19.41 with tw... |
| 19.41vvv 1947 | Version of ~ 19.41 with th... |
| 19.41vvvv 1948 | Version of ~ 19.41 with fo... |
| 19.42v 1949 | Version of ~ 19.42 with a ... |
| exdistr 1950 | Distribution of existentia... |
| exdistrv 1951 | Distribute a pair of exist... |
| 4exdistrv 1952 | Distribute two pairs of ex... |
| 19.42vv 1953 | Version of ~ 19.42 with tw... |
| exdistr2 1954 | Distribution of existentia... |
| 19.42vvv 1955 | Version of ~ 19.42 with th... |
| 3exdistr 1956 | Distribution of existentia... |
| 4exdistr 1957 | Distribution of existentia... |
| weq 1958 | Extend wff definition to i... |
| speimfw 1959 | Specialization, with addit... |
| speimfwALT 1960 | Alternate proof of ~ speim... |
| spimfw 1961 | Specialization, with addit... |
| ax12i 1962 | Inference that has ~ ax-12... |
| ax6v 1964 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1965 | At least one individual ex... |
| spimw 1966 | Specialization. Lemma 8 o... |
| spimew 1967 | Existential introduction, ... |
| speiv 1968 | Inference from existential... |
| speivw 1969 | Version of ~ spei with a d... |
| exgen 1970 | Rule of existential genera... |
| extru 1971 | There exists a variable su... |
| 19.2 1972 | Theorem 19.2 of [Margaris]... |
| 19.2d 1973 | Deduction associated with ... |
| 19.8w 1974 | Weak version of ~ 19.8a an... |
| spnfw 1975 | Weak version of ~ sp . Us... |
| spvw 1976 | Version of ~ sp when ` x `... |
| 19.3v 1977 | Version of ~ 19.3 with a d... |
| 19.8v 1978 | Version of ~ 19.8a with a ... |
| 19.9v 1979 | Version of ~ 19.9 with a d... |
| 19.39 1980 | Theorem 19.39 of [Margaris... |
| 19.24 1981 | Theorem 19.24 of [Margaris... |
| 19.34 1982 | Theorem 19.34 of [Margaris... |
| 19.36v 1983 | Version of ~ 19.36 with a ... |
| 19.12vvv 1984 | Version of ~ 19.12vv with ... |
| 19.27v 1985 | Version of ~ 19.27 with a ... |
| 19.28v 1986 | Version of ~ 19.28 with a ... |
| 19.37v 1987 | Version of ~ 19.37 with a ... |
| 19.44v 1988 | Version of ~ 19.44 with a ... |
| 19.45v 1989 | Version of ~ 19.45 with a ... |
| spimevw 1990 | Existential introduction, ... |
| spimvw 1991 | A weak form of specializat... |
| spvv 1992 | Specialization, using impl... |
| spfalw 1993 | Version of ~ sp when ` ph ... |
| chvarvv 1994 | Implicit substitution of `... |
| equs4v 1995 | Version of ~ equs4 with a ... |
| alequexv 1996 | Version of ~ equs4v with i... |
| exsbim 1997 | One direction of the equiv... |
| equsv 1998 | If a formula does not cont... |
| equsalvw 1999 | Version of ~ equsalv with ... |
| equsexvw 2000 | Version of ~ equsexv with ... |
| cbvaliw 2001 | Change bound variable. Us... |
| cbvalivw 2002 | Change bound variable. Us... |
| ax7v 2004 | Weakened version of ~ ax-7... |
| ax7v1 2005 | First of two weakened vers... |
| ax7v2 2006 | Second of two weakened ver... |
| equid 2007 | Identity law for equality.... |
| nfequid 2008 | Bound-variable hypothesis ... |
| equcomiv 2009 | Weaker form of ~ equcomi w... |
| ax6evr 2010 | A commuted form of ~ ax6ev... |
| ax7 2011 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2012 | Commutative law for equali... |
| equcom 2013 | Commutative law for equali... |
| equcomd 2014 | Deduction form of ~ equcom... |
| equcoms 2015 | An inference commuting equ... |
| equtr 2016 | A transitive law for equal... |
| equtrr 2017 | A transitive law for equal... |
| equeuclr 2018 | Commuted version of ~ eque... |
| equeucl 2019 | Equality is a left-Euclide... |
| equequ1 2020 | An equivalence law for equ... |
| equequ2 2021 | An equivalence law for equ... |
| equtr2 2022 | Equality is a left-Euclide... |
| stdpc6 2023 | One of the two equality ax... |
| equvinv 2024 | A variable introduction la... |
| equvinva 2025 | A modified version of the ... |
| equvelv 2026 | A biconditional form of ~ ... |
| ax13b 2027 | An equivalence between two... |
| spfw 2028 | Weak version of ~ sp . Us... |
| spw 2029 | Weak version of the specia... |
| cbvalw 2030 | Change bound variable. Us... |
| cbvalvw 2031 | Change bound variable. Us... |
| cbvexvw 2032 | Change bound variable. Us... |
| cbvaldvaw 2033 | Rule used to change the bo... |
| cbvexdvaw 2034 | Rule used to change the bo... |
| cbval2vw 2035 | Rule used to change bound ... |
| cbvex2vw 2036 | Rule used to change bound ... |
| cbvex4vw 2037 | Rule used to change bound ... |
| alcomiw 2038 | Weak version of ~ ax-11 . ... |
| alcomw 2039 | Weak version of ~ alcom an... |
| hbn1fw 2040 | Weak version of ~ ax-10 fr... |
| hbn1w 2041 | Weak version of ~ hbn1 . ... |
| hba1w 2042 | Weak version of ~ hba1 . ... |
| hbe1w 2043 | Weak version of ~ hbe1 . ... |
| hbalw 2044 | Weak version of ~ hbal . ... |
| 19.8aw 2045 | If a formula is true, then... |
| exexw 2046 | Existential quantification... |
| spaev 2047 | A special instance of ~ sp... |
| cbvaev 2048 | Change bound variable in a... |
| aevlem0 2049 | Lemma for ~ aevlem . Inst... |
| aevlem 2050 | Lemma for ~ aev and ~ axc1... |
| aeveq 2051 | The antecedent ` A. x x = ... |
| aev 2052 | A "distinctor elimination"... |
| aev2 2053 | A version of ~ aev with tw... |
| hbaev 2054 | All variables are effectiv... |
| naev 2055 | If some set variables can ... |
| naev2 2056 | Generalization of ~ hbnaev... |
| hbnaev 2057 | Any variable is free in ` ... |
| sbjust 2058 | Justification theorem for ... |
| sbt 2061 | A substitution into a theo... |
| sbtru 2062 | The result of substituting... |
| stdpc4 2063 | The specialization axiom o... |
| sbtALT 2064 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2065 | A double specialization us... |
| sbi1 2066 | Distribute substitution ov... |
| spsbim 2067 | Distribute substitution ov... |
| spsbbi 2068 | Biconditional property for... |
| sbimi 2069 | Distribute substitution ov... |
| sb2imi 2070 | Distribute substitution ov... |
| sbbii 2071 | Infer substitution into bo... |
| 2sbbii 2072 | Infer double substitution ... |
| sbimdv 2073 | Deduction substituting bot... |
| sbbidv 2074 | Deduction substituting bot... |
| sban 2075 | Conjunction inside and out... |
| sb3an 2076 | Threefold conjunction insi... |
| spsbe 2077 | Existential generalization... |
| sbequ 2078 | Equality property for subs... |
| sbequi 2079 | An equality theorem for su... |
| sb6 2080 | Alternate definition of su... |
| 2sb6 2081 | Equivalence for double sub... |
| sb1v 2082 | One direction of ~ sb5 , p... |
| sbv 2083 | Substitution for a variabl... |
| sbcom4 2084 | Commutativity law for subs... |
| pm11.07 2085 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2086 | Substitution in an implica... |
| sbievw 2087 | Conversion of implicit sub... |
| sbiedvw 2088 | Conversion of implicit sub... |
| 2sbievw 2089 | Conversion of double impli... |
| sbcom3vv 2090 | Substituting ` y ` for ` x... |
| sbievw2 2091 | ~ sbievw applied twice, av... |
| sbco2vv 2092 | A composition law for subs... |
| equsb3 2093 | Substitution in an equalit... |
| equsb3r 2094 | Substitution applied to th... |
| equsb1v 2095 | Substitution applied to an... |
| nsb 2096 | Any substitution in an alw... |
| sbn1 2097 | One direction of ~ sbn , u... |
| wel 2099 | Extend wff definition to i... |
| ax8v 2101 | Weakened version of ~ ax-8... |
| ax8v1 2102 | First of two weakened vers... |
| ax8v2 2103 | Second of two weakened ver... |
| ax8 2104 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2105 | An identity law for the no... |
| elsb1 2106 | Substitution for the first... |
| cleljust 2107 | When the class variables i... |
| ax9v 2109 | Weakened version of ~ ax-9... |
| ax9v1 2110 | First of two weakened vers... |
| ax9v2 2111 | Second of two weakened ver... |
| ax9 2112 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2113 | An identity law for the no... |
| elequ2g 2114 | A form of ~ elequ2 with a ... |
| elsb2 2115 | Substitution for the secon... |
| ax6dgen 2116 | Tarski's system uses the w... |
| ax10w 2117 | Weak version of ~ ax-10 fr... |
| ax11w 2118 | Weak version of ~ ax-11 fr... |
| ax11dgen 2119 | Degenerate instance of ~ a... |
| ax12wlem 2120 | Lemma for weak version of ... |
| ax12w 2121 | Weak version of ~ ax-12 fr... |
| ax12dgen 2122 | Degenerate instance of ~ a... |
| ax12wdemo 2123 | Example of an application ... |
| ax13w 2124 | Weak version (principal in... |
| ax13dgen1 2125 | Degenerate instance of ~ a... |
| ax13dgen2 2126 | Degenerate instance of ~ a... |
| ax13dgen3 2127 | Degenerate instance of ~ a... |
| ax13dgen4 2128 | Degenerate instance of ~ a... |
| hbn1 2130 | Alias for ~ ax-10 to be us... |
| hbe1 2131 | The setvar ` x ` is not fr... |
| hbe1a 2132 | Dual statement of ~ hbe1 .... |
| nf5-1 2133 | One direction of ~ nf5 can... |
| nf5i 2134 | Deduce that ` x ` is not f... |
| nf5dh 2135 | Deduce that ` x ` is not f... |
| nf5dv 2136 | Apply the definition of no... |
| nfnaew 2137 | All variables are effectiv... |
| nfnaewOLD 2138 | Obsolete version of ~ nfna... |
| nfe1 2139 | The setvar ` x ` is not fr... |
| nfa1 2140 | The setvar ` x ` is not fr... |
| nfna1 2141 | A convenience theorem part... |
| nfia1 2142 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2143 | The setvar ` x ` is not fr... |
| modal5 2144 | The analogue in our predic... |
| nfs1v 2145 | The setvar ` x ` is not fr... |
| alcoms 2147 | Swap quantifiers in an ant... |
| alcom 2148 | Theorem 19.5 of [Margaris]... |
| alrot3 2149 | Theorem *11.21 in [Whitehe... |
| alrot4 2150 | Rotate four universal quan... |
| excom 2151 | Theorem 19.11 of [Margaris... |
| excomim 2152 | One direction of Theorem 1... |
| excom13 2153 | Swap 1st and 3rd existenti... |
| exrot3 2154 | Rotate existential quantif... |
| exrot4 2155 | Rotate existential quantif... |
| hbal 2156 | If ` x ` is not free in ` ... |
| hbald 2157 | Deduction form of bound-va... |
| sbal 2158 | Move universal quantifier ... |
| sbalv 2159 | Quantify with new variable... |
| hbsbw 2160 | If ` z ` is not free in ` ... |
| hbsbwOLD 2161 | Obsolete version of ~ hbsb... |
| sbcom2 2162 | Commutativity law for subs... |
| sbco4lem 2163 | Lemma for ~ sbco4 . It re... |
| sbco4 2164 | Two ways of exchanging two... |
| nfa2 2165 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2167 | This is essentially Axiom ... |
| ax12v2 2168 | It is possible to remove a... |
| 19.8a 2169 | If a wff is true, it is tr... |
| 19.8ad 2170 | If a wff is true, it is tr... |
| sp 2171 | Specialization. A univers... |
| spi 2172 | Inference rule of universa... |
| sps 2173 | Generalization of antecede... |
| 2sp 2174 | A double specialization (s... |
| spsd 2175 | Deduction generalizing ant... |
| 19.2g 2176 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2177 | Inference form of ~ 19.21 ... |
| 19.21bbi 2178 | Inference removing two uni... |
| 19.23bi 2179 | Inference form of Theorem ... |
| nexr 2180 | Inference associated with ... |
| qexmid 2181 | Quantified excluded middle... |
| nf5r 2182 | Consequence of the definit... |
| nf5ri 2183 | Consequence of the definit... |
| nf5rd 2184 | Consequence of the definit... |
| spimedv 2185 | Deduction version of ~ spi... |
| spimefv 2186 | Version of ~ spime with a ... |
| nfim1 2187 | A closed form of ~ nfim . ... |
| nfan1 2188 | A closed form of ~ nfan . ... |
| 19.3t 2189 | Closed form of ~ 19.3 and ... |
| 19.3 2190 | A wff may be quantified wi... |
| 19.9d 2191 | A deduction version of one... |
| 19.9t 2192 | Closed form of ~ 19.9 and ... |
| 19.9 2193 | A wff may be existentially... |
| 19.21t 2194 | Closed form of Theorem 19.... |
| 19.21 2195 | Theorem 19.21 of [Margaris... |
| stdpc5 2196 | An axiom scheme of standar... |
| 19.21-2 2197 | Version of ~ 19.21 with tw... |
| 19.23t 2198 | Closed form of Theorem 19.... |
| 19.23 2199 | Theorem 19.23 of [Margaris... |
| alimd 2200 | Deduction form of Theorem ... |
| alrimi 2201 | Inference form of Theorem ... |
| alrimdd 2202 | Deduction form of Theorem ... |
| alrimd 2203 | Deduction form of Theorem ... |
| eximd 2204 | Deduction form of Theorem ... |
| exlimi 2205 | Inference associated with ... |
| exlimd 2206 | Deduction form of Theorem ... |
| exlimimdd 2207 | Existential elimination ru... |
| exlimdd 2208 | Existential elimination ru... |
| nexd 2209 | Deduction for generalizati... |
| albid 2210 | Formula-building rule for ... |
| exbid 2211 | Formula-building rule for ... |
| nfbidf 2212 | An equality theorem for ef... |
| 19.16 2213 | Theorem 19.16 of [Margaris... |
| 19.17 2214 | Theorem 19.17 of [Margaris... |
| 19.27 2215 | Theorem 19.27 of [Margaris... |
| 19.28 2216 | Theorem 19.28 of [Margaris... |
| 19.19 2217 | Theorem 19.19 of [Margaris... |
| 19.36 2218 | Theorem 19.36 of [Margaris... |
| 19.36i 2219 | Inference associated with ... |
| 19.37 2220 | Theorem 19.37 of [Margaris... |
| 19.32 2221 | Theorem 19.32 of [Margaris... |
| 19.31 2222 | Theorem 19.31 of [Margaris... |
| 19.41 2223 | Theorem 19.41 of [Margaris... |
| 19.42 2224 | Theorem 19.42 of [Margaris... |
| 19.44 2225 | Theorem 19.44 of [Margaris... |
| 19.45 2226 | Theorem 19.45 of [Margaris... |
| spimfv 2227 | Specialization, using impl... |
| chvarfv 2228 | Implicit substitution of `... |
| cbv3v2 2229 | Version of ~ cbv3 with two... |
| sbalex 2230 | Equivalence of two ways to... |
| sb4av 2231 | Version of ~ sb4a with a d... |
| sbimd 2232 | Deduction substituting bot... |
| sbbid 2233 | Deduction substituting bot... |
| 2sbbid 2234 | Deduction doubly substitut... |
| sbequ1 2235 | An equality theorem for su... |
| sbequ2 2236 | An equality theorem for su... |
| stdpc7 2237 | One of the two equality ax... |
| sbequ12 2238 | An equality theorem for su... |
| sbequ12r 2239 | An equality theorem for su... |
| sbelx 2240 | Elimination of substitutio... |
| sbequ12a 2241 | An equality theorem for su... |
| sbid 2242 | An identity theorem for su... |
| sbcov 2243 | A composition law for subs... |
| sb6a 2244 | Equivalence for substituti... |
| sbid2vw 2245 | Reverting substitution yie... |
| axc16g 2246 | Generalization of ~ axc16 ... |
| axc16 2247 | Proof of older axiom ~ ax-... |
| axc16gb 2248 | Biconditional strengthenin... |
| axc16nf 2249 | If ~ dtru is false, then t... |
| axc11v 2250 | Version of ~ axc11 with a ... |
| axc11rv 2251 | Version of ~ axc11r with a... |
| drsb2 2252 | Formula-building lemma for... |
| equsalv 2253 | An equivalence related to ... |
| equsexv 2254 | An equivalence related to ... |
| equsexvOLD 2255 | Obsolete version of ~ equs... |
| sbft 2256 | Substitution has no effect... |
| sbf 2257 | Substitution for a variabl... |
| sbf2 2258 | Substitution has no effect... |
| sbh 2259 | Substitution for a variabl... |
| hbs1 2260 | The setvar ` x ` is not fr... |
| nfs1f 2261 | If ` x ` is not free in ` ... |
| sb5 2262 | Alternate definition of su... |
| sb5OLD 2263 | Obsolete version of ~ sb5 ... |
| sb56OLD 2264 | Obsolete version of ~ sbal... |
| equs5av 2265 | A property related to subs... |
| 2sb5 2266 | Equivalence for double sub... |
| sbco4lemOLD 2267 | Obsolete version of ~ sbco... |
| dfsb7 2268 | An alternate definition of... |
| sbn 2269 | Negation inside and outsid... |
| sbex 2270 | Move existential quantifie... |
| nf5 2271 | Alternate definition of ~ ... |
| nf6 2272 | An alternate definition of... |
| nf5d 2273 | Deduce that ` x ` is not f... |
| nf5di 2274 | Since the converse holds b... |
| 19.9h 2275 | A wff may be existentially... |
| 19.21h 2276 | Theorem 19.21 of [Margaris... |
| 19.23h 2277 | Theorem 19.23 of [Margaris... |
| exlimih 2278 | Inference associated with ... |
| exlimdh 2279 | Deduction form of Theorem ... |
| equsalhw 2280 | Version of ~ equsalh with ... |
| equsexhv 2281 | An equivalence related to ... |
| hba1 2282 | The setvar ` x ` is not fr... |
| hbnt 2283 | Closed theorem version of ... |
| hbn 2284 | If ` x ` is not free in ` ... |
| hbnd 2285 | Deduction form of bound-va... |
| hbim1 2286 | A closed form of ~ hbim . ... |
| hbimd 2287 | Deduction form of bound-va... |
| hbim 2288 | If ` x ` is not free in ` ... |
| hban 2289 | If ` x ` is not free in ` ... |
| hb3an 2290 | If ` x ` is not free in ` ... |
| sbi2 2291 | Introduction of implicatio... |
| sbim 2292 | Implication inside and out... |
| sbrim 2293 | Substitution in an implica... |
| sbrimOLD 2294 | Obsolete version of ~ sbri... |
| sblim 2295 | Substitution in an implica... |
| sbor 2296 | Disjunction inside and out... |
| sbbi 2297 | Equivalence inside and out... |
| sblbis 2298 | Introduce left bicondition... |
| sbrbis 2299 | Introduce right biconditio... |
| sbrbif 2300 | Introduce right biconditio... |
| sbnf 2301 | Move nonfree predicate in ... |
| sbnfOLD 2302 | Obsolete version of ~ sbnf... |
| sbiev 2303 | Conversion of implicit sub... |
| sbiedw 2304 | Conversion of implicit sub... |
| axc7 2305 | Show that the original axi... |
| axc7e 2306 | Abbreviated version of ~ a... |
| modal-b 2307 | The analogue in our predic... |
| 19.9ht 2308 | A closed version of ~ 19.9... |
| axc4 2309 | Show that the original axi... |
| axc4i 2310 | Inference version of ~ axc... |
| nfal 2311 | If ` x ` is not free in ` ... |
| nfex 2312 | If ` x ` is not free in ` ... |
| hbex 2313 | If ` x ` is not free in ` ... |
| nfnf 2314 | If ` x ` is not free in ` ... |
| 19.12 2315 | Theorem 19.12 of [Margaris... |
| nfald 2316 | Deduction form of ~ nfal .... |
| nfexd 2317 | If ` x ` is not free in ` ... |
| nfsbv 2318 | If ` z ` is not free in ` ... |
| nfsbvOLD 2319 | Obsolete version of ~ nfsb... |
| sbco2v 2320 | A composition law for subs... |
| aaan 2321 | Distribute universal quant... |
| aaanOLD 2322 | Obsolete version of ~ aaan... |
| eeor 2323 | Distribute existential qua... |
| eeorOLD 2324 | Obsolete version of ~ eeor... |
| cbv3v 2325 | Rule used to change bound ... |
| cbv1v 2326 | Rule used to change bound ... |
| cbv2w 2327 | Rule used to change bound ... |
| cbvaldw 2328 | Deduction used to change b... |
| cbvexdw 2329 | Deduction used to change b... |
| cbv3hv 2330 | Rule used to change bound ... |
| cbvalv1 2331 | Rule used to change bound ... |
| cbvexv1 2332 | Rule used to change bound ... |
| cbval2v 2333 | Rule used to change bound ... |
| cbvex2v 2334 | Rule used to change bound ... |
| dvelimhw 2335 | Proof of ~ dvelimh without... |
| pm11.53 2336 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2337 | Special case of ~ 19.12 wh... |
| eean 2338 | Distribute existential qua... |
| eeanv 2339 | Distribute a pair of exist... |
| eeeanv 2340 | Distribute three existenti... |
| ee4anv 2341 | Distribute two pairs of ex... |
| sb8v 2342 | Substitution of variable i... |
| sb8f 2343 | Substitution of variable i... |
| sb8fOLD 2344 | Obsolete version of ~ sb8f... |
| sb8ef 2345 | Substitution of variable i... |
| 2sb8ef 2346 | An equivalent expression f... |
| sb6rfv 2347 | Reversed substitution. Ve... |
| sbnf2 2348 | Two ways of expressing " `... |
| exsb 2349 | An equivalent expression f... |
| 2exsb 2350 | An equivalent expression f... |
| sbbib 2351 | Reversal of substitution. ... |
| sbbibvv 2352 | Reversal of substitution. ... |
| cbvsbv 2353 | Change the bound variable ... |
| cbvsbvf 2354 | Change the bound variable ... |
| cleljustALT 2355 | Alternate proof of ~ clelj... |
| cleljustALT2 2356 | Alternate proof of ~ clelj... |
| equs5aALT 2357 | Alternate proof of ~ equs5... |
| equs5eALT 2358 | Alternate proof of ~ equs5... |
| axc11r 2359 | Same as ~ axc11 but with r... |
| dral1v 2360 | Formula-building lemma for... |
| dral1vOLD 2361 | Obsolete version of ~ dral... |
| drex1v 2362 | Formula-building lemma for... |
| drnf1v 2363 | Formula-building lemma for... |
| drnf1vOLD 2364 | Obsolete version of ~ drnf... |
| ax13v 2366 | A weaker version of ~ ax-1... |
| ax13lem1 2367 | A version of ~ ax13v with ... |
| ax13 2368 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2369 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2370 | An equation between setvar... |
| dveeq2 2371 | Quantifier introduction wh... |
| nfeqf1 2372 | An equation between setvar... |
| dveeq1 2373 | Quantifier introduction wh... |
| nfeqf 2374 | A variable is effectively ... |
| axc9 2375 | Derive set.mm's original ~... |
| ax6e 2376 | At least one individual ex... |
| ax6 2377 | Theorem showing that ~ ax-... |
| axc10 2378 | Show that the original axi... |
| spimt 2379 | Closed theorem form of ~ s... |
| spim 2380 | Specialization, using impl... |
| spimed 2381 | Deduction version of ~ spi... |
| spime 2382 | Existential introduction, ... |
| spimv 2383 | A version of ~ spim with a... |
| spimvALT 2384 | Alternate proof of ~ spimv... |
| spimev 2385 | Distinct-variable version ... |
| spv 2386 | Specialization, using impl... |
| spei 2387 | Inference from existential... |
| chvar 2388 | Implicit substitution of `... |
| chvarv 2389 | Implicit substitution of `... |
| cbv3 2390 | Rule used to change bound ... |
| cbval 2391 | Rule used to change bound ... |
| cbvex 2392 | Rule used to change bound ... |
| cbvalv 2393 | Rule used to change bound ... |
| cbvexv 2394 | Rule used to change bound ... |
| cbv1 2395 | Rule used to change bound ... |
| cbv2 2396 | Rule used to change bound ... |
| cbv3h 2397 | Rule used to change bound ... |
| cbv1h 2398 | Rule used to change bound ... |
| cbv2h 2399 | Rule used to change bound ... |
| cbvald 2400 | Deduction used to change b... |
| cbvexd 2401 | Deduction used to change b... |
| cbvaldva 2402 | Rule used to change the bo... |
| cbvexdva 2403 | Rule used to change the bo... |
| cbval2 2404 | Rule used to change bound ... |
| cbvex2 2405 | Rule used to change bound ... |
| cbval2vv 2406 | Rule used to change bound ... |
| cbvex2vv 2407 | Rule used to change bound ... |
| cbvex4v 2408 | Rule used to change bound ... |
| equs4 2409 | Lemma used in proofs of im... |
| equsal 2410 | An equivalence related to ... |
| equsex 2411 | An equivalence related to ... |
| equsexALT 2412 | Alternate proof of ~ equse... |
| equsalh 2413 | An equivalence related to ... |
| equsexh 2414 | An equivalence related to ... |
| axc15 2415 | Derivation of set.mm's ori... |
| ax12 2416 | Rederivation of Axiom ~ ax... |
| ax12b 2417 | A bidirectional version of... |
| ax13ALT 2418 | Alternate proof of ~ ax13 ... |
| axc11n 2419 | Derive set.mm's original ~... |
| aecom 2420 | Commutation law for identi... |
| aecoms 2421 | A commutation rule for ide... |
| naecoms 2422 | A commutation rule for dis... |
| axc11 2423 | Show that ~ ax-c11 can be ... |
| hbae 2424 | All variables are effectiv... |
| hbnae 2425 | All variables are effectiv... |
| nfae 2426 | All variables are effectiv... |
| nfnae 2427 | All variables are effectiv... |
| hbnaes 2428 | Rule that applies ~ hbnae ... |
| axc16i 2429 | Inference with ~ axc16 as ... |
| axc16nfALT 2430 | Alternate proof of ~ axc16... |
| dral2 2431 | Formula-building lemma for... |
| dral1 2432 | Formula-building lemma for... |
| dral1ALT 2433 | Alternate proof of ~ dral1... |
| drex1 2434 | Formula-building lemma for... |
| drex2 2435 | Formula-building lemma for... |
| drnf1 2436 | Formula-building lemma for... |
| drnf2 2437 | Formula-building lemma for... |
| nfald2 2438 | Variation on ~ nfald which... |
| nfexd2 2439 | Variation on ~ nfexd which... |
| exdistrf 2440 | Distribution of existentia... |
| dvelimf 2441 | Version of ~ dvelimv witho... |
| dvelimdf 2442 | Deduction form of ~ dvelim... |
| dvelimh 2443 | Version of ~ dvelim withou... |
| dvelim 2444 | This theorem can be used t... |
| dvelimv 2445 | Similar to ~ dvelim with f... |
| dvelimnf 2446 | Version of ~ dvelim using ... |
| dveeq2ALT 2447 | Alternate proof of ~ dveeq... |
| equvini 2448 | A variable introduction la... |
| equvel 2449 | A variable elimination law... |
| equs5a 2450 | A property related to subs... |
| equs5e 2451 | A property related to subs... |
| equs45f 2452 | Two ways of expressing sub... |
| equs5 2453 | Lemma used in proofs of su... |
| dveel1 2454 | Quantifier introduction wh... |
| dveel2 2455 | Quantifier introduction wh... |
| axc14 2456 | Axiom ~ ax-c14 is redundan... |
| sb6x 2457 | Equivalence involving subs... |
| sbequ5 2458 | Substitution does not chan... |
| sbequ6 2459 | Substitution does not chan... |
| sb5rf 2460 | Reversed substitution. Us... |
| sb6rf 2461 | Reversed substitution. Fo... |
| ax12vALT 2462 | Alternate proof of ~ ax12v... |
| 2ax6elem 2463 | We can always find values ... |
| 2ax6e 2464 | We can always find values ... |
| 2sb5rf 2465 | Reversed double substituti... |
| 2sb6rf 2466 | Reversed double substituti... |
| sbel2x 2467 | Elimination of double subs... |
| sb4b 2468 | Simplified definition of s... |
| sb3b 2469 | Simplified definition of s... |
| sb3 2470 | One direction of a simplif... |
| sb1 2471 | One direction of a simplif... |
| sb2 2472 | One direction of a simplif... |
| sb4a 2473 | A version of one implicati... |
| dfsb1 2474 | Alternate definition of su... |
| hbsb2 2475 | Bound-variable hypothesis ... |
| nfsb2 2476 | Bound-variable hypothesis ... |
| hbsb2a 2477 | Special case of a bound-va... |
| sb4e 2478 | One direction of a simplif... |
| hbsb2e 2479 | Special case of a bound-va... |
| hbsb3 2480 | If ` y ` is not free in ` ... |
| nfs1 2481 | If ` y ` is not free in ` ... |
| axc16ALT 2482 | Alternate proof of ~ axc16... |
| axc16gALT 2483 | Alternate proof of ~ axc16... |
| equsb1 2484 | Substitution applied to an... |
| equsb2 2485 | Substitution applied to an... |
| dfsb2 2486 | An alternate definition of... |
| dfsb3 2487 | An alternate definition of... |
| drsb1 2488 | Formula-building lemma for... |
| sb2ae 2489 | In the case of two success... |
| sb6f 2490 | Equivalence for substituti... |
| sb5f 2491 | Equivalence for substituti... |
| nfsb4t 2492 | A variable not free in a p... |
| nfsb4 2493 | A variable not free in a p... |
| sbequ8 2494 | Elimination of equality fr... |
| sbie 2495 | Conversion of implicit sub... |
| sbied 2496 | Conversion of implicit sub... |
| sbiedv 2497 | Conversion of implicit sub... |
| 2sbiev 2498 | Conversion of double impli... |
| sbcom3 2499 | Substituting ` y ` for ` x... |
| sbco 2500 | A composition law for subs... |
| sbid2 2501 | An identity law for substi... |
| sbid2v 2502 | An identity law for substi... |
| sbidm 2503 | An idempotent law for subs... |
| sbco2 2504 | A composition law for subs... |
| sbco2d 2505 | A composition law for subs... |
| sbco3 2506 | A composition law for subs... |
| sbcom 2507 | A commutativity law for su... |
| sbtrt 2508 | Partially closed form of ~... |
| sbtr 2509 | A partial converse to ~ sb... |
| sb8 2510 | Substitution of variable i... |
| sb8e 2511 | Substitution of variable i... |
| sb9 2512 | Commutation of quantificat... |
| sb9i 2513 | Commutation of quantificat... |
| sbhb 2514 | Two ways of expressing " `... |
| nfsbd 2515 | Deduction version of ~ nfs... |
| nfsb 2516 | If ` z ` is not free in ` ... |
| hbsb 2517 | If ` z ` is not free in ` ... |
| sb7f 2518 | This version of ~ dfsb7 do... |
| sb7h 2519 | This version of ~ dfsb7 do... |
| sb10f 2520 | Hao Wang's identity axiom ... |
| sbal1 2521 | Check out ~ sbal for a ver... |
| sbal2 2522 | Move quantifier in and out... |
| 2sb8e 2523 | An equivalent expression f... |
| dfmoeu 2524 | An elementary proof of ~ m... |
| dfeumo 2525 | An elementary proof showin... |
| mojust 2527 | Soundness justification th... |
| nexmo 2529 | Nonexistence implies uniqu... |
| exmo 2530 | Any proposition holds for ... |
| moabs 2531 | Absorption of existence co... |
| moim 2532 | The at-most-one quantifier... |
| moimi 2533 | The at-most-one quantifier... |
| moimdv 2534 | The at-most-one quantifier... |
| mobi 2535 | Equivalence theorem for th... |
| mobii 2536 | Formula-building rule for ... |
| mobidv 2537 | Formula-building rule for ... |
| mobid 2538 | Formula-building rule for ... |
| moa1 2539 | If an implication holds fo... |
| moan 2540 | "At most one" is still the... |
| moani 2541 | "At most one" is still tru... |
| moor 2542 | "At most one" is still the... |
| mooran1 2543 | "At most one" imports disj... |
| mooran2 2544 | "At most one" exports disj... |
| nfmo1 2545 | Bound-variable hypothesis ... |
| nfmod2 2546 | Bound-variable hypothesis ... |
| nfmodv 2547 | Bound-variable hypothesis ... |
| nfmov 2548 | Bound-variable hypothesis ... |
| nfmod 2549 | Bound-variable hypothesis ... |
| nfmo 2550 | Bound-variable hypothesis ... |
| mof 2551 | Version of ~ df-mo with di... |
| mo3 2552 | Alternate definition of th... |
| mo 2553 | Equivalent definitions of ... |
| mo4 2554 | At-most-one quantifier exp... |
| mo4f 2555 | At-most-one quantifier exp... |
| eu3v 2558 | An alternate way to expres... |
| eujust 2559 | Soundness justification th... |
| eujustALT 2560 | Alternate proof of ~ eujus... |
| eu6lem 2561 | Lemma of ~ eu6im . A diss... |
| eu6 2562 | Alternate definition of th... |
| eu6im 2563 | One direction of ~ eu6 nee... |
| euf 2564 | Version of ~ eu6 with disj... |
| euex 2565 | Existential uniqueness imp... |
| eumo 2566 | Existential uniqueness imp... |
| eumoi 2567 | Uniqueness inferred from e... |
| exmoeub 2568 | Existence implies that uni... |
| exmoeu 2569 | Existence is equivalent to... |
| moeuex 2570 | Uniqueness implies that ex... |
| moeu 2571 | Uniqueness is equivalent t... |
| eubi 2572 | Equivalence theorem for th... |
| eubii 2573 | Introduce unique existenti... |
| eubidv 2574 | Formula-building rule for ... |
| eubid 2575 | Formula-building rule for ... |
| nfeu1 2576 | Bound-variable hypothesis ... |
| nfeu1ALT 2577 | Alternate proof of ~ nfeu1... |
| nfeud2 2578 | Bound-variable hypothesis ... |
| nfeudw 2579 | Bound-variable hypothesis ... |
| nfeud 2580 | Bound-variable hypothesis ... |
| nfeuw 2581 | Bound-variable hypothesis ... |
| nfeu 2582 | Bound-variable hypothesis ... |
| dfeu 2583 | Rederive ~ df-eu from the ... |
| dfmo 2584 | Rederive ~ df-mo from the ... |
| euequ 2585 | There exists a unique set ... |
| sb8eulem 2586 | Lemma. Factor out the com... |
| sb8euv 2587 | Variable substitution in u... |
| sb8eu 2588 | Variable substitution in u... |
| sb8mo 2589 | Variable substitution for ... |
| cbvmovw 2590 | Change bound variable. Us... |
| cbvmow 2591 | Rule used to change bound ... |
| cbvmo 2592 | Rule used to change bound ... |
| cbveuvw 2593 | Change bound variable. Us... |
| cbveuw 2594 | Version of ~ cbveu with a ... |
| cbveu 2595 | Rule used to change bound ... |
| cbveuALT 2596 | Alternative proof of ~ cbv... |
| eu2 2597 | An alternate way of defini... |
| eu1 2598 | An alternate way to expres... |
| euor 2599 | Introduce a disjunct into ... |
| euorv 2600 | Introduce a disjunct into ... |
| euor2 2601 | Introduce or eliminate a d... |
| sbmo 2602 | Substitution into an at-mo... |
| eu4 2603 | Uniqueness using implicit ... |
| euimmo 2604 | Existential uniqueness imp... |
| euim 2605 | Add unique existential qua... |
| moanimlem 2606 | Factor out the common proo... |
| moanimv 2607 | Introduction of a conjunct... |
| moanim 2608 | Introduction of a conjunct... |
| euan 2609 | Introduction of a conjunct... |
| moanmo 2610 | Nested at-most-one quantif... |
| moaneu 2611 | Nested at-most-one and uni... |
| euanv 2612 | Introduction of a conjunct... |
| mopick 2613 | "At most one" picks a vari... |
| moexexlem 2614 | Factor out the proof skele... |
| 2moexv 2615 | Double quantification with... |
| moexexvw 2616 | "At most one" double quant... |
| 2moswapv 2617 | A condition allowing to sw... |
| 2euswapv 2618 | A condition allowing to sw... |
| 2euexv 2619 | Double quantification with... |
| 2exeuv 2620 | Double existential uniquen... |
| eupick 2621 | Existential uniqueness "pi... |
| eupicka 2622 | Version of ~ eupick with c... |
| eupickb 2623 | Existential uniqueness "pi... |
| eupickbi 2624 | Theorem *14.26 in [Whitehe... |
| mopick2 2625 | "At most one" can show the... |
| moexex 2626 | "At most one" double quant... |
| moexexv 2627 | "At most one" double quant... |
| 2moex 2628 | Double quantification with... |
| 2euex 2629 | Double quantification with... |
| 2eumo 2630 | Nested unique existential ... |
| 2eu2ex 2631 | Double existential uniquen... |
| 2moswap 2632 | A condition allowing to sw... |
| 2euswap 2633 | A condition allowing to sw... |
| 2exeu 2634 | Double existential uniquen... |
| 2mo2 2635 | Two ways of expressing "th... |
| 2mo 2636 | Two ways of expressing "th... |
| 2mos 2637 | Double "there exists at mo... |
| 2mosOLD 2638 | Obsolete version of ~ 2mos... |
| 2eu1 2639 | Double existential uniquen... |
| 2eu1v 2640 | Double existential uniquen... |
| 2eu2 2641 | Double existential uniquen... |
| 2eu3 2642 | Double existential uniquen... |
| 2eu4 2643 | This theorem provides us w... |
| 2eu5 2644 | An alternate definition of... |
| 2eu6 2645 | Two equivalent expressions... |
| 2eu7 2646 | Two equivalent expressions... |
| 2eu8 2647 | Two equivalent expressions... |
| euae 2648 | Two ways to express "exact... |
| exists1 2649 | Two ways to express "exact... |
| exists2 2650 | A condition implying that ... |
| barbara 2651 | "Barbara", one of the fund... |
| celarent 2652 | "Celarent", one of the syl... |
| darii 2653 | "Darii", one of the syllog... |
| dariiALT 2654 | Alternate proof of ~ darii... |
| ferio 2655 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2656 | Lemma for ~ barbari and th... |
| barbari 2657 | "Barbari", one of the syll... |
| barbariALT 2658 | Alternate proof of ~ barba... |
| celaront 2659 | "Celaront", one of the syl... |
| cesare 2660 | "Cesare", one of the syllo... |
| camestres 2661 | "Camestres", one of the sy... |
| festino 2662 | "Festino", one of the syll... |
| festinoALT 2663 | Alternate proof of ~ festi... |
| baroco 2664 | "Baroco", one of the syllo... |
| barocoALT 2665 | Alternate proof of ~ festi... |
| cesaro 2666 | "Cesaro", one of the syllo... |
| camestros 2667 | "Camestros", one of the sy... |
| datisi 2668 | "Datisi", one of the syllo... |
| disamis 2669 | "Disamis", one of the syll... |
| ferison 2670 | "Ferison", one of the syll... |
| bocardo 2671 | "Bocardo", one of the syll... |
| darapti 2672 | "Darapti", one of the syll... |
| daraptiALT 2673 | Alternate proof of ~ darap... |
| felapton 2674 | "Felapton", one of the syl... |
| calemes 2675 | "Calemes", one of the syll... |
| dimatis 2676 | "Dimatis", one of the syll... |
| fresison 2677 | "Fresison", one of the syl... |
| calemos 2678 | "Calemos", one of the syll... |
| fesapo 2679 | "Fesapo", one of the syllo... |
| bamalip 2680 | "Bamalip", one of the syll... |
| axia1 2681 | Left 'and' elimination (in... |
| axia2 2682 | Right 'and' elimination (i... |
| axia3 2683 | 'And' introduction (intuit... |
| axin1 2684 | 'Not' introduction (intuit... |
| axin2 2685 | 'Not' elimination (intuiti... |
| axio 2686 | Definition of 'or' (intuit... |
| axi4 2687 | Specialization (intuitioni... |
| axi5r 2688 | Converse of ~ axc4 (intuit... |
| axial 2689 | The setvar ` x ` is not fr... |
| axie1 2690 | The setvar ` x ` is not fr... |
| axie2 2691 | A key property of existent... |
| axi9 2692 | Axiom of existence (intuit... |
| axi10 2693 | Axiom of Quantifier Substi... |
| axi12 2694 | Axiom of Quantifier Introd... |
| axbnd 2695 | Axiom of Bundling (intuiti... |
| axexte 2697 | The axiom of extensionalit... |
| axextg 2698 | A generalization of the ax... |
| axextb 2699 | A bidirectional version of... |
| axextmo 2700 | There exists at most one s... |
| nulmo 2701 | There exists at most one e... |
| eleq1ab 2704 | Extension (in the sense of... |
| cleljustab 2705 | Extension of ~ cleljust fr... |
| abid 2706 | Simplification of class ab... |
| vexwt 2707 | A standard theorem of pred... |
| vexw 2708 | If ` ph ` is a theorem, th... |
| vextru 2709 | Every setvar is a member o... |
| nfsab1 2710 | Bound-variable hypothesis ... |
| hbab1 2711 | Bound-variable hypothesis ... |
| hbab1OLD 2712 | Obsolete version of ~ hbab... |
| hbab 2713 | Bound-variable hypothesis ... |
| hbabg 2714 | Bound-variable hypothesis ... |
| nfsab 2715 | Bound-variable hypothesis ... |
| nfsabg 2716 | Bound-variable hypothesis ... |
| dfcleq 2718 | The defining characterizat... |
| cvjust 2719 | Every set is a class. Pro... |
| ax9ALT 2720 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2721 | A weaker version of ~ eleq... |
| eqriv 2722 | Infer equality of classes ... |
| eqrdv 2723 | Deduce equality of classes... |
| eqrdav 2724 | Deduce equality of classes... |
| eqid 2725 | Law of identity (reflexivi... |
| eqidd 2726 | Class identity law with an... |
| eqeq1d 2727 | Deduction from equality to... |
| eqeq1dALT 2728 | Alternate proof of ~ eqeq1... |
| eqeq1 2729 | Equality implies equivalen... |
| eqeq1i 2730 | Inference from equality to... |
| eqcomd 2731 | Deduction from commutative... |
| eqcom 2732 | Commutative law for class ... |
| eqcoms 2733 | Inference applying commuta... |
| eqcomi 2734 | Inference from commutative... |
| neqcomd 2735 | Commute an inequality. (C... |
| eqeq2d 2736 | Deduction from equality to... |
| eqeq2 2737 | Equality implies equivalen... |
| eqeq2i 2738 | Inference from equality to... |
| eqeqan12d 2739 | A useful inference for sub... |
| eqeqan12rd 2740 | A useful inference for sub... |
| eqeq12d 2741 | A useful inference for sub... |
| eqeq12 2742 | Equality relationship amon... |
| eqeq12i 2743 | A useful inference for sub... |
| eqeq12OLD 2744 | Obsolete version of ~ eqeq... |
| eqeq12dOLD 2745 | Obsolete version of ~ eqeq... |
| eqeqan12dOLD 2746 | Obsolete version of ~ eqeq... |
| eqeqan12dALT 2747 | Alternate proof of ~ eqeqa... |
| eqtr 2748 | Transitive law for class e... |
| eqtr2 2749 | A transitive law for class... |
| eqtr2OLD 2750 | Obsolete version of eqtr2 ... |
| eqtr3 2751 | A transitive law for class... |
| eqtr3OLD 2752 | Obsolete version of ~ eqtr... |
| eqtri 2753 | An equality transitivity i... |
| eqtr2i 2754 | An equality transitivity i... |
| eqtr3i 2755 | An equality transitivity i... |
| eqtr4i 2756 | An equality transitivity i... |
| 3eqtri 2757 | An inference from three ch... |
| 3eqtrri 2758 | An inference from three ch... |
| 3eqtr2i 2759 | An inference from three ch... |
| 3eqtr2ri 2760 | An inference from three ch... |
| 3eqtr3i 2761 | An inference from three ch... |
| 3eqtr3ri 2762 | An inference from three ch... |
| 3eqtr4i 2763 | An inference from three ch... |
| 3eqtr4ri 2764 | An inference from three ch... |
| eqtrd 2765 | An equality transitivity d... |
| eqtr2d 2766 | An equality transitivity d... |
| eqtr3d 2767 | An equality transitivity e... |
| eqtr4d 2768 | An equality transitivity e... |
| 3eqtrd 2769 | A deduction from three cha... |
| 3eqtrrd 2770 | A deduction from three cha... |
| 3eqtr2d 2771 | A deduction from three cha... |
| 3eqtr2rd 2772 | A deduction from three cha... |
| 3eqtr3d 2773 | A deduction from three cha... |
| 3eqtr3rd 2774 | A deduction from three cha... |
| 3eqtr4d 2775 | A deduction from three cha... |
| 3eqtr4rd 2776 | A deduction from three cha... |
| eqtrid 2777 | An equality transitivity d... |
| eqtr2id 2778 | An equality transitivity d... |
| eqtr3id 2779 | An equality transitivity d... |
| eqtr3di 2780 | An equality transitivity d... |
| eqtrdi 2781 | An equality transitivity d... |
| eqtr2di 2782 | An equality transitivity d... |
| eqtr4di 2783 | An equality transitivity d... |
| eqtr4id 2784 | An equality transitivity d... |
| sylan9eq 2785 | An equality transitivity d... |
| sylan9req 2786 | An equality transitivity d... |
| sylan9eqr 2787 | An equality transitivity d... |
| 3eqtr3g 2788 | A chained equality inferen... |
| 3eqtr3a 2789 | A chained equality inferen... |
| 3eqtr4g 2790 | A chained equality inferen... |
| 3eqtr4a 2791 | A chained equality inferen... |
| eq2tri 2792 | A compound transitive infe... |
| abbi 2793 | Equivalent formulas yield ... |
| abbidv 2794 | Equivalent wff's yield equ... |
| abbii 2795 | Equivalent wff's yield equ... |
| abbid 2796 | Equivalent wff's yield equ... |
| abbib 2797 | Equal class abstractions r... |
| cbvabv 2798 | Rule used to change bound ... |
| cbvabw 2799 | Rule used to change bound ... |
| cbvab 2800 | Rule used to change bound ... |
| eqabbw 2801 | Version of ~ eqabb using i... |
| dfclel 2803 | Characterization of the el... |
| elex2 2804 | If a class contains anothe... |
| issetlem 2805 | Lemma for ~ elisset and ~ ... |
| elissetv 2806 | An element of a class exis... |
| elisset 2807 | An element of a class exis... |
| eleq1w 2808 | Weaker version of ~ eleq1 ... |
| eleq2w 2809 | Weaker version of ~ eleq2 ... |
| eleq1d 2810 | Deduction from equality to... |
| eleq2d 2811 | Deduction from equality to... |
| eleq2dALT 2812 | Alternate proof of ~ eleq2... |
| eleq1 2813 | Equality implies equivalen... |
| eleq2 2814 | Equality implies equivalen... |
| eleq12 2815 | Equality implies equivalen... |
| eleq1i 2816 | Inference from equality to... |
| eleq2i 2817 | Inference from equality to... |
| eleq12i 2818 | Inference from equality to... |
| eleq12d 2819 | Deduction from equality to... |
| eleq1a 2820 | A transitive-type law rela... |
| eqeltri 2821 | Substitution of equal clas... |
| eqeltrri 2822 | Substitution of equal clas... |
| eleqtri 2823 | Substitution of equal clas... |
| eleqtrri 2824 | Substitution of equal clas... |
| eqeltrd 2825 | Substitution of equal clas... |
| eqeltrrd 2826 | Deduction that substitutes... |
| eleqtrd 2827 | Deduction that substitutes... |
| eleqtrrd 2828 | Deduction that substitutes... |
| eqeltrid 2829 | A membership and equality ... |
| eqeltrrid 2830 | A membership and equality ... |
| eleqtrid 2831 | A membership and equality ... |
| eleqtrrid 2832 | A membership and equality ... |
| eqeltrdi 2833 | A membership and equality ... |
| eqeltrrdi 2834 | A membership and equality ... |
| eleqtrdi 2835 | A membership and equality ... |
| eleqtrrdi 2836 | A membership and equality ... |
| 3eltr3i 2837 | Substitution of equal clas... |
| 3eltr4i 2838 | Substitution of equal clas... |
| 3eltr3d 2839 | Substitution of equal clas... |
| 3eltr4d 2840 | Substitution of equal clas... |
| 3eltr3g 2841 | Substitution of equal clas... |
| 3eltr4g 2842 | Substitution of equal clas... |
| eleq2s 2843 | Substitution of equal clas... |
| eqneltri 2844 | If a class is not an eleme... |
| eqneltrd 2845 | If a class is not an eleme... |
| eqneltrrd 2846 | If a class is not an eleme... |
| neleqtrd 2847 | If a class is not an eleme... |
| neleqtrrd 2848 | If a class is not an eleme... |
| nelneq 2849 | A way of showing two class... |
| nelneq2 2850 | A way of showing two class... |
| eqsb1 2851 | Substitution for the left-... |
| clelsb1 2852 | Substitution for the first... |
| clelsb2 2853 | Substitution for the secon... |
| clelsb2OLD 2854 | Obsolete version of ~ clel... |
| cleqh 2855 | Establish equality between... |
| hbxfreq 2856 | A utility lemma to transfe... |
| hblem 2857 | Change the free variable o... |
| hblemg 2858 | Change the free variable o... |
| eqabdv 2859 | Deduction from a wff to a ... |
| eqabcdv 2860 | Deduction from a wff to a ... |
| eqabi 2861 | Equality of a class variab... |
| abid1 2862 | Every class is equal to a ... |
| abid2 2863 | A simplification of class ... |
| eqab 2864 | One direction of ~ eqabb i... |
| eqabb 2865 | Equality of a class variab... |
| eqabbOLD 2866 | Obsolete version of ~ eqab... |
| eqabcb 2867 | Equality of a class variab... |
| eqabrd 2868 | Equality of a class variab... |
| eqabri 2869 | Equality of a class variab... |
| eqabcri 2870 | Equality of a class variab... |
| clelab 2871 | Membership of a class vari... |
| clelabOLD 2872 | Obsolete version of ~ clel... |
| clabel 2873 | Membership of a class abst... |
| sbab 2874 | The right-hand side of the... |
| nfcjust 2876 | Justification theorem for ... |
| nfci 2878 | Deduce that a class ` A ` ... |
| nfcii 2879 | Deduce that a class ` A ` ... |
| nfcr 2880 | Consequence of the not-fre... |
| nfcrALT 2881 | Alternate version of ~ nfc... |
| nfcri 2882 | Consequence of the not-fre... |
| nfcd 2883 | Deduce that a class ` A ` ... |
| nfcrd 2884 | Consequence of the not-fre... |
| nfcriOLD 2885 | Obsolete version of ~ nfcr... |
| nfcrii 2886 | Consequence of the not-fre... |
| nfceqdf 2887 | An equality theorem for ef... |
| nfceqdfOLD 2888 | Obsolete version of ~ nfce... |
| nfceqi 2889 | Equality theorem for class... |
| nfcxfr 2890 | A utility lemma to transfe... |
| nfcxfrd 2891 | A utility lemma to transfe... |
| nfcv 2892 | If ` x ` is disjoint from ... |
| nfcvd 2893 | If ` x ` is disjoint from ... |
| nfab1 2894 | Bound-variable hypothesis ... |
| nfnfc1 2895 | The setvar ` x ` is bound ... |
| clelsb1fw 2896 | Substitution for the first... |
| clelsb1f 2897 | Substitution for the first... |
| nfab 2898 | Bound-variable hypothesis ... |
| nfabg 2899 | Bound-variable hypothesis ... |
| nfaba1 2900 | Bound-variable hypothesis ... |
| nfaba1OLD 2901 | Obsolete version of ~ nfab... |
| nfaba1g 2902 | Bound-variable hypothesis ... |
| nfeqd 2903 | Hypothesis builder for equ... |
| nfeld 2904 | Hypothesis builder for ele... |
| nfnfc 2905 | Hypothesis builder for ` F... |
| nfeq 2906 | Hypothesis builder for equ... |
| nfel 2907 | Hypothesis builder for ele... |
| nfeq1 2908 | Hypothesis builder for equ... |
| nfel1 2909 | Hypothesis builder for ele... |
| nfeq2 2910 | Hypothesis builder for equ... |
| nfel2 2911 | Hypothesis builder for ele... |
| drnfc1 2912 | Formula-building lemma for... |
| drnfc1OLD 2913 | Obsolete version of ~ drnf... |
| drnfc2 2914 | Formula-building lemma for... |
| drnfc2OLD 2915 | Obsolete version of ~ drnf... |
| nfabdw 2916 | Bound-variable hypothesis ... |
| nfabdwOLD 2917 | Obsolete version of ~ nfab... |
| nfabd 2918 | Bound-variable hypothesis ... |
| nfabd2 2919 | Bound-variable hypothesis ... |
| dvelimdc 2920 | Deduction form of ~ dvelim... |
| dvelimc 2921 | Version of ~ dvelim for cl... |
| nfcvf 2922 | If ` x ` and ` y ` are dis... |
| nfcvf2 2923 | If ` x ` and ` y ` are dis... |
| cleqf 2924 | Establish equality between... |
| eqabf 2925 | Equality of a class variab... |
| abid2f 2926 | A simplification of class ... |
| abid2fOLD 2927 | Obsolete version of ~ abid... |
| sbabel 2928 | Theorem to move a substitu... |
| sbabelOLD 2929 | Obsolete version of ~ sbab... |
| neii 2932 | Inference associated with ... |
| neir 2933 | Inference associated with ... |
| nne 2934 | Negation of inequality. (... |
| neneqd 2935 | Deduction eliminating ineq... |
| neneq 2936 | From inequality to non-equ... |
| neqned 2937 | If it is not the case that... |
| neqne 2938 | From non-equality to inequ... |
| neirr 2939 | No class is unequal to its... |
| exmidne 2940 | Excluded middle with equal... |
| eqneqall 2941 | A contradiction concerning... |
| nonconne 2942 | Law of noncontradiction wi... |
| necon3ad 2943 | Contrapositive law deducti... |
| necon3bd 2944 | Contrapositive law deducti... |
| necon2ad 2945 | Contrapositive inference f... |
| necon2bd 2946 | Contrapositive inference f... |
| necon1ad 2947 | Contrapositive deduction f... |
| necon1bd 2948 | Contrapositive deduction f... |
| necon4ad 2949 | Contrapositive inference f... |
| necon4bd 2950 | Contrapositive inference f... |
| necon3d 2951 | Contrapositive law deducti... |
| necon1d 2952 | Contrapositive law deducti... |
| necon2d 2953 | Contrapositive inference f... |
| necon4d 2954 | Contrapositive inference f... |
| necon3ai 2955 | Contrapositive inference f... |
| necon3aiOLD 2956 | Obsolete version of ~ neco... |
| necon3bi 2957 | Contrapositive inference f... |
| necon1ai 2958 | Contrapositive inference f... |
| necon1bi 2959 | Contrapositive inference f... |
| necon2ai 2960 | Contrapositive inference f... |
| necon2bi 2961 | Contrapositive inference f... |
| necon4ai 2962 | Contrapositive inference f... |
| necon3i 2963 | Contrapositive inference f... |
| necon1i 2964 | Contrapositive inference f... |
| necon2i 2965 | Contrapositive inference f... |
| necon4i 2966 | Contrapositive inference f... |
| necon3abid 2967 | Deduction from equality to... |
| necon3bbid 2968 | Deduction from equality to... |
| necon1abid 2969 | Contrapositive deduction f... |
| necon1bbid 2970 | Contrapositive inference f... |
| necon4abid 2971 | Contrapositive law deducti... |
| necon4bbid 2972 | Contrapositive law deducti... |
| necon2abid 2973 | Contrapositive deduction f... |
| necon2bbid 2974 | Contrapositive deduction f... |
| necon3bid 2975 | Deduction from equality to... |
| necon4bid 2976 | Contrapositive law deducti... |
| necon3abii 2977 | Deduction from equality to... |
| necon3bbii 2978 | Deduction from equality to... |
| necon1abii 2979 | Contrapositive inference f... |
| necon1bbii 2980 | Contrapositive inference f... |
| necon2abii 2981 | Contrapositive inference f... |
| necon2bbii 2982 | Contrapositive inference f... |
| necon3bii 2983 | Inference from equality to... |
| necom 2984 | Commutation of inequality.... |
| necomi 2985 | Inference from commutative... |
| necomd 2986 | Deduction from commutative... |
| nesym 2987 | Characterization of inequa... |
| nesymi 2988 | Inference associated with ... |
| nesymir 2989 | Inference associated with ... |
| neeq1d 2990 | Deduction for inequality. ... |
| neeq2d 2991 | Deduction for inequality. ... |
| neeq12d 2992 | Deduction for inequality. ... |
| neeq1 2993 | Equality theorem for inequ... |
| neeq2 2994 | Equality theorem for inequ... |
| neeq1i 2995 | Inference for inequality. ... |
| neeq2i 2996 | Inference for inequality. ... |
| neeq12i 2997 | Inference for inequality. ... |
| eqnetrd 2998 | Substitution of equal clas... |
| eqnetrrd 2999 | Substitution of equal clas... |
| neeqtrd 3000 | Substitution of equal clas... |
| eqnetri 3001 | Substitution of equal clas... |
| eqnetrri 3002 | Substitution of equal clas... |
| neeqtri 3003 | Substitution of equal clas... |
| neeqtrri 3004 | Substitution of equal clas... |
| neeqtrrd 3005 | Substitution of equal clas... |
| eqnetrrid 3006 | A chained equality inferen... |
| 3netr3d 3007 | Substitution of equality i... |
| 3netr4d 3008 | Substitution of equality i... |
| 3netr3g 3009 | Substitution of equality i... |
| 3netr4g 3010 | Substitution of equality i... |
| nebi 3011 | Contraposition law for ine... |
| pm13.18 3012 | Theorem *13.18 in [Whitehe... |
| pm13.181 3013 | Theorem *13.181 in [Whiteh... |
| pm13.181OLD 3014 | Obsolete version of ~ pm13... |
| pm2.61ine 3015 | Inference eliminating an i... |
| pm2.21ddne 3016 | A contradiction implies an... |
| pm2.61ne 3017 | Deduction eliminating an i... |
| pm2.61dne 3018 | Deduction eliminating an i... |
| pm2.61dane 3019 | Deduction eliminating an i... |
| pm2.61da2ne 3020 | Deduction eliminating two ... |
| pm2.61da3ne 3021 | Deduction eliminating thre... |
| pm2.61iine 3022 | Equality version of ~ pm2.... |
| mteqand 3023 | A modus tollens deduction ... |
| neor 3024 | Logical OR with an equalit... |
| neanior 3025 | A De Morgan's law for ineq... |
| ne3anior 3026 | A De Morgan's law for ineq... |
| neorian 3027 | A De Morgan's law for ineq... |
| nemtbir 3028 | An inference from an inequ... |
| nelne1 3029 | Two classes are different ... |
| nelne2 3030 | Two classes are different ... |
| nelelne 3031 | Two classes are different ... |
| neneor 3032 | If two classes are differe... |
| nfne 3033 | Bound-variable hypothesis ... |
| nfned 3034 | Bound-variable hypothesis ... |
| nabbib 3035 | Not equivalent wff's corre... |
| neli 3038 | Inference associated with ... |
| nelir 3039 | Inference associated with ... |
| nelcon3d 3040 | Contrapositive law deducti... |
| neleq12d 3041 | Equality theorem for negat... |
| neleq1 3042 | Equality theorem for negat... |
| neleq2 3043 | Equality theorem for negat... |
| nfnel 3044 | Bound-variable hypothesis ... |
| nfneld 3045 | Bound-variable hypothesis ... |
| nnel 3046 | Negation of negated member... |
| elnelne1 3047 | Two classes are different ... |
| elnelne2 3048 | Two classes are different ... |
| pm2.24nel 3049 | A contradiction concerning... |
| pm2.61danel 3050 | Deduction eliminating an e... |
| rgen 3053 | Generalization rule for re... |
| ralel 3054 | All elements of a class ar... |
| rgenw 3055 | Generalization rule for re... |
| rgen2w 3056 | Generalization rule for re... |
| mprg 3057 | Modus ponens combined with... |
| mprgbir 3058 | Modus ponens on biconditio... |
| raln 3059 | Restricted universally qua... |
| ralnex 3062 | Relationship between restr... |
| dfrex2 3063 | Relationship between restr... |
| nrex 3064 | Inference adding restricte... |
| alral 3065 | Universal quantification i... |
| rexex 3066 | Restricted existence impli... |
| rextru 3067 | Two ways of expressing tha... |
| ralimi2 3068 | Inference quantifying both... |
| reximi2 3069 | Inference quantifying both... |
| ralimia 3070 | Inference quantifying both... |
| reximia 3071 | Inference quantifying both... |
| ralimiaa 3072 | Inference quantifying both... |
| ralimi 3073 | Inference quantifying both... |
| reximi 3074 | Inference quantifying both... |
| ral2imi 3075 | Inference quantifying ante... |
| ralim 3076 | Distribution of restricted... |
| rexim 3077 | Theorem 19.22 of [Margaris... |
| reximiaOLD 3078 | Obsolete version of ~ rexi... |
| ralbii2 3079 | Inference adding different... |
| rexbii2 3080 | Inference adding different... |
| ralbiia 3081 | Inference adding restricte... |
| rexbiia 3082 | Inference adding restricte... |
| ralbii 3083 | Inference adding restricte... |
| rexbii 3084 | Inference adding restricte... |
| ralanid 3085 | Cancellation law for restr... |
| rexanid 3086 | Cancellation law for restr... |
| ralcom3 3087 | A commutation law for rest... |
| ralcom3OLD 3088 | Obsolete version of ~ ralc... |
| dfral2 3089 | Relationship between restr... |
| rexnal 3090 | Relationship between restr... |
| ralinexa 3091 | A transformation of restri... |
| rexanali 3092 | A transformation of restri... |
| ralbi 3093 | Distribute a restricted un... |
| rexbi 3094 | Distribute restricted quan... |
| rexbiOLD 3095 | Obsolete version of ~ rexb... |
| ralrexbid 3096 | Formula-building rule for ... |
| ralrexbidOLD 3097 | Obsolete version of ~ ralr... |
| r19.35 3098 | Restricted quantifier vers... |
| r19.35OLD 3099 | Obsolete version of ~ 19.3... |
| r19.26m 3100 | Version of ~ 19.26 and ~ r... |
| r19.26 3101 | Restricted quantifier vers... |
| r19.26-3 3102 | Version of ~ r19.26 with t... |
| ralbiim 3103 | Split a biconditional and ... |
| r19.29 3104 | Restricted quantifier vers... |
| r19.29OLD 3105 | Obsolete version of ~ r19.... |
| r19.29r 3106 | Restricted quantifier vers... |
| r19.29rOLD 3107 | Obsolete version of ~ r19.... |
| r19.29imd 3108 | Theorem 19.29 of [Margaris... |
| r19.40 3109 | Restricted quantifier vers... |
| r19.30 3110 | Restricted quantifier vers... |
| r19.30OLD 3111 | Obsolete version of ~ 19.3... |
| r19.43 3112 | Restricted quantifier vers... |
| 2ralimi 3113 | Inference quantifying both... |
| 3ralimi 3114 | Inference quantifying both... |
| 4ralimi 3115 | Inference quantifying both... |
| 5ralimi 3116 | Inference quantifying both... |
| 6ralimi 3117 | Inference quantifying both... |
| 2ralbii 3118 | Inference adding two restr... |
| 2rexbii 3119 | Inference adding two restr... |
| 3ralbii 3120 | Inference adding three res... |
| 4ralbii 3121 | Inference adding four rest... |
| 2ralbiim 3122 | Split a biconditional and ... |
| ralnex2 3123 | Relationship between two r... |
| ralnex3 3124 | Relationship between three... |
| rexnal2 3125 | Relationship between two r... |
| rexnal3 3126 | Relationship between three... |
| nrexralim 3127 | Negation of a complex pred... |
| r19.26-2 3128 | Restricted quantifier vers... |
| 2r19.29 3129 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3130 | Theorem 19.29 of [Margaris... |
| r19.29d2rOLD 3131 | Obsolete version of ~ r19.... |
| r2allem 3132 | Lemma factoring out common... |
| r2exlem 3133 | Lemma factoring out common... |
| hbralrimi 3134 | Inference from Theorem 19.... |
| ralrimiv 3135 | Inference from Theorem 19.... |
| ralrimiva 3136 | Inference from Theorem 19.... |
| rexlimiva 3137 | Inference from Theorem 19.... |
| rexlimiv 3138 | Inference from Theorem 19.... |
| nrexdv 3139 | Deduction adding restricte... |
| ralrimivw 3140 | Inference from Theorem 19.... |
| rexlimivw 3141 | Weaker version of ~ rexlim... |
| ralrimdv 3142 | Inference from Theorem 19.... |
| rexlimdv 3143 | Inference from Theorem 19.... |
| ralrimdva 3144 | Inference from Theorem 19.... |
| rexlimdva 3145 | Inference from Theorem 19.... |
| rexlimdvaa 3146 | Inference from Theorem 19.... |
| rexlimdva2 3147 | Inference from Theorem 19.... |
| r19.29an 3148 | A commonly used pattern in... |
| rexlimdv3a 3149 | Inference from Theorem 19.... |
| rexlimdvw 3150 | Inference from Theorem 19.... |
| rexlimddv 3151 | Restricted existential eli... |
| r19.29a 3152 | A commonly used pattern in... |
| ralimdv2 3153 | Inference quantifying both... |
| reximdv2 3154 | Deduction quantifying both... |
| reximdvai 3155 | Deduction quantifying both... |
| reximdvaiOLD 3156 | Obsolete version of ~ rexi... |
| ralimdva 3157 | Deduction quantifying both... |
| reximdva 3158 | Deduction quantifying both... |
| ralimdv 3159 | Deduction quantifying both... |
| reximdv 3160 | Deduction from Theorem 19.... |
| reximddv 3161 | Deduction from Theorem 19.... |
| reximddv3 3162 | Deduction from Theorem 19.... |
| reximssdv 3163 | Derivation of a restricted... |
| ralbidv2 3164 | Formula-building rule for ... |
| rexbidv2 3165 | Formula-building rule for ... |
| ralbidva 3166 | Formula-building rule for ... |
| rexbidva 3167 | Formula-building rule for ... |
| ralbidv 3168 | Formula-building rule for ... |
| rexbidv 3169 | Formula-building rule for ... |
| r19.21v 3170 | Restricted quantifier vers... |
| r19.21vOLD 3171 | Obsolete version of ~ r19.... |
| r19.37v 3172 | Restricted quantifier vers... |
| r19.23v 3173 | Restricted quantifier vers... |
| r19.36v 3174 | Restricted quantifier vers... |
| rexlimivOLD 3175 | Obsolete version of ~ rexl... |
| rexlimivaOLD 3176 | Obsolete version of ~ rexl... |
| rexlimivwOLD 3177 | Obsolete version of ~ rexl... |
| r19.27v 3178 | Restricted quantitifer ver... |
| r19.41v 3179 | Restricted quantifier vers... |
| r19.28v 3180 | Restricted quantifier vers... |
| r19.42v 3181 | Restricted quantifier vers... |
| r19.32v 3182 | Restricted quantifier vers... |
| r19.45v 3183 | Restricted quantifier vers... |
| r19.44v 3184 | One direction of a restric... |
| r2al 3185 | Double restricted universa... |
| r2ex 3186 | Double restricted existent... |
| r3al 3187 | Triple restricted universa... |
| rgen2 3188 | Generalization rule for re... |
| ralrimivv 3189 | Inference from Theorem 19.... |
| rexlimivv 3190 | Inference from Theorem 19.... |
| ralrimivva 3191 | Inference from Theorem 19.... |
| ralrimdvv 3192 | Inference from Theorem 19.... |
| rgen3 3193 | Generalization rule for re... |
| ralrimivvva 3194 | Inference from Theorem 19.... |
| ralimdvva 3195 | Deduction doubly quantifyi... |
| reximdvva 3196 | Deduction doubly quantifyi... |
| ralimdvv 3197 | Deduction doubly quantifyi... |
| ralimd4v 3198 | Deduction quadrupally quan... |
| ralimd6v 3199 | Deduction sextupally quant... |
| ralrimdvva 3200 | Inference from Theorem 19.... |
| rexlimdvv 3201 | Inference from Theorem 19.... |
| rexlimdvva 3202 | Inference from Theorem 19.... |
| reximddv2 3203 | Double deduction from Theo... |
| r19.29vva 3204 | A commonly used pattern ba... |
| r19.29vvaOLD 3205 | Obsolete version of ~ r19.... |
| 2rexbiia 3206 | Inference adding two restr... |
| 2ralbidva 3207 | Formula-building rule for ... |
| 2rexbidva 3208 | Formula-building rule for ... |
| 2ralbidv 3209 | Formula-building rule for ... |
| 2rexbidv 3210 | Formula-building rule for ... |
| rexralbidv 3211 | Formula-building rule for ... |
| 3ralbidv 3212 | Formula-building rule for ... |
| 4ralbidv 3213 | Formula-building rule for ... |
| 6ralbidv 3214 | Formula-building rule for ... |
| r19.41vv 3215 | Version of ~ r19.41v with ... |
| reeanlem 3216 | Lemma factoring out common... |
| reeanv 3217 | Rearrange restricted exist... |
| 3reeanv 3218 | Rearrange three restricted... |
| 2ralor 3219 | Distribute restricted univ... |
| 2ralorOLD 3220 | Obsolete version of ~ 2ral... |
| risset 3221 | Two ways to say " ` A ` be... |
| nelb 3222 | A definition of ` -. A e. ... |
| nelbOLD 3223 | Obsolete version of ~ nelb... |
| rspw 3224 | Restricted specialization.... |
| cbvralvw 3225 | Change the bound variable ... |
| cbvrexvw 3226 | Change the bound variable ... |
| cbvraldva 3227 | Rule used to change the bo... |
| cbvrexdva 3228 | Rule used to change the bo... |
| cbvral2vw 3229 | Change bound variables of ... |
| cbvrex2vw 3230 | Change bound variables of ... |
| cbvral3vw 3231 | Change bound variables of ... |
| cbvral4vw 3232 | Change bound variables of ... |
| cbvral6vw 3233 | Change bound variables of ... |
| cbvral8vw 3234 | Change bound variables of ... |
| rsp 3235 | Restricted specialization.... |
| rspa 3236 | Restricted specialization.... |
| rspe 3237 | Restricted specialization.... |
| rspec 3238 | Specialization rule for re... |
| r19.21bi 3239 | Inference from Theorem 19.... |
| r19.21be 3240 | Inference from Theorem 19.... |
| r19.21t 3241 | Restricted quantifier vers... |
| r19.21 3242 | Restricted quantifier vers... |
| r19.23t 3243 | Closed theorem form of ~ r... |
| r19.23 3244 | Restricted quantifier vers... |
| ralrimi 3245 | Inference from Theorem 19.... |
| ralrimia 3246 | Inference from Theorem 19.... |
| rexlimi 3247 | Restricted quantifier vers... |
| ralimdaa 3248 | Deduction quantifying both... |
| reximdai 3249 | Deduction from Theorem 19.... |
| r19.37 3250 | Restricted quantifier vers... |
| r19.41 3251 | Restricted quantifier vers... |
| ralrimd 3252 | Inference from Theorem 19.... |
| rexlimd2 3253 | Version of ~ rexlimd with ... |
| rexlimd 3254 | Deduction form of ~ rexlim... |
| r19.29af2 3255 | A commonly used pattern ba... |
| r19.29af 3256 | A commonly used pattern ba... |
| reximd2a 3257 | Deduction quantifying both... |
| ralbida 3258 | Formula-building rule for ... |
| ralbidaOLD 3259 | Obsolete version of ~ ralb... |
| rexbida 3260 | Formula-building rule for ... |
| ralbid 3261 | Formula-building rule for ... |
| rexbid 3262 | Formula-building rule for ... |
| rexbidvALT 3263 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3264 | Alternate proof of ~ rexbi... |
| rsp2 3265 | Restricted specialization,... |
| rsp2e 3266 | Restricted specialization.... |
| rspec2 3267 | Specialization rule for re... |
| rspec3 3268 | Specialization rule for re... |
| r2alf 3269 | Double restricted universa... |
| r2exf 3270 | Double restricted existent... |
| 2ralbida 3271 | Formula-building rule for ... |
| nfra1 3272 | The setvar ` x ` is not fr... |
| nfre1 3273 | The setvar ` x ` is not fr... |
| ralcom4 3274 | Commutation of restricted ... |
| ralcom4OLD 3275 | Obsolete version of ~ ralc... |
| rexcom4 3276 | Commutation of restricted ... |
| ralcom 3277 | Commutation of restricted ... |
| rexcom 3278 | Commutation of restricted ... |
| rexcomOLD 3279 | Obsolete version of ~ rexc... |
| rexcom4a 3280 | Specialized existential co... |
| ralrot3 3281 | Rotate three restricted un... |
| ralcom13 3282 | Swap first and third restr... |
| ralcom13OLD 3283 | Obsolete version of ~ ralc... |
| rexcom13 3284 | Swap first and third restr... |
| rexrot4 3285 | Rotate four restricted exi... |
| 2ex2rexrot 3286 | Rotate two existential qua... |
| nfra2w 3287 | Similar to Lemma 24 of [Mo... |
| nfra2wOLD 3288 | Obsolete version of ~ nfra... |
| hbra1 3289 | The setvar ` x ` is not fr... |
| ralcomf 3290 | Commutation of restricted ... |
| rexcomf 3291 | Commutation of restricted ... |
| cbvralfw 3292 | Rule used to change bound ... |
| cbvrexfw 3293 | Rule used to change bound ... |
| cbvralw 3294 | Rule used to change bound ... |
| cbvrexw 3295 | Rule used to change bound ... |
| hbral 3296 | Bound-variable hypothesis ... |
| nfraldw 3297 | Deduction version of ~ nfr... |
| nfrexdw 3298 | Deduction version of ~ nfr... |
| nfralw 3299 | Bound-variable hypothesis ... |
| nfralwOLD 3300 | Obsolete version of ~ nfra... |
| nfrexw 3301 | Bound-variable hypothesis ... |
| r19.12 3302 | Restricted quantifier vers... |
| r19.12OLD 3303 | Obsolete version of ~ 19.1... |
| reean 3304 | Rearrange restricted exist... |
| cbvralsvw 3305 | Change bound variable by u... |
| cbvrexsvw 3306 | Change bound variable by u... |
| cbvralsvwOLD 3307 | Obsolete version of ~ cbvr... |
| cbvrexsvwOLD 3308 | Obsolete version of ~ cbvr... |
| nfraldwOLD 3309 | Obsolete version of ~ nfra... |
| nfra2wOLDOLD 3310 | Obsolete version of ~ nfra... |
| rexeq 3311 | Equality theorem for restr... |
| raleq 3312 | Equality theorem for restr... |
| raleqi 3313 | Equality inference for res... |
| rexeqi 3314 | Equality inference for res... |
| raleqdv 3315 | Equality deduction for res... |
| rexeqdv 3316 | Equality deduction for res... |
| raleqbidva 3317 | Equality deduction for res... |
| rexeqbidva 3318 | Equality deduction for res... |
| raleqbidvv 3319 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3320 | Obsolete version of ~ rale... |
| rexeqbidvv 3321 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3322 | Obsolete version of ~ rexe... |
| raleqbi1dv 3323 | Equality deduction for res... |
| rexeqbi1dv 3324 | Equality deduction for res... |
| raleqOLD 3325 | Obsolete version of ~ rale... |
| rexeqOLD 3326 | Obsolete version of ~ rale... |
| raleleq 3327 | All elements of a class ar... |
| raleqbii 3328 | Equality deduction for res... |
| rexeqbii 3329 | Equality deduction for res... |
| raleqbidv 3330 | Equality deduction for res... |
| rexeqbidv 3331 | Equality deduction for res... |
| cbvraldva2 3332 | Rule used to change the bo... |
| cbvrexdva2 3333 | Rule used to change the bo... |
| cbvrexdva2OLD 3334 | Obsolete version of ~ cbvr... |
| cbvraldvaOLD 3335 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3336 | Obsolete version of ~ cbvr... |
| raleqf 3337 | Equality theorem for restr... |
| rexeqf 3338 | Equality theorem for restr... |
| rexeqfOLD 3339 | Obsolete version of ~ rexe... |
| raleqbid 3340 | Equality deduction for res... |
| rexeqbid 3341 | Equality deduction for res... |
| sbralie 3342 | Implicit to explicit subst... |
| sbralieALT 3343 | Alternative shorter proof ... |
| cbvralf 3344 | Rule used to change bound ... |
| cbvrexf 3345 | Rule used to change bound ... |
| cbvral 3346 | Rule used to change bound ... |
| cbvrex 3347 | Rule used to change bound ... |
| cbvralv 3348 | Change the bound variable ... |
| cbvrexv 3349 | Change the bound variable ... |
| cbvralsv 3350 | Change bound variable by u... |
| cbvrexsv 3351 | Change bound variable by u... |
| cbvral2v 3352 | Change bound variables of ... |
| cbvrex2v 3353 | Change bound variables of ... |
| cbvral3v 3354 | Change bound variables of ... |
| rgen2a 3355 | Generalization rule for re... |
| nfrald 3356 | Deduction version of ~ nfr... |
| nfrexd 3357 | Deduction version of ~ nfr... |
| nfral 3358 | Bound-variable hypothesis ... |
| nfrex 3359 | Bound-variable hypothesis ... |
| nfra2 3360 | Similar to Lemma 24 of [Mo... |
| ralcom2 3361 | Commutation of restricted ... |
| reu5 3366 | Restricted uniqueness in t... |
| reurmo 3367 | Restricted existential uni... |
| reurex 3368 | Restricted unique existenc... |
| mormo 3369 | Unrestricted "at most one"... |
| rmobiia 3370 | Formula-building rule for ... |
| reubiia 3371 | Formula-building rule for ... |
| rmobii 3372 | Formula-building rule for ... |
| reubii 3373 | Formula-building rule for ... |
| rmoanid 3374 | Cancellation law for restr... |
| reuanid 3375 | Cancellation law for restr... |
| rmoanidOLD 3376 | Obsolete version of ~ rmoa... |
| reuanidOLD 3377 | Obsolete version of ~ reua... |
| 2reu2rex 3378 | Double restricted existent... |
| rmobidva 3379 | Formula-building rule for ... |
| reubidva 3380 | Formula-building rule for ... |
| rmobidv 3381 | Formula-building rule for ... |
| reubidv 3382 | Formula-building rule for ... |
| reueubd 3383 | Restricted existential uni... |
| rmo5 3384 | Restricted "at most one" i... |
| nrexrmo 3385 | Nonexistence implies restr... |
| moel 3386 | "At most one" element in a... |
| cbvrmovw 3387 | Change the bound variable ... |
| cbvreuvw 3388 | Change the bound variable ... |
| moelOLD 3389 | Obsolete version of ~ moel... |
| rmobida 3390 | Formula-building rule for ... |
| reubida 3391 | Formula-building rule for ... |
| rmobidvaOLD 3392 | Obsolete version of ~ rmob... |
| cbvrmow 3393 | Change the bound variable ... |
| cbvreuw 3394 | Change the bound variable ... |
| nfrmo1 3395 | The setvar ` x ` is not fr... |
| nfreu1 3396 | The setvar ` x ` is not fr... |
| nfrmow 3397 | Bound-variable hypothesis ... |
| nfreuw 3398 | Bound-variable hypothesis ... |
| cbvreuwOLD 3399 | Obsolete version of ~ cbvr... |
| cbvreuvwOLD 3400 | Obsolete version of ~ cbvr... |
| rmoeq1 3401 | Equality theorem for restr... |
| reueq1 3402 | Equality theorem for restr... |
| rmoeq1OLD 3403 | Obsolete version of ~ rmoe... |
| reueq1OLD 3404 | Obsolete version of ~ reue... |
| rmoeqd 3405 | Equality deduction for res... |
| reueqd 3406 | Equality deduction for res... |
| rmoeq1f 3407 | Equality theorem for restr... |
| reueq1f 3408 | Equality theorem for restr... |
| nfreuwOLD 3409 | Obsolete version of ~ nfre... |
| nfrmowOLD 3410 | Obsolete version of ~ nfrm... |
| cbvreu 3411 | Change the bound variable ... |
| cbvrmo 3412 | Change the bound variable ... |
| cbvrmov 3413 | Change the bound variable ... |
| cbvreuv 3414 | Change the bound variable ... |
| nfrmod 3415 | Deduction version of ~ nfr... |
| nfreud 3416 | Deduction version of ~ nfr... |
| nfrmo 3417 | Bound-variable hypothesis ... |
| nfreu 3418 | Bound-variable hypothesis ... |
| rabbidva2 3421 | Equivalent wff's yield equ... |
| rabbia2 3422 | Equivalent wff's yield equ... |
| rabbiia 3423 | Equivalent formulas yield ... |
| rabbiiaOLD 3424 | Obsolete version of ~ rabb... |
| rabbii 3425 | Equivalent wff's correspon... |
| rabbidva 3426 | Equivalent wff's yield equ... |
| rabbidv 3427 | Equivalent wff's yield equ... |
| rabbieq 3428 | Equivalent wff's correspon... |
| rabswap 3429 | Swap with a membership rel... |
| cbvrabv 3430 | Rule to change the bound v... |
| rabeqcda 3431 | When ` ps ` is always true... |
| rabeqc 3432 | A restricted class abstrac... |
| rabeqi 3433 | Equality theorem for restr... |
| rabeq 3434 | Equality theorem for restr... |
| rabeqdv 3435 | Equality of restricted cla... |
| rabeqbidva 3436 | Equality of restricted cla... |
| rabeqbidv 3437 | Equality of restricted cla... |
| rabrabi 3438 | Abstract builder restricte... |
| nfrab1 3439 | The abstraction variable i... |
| rabid 3440 | An "identity" law of concr... |
| rabidim1 3441 | Membership in a restricted... |
| reqabi 3442 | Inference from equality of... |
| rabrab 3443 | Abstract builder restricte... |
| rabrabiOLD 3444 | Obsolete version of ~ rabr... |
| rabbida4 3445 | Version of ~ rabbidva2 wit... |
| rabbida 3446 | Equivalent wff's yield equ... |
| rabbid 3447 | Version of ~ rabbidv with ... |
| rabeqd 3448 | Deduction form of ~ rabeq ... |
| rabeqbida 3449 | Version of ~ rabeqbidva wi... |
| rabbi 3450 | Equivalent wff's correspon... |
| rabid2f 3451 | An "identity" law for rest... |
| rabid2im 3452 | One direction of ~ rabid2 ... |
| rabid2 3453 | An "identity" law for rest... |
| rabid2OLD 3454 | Obsolete version of ~ rabi... |
| rabeqf 3455 | Equality theorem for restr... |
| cbvrabw 3456 | Rule to change the bound v... |
| nfrabw 3457 | A variable not free in a w... |
| nfrabwOLD 3458 | Obsolete version of ~ nfra... |
| rabbidaOLD 3459 | Obsolete version of ~ rabb... |
| rabeqiOLD 3460 | Obsolete version of ~ rabe... |
| nfrab 3461 | A variable not free in a w... |
| cbvrab 3462 | Rule to change the bound v... |
| vjust 3464 | Justification theorem for ... |
| dfv2 3466 | Alternate definition of th... |
| vex 3467 | All setvar variables are s... |
| vexOLD 3468 | Obsolete version of ~ vex ... |
| elv 3469 | If a proposition is implie... |
| elvd 3470 | If a proposition is implie... |
| el2v 3471 | If a proposition is implie... |
| eqv 3472 | The universe contains ever... |
| eqvf 3473 | The universe contains ever... |
| abv 3474 | The class of sets verifyin... |
| abvALT 3475 | Alternate proof of ~ abv ,... |
| isset 3476 | Two ways to express that "... |
| issetft 3477 | Closed theorem form of ~ i... |
| issetf 3478 | A version of ~ isset that ... |
| isseti 3479 | A way to say " ` A ` is a ... |
| issetri 3480 | A way to say " ` A ` is a ... |
| eqvisset 3481 | A class equal to a variabl... |
| elex 3482 | If a class is a member of ... |
| elexOLD 3483 | Obsolete version of ~ elex... |
| elexi 3484 | If a class is a member of ... |
| elexd 3485 | If a class is a member of ... |
| elex2OLD 3486 | Obsolete version of ~ elex... |
| elex22 3487 | If two classes each contai... |
| prcnel 3488 | A proper class doesn't bel... |
| ralv 3489 | A universal quantifier res... |
| rexv 3490 | An existential quantifier ... |
| reuv 3491 | A unique existential quant... |
| rmov 3492 | An at-most-one quantifier ... |
| rabab 3493 | A class abstraction restri... |
| rexcom4b 3494 | Specialized existential co... |
| ceqsal1t 3495 | One direction of ~ ceqsalt... |
| ceqsalt 3496 | Closed theorem version of ... |
| ceqsralt 3497 | Restricted quantifier vers... |
| ceqsalg 3498 | A representation of explic... |
| ceqsalgALT 3499 | Alternate proof of ~ ceqsa... |
| ceqsal 3500 | A representation of explic... |
| ceqsalALT 3501 | A representation of explic... |
| ceqsalv 3502 | A representation of explic... |
| ceqsalvOLD 3503 | Obsolete version of ~ ceqs... |
| ceqsralv 3504 | Restricted quantifier vers... |
| ceqsralvOLD 3505 | Obsolete version of ~ ceqs... |
| gencl 3506 | Implicit substitution for ... |
| 2gencl 3507 | Implicit substitution for ... |
| 3gencl 3508 | Implicit substitution for ... |
| cgsexg 3509 | Implicit substitution infe... |
| cgsex2g 3510 | Implicit substitution infe... |
| cgsex4g 3511 | An implicit substitution i... |
| cgsex4gOLD 3512 | Obsolete version of ~ cgse... |
| cgsex4gOLDOLD 3513 | Obsolete version of ~ cgse... |
| ceqsex 3514 | Elimination of an existent... |
| ceqsexOLD 3515 | Obsolete version of ~ ceqs... |
| ceqsexv 3516 | Elimination of an existent... |
| ceqsexvOLD 3517 | Obsolete version of ~ ceqs... |
| ceqsexvOLDOLD 3518 | Obsolete version of ~ ceqs... |
| ceqsexv2d 3519 | Elimination of an existent... |
| ceqsex2 3520 | Elimination of two existen... |
| ceqsex2v 3521 | Elimination of two existen... |
| ceqsex3v 3522 | Elimination of three exist... |
| ceqsex4v 3523 | Elimination of four existe... |
| ceqsex6v 3524 | Elimination of six existen... |
| ceqsex8v 3525 | Elimination of eight exist... |
| gencbvex 3526 | Change of bound variable u... |
| gencbvex2 3527 | Restatement of ~ gencbvex ... |
| gencbval 3528 | Change of bound variable u... |
| sbhypf 3529 | Introduce an explicit subs... |
| sbhypfOLD 3530 | Obsolete version of ~ sbhy... |
| vtoclgft 3531 | Closed theorem form of ~ v... |
| vtocleg 3532 | Implicit substitution of a... |
| vtoclg 3533 | Implicit substitution of a... |
| vtocle 3534 | Implicit substitution of a... |
| vtocleOLD 3535 | Obsolete version of ~ vtoc... |
| vtoclbg 3536 | Implicit substitution of a... |
| vtocl 3537 | Implicit substitution of a... |
| vtocldf 3538 | Implicit substitution of a... |
| vtocld 3539 | Implicit substitution of a... |
| vtocldOLD 3540 | Obsolete version of ~ vtoc... |
| vtocl2d 3541 | Implicit substitution of t... |
| vtoclef 3542 | Implicit substitution of a... |
| vtoclf 3543 | Implicit substitution of a... |
| vtoclfOLD 3544 | Obsolete version of ~ vtoc... |
| vtoclALT 3545 | Alternate proof of ~ vtocl... |
| vtocl2 3546 | Implicit substitution of c... |
| vtocl3 3547 | Implicit substitution of c... |
| vtoclb 3548 | Implicit substitution of a... |
| vtoclgf 3549 | Implicit substitution of a... |
| vtoclg1f 3550 | Version of ~ vtoclgf with ... |
| vtoclgOLD 3551 | Obsolete version of ~ vtoc... |
| vtocl2gf 3552 | Implicit substitution of a... |
| vtocl3gf 3553 | Implicit substitution of a... |
| vtocl2g 3554 | Implicit substitution of 2... |
| vtocl3g 3555 | Implicit substitution of a... |
| vtoclgaf 3556 | Implicit substitution of a... |
| vtoclga 3557 | Implicit substitution of a... |
| vtocl2ga 3558 | Implicit substitution of 2... |
| vtocl2gaf 3559 | Implicit substitution of 2... |
| vtocl2gafOLD 3560 | Obsolete version of ~ vtoc... |
| vtocl3gaf 3561 | Implicit substitution of 3... |
| vtocl3gafOLD 3562 | Obsolete version of ~ vtoc... |
| vtocl3ga 3563 | Implicit substitution of 3... |
| vtocl3gaOLD 3564 | Obsolete version of ~ vtoc... |
| vtocl3gaOLDOLD 3565 | Obsolete version of ~ vtoc... |
| vtocl4g 3566 | Implicit substitution of 4... |
| vtocl4ga 3567 | Implicit substitution of 4... |
| vtocl4gaOLD 3568 | Obsolete version of ~ vtoc... |
| vtoclegft 3569 | Implicit substitution of a... |
| vtoclegftOLD 3570 | Obsolete version of ~ vtoc... |
| vtoclri 3571 | Implicit substitution of a... |
| spcimgft 3572 | A closed version of ~ spci... |
| spcgft 3573 | A closed version of ~ spcg... |
| spcimgf 3574 | Rule of specialization, us... |
| spcimegf 3575 | Existential specialization... |
| spcgf 3576 | Rule of specialization, us... |
| spcegf 3577 | Existential specialization... |
| spcimdv 3578 | Restricted specialization,... |
| spcdv 3579 | Rule of specialization, us... |
| spcimedv 3580 | Restricted existential spe... |
| spcgv 3581 | Rule of specialization, us... |
| spcegv 3582 | Existential specialization... |
| spcedv 3583 | Existential specialization... |
| spc2egv 3584 | Existential specialization... |
| spc2gv 3585 | Specialization with two qu... |
| spc2ed 3586 | Existential specialization... |
| spc2d 3587 | Specialization with 2 quan... |
| spc3egv 3588 | Existential specialization... |
| spc3gv 3589 | Specialization with three ... |
| spcv 3590 | Rule of specialization, us... |
| spcev 3591 | Existential specialization... |
| spc2ev 3592 | Existential specialization... |
| rspct 3593 | A closed version of ~ rspc... |
| rspcdf 3594 | Restricted specialization,... |
| rspc 3595 | Restricted specialization,... |
| rspce 3596 | Restricted existential spe... |
| rspcimdv 3597 | Restricted specialization,... |
| rspcimedv 3598 | Restricted existential spe... |
| rspcdv 3599 | Restricted specialization,... |
| rspcedv 3600 | Restricted existential spe... |
| rspcebdv 3601 | Restricted existential spe... |
| rspcdv2 3602 | Restricted specialization,... |
| rspcv 3603 | Restricted specialization,... |
| rspccv 3604 | Restricted specialization,... |
| rspcva 3605 | Restricted specialization,... |
| rspccva 3606 | Restricted specialization,... |
| rspcev 3607 | Restricted existential spe... |
| rspcdva 3608 | Restricted specialization,... |
| rspcedvd 3609 | Restricted existential spe... |
| rspcedvdw 3610 | Version of ~ rspcedvd wher... |
| rspceb2dv 3611 | Restricted existential spe... |
| rspcime 3612 | Prove a restricted existen... |
| rspceaimv 3613 | Restricted existential spe... |
| rspcedeq1vd 3614 | Restricted existential spe... |
| rspcedeq2vd 3615 | Restricted existential spe... |
| rspc2 3616 | Restricted specialization ... |
| rspc2gv 3617 | Restricted specialization ... |
| rspc2v 3618 | 2-variable restricted spec... |
| rspc2va 3619 | 2-variable restricted spec... |
| rspc2ev 3620 | 2-variable restricted exis... |
| 2rspcedvdw 3621 | Double application of ~ rs... |
| rspc2dv 3622 | 2-variable restricted spec... |
| rspc3v 3623 | 3-variable restricted spec... |
| rspc3ev 3624 | 3-variable restricted exis... |
| rspc3dv 3625 | 3-variable restricted spec... |
| rspc4v 3626 | 4-variable restricted spec... |
| rspc6v 3627 | 6-variable restricted spec... |
| rspc8v 3628 | 8-variable restricted spec... |
| rspceeqv 3629 | Restricted existential spe... |
| ralxpxfr2d 3630 | Transfer a universal quant... |
| rexraleqim 3631 | Statement following from e... |
| eqvincg 3632 | A variable introduction la... |
| eqvinc 3633 | A variable introduction la... |
| eqvincf 3634 | A variable introduction la... |
| alexeqg 3635 | Two ways to express substi... |
| ceqex 3636 | Equality implies equivalen... |
| ceqsexg 3637 | A representation of explic... |
| ceqsexgv 3638 | Elimination of an existent... |
| ceqsrexv 3639 | Elimination of a restricte... |
| ceqsrexbv 3640 | Elimination of a restricte... |
| ceqsralbv 3641 | Elimination of a restricte... |
| ceqsrex2v 3642 | Elimination of a restricte... |
| clel2g 3643 | Alternate definition of me... |
| clel2gOLD 3644 | Obsolete version of ~ clel... |
| clel2 3645 | Alternate definition of me... |
| clel3g 3646 | Alternate definition of me... |
| clel3 3647 | Alternate definition of me... |
| clel4g 3648 | Alternate definition of me... |
| clel4 3649 | Alternate definition of me... |
| clel4OLD 3650 | Obsolete version of ~ clel... |
| clel5 3651 | Alternate definition of cl... |
| pm13.183 3652 | Compare theorem *13.183 in... |
| rr19.3v 3653 | Restricted quantifier vers... |
| rr19.28v 3654 | Restricted quantifier vers... |
| elab6g 3655 | Membership in a class abst... |
| elabd2 3656 | Membership in a class abst... |
| elabd3 3657 | Membership in a class abst... |
| elabgt 3658 | Membership in a class abst... |
| elabgtOLD 3659 | Obsolete version of ~ elab... |
| elabgtOLDOLD 3660 | Obsolete version of ~ elab... |
| elabgf 3661 | Membership in a class abst... |
| elabf 3662 | Membership in a class abst... |
| elabg 3663 | Membership in a class abst... |
| elabgOLD 3664 | Obsolete version of ~ elab... |
| elab 3665 | Membership in a class abst... |
| elabOLD 3666 | Obsolete version of ~ elab... |
| 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... |
| elrabiOLD 3675 | Obsolete version of ~ elra... |
| elrabf 3676 | Membership in a restricted... |
| rabtru 3677 | Abstract builder using the... |
| rabeqcOLD 3678 | Obsolete version of ~ rabe... |
| elrab3t 3679 | Membership in a restricted... |
| elrab 3680 | Membership in a restricted... |
| elrab3 3681 | Membership in a restricted... |
| elrabd 3682 | Membership in a restricted... |
| elrab2 3683 | Membership in a restricted... |
| ralab 3684 | Universal quantification o... |
| ralabOLD 3685 | Obsolete version of ~ rala... |
| ralrab 3686 | Universal quantification o... |
| rexab 3687 | Existential quantification... |
| rexabOLD 3688 | Obsolete version of ~ rexa... |
| rexrab 3689 | Existential quantification... |
| ralab2 3690 | Universal quantification o... |
| ralrab2 3691 | Universal quantification o... |
| rexab2 3692 | Existential quantification... |
| rexrab2 3693 | Existential quantification... |
| reurab 3694 | Restricted existential uni... |
| abidnf 3695 | Identity used to create cl... |
| dedhb 3696 | A deduction theorem for co... |
| class2seteq 3697 | Writing a set as a class a... |
| nelrdva 3698 | Deduce negative membership... |
| eqeu 3699 | A condition which implies ... |
| moeq 3700 | There exists at most one s... |
| eueq 3701 | A class is a set if and on... |
| eueqi 3702 | There exists a unique set ... |
| eueq2 3703 | Equality has existential u... |
| eueq3 3704 | Equality has existential u... |
| moeq3 3705 | "At most one" property of ... |
| mosub 3706 | "At most one" remains true... |
| mo2icl 3707 | Theorem for inferring "at ... |
| mob2 3708 | Consequence of "at most on... |
| moi2 3709 | Consequence of "at most on... |
| mob 3710 | Equality implied by "at mo... |
| moi 3711 | Equality implied by "at mo... |
| morex 3712 | Derive membership from uni... |
| euxfr2w 3713 | Transfer existential uniqu... |
| euxfrw 3714 | Transfer existential uniqu... |
| euxfr2 3715 | Transfer existential uniqu... |
| euxfr 3716 | Transfer existential uniqu... |
| euind 3717 | Existential uniqueness via... |
| reu2 3718 | A way to express restricte... |
| reu6 3719 | A way to express restricte... |
| reu3 3720 | A way to express restricte... |
| reu6i 3721 | A condition which implies ... |
| eqreu 3722 | A condition which implies ... |
| rmo4 3723 | Restricted "at most one" u... |
| reu4 3724 | Restricted uniqueness usin... |
| reu7 3725 | Restricted uniqueness usin... |
| reu8 3726 | Restricted uniqueness usin... |
| rmo3f 3727 | Restricted "at most one" u... |
| rmo4f 3728 | Restricted "at most one" u... |
| reu2eqd 3729 | Deduce equality from restr... |
| reueq 3730 | Equality has existential u... |
| rmoeq 3731 | Equality's restricted exis... |
| rmoan 3732 | Restricted "at most one" s... |
| rmoim 3733 | Restricted "at most one" i... |
| rmoimia 3734 | Restricted "at most one" i... |
| rmoimi 3735 | Restricted "at most one" i... |
| rmoimi2 3736 | Restricted "at most one" i... |
| 2reu5a 3737 | Double restricted existent... |
| reuimrmo 3738 | Restricted uniqueness impl... |
| 2reuswap 3739 | A condition allowing swap ... |
| 2reuswap2 3740 | A condition allowing swap ... |
| reuxfrd 3741 | Transfer existential uniqu... |
| reuxfr 3742 | Transfer existential uniqu... |
| reuxfr1d 3743 | Transfer existential uniqu... |
| reuxfr1ds 3744 | Transfer existential uniqu... |
| reuxfr1 3745 | Transfer existential uniqu... |
| reuind 3746 | Existential uniqueness via... |
| 2rmorex 3747 | Double restricted quantifi... |
| 2reu5lem1 3748 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3749 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3750 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3751 | Double restricted existent... |
| 2reurmo 3752 | Double restricted quantifi... |
| 2reurex 3753 | Double restricted quantifi... |
| 2rmoswap 3754 | A condition allowing to sw... |
| 2rexreu 3755 | Double restricted existent... |
| cdeqi 3758 | Deduce conditional equalit... |
| cdeqri 3759 | Property of conditional eq... |
| cdeqth 3760 | Deduce conditional equalit... |
| cdeqnot 3761 | Distribute conditional equ... |
| cdeqal 3762 | Distribute conditional equ... |
| cdeqab 3763 | Distribute conditional equ... |
| cdeqal1 3764 | Distribute conditional equ... |
| cdeqab1 3765 | Distribute conditional equ... |
| cdeqim 3766 | Distribute conditional equ... |
| cdeqcv 3767 | Conditional equality for s... |
| cdeqeq 3768 | Distribute conditional equ... |
| cdeqel 3769 | Distribute conditional equ... |
| nfcdeq 3770 | If we have a conditional e... |
| nfccdeq 3771 | Variation of ~ nfcdeq for ... |
| rru 3772 | Relative version of Russel... |
| ru 3773 | Russell's Paradox. Propos... |
| dfsbcq 3776 | Proper substitution of a c... |
| dfsbcq2 3777 | This theorem, which is sim... |
| sbsbc 3778 | Show that ~ df-sb and ~ df... |
| sbceq1d 3779 | Equality theorem for class... |
| sbceq1dd 3780 | Equality theorem for class... |
| sbceqbid 3781 | Equality theorem for class... |
| sbc8g 3782 | This is the closest we can... |
| sbc2or 3783 | The disjunction of two equ... |
| sbcex 3784 | By our definition of prope... |
| sbceq1a 3785 | Equality theorem for class... |
| sbceq2a 3786 | Equality theorem for class... |
| spsbc 3787 | Specialization: if a formu... |
| spsbcd 3788 | Specialization: if a formu... |
| sbcth 3789 | A substitution into a theo... |
| sbcthdv 3790 | Deduction version of ~ sbc... |
| sbcid 3791 | An identity theorem for su... |
| nfsbc1d 3792 | Deduction version of ~ nfs... |
| nfsbc1 3793 | Bound-variable hypothesis ... |
| nfsbc1v 3794 | Bound-variable hypothesis ... |
| nfsbcdw 3795 | Deduction version of ~ nfs... |
| nfsbcw 3796 | Bound-variable hypothesis ... |
| sbccow 3797 | A composition law for clas... |
| nfsbcd 3798 | Deduction version of ~ nfs... |
| nfsbc 3799 | Bound-variable hypothesis ... |
| sbcco 3800 | A composition law for clas... |
| sbcco2 3801 | A composition law for clas... |
| sbc5 3802 | An equivalence for class s... |
| sbc5ALT 3803 | Alternate proof of ~ sbc5 ... |
| sbc6g 3804 | An equivalence for class s... |
| sbc6gOLD 3805 | Obsolete version of ~ sbc6... |
| sbc6 3806 | An equivalence for class s... |
| sbc7 3807 | An equivalence for class s... |
| cbvsbcw 3808 | Change bound variables in ... |
| cbvsbcvw 3809 | Change the bound variable ... |
| cbvsbc 3810 | Change bound variables in ... |
| cbvsbcv 3811 | Change the bound variable ... |
| sbciegft 3812 | Conversion of implicit sub... |
| sbciegftOLD 3813 | Obsolete version of ~ sbci... |
| sbciegf 3814 | Conversion of implicit sub... |
| sbcieg 3815 | Conversion of implicit sub... |
| sbciegOLD 3816 | Obsolete version of ~ sbci... |
| sbcie2g 3817 | Conversion of implicit sub... |
| sbcie 3818 | Conversion of implicit sub... |
| sbciedf 3819 | Conversion of implicit sub... |
| sbcied 3820 | Conversion of implicit sub... |
| sbciedOLD 3821 | Obsolete version of ~ sbci... |
| sbcied2 3822 | Conversion of implicit sub... |
| elrabsf 3823 | Membership in a restricted... |
| eqsbc1 3824 | Substitution for the left-... |
| sbcng 3825 | Move negation in and out o... |
| sbcimg 3826 | Distribution of class subs... |
| sbcan 3827 | Distribution of class subs... |
| sbcor 3828 | Distribution of class subs... |
| sbcbig 3829 | Distribution of class subs... |
| sbcn1 3830 | Move negation in and out o... |
| sbcim1 3831 | Distribution of class subs... |
| sbcim1OLD 3832 | Obsolete version of ~ sbci... |
| sbcbid 3833 | Formula-building deduction... |
| sbcbidv 3834 | Formula-building deduction... |
| sbcbii 3835 | Formula-building inference... |
| sbcbi1 3836 | Distribution of class subs... |
| sbcbi2 3837 | Substituting into equivale... |
| sbcal 3838 | Move universal quantifier ... |
| sbcex2 3839 | Move existential quantifie... |
| sbceqal 3840 | Class version of one impli... |
| sbceqalOLD 3841 | Obsolete version of ~ sbce... |
| sbeqalb 3842 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3843 | Substitution for the right... |
| sbc3an 3844 | Distribution of class subs... |
| sbcel1v 3845 | Class substitution into a ... |
| sbcel2gv 3846 | Class substitution into a ... |
| sbcel21v 3847 | Class substitution into a ... |
| sbcimdv 3848 | Substitution analogue of T... |
| sbcimdvOLD 3849 | Obsolete version of ~ sbci... |
| sbctt 3850 | Substitution for a variabl... |
| sbcgf 3851 | Substitution for a variabl... |
| sbc19.21g 3852 | Substitution for a variabl... |
| sbcg 3853 | Substitution for a variabl... |
| sbcgOLD 3854 | Obsolete version of ~ sbcg... |
| sbcgfi 3855 | Substitution for a variabl... |
| sbc2iegf 3856 | Conversion of implicit sub... |
| sbc2ie 3857 | Conversion of implicit sub... |
| sbc2ieOLD 3858 | Obsolete version of ~ sbc2... |
| sbc2iedv 3859 | Conversion of implicit sub... |
| sbc3ie 3860 | Conversion of implicit sub... |
| sbccomlem 3861 | Lemma for ~ sbccom . (Con... |
| sbccom 3862 | Commutative law for double... |
| sbcralt 3863 | Interchange class substitu... |
| sbcrext 3864 | Interchange class substitu... |
| sbcralg 3865 | Interchange class substitu... |
| sbcrex 3866 | Interchange class substitu... |
| sbcreu 3867 | Interchange class substitu... |
| reu8nf 3868 | Restricted uniqueness usin... |
| sbcabel 3869 | Interchange class substitu... |
| rspsbc 3870 | Restricted quantifier vers... |
| rspsbca 3871 | Restricted quantifier vers... |
| rspesbca 3872 | Existence form of ~ rspsbc... |
| spesbc 3873 | Existence form of ~ spsbc ... |
| spesbcd 3874 | form of ~ spsbc . (Contri... |
| sbcth2 3875 | A substitution into a theo... |
| ra4v 3876 | Version of ~ ra4 with a di... |
| ra4 3877 | Restricted quantifier vers... |
| rmo2 3878 | Alternate definition of re... |
| rmo2i 3879 | Condition implying restric... |
| rmo3 3880 | Restricted "at most one" u... |
| rmob 3881 | Consequence of "at most on... |
| rmoi 3882 | Consequence of "at most on... |
| rmob2 3883 | Consequence of "restricted... |
| rmoi2 3884 | Consequence of "restricted... |
| rmoanim 3885 | Introduction of a conjunct... |
| rmoanimALT 3886 | Alternate proof of ~ rmoan... |
| reuan 3887 | Introduction of a conjunct... |
| 2reu1 3888 | Double restricted existent... |
| 2reu2 3889 | Double restricted existent... |
| csb2 3892 | Alternate expression for t... |
| csbeq1 3893 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3894 | Equality deduction for pro... |
| csbeq2 3895 | Substituting into equivale... |
| csbeq2d 3896 | Formula-building deduction... |
| csbeq2dv 3897 | Formula-building deduction... |
| csbeq2i 3898 | Formula-building inference... |
| csbeq12dv 3899 | Formula-building inference... |
| cbvcsbw 3900 | Change bound variables in ... |
| cbvcsb 3901 | Change bound variables in ... |
| cbvcsbv 3902 | Change the bound variable ... |
| csbid 3903 | Analogue of ~ sbid for pro... |
| csbeq1a 3904 | Equality theorem for prope... |
| csbcow 3905 | Composition law for chaine... |
| csbco 3906 | Composition law for chaine... |
| csbtt 3907 | Substitution doesn't affec... |
| csbconstgf 3908 | Substitution doesn't affec... |
| csbconstg 3909 | Substitution doesn't affec... |
| csbconstgOLD 3910 | Obsolete version of ~ csbc... |
| csbgfi 3911 | Substitution for a variabl... |
| csbconstgi 3912 | The proper substitution of... |
| nfcsb1d 3913 | Bound-variable hypothesis ... |
| nfcsb1 3914 | Bound-variable hypothesis ... |
| nfcsb1v 3915 | Bound-variable hypothesis ... |
| nfcsbd 3916 | Deduction version of ~ nfc... |
| nfcsbw 3917 | Bound-variable hypothesis ... |
| nfcsb 3918 | Bound-variable hypothesis ... |
| csbhypf 3919 | Introduce an explicit subs... |
| csbiebt 3920 | Conversion of implicit sub... |
| csbiedf 3921 | Conversion of implicit sub... |
| csbieb 3922 | Bidirectional conversion b... |
| csbiebg 3923 | Bidirectional conversion b... |
| csbiegf 3924 | Conversion of implicit sub... |
| csbief 3925 | Conversion of implicit sub... |
| csbie 3926 | Conversion of implicit sub... |
| csbieOLD 3927 | Obsolete version of ~ csbi... |
| csbied 3928 | Conversion of implicit sub... |
| csbiedOLD 3929 | Obsolete version of ~ csbi... |
| csbied2 3930 | Conversion of implicit sub... |
| csbie2t 3931 | Conversion of implicit sub... |
| csbie2 3932 | Conversion of implicit sub... |
| csbie2g 3933 | Conversion of implicit sub... |
| cbvrabcsfw 3934 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3935 | A more general version of ... |
| cbvrexcsf 3936 | A more general version of ... |
| cbvreucsf 3937 | A more general version of ... |
| cbvrabcsf 3938 | A more general version of ... |
| cbvralv2 3939 | Rule used to change the bo... |
| cbvrexv2 3940 | Rule used to change the bo... |
| rspc2vd 3941 | Deduction version of 2-var... |
| difjust 3947 | Soundness justification th... |
| unjust 3949 | Soundness justification th... |
| injust 3951 | Soundness justification th... |
| dfin5 3953 | Alternate definition for t... |
| dfdif2 3954 | Alternate definition of cl... |
| eldif 3955 | Expansion of membership in... |
| eldifd 3956 | If a class is in one class... |
| eldifad 3957 | If a class is in the diffe... |
| eldifbd 3958 | If a class is in the diffe... |
| elneeldif 3959 | The elements of a set diff... |
| velcomp 3960 | Characterization of setvar... |
| elin 3961 | Expansion of membership in... |
| dfss2 3963 | Alternate definition of th... |
| dfss 3964 | Variant of subclass defini... |
| dfss3 3966 | Alternate definition of su... |
| dfss6 3967 | Alternate definition of su... |
| dfssf 3968 | Equivalence for subclass r... |
| dfss3f 3969 | Equivalence for subclass r... |
| nfss 3970 | If ` x ` is not free in ` ... |
| ssel 3971 | Membership relationships f... |
| ssel2 3972 | Membership relationships f... |
| sseli 3973 | Membership implication fro... |
| sselii 3974 | Membership inference from ... |
| sselid 3975 | Membership inference from ... |
| sseld 3976 | Membership deduction from ... |
| sselda 3977 | Membership deduction from ... |
| sseldd 3978 | Membership inference from ... |
| ssneld 3979 | If a class is not in anoth... |
| ssneldd 3980 | If an element is not in a ... |
| ssriv 3981 | Inference based on subclas... |
| ssrd 3982 | Deduction based on subclas... |
| ssrdv 3983 | Deduction based on subclas... |
| sstr2 3984 | Transitivity of subclass r... |
| sstr2OLD 3985 | Obsolete version of ~ sstr... |
| sstr 3986 | Transitivity of subclass r... |
| sstri 3987 | Subclass transitivity infe... |
| sstrd 3988 | Subclass transitivity dedu... |
| sstrid 3989 | Subclass transitivity dedu... |
| sstrdi 3990 | Subclass transitivity dedu... |
| sylan9ss 3991 | A subclass transitivity de... |
| sylan9ssr 3992 | A subclass transitivity de... |
| eqss 3993 | The subclass relationship ... |
| eqssi 3994 | Infer equality from two su... |
| eqssd 3995 | Equality deduction from tw... |
| sssseq 3996 | If a class is a subclass o... |
| eqrd 3997 | Deduce equality of classes... |
| eqri 3998 | Infer equality of classes ... |
| eqelssd 3999 | Equality deduction from su... |
| ssid 4000 | Any class is a subclass of... |
| ssidd 4001 | Weakening of ~ ssid . (Co... |
| ssv 4002 | Any class is a subclass of... |
| sseq1 4003 | Equality theorem for subcl... |
| sseq2 4004 | Equality theorem for the s... |
| sseq12 4005 | Equality theorem for the s... |
| sseq1i 4006 | An equality inference for ... |
| sseq2i 4007 | An equality inference for ... |
| sseq12i 4008 | An equality inference for ... |
| sseq1d 4009 | An equality deduction for ... |
| sseq2d 4010 | An equality deduction for ... |
| sseq12d 4011 | An equality deduction for ... |
| eqsstri 4012 | Substitution of equality i... |
| eqsstrri 4013 | Substitution of equality i... |
| sseqtri 4014 | Substitution of equality i... |
| sseqtrri 4015 | Substitution of equality i... |
| eqsstrd 4016 | Substitution of equality i... |
| eqsstrrd 4017 | Substitution of equality i... |
| sseqtrd 4018 | Substitution of equality i... |
| sseqtrrd 4019 | Substitution of equality i... |
| 3sstr3i 4020 | Substitution of equality i... |
| 3sstr4i 4021 | Substitution of equality i... |
| 3sstr3g 4022 | Substitution of equality i... |
| 3sstr4g 4023 | Substitution of equality i... |
| 3sstr3d 4024 | Substitution of equality i... |
| 3sstr4d 4025 | Substitution of equality i... |
| eqsstrid 4026 | A chained subclass and equ... |
| eqsstrrid 4027 | A chained subclass and equ... |
| sseqtrdi 4028 | A chained subclass and equ... |
| sseqtrrdi 4029 | A chained subclass and equ... |
| sseqtrid 4030 | Subclass transitivity dedu... |
| sseqtrrid 4031 | Subclass transitivity dedu... |
| eqsstrdi 4032 | A chained subclass and equ... |
| eqsstrrdi 4033 | A chained subclass and equ... |
| eqimssd 4034 | Equality implies inclusion... |
| eqimsscd 4035 | Equality implies inclusion... |
| eqimss 4036 | Equality implies inclusion... |
| eqimss2 4037 | Equality implies inclusion... |
| eqimssi 4038 | Infer subclass relationshi... |
| eqimss2i 4039 | Infer subclass relationshi... |
| nssne1 4040 | Two classes are different ... |
| nssne2 4041 | Two classes are different ... |
| nss 4042 | Negation of subclass relat... |
| nelss 4043 | Demonstrate by witnesses t... |
| ssrexf 4044 | Restricted existential qua... |
| ssrmof 4045 | "At most one" existential ... |
| ssralv 4046 | Quantification restricted ... |
| ssrexv 4047 | Existential quantification... |
| ss2ralv 4048 | Two quantifications restri... |
| ss2rexv 4049 | Two existential quantifica... |
| ssralvOLD 4050 | Obsolete version of ~ ssra... |
| ssrexvOLD 4051 | Obsolete version of ~ ssre... |
| ralss 4052 | Restricted universal quant... |
| rexss 4053 | Restricted existential qua... |
| ss2ab 4054 | Class abstractions in a su... |
| abss 4055 | Class abstraction in a sub... |
| ssab 4056 | Subclass of a class abstra... |
| ssabral 4057 | The relation for a subclas... |
| ss2abdv 4058 | Deduction of abstraction s... |
| ss2abdvOLD 4059 | Obsolete version of ~ ss2a... |
| ss2abi 4060 | Inference of abstraction s... |
| ss2abiOLD 4061 | Obsolete version of ~ ss2a... |
| abssdv 4062 | Deduction of abstraction s... |
| abssdvOLD 4063 | Obsolete version of ~ abss... |
| abssi 4064 | Inference of abstraction s... |
| ss2rab 4065 | Restricted abstraction cla... |
| rabss 4066 | Restricted class abstracti... |
| ssrab 4067 | Subclass of a restricted c... |
| ssrabdv 4068 | Subclass of a restricted c... |
| rabssdv 4069 | Subclass of a restricted c... |
| ss2rabdv 4070 | Deduction of restricted ab... |
| ss2rabi 4071 | Inference of restricted ab... |
| rabss2 4072 | Subclass law for restricte... |
| ssab2 4073 | Subclass relation for the ... |
| ssrab2 4074 | Subclass relation for a re... |
| ssrab2OLD 4075 | Obsolete version of ~ ssra... |
| rabss3d 4076 | Subclass law for restricte... |
| ssrab3 4077 | Subclass relation for a re... |
| rabssrabd 4078 | Subclass of a restricted c... |
| ssrabeq 4079 | If the restricting class o... |
| rabssab 4080 | A restricted class is a su... |
| uniiunlem 4081 | A subset relationship usef... |
| dfpss2 4082 | Alternate definition of pr... |
| dfpss3 4083 | Alternate definition of pr... |
| psseq1 4084 | Equality theorem for prope... |
| psseq2 4085 | Equality theorem for prope... |
| psseq1i 4086 | An equality inference for ... |
| psseq2i 4087 | An equality inference for ... |
| psseq12i 4088 | An equality inference for ... |
| psseq1d 4089 | An equality deduction for ... |
| psseq2d 4090 | An equality deduction for ... |
| psseq12d 4091 | An equality deduction for ... |
| pssss 4092 | A proper subclass is a sub... |
| pssne 4093 | Two classes in a proper su... |
| pssssd 4094 | Deduce subclass from prope... |
| pssned 4095 | Proper subclasses are uneq... |
| sspss 4096 | Subclass in terms of prope... |
| pssirr 4097 | Proper subclass is irrefle... |
| pssn2lp 4098 | Proper subclass has no 2-c... |
| sspsstri 4099 | Two ways of stating tricho... |
| ssnpss 4100 | Partial trichotomy law for... |
| psstr 4101 | Transitive law for proper ... |
| sspsstr 4102 | Transitive law for subclas... |
| psssstr 4103 | Transitive law for subclas... |
| psstrd 4104 | Proper subclass inclusion ... |
| sspsstrd 4105 | Transitivity involving sub... |
| psssstrd 4106 | Transitivity involving sub... |
| npss 4107 | A class is not a proper su... |
| ssnelpss 4108 | A subclass missing a membe... |
| ssnelpssd 4109 | Subclass inclusion with on... |
| ssexnelpss 4110 | If there is an element of ... |
| dfdif3 4111 | Alternate definition of cl... |
| difeq1 4112 | Equality theorem for class... |
| difeq2 4113 | Equality theorem for class... |
| difeq12 4114 | Equality theorem for class... |
| difeq1i 4115 | Inference adding differenc... |
| difeq2i 4116 | Inference adding differenc... |
| difeq12i 4117 | Equality inference for cla... |
| difeq1d 4118 | Deduction adding differenc... |
| difeq2d 4119 | Deduction adding differenc... |
| difeq12d 4120 | Equality deduction for cla... |
| difeqri 4121 | Inference from membership ... |
| nfdif 4122 | Bound-variable hypothesis ... |
| nfdifOLD 4123 | Obsolete version of ~ nfdi... |
| eldifi 4124 | Implication of membership ... |
| eldifn 4125 | Implication of membership ... |
| elndif 4126 | A set does not belong to a... |
| neldif 4127 | Implication of membership ... |
| difdif 4128 | Double class difference. ... |
| difss 4129 | Subclass relationship for ... |
| difssd 4130 | A difference of two classe... |
| difss2 4131 | If a class is contained in... |
| difss2d 4132 | If a class is contained in... |
| ssdifss 4133 | Preservation of a subclass... |
| ddif 4134 | Double complement under un... |
| ssconb 4135 | Contraposition law for sub... |
| sscon 4136 | Contraposition law for sub... |
| ssdif 4137 | Difference law for subsets... |
| ssdifd 4138 | If ` A ` is contained in `... |
| sscond 4139 | If ` A ` is contained in `... |
| ssdifssd 4140 | If ` A ` is contained in `... |
| ssdif2d 4141 | If ` A ` is contained in `... |
| raldifb 4142 | Restricted universal quant... |
| rexdifi 4143 | Restricted existential qua... |
| complss 4144 | Complementation reverses i... |
| compleq 4145 | Two classes are equal if a... |
| elun 4146 | Expansion of membership in... |
| elunnel1 4147 | A member of a union that i... |
| elunnel2 4148 | A member of a union that i... |
| uneqri 4149 | Inference from membership ... |
| unidm 4150 | Idempotent law for union o... |
| uncom 4151 | Commutative law for union ... |
| equncom 4152 | If a class equals the unio... |
| equncomi 4153 | Inference form of ~ equnco... |
| uneq1 4154 | Equality theorem for the u... |
| uneq2 4155 | Equality theorem for the u... |
| uneq12 4156 | Equality theorem for the u... |
| uneq1i 4157 | Inference adding union to ... |
| uneq2i 4158 | Inference adding union to ... |
| uneq12i 4159 | Equality inference for the... |
| uneq1d 4160 | Deduction adding union to ... |
| uneq2d 4161 | Deduction adding union to ... |
| uneq12d 4162 | Equality deduction for the... |
| nfun 4163 | Bound-variable hypothesis ... |
| nfunOLD 4164 | Obsolete version of ~ nfun... |
| unass 4165 | Associative law for union ... |
| un12 4166 | A rearrangement of union. ... |
| un23 4167 | A rearrangement of union. ... |
| un4 4168 | A rearrangement of the uni... |
| unundi 4169 | Union distributes over its... |
| unundir 4170 | Union distributes over its... |
| ssun1 4171 | Subclass relationship for ... |
| ssun2 4172 | Subclass relationship for ... |
| ssun3 4173 | Subclass law for union of ... |
| ssun4 4174 | Subclass law for union of ... |
| elun1 4175 | Membership law for union o... |
| elun2 4176 | Membership law for union o... |
| elunant 4177 | A statement is true for ev... |
| unss1 4178 | Subclass law for union of ... |
| ssequn1 4179 | A relationship between sub... |
| unss2 4180 | Subclass law for union of ... |
| unss12 4181 | Subclass law for union of ... |
| ssequn2 4182 | A relationship between sub... |
| unss 4183 | The union of two subclasse... |
| unssi 4184 | An inference showing the u... |
| unssd 4185 | A deduction showing the un... |
| unssad 4186 | If ` ( A u. B ) ` is conta... |
| unssbd 4187 | If ` ( A u. B ) ` is conta... |
| ssun 4188 | A condition that implies i... |
| rexun 4189 | Restricted existential qua... |
| ralunb 4190 | Restricted quantification ... |
| ralun 4191 | Restricted quantification ... |
| elini 4192 | Membership in an intersect... |
| elind 4193 | Deduce membership in an in... |
| elinel1 4194 | Membership in an intersect... |
| elinel2 4195 | Membership in an intersect... |
| elin2 4196 | Membership in a class defi... |
| elin1d 4197 | Elementhood in the first s... |
| elin2d 4198 | Elementhood in the first s... |
| elin3 4199 | Membership in a class defi... |
| incom 4200 | Commutative law for inters... |
| ineqcom 4201 | Two ways of expressing tha... |
| ineqcomi 4202 | Two ways of expressing tha... |
| ineqri 4203 | Inference from membership ... |
| ineq1 4204 | Equality theorem for inter... |
| ineq2 4205 | Equality theorem for inter... |
| ineq12 4206 | Equality theorem for inter... |
| ineq1i 4207 | Equality inference for int... |
| ineq2i 4208 | Equality inference for int... |
| ineq12i 4209 | Equality inference for int... |
| ineq1d 4210 | Equality deduction for int... |
| ineq2d 4211 | Equality deduction for int... |
| ineq12d 4212 | Equality deduction for int... |
| ineqan12d 4213 | Equality deduction for int... |
| sseqin2 4214 | A relationship between sub... |
| nfin 4215 | Bound-variable hypothesis ... |
| nfinOLD 4216 | Obsolete version of ~ nfin... |
| rabbi2dva 4217 | Deduction from a wff to a ... |
| inidm 4218 | Idempotent law for interse... |
| inass 4219 | Associative law for inters... |
| in12 4220 | A rearrangement of interse... |
| in32 4221 | A rearrangement of interse... |
| in13 4222 | A rearrangement of interse... |
| in31 4223 | A rearrangement of interse... |
| inrot 4224 | Rotate the intersection of... |
| in4 4225 | Rearrangement of intersect... |
| inindi 4226 | Intersection distributes o... |
| inindir 4227 | Intersection distributes o... |
| inss1 4228 | The intersection of two cl... |
| inss2 4229 | The intersection of two cl... |
| ssin 4230 | Subclass of intersection. ... |
| ssini 4231 | An inference showing that ... |
| ssind 4232 | A deduction showing that a... |
| ssrin 4233 | Add right intersection to ... |
| sslin 4234 | Add left intersection to s... |
| ssrind 4235 | Add right intersection to ... |
| ss2in 4236 | Intersection of subclasses... |
| ssinss1 4237 | Intersection preserves sub... |
| inss 4238 | Inclusion of an intersecti... |
| rexin 4239 | Restricted existential qua... |
| dfss7 4240 | Alternate definition of su... |
| symdifcom 4243 | Symmetric difference commu... |
| symdifeq1 4244 | Equality theorem for symme... |
| symdifeq2 4245 | Equality theorem for symme... |
| nfsymdif 4246 | Hypothesis builder for sym... |
| elsymdif 4247 | Membership in a symmetric ... |
| dfsymdif4 4248 | Alternate definition of th... |
| elsymdifxor 4249 | Membership in a symmetric ... |
| dfsymdif2 4250 | Alternate definition of th... |
| symdifass 4251 | Symmetric difference is as... |
| difsssymdif 4252 | The symmetric difference c... |
| difsymssdifssd 4253 | If the symmetric differenc... |
| unabs 4254 | Absorption law for union. ... |
| inabs 4255 | Absorption law for interse... |
| nssinpss 4256 | Negation of subclass expre... |
| nsspssun 4257 | Negation of subclass expre... |
| dfss4 4258 | Subclass defined in terms ... |
| dfun2 4259 | An alternate definition of... |
| dfin2 4260 | An alternate definition of... |
| difin 4261 | Difference with intersecti... |
| ssdifim 4262 | Implication of a class dif... |
| ssdifsym 4263 | Symmetric class difference... |
| dfss5 4264 | Alternate definition of su... |
| dfun3 4265 | Union defined in terms of ... |
| dfin3 4266 | Intersection defined in te... |
| dfin4 4267 | Alternate definition of th... |
| invdif 4268 | Intersection with universa... |
| indif 4269 | Intersection with class di... |
| indif2 4270 | Bring an intersection in a... |
| indif1 4271 | Bring an intersection in a... |
| indifcom 4272 | Commutation law for inters... |
| indi 4273 | Distributive law for inter... |
| undi 4274 | Distributive law for union... |
| indir 4275 | Distributive law for inter... |
| undir 4276 | Distributive law for union... |
| unineq 4277 | Infer equality from equali... |
| uneqin 4278 | Equality of union and inte... |
| difundi 4279 | Distributive law for class... |
| difundir 4280 | Distributive law for class... |
| difindi 4281 | Distributive law for class... |
| difindir 4282 | Distributive law for class... |
| indifdi 4283 | Distribute intersection ov... |
| indifdir 4284 | Distribute intersection ov... |
| indifdirOLD 4285 | Obsolete version of ~ indi... |
| difdif2 4286 | Class difference by a clas... |
| undm 4287 | De Morgan's law for union.... |
| indm 4288 | De Morgan's law for inters... |
| difun1 4289 | A relationship involving d... |
| undif3 4290 | An equality involving clas... |
| difin2 4291 | Represent a class differen... |
| dif32 4292 | Swap second and third argu... |
| difabs 4293 | Absorption-like law for cl... |
| sscon34b 4294 | Relative complementation r... |
| rcompleq 4295 | Two subclasses are equal i... |
| dfsymdif3 4296 | Alternate definition of th... |
| unabw 4297 | Union of two class abstrac... |
| unab 4298 | Union of two class abstrac... |
| inab 4299 | Intersection of two class ... |
| difab 4300 | Difference of two class ab... |
| abanssl 4301 | A class abstraction with a... |
| abanssr 4302 | A class abstraction with a... |
| notabw 4303 | A class abstraction define... |
| notab 4304 | A class abstraction define... |
| unrab 4305 | Union of two restricted cl... |
| inrab 4306 | Intersection of two restri... |
| inrab2 4307 | Intersection with a restri... |
| difrab 4308 | Difference of two restrict... |
| dfrab3 4309 | Alternate definition of re... |
| dfrab2 4310 | Alternate definition of re... |
| rabdif 4311 | Move difference in and out... |
| notrab 4312 | Complementation of restric... |
| dfrab3ss 4313 | Restricted class abstracti... |
| rabun2 4314 | Abstraction restricted to ... |
| reuun2 4315 | Transfer uniqueness to a s... |
| reuss2 4316 | Transfer uniqueness to a s... |
| reuss 4317 | Transfer uniqueness to a s... |
| reuun1 4318 | Transfer uniqueness to a s... |
| reupick 4319 | Restricted uniqueness "pic... |
| reupick3 4320 | Restricted uniqueness "pic... |
| reupick2 4321 | Restricted uniqueness "pic... |
| euelss 4322 | Transfer uniqueness of an ... |
| dfnul4 4325 | Alternate definition of th... |
| dfnul2 4326 | Alternate definition of th... |
| dfnul3 4327 | Alternate definition of th... |
| dfnul2OLD 4328 | Obsolete version of ~ dfnu... |
| dfnul3OLD 4329 | Obsolete version of ~ dfnu... |
| dfnul4OLD 4330 | Obsolete version of ~ dfnu... |
| noel 4331 | The empty set has no eleme... |
| noelOLD 4332 | Obsolete version of ~ noel... |
| nel02 4333 | The empty set has no eleme... |
| n0i 4334 | If a class has elements, t... |
| ne0i 4335 | If a class has elements, t... |
| ne0d 4336 | Deduction form of ~ ne0i .... |
| n0ii 4337 | If a class has elements, t... |
| ne0ii 4338 | If a class has elements, t... |
| vn0 4339 | The universal class is not... |
| vn0ALT 4340 | Alternate proof of ~ vn0 .... |
| eq0f 4341 | A class is equal to the em... |
| neq0f 4342 | A class is not empty if an... |
| n0f 4343 | A class is nonempty if and... |
| eq0 4344 | A class is equal to the em... |
| eq0ALT 4345 | Alternate proof of ~ eq0 .... |
| neq0 4346 | A class is not empty if an... |
| n0 4347 | A class is nonempty if and... |
| eq0OLDOLD 4348 | Obsolete version of ~ eq0 ... |
| neq0OLD 4349 | Obsolete version of ~ neq0... |
| n0OLD 4350 | Obsolete version of ~ n0 a... |
| nel0 4351 | From the general negation ... |
| reximdva0 4352 | Restricted existence deduc... |
| rspn0 4353 | Specialization for restric... |
| rspn0OLD 4354 | Obsolete version of ~ rspn... |
| n0rex 4355 | There is an element in a n... |
| ssn0rex 4356 | There is an element in a c... |
| n0moeu 4357 | A case of equivalence of "... |
| rex0 4358 | Vacuous restricted existen... |
| reu0 4359 | Vacuous restricted uniquen... |
| rmo0 4360 | Vacuous restricted at-most... |
| 0el 4361 | Membership of the empty se... |
| n0el 4362 | Negated membership of the ... |
| eqeuel 4363 | A condition which implies ... |
| ssdif0 4364 | Subclass expressed in term... |
| difn0 4365 | If the difference of two s... |
| pssdifn0 4366 | A proper subclass has a no... |
| pssdif 4367 | A proper subclass has a no... |
| ndisj 4368 | Express that an intersecti... |
| difin0ss 4369 | Difference, intersection, ... |
| inssdif0 4370 | Intersection, subclass, an... |
| difid 4371 | The difference between a c... |
| difidALT 4372 | Alternate proof of ~ difid... |
| dif0 4373 | The difference between a c... |
| ab0w 4374 | The class of sets verifyin... |
| ab0 4375 | The class of sets verifyin... |
| ab0OLD 4376 | Obsolete version of ~ ab0 ... |
| ab0ALT 4377 | Alternate proof of ~ ab0 ,... |
| dfnf5 4378 | Characterization of nonfre... |
| ab0orv 4379 | The class abstraction defi... |
| ab0orvALT 4380 | Alternate proof of ~ ab0or... |
| abn0 4381 | Nonempty class abstraction... |
| abn0OLD 4382 | Obsolete version of ~ abn0... |
| rab0 4383 | Any restricted class abstr... |
| rabeq0w 4384 | Condition for a restricted... |
| rabeq0 4385 | Condition for a restricted... |
| rabn0 4386 | Nonempty restricted class ... |
| rabxm 4387 | Law of excluded middle, in... |
| rabnc 4388 | Law of noncontradiction, i... |
| elneldisj 4389 | The set of elements ` s ` ... |
| elnelun 4390 | The union of the set of el... |
| un0 4391 | The union of a class with ... |
| in0 4392 | The intersection of a clas... |
| 0un 4393 | The union of the empty set... |
| 0in 4394 | The intersection of the em... |
| inv1 4395 | The intersection of a clas... |
| unv 4396 | The union of a class with ... |
| 0ss 4397 | The null set is a subset o... |
| ss0b 4398 | Any subset of the empty se... |
| ss0 4399 | Any subset of the empty se... |
| sseq0 4400 | A subclass of an empty cla... |
| ssn0 4401 | A class with a nonempty su... |
| 0dif 4402 | The difference between the... |
| abf 4403 | A class abstraction determ... |
| abfOLD 4404 | Obsolete version of ~ abf ... |
| eq0rdv 4405 | Deduction for equality to ... |
| eq0rdvALT 4406 | Alternate proof of ~ eq0rd... |
| csbprc 4407 | The proper substitution of... |
| csb0 4408 | The proper substitution of... |
| sbcel12 4409 | Distribute proper substitu... |
| sbceqg 4410 | Distribute proper substitu... |
| sbceqi 4411 | Distribution of class subs... |
| sbcnel12g 4412 | Distribute proper substitu... |
| sbcne12 4413 | Distribute proper substitu... |
| sbcel1g 4414 | Move proper substitution i... |
| sbceq1g 4415 | Move proper substitution t... |
| sbcel2 4416 | Move proper substitution i... |
| sbceq2g 4417 | Move proper substitution t... |
| csbcom 4418 | Commutative law for double... |
| sbcnestgfw 4419 | Nest the composition of tw... |
| csbnestgfw 4420 | Nest the composition of tw... |
| sbcnestgw 4421 | Nest the composition of tw... |
| csbnestgw 4422 | Nest the composition of tw... |
| sbcco3gw 4423 | Composition of two substit... |
| sbcnestgf 4424 | Nest the composition of tw... |
| csbnestgf 4425 | Nest the composition of tw... |
| sbcnestg 4426 | Nest the composition of tw... |
| csbnestg 4427 | Nest the composition of tw... |
| sbcco3g 4428 | Composition of two substit... |
| csbco3g 4429 | Composition of two class s... |
| csbnest1g 4430 | Nest the composition of tw... |
| csbidm 4431 | Idempotent law for class s... |
| csbvarg 4432 | The proper substitution of... |
| csbvargi 4433 | The proper substitution of... |
| sbccsb 4434 | Substitution into a wff ex... |
| sbccsb2 4435 | Substitution into a wff ex... |
| rspcsbela 4436 | Special case related to ~ ... |
| sbnfc2 4437 | Two ways of expressing " `... |
| csbab 4438 | Move substitution into a c... |
| csbun 4439 | Distribution of class subs... |
| csbin 4440 | Distribute proper substitu... |
| csbie2df 4441 | Conversion of implicit sub... |
| 2nreu 4442 | If there are two different... |
| un00 4443 | Two classes are empty iff ... |
| vss 4444 | Only the universal class h... |
| 0pss 4445 | The null set is a proper s... |
| npss0 4446 | No set is a proper subset ... |
| pssv 4447 | Any non-universal class is... |
| disj 4448 | Two ways of saying that tw... |
| disjOLD 4449 | Obsolete version of ~ disj... |
| disjr 4450 | Two ways of saying that tw... |
| disj1 4451 | Two ways of saying that tw... |
| reldisj 4452 | Two ways of saying that tw... |
| reldisjOLD 4453 | Obsolete version of ~ reld... |
| disj3 4454 | Two ways of saying that tw... |
| disjne 4455 | Members of disjoint sets a... |
| disjeq0 4456 | Two disjoint sets are equa... |
| disjel 4457 | A set can't belong to both... |
| disj2 4458 | Two ways of saying that tw... |
| disj4 4459 | Two ways of saying that tw... |
| ssdisj 4460 | Intersection with a subcla... |
| disjpss 4461 | A class is a proper subset... |
| undisj1 4462 | The union of disjoint clas... |
| undisj2 4463 | The union of disjoint clas... |
| ssindif0 4464 | Subclass expressed in term... |
| inelcm 4465 | The intersection of classe... |
| minel 4466 | A minimum element of a cla... |
| undif4 4467 | Distribute union over diff... |
| disjssun 4468 | Subset relation for disjoi... |
| vdif0 4469 | Universal class equality i... |
| difrab0eq 4470 | If the difference between ... |
| pssnel 4471 | A proper subclass has a me... |
| disjdif 4472 | A class and its relative c... |
| disjdifr 4473 | A class and its relative c... |
| difin0 4474 | The difference of a class ... |
| unvdif 4475 | The union of a class and i... |
| undif1 4476 | Absorption of difference b... |
| undif2 4477 | Absorption of difference b... |
| undifabs 4478 | Absorption of difference b... |
| inundif 4479 | The intersection and class... |
| disjdif2 4480 | The difference of a class ... |
| difun2 4481 | Absorption of union by dif... |
| undif 4482 | Union of complementary par... |
| undifr 4483 | Union of complementary par... |
| undifrOLD 4484 | Obsolete version of ~ undi... |
| undif5 4485 | An equality involving clas... |
| ssdifin0 4486 | A subset of a difference d... |
| ssdifeq0 4487 | A class is a subclass of i... |
| ssundif 4488 | A condition equivalent to ... |
| difcom 4489 | Swap the arguments of a cl... |
| pssdifcom1 4490 | Two ways to express overla... |
| pssdifcom2 4491 | Two ways to express non-co... |
| difdifdir 4492 | Distributive law for class... |
| uneqdifeq 4493 | Two ways to say that ` A `... |
| raldifeq 4494 | Equality theorem for restr... |
| r19.2z 4495 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4496 | A response to the notion t... |
| r19.3rz 4497 | Restricted quantification ... |
| r19.28z 4498 | Restricted quantifier vers... |
| r19.3rzv 4499 | Restricted quantification ... |
| r19.9rzv 4500 | Restricted quantification ... |
| r19.28zv 4501 | Restricted quantifier vers... |
| r19.37zv 4502 | Restricted quantifier vers... |
| r19.45zv 4503 | Restricted version of Theo... |
| r19.44zv 4504 | Restricted version of Theo... |
| r19.27z 4505 | Restricted quantifier vers... |
| r19.27zv 4506 | Restricted quantifier vers... |
| r19.36zv 4507 | Restricted quantifier vers... |
| ralidmw 4508 | Idempotent law for restric... |
| rzal 4509 | Vacuous quantification is ... |
| rzalALT 4510 | Alternate proof of ~ rzal ... |
| rexn0 4511 | Restricted existential qua... |
| ralidm 4512 | Idempotent law for restric... |
| ral0 4513 | Vacuous universal quantifi... |
| ralf0 4514 | The quantification of a fa... |
| rexn0OLD 4515 | Obsolete version of ~ rexn... |
| ralidmOLD 4516 | Obsolete version of ~ rali... |
| ral0OLD 4517 | Obsolete version of ~ ral0... |
| ralf0OLD 4518 | Obsolete version of ~ ralf... |
| ralnralall 4519 | A contradiction concerning... |
| falseral0 4520 | A false statement can only... |
| raaan 4521 | Rearrange restricted quant... |
| raaanv 4522 | Rearrange restricted quant... |
| sbss 4523 | Set substitution into the ... |
| sbcssg 4524 | Distribute proper substitu... |
| raaan2 4525 | Rearrange restricted quant... |
| 2reu4lem 4526 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4527 | Definition of double restr... |
| csbdif 4528 | Distribution of class subs... |
| dfif2 4531 | An alternate definition of... |
| dfif6 4532 | An alternate definition of... |
| ifeq1 4533 | Equality theorem for condi... |
| ifeq2 4534 | Equality theorem for condi... |
| iftrue 4535 | Value of the conditional o... |
| iftruei 4536 | Inference associated with ... |
| iftrued 4537 | Value of the conditional o... |
| iffalse 4538 | Value of the conditional o... |
| iffalsei 4539 | Inference associated with ... |
| iffalsed 4540 | Value of the conditional o... |
| ifnefalse 4541 | When values are unequal, b... |
| ifsb 4542 | Distribute a function over... |
| dfif3 4543 | Alternate definition of th... |
| dfif4 4544 | Alternate definition of th... |
| dfif5 4545 | Alternate definition of th... |
| ifssun 4546 | A conditional class is inc... |
| ifeq12 4547 | Equality theorem for condi... |
| ifeq1d 4548 | Equality deduction for con... |
| ifeq2d 4549 | Equality deduction for con... |
| ifeq12d 4550 | Equality deduction for con... |
| ifbi 4551 | Equivalence theorem for co... |
| ifbid 4552 | Equivalence deduction for ... |
| ifbieq1d 4553 | Equivalence/equality deduc... |
| ifbieq2i 4554 | Equivalence/equality infer... |
| ifbieq2d 4555 | Equivalence/equality deduc... |
| ifbieq12i 4556 | Equivalence deduction for ... |
| ifbieq12d 4557 | Equivalence deduction for ... |
| nfifd 4558 | Deduction form of ~ nfif .... |
| nfif 4559 | Bound-variable hypothesis ... |
| ifeq1da 4560 | Conditional equality. (Co... |
| ifeq2da 4561 | Conditional equality. (Co... |
| ifeq12da 4562 | Equivalence deduction for ... |
| ifbieq12d2 4563 | Equivalence deduction for ... |
| ifclda 4564 | Conditional closure. (Con... |
| ifeqda 4565 | Separation of the values o... |
| elimif 4566 | Elimination of a condition... |
| ifbothda 4567 | A wff ` th ` containing a ... |
| ifboth 4568 | A wff ` th ` containing a ... |
| ifid 4569 | Identical true and false a... |
| eqif 4570 | Expansion of an equality w... |
| ifval 4571 | Another expression of the ... |
| elif 4572 | Membership in a conditiona... |
| ifel 4573 | Membership of a conditiona... |
| ifcl 4574 | Membership (closure) of a ... |
| ifcld 4575 | Membership (closure) of a ... |
| ifcli 4576 | Inference associated with ... |
| ifexd 4577 | Existence of the condition... |
| ifexg 4578 | Existence of the condition... |
| ifex 4579 | Existence of the condition... |
| ifeqor 4580 | The possible values of a c... |
| ifnot 4581 | Negating the first argumen... |
| ifan 4582 | Rewrite a conjunction in a... |
| ifor 4583 | Rewrite a disjunction in a... |
| 2if2 4584 | Resolve two nested conditi... |
| ifcomnan 4585 | Commute the conditions in ... |
| csbif 4586 | Distribute proper substitu... |
| dedth 4587 | Weak deduction theorem tha... |
| dedth2h 4588 | Weak deduction theorem eli... |
| dedth3h 4589 | Weak deduction theorem eli... |
| dedth4h 4590 | Weak deduction theorem eli... |
| dedth2v 4591 | Weak deduction theorem for... |
| dedth3v 4592 | Weak deduction theorem for... |
| dedth4v 4593 | Weak deduction theorem for... |
| elimhyp 4594 | Eliminate a hypothesis con... |
| elimhyp2v 4595 | Eliminate a hypothesis con... |
| elimhyp3v 4596 | Eliminate a hypothesis con... |
| elimhyp4v 4597 | Eliminate a hypothesis con... |
| elimel 4598 | Eliminate a membership hyp... |
| elimdhyp 4599 | Version of ~ elimhyp where... |
| keephyp 4600 | Transform a hypothesis ` p... |
| keephyp2v 4601 | Keep a hypothesis containi... |
| keephyp3v 4602 | Keep a hypothesis containi... |
| pwjust 4604 | Soundness justification th... |
| elpwg 4606 | Membership in a power clas... |
| elpw 4607 | Membership in a power clas... |
| velpw 4608 | Setvar variable membership... |
| elpwd 4609 | Membership in a power clas... |
| elpwi 4610 | Subset relation implied by... |
| elpwb 4611 | Characterization of the el... |
| elpwid 4612 | An element of a power clas... |
| elelpwi 4613 | If ` A ` belongs to a part... |
| sspw 4614 | The powerclass preserves i... |
| sspwi 4615 | The powerclass preserves i... |
| sspwd 4616 | The powerclass preserves i... |
| pweq 4617 | Equality theorem for power... |
| pweqALT 4618 | Alternate proof of ~ pweq ... |
| pweqi 4619 | Equality inference for pow... |
| pweqd 4620 | Equality deduction for pow... |
| pwunss 4621 | The power class of the uni... |
| nfpw 4622 | Bound-variable hypothesis ... |
| pwidg 4623 | A set is an element of its... |
| pwidb 4624 | A class is an element of i... |
| pwid 4625 | A set is a member of its p... |
| pwss 4626 | Subclass relationship for ... |
| pwundif 4627 | Break up the power class o... |
| snjust 4628 | Soundness justification th... |
| sneq 4639 | Equality theorem for singl... |
| sneqi 4640 | Equality inference for sin... |
| sneqd 4641 | Equality deduction for sin... |
| dfsn2 4642 | Alternate definition of si... |
| elsng 4643 | There is exactly one eleme... |
| elsn 4644 | There is exactly one eleme... |
| velsn 4645 | There is only one element ... |
| elsni 4646 | There is at most one eleme... |
| rabsneq 4647 | Equality of class abstract... |
| absn 4648 | Condition for a class abst... |
| dfpr2 4649 | Alternate definition of a ... |
| dfsn2ALT 4650 | Alternate definition of si... |
| elprg 4651 | A member of a pair of clas... |
| elpri 4652 | If a class is an element o... |
| elpr 4653 | A member of a pair of clas... |
| elpr2g 4654 | A member of a pair of sets... |
| elpr2 4655 | A member of a pair of sets... |
| nelpr2 4656 | If a class is not an eleme... |
| nelpr1 4657 | If a class is not an eleme... |
| nelpri 4658 | If an element doesn't matc... |
| prneli 4659 | If an element doesn't matc... |
| nelprd 4660 | If an element doesn't matc... |
| eldifpr 4661 | Membership in a set with t... |
| rexdifpr 4662 | Restricted existential qua... |
| snidg 4663 | A set is a member of its s... |
| snidb 4664 | A class is a set iff it is... |
| snid 4665 | A set is a member of its s... |
| vsnid 4666 | A setvar variable is a mem... |
| elsn2g 4667 | There is exactly one eleme... |
| elsn2 4668 | There is exactly one eleme... |
| nelsn 4669 | If a class is not equal to... |
| rabeqsn 4670 | Conditions for a restricte... |
| rabsssn 4671 | Conditions for a restricte... |
| rabeqsnd 4672 | Conditions for a restricte... |
| ralsnsg 4673 | Substitution expressed in ... |
| rexsns 4674 | Restricted existential qua... |
| rexsngf 4675 | Restricted existential qua... |
| ralsngf 4676 | Restricted universal quant... |
| reusngf 4677 | Restricted existential uni... |
| ralsng 4678 | Substitution expressed in ... |
| rexsng 4679 | Restricted existential qua... |
| reusng 4680 | Restricted existential uni... |
| 2ralsng 4681 | Substitution expressed in ... |
| ralsngOLD 4682 | Obsolete version of ~ rals... |
| rexsngOLD 4683 | Obsolete version of ~ rexs... |
| rexreusng 4684 | Restricted existential uni... |
| exsnrex 4685 | There is a set being the e... |
| ralsn 4686 | Convert a universal quanti... |
| rexsn 4687 | Convert an existential qua... |
| elpwunsn 4688 | Membership in an extension... |
| eqoreldif 4689 | An element of a set is eit... |
| eltpg 4690 | Members of an unordered tr... |
| eldiftp 4691 | Membership in a set with t... |
| eltpi 4692 | A member of an unordered t... |
| eltp 4693 | A member of an unordered t... |
| dftp2 4694 | Alternate definition of un... |
| nfpr 4695 | Bound-variable hypothesis ... |
| ifpr 4696 | Membership of a conditiona... |
| ralprgf 4697 | Convert a restricted unive... |
| rexprgf 4698 | Convert a restricted exist... |
| ralprg 4699 | Convert a restricted unive... |
| ralprgOLD 4700 | Obsolete version of ~ ralp... |
| rexprg 4701 | Convert a restricted exist... |
| rexprgOLD 4702 | Obsolete version of ~ rexp... |
| raltpg 4703 | Convert a restricted unive... |
| rextpg 4704 | Convert a restricted exist... |
| ralpr 4705 | Convert a restricted unive... |
| rexpr 4706 | Convert a restricted exist... |
| reuprg0 4707 | Convert a restricted exist... |
| reuprg 4708 | Convert a restricted exist... |
| reurexprg 4709 | Convert a restricted exist... |
| raltp 4710 | Convert a universal quanti... |
| rextp 4711 | Convert an existential qua... |
| nfsn 4712 | Bound-variable hypothesis ... |
| csbsng 4713 | Distribute proper substitu... |
| csbprg 4714 | Distribute proper substitu... |
| elinsn 4715 | If the intersection of two... |
| disjsn 4716 | Intersection with the sing... |
| disjsn2 4717 | Two distinct singletons ar... |
| disjpr2 4718 | Two completely distinct un... |
| disjprsn 4719 | The disjoint intersection ... |
| disjtpsn 4720 | The disjoint intersection ... |
| disjtp2 4721 | Two completely distinct un... |
| snprc 4722 | The singleton of a proper ... |
| snnzb 4723 | A singleton is nonempty if... |
| rmosn 4724 | A restricted at-most-one q... |
| r19.12sn 4725 | Special case of ~ r19.12 w... |
| rabsn 4726 | Condition where a restrict... |
| rabsnifsb 4727 | A restricted class abstrac... |
| rabsnif 4728 | A restricted class abstrac... |
| rabrsn 4729 | A restricted class abstrac... |
| euabsn2 4730 | Another way to express exi... |
| euabsn 4731 | Another way to express exi... |
| reusn 4732 | A way to express restricte... |
| absneu 4733 | Restricted existential uni... |
| rabsneu 4734 | Restricted existential uni... |
| eusn 4735 | Two ways to express " ` A ... |
| rabsnt 4736 | Truth implied by equality ... |
| prcom 4737 | Commutative law for unorde... |
| preq1 4738 | Equality theorem for unord... |
| preq2 4739 | Equality theorem for unord... |
| preq12 4740 | Equality theorem for unord... |
| preq1i 4741 | Equality inference for uno... |
| preq2i 4742 | Equality inference for uno... |
| preq12i 4743 | Equality inference for uno... |
| preq1d 4744 | Equality deduction for uno... |
| preq2d 4745 | Equality deduction for uno... |
| preq12d 4746 | Equality deduction for uno... |
| tpeq1 4747 | Equality theorem for unord... |
| tpeq2 4748 | Equality theorem for unord... |
| tpeq3 4749 | Equality theorem for unord... |
| tpeq1d 4750 | Equality theorem for unord... |
| tpeq2d 4751 | Equality theorem for unord... |
| tpeq3d 4752 | Equality theorem for unord... |
| tpeq123d 4753 | Equality theorem for unord... |
| tprot 4754 | Rotation of the elements o... |
| tpcoma 4755 | Swap 1st and 2nd members o... |
| tpcomb 4756 | Swap 2nd and 3rd members o... |
| tpass 4757 | Split off the first elemen... |
| qdass 4758 | Two ways to write an unord... |
| qdassr 4759 | Two ways to write an unord... |
| tpidm12 4760 | Unordered triple ` { A , A... |
| tpidm13 4761 | Unordered triple ` { A , B... |
| tpidm23 4762 | Unordered triple ` { A , B... |
| tpidm 4763 | Unordered triple ` { A , A... |
| tppreq3 4764 | An unordered triple is an ... |
| prid1g 4765 | An unordered pair contains... |
| prid2g 4766 | An unordered pair contains... |
| prid1 4767 | An unordered pair contains... |
| prid2 4768 | An unordered pair contains... |
| ifpprsnss 4769 | An unordered pair is a sin... |
| prprc1 4770 | A proper class vanishes in... |
| prprc2 4771 | A proper class vanishes in... |
| prprc 4772 | An unordered pair containi... |
| tpid1 4773 | One of the three elements ... |
| tpid1g 4774 | Closed theorem form of ~ t... |
| tpid2 4775 | One of the three elements ... |
| tpid2g 4776 | Closed theorem form of ~ t... |
| tpid3g 4777 | Closed theorem form of ~ t... |
| tpid3 4778 | One of the three elements ... |
| snnzg 4779 | The singleton of a set is ... |
| snn0d 4780 | The singleton of a set is ... |
| snnz 4781 | The singleton of a set is ... |
| prnz 4782 | A pair containing a set is... |
| prnzg 4783 | A pair containing a set is... |
| tpnz 4784 | An unordered triple contai... |
| tpnzd 4785 | An unordered triple contai... |
| raltpd 4786 | Convert a universal quanti... |
| snssb 4787 | Characterization of the in... |
| snssg 4788 | The singleton formed on a ... |
| snssgOLD 4789 | Obsolete version of ~ snss... |
| snss 4790 | The singleton of an elemen... |
| eldifsn 4791 | Membership in a set with a... |
| ssdifsn 4792 | Subset of a set with an el... |
| elpwdifsn 4793 | A subset of a set is an el... |
| eldifsni 4794 | Membership in a set with a... |
| eldifsnneq 4795 | An element of a difference... |
| neldifsn 4796 | The class ` A ` is not in ... |
| neldifsnd 4797 | The class ` A ` is not in ... |
| rexdifsn 4798 | Restricted existential qua... |
| raldifsni 4799 | Rearrangement of a propert... |
| raldifsnb 4800 | Restricted universal quant... |
| eldifvsn 4801 | A set is an element of the... |
| difsn 4802 | An element not in a set ca... |
| difprsnss 4803 | Removal of a singleton fro... |
| difprsn1 4804 | Removal of a singleton fro... |
| difprsn2 4805 | Removal of a singleton fro... |
| diftpsn3 4806 | Removal of a singleton fro... |
| difpr 4807 | Removing two elements as p... |
| tpprceq3 4808 | An unordered triple is an ... |
| tppreqb 4809 | An unordered triple is an ... |
| difsnb 4810 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4811 | ` ( B \ { A } ) ` is a pro... |
| snssi 4812 | The singleton of an elemen... |
| snssd 4813 | The singleton of an elemen... |
| difsnid 4814 | If we remove a single elem... |
| eldifeldifsn 4815 | An element of a difference... |
| pw0 4816 | Compute the power set of t... |
| pwpw0 4817 | Compute the power set of t... |
| snsspr1 4818 | A singleton is a subset of... |
| snsspr2 4819 | A singleton is a subset of... |
| snsstp1 4820 | A singleton is a subset of... |
| snsstp2 4821 | A singleton is a subset of... |
| snsstp3 4822 | A singleton is a subset of... |
| prssg 4823 | A pair of elements of a cl... |
| prss 4824 | A pair of elements of a cl... |
| prssi 4825 | A pair of elements of a cl... |
| prssd 4826 | Deduction version of ~ prs... |
| prsspwg 4827 | An unordered pair belongs ... |
| ssprss 4828 | A pair as subset of a pair... |
| ssprsseq 4829 | A proper pair is a subset ... |
| sssn 4830 | The subsets of a singleton... |
| ssunsn2 4831 | The property of being sand... |
| ssunsn 4832 | Possible values for a set ... |
| eqsn 4833 | Two ways to express that a... |
| issn 4834 | A sufficient condition for... |
| n0snor2el 4835 | A nonempty set is either a... |
| ssunpr 4836 | Possible values for a set ... |
| sspr 4837 | The subsets of a pair. (C... |
| sstp 4838 | The subsets of an unordere... |
| tpss 4839 | An unordered triple of ele... |
| tpssi 4840 | An unordered triple of ele... |
| sneqrg 4841 | Closed form of ~ sneqr . ... |
| sneqr 4842 | If the singletons of two s... |
| snsssn 4843 | If a singleton is a subset... |
| mosneq 4844 | There exists at most one s... |
| sneqbg 4845 | Two singletons of sets are... |
| snsspw 4846 | The singleton of a class i... |
| prsspw 4847 | An unordered pair belongs ... |
| preq1b 4848 | Biconditional equality lem... |
| preq2b 4849 | Biconditional equality lem... |
| preqr1 4850 | Reverse equality lemma for... |
| preqr2 4851 | Reverse equality lemma for... |
| preq12b 4852 | Equality relationship for ... |
| opthpr 4853 | An unordered pair has the ... |
| preqr1g 4854 | Reverse equality lemma for... |
| preq12bg 4855 | Closed form of ~ preq12b .... |
| prneimg 4856 | Two pairs are not equal if... |
| prnebg 4857 | A (proper) pair is not equ... |
| pr1eqbg 4858 | A (proper) pair is equal t... |
| pr1nebg 4859 | A (proper) pair is not equ... |
| preqsnd 4860 | Equivalence for a pair equ... |
| prnesn 4861 | A proper unordered pair is... |
| prneprprc 4862 | A proper unordered pair is... |
| preqsn 4863 | Equivalence for a pair equ... |
| preq12nebg 4864 | Equality relationship for ... |
| prel12g 4865 | Equality of two unordered ... |
| opthprneg 4866 | An unordered pair has the ... |
| elpreqprlem 4867 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4868 | Equality and membership ru... |
| elpreqprb 4869 | A set is an element of an ... |
| elpr2elpr 4870 | For an element ` A ` of an... |
| dfopif 4871 | Rewrite ~ df-op using ` if... |
| dfopg 4872 | Value of the ordered pair ... |
| dfop 4873 | Value of an ordered pair w... |
| opeq1 4874 | Equality theorem for order... |
| opeq2 4875 | Equality theorem for order... |
| opeq12 4876 | Equality theorem for order... |
| opeq1i 4877 | Equality inference for ord... |
| opeq2i 4878 | Equality inference for ord... |
| opeq12i 4879 | Equality inference for ord... |
| opeq1d 4880 | Equality deduction for ord... |
| opeq2d 4881 | Equality deduction for ord... |
| opeq12d 4882 | Equality deduction for ord... |
| oteq1 4883 | Equality theorem for order... |
| oteq2 4884 | Equality theorem for order... |
| oteq3 4885 | Equality theorem for order... |
| oteq1d 4886 | Equality deduction for ord... |
| oteq2d 4887 | Equality deduction for ord... |
| oteq3d 4888 | Equality deduction for ord... |
| oteq123d 4889 | Equality deduction for ord... |
| nfop 4890 | Bound-variable hypothesis ... |
| nfopd 4891 | Deduction version of bound... |
| csbopg 4892 | Distribution of class subs... |
| opidg 4893 | The ordered pair ` <. A , ... |
| opid 4894 | The ordered pair ` <. A , ... |
| ralunsn 4895 | Restricted quantification ... |
| 2ralunsn 4896 | Double restricted quantifi... |
| opprc 4897 | Expansion of an ordered pa... |
| opprc1 4898 | Expansion of an ordered pa... |
| opprc2 4899 | Expansion of an ordered pa... |
| oprcl 4900 | If an ordered pair has an ... |
| pwsn 4901 | The power set of a singlet... |
| pwpr 4902 | The power set of an unorde... |
| pwtp 4903 | The power set of an unorde... |
| pwpwpw0 4904 | Compute the power set of t... |
| pwv 4905 | The power class of the uni... |
| prproe 4906 | For an element of a proper... |
| 3elpr2eq 4907 | If there are three element... |
| dfuni2 4910 | Alternate definition of cl... |
| eluni 4911 | Membership in class union.... |
| eluni2 4912 | Membership in class union.... |
| elunii 4913 | Membership in class union.... |
| nfunid 4914 | Deduction version of ~ nfu... |
| nfuni 4915 | Bound-variable hypothesis ... |
| uniss 4916 | Subclass relationship for ... |
| unissi 4917 | Subclass relationship for ... |
| unissd 4918 | Subclass relationship for ... |
| unieq 4919 | Equality theorem for class... |
| unieqi 4920 | Inference of equality of t... |
| unieqd 4921 | Deduction of equality of t... |
| eluniab 4922 | Membership in union of a c... |
| elunirab 4923 | Membership in union of a c... |
| uniprg 4924 | The union of a pair is the... |
| unipr 4925 | The union of a pair is the... |
| uniprOLD 4926 | Obsolete version of ~ unip... |
| uniprgOLD 4927 | Obsolete version of ~ unip... |
| unisng 4928 | A set equals the union of ... |
| unisn 4929 | A set equals the union of ... |
| unisnv 4930 | A set equals the union of ... |
| unisn3 4931 | Union of a singleton in th... |
| dfnfc2 4932 | An alternative statement o... |
| uniun 4933 | The class union of the uni... |
| uniin 4934 | The class union of the int... |
| ssuni 4935 | Subclass relationship for ... |
| uni0b 4936 | The union of a set is empt... |
| uni0c 4937 | The union of a set is empt... |
| uni0 4938 | The union of the empty set... |
| csbuni 4939 | Distribute proper substitu... |
| elssuni 4940 | An element of a class is a... |
| unissel 4941 | Condition turning a subcla... |
| unissb 4942 | Relationship involving mem... |
| unissbOLD 4943 | Obsolete version of ~ unis... |
| uniss2 4944 | A subclass condition on th... |
| unidif 4945 | If the difference ` A \ B ... |
| ssunieq 4946 | Relationship implying unio... |
| unimax 4947 | Any member of a class is t... |
| pwuni 4948 | A class is a subclass of t... |
| dfint2 4951 | Alternate definition of cl... |
| inteq 4952 | Equality law for intersect... |
| inteqi 4953 | Equality inference for cla... |
| inteqd 4954 | Equality deduction for cla... |
| elint 4955 | Membership in class inters... |
| elint2 4956 | Membership in class inters... |
| elintg 4957 | Membership in class inters... |
| elinti 4958 | Membership in class inters... |
| nfint 4959 | Bound-variable hypothesis ... |
| elintabg 4960 | Two ways of saying a set i... |
| elintab 4961 | Membership in the intersec... |
| elintabOLD 4962 | Obsolete version of ~ elin... |
| elintrab 4963 | Membership in the intersec... |
| elintrabg 4964 | Membership in the intersec... |
| int0 4965 | The intersection of the em... |
| intss1 4966 | An element of a class incl... |
| ssint 4967 | Subclass of a class inters... |
| ssintab 4968 | Subclass of the intersecti... |
| ssintub 4969 | Subclass of the least uppe... |
| ssmin 4970 | Subclass of the minimum va... |
| intmin 4971 | Any member of a class is t... |
| intss 4972 | Intersection of subclasses... |
| intssuni 4973 | The intersection of a none... |
| ssintrab 4974 | Subclass of the intersecti... |
| unissint 4975 | If the union of a class is... |
| intssuni2 4976 | Subclass relationship for ... |
| intminss 4977 | Under subset ordering, the... |
| intmin2 4978 | Any set is the smallest of... |
| intmin3 4979 | Under subset ordering, the... |
| intmin4 4980 | Elimination of a conjunct ... |
| intab 4981 | The intersection of a spec... |
| int0el 4982 | The intersection of a clas... |
| intun 4983 | The class intersection of ... |
| intprg 4984 | The intersection of a pair... |
| intpr 4985 | The intersection of a pair... |
| intprOLD 4986 | Obsolete version of ~ intp... |
| intprgOLD 4987 | Obsolete version of ~ intp... |
| intsng 4988 | Intersection of a singleto... |
| intsn 4989 | The intersection of a sing... |
| uniintsn 4990 | Two ways to express " ` A ... |
| uniintab 4991 | The union and the intersec... |
| intunsn 4992 | Theorem joining a singleto... |
| rint0 4993 | Relative intersection of a... |
| elrint 4994 | Membership in a restricted... |
| elrint2 4995 | Membership in a restricted... |
| eliun 5000 | Membership in indexed unio... |
| eliin 5001 | Membership in indexed inte... |
| eliuni 5002 | Membership in an indexed u... |
| iuncom 5003 | Commutation of indexed uni... |
| iuncom4 5004 | Commutation of union with ... |
| iunconst 5005 | Indexed union of a constan... |
| iinconst 5006 | Indexed intersection of a ... |
| iuneqconst 5007 | Indexed union of identical... |
| iuniin 5008 | Law combining indexed unio... |
| iinssiun 5009 | An indexed intersection is... |
| iunss1 5010 | Subclass theorem for index... |
| iinss1 5011 | Subclass theorem for index... |
| iuneq1 5012 | Equality theorem for index... |
| iineq1 5013 | Equality theorem for index... |
| ss2iun 5014 | Subclass theorem for index... |
| iuneq2 5015 | Equality theorem for index... |
| iineq2 5016 | Equality theorem for index... |
| iuneq2i 5017 | Equality inference for ind... |
| iineq2i 5018 | Equality inference for ind... |
| iineq2d 5019 | Equality deduction for ind... |
| iuneq2dv 5020 | Equality deduction for ind... |
| iineq2dv 5021 | Equality deduction for ind... |
| iuneq12df 5022 | Equality deduction for ind... |
| iuneq1d 5023 | Equality theorem for index... |
| iuneq12d 5024 | Equality deduction for ind... |
| iuneq2d 5025 | Equality deduction for ind... |
| nfiun 5026 | Bound-variable hypothesis ... |
| nfiin 5027 | Bound-variable hypothesis ... |
| nfiung 5028 | Bound-variable hypothesis ... |
| nfiing 5029 | Bound-variable hypothesis ... |
| nfiu1 5030 | Bound-variable hypothesis ... |
| nfiu1OLD 5031 | Obsolete version of ~ nfiu... |
| nfii1 5032 | Bound-variable hypothesis ... |
| dfiun2g 5033 | Alternate definition of in... |
| dfiun2gOLD 5034 | Obsolete version of ~ dfiu... |
| dfiin2g 5035 | Alternate definition of in... |
| dfiun2 5036 | Alternate definition of in... |
| dfiin2 5037 | Alternate definition of in... |
| dfiunv2 5038 | Define double indexed unio... |
| cbviun 5039 | Rule used to change the bo... |
| cbviin 5040 | Change bound variables in ... |
| cbviung 5041 | Rule used to change the bo... |
| cbviing 5042 | Change bound variables in ... |
| cbviunv 5043 | Rule used to change the bo... |
| cbviinv 5044 | Change bound variables in ... |
| cbviunvg 5045 | Rule used to change the bo... |
| cbviinvg 5046 | Change bound variables in ... |
| iunssf 5047 | Subset theorem for an inde... |
| iunss 5048 | Subset theorem for an inde... |
| ssiun 5049 | Subset implication for an ... |
| ssiun2 5050 | Identity law for subset of... |
| ssiun2s 5051 | Subset relationship for an... |
| iunss2 5052 | A subclass condition on th... |
| iunssd 5053 | Subset theorem for an inde... |
| iunab 5054 | The indexed union of a cla... |
| iunrab 5055 | The indexed union of a res... |
| iunxdif2 5056 | Indexed union with a class... |
| ssiinf 5057 | Subset theorem for an inde... |
| ssiin 5058 | Subset theorem for an inde... |
| iinss 5059 | Subset implication for an ... |
| iinss2 5060 | An indexed intersection is... |
| uniiun 5061 | Class union in terms of in... |
| intiin 5062 | Class intersection in term... |
| iunid 5063 | An indexed union of single... |
| iunidOLD 5064 | Obsolete version of ~ iuni... |
| iun0 5065 | An indexed union of the em... |
| 0iun 5066 | An empty indexed union is ... |
| 0iin 5067 | An empty indexed intersect... |
| viin 5068 | Indexed intersection with ... |
| iunsn 5069 | Indexed union of a singlet... |
| iunn0 5070 | There is a nonempty class ... |
| iinab 5071 | Indexed intersection of a ... |
| iinrab 5072 | Indexed intersection of a ... |
| iinrab2 5073 | Indexed intersection of a ... |
| iunin2 5074 | Indexed union of intersect... |
| iunin1 5075 | Indexed union of intersect... |
| iinun2 5076 | Indexed intersection of un... |
| iundif2 5077 | Indexed union of class dif... |
| iindif1 5078 | Indexed intersection of cl... |
| 2iunin 5079 | Rearrange indexed unions o... |
| iindif2 5080 | Indexed intersection of cl... |
| iinin2 5081 | Indexed intersection of in... |
| iinin1 5082 | Indexed intersection of in... |
| iinvdif 5083 | The indexed intersection o... |
| elriin 5084 | Elementhood in a relative ... |
| riin0 5085 | Relative intersection of a... |
| riinn0 5086 | Relative intersection of a... |
| riinrab 5087 | Relative intersection of a... |
| symdif0 5088 | Symmetric difference with ... |
| symdifv 5089 | The symmetric difference w... |
| symdifid 5090 | The symmetric difference o... |
| iinxsng 5091 | A singleton index picks ou... |
| iinxprg 5092 | Indexed intersection with ... |
| iunxsng 5093 | A singleton index picks ou... |
| iunxsn 5094 | A singleton index picks ou... |
| iunxsngf 5095 | A singleton index picks ou... |
| iunun 5096 | Separate a union in an ind... |
| iunxun 5097 | Separate a union in the in... |
| iunxdif3 5098 | An indexed union where som... |
| iunxprg 5099 | A pair index picks out two... |
| iunxiun 5100 | Separate an indexed union ... |
| iinuni 5101 | A relationship involving u... |
| iununi 5102 | A relationship involving u... |
| sspwuni 5103 | Subclass relationship for ... |
| pwssb 5104 | Two ways to express a coll... |
| elpwpw 5105 | Characterization of the el... |
| pwpwab 5106 | The double power class wri... |
| pwpwssunieq 5107 | The class of sets whose un... |
| elpwuni 5108 | Relationship for power cla... |
| iinpw 5109 | The power class of an inte... |
| iunpwss 5110 | Inclusion of an indexed un... |
| intss2 5111 | A nonempty intersection of... |
| rintn0 5112 | Relative intersection of a... |
| dfdisj2 5115 | Alternate definition for d... |
| disjss2 5116 | If each element of a colle... |
| disjeq2 5117 | Equality theorem for disjo... |
| disjeq2dv 5118 | Equality deduction for dis... |
| disjss1 5119 | A subset of a disjoint col... |
| disjeq1 5120 | Equality theorem for disjo... |
| disjeq1d 5121 | Equality theorem for disjo... |
| disjeq12d 5122 | Equality theorem for disjo... |
| cbvdisj 5123 | Change bound variables in ... |
| cbvdisjv 5124 | Change bound variables in ... |
| nfdisjw 5125 | Bound-variable hypothesis ... |
| nfdisj 5126 | Bound-variable hypothesis ... |
| nfdisj1 5127 | Bound-variable hypothesis ... |
| disjor 5128 | Two ways to say that a col... |
| disjors 5129 | Two ways to say that a col... |
| disji2 5130 | Property of a disjoint col... |
| disji 5131 | Property of a disjoint col... |
| invdisj 5132 | If there is a function ` C... |
| invdisjrabw 5133 | Version of ~ invdisjrab wi... |
| invdisjrab 5134 | The restricted class abstr... |
| disjiun 5135 | A disjoint collection yiel... |
| disjord 5136 | Conditions for a collectio... |
| disjiunb 5137 | Two ways to say that a col... |
| disjiund 5138 | Conditions for a collectio... |
| sndisj 5139 | Any collection of singleto... |
| 0disj 5140 | Any collection of empty se... |
| disjxsn 5141 | A singleton collection is ... |
| disjx0 5142 | An empty collection is dis... |
| disjprgw 5143 | Version of ~ disjprg with ... |
| disjprg 5144 | A pair collection is disjo... |
| disjxiun 5145 | An indexed union of a disj... |
| disjxun 5146 | The union of two disjoint ... |
| disjss3 5147 | Expand a disjoint collecti... |
| breq 5150 | Equality theorem for binar... |
| breq1 5151 | Equality theorem for a bin... |
| breq2 5152 | Equality theorem for a bin... |
| breq12 5153 | Equality theorem for a bin... |
| breqi 5154 | Equality inference for bin... |
| breq1i 5155 | Equality inference for a b... |
| breq2i 5156 | Equality inference for a b... |
| breq12i 5157 | Equality inference for a b... |
| breq1d 5158 | Equality deduction for a b... |
| breqd 5159 | Equality deduction for a b... |
| breq2d 5160 | Equality deduction for a b... |
| breq12d 5161 | Equality deduction for a b... |
| breq123d 5162 | Equality deduction for a b... |
| breqdi 5163 | Equality deduction for a b... |
| breqan12d 5164 | Equality deduction for a b... |
| breqan12rd 5165 | Equality deduction for a b... |
| eqnbrtrd 5166 | Substitution of equal clas... |
| nbrne1 5167 | Two classes are different ... |
| nbrne2 5168 | Two classes are different ... |
| eqbrtri 5169 | Substitution of equal clas... |
| eqbrtrd 5170 | Substitution of equal clas... |
| eqbrtrri 5171 | Substitution of equal clas... |
| eqbrtrrd 5172 | Substitution of equal clas... |
| breqtri 5173 | Substitution of equal clas... |
| breqtrd 5174 | Substitution of equal clas... |
| breqtrri 5175 | Substitution of equal clas... |
| breqtrrd 5176 | Substitution of equal clas... |
| 3brtr3i 5177 | Substitution of equality i... |
| 3brtr4i 5178 | Substitution of equality i... |
| 3brtr3d 5179 | Substitution of equality i... |
| 3brtr4d 5180 | Substitution of equality i... |
| 3brtr3g 5181 | Substitution of equality i... |
| 3brtr4g 5182 | Substitution of equality i... |
| eqbrtrid 5183 | A chained equality inferen... |
| eqbrtrrid 5184 | A chained equality inferen... |
| breqtrid 5185 | A chained equality inferen... |
| breqtrrid 5186 | A chained equality inferen... |
| eqbrtrdi 5187 | A chained equality inferen... |
| eqbrtrrdi 5188 | A chained equality inferen... |
| breqtrdi 5189 | A chained equality inferen... |
| breqtrrdi 5190 | A chained equality inferen... |
| ssbrd 5191 | Deduction from a subclass ... |
| ssbr 5192 | Implication from a subclas... |
| ssbri 5193 | Inference from a subclass ... |
| nfbrd 5194 | Deduction version of bound... |
| nfbr 5195 | Bound-variable hypothesis ... |
| brab1 5196 | Relationship between a bin... |
| br0 5197 | The empty binary relation ... |
| brne0 5198 | If two sets are in a binar... |
| brun 5199 | The union of two binary re... |
| brin 5200 | The intersection of two re... |
| brdif 5201 | The difference of two bina... |
| sbcbr123 5202 | Move substitution in and o... |
| sbcbr 5203 | Move substitution in and o... |
| sbcbr12g 5204 | Move substitution in and o... |
| sbcbr1g 5205 | Move substitution in and o... |
| sbcbr2g 5206 | Move substitution in and o... |
| brsymdif 5207 | Characterization of the sy... |
| brralrspcev 5208 | Restricted existential spe... |
| brimralrspcev 5209 | Restricted existential spe... |
| opabss 5212 | The collection of ordered ... |
| opabbid 5213 | Equivalent wff's yield equ... |
| opabbidv 5214 | Equivalent wff's yield equ... |
| opabbii 5215 | Equivalent wff's yield equ... |
| nfopabd 5216 | Bound-variable hypothesis ... |
| nfopab 5217 | Bound-variable hypothesis ... |
| nfopab1 5218 | The first abstraction vari... |
| nfopab2 5219 | The second abstraction var... |
| cbvopab 5220 | Rule used to change bound ... |
| cbvopabv 5221 | Rule used to change bound ... |
| cbvopabvOLD 5222 | Obsolete version of ~ cbvo... |
| cbvopab1 5223 | Change first bound variabl... |
| cbvopab1g 5224 | Change first bound variabl... |
| cbvopab2 5225 | Change second bound variab... |
| cbvopab1s 5226 | Change first bound variabl... |
| cbvopab1v 5227 | Rule used to change the fi... |
| cbvopab1vOLD 5228 | Obsolete version of ~ cbvo... |
| cbvopab2v 5229 | Rule used to change the se... |
| unopab 5230 | Union of two ordered pair ... |
| mpteq12da 5233 | An equality inference for ... |
| mpteq12df 5234 | An equality inference for ... |
| mpteq12dfOLD 5235 | Obsolete version of ~ mpte... |
| mpteq12f 5236 | An equality theorem for th... |
| mpteq12dva 5237 | An equality inference for ... |
| mpteq12dvaOLD 5238 | Obsolete version of ~ mpte... |
| mpteq12dv 5239 | An equality inference for ... |
| mpteq12 5240 | An equality theorem for th... |
| mpteq1 5241 | An equality theorem for th... |
| mpteq1OLD 5242 | Obsolete version of ~ mpte... |
| mpteq1d 5243 | An equality theorem for th... |
| mpteq1i 5244 | An equality theorem for th... |
| mpteq1iOLD 5245 | Obsolete version of ~ mpte... |
| mpteq2da 5246 | Slightly more general equa... |
| mpteq2daOLD 5247 | Obsolete version of ~ mpte... |
| mpteq2dva 5248 | Slightly more general equa... |
| mpteq2dvaOLD 5249 | Obsolete version of ~ mpte... |
| mpteq2dv 5250 | An equality inference for ... |
| mpteq2ia 5251 | An equality inference for ... |
| mpteq2iaOLD 5252 | Obsolete version of ~ mpte... |
| mpteq2i 5253 | An equality inference for ... |
| mpteq12i 5254 | An equality inference for ... |
| nfmpt 5255 | Bound-variable hypothesis ... |
| nfmpt1 5256 | Bound-variable hypothesis ... |
| cbvmptf 5257 | Rule to change the bound v... |
| cbvmptfg 5258 | Rule to change the bound v... |
| cbvmpt 5259 | Rule to change the bound v... |
| cbvmptg 5260 | Rule to change the bound v... |
| cbvmptv 5261 | Rule to change the bound v... |
| cbvmptvOLD 5262 | Obsolete version of ~ cbvm... |
| cbvmptvg 5263 | Rule to change the bound v... |
| mptv 5264 | Function with universal do... |
| dftr2 5267 | An alternate way of defini... |
| dftr2c 5268 | Variant of ~ dftr2 with co... |
| dftr5 5269 | An alternate way of defini... |
| dftr5OLD 5270 | Obsolete version of ~ dftr... |
| dftr3 5271 | An alternate way of defini... |
| dftr4 5272 | An alternate way of defini... |
| treq 5273 | Equality theorem for the t... |
| trel 5274 | In a transitive class, the... |
| trel3 5275 | In a transitive class, the... |
| trss 5276 | An element of a transitive... |
| trin 5277 | The intersection of transi... |
| tr0 5278 | The empty set is transitiv... |
| trv 5279 | The universe is transitive... |
| triun 5280 | An indexed union of a clas... |
| truni 5281 | The union of a class of tr... |
| triin 5282 | An indexed intersection of... |
| trint 5283 | The intersection of a clas... |
| trintss 5284 | Any nonempty transitive cl... |
| axrep1 5286 | The version of the Axiom o... |
| axreplem 5287 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5288 | Axiom of Replacement expre... |
| axrep3 5289 | Axiom of Replacement sligh... |
| axrep4 5290 | A more traditional version... |
| axrep5 5291 | Axiom of Replacement (simi... |
| axrep6 5292 | A condensed form of ~ ax-r... |
| axrep6g 5293 | ~ axrep6 in class notation... |
| zfrepclf 5294 | An inference based on the ... |
| zfrep3cl 5295 | An inference based on the ... |
| zfrep4 5296 | A version of Replacement u... |
| axsepgfromrep 5297 | A more general version ~ a... |
| axsep 5298 | Axiom scheme of separation... |
| axsepg 5300 | A more general version of ... |
| zfauscl 5301 | Separation Scheme (Aussond... |
| bm1.3ii 5302 | Convert implication to equ... |
| ax6vsep 5303 | Derive ~ ax6v (a weakened ... |
| axnulALT 5304 | Alternate proof of ~ axnul... |
| axnul 5305 | The Null Set Axiom of ZF s... |
| 0ex 5307 | The Null Set Axiom of ZF s... |
| al0ssb 5308 | The empty set is the uniqu... |
| sseliALT 5309 | Alternate proof of ~ sseli... |
| csbexg 5310 | The existence of proper su... |
| csbex 5311 | The existence of proper su... |
| unisn2 5312 | A version of ~ unisn witho... |
| nalset 5313 | No set contains all sets. ... |
| vnex 5314 | The universal class does n... |
| vprc 5315 | The universal class is not... |
| nvel 5316 | The universal class does n... |
| inex1 5317 | Separation Scheme (Aussond... |
| inex2 5318 | Separation Scheme (Aussond... |
| inex1g 5319 | Closed-form, generalized S... |
| inex2g 5320 | Sufficient condition for a... |
| ssex 5321 | The subset of a set is als... |
| ssexi 5322 | The subset of a set is als... |
| ssexg 5323 | The subset of a set is als... |
| ssexd 5324 | A subclass of a set is a s... |
| abexd 5325 | Conditions for a class abs... |
| abex 5326 | Conditions for a class abs... |
| prcssprc 5327 | The superclass of a proper... |
| sselpwd 5328 | Elementhood to a power set... |
| difexg 5329 | Existence of a difference.... |
| difexi 5330 | Existence of a difference,... |
| difexd 5331 | Existence of a difference.... |
| zfausab 5332 | Separation Scheme (Aussond... |
| rabexg 5333 | Separation Scheme in terms... |
| rabex 5334 | Separation Scheme in terms... |
| rabexd 5335 | Separation Scheme in terms... |
| rabex2 5336 | Separation Scheme in terms... |
| rab2ex 5337 | A class abstraction based ... |
| elssabg 5338 | Membership in a class abst... |
| intex 5339 | The intersection of a none... |
| intnex 5340 | If a class intersection is... |
| intexab 5341 | The intersection of a none... |
| intexrab 5342 | The intersection of a none... |
| iinexg 5343 | The existence of a class i... |
| intabs 5344 | Absorption of a redundant ... |
| inuni 5345 | The intersection of a unio... |
| elpw2g 5346 | Membership in a power clas... |
| elpw2 5347 | Membership in a power clas... |
| elpwi2 5348 | Membership in a power clas... |
| axpweq 5349 | Two equivalent ways to exp... |
| pwnss 5350 | The power set of a set is ... |
| pwne 5351 | No set equals its power se... |
| difelpw 5352 | A difference is an element... |
| rabelpw 5353 | A restricted class abstrac... |
| class2set 5354 | The class of elements of `... |
| 0elpw 5355 | Every power class contains... |
| pwne0 5356 | A power class is never emp... |
| 0nep0 5357 | The empty set and its powe... |
| 0inp0 5358 | Something cannot be equal ... |
| unidif0 5359 | The removal of the empty s... |
| eqsnuniex 5360 | If a class is equal to the... |
| iin0 5361 | An indexed intersection of... |
| notzfaus 5362 | In the Separation Scheme ~... |
| intv 5363 | The intersection of the un... |
| zfpow 5365 | Axiom of Power Sets expres... |
| axpow2 5366 | A variant of the Axiom of ... |
| axpow3 5367 | A variant of the Axiom of ... |
| elALT2 5368 | Alternate proof of ~ el us... |
| dtruALT2 5369 | Alternate proof of ~ dtru ... |
| dtrucor 5370 | Corollary of ~ dtru . Thi... |
| dtrucor2 5371 | The theorem form of the de... |
| dvdemo1 5372 | Demonstration of a theorem... |
| dvdemo2 5373 | Demonstration of a theorem... |
| nfnid 5374 | A setvar variable is not f... |
| nfcvb 5375 | The "distinctor" expressio... |
| vpwex 5376 | Power set axiom: the power... |
| pwexg 5377 | Power set axiom expressed ... |
| pwexd 5378 | Deduction version of the p... |
| pwex 5379 | Power set axiom expressed ... |
| pwel 5380 | Quantitative version of ~ ... |
| abssexg 5381 | Existence of a class of su... |
| snexALT 5382 | Alternate proof of ~ snex ... |
| p0ex 5383 | The power set of the empty... |
| p0exALT 5384 | Alternate proof of ~ p0ex ... |
| pp0ex 5385 | The power set of the power... |
| ord3ex 5386 | The ordinal number 3 is a ... |
| dtruALT 5387 | Alternate proof of ~ dtru ... |
| axc16b 5388 | This theorem shows that Ax... |
| eunex 5389 | Existential uniqueness imp... |
| eusv1 5390 | Two ways to express single... |
| eusvnf 5391 | Even if ` x ` is free in `... |
| eusvnfb 5392 | Two ways to say that ` A (... |
| eusv2i 5393 | Two ways to express single... |
| eusv2nf 5394 | Two ways to express single... |
| eusv2 5395 | Two ways to express single... |
| reusv1 5396 | Two ways to express single... |
| reusv2lem1 5397 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5398 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5399 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5400 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5401 | Lemma for ~ reusv2 . (Con... |
| reusv2 5402 | Two ways to express single... |
| reusv3i 5403 | Two ways of expressing exi... |
| reusv3 5404 | Two ways to express single... |
| eusv4 5405 | Two ways to express single... |
| alxfr 5406 | Transfer universal quantif... |
| ralxfrd 5407 | Transfer universal quantif... |
| rexxfrd 5408 | Transfer universal quantif... |
| ralxfr2d 5409 | Transfer universal quantif... |
| rexxfr2d 5410 | Transfer universal quantif... |
| ralxfrd2 5411 | Transfer universal quantif... |
| rexxfrd2 5412 | Transfer existence from a ... |
| ralxfr 5413 | Transfer universal quantif... |
| ralxfrALT 5414 | Alternate proof of ~ ralxf... |
| rexxfr 5415 | Transfer existence from a ... |
| rabxfrd 5416 | Membership in a restricted... |
| rabxfr 5417 | Membership in a restricted... |
| reuhypd 5418 | A theorem useful for elimi... |
| reuhyp 5419 | A theorem useful for elimi... |
| zfpair 5420 | The Axiom of Pairing of Ze... |
| axprALT 5421 | Alternate proof of ~ axpr ... |
| axprlem1 5422 | Lemma for ~ axpr . There ... |
| axprlem2 5423 | Lemma for ~ axpr . There ... |
| axprlem3 5424 | Lemma for ~ axpr . Elimin... |
| axprlem4 5425 | Lemma for ~ axpr . The fi... |
| axprlem5 5426 | Lemma for ~ axpr . The se... |
| axpr 5427 | Unabbreviated version of t... |
| zfpair2 5429 | Derive the abbreviated ver... |
| vsnex 5430 | A singleton built on a set... |
| snexg 5431 | A singleton built on a set... |
| snex 5432 | A singleton is a set. The... |
| prex 5433 | The Axiom of Pairing using... |
| exel 5434 | There exist two sets, one ... |
| exexneq 5435 | There exist two different ... |
| exneq 5436 | Given any set (the " ` y `... |
| dtru 5437 | Given any set (the " ` y `... |
| el 5438 | Any set is an element of s... |
| sels 5439 | If a class is a set, then ... |
| selsALT 5440 | Alternate proof of ~ sels ... |
| elALT 5441 | Alternate proof of ~ el , ... |
| dtruOLD 5442 | Obsolete proof of ~ dtru a... |
| snelpwg 5443 | A singleton of a set is a ... |
| snelpwi 5444 | If a set is a member of a ... |
| snelpwiOLD 5445 | Obsolete version of ~ snel... |
| snelpw 5446 | A singleton of a set is a ... |
| prelpw 5447 | An unordered pair of two s... |
| prelpwi 5448 | If two sets are members of... |
| rext 5449 | A theorem similar to exten... |
| sspwb 5450 | The powerclass constructio... |
| unipw 5451 | A class equals the union o... |
| univ 5452 | The union of the universe ... |
| pwtr 5453 | A class is transitive iff ... |
| ssextss 5454 | An extensionality-like pri... |
| ssext 5455 | An extensionality-like pri... |
| nssss 5456 | Negation of subclass relat... |
| pweqb 5457 | Classes are equal if and o... |
| intidg 5458 | The intersection of all se... |
| intidOLD 5459 | Obsolete version of ~ inti... |
| moabex 5460 | "At most one" existence im... |
| rmorabex 5461 | Restricted "at most one" e... |
| euabex 5462 | The abstraction of a wff w... |
| nnullss 5463 | A nonempty class (even if ... |
| exss 5464 | Restricted existence in a ... |
| opex 5465 | An ordered pair of classes... |
| otex 5466 | An ordered triple of class... |
| elopg 5467 | Characterization of the el... |
| elop 5468 | Characterization of the el... |
| opi1 5469 | One of the two elements in... |
| opi2 5470 | One of the two elements of... |
| opeluu 5471 | Each member of an ordered ... |
| op1stb 5472 | Extract the first member o... |
| brv 5473 | Two classes are always in ... |
| opnz 5474 | An ordered pair is nonempt... |
| opnzi 5475 | An ordered pair is nonempt... |
| opth1 5476 | Equality of the first memb... |
| opth 5477 | The ordered pair theorem. ... |
| opthg 5478 | Ordered pair theorem. ` C ... |
| opth1g 5479 | Equality of the first memb... |
| opthg2 5480 | Ordered pair theorem. (Co... |
| opth2 5481 | Ordered pair theorem. (Co... |
| opthneg 5482 | Two ordered pairs are not ... |
| opthne 5483 | Two ordered pairs are not ... |
| otth2 5484 | Ordered triple theorem, wi... |
| otth 5485 | Ordered triple theorem. (... |
| otthg 5486 | Ordered triple theorem, cl... |
| otthne 5487 | Contrapositive of the orde... |
| eqvinop 5488 | A variable introduction la... |
| sbcop1 5489 | The proper substitution of... |
| sbcop 5490 | The proper substitution of... |
| copsexgw 5491 | Version of ~ copsexg with ... |
| copsexg 5492 | Substitution of class ` A ... |
| copsex2t 5493 | Closed theorem form of ~ c... |
| copsex2g 5494 | Implicit substitution infe... |
| copsex2gOLD 5495 | Obsolete version of ~ cops... |
| copsex4g 5496 | An implicit substitution i... |
| 0nelop 5497 | A property of ordered pair... |
| opwo0id 5498 | An ordered pair is equal t... |
| opeqex 5499 | Equivalence of existence i... |
| oteqex2 5500 | Equivalence of existence i... |
| oteqex 5501 | Equivalence of existence i... |
| opcom 5502 | An ordered pair commutes i... |
| moop2 5503 | "At most one" property of ... |
| opeqsng 5504 | Equivalence for an ordered... |
| opeqsn 5505 | Equivalence for an ordered... |
| opeqpr 5506 | Equivalence for an ordered... |
| snopeqop 5507 | Equivalence for an ordered... |
| propeqop 5508 | Equivalence for an ordered... |
| propssopi 5509 | If a pair of ordered pairs... |
| snopeqopsnid 5510 | Equivalence for an ordered... |
| mosubopt 5511 | "At most one" remains true... |
| mosubop 5512 | "At most one" remains true... |
| euop2 5513 | Transfer existential uniqu... |
| euotd 5514 | Prove existential uniquene... |
| opthwiener 5515 | Justification theorem for ... |
| uniop 5516 | The union of an ordered pa... |
| uniopel 5517 | Ordered pair membership is... |
| opthhausdorff 5518 | Justification theorem for ... |
| opthhausdorff0 5519 | Justification theorem for ... |
| otsndisj 5520 | The singletons consisting ... |
| otiunsndisj 5521 | The union of singletons co... |
| iunopeqop 5522 | Implication of an ordered ... |
| brsnop 5523 | Binary relation for an ord... |
| brtp 5524 | A necessary and sufficient... |
| opabidw 5525 | The law of concretion. Sp... |
| opabid 5526 | The law of concretion. Sp... |
| elopabw 5527 | Membership in a class abst... |
| elopab 5528 | Membership in a class abst... |
| rexopabb 5529 | Restricted existential qua... |
| vopelopabsb 5530 | The law of concretion in t... |
| opelopabsb 5531 | The law of concretion in t... |
| brabsb 5532 | The law of concretion in t... |
| opelopabt 5533 | Closed theorem form of ~ o... |
| opelopabga 5534 | The law of concretion. Th... |
| brabga 5535 | The law of concretion for ... |
| opelopab2a 5536 | Ordered pair membership in... |
| opelopaba 5537 | The law of concretion. Th... |
| braba 5538 | The law of concretion for ... |
| opelopabg 5539 | The law of concretion. Th... |
| brabg 5540 | The law of concretion for ... |
| opelopabgf 5541 | The law of concretion. Th... |
| opelopab2 5542 | Ordered pair membership in... |
| opelopab 5543 | The law of concretion. Th... |
| brab 5544 | The law of concretion for ... |
| opelopabaf 5545 | The law of concretion. Th... |
| opelopabf 5546 | The law of concretion. Th... |
| ssopab2 5547 | Equivalence of ordered pai... |
| ssopab2bw 5548 | Equivalence of ordered pai... |
| eqopab2bw 5549 | Equivalence of ordered pai... |
| ssopab2b 5550 | Equivalence of ordered pai... |
| ssopab2i 5551 | Inference of ordered pair ... |
| ssopab2dv 5552 | Inference of ordered pair ... |
| eqopab2b 5553 | Equivalence of ordered pai... |
| opabn0 5554 | Nonempty ordered pair clas... |
| opab0 5555 | Empty ordered pair class a... |
| csbopab 5556 | Move substitution into a c... |
| csbopabgALT 5557 | Move substitution into a c... |
| csbmpt12 5558 | Move substitution into a m... |
| csbmpt2 5559 | Move substitution into the... |
| iunopab 5560 | Move indexed union inside ... |
| iunopabOLD 5561 | Obsolete version of ~ iuno... |
| elopabr 5562 | Membership in an ordered-p... |
| elopabran 5563 | Membership in an ordered-p... |
| elopabrOLD 5564 | Obsolete version of ~ elop... |
| rbropapd 5565 | Properties of a pair in an... |
| rbropap 5566 | Properties of a pair in a ... |
| 2rbropap 5567 | Properties of a pair in a ... |
| 0nelopab 5568 | The empty set is never an ... |
| 0nelopabOLD 5569 | Obsolete version of ~ 0nel... |
| brabv 5570 | If two classes are in a re... |
| pwin 5571 | The power class of the int... |
| pwssun 5572 | The power class of the uni... |
| pwun 5573 | The power class of the uni... |
| dfid4 5576 | The identity function expr... |
| dfid2 5577 | Alternate definition of th... |
| dfid3 5578 | A stronger version of ~ df... |
| dfid2OLD 5579 | Obsolete version of ~ dfid... |
| epelg 5582 | The membership relation an... |
| epeli 5583 | The membership relation an... |
| epel 5584 | The membership relation an... |
| 0sn0ep 5585 | An example for the members... |
| epn0 5586 | The membership relation is... |
| poss 5591 | Subset theorem for the par... |
| poeq1 5592 | Equality theorem for parti... |
| poeq2 5593 | Equality theorem for parti... |
| nfpo 5594 | Bound-variable hypothesis ... |
| nfso 5595 | Bound-variable hypothesis ... |
| pocl 5596 | Characteristic properties ... |
| poclOLD 5597 | Obsolete version of ~ pocl... |
| ispod 5598 | Sufficient conditions for ... |
| swopolem 5599 | Perform the substitutions ... |
| swopo 5600 | A strict weak order is a p... |
| poirr 5601 | A partial order is irrefle... |
| potr 5602 | A partial order is a trans... |
| po2nr 5603 | A partial order has no 2-c... |
| po3nr 5604 | A partial order has no 3-c... |
| po2ne 5605 | Two sets related by a part... |
| po0 5606 | Any relation is a partial ... |
| pofun 5607 | The inverse image of a par... |
| sopo 5608 | A strict linear order is a... |
| soss 5609 | Subset theorem for the str... |
| soeq1 5610 | Equality theorem for the s... |
| soeq2 5611 | Equality theorem for the s... |
| sonr 5612 | A strict order relation is... |
| sotr 5613 | A strict order relation is... |
| solin 5614 | A strict order relation is... |
| so2nr 5615 | A strict order relation ha... |
| so3nr 5616 | A strict order relation ha... |
| sotric 5617 | A strict order relation sa... |
| sotrieq 5618 | Trichotomy law for strict ... |
| sotrieq2 5619 | Trichotomy law for strict ... |
| soasym 5620 | Asymmetry law for strict o... |
| sotr2 5621 | A transitivity relation. ... |
| issod 5622 | An irreflexive, transitive... |
| issoi 5623 | An irreflexive, transitive... |
| isso2i 5624 | Deduce strict ordering fro... |
| so0 5625 | Any relation is a strict o... |
| somo 5626 | A totally ordered set has ... |
| sotrine 5627 | Trichotomy law for strict ... |
| sotr3 5628 | Transitivity law for stric... |
| dffr6 5635 | Alternate definition of ~ ... |
| frd 5636 | A nonempty subset of an ` ... |
| fri 5637 | A nonempty subset of an ` ... |
| friOLD 5638 | Obsolete version of ~ fri ... |
| seex 5639 | The ` R ` -preimage of an ... |
| exse 5640 | Any relation on a set is s... |
| dffr2 5641 | Alternate definition of we... |
| dffr2ALT 5642 | Alternate proof of ~ dffr2... |
| frc 5643 | Property of well-founded r... |
| frss 5644 | Subset theorem for the wel... |
| sess1 5645 | Subset theorem for the set... |
| sess2 5646 | Subset theorem for the set... |
| freq1 5647 | Equality theorem for the w... |
| freq2 5648 | Equality theorem for the w... |
| seeq1 5649 | Equality theorem for the s... |
| seeq2 5650 | Equality theorem for the s... |
| nffr 5651 | Bound-variable hypothesis ... |
| nfse 5652 | Bound-variable hypothesis ... |
| nfwe 5653 | Bound-variable hypothesis ... |
| frirr 5654 | A well-founded relation is... |
| fr2nr 5655 | A well-founded relation ha... |
| fr0 5656 | Any relation is well-found... |
| frminex 5657 | If an element of a well-fo... |
| efrirr 5658 | A well-founded class does ... |
| efrn2lp 5659 | A well-founded class conta... |
| epse 5660 | The membership relation is... |
| tz7.2 5661 | Similar to Theorem 7.2 of ... |
| dfepfr 5662 | An alternate way of saying... |
| epfrc 5663 | A subset of a well-founded... |
| wess 5664 | Subset theorem for the wel... |
| weeq1 5665 | Equality theorem for the w... |
| weeq2 5666 | Equality theorem for the w... |
| wefr 5667 | A well-ordering is well-fo... |
| weso 5668 | A well-ordering is a stric... |
| wecmpep 5669 | The elements of a class we... |
| wetrep 5670 | On a class well-ordered by... |
| wefrc 5671 | A nonempty subclass of a c... |
| we0 5672 | Any relation is a well-ord... |
| wereu 5673 | A nonempty subset of an ` ... |
| wereu2 5674 | A nonempty subclass of an ... |
| xpeq1 5691 | Equality theorem for Carte... |
| xpss12 5692 | Subset theorem for Cartesi... |
| xpss 5693 | A Cartesian product is inc... |
| inxpssres 5694 | Intersection with a Cartes... |
| relxp 5695 | A Cartesian product is a r... |
| xpss1 5696 | Subset relation for Cartes... |
| xpss2 5697 | Subset relation for Cartes... |
| xpeq2 5698 | Equality theorem for Carte... |
| elxpi 5699 | Membership in a Cartesian ... |
| elxp 5700 | Membership in a Cartesian ... |
| elxp2 5701 | Membership in a Cartesian ... |
| xpeq12 5702 | Equality theorem for Carte... |
| xpeq1i 5703 | Equality inference for Car... |
| xpeq2i 5704 | Equality inference for Car... |
| xpeq12i 5705 | Equality inference for Car... |
| xpeq1d 5706 | Equality deduction for Car... |
| xpeq2d 5707 | Equality deduction for Car... |
| xpeq12d 5708 | Equality deduction for Car... |
| sqxpeqd 5709 | Equality deduction for a C... |
| nfxp 5710 | Bound-variable hypothesis ... |
| 0nelxp 5711 | The empty set is not a mem... |
| 0nelelxp 5712 | A member of a Cartesian pr... |
| opelxp 5713 | Ordered pair membership in... |
| opelxpi 5714 | Ordered pair membership in... |
| opelxpii 5715 | Ordered pair membership in... |
| opelxpd 5716 | Ordered pair membership in... |
| opelvv 5717 | Ordered pair membership in... |
| opelvvg 5718 | Ordered pair membership in... |
| opelxp1 5719 | The first member of an ord... |
| opelxp2 5720 | The second member of an or... |
| otelxp 5721 | Ordered triple membership ... |
| otelxp1 5722 | The first member of an ord... |
| otel3xp 5723 | An ordered triple is an el... |
| opabssxpd 5724 | An ordered-pair class abst... |
| rabxp 5725 | Class abstraction restrict... |
| brxp 5726 | Binary relation on a Carte... |
| pwvrel 5727 | A set is a binary relation... |
| pwvabrel 5728 | The powerclass of the cart... |
| brrelex12 5729 | Two classes related by a b... |
| brrelex1 5730 | If two classes are related... |
| brrelex2 5731 | If two classes are related... |
| brrelex12i 5732 | Two classes that are relat... |
| brrelex1i 5733 | The first argument of a bi... |
| brrelex2i 5734 | The second argument of a b... |
| nprrel12 5735 | Proper classes are not rel... |
| nprrel 5736 | No proper class is related... |
| 0nelrel0 5737 | A binary relation does not... |
| 0nelrel 5738 | A binary relation does not... |
| fconstmpt 5739 | Representation of a consta... |
| vtoclr 5740 | Variable to class conversi... |
| opthprc 5741 | Justification theorem for ... |
| brel 5742 | Two things in a binary rel... |
| elxp3 5743 | Membership in a Cartesian ... |
| opeliunxp 5744 | Membership in a union of C... |
| xpundi 5745 | Distributive law for Carte... |
| xpundir 5746 | Distributive law for Carte... |
| xpiundi 5747 | Distributive law for Carte... |
| xpiundir 5748 | Distributive law for Carte... |
| iunxpconst 5749 | Membership in a union of C... |
| xpun 5750 | The Cartesian product of t... |
| elvv 5751 | Membership in universal cl... |
| elvvv 5752 | Membership in universal cl... |
| elvvuni 5753 | An ordered pair contains i... |
| brinxp2 5754 | Intersection of binary rel... |
| brinxp 5755 | Intersection of binary rel... |
| opelinxp 5756 | Ordered pair element in an... |
| poinxp 5757 | Intersection of partial or... |
| soinxp 5758 | Intersection of total orde... |
| frinxp 5759 | Intersection of well-found... |
| seinxp 5760 | Intersection of set-like r... |
| weinxp 5761 | Intersection of well-order... |
| posn 5762 | Partial ordering of a sing... |
| sosn 5763 | Strict ordering on a singl... |
| frsn 5764 | Founded relation on a sing... |
| wesn 5765 | Well-ordering of a singlet... |
| elopaelxp 5766 | Membership in an ordered-p... |
| elopaelxpOLD 5767 | Obsolete version of ~ elop... |
| bropaex12 5768 | Two classes related by an ... |
| opabssxp 5769 | An abstraction relation is... |
| brab2a 5770 | The law of concretion for ... |
| optocl 5771 | Implicit substitution of c... |
| 2optocl 5772 | Implicit substitution of c... |
| 3optocl 5773 | Implicit substitution of c... |
| opbrop 5774 | Ordered pair membership in... |
| 0xp 5775 | The Cartesian product with... |
| csbxp 5776 | Distribute proper substitu... |
| releq 5777 | Equality theorem for the r... |
| releqi 5778 | Equality inference for the... |
| releqd 5779 | Equality deduction for the... |
| nfrel 5780 | Bound-variable hypothesis ... |
| sbcrel 5781 | Distribute proper substitu... |
| relss 5782 | Subclass theorem for relat... |
| ssrel 5783 | A subclass relationship de... |
| ssrelOLD 5784 | Obsolete version of ~ ssre... |
| eqrel 5785 | Extensionality principle f... |
| ssrel2 5786 | A subclass relationship de... |
| ssrel3 5787 | Subclass relation in anoth... |
| relssi 5788 | Inference from subclass pr... |
| relssdv 5789 | Deduction from subclass pr... |
| eqrelriv 5790 | Inference from extensional... |
| eqrelriiv 5791 | Inference from extensional... |
| eqbrriv 5792 | Inference from extensional... |
| eqrelrdv 5793 | Deduce equality of relatio... |
| eqbrrdv 5794 | Deduction from extensional... |
| eqbrrdiv 5795 | Deduction from extensional... |
| eqrelrdv2 5796 | A version of ~ eqrelrdv . ... |
| ssrelrel 5797 | A subclass relationship de... |
| eqrelrel 5798 | Extensionality principle f... |
| elrel 5799 | A member of a relation is ... |
| rel0 5800 | The empty set is a relatio... |
| nrelv 5801 | The universal class is not... |
| relsng 5802 | A singleton is a relation ... |
| relsnb 5803 | An at-most-singleton is a ... |
| relsnopg 5804 | A singleton of an ordered ... |
| relsn 5805 | A singleton is a relation ... |
| relsnop 5806 | A singleton of an ordered ... |
| copsex2gb 5807 | Implicit substitution infe... |
| copsex2ga 5808 | Implicit substitution infe... |
| elopaba 5809 | Membership in an ordered-p... |
| xpsspw 5810 | A Cartesian product is inc... |
| unixpss 5811 | The double class union of ... |
| relun 5812 | The union of two relations... |
| relin1 5813 | The intersection with a re... |
| relin2 5814 | The intersection with a re... |
| relinxp 5815 | Intersection with a Cartes... |
| reldif 5816 | A difference cutting down ... |
| reliun 5817 | An indexed union is a rela... |
| reliin 5818 | An indexed intersection is... |
| reluni 5819 | The union of a class is a ... |
| relint 5820 | The intersection of a clas... |
| relopabiv 5821 | A class of ordered pairs i... |
| relopabv 5822 | A class of ordered pairs i... |
| relopabi 5823 | A class of ordered pairs i... |
| relopabiALT 5824 | Alternate proof of ~ relop... |
| relopab 5825 | A class of ordered pairs i... |
| mptrel 5826 | The maps-to notation alway... |
| reli 5827 | The identity relation is a... |
| rele 5828 | The membership relation is... |
| opabid2 5829 | A relation expressed as an... |
| inopab 5830 | Intersection of two ordere... |
| difopab 5831 | Difference of two ordered-... |
| difopabOLD 5832 | Obsolete version of ~ difo... |
| inxp 5833 | Intersection of two Cartes... |
| inxpOLD 5834 | Obsolete version of ~ inxp... |
| xpindi 5835 | Distributive law for Carte... |
| xpindir 5836 | Distributive law for Carte... |
| xpiindi 5837 | Distributive law for Carte... |
| xpriindi 5838 | Distributive law for Carte... |
| eliunxp 5839 | Membership in a union of C... |
| opeliunxp2 5840 | Membership in a union of C... |
| raliunxp 5841 | Write a double restricted ... |
| rexiunxp 5842 | Write a double restricted ... |
| ralxp 5843 | Universal quantification r... |
| rexxp 5844 | Existential quantification... |
| exopxfr 5845 | Transfer ordered-pair exis... |
| exopxfr2 5846 | Transfer ordered-pair exis... |
| djussxp 5847 | Disjoint union is a subset... |
| ralxpf 5848 | Version of ~ ralxp with bo... |
| rexxpf 5849 | Version of ~ rexxp with bo... |
| iunxpf 5850 | Indexed union on a Cartesi... |
| opabbi2dv 5851 | Deduce equality of a relat... |
| relop 5852 | A necessary and sufficient... |
| ideqg 5853 | For sets, the identity rel... |
| ideq 5854 | For sets, the identity rel... |
| ididg 5855 | A set is identical to itse... |
| issetid 5856 | Two ways of expressing set... |
| coss1 5857 | Subclass theorem for compo... |
| coss2 5858 | Subclass theorem for compo... |
| coeq1 5859 | Equality theorem for compo... |
| coeq2 5860 | Equality theorem for compo... |
| coeq1i 5861 | Equality inference for com... |
| coeq2i 5862 | Equality inference for com... |
| coeq1d 5863 | Equality deduction for com... |
| coeq2d 5864 | Equality deduction for com... |
| coeq12i 5865 | Equality inference for com... |
| coeq12d 5866 | Equality deduction for com... |
| nfco 5867 | Bound-variable hypothesis ... |
| brcog 5868 | Ordered pair membership in... |
| opelco2g 5869 | Ordered pair membership in... |
| brcogw 5870 | Ordered pair membership in... |
| eqbrrdva 5871 | Deduction from extensional... |
| brco 5872 | Binary relation on a compo... |
| opelco 5873 | Ordered pair membership in... |
| cnvss 5874 | Subset theorem for convers... |
| cnveq 5875 | Equality theorem for conve... |
| cnveqi 5876 | Equality inference for con... |
| cnveqd 5877 | Equality deduction for con... |
| elcnv 5878 | Membership in a converse r... |
| elcnv2 5879 | Membership in a converse r... |
| nfcnv 5880 | Bound-variable hypothesis ... |
| brcnvg 5881 | The converse of a binary r... |
| opelcnvg 5882 | Ordered-pair membership in... |
| opelcnv 5883 | Ordered-pair membership in... |
| brcnv 5884 | The converse of a binary r... |
| csbcnv 5885 | Move class substitution in... |
| csbcnvgALT 5886 | Move class substitution in... |
| cnvco 5887 | Distributive law of conver... |
| cnvuni 5888 | The converse of a class un... |
| dfdm3 5889 | Alternate definition of do... |
| dfrn2 5890 | Alternate definition of ra... |
| dfrn3 5891 | Alternate definition of ra... |
| elrn2g 5892 | Membership in a range. (C... |
| elrng 5893 | Membership in a range. (C... |
| elrn2 5894 | Membership in a range. (C... |
| elrn 5895 | Membership in a range. (C... |
| ssrelrn 5896 | If a relation is a subset ... |
| dfdm4 5897 | Alternate definition of do... |
| dfdmf 5898 | Definition of domain, usin... |
| csbdm 5899 | Distribute proper substitu... |
| eldmg 5900 | Domain membership. Theore... |
| eldm2g 5901 | Domain membership. Theore... |
| eldm 5902 | Membership in a domain. T... |
| eldm2 5903 | Membership in a domain. T... |
| dmss 5904 | Subset theorem for domain.... |
| dmeq 5905 | Equality theorem for domai... |
| dmeqi 5906 | Equality inference for dom... |
| dmeqd 5907 | Equality deduction for dom... |
| opeldmd 5908 | Membership of first of an ... |
| opeldm 5909 | Membership of first of an ... |
| breldm 5910 | Membership of first of a b... |
| breldmg 5911 | Membership of first of a b... |
| dmun 5912 | The domain of a union is t... |
| dmin 5913 | The domain of an intersect... |
| breldmd 5914 | Membership of first of a b... |
| dmiun 5915 | The domain of an indexed u... |
| dmuni 5916 | The domain of a union. Pa... |
| dmopab 5917 | The domain of a class of o... |
| dmopabelb 5918 | A set is an element of the... |
| dmopab2rex 5919 | The domain of an ordered p... |
| dmopabss 5920 | Upper bound for the domain... |
| dmopab3 5921 | The domain of a restricted... |
| dm0 5922 | The domain of the empty se... |
| dmi 5923 | The domain of the identity... |
| dmv 5924 | The domain of the universe... |
| dmep 5925 | The domain of the membersh... |
| dm0rn0 5926 | An empty domain is equival... |
| rn0 5927 | The range of the empty set... |
| rnep 5928 | The range of the membershi... |
| reldm0 5929 | A relation is empty iff it... |
| dmxp 5930 | The domain of a Cartesian ... |
| dmxpid 5931 | The domain of a Cartesian ... |
| dmxpin 5932 | The domain of the intersec... |
| xpid11 5933 | The Cartesian square is a ... |
| dmcnvcnv 5934 | The domain of the double c... |
| rncnvcnv 5935 | The range of the double co... |
| elreldm 5936 | The first member of an ord... |
| rneq 5937 | Equality theorem for range... |
| rneqi 5938 | Equality inference for ran... |
| rneqd 5939 | Equality deduction for ran... |
| rnss 5940 | Subset theorem for range. ... |
| rnssi 5941 | Subclass inference for ran... |
| brelrng 5942 | The second argument of a b... |
| brelrn 5943 | The second argument of a b... |
| opelrn 5944 | Membership of second membe... |
| releldm 5945 | The first argument of a bi... |
| relelrn 5946 | The second argument of a b... |
| releldmb 5947 | Membership in a domain. (... |
| relelrnb 5948 | Membership in a range. (C... |
| releldmi 5949 | The first argument of a bi... |
| relelrni 5950 | The second argument of a b... |
| dfrnf 5951 | Definition of range, using... |
| nfdm 5952 | Bound-variable hypothesis ... |
| nfrn 5953 | Bound-variable hypothesis ... |
| dmiin 5954 | Domain of an intersection.... |
| rnopab 5955 | The range of a class of or... |
| rnmpt 5956 | The range of a function in... |
| elrnmpt 5957 | The range of a function in... |
| elrnmpt1s 5958 | Elementhood in an image se... |
| elrnmpt1 5959 | Elementhood in an image se... |
| elrnmptg 5960 | Membership in the range of... |
| elrnmpti 5961 | Membership in the range of... |
| elrnmptd 5962 | The range of a function in... |
| elrnmpt1d 5963 | Elementhood in an image se... |
| elrnmptdv 5964 | Elementhood in the range o... |
| elrnmpt2d 5965 | Elementhood in the range o... |
| dfiun3g 5966 | Alternate definition of in... |
| dfiin3g 5967 | Alternate definition of in... |
| dfiun3 5968 | Alternate definition of in... |
| dfiin3 5969 | Alternate definition of in... |
| riinint 5970 | Express a relative indexed... |
| relrn0 5971 | A relation is empty iff it... |
| dmrnssfld 5972 | The domain and range of a ... |
| dmcoss 5973 | Domain of a composition. ... |
| rncoss 5974 | Range of a composition. (... |
| dmcosseq 5975 | Domain of a composition. ... |
| dmcoeq 5976 | Domain of a composition. ... |
| rncoeq 5977 | Range of a composition. (... |
| reseq1 5978 | Equality theorem for restr... |
| reseq2 5979 | Equality theorem for restr... |
| reseq1i 5980 | Equality inference for res... |
| reseq2i 5981 | Equality inference for res... |
| reseq12i 5982 | Equality inference for res... |
| reseq1d 5983 | Equality deduction for res... |
| reseq2d 5984 | Equality deduction for res... |
| reseq12d 5985 | Equality deduction for res... |
| nfres 5986 | Bound-variable hypothesis ... |
| csbres 5987 | Distribute proper substitu... |
| res0 5988 | A restriction to the empty... |
| dfres3 5989 | Alternate definition of re... |
| opelres 5990 | Ordered pair elementhood i... |
| brres 5991 | Binary relation on a restr... |
| opelresi 5992 | Ordered pair membership in... |
| brresi 5993 | Binary relation on a restr... |
| opres 5994 | Ordered pair membership in... |
| resieq 5995 | A restricted identity rela... |
| opelidres 5996 | ` <. A , A >. ` belongs to... |
| resres 5997 | The restriction of a restr... |
| resundi 5998 | Distributive law for restr... |
| resundir 5999 | Distributive law for restr... |
| resindi 6000 | Class restriction distribu... |
| resindir 6001 | Class restriction distribu... |
| inres 6002 | Move intersection into cla... |
| resdifcom 6003 | Commutative law for restri... |
| resiun1 6004 | Distribution of restrictio... |
| resiun2 6005 | Distribution of restrictio... |
| resss 6006 | A class includes its restr... |
| rescom 6007 | Commutative law for restri... |
| ssres 6008 | Subclass theorem for restr... |
| ssres2 6009 | Subclass theorem for restr... |
| relres 6010 | A restriction is a relatio... |
| resabs1 6011 | Absorption law for restric... |
| resabs1d 6012 | Absorption law for restric... |
| resabs2 6013 | Absorption law for restric... |
| residm 6014 | Idempotent law for restric... |
| dmresss 6015 | The domain of a restrictio... |
| dmres 6016 | The domain of a restrictio... |
| ssdmres 6017 | A domain restricted to a s... |
| dmresexg 6018 | The domain of a restrictio... |
| resima 6019 | A restriction to an image.... |
| resima2 6020 | Image under a restricted c... |
| rnresss 6021 | The range of a restriction... |
| xpssres 6022 | Restriction of a constant ... |
| elinxp 6023 | Membership in an intersect... |
| elres 6024 | Membership in a restrictio... |
| elsnres 6025 | Membership in restriction ... |
| relssres 6026 | Simplification law for res... |
| dmressnsn 6027 | The domain of a restrictio... |
| eldmressnsn 6028 | The element of the domain ... |
| eldmeldmressn 6029 | An element of the domain (... |
| resdm 6030 | A relation restricted to i... |
| resexg 6031 | The restriction of a set i... |
| resexd 6032 | The restriction of a set i... |
| resex 6033 | The restriction of a set i... |
| resindm 6034 | When restricting a relatio... |
| resdmdfsn 6035 | Restricting a relation to ... |
| reldisjun 6036 | Split a relation into two ... |
| relresdm1 6037 | Restriction of a disjoint ... |
| resopab 6038 | Restriction of a class abs... |
| iss 6039 | A subclass of the identity... |
| resopab2 6040 | Restriction of a class abs... |
| resmpt 6041 | Restriction of the mapping... |
| resmpt3 6042 | Unconditional restriction ... |
| resmptf 6043 | Restriction of the mapping... |
| resmptd 6044 | Restriction of the mapping... |
| dfres2 6045 | Alternate definition of th... |
| mptss 6046 | Sufficient condition for i... |
| elidinxp 6047 | Characterization of the el... |
| elidinxpid 6048 | Characterization of the el... |
| elrid 6049 | Characterization of the el... |
| idinxpres 6050 | The intersection of the id... |
| idinxpresid 6051 | The intersection of the id... |
| idssxp 6052 | A diagonal set as a subset... |
| opabresid 6053 | The restricted identity re... |
| mptresid 6054 | The restricted identity re... |
| dmresi 6055 | The domain of a restricted... |
| restidsing 6056 | Restriction of the identit... |
| iresn0n0 6057 | The identity function rest... |
| imaeq1 6058 | Equality theorem for image... |
| imaeq2 6059 | Equality theorem for image... |
| imaeq1i 6060 | Equality theorem for image... |
| imaeq2i 6061 | Equality theorem for image... |
| imaeq1d 6062 | Equality theorem for image... |
| imaeq2d 6063 | Equality theorem for image... |
| imaeq12d 6064 | Equality theorem for image... |
| dfima2 6065 | Alternate definition of im... |
| dfima3 6066 | Alternate definition of im... |
| elimag 6067 | Membership in an image. T... |
| elima 6068 | Membership in an image. T... |
| elima2 6069 | Membership in an image. T... |
| elima3 6070 | Membership in an image. T... |
| nfima 6071 | Bound-variable hypothesis ... |
| nfimad 6072 | Deduction version of bound... |
| imadmrn 6073 | The image of the domain of... |
| imassrn 6074 | The image of a class is a ... |
| mptima 6075 | Image of a function in map... |
| mptimass 6076 | Image of a function in map... |
| imai 6077 | Image under the identity r... |
| rnresi 6078 | The range of the restricte... |
| resiima 6079 | The image of a restriction... |
| ima0 6080 | Image of the empty set. T... |
| 0ima 6081 | Image under the empty rela... |
| csbima12 6082 | Move class substitution in... |
| imadisj 6083 | A class whose image under ... |
| imadisjlnd 6084 | Deduction form of one nega... |
| cnvimass 6085 | A preimage under any class... |
| cnvimarndm 6086 | The preimage of the range ... |
| imasng 6087 | The image of a singleton. ... |
| relimasn 6088 | The image of a singleton. ... |
| elrelimasn 6089 | Elementhood in the image o... |
| elimasng1 6090 | Membership in an image of ... |
| elimasn1 6091 | Membership in an image of ... |
| elimasng 6092 | Membership in an image of ... |
| elimasn 6093 | Membership in an image of ... |
| elimasngOLD 6094 | Obsolete version of ~ elim... |
| elimasni 6095 | Membership in an image of ... |
| args 6096 | Two ways to express the cl... |
| elinisegg 6097 | Membership in the inverse ... |
| eliniseg 6098 | Membership in the inverse ... |
| epin 6099 | Any set is equal to its pr... |
| epini 6100 | Any set is equal to its pr... |
| iniseg 6101 | An idiom that signifies an... |
| inisegn0 6102 | Nonemptiness of an initial... |
| dffr3 6103 | Alternate definition of we... |
| dfse2 6104 | Alternate definition of se... |
| imass1 6105 | Subset theorem for image. ... |
| imass2 6106 | Subset theorem for image. ... |
| ndmima 6107 | The image of a singleton o... |
| relcnv 6108 | A converse is a relation. ... |
| relbrcnvg 6109 | When ` R ` is a relation, ... |
| eliniseg2 6110 | Eliminate the class existe... |
| relbrcnv 6111 | When ` R ` is a relation, ... |
| relco 6112 | A composition is a relatio... |
| cotrg 6113 | Two ways of saying that th... |
| cotrgOLD 6114 | Obsolete version of ~ cotr... |
| cotrgOLDOLD 6115 | Obsolete version of ~ cotr... |
| cotr 6116 | Two ways of saying a relat... |
| idrefALT 6117 | Alternate proof of ~ idref... |
| cnvsym 6118 | Two ways of saying a relat... |
| cnvsymOLD 6119 | Obsolete proof of ~ cnvsym... |
| cnvsymOLDOLD 6120 | Obsolete proof of ~ cnvsym... |
| intasym 6121 | Two ways of saying a relat... |
| asymref 6122 | Two ways of saying a relat... |
| asymref2 6123 | Two ways of saying a relat... |
| intirr 6124 | Two ways of saying a relat... |
| brcodir 6125 | Two ways of saying that tw... |
| codir 6126 | Two ways of saying a relat... |
| qfto 6127 | A quantifier-free way of e... |
| xpidtr 6128 | A Cartesian square is a tr... |
| trin2 6129 | The intersection of two tr... |
| poirr2 6130 | A partial order is irrefle... |
| trinxp 6131 | The relation induced by a ... |
| soirri 6132 | A strict order relation is... |
| sotri 6133 | A strict order relation is... |
| son2lpi 6134 | A strict order relation ha... |
| sotri2 6135 | A transitivity relation. ... |
| sotri3 6136 | A transitivity relation. ... |
| poleloe 6137 | Express "less than or equa... |
| poltletr 6138 | Transitive law for general... |
| somin1 6139 | Property of a minimum in a... |
| somincom 6140 | Commutativity of minimum i... |
| somin2 6141 | Property of a minimum in a... |
| soltmin 6142 | Being less than a minimum,... |
| cnvopab 6143 | The converse of a class ab... |
| mptcnv 6144 | The converse of a mapping ... |
| cnv0 6145 | The converse of the empty ... |
| cnvi 6146 | The converse of the identi... |
| cnvun 6147 | The converse of a union is... |
| cnvdif 6148 | Distributive law for conve... |
| cnvin 6149 | Distributive law for conve... |
| rnun 6150 | Distributive law for range... |
| rnin 6151 | The range of an intersecti... |
| rniun 6152 | The range of an indexed un... |
| rnuni 6153 | The range of a union. Par... |
| imaundi 6154 | Distributive law for image... |
| imaundir 6155 | The image of a union. (Co... |
| cnvimassrndm 6156 | The preimage of a superset... |
| dminss 6157 | An upper bound for interse... |
| imainss 6158 | An upper bound for interse... |
| inimass 6159 | The image of an intersecti... |
| inimasn 6160 | The intersection of the im... |
| cnvxp 6161 | The converse of a Cartesia... |
| xp0 6162 | The Cartesian product with... |
| xpnz 6163 | The Cartesian product of n... |
| xpeq0 6164 | At least one member of an ... |
| xpdisj1 6165 | Cartesian products with di... |
| xpdisj2 6166 | Cartesian products with di... |
| xpsndisj 6167 | Cartesian products with tw... |
| difxp 6168 | Difference of Cartesian pr... |
| difxp1 6169 | Difference law for Cartesi... |
| difxp2 6170 | Difference law for Cartesi... |
| djudisj 6171 | Disjoint unions with disjo... |
| xpdifid 6172 | The set of distinct couple... |
| resdisj 6173 | A double restriction to di... |
| rnxp 6174 | The range of a Cartesian p... |
| dmxpss 6175 | The domain of a Cartesian ... |
| rnxpss 6176 | The range of a Cartesian p... |
| rnxpid 6177 | The range of a Cartesian s... |
| ssxpb 6178 | A Cartesian product subcla... |
| xp11 6179 | The Cartesian product of n... |
| xpcan 6180 | Cancellation law for Carte... |
| xpcan2 6181 | Cancellation law for Carte... |
| ssrnres 6182 | Two ways to express surjec... |
| rninxp 6183 | Two ways to express surjec... |
| dminxp 6184 | Two ways to express totali... |
| imainrect 6185 | Image by a restricted and ... |
| xpima 6186 | Direct image by a Cartesia... |
| xpima1 6187 | Direct image by a Cartesia... |
| xpima2 6188 | Direct image by a Cartesia... |
| xpimasn 6189 | Direct image of a singleto... |
| sossfld 6190 | The base set of a strict o... |
| sofld 6191 | The base set of a nonempty... |
| cnvcnv3 6192 | The set of all ordered pai... |
| dfrel2 6193 | Alternate definition of re... |
| dfrel4v 6194 | A relation can be expresse... |
| dfrel4 6195 | A relation can be expresse... |
| cnvcnv 6196 | The double converse of a c... |
| cnvcnv2 6197 | The double converse of a c... |
| cnvcnvss 6198 | The double converse of a c... |
| cnvrescnv 6199 | Two ways to express the co... |
| cnveqb 6200 | Equality theorem for conve... |
| cnveq0 6201 | A relation empty iff its c... |
| dfrel3 6202 | Alternate definition of re... |
| elid 6203 | Characterization of the el... |
| dmresv 6204 | The domain of a universal ... |
| rnresv 6205 | The range of a universal r... |
| dfrn4 6206 | Range defined in terms of ... |
| csbrn 6207 | Distribute proper substitu... |
| rescnvcnv 6208 | The restriction of the dou... |
| cnvcnvres 6209 | The double converse of the... |
| imacnvcnv 6210 | The image of the double co... |
| dmsnn0 6211 | The domain of a singleton ... |
| rnsnn0 6212 | The range of a singleton i... |
| dmsn0 6213 | The domain of the singleto... |
| cnvsn0 6214 | The converse of the single... |
| dmsn0el 6215 | The domain of a singleton ... |
| relsn2 6216 | A singleton is a relation ... |
| dmsnopg 6217 | The domain of a singleton ... |
| dmsnopss 6218 | The domain of a singleton ... |
| dmpropg 6219 | The domain of an unordered... |
| dmsnop 6220 | The domain of a singleton ... |
| dmprop 6221 | The domain of an unordered... |
| dmtpop 6222 | The domain of an unordered... |
| cnvcnvsn 6223 | Double converse of a singl... |
| dmsnsnsn 6224 | The domain of the singleto... |
| rnsnopg 6225 | The range of a singleton o... |
| rnpropg 6226 | The range of a pair of ord... |
| cnvsng 6227 | Converse of a singleton of... |
| rnsnop 6228 | The range of a singleton o... |
| op1sta 6229 | Extract the first member o... |
| cnvsn 6230 | Converse of a singleton of... |
| op2ndb 6231 | Extract the second member ... |
| op2nda 6232 | Extract the second member ... |
| opswap 6233 | Swap the members of an ord... |
| cnvresima 6234 | An image under the convers... |
| resdm2 6235 | A class restricted to its ... |
| resdmres 6236 | Restriction to the domain ... |
| resresdm 6237 | A restriction by an arbitr... |
| imadmres 6238 | The image of the domain of... |
| resdmss 6239 | Subset relationship for th... |
| resdifdi 6240 | Distributive law for restr... |
| resdifdir 6241 | Distributive law for restr... |
| mptpreima 6242 | The preimage of a function... |
| mptiniseg 6243 | Converse singleton image o... |
| dmmpt 6244 | The domain of the mapping ... |
| dmmptss 6245 | The domain of a mapping is... |
| dmmptg 6246 | The domain of the mapping ... |
| rnmpt0f 6247 | The range of a function in... |
| rnmptn0 6248 | The range of a function in... |
| dfco2 6249 | Alternate definition of a ... |
| dfco2a 6250 | Generalization of ~ dfco2 ... |
| coundi 6251 | Class composition distribu... |
| coundir 6252 | Class composition distribu... |
| cores 6253 | Restricted first member of... |
| resco 6254 | Associative law for the re... |
| imaco 6255 | Image of the composition o... |
| rnco 6256 | The range of the compositi... |
| rnco2 6257 | The range of the compositi... |
| dmco 6258 | The domain of a compositio... |
| coeq0 6259 | A composition of two relat... |
| coiun 6260 | Composition with an indexe... |
| cocnvcnv1 6261 | A composition is not affec... |
| cocnvcnv2 6262 | A composition is not affec... |
| cores2 6263 | Absorption of a reverse (p... |
| co02 6264 | Composition with the empty... |
| co01 6265 | Composition with the empty... |
| coi1 6266 | Composition with the ident... |
| coi2 6267 | Composition with the ident... |
| coires1 6268 | Composition with a restric... |
| coass 6269 | Associative law for class ... |
| relcnvtrg 6270 | General form of ~ relcnvtr... |
| relcnvtr 6271 | A relation is transitive i... |
| relssdmrn 6272 | A relation is included in ... |
| relssdmrnOLD 6273 | Obsolete version of ~ rels... |
| resssxp 6274 | If the ` R ` -image of a c... |
| cnvssrndm 6275 | The converse is a subset o... |
| cossxp 6276 | Composition as a subset of... |
| relrelss 6277 | Two ways to describe the s... |
| unielrel 6278 | The membership relation fo... |
| relfld 6279 | The double union of a rela... |
| relresfld 6280 | Restriction of a relation ... |
| relcoi2 6281 | Composition with the ident... |
| relcoi1 6282 | Composition with the ident... |
| unidmrn 6283 | The double union of the co... |
| relcnvfld 6284 | if ` R ` is a relation, it... |
| dfdm2 6285 | Alternate definition of do... |
| unixp 6286 | The double class union of ... |
| unixp0 6287 | A Cartesian product is emp... |
| unixpid 6288 | Field of a Cartesian squar... |
| ressn 6289 | Restriction of a class to ... |
| cnviin 6290 | The converse of an interse... |
| cnvpo 6291 | The converse of a partial ... |
| cnvso 6292 | The converse of a strict o... |
| xpco 6293 | Composition of two Cartesi... |
| xpcoid 6294 | Composition of two Cartesi... |
| elsnxp 6295 | Membership in a Cartesian ... |
| reu3op 6296 | There is a unique ordered ... |
| reuop 6297 | There is a unique ordered ... |
| opreu2reurex 6298 | There is a unique ordered ... |
| opreu2reu 6299 | If there is a unique order... |
| dfpo2 6300 | Quantifier-free definition... |
| csbcog 6301 | Distribute proper substitu... |
| snres0 6302 | Condition for restriction ... |
| imaindm 6303 | The image is unaffected by... |
| predeq123 6306 | Equality theorem for the p... |
| predeq1 6307 | Equality theorem for the p... |
| predeq2 6308 | Equality theorem for the p... |
| predeq3 6309 | Equality theorem for the p... |
| nfpred 6310 | Bound-variable hypothesis ... |
| csbpredg 6311 | Move class substitution in... |
| predpredss 6312 | If ` A ` is a subset of ` ... |
| predss 6313 | The predecessor class of `... |
| sspred 6314 | Another subset/predecessor... |
| dfpred2 6315 | An alternate definition of... |
| dfpred3 6316 | An alternate definition of... |
| dfpred3g 6317 | An alternate definition of... |
| elpredgg 6318 | Membership in a predecesso... |
| elpredg 6319 | Membership in a predecesso... |
| elpredimg 6320 | Membership in a predecesso... |
| elpredim 6321 | Membership in a predecesso... |
| elpred 6322 | Membership in a predecesso... |
| predexg 6323 | The predecessor class exis... |
| predasetexOLD 6324 | Obsolete form of ~ predexg... |
| dffr4 6325 | Alternate definition of we... |
| predel 6326 | Membership in the predeces... |
| predbrg 6327 | Closed form of ~ elpredim ... |
| predtrss 6328 | If ` R ` is transitive ove... |
| predpo 6329 | Property of the predecesso... |
| predso 6330 | Property of the predecesso... |
| setlikespec 6331 | If ` R ` is set-like in ` ... |
| predidm 6332 | Idempotent law for the pre... |
| predin 6333 | Intersection law for prede... |
| predun 6334 | Union law for predecessor ... |
| preddif 6335 | Difference law for predece... |
| predep 6336 | The predecessor under the ... |
| trpred 6337 | The class of predecessors ... |
| preddowncl 6338 | A property of classes that... |
| predpoirr 6339 | Given a partial ordering, ... |
| predfrirr 6340 | Given a well-founded relat... |
| pred0 6341 | The predecessor class over... |
| dfse3 6342 | Alternate definition of se... |
| predrelss 6343 | Subset carries from relati... |
| predprc 6344 | The predecessor of a prope... |
| predres 6345 | Predecessor class is unaff... |
| frpomin 6346 | Every nonempty (possibly p... |
| frpomin2 6347 | Every nonempty (possibly p... |
| frpoind 6348 | The principle of well-foun... |
| frpoinsg 6349 | Well-Founded Induction Sch... |
| frpoins2fg 6350 | Well-Founded Induction sch... |
| frpoins2g 6351 | Well-Founded Induction sch... |
| frpoins3g 6352 | Well-Founded Induction sch... |
| tz6.26 6353 | All nonempty subclasses of... |
| tz6.26OLD 6354 | Obsolete proof of ~ tz6.26... |
| tz6.26i 6355 | All nonempty subclasses of... |
| wfi 6356 | The Principle of Well-Orde... |
| wfiOLD 6357 | Obsolete proof of ~ wfi as... |
| wfii 6358 | The Principle of Well-Orde... |
| wfisg 6359 | Well-Ordered Induction Sch... |
| wfisgOLD 6360 | Obsolete version of ~ wfis... |
| wfis 6361 | Well-Ordered Induction Sch... |
| wfis2fg 6362 | Well-Ordered Induction Sch... |
| wfis2fgOLD 6363 | Obsolete version of ~ wfis... |
| wfis2f 6364 | Well-Ordered Induction sch... |
| wfis2g 6365 | Well-Ordered Induction Sch... |
| wfis2 6366 | Well-Ordered Induction sch... |
| wfis3 6367 | Well-Ordered Induction sch... |
| ordeq 6376 | Equality theorem for the o... |
| elong 6377 | An ordinal number is an or... |
| elon 6378 | An ordinal number is an or... |
| eloni 6379 | An ordinal number has the ... |
| elon2 6380 | An ordinal number is an or... |
| limeq 6381 | Equality theorem for the l... |
| ordwe 6382 | Membership well-orders eve... |
| ordtr 6383 | An ordinal class is transi... |
| ordfr 6384 | Membership is well-founded... |
| ordelss 6385 | An element of an ordinal c... |
| trssord 6386 | A transitive subclass of a... |
| ordirr 6387 | No ordinal class is a memb... |
| nordeq 6388 | A member of an ordinal cla... |
| ordn2lp 6389 | An ordinal class cannot be... |
| tz7.5 6390 | A nonempty subclass of an ... |
| ordelord 6391 | An element of an ordinal c... |
| tron 6392 | The class of all ordinal n... |
| ordelon 6393 | An element of an ordinal c... |
| onelon 6394 | An element of an ordinal n... |
| tz7.7 6395 | A transitive class belongs... |
| ordelssne 6396 | For ordinal classes, membe... |
| ordelpss 6397 | For ordinal classes, membe... |
| ordsseleq 6398 | For ordinal classes, inclu... |
| ordin 6399 | The intersection of two or... |
| onin 6400 | The intersection of two or... |
| ordtri3or 6401 | A trichotomy law for ordin... |
| ordtri1 6402 | A trichotomy law for ordin... |
| ontri1 6403 | A trichotomy law for ordin... |
| ordtri2 6404 | A trichotomy law for ordin... |
| ordtri3 6405 | A trichotomy law for ordin... |
| ordtri4 6406 | A trichotomy law for ordin... |
| orddisj 6407 | An ordinal class and its s... |
| onfr 6408 | The ordinal class is well-... |
| onelpss 6409 | Relationship between membe... |
| onsseleq 6410 | Relationship between subse... |
| onelss 6411 | An element of an ordinal n... |
| ordtr1 6412 | Transitive law for ordinal... |
| ordtr2 6413 | Transitive law for ordinal... |
| ordtr3 6414 | Transitive law for ordinal... |
| ontr1 6415 | Transitive law for ordinal... |
| ontr2 6416 | Transitive law for ordinal... |
| onelssex 6417 | Ordinal less than is equiv... |
| ordunidif 6418 | The union of an ordinal st... |
| ordintdif 6419 | If ` B ` is smaller than `... |
| onintss 6420 | If a property is true for ... |
| oneqmini 6421 | A way to show that an ordi... |
| ord0 6422 | The empty set is an ordina... |
| 0elon 6423 | The empty set is an ordina... |
| ord0eln0 6424 | A nonempty ordinal contain... |
| on0eln0 6425 | An ordinal number contains... |
| dflim2 6426 | An alternate definition of... |
| inton 6427 | The intersection of the cl... |
| nlim0 6428 | The empty set is not a lim... |
| limord 6429 | A limit ordinal is ordinal... |
| limuni 6430 | A limit ordinal is its own... |
| limuni2 6431 | The union of a limit ordin... |
| 0ellim 6432 | A limit ordinal contains t... |
| limelon 6433 | A limit ordinal class that... |
| onn0 6434 | The class of all ordinal n... |
| suceq 6435 | Equality of successors. (... |
| elsuci 6436 | Membership in a successor.... |
| elsucg 6437 | Membership in a successor.... |
| elsuc2g 6438 | Variant of membership in a... |
| elsuc 6439 | Membership in a successor.... |
| elsuc2 6440 | Membership in a successor.... |
| nfsuc 6441 | Bound-variable hypothesis ... |
| elelsuc 6442 | Membership in a successor.... |
| sucel 6443 | Membership of a successor ... |
| suc0 6444 | The successor of the empty... |
| sucprc 6445 | A proper class is its own ... |
| unisucs 6446 | The union of the successor... |
| unisucg 6447 | A transitive class is equa... |
| unisuc 6448 | A transitive class is equa... |
| sssucid 6449 | A class is included in its... |
| sucidg 6450 | Part of Proposition 7.23 o... |
| sucid 6451 | A set belongs to its succe... |
| nsuceq0 6452 | No successor is empty. (C... |
| eqelsuc 6453 | A set belongs to the succe... |
| iunsuc 6454 | Inductive definition for t... |
| suctr 6455 | The successor of a transit... |
| trsuc 6456 | A set whose successor belo... |
| trsucss 6457 | A member of the successor ... |
| ordsssuc 6458 | An ordinal is a subset of ... |
| onsssuc 6459 | A subset of an ordinal num... |
| ordsssuc2 6460 | An ordinal subset of an or... |
| onmindif 6461 | When its successor is subt... |
| ordnbtwn 6462 | There is no set between an... |
| onnbtwn 6463 | There is no set between an... |
| sucssel 6464 | A set whose successor is a... |
| orddif 6465 | Ordinal derived from its s... |
| orduniss 6466 | An ordinal class includes ... |
| ordtri2or 6467 | A trichotomy law for ordin... |
| ordtri2or2 6468 | A trichotomy law for ordin... |
| ordtri2or3 6469 | A consequence of total ord... |
| ordelinel 6470 | The intersection of two or... |
| ordssun 6471 | Property of a subclass of ... |
| ordequn 6472 | The maximum (i.e. union) o... |
| ordun 6473 | The maximum (i.e., union) ... |
| onunel 6474 | The union of two ordinals ... |
| ordunisssuc 6475 | A subclass relationship fo... |
| suc11 6476 | The successor operation be... |
| onun2 6477 | The union of two ordinals ... |
| ontr 6478 | An ordinal number is a tra... |
| onunisuc 6479 | An ordinal number is equal... |
| onordi 6480 | An ordinal number is an or... |
| ontrciOLD 6481 | Obsolete version of ~ ontr... |
| onirri 6482 | An ordinal number is not a... |
| oneli 6483 | A member of an ordinal num... |
| onelssi 6484 | A member of an ordinal num... |
| onssneli 6485 | An ordering law for ordina... |
| onssnel2i 6486 | An ordering law for ordina... |
| onelini 6487 | An element of an ordinal n... |
| oneluni 6488 | An ordinal number equals i... |
| onunisuci 6489 | An ordinal number is equal... |
| onsseli 6490 | Subset is equivalent to me... |
| onun2i 6491 | The union of two ordinal n... |
| unizlim 6492 | An ordinal equal to its ow... |
| on0eqel 6493 | An ordinal number either e... |
| snsn0non 6494 | The singleton of the singl... |
| onxpdisj 6495 | Ordinal numbers and ordere... |
| onnev 6496 | The class of ordinal numbe... |
| iotajust 6498 | Soundness justification th... |
| dfiota2 6500 | Alternate definition for d... |
| nfiota1 6501 | Bound-variable hypothesis ... |
| nfiotadw 6502 | Deduction version of ~ nfi... |
| nfiotaw 6503 | Bound-variable hypothesis ... |
| nfiotad 6504 | Deduction version of ~ nfi... |
| nfiota 6505 | Bound-variable hypothesis ... |
| cbviotaw 6506 | Change bound variables in ... |
| cbviotavw 6507 | Change bound variables in ... |
| cbviotavwOLD 6508 | Obsolete version of ~ cbvi... |
| cbviota 6509 | Change bound variables in ... |
| cbviotav 6510 | Change bound variables in ... |
| sb8iota 6511 | Variable substitution in d... |
| iotaeq 6512 | Equality theorem for descr... |
| iotabi 6513 | Equivalence theorem for de... |
| uniabio 6514 | Part of Theorem 8.17 in [Q... |
| iotaval2 6515 | Version of ~ iotaval using... |
| iotauni2 6516 | Version of ~ iotauni using... |
| iotanul2 6517 | Version of ~ iotanul using... |
| iotaval 6518 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6519 | The ` iota ` class is a su... |
| iotaex 6520 | Theorem 8.23 in [Quine] p.... |
| iotavalOLD 6521 | Obsolete version of ~ iota... |
| iotauni 6522 | Equivalence between two di... |
| iotaint 6523 | Equivalence between two di... |
| iota1 6524 | Property of iota. (Contri... |
| iotanul 6525 | Theorem 8.22 in [Quine] p.... |
| iotassuniOLD 6526 | Obsolete version of ~ iota... |
| iotaexOLD 6527 | Obsolete version of ~ iota... |
| iota4 6528 | Theorem *14.22 in [Whitehe... |
| iota4an 6529 | Theorem *14.23 in [Whitehe... |
| iota5 6530 | A method for computing iot... |
| iotabidv 6531 | Formula-building deduction... |
| iotabii 6532 | Formula-building deduction... |
| iotacl 6533 | Membership law for descrip... |
| iota2df 6534 | A condition that allows to... |
| iota2d 6535 | A condition that allows to... |
| iota2 6536 | The unique element such th... |
| iotan0 6537 | Representation of "the uni... |
| sniota 6538 | A class abstraction with a... |
| dfiota4 6539 | The ` iota ` operation usi... |
| csbiota 6540 | Class substitution within ... |
| dffun2 6557 | Alternate definition of a ... |
| dffun2OLD 6558 | Obsolete version of ~ dffu... |
| dffun2OLDOLD 6559 | Obsolete version of ~ dffu... |
| dffun6 6560 | Alternate definition of a ... |
| dffun3 6561 | Alternate definition of fu... |
| dffun3OLD 6562 | Obsolete version of ~ dffu... |
| dffun4 6563 | Alternate definition of a ... |
| dffun5 6564 | Alternate definition of fu... |
| dffun6f 6565 | Definition of function, us... |
| dffun6OLD 6566 | Obsolete version of ~ dffu... |
| funmo 6567 | A function has at most one... |
| funmoOLD 6568 | Obsolete version of ~ funm... |
| funrel 6569 | A function is a relation. ... |
| 0nelfun 6570 | A function does not contai... |
| funss 6571 | Subclass theorem for funct... |
| funeq 6572 | Equality theorem for funct... |
| funeqi 6573 | Equality inference for the... |
| funeqd 6574 | Equality deduction for the... |
| nffun 6575 | Bound-variable hypothesis ... |
| sbcfung 6576 | Distribute proper substitu... |
| funeu 6577 | There is exactly one value... |
| funeu2 6578 | There is exactly one value... |
| dffun7 6579 | Alternate definition of a ... |
| dffun8 6580 | Alternate definition of a ... |
| dffun9 6581 | Alternate definition of a ... |
| funfn 6582 | A class is a function if a... |
| funfnd 6583 | A function is a function o... |
| funi 6584 | The identity relation is a... |
| nfunv 6585 | The universal class is not... |
| funopg 6586 | A Kuratowski ordered pair ... |
| funopab 6587 | A class of ordered pairs i... |
| funopabeq 6588 | A class of ordered pairs o... |
| funopab4 6589 | A class of ordered pairs o... |
| funmpt 6590 | A function in maps-to nota... |
| funmpt2 6591 | Functionality of a class g... |
| funco 6592 | The composition of two fun... |
| funresfunco 6593 | Composition of two functio... |
| funres 6594 | A restriction of a functio... |
| funresd 6595 | A restriction of a functio... |
| funssres 6596 | The restriction of a funct... |
| fun2ssres 6597 | Equality of restrictions o... |
| funun 6598 | The union of functions wit... |
| fununmo 6599 | If the union of classes is... |
| fununfun 6600 | If the union of classes is... |
| fundif 6601 | A function with removed el... |
| funcnvsn 6602 | The converse singleton of ... |
| funsng 6603 | A singleton of an ordered ... |
| fnsng 6604 | Functionality and domain o... |
| funsn 6605 | A singleton of an ordered ... |
| funprg 6606 | A set of two pairs is a fu... |
| funtpg 6607 | A set of three pairs is a ... |
| funpr 6608 | A function with a domain o... |
| funtp 6609 | A function with a domain o... |
| fnsn 6610 | Functionality and domain o... |
| fnprg 6611 | Function with a domain of ... |
| fntpg 6612 | Function with a domain of ... |
| fntp 6613 | A function with a domain o... |
| funcnvpr 6614 | The converse pair of order... |
| funcnvtp 6615 | The converse triple of ord... |
| funcnvqp 6616 | The converse quadruple of ... |
| fun0 6617 | The empty set is a functio... |
| funcnv0 6618 | The converse of the empty ... |
| funcnvcnv 6619 | The double converse of a f... |
| funcnv2 6620 | A simpler equivalence for ... |
| funcnv 6621 | The converse of a class is... |
| funcnv3 6622 | A condition showing a clas... |
| fun2cnv 6623 | The double converse of a c... |
| svrelfun 6624 | A single-valued relation i... |
| fncnv 6625 | Single-rootedness (see ~ f... |
| fun11 6626 | Two ways of stating that `... |
| fununi 6627 | The union of a chain (with... |
| funin 6628 | The intersection with a fu... |
| funres11 6629 | The restriction of a one-t... |
| funcnvres 6630 | The converse of a restrict... |
| cnvresid 6631 | Converse of a restricted i... |
| funcnvres2 6632 | The converse of a restrict... |
| funimacnv 6633 | The image of the preimage ... |
| funimass1 6634 | A kind of contraposition l... |
| funimass2 6635 | A kind of contraposition l... |
| imadif 6636 | The image of a difference ... |
| imain 6637 | The image of an intersecti... |
| funimaexg 6638 | Axiom of Replacement using... |
| funimaexgOLD 6639 | Obsolete version of ~ funi... |
| funimaex 6640 | The image of a set under a... |
| isarep1 6641 | Part of a study of the Axi... |
| isarep1OLD 6642 | Obsolete version of ~ isar... |
| isarep2 6643 | Part of a study of the Axi... |
| fneq1 6644 | Equality theorem for funct... |
| fneq2 6645 | Equality theorem for funct... |
| fneq1d 6646 | Equality deduction for fun... |
| fneq2d 6647 | Equality deduction for fun... |
| fneq12d 6648 | Equality deduction for fun... |
| fneq12 6649 | Equality theorem for funct... |
| fneq1i 6650 | Equality inference for fun... |
| fneq2i 6651 | Equality inference for fun... |
| nffn 6652 | Bound-variable hypothesis ... |
| fnfun 6653 | A function with domain is ... |
| fnfund 6654 | A function with domain is ... |
| fnrel 6655 | A function with domain is ... |
| fndm 6656 | The domain of a function. ... |
| fndmi 6657 | The domain of a function. ... |
| fndmd 6658 | The domain of a function. ... |
| funfni 6659 | Inference to convert a fun... |
| fndmu 6660 | A function has a unique do... |
| fnbr 6661 | The first argument of bina... |
| fnop 6662 | The first argument of an o... |
| fneu 6663 | There is exactly one value... |
| fneu2 6664 | There is exactly one value... |
| fnunres1 6665 | Restriction of a disjoint ... |
| fnunres2 6666 | Restriction of a disjoint ... |
| fnun 6667 | The union of two functions... |
| fnund 6668 | The union of two functions... |
| fnunop 6669 | Extension of a function wi... |
| fncofn 6670 | Composition of a function ... |
| fnco 6671 | Composition of two functio... |
| fncoOLD 6672 | Obsolete version of ~ fnco... |
| fnresdm 6673 | A function does not change... |
| fnresdisj 6674 | A function restricted to a... |
| 2elresin 6675 | Membership in two function... |
| fnssresb 6676 | Restriction of a function ... |
| fnssres 6677 | Restriction of a function ... |
| fnssresd 6678 | Restriction of a function ... |
| fnresin1 6679 | Restriction of a function'... |
| fnresin2 6680 | Restriction of a function'... |
| fnres 6681 | An equivalence for functio... |
| idfn 6682 | The identity relation is a... |
| fnresi 6683 | The restricted identity re... |
| fnima 6684 | The image of a function's ... |
| fn0 6685 | A function with empty doma... |
| fnimadisj 6686 | A class that is disjoint w... |
| fnimaeq0 6687 | Images under a function ne... |
| dfmpt3 6688 | Alternate definition for t... |
| mptfnf 6689 | The maps-to notation defin... |
| fnmptf 6690 | The maps-to notation defin... |
| fnopabg 6691 | Functionality and domain o... |
| fnopab 6692 | Functionality and domain o... |
| mptfng 6693 | The maps-to notation defin... |
| fnmpt 6694 | The maps-to notation defin... |
| fnmptd 6695 | The maps-to notation defin... |
| mpt0 6696 | A mapping operation with e... |
| fnmpti 6697 | Functionality and domain o... |
| dmmpti 6698 | Domain of the mapping oper... |
| dmmptd 6699 | The domain of the mapping ... |
| mptun 6700 | Union of mappings which ar... |
| partfun 6701 | Rewrite a function defined... |
| feq1 6702 | Equality theorem for funct... |
| feq2 6703 | Equality theorem for funct... |
| feq3 6704 | Equality theorem for funct... |
| feq23 6705 | Equality theorem for funct... |
| feq1d 6706 | Equality deduction for fun... |
| feq2d 6707 | Equality deduction for fun... |
| feq3d 6708 | Equality deduction for fun... |
| feq12d 6709 | Equality deduction for fun... |
| feq123d 6710 | Equality deduction for fun... |
| feq123 6711 | Equality theorem for funct... |
| feq1i 6712 | Equality inference for fun... |
| feq2i 6713 | Equality inference for fun... |
| feq12i 6714 | Equality inference for fun... |
| feq23i 6715 | Equality inference for fun... |
| feq23d 6716 | Equality deduction for fun... |
| nff 6717 | Bound-variable hypothesis ... |
| sbcfng 6718 | Distribute proper substitu... |
| sbcfg 6719 | Distribute proper substitu... |
| elimf 6720 | Eliminate a mapping hypoth... |
| ffn 6721 | A mapping is a function wi... |
| ffnd 6722 | A mapping is a function wi... |
| dffn2 6723 | Any function is a mapping ... |
| ffun 6724 | A mapping is a function. ... |
| ffund 6725 | A mapping is a function, d... |
| frel 6726 | A mapping is a relation. ... |
| freld 6727 | A mapping is a relation. ... |
| frn 6728 | The range of a mapping. (... |
| frnd 6729 | Deduction form of ~ frn . ... |
| fdm 6730 | The domain of a mapping. ... |
| fdmd 6731 | Deduction form of ~ fdm . ... |
| fdmi 6732 | Inference associated with ... |
| dffn3 6733 | A function maps to its ran... |
| ffrn 6734 | A function maps to its ran... |
| ffrnb 6735 | Characterization of a func... |
| ffrnbd 6736 | A function maps to its ran... |
| fss 6737 | Expanding the codomain of ... |
| fssd 6738 | Expanding the codomain of ... |
| fssdmd 6739 | Expressing that a class is... |
| fssdm 6740 | Expressing that a class is... |
| fimass 6741 | The image of a class under... |
| fimassd 6742 | The image of a class is a ... |
| fimacnv 6743 | The preimage of the codoma... |
| fcof 6744 | Composition of a function ... |
| fco 6745 | Composition of two functio... |
| fcoOLD 6746 | Obsolete version of ~ fco ... |
| fcod 6747 | Composition of two mapping... |
| fco2 6748 | Functionality of a composi... |
| fssxp 6749 | A mapping is a class of or... |
| funssxp 6750 | Two ways of specifying a p... |
| ffdm 6751 | A mapping is a partial fun... |
| ffdmd 6752 | The domain of a function. ... |
| fdmrn 6753 | A different way to write `... |
| funcofd 6754 | Composition of two functio... |
| fco3OLD 6755 | Obsolete version of ~ func... |
| opelf 6756 | The members of an ordered ... |
| fun 6757 | The union of two functions... |
| fun2 6758 | The union of two functions... |
| fun2d 6759 | The union of functions wit... |
| fnfco 6760 | Composition of two functio... |
| fssres 6761 | Restriction of a function ... |
| fssresd 6762 | Restriction of a function ... |
| fssres2 6763 | Restriction of a restricte... |
| fresin 6764 | An identity for the mappin... |
| resasplit 6765 | If two functions agree on ... |
| fresaun 6766 | The union of two functions... |
| fresaunres2 6767 | From the union of two func... |
| fresaunres1 6768 | From the union of two func... |
| fcoi1 6769 | Composition of a mapping a... |
| fcoi2 6770 | Composition of restricted ... |
| feu 6771 | There is exactly one value... |
| fcnvres 6772 | The converse of a restrict... |
| fimacnvdisj 6773 | The preimage of a class di... |
| fint 6774 | Function into an intersect... |
| fin 6775 | Mapping into an intersecti... |
| f0 6776 | The empty function. (Cont... |
| f00 6777 | A class is a function with... |
| f0bi 6778 | A function with empty doma... |
| f0dom0 6779 | A function is empty iff it... |
| f0rn0 6780 | If there is no element in ... |
| fconst 6781 | A Cartesian product with a... |
| fconstg 6782 | A Cartesian product with a... |
| fnconstg 6783 | A Cartesian product with a... |
| fconst6g 6784 | Constant function with loo... |
| fconst6 6785 | A constant function as a m... |
| f1eq1 6786 | Equality theorem for one-t... |
| f1eq2 6787 | Equality theorem for one-t... |
| f1eq3 6788 | Equality theorem for one-t... |
| nff1 6789 | Bound-variable hypothesis ... |
| dff12 6790 | Alternate definition of a ... |
| f1f 6791 | A one-to-one mapping is a ... |
| f1fn 6792 | A one-to-one mapping is a ... |
| f1fun 6793 | A one-to-one mapping is a ... |
| f1rel 6794 | A one-to-one onto mapping ... |
| f1dm 6795 | The domain of a one-to-one... |
| f1ss 6796 | A function that is one-to-... |
| f1ssr 6797 | A function that is one-to-... |
| f1ssres 6798 | A function that is one-to-... |
| f1resf1 6799 | The restriction of an inje... |
| f1cnvcnv 6800 | Two ways to express that a... |
| f1cof1 6801 | Composition of two one-to-... |
| f1co 6802 | Composition of one-to-one ... |
| f1coOLD 6803 | Obsolete version of ~ f1co... |
| foeq1 6804 | Equality theorem for onto ... |
| foeq2 6805 | Equality theorem for onto ... |
| foeq3 6806 | Equality theorem for onto ... |
| nffo 6807 | Bound-variable hypothesis ... |
| fof 6808 | An onto mapping is a mappi... |
| fofun 6809 | An onto mapping is a funct... |
| fofn 6810 | An onto mapping is a funct... |
| forn 6811 | The codomain of an onto fu... |
| dffo2 6812 | Alternate definition of an... |
| foima 6813 | The image of the domain of... |
| dffn4 6814 | A function maps onto its r... |
| funforn 6815 | A function maps its domain... |
| fodmrnu 6816 | An onto function has uniqu... |
| fimadmfo 6817 | A function is a function o... |
| fores 6818 | Restriction of an onto fun... |
| fimadmfoALT 6819 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6820 | The preimage of the codoma... |
| focofo 6821 | Composition of onto functi... |
| foco 6822 | Composition of onto functi... |
| foconst 6823 | A nonzero constant functio... |
| f1oeq1 6824 | Equality theorem for one-t... |
| f1oeq2 6825 | Equality theorem for one-t... |
| f1oeq3 6826 | Equality theorem for one-t... |
| f1oeq23 6827 | Equality theorem for one-t... |
| f1eq123d 6828 | Equality deduction for one... |
| foeq123d 6829 | Equality deduction for ont... |
| f1oeq123d 6830 | Equality deduction for one... |
| f1oeq1d 6831 | Equality deduction for one... |
| f1oeq2d 6832 | Equality deduction for one... |
| f1oeq3d 6833 | Equality deduction for one... |
| nff1o 6834 | Bound-variable hypothesis ... |
| f1of1 6835 | A one-to-one onto mapping ... |
| f1of 6836 | A one-to-one onto mapping ... |
| f1ofn 6837 | A one-to-one onto mapping ... |
| f1ofun 6838 | A one-to-one onto mapping ... |
| f1orel 6839 | A one-to-one onto mapping ... |
| f1odm 6840 | The domain of a one-to-one... |
| dff1o2 6841 | Alternate definition of on... |
| dff1o3 6842 | Alternate definition of on... |
| f1ofo 6843 | A one-to-one onto function... |
| dff1o4 6844 | Alternate definition of on... |
| dff1o5 6845 | Alternate definition of on... |
| f1orn 6846 | A one-to-one function maps... |
| f1f1orn 6847 | A one-to-one function maps... |
| f1ocnv 6848 | The converse of a one-to-o... |
| f1ocnvb 6849 | A relation is a one-to-one... |
| f1ores 6850 | The restriction of a one-t... |
| f1orescnv 6851 | The converse of a one-to-o... |
| f1imacnv 6852 | Preimage of an image. (Co... |
| foimacnv 6853 | A reverse version of ~ f1i... |
| foun 6854 | The union of two onto func... |
| f1oun 6855 | The union of two one-to-on... |
| f1un 6856 | The union of two one-to-on... |
| resdif 6857 | The restriction of a one-t... |
| resin 6858 | The restriction of a one-t... |
| f1oco 6859 | Composition of one-to-one ... |
| f1cnv 6860 | The converse of an injecti... |
| funcocnv2 6861 | Composition with the conve... |
| fococnv2 6862 | The composition of an onto... |
| f1ococnv2 6863 | The composition of a one-t... |
| f1cocnv2 6864 | Composition of an injectiv... |
| f1ococnv1 6865 | The composition of a one-t... |
| f1cocnv1 6866 | Composition of an injectiv... |
| funcoeqres 6867 | Express a constraint on a ... |
| f1ssf1 6868 | A subset of an injective f... |
| f10 6869 | The empty set maps one-to-... |
| f10d 6870 | The empty set maps one-to-... |
| f1o00 6871 | One-to-one onto mapping of... |
| fo00 6872 | Onto mapping of the empty ... |
| f1o0 6873 | One-to-one onto mapping of... |
| f1oi 6874 | A restriction of the ident... |
| f1ovi 6875 | The identity relation is a... |
| f1osn 6876 | A singleton of an ordered ... |
| f1osng 6877 | A singleton of an ordered ... |
| f1sng 6878 | A singleton of an ordered ... |
| fsnd 6879 | A singleton of an ordered ... |
| f1oprswap 6880 | A two-element swap is a bi... |
| f1oprg 6881 | An unordered pair of order... |
| tz6.12-2 6882 | Function value when ` F ` ... |
| fveu 6883 | The value of a function at... |
| brprcneu 6884 | If ` A ` is a proper class... |
| brprcneuALT 6885 | Alternate proof of ~ brprc... |
| fvprc 6886 | A function's value at a pr... |
| fvprcALT 6887 | Alternate proof of ~ fvprc... |
| rnfvprc 6888 | The range of a function va... |
| fv2 6889 | Alternate definition of fu... |
| dffv3 6890 | A definition of function v... |
| dffv4 6891 | The previous definition of... |
| elfv 6892 | Membership in a function v... |
| fveq1 6893 | Equality theorem for funct... |
| fveq2 6894 | Equality theorem for funct... |
| fveq1i 6895 | Equality inference for fun... |
| fveq1d 6896 | Equality deduction for fun... |
| fveq2i 6897 | Equality inference for fun... |
| fveq2d 6898 | Equality deduction for fun... |
| 2fveq3 6899 | Equality theorem for neste... |
| fveq12i 6900 | Equality deduction for fun... |
| fveq12d 6901 | Equality deduction for fun... |
| fveqeq2d 6902 | Equality deduction for fun... |
| fveqeq2 6903 | Equality deduction for fun... |
| nffv 6904 | Bound-variable hypothesis ... |
| nffvmpt1 6905 | Bound-variable hypothesis ... |
| nffvd 6906 | Deduction version of bound... |
| fvex 6907 | The value of a class exist... |
| fvexi 6908 | The value of a class exist... |
| fvexd 6909 | The value of a class exist... |
| fvif 6910 | Move a conditional outside... |
| iffv 6911 | Move a conditional outside... |
| fv3 6912 | Alternate definition of th... |
| fvres 6913 | The value of a restricted ... |
| fvresd 6914 | The value of a restricted ... |
| funssfv 6915 | The value of a member of t... |
| tz6.12c 6916 | Corollary of Theorem 6.12(... |
| tz6.12-1 6917 | Function value. Theorem 6... |
| tz6.12-1OLD 6918 | Obsolete version of ~ tz6.... |
| tz6.12 6919 | Function value. Theorem 6... |
| tz6.12f 6920 | Function value, using boun... |
| tz6.12cOLD 6921 | Obsolete version of ~ tz6.... |
| tz6.12i 6922 | Corollary of Theorem 6.12(... |
| fvbr0 6923 | Two possibilities for the ... |
| fvrn0 6924 | A function value is a memb... |
| fvn0fvelrn 6925 | If the value of a function... |
| elfvunirn 6926 | A function value is a subs... |
| fvssunirn 6927 | The result of a function v... |
| fvssunirnOLD 6928 | Obsolete version of ~ fvss... |
| ndmfv 6929 | The value of a class outsi... |
| ndmfvrcl 6930 | Reverse closure law for fu... |
| elfvdm 6931 | If a function value has a ... |
| elfvex 6932 | If a function value has a ... |
| elfvexd 6933 | If a function value has a ... |
| eliman0 6934 | A nonempty function value ... |
| nfvres 6935 | The value of a non-member ... |
| nfunsn 6936 | If the restriction of a cl... |
| fvfundmfvn0 6937 | If the "value of a class" ... |
| 0fv 6938 | Function value of the empt... |
| fv2prc 6939 | A function value of a func... |
| elfv2ex 6940 | If a function value of a f... |
| fveqres 6941 | Equal values imply equal v... |
| csbfv12 6942 | Move class substitution in... |
| csbfv2g 6943 | Move class substitution in... |
| csbfv 6944 | Substitution for a functio... |
| funbrfv 6945 | The second argument of a b... |
| funopfv 6946 | The second element in an o... |
| fnbrfvb 6947 | Equivalence of function va... |
| fnopfvb 6948 | Equivalence of function va... |
| funbrfvb 6949 | Equivalence of function va... |
| funopfvb 6950 | Equivalence of function va... |
| fnbrfvb2 6951 | Version of ~ fnbrfvb for f... |
| fdmeu 6952 | There is exactly one codom... |
| funbrfv2b 6953 | Function value in terms of... |
| dffn5 6954 | Representation of a functi... |
| fnrnfv 6955 | The range of a function ex... |
| fvelrnb 6956 | A member of a function's r... |
| foelcdmi 6957 | A member of a surjective f... |
| dfimafn 6958 | Alternate definition of th... |
| dfimafn2 6959 | Alternate definition of th... |
| funimass4 6960 | Membership relation for th... |
| fvelima 6961 | Function value in an image... |
| funimassd 6962 | Sufficient condition for t... |
| fvelimad 6963 | Function value in an image... |
| feqmptd 6964 | Deduction form of ~ dffn5 ... |
| feqresmpt 6965 | Express a restricted funct... |
| feqmptdf 6966 | Deduction form of ~ dffn5f... |
| dffn5f 6967 | Representation of a functi... |
| fvelimab 6968 | Function value in an image... |
| fvelimabd 6969 | Deduction form of ~ fvelim... |
| unima 6970 | Image of a union. (Contri... |
| fvi 6971 | The value of the identity ... |
| fviss 6972 | The value of the identity ... |
| fniinfv 6973 | The indexed intersection o... |
| fnsnfv 6974 | Singleton of function valu... |
| fnsnfvOLD 6975 | Obsolete version of ~ fnsn... |
| opabiotafun 6976 | Define a function whose va... |
| opabiotadm 6977 | Define a function whose va... |
| opabiota 6978 | Define a function whose va... |
| fnimapr 6979 | The image of a pair under ... |
| ssimaex 6980 | The existence of a subimag... |
| ssimaexg 6981 | The existence of a subimag... |
| funfv 6982 | A simplified expression fo... |
| funfv2 6983 | The value of a function. ... |
| funfv2f 6984 | The value of a function. ... |
| fvun 6985 | Value of the union of two ... |
| fvun1 6986 | The value of a union when ... |
| fvun2 6987 | The value of a union when ... |
| fvun1d 6988 | The value of a union when ... |
| fvun2d 6989 | The value of a union when ... |
| dffv2 6990 | Alternate definition of fu... |
| dmfco 6991 | Domains of a function comp... |
| fvco2 6992 | Value of a function compos... |
| fvco 6993 | Value of a function compos... |
| fvco3 6994 | Value of a function compos... |
| fvco3d 6995 | Value of a function compos... |
| fvco4i 6996 | Conditions for a compositi... |
| fvopab3g 6997 | Value of a function given ... |
| fvopab3ig 6998 | Value of a function given ... |
| brfvopabrbr 6999 | The binary relation of a f... |
| fvmptg 7000 | Value of a function given ... |
| fvmpti 7001 | Value of a function given ... |
| fvmpt 7002 | Value of a function given ... |
| fvmpt2f 7003 | Value of a function given ... |
| fvtresfn 7004 | Functionality of a tuple-r... |
| fvmpts 7005 | Value of a function given ... |
| fvmpt3 7006 | Value of a function given ... |
| fvmpt3i 7007 | Value of a function given ... |
| fvmptdf 7008 | Deduction version of ~ fvm... |
| fvmptd 7009 | Deduction version of ~ fvm... |
| fvmptd2 7010 | Deduction version of ~ fvm... |
| mptrcl 7011 | Reverse closure for a mapp... |
| fvmpt2i 7012 | Value of a function given ... |
| fvmpt2 7013 | Value of a function given ... |
| fvmptss 7014 | If all the values of the m... |
| fvmpt2d 7015 | Deduction version of ~ fvm... |
| fvmptex 7016 | Express a function ` F ` w... |
| fvmptd3f 7017 | Alternate deduction versio... |
| fvmptd2f 7018 | Alternate deduction versio... |
| fvmptdv 7019 | Alternate deduction versio... |
| fvmptdv2 7020 | Alternate deduction versio... |
| mpteqb 7021 | Bidirectional equality the... |
| fvmptt 7022 | Closed theorem form of ~ f... |
| fvmptf 7023 | Value of a function given ... |
| fvmptnf 7024 | The value of a function gi... |
| fvmptd3 7025 | Deduction version of ~ fvm... |
| fvmptd4 7026 | Deduction version of ~ fvm... |
| fvmptn 7027 | This somewhat non-intuitiv... |
| fvmptss2 7028 | A mapping always evaluates... |
| elfvmptrab1w 7029 | Implications for the value... |
| elfvmptrab1 7030 | Implications for the value... |
| elfvmptrab 7031 | Implications for the value... |
| fvopab4ndm 7032 | Value of a function given ... |
| fvmptndm 7033 | Value of a function given ... |
| fvmptrabfv 7034 | Value of a function mappin... |
| fvopab5 7035 | The value of a function th... |
| fvopab6 7036 | Value of a function given ... |
| eqfnfv 7037 | Equality of functions is d... |
| eqfnfv2 7038 | Equality of functions is d... |
| eqfnfv3 7039 | Derive equality of functio... |
| eqfnfvd 7040 | Deduction for equality of ... |
| eqfnfv2f 7041 | Equality of functions is d... |
| eqfunfv 7042 | Equality of functions is d... |
| eqfnun 7043 | Two functions on ` A u. B ... |
| fvreseq0 7044 | Equality of restricted fun... |
| fvreseq1 7045 | Equality of a function res... |
| fvreseq 7046 | Equality of restricted fun... |
| fnmptfvd 7047 | A function with a given do... |
| fndmdif 7048 | Two ways to express the lo... |
| fndmdifcom 7049 | The difference set between... |
| fndmdifeq0 7050 | The difference set of two ... |
| fndmin 7051 | Two ways to express the lo... |
| fneqeql 7052 | Two functions are equal if... |
| fneqeql2 7053 | Two functions are equal if... |
| fnreseql 7054 | Two functions are equal on... |
| chfnrn 7055 | The range of a choice func... |
| funfvop 7056 | Ordered pair with function... |
| funfvbrb 7057 | Two ways to say that ` A `... |
| fvimacnvi 7058 | A member of a preimage is ... |
| fvimacnv 7059 | The argument of a function... |
| funimass3 7060 | A kind of contraposition l... |
| funimass5 7061 | A subclass of a preimage i... |
| funconstss 7062 | Two ways of specifying tha... |
| fvimacnvALT 7063 | Alternate proof of ~ fvima... |
| elpreima 7064 | Membership in the preimage... |
| elpreimad 7065 | Membership in the preimage... |
| fniniseg 7066 | Membership in the preimage... |
| fncnvima2 7067 | Inverse images under funct... |
| fniniseg2 7068 | Inverse point images under... |
| unpreima 7069 | Preimage of a union. (Con... |
| inpreima 7070 | Preimage of an intersectio... |
| difpreima 7071 | Preimage of a difference. ... |
| respreima 7072 | The preimage of a restrict... |
| cnvimainrn 7073 | The preimage of the inters... |
| sspreima 7074 | The preimage of a subset i... |
| iinpreima 7075 | Preimage of an intersectio... |
| intpreima 7076 | Preimage of an intersectio... |
| fimacnvOLD 7077 | Obsolete version of ~ fima... |
| fimacnvinrn 7078 | Taking the converse image ... |
| fimacnvinrn2 7079 | Taking the converse image ... |
| rescnvimafod 7080 | The restriction of a funct... |
| fvn0ssdmfun 7081 | If a class' function value... |
| fnopfv 7082 | Ordered pair with function... |
| fvelrn 7083 | A function's value belongs... |
| nelrnfvne 7084 | A function value cannot be... |
| fveqdmss 7085 | If the empty set is not co... |
| fveqressseq 7086 | If the empty set is not co... |
| fnfvelrn 7087 | A function's value belongs... |
| ffvelcdm 7088 | A function's value belongs... |
| fnfvelrnd 7089 | A function's value belongs... |
| ffvelcdmi 7090 | A function's value belongs... |
| ffvelcdmda 7091 | A function's value belongs... |
| ffvelcdmd 7092 | A function's value belongs... |
| feldmfvelcdm 7093 | A class is an element of t... |
| rexrn 7094 | Restricted existential qua... |
| ralrn 7095 | Restricted universal quant... |
| elrnrexdm 7096 | For any element in the ran... |
| elrnrexdmb 7097 | For any element in the ran... |
| eldmrexrn 7098 | For any element in the dom... |
| eldmrexrnb 7099 | For any element in the dom... |
| fvcofneq 7100 | The values of two function... |
| ralrnmptw 7101 | A restricted quantifier ov... |
| rexrnmptw 7102 | A restricted quantifier ov... |
| ralrnmpt 7103 | A restricted quantifier ov... |
| rexrnmpt 7104 | A restricted quantifier ov... |
| f0cli 7105 | Unconditional closure of a... |
| dff2 7106 | Alternate definition of a ... |
| dff3 7107 | Alternate definition of a ... |
| dff4 7108 | Alternate definition of a ... |
| dffo3 7109 | An onto mapping expressed ... |
| dffo4 7110 | Alternate definition of an... |
| dffo5 7111 | Alternate definition of an... |
| exfo 7112 | A relation equivalent to t... |
| dffo3f 7113 | An onto mapping expressed ... |
| foelrn 7114 | Property of a surjective f... |
| foelrnf 7115 | Property of a surjective f... |
| foco2 7116 | If a composition of two fu... |
| fmpt 7117 | Functionality of the mappi... |
| f1ompt 7118 | Express bijection for a ma... |
| fmpti 7119 | Functionality of the mappi... |
| fvmptelcdm 7120 | The value of a function at... |
| fmptd 7121 | Domain and codomain of the... |
| fmpttd 7122 | Version of ~ fmptd with in... |
| fmpt3d 7123 | Domain and codomain of the... |
| fmptdf 7124 | A version of ~ fmptd using... |
| fompt 7125 | Express being onto for a m... |
| ffnfv 7126 | A function maps to a class... |
| ffnfvf 7127 | A function maps to a class... |
| fnfvrnss 7128 | An upper bound for range d... |
| fcdmssb 7129 | A function is a function i... |
| rnmptss 7130 | The range of an operation ... |
| fmpt2d 7131 | Domain and codomain of the... |
| ffvresb 7132 | A necessary and sufficient... |
| fssrescdmd 7133 | Restriction of a function ... |
| f1oresrab 7134 | Build a bijection between ... |
| f1ossf1o 7135 | Restricting a bijection, w... |
| fmptco 7136 | Composition of two functio... |
| fmptcof 7137 | Version of ~ fmptco where ... |
| fmptcos 7138 | Composition of two functio... |
| cofmpt 7139 | Express composition of a m... |
| fcompt 7140 | Express composition of two... |
| fcoconst 7141 | Composition with a constan... |
| fsn 7142 | A function maps a singleto... |
| fsn2 7143 | A function that maps a sin... |
| fsng 7144 | A function maps a singleto... |
| fsn2g 7145 | A function that maps a sin... |
| xpsng 7146 | The Cartesian product of t... |
| xpprsng 7147 | The Cartesian product of a... |
| xpsn 7148 | The Cartesian product of t... |
| f1o2sn 7149 | A singleton consisting in ... |
| residpr 7150 | Restriction of the identit... |
| dfmpt 7151 | Alternate definition for t... |
| fnasrn 7152 | A function expressed as th... |
| idref 7153 | Two ways to state that a r... |
| funiun 7154 | A function is a union of s... |
| funopsn 7155 | If a function is an ordere... |
| funop 7156 | An ordered pair is a funct... |
| funopdmsn 7157 | The domain of a function w... |
| funsndifnop 7158 | A singleton of an ordered ... |
| funsneqopb 7159 | A singleton of an ordered ... |
| ressnop0 7160 | If ` A ` is not in ` C ` ,... |
| fpr 7161 | A function with a domain o... |
| fprg 7162 | A function with a domain o... |
| ftpg 7163 | A function with a domain o... |
| ftp 7164 | A function with a domain o... |
| fnressn 7165 | A function restricted to a... |
| funressn 7166 | A function restricted to a... |
| fressnfv 7167 | The value of a function re... |
| fvrnressn 7168 | If the value of a function... |
| fvressn 7169 | The value of a function re... |
| fvn0fvelrnOLD 7170 | Obsolete version of ~ fvn0... |
| fvconst 7171 | The value of a constant fu... |
| fnsnr 7172 | If a class belongs to a fu... |
| fnsnb 7173 | A function whose domain is... |
| fmptsn 7174 | Express a singleton functi... |
| fmptsng 7175 | Express a singleton functi... |
| fmptsnd 7176 | Express a singleton functi... |
| fmptap 7177 | Append an additional value... |
| fmptapd 7178 | Append an additional value... |
| fmptpr 7179 | Express a pair function in... |
| fvresi 7180 | The value of a restricted ... |
| fninfp 7181 | Express the class of fixed... |
| fnelfp 7182 | Property of a fixed point ... |
| fndifnfp 7183 | Express the class of non-f... |
| fnelnfp 7184 | Property of a non-fixed po... |
| fnnfpeq0 7185 | A function is the identity... |
| fvunsn 7186 | Remove an ordered pair not... |
| fvsng 7187 | The value of a singleton o... |
| fvsn 7188 | The value of a singleton o... |
| fvsnun1 7189 | The value of a function wi... |
| fvsnun2 7190 | The value of a function wi... |
| fnsnsplit 7191 | Split a function into a si... |
| fsnunf 7192 | Adjoining a point to a fun... |
| fsnunf2 7193 | Adjoining a point to a pun... |
| fsnunfv 7194 | Recover the added point fr... |
| fsnunres 7195 | Recover the original funct... |
| funresdfunsn 7196 | Restricting a function to ... |
| fvpr1g 7197 | The value of a function wi... |
| fvpr2g 7198 | The value of a function wi... |
| fvpr2gOLD 7199 | Obsolete version of ~ fvpr... |
| fvpr1 7200 | The value of a function wi... |
| fvpr1OLD 7201 | Obsolete version of ~ fvpr... |
| fvpr2 7202 | The value of a function wi... |
| fvpr2OLD 7203 | Obsolete version of ~ fvpr... |
| fprb 7204 | A condition for functionho... |
| fvtp1 7205 | The first value of a funct... |
| fvtp2 7206 | The second value of a func... |
| fvtp3 7207 | The third value of a funct... |
| fvtp1g 7208 | The value of a function wi... |
| fvtp2g 7209 | The value of a function wi... |
| fvtp3g 7210 | The value of a function wi... |
| tpres 7211 | An unordered triple of ord... |
| fvconst2g 7212 | The value of a constant fu... |
| fconst2g 7213 | A constant function expres... |
| fvconst2 7214 | The value of a constant fu... |
| fconst2 7215 | A constant function expres... |
| fconst5 7216 | Two ways to express that a... |
| rnmptc 7217 | Range of a constant functi... |
| fnprb 7218 | A function whose domain ha... |
| fntpb 7219 | A function whose domain ha... |
| fnpr2g 7220 | A function whose domain ha... |
| fpr2g 7221 | A function that maps a pai... |
| fconstfv 7222 | A constant function expres... |
| fconst3 7223 | Two ways to express a cons... |
| fconst4 7224 | Two ways to express a cons... |
| resfunexg 7225 | The restriction of a funct... |
| resiexd 7226 | The restriction of the ide... |
| fnex 7227 | If the domain of a functio... |
| fnexd 7228 | If the domain of a functio... |
| funex 7229 | If the domain of a functio... |
| opabex 7230 | Existence of a function ex... |
| mptexg 7231 | If the domain of a functio... |
| mptexgf 7232 | If the domain of a functio... |
| mptex 7233 | If the domain of a functio... |
| mptexd 7234 | If the domain of a functio... |
| mptrabex 7235 | If the domain of a functio... |
| fex 7236 | If the domain of a mapping... |
| fexd 7237 | If the domain of a mapping... |
| mptfvmpt 7238 | A function in maps-to nota... |
| eufnfv 7239 | A function is uniquely det... |
| funfvima 7240 | A function's value in a pr... |
| funfvima2 7241 | A function's value in an i... |
| funfvima2d 7242 | A function's value in a pr... |
| fnfvima 7243 | The function value of an o... |
| fnfvimad 7244 | A function's value belongs... |
| resfvresima 7245 | The value of the function ... |
| funfvima3 7246 | A class including a functi... |
| rexima 7247 | Existential quantification... |
| ralima 7248 | Universal quantification u... |
| fvclss 7249 | Upper bound for the class ... |
| elabrex 7250 | Elementhood in an image se... |
| elabrexg 7251 | Elementhood in an image se... |
| abrexco 7252 | Composition of two image m... |
| imaiun 7253 | The image of an indexed un... |
| imauni 7254 | The image of a union is th... |
| fniunfv 7255 | The indexed union of a fun... |
| funiunfv 7256 | The indexed union of a fun... |
| funiunfvf 7257 | The indexed union of a fun... |
| eluniima 7258 | Membership in the union of... |
| elunirn 7259 | Membership in the union of... |
| elunirnALT 7260 | Alternate proof of ~ eluni... |
| elunirn2OLD 7261 | Obsolete version of ~ elfv... |
| fnunirn 7262 | Membership in a union of s... |
| dff13 7263 | A one-to-one function in t... |
| dff13f 7264 | A one-to-one function in t... |
| f1veqaeq 7265 | If the values of a one-to-... |
| f1cofveqaeq 7266 | If the values of a composi... |
| f1cofveqaeqALT 7267 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7268 | If two one-to-one function... |
| f1mpt 7269 | Express injection for a ma... |
| f1fveq 7270 | Equality of function value... |
| f1elima 7271 | Membership in the image of... |
| f1imass 7272 | Taking images under a one-... |
| f1imaeq 7273 | Taking images under a one-... |
| f1imapss 7274 | Taking images under a one-... |
| fpropnf1 7275 | A function, given by an un... |
| f1dom3fv3dif 7276 | The function values for a ... |
| f1dom3el3dif 7277 | The codomain of a 1-1 func... |
| dff14a 7278 | A one-to-one function in t... |
| dff14b 7279 | A one-to-one function in t... |
| f12dfv 7280 | A one-to-one function with... |
| f13dfv 7281 | A one-to-one function with... |
| dff1o6 7282 | A one-to-one onto function... |
| f1ocnvfv1 7283 | The converse value of the ... |
| f1ocnvfv2 7284 | The value of the converse ... |
| f1ocnvfv 7285 | Relationship between the v... |
| f1ocnvfvb 7286 | Relationship between the v... |
| nvof1o 7287 | An involution is a bijecti... |
| nvocnv 7288 | The converse of an involut... |
| f1cdmsn 7289 | If a one-to-one function w... |
| fsnex 7290 | Relate a function with a s... |
| f1prex 7291 | Relate a one-to-one functi... |
| f1ocnvdm 7292 | The value of the converse ... |
| f1ocnvfvrneq 7293 | If the values of a one-to-... |
| fcof1 7294 | An application is injectiv... |
| fcofo 7295 | An application is surjecti... |
| cbvfo 7296 | Change bound variable betw... |
| cbvexfo 7297 | Change bound variable betw... |
| cocan1 7298 | An injection is left-cance... |
| cocan2 7299 | A surjection is right-canc... |
| fcof1oinvd 7300 | Show that a function is th... |
| fcof1od 7301 | A function is bijective if... |
| 2fcoidinvd 7302 | Show that a function is th... |
| fcof1o 7303 | Show that two functions ar... |
| 2fvcoidd 7304 | Show that the composition ... |
| 2fvidf1od 7305 | A function is bijective if... |
| 2fvidinvd 7306 | Show that two functions ar... |
| foeqcnvco 7307 | Condition for function equ... |
| f1eqcocnv 7308 | Condition for function equ... |
| fveqf1o 7309 | Given a bijection ` F ` , ... |
| nf1const 7310 | A constant function from a... |
| nf1oconst 7311 | A constant function from a... |
| f1ofvswap 7312 | Swapping two values in a b... |
| fliftrel 7313 | ` F ` , a function lift, i... |
| fliftel 7314 | Elementhood in the relatio... |
| fliftel1 7315 | Elementhood in the relatio... |
| fliftcnv 7316 | Converse of the relation `... |
| fliftfun 7317 | The function ` F ` is the ... |
| fliftfund 7318 | The function ` F ` is the ... |
| fliftfuns 7319 | The function ` F ` is the ... |
| fliftf 7320 | The domain and range of th... |
| fliftval 7321 | The value of the function ... |
| isoeq1 7322 | Equality theorem for isomo... |
| isoeq2 7323 | Equality theorem for isomo... |
| isoeq3 7324 | Equality theorem for isomo... |
| isoeq4 7325 | Equality theorem for isomo... |
| isoeq5 7326 | Equality theorem for isomo... |
| nfiso 7327 | Bound-variable hypothesis ... |
| isof1o 7328 | An isomorphism is a one-to... |
| isof1oidb 7329 | A function is a bijection ... |
| isof1oopb 7330 | A function is a bijection ... |
| isorel 7331 | An isomorphism connects bi... |
| soisores 7332 | Express the condition of i... |
| soisoi 7333 | Infer isomorphism from one... |
| isoid 7334 | Identity law for isomorphi... |
| isocnv 7335 | Converse law for isomorphi... |
| isocnv2 7336 | Converse law for isomorphi... |
| isocnv3 7337 | Complementation law for is... |
| isores2 7338 | An isomorphism from one we... |
| isores1 7339 | An isomorphism from one we... |
| isores3 7340 | Induced isomorphism on a s... |
| isotr 7341 | Composition (transitive) l... |
| isomin 7342 | Isomorphisms preserve mini... |
| isoini 7343 | Isomorphisms preserve init... |
| isoini2 7344 | Isomorphisms are isomorphi... |
| isofrlem 7345 | Lemma for ~ isofr . (Cont... |
| isoselem 7346 | Lemma for ~ isose . (Cont... |
| isofr 7347 | An isomorphism preserves w... |
| isose 7348 | An isomorphism preserves s... |
| isofr2 7349 | A weak form of ~ isofr tha... |
| isopolem 7350 | Lemma for ~ isopo . (Cont... |
| isopo 7351 | An isomorphism preserves t... |
| isosolem 7352 | Lemma for ~ isoso . (Cont... |
| isoso 7353 | An isomorphism preserves t... |
| isowe 7354 | An isomorphism preserves t... |
| isowe2 7355 | A weak form of ~ isowe tha... |
| f1oiso 7356 | Any one-to-one onto functi... |
| f1oiso2 7357 | Any one-to-one onto functi... |
| f1owe 7358 | Well-ordering of isomorphi... |
| weniso 7359 | A set-like well-ordering h... |
| weisoeq 7360 | Thus, there is at most one... |
| weisoeq2 7361 | Thus, there is at most one... |
| knatar 7362 | The Knaster-Tarski theorem... |
| fvresval 7363 | The value of a restricted ... |
| funeldmb 7364 | If ` (/) ` is not part of ... |
| eqfunresadj 7365 | Law for adjoining an eleme... |
| eqfunressuc 7366 | Law for equality of restri... |
| fnssintima 7367 | Condition for subset of an... |
| imaeqsexv 7368 | Substitute a function valu... |
| imaeqsalv 7369 | Substitute a function valu... |
| canth 7370 | No set ` A ` is equinumero... |
| ncanth 7371 | Cantor's theorem fails for... |
| riotaeqdv 7374 | Formula-building deduction... |
| riotabidv 7375 | Formula-building deduction... |
| riotaeqbidv 7376 | Equality deduction for res... |
| riotaex 7377 | Restricted iota is a set. ... |
| riotav 7378 | An iota restricted to the ... |
| riotauni 7379 | Restricted iota in terms o... |
| nfriota1 7380 | The abstraction variable i... |
| nfriotadw 7381 | Deduction version of ~ nfr... |
| cbvriotaw 7382 | Change bound variable in a... |
| cbvriotavw 7383 | Change bound variable in a... |
| cbvriotavwOLD 7384 | Obsolete version of ~ cbvr... |
| nfriotad 7385 | Deduction version of ~ nfr... |
| nfriota 7386 | A variable not free in a w... |
| cbvriota 7387 | Change bound variable in a... |
| cbvriotav 7388 | Change bound variable in a... |
| csbriota 7389 | Interchange class substitu... |
| riotacl2 7390 | Membership law for "the un... |
| riotacl 7391 | Closure of restricted iota... |
| riotasbc 7392 | Substitution law for descr... |
| riotabidva 7393 | Equivalent wff's yield equ... |
| riotabiia 7394 | Equivalent wff's yield equ... |
| riota1 7395 | Property of restricted iot... |
| riota1a 7396 | Property of iota. (Contri... |
| riota2df 7397 | A deduction version of ~ r... |
| riota2f 7398 | This theorem shows a condi... |
| riota2 7399 | This theorem shows a condi... |
| riotaeqimp 7400 | If two restricted iota des... |
| riotaprop 7401 | Properties of a restricted... |
| riota5f 7402 | A method for computing res... |
| riota5 7403 | A method for computing res... |
| riotass2 7404 | Restriction of a unique el... |
| riotass 7405 | Restriction of a unique el... |
| moriotass 7406 | Restriction of a unique el... |
| snriota 7407 | A restricted class abstrac... |
| riotaxfrd 7408 | Change the variable ` x ` ... |
| eusvobj2 7409 | Specify the same property ... |
| eusvobj1 7410 | Specify the same object in... |
| f1ofveu 7411 | There is one domain elemen... |
| f1ocnvfv3 7412 | Value of the converse of a... |
| riotaund 7413 | Restricted iota equals the... |
| riotassuni 7414 | The restricted iota class ... |
| riotaclb 7415 | Bidirectional closure of r... |
| riotarab 7416 | Restricted iota of a restr... |
| oveq 7423 | Equality theorem for opera... |
| oveq1 7424 | Equality theorem for opera... |
| oveq2 7425 | Equality theorem for opera... |
| oveq12 7426 | Equality theorem for opera... |
| oveq1i 7427 | Equality inference for ope... |
| oveq2i 7428 | Equality inference for ope... |
| oveq12i 7429 | Equality inference for ope... |
| oveqi 7430 | Equality inference for ope... |
| oveq123i 7431 | Equality inference for ope... |
| oveq1d 7432 | Equality deduction for ope... |
| oveq2d 7433 | Equality deduction for ope... |
| oveqd 7434 | Equality deduction for ope... |
| oveq12d 7435 | Equality deduction for ope... |
| oveqan12d 7436 | Equality deduction for ope... |
| oveqan12rd 7437 | Equality deduction for ope... |
| oveq123d 7438 | Equality deduction for ope... |
| fvoveq1d 7439 | Equality deduction for nes... |
| fvoveq1 7440 | Equality theorem for neste... |
| ovanraleqv 7441 | Equality theorem for a con... |
| imbrov2fvoveq 7442 | Equality theorem for neste... |
| ovrspc2v 7443 | If an operation value is e... |
| oveqrspc2v 7444 | Restricted specialization ... |
| oveqdr 7445 | Equality of two operations... |
| nfovd 7446 | Deduction version of bound... |
| nfov 7447 | Bound-variable hypothesis ... |
| oprabidw 7448 | The law of concretion. Sp... |
| oprabid 7449 | The law of concretion. Sp... |
| ovex 7450 | The result of an operation... |
| ovexi 7451 | The result of an operation... |
| ovexd 7452 | The result of an operation... |
| ovssunirn 7453 | The result of an operation... |
| 0ov 7454 | Operation value of the emp... |
| ovprc 7455 | The value of an operation ... |
| ovprc1 7456 | The value of an operation ... |
| ovprc2 7457 | The value of an operation ... |
| ovrcl 7458 | Reverse closure for an ope... |
| elfvov1 7459 | Utility theorem: reverse c... |
| csbov123 7460 | Move class substitution in... |
| csbov 7461 | Move class substitution in... |
| csbov12g 7462 | Move class substitution in... |
| csbov1g 7463 | Move class substitution in... |
| csbov2g 7464 | Move class substitution in... |
| rspceov 7465 | A frequently used special ... |
| elovimad 7466 | Elementhood of the image s... |
| fnbrovb 7467 | Value of a binary operatio... |
| fnotovb 7468 | Equivalence of operation v... |
| opabbrex 7469 | A collection of ordered pa... |
| opabresex2 7470 | Restrictions of a collecti... |
| opabresex2d 7471 | Obsolete version of ~ opab... |
| fvmptopab 7472 | The function value of a ma... |
| fvmptopabOLD 7473 | Obsolete version of ~ fvmp... |
| f1opr 7474 | Condition for an operation... |
| brfvopab 7475 | The classes involved in a ... |
| dfoprab2 7476 | Class abstraction for oper... |
| reloprab 7477 | An operation class abstrac... |
| oprabv 7478 | If a pair and a class are ... |
| nfoprab1 7479 | The abstraction variables ... |
| nfoprab2 7480 | The abstraction variables ... |
| nfoprab3 7481 | The abstraction variables ... |
| nfoprab 7482 | Bound-variable hypothesis ... |
| oprabbid 7483 | Equivalent wff's yield equ... |
| oprabbidv 7484 | Equivalent wff's yield equ... |
| oprabbii 7485 | Equivalent wff's yield equ... |
| ssoprab2 7486 | Equivalence of ordered pai... |
| ssoprab2b 7487 | Equivalence of ordered pai... |
| eqoprab2bw 7488 | Equivalence of ordered pai... |
| eqoprab2b 7489 | Equivalence of ordered pai... |
| mpoeq123 7490 | An equality theorem for th... |
| mpoeq12 7491 | An equality theorem for th... |
| mpoeq123dva 7492 | An equality deduction for ... |
| mpoeq123dv 7493 | An equality deduction for ... |
| mpoeq123i 7494 | An equality inference for ... |
| mpoeq3dva 7495 | Slightly more general equa... |
| mpoeq3ia 7496 | An equality inference for ... |
| mpoeq3dv 7497 | An equality deduction for ... |
| nfmpo1 7498 | Bound-variable hypothesis ... |
| nfmpo2 7499 | Bound-variable hypothesis ... |
| nfmpo 7500 | Bound-variable hypothesis ... |
| 0mpo0 7501 | A mapping operation with e... |
| mpo0v 7502 | A mapping operation with e... |
| mpo0 7503 | A mapping operation with e... |
| oprab4 7504 | Two ways to state the doma... |
| cbvoprab1 7505 | Rule used to change first ... |
| cbvoprab2 7506 | Change the second bound va... |
| cbvoprab12 7507 | Rule used to change first ... |
| cbvoprab12v 7508 | Rule used to change first ... |
| cbvoprab3 7509 | Rule used to change the th... |
| cbvoprab3v 7510 | Rule used to change the th... |
| cbvmpox 7511 | Rule to change the bound v... |
| cbvmpo 7512 | Rule to change the bound v... |
| cbvmpov 7513 | Rule to change the bound v... |
| elimdelov 7514 | Eliminate a hypothesis whi... |
| brif1 7515 | Move a relation inside and... |
| ovif 7516 | Move a conditional outside... |
| ovif2 7517 | Move a conditional outside... |
| ovif12 7518 | Move a conditional outside... |
| ifov 7519 | Move a conditional outside... |
| dmoprab 7520 | The domain of an operation... |
| dmoprabss 7521 | The domain of an operation... |
| rnoprab 7522 | The range of an operation ... |
| rnoprab2 7523 | The range of a restricted ... |
| reldmoprab 7524 | The domain of an operation... |
| oprabss 7525 | Structure of an operation ... |
| eloprabga 7526 | The law of concretion for ... |
| eloprabgaOLD 7527 | Obsolete version of ~ elop... |
| eloprabg 7528 | The law of concretion for ... |
| ssoprab2i 7529 | Inference of operation cla... |
| mpov 7530 | Operation with universal d... |
| mpomptx 7531 | Express a two-argument fun... |
| mpompt 7532 | Express a two-argument fun... |
| mpodifsnif 7533 | A mapping with two argumen... |
| mposnif 7534 | A mapping with two argumen... |
| fconstmpo 7535 | Representation of a consta... |
| resoprab 7536 | Restriction of an operatio... |
| resoprab2 7537 | Restriction of an operator... |
| resmpo 7538 | Restriction of the mapping... |
| funoprabg 7539 | "At most one" is a suffici... |
| funoprab 7540 | "At most one" is a suffici... |
| fnoprabg 7541 | Functionality and domain o... |
| mpofun 7542 | The maps-to notation for a... |
| mpofunOLD 7543 | Obsolete version of ~ mpof... |
| fnoprab 7544 | Functionality and domain o... |
| ffnov 7545 | An operation maps to a cla... |
| fovcld 7546 | Closure law for an operati... |
| fovcl 7547 | Closure law for an operati... |
| eqfnov 7548 | Equality of two operations... |
| eqfnov2 7549 | Two operators with the sam... |
| fnov 7550 | Representation of a functi... |
| mpo2eqb 7551 | Bidirectional equality the... |
| rnmpo 7552 | The range of an operation ... |
| reldmmpo 7553 | The domain of an operation... |
| elrnmpog 7554 | Membership in the range of... |
| elrnmpo 7555 | Membership in the range of... |
| elimampo 7556 | Membership in the image of... |
| elrnmpores 7557 | Membership in the range of... |
| ralrnmpo 7558 | A restricted quantifier ov... |
| rexrnmpo 7559 | A restricted quantifier ov... |
| ovid 7560 | The value of an operation ... |
| ovidig 7561 | The value of an operation ... |
| ovidi 7562 | The value of an operation ... |
| ov 7563 | The value of an operation ... |
| ovigg 7564 | The value of an operation ... |
| ovig 7565 | The value of an operation ... |
| ovmpt4g 7566 | Value of a function given ... |
| ovmpos 7567 | Value of a function given ... |
| ov2gf 7568 | The value of an operation ... |
| ovmpodxf 7569 | Value of an operation give... |
| ovmpodx 7570 | Value of an operation give... |
| ovmpod 7571 | Value of an operation give... |
| ovmpox 7572 | The value of an operation ... |
| ovmpoga 7573 | Value of an operation give... |
| ovmpoa 7574 | Value of an operation give... |
| ovmpodf 7575 | Alternate deduction versio... |
| ovmpodv 7576 | Alternate deduction versio... |
| ovmpodv2 7577 | Alternate deduction versio... |
| ovmpog 7578 | Value of an operation give... |
| ovmpo 7579 | Value of an operation give... |
| ovmpot 7580 | The value of an operation ... |
| fvmpopr2d 7581 | Value of an operation give... |
| ov3 7582 | The value of an operation ... |
| ov6g 7583 | The value of an operation ... |
| ovg 7584 | The value of an operation ... |
| ovres 7585 | The value of a restricted ... |
| ovresd 7586 | Lemma for converting metri... |
| oprres 7587 | The restriction of an oper... |
| oprssov 7588 | The value of a member of t... |
| fovcdm 7589 | An operation's value belon... |
| fovcdmda 7590 | An operation's value belon... |
| fovcdmd 7591 | An operation's value belon... |
| fnrnov 7592 | The range of an operation ... |
| foov 7593 | An onto mapping of an oper... |
| fnovrn 7594 | An operation's value belon... |
| ovelrn 7595 | A member of an operation's... |
| funimassov 7596 | Membership relation for th... |
| ovelimab 7597 | Operation value in an imag... |
| ovima0 7598 | An operation value is a me... |
| ovconst2 7599 | The value of a constant op... |
| oprssdm 7600 | Domain of closure of an op... |
| nssdmovg 7601 | The value of an operation ... |
| ndmovg 7602 | The value of an operation ... |
| ndmov 7603 | The value of an operation ... |
| ndmovcl 7604 | The closure of an operatio... |
| ndmovrcl 7605 | Reverse closure law, when ... |
| ndmovcom 7606 | Any operation is commutati... |
| ndmovass 7607 | Any operation is associati... |
| ndmovdistr 7608 | Any operation is distribut... |
| ndmovord 7609 | Elimination of redundant a... |
| ndmovordi 7610 | Elimination of redundant a... |
| caovclg 7611 | Convert an operation closu... |
| caovcld 7612 | Convert an operation closu... |
| caovcl 7613 | Convert an operation closu... |
| caovcomg 7614 | Convert an operation commu... |
| caovcomd 7615 | Convert an operation commu... |
| caovcom 7616 | Convert an operation commu... |
| caovassg 7617 | Convert an operation assoc... |
| caovassd 7618 | Convert an operation assoc... |
| caovass 7619 | Convert an operation assoc... |
| caovcang 7620 | Convert an operation cance... |
| caovcand 7621 | Convert an operation cance... |
| caovcanrd 7622 | Commute the arguments of a... |
| caovcan 7623 | Convert an operation cance... |
| caovordig 7624 | Convert an operation order... |
| caovordid 7625 | Convert an operation order... |
| caovordg 7626 | Convert an operation order... |
| caovordd 7627 | Convert an operation order... |
| caovord2d 7628 | Operation ordering law wit... |
| caovord3d 7629 | Ordering law. (Contribute... |
| caovord 7630 | Convert an operation order... |
| caovord2 7631 | Operation ordering law wit... |
| caovord3 7632 | Ordering law. (Contribute... |
| caovdig 7633 | Convert an operation distr... |
| caovdid 7634 | Convert an operation distr... |
| caovdir2d 7635 | Convert an operation distr... |
| caovdirg 7636 | Convert an operation rever... |
| caovdird 7637 | Convert an operation distr... |
| caovdi 7638 | Convert an operation distr... |
| caov32d 7639 | Rearrange arguments in a c... |
| caov12d 7640 | Rearrange arguments in a c... |
| caov31d 7641 | Rearrange arguments in a c... |
| caov13d 7642 | Rearrange arguments in a c... |
| caov4d 7643 | Rearrange arguments in a c... |
| caov411d 7644 | Rearrange arguments in a c... |
| caov42d 7645 | Rearrange arguments in a c... |
| caov32 7646 | Rearrange arguments in a c... |
| caov12 7647 | Rearrange arguments in a c... |
| caov31 7648 | Rearrange arguments in a c... |
| caov13 7649 | Rearrange arguments in a c... |
| caov4 7650 | Rearrange arguments in a c... |
| caov411 7651 | Rearrange arguments in a c... |
| caov42 7652 | Rearrange arguments in a c... |
| caovdir 7653 | Reverse distributive law. ... |
| caovdilem 7654 | Lemma used by real number ... |
| caovlem2 7655 | Lemma used in real number ... |
| caovmo 7656 | Uniqueness of inverse elem... |
| imaeqexov 7657 | Substitute an operation va... |
| imaeqalov 7658 | Substitute an operation va... |
| mpondm0 7659 | The value of an operation ... |
| elmpocl 7660 | If a two-parameter class i... |
| elmpocl1 7661 | If a two-parameter class i... |
| elmpocl2 7662 | If a two-parameter class i... |
| elovmpod 7663 | Utility lemma for two-para... |
| elovmpo 7664 | Utility lemma for two-para... |
| elovmporab 7665 | Implications for the value... |
| elovmporab1w 7666 | Implications for the value... |
| elovmporab1 7667 | Implications for the value... |
| 2mpo0 7668 | If the operation value of ... |
| relmptopab 7669 | Any function to sets of or... |
| f1ocnvd 7670 | Describe an implicit one-t... |
| f1od 7671 | Describe an implicit one-t... |
| f1ocnv2d 7672 | Describe an implicit one-t... |
| f1o2d 7673 | Describe an implicit one-t... |
| f1opw2 7674 | A one-to-one mapping induc... |
| f1opw 7675 | A one-to-one mapping induc... |
| elovmpt3imp 7676 | If the value of a function... |
| ovmpt3rab1 7677 | The value of an operation ... |
| ovmpt3rabdm 7678 | If the value of a function... |
| elovmpt3rab1 7679 | Implications for the value... |
| elovmpt3rab 7680 | Implications for the value... |
| ofeqd 7685 | Equality theorem for funct... |
| ofeq 7686 | Equality theorem for funct... |
| ofreq 7687 | Equality theorem for funct... |
| ofexg 7688 | A function operation restr... |
| nfof 7689 | Hypothesis builder for fun... |
| nfofr 7690 | Hypothesis builder for fun... |
| ofrfvalg 7691 | Value of a relation applie... |
| offval 7692 | Value of an operation appl... |
| ofrfval 7693 | Value of a relation applie... |
| ofval 7694 | Evaluate a function operat... |
| ofrval 7695 | Exhibit a function relatio... |
| offn 7696 | The function operation pro... |
| offun 7697 | The function operation pro... |
| offval2f 7698 | The function operation exp... |
| ofmresval 7699 | Value of a restriction of ... |
| fnfvof 7700 | Function value of a pointw... |
| off 7701 | The function operation pro... |
| ofres 7702 | Restrict the operands of a... |
| offval2 7703 | The function operation exp... |
| ofrfval2 7704 | The function relation acti... |
| ofmpteq 7705 | Value of a pointwise opera... |
| coof 7706 | The composition of a _homo... |
| ofco 7707 | The composition of a funct... |
| offveq 7708 | Convert an identity of the... |
| offveqb 7709 | Equivalent expressions for... |
| ofc1 7710 | Left operation by a consta... |
| ofc2 7711 | Right operation by a const... |
| ofc12 7712 | Function operation on two ... |
| caofref 7713 | Transfer a reflexive law t... |
| caofinvl 7714 | Transfer a left inverse la... |
| caofid0l 7715 | Transfer a left identity l... |
| caofid0r 7716 | Transfer a right identity ... |
| caofid1 7717 | Transfer a right absorptio... |
| caofid2 7718 | Transfer a right absorptio... |
| caofcom 7719 | Transfer a commutative law... |
| caofrss 7720 | Transfer a relation subset... |
| caofass 7721 | Transfer an associative la... |
| caoftrn 7722 | Transfer a transitivity la... |
| caofdi 7723 | Transfer a distributive la... |
| caofdir 7724 | Transfer a reverse distrib... |
| caonncan 7725 | Transfer ~ nncan -shaped l... |
| relrpss 7728 | The proper subset relation... |
| brrpssg 7729 | The proper subset relation... |
| brrpss 7730 | The proper subset relation... |
| porpss 7731 | Every class is partially o... |
| sorpss 7732 | Express strict ordering un... |
| sorpssi 7733 | Property of a chain of set... |
| sorpssun 7734 | A chain of sets is closed ... |
| sorpssin 7735 | A chain of sets is closed ... |
| sorpssuni 7736 | In a chain of sets, a maxi... |
| sorpssint 7737 | In a chain of sets, a mini... |
| sorpsscmpl 7738 | The componentwise compleme... |
| zfun 7740 | Axiom of Union expressed w... |
| axun2 7741 | A variant of the Axiom of ... |
| uniex2 7742 | The Axiom of Union using t... |
| vuniex 7743 | The union of a setvar is a... |
| uniexg 7744 | The ZF Axiom of Union in c... |
| uniex 7745 | The Axiom of Union in clas... |
| uniexd 7746 | Deduction version of the Z... |
| unex 7747 | The union of two sets is a... |
| tpex 7748 | An unordered triple of cla... |
| unexb 7749 | Existence of union is equi... |
| unexg 7750 | A union of two sets is a s... |
| xpexg 7751 | The Cartesian product of t... |
| xpexd 7752 | The Cartesian product of t... |
| 3xpexg 7753 | The Cartesian product of t... |
| xpex 7754 | The Cartesian product of t... |
| unexd 7755 | The union of two sets is a... |
| sqxpexg 7756 | The Cartesian square of a ... |
| abnexg 7757 | Sufficient condition for a... |
| abnex 7758 | Sufficient condition for a... |
| snnex 7759 | The class of all singleton... |
| pwnex 7760 | The class of all power set... |
| difex2 7761 | If the subtrahend of a cla... |
| difsnexi 7762 | If the difference of a cla... |
| uniuni 7763 | Expression for double unio... |
| uniexr 7764 | Converse of the Axiom of U... |
| uniexb 7765 | The Axiom of Union and its... |
| pwexr 7766 | Converse of the Axiom of P... |
| pwexb 7767 | The Axiom of Power Sets an... |
| elpwpwel 7768 | A class belongs to a doubl... |
| eldifpw 7769 | Membership in a power clas... |
| elpwun 7770 | Membership in the power cl... |
| pwuncl 7771 | Power classes are closed u... |
| iunpw 7772 | An indexed union of a powe... |
| fr3nr 7773 | A well-founded relation ha... |
| epne3 7774 | A well-founded class conta... |
| dfwe2 7775 | Alternate definition of we... |
| epweon 7776 | The membership relation we... |
| epweonALT 7777 | Alternate proof of ~ epweo... |
| ordon 7778 | The class of all ordinal n... |
| onprc 7779 | No set contains all ordina... |
| ssorduni 7780 | The union of a class of or... |
| ssonuni 7781 | The union of a set of ordi... |
| ssonunii 7782 | The union of a set of ordi... |
| ordeleqon 7783 | A way to express the ordin... |
| ordsson 7784 | Any ordinal class is a sub... |
| dford5 7785 | A class is ordinal iff it ... |
| onss 7786 | An ordinal number is a sub... |
| predon 7787 | The predecessor of an ordi... |
| predonOLD 7788 | Obsolete version of ~ pred... |
| ssonprc 7789 | Two ways of saying a class... |
| onuni 7790 | The union of an ordinal nu... |
| orduni 7791 | The union of an ordinal cl... |
| onint 7792 | The intersection (infimum)... |
| onint0 7793 | The intersection of a clas... |
| onssmin 7794 | A nonempty class of ordina... |
| onminesb 7795 | If a property is true for ... |
| onminsb 7796 | If a property is true for ... |
| oninton 7797 | The intersection of a none... |
| onintrab 7798 | The intersection of a clas... |
| onintrab2 7799 | An existence condition equ... |
| onnmin 7800 | No member of a set of ordi... |
| onnminsb 7801 | An ordinal number smaller ... |
| oneqmin 7802 | A way to show that an ordi... |
| uniordint 7803 | The union of a set of ordi... |
| onminex 7804 | If a wff is true for an or... |
| sucon 7805 | The class of all ordinal n... |
| sucexb 7806 | A successor exists iff its... |
| sucexg 7807 | The successor of a set is ... |
| sucex 7808 | The successor of a set is ... |
| onmindif2 7809 | The minimum of a class of ... |
| ordsuci 7810 | The successor of an ordina... |
| sucexeloni 7811 | If the successor of an ord... |
| sucexeloniOLD 7812 | Obsolete version of ~ suce... |
| onsuc 7813 | The successor of an ordina... |
| suceloniOLD 7814 | Obsolete version of ~ onsu... |
| ordsuc 7815 | A class is ordinal if and ... |
| ordsucOLD 7816 | Obsolete version of ~ ords... |
| ordpwsuc 7817 | The collection of ordinals... |
| onpwsuc 7818 | The collection of ordinal ... |
| onsucb 7819 | A class is an ordinal numb... |
| ordsucss 7820 | The successor of an elemen... |
| onpsssuc 7821 | An ordinal number is a pro... |
| ordelsuc 7822 | A set belongs to an ordina... |
| onsucmin 7823 | The successor of an ordina... |
| ordsucelsuc 7824 | Membership is inherited by... |
| ordsucsssuc 7825 | The subclass relationship ... |
| ordsucuniel 7826 | Given an element ` A ` of ... |
| ordsucun 7827 | The successor of the maxim... |
| ordunpr 7828 | The maximum of two ordinal... |
| ordunel 7829 | The maximum of two ordinal... |
| onsucuni 7830 | A class of ordinal numbers... |
| ordsucuni 7831 | An ordinal class is a subc... |
| orduniorsuc 7832 | An ordinal class is either... |
| unon 7833 | The class of all ordinal n... |
| ordunisuc 7834 | An ordinal class is equal ... |
| orduniss2 7835 | The union of the ordinal s... |
| onsucuni2 7836 | A successor ordinal is the... |
| 0elsuc 7837 | The successor of an ordina... |
| limon 7838 | The class of ordinal numbe... |
| onuniorsuc 7839 | An ordinal number is eithe... |
| onssi 7840 | An ordinal number is a sub... |
| onsuci 7841 | The successor of an ordina... |
| onuniorsuciOLD 7842 | Obsolete version of ~ onun... |
| onuninsuci 7843 | An ordinal is equal to its... |
| onsucssi 7844 | A set belongs to an ordina... |
| nlimsucg 7845 | A successor is not a limit... |
| orduninsuc 7846 | An ordinal class is equal ... |
| ordunisuc2 7847 | An ordinal equal to its un... |
| ordzsl 7848 | An ordinal is zero, a succ... |
| onzsl 7849 | An ordinal number is zero,... |
| dflim3 7850 | An alternate definition of... |
| dflim4 7851 | An alternate definition of... |
| limsuc 7852 | The successor of a member ... |
| limsssuc 7853 | A class includes a limit o... |
| nlimon 7854 | Two ways to express the cl... |
| limuni3 7855 | The union of a nonempty cl... |
| tfi 7856 | The Principle of Transfini... |
| tfisg 7857 | A closed form of ~ tfis . ... |
| tfis 7858 | Transfinite Induction Sche... |
| tfis2f 7859 | Transfinite Induction Sche... |
| tfis2 7860 | Transfinite Induction Sche... |
| tfis3 7861 | Transfinite Induction Sche... |
| tfisi 7862 | A transfinite induction sc... |
| tfinds 7863 | Principle of Transfinite I... |
| tfindsg 7864 | Transfinite Induction (inf... |
| tfindsg2 7865 | Transfinite Induction (inf... |
| tfindes 7866 | Transfinite Induction with... |
| tfinds2 7867 | Transfinite Induction (inf... |
| tfinds3 7868 | Principle of Transfinite I... |
| dfom2 7871 | An alternate definition of... |
| elom 7872 | Membership in omega. The ... |
| omsson 7873 | Omega is a subset of ` On ... |
| limomss 7874 | The class of natural numbe... |
| nnon 7875 | A natural number is an ord... |
| nnoni 7876 | A natural number is an ord... |
| nnord 7877 | A natural number is ordina... |
| trom 7878 | The class of finite ordina... |
| ordom 7879 | The class of finite ordina... |
| elnn 7880 | A member of a natural numb... |
| omon 7881 | The class of natural numbe... |
| omelon2 7882 | Omega is an ordinal number... |
| nnlim 7883 | A natural number is not a ... |
| omssnlim 7884 | The class of natural numbe... |
| limom 7885 | Omega is a limit ordinal. ... |
| peano2b 7886 | A class belongs to omega i... |
| nnsuc 7887 | A nonzero natural number i... |
| omsucne 7888 | A natural number is not th... |
| ssnlim 7889 | An ordinal subclass of non... |
| omsinds 7890 | Strong (or "total") induct... |
| omsindsOLD 7891 | Obsolete version of ~ omsi... |
| omun 7892 | The union of two finite or... |
| peano1 7893 | Zero is a natural number. ... |
| peano1OLD 7894 | Obsolete version of ~ pean... |
| peano2 7895 | The successor of any natur... |
| peano3 7896 | The successor of any natur... |
| peano4 7897 | Two natural numbers are eq... |
| peano5 7898 | The induction postulate: a... |
| peano5OLD 7899 | Obsolete version of ~ pean... |
| nn0suc 7900 | A natural number is either... |
| find 7901 | The Principle of Finite In... |
| finds 7902 | Principle of Finite Induct... |
| findsg 7903 | Principle of Finite Induct... |
| finds2 7904 | Principle of Finite Induct... |
| finds1 7905 | Principle of Finite Induct... |
| findes 7906 | Finite induction with expl... |
| dmexg 7907 | The domain of a set is a s... |
| rnexg 7908 | The range of a set is a se... |
| dmexd 7909 | The domain of a set is a s... |
| fndmexd 7910 | If a function is a set, it... |
| dmfex 7911 | If a mapping is a set, its... |
| fndmexb 7912 | The domain of a function i... |
| fdmexb 7913 | The domain of a function i... |
| dmfexALT 7914 | Alternate proof of ~ dmfex... |
| dmex 7915 | The domain of a set is a s... |
| rnex 7916 | The range of a set is a se... |
| iprc 7917 | The identity function is a... |
| resiexg 7918 | The existence of a restric... |
| imaexg 7919 | The image of a set is a se... |
| imaex 7920 | The image of a set is a se... |
| rnexd 7921 | The range of a set is a se... |
| imaexd 7922 | The image of a set is a se... |
| exse2 7923 | Any set relation is set-li... |
| xpexr 7924 | If a Cartesian product is ... |
| xpexr2 7925 | If a nonempty Cartesian pr... |
| xpexcnv 7926 | A condition where the conv... |
| soex 7927 | If the relation in a stric... |
| elxp4 7928 | Membership in a Cartesian ... |
| elxp5 7929 | Membership in a Cartesian ... |
| cnvexg 7930 | The converse of a set is a... |
| cnvex 7931 | The converse of a set is a... |
| relcnvexb 7932 | A relation is a set iff it... |
| f1oexrnex 7933 | If the range of a 1-1 onto... |
| f1oexbi 7934 | There is a one-to-one onto... |
| coexg 7935 | The composition of two set... |
| coex 7936 | The composition of two set... |
| coexd 7937 | The composition of two set... |
| funcnvuni 7938 | The union of a chain (with... |
| fun11uni 7939 | The union of a chain (with... |
| fex2 7940 | A function with bounded do... |
| fabexg 7941 | Existence of a set of func... |
| fabex 7942 | Existence of a set of func... |
| f1oabexg 7943 | The class of all 1-1-onto ... |
| fiunlem 7944 | Lemma for ~ fiun and ~ f1i... |
| fiun 7945 | The union of a chain (with... |
| f1iun 7946 | The union of a chain (with... |
| fviunfun 7947 | The function value of an i... |
| ffoss 7948 | Relationship between a map... |
| f11o 7949 | Relationship between one-t... |
| resfunexgALT 7950 | Alternate proof of ~ resfu... |
| cofunexg 7951 | Existence of a composition... |
| cofunex2g 7952 | Existence of a composition... |
| fnexALT 7953 | Alternate proof of ~ fnex ... |
| funexw 7954 | Weak version of ~ funex th... |
| mptexw 7955 | Weak version of ~ mptex th... |
| funrnex 7956 | If the domain of a functio... |
| zfrep6 7957 | A version of the Axiom of ... |
| focdmex 7958 | If the domain of an onto f... |
| f1dmex 7959 | If the codomain of a one-t... |
| f1ovv 7960 | The codomain/range of a 1-... |
| fvclex 7961 | Existence of the class of ... |
| fvresex 7962 | Existence of the class of ... |
| abrexexg 7963 | Existence of a class abstr... |
| abrexexgOLD 7964 | Obsolete version of ~ abre... |
| abrexex 7965 | Existence of a class abstr... |
| iunexg 7966 | The existence of an indexe... |
| abrexex2g 7967 | Existence of an existentia... |
| opabex3d 7968 | Existence of an ordered pa... |
| opabex3rd 7969 | Existence of an ordered pa... |
| opabex3 7970 | Existence of an ordered pa... |
| iunex 7971 | The existence of an indexe... |
| abrexex2 7972 | Existence of an existentia... |
| abexssex 7973 | Existence of a class abstr... |
| abexex 7974 | A condition where a class ... |
| f1oweALT 7975 | Alternate proof of ~ f1owe... |
| wemoiso 7976 | Thus, there is at most one... |
| wemoiso2 7977 | Thus, there is at most one... |
| oprabexd 7978 | Existence of an operator a... |
| oprabex 7979 | Existence of an operation ... |
| oprabex3 7980 | Existence of an operation ... |
| oprabrexex2 7981 | Existence of an existentia... |
| ab2rexex 7982 | Existence of a class abstr... |
| ab2rexex2 7983 | Existence of an existentia... |
| xpexgALT 7984 | Alternate proof of ~ xpexg... |
| offval3 7985 | General value of ` ( F oF ... |
| offres 7986 | Pointwise combination comm... |
| ofmres 7987 | Equivalent expressions for... |
| ofmresex 7988 | Existence of a restriction... |
| mptcnfimad 7989 | The converse of a mapping ... |
| 1stval 7994 | The value of the function ... |
| 2ndval 7995 | The value of the function ... |
| 1stnpr 7996 | Value of the first-member ... |
| 2ndnpr 7997 | Value of the second-member... |
| 1st0 7998 | The value of the first-mem... |
| 2nd0 7999 | The value of the second-me... |
| op1st 8000 | Extract the first member o... |
| op2nd 8001 | Extract the second member ... |
| op1std 8002 | Extract the first member o... |
| op2ndd 8003 | Extract the second member ... |
| op1stg 8004 | Extract the first member o... |
| op2ndg 8005 | Extract the second member ... |
| ot1stg 8006 | Extract the first member o... |
| ot2ndg 8007 | Extract the second member ... |
| ot3rdg 8008 | Extract the third member o... |
| 1stval2 8009 | Alternate value of the fun... |
| 2ndval2 8010 | Alternate value of the fun... |
| oteqimp 8011 | The components of an order... |
| fo1st 8012 | The ` 1st ` function maps ... |
| fo2nd 8013 | The ` 2nd ` function maps ... |
| br1steqg 8014 | Uniqueness condition for t... |
| br2ndeqg 8015 | Uniqueness condition for t... |
| f1stres 8016 | Mapping of a restriction o... |
| f2ndres 8017 | Mapping of a restriction o... |
| fo1stres 8018 | Onto mapping of a restrict... |
| fo2ndres 8019 | Onto mapping of a restrict... |
| 1st2val 8020 | Value of an alternate defi... |
| 2nd2val 8021 | Value of an alternate defi... |
| 1stcof 8022 | Composition of the first m... |
| 2ndcof 8023 | Composition of the second ... |
| xp1st 8024 | Location of the first elem... |
| xp2nd 8025 | Location of the second ele... |
| elxp6 8026 | Membership in a Cartesian ... |
| elxp7 8027 | Membership in a Cartesian ... |
| eqopi 8028 | Equality with an ordered p... |
| xp2 8029 | Representation of Cartesia... |
| unielxp 8030 | The membership relation fo... |
| 1st2nd2 8031 | Reconstruction of a member... |
| 1st2ndb 8032 | Reconstruction of an order... |
| xpopth 8033 | An ordered pair theorem fo... |
| eqop 8034 | Two ways to express equali... |
| eqop2 8035 | Two ways to express equali... |
| op1steq 8036 | Two ways of expressing tha... |
| opreuopreu 8037 | There is a unique ordered ... |
| el2xptp 8038 | A member of a nested Carte... |
| el2xptp0 8039 | A member of a nested Carte... |
| el2xpss 8040 | Version of ~ elrel for tri... |
| 2nd1st 8041 | Swap the members of an ord... |
| 1st2nd 8042 | Reconstruction of a member... |
| 1stdm 8043 | The first ordered pair com... |
| 2ndrn 8044 | The second ordered pair co... |
| 1st2ndbr 8045 | Express an element of a re... |
| releldm2 8046 | Two ways of expressing mem... |
| reldm 8047 | An expression for the doma... |
| releldmdifi 8048 | One way of expressing memb... |
| funfv1st2nd 8049 | The function value for the... |
| funelss 8050 | If the first component of ... |
| funeldmdif 8051 | Two ways of expressing mem... |
| sbcopeq1a 8052 | Equality theorem for subst... |
| csbopeq1a 8053 | Equality theorem for subst... |
| sbcoteq1a 8054 | Equality theorem for subst... |
| dfopab2 8055 | A way to define an ordered... |
| dfoprab3s 8056 | A way to define an operati... |
| dfoprab3 8057 | Operation class abstractio... |
| dfoprab4 8058 | Operation class abstractio... |
| dfoprab4f 8059 | Operation class abstractio... |
| opabex2 8060 | Condition for an operation... |
| opabn1stprc 8061 | An ordered-pair class abst... |
| opiota 8062 | The property of a uniquely... |
| cnvoprab 8063 | The converse of a class ab... |
| dfxp3 8064 | Define the Cartesian produ... |
| elopabi 8065 | A consequence of membershi... |
| eloprabi 8066 | A consequence of membershi... |
| mpomptsx 8067 | Express a two-argument fun... |
| mpompts 8068 | Express a two-argument fun... |
| dmmpossx 8069 | The domain of a mapping is... |
| fmpox 8070 | Functionality, domain and ... |
| fmpo 8071 | Functionality, domain and ... |
| fnmpo 8072 | Functionality and domain o... |
| fnmpoi 8073 | Functionality and domain o... |
| dmmpo 8074 | Domain of a class given by... |
| ovmpoelrn 8075 | An operation's value belon... |
| dmmpoga 8076 | Domain of an operation giv... |
| dmmpog 8077 | Domain of an operation giv... |
| mpoexxg 8078 | Existence of an operation ... |
| mpoexg 8079 | Existence of an operation ... |
| mpoexga 8080 | If the domain of an operat... |
| mpoexw 8081 | Weak version of ~ mpoex th... |
| mpoex 8082 | If the domain of an operat... |
| mptmpoopabbrd 8083 | The operation value of a f... |
| mptmpoopabbrdOLD 8084 | Obsolete version of ~ mptm... |
| mptmpoopabovd 8085 | The operation value of a f... |
| mptmpoopabbrdOLDOLD 8086 | Obsolete version of ~ mptm... |
| mptmpoopabovdOLD 8087 | Obsolete version of ~ mptm... |
| el2mpocsbcl 8088 | If the operation value of ... |
| el2mpocl 8089 | If the operation value of ... |
| fnmpoovd 8090 | A function with a Cartesia... |
| offval22 8091 | The function operation exp... |
| brovpreldm 8092 | If a binary relation holds... |
| bropopvvv 8093 | If a binary relation holds... |
| bropfvvvvlem 8094 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8095 | If a binary relation holds... |
| ovmptss 8096 | If all the values of the m... |
| relmpoopab 8097 | Any function to sets of or... |
| fmpoco 8098 | Composition of two functio... |
| oprabco 8099 | Composition of a function ... |
| oprab2co 8100 | Composition of operator ab... |
| df1st2 8101 | An alternate possible defi... |
| df2nd2 8102 | An alternate possible defi... |
| 1stconst 8103 | The mapping of a restricti... |
| 2ndconst 8104 | The mapping of a restricti... |
| dfmpo 8105 | Alternate definition for t... |
| mposn 8106 | An operation (in maps-to n... |
| curry1 8107 | Composition with ` ``' ( 2... |
| curry1val 8108 | The value of a curried fun... |
| curry1f 8109 | Functionality of a curried... |
| curry2 8110 | Composition with ` ``' ( 1... |
| curry2f 8111 | Functionality of a curried... |
| curry2val 8112 | The value of a curried fun... |
| cnvf1olem 8113 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8114 | Describe a function that m... |
| fparlem1 8115 | Lemma for ~ fpar . (Contr... |
| fparlem2 8116 | Lemma for ~ fpar . (Contr... |
| fparlem3 8117 | Lemma for ~ fpar . (Contr... |
| fparlem4 8118 | Lemma for ~ fpar . (Contr... |
| fpar 8119 | Merge two functions in par... |
| fsplit 8120 | A function that can be use... |
| fsplitfpar 8121 | Merge two functions with a... |
| offsplitfpar 8122 | Express the function opera... |
| f2ndf 8123 | The ` 2nd ` (second compon... |
| fo2ndf 8124 | The ` 2nd ` (second compon... |
| f1o2ndf1 8125 | The ` 2nd ` (second compon... |
| opco1 8126 | Value of an operation prec... |
| opco2 8127 | Value of an operation prec... |
| opco1i 8128 | Inference form of ~ opco1 ... |
| frxp 8129 | A lexicographical ordering... |
| xporderlem 8130 | Lemma for lexicographical ... |
| poxp 8131 | A lexicographical ordering... |
| soxp 8132 | A lexicographical ordering... |
| wexp 8133 | A lexicographical ordering... |
| fnwelem 8134 | Lemma for ~ fnwe . (Contr... |
| fnwe 8135 | A variant on lexicographic... |
| fnse 8136 | Condition for the well-ord... |
| fvproj 8137 | Value of a function on ord... |
| fimaproj 8138 | Image of a cartesian produ... |
| ralxpes 8139 | A version of ~ ralxp with ... |
| ralxp3f 8140 | Restricted for all over a ... |
| ralxp3 8141 | Restricted for all over a ... |
| ralxp3es 8142 | Restricted for-all over a ... |
| frpoins3xpg 8143 | Special case of founded pa... |
| frpoins3xp3g 8144 | Special case of founded pa... |
| xpord2lem 8145 | Lemma for Cartesian produc... |
| poxp2 8146 | Another way of partially o... |
| frxp2 8147 | Another way of giving a we... |
| xpord2pred 8148 | Calculate the predecessor ... |
| sexp2 8149 | Condition for the relation... |
| xpord2indlem 8150 | Induction over the Cartesi... |
| xpord2ind 8151 | Induction over the Cartesi... |
| xpord3lem 8152 | Lemma for triple ordering.... |
| poxp3 8153 | Triple Cartesian product p... |
| frxp3 8154 | Give well-foundedness over... |
| xpord3pred 8155 | Calculate the predecsessor... |
| sexp3 8156 | Show that the triple order... |
| xpord3inddlem 8157 | Induction over the triple ... |
| xpord3indd 8158 | Induction over the triple ... |
| xpord3ind 8159 | Induction over the triple ... |
| orderseqlem 8160 | Lemma for ~ poseq and ~ so... |
| poseq 8161 | A partial ordering of ordi... |
| soseq 8162 | A linear ordering of ordin... |
| suppval 8165 | The value of the operation... |
| supp0prc 8166 | The support of a class is ... |
| suppvalbr 8167 | The value of the operation... |
| supp0 8168 | The support of the empty s... |
| suppval1 8169 | The value of the operation... |
| suppvalfng 8170 | The value of the operation... |
| suppvalfn 8171 | The value of the operation... |
| elsuppfng 8172 | An element of the support ... |
| elsuppfn 8173 | An element of the support ... |
| fvdifsupp 8174 | Function value is zero out... |
| cnvimadfsn 8175 | The support of functions "... |
| suppimacnvss 8176 | The support of functions "... |
| suppimacnv 8177 | Support sets of functions ... |
| fsuppeq 8178 | Two ways of writing the su... |
| fsuppeqg 8179 | Version of ~ fsuppeq avoid... |
| suppssdm 8180 | The support of a function ... |
| suppsnop 8181 | The support of a singleton... |
| snopsuppss 8182 | The support of a singleton... |
| fvn0elsupp 8183 | If the function value for ... |
| fvn0elsuppb 8184 | The function value for a g... |
| rexsupp 8185 | Existential quantification... |
| ressuppss 8186 | The support of the restric... |
| suppun 8187 | The support of a class/fun... |
| ressuppssdif 8188 | The support of the restric... |
| mptsuppdifd 8189 | The support of a function ... |
| mptsuppd 8190 | The support of a function ... |
| extmptsuppeq 8191 | The support of an extended... |
| suppfnss 8192 | The support of a function ... |
| funsssuppss 8193 | The support of a function ... |
| fnsuppres 8194 | Two ways to express restri... |
| fnsuppeq0 8195 | The support of a function ... |
| fczsupp0 8196 | The support of a constant ... |
| suppss 8197 | Show that the support of a... |
| suppssOLD 8198 | Obsolete version of ~ supp... |
| suppssr 8199 | A function is zero outside... |
| suppssrg 8200 | A function is zero outside... |
| suppssov1 8201 | Formula building theorem f... |
| suppssov2 8202 | Formula building theorem f... |
| suppssof1 8203 | Formula building theorem f... |
| suppss2 8204 | Show that the support of a... |
| suppsssn 8205 | Show that the support of a... |
| suppssfv 8206 | Formula building theorem f... |
| suppofssd 8207 | Condition for the support ... |
| suppofss1d 8208 | Condition for the support ... |
| suppofss2d 8209 | Condition for the support ... |
| suppco 8210 | The support of the composi... |
| suppcoss 8211 | The support of the composi... |
| supp0cosupp0 8212 | The support of the composi... |
| imacosupp 8213 | The image of the support o... |
| opeliunxp2f 8214 | Membership in a union of C... |
| mpoxeldm 8215 | If there is an element of ... |
| mpoxneldm 8216 | If the first argument of a... |
| mpoxopn0yelv 8217 | If there is an element of ... |
| mpoxopynvov0g 8218 | If the second argument of ... |
| mpoxopxnop0 8219 | If the first argument of a... |
| mpoxopx0ov0 8220 | If the first argument of a... |
| mpoxopxprcov0 8221 | If the components of the f... |
| mpoxopynvov0 8222 | If the second argument of ... |
| mpoxopoveq 8223 | Value of an operation give... |
| mpoxopovel 8224 | Element of the value of an... |
| mpoxopoveqd 8225 | Value of an operation give... |
| brovex 8226 | A binary relation of the v... |
| brovmpoex 8227 | A binary relation of the v... |
| sprmpod 8228 | The extension of a binary ... |
| tposss 8231 | Subset theorem for transpo... |
| tposeq 8232 | Equality theorem for trans... |
| tposeqd 8233 | Equality theorem for trans... |
| tposssxp 8234 | The transposition is a sub... |
| reltpos 8235 | The transposition is a rel... |
| brtpos2 8236 | Value of the transposition... |
| brtpos0 8237 | The behavior of ` tpos ` w... |
| reldmtpos 8238 | Necessary and sufficient c... |
| brtpos 8239 | The transposition swaps ar... |
| ottpos 8240 | The transposition swaps th... |
| relbrtpos 8241 | The transposition swaps ar... |
| dmtpos 8242 | The domain of ` tpos F ` w... |
| rntpos 8243 | The range of ` tpos F ` wh... |
| tposexg 8244 | The transposition of a set... |
| ovtpos 8245 | The transposition swaps th... |
| tposfun 8246 | The transposition of a fun... |
| dftpos2 8247 | Alternate definition of ` ... |
| dftpos3 8248 | Alternate definition of ` ... |
| dftpos4 8249 | Alternate definition of ` ... |
| tpostpos 8250 | Value of the double transp... |
| tpostpos2 8251 | Value of the double transp... |
| tposfn2 8252 | The domain of a transposit... |
| tposfo2 8253 | Condition for a surjective... |
| tposf2 8254 | The domain and codomain of... |
| tposf12 8255 | Condition for an injective... |
| tposf1o2 8256 | Condition of a bijective t... |
| tposfo 8257 | The domain and codomain/ra... |
| tposf 8258 | The domain and codomain of... |
| tposfn 8259 | Functionality of a transpo... |
| tpos0 8260 | Transposition of the empty... |
| tposco 8261 | Transposition of a composi... |
| tpossym 8262 | Two ways to say a function... |
| tposeqi 8263 | Equality theorem for trans... |
| tposex 8264 | A transposition is a set. ... |
| nftpos 8265 | Hypothesis builder for tra... |
| tposoprab 8266 | Transposition of a class o... |
| tposmpo 8267 | Transposition of a two-arg... |
| tposconst 8268 | The transposition of a con... |
| mpocurryd 8273 | The currying of an operati... |
| mpocurryvald 8274 | The value of a curried ope... |
| fvmpocurryd 8275 | The value of the value of ... |
| pwuninel2 8278 | Direct proof of ~ pwuninel... |
| pwuninel 8279 | The power set of the union... |
| undefval 8280 | Value of the undefined val... |
| undefnel2 8281 | The undefined value genera... |
| undefnel 8282 | The undefined value genera... |
| undefne0 8283 | The undefined value genera... |
| frecseq123 8286 | Equality theorem for the w... |
| nffrecs 8287 | Bound-variable hypothesis ... |
| csbfrecsg 8288 | Move class substitution in... |
| fpr3g 8289 | Functions defined by well-... |
| frrlem1 8290 | Lemma for well-founded rec... |
| frrlem2 8291 | Lemma for well-founded rec... |
| frrlem3 8292 | Lemma for well-founded rec... |
| frrlem4 8293 | Lemma for well-founded rec... |
| frrlem5 8294 | Lemma for well-founded rec... |
| frrlem6 8295 | Lemma for well-founded rec... |
| frrlem7 8296 | Lemma for well-founded rec... |
| frrlem8 8297 | Lemma for well-founded rec... |
| frrlem9 8298 | Lemma for well-founded rec... |
| frrlem10 8299 | Lemma for well-founded rec... |
| frrlem11 8300 | Lemma for well-founded rec... |
| frrlem12 8301 | Lemma for well-founded rec... |
| frrlem13 8302 | Lemma for well-founded rec... |
| frrlem14 8303 | Lemma for well-founded rec... |
| fprlem1 8304 | Lemma for well-founded rec... |
| fprlem2 8305 | Lemma for well-founded rec... |
| fpr2a 8306 | Weak version of ~ fpr2 whi... |
| fpr1 8307 | Law of well-founded recurs... |
| fpr2 8308 | Law of well-founded recurs... |
| fpr3 8309 | Law of well-founded recurs... |
| frrrel 8310 | Show without using the axi... |
| frrdmss 8311 | Show without using the axi... |
| frrdmcl 8312 | Show without using the axi... |
| fprfung 8313 | A "function" defined by we... |
| fprresex 8314 | The restriction of a funct... |
| dfwrecsOLD 8317 | Obsolete definition of the... |
| wrecseq123 8318 | General equality theorem f... |
| wrecseq123OLD 8319 | Obsolete version of ~ wrec... |
| nfwrecs 8320 | Bound-variable hypothesis ... |
| nfwrecsOLD 8321 | Obsolete proof of ~ nfwrec... |
| wrecseq1 8322 | Equality theorem for the w... |
| wrecseq2 8323 | Equality theorem for the w... |
| wrecseq3 8324 | Equality theorem for the w... |
| csbwrecsg 8325 | Move class substitution in... |
| wfr3g 8326 | Functions defined by well-... |
| wfrlem1OLD 8327 | Lemma for well-ordered rec... |
| wfrlem2OLD 8328 | Lemma for well-ordered rec... |
| wfrlem3OLD 8329 | Lemma for well-ordered rec... |
| wfrlem3OLDa 8330 | Lemma for well-ordered rec... |
| wfrlem4OLD 8331 | Lemma for well-ordered rec... |
| wfrlem5OLD 8332 | Lemma for well-ordered rec... |
| wfrrelOLD 8333 | Obsolete proof of ~ wfrrel... |
| wfrdmssOLD 8334 | Obsolete proof of ~ wfrdms... |
| wfrlem8OLD 8335 | Lemma for well-ordered rec... |
| wfrdmclOLD 8336 | Obsolete version of ~ wfrd... |
| wfrlem10OLD 8337 | Lemma for well-ordered rec... |
| wfrfunOLD 8338 | Obsolete proof of ~ wfrfun... |
| wfrlem12OLD 8339 | Lemma for well-ordered rec... |
| wfrlem13OLD 8340 | Lemma for well-ordered rec... |
| wfrlem14OLD 8341 | Lemma for well-ordered rec... |
| wfrlem15OLD 8342 | Lemma for well-ordered rec... |
| wfrlem16OLD 8343 | Lemma for well-ordered rec... |
| wfrlem17OLD 8344 | Without using ~ ax-rep , s... |
| wfr2aOLD 8345 | Obsolete version of ~ wfr2... |
| wfr1OLD 8346 | Obsolete version of ~ wfr1... |
| wfr2OLD 8347 | Obsolete version of ~ wfr2... |
| wfrrel 8348 | The well-ordered recursion... |
| wfrdmss 8349 | The domain of the well-ord... |
| wfrdmcl 8350 | The predecessor class of a... |
| wfrfun 8351 | The "function" generated b... |
| wfrresex 8352 | Show without using the axi... |
| wfr2a 8353 | A weak version of ~ wfr2 w... |
| wfr1 8354 | The Principle of Well-Orde... |
| wfr2 8355 | The Principle of Well-Orde... |
| wfr3 8356 | The principle of Well-Orde... |
| wfr3OLD 8357 | Obsolete form of ~ wfr3 as... |
| iunon 8358 | The indexed union of a set... |
| iinon 8359 | The nonempty indexed inter... |
| onfununi 8360 | A property of functions on... |
| onovuni 8361 | A variant of ~ onfununi fo... |
| onoviun 8362 | A variant of ~ onovuni wit... |
| onnseq 8363 | There are no length ` _om ... |
| dfsmo2 8366 | Alternate definition of a ... |
| issmo 8367 | Conditions for which ` A `... |
| issmo2 8368 | Alternate definition of a ... |
| smoeq 8369 | Equality theorem for stric... |
| smodm 8370 | The domain of a strictly m... |
| smores 8371 | A strictly monotone functi... |
| smores3 8372 | A strictly monotone functi... |
| smores2 8373 | A strictly monotone ordina... |
| smodm2 8374 | The domain of a strictly m... |
| smofvon2 8375 | The function values of a s... |
| iordsmo 8376 | The identity relation rest... |
| smo0 8377 | The null set is a strictly... |
| smofvon 8378 | If ` B ` is a strictly mon... |
| smoel 8379 | If ` x ` is less than ` y ... |
| smoiun 8380 | The value of a strictly mo... |
| smoiso 8381 | If ` F ` is an isomorphism... |
| smoel2 8382 | A strictly monotone ordina... |
| smo11 8383 | A strictly monotone ordina... |
| smoord 8384 | A strictly monotone ordina... |
| smoword 8385 | A strictly monotone ordina... |
| smogt 8386 | A strictly monotone ordina... |
| smocdmdom 8387 | The codomain of a strictly... |
| smoiso2 8388 | The strictly monotone ordi... |
| dfrecs3 8391 | The old definition of tran... |
| dfrecs3OLD 8392 | Obsolete version of ~ dfre... |
| recseq 8393 | Equality theorem for ` rec... |
| nfrecs 8394 | Bound-variable hypothesis ... |
| tfrlem1 8395 | A technical lemma for tran... |
| tfrlem3a 8396 | Lemma for transfinite recu... |
| tfrlem3 8397 | Lemma for transfinite recu... |
| tfrlem4 8398 | Lemma for transfinite recu... |
| tfrlem5 8399 | Lemma for transfinite recu... |
| recsfval 8400 | Lemma for transfinite recu... |
| tfrlem6 8401 | Lemma for transfinite recu... |
| tfrlem7 8402 | Lemma for transfinite recu... |
| tfrlem8 8403 | Lemma for transfinite recu... |
| tfrlem9 8404 | Lemma for transfinite recu... |
| tfrlem9a 8405 | Lemma for transfinite recu... |
| tfrlem10 8406 | Lemma for transfinite recu... |
| tfrlem11 8407 | Lemma for transfinite recu... |
| tfrlem12 8408 | Lemma for transfinite recu... |
| tfrlem13 8409 | Lemma for transfinite recu... |
| tfrlem14 8410 | Lemma for transfinite recu... |
| tfrlem15 8411 | Lemma for transfinite recu... |
| tfrlem16 8412 | Lemma for finite recursion... |
| tfr1a 8413 | A weak version of ~ tfr1 w... |
| tfr2a 8414 | A weak version of ~ tfr2 w... |
| tfr2b 8415 | Without assuming ~ ax-rep ... |
| tfr1 8416 | Principle of Transfinite R... |
| tfr2 8417 | Principle of Transfinite R... |
| tfr3 8418 | Principle of Transfinite R... |
| tfr1ALT 8419 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8420 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8421 | Alternate proof of ~ tfr3 ... |
| recsfnon 8422 | Strong transfinite recursi... |
| recsval 8423 | Strong transfinite recursi... |
| tz7.44lem1 8424 | The ordered pair abstracti... |
| tz7.44-1 8425 | The value of ` F ` at ` (/... |
| tz7.44-2 8426 | The value of ` F ` at a su... |
| tz7.44-3 8427 | The value of ` F ` at a li... |
| rdgeq1 8430 | Equality theorem for the r... |
| rdgeq2 8431 | Equality theorem for the r... |
| rdgeq12 8432 | Equality theorem for the r... |
| nfrdg 8433 | Bound-variable hypothesis ... |
| rdglem1 8434 | Lemma used with the recurs... |
| rdgfun 8435 | The recursive definition g... |
| rdgdmlim 8436 | The domain of the recursiv... |
| rdgfnon 8437 | The recursive definition g... |
| rdgvalg 8438 | Value of the recursive def... |
| rdgval 8439 | Value of the recursive def... |
| rdg0 8440 | The initial value of the r... |
| rdgseg 8441 | The initial segments of th... |
| rdgsucg 8442 | The value of the recursive... |
| rdgsuc 8443 | The value of the recursive... |
| rdglimg 8444 | The value of the recursive... |
| rdglim 8445 | The value of the recursive... |
| rdg0g 8446 | The initial value of the r... |
| rdgsucmptf 8447 | The value of the recursive... |
| rdgsucmptnf 8448 | The value of the recursive... |
| rdgsucmpt2 8449 | This version of ~ rdgsucmp... |
| rdgsucmpt 8450 | The value of the recursive... |
| rdglim2 8451 | The value of the recursive... |
| rdglim2a 8452 | The value of the recursive... |
| rdg0n 8453 | If ` A ` is a proper class... |
| frfnom 8454 | The function generated by ... |
| fr0g 8455 | The initial value resultin... |
| frsuc 8456 | The successor value result... |
| frsucmpt 8457 | The successor value result... |
| frsucmptn 8458 | The value of the finite re... |
| frsucmpt2 8459 | The successor value result... |
| tz7.48lem 8460 | A way of showing an ordina... |
| tz7.48-2 8461 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8462 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8463 | Proposition 7.48(3) of [Ta... |
| tz7.49 8464 | Proposition 7.49 of [Takeu... |
| tz7.49c 8465 | Corollary of Proposition 7... |
| seqomlem0 8468 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8469 | Lemma for ` seqom ` . The... |
| seqomlem2 8470 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8471 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8472 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8473 | Equality theorem for ` seq... |
| fnseqom 8474 | An index-aware recursive d... |
| seqom0g 8475 | Value of an index-aware re... |
| seqomsuc 8476 | Value of an index-aware re... |
| omsucelsucb 8477 | Membership is inherited by... |
| df1o2 8492 | Expanded value of the ordi... |
| df2o3 8493 | Expanded value of the ordi... |
| df2o2 8494 | Expanded value of the ordi... |
| 1oex 8495 | Ordinal 1 is a set. (Cont... |
| 2oex 8496 | ` 2o ` is a set. (Contrib... |
| 1on 8497 | Ordinal 1 is an ordinal nu... |
| 1onOLD 8498 | Obsolete version of ~ 1on ... |
| 2on 8499 | Ordinal 2 is an ordinal nu... |
| 2onOLD 8500 | Obsolete version of ~ 2on ... |
| 2on0 8501 | Ordinal two is not zero. ... |
| ord3 8502 | Ordinal 3 is an ordinal cl... |
| 3on 8503 | Ordinal 3 is an ordinal nu... |
| 4on 8504 | Ordinal 4 is an ordinal nu... |
| 1oexOLD 8505 | Obsolete version of ~ 1oex... |
| 2oexOLD 8506 | Obsolete version of ~ 2oex... |
| 1n0 8507 | Ordinal one is not equal t... |
| nlim1 8508 | 1 is not a limit ordinal. ... |
| nlim2 8509 | 2 is not a limit ordinal. ... |
| xp01disj 8510 | Cartesian products with th... |
| xp01disjl 8511 | Cartesian products with th... |
| ordgt0ge1 8512 | Two ways to express that a... |
| ordge1n0 8513 | An ordinal greater than or... |
| el1o 8514 | Membership in ordinal one.... |
| ord1eln01 8515 | An ordinal that is not 0 o... |
| ord2eln012 8516 | An ordinal that is not 0, ... |
| 1ellim 8517 | A limit ordinal contains 1... |
| 2ellim 8518 | A limit ordinal contains 2... |
| dif1o 8519 | Two ways to say that ` A `... |
| ondif1 8520 | Two ways to say that ` A `... |
| ondif2 8521 | Two ways to say that ` A `... |
| 2oconcl 8522 | Closure of the pair swappi... |
| 0lt1o 8523 | Ordinal zero is less than ... |
| dif20el 8524 | An ordinal greater than on... |
| 0we1 8525 | The empty set is a well-or... |
| brwitnlem 8526 | Lemma for relations which ... |
| fnoa 8527 | Functionality and domain o... |
| fnom 8528 | Functionality and domain o... |
| fnoe 8529 | Functionality and domain o... |
| oav 8530 | Value of ordinal addition.... |
| omv 8531 | Value of ordinal multiplic... |
| oe0lem 8532 | A helper lemma for ~ oe0 a... |
| oev 8533 | Value of ordinal exponenti... |
| oevn0 8534 | Value of ordinal exponenti... |
| oa0 8535 | Addition with zero. Propo... |
| om0 8536 | Ordinal multiplication wit... |
| oe0m 8537 | Value of zero raised to an... |
| om0x 8538 | Ordinal multiplication wit... |
| oe0m0 8539 | Ordinal exponentiation wit... |
| oe0m1 8540 | Ordinal exponentiation wit... |
| oe0 8541 | Ordinal exponentiation wit... |
| oev2 8542 | Alternate value of ordinal... |
| oasuc 8543 | Addition with successor. ... |
| oesuclem 8544 | Lemma for ~ oesuc . (Cont... |
| omsuc 8545 | Multiplication with succes... |
| oesuc 8546 | Ordinal exponentiation wit... |
| onasuc 8547 | Addition with successor. ... |
| onmsuc 8548 | Multiplication with succes... |
| onesuc 8549 | Exponentiation with a succ... |
| oa1suc 8550 | Addition with 1 is same as... |
| oalim 8551 | Ordinal addition with a li... |
| omlim 8552 | Ordinal multiplication wit... |
| oelim 8553 | Ordinal exponentiation wit... |
| oacl 8554 | Closure law for ordinal ad... |
| omcl 8555 | Closure law for ordinal mu... |
| oecl 8556 | Closure law for ordinal ex... |
| oa0r 8557 | Ordinal addition with zero... |
| om0r 8558 | Ordinal multiplication wit... |
| o1p1e2 8559 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8560 | 2 + 2 = 4 for ordinal numb... |
| om1 8561 | Ordinal multiplication wit... |
| om1r 8562 | Ordinal multiplication wit... |
| oe1 8563 | Ordinal exponentiation wit... |
| oe1m 8564 | Ordinal exponentiation wit... |
| oaordi 8565 | Ordering property of ordin... |
| oaord 8566 | Ordering property of ordin... |
| oacan 8567 | Left cancellation law for ... |
| oaword 8568 | Weak ordering property of ... |
| oawordri 8569 | Weak ordering property of ... |
| oaord1 8570 | An ordinal is less than it... |
| oaword1 8571 | An ordinal is less than or... |
| oaword2 8572 | An ordinal is less than or... |
| oawordeulem 8573 | Lemma for ~ oawordex . (C... |
| oawordeu 8574 | Existence theorem for weak... |
| oawordexr 8575 | Existence theorem for weak... |
| oawordex 8576 | Existence theorem for weak... |
| oaordex 8577 | Existence theorem for orde... |
| oa00 8578 | An ordinal sum is zero iff... |
| oalimcl 8579 | The ordinal sum with a lim... |
| oaass 8580 | Ordinal addition is associ... |
| oarec 8581 | Recursive definition of or... |
| oaf1o 8582 | Left addition by a constan... |
| oacomf1olem 8583 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8584 | Define a bijection from ` ... |
| omordi 8585 | Ordering property of ordin... |
| omord2 8586 | Ordering property of ordin... |
| omord 8587 | Ordering property of ordin... |
| omcan 8588 | Left cancellation law for ... |
| omword 8589 | Weak ordering property of ... |
| omwordi 8590 | Weak ordering property of ... |
| omwordri 8591 | Weak ordering property of ... |
| omword1 8592 | An ordinal is less than or... |
| omword2 8593 | An ordinal is less than or... |
| om00 8594 | The product of two ordinal... |
| om00el 8595 | The product of two nonzero... |
| omordlim 8596 | Ordering involving the pro... |
| omlimcl 8597 | The product of any nonzero... |
| odi 8598 | Distributive law for ordin... |
| omass 8599 | Multiplication of ordinal ... |
| oneo 8600 | If an ordinal number is ev... |
| omeulem1 8601 | Lemma for ~ omeu : existen... |
| omeulem2 8602 | Lemma for ~ omeu : uniquen... |
| omopth2 8603 | An ordered pair-like theor... |
| omeu 8604 | The division algorithm for... |
| oen0 8605 | Ordinal exponentiation wit... |
| oeordi 8606 | Ordering law for ordinal e... |
| oeord 8607 | Ordering property of ordin... |
| oecan 8608 | Left cancellation law for ... |
| oeword 8609 | Weak ordering property of ... |
| oewordi 8610 | Weak ordering property of ... |
| oewordri 8611 | Weak ordering property of ... |
| oeworde 8612 | Ordinal exponentiation com... |
| oeordsuc 8613 | Ordering property of ordin... |
| oelim2 8614 | Ordinal exponentiation wit... |
| oeoalem 8615 | Lemma for ~ oeoa . (Contr... |
| oeoa 8616 | Sum of exponents law for o... |
| oeoelem 8617 | Lemma for ~ oeoe . (Contr... |
| oeoe 8618 | Product of exponents law f... |
| oelimcl 8619 | The ordinal exponential wi... |
| oeeulem 8620 | Lemma for ~ oeeu . (Contr... |
| oeeui 8621 | The division algorithm for... |
| oeeu 8622 | The division algorithm for... |
| nna0 8623 | Addition with zero. Theor... |
| nnm0 8624 | Multiplication with zero. ... |
| nnasuc 8625 | Addition with successor. ... |
| nnmsuc 8626 | Multiplication with succes... |
| nnesuc 8627 | Exponentiation with a succ... |
| nna0r 8628 | Addition to zero. Remark ... |
| nnm0r 8629 | Multiplication with zero. ... |
| nnacl 8630 | Closure of addition of nat... |
| nnmcl 8631 | Closure of multiplication ... |
| nnecl 8632 | Closure of exponentiation ... |
| nnacli 8633 | ` _om ` is closed under ad... |
| nnmcli 8634 | ` _om ` is closed under mu... |
| nnarcl 8635 | Reverse closure law for ad... |
| nnacom 8636 | Addition of natural number... |
| nnaordi 8637 | Ordering property of addit... |
| nnaord 8638 | Ordering property of addit... |
| nnaordr 8639 | Ordering property of addit... |
| nnawordi 8640 | Adding to both sides of an... |
| nnaass 8641 | Addition of natural number... |
| nndi 8642 | Distributive law for natur... |
| nnmass 8643 | Multiplication of natural ... |
| nnmsucr 8644 | Multiplication with succes... |
| nnmcom 8645 | Multiplication of natural ... |
| nnaword 8646 | Weak ordering property of ... |
| nnacan 8647 | Cancellation law for addit... |
| nnaword1 8648 | Weak ordering property of ... |
| nnaword2 8649 | Weak ordering property of ... |
| nnmordi 8650 | Ordering property of multi... |
| nnmord 8651 | Ordering property of multi... |
| nnmword 8652 | Weak ordering property of ... |
| nnmcan 8653 | Cancellation law for multi... |
| nnmwordi 8654 | Weak ordering property of ... |
| nnmwordri 8655 | Weak ordering property of ... |
| nnawordex 8656 | Equivalence for weak order... |
| nnaordex 8657 | Equivalence for ordering. ... |
| nnaordex2 8658 | Equivalence for ordering. ... |
| 1onn 8659 | The ordinal 1 is a natural... |
| 1onnALT 8660 | Shorter proof of ~ 1onn us... |
| 2onn 8661 | The ordinal 2 is a natural... |
| 2onnALT 8662 | Shorter proof of ~ 2onn us... |
| 3onn 8663 | The ordinal 3 is a natural... |
| 4onn 8664 | The ordinal 4 is a natural... |
| 1one2o 8665 | Ordinal one is not ordinal... |
| oaabslem 8666 | Lemma for ~ oaabs . (Cont... |
| oaabs 8667 | Ordinal addition absorbs a... |
| oaabs2 8668 | The absorption law ~ oaabs... |
| omabslem 8669 | Lemma for ~ omabs . (Cont... |
| omabs 8670 | Ordinal multiplication is ... |
| nnm1 8671 | Multiply an element of ` _... |
| nnm2 8672 | Multiply an element of ` _... |
| nn2m 8673 | Multiply an element of ` _... |
| nnneo 8674 | If a natural number is eve... |
| nneob 8675 | A natural number is even i... |
| omsmolem 8676 | Lemma for ~ omsmo . (Cont... |
| omsmo 8677 | A strictly monotonic ordin... |
| omopthlem1 8678 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8679 | Lemma for ~ omopthi . (Co... |
| omopthi 8680 | An ordered pair theorem fo... |
| omopth 8681 | An ordered pair theorem fo... |
| nnasmo 8682 | There is at most one left ... |
| eldifsucnn 8683 | Condition for membership i... |
| on2recsfn 8686 | Show that double recursion... |
| on2recsov 8687 | Calculate the value of the... |
| on2ind 8688 | Double induction over ordi... |
| on3ind 8689 | Triple induction over ordi... |
| coflton 8690 | Cofinality theorem for ord... |
| cofon1 8691 | Cofinality theorem for ord... |
| cofon2 8692 | Cofinality theorem for ord... |
| cofonr 8693 | Inverse cofinality law for... |
| naddfn 8694 | Natural addition is a func... |
| naddcllem 8695 | Lemma for ordinal addition... |
| naddcl 8696 | Closure law for natural ad... |
| naddov 8697 | The value of natural addit... |
| naddov2 8698 | Alternate expression for n... |
| naddov3 8699 | Alternate expression for n... |
| naddf 8700 | Function statement for nat... |
| naddcom 8701 | Natural addition commutes.... |
| naddrid 8702 | Ordinal zero is the additi... |
| naddlid 8703 | Ordinal zero is the additi... |
| naddssim 8704 | Ordinal less-than-or-equal... |
| naddelim 8705 | Ordinal less-than is prese... |
| naddel1 8706 | Ordinal less-than is not a... |
| naddel2 8707 | Ordinal less-than is not a... |
| naddss1 8708 | Ordinal less-than-or-equal... |
| naddss2 8709 | Ordinal less-than-or-equal... |
| naddword1 8710 | Weak-ordering principle fo... |
| naddword2 8711 | Weak-ordering principle fo... |
| naddunif 8712 | Uniformity theorem for nat... |
| naddasslem1 8713 | Lemma for ~ naddass . Exp... |
| naddasslem2 8714 | Lemma for ~ naddass . Exp... |
| naddass 8715 | Natural ordinal addition i... |
| nadd32 8716 | Commutative/associative la... |
| nadd4 8717 | Rearragement of terms in a... |
| nadd42 8718 | Rearragement of terms in a... |
| naddel12 8719 | Natural addition to both s... |
| dfer2 8724 | Alternate definition of eq... |
| dfec2 8726 | Alternate definition of ` ... |
| ecexg 8727 | An equivalence class modul... |
| ecexr 8728 | A nonempty equivalence cla... |
| ereq1 8730 | Equality theorem for equiv... |
| ereq2 8731 | Equality theorem for equiv... |
| errel 8732 | An equivalence relation is... |
| erdm 8733 | The domain of an equivalen... |
| ercl 8734 | Elementhood in the field o... |
| ersym 8735 | An equivalence relation is... |
| ercl2 8736 | Elementhood in the field o... |
| ersymb 8737 | An equivalence relation is... |
| ertr 8738 | An equivalence relation is... |
| ertrd 8739 | A transitivity relation fo... |
| ertr2d 8740 | A transitivity relation fo... |
| ertr3d 8741 | A transitivity relation fo... |
| ertr4d 8742 | A transitivity relation fo... |
| erref 8743 | An equivalence relation is... |
| ercnv 8744 | The converse of an equival... |
| errn 8745 | The range and domain of an... |
| erssxp 8746 | An equivalence relation is... |
| erex 8747 | An equivalence relation is... |
| erexb 8748 | An equivalence relation is... |
| iserd 8749 | A reflexive, symmetric, tr... |
| iseri 8750 | A reflexive, symmetric, tr... |
| iseriALT 8751 | Alternate proof of ~ iseri... |
| brdifun 8752 | Evaluate the incomparabili... |
| swoer 8753 | Incomparability under a st... |
| swoord1 8754 | The incomparability equiva... |
| swoord2 8755 | The incomparability equiva... |
| swoso 8756 | If the incomparability rel... |
| eqerlem 8757 | Lemma for ~ eqer . (Contr... |
| eqer 8758 | Equivalence relation invol... |
| ider 8759 | The identity relation is a... |
| 0er 8760 | The empty set is an equiva... |
| eceq1 8761 | Equality theorem for equiv... |
| eceq1d 8762 | Equality theorem for equiv... |
| eceq2 8763 | Equality theorem for equiv... |
| eceq2i 8764 | Equality theorem for the `... |
| eceq2d 8765 | Equality theorem for the `... |
| elecg 8766 | Membership in an equivalen... |
| ecref 8767 | All elements are in their ... |
| elec 8768 | Membership in an equivalen... |
| relelec 8769 | Membership in an equivalen... |
| ecss 8770 | An equivalence class is a ... |
| ecdmn0 8771 | A representative of a none... |
| ereldm 8772 | Equality of equivalence cl... |
| erth 8773 | Basic property of equivale... |
| erth2 8774 | Basic property of equivale... |
| erthi 8775 | Basic property of equivale... |
| erdisj 8776 | Equivalence classes do not... |
| ecidsn 8777 | An equivalence class modul... |
| qseq1 8778 | Equality theorem for quoti... |
| qseq2 8779 | Equality theorem for quoti... |
| qseq2i 8780 | Equality theorem for quoti... |
| qseq1d 8781 | Equality theorem for quoti... |
| qseq2d 8782 | Equality theorem for quoti... |
| qseq12 8783 | Equality theorem for quoti... |
| 0qs 8784 | Quotient set with the empt... |
| elqsg 8785 | Closed form of ~ elqs . (... |
| elqs 8786 | Membership in a quotient s... |
| elqsi 8787 | Membership in a quotient s... |
| elqsecl 8788 | Membership in a quotient s... |
| ecelqsg 8789 | Membership of an equivalen... |
| ecelqsi 8790 | Membership of an equivalen... |
| ecopqsi 8791 | "Closure" law for equivale... |
| qsexg 8792 | A quotient set exists. (C... |
| qsex 8793 | A quotient set exists. (C... |
| uniqs 8794 | The union of a quotient se... |
| qsss 8795 | A quotient set is a set of... |
| uniqs2 8796 | The union of a quotient se... |
| snec 8797 | The singleton of an equiva... |
| ecqs 8798 | Equivalence class in terms... |
| ecid 8799 | A set is equal to its cose... |
| qsid 8800 | A set is equal to its quot... |
| ectocld 8801 | Implicit substitution of c... |
| ectocl 8802 | Implicit substitution of c... |
| elqsn0 8803 | A quotient set does not co... |
| ecelqsdm 8804 | Membership of an equivalen... |
| xpider 8805 | A Cartesian square is an e... |
| iiner 8806 | The intersection of a none... |
| riiner 8807 | The relative intersection ... |
| erinxp 8808 | A restricted equivalence r... |
| ecinxp 8809 | Restrict the relation in a... |
| qsinxp 8810 | Restrict the equivalence r... |
| qsdisj 8811 | Members of a quotient set ... |
| qsdisj2 8812 | A quotient set is a disjoi... |
| qsel 8813 | If an element of a quotien... |
| uniinqs 8814 | Class union distributes ov... |
| qliftlem 8815 | Lemma for theorems about a... |
| qliftrel 8816 | ` F ` , a function lift, i... |
| qliftel 8817 | Elementhood in the relatio... |
| qliftel1 8818 | Elementhood in the relatio... |
| qliftfun 8819 | The function ` F ` is the ... |
| qliftfund 8820 | The function ` F ` is the ... |
| qliftfuns 8821 | The function ` F ` is the ... |
| qliftf 8822 | The domain and codomain of... |
| qliftval 8823 | The value of the function ... |
| ecoptocl 8824 | Implicit substitution of c... |
| 2ecoptocl 8825 | Implicit substitution of c... |
| 3ecoptocl 8826 | Implicit substitution of c... |
| brecop 8827 | Binary relation on a quoti... |
| brecop2 8828 | Binary relation on a quoti... |
| eroveu 8829 | Lemma for ~ erov and ~ ero... |
| erovlem 8830 | Lemma for ~ erov and ~ ero... |
| erov 8831 | The value of an operation ... |
| eroprf 8832 | Functionality of an operat... |
| erov2 8833 | The value of an operation ... |
| eroprf2 8834 | Functionality of an operat... |
| ecopoveq 8835 | This is the first of sever... |
| ecopovsym 8836 | Assuming the operation ` F... |
| ecopovtrn 8837 | Assuming that operation ` ... |
| ecopover 8838 | Assuming that operation ` ... |
| eceqoveq 8839 | Equality of equivalence re... |
| ecovcom 8840 | Lemma used to transfer a c... |
| ecovass 8841 | Lemma used to transfer an ... |
| ecovdi 8842 | Lemma used to transfer a d... |
| mapprc 8847 | When ` A ` is a proper cla... |
| pmex 8848 | The class of all partial f... |
| mapex 8849 | The class of all functions... |
| fnmap 8850 | Set exponentiation has a u... |
| fnpm 8851 | Partial function exponenti... |
| reldmmap 8852 | Set exponentiation is a we... |
| mapvalg 8853 | The value of set exponenti... |
| pmvalg 8854 | The value of the partial m... |
| mapval 8855 | The value of set exponenti... |
| elmapg 8856 | Membership relation for se... |
| elmapd 8857 | Deduction form of ~ elmapg... |
| elmapdd 8858 | Deduction associated with ... |
| mapdm0 8859 | The empty set is the only ... |
| elpmg 8860 | The predicate "is a partia... |
| elpm2g 8861 | The predicate "is a partia... |
| elpm2r 8862 | Sufficient condition for b... |
| elpmi 8863 | A partial function is a fu... |
| pmfun 8864 | A partial function is a fu... |
| elmapex 8865 | Eliminate antecedent for m... |
| elmapi 8866 | A mapping is a function, f... |
| mapfset 8867 | If ` B ` is a set, the val... |
| mapssfset 8868 | The value of the set expon... |
| mapfoss 8869 | The value of the set expon... |
| fsetsspwxp 8870 | The class of all functions... |
| fset0 8871 | The set of functions from ... |
| fsetdmprc0 8872 | The set of functions with ... |
| fsetex 8873 | The set of functions betwe... |
| f1setex 8874 | The set of injections betw... |
| fosetex 8875 | The set of surjections bet... |
| f1osetex 8876 | The set of bijections betw... |
| fsetfcdm 8877 | The class of functions wit... |
| fsetfocdm 8878 | The class of functions wit... |
| fsetprcnex 8879 | The class of all functions... |
| fsetcdmex 8880 | The class of all functions... |
| fsetexb 8881 | The class of all functions... |
| elmapfn 8882 | A mapping is a function wi... |
| elmapfun 8883 | A mapping is always a func... |
| elmapssres 8884 | A restricted mapping is a ... |
| fpmg 8885 | A total function is a part... |
| pmss12g 8886 | Subset relation for the se... |
| pmresg 8887 | Elementhood of a restricte... |
| elmap 8888 | Membership relation for se... |
| mapval2 8889 | Alternate expression for t... |
| elpm 8890 | The predicate "is a partia... |
| elpm2 8891 | The predicate "is a partia... |
| fpm 8892 | A total function is a part... |
| mapsspm 8893 | Set exponentiation is a su... |
| pmsspw 8894 | Partial maps are a subset ... |
| mapsspw 8895 | Set exponentiation is a su... |
| mapfvd 8896 | The value of a function th... |
| elmapresaun 8897 | ~ fresaun transposed to ma... |
| fvmptmap 8898 | Special case of ~ fvmpt fo... |
| map0e 8899 | Set exponentiation with an... |
| map0b 8900 | Set exponentiation with an... |
| map0g 8901 | Set exponentiation is empt... |
| 0map0sn0 8902 | The set of mappings of the... |
| mapsnd 8903 | The value of set exponenti... |
| map0 8904 | Set exponentiation is empt... |
| mapsn 8905 | The value of set exponenti... |
| mapss 8906 | Subset inheritance for set... |
| fdiagfn 8907 | Functionality of the diago... |
| fvdiagfn 8908 | Functionality of the diago... |
| mapsnconst 8909 | Every singleton map is a c... |
| mapsncnv 8910 | Expression for the inverse... |
| mapsnf1o2 8911 | Explicit bijection between... |
| mapsnf1o3 8912 | Explicit bijection in the ... |
| ralxpmap 8913 | Quantification over functi... |
| dfixp 8916 | Eliminate the expression `... |
| ixpsnval 8917 | The value of an infinite C... |
| elixp2 8918 | Membership in an infinite ... |
| fvixp 8919 | Projection of a factor of ... |
| ixpfn 8920 | A nuple is a function. (C... |
| elixp 8921 | Membership in an infinite ... |
| elixpconst 8922 | Membership in an infinite ... |
| ixpconstg 8923 | Infinite Cartesian product... |
| ixpconst 8924 | Infinite Cartesian product... |
| ixpeq1 8925 | Equality theorem for infin... |
| ixpeq1d 8926 | Equality theorem for infin... |
| ss2ixp 8927 | Subclass theorem for infin... |
| ixpeq2 8928 | Equality theorem for infin... |
| ixpeq2dva 8929 | Equality theorem for infin... |
| ixpeq2dv 8930 | Equality theorem for infin... |
| cbvixp 8931 | Change bound variable in a... |
| cbvixpv 8932 | Change bound variable in a... |
| nfixpw 8933 | Bound-variable hypothesis ... |
| nfixp 8934 | Bound-variable hypothesis ... |
| nfixp1 8935 | The index variable in an i... |
| ixpprc 8936 | A cartesian product of pro... |
| ixpf 8937 | A member of an infinite Ca... |
| uniixp 8938 | The union of an infinite C... |
| ixpexg 8939 | The existence of an infini... |
| ixpin 8940 | The intersection of two in... |
| ixpiin 8941 | The indexed intersection o... |
| ixpint 8942 | The intersection of a coll... |
| ixp0x 8943 | An infinite Cartesian prod... |
| ixpssmap2g 8944 | An infinite Cartesian prod... |
| ixpssmapg 8945 | An infinite Cartesian prod... |
| 0elixp 8946 | Membership of the empty se... |
| ixpn0 8947 | The infinite Cartesian pro... |
| ixp0 8948 | The infinite Cartesian pro... |
| ixpssmap 8949 | An infinite Cartesian prod... |
| resixp 8950 | Restriction of an element ... |
| undifixp 8951 | Union of two projections o... |
| mptelixpg 8952 | Condition for an explicit ... |
| resixpfo 8953 | Restriction of elements of... |
| elixpsn 8954 | Membership in a class of s... |
| ixpsnf1o 8955 | A bijection between a clas... |
| mapsnf1o 8956 | A bijection between a set ... |
| boxriin 8957 | A rectangular subset of a ... |
| boxcutc 8958 | The relative complement of... |
| relen 8967 | Equinumerosity is a relati... |
| reldom 8968 | Dominance is a relation. ... |
| relsdom 8969 | Strict dominance is a rela... |
| encv 8970 | If two classes are equinum... |
| breng 8971 | Equinumerosity relation. ... |
| bren 8972 | Equinumerosity relation. ... |
| brenOLD 8973 | Obsolete version of ~ bren... |
| brdom2g 8974 | Dominance relation. This ... |
| brdomg 8975 | Dominance relation. (Cont... |
| brdomgOLD 8976 | Obsolete version of ~ brdo... |
| brdomi 8977 | Dominance relation. (Cont... |
| brdomiOLD 8978 | Obsolete version of ~ brdo... |
| brdom 8979 | Dominance relation. (Cont... |
| domen 8980 | Dominance in terms of equi... |
| domeng 8981 | Dominance in terms of equi... |
| ctex 8982 | A countable set is a set. ... |
| f1oen4g 8983 | The domain and range of a ... |
| f1dom4g 8984 | The domain of a one-to-one... |
| f1oen3g 8985 | The domain and range of a ... |
| f1dom3g 8986 | The domain of a one-to-one... |
| f1oen2g 8987 | The domain and range of a ... |
| f1dom2g 8988 | The domain of a one-to-one... |
| f1dom2gOLD 8989 | Obsolete version of ~ f1do... |
| f1oeng 8990 | The domain and range of a ... |
| f1domg 8991 | The domain of a one-to-one... |
| f1oen 8992 | The domain and range of a ... |
| f1dom 8993 | The domain of a one-to-one... |
| brsdom 8994 | Strict dominance relation,... |
| isfi 8995 | Express " ` A ` is finite"... |
| enssdom 8996 | Equinumerosity implies dom... |
| dfdom2 8997 | Alternate definition of do... |
| endom 8998 | Equinumerosity implies dom... |
| sdomdom 8999 | Strict dominance implies d... |
| sdomnen 9000 | Strict dominance implies n... |
| brdom2 9001 | Dominance in terms of stri... |
| bren2 9002 | Equinumerosity expressed i... |
| enrefg 9003 | Equinumerosity is reflexiv... |
| enref 9004 | Equinumerosity is reflexiv... |
| eqeng 9005 | Equality implies equinumer... |
| domrefg 9006 | Dominance is reflexive. (... |
| en2d 9007 | Equinumerosity inference f... |
| en3d 9008 | Equinumerosity inference f... |
| en2i 9009 | Equinumerosity inference f... |
| en3i 9010 | Equinumerosity inference f... |
| dom2lem 9011 | A mapping (first hypothesi... |
| dom2d 9012 | A mapping (first hypothesi... |
| dom3d 9013 | A mapping (first hypothesi... |
| dom2 9014 | A mapping (first hypothesi... |
| dom3 9015 | A mapping (first hypothesi... |
| idssen 9016 | Equality implies equinumer... |
| domssl 9017 | If ` A ` is a subset of ` ... |
| domssr 9018 | If ` C ` is a superset of ... |
| ssdomg 9019 | A set dominates its subset... |
| ener 9020 | Equinumerosity is an equiv... |
| ensymb 9021 | Symmetry of equinumerosity... |
| ensym 9022 | Symmetry of equinumerosity... |
| ensymi 9023 | Symmetry of equinumerosity... |
| ensymd 9024 | Symmetry of equinumerosity... |
| entr 9025 | Transitivity of equinumero... |
| domtr 9026 | Transitivity of dominance ... |
| entri 9027 | A chained equinumerosity i... |
| entr2i 9028 | A chained equinumerosity i... |
| entr3i 9029 | A chained equinumerosity i... |
| entr4i 9030 | A chained equinumerosity i... |
| endomtr 9031 | Transitivity of equinumero... |
| domentr 9032 | Transitivity of dominance ... |
| f1imaeng 9033 | If a function is one-to-on... |
| f1imaen2g 9034 | If a function is one-to-on... |
| f1imaen 9035 | If a function is one-to-on... |
| en0 9036 | The empty set is equinumer... |
| en0OLD 9037 | Obsolete version of ~ en0 ... |
| en0ALT 9038 | Shorter proof of ~ en0 , d... |
| en0r 9039 | The empty set is equinumer... |
| ensn1 9040 | A singleton is equinumerou... |
| ensn1OLD 9041 | Obsolete version of ~ ensn... |
| ensn1g 9042 | A singleton is equinumerou... |
| enpr1g 9043 | ` { A , A } ` has only one... |
| en1 9044 | A set is equinumerous to o... |
| en1OLD 9045 | Obsolete version of ~ en1 ... |
| en1b 9046 | A set is equinumerous to o... |
| en1bOLD 9047 | Obsolete version of ~ en1b... |
| reuen1 9048 | Two ways to express "exact... |
| euen1 9049 | Two ways to express "exact... |
| euen1b 9050 | Two ways to express " ` A ... |
| en1uniel 9051 | A singleton contains its s... |
| en1unielOLD 9052 | Obsolete version of ~ en1u... |
| 2dom 9053 | A set that dominates ordin... |
| fundmen 9054 | A function is equinumerous... |
| fundmeng 9055 | A function is equinumerous... |
| cnven 9056 | A relational set is equinu... |
| cnvct 9057 | If a set is countable, so ... |
| fndmeng 9058 | A function is equinumerate... |
| mapsnend 9059 | Set exponentiation to a si... |
| mapsnen 9060 | Set exponentiation to a si... |
| snmapen 9061 | Set exponentiation: a sing... |
| snmapen1 9062 | Set exponentiation: a sing... |
| map1 9063 | Set exponentiation: ordina... |
| en2sn 9064 | Two singletons are equinum... |
| en2snOLD 9065 | Obsolete version of ~ en2s... |
| en2snOLDOLD 9066 | Obsolete version of ~ en2s... |
| snfi 9067 | A singleton is finite. (C... |
| fiprc 9068 | The class of finite sets i... |
| unen 9069 | Equinumerosity of union of... |
| enrefnn 9070 | Equinumerosity is reflexiv... |
| en2prd 9071 | Two unordered pairs are eq... |
| enpr2d 9072 | A pair with distinct eleme... |
| enpr2dOLD 9073 | Obsolete version of ~ enpr... |
| ssct 9074 | Any subset of a countable ... |
| ssctOLD 9075 | Obsolete version of ~ ssct... |
| difsnen 9076 | All decrements of a set ar... |
| domdifsn 9077 | Dominance over a set with ... |
| xpsnen 9078 | A set is equinumerous to i... |
| xpsneng 9079 | A set is equinumerous to i... |
| xp1en 9080 | One times a cardinal numbe... |
| endisj 9081 | Any two sets are equinumer... |
| undom 9082 | Dominance law for union. ... |
| undomOLD 9083 | Obsolete version of ~ undo... |
| xpcomf1o 9084 | The canonical bijection fr... |
| xpcomco 9085 | Composition with the bijec... |
| xpcomen 9086 | Commutative law for equinu... |
| xpcomeng 9087 | Commutative law for equinu... |
| xpsnen2g 9088 | A set is equinumerous to i... |
| xpassen 9089 | Associative law for equinu... |
| xpdom2 9090 | Dominance law for Cartesia... |
| xpdom2g 9091 | Dominance law for Cartesia... |
| xpdom1g 9092 | Dominance law for Cartesia... |
| xpdom3 9093 | A set is dominated by its ... |
| xpdom1 9094 | Dominance law for Cartesia... |
| domunsncan 9095 | A singleton cancellation l... |
| omxpenlem 9096 | Lemma for ~ omxpen . (Con... |
| omxpen 9097 | The cardinal and ordinal p... |
| omf1o 9098 | Construct an explicit bije... |
| pw2f1olem 9099 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9100 | The power set of a set is ... |
| pw2eng 9101 | The power set of a set is ... |
| pw2en 9102 | The power set of a set is ... |
| fopwdom 9103 | Covering implies injection... |
| enfixsn 9104 | Given two equipollent sets... |
| sucdom2OLD 9105 | Obsolete version of ~ sucd... |
| sbthlem1 9106 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9107 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9108 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9109 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9110 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9111 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9112 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9113 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9114 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9115 | Lemma for ~ sbth . (Contr... |
| sbth 9116 | Schroeder-Bernstein Theore... |
| sbthb 9117 | Schroeder-Bernstein Theore... |
| sbthcl 9118 | Schroeder-Bernstein Theore... |
| dfsdom2 9119 | Alternate definition of st... |
| brsdom2 9120 | Alternate definition of st... |
| sdomnsym 9121 | Strict dominance is asymme... |
| domnsym 9122 | Theorem 22(i) of [Suppes] ... |
| 0domg 9123 | Any set dominates the empt... |
| 0domgOLD 9124 | Obsolete version of ~ 0dom... |
| dom0 9125 | A set dominated by the emp... |
| dom0OLD 9126 | Obsolete version of ~ dom0... |
| 0sdomg 9127 | A set strictly dominates t... |
| 0sdomgOLD 9128 | Obsolete version of ~ 0sdo... |
| 0dom 9129 | Any set dominates the empt... |
| 0sdom 9130 | A set strictly dominates t... |
| sdom0 9131 | The empty set does not str... |
| sdom0OLD 9132 | Obsolete version of ~ sdom... |
| sdomdomtr 9133 | Transitivity of strict dom... |
| sdomentr 9134 | Transitivity of strict dom... |
| domsdomtr 9135 | Transitivity of dominance ... |
| ensdomtr 9136 | Transitivity of equinumero... |
| sdomirr 9137 | Strict dominance is irrefl... |
| sdomtr 9138 | Strict dominance is transi... |
| sdomn2lp 9139 | Strict dominance has no 2-... |
| enen1 9140 | Equality-like theorem for ... |
| enen2 9141 | Equality-like theorem for ... |
| domen1 9142 | Equality-like theorem for ... |
| domen2 9143 | Equality-like theorem for ... |
| sdomen1 9144 | Equality-like theorem for ... |
| sdomen2 9145 | Equality-like theorem for ... |
| domtriord 9146 | Dominance is trichotomous ... |
| sdomel 9147 | For ordinals, strict domin... |
| sdomdif 9148 | The difference of a set fr... |
| onsdominel 9149 | An ordinal with more eleme... |
| domunsn 9150 | Dominance over a set with ... |
| fodomr 9151 | There exists a mapping fro... |
| pwdom 9152 | Injection of sets implies ... |
| canth2 9153 | Cantor's Theorem. No set ... |
| canth2g 9154 | Cantor's theorem with the ... |
| 2pwuninel 9155 | The power set of the power... |
| 2pwne 9156 | No set equals the power se... |
| disjen 9157 | A stronger form of ~ pwuni... |
| disjenex 9158 | Existence version of ~ dis... |
| domss2 9159 | A corollary of ~ disjenex ... |
| domssex2 9160 | A corollary of ~ disjenex ... |
| domssex 9161 | Weakening of ~ domssex2 to... |
| xpf1o 9162 | Construct a bijection on a... |
| xpen 9163 | Equinumerosity law for Car... |
| mapen 9164 | Two set exponentiations ar... |
| mapdom1 9165 | Order-preserving property ... |
| mapxpen 9166 | Equinumerosity law for dou... |
| xpmapenlem 9167 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9168 | Equinumerosity law for set... |
| mapunen 9169 | Equinumerosity law for set... |
| map2xp 9170 | A cardinal power with expo... |
| mapdom2 9171 | Order-preserving property ... |
| mapdom3 9172 | Set exponentiation dominat... |
| pwen 9173 | If two sets are equinumero... |
| ssenen 9174 | Equinumerosity of equinume... |
| limenpsi 9175 | A limit ordinal is equinum... |
| limensuci 9176 | A limit ordinal is equinum... |
| limensuc 9177 | A limit ordinal is equinum... |
| infensuc 9178 | Any infinite ordinal is eq... |
| dif1enlem 9179 | Lemma for ~ rexdif1en and ... |
| dif1enlemOLD 9180 | Obsolete version of ~ dif1... |
| rexdif1en 9181 | If a set is equinumerous t... |
| rexdif1enOLD 9182 | Obsolete version of ~ rexd... |
| dif1en 9183 | If a set ` A ` is equinume... |
| dif1ennn 9184 | If a set ` A ` is equinume... |
| dif1enOLD 9185 | Obsolete version of ~ dif1... |
| findcard 9186 | Schema for induction on th... |
| findcard2 9187 | Schema for induction on th... |
| findcard2s 9188 | Variation of ~ findcard2 r... |
| findcard2d 9189 | Deduction version of ~ fin... |
| nnfi 9190 | Natural numbers are finite... |
| pssnn 9191 | A proper subset of a natur... |
| ssnnfi 9192 | A subset of a natural numb... |
| ssnnfiOLD 9193 | Obsolete version of ~ ssnn... |
| 0fin 9194 | The empty set is finite. ... |
| unfi 9195 | The union of two finite se... |
| ssfi 9196 | A subset of a finite set i... |
| ssfiALT 9197 | Shorter proof of ~ ssfi us... |
| imafi 9198 | Images of finite sets are ... |
| pwfir 9199 | If the power set of a set ... |
| pwfilem 9200 | Lemma for ~ pwfi . (Contr... |
| pwfi 9201 | The power set of a finite ... |
| diffi 9202 | If ` A ` is finite, ` ( A ... |
| cnvfi 9203 | If a set is finite, its co... |
| fnfi 9204 | A version of ~ fnex for fi... |
| f1oenfi 9205 | If the domain of a one-to-... |
| f1oenfirn 9206 | If the range of a one-to-o... |
| f1domfi 9207 | If the codomain of a one-t... |
| f1domfi2 9208 | If the domain of a one-to-... |
| enreffi 9209 | Equinumerosity is reflexiv... |
| ensymfib 9210 | Symmetry of equinumerosity... |
| entrfil 9211 | Transitivity of equinumero... |
| enfii 9212 | A set equinumerous to a fi... |
| enfi 9213 | Equinumerous sets have the... |
| enfiALT 9214 | Shorter proof of ~ enfi us... |
| domfi 9215 | A set dominated by a finit... |
| entrfi 9216 | Transitivity of equinumero... |
| entrfir 9217 | Transitivity of equinumero... |
| domtrfil 9218 | Transitivity of dominance ... |
| domtrfi 9219 | Transitivity of dominance ... |
| domtrfir 9220 | Transitivity of dominance ... |
| f1imaenfi 9221 | If a function is one-to-on... |
| ssdomfi 9222 | A finite set dominates its... |
| ssdomfi2 9223 | A set dominates its finite... |
| sbthfilem 9224 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9225 | Schroeder-Bernstein Theore... |
| domnsymfi 9226 | If a set dominates a finit... |
| sdomdomtrfi 9227 | Transitivity of strict dom... |
| domsdomtrfi 9228 | Transitivity of dominance ... |
| sucdom2 9229 | Strict dominance of a set ... |
| phplem1 9230 | Lemma for Pigeonhole Princ... |
| phplem2 9231 | Lemma for Pigeonhole Princ... |
| nneneq 9232 | Two equinumerous natural n... |
| php 9233 | Pigeonhole Principle. A n... |
| php2 9234 | Corollary of Pigeonhole Pr... |
| php3 9235 | Corollary of Pigeonhole Pr... |
| php4 9236 | Corollary of the Pigeonhol... |
| php5 9237 | Corollary of the Pigeonhol... |
| phpeqd 9238 | Corollary of the Pigeonhol... |
| nndomog 9239 | Cardinal ordering agrees w... |
| phplem1OLD 9240 | Obsolete lemma for ~ php a... |
| phplem2OLD 9241 | Obsolete lemma for ~ php a... |
| phplem3OLD 9242 | Obsolete version of ~ phpl... |
| phplem4OLD 9243 | Obsolete version of ~ phpl... |
| nneneqOLD 9244 | Obsolete version of ~ nnen... |
| phpOLD 9245 | Obsolete version of ~ php ... |
| php2OLD 9246 | Obsolete version of ~ php2... |
| php3OLD 9247 | Obsolete version of ~ php3... |
| phpeqdOLD 9248 | Obsolete version of ~ phpe... |
| nndomogOLD 9249 | Obsolete version of ~ nndo... |
| snnen2oOLD 9250 | Obsolete version of ~ snne... |
| onomeneq 9251 | An ordinal number equinume... |
| onomeneqOLD 9252 | Obsolete version of ~ onom... |
| onfin 9253 | An ordinal number is finit... |
| onfin2 9254 | A set is a natural number ... |
| nnfiOLD 9255 | Obsolete version of ~ nnfi... |
| nndomo 9256 | Cardinal ordering agrees w... |
| nnsdomo 9257 | Cardinal ordering agrees w... |
| sucdom 9258 | Strict dominance of a set ... |
| sucdomOLD 9259 | Obsolete version of ~ sucd... |
| snnen2o 9260 | A singleton ` { A } ` is n... |
| 0sdom1dom 9261 | Strict dominance over 0 is... |
| 0sdom1domALT 9262 | Alternate proof of ~ 0sdom... |
| 1sdom2 9263 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9264 | Alternate proof of ~ 1sdom... |
| sdom1 9265 | A set has less than one me... |
| sdom1OLD 9266 | Obsolete version of ~ sdom... |
| modom 9267 | Two ways to express "at mo... |
| modom2 9268 | Two ways to express "at mo... |
| rex2dom 9269 | A set that has at least 2 ... |
| 1sdom2dom 9270 | Strict dominance over 1 is... |
| 1sdom 9271 | A set that strictly domina... |
| 1sdomOLD 9272 | Obsolete version of ~ 1sdo... |
| unxpdomlem1 9273 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9274 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9275 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9276 | Cartesian product dominate... |
| unxpdom2 9277 | Corollary of ~ unxpdom . ... |
| sucxpdom 9278 | Cartesian product dominate... |
| pssinf 9279 | A set equinumerous to a pr... |
| fisseneq 9280 | A finite set is equal to i... |
| ominf 9281 | The set of natural numbers... |
| ominfOLD 9282 | Obsolete version of ~ omin... |
| isinf 9283 | Any set that is not finite... |
| isinfOLD 9284 | Obsolete version of ~ isin... |
| fineqvlem 9285 | Lemma for ~ fineqv . (Con... |
| fineqv 9286 | If the Axiom of Infinity i... |
| enfiiOLD 9287 | Obsolete version of ~ enfi... |
| pssnnOLD 9288 | Obsolete version of ~ pssn... |
| xpfir 9289 | The components of a nonemp... |
| ssfid 9290 | A subset of a finite set i... |
| infi 9291 | The intersection of two se... |
| rabfi 9292 | A restricted class built f... |
| finresfin 9293 | The restriction of a finit... |
| f1finf1o 9294 | Any injection from one fin... |
| f1finf1oOLD 9295 | Obsolete version of ~ f1fi... |
| nfielex 9296 | If a class is not finite, ... |
| en1eqsn 9297 | A set with one element is ... |
| en1eqsnOLD 9298 | Obsolete version of ~ en1e... |
| en1eqsnbi 9299 | A set containing an elemen... |
| dif1ennnALT 9300 | Alternate proof of ~ dif1e... |
| enp1ilem 9301 | Lemma for uses of ~ enp1i ... |
| enp1i 9302 | Proof induction for ~ en2 ... |
| enp1iOLD 9303 | Obsolete version of ~ enp1... |
| en2 9304 | A set equinumerous to ordi... |
| en3 9305 | A set equinumerous to ordi... |
| en4 9306 | A set equinumerous to ordi... |
| findcard2OLD 9307 | Obsolete version of ~ find... |
| findcard3 9308 | Schema for strong inductio... |
| findcard3OLD 9309 | Obsolete version of ~ find... |
| ac6sfi 9310 | A version of ~ ac6s for fi... |
| frfi 9311 | A partial order is well-fo... |
| fimax2g 9312 | A finite set has a maximum... |
| fimaxg 9313 | A finite set has a maximum... |
| fisupg 9314 | Lemma showing existence an... |
| wofi 9315 | A total order on a finite ... |
| ordunifi 9316 | The maximum of a finite co... |
| nnunifi 9317 | The union (supremum) of a ... |
| unblem1 9318 | Lemma for ~ unbnn . After... |
| unblem2 9319 | Lemma for ~ unbnn . The v... |
| unblem3 9320 | Lemma for ~ unbnn . The v... |
| unblem4 9321 | Lemma for ~ unbnn . The f... |
| unbnn 9322 | Any unbounded subset of na... |
| unbnn2 9323 | Version of ~ unbnn that do... |
| isfinite2 9324 | Any set strictly dominated... |
| nnsdomg 9325 | Omega strictly dominates a... |
| nnsdomgOLD 9326 | Obsolete version of ~ nnsd... |
| isfiniteg 9327 | A set is finite iff it is ... |
| infsdomnn 9328 | An infinite set strictly d... |
| infsdomnnOLD 9329 | Obsolete version of ~ infs... |
| infn0 9330 | An infinite set is not emp... |
| infn0ALT 9331 | Shorter proof of ~ infn0 u... |
| fin2inf 9332 | This (useless) theorem, wh... |
| unfilem1 9333 | Lemma for proving that the... |
| unfilem2 9334 | Lemma for proving that the... |
| unfilem3 9335 | Lemma for proving that the... |
| unfiOLD 9336 | Obsolete version of ~ unfi... |
| unfir 9337 | If a union is finite, the ... |
| unfib 9338 | A union is finite if and o... |
| unfi2 9339 | The union of two finite se... |
| difinf 9340 | An infinite set ` A ` minu... |
| xpfi 9341 | The Cartesian product of t... |
| xpfiOLD 9342 | Obsolete version of ~ xpfi... |
| 3xpfi 9343 | The Cartesian product of t... |
| domunfican 9344 | A finite set union cancell... |
| infcntss 9345 | Every infinite set has a d... |
| prfi 9346 | An unordered pair is finit... |
| tpfi 9347 | An unordered triple is fin... |
| fiint 9348 | Equivalent ways of stating... |
| fodomfi 9349 | An onto function implies d... |
| fodomfib 9350 | Equivalence of an onto map... |
| fofinf1o 9351 | Any surjection from one fi... |
| rneqdmfinf1o 9352 | Any function from a finite... |
| fidomdm 9353 | Any finite set dominates i... |
| dmfi 9354 | The domain of a finite set... |
| fundmfibi 9355 | A function is finite if an... |
| resfnfinfin 9356 | The restriction of a funct... |
| residfi 9357 | A restricted identity func... |
| cnvfiALT 9358 | Shorter proof of ~ cnvfi u... |
| rnfi 9359 | The range of a finite set ... |
| f1dmvrnfibi 9360 | A one-to-one function whos... |
| f1vrnfibi 9361 | A one-to-one function whic... |
| fofi 9362 | If an onto function has a ... |
| f1fi 9363 | If a 1-to-1 function has a... |
| iunfi 9364 | The finite union of finite... |
| unifi 9365 | The finite union of finite... |
| unifi2 9366 | The finite union of finite... |
| infssuni 9367 | If an infinite set ` A ` i... |
| unirnffid 9368 | The union of the range of ... |
| imafiALT 9369 | Shorter proof of ~ imafi u... |
| pwfilemOLD 9370 | Obsolete version of ~ pwfi... |
| pwfiOLD 9371 | Obsolete version of ~ pwfi... |
| mapfi 9372 | Set exponentiation of fini... |
| ixpfi 9373 | A Cartesian product of fin... |
| ixpfi2 9374 | A Cartesian product of fin... |
| mptfi 9375 | A finite mapping set is fi... |
| abrexfi 9376 | An image set from a finite... |
| cnvimamptfin 9377 | A preimage of a mapping wi... |
| elfpw 9378 | Membership in a class of f... |
| unifpw 9379 | A set is the union of its ... |
| f1opwfi 9380 | A one-to-one mapping induc... |
| fissuni 9381 | A finite subset of a union... |
| fipreima 9382 | Given a finite subset ` A ... |
| finsschain 9383 | A finite subset of the uni... |
| indexfi 9384 | If for every element of a ... |
| relfsupp 9387 | The property of a function... |
| relprcnfsupp 9388 | A proper class is never fi... |
| isfsupp 9389 | The property of a class to... |
| isfsuppd 9390 | Deduction form of ~ isfsup... |
| funisfsupp 9391 | The property of a function... |
| fsuppimp 9392 | Implications of a class be... |
| fsuppimpd 9393 | A finitely supported funct... |
| fsuppfund 9394 | A finitely supported funct... |
| fisuppfi 9395 | A function on a finite set... |
| fidmfisupp 9396 | A function with a finite d... |
| fdmfisuppfi 9397 | The support of a function ... |
| fdmfifsupp 9398 | A function with a finite d... |
| fsuppmptdm 9399 | A mapping with a finite do... |
| fndmfisuppfi 9400 | The support of a function ... |
| fndmfifsupp 9401 | A function with a finite d... |
| suppeqfsuppbi 9402 | If two functions have the ... |
| suppssfifsupp 9403 | If the support of a functi... |
| fsuppsssupp 9404 | If the support of a functi... |
| fsuppsssuppgd 9405 | If the support of a functi... |
| fsuppss 9406 | A subset of a finitely sup... |
| fsuppssov1 9407 | Formula building theorem f... |
| fsuppxpfi 9408 | The cartesian product of t... |
| fczfsuppd 9409 | A constant function with v... |
| fsuppun 9410 | The union of two finitely ... |
| fsuppunfi 9411 | The union of the support o... |
| fsuppunbi 9412 | If the union of two classe... |
| 0fsupp 9413 | The empty set is a finitel... |
| snopfsupp 9414 | A singleton containing an ... |
| funsnfsupp 9415 | Finite support for a funct... |
| fsuppres 9416 | The restriction of a finit... |
| fmptssfisupp 9417 | The restriction of a mappi... |
| ressuppfi 9418 | If the support of the rest... |
| resfsupp 9419 | If the restriction of a fu... |
| resfifsupp 9420 | The restriction of a funct... |
| ffsuppbi 9421 | Two ways of saying that a ... |
| fsuppmptif 9422 | A function mapping an argu... |
| sniffsupp 9423 | A function mapping all but... |
| fsuppcolem 9424 | Lemma for ~ fsuppco . For... |
| fsuppco 9425 | The composition of a 1-1 f... |
| fsuppco2 9426 | The composition of a funct... |
| fsuppcor 9427 | The composition of a funct... |
| mapfienlem1 9428 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9429 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9430 | Lemma 3 for ~ mapfien . (... |
| mapfien 9431 | A bijection of the base se... |
| mapfien2 9432 | Equinumerousity relation f... |
| fival 9435 | The set of all the finite ... |
| elfi 9436 | Specific properties of an ... |
| elfi2 9437 | The empty intersection nee... |
| elfir 9438 | Sufficient condition for a... |
| intrnfi 9439 | Sufficient condition for t... |
| iinfi 9440 | An indexed intersection of... |
| inelfi 9441 | The intersection of two se... |
| ssfii 9442 | Any element of a set ` A `... |
| fi0 9443 | The set of finite intersec... |
| fieq0 9444 | A set is empty iff the cla... |
| fiin 9445 | The elements of ` ( fi `` ... |
| dffi2 9446 | The set of finite intersec... |
| fiss 9447 | Subset relationship for fu... |
| inficl 9448 | A set which is closed unde... |
| fipwuni 9449 | The set of finite intersec... |
| fisn 9450 | A singleton is closed unde... |
| fiuni 9451 | The union of the finite in... |
| fipwss 9452 | If a set is a family of su... |
| elfiun 9453 | A finite intersection of e... |
| dffi3 9454 | The set of finite intersec... |
| fifo 9455 | Describe a surjection from... |
| marypha1lem 9456 | Core induction for Philip ... |
| marypha1 9457 | (Philip) Hall's marriage t... |
| marypha2lem1 9458 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9459 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9460 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9461 | Lemma for ~ marypha2 . Pr... |
| marypha2 9462 | Version of ~ marypha1 usin... |
| dfsup2 9467 | Quantifier-free definition... |
| supeq1 9468 | Equality theorem for supre... |
| supeq1d 9469 | Equality deduction for sup... |
| supeq1i 9470 | Equality inference for sup... |
| supeq2 9471 | Equality theorem for supre... |
| supeq3 9472 | Equality theorem for supre... |
| supeq123d 9473 | Equality deduction for sup... |
| nfsup 9474 | Hypothesis builder for sup... |
| supmo 9475 | Any class ` B ` has at mos... |
| supexd 9476 | A supremum is a set. (Con... |
| supeu 9477 | A supremum is unique. Sim... |
| supval2 9478 | Alternate expression for t... |
| eqsup 9479 | Sufficient condition for a... |
| eqsupd 9480 | Sufficient condition for a... |
| supcl 9481 | A supremum belongs to its ... |
| supub 9482 | A supremum is an upper bou... |
| suplub 9483 | A supremum is the least up... |
| suplub2 9484 | Bidirectional form of ~ su... |
| supnub 9485 | An upper bound is not less... |
| supex 9486 | A supremum is a set. (Con... |
| sup00 9487 | The supremum under an empt... |
| sup0riota 9488 | The supremum of an empty s... |
| sup0 9489 | The supremum of an empty s... |
| supmax 9490 | The greatest element of a ... |
| fisup2g 9491 | A finite set satisfies the... |
| fisupcl 9492 | A nonempty finite set cont... |
| supgtoreq 9493 | The supremum of a finite s... |
| suppr 9494 | The supremum of a pair. (... |
| supsn 9495 | The supremum of a singleto... |
| supisolem 9496 | Lemma for ~ supiso . (Con... |
| supisoex 9497 | Lemma for ~ supiso . (Con... |
| supiso 9498 | Image of a supremum under ... |
| infeq1 9499 | Equality theorem for infim... |
| infeq1d 9500 | Equality deduction for inf... |
| infeq1i 9501 | Equality inference for inf... |
| infeq2 9502 | Equality theorem for infim... |
| infeq3 9503 | Equality theorem for infim... |
| infeq123d 9504 | Equality deduction for inf... |
| nfinf 9505 | Hypothesis builder for inf... |
| infexd 9506 | An infimum is a set. (Con... |
| eqinf 9507 | Sufficient condition for a... |
| eqinfd 9508 | Sufficient condition for a... |
| infval 9509 | Alternate expression for t... |
| infcllem 9510 | Lemma for ~ infcl , ~ infl... |
| infcl 9511 | An infimum belongs to its ... |
| inflb 9512 | An infimum is a lower boun... |
| infglb 9513 | An infimum is the greatest... |
| infglbb 9514 | Bidirectional form of ~ in... |
| infnlb 9515 | A lower bound is not great... |
| infex 9516 | An infimum is a set. (Con... |
| infmin 9517 | The smallest element of a ... |
| infmo 9518 | Any class ` B ` has at mos... |
| infeu 9519 | An infimum is unique. (Co... |
| fimin2g 9520 | A finite set has a minimum... |
| fiming 9521 | A finite set has a minimum... |
| fiinfg 9522 | Lemma showing existence an... |
| fiinf2g 9523 | A finite set satisfies the... |
| fiinfcl 9524 | A nonempty finite set cont... |
| infltoreq 9525 | The infimum of a finite se... |
| infpr 9526 | The infimum of a pair. (C... |
| infsupprpr 9527 | The infimum of a proper pa... |
| infsn 9528 | The infimum of a singleton... |
| inf00 9529 | The infimum regarding an e... |
| infempty 9530 | The infimum of an empty se... |
| infiso 9531 | Image of an infimum under ... |
| dfoi 9534 | Rewrite ~ df-oi with abbre... |
| oieq1 9535 | Equality theorem for ordin... |
| oieq2 9536 | Equality theorem for ordin... |
| nfoi 9537 | Hypothesis builder for ord... |
| ordiso2 9538 | Generalize ~ ordiso to pro... |
| ordiso 9539 | Order-isomorphic ordinal n... |
| ordtypecbv 9540 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9541 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9542 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9543 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9544 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9545 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9546 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9547 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9548 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9549 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9550 | Lemma for ~ ordtype . Usi... |
| oi0 9551 | Definition of the ordinal ... |
| oicl 9552 | The order type of the well... |
| oif 9553 | The order isomorphism of t... |
| oiiso2 9554 | The order isomorphism of t... |
| ordtype 9555 | For any set-like well-orde... |
| oiiniseg 9556 | ` ran F ` is an initial se... |
| ordtype2 9557 | For any set-like well-orde... |
| oiexg 9558 | The order isomorphism on a... |
| oion 9559 | The order type of the well... |
| oiiso 9560 | The order isomorphism of t... |
| oien 9561 | The order type of a well-o... |
| oieu 9562 | Uniqueness of the unique o... |
| oismo 9563 | When ` A ` is a subclass o... |
| oiid 9564 | The order type of an ordin... |
| hartogslem1 9565 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9566 | Lemma for ~ hartogs . (Co... |
| hartogs 9567 | The class of ordinals domi... |
| wofib 9568 | The only sets which are we... |
| wemaplem1 9569 | Value of the lexicographic... |
| wemaplem2 9570 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9571 | Lemma for ~ wemapso . Tra... |
| wemappo 9572 | Construct lexicographic or... |
| wemapsolem 9573 | Lemma for ~ wemapso . (Co... |
| wemapso 9574 | Construct lexicographic or... |
| wemapso2lem 9575 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9576 | An alternative to having a... |
| card2on 9577 | The alternate definition o... |
| card2inf 9578 | The alternate definition o... |
| harf 9581 | Functionality of the Harto... |
| harcl 9582 | Values of the Hartogs func... |
| harval 9583 | Function value of the Hart... |
| elharval 9584 | The Hartogs number of a se... |
| harndom 9585 | The Hartogs number of a se... |
| harword 9586 | Weak ordering property of ... |
| relwdom 9589 | Weak dominance is a relati... |
| brwdom 9590 | Property of weak dominance... |
| brwdomi 9591 | Property of weak dominance... |
| brwdomn0 9592 | Weak dominance over nonemp... |
| 0wdom 9593 | Any set weakly dominates t... |
| fowdom 9594 | An onto function implies w... |
| wdomref 9595 | Reflexivity of weak domina... |
| brwdom2 9596 | Alternate characterization... |
| domwdom 9597 | Weak dominance is implied ... |
| wdomtr 9598 | Transitivity of weak domin... |
| wdomen1 9599 | Equality-like theorem for ... |
| wdomen2 9600 | Equality-like theorem for ... |
| wdompwdom 9601 | Weak dominance strengthens... |
| canthwdom 9602 | Cantor's Theorem, stated u... |
| wdom2d 9603 | Deduce weak dominance from... |
| wdomd 9604 | Deduce weak dominance from... |
| brwdom3 9605 | Condition for weak dominan... |
| brwdom3i 9606 | Weak dominance implies exi... |
| unwdomg 9607 | Weak dominance of a (disjo... |
| xpwdomg 9608 | Weak dominance of a Cartes... |
| wdomima2g 9609 | A set is weakly dominant o... |
| wdomimag 9610 | A set is weakly dominant o... |
| unxpwdom2 9611 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9612 | If a Cartesian product is ... |
| ixpiunwdom 9613 | Describe an onto function ... |
| harwdom 9614 | The value of the Hartogs f... |
| axreg2 9616 | Axiom of Regularity expres... |
| zfregcl 9617 | The Axiom of Regularity wi... |
| zfreg 9618 | The Axiom of Regularity us... |
| elirrv 9619 | The membership relation is... |
| elirr 9620 | No class is a member of it... |
| elneq 9621 | A class is not equal to an... |
| nelaneq 9622 | A class is not an element ... |
| epinid0 9623 | The membership relation an... |
| sucprcreg 9624 | A class is equal to its su... |
| ruv 9625 | The Russell class is equal... |
| ruALT 9626 | Alternate proof of ~ ru , ... |
| disjcsn 9627 | A class is disjoint from i... |
| zfregfr 9628 | The membership relation is... |
| en2lp 9629 | No class has 2-cycle membe... |
| elnanel 9630 | Two classes are not elemen... |
| cnvepnep 9631 | The membership (epsilon) r... |
| epnsym 9632 | The membership (epsilon) r... |
| elnotel 9633 | A class cannot be an eleme... |
| elnel 9634 | A class cannot be an eleme... |
| en3lplem1 9635 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9636 | Lemma for ~ en3lp . (Cont... |
| en3lp 9637 | No class has 3-cycle membe... |
| preleqg 9638 | Equality of two unordered ... |
| preleq 9639 | Equality of two unordered ... |
| preleqALT 9640 | Alternate proof of ~ prele... |
| opthreg 9641 | Theorem for alternate repr... |
| suc11reg 9642 | The successor operation be... |
| dford2 9643 | Assuming ~ ax-reg , an ord... |
| inf0 9644 | Existence of ` _om ` impli... |
| inf1 9645 | Variation of Axiom of Infi... |
| inf2 9646 | Variation of Axiom of Infi... |
| inf3lema 9647 | Lemma for our Axiom of Inf... |
| inf3lemb 9648 | Lemma for our Axiom of Inf... |
| inf3lemc 9649 | Lemma for our Axiom of Inf... |
| inf3lemd 9650 | Lemma for our Axiom of Inf... |
| inf3lem1 9651 | Lemma for our Axiom of Inf... |
| inf3lem2 9652 | Lemma for our Axiom of Inf... |
| inf3lem3 9653 | Lemma for our Axiom of Inf... |
| inf3lem4 9654 | Lemma for our Axiom of Inf... |
| inf3lem5 9655 | Lemma for our Axiom of Inf... |
| inf3lem6 9656 | Lemma for our Axiom of Inf... |
| inf3lem7 9657 | Lemma for our Axiom of Inf... |
| inf3 9658 | Our Axiom of Infinity ~ ax... |
| infeq5i 9659 | Half of ~ infeq5 . (Contr... |
| infeq5 9660 | The statement "there exist... |
| zfinf 9662 | Axiom of Infinity expresse... |
| axinf2 9663 | A standard version of Axio... |
| zfinf2 9665 | A standard version of the ... |
| omex 9666 | The existence of omega (th... |
| axinf 9667 | The first version of the A... |
| inf5 9668 | The statement "there exist... |
| omelon 9669 | Omega is an ordinal number... |
| dfom3 9670 | The class of natural numbe... |
| elom3 9671 | A simplification of ~ elom... |
| dfom4 9672 | A simplification of ~ df-o... |
| dfom5 9673 | ` _om ` is the smallest li... |
| oancom 9674 | Ordinal addition is not co... |
| isfinite 9675 | A set is finite iff it is ... |
| fict 9676 | A finite set is countable ... |
| nnsdom 9677 | A natural number is strict... |
| omenps 9678 | Omega is equinumerous to a... |
| omensuc 9679 | The set of natural numbers... |
| infdifsn 9680 | Removing a singleton from ... |
| infdiffi 9681 | Removing a finite set from... |
| unbnn3 9682 | Any unbounded subset of na... |
| noinfep 9683 | Using the Axiom of Regular... |
| cantnffval 9686 | The value of the Cantor no... |
| cantnfdm 9687 | The domain of the Cantor n... |
| cantnfvalf 9688 | Lemma for ~ cantnf . The ... |
| cantnfs 9689 | Elementhood in the set of ... |
| cantnfcl 9690 | Basic properties of the or... |
| cantnfval 9691 | The value of the Cantor no... |
| cantnfval2 9692 | Alternate expression for t... |
| cantnfsuc 9693 | The value of the recursive... |
| cantnfle 9694 | A lower bound on the ` CNF... |
| cantnflt 9695 | An upper bound on the part... |
| cantnflt2 9696 | An upper bound on the ` CN... |
| cantnff 9697 | The ` CNF ` function is a ... |
| cantnf0 9698 | The value of the zero func... |
| cantnfrescl 9699 | A function is finitely sup... |
| cantnfres 9700 | The ` CNF ` function respe... |
| cantnfp1lem1 9701 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9702 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9703 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9704 | If ` F ` is created by add... |
| oemapso 9705 | The relation ` T ` is a st... |
| oemapval 9706 | Value of the relation ` T ... |
| oemapvali 9707 | If ` F < G ` , then there ... |
| cantnflem1a 9708 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9709 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9710 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9711 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9712 | Lemma for ~ cantnf . This... |
| cantnflem2 9713 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9714 | Lemma for ~ cantnf . Here... |
| cantnflem4 9715 | Lemma for ~ cantnf . Comp... |
| cantnf 9716 | The Cantor Normal Form the... |
| oemapwe 9717 | The lexicographic order on... |
| cantnffval2 9718 | An alternate definition of... |
| cantnff1o 9719 | Simplify the isomorphism o... |
| wemapwe 9720 | Construct lexicographic or... |
| oef1o 9721 | A bijection of the base se... |
| cnfcomlem 9722 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9723 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9724 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9725 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9726 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9727 | Any infinite ordinal ` B `... |
| cnfcom3clem 9728 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9729 | Wrap the construction of ~... |
| ttrcleq 9732 | Equality theorem for trans... |
| nfttrcld 9733 | Bound variable hypothesis ... |
| nfttrcl 9734 | Bound variable hypothesis ... |
| relttrcl 9735 | The transitive closure of ... |
| brttrcl 9736 | Characterization of elemen... |
| brttrcl2 9737 | Characterization of elemen... |
| ssttrcl 9738 | If ` R ` is a relation, th... |
| ttrcltr 9739 | The transitive closure of ... |
| ttrclresv 9740 | The transitive closure of ... |
| ttrclco 9741 | Composition law for the tr... |
| cottrcl 9742 | Composition law for the tr... |
| ttrclss 9743 | If ` R ` is a subclass of ... |
| dmttrcl 9744 | The domain of a transitive... |
| rnttrcl 9745 | The range of a transitive ... |
| ttrclexg 9746 | If ` R ` is a set, then so... |
| dfttrcl2 9747 | When ` R ` is a set and a ... |
| ttrclselem1 9748 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9749 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9750 | If ` R ` is set-like over ... |
| trcl 9751 | For any set ` A ` , show t... |
| tz9.1 9752 | Every set has a transitive... |
| tz9.1c 9753 | Alternate expression for t... |
| epfrs 9754 | The strong form of the Axi... |
| zfregs 9755 | The strong form of the Axi... |
| zfregs2 9756 | Alternate strong form of t... |
| setind 9757 | Set (epsilon) induction. ... |
| setind2 9758 | Set (epsilon) induction, s... |
| tcvalg 9761 | Value of the transitive cl... |
| tcid 9762 | Defining property of the t... |
| tctr 9763 | Defining property of the t... |
| tcmin 9764 | Defining property of the t... |
| tc2 9765 | A variant of the definitio... |
| tcsni 9766 | The transitive closure of ... |
| tcss 9767 | The transitive closure fun... |
| tcel 9768 | The transitive closure fun... |
| tcidm 9769 | The transitive closure fun... |
| tc0 9770 | The transitive closure of ... |
| tc00 9771 | The transitive closure is ... |
| frmin 9772 | Every (possibly proper) su... |
| frind 9773 | A subclass of a well-found... |
| frinsg 9774 | Well-Founded Induction Sch... |
| frins 9775 | Well-Founded Induction Sch... |
| frins2f 9776 | Well-Founded Induction sch... |
| frins2 9777 | Well-Founded Induction sch... |
| frins3 9778 | Well-Founded Induction sch... |
| frr3g 9779 | Functions defined by well-... |
| frrlem15 9780 | Lemma for general well-fou... |
| frrlem16 9781 | Lemma for general well-fou... |
| frr1 9782 | Law of general well-founde... |
| frr2 9783 | Law of general well-founde... |
| frr3 9784 | Law of general well-founde... |
| r1funlim 9789 | The cumulative hierarchy o... |
| r1fnon 9790 | The cumulative hierarchy o... |
| r10 9791 | Value of the cumulative hi... |
| r1sucg 9792 | Value of the cumulative hi... |
| r1suc 9793 | Value of the cumulative hi... |
| r1limg 9794 | Value of the cumulative hi... |
| r1lim 9795 | Value of the cumulative hi... |
| r1fin 9796 | The first ` _om ` levels o... |
| r1sdom 9797 | Each stage in the cumulati... |
| r111 9798 | The cumulative hierarchy i... |
| r1tr 9799 | The cumulative hierarchy o... |
| r1tr2 9800 | The union of a cumulative ... |
| r1ordg 9801 | Ordering relation for the ... |
| r1ord3g 9802 | Ordering relation for the ... |
| r1ord 9803 | Ordering relation for the ... |
| r1ord2 9804 | Ordering relation for the ... |
| r1ord3 9805 | Ordering relation for the ... |
| r1sssuc 9806 | The value of the cumulativ... |
| r1pwss 9807 | Each set of the cumulative... |
| r1sscl 9808 | Each set of the cumulative... |
| r1val1 9809 | The value of the cumulativ... |
| tz9.12lem1 9810 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9811 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9812 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9813 | A set is well-founded if a... |
| tz9.13 9814 | Every set is well-founded,... |
| tz9.13g 9815 | Every set is well-founded,... |
| rankwflemb 9816 | Two ways of saying a set i... |
| rankf 9817 | The domain and codomain of... |
| rankon 9818 | The rank of a set is an or... |
| r1elwf 9819 | Any member of the cumulati... |
| rankvalb 9820 | Value of the rank function... |
| rankr1ai 9821 | One direction of ~ rankr1a... |
| rankvaln 9822 | Value of the rank function... |
| rankidb 9823 | Identity law for the rank ... |
| rankdmr1 9824 | A rank is a member of the ... |
| rankr1ag 9825 | A version of ~ rankr1a tha... |
| rankr1bg 9826 | A relationship between ran... |
| r1rankidb 9827 | Any set is a subset of the... |
| r1elssi 9828 | The range of the ` R1 ` fu... |
| r1elss 9829 | The range of the ` R1 ` fu... |
| pwwf 9830 | A power set is well-founde... |
| sswf 9831 | A subset of a well-founded... |
| snwf 9832 | A singleton is well-founde... |
| unwf 9833 | A binary union is well-fou... |
| prwf 9834 | An unordered pair is well-... |
| opwf 9835 | An ordered pair is well-fo... |
| unir1 9836 | The cumulative hierarchy o... |
| jech9.3 9837 | Every set belongs to some ... |
| rankwflem 9838 | Every set is well-founded,... |
| rankval 9839 | Value of the rank function... |
| rankvalg 9840 | Value of the rank function... |
| rankval2 9841 | Value of an alternate defi... |
| uniwf 9842 | A union is well-founded if... |
| rankr1clem 9843 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9844 | A relationship between the... |
| rankidn 9845 | A relationship between the... |
| rankpwi 9846 | The rank of a power set. ... |
| rankelb 9847 | The membership relation is... |
| wfelirr 9848 | A well-founded set is not ... |
| rankval3b 9849 | The value of the rank func... |
| ranksnb 9850 | The rank of a singleton. ... |
| rankonidlem 9851 | Lemma for ~ rankonid . (C... |
| rankonid 9852 | The rank of an ordinal num... |
| onwf 9853 | The ordinals are all well-... |
| onssr1 9854 | Initial segments of the or... |
| rankr1g 9855 | A relationship between the... |
| rankid 9856 | Identity law for the rank ... |
| rankr1 9857 | A relationship between the... |
| ssrankr1 9858 | A relationship between an ... |
| rankr1a 9859 | A relationship between ran... |
| r1val2 9860 | The value of the cumulativ... |
| r1val3 9861 | The value of the cumulativ... |
| rankel 9862 | The membership relation is... |
| rankval3 9863 | The value of the rank func... |
| bndrank 9864 | Any class whose elements h... |
| unbndrank 9865 | The elements of a proper c... |
| rankpw 9866 | The rank of a power set. ... |
| ranklim 9867 | The rank of a set belongs ... |
| r1pw 9868 | A stronger property of ` R... |
| r1pwALT 9869 | Alternate shorter proof of... |
| r1pwcl 9870 | The cumulative hierarchy o... |
| rankssb 9871 | The subset relation is inh... |
| rankss 9872 | The subset relation is inh... |
| rankunb 9873 | The rank of the union of t... |
| rankprb 9874 | The rank of an unordered p... |
| rankopb 9875 | The rank of an ordered pai... |
| rankuni2b 9876 | The value of the rank func... |
| ranksn 9877 | The rank of a singleton. ... |
| rankuni2 9878 | The rank of a union. Part... |
| rankun 9879 | The rank of the union of t... |
| rankpr 9880 | The rank of an unordered p... |
| rankop 9881 | The rank of an ordered pai... |
| r1rankid 9882 | Any set is a subset of the... |
| rankeq0b 9883 | A set is empty iff its ran... |
| rankeq0 9884 | A set is empty iff its ran... |
| rankr1id 9885 | The rank of the hierarchy ... |
| rankuni 9886 | The rank of a union. Part... |
| rankr1b 9887 | A relationship between ran... |
| ranksuc 9888 | The rank of a successor. ... |
| rankuniss 9889 | Upper bound of the rank of... |
| rankval4 9890 | The rank of a set is the s... |
| rankbnd 9891 | The rank of a set is bound... |
| rankbnd2 9892 | The rank of a set is bound... |
| rankc1 9893 | A relationship that can be... |
| rankc2 9894 | A relationship that can be... |
| rankelun 9895 | Rank membership is inherit... |
| rankelpr 9896 | Rank membership is inherit... |
| rankelop 9897 | Rank membership is inherit... |
| rankxpl 9898 | A lower bound on the rank ... |
| rankxpu 9899 | An upper bound on the rank... |
| rankfu 9900 | An upper bound on the rank... |
| rankmapu 9901 | An upper bound on the rank... |
| rankxplim 9902 | The rank of a Cartesian pr... |
| rankxplim2 9903 | If the rank of a Cartesian... |
| rankxplim3 9904 | The rank of a Cartesian pr... |
| rankxpsuc 9905 | The rank of a Cartesian pr... |
| tcwf 9906 | The transitive closure fun... |
| tcrank 9907 | This theorem expresses two... |
| scottex 9908 | Scott's trick collects all... |
| scott0 9909 | Scott's trick collects all... |
| scottexs 9910 | Theorem scheme version of ... |
| scott0s 9911 | Theorem scheme version of ... |
| cplem1 9912 | Lemma for the Collection P... |
| cplem2 9913 | Lemma for the Collection P... |
| cp 9914 | Collection Principle. Thi... |
| bnd 9915 | A very strong generalizati... |
| bnd2 9916 | A variant of the Boundedne... |
| kardex 9917 | The collection of all sets... |
| karden 9918 | If we allow the Axiom of R... |
| htalem 9919 | Lemma for defining an emul... |
| hta 9920 | A ZFC emulation of Hilbert... |
| djueq12 9927 | Equality theorem for disjo... |
| djueq1 9928 | Equality theorem for disjo... |
| djueq2 9929 | Equality theorem for disjo... |
| nfdju 9930 | Bound-variable hypothesis ... |
| djuex 9931 | The disjoint union of sets... |
| djuexb 9932 | The disjoint union of two ... |
| djulcl 9933 | Left closure of disjoint u... |
| djurcl 9934 | Right closure of disjoint ... |
| djulf1o 9935 | The left injection functio... |
| djurf1o 9936 | The right injection functi... |
| inlresf 9937 | The left injection restric... |
| inlresf1 9938 | The left injection restric... |
| inrresf 9939 | The right injection restri... |
| inrresf1 9940 | The right injection restri... |
| djuin 9941 | The images of any classes ... |
| djur 9942 | A member of a disjoint uni... |
| djuss 9943 | A disjoint union is a subc... |
| djuunxp 9944 | The union of a disjoint un... |
| djuexALT 9945 | Alternate proof of ~ djuex... |
| eldju1st 9946 | The first component of an ... |
| eldju2ndl 9947 | The second component of an... |
| eldju2ndr 9948 | The second component of an... |
| djuun 9949 | The disjoint union of two ... |
| 1stinl 9950 | The first component of the... |
| 2ndinl 9951 | The second component of th... |
| 1stinr 9952 | The first component of the... |
| 2ndinr 9953 | The second component of th... |
| updjudhf 9954 | The mapping of an element ... |
| updjudhcoinlf 9955 | The composition of the map... |
| updjudhcoinrg 9956 | The composition of the map... |
| updjud 9957 | Universal property of the ... |
| cardf2 9966 | The cardinality function i... |
| cardon 9967 | The cardinal number of a s... |
| isnum2 9968 | A way to express well-orde... |
| isnumi 9969 | A set equinumerous to an o... |
| ennum 9970 | Equinumerous sets are equi... |
| finnum 9971 | Every finite set is numera... |
| onenon 9972 | Every ordinal number is nu... |
| tskwe 9973 | A Tarski set is well-order... |
| xpnum 9974 | The cartesian product of n... |
| cardval3 9975 | An alternate definition of... |
| cardid2 9976 | Any numerable set is equin... |
| isnum3 9977 | A set is numerable iff it ... |
| oncardval 9978 | The value of the cardinal ... |
| oncardid 9979 | Any ordinal number is equi... |
| cardonle 9980 | The cardinal of an ordinal... |
| card0 9981 | The cardinality of the emp... |
| cardidm 9982 | The cardinality function i... |
| oncard 9983 | A set is a cardinal number... |
| ficardom 9984 | The cardinal number of a f... |
| ficardid 9985 | A finite set is equinumero... |
| cardnn 9986 | The cardinality of a natur... |
| cardnueq0 9987 | The empty set is the only ... |
| cardne 9988 | No member of a cardinal nu... |
| carden2a 9989 | If two sets have equal non... |
| carden2b 9990 | If two sets are equinumero... |
| card1 9991 | A set has cardinality one ... |
| cardsn 9992 | A singleton has cardinalit... |
| carddomi2 9993 | Two sets have the dominanc... |
| sdomsdomcardi 9994 | A set strictly dominates i... |
| cardlim 9995 | An infinite cardinal is a ... |
| cardsdomelir 9996 | A cardinal strictly domina... |
| cardsdomel 9997 | A cardinal strictly domina... |
| iscard 9998 | Two ways to express the pr... |
| iscard2 9999 | Two ways to express the pr... |
| carddom2 10000 | Two numerable sets have th... |
| harcard 10001 | The class of ordinal numbe... |
| cardprclem 10002 | Lemma for ~ cardprc . (Co... |
| cardprc 10003 | The class of all cardinal ... |
| carduni 10004 | The union of a set of card... |
| cardiun 10005 | The indexed union of a set... |
| cardennn 10006 | If ` A ` is equinumerous t... |
| cardsucinf 10007 | The cardinality of the suc... |
| cardsucnn 10008 | The cardinality of the suc... |
| cardom 10009 | The set of natural numbers... |
| carden2 10010 | Two numerable sets are equ... |
| cardsdom2 10011 | A numerable set is strictl... |
| domtri2 10012 | Trichotomy of dominance fo... |
| nnsdomel 10013 | Strict dominance and eleme... |
| cardval2 10014 | An alternate version of th... |
| isinffi 10015 | An infinite set contains s... |
| fidomtri 10016 | Trichotomy of dominance wi... |
| fidomtri2 10017 | Trichotomy of dominance wi... |
| harsdom 10018 | The Hartogs number of a we... |
| onsdom 10019 | Any well-orderable set is ... |
| harval2 10020 | An alternate expression fo... |
| harsucnn 10021 | The next cardinal after a ... |
| cardmin2 10022 | The smallest ordinal that ... |
| pm54.43lem 10023 | In Theorem *54.43 of [Whit... |
| pm54.43 10024 | Theorem *54.43 of [Whitehe... |
| enpr2 10025 | An unordered pair with dis... |
| pr2nelemOLD 10026 | Obsolete version of ~ enpr... |
| pr2ne 10027 | If an unordered pair has t... |
| pr2neOLD 10028 | Obsolete version of ~ pr2n... |
| prdom2 10029 | An unordered pair has at m... |
| en2eqpr 10030 | Building a set with two el... |
| en2eleq 10031 | Express a set of pair card... |
| en2other2 10032 | Taking the other element t... |
| dif1card 10033 | The cardinality of a nonem... |
| leweon 10034 | Lexicographical order is a... |
| r0weon 10035 | A set-like well-ordering o... |
| infxpenlem 10036 | Lemma for ~ infxpen . (Co... |
| infxpen 10037 | Every infinite ordinal is ... |
| xpomen 10038 | The Cartesian product of o... |
| xpct 10039 | The cartesian product of t... |
| infxpidm2 10040 | Every infinite well-ordera... |
| infxpenc 10041 | A canonical version of ~ i... |
| infxpenc2lem1 10042 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10043 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10044 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10045 | Existence form of ~ infxpe... |
| iunmapdisj 10046 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10047 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10048 | Lemma for ~ fseqen . (Con... |
| fseqdom 10049 | One half of ~ fseqen . (C... |
| fseqen 10050 | A set that is equinumerous... |
| infpwfidom 10051 | The collection of finite s... |
| dfac8alem 10052 | Lemma for ~ dfac8a . If t... |
| dfac8a 10053 | Numeration theorem: every ... |
| dfac8b 10054 | The well-ordering theorem:... |
| dfac8clem 10055 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10056 | If the union of a set is w... |
| ac10ct 10057 | A proof of the well-orderi... |
| ween 10058 | A set is numerable iff it ... |
| ac5num 10059 | A version of ~ ac5b with t... |
| ondomen 10060 | If a set is dominated by a... |
| numdom 10061 | A set dominated by a numer... |
| ssnum 10062 | A subset of a numerable se... |
| onssnum 10063 | All subsets of the ordinal... |
| indcardi 10064 | Indirect strong induction ... |
| acnrcl 10065 | Reverse closure for the ch... |
| acneq 10066 | Equality theorem for the c... |
| isacn 10067 | The property of being a ch... |
| acni 10068 | The property of being a ch... |
| acni2 10069 | The property of being a ch... |
| acni3 10070 | The property of being a ch... |
| acnlem 10071 | Construct a mapping satisf... |
| numacn 10072 | A well-orderable set has c... |
| finacn 10073 | Every set has finite choic... |
| acndom 10074 | A set with long choice seq... |
| acnnum 10075 | A set ` X ` which has choi... |
| acnen 10076 | The class of choice sets o... |
| acndom2 10077 | A set smaller than one wit... |
| acnen2 10078 | The class of sets with cho... |
| fodomacn 10079 | A version of ~ fodom that ... |
| fodomnum 10080 | A version of ~ fodom that ... |
| fonum 10081 | A surjection maps numerabl... |
| numwdom 10082 | A surjection maps numerabl... |
| fodomfi2 10083 | Onto functions define domi... |
| wdomfil 10084 | Weak dominance agrees with... |
| infpwfien 10085 | Any infinite well-orderabl... |
| inffien 10086 | The set of finite intersec... |
| wdomnumr 10087 | Weak dominance agrees with... |
| alephfnon 10088 | The aleph function is a fu... |
| aleph0 10089 | The first infinite cardina... |
| alephlim 10090 | Value of the aleph functio... |
| alephsuc 10091 | Value of the aleph functio... |
| alephon 10092 | An aleph is an ordinal num... |
| alephcard 10093 | Every aleph is a cardinal ... |
| alephnbtwn 10094 | No cardinal can be sandwic... |
| alephnbtwn2 10095 | No set has equinumerosity ... |
| alephordilem1 10096 | Lemma for ~ alephordi . (... |
| alephordi 10097 | Strict ordering property o... |
| alephord 10098 | Ordering property of the a... |
| alephord2 10099 | Ordering property of the a... |
| alephord2i 10100 | Ordering property of the a... |
| alephord3 10101 | Ordering property of the a... |
| alephsucdom 10102 | A set dominated by an alep... |
| alephsuc2 10103 | An alternate representatio... |
| alephdom 10104 | Relationship between inclu... |
| alephgeom 10105 | Every aleph is greater tha... |
| alephislim 10106 | Every aleph is a limit ord... |
| aleph11 10107 | The aleph function is one-... |
| alephf1 10108 | The aleph function is a on... |
| alephsdom 10109 | If an ordinal is smaller t... |
| alephdom2 10110 | A dominated initial ordina... |
| alephle 10111 | The argument of the aleph ... |
| cardaleph 10112 | Given any transfinite card... |
| cardalephex 10113 | Every transfinite cardinal... |
| infenaleph 10114 | An infinite numerable set ... |
| isinfcard 10115 | Two ways to express the pr... |
| iscard3 10116 | Two ways to express the pr... |
| cardnum 10117 | Two ways to express the cl... |
| alephinit 10118 | An infinite initial ordina... |
| carduniima 10119 | The union of the image of ... |
| cardinfima 10120 | If a mapping to cardinals ... |
| alephiso 10121 | Aleph is an order isomorph... |
| alephprc 10122 | The class of all transfini... |
| alephsson 10123 | The class of transfinite c... |
| unialeph 10124 | The union of the class of ... |
| alephsmo 10125 | The aleph function is stri... |
| alephf1ALT 10126 | Alternate proof of ~ aleph... |
| alephfplem1 10127 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10128 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10129 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10130 | Lemma for ~ alephfp . (Co... |
| alephfp 10131 | The aleph function has a f... |
| alephfp2 10132 | The aleph function has at ... |
| alephval3 10133 | An alternate way to expres... |
| alephsucpw2 10134 | The power set of an aleph ... |
| mappwen 10135 | Power rule for cardinal ar... |
| finnisoeu 10136 | A finite totally ordered s... |
| iunfictbso 10137 | Countability of a countabl... |
| aceq1 10140 | Equivalence of two version... |
| aceq0 10141 | Equivalence of two version... |
| aceq2 10142 | Equivalence of two version... |
| aceq3lem 10143 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10144 | Equivalence of two version... |
| dfac4 10145 | Equivalence of two version... |
| dfac5lem1 10146 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10147 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10148 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10149 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10150 | Lemma for ~ dfac5 . (Cont... |
| dfac5 10151 | Equivalence of two version... |
| dfac2a 10152 | Our Axiom of Choice (in th... |
| dfac2b 10153 | Axiom of Choice (first for... |
| dfac2 10154 | Axiom of Choice (first for... |
| dfac7 10155 | Equivalence of the Axiom o... |
| dfac0 10156 | Equivalence of two version... |
| dfac1 10157 | Equivalence of two version... |
| dfac8 10158 | A proof of the equivalency... |
| dfac9 10159 | Equivalence of the axiom o... |
| dfac10 10160 | Axiom of Choice equivalent... |
| dfac10c 10161 | Axiom of Choice equivalent... |
| dfac10b 10162 | Axiom of Choice equivalent... |
| acacni 10163 | A choice equivalent: every... |
| dfacacn 10164 | A choice equivalent: every... |
| dfac13 10165 | The axiom of choice holds ... |
| dfac12lem1 10166 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10167 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10168 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10169 | The axiom of choice holds ... |
| dfac12k 10170 | Equivalence of ~ dfac12 an... |
| dfac12a 10171 | The axiom of choice holds ... |
| dfac12 10172 | The axiom of choice holds ... |
| kmlem1 10173 | Lemma for 5-quantifier AC ... |
| kmlem2 10174 | Lemma for 5-quantifier AC ... |
| kmlem3 10175 | Lemma for 5-quantifier AC ... |
| kmlem4 10176 | Lemma for 5-quantifier AC ... |
| kmlem5 10177 | Lemma for 5-quantifier AC ... |
| kmlem6 10178 | Lemma for 5-quantifier AC ... |
| kmlem7 10179 | Lemma for 5-quantifier AC ... |
| kmlem8 10180 | Lemma for 5-quantifier AC ... |
| kmlem9 10181 | Lemma for 5-quantifier AC ... |
| kmlem10 10182 | Lemma for 5-quantifier AC ... |
| kmlem11 10183 | Lemma for 5-quantifier AC ... |
| kmlem12 10184 | Lemma for 5-quantifier AC ... |
| kmlem13 10185 | Lemma for 5-quantifier AC ... |
| kmlem14 10186 | Lemma for 5-quantifier AC ... |
| kmlem15 10187 | Lemma for 5-quantifier AC ... |
| kmlem16 10188 | Lemma for 5-quantifier AC ... |
| dfackm 10189 | Equivalence of the Axiom o... |
| undjudom 10190 | Cardinal addition dominate... |
| endjudisj 10191 | Equinumerosity of a disjoi... |
| djuen 10192 | Disjoint unions of equinum... |
| djuenun 10193 | Disjoint union is equinume... |
| dju1en 10194 | Cardinal addition with car... |
| dju1dif 10195 | Adding and subtracting one... |
| dju1p1e2 10196 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10197 | Alternate proof of ~ dju1p... |
| dju0en 10198 | Cardinal addition with car... |
| xp2dju 10199 | Two times a cardinal numbe... |
| djucomen 10200 | Commutative law for cardin... |
| djuassen 10201 | Associative law for cardin... |
| xpdjuen 10202 | Cardinal multiplication di... |
| mapdjuen 10203 | Sum of exponents law for c... |
| pwdjuen 10204 | Sum of exponents law for c... |
| djudom1 10205 | Ordering law for cardinal ... |
| djudom2 10206 | Ordering law for cardinal ... |
| djudoml 10207 | A set is dominated by its ... |
| djuxpdom 10208 | Cartesian product dominate... |
| djufi 10209 | The disjoint union of two ... |
| cdainflem 10210 | Any partition of omega int... |
| djuinf 10211 | A set is infinite iff the ... |
| infdju1 10212 | An infinite set is equinum... |
| pwdju1 10213 | The sum of a powerset with... |
| pwdjuidm 10214 | If the natural numbers inj... |
| djulepw 10215 | If ` A ` is idempotent und... |
| onadju 10216 | The cardinal and ordinal s... |
| cardadju 10217 | The cardinal sum is equinu... |
| djunum 10218 | The disjoint union of two ... |
| unnum 10219 | The union of two numerable... |
| nnadju 10220 | The cardinal and ordinal s... |
| nnadjuALT 10221 | Shorter proof of ~ nnadju ... |
| ficardadju 10222 | The disjoint union of fini... |
| ficardun 10223 | The cardinality of the uni... |
| ficardunOLD 10224 | Obsolete version of ~ fica... |
| ficardun2 10225 | The cardinality of the uni... |
| ficardun2OLD 10226 | Obsolete version of ~ fica... |
| pwsdompw 10227 | Lemma for ~ domtriom . Th... |
| unctb 10228 | The union of two countable... |
| infdjuabs 10229 | Absorption law for additio... |
| infunabs 10230 | An infinite set is equinum... |
| infdju 10231 | The sum of two cardinal nu... |
| infdif 10232 | The cardinality of an infi... |
| infdif2 10233 | Cardinality ordering for a... |
| infxpdom 10234 | Dominance law for multipli... |
| infxpabs 10235 | Absorption law for multipl... |
| infunsdom1 10236 | The union of two sets that... |
| infunsdom 10237 | The union of two sets that... |
| infxp 10238 | Absorption law for multipl... |
| pwdjudom 10239 | A property of dominance ov... |
| infpss 10240 | Every infinite set has an ... |
| infmap2 10241 | An exponentiation law for ... |
| ackbij2lem1 10242 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10243 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10244 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10245 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10246 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10247 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10248 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10249 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10250 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10251 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10252 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10253 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10254 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10255 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10256 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10257 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10258 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10259 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10260 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10261 | The Ackermann bijection, p... |
| ackbij1b 10262 | The Ackermann bijection, p... |
| ackbij2lem2 10263 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10264 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10265 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10266 | The Ackermann bijection, p... |
| r1om 10267 | The set of hereditarily fi... |
| fictb 10268 | A set is countable iff its... |
| cflem 10269 | A lemma used to simplify c... |
| cfval 10270 | Value of the cofinality fu... |
| cff 10271 | Cofinality is a function o... |
| cfub 10272 | An upper bound on cofinali... |
| cflm 10273 | Value of the cofinality fu... |
| cf0 10274 | Value of the cofinality fu... |
| cardcf 10275 | Cofinality is a cardinal n... |
| cflecard 10276 | Cofinality is bounded by t... |
| cfle 10277 | Cofinality is bounded by i... |
| cfon 10278 | The cofinality of any set ... |
| cfeq0 10279 | Only the ordinal zero has ... |
| cfsuc 10280 | Value of the cofinality fu... |
| cff1 10281 | There is always a map from... |
| cfflb 10282 | If there is a cofinal map ... |
| cfval2 10283 | Another expression for the... |
| coflim 10284 | A simpler expression for t... |
| cflim3 10285 | Another expression for the... |
| cflim2 10286 | The cofinality function is... |
| cfom 10287 | Value of the cofinality fu... |
| cfss 10288 | There is a cofinal subset ... |
| cfslb 10289 | Any cofinal subset of ` A ... |
| cfslbn 10290 | Any subset of ` A ` smalle... |
| cfslb2n 10291 | Any small collection of sm... |
| cofsmo 10292 | Any cofinal map implies th... |
| cfsmolem 10293 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10294 | The map in ~ cff1 can be a... |
| cfcoflem 10295 | Lemma for ~ cfcof , showin... |
| coftr 10296 | If there is a cofinal map ... |
| cfcof 10297 | If there is a cofinal map ... |
| cfidm 10298 | The cofinality function is... |
| alephsing 10299 | The cofinality of a limit ... |
| sornom 10300 | The range of a single-step... |
| isfin1a 10315 | Definition of a Ia-finite ... |
| fin1ai 10316 | Property of a Ia-finite se... |
| isfin2 10317 | Definition of a II-finite ... |
| fin2i 10318 | Property of a II-finite se... |
| isfin3 10319 | Definition of a III-finite... |
| isfin4 10320 | Definition of a IV-finite ... |
| fin4i 10321 | Infer that a set is IV-inf... |
| isfin5 10322 | Definition of a V-finite s... |
| isfin6 10323 | Definition of a VI-finite ... |
| isfin7 10324 | Definition of a VII-finite... |
| sdom2en01 10325 | A set with less than two e... |
| infpssrlem1 10326 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10327 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10328 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10329 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10330 | Lemma for ~ infpssr . (Co... |
| infpssr 10331 | Dedekind infinity implies ... |
| fin4en1 10332 | Dedekind finite is a cardi... |
| ssfin4 10333 | Dedekind finite sets have ... |
| domfin4 10334 | A set dominated by a Dedek... |
| ominf4 10335 | ` _om ` is Dedekind infini... |
| infpssALT 10336 | Alternate proof of ~ infps... |
| isfin4-2 10337 | Alternate definition of IV... |
| isfin4p1 10338 | Alternate definition of IV... |
| fin23lem7 10339 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10340 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10341 | A II-finite set contains m... |
| isfin2-2 10342 | ` Fin2 ` expressed in term... |
| ssfin2 10343 | A subset of a II-finite se... |
| enfin2i 10344 | II-finiteness is a cardina... |
| fin23lem24 10345 | Lemma for ~ fin23 . In a ... |
| fincssdom 10346 | In a chain of finite sets,... |
| fin23lem25 10347 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10348 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10349 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10350 | Lemma for ~ fin23 but coul... |
| fin23lem27 10351 | The mapping constructed in... |
| isfin3ds 10352 | Property of a III-finite s... |
| ssfin3ds 10353 | A subset of a III-finite s... |
| fin23lem12 10354 | The beginning of the proof... |
| fin23lem13 10355 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10356 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10357 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10358 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10359 | Lemma for ~ fin23 . The f... |
| fin23lem20 10360 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10361 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10362 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10363 | Lemma for ~ fin23 . The r... |
| fin23lem29 10364 | Lemma for ~ fin23 . The r... |
| fin23lem30 10365 | Lemma for ~ fin23 . The r... |
| fin23lem31 10366 | Lemma for ~ fin23 . The r... |
| fin23lem32 10367 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10368 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10369 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10370 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10371 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10372 | Lemma for ~ fin23 . The c... |
| fin23lem39 10373 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10374 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10375 | Lemma for ~ fin23 . A set... |
| isf32lem1 10376 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10377 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10378 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10379 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10380 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10381 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10382 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10383 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10384 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10385 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10386 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10387 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10388 | One half of ~ isfin3-2 . ... |
| isf33lem 10389 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10390 | Weakly Dedekind-infinite s... |
| isfin3-3 10391 | Weakly Dedekind-infinite s... |
| fin33i 10392 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10393 | Lemma for ~ compsscnv . (... |
| compsscnv 10394 | Complementation on a power... |
| isf34lem1 10395 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10396 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10397 | Complementation is an anti... |
| isf34lem3 10398 | Lemma for ~ isfin3-4 . (C... |
| compss 10399 | Express image under of the... |
| isf34lem4 10400 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10401 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10402 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10403 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10404 | Inference from ~ isfin3-4 ... |
| isfin3-4 10405 | Weakly Dedekind-infinite s... |
| fin11a 10406 | Every I-finite set is Ia-f... |
| enfin1ai 10407 | Ia-finiteness is a cardina... |
| isfin1-2 10408 | A set is finite in the usu... |
| isfin1-3 10409 | A set is I-finite iff ever... |
| isfin1-4 10410 | A set is I-finite iff ever... |
| dffin1-5 10411 | Compact quantifier-free ve... |
| fin23 10412 | Every II-finite set (every... |
| fin34 10413 | Every III-finite set is IV... |
| isfin5-2 10414 | Alternate definition of V-... |
| fin45 10415 | Every IV-finite set is V-f... |
| fin56 10416 | Every V-finite set is VI-f... |
| fin17 10417 | Every I-finite set is VII-... |
| fin67 10418 | Every VI-finite set is VII... |
| isfin7-2 10419 | A set is VII-finite iff it... |
| fin71num 10420 | A well-orderable set is VI... |
| dffin7-2 10421 | Class form of ~ isfin7-2 .... |
| dfacfin7 10422 | Axiom of Choice equivalent... |
| fin1a2lem1 10423 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10424 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10425 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10426 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10427 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10428 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10429 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10430 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10431 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10432 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10433 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10434 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10435 | Lemma for ~ fin1a2 . (Con... |
| fin12 10436 | Weak theorem which skips I... |
| fin1a2s 10437 | An II-infinite set can hav... |
| fin1a2 10438 | Every Ia-finite set is II-... |
| itunifval 10439 | Function value of iterated... |
| itunifn 10440 | Functionality of the itera... |
| ituni0 10441 | A zero-fold iterated union... |
| itunisuc 10442 | Successor iterated union. ... |
| itunitc1 10443 | Each union iterate is a me... |
| itunitc 10444 | The union of all union ite... |
| ituniiun 10445 | Unwrap an iterated union f... |
| hsmexlem7 10446 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10447 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10448 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10449 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10450 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10451 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10452 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10453 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10454 | Lemma for ~ hsmex . (Cont... |
| hsmex 10455 | The collection of heredita... |
| hsmex2 10456 | The set of hereditary size... |
| hsmex3 10457 | The set of hereditary size... |
| axcc2lem 10459 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10460 | A possibly more useful ver... |
| axcc3 10461 | A possibly more useful ver... |
| axcc4 10462 | A version of ~ axcc3 that ... |
| acncc 10463 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10464 | Relax the constraint on ~ ... |
| domtriomlem 10465 | Lemma for ~ domtriom . (C... |
| domtriom 10466 | Trichotomy of equinumerosi... |
| fin41 10467 | Under countable choice, th... |
| dominf 10468 | A nonempty set that is a s... |
| dcomex 10470 | The Axiom of Dependent Cho... |
| axdc2lem 10471 | Lemma for ~ axdc2 . We co... |
| axdc2 10472 | An apparent strengthening ... |
| axdc3lem 10473 | The class ` S ` of finite ... |
| axdc3lem2 10474 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10475 | Simple substitution lemma ... |
| axdc3lem4 10476 | Lemma for ~ axdc3 . We ha... |
| axdc3 10477 | Dependent Choice. Axiom D... |
| axdc4lem 10478 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10479 | A more general version of ... |
| axcclem 10480 | Lemma for ~ axcc . (Contr... |
| axcc 10481 | Although CC can be proven ... |
| zfac 10483 | Axiom of Choice expressed ... |
| ac2 10484 | Axiom of Choice equivalent... |
| ac3 10485 | Axiom of Choice using abbr... |
| axac3 10487 | This theorem asserts that ... |
| ackm 10488 | A remarkable equivalent to... |
| axac2 10489 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10490 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10491 | Apply a choice equivalent.... |
| cardeqv 10492 | All sets are well-orderabl... |
| numth3 10493 | All sets are well-orderabl... |
| numth2 10494 | Numeration theorem: any se... |
| numth 10495 | Numeration theorem: every ... |
| ac7 10496 | An Axiom of Choice equival... |
| ac7g 10497 | An Axiom of Choice equival... |
| ac4 10498 | Equivalent of Axiom of Cho... |
| ac4c 10499 | Equivalent of Axiom of Cho... |
| ac5 10500 | An Axiom of Choice equival... |
| ac5b 10501 | Equivalent of Axiom of Cho... |
| ac6num 10502 | A version of ~ ac6 which t... |
| ac6 10503 | Equivalent of Axiom of Cho... |
| ac6c4 10504 | Equivalent of Axiom of Cho... |
| ac6c5 10505 | Equivalent of Axiom of Cho... |
| ac9 10506 | An Axiom of Choice equival... |
| ac6s 10507 | Equivalent of Axiom of Cho... |
| ac6n 10508 | Equivalent of Axiom of Cho... |
| ac6s2 10509 | Generalization of the Axio... |
| ac6s3 10510 | Generalization of the Axio... |
| ac6sg 10511 | ~ ac6s with sethood as ant... |
| ac6sf 10512 | Version of ~ ac6 with boun... |
| ac6s4 10513 | Generalization of the Axio... |
| ac6s5 10514 | Generalization of the Axio... |
| ac8 10515 | An Axiom of Choice equival... |
| ac9s 10516 | An Axiom of Choice equival... |
| numthcor 10517 | Any set is strictly domina... |
| weth 10518 | Well-ordering theorem: any... |
| zorn2lem1 10519 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10520 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10521 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10522 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10523 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10524 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10525 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10526 | Zorn's Lemma of [Monk1] p.... |
| zorng 10527 | Zorn's Lemma. If the unio... |
| zornn0g 10528 | Variant of Zorn's lemma ~ ... |
| zorn2 10529 | Zorn's Lemma of [Monk1] p.... |
| zorn 10530 | Zorn's Lemma. If the unio... |
| zornn0 10531 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10532 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10533 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10534 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10535 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10536 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10537 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10538 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10539 | The Teichmüller-Tukey... |
| ttukeyg 10540 | The Teichmüller-Tukey... |
| ttukey 10541 | The Teichmüller-Tukey... |
| axdclem 10542 | Lemma for ~ axdc . (Contr... |
| axdclem2 10543 | Lemma for ~ axdc . Using ... |
| axdc 10544 | This theorem derives ~ ax-... |
| fodomg 10545 | An onto function implies d... |
| fodom 10546 | An onto function implies d... |
| dmct 10547 | The domain of a countable ... |
| rnct 10548 | The range of a countable s... |
| fodomb 10549 | Equivalence of an onto map... |
| wdomac 10550 | When assuming AC, weak and... |
| brdom3 10551 | Equivalence to a dominance... |
| brdom5 10552 | An equivalence to a domina... |
| brdom4 10553 | An equivalence to a domina... |
| brdom7disj 10554 | An equivalence to a domina... |
| brdom6disj 10555 | An equivalence to a domina... |
| fin71ac 10556 | Once we allow AC, the "str... |
| imadomg 10557 | An image of a function und... |
| fimact 10558 | The image by a function of... |
| fnrndomg 10559 | The range of a function is... |
| fnct 10560 | If the domain of a functio... |
| mptct 10561 | A countable mapping set is... |
| iunfo 10562 | Existence of an onto funct... |
| iundom2g 10563 | An upper bound for the car... |
| iundomg 10564 | An upper bound for the car... |
| iundom 10565 | An upper bound for the car... |
| unidom 10566 | An upper bound for the car... |
| uniimadom 10567 | An upper bound for the car... |
| uniimadomf 10568 | An upper bound for the car... |
| cardval 10569 | The value of the cardinal ... |
| cardid 10570 | Any set is equinumerous to... |
| cardidg 10571 | Any set is equinumerous to... |
| cardidd 10572 | Any set is equinumerous to... |
| cardf 10573 | The cardinality function i... |
| carden 10574 | Two sets are equinumerous ... |
| cardeq0 10575 | Only the empty set has car... |
| unsnen 10576 | Equinumerosity of a set wi... |
| carddom 10577 | Two sets have the dominanc... |
| cardsdom 10578 | Two sets have the strict d... |
| domtri 10579 | Trichotomy law for dominan... |
| entric 10580 | Trichotomy of equinumerosi... |
| entri2 10581 | Trichotomy of dominance an... |
| entri3 10582 | Trichotomy of dominance. ... |
| sdomsdomcard 10583 | A set strictly dominates i... |
| canth3 10584 | Cantor's theorem in terms ... |
| infxpidm 10585 | Every infinite class is eq... |
| ondomon 10586 | The class of ordinals domi... |
| cardmin 10587 | The smallest ordinal that ... |
| ficard 10588 | A set is finite iff its ca... |
| infinf 10589 | Equivalence between two in... |
| unirnfdomd 10590 | The union of the range of ... |
| konigthlem 10591 | Lemma for ~ konigth . (Co... |
| konigth 10592 | Konig's Theorem. If ` m (... |
| alephsucpw 10593 | The power set of an aleph ... |
| aleph1 10594 | The set exponentiation of ... |
| alephval2 10595 | An alternate way to expres... |
| dominfac 10596 | A nonempty set that is a s... |
| iunctb 10597 | The countable union of cou... |
| unictb 10598 | The countable union of cou... |
| infmap 10599 | An exponentiation law for ... |
| alephadd 10600 | The sum of two alephs is t... |
| alephmul 10601 | The product of two alephs ... |
| alephexp1 10602 | An exponentiation law for ... |
| alephsuc3 10603 | An alternate representatio... |
| alephexp2 10604 | An expression equinumerous... |
| alephreg 10605 | A successor aleph is regul... |
| pwcfsdom 10606 | A corollary of Konig's The... |
| cfpwsdom 10607 | A corollary of Konig's The... |
| alephom 10608 | From ~ canth2 , we know th... |
| smobeth 10609 | The beth function is stric... |
| nd1 10610 | A lemma for proving condit... |
| nd2 10611 | A lemma for proving condit... |
| nd3 10612 | A lemma for proving condit... |
| nd4 10613 | A lemma for proving condit... |
| axextnd 10614 | A version of the Axiom of ... |
| axrepndlem1 10615 | Lemma for the Axiom of Rep... |
| axrepndlem2 10616 | Lemma for the Axiom of Rep... |
| axrepnd 10617 | A version of the Axiom of ... |
| axunndlem1 10618 | Lemma for the Axiom of Uni... |
| axunnd 10619 | A version of the Axiom of ... |
| axpowndlem1 10620 | Lemma for the Axiom of Pow... |
| axpowndlem2 10621 | Lemma for the Axiom of Pow... |
| axpowndlem3 10622 | Lemma for the Axiom of Pow... |
| axpowndlem4 10623 | Lemma for the Axiom of Pow... |
| axpownd 10624 | A version of the Axiom of ... |
| axregndlem1 10625 | Lemma for the Axiom of Reg... |
| axregndlem2 10626 | Lemma for the Axiom of Reg... |
| axregnd 10627 | A version of the Axiom of ... |
| axinfndlem1 10628 | Lemma for the Axiom of Inf... |
| axinfnd 10629 | A version of the Axiom of ... |
| axacndlem1 10630 | Lemma for the Axiom of Cho... |
| axacndlem2 10631 | Lemma for the Axiom of Cho... |
| axacndlem3 10632 | Lemma for the Axiom of Cho... |
| axacndlem4 10633 | Lemma for the Axiom of Cho... |
| axacndlem5 10634 | Lemma for the Axiom of Cho... |
| axacnd 10635 | A version of the Axiom of ... |
| zfcndext 10636 | Axiom of Extensionality ~ ... |
| zfcndrep 10637 | Axiom of Replacement ~ ax-... |
| zfcndun 10638 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10639 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10640 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10641 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10642 | Axiom of Choice ~ ax-ac , ... |
| elgch 10645 | Elementhood in the collect... |
| fingch 10646 | A finite set is a GCH-set.... |
| gchi 10647 | The only GCH-sets which ha... |
| gchen1 10648 | If ` A <_ B < ~P A ` , and... |
| gchen2 10649 | If ` A < B <_ ~P A ` , and... |
| gchor 10650 | If ` A <_ B <_ ~P A ` , an... |
| engch 10651 | The property of being a GC... |
| gchdomtri 10652 | Under certain conditions, ... |
| fpwwe2cbv 10653 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10654 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10655 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10656 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10657 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10658 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10659 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10660 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10661 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10662 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10663 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10664 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10665 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10666 | Given any function ` F ` f... |
| fpwwecbv 10667 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10668 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10669 | Given any function ` F ` f... |
| canth4 10670 | An "effective" form of Can... |
| canthnumlem 10671 | Lemma for ~ canthnum . (C... |
| canthnum 10672 | The set of well-orderable ... |
| canthwelem 10673 | Lemma for ~ canthwe . (Co... |
| canthwe 10674 | The set of well-orders of ... |
| canthp1lem1 10675 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10676 | Lemma for ~ canthp1 . (Co... |
| canthp1 10677 | A slightly stronger form o... |
| finngch 10678 | The exclusion of finite se... |
| gchdju1 10679 | An infinite GCH-set is ide... |
| gchinf 10680 | An infinite GCH-set is Ded... |
| pwfseqlem1 10681 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10682 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10683 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10684 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10685 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10686 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10687 | The powerset of a Dedekind... |
| pwxpndom2 10688 | The powerset of a Dedekind... |
| pwxpndom 10689 | The powerset of a Dedekind... |
| pwdjundom 10690 | The powerset of a Dedekind... |
| gchdjuidm 10691 | An infinite GCH-set is ide... |
| gchxpidm 10692 | An infinite GCH-set is ide... |
| gchpwdom 10693 | A relationship between dom... |
| gchaleph 10694 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10695 | If ` ( aleph `` A ) ` and ... |
| hargch 10696 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10697 | If ` ( aleph `` suc A ) ` ... |
| gch2 10698 | It is sufficient to requir... |
| gch3 10699 | An equivalent formulation ... |
| gch-kn 10700 | The equivalence of two ver... |
| gchaclem 10701 | Lemma for ~ gchac (obsolet... |
| gchhar 10702 | A "local" form of ~ gchac ... |
| gchacg 10703 | A "local" form of ~ gchac ... |
| gchac 10704 | The Generalized Continuum ... |
| elwina 10709 | Conditions of weak inacces... |
| elina 10710 | Conditions of strong inacc... |
| winaon 10711 | A weakly inaccessible card... |
| inawinalem 10712 | Lemma for ~ inawina . (Co... |
| inawina 10713 | Every strongly inaccessibl... |
| omina 10714 | ` _om ` is a strongly inac... |
| winacard 10715 | A weakly inaccessible card... |
| winainflem 10716 | A weakly inaccessible card... |
| winainf 10717 | A weakly inaccessible card... |
| winalim 10718 | A weakly inaccessible card... |
| winalim2 10719 | A nontrivial weakly inacce... |
| winafp 10720 | A nontrivial weakly inacce... |
| winafpi 10721 | This theorem, which states... |
| gchina 10722 | Assuming the GCH, weakly a... |
| iswun 10727 | Properties of a weak unive... |
| wuntr 10728 | A weak universe is transit... |
| wununi 10729 | A weak universe is closed ... |
| wunpw 10730 | A weak universe is closed ... |
| wunelss 10731 | The elements of a weak uni... |
| wunpr 10732 | A weak universe is closed ... |
| wunun 10733 | A weak universe is closed ... |
| wuntp 10734 | A weak universe is closed ... |
| wunss 10735 | A weak universe is closed ... |
| wunin 10736 | A weak universe is closed ... |
| wundif 10737 | A weak universe is closed ... |
| wunint 10738 | A weak universe is closed ... |
| wunsn 10739 | A weak universe is closed ... |
| wunsuc 10740 | A weak universe is closed ... |
| wun0 10741 | A weak universe contains t... |
| wunr1om 10742 | A weak universe is infinit... |
| wunom 10743 | A weak universe contains a... |
| wunfi 10744 | A weak universe contains a... |
| wunop 10745 | A weak universe is closed ... |
| wunot 10746 | A weak universe is closed ... |
| wunxp 10747 | A weak universe is closed ... |
| wunpm 10748 | A weak universe is closed ... |
| wunmap 10749 | A weak universe is closed ... |
| wunf 10750 | A weak universe is closed ... |
| wundm 10751 | A weak universe is closed ... |
| wunrn 10752 | A weak universe is closed ... |
| wuncnv 10753 | A weak universe is closed ... |
| wunres 10754 | A weak universe is closed ... |
| wunfv 10755 | A weak universe is closed ... |
| wunco 10756 | A weak universe is closed ... |
| wuntpos 10757 | A weak universe is closed ... |
| intwun 10758 | The intersection of a coll... |
| r1limwun 10759 | Each limit stage in the cu... |
| r1wunlim 10760 | The weak universes in the ... |
| wunex2 10761 | Construct a weak universe ... |
| wunex 10762 | Construct a weak universe ... |
| uniwun 10763 | Every set is contained in ... |
| wunex3 10764 | Construct a weak universe ... |
| wuncval 10765 | Value of the weak universe... |
| wuncid 10766 | The weak universe closure ... |
| wunccl 10767 | The weak universe closure ... |
| wuncss 10768 | The weak universe closure ... |
| wuncidm 10769 | The weak universe closure ... |
| wuncval2 10770 | Our earlier expression for... |
| eltskg 10773 | Properties of a Tarski cla... |
| eltsk2g 10774 | Properties of a Tarski cla... |
| tskpwss 10775 | First axiom of a Tarski cl... |
| tskpw 10776 | Second axiom of a Tarski c... |
| tsken 10777 | Third axiom of a Tarski cl... |
| 0tsk 10778 | The empty set is a (transi... |
| tsksdom 10779 | An element of a Tarski cla... |
| tskssel 10780 | A part of a Tarski class s... |
| tskss 10781 | The subsets of an element ... |
| tskin 10782 | The intersection of two el... |
| tsksn 10783 | A singleton of an element ... |
| tsktrss 10784 | A transitive element of a ... |
| tsksuc 10785 | If an element of a Tarski ... |
| tsk0 10786 | A nonempty Tarski class co... |
| tsk1 10787 | One is an element of a non... |
| tsk2 10788 | Two is an element of a non... |
| 2domtsk 10789 | If a Tarski class is not e... |
| tskr1om 10790 | A nonempty Tarski class is... |
| tskr1om2 10791 | A nonempty Tarski class co... |
| tskinf 10792 | A nonempty Tarski class is... |
| tskpr 10793 | If ` A ` and ` B ` are mem... |
| tskop 10794 | If ` A ` and ` B ` are mem... |
| tskxpss 10795 | A Cartesian product of two... |
| tskwe2 10796 | A Tarski class is well-ord... |
| inttsk 10797 | The intersection of a coll... |
| inar1 10798 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10799 | Alternate proof of ~ r1om ... |
| rankcf 10800 | Any set must be at least a... |
| inatsk 10801 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10802 | The set of hereditarily fi... |
| tskord 10803 | A Tarski class contains al... |
| tskcard 10804 | An even more direct relati... |
| r1tskina 10805 | There is a direct relation... |
| tskuni 10806 | The union of an element of... |
| tskwun 10807 | A nonempty transitive Tars... |
| tskint 10808 | The intersection of an ele... |
| tskun 10809 | The union of two elements ... |
| tskxp 10810 | The Cartesian product of t... |
| tskmap 10811 | Set exponentiation is an e... |
| tskurn 10812 | A transitive Tarski class ... |
| elgrug 10815 | Properties of a Grothendie... |
| grutr 10816 | A Grothendieck universe is... |
| gruelss 10817 | A Grothendieck universe is... |
| grupw 10818 | A Grothendieck universe co... |
| gruss 10819 | Any subset of an element o... |
| grupr 10820 | A Grothendieck universe co... |
| gruurn 10821 | A Grothendieck universe co... |
| gruiun 10822 | If ` B ( x ) ` is a family... |
| gruuni 10823 | A Grothendieck universe co... |
| grurn 10824 | A Grothendieck universe co... |
| gruima 10825 | A Grothendieck universe co... |
| gruel 10826 | Any element of an element ... |
| grusn 10827 | A Grothendieck universe co... |
| gruop 10828 | A Grothendieck universe co... |
| gruun 10829 | A Grothendieck universe co... |
| gruxp 10830 | A Grothendieck universe co... |
| grumap 10831 | A Grothendieck universe co... |
| gruixp 10832 | A Grothendieck universe co... |
| gruiin 10833 | A Grothendieck universe co... |
| gruf 10834 | A Grothendieck universe co... |
| gruen 10835 | A Grothendieck universe co... |
| gruwun 10836 | A nonempty Grothendieck un... |
| intgru 10837 | The intersection of a fami... |
| ingru 10838 | The intersection of a univ... |
| wfgru 10839 | The wellfounded part of a ... |
| grudomon 10840 | Each ordinal that is compa... |
| gruina 10841 | If a Grothendieck universe... |
| grur1a 10842 | A characterization of Grot... |
| grur1 10843 | A characterization of Grot... |
| grutsk1 10844 | Grothendieck universes are... |
| grutsk 10845 | Grothendieck universes are... |
| axgroth5 10847 | The Tarski-Grothendieck ax... |
| axgroth2 10848 | Alternate version of the T... |
| grothpw 10849 | Derive the Axiom of Power ... |
| grothpwex 10850 | Derive the Axiom of Power ... |
| axgroth6 10851 | The Tarski-Grothendieck ax... |
| grothomex 10852 | The Tarski-Grothendieck Ax... |
| grothac 10853 | The Tarski-Grothendieck Ax... |
| axgroth3 10854 | Alternate version of the T... |
| axgroth4 10855 | Alternate version of the T... |
| grothprimlem 10856 | Lemma for ~ grothprim . E... |
| grothprim 10857 | The Tarski-Grothendieck Ax... |
| grothtsk 10858 | The Tarski-Grothendieck Ax... |
| inaprc 10859 | An equivalent to the Tarsk... |
| tskmval 10862 | Value of our tarski map. ... |
| tskmid 10863 | The set ` A ` is an elemen... |
| tskmcl 10864 | A Tarski class that contai... |
| sstskm 10865 | Being a part of ` ( tarski... |
| eltskm 10866 | Belonging to ` ( tarskiMap... |
| elni 10899 | Membership in the class of... |
| elni2 10900 | Membership in the class of... |
| pinn 10901 | A positive integer is a na... |
| pion 10902 | A positive integer is an o... |
| piord 10903 | A positive integer is ordi... |
| niex 10904 | The class of positive inte... |
| 0npi 10905 | The empty set is not a pos... |
| 1pi 10906 | Ordinal 'one' is a positiv... |
| addpiord 10907 | Positive integer addition ... |
| mulpiord 10908 | Positive integer multiplic... |
| mulidpi 10909 | 1 is an identity element f... |
| ltpiord 10910 | Positive integer 'less tha... |
| ltsopi 10911 | Positive integer 'less tha... |
| ltrelpi 10912 | Positive integer 'less tha... |
| dmaddpi 10913 | Domain of addition on posi... |
| dmmulpi 10914 | Domain of multiplication o... |
| addclpi 10915 | Closure of addition of pos... |
| mulclpi 10916 | Closure of multiplication ... |
| addcompi 10917 | Addition of positive integ... |
| addasspi 10918 | Addition of positive integ... |
| mulcompi 10919 | Multiplication of positive... |
| mulasspi 10920 | Multiplication of positive... |
| distrpi 10921 | Multiplication of positive... |
| addcanpi 10922 | Addition cancellation law ... |
| mulcanpi 10923 | Multiplication cancellatio... |
| addnidpi 10924 | There is no identity eleme... |
| ltexpi 10925 | Ordering on positive integ... |
| ltapi 10926 | Ordering property of addit... |
| ltmpi 10927 | Ordering property of multi... |
| 1lt2pi 10928 | One is less than two (one ... |
| nlt1pi 10929 | No positive integer is les... |
| indpi 10930 | Principle of Finite Induct... |
| enqbreq 10942 | Equivalence relation for p... |
| enqbreq2 10943 | Equivalence relation for p... |
| enqer 10944 | The equivalence relation f... |
| enqex 10945 | The equivalence relation f... |
| nqex 10946 | The class of positive frac... |
| 0nnq 10947 | The empty set is not a pos... |
| elpqn 10948 | Each positive fraction is ... |
| ltrelnq 10949 | Positive fraction 'less th... |
| pinq 10950 | The representatives of pos... |
| 1nq 10951 | The positive fraction 'one... |
| nqereu 10952 | There is a unique element ... |
| nqerf 10953 | Corollary of ~ nqereu : th... |
| nqercl 10954 | Corollary of ~ nqereu : cl... |
| nqerrel 10955 | Any member of ` ( N. X. N.... |
| nqerid 10956 | Corollary of ~ nqereu : th... |
| enqeq 10957 | Corollary of ~ nqereu : if... |
| nqereq 10958 | The function ` /Q ` acts a... |
| addpipq2 10959 | Addition of positive fract... |
| addpipq 10960 | Addition of positive fract... |
| addpqnq 10961 | Addition of positive fract... |
| mulpipq2 10962 | Multiplication of positive... |
| mulpipq 10963 | Multiplication of positive... |
| mulpqnq 10964 | Multiplication of positive... |
| ordpipq 10965 | Ordering of positive fract... |
| ordpinq 10966 | Ordering of positive fract... |
| addpqf 10967 | Closure of addition on pos... |
| addclnq 10968 | Closure of addition on pos... |
| mulpqf 10969 | Closure of multiplication ... |
| mulclnq 10970 | Closure of multiplication ... |
| addnqf 10971 | Domain of addition on posi... |
| mulnqf 10972 | Domain of multiplication o... |
| addcompq 10973 | Addition of positive fract... |
| addcomnq 10974 | Addition of positive fract... |
| mulcompq 10975 | Multiplication of positive... |
| mulcomnq 10976 | Multiplication of positive... |
| adderpqlem 10977 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10978 | Lemma for ~ mulerpq . (Co... |
| adderpq 10979 | Addition is compatible wit... |
| mulerpq 10980 | Multiplication is compatib... |
| addassnq 10981 | Addition of positive fract... |
| mulassnq 10982 | Multiplication of positive... |
| mulcanenq 10983 | Lemma for distributive law... |
| distrnq 10984 | Multiplication of positive... |
| 1nqenq 10985 | The equivalence class of r... |
| mulidnq 10986 | Multiplication identity el... |
| recmulnq 10987 | Relationship between recip... |
| recidnq 10988 | A positive fraction times ... |
| recclnq 10989 | Closure law for positive f... |
| recrecnq 10990 | Reciprocal of reciprocal o... |
| dmrecnq 10991 | Domain of reciprocal on po... |
| ltsonq 10992 | 'Less than' is a strict or... |
| lterpq 10993 | Compatibility of ordering ... |
| ltanq 10994 | Ordering property of addit... |
| ltmnq 10995 | Ordering property of multi... |
| 1lt2nq 10996 | One is less than two (one ... |
| ltaddnq 10997 | The sum of two fractions i... |
| ltexnq 10998 | Ordering on positive fract... |
| halfnq 10999 | One-half of any positive f... |
| nsmallnq 11000 | The is no smallest positiv... |
| ltbtwnnq 11001 | There exists a number betw... |
| ltrnq 11002 | Ordering property of recip... |
| archnq 11003 | For any fraction, there is... |
| npex 11009 | The class of positive real... |
| elnp 11010 | Membership in positive rea... |
| elnpi 11011 | Membership in positive rea... |
| prn0 11012 | A positive real is not emp... |
| prpssnq 11013 | A positive real is a subse... |
| elprnq 11014 | A positive real is a set o... |
| 0npr 11015 | The empty set is not a pos... |
| prcdnq 11016 | A positive real is closed ... |
| prub 11017 | A positive fraction not in... |
| prnmax 11018 | A positive real has no lar... |
| npomex 11019 | A simplifying observation,... |
| prnmadd 11020 | A positive real has no lar... |
| ltrelpr 11021 | Positive real 'less than' ... |
| genpv 11022 | Value of general operation... |
| genpelv 11023 | Membership in value of gen... |
| genpprecl 11024 | Pre-closure law for genera... |
| genpdm 11025 | Domain of general operatio... |
| genpn0 11026 | The result of an operation... |
| genpss 11027 | The result of an operation... |
| genpnnp 11028 | The result of an operation... |
| genpcd 11029 | Downward closure of an ope... |
| genpnmax 11030 | An operation on positive r... |
| genpcl 11031 | Closure of an operation on... |
| genpass 11032 | Associativity of an operat... |
| plpv 11033 | Value of addition on posit... |
| mpv 11034 | Value of multiplication on... |
| dmplp 11035 | Domain of addition on posi... |
| dmmp 11036 | Domain of multiplication o... |
| nqpr 11037 | The canonical embedding of... |
| 1pr 11038 | The positive real number '... |
| addclprlem1 11039 | Lemma to prove downward cl... |
| addclprlem2 11040 | Lemma to prove downward cl... |
| addclpr 11041 | Closure of addition on pos... |
| mulclprlem 11042 | Lemma to prove downward cl... |
| mulclpr 11043 | Closure of multiplication ... |
| addcompr 11044 | Addition of positive reals... |
| addasspr 11045 | Addition of positive reals... |
| mulcompr 11046 | Multiplication of positive... |
| mulasspr 11047 | Multiplication of positive... |
| distrlem1pr 11048 | Lemma for distributive law... |
| distrlem4pr 11049 | Lemma for distributive law... |
| distrlem5pr 11050 | Lemma for distributive law... |
| distrpr 11051 | Multiplication of positive... |
| 1idpr 11052 | 1 is an identity element f... |
| ltprord 11053 | Positive real 'less than' ... |
| psslinpr 11054 | Proper subset is a linear ... |
| ltsopr 11055 | Positive real 'less than' ... |
| prlem934 11056 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11057 | The sum of two positive re... |
| ltaddpr2 11058 | The sum of two positive re... |
| ltexprlem1 11059 | Lemma for Proposition 9-3.... |
| ltexprlem2 11060 | Lemma for Proposition 9-3.... |
| ltexprlem3 11061 | Lemma for Proposition 9-3.... |
| ltexprlem4 11062 | Lemma for Proposition 9-3.... |
| ltexprlem5 11063 | Lemma for Proposition 9-3.... |
| ltexprlem6 11064 | Lemma for Proposition 9-3.... |
| ltexprlem7 11065 | Lemma for Proposition 9-3.... |
| ltexpri 11066 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11067 | Lemma for Proposition 9-3.... |
| ltapr 11068 | Ordering property of addit... |
| addcanpr 11069 | Addition cancellation law ... |
| prlem936 11070 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11071 | Lemma for Proposition 9-3.... |
| reclem3pr 11072 | Lemma for Proposition 9-3.... |
| reclem4pr 11073 | Lemma for Proposition 9-3.... |
| recexpr 11074 | The reciprocal of a positi... |
| suplem1pr 11075 | The union of a nonempty, b... |
| suplem2pr 11076 | The union of a set of posi... |
| supexpr 11077 | The union of a nonempty, b... |
| enrer 11086 | The equivalence relation f... |
| nrex1 11087 | The class of signed reals ... |
| enrbreq 11088 | Equivalence relation for s... |
| enreceq 11089 | Equivalence class equality... |
| enrex 11090 | The equivalence relation f... |
| ltrelsr 11091 | Signed real 'less than' is... |
| addcmpblnr 11092 | Lemma showing compatibilit... |
| mulcmpblnrlem 11093 | Lemma used in lemma showin... |
| mulcmpblnr 11094 | Lemma showing compatibilit... |
| prsrlem1 11095 | Decomposing signed reals i... |
| addsrmo 11096 | There is at most one resul... |
| mulsrmo 11097 | There is at most one resul... |
| addsrpr 11098 | Addition of signed reals i... |
| mulsrpr 11099 | Multiplication of signed r... |
| ltsrpr 11100 | Ordering of signed reals i... |
| gt0srpr 11101 | Greater than zero in terms... |
| 0nsr 11102 | The empty set is not a sig... |
| 0r 11103 | The constant ` 0R ` is a s... |
| 1sr 11104 | The constant ` 1R ` is a s... |
| m1r 11105 | The constant ` -1R ` is a ... |
| addclsr 11106 | Closure of addition on sig... |
| mulclsr 11107 | Closure of multiplication ... |
| dmaddsr 11108 | Domain of addition on sign... |
| dmmulsr 11109 | Domain of multiplication o... |
| addcomsr 11110 | Addition of signed reals i... |
| addasssr 11111 | Addition of signed reals i... |
| mulcomsr 11112 | Multiplication of signed r... |
| mulasssr 11113 | Multiplication of signed r... |
| distrsr 11114 | Multiplication of signed r... |
| m1p1sr 11115 | Minus one plus one is zero... |
| m1m1sr 11116 | Minus one times minus one ... |
| ltsosr 11117 | Signed real 'less than' is... |
| 0lt1sr 11118 | 0 is less than 1 for signe... |
| 1ne0sr 11119 | 1 and 0 are distinct for s... |
| 0idsr 11120 | The signed real number 0 i... |
| 1idsr 11121 | 1 is an identity element f... |
| 00sr 11122 | A signed real times 0 is 0... |
| ltasr 11123 | Ordering property of addit... |
| pn0sr 11124 | A signed real plus its neg... |
| negexsr 11125 | Existence of negative sign... |
| recexsrlem 11126 | The reciprocal of a positi... |
| addgt0sr 11127 | The sum of two positive si... |
| mulgt0sr 11128 | The product of two positiv... |
| sqgt0sr 11129 | The square of a nonzero si... |
| recexsr 11130 | The reciprocal of a nonzer... |
| mappsrpr 11131 | Mapping from positive sign... |
| ltpsrpr 11132 | Mapping of order from posi... |
| map2psrpr 11133 | Equivalence for positive s... |
| supsrlem 11134 | Lemma for supremum theorem... |
| supsr 11135 | A nonempty, bounded set of... |
| opelcn 11152 | Ordered pair membership in... |
| opelreal 11153 | Ordered pair membership in... |
| elreal 11154 | Membership in class of rea... |
| elreal2 11155 | Ordered pair membership in... |
| 0ncn 11156 | The empty set is not a com... |
| ltrelre 11157 | 'Less than' is a relation ... |
| addcnsr 11158 | Addition of complex number... |
| mulcnsr 11159 | Multiplication of complex ... |
| eqresr 11160 | Equality of real numbers i... |
| addresr 11161 | Addition of real numbers i... |
| mulresr 11162 | Multiplication of real num... |
| ltresr 11163 | Ordering of real subset of... |
| ltresr2 11164 | Ordering of real subset of... |
| dfcnqs 11165 | Technical trick to permit ... |
| addcnsrec 11166 | Technical trick to permit ... |
| mulcnsrec 11167 | Technical trick to permit ... |
| axaddf 11168 | Addition is an operation o... |
| axmulf 11169 | Multiplication is an opera... |
| axcnex 11170 | The complex numbers form a... |
| axresscn 11171 | The real numbers are a sub... |
| ax1cn 11172 | 1 is a complex number. Ax... |
| axicn 11173 | ` _i ` is a complex number... |
| axaddcl 11174 | Closure law for addition o... |
| axaddrcl 11175 | Closure law for addition i... |
| axmulcl 11176 | Closure law for multiplica... |
| axmulrcl 11177 | Closure law for multiplica... |
| axmulcom 11178 | Multiplication of complex ... |
| axaddass 11179 | Addition of complex number... |
| axmulass 11180 | Multiplication of complex ... |
| axdistr 11181 | Distributive law for compl... |
| axi2m1 11182 | i-squared equals -1 (expre... |
| ax1ne0 11183 | 1 and 0 are distinct. Axi... |
| ax1rid 11184 | ` 1 ` is an identity eleme... |
| axrnegex 11185 | Existence of negative of r... |
| axrrecex 11186 | Existence of reciprocal of... |
| axcnre 11187 | A complex number can be ex... |
| axpre-lttri 11188 | Ordering on reals satisfie... |
| axpre-lttrn 11189 | Ordering on reals is trans... |
| axpre-ltadd 11190 | Ordering property of addit... |
| axpre-mulgt0 11191 | The product of two positiv... |
| axpre-sup 11192 | A nonempty, bounded-above ... |
| wuncn 11193 | A weak universe containing... |
| cnex 11219 | Alias for ~ ax-cnex . See... |
| addcl 11220 | Alias for ~ ax-addcl , for... |
| readdcl 11221 | Alias for ~ ax-addrcl , fo... |
| mulcl 11222 | Alias for ~ ax-mulcl , for... |
| remulcl 11223 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11224 | Alias for ~ ax-mulcom , fo... |
| addass 11225 | Alias for ~ ax-addass , fo... |
| mulass 11226 | Alias for ~ ax-mulass , fo... |
| adddi 11227 | Alias for ~ ax-distr , for... |
| recn 11228 | A real number is a complex... |
| reex 11229 | The real numbers form a se... |
| reelprrecn 11230 | Reals are a subset of the ... |
| cnelprrecn 11231 | Complex numbers are a subs... |
| mpoaddf 11232 | Addition is an operation o... |
| mpomulf 11233 | Multiplication is an opera... |
| elimne0 11234 | Hypothesis for weak deduct... |
| adddir 11235 | Distributive law for compl... |
| 0cn 11236 | Zero is a complex number. ... |
| 0cnd 11237 | Zero is a complex number, ... |
| c0ex 11238 | Zero is a set. (Contribut... |
| 1cnd 11239 | One is a complex number, d... |
| 1ex 11240 | One is a set. (Contribute... |
| cnre 11241 | Alias for ~ ax-cnre , for ... |
| mulrid 11242 | The number 1 is an identit... |
| mullid 11243 | Identity law for multiplic... |
| 1re 11244 | The number 1 is real. Thi... |
| 1red 11245 | The number 1 is real, dedu... |
| 0re 11246 | The number 0 is real. Rem... |
| 0red 11247 | The number 0 is real, dedu... |
| mulridi 11248 | Identity law for multiplic... |
| mullidi 11249 | Identity law for multiplic... |
| addcli 11250 | Closure law for addition. ... |
| mulcli 11251 | Closure law for multiplica... |
| mulcomi 11252 | Commutative law for multip... |
| mulcomli 11253 | Commutative law for multip... |
| addassi 11254 | Associative law for additi... |
| mulassi 11255 | Associative law for multip... |
| adddii 11256 | Distributive law (left-dis... |
| adddiri 11257 | Distributive law (right-di... |
| recni 11258 | A real number is a complex... |
| readdcli 11259 | Closure law for addition o... |
| remulcli 11260 | Closure law for multiplica... |
| mulridd 11261 | Identity law for multiplic... |
| mullidd 11262 | Identity law for multiplic... |
| addcld 11263 | Closure law for addition. ... |
| mulcld 11264 | Closure law for multiplica... |
| mulcomd 11265 | Commutative law for multip... |
| addassd 11266 | Associative law for additi... |
| mulassd 11267 | Associative law for multip... |
| adddid 11268 | Distributive law (left-dis... |
| adddird 11269 | Distributive law (right-di... |
| adddirp1d 11270 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11271 | Join AB+CB into (A+C) on L... |
| recnd 11272 | Deduction from real number... |
| readdcld 11273 | Closure law for addition o... |
| remulcld 11274 | Closure law for multiplica... |
| pnfnre 11285 | Plus infinity is not a rea... |
| pnfnre2 11286 | Plus infinity is not a rea... |
| mnfnre 11287 | Minus infinity is not a re... |
| ressxr 11288 | The standard reals are a s... |
| rexpssxrxp 11289 | The Cartesian product of s... |
| rexr 11290 | A standard real is an exte... |
| 0xr 11291 | Zero is an extended real. ... |
| renepnf 11292 | No (finite) real equals pl... |
| renemnf 11293 | No real equals minus infin... |
| rexrd 11294 | A standard real is an exte... |
| renepnfd 11295 | No (finite) real equals pl... |
| renemnfd 11296 | No real equals minus infin... |
| pnfex 11297 | Plus infinity exists. (Co... |
| pnfxr 11298 | Plus infinity belongs to t... |
| pnfnemnf 11299 | Plus and minus infinity ar... |
| mnfnepnf 11300 | Minus and plus infinity ar... |
| mnfxr 11301 | Minus infinity belongs to ... |
| rexri 11302 | A standard real is an exte... |
| 1xr 11303 | ` 1 ` is an extended real ... |
| renfdisj 11304 | The reals and the infiniti... |
| ltrelxr 11305 | "Less than" is a relation ... |
| ltrel 11306 | "Less than" is a relation.... |
| lerelxr 11307 | "Less than or equal to" is... |
| lerel 11308 | "Less than or equal to" is... |
| xrlenlt 11309 | "Less than or equal to" ex... |
| xrlenltd 11310 | "Less than or equal to" ex... |
| xrltnle 11311 | "Less than" expressed in t... |
| xrnltled 11312 | "Not less than" implies "l... |
| ssxr 11313 | The three (non-exclusive) ... |
| ltxrlt 11314 | The standard less-than ` <... |
| axlttri 11315 | Ordering on reals satisfie... |
| axlttrn 11316 | Ordering on reals is trans... |
| axltadd 11317 | Ordering property of addit... |
| axmulgt0 11318 | The product of two positiv... |
| axsup 11319 | A nonempty, bounded-above ... |
| lttr 11320 | Alias for ~ axlttrn , for ... |
| mulgt0 11321 | The product of two positiv... |
| lenlt 11322 | 'Less than or equal to' ex... |
| ltnle 11323 | 'Less than' expressed in t... |
| ltso 11324 | 'Less than' is a strict or... |
| gtso 11325 | 'Greater than' is a strict... |
| lttri2 11326 | Consequence of trichotomy.... |
| lttri3 11327 | Trichotomy law for 'less t... |
| lttri4 11328 | Trichotomy law for 'less t... |
| letri3 11329 | Trichotomy law. (Contribu... |
| leloe 11330 | 'Less than or equal to' ex... |
| eqlelt 11331 | Equality in terms of 'less... |
| ltle 11332 | 'Less than' implies 'less ... |
| leltne 11333 | 'Less than or equal to' im... |
| lelttr 11334 | Transitive law. (Contribu... |
| leltletr 11335 | Transitive law, weaker for... |
| ltletr 11336 | Transitive law. (Contribu... |
| ltleletr 11337 | Transitive law, weaker for... |
| letr 11338 | Transitive law. (Contribu... |
| ltnr 11339 | 'Less than' is irreflexive... |
| leid 11340 | 'Less than or equal to' is... |
| ltne 11341 | 'Less than' implies not eq... |
| ltnsym 11342 | 'Less than' is not symmetr... |
| ltnsym2 11343 | 'Less than' is antisymmetr... |
| letric 11344 | Trichotomy law. (Contribu... |
| ltlen 11345 | 'Less than' expressed in t... |
| eqle 11346 | Equality implies 'less tha... |
| eqled 11347 | Equality implies 'less tha... |
| ltadd2 11348 | Addition to both sides of ... |
| ne0gt0 11349 | A nonzero nonnegative numb... |
| lecasei 11350 | Ordering elimination by ca... |
| lelttric 11351 | Trichotomy law. (Contribu... |
| ltlecasei 11352 | Ordering elimination by ca... |
| ltnri 11353 | 'Less than' is irreflexive... |
| eqlei 11354 | Equality implies 'less tha... |
| eqlei2 11355 | Equality implies 'less tha... |
| gtneii 11356 | 'Less than' implies not eq... |
| ltneii 11357 | 'Greater than' implies not... |
| lttri2i 11358 | Consequence of trichotomy.... |
| lttri3i 11359 | Consequence of trichotomy.... |
| letri3i 11360 | Consequence of trichotomy.... |
| leloei 11361 | 'Less than or equal to' in... |
| ltleni 11362 | 'Less than' expressed in t... |
| ltnsymi 11363 | 'Less than' is not symmetr... |
| lenlti 11364 | 'Less than or equal to' in... |
| ltnlei 11365 | 'Less than' in terms of 'l... |
| ltlei 11366 | 'Less than' implies 'less ... |
| ltleii 11367 | 'Less than' implies 'less ... |
| ltnei 11368 | 'Less than' implies not eq... |
| letrii 11369 | Trichotomy law for 'less t... |
| lttri 11370 | 'Less than' is transitive.... |
| lelttri 11371 | 'Less than or equal to', '... |
| ltletri 11372 | 'Less than', 'less than or... |
| letri 11373 | 'Less than or equal to' is... |
| le2tri3i 11374 | Extended trichotomy law fo... |
| ltadd2i 11375 | Addition to both sides of ... |
| mulgt0i 11376 | The product of two positiv... |
| mulgt0ii 11377 | The product of two positiv... |
| ltnrd 11378 | 'Less than' is irreflexive... |
| gtned 11379 | 'Less than' implies not eq... |
| ltned 11380 | 'Greater than' implies not... |
| ne0gt0d 11381 | A nonzero nonnegative numb... |
| lttrid 11382 | Ordering on reals satisfie... |
| lttri2d 11383 | Consequence of trichotomy.... |
| lttri3d 11384 | Consequence of trichotomy.... |
| lttri4d 11385 | Trichotomy law for 'less t... |
| letri3d 11386 | Consequence of trichotomy.... |
| leloed 11387 | 'Less than or equal to' in... |
| eqleltd 11388 | Equality in terms of 'less... |
| ltlend 11389 | 'Less than' expressed in t... |
| lenltd 11390 | 'Less than or equal to' in... |
| ltnled 11391 | 'Less than' in terms of 'l... |
| ltled 11392 | 'Less than' implies 'less ... |
| ltnsymd 11393 | 'Less than' implies 'less ... |
| nltled 11394 | 'Not less than ' implies '... |
| lensymd 11395 | 'Less than or equal to' im... |
| letrid 11396 | Trichotomy law for 'less t... |
| leltned 11397 | 'Less than or equal to' im... |
| leneltd 11398 | 'Less than or equal to' an... |
| mulgt0d 11399 | The product of two positiv... |
| ltadd2d 11400 | Addition to both sides of ... |
| letrd 11401 | Transitive law deduction f... |
| lelttrd 11402 | Transitive law deduction f... |
| ltadd2dd 11403 | Addition to both sides of ... |
| ltletrd 11404 | Transitive law deduction f... |
| lttrd 11405 | Transitive law deduction f... |
| lelttrdi 11406 | If a number is less than a... |
| dedekind 11407 | The Dedekind cut theorem. ... |
| dedekindle 11408 | The Dedekind cut theorem, ... |
| mul12 11409 | Commutative/associative la... |
| mul32 11410 | Commutative/associative la... |
| mul31 11411 | Commutative/associative la... |
| mul4 11412 | Rearrangement of 4 factors... |
| mul4r 11413 | Rearrangement of 4 factors... |
| muladd11 11414 | A simple product of sums e... |
| 1p1times 11415 | Two times a number. (Cont... |
| peano2cn 11416 | A theorem for complex numb... |
| peano2re 11417 | A theorem for reals analog... |
| readdcan 11418 | Cancellation law for addit... |
| 00id 11419 | ` 0 ` is its own additive ... |
| mul02lem1 11420 | Lemma for ~ mul02 . If an... |
| mul02lem2 11421 | Lemma for ~ mul02 . Zero ... |
| mul02 11422 | Multiplication by ` 0 ` . ... |
| mul01 11423 | Multiplication by ` 0 ` . ... |
| addrid 11424 | ` 0 ` is an additive ident... |
| cnegex 11425 | Existence of the negative ... |
| cnegex2 11426 | Existence of a left invers... |
| addlid 11427 | ` 0 ` is a left identity f... |
| addcan 11428 | Cancellation law for addit... |
| addcan2 11429 | Cancellation law for addit... |
| addcom 11430 | Addition commutes. This u... |
| addridi 11431 | ` 0 ` is an additive ident... |
| addlidi 11432 | ` 0 ` is a left identity f... |
| mul02i 11433 | Multiplication by 0. Theo... |
| mul01i 11434 | Multiplication by ` 0 ` . ... |
| addcomi 11435 | Addition commutes. Based ... |
| addcomli 11436 | Addition commutes. (Contr... |
| addcani 11437 | Cancellation law for addit... |
| addcan2i 11438 | Cancellation law for addit... |
| mul12i 11439 | Commutative/associative la... |
| mul32i 11440 | Commutative/associative la... |
| mul4i 11441 | Rearrangement of 4 factors... |
| mul02d 11442 | Multiplication by 0. Theo... |
| mul01d 11443 | Multiplication by ` 0 ` . ... |
| addridd 11444 | ` 0 ` is an additive ident... |
| addlidd 11445 | ` 0 ` is a left identity f... |
| addcomd 11446 | Addition commutes. Based ... |
| addcand 11447 | Cancellation law for addit... |
| addcan2d 11448 | Cancellation law for addit... |
| addcanad 11449 | Cancelling a term on the l... |
| addcan2ad 11450 | Cancelling a term on the r... |
| addneintrd 11451 | Introducing a term on the ... |
| addneintr2d 11452 | Introducing a term on the ... |
| mul12d 11453 | Commutative/associative la... |
| mul32d 11454 | Commutative/associative la... |
| mul31d 11455 | Commutative/associative la... |
| mul4d 11456 | Rearrangement of 4 factors... |
| muladd11r 11457 | A simple product of sums e... |
| comraddd 11458 | Commute RHS addition, in d... |
| ltaddneg 11459 | Adding a negative number t... |
| ltaddnegr 11460 | Adding a negative number t... |
| add12 11461 | Commutative/associative la... |
| add32 11462 | Commutative/associative la... |
| add32r 11463 | Commutative/associative la... |
| add4 11464 | Rearrangement of 4 terms i... |
| add42 11465 | Rearrangement of 4 terms i... |
| add12i 11466 | Commutative/associative la... |
| add32i 11467 | Commutative/associative la... |
| add4i 11468 | Rearrangement of 4 terms i... |
| add42i 11469 | Rearrangement of 4 terms i... |
| add12d 11470 | Commutative/associative la... |
| add32d 11471 | Commutative/associative la... |
| add4d 11472 | Rearrangement of 4 terms i... |
| add42d 11473 | Rearrangement of 4 terms i... |
| 0cnALT 11478 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11479 | Alternate proof of ~ 0cnAL... |
| negeu 11480 | Existential uniqueness of ... |
| subval 11481 | Value of subtraction, whic... |
| negeq 11482 | Equality theorem for negat... |
| negeqi 11483 | Equality inference for neg... |
| negeqd 11484 | Equality deduction for neg... |
| nfnegd 11485 | Deduction version of ~ nfn... |
| nfneg 11486 | Bound-variable hypothesis ... |
| csbnegg 11487 | Move class substitution in... |
| negex 11488 | A negative is a set. (Con... |
| subcl 11489 | Closure law for subtractio... |
| negcl 11490 | Closure law for negative. ... |
| negicn 11491 | ` -u _i ` is a complex num... |
| subf 11492 | Subtraction is an operatio... |
| subadd 11493 | Relationship between subtr... |
| subadd2 11494 | Relationship between subtr... |
| subsub23 11495 | Swap subtrahend and result... |
| pncan 11496 | Cancellation law for subtr... |
| pncan2 11497 | Cancellation law for subtr... |
| pncan3 11498 | Subtraction and addition o... |
| npcan 11499 | Cancellation law for subtr... |
| addsubass 11500 | Associative-type law for a... |
| addsub 11501 | Law for addition and subtr... |
| subadd23 11502 | Commutative/associative la... |
| addsub12 11503 | Commutative/associative la... |
| 2addsub 11504 | Law for subtraction and ad... |
| addsubeq4 11505 | Relation between sums and ... |
| pncan3oi 11506 | Subtraction and addition o... |
| mvrraddi 11507 | Move the right term in a s... |
| mvlladdi 11508 | Move the left term in a su... |
| subid 11509 | Subtraction of a number fr... |
| subid1 11510 | Identity law for subtracti... |
| npncan 11511 | Cancellation law for subtr... |
| nppcan 11512 | Cancellation law for subtr... |
| nnpcan 11513 | Cancellation law for subtr... |
| nppcan3 11514 | Cancellation law for subtr... |
| subcan2 11515 | Cancellation law for subtr... |
| subeq0 11516 | If the difference between ... |
| npncan2 11517 | Cancellation law for subtr... |
| subsub2 11518 | Law for double subtraction... |
| nncan 11519 | Cancellation law for subtr... |
| subsub 11520 | Law for double subtraction... |
| nppcan2 11521 | Cancellation law for subtr... |
| subsub3 11522 | Law for double subtraction... |
| subsub4 11523 | Law for double subtraction... |
| sub32 11524 | Swap the second and third ... |
| nnncan 11525 | Cancellation law for subtr... |
| nnncan1 11526 | Cancellation law for subtr... |
| nnncan2 11527 | Cancellation law for subtr... |
| npncan3 11528 | Cancellation law for subtr... |
| pnpcan 11529 | Cancellation law for mixed... |
| pnpcan2 11530 | Cancellation law for mixed... |
| pnncan 11531 | Cancellation law for mixed... |
| ppncan 11532 | Cancellation law for mixed... |
| addsub4 11533 | Rearrangement of 4 terms i... |
| subadd4 11534 | Rearrangement of 4 terms i... |
| sub4 11535 | Rearrangement of 4 terms i... |
| neg0 11536 | Minus 0 equals 0. (Contri... |
| negid 11537 | Addition of a number and i... |
| negsub 11538 | Relationship between subtr... |
| subneg 11539 | Relationship between subtr... |
| negneg 11540 | A number is equal to the n... |
| neg11 11541 | Negative is one-to-one. (... |
| negcon1 11542 | Negative contraposition la... |
| negcon2 11543 | Negative contraposition la... |
| negeq0 11544 | A number is zero iff its n... |
| subcan 11545 | Cancellation law for subtr... |
| negsubdi 11546 | Distribution of negative o... |
| negdi 11547 | Distribution of negative o... |
| negdi2 11548 | Distribution of negative o... |
| negsubdi2 11549 | Distribution of negative o... |
| neg2sub 11550 | Relationship between subtr... |
| renegcli 11551 | Closure law for negative o... |
| resubcli 11552 | Closure law for subtractio... |
| renegcl 11553 | Closure law for negative o... |
| resubcl 11554 | Closure law for subtractio... |
| negreb 11555 | The negative of a real is ... |
| peano2cnm 11556 | "Reverse" second Peano pos... |
| peano2rem 11557 | "Reverse" second Peano pos... |
| negcli 11558 | Closure law for negative. ... |
| negidi 11559 | Addition of a number and i... |
| negnegi 11560 | A number is equal to the n... |
| subidi 11561 | Subtraction of a number fr... |
| subid1i 11562 | Identity law for subtracti... |
| negne0bi 11563 | A number is nonzero iff it... |
| negrebi 11564 | The negative of a real is ... |
| negne0i 11565 | The negative of a nonzero ... |
| subcli 11566 | Closure law for subtractio... |
| pncan3i 11567 | Subtraction and addition o... |
| negsubi 11568 | Relationship between subtr... |
| subnegi 11569 | Relationship between subtr... |
| subeq0i 11570 | If the difference between ... |
| neg11i 11571 | Negative is one-to-one. (... |
| negcon1i 11572 | Negative contraposition la... |
| negcon2i 11573 | Negative contraposition la... |
| negdii 11574 | Distribution of negative o... |
| negsubdii 11575 | Distribution of negative o... |
| negsubdi2i 11576 | Distribution of negative o... |
| subaddi 11577 | Relationship between subtr... |
| subadd2i 11578 | Relationship between subtr... |
| subaddrii 11579 | Relationship between subtr... |
| subsub23i 11580 | Swap subtrahend and result... |
| addsubassi 11581 | Associative-type law for s... |
| addsubi 11582 | Law for subtraction and ad... |
| subcani 11583 | Cancellation law for subtr... |
| subcan2i 11584 | Cancellation law for subtr... |
| pnncani 11585 | Cancellation law for mixed... |
| addsub4i 11586 | Rearrangement of 4 terms i... |
| 0reALT 11587 | Alternate proof of ~ 0re .... |
| negcld 11588 | Closure law for negative. ... |
| subidd 11589 | Subtraction of a number fr... |
| subid1d 11590 | Identity law for subtracti... |
| negidd 11591 | Addition of a number and i... |
| negnegd 11592 | A number is equal to the n... |
| negeq0d 11593 | A number is zero iff its n... |
| negne0bd 11594 | A number is nonzero iff it... |
| negcon1d 11595 | Contraposition law for una... |
| negcon1ad 11596 | Contraposition law for una... |
| neg11ad 11597 | The negatives of two compl... |
| negned 11598 | If two complex numbers are... |
| negne0d 11599 | The negative of a nonzero ... |
| negrebd 11600 | The negative of a real is ... |
| subcld 11601 | Closure law for subtractio... |
| pncand 11602 | Cancellation law for subtr... |
| pncan2d 11603 | Cancellation law for subtr... |
| pncan3d 11604 | Subtraction and addition o... |
| npcand 11605 | Cancellation law for subtr... |
| nncand 11606 | Cancellation law for subtr... |
| negsubd 11607 | Relationship between subtr... |
| subnegd 11608 | Relationship between subtr... |
| subeq0d 11609 | If the difference between ... |
| subne0d 11610 | Two unequal numbers have n... |
| subeq0ad 11611 | The difference of two comp... |
| subne0ad 11612 | If the difference of two c... |
| neg11d 11613 | If the difference between ... |
| negdid 11614 | Distribution of negative o... |
| negdi2d 11615 | Distribution of negative o... |
| negsubdid 11616 | Distribution of negative o... |
| negsubdi2d 11617 | Distribution of negative o... |
| neg2subd 11618 | Relationship between subtr... |
| subaddd 11619 | Relationship between subtr... |
| subadd2d 11620 | Relationship between subtr... |
| addsubassd 11621 | Associative-type law for s... |
| addsubd 11622 | Law for subtraction and ad... |
| subadd23d 11623 | Commutative/associative la... |
| addsub12d 11624 | Commutative/associative la... |
| npncand 11625 | Cancellation law for subtr... |
| nppcand 11626 | Cancellation law for subtr... |
| nppcan2d 11627 | Cancellation law for subtr... |
| nppcan3d 11628 | Cancellation law for subtr... |
| subsubd 11629 | Law for double subtraction... |
| subsub2d 11630 | Law for double subtraction... |
| subsub3d 11631 | Law for double subtraction... |
| subsub4d 11632 | Law for double subtraction... |
| sub32d 11633 | Swap the second and third ... |
| nnncand 11634 | Cancellation law for subtr... |
| nnncan1d 11635 | Cancellation law for subtr... |
| nnncan2d 11636 | Cancellation law for subtr... |
| npncan3d 11637 | Cancellation law for subtr... |
| pnpcand 11638 | Cancellation law for mixed... |
| pnpcan2d 11639 | Cancellation law for mixed... |
| pnncand 11640 | Cancellation law for mixed... |
| ppncand 11641 | Cancellation law for mixed... |
| subcand 11642 | Cancellation law for subtr... |
| subcan2d 11643 | Cancellation law for subtr... |
| subcanad 11644 | Cancellation law for subtr... |
| subneintrd 11645 | Introducing subtraction on... |
| subcan2ad 11646 | Cancellation law for subtr... |
| subneintr2d 11647 | Introducing subtraction on... |
| addsub4d 11648 | Rearrangement of 4 terms i... |
| subadd4d 11649 | Rearrangement of 4 terms i... |
| sub4d 11650 | Rearrangement of 4 terms i... |
| 2addsubd 11651 | Law for subtraction and ad... |
| addsubeq4d 11652 | Relation between sums and ... |
| subeqxfrd 11653 | Transfer two terms of a su... |
| mvlraddd 11654 | Move the right term in a s... |
| mvlladdd 11655 | Move the left term in a su... |
| mvrraddd 11656 | Move the right term in a s... |
| mvrladdd 11657 | Move the left term in a su... |
| assraddsubd 11658 | Associate RHS addition-sub... |
| subaddeqd 11659 | Transfer two terms of a su... |
| addlsub 11660 | Left-subtraction: Subtrac... |
| addrsub 11661 | Right-subtraction: Subtra... |
| subexsub 11662 | A subtraction law: Exchan... |
| addid0 11663 | If adding a number to a an... |
| addn0nid 11664 | Adding a nonzero number to... |
| pnpncand 11665 | Addition/subtraction cance... |
| subeqrev 11666 | Reverse the order of subtr... |
| addeq0 11667 | Two complex numbers add up... |
| pncan1 11668 | Cancellation law for addit... |
| npcan1 11669 | Cancellation law for subtr... |
| subeq0bd 11670 | If two complex numbers are... |
| renegcld 11671 | Closure law for negative o... |
| resubcld 11672 | Closure law for subtractio... |
| negn0 11673 | The image under negation o... |
| negf1o 11674 | Negation is an isomorphism... |
| kcnktkm1cn 11675 | k times k minus 1 is a com... |
| muladd 11676 | Product of two sums. (Con... |
| subdi 11677 | Distribution of multiplica... |
| subdir 11678 | Distribution of multiplica... |
| ine0 11679 | The imaginary unit ` _i ` ... |
| mulneg1 11680 | Product with negative is n... |
| mulneg2 11681 | The product with a negativ... |
| mulneg12 11682 | Swap the negative sign in ... |
| mul2neg 11683 | Product of two negatives. ... |
| submul2 11684 | Convert a subtraction to a... |
| mulm1 11685 | Product with minus one is ... |
| addneg1mul 11686 | Addition with product with... |
| mulsub 11687 | Product of two differences... |
| mulsub2 11688 | Swap the order of subtract... |
| mulm1i 11689 | Product with minus one is ... |
| mulneg1i 11690 | Product with negative is n... |
| mulneg2i 11691 | Product with negative is n... |
| mul2negi 11692 | Product of two negatives. ... |
| subdii 11693 | Distribution of multiplica... |
| subdiri 11694 | Distribution of multiplica... |
| muladdi 11695 | Product of two sums. (Con... |
| mulm1d 11696 | Product with minus one is ... |
| mulneg1d 11697 | Product with negative is n... |
| mulneg2d 11698 | Product with negative is n... |
| mul2negd 11699 | Product of two negatives. ... |
| subdid 11700 | Distribution of multiplica... |
| subdird 11701 | Distribution of multiplica... |
| muladdd 11702 | Product of two sums. (Con... |
| mulsubd 11703 | Product of two differences... |
| muls1d 11704 | Multiplication by one minu... |
| mulsubfacd 11705 | Multiplication followed by... |
| addmulsub 11706 | The product of a sum and a... |
| subaddmulsub 11707 | The difference with a prod... |
| mulsubaddmulsub 11708 | A special difference of a ... |
| gt0ne0 11709 | Positive implies nonzero. ... |
| lt0ne0 11710 | A number which is less tha... |
| ltadd1 11711 | Addition to both sides of ... |
| leadd1 11712 | Addition to both sides of ... |
| leadd2 11713 | Addition to both sides of ... |
| ltsubadd 11714 | 'Less than' relationship b... |
| ltsubadd2 11715 | 'Less than' relationship b... |
| lesubadd 11716 | 'Less than or equal to' re... |
| lesubadd2 11717 | 'Less than or equal to' re... |
| ltaddsub 11718 | 'Less than' relationship b... |
| ltaddsub2 11719 | 'Less than' relationship b... |
| leaddsub 11720 | 'Less than or equal to' re... |
| leaddsub2 11721 | 'Less than or equal to' re... |
| suble 11722 | Swap subtrahends in an ine... |
| lesub 11723 | Swap subtrahends in an ine... |
| ltsub23 11724 | 'Less than' relationship b... |
| ltsub13 11725 | 'Less than' relationship b... |
| le2add 11726 | Adding both sides of two '... |
| ltleadd 11727 | Adding both sides of two o... |
| leltadd 11728 | Adding both sides of two o... |
| lt2add 11729 | Adding both sides of two '... |
| addgt0 11730 | The sum of 2 positive numb... |
| addgegt0 11731 | The sum of nonnegative and... |
| addgtge0 11732 | The sum of nonnegative and... |
| addge0 11733 | The sum of 2 nonnegative n... |
| ltaddpos 11734 | Adding a positive number t... |
| ltaddpos2 11735 | Adding a positive number t... |
| ltsubpos 11736 | Subtracting a positive num... |
| posdif 11737 | Comparison of two numbers ... |
| lesub1 11738 | Subtraction from both side... |
| lesub2 11739 | Subtraction of both sides ... |
| ltsub1 11740 | Subtraction from both side... |
| ltsub2 11741 | Subtraction of both sides ... |
| lt2sub 11742 | Subtracting both sides of ... |
| le2sub 11743 | Subtracting both sides of ... |
| ltneg 11744 | Negative of both sides of ... |
| ltnegcon1 11745 | Contraposition of negative... |
| ltnegcon2 11746 | Contraposition of negative... |
| leneg 11747 | Negative of both sides of ... |
| lenegcon1 11748 | Contraposition of negative... |
| lenegcon2 11749 | Contraposition of negative... |
| lt0neg1 11750 | Comparison of a number and... |
| lt0neg2 11751 | Comparison of a number and... |
| le0neg1 11752 | Comparison of a number and... |
| le0neg2 11753 | Comparison of a number and... |
| addge01 11754 | A number is less than or e... |
| addge02 11755 | A number is less than or e... |
| add20 11756 | Two nonnegative numbers ar... |
| subge0 11757 | Nonnegative subtraction. ... |
| suble0 11758 | Nonpositive subtraction. ... |
| leaddle0 11759 | The sum of a real number a... |
| subge02 11760 | Nonnegative subtraction. ... |
| lesub0 11761 | Lemma to show a nonnegativ... |
| mulge0 11762 | The product of two nonnega... |
| mullt0 11763 | The product of two negativ... |
| msqgt0 11764 | A nonzero square is positi... |
| msqge0 11765 | A square is nonnegative. ... |
| 0lt1 11766 | 0 is less than 1. Theorem... |
| 0le1 11767 | 0 is less than or equal to... |
| relin01 11768 | An interval law for less t... |
| ltordlem 11769 | Lemma for ~ ltord1 . (Con... |
| ltord1 11770 | Infer an ordering relation... |
| leord1 11771 | Infer an ordering relation... |
| eqord1 11772 | A strictly increasing real... |
| ltord2 11773 | Infer an ordering relation... |
| leord2 11774 | Infer an ordering relation... |
| eqord2 11775 | A strictly decreasing real... |
| wloglei 11776 | Form of ~ wlogle where bot... |
| wlogle 11777 | If the predicate ` ch ( x ... |
| leidi 11778 | 'Less than or equal to' is... |
| gt0ne0i 11779 | Positive means nonzero (us... |
| gt0ne0ii 11780 | Positive implies nonzero. ... |
| msqgt0i 11781 | A nonzero square is positi... |
| msqge0i 11782 | A square is nonnegative. ... |
| addgt0i 11783 | Addition of 2 positive num... |
| addge0i 11784 | Addition of 2 nonnegative ... |
| addgegt0i 11785 | Addition of nonnegative an... |
| addgt0ii 11786 | Addition of 2 positive num... |
| add20i 11787 | Two nonnegative numbers ar... |
| ltnegi 11788 | Negative of both sides of ... |
| lenegi 11789 | Negative of both sides of ... |
| ltnegcon2i 11790 | Contraposition of negative... |
| mulge0i 11791 | The product of two nonnega... |
| lesub0i 11792 | Lemma to show a nonnegativ... |
| ltaddposi 11793 | Adding a positive number t... |
| posdifi 11794 | Comparison of two numbers ... |
| ltnegcon1i 11795 | Contraposition of negative... |
| lenegcon1i 11796 | Contraposition of negative... |
| subge0i 11797 | Nonnegative subtraction. ... |
| ltadd1i 11798 | Addition to both sides of ... |
| leadd1i 11799 | Addition to both sides of ... |
| leadd2i 11800 | Addition to both sides of ... |
| ltsubaddi 11801 | 'Less than' relationship b... |
| lesubaddi 11802 | 'Less than or equal to' re... |
| ltsubadd2i 11803 | 'Less than' relationship b... |
| lesubadd2i 11804 | 'Less than or equal to' re... |
| ltaddsubi 11805 | 'Less than' relationship b... |
| lt2addi 11806 | Adding both side of two in... |
| le2addi 11807 | Adding both side of two in... |
| gt0ne0d 11808 | Positive implies nonzero. ... |
| lt0ne0d 11809 | Something less than zero i... |
| leidd 11810 | 'Less than or equal to' is... |
| msqgt0d 11811 | A nonzero square is positi... |
| msqge0d 11812 | A square is nonnegative. ... |
| lt0neg1d 11813 | Comparison of a number and... |
| lt0neg2d 11814 | Comparison of a number and... |
| le0neg1d 11815 | Comparison of a number and... |
| le0neg2d 11816 | Comparison of a number and... |
| addgegt0d 11817 | Addition of nonnegative an... |
| addgtge0d 11818 | Addition of positive and n... |
| addgt0d 11819 | Addition of 2 positive num... |
| addge0d 11820 | Addition of 2 nonnegative ... |
| mulge0d 11821 | The product of two nonnega... |
| ltnegd 11822 | Negative of both sides of ... |
| lenegd 11823 | Negative of both sides of ... |
| ltnegcon1d 11824 | Contraposition of negative... |
| ltnegcon2d 11825 | Contraposition of negative... |
| lenegcon1d 11826 | Contraposition of negative... |
| lenegcon2d 11827 | Contraposition of negative... |
| ltaddposd 11828 | Adding a positive number t... |
| ltaddpos2d 11829 | Adding a positive number t... |
| ltsubposd 11830 | Subtracting a positive num... |
| posdifd 11831 | Comparison of two numbers ... |
| addge01d 11832 | A number is less than or e... |
| addge02d 11833 | A number is less than or e... |
| subge0d 11834 | Nonnegative subtraction. ... |
| suble0d 11835 | Nonpositive subtraction. ... |
| subge02d 11836 | Nonnegative subtraction. ... |
| ltadd1d 11837 | Addition to both sides of ... |
| leadd1d 11838 | Addition to both sides of ... |
| leadd2d 11839 | Addition to both sides of ... |
| ltsubaddd 11840 | 'Less than' relationship b... |
| lesubaddd 11841 | 'Less than or equal to' re... |
| ltsubadd2d 11842 | 'Less than' relationship b... |
| lesubadd2d 11843 | 'Less than or equal to' re... |
| ltaddsubd 11844 | 'Less than' relationship b... |
| ltaddsub2d 11845 | 'Less than' relationship b... |
| leaddsub2d 11846 | 'Less than or equal to' re... |
| subled 11847 | Swap subtrahends in an ine... |
| lesubd 11848 | Swap subtrahends in an ine... |
| ltsub23d 11849 | 'Less than' relationship b... |
| ltsub13d 11850 | 'Less than' relationship b... |
| lesub1d 11851 | Subtraction from both side... |
| lesub2d 11852 | Subtraction of both sides ... |
| ltsub1d 11853 | Subtraction from both side... |
| ltsub2d 11854 | Subtraction of both sides ... |
| ltadd1dd 11855 | Addition to both sides of ... |
| ltsub1dd 11856 | Subtraction from both side... |
| ltsub2dd 11857 | Subtraction of both sides ... |
| leadd1dd 11858 | Addition to both sides of ... |
| leadd2dd 11859 | Addition to both sides of ... |
| lesub1dd 11860 | Subtraction from both side... |
| lesub2dd 11861 | Subtraction of both sides ... |
| lesub3d 11862 | The result of subtracting ... |
| le2addd 11863 | Adding both side of two in... |
| le2subd 11864 | Subtracting both sides of ... |
| ltleaddd 11865 | Adding both sides of two o... |
| leltaddd 11866 | Adding both sides of two o... |
| lt2addd 11867 | Adding both side of two in... |
| lt2subd 11868 | Subtracting both sides of ... |
| possumd 11869 | Condition for a positive s... |
| sublt0d 11870 | When a subtraction gives a... |
| ltaddsublt 11871 | Addition and subtraction o... |
| 1le1 11872 | One is less than or equal ... |
| ixi 11873 | ` _i ` times itself is min... |
| recextlem1 11874 | Lemma for ~ recex . (Cont... |
| recextlem2 11875 | Lemma for ~ recex . (Cont... |
| recex 11876 | Existence of reciprocal of... |
| mulcand 11877 | Cancellation law for multi... |
| mulcan2d 11878 | Cancellation law for multi... |
| mulcanad 11879 | Cancellation of a nonzero ... |
| mulcan2ad 11880 | Cancellation of a nonzero ... |
| mulcan 11881 | Cancellation law for multi... |
| mulcan2 11882 | Cancellation law for multi... |
| mulcani 11883 | Cancellation law for multi... |
| mul0or 11884 | If a product is zero, one ... |
| mulne0b 11885 | The product of two nonzero... |
| mulne0 11886 | The product of two nonzero... |
| mulne0i 11887 | The product of two nonzero... |
| muleqadd 11888 | Property of numbers whose ... |
| receu 11889 | Existential uniqueness of ... |
| mulnzcnf 11890 | Multiplication maps nonzer... |
| msq0i 11891 | A number is zero iff its s... |
| mul0ori 11892 | If a product is zero, one ... |
| msq0d 11893 | A number is zero iff its s... |
| mul0ord 11894 | If a product is zero, one ... |
| mulne0bd 11895 | The product of two nonzero... |
| mulne0d 11896 | The product of two nonzero... |
| mulcan1g 11897 | A generalized form of the ... |
| mulcan2g 11898 | A generalized form of the ... |
| mulne0bad 11899 | A factor of a nonzero comp... |
| mulne0bbd 11900 | A factor of a nonzero comp... |
| 1div0 11903 | You can't divide by zero, ... |
| divval 11904 | Value of division: if ` A ... |
| divmul 11905 | Relationship between divis... |
| divmul2 11906 | Relationship between divis... |
| divmul3 11907 | Relationship between divis... |
| divcl 11908 | Closure law for division. ... |
| reccl 11909 | Closure law for reciprocal... |
| divcan2 11910 | A cancellation law for div... |
| divcan1 11911 | A cancellation law for div... |
| diveq0 11912 | A ratio is zero iff the nu... |
| divne0b 11913 | The ratio of nonzero numbe... |
| divne0 11914 | The ratio of nonzero numbe... |
| recne0 11915 | The reciprocal of a nonzer... |
| recid 11916 | Multiplication of a number... |
| recid2 11917 | Multiplication of a number... |
| divrec 11918 | Relationship between divis... |
| divrec2 11919 | Relationship between divis... |
| divass 11920 | An associative law for div... |
| div23 11921 | A commutative/associative ... |
| div32 11922 | A commutative/associative ... |
| div13 11923 | A commutative/associative ... |
| div12 11924 | A commutative/associative ... |
| divmulass 11925 | An associative law for div... |
| divmulasscom 11926 | An associative/commutative... |
| divdir 11927 | Distribution of division o... |
| divcan3 11928 | A cancellation law for div... |
| divcan4 11929 | A cancellation law for div... |
| div11 11930 | One-to-one relationship fo... |
| divid 11931 | A number divided by itself... |
| div0 11932 | Division into zero is zero... |
| div1 11933 | A number divided by 1 is i... |
| 1div1e1 11934 | 1 divided by 1 is 1. (Con... |
| diveq1 11935 | Equality in terms of unit ... |
| divneg 11936 | Move negative sign inside ... |
| muldivdir 11937 | Distribution of division o... |
| divsubdir 11938 | Distribution of division o... |
| subdivcomb1 11939 | Bring a term in a subtract... |
| subdivcomb2 11940 | Bring a term in a subtract... |
| recrec 11941 | A number is equal to the r... |
| rec11 11942 | Reciprocal is one-to-one. ... |
| rec11r 11943 | Mutual reciprocals. (Cont... |
| divmuldiv 11944 | Multiplication of two rati... |
| divdivdiv 11945 | Division of two ratios. T... |
| divcan5 11946 | Cancellation of common fac... |
| divmul13 11947 | Swap the denominators in t... |
| divmul24 11948 | Swap the numerators in the... |
| divmuleq 11949 | Cross-multiply in an equal... |
| recdiv 11950 | The reciprocal of a ratio.... |
| divcan6 11951 | Cancellation of inverted f... |
| divdiv32 11952 | Swap denominators in a div... |
| divcan7 11953 | Cancel equal divisors in a... |
| dmdcan 11954 | Cancellation law for divis... |
| divdiv1 11955 | Division into a fraction. ... |
| divdiv2 11956 | Division by a fraction. (... |
| recdiv2 11957 | Division into a reciprocal... |
| ddcan 11958 | Cancellation in a double d... |
| divadddiv 11959 | Addition of two ratios. T... |
| divsubdiv 11960 | Subtraction of two ratios.... |
| conjmul 11961 | Two numbers whose reciproc... |
| rereccl 11962 | Closure law for reciprocal... |
| redivcl 11963 | Closure law for division o... |
| eqneg 11964 | A number equal to its nega... |
| eqnegd 11965 | A complex number equals it... |
| eqnegad 11966 | If a complex number equals... |
| div2neg 11967 | Quotient of two negatives.... |
| divneg2 11968 | Move negative sign inside ... |
| recclzi 11969 | Closure law for reciprocal... |
| recne0zi 11970 | The reciprocal of a nonzer... |
| recidzi 11971 | Multiplication of a number... |
| div1i 11972 | A number divided by 1 is i... |
| eqnegi 11973 | A number equal to its nega... |
| reccli 11974 | Closure law for reciprocal... |
| recidi 11975 | Multiplication of a number... |
| recreci 11976 | A number is equal to the r... |
| dividi 11977 | A number divided by itself... |
| div0i 11978 | Division into zero is zero... |
| divclzi 11979 | Closure law for division. ... |
| divcan1zi 11980 | A cancellation law for div... |
| divcan2zi 11981 | A cancellation law for div... |
| divreczi 11982 | Relationship between divis... |
| divcan3zi 11983 | A cancellation law for div... |
| divcan4zi 11984 | A cancellation law for div... |
| rec11i 11985 | Reciprocal is one-to-one. ... |
| divcli 11986 | Closure law for division. ... |
| divcan2i 11987 | A cancellation law for div... |
| divcan1i 11988 | A cancellation law for div... |
| divreci 11989 | Relationship between divis... |
| divcan3i 11990 | A cancellation law for div... |
| divcan4i 11991 | A cancellation law for div... |
| divne0i 11992 | The ratio of nonzero numbe... |
| rec11ii 11993 | Reciprocal is one-to-one. ... |
| divasszi 11994 | An associative law for div... |
| divmulzi 11995 | Relationship between divis... |
| divdirzi 11996 | Distribution of division o... |
| divdiv23zi 11997 | Swap denominators in a div... |
| divmuli 11998 | Relationship between divis... |
| divdiv32i 11999 | Swap denominators in a div... |
| divassi 12000 | An associative law for div... |
| divdiri 12001 | Distribution of division o... |
| div23i 12002 | A commutative/associative ... |
| div11i 12003 | One-to-one relationship fo... |
| divmuldivi 12004 | Multiplication of two rati... |
| divmul13i 12005 | Swap denominators of two r... |
| divadddivi 12006 | Addition of two ratios. T... |
| divdivdivi 12007 | Division of two ratios. T... |
| rerecclzi 12008 | Closure law for reciprocal... |
| rereccli 12009 | Closure law for reciprocal... |
| redivclzi 12010 | Closure law for division o... |
| redivcli 12011 | Closure law for division o... |
| div1d 12012 | A number divided by 1 is i... |
| reccld 12013 | Closure law for reciprocal... |
| recne0d 12014 | The reciprocal of a nonzer... |
| recidd 12015 | Multiplication of a number... |
| recid2d 12016 | Multiplication of a number... |
| recrecd 12017 | A number is equal to the r... |
| dividd 12018 | A number divided by itself... |
| div0d 12019 | Division into zero is zero... |
| divcld 12020 | Closure law for division. ... |
| divcan1d 12021 | A cancellation law for div... |
| divcan2d 12022 | A cancellation law for div... |
| divrecd 12023 | Relationship between divis... |
| divrec2d 12024 | Relationship between divis... |
| divcan3d 12025 | A cancellation law for div... |
| divcan4d 12026 | A cancellation law for div... |
| diveq0d 12027 | A ratio is zero iff the nu... |
| diveq1d 12028 | Equality in terms of unit ... |
| diveq1ad 12029 | The quotient of two comple... |
| diveq0ad 12030 | A fraction of complex numb... |
| divne1d 12031 | If two complex numbers are... |
| divne0bd 12032 | A ratio is zero iff the nu... |
| divnegd 12033 | Move negative sign inside ... |
| divneg2d 12034 | Move negative sign inside ... |
| div2negd 12035 | Quotient of two negatives.... |
| divne0d 12036 | The ratio of nonzero numbe... |
| recdivd 12037 | The reciprocal of a ratio.... |
| recdiv2d 12038 | Division into a reciprocal... |
| divcan6d 12039 | Cancellation of inverted f... |
| ddcand 12040 | Cancellation in a double d... |
| rec11d 12041 | Reciprocal is one-to-one. ... |
| divmuld 12042 | Relationship between divis... |
| div32d 12043 | A commutative/associative ... |
| div13d 12044 | A commutative/associative ... |
| divdiv32d 12045 | Swap denominators in a div... |
| divcan5d 12046 | Cancellation of common fac... |
| divcan5rd 12047 | Cancellation of common fac... |
| divcan7d 12048 | Cancel equal divisors in a... |
| dmdcand 12049 | Cancellation law for divis... |
| dmdcan2d 12050 | Cancellation law for divis... |
| divdiv1d 12051 | Division into a fraction. ... |
| divdiv2d 12052 | Division by a fraction. (... |
| divmul2d 12053 | Relationship between divis... |
| divmul3d 12054 | Relationship between divis... |
| divassd 12055 | An associative law for div... |
| div12d 12056 | A commutative/associative ... |
| div23d 12057 | A commutative/associative ... |
| divdird 12058 | Distribution of division o... |
| divsubdird 12059 | Distribution of division o... |
| div11d 12060 | One-to-one relationship fo... |
| divmuldivd 12061 | Multiplication of two rati... |
| divmul13d 12062 | Swap denominators of two r... |
| divmul24d 12063 | Swap the numerators in the... |
| divadddivd 12064 | Addition of two ratios. T... |
| divsubdivd 12065 | Subtraction of two ratios.... |
| divmuleqd 12066 | Cross-multiply in an equal... |
| divdivdivd 12067 | Division of two ratios. T... |
| diveq1bd 12068 | If two complex numbers are... |
| div2sub 12069 | Swap the order of subtract... |
| div2subd 12070 | Swap subtrahend and minuen... |
| rereccld 12071 | Closure law for reciprocal... |
| redivcld 12072 | Closure law for division o... |
| subrec 12073 | Subtraction of reciprocals... |
| subreci 12074 | Subtraction of reciprocals... |
| subrecd 12075 | Subtraction of reciprocals... |
| mvllmuld 12076 | Move the left term in a pr... |
| mvllmuli 12077 | Move the left term in a pr... |
| ldiv 12078 | Left-division. (Contribut... |
| rdiv 12079 | Right-division. (Contribu... |
| mdiv 12080 | A division law. (Contribu... |
| lineq 12081 | Solution of a (scalar) lin... |
| elimgt0 12082 | Hypothesis for weak deduct... |
| elimge0 12083 | Hypothesis for weak deduct... |
| ltp1 12084 | A number is less than itse... |
| lep1 12085 | A number is less than or e... |
| ltm1 12086 | A number minus 1 is less t... |
| lem1 12087 | A number minus 1 is less t... |
| letrp1 12088 | A transitive property of '... |
| p1le 12089 | A transitive property of p... |
| recgt0 12090 | The reciprocal of a positi... |
| prodgt0 12091 | Infer that a multiplicand ... |
| prodgt02 12092 | Infer that a multiplier is... |
| ltmul1a 12093 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12094 | Multiplication of both sid... |
| ltmul2 12095 | Multiplication of both sid... |
| lemul1 12096 | Multiplication of both sid... |
| lemul2 12097 | Multiplication of both sid... |
| lemul1a 12098 | Multiplication of both sid... |
| lemul2a 12099 | Multiplication of both sid... |
| ltmul12a 12100 | Comparison of product of t... |
| lemul12b 12101 | Comparison of product of t... |
| lemul12a 12102 | Comparison of product of t... |
| mulgt1 12103 | The product of two numbers... |
| ltmulgt11 12104 | Multiplication by a number... |
| ltmulgt12 12105 | Multiplication by a number... |
| lemulge11 12106 | Multiplication by a number... |
| lemulge12 12107 | Multiplication by a number... |
| ltdiv1 12108 | Division of both sides of ... |
| lediv1 12109 | Division of both sides of ... |
| gt0div 12110 | Division of a positive num... |
| ge0div 12111 | Division of a nonnegative ... |
| divgt0 12112 | The ratio of two positive ... |
| divge0 12113 | The ratio of nonnegative a... |
| mulge0b 12114 | A condition for multiplica... |
| mulle0b 12115 | A condition for multiplica... |
| mulsuble0b 12116 | A condition for multiplica... |
| ltmuldiv 12117 | 'Less than' relationship b... |
| ltmuldiv2 12118 | 'Less than' relationship b... |
| ltdivmul 12119 | 'Less than' relationship b... |
| ledivmul 12120 | 'Less than or equal to' re... |
| ltdivmul2 12121 | 'Less than' relationship b... |
| lt2mul2div 12122 | 'Less than' relationship b... |
| ledivmul2 12123 | 'Less than or equal to' re... |
| lemuldiv 12124 | 'Less than or equal' relat... |
| lemuldiv2 12125 | 'Less than or equal' relat... |
| ltrec 12126 | The reciprocal of both sid... |
| lerec 12127 | The reciprocal of both sid... |
| lt2msq1 12128 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12129 | Two nonnegative numbers co... |
| ltdiv2 12130 | Division of a positive num... |
| ltrec1 12131 | Reciprocal swap in a 'less... |
| lerec2 12132 | Reciprocal swap in a 'less... |
| ledivdiv 12133 | Invert ratios of positive ... |
| lediv2 12134 | Division of a positive num... |
| ltdiv23 12135 | Swap denominator with othe... |
| lediv23 12136 | Swap denominator with othe... |
| lediv12a 12137 | Comparison of ratio of two... |
| lediv2a 12138 | Division of both sides of ... |
| reclt1 12139 | The reciprocal of a positi... |
| recgt1 12140 | The reciprocal of a positi... |
| recgt1i 12141 | The reciprocal of a number... |
| recp1lt1 12142 | Construct a number less th... |
| recreclt 12143 | Given a positive number ` ... |
| le2msq 12144 | The square function on non... |
| msq11 12145 | The square of a nonnegativ... |
| ledivp1 12146 | "Less than or equal to" an... |
| squeeze0 12147 | If a nonnegative number is... |
| ltp1i 12148 | A number is less than itse... |
| recgt0i 12149 | The reciprocal of a positi... |
| recgt0ii 12150 | The reciprocal of a positi... |
| prodgt0i 12151 | Infer that a multiplicand ... |
| divgt0i 12152 | The ratio of two positive ... |
| divge0i 12153 | The ratio of nonnegative a... |
| ltreci 12154 | The reciprocal of both sid... |
| lereci 12155 | The reciprocal of both sid... |
| lt2msqi 12156 | The square function on non... |
| le2msqi 12157 | The square function on non... |
| msq11i 12158 | The square of a nonnegativ... |
| divgt0i2i 12159 | The ratio of two positive ... |
| ltrecii 12160 | The reciprocal of both sid... |
| divgt0ii 12161 | The ratio of two positive ... |
| ltmul1i 12162 | Multiplication of both sid... |
| ltdiv1i 12163 | Division of both sides of ... |
| ltmuldivi 12164 | 'Less than' relationship b... |
| ltmul2i 12165 | Multiplication of both sid... |
| lemul1i 12166 | Multiplication of both sid... |
| lemul2i 12167 | Multiplication of both sid... |
| ltdiv23i 12168 | Swap denominator with othe... |
| ledivp1i 12169 | "Less than or equal to" an... |
| ltdivp1i 12170 | Less-than and division rel... |
| ltdiv23ii 12171 | Swap denominator with othe... |
| ltmul1ii 12172 | Multiplication of both sid... |
| ltdiv1ii 12173 | Division of both sides of ... |
| ltp1d 12174 | A number is less than itse... |
| lep1d 12175 | A number is less than or e... |
| ltm1d 12176 | A number minus 1 is less t... |
| lem1d 12177 | A number minus 1 is less t... |
| recgt0d 12178 | The reciprocal of a positi... |
| divgt0d 12179 | The ratio of two positive ... |
| mulgt1d 12180 | The product of two numbers... |
| lemulge11d 12181 | Multiplication by a number... |
| lemulge12d 12182 | Multiplication by a number... |
| lemul1ad 12183 | Multiplication of both sid... |
| lemul2ad 12184 | Multiplication of both sid... |
| ltmul12ad 12185 | Comparison of product of t... |
| lemul12ad 12186 | Comparison of product of t... |
| lemul12bd 12187 | Comparison of product of t... |
| fimaxre 12188 | A finite set of real numbe... |
| fimaxre2 12189 | A nonempty finite set of r... |
| fimaxre3 12190 | A nonempty finite set of r... |
| fiminre 12191 | A nonempty finite set of r... |
| fiminre2 12192 | A nonempty finite set of r... |
| negfi 12193 | The negation of a finite s... |
| lbreu 12194 | If a set of reals contains... |
| lbcl 12195 | If a set of reals contains... |
| lble 12196 | If a set of reals contains... |
| lbinf 12197 | If a set of reals contains... |
| lbinfcl 12198 | If a set of reals contains... |
| lbinfle 12199 | If a set of reals contains... |
| sup2 12200 | A nonempty, bounded-above ... |
| sup3 12201 | A version of the completen... |
| infm3lem 12202 | Lemma for ~ infm3 . (Cont... |
| infm3 12203 | The completeness axiom for... |
| suprcl 12204 | Closure of supremum of a n... |
| suprub 12205 | A member of a nonempty bou... |
| suprubd 12206 | Natural deduction form of ... |
| suprcld 12207 | Natural deduction form of ... |
| suprlub 12208 | The supremum of a nonempty... |
| suprnub 12209 | An upper bound is not less... |
| suprleub 12210 | The supremum of a nonempty... |
| supaddc 12211 | The supremum function dist... |
| supadd 12212 | The supremum function dist... |
| supmul1 12213 | The supremum function dist... |
| supmullem1 12214 | Lemma for ~ supmul . (Con... |
| supmullem2 12215 | Lemma for ~ supmul . (Con... |
| supmul 12216 | The supremum function dist... |
| sup3ii 12217 | A version of the completen... |
| suprclii 12218 | Closure of supremum of a n... |
| suprubii 12219 | A member of a nonempty bou... |
| suprlubii 12220 | The supremum of a nonempty... |
| suprnubii 12221 | An upper bound is not less... |
| suprleubii 12222 | The supremum of a nonempty... |
| riotaneg 12223 | The negative of the unique... |
| negiso 12224 | Negation is an order anti-... |
| dfinfre 12225 | The infimum of a set of re... |
| infrecl 12226 | Closure of infimum of a no... |
| infrenegsup 12227 | The infimum of a set of re... |
| infregelb 12228 | Any lower bound of a nonem... |
| infrelb 12229 | If a nonempty set of real ... |
| infrefilb 12230 | The infimum of a finite se... |
| supfirege 12231 | The supremum of a finite s... |
| inelr 12232 | The imaginary unit ` _i ` ... |
| rimul 12233 | A real number times the im... |
| cru 12234 | The representation of comp... |
| crne0 12235 | The real representation of... |
| creur 12236 | The real part of a complex... |
| creui 12237 | The imaginary part of a co... |
| cju 12238 | The complex conjugate of a... |
| ofsubeq0 12239 | Function analogue of ~ sub... |
| ofnegsub 12240 | Function analogue of ~ neg... |
| ofsubge0 12241 | Function analogue of ~ sub... |
| nnexALT 12244 | Alternate proof of ~ nnex ... |
| peano5nni 12245 | Peano's inductive postulat... |
| nnssre 12246 | The positive integers are ... |
| nnsscn 12247 | The positive integers are ... |
| nnex 12248 | The set of positive intege... |
| nnre 12249 | A positive integer is a re... |
| nncn 12250 | A positive integer is a co... |
| nnrei 12251 | A positive integer is a re... |
| nncni 12252 | A positive integer is a co... |
| 1nn 12253 | Peano postulate: 1 is a po... |
| peano2nn 12254 | Peano postulate: a success... |
| dfnn2 12255 | Alternate definition of th... |
| dfnn3 12256 | Alternate definition of th... |
| nnred 12257 | A positive integer is a re... |
| nncnd 12258 | A positive integer is a co... |
| peano2nnd 12259 | Peano postulate: a success... |
| nnind 12260 | Principle of Mathematical ... |
| nnindALT 12261 | Principle of Mathematical ... |
| nnindd 12262 | Principle of Mathematical ... |
| nn1m1nn 12263 | Every positive integer is ... |
| nn1suc 12264 | If a statement holds for 1... |
| nnaddcl 12265 | Closure of addition of pos... |
| nnmulcl 12266 | Closure of multiplication ... |
| nnmulcli 12267 | Closure of multiplication ... |
| nnmtmip 12268 | "Minus times minus is plus... |
| nn2ge 12269 | There exists a positive in... |
| nnge1 12270 | A positive integer is one ... |
| nngt1ne1 12271 | A positive integer is grea... |
| nnle1eq1 12272 | A positive integer is less... |
| nngt0 12273 | A positive integer is posi... |
| nnnlt1 12274 | A positive integer is not ... |
| nnnle0 12275 | A positive integer is not ... |
| nnne0 12276 | A positive integer is nonz... |
| nnneneg 12277 | No positive integer is equ... |
| 0nnn 12278 | Zero is not a positive int... |
| 0nnnALT 12279 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12280 | Alternate version of ~ nnn... |
| nngt0i 12281 | A positive integer is posi... |
| nnne0i 12282 | A positive integer is nonz... |
| nndivre 12283 | The quotient of a real and... |
| nnrecre 12284 | The reciprocal of a positi... |
| nnrecgt0 12285 | The reciprocal of a positi... |
| nnsub 12286 | Subtraction of positive in... |
| nnsubi 12287 | Subtraction of positive in... |
| nndiv 12288 | Two ways to express " ` A ... |
| nndivtr 12289 | Transitive property of div... |
| nnge1d 12290 | A positive integer is one ... |
| nngt0d 12291 | A positive integer is posi... |
| nnne0d 12292 | A positive integer is nonz... |
| nnrecred 12293 | The reciprocal of a positi... |
| nnaddcld 12294 | Closure of addition of pos... |
| nnmulcld 12295 | Closure of multiplication ... |
| nndivred 12296 | A positive integer is one ... |
| 0ne1 12313 | Zero is different from one... |
| 1m1e0 12314 | One minus one equals zero.... |
| 2nn 12315 | 2 is a positive integer. ... |
| 2re 12316 | The number 2 is real. (Co... |
| 2cn 12317 | The number 2 is a complex ... |
| 2cnALT 12318 | Alternate proof of ~ 2cn .... |
| 2ex 12319 | The number 2 is a set. (C... |
| 2cnd 12320 | The number 2 is a complex ... |
| 3nn 12321 | 3 is a positive integer. ... |
| 3re 12322 | The number 3 is real. (Co... |
| 3cn 12323 | The number 3 is a complex ... |
| 3ex 12324 | The number 3 is a set. (C... |
| 4nn 12325 | 4 is a positive integer. ... |
| 4re 12326 | The number 4 is real. (Co... |
| 4cn 12327 | The number 4 is a complex ... |
| 5nn 12328 | 5 is a positive integer. ... |
| 5re 12329 | The number 5 is real. (Co... |
| 5cn 12330 | The number 5 is a complex ... |
| 6nn 12331 | 6 is a positive integer. ... |
| 6re 12332 | The number 6 is real. (Co... |
| 6cn 12333 | The number 6 is a complex ... |
| 7nn 12334 | 7 is a positive integer. ... |
| 7re 12335 | The number 7 is real. (Co... |
| 7cn 12336 | The number 7 is a complex ... |
| 8nn 12337 | 8 is a positive integer. ... |
| 8re 12338 | The number 8 is real. (Co... |
| 8cn 12339 | The number 8 is a complex ... |
| 9nn 12340 | 9 is a positive integer. ... |
| 9re 12341 | The number 9 is real. (Co... |
| 9cn 12342 | The number 9 is a complex ... |
| 0le0 12343 | Zero is nonnegative. (Con... |
| 0le2 12344 | The number 0 is less than ... |
| 2pos 12345 | The number 2 is positive. ... |
| 2ne0 12346 | The number 2 is nonzero. ... |
| 3pos 12347 | The number 3 is positive. ... |
| 3ne0 12348 | The number 3 is nonzero. ... |
| 4pos 12349 | The number 4 is positive. ... |
| 4ne0 12350 | The number 4 is nonzero. ... |
| 5pos 12351 | The number 5 is positive. ... |
| 6pos 12352 | The number 6 is positive. ... |
| 7pos 12353 | The number 7 is positive. ... |
| 8pos 12354 | The number 8 is positive. ... |
| 9pos 12355 | The number 9 is positive. ... |
| neg1cn 12356 | -1 is a complex number. (... |
| neg1rr 12357 | -1 is a real number. (Con... |
| neg1ne0 12358 | -1 is nonzero. (Contribut... |
| neg1lt0 12359 | -1 is less than 0. (Contr... |
| negneg1e1 12360 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 12361 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12362 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12363 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12364 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12365 | Function value at ` N + 1 ... |
| 1p0e1 12366 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12367 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12368 | 2 - 1 = 1. The result is ... |
| 1e2m1 12369 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12370 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12371 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12372 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12373 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12374 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12375 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12376 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12377 | Two plus two equals four. ... |
| 2times 12378 | Two times a number. (Cont... |
| times2 12379 | A number times 2. (Contri... |
| 2timesi 12380 | Two times a number. (Cont... |
| times2i 12381 | A number times 2. (Contri... |
| 2txmxeqx 12382 | Two times a complex number... |
| 2div2e1 12383 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12384 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12385 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12386 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12387 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12388 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12389 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12390 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12391 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12392 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12393 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12394 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12395 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12396 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12397 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12398 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12399 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12400 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12401 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12402 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12403 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12404 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12405 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12406 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12407 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12408 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12409 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12410 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12411 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12412 | One half of four is two. ... |
| 1lt2 12413 | 1 is less than 2. (Contri... |
| 2lt3 12414 | 2 is less than 3. (Contri... |
| 1lt3 12415 | 1 is less than 3. (Contri... |
| 3lt4 12416 | 3 is less than 4. (Contri... |
| 2lt4 12417 | 2 is less than 4. (Contri... |
| 1lt4 12418 | 1 is less than 4. (Contri... |
| 4lt5 12419 | 4 is less than 5. (Contri... |
| 3lt5 12420 | 3 is less than 5. (Contri... |
| 2lt5 12421 | 2 is less than 5. (Contri... |
| 1lt5 12422 | 1 is less than 5. (Contri... |
| 5lt6 12423 | 5 is less than 6. (Contri... |
| 4lt6 12424 | 4 is less than 6. (Contri... |
| 3lt6 12425 | 3 is less than 6. (Contri... |
| 2lt6 12426 | 2 is less than 6. (Contri... |
| 1lt6 12427 | 1 is less than 6. (Contri... |
| 6lt7 12428 | 6 is less than 7. (Contri... |
| 5lt7 12429 | 5 is less than 7. (Contri... |
| 4lt7 12430 | 4 is less than 7. (Contri... |
| 3lt7 12431 | 3 is less than 7. (Contri... |
| 2lt7 12432 | 2 is less than 7. (Contri... |
| 1lt7 12433 | 1 is less than 7. (Contri... |
| 7lt8 12434 | 7 is less than 8. (Contri... |
| 6lt8 12435 | 6 is less than 8. (Contri... |
| 5lt8 12436 | 5 is less than 8. (Contri... |
| 4lt8 12437 | 4 is less than 8. (Contri... |
| 3lt8 12438 | 3 is less than 8. (Contri... |
| 2lt8 12439 | 2 is less than 8. (Contri... |
| 1lt8 12440 | 1 is less than 8. (Contri... |
| 8lt9 12441 | 8 is less than 9. (Contri... |
| 7lt9 12442 | 7 is less than 9. (Contri... |
| 6lt9 12443 | 6 is less than 9. (Contri... |
| 5lt9 12444 | 5 is less than 9. (Contri... |
| 4lt9 12445 | 4 is less than 9. (Contri... |
| 3lt9 12446 | 3 is less than 9. (Contri... |
| 2lt9 12447 | 2 is less than 9. (Contri... |
| 1lt9 12448 | 1 is less than 9. (Contri... |
| 0ne2 12449 | 0 is not equal to 2. (Con... |
| 1ne2 12450 | 1 is not equal to 2. (Con... |
| 1le2 12451 | 1 is less than or equal to... |
| 2cnne0 12452 | 2 is a nonzero complex num... |
| 2rene0 12453 | 2 is a nonzero real number... |
| 1le3 12454 | 1 is less than or equal to... |
| neg1mulneg1e1 12455 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12456 | One-half is real. (Contri... |
| halfcn 12457 | One-half is a complex numb... |
| halfgt0 12458 | One-half is greater than z... |
| halfge0 12459 | One-half is not negative. ... |
| halflt1 12460 | One-half is less than one.... |
| 1mhlfehlf 12461 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12462 | An eighth of four thirds i... |
| halfpm6th 12463 | One half plus or minus one... |
| it0e0 12464 | i times 0 equals 0. (Cont... |
| 2mulicn 12465 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12466 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12467 | Closure of half of a numbe... |
| rehalfcl 12468 | Real closure of half. (Co... |
| half0 12469 | Half of a number is zero i... |
| 2halves 12470 | Two halves make a whole. ... |
| halfpos2 12471 | A number is positive iff i... |
| halfpos 12472 | A positive number is great... |
| halfnneg2 12473 | A number is nonnegative if... |
| halfaddsubcl 12474 | Closure of half-sum and ha... |
| halfaddsub 12475 | Sum and difference of half... |
| subhalfhalf 12476 | Subtracting the half of a ... |
| lt2halves 12477 | A sum is less than the who... |
| addltmul 12478 | Sum is less than product f... |
| nominpos 12479 | There is no smallest posit... |
| avglt1 12480 | Ordering property for aver... |
| avglt2 12481 | Ordering property for aver... |
| avgle1 12482 | Ordering property for aver... |
| avgle2 12483 | Ordering property for aver... |
| avgle 12484 | The average of two numbers... |
| 2timesd 12485 | Two times a number. (Cont... |
| times2d 12486 | A number times 2. (Contri... |
| halfcld 12487 | Closure of half of a numbe... |
| 2halvesd 12488 | Two halves make a whole. ... |
| rehalfcld 12489 | Real closure of half. (Co... |
| lt2halvesd 12490 | A sum is less than the who... |
| rehalfcli 12491 | Half a real number is real... |
| lt2addmuld 12492 | If two real numbers are le... |
| add1p1 12493 | Adding two times 1 to a nu... |
| sub1m1 12494 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12495 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12496 | A complex number increased... |
| div4p1lem1div2 12497 | An integer greater than 5,... |
| nnunb 12498 | The set of positive intege... |
| arch 12499 | Archimedean property of re... |
| nnrecl 12500 | There exists a positive in... |
| bndndx 12501 | A bounded real sequence ` ... |
| elnn0 12504 | Nonnegative integers expre... |
| nnssnn0 12505 | Positive naturals are a su... |
| nn0ssre 12506 | Nonnegative integers are a... |
| nn0sscn 12507 | Nonnegative integers are a... |
| nn0ex 12508 | The set of nonnegative int... |
| nnnn0 12509 | A positive integer is a no... |
| nnnn0i 12510 | A positive integer is a no... |
| nn0re 12511 | A nonnegative integer is a... |
| nn0cn 12512 | A nonnegative integer is a... |
| nn0rei 12513 | A nonnegative integer is a... |
| nn0cni 12514 | A nonnegative integer is a... |
| dfn2 12515 | The set of positive intege... |
| elnnne0 12516 | The positive integer prope... |
| 0nn0 12517 | 0 is a nonnegative integer... |
| 1nn0 12518 | 1 is a nonnegative integer... |
| 2nn0 12519 | 2 is a nonnegative integer... |
| 3nn0 12520 | 3 is a nonnegative integer... |
| 4nn0 12521 | 4 is a nonnegative integer... |
| 5nn0 12522 | 5 is a nonnegative integer... |
| 6nn0 12523 | 6 is a nonnegative integer... |
| 7nn0 12524 | 7 is a nonnegative integer... |
| 8nn0 12525 | 8 is a nonnegative integer... |
| 9nn0 12526 | 9 is a nonnegative integer... |
| nn0ge0 12527 | A nonnegative integer is g... |
| nn0nlt0 12528 | A nonnegative integer is n... |
| nn0ge0i 12529 | Nonnegative integers are n... |
| nn0le0eq0 12530 | A nonnegative integer is l... |
| nn0p1gt0 12531 | A nonnegative integer incr... |
| nnnn0addcl 12532 | A positive integer plus a ... |
| nn0nnaddcl 12533 | A nonnegative integer plus... |
| 0mnnnnn0 12534 | The result of subtracting ... |
| un0addcl 12535 | If ` S ` is closed under a... |
| un0mulcl 12536 | If ` S ` is closed under m... |
| nn0addcl 12537 | Closure of addition of non... |
| nn0mulcl 12538 | Closure of multiplication ... |
| nn0addcli 12539 | Closure of addition of non... |
| nn0mulcli 12540 | Closure of multiplication ... |
| nn0p1nn 12541 | A nonnegative integer plus... |
| peano2nn0 12542 | Second Peano postulate for... |
| nnm1nn0 12543 | A positive integer minus 1... |
| elnn0nn 12544 | The nonnegative integer pr... |
| elnnnn0 12545 | The positive integer prope... |
| elnnnn0b 12546 | The positive integer prope... |
| elnnnn0c 12547 | The positive integer prope... |
| nn0addge1 12548 | A number is less than or e... |
| nn0addge2 12549 | A number is less than or e... |
| nn0addge1i 12550 | A number is less than or e... |
| nn0addge2i 12551 | A number is less than or e... |
| nn0sub 12552 | Subtraction of nonnegative... |
| ltsubnn0 12553 | Subtracting a nonnegative ... |
| nn0negleid 12554 | A nonnegative integer is g... |
| difgtsumgt 12555 | If the difference of a rea... |
| nn0le2xi 12556 | A nonnegative integer is l... |
| nn0lele2xi 12557 | 'Less than or equal to' im... |
| fcdmnn0supp 12558 | Two ways to write the supp... |
| fcdmnn0fsupp 12559 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12560 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12561 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12562 | A positive integer is a no... |
| nn0red 12563 | A nonnegative integer is a... |
| nn0cnd 12564 | A nonnegative integer is a... |
| nn0ge0d 12565 | A nonnegative integer is g... |
| nn0addcld 12566 | Closure of addition of non... |
| nn0mulcld 12567 | Closure of multiplication ... |
| nn0readdcl 12568 | Closure law for addition o... |
| nn0n0n1ge2 12569 | A nonnegative integer whic... |
| nn0n0n1ge2b 12570 | A nonnegative integer is n... |
| nn0ge2m1nn 12571 | If a nonnegative integer i... |
| nn0ge2m1nn0 12572 | If a nonnegative integer i... |
| nn0nndivcl 12573 | Closure law for dividing o... |
| elxnn0 12576 | An extended nonnegative in... |
| nn0ssxnn0 12577 | The standard nonnegative i... |
| nn0xnn0 12578 | A standard nonnegative int... |
| xnn0xr 12579 | An extended nonnegative in... |
| 0xnn0 12580 | Zero is an extended nonneg... |
| pnf0xnn0 12581 | Positive infinity is an ex... |
| nn0nepnf 12582 | No standard nonnegative in... |
| nn0xnn0d 12583 | A standard nonnegative int... |
| nn0nepnfd 12584 | No standard nonnegative in... |
| xnn0nemnf 12585 | No extended nonnegative in... |
| xnn0xrnemnf 12586 | The extended nonnegative i... |
| xnn0nnn0pnf 12587 | An extended nonnegative in... |
| elz 12590 | Membership in the set of i... |
| nnnegz 12591 | The negative of a positive... |
| zre 12592 | An integer is a real. (Co... |
| zcn 12593 | An integer is a complex nu... |
| zrei 12594 | An integer is a real numbe... |
| zssre 12595 | The integers are a subset ... |
| zsscn 12596 | The integers are a subset ... |
| zex 12597 | The set of integers exists... |
| elnnz 12598 | Positive integer property ... |
| 0z 12599 | Zero is an integer. (Cont... |
| 0zd 12600 | Zero is an integer, deduct... |
| elnn0z 12601 | Nonnegative integer proper... |
| elznn0nn 12602 | Integer property expressed... |
| elznn0 12603 | Integer property expressed... |
| elznn 12604 | Integer property expressed... |
| zle0orge1 12605 | There is no integer in the... |
| elz2 12606 | Membership in the set of i... |
| dfz2 12607 | Alternative definition of ... |
| zexALT 12608 | Alternate proof of ~ zex .... |
| nnz 12609 | A positive integer is an i... |
| nnssz 12610 | Positive integers are a su... |
| nn0ssz 12611 | Nonnegative integers are a... |
| nnzOLD 12612 | Obsolete version of ~ nnz ... |
| nn0z 12613 | A nonnegative integer is a... |
| nn0zd 12614 | A nonnegative integer is a... |
| nnzd 12615 | A positive integer is an i... |
| nnzi 12616 | A positive integer is an i... |
| nn0zi 12617 | A nonnegative integer is a... |
| elnnz1 12618 | Positive integer property ... |
| znnnlt1 12619 | An integer is not a positi... |
| nnzrab 12620 | Positive integers expresse... |
| nn0zrab 12621 | Nonnegative integers expre... |
| 1z 12622 | One is an integer. (Contr... |
| 1zzd 12623 | One is an integer, deducti... |
| 2z 12624 | 2 is an integer. (Contrib... |
| 3z 12625 | 3 is an integer. (Contrib... |
| 4z 12626 | 4 is an integer. (Contrib... |
| znegcl 12627 | Closure law for negative i... |
| neg1z 12628 | -1 is an integer. (Contri... |
| znegclb 12629 | A complex number is an int... |
| nn0negz 12630 | The negative of a nonnegat... |
| nn0negzi 12631 | The negative of a nonnegat... |
| zaddcl 12632 | Closure of addition of int... |
| peano2z 12633 | Second Peano postulate gen... |
| zsubcl 12634 | Closure of subtraction of ... |
| peano2zm 12635 | "Reverse" second Peano pos... |
| zletr 12636 | Transitive law of ordering... |
| zrevaddcl 12637 | Reverse closure law for ad... |
| znnsub 12638 | The positive difference of... |
| znn0sub 12639 | The nonnegative difference... |
| nzadd 12640 | The sum of a real number n... |
| zmulcl 12641 | Closure of multiplication ... |
| zltp1le 12642 | Integer ordering relation.... |
| zleltp1 12643 | Integer ordering relation.... |
| zlem1lt 12644 | Integer ordering relation.... |
| zltlem1 12645 | Integer ordering relation.... |
| zgt0ge1 12646 | An integer greater than ` ... |
| nnleltp1 12647 | Positive integer ordering ... |
| nnltp1le 12648 | Positive integer ordering ... |
| nnaddm1cl 12649 | Closure of addition of pos... |
| nn0ltp1le 12650 | Nonnegative integer orderi... |
| nn0leltp1 12651 | Nonnegative integer orderi... |
| nn0ltlem1 12652 | Nonnegative integer orderi... |
| nn0sub2 12653 | Subtraction of nonnegative... |
| nn0lt10b 12654 | A nonnegative integer less... |
| nn0lt2 12655 | A nonnegative integer less... |
| nn0le2is012 12656 | A nonnegative integer whic... |
| nn0lem1lt 12657 | Nonnegative integer orderi... |
| nnlem1lt 12658 | Positive integer ordering ... |
| nnltlem1 12659 | Positive integer ordering ... |
| nnm1ge0 12660 | A positive integer decreas... |
| nn0ge0div 12661 | Division of a nonnegative ... |
| zdiv 12662 | Two ways to express " ` M ... |
| zdivadd 12663 | Property of divisibility: ... |
| zdivmul 12664 | Property of divisibility: ... |
| zextle 12665 | An extensionality-like pro... |
| zextlt 12666 | An extensionality-like pro... |
| recnz 12667 | The reciprocal of a number... |
| btwnnz 12668 | A number between an intege... |
| gtndiv 12669 | A larger number does not d... |
| halfnz 12670 | One-half is not an integer... |
| 3halfnz 12671 | Three halves is not an int... |
| suprzcl 12672 | The supremum of a bounded-... |
| prime 12673 | Two ways to express " ` A ... |
| msqznn 12674 | The square of a nonzero in... |
| zneo 12675 | No even integer equals an ... |
| nneo 12676 | A positive integer is even... |
| nneoi 12677 | A positive integer is even... |
| zeo 12678 | An integer is even or odd.... |
| zeo2 12679 | An integer is even or odd ... |
| peano2uz2 12680 | Second Peano postulate for... |
| peano5uzi 12681 | Peano's inductive postulat... |
| peano5uzti 12682 | Peano's inductive postulat... |
| dfuzi 12683 | An expression for the uppe... |
| uzind 12684 | Induction on the upper int... |
| uzind2 12685 | Induction on the upper int... |
| uzind3 12686 | Induction on the upper int... |
| nn0ind 12687 | Principle of Mathematical ... |
| nn0indALT 12688 | Principle of Mathematical ... |
| nn0indd 12689 | Principle of Mathematical ... |
| fzind 12690 | Induction on the integers ... |
| fnn0ind 12691 | Induction on the integers ... |
| nn0ind-raph 12692 | Principle of Mathematical ... |
| zindd 12693 | Principle of Mathematical ... |
| fzindd 12694 | Induction on the integers ... |
| btwnz 12695 | Any real number can be san... |
| zred 12696 | An integer is a real numbe... |
| zcnd 12697 | An integer is a complex nu... |
| znegcld 12698 | Closure law for negative i... |
| peano2zd 12699 | Deduction from second Pean... |
| zaddcld 12700 | Closure of addition of int... |
| zsubcld 12701 | Closure of subtraction of ... |
| zmulcld 12702 | Closure of multiplication ... |
| znnn0nn 12703 | The negative of a negative... |
| zadd2cl 12704 | Increasing an integer by 2... |
| zriotaneg 12705 | The negative of the unique... |
| suprfinzcl 12706 | The supremum of a nonempty... |
| 9p1e10 12709 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12710 | Version of the definition ... |
| decex 12711 | A decimal number is a set.... |
| deceq1 12712 | Equality theorem for the d... |
| deceq2 12713 | Equality theorem for the d... |
| deceq1i 12714 | Equality theorem for the d... |
| deceq2i 12715 | Equality theorem for the d... |
| deceq12i 12716 | Equality theorem for the d... |
| numnncl 12717 | Closure for a numeral (wit... |
| num0u 12718 | Add a zero in the units pl... |
| num0h 12719 | Add a zero in the higher p... |
| numcl 12720 | Closure for a decimal inte... |
| numsuc 12721 | The successor of a decimal... |
| deccl 12722 | Closure for a numeral. (C... |
| 10nn 12723 | 10 is a positive integer. ... |
| 10pos 12724 | The number 10 is positive.... |
| 10nn0 12725 | 10 is a nonnegative intege... |
| 10re 12726 | The number 10 is real. (C... |
| decnncl 12727 | Closure for a numeral. (C... |
| dec0u 12728 | Add a zero in the units pl... |
| dec0h 12729 | Add a zero in the higher p... |
| numnncl2 12730 | Closure for a decimal inte... |
| decnncl2 12731 | Closure for a decimal inte... |
| numlt 12732 | Comparing two decimal inte... |
| numltc 12733 | Comparing two decimal inte... |
| le9lt10 12734 | A "decimal digit" (i.e. a ... |
| declt 12735 | Comparing two decimal inte... |
| decltc 12736 | Comparing two decimal inte... |
| declth 12737 | Comparing two decimal inte... |
| decsuc 12738 | The successor of a decimal... |
| 3declth 12739 | Comparing two decimal inte... |
| 3decltc 12740 | Comparing two decimal inte... |
| decle 12741 | Comparing two decimal inte... |
| decleh 12742 | Comparing two decimal inte... |
| declei 12743 | Comparing a digit to a dec... |
| numlti 12744 | Comparing a digit to a dec... |
| declti 12745 | Comparing a digit to a dec... |
| decltdi 12746 | Comparing a digit to a dec... |
| numsucc 12747 | The successor of a decimal... |
| decsucc 12748 | The successor of a decimal... |
| 1e0p1 12749 | The successor of zero. (C... |
| dec10p 12750 | Ten plus an integer. (Con... |
| numma 12751 | Perform a multiply-add of ... |
| nummac 12752 | Perform a multiply-add of ... |
| numma2c 12753 | Perform a multiply-add of ... |
| numadd 12754 | Add two decimal integers `... |
| numaddc 12755 | Add two decimal integers `... |
| nummul1c 12756 | The product of a decimal i... |
| nummul2c 12757 | The product of a decimal i... |
| decma 12758 | Perform a multiply-add of ... |
| decmac 12759 | Perform a multiply-add of ... |
| decma2c 12760 | Perform a multiply-add of ... |
| decadd 12761 | Add two numerals ` M ` and... |
| decaddc 12762 | Add two numerals ` M ` and... |
| decaddc2 12763 | Add two numerals ` M ` and... |
| decrmanc 12764 | Perform a multiply-add of ... |
| decrmac 12765 | Perform a multiply-add of ... |
| decaddm10 12766 | The sum of two multiples o... |
| decaddi 12767 | Add two numerals ` M ` and... |
| decaddci 12768 | Add two numerals ` M ` and... |
| decaddci2 12769 | Add two numerals ` M ` and... |
| decsubi 12770 | Difference between a numer... |
| decmul1 12771 | The product of a numeral w... |
| decmul1c 12772 | The product of a numeral w... |
| decmul2c 12773 | The product of a numeral w... |
| decmulnc 12774 | The product of a numeral w... |
| 11multnc 12775 | The product of 11 (as nume... |
| decmul10add 12776 | A multiplication of a numb... |
| 6p5lem 12777 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12778 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12779 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12780 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12781 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12782 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12783 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12784 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12785 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12786 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12787 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12788 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12789 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12790 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12791 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12792 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12793 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12794 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12795 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12796 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12797 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12798 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12799 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12800 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12801 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12802 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12803 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12804 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12805 | 4 times 3 equals 12. (Con... |
| 4t4e16 12806 | 4 times 4 equals 16. (Con... |
| 5t2e10 12807 | 5 times 2 equals 10. (Con... |
| 5t3e15 12808 | 5 times 3 equals 15. (Con... |
| 5t4e20 12809 | 5 times 4 equals 20. (Con... |
| 5t5e25 12810 | 5 times 5 equals 25. (Con... |
| 6t2e12 12811 | 6 times 2 equals 12. (Con... |
| 6t3e18 12812 | 6 times 3 equals 18. (Con... |
| 6t4e24 12813 | 6 times 4 equals 24. (Con... |
| 6t5e30 12814 | 6 times 5 equals 30. (Con... |
| 6t6e36 12815 | 6 times 6 equals 36. (Con... |
| 7t2e14 12816 | 7 times 2 equals 14. (Con... |
| 7t3e21 12817 | 7 times 3 equals 21. (Con... |
| 7t4e28 12818 | 7 times 4 equals 28. (Con... |
| 7t5e35 12819 | 7 times 5 equals 35. (Con... |
| 7t6e42 12820 | 7 times 6 equals 42. (Con... |
| 7t7e49 12821 | 7 times 7 equals 49. (Con... |
| 8t2e16 12822 | 8 times 2 equals 16. (Con... |
| 8t3e24 12823 | 8 times 3 equals 24. (Con... |
| 8t4e32 12824 | 8 times 4 equals 32. (Con... |
| 8t5e40 12825 | 8 times 5 equals 40. (Con... |
| 8t6e48 12826 | 8 times 6 equals 48. (Con... |
| 8t7e56 12827 | 8 times 7 equals 56. (Con... |
| 8t8e64 12828 | 8 times 8 equals 64. (Con... |
| 9t2e18 12829 | 9 times 2 equals 18. (Con... |
| 9t3e27 12830 | 9 times 3 equals 27. (Con... |
| 9t4e36 12831 | 9 times 4 equals 36. (Con... |
| 9t5e45 12832 | 9 times 5 equals 45. (Con... |
| 9t6e54 12833 | 9 times 6 equals 54. (Con... |
| 9t7e63 12834 | 9 times 7 equals 63. (Con... |
| 9t8e72 12835 | 9 times 8 equals 72. (Con... |
| 9t9e81 12836 | 9 times 9 equals 81. (Con... |
| 9t11e99 12837 | 9 times 11 equals 99. (Co... |
| 9lt10 12838 | 9 is less than 10. (Contr... |
| 8lt10 12839 | 8 is less than 10. (Contr... |
| 7lt10 12840 | 7 is less than 10. (Contr... |
| 6lt10 12841 | 6 is less than 10. (Contr... |
| 5lt10 12842 | 5 is less than 10. (Contr... |
| 4lt10 12843 | 4 is less than 10. (Contr... |
| 3lt10 12844 | 3 is less than 10. (Contr... |
| 2lt10 12845 | 2 is less than 10. (Contr... |
| 1lt10 12846 | 1 is less than 10. (Contr... |
| decbin0 12847 | Decompose base 4 into base... |
| decbin2 12848 | Decompose base 4 into base... |
| decbin3 12849 | Decompose base 4 into base... |
| halfthird 12850 | Half minus a third. (Cont... |
| 5recm6rec 12851 | One fifth minus one sixth.... |
| uzval 12854 | The value of the upper int... |
| uzf 12855 | The domain and codomain of... |
| eluz1 12856 | Membership in the upper se... |
| eluzel2 12857 | Implication of membership ... |
| eluz2 12858 | Membership in an upper set... |
| eluzmn 12859 | Membership in an earlier u... |
| eluz1i 12860 | Membership in an upper set... |
| eluzuzle 12861 | An integer in an upper set... |
| eluzelz 12862 | A member of an upper set o... |
| eluzelre 12863 | A member of an upper set o... |
| eluzelcn 12864 | A member of an upper set o... |
| eluzle 12865 | Implication of membership ... |
| eluz 12866 | Membership in an upper set... |
| uzid 12867 | Membership of the least me... |
| uzidd 12868 | Membership of the least me... |
| uzn0 12869 | The upper integers are all... |
| uztrn 12870 | Transitive law for sets of... |
| uztrn2 12871 | Transitive law for sets of... |
| uzneg 12872 | Contraposition law for upp... |
| uzssz 12873 | An upper set of integers i... |
| uzssre 12874 | An upper set of integers i... |
| uzss 12875 | Subset relationship for tw... |
| uztric 12876 | Totality of the ordering r... |
| uz11 12877 | The upper integers functio... |
| eluzp1m1 12878 | Membership in the next upp... |
| eluzp1l 12879 | Strict ordering implied by... |
| eluzp1p1 12880 | Membership in the next upp... |
| eluzadd 12881 | Membership in a later uppe... |
| eluzsub 12882 | Membership in an earlier u... |
| eluzaddi 12883 | Membership in a later uppe... |
| eluzaddiOLD 12884 | Obsolete version of ~ eluz... |
| eluzsubi 12885 | Membership in an earlier u... |
| eluzsubiOLD 12886 | Obsolete version of ~ eluz... |
| eluzaddOLD 12887 | Obsolete version of ~ eluz... |
| eluzsubOLD 12888 | Obsolete version of ~ eluz... |
| subeluzsub 12889 | Membership of a difference... |
| uzm1 12890 | Choices for an element of ... |
| uznn0sub 12891 | The nonnegative difference... |
| uzin 12892 | Intersection of two upper ... |
| uzp1 12893 | Choices for an element of ... |
| nn0uz 12894 | Nonnegative integers expre... |
| nnuz 12895 | Positive integers expresse... |
| elnnuz 12896 | A positive integer express... |
| elnn0uz 12897 | A nonnegative integer expr... |
| eluz2nn 12898 | An integer greater than or... |
| eluz4eluz2 12899 | An integer greater than or... |
| eluz4nn 12900 | An integer greater than or... |
| eluzge2nn0 12901 | If an integer is greater t... |
| eluz2n0 12902 | An integer greater than or... |
| uzuzle23 12903 | An integer in the upper se... |
| eluzge3nn 12904 | If an integer is greater t... |
| uz3m2nn 12905 | An integer greater than or... |
| 1eluzge0 12906 | 1 is an integer greater th... |
| 2eluzge0 12907 | 2 is an integer greater th... |
| 2eluzge1 12908 | 2 is an integer greater th... |
| uznnssnn 12909 | The upper integers startin... |
| raluz 12910 | Restricted universal quant... |
| raluz2 12911 | Restricted universal quant... |
| rexuz 12912 | Restricted existential qua... |
| rexuz2 12913 | Restricted existential qua... |
| 2rexuz 12914 | Double existential quantif... |
| peano2uz 12915 | Second Peano postulate for... |
| peano2uzs 12916 | Second Peano postulate for... |
| peano2uzr 12917 | Reversed second Peano axio... |
| uzaddcl 12918 | Addition closure law for a... |
| nn0pzuz 12919 | The sum of a nonnegative i... |
| uzind4 12920 | Induction on the upper set... |
| uzind4ALT 12921 | Induction on the upper set... |
| uzind4s 12922 | Induction on the upper set... |
| uzind4s2 12923 | Induction on the upper set... |
| uzind4i 12924 | Induction on the upper int... |
| uzwo 12925 | Well-ordering principle: a... |
| uzwo2 12926 | Well-ordering principle: a... |
| nnwo 12927 | Well-ordering principle: a... |
| nnwof 12928 | Well-ordering principle: a... |
| nnwos 12929 | Well-ordering principle: a... |
| indstr 12930 | Strong Mathematical Induct... |
| eluznn0 12931 | Membership in a nonnegativ... |
| eluznn 12932 | Membership in a positive u... |
| eluz2b1 12933 | Two ways to say "an intege... |
| eluz2gt1 12934 | An integer greater than or... |
| eluz2b2 12935 | Two ways to say "an intege... |
| eluz2b3 12936 | Two ways to say "an intege... |
| uz2m1nn 12937 | One less than an integer g... |
| 1nuz2 12938 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12939 | A positive integer is eith... |
| uz2mulcl 12940 | Closure of multiplication ... |
| indstr2 12941 | Strong Mathematical Induct... |
| uzinfi 12942 | Extract the lower bound of... |
| nninf 12943 | The infimum of the set of ... |
| nn0inf 12944 | The infimum of the set of ... |
| infssuzle 12945 | The infimum of a subset of... |
| infssuzcl 12946 | The infimum of a subset of... |
| ublbneg 12947 | The image under negation o... |
| eqreznegel 12948 | Two ways to express the im... |
| supminf 12949 | The supremum of a bounded-... |
| lbzbi 12950 | If a set of reals is bound... |
| zsupss 12951 | Any nonempty bounded subse... |
| suprzcl2 12952 | The supremum of a bounded-... |
| suprzub 12953 | The supremum of a bounded-... |
| uzsupss 12954 | Any bounded subset of an u... |
| nn01to3 12955 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12956 | Alternate proof of ~ nn0ge... |
| uzwo3 12957 | Well-ordering principle: a... |
| zmin 12958 | There is a unique smallest... |
| zmax 12959 | There is a unique largest ... |
| zbtwnre 12960 | There is a unique integer ... |
| rebtwnz 12961 | There is a unique greatest... |
| elq 12964 | Membership in the set of r... |
| qmulz 12965 | If ` A ` is rational, then... |
| znq 12966 | The ratio of an integer an... |
| qre 12967 | A rational number is a rea... |
| zq 12968 | An integer is a rational n... |
| qred 12969 | A rational number is a rea... |
| zssq 12970 | The integers are a subset ... |
| nn0ssq 12971 | The nonnegative integers a... |
| nnssq 12972 | The positive integers are ... |
| qssre 12973 | The rationals are a subset... |
| qsscn 12974 | The rationals are a subset... |
| qex 12975 | The set of rational number... |
| nnq 12976 | A positive integer is rati... |
| qcn 12977 | A rational number is a com... |
| qexALT 12978 | Alternate proof of ~ qex .... |
| qaddcl 12979 | Closure of addition of rat... |
| qnegcl 12980 | Closure law for the negati... |
| qmulcl 12981 | Closure of multiplication ... |
| qsubcl 12982 | Closure of subtraction of ... |
| qreccl 12983 | Closure of reciprocal of r... |
| qdivcl 12984 | Closure of division of rat... |
| qrevaddcl 12985 | Reverse closure law for ad... |
| nnrecq 12986 | The reciprocal of a positi... |
| irradd 12987 | The sum of an irrational n... |
| irrmul 12988 | The product of an irration... |
| elpq 12989 | A positive rational is the... |
| elpqb 12990 | A class is a positive rati... |
| rpnnen1lem2 12991 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12992 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12993 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12994 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12995 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12996 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12997 | One half of ~ rpnnen , whe... |
| reexALT 12998 | Alternate proof of ~ reex ... |
| cnref1o 12999 | There is a natural one-to-... |
| cnexALT 13000 | The set of complex numbers... |
| xrex 13001 | The set of extended reals ... |
| mpoaddex 13002 | The addition operation is ... |
| addex 13003 | The addition operation is ... |
| mpomulex 13004 | The multiplication operati... |
| mulex 13005 | The multiplication operati... |
| elrp 13008 | Membership in the set of p... |
| elrpii 13009 | Membership in the set of p... |
| 1rp 13010 | 1 is a positive real. (Co... |
| 2rp 13011 | 2 is a positive real. (Co... |
| 3rp 13012 | 3 is a positive real. (Co... |
| rpssre 13013 | The positive reals are a s... |
| rpre 13014 | A positive real is a real.... |
| rpxr 13015 | A positive real is an exte... |
| rpcn 13016 | A positive real is a compl... |
| nnrp 13017 | A positive integer is a po... |
| rpgt0 13018 | A positive real is greater... |
| rpge0 13019 | A positive real is greater... |
| rpregt0 13020 | A positive real is a posit... |
| rprege0 13021 | A positive real is a nonne... |
| rpne0 13022 | A positive real is nonzero... |
| rprene0 13023 | A positive real is a nonze... |
| rpcnne0 13024 | A positive real is a nonze... |
| rpcndif0 13025 | A positive real number is ... |
| ralrp 13026 | Quantification over positi... |
| rexrp 13027 | Quantification over positi... |
| rpaddcl 13028 | Closure law for addition o... |
| rpmulcl 13029 | Closure law for multiplica... |
| rpmtmip 13030 | "Minus times minus is plus... |
| rpdivcl 13031 | Closure law for division o... |
| rpreccl 13032 | Closure law for reciprocat... |
| rphalfcl 13033 | Closure law for half of a ... |
| rpgecl 13034 | A number greater than or e... |
| rphalflt 13035 | Half of a positive real is... |
| rerpdivcl 13036 | Closure law for division o... |
| ge0p1rp 13037 | A nonnegative number plus ... |
| rpneg 13038 | Either a nonzero real or i... |
| negelrp 13039 | Elementhood of a negation ... |
| negelrpd 13040 | The negation of a negative... |
| 0nrp 13041 | Zero is not a positive rea... |
| ltsubrp 13042 | Subtracting a positive rea... |
| ltaddrp 13043 | Adding a positive number t... |
| difrp 13044 | Two ways to say one number... |
| elrpd 13045 | Membership in the set of p... |
| nnrpd 13046 | A positive integer is a po... |
| zgt1rpn0n1 13047 | An integer greater than 1 ... |
| rpred 13048 | A positive real is a real.... |
| rpxrd 13049 | A positive real is an exte... |
| rpcnd 13050 | A positive real is a compl... |
| rpgt0d 13051 | A positive real is greater... |
| rpge0d 13052 | A positive real is greater... |
| rpne0d 13053 | A positive real is nonzero... |
| rpregt0d 13054 | A positive real is real an... |
| rprege0d 13055 | A positive real is real an... |
| rprene0d 13056 | A positive real is a nonze... |
| rpcnne0d 13057 | A positive real is a nonze... |
| rpreccld 13058 | Closure law for reciprocat... |
| rprecred 13059 | Closure law for reciprocat... |
| rphalfcld 13060 | Closure law for half of a ... |
| reclt1d 13061 | The reciprocal of a positi... |
| recgt1d 13062 | The reciprocal of a positi... |
| rpaddcld 13063 | Closure law for addition o... |
| rpmulcld 13064 | Closure law for multiplica... |
| rpdivcld 13065 | Closure law for division o... |
| ltrecd 13066 | The reciprocal of both sid... |
| lerecd 13067 | The reciprocal of both sid... |
| ltrec1d 13068 | Reciprocal swap in a 'less... |
| lerec2d 13069 | Reciprocal swap in a 'less... |
| lediv2ad 13070 | Division of both sides of ... |
| ltdiv2d 13071 | Division of a positive num... |
| lediv2d 13072 | Division of a positive num... |
| ledivdivd 13073 | Invert ratios of positive ... |
| divge1 13074 | The ratio of a number over... |
| divlt1lt 13075 | A real number divided by a... |
| divle1le 13076 | A real number divided by a... |
| ledivge1le 13077 | If a number is less than o... |
| ge0p1rpd 13078 | A nonnegative number plus ... |
| rerpdivcld 13079 | Closure law for division o... |
| ltsubrpd 13080 | Subtracting a positive rea... |
| ltaddrpd 13081 | Adding a positive number t... |
| ltaddrp2d 13082 | Adding a positive number t... |
| ltmulgt11d 13083 | Multiplication by a number... |
| ltmulgt12d 13084 | Multiplication by a number... |
| gt0divd 13085 | Division of a positive num... |
| ge0divd 13086 | Division of a nonnegative ... |
| rpgecld 13087 | A number greater than or e... |
| divge0d 13088 | The ratio of nonnegative a... |
| ltmul1d 13089 | The ratio of nonnegative a... |
| ltmul2d 13090 | Multiplication of both sid... |
| lemul1d 13091 | Multiplication of both sid... |
| lemul2d 13092 | Multiplication of both sid... |
| ltdiv1d 13093 | Division of both sides of ... |
| lediv1d 13094 | Division of both sides of ... |
| ltmuldivd 13095 | 'Less than' relationship b... |
| ltmuldiv2d 13096 | 'Less than' relationship b... |
| lemuldivd 13097 | 'Less than or equal to' re... |
| lemuldiv2d 13098 | 'Less than or equal to' re... |
| ltdivmuld 13099 | 'Less than' relationship b... |
| ltdivmul2d 13100 | 'Less than' relationship b... |
| ledivmuld 13101 | 'Less than or equal to' re... |
| ledivmul2d 13102 | 'Less than or equal to' re... |
| ltmul1dd 13103 | The ratio of nonnegative a... |
| ltmul2dd 13104 | Multiplication of both sid... |
| ltdiv1dd 13105 | Division of both sides of ... |
| lediv1dd 13106 | Division of both sides of ... |
| lediv12ad 13107 | Comparison of ratio of two... |
| mul2lt0rlt0 13108 | If the result of a multipl... |
| mul2lt0rgt0 13109 | If the result of a multipl... |
| mul2lt0llt0 13110 | If the result of a multipl... |
| mul2lt0lgt0 13111 | If the result of a multipl... |
| mul2lt0bi 13112 | If the result of a multipl... |
| prodge0rd 13113 | Infer that a multiplicand ... |
| prodge0ld 13114 | Infer that a multiplier is... |
| ltdiv23d 13115 | Swap denominator with othe... |
| lediv23d 13116 | Swap denominator with othe... |
| lt2mul2divd 13117 | The ratio of nonnegative a... |
| nnledivrp 13118 | Division of a positive int... |
| nn0ledivnn 13119 | Division of a nonnegative ... |
| addlelt 13120 | If the sum of a real numbe... |
| ltxr 13127 | The 'less than' binary rel... |
| elxr 13128 | Membership in the set of e... |
| xrnemnf 13129 | An extended real other tha... |
| xrnepnf 13130 | An extended real other tha... |
| xrltnr 13131 | The extended real 'less th... |
| ltpnf 13132 | Any (finite) real is less ... |
| ltpnfd 13133 | Any (finite) real is less ... |
| 0ltpnf 13134 | Zero is less than plus inf... |
| mnflt 13135 | Minus infinity is less tha... |
| mnfltd 13136 | Minus infinity is less tha... |
| mnflt0 13137 | Minus infinity is less tha... |
| mnfltpnf 13138 | Minus infinity is less tha... |
| mnfltxr 13139 | Minus infinity is less tha... |
| pnfnlt 13140 | No extended real is greate... |
| nltmnf 13141 | No extended real is less t... |
| pnfge 13142 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13143 | An extended nonnegative in... |
| 0lepnf 13144 | 0 less than or equal to po... |
| xnn0ge0 13145 | An extended nonnegative in... |
| mnfle 13146 | Minus infinity is less tha... |
| mnfled 13147 | Minus infinity is less tha... |
| xrltnsym 13148 | Ordering on the extended r... |
| xrltnsym2 13149 | 'Less than' is antisymmetr... |
| xrlttri 13150 | Ordering on the extended r... |
| xrlttr 13151 | Ordering on the extended r... |
| xrltso 13152 | 'Less than' is a strict or... |
| xrlttri2 13153 | Trichotomy law for 'less t... |
| xrlttri3 13154 | Trichotomy law for 'less t... |
| xrleloe 13155 | 'Less than or equal' expre... |
| xrleltne 13156 | 'Less than or equal to' im... |
| xrltlen 13157 | 'Less than' expressed in t... |
| dfle2 13158 | Alternative definition of ... |
| dflt2 13159 | Alternative definition of ... |
| xrltle 13160 | 'Less than' implies 'less ... |
| xrltled 13161 | 'Less than' implies 'less ... |
| xrleid 13162 | 'Less than or equal to' is... |
| xrleidd 13163 | 'Less than or equal to' is... |
| xrletri 13164 | Trichotomy law for extende... |
| xrletri3 13165 | Trichotomy law for extende... |
| xrletrid 13166 | Trichotomy law for extende... |
| xrlelttr 13167 | Transitive law for orderin... |
| xrltletr 13168 | Transitive law for orderin... |
| xrletr 13169 | Transitive law for orderin... |
| xrlttrd 13170 | Transitive law for orderin... |
| xrlelttrd 13171 | Transitive law for orderin... |
| xrltletrd 13172 | Transitive law for orderin... |
| xrletrd 13173 | Transitive law for orderin... |
| xrltne 13174 | 'Less than' implies not eq... |
| nltpnft 13175 | An extended real is not le... |
| xgepnf 13176 | An extended real which is ... |
| ngtmnft 13177 | An extended real is not gr... |
| xlemnf 13178 | An extended real which is ... |
| xrrebnd 13179 | An extended real is real i... |
| xrre 13180 | A way of proving that an e... |
| xrre2 13181 | An extended real between t... |
| xrre3 13182 | A way of proving that an e... |
| ge0gtmnf 13183 | A nonnegative extended rea... |
| ge0nemnf 13184 | A nonnegative extended rea... |
| xrrege0 13185 | A nonnegative extended rea... |
| xrmax1 13186 | An extended real is less t... |
| xrmax2 13187 | An extended real is less t... |
| xrmin1 13188 | The minimum of two extende... |
| xrmin2 13189 | The minimum of two extende... |
| xrmaxeq 13190 | The maximum of two extende... |
| xrmineq 13191 | The minimum of two extende... |
| xrmaxlt 13192 | Two ways of saying the max... |
| xrltmin 13193 | Two ways of saying an exte... |
| xrmaxle 13194 | Two ways of saying the max... |
| xrlemin 13195 | Two ways of saying a numbe... |
| max1 13196 | A number is less than or e... |
| max1ALT 13197 | A number is less than or e... |
| max2 13198 | A number is less than or e... |
| 2resupmax 13199 | The supremum of two real n... |
| min1 13200 | The minimum of two numbers... |
| min2 13201 | The minimum of two numbers... |
| maxle 13202 | Two ways of saying the max... |
| lemin 13203 | Two ways of saying a numbe... |
| maxlt 13204 | Two ways of saying the max... |
| ltmin 13205 | Two ways of saying a numbe... |
| lemaxle 13206 | A real number which is les... |
| max0sub 13207 | Decompose a real number in... |
| ifle 13208 | An if statement transforms... |
| z2ge 13209 | There exists an integer gr... |
| qbtwnre 13210 | The rational numbers are d... |
| qbtwnxr 13211 | The rational numbers are d... |
| qsqueeze 13212 | If a nonnegative real is l... |
| qextltlem 13213 | Lemma for ~ qextlt and qex... |
| qextlt 13214 | An extensionality-like pro... |
| qextle 13215 | An extensionality-like pro... |
| xralrple 13216 | Show that ` A ` is less th... |
| alrple 13217 | Show that ` A ` is less th... |
| xnegeq 13218 | Equality of two extended n... |
| xnegex 13219 | A negative extended real e... |
| xnegpnf 13220 | Minus ` +oo ` . Remark of... |
| xnegmnf 13221 | Minus ` -oo ` . Remark of... |
| rexneg 13222 | Minus a real number. Rema... |
| xneg0 13223 | The negative of zero. (Co... |
| xnegcl 13224 | Closure of extended real n... |
| xnegneg 13225 | Extended real version of ~... |
| xneg11 13226 | Extended real version of ~... |
| xltnegi 13227 | Forward direction of ~ xlt... |
| xltneg 13228 | Extended real version of ~... |
| xleneg 13229 | Extended real version of ~... |
| xlt0neg1 13230 | Extended real version of ~... |
| xlt0neg2 13231 | Extended real version of ~... |
| xle0neg1 13232 | Extended real version of ~... |
| xle0neg2 13233 | Extended real version of ~... |
| xaddval 13234 | Value of the extended real... |
| xaddf 13235 | The extended real addition... |
| xmulval 13236 | Value of the extended real... |
| xaddpnf1 13237 | Addition of positive infin... |
| xaddpnf2 13238 | Addition of positive infin... |
| xaddmnf1 13239 | Addition of negative infin... |
| xaddmnf2 13240 | Addition of negative infin... |
| pnfaddmnf 13241 | Addition of positive and n... |
| mnfaddpnf 13242 | Addition of negative and p... |
| rexadd 13243 | The extended real addition... |
| rexsub 13244 | Extended real subtraction ... |
| rexaddd 13245 | The extended real addition... |
| xnn0xaddcl 13246 | The extended nonnegative i... |
| xaddnemnf 13247 | Closure of extended real a... |
| xaddnepnf 13248 | Closure of extended real a... |
| xnegid 13249 | Extended real version of ~... |
| xaddcl 13250 | The extended real addition... |
| xaddcom 13251 | The extended real addition... |
| xaddrid 13252 | Extended real version of ~... |
| xaddlid 13253 | Extended real version of ~... |
| xaddridd 13254 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13255 | Extended nonnegative integ... |
| xnn0lenn0nn0 13256 | An extended nonnegative in... |
| xnn0le2is012 13257 | An extended nonnegative in... |
| xnn0xadd0 13258 | The sum of two extended no... |
| xnegdi 13259 | Extended real version of ~... |
| xaddass 13260 | Associativity of extended ... |
| xaddass2 13261 | Associativity of extended ... |
| xpncan 13262 | Extended real version of ~... |
| xnpcan 13263 | Extended real version of ~... |
| xleadd1a 13264 | Extended real version of ~... |
| xleadd2a 13265 | Commuted form of ~ xleadd1... |
| xleadd1 13266 | Weakened version of ~ xlea... |
| xltadd1 13267 | Extended real version of ~... |
| xltadd2 13268 | Extended real version of ~... |
| xaddge0 13269 | The sum of nonnegative ext... |
| xle2add 13270 | Extended real version of ~... |
| xlt2add 13271 | Extended real version of ~... |
| xsubge0 13272 | Extended real version of ~... |
| xposdif 13273 | Extended real version of ~... |
| xlesubadd 13274 | Under certain conditions, ... |
| xmullem 13275 | Lemma for ~ rexmul . (Con... |
| xmullem2 13276 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13277 | Extended real multiplicati... |
| xmul01 13278 | Extended real version of ~... |
| xmul02 13279 | Extended real version of ~... |
| xmulneg1 13280 | Extended real version of ~... |
| xmulneg2 13281 | Extended real version of ~... |
| rexmul 13282 | The extended real multipli... |
| xmulf 13283 | The extended real multipli... |
| xmulcl 13284 | Closure of extended real m... |
| xmulpnf1 13285 | Multiplication by plus inf... |
| xmulpnf2 13286 | Multiplication by plus inf... |
| xmulmnf1 13287 | Multiplication by minus in... |
| xmulmnf2 13288 | Multiplication by minus in... |
| xmulpnf1n 13289 | Multiplication by plus inf... |
| xmulrid 13290 | Extended real version of ~... |
| xmullid 13291 | Extended real version of ~... |
| xmulm1 13292 | Extended real version of ~... |
| xmulasslem2 13293 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13294 | Extended real version of ~... |
| xmulge0 13295 | Extended real version of ~... |
| xmulasslem 13296 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13297 | Lemma for ~ xmulass . (Co... |
| xmulass 13298 | Associativity of the exten... |
| xlemul1a 13299 | Extended real version of ~... |
| xlemul2a 13300 | Extended real version of ~... |
| xlemul1 13301 | Extended real version of ~... |
| xlemul2 13302 | Extended real version of ~... |
| xltmul1 13303 | Extended real version of ~... |
| xltmul2 13304 | Extended real version of ~... |
| xadddilem 13305 | Lemma for ~ xadddi . (Con... |
| xadddi 13306 | Distributive property for ... |
| xadddir 13307 | Commuted version of ~ xadd... |
| xadddi2 13308 | The assumption that the mu... |
| xadddi2r 13309 | Commuted version of ~ xadd... |
| x2times 13310 | Extended real version of ~... |
| xnegcld 13311 | Closure of extended real n... |
| xaddcld 13312 | The extended real addition... |
| xmulcld 13313 | Closure of extended real m... |
| xadd4d 13314 | Rearrangement of 4 terms i... |
| xnn0add4d 13315 | Rearrangement of 4 terms i... |
| xrsupexmnf 13316 | Adding minus infinity to a... |
| xrinfmexpnf 13317 | Adding plus infinity to a ... |
| xrsupsslem 13318 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13319 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13320 | Any subset of extended rea... |
| xrinfmss 13321 | Any subset of extended rea... |
| xrinfmss2 13322 | Any subset of extended rea... |
| xrub 13323 | By quantifying only over r... |
| supxr 13324 | The supremum of a set of e... |
| supxr2 13325 | The supremum of a set of e... |
| supxrcl 13326 | The supremum of an arbitra... |
| supxrun 13327 | The supremum of the union ... |
| supxrmnf 13328 | Adding minus infinity to a... |
| supxrpnf 13329 | The supremum of a set of e... |
| supxrunb1 13330 | The supremum of an unbound... |
| supxrunb2 13331 | The supremum of an unbound... |
| supxrbnd1 13332 | The supremum of a bounded-... |
| supxrbnd2 13333 | The supremum of a bounded-... |
| xrsup0 13334 | The supremum of an empty s... |
| supxrub 13335 | A member of a set of exten... |
| supxrlub 13336 | The supremum of a set of e... |
| supxrleub 13337 | The supremum of a set of e... |
| supxrre 13338 | The real and extended real... |
| supxrbnd 13339 | The supremum of a bounded-... |
| supxrgtmnf 13340 | The supremum of a nonempty... |
| supxrre1 13341 | The supremum of a nonempty... |
| supxrre2 13342 | The supremum of a nonempty... |
| supxrss 13343 | Smaller sets of extended r... |
| infxrcl 13344 | The infimum of an arbitrar... |
| infxrlb 13345 | A member of a set of exten... |
| infxrgelb 13346 | The infimum of a set of ex... |
| infxrre 13347 | The real and extended real... |
| infxrmnf 13348 | The infinimum of a set of ... |
| xrinf0 13349 | The infimum of the empty s... |
| infxrss 13350 | Larger sets of extended re... |
| reltre 13351 | For all real numbers there... |
| rpltrp 13352 | For all positive real numb... |
| reltxrnmnf 13353 | For all extended real numb... |
| infmremnf 13354 | The infimum of the reals i... |
| infmrp1 13355 | The infimum of the positiv... |
| ixxval 13364 | Value of the interval func... |
| elixx1 13365 | Membership in an interval ... |
| ixxf 13366 | The set of intervals of ex... |
| ixxex 13367 | The set of intervals of ex... |
| ixxssxr 13368 | The set of intervals of ex... |
| elixx3g 13369 | Membership in a set of ope... |
| ixxssixx 13370 | An interval is a subset of... |
| ixxdisj 13371 | Split an interval into dis... |
| ixxun 13372 | Split an interval into two... |
| ixxin 13373 | Intersection of two interv... |
| ixxss1 13374 | Subset relationship for in... |
| ixxss2 13375 | Subset relationship for in... |
| ixxss12 13376 | Subset relationship for in... |
| ixxub 13377 | Extract the upper bound of... |
| ixxlb 13378 | Extract the lower bound of... |
| iooex 13379 | The set of open intervals ... |
| iooval 13380 | Value of the open interval... |
| ioo0 13381 | An empty open interval of ... |
| ioon0 13382 | An open interval of extend... |
| ndmioo 13383 | The open interval function... |
| iooid 13384 | An open interval with iden... |
| elioo3g 13385 | Membership in a set of ope... |
| elioore 13386 | A member of an open interv... |
| lbioo 13387 | An open interval does not ... |
| ubioo 13388 | An open interval does not ... |
| iooval2 13389 | Value of the open interval... |
| iooin 13390 | Intersection of two open i... |
| iooss1 13391 | Subset relationship for op... |
| iooss2 13392 | Subset relationship for op... |
| iocval 13393 | Value of the open-below, c... |
| icoval 13394 | Value of the closed-below,... |
| iccval 13395 | Value of the closed interv... |
| elioo1 13396 | Membership in an open inte... |
| elioo2 13397 | Membership in an open inte... |
| elioc1 13398 | Membership in an open-belo... |
| elico1 13399 | Membership in a closed-bel... |
| elicc1 13400 | Membership in a closed int... |
| iccid 13401 | A closed interval with ide... |
| ico0 13402 | An empty open interval of ... |
| ioc0 13403 | An empty open interval of ... |
| icc0 13404 | An empty closed interval o... |
| dfrp2 13405 | Alternate definition of th... |
| elicod 13406 | Membership in a left-close... |
| icogelb 13407 | An element of a left-close... |
| elicore 13408 | A member of a left-closed ... |
| ubioc1 13409 | The upper bound belongs to... |
| lbico1 13410 | The lower bound belongs to... |
| iccleub 13411 | An element of a closed int... |
| iccgelb 13412 | An element of a closed int... |
| elioo5 13413 | Membership in an open inte... |
| eliooxr 13414 | A nonempty open interval s... |
| eliooord 13415 | Ordering implied by a memb... |
| elioo4g 13416 | Membership in an open inte... |
| ioossre 13417 | An open interval is a set ... |
| ioosscn 13418 | An open interval is a set ... |
| elioc2 13419 | Membership in an open-belo... |
| elico2 13420 | Membership in a closed-bel... |
| elicc2 13421 | Membership in a closed rea... |
| elicc2i 13422 | Inference for membership i... |
| elicc4 13423 | Membership in a closed rea... |
| iccss 13424 | Condition for a closed int... |
| iccssioo 13425 | Condition for a closed int... |
| icossico 13426 | Condition for a closed-bel... |
| iccss2 13427 | Condition for a closed int... |
| iccssico 13428 | Condition for a closed int... |
| iccssioo2 13429 | Condition for a closed int... |
| iccssico2 13430 | Condition for a closed int... |
| ioomax 13431 | The open interval from min... |
| iccmax 13432 | The closed interval from m... |
| ioopos 13433 | The set of positive reals ... |
| ioorp 13434 | The set of positive reals ... |
| iooshf 13435 | Shift the arguments of the... |
| iocssre 13436 | A closed-above interval wi... |
| icossre 13437 | A closed-below interval wi... |
| iccssre 13438 | A closed real interval is ... |
| iccssxr 13439 | A closed interval is a set... |
| iocssxr 13440 | An open-below, closed-abov... |
| icossxr 13441 | A closed-below, open-above... |
| ioossicc 13442 | An open interval is a subs... |
| iccssred 13443 | A closed real interval is ... |
| eliccxr 13444 | A member of a closed inter... |
| icossicc 13445 | A closed-below, open-above... |
| iocssicc 13446 | A closed-above, open-below... |
| ioossico 13447 | An open interval is a subs... |
| iocssioo 13448 | Condition for a closed int... |
| icossioo 13449 | Condition for a closed int... |
| ioossioo 13450 | Condition for an open inte... |
| iccsupr 13451 | A nonempty subset of a clo... |
| elioopnf 13452 | Membership in an unbounded... |
| elioomnf 13453 | Membership in an unbounded... |
| elicopnf 13454 | Membership in a closed unb... |
| repos 13455 | Two ways of saying that a ... |
| ioof 13456 | The set of open intervals ... |
| iccf 13457 | The set of closed interval... |
| unirnioo 13458 | The union of the range of ... |
| dfioo2 13459 | Alternate definition of th... |
| ioorebas 13460 | Open intervals are element... |
| xrge0neqmnf 13461 | A nonnegative extended rea... |
| xrge0nre 13462 | An extended real which is ... |
| elrege0 13463 | The predicate "is a nonneg... |
| nn0rp0 13464 | A nonnegative integer is a... |
| rge0ssre 13465 | Nonnegative real numbers a... |
| elxrge0 13466 | Elementhood in the set of ... |
| 0e0icopnf 13467 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13468 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13469 | The nonnegative reals are ... |
| ge0mulcl 13470 | The nonnegative reals are ... |
| ge0xaddcl 13471 | The nonnegative reals are ... |
| ge0xmulcl 13472 | The nonnegative extended r... |
| lbicc2 13473 | The lower bound of a close... |
| ubicc2 13474 | The upper bound of a close... |
| elicc01 13475 | Membership in the closed r... |
| elunitrn 13476 | The closed unit interval i... |
| elunitcn 13477 | The closed unit interval i... |
| 0elunit 13478 | Zero is an element of the ... |
| 1elunit 13479 | One is an element of the c... |
| iooneg 13480 | Membership in a negated op... |
| iccneg 13481 | Membership in a negated cl... |
| icoshft 13482 | A shifted real is a member... |
| icoshftf1o 13483 | Shifting a closed-below, o... |
| icoun 13484 | The union of two adjacent ... |
| icodisj 13485 | Adjacent left-closed right... |
| ioounsn 13486 | The union of an open inter... |
| snunioo 13487 | The closure of one end of ... |
| snunico 13488 | The closure of the open en... |
| snunioc 13489 | The closure of the open en... |
| prunioo 13490 | The closure of an open rea... |
| ioodisj 13491 | If the upper bound of one ... |
| ioojoin 13492 | Join two open intervals to... |
| difreicc 13493 | The class difference of ` ... |
| iccsplit 13494 | Split a closed interval in... |
| iccshftr 13495 | Membership in a shifted in... |
| iccshftri 13496 | Membership in a shifted in... |
| iccshftl 13497 | Membership in a shifted in... |
| iccshftli 13498 | Membership in a shifted in... |
| iccdil 13499 | Membership in a dilated in... |
| iccdili 13500 | Membership in a dilated in... |
| icccntr 13501 | Membership in a contracted... |
| icccntri 13502 | Membership in a contracted... |
| divelunit 13503 | A condition for a ratio to... |
| lincmb01cmp 13504 | A linear combination of tw... |
| iccf1o 13505 | Describe a bijection from ... |
| iccen 13506 | Any nontrivial closed inte... |
| xov1plusxeqvd 13507 | A complex number ` X ` is ... |
| unitssre 13508 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13509 | The closed unit interval i... |
| supicc 13510 | Supremum of a bounded set ... |
| supiccub 13511 | The supremum of a bounded ... |
| supicclub 13512 | The supremum of a bounded ... |
| supicclub2 13513 | The supremum of a bounded ... |
| zltaddlt1le 13514 | The sum of an integer and ... |
| xnn0xrge0 13515 | An extended nonnegative in... |
| fzval 13518 | The value of a finite set ... |
| fzval2 13519 | An alternative way of expr... |
| fzf 13520 | Establish the domain and c... |
| elfz1 13521 | Membership in a finite set... |
| elfz 13522 | Membership in a finite set... |
| elfz2 13523 | Membership in a finite set... |
| elfzd 13524 | Membership in a finite set... |
| elfz5 13525 | Membership in a finite set... |
| elfz4 13526 | Membership in a finite set... |
| elfzuzb 13527 | Membership in a finite set... |
| eluzfz 13528 | Membership in a finite set... |
| elfzuz 13529 | A member of a finite set o... |
| elfzuz3 13530 | Membership in a finite set... |
| elfzel2 13531 | Membership in a finite set... |
| elfzel1 13532 | Membership in a finite set... |
| elfzelz 13533 | A member of a finite set o... |
| elfzelzd 13534 | A member of a finite set o... |
| fzssz 13535 | A finite sequence of integ... |
| elfzle1 13536 | A member of a finite set o... |
| elfzle2 13537 | A member of a finite set o... |
| elfzuz2 13538 | Implication of membership ... |
| elfzle3 13539 | Membership in a finite set... |
| eluzfz1 13540 | Membership in a finite set... |
| eluzfz2 13541 | Membership in a finite set... |
| eluzfz2b 13542 | Membership in a finite set... |
| elfz3 13543 | Membership in a finite set... |
| elfz1eq 13544 | Membership in a finite set... |
| elfzubelfz 13545 | If there is a member in a ... |
| peano2fzr 13546 | A Peano-postulate-like the... |
| fzn0 13547 | Properties of a finite int... |
| fz0 13548 | A finite set of sequential... |
| fzn 13549 | A finite set of sequential... |
| fzen 13550 | A shifted finite set of se... |
| fz1n 13551 | A 1-based finite set of se... |
| 0nelfz1 13552 | 0 is not an element of a f... |
| 0fz1 13553 | Two ways to say a finite 1... |
| fz10 13554 | There are no integers betw... |
| uzsubsubfz 13555 | Membership of an integer g... |
| uzsubsubfz1 13556 | Membership of an integer g... |
| ige3m2fz 13557 | Membership of an integer g... |
| fzsplit2 13558 | Split a finite interval of... |
| fzsplit 13559 | Split a finite interval of... |
| fzdisj 13560 | Condition for two finite i... |
| fz01en 13561 | 0-based and 1-based finite... |
| elfznn 13562 | A member of a finite set o... |
| elfz1end 13563 | A nonempty finite range of... |
| fz1ssnn 13564 | A finite set of positive i... |
| fznn0sub 13565 | Subtraction closure for a ... |
| fzmmmeqm 13566 | Subtracting the difference... |
| fzaddel 13567 | Membership of a sum in a f... |
| fzadd2 13568 | Membership of a sum in a f... |
| fzsubel 13569 | Membership of a difference... |
| fzopth 13570 | A finite set of sequential... |
| fzass4 13571 | Two ways to express a nond... |
| fzss1 13572 | Subset relationship for fi... |
| fzss2 13573 | Subset relationship for fi... |
| fzssuz 13574 | A finite set of sequential... |
| fzsn 13575 | A finite interval of integ... |
| fzssp1 13576 | Subset relationship for fi... |
| fzssnn 13577 | Finite sets of sequential ... |
| ssfzunsnext 13578 | A subset of a finite seque... |
| ssfzunsn 13579 | A subset of a finite seque... |
| fzsuc 13580 | Join a successor to the en... |
| fzpred 13581 | Join a predecessor to the ... |
| fzpreddisj 13582 | A finite set of sequential... |
| elfzp1 13583 | Append an element to a fin... |
| fzp1ss 13584 | Subset relationship for fi... |
| fzelp1 13585 | Membership in a set of seq... |
| fzp1elp1 13586 | Add one to an element of a... |
| fznatpl1 13587 | Shift membership in a fini... |
| fzpr 13588 | A finite interval of integ... |
| fztp 13589 | A finite interval of integ... |
| fz12pr 13590 | An integer range between 1... |
| fzsuc2 13591 | Join a successor to the en... |
| fzp1disj 13592 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13593 | Remove a successor from th... |
| fzprval 13594 | Two ways of defining the f... |
| fztpval 13595 | Two ways of defining the f... |
| fzrev 13596 | Reversal of start and end ... |
| fzrev2 13597 | Reversal of start and end ... |
| fzrev2i 13598 | Reversal of start and end ... |
| fzrev3 13599 | The "complement" of a memb... |
| fzrev3i 13600 | The "complement" of a memb... |
| fznn 13601 | Finite set of sequential i... |
| elfz1b 13602 | Membership in a 1-based fi... |
| elfz1uz 13603 | Membership in a 1-based fi... |
| elfzm11 13604 | Membership in a finite set... |
| uzsplit 13605 | Express an upper integer s... |
| uzdisj 13606 | The first ` N ` elements o... |
| fseq1p1m1 13607 | Add/remove an item to/from... |
| fseq1m1p1 13608 | Add/remove an item to/from... |
| fz1sbc 13609 | Quantification over a one-... |
| elfzp1b 13610 | An integer is a member of ... |
| elfzm1b 13611 | An integer is a member of ... |
| elfzp12 13612 | Options for membership in ... |
| fzm1 13613 | Choices for an element of ... |
| fzneuz 13614 | No finite set of sequentia... |
| fznuz 13615 | Disjointness of the upper ... |
| uznfz 13616 | Disjointness of the upper ... |
| fzp1nel 13617 | One plus the upper bound o... |
| fzrevral 13618 | Reversal of scanning order... |
| fzrevral2 13619 | Reversal of scanning order... |
| fzrevral3 13620 | Reversal of scanning order... |
| fzshftral 13621 | Shift the scanning order i... |
| ige2m1fz1 13622 | Membership of an integer g... |
| ige2m1fz 13623 | Membership in a 0-based fi... |
| elfz2nn0 13624 | Membership in a finite set... |
| fznn0 13625 | Characterization of a fini... |
| elfznn0 13626 | A member of a finite set o... |
| elfz3nn0 13627 | The upper bound of a nonem... |
| fz0ssnn0 13628 | Finite sets of sequential ... |
| fz1ssfz0 13629 | Subset relationship for fi... |
| 0elfz 13630 | 0 is an element of a finit... |
| nn0fz0 13631 | A nonnegative integer is a... |
| elfz0add 13632 | An element of a finite set... |
| fz0sn 13633 | An integer range from 0 to... |
| fz0tp 13634 | An integer range from 0 to... |
| fz0to3un2pr 13635 | An integer range from 0 to... |
| fz0to4untppr 13636 | An integer range from 0 to... |
| elfz0ubfz0 13637 | An element of a finite set... |
| elfz0fzfz0 13638 | A member of a finite set o... |
| fz0fzelfz0 13639 | If a member of a finite se... |
| fznn0sub2 13640 | Subtraction closure for a ... |
| uzsubfz0 13641 | Membership of an integer g... |
| fz0fzdiffz0 13642 | The difference of an integ... |
| elfzmlbm 13643 | Subtracting the lower boun... |
| elfzmlbp 13644 | Subtracting the lower boun... |
| fzctr 13645 | Lemma for theorems about t... |
| difelfzle 13646 | The difference of two inte... |
| difelfznle 13647 | The difference of two inte... |
| nn0split 13648 | Express the set of nonnega... |
| nn0disj 13649 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13650 | A finite set of sequential... |
| fvffz0 13651 | The function value of a fu... |
| 1fv 13652 | A function on a singleton.... |
| 4fvwrd4 13653 | The first four function va... |
| 2ffzeq 13654 | Two functions over 0-based... |
| preduz 13655 | The value of the predecess... |
| prednn 13656 | The value of the predecess... |
| prednn0 13657 | The value of the predecess... |
| predfz 13658 | Calculate the predecessor ... |
| fzof 13661 | Functionality of the half-... |
| elfzoel1 13662 | Reverse closure for half-o... |
| elfzoel2 13663 | Reverse closure for half-o... |
| elfzoelz 13664 | Reverse closure for half-o... |
| fzoval 13665 | Value of the half-open int... |
| elfzo 13666 | Membership in a half-open ... |
| elfzo2 13667 | Membership in a half-open ... |
| elfzouz 13668 | Membership in a half-open ... |
| nelfzo 13669 | An integer not being a mem... |
| fzolb 13670 | The left endpoint of a hal... |
| fzolb2 13671 | The left endpoint of a hal... |
| elfzole1 13672 | A member in a half-open in... |
| elfzolt2 13673 | A member in a half-open in... |
| elfzolt3 13674 | Membership in a half-open ... |
| elfzolt2b 13675 | A member in a half-open in... |
| elfzolt3b 13676 | Membership in a half-open ... |
| elfzop1le2 13677 | A member in a half-open in... |
| fzonel 13678 | A half-open range does not... |
| elfzouz2 13679 | The upper bound of a half-... |
| elfzofz 13680 | A half-open range is conta... |
| elfzo3 13681 | Express membership in a ha... |
| fzon0 13682 | A half-open integer interv... |
| fzossfz 13683 | A half-open range is conta... |
| fzossz 13684 | A half-open integer interv... |
| fzon 13685 | A half-open set of sequent... |
| fzo0n 13686 | A half-open range of nonne... |
| fzonlt0 13687 | A half-open integer range ... |
| fzo0 13688 | Half-open sets with equal ... |
| fzonnsub 13689 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13690 | If ` M < N ` then ` N - M ... |
| fzoss1 13691 | Subset relationship for ha... |
| fzoss2 13692 | Subset relationship for ha... |
| fzossrbm1 13693 | Subset of a half-open rang... |
| fzo0ss1 13694 | Subset relationship for ha... |
| fzossnn0 13695 | A half-open integer range ... |
| fzospliti 13696 | One direction of splitting... |
| fzosplit 13697 | Split a half-open integer ... |
| fzodisj 13698 | Abutting half-open integer... |
| fzouzsplit 13699 | Split an upper integer set... |
| fzouzdisj 13700 | A half-open integer range ... |
| fzoun 13701 | A half-open integer range ... |
| fzodisjsn 13702 | A half-open integer range ... |
| prinfzo0 13703 | The intersection of a half... |
| lbfzo0 13704 | An integer is strictly gre... |
| elfzo0 13705 | Membership in a half-open ... |
| elfzo0z 13706 | Membership in a half-open ... |
| nn0p1elfzo 13707 | A nonnegative integer incr... |
| elfzo0le 13708 | A member in a half-open ra... |
| elfzonn0 13709 | A member of a half-open ra... |
| fzonmapblen 13710 | The result of subtracting ... |
| fzofzim 13711 | If a nonnegative integer i... |
| fz1fzo0m1 13712 | Translation of one between... |
| fzossnn 13713 | Half-open integer ranges s... |
| elfzo1 13714 | Membership in a half-open ... |
| fzo1fzo0n0 13715 | An integer between 1 and a... |
| fzo0n0 13716 | A half-open integer range ... |
| fzoaddel 13717 | Translate membership in a ... |
| fzo0addel 13718 | Translate membership in a ... |
| fzo0addelr 13719 | Translate membership in a ... |
| fzoaddel2 13720 | Translate membership in a ... |
| elfzoext 13721 | Membership of an integer i... |
| elincfzoext 13722 | Membership of an increased... |
| fzosubel 13723 | Translate membership in a ... |
| fzosubel2 13724 | Membership in a translated... |
| fzosubel3 13725 | Membership in a translated... |
| eluzgtdifelfzo 13726 | Membership of the differen... |
| ige2m2fzo 13727 | Membership of an integer g... |
| fzocatel 13728 | Translate membership in a ... |
| ubmelfzo 13729 | If an integer in a 1-based... |
| elfzodifsumelfzo 13730 | If an integer is in a half... |
| elfzom1elp1fzo 13731 | Membership of an integer i... |
| elfzom1elfzo 13732 | Membership in a half-open ... |
| fzval3 13733 | Expressing a closed intege... |
| fz0add1fz1 13734 | Translate membership in a ... |
| fzosn 13735 | Expressing a singleton as ... |
| elfzomin 13736 | Membership of an integer i... |
| zpnn0elfzo 13737 | Membership of an integer i... |
| zpnn0elfzo1 13738 | Membership of an integer i... |
| fzosplitsnm1 13739 | Removing a singleton from ... |
| elfzonlteqm1 13740 | If an element of a half-op... |
| fzonn0p1 13741 | A nonnegative integer is e... |
| fzossfzop1 13742 | A half-open range of nonne... |
| fzonn0p1p1 13743 | If a nonnegative integer i... |
| elfzom1p1elfzo 13744 | Increasing an element of a... |
| fzo0ssnn0 13745 | Half-open integer ranges s... |
| fzo01 13746 | Expressing the singleton o... |
| fzo12sn 13747 | A 1-based half-open intege... |
| fzo13pr 13748 | A 1-based half-open intege... |
| fzo0to2pr 13749 | A half-open integer range ... |
| fzo0to3tp 13750 | A half-open integer range ... |
| fzo0to42pr 13751 | A half-open integer range ... |
| fzo1to4tp 13752 | A half-open integer range ... |
| fzo0sn0fzo1 13753 | A half-open range of nonne... |
| elfzo0l 13754 | A member of a half-open ra... |
| fzoend 13755 | The endpoint of a half-ope... |
| fzo0end 13756 | The endpoint of a zero-bas... |
| ssfzo12 13757 | Subset relationship for ha... |
| ssfzoulel 13758 | If a half-open integer ran... |
| ssfzo12bi 13759 | Subset relationship for ha... |
| ubmelm1fzo 13760 | The result of subtracting ... |
| fzofzp1 13761 | If a point is in a half-op... |
| fzofzp1b 13762 | If a point is in a half-op... |
| elfzom1b 13763 | An integer is a member of ... |
| elfzom1elp1fzo1 13764 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13765 | Membership of a positive i... |
| elfzonelfzo 13766 | If an element of a half-op... |
| fzonfzoufzol 13767 | If an element of a half-op... |
| elfzomelpfzo 13768 | An integer increased by an... |
| elfznelfzo 13769 | A value in a finite set of... |
| elfznelfzob 13770 | A value in a finite set of... |
| peano2fzor 13771 | A Peano-postulate-like the... |
| fzosplitsn 13772 | Extending a half-open rang... |
| fzosplitpr 13773 | Extending a half-open inte... |
| fzosplitprm1 13774 | Extending a half-open inte... |
| fzosplitsni 13775 | Membership in a half-open ... |
| fzisfzounsn 13776 | A finite interval of integ... |
| elfzr 13777 | A member of a finite inter... |
| elfzlmr 13778 | A member of a finite inter... |
| elfz0lmr 13779 | A member of a finite inter... |
| fzostep1 13780 | Two possibilities for a nu... |
| fzoshftral 13781 | Shift the scanning order i... |
| fzind2 13782 | Induction on the integers ... |
| fvinim0ffz 13783 | The function values for th... |
| injresinjlem 13784 | Lemma for ~ injresinj . (... |
| injresinj 13785 | A function whose restricti... |
| subfzo0 13786 | The difference between two... |
| flval 13791 | Value of the floor (greate... |
| flcl 13792 | The floor (greatest intege... |
| reflcl 13793 | The floor (greatest intege... |
| fllelt 13794 | A basic property of the fl... |
| flcld 13795 | The floor (greatest intege... |
| flle 13796 | A basic property of the fl... |
| flltp1 13797 | A basic property of the fl... |
| fllep1 13798 | A basic property of the fl... |
| fraclt1 13799 | The fractional part of a r... |
| fracle1 13800 | The fractional part of a r... |
| fracge0 13801 | The fractional part of a r... |
| flge 13802 | The floor function value i... |
| fllt 13803 | The floor function value i... |
| flflp1 13804 | Move floor function betwee... |
| flid 13805 | An integer is its own floo... |
| flidm 13806 | The floor function is idem... |
| flidz 13807 | A real number equals its f... |
| flltnz 13808 | The floor of a non-integer... |
| flwordi 13809 | Ordering relation for the ... |
| flword2 13810 | Ordering relation for the ... |
| flval2 13811 | An alternate way to define... |
| flval3 13812 | An alternate way to define... |
| flbi 13813 | A condition equivalent to ... |
| flbi2 13814 | A condition equivalent to ... |
| adddivflid 13815 | The floor of a sum of an i... |
| ico01fl0 13816 | The floor of a real number... |
| flge0nn0 13817 | The floor of a number grea... |
| flge1nn 13818 | The floor of a number grea... |
| fldivnn0 13819 | The floor function of a di... |
| refldivcl 13820 | The floor function of a di... |
| divfl0 13821 | The floor of a fraction is... |
| fladdz 13822 | An integer can be moved in... |
| flzadd 13823 | An integer can be moved in... |
| flmulnn0 13824 | Move a nonnegative integer... |
| btwnzge0 13825 | A real bounded between an ... |
| 2tnp1ge0ge0 13826 | Two times an integer plus ... |
| flhalf 13827 | Ordering relation for the ... |
| fldivle 13828 | The floor function of a di... |
| fldivnn0le 13829 | The floor function of a di... |
| flltdivnn0lt 13830 | The floor function of a di... |
| ltdifltdiv 13831 | If the dividend of a divis... |
| fldiv4p1lem1div2 13832 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13833 | The floor of an integer gr... |
| fldiv4lem1div2 13834 | The floor of a positive in... |
| ceilval 13835 | The value of the ceiling f... |
| dfceil2 13836 | Alternative definition of ... |
| ceilval2 13837 | The value of the ceiling f... |
| ceicl 13838 | The ceiling function retur... |
| ceilcl 13839 | Closure of the ceiling fun... |
| ceilcld 13840 | Closure of the ceiling fun... |
| ceige 13841 | The ceiling of a real numb... |
| ceilge 13842 | The ceiling of a real numb... |
| ceilged 13843 | The ceiling of a real numb... |
| ceim1l 13844 | One less than the ceiling ... |
| ceilm1lt 13845 | One less than the ceiling ... |
| ceile 13846 | The ceiling of a real numb... |
| ceille 13847 | The ceiling of a real numb... |
| ceilid 13848 | An integer is its own ceil... |
| ceilidz 13849 | A real number equals its c... |
| flleceil 13850 | The floor of a real number... |
| fleqceilz 13851 | A real number is an intege... |
| quoremz 13852 | Quotient and remainder of ... |
| quoremnn0 13853 | Quotient and remainder of ... |
| quoremnn0ALT 13854 | Alternate proof of ~ quore... |
| intfrac2 13855 | Decompose a real into inte... |
| intfracq 13856 | Decompose a rational numbe... |
| fldiv 13857 | Cancellation of the embedd... |
| fldiv2 13858 | Cancellation of an embedde... |
| fznnfl 13859 | Finite set of sequential i... |
| uzsup 13860 | An upper set of integers i... |
| ioopnfsup 13861 | An upper set of reals is u... |
| icopnfsup 13862 | An upper set of reals is u... |
| rpsup 13863 | The positive reals are unb... |
| resup 13864 | The real numbers are unbou... |
| xrsup 13865 | The extended real numbers ... |
| modval 13868 | The value of the modulo op... |
| modvalr 13869 | The value of the modulo op... |
| modcl 13870 | Closure law for the modulo... |
| flpmodeq 13871 | Partition of a division in... |
| modcld 13872 | Closure law for the modulo... |
| mod0 13873 | ` A mod B ` is zero iff ` ... |
| mulmod0 13874 | The product of an integer ... |
| negmod0 13875 | ` A ` is divisible by ` B ... |
| modge0 13876 | The modulo operation is no... |
| modlt 13877 | The modulo operation is le... |
| modelico 13878 | Modular reduction produces... |
| moddiffl 13879 | Value of the modulo operat... |
| moddifz 13880 | The modulo operation diffe... |
| modfrac 13881 | The fractional part of a n... |
| flmod 13882 | The floor function express... |
| intfrac 13883 | Break a number into its in... |
| zmod10 13884 | An integer modulo 1 is 0. ... |
| zmod1congr 13885 | Two arbitrary integers are... |
| modmulnn 13886 | Move a positive integer in... |
| modvalp1 13887 | The value of the modulo op... |
| zmodcl 13888 | Closure law for the modulo... |
| zmodcld 13889 | Closure law for the modulo... |
| zmodfz 13890 | An integer mod ` B ` lies ... |
| zmodfzo 13891 | An integer mod ` B ` lies ... |
| zmodfzp1 13892 | An integer mod ` B ` lies ... |
| modid 13893 | Identity law for modulo. ... |
| modid0 13894 | A positive real number mod... |
| modid2 13895 | Identity law for modulo. ... |
| zmodid2 13896 | Identity law for modulo re... |
| zmodidfzo 13897 | Identity law for modulo re... |
| zmodidfzoimp 13898 | Identity law for modulo re... |
| 0mod 13899 | Special case: 0 modulo a p... |
| 1mod 13900 | Special case: 1 modulo a r... |
| modabs 13901 | Absorption law for modulo.... |
| modabs2 13902 | Absorption law for modulo.... |
| modcyc 13903 | The modulo operation is pe... |
| modcyc2 13904 | The modulo operation is pe... |
| modadd1 13905 | Addition property of the m... |
| modaddabs 13906 | Absorption law for modulo.... |
| modaddmod 13907 | The sum of a real number m... |
| muladdmodid 13908 | The sum of a positive real... |
| mulp1mod1 13909 | The product of an integer ... |
| modmuladd 13910 | Decomposition of an intege... |
| modmuladdim 13911 | Implication of a decomposi... |
| modmuladdnn0 13912 | Implication of a decomposi... |
| negmod 13913 | The negation of a number m... |
| m1modnnsub1 13914 | Minus one modulo a positiv... |
| m1modge3gt1 13915 | Minus one modulo an intege... |
| addmodid 13916 | The sum of a positive inte... |
| addmodidr 13917 | The sum of a positive inte... |
| modadd2mod 13918 | The sum of a real number m... |
| modm1p1mod0 13919 | If a real number modulo a ... |
| modltm1p1mod 13920 | If a real number modulo a ... |
| modmul1 13921 | Multiplication property of... |
| modmul12d 13922 | Multiplication property of... |
| modnegd 13923 | Negation property of the m... |
| modadd12d 13924 | Additive property of the m... |
| modsub12d 13925 | Subtraction property of th... |
| modsubmod 13926 | The difference of a real n... |
| modsubmodmod 13927 | The difference of a real n... |
| 2txmodxeq0 13928 | Two times a positive real ... |
| 2submod 13929 | If a real number is betwee... |
| modifeq2int 13930 | If a nonnegative integer i... |
| modaddmodup 13931 | The sum of an integer modu... |
| modaddmodlo 13932 | The sum of an integer modu... |
| modmulmod 13933 | The product of a real numb... |
| modmulmodr 13934 | The product of an integer ... |
| modaddmulmod 13935 | The sum of a real number a... |
| moddi 13936 | Distribute multiplication ... |
| modsubdir 13937 | Distribute the modulo oper... |
| modeqmodmin 13938 | A real number equals the d... |
| modirr 13939 | A number modulo an irratio... |
| modfzo0difsn 13940 | For a number within a half... |
| modsumfzodifsn 13941 | The sum of a number within... |
| modlteq 13942 | Two nonnegative integers l... |
| addmodlteq 13943 | Two nonnegative integers l... |
| om2uz0i 13944 | The mapping ` G ` is a one... |
| om2uzsuci 13945 | The value of ` G ` (see ~ ... |
| om2uzuzi 13946 | The value ` G ` (see ~ om2... |
| om2uzlti 13947 | Less-than relation for ` G... |
| om2uzlt2i 13948 | The mapping ` G ` (see ~ o... |
| om2uzrani 13949 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13950 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13951 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13952 | An alternative definition ... |
| om2uzrdg 13953 | A helper lemma for the val... |
| uzrdglem 13954 | A helper lemma for the val... |
| uzrdgfni 13955 | The recursive definition g... |
| uzrdg0i 13956 | Initial value of a recursi... |
| uzrdgsuci 13957 | Successor value of a recur... |
| ltweuz 13958 | ` < ` is a well-founded re... |
| ltwenn 13959 | Less than well-orders the ... |
| ltwefz 13960 | Less than well-orders a se... |
| uzenom 13961 | An upper integer set is de... |
| uzinf 13962 | An upper integer set is in... |
| nnnfi 13963 | The set of positive intege... |
| uzrdgxfr 13964 | Transfer the value of the ... |
| fzennn 13965 | The cardinality of a finit... |
| fzen2 13966 | The cardinality of a finit... |
| cardfz 13967 | The cardinality of a finit... |
| hashgf1o 13968 | ` G ` maps ` _om ` one-to-... |
| fzfi 13969 | A finite interval of integ... |
| fzfid 13970 | Commonly used special case... |
| fzofi 13971 | Half-open integer sets are... |
| fsequb 13972 | The values of a finite rea... |
| fsequb2 13973 | The values of a finite rea... |
| fseqsupcl 13974 | The values of a finite rea... |
| fseqsupubi 13975 | The values of a finite rea... |
| nn0ennn 13976 | The nonnegative integers a... |
| nnenom 13977 | The set of positive intege... |
| nnct 13978 | ` NN ` is countable. (Con... |
| uzindi 13979 | Indirect strong induction ... |
| axdc4uzlem 13980 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13981 | A version of ~ axdc4 that ... |
| ssnn0fi 13982 | A subset of the nonnegativ... |
| rabssnn0fi 13983 | A subset of the nonnegativ... |
| uzsinds 13984 | Strong (or "total") induct... |
| nnsinds 13985 | Strong (or "total") induct... |
| nn0sinds 13986 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13987 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13988 | If all functions of a fini... |
| fsuppmapnn0fiubex 13989 | If all functions of a fini... |
| fsuppmapnn0fiub0 13990 | If all functions of a fini... |
| suppssfz 13991 | Condition for a function o... |
| fsuppmapnn0ub 13992 | If a function over the non... |
| fsuppmapnn0fz 13993 | If a function over the non... |
| mptnn0fsupp 13994 | A mapping from the nonnega... |
| mptnn0fsuppd 13995 | A mapping from the nonnega... |
| mptnn0fsuppr 13996 | A finitely supported mappi... |
| f13idfv 13997 | A one-to-one function with... |
| seqex 14000 | Existence of the sequence ... |
| seqeq1 14001 | Equality theorem for the s... |
| seqeq2 14002 | Equality theorem for the s... |
| seqeq3 14003 | Equality theorem for the s... |
| seqeq1d 14004 | Equality deduction for the... |
| seqeq2d 14005 | Equality deduction for the... |
| seqeq3d 14006 | Equality deduction for the... |
| seqeq123d 14007 | Equality deduction for the... |
| nfseq 14008 | Hypothesis builder for the... |
| seqval 14009 | Value of the sequence buil... |
| seqfn 14010 | The sequence builder funct... |
| seq1 14011 | Value of the sequence buil... |
| seq1i 14012 | Value of the sequence buil... |
| seqp1 14013 | Value of the sequence buil... |
| seqexw 14014 | Weak version of ~ seqex th... |
| seqp1d 14015 | Value of the sequence buil... |
| seqm1 14016 | Value of the sequence buil... |
| seqcl2 14017 | Closure properties of the ... |
| seqf2 14018 | Range of the recursive seq... |
| seqcl 14019 | Closure properties of the ... |
| seqf 14020 | Range of the recursive seq... |
| seqfveq2 14021 | Equality of sequences. (C... |
| seqfeq2 14022 | Equality of sequences. (C... |
| seqfveq 14023 | Equality of sequences. (C... |
| seqfeq 14024 | Equality of sequences. (C... |
| seqshft2 14025 | Shifting the index set of ... |
| seqres 14026 | Restricting its characteri... |
| serf 14027 | An infinite series of comp... |
| serfre 14028 | An infinite series of real... |
| monoord 14029 | Ordering relation for a mo... |
| monoord2 14030 | Ordering relation for a mo... |
| sermono 14031 | The partial sums in an inf... |
| seqsplit 14032 | Split a sequence into two ... |
| seq1p 14033 | Removing the first term fr... |
| seqcaopr3 14034 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14035 | The sum of two infinite se... |
| seqcaopr 14036 | The sum of two infinite se... |
| seqf1olem2a 14037 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14038 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14039 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14040 | Rearrange a sum via an arb... |
| seradd 14041 | The sum of two infinite se... |
| sersub 14042 | The difference of two infi... |
| seqid3 14043 | A sequence that consists e... |
| seqid 14044 | Discarding the first few t... |
| seqid2 14045 | The last few partial sums ... |
| seqhomo 14046 | Apply a homomorphism to a ... |
| seqz 14047 | If the operation ` .+ ` ha... |
| seqfeq4 14048 | Equality of series under d... |
| seqfeq3 14049 | Equality of series under d... |
| seqdistr 14050 | The distributive property ... |
| ser0 14051 | The value of the partial s... |
| ser0f 14052 | A zero-valued infinite ser... |
| serge0 14053 | A finite sum of nonnegativ... |
| serle 14054 | Comparison of partial sums... |
| ser1const 14055 | Value of the partial serie... |
| seqof 14056 | Distribute function operat... |
| seqof2 14057 | Distribute function operat... |
| expval 14060 | Value of exponentiation to... |
| expnnval 14061 | Value of exponentiation to... |
| exp0 14062 | Value of a complex number ... |
| 0exp0e1 14063 | The zeroth power of zero e... |
| exp1 14064 | Value of a complex number ... |
| expp1 14065 | Value of a complex number ... |
| expneg 14066 | Value of a complex number ... |
| expneg2 14067 | Value of a complex number ... |
| expn1 14068 | A complex number raised to... |
| expcllem 14069 | Lemma for proving nonnegat... |
| expcl2lem 14070 | Lemma for proving integer ... |
| nnexpcl 14071 | Closure of exponentiation ... |
| nn0expcl 14072 | Closure of exponentiation ... |
| zexpcl 14073 | Closure of exponentiation ... |
| qexpcl 14074 | Closure of exponentiation ... |
| reexpcl 14075 | Closure of exponentiation ... |
| expcl 14076 | Closure law for nonnegativ... |
| rpexpcl 14077 | Closure law for integer ex... |
| qexpclz 14078 | Closure of integer exponen... |
| reexpclz 14079 | Closure of integer exponen... |
| expclzlem 14080 | Lemma for ~ expclz . (Con... |
| expclz 14081 | Closure law for integer ex... |
| m1expcl2 14082 | Closure of integer exponen... |
| m1expcl 14083 | Closure of exponentiation ... |
| zexpcld 14084 | Closure of exponentiation ... |
| nn0expcli 14085 | Closure of exponentiation ... |
| nn0sqcl 14086 | The square of a nonnegativ... |
| expm1t 14087 | Exponentiation in terms of... |
| 1exp 14088 | Value of 1 raised to an in... |
| expeq0 14089 | A positive integer power i... |
| expne0 14090 | A positive integer power i... |
| expne0i 14091 | An integer power is nonzer... |
| expgt0 14092 | A positive real raised to ... |
| expnegz 14093 | Value of a nonzero complex... |
| 0exp 14094 | Value of zero raised to a ... |
| expge0 14095 | A nonnegative real raised ... |
| expge1 14096 | A real greater than or equ... |
| expgt1 14097 | A real greater than 1 rais... |
| mulexp 14098 | Nonnegative integer expone... |
| mulexpz 14099 | Integer exponentiation of ... |
| exprec 14100 | Integer exponentiation of ... |
| expadd 14101 | Sum of exponents law for n... |
| expaddzlem 14102 | Lemma for ~ expaddz . (Co... |
| expaddz 14103 | Sum of exponents law for i... |
| expmul 14104 | Product of exponents law f... |
| expmulz 14105 | Product of exponents law f... |
| m1expeven 14106 | Exponentiation of negative... |
| expsub 14107 | Exponent subtraction law f... |
| expp1z 14108 | Value of a nonzero complex... |
| expm1 14109 | Value of a nonzero complex... |
| expdiv 14110 | Nonnegative integer expone... |
| sqval 14111 | Value of the square of a c... |
| sqneg 14112 | The square of the negative... |
| sqsubswap 14113 | Swap the order of subtract... |
| sqcl 14114 | Closure of square. (Contr... |
| sqmul 14115 | Distribution of squaring o... |
| sqeq0 14116 | A complex number is zero i... |
| sqdiv 14117 | Distribution of squaring o... |
| sqdivid 14118 | The square of a nonzero co... |
| sqne0 14119 | A complex number is nonzer... |
| resqcl 14120 | Closure of squaring in rea... |
| resqcld 14121 | Closure of squaring in rea... |
| sqgt0 14122 | The square of a nonzero re... |
| sqn0rp 14123 | The square of a nonzero re... |
| nnsqcl 14124 | The positive naturals are ... |
| zsqcl 14125 | Integers are closed under ... |
| qsqcl 14126 | The square of a rational i... |
| sq11 14127 | The square function is one... |
| nn0sq11 14128 | The square function is one... |
| lt2sq 14129 | The square function is inc... |
| le2sq 14130 | The square function is non... |
| le2sq2 14131 | The square function is non... |
| sqge0 14132 | The square of a real is no... |
| sqge0d 14133 | The square of a real is no... |
| zsqcl2 14134 | The square of an integer i... |
| 0expd 14135 | Value of zero raised to a ... |
| exp0d 14136 | Value of a complex number ... |
| exp1d 14137 | Value of a complex number ... |
| expeq0d 14138 | If a positive integer powe... |
| sqvald 14139 | Value of square. Inferenc... |
| sqcld 14140 | Closure of square. (Contr... |
| sqeq0d 14141 | A number is zero iff its s... |
| expcld 14142 | Closure law for nonnegativ... |
| expp1d 14143 | Value of a complex number ... |
| expaddd 14144 | Sum of exponents law for n... |
| expmuld 14145 | Product of exponents law f... |
| sqrecd 14146 | Square of reciprocal is re... |
| expclzd 14147 | Closure law for integer ex... |
| expne0d 14148 | A nonnegative integer powe... |
| expnegd 14149 | Value of a nonzero complex... |
| exprecd 14150 | An integer power of a reci... |
| expp1zd 14151 | Value of a nonzero complex... |
| expm1d 14152 | Value of a nonzero complex... |
| expsubd 14153 | Exponent subtraction law f... |
| sqmuld 14154 | Distribution of squaring o... |
| sqdivd 14155 | Distribution of squaring o... |
| expdivd 14156 | Nonnegative integer expone... |
| mulexpd 14157 | Nonnegative integer expone... |
| znsqcld 14158 | The square of a nonzero in... |
| reexpcld 14159 | Closure of exponentiation ... |
| expge0d 14160 | A nonnegative real raised ... |
| expge1d 14161 | A real greater than or equ... |
| ltexp2a 14162 | Exponent ordering relation... |
| expmordi 14163 | Base ordering relationship... |
| rpexpmord 14164 | Base ordering relationship... |
| expcan 14165 | Cancellation law for integ... |
| ltexp2 14166 | Strict ordering law for ex... |
| leexp2 14167 | Ordering law for exponenti... |
| leexp2a 14168 | Weak ordering relationship... |
| ltexp2r 14169 | The integer powers of a fi... |
| leexp2r 14170 | Weak ordering relationship... |
| leexp1a 14171 | Weak base ordering relatio... |
| exple1 14172 | A real between 0 and 1 inc... |
| expubnd 14173 | An upper bound on ` A ^ N ... |
| sumsqeq0 14174 | The sum of two squres of r... |
| sqvali 14175 | Value of square. Inferenc... |
| sqcli 14176 | Closure of square. (Contr... |
| sqeq0i 14177 | A complex number is zero i... |
| sqrecii 14178 | The square of a reciprocal... |
| sqmuli 14179 | Distribution of squaring o... |
| sqdivi 14180 | Distribution of squaring o... |
| resqcli 14181 | Closure of square in reals... |
| sqgt0i 14182 | The square of a nonzero re... |
| sqge0i 14183 | The square of a real is no... |
| lt2sqi 14184 | The square function on non... |
| le2sqi 14185 | The square function on non... |
| sq11i 14186 | The square function is one... |
| sq0 14187 | The square of 0 is 0. (Co... |
| sq0i 14188 | If a number is zero, then ... |
| sq0id 14189 | If a number is zero, then ... |
| sq1 14190 | The square of 1 is 1. (Co... |
| neg1sqe1 14191 | The square of ` -u 1 ` is ... |
| sq2 14192 | The square of 2 is 4. (Co... |
| sq3 14193 | The square of 3 is 9. (Co... |
| sq4e2t8 14194 | The square of 4 is 2 times... |
| cu2 14195 | The cube of 2 is 8. (Cont... |
| irec 14196 | The reciprocal of ` _i ` .... |
| i2 14197 | ` _i ` squared. (Contribu... |
| i3 14198 | ` _i ` cubed. (Contribute... |
| i4 14199 | ` _i ` to the fourth power... |
| nnlesq 14200 | A positive integer is less... |
| zzlesq 14201 | An integer is less than or... |
| iexpcyc 14202 | Taking ` _i ` to the ` K `... |
| expnass 14203 | A counterexample showing t... |
| sqlecan 14204 | Cancel one factor of a squ... |
| subsq 14205 | Factor the difference of t... |
| subsq2 14206 | Express the difference of ... |
| binom2i 14207 | The square of a binomial. ... |
| subsqi 14208 | Factor the difference of t... |
| sqeqori 14209 | The squares of two complex... |
| subsq0i 14210 | The two solutions to the d... |
| sqeqor 14211 | The squares of two complex... |
| binom2 14212 | The square of a binomial. ... |
| binom21 14213 | Special case of ~ binom2 w... |
| binom2sub 14214 | Expand the square of a sub... |
| binom2sub1 14215 | Special case of ~ binom2su... |
| binom2subi 14216 | Expand the square of a sub... |
| mulbinom2 14217 | The square of a binomial w... |
| binom3 14218 | The cube of a binomial. (... |
| sq01 14219 | If a complex number equals... |
| zesq 14220 | An integer is even iff its... |
| nnesq 14221 | A positive integer is even... |
| crreczi 14222 | Reciprocal of a complex nu... |
| bernneq 14223 | Bernoulli's inequality, du... |
| bernneq2 14224 | Variation of Bernoulli's i... |
| bernneq3 14225 | A corollary of ~ bernneq .... |
| expnbnd 14226 | Exponentiation with a base... |
| expnlbnd 14227 | The reciprocal of exponent... |
| expnlbnd2 14228 | The reciprocal of exponent... |
| expmulnbnd 14229 | Exponentiation with a base... |
| digit2 14230 | Two ways to express the ` ... |
| digit1 14231 | Two ways to express the ` ... |
| modexp 14232 | Exponentiation property of... |
| discr1 14233 | A nonnegative quadratic fo... |
| discr 14234 | If a quadratic polynomial ... |
| expnngt1 14235 | If an integer power with a... |
| expnngt1b 14236 | An integer power with an i... |
| sqoddm1div8 14237 | A squared odd number minus... |
| nnsqcld 14238 | The naturals are closed un... |
| nnexpcld 14239 | Closure of exponentiation ... |
| nn0expcld 14240 | Closure of exponentiation ... |
| rpexpcld 14241 | Closure law for exponentia... |
| ltexp2rd 14242 | The power of a positive nu... |
| reexpclzd 14243 | Closure of exponentiation ... |
| sqgt0d 14244 | The square of a nonzero re... |
| ltexp2d 14245 | Ordering relationship for ... |
| leexp2d 14246 | Ordering law for exponenti... |
| expcand 14247 | Ordering relationship for ... |
| leexp2ad 14248 | Ordering relationship for ... |
| leexp2rd 14249 | Ordering relationship for ... |
| lt2sqd 14250 | The square function on non... |
| le2sqd 14251 | The square function on non... |
| sq11d 14252 | The square function is one... |
| mulsubdivbinom2 14253 | The square of a binomial w... |
| muldivbinom2 14254 | The square of a binomial w... |
| sq10 14255 | The square of 10 is 100. ... |
| sq10e99m1 14256 | The square of 10 is 99 plu... |
| 3dec 14257 | A "decimal constructor" wh... |
| nn0le2msqi 14258 | The square function on non... |
| nn0opthlem1 14259 | A rather pretty lemma for ... |
| nn0opthlem2 14260 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14261 | An ordered pair theorem fo... |
| nn0opth2i 14262 | An ordered pair theorem fo... |
| nn0opth2 14263 | An ordered pair theorem fo... |
| facnn 14266 | Value of the factorial fun... |
| fac0 14267 | The factorial of 0. (Cont... |
| fac1 14268 | The factorial of 1. (Cont... |
| facp1 14269 | The factorial of a success... |
| fac2 14270 | The factorial of 2. (Cont... |
| fac3 14271 | The factorial of 3. (Cont... |
| fac4 14272 | The factorial of 4. (Cont... |
| facnn2 14273 | Value of the factorial fun... |
| faccl 14274 | Closure of the factorial f... |
| faccld 14275 | Closure of the factorial f... |
| facmapnn 14276 | The factorial function res... |
| facne0 14277 | The factorial function is ... |
| facdiv 14278 | A positive integer divides... |
| facndiv 14279 | No positive integer (great... |
| facwordi 14280 | Ordering property of facto... |
| faclbnd 14281 | A lower bound for the fact... |
| faclbnd2 14282 | A lower bound for the fact... |
| faclbnd3 14283 | A lower bound for the fact... |
| faclbnd4lem1 14284 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14285 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14286 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14287 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14288 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14289 | The factorial function gro... |
| faclbnd6 14290 | Geometric lower bound for ... |
| facubnd 14291 | An upper bound for the fac... |
| facavg 14292 | The product of two factori... |
| bcval 14295 | Value of the binomial coef... |
| bcval2 14296 | Value of the binomial coef... |
| bcval3 14297 | Value of the binomial coef... |
| bcval4 14298 | Value of the binomial coef... |
| bcrpcl 14299 | Closure of the binomial co... |
| bccmpl 14300 | "Complementing" its second... |
| bcn0 14301 | ` N ` choose 0 is 1. Rema... |
| bc0k 14302 | The binomial coefficient "... |
| bcnn 14303 | ` N ` choose ` N ` is 1. ... |
| bcn1 14304 | Binomial coefficient: ` N ... |
| bcnp1n 14305 | Binomial coefficient: ` N ... |
| bcm1k 14306 | The proportion of one bino... |
| bcp1n 14307 | The proportion of one bino... |
| bcp1nk 14308 | The proportion of one bino... |
| bcval5 14309 | Write out the top and bott... |
| bcn2 14310 | Binomial coefficient: ` N ... |
| bcp1m1 14311 | Compute the binomial coeff... |
| bcpasc 14312 | Pascal's rule for the bino... |
| bccl 14313 | A binomial coefficient, in... |
| bccl2 14314 | A binomial coefficient, in... |
| bcn2m1 14315 | Compute the binomial coeff... |
| bcn2p1 14316 | Compute the binomial coeff... |
| permnn 14317 | The number of permutations... |
| bcnm1 14318 | The binomial coefficent of... |
| 4bc3eq4 14319 | The value of four choose t... |
| 4bc2eq6 14320 | The value of four choose t... |
| hashkf 14323 | The finite part of the siz... |
| hashgval 14324 | The value of the ` # ` fun... |
| hashginv 14325 | The converse of ` G ` maps... |
| hashinf 14326 | The value of the ` # ` fun... |
| hashbnd 14327 | If ` A ` has size bounded ... |
| hashfxnn0 14328 | The size function is a fun... |
| hashf 14329 | The size function maps all... |
| hashxnn0 14330 | The value of the hash func... |
| hashresfn 14331 | Restriction of the domain ... |
| dmhashres 14332 | Restriction of the domain ... |
| hashnn0pnf 14333 | The value of the hash func... |
| hashnnn0genn0 14334 | If the size of a set is no... |
| hashnemnf 14335 | The size of a set is never... |
| hashv01gt1 14336 | The size of a set is eithe... |
| hashfz1 14337 | The set ` ( 1 ... N ) ` ha... |
| hashen 14338 | Two finite sets have the s... |
| hasheni 14339 | Equinumerous sets have the... |
| hasheqf1o 14340 | The size of two finite set... |
| fiinfnf1o 14341 | There is no bijection betw... |
| hasheqf1oi 14342 | The size of two sets is eq... |
| hashf1rn 14343 | The size of a finite set w... |
| hasheqf1od 14344 | The size of two sets is eq... |
| fz1eqb 14345 | Two possibly-empty 1-based... |
| hashcard 14346 | The size function of the c... |
| hashcl 14347 | Closure of the ` # ` funct... |
| hashxrcl 14348 | Extended real closure of t... |
| hashclb 14349 | Reverse closure of the ` #... |
| nfile 14350 | The size of any infinite s... |
| hashvnfin 14351 | A set of finite size is a ... |
| hashnfinnn0 14352 | The size of an infinite se... |
| isfinite4 14353 | A finite set is equinumero... |
| hasheq0 14354 | Two ways of saying a set i... |
| hashneq0 14355 | Two ways of saying a set i... |
| hashgt0n0 14356 | If the size of a set is gr... |
| hashnncl 14357 | Positive natural closure o... |
| hash0 14358 | The empty set has size zer... |
| hashelne0d 14359 | A set with an element has ... |
| hashsng 14360 | The size of a singleton. ... |
| hashen1 14361 | A set has size 1 if and on... |
| hash1elsn 14362 | A set of size 1 with a kno... |
| hashrabrsn 14363 | The size of a restricted c... |
| hashrabsn01 14364 | The size of a restricted c... |
| hashrabsn1 14365 | If the size of a restricte... |
| hashfn 14366 | A function is equinumerous... |
| fseq1hash 14367 | The value of the size func... |
| hashgadd 14368 | ` G ` maps ordinal additio... |
| hashgval2 14369 | A short expression for the... |
| hashdom 14370 | Dominance relation for the... |
| hashdomi 14371 | Non-strict order relation ... |
| hashsdom 14372 | Strict dominance relation ... |
| hashun 14373 | The size of the union of d... |
| hashun2 14374 | The size of the union of f... |
| hashun3 14375 | The size of the union of f... |
| hashinfxadd 14376 | The extended real addition... |
| hashunx 14377 | The size of the union of d... |
| hashge0 14378 | The cardinality of a set i... |
| hashgt0 14379 | The cardinality of a nonem... |
| hashge1 14380 | The cardinality of a nonem... |
| 1elfz0hash 14381 | 1 is an element of the fin... |
| hashnn0n0nn 14382 | If a nonnegative integer i... |
| hashunsng 14383 | The size of the union of a... |
| hashunsngx 14384 | The size of the union of a... |
| hashunsnggt 14385 | The size of a set is great... |
| hashprg 14386 | The size of an unordered p... |
| elprchashprn2 14387 | If one element of an unord... |
| hashprb 14388 | The size of an unordered p... |
| hashprdifel 14389 | The elements of an unorder... |
| prhash2ex 14390 | There is (at least) one se... |
| hashle00 14391 | If the size of a set is le... |
| hashgt0elex 14392 | If the size of a set is gr... |
| hashgt0elexb 14393 | The size of a set is great... |
| hashp1i 14394 | Size of a finite ordinal. ... |
| hash1 14395 | Size of a finite ordinal. ... |
| hash2 14396 | Size of a finite ordinal. ... |
| hash3 14397 | Size of a finite ordinal. ... |
| hash4 14398 | Size of a finite ordinal. ... |
| pr0hash2ex 14399 | There is (at least) one se... |
| hashss 14400 | The size of a subset is le... |
| prsshashgt1 14401 | The size of a superset of ... |
| hashin 14402 | The size of the intersecti... |
| hashssdif 14403 | The size of the difference... |
| hashdif 14404 | The size of the difference... |
| hashdifsn 14405 | The size of the difference... |
| hashdifpr 14406 | The size of the difference... |
| hashsn01 14407 | The size of a singleton is... |
| hashsnle1 14408 | The size of a singleton is... |
| hashsnlei 14409 | Get an upper bound on a co... |
| hash1snb 14410 | The size of a set is 1 if ... |
| euhash1 14411 | The size of a set is 1 in ... |
| hash1n0 14412 | If the size of a set is 1 ... |
| hashgt12el 14413 | In a set with more than on... |
| hashgt12el2 14414 | In a set with more than on... |
| hashgt23el 14415 | A set with more than two e... |
| hashunlei 14416 | Get an upper bound on a co... |
| hashsslei 14417 | Get an upper bound on a co... |
| hashfz 14418 | Value of the numeric cardi... |
| fzsdom2 14419 | Condition for finite range... |
| hashfzo 14420 | Cardinality of a half-open... |
| hashfzo0 14421 | Cardinality of a half-open... |
| hashfzp1 14422 | Value of the numeric cardi... |
| hashfz0 14423 | Value of the numeric cardi... |
| hashxplem 14424 | Lemma for ~ hashxp . (Con... |
| hashxp 14425 | The size of the Cartesian ... |
| hashmap 14426 | The size of the set expone... |
| hashpw 14427 | The size of the power set ... |
| hashfun 14428 | A finite set is a function... |
| hashres 14429 | The number of elements of ... |
| hashreshashfun 14430 | The number of elements of ... |
| hashimarn 14431 | The size of the image of a... |
| hashimarni 14432 | If the size of the image o... |
| hashfundm 14433 | The size of a set function... |
| hashf1dmrn 14434 | The size of the domain of ... |
| hashf1dmcdm 14435 | The size of the domain of ... |
| resunimafz0 14436 | TODO-AV: Revise using ` F... |
| fnfz0hash 14437 | The size of a function on ... |
| ffz0hash 14438 | The size of a function on ... |
| fnfz0hashnn0 14439 | The size of a function on ... |
| ffzo0hash 14440 | The size of a function on ... |
| fnfzo0hash 14441 | The size of a function on ... |
| fnfzo0hashnn0 14442 | The value of the size func... |
| hashbclem 14443 | Lemma for ~ hashbc : induc... |
| hashbc 14444 | The binomial coefficient c... |
| hashfacen 14445 | The number of bijections b... |
| hashfacenOLD 14446 | Obsolete version of ~ hash... |
| hashf1lem1 14447 | Lemma for ~ hashf1 . (Con... |
| hashf1lem1OLD 14448 | Obsolete version of ~ hash... |
| hashf1lem2 14449 | Lemma for ~ hashf1 . (Con... |
| hashf1 14450 | The permutation number ` |... |
| hashfac 14451 | A factorial counts the num... |
| leiso 14452 | Two ways to write a strict... |
| leisorel 14453 | Version of ~ isorel for st... |
| fz1isolem 14454 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14455 | Any finite ordered set has... |
| ishashinf 14456 | Any set that is not finite... |
| seqcoll 14457 | The function ` F ` contain... |
| seqcoll2 14458 | The function ` F ` contain... |
| phphashd 14459 | Corollary of the Pigeonhol... |
| phphashrd 14460 | Corollary of the Pigeonhol... |
| hashprlei 14461 | An unordered pair has at m... |
| hash2pr 14462 | A set of size two is an un... |
| hash2prde 14463 | A set of size two is an un... |
| hash2exprb 14464 | A set of size two is an un... |
| hash2prb 14465 | A set of size two is a pro... |
| prprrab 14466 | The set of proper pairs of... |
| nehash2 14467 | The cardinality of a set w... |
| hash2prd 14468 | A set of size two is an un... |
| hash2pwpr 14469 | If the size of a subset of... |
| hashle2pr 14470 | A nonempty set of size les... |
| hashle2prv 14471 | A nonempty subset of a pow... |
| pr2pwpr 14472 | The set of subsets of a pa... |
| hashge2el2dif 14473 | A set with size at least 2... |
| hashge2el2difr 14474 | A set with at least 2 diff... |
| hashge2el2difb 14475 | A set has size at least 2 ... |
| hashdmpropge2 14476 | The size of the domain of ... |
| hashtplei 14477 | An unordered triple has at... |
| hashtpg 14478 | The size of an unordered t... |
| hashge3el3dif 14479 | A set with size at least 3... |
| elss2prb 14480 | An element of the set of s... |
| hash2sspr 14481 | A subset of size two is an... |
| exprelprel 14482 | If there is an element of ... |
| hash3tr 14483 | A set of size three is an ... |
| hash1to3 14484 | If the size of a set is be... |
| fundmge2nop0 14485 | A function with a domain c... |
| fundmge2nop 14486 | A function with a domain c... |
| fun2dmnop0 14487 | A function with a domain c... |
| fun2dmnop 14488 | A function with a domain c... |
| hashdifsnp1 14489 | If the size of a set is a ... |
| fi1uzind 14490 | Properties of an ordered p... |
| brfi1uzind 14491 | Properties of a binary rel... |
| brfi1ind 14492 | Properties of a binary rel... |
| brfi1indALT 14493 | Alternate proof of ~ brfi1... |
| opfi1uzind 14494 | Properties of an ordered p... |
| opfi1ind 14495 | Properties of an ordered p... |
| iswrd 14498 | Property of being a word o... |
| wrdval 14499 | Value of the set of words ... |
| iswrdi 14500 | A zero-based sequence is a... |
| wrdf 14501 | A word is a zero-based seq... |
| iswrdb 14502 | A word over an alphabet is... |
| wrddm 14503 | The indices of a word (i.e... |
| sswrd 14504 | The set of words respects ... |
| snopiswrd 14505 | A singleton of an ordered ... |
| wrdexg 14506 | The set of words over a se... |
| wrdexb 14507 | The set of words over a se... |
| wrdexi 14508 | The set of words over a se... |
| wrdsymbcl 14509 | A symbol within a word ove... |
| wrdfn 14510 | A word is a function with ... |
| wrdv 14511 | A word over an alphabet is... |
| wrdlndm 14512 | The length of a word is no... |
| iswrdsymb 14513 | An arbitrary word is a wor... |
| wrdfin 14514 | A word is a finite set. (... |
| lencl 14515 | The length of a word is a ... |
| lennncl 14516 | The length of a nonempty w... |
| wrdffz 14517 | A word is a function from ... |
| wrdeq 14518 | Equality theorem for the s... |
| wrdeqi 14519 | Equality theorem for the s... |
| iswrddm0 14520 | A function with empty doma... |
| wrd0 14521 | The empty set is a word (t... |
| 0wrd0 14522 | The empty word is the only... |
| ffz0iswrd 14523 | A sequence with zero-based... |
| wrdsymb 14524 | A word is a word over the ... |
| nfwrd 14525 | Hypothesis builder for ` W... |
| csbwrdg 14526 | Class substitution for the... |
| wrdnval 14527 | Words of a fixed length ar... |
| wrdmap 14528 | Words as a mapping. (Cont... |
| hashwrdn 14529 | If there is only a finite ... |
| wrdnfi 14530 | If there is only a finite ... |
| wrdsymb0 14531 | A symbol at a position "ou... |
| wrdlenge1n0 14532 | A word with length at leas... |
| len0nnbi 14533 | The length of a word is a ... |
| wrdlenge2n0 14534 | A word with length at leas... |
| wrdsymb1 14535 | The first symbol of a none... |
| wrdlen1 14536 | A word of length 1 starts ... |
| fstwrdne 14537 | The first symbol of a none... |
| fstwrdne0 14538 | The first symbol of a none... |
| eqwrd 14539 | Two words are equal iff th... |
| elovmpowrd 14540 | Implications for the value... |
| elovmptnn0wrd 14541 | Implications for the value... |
| wrdred1 14542 | A word truncated by a symb... |
| wrdred1hash 14543 | The length of a word trunc... |
| lsw 14546 | Extract the last symbol of... |
| lsw0 14547 | The last symbol of an empt... |
| lsw0g 14548 | The last symbol of an empt... |
| lsw1 14549 | The last symbol of a word ... |
| lswcl 14550 | Closure of the last symbol... |
| lswlgt0cl 14551 | The last symbol of a nonem... |
| ccatfn 14554 | The concatenation operator... |
| ccatfval 14555 | Value of the concatenation... |
| ccatcl 14556 | The concatenation of two w... |
| ccatlen 14557 | The length of a concatenat... |
| ccat0 14558 | The concatenation of two w... |
| ccatval1 14559 | Value of a symbol in the l... |
| ccatval2 14560 | Value of a symbol in the r... |
| ccatval3 14561 | Value of a symbol in the r... |
| elfzelfzccat 14562 | An element of a finite set... |
| ccatvalfn 14563 | The concatenation of two w... |
| ccatsymb 14564 | The symbol at a given posi... |
| ccatfv0 14565 | The first symbol of a conc... |
| ccatval1lsw 14566 | The last symbol of the lef... |
| ccatval21sw 14567 | The first symbol of the ri... |
| ccatlid 14568 | Concatenation of a word by... |
| ccatrid 14569 | Concatenation of a word by... |
| ccatass 14570 | Associative law for concat... |
| ccatrn 14571 | The range of a concatenate... |
| ccatidid 14572 | Concatenation of the empty... |
| lswccatn0lsw 14573 | The last symbol of a word ... |
| lswccat0lsw 14574 | The last symbol of a word ... |
| ccatalpha 14575 | A concatenation of two arb... |
| ccatrcl1 14576 | Reverse closure of a conca... |
| ids1 14579 | Identity function protecti... |
| s1val 14580 | Value of a singleton word.... |
| s1rn 14581 | The range of a singleton w... |
| s1eq 14582 | Equality theorem for a sin... |
| s1eqd 14583 | Equality theorem for a sin... |
| s1cl 14584 | A singleton word is a word... |
| s1cld 14585 | A singleton word is a word... |
| s1prc 14586 | Value of a singleton word ... |
| s1cli 14587 | A singleton word is a word... |
| s1len 14588 | Length of a singleton word... |
| s1nz 14589 | A singleton word is not th... |
| s1dm 14590 | The domain of a singleton ... |
| s1dmALT 14591 | Alternate version of ~ s1d... |
| s1fv 14592 | Sole symbol of a singleton... |
| lsws1 14593 | The last symbol of a singl... |
| eqs1 14594 | A word of length 1 is a si... |
| wrdl1exs1 14595 | A word of length 1 is a si... |
| wrdl1s1 14596 | A word of length 1 is a si... |
| s111 14597 | The singleton word functio... |
| ccatws1cl 14598 | The concatenation of a wor... |
| ccatws1clv 14599 | The concatenation of a wor... |
| ccat2s1cl 14600 | The concatenation of two s... |
| ccats1alpha 14601 | A concatenation of a word ... |
| ccatws1len 14602 | The length of the concaten... |
| ccatws1lenp1b 14603 | The length of a word is ` ... |
| wrdlenccats1lenm1 14604 | The length of a word is th... |
| ccat2s1len 14605 | The length of the concaten... |
| ccatw2s1cl 14606 | The concatenation of a wor... |
| ccatw2s1len 14607 | The length of the concaten... |
| ccats1val1 14608 | Value of a symbol in the l... |
| ccats1val2 14609 | Value of the symbol concat... |
| ccat1st1st 14610 | The first symbol of a word... |
| ccat2s1p1 14611 | Extract the first of two c... |
| ccat2s1p2 14612 | Extract the second of two ... |
| ccatw2s1ass 14613 | Associative law for a conc... |
| ccatws1n0 14614 | The concatenation of a wor... |
| ccatws1ls 14615 | The last symbol of the con... |
| lswccats1 14616 | The last symbol of a word ... |
| lswccats1fst 14617 | The last symbol of a nonem... |
| ccatw2s1p1 14618 | Extract the symbol of the ... |
| ccatw2s1p2 14619 | Extract the second of two ... |
| ccat2s1fvw 14620 | Extract a symbol of a word... |
| ccat2s1fst 14621 | The first symbol of the co... |
| swrdnznd 14624 | The value of a subword ope... |
| swrdval 14625 | Value of a subword. (Cont... |
| swrd00 14626 | A zero length substring. ... |
| swrdcl 14627 | Closure of the subword ext... |
| swrdval2 14628 | Value of the subword extra... |
| swrdlen 14629 | Length of an extracted sub... |
| swrdfv 14630 | A symbol in an extracted s... |
| swrdfv0 14631 | The first symbol in an ext... |
| swrdf 14632 | A subword of a word is a f... |
| swrdvalfn 14633 | Value of the subword extra... |
| swrdrn 14634 | The range of a subword of ... |
| swrdlend 14635 | The value of the subword e... |
| swrdnd 14636 | The value of the subword e... |
| swrdnd2 14637 | Value of the subword extra... |
| swrdnnn0nd 14638 | The value of a subword ope... |
| swrdnd0 14639 | The value of a subword ope... |
| swrd0 14640 | A subword of an empty set ... |
| swrdrlen 14641 | Length of a right-anchored... |
| swrdlen2 14642 | Length of an extracted sub... |
| swrdfv2 14643 | A symbol in an extracted s... |
| swrdwrdsymb 14644 | A subword is a word over t... |
| swrdsb0eq 14645 | Two subwords with the same... |
| swrdsbslen 14646 | Two subwords with the same... |
| swrdspsleq 14647 | Two words have a common su... |
| swrds1 14648 | Extract a single symbol fr... |
| swrdlsw 14649 | Extract the last single sy... |
| ccatswrd 14650 | Joining two adjacent subwo... |
| swrdccat2 14651 | Recover the right half of ... |
| pfxnndmnd 14654 | The value of a prefix oper... |
| pfxval 14655 | Value of a prefix operatio... |
| pfx00 14656 | The zero length prefix is ... |
| pfx0 14657 | A prefix of an empty set i... |
| pfxval0 14658 | Value of a prefix operatio... |
| pfxcl 14659 | Closure of the prefix extr... |
| pfxmpt 14660 | Value of the prefix extrac... |
| pfxres 14661 | Value of the subword extra... |
| pfxf 14662 | A prefix of a word is a fu... |
| pfxfn 14663 | Value of the prefix extrac... |
| pfxfv 14664 | A symbol in a prefix of a ... |
| pfxlen 14665 | Length of a prefix. (Cont... |
| pfxid 14666 | A word is a prefix of itse... |
| pfxrn 14667 | The range of a prefix of a... |
| pfxn0 14668 | A prefix consisting of at ... |
| pfxnd 14669 | The value of a prefix oper... |
| pfxnd0 14670 | The value of a prefix oper... |
| pfxwrdsymb 14671 | A prefix of a word is a wo... |
| addlenrevpfx 14672 | The sum of the lengths of ... |
| addlenpfx 14673 | The sum of the lengths of ... |
| pfxfv0 14674 | The first symbol of a pref... |
| pfxtrcfv 14675 | A symbol in a word truncat... |
| pfxtrcfv0 14676 | The first symbol in a word... |
| pfxfvlsw 14677 | The last symbol in a nonem... |
| pfxeq 14678 | The prefixes of two words ... |
| pfxtrcfvl 14679 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14680 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14681 | Two (nonempty) words are e... |
| disjwrdpfx 14682 | Sets of words are disjoint... |
| ccatpfx 14683 | Concatenating a prefix wit... |
| pfxccat1 14684 | Recover the left half of a... |
| pfx1 14685 | The prefix of length one o... |
| swrdswrdlem 14686 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14687 | A subword of a subword is ... |
| pfxswrd 14688 | A prefix of a subword is a... |
| swrdpfx 14689 | A subword of a prefix is a... |
| pfxpfx 14690 | A prefix of a prefix is a ... |
| pfxpfxid 14691 | A prefix of a prefix with ... |
| pfxcctswrd 14692 | The concatenation of the p... |
| lenpfxcctswrd 14693 | The length of the concaten... |
| lenrevpfxcctswrd 14694 | The length of the concaten... |
| pfxlswccat 14695 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14696 | The last symbol of a word ... |
| ccats1pfxeqrex 14697 | There exists a symbol such... |
| ccatopth 14698 | An ~ opth -like theorem fo... |
| ccatopth2 14699 | An ~ opth -like theorem fo... |
| ccatlcan 14700 | Concatenation of words is ... |
| ccatrcan 14701 | Concatenation of words is ... |
| wrdeqs1cat 14702 | Decompose a nonempty word ... |
| cats1un 14703 | Express a word with an ext... |
| wrdind 14704 | Perform induction over the... |
| wrd2ind 14705 | Perform induction over the... |
| swrdccatfn 14706 | The subword of a concatena... |
| swrdccatin1 14707 | The subword of a concatena... |
| pfxccatin12lem4 14708 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14709 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14710 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14711 | The subword of a concatena... |
| pfxccatin12lem2c 14712 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14713 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14714 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14715 | The subword of a concatena... |
| pfxccat3 14716 | The subword of a concatena... |
| swrdccat 14717 | The subword of a concatena... |
| pfxccatpfx1 14718 | A prefix of a concatenatio... |
| pfxccatpfx2 14719 | A prefix of a concatenatio... |
| pfxccat3a 14720 | A prefix of a concatenatio... |
| swrdccat3blem 14721 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14722 | A suffix of a concatenatio... |
| pfxccatid 14723 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14724 | A word is a prefix of a wo... |
| swrdccatin1d 14725 | The subword of a concatena... |
| swrdccatin2d 14726 | The subword of a concatena... |
| pfxccatin12d 14727 | The subword of a concatena... |
| reuccatpfxs1lem 14728 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14729 | There is a unique word hav... |
| reuccatpfxs1v 14730 | There is a unique word hav... |
| splval 14733 | Value of the substring rep... |
| splcl 14734 | Closure of the substring r... |
| splid 14735 | Splicing a subword for the... |
| spllen 14736 | The length of a splice. (... |
| splfv1 14737 | Symbols to the left of a s... |
| splfv2a 14738 | Symbols within the replace... |
| splval2 14739 | Value of a splice, assumin... |
| revval 14742 | Value of the word reversin... |
| revcl 14743 | The reverse of a word is a... |
| revlen 14744 | The reverse of a word has ... |
| revfv 14745 | Reverse of a word at a poi... |
| rev0 14746 | The empty word is its own ... |
| revs1 14747 | Singleton words are their ... |
| revccat 14748 | Antiautomorphic property o... |
| revrev 14749 | Reversal is an involution ... |
| reps 14752 | Construct a function mappi... |
| repsundef 14753 | A function mapping a half-... |
| repsconst 14754 | Construct a function mappi... |
| repsf 14755 | The constructed function m... |
| repswsymb 14756 | The symbols of a "repeated... |
| repsw 14757 | A function mapping a half-... |
| repswlen 14758 | The length of a "repeated ... |
| repsw0 14759 | The "repeated symbol word"... |
| repsdf2 14760 | Alternative definition of ... |
| repswsymball 14761 | All the symbols of a "repe... |
| repswsymballbi 14762 | A word is a "repeated symb... |
| repswfsts 14763 | The first symbol of a none... |
| repswlsw 14764 | The last symbol of a nonem... |
| repsw1 14765 | The "repeated symbol word"... |
| repswswrd 14766 | A subword of a "repeated s... |
| repswpfx 14767 | A prefix of a repeated sym... |
| repswccat 14768 | The concatenation of two "... |
| repswrevw 14769 | The reverse of a "repeated... |
| cshfn 14772 | Perform a cyclical shift f... |
| cshword 14773 | Perform a cyclical shift f... |
| cshnz 14774 | A cyclical shift is the em... |
| 0csh0 14775 | Cyclically shifting an emp... |
| cshw0 14776 | A word cyclically shifted ... |
| cshwmodn 14777 | Cyclically shifting a word... |
| cshwsublen 14778 | Cyclically shifting a word... |
| cshwn 14779 | A word cyclically shifted ... |
| cshwcl 14780 | A cyclically shifted word ... |
| cshwlen 14781 | The length of a cyclically... |
| cshwf 14782 | A cyclically shifted word ... |
| cshwfn 14783 | A cyclically shifted word ... |
| cshwrn 14784 | The range of a cyclically ... |
| cshwidxmod 14785 | The symbol at a given inde... |
| cshwidxmodr 14786 | The symbol at a given inde... |
| cshwidx0mod 14787 | The symbol at index 0 of a... |
| cshwidx0 14788 | The symbol at index 0 of a... |
| cshwidxm1 14789 | The symbol at index ((n-N)... |
| cshwidxm 14790 | The symbol at index (n-N) ... |
| cshwidxn 14791 | The symbol at index (n-1) ... |
| cshf1 14792 | Cyclically shifting a word... |
| cshinj 14793 | If a word is injectiv (reg... |
| repswcshw 14794 | A cyclically shifted "repe... |
| 2cshw 14795 | Cyclically shifting a word... |
| 2cshwid 14796 | Cyclically shifting a word... |
| lswcshw 14797 | The last symbol of a word ... |
| 2cshwcom 14798 | Cyclically shifting a word... |
| cshwleneq 14799 | If the results of cyclical... |
| 3cshw 14800 | Cyclically shifting a word... |
| cshweqdif2 14801 | If cyclically shifting two... |
| cshweqdifid 14802 | If cyclically shifting a w... |
| cshweqrep 14803 | If cyclically shifting a w... |
| cshw1 14804 | If cyclically shifting a w... |
| cshw1repsw 14805 | If cyclically shifting a w... |
| cshwsexa 14806 | The class of (different!) ... |
| cshwsexaOLD 14807 | Obsolete version of ~ cshw... |
| 2cshwcshw 14808 | If a word is a cyclically ... |
| scshwfzeqfzo 14809 | For a nonempty word the se... |
| cshwcshid 14810 | A cyclically shifted word ... |
| cshwcsh2id 14811 | A cyclically shifted word ... |
| cshimadifsn 14812 | The image of a cyclically ... |
| cshimadifsn0 14813 | The image of a cyclically ... |
| wrdco 14814 | Mapping a word by a functi... |
| lenco 14815 | Length of a mapped word is... |
| s1co 14816 | Mapping of a singleton wor... |
| revco 14817 | Mapping of words (i.e., a ... |
| ccatco 14818 | Mapping of words commutes ... |
| cshco 14819 | Mapping of words commutes ... |
| swrdco 14820 | Mapping of words commutes ... |
| pfxco 14821 | Mapping of words commutes ... |
| lswco 14822 | Mapping of (nonempty) word... |
| repsco 14823 | Mapping of words commutes ... |
| cats1cld 14838 | Closure of concatenation w... |
| cats1co 14839 | Closure of concatenation w... |
| cats1cli 14840 | Closure of concatenation w... |
| cats1fvn 14841 | The last symbol of a conca... |
| cats1fv 14842 | A symbol other than the la... |
| cats1len 14843 | The length of concatenatio... |
| cats1cat 14844 | Closure of concatenation w... |
| cats2cat 14845 | Closure of concatenation o... |
| s2eqd 14846 | Equality theorem for a dou... |
| s3eqd 14847 | Equality theorem for a len... |
| s4eqd 14848 | Equality theorem for a len... |
| s5eqd 14849 | Equality theorem for a len... |
| s6eqd 14850 | Equality theorem for a len... |
| s7eqd 14851 | Equality theorem for a len... |
| s8eqd 14852 | Equality theorem for a len... |
| s3eq2 14853 | Equality theorem for a len... |
| s2cld 14854 | A doubleton word is a word... |
| s3cld 14855 | A length 3 string is a wor... |
| s4cld 14856 | A length 4 string is a wor... |
| s5cld 14857 | A length 5 string is a wor... |
| s6cld 14858 | A length 6 string is a wor... |
| s7cld 14859 | A length 7 string is a wor... |
| s8cld 14860 | A length 7 string is a wor... |
| s2cl 14861 | A doubleton word is a word... |
| s3cl 14862 | A length 3 string is a wor... |
| s2cli 14863 | A doubleton word is a word... |
| s3cli 14864 | A length 3 string is a wor... |
| s4cli 14865 | A length 4 string is a wor... |
| s5cli 14866 | A length 5 string is a wor... |
| s6cli 14867 | A length 6 string is a wor... |
| s7cli 14868 | A length 7 string is a wor... |
| s8cli 14869 | A length 8 string is a wor... |
| s2fv0 14870 | Extract the first symbol f... |
| s2fv1 14871 | Extract the second symbol ... |
| s2len 14872 | The length of a doubleton ... |
| s2dm 14873 | The domain of a doubleton ... |
| s3fv0 14874 | Extract the first symbol f... |
| s3fv1 14875 | Extract the second symbol ... |
| s3fv2 14876 | Extract the third symbol f... |
| s3len 14877 | The length of a length 3 s... |
| s4fv0 14878 | Extract the first symbol f... |
| s4fv1 14879 | Extract the second symbol ... |
| s4fv2 14880 | Extract the third symbol f... |
| s4fv3 14881 | Extract the fourth symbol ... |
| s4len 14882 | The length of a length 4 s... |
| s5len 14883 | The length of a length 5 s... |
| s6len 14884 | The length of a length 6 s... |
| s7len 14885 | The length of a length 7 s... |
| s8len 14886 | The length of a length 8 s... |
| lsws2 14887 | The last symbol of a doubl... |
| lsws3 14888 | The last symbol of a 3 let... |
| lsws4 14889 | The last symbol of a 4 let... |
| s2prop 14890 | A length 2 word is an unor... |
| s2dmALT 14891 | Alternate version of ~ s2d... |
| s3tpop 14892 | A length 3 word is an unor... |
| s4prop 14893 | A length 4 word is a union... |
| s3fn 14894 | A length 3 word is a funct... |
| funcnvs1 14895 | The converse of a singleto... |
| funcnvs2 14896 | The converse of a length 2... |
| funcnvs3 14897 | The converse of a length 3... |
| funcnvs4 14898 | The converse of a length 4... |
| s2f1o 14899 | A length 2 word with mutua... |
| f1oun2prg 14900 | A union of unordered pairs... |
| s4f1o 14901 | A length 4 word with mutua... |
| s4dom 14902 | The domain of a length 4 w... |
| s2co 14903 | Mapping a doubleton word b... |
| s3co 14904 | Mapping a length 3 string ... |
| s0s1 14905 | Concatenation of fixed len... |
| s1s2 14906 | Concatenation of fixed len... |
| s1s3 14907 | Concatenation of fixed len... |
| s1s4 14908 | Concatenation of fixed len... |
| s1s5 14909 | Concatenation of fixed len... |
| s1s6 14910 | Concatenation of fixed len... |
| s1s7 14911 | Concatenation of fixed len... |
| s2s2 14912 | Concatenation of fixed len... |
| s4s2 14913 | Concatenation of fixed len... |
| s4s3 14914 | Concatenation of fixed len... |
| s4s4 14915 | Concatenation of fixed len... |
| s3s4 14916 | Concatenation of fixed len... |
| s2s5 14917 | Concatenation of fixed len... |
| s5s2 14918 | Concatenation of fixed len... |
| s2eq2s1eq 14919 | Two length 2 words are equ... |
| s2eq2seq 14920 | Two length 2 words are equ... |
| s3eqs2s1eq 14921 | Two length 3 words are equ... |
| s3eq3seq 14922 | Two length 3 words are equ... |
| swrds2 14923 | Extract two adjacent symbo... |
| swrds2m 14924 | Extract two adjacent symbo... |
| wrdlen2i 14925 | Implications of a word of ... |
| wrd2pr2op 14926 | A word of length two repre... |
| wrdlen2 14927 | A word of length two. (Co... |
| wrdlen2s2 14928 | A word of length two as do... |
| wrdl2exs2 14929 | A word of length two is a ... |
| pfx2 14930 | A prefix of length two. (... |
| wrd3tpop 14931 | A word of length three rep... |
| wrdlen3s3 14932 | A word of length three as ... |
| repsw2 14933 | The "repeated symbol word"... |
| repsw3 14934 | The "repeated symbol word"... |
| swrd2lsw 14935 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14936 | Two words of length at lea... |
| ccatw2s1ccatws2 14937 | The concatenation of a wor... |
| ccat2s1fvwALT 14938 | Alternate proof of ~ ccat2... |
| wwlktovf 14939 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14940 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14941 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14942 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14943 | There is a bijection betwe... |
| eqwrds3 14944 | A word is equal with a len... |
| wrdl3s3 14945 | A word of length 3 is a le... |
| s3sndisj 14946 | The singletons consisting ... |
| s3iunsndisj 14947 | The union of singletons co... |
| ofccat 14948 | Letterwise operations on w... |
| ofs1 14949 | Letterwise operations on a... |
| ofs2 14950 | Letterwise operations on a... |
| coss12d 14951 | Subset deduction for compo... |
| trrelssd 14952 | The composition of subclas... |
| xpcogend 14953 | The most interesting case ... |
| xpcoidgend 14954 | If two classes are not dis... |
| cotr2g 14955 | Two ways of saying that th... |
| cotr2 14956 | Two ways of saying a relat... |
| cotr3 14957 | Two ways of saying a relat... |
| coemptyd 14958 | Deduction about compositio... |
| xptrrel 14959 | The cross product is alway... |
| 0trrel 14960 | The empty class is a trans... |
| cleq1lem 14961 | Equality implies bijection... |
| cleq1 14962 | Equality of relations impl... |
| clsslem 14963 | The closure of a subclass ... |
| trcleq1 14968 | Equality of relations impl... |
| trclsslem 14969 | The transitive closure (as... |
| trcleq2lem 14970 | Equality implies bijection... |
| cvbtrcl 14971 | Change of bound variable i... |
| trcleq12lem 14972 | Equality implies bijection... |
| trclexlem 14973 | Existence of relation impl... |
| trclublem 14974 | If a relation exists then ... |
| trclubi 14975 | The Cartesian product of t... |
| trclubgi 14976 | The union with the Cartesi... |
| trclub 14977 | The Cartesian product of t... |
| trclubg 14978 | The union with the Cartesi... |
| trclfv 14979 | The transitive closure of ... |
| brintclab 14980 | Two ways to express a bina... |
| brtrclfv 14981 | Two ways of expressing the... |
| brcnvtrclfv 14982 | Two ways of expressing the... |
| brtrclfvcnv 14983 | Two ways of expressing the... |
| brcnvtrclfvcnv 14984 | Two ways of expressing the... |
| trclfvss 14985 | The transitive closure (as... |
| trclfvub 14986 | The transitive closure of ... |
| trclfvlb 14987 | The transitive closure of ... |
| trclfvcotr 14988 | The transitive closure of ... |
| trclfvlb2 14989 | The transitive closure of ... |
| trclfvlb3 14990 | The transitive closure of ... |
| cotrtrclfv 14991 | The transitive closure of ... |
| trclidm 14992 | The transitive closure of ... |
| trclun 14993 | Transitive closure of a un... |
| trclfvg 14994 | The value of the transitiv... |
| trclfvcotrg 14995 | The value of the transitiv... |
| reltrclfv 14996 | The transitive closure of ... |
| dmtrclfv 14997 | The domain of the transiti... |
| reldmrelexp 15000 | The domain of the repeated... |
| relexp0g 15001 | A relation composed zero t... |
| relexp0 15002 | A relation composed zero t... |
| relexp0d 15003 | A relation composed zero t... |
| relexpsucnnr 15004 | A reduction for relation e... |
| relexp1g 15005 | A relation composed once i... |
| dfid5 15006 | Identity relation is equal... |
| dfid6 15007 | Identity relation expresse... |
| relexp1d 15008 | A relation composed once i... |
| relexpsucnnl 15009 | A reduction for relation e... |
| relexpsucl 15010 | A reduction for relation e... |
| relexpsucr 15011 | A reduction for relation e... |
| relexpsucrd 15012 | A reduction for relation e... |
| relexpsucld 15013 | A reduction for relation e... |
| relexpcnv 15014 | Commutation of converse an... |
| relexpcnvd 15015 | Commutation of converse an... |
| relexp0rel 15016 | The exponentiation of a cl... |
| relexprelg 15017 | The exponentiation of a cl... |
| relexprel 15018 | The exponentiation of a re... |
| relexpreld 15019 | The exponentiation of a re... |
| relexpnndm 15020 | The domain of an exponenti... |
| relexpdmg 15021 | The domain of an exponenti... |
| relexpdm 15022 | The domain of an exponenti... |
| relexpdmd 15023 | The domain of an exponenti... |
| relexpnnrn 15024 | The range of an exponentia... |
| relexprng 15025 | The range of an exponentia... |
| relexprn 15026 | The range of an exponentia... |
| relexprnd 15027 | The range of an exponentia... |
| relexpfld 15028 | The field of an exponentia... |
| relexpfldd 15029 | The field of an exponentia... |
| relexpaddnn 15030 | Relation composition becom... |
| relexpuzrel 15031 | The exponentiation of a cl... |
| relexpaddg 15032 | Relation composition becom... |
| relexpaddd 15033 | Relation composition becom... |
| rtrclreclem1 15036 | The reflexive, transitive ... |
| dfrtrclrec2 15037 | If two elements are connec... |
| rtrclreclem2 15038 | The reflexive, transitive ... |
| rtrclreclem3 15039 | The reflexive, transitive ... |
| rtrclreclem4 15040 | The reflexive, transitive ... |
| dfrtrcl2 15041 | The two definitions ` t* `... |
| relexpindlem 15042 | Principle of transitive in... |
| relexpind 15043 | Principle of transitive in... |
| rtrclind 15044 | Principle of transitive in... |
| shftlem 15047 | Two ways to write a shifte... |
| shftuz 15048 | A shift of the upper integ... |
| shftfval 15049 | The value of the sequence ... |
| shftdm 15050 | Domain of a relation shift... |
| shftfib 15051 | Value of a fiber of the re... |
| shftfn 15052 | Functionality and domain o... |
| shftval 15053 | Value of a sequence shifte... |
| shftval2 15054 | Value of a sequence shifte... |
| shftval3 15055 | Value of a sequence shifte... |
| shftval4 15056 | Value of a sequence shifte... |
| shftval5 15057 | Value of a shifted sequenc... |
| shftf 15058 | Functionality of a shifted... |
| 2shfti 15059 | Composite shift operations... |
| shftidt2 15060 | Identity law for the shift... |
| shftidt 15061 | Identity law for the shift... |
| shftcan1 15062 | Cancellation law for the s... |
| shftcan2 15063 | Cancellation law for the s... |
| seqshft 15064 | Shifting the index set of ... |
| sgnval 15067 | Value of the signum functi... |
| sgn0 15068 | The signum of 0 is 0. (Co... |
| sgnp 15069 | The signum of a positive e... |
| sgnrrp 15070 | The signum of a positive r... |
| sgn1 15071 | The signum of 1 is 1. (Co... |
| sgnpnf 15072 | The signum of ` +oo ` is 1... |
| sgnn 15073 | The signum of a negative e... |
| sgnmnf 15074 | The signum of ` -oo ` is -... |
| cjval 15081 | The value of the conjugate... |
| cjth 15082 | The defining property of t... |
| cjf 15083 | Domain and codomain of the... |
| cjcl 15084 | The conjugate of a complex... |
| reval 15085 | The value of the real part... |
| imval 15086 | The value of the imaginary... |
| imre 15087 | The imaginary part of a co... |
| reim 15088 | The real part of a complex... |
| recl 15089 | The real part of a complex... |
| imcl 15090 | The imaginary part of a co... |
| ref 15091 | Domain and codomain of the... |
| imf 15092 | Domain and codomain of the... |
| crre 15093 | The real part of a complex... |
| crim 15094 | The real part of a complex... |
| replim 15095 | Reconstruct a complex numb... |
| remim 15096 | Value of the conjugate of ... |
| reim0 15097 | The imaginary part of a re... |
| reim0b 15098 | A number is real iff its i... |
| rereb 15099 | A number is real iff it eq... |
| mulre 15100 | A product with a nonzero r... |
| rere 15101 | A real number equals its r... |
| cjreb 15102 | A number is real iff it eq... |
| recj 15103 | Real part of a complex con... |
| reneg 15104 | Real part of negative. (C... |
| readd 15105 | Real part distributes over... |
| resub 15106 | Real part distributes over... |
| remullem 15107 | Lemma for ~ remul , ~ immu... |
| remul 15108 | Real part of a product. (... |
| remul2 15109 | Real part of a product. (... |
| rediv 15110 | Real part of a division. ... |
| imcj 15111 | Imaginary part of a comple... |
| imneg 15112 | The imaginary part of a ne... |
| imadd 15113 | Imaginary part distributes... |
| imsub 15114 | Imaginary part distributes... |
| immul 15115 | Imaginary part of a produc... |
| immul2 15116 | Imaginary part of a produc... |
| imdiv 15117 | Imaginary part of a divisi... |
| cjre 15118 | A real number equals its c... |
| cjcj 15119 | The conjugate of the conju... |
| cjadd 15120 | Complex conjugate distribu... |
| cjmul 15121 | Complex conjugate distribu... |
| ipcnval 15122 | Standard inner product on ... |
| cjmulrcl 15123 | A complex number times its... |
| cjmulval 15124 | A complex number times its... |
| cjmulge0 15125 | A complex number times its... |
| cjneg 15126 | Complex conjugate of negat... |
| addcj 15127 | A number plus its conjugat... |
| cjsub 15128 | Complex conjugate distribu... |
| cjexp 15129 | Complex conjugate of posit... |
| imval2 15130 | The imaginary part of a nu... |
| re0 15131 | The real part of zero. (C... |
| im0 15132 | The imaginary part of zero... |
| re1 15133 | The real part of one. (Co... |
| im1 15134 | The imaginary part of one.... |
| rei 15135 | The real part of ` _i ` . ... |
| imi 15136 | The imaginary part of ` _i... |
| cj0 15137 | The conjugate of zero. (C... |
| cji 15138 | The complex conjugate of t... |
| cjreim 15139 | The conjugate of a represe... |
| cjreim2 15140 | The conjugate of the repre... |
| cj11 15141 | Complex conjugate is a one... |
| cjne0 15142 | A number is nonzero iff it... |
| cjdiv 15143 | Complex conjugate distribu... |
| cnrecnv 15144 | The inverse to the canonic... |
| sqeqd 15145 | A deduction for showing tw... |
| recli 15146 | The real part of a complex... |
| imcli 15147 | The imaginary part of a co... |
| cjcli 15148 | Closure law for complex co... |
| replimi 15149 | Construct a complex number... |
| cjcji 15150 | The conjugate of the conju... |
| reim0bi 15151 | A number is real iff its i... |
| rerebi 15152 | A real number equals its r... |
| cjrebi 15153 | A number is real iff it eq... |
| recji 15154 | Real part of a complex con... |
| imcji 15155 | Imaginary part of a comple... |
| cjmulrcli 15156 | A complex number times its... |
| cjmulvali 15157 | A complex number times its... |
| cjmulge0i 15158 | A complex number times its... |
| renegi 15159 | Real part of negative. (C... |
| imnegi 15160 | Imaginary part of negative... |
| cjnegi 15161 | Complex conjugate of negat... |
| addcji 15162 | A number plus its conjugat... |
| readdi 15163 | Real part distributes over... |
| imaddi 15164 | Imaginary part distributes... |
| remuli 15165 | Real part of a product. (... |
| immuli 15166 | Imaginary part of a produc... |
| cjaddi 15167 | Complex conjugate distribu... |
| cjmuli 15168 | Complex conjugate distribu... |
| ipcni 15169 | Standard inner product on ... |
| cjdivi 15170 | Complex conjugate distribu... |
| crrei 15171 | The real part of a complex... |
| crimi 15172 | The imaginary part of a co... |
| recld 15173 | The real part of a complex... |
| imcld 15174 | The imaginary part of a co... |
| cjcld 15175 | Closure law for complex co... |
| replimd 15176 | Construct a complex number... |
| remimd 15177 | Value of the conjugate of ... |
| cjcjd 15178 | The conjugate of the conju... |
| reim0bd 15179 | A number is real iff its i... |
| rerebd 15180 | A real number equals its r... |
| cjrebd 15181 | A number is real iff it eq... |
| cjne0d 15182 | A number is nonzero iff it... |
| recjd 15183 | Real part of a complex con... |
| imcjd 15184 | Imaginary part of a comple... |
| cjmulrcld 15185 | A complex number times its... |
| cjmulvald 15186 | A complex number times its... |
| cjmulge0d 15187 | A complex number times its... |
| renegd 15188 | Real part of negative. (C... |
| imnegd 15189 | Imaginary part of negative... |
| cjnegd 15190 | Complex conjugate of negat... |
| addcjd 15191 | A number plus its conjugat... |
| cjexpd 15192 | Complex conjugate of posit... |
| readdd 15193 | Real part distributes over... |
| imaddd 15194 | Imaginary part distributes... |
| resubd 15195 | Real part distributes over... |
| imsubd 15196 | Imaginary part distributes... |
| remuld 15197 | Real part of a product. (... |
| immuld 15198 | Imaginary part of a produc... |
| cjaddd 15199 | Complex conjugate distribu... |
| cjmuld 15200 | Complex conjugate distribu... |
| ipcnd 15201 | Standard inner product on ... |
| cjdivd 15202 | Complex conjugate distribu... |
| rered 15203 | A real number equals its r... |
| reim0d 15204 | The imaginary part of a re... |
| cjred 15205 | A real number equals its c... |
| remul2d 15206 | Real part of a product. (... |
| immul2d 15207 | Imaginary part of a produc... |
| redivd 15208 | Real part of a division. ... |
| imdivd 15209 | Imaginary part of a divisi... |
| crred 15210 | The real part of a complex... |
| crimd 15211 | The imaginary part of a co... |
| sqrtval 15216 | Value of square root funct... |
| absval 15217 | The absolute value (modulu... |
| rennim 15218 | A real number does not lie... |
| cnpart 15219 | The specification of restr... |
| sqrt0 15220 | The square root of zero is... |
| 01sqrexlem1 15221 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15222 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15223 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15224 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15225 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15226 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15227 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15228 | Existence of a square root... |
| resqrex 15229 | Existence of a square root... |
| sqrmo 15230 | Uniqueness for the square ... |
| resqreu 15231 | Existence and uniqueness f... |
| resqrtcl 15232 | Closure of the square root... |
| resqrtthlem 15233 | Lemma for ~ resqrtth . (C... |
| resqrtth 15234 | Square root theorem over t... |
| remsqsqrt 15235 | Square of square root. (C... |
| sqrtge0 15236 | The square root function i... |
| sqrtgt0 15237 | The square root function i... |
| sqrtmul 15238 | Square root distributes ov... |
| sqrtle 15239 | Square root is monotonic. ... |
| sqrtlt 15240 | Square root is strictly mo... |
| sqrt11 15241 | The square root function i... |
| sqrt00 15242 | A square root is zero iff ... |
| rpsqrtcl 15243 | The square root of a posit... |
| sqrtdiv 15244 | Square root distributes ov... |
| sqrtneglem 15245 | The square root of a negat... |
| sqrtneg 15246 | The square root of a negat... |
| sqrtsq2 15247 | Relationship between squar... |
| sqrtsq 15248 | Square root of square. (C... |
| sqrtmsq 15249 | Square root of square. (C... |
| sqrt1 15250 | The square root of 1 is 1.... |
| sqrt4 15251 | The square root of 4 is 2.... |
| sqrt9 15252 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15253 | The square root of 2 is bo... |
| sqrtm1 15254 | The imaginary unit is the ... |
| nn0sqeq1 15255 | A natural number with squa... |
| absneg 15256 | Absolute value of the nega... |
| abscl 15257 | Real closure of absolute v... |
| abscj 15258 | The absolute value of a nu... |
| absvalsq 15259 | Square of value of absolut... |
| absvalsq2 15260 | Square of value of absolut... |
| sqabsadd 15261 | Square of absolute value o... |
| sqabssub 15262 | Square of absolute value o... |
| absval2 15263 | Value of absolute value fu... |
| abs0 15264 | The absolute value of 0. ... |
| absi 15265 | The absolute value of the ... |
| absge0 15266 | Absolute value is nonnegat... |
| absrpcl 15267 | The absolute value of a no... |
| abs00 15268 | The absolute value of a nu... |
| abs00ad 15269 | A complex number is zero i... |
| abs00bd 15270 | If a complex number is zer... |
| absreimsq 15271 | Square of the absolute val... |
| absreim 15272 | Absolute value of a number... |
| absmul 15273 | Absolute value distributes... |
| absdiv 15274 | Absolute value distributes... |
| absid 15275 | A nonnegative number is it... |
| abs1 15276 | The absolute value of one ... |
| absnid 15277 | For a negative number, its... |
| leabs 15278 | A real number is less than... |
| absor 15279 | The absolute value of a re... |
| absre 15280 | Absolute value of a real n... |
| absresq 15281 | Square of the absolute val... |
| absmod0 15282 | ` A ` is divisible by ` B ... |
| absexp 15283 | Absolute value of positive... |
| absexpz 15284 | Absolute value of integer ... |
| abssq 15285 | Square can be moved in and... |
| sqabs 15286 | The squares of two reals a... |
| absrele 15287 | The absolute value of a co... |
| absimle 15288 | The absolute value of a co... |
| max0add 15289 | The sum of the positive an... |
| absz 15290 | A real number is an intege... |
| nn0abscl 15291 | The absolute value of an i... |
| zabscl 15292 | The absolute value of an i... |
| abslt 15293 | Absolute value and 'less t... |
| absle 15294 | Absolute value and 'less t... |
| abssubne0 15295 | If the absolute value of a... |
| absdiflt 15296 | The absolute value of a di... |
| absdifle 15297 | The absolute value of a di... |
| elicc4abs 15298 | Membership in a symmetric ... |
| lenegsq 15299 | Comparison to a nonnegativ... |
| releabs 15300 | The real part of a number ... |
| recval 15301 | Reciprocal expressed with ... |
| absidm 15302 | The absolute value functio... |
| absgt0 15303 | The absolute value of a no... |
| nnabscl 15304 | The absolute value of a no... |
| abssub 15305 | Swapping order of subtract... |
| abssubge0 15306 | Absolute value of a nonneg... |
| abssuble0 15307 | Absolute value of a nonpos... |
| absmax 15308 | The maximum of two numbers... |
| abstri 15309 | Triangle inequality for ab... |
| abs3dif 15310 | Absolute value of differen... |
| abs2dif 15311 | Difference of absolute val... |
| abs2dif2 15312 | Difference of absolute val... |
| abs2difabs 15313 | Absolute value of differen... |
| abs1m 15314 | For any complex number, th... |
| recan 15315 | Cancellation law involving... |
| absf 15316 | Mapping domain and codomai... |
| abs3lem 15317 | Lemma involving absolute v... |
| abslem2 15318 | Lemma involving absolute v... |
| rddif 15319 | The difference between a r... |
| absrdbnd 15320 | Bound on the absolute valu... |
| fzomaxdiflem 15321 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15322 | A bound on the separation ... |
| uzin2 15323 | The upper integers are clo... |
| rexanuz 15324 | Combine two different uppe... |
| rexanre 15325 | Combine two different uppe... |
| rexfiuz 15326 | Combine finitely many diff... |
| rexuz3 15327 | Restrict the base of the u... |
| rexanuz2 15328 | Combine two different uppe... |
| r19.29uz 15329 | A version of ~ 19.29 for u... |
| r19.2uz 15330 | A version of ~ r19.2z for ... |
| rexuzre 15331 | Convert an upper real quan... |
| rexico 15332 | Restrict the base of an up... |
| cau3lem 15333 | Lemma for ~ cau3 . (Contr... |
| cau3 15334 | Convert between three-quan... |
| cau4 15335 | Change the base of a Cauch... |
| caubnd2 15336 | A Cauchy sequence of compl... |
| caubnd 15337 | A Cauchy sequence of compl... |
| sqreulem 15338 | Lemma for ~ sqreu : write ... |
| sqreu 15339 | Existence and uniqueness f... |
| sqrtcl 15340 | Closure of the square root... |
| sqrtthlem 15341 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15342 | Mapping domain and codomai... |
| sqrtth 15343 | Square root theorem over t... |
| sqrtrege0 15344 | The square root function m... |
| eqsqrtor 15345 | Solve an equation containi... |
| eqsqrtd 15346 | A deduction for showing th... |
| eqsqrt2d 15347 | A deduction for showing th... |
| amgm2 15348 | Arithmetic-geometric mean ... |
| sqrtthi 15349 | Square root theorem. Theo... |
| sqrtcli 15350 | The square root of a nonne... |
| sqrtgt0i 15351 | The square root of a posit... |
| sqrtmsqi 15352 | Square root of square. (C... |
| sqrtsqi 15353 | Square root of square. (C... |
| sqsqrti 15354 | Square of square root. (C... |
| sqrtge0i 15355 | The square root of a nonne... |
| absidi 15356 | A nonnegative number is it... |
| absnidi 15357 | A negative number is the n... |
| leabsi 15358 | A real number is less than... |
| absori 15359 | The absolute value of a re... |
| absrei 15360 | Absolute value of a real n... |
| sqrtpclii 15361 | The square root of a posit... |
| sqrtgt0ii 15362 | The square root of a posit... |
| sqrt11i 15363 | The square root function i... |
| sqrtmuli 15364 | Square root distributes ov... |
| sqrtmulii 15365 | Square root distributes ov... |
| sqrtmsq2i 15366 | Relationship between squar... |
| sqrtlei 15367 | Square root is monotonic. ... |
| sqrtlti 15368 | Square root is strictly mo... |
| abslti 15369 | Absolute value and 'less t... |
| abslei 15370 | Absolute value and 'less t... |
| cnsqrt00 15371 | A square root of a complex... |
| absvalsqi 15372 | Square of value of absolut... |
| absvalsq2i 15373 | Square of value of absolut... |
| abscli 15374 | Real closure of absolute v... |
| absge0i 15375 | Absolute value is nonnegat... |
| absval2i 15376 | Value of absolute value fu... |
| abs00i 15377 | The absolute value of a nu... |
| absgt0i 15378 | The absolute value of a no... |
| absnegi 15379 | Absolute value of negative... |
| abscji 15380 | The absolute value of a nu... |
| releabsi 15381 | The real part of a number ... |
| abssubi 15382 | Swapping order of subtract... |
| absmuli 15383 | Absolute value distributes... |
| sqabsaddi 15384 | Square of absolute value o... |
| sqabssubi 15385 | Square of absolute value o... |
| absdivzi 15386 | Absolute value distributes... |
| abstrii 15387 | Triangle inequality for ab... |
| abs3difi 15388 | Absolute value of differen... |
| abs3lemi 15389 | Lemma involving absolute v... |
| rpsqrtcld 15390 | The square root of a posit... |
| sqrtgt0d 15391 | The square root of a posit... |
| absnidd 15392 | A negative number is the n... |
| leabsd 15393 | A real number is less than... |
| absord 15394 | The absolute value of a re... |
| absred 15395 | Absolute value of a real n... |
| resqrtcld 15396 | The square root of a nonne... |
| sqrtmsqd 15397 | Square root of square. (C... |
| sqrtsqd 15398 | Square root of square. (C... |
| sqrtge0d 15399 | The square root of a nonne... |
| sqrtnegd 15400 | The square root of a negat... |
| absidd 15401 | A nonnegative number is it... |
| sqrtdivd 15402 | Square root distributes ov... |
| sqrtmuld 15403 | Square root distributes ov... |
| sqrtsq2d 15404 | Relationship between squar... |
| sqrtled 15405 | Square root is monotonic. ... |
| sqrtltd 15406 | Square root is strictly mo... |
| sqr11d 15407 | The square root function i... |
| absltd 15408 | Absolute value and 'less t... |
| absled 15409 | Absolute value and 'less t... |
| abssubge0d 15410 | Absolute value of a nonneg... |
| abssuble0d 15411 | Absolute value of a nonpos... |
| absdifltd 15412 | The absolute value of a di... |
| absdifled 15413 | The absolute value of a di... |
| icodiamlt 15414 | Two elements in a half-ope... |
| abscld 15415 | Real closure of absolute v... |
| sqrtcld 15416 | Closure of the square root... |
| sqrtrege0d 15417 | The real part of the squar... |
| sqsqrtd 15418 | Square root theorem. Theo... |
| msqsqrtd 15419 | Square root theorem. Theo... |
| sqr00d 15420 | A square root is zero iff ... |
| absvalsqd 15421 | Square of value of absolut... |
| absvalsq2d 15422 | Square of value of absolut... |
| absge0d 15423 | Absolute value is nonnegat... |
| absval2d 15424 | Value of absolute value fu... |
| abs00d 15425 | The absolute value of a nu... |
| absne0d 15426 | The absolute value of a nu... |
| absrpcld 15427 | The absolute value of a no... |
| absnegd 15428 | Absolute value of negative... |
| abscjd 15429 | The absolute value of a nu... |
| releabsd 15430 | The real part of a number ... |
| absexpd 15431 | Absolute value of positive... |
| abssubd 15432 | Swapping order of subtract... |
| absmuld 15433 | Absolute value distributes... |
| absdivd 15434 | Absolute value distributes... |
| abstrid 15435 | Triangle inequality for ab... |
| abs2difd 15436 | Difference of absolute val... |
| abs2dif2d 15437 | Difference of absolute val... |
| abs2difabsd 15438 | Absolute value of differen... |
| abs3difd 15439 | Absolute value of differen... |
| abs3lemd 15440 | Lemma involving absolute v... |
| reusq0 15441 | A complex number is the sq... |
| bhmafibid1cn 15442 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15443 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15444 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15445 | The Brahmagupta-Fibonacci ... |
| limsupgord 15448 | Ordering property of the s... |
| limsupcl 15449 | Closure of the superior li... |
| limsupval 15450 | The superior limit of an i... |
| limsupgf 15451 | Closure of the superior li... |
| limsupgval 15452 | Value of the superior limi... |
| limsupgle 15453 | The defining property of t... |
| limsuple 15454 | The defining property of t... |
| limsuplt 15455 | The defining property of t... |
| limsupval2 15456 | The superior limit, relati... |
| limsupgre 15457 | If a sequence of real numb... |
| limsupbnd1 15458 | If a sequence is eventuall... |
| limsupbnd2 15459 | If a sequence is eventuall... |
| climrel 15468 | The limit relation is a re... |
| rlimrel 15469 | The limit relation is a re... |
| clim 15470 | Express the predicate: Th... |
| rlim 15471 | Express the predicate: Th... |
| rlim2 15472 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15473 | Use strictly less-than in ... |
| rlim3 15474 | Restrict the range of the ... |
| climcl 15475 | Closure of the limit of a ... |
| rlimpm 15476 | Closure of a function with... |
| rlimf 15477 | Closure of a function with... |
| rlimss 15478 | Domain closure of a functi... |
| rlimcl 15479 | Closure of the limit of a ... |
| clim2 15480 | Express the predicate: Th... |
| clim2c 15481 | Express the predicate ` F ... |
| clim0 15482 | Express the predicate ` F ... |
| clim0c 15483 | Express the predicate ` F ... |
| rlim0 15484 | Express the predicate ` B ... |
| rlim0lt 15485 | Use strictly less-than in ... |
| climi 15486 | Convergence of a sequence ... |
| climi2 15487 | Convergence of a sequence ... |
| climi0 15488 | Convergence of a sequence ... |
| rlimi 15489 | Convergence at infinity of... |
| rlimi2 15490 | Convergence at infinity of... |
| ello1 15491 | Elementhood in the set of ... |
| ello12 15492 | Elementhood in the set of ... |
| ello12r 15493 | Sufficient condition for e... |
| lo1f 15494 | An eventually upper bounde... |
| lo1dm 15495 | An eventually upper bounde... |
| lo1bdd 15496 | The defining property of a... |
| ello1mpt 15497 | Elementhood in the set of ... |
| ello1mpt2 15498 | Elementhood in the set of ... |
| ello1d 15499 | Sufficient condition for e... |
| lo1bdd2 15500 | If an eventually bounded f... |
| lo1bddrp 15501 | Refine ~ o1bdd2 to give a ... |
| elo1 15502 | Elementhood in the set of ... |
| elo12 15503 | Elementhood in the set of ... |
| elo12r 15504 | Sufficient condition for e... |
| o1f 15505 | An eventually bounded func... |
| o1dm 15506 | An eventually bounded func... |
| o1bdd 15507 | The defining property of a... |
| lo1o1 15508 | A function is eventually b... |
| lo1o12 15509 | A function is eventually b... |
| elo1mpt 15510 | Elementhood in the set of ... |
| elo1mpt2 15511 | Elementhood in the set of ... |
| elo1d 15512 | Sufficient condition for e... |
| o1lo1 15513 | A real function is eventua... |
| o1lo12 15514 | A lower bounded real funct... |
| o1lo1d 15515 | A real eventually bounded ... |
| icco1 15516 | Derive eventual boundednes... |
| o1bdd2 15517 | If an eventually bounded f... |
| o1bddrp 15518 | Refine ~ o1bdd2 to give a ... |
| climconst 15519 | An (eventually) constant s... |
| rlimconst 15520 | A constant sequence conver... |
| rlimclim1 15521 | Forward direction of ~ rli... |
| rlimclim 15522 | A sequence on an upper int... |
| climrlim2 15523 | Produce a real limit from ... |
| climconst2 15524 | A constant sequence conver... |
| climz 15525 | The zero sequence converge... |
| rlimuni 15526 | A real function whose doma... |
| rlimdm 15527 | Two ways to express that a... |
| climuni 15528 | An infinite sequence of co... |
| fclim 15529 | The limit relation is func... |
| climdm 15530 | Two ways to express that a... |
| climeu 15531 | An infinite sequence of co... |
| climreu 15532 | An infinite sequence of co... |
| climmo 15533 | An infinite sequence of co... |
| rlimres 15534 | The restriction of a funct... |
| lo1res 15535 | The restriction of an even... |
| o1res 15536 | The restriction of an even... |
| rlimres2 15537 | The restriction of a funct... |
| lo1res2 15538 | The restriction of a funct... |
| o1res2 15539 | The restriction of a funct... |
| lo1resb 15540 | The restriction of a funct... |
| rlimresb 15541 | The restriction of a funct... |
| o1resb 15542 | The restriction of a funct... |
| climeq 15543 | Two functions that are eve... |
| lo1eq 15544 | Two functions that are eve... |
| rlimeq 15545 | Two functions that are eve... |
| o1eq 15546 | Two functions that are eve... |
| climmpt 15547 | Exhibit a function ` G ` w... |
| 2clim 15548 | If two sequences converge ... |
| climmpt2 15549 | Relate an integer limit on... |
| climshftlem 15550 | A shifted function converg... |
| climres 15551 | A function restricted to u... |
| climshft 15552 | A shifted function converg... |
| serclim0 15553 | The zero series converges ... |
| rlimcld2 15554 | If ` D ` is a closed set i... |
| rlimrege0 15555 | The limit of a sequence of... |
| rlimrecl 15556 | The limit of a real sequen... |
| rlimge0 15557 | The limit of a sequence of... |
| climshft2 15558 | A shifted function converg... |
| climrecl 15559 | The limit of a convergent ... |
| climge0 15560 | A nonnegative sequence con... |
| climabs0 15561 | Convergence to zero of the... |
| o1co 15562 | Sufficient condition for t... |
| o1compt 15563 | Sufficient condition for t... |
| rlimcn1 15564 | Image of a limit under a c... |
| rlimcn1b 15565 | Image of a limit under a c... |
| rlimcn3 15566 | Image of a limit under a c... |
| rlimcn2 15567 | Image of a limit under a c... |
| climcn1 15568 | Image of a limit under a c... |
| climcn2 15569 | Image of a limit under a c... |
| addcn2 15570 | Complex number addition is... |
| subcn2 15571 | Complex number subtraction... |
| mulcn2 15572 | Complex number multiplicat... |
| reccn2 15573 | The reciprocal function is... |
| cn1lem 15574 | A sufficient condition for... |
| abscn2 15575 | The absolute value functio... |
| cjcn2 15576 | The complex conjugate func... |
| recn2 15577 | The real part function is ... |
| imcn2 15578 | The imaginary part functio... |
| climcn1lem 15579 | The limit of a continuous ... |
| climabs 15580 | Limit of the absolute valu... |
| climcj 15581 | Limit of the complex conju... |
| climre 15582 | Limit of the real part of ... |
| climim 15583 | Limit of the imaginary par... |
| rlimmptrcl 15584 | Reverse closure for a real... |
| rlimabs 15585 | Limit of the absolute valu... |
| rlimcj 15586 | Limit of the complex conju... |
| rlimre 15587 | Limit of the real part of ... |
| rlimim 15588 | Limit of the imaginary par... |
| o1of2 15589 | Show that a binary operati... |
| o1add 15590 | The sum of two eventually ... |
| o1mul 15591 | The product of two eventua... |
| o1sub 15592 | The difference of two even... |
| rlimo1 15593 | Any function with a finite... |
| rlimdmo1 15594 | A convergent function is e... |
| o1rlimmul 15595 | The product of an eventual... |
| o1const 15596 | A constant function is eve... |
| lo1const 15597 | A constant function is eve... |
| lo1mptrcl 15598 | Reverse closure for an eve... |
| o1mptrcl 15599 | Reverse closure for an eve... |
| o1add2 15600 | The sum of two eventually ... |
| o1mul2 15601 | The product of two eventua... |
| o1sub2 15602 | The product of two eventua... |
| lo1add 15603 | The sum of two eventually ... |
| lo1mul 15604 | The product of an eventual... |
| lo1mul2 15605 | The product of an eventual... |
| o1dif 15606 | If the difference of two f... |
| lo1sub 15607 | The difference of an event... |
| climadd 15608 | Limit of the sum of two co... |
| climmul 15609 | Limit of the product of tw... |
| climsub 15610 | Limit of the difference of... |
| climaddc1 15611 | Limit of a constant ` C ` ... |
| climaddc2 15612 | Limit of a constant ` C ` ... |
| climmulc2 15613 | Limit of a sequence multip... |
| climsubc1 15614 | Limit of a constant ` C ` ... |
| climsubc2 15615 | Limit of a constant ` C ` ... |
| climle 15616 | Comparison of the limits o... |
| climsqz 15617 | Convergence of a sequence ... |
| climsqz2 15618 | Convergence of a sequence ... |
| rlimadd 15619 | Limit of the sum of two co... |
| rlimaddOLD 15620 | Obsolete version of ~ rlim... |
| rlimsub 15621 | Limit of the difference of... |
| rlimmul 15622 | Limit of the product of tw... |
| rlimmulOLD 15623 | Obsolete version of ~ rlim... |
| rlimdiv 15624 | Limit of the quotient of t... |
| rlimneg 15625 | Limit of the negative of a... |
| rlimle 15626 | Comparison of the limits o... |
| rlimsqzlem 15627 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15628 | Convergence of a sequence ... |
| rlimsqz2 15629 | Convergence of a sequence ... |
| lo1le 15630 | Transfer eventual upper bo... |
| o1le 15631 | Transfer eventual boundedn... |
| rlimno1 15632 | A function whose inverse c... |
| clim2ser 15633 | The limit of an infinite s... |
| clim2ser2 15634 | The limit of an infinite s... |
| iserex 15635 | An infinite series converg... |
| isermulc2 15636 | Multiplication of an infin... |
| climlec2 15637 | Comparison of a constant t... |
| iserle 15638 | Comparison of the limits o... |
| iserge0 15639 | The limit of an infinite s... |
| climub 15640 | The limit of a monotonic s... |
| climserle 15641 | The partial sums of a conv... |
| isershft 15642 | Index shift of the limit o... |
| isercolllem1 15643 | Lemma for ~ isercoll . (C... |
| isercolllem2 15644 | Lemma for ~ isercoll . (C... |
| isercolllem3 15645 | Lemma for ~ isercoll . (C... |
| isercoll 15646 | Rearrange an infinite seri... |
| isercoll2 15647 | Generalize ~ isercoll so t... |
| climsup 15648 | A bounded monotonic sequen... |
| climcau 15649 | A converging sequence of c... |
| climbdd 15650 | A converging sequence of c... |
| caucvgrlem 15651 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15652 | A Cauchy sequence of real ... |
| caucvgrlem2 15653 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15654 | A Cauchy sequence of compl... |
| caurcvg 15655 | A Cauchy sequence of real ... |
| caurcvg2 15656 | A Cauchy sequence of real ... |
| caucvg 15657 | A Cauchy sequence of compl... |
| caucvgb 15658 | A function is convergent i... |
| serf0 15659 | If an infinite series conv... |
| iseraltlem1 15660 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15661 | Lemma for ~ iseralt . The... |
| iseraltlem3 15662 | Lemma for ~ iseralt . Fro... |
| iseralt 15663 | The alternating series tes... |
| sumex 15666 | A sum is a set. (Contribu... |
| sumeq1 15667 | Equality theorem for a sum... |
| nfsum1 15668 | Bound-variable hypothesis ... |
| nfsum 15669 | Bound-variable hypothesis ... |
| sumeq2w 15670 | Equality theorem for sum, ... |
| sumeq2ii 15671 | Equality theorem for sum, ... |
| sumeq2 15672 | Equality theorem for sum. ... |
| cbvsum 15673 | Change bound variable in a... |
| cbvsumv 15674 | Change bound variable in a... |
| cbvsumi 15675 | Change bound variable in a... |
| sumeq1i 15676 | Equality inference for sum... |
| sumeq2i 15677 | Equality inference for sum... |
| sumeq12i 15678 | Equality inference for sum... |
| sumeq1d 15679 | Equality deduction for sum... |
| sumeq2d 15680 | Equality deduction for sum... |
| sumeq2dv 15681 | Equality deduction for sum... |
| sumeq2sdv 15682 | Equality deduction for sum... |
| 2sumeq2dv 15683 | Equality deduction for dou... |
| sumeq12dv 15684 | Equality deduction for sum... |
| sumeq12rdv 15685 | Equality deduction for sum... |
| sum2id 15686 | The second class argument ... |
| sumfc 15687 | A lemma to facilitate conv... |
| fz1f1o 15688 | A lemma for working with f... |
| sumrblem 15689 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15690 | The sequence of partial su... |
| sumrb 15691 | Rebase the starting point ... |
| summolem3 15692 | Lemma for ~ summo . (Cont... |
| summolem2a 15693 | Lemma for ~ summo . (Cont... |
| summolem2 15694 | Lemma for ~ summo . (Cont... |
| summo 15695 | A sum has at most one limi... |
| zsum 15696 | Series sum with index set ... |
| isum 15697 | Series sum with an upper i... |
| fsum 15698 | The value of a sum over a ... |
| sum0 15699 | Any sum over the empty set... |
| sumz 15700 | Any sum of zero over a sum... |
| fsumf1o 15701 | Re-index a finite sum usin... |
| sumss 15702 | Change the index set to a ... |
| fsumss 15703 | Change the index set to a ... |
| sumss2 15704 | Change the index set of a ... |
| fsumcvg2 15705 | The sequence of partial su... |
| fsumsers 15706 | Special case of series sum... |
| fsumcvg3 15707 | A finite sum is convergent... |
| fsumser 15708 | A finite sum expressed in ... |
| fsumcl2lem 15709 | - Lemma for finite sum clo... |
| fsumcllem 15710 | - Lemma for finite sum clo... |
| fsumcl 15711 | Closure of a finite sum of... |
| fsumrecl 15712 | Closure of a finite sum of... |
| fsumzcl 15713 | Closure of a finite sum of... |
| fsumnn0cl 15714 | Closure of a finite sum of... |
| fsumrpcl 15715 | Closure of a finite sum of... |
| fsumclf 15716 | Closure of a finite sum of... |
| fsumzcl2 15717 | A finite sum with integer ... |
| fsumadd 15718 | The sum of two finite sums... |
| fsumsplit 15719 | Split a sum into two parts... |
| fsumsplitf 15720 | Split a sum into two parts... |
| sumsnf 15721 | A sum of a singleton is th... |
| fsumsplitsn 15722 | Separate out a term in a f... |
| fsumsplit1 15723 | Separate out a term in a f... |
| sumsn 15724 | A sum of a singleton is th... |
| fsum1 15725 | The finite sum of ` A ( k ... |
| sumpr 15726 | A sum over a pair is the s... |
| sumtp 15727 | A sum over a triple is the... |
| sumsns 15728 | A sum of a singleton is th... |
| fsumm1 15729 | Separate out the last term... |
| fzosump1 15730 | Separate out the last term... |
| fsum1p 15731 | Separate out the first ter... |
| fsummsnunz 15732 | A finite sum all of whose ... |
| fsumsplitsnun 15733 | Separate out a term in a f... |
| fsump1 15734 | The addition of the next t... |
| isumclim 15735 | An infinite sum equals the... |
| isumclim2 15736 | A converging series conver... |
| isumclim3 15737 | The sequence of partial fi... |
| sumnul 15738 | The sum of a non-convergen... |
| isumcl 15739 | The sum of a converging in... |
| isummulc2 15740 | An infinite sum multiplied... |
| isummulc1 15741 | An infinite sum multiplied... |
| isumdivc 15742 | An infinite sum divided by... |
| isumrecl 15743 | The sum of a converging in... |
| isumge0 15744 | An infinite sum of nonnega... |
| isumadd 15745 | Addition of infinite sums.... |
| sumsplit 15746 | Split a sum into two parts... |
| fsump1i 15747 | Optimized version of ~ fsu... |
| fsum2dlem 15748 | Lemma for ~ fsum2d - induc... |
| fsum2d 15749 | Write a double sum as a su... |
| fsumxp 15750 | Combine two sums into a si... |
| fsumcnv 15751 | Transform a region of summ... |
| fsumcom2 15752 | Interchange order of summa... |
| fsumcom 15753 | Interchange order of summa... |
| fsum0diaglem 15754 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15755 | Two ways to express "the s... |
| mptfzshft 15756 | 1-1 onto function in maps-... |
| fsumrev 15757 | Reversal of a finite sum. ... |
| fsumshft 15758 | Index shift of a finite su... |
| fsumshftm 15759 | Negative index shift of a ... |
| fsumrev2 15760 | Reversal of a finite sum. ... |
| fsum0diag2 15761 | Two ways to express "the s... |
| fsummulc2 15762 | A finite sum multiplied by... |
| fsummulc1 15763 | A finite sum multiplied by... |
| fsumdivc 15764 | A finite sum divided by a ... |
| fsumneg 15765 | Negation of a finite sum. ... |
| fsumsub 15766 | Split a finite sum over a ... |
| fsum2mul 15767 | Separate the nested sum of... |
| fsumconst 15768 | The sum of constant terms ... |
| fsumdifsnconst 15769 | The sum of constant terms ... |
| modfsummodslem1 15770 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15771 | Induction step for ~ modfs... |
| modfsummod 15772 | A finite sum modulo a posi... |
| fsumge0 15773 | If all of the terms of a f... |
| fsumless 15774 | A shorter sum of nonnegati... |
| fsumge1 15775 | A sum of nonnegative numbe... |
| fsum00 15776 | A sum of nonnegative numbe... |
| fsumle 15777 | If all of the terms of fin... |
| fsumlt 15778 | If every term in one finit... |
| fsumabs 15779 | Generalized triangle inequ... |
| telfsumo 15780 | Sum of a telescoping serie... |
| telfsumo2 15781 | Sum of a telescoping serie... |
| telfsum 15782 | Sum of a telescoping serie... |
| telfsum2 15783 | Sum of a telescoping serie... |
| fsumparts 15784 | Summation by parts. (Cont... |
| fsumrelem 15785 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15786 | The real part of a sum. (... |
| fsumim 15787 | The imaginary part of a su... |
| fsumcj 15788 | The complex conjugate of a... |
| fsumrlim 15789 | Limit of a finite sum of c... |
| fsumo1 15790 | The finite sum of eventual... |
| o1fsum 15791 | If ` A ( k ) ` is O(1), th... |
| seqabs 15792 | Generalized triangle inequ... |
| iserabs 15793 | Generalized triangle inequ... |
| cvgcmp 15794 | A comparison test for conv... |
| cvgcmpub 15795 | An upper bound for the lim... |
| cvgcmpce 15796 | A comparison test for conv... |
| abscvgcvg 15797 | An absolutely convergent s... |
| climfsum 15798 | Limit of a finite sum of c... |
| fsumiun 15799 | Sum over a disjoint indexe... |
| hashiun 15800 | The cardinality of a disjo... |
| hash2iun 15801 | The cardinality of a neste... |
| hash2iun1dif1 15802 | The cardinality of a neste... |
| hashrabrex 15803 | The number of elements in ... |
| hashuni 15804 | The cardinality of a disjo... |
| qshash 15805 | The cardinality of a set w... |
| ackbijnn 15806 | Translate the Ackermann bi... |
| binomlem 15807 | Lemma for ~ binom (binomia... |
| binom 15808 | The binomial theorem: ` ( ... |
| binom1p 15809 | Special case of the binomi... |
| binom11 15810 | Special case of the binomi... |
| binom1dif 15811 | A summation for the differ... |
| bcxmaslem1 15812 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15813 | Parallel summation (Christ... |
| incexclem 15814 | Lemma for ~ incexc . (Con... |
| incexc 15815 | The inclusion/exclusion pr... |
| incexc2 15816 | The inclusion/exclusion pr... |
| isumshft 15817 | Index shift of an infinite... |
| isumsplit 15818 | Split off the first ` N ` ... |
| isum1p 15819 | The infinite sum of a conv... |
| isumnn0nn 15820 | Sum from 0 to infinity in ... |
| isumrpcl 15821 | The infinite sum of positi... |
| isumle 15822 | Comparison of two infinite... |
| isumless 15823 | A finite sum of nonnegativ... |
| isumsup2 15824 | An infinite sum of nonnega... |
| isumsup 15825 | An infinite sum of nonnega... |
| isumltss 15826 | A partial sum of a series ... |
| climcndslem1 15827 | Lemma for ~ climcnds : bou... |
| climcndslem2 15828 | Lemma for ~ climcnds : bou... |
| climcnds 15829 | The Cauchy condensation te... |
| divrcnv 15830 | The sequence of reciprocal... |
| divcnv 15831 | The sequence of reciprocal... |
| flo1 15832 | The floor function satisfi... |
| divcnvshft 15833 | Limit of a ratio function.... |
| supcvg 15834 | Extract a sequence ` f ` i... |
| infcvgaux1i 15835 | Auxiliary theorem for appl... |
| infcvgaux2i 15836 | Auxiliary theorem for appl... |
| harmonic 15837 | The harmonic series ` H ` ... |
| arisum 15838 | Arithmetic series sum of t... |
| arisum2 15839 | Arithmetic series sum of t... |
| trireciplem 15840 | Lemma for ~ trirecip . Sh... |
| trirecip 15841 | The sum of the reciprocals... |
| expcnv 15842 | A sequence of powers of a ... |
| explecnv 15843 | A sequence of terms conver... |
| geoserg 15844 | The value of the finite ge... |
| geoser 15845 | The value of the finite ge... |
| pwdif 15846 | The difference of two numb... |
| pwm1geoser 15847 | The n-th power of a number... |
| geolim 15848 | The partial sums in the in... |
| geolim2 15849 | The partial sums in the ge... |
| georeclim 15850 | The limit of a geometric s... |
| geo2sum 15851 | The value of the finite ge... |
| geo2sum2 15852 | The value of the finite ge... |
| geo2lim 15853 | The value of the infinite ... |
| geomulcvg 15854 | The geometric series conve... |
| geoisum 15855 | The infinite sum of ` 1 + ... |
| geoisumr 15856 | The infinite sum of recipr... |
| geoisum1 15857 | The infinite sum of ` A ^ ... |
| geoisum1c 15858 | The infinite sum of ` A x.... |
| 0.999... 15859 | The recurring decimal 0.99... |
| geoihalfsum 15860 | Prove that the infinite ge... |
| cvgrat 15861 | Ratio test for convergence... |
| mertenslem1 15862 | Lemma for ~ mertens . (Co... |
| mertenslem2 15863 | Lemma for ~ mertens . (Co... |
| mertens 15864 | Mertens' theorem. If ` A ... |
| prodf 15865 | An infinite product of com... |
| clim2prod 15866 | The limit of an infinite p... |
| clim2div 15867 | The limit of an infinite p... |
| prodfmul 15868 | The product of two infinit... |
| prodf1 15869 | The value of the partial p... |
| prodf1f 15870 | A one-valued infinite prod... |
| prodfclim1 15871 | The constant one product c... |
| prodfn0 15872 | No term of a nonzero infin... |
| prodfrec 15873 | The reciprocal of an infin... |
| prodfdiv 15874 | The quotient of two infini... |
| ntrivcvg 15875 | A non-trivially converging... |
| ntrivcvgn0 15876 | A product that converges t... |
| ntrivcvgfvn0 15877 | Any value of a product seq... |
| ntrivcvgtail 15878 | A tail of a non-trivially ... |
| ntrivcvgmullem 15879 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15880 | The product of two non-tri... |
| prodex 15883 | A product is a set. (Cont... |
| prodeq1f 15884 | Equality theorem for a pro... |
| prodeq1 15885 | Equality theorem for a pro... |
| nfcprod1 15886 | Bound-variable hypothesis ... |
| nfcprod 15887 | Bound-variable hypothesis ... |
| prodeq2w 15888 | Equality theorem for produ... |
| prodeq2ii 15889 | Equality theorem for produ... |
| prodeq2 15890 | Equality theorem for produ... |
| cbvprod 15891 | Change bound variable in a... |
| cbvprodv 15892 | Change bound variable in a... |
| cbvprodi 15893 | Change bound variable in a... |
| prodeq1i 15894 | Equality inference for pro... |
| prodeq2i 15895 | Equality inference for pro... |
| prodeq12i 15896 | Equality inference for pro... |
| prodeq1d 15897 | Equality deduction for pro... |
| prodeq2d 15898 | Equality deduction for pro... |
| prodeq2dv 15899 | Equality deduction for pro... |
| prodeq2sdv 15900 | Equality deduction for pro... |
| 2cprodeq2dv 15901 | Equality deduction for dou... |
| prodeq12dv 15902 | Equality deduction for pro... |
| prodeq12rdv 15903 | Equality deduction for pro... |
| prod2id 15904 | The second class argument ... |
| prodrblem 15905 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15906 | The sequence of partial pr... |
| prodrblem2 15907 | Lemma for ~ prodrb . (Con... |
| prodrb 15908 | Rebase the starting point ... |
| prodmolem3 15909 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15910 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15911 | Lemma for ~ prodmo . (Con... |
| prodmo 15912 | A product has at most one ... |
| zprod 15913 | Series product with index ... |
| iprod 15914 | Series product with an upp... |
| zprodn0 15915 | Nonzero series product wit... |
| iprodn0 15916 | Nonzero series product wit... |
| fprod 15917 | The value of a product ove... |
| fprodntriv 15918 | A non-triviality lemma for... |
| prod0 15919 | A product over the empty s... |
| prod1 15920 | Any product of one over a ... |
| prodfc 15921 | A lemma to facilitate conv... |
| fprodf1o 15922 | Re-index a finite product ... |
| prodss 15923 | Change the index set to a ... |
| fprodss 15924 | Change the index set to a ... |
| fprodser 15925 | A finite product expressed... |
| fprodcl2lem 15926 | Finite product closure lem... |
| fprodcllem 15927 | Finite product closure lem... |
| fprodcl 15928 | Closure of a finite produc... |
| fprodrecl 15929 | Closure of a finite produc... |
| fprodzcl 15930 | Closure of a finite produc... |
| fprodnncl 15931 | Closure of a finite produc... |
| fprodrpcl 15932 | Closure of a finite produc... |
| fprodnn0cl 15933 | Closure of a finite produc... |
| fprodcllemf 15934 | Finite product closure lem... |
| fprodreclf 15935 | Closure of a finite produc... |
| fprodmul 15936 | The product of two finite ... |
| fproddiv 15937 | The quotient of two finite... |
| prodsn 15938 | A product of a singleton i... |
| fprod1 15939 | A finite product of only o... |
| prodsnf 15940 | A product of a singleton i... |
| climprod1 15941 | The limit of a product ove... |
| fprodsplit 15942 | Split a finite product int... |
| fprodm1 15943 | Separate out the last term... |
| fprod1p 15944 | Separate out the first ter... |
| fprodp1 15945 | Multiply in the last term ... |
| fprodm1s 15946 | Separate out the last term... |
| fprodp1s 15947 | Multiply in the last term ... |
| prodsns 15948 | A product of the singleton... |
| fprodfac 15949 | Factorial using product no... |
| fprodabs 15950 | The absolute value of a fi... |
| fprodeq0 15951 | Any finite product contain... |
| fprodshft 15952 | Shift the index of a finit... |
| fprodrev 15953 | Reversal of a finite produ... |
| fprodconst 15954 | The product of constant te... |
| fprodn0 15955 | A finite product of nonzer... |
| fprod2dlem 15956 | Lemma for ~ fprod2d - indu... |
| fprod2d 15957 | Write a double product as ... |
| fprodxp 15958 | Combine two products into ... |
| fprodcnv 15959 | Transform a product region... |
| fprodcom2 15960 | Interchange order of multi... |
| fprodcom 15961 | Interchange product order.... |
| fprod0diag 15962 | Two ways to express "the p... |
| fproddivf 15963 | The quotient of two finite... |
| fprodsplitf 15964 | Split a finite product int... |
| fprodsplitsn 15965 | Separate out a term in a f... |
| fprodsplit1f 15966 | Separate out a term in a f... |
| fprodn0f 15967 | A finite product of nonzer... |
| fprodclf 15968 | Closure of a finite produc... |
| fprodge0 15969 | If all the terms of a fini... |
| fprodeq0g 15970 | Any finite product contain... |
| fprodge1 15971 | If all of the terms of a f... |
| fprodle 15972 | If all the terms of two fi... |
| fprodmodd 15973 | If all factors of two fini... |
| iprodclim 15974 | An infinite product equals... |
| iprodclim2 15975 | A converging product conve... |
| iprodclim3 15976 | The sequence of partial fi... |
| iprodcl 15977 | The product of a non-trivi... |
| iprodrecl 15978 | The product of a non-trivi... |
| iprodmul 15979 | Multiplication of infinite... |
| risefacval 15984 | The value of the rising fa... |
| fallfacval 15985 | The value of the falling f... |
| risefacval2 15986 | One-based value of rising ... |
| fallfacval2 15987 | One-based value of falling... |
| fallfacval3 15988 | A product representation o... |
| risefaccllem 15989 | Lemma for rising factorial... |
| fallfaccllem 15990 | Lemma for falling factoria... |
| risefaccl 15991 | Closure law for rising fac... |
| fallfaccl 15992 | Closure law for falling fa... |
| rerisefaccl 15993 | Closure law for rising fac... |
| refallfaccl 15994 | Closure law for falling fa... |
| nnrisefaccl 15995 | Closure law for rising fac... |
| zrisefaccl 15996 | Closure law for rising fac... |
| zfallfaccl 15997 | Closure law for falling fa... |
| nn0risefaccl 15998 | Closure law for rising fac... |
| rprisefaccl 15999 | Closure law for rising fac... |
| risefallfac 16000 | A relationship between ris... |
| fallrisefac 16001 | A relationship between fal... |
| risefall0lem 16002 | Lemma for ~ risefac0 and ~... |
| risefac0 16003 | The value of the rising fa... |
| fallfac0 16004 | The value of the falling f... |
| risefacp1 16005 | The value of the rising fa... |
| fallfacp1 16006 | The value of the falling f... |
| risefacp1d 16007 | The value of the rising fa... |
| fallfacp1d 16008 | The value of the falling f... |
| risefac1 16009 | The value of rising factor... |
| fallfac1 16010 | The value of falling facto... |
| risefacfac 16011 | Relate rising factorial to... |
| fallfacfwd 16012 | The forward difference of ... |
| 0fallfac 16013 | The value of the zero fall... |
| 0risefac 16014 | The value of the zero risi... |
| binomfallfaclem1 16015 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16016 | Lemma for ~ binomfallfac .... |
| binomfallfac 16017 | A version of the binomial ... |
| binomrisefac 16018 | A version of the binomial ... |
| fallfacval4 16019 | Represent the falling fact... |
| bcfallfac 16020 | Binomial coefficient in te... |
| fallfacfac 16021 | Relate falling factorial t... |
| bpolylem 16024 | Lemma for ~ bpolyval . (C... |
| bpolyval 16025 | The value of the Bernoulli... |
| bpoly0 16026 | The value of the Bernoulli... |
| bpoly1 16027 | The value of the Bernoulli... |
| bpolycl 16028 | Closure law for Bernoulli ... |
| bpolysum 16029 | A sum for Bernoulli polyno... |
| bpolydiflem 16030 | Lemma for ~ bpolydif . (C... |
| bpolydif 16031 | Calculate the difference b... |
| fsumkthpow 16032 | A closed-form expression f... |
| bpoly2 16033 | The Bernoulli polynomials ... |
| bpoly3 16034 | The Bernoulli polynomials ... |
| bpoly4 16035 | The Bernoulli polynomials ... |
| fsumcube 16036 | Express the sum of cubes i... |
| eftcl 16049 | Closure of a term in the s... |
| reeftcl 16050 | The terms of the series ex... |
| eftabs 16051 | The absolute value of a te... |
| eftval 16052 | The value of a term in the... |
| efcllem 16053 | Lemma for ~ efcl . The se... |
| ef0lem 16054 | The series defining the ex... |
| efval 16055 | Value of the exponential f... |
| esum 16056 | Value of Euler's constant ... |
| eff 16057 | Domain and codomain of the... |
| efcl 16058 | Closure law for the expone... |
| efcld 16059 | Closure law for the expone... |
| efval2 16060 | Value of the exponential f... |
| efcvg 16061 | The series that defines th... |
| efcvgfsum 16062 | Exponential function conve... |
| reefcl 16063 | The exponential function i... |
| reefcld 16064 | The exponential function i... |
| ere 16065 | Euler's constant ` _e ` = ... |
| ege2le3 16066 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16067 | Value of the exponential f... |
| efcj 16068 | The exponential of a compl... |
| efaddlem 16069 | Lemma for ~ efadd (exponen... |
| efadd 16070 | Sum of exponents law for e... |
| fprodefsum 16071 | Move the exponential funct... |
| efcan 16072 | Cancellation law for expon... |
| efne0 16073 | The exponential of a compl... |
| efneg 16074 | The exponential of the opp... |
| eff2 16075 | The exponential function m... |
| efsub 16076 | Difference of exponents la... |
| efexp 16077 | The exponential of an inte... |
| efzval 16078 | Value of the exponential f... |
| efgt0 16079 | The exponential of a real ... |
| rpefcl 16080 | The exponential of a real ... |
| rpefcld 16081 | The exponential of a real ... |
| eftlcvg 16082 | The tail series of the exp... |
| eftlcl 16083 | Closure of the sum of an i... |
| reeftlcl 16084 | Closure of the sum of an i... |
| eftlub 16085 | An upper bound on the abso... |
| efsep 16086 | Separate out the next term... |
| effsumlt 16087 | The partial sums of the se... |
| eft0val 16088 | The value of the first ter... |
| ef4p 16089 | Separate out the first fou... |
| efgt1p2 16090 | The exponential of a posit... |
| efgt1p 16091 | The exponential of a posit... |
| efgt1 16092 | The exponential of a posit... |
| eflt 16093 | The exponential function o... |
| efle 16094 | The exponential function o... |
| reef11 16095 | The exponential function o... |
| reeff1 16096 | The exponential function m... |
| eflegeo 16097 | The exponential function o... |
| sinval 16098 | Value of the sine function... |
| cosval 16099 | Value of the cosine functi... |
| sinf 16100 | Domain and codomain of the... |
| cosf 16101 | Domain and codomain of the... |
| sincl 16102 | Closure of the sine functi... |
| coscl 16103 | Closure of the cosine func... |
| tanval 16104 | Value of the tangent funct... |
| tancl 16105 | The closure of the tangent... |
| sincld 16106 | Closure of the sine functi... |
| coscld 16107 | Closure of the cosine func... |
| tancld 16108 | Closure of the tangent fun... |
| tanval2 16109 | Express the tangent functi... |
| tanval3 16110 | Express the tangent functi... |
| resinval 16111 | The sine of a real number ... |
| recosval 16112 | The cosine of a real numbe... |
| efi4p 16113 | Separate out the first fou... |
| resin4p 16114 | Separate out the first fou... |
| recos4p 16115 | Separate out the first fou... |
| resincl 16116 | The sine of a real number ... |
| recoscl 16117 | The cosine of a real numbe... |
| retancl 16118 | The closure of the tangent... |
| resincld 16119 | Closure of the sine functi... |
| recoscld 16120 | Closure of the cosine func... |
| retancld 16121 | Closure of the tangent fun... |
| sinneg 16122 | The sine of a negative is ... |
| cosneg 16123 | The cosines of a number an... |
| tanneg 16124 | The tangent of a negative ... |
| sin0 16125 | Value of the sine function... |
| cos0 16126 | Value of the cosine functi... |
| tan0 16127 | The value of the tangent f... |
| efival 16128 | The exponential function i... |
| efmival 16129 | The exponential function i... |
| sinhval 16130 | Value of the hyperbolic si... |
| coshval 16131 | Value of the hyperbolic co... |
| resinhcl 16132 | The hyperbolic sine of a r... |
| rpcoshcl 16133 | The hyperbolic cosine of a... |
| recoshcl 16134 | The hyperbolic cosine of a... |
| retanhcl 16135 | The hyperbolic tangent of ... |
| tanhlt1 16136 | The hyperbolic tangent of ... |
| tanhbnd 16137 | The hyperbolic tangent of ... |
| efeul 16138 | Eulerian representation of... |
| efieq 16139 | The exponentials of two im... |
| sinadd 16140 | Addition formula for sine.... |
| cosadd 16141 | Addition formula for cosin... |
| tanaddlem 16142 | A useful intermediate step... |
| tanadd 16143 | Addition formula for tange... |
| sinsub 16144 | Sine of difference. (Cont... |
| cossub 16145 | Cosine of difference. (Co... |
| addsin 16146 | Sum of sines. (Contribute... |
| subsin 16147 | Difference of sines. (Con... |
| sinmul 16148 | Product of sines can be re... |
| cosmul 16149 | Product of cosines can be ... |
| addcos 16150 | Sum of cosines. (Contribu... |
| subcos 16151 | Difference of cosines. (C... |
| sincossq 16152 | Sine squared plus cosine s... |
| sin2t 16153 | Double-angle formula for s... |
| cos2t 16154 | Double-angle formula for c... |
| cos2tsin 16155 | Double-angle formula for c... |
| sinbnd 16156 | The sine of a real number ... |
| cosbnd 16157 | The cosine of a real numbe... |
| sinbnd2 16158 | The sine of a real number ... |
| cosbnd2 16159 | The cosine of a real numbe... |
| ef01bndlem 16160 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16161 | Bounds on the sine of a po... |
| cos01bnd 16162 | Bounds on the cosine of a ... |
| cos1bnd 16163 | Bounds on the cosine of 1.... |
| cos2bnd 16164 | Bounds on the cosine of 2.... |
| sinltx 16165 | The sine of a positive rea... |
| sin01gt0 16166 | The sine of a positive rea... |
| cos01gt0 16167 | The cosine of a positive r... |
| sin02gt0 16168 | The sine of a positive rea... |
| sincos1sgn 16169 | The signs of the sine and ... |
| sincos2sgn 16170 | The signs of the sine and ... |
| sin4lt0 16171 | The sine of 4 is negative.... |
| absefi 16172 | The absolute value of the ... |
| absef 16173 | The absolute value of the ... |
| absefib 16174 | A complex number is real i... |
| efieq1re 16175 | A number whose imaginary e... |
| demoivre 16176 | De Moivre's Formula. Proo... |
| demoivreALT 16177 | Alternate proof of ~ demoi... |
| eirrlem 16180 | Lemma for ~ eirr . (Contr... |
| eirr 16181 | ` _e ` is irrational. (Co... |
| egt2lt3 16182 | Euler's constant ` _e ` = ... |
| epos 16183 | Euler's constant ` _e ` is... |
| epr 16184 | Euler's constant ` _e ` is... |
| ene0 16185 | ` _e ` is not 0. (Contrib... |
| ene1 16186 | ` _e ` is not 1. (Contrib... |
| xpnnen 16187 | The Cartesian product of t... |
| znnen 16188 | The set of integers and th... |
| qnnen 16189 | The rational numbers are c... |
| rpnnen2lem1 16190 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16191 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16192 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16193 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16194 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16195 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16196 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16197 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16198 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16199 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16200 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16201 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16202 | The other half of ~ rpnnen... |
| rpnnen 16203 | The cardinality of the con... |
| rexpen 16204 | The real numbers are equin... |
| cpnnen 16205 | The complex numbers are eq... |
| rucALT 16206 | Alternate proof of ~ ruc .... |
| ruclem1 16207 | Lemma for ~ ruc (the reals... |
| ruclem2 16208 | Lemma for ~ ruc . Orderin... |
| ruclem3 16209 | Lemma for ~ ruc . The con... |
| ruclem4 16210 | Lemma for ~ ruc . Initial... |
| ruclem6 16211 | Lemma for ~ ruc . Domain ... |
| ruclem7 16212 | Lemma for ~ ruc . Success... |
| ruclem8 16213 | Lemma for ~ ruc . The int... |
| ruclem9 16214 | Lemma for ~ ruc . The fir... |
| ruclem10 16215 | Lemma for ~ ruc . Every f... |
| ruclem11 16216 | Lemma for ~ ruc . Closure... |
| ruclem12 16217 | Lemma for ~ ruc . The sup... |
| ruclem13 16218 | Lemma for ~ ruc . There i... |
| ruc 16219 | The set of positive intege... |
| resdomq 16220 | The set of rationals is st... |
| aleph1re 16221 | There are at least aleph-o... |
| aleph1irr 16222 | There are at least aleph-o... |
| cnso 16223 | The complex numbers can be... |
| sqrt2irrlem 16224 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16225 | The square root of 2 is ir... |
| sqrt2re 16226 | The square root of 2 exist... |
| sqrt2irr0 16227 | The square root of 2 is an... |
| nthruc 16228 | The sequence ` NN ` , ` ZZ... |
| nthruz 16229 | The sequence ` NN ` , ` NN... |
| divides 16232 | Define the divides relatio... |
| dvdsval2 16233 | One nonzero integer divide... |
| dvdsval3 16234 | One nonzero integer divide... |
| dvdszrcl 16235 | Reverse closure for the di... |
| dvdsmod0 16236 | If a positive integer divi... |
| p1modz1 16237 | If a number greater than 1... |
| dvdsmodexp 16238 | If a positive integer divi... |
| nndivdvds 16239 | Strong form of ~ dvdsval2 ... |
| nndivides 16240 | Definition of the divides ... |
| moddvds 16241 | Two ways to say ` A == B `... |
| modm1div 16242 | An integer greater than on... |
| dvds0lem 16243 | A lemma to assist theorems... |
| dvds1lem 16244 | A lemma to assist theorems... |
| dvds2lem 16245 | A lemma to assist theorems... |
| iddvds 16246 | An integer divides itself.... |
| 1dvds 16247 | 1 divides any integer. Th... |
| dvds0 16248 | Any integer divides 0. Th... |
| negdvdsb 16249 | An integer divides another... |
| dvdsnegb 16250 | An integer divides another... |
| absdvdsb 16251 | An integer divides another... |
| dvdsabsb 16252 | An integer divides another... |
| 0dvds 16253 | Only 0 is divisible by 0. ... |
| dvdsmul1 16254 | An integer divides a multi... |
| dvdsmul2 16255 | An integer divides a multi... |
| iddvdsexp 16256 | An integer divides a posit... |
| muldvds1 16257 | If a product divides an in... |
| muldvds2 16258 | If a product divides an in... |
| dvdscmul 16259 | Multiplication by a consta... |
| dvdsmulc 16260 | Multiplication by a consta... |
| dvdscmulr 16261 | Cancellation law for the d... |
| dvdsmulcr 16262 | Cancellation law for the d... |
| summodnegmod 16263 | The sum of two integers mo... |
| modmulconst 16264 | Constant multiplication in... |
| dvds2ln 16265 | If an integer divides each... |
| dvds2add 16266 | If an integer divides each... |
| dvds2sub 16267 | If an integer divides each... |
| dvds2addd 16268 | Deduction form of ~ dvds2a... |
| dvds2subd 16269 | Deduction form of ~ dvds2s... |
| dvdstr 16270 | The divides relation is tr... |
| dvdstrd 16271 | The divides relation is tr... |
| dvdsmultr1 16272 | If an integer divides anot... |
| dvdsmultr1d 16273 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16274 | If an integer divides anot... |
| dvdsmultr2d 16275 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16276 | If an integer divides eith... |
| dvdssub2 16277 | If an integer divides a di... |
| dvdsadd 16278 | An integer divides another... |
| dvdsaddr 16279 | An integer divides another... |
| dvdssub 16280 | An integer divides another... |
| dvdssubr 16281 | An integer divides another... |
| dvdsadd2b 16282 | Adding a multiple of the b... |
| dvdsaddre2b 16283 | Adding a multiple of the b... |
| fsumdvds 16284 | If every term in a sum is ... |
| dvdslelem 16285 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16286 | The divisors of a positive... |
| dvdsleabs 16287 | The divisors of a nonzero ... |
| dvdsleabs2 16288 | Transfer divisibility to a... |
| dvdsabseq 16289 | If two integers divide eac... |
| dvdseq 16290 | If two nonnegative integer... |
| divconjdvds 16291 | If a nonzero integer ` M `... |
| dvdsdivcl 16292 | The complement of a diviso... |
| dvdsflip 16293 | An involution of the divis... |
| dvdsssfz1 16294 | The set of divisors of a n... |
| dvds1 16295 | The only nonnegative integ... |
| alzdvds 16296 | Only 0 is divisible by all... |
| dvdsext 16297 | Poset extensionality for d... |
| fzm1ndvds 16298 | No number between ` 1 ` an... |
| fzo0dvdseq 16299 | Zero is the only one of th... |
| fzocongeq 16300 | Two different elements of ... |
| addmodlteqALT 16301 | Two nonnegative integers l... |
| dvdsfac 16302 | A positive integer divides... |
| dvdsexp2im 16303 | If an integer divides anot... |
| dvdsexp 16304 | A power divides a power wi... |
| dvdsmod 16305 | Any number ` K ` whose mod... |
| mulmoddvds 16306 | If an integer is divisible... |
| 3dvds 16307 | A rule for divisibility by... |
| 3dvdsdec 16308 | A decimal number is divisi... |
| 3dvds2dec 16309 | A decimal number is divisi... |
| fprodfvdvdsd 16310 | A finite product of intege... |
| fproddvdsd 16311 | A finite product of intege... |
| evenelz 16312 | An even number is an integ... |
| zeo3 16313 | An integer is even or odd.... |
| zeo4 16314 | An integer is even or odd ... |
| zeneo 16315 | No even integer equals an ... |
| odd2np1lem 16316 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16317 | An integer is odd iff it i... |
| even2n 16318 | An integer is even iff it ... |
| oddm1even 16319 | An integer is odd iff its ... |
| oddp1even 16320 | An integer is odd iff its ... |
| oexpneg 16321 | The exponential of the neg... |
| mod2eq0even 16322 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16323 | An integer is 1 modulo 2 i... |
| oddnn02np1 16324 | A nonnegative integer is o... |
| oddge22np1 16325 | An integer greater than on... |
| evennn02n 16326 | A nonnegative integer is e... |
| evennn2n 16327 | A positive integer is even... |
| 2tp1odd 16328 | A number which is twice an... |
| mulsucdiv2z 16329 | An integer multiplied with... |
| sqoddm1div8z 16330 | A squared odd number minus... |
| 2teven 16331 | A number which is twice an... |
| zeo5 16332 | An integer is either even ... |
| evend2 16333 | An integer is even iff its... |
| oddp1d2 16334 | An integer is odd iff its ... |
| zob 16335 | Alternate characterization... |
| oddm1d2 16336 | An integer is odd iff its ... |
| ltoddhalfle 16337 | An integer is less than ha... |
| halfleoddlt 16338 | An integer is greater than... |
| opoe 16339 | The sum of two odds is eve... |
| omoe 16340 | The difference of two odds... |
| opeo 16341 | The sum of an odd and an e... |
| omeo 16342 | The difference of an odd a... |
| z0even 16343 | 2 divides 0. That means 0... |
| n2dvds1 16344 | 2 does not divide 1. That... |
| n2dvdsm1 16345 | 2 does not divide -1. Tha... |
| z2even 16346 | 2 divides 2. That means 2... |
| n2dvds3 16347 | 2 does not divide 3. That... |
| z4even 16348 | 2 divides 4. That means 4... |
| 4dvdseven 16349 | An integer which is divisi... |
| m1expe 16350 | Exponentiation of -1 by an... |
| m1expo 16351 | Exponentiation of -1 by an... |
| m1exp1 16352 | Exponentiation of negative... |
| nn0enne 16353 | A positive integer is an e... |
| nn0ehalf 16354 | The half of an even nonneg... |
| nnehalf 16355 | The half of an even positi... |
| nn0onn 16356 | An odd nonnegative integer... |
| nn0o1gt2 16357 | An odd nonnegative integer... |
| nno 16358 | An alternate characterizat... |
| nn0o 16359 | An alternate characterizat... |
| nn0ob 16360 | Alternate characterization... |
| nn0oddm1d2 16361 | A positive integer is odd ... |
| nnoddm1d2 16362 | A positive integer is odd ... |
| sumeven 16363 | If every term in a sum is ... |
| sumodd 16364 | If every term in a sum is ... |
| evensumodd 16365 | If every term in a sum wit... |
| oddsumodd 16366 | If every term in a sum wit... |
| pwp1fsum 16367 | The n-th power of a number... |
| oddpwp1fsum 16368 | An odd power of a number i... |
| divalglem0 16369 | Lemma for ~ divalg . (Con... |
| divalglem1 16370 | Lemma for ~ divalg . (Con... |
| divalglem2 16371 | Lemma for ~ divalg . (Con... |
| divalglem4 16372 | Lemma for ~ divalg . (Con... |
| divalglem5 16373 | Lemma for ~ divalg . (Con... |
| divalglem6 16374 | Lemma for ~ divalg . (Con... |
| divalglem7 16375 | Lemma for ~ divalg . (Con... |
| divalglem8 16376 | Lemma for ~ divalg . (Con... |
| divalglem9 16377 | Lemma for ~ divalg . (Con... |
| divalglem10 16378 | Lemma for ~ divalg . (Con... |
| divalg 16379 | The division algorithm (th... |
| divalgb 16380 | Express the division algor... |
| divalg2 16381 | The division algorithm (th... |
| divalgmod 16382 | The result of the ` mod ` ... |
| divalgmodcl 16383 | The result of the ` mod ` ... |
| modremain 16384 | The result of the modulo o... |
| ndvdssub 16385 | Corollary of the division ... |
| ndvdsadd 16386 | Corollary of the division ... |
| ndvdsp1 16387 | Special case of ~ ndvdsadd... |
| ndvdsi 16388 | A quick test for non-divis... |
| flodddiv4 16389 | The floor of an odd intege... |
| fldivndvdslt 16390 | The floor of an integer di... |
| flodddiv4lt 16391 | The floor of an odd number... |
| flodddiv4t2lthalf 16392 | The floor of an odd number... |
| bitsfval 16397 | Expand the definition of t... |
| bitsval 16398 | Expand the definition of t... |
| bitsval2 16399 | Expand the definition of t... |
| bitsss 16400 | The set of bits of an inte... |
| bitsf 16401 | The ` bits ` function is a... |
| bits0 16402 | Value of the zeroth bit. ... |
| bits0e 16403 | The zeroth bit of an even ... |
| bits0o 16404 | The zeroth bit of an odd n... |
| bitsp1 16405 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16406 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16407 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16408 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16409 | The bits of a number are a... |
| bitsmod 16410 | Truncating the bit sequenc... |
| bitsfi 16411 | Every number is associated... |
| bitscmp 16412 | The bit complement of ` N ... |
| 0bits 16413 | The bits of zero. (Contri... |
| m1bits 16414 | The bits of negative one. ... |
| bitsinv1lem 16415 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16416 | There is an explicit inver... |
| bitsinv2 16417 | There is an explicit inver... |
| bitsf1ocnv 16418 | The ` bits ` function rest... |
| bitsf1o 16419 | The ` bits ` function rest... |
| bitsf1 16420 | The ` bits ` function is a... |
| 2ebits 16421 | The bits of a power of two... |
| bitsinv 16422 | The inverse of the ` bits ... |
| bitsinvp1 16423 | Recursive definition of th... |
| sadadd2lem2 16424 | The core of the proof of ~... |
| sadfval 16426 | Define the addition of two... |
| sadcf 16427 | The carry sequence is a se... |
| sadc0 16428 | The initial element of the... |
| sadcp1 16429 | The carry sequence (which ... |
| sadval 16430 | The full adder sequence is... |
| sadcaddlem 16431 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16432 | Non-recursive definition o... |
| sadadd2lem 16433 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16434 | Sum of initial segments of... |
| sadadd3 16435 | Sum of initial segments of... |
| sadcl 16436 | The sum of two sequences i... |
| sadcom 16437 | The adder sequence functio... |
| saddisjlem 16438 | Lemma for ~ sadadd . (Con... |
| saddisj 16439 | The sum of disjoint sequen... |
| sadaddlem 16440 | Lemma for ~ sadadd . (Con... |
| sadadd 16441 | For sequences that corresp... |
| sadid1 16442 | The adder sequence functio... |
| sadid2 16443 | The adder sequence functio... |
| sadasslem 16444 | Lemma for ~ sadass . (Con... |
| sadass 16445 | Sequence addition is assoc... |
| sadeq 16446 | Any element of a sequence ... |
| bitsres 16447 | Restrict the bits of a num... |
| bitsuz 16448 | The bits of a number are a... |
| bitsshft 16449 | Shifting a bit sequence to... |
| smufval 16451 | The multiplication of two ... |
| smupf 16452 | The sequence of partial su... |
| smup0 16453 | The initial element of the... |
| smupp1 16454 | The initial element of the... |
| smuval 16455 | Define the addition of two... |
| smuval2 16456 | The partial sum sequence s... |
| smupvallem 16457 | If ` A ` only has elements... |
| smucl 16458 | The product of two sequenc... |
| smu01lem 16459 | Lemma for ~ smu01 and ~ sm... |
| smu01 16460 | Multiplication of a sequen... |
| smu02 16461 | Multiplication of a sequen... |
| smupval 16462 | Rewrite the elements of th... |
| smup1 16463 | Rewrite ~ smupp1 using onl... |
| smueqlem 16464 | Any element of a sequence ... |
| smueq 16465 | Any element of a sequence ... |
| smumullem 16466 | Lemma for ~ smumul . (Con... |
| smumul 16467 | For sequences that corresp... |
| gcdval 16470 | The value of the ` gcd ` o... |
| gcd0val 16471 | The value, by convention, ... |
| gcdn0val 16472 | The value of the ` gcd ` o... |
| gcdcllem1 16473 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16474 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16475 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16476 | Closure of the ` gcd ` ope... |
| gcddvds 16477 | The gcd of two integers di... |
| dvdslegcd 16478 | An integer which divides b... |
| nndvdslegcd 16479 | A positive integer which d... |
| gcdcl 16480 | Closure of the ` gcd ` ope... |
| gcdnncl 16481 | Closure of the ` gcd ` ope... |
| gcdcld 16482 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16483 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16484 | An integer is the product ... |
| divgcdz 16485 | An integer divided by the ... |
| gcdf 16486 | Domain and codomain of the... |
| gcdcom 16487 | The ` gcd ` operator is co... |
| gcdcomd 16488 | The ` gcd ` operator is co... |
| divgcdnn 16489 | A positive integer divided... |
| divgcdnnr 16490 | A positive integer divided... |
| gcdeq0 16491 | The gcd of two integers is... |
| gcdn0gt0 16492 | The gcd of two integers is... |
| gcd0id 16493 | The gcd of 0 and an intege... |
| gcdid0 16494 | The gcd of an integer and ... |
| nn0gcdid0 16495 | The gcd of a nonnegative i... |
| gcdneg 16496 | Negating one operand of th... |
| neggcd 16497 | Negating one operand of th... |
| gcdaddmlem 16498 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16499 | Adding a multiple of one o... |
| gcdadd 16500 | The GCD of two numbers is ... |
| gcdid 16501 | The gcd of a number and it... |
| gcd1 16502 | The gcd of a number with 1... |
| gcdabs1 16503 | ` gcd ` of the absolute va... |
| gcdabs2 16504 | ` gcd ` of the absolute va... |
| gcdabs 16505 | The gcd of two integers is... |
| gcdabsOLD 16506 | Obsolete version of ~ gcda... |
| modgcd 16507 | The gcd remains unchanged ... |
| 1gcd 16508 | The GCD of one and an inte... |
| gcdmultipled 16509 | The greatest common diviso... |
| gcdmultiplez 16510 | The GCD of a multiple of a... |
| gcdmultiple 16511 | The GCD of a multiple of a... |
| dvdsgcdidd 16512 | The greatest common diviso... |
| 6gcd4e2 16513 | The greatest common diviso... |
| bezoutlem1 16514 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16515 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16516 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16517 | Lemma for ~ bezout . (Con... |
| bezout 16518 | Bézout's identity: ... |
| dvdsgcd 16519 | An integer which divides e... |
| dvdsgcdb 16520 | Biconditional form of ~ dv... |
| dfgcd2 16521 | Alternate definition of th... |
| gcdass 16522 | Associative law for ` gcd ... |
| mulgcd 16523 | Distribute multiplication ... |
| absmulgcd 16524 | Distribute absolute value ... |
| mulgcdr 16525 | Reverse distribution law f... |
| gcddiv 16526 | Division law for GCD. (Con... |
| gcdzeq 16527 | A positive integer ` A ` i... |
| gcdeq 16528 | ` A ` is equal to its gcd ... |
| dvdssqim 16529 | Unidirectional form of ~ d... |
| dvdsmulgcd 16530 | A divisibility equivalent ... |
| rpmulgcd 16531 | If ` K ` and ` M ` are rel... |
| rplpwr 16532 | If ` A ` and ` B ` are rel... |
| rprpwr 16533 | If ` A ` and ` B ` are rel... |
| rppwr 16534 | If ` A ` and ` B ` are rel... |
| sqgcd 16535 | Square distributes over gc... |
| dvdssqlem 16536 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16537 | Two numbers are divisible ... |
| bezoutr 16538 | Partial converse to ~ bezo... |
| bezoutr1 16539 | Converse of ~ bezout for w... |
| nn0seqcvgd 16540 | A strictly-decreasing nonn... |
| seq1st 16541 | A sequence whose iteration... |
| algr0 16542 | The value of the algorithm... |
| algrf 16543 | An algorithm is a step fun... |
| algrp1 16544 | The value of the algorithm... |
| alginv 16545 | If ` I ` is an invariant o... |
| algcvg 16546 | One way to prove that an a... |
| algcvgblem 16547 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16548 | Two ways of expressing tha... |
| algcvga 16549 | The countdown function ` C... |
| algfx 16550 | If ` F ` reaches a fixed p... |
| eucalgval2 16551 | The value of the step func... |
| eucalgval 16552 | Euclid's Algorithm ~ eucal... |
| eucalgf 16553 | Domain and codomain of the... |
| eucalginv 16554 | The invariant of the step ... |
| eucalglt 16555 | The second member of the s... |
| eucalgcvga 16556 | Once Euclid's Algorithm ha... |
| eucalg 16557 | Euclid's Algorithm compute... |
| lcmval 16562 | Value of the ` lcm ` opera... |
| lcmcom 16563 | The ` lcm ` operator is co... |
| lcm0val 16564 | The value, by convention, ... |
| lcmn0val 16565 | The value of the ` lcm ` o... |
| lcmcllem 16566 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16567 | Closure of the ` lcm ` ope... |
| dvdslcm 16568 | The lcm of two integers is... |
| lcmledvds 16569 | A positive integer which b... |
| lcmeq0 16570 | The lcm of two integers is... |
| lcmcl 16571 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16572 | The greatest common diviso... |
| lcmneg 16573 | Negating one operand of th... |
| neglcm 16574 | Negating one operand of th... |
| lcmabs 16575 | The lcm of two integers is... |
| lcmgcdlem 16576 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16577 | The product of two numbers... |
| lcmdvds 16578 | The lcm of two integers di... |
| lcmid 16579 | The lcm of an integer and ... |
| lcm1 16580 | The lcm of an integer and ... |
| lcmgcdnn 16581 | The product of two positiv... |
| lcmgcdeq 16582 | Two integers' absolute val... |
| lcmdvdsb 16583 | Biconditional form of ~ lc... |
| lcmass 16584 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16585 | The least common multiple ... |
| 6lcm4e12 16586 | The least common multiple ... |
| absproddvds 16587 | The absolute value of the ... |
| absprodnn 16588 | The absolute value of the ... |
| fissn0dvds 16589 | For each finite subset of ... |
| fissn0dvdsn0 16590 | For each finite subset of ... |
| lcmfval 16591 | Value of the ` _lcm ` func... |
| lcmf0val 16592 | The value, by convention, ... |
| lcmfn0val 16593 | The value of the ` _lcm ` ... |
| lcmfnnval 16594 | The value of the ` _lcm ` ... |
| lcmfcllem 16595 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16596 | Closure of the ` _lcm ` fu... |
| lcmfpr 16597 | The value of the ` _lcm ` ... |
| lcmfcl 16598 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16599 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16600 | The least common multiple ... |
| dvdslcmf 16601 | The least common multiple ... |
| lcmfledvds 16602 | A positive integer which i... |
| lcmf 16603 | Characterization of the le... |
| lcmf0 16604 | The least common multiple ... |
| lcmfsn 16605 | The least common multiple ... |
| lcmftp 16606 | The least common multiple ... |
| lcmfunsnlem1 16607 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16608 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16609 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16610 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16611 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16612 | The least common multiple ... |
| lcmfdvdsb 16613 | Biconditional form of ~ lc... |
| lcmfunsn 16614 | The ` _lcm ` function for ... |
| lcmfun 16615 | The ` _lcm ` function for ... |
| lcmfass 16616 | Associative law for the ` ... |
| lcmf2a3a4e12 16617 | The least common multiple ... |
| lcmflefac 16618 | The least common multiple ... |
| coprmgcdb 16619 | Two positive integers are ... |
| ncoprmgcdne1b 16620 | Two positive integers are ... |
| ncoprmgcdgt1b 16621 | Two positive integers are ... |
| coprmdvds1 16622 | If two positive integers a... |
| coprmdvds 16623 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16624 | If an integer is divisible... |
| mulgcddvds 16625 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16626 | If ` M ` is relatively pri... |
| qredeq 16627 | Two equal reduced fraction... |
| qredeu 16628 | Every rational number has ... |
| rpmul 16629 | If ` K ` is relatively pri... |
| rpdvds 16630 | If ` K ` is relatively pri... |
| coprmprod 16631 | The product of the element... |
| coprmproddvdslem 16632 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16633 | If a positive integer is d... |
| congr 16634 | Definition of congruence b... |
| divgcdcoprm0 16635 | Integers divided by gcd ar... |
| divgcdcoprmex 16636 | Integers divided by gcd ar... |
| cncongr1 16637 | One direction of the bicon... |
| cncongr2 16638 | The other direction of the... |
| cncongr 16639 | Cancellability of Congruen... |
| cncongrcoprm 16640 | Corollary 1 of Cancellabil... |
| isprm 16643 | The predicate "is a prime ... |
| prmnn 16644 | A prime number is a positi... |
| prmz 16645 | A prime number is an integ... |
| prmssnn 16646 | The prime numbers are a su... |
| prmex 16647 | The set of prime numbers e... |
| 0nprm 16648 | 0 is not a prime number. ... |
| 1nprm 16649 | 1 is not a prime number. ... |
| 1idssfct 16650 | The positive divisors of a... |
| isprm2lem 16651 | Lemma for ~ isprm2 . (Con... |
| isprm2 16652 | The predicate "is a prime ... |
| isprm3 16653 | The predicate "is a prime ... |
| isprm4 16654 | The predicate "is a prime ... |
| prmind2 16655 | A variation on ~ prmind as... |
| prmind 16656 | Perform induction over the... |
| dvdsprime 16657 | If ` M ` divides a prime, ... |
| nprm 16658 | A product of two integers ... |
| nprmi 16659 | An inference for composite... |
| dvdsnprmd 16660 | If a number is divisible b... |
| prm2orodd 16661 | A prime number is either 2... |
| 2prm 16662 | 2 is a prime number. (Con... |
| 2mulprm 16663 | A multiple of two is prime... |
| 3prm 16664 | 3 is a prime number. (Con... |
| 4nprm 16665 | 4 is not a prime number. ... |
| prmuz2 16666 | A prime number is an integ... |
| prmgt1 16667 | A prime number is an integ... |
| prmm2nn0 16668 | Subtracting 2 from a prime... |
| oddprmgt2 16669 | An odd prime is greater th... |
| oddprmge3 16670 | An odd prime is greater th... |
| ge2nprmge4 16671 | A composite integer greate... |
| sqnprm 16672 | A square is never prime. ... |
| dvdsprm 16673 | An integer greater than or... |
| exprmfct 16674 | Every integer greater than... |
| prmdvdsfz 16675 | Each integer greater than ... |
| nprmdvds1 16676 | No prime number divides 1.... |
| isprm5 16677 | One need only check prime ... |
| isprm7 16678 | One need only check prime ... |
| maxprmfct 16679 | The set of prime factors o... |
| divgcdodd 16680 | Either ` A / ( A gcd B ) `... |
| coprm 16681 | A prime number either divi... |
| prmrp 16682 | Unequal prime numbers are ... |
| euclemma 16683 | Euclid's lemma. A prime n... |
| isprm6 16684 | A number is prime iff it s... |
| prmdvdsexp 16685 | A prime divides a positive... |
| prmdvdsexpb 16686 | A prime divides a positive... |
| prmdvdsexpr 16687 | If a prime divides a nonne... |
| prmdvdssq 16688 | Condition for a prime divi... |
| prmdvdssqOLD 16689 | Obsolete version of ~ prmd... |
| prmexpb 16690 | Two positive prime powers ... |
| prmfac1 16691 | The factorial of a number ... |
| dvdszzq 16692 | Divisibility for an intege... |
| rpexp 16693 | If two numbers ` A ` and `... |
| rpexp1i 16694 | Relative primality passes ... |
| rpexp12i 16695 | Relative primality passes ... |
| prmndvdsfaclt 16696 | A prime number does not di... |
| prmdvdsbc 16697 | Condition for a prime numb... |
| prmdvdsncoprmbd 16698 | Two positive integers are ... |
| ncoprmlnprm 16699 | If two positive integers a... |
| cncongrprm 16700 | Corollary 2 of Cancellabil... |
| isevengcd2 16701 | The predicate "is an even ... |
| isoddgcd1 16702 | The predicate "is an odd n... |
| 3lcm2e6 16703 | The least common multiple ... |
| qnumval 16708 | Value of the canonical num... |
| qdenval 16709 | Value of the canonical den... |
| qnumdencl 16710 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16711 | The canonical numerator of... |
| qdencl 16712 | The canonical denominator ... |
| fnum 16713 | Canonical numerator define... |
| fden 16714 | Canonical denominator defi... |
| qnumdenbi 16715 | Two numbers are the canoni... |
| qnumdencoprm 16716 | The canonical representati... |
| qeqnumdivden 16717 | Recover a rational number ... |
| qmuldeneqnum 16718 | Multiplying a rational by ... |
| divnumden 16719 | Calculate the reduced form... |
| divdenle 16720 | Reducing a quotient never ... |
| qnumgt0 16721 | A rational is positive iff... |
| qgt0numnn 16722 | A rational is positive iff... |
| nn0gcdsq 16723 | Squaring commutes with GCD... |
| zgcdsq 16724 | ~ nn0gcdsq extended to int... |
| numdensq 16725 | Squaring a rational square... |
| numsq 16726 | Square commutes with canon... |
| densq 16727 | Square commutes with canon... |
| qden1elz 16728 | A rational is an integer i... |
| zsqrtelqelz 16729 | If an integer has a ration... |
| nonsq 16730 | Any integer strictly betwe... |
| phival 16735 | Value of the Euler ` phi `... |
| phicl2 16736 | Bounds and closure for the... |
| phicl 16737 | Closure for the value of t... |
| phibndlem 16738 | Lemma for ~ phibnd . (Con... |
| phibnd 16739 | A slightly tighter bound o... |
| phicld 16740 | Closure for the value of t... |
| phi1 16741 | Value of the Euler ` phi `... |
| dfphi2 16742 | Alternate definition of th... |
| hashdvds 16743 | The number of numbers in a... |
| phiprmpw 16744 | Value of the Euler ` phi `... |
| phiprm 16745 | Value of the Euler ` phi `... |
| crth 16746 | The Chinese Remainder Theo... |
| phimullem 16747 | Lemma for ~ phimul . (Con... |
| phimul 16748 | The Euler ` phi ` function... |
| eulerthlem1 16749 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16750 | Lemma for ~ eulerth . (Co... |
| eulerth 16751 | Euler's theorem, a general... |
| fermltl 16752 | Fermat's little theorem. ... |
| prmdiv 16753 | Show an explicit expressio... |
| prmdiveq 16754 | The modular inverse of ` A... |
| prmdivdiv 16755 | The (modular) inverse of t... |
| hashgcdlem 16756 | A correspondence between e... |
| hashgcdeq 16757 | Number of initial positive... |
| phisum 16758 | The divisor sum identity o... |
| odzval 16759 | Value of the order functio... |
| odzcllem 16760 | - Lemma for ~ odzcl , show... |
| odzcl 16761 | The order of a group eleme... |
| odzid 16762 | Any element raised to the ... |
| odzdvds 16763 | The only powers of ` A ` t... |
| odzphi 16764 | The order of any group ele... |
| modprm1div 16765 | A prime number divides an ... |
| m1dvdsndvds 16766 | If an integer minus 1 is d... |
| modprminv 16767 | Show an explicit expressio... |
| modprminveq 16768 | The modular inverse of ` A... |
| vfermltl 16769 | Variant of Fermat's little... |
| vfermltlALT 16770 | Alternate proof of ~ vferm... |
| powm2modprm 16771 | If an integer minus 1 is d... |
| reumodprminv 16772 | For any prime number and f... |
| modprm0 16773 | For two positive integers ... |
| nnnn0modprm0 16774 | For a positive integer and... |
| modprmn0modprm0 16775 | For an integer not being 0... |
| coprimeprodsq 16776 | If three numbers are copri... |
| coprimeprodsq2 16777 | If three numbers are copri... |
| oddprm 16778 | A prime not equal to ` 2 `... |
| nnoddn2prm 16779 | A prime not equal to ` 2 `... |
| oddn2prm 16780 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16781 | A number is a prime number... |
| prm23lt5 16782 | A prime less than 5 is eit... |
| prm23ge5 16783 | A prime is either 2 or 3 o... |
| pythagtriplem1 16784 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16785 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16786 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16787 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16788 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16789 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16790 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16791 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16792 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16793 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16794 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16795 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16796 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16797 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16798 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16799 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16800 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16801 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16802 | Parameterize the Pythagore... |
| iserodd 16803 | Collect the odd terms in a... |
| pclem 16806 | - Lemma for the prime powe... |
| pcprecl 16807 | Closure of the prime power... |
| pcprendvds 16808 | Non-divisibility property ... |
| pcprendvds2 16809 | Non-divisibility property ... |
| pcpre1 16810 | Value of the prime power p... |
| pcpremul 16811 | Multiplicative property of... |
| pcval 16812 | The value of the prime pow... |
| pceulem 16813 | Lemma for ~ pceu . (Contr... |
| pceu 16814 | Uniqueness for the prime p... |
| pczpre 16815 | Connect the prime count pr... |
| pczcl 16816 | Closure of the prime power... |
| pccl 16817 | Closure of the prime power... |
| pccld 16818 | Closure of the prime power... |
| pcmul 16819 | Multiplication property of... |
| pcdiv 16820 | Division property of the p... |
| pcqmul 16821 | Multiplication property of... |
| pc0 16822 | The value of the prime pow... |
| pc1 16823 | Value of the prime count f... |
| pcqcl 16824 | Closure of the general pri... |
| pcqdiv 16825 | Division property of the p... |
| pcrec 16826 | Prime power of a reciproca... |
| pcexp 16827 | Prime power of an exponent... |
| pcxnn0cl 16828 | Extended nonnegative integ... |
| pcxcl 16829 | Extended real closure of t... |
| pcge0 16830 | The prime count of an inte... |
| pczdvds 16831 | Defining property of the p... |
| pcdvds 16832 | Defining property of the p... |
| pczndvds 16833 | Defining property of the p... |
| pcndvds 16834 | Defining property of the p... |
| pczndvds2 16835 | The remainder after dividi... |
| pcndvds2 16836 | The remainder after dividi... |
| pcdvdsb 16837 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16838 | There are a positive numbe... |
| pceq0 16839 | There are zero powers of a... |
| pcidlem 16840 | The prime count of a prime... |
| pcid 16841 | The prime count of a prime... |
| pcneg 16842 | The prime count of a negat... |
| pcabs 16843 | The prime count of an abso... |
| pcdvdstr 16844 | The prime count increases ... |
| pcgcd1 16845 | The prime count of a GCD i... |
| pcgcd 16846 | The prime count of a GCD i... |
| pc2dvds 16847 | A characterization of divi... |
| pc11 16848 | The prime count function, ... |
| pcz 16849 | The prime count function c... |
| pcprmpw2 16850 | Self-referential expressio... |
| pcprmpw 16851 | Self-referential expressio... |
| dvdsprmpweq 16852 | If a positive integer divi... |
| dvdsprmpweqnn 16853 | If an integer greater than... |
| dvdsprmpweqle 16854 | If a positive integer divi... |
| difsqpwdvds 16855 | If the difference of two s... |
| pcaddlem 16856 | Lemma for ~ pcadd . The o... |
| pcadd 16857 | An inequality for the prim... |
| pcadd2 16858 | The inequality of ~ pcadd ... |
| pcmptcl 16859 | Closure for the prime powe... |
| pcmpt 16860 | Construct a function with ... |
| pcmpt2 16861 | Dividing two prime count m... |
| pcmptdvds 16862 | The partial products of th... |
| pcprod 16863 | The product of the primes ... |
| sumhash 16864 | The sum of 1 over a set is... |
| fldivp1 16865 | The difference between the... |
| pcfaclem 16866 | Lemma for ~ pcfac . (Cont... |
| pcfac 16867 | Calculate the prime count ... |
| pcbc 16868 | Calculate the prime count ... |
| qexpz 16869 | If a power of a rational n... |
| expnprm 16870 | A second or higher power o... |
| oddprmdvds 16871 | Every positive integer whi... |
| prmpwdvds 16872 | A relation involving divis... |
| pockthlem 16873 | Lemma for ~ pockthg . (Co... |
| pockthg 16874 | The generalized Pocklingto... |
| pockthi 16875 | Pocklington's theorem, whi... |
| unbenlem 16876 | Lemma for ~ unben . (Cont... |
| unben 16877 | An unbounded set of positi... |
| infpnlem1 16878 | Lemma for ~ infpn . The s... |
| infpnlem2 16879 | Lemma for ~ infpn . For a... |
| infpn 16880 | There exist infinitely man... |
| infpn2 16881 | There exist infinitely man... |
| prmunb 16882 | The primes are unbounded. ... |
| prminf 16883 | There are an infinite numb... |
| prmreclem1 16884 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16885 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16886 | Lemma for ~ prmrec . The ... |
| prmreclem4 16887 | Lemma for ~ prmrec . Show... |
| prmreclem5 16888 | Lemma for ~ prmrec . Here... |
| prmreclem6 16889 | Lemma for ~ prmrec . If t... |
| prmrec 16890 | The sum of the reciprocals... |
| 1arithlem1 16891 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16892 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16893 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16894 | Lemma for ~ 1arith . (Con... |
| 1arith 16895 | Fundamental theorem of ari... |
| 1arith2 16896 | Fundamental theorem of ari... |
| elgz 16899 | Elementhood in the gaussia... |
| gzcn 16900 | A gaussian integer is a co... |
| zgz 16901 | An integer is a gaussian i... |
| igz 16902 | ` _i ` is a gaussian integ... |
| gznegcl 16903 | The gaussian integers are ... |
| gzcjcl 16904 | The gaussian integers are ... |
| gzaddcl 16905 | The gaussian integers are ... |
| gzmulcl 16906 | The gaussian integers are ... |
| gzreim 16907 | Construct a gaussian integ... |
| gzsubcl 16908 | The gaussian integers are ... |
| gzabssqcl 16909 | The squared norm of a gaus... |
| 4sqlem5 16910 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16911 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16912 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16913 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16914 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16915 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16916 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16917 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16918 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16919 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16920 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16921 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16922 | Euler's four-square identi... |
| 4sqlem11 16923 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16924 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16925 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16926 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16927 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16928 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16929 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16930 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16931 | Lemma for ~ 4sq . The pro... |
| 4sq 16932 | Lagrange's four-square the... |
| vdwapfval 16939 | Define the arithmetic prog... |
| vdwapf 16940 | The arithmetic progression... |
| vdwapval 16941 | Value of the arithmetic pr... |
| vdwapun 16942 | Remove the first element o... |
| vdwapid1 16943 | The first element of an ar... |
| vdwap0 16944 | Value of a length-1 arithm... |
| vdwap1 16945 | Value of a length-1 arithm... |
| vdwmc 16946 | The predicate " The ` <. R... |
| vdwmc2 16947 | Expand out the definition ... |
| vdwpc 16948 | The predicate " The colori... |
| vdwlem1 16949 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16950 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16951 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16952 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16953 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16954 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16955 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16956 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16957 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16958 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16959 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16960 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16961 | Lemma for ~ vdw . Main in... |
| vdw 16962 | Van der Waerden's theorem.... |
| vdwnnlem1 16963 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16964 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16965 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16966 | Van der Waerden's theorem,... |
| ramtlecl 16968 | The set ` T ` of numbers w... |
| hashbcval 16970 | Value of the "binomial set... |
| hashbccl 16971 | The binomial set is a fini... |
| hashbcss 16972 | Subset relation for the bi... |
| hashbc0 16973 | The set of subsets of size... |
| hashbc2 16974 | The size of the binomial s... |
| 0hashbc 16975 | There are no subsets of th... |
| ramval 16976 | The value of the Ramsey nu... |
| ramcl2lem 16977 | Lemma for extended real cl... |
| ramtcl 16978 | The Ramsey number has the ... |
| ramtcl2 16979 | The Ramsey number is an in... |
| ramtub 16980 | The Ramsey number is a low... |
| ramub 16981 | The Ramsey number is a low... |
| ramub2 16982 | It is sufficient to check ... |
| rami 16983 | The defining property of a... |
| ramcl2 16984 | The Ramsey number is eithe... |
| ramxrcl 16985 | The Ramsey number is an ex... |
| ramubcl 16986 | If the Ramsey number is up... |
| ramlb 16987 | Establish a lower bound on... |
| 0ram 16988 | The Ramsey number when ` M... |
| 0ram2 16989 | The Ramsey number when ` M... |
| ram0 16990 | The Ramsey number when ` R... |
| 0ramcl 16991 | Lemma for ~ ramcl : Exist... |
| ramz2 16992 | The Ramsey number when ` F... |
| ramz 16993 | The Ramsey number when ` F... |
| ramub1lem1 16994 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16995 | Lemma for ~ ramub1 . (Con... |
| ramub1 16996 | Inductive step for Ramsey'... |
| ramcl 16997 | Ramsey's theorem: the Rams... |
| ramsey 16998 | Ramsey's theorem with the ... |
| prmoval 17001 | Value of the primorial fun... |
| prmocl 17002 | Closure of the primorial f... |
| prmone0 17003 | The primorial function is ... |
| prmo0 17004 | The primorial of 0. (Cont... |
| prmo1 17005 | The primorial of 1. (Cont... |
| prmop1 17006 | The primorial of a success... |
| prmonn2 17007 | Value of the primorial fun... |
| prmo2 17008 | The primorial of 2. (Cont... |
| prmo3 17009 | The primorial of 3. (Cont... |
| prmdvdsprmo 17010 | The primorial of a number ... |
| prmdvdsprmop 17011 | The primorial of a number ... |
| fvprmselelfz 17012 | The value of the prime sel... |
| fvprmselgcd1 17013 | The greatest common diviso... |
| prmolefac 17014 | The primorial of a positiv... |
| prmodvdslcmf 17015 | The primorial of a nonnega... |
| prmolelcmf 17016 | The primorial of a positiv... |
| prmgaplem1 17017 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17018 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17019 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17020 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17021 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17022 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17023 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17024 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17025 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17026 | Lemma for ~ prmgap . (Con... |
| prmgap 17027 | The prime gap theorem: for... |
| prmgaplcm 17028 | Alternate proof of ~ prmga... |
| prmgapprmolem 17029 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17030 | Alternate proof of ~ prmga... |
| dec2dvds 17031 | Divisibility by two is obv... |
| dec5dvds 17032 | Divisibility by five is ob... |
| dec5dvds2 17033 | Divisibility by five is ob... |
| dec5nprm 17034 | Divisibility by five is ob... |
| dec2nprm 17035 | Divisibility by two is obv... |
| modxai 17036 | Add exponents in a power m... |
| mod2xi 17037 | Double exponents in a powe... |
| modxp1i 17038 | Add one to an exponent in ... |
| mod2xnegi 17039 | Version of ~ mod2xi with a... |
| modsubi 17040 | Subtract from within a mod... |
| gcdi 17041 | Calculate a GCD via Euclid... |
| gcdmodi 17042 | Calculate a GCD via Euclid... |
| decexp2 17043 | Calculate a power of two. ... |
| numexp0 17044 | Calculate an integer power... |
| numexp1 17045 | Calculate an integer power... |
| numexpp1 17046 | Calculate an integer power... |
| numexp2x 17047 | Double an integer power. ... |
| decsplit0b 17048 | Split a decimal number int... |
| decsplit0 17049 | Split a decimal number int... |
| decsplit1 17050 | Split a decimal number int... |
| decsplit 17051 | Split a decimal number int... |
| karatsuba 17052 | The Karatsuba multiplicati... |
| 2exp4 17053 | Two to the fourth power is... |
| 2exp5 17054 | Two to the fifth power is ... |
| 2exp6 17055 | Two to the sixth power is ... |
| 2exp7 17056 | Two to the seventh power i... |
| 2exp8 17057 | Two to the eighth power is... |
| 2exp11 17058 | Two to the eleventh power ... |
| 2exp16 17059 | Two to the sixteenth power... |
| 3exp3 17060 | Three to the third power i... |
| 2expltfac 17061 | The factorial grows faster... |
| cshwsidrepsw 17062 | If cyclically shifting a w... |
| cshwsidrepswmod0 17063 | If cyclically shifting a w... |
| cshwshashlem1 17064 | If cyclically shifting a w... |
| cshwshashlem2 17065 | If cyclically shifting a w... |
| cshwshashlem3 17066 | If cyclically shifting a w... |
| cshwsdisj 17067 | The singletons resulting b... |
| cshwsiun 17068 | The set of (different!) wo... |
| cshwsex 17069 | The class of (different!) ... |
| cshws0 17070 | The size of the set of (di... |
| cshwrepswhash1 17071 | The size of the set of (di... |
| cshwshashnsame 17072 | If a word (not consisting ... |
| cshwshash 17073 | If a word has a length bei... |
| prmlem0 17074 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17075 | A quick proof skeleton to ... |
| prmlem1 17076 | A quick proof skeleton to ... |
| 5prm 17077 | 5 is a prime number. (Con... |
| 6nprm 17078 | 6 is not a prime number. ... |
| 7prm 17079 | 7 is a prime number. (Con... |
| 8nprm 17080 | 8 is not a prime number. ... |
| 9nprm 17081 | 9 is not a prime number. ... |
| 10nprm 17082 | 10 is not a prime number. ... |
| 11prm 17083 | 11 is a prime number. (Co... |
| 13prm 17084 | 13 is a prime number. (Co... |
| 17prm 17085 | 17 is a prime number. (Co... |
| 19prm 17086 | 19 is a prime number. (Co... |
| 23prm 17087 | 23 is a prime number. (Co... |
| prmlem2 17088 | Our last proving session g... |
| 37prm 17089 | 37 is a prime number. (Co... |
| 43prm 17090 | 43 is a prime number. (Co... |
| 83prm 17091 | 83 is a prime number. (Co... |
| 139prm 17092 | 139 is a prime number. (C... |
| 163prm 17093 | 163 is a prime number. (C... |
| 317prm 17094 | 317 is a prime number. (C... |
| 631prm 17095 | 631 is a prime number. (C... |
| prmo4 17096 | The primorial of 4. (Cont... |
| prmo5 17097 | The primorial of 5. (Cont... |
| prmo6 17098 | The primorial of 6. (Cont... |
| 1259lem1 17099 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17100 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17101 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17102 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17103 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17104 | 1259 is a prime number. (... |
| 2503lem1 17105 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17106 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17107 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17108 | 2503 is a prime number. (... |
| 4001lem1 17109 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17110 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17111 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17112 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17113 | 4001 is a prime number. (... |
| brstruct 17116 | The structure relation is ... |
| isstruct2 17117 | The property of being a st... |
| structex 17118 | A structure is a set. (Co... |
| structn0fun 17119 | A structure without the em... |
| isstruct 17120 | The property of being a st... |
| structcnvcnv 17121 | Two ways to express the re... |
| structfung 17122 | The converse of the conver... |
| structfun 17123 | Convert between two kinds ... |
| structfn 17124 | Convert between two kinds ... |
| strleun 17125 | Combine two structures int... |
| strle1 17126 | Make a structure from a si... |
| strle2 17127 | Make a structure from a pa... |
| strle3 17128 | Make a structure from a tr... |
| sbcie2s 17129 | A special version of class... |
| sbcie3s 17130 | A special version of class... |
| reldmsets 17133 | The structure override ope... |
| setsvalg 17134 | Value of the structure rep... |
| setsval 17135 | Value of the structure rep... |
| fvsetsid 17136 | The value of the structure... |
| fsets 17137 | The structure replacement ... |
| setsdm 17138 | The domain of a structure ... |
| setsfun 17139 | A structure with replaceme... |
| setsfun0 17140 | A structure with replaceme... |
| setsn0fun 17141 | The value of the structure... |
| setsstruct2 17142 | An extensible structure wi... |
| setsexstruct2 17143 | An extensible structure wi... |
| setsstruct 17144 | An extensible structure wi... |
| wunsets 17145 | Closure of structure repla... |
| setsres 17146 | The structure replacement ... |
| setsabs 17147 | Replacing the same compone... |
| setscom 17148 | Different components can b... |
| sloteq 17151 | Equality theorem for the `... |
| slotfn 17152 | A slot is a function on se... |
| strfvnd 17153 | Deduction version of ~ str... |
| strfvn 17154 | Value of a structure compo... |
| strfvss 17155 | A structure component extr... |
| wunstr 17156 | Closure of a structure ind... |
| str0 17157 | All components of the empt... |
| strfvi 17158 | Structure slot extractors ... |
| fveqprc 17159 | Lemma for showing the equa... |
| oveqprc 17160 | Lemma for showing the equa... |
| wunndx 17163 | Closure of the index extra... |
| ndxarg 17164 | Get the numeric argument f... |
| ndxid 17165 | A structure component extr... |
| strndxid 17166 | The value of a structure c... |
| setsidvald 17167 | Value of the structure rep... |
| setsidvaldOLD 17168 | Obsolete version of ~ sets... |
| strfvd 17169 | Deduction version of ~ str... |
| strfv2d 17170 | Deduction version of ~ str... |
| strfv2 17171 | A variation on ~ strfv to ... |
| strfv 17172 | Extract a structure compon... |
| strfv3 17173 | Variant on ~ strfv for lar... |
| strssd 17174 | Deduction version of ~ str... |
| strss 17175 | Propagate component extrac... |
| setsid 17176 | Value of the structure rep... |
| setsnid 17177 | Value of the structure rep... |
| setsnidOLD 17178 | Obsolete proof of ~ setsni... |
| baseval 17181 | Value of the base set extr... |
| baseid 17182 | Utility theorem: index-ind... |
| basfn 17183 | The base set extractor is ... |
| base0 17184 | The base set of the empty ... |
| elbasfv 17185 | Utility theorem: reverse c... |
| elbasov 17186 | Utility theorem: reverse c... |
| strov2rcl 17187 | Partial reverse closure fo... |
| basendx 17188 | Index value of the base se... |
| basendxnn 17189 | The index value of the bas... |
| basendxnnOLD 17190 | Obsolete proof of ~ basend... |
| basndxelwund 17191 | The index of the base set ... |
| basprssdmsets 17192 | The pair of the base index... |
| opelstrbas 17193 | The base set of a structur... |
| 1strstr 17194 | A constructed one-slot str... |
| 1strstr1 17195 | A constructed one-slot str... |
| 1strbas 17196 | The base set of a construc... |
| 1strbasOLD 17197 | Obsolete proof of ~ 1strba... |
| 1strwunbndx 17198 | A constructed one-slot str... |
| 1strwun 17199 | A constructed one-slot str... |
| 1strwunOLD 17200 | Obsolete version of ~ 1str... |
| 2strstr 17201 | A constructed two-slot str... |
| 2strbas 17202 | The base set of a construc... |
| 2strop 17203 | The other slot of a constr... |
| 2strstr1 17204 | A constructed two-slot str... |
| 2strstr1OLD 17205 | Obsolete version of ~ 2str... |
| 2strbas1 17206 | The base set of a construc... |
| 2strop1 17207 | The other slot of a constr... |
| reldmress 17210 | The structure restriction ... |
| ressval 17211 | Value of structure restric... |
| ressid2 17212 | General behavior of trivia... |
| ressval2 17213 | Value of nontrivial struct... |
| ressbas 17214 | Base set of a structure re... |
| ressbasOLD 17215 | Obsolete proof of ~ ressba... |
| ressbasssg 17216 | The base set of a restrict... |
| ressbas2 17217 | Base set of a structure re... |
| ressbasss 17218 | The base set of a restrict... |
| ressbasssOLD 17219 | Obsolete proof of ~ ressba... |
| ressbasss2 17220 | The base set of a restrict... |
| resseqnbas 17221 | The components of an exten... |
| resslemOLD 17222 | Obsolete version of ~ ress... |
| ress0 17223 | All restrictions of the nu... |
| ressid 17224 | Behavior of trivial restri... |
| ressinbas 17225 | Restriction only cares abo... |
| ressval3d 17226 | Value of structure restric... |
| ressval3dOLD 17227 | Obsolete version of ~ ress... |
| ressress 17228 | Restriction composition la... |
| ressabs 17229 | Restriction absorption law... |
| wunress 17230 | Closure of structure restr... |
| wunressOLD 17231 | Obsolete proof of ~ wunres... |
| plusgndx 17258 | Index value of the ~ df-pl... |
| plusgid 17259 | Utility theorem: index-ind... |
| plusgndxnn 17260 | The index of the slot for ... |
| basendxltplusgndx 17261 | The index of the slot for ... |
| basendxnplusgndx 17262 | The slot for the base set ... |
| basendxnplusgndxOLD 17263 | Obsolete version of ~ base... |
| grpstr 17264 | A constructed group is a s... |
| grpstrndx 17265 | A constructed group is a s... |
| grpbase 17266 | The base set of a construc... |
| grpbaseOLD 17267 | Obsolete version of ~ grpb... |
| grpplusg 17268 | The operation of a constru... |
| grpplusgOLD 17269 | Obsolete version of ~ grpp... |
| ressplusg 17270 | ` +g ` is unaffected by re... |
| grpbasex 17271 | The base of an explicitly ... |
| grpplusgx 17272 | The operation of an explic... |
| mulrndx 17273 | Index value of the ~ df-mu... |
| mulridx 17274 | Utility theorem: index-ind... |
| basendxnmulrndx 17275 | The slot for the base set ... |
| basendxnmulrndxOLD 17276 | Obsolete proof of ~ basend... |
| plusgndxnmulrndx 17277 | The slot for the group (ad... |
| rngstr 17278 | A constructed ring is a st... |
| rngbase 17279 | The base set of a construc... |
| rngplusg 17280 | The additive operation of ... |
| rngmulr 17281 | The multiplicative operati... |
| starvndx 17282 | Index value of the ~ df-st... |
| starvid 17283 | Utility theorem: index-ind... |
| starvndxnbasendx 17284 | The slot for the involutio... |
| starvndxnplusgndx 17285 | The slot for the involutio... |
| starvndxnmulrndx 17286 | The slot for the involutio... |
| ressmulr 17287 | ` .r ` is unaffected by re... |
| ressstarv 17288 | ` *r ` is unaffected by re... |
| srngstr 17289 | A constructed star ring is... |
| srngbase 17290 | The base set of a construc... |
| srngplusg 17291 | The addition operation of ... |
| srngmulr 17292 | The multiplication operati... |
| srnginvl 17293 | The involution function of... |
| scandx 17294 | Index value of the ~ df-sc... |
| scaid 17295 | Utility theorem: index-ind... |
| scandxnbasendx 17296 | The slot for the scalar is... |
| scandxnplusgndx 17297 | The slot for the scalar fi... |
| scandxnmulrndx 17298 | The slot for the scalar fi... |
| vscandx 17299 | Index value of the ~ df-vs... |
| vscaid 17300 | Utility theorem: index-ind... |
| vscandxnbasendx 17301 | The slot for the scalar pr... |
| vscandxnplusgndx 17302 | The slot for the scalar pr... |
| vscandxnmulrndx 17303 | The slot for the scalar pr... |
| vscandxnscandx 17304 | The slot for the scalar pr... |
| lmodstr 17305 | A constructed left module ... |
| lmodbase 17306 | The base set of a construc... |
| lmodplusg 17307 | The additive operation of ... |
| lmodsca 17308 | The set of scalars of a co... |
| lmodvsca 17309 | The scalar product operati... |
| ipndx 17310 | Index value of the ~ df-ip... |
| ipid 17311 | Utility theorem: index-ind... |
| ipndxnbasendx 17312 | The slot for the inner pro... |
| ipndxnplusgndx 17313 | The slot for the inner pro... |
| ipndxnmulrndx 17314 | The slot for the inner pro... |
| slotsdifipndx 17315 | The slot for the scalar is... |
| ipsstr 17316 | Lemma to shorten proofs of... |
| ipsbase 17317 | The base set of a construc... |
| ipsaddg 17318 | The additive operation of ... |
| ipsmulr 17319 | The multiplicative operati... |
| ipssca 17320 | The set of scalars of a co... |
| ipsvsca 17321 | The scalar product operati... |
| ipsip 17322 | The multiplicative operati... |
| resssca 17323 | ` Scalar ` is unaffected b... |
| ressvsca 17324 | ` .s ` is unaffected by re... |
| ressip 17325 | The inner product is unaff... |
| phlstr 17326 | A constructed pre-Hilbert ... |
| phlbase 17327 | The base set of a construc... |
| phlplusg 17328 | The additive operation of ... |
| phlsca 17329 | The ring of scalars of a c... |
| phlvsca 17330 | The scalar product operati... |
| phlip 17331 | The inner product (Hermiti... |
| tsetndx 17332 | Index value of the ~ df-ts... |
| tsetid 17333 | Utility theorem: index-ind... |
| tsetndxnn 17334 | The index of the slot for ... |
| basendxlttsetndx 17335 | The index of the slot for ... |
| tsetndxnbasendx 17336 | The slot for the topology ... |
| tsetndxnplusgndx 17337 | The slot for the topology ... |
| tsetndxnmulrndx 17338 | The slot for the topology ... |
| tsetndxnstarvndx 17339 | The slot for the topology ... |
| slotstnscsi 17340 | The slots ` Scalar ` , ` .... |
| topgrpstr 17341 | A constructed topological ... |
| topgrpbas 17342 | The base set of a construc... |
| topgrpplusg 17343 | The additive operation of ... |
| topgrptset 17344 | The topology of a construc... |
| resstset 17345 | ` TopSet ` is unaffected b... |
| plendx 17346 | Index value of the ~ df-pl... |
| pleid 17347 | Utility theorem: self-refe... |
| plendxnn 17348 | The index value of the ord... |
| basendxltplendx 17349 | The index value of the ` B... |
| plendxnbasendx 17350 | The slot for the order is ... |
| plendxnplusgndx 17351 | The slot for the "less tha... |
| plendxnmulrndx 17352 | The slot for the "less tha... |
| plendxnscandx 17353 | The slot for the "less tha... |
| plendxnvscandx 17354 | The slot for the "less tha... |
| slotsdifplendx 17355 | The index of the slot for ... |
| otpsstr 17356 | Functionality of a topolog... |
| otpsbas 17357 | The base set of a topologi... |
| otpstset 17358 | The open sets of a topolog... |
| otpsle 17359 | The order of a topological... |
| ressle 17360 | ` le ` is unaffected by re... |
| ocndx 17361 | Index value of the ~ df-oc... |
| ocid 17362 | Utility theorem: index-ind... |
| basendxnocndx 17363 | The slot for the orthocomp... |
| plendxnocndx 17364 | The slot for the orthocomp... |
| dsndx 17365 | Index value of the ~ df-ds... |
| dsid 17366 | Utility theorem: index-ind... |
| dsndxnn 17367 | The index of the slot for ... |
| basendxltdsndx 17368 | The index of the slot for ... |
| dsndxnbasendx 17369 | The slot for the distance ... |
| dsndxnplusgndx 17370 | The slot for the distance ... |
| dsndxnmulrndx 17371 | The slot for the distance ... |
| slotsdnscsi 17372 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17373 | The slot for the distance ... |
| slotsdifdsndx 17374 | The index of the slot for ... |
| unifndx 17375 | Index value of the ~ df-un... |
| unifid 17376 | Utility theorem: index-ind... |
| unifndxnn 17377 | The index of the slot for ... |
| basendxltunifndx 17378 | The index of the slot for ... |
| unifndxnbasendx 17379 | The slot for the uniform s... |
| unifndxntsetndx 17380 | The slot for the uniform s... |
| slotsdifunifndx 17381 | The index of the slot for ... |
| ressunif 17382 | ` UnifSet ` is unaffected ... |
| odrngstr 17383 | Functionality of an ordere... |
| odrngbas 17384 | The base set of an ordered... |
| odrngplusg 17385 | The addition operation of ... |
| odrngmulr 17386 | The multiplication operati... |
| odrngtset 17387 | The open sets of an ordere... |
| odrngle 17388 | The order of an ordered me... |
| odrngds 17389 | The metric of an ordered m... |
| ressds 17390 | ` dist ` is unaffected by ... |
| homndx 17391 | Index value of the ~ df-ho... |
| homid 17392 | Utility theorem: index-ind... |
| ccondx 17393 | Index value of the ~ df-cc... |
| ccoid 17394 | Utility theorem: index-ind... |
| slotsbhcdif 17395 | The slots ` Base ` , ` Hom... |
| slotsbhcdifOLD 17396 | Obsolete proof of ~ slotsb... |
| slotsdifplendx2 17397 | The index of the slot for ... |
| slotsdifocndx 17398 | The index of the slot for ... |
| resshom 17399 | ` Hom ` is unaffected by r... |
| ressco 17400 | ` comp ` is unaffected by ... |
| restfn 17405 | The subspace topology oper... |
| topnfn 17406 | The topology extractor fun... |
| restval 17407 | The subspace topology indu... |
| elrest 17408 | The predicate "is an open ... |
| elrestr 17409 | Sufficient condition for b... |
| 0rest 17410 | Value of the structure res... |
| restid2 17411 | The subspace topology over... |
| restsspw 17412 | The subspace topology is a... |
| firest 17413 | The finite intersections o... |
| restid 17414 | The subspace topology of t... |
| topnval 17415 | Value of the topology extr... |
| topnid 17416 | Value of the topology extr... |
| topnpropd 17417 | The topology extractor fun... |
| reldmprds 17429 | The structure product is a... |
| prdsbasex 17431 | Lemma for structure produc... |
| imasvalstr 17432 | An image structure value i... |
| prdsvalstr 17433 | Structure product value is... |
| prdsbaslem 17434 | Lemma for ~ prdsbas and si... |
| prdsvallem 17435 | Lemma for ~ prdsval . (Co... |
| prdsval 17436 | Value of the structure pro... |
| prdssca 17437 | Scalar ring of a structure... |
| prdsbas 17438 | Base set of a structure pr... |
| prdsplusg 17439 | Addition in a structure pr... |
| prdsmulr 17440 | Multiplication in a struct... |
| prdsvsca 17441 | Scalar multiplication in a... |
| prdsip 17442 | Inner product in a structu... |
| prdsle 17443 | Structure product weak ord... |
| prdsless 17444 | Closure of the order relat... |
| prdsds 17445 | Structure product distance... |
| prdsdsfn 17446 | Structure product distance... |
| prdstset 17447 | Structure product topology... |
| prdshom 17448 | Structure product hom-sets... |
| prdsco 17449 | Structure product composit... |
| prdsbas2 17450 | The base set of a structur... |
| prdsbasmpt 17451 | A constructed tuple is a p... |
| prdsbasfn 17452 | Points in the structure pr... |
| prdsbasprj 17453 | Each point in a structure ... |
| prdsplusgval 17454 | Value of a componentwise s... |
| prdsplusgfval 17455 | Value of a structure produ... |
| prdsmulrval 17456 | Value of a componentwise r... |
| prdsmulrfval 17457 | Value of a structure produ... |
| prdsleval 17458 | Value of the product order... |
| prdsdsval 17459 | Value of the metric in a s... |
| prdsvscaval 17460 | Scalar multiplication in a... |
| prdsvscafval 17461 | Scalar multiplication of a... |
| prdsbas3 17462 | The base set of an indexed... |
| prdsbasmpt2 17463 | A constructed tuple is a p... |
| prdsbascl 17464 | An element of the base has... |
| prdsdsval2 17465 | Value of the metric in a s... |
| prdsdsval3 17466 | Value of the metric in a s... |
| pwsval 17467 | Value of a structure power... |
| pwsbas 17468 | Base set of a structure po... |
| pwselbasb 17469 | Membership in the base set... |
| pwselbas 17470 | An element of a structure ... |
| pwsplusgval 17471 | Value of addition in a str... |
| pwsmulrval 17472 | Value of multiplication in... |
| pwsle 17473 | Ordering in a structure po... |
| pwsleval 17474 | Ordering in a structure po... |
| pwsvscafval 17475 | Scalar multiplication in a... |
| pwsvscaval 17476 | Scalar multiplication of a... |
| pwssca 17477 | The ring of scalars of a s... |
| pwsdiagel 17478 | Membership of diagonal ele... |
| pwssnf1o 17479 | Triviality of singleton po... |
| imasval 17492 | Value of an image structur... |
| imasbas 17493 | The base set of an image s... |
| imasds 17494 | The distance function of a... |
| imasdsfn 17495 | The distance function is a... |
| imasdsval 17496 | The distance function of a... |
| imasdsval2 17497 | The distance function of a... |
| imasplusg 17498 | The group operation in an ... |
| imasmulr 17499 | The ring multiplication in... |
| imassca 17500 | The scalar field of an ima... |
| imasvsca 17501 | The scalar multiplication ... |
| imasip 17502 | The inner product of an im... |
| imastset 17503 | The topology of an image s... |
| imasle 17504 | The ordering of an image s... |
| f1ocpbllem 17505 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17506 | An injection is compatible... |
| f1ovscpbl 17507 | An injection is compatible... |
| f1olecpbl 17508 | An injection is compatible... |
| imasaddfnlem 17509 | The image structure operat... |
| imasaddvallem 17510 | The operation of an image ... |
| imasaddflem 17511 | The image set operations a... |
| imasaddfn 17512 | The image structure's grou... |
| imasaddval 17513 | The value of an image stru... |
| imasaddf 17514 | The image structure's grou... |
| imasmulfn 17515 | The image structure's ring... |
| imasmulval 17516 | The value of an image stru... |
| imasmulf 17517 | The image structure's ring... |
| imasvscafn 17518 | The image structure's scal... |
| imasvscaval 17519 | The value of an image stru... |
| imasvscaf 17520 | The image structure's scal... |
| imasless 17521 | The order relation defined... |
| imasleval 17522 | The value of the image str... |
| qusval 17523 | Value of a quotient struct... |
| quslem 17524 | The function in ~ qusval i... |
| qusin 17525 | Restrict the equivalence r... |
| qusbas 17526 | Base set of a quotient str... |
| quss 17527 | The scalar field of a quot... |
| divsfval 17528 | Value of the function in ~... |
| ercpbllem 17529 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17530 | Translate the function com... |
| erlecpbl 17531 | Translate the relation com... |
| qusaddvallem 17532 | Value of an operation defi... |
| qusaddflem 17533 | The operation of a quotien... |
| qusaddval 17534 | The addition in a quotient... |
| qusaddf 17535 | The addition in a quotient... |
| qusmulval 17536 | The multiplication in a qu... |
| qusmulf 17537 | The multiplication in a qu... |
| fnpr2o 17538 | Function with a domain of ... |
| fnpr2ob 17539 | Biconditional version of ~... |
| fvpr0o 17540 | The value of a function wi... |
| fvpr1o 17541 | The value of a function wi... |
| fvprif 17542 | The value of the pair func... |
| xpsfrnel 17543 | Elementhood in the target ... |
| xpsfeq 17544 | A function on ` 2o ` is de... |
| xpsfrnel2 17545 | Elementhood in the target ... |
| xpscf 17546 | Equivalent condition for t... |
| xpsfval 17547 | The value of the function ... |
| xpsff1o 17548 | The function appearing in ... |
| xpsfrn 17549 | A short expression for the... |
| xpsff1o2 17550 | The function appearing in ... |
| xpsval 17551 | Value of the binary struct... |
| xpsrnbas 17552 | The indexed structure prod... |
| xpsbas 17553 | The base set of the binary... |
| xpsaddlem 17554 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17555 | Value of the addition oper... |
| xpsmul 17556 | Value of the multiplicatio... |
| xpssca 17557 | Value of the scalar field ... |
| xpsvsca 17558 | Value of the scalar multip... |
| xpsless 17559 | Closure of the ordering in... |
| xpsle 17560 | Value of the ordering in a... |
| ismre 17569 | Property of being a Moore ... |
| fnmre 17570 | The Moore collection gener... |
| mresspw 17571 | A Moore collection is a su... |
| mress 17572 | A Moore-closed subset is a... |
| mre1cl 17573 | In any Moore collection th... |
| mreintcl 17574 | A nonempty collection of c... |
| mreiincl 17575 | A nonempty indexed interse... |
| mrerintcl 17576 | The relative intersection ... |
| mreriincl 17577 | The relative intersection ... |
| mreincl 17578 | Two closed sets have a clo... |
| mreuni 17579 | Since the entire base set ... |
| mreunirn 17580 | Two ways to express the no... |
| ismred 17581 | Properties that determine ... |
| ismred2 17582 | Properties that determine ... |
| mremre 17583 | The Moore collections of s... |
| submre 17584 | The subcollection of a clo... |
| mrcflem 17585 | The domain and codomain of... |
| fnmrc 17586 | Moore-closure is a well-be... |
| mrcfval 17587 | Value of the function expr... |
| mrcf 17588 | The Moore closure is a fun... |
| mrcval 17589 | Evaluation of the Moore cl... |
| mrccl 17590 | The Moore closure of a set... |
| mrcsncl 17591 | The Moore closure of a sin... |
| mrcid 17592 | The closure of a closed se... |
| mrcssv 17593 | The closure of a set is a ... |
| mrcidb 17594 | A set is closed iff it is ... |
| mrcss 17595 | Closure preserves subset o... |
| mrcssid 17596 | The closure of a set is a ... |
| mrcidb2 17597 | A set is closed iff it con... |
| mrcidm 17598 | The closure operation is i... |
| mrcsscl 17599 | The closure is the minimal... |
| mrcuni 17600 | Idempotence of closure und... |
| mrcun 17601 | Idempotence of closure und... |
| mrcssvd 17602 | The Moore closure of a set... |
| mrcssd 17603 | Moore closure preserves su... |
| mrcssidd 17604 | A set is contained in its ... |
| mrcidmd 17605 | Moore closure is idempoten... |
| mressmrcd 17606 | In a Moore system, if a se... |
| submrc 17607 | In a closure system which ... |
| mrieqvlemd 17608 | In a Moore system, if ` Y ... |
| mrisval 17609 | Value of the set of indepe... |
| ismri 17610 | Criterion for a set to be ... |
| ismri2 17611 | Criterion for a subset of ... |
| ismri2d 17612 | Criterion for a subset of ... |
| ismri2dd 17613 | Definition of independence... |
| mriss 17614 | An independent set of a Mo... |
| mrissd 17615 | An independent set of a Mo... |
| ismri2dad 17616 | Consequence of a set in a ... |
| mrieqvd 17617 | In a Moore system, a set i... |
| mrieqv2d 17618 | In a Moore system, a set i... |
| mrissmrcd 17619 | In a Moore system, if an i... |
| mrissmrid 17620 | In a Moore system, subsets... |
| mreexd 17621 | In a Moore system, the clo... |
| mreexmrid 17622 | In a Moore system whose cl... |
| mreexexlemd 17623 | This lemma is used to gene... |
| mreexexlem2d 17624 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17625 | Base case of the induction... |
| mreexexlem4d 17626 | Induction step of the indu... |
| mreexexd 17627 | Exchange-type theorem. In... |
| mreexdomd 17628 | In a Moore system whose cl... |
| mreexfidimd 17629 | In a Moore system whose cl... |
| isacs 17630 | A set is an algebraic clos... |
| acsmre 17631 | Algebraic closure systems ... |
| isacs2 17632 | In the definition of an al... |
| acsfiel 17633 | A set is closed in an alge... |
| acsfiel2 17634 | A set is closed in an alge... |
| acsmred 17635 | An algebraic closure syste... |
| isacs1i 17636 | A closure system determine... |
| mreacs 17637 | Algebraicity is a composab... |
| acsfn 17638 | Algebraicity of a conditio... |
| acsfn0 17639 | Algebraicity of a point cl... |
| acsfn1 17640 | Algebraicity of a one-argu... |
| acsfn1c 17641 | Algebraicity of a one-argu... |
| acsfn2 17642 | Algebraicity of a two-argu... |
| iscat 17651 | The predicate "is a catego... |
| iscatd 17652 | Properties that determine ... |
| catidex 17653 | Each object in a category ... |
| catideu 17654 | Each object in a category ... |
| cidfval 17655 | Each object in a category ... |
| cidval 17656 | Each object in a category ... |
| cidffn 17657 | The identity arrow constru... |
| cidfn 17658 | The identity arrow operato... |
| catidd 17659 | Deduce the identity arrow ... |
| iscatd2 17660 | Version of ~ iscatd with a... |
| catidcl 17661 | Each object in a category ... |
| catlid 17662 | Left identity property of ... |
| catrid 17663 | Right identity property of... |
| catcocl 17664 | Closure of a composition a... |
| catass 17665 | Associativity of compositi... |
| catcone0 17666 | Composition of non-empty h... |
| 0catg 17667 | Any structure with an empt... |
| 0cat 17668 | The empty set is a categor... |
| homffval 17669 | Value of the functionalize... |
| fnhomeqhomf 17670 | If the Hom-set operation i... |
| homfval 17671 | Value of the functionalize... |
| homffn 17672 | The functionalized Hom-set... |
| homfeq 17673 | Condition for two categori... |
| homfeqd 17674 | If two structures have the... |
| homfeqbas 17675 | Deduce equality of base se... |
| homfeqval 17676 | Value of the functionalize... |
| comfffval 17677 | Value of the functionalize... |
| comffval 17678 | Value of the functionalize... |
| comfval 17679 | Value of the functionalize... |
| comfffval2 17680 | Value of the functionalize... |
| comffval2 17681 | Value of the functionalize... |
| comfval2 17682 | Value of the functionalize... |
| comfffn 17683 | The functionalized composi... |
| comffn 17684 | The functionalized composi... |
| comfeq 17685 | Condition for two categori... |
| comfeqd 17686 | Condition for two categori... |
| comfeqval 17687 | Equality of two compositio... |
| catpropd 17688 | Two structures with the sa... |
| cidpropd 17689 | Two structures with the sa... |
| oppcval 17692 | Value of the opposite cate... |
| oppchomfval 17693 | Hom-sets of the opposite c... |
| oppchomfvalOLD 17694 | Obsolete proof of ~ oppcho... |
| oppchom 17695 | Hom-sets of the opposite c... |
| oppccofval 17696 | Composition in the opposit... |
| oppcco 17697 | Composition in the opposit... |
| oppcbas 17698 | Base set of an opposite ca... |
| oppcbasOLD 17699 | Obsolete version of ~ oppc... |
| oppccatid 17700 | Lemma for ~ oppccat . (Co... |
| oppchomf 17701 | Hom-sets of the opposite c... |
| oppcid 17702 | Identity function of an op... |
| oppccat 17703 | An opposite category is a ... |
| 2oppcbas 17704 | The double opposite catego... |
| 2oppchomf 17705 | The double opposite catego... |
| 2oppccomf 17706 | The double opposite catego... |
| oppchomfpropd 17707 | If two categories have the... |
| oppccomfpropd 17708 | If two categories have the... |
| oppccatf 17709 | ` oppCat ` restricted to `... |
| monfval 17714 | Definition of a monomorphi... |
| ismon 17715 | Definition of a monomorphi... |
| ismon2 17716 | Write out the monomorphism... |
| monhom 17717 | A monomorphism is a morphi... |
| moni 17718 | Property of a monomorphism... |
| monpropd 17719 | If two categories have the... |
| oppcmon 17720 | A monomorphism in the oppo... |
| oppcepi 17721 | An epimorphism in the oppo... |
| isepi 17722 | Definition of an epimorphi... |
| isepi2 17723 | Write out the epimorphism ... |
| epihom 17724 | An epimorphism is a morphi... |
| epii 17725 | Property of an epimorphism... |
| sectffval 17732 | Value of the section opera... |
| sectfval 17733 | Value of the section relat... |
| sectss 17734 | The section relation is a ... |
| issect 17735 | The property " ` F ` is a ... |
| issect2 17736 | Property of being a sectio... |
| sectcan 17737 | If ` G ` is a section of `... |
| sectco 17738 | Composition of two section... |
| isofval 17739 | Function value of the func... |
| invffval 17740 | Value of the inverse relat... |
| invfval 17741 | Value of the inverse relat... |
| isinv 17742 | Value of the inverse relat... |
| invss 17743 | The inverse relation is a ... |
| invsym 17744 | The inverse relation is sy... |
| invsym2 17745 | The inverse relation is sy... |
| invfun 17746 | The inverse relation is a ... |
| isoval 17747 | The isomorphisms are the d... |
| inviso1 17748 | If ` G ` is an inverse to ... |
| inviso2 17749 | If ` G ` is an inverse to ... |
| invf 17750 | The inverse relation is a ... |
| invf1o 17751 | The inverse relation is a ... |
| invinv 17752 | The inverse of the inverse... |
| invco 17753 | The composition of two iso... |
| dfiso2 17754 | Alternate definition of an... |
| dfiso3 17755 | Alternate definition of an... |
| inveq 17756 | If there are two inverses ... |
| isofn 17757 | The function value of the ... |
| isohom 17758 | An isomorphism is a homomo... |
| isoco 17759 | The composition of two iso... |
| oppcsect 17760 | A section in the opposite ... |
| oppcsect2 17761 | A section in the opposite ... |
| oppcinv 17762 | An inverse in the opposite... |
| oppciso 17763 | An isomorphism in the oppo... |
| sectmon 17764 | If ` F ` is a section of `... |
| monsect 17765 | If ` F ` is a monomorphism... |
| sectepi 17766 | If ` F ` is a section of `... |
| episect 17767 | If ` F ` is an epimorphism... |
| sectid 17768 | The identity is a section ... |
| invid 17769 | The inverse of the identit... |
| idiso 17770 | The identity is an isomorp... |
| idinv 17771 | The inverse of the identit... |
| invisoinvl 17772 | The inverse of an isomorph... |
| invisoinvr 17773 | The inverse of an isomorph... |
| invcoisoid 17774 | The inverse of an isomorph... |
| isocoinvid 17775 | The inverse of an isomorph... |
| rcaninv 17776 | Right cancellation of an i... |
| cicfval 17779 | The set of isomorphic obje... |
| brcic 17780 | The relation "is isomorphi... |
| cic 17781 | Objects ` X ` and ` Y ` in... |
| brcici 17782 | Prove that two objects are... |
| cicref 17783 | Isomorphism is reflexive. ... |
| ciclcl 17784 | Isomorphism implies the le... |
| cicrcl 17785 | Isomorphism implies the ri... |
| cicsym 17786 | Isomorphism is symmetric. ... |
| cictr 17787 | Isomorphism is transitive.... |
| cicer 17788 | Isomorphism is an equivale... |
| sscrel 17795 | The subcategory subset rel... |
| brssc 17796 | The subcategory subset rel... |
| sscpwex 17797 | An analogue of ~ pwex for ... |
| subcrcl 17798 | Reverse closure for the su... |
| sscfn1 17799 | The subcategory subset rel... |
| sscfn2 17800 | The subcategory subset rel... |
| ssclem 17801 | Lemma for ~ ssc1 and simil... |
| isssc 17802 | Value of the subcategory s... |
| ssc1 17803 | Infer subset relation on o... |
| ssc2 17804 | Infer subset relation on m... |
| sscres 17805 | Any function restricted to... |
| sscid 17806 | The subcategory subset rel... |
| ssctr 17807 | The subcategory subset rel... |
| ssceq 17808 | The subcategory subset rel... |
| rescval 17809 | Value of the category rest... |
| rescval2 17810 | Value of the category rest... |
| rescbas 17811 | Base set of the category r... |
| rescbasOLD 17812 | Obsolete version of ~ resc... |
| reschom 17813 | Hom-sets of the category r... |
| reschomf 17814 | Hom-sets of the category r... |
| rescco 17815 | Composition in the categor... |
| resccoOLD 17816 | Obsolete proof of ~ rescco... |
| rescabs 17817 | Restriction absorption law... |
| rescabsOLD 17818 | Obsolete proof of ~ seqp1d... |
| rescabs2 17819 | Restriction absorption law... |
| issubc 17820 | Elementhood in the set of ... |
| issubc2 17821 | Elementhood in the set of ... |
| 0ssc 17822 | For any category ` C ` , t... |
| 0subcat 17823 | For any category ` C ` , t... |
| catsubcat 17824 | For any category ` C ` , `... |
| subcssc 17825 | An element in the set of s... |
| subcfn 17826 | An element in the set of s... |
| subcss1 17827 | The objects of a subcatego... |
| subcss2 17828 | The morphisms of a subcate... |
| subcidcl 17829 | The identity of the origin... |
| subccocl 17830 | A subcategory is closed un... |
| subccatid 17831 | A subcategory is a categor... |
| subcid 17832 | The identity in a subcateg... |
| subccat 17833 | A subcategory is a categor... |
| issubc3 17834 | Alternate definition of a ... |
| fullsubc 17835 | The full subcategory gener... |
| fullresc 17836 | The category formed by str... |
| resscat 17837 | A category restricted to a... |
| subsubc 17838 | A subcategory of a subcate... |
| relfunc 17847 | The set of functors is a r... |
| funcrcl 17848 | Reverse closure for a func... |
| isfunc 17849 | Value of the set of functo... |
| isfuncd 17850 | Deduce that an operation i... |
| funcf1 17851 | The object part of a funct... |
| funcixp 17852 | The morphism part of a fun... |
| funcf2 17853 | The morphism part of a fun... |
| funcfn2 17854 | The morphism part of a fun... |
| funcid 17855 | A functor maps each identi... |
| funcco 17856 | A functor maps composition... |
| funcsect 17857 | The image of a section und... |
| funcinv 17858 | The image of an inverse un... |
| funciso 17859 | The image of an isomorphis... |
| funcoppc 17860 | A functor on categories yi... |
| idfuval 17861 | Value of the identity func... |
| idfu2nd 17862 | Value of the morphism part... |
| idfu2 17863 | Value of the morphism part... |
| idfu1st 17864 | Value of the object part o... |
| idfu1 17865 | Value of the object part o... |
| idfucl 17866 | The identity functor is a ... |
| cofuval 17867 | Value of the composition o... |
| cofu1st 17868 | Value of the object part o... |
| cofu1 17869 | Value of the object part o... |
| cofu2nd 17870 | Value of the morphism part... |
| cofu2 17871 | Value of the morphism part... |
| cofuval2 17872 | Value of the composition o... |
| cofucl 17873 | The composition of two fun... |
| cofuass 17874 | Functor composition is ass... |
| cofulid 17875 | The identity functor is a ... |
| cofurid 17876 | The identity functor is a ... |
| resfval 17877 | Value of the functor restr... |
| resfval2 17878 | Value of the functor restr... |
| resf1st 17879 | Value of the functor restr... |
| resf2nd 17880 | Value of the functor restr... |
| funcres 17881 | A functor restricted to a ... |
| funcres2b 17882 | Condition for a functor to... |
| funcres2 17883 | A functor into a restricte... |
| idfusubc0 17884 | The identity functor for a... |
| idfusubc 17885 | The identity functor for a... |
| wunfunc 17886 | A weak universe is closed ... |
| wunfuncOLD 17887 | Obsolete proof of ~ wunfun... |
| funcpropd 17888 | If two categories have the... |
| funcres2c 17889 | Condition for a functor to... |
| fullfunc 17894 | A full functor is a functo... |
| fthfunc 17895 | A faithful functor is a fu... |
| relfull 17896 | The set of full functors i... |
| relfth 17897 | The set of faithful functo... |
| isfull 17898 | Value of the set of full f... |
| isfull2 17899 | Equivalent condition for a... |
| fullfo 17900 | The morphism map of a full... |
| fulli 17901 | The morphism map of a full... |
| isfth 17902 | Value of the set of faithf... |
| isfth2 17903 | Equivalent condition for a... |
| isffth2 17904 | A fully faithful functor i... |
| fthf1 17905 | The morphism map of a fait... |
| fthi 17906 | The morphism map of a fait... |
| ffthf1o 17907 | The morphism map of a full... |
| fullpropd 17908 | If two categories have the... |
| fthpropd 17909 | If two categories have the... |
| fulloppc 17910 | The opposite functor of a ... |
| fthoppc 17911 | The opposite functor of a ... |
| ffthoppc 17912 | The opposite functor of a ... |
| fthsect 17913 | A faithful functor reflect... |
| fthinv 17914 | A faithful functor reflect... |
| fthmon 17915 | A faithful functor reflect... |
| fthepi 17916 | A faithful functor reflect... |
| ffthiso 17917 | A fully faithful functor r... |
| fthres2b 17918 | Condition for a faithful f... |
| fthres2c 17919 | Condition for a faithful f... |
| fthres2 17920 | A faithful functor into a ... |
| idffth 17921 | The identity functor is a ... |
| cofull 17922 | The composition of two ful... |
| cofth 17923 | The composition of two fai... |
| coffth 17924 | The composition of two ful... |
| rescfth 17925 | The inclusion functor from... |
| ressffth 17926 | The inclusion functor from... |
| fullres2c 17927 | Condition for a full funct... |
| ffthres2c 17928 | Condition for a fully fait... |
| inclfusubc 17929 | The "inclusion functor" fr... |
| fnfuc 17934 | The ` FuncCat ` operation ... |
| natfval 17935 | Value of the function givi... |
| isnat 17936 | Property of being a natura... |
| isnat2 17937 | Property of being a natura... |
| natffn 17938 | The natural transformation... |
| natrcl 17939 | Reverse closure for a natu... |
| nat1st2nd 17940 | Rewrite the natural transf... |
| natixp 17941 | A natural transformation i... |
| natcl 17942 | A component of a natural t... |
| natfn 17943 | A natural transformation i... |
| nati 17944 | Naturality property of a n... |
| wunnat 17945 | A weak universe is closed ... |
| wunnatOLD 17946 | Obsolete proof of ~ wunnat... |
| catstr 17947 | A category structure is a ... |
| fucval 17948 | Value of the functor categ... |
| fuccofval 17949 | Value of the functor categ... |
| fucbas 17950 | The objects of the functor... |
| fuchom 17951 | The morphisms in the funct... |
| fuchomOLD 17952 | Obsolete proof of ~ fuchom... |
| fucco 17953 | Value of the composition o... |
| fuccoval 17954 | Value of the functor categ... |
| fuccocl 17955 | The composition of two nat... |
| fucidcl 17956 | The identity natural trans... |
| fuclid 17957 | Left identity of natural t... |
| fucrid 17958 | Right identity of natural ... |
| fucass 17959 | Associativity of natural t... |
| fuccatid 17960 | The functor category is a ... |
| fuccat 17961 | The functor category is a ... |
| fucid 17962 | The identity morphism in t... |
| fucsect 17963 | Two natural transformation... |
| fucinv 17964 | Two natural transformation... |
| invfuc 17965 | If ` V ( x ) ` is an inver... |
| fuciso 17966 | A natural transformation i... |
| natpropd 17967 | If two categories have the... |
| fucpropd 17968 | If two categories have the... |
| initofn 17975 | ` InitO ` is a function on... |
| termofn 17976 | ` TermO ` is a function on... |
| zeroofn 17977 | ` ZeroO ` is a function on... |
| initorcl 17978 | Reverse closure for an ini... |
| termorcl 17979 | Reverse closure for a term... |
| zeroorcl 17980 | Reverse closure for a zero... |
| initoval 17981 | The value of the initial o... |
| termoval 17982 | The value of the terminal ... |
| zerooval 17983 | The value of the zero obje... |
| isinito 17984 | The predicate "is an initi... |
| istermo 17985 | The predicate "is a termin... |
| iszeroo 17986 | The predicate "is a zero o... |
| isinitoi 17987 | Implication of a class bei... |
| istermoi 17988 | Implication of a class bei... |
| initoid 17989 | For an initial object, the... |
| termoid 17990 | For a terminal object, the... |
| dfinito2 17991 | An initial object is a ter... |
| dftermo2 17992 | A terminal object is an in... |
| dfinito3 17993 | An alternate definition of... |
| dftermo3 17994 | An alternate definition of... |
| initoo 17995 | An initial object is an ob... |
| termoo 17996 | A terminal object is an ob... |
| iszeroi 17997 | Implication of a class bei... |
| 2initoinv 17998 | Morphisms between two init... |
| initoeu1 17999 | Initial objects are essent... |
| initoeu1w 18000 | Initial objects are essent... |
| initoeu2lem0 18001 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 18002 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 18003 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 18004 | Initial objects are essent... |
| 2termoinv 18005 | Morphisms between two term... |
| termoeu1 18006 | Terminal objects are essen... |
| termoeu1w 18007 | Terminal objects are essen... |
| homarcl 18016 | Reverse closure for an arr... |
| homafval 18017 | Value of the disjointified... |
| homaf 18018 | Functionality of the disjo... |
| homaval 18019 | Value of the disjointified... |
| elhoma 18020 | Value of the disjointified... |
| elhomai 18021 | Produce an arrow from a mo... |
| elhomai2 18022 | Produce an arrow from a mo... |
| homarcl2 18023 | Reverse closure for the do... |
| homarel 18024 | An arrow is an ordered pai... |
| homa1 18025 | The first component of an ... |
| homahom2 18026 | The second component of an... |
| homahom 18027 | The second component of an... |
| homadm 18028 | The domain of an arrow wit... |
| homacd 18029 | The codomain of an arrow w... |
| homadmcd 18030 | Decompose an arrow into do... |
| arwval 18031 | The set of arrows is the u... |
| arwrcl 18032 | The first component of an ... |
| arwhoma 18033 | An arrow is contained in t... |
| homarw 18034 | A hom-set is a subset of t... |
| arwdm 18035 | The domain of an arrow is ... |
| arwcd 18036 | The codomain of an arrow i... |
| dmaf 18037 | The domain function is a f... |
| cdaf 18038 | The codomain function is a... |
| arwhom 18039 | The second component of an... |
| arwdmcd 18040 | Decompose an arrow into do... |
| idafval 18045 | Value of the identity arro... |
| idaval 18046 | Value of the identity arro... |
| ida2 18047 | Morphism part of the ident... |
| idahom 18048 | Domain and codomain of the... |
| idadm 18049 | Domain of the identity arr... |
| idacd 18050 | Codomain of the identity a... |
| idaf 18051 | The identity arrow functio... |
| coafval 18052 | The value of the compositi... |
| eldmcoa 18053 | A pair ` <. G , F >. ` is ... |
| dmcoass 18054 | The domain of composition ... |
| homdmcoa 18055 | If ` F : X --> Y ` and ` G... |
| coaval 18056 | Value of composition for c... |
| coa2 18057 | The morphism part of arrow... |
| coahom 18058 | The composition of two com... |
| coapm 18059 | Composition of arrows is a... |
| arwlid 18060 | Left identity of a categor... |
| arwrid 18061 | Right identity of a catego... |
| arwass 18062 | Associativity of compositi... |
| setcval 18065 | Value of the category of s... |
| setcbas 18066 | Set of objects of the cate... |
| setchomfval 18067 | Set of arrows of the categ... |
| setchom 18068 | Set of arrows of the categ... |
| elsetchom 18069 | A morphism of sets is a fu... |
| setccofval 18070 | Composition in the categor... |
| setcco 18071 | Composition in the categor... |
| setccatid 18072 | Lemma for ~ setccat . (Co... |
| setccat 18073 | The category of sets is a ... |
| setcid 18074 | The identity arrow in the ... |
| setcmon 18075 | A monomorphism of sets is ... |
| setcepi 18076 | An epimorphism of sets is ... |
| setcsect 18077 | A section in the category ... |
| setcinv 18078 | An inverse in the category... |
| setciso 18079 | An isomorphism in the cate... |
| resssetc 18080 | The restriction of the cat... |
| funcsetcres2 18081 | A functor into a smaller c... |
| setc2obas 18082 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18083 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18084 | The category of sets in a ... |
| cat1 18085 | The definition of category... |
| catcval 18088 | Value of the category of c... |
| catcbas 18089 | Set of objects of the cate... |
| catchomfval 18090 | Set of arrows of the categ... |
| catchom 18091 | Set of arrows of the categ... |
| catccofval 18092 | Composition in the categor... |
| catcco 18093 | Composition in the categor... |
| catccatid 18094 | Lemma for ~ catccat . (Co... |
| catcid 18095 | The identity arrow in the ... |
| catccat 18096 | The category of categories... |
| resscatc 18097 | The restriction of the cat... |
| catcisolem 18098 | Lemma for ~ catciso . (Co... |
| catciso 18099 | A functor is an isomorphis... |
| catcbascl 18100 | An element of the base set... |
| catcslotelcl 18101 | A slot entry of an element... |
| catcbaselcl 18102 | The base set of an element... |
| catchomcl 18103 | The Hom-set of an element ... |
| catcccocl 18104 | The composition operation ... |
| catcoppccl 18105 | The category of categories... |
| catcoppcclOLD 18106 | Obsolete proof of ~ catcop... |
| catcfuccl 18107 | The category of categories... |
| catcfucclOLD 18108 | Obsolete proof of ~ catcfu... |
| fncnvimaeqv 18109 | The inverse images of the ... |
| bascnvimaeqv 18110 | The inverse image of the u... |
| estrcval 18113 | Value of the category of e... |
| estrcbas 18114 | Set of objects of the cate... |
| estrchomfval 18115 | Set of morphisms ("arrows"... |
| estrchom 18116 | The morphisms between exte... |
| elestrchom 18117 | A morphism between extensi... |
| estrccofval 18118 | Composition in the categor... |
| estrcco 18119 | Composition in the categor... |
| estrcbasbas 18120 | An element of the base set... |
| estrccatid 18121 | Lemma for ~ estrccat . (C... |
| estrccat 18122 | The category of extensible... |
| estrcid 18123 | The identity arrow in the ... |
| estrchomfn 18124 | The Hom-set operation in t... |
| estrchomfeqhom 18125 | The functionalized Hom-set... |
| estrreslem1 18126 | Lemma 1 for ~ estrres . (... |
| estrreslem1OLD 18127 | Obsolete version of ~ estr... |
| estrreslem2 18128 | Lemma 2 for ~ estrres . (... |
| estrres 18129 | Any restriction of a categ... |
| funcestrcsetclem1 18130 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18131 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18132 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18133 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18134 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18135 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18136 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18137 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18138 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18139 | The "natural forgetful fun... |
| fthestrcsetc 18140 | The "natural forgetful fun... |
| fullestrcsetc 18141 | The "natural forgetful fun... |
| equivestrcsetc 18142 | The "natural forgetful fun... |
| setc1strwun 18143 | A constructed one-slot str... |
| funcsetcestrclem1 18144 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18145 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18146 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18147 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18148 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18149 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18150 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18151 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18152 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18153 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18154 | The "embedding functor" fr... |
| fthsetcestrc 18155 | The "embedding functor" fr... |
| fullsetcestrc 18156 | The "embedding functor" fr... |
| embedsetcestrc 18157 | The "embedding functor" fr... |
| fnxpc 18166 | The binary product of cate... |
| xpcval 18167 | Value of the binary produc... |
| xpcbas 18168 | Set of objects of the bina... |
| xpchomfval 18169 | Set of morphisms of the bi... |
| xpchom 18170 | Set of morphisms of the bi... |
| relxpchom 18171 | A hom-set in the binary pr... |
| xpccofval 18172 | Value of composition in th... |
| xpcco 18173 | Value of composition in th... |
| xpcco1st 18174 | Value of composition in th... |
| xpcco2nd 18175 | Value of composition in th... |
| xpchom2 18176 | Value of the set of morphi... |
| xpcco2 18177 | Value of composition in th... |
| xpccatid 18178 | The product of two categor... |
| xpcid 18179 | The identity morphism in t... |
| xpccat 18180 | The product of two categor... |
| 1stfval 18181 | Value of the first project... |
| 1stf1 18182 | Value of the first project... |
| 1stf2 18183 | Value of the first project... |
| 2ndfval 18184 | Value of the first project... |
| 2ndf1 18185 | Value of the first project... |
| 2ndf2 18186 | Value of the first project... |
| 1stfcl 18187 | The first projection funct... |
| 2ndfcl 18188 | The second projection func... |
| prfval 18189 | Value of the pairing funct... |
| prf1 18190 | Value of the pairing funct... |
| prf2fval 18191 | Value of the pairing funct... |
| prf2 18192 | Value of the pairing funct... |
| prfcl 18193 | The pairing of functors ` ... |
| prf1st 18194 | Cancellation of pairing wi... |
| prf2nd 18195 | Cancellation of pairing wi... |
| 1st2ndprf 18196 | Break a functor into a pro... |
| catcxpccl 18197 | The category of categories... |
| catcxpcclOLD 18198 | Obsolete proof of ~ catcxp... |
| xpcpropd 18199 | If two categories have the... |
| evlfval 18208 | Value of the evaluation fu... |
| evlf2 18209 | Value of the evaluation fu... |
| evlf2val 18210 | Value of the evaluation na... |
| evlf1 18211 | Value of the evaluation fu... |
| evlfcllem 18212 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18213 | The evaluation functor is ... |
| curfval 18214 | Value of the curry functor... |
| curf1fval 18215 | Value of the object part o... |
| curf1 18216 | Value of the object part o... |
| curf11 18217 | Value of the double evalua... |
| curf12 18218 | The partially evaluated cu... |
| curf1cl 18219 | The partially evaluated cu... |
| curf2 18220 | Value of the curry functor... |
| curf2val 18221 | Value of a component of th... |
| curf2cl 18222 | The curry functor at a mor... |
| curfcl 18223 | The curry functor of a fun... |
| curfpropd 18224 | If two categories have the... |
| uncfval 18225 | Value of the uncurry funct... |
| uncfcl 18226 | The uncurry operation take... |
| uncf1 18227 | Value of the uncurry funct... |
| uncf2 18228 | Value of the uncurry funct... |
| curfuncf 18229 | Cancellation of curry with... |
| uncfcurf 18230 | Cancellation of uncurry wi... |
| diagval 18231 | Define the diagonal functo... |
| diagcl 18232 | The diagonal functor is a ... |
| diag1cl 18233 | The constant functor of ` ... |
| diag11 18234 | Value of the constant func... |
| diag12 18235 | Value of the constant func... |
| diag2 18236 | Value of the diagonal func... |
| diag2cl 18237 | The diagonal functor at a ... |
| curf2ndf 18238 | As shown in ~ diagval , th... |
| hofval 18243 | Value of the Hom functor, ... |
| hof1fval 18244 | The object part of the Hom... |
| hof1 18245 | The object part of the Hom... |
| hof2fval 18246 | The morphism part of the H... |
| hof2val 18247 | The morphism part of the H... |
| hof2 18248 | The morphism part of the H... |
| hofcllem 18249 | Lemma for ~ hofcl . (Cont... |
| hofcl 18250 | Closure of the Hom functor... |
| oppchofcl 18251 | Closure of the opposite Ho... |
| yonval 18252 | Value of the Yoneda embedd... |
| yoncl 18253 | The Yoneda embedding is a ... |
| yon1cl 18254 | The Yoneda embedding at an... |
| yon11 18255 | Value of the Yoneda embedd... |
| yon12 18256 | Value of the Yoneda embedd... |
| yon2 18257 | Value of the Yoneda embedd... |
| hofpropd 18258 | If two categories have the... |
| yonpropd 18259 | If two categories have the... |
| oppcyon 18260 | Value of the opposite Yone... |
| oyoncl 18261 | The opposite Yoneda embedd... |
| oyon1cl 18262 | The opposite Yoneda embedd... |
| yonedalem1 18263 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18264 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18265 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18266 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18267 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18268 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18269 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18270 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18271 | Lemma for ~ yoneda . (Con... |
| yonedainv 18272 | The Yoneda Lemma with expl... |
| yonffthlem 18273 | Lemma for ~ yonffth . (Co... |
| yoneda 18274 | The Yoneda Lemma. There i... |
| yonffth 18275 | The Yoneda Lemma. The Yon... |
| yoniso 18276 | If the codomain is recover... |
| oduval 18279 | Value of an order dual str... |
| oduleval 18280 | Value of the less-equal re... |
| oduleg 18281 | Truth of the less-equal re... |
| odubas 18282 | Base set of an order dual ... |
| odubasOLD 18283 | Obsolete proof of ~ odubas... |
| isprs 18288 | Property of being a preord... |
| prslem 18289 | Lemma for ~ prsref and ~ p... |
| prsref 18290 | "Less than or equal to" is... |
| prstr 18291 | "Less than or equal to" is... |
| isdrs 18292 | Property of being a direct... |
| drsdir 18293 | Direction of a directed se... |
| drsprs 18294 | A directed set is a proset... |
| drsbn0 18295 | The base of a directed set... |
| drsdirfi 18296 | Any _finite_ number of ele... |
| isdrs2 18297 | Directed sets may be defin... |
| ispos 18305 | The predicate "is a poset"... |
| ispos2 18306 | A poset is an antisymmetri... |
| posprs 18307 | A poset is a proset. (Con... |
| posi 18308 | Lemma for poset properties... |
| posref 18309 | A poset ordering is reflex... |
| posasymb 18310 | A poset ordering is asymme... |
| postr 18311 | A poset ordering is transi... |
| 0pos 18312 | Technical lemma to simplif... |
| 0posOLD 18313 | Obsolete proof of ~ 0pos a... |
| isposd 18314 | Properties that determine ... |
| isposi 18315 | Properties that determine ... |
| isposix 18316 | Properties that determine ... |
| isposixOLD 18317 | Obsolete proof of ~ isposi... |
| pospropd 18318 | Posethood is determined on... |
| odupos 18319 | Being a poset is a self-du... |
| oduposb 18320 | Being a poset is a self-du... |
| pltfval 18322 | Value of the less-than rel... |
| pltval 18323 | Less-than relation. ( ~ d... |
| pltle 18324 | "Less than" implies "less ... |
| pltne 18325 | The "less than" relation i... |
| pltirr 18326 | The "less than" relation i... |
| pleval2i 18327 | One direction of ~ pleval2... |
| pleval2 18328 | "Less than or equal to" in... |
| pltnle 18329 | "Less than" implies not co... |
| pltval3 18330 | Alternate expression for t... |
| pltnlt 18331 | The less-than relation imp... |
| pltn2lp 18332 | The less-than relation has... |
| plttr 18333 | The less-than relation is ... |
| pltletr 18334 | Transitive law for chained... |
| plelttr 18335 | Transitive law for chained... |
| pospo 18336 | Write a poset structure in... |
| lubfval 18341 | Value of the least upper b... |
| lubdm 18342 | Domain of the least upper ... |
| lubfun 18343 | The LUB is a function. (C... |
| lubeldm 18344 | Member of the domain of th... |
| lubelss 18345 | A member of the domain of ... |
| lubeu 18346 | Unique existence proper of... |
| lubval 18347 | Value of the least upper b... |
| lubcl 18348 | The least upper bound func... |
| lubprop 18349 | Properties of greatest low... |
| luble 18350 | The greatest lower bound i... |
| lublecllem 18351 | Lemma for ~ lublecl and ~ ... |
| lublecl 18352 | The set of all elements le... |
| lubid 18353 | The LUB of elements less t... |
| glbfval 18354 | Value of the greatest lowe... |
| glbdm 18355 | Domain of the greatest low... |
| glbfun 18356 | The GLB is a function. (C... |
| glbeldm 18357 | Member of the domain of th... |
| glbelss 18358 | A member of the domain of ... |
| glbeu 18359 | Unique existence proper of... |
| glbval 18360 | Value of the greatest lowe... |
| glbcl 18361 | The least upper bound func... |
| glbprop 18362 | Properties of greatest low... |
| glble 18363 | The greatest lower bound i... |
| joinfval 18364 | Value of join function for... |
| joinfval2 18365 | Value of join function for... |
| joindm 18366 | Domain of join function fo... |
| joindef 18367 | Two ways to say that a joi... |
| joinval 18368 | Join value. Since both si... |
| joincl 18369 | Closure of join of element... |
| joindmss 18370 | Subset property of domain ... |
| joinval2lem 18371 | Lemma for ~ joinval2 and ~... |
| joinval2 18372 | Value of join for a poset ... |
| joineu 18373 | Uniqueness of join of elem... |
| joinlem 18374 | Lemma for join properties.... |
| lejoin1 18375 | A join's first argument is... |
| lejoin2 18376 | A join's second argument i... |
| joinle 18377 | A join is less than or equ... |
| meetfval 18378 | Value of meet function for... |
| meetfval2 18379 | Value of meet function for... |
| meetdm 18380 | Domain of meet function fo... |
| meetdef 18381 | Two ways to say that a mee... |
| meetval 18382 | Meet value. Since both si... |
| meetcl 18383 | Closure of meet of element... |
| meetdmss 18384 | Subset property of domain ... |
| meetval2lem 18385 | Lemma for ~ meetval2 and ~... |
| meetval2 18386 | Value of meet for a poset ... |
| meeteu 18387 | Uniqueness of meet of elem... |
| meetlem 18388 | Lemma for meet properties.... |
| lemeet1 18389 | A meet's first argument is... |
| lemeet2 18390 | A meet's second argument i... |
| meetle 18391 | A meet is less than or equ... |
| joincomALT 18392 | The join of a poset is com... |
| joincom 18393 | The join of a poset is com... |
| meetcomALT 18394 | The meet of a poset is com... |
| meetcom 18395 | The meet of a poset is com... |
| join0 18396 | Lemma for ~ odumeet . (Co... |
| meet0 18397 | Lemma for ~ odujoin . (Co... |
| odulub 18398 | Least upper bounds in a du... |
| odujoin 18399 | Joins in a dual order are ... |
| oduglb 18400 | Greatest lower bounds in a... |
| odumeet 18401 | Meets in a dual order are ... |
| poslubmo 18402 | Least upper bounds in a po... |
| posglbmo 18403 | Greatest lower bounds in a... |
| poslubd 18404 | Properties which determine... |
| poslubdg 18405 | Properties which determine... |
| posglbdg 18406 | Properties which determine... |
| istos 18409 | The predicate "is a toset"... |
| tosso 18410 | Write the totally ordered ... |
| tospos 18411 | A Toset is a Poset. (Cont... |
| tleile 18412 | In a Toset, any two elemen... |
| tltnle 18413 | In a Toset, "less than" is... |
| p0val 18418 | Value of poset zero. (Con... |
| p1val 18419 | Value of poset zero. (Con... |
| p0le 18420 | Any element is less than o... |
| ple1 18421 | Any element is less than o... |
| islat 18424 | The predicate "is a lattic... |
| odulatb 18425 | Being a lattice is self-du... |
| odulat 18426 | Being a lattice is self-du... |
| latcl2 18427 | The join and meet of any t... |
| latlem 18428 | Lemma for lattice properti... |
| latpos 18429 | A lattice is a poset. (Co... |
| latjcl 18430 | Closure of join operation ... |
| latmcl 18431 | Closure of meet operation ... |
| latref 18432 | A lattice ordering is refl... |
| latasymb 18433 | A lattice ordering is asym... |
| latasym 18434 | A lattice ordering is asym... |
| lattr 18435 | A lattice ordering is tran... |
| latasymd 18436 | Deduce equality from latti... |
| lattrd 18437 | A lattice ordering is tran... |
| latjcom 18438 | The join of a lattice comm... |
| latlej1 18439 | A join's first argument is... |
| latlej2 18440 | A join's second argument i... |
| latjle12 18441 | A join is less than or equ... |
| latleeqj1 18442 | "Less than or equal to" in... |
| latleeqj2 18443 | "Less than or equal to" in... |
| latjlej1 18444 | Add join to both sides of ... |
| latjlej2 18445 | Add join to both sides of ... |
| latjlej12 18446 | Add join to both sides of ... |
| latnlej 18447 | An idiom to express that a... |
| latnlej1l 18448 | An idiom to express that a... |
| latnlej1r 18449 | An idiom to express that a... |
| latnlej2 18450 | An idiom to express that a... |
| latnlej2l 18451 | An idiom to express that a... |
| latnlej2r 18452 | An idiom to express that a... |
| latjidm 18453 | Lattice join is idempotent... |
| latmcom 18454 | The join of a lattice comm... |
| latmle1 18455 | A meet is less than or equ... |
| latmle2 18456 | A meet is less than or equ... |
| latlem12 18457 | An element is less than or... |
| latleeqm1 18458 | "Less than or equal to" in... |
| latleeqm2 18459 | "Less than or equal to" in... |
| latmlem1 18460 | Add meet to both sides of ... |
| latmlem2 18461 | Add meet to both sides of ... |
| latmlem12 18462 | Add join to both sides of ... |
| latnlemlt 18463 | Negation of "less than or ... |
| latnle 18464 | Equivalent expressions for... |
| latmidm 18465 | Lattice meet is idempotent... |
| latabs1 18466 | Lattice absorption law. F... |
| latabs2 18467 | Lattice absorption law. F... |
| latledi 18468 | An ortholattice is distrib... |
| latmlej11 18469 | Ordering of a meet and joi... |
| latmlej12 18470 | Ordering of a meet and joi... |
| latmlej21 18471 | Ordering of a meet and joi... |
| latmlej22 18472 | Ordering of a meet and joi... |
| lubsn 18473 | The least upper bound of a... |
| latjass 18474 | Lattice join is associativ... |
| latj12 18475 | Swap 1st and 2nd members o... |
| latj32 18476 | Swap 2nd and 3rd members o... |
| latj13 18477 | Swap 1st and 3rd members o... |
| latj31 18478 | Swap 2nd and 3rd members o... |
| latjrot 18479 | Rotate lattice join of 3 c... |
| latj4 18480 | Rearrangement of lattice j... |
| latj4rot 18481 | Rotate lattice join of 4 c... |
| latjjdi 18482 | Lattice join distributes o... |
| latjjdir 18483 | Lattice join distributes o... |
| mod1ile 18484 | The weak direction of the ... |
| mod2ile 18485 | The weak direction of the ... |
| latmass 18486 | Lattice meet is associativ... |
| latdisdlem 18487 | Lemma for ~ latdisd . (Co... |
| latdisd 18488 | In a lattice, joins distri... |
| isclat 18491 | The predicate "is a comple... |
| clatpos 18492 | A complete lattice is a po... |
| clatlem 18493 | Lemma for properties of a ... |
| clatlubcl 18494 | Any subset of the base set... |
| clatlubcl2 18495 | Any subset of the base set... |
| clatglbcl 18496 | Any subset of the base set... |
| clatglbcl2 18497 | Any subset of the base set... |
| oduclatb 18498 | Being a complete lattice i... |
| clatl 18499 | A complete lattice is a la... |
| isglbd 18500 | Properties that determine ... |
| lublem 18501 | Lemma for the least upper ... |
| lubub 18502 | The LUB of a complete latt... |
| lubl 18503 | The LUB of a complete latt... |
| lubss 18504 | Subset law for least upper... |
| lubel 18505 | An element of a set is les... |
| lubun 18506 | The LUB of a union. (Cont... |
| clatglb 18507 | Properties of greatest low... |
| clatglble 18508 | The greatest lower bound i... |
| clatleglb 18509 | Two ways of expressing "le... |
| clatglbss 18510 | Subset law for greatest lo... |
| isdlat 18513 | Property of being a distri... |
| dlatmjdi 18514 | In a distributive lattice,... |
| dlatl 18515 | A distributive lattice is ... |
| odudlatb 18516 | The dual of a distributive... |
| dlatjmdi 18517 | In a distributive lattice,... |
| ipostr 18520 | The structure of ~ df-ipo ... |
| ipoval 18521 | Value of the inclusion pos... |
| ipobas 18522 | Base set of the inclusion ... |
| ipolerval 18523 | Relation of the inclusion ... |
| ipotset 18524 | Topology of the inclusion ... |
| ipole 18525 | Weak order condition of th... |
| ipolt 18526 | Strict order condition of ... |
| ipopos 18527 | The inclusion poset on a f... |
| isipodrs 18528 | Condition for a family of ... |
| ipodrscl 18529 | Direction by inclusion as ... |
| ipodrsfi 18530 | Finite upper bound propert... |
| fpwipodrs 18531 | The finite subsets of any ... |
| ipodrsima 18532 | The monotone image of a di... |
| isacs3lem 18533 | An algebraic closure syste... |
| acsdrsel 18534 | An algebraic closure syste... |
| isacs4lem 18535 | In a closure system in whi... |
| isacs5lem 18536 | If closure commutes with d... |
| acsdrscl 18537 | In an algebraic closure sy... |
| acsficl 18538 | A closure in an algebraic ... |
| isacs5 18539 | A closure system is algebr... |
| isacs4 18540 | A closure system is algebr... |
| isacs3 18541 | A closure system is algebr... |
| acsficld 18542 | In an algebraic closure sy... |
| acsficl2d 18543 | In an algebraic closure sy... |
| acsfiindd 18544 | In an algebraic closure sy... |
| acsmapd 18545 | In an algebraic closure sy... |
| acsmap2d 18546 | In an algebraic closure sy... |
| acsinfd 18547 | In an algebraic closure sy... |
| acsdomd 18548 | In an algebraic closure sy... |
| acsinfdimd 18549 | In an algebraic closure sy... |
| acsexdimd 18550 | In an algebraic closure sy... |
| mrelatglb 18551 | Greatest lower bounds in a... |
| mrelatglb0 18552 | The empty intersection in ... |
| mrelatlub 18553 | Least upper bounds in a Mo... |
| mreclatBAD 18554 | A Moore space is a complet... |
| isps 18559 | The predicate "is a poset"... |
| psrel 18560 | A poset is a relation. (C... |
| psref2 18561 | A poset is antisymmetric a... |
| pstr2 18562 | A poset is transitive. (C... |
| pslem 18563 | Lemma for ~ psref and othe... |
| psdmrn 18564 | The domain and range of a ... |
| psref 18565 | A poset is reflexive. (Co... |
| psrn 18566 | The range of a poset equal... |
| psasym 18567 | A poset is antisymmetric. ... |
| pstr 18568 | A poset is transitive. (C... |
| cnvps 18569 | The converse of a poset is... |
| cnvpsb 18570 | The converse of a poset is... |
| psss 18571 | Any subset of a partially ... |
| psssdm2 18572 | Field of a subposet. (Con... |
| psssdm 18573 | Field of a subposet. (Con... |
| istsr 18574 | The predicate is a toset. ... |
| istsr2 18575 | The predicate is a toset. ... |
| tsrlin 18576 | A toset is a linear order.... |
| tsrlemax 18577 | Two ways of saying a numbe... |
| tsrps 18578 | A toset is a poset. (Cont... |
| cnvtsr 18579 | The converse of a toset is... |
| tsrss 18580 | Any subset of a totally or... |
| ledm 18581 | The domain of ` <_ ` is ` ... |
| lern 18582 | The range of ` <_ ` is ` R... |
| lefld 18583 | The field of the 'less or ... |
| letsr 18584 | The "less than or equal to... |
| isdir 18589 | A condition for a relation... |
| reldir 18590 | A direction is a relation.... |
| dirdm 18591 | A direction's domain is eq... |
| dirref 18592 | A direction is reflexive. ... |
| dirtr 18593 | A direction is transitive.... |
| dirge 18594 | For any two elements of a ... |
| tsrdir 18595 | A totally ordered set is a... |
| ismgm 18600 | The predicate "is a magma"... |
| ismgmn0 18601 | The predicate "is a magma"... |
| mgmcl 18602 | Closure of the operation o... |
| isnmgm 18603 | A condition for a structur... |
| mgmsscl 18604 | If the base set of a magma... |
| plusffval 18605 | The group addition operati... |
| plusfval 18606 | The group addition operati... |
| plusfeq 18607 | If the addition operation ... |
| plusffn 18608 | The group addition operati... |
| mgmplusf 18609 | The group addition functio... |
| mgmpropd 18610 | If two structures have the... |
| ismgmd 18611 | Deduce a magma from its pr... |
| issstrmgm 18612 | Characterize a substructur... |
| intopsn 18613 | The internal operation for... |
| mgmb1mgm1 18614 | The only magma with a base... |
| mgm0 18615 | Any set with an empty base... |
| mgm0b 18616 | The structure with an empt... |
| mgm1 18617 | The structure with one ele... |
| opifismgm 18618 | A structure with a group a... |
| mgmidmo 18619 | A two-sided identity eleme... |
| grpidval 18620 | The value of the identity ... |
| grpidpropd 18621 | If two structures have the... |
| fn0g 18622 | The group zero extractor i... |
| 0g0 18623 | The identity element funct... |
| ismgmid 18624 | The identity element of a ... |
| mgmidcl 18625 | The identity element of a ... |
| mgmlrid 18626 | The identity element of a ... |
| ismgmid2 18627 | Show that a given element ... |
| lidrideqd 18628 | If there is a left and rig... |
| lidrididd 18629 | If there is a left and rig... |
| grpidd 18630 | Deduce the identity elemen... |
| mgmidsssn0 18631 | Property of the set of ide... |
| grpinvalem 18632 | Lemma for ~ grpinva . (Co... |
| grpinva 18633 | Deduce right inverse from ... |
| grprida 18634 | Deduce right identity from... |
| gsumvalx 18635 | Expand out the substitutio... |
| gsumval 18636 | Expand out the substitutio... |
| gsumpropd 18637 | The group sum depends only... |
| gsumpropd2lem 18638 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18639 | A stronger version of ~ gs... |
| gsummgmpropd 18640 | A stronger version of ~ gs... |
| gsumress 18641 | The group sum in a substru... |
| gsumval1 18642 | Value of the group sum ope... |
| gsum0 18643 | Value of the empty group s... |
| gsumval2a 18644 | Value of the group sum ope... |
| gsumval2 18645 | Value of the group sum ope... |
| gsumsplit1r 18646 | Splitting off the rightmos... |
| gsumprval 18647 | Value of the group sum ope... |
| gsumpr12val 18648 | Value of the group sum ope... |
| mgmhmrcl 18653 | Reverse closure of a magma... |
| submgmrcl 18654 | Reverse closure for submag... |
| ismgmhm 18655 | Property of a magma homomo... |
| mgmhmf 18656 | A magma homomorphism is a ... |
| mgmhmpropd 18657 | Magma homomorphism depends... |
| mgmhmlin 18658 | A magma homomorphism prese... |
| mgmhmf1o 18659 | A magma homomorphism is bi... |
| idmgmhm 18660 | The identity homomorphism ... |
| issubmgm 18661 | Expand definition of a sub... |
| issubmgm2 18662 | Submagmas are subsets that... |
| rabsubmgmd 18663 | Deduction for proving that... |
| submgmss 18664 | Submagmas are subsets of t... |
| submgmid 18665 | Every magma is trivially a... |
| submgmcl 18666 | Submagmas are closed under... |
| submgmmgm 18667 | Submagmas are themselves m... |
| submgmbas 18668 | The base set of a submagma... |
| subsubmgm 18669 | A submagma of a submagma i... |
| resmgmhm 18670 | Restriction of a magma hom... |
| resmgmhm2 18671 | One direction of ~ resmgmh... |
| resmgmhm2b 18672 | Restriction of the codomai... |
| mgmhmco 18673 | The composition of magma h... |
| mgmhmima 18674 | The homomorphic image of a... |
| mgmhmeql 18675 | The equalizer of two magma... |
| submgmacs 18676 | Submagmas are an algebraic... |
| issgrp 18679 | The predicate "is a semigr... |
| issgrpv 18680 | The predicate "is a semigr... |
| issgrpn0 18681 | The predicate "is a semigr... |
| isnsgrp 18682 | A condition for a structur... |
| sgrpmgm 18683 | A semigroup is a magma. (... |
| sgrpass 18684 | A semigroup operation is a... |
| sgrpcl 18685 | Closure of the operation o... |
| sgrp0 18686 | Any set with an empty base... |
| sgrp0b 18687 | The structure with an empt... |
| sgrp1 18688 | The structure with one ele... |
| issgrpd 18689 | Deduce a semigroup from it... |
| sgrppropd 18690 | If two structures are sets... |
| prdsplusgsgrpcl 18691 | Structure product pointwis... |
| prdssgrpd 18692 | The product of a family of... |
| ismnddef 18695 | The predicate "is a monoid... |
| ismnd 18696 | The predicate "is a monoid... |
| isnmnd 18697 | A condition for a structur... |
| sgrpidmnd 18698 | A semigroup with an identi... |
| mndsgrp 18699 | A monoid is a semigroup. ... |
| mndmgm 18700 | A monoid is a magma. (Con... |
| mndcl 18701 | Closure of the operation o... |
| mndass 18702 | A monoid operation is asso... |
| mndid 18703 | A monoid has a two-sided i... |
| mndideu 18704 | The two-sided identity ele... |
| mnd32g 18705 | Commutative/associative la... |
| mnd12g 18706 | Commutative/associative la... |
| mnd4g 18707 | Commutative/associative la... |
| mndidcl 18708 | The identity element of a ... |
| mndbn0 18709 | The base set of a monoid i... |
| hashfinmndnn 18710 | A finite monoid has positi... |
| mndplusf 18711 | The group addition operati... |
| mndlrid 18712 | A monoid's identity elemen... |
| mndlid 18713 | The identity element of a ... |
| mndrid 18714 | The identity element of a ... |
| ismndd 18715 | Deduce a monoid from its p... |
| mndpfo 18716 | The addition operation of ... |
| mndfo 18717 | The addition operation of ... |
| mndpropd 18718 | If two structures have the... |
| mndprop 18719 | If two structures have the... |
| issubmnd 18720 | Characterize a submonoid b... |
| ress0g 18721 | ` 0g ` is unaffected by re... |
| submnd0 18722 | The zero of a submonoid is... |
| mndinvmod 18723 | Uniqueness of an inverse e... |
| prdsplusgcl 18724 | Structure product pointwis... |
| prdsidlem 18725 | Characterization of identi... |
| prdsmndd 18726 | The product of a family of... |
| prds0g 18727 | Zero in a product of monoi... |
| pwsmnd 18728 | The structure power of a m... |
| pws0g 18729 | Zero in a structure power ... |
| imasmnd2 18730 | The image structure of a m... |
| imasmnd 18731 | The image structure of a m... |
| imasmndf1 18732 | The image of a monoid unde... |
| xpsmnd 18733 | The binary product of mono... |
| xpsmnd0 18734 | The identity element of a ... |
| mnd1 18735 | The (smallest) structure r... |
| mnd1id 18736 | The singleton element of a... |
| ismhm 18741 | Property of a monoid homom... |
| ismhmd 18742 | Deduction version of ~ ism... |
| mhmrcl1 18743 | Reverse closure of a monoi... |
| mhmrcl2 18744 | Reverse closure of a monoi... |
| mhmf 18745 | A monoid homomorphism is a... |
| ismhm0 18746 | Property of a monoid homom... |
| mhmismgmhm 18747 | Each monoid homomorphism i... |
| mhmpropd 18748 | Monoid homomorphism depend... |
| mhmlin 18749 | A monoid homomorphism comm... |
| mhm0 18750 | A monoid homomorphism pres... |
| idmhm 18751 | The identity homomorphism ... |
| mhmf1o 18752 | A monoid homomorphism is b... |
| mndvcl 18753 | Tuple-wise additive closur... |
| mndvass 18754 | Tuple-wise associativity i... |
| mndvlid 18755 | Tuple-wise left identity i... |
| mndvrid 18756 | Tuple-wise right identity ... |
| mhmvlin 18757 | Tuple extension of monoid ... |
| submrcl 18758 | Reverse closure for submon... |
| issubm 18759 | Expand definition of a sub... |
| issubm2 18760 | Submonoids are subsets tha... |
| issubmndb 18761 | The submonoid predicate. ... |
| issubmd 18762 | Deduction for proving a su... |
| mndissubm 18763 | If the base set of a monoi... |
| resmndismnd 18764 | If the base set of a monoi... |
| submss 18765 | Submonoids are subsets of ... |
| submid 18766 | Every monoid is trivially ... |
| subm0cl 18767 | Submonoids contain zero. ... |
| submcl 18768 | Submonoids are closed unde... |
| submmnd 18769 | Submonoids are themselves ... |
| submbas 18770 | The base set of a submonoi... |
| subm0 18771 | Submonoids have the same i... |
| subsubm 18772 | A submonoid of a submonoid... |
| 0subm 18773 | The zero submonoid of an a... |
| insubm 18774 | The intersection of two su... |
| 0mhm 18775 | The constant zero linear f... |
| resmhm 18776 | Restriction of a monoid ho... |
| resmhm2 18777 | One direction of ~ resmhm2... |
| resmhm2b 18778 | Restriction of the codomai... |
| mhmco 18779 | The composition of monoid ... |
| mhmimalem 18780 | Lemma for ~ mhmima and sim... |
| mhmima 18781 | The homomorphic image of a... |
| mhmeql 18782 | The equalizer of two monoi... |
| submacs 18783 | Submonoids are an algebrai... |
| mndind 18784 | Induction in a monoid. In... |
| prdspjmhm 18785 | A projection from a produc... |
| pwspjmhm 18786 | A projection from a struct... |
| pwsdiagmhm 18787 | Diagonal monoid homomorphi... |
| pwsco1mhm 18788 | Right composition with a f... |
| pwsco2mhm 18789 | Left composition with a mo... |
| gsumvallem2 18790 | Lemma for properties of th... |
| gsumsubm 18791 | Evaluate a group sum in a ... |
| gsumz 18792 | Value of a group sum over ... |
| gsumwsubmcl 18793 | Closure of the composite i... |
| gsumws1 18794 | A singleton composite reco... |
| gsumwcl 18795 | Closure of the composite o... |
| gsumsgrpccat 18796 | Homomorphic property of no... |
| gsumccat 18797 | Homomorphic property of co... |
| gsumws2 18798 | Valuation of a pair in a m... |
| gsumccatsn 18799 | Homomorphic property of co... |
| gsumspl 18800 | The primary purpose of the... |
| gsumwmhm 18801 | Behavior of homomorphisms ... |
| gsumwspan 18802 | The submonoid generated by... |
| frmdval 18807 | Value of the free monoid c... |
| frmdbas 18808 | The base set of a free mon... |
| frmdelbas 18809 | An element of the base set... |
| frmdplusg 18810 | The monoid operation of a ... |
| frmdadd 18811 | Value of the monoid operat... |
| vrmdfval 18812 | The canonical injection fr... |
| vrmdval 18813 | The value of the generatin... |
| vrmdf 18814 | The mapping from the index... |
| frmdmnd 18815 | A free monoid is a monoid.... |
| frmd0 18816 | The identity of the free m... |
| frmdsssubm 18817 | The set of words taking va... |
| frmdgsum 18818 | Any word in a free monoid ... |
| frmdss2 18819 | A subset of generators is ... |
| frmdup1 18820 | Any assignment of the gene... |
| frmdup2 18821 | The evaluation map has the... |
| frmdup3lem 18822 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18823 | Universal property of the ... |
| efmnd 18826 | The monoid of endofunction... |
| efmndbas 18827 | The base set of the monoid... |
| efmndbasabf 18828 | The base set of the monoid... |
| elefmndbas 18829 | Two ways of saying a funct... |
| elefmndbas2 18830 | Two ways of saying a funct... |
| efmndbasf 18831 | Elements in the monoid of ... |
| efmndhash 18832 | The monoid of endofunction... |
| efmndbasfi 18833 | The monoid of endofunction... |
| efmndfv 18834 | The function value of an e... |
| efmndtset 18835 | The topology of the monoid... |
| efmndplusg 18836 | The group operation of a m... |
| efmndov 18837 | The value of the group ope... |
| efmndcl 18838 | The group operation of the... |
| efmndtopn 18839 | The topology of the monoid... |
| symggrplem 18840 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18841 | The monoid of endofunction... |
| efmndsgrp 18842 | The monoid of endofunction... |
| ielefmnd 18843 | The identity function rest... |
| efmndid 18844 | The identity function rest... |
| efmndmnd 18845 | The monoid of endofunction... |
| efmnd0nmnd 18846 | Even the monoid of endofun... |
| efmndbas0 18847 | The base set of the monoid... |
| efmnd1hash 18848 | The monoid of endofunction... |
| efmnd1bas 18849 | The monoid of endofunction... |
| efmnd2hash 18850 | The monoid of endofunction... |
| submefmnd 18851 | If the base set of a monoi... |
| sursubmefmnd 18852 | The set of surjective endo... |
| injsubmefmnd 18853 | The set of injective endof... |
| idressubmefmnd 18854 | The singleton containing o... |
| idresefmnd 18855 | The structure with the sin... |
| smndex1ibas 18856 | The modulo function ` I ` ... |
| smndex1iidm 18857 | The modulo function ` I ` ... |
| smndex1gbas 18858 | The constant functions ` (... |
| smndex1gid 18859 | The composition of a const... |
| smndex1igid 18860 | The composition of the mod... |
| smndex1basss 18861 | The modulo function ` I ` ... |
| smndex1bas 18862 | The base set of the monoid... |
| smndex1mgm 18863 | The monoid of endofunction... |
| smndex1sgrp 18864 | The monoid of endofunction... |
| smndex1mndlem 18865 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18866 | The monoid of endofunction... |
| smndex1id 18867 | The modulo function ` I ` ... |
| smndex1n0mnd 18868 | The identity of the monoid... |
| nsmndex1 18869 | The base set ` B ` of the ... |
| smndex2dbas 18870 | The doubling function ` D ... |
| smndex2dnrinv 18871 | The doubling function ` D ... |
| smndex2hbas 18872 | The halving functions ` H ... |
| smndex2dlinvh 18873 | The halving functions ` H ... |
| mgm2nsgrplem1 18874 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18875 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18876 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18877 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18878 | A small magma (with two el... |
| sgrp2nmndlem1 18879 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18880 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18881 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18882 | A small semigroup (with tw... |
| sgrp2rid2ex 18883 | A small semigroup (with tw... |
| sgrp2nmndlem4 18884 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18885 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18886 | A small semigroup (with tw... |
| mgmnsgrpex 18887 | There is a magma which is ... |
| sgrpnmndex 18888 | There is a semigroup which... |
| sgrpssmgm 18889 | The class of all semigroup... |
| mndsssgrp 18890 | The class of all monoids i... |
| pwmndgplus 18891 | The operation of the monoi... |
| pwmndid 18892 | The identity of the monoid... |
| pwmnd 18893 | The power set of a class `... |
| isgrp 18900 | The predicate "is a group"... |
| grpmnd 18901 | A group is a monoid. (Con... |
| grpcl 18902 | Closure of the operation o... |
| grpass 18903 | A group operation is assoc... |
| grpinvex 18904 | Every member of a group ha... |
| grpideu 18905 | The two-sided identity ele... |
| grpassd 18906 | A group operation is assoc... |
| grpmndd 18907 | A group is a monoid. (Con... |
| grpcld 18908 | Closure of the operation o... |
| grpplusf 18909 | The group addition operati... |
| grpplusfo 18910 | The group addition operati... |
| resgrpplusfrn 18911 | The underlying set of a gr... |
| grppropd 18912 | If two structures have the... |
| grpprop 18913 | If two structures have the... |
| grppropstr 18914 | Generalize a specific 2-el... |
| grpss 18915 | Show that a structure exte... |
| isgrpd2e 18916 | Deduce a group from its pr... |
| isgrpd2 18917 | Deduce a group from its pr... |
| isgrpde 18918 | Deduce a group from its pr... |
| isgrpd 18919 | Deduce a group from its pr... |
| isgrpi 18920 | Properties that determine ... |
| grpsgrp 18921 | A group is a semigroup. (... |
| grpmgmd 18922 | A group is a magma, deduct... |
| dfgrp2 18923 | Alternate definition of a ... |
| dfgrp2e 18924 | Alternate definition of a ... |
| isgrpix 18925 | Properties that determine ... |
| grpidcl 18926 | The identity element of a ... |
| grpbn0 18927 | The base set of a group is... |
| grplid 18928 | The identity element of a ... |
| grprid 18929 | The identity element of a ... |
| grplidd 18930 | The identity element of a ... |
| grpridd 18931 | The identity element of a ... |
| grpn0 18932 | A group is not empty. (Co... |
| hashfingrpnn 18933 | A finite group has positiv... |
| grprcan 18934 | Right cancellation law for... |
| grpinveu 18935 | The left inverse element o... |
| grpid 18936 | Two ways of saying that an... |
| isgrpid2 18937 | Properties showing that an... |
| grpidd2 18938 | Deduce the identity elemen... |
| grpinvfval 18939 | The inverse function of a ... |
| grpinvfvalALT 18940 | Shorter proof of ~ grpinvf... |
| grpinvval 18941 | The inverse of a group ele... |
| grpinvfn 18942 | Functionality of the group... |
| grpinvfvi 18943 | The group inverse function... |
| grpsubfval 18944 | Group subtraction (divisio... |
| grpsubfvalALT 18945 | Shorter proof of ~ grpsubf... |
| grpsubval 18946 | Group subtraction (divisio... |
| grpinvf 18947 | The group inversion operat... |
| grpinvcl 18948 | A group element's inverse ... |
| grpinvcld 18949 | A group element's inverse ... |
| grplinv 18950 | The left inverse of a grou... |
| grprinv 18951 | The right inverse of a gro... |
| grpinvid1 18952 | The inverse of a group ele... |
| grpinvid2 18953 | The inverse of a group ele... |
| isgrpinv 18954 | Properties showing that a ... |
| grplinvd 18955 | The left inverse of a grou... |
| grprinvd 18956 | The right inverse of a gro... |
| grplrinv 18957 | In a group, every member h... |
| grpidinv2 18958 | A group's properties using... |
| grpidinv 18959 | A group has a left and rig... |
| grpinvid 18960 | The inverse of the identit... |
| grplcan 18961 | Left cancellation law for ... |
| grpasscan1 18962 | An associative cancellatio... |
| grpasscan2 18963 | An associative cancellatio... |
| grpidrcan 18964 | If right adding an element... |
| grpidlcan 18965 | If left adding an element ... |
| grpinvinv 18966 | Double inverse law for gro... |
| grpinvcnv 18967 | The group inverse is its o... |
| grpinv11 18968 | The group inverse is one-t... |
| grpinvf1o 18969 | The group inverse is a one... |
| grpinvnz 18970 | The inverse of a nonzero g... |
| grpinvnzcl 18971 | The inverse of a nonzero g... |
| grpsubinv 18972 | Subtraction of an inverse.... |
| grplmulf1o 18973 | Left multiplication by a g... |
| grpraddf1o 18974 | Right addition by a group ... |
| grpinvpropd 18975 | If two structures have the... |
| grpidssd 18976 | If the base set of a group... |
| grpinvssd 18977 | If the base set of a group... |
| grpinvadd 18978 | The inverse of the group o... |
| grpsubf 18979 | Functionality of group sub... |
| grpsubcl 18980 | Closure of group subtracti... |
| grpsubrcan 18981 | Right cancellation law for... |
| grpinvsub 18982 | Inverse of a group subtrac... |
| grpinvval2 18983 | A ~ df-neg -like equation ... |
| grpsubid 18984 | Subtraction of a group ele... |
| grpsubid1 18985 | Subtraction of the identit... |
| grpsubeq0 18986 | If the difference between ... |
| grpsubadd0sub 18987 | Subtraction expressed as a... |
| grpsubadd 18988 | Relationship between group... |
| grpsubsub 18989 | Double group subtraction. ... |
| grpaddsubass 18990 | Associative-type law for g... |
| grppncan 18991 | Cancellation law for subtr... |
| grpnpcan 18992 | Cancellation law for subtr... |
| grpsubsub4 18993 | Double group subtraction (... |
| grppnpcan2 18994 | Cancellation law for mixed... |
| grpnpncan 18995 | Cancellation law for group... |
| grpnpncan0 18996 | Cancellation law for group... |
| grpnnncan2 18997 | Cancellation law for group... |
| dfgrp3lem 18998 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18999 | Alternate definition of a ... |
| dfgrp3e 19000 | Alternate definition of a ... |
| grplactfval 19001 | The left group action of e... |
| grplactval 19002 | The value of the left grou... |
| grplactcnv 19003 | The left group action of e... |
| grplactf1o 19004 | The left group action of e... |
| grpsubpropd 19005 | Weak property deduction fo... |
| grpsubpropd2 19006 | Strong property deduction ... |
| grp1 19007 | The (smallest) structure r... |
| grp1inv 19008 | The inverse function of th... |
| prdsinvlem 19009 | Characterization of invers... |
| prdsgrpd 19010 | The product of a family of... |
| prdsinvgd 19011 | Negation in a product of g... |
| pwsgrp 19012 | A structure power of a gro... |
| pwsinvg 19013 | Negation in a group power.... |
| pwssub 19014 | Subtraction in a group pow... |
| imasgrp2 19015 | The image structure of a g... |
| imasgrp 19016 | The image structure of a g... |
| imasgrpf1 19017 | The image of a group under... |
| qusgrp2 19018 | Prove that a quotient stru... |
| xpsgrp 19019 | The binary product of grou... |
| xpsinv 19020 | Value of the negation oper... |
| xpsgrpsub 19021 | Value of the subtraction o... |
| mhmlem 19022 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19023 | A surjective monoid morphi... |
| mhmmnd 19024 | The image of a monoid ` G ... |
| mhmfmhm 19025 | The function fulfilling th... |
| ghmgrp 19026 | The image of a group ` G `... |
| mulgfval 19029 | Group multiple (exponentia... |
| mulgfvalALT 19030 | Shorter proof of ~ mulgfva... |
| mulgval 19031 | Value of the group multipl... |
| mulgfn 19032 | Functionality of the group... |
| mulgfvi 19033 | The group multiple operati... |
| mulg0 19034 | Group multiple (exponentia... |
| mulgnn 19035 | Group multiple (exponentia... |
| ressmulgnn 19036 | Values for the group multi... |
| ressmulgnn0 19037 | Values for the group multi... |
| mulgnngsum 19038 | Group multiple (exponentia... |
| mulgnn0gsum 19039 | Group multiple (exponentia... |
| mulg1 19040 | Group multiple (exponentia... |
| mulgnnp1 19041 | Group multiple (exponentia... |
| mulg2 19042 | Group multiple (exponentia... |
| mulgnegnn 19043 | Group multiple (exponentia... |
| mulgnn0p1 19044 | Group multiple (exponentia... |
| mulgnnsubcl 19045 | Closure of the group multi... |
| mulgnn0subcl 19046 | Closure of the group multi... |
| mulgsubcl 19047 | Closure of the group multi... |
| mulgnncl 19048 | Closure of the group multi... |
| mulgnn0cl 19049 | Closure of the group multi... |
| mulgcl 19050 | Closure of the group multi... |
| mulgneg 19051 | Group multiple (exponentia... |
| mulgnegneg 19052 | The inverse of a negative ... |
| mulgm1 19053 | Group multiple (exponentia... |
| mulgnn0cld 19054 | Closure of the group multi... |
| mulgcld 19055 | Deduction associated with ... |
| mulgaddcomlem 19056 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19057 | The group multiple operato... |
| mulginvcom 19058 | The group multiple operato... |
| mulginvinv 19059 | The group multiple operato... |
| mulgnn0z 19060 | A group multiple of the id... |
| mulgz 19061 | A group multiple of the id... |
| mulgnndir 19062 | Sum of group multiples, fo... |
| mulgnn0dir 19063 | Sum of group multiples, ge... |
| mulgdirlem 19064 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19065 | Sum of group multiples, ge... |
| mulgp1 19066 | Group multiple (exponentia... |
| mulgneg2 19067 | Group multiple (exponentia... |
| mulgnnass 19068 | Product of group multiples... |
| mulgnn0ass 19069 | Product of group multiples... |
| mulgass 19070 | Product of group multiples... |
| mulgassr 19071 | Reversed product of group ... |
| mulgmodid 19072 | Casting out multiples of t... |
| mulgsubdir 19073 | Distribution of group mult... |
| mhmmulg 19074 | A homomorphism of monoids ... |
| mulgpropd 19075 | Two structures with the sa... |
| submmulgcl 19076 | Closure of the group multi... |
| submmulg 19077 | A group multiple is the sa... |
| pwsmulg 19078 | Value of a group multiple ... |
| issubg 19085 | The subgroup predicate. (... |
| subgss 19086 | A subgroup is a subset. (... |
| subgid 19087 | A group is a subgroup of i... |
| subggrp 19088 | A subgroup is a group. (C... |
| subgbas 19089 | The base of the restricted... |
| subgrcl 19090 | Reverse closure for the su... |
| subg0 19091 | A subgroup of a group must... |
| subginv 19092 | The inverse of an element ... |
| subg0cl 19093 | The group identity is an e... |
| subginvcl 19094 | The inverse of an element ... |
| subgcl 19095 | A subgroup is closed under... |
| subgsubcl 19096 | A subgroup is closed under... |
| subgsub 19097 | The subtraction of element... |
| subgmulgcl 19098 | Closure of the group multi... |
| subgmulg 19099 | A group multiple is the sa... |
| issubg2 19100 | Characterize the subgroups... |
| issubgrpd2 19101 | Prove a subgroup by closur... |
| issubgrpd 19102 | Prove a subgroup by closur... |
| issubg3 19103 | A subgroup is a symmetric ... |
| issubg4 19104 | A subgroup is a nonempty s... |
| grpissubg 19105 | If the base set of a group... |
| resgrpisgrp 19106 | If the base set of a group... |
| subgsubm 19107 | A subgroup is a submonoid.... |
| subsubg 19108 | A subgroup of a subgroup i... |
| subgint 19109 | The intersection of a none... |
| 0subg 19110 | The zero subgroup of an ar... |
| 0subgOLD 19111 | Obsolete version of ~ 0sub... |
| trivsubgd 19112 | The only subgroup of a tri... |
| trivsubgsnd 19113 | The only subgroup of a tri... |
| isnsg 19114 | Property of being a normal... |
| isnsg2 19115 | Weaken the condition of ~ ... |
| nsgbi 19116 | Defining property of a nor... |
| nsgsubg 19117 | A normal subgroup is a sub... |
| nsgconj 19118 | The conjugation of an elem... |
| isnsg3 19119 | A subgroup is normal iff t... |
| subgacs 19120 | Subgroups are an algebraic... |
| nsgacs 19121 | Normal subgroups form an a... |
| elnmz 19122 | Elementhood in the normali... |
| nmzbi 19123 | Defining property of the n... |
| nmzsubg 19124 | The normalizer N_G(S) of a... |
| ssnmz 19125 | A subgroup is a subset of ... |
| isnsg4 19126 | A subgroup is normal iff i... |
| nmznsg 19127 | Any subgroup is a normal s... |
| 0nsg 19128 | The zero subgroup is norma... |
| nsgid 19129 | The whole group is a norma... |
| 0idnsgd 19130 | The whole group and the ze... |
| trivnsgd 19131 | The only normal subgroup o... |
| triv1nsgd 19132 | A trivial group has exactl... |
| 1nsgtrivd 19133 | A group with exactly one n... |
| releqg 19134 | The left coset equivalence... |
| eqgfval 19135 | Value of the subgroup left... |
| eqgval 19136 | Value of the subgroup left... |
| eqger 19137 | The subgroup coset equival... |
| eqglact 19138 | A left coset can be expres... |
| eqgid 19139 | The left coset containing ... |
| eqgen 19140 | Each coset is equipotent t... |
| eqgcpbl 19141 | The subgroup coset equival... |
| eqg0el 19142 | Equivalence class of a quo... |
| quselbas 19143 | Membership in the base set... |
| quseccl0 19144 | Closure of the quotient ma... |
| qusgrp 19145 | If ` Y ` is a normal subgr... |
| quseccl 19146 | Closure of the quotient ma... |
| qusadd 19147 | Value of the group operati... |
| qus0 19148 | Value of the group identit... |
| qusinv 19149 | Value of the group inverse... |
| qussub 19150 | Value of the group subtrac... |
| ecqusaddd 19151 | Addition of equivalence cl... |
| ecqusaddcl 19152 | Closure of the addition in... |
| lagsubg2 19153 | Lagrange's theorem for fin... |
| lagsubg 19154 | Lagrange's theorem for Gro... |
| eqg0subg 19155 | The coset equivalence rela... |
| eqg0subgecsn 19156 | The equivalence classes mo... |
| qus0subgbas 19157 | The base set of a quotient... |
| qus0subgadd 19158 | The addition in a quotient... |
| cycsubmel 19159 | Characterization of an ele... |
| cycsubmcl 19160 | The set of nonnegative int... |
| cycsubm 19161 | The set of nonnegative int... |
| cyccom 19162 | Condition for an operation... |
| cycsubmcom 19163 | The operation of a monoid ... |
| cycsubggend 19164 | The cyclic subgroup genera... |
| cycsubgcl 19165 | The set of integer powers ... |
| cycsubgss 19166 | The cyclic subgroup genera... |
| cycsubg 19167 | The cyclic group generated... |
| cycsubgcld 19168 | The cyclic subgroup genera... |
| cycsubg2 19169 | The subgroup generated by ... |
| cycsubg2cl 19170 | Any multiple of an element... |
| reldmghm 19173 | Lemma for group homomorphi... |
| isghm 19174 | Property of being a homomo... |
| isghm3 19175 | Property of a group homomo... |
| ghmgrp1 19176 | A group homomorphism is on... |
| ghmgrp2 19177 | A group homomorphism is on... |
| ghmf 19178 | A group homomorphism is a ... |
| ghmlin 19179 | A homomorphism of groups i... |
| ghmid 19180 | A homomorphism of groups p... |
| ghminv 19181 | A homomorphism of groups p... |
| ghmsub 19182 | Linearity of subtraction t... |
| isghmd 19183 | Deduction for a group homo... |
| ghmmhm 19184 | A group homomorphism is a ... |
| ghmmhmb 19185 | Group homomorphisms and mo... |
| ghmmulg 19186 | A group homomorphism prese... |
| ghmrn 19187 | The range of a homomorphis... |
| 0ghm 19188 | The constant zero linear f... |
| idghm 19189 | The identity homomorphism ... |
| resghm 19190 | Restriction of a homomorph... |
| resghm2 19191 | One direction of ~ resghm2... |
| resghm2b 19192 | Restriction of the codomai... |
| ghmghmrn 19193 | A group homomorphism from ... |
| ghmco 19194 | The composition of group h... |
| ghmima 19195 | The image of a subgroup un... |
| ghmpreima 19196 | The inverse image of a sub... |
| ghmeql 19197 | The equalizer of two group... |
| ghmnsgima 19198 | The image of a normal subg... |
| ghmnsgpreima 19199 | The inverse image of a nor... |
| ghmker 19200 | The kernel of a homomorphi... |
| ghmeqker 19201 | Two source points map to t... |
| pwsdiagghm 19202 | Diagonal homomorphism into... |
| f1ghm0to0 19203 | If a group homomorphism ` ... |
| ghmf1 19204 | Two ways of saying a group... |
| kerf1ghm 19205 | A group homomorphism ` F `... |
| ghmf1o 19206 | A bijective group homomorp... |
| conjghm 19207 | Conjugation is an automorp... |
| conjsubg 19208 | A conjugated subgroup is a... |
| conjsubgen 19209 | A conjugated subgroup is e... |
| conjnmz 19210 | A subgroup is unchanged un... |
| conjnmzb 19211 | Alternative condition for ... |
| conjnsg 19212 | A normal subgroup is uncha... |
| qusghm 19213 | If ` Y ` is a normal subgr... |
| ghmpropd 19214 | Group homomorphism depends... |
| gimfn 19219 | The group isomorphism func... |
| isgim 19220 | An isomorphism of groups i... |
| gimf1o 19221 | An isomorphism of groups i... |
| gimghm 19222 | An isomorphism of groups i... |
| isgim2 19223 | A group isomorphism is a h... |
| subggim 19224 | Behavior of subgroups unde... |
| gimcnv 19225 | The converse of a group is... |
| gimco 19226 | The composition of group i... |
| gim0to0 19227 | A group isomorphism maps t... |
| brgic 19228 | The relation "is isomorphi... |
| brgici 19229 | Prove isomorphic by an exp... |
| gicref 19230 | Isomorphism is reflexive. ... |
| giclcl 19231 | Isomorphism implies the le... |
| gicrcl 19232 | Isomorphism implies the ri... |
| gicsym 19233 | Isomorphism is symmetric. ... |
| gictr 19234 | Isomorphism is transitive.... |
| gicer 19235 | Isomorphism is an equivale... |
| gicen 19236 | Isomorphic groups have equ... |
| gicsubgen 19237 | A less trivial example of ... |
| ghmquskerlem1 19238 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19239 | In the case of theorem ~ g... |
| ghmquskerlem2 19240 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19241 | The mapping ` H ` induced ... |
| ghmqusker 19242 | A surjective group homomor... |
| gicqusker 19243 | The image ` H ` of a group... |
| isga 19246 | The predicate "is a (left)... |
| gagrp 19247 | The left argument of a gro... |
| gaset 19248 | The right argument of a gr... |
| gagrpid 19249 | The identity of the group ... |
| gaf 19250 | The mapping of the group a... |
| gafo 19251 | A group action is onto its... |
| gaass 19252 | An "associative" property ... |
| ga0 19253 | The action of a group on t... |
| gaid 19254 | The trivial action of a gr... |
| subgga 19255 | A subgroup acts on its par... |
| gass 19256 | A subset of a group action... |
| gasubg 19257 | The restriction of a group... |
| gaid2 19258 | A group operation is a lef... |
| galcan 19259 | The action of a particular... |
| gacan 19260 | Group inverses cancel in a... |
| gapm 19261 | The action of a particular... |
| gaorb 19262 | The orbit equivalence rela... |
| gaorber 19263 | The orbit equivalence rela... |
| gastacl 19264 | The stabilizer subgroup in... |
| gastacos 19265 | Write the coset relation f... |
| orbstafun 19266 | Existence and uniqueness f... |
| orbstaval 19267 | Value of the function at a... |
| orbsta 19268 | The Orbit-Stabilizer theor... |
| orbsta2 19269 | Relation between the size ... |
| cntrval 19274 | Substitute definition of t... |
| cntzfval 19275 | First level substitution f... |
| cntzval 19276 | Definition substitution fo... |
| elcntz 19277 | Elementhood in the central... |
| cntzel 19278 | Membership in a centralize... |
| cntzsnval 19279 | Special substitution for t... |
| elcntzsn 19280 | Value of the centralizer o... |
| sscntz 19281 | A centralizer expression f... |
| cntzrcl 19282 | Reverse closure for elemen... |
| cntzssv 19283 | The centralizer is uncondi... |
| cntzi 19284 | Membership in a centralize... |
| elcntr 19285 | Elementhood in the center ... |
| cntrss 19286 | The center is a subset of ... |
| cntri 19287 | Defining property of the c... |
| resscntz 19288 | Centralizer in a substruct... |
| cntzsgrpcl 19289 | Centralizers are closed un... |
| cntz2ss 19290 | Centralizers reverse the s... |
| cntzrec 19291 | Reciprocity relationship f... |
| cntziinsn 19292 | Express any centralizer as... |
| cntzsubm 19293 | Centralizers in a monoid a... |
| cntzsubg 19294 | Centralizers in a group ar... |
| cntzidss 19295 | If the elements of ` S ` c... |
| cntzmhm 19296 | Centralizers in a monoid a... |
| cntzmhm2 19297 | Centralizers in a monoid a... |
| cntrsubgnsg 19298 | A central subgroup is norm... |
| cntrnsg 19299 | The center of a group is a... |
| oppgval 19302 | Value of the opposite grou... |
| oppgplusfval 19303 | Value of the addition oper... |
| oppgplus 19304 | Value of the addition oper... |
| setsplusg 19305 | The other components of an... |
| oppglemOLD 19306 | Obsolete version of ~ sets... |
| oppgbas 19307 | Base set of an opposite gr... |
| oppgbasOLD 19308 | Obsolete version of ~ oppg... |
| oppgtset 19309 | Topology of an opposite gr... |
| oppgtsetOLD 19310 | Obsolete version of ~ oppg... |
| oppgtopn 19311 | Topology of an opposite gr... |
| oppgmnd 19312 | The opposite of a monoid i... |
| oppgmndb 19313 | Bidirectional form of ~ op... |
| oppgid 19314 | Zero in a monoid is a symm... |
| oppggrp 19315 | The opposite of a group is... |
| oppggrpb 19316 | Bidirectional form of ~ op... |
| oppginv 19317 | Inverses in a group are a ... |
| invoppggim 19318 | The inverse is an antiauto... |
| oppggic 19319 | Every group is (naturally)... |
| oppgsubm 19320 | Being a submonoid is a sym... |
| oppgsubg 19321 | Being a subgroup is a symm... |
| oppgcntz 19322 | A centralizer in a group i... |
| oppgcntr 19323 | The center of a group is t... |
| gsumwrev 19324 | A sum in an opposite monoi... |
| symgval 19327 | The value of the symmetric... |
| permsetexOLD 19328 | Obsolete version of ~ f1os... |
| symgbas 19329 | The base set of the symmet... |
| symgbasexOLD 19330 | Obsolete as of 8-Aug-2024.... |
| elsymgbas2 19331 | Two ways of saying a funct... |
| elsymgbas 19332 | Two ways of saying a funct... |
| symgbasf1o 19333 | Elements in the symmetric ... |
| symgbasf 19334 | A permutation (element of ... |
| symgbasmap 19335 | A permutation (element of ... |
| symghash 19336 | The symmetric group on ` n... |
| symgbasfi 19337 | The symmetric group on a f... |
| symgfv 19338 | The function value of a pe... |
| symgfvne 19339 | The function values of a p... |
| symgressbas 19340 | The symmetric group on ` A... |
| symgplusg 19341 | The group operation of a s... |
| symgov 19342 | The value of the group ope... |
| symgcl 19343 | The group operation of the... |
| idresperm 19344 | The identity function rest... |
| symgmov1 19345 | For a permutation of a set... |
| symgmov2 19346 | For a permutation of a set... |
| symgbas0 19347 | The base set of the symmet... |
| symg1hash 19348 | The symmetric group on a s... |
| symg1bas 19349 | The symmetric group on a s... |
| symg2hash 19350 | The symmetric group on a (... |
| symg2bas 19351 | The symmetric group on a p... |
| 0symgefmndeq 19352 | The symmetric group on the... |
| snsymgefmndeq 19353 | The symmetric group on a s... |
| symgpssefmnd 19354 | For a set ` A ` with more ... |
| symgvalstruct 19355 | The value of the symmetric... |
| symgvalstructOLD 19356 | Obsolete proof of ~ symgva... |
| symgsubmefmnd 19357 | The symmetric group on a s... |
| symgtset 19358 | The topology of the symmet... |
| symggrp 19359 | The symmetric group on a s... |
| symgid 19360 | The group identity element... |
| symginv 19361 | The group inverse in the s... |
| symgsubmefmndALT 19362 | The symmetric group on a s... |
| galactghm 19363 | The currying of a group ac... |
| lactghmga 19364 | The converse of ~ galactgh... |
| symgtopn 19365 | The topology of the symmet... |
| symgga 19366 | The symmetric group induce... |
| pgrpsubgsymgbi 19367 | Every permutation group is... |
| pgrpsubgsymg 19368 | Every permutation group is... |
| idressubgsymg 19369 | The singleton containing o... |
| idrespermg 19370 | The structure with the sin... |
| cayleylem1 19371 | Lemma for ~ cayley . (Con... |
| cayleylem2 19372 | Lemma for ~ cayley . (Con... |
| cayley 19373 | Cayley's Theorem (construc... |
| cayleyth 19374 | Cayley's Theorem (existenc... |
| symgfix2 19375 | If a permutation does not ... |
| symgextf 19376 | The extension of a permuta... |
| symgextfv 19377 | The function value of the ... |
| symgextfve 19378 | The function value of the ... |
| symgextf1lem 19379 | Lemma for ~ symgextf1 . (... |
| symgextf1 19380 | The extension of a permuta... |
| symgextfo 19381 | The extension of a permuta... |
| symgextf1o 19382 | The extension of a permuta... |
| symgextsymg 19383 | The extension of a permuta... |
| symgextres 19384 | The restriction of the ext... |
| gsumccatsymgsn 19385 | Homomorphic property of co... |
| gsmsymgrfixlem1 19386 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19387 | The composition of permuta... |
| fvcosymgeq 19388 | The values of two composit... |
| gsmsymgreqlem1 19389 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19390 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19391 | Two combination of permuta... |
| symgfixelq 19392 | A permutation of a set fix... |
| symgfixels 19393 | The restriction of a permu... |
| symgfixelsi 19394 | The restriction of a permu... |
| symgfixf 19395 | The mapping of a permutati... |
| symgfixf1 19396 | The mapping of a permutati... |
| symgfixfolem1 19397 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19398 | The mapping of a permutati... |
| symgfixf1o 19399 | The mapping of a permutati... |
| f1omvdmvd 19402 | A permutation of any class... |
| f1omvdcnv 19403 | A permutation and its inve... |
| mvdco 19404 | Composing two permutations... |
| f1omvdconj 19405 | Conjugation of a permutati... |
| f1otrspeq 19406 | A transposition is charact... |
| f1omvdco2 19407 | If exactly one of two perm... |
| f1omvdco3 19408 | If a point is moved by exa... |
| pmtrfval 19409 | The function generating tr... |
| pmtrval 19410 | A generated transposition,... |
| pmtrfv 19411 | General value of mapping a... |
| pmtrprfv 19412 | In a transposition of two ... |
| pmtrprfv3 19413 | In a transposition of two ... |
| pmtrf 19414 | Functionality of a transpo... |
| pmtrmvd 19415 | A transposition moves prec... |
| pmtrrn 19416 | Transposing two points giv... |
| pmtrfrn 19417 | A transposition (as a kind... |
| pmtrffv 19418 | Mapping of a point under a... |
| pmtrrn2 19419 | For any transposition ther... |
| pmtrfinv 19420 | A transposition function i... |
| pmtrfmvdn0 19421 | A transposition moves at l... |
| pmtrff1o 19422 | A transposition function i... |
| pmtrfcnv 19423 | A transposition function i... |
| pmtrfb 19424 | An intrinsic characterizat... |
| pmtrfconj 19425 | Any conjugate of a transpo... |
| symgsssg 19426 | The symmetric group has su... |
| symgfisg 19427 | The symmetric group has a ... |
| symgtrf 19428 | Transpositions are element... |
| symggen 19429 | The span of the transposit... |
| symggen2 19430 | A finite permutation group... |
| symgtrinv 19431 | To invert a permutation re... |
| pmtr3ncomlem1 19432 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19433 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19434 | Transpositions over sets w... |
| pmtrdifellem1 19435 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19436 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19437 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19438 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19439 | A transposition of element... |
| pmtrdifwrdellem1 19440 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19441 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19442 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19443 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19444 | A sequence of transpositio... |
| pmtrdifwrdel2 19445 | A sequence of transpositio... |
| pmtrprfval 19446 | The transpositions on a pa... |
| pmtrprfvalrn 19447 | The range of the transposi... |
| psgnunilem1 19452 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19453 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19454 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19455 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19456 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19457 | Addition and subtraction o... |
| psgnuni 19458 | If the same permutation ca... |
| psgnfval 19459 | Function definition of the... |
| psgnfn 19460 | Functionality and domain o... |
| psgndmsubg 19461 | The finitary permutations ... |
| psgneldm 19462 | Property of being a finita... |
| psgneldm2 19463 | The finitary permutations ... |
| psgneldm2i 19464 | A sequence of transpositio... |
| psgneu 19465 | A finitary permutation has... |
| psgnval 19466 | Value of the permutation s... |
| psgnvali 19467 | A finitary permutation has... |
| psgnvalii 19468 | Any representation of a pe... |
| psgnpmtr 19469 | All transpositions are odd... |
| psgn0fv0 19470 | The permutation sign funct... |
| sygbasnfpfi 19471 | The class of non-fixed poi... |
| psgnfvalfi 19472 | Function definition of the... |
| psgnvalfi 19473 | Value of the permutation s... |
| psgnran 19474 | The range of the permutati... |
| gsmtrcl 19475 | The group sum of transposi... |
| psgnfitr 19476 | A permutation of a finite ... |
| psgnfieu 19477 | A permutation of a finite ... |
| pmtrsn 19478 | The value of the transposi... |
| psgnsn 19479 | The permutation sign funct... |
| psgnprfval 19480 | The permutation sign funct... |
| psgnprfval1 19481 | The permutation sign of th... |
| psgnprfval2 19482 | The permutation sign of th... |
| odfval 19491 | Value of the order functio... |
| odfvalALT 19492 | Shorter proof of ~ odfval ... |
| odval 19493 | Second substitution for th... |
| odlem1 19494 | The group element order is... |
| odcl 19495 | The order of a group eleme... |
| odf 19496 | Functionality of the group... |
| odid 19497 | Any element to the power o... |
| odlem2 19498 | Any positive annihilator o... |
| odmodnn0 19499 | Reduce the argument of a g... |
| mndodconglem 19500 | Lemma for ~ mndodcong . (... |
| mndodcong 19501 | If two multipliers are con... |
| mndodcongi 19502 | If two multipliers are con... |
| oddvdsnn0 19503 | The only multiples of ` A ... |
| odnncl 19504 | If a nonzero multiple of a... |
| odmod 19505 | Reduce the argument of a g... |
| oddvds 19506 | The only multiples of ` A ... |
| oddvdsi 19507 | Any group element is annih... |
| odcong 19508 | If two multipliers are con... |
| odeq 19509 | The ~ oddvds property uniq... |
| odval2 19510 | A non-conditional definiti... |
| odcld 19511 | The order of a group eleme... |
| odm1inv 19512 | The (order-1)th multiple o... |
| odmulgid 19513 | A relationship between the... |
| odmulg2 19514 | The order of a multiple di... |
| odmulg 19515 | Relationship between the o... |
| odmulgeq 19516 | A multiple of a point of f... |
| odbezout 19517 | If ` N ` is coprime to the... |
| od1 19518 | The order of the group ide... |
| odeq1 19519 | The group identity is the ... |
| odinv 19520 | The order of the inverse o... |
| odf1 19521 | The multiples of an elemen... |
| odinf 19522 | The multiples of an elemen... |
| dfod2 19523 | An alternative definition ... |
| odcl2 19524 | The order of an element of... |
| oddvds2 19525 | The order of an element of... |
| finodsubmsubg 19526 | A submonoid whose elements... |
| 0subgALT 19527 | A shorter proof of ~ 0subg... |
| submod 19528 | The order of an element is... |
| subgod 19529 | The order of an element is... |
| odsubdvds 19530 | The order of an element of... |
| odf1o1 19531 | An element with zero order... |
| odf1o2 19532 | An element with nonzero or... |
| odhash 19533 | An element of zero order g... |
| odhash2 19534 | If an element has nonzero ... |
| odhash3 19535 | An element which generates... |
| odngen 19536 | A cyclic subgroup of size ... |
| gexval 19537 | Value of the exponent of a... |
| gexlem1 19538 | The group element order is... |
| gexcl 19539 | The exponent of a group is... |
| gexid 19540 | Any element to the power o... |
| gexlem2 19541 | Any positive annihilator o... |
| gexdvdsi 19542 | Any group element is annih... |
| gexdvds 19543 | The only ` N ` that annihi... |
| gexdvds2 19544 | An integer divides the gro... |
| gexod 19545 | Any group element is annih... |
| gexcl3 19546 | If the order of every grou... |
| gexnnod 19547 | Every group element has fi... |
| gexcl2 19548 | The exponent of a finite g... |
| gexdvds3 19549 | The exponent of a finite g... |
| gex1 19550 | A group or monoid has expo... |
| ispgp 19551 | A group is a ` P ` -group ... |
| pgpprm 19552 | Reverse closure for the fi... |
| pgpgrp 19553 | Reverse closure for the se... |
| pgpfi1 19554 | A finite group with order ... |
| pgp0 19555 | The identity subgroup is a... |
| subgpgp 19556 | A subgroup of a p-group is... |
| sylow1lem1 19557 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19558 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19559 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19560 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19561 | Lemma for ~ sylow1 . Usin... |
| sylow1 19562 | Sylow's first theorem. If... |
| odcau 19563 | Cauchy's theorem for the o... |
| pgpfi 19564 | The converse to ~ pgpfi1 .... |
| pgpfi2 19565 | Alternate version of ~ pgp... |
| pgphash 19566 | The order of a p-group. (... |
| isslw 19567 | The property of being a Sy... |
| slwprm 19568 | Reverse closure for the fi... |
| slwsubg 19569 | A Sylow ` P ` -subgroup is... |
| slwispgp 19570 | Defining property of a Syl... |
| slwpss 19571 | A proper superset of a Syl... |
| slwpgp 19572 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19573 | Every ` P ` -subgroup is c... |
| slwn0 19574 | Every finite group contain... |
| subgslw 19575 | A Sylow subgroup that is c... |
| sylow2alem1 19576 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19577 | Lemma for ~ sylow2a . All... |
| sylow2a 19578 | A named lemma of Sylow's s... |
| sylow2blem1 19579 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19580 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19581 | Sylow's second theorem. P... |
| sylow2b 19582 | Sylow's second theorem. A... |
| slwhash 19583 | A sylow subgroup has cardi... |
| fislw 19584 | The sylow subgroups of a f... |
| sylow2 19585 | Sylow's second theorem. S... |
| sylow3lem1 19586 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19587 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19588 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19589 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19590 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19591 | Lemma for ~ sylow3 , secon... |
| sylow3 19592 | Sylow's third theorem. Th... |
| lsmfval 19597 | The subgroup sum function ... |
| lsmvalx 19598 | Subspace sum value (for a ... |
| lsmelvalx 19599 | Subspace sum membership (f... |
| lsmelvalix 19600 | Subspace sum membership (f... |
| oppglsm 19601 | The subspace sum operation... |
| lsmssv 19602 | Subgroup sum is a subset o... |
| lsmless1x 19603 | Subset implies subgroup su... |
| lsmless2x 19604 | Subset implies subgroup su... |
| lsmub1x 19605 | Subgroup sum is an upper b... |
| lsmub2x 19606 | Subgroup sum is an upper b... |
| lsmval 19607 | Subgroup sum value (for a ... |
| lsmelval 19608 | Subgroup sum membership (f... |
| lsmelvali 19609 | Subgroup sum membership (f... |
| lsmelvalm 19610 | Subgroup sum membership an... |
| lsmelvalmi 19611 | Membership of vector subtr... |
| lsmsubm 19612 | The sum of two commuting s... |
| lsmsubg 19613 | The sum of two commuting s... |
| lsmcom2 19614 | Subgroup sum commutes. (C... |
| smndlsmidm 19615 | The direct product is idem... |
| lsmub1 19616 | Subgroup sum is an upper b... |
| lsmub2 19617 | Subgroup sum is an upper b... |
| lsmunss 19618 | Union of subgroups is a su... |
| lsmless1 19619 | Subset implies subgroup su... |
| lsmless2 19620 | Subset implies subgroup su... |
| lsmless12 19621 | Subset implies subgroup su... |
| lsmidm 19622 | Subgroup sum is idempotent... |
| lsmlub 19623 | The least upper bound prop... |
| lsmss1 19624 | Subgroup sum with a subset... |
| lsmss1b 19625 | Subgroup sum with a subset... |
| lsmss2 19626 | Subgroup sum with a subset... |
| lsmss2b 19627 | Subgroup sum with a subset... |
| lsmass 19628 | Subgroup sum is associativ... |
| mndlsmidm 19629 | Subgroup sum is idempotent... |
| lsm01 19630 | Subgroup sum with the zero... |
| lsm02 19631 | Subgroup sum with the zero... |
| subglsm 19632 | The subgroup sum evaluated... |
| lssnle 19633 | Equivalent expressions for... |
| lsmmod 19634 | The modular law holds for ... |
| lsmmod2 19635 | Modular law dual for subgr... |
| lsmpropd 19636 | If two structures have the... |
| cntzrecd 19637 | Commute the "subgroups com... |
| lsmcntz 19638 | The "subgroups commute" pr... |
| lsmcntzr 19639 | The "subgroups commute" pr... |
| lsmdisj 19640 | Disjointness from a subgro... |
| lsmdisj2 19641 | Association of the disjoin... |
| lsmdisj3 19642 | Association of the disjoin... |
| lsmdisjr 19643 | Disjointness from a subgro... |
| lsmdisj2r 19644 | Association of the disjoin... |
| lsmdisj3r 19645 | Association of the disjoin... |
| lsmdisj2a 19646 | Association of the disjoin... |
| lsmdisj2b 19647 | Association of the disjoin... |
| lsmdisj3a 19648 | Association of the disjoin... |
| lsmdisj3b 19649 | Association of the disjoin... |
| subgdisj1 19650 | Vectors belonging to disjo... |
| subgdisj2 19651 | Vectors belonging to disjo... |
| subgdisjb 19652 | Vectors belonging to disjo... |
| pj1fval 19653 | The left projection functi... |
| pj1val 19654 | The left projection functi... |
| pj1eu 19655 | Uniqueness of a left proje... |
| pj1f 19656 | The left projection functi... |
| pj2f 19657 | The right projection funct... |
| pj1id 19658 | Any element of a direct su... |
| pj1eq 19659 | Any element of a direct su... |
| pj1lid 19660 | The left projection functi... |
| pj1rid 19661 | The left projection functi... |
| pj1ghm 19662 | The left projection functi... |
| pj1ghm2 19663 | The left projection functi... |
| lsmhash 19664 | The order of the direct pr... |
| efgmval 19671 | Value of the formal invers... |
| efgmf 19672 | The formal inverse operati... |
| efgmnvl 19673 | The inversion function on ... |
| efgrcl 19674 | Lemma for ~ efgval . (Con... |
| efglem 19675 | Lemma for ~ efgval . (Con... |
| efgval 19676 | Value of the free group co... |
| efger 19677 | Value of the free group co... |
| efgi 19678 | Value of the free group co... |
| efgi0 19679 | Value of the free group co... |
| efgi1 19680 | Value of the free group co... |
| efgtf 19681 | Value of the free group co... |
| efgtval 19682 | Value of the extension fun... |
| efgval2 19683 | Value of the free group co... |
| efgi2 19684 | Value of the free group co... |
| efgtlen 19685 | Value of the free group co... |
| efginvrel2 19686 | The inverse of the reverse... |
| efginvrel1 19687 | The inverse of the reverse... |
| efgsf 19688 | Value of the auxiliary fun... |
| efgsdm 19689 | Elementhood in the domain ... |
| efgsval 19690 | Value of the auxiliary fun... |
| efgsdmi 19691 | Property of the last link ... |
| efgsval2 19692 | Value of the auxiliary fun... |
| efgsrel 19693 | The start and end of any e... |
| efgs1 19694 | A singleton of an irreduci... |
| efgs1b 19695 | Every extension sequence e... |
| efgsp1 19696 | If ` F ` is an extension s... |
| efgsres 19697 | An initial segment of an e... |
| efgsfo 19698 | For any word, there is a s... |
| efgredlema 19699 | The reduced word that form... |
| efgredlemf 19700 | Lemma for ~ efgredleme . ... |
| efgredlemg 19701 | Lemma for ~ efgred . (Con... |
| efgredleme 19702 | Lemma for ~ efgred . (Con... |
| efgredlemd 19703 | The reduced word that form... |
| efgredlemc 19704 | The reduced word that form... |
| efgredlemb 19705 | The reduced word that form... |
| efgredlem 19706 | The reduced word that form... |
| efgred 19707 | The reduced word that form... |
| efgrelexlema 19708 | If two words ` A , B ` are... |
| efgrelexlemb 19709 | If two words ` A , B ` are... |
| efgrelex 19710 | If two words ` A , B ` are... |
| efgredeu 19711 | There is a unique reduced ... |
| efgred2 19712 | Two extension sequences ha... |
| efgcpbllema 19713 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19714 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19715 | Two extension sequences ha... |
| efgcpbl2 19716 | Two extension sequences ha... |
| frgpval 19717 | Value of the free group co... |
| frgpcpbl 19718 | Compatibility of the group... |
| frgp0 19719 | The free group is a group.... |
| frgpeccl 19720 | Closure of the quotient ma... |
| frgpgrp 19721 | The free group is a group.... |
| frgpadd 19722 | Addition in the free group... |
| frgpinv 19723 | The inverse of an element ... |
| frgpmhm 19724 | The "natural map" from wor... |
| vrgpfval 19725 | The canonical injection fr... |
| vrgpval 19726 | The value of the generatin... |
| vrgpf 19727 | The mapping from the index... |
| vrgpinv 19728 | The inverse of a generatin... |
| frgpuptf 19729 | Any assignment of the gene... |
| frgpuptinv 19730 | Any assignment of the gene... |
| frgpuplem 19731 | Any assignment of the gene... |
| frgpupf 19732 | Any assignment of the gene... |
| frgpupval 19733 | Any assignment of the gene... |
| frgpup1 19734 | Any assignment of the gene... |
| frgpup2 19735 | The evaluation map has the... |
| frgpup3lem 19736 | The evaluation map has the... |
| frgpup3 19737 | Universal property of the ... |
| 0frgp 19738 | The free group on zero gen... |
| isabl 19743 | The predicate "is an Abeli... |
| ablgrp 19744 | An Abelian group is a grou... |
| ablgrpd 19745 | An Abelian group is a grou... |
| ablcmn 19746 | An Abelian group is a comm... |
| ablcmnd 19747 | An Abelian group is a comm... |
| iscmn 19748 | The predicate "is a commut... |
| isabl2 19749 | The predicate "is an Abeli... |
| cmnpropd 19750 | If two structures have the... |
| ablpropd 19751 | If two structures have the... |
| ablprop 19752 | If two structures have the... |
| iscmnd 19753 | Properties that determine ... |
| isabld 19754 | Properties that determine ... |
| isabli 19755 | Properties that determine ... |
| cmnmnd 19756 | A commutative monoid is a ... |
| cmncom 19757 | A commutative monoid is co... |
| ablcom 19758 | An Abelian group operation... |
| cmn32 19759 | Commutative/associative la... |
| cmn4 19760 | Commutative/associative la... |
| cmn12 19761 | Commutative/associative la... |
| abl32 19762 | Commutative/associative la... |
| cmnmndd 19763 | A commutative monoid is a ... |
| cmnbascntr 19764 | The base set of a commutat... |
| rinvmod 19765 | Uniqueness of a right inve... |
| ablinvadd 19766 | The inverse of an Abelian ... |
| ablsub2inv 19767 | Abelian group subtraction ... |
| ablsubadd 19768 | Relationship between Abeli... |
| ablsub4 19769 | Commutative/associative su... |
| abladdsub4 19770 | Abelian group addition/sub... |
| abladdsub 19771 | Associative-type law for g... |
| ablsubadd23 19772 | Commutative/associative la... |
| ablsubaddsub 19773 | Double subtraction and add... |
| ablpncan2 19774 | Cancellation law for subtr... |
| ablpncan3 19775 | A cancellation law for Abe... |
| ablsubsub 19776 | Law for double subtraction... |
| ablsubsub4 19777 | Law for double subtraction... |
| ablpnpcan 19778 | Cancellation law for mixed... |
| ablnncan 19779 | Cancellation law for group... |
| ablsub32 19780 | Swap the second and third ... |
| ablnnncan 19781 | Cancellation law for group... |
| ablnnncan1 19782 | Cancellation law for group... |
| ablsubsub23 19783 | Swap subtrahend and result... |
| mulgnn0di 19784 | Group multiple of a sum, f... |
| mulgdi 19785 | Group multiple of a sum. ... |
| mulgmhm 19786 | The map from ` x ` to ` n ... |
| mulgghm 19787 | The map from ` x ` to ` n ... |
| mulgsubdi 19788 | Group multiple of a differ... |
| ghmfghm 19789 | The function fulfilling th... |
| ghmcmn 19790 | The image of a commutative... |
| ghmabl 19791 | The image of an abelian gr... |
| invghm 19792 | The inversion map is a gro... |
| eqgabl 19793 | Value of the subgroup cose... |
| qusecsub 19794 | Two subgroup cosets are eq... |
| subgabl 19795 | A subgroup of an abelian g... |
| subcmn 19796 | A submonoid of a commutati... |
| submcmn 19797 | A submonoid of a commutati... |
| submcmn2 19798 | A submonoid is commutative... |
| cntzcmn 19799 | The centralizer of any sub... |
| cntzcmnss 19800 | Any subset in a commutativ... |
| cntrcmnd 19801 | The center of a monoid is ... |
| cntrabl 19802 | The center of a group is a... |
| cntzspan 19803 | If the generators commute,... |
| cntzcmnf 19804 | Discharge the centralizer ... |
| ghmplusg 19805 | The pointwise sum of two l... |
| ablnsg 19806 | Every subgroup of an abeli... |
| odadd1 19807 | The order of a product in ... |
| odadd2 19808 | The order of a product in ... |
| odadd 19809 | The order of a product is ... |
| gex2abl 19810 | A group with exponent 2 (o... |
| gexexlem 19811 | Lemma for ~ gexex . (Cont... |
| gexex 19812 | In an abelian group with f... |
| torsubg 19813 | The set of all elements of... |
| oddvdssubg 19814 | The set of all elements wh... |
| lsmcomx 19815 | Subgroup sum commutes (ext... |
| ablcntzd 19816 | All subgroups in an abelia... |
| lsmcom 19817 | Subgroup sum commutes. (C... |
| lsmsubg2 19818 | The sum of two subgroups i... |
| lsm4 19819 | Commutative/associative la... |
| prdscmnd 19820 | The product of a family of... |
| prdsabld 19821 | The product of a family of... |
| pwscmn 19822 | The structure power on a c... |
| pwsabl 19823 | The structure power on an ... |
| qusabl 19824 | If ` Y ` is a subgroup of ... |
| abl1 19825 | The (smallest) structure r... |
| abln0 19826 | Abelian groups (and theref... |
| cnaddablx 19827 | The complex numbers are an... |
| cnaddabl 19828 | The complex numbers are an... |
| cnaddid 19829 | The group identity element... |
| cnaddinv 19830 | Value of the group inverse... |
| zaddablx 19831 | The integers are an Abelia... |
| frgpnabllem1 19832 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19833 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19834 | The free group on two or m... |
| imasabl 19835 | The image structure of an ... |
| iscyg 19838 | Definition of a cyclic gro... |
| iscyggen 19839 | The property of being a cy... |
| iscyggen2 19840 | The property of being a cy... |
| iscyg2 19841 | A cyclic group is a group ... |
| cyggeninv 19842 | The inverse of a cyclic ge... |
| cyggenod 19843 | An element is the generato... |
| cyggenod2 19844 | In an infinite cyclic grou... |
| iscyg3 19845 | Definition of a cyclic gro... |
| iscygd 19846 | Definition of a cyclic gro... |
| iscygodd 19847 | Show that a group with an ... |
| cycsubmcmn 19848 | The set of nonnegative int... |
| cyggrp 19849 | A cyclic group is a group.... |
| cygabl 19850 | A cyclic group is abelian.... |
| cygctb 19851 | A cyclic group is countabl... |
| 0cyg 19852 | The trivial group is cycli... |
| prmcyg 19853 | A group with prime order i... |
| lt6abl 19854 | A group with fewer than ` ... |
| ghmcyg 19855 | The image of a cyclic grou... |
| cyggex2 19856 | The exponent of a cyclic g... |
| cyggex 19857 | The exponent of a finite c... |
| cyggexb 19858 | A finite abelian group is ... |
| giccyg 19859 | Cyclicity is a group prope... |
| cycsubgcyg 19860 | The cyclic subgroup genera... |
| cycsubgcyg2 19861 | The cyclic subgroup genera... |
| gsumval3a 19862 | Value of the group sum ope... |
| gsumval3eu 19863 | The group sum as defined i... |
| gsumval3lem1 19864 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19865 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19866 | Value of the group sum ope... |
| gsumcllem 19867 | Lemma for ~ gsumcl and rel... |
| gsumzres 19868 | Extend a finite group sum ... |
| gsumzcl2 19869 | Closure of a finite group ... |
| gsumzcl 19870 | Closure of a finite group ... |
| gsumzf1o 19871 | Re-index a finite group su... |
| gsumres 19872 | Extend a finite group sum ... |
| gsumcl2 19873 | Closure of a finite group ... |
| gsumcl 19874 | Closure of a finite group ... |
| gsumf1o 19875 | Re-index a finite group su... |
| gsumreidx 19876 | Re-index a finite group su... |
| gsumzsubmcl 19877 | Closure of a group sum in ... |
| gsumsubmcl 19878 | Closure of a group sum in ... |
| gsumsubgcl 19879 | Closure of a group sum in ... |
| gsumzaddlem 19880 | The sum of two group sums.... |
| gsumzadd 19881 | The sum of two group sums.... |
| gsumadd 19882 | The sum of two group sums.... |
| gsummptfsadd 19883 | The sum of two group sums ... |
| gsummptfidmadd 19884 | The sum of two group sums ... |
| gsummptfidmadd2 19885 | The sum of two group sums ... |
| gsumzsplit 19886 | Split a group sum into two... |
| gsumsplit 19887 | Split a group sum into two... |
| gsumsplit2 19888 | Split a group sum into two... |
| gsummptfidmsplit 19889 | Split a group sum expresse... |
| gsummptfidmsplitres 19890 | Split a group sum expresse... |
| gsummptfzsplit 19891 | Split a group sum expresse... |
| gsummptfzsplitl 19892 | Split a group sum expresse... |
| gsumconst 19893 | Sum of a constant series. ... |
| gsumconstf 19894 | Sum of a constant series. ... |
| gsummptshft 19895 | Index shift of a finite gr... |
| gsumzmhm 19896 | Apply a group homomorphism... |
| gsummhm 19897 | Apply a group homomorphism... |
| gsummhm2 19898 | Apply a group homomorphism... |
| gsummptmhm 19899 | Apply a group homomorphism... |
| gsummulglem 19900 | Lemma for ~ gsummulg and ~... |
| gsummulg 19901 | Nonnegative multiple of a ... |
| gsummulgz 19902 | Integer multiple of a grou... |
| gsumzoppg 19903 | The opposite of a group su... |
| gsumzinv 19904 | Inverse of a group sum. (... |
| gsuminv 19905 | Inverse of a group sum. (... |
| gsummptfidminv 19906 | Inverse of a group sum exp... |
| gsumsub 19907 | The difference of two grou... |
| gsummptfssub 19908 | The difference of two grou... |
| gsummptfidmsub 19909 | The difference of two grou... |
| gsumsnfd 19910 | Group sum of a singleton, ... |
| gsumsnd 19911 | Group sum of a singleton, ... |
| gsumsnf 19912 | Group sum of a singleton, ... |
| gsumsn 19913 | Group sum of a singleton. ... |
| gsumpr 19914 | Group sum of a pair. (Con... |
| gsumzunsnd 19915 | Append an element to a fin... |
| gsumunsnfd 19916 | Append an element to a fin... |
| gsumunsnd 19917 | Append an element to a fin... |
| gsumunsnf 19918 | Append an element to a fin... |
| gsumunsn 19919 | Append an element to a fin... |
| gsumdifsnd 19920 | Extract a summand from a f... |
| gsumpt 19921 | Sum of a family that is no... |
| gsummptf1o 19922 | Re-index a finite group su... |
| gsummptun 19923 | Group sum of a disjoint un... |
| gsummpt1n0 19924 | If only one summand in a f... |
| gsummptif1n0 19925 | If only one summand in a f... |
| gsummptcl 19926 | Closure of a finite group ... |
| gsummptfif1o 19927 | Re-index a finite group su... |
| gsummptfzcl 19928 | Closure of a finite group ... |
| gsum2dlem1 19929 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19930 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19931 | Write a sum over a two-dim... |
| gsum2d2lem 19932 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19933 | Write a group sum over a t... |
| gsumcom2 19934 | Two-dimensional commutatio... |
| gsumxp 19935 | Write a group sum over a c... |
| gsumcom 19936 | Commute the arguments of a... |
| gsumcom3 19937 | A commutative law for fini... |
| gsumcom3fi 19938 | A commutative law for fini... |
| gsumxp2 19939 | Write a group sum over a c... |
| prdsgsum 19940 | Finite commutative sums in... |
| pwsgsum 19941 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19942 | Replacing a finitely suppo... |
| nn0gsumfz 19943 | Replacing a finitely suppo... |
| nn0gsumfz0 19944 | Replacing a finitely suppo... |
| gsummptnn0fz 19945 | A final group sum over a f... |
| gsummptnn0fzfv 19946 | A final group sum over a f... |
| telgsumfzslem 19947 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19948 | Telescoping group sum rang... |
| telgsumfz 19949 | Telescoping group sum rang... |
| telgsumfz0s 19950 | Telescoping finite group s... |
| telgsumfz0 19951 | Telescoping finite group s... |
| telgsums 19952 | Telescoping finitely suppo... |
| telgsum 19953 | Telescoping finitely suppo... |
| reldmdprd 19958 | The domain of the internal... |
| dmdprd 19959 | The domain of definition o... |
| dmdprdd 19960 | Show that a given family i... |
| dprddomprc 19961 | A family of subgroups inde... |
| dprddomcld 19962 | If a family of subgroups i... |
| dprdval0prc 19963 | The internal direct produc... |
| dprdval 19964 | The value of the internal ... |
| eldprd 19965 | A class ` A ` is an intern... |
| dprdgrp 19966 | Reverse closure for the in... |
| dprdf 19967 | The function ` S ` is a fa... |
| dprdf2 19968 | The function ` S ` is a fa... |
| dprdcntz 19969 | The function ` S ` is a fa... |
| dprddisj 19970 | The function ` S ` is a fa... |
| dprdw 19971 | The property of being a fi... |
| dprdwd 19972 | A mapping being a finitely... |
| dprdff 19973 | A finitely supported funct... |
| dprdfcl 19974 | A finitely supported funct... |
| dprdffsupp 19975 | A finitely supported funct... |
| dprdfcntz 19976 | A function on the elements... |
| dprdssv 19977 | The internal direct produc... |
| dprdfid 19978 | A function mapping all but... |
| eldprdi 19979 | The domain of definition o... |
| dprdfinv 19980 | Take the inverse of a grou... |
| dprdfadd 19981 | Take the sum of group sums... |
| dprdfsub 19982 | Take the difference of gro... |
| dprdfeq0 19983 | The zero function is the o... |
| dprdf11 19984 | Two group sums over a dire... |
| dprdsubg 19985 | The internal direct produc... |
| dprdub 19986 | Each factor is a subset of... |
| dprdlub 19987 | The direct product is smal... |
| dprdspan 19988 | The direct product is the ... |
| dprdres 19989 | Restriction of a direct pr... |
| dprdss 19990 | Create a direct product by... |
| dprdz 19991 | A family consisting entire... |
| dprd0 19992 | The empty family is an int... |
| dprdf1o 19993 | Rearrange the index set of... |
| dprdf1 19994 | Rearrange the index set of... |
| subgdmdprd 19995 | A direct product in a subg... |
| subgdprd 19996 | A direct product in a subg... |
| dprdsn 19997 | A singleton family is an i... |
| dmdprdsplitlem 19998 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19999 | The function ` S ` is a fa... |
| dprddisj2 20000 | The function ` S ` is a fa... |
| dprd2dlem2 20001 | The direct product of a co... |
| dprd2dlem1 20002 | The direct product of a co... |
| dprd2da 20003 | The direct product of a co... |
| dprd2db 20004 | The direct product of a co... |
| dprd2d2 20005 | The direct product of a co... |
| dmdprdsplit2lem 20006 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 20007 | The direct product splits ... |
| dmdprdsplit 20008 | The direct product splits ... |
| dprdsplit 20009 | The direct product is the ... |
| dmdprdpr 20010 | A singleton family is an i... |
| dprdpr 20011 | A singleton family is an i... |
| dpjlem 20012 | Lemma for theorems about d... |
| dpjcntz 20013 | The two subgroups that app... |
| dpjdisj 20014 | The two subgroups that app... |
| dpjlsm 20015 | The two subgroups that app... |
| dpjfval 20016 | Value of the direct produc... |
| dpjval 20017 | Value of the direct produc... |
| dpjf 20018 | The ` X ` -th index projec... |
| dpjidcl 20019 | The key property of projec... |
| dpjeq 20020 | Decompose a group sum into... |
| dpjid 20021 | The key property of projec... |
| dpjlid 20022 | The ` X ` -th index projec... |
| dpjrid 20023 | The ` Y ` -th index projec... |
| dpjghm 20024 | The direct product is the ... |
| dpjghm2 20025 | The direct product is the ... |
| ablfacrplem 20026 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20027 | A finite abelian group who... |
| ablfacrp2 20028 | The factors ` K , L ` of ~... |
| ablfac1lem 20029 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20030 | The factors of ~ ablfac1b ... |
| ablfac1b 20031 | Any abelian group is the d... |
| ablfac1c 20032 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20033 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20034 | The factorization of ~ abl... |
| pgpfac1lem1 20035 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20036 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20037 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20038 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20039 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20040 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20041 | Factorization of a finite ... |
| pgpfaclem1 20042 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20043 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20044 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20045 | Full factorization of a fi... |
| ablfaclem1 20046 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20047 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20048 | Lemma for ~ ablfac . (Con... |
| ablfac 20049 | The Fundamental Theorem of... |
| ablfac2 20050 | Choose generators for each... |
| issimpg 20053 | The predicate "is a simple... |
| issimpgd 20054 | Deduce a simple group from... |
| simpggrp 20055 | A simple group is a group.... |
| simpggrpd 20056 | A simple group is a group.... |
| simpg2nsg 20057 | A simple group has two nor... |
| trivnsimpgd 20058 | Trivial groups are not sim... |
| simpgntrivd 20059 | Simple groups are nontrivi... |
| simpgnideld 20060 | A simple group contains a ... |
| simpgnsgd 20061 | The only normal subgroups ... |
| simpgnsgeqd 20062 | A normal subgroup of a sim... |
| 2nsgsimpgd 20063 | If any normal subgroup of ... |
| simpgnsgbid 20064 | A nontrivial group is simp... |
| ablsimpnosubgd 20065 | A subgroup of an abelian s... |
| ablsimpg1gend 20066 | An abelian simple group is... |
| ablsimpgcygd 20067 | An abelian simple group is... |
| ablsimpgfindlem1 20068 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20069 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20070 | Relationship between the o... |
| ablsimpgfind 20071 | An abelian simple group is... |
| fincygsubgd 20072 | The subgroup referenced in... |
| fincygsubgodd 20073 | Calculate the order of a s... |
| fincygsubgodexd 20074 | A finite cyclic group has ... |
| prmgrpsimpgd 20075 | A group of prime order is ... |
| ablsimpgprmd 20076 | An abelian simple group ha... |
| ablsimpgd 20077 | An abelian group is simple... |
| fnmgp 20080 | The multiplicative group o... |
| mgpval 20081 | Value of the multiplicatio... |
| mgpplusg 20082 | Value of the group operati... |
| mgplemOLD 20083 | Obsolete version of ~ sets... |
| mgpbas 20084 | Base set of the multiplica... |
| mgpbasOLD 20085 | Obsolete version of ~ mgpb... |
| mgpsca 20086 | The multiplication monoid ... |
| mgpscaOLD 20087 | Obsolete version of ~ mgps... |
| mgptset 20088 | Topology component of the ... |
| mgptsetOLD 20089 | Obsolete version of ~ mgpt... |
| mgptopn 20090 | Topology of the multiplica... |
| mgpds 20091 | Distance function of the m... |
| mgpdsOLD 20092 | Obsolete version of ~ mgpd... |
| mgpress 20093 | Subgroup commutes with the... |
| mgpressOLD 20094 | Obsolete version of ~ mgpr... |
| prdsmgp 20095 | The multiplicative monoid ... |
| isrng 20098 | The predicate "is a non-un... |
| rngabl 20099 | A non-unital ring is an (a... |
| rngmgp 20100 | A non-unital ring is a sem... |
| rngmgpf 20101 | Restricted functionality o... |
| rnggrp 20102 | A non-unital ring is a (ad... |
| rngass 20103 | Associative law for the mu... |
| rngdi 20104 | Distributive law for the m... |
| rngdir 20105 | Distributive law for the m... |
| rngacl 20106 | Closure of the addition op... |
| rng0cl 20107 | The zero element of a non-... |
| rngcl 20108 | Closure of the multiplicat... |
| rnglz 20109 | The zero of a non-unital r... |
| rngrz 20110 | The zero of a non-unital r... |
| rngmneg1 20111 | Negation of a product in a... |
| rngmneg2 20112 | Negation of a product in a... |
| rngm2neg 20113 | Double negation of a produ... |
| rngansg 20114 | Every additive subgroup of... |
| rngsubdi 20115 | Ring multiplication distri... |
| rngsubdir 20116 | Ring multiplication distri... |
| isrngd 20117 | Properties that determine ... |
| rngpropd 20118 | If two structures have the... |
| prdsmulrngcl 20119 | Closure of the multiplicat... |
| prdsrngd 20120 | A product of non-unital ri... |
| imasrng 20121 | The image structure of a n... |
| imasrngf1 20122 | The image of a non-unital ... |
| xpsrngd 20123 | A product of two non-unita... |
| qusrng 20124 | The quotient structure of ... |
| ringidval 20127 | The value of the unity ele... |
| dfur2 20128 | The multiplicative identit... |
| ringurd 20129 | Deduce the unity element o... |
| issrg 20132 | The predicate "is a semiri... |
| srgcmn 20133 | A semiring is a commutativ... |
| srgmnd 20134 | A semiring is a monoid. (... |
| srgmgp 20135 | A semiring is a monoid und... |
| srgdilem 20136 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20137 | Closure of the multiplicat... |
| srgass 20138 | Associative law for the mu... |
| srgideu 20139 | The unity element of a sem... |
| srgfcl 20140 | Functionality of the multi... |
| srgdi 20141 | Distributive law for the m... |
| srgdir 20142 | Distributive law for the m... |
| srgidcl 20143 | The unity element of a sem... |
| srg0cl 20144 | The zero element of a semi... |
| srgidmlem 20145 | Lemma for ~ srglidm and ~ ... |
| srglidm 20146 | The unity element of a sem... |
| srgridm 20147 | The unity element of a sem... |
| issrgid 20148 | Properties showing that an... |
| srgacl 20149 | Closure of the addition op... |
| srgcom 20150 | Commutativity of the addit... |
| srgrz 20151 | The zero of a semiring is ... |
| srglz 20152 | The zero of a semiring is ... |
| srgisid 20153 | In a semiring, the only le... |
| o2timesd 20154 | An element of a ring-like ... |
| rglcom4d 20155 | Restricted commutativity o... |
| srgo2times 20156 | A semiring element plus it... |
| srgcom4lem 20157 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20158 | Restricted commutativity o... |
| srg1zr 20159 | The only semiring with a b... |
| srgen1zr 20160 | The only semiring with one... |
| srgmulgass 20161 | An associative property be... |
| srgpcomp 20162 | If two elements of a semir... |
| srgpcompp 20163 | If two elements of a semir... |
| srgpcomppsc 20164 | If two elements of a semir... |
| srglmhm 20165 | Left-multiplication in a s... |
| srgrmhm 20166 | Right-multiplication in a ... |
| srgsummulcr 20167 | A finite semiring sum mult... |
| sgsummulcl 20168 | A finite semiring sum mult... |
| srg1expzeq1 20169 | The exponentiation (by a n... |
| srgbinomlem1 20170 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20171 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20172 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20173 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20174 | Lemma for ~ srgbinom . In... |
| srgbinom 20175 | The binomial theorem for c... |
| csrgbinom 20176 | The binomial theorem for c... |
| isring 20181 | The predicate "is a (unita... |
| ringgrp 20182 | A ring is a group. (Contr... |
| ringmgp 20183 | A ring is a monoid under m... |
| iscrng 20184 | A commutative ring is a ri... |
| crngmgp 20185 | A commutative ring's multi... |
| ringgrpd 20186 | A ring is a group. (Contr... |
| ringmnd 20187 | A ring is a monoid under a... |
| ringmgm 20188 | A ring is a magma. (Contr... |
| crngring 20189 | A commutative ring is a ri... |
| crngringd 20190 | A commutative ring is a ri... |
| crnggrpd 20191 | A commutative ring is a gr... |
| mgpf 20192 | Restricted functionality o... |
| ringdilem 20193 | Properties of a unital rin... |
| ringcl 20194 | Closure of the multiplicat... |
| crngcom 20195 | A commutative ring's multi... |
| iscrng2 20196 | A commutative ring is a ri... |
| ringass 20197 | Associative law for multip... |
| ringideu 20198 | The unity element of a rin... |
| crngcomd 20199 | Multiplication is commutat... |
| crngbascntr 20200 | The base set of a commutat... |
| ringassd 20201 | Associative law for multip... |
| crng12d 20202 | Commutative/associative la... |
| ringcld 20203 | Closure of the multiplicat... |
| ringdi 20204 | Distributive law for the m... |
| ringdir 20205 | Distributive law for the m... |
| ringidcl 20206 | The unity element of a rin... |
| ring0cl 20207 | The zero element of a ring... |
| ringidmlem 20208 | Lemma for ~ ringlidm and ~... |
| ringlidm 20209 | The unity element of a rin... |
| ringridm 20210 | The unity element of a rin... |
| isringid 20211 | Properties showing that an... |
| ringlidmd 20212 | The unity element of a rin... |
| ringridmd 20213 | The unity element of a rin... |
| ringid 20214 | The multiplication operati... |
| ringo2times 20215 | A ring element plus itself... |
| ringadd2 20216 | A ring element plus itself... |
| ringidss 20217 | A subset of the multiplica... |
| ringacl 20218 | Closure of the addition op... |
| ringcomlem 20219 | Lemma for ~ ringcom . Thi... |
| ringcom 20220 | Commutativity of the addit... |
| ringabl 20221 | A ring is an Abelian group... |
| ringcmn 20222 | A ring is a commutative mo... |
| ringabld 20223 | A ring is an Abelian group... |
| ringcmnd 20224 | A ring is a commutative mo... |
| ringrng 20225 | A unital ring is a non-uni... |
| ringssrng 20226 | The unital rings are non-u... |
| isringrng 20227 | The predicate "is a unital... |
| ringpropd 20228 | If two structures have the... |
| crngpropd 20229 | If two structures have the... |
| ringprop 20230 | If two structures have the... |
| isringd 20231 | Properties that determine ... |
| iscrngd 20232 | Properties that determine ... |
| ringlz 20233 | The zero of a unital ring ... |
| ringrz 20234 | The zero of a unital ring ... |
| ringlzd 20235 | The zero of a unital ring ... |
| ringrzd 20236 | The zero of a unital ring ... |
| ringsrg 20237 | Any ring is also a semirin... |
| ring1eq0 20238 | If one and zero are equal,... |
| ring1ne0 20239 | If a ring has at least two... |
| ringinvnz1ne0 20240 | In a unital ring, a left i... |
| ringinvnzdiv 20241 | In a unital ring, a left i... |
| ringnegl 20242 | Negation in a ring is the ... |
| ringnegr 20243 | Negation in a ring is the ... |
| ringmneg1 20244 | Negation of a product in a... |
| ringmneg2 20245 | Negation of a product in a... |
| ringm2neg 20246 | Double negation of a produ... |
| ringsubdi 20247 | Ring multiplication distri... |
| ringsubdir 20248 | Ring multiplication distri... |
| mulgass2 20249 | An associative property be... |
| ring1 20250 | The (smallest) structure r... |
| ringn0 20251 | Rings exist. (Contributed... |
| ringlghm 20252 | Left-multiplication in a r... |
| ringrghm 20253 | Right-multiplication in a ... |
| gsummulc1OLD 20254 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20255 | Obsolete version of ~ gsum... |
| gsummulc1 20256 | A finite ring sum multipli... |
| gsummulc2 20257 | A finite ring sum multipli... |
| gsummgp0 20258 | If one factor in a finite ... |
| gsumdixp 20259 | Distribute a binary produc... |
| prdsmulrcl 20260 | A structure product of rin... |
| prdsringd 20261 | A product of rings is a ri... |
| prdscrngd 20262 | A product of commutative r... |
| prds1 20263 | Value of the ring unity in... |
| pwsring 20264 | A structure power of a rin... |
| pws1 20265 | Value of the ring unity in... |
| pwscrng 20266 | A structure power of a com... |
| pwsmgp 20267 | The multiplicative group o... |
| pwspjmhmmgpd 20268 | The projection given by ~ ... |
| pwsexpg 20269 | Value of a group exponenti... |
| imasring 20270 | The image structure of a r... |
| imasringf1 20271 | The image of a ring under ... |
| xpsringd 20272 | A product of two rings is ... |
| xpsring1d 20273 | The multiplicative identit... |
| qusring2 20274 | The quotient structure of ... |
| crngbinom 20275 | The binomial theorem for c... |
| opprval 20278 | Value of the opposite ring... |
| opprmulfval 20279 | Value of the multiplicatio... |
| opprmul 20280 | Value of the multiplicatio... |
| crngoppr 20281 | In a commutative ring, the... |
| opprlem 20282 | Lemma for ~ opprbas and ~ ... |
| opprlemOLD 20283 | Obsolete version of ~ oppr... |
| opprbas 20284 | Base set of an opposite ri... |
| opprbasOLD 20285 | Obsolete proof of ~ opprba... |
| oppradd 20286 | Addition operation of an o... |
| oppraddOLD 20287 | Obsolete proof of ~ opprba... |
| opprrng 20288 | An opposite non-unital rin... |
| opprrngb 20289 | A class is a non-unital ri... |
| opprring 20290 | An opposite ring is a ring... |
| opprringb 20291 | Bidirectional form of ~ op... |
| oppr0 20292 | Additive identity of an op... |
| oppr1 20293 | Multiplicative identity of... |
| opprneg 20294 | The negative function in a... |
| opprsubg 20295 | Being a subgroup is a symm... |
| mulgass3 20296 | An associative property be... |
| reldvdsr 20303 | The divides relation is a ... |
| dvdsrval 20304 | Value of the divides relat... |
| dvdsr 20305 | Value of the divides relat... |
| dvdsr2 20306 | Value of the divides relat... |
| dvdsrmul 20307 | A left-multiple of ` X ` i... |
| dvdsrcl 20308 | Closure of a dividing elem... |
| dvdsrcl2 20309 | Closure of a dividing elem... |
| dvdsrid 20310 | An element in a (unital) r... |
| dvdsrtr 20311 | Divisibility is transitive... |
| dvdsrmul1 20312 | The divisibility relation ... |
| dvdsrneg 20313 | An element divides its neg... |
| dvdsr01 20314 | In a ring, zero is divisib... |
| dvdsr02 20315 | Only zero is divisible by ... |
| isunit 20316 | Property of being a unit o... |
| 1unit 20317 | The multiplicative identit... |
| unitcl 20318 | A unit is an element of th... |
| unitss 20319 | The set of units is contai... |
| opprunit 20320 | Being a unit is a symmetri... |
| crngunit 20321 | Property of being a unit i... |
| dvdsunit 20322 | A divisor of a unit is a u... |
| unitmulcl 20323 | The product of units is a ... |
| unitmulclb 20324 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20325 | The base set of the group ... |
| unitgrp 20326 | The group of units is a gr... |
| unitabl 20327 | The group of units of a co... |
| unitgrpid 20328 | The identity of the group ... |
| unitsubm 20329 | The group of units is a su... |
| invrfval 20332 | Multiplicative inverse fun... |
| unitinvcl 20333 | The inverse of a unit exis... |
| unitinvinv 20334 | The inverse of the inverse... |
| ringinvcl 20335 | The inverse of a unit is a... |
| unitlinv 20336 | A unit times its inverse i... |
| unitrinv 20337 | A unit times its inverse i... |
| 1rinv 20338 | The inverse of the ring un... |
| 0unit 20339 | The additive identity is a... |
| unitnegcl 20340 | The negative of a unit is ... |
| ringunitnzdiv 20341 | In a unitary ring, a unit ... |
| ring1nzdiv 20342 | In a unitary ring, the rin... |
| dvrfval 20345 | Division operation in a ri... |
| dvrval 20346 | Division operation in a ri... |
| dvrcl 20347 | Closure of division operat... |
| unitdvcl 20348 | The units are closed under... |
| dvrid 20349 | A ring element divided by ... |
| dvr1 20350 | A ring element divided by ... |
| dvrass 20351 | An associative law for div... |
| dvrcan1 20352 | A cancellation law for div... |
| dvrcan3 20353 | A cancellation law for div... |
| dvreq1 20354 | Equality in terms of ratio... |
| dvrdir 20355 | Distributive law for the d... |
| rdivmuldivd 20356 | Multiplication of two rati... |
| ringinvdv 20357 | Write the inverse function... |
| rngidpropd 20358 | The ring unity depends onl... |
| dvdsrpropd 20359 | The divisibility relation ... |
| unitpropd 20360 | The set of units depends o... |
| invrpropd 20361 | The ring inverse function ... |
| isirred 20362 | An irreducible element of ... |
| isnirred 20363 | The property of being a no... |
| isirred2 20364 | Expand out the class diffe... |
| opprirred 20365 | Irreducibility is symmetri... |
| irredn0 20366 | The additive identity is n... |
| irredcl 20367 | An irreducible element is ... |
| irrednu 20368 | An irreducible element is ... |
| irredn1 20369 | The multiplicative identit... |
| irredrmul 20370 | The product of an irreduci... |
| irredlmul 20371 | The product of a unit and ... |
| irredmul 20372 | If product of two elements... |
| irredneg 20373 | The negative of an irreduc... |
| irrednegb 20374 | An element is irreducible ... |
| rnghmrcl 20381 | Reverse closure of a non-u... |
| rnghmfn 20382 | The mapping of two non-uni... |
| rnghmval 20383 | The set of the non-unital ... |
| isrnghm 20384 | A function is a non-unital... |
| isrnghmmul 20385 | A function is a non-unital... |
| rnghmmgmhm 20386 | A non-unital ring homomorp... |
| rnghmval2 20387 | The non-unital ring homomo... |
| isrngim 20388 | An isomorphism of non-unit... |
| rngimrcl 20389 | Reverse closure for an iso... |
| rnghmghm 20390 | A non-unital ring homomorp... |
| rnghmf 20391 | A ring homomorphism is a f... |
| rnghmmul 20392 | A homomorphism of non-unit... |
| isrnghm2d 20393 | Demonstration of non-unita... |
| isrnghmd 20394 | Demonstration of non-unita... |
| rnghmf1o 20395 | A non-unital ring homomorp... |
| isrngim2 20396 | An isomorphism of non-unit... |
| rngimf1o 20397 | An isomorphism of non-unit... |
| rngimrnghm 20398 | An isomorphism of non-unit... |
| rngimcnv 20399 | The converse of an isomorp... |
| rnghmco 20400 | The composition of non-uni... |
| idrnghm 20401 | The identity homomorphism ... |
| c0mgm 20402 | The constant mapping to ze... |
| c0mhm 20403 | The constant mapping to ze... |
| c0ghm 20404 | The constant mapping to ze... |
| c0snmgmhm 20405 | The constant mapping to ze... |
| c0snmhm 20406 | The constant mapping to ze... |
| c0snghm 20407 | The constant mapping to ze... |
| rngisomfv1 20408 | If there is a non-unital r... |
| rngisom1 20409 | If there is a non-unital r... |
| rngisomring 20410 | If there is a non-unital r... |
| rngisomring1 20411 | If there is a non-unital r... |
| dfrhm2 20417 | The property of a ring hom... |
| rhmrcl1 20419 | Reverse closure of a ring ... |
| rhmrcl2 20420 | Reverse closure of a ring ... |
| isrhm 20421 | A function is a ring homom... |
| rhmmhm 20422 | A ring homomorphism is a h... |
| rhmisrnghm 20423 | Each unital ring homomorph... |
| isrim0OLD 20424 | Obsolete version of ~ isri... |
| rimrcl 20425 | Reverse closure for an iso... |
| isrim0 20426 | A ring isomorphism is a ho... |
| rhmghm 20427 | A ring homomorphism is an ... |
| rhmf 20428 | A ring homomorphism is a f... |
| rhmmul 20429 | A homomorphism of rings pr... |
| isrhm2d 20430 | Demonstration of ring homo... |
| isrhmd 20431 | Demonstration of ring homo... |
| rhm1 20432 | Ring homomorphisms are req... |
| idrhm 20433 | The identity homomorphism ... |
| rhmf1o 20434 | A ring homomorphism is bij... |
| isrim 20435 | An isomorphism of rings is... |
| isrimOLD 20436 | Obsolete version of ~ isri... |
| rimf1o 20437 | An isomorphism of rings is... |
| rimrhmOLD 20438 | Obsolete version of ~ rimr... |
| rimrhm 20439 | A ring isomorphism is a ho... |
| rimgim 20440 | An isomorphism of rings is... |
| rimisrngim 20441 | Each unital ring isomorphi... |
| rhmfn 20442 | The mapping of two rings t... |
| rhmval 20443 | The ring homomorphisms bet... |
| rhmco 20444 | The composition of ring ho... |
| pwsco1rhm 20445 | Right composition with a f... |
| pwsco2rhm 20446 | Left composition with a ri... |
| brric 20447 | The relation "is isomorphi... |
| brrici 20448 | Prove isomorphic by an exp... |
| brric2 20449 | The relation "is isomorphi... |
| ricgic 20450 | If two rings are (ring) is... |
| rhmdvdsr 20451 | A ring homomorphism preser... |
| rhmopp 20452 | A ring homomorphism is als... |
| elrhmunit 20453 | Ring homomorphisms preserv... |
| rhmunitinv 20454 | Ring homomorphisms preserv... |
| isnzr 20457 | Property of a nonzero ring... |
| nzrnz 20458 | One and zero are different... |
| nzrring 20459 | A nonzero ring is a ring. ... |
| nzrringOLD 20460 | Obsolete version of ~ nzrr... |
| isnzr2 20461 | Equivalent characterizatio... |
| isnzr2hash 20462 | Equivalent characterizatio... |
| opprnzr 20463 | The opposite of a nonzero ... |
| ringelnzr 20464 | A ring is nonzero if it ha... |
| nzrunit 20465 | A unit is nonzero in any n... |
| 0ringnnzr 20466 | A ring is a zero ring iff ... |
| 0ring 20467 | If a ring has only one ele... |
| 0ringdif 20468 | A zero ring is a ring whic... |
| 0ringbas 20469 | The base set of a zero rin... |
| 0ring01eq 20470 | In a ring with only one el... |
| 01eq0ring 20471 | If the zero and the identi... |
| 01eq0ringOLD 20472 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20473 | In a unital ring the zero ... |
| 0ring1eq0 20474 | In a zero ring, a ring whi... |
| c0rhm 20475 | The constant mapping to ze... |
| c0rnghm 20476 | The constant mapping to ze... |
| zrrnghm 20477 | The constant mapping to ze... |
| nrhmzr 20478 | There is no ring homomorph... |
| islring 20481 | The predicate "is a local ... |
| lringnzr 20482 | A local ring is a nonzero ... |
| lringring 20483 | A local ring is a ring. (... |
| lringnz 20484 | A local ring is a nonzero ... |
| lringuplu 20485 | If the sum of two elements... |
| issubrng 20488 | The subring of non-unital ... |
| subrngss 20489 | A subring is a subset. (C... |
| subrngid 20490 | Every non-unital ring is a... |
| subrngrng 20491 | A subring is a non-unital ... |
| subrngrcl 20492 | Reverse closure for a subr... |
| subrngsubg 20493 | A subring is a subgroup. ... |
| subrngringnsg 20494 | A subring is a normal subg... |
| subrngbas 20495 | Base set of a subring stru... |
| subrng0 20496 | A subring always has the s... |
| subrngacl 20497 | A subring is closed under ... |
| subrngmcl 20498 | A subgroup is closed under... |
| issubrng2 20499 | Characterize the subrings ... |
| opprsubrng 20500 | Being a subring is a symme... |
| subrngint 20501 | The intersection of a none... |
| subrngin 20502 | The intersection of two su... |
| subrngmre 20503 | The subrings of a non-unit... |
| subsubrng 20504 | A subring of a subring is ... |
| subsubrng2 20505 | The set of subrings of a s... |
| rhmimasubrnglem 20506 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20507 | The homomorphic image of a... |
| cntzsubrng 20508 | Centralizers in a non-unit... |
| subrngpropd 20509 | If two structures have the... |
| issubrg 20514 | The subring predicate. (C... |
| subrgss 20515 | A subring is a subset. (C... |
| subrgid 20516 | Every ring is a subring of... |
| subrgring 20517 | A subring is a ring. (Con... |
| subrgcrng 20518 | A subring of a commutative... |
| subrgrcl 20519 | Reverse closure for a subr... |
| subrgsubg 20520 | A subring is a subgroup. ... |
| subrgsubrng 20521 | A subring of a unital ring... |
| subrg0 20522 | A subring always has the s... |
| subrg1cl 20523 | A subring contains the mul... |
| subrgbas 20524 | Base set of a subring stru... |
| subrg1 20525 | A subring always has the s... |
| subrgacl 20526 | A subring is closed under ... |
| subrgmcl 20527 | A subgroup is closed under... |
| subrgsubm 20528 | A subring is a submonoid o... |
| subrgdvds 20529 | If an element divides anot... |
| subrguss 20530 | A unit of a subring is a u... |
| subrginv 20531 | A subring always has the s... |
| subrgdv 20532 | A subring always has the s... |
| subrgunit 20533 | An element of a ring is a ... |
| subrgugrp 20534 | The units of a subring for... |
| issubrg2 20535 | Characterize the subrings ... |
| opprsubrg 20536 | Being a subring is a symme... |
| subrgnzr 20537 | A subring of a nonzero rin... |
| subrgint 20538 | The intersection of a none... |
| subrgin 20539 | The intersection of two su... |
| subrgmre 20540 | The subrings of a ring are... |
| subsubrg 20541 | A subring of a subring is ... |
| subsubrg2 20542 | The set of subrings of a s... |
| issubrg3 20543 | A subring is an additive s... |
| resrhm 20544 | Restriction of a ring homo... |
| resrhm2b 20545 | Restriction of the codomai... |
| rhmeql 20546 | The equalizer of two ring ... |
| rhmima 20547 | The homomorphic image of a... |
| rnrhmsubrg 20548 | The range of a ring homomo... |
| cntzsubr 20549 | Centralizers in a ring are... |
| pwsdiagrhm 20550 | Diagonal homomorphism into... |
| subrgpropd 20551 | If two structures have the... |
| rhmpropd 20552 | Ring homomorphism depends ... |
| rngcval 20555 | Value of the category of n... |
| rnghmresfn 20556 | The class of non-unital ri... |
| rnghmresel 20557 | An element of the non-unit... |
| rngcbas 20558 | Set of objects of the cate... |
| rngchomfval 20559 | Set of arrows of the categ... |
| rngchom 20560 | Set of arrows of the categ... |
| elrngchom 20561 | A morphism of non-unital r... |
| rngchomfeqhom 20562 | The functionalized Hom-set... |
| rngccofval 20563 | Composition in the categor... |
| rngcco 20564 | Composition in the categor... |
| dfrngc2 20565 | Alternate definition of th... |
| rnghmsscmap2 20566 | The non-unital ring homomo... |
| rnghmsscmap 20567 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20568 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20569 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20570 | The non-unital ring homomo... |
| rngccat 20571 | The category of non-unital... |
| rngcid 20572 | The identity arrow in the ... |
| rngcsect 20573 | A section in the category ... |
| rngcinv 20574 | An inverse in the category... |
| rngciso 20575 | An isomorphism in the cate... |
| rngcifuestrc 20576 | The "inclusion functor" fr... |
| funcrngcsetc 20577 | The "natural forgetful fun... |
| funcrngcsetcALT 20578 | Alternate proof of ~ funcr... |
| zrinitorngc 20579 | The zero ring is an initia... |
| zrtermorngc 20580 | The zero ring is a termina... |
| zrzeroorngc 20581 | The zero ring is a zero ob... |
| ringcval 20584 | Value of the category of u... |
| rhmresfn 20585 | The class of unital ring h... |
| rhmresel 20586 | An element of the unital r... |
| ringcbas 20587 | Set of objects of the cate... |
| ringchomfval 20588 | Set of arrows of the categ... |
| ringchom 20589 | Set of arrows of the categ... |
| elringchom 20590 | A morphism of unital rings... |
| ringchomfeqhom 20591 | The functionalized Hom-set... |
| ringccofval 20592 | Composition in the categor... |
| ringcco 20593 | Composition in the categor... |
| dfringc2 20594 | Alternate definition of th... |
| rhmsscmap2 20595 | The unital ring homomorphi... |
| rhmsscmap 20596 | The unital ring homomorphi... |
| rhmsubcsetclem1 20597 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20598 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20599 | The unital ring homomorphi... |
| ringccat 20600 | The category of unital rin... |
| ringcid 20601 | The identity arrow in the ... |
| rhmsscrnghm 20602 | The unital ring homomorphi... |
| rhmsubcrngclem1 20603 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20604 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20605 | The unital ring homomorphi... |
| rngcresringcat 20606 | The restriction of the cat... |
| ringcsect 20607 | A section in the category ... |
| ringcinv 20608 | An inverse in the category... |
| ringciso 20609 | An isomorphism in the cate... |
| ringcbasbas 20610 | An element of the base set... |
| funcringcsetc 20611 | The "natural forgetful fun... |
| zrtermoringc 20612 | The zero ring is a termina... |
| zrninitoringc 20613 | The zero ring is not an in... |
| srhmsubclem1 20614 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20615 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20616 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20617 | According to ~ df-subc , t... |
| sringcat 20618 | The restriction of the cat... |
| crhmsubc 20619 | According to ~ df-subc , t... |
| cringcat 20620 | The restriction of the cat... |
| rngcrescrhm 20621 | The category of non-unital... |
| rhmsubclem1 20622 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20623 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20624 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20625 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20626 | According to ~ df-subc , t... |
| rhmsubccat 20627 | The restriction of the cat... |
| isdrng 20632 | The predicate "is a divisi... |
| drngunit 20633 | Elementhood in the set of ... |
| drngui 20634 | The set of units of a divi... |
| drngring 20635 | A division ring is a ring.... |
| drngringd 20636 | A division ring is a ring.... |
| drnggrpd 20637 | A division ring is a group... |
| drnggrp 20638 | A division ring is a group... |
| isfld 20639 | A field is a commutative d... |
| flddrngd 20640 | A field is a division ring... |
| fldcrngd 20641 | A field is a commutative r... |
| isdrng2 20642 | A division ring can equiva... |
| drngprop 20643 | If two structures have the... |
| drngmgp 20644 | A division ring contains a... |
| drngmcl 20645 | The product of two nonzero... |
| drngid 20646 | A division ring's unity is... |
| drngunz 20647 | A division ring's unity is... |
| drngnzr 20648 | All division rings are non... |
| drngid2 20649 | Properties showing that an... |
| drnginvrcl 20650 | Closure of the multiplicat... |
| drnginvrn0 20651 | The multiplicative inverse... |
| drnginvrcld 20652 | Closure of the multiplicat... |
| drnginvrl 20653 | Property of the multiplica... |
| drnginvrr 20654 | Property of the multiplica... |
| drnginvrld 20655 | Property of the multiplica... |
| drnginvrrd 20656 | Property of the multiplica... |
| drngmul0or 20657 | A product is zero iff one ... |
| drngmulne0 20658 | A product is nonzero iff b... |
| drngmuleq0 20659 | An element is zero iff its... |
| opprdrng 20660 | The opposite of a division... |
| isdrngd 20661 | Properties that characteri... |
| isdrngrd 20662 | Properties that characteri... |
| isdrngdOLD 20663 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20664 | Obsolete version of ~ isdr... |
| drngpropd 20665 | If two structures have the... |
| fldpropd 20666 | If two structures have the... |
| rng1nnzr 20667 | The (smallest) structure r... |
| ring1zr 20668 | The only (unital) ring wit... |
| rngen1zr 20669 | The only (unital) ring wit... |
| ringen1zr 20670 | The only unital ring with ... |
| rng1nfld 20671 | The zero ring is not a fie... |
| issubdrg 20672 | Characterize the subfields... |
| drhmsubc 20673 | According to ~ df-subc , t... |
| drngcat 20674 | The restriction of the cat... |
| fldcat 20675 | The restriction of the cat... |
| fldc 20676 | The restriction of the cat... |
| fldhmsubc 20677 | According to ~ df-subc , t... |
| issdrg 20680 | Property of a division sub... |
| sdrgrcl 20681 | Reverse closure for a sub-... |
| sdrgdrng 20682 | A sub-division-ring is a d... |
| sdrgsubrg 20683 | A sub-division-ring is a s... |
| sdrgid 20684 | Every division ring is a d... |
| sdrgss 20685 | A division subring is a su... |
| sdrgbas 20686 | Base set of a sub-division... |
| issdrg2 20687 | Property of a division sub... |
| sdrgunit 20688 | A unit of a sub-division-r... |
| imadrhmcl 20689 | The image of a (nontrivial... |
| fldsdrgfld 20690 | A sub-division-ring of a f... |
| acsfn1p 20691 | Construction of a closure ... |
| subrgacs 20692 | Closure property of subrin... |
| sdrgacs 20693 | Closure property of divisi... |
| cntzsdrg 20694 | Centralizers in division r... |
| subdrgint 20695 | The intersection of a none... |
| sdrgint 20696 | The intersection of a none... |
| primefld 20697 | The smallest sub division ... |
| primefld0cl 20698 | The prime field contains t... |
| primefld1cl 20699 | The prime field contains t... |
| abvfval 20702 | Value of the set of absolu... |
| isabv 20703 | Elementhood in the set of ... |
| isabvd 20704 | Properties that determine ... |
| abvrcl 20705 | Reverse closure for the ab... |
| abvfge0 20706 | An absolute value is a fun... |
| abvf 20707 | An absolute value is a fun... |
| abvcl 20708 | An absolute value is a fun... |
| abvge0 20709 | The absolute value of a nu... |
| abveq0 20710 | The value of an absolute v... |
| abvne0 20711 | The absolute value of a no... |
| abvgt0 20712 | The absolute value of a no... |
| abvmul 20713 | An absolute value distribu... |
| abvtri 20714 | An absolute value satisfie... |
| abv0 20715 | The absolute value of zero... |
| abv1z 20716 | The absolute value of one ... |
| abv1 20717 | The absolute value of one ... |
| abvneg 20718 | The absolute value of a ne... |
| abvsubtri 20719 | An absolute value satisfie... |
| abvrec 20720 | The absolute value distrib... |
| abvdiv 20721 | The absolute value distrib... |
| abvdom 20722 | Any ring with an absolute ... |
| abvres 20723 | The restriction of an abso... |
| abvtrivd 20724 | The trivial absolute value... |
| abvtriv 20725 | The trivial absolute value... |
| abvpropd 20726 | If two structures have the... |
| staffval 20731 | The functionalization of t... |
| stafval 20732 | The functionalization of t... |
| staffn 20733 | The functionalization is e... |
| issrng 20734 | The predicate "is a star r... |
| srngrhm 20735 | The involution function in... |
| srngring 20736 | A star ring is a ring. (C... |
| srngcnv 20737 | The involution function in... |
| srngf1o 20738 | The involution function in... |
| srngcl 20739 | The involution function in... |
| srngnvl 20740 | The involution function in... |
| srngadd 20741 | The involution function in... |
| srngmul 20742 | The involution function in... |
| srng1 20743 | The conjugate of the ring ... |
| srng0 20744 | The conjugate of the ring ... |
| issrngd 20745 | Properties that determine ... |
| idsrngd 20746 | A commutative ring is a st... |
| islmod 20751 | The predicate "is a left m... |
| lmodlema 20752 | Lemma for properties of a ... |
| islmodd 20753 | Properties that determine ... |
| lmodgrp 20754 | A left module is a group. ... |
| lmodring 20755 | The scalar component of a ... |
| lmodfgrp 20756 | The scalar component of a ... |
| lmodgrpd 20757 | A left module is a group. ... |
| lmodbn0 20758 | The base set of a left mod... |
| lmodacl 20759 | Closure of ring addition f... |
| lmodmcl 20760 | Closure of ring multiplica... |
| lmodsn0 20761 | The set of scalars in a le... |
| lmodvacl 20762 | Closure of vector addition... |
| lmodass 20763 | Left module vector sum is ... |
| lmodlcan 20764 | Left cancellation law for ... |
| lmodvscl 20765 | Closure of scalar product ... |
| lmodvscld 20766 | Closure of scalar product ... |
| scaffval 20767 | The scalar multiplication ... |
| scafval 20768 | The scalar multiplication ... |
| scafeq 20769 | If the scalar multiplicati... |
| scaffn 20770 | The scalar multiplication ... |
| lmodscaf 20771 | The scalar multiplication ... |
| lmodvsdi 20772 | Distributive law for scala... |
| lmodvsdir 20773 | Distributive law for scala... |
| lmodvsass 20774 | Associative law for scalar... |
| lmod0cl 20775 | The ring zero in a left mo... |
| lmod1cl 20776 | The ring unity in a left m... |
| lmodvs1 20777 | Scalar product with the ri... |
| lmod0vcl 20778 | The zero vector is a vecto... |
| lmod0vlid 20779 | Left identity law for the ... |
| lmod0vrid 20780 | Right identity law for the... |
| lmod0vid 20781 | Identity equivalent to the... |
| lmod0vs 20782 | Zero times a vector is the... |
| lmodvs0 20783 | Anything times the zero ve... |
| lmodvsmmulgdi 20784 | Distributive law for a gro... |
| lmodfopnelem1 20785 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20786 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20787 | The (functionalized) opera... |
| lcomf 20788 | A linear-combination sum i... |
| lcomfsupp 20789 | A linear-combination sum i... |
| lmodvnegcl 20790 | Closure of vector negative... |
| lmodvnegid 20791 | Addition of a vector with ... |
| lmodvneg1 20792 | Minus 1 times a vector is ... |
| lmodvsneg 20793 | Multiplication of a vector... |
| lmodvsubcl 20794 | Closure of vector subtract... |
| lmodcom 20795 | Left module vector sum is ... |
| lmodabl 20796 | A left module is an abelia... |
| lmodcmn 20797 | A left module is a commuta... |
| lmodnegadd 20798 | Distribute negation throug... |
| lmod4 20799 | Commutative/associative la... |
| lmodvsubadd 20800 | Relationship between vecto... |
| lmodvaddsub4 20801 | Vector addition/subtractio... |
| lmodvpncan 20802 | Addition/subtraction cance... |
| lmodvnpcan 20803 | Cancellation law for vecto... |
| lmodvsubval2 20804 | Value of vector subtractio... |
| lmodsubvs 20805 | Subtraction of a scalar pr... |
| lmodsubdi 20806 | Scalar multiplication dist... |
| lmodsubdir 20807 | Scalar multiplication dist... |
| lmodsubeq0 20808 | If the difference between ... |
| lmodsubid 20809 | Subtraction of a vector fr... |
| lmodvsghm 20810 | Scalar multiplication of t... |
| lmodprop2d 20811 | If two structures have the... |
| lmodpropd 20812 | If two structures have the... |
| gsumvsmul 20813 | Pull a scalar multiplicati... |
| mptscmfsupp0 20814 | A mapping to a scalar prod... |
| mptscmfsuppd 20815 | A function mapping to a sc... |
| rmodislmodlem 20816 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20817 | The right module ` R ` ind... |
| rmodislmodOLD 20818 | Obsolete version of ~ rmod... |
| lssset 20821 | The set of all (not necess... |
| islss 20822 | The predicate "is a subspa... |
| islssd 20823 | Properties that determine ... |
| lssss 20824 | A subspace is a set of vec... |
| lssel 20825 | A subspace member is a vec... |
| lss1 20826 | The set of vectors in a le... |
| lssuni 20827 | The union of all subspaces... |
| lssn0 20828 | A subspace is not empty. ... |
| 00lss 20829 | The empty structure has no... |
| lsscl 20830 | Closure property of a subs... |
| lssvacl 20831 | Closure of vector addition... |
| lssvsubcl 20832 | Closure of vector subtract... |
| lssvancl1 20833 | Non-closure: if one vector... |
| lssvancl2 20834 | Non-closure: if one vector... |
| lss0cl 20835 | The zero vector belongs to... |
| lsssn0 20836 | The singleton of the zero ... |
| lss0ss 20837 | The zero subspace is inclu... |
| lssle0 20838 | No subspace is smaller tha... |
| lssne0 20839 | A nonzero subspace has a n... |
| lssvneln0 20840 | A vector ` X ` which doesn... |
| lssneln0 20841 | A vector ` X ` which doesn... |
| lssssr 20842 | Conclude subspace ordering... |
| lssvscl 20843 | Closure of scalar product ... |
| lssvnegcl 20844 | Closure of negative vector... |
| lsssubg 20845 | All subspaces are subgroup... |
| lsssssubg 20846 | All subspaces are subgroup... |
| islss3 20847 | A linear subspace of a mod... |
| lsslmod 20848 | A submodule is a module. ... |
| lsslss 20849 | The subspaces of a subspac... |
| islss4 20850 | A linear subspace is a sub... |
| lss1d 20851 | One-dimensional subspace (... |
| lssintcl 20852 | The intersection of a none... |
| lssincl 20853 | The intersection of two su... |
| lssmre 20854 | The subspaces of a module ... |
| lssacs 20855 | Submodules are an algebrai... |
| prdsvscacl 20856 | Pointwise scalar multiplic... |
| prdslmodd 20857 | The product of a family of... |
| pwslmod 20858 | A structure power of a lef... |
| lspfval 20861 | The span function for a le... |
| lspf 20862 | The span function on a lef... |
| lspval 20863 | The span of a set of vecto... |
| lspcl 20864 | The span of a set of vecto... |
| lspsncl 20865 | The span of a singleton is... |
| lspprcl 20866 | The span of a pair is a su... |
| lsptpcl 20867 | The span of an unordered t... |
| lspsnsubg 20868 | The span of a singleton is... |
| 00lsp 20869 | ~ fvco4i lemma for linear ... |
| lspid 20870 | The span of a subspace is ... |
| lspssv 20871 | A span is a set of vectors... |
| lspss 20872 | Span preserves subset orde... |
| lspssid 20873 | A set of vectors is a subs... |
| lspidm 20874 | The span of a set of vecto... |
| lspun 20875 | The span of union is the s... |
| lspssp 20876 | If a set of vectors is a s... |
| mrclsp 20877 | Moore closure generalizes ... |
| lspsnss 20878 | The span of the singleton ... |
| lspsnel3 20879 | A member of the span of th... |
| lspprss 20880 | The span of a pair of vect... |
| lspsnid 20881 | A vector belongs to the sp... |
| lspsnel6 20882 | Relationship between a vec... |
| lspsnel5 20883 | Relationship between a vec... |
| lspsnel5a 20884 | Relationship between a vec... |
| lspprid1 20885 | A member of a pair of vect... |
| lspprid2 20886 | A member of a pair of vect... |
| lspprvacl 20887 | The sum of two vectors bel... |
| lssats2 20888 | A way to express atomistic... |
| lspsneli 20889 | A scalar product with a ve... |
| lspsn 20890 | Span of the singleton of a... |
| lspsnel 20891 | Member of span of the sing... |
| lspsnvsi 20892 | Span of a scalar product o... |
| lspsnss2 20893 | Comparable spans of single... |
| lspsnneg 20894 | Negation does not change t... |
| lspsnsub 20895 | Swapping subtraction order... |
| lspsn0 20896 | Span of the singleton of t... |
| lsp0 20897 | Span of the empty set. (C... |
| lspuni0 20898 | Union of the span of the e... |
| lspun0 20899 | The span of a union with t... |
| lspsneq0 20900 | Span of the singleton is t... |
| lspsneq0b 20901 | Equal singleton spans impl... |
| lmodindp1 20902 | Two independent (non-colin... |
| lsslsp 20903 | Spans in submodules corres... |
| lsslspOLD 20904 | Obsolete version of ~ lssl... |
| lss0v 20905 | The zero vector in a submo... |
| lsspropd 20906 | If two structures have the... |
| lsppropd 20907 | If two structures have the... |
| reldmlmhm 20914 | Lemma for module homomorph... |
| lmimfn 20915 | Lemma for module isomorphi... |
| islmhm 20916 | Property of being a homomo... |
| islmhm3 20917 | Property of a module homom... |
| lmhmlem 20918 | Non-quantified consequence... |
| lmhmsca 20919 | A homomorphism of left mod... |
| lmghm 20920 | A homomorphism of left mod... |
| lmhmlmod2 20921 | A homomorphism of left mod... |
| lmhmlmod1 20922 | A homomorphism of left mod... |
| lmhmf 20923 | A homomorphism of left mod... |
| lmhmlin 20924 | A homomorphism of left mod... |
| lmodvsinv 20925 | Multiplication of a vector... |
| lmodvsinv2 20926 | Multiplying a negated vect... |
| islmhm2 20927 | A one-equation proof of li... |
| islmhmd 20928 | Deduction for a module hom... |
| 0lmhm 20929 | The constant zero linear f... |
| idlmhm 20930 | The identity function on a... |
| invlmhm 20931 | The negative function on a... |
| lmhmco 20932 | The composition of two mod... |
| lmhmplusg 20933 | The pointwise sum of two l... |
| lmhmvsca 20934 | The pointwise scalar produ... |
| lmhmf1o 20935 | A bijective module homomor... |
| lmhmima 20936 | The image of a subspace un... |
| lmhmpreima 20937 | The inverse image of a sub... |
| lmhmlsp 20938 | Homomorphisms preserve spa... |
| lmhmrnlss 20939 | The range of a homomorphis... |
| lmhmkerlss 20940 | The kernel of a homomorphi... |
| reslmhm 20941 | Restriction of a homomorph... |
| reslmhm2 20942 | Expansion of the codomain ... |
| reslmhm2b 20943 | Expansion of the codomain ... |
| lmhmeql 20944 | The equalizer of two modul... |
| lspextmo 20945 | A linear function is compl... |
| pwsdiaglmhm 20946 | Diagonal homomorphism into... |
| pwssplit0 20947 | Splitting for structure po... |
| pwssplit1 20948 | Splitting for structure po... |
| pwssplit2 20949 | Splitting for structure po... |
| pwssplit3 20950 | Splitting for structure po... |
| islmim 20951 | An isomorphism of left mod... |
| lmimf1o 20952 | An isomorphism of left mod... |
| lmimlmhm 20953 | An isomorphism of modules ... |
| lmimgim 20954 | An isomorphism of modules ... |
| islmim2 20955 | An isomorphism of left mod... |
| lmimcnv 20956 | The converse of a bijectiv... |
| brlmic 20957 | The relation "is isomorphi... |
| brlmici 20958 | Prove isomorphic by an exp... |
| lmiclcl 20959 | Isomorphism implies the le... |
| lmicrcl 20960 | Isomorphism implies the ri... |
| lmicsym 20961 | Module isomorphism is symm... |
| lmhmpropd 20962 | Module homomorphism depend... |
| islbs 20965 | The predicate " ` B ` is a... |
| lbsss 20966 | A basis is a set of vector... |
| lbsel 20967 | An element of a basis is a... |
| lbssp 20968 | The span of a basis is the... |
| lbsind 20969 | A basis is linearly indepe... |
| lbsind2 20970 | A basis is linearly indepe... |
| lbspss 20971 | No proper subset of a basi... |
| lsmcl 20972 | The sum of two subspaces i... |
| lsmspsn 20973 | Member of subspace sum of ... |
| lsmelval2 20974 | Subspace sum membership in... |
| lsmsp 20975 | Subspace sum in terms of s... |
| lsmsp2 20976 | Subspace sum of spans of s... |
| lsmssspx 20977 | Subspace sum (in its exten... |
| lsmpr 20978 | The span of a pair of vect... |
| lsppreli 20979 | A vector expressed as a su... |
| lsmelpr 20980 | Two ways to say that a vec... |
| lsppr0 20981 | The span of a vector paire... |
| lsppr 20982 | Span of a pair of vectors.... |
| lspprel 20983 | Member of the span of a pa... |
| lspprabs 20984 | Absorption of vector sum i... |
| lspvadd 20985 | The span of a vector sum i... |
| lspsntri 20986 | Triangle-type inequality f... |
| lspsntrim 20987 | Triangle-type inequality f... |
| lbspropd 20988 | If two structures have the... |
| pj1lmhm 20989 | The left projection functi... |
| pj1lmhm2 20990 | The left projection functi... |
| islvec 20993 | The predicate "is a left v... |
| lvecdrng 20994 | The set of scalars of a le... |
| lveclmod 20995 | A left vector space is a l... |
| lveclmodd 20996 | A vector space is a left m... |
| lvecgrpd 20997 | A vector space is a group.... |
| lsslvec 20998 | A vector subspace is a vec... |
| lmhmlvec 20999 | The property for modules t... |
| lvecvs0or 21000 | If a scalar product is zer... |
| lvecvsn0 21001 | A scalar product is nonzer... |
| lssvs0or 21002 | If a scalar product belong... |
| lvecvscan 21003 | Cancellation law for scala... |
| lvecvscan2 21004 | Cancellation law for scala... |
| lvecinv 21005 | Invert coefficient of scal... |
| lspsnvs 21006 | A nonzero scalar product d... |
| lspsneleq 21007 | Membership relation that i... |
| lspsncmp 21008 | Comparable spans of nonzer... |
| lspsnne1 21009 | Two ways to express that v... |
| lspsnne2 21010 | Two ways to express that v... |
| lspsnnecom 21011 | Swap two vectors with diff... |
| lspabs2 21012 | Absorption law for span of... |
| lspabs3 21013 | Absorption law for span of... |
| lspsneq 21014 | Equal spans of singletons ... |
| lspsneu 21015 | Nonzero vectors with equal... |
| lspsnel4 21016 | A member of the span of th... |
| lspdisj 21017 | The span of a vector not i... |
| lspdisjb 21018 | A nonzero vector is not in... |
| lspdisj2 21019 | Unequal spans are disjoint... |
| lspfixed 21020 | Show membership in the spa... |
| lspexch 21021 | Exchange property for span... |
| lspexchn1 21022 | Exchange property for span... |
| lspexchn2 21023 | Exchange property for span... |
| lspindpi 21024 | Partial independence prope... |
| lspindp1 21025 | Alternate way to say 3 vec... |
| lspindp2l 21026 | Alternate way to say 3 vec... |
| lspindp2 21027 | Alternate way to say 3 vec... |
| lspindp3 21028 | Independence of 2 vectors ... |
| lspindp4 21029 | (Partial) independence of ... |
| lvecindp 21030 | Compute the ` X ` coeffici... |
| lvecindp2 21031 | Sums of independent vector... |
| lspsnsubn0 21032 | Unequal singleton spans im... |
| lsmcv 21033 | Subspace sum has the cover... |
| lspsolvlem 21034 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21035 | If ` X ` is in the span of... |
| lssacsex 21036 | In a vector space, subspac... |
| lspsnat 21037 | There is no subspace stric... |
| lspsncv0 21038 | The span of a singleton co... |
| lsppratlem1 21039 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21040 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21041 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21042 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21043 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21044 | Lemma for ~ lspprat . Neg... |
| lspprat 21045 | A proper subspace of the s... |
| islbs2 21046 | An equivalent formulation ... |
| islbs3 21047 | An equivalent formulation ... |
| lbsacsbs 21048 | Being a basis in a vector ... |
| lvecdim 21049 | The dimension theorem for ... |
| lbsextlem1 21050 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21051 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21052 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21053 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21054 | For any linearly independe... |
| lbsext 21055 | For any linearly independe... |
| lbsexg 21056 | Every vector space has a b... |
| lbsex 21057 | Every vector space has a b... |
| lvecprop2d 21058 | If two structures have the... |
| lvecpropd 21059 | If two structures have the... |
| sraval 21064 | Lemma for ~ srabase throug... |
| sralem 21065 | Lemma for ~ srabase and si... |
| sralemOLD 21066 | Obsolete version of ~ sral... |
| srabase 21067 | Base set of a subring alge... |
| srabaseOLD 21068 | Obsolete proof of ~ srabas... |
| sraaddg 21069 | Additive operation of a su... |
| sraaddgOLD 21070 | Obsolete proof of ~ sraadd... |
| sramulr 21071 | Multiplicative operation o... |
| sramulrOLD 21072 | Obsolete proof of ~ sramul... |
| srasca 21073 | The set of scalars of a su... |
| srascaOLD 21074 | Obsolete proof of ~ srasca... |
| sravsca 21075 | The scalar product operati... |
| sravscaOLD 21076 | Obsolete proof of ~ sravsc... |
| sraip 21077 | The inner product operatio... |
| sratset 21078 | Topology component of a su... |
| sratsetOLD 21079 | Obsolete proof of ~ sratse... |
| sratopn 21080 | Topology component of a su... |
| srads 21081 | Distance function of a sub... |
| sradsOLD 21082 | Obsolete proof of ~ srads ... |
| sraring 21083 | Condition for a subring al... |
| sralmod 21084 | The subring algebra is a l... |
| sralmod0 21085 | The subring module inherit... |
| issubrgd 21086 | Prove a subring by closure... |
| rlmfn 21087 | ` ringLMod ` is a function... |
| rlmval 21088 | Value of the ring module. ... |
| rlmval2 21089 | Value of the ring module e... |
| rlmbas 21090 | Base set of the ring modul... |
| rlmplusg 21091 | Vector addition in the rin... |
| rlm0 21092 | Zero vector in the ring mo... |
| rlmsub 21093 | Subtraction in the ring mo... |
| rlmmulr 21094 | Ring multiplication in the... |
| rlmsca 21095 | Scalars in the ring module... |
| rlmsca2 21096 | Scalars in the ring module... |
| rlmvsca 21097 | Scalar multiplication in t... |
| rlmtopn 21098 | Topology component of the ... |
| rlmds 21099 | Metric component of the ri... |
| rlmlmod 21100 | The ring module is a modul... |
| rlmlvec 21101 | The ring module over a div... |
| rlmlsm 21102 | Subgroup sum of the ring m... |
| rlmvneg 21103 | Vector negation in the rin... |
| rlmscaf 21104 | Functionalized scalar mult... |
| ixpsnbasval 21105 | The value of an infinite C... |
| lidlval 21110 | Value of the set of ring i... |
| rspval 21111 | Value of the ring span fun... |
| lidlss 21112 | An ideal is a subset of th... |
| lidlssbas 21113 | The base set of the restri... |
| lidlbas 21114 | A (left) ideal of a ring i... |
| islidl 21115 | Predicate of being a (left... |
| rnglidlmcl 21116 | A (left) ideal containing ... |
| rngridlmcl 21117 | A right ideal (which is a ... |
| dflidl2rng 21118 | Alternate (the usual textb... |
| isridlrng 21119 | A right ideal is a left id... |
| lidl0cl 21120 | An ideal contains 0. (Con... |
| lidlacl 21121 | An ideal is closed under a... |
| lidlnegcl 21122 | An ideal contains negative... |
| lidlsubg 21123 | An ideal is a subgroup of ... |
| lidlsubcl 21124 | An ideal is closed under s... |
| lidlmcl 21125 | An ideal is closed under l... |
| lidl1el 21126 | An ideal contains 1 iff it... |
| dflidl2 21127 | Alternate (the usual textb... |
| lidl0ALT 21128 | Alternate proof for ~ lidl... |
| rnglidl0 21129 | Every non-unital ring cont... |
| lidl0 21130 | Every ring contains a zero... |
| lidl1ALT 21131 | Alternate proof for ~ lidl... |
| rnglidl1 21132 | The base set of every non-... |
| lidl1 21133 | Every ring contains a unit... |
| lidlacs 21134 | The ideal system is an alg... |
| rspcl 21135 | The span of a set of ring ... |
| rspssid 21136 | The span of a set of ring ... |
| rsp1 21137 | The span of the identity e... |
| rsp0 21138 | The span of the zero eleme... |
| rspssp 21139 | The ideal span of a set of... |
| mrcrsp 21140 | Moore closure generalizes ... |
| lidlnz 21141 | A nonzero ideal contains a... |
| drngnidl 21142 | A division ring has only t... |
| lidlrsppropd 21143 | The left ideals and ring s... |
| rnglidlmmgm 21144 | The multiplicative group o... |
| rnglidlmsgrp 21145 | The multiplicative group o... |
| rnglidlrng 21146 | A (left) ideal of a non-un... |
| 2idlval 21149 | Definition of a two-sided ... |
| isridl 21150 | A right ideal is a left id... |
| 2idlelb 21151 | Membership in a two-sided ... |
| 2idllidld 21152 | A two-sided ideal is a lef... |
| 2idlridld 21153 | A two-sided ideal is a rig... |
| df2idl2rng 21154 | Alternate (the usual textb... |
| df2idl2 21155 | Alternate (the usual textb... |
| ridl0 21156 | Every ring contains a zero... |
| ridl1 21157 | Every ring contains a unit... |
| 2idl0 21158 | Every ring contains a zero... |
| 2idl1 21159 | Every ring contains a unit... |
| 2idlss 21160 | A two-sided ideal is a sub... |
| 2idlbas 21161 | The base set of a two-side... |
| 2idlelbas 21162 | The base set of a two-side... |
| rng2idlsubrng 21163 | A two-sided ideal of a non... |
| rng2idlnsg 21164 | A two-sided ideal of a non... |
| rng2idl0 21165 | The zero (additive identit... |
| rng2idlsubgsubrng 21166 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21167 | A two-sided ideal of a non... |
| rng2idlsubg0 21168 | The zero (additive identit... |
| 2idlcpblrng 21169 | The coset equivalence rela... |
| 2idlcpbl 21170 | The coset equivalence rela... |
| qus2idrng 21171 | The quotient of a non-unit... |
| qus1 21172 | The multiplicative identit... |
| qusring 21173 | If ` S ` is a two-sided id... |
| qusrhm 21174 | If ` S ` is a two-sided id... |
| qusmul2 21175 | Value of the ring operatio... |
| crngridl 21176 | In a commutative ring, the... |
| crng2idl 21177 | In a commutative ring, a t... |
| qusmulrng 21178 | Value of the multiplicatio... |
| quscrng 21179 | The quotient of a commutat... |
| rngqiprng1elbas 21180 | The ring unity of a two-si... |
| rngqiprngghmlem1 21181 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21182 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21183 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21184 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21185 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21186 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21187 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21188 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21189 | The base set of the produc... |
| rngqiprng 21190 | The product of the quotien... |
| rngqiprngimf 21191 | ` F ` is a function from (... |
| rngqiprngimfv 21192 | The value of the function ... |
| rngqiprngghm 21193 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21194 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21195 | ` F ` is a function from (... |
| rngqiprnglin 21196 | ` F ` is linear with respe... |
| rngqiprngho 21197 | ` F ` is a homomorphism of... |
| rngqiprngim 21198 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21199 | The unity of a two-sided i... |
| rngringbdlem1 21200 | In a unital ring, the quot... |
| rngringbdlem2 21201 | A non-unital ring is unita... |
| rngringbd 21202 | A non-unital ring is unita... |
| ring2idlqus 21203 | For every unital ring ther... |
| ring2idlqusb 21204 | A non-unital ring is unita... |
| rngqiprngfulem1 21205 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21206 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21207 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21208 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21209 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21210 | The ring unity of the prod... |
| rngqiprngfu 21211 | The function value of ` F ... |
| rngqiprngu 21212 | If a non-unital ring has a... |
| ring2idlqus1 21213 | If a non-unital ring has a... |
| lpival 21218 | Value of the set of princi... |
| islpidl 21219 | Property of being a princi... |
| lpi0 21220 | The zero ideal is always p... |
| lpi1 21221 | The unit ideal is always p... |
| islpir 21222 | Principal ideal rings are ... |
| lpiss 21223 | Principal ideals are a sub... |
| islpir2 21224 | Principal ideal rings are ... |
| lpirring 21225 | Principal ideal rings are ... |
| drnglpir 21226 | Division rings are princip... |
| rspsn 21227 | Membership in principal id... |
| lidldvgen 21228 | An element generates an id... |
| lpigen 21229 | An ideal is principal iff ... |
| rrgval 21238 | Value of the set or left-r... |
| isrrg 21239 | Membership in the set of l... |
| rrgeq0i 21240 | Property of a left-regular... |
| rrgeq0 21241 | Left-multiplication by a l... |
| rrgsupp 21242 | Left multiplication by a l... |
| rrgss 21243 | Left-regular elements are ... |
| unitrrg 21244 | Units are regular elements... |
| isdomn 21245 | Expand definition of a dom... |
| domnnzr 21246 | A domain is a nonzero ring... |
| domnring 21247 | A domain is a ring. (Cont... |
| domneq0 21248 | In a domain, a product is ... |
| domnmuln0 21249 | In a domain, a product of ... |
| isdomn2 21250 | A ring is a domain iff all... |
| domnrrg 21251 | In a domain, any nonzero e... |
| isdomn3 21252 | Nonzero elements form a mu... |
| isdomn5 21253 | The right conjunct in the ... |
| isdomn4 21254 | A ring is a domain iff it ... |
| opprdomn 21255 | The opposite of a domain i... |
| abvn0b 21256 | Another characterization o... |
| drngdomn 21257 | A division ring is a domai... |
| isidom 21258 | An integral domain is a co... |
| idomdomd 21259 | An integral domain is a do... |
| idomcringd 21260 | An integral domain is a co... |
| idomringd 21261 | An integral domain is a ri... |
| fldidom 21262 | A field is an integral dom... |
| fldidomOLD 21263 | Obsolete version of ~ fldi... |
| fidomndrnglem 21264 | Lemma for ~ fidomndrng . ... |
| fidomndrng 21265 | A finite domain is a divis... |
| fiidomfld 21266 | A finite integral domain i... |
| cnfldstr 21285 | The field of complex numbe... |
| cnfldex 21286 | The field of complex numbe... |
| cnfldbas 21287 | The base set of the field ... |
| mpocnfldadd 21288 | The addition operation of ... |
| cnfldadd 21289 | The addition operation of ... |
| mpocnfldmul 21290 | The multiplication operati... |
| cnfldmul 21291 | The multiplication operati... |
| cnfldcj 21292 | The conjugation operation ... |
| cnfldtset 21293 | The topology component of ... |
| cnfldle 21294 | The ordering of the field ... |
| cnfldds 21295 | The metric of the field of... |
| cnfldunif 21296 | The uniform structure comp... |
| cnfldfun 21297 | The field of complex numbe... |
| cnfldfunALT 21298 | The field of complex numbe... |
| dfcnfldOLD 21299 | Obsolete version of ~ df-c... |
| cnfldstrOLD 21300 | Obsolete version of ~ cnfl... |
| cnfldexOLD 21301 | Obsolete version of ~ cnfl... |
| cnfldbasOLD 21302 | Obsolete version of ~ cnfl... |
| cnfldaddOLD 21303 | Obsolete version of ~ cnfl... |
| cnfldmulOLD 21304 | Obsolete version of ~ cnfl... |
| cnfldcjOLD 21305 | Obsolete version of ~ cnfl... |
| cnfldtsetOLD 21306 | Obsolete version of ~ cnfl... |
| cnfldleOLD 21307 | Obsolete version of ~ cnfl... |
| cnflddsOLD 21308 | Obsolete version of ~ cnfl... |
| cnfldunifOLD 21309 | Obsolete version of ~ cnfl... |
| cnfldfunOLD 21310 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLD 21311 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLDOLD 21312 | Obsolete proof of ~ cnfldf... |
| xrsstr 21313 | The extended real structur... |
| xrsex 21314 | The extended real structur... |
| xrsbas 21315 | The base set of the extend... |
| xrsadd 21316 | The addition operation of ... |
| xrsmul 21317 | The multiplication operati... |
| xrstset 21318 | The topology component of ... |
| xrsle 21319 | The ordering of the extend... |
| cncrng 21320 | The complex numbers form a... |
| cncrngOLD 21321 | Obsolete version of ~ cncr... |
| cnring 21322 | The complex numbers form a... |
| xrsmcmn 21323 | The "multiplicative group"... |
| cnfld0 21324 | Zero is the zero element o... |
| cnfld1 21325 | One is the unity element o... |
| cnfld1OLD 21326 | Obsolete version of ~ cnfl... |
| cnfldneg 21327 | The additive inverse in th... |
| cnfldplusf 21328 | The functionalized additio... |
| cnfldsub 21329 | The subtraction operator i... |
| cndrng 21330 | The complex numbers form a... |
| cndrngOLD 21331 | Obsolete version of ~ cndr... |
| cnflddiv 21332 | The division operation in ... |
| cnflddivOLD 21333 | Obsolete version of ~ cnfl... |
| cnfldinv 21334 | The multiplicative inverse... |
| cnfldmulg 21335 | The group multiple functio... |
| cnfldexp 21336 | The exponentiation operato... |
| cnsrng 21337 | The complex numbers form a... |
| xrsmgm 21338 | The "additive group" of th... |
| xrsnsgrp 21339 | The "additive group" of th... |
| xrsmgmdifsgrp 21340 | The "additive group" of th... |
| xrs1mnd 21341 | The extended real numbers,... |
| xrs10 21342 | The zero of the extended r... |
| xrs1cmn 21343 | The extended real numbers ... |
| xrge0subm 21344 | The nonnegative extended r... |
| xrge0cmn 21345 | The nonnegative extended r... |
| xrsds 21346 | The metric of the extended... |
| xrsdsval 21347 | The metric of the extended... |
| xrsdsreval 21348 | The metric of the extended... |
| xrsdsreclblem 21349 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21350 | The metric of the extended... |
| cnsubmlem 21351 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21352 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21353 | Lemma for ~ resubdrg and f... |
| cnsubrglemOLD 21354 | Obsolete version of ~ cnsu... |
| cnsubdrglem 21355 | Lemma for ~ resubdrg and f... |
| qsubdrg 21356 | The rational numbers form ... |
| zsubrg 21357 | The integers form a subrin... |
| gzsubrg 21358 | The gaussian integers form... |
| nn0subm 21359 | The nonnegative integers f... |
| rege0subm 21360 | The nonnegative reals form... |
| absabv 21361 | The regular absolute value... |
| zsssubrg 21362 | The integers are a subset ... |
| qsssubdrg 21363 | The rational numbers are a... |
| cnsubrg 21364 | There are no subrings of t... |
| cnmgpabl 21365 | The unit group of the comp... |
| cnmgpid 21366 | The group identity element... |
| cnmsubglem 21367 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21368 | The positive reals form a ... |
| gzrngunitlem 21369 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21370 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21371 | Relate a group sum on ` CC... |
| regsumfsum 21372 | Relate a group sum on ` ( ... |
| expmhm 21373 | Exponentiation is a monoid... |
| nn0srg 21374 | The nonnegative integers f... |
| rge0srg 21375 | The nonnegative real numbe... |
| zringcrng 21378 | The ring of integers is a ... |
| zringring 21379 | The ring of integers is a ... |
| zringrng 21380 | The ring of integers is a ... |
| zringabl 21381 | The ring of integers is an... |
| zringgrp 21382 | The ring of integers is an... |
| zringbas 21383 | The integers are the base ... |
| zringplusg 21384 | The addition operation of ... |
| zringsub 21385 | The subtraction of element... |
| zringmulg 21386 | The multiplication (group ... |
| zringmulr 21387 | The multiplication operati... |
| zring0 21388 | The zero element of the ri... |
| zring1 21389 | The unity element of the r... |
| zringnzr 21390 | The ring of integers is a ... |
| dvdsrzring 21391 | Ring divisibility in the r... |
| zringlpirlem1 21392 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21393 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21394 | Lemma for ~ zringlpir . A... |
| zringinvg 21395 | The additive inverse of an... |
| zringunit 21396 | The units of ` ZZ ` are th... |
| zringlpir 21397 | The integers are a princip... |
| zringndrg 21398 | The integers are not a div... |
| zringcyg 21399 | The integers are a cyclic ... |
| zringsubgval 21400 | Subtraction in the ring of... |
| zringmpg 21401 | The multiplicative group o... |
| prmirredlem 21402 | A positive integer is irre... |
| dfprm2 21403 | The positive irreducible e... |
| prmirred 21404 | The irreducible elements o... |
| expghm 21405 | Exponentiation is a group ... |
| mulgghm2 21406 | The powers of a group elem... |
| mulgrhm 21407 | The powers of the element ... |
| mulgrhm2 21408 | The powers of the element ... |
| irinitoringc 21409 | The ring of integers is an... |
| nzerooringczr 21410 | There is no zero object in... |
| pzriprnglem1 21411 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21412 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21413 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21414 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21415 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21416 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21417 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21418 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21419 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21420 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21421 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21422 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21423 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21424 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21425 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21426 | The ring unity of the ring... |
| pzriprng 21427 | The non-unital ring ` ( ZZ... |
| pzriprng1 21428 | The ring unity of the ring... |
| zrhval 21437 | Define the unique homomorp... |
| zrhval2 21438 | Alternate value of the ` Z... |
| zrhmulg 21439 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21440 | The ` ZRHom ` homomorphism... |
| zrhrhm 21441 | The ` ZRHom ` homomorphism... |
| zrh1 21442 | Interpretation of 1 in a r... |
| zrh0 21443 | Interpretation of 0 in a r... |
| zrhpropd 21444 | The ` ZZ ` ring homomorphi... |
| zlmval 21445 | Augment an abelian group w... |
| zlmlem 21446 | Lemma for ~ zlmbas and ~ z... |
| zlmlemOLD 21447 | Obsolete version of ~ zlml... |
| zlmbas 21448 | Base set of a ` ZZ ` -modu... |
| zlmbasOLD 21449 | Obsolete version of ~ zlmb... |
| zlmplusg 21450 | Group operation of a ` ZZ ... |
| zlmplusgOLD 21451 | Obsolete version of ~ zlmb... |
| zlmmulr 21452 | Ring operation of a ` ZZ `... |
| zlmmulrOLD 21453 | Obsolete version of ~ zlmb... |
| zlmsca 21454 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21455 | Scalar multiplication oper... |
| zlmlmod 21456 | The ` ZZ ` -module operati... |
| chrval 21457 | Definition substitution of... |
| chrcl 21458 | Closure of the characteris... |
| chrid 21459 | The canonical ` ZZ ` ring ... |
| chrdvds 21460 | The ` ZZ ` ring homomorphi... |
| chrcong 21461 | If two integers are congru... |
| dvdschrmulg 21462 | In a ring, any multiple of... |
| fermltlchr 21463 | A generalization of Fermat... |
| chrnzr 21464 | Nonzero rings are precisel... |
| chrrhm 21465 | The characteristic restric... |
| domnchr 21466 | The characteristic of a do... |
| znlidl 21467 | The set ` n ZZ ` is an ide... |
| zncrng2 21468 | Making a commutative ring ... |
| znval 21469 | The value of the ` Z/nZ ` ... |
| znle 21470 | The value of the ` Z/nZ ` ... |
| znval2 21471 | Self-referential expressio... |
| znbaslem 21472 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 21473 | Obsolete version of ~ znba... |
| znbas2 21474 | The base set of ` Z/nZ ` i... |
| znbas2OLD 21475 | Obsolete version of ~ znba... |
| znadd 21476 | The additive structure of ... |
| znaddOLD 21477 | Obsolete version of ~ znad... |
| znmul 21478 | The multiplicative structu... |
| znmulOLD 21479 | Obsolete version of ~ znad... |
| znzrh 21480 | The ` ZZ ` ring homomorphi... |
| znbas 21481 | The base set of ` Z/nZ ` s... |
| zncrng 21482 | ` Z/nZ ` is a commutative ... |
| znzrh2 21483 | The ` ZZ ` ring homomorphi... |
| znzrhval 21484 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21485 | The ` ZZ ` ring homomorphi... |
| zncyg 21486 | The group ` ZZ / n ZZ ` is... |
| zndvds 21487 | Express equality of equiva... |
| zndvds0 21488 | Special case of ~ zndvds w... |
| znf1o 21489 | The function ` F ` enumera... |
| zzngim 21490 | The ` ZZ ` ring homomorphi... |
| znle2 21491 | The ordering of the ` Z/nZ... |
| znleval 21492 | The ordering of the ` Z/nZ... |
| znleval2 21493 | The ordering of the ` Z/nZ... |
| zntoslem 21494 | Lemma for ~ zntos . (Cont... |
| zntos 21495 | The ` Z/nZ ` structure is ... |
| znhash 21496 | The ` Z/nZ ` structure has... |
| znfi 21497 | The ` Z/nZ ` structure is ... |
| znfld 21498 | The ` Z/nZ ` structure is ... |
| znidomb 21499 | The ` Z/nZ ` structure is ... |
| znchr 21500 | Cyclic rings are defined b... |
| znunit 21501 | The units of ` Z/nZ ` are ... |
| znunithash 21502 | The size of the unit group... |
| znrrg 21503 | The regular elements of ` ... |
| cygznlem1 21504 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21505 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21506 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21507 | A cyclic group with ` n ` ... |
| cygzn 21508 | A cyclic group with ` n ` ... |
| cygth 21509 | The "fundamental theorem o... |
| cyggic 21510 | Cyclic groups are isomorph... |
| frgpcyg 21511 | A free group is cyclic iff... |
| freshmansdream 21512 | For a prime number ` P ` ,... |
| cnmsgnsubg 21513 | The signs form a multiplic... |
| cnmsgnbas 21514 | The base set of the sign s... |
| cnmsgngrp 21515 | The group of signs under m... |
| psgnghm 21516 | The sign is a homomorphism... |
| psgnghm2 21517 | The sign is a homomorphism... |
| psgninv 21518 | The sign of a permutation ... |
| psgnco 21519 | Multiplicativity of the pe... |
| zrhpsgnmhm 21520 | Embedding of permutation s... |
| zrhpsgninv 21521 | The embedded sign of a per... |
| evpmss 21522 | Even permutations are perm... |
| psgnevpmb 21523 | A class is an even permuta... |
| psgnodpm 21524 | A permutation which is odd... |
| psgnevpm 21525 | A permutation which is eve... |
| psgnodpmr 21526 | If a permutation has sign ... |
| zrhpsgnevpm 21527 | The sign of an even permut... |
| zrhpsgnodpm 21528 | The sign of an odd permuta... |
| cofipsgn 21529 | Composition of any class `... |
| zrhpsgnelbas 21530 | Embedding of permutation s... |
| zrhcopsgnelbas 21531 | Embedding of permutation s... |
| evpmodpmf1o 21532 | The function for performin... |
| pmtrodpm 21533 | A transposition is an odd ... |
| psgnfix1 21534 | A permutation of a finite ... |
| psgnfix2 21535 | A permutation of a finite ... |
| psgndiflemB 21536 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21537 | Lemma 2 for ~ psgndif . (... |
| psgndif 21538 | Embedding of permutation s... |
| copsgndif 21539 | Embedding of permutation s... |
| rebase 21542 | The base of the field of r... |
| remulg 21543 | The multiplication (group ... |
| resubdrg 21544 | The real numbers form a di... |
| resubgval 21545 | Subtraction in the field o... |
| replusg 21546 | The addition operation of ... |
| remulr 21547 | The multiplication operati... |
| re0g 21548 | The zero element of the fi... |
| re1r 21549 | The unity element of the f... |
| rele2 21550 | The ordering relation of t... |
| relt 21551 | The ordering relation of t... |
| reds 21552 | The distance of the field ... |
| redvr 21553 | The division operation of ... |
| retos 21554 | The real numbers are a tot... |
| refld 21555 | The real numbers form a fi... |
| refldcj 21556 | The conjugation operation ... |
| resrng 21557 | The real numbers form a st... |
| regsumsupp 21558 | The group sum over the rea... |
| rzgrp 21559 | The quotient group ` RR / ... |
| isphl 21564 | The predicate "is a genera... |
| phllvec 21565 | A pre-Hilbert space is a l... |
| phllmod 21566 | A pre-Hilbert space is a l... |
| phlsrng 21567 | The scalar ring of a pre-H... |
| phllmhm 21568 | The inner product of a pre... |
| ipcl 21569 | Closure of the inner produ... |
| ipcj 21570 | Conjugate of an inner prod... |
| iporthcom 21571 | Orthogonality (meaning inn... |
| ip0l 21572 | Inner product with a zero ... |
| ip0r 21573 | Inner product with a zero ... |
| ipeq0 21574 | The inner product of a vec... |
| ipdir 21575 | Distributive law for inner... |
| ipdi 21576 | Distributive law for inner... |
| ip2di 21577 | Distributive law for inner... |
| ipsubdir 21578 | Distributive law for inner... |
| ipsubdi 21579 | Distributive law for inner... |
| ip2subdi 21580 | Distributive law for inner... |
| ipass 21581 | Associative law for inner ... |
| ipassr 21582 | "Associative" law for seco... |
| ipassr2 21583 | "Associative" law for inne... |
| ipffval 21584 | The inner product operatio... |
| ipfval 21585 | The inner product operatio... |
| ipfeq 21586 | If the inner product opera... |
| ipffn 21587 | The inner product operatio... |
| phlipf 21588 | The inner product operatio... |
| ip2eq 21589 | Two vectors are equal iff ... |
| isphld 21590 | Properties that determine ... |
| phlpropd 21591 | If two structures have the... |
| ssipeq 21592 | The inner product on a sub... |
| phssipval 21593 | The inner product on a sub... |
| phssip 21594 | The inner product (as a fu... |
| phlssphl 21595 | A subspace of an inner pro... |
| ocvfval 21602 | The orthocomplement operat... |
| ocvval 21603 | Value of the orthocompleme... |
| elocv 21604 | Elementhood in the orthoco... |
| ocvi 21605 | Property of a member of th... |
| ocvss 21606 | The orthocomplement of a s... |
| ocvocv 21607 | A set is contained in its ... |
| ocvlss 21608 | The orthocomplement of a s... |
| ocv2ss 21609 | Orthocomplements reverse s... |
| ocvin 21610 | An orthocomplement has tri... |
| ocvsscon 21611 | Two ways to say that ` S `... |
| ocvlsp 21612 | The orthocomplement of a l... |
| ocv0 21613 | The orthocomplement of the... |
| ocvz 21614 | The orthocomplement of the... |
| ocv1 21615 | The orthocomplement of the... |
| unocv 21616 | The orthocomplement of a u... |
| iunocv 21617 | The orthocomplement of an ... |
| cssval 21618 | The set of closed subspace... |
| iscss 21619 | The predicate "is a closed... |
| cssi 21620 | Property of a closed subsp... |
| cssss 21621 | A closed subspace is a sub... |
| iscss2 21622 | It is sufficient to prove ... |
| ocvcss 21623 | The orthocomplement of any... |
| cssincl 21624 | The zero subspace is a clo... |
| css0 21625 | The zero subspace is a clo... |
| css1 21626 | The whole space is a close... |
| csslss 21627 | A closed subspace of a pre... |
| lsmcss 21628 | A subset of a pre-Hilbert ... |
| cssmre 21629 | The closed subspaces of a ... |
| mrccss 21630 | The Moore closure correspo... |
| thlval 21631 | Value of the Hilbert latti... |
| thlbas 21632 | Base set of the Hilbert la... |
| thlbasOLD 21633 | Obsolete proof of ~ thlbas... |
| thlle 21634 | Ordering on the Hilbert la... |
| thlleOLD 21635 | Obsolete proof of ~ thlle ... |
| thlleval 21636 | Ordering on the Hilbert la... |
| thloc 21637 | Orthocomplement on the Hil... |
| pjfval 21644 | The value of the projectio... |
| pjdm 21645 | A subspace is in the domai... |
| pjpm 21646 | The projection map is a pa... |
| pjfval2 21647 | Value of the projection ma... |
| pjval 21648 | Value of the projection ma... |
| pjdm2 21649 | A subspace is in the domai... |
| pjff 21650 | A projection is a linear o... |
| pjf 21651 | A projection is a function... |
| pjf2 21652 | A projection is a function... |
| pjfo 21653 | A projection is a surjecti... |
| pjcss 21654 | A projection subspace is a... |
| ocvpj 21655 | The orthocomplement of a p... |
| ishil 21656 | The predicate "is a Hilber... |
| ishil2 21657 | The predicate "is a Hilber... |
| isobs 21658 | The predicate "is an ortho... |
| obsip 21659 | The inner product of two e... |
| obsipid 21660 | A basis element has length... |
| obsrcl 21661 | Reverse closure for an ort... |
| obsss 21662 | An orthonormal basis is a ... |
| obsne0 21663 | A basis element is nonzero... |
| obsocv 21664 | An orthonormal basis has t... |
| obs2ocv 21665 | The double orthocomplement... |
| obselocv 21666 | A basis element is in the ... |
| obs2ss 21667 | A basis has no proper subs... |
| obslbs 21668 | An orthogonal basis is a l... |
| reldmdsmm 21671 | The direct sum is a well-b... |
| dsmmval 21672 | Value of the module direct... |
| dsmmbase 21673 | Base set of the module dir... |
| dsmmval2 21674 | Self-referential definitio... |
| dsmmbas2 21675 | Base set of the direct sum... |
| dsmmfi 21676 | For finite products, the d... |
| dsmmelbas 21677 | Membership in the finitely... |
| dsmm0cl 21678 | The all-zero vector is con... |
| dsmmacl 21679 | The finite hull is closed ... |
| prdsinvgd2 21680 | Negation of a single coord... |
| dsmmsubg 21681 | The finite hull of a produ... |
| dsmmlss 21682 | The finite hull of a produ... |
| dsmmlmod 21683 | The direct sum of a family... |
| frlmval 21686 | Value of the "free module"... |
| frlmlmod 21687 | The free module is a modul... |
| frlmpws 21688 | The free module as a restr... |
| frlmlss 21689 | The base set of the free m... |
| frlmpwsfi 21690 | The finite free module is ... |
| frlmsca 21691 | The ring of scalars of a f... |
| frlm0 21692 | Zero in a free module (rin... |
| frlmbas 21693 | Base set of the free modul... |
| frlmelbas 21694 | Membership in the base set... |
| frlmrcl 21695 | If a free module is inhabi... |
| frlmbasfsupp 21696 | Elements of the free modul... |
| frlmbasmap 21697 | Elements of the free modul... |
| frlmbasf 21698 | Elements of the free modul... |
| frlmlvec 21699 | The free module over a div... |
| frlmfibas 21700 | The base set of the finite... |
| elfrlmbasn0 21701 | If the dimension of a free... |
| frlmplusgval 21702 | Addition in a free module.... |
| frlmsubgval 21703 | Subtraction in a free modu... |
| frlmvscafval 21704 | Scalar multiplication in a... |
| frlmvplusgvalc 21705 | Coordinates of a sum with ... |
| frlmvscaval 21706 | Coordinates of a scalar mu... |
| frlmplusgvalb 21707 | Addition in a free module ... |
| frlmvscavalb 21708 | Scalar multiplication in a... |
| frlmvplusgscavalb 21709 | Addition combined with sca... |
| frlmgsum 21710 | Finite commutative sums in... |
| frlmsplit2 21711 | Restriction is homomorphic... |
| frlmsslss 21712 | A subset of a free module ... |
| frlmsslss2 21713 | A subset of a free module ... |
| frlmbas3 21714 | An element of the base set... |
| mpofrlmd 21715 | Elements of the free modul... |
| frlmip 21716 | The inner product of a fre... |
| frlmipval 21717 | The inner product of a fre... |
| frlmphllem 21718 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21719 | Conditions for a free modu... |
| uvcfval 21722 | Value of the unit-vector g... |
| uvcval 21723 | Value of a single unit vec... |
| uvcvval 21724 | Value of a unit vector coo... |
| uvcvvcl 21725 | A coordinate of a unit vec... |
| uvcvvcl2 21726 | A unit vector coordinate i... |
| uvcvv1 21727 | The unit vector is one at ... |
| uvcvv0 21728 | The unit vector is zero at... |
| uvcff 21729 | Domain and codomain of the... |
| uvcf1 21730 | In a nonzero ring, each un... |
| uvcresum 21731 | Any element of a free modu... |
| frlmssuvc1 21732 | A scalar multiple of a uni... |
| frlmssuvc2 21733 | A nonzero scalar multiple ... |
| frlmsslsp 21734 | A subset of a free module ... |
| frlmlbs 21735 | The unit vectors comprise ... |
| frlmup1 21736 | Any assignment of unit vec... |
| frlmup2 21737 | The evaluation map has the... |
| frlmup3 21738 | The range of such an evalu... |
| frlmup4 21739 | Universal property of the ... |
| ellspd 21740 | The elements of the span o... |
| elfilspd 21741 | Simplified version of ~ el... |
| rellindf 21746 | The independent-family pre... |
| islinds 21747 | Property of an independent... |
| linds1 21748 | An independent set of vect... |
| linds2 21749 | An independent set of vect... |
| islindf 21750 | Property of an independent... |
| islinds2 21751 | Expanded property of an in... |
| islindf2 21752 | Property of an independent... |
| lindff 21753 | Functional property of a l... |
| lindfind 21754 | A linearly independent fam... |
| lindsind 21755 | A linearly independent set... |
| lindfind2 21756 | In a linearly independent ... |
| lindsind2 21757 | In a linearly independent ... |
| lindff1 21758 | A linearly independent fam... |
| lindfrn 21759 | The range of an independen... |
| f1lindf 21760 | Rearranging and deleting e... |
| lindfres 21761 | Any restriction of an inde... |
| lindsss 21762 | Any subset of an independe... |
| f1linds 21763 | A family constructed from ... |
| islindf3 21764 | In a nonzero ring, indepen... |
| lindfmm 21765 | Linear independence of a f... |
| lindsmm 21766 | Linear independence of a s... |
| lindsmm2 21767 | The monomorphic image of a... |
| lsslindf 21768 | Linear independence is unc... |
| lsslinds 21769 | Linear independence is unc... |
| islbs4 21770 | A basis is an independent ... |
| lbslinds 21771 | A basis is independent. (... |
| islinds3 21772 | A subset is linearly indep... |
| islinds4 21773 | A set is independent in a ... |
| lmimlbs 21774 | The isomorphic image of a ... |
| lmiclbs 21775 | Having a basis is an isomo... |
| islindf4 21776 | A family is independent if... |
| islindf5 21777 | A family is independent if... |
| indlcim 21778 | An independent, spanning f... |
| lbslcic 21779 | A module with a basis is i... |
| lmisfree 21780 | A module has a basis iff i... |
| lvecisfrlm 21781 | Every vector space is isom... |
| lmimco 21782 | The composition of two iso... |
| lmictra 21783 | Module isomorphism is tran... |
| uvcf1o 21784 | In a nonzero ring, the map... |
| uvcendim 21785 | In a nonzero ring, the num... |
| frlmisfrlm 21786 | A free module is isomorphi... |
| frlmiscvec 21787 | Every free module is isomo... |
| isassa 21794 | The properties of an assoc... |
| assalem 21795 | The properties of an assoc... |
| assaass 21796 | Left-associative property ... |
| assaassr 21797 | Right-associative property... |
| assalmod 21798 | An associative algebra is ... |
| assaring 21799 | An associative algebra is ... |
| assasca 21800 | The scalars of an associat... |
| assa2ass 21801 | Left- and right-associativ... |
| isassad 21802 | Sufficient condition for b... |
| issubassa3 21803 | A subring that is also a s... |
| issubassa 21804 | The subalgebras of an asso... |
| sraassab 21805 | A subring algebra is an as... |
| sraassa 21806 | The subring algebra over a... |
| sraassaOLD 21807 | Obsolete version of ~ sraa... |
| rlmassa 21808 | The ring module over a com... |
| assapropd 21809 | If two structures have the... |
| aspval 21810 | Value of the algebraic clo... |
| asplss 21811 | The algebraic span of a se... |
| aspid 21812 | The algebraic span of a su... |
| aspsubrg 21813 | The algebraic span of a se... |
| aspss 21814 | Span preserves subset orde... |
| aspssid 21815 | A set of vectors is a subs... |
| asclfval 21816 | Function value of the alge... |
| asclval 21817 | Value of a mapped algebra ... |
| asclfn 21818 | Unconditional functionalit... |
| asclf 21819 | The algebra scalars functi... |
| asclghm 21820 | The algebra scalars functi... |
| ascl0 21821 | The scalar 0 embedded into... |
| ascl1 21822 | The scalar 1 embedded into... |
| asclmul1 21823 | Left multiplication by a l... |
| asclmul2 21824 | Right multiplication by a ... |
| ascldimul 21825 | The algebra scalars functi... |
| asclinvg 21826 | The group inverse (negatio... |
| asclrhm 21827 | The scalar injection is a ... |
| rnascl 21828 | The set of injected scalar... |
| issubassa2 21829 | A subring of a unital alge... |
| rnasclsubrg 21830 | The scalar multiples of th... |
| rnasclmulcl 21831 | (Vector) multiplication is... |
| rnasclassa 21832 | The scalar multiples of th... |
| ressascl 21833 | The injection of scalars i... |
| asclpropd 21834 | If two structures have the... |
| aspval2 21835 | The algebraic closure is t... |
| assamulgscmlem1 21836 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21837 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21838 | Exponentiation of a scalar... |
| asclmulg 21839 | Apply group multiplication... |
| zlmassa 21840 | The ` ZZ ` -module operati... |
| reldmpsr 21851 | The multivariate power ser... |
| psrval 21852 | Value of the multivariate ... |
| psrvalstr 21853 | The multivariate power ser... |
| psrbag 21854 | Elementhood in the set of ... |
| psrbagf 21855 | A finite bag is a function... |
| psrbagfOLD 21856 | Obsolete version of ~ psrb... |
| psrbagfsupp 21857 | Finite bags have finite su... |
| psrbagfsuppOLD 21858 | Obsolete version of ~ psrb... |
| snifpsrbag 21859 | A bag containing one eleme... |
| fczpsrbag 21860 | The constant function equa... |
| psrbaglesupp 21861 | The support of a dominated... |
| psrbaglesuppOLD 21862 | Obsolete version of ~ psrb... |
| psrbaglecl 21863 | The set of finite bags is ... |
| psrbagleclOLD 21864 | Obsolete version of ~ psrb... |
| psrbagaddcl 21865 | The sum of two finite bags... |
| psrbagaddclOLD 21866 | Obsolete version of ~ psrb... |
| psrbagcon 21867 | The analogue of the statem... |
| psrbagconOLD 21868 | Obsolete version of ~ psrb... |
| psrbaglefi 21869 | There are finitely many ba... |
| psrbaglefiOLD 21870 | Obsolete version of ~ psrb... |
| psrbagconcl 21871 | The complement of a bag is... |
| psrbagconclOLD 21872 | Obsolete version of ~ psrb... |
| psrbagleadd1 21873 | The analogue of " ` X <_ F... |
| psrbagconf1o 21874 | Bag complementation is a b... |
| psrbagconf1oOLD 21875 | Obsolete version of ~ psrb... |
| gsumbagdiaglemOLD 21876 | Obsolete version of ~ gsum... |
| gsumbagdiagOLD 21877 | Obsolete version of ~ gsum... |
| psrass1lemOLD 21878 | Obsolete version of ~ psra... |
| gsumbagdiaglem 21879 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21880 | Two-dimensional commutatio... |
| psrass1lem 21881 | A group sum commutation us... |
| psrbas 21882 | The base set of the multiv... |
| psrelbas 21883 | An element of the set of p... |
| psrelbasfun 21884 | An element of the set of p... |
| psrplusg 21885 | The addition operation of ... |
| psradd 21886 | The addition operation of ... |
| psraddcl 21887 | Closure of the power serie... |
| psraddclOLD 21888 | Obsolete version of ~ psra... |
| rhmpsrlem1 21889 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21890 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21891 | The multiplication operati... |
| psrmulfval 21892 | The multiplication operati... |
| psrmulval 21893 | The multiplication operati... |
| psrmulcllem 21894 | Closure of the power serie... |
| psrmulcl 21895 | Closure of the power serie... |
| psrsca 21896 | The scalar field of the mu... |
| psrvscafval 21897 | The scalar multiplication ... |
| psrvsca 21898 | The scalar multiplication ... |
| psrvscaval 21899 | The scalar multiplication ... |
| psrvscacl 21900 | Closure of the power serie... |
| psr0cl 21901 | The zero element of the ri... |
| psr0lid 21902 | The zero element of the ri... |
| psrnegcl 21903 | The negative function in t... |
| psrlinv 21904 | The negative function in t... |
| psrgrp 21905 | The ring of power series i... |
| psrgrpOLD 21906 | Obsolete proof of ~ psrgrp... |
| psr0 21907 | The zero element of the ri... |
| psrneg 21908 | The negative function of t... |
| psrlmod 21909 | The ring of power series i... |
| psr1cl 21910 | The identity element of th... |
| psrlidm 21911 | The identity element of th... |
| psrridm 21912 | The identity element of th... |
| psrass1 21913 | Associative identity for t... |
| psrdi 21914 | Distributive law for the r... |
| psrdir 21915 | Distributive law for the r... |
| psrass23l 21916 | Associative identity for t... |
| psrcom 21917 | Commutative law for the ri... |
| psrass23 21918 | Associative identities for... |
| psrring 21919 | The ring of power series i... |
| psr1 21920 | The identity element of th... |
| psrcrng 21921 | The ring of power series i... |
| psrassa 21922 | The ring of power series i... |
| resspsrbas 21923 | A restricted power series ... |
| resspsradd 21924 | A restricted power series ... |
| resspsrmul 21925 | A restricted power series ... |
| resspsrvsca 21926 | A restricted power series ... |
| subrgpsr 21927 | A subring of the base ring... |
| psrascl 21928 | Value of the scalar inject... |
| psrasclcl 21929 | A scalar is lifted into a ... |
| mvrfval 21930 | Value of the generating el... |
| mvrval 21931 | Value of the generating el... |
| mvrval2 21932 | Value of the generating el... |
| mvrid 21933 | The ` X i ` -th coefficien... |
| mvrf 21934 | The power series variable ... |
| mvrf1 21935 | The power series variable ... |
| mvrcl2 21936 | A power series variable is... |
| reldmmpl 21937 | The multivariate polynomia... |
| mplval 21938 | Value of the set of multiv... |
| mplbas 21939 | Base set of the set of mul... |
| mplelbas 21940 | Property of being a polyno... |
| mvrcl 21941 | A power series variable is... |
| mvrf2 21942 | The power series/polynomia... |
| mplrcl 21943 | Reverse closure for the po... |
| mplelsfi 21944 | A polynomial treated as a ... |
| mplval2 21945 | Self-referential expressio... |
| mplbasss 21946 | The set of polynomials is ... |
| mplelf 21947 | A polynomial is defined as... |
| mplsubglem 21948 | If ` A ` is an ideal of se... |
| mpllsslem 21949 | If ` A ` is an ideal of su... |
| mplsubglem2 21950 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21951 | The set of polynomials is ... |
| mpllss 21952 | The set of polynomials is ... |
| mplsubrglem 21953 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21954 | The set of polynomials is ... |
| mpl0 21955 | The zero polynomial. (Con... |
| mplplusg 21956 | Value of addition in a pol... |
| mplmulr 21957 | Value of multiplication in... |
| mpladd 21958 | The addition operation on ... |
| mplneg 21959 | The negative function on m... |
| mplmul 21960 | The multiplication operati... |
| mpl1 21961 | The identity element of th... |
| mplsca 21962 | The scalar field of a mult... |
| mplvsca2 21963 | The scalar multiplication ... |
| mplvsca 21964 | The scalar multiplication ... |
| mplvscaval 21965 | The scalar multiplication ... |
| mplgrp 21966 | The polynomial ring is a g... |
| mpllmod 21967 | The polynomial ring is a l... |
| mplring 21968 | The polynomial ring is a r... |
| mpllvec 21969 | The polynomial ring is a v... |
| mplcrng 21970 | The polynomial ring is a c... |
| mplassa 21971 | The polynomial ring is an ... |
| ressmplbas2 21972 | The base set of a restrict... |
| ressmplbas 21973 | A restricted polynomial al... |
| ressmpladd 21974 | A restricted polynomial al... |
| ressmplmul 21975 | A restricted polynomial al... |
| ressmplvsca 21976 | A restricted power series ... |
| subrgmpl 21977 | A subring of the base ring... |
| subrgmvr 21978 | The variables in a subring... |
| subrgmvrf 21979 | The variables in a polynom... |
| mplmon 21980 | A monomial is a polynomial... |
| mplmonmul 21981 | The product of two monomia... |
| mplcoe1 21982 | Decompose a polynomial int... |
| mplcoe3 21983 | Decompose a monomial in on... |
| mplcoe5lem 21984 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21985 | Decompose a monomial into ... |
| mplcoe2 21986 | Decompose a monomial into ... |
| mplbas2 21987 | An alternative expression ... |
| ltbval 21988 | Value of the well-order on... |
| ltbwe 21989 | The finite bag order is a ... |
| reldmopsr 21990 | Lemma for ordered power se... |
| opsrval 21991 | The value of the "ordered ... |
| opsrle 21992 | An alternative expression ... |
| opsrval2 21993 | Self-referential expressio... |
| opsrbaslem 21994 | Get a component of the ord... |
| opsrbaslemOLD 21995 | Obsolete version of ~ opsr... |
| opsrbas 21996 | The base set of the ordere... |
| opsrbasOLD 21997 | Obsolete version of ~ opsr... |
| opsrplusg 21998 | The addition operation of ... |
| opsrplusgOLD 21999 | Obsolete version of ~ opsr... |
| opsrmulr 22000 | The multiplication operati... |
| opsrmulrOLD 22001 | Obsolete version of ~ opsr... |
| opsrvsca 22002 | The scalar product operati... |
| opsrvscaOLD 22003 | Obsolete version of ~ opsr... |
| opsrsca 22004 | The scalar ring of the ord... |
| opsrscaOLD 22005 | Obsolete version of ~ opsr... |
| opsrtoslem1 22006 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22007 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22008 | The ordered power series s... |
| opsrso 22009 | The ordered power series s... |
| opsrcrng 22010 | The ring of ordered power ... |
| opsrassa 22011 | The ring of ordered power ... |
| mplmon2 22012 | Express a scaled monomial.... |
| psrbag0 22013 | The empty bag is a bag. (... |
| psrbagsn 22014 | A singleton bag is a bag. ... |
| mplascl 22015 | Value of the scalar inject... |
| mplasclf 22016 | The scalar injection is a ... |
| subrgascl 22017 | The scalar injection funct... |
| subrgasclcl 22018 | The scalars in a polynomia... |
| mplmon2cl 22019 | A scaled monomial is a pol... |
| mplmon2mul 22020 | Product of scaled monomial... |
| mplind 22021 | Prove a property of polyno... |
| mplcoe4 22022 | Decompose a polynomial int... |
| evlslem4 22027 | The support of a tensor pr... |
| psrbagev1 22028 | A bag of multipliers provi... |
| psrbagev1OLD 22029 | Obsolete version of ~ psrb... |
| psrbagev2 22030 | Closure of a sum using a b... |
| psrbagev2OLD 22031 | Obsolete version of ~ psrb... |
| evlslem2 22032 | A linear function on the p... |
| evlslem3 22033 | Lemma for ~ evlseu . Poly... |
| evlslem6 22034 | Lemma for ~ evlseu . Fini... |
| evlslem1 22035 | Lemma for ~ evlseu , give ... |
| evlseu 22036 | For a given interpretation... |
| reldmevls 22037 | Well-behaved binary operat... |
| mpfrcl 22038 | Reverse closure for the se... |
| evlsval 22039 | Value of the polynomial ev... |
| evlsval2 22040 | Characterizing properties ... |
| evlsrhm 22041 | Polynomial evaluation is a... |
| evlssca 22042 | Polynomial evaluation maps... |
| evlsvar 22043 | Polynomial evaluation maps... |
| evlsgsumadd 22044 | Polynomial evaluation maps... |
| evlsgsummul 22045 | Polynomial evaluation maps... |
| evlspw 22046 | Polynomial evaluation for ... |
| evlsvarpw 22047 | Polynomial evaluation for ... |
| evlval 22048 | Value of the simple/same r... |
| evlrhm 22049 | The simple evaluation map ... |
| evlsscasrng 22050 | The evaluation of a scalar... |
| evlsca 22051 | Simple polynomial evaluati... |
| evlsvarsrng 22052 | The evaluation of the vari... |
| evlvar 22053 | Simple polynomial evaluati... |
| mpfconst 22054 | Constants are multivariate... |
| mpfproj 22055 | Projections are multivaria... |
| mpfsubrg 22056 | Polynomial functions are a... |
| mpff 22057 | Polynomial functions are f... |
| mpfaddcl 22058 | The sum of multivariate po... |
| mpfmulcl 22059 | The product of multivariat... |
| mpfind 22060 | Prove a property of polyno... |
| selvffval 22066 | Value of the "variable sel... |
| selvfval 22067 | Value of the "variable sel... |
| selvval 22068 | Value of the "variable sel... |
| reldmmhp 22070 | The domain of the homogene... |
| mhpfval 22071 | Value of the "homogeneous ... |
| mhpval 22072 | Value of the "homogeneous ... |
| ismhp 22073 | Property of being a homoge... |
| ismhp2 22074 | Deduce a homogeneous polyn... |
| ismhp3 22075 | A polynomial is homogeneou... |
| mhpmpl 22076 | A homogeneous polynomial i... |
| mhpdeg 22077 | All nonzero terms of a hom... |
| mhp0cl 22078 | The zero polynomial is hom... |
| mhpsclcl 22079 | A scalar (or constant) pol... |
| mhpvarcl 22080 | A power series variable is... |
| mhpmulcl 22081 | A product of homogeneous p... |
| mhppwdeg 22082 | Degree of a homogeneous po... |
| mhpaddcl 22083 | Homogeneous polynomials ar... |
| mhpinvcl 22084 | Homogeneous polynomials ar... |
| mhpsubg 22085 | Homogeneous polynomials fo... |
| mhpvscacl 22086 | Homogeneous polynomials ar... |
| mhplss 22087 | Homogeneous polynomials fo... |
| psdffval 22089 | Value of the power series ... |
| psdfval 22090 | Give a map between power s... |
| psdval 22091 | Evaluate the partial deriv... |
| psdcoef 22092 | Coefficient of a term of t... |
| psdcl 22093 | The derivative of a power ... |
| psdmplcl 22094 | The derivative of a polyno... |
| psdadd 22095 | The derivative of a sum is... |
| psdvsca 22096 | The derivative of a scaled... |
| psdmullem 22097 | Lemma for ~ psdmul . Tran... |
| psdmul 22098 | Product rule for power ser... |
| psd1 22099 | The derivative of one is z... |
| psdascl 22100 | The derivative of a consta... |
| psr1baslem 22112 | The set of finite bags on ... |
| psr1val 22113 | Value of the ring of univa... |
| psr1crng 22114 | The ring of univariate pow... |
| psr1assa 22115 | The ring of univariate pow... |
| psr1tos 22116 | The ordered power series s... |
| psr1bas2 22117 | The base set of the ring o... |
| psr1bas 22118 | The base set of the ring o... |
| vr1val 22119 | The value of the generator... |
| vr1cl2 22120 | The variable ` X ` is a me... |
| ply1val 22121 | The value of the set of un... |
| ply1bas 22122 | The value of the base set ... |
| ply1basOLD 22123 | Obsolete version of ~ ply1... |
| ply1lss 22124 | Univariate polynomials for... |
| ply1subrg 22125 | Univariate polynomials for... |
| ply1crng 22126 | The ring of univariate pol... |
| ply1assa 22127 | The ring of univariate pol... |
| psr1bascl 22128 | A univariate power series ... |
| psr1basf 22129 | Univariate power series ba... |
| ply1basf 22130 | Univariate polynomial base... |
| ply1bascl 22131 | A univariate polynomial is... |
| ply1bascl2 22132 | A univariate polynomial is... |
| coe1fval 22133 | Value of the univariate po... |
| coe1fv 22134 | Value of an evaluated coef... |
| fvcoe1 22135 | Value of a multivariate co... |
| coe1fval3 22136 | Univariate power series co... |
| coe1f2 22137 | Functionality of univariat... |
| coe1fval2 22138 | Univariate polynomial coef... |
| coe1f 22139 | Functionality of univariat... |
| coe1fvalcl 22140 | A coefficient of a univari... |
| coe1sfi 22141 | Finite support of univaria... |
| coe1fsupp 22142 | The coefficient vector of ... |
| mptcoe1fsupp 22143 | A mapping involving coeffi... |
| coe1ae0 22144 | The coefficient vector of ... |
| vr1cl 22145 | The generator of a univari... |
| opsr0 22146 | Zero in the ordered power ... |
| opsr1 22147 | One in the ordered power s... |
| psr1plusg 22148 | Value of addition in a uni... |
| psr1vsca 22149 | Value of scalar multiplica... |
| psr1mulr 22150 | Value of multiplication in... |
| ply1plusg 22151 | Value of addition in a uni... |
| ply1vsca 22152 | Value of scalar multiplica... |
| ply1mulr 22153 | Value of multiplication in... |
| ply1ass23l 22154 | Associative identity with ... |
| ressply1bas2 22155 | The base set of a restrict... |
| ressply1bas 22156 | A restricted polynomial al... |
| ressply1add 22157 | A restricted polynomial al... |
| ressply1mul 22158 | A restricted polynomial al... |
| ressply1vsca 22159 | A restricted power series ... |
| subrgply1 22160 | A subring of the base ring... |
| gsumply1subr 22161 | Evaluate a group sum in a ... |
| psrbaspropd 22162 | Property deduction for pow... |
| psrplusgpropd 22163 | Property deduction for pow... |
| mplbaspropd 22164 | Property deduction for pol... |
| psropprmul 22165 | Reversing multiplication i... |
| ply1opprmul 22166 | Reversing multiplication i... |
| 00ply1bas 22167 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22168 | Protection compatibility o... |
| ply1plusgfvi 22169 | Protection compatibility o... |
| ply1baspropd 22170 | Property deduction for uni... |
| ply1plusgpropd 22171 | Property deduction for uni... |
| opsrring 22172 | Ordered power series form ... |
| opsrlmod 22173 | Ordered power series form ... |
| psr1ring 22174 | Univariate power series fo... |
| ply1ring 22175 | Univariate polynomials for... |
| psr1lmod 22176 | Univariate power series fo... |
| psr1sca 22177 | Scalars of a univariate po... |
| psr1sca2 22178 | Scalars of a univariate po... |
| ply1lmod 22179 | Univariate polynomials for... |
| ply1sca 22180 | Scalars of a univariate po... |
| ply1sca2 22181 | Scalars of a univariate po... |
| ply1ascl0 22182 | The zero scalar as a polyn... |
| ply1mpl0 22183 | The univariate polynomial ... |
| ply10s0 22184 | Zero times a univariate po... |
| ply1mpl1 22185 | The univariate polynomial ... |
| ply1ascl 22186 | The univariate polynomial ... |
| subrg1ascl 22187 | The scalar injection funct... |
| subrg1asclcl 22188 | The scalars in a polynomia... |
| subrgvr1 22189 | The variables in a subring... |
| subrgvr1cl 22190 | The variables in a polynom... |
| coe1z 22191 | The coefficient vector of ... |
| coe1add 22192 | The coefficient vector of ... |
| coe1addfv 22193 | A particular coefficient o... |
| coe1subfv 22194 | A particular coefficient o... |
| coe1mul2lem1 22195 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22196 | An equivalence for ~ coe1m... |
| coe1mul2 22197 | The coefficient vector of ... |
| coe1mul 22198 | The coefficient vector of ... |
| ply1moncl 22199 | Closure of the expression ... |
| ply1tmcl 22200 | Closure of the expression ... |
| coe1tm 22201 | Coefficient vector of a po... |
| coe1tmfv1 22202 | Nonzero coefficient of a p... |
| coe1tmfv2 22203 | Zero coefficient of a poly... |
| coe1tmmul2 22204 | Coefficient vector of a po... |
| coe1tmmul 22205 | Coefficient vector of a po... |
| coe1tmmul2fv 22206 | Function value of a right-... |
| coe1pwmul 22207 | Coefficient vector of a po... |
| coe1pwmulfv 22208 | Function value of a right-... |
| ply1scltm 22209 | A scalar is a term with ze... |
| coe1sclmul 22210 | Coefficient vector of a po... |
| coe1sclmulfv 22211 | A single coefficient of a ... |
| coe1sclmul2 22212 | Coefficient vector of a po... |
| ply1sclf 22213 | A scalar polynomial is a p... |
| ply1sclcl 22214 | The value of the algebra s... |
| coe1scl 22215 | Coefficient vector of a sc... |
| ply1sclid 22216 | Recover the base scalar fr... |
| ply1sclf1 22217 | The polynomial scalar func... |
| ply1scl0 22218 | The zero scalar is zero. ... |
| ply1scl0OLD 22219 | Obsolete version of ~ ply1... |
| ply1scln0 22220 | Nonzero scalars create non... |
| ply1scl1 22221 | The one scalar is the unit... |
| ply1scl1OLD 22222 | Obsolete version of ~ ply1... |
| ply1idvr1 22223 | The identity of a polynomi... |
| cply1mul 22224 | The product of two constan... |
| ply1coefsupp 22225 | The decomposition of a uni... |
| ply1coe 22226 | Decompose a univariate pol... |
| eqcoe1ply1eq 22227 | Two polynomials over the s... |
| ply1coe1eq 22228 | Two polynomials over the s... |
| cply1coe0 22229 | All but the first coeffici... |
| cply1coe0bi 22230 | A polynomial is constant (... |
| coe1fzgsumdlem 22231 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22232 | Value of an evaluated coef... |
| ply1scleq 22233 | Equality of a constant pol... |
| ply1chr 22234 | The characteristic of a po... |
| gsumsmonply1 22235 | A finite group sum of scal... |
| gsummoncoe1 22236 | A coefficient of the polyn... |
| gsumply1eq 22237 | Two univariate polynomials... |
| lply1binom 22238 | The binomial theorem for l... |
| lply1binomsc 22239 | The binomial theorem for l... |
| ply1fermltlchr 22240 | Fermat's little theorem fo... |
| reldmevls1 22245 | Well-behaved binary operat... |
| ply1frcl 22246 | Reverse closure for the se... |
| evls1fval 22247 | Value of the univariate po... |
| evls1val 22248 | Value of the univariate po... |
| evls1rhmlem 22249 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22250 | Polynomial evaluation is a... |
| evls1sca 22251 | Univariate polynomial eval... |
| evls1gsumadd 22252 | Univariate polynomial eval... |
| evls1gsummul 22253 | Univariate polynomial eval... |
| evls1pw 22254 | Univariate polynomial eval... |
| evls1varpw 22255 | Univariate polynomial eval... |
| evl1fval 22256 | Value of the simple/same r... |
| evl1val 22257 | Value of the simple/same r... |
| evl1fval1lem 22258 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22259 | Value of the simple/same r... |
| evl1rhm 22260 | Polynomial evaluation is a... |
| fveval1fvcl 22261 | The function value of the ... |
| evl1sca 22262 | Polynomial evaluation maps... |
| evl1scad 22263 | Polynomial evaluation buil... |
| evl1var 22264 | Polynomial evaluation maps... |
| evl1vard 22265 | Polynomial evaluation buil... |
| evls1var 22266 | Univariate polynomial eval... |
| evls1scasrng 22267 | The evaluation of a scalar... |
| evls1varsrng 22268 | The evaluation of the vari... |
| evl1addd 22269 | Polynomial evaluation buil... |
| evl1subd 22270 | Polynomial evaluation buil... |
| evl1muld 22271 | Polynomial evaluation buil... |
| evl1vsd 22272 | Polynomial evaluation buil... |
| evl1expd 22273 | Polynomial evaluation buil... |
| pf1const 22274 | Constants are polynomial f... |
| pf1id 22275 | The identity is a polynomi... |
| pf1subrg 22276 | Polynomial functions are a... |
| pf1rcl 22277 | Reverse closure for the se... |
| pf1f 22278 | Polynomial functions are f... |
| mpfpf1 22279 | Convert a multivariate pol... |
| pf1mpf 22280 | Convert a univariate polyn... |
| pf1addcl 22281 | The sum of multivariate po... |
| pf1mulcl 22282 | The product of multivariat... |
| pf1ind 22283 | Prove a property of polyno... |
| evl1gsumdlem 22284 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22285 | Polynomial evaluation buil... |
| evl1gsumadd 22286 | Univariate polynomial eval... |
| evl1gsumaddval 22287 | Value of a univariate poly... |
| evl1gsummul 22288 | Univariate polynomial eval... |
| evl1varpw 22289 | Univariate polynomial eval... |
| evl1varpwval 22290 | Value of a univariate poly... |
| evl1scvarpw 22291 | Univariate polynomial eval... |
| evl1scvarpwval 22292 | Value of a univariate poly... |
| evl1gsummon 22293 | Value of a univariate poly... |
| evls1scafv 22294 | Value of the univariate po... |
| evls1expd 22295 | Univariate polynomial eval... |
| evls1varpwval 22296 | Univariate polynomial eval... |
| evls1fpws 22297 | Evaluation of a univariate... |
| ressply1evl 22298 | Evaluation of a univariate... |
| evls1addd 22299 | Univariate polynomial eval... |
| evls1muld 22300 | Univariate polynomial eval... |
| evls1vsca 22301 | Univariate polynomial eval... |
| asclply1subcl 22302 | Closure of the algebra sca... |
| evls1fvcl 22303 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22304 | The function ` F ` mapping... |
| evls1maplmhm 22305 | The function ` F ` mapping... |
| evls1maprnss 22306 | The function ` F ` mapping... |
| evl1maprhm 22307 | The function ` F ` mapping... |
| mplringd 22308 | The polynomial ring is a r... |
| mpllmodd 22309 | The polynomial ring is a l... |
| mhmcompl 22310 | The composition of a monoi... |
| mhmcoaddmpl 22311 | Show that the ring homomor... |
| rhmcomulmpl 22312 | Show that the ring homomor... |
| rhmmpl 22313 | Provide a ring homomorphis... |
| ply1vscl 22314 | Closure of scalar multipli... |
| mhmcoply1 22315 | The composition of a monoi... |
| rhmply1 22316 | Provide a ring homomorphis... |
| rhmply1vr1 22317 | A ring homomorphism betwee... |
| rhmply1vsca 22318 | Apply a ring homomorphism ... |
| rhmply1mon 22319 | Apply a ring homomorphism ... |
| mamufval 22322 | Functional value of the ma... |
| mamuval 22323 | Multiplication of two matr... |
| mamufv 22324 | A cell in the multiplicati... |
| mamudm 22325 | The domain of the matrix m... |
| mamufacex 22326 | Every solution of the equa... |
| mamures 22327 | Rows in a matrix product a... |
| grpvlinv 22328 | Tuple-wise left inverse in... |
| grpvrinv 22329 | Tuple-wise right inverse i... |
| ringvcl 22330 | Tuple-wise multiplication ... |
| mamucl 22331 | Operation closure of matri... |
| mamuass 22332 | Matrix multiplication is a... |
| mamudi 22333 | Matrix multiplication dist... |
| mamudir 22334 | Matrix multiplication dist... |
| mamuvs1 22335 | Matrix multiplication dist... |
| mamuvs2 22336 | Matrix multiplication dist... |
| matbas0pc 22339 | There is no matrix with a ... |
| matbas0 22340 | There is no matrix for a n... |
| matval 22341 | Value of the matrix algebr... |
| matrcl 22342 | Reverse closure for the ma... |
| matbas 22343 | The matrix ring has the sa... |
| matplusg 22344 | The matrix ring has the sa... |
| matsca 22345 | The matrix ring has the sa... |
| matscaOLD 22346 | Obsolete proof of ~ matsca... |
| matvsca 22347 | The matrix ring has the sa... |
| matvscaOLD 22348 | Obsolete proof of ~ matvsc... |
| mat0 22349 | The matrix ring has the sa... |
| matinvg 22350 | The matrix ring has the sa... |
| mat0op 22351 | Value of a zero matrix as ... |
| matsca2 22352 | The scalars of the matrix ... |
| matbas2 22353 | The base set of the matrix... |
| matbas2i 22354 | A matrix is a function. (... |
| matbas2d 22355 | The base set of the matrix... |
| eqmat 22356 | Two square matrices of the... |
| matecl 22357 | Each entry (according to W... |
| matecld 22358 | Each entry (according to W... |
| matplusg2 22359 | Addition in the matrix rin... |
| matvsca2 22360 | Scalar multiplication in t... |
| matlmod 22361 | The matrix ring is a linea... |
| matgrp 22362 | The matrix ring is a group... |
| matvscl 22363 | Closure of the scalar mult... |
| matsubg 22364 | The matrix ring has the sa... |
| matplusgcell 22365 | Addition in the matrix rin... |
| matsubgcell 22366 | Subtraction in the matrix ... |
| matinvgcell 22367 | Additive inversion in the ... |
| matvscacell 22368 | Scalar multiplication in t... |
| matgsum 22369 | Finite commutative sums in... |
| matmulr 22370 | Multiplication in the matr... |
| mamumat1cl 22371 | The identity matrix (as op... |
| mat1comp 22372 | The components of the iden... |
| mamulid 22373 | The identity matrix (as op... |
| mamurid 22374 | The identity matrix (as op... |
| matring 22375 | Existence of the matrix ri... |
| matassa 22376 | Existence of the matrix al... |
| matmulcell 22377 | Multiplication in the matr... |
| mpomatmul 22378 | Multiplication of two N x ... |
| mat1 22379 | Value of an identity matri... |
| mat1ov 22380 | Entries of an identity mat... |
| mat1bas 22381 | The identity matrix is a m... |
| matsc 22382 | The identity matrix multip... |
| ofco2 22383 | Distribution law for the f... |
| oftpos 22384 | The transposition of the v... |
| mattposcl 22385 | The transpose of a square ... |
| mattpostpos 22386 | The transpose of the trans... |
| mattposvs 22387 | The transposition of a mat... |
| mattpos1 22388 | The transposition of the i... |
| tposmap 22389 | The transposition of an I ... |
| mamutpos 22390 | Behavior of transposes in ... |
| mattposm 22391 | Multiplying two transposed... |
| matgsumcl 22392 | Closure of a group sum ove... |
| madetsumid 22393 | The identity summand in th... |
| matepmcl 22394 | Each entry of a matrix wit... |
| matepm2cl 22395 | Each entry of a matrix wit... |
| madetsmelbas 22396 | A summand of the determina... |
| madetsmelbas2 22397 | A summand of the determina... |
| mat0dimbas0 22398 | The empty set is the one a... |
| mat0dim0 22399 | The zero of the algebra of... |
| mat0dimid 22400 | The identity of the algebr... |
| mat0dimscm 22401 | The scalar multiplication ... |
| mat0dimcrng 22402 | The algebra of matrices wi... |
| mat1dimelbas 22403 | A matrix with dimension 1 ... |
| mat1dimbas 22404 | A matrix with dimension 1 ... |
| mat1dim0 22405 | The zero of the algebra of... |
| mat1dimid 22406 | The identity of the algebr... |
| mat1dimscm 22407 | The scalar multiplication ... |
| mat1dimmul 22408 | The ring multiplication in... |
| mat1dimcrng 22409 | The algebra of matrices wi... |
| mat1f1o 22410 | There is a 1-1 function fr... |
| mat1rhmval 22411 | The value of the ring homo... |
| mat1rhmelval 22412 | The value of the ring homo... |
| mat1rhmcl 22413 | The value of the ring homo... |
| mat1f 22414 | There is a function from a... |
| mat1ghm 22415 | There is a group homomorph... |
| mat1mhm 22416 | There is a monoid homomorp... |
| mat1rhm 22417 | There is a ring homomorphi... |
| mat1rngiso 22418 | There is a ring isomorphis... |
| mat1ric 22419 | A ring is isomorphic to th... |
| dmatval 22424 | The set of ` N ` x ` N ` d... |
| dmatel 22425 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22426 | An ` N ` x ` N ` diagonal ... |
| dmatid 22427 | The identity matrix is a d... |
| dmatelnd 22428 | An extradiagonal entry of ... |
| dmatmul 22429 | The product of two diagona... |
| dmatsubcl 22430 | The difference of two diag... |
| dmatsgrp 22431 | The set of diagonal matric... |
| dmatmulcl 22432 | The product of two diagona... |
| dmatsrng 22433 | The set of diagonal matric... |
| dmatcrng 22434 | The subring of diagonal ma... |
| dmatscmcl 22435 | The multiplication of a di... |
| scmatval 22436 | The set of ` N ` x ` N ` s... |
| scmatel 22437 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22438 | A scalar matrix can be exp... |
| scmatscmide 22439 | An entry of a scalar matri... |
| scmatscmiddistr 22440 | Distributive law for scala... |
| scmatmat 22441 | An ` N ` x ` N ` scalar ma... |
| scmate 22442 | An entry of an ` N ` x ` N... |
| scmatmats 22443 | The set of an ` N ` x ` N ... |
| scmateALT 22444 | Alternate proof of ~ scmat... |
| scmatscm 22445 | The multiplication of a ma... |
| scmatid 22446 | The identity matrix is a s... |
| scmatdmat 22447 | A scalar matrix is a diago... |
| scmataddcl 22448 | The sum of two scalar matr... |
| scmatsubcl 22449 | The difference of two scal... |
| scmatmulcl 22450 | The product of two scalar ... |
| scmatsgrp 22451 | The set of scalar matrices... |
| scmatsrng 22452 | The set of scalar matrices... |
| scmatcrng 22453 | The subring of scalar matr... |
| scmatsgrp1 22454 | The set of scalar matrices... |
| scmatsrng1 22455 | The set of scalar matrices... |
| smatvscl 22456 | Closure of the scalar mult... |
| scmatlss 22457 | The set of scalar matrices... |
| scmatstrbas 22458 | The set of scalar matrices... |
| scmatrhmval 22459 | The value of the ring homo... |
| scmatrhmcl 22460 | The value of the ring homo... |
| scmatf 22461 | There is a function from a... |
| scmatfo 22462 | There is a function from a... |
| scmatf1 22463 | There is a 1-1 function fr... |
| scmatf1o 22464 | There is a bijection betwe... |
| scmatghm 22465 | There is a group homomorph... |
| scmatmhm 22466 | There is a monoid homomorp... |
| scmatrhm 22467 | There is a ring homomorphi... |
| scmatrngiso 22468 | There is a ring isomorphis... |
| scmatric 22469 | A ring is isomorphic to ev... |
| mat0scmat 22470 | The empty matrix over a ri... |
| mat1scmat 22471 | A 1-dimensional matrix ove... |
| mvmulfval 22474 | Functional value of the ma... |
| mvmulval 22475 | Multiplication of a vector... |
| mvmulfv 22476 | A cell/element in the vect... |
| mavmulval 22477 | Multiplication of a vector... |
| mavmulfv 22478 | A cell/element in the vect... |
| mavmulcl 22479 | Multiplication of an NxN m... |
| 1mavmul 22480 | Multiplication of the iden... |
| mavmulass 22481 | Associativity of the multi... |
| mavmuldm 22482 | The domain of the matrix v... |
| mavmulsolcl 22483 | Every solution of the equa... |
| mavmul0 22484 | Multiplication of a 0-dime... |
| mavmul0g 22485 | The result of the 0-dimens... |
| mvmumamul1 22486 | The multiplication of an M... |
| mavmumamul1 22487 | The multiplication of an N... |
| marrepfval 22492 | First substitution for the... |
| marrepval0 22493 | Second substitution for th... |
| marrepval 22494 | Third substitution for the... |
| marrepeval 22495 | An entry of a matrix with ... |
| marrepcl 22496 | Closure of the row replace... |
| marepvfval 22497 | First substitution for the... |
| marepvval0 22498 | Second substitution for th... |
| marepvval 22499 | Third substitution for the... |
| marepveval 22500 | An entry of a matrix with ... |
| marepvcl 22501 | Closure of the column repl... |
| ma1repvcl 22502 | Closure of the column repl... |
| ma1repveval 22503 | An entry of an identity ma... |
| mulmarep1el 22504 | Element by element multipl... |
| mulmarep1gsum1 22505 | The sum of element by elem... |
| mulmarep1gsum2 22506 | The sum of element by elem... |
| 1marepvmarrepid 22507 | Replacing the ith row by 0... |
| submabas 22510 | Any subset of the index se... |
| submafval 22511 | First substitution for a s... |
| submaval0 22512 | Second substitution for a ... |
| submaval 22513 | Third substitution for a s... |
| submaeval 22514 | An entry of a submatrix of... |
| 1marepvsma1 22515 | The submatrix of the ident... |
| mdetfval 22518 | First substitution for the... |
| mdetleib 22519 | Full substitution of our d... |
| mdetleib2 22520 | Leibniz' formula can also ... |
| nfimdetndef 22521 | The determinant is not def... |
| mdetfval1 22522 | First substitution of an a... |
| mdetleib1 22523 | Full substitution of an al... |
| mdet0pr 22524 | The determinant function f... |
| mdet0f1o 22525 | The determinant function f... |
| mdet0fv0 22526 | The determinant of the emp... |
| mdetf 22527 | Functionality of the deter... |
| mdetcl 22528 | The determinant evaluates ... |
| m1detdiag 22529 | The determinant of a 1-dim... |
| mdetdiaglem 22530 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22531 | The determinant of a diago... |
| mdetdiagid 22532 | The determinant of a diago... |
| mdet1 22533 | The determinant of the ide... |
| mdetrlin 22534 | The determinant function i... |
| mdetrsca 22535 | The determinant function i... |
| mdetrsca2 22536 | The determinant function i... |
| mdetr0 22537 | The determinant of a matri... |
| mdet0 22538 | The determinant of the zer... |
| mdetrlin2 22539 | The determinant function i... |
| mdetralt 22540 | The determinant function i... |
| mdetralt2 22541 | The determinant function i... |
| mdetero 22542 | The determinant function i... |
| mdettpos 22543 | Determinant is invariant u... |
| mdetunilem1 22544 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22545 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22546 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22547 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22548 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22549 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22550 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22551 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22552 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22553 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22554 | According to the definitio... |
| mdetmul 22555 | Multiplicativity of the de... |
| m2detleiblem1 22556 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22557 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22558 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22559 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22560 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22561 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22562 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22563 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22568 | Lemma for ~ maducoeval2 . ... |
| madufval 22569 | First substitution for the... |
| maduval 22570 | Second substitution for th... |
| maducoeval 22571 | An entry of the adjunct (c... |
| maducoeval2 22572 | An entry of the adjunct (c... |
| maduf 22573 | Creating the adjunct of ma... |
| madutpos 22574 | The adjuct of a transposed... |
| madugsum 22575 | The determinant of a matri... |
| madurid 22576 | Multiplying a matrix with ... |
| madulid 22577 | Multiplying the adjunct of... |
| minmar1fval 22578 | First substitution for the... |
| minmar1val0 22579 | Second substitution for th... |
| minmar1val 22580 | Third substitution for the... |
| minmar1eval 22581 | An entry of a matrix for a... |
| minmar1marrep 22582 | The minor matrix is a spec... |
| minmar1cl 22583 | Closure of the row replace... |
| maducoevalmin1 22584 | The coefficients of an adj... |
| symgmatr01lem 22585 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22586 | Applying a permutation tha... |
| gsummatr01lem1 22587 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22588 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22589 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22590 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22591 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22592 | Replacing a row of a squar... |
| smadiadetlem0 22593 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22594 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22595 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22596 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22597 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22598 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22599 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22600 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22601 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22602 | The determinant of a subma... |
| smadiadetglem1 22603 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22604 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22605 | The determinant of a squar... |
| smadiadetg0 22606 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22607 | The determinant of a squar... |
| invrvald 22608 | If a matrix multiplied wit... |
| matinv 22609 | The inverse of a matrix is... |
| matunit 22610 | A matrix is a unit in the ... |
| slesolvec 22611 | Every solution of a system... |
| slesolinv 22612 | The solution of a system o... |
| slesolinvbi 22613 | The solution of a system o... |
| slesolex 22614 | Every system of linear equ... |
| cramerimplem1 22615 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22616 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22617 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22618 | One direction of Cramer's ... |
| cramerlem1 22619 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22620 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22621 | Lemma 3 for ~ cramer . (C... |
| cramer0 22622 | Special case of Cramer's r... |
| cramer 22623 | Cramer's rule. According ... |
| pmatring 22624 | The set of polynomial matr... |
| pmatlmod 22625 | The set of polynomial matr... |
| pmatassa 22626 | The set of polynomial matr... |
| pmat0op 22627 | The zero polynomial matrix... |
| pmat1op 22628 | The identity polynomial ma... |
| pmat1ovd 22629 | Entries of the identity po... |
| pmat0opsc 22630 | The zero polynomial matrix... |
| pmat1opsc 22631 | The identity polynomial ma... |
| pmat1ovscd 22632 | Entries of the identity po... |
| pmatcoe1fsupp 22633 | For a polynomial matrix th... |
| 1pmatscmul 22634 | The scalar product of the ... |
| cpmat 22641 | Value of the constructor o... |
| cpmatpmat 22642 | A constant polynomial matr... |
| cpmatel 22643 | Property of a constant pol... |
| cpmatelimp 22644 | Implication of a set being... |
| cpmatel2 22645 | Another property of a cons... |
| cpmatelimp2 22646 | Another implication of a s... |
| 1elcpmat 22647 | The identity of the ring o... |
| cpmatacl 22648 | The set of all constant po... |
| cpmatinvcl 22649 | The set of all constant po... |
| cpmatmcllem 22650 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22651 | The set of all constant po... |
| cpmatsubgpmat 22652 | The set of all constant po... |
| cpmatsrgpmat 22653 | The set of all constant po... |
| 0elcpmat 22654 | The zero of the ring of al... |
| mat2pmatfval 22655 | Value of the matrix transf... |
| mat2pmatval 22656 | The result of a matrix tra... |
| mat2pmatvalel 22657 | A (matrix) element of the ... |
| mat2pmatbas 22658 | The result of a matrix tra... |
| mat2pmatbas0 22659 | The result of a matrix tra... |
| mat2pmatf 22660 | The matrix transformation ... |
| mat2pmatf1 22661 | The matrix transformation ... |
| mat2pmatghm 22662 | The transformation of matr... |
| mat2pmatmul 22663 | The transformation of matr... |
| mat2pmat1 22664 | The transformation of the ... |
| mat2pmatmhm 22665 | The transformation of matr... |
| mat2pmatrhm 22666 | The transformation of matr... |
| mat2pmatlin 22667 | The transformation of matr... |
| 0mat2pmat 22668 | The transformed zero matri... |
| idmatidpmat 22669 | The transformed identity m... |
| d0mat2pmat 22670 | The transformed empty set ... |
| d1mat2pmat 22671 | The transformation of a ma... |
| mat2pmatscmxcl 22672 | A transformed matrix multi... |
| m2cpm 22673 | The result of a matrix tra... |
| m2cpmf 22674 | The matrix transformation ... |
| m2cpmf1 22675 | The matrix transformation ... |
| m2cpmghm 22676 | The transformation of matr... |
| m2cpmmhm 22677 | The transformation of matr... |
| m2cpmrhm 22678 | The transformation of matr... |
| m2pmfzmap 22679 | The transformed values of ... |
| m2pmfzgsumcl 22680 | Closure of the sum of scal... |
| cpm2mfval 22681 | Value of the inverse matri... |
| cpm2mval 22682 | The result of an inverse m... |
| cpm2mvalel 22683 | A (matrix) element of the ... |
| cpm2mf 22684 | The inverse matrix transfo... |
| m2cpminvid 22685 | The inverse transformation... |
| m2cpminvid2lem 22686 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22687 | The transformation applied... |
| m2cpmfo 22688 | The matrix transformation ... |
| m2cpmf1o 22689 | The matrix transformation ... |
| m2cpmrngiso 22690 | The transformation of matr... |
| matcpmric 22691 | The ring of matrices over ... |
| m2cpminv 22692 | The inverse matrix transfo... |
| m2cpminv0 22693 | The inverse matrix transfo... |
| decpmatval0 22696 | The matrix consisting of t... |
| decpmatval 22697 | The matrix consisting of t... |
| decpmate 22698 | An entry of the matrix con... |
| decpmatcl 22699 | Closure of the decompositi... |
| decpmataa0 22700 | The matrix consisting of t... |
| decpmatfsupp 22701 | The mapping to the matrice... |
| decpmatid 22702 | The matrix consisting of t... |
| decpmatmullem 22703 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22704 | The matrix consisting of t... |
| decpmatmulsumfsupp 22705 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22706 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22707 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22708 | Write a polynomial matrix ... |
| pmatcollpw2lem 22709 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22710 | Write a polynomial matrix ... |
| monmatcollpw 22711 | The matrix consisting of t... |
| pmatcollpwlem 22712 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22713 | Write a polynomial matrix ... |
| pmatcollpwfi 22714 | Write a polynomial matrix ... |
| pmatcollpw3lem 22715 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22716 | Write a polynomial matrix ... |
| pmatcollpw3fi 22717 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22718 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22719 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22720 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22721 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22722 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22723 | Write a scalar matrix over... |
| pm2mpf1lem 22726 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22727 | Value of the transformatio... |
| pm2mpfval 22728 | A polynomial matrix transf... |
| pm2mpcl 22729 | The transformation of poly... |
| pm2mpf 22730 | The transformation of poly... |
| pm2mpf1 22731 | The transformation of poly... |
| pm2mpcoe1 22732 | A coefficient of the polyn... |
| idpm2idmp 22733 | The transformation of the ... |
| mptcoe1matfsupp 22734 | The mapping extracting the... |
| mply1topmatcllem 22735 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22736 | A polynomial over matrices... |
| mply1topmatcl 22737 | A polynomial over matrices... |
| mp2pm2mplem1 22738 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22739 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22740 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22741 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22742 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22743 | A polynomial over matrices... |
| pm2mpghmlem2 22744 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22745 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22746 | The transformation of poly... |
| pm2mpf1o 22747 | The transformation of poly... |
| pm2mpghm 22748 | The transformation of poly... |
| pm2mpgrpiso 22749 | The transformation of poly... |
| pm2mpmhmlem1 22750 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22751 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22752 | The transformation of poly... |
| pm2mprhm 22753 | The transformation of poly... |
| pm2mprngiso 22754 | The transformation of poly... |
| pmmpric 22755 | The ring of polynomial mat... |
| monmat2matmon 22756 | The transformation of a po... |
| pm2mp 22757 | The transformation of a su... |
| chmatcl 22760 | Closure of the characteris... |
| chmatval 22761 | The entries of the charact... |
| chpmatfval 22762 | Value of the characteristi... |
| chpmatval 22763 | The characteristic polynom... |
| chpmatply1 22764 | The characteristic polynom... |
| chpmatval2 22765 | The characteristic polynom... |
| chpmat0d 22766 | The characteristic polynom... |
| chpmat1dlem 22767 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22768 | The characteristic polynom... |
| chpdmatlem0 22769 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22770 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22771 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22772 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22773 | The characteristic polynom... |
| chpscmat 22774 | The characteristic polynom... |
| chpscmat0 22775 | The characteristic polynom... |
| chpscmatgsumbin 22776 | The characteristic polynom... |
| chpscmatgsummon 22777 | The characteristic polynom... |
| chp0mat 22778 | The characteristic polynom... |
| chpidmat 22779 | The characteristic polynom... |
| chmaidscmat 22780 | The characteristic polynom... |
| fvmptnn04if 22781 | The function values of a m... |
| fvmptnn04ifa 22782 | The function value of a ma... |
| fvmptnn04ifb 22783 | The function value of a ma... |
| fvmptnn04ifc 22784 | The function value of a ma... |
| fvmptnn04ifd 22785 | The function value of a ma... |
| chfacfisf 22786 | The "characteristic factor... |
| chfacfisfcpmat 22787 | The "characteristic factor... |
| chfacffsupp 22788 | The "characteristic factor... |
| chfacfscmulcl 22789 | Closure of a scaled value ... |
| chfacfscmul0 22790 | A scaled value of the "cha... |
| chfacfscmulfsupp 22791 | A mapping of scaled values... |
| chfacfscmulgsum 22792 | Breaking up a sum of value... |
| chfacfpmmulcl 22793 | Closure of the value of th... |
| chfacfpmmul0 22794 | The value of the "characte... |
| chfacfpmmulfsupp 22795 | A mapping of values of the... |
| chfacfpmmulgsum 22796 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22797 | Breaking up a sum of value... |
| cayhamlem1 22798 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22799 | The right-hand fundamental... |
| cpmidgsum 22800 | Representation of the iden... |
| cpmidgsumm2pm 22801 | Representation of the iden... |
| cpmidpmatlem1 22802 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22803 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22804 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22805 | Representation of the iden... |
| cpmadugsumlemB 22806 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22807 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22808 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22809 | The product of the charact... |
| cpmadugsum 22810 | The product of the charact... |
| cpmidgsum2 22811 | Representation of the iden... |
| cpmidg2sum 22812 | Equality of two sums repre... |
| cpmadumatpolylem1 22813 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22814 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22815 | The product of the charact... |
| cayhamlem2 22816 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22817 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22818 | The coefficients of the ch... |
| cayhamlem3 22819 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22820 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22821 | The Cayley-Hamilton theore... |
| cayleyhamilton 22822 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22823 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22824 | The Cayley-Hamilton theore... |
| istopg 22827 | Express the predicate " ` ... |
| istop2g 22828 | Express the predicate " ` ... |
| uniopn 22829 | The union of a subset of a... |
| iunopn 22830 | The indexed union of a sub... |
| inopn 22831 | The intersection of two op... |
| fitop 22832 | A topology is closed under... |
| fiinopn 22833 | The intersection of a none... |
| iinopn 22834 | The intersection of a none... |
| unopn 22835 | The union of two open sets... |
| 0opn 22836 | The empty set is an open s... |
| 0ntop 22837 | The empty set is not a top... |
| topopn 22838 | The underlying set of a to... |
| eltopss 22839 | A member of a topology is ... |
| riinopn 22840 | A finite indexed relative ... |
| rintopn 22841 | A finite relative intersec... |
| istopon 22844 | Property of being a topolo... |
| topontop 22845 | A topology on a given base... |
| toponuni 22846 | The base set of a topology... |
| topontopi 22847 | A topology on a given base... |
| toponunii 22848 | The base set of a topology... |
| toptopon 22849 | Alternative definition of ... |
| toptopon2 22850 | A topology is the same thi... |
| topontopon 22851 | A topology on a set is a t... |
| funtopon 22852 | The class ` TopOn ` is a f... |
| toponrestid 22853 | Given a topology on a set,... |
| toponsspwpw 22854 | The set of topologies on a... |
| dmtopon 22855 | The domain of ` TopOn ` is... |
| fntopon 22856 | The class ` TopOn ` is a f... |
| toprntopon 22857 | A topology is the same thi... |
| toponmax 22858 | The base set of a topology... |
| toponss 22859 | A member of a topology is ... |
| toponcom 22860 | If ` K ` is a topology on ... |
| toponcomb 22861 | Biconditional form of ~ to... |
| topgele 22862 | The topologies over the sa... |
| topsn 22863 | The only topology on a sin... |
| istps 22866 | Express the predicate "is ... |
| istps2 22867 | Express the predicate "is ... |
| tpsuni 22868 | The base set of a topologi... |
| tpstop 22869 | The topology extractor on ... |
| tpspropd 22870 | A topological space depend... |
| tpsprop2d 22871 | A topological space depend... |
| topontopn 22872 | Express the predicate "is ... |
| tsettps 22873 | If the topology component ... |
| istpsi 22874 | Properties that determine ... |
| eltpsg 22875 | Properties that determine ... |
| eltpsgOLD 22876 | Obsolete version of ~ eltp... |
| eltpsi 22877 | Properties that determine ... |
| isbasisg 22880 | Express the predicate "the... |
| isbasis2g 22881 | Express the predicate "the... |
| isbasis3g 22882 | Express the predicate "the... |
| basis1 22883 | Property of a basis. (Con... |
| basis2 22884 | Property of a basis. (Con... |
| fiinbas 22885 | If a set is closed under f... |
| basdif0 22886 | A basis is not affected by... |
| baspartn 22887 | A disjoint system of sets ... |
| tgval 22888 | The topology generated by ... |
| tgval2 22889 | Definition of a topology g... |
| eltg 22890 | Membership in a topology g... |
| eltg2 22891 | Membership in a topology g... |
| eltg2b 22892 | Membership in a topology g... |
| eltg4i 22893 | An open set in a topology ... |
| eltg3i 22894 | The union of a set of basi... |
| eltg3 22895 | Membership in a topology g... |
| tgval3 22896 | Alternate expression for t... |
| tg1 22897 | Property of a member of a ... |
| tg2 22898 | Property of a member of a ... |
| bastg 22899 | A member of a basis is a s... |
| unitg 22900 | The topology generated by ... |
| tgss 22901 | Subset relation for genera... |
| tgcl 22902 | Show that a basis generate... |
| tgclb 22903 | The property ~ tgcl can be... |
| tgtopon 22904 | A basis generates a topolo... |
| topbas 22905 | A topology is its own basi... |
| tgtop 22906 | A topology is its own basi... |
| eltop 22907 | Membership in a topology, ... |
| eltop2 22908 | Membership in a topology. ... |
| eltop3 22909 | Membership in a topology. ... |
| fibas 22910 | A collection of finite int... |
| tgdom 22911 | A space has no more open s... |
| tgiun 22912 | The indexed union of a set... |
| tgidm 22913 | The topology generator fun... |
| bastop 22914 | Two ways to express that a... |
| tgtop11 22915 | The topology generation fu... |
| 0top 22916 | The singleton of the empty... |
| en1top 22917 | ` { (/) } ` is the only to... |
| en2top 22918 | If a topology has two elem... |
| tgss3 22919 | A criterion for determinin... |
| tgss2 22920 | A criterion for determinin... |
| basgen 22921 | Given a topology ` J ` , s... |
| basgen2 22922 | Given a topology ` J ` , s... |
| 2basgen 22923 | Conditions that determine ... |
| tgfiss 22924 | If a subbase is included i... |
| tgdif0 22925 | A generated topology is no... |
| bastop1 22926 | A subset of a topology is ... |
| bastop2 22927 | A version of ~ bastop1 tha... |
| distop 22928 | The discrete topology on a... |
| topnex 22929 | The class of all topologie... |
| distopon 22930 | The discrete topology on a... |
| sn0topon 22931 | The singleton of the empty... |
| sn0top 22932 | The singleton of the empty... |
| indislem 22933 | A lemma to eliminate some ... |
| indistopon 22934 | The indiscrete topology on... |
| indistop 22935 | The indiscrete topology on... |
| indisuni 22936 | The base set of the indisc... |
| fctop 22937 | The finite complement topo... |
| fctop2 22938 | The finite complement topo... |
| cctop 22939 | The countable complement t... |
| ppttop 22940 | The particular point topol... |
| pptbas 22941 | The particular point topol... |
| epttop 22942 | The excluded point topolog... |
| indistpsx 22943 | The indiscrete topology on... |
| indistps 22944 | The indiscrete topology on... |
| indistps2 22945 | The indiscrete topology on... |
| indistpsALT 22946 | The indiscrete topology on... |
| indistpsALTOLD 22947 | Obsolete version of ~ indi... |
| indistps2ALT 22948 | The indiscrete topology on... |
| distps 22949 | The discrete topology on a... |
| fncld 22956 | The closed-set generator i... |
| cldval 22957 | The set of closed sets of ... |
| ntrfval 22958 | The interior function on t... |
| clsfval 22959 | The closure function on th... |
| cldrcl 22960 | Reverse closure of the clo... |
| iscld 22961 | The predicate "the class `... |
| iscld2 22962 | A subset of the underlying... |
| cldss 22963 | A closed set is a subset o... |
| cldss2 22964 | The set of closed sets is ... |
| cldopn 22965 | The complement of a closed... |
| isopn2 22966 | A subset of the underlying... |
| opncld 22967 | The complement of an open ... |
| difopn 22968 | The difference of a closed... |
| topcld 22969 | The underlying set of a to... |
| ntrval 22970 | The interior of a subset o... |
| clsval 22971 | The closure of a subset of... |
| 0cld 22972 | The empty set is closed. ... |
| iincld 22973 | The indexed intersection o... |
| intcld 22974 | The intersection of a set ... |
| uncld 22975 | The union of two closed se... |
| cldcls 22976 | A closed subset equals its... |
| incld 22977 | The intersection of two cl... |
| riincld 22978 | An indexed relative inters... |
| iuncld 22979 | A finite indexed union of ... |
| unicld 22980 | A finite union of closed s... |
| clscld 22981 | The closure of a subset of... |
| clsf 22982 | The closure function is a ... |
| ntropn 22983 | The interior of a subset o... |
| clsval2 22984 | Express closure in terms o... |
| ntrval2 22985 | Interior expressed in term... |
| ntrdif 22986 | An interior of a complemen... |
| clsdif 22987 | A closure of a complement ... |
| clsss 22988 | Subset relationship for cl... |
| ntrss 22989 | Subset relationship for in... |
| sscls 22990 | A subset of a topology's u... |
| ntrss2 22991 | A subset includes its inte... |
| ssntr 22992 | An open subset of a set is... |
| clsss3 22993 | The closure of a subset of... |
| ntrss3 22994 | The interior of a subset o... |
| ntrin 22995 | A pairwise intersection of... |
| cmclsopn 22996 | The complement of a closur... |
| cmntrcld 22997 | The complement of an inter... |
| iscld3 22998 | A subset is closed iff it ... |
| iscld4 22999 | A subset is closed iff it ... |
| isopn3 23000 | A subset is open iff it eq... |
| clsidm 23001 | The closure operation is i... |
| ntridm 23002 | The interior operation is ... |
| clstop 23003 | The closure of a topology'... |
| ntrtop 23004 | The interior of a topology... |
| 0ntr 23005 | A subset with an empty int... |
| clsss2 23006 | If a subset is included in... |
| elcls 23007 | Membership in a closure. ... |
| elcls2 23008 | Membership in a closure. ... |
| clsndisj 23009 | Any open set containing a ... |
| ntrcls0 23010 | A subset whose closure has... |
| ntreq0 23011 | Two ways to say that a sub... |
| cldmre 23012 | The closed sets of a topol... |
| mrccls 23013 | Moore closure generalizes ... |
| cls0 23014 | The closure of the empty s... |
| ntr0 23015 | The interior of the empty ... |
| isopn3i 23016 | An open subset equals its ... |
| elcls3 23017 | Membership in a closure in... |
| opncldf1 23018 | A bijection useful for con... |
| opncldf2 23019 | The values of the open-clo... |
| opncldf3 23020 | The values of the converse... |
| isclo 23021 | A set ` A ` is clopen iff ... |
| isclo2 23022 | A set ` A ` is clopen iff ... |
| discld 23023 | The open sets of a discret... |
| sn0cld 23024 | The closed sets of the top... |
| indiscld 23025 | The closed sets of an indi... |
| mretopd 23026 | A Moore collection which i... |
| toponmre 23027 | The topologies over a give... |
| cldmreon 23028 | The closed sets of a topol... |
| iscldtop 23029 | A family is the closed set... |
| mreclatdemoBAD 23030 | The closed subspaces of a ... |
| neifval 23033 | Value of the neighborhood ... |
| neif 23034 | The neighborhood function ... |
| neiss2 23035 | A set with a neighborhood ... |
| neival 23036 | Value of the set of neighb... |
| isnei 23037 | The predicate "the class `... |
| neiint 23038 | An intuitive definition of... |
| isneip 23039 | The predicate "the class `... |
| neii1 23040 | A neighborhood is included... |
| neisspw 23041 | The neighborhoods of any s... |
| neii2 23042 | Property of a neighborhood... |
| neiss 23043 | Any neighborhood of a set ... |
| ssnei 23044 | A set is included in any o... |
| elnei 23045 | A point belongs to any of ... |
| 0nnei 23046 | The empty set is not a nei... |
| neips 23047 | A neighborhood of a set is... |
| opnneissb 23048 | An open set is a neighborh... |
| opnssneib 23049 | Any superset of an open se... |
| ssnei2 23050 | Any subset ` M ` of ` X ` ... |
| neindisj 23051 | Any neighborhood of an ele... |
| opnneiss 23052 | An open set is a neighborh... |
| opnneip 23053 | An open set is a neighborh... |
| opnnei 23054 | A set is open iff it is a ... |
| tpnei 23055 | The underlying set of a to... |
| neiuni 23056 | The union of the neighborh... |
| neindisj2 23057 | A point ` P ` belongs to t... |
| topssnei 23058 | A finer topology has more ... |
| innei 23059 | The intersection of two ne... |
| opnneiid 23060 | Only an open set is a neig... |
| neissex 23061 | For any neighborhood ` N `... |
| 0nei 23062 | The empty set is a neighbo... |
| neipeltop 23063 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23064 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23065 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23066 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23067 | If, to each element ` P ` ... |
| lpfval 23072 | The limit point function o... |
| lpval 23073 | The set of limit points of... |
| islp 23074 | The predicate "the class `... |
| lpsscls 23075 | The limit points of a subs... |
| lpss 23076 | The limit points of a subs... |
| lpdifsn 23077 | ` P ` is a limit point of ... |
| lpss3 23078 | Subset relationship for li... |
| islp2 23079 | The predicate " ` P ` is a... |
| islp3 23080 | The predicate " ` P ` is a... |
| maxlp 23081 | A point is a limit point o... |
| clslp 23082 | The closure of a subset of... |
| islpi 23083 | A point belonging to a set... |
| cldlp 23084 | A subset of a topological ... |
| isperf 23085 | Definition of a perfect sp... |
| isperf2 23086 | Definition of a perfect sp... |
| isperf3 23087 | A perfect space is a topol... |
| perflp 23088 | The limit points of a perf... |
| perfi 23089 | Property of a perfect spac... |
| perftop 23090 | A perfect space is a topol... |
| restrcl 23091 | Reverse closure for the su... |
| restbas 23092 | A subspace topology basis ... |
| tgrest 23093 | A subspace can be generate... |
| resttop 23094 | A subspace topology is a t... |
| resttopon 23095 | A subspace topology is a t... |
| restuni 23096 | The underlying set of a su... |
| stoig 23097 | The topological space buil... |
| restco 23098 | Composition of subspaces. ... |
| restabs 23099 | Equivalence of being a sub... |
| restin 23100 | When the subspace region i... |
| restuni2 23101 | The underlying set of a su... |
| resttopon2 23102 | The underlying set of a su... |
| rest0 23103 | The subspace topology indu... |
| restsn 23104 | The only subspace topology... |
| restsn2 23105 | The subspace topology indu... |
| restcld 23106 | A closed set of a subspace... |
| restcldi 23107 | A closed set is closed in ... |
| restcldr 23108 | A set which is closed in t... |
| restopnb 23109 | If ` B ` is an open subset... |
| ssrest 23110 | If ` K ` is a finer topolo... |
| restopn2 23111 | If ` A ` is open, then ` B... |
| restdis 23112 | A subspace of a discrete t... |
| restfpw 23113 | The restriction of the set... |
| neitr 23114 | The neighborhood of a trac... |
| restcls 23115 | A closure in a subspace to... |
| restntr 23116 | An interior in a subspace ... |
| restlp 23117 | The limit points of a subs... |
| restperf 23118 | Perfection of a subspace. ... |
| perfopn 23119 | An open subset of a perfec... |
| resstopn 23120 | The topology of a restrict... |
| resstps 23121 | A restricted topological s... |
| ordtbaslem 23122 | Lemma for ~ ordtbas . In ... |
| ordtval 23123 | Value of the order topolog... |
| ordtuni 23124 | Value of the order topolog... |
| ordtbas2 23125 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23126 | In a total order, the fini... |
| ordttopon 23127 | Value of the order topolog... |
| ordtopn1 23128 | An upward ray ` ( P , +oo ... |
| ordtopn2 23129 | A downward ray ` ( -oo , P... |
| ordtopn3 23130 | An open interval ` ( A , B... |
| ordtcld1 23131 | A downward ray ` ( -oo , P... |
| ordtcld2 23132 | An upward ray ` [ P , +oo ... |
| ordtcld3 23133 | A closed interval ` [ A , ... |
| ordttop 23134 | The order topology is a to... |
| ordtcnv 23135 | The order dual generates t... |
| ordtrest 23136 | The subspace topology of a... |
| ordtrest2lem 23137 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23138 | An interval-closed set ` A... |
| letopon 23139 | The topology of the extend... |
| letop 23140 | The topology of the extend... |
| letopuni 23141 | The topology of the extend... |
| xrstopn 23142 | The topology component of ... |
| xrstps 23143 | The extended real number s... |
| leordtvallem1 23144 | Lemma for ~ leordtval . (... |
| leordtvallem2 23145 | Lemma for ~ leordtval . (... |
| leordtval2 23146 | The topology of the extend... |
| leordtval 23147 | The topology of the extend... |
| iccordt 23148 | A closed interval is close... |
| iocpnfordt 23149 | An unbounded above open in... |
| icomnfordt 23150 | An unbounded above open in... |
| iooordt 23151 | An open interval is open i... |
| reordt 23152 | The real numbers are an op... |
| lecldbas 23153 | The set of closed interval... |
| pnfnei 23154 | A neighborhood of ` +oo ` ... |
| mnfnei 23155 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23156 | The restriction of the les... |
| ordtresticc 23157 | The restriction of the les... |
| lmrel 23164 | The topological space conv... |
| lmrcl 23165 | Reverse closure for the co... |
| lmfval 23166 | The relation "sequence ` f... |
| cnfval 23167 | The set of all continuous ... |
| cnpfval 23168 | The function mapping the p... |
| iscn 23169 | The predicate "the class `... |
| cnpval 23170 | The set of all functions f... |
| iscnp 23171 | The predicate "the class `... |
| iscn2 23172 | The predicate "the class `... |
| iscnp2 23173 | The predicate "the class `... |
| cntop1 23174 | Reverse closure for a cont... |
| cntop2 23175 | Reverse closure for a cont... |
| cnptop1 23176 | Reverse closure for a func... |
| cnptop2 23177 | Reverse closure for a func... |
| iscnp3 23178 | The predicate "the class `... |
| cnprcl 23179 | Reverse closure for a func... |
| cnf 23180 | A continuous function is a... |
| cnpf 23181 | A continuous function at p... |
| cnpcl 23182 | The value of a continuous ... |
| cnf2 23183 | A continuous function is a... |
| cnpf2 23184 | A continuous function at p... |
| cnprcl2 23185 | Reverse closure for a func... |
| tgcn 23186 | The continuity predicate w... |
| tgcnp 23187 | The "continuous at a point... |
| subbascn 23188 | The continuity predicate w... |
| ssidcn 23189 | The identity function is a... |
| cnpimaex 23190 | Property of a function con... |
| idcn 23191 | A restricted identity func... |
| lmbr 23192 | Express the binary relatio... |
| lmbr2 23193 | Express the binary relatio... |
| lmbrf 23194 | Express the binary relatio... |
| lmconst 23195 | A constant sequence conver... |
| lmcvg 23196 | Convergence property of a ... |
| iscnp4 23197 | The predicate "the class `... |
| cnpnei 23198 | A condition for continuity... |
| cnima 23199 | An open subset of the codo... |
| cnco 23200 | The composition of two con... |
| cnpco 23201 | The composition of a funct... |
| cnclima 23202 | A closed subset of the cod... |
| iscncl 23203 | A characterization of a co... |
| cncls2i 23204 | Property of the preimage o... |
| cnntri 23205 | Property of the preimage o... |
| cnclsi 23206 | Property of the image of a... |
| cncls2 23207 | Continuity in terms of clo... |
| cncls 23208 | Continuity in terms of clo... |
| cnntr 23209 | Continuity in terms of int... |
| cnss1 23210 | If the topology ` K ` is f... |
| cnss2 23211 | If the topology ` K ` is f... |
| cncnpi 23212 | A continuous function is c... |
| cnsscnp 23213 | The set of continuous func... |
| cncnp 23214 | A continuous function is c... |
| cncnp2 23215 | A continuous function is c... |
| cnnei 23216 | Continuity in terms of nei... |
| cnconst2 23217 | A constant function is con... |
| cnconst 23218 | A constant function is con... |
| cnrest 23219 | Continuity of a restrictio... |
| cnrest2 23220 | Equivalence of continuity ... |
| cnrest2r 23221 | Equivalence of continuity ... |
| cnpresti 23222 | One direction of ~ cnprest... |
| cnprest 23223 | Equivalence of continuity ... |
| cnprest2 23224 | Equivalence of point-conti... |
| cndis 23225 | Every function is continuo... |
| cnindis 23226 | Every function is continuo... |
| cnpdis 23227 | If ` A ` is an isolated po... |
| paste 23228 | Pasting lemma. If ` A ` a... |
| lmfpm 23229 | If ` F ` converges, then `... |
| lmfss 23230 | Inclusion of a function ha... |
| lmcl 23231 | Closure of a limit. (Cont... |
| lmss 23232 | Limit on a subspace. (Con... |
| sslm 23233 | A finer topology has fewer... |
| lmres 23234 | A function converges iff i... |
| lmff 23235 | If ` F ` converges, there ... |
| lmcls 23236 | Any convergent sequence of... |
| lmcld 23237 | Any convergent sequence of... |
| lmcnp 23238 | The image of a convergent ... |
| lmcn 23239 | The image of a convergent ... |
| ist0 23254 | The predicate "is a T_0 sp... |
| ist1 23255 | The predicate "is a T_1 sp... |
| ishaus 23256 | The predicate "is a Hausdo... |
| iscnrm 23257 | The property of being comp... |
| t0sep 23258 | Any two topologically indi... |
| t0dist 23259 | Any two distinct points in... |
| t1sncld 23260 | In a T_1 space, singletons... |
| t1ficld 23261 | In a T_1 space, finite set... |
| hausnei 23262 | Neighborhood property of a... |
| t0top 23263 | A T_0 space is a topologic... |
| t1top 23264 | A T_1 space is a topologic... |
| haustop 23265 | A Hausdorff space is a top... |
| isreg 23266 | The predicate "is a regula... |
| regtop 23267 | A regular space is a topol... |
| regsep 23268 | In a regular space, every ... |
| isnrm 23269 | The predicate "is a normal... |
| nrmtop 23270 | A normal space is a topolo... |
| cnrmtop 23271 | A completely normal space ... |
| iscnrm2 23272 | The property of being comp... |
| ispnrm 23273 | The property of being perf... |
| pnrmnrm 23274 | A perfectly normal space i... |
| pnrmtop 23275 | A perfectly normal space i... |
| pnrmcld 23276 | A closed set in a perfectl... |
| pnrmopn 23277 | An open set in a perfectly... |
| ist0-2 23278 | The predicate "is a T_0 sp... |
| ist0-3 23279 | The predicate "is a T_0 sp... |
| cnt0 23280 | The preimage of a T_0 topo... |
| ist1-2 23281 | An alternate characterizat... |
| t1t0 23282 | A T_1 space is a T_0 space... |
| ist1-3 23283 | A space is T_1 iff every p... |
| cnt1 23284 | The preimage of a T_1 topo... |
| ishaus2 23285 | Express the predicate " ` ... |
| haust1 23286 | A Hausdorff space is a T_1... |
| hausnei2 23287 | The Hausdorff condition st... |
| cnhaus 23288 | The preimage of a Hausdorf... |
| nrmsep3 23289 | In a normal space, given a... |
| nrmsep2 23290 | In a normal space, any two... |
| nrmsep 23291 | In a normal space, disjoin... |
| isnrm2 23292 | An alternate characterizat... |
| isnrm3 23293 | A topological space is nor... |
| cnrmi 23294 | A subspace of a completely... |
| cnrmnrm 23295 | A completely normal space ... |
| restcnrm 23296 | A subspace of a completely... |
| resthauslem 23297 | Lemma for ~ resthaus and s... |
| lpcls 23298 | The limit points of the cl... |
| perfcls 23299 | A subset of a perfect spac... |
| restt0 23300 | A subspace of a T_0 topolo... |
| restt1 23301 | A subspace of a T_1 topolo... |
| resthaus 23302 | A subspace of a Hausdorff ... |
| t1sep2 23303 | Any two points in a T_1 sp... |
| t1sep 23304 | Any two distinct points in... |
| sncld 23305 | A singleton is closed in a... |
| sshauslem 23306 | Lemma for ~ sshaus and sim... |
| sst0 23307 | A topology finer than a T_... |
| sst1 23308 | A topology finer than a T_... |
| sshaus 23309 | A topology finer than a Ha... |
| regsep2 23310 | In a regular space, a clos... |
| isreg2 23311 | A topological space is reg... |
| dnsconst 23312 | If a continuous mapping to... |
| ordtt1 23313 | The order topology is T_1 ... |
| lmmo 23314 | A sequence in a Hausdorff ... |
| lmfun 23315 | The convergence relation i... |
| dishaus 23316 | A discrete topology is Hau... |
| ordthauslem 23317 | Lemma for ~ ordthaus . (C... |
| ordthaus 23318 | The order topology of a to... |
| xrhaus 23319 | The topology of the extend... |
| iscmp 23322 | The predicate "is a compac... |
| cmpcov 23323 | An open cover of a compact... |
| cmpcov2 23324 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23325 | Combine ~ cmpcov with ~ ac... |
| cncmp 23326 | Compactness is respected b... |
| fincmp 23327 | A finite topology is compa... |
| 0cmp 23328 | The singleton of the empty... |
| cmptop 23329 | A compact topology is a to... |
| rncmp 23330 | The image of a compact set... |
| imacmp 23331 | The image of a compact set... |
| discmp 23332 | A discrete topology is com... |
| cmpsublem 23333 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23334 | Two equivalent ways of des... |
| tgcmp 23335 | A topology generated by a ... |
| cmpcld 23336 | A closed subset of a compa... |
| uncmp 23337 | The union of two compact s... |
| fiuncmp 23338 | A finite union of compact ... |
| sscmp 23339 | A subset of a compact topo... |
| hauscmplem 23340 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23341 | A compact subspace of a T2... |
| cmpfi 23342 | If a topology is compact a... |
| cmpfii 23343 | In a compact topology, a s... |
| bwth 23344 | The glorious Bolzano-Weier... |
| isconn 23347 | The predicate ` J ` is a c... |
| isconn2 23348 | The predicate ` J ` is a c... |
| connclo 23349 | The only nonempty clopen s... |
| conndisj 23350 | If a topology is connected... |
| conntop 23351 | A connected topology is a ... |
| indisconn 23352 | The indiscrete topology (o... |
| dfconn2 23353 | An alternate definition of... |
| connsuba 23354 | Connectedness for a subspa... |
| connsub 23355 | Two equivalent ways of say... |
| cnconn 23356 | Connectedness is respected... |
| nconnsubb 23357 | Disconnectedness for a sub... |
| connsubclo 23358 | If a clopen set meets a co... |
| connima 23359 | The image of a connected s... |
| conncn 23360 | A continuous function from... |
| iunconnlem 23361 | Lemma for ~ iunconn . (Co... |
| iunconn 23362 | The indexed union of conne... |
| unconn 23363 | The union of two connected... |
| clsconn 23364 | The closure of a connected... |
| conncompid 23365 | The connected component co... |
| conncompconn 23366 | The connected component co... |
| conncompss 23367 | The connected component co... |
| conncompcld 23368 | The connected component co... |
| conncompclo 23369 | The connected component co... |
| t1connperf 23370 | A connected T_1 space is p... |
| is1stc 23375 | The predicate "is a first-... |
| is1stc2 23376 | An equivalent way of sayin... |
| 1stctop 23377 | A first-countable topology... |
| 1stcclb 23378 | A property of points in a ... |
| 1stcfb 23379 | For any point ` A ` in a f... |
| is2ndc 23380 | The property of being seco... |
| 2ndctop 23381 | A second-countable topolog... |
| 2ndci 23382 | A countable basis generate... |
| 2ndcsb 23383 | Having a countable subbase... |
| 2ndcredom 23384 | A second-countable space h... |
| 2ndc1stc 23385 | A second-countable space i... |
| 1stcrestlem 23386 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23387 | A subspace of a first-coun... |
| 2ndcrest 23388 | A subspace of a second-cou... |
| 2ndcctbss 23389 | If a topology is second-co... |
| 2ndcdisj 23390 | Any disjoint family of ope... |
| 2ndcdisj2 23391 | Any disjoint collection of... |
| 2ndcomap 23392 | A surjective continuous op... |
| 2ndcsep 23393 | A second-countable topolog... |
| dis2ndc 23394 | A discrete space is second... |
| 1stcelcls 23395 | A point belongs to the clo... |
| 1stccnp 23396 | A mapping is continuous at... |
| 1stccn 23397 | A mapping ` X --> Y ` , wh... |
| islly 23402 | The property of being a lo... |
| isnlly 23403 | The property of being an n... |
| llyeq 23404 | Equality theorem for the `... |
| nllyeq 23405 | Equality theorem for the `... |
| llytop 23406 | A locally ` A ` space is a... |
| nllytop 23407 | A locally ` A ` space is a... |
| llyi 23408 | The property of a locally ... |
| nllyi 23409 | The property of an n-local... |
| nlly2i 23410 | Eliminate the neighborhood... |
| llynlly 23411 | A locally ` A ` space is n... |
| llyssnlly 23412 | A locally ` A ` space is n... |
| llyss 23413 | The "locally" predicate re... |
| nllyss 23414 | The "n-locally" predicate ... |
| subislly 23415 | The property of a subspace... |
| restnlly 23416 | If the property ` A ` pass... |
| restlly 23417 | If the property ` A ` pass... |
| islly2 23418 | An alternative expression ... |
| llyrest 23419 | An open subspace of a loca... |
| nllyrest 23420 | An open subspace of an n-l... |
| loclly 23421 | If ` A ` is a local proper... |
| llyidm 23422 | Idempotence of the "locall... |
| nllyidm 23423 | Idempotence of the "n-loca... |
| toplly 23424 | A topology is locally a to... |
| topnlly 23425 | A topology is n-locally a ... |
| hauslly 23426 | A Hausdorff space is local... |
| hausnlly 23427 | A Hausdorff space is n-loc... |
| hausllycmp 23428 | A compact Hausdorff space ... |
| cldllycmp 23429 | A closed subspace of a loc... |
| lly1stc 23430 | First-countability is a lo... |
| dislly 23431 | The discrete space ` ~P X ... |
| disllycmp 23432 | A discrete space is locall... |
| dis1stc 23433 | A discrete space is first-... |
| hausmapdom 23434 | If ` X ` is a first-counta... |
| hauspwdom 23435 | Simplify the cardinal ` A ... |
| refrel 23442 | Refinement is a relation. ... |
| isref 23443 | The property of being a re... |
| refbas 23444 | A refinement covers the sa... |
| refssex 23445 | Every set in a refinement ... |
| ssref 23446 | A subcover is a refinement... |
| refref 23447 | Reflexivity of refinement.... |
| reftr 23448 | Refinement is transitive. ... |
| refun0 23449 | Adding the empty set prese... |
| isptfin 23450 | The statement "is a point-... |
| islocfin 23451 | The statement "is a locall... |
| finptfin 23452 | A finite cover is a point-... |
| ptfinfin 23453 | A point covered by a point... |
| finlocfin 23454 | A finite cover of a topolo... |
| locfintop 23455 | A locally finite cover cov... |
| locfinbas 23456 | A locally finite cover mus... |
| locfinnei 23457 | A point covered by a local... |
| lfinpfin 23458 | A locally finite cover is ... |
| lfinun 23459 | Adding a finite set preser... |
| locfincmp 23460 | For a compact space, the l... |
| unisngl 23461 | Taking the union of the se... |
| dissnref 23462 | The set of singletons is a... |
| dissnlocfin 23463 | The set of singletons is l... |
| locfindis 23464 | The locally finite covers ... |
| locfincf 23465 | A locally finite cover in ... |
| comppfsc 23466 | A space where every open c... |
| kgenval 23469 | Value of the compact gener... |
| elkgen 23470 | Value of the compact gener... |
| kgeni 23471 | Property of the open sets ... |
| kgentopon 23472 | The compact generator gene... |
| kgenuni 23473 | The base set of the compac... |
| kgenftop 23474 | The compact generator gene... |
| kgenf 23475 | The compact generator is a... |
| kgentop 23476 | A compactly generated spac... |
| kgenss 23477 | The compact generator gene... |
| kgenhaus 23478 | The compact generator gene... |
| kgencmp 23479 | The compact generator topo... |
| kgencmp2 23480 | The compact generator topo... |
| kgenidm 23481 | The compact generator is i... |
| iskgen2 23482 | A space is compactly gener... |
| iskgen3 23483 | Derive the usual definitio... |
| llycmpkgen2 23484 | A locally compact space is... |
| cmpkgen 23485 | A compact space is compact... |
| llycmpkgen 23486 | A locally compact space is... |
| 1stckgenlem 23487 | The one-point compactifica... |
| 1stckgen 23488 | A first-countable space is... |
| kgen2ss 23489 | The compact generator pres... |
| kgencn 23490 | A function from a compactl... |
| kgencn2 23491 | A function ` F : J --> K `... |
| kgencn3 23492 | The set of continuous func... |
| kgen2cn 23493 | A continuous function is a... |
| txval 23498 | Value of the binary topolo... |
| txuni2 23499 | The underlying set of the ... |
| txbasex 23500 | The basis for the product ... |
| txbas 23501 | The set of Cartesian produ... |
| eltx 23502 | A set in a product is open... |
| txtop 23503 | The product of two topolog... |
| ptval 23504 | The value of the product t... |
| ptpjpre1 23505 | The preimage of a projecti... |
| elpt 23506 | Elementhood in the bases o... |
| elptr 23507 | A basic open set in the pr... |
| elptr2 23508 | A basic open set in the pr... |
| ptbasid 23509 | The base set of the produc... |
| ptuni2 23510 | The base set for the produ... |
| ptbasin 23511 | The basis for a product to... |
| ptbasin2 23512 | The basis for a product to... |
| ptbas 23513 | The basis for a product to... |
| ptpjpre2 23514 | The basis for a product to... |
| ptbasfi 23515 | The basis for the product ... |
| pttop 23516 | The product topology is a ... |
| ptopn 23517 | A basic open set in the pr... |
| ptopn2 23518 | A sub-basic open set in th... |
| xkotf 23519 | Functionality of function ... |
| xkobval 23520 | Alternative expression for... |
| xkoval 23521 | Value of the compact-open ... |
| xkotop 23522 | The compact-open topology ... |
| xkoopn 23523 | A basic open set of the co... |
| txtopi 23524 | The product of two topolog... |
| txtopon 23525 | The underlying set of the ... |
| txuni 23526 | The underlying set of the ... |
| txunii 23527 | The underlying set of the ... |
| ptuni 23528 | The base set for the produ... |
| ptunimpt 23529 | Base set of a product topo... |
| pttopon 23530 | The base set for the produ... |
| pttoponconst 23531 | The base set for a product... |
| ptuniconst 23532 | The base set for a product... |
| xkouni 23533 | The base set of the compac... |
| xkotopon 23534 | The base set of the compac... |
| ptval2 23535 | The value of the product t... |
| txopn 23536 | The product of two open se... |
| txcld 23537 | The product of two closed ... |
| txcls 23538 | Closure of a rectangle in ... |
| txss12 23539 | Subset property of the top... |
| txbasval 23540 | It is sufficient to consid... |
| neitx 23541 | The Cartesian product of t... |
| txcnpi 23542 | Continuity of a two-argume... |
| tx1cn 23543 | Continuity of the first pr... |
| tx2cn 23544 | Continuity of the second p... |
| ptpjcn 23545 | Continuity of a projection... |
| ptpjopn 23546 | The projection map is an o... |
| ptcld 23547 | A closed box in the produc... |
| ptcldmpt 23548 | A closed box in the produc... |
| ptclsg 23549 | The closure of a box in th... |
| ptcls 23550 | The closure of a box in th... |
| dfac14lem 23551 | Lemma for ~ dfac14 . By e... |
| dfac14 23552 | Theorem ~ ptcls is an equi... |
| xkoccn 23553 | The "constant function" fu... |
| txcnp 23554 | If two functions are conti... |
| ptcnplem 23555 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23556 | If every projection of a f... |
| upxp 23557 | Universal property of the ... |
| txcnmpt 23558 | A map into the product of ... |
| uptx 23559 | Universal property of the ... |
| txcn 23560 | A map into the product of ... |
| ptcn 23561 | If every projection of a f... |
| prdstopn 23562 | Topology of a structure pr... |
| prdstps 23563 | A structure product of top... |
| pwstps 23564 | A structure power of a top... |
| txrest 23565 | The subspace of a topologi... |
| txdis 23566 | The topological product of... |
| txindislem 23567 | Lemma for ~ txindis . (Co... |
| txindis 23568 | The topological product of... |
| txdis1cn 23569 | A function is jointly cont... |
| txlly 23570 | If the property ` A ` is p... |
| txnlly 23571 | If the property ` A ` is p... |
| pthaus 23572 | The product of a collectio... |
| ptrescn 23573 | Restriction is a continuou... |
| txtube 23574 | The "tube lemma". If ` X ... |
| txcmplem1 23575 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23576 | Lemma for ~ txcmp . (Cont... |
| txcmp 23577 | The topological product of... |
| txcmpb 23578 | The topological product of... |
| hausdiag 23579 | A topology is Hausdorff if... |
| hauseqlcld 23580 | In a Hausdorff topology, t... |
| txhaus 23581 | The topological product of... |
| txlm 23582 | Two sequences converge iff... |
| lmcn2 23583 | The image of a convergent ... |
| tx1stc 23584 | The topological product of... |
| tx2ndc 23585 | The topological product of... |
| txkgen 23586 | The topological product of... |
| xkohaus 23587 | If the codomain space is H... |
| xkoptsub 23588 | The compact-open topology ... |
| xkopt 23589 | The compact-open topology ... |
| xkopjcn 23590 | Continuity of a projection... |
| xkoco1cn 23591 | If ` F ` is a continuous f... |
| xkoco2cn 23592 | If ` F ` is a continuous f... |
| xkococnlem 23593 | Continuity of the composit... |
| xkococn 23594 | Continuity of the composit... |
| cnmptid 23595 | The identity function is c... |
| cnmptc 23596 | A constant function is con... |
| cnmpt11 23597 | The composition of continu... |
| cnmpt11f 23598 | The composition of continu... |
| cnmpt1t 23599 | The composition of continu... |
| cnmpt12f 23600 | The composition of continu... |
| cnmpt12 23601 | The composition of continu... |
| cnmpt1st 23602 | The projection onto the fi... |
| cnmpt2nd 23603 | The projection onto the se... |
| cnmpt2c 23604 | A constant function is con... |
| cnmpt21 23605 | The composition of continu... |
| cnmpt21f 23606 | The composition of continu... |
| cnmpt2t 23607 | The composition of continu... |
| cnmpt22 23608 | The composition of continu... |
| cnmpt22f 23609 | The composition of continu... |
| cnmpt1res 23610 | The restriction of a conti... |
| cnmpt2res 23611 | The restriction of a conti... |
| cnmptcom 23612 | The argument converse of a... |
| cnmptkc 23613 | The curried first projecti... |
| cnmptkp 23614 | The evaluation of the inne... |
| cnmptk1 23615 | The composition of a curri... |
| cnmpt1k 23616 | The composition of a one-a... |
| cnmptkk 23617 | The composition of two cur... |
| xkofvcn 23618 | Joint continuity of the fu... |
| cnmptk1p 23619 | The evaluation of a currie... |
| cnmptk2 23620 | The uncurrying of a currie... |
| xkoinjcn 23621 | Continuity of "injection",... |
| cnmpt2k 23622 | The currying of a two-argu... |
| txconn 23623 | The topological product of... |
| imasnopn 23624 | If a relation graph is ope... |
| imasncld 23625 | If a relation graph is clo... |
| imasncls 23626 | If a relation graph is clo... |
| qtopval 23629 | Value of the quotient topo... |
| qtopval2 23630 | Value of the quotient topo... |
| elqtop 23631 | Value of the quotient topo... |
| qtopres 23632 | The quotient topology is u... |
| qtoptop2 23633 | The quotient topology is a... |
| qtoptop 23634 | The quotient topology is a... |
| elqtop2 23635 | Value of the quotient topo... |
| qtopuni 23636 | The base set of the quotie... |
| elqtop3 23637 | Value of the quotient topo... |
| qtoptopon 23638 | The base set of the quotie... |
| qtopid 23639 | A quotient map is a contin... |
| idqtop 23640 | The quotient topology indu... |
| qtopcmplem 23641 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23642 | A quotient of a compact sp... |
| qtopconn 23643 | A quotient of a connected ... |
| qtopkgen 23644 | A quotient of a compactly ... |
| basqtop 23645 | An injection maps bases to... |
| tgqtop 23646 | An injection maps generate... |
| qtopcld 23647 | The property of being a cl... |
| qtopcn 23648 | Universal property of a qu... |
| qtopss 23649 | A surjective continuous fu... |
| qtopeu 23650 | Universal property of the ... |
| qtoprest 23651 | If ` A ` is a saturated op... |
| qtopomap 23652 | If ` F ` is a surjective c... |
| qtopcmap 23653 | If ` F ` is a surjective c... |
| imastopn 23654 | The topology of an image s... |
| imastps 23655 | The image of a topological... |
| qustps 23656 | A quotient structure is a ... |
| kqfval 23657 | Value of the function appe... |
| kqfeq 23658 | Two points in the Kolmogor... |
| kqffn 23659 | The topological indistingu... |
| kqval 23660 | Value of the quotient topo... |
| kqtopon 23661 | The Kolmogorov quotient is... |
| kqid 23662 | The topological indistingu... |
| ist0-4 23663 | The topological indistingu... |
| kqfvima 23664 | When the image set is open... |
| kqsat 23665 | Any open set is saturated ... |
| kqdisj 23666 | A version of ~ imain for t... |
| kqcldsat 23667 | Any closed set is saturate... |
| kqopn 23668 | The topological indistingu... |
| kqcld 23669 | The topological indistingu... |
| kqt0lem 23670 | Lemma for ~ kqt0 . (Contr... |
| isr0 23671 | The property " ` J ` is an... |
| r0cld 23672 | The analogue of the T_1 ax... |
| regr1lem 23673 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23674 | A Kolmogorov quotient of a... |
| kqreglem1 23675 | A Kolmogorov quotient of a... |
| kqreglem2 23676 | If the Kolmogorov quotient... |
| kqnrmlem1 23677 | A Kolmogorov quotient of a... |
| kqnrmlem2 23678 | If the Kolmogorov quotient... |
| kqtop 23679 | The Kolmogorov quotient is... |
| kqt0 23680 | The Kolmogorov quotient is... |
| kqf 23681 | The Kolmogorov quotient is... |
| r0sep 23682 | The separation property of... |
| nrmr0reg 23683 | A normal R_0 space is also... |
| regr1 23684 | A regular space is R_1, wh... |
| kqreg 23685 | The Kolmogorov quotient of... |
| kqnrm 23686 | The Kolmogorov quotient of... |
| hmeofn 23691 | The set of homeomorphisms ... |
| hmeofval 23692 | The set of all the homeomo... |
| ishmeo 23693 | The predicate F is a homeo... |
| hmeocn 23694 | A homeomorphism is continu... |
| hmeocnvcn 23695 | The converse of a homeomor... |
| hmeocnv 23696 | The converse of a homeomor... |
| hmeof1o2 23697 | A homeomorphism is a 1-1-o... |
| hmeof1o 23698 | A homeomorphism is a 1-1-o... |
| hmeoima 23699 | The image of an open set b... |
| hmeoopn 23700 | Homeomorphisms preserve op... |
| hmeocld 23701 | Homeomorphisms preserve cl... |
| hmeocls 23702 | Homeomorphisms preserve cl... |
| hmeontr 23703 | Homeomorphisms preserve in... |
| hmeoimaf1o 23704 | The function mapping open ... |
| hmeores 23705 | The restriction of a homeo... |
| hmeoco 23706 | The composite of two homeo... |
| idhmeo 23707 | The identity function is a... |
| hmeocnvb 23708 | The converse of a homeomor... |
| hmeoqtop 23709 | A homeomorphism is a quoti... |
| hmph 23710 | Express the predicate ` J ... |
| hmphi 23711 | If there is a homeomorphis... |
| hmphtop 23712 | Reverse closure for the ho... |
| hmphtop1 23713 | The relation "being homeom... |
| hmphtop2 23714 | The relation "being homeom... |
| hmphref 23715 | "Is homeomorphic to" is re... |
| hmphsym 23716 | "Is homeomorphic to" is sy... |
| hmphtr 23717 | "Is homeomorphic to" is tr... |
| hmpher 23718 | "Is homeomorphic to" is an... |
| hmphen 23719 | Homeomorphisms preserve th... |
| hmphsymb 23720 | "Is homeomorphic to" is sy... |
| haushmphlem 23721 | Lemma for ~ haushmph and s... |
| cmphmph 23722 | Compactness is a topologic... |
| connhmph 23723 | Connectedness is a topolog... |
| t0hmph 23724 | T_0 is a topological prope... |
| t1hmph 23725 | T_1 is a topological prope... |
| haushmph 23726 | Hausdorff-ness is a topolo... |
| reghmph 23727 | Regularity is a topologica... |
| nrmhmph 23728 | Normality is a topological... |
| hmph0 23729 | A topology homeomorphic to... |
| hmphdis 23730 | Homeomorphisms preserve to... |
| hmphindis 23731 | Homeomorphisms preserve to... |
| indishmph 23732 | Equinumerous sets equipped... |
| hmphen2 23733 | Homeomorphisms preserve th... |
| cmphaushmeo 23734 | A continuous bijection fro... |
| ordthmeolem 23735 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23736 | An order isomorphism is a ... |
| txhmeo 23737 | Lift a pair of homeomorphi... |
| txswaphmeolem 23738 | Show inverse for the "swap... |
| txswaphmeo 23739 | There is a homeomorphism f... |
| pt1hmeo 23740 | The canonical homeomorphis... |
| ptuncnv 23741 | Exhibit the converse funct... |
| ptunhmeo 23742 | Define a homeomorphism fro... |
| xpstopnlem1 23743 | The function ` F ` used in... |
| xpstps 23744 | A binary product of topolo... |
| xpstopnlem2 23745 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23746 | The topology on a binary p... |
| ptcmpfi 23747 | A topological product of f... |
| xkocnv 23748 | The inverse of the "curryi... |
| xkohmeo 23749 | The Exponential Law for to... |
| qtopf1 23750 | If a quotient map is injec... |
| qtophmeo 23751 | If two functions on a base... |
| t0kq 23752 | A topological space is T_0... |
| kqhmph 23753 | A topological space is T_0... |
| ist1-5lem 23754 | Lemma for ~ ist1-5 and sim... |
| t1r0 23755 | A T_1 space is R_0. That ... |
| ist1-5 23756 | A topological space is T_1... |
| ishaus3 23757 | A topological space is Hau... |
| nrmreg 23758 | A normal T_1 space is regu... |
| reghaus 23759 | A regular T_0 space is Hau... |
| nrmhaus 23760 | A T_1 normal space is Haus... |
| elmptrab 23761 | Membership in a one-parame... |
| elmptrab2 23762 | Membership in a one-parame... |
| isfbas 23763 | The predicate " ` F ` is a... |
| fbasne0 23764 | There are no empty filter ... |
| 0nelfb 23765 | No filter base contains th... |
| fbsspw 23766 | A filter base on a set is ... |
| fbelss 23767 | An element of the filter b... |
| fbdmn0 23768 | The domain of a filter bas... |
| isfbas2 23769 | The predicate " ` F ` is a... |
| fbasssin 23770 | A filter base contains sub... |
| fbssfi 23771 | A filter base contains sub... |
| fbssint 23772 | A filter base contains sub... |
| fbncp 23773 | A filter base does not con... |
| fbun 23774 | A necessary and sufficient... |
| fbfinnfr 23775 | No filter base containing ... |
| opnfbas 23776 | The collection of open sup... |
| trfbas2 23777 | Conditions for the trace o... |
| trfbas 23778 | Conditions for the trace o... |
| isfil 23781 | The predicate "is a filter... |
| filfbas 23782 | A filter is a filter base.... |
| 0nelfil 23783 | The empty set doesn't belo... |
| fileln0 23784 | An element of a filter is ... |
| filsspw 23785 | A filter is a subset of th... |
| filelss 23786 | An element of a filter is ... |
| filss 23787 | A filter is closed under t... |
| filin 23788 | A filter is closed under t... |
| filtop 23789 | The underlying set belongs... |
| isfil2 23790 | Derive the standard axioms... |
| isfildlem 23791 | Lemma for ~ isfild . (Con... |
| isfild 23792 | Sufficient condition for a... |
| filfi 23793 | A filter is closed under t... |
| filinn0 23794 | The intersection of two el... |
| filintn0 23795 | A filter has the finite in... |
| filn0 23796 | The empty set is not a fil... |
| infil 23797 | The intersection of two fi... |
| snfil 23798 | A singleton is a filter. ... |
| fbasweak 23799 | A filter base on any set i... |
| snfbas 23800 | Condition for a singleton ... |
| fsubbas 23801 | A condition for a set to g... |
| fbasfip 23802 | A filter base has the fini... |
| fbunfip 23803 | A helpful lemma for showin... |
| fgval 23804 | The filter generating clas... |
| elfg 23805 | A condition for elements o... |
| ssfg 23806 | A filter base is a subset ... |
| fgss 23807 | A bigger base generates a ... |
| fgss2 23808 | A condition for a filter t... |
| fgfil 23809 | A filter generates itself.... |
| elfilss 23810 | An element belongs to a fi... |
| filfinnfr 23811 | No filter containing a fin... |
| fgcl 23812 | A generated filter is a fi... |
| fgabs 23813 | Absorption law for filter ... |
| neifil 23814 | The neighborhoods of a non... |
| filunibas 23815 | Recover the base set from ... |
| filunirn 23816 | Two ways to express a filt... |
| filconn 23817 | A filter gives rise to a c... |
| fbasrn 23818 | Given a filter on a domain... |
| filuni 23819 | The union of a nonempty se... |
| trfil1 23820 | Conditions for the trace o... |
| trfil2 23821 | Conditions for the trace o... |
| trfil3 23822 | Conditions for the trace o... |
| trfilss 23823 | If ` A ` is a member of th... |
| fgtr 23824 | If ` A ` is a member of th... |
| trfg 23825 | The trace operation and th... |
| trnei 23826 | The trace, over a set ` A ... |
| cfinfil 23827 | Relative complements of th... |
| csdfil 23828 | The set of all elements wh... |
| supfil 23829 | The supersets of a nonempt... |
| zfbas 23830 | The set of upper sets of i... |
| uzrest 23831 | The restriction of the set... |
| uzfbas 23832 | The set of upper sets of i... |
| isufil 23837 | The property of being an u... |
| ufilfil 23838 | An ultrafilter is a filter... |
| ufilss 23839 | For any subset of the base... |
| ufilb 23840 | The complement is in an ul... |
| ufilmax 23841 | Any filter finer than an u... |
| isufil2 23842 | The maximal property of an... |
| ufprim 23843 | An ultrafilter is a prime ... |
| trufil 23844 | Conditions for the trace o... |
| filssufilg 23845 | A filter is contained in s... |
| filssufil 23846 | A filter is contained in s... |
| isufl 23847 | Define the (strong) ultraf... |
| ufli 23848 | Property of a set that sat... |
| numufl 23849 | Consequence of ~ filssufil... |
| fiufl 23850 | A finite set satisfies the... |
| acufl 23851 | The axiom of choice implie... |
| ssufl 23852 | If ` Y ` is a subset of ` ... |
| ufileu 23853 | If the ultrafilter contain... |
| filufint 23854 | A filter is equal to the i... |
| uffix 23855 | Lemma for ~ fixufil and ~ ... |
| fixufil 23856 | The condition describing a... |
| uffixfr 23857 | An ultrafilter is either f... |
| uffix2 23858 | A classification of fixed ... |
| uffixsn 23859 | The singleton of the gener... |
| ufildom1 23860 | An ultrafilter is generate... |
| uffinfix 23861 | An ultrafilter containing ... |
| cfinufil 23862 | An ultrafilter is free iff... |
| ufinffr 23863 | An infinite subset is cont... |
| ufilen 23864 | Any infinite set has an ul... |
| ufildr 23865 | An ultrafilter gives rise ... |
| fin1aufil 23866 | There are no definable fre... |
| fmval 23877 | Introduce a function that ... |
| fmfil 23878 | A mapping filter is a filt... |
| fmf 23879 | Pushing-forward via a func... |
| fmss 23880 | A finer filter produces a ... |
| elfm 23881 | An element of a mapping fi... |
| elfm2 23882 | An element of a mapping fi... |
| fmfg 23883 | The image filter of a filt... |
| elfm3 23884 | An alternate formulation o... |
| imaelfm 23885 | An image of a filter eleme... |
| rnelfmlem 23886 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23887 | A condition for a filter t... |
| fmfnfmlem1 23888 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23889 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23890 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23891 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23892 | A filter finer than an ima... |
| fmufil 23893 | An image filter of an ultr... |
| fmid 23894 | The filter map applied to ... |
| fmco 23895 | Composition of image filte... |
| ufldom 23896 | The ultrafilter lemma prop... |
| flimval 23897 | The set of limit points of... |
| elflim2 23898 | The predicate "is a limit ... |
| flimtop 23899 | Reverse closure for the li... |
| flimneiss 23900 | A filter contains the neig... |
| flimnei 23901 | A filter contains all of t... |
| flimelbas 23902 | A limit point of a filter ... |
| flimfil 23903 | Reverse closure for the li... |
| flimtopon 23904 | Reverse closure for the li... |
| elflim 23905 | The predicate "is a limit ... |
| flimss2 23906 | A limit point of a filter ... |
| flimss1 23907 | A limit point of a filter ... |
| neiflim 23908 | A point is a limit point o... |
| flimopn 23909 | The condition for being a ... |
| fbflim 23910 | A condition for a filter t... |
| fbflim2 23911 | A condition for a filter b... |
| flimclsi 23912 | The convergent points of a... |
| hausflimlem 23913 | If ` A ` and ` B ` are bot... |
| hausflimi 23914 | One direction of ~ hausfli... |
| hausflim 23915 | A condition for a topology... |
| flimcf 23916 | Fineness is properly chara... |
| flimrest 23917 | The set of limit points in... |
| flimclslem 23918 | Lemma for ~ flimcls . (Co... |
| flimcls 23919 | Closure in terms of filter... |
| flimsncls 23920 | If ` A ` is a limit point ... |
| hauspwpwf1 23921 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23922 | If ` X ` is a Hausdorff sp... |
| flffval 23923 | Given a topology and a fil... |
| flfval 23924 | Given a function from a fi... |
| flfnei 23925 | The property of being a li... |
| flfneii 23926 | A neighborhood of a limit ... |
| isflf 23927 | The property of being a li... |
| flfelbas 23928 | A limit point of a functio... |
| flffbas 23929 | Limit points of a function... |
| flftg 23930 | Limit points of a function... |
| hausflf 23931 | If a function has its valu... |
| hausflf2 23932 | If a convergent function h... |
| cnpflfi 23933 | Forward direction of ~ cnp... |
| cnpflf2 23934 | ` F ` is continuous at poi... |
| cnpflf 23935 | Continuity of a function a... |
| cnflf 23936 | A function is continuous i... |
| cnflf2 23937 | A function is continuous i... |
| flfcnp 23938 | A continuous function pres... |
| lmflf 23939 | The topological limit rela... |
| txflf 23940 | Two sequences converge in ... |
| flfcnp2 23941 | The image of a convergent ... |
| fclsval 23942 | The set of all cluster poi... |
| isfcls 23943 | A cluster point of a filte... |
| fclsfil 23944 | Reverse closure for the cl... |
| fclstop 23945 | Reverse closure for the cl... |
| fclstopon 23946 | Reverse closure for the cl... |
| isfcls2 23947 | A cluster point of a filte... |
| fclsopn 23948 | Write the cluster point co... |
| fclsopni 23949 | An open neighborhood of a ... |
| fclselbas 23950 | A cluster point is in the ... |
| fclsneii 23951 | A neighborhood of a cluste... |
| fclssscls 23952 | The set of cluster points ... |
| fclsnei 23953 | Cluster points in terms of... |
| supnfcls 23954 | The filter of supersets of... |
| fclsbas 23955 | Cluster points in terms of... |
| fclsss1 23956 | A finer topology has fewer... |
| fclsss2 23957 | A finer filter has fewer c... |
| fclsrest 23958 | The set of cluster points ... |
| fclscf 23959 | Characterization of finene... |
| flimfcls 23960 | A limit point is a cluster... |
| fclsfnflim 23961 | A filter clusters at a poi... |
| flimfnfcls 23962 | A filter converges to a po... |
| fclscmpi 23963 | Forward direction of ~ fcl... |
| fclscmp 23964 | A space is compact iff eve... |
| uffclsflim 23965 | The cluster points of an u... |
| ufilcmp 23966 | A space is compact iff eve... |
| fcfval 23967 | The set of cluster points ... |
| isfcf 23968 | The property of being a cl... |
| fcfnei 23969 | The property of being a cl... |
| fcfelbas 23970 | A cluster point of a funct... |
| fcfneii 23971 | A neighborhood of a cluste... |
| flfssfcf 23972 | A limit point of a functio... |
| uffcfflf 23973 | If the domain filter is an... |
| cnpfcfi 23974 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23975 | A function ` F ` is contin... |
| cnfcf 23976 | Continuity of a function i... |
| flfcntr 23977 | A continuous function's va... |
| alexsublem 23978 | Lemma for ~ alexsub . (Co... |
| alexsub 23979 | The Alexander Subbase Theo... |
| alexsubb 23980 | Biconditional form of the ... |
| alexsubALTlem1 23981 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23982 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23983 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23984 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23985 | The Alexander Subbase Theo... |
| ptcmplem1 23986 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23987 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23988 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23989 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23990 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 23991 | Tychonoff's theorem: The ... |
| ptcmp 23992 | Tychonoff's theorem: The ... |
| cnextval 23995 | The function applying cont... |
| cnextfval 23996 | The continuous extension o... |
| cnextrel 23997 | In the general case, a con... |
| cnextfun 23998 | If the target space is Hau... |
| cnextfvval 23999 | The value of the continuou... |
| cnextf 24000 | Extension by continuity. ... |
| cnextcn 24001 | Extension by continuity. ... |
| cnextfres1 24002 | ` F ` and its extension by... |
| cnextfres 24003 | ` F ` and its extension by... |
| istmd 24008 | The predicate "is a topolo... |
| tmdmnd 24009 | A topological monoid is a ... |
| tmdtps 24010 | A topological monoid is a ... |
| istgp 24011 | The predicate "is a topolo... |
| tgpgrp 24012 | A topological group is a g... |
| tgptmd 24013 | A topological group is a t... |
| tgptps 24014 | A topological group is a t... |
| tmdtopon 24015 | The topology of a topologi... |
| tgptopon 24016 | The topology of a topologi... |
| tmdcn 24017 | In a topological monoid, t... |
| tgpcn 24018 | In a topological group, th... |
| tgpinv 24019 | In a topological group, th... |
| grpinvhmeo 24020 | The inverse function in a ... |
| cnmpt1plusg 24021 | Continuity of the group su... |
| cnmpt2plusg 24022 | Continuity of the group su... |
| tmdcn2 24023 | Write out the definition o... |
| tgpsubcn 24024 | In a topological group, th... |
| istgp2 24025 | A group with a topology is... |
| tmdmulg 24026 | In a topological monoid, t... |
| tgpmulg 24027 | In a topological group, th... |
| tgpmulg2 24028 | In a topological monoid, t... |
| tmdgsum 24029 | In a topological monoid, t... |
| tmdgsum2 24030 | For any neighborhood ` U `... |
| oppgtmd 24031 | The opposite of a topologi... |
| oppgtgp 24032 | The opposite of a topologi... |
| distgp 24033 | Any group equipped with th... |
| indistgp 24034 | Any group equipped with th... |
| efmndtmd 24035 | The monoid of endofunction... |
| tmdlactcn 24036 | The left group action of e... |
| tgplacthmeo 24037 | The left group action of e... |
| submtmd 24038 | A submonoid of a topologic... |
| subgtgp 24039 | A subgroup of a topologica... |
| symgtgp 24040 | The symmetric group is a t... |
| subgntr 24041 | A subgroup of a topologica... |
| opnsubg 24042 | An open subgroup of a topo... |
| clssubg 24043 | The closure of a subgroup ... |
| clsnsg 24044 | The closure of a normal su... |
| cldsubg 24045 | A subgroup of finite index... |
| tgpconncompeqg 24046 | The connected component co... |
| tgpconncomp 24047 | The identity component, th... |
| tgpconncompss 24048 | The identity component is ... |
| ghmcnp 24049 | A group homomorphism on to... |
| snclseqg 24050 | The coset of the closure o... |
| tgphaus 24051 | A topological group is Hau... |
| tgpt1 24052 | Hausdorff and T1 are equiv... |
| tgpt0 24053 | Hausdorff and T0 are equiv... |
| qustgpopn 24054 | A quotient map in a topolo... |
| qustgplem 24055 | Lemma for ~ qustgp . (Con... |
| qustgp 24056 | The quotient of a topologi... |
| qustgphaus 24057 | The quotient of a topologi... |
| prdstmdd 24058 | The product of a family of... |
| prdstgpd 24059 | The product of a family of... |
| tsmsfbas 24062 | The collection of all sets... |
| tsmslem1 24063 | The finite partial sums of... |
| tsmsval2 24064 | Definition of the topologi... |
| tsmsval 24065 | Definition of the topologi... |
| tsmspropd 24066 | The group sum depends only... |
| eltsms 24067 | The property of being a su... |
| tsmsi 24068 | The property of being a su... |
| tsmscl 24069 | A sum in a topological gro... |
| haustsms 24070 | In a Hausdorff topological... |
| haustsms2 24071 | In a Hausdorff topological... |
| tsmscls 24072 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24073 | The convergent points of a... |
| tsmsid 24074 | If a sum is finite, the us... |
| haustsmsid 24075 | In a Hausdorff topological... |
| tsms0 24076 | The sum of zero is zero. ... |
| tsmssubm 24077 | Evaluate an infinite group... |
| tsmsres 24078 | Extend an infinite group s... |
| tsmsf1o 24079 | Re-index an infinite group... |
| tsmsmhm 24080 | Apply a continuous group h... |
| tsmsadd 24081 | The sum of two infinite gr... |
| tsmsinv 24082 | Inverse of an infinite gro... |
| tsmssub 24083 | The difference of two infi... |
| tgptsmscls 24084 | A sum in a topological gro... |
| tgptsmscld 24085 | The set of limit points to... |
| tsmssplit 24086 | Split a topological group ... |
| tsmsxplem1 24087 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24088 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24089 | Write a sum over a two-dim... |
| istrg 24098 | Express the predicate " ` ... |
| trgtmd 24099 | The multiplicative monoid ... |
| istdrg 24100 | Express the predicate " ` ... |
| tdrgunit 24101 | The unit group of a topolo... |
| trgtgp 24102 | A topological ring is a to... |
| trgtmd2 24103 | A topological ring is a to... |
| trgtps 24104 | A topological ring is a to... |
| trgring 24105 | A topological ring is a ri... |
| trggrp 24106 | A topological ring is a gr... |
| tdrgtrg 24107 | A topological division rin... |
| tdrgdrng 24108 | A topological division rin... |
| tdrgring 24109 | A topological division rin... |
| tdrgtmd 24110 | A topological division rin... |
| tdrgtps 24111 | A topological division rin... |
| istdrg2 24112 | A topological-ring divisio... |
| mulrcn 24113 | The functionalization of t... |
| invrcn2 24114 | The multiplicative inverse... |
| invrcn 24115 | The multiplicative inverse... |
| cnmpt1mulr 24116 | Continuity of ring multipl... |
| cnmpt2mulr 24117 | Continuity of ring multipl... |
| dvrcn 24118 | The division function is c... |
| istlm 24119 | The predicate " ` W ` is a... |
| vscacn 24120 | The scalar multiplication ... |
| tlmtmd 24121 | A topological module is a ... |
| tlmtps 24122 | A topological module is a ... |
| tlmlmod 24123 | A topological module is a ... |
| tlmtrg 24124 | The scalar ring of a topol... |
| tlmscatps 24125 | The scalar ring of a topol... |
| istvc 24126 | A topological vector space... |
| tvctdrg 24127 | The scalar field of a topo... |
| cnmpt1vsca 24128 | Continuity of scalar multi... |
| cnmpt2vsca 24129 | Continuity of scalar multi... |
| tlmtgp 24130 | A topological vector space... |
| tvctlm 24131 | A topological vector space... |
| tvclmod 24132 | A topological vector space... |
| tvclvec 24133 | A topological vector space... |
| ustfn 24136 | The defined uniform struct... |
| ustval 24137 | The class of all uniform s... |
| isust 24138 | The predicate " ` U ` is a... |
| ustssxp 24139 | Entourages are subsets of ... |
| ustssel 24140 | A uniform structure is upw... |
| ustbasel 24141 | The full set is always an ... |
| ustincl 24142 | A uniform structure is clo... |
| ustdiag 24143 | The diagonal set is includ... |
| ustinvel 24144 | If ` V ` is an entourage, ... |
| ustexhalf 24145 | For each entourage ` V ` t... |
| ustrel 24146 | The elements of uniform st... |
| ustfilxp 24147 | A uniform structure on a n... |
| ustne0 24148 | A uniform structure cannot... |
| ustssco 24149 | In an uniform structure, a... |
| ustexsym 24150 | In an uniform structure, f... |
| ustex2sym 24151 | In an uniform structure, f... |
| ustex3sym 24152 | In an uniform structure, f... |
| ustref 24153 | Any element of the base se... |
| ust0 24154 | The unique uniform structu... |
| ustn0 24155 | The empty set is not an un... |
| ustund 24156 | If two intersecting sets `... |
| ustelimasn 24157 | Any point ` A ` is near en... |
| ustneism 24158 | For a point ` A ` in ` X `... |
| elrnustOLD 24159 | Obsolete version of ~ elfv... |
| ustbas2 24160 | Second direction for ~ ust... |
| ustuni 24161 | The set union of a uniform... |
| ustbas 24162 | Recover the base of an uni... |
| ustimasn 24163 | Lemma for ~ ustuqtop . (C... |
| trust 24164 | The trace of a uniform str... |
| utopval 24167 | The topology induced by a ... |
| elutop 24168 | Open sets in the topology ... |
| utoptop 24169 | The topology induced by a ... |
| utopbas 24170 | The base of the topology i... |
| utoptopon 24171 | Topology induced by a unif... |
| restutop 24172 | Restriction of a topology ... |
| restutopopn 24173 | The restriction of the top... |
| ustuqtoplem 24174 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24175 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24176 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24177 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24178 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24179 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24180 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24181 | For a given uniform struct... |
| utopsnneiplem 24182 | The neighborhoods of a poi... |
| utopsnneip 24183 | The neighborhoods of a poi... |
| utopsnnei 24184 | Images of singletons by en... |
| utop2nei 24185 | For any symmetrical entour... |
| utop3cls 24186 | Relation between a topolog... |
| utopreg 24187 | All Hausdorff uniform spac... |
| ussval 24194 | The uniform structure on u... |
| ussid 24195 | In case the base of the ` ... |
| isusp 24196 | The predicate ` W ` is a u... |
| ressuss 24197 | Value of the uniform struc... |
| ressust 24198 | The uniform structure of a... |
| ressusp 24199 | The restriction of a unifo... |
| tusval 24200 | The value of the uniform s... |
| tuslem 24201 | Lemma for ~ tusbas , ~ tus... |
| tuslemOLD 24202 | Obsolete proof of ~ tuslem... |
| tusbas 24203 | The base set of a construc... |
| tusunif 24204 | The uniform structure of a... |
| tususs 24205 | The uniform structure of a... |
| tustopn 24206 | The topology induced by a ... |
| tususp 24207 | A constructed uniform spac... |
| tustps 24208 | A constructed uniform spac... |
| uspreg 24209 | If a uniform space is Haus... |
| ucnval 24212 | The set of all uniformly c... |
| isucn 24213 | The predicate " ` F ` is a... |
| isucn2 24214 | The predicate " ` F ` is a... |
| ucnimalem 24215 | Reformulate the ` G ` func... |
| ucnima 24216 | An equivalent statement of... |
| ucnprima 24217 | The preimage by a uniforml... |
| iducn 24218 | The identity is uniformly ... |
| cstucnd 24219 | A constant function is uni... |
| ucncn 24220 | Uniform continuity implies... |
| iscfilu 24223 | The predicate " ` F ` is a... |
| cfilufbas 24224 | A Cauchy filter base is a ... |
| cfiluexsm 24225 | For a Cauchy filter base a... |
| fmucndlem 24226 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24227 | The image of a Cauchy filt... |
| cfilufg 24228 | The filter generated by a ... |
| trcfilu 24229 | Condition for the trace of... |
| cfiluweak 24230 | A Cauchy filter base is al... |
| neipcfilu 24231 | In an uniform space, a nei... |
| iscusp 24234 | The predicate " ` W ` is a... |
| cuspusp 24235 | A complete uniform space i... |
| cuspcvg 24236 | In a complete uniform spac... |
| iscusp2 24237 | The predicate " ` W ` is a... |
| cnextucn 24238 | Extension by continuity. ... |
| ucnextcn 24239 | Extension by continuity. ... |
| ispsmet 24240 | Express the predicate " ` ... |
| psmetdmdm 24241 | Recover the base set from ... |
| psmetf 24242 | The distance function of a... |
| psmetcl 24243 | Closure of the distance fu... |
| psmet0 24244 | The distance function of a... |
| psmettri2 24245 | Triangle inequality for th... |
| psmetsym 24246 | The distance function of a... |
| psmettri 24247 | Triangle inequality for th... |
| psmetge0 24248 | The distance function of a... |
| psmetxrge0 24249 | The distance function of a... |
| psmetres2 24250 | Restriction of a pseudomet... |
| psmetlecl 24251 | Real closure of an extende... |
| distspace 24252 | A set ` X ` together with ... |
| ismet 24259 | Express the predicate " ` ... |
| isxmet 24260 | Express the predicate " ` ... |
| ismeti 24261 | Properties that determine ... |
| isxmetd 24262 | Properties that determine ... |
| isxmet2d 24263 | It is safe to only require... |
| metflem 24264 | Lemma for ~ metf and other... |
| xmetf 24265 | Mapping of the distance fu... |
| metf 24266 | Mapping of the distance fu... |
| xmetcl 24267 | Closure of the distance fu... |
| metcl 24268 | Closure of the distance fu... |
| ismet2 24269 | An extended metric is a me... |
| metxmet 24270 | A metric is an extended me... |
| xmetdmdm 24271 | Recover the base set from ... |
| metdmdm 24272 | Recover the base set from ... |
| xmetunirn 24273 | Two ways to express an ext... |
| xmeteq0 24274 | The value of an extended m... |
| meteq0 24275 | The value of a metric is z... |
| xmettri2 24276 | Triangle inequality for th... |
| mettri2 24277 | Triangle inequality for th... |
| xmet0 24278 | The distance function of a... |
| met0 24279 | The distance function of a... |
| xmetge0 24280 | The distance function of a... |
| metge0 24281 | The distance function of a... |
| xmetlecl 24282 | Real closure of an extende... |
| xmetsym 24283 | The distance function of a... |
| xmetpsmet 24284 | An extended metric is a ps... |
| xmettpos 24285 | The distance function of a... |
| metsym 24286 | The distance function of a... |
| xmettri 24287 | Triangle inequality for th... |
| mettri 24288 | Triangle inequality for th... |
| xmettri3 24289 | Triangle inequality for th... |
| mettri3 24290 | Triangle inequality for th... |
| xmetrtri 24291 | One half of the reverse tr... |
| xmetrtri2 24292 | The reverse triangle inequ... |
| metrtri 24293 | Reverse triangle inequalit... |
| xmetgt0 24294 | The distance function of a... |
| metgt0 24295 | The distance function of a... |
| metn0 24296 | A metric space is nonempty... |
| xmetres2 24297 | Restriction of an extended... |
| metreslem 24298 | Lemma for ~ metres . (Con... |
| metres2 24299 | Lemma for ~ metres . (Con... |
| xmetres 24300 | A restriction of an extend... |
| metres 24301 | A restriction of a metric ... |
| 0met 24302 | The empty metric. (Contri... |
| prdsdsf 24303 | The product metric is a fu... |
| prdsxmetlem 24304 | The product metric is an e... |
| prdsxmet 24305 | The product metric is an e... |
| prdsmet 24306 | The product metric is a me... |
| ressprdsds 24307 | Restriction of a product m... |
| resspwsds 24308 | Restriction of a power met... |
| imasdsf1olem 24309 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24310 | The distance function is t... |
| imasf1oxmet 24311 | The image of an extended m... |
| imasf1omet 24312 | The image of a metric is a... |
| xpsdsfn 24313 | Closure of the metric in a... |
| xpsdsfn2 24314 | Closure of the metric in a... |
| xpsxmetlem 24315 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24316 | A product metric of extend... |
| xpsdsval 24317 | Value of the metric in a b... |
| xpsmet 24318 | The direct product of two ... |
| blfvalps 24319 | The value of the ball func... |
| blfval 24320 | The value of the ball func... |
| blvalps 24321 | The ball around a point ` ... |
| blval 24322 | The ball around a point ` ... |
| elblps 24323 | Membership in a ball. (Co... |
| elbl 24324 | Membership in a ball. (Co... |
| elbl2ps 24325 | Membership in a ball. (Co... |
| elbl2 24326 | Membership in a ball. (Co... |
| elbl3ps 24327 | Membership in a ball, with... |
| elbl3 24328 | Membership in a ball, with... |
| blcomps 24329 | Commute the arguments to t... |
| blcom 24330 | Commute the arguments to t... |
| xblpnfps 24331 | The infinity ball in an ex... |
| xblpnf 24332 | The infinity ball in an ex... |
| blpnf 24333 | The infinity ball in a sta... |
| bldisj 24334 | Two balls are disjoint if ... |
| blgt0 24335 | A nonempty ball implies th... |
| bl2in 24336 | Two balls are disjoint if ... |
| xblss2ps 24337 | One ball is contained in a... |
| xblss2 24338 | One ball is contained in a... |
| blss2ps 24339 | One ball is contained in a... |
| blss2 24340 | One ball is contained in a... |
| blhalf 24341 | A ball of radius ` R / 2 `... |
| blfps 24342 | Mapping of a ball. (Contr... |
| blf 24343 | Mapping of a ball. (Contr... |
| blrnps 24344 | Membership in the range of... |
| blrn 24345 | Membership in the range of... |
| xblcntrps 24346 | A ball contains its center... |
| xblcntr 24347 | A ball contains its center... |
| blcntrps 24348 | A ball contains its center... |
| blcntr 24349 | A ball contains its center... |
| xbln0 24350 | A ball is nonempty iff the... |
| bln0 24351 | A ball is not empty. (Con... |
| blelrnps 24352 | A ball belongs to the set ... |
| blelrn 24353 | A ball belongs to the set ... |
| blssm 24354 | A ball is a subset of the ... |
| unirnblps 24355 | The union of the set of ba... |
| unirnbl 24356 | The union of the set of ba... |
| blin 24357 | The intersection of two ba... |
| ssblps 24358 | The size of a ball increas... |
| ssbl 24359 | The size of a ball increas... |
| blssps 24360 | Any point ` P ` in a ball ... |
| blss 24361 | Any point ` P ` in a ball ... |
| blssexps 24362 | Two ways to express the ex... |
| blssex 24363 | Two ways to express the ex... |
| ssblex 24364 | A nested ball exists whose... |
| blin2 24365 | Given any two balls and a ... |
| blbas 24366 | The balls of a metric spac... |
| blres 24367 | A ball in a restricted met... |
| xmeterval 24368 | Value of the "finitely sep... |
| xmeter 24369 | The "finitely separated" r... |
| xmetec 24370 | The equivalence classes un... |
| blssec 24371 | A ball centered at ` P ` i... |
| blpnfctr 24372 | The infinity ball in an ex... |
| xmetresbl 24373 | An extended metric restric... |
| mopnval 24374 | An open set is a subset of... |
| mopntopon 24375 | The set of open sets of a ... |
| mopntop 24376 | The set of open sets of a ... |
| mopnuni 24377 | The union of all open sets... |
| elmopn 24378 | The defining property of a... |
| mopnfss 24379 | The family of open sets of... |
| mopnm 24380 | The base set of a metric s... |
| elmopn2 24381 | A defining property of an ... |
| mopnss 24382 | An open set of a metric sp... |
| isxms 24383 | Express the predicate " ` ... |
| isxms2 24384 | Express the predicate " ` ... |
| isms 24385 | Express the predicate " ` ... |
| isms2 24386 | Express the predicate " ` ... |
| xmstopn 24387 | The topology component of ... |
| mstopn 24388 | The topology component of ... |
| xmstps 24389 | An extended metric space i... |
| msxms 24390 | A metric space is an exten... |
| mstps 24391 | A metric space is a topolo... |
| xmsxmet 24392 | The distance function, sui... |
| msmet 24393 | The distance function, sui... |
| msf 24394 | The distance function of a... |
| xmsxmet2 24395 | The distance function, sui... |
| msmet2 24396 | The distance function, sui... |
| mscl 24397 | Closure of the distance fu... |
| xmscl 24398 | Closure of the distance fu... |
| xmsge0 24399 | The distance function in a... |
| xmseq0 24400 | The distance between two p... |
| xmssym 24401 | The distance function in a... |
| xmstri2 24402 | Triangle inequality for th... |
| mstri2 24403 | Triangle inequality for th... |
| xmstri 24404 | Triangle inequality for th... |
| mstri 24405 | Triangle inequality for th... |
| xmstri3 24406 | Triangle inequality for th... |
| mstri3 24407 | Triangle inequality for th... |
| msrtri 24408 | Reverse triangle inequalit... |
| xmspropd 24409 | Property deduction for an ... |
| mspropd 24410 | Property deduction for a m... |
| setsmsbas 24411 | The base set of a construc... |
| setsmsbasOLD 24412 | Obsolete proof of ~ setsms... |
| setsmsds 24413 | The distance function of a... |
| setsmsdsOLD 24414 | Obsolete proof of ~ setsms... |
| setsmstset 24415 | The topology of a construc... |
| setsmstopn 24416 | The topology of a construc... |
| setsxms 24417 | The constructed metric spa... |
| setsms 24418 | The constructed metric spa... |
| tmsval 24419 | For any metric there is an... |
| tmslem 24420 | Lemma for ~ tmsbas , ~ tms... |
| tmslemOLD 24421 | Obsolete version of ~ tmsl... |
| tmsbas 24422 | The base set of a construc... |
| tmsds 24423 | The metric of a constructe... |
| tmstopn 24424 | The topology of a construc... |
| tmsxms 24425 | The constructed metric spa... |
| tmsms 24426 | The constructed metric spa... |
| imasf1obl 24427 | The image of a metric spac... |
| imasf1oxms 24428 | The image of a metric spac... |
| imasf1oms 24429 | The image of a metric spac... |
| prdsbl 24430 | A ball in the product metr... |
| mopni 24431 | An open set of a metric sp... |
| mopni2 24432 | An open set of a metric sp... |
| mopni3 24433 | An open set of a metric sp... |
| blssopn 24434 | The balls of a metric spac... |
| unimopn 24435 | The union of a collection ... |
| mopnin 24436 | The intersection of two op... |
| mopn0 24437 | The empty set is an open s... |
| rnblopn 24438 | A ball of a metric space i... |
| blopn 24439 | A ball of a metric space i... |
| neibl 24440 | The neighborhoods around a... |
| blnei 24441 | A ball around a point is a... |
| lpbl 24442 | Every ball around a limit ... |
| blsscls2 24443 | A smaller closed ball is c... |
| blcld 24444 | A "closed ball" in a metri... |
| blcls 24445 | The closure of an open bal... |
| blsscls 24446 | If two concentric balls ha... |
| metss 24447 | Two ways of saying that me... |
| metequiv 24448 | Two ways of saying that tw... |
| metequiv2 24449 | If there is a sequence of ... |
| metss2lem 24450 | Lemma for ~ metss2 . (Con... |
| metss2 24451 | If the metric ` D ` is "st... |
| comet 24452 | The composition of an exte... |
| stdbdmetval 24453 | Value of the standard boun... |
| stdbdxmet 24454 | The standard bounded metri... |
| stdbdmet 24455 | The standard bounded metri... |
| stdbdbl 24456 | The standard bounded metri... |
| stdbdmopn 24457 | The standard bounded metri... |
| mopnex 24458 | The topology generated by ... |
| methaus 24459 | The topology generated by ... |
| met1stc 24460 | The topology generated by ... |
| met2ndci 24461 | A separable metric space (... |
| met2ndc 24462 | A metric space is second-c... |
| metrest 24463 | Two alternate formulations... |
| ressxms 24464 | The restriction of a metri... |
| ressms 24465 | The restriction of a metri... |
| prdsmslem1 24466 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24467 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24468 | Lemma for ~ prdsxms . The... |
| prdsxms 24469 | The indexed product struct... |
| prdsms 24470 | The indexed product struct... |
| pwsxms 24471 | A power of an extended met... |
| pwsms 24472 | A power of a metric space ... |
| xpsxms 24473 | A binary product of metric... |
| xpsms 24474 | A binary product of metric... |
| tmsxps 24475 | Express the product of two... |
| tmsxpsmopn 24476 | Express the product of two... |
| tmsxpsval 24477 | Value of the product of tw... |
| tmsxpsval2 24478 | Value of the product of tw... |
| metcnp3 24479 | Two ways to express that `... |
| metcnp 24480 | Two ways to say a mapping ... |
| metcnp2 24481 | Two ways to say a mapping ... |
| metcn 24482 | Two ways to say a mapping ... |
| metcnpi 24483 | Epsilon-delta property of ... |
| metcnpi2 24484 | Epsilon-delta property of ... |
| metcnpi3 24485 | Epsilon-delta property of ... |
| txmetcnp 24486 | Continuity of a binary ope... |
| txmetcn 24487 | Continuity of a binary ope... |
| metuval 24488 | Value of the uniform struc... |
| metustel 24489 | Define a filter base ` F `... |
| metustss 24490 | Range of the elements of t... |
| metustrel 24491 | Elements of the filter bas... |
| metustto 24492 | Any two elements of the fi... |
| metustid 24493 | The identity diagonal is i... |
| metustsym 24494 | Elements of the filter bas... |
| metustexhalf 24495 | For any element ` A ` of t... |
| metustfbas 24496 | The filter base generated ... |
| metust 24497 | The uniform structure gene... |
| cfilucfil 24498 | Given a metric ` D ` and a... |
| metuust 24499 | The uniform structure gene... |
| cfilucfil2 24500 | Given a metric ` D ` and a... |
| blval2 24501 | The ball around a point ` ... |
| elbl4 24502 | Membership in a ball, alte... |
| metuel 24503 | Elementhood in the uniform... |
| metuel2 24504 | Elementhood in the uniform... |
| metustbl 24505 | The "section" image of an ... |
| psmetutop 24506 | The topology induced by a ... |
| xmetutop 24507 | The topology induced by a ... |
| xmsusp 24508 | If the uniform set of a me... |
| restmetu 24509 | The uniform structure gene... |
| metucn 24510 | Uniform continuity in metr... |
| dscmet 24511 | The discrete metric on any... |
| dscopn 24512 | The discrete metric genera... |
| nrmmetd 24513 | Show that a group norm gen... |
| abvmet 24514 | An absolute value ` F ` ge... |
| nmfval 24527 | The value of the norm func... |
| nmval 24528 | The value of the norm as t... |
| nmfval0 24529 | The value of the norm func... |
| nmfval2 24530 | The value of the norm func... |
| nmval2 24531 | The value of the norm on a... |
| nmf2 24532 | The norm on a metric group... |
| nmpropd 24533 | Weak property deduction fo... |
| nmpropd2 24534 | Strong property deduction ... |
| isngp 24535 | The property of being a no... |
| isngp2 24536 | The property of being a no... |
| isngp3 24537 | The property of being a no... |
| ngpgrp 24538 | A normed group is a group.... |
| ngpms 24539 | A normed group is a metric... |
| ngpxms 24540 | A normed group is an exten... |
| ngptps 24541 | A normed group is a topolo... |
| ngpmet 24542 | The (induced) metric of a ... |
| ngpds 24543 | Value of the distance func... |
| ngpdsr 24544 | Value of the distance func... |
| ngpds2 24545 | Write the distance between... |
| ngpds2r 24546 | Write the distance between... |
| ngpds3 24547 | Write the distance between... |
| ngpds3r 24548 | Write the distance between... |
| ngprcan 24549 | Cancel right addition insi... |
| ngplcan 24550 | Cancel left addition insid... |
| isngp4 24551 | Express the property of be... |
| ngpinvds 24552 | Two elements are the same ... |
| ngpsubcan 24553 | Cancel right subtraction i... |
| nmf 24554 | The norm on a normed group... |
| nmcl 24555 | The norm of a normed group... |
| nmge0 24556 | The norm of a normed group... |
| nmeq0 24557 | The identity is the only e... |
| nmne0 24558 | The norm of a nonzero elem... |
| nmrpcl 24559 | The norm of a nonzero elem... |
| nminv 24560 | The norm of a negated elem... |
| nmmtri 24561 | The triangle inequality fo... |
| nmsub 24562 | The norm of the difference... |
| nmrtri 24563 | Reverse triangle inequalit... |
| nm2dif 24564 | Inequality for the differe... |
| nmtri 24565 | The triangle inequality fo... |
| nmtri2 24566 | Triangle inequality for th... |
| ngpi 24567 | The properties of a normed... |
| nm0 24568 | Norm of the identity eleme... |
| nmgt0 24569 | The norm of a nonzero elem... |
| sgrim 24570 | The induced metric on a su... |
| sgrimval 24571 | The induced metric on a su... |
| subgnm 24572 | The norm in a subgroup. (... |
| subgnm2 24573 | A substructure assigns the... |
| subgngp 24574 | A normed group restricted ... |
| ngptgp 24575 | A normed abelian group is ... |
| ngppropd 24576 | Property deduction for a n... |
| reldmtng 24577 | The function ` toNrmGrp ` ... |
| tngval 24578 | Value of the function whic... |
| tnglem 24579 | Lemma for ~ tngbas and sim... |
| tnglemOLD 24580 | Obsolete version of ~ tngl... |
| tngbas 24581 | The base set of a structur... |
| tngbasOLD 24582 | Obsolete proof of ~ tngbas... |
| tngplusg 24583 | The group addition of a st... |
| tngplusgOLD 24584 | Obsolete proof of ~ tngplu... |
| tng0 24585 | The group identity of a st... |
| tngmulr 24586 | The ring multiplication of... |
| tngmulrOLD 24587 | Obsolete proof of ~ tngmul... |
| tngsca 24588 | The scalar ring of a struc... |
| tngscaOLD 24589 | Obsolete proof of ~ tngsca... |
| tngvsca 24590 | The scalar multiplication ... |
| tngvscaOLD 24591 | Obsolete proof of ~ tngvsc... |
| tngip 24592 | The inner product operatio... |
| tngipOLD 24593 | Obsolete proof of ~ tngip ... |
| tngds 24594 | The metric function of a s... |
| tngdsOLD 24595 | Obsolete proof of ~ tngds ... |
| tngtset 24596 | The topology generated by ... |
| tngtopn 24597 | The topology generated by ... |
| tngnm 24598 | The topology generated by ... |
| tngngp2 24599 | A norm turns a group into ... |
| tngngpd 24600 | Derive the axioms for a no... |
| tngngp 24601 | Derive the axioms for a no... |
| tnggrpr 24602 | If a structure equipped wi... |
| tngngp3 24603 | Alternate definition of a ... |
| nrmtngdist 24604 | The augmentation of a norm... |
| nrmtngnrm 24605 | The augmentation of a norm... |
| tngngpim 24606 | The induced metric of a no... |
| isnrg 24607 | A normed ring is a ring wi... |
| nrgabv 24608 | The norm of a normed ring ... |
| nrgngp 24609 | A normed ring is a normed ... |
| nrgring 24610 | A normed ring is a ring. ... |
| nmmul 24611 | The norm of a product in a... |
| nrgdsdi 24612 | Distribute a distance calc... |
| nrgdsdir 24613 | Distribute a distance calc... |
| nm1 24614 | The norm of one in a nonze... |
| unitnmn0 24615 | The norm of a unit is nonz... |
| nminvr 24616 | The norm of an inverse in ... |
| nmdvr 24617 | The norm of a division in ... |
| nrgdomn 24618 | A nonzero normed ring is a... |
| nrgtgp 24619 | A normed ring is a topolog... |
| subrgnrg 24620 | A normed ring restricted t... |
| tngnrg 24621 | Given any absolute value o... |
| isnlm 24622 | A normed (left) module is ... |
| nmvs 24623 | Defining property of a nor... |
| nlmngp 24624 | A normed module is a norme... |
| nlmlmod 24625 | A normed module is a left ... |
| nlmnrg 24626 | The scalar component of a ... |
| nlmngp2 24627 | The scalar component of a ... |
| nlmdsdi 24628 | Distribute a distance calc... |
| nlmdsdir 24629 | Distribute a distance calc... |
| nlmmul0or 24630 | If a scalar product is zer... |
| sranlm 24631 | The subring algebra over a... |
| nlmvscnlem2 24632 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24633 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24634 | The scalar multiplication ... |
| rlmnlm 24635 | The ring module over a nor... |
| rlmnm 24636 | The norm function in the r... |
| nrgtrg 24637 | A normed ring is a topolog... |
| nrginvrcnlem 24638 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24639 | The ring inverse function ... |
| nrgtdrg 24640 | A normed division ring is ... |
| nlmtlm 24641 | A normed module is a topol... |
| isnvc 24642 | A normed vector space is j... |
| nvcnlm 24643 | A normed vector space is a... |
| nvclvec 24644 | A normed vector space is a... |
| nvclmod 24645 | A normed vector space is a... |
| isnvc2 24646 | A normed vector space is j... |
| nvctvc 24647 | A normed vector space is a... |
| lssnlm 24648 | A subspace of a normed mod... |
| lssnvc 24649 | A subspace of a normed vec... |
| rlmnvc 24650 | The ring module over a nor... |
| ngpocelbl 24651 | Membership of an off-cente... |
| nmoffn 24658 | The function producing ope... |
| reldmnghm 24659 | Lemma for normed group hom... |
| reldmnmhm 24660 | Lemma for module homomorph... |
| nmofval 24661 | Value of the operator norm... |
| nmoval 24662 | Value of the operator norm... |
| nmogelb 24663 | Property of the operator n... |
| nmolb 24664 | Any upper bound on the val... |
| nmolb2d 24665 | Any upper bound on the val... |
| nmof 24666 | The operator norm is a fun... |
| nmocl 24667 | The operator norm of an op... |
| nmoge0 24668 | The operator norm of an op... |
| nghmfval 24669 | A normed group homomorphis... |
| isnghm 24670 | A normed group homomorphis... |
| isnghm2 24671 | A normed group homomorphis... |
| isnghm3 24672 | A normed group homomorphis... |
| bddnghm 24673 | A bounded group homomorphi... |
| nghmcl 24674 | A normed group homomorphis... |
| nmoi 24675 | The operator norm achieves... |
| nmoix 24676 | The operator norm is a bou... |
| nmoi2 24677 | The operator norm is a bou... |
| nmoleub 24678 | The operator norm, defined... |
| nghmrcl1 24679 | Reverse closure for a norm... |
| nghmrcl2 24680 | Reverse closure for a norm... |
| nghmghm 24681 | A normed group homomorphis... |
| nmo0 24682 | The operator norm of the z... |
| nmoeq0 24683 | The operator norm is zero ... |
| nmoco 24684 | An upper bound on the oper... |
| nghmco 24685 | The composition of normed ... |
| nmotri 24686 | Triangle inequality for th... |
| nghmplusg 24687 | The sum of two bounded lin... |
| 0nghm 24688 | The zero operator is a nor... |
| nmoid 24689 | The operator norm of the i... |
| idnghm 24690 | The identity operator is a... |
| nmods 24691 | Upper bound for the distan... |
| nghmcn 24692 | A normed group homomorphis... |
| isnmhm 24693 | A normed module homomorphi... |
| nmhmrcl1 24694 | Reverse closure for a norm... |
| nmhmrcl2 24695 | Reverse closure for a norm... |
| nmhmlmhm 24696 | A normed module homomorphi... |
| nmhmnghm 24697 | A normed module homomorphi... |
| nmhmghm 24698 | A normed module homomorphi... |
| isnmhm2 24699 | A normed module homomorphi... |
| nmhmcl 24700 | A normed module homomorphi... |
| idnmhm 24701 | The identity operator is a... |
| 0nmhm 24702 | The zero operator is a bou... |
| nmhmco 24703 | The composition of bounded... |
| nmhmplusg 24704 | The sum of two bounded lin... |
| qtopbaslem 24705 | The set of open intervals ... |
| qtopbas 24706 | The set of open intervals ... |
| retopbas 24707 | A basis for the standard t... |
| retop 24708 | The standard topology on t... |
| uniretop 24709 | The underlying set of the ... |
| retopon 24710 | The standard topology on t... |
| retps 24711 | The standard topological s... |
| iooretop 24712 | Open intervals are open se... |
| icccld 24713 | Closed intervals are close... |
| icopnfcld 24714 | Right-unbounded closed int... |
| iocmnfcld 24715 | Left-unbounded closed inte... |
| qdensere 24716 | ` QQ ` is dense in the sta... |
| cnmetdval 24717 | Value of the distance func... |
| cnmet 24718 | The absolute value metric ... |
| cnxmet 24719 | The absolute value metric ... |
| cnbl0 24720 | Two ways to write the open... |
| cnblcld 24721 | Two ways to write the clos... |
| cnfldms 24722 | The complex number field i... |
| cnfldxms 24723 | The complex number field i... |
| cnfldtps 24724 | The complex number field i... |
| cnfldnm 24725 | The norm of the field of c... |
| cnngp 24726 | The complex numbers form a... |
| cnnrg 24727 | The complex numbers form a... |
| cnfldtopn 24728 | The topology of the comple... |
| cnfldtopon 24729 | The topology of the comple... |
| cnfldtop 24730 | The topology of the comple... |
| cnfldhaus 24731 | The topology of the comple... |
| unicntop 24732 | The underlying set of the ... |
| cnopn 24733 | The set of complex numbers... |
| zringnrg 24734 | The ring of integers is a ... |
| remetdval 24735 | Value of the distance func... |
| remet 24736 | The absolute value metric ... |
| rexmet 24737 | The absolute value metric ... |
| bl2ioo 24738 | A ball in terms of an open... |
| ioo2bl 24739 | An open interval of reals ... |
| ioo2blex 24740 | An open interval of reals ... |
| blssioo 24741 | The balls of the standard ... |
| tgioo 24742 | The topology generated by ... |
| qdensere2 24743 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24744 | An open ball in the comple... |
| rehaus 24745 | The standard topology on t... |
| tgqioo 24746 | The topology generated by ... |
| re2ndc 24747 | The standard topology on t... |
| resubmet 24748 | The subspace topology indu... |
| tgioo2 24749 | The standard topology on t... |
| rerest 24750 | The subspace topology indu... |
| tgioo3 24751 | The standard topology on t... |
| xrtgioo 24752 | The topology on the extend... |
| xrrest 24753 | The subspace topology indu... |
| xrrest2 24754 | The subspace topology indu... |
| xrsxmet 24755 | The metric on the extended... |
| xrsdsre 24756 | The metric on the extended... |
| xrsblre 24757 | Any ball of the metric of ... |
| xrsmopn 24758 | The metric on the extended... |
| zcld 24759 | The integers are a closed ... |
| recld2 24760 | The real numbers are a clo... |
| zcld2 24761 | The integers are a closed ... |
| zdis 24762 | The integers are a discret... |
| sszcld 24763 | Every subset of the intege... |
| reperflem 24764 | A subset of the real numbe... |
| reperf 24765 | The real numbers are a per... |
| cnperf 24766 | The complex numbers are a ... |
| iccntr 24767 | The interior of a closed i... |
| icccmplem1 24768 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24769 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24770 | Lemma for ~ icccmp . (Con... |
| icccmp 24771 | A closed interval in ` RR ... |
| reconnlem1 24772 | Lemma for ~ reconn . Conn... |
| reconnlem2 24773 | Lemma for ~ reconn . (Con... |
| reconn 24774 | A subset of the reals is c... |
| retopconn 24775 | Corollary of ~ reconn . T... |
| iccconn 24776 | A closed interval is conne... |
| opnreen 24777 | Every nonempty open set is... |
| rectbntr0 24778 | A countable subset of the ... |
| xrge0gsumle 24779 | A finite sum in the nonneg... |
| xrge0tsms 24780 | Any finite or infinite sum... |
| xrge0tsms2 24781 | Any finite or infinite sum... |
| metdcnlem 24782 | The metric function of a m... |
| xmetdcn2 24783 | The metric function of an ... |
| xmetdcn 24784 | The metric function of an ... |
| metdcn2 24785 | The metric function of a m... |
| metdcn 24786 | The metric function of a m... |
| msdcn 24787 | The metric function of a m... |
| cnmpt1ds 24788 | Continuity of the metric f... |
| cnmpt2ds 24789 | Continuity of the metric f... |
| nmcn 24790 | The norm of a normed group... |
| ngnmcncn 24791 | The norm of a normed group... |
| abscn 24792 | The absolute value functio... |
| metdsval 24793 | Value of the "distance to ... |
| metdsf 24794 | The distance from a point ... |
| metdsge 24795 | The distance from the poin... |
| metds0 24796 | If a point is in a set, it... |
| metdstri 24797 | A generalization of the tr... |
| metdsle 24798 | The distance from a point ... |
| metdsre 24799 | The distance from a point ... |
| metdseq0 24800 | The distance from a point ... |
| metdscnlem 24801 | Lemma for ~ metdscn . (Co... |
| metdscn 24802 | The function ` F ` which g... |
| metdscn2 24803 | The function ` F ` which g... |
| metnrmlem1a 24804 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24805 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24806 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24807 | Lemma for ~ metnrm . (Con... |
| metnrm 24808 | A metric space is normal. ... |
| metreg 24809 | A metric space is regular.... |
| addcnlem 24810 | Lemma for ~ addcn , ~ subc... |
| addcn 24811 | Complex number addition is... |
| subcn 24812 | Complex number subtraction... |
| mulcn 24813 | Complex number multiplicat... |
| divcnOLD 24814 | Obsolete version of ~ divc... |
| mpomulcn 24815 | Complex number multiplicat... |
| divcn 24816 | Complex number division is... |
| cnfldtgp 24817 | The complex numbers form a... |
| fsumcn 24818 | A finite sum of functions ... |
| fsum2cn 24819 | Version of ~ fsumcn for tw... |
| expcn 24820 | The power function on comp... |
| divccn 24821 | Division by a nonzero cons... |
| expcnOLD 24822 | Obsolete version of ~ expc... |
| divccnOLD 24823 | Obsolete version of ~ divc... |
| sqcn 24824 | The square function on com... |
| iitopon 24829 | The unit interval is a top... |
| iitop 24830 | The unit interval is a top... |
| iiuni 24831 | The base set of the unit i... |
| dfii2 24832 | Alternate definition of th... |
| dfii3 24833 | Alternate definition of th... |
| dfii4 24834 | Alternate definition of th... |
| dfii5 24835 | The unit interval expresse... |
| iicmp 24836 | The unit interval is compa... |
| iiconn 24837 | The unit interval is conne... |
| cncfval 24838 | The value of the continuou... |
| elcncf 24839 | Membership in the set of c... |
| elcncf2 24840 | Version of ~ elcncf with a... |
| cncfrss 24841 | Reverse closure of the con... |
| cncfrss2 24842 | Reverse closure of the con... |
| cncff 24843 | A continuous complex funct... |
| cncfi 24844 | Defining property of a con... |
| elcncf1di 24845 | Membership in the set of c... |
| elcncf1ii 24846 | Membership in the set of c... |
| rescncf 24847 | A continuous complex funct... |
| cncfcdm 24848 | Change the codomain of a c... |
| cncfss 24849 | The set of continuous func... |
| climcncf 24850 | Image of a limit under a c... |
| abscncf 24851 | Absolute value is continuo... |
| recncf 24852 | Real part is continuous. ... |
| imcncf 24853 | Imaginary part is continuo... |
| cjcncf 24854 | Complex conjugate is conti... |
| mulc1cncf 24855 | Multiplication by a consta... |
| divccncf 24856 | Division by a constant is ... |
| cncfco 24857 | The composition of two con... |
| cncfcompt2 24858 | Composition of continuous ... |
| cncfmet 24859 | Relate complex function co... |
| cncfcn 24860 | Relate complex function co... |
| cncfcn1 24861 | Relate complex function co... |
| cncfmptc 24862 | A constant function is a c... |
| cncfmptid 24863 | The identity function is a... |
| cncfmpt1f 24864 | Composition of continuous ... |
| cncfmpt2f 24865 | Composition of continuous ... |
| cncfmpt2ss 24866 | Composition of continuous ... |
| addccncf 24867 | Adding a constant is a con... |
| idcncf 24868 | The identity function is a... |
| sub1cncf 24869 | Subtracting a constant is ... |
| sub2cncf 24870 | Subtraction from a constan... |
| cdivcncf 24871 | Division with a constant n... |
| negcncf 24872 | The negative function is c... |
| negcncfOLD 24873 | Obsolete version of ~ negc... |
| negfcncf 24874 | The negative of a continuo... |
| abscncfALT 24875 | Absolute value is continuo... |
| cncfcnvcn 24876 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24877 | The power function on comp... |
| cnmptre 24878 | Lemma for ~ iirevcn and re... |
| cnmpopc 24879 | Piecewise definition of a ... |
| iirev 24880 | Reverse the unit interval.... |
| iirevcn 24881 | The reversion function is ... |
| iihalf1 24882 | Map the first half of ` II... |
| iihalf1cn 24883 | The first half function is... |
| iihalf1cnOLD 24884 | Obsolete version of ~ iiha... |
| iihalf2 24885 | Map the second half of ` I... |
| iihalf2cn 24886 | The second half function i... |
| iihalf2cnOLD 24887 | Obsolete version of ~ iiha... |
| elii1 24888 | Divide the unit interval i... |
| elii2 24889 | Divide the unit interval i... |
| iimulcl 24890 | The unit interval is close... |
| iimulcn 24891 | Multiplication is a contin... |
| iimulcnOLD 24892 | Obsolete version of ~ iimu... |
| icoopnst 24893 | A half-open interval start... |
| iocopnst 24894 | A half-open interval endin... |
| icchmeo 24895 | The natural bijection from... |
| icchmeoOLD 24896 | Obsolete version of ~ icch... |
| icopnfcnv 24897 | Define a bijection from ` ... |
| icopnfhmeo 24898 | The defined bijection from... |
| iccpnfcnv 24899 | Define a bijection from ` ... |
| iccpnfhmeo 24900 | The defined bijection from... |
| xrhmeo 24901 | The bijection from ` [ -u ... |
| xrhmph 24902 | The extended reals are hom... |
| xrcmp 24903 | The topology of the extend... |
| xrconn 24904 | The topology of the extend... |
| icccvx 24905 | A linear combination of tw... |
| oprpiece1res1 24906 | Restriction to the first p... |
| oprpiece1res2 24907 | Restriction to the second ... |
| cnrehmeo 24908 | The canonical bijection fr... |
| cnrehmeoOLD 24909 | Obsolete version of ~ cnre... |
| cnheiborlem 24910 | Lemma for ~ cnheibor . (C... |
| cnheibor 24911 | Heine-Borel theorem for co... |
| cnllycmp 24912 | The topology on the comple... |
| rellycmp 24913 | The topology on the reals ... |
| bndth 24914 | The Boundedness Theorem. ... |
| evth 24915 | The Extreme Value Theorem.... |
| evth2 24916 | The Extreme Value Theorem,... |
| lebnumlem1 24917 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24918 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24919 | Lemma for ~ lebnum . By t... |
| lebnum 24920 | The Lebesgue number lemma,... |
| xlebnum 24921 | Generalize ~ lebnum to ext... |
| lebnumii 24922 | Specialize the Lebesgue nu... |
| ishtpy 24928 | Membership in the class of... |
| htpycn 24929 | A homotopy is a continuous... |
| htpyi 24930 | A homotopy evaluated at it... |
| ishtpyd 24931 | Deduction for membership i... |
| htpycom 24932 | Given a homotopy from ` F ... |
| htpyid 24933 | A homotopy from a function... |
| htpyco1 24934 | Compose a homotopy with a ... |
| htpyco2 24935 | Compose a homotopy with a ... |
| htpycc 24936 | Concatenate two homotopies... |
| isphtpy 24937 | Membership in the class of... |
| phtpyhtpy 24938 | A path homotopy is a homot... |
| phtpycn 24939 | A path homotopy is a conti... |
| phtpyi 24940 | Membership in the class of... |
| phtpy01 24941 | Two path-homotopic paths h... |
| isphtpyd 24942 | Deduction for membership i... |
| isphtpy2d 24943 | Deduction for membership i... |
| phtpycom 24944 | Given a homotopy from ` F ... |
| phtpyid 24945 | A homotopy from a path to ... |
| phtpyco2 24946 | Compose a path homotopy wi... |
| phtpycc 24947 | Concatenate two path homot... |
| phtpcrel 24949 | The path homotopy relation... |
| isphtpc 24950 | The relation "is path homo... |
| phtpcer 24951 | Path homotopy is an equiva... |
| phtpc01 24952 | Path homotopic paths have ... |
| reparphti 24953 | Lemma for ~ reparpht . (C... |
| reparphtiOLD 24954 | Obsolete version of ~ repa... |
| reparpht 24955 | Reparametrization lemma. ... |
| phtpcco2 24956 | Compose a path homotopy wi... |
| pcofval 24967 | The value of the path conc... |
| pcoval 24968 | The concatenation of two p... |
| pcovalg 24969 | Evaluate the concatenation... |
| pcoval1 24970 | Evaluate the concatenation... |
| pco0 24971 | The starting point of a pa... |
| pco1 24972 | The ending point of a path... |
| pcoval2 24973 | Evaluate the concatenation... |
| pcocn 24974 | The concatenation of two p... |
| copco 24975 | The composition of a conca... |
| pcohtpylem 24976 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24977 | Homotopy invariance of pat... |
| pcoptcl 24978 | A constant function is a p... |
| pcopt 24979 | Concatenation with a point... |
| pcopt2 24980 | Concatenation with a point... |
| pcoass 24981 | Order of concatenation doe... |
| pcorevcl 24982 | Closure for a reversed pat... |
| pcorevlem 24983 | Lemma for ~ pcorev . Prov... |
| pcorev 24984 | Concatenation with the rev... |
| pcorev2 24985 | Concatenation with the rev... |
| pcophtb 24986 | The path homotopy equivale... |
| om1val 24987 | The definition of the loop... |
| om1bas 24988 | The base set of the loop s... |
| om1elbas 24989 | Elementhood in the base se... |
| om1addcl 24990 | Closure of the group opera... |
| om1plusg 24991 | The group operation (which... |
| om1tset 24992 | The topology of the loop s... |
| om1opn 24993 | The topology of the loop s... |
| pi1val 24994 | The definition of the fund... |
| pi1bas 24995 | The base set of the fundam... |
| pi1blem 24996 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24997 | Another way to write the l... |
| pi1bas2 24998 | The base set of the fundam... |
| pi1eluni 24999 | Elementhood in the base se... |
| pi1bas3 25000 | The base set of the fundam... |
| pi1cpbl 25001 | The group operation, loop ... |
| elpi1 25002 | The elements of the fundam... |
| elpi1i 25003 | The elements of the fundam... |
| pi1addf 25004 | The group operation of ` p... |
| pi1addval 25005 | The concatenation of two p... |
| pi1grplem 25006 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25007 | The fundamental group is a... |
| pi1id 25008 | The identity element of th... |
| pi1inv 25009 | An inverse in the fundamen... |
| pi1xfrf 25010 | Functionality of the loop ... |
| pi1xfrval 25011 | The value of the loop tran... |
| pi1xfr 25012 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25013 | Given a path ` F ` between... |
| pi1xfrcnv 25014 | Given a path ` F ` between... |
| pi1xfrgim 25015 | The mapping ` G ` between ... |
| pi1cof 25016 | Functionality of the loop ... |
| pi1coval 25017 | The value of the loop tran... |
| pi1coghm 25018 | The mapping ` G ` between ... |
| isclm 25021 | A subcomplex module is a l... |
| clmsca 25022 | The ring of scalars ` F ` ... |
| clmsubrg 25023 | The base set of the ring o... |
| clmlmod 25024 | A subcomplex module is a l... |
| clmgrp 25025 | A subcomplex module is an ... |
| clmabl 25026 | A subcomplex module is an ... |
| clmring 25027 | The scalar ring of a subco... |
| clmfgrp 25028 | The scalar ring of a subco... |
| clm0 25029 | The zero of the scalar rin... |
| clm1 25030 | The identity of the scalar... |
| clmadd 25031 | The addition of the scalar... |
| clmmul 25032 | The multiplication of the ... |
| clmcj 25033 | The conjugation of the sca... |
| isclmi 25034 | Reverse direction of ~ isc... |
| clmzss 25035 | The scalar ring of a subco... |
| clmsscn 25036 | The scalar ring of a subco... |
| clmsub 25037 | Subtraction in the scalar ... |
| clmneg 25038 | Negation in the scalar rin... |
| clmneg1 25039 | Minus one is in the scalar... |
| clmabs 25040 | Norm in the scalar ring of... |
| clmacl 25041 | Closure of ring addition f... |
| clmmcl 25042 | Closure of ring multiplica... |
| clmsubcl 25043 | Closure of ring subtractio... |
| lmhmclm 25044 | The domain of a linear ope... |
| clmvscl 25045 | Closure of scalar product ... |
| clmvsass 25046 | Associative law for scalar... |
| clmvscom 25047 | Commutative law for the sc... |
| clmvsdir 25048 | Distributive law for scala... |
| clmvsdi 25049 | Distributive law for scala... |
| clmvs1 25050 | Scalar product with ring u... |
| clmvs2 25051 | A vector plus itself is tw... |
| clm0vs 25052 | Zero times a vector is the... |
| clmopfne 25053 | The (functionalized) opera... |
| isclmp 25054 | The predicate "is a subcom... |
| isclmi0 25055 | Properties that determine ... |
| clmvneg1 25056 | Minus 1 times a vector is ... |
| clmvsneg 25057 | Multiplication of a vector... |
| clmmulg 25058 | The group multiple functio... |
| clmsubdir 25059 | Scalar multiplication dist... |
| clmpm1dir 25060 | Subtractive distributive l... |
| clmnegneg 25061 | Double negative of a vecto... |
| clmnegsubdi2 25062 | Distribution of negative o... |
| clmsub4 25063 | Rearrangement of 4 terms i... |
| clmvsrinv 25064 | A vector minus itself. (C... |
| clmvslinv 25065 | Minus a vector plus itself... |
| clmvsubval 25066 | Value of vector subtractio... |
| clmvsubval2 25067 | Value of vector subtractio... |
| clmvz 25068 | Two ways to express the ne... |
| zlmclm 25069 | The ` ZZ ` -module operati... |
| clmzlmvsca 25070 | The scalar product of a su... |
| nmoleub2lem 25071 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25072 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25073 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25074 | The operator norm is the s... |
| nmoleub2b 25075 | The operator norm is the s... |
| nmoleub3 25076 | The operator norm is the s... |
| nmhmcn 25077 | A linear operator over a n... |
| cmodscexp 25078 | The powers of ` _i ` belon... |
| cmodscmulexp 25079 | The scalar product of a ve... |
| cvslvec 25082 | A subcomplex vector space ... |
| cvsclm 25083 | A subcomplex vector space ... |
| iscvs 25084 | A subcomplex vector space ... |
| iscvsp 25085 | The predicate "is a subcom... |
| iscvsi 25086 | Properties that determine ... |
| cvsi 25087 | The properties of a subcom... |
| cvsunit 25088 | Unit group of the scalar r... |
| cvsdiv 25089 | Division of the scalar rin... |
| cvsdivcl 25090 | The scalar field of a subc... |
| cvsmuleqdivd 25091 | An equality involving rati... |
| cvsdiveqd 25092 | An equality involving rati... |
| cnlmodlem1 25093 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25094 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25095 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25096 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25097 | The set of complex numbers... |
| cnstrcvs 25098 | The set of complex numbers... |
| cnrbas 25099 | The set of complex numbers... |
| cnrlmod 25100 | The complex left module of... |
| cnrlvec 25101 | The complex left module of... |
| cncvs 25102 | The complex left module of... |
| recvs 25103 | The field of the real numb... |
| recvsOLD 25104 | Obsolete version of ~ recv... |
| qcvs 25105 | The field of rational numb... |
| zclmncvs 25106 | The ring of integers as le... |
| isncvsngp 25107 | A normed subcomplex vector... |
| isncvsngpd 25108 | Properties that determine ... |
| ncvsi 25109 | The properties of a normed... |
| ncvsprp 25110 | Proportionality property o... |
| ncvsge0 25111 | The norm of a scalar produ... |
| ncvsm1 25112 | The norm of the opposite o... |
| ncvsdif 25113 | The norm of the difference... |
| ncvspi 25114 | The norm of a vector plus ... |
| ncvs1 25115 | From any nonzero vector of... |
| cnrnvc 25116 | The module of complex numb... |
| cnncvs 25117 | The module of complex numb... |
| cnnm 25118 | The norm of the normed sub... |
| ncvspds 25119 | Value of the distance func... |
| cnindmet 25120 | The metric induced on the ... |
| cnncvsaddassdemo 25121 | Derive the associative law... |
| cnncvsmulassdemo 25122 | Derive the associative law... |
| cnncvsabsnegdemo 25123 | Derive the absolute value ... |
| iscph 25128 | A subcomplex pre-Hilbert s... |
| cphphl 25129 | A subcomplex pre-Hilbert s... |
| cphnlm 25130 | A subcomplex pre-Hilbert s... |
| cphngp 25131 | A subcomplex pre-Hilbert s... |
| cphlmod 25132 | A subcomplex pre-Hilbert s... |
| cphlvec 25133 | A subcomplex pre-Hilbert s... |
| cphnvc 25134 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25135 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25136 | Lemma for ~ cphreccl . (C... |
| cphsca 25137 | A subcomplex pre-Hilbert s... |
| cphsubrg 25138 | The scalar field of a subc... |
| cphreccl 25139 | The scalar field of a subc... |
| cphdivcl 25140 | The scalar field of a subc... |
| cphcjcl 25141 | The scalar field of a subc... |
| cphsqrtcl 25142 | The scalar field of a subc... |
| cphabscl 25143 | The scalar field of a subc... |
| cphsqrtcl2 25144 | The scalar field of a subc... |
| cphsqrtcl3 25145 | If the scalar field of a s... |
| cphqss 25146 | The scalar field of a subc... |
| cphclm 25147 | A subcomplex pre-Hilbert s... |
| cphnmvs 25148 | Norm of a scalar product. ... |
| cphipcl 25149 | An inner product is a memb... |
| cphnmfval 25150 | The value of the norm in a... |
| cphnm 25151 | The square of the norm is ... |
| nmsq 25152 | The square of the norm is ... |
| cphnmf 25153 | The norm of a vector is a ... |
| cphnmcl 25154 | The norm of a vector is a ... |
| reipcl 25155 | An inner product of an ele... |
| ipge0 25156 | The inner product in a sub... |
| cphipcj 25157 | Conjugate of an inner prod... |
| cphipipcj 25158 | An inner product times its... |
| cphorthcom 25159 | Orthogonality (meaning inn... |
| cphip0l 25160 | Inner product with a zero ... |
| cphip0r 25161 | Inner product with a zero ... |
| cphipeq0 25162 | The inner product of a vec... |
| cphdir 25163 | Distributive law for inner... |
| cphdi 25164 | Distributive law for inner... |
| cph2di 25165 | Distributive law for inner... |
| cphsubdir 25166 | Distributive law for inner... |
| cphsubdi 25167 | Distributive law for inner... |
| cph2subdi 25168 | Distributive law for inner... |
| cphass 25169 | Associative law for inner ... |
| cphassr 25170 | "Associative" law for seco... |
| cph2ass 25171 | Move scalar multiplication... |
| cphassi 25172 | Associative law for the fi... |
| cphassir 25173 | "Associative" law for the ... |
| cphpyth 25174 | The pythagorean theorem fo... |
| tcphex 25175 | Lemma for ~ tcphbas and si... |
| tcphval 25176 | Define a function to augme... |
| tcphbas 25177 | The base set of a subcompl... |
| tchplusg 25178 | The addition operation of ... |
| tcphsub 25179 | The subtraction operation ... |
| tcphmulr 25180 | The ring operation of a su... |
| tcphsca 25181 | The scalar field of a subc... |
| tcphvsca 25182 | The scalar multiplication ... |
| tcphip 25183 | The inner product of a sub... |
| tcphtopn 25184 | The topology of a subcompl... |
| tcphphl 25185 | Augmentation of a subcompl... |
| tchnmfval 25186 | The norm of a subcomplex p... |
| tcphnmval 25187 | The norm of a subcomplex p... |
| cphtcphnm 25188 | The norm of a norm-augment... |
| tcphds 25189 | The distance of a pre-Hilb... |
| phclm 25190 | A pre-Hilbert space whose ... |
| tcphcphlem3 25191 | Lemma for ~ tcphcph : real... |
| ipcau2 25192 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25193 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25194 | Lemma for ~ tcphcph : homo... |
| tcphcph 25195 | The standard definition of... |
| ipcau 25196 | The Cauchy-Schwarz inequal... |
| nmparlem 25197 | Lemma for ~ nmpar . (Cont... |
| nmpar 25198 | A subcomplex pre-Hilbert s... |
| cphipval2 25199 | Value of the inner product... |
| 4cphipval2 25200 | Four times the inner produ... |
| cphipval 25201 | Value of the inner product... |
| ipcnlem2 25202 | The inner product operatio... |
| ipcnlem1 25203 | The inner product operatio... |
| ipcn 25204 | The inner product operatio... |
| cnmpt1ip 25205 | Continuity of inner produc... |
| cnmpt2ip 25206 | Continuity of inner produc... |
| csscld 25207 | A "closed subspace" in a s... |
| clsocv 25208 | The orthogonal complement ... |
| cphsscph 25209 | A subspace of a subcomplex... |
| lmmbr 25216 | Express the binary relatio... |
| lmmbr2 25217 | Express the binary relatio... |
| lmmbr3 25218 | Express the binary relatio... |
| lmmcvg 25219 | Convergence property of a ... |
| lmmbrf 25220 | Express the binary relatio... |
| lmnn 25221 | A condition that implies c... |
| cfilfval 25222 | The set of Cauchy filters ... |
| iscfil 25223 | The property of being a Ca... |
| iscfil2 25224 | The property of being a Ca... |
| cfilfil 25225 | A Cauchy filter is a filte... |
| cfili 25226 | Property of a Cauchy filte... |
| cfil3i 25227 | A Cauchy filter contains b... |
| cfilss 25228 | A filter finer than a Cauc... |
| fgcfil 25229 | The Cauchy filter conditio... |
| fmcfil 25230 | The Cauchy filter conditio... |
| iscfil3 25231 | A filter is Cauchy iff it ... |
| cfilfcls 25232 | Similar to ultrafilters ( ... |
| caufval 25233 | The set of Cauchy sequence... |
| iscau 25234 | Express the property " ` F... |
| iscau2 25235 | Express the property " ` F... |
| iscau3 25236 | Express the Cauchy sequenc... |
| iscau4 25237 | Express the property " ` F... |
| iscauf 25238 | Express the property " ` F... |
| caun0 25239 | A metric with a Cauchy seq... |
| caufpm 25240 | Inclusion of a Cauchy sequ... |
| caucfil 25241 | A Cauchy sequence predicat... |
| iscmet 25242 | The property " ` D ` is a ... |
| cmetcvg 25243 | The convergence of a Cauch... |
| cmetmet 25244 | A complete metric space is... |
| cmetmeti 25245 | A complete metric space is... |
| cmetcaulem 25246 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25247 | The convergence of a Cauch... |
| iscmet3lem3 25248 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25249 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25250 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25251 | The property " ` D ` is a ... |
| iscmet2 25252 | A metric ` D ` is complete... |
| cfilresi 25253 | A Cauchy filter on a metri... |
| cfilres 25254 | Cauchy filter on a metric ... |
| caussi 25255 | Cauchy sequence on a metri... |
| causs 25256 | Cauchy sequence on a metri... |
| equivcfil 25257 | If the metric ` D ` is "st... |
| equivcau 25258 | If the metric ` D ` is "st... |
| lmle 25259 | If the distance from each ... |
| nglmle 25260 | If the norm of each member... |
| lmclim 25261 | Relate a limit on the metr... |
| lmclimf 25262 | Relate a limit on the metr... |
| metelcls 25263 | A point belongs to the clo... |
| metcld 25264 | A subset of a metric space... |
| metcld2 25265 | A subset of a metric space... |
| caubl 25266 | Sufficient condition to en... |
| caublcls 25267 | The convergent point of a ... |
| metcnp4 25268 | Two ways to say a mapping ... |
| metcn4 25269 | Two ways to say a mapping ... |
| iscmet3i 25270 | Properties that determine ... |
| lmcau 25271 | Every convergent sequence ... |
| flimcfil 25272 | Every convergent filter in... |
| metsscmetcld 25273 | A complete subspace of a m... |
| cmetss 25274 | A subspace of a complete m... |
| equivcmet 25275 | If two metrics are strongl... |
| relcmpcmet 25276 | If ` D ` is a metric space... |
| cmpcmet 25277 | A compact metric space is ... |
| cfilucfil3 25278 | Given a metric ` D ` and a... |
| cfilucfil4 25279 | Given a metric ` D ` and a... |
| cncmet 25280 | The set of complex numbers... |
| recmet 25281 | The real numbers are a com... |
| bcthlem1 25282 | Lemma for ~ bcth . Substi... |
| bcthlem2 25283 | Lemma for ~ bcth . The ba... |
| bcthlem3 25284 | Lemma for ~ bcth . The li... |
| bcthlem4 25285 | Lemma for ~ bcth . Given ... |
| bcthlem5 25286 | Lemma for ~ bcth . The pr... |
| bcth 25287 | Baire's Category Theorem. ... |
| bcth2 25288 | Baire's Category Theorem, ... |
| bcth3 25289 | Baire's Category Theorem, ... |
| isbn 25296 | A Banach space is a normed... |
| bnsca 25297 | The scalar field of a Bana... |
| bnnvc 25298 | A Banach space is a normed... |
| bnnlm 25299 | A Banach space is a normed... |
| bnngp 25300 | A Banach space is a normed... |
| bnlmod 25301 | A Banach space is a left m... |
| bncms 25302 | A Banach space is a comple... |
| iscms 25303 | A complete metric space is... |
| cmscmet 25304 | The induced metric on a co... |
| bncmet 25305 | The induced metric on Bana... |
| cmsms 25306 | A complete metric space is... |
| cmspropd 25307 | Property deduction for a c... |
| cmssmscld 25308 | The restriction of a metri... |
| cmsss 25309 | The restriction of a compl... |
| lssbn 25310 | A subspace of a Banach spa... |
| cmetcusp1 25311 | If the uniform set of a co... |
| cmetcusp 25312 | The uniform space generate... |
| cncms 25313 | The field of complex numbe... |
| cnflduss 25314 | The uniform structure of t... |
| cnfldcusp 25315 | The field of complex numbe... |
| resscdrg 25316 | The real numbers are a sub... |
| cncdrg 25317 | The only complete subfield... |
| srabn 25318 | The subring algebra over a... |
| rlmbn 25319 | The ring module over a com... |
| ishl 25320 | The predicate "is a subcom... |
| hlbn 25321 | Every subcomplex Hilbert s... |
| hlcph 25322 | Every subcomplex Hilbert s... |
| hlphl 25323 | Every subcomplex Hilbert s... |
| hlcms 25324 | Every subcomplex Hilbert s... |
| hlprlem 25325 | Lemma for ~ hlpr . (Contr... |
| hlress 25326 | The scalar field of a subc... |
| hlpr 25327 | The scalar field of a subc... |
| ishl2 25328 | A Hilbert space is a compl... |
| cphssphl 25329 | A Banach subspace of a sub... |
| cmslssbn 25330 | A complete linear subspace... |
| cmscsscms 25331 | A closed subspace of a com... |
| bncssbn 25332 | A closed subspace of a Ban... |
| cssbn 25333 | A complete subspace of a n... |
| csschl 25334 | A complete subspace of a c... |
| cmslsschl 25335 | A complete linear subspace... |
| chlcsschl 25336 | A closed subspace of a sub... |
| retopn 25337 | The topology of the real n... |
| recms 25338 | The real numbers form a co... |
| reust 25339 | The Uniform structure of t... |
| recusp 25340 | The real numbers form a co... |
| rrxval 25345 | Value of the generalized E... |
| rrxbase 25346 | The base of the generalize... |
| rrxprds 25347 | Expand the definition of t... |
| rrxip 25348 | The inner product of the g... |
| rrxnm 25349 | The norm of the generalize... |
| rrxcph 25350 | Generalized Euclidean real... |
| rrxds 25351 | The distance over generali... |
| rrxvsca 25352 | The scalar product over ge... |
| rrxplusgvscavalb 25353 | The result of the addition... |
| rrxsca 25354 | The field of real numbers ... |
| rrx0 25355 | The zero ("origin") in a g... |
| rrx0el 25356 | The zero ("origin") in a g... |
| csbren 25357 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25358 | Triangle inequality in R^n... |
| rrxf 25359 | Euclidean vectors as funct... |
| rrxfsupp 25360 | Euclidean vectors are of f... |
| rrxsuppss 25361 | Support of Euclidean vecto... |
| rrxmvallem 25362 | Support of the function us... |
| rrxmval 25363 | The value of the Euclidean... |
| rrxmfval 25364 | The value of the Euclidean... |
| rrxmetlem 25365 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25366 | Euclidean space is a metri... |
| rrxdstprj1 25367 | The distance between two p... |
| rrxbasefi 25368 | The base of the generalize... |
| rrxdsfi 25369 | The distance over generali... |
| rrxmetfi 25370 | Euclidean space is a metri... |
| rrxdsfival 25371 | The value of the Euclidean... |
| ehlval 25372 | Value of the Euclidean spa... |
| ehlbase 25373 | The base of the Euclidean ... |
| ehl0base 25374 | The base of the Euclidean ... |
| ehl0 25375 | The Euclidean space of dim... |
| ehleudis 25376 | The Euclidean distance fun... |
| ehleudisval 25377 | The value of the Euclidean... |
| ehl1eudis 25378 | The Euclidean distance fun... |
| ehl1eudisval 25379 | The value of the Euclidean... |
| ehl2eudis 25380 | The Euclidean distance fun... |
| ehl2eudisval 25381 | The value of the Euclidean... |
| minveclem1 25382 | Lemma for ~ minvec . The ... |
| minveclem4c 25383 | Lemma for ~ minvec . The ... |
| minveclem2 25384 | Lemma for ~ minvec . Any ... |
| minveclem3a 25385 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25386 | Lemma for ~ minvec . The ... |
| minveclem3 25387 | Lemma for ~ minvec . The ... |
| minveclem4a 25388 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25389 | Lemma for ~ minvec . The ... |
| minveclem4 25390 | Lemma for ~ minvec . The ... |
| minveclem5 25391 | Lemma for ~ minvec . Disc... |
| minveclem6 25392 | Lemma for ~ minvec . Any ... |
| minveclem7 25393 | Lemma for ~ minvec . Sinc... |
| minvec 25394 | Minimizing vector theorem,... |
| pjthlem1 25395 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25396 | Lemma for ~ pjth . (Contr... |
| pjth 25397 | Projection Theorem: Any H... |
| pjth2 25398 | Projection Theorem with ab... |
| cldcss 25399 | Corollary of the Projectio... |
| cldcss2 25400 | Corollary of the Projectio... |
| hlhil 25401 | Corollary of the Projectio... |
| addcncf 25402 | The addition of two contin... |
| subcncf 25403 | The addition of two contin... |
| mulcncf 25404 | The multiplication of two ... |
| mulcncfOLD 25405 | Obsolete version of ~ mulc... |
| divcncf 25406 | The quotient of two contin... |
| pmltpclem1 25407 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25408 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25409 | Any function on the reals ... |
| ivthlem1 25410 | Lemma for ~ ivth . The se... |
| ivthlem2 25411 | Lemma for ~ ivth . Show t... |
| ivthlem3 25412 | Lemma for ~ ivth , the int... |
| ivth 25413 | The intermediate value the... |
| ivth2 25414 | The intermediate value the... |
| ivthle 25415 | The intermediate value the... |
| ivthle2 25416 | The intermediate value the... |
| ivthicc 25417 | The interval between any t... |
| evthicc 25418 | Specialization of the Extr... |
| evthicc2 25419 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25420 | A continuous function on a... |
| ovolfcl 25425 | Closure for the interval e... |
| ovolfioo 25426 | Unpack the interval coveri... |
| ovolficc 25427 | Unpack the interval coveri... |
| ovolficcss 25428 | Any (closed) interval cove... |
| ovolfsval 25429 | The value of the interval ... |
| ovolfsf 25430 | Closure for the interval l... |
| ovolsf 25431 | Closure for the partial su... |
| ovolval 25432 | The value of the outer mea... |
| elovolmlem 25433 | Lemma for ~ elovolm and re... |
| elovolm 25434 | Elementhood in the set ` M... |
| elovolmr 25435 | Sufficient condition for e... |
| ovolmge0 25436 | The set ` M ` is composed ... |
| ovolcl 25437 | The volume of a set is an ... |
| ovollb 25438 | The outer volume is a lowe... |
| ovolgelb 25439 | The outer volume is the gr... |
| ovolge0 25440 | The volume of a set is alw... |
| ovolf 25441 | The domain and codomain of... |
| ovollecl 25442 | If an outer volume is boun... |
| ovolsslem 25443 | Lemma for ~ ovolss . (Con... |
| ovolss 25444 | The volume of a set is mon... |
| ovolsscl 25445 | If a set is contained in a... |
| ovolssnul 25446 | A subset of a nullset is n... |
| ovollb2lem 25447 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25448 | It is often more convenien... |
| ovolctb 25449 | The volume of a denumerabl... |
| ovolq 25450 | The rational numbers have ... |
| ovolctb2 25451 | The volume of a countable ... |
| ovol0 25452 | The empty set has 0 outer ... |
| ovolfi 25453 | A finite set has 0 outer L... |
| ovolsn 25454 | A singleton has 0 outer Le... |
| ovolunlem1a 25455 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25456 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25457 | Lemma for ~ ovolun . (Con... |
| ovolun 25458 | The Lebesgue outer measure... |
| ovolunnul 25459 | Adding a nullset does not ... |
| ovolfiniun 25460 | The Lebesgue outer measure... |
| ovoliunlem1 25461 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25462 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25463 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25464 | The Lebesgue outer measure... |
| ovoliun2 25465 | The Lebesgue outer measure... |
| ovoliunnul 25466 | A countable union of nulls... |
| shft2rab 25467 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25468 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25469 | Lemma for ~ ovolshft . (C... |
| ovolshft 25470 | The Lebesgue outer measure... |
| sca2rab 25471 | If ` B ` is a scale of ` A... |
| ovolscalem1 25472 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25473 | Lemma for ~ ovolshft . (C... |
| ovolsca 25474 | The Lebesgue outer measure... |
| ovolicc1 25475 | The measure of a closed in... |
| ovolicc2lem1 25476 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25477 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25478 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25479 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25480 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25481 | The measure of a closed in... |
| ovolicc 25482 | The measure of a closed in... |
| ovolicopnf 25483 | The measure of a right-unb... |
| ovolre 25484 | The measure of the real nu... |
| ismbl 25485 | The predicate " ` A ` is L... |
| ismbl2 25486 | From ~ ovolun , it suffice... |
| volres 25487 | A self-referencing abbrevi... |
| volf 25488 | The domain and codomain of... |
| mblvol 25489 | The volume of a measurable... |
| mblss 25490 | A measurable set is a subs... |
| mblsplit 25491 | The defining property of m... |
| volss 25492 | The Lebesgue measure is mo... |
| cmmbl 25493 | The complement of a measur... |
| nulmbl 25494 | A nullset is measurable. ... |
| nulmbl2 25495 | A set of outer measure zer... |
| unmbl 25496 | A union of measurable sets... |
| shftmbl 25497 | A shift of a measurable se... |
| 0mbl 25498 | The empty set is measurabl... |
| rembl 25499 | The set of all real number... |
| unidmvol 25500 | The union of the Lebesgue ... |
| inmbl 25501 | An intersection of measura... |
| difmbl 25502 | A difference of measurable... |
| finiunmbl 25503 | A finite union of measurab... |
| volun 25504 | The Lebesgue measure funct... |
| volinun 25505 | Addition of non-disjoint s... |
| volfiniun 25506 | The volume of a disjoint f... |
| iundisj 25507 | Rewrite a countable union ... |
| iundisj2 25508 | A disjoint union is disjoi... |
| voliunlem1 25509 | Lemma for ~ voliun . (Con... |
| voliunlem2 25510 | Lemma for ~ voliun . (Con... |
| voliunlem3 25511 | Lemma for ~ voliun . (Con... |
| iunmbl 25512 | The measurable sets are cl... |
| voliun 25513 | The Lebesgue measure funct... |
| volsuplem 25514 | Lemma for ~ volsup . (Con... |
| volsup 25515 | The volume of the limit of... |
| iunmbl2 25516 | The measurable sets are cl... |
| ioombl1lem1 25517 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25518 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25519 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25520 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25521 | An open right-unbounded in... |
| icombl1 25522 | A closed unbounded-above i... |
| icombl 25523 | A closed-below, open-above... |
| ioombl 25524 | An open real interval is m... |
| iccmbl 25525 | A closed real interval is ... |
| iccvolcl 25526 | A closed real interval has... |
| ovolioo 25527 | The measure of an open int... |
| volioo 25528 | The measure of an open int... |
| ioovolcl 25529 | An open real interval has ... |
| ovolfs2 25530 | Alternative expression for... |
| ioorcl2 25531 | An open interval with fini... |
| ioorf 25532 | Define a function from ope... |
| ioorval 25533 | Define a function from ope... |
| ioorinv2 25534 | The function ` F ` is an "... |
| ioorinv 25535 | The function ` F ` is an "... |
| ioorcl 25536 | The function ` F ` does no... |
| uniiccdif 25537 | A union of closed interval... |
| uniioovol 25538 | A disjoint union of open i... |
| uniiccvol 25539 | An almost-disjoint union o... |
| uniioombllem1 25540 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25541 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25542 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25543 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25544 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25545 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25546 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25547 | Lemma for ~ uniioombl . (... |
| uniioombl 25548 | A disjoint union of open i... |
| uniiccmbl 25549 | An almost-disjoint union o... |
| dyadf 25550 | The function ` F ` returns... |
| dyadval 25551 | Value of the dyadic ration... |
| dyadovol 25552 | Volume of a dyadic rationa... |
| dyadss 25553 | Two closed dyadic rational... |
| dyaddisjlem 25554 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25555 | Two closed dyadic rational... |
| dyadmaxlem 25556 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25557 | Any nonempty set of dyadic... |
| dyadmbllem 25558 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25559 | Any union of dyadic ration... |
| opnmbllem 25560 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25561 | All open sets are measurab... |
| opnmblALT 25562 | All open sets are measurab... |
| subopnmbl 25563 | Sets which are open in a m... |
| volsup2 25564 | The volume of ` A ` is the... |
| volcn 25565 | The function formed by res... |
| volivth 25566 | The Intermediate Value The... |
| vitalilem1 25567 | Lemma for ~ vitali . (Con... |
| vitalilem2 25568 | Lemma for ~ vitali . (Con... |
| vitalilem3 25569 | Lemma for ~ vitali . (Con... |
| vitalilem4 25570 | Lemma for ~ vitali . (Con... |
| vitalilem5 25571 | Lemma for ~ vitali . (Con... |
| vitali 25572 | If the reals can be well-o... |
| ismbf1 25583 | The predicate " ` F ` is a... |
| mbff 25584 | A measurable function is a... |
| mbfdm 25585 | The domain of a measurable... |
| mbfconstlem 25586 | Lemma for ~ mbfconst and r... |
| ismbf 25587 | The predicate " ` F ` is a... |
| ismbfcn 25588 | A complex function is meas... |
| mbfima 25589 | Definitional property of a... |
| mbfimaicc 25590 | The preimage of any closed... |
| mbfimasn 25591 | The preimage of a point un... |
| mbfconst 25592 | A constant function is mea... |
| mbf0 25593 | The empty function is meas... |
| mbfid 25594 | The identity function is m... |
| mbfmptcl 25595 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25596 | The domain of a measurable... |
| ismbfcn2 25597 | A complex function is meas... |
| ismbfd 25598 | Deduction to prove measura... |
| ismbf2d 25599 | Deduction to prove measura... |
| mbfeqalem1 25600 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25601 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25602 | If two functions are equal... |
| mbfres 25603 | The restriction of a measu... |
| mbfres2 25604 | Measurability of a piecewi... |
| mbfss 25605 | Change the domain of a mea... |
| mbfmulc2lem 25606 | Multiplication by a consta... |
| mbfmulc2re 25607 | Multiplication by a consta... |
| mbfmax 25608 | The maximum of two functio... |
| mbfneg 25609 | The negative of a measurab... |
| mbfpos 25610 | The positive part of a mea... |
| mbfposr 25611 | Converse to ~ mbfpos . (C... |
| mbfposb 25612 | A function is measurable i... |
| ismbf3d 25613 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25614 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25615 | The preimage of any open s... |
| mbfimaopn2 25616 | The preimage of any set op... |
| cncombf 25617 | The composition of a conti... |
| cnmbf 25618 | A continuous function is m... |
| mbfaddlem 25619 | The sum of two measurable ... |
| mbfadd 25620 | The sum of two measurable ... |
| mbfsub 25621 | The difference of two meas... |
| mbfmulc2 25622 | A complex constant times a... |
| mbfsup 25623 | The supremum of a sequence... |
| mbfinf 25624 | The infimum of a sequence ... |
| mbflimsup 25625 | The limit supremum of a se... |
| mbflimlem 25626 | The pointwise limit of a s... |
| mbflim 25627 | The pointwise limit of a s... |
| 0pval 25630 | The zero function evaluate... |
| 0plef 25631 | Two ways to say that the f... |
| 0pledm 25632 | Adjust the domain of the l... |
| isi1f 25633 | The predicate " ` F ` is a... |
| i1fmbf 25634 | Simple functions are measu... |
| i1ff 25635 | A simple function is a fun... |
| i1frn 25636 | A simple function has fini... |
| i1fima 25637 | Any preimage of a simple f... |
| i1fima2 25638 | Any preimage of a simple f... |
| i1fima2sn 25639 | Preimage of a singleton. ... |
| i1fd 25640 | A simplified set of assump... |
| i1f0rn 25641 | Any simple function takes ... |
| itg1val 25642 | The value of the integral ... |
| itg1val2 25643 | The value of the integral ... |
| itg1cl 25644 | Closure of the integral on... |
| itg1ge0 25645 | Closure of the integral on... |
| i1f0 25646 | The zero function is simpl... |
| itg10 25647 | The zero function has zero... |
| i1f1lem 25648 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25649 | Base case simple functions... |
| itg11 25650 | The integral of an indicat... |
| itg1addlem1 25651 | Decompose a preimage, whic... |
| i1faddlem 25652 | Decompose the preimage of ... |
| i1fmullem 25653 | Decompose the preimage of ... |
| i1fadd 25654 | The sum of two simple func... |
| i1fmul 25655 | The pointwise product of t... |
| itg1addlem2 25656 | Lemma for ~ itg1add . The... |
| itg1addlem3 25657 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25658 | Lemma for ~ itg1add . (Co... |
| itg1addlem4OLD 25659 | Obsolete version of ~ itg1... |
| itg1addlem5 25660 | Lemma for ~ itg1add . (Co... |
| itg1add 25661 | The integral of a sum of s... |
| i1fmulclem 25662 | Decompose the preimage of ... |
| i1fmulc 25663 | A nonnegative constant tim... |
| itg1mulc 25664 | The integral of a constant... |
| i1fres 25665 | The "restriction" of a sim... |
| i1fpos 25666 | The positive part of a sim... |
| i1fposd 25667 | Deduction form of ~ i1fpos... |
| i1fsub 25668 | The difference of two simp... |
| itg1sub 25669 | The integral of a differen... |
| itg10a 25670 | The integral of a simple f... |
| itg1ge0a 25671 | The integral of an almost ... |
| itg1lea 25672 | Approximate version of ~ i... |
| itg1le 25673 | If one simple function dom... |
| itg1climres 25674 | Restricting the simple fun... |
| mbfi1fseqlem1 25675 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25676 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25677 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25678 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25679 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25680 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25681 | A characterization of meas... |
| mbfi1flimlem 25682 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25683 | Any real measurable functi... |
| mbfmullem2 25684 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25685 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25686 | The product of two measura... |
| itg2lcl 25687 | The set of lower sums is a... |
| itg2val 25688 | Value of the integral on n... |
| itg2l 25689 | Elementhood in the set ` L... |
| itg2lr 25690 | Sufficient condition for e... |
| xrge0f 25691 | A real function is a nonne... |
| itg2cl 25692 | The integral of a nonnegat... |
| itg2ub 25693 | The integral of a nonnegat... |
| itg2leub 25694 | Any upper bound on the int... |
| itg2ge0 25695 | The integral of a nonnegat... |
| itg2itg1 25696 | The integral of a nonnegat... |
| itg20 25697 | The integral of the zero f... |
| itg2lecl 25698 | If an ` S.2 ` integral is ... |
| itg2le 25699 | If one function dominates ... |
| itg2const 25700 | Integral of a constant fun... |
| itg2const2 25701 | When the base set of a con... |
| itg2seq 25702 | Definitional property of t... |
| itg2uba 25703 | Approximate version of ~ i... |
| itg2lea 25704 | Approximate version of ~ i... |
| itg2eqa 25705 | Approximate equality of in... |
| itg2mulclem 25706 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25707 | The integral of a nonnegat... |
| itg2splitlem 25708 | Lemma for ~ itg2split . (... |
| itg2split 25709 | The ` S.2 ` integral split... |
| itg2monolem1 25710 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25711 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25712 | Lemma for ~ itg2mono . (C... |
| itg2mono 25713 | The Monotone Convergence T... |
| itg2i1fseqle 25714 | Subject to the conditions ... |
| itg2i1fseq 25715 | Subject to the conditions ... |
| itg2i1fseq2 25716 | In an extension to the res... |
| itg2i1fseq3 25717 | Special case of ~ itg2i1fs... |
| itg2addlem 25718 | Lemma for ~ itg2add . (Co... |
| itg2add 25719 | The ` S.2 ` integral is li... |
| itg2gt0 25720 | If the function ` F ` is s... |
| itg2cnlem1 25721 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25722 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25723 | A sort of absolute continu... |
| ibllem 25724 | Conditioned equality theor... |
| isibl 25725 | The predicate " ` F ` is i... |
| isibl2 25726 | The predicate " ` F ` is i... |
| iblmbf 25727 | An integrable function is ... |
| iblitg 25728 | If a function is integrabl... |
| dfitg 25729 | Evaluate the class substit... |
| itgex 25730 | An integral is a set. (Co... |
| itgeq1f 25731 | Equality theorem for an in... |
| itgeq1 25732 | Equality theorem for an in... |
| nfitg1 25733 | Bound-variable hypothesis ... |
| nfitg 25734 | Bound-variable hypothesis ... |
| cbvitg 25735 | Change bound variable in a... |
| cbvitgv 25736 | Change bound variable in a... |
| itgeq2 25737 | Equality theorem for an in... |
| itgresr 25738 | The domain of an integral ... |
| itg0 25739 | The integral of anything o... |
| itgz 25740 | The integral of zero on an... |
| itgeq2dv 25741 | Equality theorem for an in... |
| itgmpt 25742 | Change bound variable in a... |
| itgcl 25743 | The integral of an integra... |
| itgvallem 25744 | Substitution lemma. (Cont... |
| itgvallem3 25745 | Lemma for ~ itgposval and ... |
| ibl0 25746 | The zero function is integ... |
| iblcnlem1 25747 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25748 | Expand out the universal q... |
| itgcnlem 25749 | Expand out the sum in ~ df... |
| iblrelem 25750 | Integrability of a real fu... |
| iblposlem 25751 | Lemma for ~ iblpos . (Con... |
| iblpos 25752 | Integrability of a nonnega... |
| iblre 25753 | Integrability of a real fu... |
| itgrevallem1 25754 | Lemma for ~ itgposval and ... |
| itgposval 25755 | The integral of a nonnegat... |
| itgreval 25756 | Decompose the integral of ... |
| itgrecl 25757 | Real closure of an integra... |
| iblcn 25758 | Integrability of a complex... |
| itgcnval 25759 | Decompose the integral of ... |
| itgre 25760 | Real part of an integral. ... |
| itgim 25761 | Imaginary part of an integ... |
| iblneg 25762 | The negative of an integra... |
| itgneg 25763 | Negation of an integral. ... |
| iblss 25764 | A subset of an integrable ... |
| iblss2 25765 | Change the domain of an in... |
| itgitg2 25766 | Transfer an integral using... |
| i1fibl 25767 | A simple function is integ... |
| itgitg1 25768 | Transfer an integral using... |
| itgle 25769 | Monotonicity of an integra... |
| itgge0 25770 | The integral of a positive... |
| itgss 25771 | Expand the set of an integ... |
| itgss2 25772 | Expand the set of an integ... |
| itgeqa 25773 | Approximate equality of in... |
| itgss3 25774 | Expand the set of an integ... |
| itgioo 25775 | Equality of integrals on o... |
| itgless 25776 | Expand the integral of a n... |
| iblconst 25777 | A constant function is int... |
| itgconst 25778 | Integral of a constant fun... |
| ibladdlem 25779 | Lemma for ~ ibladd . (Con... |
| ibladd 25780 | Add two integrals over the... |
| iblsub 25781 | Subtract two integrals ove... |
| itgaddlem1 25782 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25783 | Lemma for ~ itgadd . (Con... |
| itgadd 25784 | Add two integrals over the... |
| itgsub 25785 | Subtract two integrals ove... |
| itgfsum 25786 | Take a finite sum of integ... |
| iblabslem 25787 | Lemma for ~ iblabs . (Con... |
| iblabs 25788 | The absolute value of an i... |
| iblabsr 25789 | A measurable function is i... |
| iblmulc2 25790 | Multiply an integral by a ... |
| itgmulc2lem1 25791 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25792 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25793 | Multiply an integral by a ... |
| itgabs 25794 | The triangle inequality fo... |
| itgsplit 25795 | The ` S. ` integral splits... |
| itgspliticc 25796 | The ` S. ` integral splits... |
| itgsplitioo 25797 | The ` S. ` integral splits... |
| bddmulibl 25798 | A bounded function times a... |
| bddibl 25799 | A bounded function is inte... |
| cniccibl 25800 | A continuous function on a... |
| bddiblnc 25801 | Choice-free proof of ~ bdd... |
| cnicciblnc 25802 | Choice-free proof of ~ cni... |
| itggt0 25803 | The integral of a strictly... |
| itgcn 25804 | Transfer ~ itg2cn to the f... |
| ditgeq1 25807 | Equality theorem for the d... |
| ditgeq2 25808 | Equality theorem for the d... |
| ditgeq3 25809 | Equality theorem for the d... |
| ditgeq3dv 25810 | Equality theorem for the d... |
| ditgex 25811 | A directed integral is a s... |
| ditg0 25812 | Value of the directed inte... |
| cbvditg 25813 | Change bound variable in a... |
| cbvditgv 25814 | Change bound variable in a... |
| ditgpos 25815 | Value of the directed inte... |
| ditgneg 25816 | Value of the directed inte... |
| ditgcl 25817 | Closure of a directed inte... |
| ditgswap 25818 | Reverse a directed integra... |
| ditgsplitlem 25819 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25820 | This theorem is the raison... |
| reldv 25829 | The derivative function is... |
| limcvallem 25830 | Lemma for ~ ellimc . (Con... |
| limcfval 25831 | Value and set bounds on th... |
| ellimc 25832 | Value of the limit predica... |
| limcrcl 25833 | Reverse closure for the li... |
| limccl 25834 | Closure of the limit opera... |
| limcdif 25835 | It suffices to consider fu... |
| ellimc2 25836 | Write the definition of a ... |
| limcnlp 25837 | If ` B ` is not a limit po... |
| ellimc3 25838 | Write the epsilon-delta de... |
| limcflflem 25839 | Lemma for ~ limcflf . (Co... |
| limcflf 25840 | The limit operator can be ... |
| limcmo 25841 | If ` B ` is a limit point ... |
| limcmpt 25842 | Express the limit operator... |
| limcmpt2 25843 | Express the limit operator... |
| limcresi 25844 | Any limit of ` F ` is also... |
| limcres 25845 | If ` B ` is an interior po... |
| cnplimc 25846 | A function is continuous a... |
| cnlimc 25847 | ` F ` is a continuous func... |
| cnlimci 25848 | If ` F ` is a continuous f... |
| cnmptlimc 25849 | If ` F ` is a continuous f... |
| limccnp 25850 | If the limit of ` F ` at `... |
| limccnp2 25851 | The image of a convergent ... |
| limcco 25852 | Composition of two limits.... |
| limciun 25853 | A point is a limit of ` F ... |
| limcun 25854 | A point is a limit of ` F ... |
| dvlem 25855 | Closure for a difference q... |
| dvfval 25856 | Value and set bounds on th... |
| eldv 25857 | The differentiable predica... |
| dvcl 25858 | The derivative function ta... |
| dvbssntr 25859 | The set of differentiable ... |
| dvbss 25860 | The set of differentiable ... |
| dvbsss 25861 | The set of differentiable ... |
| perfdvf 25862 | The derivative is a functi... |
| recnprss 25863 | Both ` RR ` and ` CC ` are... |
| recnperf 25864 | Both ` RR ` and ` CC ` are... |
| dvfg 25865 | Explicitly write out the f... |
| dvf 25866 | The derivative is a functi... |
| dvfcn 25867 | The derivative is a functi... |
| dvreslem 25868 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25869 | Lemma for ~ dvres2 . (Con... |
| dvres 25870 | Restriction of a derivativ... |
| dvres2 25871 | Restriction of the base se... |
| dvres3 25872 | Restriction of a complex d... |
| dvres3a 25873 | Restriction of a complex d... |
| dvidlem 25874 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25875 | Derivative of a function r... |
| dvconst 25876 | Derivative of a constant f... |
| dvid 25877 | Derivative of the identity... |
| dvcnp 25878 | The difference quotient is... |
| dvcnp2 25879 | A function is continuous a... |
| dvcnp2OLD 25880 | Obsolete version of ~ dvcn... |
| dvcn 25881 | A differentiable function ... |
| dvnfval 25882 | Value of the iterated deri... |
| dvnff 25883 | The iterated derivative is... |
| dvn0 25884 | Zero times iterated deriva... |
| dvnp1 25885 | Successor iterated derivat... |
| dvn1 25886 | One times iterated derivat... |
| dvnf 25887 | The N-times derivative is ... |
| dvnbss 25888 | The set of N-times differe... |
| dvnadd 25889 | The ` N ` -th derivative o... |
| dvn2bss 25890 | An N-times differentiable ... |
| dvnres 25891 | Multiple derivative versio... |
| cpnfval 25892 | Condition for n-times cont... |
| fncpn 25893 | The ` C^n ` object is a fu... |
| elcpn 25894 | Condition for n-times cont... |
| cpnord 25895 | ` C^n ` conditions are ord... |
| cpncn 25896 | A ` C^n ` function is cont... |
| cpnres 25897 | The restriction of a ` C^n... |
| dvaddbr 25898 | The sum rule for derivativ... |
| dvmulbr 25899 | The product rule for deriv... |
| dvmulbrOLD 25900 | Obsolete version of ~ dvmu... |
| dvadd 25901 | The sum rule for derivativ... |
| dvmul 25902 | The product rule for deriv... |
| dvaddf 25903 | The sum rule for everywher... |
| dvmulf 25904 | The product rule for every... |
| dvcmul 25905 | The product rule when one ... |
| dvcmulf 25906 | The product rule when one ... |
| dvcobr 25907 | The chain rule for derivat... |
| dvcobrOLD 25908 | Obsolete version of ~ dvco... |
| dvco 25909 | The chain rule for derivat... |
| dvcof 25910 | The chain rule for everywh... |
| dvcjbr 25911 | The derivative of the conj... |
| dvcj 25912 | The derivative of the conj... |
| dvfre 25913 | The derivative of a real f... |
| dvnfre 25914 | The ` N ` -th derivative o... |
| dvexp 25915 | Derivative of a power func... |
| dvexp2 25916 | Derivative of an exponenti... |
| dvrec 25917 | Derivative of the reciproc... |
| dvmptres3 25918 | Function-builder for deriv... |
| dvmptid 25919 | Function-builder for deriv... |
| dvmptc 25920 | Function-builder for deriv... |
| dvmptcl 25921 | Closure lemma for ~ dvmptc... |
| dvmptadd 25922 | Function-builder for deriv... |
| dvmptmul 25923 | Function-builder for deriv... |
| dvmptres2 25924 | Function-builder for deriv... |
| dvmptres 25925 | Function-builder for deriv... |
| dvmptcmul 25926 | Function-builder for deriv... |
| dvmptdivc 25927 | Function-builder for deriv... |
| dvmptneg 25928 | Function-builder for deriv... |
| dvmptsub 25929 | Function-builder for deriv... |
| dvmptcj 25930 | Function-builder for deriv... |
| dvmptre 25931 | Function-builder for deriv... |
| dvmptim 25932 | Function-builder for deriv... |
| dvmptntr 25933 | Function-builder for deriv... |
| dvmptco 25934 | Function-builder for deriv... |
| dvrecg 25935 | Derivative of the reciproc... |
| dvmptdiv 25936 | Function-builder for deriv... |
| dvmptfsum 25937 | Function-builder for deriv... |
| dvcnvlem 25938 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25939 | A weak version of ~ dvcnvr... |
| dvexp3 25940 | Derivative of an exponenti... |
| dveflem 25941 | Derivative of the exponent... |
| dvef 25942 | Derivative of the exponent... |
| dvsincos 25943 | Derivative of the sine and... |
| dvsin 25944 | Derivative of the sine fun... |
| dvcos 25945 | Derivative of the cosine f... |
| dvferm1lem 25946 | Lemma for ~ dvferm . (Con... |
| dvferm1 25947 | One-sided version of ~ dvf... |
| dvferm2lem 25948 | Lemma for ~ dvferm . (Con... |
| dvferm2 25949 | One-sided version of ~ dvf... |
| dvferm 25950 | Fermat's theorem on statio... |
| rollelem 25951 | Lemma for ~ rolle . (Cont... |
| rolle 25952 | Rolle's theorem. If ` F `... |
| cmvth 25953 | Cauchy's Mean Value Theore... |
| cmvthOLD 25954 | Obsolete version of ~ cmvt... |
| mvth 25955 | The Mean Value Theorem. I... |
| dvlip 25956 | A function with derivative... |
| dvlipcn 25957 | A complex function with de... |
| dvlip2 25958 | Combine the results of ~ d... |
| c1liplem1 25959 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25960 | C^1 functions are Lipschit... |
| c1lip2 25961 | C^1 functions are Lipschit... |
| c1lip3 25962 | C^1 functions are Lipschit... |
| dveq0 25963 | If a continuous function h... |
| dv11cn 25964 | Two functions defined on a... |
| dvgt0lem1 25965 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25966 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25967 | A function on a closed int... |
| dvlt0 25968 | A function on a closed int... |
| dvge0 25969 | A function on a closed int... |
| dvle 25970 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25971 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25972 | Lemma for ~ dvivth . (Con... |
| dvivth 25973 | Darboux' theorem, or the i... |
| dvne0 25974 | A function on a closed int... |
| dvne0f1 25975 | A function on a closed int... |
| lhop1lem 25976 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25977 | L'Hôpital's Rule for... |
| lhop2 25978 | L'Hôpital's Rule for... |
| lhop 25979 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25980 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25981 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25982 | The derivative rule for in... |
| dvcvx 25983 | A real function with stric... |
| dvfsumle 25984 | Compare a finite sum to an... |
| dvfsumleOLD 25985 | Obsolete version of ~ dvfs... |
| dvfsumge 25986 | Compare a finite sum to an... |
| dvfsumabs 25987 | Compare a finite sum to an... |
| dvmptrecl 25988 | Real closure of a derivati... |
| dvfsumrlimf 25989 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25990 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25991 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2OLD 25992 | Obsolete version of ~ dvfs... |
| dvfsumlem3 25993 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25994 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25995 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25996 | Compare a finite sum to an... |
| dvfsumrlim2 25997 | Compare a finite sum to an... |
| dvfsumrlim3 25998 | Conjoin the statements of ... |
| dvfsum2 25999 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26000 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26001 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26002 | The Fundamental Theorem of... |
| ftc1lem3 26003 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26004 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26005 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26006 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26007 | The Fundamental Theorem of... |
| ftc1cn 26008 | Strengthen the assumptions... |
| ftc2 26009 | The Fundamental Theorem of... |
| ftc2ditglem 26010 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26011 | Directed integral analogue... |
| itgparts 26012 | Integration by parts. If ... |
| itgsubstlem 26013 | Lemma for ~ itgsubst . (C... |
| itgsubst 26014 | Integration by ` u ` -subs... |
| itgpowd 26015 | The integral of a monomial... |
| reldmmdeg 26020 | Multivariate degree is a b... |
| tdeglem1 26021 | Functionality of the total... |
| tdeglem1OLD 26022 | Obsolete version of ~ tdeg... |
| tdeglem3 26023 | Additivity of the total de... |
| tdeglem3OLD 26024 | Obsolete version of ~ tdeg... |
| tdeglem4 26025 | There is only one multi-in... |
| tdeglem4OLD 26026 | Obsolete version of ~ tdeg... |
| tdeglem2 26027 | Simplification of total de... |
| mdegfval 26028 | Value of the multivariate ... |
| mdegval 26029 | Value of the multivariate ... |
| mdegleb 26030 | Property of being of limit... |
| mdeglt 26031 | If there is an upper limit... |
| mdegldg 26032 | A nonzero polynomial has s... |
| mdegxrcl 26033 | Closure of polynomial degr... |
| mdegxrf 26034 | Functionality of polynomia... |
| mdegcl 26035 | Sharp closure for multivar... |
| mdeg0 26036 | Degree of the zero polynom... |
| mdegnn0cl 26037 | Degree of a nonzero polyno... |
| degltlem1 26038 | Theorem on arithmetic of e... |
| degltp1le 26039 | Theorem on arithmetic of e... |
| mdegaddle 26040 | The degree of a sum is at ... |
| mdegvscale 26041 | The degree of a scalar mul... |
| mdegvsca 26042 | The degree of a scalar mul... |
| mdegle0 26043 | A polynomial has nonpositi... |
| mdegmullem 26044 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26045 | The multivariate degree of... |
| deg1fval 26046 | Relate univariate polynomi... |
| deg1xrf 26047 | Functionality of univariat... |
| deg1xrcl 26048 | Closure of univariate poly... |
| deg1cl 26049 | Sharp closure of univariat... |
| mdegpropd 26050 | Property deduction for pol... |
| deg1fvi 26051 | Univariate polynomial degr... |
| deg1propd 26052 | Property deduction for pol... |
| deg1z 26053 | Degree of the zero univari... |
| deg1nn0cl 26054 | Degree of a nonzero univar... |
| deg1n0ima 26055 | Degree image of a set of p... |
| deg1nn0clb 26056 | A polynomial is nonzero if... |
| deg1lt0 26057 | A polynomial is zero iff i... |
| deg1ldg 26058 | A nonzero univariate polyn... |
| deg1ldgn 26059 | An index at which a polyno... |
| deg1ldgdomn 26060 | A nonzero univariate polyn... |
| deg1leb 26061 | Property of being of limit... |
| deg1val 26062 | Value of the univariate de... |
| deg1lt 26063 | If the degree of a univari... |
| deg1ge 26064 | Conversely, a nonzero coef... |
| coe1mul3 26065 | The coefficient vector of ... |
| coe1mul4 26066 | Value of the "leading" coe... |
| deg1addle 26067 | The degree of a sum is at ... |
| deg1addle2 26068 | If both factors have degre... |
| deg1add 26069 | Exact degree of a sum of t... |
| deg1vscale 26070 | The degree of a scalar tim... |
| deg1vsca 26071 | The degree of a scalar tim... |
| deg1invg 26072 | The degree of the negated ... |
| deg1suble 26073 | The degree of a difference... |
| deg1sub 26074 | Exact degree of a differen... |
| deg1mulle2 26075 | Produce a bound on the pro... |
| deg1sublt 26076 | Subtraction of two polynom... |
| deg1le0 26077 | A polynomial has nonpositi... |
| deg1sclle 26078 | A scalar polynomial has no... |
| deg1scl 26079 | A nonzero scalar polynomia... |
| deg1mul2 26080 | Degree of multiplication o... |
| deg1mul3 26081 | Degree of multiplication o... |
| deg1mul3le 26082 | Degree of multiplication o... |
| deg1tmle 26083 | Limiting degree of a polyn... |
| deg1tm 26084 | Exact degree of a polynomi... |
| deg1pwle 26085 | Limiting degree of a varia... |
| deg1pw 26086 | Exact degree of a variable... |
| ply1nz 26087 | Univariate polynomials ove... |
| ply1nzb 26088 | Univariate polynomials are... |
| ply1domn 26089 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26090 | The ring of univariate pol... |
| ply1divmo 26101 | Uniqueness of a quotient i... |
| ply1divex 26102 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26103 | The division algorithm for... |
| ply1divalg2 26104 | Reverse the order of multi... |
| uc1pval 26105 | Value of the set of unitic... |
| isuc1p 26106 | Being a unitic polynomial.... |
| mon1pval 26107 | Value of the set of monic ... |
| ismon1p 26108 | Being a monic polynomial. ... |
| uc1pcl 26109 | Unitic polynomials are pol... |
| mon1pcl 26110 | Monic polynomials are poly... |
| uc1pn0 26111 | Unitic polynomials are not... |
| mon1pn0 26112 | Monic polynomials are not ... |
| uc1pdeg 26113 | Unitic polynomials have no... |
| uc1pldg 26114 | Unitic polynomials have un... |
| mon1pldg 26115 | Unitic polynomials have on... |
| mon1puc1p 26116 | Monic polynomials are unit... |
| uc1pmon1p 26117 | Make a unitic polynomial m... |
| deg1submon1p 26118 | The difference of two moni... |
| mon1pid 26119 | Monicity and degree of the... |
| q1pval 26120 | Value of the univariate po... |
| q1peqb 26121 | Characterizing property of... |
| q1pcl 26122 | Closure of the quotient by... |
| r1pval 26123 | Value of the polynomial re... |
| r1pcl 26124 | Closure of remainder follo... |
| r1pdeglt 26125 | The remainder has a degree... |
| r1pid 26126 | Express the original polyn... |
| dvdsq1p 26127 | Divisibility in a polynomi... |
| dvdsr1p 26128 | Divisibility in a polynomi... |
| ply1remlem 26129 | A term of the form ` x - N... |
| ply1rem 26130 | The polynomial remainder t... |
| facth1 26131 | The factor theorem and its... |
| fta1glem1 26132 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26133 | Lemma for ~ fta1g . (Cont... |
| fta1g 26134 | The one-sided fundamental ... |
| fta1blem 26135 | Lemma for ~ fta1b . (Cont... |
| fta1b 26136 | The assumption that ` R ` ... |
| idomrootle 26137 | No element of an integral ... |
| drnguc1p 26138 | Over a division ring, all ... |
| ig1peu 26139 | There is a unique monic po... |
| ig1pval 26140 | Substitutions for the poly... |
| ig1pval2 26141 | Generator of the zero idea... |
| ig1pval3 26142 | Characterizing properties ... |
| ig1pcl 26143 | The monic generator of an ... |
| ig1pdvds 26144 | The monic generator of an ... |
| ig1prsp 26145 | Any ideal of polynomials o... |
| ply1lpir 26146 | The ring of polynomials ov... |
| ply1pid 26147 | The polynomials over a fie... |
| plyco0 26156 | Two ways to say that a fun... |
| plyval 26157 | Value of the polynomial se... |
| plybss 26158 | Reverse closure of the par... |
| elply 26159 | Definition of a polynomial... |
| elply2 26160 | The coefficient function c... |
| plyun0 26161 | The set of polynomials is ... |
| plyf 26162 | The polynomial is a functi... |
| plyss 26163 | The polynomial set functio... |
| plyssc 26164 | Every polynomial ring is c... |
| elplyr 26165 | Sufficient condition for e... |
| elplyd 26166 | Sufficient condition for e... |
| ply1termlem 26167 | Lemma for ~ ply1term . (C... |
| ply1term 26168 | A one-term polynomial. (C... |
| plypow 26169 | A power is a polynomial. ... |
| plyconst 26170 | A constant function is a p... |
| ne0p 26171 | A test to show that a poly... |
| ply0 26172 | The zero function is a pol... |
| plyid 26173 | The identity function is a... |
| plyeq0lem 26174 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26175 | If a polynomial is zero at... |
| plypf1 26176 | Write the set of complex p... |
| plyaddlem1 26177 | Derive the coefficient fun... |
| plymullem1 26178 | Derive the coefficient fun... |
| plyaddlem 26179 | Lemma for ~ plyadd . (Con... |
| plymullem 26180 | Lemma for ~ plymul . (Con... |
| plyadd 26181 | The sum of two polynomials... |
| plymul 26182 | The product of two polynom... |
| plysub 26183 | The difference of two poly... |
| plyaddcl 26184 | The sum of two polynomials... |
| plymulcl 26185 | The product of two polynom... |
| plysubcl 26186 | The difference of two poly... |
| coeval 26187 | Value of the coefficient f... |
| coeeulem 26188 | Lemma for ~ coeeu . (Cont... |
| coeeu 26189 | Uniqueness of the coeffici... |
| coelem 26190 | Lemma for properties of th... |
| coeeq 26191 | If ` A ` satisfies the pro... |
| dgrval 26192 | Value of the degree functi... |
| dgrlem 26193 | Lemma for ~ dgrcl and simi... |
| coef 26194 | The domain and codomain of... |
| coef2 26195 | The domain and codomain of... |
| coef3 26196 | The domain and codomain of... |
| dgrcl 26197 | The degree of any polynomi... |
| dgrub 26198 | If the ` M ` -th coefficie... |
| dgrub2 26199 | All the coefficients above... |
| dgrlb 26200 | If all the coefficients ab... |
| coeidlem 26201 | Lemma for ~ coeid . (Cont... |
| coeid 26202 | Reconstruct a polynomial a... |
| coeid2 26203 | Reconstruct a polynomial a... |
| coeid3 26204 | Reconstruct a polynomial a... |
| plyco 26205 | The composition of two pol... |
| coeeq2 26206 | Compute the coefficient fu... |
| dgrle 26207 | Given an explicit expressi... |
| dgreq 26208 | If the highest term in a p... |
| 0dgr 26209 | A constant function has de... |
| 0dgrb 26210 | A function has degree zero... |
| dgrnznn 26211 | A nonzero polynomial with ... |
| coefv0 26212 | The result of evaluating a... |
| coeaddlem 26213 | Lemma for ~ coeadd and ~ d... |
| coemullem 26214 | Lemma for ~ coemul and ~ d... |
| coeadd 26215 | The coefficient function o... |
| coemul 26216 | A coefficient of a product... |
| coe11 26217 | The coefficient function i... |
| coemulhi 26218 | The leading coefficient of... |
| coemulc 26219 | The coefficient function i... |
| coe0 26220 | The coefficients of the ze... |
| coesub 26221 | The coefficient function o... |
| coe1termlem 26222 | The coefficient function o... |
| coe1term 26223 | The coefficient function o... |
| dgr1term 26224 | The degree of a monomial. ... |
| plycn 26225 | A polynomial is a continuo... |
| plycnOLD 26226 | Obsolete version of ~ plyc... |
| dgr0 26227 | The degree of the zero pol... |
| coeidp 26228 | The coefficients of the id... |
| dgrid 26229 | The degree of the identity... |
| dgreq0 26230 | The leading coefficient of... |
| dgrlt 26231 | Two ways to say that the d... |
| dgradd 26232 | The degree of a sum of pol... |
| dgradd2 26233 | The degree of a sum of pol... |
| dgrmul2 26234 | The degree of a product of... |
| dgrmul 26235 | The degree of a product of... |
| dgrmulc 26236 | Scalar multiplication by a... |
| dgrsub 26237 | The degree of a difference... |
| dgrcolem1 26238 | The degree of a compositio... |
| dgrcolem2 26239 | Lemma for ~ dgrco . (Cont... |
| dgrco 26240 | The degree of a compositio... |
| plycjlem 26241 | Lemma for ~ plycj and ~ co... |
| plycj 26242 | The double conjugation of ... |
| coecj 26243 | Double conjugation of a po... |
| plyrecj 26244 | A polynomial with real coe... |
| plymul0or 26245 | Polynomial multiplication ... |
| ofmulrt 26246 | The set of roots of a prod... |
| plyreres 26247 | Real-coefficient polynomia... |
| dvply1 26248 | Derivative of a polynomial... |
| dvply2g 26249 | The derivative of a polyno... |
| dvply2gOLD 26250 | Obsolete version of ~ dvpl... |
| dvply2 26251 | The derivative of a polyno... |
| dvnply2 26252 | Polynomials have polynomia... |
| dvnply 26253 | Polynomials have polynomia... |
| plycpn 26254 | Polynomials are smooth. (... |
| quotval 26257 | Value of the quotient func... |
| plydivlem1 26258 | Lemma for ~ plydivalg . (... |
| plydivlem2 26259 | Lemma for ~ plydivalg . (... |
| plydivlem3 26260 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26261 | Lemma for ~ plydivex . In... |
| plydivex 26262 | Lemma for ~ plydivalg . (... |
| plydiveu 26263 | Lemma for ~ plydivalg . (... |
| plydivalg 26264 | The division algorithm on ... |
| quotlem 26265 | Lemma for properties of th... |
| quotcl 26266 | The quotient of two polyno... |
| quotcl2 26267 | Closure of the quotient fu... |
| quotdgr 26268 | Remainder property of the ... |
| plyremlem 26269 | Closure of a linear factor... |
| plyrem 26270 | The polynomial remainder t... |
| facth 26271 | The factor theorem. If a ... |
| fta1lem 26272 | Lemma for ~ fta1 . (Contr... |
| fta1 26273 | The easy direction of the ... |
| quotcan 26274 | Exact division with a mult... |
| vieta1lem1 26275 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26276 | Lemma for ~ vieta1 : induc... |
| vieta1 26277 | The first-order Vieta's fo... |
| plyexmo 26278 | An infinite set of values ... |
| elaa 26281 | Elementhood in the set of ... |
| aacn 26282 | An algebraic number is a c... |
| aasscn 26283 | The algebraic numbers are ... |
| elqaalem1 26284 | Lemma for ~ elqaa . The f... |
| elqaalem2 26285 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26286 | Lemma for ~ elqaa . (Cont... |
| elqaa 26287 | The set of numbers generat... |
| qaa 26288 | Every rational number is a... |
| qssaa 26289 | The rational numbers are c... |
| iaa 26290 | The imaginary unit is alge... |
| aareccl 26291 | The reciprocal of an algeb... |
| aacjcl 26292 | The conjugate of an algebr... |
| aannenlem1 26293 | Lemma for ~ aannen . (Con... |
| aannenlem2 26294 | Lemma for ~ aannen . (Con... |
| aannenlem3 26295 | The algebraic numbers are ... |
| aannen 26296 | The algebraic numbers are ... |
| aalioulem1 26297 | Lemma for ~ aaliou . An i... |
| aalioulem2 26298 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26299 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26300 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26301 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26302 | Lemma for ~ aaliou . (Con... |
| aaliou 26303 | Liouville's theorem on dio... |
| geolim3 26304 | Geometric series convergen... |
| aaliou2 26305 | Liouville's approximation ... |
| aaliou2b 26306 | Liouville's approximation ... |
| aaliou3lem1 26307 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26308 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26309 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26310 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26311 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26312 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26313 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26314 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26315 | Example of a "Liouville nu... |
| aaliou3 26316 | Example of a "Liouville nu... |
| taylfvallem1 26321 | Lemma for ~ taylfval . (C... |
| taylfvallem 26322 | Lemma for ~ taylfval . (C... |
| taylfval 26323 | Define the Taylor polynomi... |
| eltayl 26324 | Value of the Taylor series... |
| taylf 26325 | The Taylor series defines ... |
| tayl0 26326 | The Taylor series is alway... |
| taylplem1 26327 | Lemma for ~ taylpfval and ... |
| taylplem2 26328 | Lemma for ~ taylpfval and ... |
| taylpfval 26329 | Define the Taylor polynomi... |
| taylpf 26330 | The Taylor polynomial is a... |
| taylpval 26331 | Value of the Taylor polyno... |
| taylply2 26332 | The Taylor polynomial is a... |
| taylply2OLD 26333 | Obsolete version of ~ tayl... |
| taylply 26334 | The Taylor polynomial is a... |
| dvtaylp 26335 | The derivative of the Tayl... |
| dvntaylp 26336 | The ` M ` -th derivative o... |
| dvntaylp0 26337 | The first ` N ` derivative... |
| taylthlem1 26338 | Lemma for ~ taylth . This... |
| taylthlem2 26339 | Lemma for ~ taylth . (Con... |
| taylthlem2OLD 26340 | Obsolete version of ~ tayl... |
| taylth 26341 | Taylor's theorem. The Tay... |
| ulmrel 26344 | The uniform limit relation... |
| ulmscl 26345 | Closure of the base set in... |
| ulmval 26346 | Express the predicate: Th... |
| ulmcl 26347 | Closure of a uniform limit... |
| ulmf 26348 | Closure of a uniform limit... |
| ulmpm 26349 | Closure of a uniform limit... |
| ulmf2 26350 | Closure of a uniform limit... |
| ulm2 26351 | Simplify ~ ulmval when ` F... |
| ulmi 26352 | The uniform limit property... |
| ulmclm 26353 | A uniform limit of functio... |
| ulmres 26354 | A sequence of functions co... |
| ulmshftlem 26355 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26356 | A sequence of functions co... |
| ulm0 26357 | Every function converges u... |
| ulmuni 26358 | A sequence of functions un... |
| ulmdm 26359 | Two ways to express that a... |
| ulmcaulem 26360 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26361 | A sequence of functions co... |
| ulmcau2 26362 | A sequence of functions co... |
| ulmss 26363 | A uniform limit of functio... |
| ulmbdd 26364 | A uniform limit of bounded... |
| ulmcn 26365 | A uniform limit of continu... |
| ulmdvlem1 26366 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26367 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26368 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26369 | If ` F ` is a sequence of ... |
| mtest 26370 | The Weierstrass M-test. I... |
| mtestbdd 26371 | Given the hypotheses of th... |
| mbfulm 26372 | A uniform limit of measura... |
| iblulm 26373 | A uniform limit of integra... |
| itgulm 26374 | A uniform limit of integra... |
| itgulm2 26375 | A uniform limit of integra... |
| pserval 26376 | Value of the function ` G ... |
| pserval2 26377 | Value of the function ` G ... |
| psergf 26378 | The sequence of terms in t... |
| radcnvlem1 26379 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26380 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26381 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26382 | Zero is always a convergen... |
| radcnvcl 26383 | The radius of convergence ... |
| radcnvlt1 26384 | If ` X ` is within the ope... |
| radcnvlt2 26385 | If ` X ` is within the ope... |
| radcnvle 26386 | If ` X ` is a convergent p... |
| dvradcnv 26387 | The radius of convergence ... |
| pserulm 26388 | If ` S ` is a region conta... |
| psercn2 26389 | Since by ~ pserulm the ser... |
| psercn2OLD 26390 | Obsolete version of ~ pser... |
| psercnlem2 26391 | Lemma for ~ psercn . (Con... |
| psercnlem1 26392 | Lemma for ~ psercn . (Con... |
| psercn 26393 | An infinite series converg... |
| pserdvlem1 26394 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26395 | Lemma for ~ pserdv . (Con... |
| pserdv 26396 | The derivative of a power ... |
| pserdv2 26397 | The derivative of a power ... |
| abelthlem1 26398 | Lemma for ~ abelth . (Con... |
| abelthlem2 26399 | Lemma for ~ abelth . The ... |
| abelthlem3 26400 | Lemma for ~ abelth . (Con... |
| abelthlem4 26401 | Lemma for ~ abelth . (Con... |
| abelthlem5 26402 | Lemma for ~ abelth . (Con... |
| abelthlem6 26403 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26404 | Lemma for ~ abelth . (Con... |
| abelthlem7 26405 | Lemma for ~ abelth . (Con... |
| abelthlem8 26406 | Lemma for ~ abelth . (Con... |
| abelthlem9 26407 | Lemma for ~ abelth . By a... |
| abelth 26408 | Abel's theorem. If the po... |
| abelth2 26409 | Abel's theorem, restricted... |
| efcn 26410 | The exponential function i... |
| sincn 26411 | Sine is continuous. (Cont... |
| coscn 26412 | Cosine is continuous. (Co... |
| reeff1olem 26413 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26414 | The real exponential funct... |
| reefiso 26415 | The exponential function o... |
| efcvx 26416 | The exponential function o... |
| reefgim 26417 | The exponential function i... |
| pilem1 26418 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26419 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26420 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26421 | ` _pi ` is between 2 and 4... |
| sinpi 26422 | The sine of ` _pi ` is 0. ... |
| pire 26423 | ` _pi ` is a real number. ... |
| picn 26424 | ` _pi ` is a complex numbe... |
| pipos 26425 | ` _pi ` is positive. (Con... |
| pirp 26426 | ` _pi ` is a positive real... |
| negpicn 26427 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26428 | Lemma for ~ sinhalfpi and ... |
| halfpire 26429 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26430 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26431 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26432 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26433 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26434 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26435 | The cosine of ` -u _pi / 2... |
| efhalfpi 26436 | The exponential of ` _i _p... |
| cospi 26437 | The cosine of ` _pi ` is `... |
| efipi 26438 | The exponential of ` _i x.... |
| eulerid 26439 | Euler's identity. (Contri... |
| sin2pi 26440 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26441 | The cosine of ` 2 _pi ` is... |
| ef2pi 26442 | The exponential of ` 2 _pi... |
| ef2kpi 26443 | If ` K ` is an integer, th... |
| efper 26444 | The exponential function i... |
| sinperlem 26445 | Lemma for ~ sinper and ~ c... |
| sinper 26446 | The sine function is perio... |
| cosper 26447 | The cosine function is per... |
| sin2kpi 26448 | If ` K ` is an integer, th... |
| cos2kpi 26449 | If ` K ` is an integer, th... |
| sin2pim 26450 | Sine of a number subtracte... |
| cos2pim 26451 | Cosine of a number subtrac... |
| sinmpi 26452 | Sine of a number less ` _p... |
| cosmpi 26453 | Cosine of a number less ` ... |
| sinppi 26454 | Sine of a number plus ` _p... |
| cosppi 26455 | Cosine of a number plus ` ... |
| efimpi 26456 | The exponential function a... |
| sinhalfpip 26457 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26458 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26459 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26460 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26461 | Ptolemy's Theorem. This t... |
| sincosq1lem 26462 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26463 | The signs of the sine and ... |
| sincosq2sgn 26464 | The signs of the sine and ... |
| sincosq3sgn 26465 | The signs of the sine and ... |
| sincosq4sgn 26466 | The signs of the sine and ... |
| coseq00topi 26467 | Location of the zeroes of ... |
| coseq0negpitopi 26468 | Location of the zeroes of ... |
| tanrpcl 26469 | Positive real closure of t... |
| tangtx 26470 | The tangent function is gr... |
| tanabsge 26471 | The tangent function is gr... |
| sinq12gt0 26472 | The sine of a number stric... |
| sinq12ge0 26473 | The sine of a number betwe... |
| sinq34lt0t 26474 | The sine of a number stric... |
| cosq14gt0 26475 | The cosine of a number str... |
| cosq14ge0 26476 | The cosine of a number bet... |
| sincosq1eq 26477 | Complementarity of the sin... |
| sincos4thpi 26478 | The sine and cosine of ` _... |
| tan4thpi 26479 | The tangent of ` _pi / 4 `... |
| sincos6thpi 26480 | The sine and cosine of ` _... |
| sincos3rdpi 26481 | The sine and cosine of ` _... |
| pigt3 26482 | ` _pi ` is greater than 3.... |
| pige3 26483 | ` _pi ` is greater than or... |
| pige3ALT 26484 | Alternate proof of ~ pige3... |
| abssinper 26485 | The absolute value of sine... |
| sinkpi 26486 | The sine of an integer mul... |
| coskpi 26487 | The absolute value of the ... |
| sineq0 26488 | A complex number whose sin... |
| coseq1 26489 | A complex number whose cos... |
| cos02pilt1 26490 | Cosine is less than one be... |
| cosq34lt1 26491 | Cosine is less than one in... |
| efeq1 26492 | A complex number whose exp... |
| cosne0 26493 | The cosine function has no... |
| cosordlem 26494 | Lemma for ~ cosord . (Con... |
| cosord 26495 | Cosine is decreasing over ... |
| cos0pilt1 26496 | Cosine is between minus on... |
| cos11 26497 | Cosine is one-to-one over ... |
| sinord 26498 | Sine is increasing over th... |
| recosf1o 26499 | The cosine function is a b... |
| resinf1o 26500 | The sine function is a bij... |
| tanord1 26501 | The tangent function is st... |
| tanord 26502 | The tangent function is st... |
| tanregt0 26503 | The real part of the tange... |
| negpitopissre 26504 | The interval ` ( -u _pi (,... |
| efgh 26505 | The exponential function o... |
| efif1olem1 26506 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26507 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26508 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26509 | The exponential function o... |
| efif1o 26510 | The exponential function o... |
| efifo 26511 | The exponential function o... |
| eff1olem 26512 | The exponential function m... |
| eff1o 26513 | The exponential function m... |
| efabl 26514 | The image of a subgroup of... |
| efsubm 26515 | The image of a subgroup of... |
| circgrp 26516 | The circle group ` T ` is ... |
| circsubm 26517 | The circle group ` T ` is ... |
| logrn 26522 | The range of the natural l... |
| ellogrn 26523 | Write out the property ` A... |
| dflog2 26524 | The natural logarithm func... |
| relogrn 26525 | The range of the natural l... |
| logrncn 26526 | The range of the natural l... |
| eff1o2 26527 | The exponential function r... |
| logf1o 26528 | The natural logarithm func... |
| dfrelog 26529 | The natural logarithm func... |
| relogf1o 26530 | The natural logarithm func... |
| logrncl 26531 | Closure of the natural log... |
| logcl 26532 | Closure of the natural log... |
| logimcl 26533 | Closure of the imaginary p... |
| logcld 26534 | The logarithm of a nonzero... |
| logimcld 26535 | The imaginary part of the ... |
| logimclad 26536 | The imaginary part of the ... |
| abslogimle 26537 | The imaginary part of the ... |
| logrnaddcl 26538 | The range of the natural l... |
| relogcl 26539 | Closure of the natural log... |
| eflog 26540 | Relationship between the n... |
| logeq0im1 26541 | If the logarithm of a numb... |
| logccne0 26542 | The logarithm isn't 0 if i... |
| logne0 26543 | Logarithm of a non-1 posit... |
| reeflog 26544 | Relationship between the n... |
| logef 26545 | Relationship between the n... |
| relogef 26546 | Relationship between the n... |
| logeftb 26547 | Relationship between the n... |
| relogeftb 26548 | Relationship between the n... |
| log1 26549 | The natural logarithm of `... |
| loge 26550 | The natural logarithm of `... |
| logi 26551 | The natural logarithm of `... |
| logneg 26552 | The natural logarithm of a... |
| logm1 26553 | The natural logarithm of n... |
| lognegb 26554 | If a number has imaginary ... |
| relogoprlem 26555 | Lemma for ~ relogmul and ~... |
| relogmul 26556 | The natural logarithm of t... |
| relogdiv 26557 | The natural logarithm of t... |
| explog 26558 | Exponentiation of a nonzer... |
| reexplog 26559 | Exponentiation of a positi... |
| relogexp 26560 | The natural logarithm of p... |
| relog 26561 | Real part of a logarithm. ... |
| relogiso 26562 | The natural logarithm func... |
| reloggim 26563 | The natural logarithm is a... |
| logltb 26564 | The natural logarithm func... |
| logfac 26565 | The logarithm of a factori... |
| eflogeq 26566 | Solve an equation involvin... |
| logleb 26567 | Natural logarithm preserve... |
| rplogcl 26568 | Closure of the logarithm f... |
| logge0 26569 | The logarithm of a number ... |
| logcj 26570 | The natural logarithm dist... |
| efiarg 26571 | The exponential of the "ar... |
| cosargd 26572 | The cosine of the argument... |
| cosarg0d 26573 | The cosine of the argument... |
| argregt0 26574 | Closure of the argument of... |
| argrege0 26575 | Closure of the argument of... |
| argimgt0 26576 | Closure of the argument of... |
| argimlt0 26577 | Closure of the argument of... |
| logimul 26578 | Multiplying a number by ` ... |
| logneg2 26579 | The logarithm of the negat... |
| logmul2 26580 | Generalization of ~ relogm... |
| logdiv2 26581 | Generalization of ~ relogd... |
| abslogle 26582 | Bound on the magnitude of ... |
| tanarg 26583 | The basic relation between... |
| logdivlti 26584 | The ` log x / x ` function... |
| logdivlt 26585 | The ` log x / x ` function... |
| logdivle 26586 | The ` log x / x ` function... |
| relogcld 26587 | Closure of the natural log... |
| reeflogd 26588 | Relationship between the n... |
| relogmuld 26589 | The natural logarithm of t... |
| relogdivd 26590 | The natural logarithm of t... |
| logled 26591 | Natural logarithm preserve... |
| relogefd 26592 | Relationship between the n... |
| rplogcld 26593 | Closure of the logarithm f... |
| logge0d 26594 | The logarithm of a number ... |
| logge0b 26595 | The logarithm of a number ... |
| loggt0b 26596 | The logarithm of a number ... |
| logle1b 26597 | The logarithm of a number ... |
| loglt1b 26598 | The logarithm of a number ... |
| divlogrlim 26599 | The inverse logarithm func... |
| logno1 26600 | The logarithm function is ... |
| dvrelog 26601 | The derivative of the real... |
| relogcn 26602 | The real logarithm functio... |
| ellogdm 26603 | Elementhood in the "contin... |
| logdmn0 26604 | A number in the continuous... |
| logdmnrp 26605 | A number in the continuous... |
| logdmss 26606 | The continuity domain of `... |
| logcnlem2 26607 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26608 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26609 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26610 | Lemma for ~ logcn . (Cont... |
| logcn 26611 | The logarithm function is ... |
| dvloglem 26612 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26613 | The "continuous domain" of... |
| logf1o2 26614 | The logarithm maps its con... |
| dvlog 26615 | The derivative of the comp... |
| dvlog2lem 26616 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26617 | The derivative of the comp... |
| advlog 26618 | The antiderivative of the ... |
| advlogexp 26619 | The antiderivative of a po... |
| efopnlem1 26620 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26621 | Lemma for ~ efopn . (Cont... |
| efopn 26622 | The exponential map is an ... |
| logtayllem 26623 | Lemma for ~ logtayl . (Co... |
| logtayl 26624 | The Taylor series for ` -u... |
| logtaylsum 26625 | The Taylor series for ` -u... |
| logtayl2 26626 | Power series expression fo... |
| logccv 26627 | The natural logarithm func... |
| cxpval 26628 | Value of the complex power... |
| cxpef 26629 | Value of the complex power... |
| 0cxp 26630 | Value of the complex power... |
| cxpexpz 26631 | Relate the complex power f... |
| cxpexp 26632 | Relate the complex power f... |
| logcxp 26633 | Logarithm of a complex pow... |
| cxp0 26634 | Value of the complex power... |
| cxp1 26635 | Value of the complex power... |
| 1cxp 26636 | Value of the complex power... |
| ecxp 26637 | Write the exponential func... |
| cxpcl 26638 | Closure of the complex pow... |
| recxpcl 26639 | Real closure of the comple... |
| rpcxpcl 26640 | Positive real closure of t... |
| cxpne0 26641 | Complex exponentiation is ... |
| cxpeq0 26642 | Complex exponentiation is ... |
| cxpadd 26643 | Sum of exponents law for c... |
| cxpp1 26644 | Value of a nonzero complex... |
| cxpneg 26645 | Value of a complex number ... |
| cxpsub 26646 | Exponent subtraction law f... |
| cxpge0 26647 | Nonnegative exponentiation... |
| mulcxplem 26648 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26649 | Complex exponentiation of ... |
| cxprec 26650 | Complex exponentiation of ... |
| divcxp 26651 | Complex exponentiation of ... |
| cxpmul 26652 | Product of exponents law f... |
| cxpmul2 26653 | Product of exponents law f... |
| cxproot 26654 | The complex power function... |
| cxpmul2z 26655 | Generalize ~ cxpmul2 to ne... |
| abscxp 26656 | Absolute value of a power,... |
| abscxp2 26657 | Absolute value of a power,... |
| cxplt 26658 | Ordering property for comp... |
| cxple 26659 | Ordering property for comp... |
| cxplea 26660 | Ordering property for comp... |
| cxple2 26661 | Ordering property for comp... |
| cxplt2 26662 | Ordering property for comp... |
| cxple2a 26663 | Ordering property for comp... |
| cxplt3 26664 | Ordering property for comp... |
| cxple3 26665 | Ordering property for comp... |
| cxpsqrtlem 26666 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26667 | The complex exponential fu... |
| logsqrt 26668 | Logarithm of a square root... |
| cxp0d 26669 | Value of the complex power... |
| cxp1d 26670 | Value of the complex power... |
| 1cxpd 26671 | Value of the complex power... |
| cxpcld 26672 | Closure of the complex pow... |
| cxpmul2d 26673 | Product of exponents law f... |
| 0cxpd 26674 | Value of the complex power... |
| cxpexpzd 26675 | Relate the complex power f... |
| cxpefd 26676 | Value of the complex power... |
| cxpne0d 26677 | Complex exponentiation is ... |
| cxpp1d 26678 | Value of a nonzero complex... |
| cxpnegd 26679 | Value of a complex number ... |
| cxpmul2zd 26680 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26681 | Sum of exponents law for c... |
| cxpsubd 26682 | Exponent subtraction law f... |
| cxpltd 26683 | Ordering property for comp... |
| cxpled 26684 | Ordering property for comp... |
| cxplead 26685 | Ordering property for comp... |
| divcxpd 26686 | Complex exponentiation of ... |
| recxpcld 26687 | Positive real closure of t... |
| cxpge0d 26688 | Nonnegative exponentiation... |
| cxple2ad 26689 | Ordering property for comp... |
| cxplt2d 26690 | Ordering property for comp... |
| cxple2d 26691 | Ordering property for comp... |
| mulcxpd 26692 | Complex exponentiation of ... |
| recxpf1lem 26693 | Complex exponentiation on ... |
| cxpsqrtth 26694 | Square root theorem over t... |
| 2irrexpq 26695 | There exist irrational num... |
| cxprecd 26696 | Complex exponentiation of ... |
| rpcxpcld 26697 | Positive real closure of t... |
| logcxpd 26698 | Logarithm of a complex pow... |
| cxplt3d 26699 | Ordering property for comp... |
| cxple3d 26700 | Ordering property for comp... |
| cxpmuld 26701 | Product of exponents law f... |
| cxpgt0d 26702 | A positive real raised to ... |
| cxpcom 26703 | Commutative law for real e... |
| dvcxp1 26704 | The derivative of a comple... |
| dvcxp2 26705 | The derivative of a comple... |
| dvsqrt 26706 | The derivative of the real... |
| dvcncxp1 26707 | Derivative of complex powe... |
| dvcnsqrt 26708 | Derivative of square root ... |
| cxpcn 26709 | Domain of continuity of th... |
| cxpcnOLD 26710 | Obsolete version of ~ cxpc... |
| cxpcn2 26711 | Continuity of the complex ... |
| cxpcn3lem 26712 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26713 | Extend continuity of the c... |
| resqrtcn 26714 | Continuity of the real squ... |
| sqrtcn 26715 | Continuity of the square r... |
| cxpaddlelem 26716 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26717 | Ordering property for comp... |
| abscxpbnd 26718 | Bound on the absolute valu... |
| root1id 26719 | Property of an ` N ` -th r... |
| root1eq1 26720 | The only powers of an ` N ... |
| root1cj 26721 | Within the ` N ` -th roots... |
| cxpeq 26722 | Solve an equation involvin... |
| loglesqrt 26723 | An upper bound on the loga... |
| logreclem 26724 | Symmetry of the natural lo... |
| logrec 26725 | Logarithm of a reciprocal ... |
| logbval 26728 | Define the value of the ` ... |
| logbcl 26729 | General logarithm closure.... |
| logbid1 26730 | General logarithm is 1 whe... |
| logb1 26731 | The logarithm of ` 1 ` to ... |
| elogb 26732 | The general logarithm of a... |
| logbchbase 26733 | Change of base for logarit... |
| relogbval 26734 | Value of the general logar... |
| relogbcl 26735 | Closure of the general log... |
| relogbzcl 26736 | Closure of the general log... |
| relogbreexp 26737 | Power law for the general ... |
| relogbzexp 26738 | Power law for the general ... |
| relogbmul 26739 | The logarithm of the produ... |
| relogbmulexp 26740 | The logarithm of the produ... |
| relogbdiv 26741 | The logarithm of the quoti... |
| relogbexp 26742 | Identity law for general l... |
| nnlogbexp 26743 | Identity law for general l... |
| logbrec 26744 | Logarithm of a reciprocal ... |
| logbleb 26745 | The general logarithm func... |
| logblt 26746 | The general logarithm func... |
| relogbcxp 26747 | Identity law for the gener... |
| cxplogb 26748 | Identity law for the gener... |
| relogbcxpb 26749 | The logarithm is the inver... |
| logbmpt 26750 | The general logarithm to a... |
| logbf 26751 | The general logarithm to a... |
| logbfval 26752 | The general logarithm of a... |
| relogbf 26753 | The general logarithm to a... |
| logblog 26754 | The general logarithm to t... |
| logbgt0b 26755 | The logarithm of a positiv... |
| logbgcd1irr 26756 | The logarithm of an intege... |
| 2logb9irr 26757 | Example for ~ logbgcd1irr ... |
| logbprmirr 26758 | The logarithm of a prime t... |
| 2logb3irr 26759 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26760 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26761 | The square root of two to ... |
| 2irrexpqALT 26762 | Alternate proof of ~ 2irre... |
| angval 26763 | Define the angle function,... |
| angcan 26764 | Cancel a constant multipli... |
| angneg 26765 | Cancel a negative sign in ... |
| angvald 26766 | The (signed) angle between... |
| angcld 26767 | The (signed) angle between... |
| angrteqvd 26768 | Two vectors are at a right... |
| cosangneg2d 26769 | The cosine of the angle be... |
| angrtmuld 26770 | Perpendicularity of two ve... |
| ang180lem1 26771 | Lemma for ~ ang180 . Show... |
| ang180lem2 26772 | Lemma for ~ ang180 . Show... |
| ang180lem3 26773 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26774 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26775 | Lemma for ~ ang180 : Redu... |
| ang180 26776 | The sum of angles ` m A B ... |
| lawcoslem1 26777 | Lemma for ~ lawcos . Here... |
| lawcos 26778 | Law of cosines (also known... |
| pythag 26779 | Pythagorean theorem. Give... |
| isosctrlem1 26780 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26781 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26782 | Lemma for ~ isosctr . Cor... |
| isosctr 26783 | Isosceles triangle theorem... |
| ssscongptld 26784 | If two triangles have equa... |
| affineequiv 26785 | Equivalence between two wa... |
| affineequiv2 26786 | Equivalence between two wa... |
| affineequiv3 26787 | Equivalence between two wa... |
| affineequiv4 26788 | Equivalence between two wa... |
| affineequivne 26789 | Equivalence between two wa... |
| angpieqvdlem 26790 | Equivalence used in the pr... |
| angpieqvdlem2 26791 | Equivalence used in ~ angp... |
| angpined 26792 | If the angle at ABC is ` _... |
| angpieqvd 26793 | The angle ABC is ` _pi ` i... |
| chordthmlem 26794 | If ` M ` is the midpoint o... |
| chordthmlem2 26795 | If M is the midpoint of AB... |
| chordthmlem3 26796 | If M is the midpoint of AB... |
| chordthmlem4 26797 | If P is on the segment AB ... |
| chordthmlem5 26798 | If P is on the segment AB ... |
| chordthm 26799 | The intersecting chords th... |
| heron 26800 | Heron's formula gives the ... |
| quad2 26801 | The quadratic equation, wi... |
| quad 26802 | The quadratic equation. (... |
| 1cubrlem 26803 | The cube roots of unity. ... |
| 1cubr 26804 | The cube roots of unity. ... |
| dcubic1lem 26805 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26806 | Reverse direction of ~ dcu... |
| dcubic1 26807 | Forward direction of ~ dcu... |
| dcubic 26808 | Solutions to the depressed... |
| mcubic 26809 | Solutions to a monic cubic... |
| cubic2 26810 | The solution to the genera... |
| cubic 26811 | The cubic equation, which ... |
| binom4 26812 | Work out a quartic binomia... |
| dquartlem1 26813 | Lemma for ~ dquart . (Con... |
| dquartlem2 26814 | Lemma for ~ dquart . (Con... |
| dquart 26815 | Solve a depressed quartic ... |
| quart1cl 26816 | Closure lemmas for ~ quart... |
| quart1lem 26817 | Lemma for ~ quart1 . (Con... |
| quart1 26818 | Depress a quartic equation... |
| quartlem1 26819 | Lemma for ~ quart . (Cont... |
| quartlem2 26820 | Closure lemmas for ~ quart... |
| quartlem3 26821 | Closure lemmas for ~ quart... |
| quartlem4 26822 | Closure lemmas for ~ quart... |
| quart 26823 | The quartic equation, writ... |
| asinlem 26830 | The argument to the logari... |
| asinlem2 26831 | The argument to the logari... |
| asinlem3a 26832 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26833 | The argument to the logari... |
| asinf 26834 | Domain and codomain of the... |
| asincl 26835 | Closure for the arcsin fun... |
| acosf 26836 | Domain and codoamin of the... |
| acoscl 26837 | Closure for the arccos fun... |
| atandm 26838 | Since the property is a li... |
| atandm2 26839 | This form of ~ atandm is a... |
| atandm3 26840 | A compact form of ~ atandm... |
| atandm4 26841 | A compact form of ~ atandm... |
| atanf 26842 | Domain and codoamin of the... |
| atancl 26843 | Closure for the arctan fun... |
| asinval 26844 | Value of the arcsin functi... |
| acosval 26845 | Value of the arccos functi... |
| atanval 26846 | Value of the arctan functi... |
| atanre 26847 | A real number is in the do... |
| asinneg 26848 | The arcsine function is od... |
| acosneg 26849 | The negative symmetry rela... |
| efiasin 26850 | The exponential of the arc... |
| sinasin 26851 | The arcsine function is an... |
| cosacos 26852 | The arccosine function is ... |
| asinsinlem 26853 | Lemma for ~ asinsin . (Co... |
| asinsin 26854 | The arcsine function compo... |
| acoscos 26855 | The arccosine function is ... |
| asin1 26856 | The arcsine of ` 1 ` is ` ... |
| acos1 26857 | The arccosine of ` 1 ` is ... |
| reasinsin 26858 | The arcsine function compo... |
| asinsinb 26859 | Relationship between sine ... |
| acoscosb 26860 | Relationship between cosin... |
| asinbnd 26861 | The arcsine function has r... |
| acosbnd 26862 | The arccosine function has... |
| asinrebnd 26863 | Bounds on the arcsine func... |
| asinrecl 26864 | The arcsine function is re... |
| acosrecl 26865 | The arccosine function is ... |
| cosasin 26866 | The cosine of the arcsine ... |
| sinacos 26867 | The sine of the arccosine ... |
| atandmneg 26868 | The domain of the arctange... |
| atanneg 26869 | The arctangent function is... |
| atan0 26870 | The arctangent of zero is ... |
| atandmcj 26871 | The arctangent function di... |
| atancj 26872 | The arctangent function di... |
| atanrecl 26873 | The arctangent function is... |
| efiatan 26874 | Value of the exponential o... |
| atanlogaddlem 26875 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26876 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26877 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26878 | A variation on ~ atanlogad... |
| efiatan2 26879 | Value of the exponential o... |
| 2efiatan 26880 | Value of the exponential o... |
| tanatan 26881 | The arctangent function is... |
| atandmtan 26882 | The tangent function has r... |
| cosatan 26883 | The cosine of an arctangen... |
| cosatanne0 26884 | The arctangent function ha... |
| atantan 26885 | The arctangent function is... |
| atantanb 26886 | Relationship between tange... |
| atanbndlem 26887 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26888 | The arctangent function is... |
| atanord 26889 | The arctangent function is... |
| atan1 26890 | The arctangent of ` 1 ` is... |
| bndatandm 26891 | A point in the open unit d... |
| atans 26892 | The "domain of continuity"... |
| atans2 26893 | It suffices to show that `... |
| atansopn 26894 | The domain of continuity o... |
| atansssdm 26895 | The domain of continuity o... |
| ressatans 26896 | The real number line is a ... |
| dvatan 26897 | The derivative of the arct... |
| atancn 26898 | The arctangent is a contin... |
| atantayl 26899 | The Taylor series for ` ar... |
| atantayl2 26900 | The Taylor series for ` ar... |
| atantayl3 26901 | The Taylor series for ` ar... |
| leibpilem1 26902 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26903 | The Leibniz formula for ` ... |
| leibpi 26904 | The Leibniz formula for ` ... |
| leibpisum 26905 | The Leibniz formula for ` ... |
| log2cnv 26906 | Using the Taylor series fo... |
| log2tlbnd 26907 | Bound the error term in th... |
| log2ublem1 26908 | Lemma for ~ log2ub . The ... |
| log2ublem2 26909 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26910 | Lemma for ~ log2ub . In d... |
| log2ub 26911 | ` log 2 ` is less than ` 2... |
| log2le1 26912 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26913 | Lemma for ~ birthday . (C... |
| birthdaylem2 26914 | For general ` N ` and ` K ... |
| birthdaylem3 26915 | For general ` N ` and ` K ... |
| birthday 26916 | The Birthday Problem. The... |
| dmarea 26919 | The domain of the area fun... |
| areambl 26920 | The fibers of a measurable... |
| areass 26921 | A measurable region is a s... |
| dfarea 26922 | Rewrite ~ df-area self-ref... |
| areaf 26923 | Area measurement is a func... |
| areacl 26924 | The area of a measurable r... |
| areage0 26925 | The area of a measurable r... |
| areaval 26926 | The area of a measurable r... |
| rlimcnp 26927 | Relate a limit of a real-v... |
| rlimcnp2 26928 | Relate a limit of a real-v... |
| rlimcnp3 26929 | Relate a limit of a real-v... |
| xrlimcnp 26930 | Relate a limit of a real-v... |
| efrlim 26931 | The limit of the sequence ... |
| efrlimOLD 26932 | Obsolete version of ~ efrl... |
| dfef2 26933 | The limit of the sequence ... |
| cxplim 26934 | A power to a negative expo... |
| sqrtlim 26935 | The inverse square root fu... |
| rlimcxp 26936 | Any power to a positive ex... |
| o1cxp 26937 | An eventually bounded func... |
| cxp2limlem 26938 | A linear factor grows slow... |
| cxp2lim 26939 | Any power grows slower tha... |
| cxploglim 26940 | The logarithm grows slower... |
| cxploglim2 26941 | Every power of the logarit... |
| divsqrtsumlem 26942 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26943 | The function ` F ` used in... |
| divsqrsum 26944 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26945 | A bound on the distance of... |
| divsqrtsumo1 26946 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26947 | Closure of a 0-1 linear co... |
| scvxcvx 26948 | A strictly convex function... |
| jensenlem1 26949 | Lemma for ~ jensen . (Con... |
| jensenlem2 26950 | Lemma for ~ jensen . (Con... |
| jensen 26951 | Jensen's inequality, a fin... |
| amgmlem 26952 | Lemma for ~ amgm . (Contr... |
| amgm 26953 | Inequality of arithmetic a... |
| logdifbnd 26956 | Bound on the difference of... |
| logdiflbnd 26957 | Lower bound on the differe... |
| emcllem1 26958 | Lemma for ~ emcl . The se... |
| emcllem2 26959 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26960 | Lemma for ~ emcl . The fu... |
| emcllem4 26961 | Lemma for ~ emcl . The di... |
| emcllem5 26962 | Lemma for ~ emcl . The pa... |
| emcllem6 26963 | Lemma for ~ emcl . By the... |
| emcllem7 26964 | Lemma for ~ emcl and ~ har... |
| emcl 26965 | Closure and bounds for the... |
| harmonicbnd 26966 | A bound on the harmonic se... |
| harmonicbnd2 26967 | A bound on the harmonic se... |
| emre 26968 | The Euler-Mascheroni const... |
| emgt0 26969 | The Euler-Mascheroni const... |
| harmonicbnd3 26970 | A bound on the harmonic se... |
| harmoniclbnd 26971 | A bound on the harmonic se... |
| harmonicubnd 26972 | A bound on the harmonic se... |
| harmonicbnd4 26973 | The asymptotic behavior of... |
| fsumharmonic 26974 | Bound a finite sum based o... |
| zetacvg 26977 | The zeta series is converg... |
| eldmgm 26984 | Elementhood in the set of ... |
| dmgmaddn0 26985 | If ` A ` is not a nonposit... |
| dmlogdmgm 26986 | If ` A ` is in the continu... |
| rpdmgm 26987 | A positive real number is ... |
| dmgmn0 26988 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26989 | If ` A ` is not a nonposit... |
| dmgmdivn0 26990 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26991 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26992 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26993 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26994 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26995 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26996 | The series ` G ` is unifor... |
| lgamgulm 26997 | The series ` G ` is unifor... |
| lgamgulm2 26998 | Rewrite the limit of the s... |
| lgambdd 26999 | The log-Gamma function is ... |
| lgamucov 27000 | The ` U ` regions used in ... |
| lgamucov2 27001 | The ` U ` regions used in ... |
| lgamcvglem 27002 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27003 | The log-Gamma function is ... |
| lgamf 27004 | The log-Gamma function is ... |
| gamf 27005 | The Gamma function is a co... |
| gamcl 27006 | The exponential of the log... |
| eflgam 27007 | The exponential of the log... |
| gamne0 27008 | The Gamma function is neve... |
| igamval 27009 | Value of the inverse Gamma... |
| igamz 27010 | Value of the inverse Gamma... |
| igamgam 27011 | Value of the inverse Gamma... |
| igamlgam 27012 | Value of the inverse Gamma... |
| igamf 27013 | Closure of the inverse Gam... |
| igamcl 27014 | Closure of the inverse Gam... |
| gamigam 27015 | The Gamma function is the ... |
| lgamcvg 27016 | The series ` G ` converges... |
| lgamcvg2 27017 | The series ` G ` converges... |
| gamcvg 27018 | The pointwise exponential ... |
| lgamp1 27019 | The functional equation of... |
| gamp1 27020 | The functional equation of... |
| gamcvg2lem 27021 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27022 | An infinite product expres... |
| regamcl 27023 | The Gamma function is real... |
| relgamcl 27024 | The log-Gamma function is ... |
| rpgamcl 27025 | The log-Gamma function is ... |
| lgam1 27026 | The log-Gamma function at ... |
| gam1 27027 | The log-Gamma function at ... |
| facgam 27028 | The Gamma function general... |
| gamfac 27029 | The Gamma function general... |
| wilthlem1 27030 | The only elements that are... |
| wilthlem2 27031 | Lemma for ~ wilth : induct... |
| wilthlem3 27032 | Lemma for ~ wilth . Here ... |
| wilth 27033 | Wilson's theorem. A numbe... |
| wilthimp 27034 | The forward implication of... |
| ftalem1 27035 | Lemma for ~ fta : "growth... |
| ftalem2 27036 | Lemma for ~ fta . There e... |
| ftalem3 27037 | Lemma for ~ fta . There e... |
| ftalem4 27038 | Lemma for ~ fta : Closure... |
| ftalem5 27039 | Lemma for ~ fta : Main pr... |
| ftalem6 27040 | Lemma for ~ fta : Dischar... |
| ftalem7 27041 | Lemma for ~ fta . Shift t... |
| fta 27042 | The Fundamental Theorem of... |
| basellem1 27043 | Lemma for ~ basel . Closu... |
| basellem2 27044 | Lemma for ~ basel . Show ... |
| basellem3 27045 | Lemma for ~ basel . Using... |
| basellem4 27046 | Lemma for ~ basel . By ~ ... |
| basellem5 27047 | Lemma for ~ basel . Using... |
| basellem6 27048 | Lemma for ~ basel . The f... |
| basellem7 27049 | Lemma for ~ basel . The f... |
| basellem8 27050 | Lemma for ~ basel . The f... |
| basellem9 27051 | Lemma for ~ basel . Since... |
| basel 27052 | The sum of the inverse squ... |
| efnnfsumcl 27065 | Finite sum closure in the ... |
| ppisval 27066 | The set of primes less tha... |
| ppisval2 27067 | The set of primes less tha... |
| ppifi 27068 | The set of primes less tha... |
| prmdvdsfi 27069 | The set of prime divisors ... |
| chtf 27070 | Domain and codoamin of the... |
| chtcl 27071 | Real closure of the Chebys... |
| chtval 27072 | Value of the Chebyshev fun... |
| efchtcl 27073 | The Chebyshev function is ... |
| chtge0 27074 | The Chebyshev function is ... |
| vmaval 27075 | Value of the von Mangoldt ... |
| isppw 27076 | Two ways to say that ` A `... |
| isppw2 27077 | Two ways to say that ` A `... |
| vmappw 27078 | Value of the von Mangoldt ... |
| vmaprm 27079 | Value of the von Mangoldt ... |
| vmacl 27080 | Closure for the von Mangol... |
| vmaf 27081 | Functionality of the von M... |
| efvmacl 27082 | The von Mangoldt is closed... |
| vmage0 27083 | The von Mangoldt function ... |
| chpval 27084 | Value of the second Chebys... |
| chpf 27085 | Functionality of the secon... |
| chpcl 27086 | Closure for the second Che... |
| efchpcl 27087 | The second Chebyshev funct... |
| chpge0 27088 | The second Chebyshev funct... |
| ppival 27089 | Value of the prime-countin... |
| ppival2 27090 | Value of the prime-countin... |
| ppival2g 27091 | Value of the prime-countin... |
| ppif 27092 | Domain and codomain of the... |
| ppicl 27093 | Real closure of the prime-... |
| muval 27094 | The value of the Möbi... |
| muval1 27095 | The value of the Möbi... |
| muval2 27096 | The value of the Möbi... |
| isnsqf 27097 | Two ways to say that a num... |
| issqf 27098 | Two ways to say that a num... |
| sqfpc 27099 | The prime count of a squar... |
| dvdssqf 27100 | A divisor of a squarefree ... |
| sqf11 27101 | A squarefree number is com... |
| muf 27102 | The Möbius function i... |
| mucl 27103 | Closure of the Möbius... |
| sgmval 27104 | The value of the divisor f... |
| sgmval2 27105 | The value of the divisor f... |
| 0sgm 27106 | The value of the sum-of-di... |
| sgmf 27107 | The divisor function is a ... |
| sgmcl 27108 | Closure of the divisor fun... |
| sgmnncl 27109 | Closure of the divisor fun... |
| mule1 27110 | The Möbius function t... |
| chtfl 27111 | The Chebyshev function doe... |
| chpfl 27112 | The second Chebyshev funct... |
| ppiprm 27113 | The prime-counting functio... |
| ppinprm 27114 | The prime-counting functio... |
| chtprm 27115 | The Chebyshev function at ... |
| chtnprm 27116 | The Chebyshev function at ... |
| chpp1 27117 | The second Chebyshev funct... |
| chtwordi 27118 | The Chebyshev function is ... |
| chpwordi 27119 | The second Chebyshev funct... |
| chtdif 27120 | The difference of the Cheb... |
| efchtdvds 27121 | The exponentiated Chebyshe... |
| ppifl 27122 | The prime-counting functio... |
| ppip1le 27123 | The prime-counting functio... |
| ppiwordi 27124 | The prime-counting functio... |
| ppidif 27125 | The difference of the prim... |
| ppi1 27126 | The prime-counting functio... |
| cht1 27127 | The Chebyshev function at ... |
| vma1 27128 | The von Mangoldt function ... |
| chp1 27129 | The second Chebyshev funct... |
| ppi1i 27130 | Inference form of ~ ppiprm... |
| ppi2i 27131 | Inference form of ~ ppinpr... |
| ppi2 27132 | The prime-counting functio... |
| ppi3 27133 | The prime-counting functio... |
| cht2 27134 | The Chebyshev function at ... |
| cht3 27135 | The Chebyshev function at ... |
| ppinncl 27136 | Closure of the prime-count... |
| chtrpcl 27137 | Closure of the Chebyshev f... |
| ppieq0 27138 | The prime-counting functio... |
| ppiltx 27139 | The prime-counting functio... |
| prmorcht 27140 | Relate the primorial (prod... |
| mumullem1 27141 | Lemma for ~ mumul . A mul... |
| mumullem2 27142 | Lemma for ~ mumul . The p... |
| mumul 27143 | The Möbius function i... |
| sqff1o 27144 | There is a bijection from ... |
| fsumdvdsdiaglem 27145 | A "diagonal commutation" o... |
| fsumdvdsdiag 27146 | A "diagonal commutation" o... |
| fsumdvdscom 27147 | A double commutation of di... |
| dvdsppwf1o 27148 | A bijection from the divis... |
| dvdsflf1o 27149 | A bijection from the numbe... |
| dvdsflsumcom 27150 | A sum commutation from ` s... |
| fsumfldivdiaglem 27151 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27152 | The right-hand side of ~ d... |
| musum 27153 | The sum of the Möbius... |
| musumsum 27154 | Evaluate a collapsing sum ... |
| muinv 27155 | The Möbius inversion ... |
| mpodvdsmulf1o 27156 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27157 | Product of two divisor sum... |
| dvdsmulf1o 27158 | If ` M ` and ` N ` are two... |
| fsumdvdsmulOLD 27159 | Obsolete version of ~ fsum... |
| sgmppw 27160 | The value of the divisor f... |
| 0sgmppw 27161 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27162 | The sum of divisors for a ... |
| 1sgm2ppw 27163 | The sum of the divisors of... |
| sgmmul 27164 | The divisor function for f... |
| ppiublem1 27165 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27166 | A prime greater than ` 3 `... |
| ppiub 27167 | An upper bound on the prim... |
| vmalelog 27168 | The von Mangoldt function ... |
| chtlepsi 27169 | The first Chebyshev functi... |
| chprpcl 27170 | Closure of the second Cheb... |
| chpeq0 27171 | The second Chebyshev funct... |
| chteq0 27172 | The first Chebyshev functi... |
| chtleppi 27173 | Upper bound on the ` theta... |
| chtublem 27174 | Lemma for ~ chtub . (Cont... |
| chtub 27175 | An upper bound on the Cheb... |
| fsumvma 27176 | Rewrite a sum over the von... |
| fsumvma2 27177 | Apply ~ fsumvma for the co... |
| pclogsum 27178 | The logarithmic analogue o... |
| vmasum 27179 | The sum of the von Mangold... |
| logfac2 27180 | Another expression for the... |
| chpval2 27181 | Express the second Chebysh... |
| chpchtsum 27182 | The second Chebyshev funct... |
| chpub 27183 | An upper bound on the seco... |
| logfacubnd 27184 | A simple upper bound on th... |
| logfaclbnd 27185 | A lower bound on the logar... |
| logfacbnd3 27186 | Show the stronger statemen... |
| logfacrlim 27187 | Combine the estimates ~ lo... |
| logexprlim 27188 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27189 | Write out ~ logfacrlim as ... |
| mersenne 27190 | A Mersenne prime is a prim... |
| perfect1 27191 | Euclid's contribution to t... |
| perfectlem1 27192 | Lemma for ~ perfect . (Co... |
| perfectlem2 27193 | Lemma for ~ perfect . (Co... |
| perfect 27194 | The Euclid-Euler theorem, ... |
| dchrval 27197 | Value of the group of Diri... |
| dchrbas 27198 | Base set of the group of D... |
| dchrelbas 27199 | A Dirichlet character is a... |
| dchrelbas2 27200 | A Dirichlet character is a... |
| dchrelbas3 27201 | A Dirichlet character is a... |
| dchrelbasd 27202 | A Dirichlet character is a... |
| dchrrcl 27203 | Reverse closure for a Diri... |
| dchrmhm 27204 | A Dirichlet character is a... |
| dchrf 27205 | A Dirichlet character is a... |
| dchrelbas4 27206 | A Dirichlet character is a... |
| dchrzrh1 27207 | Value of a Dirichlet chara... |
| dchrzrhcl 27208 | A Dirichlet character take... |
| dchrzrhmul 27209 | A Dirichlet character is c... |
| dchrplusg 27210 | Group operation on the gro... |
| dchrmul 27211 | Group operation on the gro... |
| dchrmulcl 27212 | Closure of the group opera... |
| dchrn0 27213 | A Dirichlet character is n... |
| dchr1cl 27214 | Closure of the principal D... |
| dchrmullid 27215 | Left identity for the prin... |
| dchrinvcl 27216 | Closure of the group inver... |
| dchrabl 27217 | The set of Dirichlet chara... |
| dchrfi 27218 | The group of Dirichlet cha... |
| dchrghm 27219 | A Dirichlet character rest... |
| dchr1 27220 | Value of the principal Dir... |
| dchreq 27221 | A Dirichlet character is d... |
| dchrresb 27222 | A Dirichlet character is d... |
| dchrabs 27223 | A Dirichlet character take... |
| dchrinv 27224 | The inverse of a Dirichlet... |
| dchrabs2 27225 | A Dirichlet character take... |
| dchr1re 27226 | The principal Dirichlet ch... |
| dchrptlem1 27227 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27228 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27229 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27230 | For any element other than... |
| dchrsum2 27231 | An orthogonality relation ... |
| dchrsum 27232 | An orthogonality relation ... |
| sumdchr2 27233 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27234 | There are exactly ` phi ( ... |
| sumdchr 27235 | An orthogonality relation ... |
| dchr2sum 27236 | An orthogonality relation ... |
| sum2dchr 27237 | An orthogonality relation ... |
| bcctr 27238 | Value of the central binom... |
| pcbcctr 27239 | Prime count of a central b... |
| bcmono 27240 | The binomial coefficient i... |
| bcmax 27241 | The binomial coefficient t... |
| bcp1ctr 27242 | Ratio of two central binom... |
| bclbnd 27243 | A bound on the binomial co... |
| efexple 27244 | Convert a bound on a power... |
| bpos1lem 27245 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27246 | Bertrand's postulate, chec... |
| bposlem1 27247 | An upper bound on the prim... |
| bposlem2 27248 | There are no odd primes in... |
| bposlem3 27249 | Lemma for ~ bpos . Since ... |
| bposlem4 27250 | Lemma for ~ bpos . (Contr... |
| bposlem5 27251 | Lemma for ~ bpos . Bound ... |
| bposlem6 27252 | Lemma for ~ bpos . By usi... |
| bposlem7 27253 | Lemma for ~ bpos . The fu... |
| bposlem8 27254 | Lemma for ~ bpos . Evalua... |
| bposlem9 27255 | Lemma for ~ bpos . Derive... |
| bpos 27256 | Bertrand's postulate: ther... |
| zabsle1 27259 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27260 | When ` a ` is coprime to t... |
| lgslem2 27261 | The set ` Z ` of all integ... |
| lgslem3 27262 | The set ` Z ` of all integ... |
| lgslem4 27263 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27264 | Value of the Legendre symb... |
| lgsfval 27265 | Value of the function ` F ... |
| lgsfcl2 27266 | The function ` F ` is clos... |
| lgscllem 27267 | The Legendre symbol is an ... |
| lgsfcl 27268 | Closure of the function ` ... |
| lgsfle1 27269 | The function ` F ` has mag... |
| lgsval2lem 27270 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27271 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27272 | The Legendre symbol is an ... |
| lgs0 27273 | The Legendre symbol when t... |
| lgscl 27274 | The Legendre symbol is an ... |
| lgsle1 27275 | The Legendre symbol has ab... |
| lgsval2 27276 | The Legendre symbol at a p... |
| lgs2 27277 | The Legendre symbol at ` 2... |
| lgsval3 27278 | The Legendre symbol at an ... |
| lgsvalmod 27279 | The Legendre symbol is equ... |
| lgsval4 27280 | Restate ~ lgsval for nonze... |
| lgsfcl3 27281 | Closure of the function ` ... |
| lgsval4a 27282 | Same as ~ lgsval4 for posi... |
| lgscl1 27283 | The value of the Legendre ... |
| lgsneg 27284 | The Legendre symbol is eit... |
| lgsneg1 27285 | The Legendre symbol for no... |
| lgsmod 27286 | The Legendre (Jacobi) symb... |
| lgsdilem 27287 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27288 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27289 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27290 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27291 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27292 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27293 | The Legendre symbol is com... |
| lgsdirprm 27294 | The Legendre symbol is com... |
| lgsdir 27295 | The Legendre symbol is com... |
| lgsdilem2 27296 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27297 | The Legendre symbol is com... |
| lgsne0 27298 | The Legendre symbol is non... |
| lgsabs1 27299 | The Legendre symbol is non... |
| lgssq 27300 | The Legendre symbol at a s... |
| lgssq2 27301 | The Legendre symbol at a s... |
| lgsprme0 27302 | The Legendre symbol at any... |
| 1lgs 27303 | The Legendre symbol at ` 1... |
| lgs1 27304 | The Legendre symbol at ` 1... |
| lgsmodeq 27305 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27306 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27307 | Variation on ~ lgsdir vali... |
| lgsdinn0 27308 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27309 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27310 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27311 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27312 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27313 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27314 | The Legendre symbol for od... |
| lgsqrmod 27315 | If the Legendre symbol of ... |
| lgsqrmodndvds 27316 | If the Legendre symbol of ... |
| lgsdchrval 27317 | The Legendre symbol functi... |
| lgsdchr 27318 | The Legendre symbol functi... |
| gausslemma2dlem0a 27319 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27320 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27321 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27322 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27323 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27324 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27325 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27326 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27327 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27328 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27329 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27330 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27331 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27332 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27333 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27334 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27335 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27336 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27337 | Gauss' Lemma (see also the... |
| lgseisenlem1 27338 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27339 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27340 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27341 | Lemma for ~ lgseisen . Th... |
| lgseisen 27342 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27343 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27344 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27345 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27346 | The Law of Quadratic Recip... |
| lgsquad2lem1 27347 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27348 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27349 | Extend ~ lgsquad to coprim... |
| lgsquad3 27350 | Extend ~ lgsquad2 to integ... |
| m1lgs 27351 | The first supplement to th... |
| 2lgslem1a1 27352 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27353 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27354 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27355 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27356 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27357 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27358 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27359 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27360 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27361 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27362 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27363 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27364 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27365 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27366 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27367 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27368 | The Legendre symbol for ` ... |
| 2lgslem4 27369 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27370 | The second supplement to t... |
| 2lgsoddprmlem1 27371 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27372 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27373 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27374 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27375 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27376 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27377 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27378 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27379 | The second supplement to t... |
| 2sqlem1 27380 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27381 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27382 | Fibonacci's identity (actu... |
| 2sqlem3 27383 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27384 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27385 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27386 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27387 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27388 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27389 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27390 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27391 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27392 | Lemma for ~ 2sq . (Contri... |
| 2sq 27393 | All primes of the form ` 4... |
| 2sqblem 27394 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27395 | The converse to ~ 2sq . (... |
| 2sq2 27396 | ` 2 ` is the sum of square... |
| 2sqn0 27397 | If the sum of two squares ... |
| 2sqcoprm 27398 | If the sum of two squares ... |
| 2sqmod 27399 | Given two decompositions o... |
| 2sqmo 27400 | There exists at most one d... |
| 2sqnn0 27401 | All primes of the form ` 4... |
| 2sqnn 27402 | All primes of the form ` 4... |
| addsq2reu 27403 | For each complex number ` ... |
| addsqn2reu 27404 | For each complex number ` ... |
| addsqrexnreu 27405 | For each complex number, t... |
| addsqnreup 27406 | There is no unique decompo... |
| addsq2nreurex 27407 | For each complex number ` ... |
| addsqn2reurex2 27408 | For each complex number ` ... |
| 2sqreulem1 27409 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27410 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27411 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27412 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27413 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27414 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27415 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27416 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27417 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27418 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27419 | There exists a unique deco... |
| 2sqreunn 27420 | There exists a unique deco... |
| 2sqreult 27421 | There exists a unique deco... |
| 2sqreultb 27422 | There exists a unique deco... |
| 2sqreunnlt 27423 | There exists a unique deco... |
| 2sqreunnltb 27424 | There exists a unique deco... |
| 2sqreuop 27425 | There exists a unique deco... |
| 2sqreuopnn 27426 | There exists a unique deco... |
| 2sqreuoplt 27427 | There exists a unique deco... |
| 2sqreuopltb 27428 | There exists a unique deco... |
| 2sqreuopnnlt 27429 | There exists a unique deco... |
| 2sqreuopnnltb 27430 | There exists a unique deco... |
| 2sqreuopb 27431 | There exists a unique deco... |
| chebbnd1lem1 27432 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27433 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27434 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27435 | The Chebyshev bound: The ... |
| chtppilimlem1 27436 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27437 | Lemma for ~ chtppilim . (... |
| chtppilim 27438 | The ` theta ` function is ... |
| chto1ub 27439 | The ` theta ` function is ... |
| chebbnd2 27440 | The Chebyshev bound, part ... |
| chto1lb 27441 | The ` theta ` function is ... |
| chpchtlim 27442 | The ` psi ` and ` theta ` ... |
| chpo1ub 27443 | The ` psi ` function is up... |
| chpo1ubb 27444 | The ` psi ` function is up... |
| vmadivsum 27445 | The sum of the von Mangold... |
| vmadivsumb 27446 | Give a total bound on the ... |
| rplogsumlem1 27447 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27448 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27449 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27450 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27451 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27452 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27453 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27454 | Lemma for ~ dchrisum . Le... |
| dchrisum 27455 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27456 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27457 | The sum of the Möbius... |
| dchrvmasumlem1 27458 | An alternative expression ... |
| dchrvmasum2lem 27459 | Give an expression for ` l... |
| dchrvmasum2if 27460 | Combine the results of ~ d... |
| dchrvmasumlem2 27461 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27462 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27463 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27464 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27465 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27466 | An asymptotic approximatio... |
| dchrvmaeq0 27467 | The set ` W ` is the colle... |
| dchrisum0fval 27468 | Value of the function ` F ... |
| dchrisum0fmul 27469 | The function ` F ` , the d... |
| dchrisum0ff 27470 | The function ` F ` is a re... |
| dchrisum0flblem1 27471 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27472 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27473 | The divisor sum of a real ... |
| dchrisum0fno1 27474 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27475 | A partial result along the... |
| dchrisum0re 27476 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27477 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27478 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27479 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27480 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27481 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27482 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27483 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27484 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27485 | The sum of the Möbius... |
| dchrvmasumlem 27486 | The sum of the Möbius... |
| dchrmusum 27487 | The sum of the Möbius... |
| dchrvmasum 27488 | The sum of the von Mangold... |
| rpvmasum 27489 | The sum of the von Mangold... |
| rplogsum 27490 | The sum of ` log p / p ` o... |
| dirith2 27491 | Dirichlet's theorem: there... |
| dirith 27492 | Dirichlet's theorem: there... |
| mudivsum 27493 | Asymptotic formula for ` s... |
| mulogsumlem 27494 | Lemma for ~ mulogsum . (C... |
| mulogsum 27495 | Asymptotic formula for ... |
| logdivsum 27496 | Asymptotic analysis of ... |
| mulog2sumlem1 27497 | Asymptotic formula for ... |
| mulog2sumlem2 27498 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27499 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27500 | Asymptotic formula for ... |
| vmalogdivsum2 27501 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27502 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27503 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27504 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27505 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27506 | The Möbius inverse of... |
| log2sumbnd 27507 | Bound on the difference be... |
| selberglem1 27508 | Lemma for ~ selberg . Est... |
| selberglem2 27509 | Lemma for ~ selberg . (Co... |
| selberglem3 27510 | Lemma for ~ selberg . Est... |
| selberg 27511 | Selberg's symmetry formula... |
| selbergb 27512 | Convert eventual boundedne... |
| selberg2lem 27513 | Lemma for ~ selberg2 . Eq... |
| selberg2 27514 | Selberg's symmetry formula... |
| selberg2b 27515 | Convert eventual boundedne... |
| chpdifbndlem1 27516 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27517 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27518 | A bound on the difference ... |
| logdivbnd 27519 | A bound on a sum of logs, ... |
| selberg3lem1 27520 | Introduce a log weighting ... |
| selberg3lem2 27521 | Lemma for ~ selberg3 . Eq... |
| selberg3 27522 | Introduce a log weighting ... |
| selberg4lem1 27523 | Lemma for ~ selberg4 . Eq... |
| selberg4 27524 | The Selberg symmetry formu... |
| pntrval 27525 | Define the residual of the... |
| pntrf 27526 | Functionality of the resid... |
| pntrmax 27527 | There is a bound on the re... |
| pntrsumo1 27528 | A bound on a sum over ` R ... |
| pntrsumbnd 27529 | A bound on a sum over ` R ... |
| pntrsumbnd2 27530 | A bound on a sum over ` R ... |
| selbergr 27531 | Selberg's symmetry formula... |
| selberg3r 27532 | Selberg's symmetry formula... |
| selberg4r 27533 | Selberg's symmetry formula... |
| selberg34r 27534 | The sum of ~ selberg3r and... |
| pntsval 27535 | Define the "Selberg functi... |
| pntsf 27536 | Functionality of the Selbe... |
| selbergs 27537 | Selberg's symmetry formula... |
| selbergsb 27538 | Selberg's symmetry formula... |
| pntsval2 27539 | The Selberg function can b... |
| pntrlog2bndlem1 27540 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27541 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27542 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27543 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27544 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27545 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27546 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27547 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27548 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27549 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27550 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27551 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27552 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27553 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27554 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27555 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27556 | Lemma for ~ pnt . Establi... |
| pntlemd 27557 | Lemma for ~ pnt . Closure... |
| pntlemc 27558 | Lemma for ~ pnt . Closure... |
| pntlema 27559 | Lemma for ~ pnt . Closure... |
| pntlemb 27560 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27561 | Lemma for ~ pnt . Closure... |
| pntlemh 27562 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27563 | Lemma for ~ pnt . The "na... |
| pntlemq 27564 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27565 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27566 | Lemma for ~ pnt . The ind... |
| pntlemi 27567 | Lemma for ~ pnt . Elimina... |
| pntlemf 27568 | Lemma for ~ pnt . Add up ... |
| pntlemk 27569 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27570 | Lemma for ~ pnt . Combine... |
| pntleme 27571 | Lemma for ~ pnt . Package... |
| pntlem3 27572 | Lemma for ~ pnt . Equatio... |
| pntlemp 27573 | Lemma for ~ pnt . Wrappin... |
| pntleml 27574 | Lemma for ~ pnt . Equatio... |
| pnt3 27575 | The Prime Number Theorem, ... |
| pnt2 27576 | The Prime Number Theorem, ... |
| pnt 27577 | The Prime Number Theorem: ... |
| abvcxp 27578 | Raising an absolute value ... |
| padicfval 27579 | Value of the p-adic absolu... |
| padicval 27580 | Value of the p-adic absolu... |
| ostth2lem1 27581 | Lemma for ~ ostth2 , altho... |
| qrngbas 27582 | The base set of the field ... |
| qdrng 27583 | The rationals form a divis... |
| qrng0 27584 | The zero element of the fi... |
| qrng1 27585 | The unity element of the f... |
| qrngneg 27586 | The additive inverse in th... |
| qrngdiv 27587 | The division operation in ... |
| qabvle 27588 | By using induction on ` N ... |
| qabvexp 27589 | Induct the product rule ~ ... |
| ostthlem1 27590 | Lemma for ~ ostth . If tw... |
| ostthlem2 27591 | Lemma for ~ ostth . Refin... |
| qabsabv 27592 | The regular absolute value... |
| padicabv 27593 | The p-adic absolute value ... |
| padicabvf 27594 | The p-adic absolute value ... |
| padicabvcxp 27595 | All positive powers of the... |
| ostth1 27596 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27597 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27598 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27599 | Lemma for ~ ostth2 . (Con... |
| ostth2 27600 | - Lemma for ~ ostth : regu... |
| ostth3 27601 | - Lemma for ~ ostth : p-ad... |
| ostth 27602 | Ostrowski's theorem, which... |
| elno 27609 | Membership in the surreals... |
| sltval 27610 | The value of the surreal l... |
| bdayval 27611 | The value of the birthday ... |
| nofun 27612 | A surreal is a function. ... |
| nodmon 27613 | The domain of a surreal is... |
| norn 27614 | The range of a surreal is ... |
| nofnbday 27615 | A surreal is a function ov... |
| nodmord 27616 | The domain of a surreal ha... |
| elno2 27617 | An alternative condition f... |
| elno3 27618 | Another condition for memb... |
| sltval2 27619 | Alternate expression for s... |
| nofv 27620 | The function value of a su... |
| nosgnn0 27621 | ` (/) ` is not a surreal s... |
| nosgnn0i 27622 | If ` X ` is a surreal sign... |
| noreson 27623 | The restriction of a surre... |
| sltintdifex 27624 |
If ` A |
| sltres 27625 | If the restrictions of two... |
| noxp1o 27626 | The Cartesian product of a... |
| noseponlem 27627 | Lemma for ~ nosepon . Con... |
| nosepon 27628 | Given two unequal surreals... |
| noextend 27629 | Extending a surreal by one... |
| noextendseq 27630 | Extend a surreal by a sequ... |
| noextenddif 27631 | Calculate the place where ... |
| noextendlt 27632 | Extending a surreal with a... |
| noextendgt 27633 | Extending a surreal with a... |
| nolesgn2o 27634 | Given ` A ` less-than or e... |
| nolesgn2ores 27635 | Given ` A ` less-than or e... |
| nogesgn1o 27636 | Given ` A ` greater than o... |
| nogesgn1ores 27637 | Given ` A ` greater than o... |
| sltsolem1 27638 | Lemma for ~ sltso . The "... |
| sltso 27639 | Less-than totally orders t... |
| bdayfo 27640 | The birthday function maps... |
| fvnobday 27641 | The value of a surreal at ... |
| nosepnelem 27642 | Lemma for ~ nosepne . (Co... |
| nosepne 27643 | The value of two non-equal... |
| nosep1o 27644 | If the value of a surreal ... |
| nosep2o 27645 | If the value of a surreal ... |
| nosepdmlem 27646 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27647 | The first place two surrea... |
| nosepeq 27648 | The values of two surreals... |
| nosepssdm 27649 | Given two non-equal surrea... |
| nodenselem4 27650 | Lemma for ~ nodense . Sho... |
| nodenselem5 27651 | Lemma for ~ nodense . If ... |
| nodenselem6 27652 | The restriction of a surre... |
| nodenselem7 27653 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27654 | Lemma for ~ nodense . Giv... |
| nodense 27655 | Given two distinct surreal... |
| bdayimaon 27656 | Lemma for full-eta propert... |
| nolt02olem 27657 | Lemma for ~ nolt02o . If ... |
| nolt02o 27658 | Given ` A ` less-than ` B ... |
| nogt01o 27659 | Given ` A ` greater than `... |
| noresle 27660 | Restriction law for surrea... |
| nomaxmo 27661 | A class of surreals has at... |
| nominmo 27662 | A class of surreals has at... |
| nosupprefixmo 27663 | In any class of surreals, ... |
| noinfprefixmo 27664 | In any class of surreals, ... |
| nosupcbv 27665 | Lemma to change bound vari... |
| nosupno 27666 | The next several theorems ... |
| nosupdm 27667 | The domain of the surreal ... |
| nosupbday 27668 | Birthday bounding law for ... |
| nosupfv 27669 | The value of surreal supre... |
| nosupres 27670 | A restriction law for surr... |
| nosupbnd1lem1 27671 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27672 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27673 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27674 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27675 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27676 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27677 | Bounding law from below fo... |
| nosupbnd2lem1 27678 | Bounding law from above wh... |
| nosupbnd2 27679 | Bounding law from above fo... |
| noinfcbv 27680 | Change bound variables for... |
| noinfno 27681 | The next several theorems ... |
| noinfdm 27682 | Next, we calculate the dom... |
| noinfbday 27683 | Birthday bounding law for ... |
| noinffv 27684 | The value of surreal infim... |
| noinfres 27685 | The restriction of surreal... |
| noinfbnd1lem1 27686 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27687 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27688 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27689 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27690 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27691 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27692 | Bounding law from above fo... |
| noinfbnd2lem1 27693 | Bounding law from below wh... |
| noinfbnd2 27694 | Bounding law from below fo... |
| nosupinfsep 27695 | Given two sets of surreals... |
| noetasuplem1 27696 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27697 | Lemma for ~ noeta . The r... |
| noetasuplem3 27698 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27699 | Lemma for ~ noeta . When ... |
| noetainflem1 27700 | Lemma for ~ noeta . Estab... |
| noetainflem2 27701 | Lemma for ~ noeta . The r... |
| noetainflem3 27702 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27703 | Lemma for ~ noeta . If ` ... |
| noetalem1 27704 | Lemma for ~ noeta . Eithe... |
| noetalem2 27705 | Lemma for ~ noeta . The f... |
| noeta 27706 | The full-eta axiom for the... |
| sltirr 27709 | Surreal less-than is irref... |
| slttr 27710 | Surreal less-than is trans... |
| sltasym 27711 | Surreal less-than is asymm... |
| sltlin 27712 | Surreal less-than obeys tr... |
| slttrieq2 27713 | Trichotomy law for surreal... |
| slttrine 27714 | Trichotomy law for surreal... |
| slenlt 27715 | Surreal less-than or equal... |
| sltnle 27716 | Surreal less-than in terms... |
| sleloe 27717 | Surreal less-than or equal... |
| sletri3 27718 | Trichotomy law for surreal... |
| sltletr 27719 | Surreal transitive law. (... |
| slelttr 27720 | Surreal transitive law. (... |
| sletr 27721 | Surreal transitive law. (... |
| slttrd 27722 | Surreal less-than is trans... |
| sltletrd 27723 | Surreal less-than is trans... |
| slelttrd 27724 | Surreal less-than is trans... |
| sletrd 27725 | Surreal less-than or equal... |
| slerflex 27726 | Surreal less-than or equal... |
| sletric 27727 | Surreal trichotomy law. (... |
| maxs1 27728 | A surreal is less than or ... |
| maxs2 27729 | A surreal is less than or ... |
| mins1 27730 | The minimum of two surreal... |
| mins2 27731 | The minimum of two surreal... |
| sltled 27732 | Surreal less-than implies ... |
| sltne 27733 | Surreal less-than implies ... |
| sltlend 27734 | Surreal less-than in terms... |
| bdayfun 27735 | The birthday function is a... |
| bdayfn 27736 | The birthday function is a... |
| bdaydm 27737 | The birthday function's do... |
| bdayrn 27738 | The birthday function's ra... |
| bdayelon 27739 | The value of the birthday ... |
| nocvxminlem 27740 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27741 | Given a nonempty convex cl... |
| noprc 27742 | The surreal numbers are a ... |
| noeta2 27747 | A version of ~ noeta with ... |
| brsslt 27748 | Binary relation form of th... |
| ssltex1 27749 | The first argument of surr... |
| ssltex2 27750 | The second argument of sur... |
| ssltss1 27751 | The first argument of surr... |
| ssltss2 27752 | The second argument of sur... |
| ssltsep 27753 | The separation property of... |
| ssltd 27754 | Deduce surreal set less-th... |
| ssltsn 27755 | Surreal set less-than of t... |
| ssltsepc 27756 | Two elements of separated ... |
| ssltsepcd 27757 | Two elements of separated ... |
| sssslt1 27758 | Relation between surreal s... |
| sssslt2 27759 | Relation between surreal s... |
| nulsslt 27760 | The empty set is less-than... |
| nulssgt 27761 | The empty set is greater t... |
| conway 27762 | Conway's Simplicity Theore... |
| scutval 27763 | The value of the surreal c... |
| scutcut 27764 | Cut properties of the surr... |
| scutcl 27765 | Closure law for surreal cu... |
| scutcld 27766 | Closure law for surreal cu... |
| scutbday 27767 | The birthday of the surrea... |
| eqscut 27768 | Condition for equality to ... |
| eqscut2 27769 | Condition for equality to ... |
| sslttr 27770 | Transitive law for surreal... |
| ssltun1 27771 | Union law for surreal set ... |
| ssltun2 27772 | Union law for surreal set ... |
| scutun12 27773 | Union law for surreal cuts... |
| dmscut 27774 | The domain of the surreal ... |
| scutf 27775 | Functionality statement fo... |
| etasslt 27776 | A restatement of ~ noeta u... |
| etasslt2 27777 | A version of ~ etasslt wit... |
| scutbdaybnd 27778 | An upper bound on the birt... |
| scutbdaybnd2 27779 | An upper bound on the birt... |
| scutbdaybnd2lim 27780 | An upper bound on the birt... |
| scutbdaylt 27781 | If a surreal lies in a gap... |
| slerec 27782 | A comparison law for surre... |
| sltrec 27783 | A comparison law for surre... |
| ssltdisj 27784 | If ` A ` preceeds ` B ` , ... |
| 0sno 27789 | Surreal zero is a surreal.... |
| 1sno 27790 | Surreal one is a surreal. ... |
| bday0s 27791 | Calculate the birthday of ... |
| 0slt1s 27792 | Surreal zero is less than ... |
| bday0b 27793 | The only surreal with birt... |
| bday1s 27794 | The birthday of surreal on... |
| cuteq0 27795 | Condition for a surreal cu... |
| cuteq1 27796 | Condition for a surreal cu... |
| sgt0ne0 27797 | A positive surreal is not ... |
| sgt0ne0d 27798 | A positive surreal is not ... |
| madeval 27809 | The value of the made by f... |
| madeval2 27810 | Alternative characterizati... |
| oldval 27811 | The value of the old optio... |
| newval 27812 | The value of the new optio... |
| madef 27813 | The made function is a fun... |
| oldf 27814 | The older function is a fu... |
| newf 27815 | The new function is a func... |
| old0 27816 | No surreal is older than `... |
| madessno 27817 | Made sets are surreals. (... |
| oldssno 27818 | Old sets are surreals. (C... |
| newssno 27819 | New sets are surreals. (C... |
| leftval 27820 | The value of the left opti... |
| rightval 27821 | The value of the right opt... |
| leftf 27822 | The functionality of the l... |
| rightf 27823 | The functionality of the r... |
| elmade 27824 | Membership in the made fun... |
| elmade2 27825 | Membership in the made fun... |
| elold 27826 | Membership in an old set. ... |
| ssltleft 27827 | A surreal is greater than ... |
| ssltright 27828 | A surreal is less than its... |
| lltropt 27829 | The left options of a surr... |
| made0 27830 | The only surreal made on d... |
| new0 27831 | The only surreal new on da... |
| old1 27832 | The only surreal older tha... |
| madess 27833 | If ` A ` is less than or e... |
| oldssmade 27834 | The older-than set is a su... |
| leftssold 27835 | The left options are a sub... |
| rightssold 27836 | The right options are a su... |
| leftssno 27837 | The left set of a surreal ... |
| rightssno 27838 | The right set of a surreal... |
| madecut 27839 | Given a section that is a ... |
| madeun 27840 | The made set is the union ... |
| madeoldsuc 27841 | The made set is the old se... |
| oldsuc 27842 | The value of the old set a... |
| oldlim 27843 | The value of the old set a... |
| madebdayim 27844 | If a surreal is a member o... |
| oldbdayim 27845 | If ` X ` is in the old set... |
| oldirr 27846 | No surreal is a member of ... |
| leftirr 27847 | No surreal is a member of ... |
| rightirr 27848 | No surreal is a member of ... |
| left0s 27849 | The left set of ` 0s ` is ... |
| right0s 27850 | The right set of ` 0s ` is... |
| left1s 27851 | The left set of ` 1s ` is ... |
| right1s 27852 | The right set of ` 1s ` is... |
| lrold 27853 | The union of the left and ... |
| madebdaylemold 27854 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27855 | Lemma for ~ madebday . If... |
| madebday 27856 | A surreal is part of the s... |
| oldbday 27857 | A surreal is part of the s... |
| newbday 27858 | A surreal is an element of... |
| lrcut 27859 | A surreal is equal to the ... |
| scutfo 27860 | The surreal cut function i... |
| sltn0 27861 | If ` X ` is less than ` Y ... |
| lruneq 27862 | If two surreals share a bi... |
| sltlpss 27863 | If two surreals share a bi... |
| slelss 27864 | If two surreals ` A ` and ... |
| 0elold 27865 | Zero is in the old set of ... |
| 0elleft 27866 | Zero is in the left set of... |
| 0elright 27867 | Zero is in the right set o... |
| cofsslt 27868 | If every element of ` A ` ... |
| coinitsslt 27869 | If ` B ` is coinitial with... |
| cofcut1 27870 | If ` C ` is cofinal with `... |
| cofcut1d 27871 | If ` C ` is cofinal with `... |
| cofcut2 27872 | If ` A ` and ` C ` are mut... |
| cofcut2d 27873 | If ` A ` and ` C ` are mut... |
| cofcutr 27874 | If ` X ` is the cut of ` A... |
| cofcutr1d 27875 | If ` X ` is the cut of ` A... |
| cofcutr2d 27876 | If ` X ` is the cut of ` A... |
| cofcutrtime 27877 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27878 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27879 | If ` X ` is a timely cut o... |
| cofss 27880 | Cofinality for a subset. ... |
| coiniss 27881 | Coinitiality for a subset.... |
| cutlt 27882 | Eliminating all elements b... |
| cutpos 27883 | Reduce the elements of a c... |
| lrrecval 27886 | The next step in the devel... |
| lrrecval2 27887 | Next, we establish an alte... |
| lrrecpo 27888 | Now, we establish that ` R... |
| lrrecse 27889 | Next, we show that ` R ` i... |
| lrrecfr 27890 | Now we show that ` R ` is ... |
| lrrecpred 27891 | Finally, we calculate the ... |
| noinds 27892 | Induction principle for a ... |
| norecfn 27893 | Surreal recursion over one... |
| norecov 27894 | Calculate the value of the... |
| noxpordpo 27897 | To get through most of the... |
| noxpordfr 27898 | Next we establish the foun... |
| noxpordse 27899 | Next we establish the set-... |
| noxpordpred 27900 | Next we calculate the pred... |
| no2indslem 27901 | Double induction on surrea... |
| no2inds 27902 | Double induction on surrea... |
| norec2fn 27903 | The double-recursion opera... |
| norec2ov 27904 | The value of the double-re... |
| no3inds 27905 | Triple induction over surr... |
| addsfn 27908 | Surreal addition is a func... |
| addsval 27909 | The value of surreal addit... |
| addsval2 27910 | The value of surreal addit... |
| addsrid 27911 | Surreal addition to zero i... |
| addsridd 27912 | Surreal addition to zero i... |
| addscom 27913 | Surreal addition commutes.... |
| addscomd 27914 | Surreal addition commutes.... |
| addslid 27915 | Surreal addition to zero i... |
| addsproplem1 27916 | Lemma for surreal addition... |
| addsproplem2 27917 | Lemma for surreal addition... |
| addsproplem3 27918 | Lemma for surreal addition... |
| addsproplem4 27919 | Lemma for surreal addition... |
| addsproplem5 27920 | Lemma for surreal addition... |
| addsproplem6 27921 | Lemma for surreal addition... |
| addsproplem7 27922 | Lemma for surreal addition... |
| addsprop 27923 | Inductively show that surr... |
| addscutlem 27924 | Lemma for ~ addscut . Sho... |
| addscut 27925 | Demonstrate the cut proper... |
| addscut2 27926 | Show that the cut involved... |
| addscld 27927 | Surreal numbers are closed... |
| addscl 27928 | Surreal numbers are closed... |
| addsf 27929 | Function statement for sur... |
| addsfo 27930 | Surreal addition is onto. ... |
| peano2no 27931 | A theorem for surreals tha... |
| sltadd1im 27932 | Surreal less-than is prese... |
| sltadd2im 27933 | Surreal less-than is prese... |
| sleadd1im 27934 | Surreal less-than or equal... |
| sleadd2im 27935 | Surreal less-than or equal... |
| sleadd1 27936 | Addition to both sides of ... |
| sleadd2 27937 | Addition to both sides of ... |
| sltadd2 27938 | Addition to both sides of ... |
| sltadd1 27939 | Addition to both sides of ... |
| addscan2 27940 | Cancellation law for surre... |
| addscan1 27941 | Cancellation law for surre... |
| sleadd1d 27942 | Addition to both sides of ... |
| sleadd2d 27943 | Addition to both sides of ... |
| sltadd2d 27944 | Addition to both sides of ... |
| sltadd1d 27945 | Addition to both sides of ... |
| addscan2d 27946 | Cancellation law for surre... |
| addscan1d 27947 | Cancellation law for surre... |
| addsuniflem 27948 | Lemma for ~ addsunif . St... |
| addsunif 27949 | Uniformity theorem for sur... |
| addsasslem1 27950 | Lemma for addition associa... |
| addsasslem2 27951 | Lemma for addition associa... |
| addsass 27952 | Surreal addition is associ... |
| addsassd 27953 | Surreal addition is associ... |
| adds32d 27954 | Commutative/associative la... |
| adds12d 27955 | Commutative/associative la... |
| adds4d 27956 | Rearrangement of four term... |
| adds42d 27957 | Rearrangement of four term... |
| sltaddpos1d 27958 | Addition of a positive num... |
| sltaddpos2d 27959 | Addition of a positive num... |
| slt2addd 27960 | Adding both sides of two s... |
| addsgt0d 27961 | The sum of two positive su... |
| negsfn 27966 | Surreal negation is a func... |
| subsfn 27967 | Surreal subtraction is a f... |
| negsval 27968 | The value of the surreal n... |
| negs0s 27969 | Negative surreal zero is s... |
| negsproplem1 27970 | Lemma for surreal negation... |
| negsproplem2 27971 | Lemma for surreal negation... |
| negsproplem3 27972 | Lemma for surreal negation... |
| negsproplem4 27973 | Lemma for surreal negation... |
| negsproplem5 27974 | Lemma for surreal negation... |
| negsproplem6 27975 | Lemma for surreal negation... |
| negsproplem7 27976 | Lemma for surreal negation... |
| negsprop 27977 | Show closure and ordering ... |
| negscl 27978 | The surreals are closed un... |
| negscld 27979 | The surreals are closed un... |
| sltnegim 27980 | The forward direction of t... |
| negscut 27981 | The cut properties of surr... |
| negscut2 27982 | The cut that defines surre... |
| negsid 27983 | Surreal addition of a numb... |
| negsidd 27984 | Surreal addition of a numb... |
| negsex 27985 | Every surreal has a negati... |
| negnegs 27986 | A surreal is equal to the ... |
| sltneg 27987 | Negative of both sides of ... |
| sleneg 27988 | Negative of both sides of ... |
| sltnegd 27989 | Negative of both sides of ... |
| slenegd 27990 | Negative of both sides of ... |
| negs11 27991 | Surreal negation is one-to... |
| negsdi 27992 | Distribution of surreal ne... |
| slt0neg2d 27993 | Comparison of a surreal an... |
| negsf 27994 | Function statement for sur... |
| negsfo 27995 | Function statement for sur... |
| negsf1o 27996 | Surreal negation is a bije... |
| negsunif 27997 | Uniformity property for su... |
| negsbdaylem 27998 | Lemma for ~ negsbday . Bo... |
| negsbday 27999 | Negation of a surreal numb... |
| subsval 28000 | The value of surreal subtr... |
| subsvald 28001 | The value of surreal subtr... |
| subscl 28002 | Closure law for surreal su... |
| subscld 28003 | Closure law for surreal su... |
| subsf 28004 | Function statement for sur... |
| subsfo 28005 | Surreal subtraction is an ... |
| negsval2 28006 | Surreal negation in terms ... |
| negsval2d 28007 | Surreal negation in terms ... |
| subsid1 28008 | Identity law for subtracti... |
| subsid 28009 | Subtraction of a surreal f... |
| subadds 28010 | Relationship between addit... |
| subaddsd 28011 | Relationship between addit... |
| pncans 28012 | Cancellation law for surre... |
| pncan3s 28013 | Subtraction and addition o... |
| pncan2s 28014 | Cancellation law for surre... |
| npcans 28015 | Cancellation law for surre... |
| sltsub1 28016 | Subtraction from both side... |
| sltsub2 28017 | Subtraction from both side... |
| sltsub1d 28018 | Subtraction from both side... |
| sltsub2d 28019 | Subtraction from both side... |
| negsubsdi2d 28020 | Distribution of negative o... |
| addsubsassd 28021 | Associative-type law for s... |
| addsubsd 28022 | Law for surreal addition a... |
| sltsubsubbd 28023 | Equivalence for the surrea... |
| sltsubsub2bd 28024 | Equivalence for the surrea... |
| sltsubsub3bd 28025 | Equivalence for the surrea... |
| slesubsubbd 28026 | Equivalence for the surrea... |
| slesubsub2bd 28027 | Equivalence for the surrea... |
| slesubsub3bd 28028 | Equivalence for the surrea... |
| sltsubaddd 28029 | Surreal less-than relation... |
| sltsubadd2d 28030 | Surreal less-than relation... |
| sltaddsubd 28031 | Surreal less-than relation... |
| sltaddsub2d 28032 | Surreal less-than relation... |
| slesubaddd 28033 | Surreal less-than or equal... |
| subsubs4d 28034 | Law for double surreal sub... |
| subsubs2d 28035 | Law for double surreal sub... |
| nncansd 28036 | Cancellation law for surre... |
| posdifsd 28037 | Comparison of two surreals... |
| sltsubposd 28038 | Subtraction of a positive ... |
| subsge0d 28039 | Non-negative subtraction. ... |
| mulsfn 28042 | Surreal multiplication is ... |
| mulsval 28043 | The value of surreal multi... |
| mulsval2lem 28044 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28045 | The value of surreal multi... |
| muls01 28046 | Surreal multiplication by ... |
| mulsrid 28047 | Surreal one is a right ide... |
| mulsridd 28048 | Surreal one is a right ide... |
| mulsproplemcbv 28049 | Lemma for surreal multipli... |
| mulsproplem1 28050 | Lemma for surreal multipli... |
| mulsproplem2 28051 | Lemma for surreal multipli... |
| mulsproplem3 28052 | Lemma for surreal multipli... |
| mulsproplem4 28053 | Lemma for surreal multipli... |
| mulsproplem5 28054 | Lemma for surreal multipli... |
| mulsproplem6 28055 | Lemma for surreal multipli... |
| mulsproplem7 28056 | Lemma for surreal multipli... |
| mulsproplem8 28057 | Lemma for surreal multipli... |
| mulsproplem9 28058 | Lemma for surreal multipli... |
| mulsproplem10 28059 | Lemma for surreal multipli... |
| mulsproplem11 28060 | Lemma for surreal multipli... |
| mulsproplem12 28061 | Lemma for surreal multipli... |
| mulsproplem13 28062 | Lemma for surreal multipli... |
| mulsproplem14 28063 | Lemma for surreal multipli... |
| mulsprop 28064 | Surreals are closed under ... |
| mulscutlem 28065 | Lemma for ~ mulscut . Sta... |
| mulscut 28066 | Show the cut properties of... |
| mulscut2 28067 | Show that the cut involved... |
| mulscl 28068 | The surreals are closed un... |
| mulscld 28069 | The surreals are closed un... |
| sltmul 28070 | An ordering relationship f... |
| sltmuld 28071 | An ordering relationship f... |
| slemuld 28072 | An ordering relationship f... |
| mulscom 28073 | Surreal multiplication com... |
| mulscomd 28074 | Surreal multiplication com... |
| muls02 28075 | Surreal multiplication by ... |
| mulslid 28076 | Surreal one is a left iden... |
| mulslidd 28077 | Surreal one is a left iden... |
| mulsgt0 28078 | The product of two positiv... |
| mulsgt0d 28079 | The product of two positiv... |
| mulsge0d 28080 | The product of two non-neg... |
| ssltmul1 28081 | One surreal set less-than ... |
| ssltmul2 28082 | One surreal set less-than ... |
| mulsuniflem 28083 | Lemma for ~ mulsunif . St... |
| mulsunif 28084 | Surreal multiplication has... |
| addsdilem1 28085 | Lemma for surreal distribu... |
| addsdilem2 28086 | Lemma for surreal distribu... |
| addsdilem3 28087 | Lemma for ~ addsdi . Show... |
| addsdilem4 28088 | Lemma for ~ addsdi . Show... |
| addsdi 28089 | Distributive law for surre... |
| addsdid 28090 | Distributive law for surre... |
| addsdird 28091 | Distributive law for surre... |
| subsdid 28092 | Distribution of surreal mu... |
| subsdird 28093 | Distribution of surreal mu... |
| mulnegs1d 28094 | Product with negative is n... |
| mulnegs2d 28095 | Product with negative is n... |
| mul2negsd 28096 | Surreal product of two neg... |
| mulsasslem1 28097 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28098 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28099 | Lemma for ~ mulsass . Dem... |
| mulsass 28100 | Associative law for surrea... |
| mulsassd 28101 | Associative law for surrea... |
| muls4d 28102 | Rearrangement of four surr... |
| mulsunif2lem 28103 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28104 | Alternate expression for s... |
| sltmul2 28105 | Multiplication of both sid... |
| sltmul2d 28106 | Multiplication of both sid... |
| sltmul1d 28107 | Multiplication of both sid... |
| slemul2d 28108 | Multiplication of both sid... |
| slemul1d 28109 | Multiplication of both sid... |
| sltmulneg1d 28110 | Multiplication of both sid... |
| sltmulneg2d 28111 | Multiplication of both sid... |
| mulscan2dlem 28112 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28113 | Cancellation of surreal mu... |
| mulscan1d 28114 | Cancellation of surreal mu... |
| muls12d 28115 | Commutative/associative la... |
| slemul1ad 28116 | Multiplication of both sid... |
| sltmul12ad 28117 | Comparison of the product ... |
| divsmo 28118 | Uniqueness of surreal inve... |
| muls0ord 28119 | If a surreal product is ze... |
| mulsne0bd 28120 | The product of two non-zer... |
| divsval 28123 | The value of surreal divis... |
| norecdiv 28124 | If a surreal has a recipro... |
| noreceuw 28125 | If a surreal has a recipro... |
| divsmulw 28126 | Relationship between surre... |
| divsmulwd 28127 | Relationship between surre... |
| divsclw 28128 | Weak division closure law.... |
| divsclwd 28129 | Weak division closure law.... |
| divscan2wd 28130 | A weak cancellation law fo... |
| divscan1wd 28131 | A weak cancellation law fo... |
| sltdivmulwd 28132 | Surreal less-than relation... |
| sltdivmul2wd 28133 | Surreal less-than relation... |
| sltmuldivwd 28134 | Surreal less-than relation... |
| sltmuldiv2wd 28135 | Surreal less-than relation... |
| divsasswd 28136 | An associative law for sur... |
| divs1 28137 | A surreal divided by one i... |
| precsexlemcbv 28138 | Lemma for surreal reciproc... |
| precsexlem1 28139 | Lemma for surreal reciproc... |
| precsexlem2 28140 | Lemma for surreal reciproc... |
| precsexlem3 28141 | Lemma for surreal reciproc... |
| precsexlem4 28142 | Lemma for surreal reciproc... |
| precsexlem5 28143 | Lemma for surreal reciproc... |
| precsexlem6 28144 | Lemma for surreal reciproc... |
| precsexlem7 28145 | Lemma for surreal reciproc... |
| precsexlem8 28146 | Lemma for surreal reciproc... |
| precsexlem9 28147 | Lemma for surreal reciproc... |
| precsexlem10 28148 | Lemma for surreal reciproc... |
| precsexlem11 28149 | Lemma for surreal reciproc... |
| precsex 28150 | Every positive surreal has... |
| recsex 28151 | A non-zero surreal has a r... |
| recsexd 28152 | A non-zero surreal has a r... |
| divsmul 28153 | Relationship between surre... |
| divsmuld 28154 | Relationship between surre... |
| divscl 28155 | Surreal division closure l... |
| divscld 28156 | Surreal division closure l... |
| divscan2d 28157 | A cancellation law for sur... |
| divscan1d 28158 | A cancellation law for sur... |
| sltdivmuld 28159 | Surreal less-than relation... |
| sltdivmul2d 28160 | Surreal less-than relation... |
| sltmuldivd 28161 | Surreal less-than relation... |
| sltmuldiv2d 28162 | Surreal less-than relation... |
| divsassd 28163 | An associative law for sur... |
| divmuldivsd 28164 | Multiplication of two surr... |
| abssval 28167 | The value of surreal absol... |
| absscl 28168 | Closure law for surreal ab... |
| abssid 28169 | The absolute value of a no... |
| abs0s 28170 | The absolute value of surr... |
| abssnid 28171 | For a negative surreal, it... |
| absmuls 28172 | Surreal absolute value dis... |
| abssge0 28173 | The absolute value of a su... |
| abssor 28174 | The absolute value of a su... |
| abssneg 28175 | Surreal absolute value of ... |
| sleabs 28176 | A surreal is less than or ... |
| absslt 28177 | Surreal absolute value and... |
| elons 28180 | Membership in the class of... |
| onssno 28181 | The surreal ordinals are a... |
| onsno 28182 | A surreal ordinal is a sur... |
| 0ons 28183 | Surreal zero is a surreal ... |
| 1ons 28184 | Surreal one is a surreal o... |
| elons2 28185 | A surreal is ordinal iff i... |
| elons2d 28186 | The cut of any set of surr... |
| sltonold 28187 | The class of ordinals less... |
| sltonex 28188 | The class of ordinals less... |
| onscutleft 28189 | A surreal ordinal is equal... |
| seqsex 28192 | Existence of the surreal s... |
| seqseq123d 28193 | Equality deduction for the... |
| nfseqs 28194 | Hypothesis builder for the... |
| seqsval 28195 | The value of the surreal s... |
| noseqex 28196 | The next several theorems ... |
| noseq0 28197 | The surreal ` A ` is a mem... |
| noseqp1 28198 | One plus an element of ` Z... |
| noseqind 28199 | Peano's inductive postulat... |
| noseqinds 28200 | Induction schema for surre... |
| noseqssno 28201 | A surreal sequence is a su... |
| noseqno 28202 | An element of a surreal se... |
| om2noseq0 28203 | The mapping ` G ` is a one... |
| om2noseqsuc 28204 | The value of ` G ` at a su... |
| om2noseqfo 28205 | Function statement for ` G... |
| om2noseqlt 28206 | Surreal less-than relation... |
| om2noseqlt2 28207 | The mapping ` G ` preserve... |
| om2noseqf1o 28208 | ` G ` is a bijection. (Co... |
| om2noseqiso 28209 | ` G ` is an isomorphism fr... |
| om2noseqoi 28210 | An alternative definition ... |
| om2noseqrdg 28211 | A helper lemma for the val... |
| noseqrdglem 28212 | A helper lemma for the val... |
| noseqrdgfn 28213 | The recursive definition g... |
| noseqrdg0 28214 | Initial value of a recursi... |
| noseqrdgsuc 28215 | Successor value of a recur... |
| seqsfn 28216 | The surreal sequence build... |
| seqs1 28217 | The value of the surreal s... |
| seqsp1 28218 | The value of the surreal s... |
| n0sex 28223 | The set of all non-negativ... |
| nnsex 28224 | The set of all positive su... |
| peano5n0s 28225 | Peano's inductive postulat... |
| n0ssno 28226 | The non-negative surreal i... |
| nnssn0s 28227 | The positive surreal integ... |
| nnssno 28228 | The positive surreal integ... |
| n0sno 28229 | A non-negative surreal int... |
| nnsno 28230 | A positive surreal integer... |
| n0snod 28231 | A non-negative surreal int... |
| nnsnod 28232 | A positive surreal integer... |
| nnn0s 28233 | A positive surreal integer... |
| nnn0sd 28234 | A positive surreal integer... |
| 0n0s 28235 | Peano postulate: ` 0s ` is... |
| peano2n0s 28236 | Peano postulate: the succe... |
| dfn0s2 28237 | Alternate definition of th... |
| n0sind 28238 | Principle of Mathematical ... |
| n0scut 28239 | A cut form for surreal nat... |
| n0ons 28240 | A surreal natural is a sur... |
| nnne0s 28241 | A surreal positive integer... |
| n0sge0 28242 | A non-negative integer is ... |
| nnsgt0 28243 | A positive integer is grea... |
| elnns 28244 | Membership in the positive... |
| elnns2 28245 | A positive surreal integer... |
| n0addscl 28246 | The non-negative surreal i... |
| n0mulscl 28247 | The non-negative surreal i... |
| nnaddscl 28248 | The positive surreal integ... |
| nnmulscl 28249 | The positive surreal integ... |
| 1n0s 28250 | Surreal one is a non-negat... |
| 1nns 28251 | Surreal one is a positive ... |
| peano2nns 28252 | Peano postulate for positi... |
| n0sbday 28253 | A non-negative surreal int... |
| n0ssold 28254 | The non-negative surreal i... |
| nnsrecgt0d 28255 | The reciprocal of a positi... |
| seqn0sfn 28256 | The surreal sequence build... |
| eln0s 28257 | A non-negative surreal int... |
| n0s0m1 28258 | Every non-negative surreal... |
| n0subs 28259 | Subtraction of non-negativ... |
| n0p1nns 28260 | One plus a non-negative su... |
| zsex 28263 | The surreal integers form ... |
| zssno 28264 | The surreal integers are a... |
| zno 28265 | A surreal integer is a sur... |
| znod 28266 | A surreal integer is a sur... |
| elzs 28267 | Membership in the set of s... |
| nnzs 28268 | A positive surreal integer... |
| nnzsd 28269 | A positive surreal integer... |
| 0zs 28270 | Zero is a surreal integer.... |
| n0zs 28271 | A non-negative surreal int... |
| n0zsd 28272 | A non-negative surreal int... |
| znegscl 28273 | The surreal integers are c... |
| znegscld 28274 | The surreal integers are c... |
| elzn0s 28275 | A surreal integer is a sur... |
| zsbday 28276 | A surreal integer has a fi... |
| elreno 28279 | Membership in the set of s... |
| recut 28280 | The cut involved in defini... |
| 0reno 28281 | Surreal zero is a surreal ... |
| renegscl 28282 | The surreal reals are clos... |
| readdscl 28283 | The surreal reals are clos... |
| remulscllem1 28284 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28285 | Lemma for ~ remulscl . Bo... |
| remulscl 28286 | The surreal reals are clos... |
| itvndx 28297 | Index value of the Interva... |
| lngndx 28298 | Index value of the "line" ... |
| itvid 28299 | Utility theorem: index-ind... |
| lngid 28300 | Utility theorem: index-ind... |
| slotsinbpsd 28301 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28302 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28303 | The slot for the line is n... |
| trkgstr 28304 | Functionality of a Tarski ... |
| trkgbas 28305 | The base set of a Tarski g... |
| trkgdist 28306 | The measure of a distance ... |
| trkgitv 28307 | The congruence relation in... |
| istrkgc 28314 | Property of being a Tarski... |
| istrkgb 28315 | Property of being a Tarski... |
| istrkgcb 28316 | Property of being a Tarski... |
| istrkge 28317 | Property of fulfilling Euc... |
| istrkgl 28318 | Building lines from the se... |
| istrkgld 28319 | Property of fulfilling the... |
| istrkg2ld 28320 | Property of fulfilling the... |
| istrkg3ld 28321 | Property of fulfilling the... |
| axtgcgrrflx 28322 | Axiom of reflexivity of co... |
| axtgcgrid 28323 | Axiom of identity of congr... |
| axtgsegcon 28324 | Axiom of segment construct... |
| axtg5seg 28325 | Five segments axiom, Axiom... |
| axtgbtwnid 28326 | Identity of Betweenness. ... |
| axtgpasch 28327 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28328 | Axiom of Continuity. Axio... |
| axtgcont 28329 | Axiom of Continuity. Axio... |
| axtglowdim2 28330 | Lower dimension axiom for ... |
| axtgupdim2 28331 | Upper dimension axiom for ... |
| axtgeucl 28332 | Euclid's Axiom. Axiom A10... |
| tgjustf 28333 | Given any function ` F ` ,... |
| tgjustr 28334 | Given any equivalence rela... |
| tgjustc1 28335 | A justification for using ... |
| tgjustc2 28336 | A justification for using ... |
| tgcgrcomimp 28337 | Congruence commutes on the... |
| tgcgrcomr 28338 | Congruence commutes on the... |
| tgcgrcoml 28339 | Congruence commutes on the... |
| tgcgrcomlr 28340 | Congruence commutes on bot... |
| tgcgreqb 28341 | Congruence and equality. ... |
| tgcgreq 28342 | Congruence and equality. ... |
| tgcgrneq 28343 | Congruence and equality. ... |
| tgcgrtriv 28344 | Degenerate segments are co... |
| tgcgrextend 28345 | Link congruence over a pai... |
| tgsegconeq 28346 | Two points that satisfy th... |
| tgbtwntriv2 28347 | Betweenness always holds f... |
| tgbtwncom 28348 | Betweenness commutes. The... |
| tgbtwncomb 28349 | Betweenness commutes, bico... |
| tgbtwnne 28350 | Betweenness and inequality... |
| tgbtwntriv1 28351 | Betweenness always holds f... |
| tgbtwnswapid 28352 | If you can swap the first ... |
| tgbtwnintr 28353 | Inner transitivity law for... |
| tgbtwnexch3 28354 | Exchange the first endpoin... |
| tgbtwnouttr2 28355 | Outer transitivity law for... |
| tgbtwnexch2 28356 | Exchange the outer point o... |
| tgbtwnouttr 28357 | Outer transitivity law for... |
| tgbtwnexch 28358 | Outer transitivity law for... |
| tgtrisegint 28359 | A line segment between two... |
| tglowdim1 28360 | Lower dimension axiom for ... |
| tglowdim1i 28361 | Lower dimension axiom for ... |
| tgldimor 28362 | Excluded-middle like state... |
| tgldim0eq 28363 | In dimension zero, any two... |
| tgldim0itv 28364 | In dimension zero, any two... |
| tgldim0cgr 28365 | In dimension zero, any two... |
| tgbtwndiff 28366 | There is always a ` c ` di... |
| tgdim01 28367 | In geometries of dimension... |
| tgifscgr 28368 | Inner five segment congrue... |
| tgcgrsub 28369 | Removing identical parts f... |
| iscgrg 28372 | The congruence property fo... |
| iscgrgd 28373 | The property for two seque... |
| iscgrglt 28374 | The property for two seque... |
| trgcgrg 28375 | The property for two trian... |
| trgcgr 28376 | Triangle congruence. (Con... |
| ercgrg 28377 | The shape congruence relat... |
| tgcgrxfr 28378 | A line segment can be divi... |
| cgr3id 28379 | Reflexivity law for three-... |
| cgr3simp1 28380 | Deduce segment congruence ... |
| cgr3simp2 28381 | Deduce segment congruence ... |
| cgr3simp3 28382 | Deduce segment congruence ... |
| cgr3swap12 28383 | Permutation law for three-... |
| cgr3swap23 28384 | Permutation law for three-... |
| cgr3swap13 28385 | Permutation law for three-... |
| cgr3rotr 28386 | Permutation law for three-... |
| cgr3rotl 28387 | Permutation law for three-... |
| trgcgrcom 28388 | Commutative law for three-... |
| cgr3tr 28389 | Transitivity law for three... |
| tgbtwnxfr 28390 | A condition for extending ... |
| tgcgr4 28391 | Two quadrilaterals to be c... |
| isismt 28394 | Property of being an isome... |
| ismot 28395 | Property of being an isome... |
| motcgr 28396 | Property of a motion: dist... |
| idmot 28397 | The identity is a motion. ... |
| motf1o 28398 | Motions are bijections. (... |
| motcl 28399 | Closure of motions. (Cont... |
| motco 28400 | The composition of two mot... |
| cnvmot 28401 | The converse of a motion i... |
| motplusg 28402 | The operation for motions ... |
| motgrp 28403 | The motions of a geometry ... |
| motcgrg 28404 | Property of a motion: dist... |
| motcgr3 28405 | Property of a motion: dist... |
| tglng 28406 | Lines of a Tarski Geometry... |
| tglnfn 28407 | Lines as functions. (Cont... |
| tglnunirn 28408 | Lines are sets of points. ... |
| tglnpt 28409 | Lines are sets of points. ... |
| tglngne 28410 | It takes two different poi... |
| tglngval 28411 | The line going through poi... |
| tglnssp 28412 | Lines are subset of the ge... |
| tgellng 28413 | Property of lying on the l... |
| tgcolg 28414 | We choose the notation ` (... |
| btwncolg1 28415 | Betweenness implies coline... |
| btwncolg2 28416 | Betweenness implies coline... |
| btwncolg3 28417 | Betweenness implies coline... |
| colcom 28418 | Swapping the points defini... |
| colrot1 28419 | Rotating the points defini... |
| colrot2 28420 | Rotating the points defini... |
| ncolcom 28421 | Swapping non-colinear poin... |
| ncolrot1 28422 | Rotating non-colinear poin... |
| ncolrot2 28423 | Rotating non-colinear poin... |
| tgdim01ln 28424 | In geometries of dimension... |
| ncoltgdim2 28425 | If there are three non-col... |
| lnxfr 28426 | Transfer law for colineari... |
| lnext 28427 | Extend a line with a missi... |
| tgfscgr 28428 | Congruence law for the gen... |
| lncgr 28429 | Congruence rule for lines.... |
| lnid 28430 | Identity law for points on... |
| tgidinside 28431 | Law for finding a point in... |
| tgbtwnconn1lem1 28432 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28433 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28434 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28435 | Connectivity law for betwe... |
| tgbtwnconn2 28436 | Another connectivity law f... |
| tgbtwnconn3 28437 | Inner connectivity law for... |
| tgbtwnconnln3 28438 | Derive colinearity from be... |
| tgbtwnconn22 28439 | Double connectivity law fo... |
| tgbtwnconnln1 28440 | Derive colinearity from be... |
| tgbtwnconnln2 28441 | Derive colinearity from be... |
| legval 28444 | Value of the less-than rel... |
| legov 28445 | Value of the less-than rel... |
| legov2 28446 | An equivalent definition o... |
| legid 28447 | Reflexivity of the less-th... |
| btwnleg 28448 | Betweenness implies less-t... |
| legtrd 28449 | Transitivity of the less-t... |
| legtri3 28450 | Equality from the less-tha... |
| legtrid 28451 | Trichotomy law for the les... |
| leg0 28452 | Degenerated (zero-length) ... |
| legeq 28453 | Deduce equality from "less... |
| legbtwn 28454 | Deduce betweenness from "l... |
| tgcgrsub2 28455 | Removing identical parts f... |
| ltgseg 28456 | The set ` E ` denotes the ... |
| ltgov 28457 | Strict "shorter than" geom... |
| legov3 28458 | An equivalent definition o... |
| legso 28459 | The "shorter than" relatio... |
| ishlg 28462 | Rays : Definition 6.1 of ... |
| hlcomb 28463 | The half-line relation com... |
| hlcomd 28464 | The half-line relation com... |
| hlne1 28465 | The half-line relation imp... |
| hlne2 28466 | The half-line relation imp... |
| hlln 28467 | The half-line relation imp... |
| hleqnid 28468 | The endpoint does not belo... |
| hlid 28469 | The half-line relation is ... |
| hltr 28470 | The half-line relation is ... |
| hlbtwn 28471 | Betweenness is a sufficien... |
| btwnhl1 28472 | Deduce half-line from betw... |
| btwnhl2 28473 | Deduce half-line from betw... |
| btwnhl 28474 | Swap betweenness for a hal... |
| lnhl 28475 | Either a point ` C ` on th... |
| hlcgrex 28476 | Construct a point on a hal... |
| hlcgreulem 28477 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28478 | The point constructed in ~... |
| btwnlng1 28479 | Betweenness implies coline... |
| btwnlng2 28480 | Betweenness implies coline... |
| btwnlng3 28481 | Betweenness implies coline... |
| lncom 28482 | Swapping the points defini... |
| lnrot1 28483 | Rotating the points defini... |
| lnrot2 28484 | Rotating the points defini... |
| ncolne1 28485 | Non-colinear points are di... |
| ncolne2 28486 | Non-colinear points are di... |
| tgisline 28487 | The property of being a pr... |
| tglnne 28488 | It takes two different poi... |
| tglndim0 28489 | There are no lines in dime... |
| tgelrnln 28490 | The property of being a pr... |
| tglineeltr 28491 | Transitivity law for lines... |
| tglineelsb2 28492 | If ` S ` lies on PQ , then... |
| tglinerflx1 28493 | Reflexivity law for line m... |
| tglinerflx2 28494 | Reflexivity law for line m... |
| tglinecom 28495 | Commutativity law for line... |
| tglinethru 28496 | If ` A ` is a line contain... |
| tghilberti1 28497 | There is a line through an... |
| tghilberti2 28498 | There is at most one line ... |
| tglinethrueu 28499 | There is a unique line goi... |
| tglnne0 28500 | A line ` A ` has at least ... |
| tglnpt2 28501 | Find a second point on a l... |
| tglineintmo 28502 | Two distinct lines interse... |
| tglineineq 28503 | Two distinct lines interse... |
| tglineneq 28504 | Given three non-colinear p... |
| tglineinteq 28505 | Two distinct lines interse... |
| ncolncol 28506 | Deduce non-colinearity fro... |
| coltr 28507 | A transitivity law for col... |
| coltr3 28508 | A transitivity law for col... |
| colline 28509 | Three points are colinear ... |
| tglowdim2l 28510 | Reformulation of the lower... |
| tglowdim2ln 28511 | There is always one point ... |
| mirreu3 28514 | Existential uniqueness of ... |
| mirval 28515 | Value of the point inversi... |
| mirfv 28516 | Value of the point inversi... |
| mircgr 28517 | Property of the image by t... |
| mirbtwn 28518 | Property of the image by t... |
| ismir 28519 | Property of the image by t... |
| mirf 28520 | Point inversion as functio... |
| mircl 28521 | Closure of the point inver... |
| mirmir 28522 | The point inversion functi... |
| mircom 28523 | Variation on ~ mirmir . (... |
| mirreu 28524 | Any point has a unique ant... |
| mireq 28525 | Equality deduction for poi... |
| mirinv 28526 | The only invariant point o... |
| mirne 28527 | Mirror of non-center point... |
| mircinv 28528 | The center point is invari... |
| mirf1o 28529 | The point inversion functi... |
| miriso 28530 | The point inversion functi... |
| mirbtwni 28531 | Point inversion preserves ... |
| mirbtwnb 28532 | Point inversion preserves ... |
| mircgrs 28533 | Point inversion preserves ... |
| mirmir2 28534 | Point inversion of a point... |
| mirmot 28535 | Point investion is a motio... |
| mirln 28536 | If two points are on the s... |
| mirln2 28537 | If a point and its mirror ... |
| mirconn 28538 | Point inversion of connect... |
| mirhl 28539 | If two points ` X ` and ` ... |
| mirbtwnhl 28540 | If the center of the point... |
| mirhl2 28541 | Deduce half-line relation ... |
| mircgrextend 28542 | Link congruence over a pai... |
| mirtrcgr 28543 | Point inversion of one poi... |
| mirauto 28544 | Point inversion preserves ... |
| miduniq 28545 | Uniqueness of the middle p... |
| miduniq1 28546 | Uniqueness of the middle p... |
| miduniq2 28547 | If two point inversions co... |
| colmid 28548 | Colinearity and equidistan... |
| symquadlem 28549 | Lemma of the symetrial qua... |
| krippenlem 28550 | Lemma for ~ krippen . We ... |
| krippen 28551 | Krippenlemma (German for c... |
| midexlem 28552 | Lemma for the existence of... |
| israg 28557 | Property for 3 points A, B... |
| ragcom 28558 | Commutative rule for right... |
| ragcol 28559 | The right angle property i... |
| ragmir 28560 | Right angle property is pr... |
| mirrag 28561 | Right angle is conserved b... |
| ragtrivb 28562 | Trivial right angle. Theo... |
| ragflat2 28563 | Deduce equality from two r... |
| ragflat 28564 | Deduce equality from two r... |
| ragtriva 28565 | Trivial right angle. Theo... |
| ragflat3 28566 | Right angle and colinearit... |
| ragcgr 28567 | Right angle and colinearit... |
| motrag 28568 | Right angles are preserved... |
| ragncol 28569 | Right angle implies non-co... |
| perpln1 28570 | Derive a line from perpend... |
| perpln2 28571 | Derive a line from perpend... |
| isperp 28572 | Property for 2 lines A, B ... |
| perpcom 28573 | The "perpendicular" relati... |
| perpneq 28574 | Two perpendicular lines ar... |
| isperp2 28575 | Property for 2 lines A, B,... |
| isperp2d 28576 | One direction of ~ isperp2... |
| ragperp 28577 | Deduce that two lines are ... |
| footexALT 28578 | Alternative version of ~ f... |
| footexlem1 28579 | Lemma for ~ footex . (Con... |
| footexlem2 28580 | Lemma for ~ footex . (Con... |
| footex 28581 | From a point ` C ` outside... |
| foot 28582 | From a point ` C ` outside... |
| footne 28583 | Uniqueness of the foot poi... |
| footeq 28584 | Uniqueness of the foot poi... |
| hlperpnel 28585 | A point on a half-line whi... |
| perprag 28586 | Deduce a right angle from ... |
| perpdragALT 28587 | Deduce a right angle from ... |
| perpdrag 28588 | Deduce a right angle from ... |
| colperp 28589 | Deduce a perpendicularity ... |
| colperpexlem1 28590 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28591 | Lemma for ~ colperpex . S... |
| colperpexlem3 28592 | Lemma for ~ colperpex . C... |
| colperpex 28593 | In dimension 2 and above, ... |
| mideulem2 28594 | Lemma for ~ opphllem , whi... |
| opphllem 28595 | Lemma 8.24 of [Schwabhause... |
| mideulem 28596 | Lemma for ~ mideu . We ca... |
| midex 28597 | Existence of the midpoint,... |
| mideu 28598 | Existence and uniqueness o... |
| islnopp 28599 | The property for two point... |
| islnoppd 28600 | Deduce that ` A ` and ` B ... |
| oppne1 28601 | Points lying on opposite s... |
| oppne2 28602 | Points lying on opposite s... |
| oppne3 28603 | Points lying on opposite s... |
| oppcom 28604 | Commutativity rule for "op... |
| opptgdim2 28605 | If two points opposite to ... |
| oppnid 28606 | The "opposite to a line" r... |
| opphllem1 28607 | Lemma for ~ opphl . (Cont... |
| opphllem2 28608 | Lemma for ~ opphl . Lemma... |
| opphllem3 28609 | Lemma for ~ opphl : We as... |
| opphllem4 28610 | Lemma for ~ opphl . (Cont... |
| opphllem5 28611 | Second part of Lemma 9.4 o... |
| opphllem6 28612 | First part of Lemma 9.4 of... |
| oppperpex 28613 | Restating ~ colperpex usin... |
| opphl 28614 | If two points ` A ` and ` ... |
| outpasch 28615 | Axiom of Pasch, outer form... |
| hlpasch 28616 | An application of the axio... |
| ishpg 28619 | Value of the half-plane re... |
| hpgbr 28620 | Half-planes : property for... |
| hpgne1 28621 | Points on the open half pl... |
| hpgne2 28622 | Points on the open half pl... |
| lnopp2hpgb 28623 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28624 | If two points lie on the o... |
| hpgerlem 28625 | Lemma for the proof that t... |
| hpgid 28626 | The half-plane relation is... |
| hpgcom 28627 | The half-plane relation co... |
| hpgtr 28628 | The half-plane relation is... |
| colopp 28629 | Opposite sides of a line f... |
| colhp 28630 | Half-plane relation for co... |
| hphl 28631 | If two points are on the s... |
| midf 28636 | Midpoint as a function. (... |
| midcl 28637 | Closure of the midpoint. ... |
| ismidb 28638 | Property of the midpoint. ... |
| midbtwn 28639 | Betweenness of midpoint. ... |
| midcgr 28640 | Congruence of midpoint. (... |
| midid 28641 | Midpoint of a null segment... |
| midcom 28642 | Commutativity rule for the... |
| mirmid 28643 | Point inversion preserves ... |
| lmieu 28644 | Uniqueness of the line mir... |
| lmif 28645 | Line mirror as a function.... |
| lmicl 28646 | Closure of the line mirror... |
| islmib 28647 | Property of the line mirro... |
| lmicom 28648 | The line mirroring functio... |
| lmilmi 28649 | Line mirroring is an invol... |
| lmireu 28650 | Any point has a unique ant... |
| lmieq 28651 | Equality deduction for lin... |
| lmiinv 28652 | The invariants of the line... |
| lmicinv 28653 | The mirroring line is an i... |
| lmimid 28654 | If we have a right angle, ... |
| lmif1o 28655 | The line mirroring functio... |
| lmiisolem 28656 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28657 | The line mirroring functio... |
| lmimot 28658 | Line mirroring is a motion... |
| hypcgrlem1 28659 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28660 | Lemma for ~ hypcgr , case ... |
| hypcgr 28661 | If the catheti of two righ... |
| lmiopp 28662 | Line mirroring produces po... |
| lnperpex 28663 | Existence of a perpendicul... |
| trgcopy 28664 | Triangle construction: a c... |
| trgcopyeulem 28665 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28666 | Triangle construction: a c... |
| iscgra 28669 | Property for two angles AB... |
| iscgra1 28670 | A special version of ~ isc... |
| iscgrad 28671 | Sufficient conditions for ... |
| cgrane1 28672 | Angles imply inequality. ... |
| cgrane2 28673 | Angles imply inequality. ... |
| cgrane3 28674 | Angles imply inequality. ... |
| cgrane4 28675 | Angles imply inequality. ... |
| cgrahl1 28676 | Angle congruence is indepe... |
| cgrahl2 28677 | Angle congruence is indepe... |
| cgracgr 28678 | First direction of proposi... |
| cgraid 28679 | Angle congruence is reflex... |
| cgraswap 28680 | Swap rays in a congruence ... |
| cgrcgra 28681 | Triangle congruence implie... |
| cgracom 28682 | Angle congruence commutes.... |
| cgratr 28683 | Angle congruence is transi... |
| flatcgra 28684 | Flat angles are congruent.... |
| cgraswaplr 28685 | Swap both side of angle co... |
| cgrabtwn 28686 | Angle congruence preserves... |
| cgrahl 28687 | Angle congruence preserves... |
| cgracol 28688 | Angle congruence preserves... |
| cgrancol 28689 | Angle congruence preserves... |
| dfcgra2 28690 | This is the full statement... |
| sacgr 28691 | Supplementary angles of co... |
| oacgr 28692 | Vertical angle theorem. V... |
| acopy 28693 | Angle construction. Theor... |
| acopyeu 28694 | Angle construction. Theor... |
| isinag 28698 | Property for point ` X ` t... |
| isinagd 28699 | Sufficient conditions for ... |
| inagflat 28700 | Any point lies in a flat a... |
| inagswap 28701 | Swap the order of the half... |
| inagne1 28702 | Deduce inequality from the... |
| inagne2 28703 | Deduce inequality from the... |
| inagne3 28704 | Deduce inequality from the... |
| inaghl 28705 | The "point lie in angle" r... |
| isleag 28707 | Geometrical "less than" pr... |
| isleagd 28708 | Sufficient condition for "... |
| leagne1 28709 | Deduce inequality from the... |
| leagne2 28710 | Deduce inequality from the... |
| leagne3 28711 | Deduce inequality from the... |
| leagne4 28712 | Deduce inequality from the... |
| cgrg3col4 28713 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28714 | First congruence theorem: ... |
| tgsas 28715 | First congruence theorem: ... |
| tgsas2 28716 | First congruence theorem: ... |
| tgsas3 28717 | First congruence theorem: ... |
| tgasa1 28718 | Second congruence theorem:... |
| tgasa 28719 | Second congruence theorem:... |
| tgsss1 28720 | Third congruence theorem: ... |
| tgsss2 28721 | Third congruence theorem: ... |
| tgsss3 28722 | Third congruence theorem: ... |
| dfcgrg2 28723 | Congruence for two triangl... |
| isoas 28724 | Congruence theorem for iso... |
| iseqlg 28727 | Property of a triangle bei... |
| iseqlgd 28728 | Condition for a triangle t... |
| f1otrgds 28729 | Convenient lemma for ~ f1o... |
| f1otrgitv 28730 | Convenient lemma for ~ f1o... |
| f1otrg 28731 | A bijection between bases ... |
| f1otrge 28732 | A bijection between bases ... |
| ttgval 28735 | Define a function to augme... |
| ttgvalOLD 28736 | Obsolete proof of ~ ttgval... |
| ttglem 28737 | Lemma for ~ ttgbas , ~ ttg... |
| ttglemOLD 28738 | Obsolete version of ~ ttgl... |
| ttgbas 28739 | The base set of a subcompl... |
| ttgbasOLD 28740 | Obsolete proof of ~ ttgbas... |
| ttgplusg 28741 | The addition operation of ... |
| ttgplusgOLD 28742 | Obsolete proof of ~ ttgplu... |
| ttgsub 28743 | The subtraction operation ... |
| ttgvsca 28744 | The scalar product of a su... |
| ttgvscaOLD 28745 | Obsolete proof of ~ ttgvsc... |
| ttgds 28746 | The metric of a subcomplex... |
| ttgdsOLD 28747 | Obsolete proof of ~ ttgds ... |
| ttgitvval 28748 | Betweenness for a subcompl... |
| ttgelitv 28749 | Betweenness for a subcompl... |
| ttgbtwnid 28750 | Any subcomplex module equi... |
| ttgcontlem1 28751 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28752 | Any metric space fulfills ... |
| cchhllem 28753 | Lemma for chlbas and chlvs... |
| cchhllemOLD 28754 | Obsolete version of ~ cchh... |
| elee 28761 | Membership in a Euclidean ... |
| mptelee 28762 | A condition for a mapping ... |
| eleenn 28763 | If ` A ` is in ` ( EE `` N... |
| eleei 28764 | The forward direction of ~... |
| eedimeq 28765 | A point belongs to at most... |
| brbtwn 28766 | The binary relation form o... |
| brcgr 28767 | The binary relation form o... |
| fveere 28768 | The function value of a po... |
| fveecn 28769 | The function value of a po... |
| eqeefv 28770 | Two points are equal iff t... |
| eqeelen 28771 | Two points are equal iff t... |
| brbtwn2 28772 | Alternate characterization... |
| colinearalglem1 28773 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28774 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28775 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28776 | Lemma for ~ colinearalg . ... |
| colinearalg 28777 | An algebraic characterizat... |
| eleesub 28778 | Membership of a subtractio... |
| eleesubd 28779 | Membership of a subtractio... |
| axdimuniq 28780 | The unique dimension axiom... |
| axcgrrflx 28781 | ` A ` is as far from ` B `... |
| axcgrtr 28782 | Congruence is transitive. ... |
| axcgrid 28783 | If there is no distance be... |
| axsegconlem1 28784 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28785 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28786 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28787 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28788 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28789 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28790 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28791 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28792 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28793 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28794 | Any segment ` A B ` can be... |
| ax5seglem1 28795 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28796 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28797 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28798 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28799 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28800 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28801 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28802 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28803 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28804 | Lemma for ~ ax5seg . Take... |
| ax5seg 28805 | The five segment axiom. T... |
| axbtwnid 28806 | Points are indivisible. T... |
| axpaschlem 28807 | Lemma for ~ axpasch . Set... |
| axpasch 28808 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28809 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 28810 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 28811 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 28812 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 28813 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 28814 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 28815 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 28816 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 28817 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 28818 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 28819 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 28820 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 28821 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 28822 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 28823 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 28824 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 28825 | Lemma for ~ axlowdim . Es... |
| axlowdim1 28826 | The lower dimension axiom ... |
| axlowdim2 28827 | The lower two-dimensional ... |
| axlowdim 28828 | The general lower dimensio... |
| axeuclidlem 28829 | Lemma for ~ axeuclid . Ha... |
| axeuclid 28830 | Euclid's axiom. Take an a... |
| axcontlem1 28831 | Lemma for ~ axcont . Chan... |
| axcontlem2 28832 | Lemma for ~ axcont . The ... |
| axcontlem3 28833 | Lemma for ~ axcont . Give... |
| axcontlem4 28834 | Lemma for ~ axcont . Give... |
| axcontlem5 28835 | Lemma for ~ axcont . Comp... |
| axcontlem6 28836 | Lemma for ~ axcont . Stat... |
| axcontlem7 28837 | Lemma for ~ axcont . Give... |
| axcontlem8 28838 | Lemma for ~ axcont . A po... |
| axcontlem9 28839 | Lemma for ~ axcont . Give... |
| axcontlem10 28840 | Lemma for ~ axcont . Give... |
| axcontlem11 28841 | Lemma for ~ axcont . Elim... |
| axcontlem12 28842 | Lemma for ~ axcont . Elim... |
| axcont 28843 | The axiom of continuity. ... |
| eengv 28846 | The value of the Euclidean... |
| eengstr 28847 | The Euclidean geometry as ... |
| eengbas 28848 | The Base of the Euclidean ... |
| ebtwntg 28849 | The betweenness relation u... |
| ecgrtg 28850 | The congruence relation us... |
| elntg 28851 | The line definition in the... |
| elntg2 28852 | The line definition in the... |
| eengtrkg 28853 | The geometry structure for... |
| eengtrkge 28854 | The geometry structure for... |
| edgfid 28857 | Utility theorem: index-ind... |
| edgfndx 28858 | Index value of the ~ df-ed... |
| edgfndxnn 28859 | The index value of the edg... |
| edgfndxid 28860 | The value of the edge func... |
| edgfndxidOLD 28861 | Obsolete version of ~ edgf... |
| basendxltedgfndx 28862 | The index value of the ` B... |
| baseltedgfOLD 28863 | Obsolete proof of ~ basend... |
| basendxnedgfndx 28864 | The slots ` Base ` and ` .... |
| vtxval 28869 | The set of vertices of a g... |
| iedgval 28870 | The set of indexed edges o... |
| 1vgrex 28871 | A graph with at least one ... |
| opvtxval 28872 | The set of vertices of a g... |
| opvtxfv 28873 | The set of vertices of a g... |
| opvtxov 28874 | The set of vertices of a g... |
| opiedgval 28875 | The set of indexed edges o... |
| opiedgfv 28876 | The set of indexed edges o... |
| opiedgov 28877 | The set of indexed edges o... |
| opvtxfvi 28878 | The set of vertices of a g... |
| opiedgfvi 28879 | The set of indexed edges o... |
| funvtxdmge2val 28880 | The set of vertices of an ... |
| funiedgdmge2val 28881 | The set of indexed edges o... |
| funvtxdm2val 28882 | The set of vertices of an ... |
| funiedgdm2val 28883 | The set of indexed edges o... |
| funvtxval0 28884 | The set of vertices of an ... |
| basvtxval 28885 | The set of vertices of a g... |
| edgfiedgval 28886 | The set of indexed edges o... |
| funvtxval 28887 | The set of vertices of a g... |
| funiedgval 28888 | The set of indexed edges o... |
| structvtxvallem 28889 | Lemma for ~ structvtxval a... |
| structvtxval 28890 | The set of vertices of an ... |
| structiedg0val 28891 | The set of indexed edges o... |
| structgrssvtxlem 28892 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 28893 | The set of vertices of a g... |
| structgrssiedg 28894 | The set of indexed edges o... |
| struct2grstr 28895 | A graph represented as an ... |
| struct2grvtx 28896 | The set of vertices of a g... |
| struct2griedg 28897 | The set of indexed edges o... |
| graop 28898 | Any representation of a gr... |
| grastruct 28899 | Any representation of a gr... |
| gropd 28900 | If any representation of a... |
| grstructd 28901 | If any representation of a... |
| gropeld 28902 | If any representation of a... |
| grstructeld 28903 | If any representation of a... |
| setsvtx 28904 | The vertices of a structur... |
| setsiedg 28905 | The (indexed) edges of a s... |
| snstrvtxval 28906 | The set of vertices of a g... |
| snstriedgval 28907 | The set of indexed edges o... |
| vtxval0 28908 | Degenerated case 1 for ver... |
| iedgval0 28909 | Degenerated case 1 for edg... |
| vtxvalsnop 28910 | Degenerated case 2 for ver... |
| iedgvalsnop 28911 | Degenerated case 2 for edg... |
| vtxval3sn 28912 | Degenerated case 3 for ver... |
| iedgval3sn 28913 | Degenerated case 3 for edg... |
| vtxvalprc 28914 | Degenerated case 4 for ver... |
| iedgvalprc 28915 | Degenerated case 4 for edg... |
| edgval 28918 | The edges of a graph. (Co... |
| iedgedg 28919 | An indexed edge is an edge... |
| edgopval 28920 | The edges of a graph repre... |
| edgov 28921 | The edges of a graph repre... |
| edgstruct 28922 | The edges of a graph repre... |
| edgiedgb 28923 | A set is an edge iff it is... |
| edg0iedg0 28924 | There is no edge in a grap... |
| isuhgr 28929 | The predicate "is an undir... |
| isushgr 28930 | The predicate "is an undir... |
| uhgrf 28931 | The edge function of an un... |
| ushgrf 28932 | The edge function of an un... |
| uhgrss 28933 | An edge is a subset of ver... |
| uhgreq12g 28934 | If two sets have the same ... |
| uhgrfun 28935 | The edge function of an un... |
| uhgrn0 28936 | An edge is a nonempty subs... |
| lpvtx 28937 | The endpoints of a loop (w... |
| ushgruhgr 28938 | An undirected simple hyper... |
| isuhgrop 28939 | The property of being an u... |
| uhgr0e 28940 | The empty graph, with vert... |
| uhgr0vb 28941 | The null graph, with no ve... |
| uhgr0 28942 | The null graph represented... |
| uhgrun 28943 | The union ` U ` of two (un... |
| uhgrunop 28944 | The union of two (undirect... |
| ushgrun 28945 | The union ` U ` of two (un... |
| ushgrunop 28946 | The union of two (undirect... |
| uhgrstrrepe 28947 | Replacing (or adding) the ... |
| incistruhgr 28948 | An _incidence structure_ `... |
| isupgr 28953 | The property of being an u... |
| wrdupgr 28954 | The property of being an u... |
| upgrf 28955 | The edge function of an un... |
| upgrfn 28956 | The edge function of an un... |
| upgrss 28957 | An edge is a subset of ver... |
| upgrn0 28958 | An edge is a nonempty subs... |
| upgrle 28959 | An edge of an undirected p... |
| upgrfi 28960 | An edge is a finite subset... |
| upgrex 28961 | An edge is an unordered pa... |
| upgrbi 28962 | Show that an unordered pai... |
| upgrop 28963 | A pseudograph represented ... |
| isumgr 28964 | The property of being an u... |
| isumgrs 28965 | The simplified property of... |
| wrdumgr 28966 | The property of being an u... |
| umgrf 28967 | The edge function of an un... |
| umgrfn 28968 | The edge function of an un... |
| umgredg2 28969 | An edge of a multigraph ha... |
| umgrbi 28970 | Show that an unordered pai... |
| upgruhgr 28971 | An undirected pseudograph ... |
| umgrupgr 28972 | An undirected multigraph i... |
| umgruhgr 28973 | An undirected multigraph i... |
| upgrle2 28974 | An edge of an undirected p... |
| umgrnloopv 28975 | In a multigraph, there is ... |
| umgredgprv 28976 | In a multigraph, an edge i... |
| umgrnloop 28977 | In a multigraph, there is ... |
| umgrnloop0 28978 | A multigraph has no loops.... |
| umgr0e 28979 | The empty graph, with vert... |
| upgr0e 28980 | The empty graph, with vert... |
| upgr1elem 28981 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 28982 | A pseudograph with one edg... |
| upgr0eop 28983 | The empty graph, with vert... |
| upgr1eop 28984 | A pseudograph with one edg... |
| upgr0eopALT 28985 | Alternate proof of ~ upgr0... |
| upgr1eopALT 28986 | Alternate proof of ~ upgr1... |
| upgrun 28987 | The union ` U ` of two pse... |
| upgrunop 28988 | The union of two pseudogra... |
| umgrun 28989 | The union ` U ` of two mul... |
| umgrunop 28990 | The union of two multigrap... |
| umgrislfupgrlem 28991 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 28992 | A multigraph is a loop-fre... |
| lfgredgge2 28993 | An edge of a loop-free gra... |
| lfgrnloop 28994 | A loop-free graph has no l... |
| uhgredgiedgb 28995 | In a hypergraph, a set is ... |
| uhgriedg0edg0 28996 | A hypergraph has no edges ... |
| uhgredgn0 28997 | An edge of a hypergraph is... |
| edguhgr 28998 | An edge of a hypergraph is... |
| uhgredgrnv 28999 | An edge of a hypergraph co... |
| uhgredgss 29000 | The set of edges of a hype... |
| upgredgss 29001 | The set of edges of a pseu... |
| umgredgss 29002 | The set of edges of a mult... |
| edgupgr 29003 | Properties of an edge of a... |
| edgumgr 29004 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29005 | In a hypergraph, a vertex ... |
| upgredg 29006 | For each edge in a pseudog... |
| umgredg 29007 | For each edge in a multigr... |
| upgrpredgv 29008 | An edge of a pseudograph a... |
| umgrpredgv 29009 | An edge of a multigraph al... |
| upgredg2vtx 29010 | For a vertex incident to a... |
| upgredgpr 29011 | If a proper pair (of verti... |
| edglnl 29012 | The edges incident with a ... |
| numedglnl 29013 | The number of edges incide... |
| umgredgne 29014 | An edge of a multigraph al... |
| umgrnloop2 29015 | A multigraph has no loops.... |
| umgredgnlp 29016 | An edge of a multigraph is... |
| isuspgr 29021 | The property of being a si... |
| isusgr 29022 | The property of being a si... |
| uspgrf 29023 | The edge function of a sim... |
| usgrf 29024 | The edge function of a sim... |
| isusgrs 29025 | The property of being a si... |
| usgrfs 29026 | The edge function of a sim... |
| usgrfun 29027 | The edge function of a sim... |
| usgredgss 29028 | The set of edges of a simp... |
| edgusgr 29029 | An edge of a simple graph ... |
| isuspgrop 29030 | The property of being an u... |
| isusgrop 29031 | The property of being an u... |
| usgrop 29032 | A simple graph represented... |
| isausgr 29033 | The property of an unorder... |
| ausgrusgrb 29034 | The equivalence of the def... |
| usgrausgri 29035 | A simple graph represented... |
| ausgrumgri 29036 | If an alternatively define... |
| ausgrusgri 29037 | The equivalence of the def... |
| usgrausgrb 29038 | The equivalence of the def... |
| usgredgop 29039 | An edge of a simple graph ... |
| usgrf1o 29040 | The edge function of a sim... |
| usgrf1 29041 | The edge function of a sim... |
| uspgrf1oedg 29042 | The edge function of a sim... |
| usgrss 29043 | An edge is a subset of ver... |
| uspgredgiedg 29044 | In a simple pseudograph, f... |
| uspgriedgedg 29045 | In a simple pseudograph, f... |
| uspgrushgr 29046 | A simple pseudograph is an... |
| uspgrupgr 29047 | A simple pseudograph is an... |
| uspgrupgrushgr 29048 | A graph is a simple pseudo... |
| usgruspgr 29049 | A simple graph is a simple... |
| usgrumgr 29050 | A simple graph is an undir... |
| usgrumgruspgr 29051 | A graph is a simple graph ... |
| usgruspgrb 29052 | A class is a simple graph ... |
| uspgruhgr 29053 | An undirected simple pseud... |
| usgrupgr 29054 | A simple graph is an undir... |
| usgruhgr 29055 | A simple graph is an undir... |
| usgrislfuspgr 29056 | A simple graph is a loop-f... |
| uspgrun 29057 | The union ` U ` of two sim... |
| uspgrunop 29058 | The union of two simple ps... |
| usgrun 29059 | The union ` U ` of two sim... |
| usgrunop 29060 | The union of two simple gr... |
| usgredg2 29061 | The value of the "edge fun... |
| usgredg2ALT 29062 | Alternate proof of ~ usgre... |
| usgredgprv 29063 | In a simple graph, an edge... |
| usgredgprvALT 29064 | Alternate proof of ~ usgre... |
| usgredgppr 29065 | An edge of a simple graph ... |
| usgrpredgv 29066 | An edge of a simple graph ... |
| edgssv2 29067 | An edge of a simple graph ... |
| usgredg 29068 | For each edge in a simple ... |
| usgrnloopv 29069 | In a simple graph, there i... |
| usgrnloopvALT 29070 | Alternate proof of ~ usgrn... |
| usgrnloop 29071 | In a simple graph, there i... |
| usgrnloopALT 29072 | Alternate proof of ~ usgrn... |
| usgrnloop0 29073 | A simple graph has no loop... |
| usgrnloop0ALT 29074 | Alternate proof of ~ usgrn... |
| usgredgne 29075 | An edge of a simple graph ... |
| usgrf1oedg 29076 | The edge function of a sim... |
| uhgr2edg 29077 | If a vertex is adjacent to... |
| umgr2edg 29078 | If a vertex is adjacent to... |
| usgr2edg 29079 | If a vertex is adjacent to... |
| umgr2edg1 29080 | If a vertex is adjacent to... |
| usgr2edg1 29081 | If a vertex is adjacent to... |
| umgrvad2edg 29082 | If a vertex is adjacent to... |
| umgr2edgneu 29083 | If a vertex is adjacent to... |
| usgrsizedg 29084 | In a simple graph, the siz... |
| usgredg3 29085 | The value of the "edge fun... |
| usgredg4 29086 | For a vertex incident to a... |
| usgredgreu 29087 | For a vertex incident to a... |
| usgredg2vtx 29088 | For a vertex incident to a... |
| uspgredg2vtxeu 29089 | For a vertex incident to a... |
| usgredg2vtxeu 29090 | For a vertex incident to a... |
| usgredg2vtxeuALT 29091 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29092 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29093 | In a simple pseudograph, t... |
| usgredg2vlem1 29094 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29095 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29096 | In a simple graph, the map... |
| usgriedgleord 29097 | Alternate version of ~ usg... |
| ushgredgedg 29098 | In a simple hypergraph the... |
| usgredgedg 29099 | In a simple graph there is... |
| ushgredgedgloop 29100 | In a simple hypergraph the... |
| uspgredgleord 29101 | In a simple pseudograph th... |
| usgredgleord 29102 | In a simple graph the numb... |
| usgredgleordALT 29103 | Alternate proof for ~ usgr... |
| usgrstrrepe 29104 | Replacing (or adding) the ... |
| usgr0e 29105 | The empty graph, with vert... |
| usgr0vb 29106 | The null graph, with no ve... |
| uhgr0v0e 29107 | The null graph, with no ve... |
| uhgr0vsize0 29108 | The size of a hypergraph w... |
| uhgr0edgfi 29109 | A graph of order 0 (i.e. w... |
| usgr0v 29110 | The null graph, with no ve... |
| uhgr0vusgr 29111 | The null graph, with no ve... |
| usgr0 29112 | The null graph represented... |
| uspgr1e 29113 | A simple pseudograph with ... |
| usgr1e 29114 | A simple graph with one ed... |
| usgr0eop 29115 | The empty graph, with vert... |
| uspgr1eop 29116 | A simple pseudograph with ... |
| uspgr1ewop 29117 | A simple pseudograph with ... |
| uspgr1v1eop 29118 | A simple pseudograph with ... |
| usgr1eop 29119 | A simple graph with (at le... |
| uspgr2v1e2w 29120 | A simple pseudograph with ... |
| usgr2v1e2w 29121 | A simple graph with two ve... |
| edg0usgr 29122 | A class without edges is a... |
| lfuhgr1v0e 29123 | A loop-free hypergraph wit... |
| usgr1vr 29124 | A simple graph with one ve... |
| usgr1v 29125 | A class with one (or no) v... |
| usgr1v0edg 29126 | A class with one (or no) v... |
| usgrexmpldifpr 29127 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29128 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29129 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29130 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29131 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29132 | ` G ` is a simple graph of... |
| griedg0prc 29133 | The class of empty graphs ... |
| griedg0ssusgr 29134 | The class of all simple gr... |
| usgrprc 29135 | The class of simple graphs... |
| relsubgr 29138 | The class of the subgraph ... |
| subgrv 29139 | If a class is a subgraph o... |
| issubgr 29140 | The property of a set to b... |
| issubgr2 29141 | The property of a set to b... |
| subgrprop 29142 | The properties of a subgra... |
| subgrprop2 29143 | The properties of a subgra... |
| uhgrissubgr 29144 | The property of a hypergra... |
| subgrprop3 29145 | The properties of a subgra... |
| egrsubgr 29146 | An empty graph consisting ... |
| 0grsubgr 29147 | The null graph (represente... |
| 0uhgrsubgr 29148 | The null graph (as hypergr... |
| uhgrsubgrself 29149 | A hypergraph is a subgraph... |
| subgrfun 29150 | The edge function of a sub... |
| subgruhgrfun 29151 | The edge function of a sub... |
| subgreldmiedg 29152 | An element of the domain o... |
| subgruhgredgd 29153 | An edge of a subgraph of a... |
| subumgredg2 29154 | An edge of a subgraph of a... |
| subuhgr 29155 | A subgraph of a hypergraph... |
| subupgr 29156 | A subgraph of a pseudograp... |
| subumgr 29157 | A subgraph of a multigraph... |
| subusgr 29158 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29159 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29160 | A spanning subgraph ` S ` ... |
| uhgrspan 29161 | A spanning subgraph ` S ` ... |
| upgrspan 29162 | A spanning subgraph ` S ` ... |
| umgrspan 29163 | A spanning subgraph ` S ` ... |
| usgrspan 29164 | A spanning subgraph ` S ` ... |
| uhgrspanop 29165 | A spanning subgraph of a h... |
| upgrspanop 29166 | A spanning subgraph of a p... |
| umgrspanop 29167 | A spanning subgraph of a m... |
| usgrspanop 29168 | A spanning subgraph of a s... |
| uhgrspan1lem1 29169 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29170 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29171 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29172 | The induced subgraph ` S `... |
| upgrreslem 29173 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29174 | Lemma for ~ umgrres and ~ ... |
| upgrres 29175 | A subgraph obtained by rem... |
| umgrres 29176 | A subgraph obtained by rem... |
| usgrres 29177 | A subgraph obtained by rem... |
| upgrres1lem1 29178 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29179 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29180 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29181 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29182 | A pseudograph obtained by ... |
| umgrres1 29183 | A multigraph obtained by r... |
| usgrres1 29184 | Restricting a simple graph... |
| isfusgr 29187 | The property of being a fi... |
| fusgrvtxfi 29188 | A finite simple graph has ... |
| isfusgrf1 29189 | The property of being a fi... |
| isfusgrcl 29190 | The property of being a fi... |
| fusgrusgr 29191 | A finite simple graph is a... |
| opfusgr 29192 | A finite simple graph repr... |
| usgredgffibi 29193 | The number of edges in a s... |
| fusgredgfi 29194 | In a finite simple graph t... |
| usgr1v0e 29195 | The size of a (finite) sim... |
| usgrfilem 29196 | In a finite simple graph, ... |
| fusgrfisbase 29197 | Induction base for ~ fusgr... |
| fusgrfisstep 29198 | Induction step in ~ fusgrf... |
| fusgrfis 29199 | A finite simple graph is o... |
| fusgrfupgrfs 29200 | A finite simple graph is a... |
| nbgrprc0 29203 | The set of neighbors is em... |
| nbgrcl 29204 | If a class ` X ` has at le... |
| nbgrval 29205 | The set of neighbors of a ... |
| dfnbgr2 29206 | Alternate definition of th... |
| dfnbgr3 29207 | Alternate definition of th... |
| nbgrnvtx0 29208 | If a class ` X ` is not a ... |
| nbgrel 29209 | Characterization of a neig... |
| nbgrisvtx 29210 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29211 | The neighbors of a vertex ... |
| nbuhgr 29212 | The set of neighbors of a ... |
| nbupgr 29213 | The set of neighbors of a ... |
| nbupgrel 29214 | A neighbor of a vertex in ... |
| nbumgrvtx 29215 | The set of neighbors of a ... |
| nbumgr 29216 | The set of neighbors of an... |
| nbusgrvtx 29217 | The set of neighbors of a ... |
| nbusgr 29218 | The set of neighbors of an... |
| nbgr2vtx1edg 29219 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29220 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29221 | If a hypergraph has two ve... |
| nbusgreledg 29222 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29223 | A vertex which is not endp... |
| nbgr0vtx 29224 | In a null graph (with no v... |
| nbgr0edglem 29225 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29226 | In an empty graph (with no... |
| nbgr1vtx 29227 | In a graph with one vertex... |
| nbgrnself 29228 | A vertex in a graph is not... |
| nbgrnself2 29229 | A class ` X ` is not a nei... |
| nbgrssovtx 29230 | The neighbors of a vertex ... |
| nbgrssvwo2 29231 | The neighbors of a vertex ... |
| nbgrsym 29232 | In a graph, the neighborho... |
| nbupgrres 29233 | The neighborhood of a vert... |
| usgrnbcnvfv 29234 | Applying the edge function... |
| nbusgredgeu 29235 | For each neighbor of a ver... |
| edgnbusgreu 29236 | For each edge incident to ... |
| nbusgredgeu0 29237 | For each neighbor of a ver... |
| nbusgrf1o0 29238 | The mapping of neighbors o... |
| nbusgrf1o1 29239 | The set of neighbors of a ... |
| nbusgrf1o 29240 | The set of neighbors of a ... |
| nbedgusgr 29241 | The number of neighbors of... |
| edgusgrnbfin 29242 | The number of neighbors of... |
| nbusgrfi 29243 | The class of neighbors of ... |
| nbfiusgrfi 29244 | The class of neighbors of ... |
| hashnbusgrnn0 29245 | The number of neighbors of... |
| nbfusgrlevtxm1 29246 | The number of neighbors of... |
| nbfusgrlevtxm2 29247 | If there is a vertex which... |
| nbusgrvtxm1 29248 | If the number of neighbors... |
| nb3grprlem1 29249 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29250 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29251 | The neighbors of a vertex ... |
| nb3grpr2 29252 | The neighbors of a vertex ... |
| nb3gr2nb 29253 | If the neighbors of two ve... |
| uvtxval 29256 | The set of all universal v... |
| uvtxel 29257 | A universal vertex, i.e. a... |
| uvtxisvtx 29258 | A universal vertex is a ve... |
| uvtxssvtx 29259 | The set of the universal v... |
| vtxnbuvtx 29260 | A universal vertex has all... |
| uvtxnbgrss 29261 | A universal vertex has all... |
| uvtxnbgrvtx 29262 | A universal vertex is neig... |
| uvtx0 29263 | There is no universal vert... |
| isuvtx 29264 | The set of all universal v... |
| uvtxel1 29265 | Characterization of a univ... |
| uvtx01vtx 29266 | If a graph/class has no ed... |
| uvtx2vtx1edg 29267 | If a graph has two vertice... |
| uvtx2vtx1edgb 29268 | If a hypergraph has two ve... |
| uvtxnbgr 29269 | A universal vertex has all... |
| uvtxnbgrb 29270 | A vertex is universal iff ... |
| uvtxusgr 29271 | The set of all universal v... |
| uvtxusgrel 29272 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29273 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29274 | If the number of neighbors... |
| uvtxnbvtxm1 29275 | A universal vertex has ` n... |
| nbupgruvtxres 29276 | The neighborhood of a univ... |
| uvtxupgrres 29277 | A universal vertex is univ... |
| cplgruvtxb 29282 | A graph ` G ` is complete ... |
| prcliscplgr 29283 | A proper class (representi... |
| iscplgr 29284 | The property of being a co... |
| iscplgrnb 29285 | A graph is complete iff al... |
| iscplgredg 29286 | A graph ` G ` is complete ... |
| iscusgr 29287 | The property of being a co... |
| cusgrusgr 29288 | A complete simple graph is... |
| cusgrcplgr 29289 | A complete simple graph is... |
| iscusgrvtx 29290 | A simple graph is complete... |
| cusgruvtxb 29291 | A simple graph is complete... |
| iscusgredg 29292 | A simple graph is complete... |
| cusgredg 29293 | In a complete simple graph... |
| cplgr0 29294 | The null graph (with no ve... |
| cusgr0 29295 | The null graph (with no ve... |
| cplgr0v 29296 | A null graph (with no vert... |
| cusgr0v 29297 | A graph with no vertices a... |
| cplgr1vlem 29298 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29299 | A graph with one vertex is... |
| cusgr1v 29300 | A graph with one vertex an... |
| cplgr2v 29301 | An undirected hypergraph w... |
| cplgr2vpr 29302 | An undirected hypergraph w... |
| nbcplgr 29303 | In a complete graph, each ... |
| cplgr3v 29304 | A pseudograph with three (... |
| cusgr3vnbpr 29305 | The neighbors of a vertex ... |
| cplgrop 29306 | A complete graph represent... |
| cusgrop 29307 | A complete simple graph re... |
| cusgrexilem1 29308 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29309 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29310 | An arbitrary set regarded ... |
| cusgrexilem2 29311 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29312 | An arbitrary set ` V ` reg... |
| cusgrexg 29313 | For each set there is a se... |
| structtousgr 29314 | Any (extensible) structure... |
| structtocusgr 29315 | Any (extensible) structure... |
| cffldtocusgr 29316 | The field of complex numbe... |
| cffldtocusgrOLD 29317 | Obsolete version of ~ cffl... |
| cusgrres 29318 | Restricting a complete sim... |
| cusgrsizeindb0 29319 | Base case of the induction... |
| cusgrsizeindb1 29320 | Base case of the induction... |
| cusgrsizeindslem 29321 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29322 | Part 1 of induction step i... |
| cusgrsize2inds 29323 | Induction step in ~ cusgrs... |
| cusgrsize 29324 | The size of a finite compl... |
| cusgrfilem1 29325 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29326 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29327 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29328 | If the size of a complete ... |
| usgredgsscusgredg 29329 | A simple graph is a subgra... |
| usgrsscusgr 29330 | A simple graph is a subgra... |
| sizusglecusglem1 29331 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29332 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29333 | The size of a simple graph... |
| fusgrmaxsize 29334 | The maximum size of a fini... |
| vtxdgfval 29337 | The value of the vertex de... |
| vtxdgval 29338 | The degree of a vertex. (... |
| vtxdgfival 29339 | The degree of a vertex for... |
| vtxdgop 29340 | The vertex degree expresse... |
| vtxdgf 29341 | The vertex degree function... |
| vtxdgelxnn0 29342 | The degree of a vertex is ... |
| vtxdg0v 29343 | The degree of a vertex in ... |
| vtxdg0e 29344 | The degree of a vertex in ... |
| vtxdgfisnn0 29345 | The degree of a vertex in ... |
| vtxdgfisf 29346 | The vertex degree function... |
| vtxdeqd 29347 | Equality theorem for the v... |
| vtxduhgr0e 29348 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29349 | The degree of the vertex i... |
| vdumgr0 29350 | A vertex in a multigraph h... |
| vtxdun 29351 | The degree of a vertex in ... |
| vtxdfiun 29352 | The degree of a vertex in ... |
| vtxduhgrun 29353 | The degree of a vertex in ... |
| vtxduhgrfiun 29354 | The degree of a vertex in ... |
| vtxdlfgrval 29355 | The value of the vertex de... |
| vtxdumgrval 29356 | The value of the vertex de... |
| vtxdusgrval 29357 | The value of the vertex de... |
| vtxd0nedgb 29358 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29359 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29360 | The value of the vertex de... |
| vtxdusgrfvedg 29361 | The value of the vertex de... |
| vtxduhgr0nedg 29362 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29363 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29364 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29365 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29366 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29367 | The vertex degree function... |
| vtxdgfusgr 29368 | In a finite simple graph, ... |
| fusgrn0degnn0 29369 | In a nonempty, finite grap... |
| 1loopgruspgr 29370 | A graph with one edge whic... |
| 1loopgredg 29371 | The set of edges in a grap... |
| 1loopgrnb0 29372 | In a graph (simple pseudog... |
| 1loopgrvd2 29373 | The vertex degree of a one... |
| 1loopgrvd0 29374 | The vertex degree of a one... |
| 1hevtxdg0 29375 | The vertex degree of verte... |
| 1hevtxdg1 29376 | The vertex degree of verte... |
| 1hegrvtxdg1 29377 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29378 | The vertex degree of a gra... |
| 1egrvtxdg1 29379 | The vertex degree of a one... |
| 1egrvtxdg1r 29380 | The vertex degree of a one... |
| 1egrvtxdg0 29381 | The vertex degree of a one... |
| p1evtxdeqlem 29382 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29383 | If an edge ` E ` which doe... |
| p1evtxdp1 29384 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29385 | The set of vertices in a g... |
| uspgrloopvtxel 29386 | A vertex in a graph (simpl... |
| uspgrloopiedg 29387 | The set of edges in a grap... |
| uspgrloopedg 29388 | The set of edges in a grap... |
| uspgrloopnb0 29389 | In a graph (simple pseudog... |
| uspgrloopvd2 29390 | The vertex degree of a one... |
| umgr2v2evtx 29391 | The set of vertices in a m... |
| umgr2v2evtxel 29392 | A vertex in a multigraph w... |
| umgr2v2eiedg 29393 | The edge function in a mul... |
| umgr2v2eedg 29394 | The set of edges in a mult... |
| umgr2v2e 29395 | A multigraph with two edge... |
| umgr2v2enb1 29396 | In a multigraph with two e... |
| umgr2v2evd2 29397 | In a multigraph with two e... |
| hashnbusgrvd 29398 | In a simple graph, the num... |
| usgruvtxvdb 29399 | In a finite simple graph w... |
| vdiscusgrb 29400 | A finite simple graph with... |
| vdiscusgr 29401 | In a finite complete simpl... |
| vtxdusgradjvtx 29402 | The degree of a vertex in ... |
| usgrvd0nedg 29403 | If a vertex in a simple gr... |
| uhgrvd00 29404 | If every vertex in a hyper... |
| usgrvd00 29405 | If every vertex in a simpl... |
| vdegp1ai 29406 | The induction step for a v... |
| vdegp1bi 29407 | The induction step for a v... |
| vdegp1ci 29408 | The induction step for a v... |
| vtxdginducedm1lem1 29409 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29410 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29411 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29412 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29413 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29414 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29415 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29416 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29417 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29418 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29419 | Induction step of ~ finsum... |
| finsumvtxdg2size 29420 | The sum of the degrees of ... |
| fusgr1th 29421 | The sum of the degrees of ... |
| finsumvtxdgeven 29422 | The sum of the degrees of ... |
| vtxdgoddnumeven 29423 | The number of vertices of ... |
| fusgrvtxdgonume 29424 | The number of vertices of ... |
| isrgr 29429 | The property of a class be... |
| rgrprop 29430 | The properties of a k-regu... |
| isrusgr 29431 | The property of being a k-... |
| rusgrprop 29432 | The properties of a k-regu... |
| rusgrrgr 29433 | A k-regular simple graph i... |
| rusgrusgr 29434 | A k-regular simple graph i... |
| finrusgrfusgr 29435 | A finite regular simple gr... |
| isrusgr0 29436 | The property of being a k-... |
| rusgrprop0 29437 | The properties of a k-regu... |
| usgreqdrusgr 29438 | If all vertices in a simpl... |
| fusgrregdegfi 29439 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29440 | If all vertices in a nonem... |
| frusgrnn0 29441 | In a nonempty finite k-reg... |
| 0edg0rgr 29442 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29443 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29444 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29445 | A simple graph is 0-regula... |
| 0vtxrgr 29446 | A null graph (with no vert... |
| 0vtxrusgr 29447 | A graph with no vertices a... |
| 0uhgrrusgr 29448 | The null graph as hypergra... |
| 0grrusgr 29449 | The null graph represented... |
| 0grrgr 29450 | The null graph represented... |
| cusgrrusgr 29451 | A complete simple graph wi... |
| cusgrm1rusgr 29452 | A finite simple graph with... |
| rusgrpropnb 29453 | The properties of a k-regu... |
| rusgrpropedg 29454 | The properties of a k-regu... |
| rusgrpropadjvtx 29455 | The properties of a k-regu... |
| rusgrnumwrdl2 29456 | In a k-regular simple grap... |
| rusgr1vtxlem 29457 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29458 | If a k-regular simple grap... |
| rgrusgrprc 29459 | The class of 0-regular sim... |
| rusgrprc 29460 | The class of 0-regular sim... |
| rgrprc 29461 | The class of 0-regular gra... |
| rgrprcx 29462 | The class of 0-regular gra... |
| rgrx0ndm 29463 | 0 is not in the domain of ... |
| rgrx0nd 29464 | The potentially alternativ... |
| ewlksfval 29471 | The set of s-walks of edge... |
| isewlk 29472 | Conditions for a function ... |
| ewlkprop 29473 | Properties of an s-walk of... |
| ewlkinedg 29474 | The intersection (common v... |
| ewlkle 29475 | An s-walk of edges is also... |
| upgrewlkle2 29476 | In a pseudograph, there is... |
| wkslem1 29477 | Lemma 1 for walks to subst... |
| wkslem2 29478 | Lemma 2 for walks to subst... |
| wksfval 29479 | The set of walks (in an un... |
| iswlk 29480 | Properties of a pair of fu... |
| wlkprop 29481 | Properties of a walk. (Co... |
| wlkv 29482 | The classes involved in a ... |
| iswlkg 29483 | Generalization of ~ iswlk ... |
| wlkf 29484 | The mapping enumerating th... |
| wlkcl 29485 | A walk has length ` # ( F ... |
| wlkp 29486 | The mapping enumerating th... |
| wlkpwrd 29487 | The sequence of vertices o... |
| wlklenvp1 29488 | The number of vertices of ... |
| wksv 29489 | The class of walks is a se... |
| wksvOLD 29490 | Obsolete version of ~ wksv... |
| wlkn0 29491 | The sequence of vertices o... |
| wlklenvm1 29492 | The number of edges of a w... |
| ifpsnprss 29493 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29494 | Each pair of adjacent vert... |
| wlkvtxiedg 29495 | The vertices of a walk are... |
| relwlk 29496 | The set ` ( Walks `` G ) `... |
| wlkvv 29497 | If there is at least one w... |
| wlkop 29498 | A walk is an ordered pair.... |
| wlkcpr 29499 | A walk as class with two c... |
| wlk2f 29500 | If there is a walk ` W ` t... |
| wlkcomp 29501 | A walk expressed by proper... |
| wlkcompim 29502 | Implications for the prope... |
| wlkelwrd 29503 | The components of a walk a... |
| wlkeq 29504 | Conditions for two walks (... |
| edginwlk 29505 | The value of the edge func... |
| upgredginwlk 29506 | The value of the edge func... |
| iedginwlk 29507 | The value of the edge func... |
| wlkl1loop 29508 | A walk of length 1 from a ... |
| wlk1walk 29509 | A walk is a 1-walk "on the... |
| wlk1ewlk 29510 | A walk is an s-walk "on th... |
| upgriswlk 29511 | Properties of a pair of fu... |
| upgrwlkedg 29512 | The edges of a walk in a p... |
| upgrwlkcompim 29513 | Implications for the prope... |
| wlkvtxedg 29514 | The vertices of a walk are... |
| upgrwlkvtxedg 29515 | The pairs of connected ver... |
| uspgr2wlkeq 29516 | Conditions for two walks w... |
| uspgr2wlkeq2 29517 | Conditions for two walks w... |
| uspgr2wlkeqi 29518 | Conditions for two walks w... |
| umgrwlknloop 29519 | In a multigraph, each walk... |
| wlkResOLD 29520 | Obsolete version of ~ opab... |
| wlkv0 29521 | If there is a walk in the ... |
| g0wlk0 29522 | There is no walk in a null... |
| 0wlk0 29523 | There is no walk for the e... |
| wlk0prc 29524 | There is no walk in a null... |
| wlklenvclwlk 29525 | The number of vertices in ... |
| wlkson 29526 | The set of walks between t... |
| iswlkon 29527 | Properties of a pair of fu... |
| wlkonprop 29528 | Properties of a walk betwe... |
| wlkpvtx 29529 | A walk connects vertices. ... |
| wlkepvtx 29530 | The endpoints of a walk ar... |
| wlkoniswlk 29531 | A walk between two vertice... |
| wlkonwlk 29532 | A walk is a walk between i... |
| wlkonwlk1l 29533 | A walk is a walk from its ... |
| wlksoneq1eq2 29534 | Two walks with identical s... |
| wlkonl1iedg 29535 | If there is a walk between... |
| wlkon2n0 29536 | The length of a walk betwe... |
| 2wlklem 29537 | Lemma for theorems for wal... |
| upgr2wlk 29538 | Properties of a pair of fu... |
| wlkreslem 29539 | Lemma for ~ wlkres . (Con... |
| wlkres 29540 | The restriction ` <. H , Q... |
| redwlklem 29541 | Lemma for ~ redwlk . (Con... |
| redwlk 29542 | A walk ending at the last ... |
| wlkp1lem1 29543 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29544 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29545 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29546 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29547 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29548 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29549 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29550 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29551 | Append one path segment (e... |
| wlkdlem1 29552 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29553 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29554 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29555 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29556 | Two words representing a w... |
| lfgrwlkprop 29557 | Two adjacent vertices in a... |
| lfgriswlk 29558 | Conditions for a pair of f... |
| lfgrwlknloop 29559 | In a loop-free graph, each... |
| reltrls 29564 | The set ` ( Trails `` G ) ... |
| trlsfval 29565 | The set of trails (in an u... |
| istrl 29566 | Conditions for a pair of c... |
| trliswlk 29567 | A trail is a walk. (Contr... |
| trlf1 29568 | The enumeration ` F ` of a... |
| trlreslem 29569 | Lemma for ~ trlres . Form... |
| trlres 29570 | The restriction ` <. H , Q... |
| upgrtrls 29571 | The set of trails in a pse... |
| upgristrl 29572 | Properties of a pair of fu... |
| upgrf1istrl 29573 | Properties of a pair of a ... |
| wksonproplem 29574 | Lemma for theorems for pro... |
| wksonproplemOLD 29575 | Obsolete version of ~ wkso... |
| trlsonfval 29576 | The set of trails between ... |
| istrlson 29577 | Properties of a pair of fu... |
| trlsonprop 29578 | Properties of a trail betw... |
| trlsonistrl 29579 | A trail between two vertic... |
| trlsonwlkon 29580 | A trail between two vertic... |
| trlontrl 29581 | A trail is a trail between... |
| relpths 29590 | The set ` ( Paths `` G ) `... |
| pthsfval 29591 | The set of paths (in an un... |
| spthsfval 29592 | The set of simple paths (i... |
| ispth 29593 | Conditions for a pair of c... |
| isspth 29594 | Conditions for a pair of c... |
| pthistrl 29595 | A path is a trail (in an u... |
| spthispth 29596 | A simple path is a path (i... |
| pthiswlk 29597 | A path is a walk (in an un... |
| spthiswlk 29598 | A simple path is a walk (i... |
| pthdivtx 29599 | The inner vertices of a pa... |
| pthdadjvtx 29600 | The adjacent vertices of a... |
| 2pthnloop 29601 | A path of length at least ... |
| upgr2pthnlp 29602 | A path of length at least ... |
| spthdifv 29603 | The vertices of a simple p... |
| spthdep 29604 | A simple path (at least of... |
| pthdepisspth 29605 | A path with different star... |
| upgrwlkdvdelem 29606 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29607 | In a pseudograph, all edge... |
| upgrspthswlk 29608 | The set of simple paths in... |
| upgrwlkdvspth 29609 | A walk consisting of diffe... |
| pthsonfval 29610 | The set of paths between t... |
| spthson 29611 | The set of simple paths be... |
| ispthson 29612 | Properties of a pair of fu... |
| isspthson 29613 | Properties of a pair of fu... |
| pthsonprop 29614 | Properties of a path betwe... |
| spthonprop 29615 | Properties of a simple pat... |
| pthonispth 29616 | A path between two vertice... |
| pthontrlon 29617 | A path between two vertice... |
| pthonpth 29618 | A path is a path between i... |
| isspthonpth 29619 | A pair of functions is a s... |
| spthonisspth 29620 | A simple path between to v... |
| spthonpthon 29621 | A simple path between two ... |
| spthonepeq 29622 | The endpoints of a simple ... |
| uhgrwkspthlem1 29623 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29624 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29625 | Any walk of length 1 betwe... |
| usgr2wlkneq 29626 | The vertices and edges are... |
| usgr2wlkspthlem1 29627 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29628 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29629 | In a simple graph, any wal... |
| usgr2trlncl 29630 | In a simple graph, any tra... |
| usgr2trlspth 29631 | In a simple graph, any tra... |
| usgr2pthspth 29632 | In a simple graph, any pat... |
| usgr2pthlem 29633 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29634 | In a simple graph, there i... |
| usgr2pth0 29635 | In a simply graph, there i... |
| pthdlem1 29636 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29637 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29638 | Lemma 2 for ~ pthd . (Con... |
| pthd 29639 | Two words representing a t... |
| clwlks 29642 | The set of closed walks (i... |
| isclwlk 29643 | A pair of functions repres... |
| clwlkiswlk 29644 | A closed walk is a walk (i... |
| clwlkwlk 29645 | Closed walks are walks (in... |
| clwlkswks 29646 | Closed walks are walks (in... |
| isclwlke 29647 | Properties of a pair of fu... |
| isclwlkupgr 29648 | Properties of a pair of fu... |
| clwlkcomp 29649 | A closed walk expressed by... |
| clwlkcompim 29650 | Implications for the prope... |
| upgrclwlkcompim 29651 | Implications for the prope... |
| clwlkcompbp 29652 | Basic properties of the co... |
| clwlkl1loop 29653 | A closed walk of length 1 ... |
| crcts 29658 | The set of circuits (in an... |
| cycls 29659 | The set of cycles (in an u... |
| iscrct 29660 | Sufficient and necessary c... |
| iscycl 29661 | Sufficient and necessary c... |
| crctprop 29662 | The properties of a circui... |
| cyclprop 29663 | The properties of a cycle:... |
| crctisclwlk 29664 | A circuit is a closed walk... |
| crctistrl 29665 | A circuit is a trail. (Co... |
| crctiswlk 29666 | A circuit is a walk. (Con... |
| cyclispth 29667 | A cycle is a path. (Contr... |
| cycliswlk 29668 | A cycle is a walk. (Contr... |
| cycliscrct 29669 | A cycle is a circuit. (Co... |
| cyclnspth 29670 | A (non-trivial) cycle is n... |
| cyclispthon 29671 | A cycle is a path starting... |
| lfgrn1cycl 29672 | In a loop-free graph there... |
| usgr2trlncrct 29673 | In a simple graph, any tra... |
| umgrn1cycl 29674 | In a multigraph graph (wit... |
| uspgrn2crct 29675 | In a simple pseudograph th... |
| usgrn2cycl 29676 | In a simple graph there ar... |
| crctcshwlkn0lem1 29677 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29678 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29679 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29680 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29681 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29682 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29683 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29684 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29685 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29686 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29687 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29688 | Cyclically shifting the in... |
| crctcshwlk 29689 | Cyclically shifting the in... |
| crctcshtrl 29690 | Cyclically shifting the in... |
| crctcsh 29691 | Cyclically shifting the in... |
| wwlks 29702 | The set of walks (in an un... |
| iswwlks 29703 | A word over the set of ver... |
| wwlksn 29704 | The set of walks (in an un... |
| iswwlksn 29705 | A word over the set of ver... |
| wwlksnprcl 29706 | Derivation of the length o... |
| iswwlksnx 29707 | Properties of a word to re... |
| wwlkbp 29708 | Basic properties of a walk... |
| wwlknbp 29709 | Basic properties of a walk... |
| wwlknp 29710 | Properties of a set being ... |
| wwlknbp1 29711 | Other basic properties of ... |
| wwlknvtx 29712 | The symbols of a word ` W ... |
| wwlknllvtx 29713 | If a word ` W ` represents... |
| wwlknlsw 29714 | If a word represents a wal... |
| wspthsn 29715 | The set of simple paths of... |
| iswspthn 29716 | An element of the set of s... |
| wspthnp 29717 | Properties of a set being ... |
| wwlksnon 29718 | The set of walks of a fixe... |
| wspthsnon 29719 | The set of simple paths of... |
| iswwlksnon 29720 | The set of walks of a fixe... |
| wwlksnon0 29721 | Sufficient conditions for ... |
| wwlksonvtx 29722 | If a word ` W ` represents... |
| iswspthsnon 29723 | The set of simple paths of... |
| wwlknon 29724 | An element of the set of w... |
| wspthnon 29725 | An element of the set of s... |
| wspthnonp 29726 | Properties of a set being ... |
| wspthneq1eq2 29727 | Two simple paths with iden... |
| wwlksn0s 29728 | The set of all walks as wo... |
| wwlkssswrd 29729 | Walks (represented by word... |
| wwlksn0 29730 | A walk of length 0 is repr... |
| 0enwwlksnge1 29731 | In graphs without edges, t... |
| wwlkswwlksn 29732 | A walk of a fixed length a... |
| wwlkssswwlksn 29733 | The walks of a fixed lengt... |
| wlkiswwlks1 29734 | The sequence of vertices i... |
| wlklnwwlkln1 29735 | The sequence of vertices i... |
| wlkiswwlks2lem1 29736 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29737 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29738 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29739 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29740 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29741 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29742 | A walk as word corresponds... |
| wlkiswwlks 29743 | A walk as word corresponds... |
| wlkiswwlksupgr2 29744 | A walk as word corresponds... |
| wlkiswwlkupgr 29745 | A walk as word corresponds... |
| wlkswwlksf1o 29746 | The mapping of (ordinary) ... |
| wlkswwlksen 29747 | The set of walks as words ... |
| wwlksm1edg 29748 | Removing the trailing edge... |
| wlklnwwlkln2lem 29749 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29750 | A walk of length ` N ` as ... |
| wlklnwwlkn 29751 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29752 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29753 | A walk of length ` N ` as ... |
| wlknewwlksn 29754 | If a walk in a pseudograph... |
| wlknwwlksnbij 29755 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29756 | In a simple pseudograph, t... |
| wlknwwlksneqs 29757 | The set of walks of a fixe... |
| wwlkseq 29758 | Equality of two walks (as ... |
| wwlksnred 29759 | Reduction of a walk (as wo... |
| wwlksnext 29760 | Extension of a walk (as wo... |
| wwlksnextbi 29761 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29762 | For each walk (as word) of... |
| wwlksnredwwlkn0 29763 | For each walk (as word) of... |
| wwlksnextwrd 29764 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29765 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29766 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29767 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29768 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29769 | There is a bijection betwe... |
| wwlksnexthasheq 29770 | The number of the extensio... |
| disjxwwlksn 29771 | Sets of walks (as words) e... |
| wwlksnndef 29772 | Conditions for ` WWalksN `... |
| wwlksnfi 29773 | The number of walks repres... |
| wlksnfi 29774 | The number of walks of fix... |
| wlksnwwlknvbij 29775 | There is a bijection betwe... |
| wwlksnextproplem1 29776 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29777 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29778 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29779 | Adding additional properti... |
| disjxwwlkn 29780 | Sets of walks (as words) e... |
| hashwwlksnext 29781 | Number of walks (as words)... |
| wwlksnwwlksnon 29782 | A walk of fixed length is ... |
| wspthsnwspthsnon 29783 | A simple path of fixed len... |
| wspthsnonn0vne 29784 | If the set of simple paths... |
| wspthsswwlkn 29785 | The set of simple paths of... |
| wspthnfi 29786 | In a finite graph, the set... |
| wwlksnonfi 29787 | In a finite graph, the set... |
| wspthsswwlknon 29788 | The set of simple paths of... |
| wspthnonfi 29789 | In a finite graph, the set... |
| wspniunwspnon 29790 | The set of nonempty simple... |
| wspn0 29791 | If there are no vertices, ... |
| 2wlkdlem1 29792 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29793 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29794 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29795 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29796 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29797 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29798 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29799 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29800 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29801 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29802 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29803 | Construction of a walk fro... |
| 2wlkond 29804 | A walk of length 2 from on... |
| 2trld 29805 | Construction of a trail fr... |
| 2trlond 29806 | A trail of length 2 from o... |
| 2pthd 29807 | A path of length 2 from on... |
| 2spthd 29808 | A simple path of length 2 ... |
| 2pthond 29809 | A simple path of length 2 ... |
| 2pthon3v 29810 | For a vertex adjacent to t... |
| umgr2adedgwlklem 29811 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 29812 | In a multigraph, two adjac... |
| umgr2adedgwlkon 29813 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 29814 | Alternate proof for ~ umgr... |
| umgr2adedgspth 29815 | In a multigraph, two adjac... |
| umgr2wlk 29816 | In a multigraph, there is ... |
| umgr2wlkon 29817 | For each pair of adjacent ... |
| elwwlks2s3 29818 | A walk of length 2 as word... |
| midwwlks2s3 29819 | There is a vertex between ... |
| wwlks2onv 29820 | If a length 3 string repre... |
| elwwlks2ons3im 29821 | A walk as word of length 2... |
| elwwlks2ons3 29822 | For each walk of length 2 ... |
| s3wwlks2on 29823 | A length 3 string which re... |
| umgrwwlks2on 29824 | A walk of length 2 between... |
| wwlks2onsym 29825 | There is a walk of length ... |
| elwwlks2on 29826 | A walk of length 2 between... |
| elwspths2on 29827 | A simple path of length 2 ... |
| wpthswwlks2on 29828 | For two different vertices... |
| 2wspdisj 29829 | All simple paths of length... |
| 2wspiundisj 29830 | All simple paths of length... |
| usgr2wspthons3 29831 | A simple path of length 2 ... |
| usgr2wspthon 29832 | A simple path of length 2 ... |
| elwwlks2 29833 | A walk of length 2 between... |
| elwspths2spth 29834 | A simple path of length 2 ... |
| rusgrnumwwlkl1 29835 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 29836 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 29837 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 29838 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 29839 | Induction base 1 for ~ rus... |
| rusgr0edg 29840 | Special case for graphs wi... |
| rusgrnumwwlks 29841 | Induction step for ~ rusgr... |
| rusgrnumwwlk 29842 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 29843 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 29844 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 29845 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 29846 | In a ` K `-regular graph, ... |
| clwwlk 29849 | The set of closed walks (i... |
| isclwwlk 29850 | Properties of a word to re... |
| clwwlkbp 29851 | Basic properties of a clos... |
| clwwlkgt0 29852 | There is no empty closed w... |
| clwwlksswrd 29853 | Closed walks (represented ... |
| clwwlk1loop 29854 | A closed walk of length 1 ... |
| clwwlkccatlem 29855 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 29856 | The concatenation of two w... |
| umgrclwwlkge2 29857 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 29858 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 29859 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 29860 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 29861 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 29862 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 29863 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 29864 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 29865 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 29866 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 29867 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 29868 | A closed walk as word of l... |
| clwlkclwwlk2 29869 | A closed walk corresponds ... |
| clwlkclwwlkflem 29870 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 29871 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 29872 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 29873 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 29874 | ` F ` is a function from t... |
| clwlkclwwlkfo 29875 | ` F ` is a function from t... |
| clwlkclwwlkf1 29876 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 29877 | ` F ` is a bijection betwe... |
| clwlkclwwlken 29878 | The set of the nonempty cl... |
| clwwisshclwwslemlem 29879 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 29880 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 29881 | Cyclically shifting a clos... |
| clwwisshclwwsn 29882 | Cyclically shifting a clos... |
| erclwwlkrel 29883 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 29884 | Two classes are equivalent... |
| erclwwlkeqlen 29885 | If two classes are equival... |
| erclwwlkref 29886 | ` .~ ` is a reflexive rela... |
| erclwwlksym 29887 | ` .~ ` is a symmetric rela... |
| erclwwlktr 29888 | ` .~ ` is a transitive rel... |
| erclwwlk 29889 | ` .~ ` is an equivalence r... |
| clwwlkn 29892 | The set of closed walks of... |
| isclwwlkn 29893 | A word over the set of ver... |
| clwwlkn0 29894 | There is no closed walk of... |
| clwwlkneq0 29895 | Sufficient conditions for ... |
| clwwlkclwwlkn 29896 | A closed walk of a fixed l... |
| clwwlksclwwlkn 29897 | The closed walks of a fixe... |
| clwwlknlen 29898 | The length of a word repre... |
| clwwlknnn 29899 | The length of a closed wal... |
| clwwlknwrd 29900 | A closed walk of a fixed l... |
| clwwlknbp 29901 | Basic properties of a clos... |
| isclwwlknx 29902 | Characterization of a word... |
| clwwlknp 29903 | Properties of a set being ... |
| clwwlknwwlksn 29904 | A word representing a clos... |
| clwwlknlbonbgr1 29905 | The last but one vertex in... |
| clwwlkinwwlk 29906 | If the initial vertex of a... |
| clwwlkn1 29907 | A closed walk of length 1 ... |
| loopclwwlkn1b 29908 | The singleton word consist... |
| clwwlkn1loopb 29909 | A word represents a closed... |
| clwwlkn2 29910 | A closed walk of length 2 ... |
| clwwlknfi 29911 | If there is only a finite ... |
| clwwlkel 29912 | Obtaining a closed walk (a... |
| clwwlkf 29913 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 29914 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 29915 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 29916 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 29917 | F is a 1-1 onto function, ... |
| clwwlken 29918 | The set of closed walks of... |
| clwwlknwwlkncl 29919 | Obtaining a closed walk (a... |
| clwwlkwwlksb 29920 | A nonempty word over verti... |
| clwwlknwwlksnb 29921 | A word over vertices repre... |
| clwwlkext2edg 29922 | If a word concatenated wit... |
| wwlksext2clwwlk 29923 | If a word represents a wal... |
| wwlksubclwwlk 29924 | Any prefix of a word repre... |
| clwwnisshclwwsn 29925 | Cyclically shifting a clos... |
| eleclclwwlknlem1 29926 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 29927 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 29928 | The set of cyclical shifts... |
| clwwlknccat 29929 | The concatenation of two w... |
| umgr2cwwk2dif 29930 | If a word represents a clo... |
| umgr2cwwkdifex 29931 | If a word represents a clo... |
| erclwwlknrel 29932 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 29933 | Two classes are equivalent... |
| erclwwlkneqlen 29934 | If two classes are equival... |
| erclwwlknref 29935 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 29936 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 29937 | ` .~ ` is a transitive rel... |
| erclwwlkn 29938 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 29939 | The quotient set of the se... |
| hashclwwlkn0 29940 | The number of closed walks... |
| eclclwwlkn1 29941 | An equivalence class accor... |
| eleclclwwlkn 29942 | A member of an equivalence... |
| hashecclwwlkn1 29943 | The size of every equivale... |
| umgrhashecclwwlk 29944 | The size of every equivale... |
| fusgrhashclwwlkn 29945 | The size of the set of clo... |
| clwwlkndivn 29946 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 29947 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 29948 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 29949 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 29950 | There is a one-to-one onto... |
| clwlkssizeeq 29951 | The size of the set of clo... |
| clwlksndivn 29952 | The size of the set of clo... |
| clwwlknonmpo 29955 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 29956 | The set of closed walks on... |
| isclwwlknon 29957 | A word over the set of ver... |
| clwwlk0on0 29958 | There is no word over the ... |
| clwwlknon0 29959 | Sufficient conditions for ... |
| clwwlknonfin 29960 | In a finite graph ` G ` , ... |
| clwwlknonel 29961 | Characterization of a word... |
| clwwlknonccat 29962 | The concatenation of two w... |
| clwwlknon1 29963 | The set of closed walks on... |
| clwwlknon1loop 29964 | If there is a loop at vert... |
| clwwlknon1nloop 29965 | If there is no loop at ver... |
| clwwlknon1sn 29966 | The set of (closed) walks ... |
| clwwlknon1le1 29967 | There is at most one (clos... |
| clwwlknon2 29968 | The set of closed walks on... |
| clwwlknon2x 29969 | The set of closed walks on... |
| s2elclwwlknon2 29970 | Sufficient conditions of a... |
| clwwlknon2num 29971 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 29972 | A word over vertices repre... |
| clwwlknonex2lem1 29973 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 29974 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 29975 | Extending a closed walk ` ... |
| clwwlknonex2e 29976 | Extending a closed walk ` ... |
| clwwlknondisj 29977 | The sets of closed walks o... |
| clwwlknun 29978 | The set of closed walks of... |
| clwwlkvbij 29979 | There is a bijection betwe... |
| 0ewlk 29980 | The empty set (empty seque... |
| 1ewlk 29981 | A sequence of 1 edge is an... |
| 0wlk 29982 | A pair of an empty set (of... |
| is0wlk 29983 | A pair of an empty set (of... |
| 0wlkonlem1 29984 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 29985 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 29986 | A walk of length 0 from a ... |
| 0wlkons1 29987 | A walk of length 0 from a ... |
| 0trl 29988 | A pair of an empty set (of... |
| is0trl 29989 | A pair of an empty set (of... |
| 0trlon 29990 | A trail of length 0 from a... |
| 0pth 29991 | A pair of an empty set (of... |
| 0spth 29992 | A pair of an empty set (of... |
| 0pthon 29993 | A path of length 0 from a ... |
| 0pthon1 29994 | A path of length 0 from a ... |
| 0pthonv 29995 | For each vertex there is a... |
| 0clwlk 29996 | A pair of an empty set (of... |
| 0clwlkv 29997 | Any vertex (more precisely... |
| 0clwlk0 29998 | There is no closed walk in... |
| 0crct 29999 | A pair of an empty set (of... |
| 0cycl 30000 | A pair of an empty set (of... |
| 1pthdlem1 30001 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30002 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30003 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30004 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30005 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30006 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30007 | In a graph with two vertic... |
| 1trld 30008 | In a graph with two vertic... |
| 1pthd 30009 | In a graph with two vertic... |
| 1pthond 30010 | In a graph with two vertic... |
| upgr1wlkdlem1 30011 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30012 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30013 | In a pseudograph with two ... |
| upgr1trld 30014 | In a pseudograph with two ... |
| upgr1pthd 30015 | In a pseudograph with two ... |
| upgr1pthond 30016 | In a pseudograph with two ... |
| lppthon 30017 | A loop (which is an edge a... |
| lp1cycl 30018 | A loop (which is an edge a... |
| 1pthon2v 30019 | For each pair of adjacent ... |
| 1pthon2ve 30020 | For each pair of adjacent ... |
| wlk2v2elem1 30021 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30022 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30023 | In a graph with two vertic... |
| ntrl2v2e 30024 | A walk which is not a trai... |
| 3wlkdlem1 30025 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30026 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30027 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30028 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30029 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30030 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30031 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30032 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30033 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30034 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30035 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30036 | Construction of a walk fro... |
| 3wlkond 30037 | A walk of length 3 from on... |
| 3trld 30038 | Construction of a trail fr... |
| 3trlond 30039 | A trail of length 3 from o... |
| 3pthd 30040 | A path of length 3 from on... |
| 3pthond 30041 | A path of length 3 from on... |
| 3spthd 30042 | A simple path of length 3 ... |
| 3spthond 30043 | A simple path of length 3 ... |
| 3cycld 30044 | Construction of a 3-cycle ... |
| 3cyclpd 30045 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30046 | If there is a cycle of len... |
| uhgr3cyclexlem 30047 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30048 | If there are three differe... |
| umgr3cyclex 30049 | If there are three (differ... |
| umgr3v3e3cycl 30050 | If and only if there is a ... |
| upgr4cycl4dv4e 30051 | If there is a cycle of len... |
| dfconngr1 30054 | Alternative definition of ... |
| isconngr 30055 | The property of being a co... |
| isconngr1 30056 | The property of being a co... |
| cusconngr 30057 | A complete hypergraph is c... |
| 0conngr 30058 | A graph without vertices i... |
| 0vconngr 30059 | A graph without vertices i... |
| 1conngr 30060 | A graph with (at most) one... |
| conngrv2edg 30061 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30062 | A vertex in a connected mu... |
| releupth 30065 | The set ` ( EulerPaths `` ... |
| eupths 30066 | The Eulerian paths on the ... |
| iseupth 30067 | The property " ` <. F , P ... |
| iseupthf1o 30068 | The property " ` <. F , P ... |
| eupthi 30069 | Properties of an Eulerian ... |
| eupthf1o 30070 | The ` F ` function in an E... |
| eupthfi 30071 | Any graph with an Eulerian... |
| eupthseg 30072 | The ` N ` -th edge in an e... |
| upgriseupth 30073 | The property " ` <. F , P ... |
| upgreupthi 30074 | Properties of an Eulerian ... |
| upgreupthseg 30075 | The ` N ` -th edge in an e... |
| eupthcl 30076 | An Eulerian path has lengt... |
| eupthistrl 30077 | An Eulerian path is a trai... |
| eupthiswlk 30078 | An Eulerian path is a walk... |
| eupthpf 30079 | The ` P ` function in an E... |
| eupth0 30080 | There is an Eulerian path ... |
| eupthres 30081 | The restriction ` <. H , Q... |
| eupthp1 30082 | Append one path segment to... |
| eupth2eucrct 30083 | Append one path segment to... |
| eupth2lem1 30084 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30085 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30086 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30087 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30088 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30089 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30090 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30091 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30092 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30093 | Formerly part of proof of ... |
| eupth2lem3lem1 30094 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30095 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30096 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30097 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30098 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30099 | Formerly part of proof of ... |
| eupth2lem3lem7 30100 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30101 | Formerly part of proof of ... |
| eupth2lem3 30102 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30103 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30104 | Lemma for ~ eupth2 (induct... |
| eupth2 30105 | The only vertices of odd d... |
| eulerpathpr 30106 | A graph with an Eulerian p... |
| eulerpath 30107 | A pseudograph with an Eule... |
| eulercrct 30108 | A pseudograph with an Eule... |
| eucrctshift 30109 | Cyclically shifting the in... |
| eucrct2eupth1 30110 | Removing one edge ` ( I ``... |
| eucrct2eupth 30111 | Removing one edge ` ( I ``... |
| konigsbergvtx 30112 | The set of vertices of the... |
| konigsbergiedg 30113 | The indexed edges of the K... |
| konigsbergiedgw 30114 | The indexed edges of the K... |
| konigsbergssiedgwpr 30115 | Each subset of the indexed... |
| konigsbergssiedgw 30116 | Each subset of the indexed... |
| konigsbergumgr 30117 | The Königsberg graph ... |
| konigsberglem1 30118 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30119 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30120 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30121 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30122 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30123 | The Königsberg Bridge... |
| isfrgr 30126 | The property of being a fr... |
| frgrusgr 30127 | A friendship graph is a si... |
| frgr0v 30128 | Any null graph (set with n... |
| frgr0vb 30129 | Any null graph (without ve... |
| frgruhgr0v 30130 | Any null graph (without ve... |
| frgr0 30131 | The null graph (graph with... |
| frcond1 30132 | The friendship condition: ... |
| frcond2 30133 | The friendship condition: ... |
| frgreu 30134 | Variant of ~ frcond2 : An... |
| frcond3 30135 | The friendship condition, ... |
| frcond4 30136 | The friendship condition, ... |
| frgr1v 30137 | Any graph with (at most) o... |
| nfrgr2v 30138 | Any graph with two (differ... |
| frgr3vlem1 30139 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30140 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30141 | Any graph with three verti... |
| 1vwmgr 30142 | Every graph with one verte... |
| 3vfriswmgrlem 30143 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30144 | Every friendship graph wit... |
| 1to2vfriswmgr 30145 | Every friendship graph wit... |
| 1to3vfriswmgr 30146 | Every friendship graph wit... |
| 1to3vfriendship 30147 | The friendship theorem for... |
| 2pthfrgrrn 30148 | Between any two (different... |
| 2pthfrgrrn2 30149 | Between any two (different... |
| 2pthfrgr 30150 | Between any two (different... |
| 3cyclfrgrrn1 30151 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30152 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30153 | Every vertex in a friendsh... |
| 3cyclfrgr 30154 | Every vertex in a friendsh... |
| 4cycl2v2nb 30155 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30156 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30157 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30158 | A graph with a 4-cycle is ... |
| frgrnbnb 30159 | If two neighbors ` U ` and... |
| frgrconngr 30160 | A friendship graph is conn... |
| vdgn0frgrv2 30161 | A vertex in a friendship g... |
| vdgn1frgrv2 30162 | Any vertex in a friendship... |
| vdgn1frgrv3 30163 | Any vertex in a friendship... |
| vdgfrgrgt2 30164 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30165 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30166 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30167 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30168 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30169 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30170 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30171 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30172 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30173 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30174 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30175 | In a friendship graph, two... |
| frgrwopreglem4a 30176 | In a friendship graph any ... |
| frgrwopreglem5a 30177 | If a friendship graph has ... |
| frgrwopreglem1 30178 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30179 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30180 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30181 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30182 | According to statement 5 i... |
| frgrwopregbsn 30183 | According to statement 5 i... |
| frgrwopreg1 30184 | According to statement 5 i... |
| frgrwopreg2 30185 | According to statement 5 i... |
| frgrwopreglem5lem 30186 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30187 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30188 | Alternate direct proof of ... |
| frgrwopreg 30189 | In a friendship graph ther... |
| frgrregorufr0 30190 | In a friendship graph ther... |
| frgrregorufr 30191 | If there is a vertex havin... |
| frgrregorufrg 30192 | If there is a vertex havin... |
| frgr2wwlkeu 30193 | For two different vertices... |
| frgr2wwlkn0 30194 | In a friendship graph, the... |
| frgr2wwlk1 30195 | In a friendship graph, the... |
| frgr2wsp1 30196 | In a friendship graph, the... |
| frgr2wwlkeqm 30197 | If there is a (simple) pat... |
| frgrhash2wsp 30198 | The number of simple paths... |
| fusgreg2wsplem 30199 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30200 | The set of paths of length... |
| fusgreghash2wspv 30201 | According to statement 7 i... |
| fusgreg2wsp 30202 | In a finite simple graph, ... |
| 2wspmdisj 30203 | The sets of paths of lengt... |
| fusgreghash2wsp 30204 | In a finite k-regular grap... |
| frrusgrord0lem 30205 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30206 | If a nonempty finite frien... |
| frrusgrord 30207 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30208 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30209 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30210 | If the initial vertex of a... |
| clwwnonrepclwwnon 30211 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30212 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30213 | Value of operation ` C ` ,... |
| 2clwwlk2 30214 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30215 | Characterization of an ele... |
| 2clwwlk2clwwlk 30216 | An element of the value of... |
| numclwwlk1lem2foalem 30217 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30218 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30219 | Characterization of an ele... |
| numclwwlk1lem2foa 30220 | Going forth and back from ... |
| numclwwlk1lem2f 30221 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30222 | Value of the function ` T ... |
| numclwwlk1lem2f1 30223 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30224 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30225 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30226 | The set of double loops of... |
| numclwwlk1 30227 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30228 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30229 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30230 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30231 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30232 | The sets of the two repres... |
| wlkl0 30233 | There is exactly one walk ... |
| clwlknon2num 30234 | There are k walks of lengt... |
| numclwlk1lem1 30235 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30236 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30237 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30238 | Value of operation ` H ` ,... |
| numclwwlkovh 30239 | Value of operation ` H ` ,... |
| numclwwlkovq 30240 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30241 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30242 | In a friendship graph, for... |
| numclwlk2lem2f 30243 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30244 | Value of the function ` R ... |
| numclwlk2lem2f1o 30245 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30246 | In a friendship graph, the... |
| numclwwlk2 30247 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30248 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30249 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30250 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30251 | Statement 12 in [Huneke] p... |
| numclwwlk4 30252 | The total number of closed... |
| numclwwlk5lem 30253 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30254 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30255 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30256 | For a prime divisor ` P ` ... |
| numclwwlk7 30257 | Statement 14 in [Huneke] p... |
| numclwwlk8 30258 | The size of the set of clo... |
| frgrreggt1 30259 | If a finite nonempty frien... |
| frgrreg 30260 | If a finite nonempty frien... |
| frgrregord013 30261 | If a finite friendship gra... |
| frgrregord13 30262 | If a nonempty finite frien... |
| frgrogt3nreg 30263 | If a finite friendship gra... |
| friendshipgt3 30264 | The friendship theorem for... |
| friendship 30265 | The friendship theorem: I... |
| conventions 30266 |
H... |
| conventions-labels 30267 |
... |
| conventions-comments 30268 |
... |
| natded 30269 | Here are typical n... |
| ex-natded5.2 30270 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30271 | A more efficient proof of ... |
| ex-natded5.2i 30272 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30273 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30274 | A more efficient proof of ... |
| ex-natded5.3i 30275 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30276 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30277 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30278 | A more efficient proof of ... |
| ex-natded5.8 30279 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30280 | A more efficient proof of ... |
| ex-natded5.13 30281 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30282 | A more efficient proof of ... |
| ex-natded9.20 30283 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30284 | A more efficient proof of ... |
| ex-natded9.26 30285 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30286 | A more efficient proof of ... |
| ex-or 30287 | Example for ~ df-or . Exa... |
| ex-an 30288 | Example for ~ df-an . Exa... |
| ex-dif 30289 | Example for ~ df-dif . Ex... |
| ex-un 30290 | Example for ~ df-un . Exa... |
| ex-in 30291 | Example for ~ df-in . Exa... |
| ex-uni 30292 | Example for ~ df-uni . Ex... |
| ex-ss 30293 | Example for ~ df-ss . Exa... |
| ex-pss 30294 | Example for ~ df-pss . Ex... |
| ex-pw 30295 | Example for ~ df-pw . Exa... |
| ex-pr 30296 | Example for ~ df-pr . (Co... |
| ex-br 30297 | Example for ~ df-br . Exa... |
| ex-opab 30298 | Example for ~ df-opab . E... |
| ex-eprel 30299 | Example for ~ df-eprel . ... |
| ex-id 30300 | Example for ~ df-id . Exa... |
| ex-po 30301 | Example for ~ df-po . Exa... |
| ex-xp 30302 | Example for ~ df-xp . Exa... |
| ex-cnv 30303 | Example for ~ df-cnv . Ex... |
| ex-co 30304 | Example for ~ df-co . Exa... |
| ex-dm 30305 | Example for ~ df-dm . Exa... |
| ex-rn 30306 | Example for ~ df-rn . Exa... |
| ex-res 30307 | Example for ~ df-res . Ex... |
| ex-ima 30308 | Example for ~ df-ima . Ex... |
| ex-fv 30309 | Example for ~ df-fv . Exa... |
| ex-1st 30310 | Example for ~ df-1st . Ex... |
| ex-2nd 30311 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30312 | Example for ~ df-dec , 100... |
| ex-fl 30313 | Example for ~ df-fl . Exa... |
| ex-ceil 30314 | Example for ~ df-ceil . (... |
| ex-mod 30315 | Example for ~ df-mod . (C... |
| ex-exp 30316 | Example for ~ df-exp . (C... |
| ex-fac 30317 | Example for ~ df-fac . (C... |
| ex-bc 30318 | Example for ~ df-bc . (Co... |
| ex-hash 30319 | Example for ~ df-hash . (... |
| ex-sqrt 30320 | Example for ~ df-sqrt . (... |
| ex-abs 30321 | Example for ~ df-abs . (C... |
| ex-dvds 30322 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30323 | Example for ~ df-gcd . (C... |
| ex-lcm 30324 | Example for ~ df-lcm . (C... |
| ex-prmo 30325 | Example for ~ df-prmo : ` ... |
| aevdemo 30326 | Proof illustrating the com... |
| ex-ind-dvds 30327 | Example of a proof by indu... |
| ex-fpar 30328 | Formalized example provide... |
| avril1 30329 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30330 | The law of excluded middle... |
| helloworld 30331 | The classic "Hello world" ... |
| 1p1e2apr1 30332 | One plus one equals two. ... |
| eqid1 30333 | Law of identity (reflexivi... |
| 1div0apr 30334 | Division by zero is forbid... |
| topnfbey 30335 | Nothing seems to be imposs... |
| 9p10ne21 30336 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30337 | 9 + 10 equals 21. This as... |
| nrt2irr 30339 | The ` N ` -th root of 2 is... |
| isplig 30342 | The predicate "is a planar... |
| ispligb 30343 | The predicate "is a planar... |
| tncp 30344 | In any planar incidence ge... |
| l2p 30345 | For any line in a planar i... |
| lpni 30346 | For any line in a planar i... |
| nsnlplig 30347 | There is no "one-point lin... |
| nsnlpligALT 30348 | Alternate version of ~ nsn... |
| n0lplig 30349 | There is no "empty line" i... |
| n0lpligALT 30350 | Alternate version of ~ n0l... |
| eulplig 30351 | Through two distinct point... |
| pliguhgr 30352 | Any planar incidence geome... |
| dummylink 30353 | Alias for ~ a1ii that may ... |
| id1 30354 | Alias for ~ idALT that may... |
| isgrpo 30363 | The predicate "is a group ... |
| isgrpoi 30364 | Properties that determine ... |
| grpofo 30365 | A group operation maps ont... |
| grpocl 30366 | Closure law for a group op... |
| grpolidinv 30367 | A group has a left identit... |
| grpon0 30368 | The base set of a group is... |
| grpoass 30369 | A group operation is assoc... |
| grpoidinvlem1 30370 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30371 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30372 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30373 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30374 | A group has a left and rig... |
| grpoideu 30375 | The left identity element ... |
| grporndm 30376 | A group's range in terms o... |
| 0ngrp 30377 | The empty set is not a gro... |
| gidval 30378 | The value of the identity ... |
| grpoidval 30379 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30380 | The identity element of a ... |
| grpoidinv2 30381 | A group's properties using... |
| grpolid 30382 | The identity element of a ... |
| grporid 30383 | The identity element of a ... |
| grporcan 30384 | Right cancellation law for... |
| grpoinveu 30385 | The left inverse element o... |
| grpoid 30386 | Two ways of saying that an... |
| grporn 30387 | The range of a group opera... |
| grpoinvfval 30388 | The inverse function of a ... |
| grpoinvval 30389 | The inverse of a group ele... |
| grpoinvcl 30390 | A group element's inverse ... |
| grpoinv 30391 | The properties of a group ... |
| grpolinv 30392 | The left inverse of a grou... |
| grporinv 30393 | The right inverse of a gro... |
| grpoinvid1 30394 | The inverse of a group ele... |
| grpoinvid2 30395 | The inverse of a group ele... |
| grpolcan 30396 | Left cancellation law for ... |
| grpo2inv 30397 | Double inverse law for gro... |
| grpoinvf 30398 | Mapping of the inverse fun... |
| grpoinvop 30399 | The inverse of the group o... |
| grpodivfval 30400 | Group division (or subtrac... |
| grpodivval 30401 | Group division (or subtrac... |
| grpodivinv 30402 | Group division by an inver... |
| grpoinvdiv 30403 | Inverse of a group divisio... |
| grpodivf 30404 | Mapping for group division... |
| grpodivcl 30405 | Closure of group division ... |
| grpodivdiv 30406 | Double group division. (C... |
| grpomuldivass 30407 | Associative-type law for m... |
| grpodivid 30408 | Division of a group member... |
| grponpcan 30409 | Cancellation law for group... |
| isablo 30412 | The predicate "is an Abeli... |
| ablogrpo 30413 | An Abelian group operation... |
| ablocom 30414 | An Abelian group operation... |
| ablo32 30415 | Commutative/associative la... |
| ablo4 30416 | Commutative/associative la... |
| isabloi 30417 | Properties that determine ... |
| ablomuldiv 30418 | Law for group multiplicati... |
| ablodivdiv 30419 | Law for double group divis... |
| ablodivdiv4 30420 | Law for double group divis... |
| ablodiv32 30421 | Swap the second and third ... |
| ablonncan 30422 | Cancellation law for group... |
| ablonnncan1 30423 | Cancellation law for group... |
| vcrel 30426 | The class of all complex v... |
| vciOLD 30427 | Obsolete version of ~ cvsi... |
| vcsm 30428 | Functionality of th scalar... |
| vccl 30429 | Closure of the scalar prod... |
| vcidOLD 30430 | Identity element for the s... |
| vcdi 30431 | Distributive law for the s... |
| vcdir 30432 | Distributive law for the s... |
| vcass 30433 | Associative law for the sc... |
| vc2OLD 30434 | A vector plus itself is tw... |
| vcablo 30435 | Vector addition is an Abel... |
| vcgrp 30436 | Vector addition is a group... |
| vclcan 30437 | Left cancellation law for ... |
| vczcl 30438 | The zero vector is a vecto... |
| vc0rid 30439 | The zero vector is a right... |
| vc0 30440 | Zero times a vector is the... |
| vcz 30441 | Anything times the zero ve... |
| vcm 30442 | Minus 1 times a vector is ... |
| isvclem 30443 | Lemma for ~ isvcOLD . (Co... |
| vcex 30444 | The components of a comple... |
| isvcOLD 30445 | The predicate "is a comple... |
| isvciOLD 30446 | Properties that determine ... |
| cnaddabloOLD 30447 | Obsolete version of ~ cnad... |
| cnidOLD 30448 | Obsolete version of ~ cnad... |
| cncvcOLD 30449 | Obsolete version of ~ cncv... |
| nvss 30459 | Structure of the class of ... |
| nvvcop 30460 | A normed complex vector sp... |
| nvrel 30468 | The class of all normed co... |
| vafval 30469 | Value of the function for ... |
| bafval 30470 | Value of the function for ... |
| smfval 30471 | Value of the function for ... |
| 0vfval 30472 | Value of the function for ... |
| nmcvfval 30473 | Value of the norm function... |
| nvop2 30474 | A normed complex vector sp... |
| nvvop 30475 | The vector space component... |
| isnvlem 30476 | Lemma for ~ isnv . (Contr... |
| nvex 30477 | The components of a normed... |
| isnv 30478 | The predicate "is a normed... |
| isnvi 30479 | Properties that determine ... |
| nvi 30480 | The properties of a normed... |
| nvvc 30481 | The vector space component... |
| nvablo 30482 | The vector addition operat... |
| nvgrp 30483 | The vector addition operat... |
| nvgf 30484 | Mapping for the vector add... |
| nvsf 30485 | Mapping for the scalar mul... |
| nvgcl 30486 | Closure law for the vector... |
| nvcom 30487 | The vector addition (group... |
| nvass 30488 | The vector addition (group... |
| nvadd32 30489 | Commutative/associative la... |
| nvrcan 30490 | Right cancellation law for... |
| nvadd4 30491 | Rearrangement of 4 terms i... |
| nvscl 30492 | Closure law for the scalar... |
| nvsid 30493 | Identity element for the s... |
| nvsass 30494 | Associative law for the sc... |
| nvscom 30495 | Commutative law for the sc... |
| nvdi 30496 | Distributive law for the s... |
| nvdir 30497 | Distributive law for the s... |
| nv2 30498 | A vector plus itself is tw... |
| vsfval 30499 | Value of the function for ... |
| nvzcl 30500 | Closure law for the zero v... |
| nv0rid 30501 | The zero vector is a right... |
| nv0lid 30502 | The zero vector is a left ... |
| nv0 30503 | Zero times a vector is the... |
| nvsz 30504 | Anything times the zero ve... |
| nvinv 30505 | Minus 1 times a vector is ... |
| nvinvfval 30506 | Function for the negative ... |
| nvm 30507 | Vector subtraction in term... |
| nvmval 30508 | Value of vector subtractio... |
| nvmval2 30509 | Value of vector subtractio... |
| nvmfval 30510 | Value of the function for ... |
| nvmf 30511 | Mapping for the vector sub... |
| nvmcl 30512 | Closure law for the vector... |
| nvnnncan1 30513 | Cancellation law for vecto... |
| nvmdi 30514 | Distributive law for scala... |
| nvnegneg 30515 | Double negative of a vecto... |
| nvmul0or 30516 | If a scalar product is zer... |
| nvrinv 30517 | A vector minus itself. (C... |
| nvlinv 30518 | Minus a vector plus itself... |
| nvpncan2 30519 | Cancellation law for vecto... |
| nvpncan 30520 | Cancellation law for vecto... |
| nvaddsub 30521 | Commutative/associative la... |
| nvnpcan 30522 | Cancellation law for a nor... |
| nvaddsub4 30523 | Rearrangement of 4 terms i... |
| nvmeq0 30524 | The difference between two... |
| nvmid 30525 | A vector minus itself is t... |
| nvf 30526 | Mapping for the norm funct... |
| nvcl 30527 | The norm of a normed compl... |
| nvcli 30528 | The norm of a normed compl... |
| nvs 30529 | Proportionality property o... |
| nvsge0 30530 | The norm of a scalar produ... |
| nvm1 30531 | The norm of the negative o... |
| nvdif 30532 | The norm of the difference... |
| nvpi 30533 | The norm of a vector plus ... |
| nvz0 30534 | The norm of a zero vector ... |
| nvz 30535 | The norm of a vector is ze... |
| nvtri 30536 | Triangle inequality for th... |
| nvmtri 30537 | Triangle inequality for th... |
| nvabs 30538 | Norm difference property o... |
| nvge0 30539 | The norm of a normed compl... |
| nvgt0 30540 | A nonzero norm is positive... |
| nv1 30541 | From any nonzero vector, c... |
| nvop 30542 | A complex inner product sp... |
| cnnv 30543 | The set of complex numbers... |
| cnnvg 30544 | The vector addition (group... |
| cnnvba 30545 | The base set of the normed... |
| cnnvs 30546 | The scalar product operati... |
| cnnvnm 30547 | The norm operation of the ... |
| cnnvm 30548 | The vector subtraction ope... |
| elimnv 30549 | Hypothesis elimination lem... |
| elimnvu 30550 | Hypothesis elimination lem... |
| imsval 30551 | Value of the induced metri... |
| imsdval 30552 | Value of the induced metri... |
| imsdval2 30553 | Value of the distance func... |
| nvnd 30554 | The norm of a normed compl... |
| imsdf 30555 | Mapping for the induced me... |
| imsmetlem 30556 | Lemma for ~ imsmet . (Con... |
| imsmet 30557 | The induced metric of a no... |
| imsxmet 30558 | The induced metric of a no... |
| cnims 30559 | The metric induced on the ... |
| vacn 30560 | Vector addition is jointly... |
| nmcvcn 30561 | The norm of a normed compl... |
| nmcnc 30562 | The norm of a normed compl... |
| smcnlem 30563 | Lemma for ~ smcn . (Contr... |
| smcn 30564 | Scalar multiplication is j... |
| vmcn 30565 | Vector subtraction is join... |
| dipfval 30568 | The inner product function... |
| ipval 30569 | Value of the inner product... |
| ipval2lem2 30570 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30571 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30572 | Lemma for ~ ipval3 . (Con... |
| ipval2 30573 | Expansion of the inner pro... |
| 4ipval2 30574 | Four times the inner produ... |
| ipval3 30575 | Expansion of the inner pro... |
| ipidsq 30576 | The inner product of a vec... |
| ipnm 30577 | Norm expressed in terms of... |
| dipcl 30578 | An inner product is a comp... |
| ipf 30579 | Mapping for the inner prod... |
| dipcj 30580 | The complex conjugate of a... |
| ipipcj 30581 | An inner product times its... |
| diporthcom 30582 | Orthogonality (meaning inn... |
| dip0r 30583 | Inner product with a zero ... |
| dip0l 30584 | Inner product with a zero ... |
| ipz 30585 | The inner product of a vec... |
| dipcn 30586 | Inner product is jointly c... |
| sspval 30589 | The set of all subspaces o... |
| isssp 30590 | The predicate "is a subspa... |
| sspid 30591 | A normed complex vector sp... |
| sspnv 30592 | A subspace is a normed com... |
| sspba 30593 | The base set of a subspace... |
| sspg 30594 | Vector addition on a subsp... |
| sspgval 30595 | Vector addition on a subsp... |
| ssps 30596 | Scalar multiplication on a... |
| sspsval 30597 | Scalar multiplication on a... |
| sspmlem 30598 | Lemma for ~ sspm and other... |
| sspmval 30599 | Vector addition on a subsp... |
| sspm 30600 | Vector subtraction on a su... |
| sspz 30601 | The zero vector of a subsp... |
| sspn 30602 | The norm on a subspace is ... |
| sspnval 30603 | The norm on a subspace in ... |
| sspimsval 30604 | The induced metric on a su... |
| sspims 30605 | The induced metric on a su... |
| lnoval 30618 | The set of linear operator... |
| islno 30619 | The predicate "is a linear... |
| lnolin 30620 | Basic linearity property o... |
| lnof 30621 | A linear operator is a map... |
| lno0 30622 | The value of a linear oper... |
| lnocoi 30623 | The composition of two lin... |
| lnoadd 30624 | Addition property of a lin... |
| lnosub 30625 | Subtraction property of a ... |
| lnomul 30626 | Scalar multiplication prop... |
| nvo00 30627 | Two ways to express a zero... |
| nmoofval 30628 | The operator norm function... |
| nmooval 30629 | The operator norm function... |
| nmosetre 30630 | The set in the supremum of... |
| nmosetn0 30631 | The set in the supremum of... |
| nmoxr 30632 | The norm of an operator is... |
| nmooge0 30633 | The norm of an operator is... |
| nmorepnf 30634 | The norm of an operator is... |
| nmoreltpnf 30635 | The norm of any operator i... |
| nmogtmnf 30636 | The norm of an operator is... |
| nmoolb 30637 | A lower bound for an opera... |
| nmoubi 30638 | An upper bound for an oper... |
| nmoub3i 30639 | An upper bound for an oper... |
| nmoub2i 30640 | An upper bound for an oper... |
| nmobndi 30641 | Two ways to express that a... |
| nmounbi 30642 | Two ways two express that ... |
| nmounbseqi 30643 | An unbounded operator dete... |
| nmounbseqiALT 30644 | Alternate shorter proof of... |
| nmobndseqi 30645 | A bounded sequence determi... |
| nmobndseqiALT 30646 | Alternate shorter proof of... |
| bloval 30647 | The class of bounded linea... |
| isblo 30648 | The predicate "is a bounde... |
| isblo2 30649 | The predicate "is a bounde... |
| bloln 30650 | A bounded operator is a li... |
| blof 30651 | A bounded operator is an o... |
| nmblore 30652 | The norm of a bounded oper... |
| 0ofval 30653 | The zero operator between ... |
| 0oval 30654 | Value of the zero operator... |
| 0oo 30655 | The zero operator is an op... |
| 0lno 30656 | The zero operator is linea... |
| nmoo0 30657 | The operator norm of the z... |
| 0blo 30658 | The zero operator is a bou... |
| nmlno0lem 30659 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30660 | The norm of a linear opera... |
| nmlno0 30661 | The norm of a linear opera... |
| nmlnoubi 30662 | An upper bound for the ope... |
| nmlnogt0 30663 | The norm of a nonzero line... |
| lnon0 30664 | The domain of a nonzero li... |
| nmblolbii 30665 | A lower bound for the norm... |
| nmblolbi 30666 | A lower bound for the norm... |
| isblo3i 30667 | The predicate "is a bounde... |
| blo3i 30668 | Properties that determine ... |
| blometi 30669 | Upper bound for the distan... |
| blocnilem 30670 | Lemma for ~ blocni and ~ l... |
| blocni 30671 | A linear operator is conti... |
| lnocni 30672 | If a linear operator is co... |
| blocn 30673 | A linear operator is conti... |
| blocn2 30674 | A bounded linear operator ... |
| ajfval 30675 | The adjoint function. (Co... |
| hmoval 30676 | The set of Hermitian (self... |
| ishmo 30677 | The predicate "is a hermit... |
| phnv 30680 | Every complex inner produc... |
| phrel 30681 | The class of all complex i... |
| phnvi 30682 | Every complex inner produc... |
| isphg 30683 | The predicate "is a comple... |
| phop 30684 | A complex inner product sp... |
| cncph 30685 | The set of complex numbers... |
| elimph 30686 | Hypothesis elimination lem... |
| elimphu 30687 | Hypothesis elimination lem... |
| isph 30688 | The predicate "is an inner... |
| phpar2 30689 | The parallelogram law for ... |
| phpar 30690 | The parallelogram law for ... |
| ip0i 30691 | A slight variant of Equati... |
| ip1ilem 30692 | Lemma for ~ ip1i . (Contr... |
| ip1i 30693 | Equation 6.47 of [Ponnusam... |
| ip2i 30694 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30695 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30696 | Distributive law for inner... |
| ipasslem1 30697 | Lemma for ~ ipassi . Show... |
| ipasslem2 30698 | Lemma for ~ ipassi . Show... |
| ipasslem3 30699 | Lemma for ~ ipassi . Show... |
| ipasslem4 30700 | Lemma for ~ ipassi . Show... |
| ipasslem5 30701 | Lemma for ~ ipassi . Show... |
| ipasslem7 30702 | Lemma for ~ ipassi . Show... |
| ipasslem8 30703 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30704 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30705 | Lemma for ~ ipassi . Show... |
| ipasslem11 30706 | Lemma for ~ ipassi . Show... |
| ipassi 30707 | Associative law for inner ... |
| dipdir 30708 | Distributive law for inner... |
| dipdi 30709 | Distributive law for inner... |
| ip2dii 30710 | Inner product of two sums.... |
| dipass 30711 | Associative law for inner ... |
| dipassr 30712 | "Associative" law for seco... |
| dipassr2 30713 | "Associative" law for inne... |
| dipsubdir 30714 | Distributive law for inner... |
| dipsubdi 30715 | Distributive law for inner... |
| pythi 30716 | The Pythagorean theorem fo... |
| siilem1 30717 | Lemma for ~ sii . (Contri... |
| siilem2 30718 | Lemma for ~ sii . (Contri... |
| siii 30719 | Inference from ~ sii . (C... |
| sii 30720 | Obsolete version of ~ ipca... |
| ipblnfi 30721 | A function ` F ` generated... |
| ip2eqi 30722 | Two vectors are equal iff ... |
| phoeqi 30723 | A condition implying that ... |
| ajmoi 30724 | Every operator has at most... |
| ajfuni 30725 | The adjoint function is a ... |
| ajfun 30726 | The adjoint function is a ... |
| ajval 30727 | Value of the adjoint funct... |
| iscbn 30730 | A complex Banach space is ... |
| cbncms 30731 | The induced metric on comp... |
| bnnv 30732 | Every complex Banach space... |
| bnrel 30733 | The class of all complex B... |
| bnsscmcl 30734 | A subspace of a Banach spa... |
| cnbn 30735 | The set of complex numbers... |
| ubthlem1 30736 | Lemma for ~ ubth . The fu... |
| ubthlem2 30737 | Lemma for ~ ubth . Given ... |
| ubthlem3 30738 | Lemma for ~ ubth . Prove ... |
| ubth 30739 | Uniform Boundedness Theore... |
| minvecolem1 30740 | Lemma for ~ minveco . The... |
| minvecolem2 30741 | Lemma for ~ minveco . Any... |
| minvecolem3 30742 | Lemma for ~ minveco . The... |
| minvecolem4a 30743 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30744 | Lemma for ~ minveco . The... |
| minvecolem4c 30745 | Lemma for ~ minveco . The... |
| minvecolem4 30746 | Lemma for ~ minveco . The... |
| minvecolem5 30747 | Lemma for ~ minveco . Dis... |
| minvecolem6 30748 | Lemma for ~ minveco . Any... |
| minvecolem7 30749 | Lemma for ~ minveco . Sin... |
| minveco 30750 | Minimizing vector theorem,... |
| ishlo 30753 | The predicate "is a comple... |
| hlobn 30754 | Every complex Hilbert spac... |
| hlph 30755 | Every complex Hilbert spac... |
| hlrel 30756 | The class of all complex H... |
| hlnv 30757 | Every complex Hilbert spac... |
| hlnvi 30758 | Every complex Hilbert spac... |
| hlvc 30759 | Every complex Hilbert spac... |
| hlcmet 30760 | The induced metric on a co... |
| hlmet 30761 | The induced metric on a co... |
| hlpar2 30762 | The parallelogram law sati... |
| hlpar 30763 | The parallelogram law sati... |
| hlex 30764 | The base set of a Hilbert ... |
| hladdf 30765 | Mapping for Hilbert space ... |
| hlcom 30766 | Hilbert space vector addit... |
| hlass 30767 | Hilbert space vector addit... |
| hl0cl 30768 | The Hilbert space zero vec... |
| hladdid 30769 | Hilbert space addition wit... |
| hlmulf 30770 | Mapping for Hilbert space ... |
| hlmulid 30771 | Hilbert space scalar multi... |
| hlmulass 30772 | Hilbert space scalar multi... |
| hldi 30773 | Hilbert space scalar multi... |
| hldir 30774 | Hilbert space scalar multi... |
| hlmul0 30775 | Hilbert space scalar multi... |
| hlipf 30776 | Mapping for Hilbert space ... |
| hlipcj 30777 | Conjugate law for Hilbert ... |
| hlipdir 30778 | Distributive law for Hilbe... |
| hlipass 30779 | Associative law for Hilber... |
| hlipgt0 30780 | The inner product of a Hil... |
| hlcompl 30781 | Completeness of a Hilbert ... |
| cnchl 30782 | The set of complex numbers... |
| htthlem 30783 | Lemma for ~ htth . The co... |
| htth 30784 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 30840 | The group (addition) opera... |
| h2hsm 30841 | The scalar product operati... |
| h2hnm 30842 | The norm function of Hilbe... |
| h2hvs 30843 | The vector subtraction ope... |
| h2hmetdval 30844 | Value of the distance func... |
| h2hcau 30845 | The Cauchy sequences of Hi... |
| h2hlm 30846 | The limit sequences of Hil... |
| axhilex-zf 30847 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 30848 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 30849 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 30850 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 30851 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 30852 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 30853 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 30854 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 30855 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 30856 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 30857 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 30858 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 30859 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 30860 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 30861 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 30862 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 30863 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 30864 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 30877 | The Hilbert space scalar p... |
| hvaddcl 30878 | Closure of vector addition... |
| hvmulcl 30879 | Closure of scalar multipli... |
| hvmulcli 30880 | Closure inference for scal... |
| hvsubf 30881 | Mapping domain and codomai... |
| hvsubval 30882 | Value of vector subtractio... |
| hvsubcl 30883 | Closure of vector subtract... |
| hvaddcli 30884 | Closure of vector addition... |
| hvcomi 30885 | Commutation of vector addi... |
| hvsubvali 30886 | Value of vector subtractio... |
| hvsubcli 30887 | Closure of vector subtract... |
| ifhvhv0 30888 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 30889 | Addition with the zero vec... |
| hvmul0 30890 | Scalar multiplication with... |
| hvmul0or 30891 | If a scalar product is zer... |
| hvsubid 30892 | Subtraction of a vector fr... |
| hvnegid 30893 | Addition of negative of a ... |
| hv2neg 30894 | Two ways to express the ne... |
| hvaddlidi 30895 | Addition with the zero vec... |
| hvnegidi 30896 | Addition of negative of a ... |
| hv2negi 30897 | Two ways to express the ne... |
| hvm1neg 30898 | Convert minus one times a ... |
| hvaddsubval 30899 | Value of vector addition i... |
| hvadd32 30900 | Commutative/associative la... |
| hvadd12 30901 | Commutative/associative la... |
| hvadd4 30902 | Hilbert vector space addit... |
| hvsub4 30903 | Hilbert vector space addit... |
| hvaddsub12 30904 | Commutative/associative la... |
| hvpncan 30905 | Addition/subtraction cance... |
| hvpncan2 30906 | Addition/subtraction cance... |
| hvaddsubass 30907 | Associativity of sum and d... |
| hvpncan3 30908 | Subtraction and addition o... |
| hvmulcom 30909 | Scalar multiplication comm... |
| hvsubass 30910 | Hilbert vector space assoc... |
| hvsub32 30911 | Hilbert vector space commu... |
| hvmulassi 30912 | Scalar multiplication asso... |
| hvmulcomi 30913 | Scalar multiplication comm... |
| hvmul2negi 30914 | Double negative in scalar ... |
| hvsubdistr1 30915 | Scalar multiplication dist... |
| hvsubdistr2 30916 | Scalar multiplication dist... |
| hvdistr1i 30917 | Scalar multiplication dist... |
| hvsubdistr1i 30918 | Scalar multiplication dist... |
| hvassi 30919 | Hilbert vector space assoc... |
| hvadd32i 30920 | Hilbert vector space commu... |
| hvsubassi 30921 | Hilbert vector space assoc... |
| hvsub32i 30922 | Hilbert vector space commu... |
| hvadd12i 30923 | Hilbert vector space commu... |
| hvadd4i 30924 | Hilbert vector space addit... |
| hvsubsub4i 30925 | Hilbert vector space addit... |
| hvsubsub4 30926 | Hilbert vector space addit... |
| hv2times 30927 | Two times a vector. (Cont... |
| hvnegdii 30928 | Distribution of negative o... |
| hvsubeq0i 30929 | If the difference between ... |
| hvsubcan2i 30930 | Vector cancellation law. ... |
| hvaddcani 30931 | Cancellation law for vecto... |
| hvsubaddi 30932 | Relationship between vecto... |
| hvnegdi 30933 | Distribution of negative o... |
| hvsubeq0 30934 | If the difference between ... |
| hvaddeq0 30935 | If the sum of two vectors ... |
| hvaddcan 30936 | Cancellation law for vecto... |
| hvaddcan2 30937 | Cancellation law for vecto... |
| hvmulcan 30938 | Cancellation law for scala... |
| hvmulcan2 30939 | Cancellation law for scala... |
| hvsubcan 30940 | Cancellation law for vecto... |
| hvsubcan2 30941 | Cancellation law for vecto... |
| hvsub0 30942 | Subtraction of a zero vect... |
| hvsubadd 30943 | Relationship between vecto... |
| hvaddsub4 30944 | Hilbert vector space addit... |
| hicl 30946 | Closure of inner product. ... |
| hicli 30947 | Closure inference for inne... |
| his5 30952 | Associative law for inner ... |
| his52 30953 | Associative law for inner ... |
| his35 30954 | Move scalar multiplication... |
| his35i 30955 | Move scalar multiplication... |
| his7 30956 | Distributive law for inner... |
| hiassdi 30957 | Distributive/associative l... |
| his2sub 30958 | Distributive law for inner... |
| his2sub2 30959 | Distributive law for inner... |
| hire 30960 | A necessary and sufficient... |
| hiidrcl 30961 | Real closure of inner prod... |
| hi01 30962 | Inner product with the 0 v... |
| hi02 30963 | Inner product with the 0 v... |
| hiidge0 30964 | Inner product with self is... |
| his6 30965 | Zero inner product with se... |
| his1i 30966 | Conjugate law for inner pr... |
| abshicom 30967 | Commuted inner products ha... |
| hial0 30968 | A vector whose inner produ... |
| hial02 30969 | A vector whose inner produ... |
| hisubcomi 30970 | Two vector subtractions si... |
| hi2eq 30971 | Lemma used to prove equali... |
| hial2eq 30972 | Two vectors whose inner pr... |
| hial2eq2 30973 | Two vectors whose inner pr... |
| orthcom 30974 | Orthogonality commutes. (... |
| normlem0 30975 | Lemma used to derive prope... |
| normlem1 30976 | Lemma used to derive prope... |
| normlem2 30977 | Lemma used to derive prope... |
| normlem3 30978 | Lemma used to derive prope... |
| normlem4 30979 | Lemma used to derive prope... |
| normlem5 30980 | Lemma used to derive prope... |
| normlem6 30981 | Lemma used to derive prope... |
| normlem7 30982 | Lemma used to derive prope... |
| normlem8 30983 | Lemma used to derive prope... |
| normlem9 30984 | Lemma used to derive prope... |
| normlem7tALT 30985 | Lemma used to derive prope... |
| bcseqi 30986 | Equality case of Bunjakova... |
| normlem9at 30987 | Lemma used to derive prope... |
| dfhnorm2 30988 | Alternate definition of th... |
| normf 30989 | The norm function maps fro... |
| normval 30990 | The value of the norm of a... |
| normcl 30991 | Real closure of the norm o... |
| normge0 30992 | The norm of a vector is no... |
| normgt0 30993 | The norm of nonzero vector... |
| norm0 30994 | The norm of a zero vector.... |
| norm-i 30995 | Theorem 3.3(i) of [Beran] ... |
| normne0 30996 | A norm is nonzero iff its ... |
| normcli 30997 | Real closure of the norm o... |
| normsqi 30998 | The square of a norm. (Co... |
| norm-i-i 30999 | Theorem 3.3(i) of [Beran] ... |
| normsq 31000 | The square of a norm. (Co... |
| normsub0i 31001 | Two vectors are equal iff ... |
| normsub0 31002 | Two vectors are equal iff ... |
| norm-ii-i 31003 | Triangle inequality for no... |
| norm-ii 31004 | Triangle inequality for no... |
| norm-iii-i 31005 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31006 | Theorem 3.3(iii) of [Beran... |
| normsubi 31007 | Negative doesn't change th... |
| normpythi 31008 | Analogy to Pythagorean the... |
| normsub 31009 | Swapping order of subtract... |
| normneg 31010 | The norm of a vector equal... |
| normpyth 31011 | Analogy to Pythagorean the... |
| normpyc 31012 | Corollary to Pythagorean t... |
| norm3difi 31013 | Norm of differences around... |
| norm3adifii 31014 | Norm of differences around... |
| norm3lem 31015 | Lemma involving norm of di... |
| norm3dif 31016 | Norm of differences around... |
| norm3dif2 31017 | Norm of differences around... |
| norm3lemt 31018 | Lemma involving norm of di... |
| norm3adifi 31019 | Norm of differences around... |
| normpari 31020 | Parallelogram law for norm... |
| normpar 31021 | Parallelogram law for norm... |
| normpar2i 31022 | Corollary of parallelogram... |
| polid2i 31023 | Generalized polarization i... |
| polidi 31024 | Polarization identity. Re... |
| polid 31025 | Polarization identity. Re... |
| hilablo 31026 | Hilbert space vector addit... |
| hilid 31027 | The group identity element... |
| hilvc 31028 | Hilbert space is a complex... |
| hilnormi 31029 | Hilbert space norm in term... |
| hilhhi 31030 | Deduce the structure of Hi... |
| hhnv 31031 | Hilbert space is a normed ... |
| hhva 31032 | The group (addition) opera... |
| hhba 31033 | The base set of Hilbert sp... |
| hh0v 31034 | The zero vector of Hilbert... |
| hhsm 31035 | The scalar product operati... |
| hhvs 31036 | The vector subtraction ope... |
| hhnm 31037 | The norm function of Hilbe... |
| hhims 31038 | The induced metric of Hilb... |
| hhims2 31039 | Hilbert space distance met... |
| hhmet 31040 | The induced metric of Hilb... |
| hhxmet 31041 | The induced metric of Hilb... |
| hhmetdval 31042 | Value of the distance func... |
| hhip 31043 | The inner product operatio... |
| hhph 31044 | The Hilbert space of the H... |
| bcsiALT 31045 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31046 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31047 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31048 | Corollary of the Bunjakova... |
| bcs3 31049 | Corollary of the Bunjakova... |
| hcau 31050 | Member of the set of Cauch... |
| hcauseq 31051 | A Cauchy sequences on a Hi... |
| hcaucvg 31052 | A Cauchy sequence on a Hil... |
| seq1hcau 31053 | A sequence on a Hilbert sp... |
| hlimi 31054 | Express the predicate: Th... |
| hlimseqi 31055 | A sequence with a limit on... |
| hlimveci 31056 | Closure of the limit of a ... |
| hlimconvi 31057 | Convergence of a sequence ... |
| hlim2 31058 | The limit of a sequence on... |
| hlimadd 31059 | Limit of the sum of two se... |
| hilmet 31060 | The Hilbert space norm det... |
| hilxmet 31061 | The Hilbert space norm det... |
| hilmetdval 31062 | Value of the distance func... |
| hilims 31063 | Hilbert space distance met... |
| hhcau 31064 | The Cauchy sequences of Hi... |
| hhlm 31065 | The limit sequences of Hil... |
| hhcmpl 31066 | Lemma used for derivation ... |
| hilcompl 31067 | Lemma used for derivation ... |
| hhcms 31069 | The Hilbert space induced ... |
| hhhl 31070 | The Hilbert space structur... |
| hilcms 31071 | The Hilbert space norm det... |
| hilhl 31072 | The Hilbert space of the H... |
| issh 31074 | Subspace ` H ` of a Hilber... |
| issh2 31075 | Subspace ` H ` of a Hilber... |
| shss 31076 | A subspace is a subset of ... |
| shel 31077 | A member of a subspace of ... |
| shex 31078 | The set of subspaces of a ... |
| shssii 31079 | A closed subspace of a Hil... |
| sheli 31080 | A member of a subspace of ... |
| shelii 31081 | A member of a subspace of ... |
| sh0 31082 | The zero vector belongs to... |
| shaddcl 31083 | Closure of vector addition... |
| shmulcl 31084 | Closure of vector scalar m... |
| issh3 31085 | Subspace ` H ` of a Hilber... |
| shsubcl 31086 | Closure of vector subtract... |
| isch 31088 | Closed subspace ` H ` of a... |
| isch2 31089 | Closed subspace ` H ` of a... |
| chsh 31090 | A closed subspace is a sub... |
| chsssh 31091 | Closed subspaces are subsp... |
| chex 31092 | The set of closed subspace... |
| chshii 31093 | A closed subspace is a sub... |
| ch0 31094 | The zero vector belongs to... |
| chss 31095 | A closed subspace of a Hil... |
| chel 31096 | A member of a closed subsp... |
| chssii 31097 | A closed subspace of a Hil... |
| cheli 31098 | A member of a closed subsp... |
| chelii 31099 | A member of a closed subsp... |
| chlimi 31100 | The limit property of a cl... |
| hlim0 31101 | The zero sequence in Hilbe... |
| hlimcaui 31102 | If a sequence in Hilbert s... |
| hlimf 31103 | Function-like behavior of ... |
| hlimuni 31104 | A Hilbert space sequence c... |
| hlimreui 31105 | The limit of a Hilbert spa... |
| hlimeui 31106 | The limit of a Hilbert spa... |
| isch3 31107 | A Hilbert subspace is clos... |
| chcompl 31108 | Completeness of a closed s... |
| helch 31109 | The Hilbert lattice one (w... |
| ifchhv 31110 | Prove ` if ( A e. CH , A ,... |
| helsh 31111 | Hilbert space is a subspac... |
| shsspwh 31112 | Subspaces are subsets of H... |
| chsspwh 31113 | Closed subspaces are subse... |
| hsn0elch 31114 | The zero subspace belongs ... |
| norm1 31115 | From any nonzero Hilbert s... |
| norm1exi 31116 | A normalized vector exists... |
| norm1hex 31117 | A normalized vector can ex... |
| elch0 31120 | Membership in zero for clo... |
| h0elch 31121 | The zero subspace is a clo... |
| h0elsh 31122 | The zero subspace is a sub... |
| hhssva 31123 | The vector addition operat... |
| hhsssm 31124 | The scalar multiplication ... |
| hhssnm 31125 | The norm operation on a su... |
| issubgoilem 31126 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31127 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31128 | Abelian group property of ... |
| hhssablo 31129 | Abelian group property of ... |
| hhssnv 31130 | Normed complex vector spac... |
| hhssnvt 31131 | Normed complex vector spac... |
| hhsst 31132 | A member of ` SH ` is a su... |
| hhshsslem1 31133 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31134 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31135 | The predicate " ` H ` is a... |
| hhsssh2 31136 | The predicate " ` H ` is a... |
| hhssba 31137 | The base set of a subspace... |
| hhssvs 31138 | The vector subtraction ope... |
| hhssvsf 31139 | Mapping of the vector subt... |
| hhssims 31140 | Induced metric of a subspa... |
| hhssims2 31141 | Induced metric of a subspa... |
| hhssmet 31142 | Induced metric of a subspa... |
| hhssmetdval 31143 | Value of the distance func... |
| hhsscms 31144 | The induced metric of a cl... |
| hhssbnOLD 31145 | Obsolete version of ~ cssb... |
| ocval 31146 | Value of orthogonal comple... |
| ocel 31147 | Membership in orthogonal c... |
| shocel 31148 | Membership in orthogonal c... |
| ocsh 31149 | The orthogonal complement ... |
| shocsh 31150 | The orthogonal complement ... |
| ocss 31151 | An orthogonal complement i... |
| shocss 31152 | An orthogonal complement i... |
| occon 31153 | Contraposition law for ort... |
| occon2 31154 | Double contraposition for ... |
| occon2i 31155 | Double contraposition for ... |
| oc0 31156 | The zero vector belongs to... |
| ocorth 31157 | Members of a subset and it... |
| shocorth 31158 | Members of a subspace and ... |
| ococss 31159 | Inclusion in complement of... |
| shococss 31160 | Inclusion in complement of... |
| shorth 31161 | Members of orthogonal subs... |
| ocin 31162 | Intersection of a Hilbert ... |
| occon3 31163 | Hilbert lattice contraposi... |
| ocnel 31164 | A nonzero vector in the co... |
| chocvali 31165 | Value of the orthogonal co... |
| shuni 31166 | Two subspaces with trivial... |
| chocunii 31167 | Lemma for uniqueness part ... |
| pjhthmo 31168 | Projection Theorem, unique... |
| occllem 31169 | Lemma for ~ occl . (Contr... |
| occl 31170 | Closure of complement of H... |
| shoccl 31171 | Closure of complement of H... |
| choccl 31172 | Closure of complement of H... |
| choccli 31173 | Closure of ` CH ` orthocom... |
| shsval 31178 | Value of subspace sum of t... |
| shsss 31179 | The subspace sum is a subs... |
| shsel 31180 | Membership in the subspace... |
| shsel3 31181 | Membership in the subspace... |
| shseli 31182 | Membership in subspace sum... |
| shscli 31183 | Closure of subspace sum. ... |
| shscl 31184 | Closure of subspace sum. ... |
| shscom 31185 | Commutative law for subspa... |
| shsva 31186 | Vector sum belongs to subs... |
| shsel1 31187 | A subspace sum contains a ... |
| shsel2 31188 | A subspace sum contains a ... |
| shsvs 31189 | Vector subtraction belongs... |
| shsub1 31190 | Subspace sum is an upper b... |
| shsub2 31191 | Subspace sum is an upper b... |
| choc0 31192 | The orthocomplement of the... |
| choc1 31193 | The orthocomplement of the... |
| chocnul 31194 | Orthogonal complement of t... |
| shintcli 31195 | Closure of intersection of... |
| shintcl 31196 | The intersection of a none... |
| chintcli 31197 | The intersection of a none... |
| chintcl 31198 | The intersection (infimum)... |
| spanval 31199 | Value of the linear span o... |
| hsupval 31200 | Value of supremum of set o... |
| chsupval 31201 | The value of the supremum ... |
| spancl 31202 | The span of a subset of Hi... |
| elspancl 31203 | A member of a span is a ve... |
| shsupcl 31204 | Closure of the subspace su... |
| hsupcl 31205 | Closure of supremum of set... |
| chsupcl 31206 | Closure of supremum of sub... |
| hsupss 31207 | Subset relation for suprem... |
| chsupss 31208 | Subset relation for suprem... |
| hsupunss 31209 | The union of a set of Hilb... |
| chsupunss 31210 | The union of a set of clos... |
| spanss2 31211 | A subset of Hilbert space ... |
| shsupunss 31212 | The union of a set of subs... |
| spanid 31213 | A subspace of Hilbert spac... |
| spanss 31214 | Ordering relationship for ... |
| spanssoc 31215 | The span of a subset of Hi... |
| sshjval 31216 | Value of join for subsets ... |
| shjval 31217 | Value of join in ` SH ` . ... |
| chjval 31218 | Value of join in ` CH ` . ... |
| chjvali 31219 | Value of join in ` CH ` . ... |
| sshjval3 31220 | Value of join for subsets ... |
| sshjcl 31221 | Closure of join for subset... |
| shjcl 31222 | Closure of join in ` SH ` ... |
| chjcl 31223 | Closure of join in ` CH ` ... |
| shjcom 31224 | Commutative law for Hilber... |
| shless 31225 | Subset implies subset of s... |
| shlej1 31226 | Add disjunct to both sides... |
| shlej2 31227 | Add disjunct to both sides... |
| shincli 31228 | Closure of intersection of... |
| shscomi 31229 | Commutative law for subspa... |
| shsvai 31230 | Vector sum belongs to subs... |
| shsel1i 31231 | A subspace sum contains a ... |
| shsel2i 31232 | A subspace sum contains a ... |
| shsvsi 31233 | Vector subtraction belongs... |
| shunssi 31234 | Union is smaller than subs... |
| shunssji 31235 | Union is smaller than Hilb... |
| shsleji 31236 | Subspace sum is smaller th... |
| shjcomi 31237 | Commutative law for join i... |
| shsub1i 31238 | Subspace sum is an upper b... |
| shsub2i 31239 | Subspace sum is an upper b... |
| shub1i 31240 | Hilbert lattice join is an... |
| shjcli 31241 | Closure of ` CH ` join. (... |
| shjshcli 31242 | ` SH ` closure of join. (... |
| shlessi 31243 | Subset implies subset of s... |
| shlej1i 31244 | Add disjunct to both sides... |
| shlej2i 31245 | Add disjunct to both sides... |
| shslej 31246 | Subspace sum is smaller th... |
| shincl 31247 | Closure of intersection of... |
| shub1 31248 | Hilbert lattice join is an... |
| shub2 31249 | A subspace is a subset of ... |
| shsidmi 31250 | Idempotent law for Hilbert... |
| shslubi 31251 | The least upper bound law ... |
| shlesb1i 31252 | Hilbert lattice ordering i... |
| shsval2i 31253 | An alternate way to expres... |
| shsval3i 31254 | An alternate way to expres... |
| shmodsi 31255 | The modular law holds for ... |
| shmodi 31256 | The modular law is implied... |
| pjhthlem1 31257 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31258 | Lemma for ~ pjhth . (Cont... |
| pjhth 31259 | Projection Theorem: Any H... |
| pjhtheu 31260 | Projection Theorem: Any H... |
| pjhfval 31262 | The value of the projectio... |
| pjhval 31263 | Value of a projection. (C... |
| pjpreeq 31264 | Equality with a projection... |
| pjeq 31265 | Equality with a projection... |
| axpjcl 31266 | Closure of a projection in... |
| pjhcl 31267 | Closure of a projection in... |
| omlsilem 31268 | Lemma for orthomodular law... |
| omlsii 31269 | Subspace inference form of... |
| omlsi 31270 | Subspace form of orthomodu... |
| ococi 31271 | Complement of complement o... |
| ococ 31272 | Complement of complement o... |
| dfch2 31273 | Alternate definition of th... |
| ococin 31274 | The double complement is t... |
| hsupval2 31275 | Alternate definition of su... |
| chsupval2 31276 | The value of the supremum ... |
| sshjval2 31277 | Value of join in the set o... |
| chsupid 31278 | A subspace is the supremum... |
| chsupsn 31279 | Value of supremum of subse... |
| shlub 31280 | Hilbert lattice join is th... |
| shlubi 31281 | Hilbert lattice join is th... |
| pjhtheu2 31282 | Uniqueness of ` y ` for th... |
| pjcli 31283 | Closure of a projection in... |
| pjhcli 31284 | Closure of a projection in... |
| pjpjpre 31285 | Decomposition of a vector ... |
| axpjpj 31286 | Decomposition of a vector ... |
| pjclii 31287 | Closure of a projection in... |
| pjhclii 31288 | Closure of a projection in... |
| pjpj0i 31289 | Decomposition of a vector ... |
| pjpji 31290 | Decomposition of a vector ... |
| pjpjhth 31291 | Projection Theorem: Any H... |
| pjpjhthi 31292 | Projection Theorem: Any H... |
| pjop 31293 | Orthocomplement projection... |
| pjpo 31294 | Projection in terms of ort... |
| pjopi 31295 | Orthocomplement projection... |
| pjpoi 31296 | Projection in terms of ort... |
| pjoc1i 31297 | Projection of a vector in ... |
| pjchi 31298 | Projection of a vector in ... |
| pjoccl 31299 | The part of a vector that ... |
| pjoc1 31300 | Projection of a vector in ... |
| pjomli 31301 | Subspace form of orthomodu... |
| pjoml 31302 | Subspace form of orthomodu... |
| pjococi 31303 | Proof of orthocomplement t... |
| pjoc2i 31304 | Projection of a vector in ... |
| pjoc2 31305 | Projection of a vector in ... |
| sh0le 31306 | The zero subspace is the s... |
| ch0le 31307 | The zero subspace is the s... |
| shle0 31308 | No subspace is smaller tha... |
| chle0 31309 | No Hilbert lattice element... |
| chnlen0 31310 | A Hilbert lattice element ... |
| ch0pss 31311 | The zero subspace is a pro... |
| orthin 31312 | The intersection of orthog... |
| ssjo 31313 | The lattice join of a subs... |
| shne0i 31314 | A nonzero subspace has a n... |
| shs0i 31315 | Hilbert subspace sum with ... |
| shs00i 31316 | Two subspaces are zero iff... |
| ch0lei 31317 | The closed subspace zero i... |
| chle0i 31318 | No Hilbert closed subspace... |
| chne0i 31319 | A nonzero closed subspace ... |
| chocini 31320 | Intersection of a closed s... |
| chj0i 31321 | Join with lattice zero in ... |
| chm1i 31322 | Meet with lattice one in `... |
| chjcli 31323 | Closure of ` CH ` join. (... |
| chsleji 31324 | Subspace sum is smaller th... |
| chseli 31325 | Membership in subspace sum... |
| chincli 31326 | Closure of Hilbert lattice... |
| chsscon3i 31327 | Hilbert lattice contraposi... |
| chsscon1i 31328 | Hilbert lattice contraposi... |
| chsscon2i 31329 | Hilbert lattice contraposi... |
| chcon2i 31330 | Hilbert lattice contraposi... |
| chcon1i 31331 | Hilbert lattice contraposi... |
| chcon3i 31332 | Hilbert lattice contraposi... |
| chunssji 31333 | Union is smaller than ` CH... |
| chjcomi 31334 | Commutative law for join i... |
| chub1i 31335 | ` CH ` join is an upper bo... |
| chub2i 31336 | ` CH ` join is an upper bo... |
| chlubi 31337 | Hilbert lattice join is th... |
| chlubii 31338 | Hilbert lattice join is th... |
| chlej1i 31339 | Add join to both sides of ... |
| chlej2i 31340 | Add join to both sides of ... |
| chlej12i 31341 | Add join to both sides of ... |
| chlejb1i 31342 | Hilbert lattice ordering i... |
| chdmm1i 31343 | De Morgan's law for meet i... |
| chdmm2i 31344 | De Morgan's law for meet i... |
| chdmm3i 31345 | De Morgan's law for meet i... |
| chdmm4i 31346 | De Morgan's law for meet i... |
| chdmj1i 31347 | De Morgan's law for join i... |
| chdmj2i 31348 | De Morgan's law for join i... |
| chdmj3i 31349 | De Morgan's law for join i... |
| chdmj4i 31350 | De Morgan's law for join i... |
| chnlei 31351 | Equivalent expressions for... |
| chjassi 31352 | Associative law for Hilber... |
| chj00i 31353 | Two Hilbert lattice elemen... |
| chjoi 31354 | The join of a closed subsp... |
| chj1i 31355 | Join with Hilbert lattice ... |
| chm0i 31356 | Meet with Hilbert lattice ... |
| chm0 31357 | Meet with Hilbert lattice ... |
| shjshsi 31358 | Hilbert lattice join equal... |
| shjshseli 31359 | A closed subspace sum equa... |
| chne0 31360 | A nonzero closed subspace ... |
| chocin 31361 | Intersection of a closed s... |
| chssoc 31362 | A closed subspace less tha... |
| chj0 31363 | Join with Hilbert lattice ... |
| chslej 31364 | Subspace sum is smaller th... |
| chincl 31365 | Closure of Hilbert lattice... |
| chsscon3 31366 | Hilbert lattice contraposi... |
| chsscon1 31367 | Hilbert lattice contraposi... |
| chsscon2 31368 | Hilbert lattice contraposi... |
| chpsscon3 31369 | Hilbert lattice contraposi... |
| chpsscon1 31370 | Hilbert lattice contraposi... |
| chpsscon2 31371 | Hilbert lattice contraposi... |
| chjcom 31372 | Commutative law for Hilber... |
| chub1 31373 | Hilbert lattice join is gr... |
| chub2 31374 | Hilbert lattice join is gr... |
| chlub 31375 | Hilbert lattice join is th... |
| chlej1 31376 | Add join to both sides of ... |
| chlej2 31377 | Add join to both sides of ... |
| chlejb1 31378 | Hilbert lattice ordering i... |
| chlejb2 31379 | Hilbert lattice ordering i... |
| chnle 31380 | Equivalent expressions for... |
| chjo 31381 | The join of a closed subsp... |
| chabs1 31382 | Hilbert lattice absorption... |
| chabs2 31383 | Hilbert lattice absorption... |
| chabs1i 31384 | Hilbert lattice absorption... |
| chabs2i 31385 | Hilbert lattice absorption... |
| chjidm 31386 | Idempotent law for Hilbert... |
| chjidmi 31387 | Idempotent law for Hilbert... |
| chj12i 31388 | A rearrangement of Hilbert... |
| chj4i 31389 | Rearrangement of the join ... |
| chjjdiri 31390 | Hilbert lattice join distr... |
| chdmm1 31391 | De Morgan's law for meet i... |
| chdmm2 31392 | De Morgan's law for meet i... |
| chdmm3 31393 | De Morgan's law for meet i... |
| chdmm4 31394 | De Morgan's law for meet i... |
| chdmj1 31395 | De Morgan's law for join i... |
| chdmj2 31396 | De Morgan's law for join i... |
| chdmj3 31397 | De Morgan's law for join i... |
| chdmj4 31398 | De Morgan's law for join i... |
| chjass 31399 | Associative law for Hilber... |
| chj12 31400 | A rearrangement of Hilbert... |
| chj4 31401 | Rearrangement of the join ... |
| ledii 31402 | An ortholattice is distrib... |
| lediri 31403 | An ortholattice is distrib... |
| lejdii 31404 | An ortholattice is distrib... |
| lejdiri 31405 | An ortholattice is distrib... |
| ledi 31406 | An ortholattice is distrib... |
| spansn0 31407 | The span of the singleton ... |
| span0 31408 | The span of the empty set ... |
| elspani 31409 | Membership in the span of ... |
| spanuni 31410 | The span of a union is the... |
| spanun 31411 | The span of a union is the... |
| sshhococi 31412 | The join of two Hilbert sp... |
| hne0 31413 | Hilbert space has a nonzer... |
| chsup0 31414 | The supremum of the empty ... |
| h1deoi 31415 | Membership in orthocomplem... |
| h1dei 31416 | Membership in 1-dimensiona... |
| h1did 31417 | A generating vector belong... |
| h1dn0 31418 | A nonzero vector generates... |
| h1de2i 31419 | Membership in 1-dimensiona... |
| h1de2bi 31420 | Membership in 1-dimensiona... |
| h1de2ctlem 31421 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31422 | Membership in 1-dimensiona... |
| spansni 31423 | The span of a singleton in... |
| elspansni 31424 | Membership in the span of ... |
| spansn 31425 | The span of a singleton in... |
| spansnch 31426 | The span of a Hilbert spac... |
| spansnsh 31427 | The span of a Hilbert spac... |
| spansnchi 31428 | The span of a singleton in... |
| spansnid 31429 | A vector belongs to the sp... |
| spansnmul 31430 | A scalar product with a ve... |
| elspansncl 31431 | A member of a span of a si... |
| elspansn 31432 | Membership in the span of ... |
| elspansn2 31433 | Membership in the span of ... |
| spansncol 31434 | The singletons of collinea... |
| spansneleqi 31435 | Membership relation implie... |
| spansneleq 31436 | Membership relation that i... |
| spansnss 31437 | The span of the singleton ... |
| elspansn3 31438 | A member of the span of th... |
| elspansn4 31439 | A span membership conditio... |
| elspansn5 31440 | A vector belonging to both... |
| spansnss2 31441 | The span of the singleton ... |
| normcan 31442 | Cancellation-type law that... |
| pjspansn 31443 | A projection on the span o... |
| spansnpji 31444 | A subset of Hilbert space ... |
| spanunsni 31445 | The span of the union of a... |
| spanpr 31446 | The span of a pair of vect... |
| h1datomi 31447 | A 1-dimensional subspace i... |
| h1datom 31448 | A 1-dimensional subspace i... |
| cmbr 31450 | Binary relation expressing... |
| pjoml2i 31451 | Variation of orthomodular ... |
| pjoml3i 31452 | Variation of orthomodular ... |
| pjoml4i 31453 | Variation of orthomodular ... |
| pjoml5i 31454 | The orthomodular law. Rem... |
| pjoml6i 31455 | An equivalent of the ortho... |
| cmbri 31456 | Binary relation expressing... |
| cmcmlem 31457 | Commutation is symmetric. ... |
| cmcmi 31458 | Commutation is symmetric. ... |
| cmcm2i 31459 | Commutation with orthocomp... |
| cmcm3i 31460 | Commutation with orthocomp... |
| cmcm4i 31461 | Commutation with orthocomp... |
| cmbr2i 31462 | Alternate definition of th... |
| cmcmii 31463 | Commutation is symmetric. ... |
| cmcm2ii 31464 | Commutation with orthocomp... |
| cmcm3ii 31465 | Commutation with orthocomp... |
| cmbr3i 31466 | Alternate definition for t... |
| cmbr4i 31467 | Alternate definition for t... |
| lecmi 31468 | Comparable Hilbert lattice... |
| lecmii 31469 | Comparable Hilbert lattice... |
| cmj1i 31470 | A Hilbert lattice element ... |
| cmj2i 31471 | A Hilbert lattice element ... |
| cmm1i 31472 | A Hilbert lattice element ... |
| cmm2i 31473 | A Hilbert lattice element ... |
| cmbr3 31474 | Alternate definition for t... |
| cm0 31475 | The zero Hilbert lattice e... |
| cmidi 31476 | The commutes relation is r... |
| pjoml2 31477 | Variation of orthomodular ... |
| pjoml3 31478 | Variation of orthomodular ... |
| pjoml5 31479 | The orthomodular law. Rem... |
| cmcm 31480 | Commutation is symmetric. ... |
| cmcm3 31481 | Commutation with orthocomp... |
| cmcm2 31482 | Commutation with orthocomp... |
| lecm 31483 | Comparable Hilbert lattice... |
| fh1 31484 | Foulis-Holland Theorem. I... |
| fh2 31485 | Foulis-Holland Theorem. I... |
| cm2j 31486 | A lattice element that com... |
| fh1i 31487 | Foulis-Holland Theorem. I... |
| fh2i 31488 | Foulis-Holland Theorem. I... |
| fh3i 31489 | Variation of the Foulis-Ho... |
| fh4i 31490 | Variation of the Foulis-Ho... |
| cm2ji 31491 | A lattice element that com... |
| cm2mi 31492 | A lattice element that com... |
| qlax1i 31493 | One of the equations showi... |
| qlax2i 31494 | One of the equations showi... |
| qlax3i 31495 | One of the equations showi... |
| qlax4i 31496 | One of the equations showi... |
| qlax5i 31497 | One of the equations showi... |
| qlaxr1i 31498 | One of the conditions show... |
| qlaxr2i 31499 | One of the conditions show... |
| qlaxr4i 31500 | One of the conditions show... |
| qlaxr5i 31501 | One of the conditions show... |
| qlaxr3i 31502 | A variation of the orthomo... |
| chscllem1 31503 | Lemma for ~ chscl . (Cont... |
| chscllem2 31504 | Lemma for ~ chscl . (Cont... |
| chscllem3 31505 | Lemma for ~ chscl . (Cont... |
| chscllem4 31506 | Lemma for ~ chscl . (Cont... |
| chscl 31507 | The subspace sum of two cl... |
| osumi 31508 | If two closed subspaces of... |
| osumcori 31509 | Corollary of ~ osumi . (C... |
| osumcor2i 31510 | Corollary of ~ osumi , sho... |
| osum 31511 | If two closed subspaces of... |
| spansnji 31512 | The subspace sum of a clos... |
| spansnj 31513 | The subspace sum of a clos... |
| spansnscl 31514 | The subspace sum of a clos... |
| sumspansn 31515 | The sum of two vectors bel... |
| spansnm0i 31516 | The meet of different one-... |
| nonbooli 31517 | A Hilbert lattice with two... |
| spansncvi 31518 | Hilbert space has the cove... |
| spansncv 31519 | Hilbert space has the cove... |
| 5oalem1 31520 | Lemma for orthoarguesian l... |
| 5oalem2 31521 | Lemma for orthoarguesian l... |
| 5oalem3 31522 | Lemma for orthoarguesian l... |
| 5oalem4 31523 | Lemma for orthoarguesian l... |
| 5oalem5 31524 | Lemma for orthoarguesian l... |
| 5oalem6 31525 | Lemma for orthoarguesian l... |
| 5oalem7 31526 | Lemma for orthoarguesian l... |
| 5oai 31527 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31528 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31529 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31530 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31531 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31532 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31533 | Lemma for 3OA (weak) ortho... |
| 3oai 31534 | 3OA (weak) orthoarguesian ... |
| pjorthi 31535 | Projection components on o... |
| pjch1 31536 | Property of identity proje... |
| pjo 31537 | The orthogonal projection.... |
| pjcompi 31538 | Component of a projection.... |
| pjidmi 31539 | A projection is idempotent... |
| pjadjii 31540 | A projection is self-adjoi... |
| pjaddii 31541 | Projection of vector sum i... |
| pjinormii 31542 | The inner product of a pro... |
| pjmulii 31543 | Projection of (scalar) pro... |
| pjsubii 31544 | Projection of vector diffe... |
| pjsslem 31545 | Lemma for subset relations... |
| pjss2i 31546 | Subset relationship for pr... |
| pjssmii 31547 | Projection meet property. ... |
| pjssge0ii 31548 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31549 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31550 | The projection on a subspa... |
| pjadji 31551 | A projection is self-adjoi... |
| pjaddi 31552 | Projection of vector sum i... |
| pjinormi 31553 | The inner product of a pro... |
| pjsubi 31554 | Projection of vector diffe... |
| pjmuli 31555 | Projection of scalar produ... |
| pjige0i 31556 | The inner product of a pro... |
| pjige0 31557 | The inner product of a pro... |
| pjcjt2 31558 | The projection on a subspa... |
| pj0i 31559 | The projection of the zero... |
| pjch 31560 | Projection of a vector in ... |
| pjid 31561 | The projection of a vector... |
| pjvec 31562 | The set of vectors belongi... |
| pjocvec 31563 | The set of vectors belongi... |
| pjocini 31564 | Membership of projection i... |
| pjini 31565 | Membership of projection i... |
| pjjsi 31566 | A sufficient condition for... |
| pjfni 31567 | Functionality of a project... |
| pjrni 31568 | The range of a projection.... |
| pjfoi 31569 | A projection maps onto its... |
| pjfi 31570 | The mapping of a projectio... |
| pjvi 31571 | The value of a projection ... |
| pjhfo 31572 | A projection maps onto its... |
| pjrn 31573 | The range of a projection.... |
| pjhf 31574 | The mapping of a projectio... |
| pjfn 31575 | Functionality of a project... |
| pjsumi 31576 | The projection on a subspa... |
| pj11i 31577 | One-to-one correspondence ... |
| pjdsi 31578 | Vector decomposition into ... |
| pjds3i 31579 | Vector decomposition into ... |
| pj11 31580 | One-to-one correspondence ... |
| pjmfn 31581 | Functionality of the proje... |
| pjmf1 31582 | The projector function map... |
| pjoi0 31583 | The inner product of proje... |
| pjoi0i 31584 | The inner product of proje... |
| pjopythi 31585 | Pythagorean theorem for pr... |
| pjopyth 31586 | Pythagorean theorem for pr... |
| pjnormi 31587 | The norm of the projection... |
| pjpythi 31588 | Pythagorean theorem for pr... |
| pjneli 31589 | If a vector does not belon... |
| pjnorm 31590 | The norm of the projection... |
| pjpyth 31591 | Pythagorean theorem for pr... |
| pjnel 31592 | If a vector does not belon... |
| pjnorm2 31593 | A vector belongs to the su... |
| mayete3i 31594 | Mayet's equation E_3. Par... |
| mayetes3i 31595 | Mayet's equation E^*_3, de... |
| hosmval 31601 | Value of the sum of two Hi... |
| hommval 31602 | Value of the scalar produc... |
| hodmval 31603 | Value of the difference of... |
| hfsmval 31604 | Value of the sum of two Hi... |
| hfmmval 31605 | Value of the scalar produc... |
| hosval 31606 | Value of the sum of two Hi... |
| homval 31607 | Value of the scalar produc... |
| hodval 31608 | Value of the difference of... |
| hfsval 31609 | Value of the sum of two Hi... |
| hfmval 31610 | Value of the scalar produc... |
| hoscl 31611 | Closure of the sum of two ... |
| homcl 31612 | Closure of the scalar prod... |
| hodcl 31613 | Closure of the difference ... |
| ho0val 31616 | Value of the zero Hilbert ... |
| ho0f 31617 | Functionality of the zero ... |
| df0op2 31618 | Alternate definition of Hi... |
| dfiop2 31619 | Alternate definition of Hi... |
| hoif 31620 | Functionality of the Hilbe... |
| hoival 31621 | The value of the Hilbert s... |
| hoico1 31622 | Composition with the Hilbe... |
| hoico2 31623 | Composition with the Hilbe... |
| hoaddcl 31624 | The sum of Hilbert space o... |
| homulcl 31625 | The scalar product of a Hi... |
| hoeq 31626 | Equality of Hilbert space ... |
| hoeqi 31627 | Equality of Hilbert space ... |
| hoscli 31628 | Closure of Hilbert space o... |
| hodcli 31629 | Closure of Hilbert space o... |
| hocoi 31630 | Composition of Hilbert spa... |
| hococli 31631 | Closure of composition of ... |
| hocofi 31632 | Mapping of composition of ... |
| hocofni 31633 | Functionality of compositi... |
| hoaddcli 31634 | Mapping of sum of Hilbert ... |
| hosubcli 31635 | Mapping of difference of H... |
| hoaddfni 31636 | Functionality of sum of Hi... |
| hosubfni 31637 | Functionality of differenc... |
| hoaddcomi 31638 | Commutativity of sum of Hi... |
| hosubcl 31639 | Mapping of difference of H... |
| hoaddcom 31640 | Commutativity of sum of Hi... |
| hodsi 31641 | Relationship between Hilbe... |
| hoaddassi 31642 | Associativity of sum of Hi... |
| hoadd12i 31643 | Commutative/associative la... |
| hoadd32i 31644 | Commutative/associative la... |
| hocadddiri 31645 | Distributive law for Hilbe... |
| hocsubdiri 31646 | Distributive law for Hilbe... |
| ho2coi 31647 | Double composition of Hilb... |
| hoaddass 31648 | Associativity of sum of Hi... |
| hoadd32 31649 | Commutative/associative la... |
| hoadd4 31650 | Rearrangement of 4 terms i... |
| hocsubdir 31651 | Distributive law for Hilbe... |
| hoaddridi 31652 | Sum of a Hilbert space ope... |
| hodidi 31653 | Difference of a Hilbert sp... |
| ho0coi 31654 | Composition of the zero op... |
| hoid1i 31655 | Composition of Hilbert spa... |
| hoid1ri 31656 | Composition of Hilbert spa... |
| hoaddrid 31657 | Sum of a Hilbert space ope... |
| hodid 31658 | Difference of a Hilbert sp... |
| hon0 31659 | A Hilbert space operator i... |
| hodseqi 31660 | Subtraction and addition o... |
| ho0subi 31661 | Subtraction of Hilbert spa... |
| honegsubi 31662 | Relationship between Hilbe... |
| ho0sub 31663 | Subtraction of Hilbert spa... |
| hosubid1 31664 | The zero operator subtract... |
| honegsub 31665 | Relationship between Hilbe... |
| homullid 31666 | An operator equals its sca... |
| homco1 31667 | Associative law for scalar... |
| homulass 31668 | Scalar product associative... |
| hoadddi 31669 | Scalar product distributiv... |
| hoadddir 31670 | Scalar product reverse dis... |
| homul12 31671 | Swap first and second fact... |
| honegneg 31672 | Double negative of a Hilbe... |
| hosubneg 31673 | Relationship between opera... |
| hosubdi 31674 | Scalar product distributiv... |
| honegdi 31675 | Distribution of negative o... |
| honegsubdi 31676 | Distribution of negative o... |
| honegsubdi2 31677 | Distribution of negative o... |
| hosubsub2 31678 | Law for double subtraction... |
| hosub4 31679 | Rearrangement of 4 terms i... |
| hosubadd4 31680 | Rearrangement of 4 terms i... |
| hoaddsubass 31681 | Associative-type law for a... |
| hoaddsub 31682 | Law for operator addition ... |
| hosubsub 31683 | Law for double subtraction... |
| hosubsub4 31684 | Law for double subtraction... |
| ho2times 31685 | Two times a Hilbert space ... |
| hoaddsubassi 31686 | Associativity of sum and d... |
| hoaddsubi 31687 | Law for sum and difference... |
| hosd1i 31688 | Hilbert space operator sum... |
| hosd2i 31689 | Hilbert space operator sum... |
| hopncani 31690 | Hilbert space operator can... |
| honpcani 31691 | Hilbert space operator can... |
| hosubeq0i 31692 | If the difference between ... |
| honpncani 31693 | Hilbert space operator can... |
| ho01i 31694 | A condition implying that ... |
| ho02i 31695 | A condition implying that ... |
| hoeq1 31696 | A condition implying that ... |
| hoeq2 31697 | A condition implying that ... |
| adjmo 31698 | Every Hilbert space operat... |
| adjsym 31699 | Symmetry property of an ad... |
| eigrei 31700 | A necessary and sufficient... |
| eigre 31701 | A necessary and sufficient... |
| eigposi 31702 | A sufficient condition (fi... |
| eigorthi 31703 | A necessary and sufficient... |
| eigorth 31704 | A necessary and sufficient... |
| nmopval 31722 | Value of the norm of a Hil... |
| elcnop 31723 | Property defining a contin... |
| ellnop 31724 | Property defining a linear... |
| lnopf 31725 | A linear Hilbert space ope... |
| elbdop 31726 | Property defining a bounde... |
| bdopln 31727 | A bounded linear Hilbert s... |
| bdopf 31728 | A bounded linear Hilbert s... |
| nmopsetretALT 31729 | The set in the supremum of... |
| nmopsetretHIL 31730 | The set in the supremum of... |
| nmopsetn0 31731 | The set in the supremum of... |
| nmopxr 31732 | The norm of a Hilbert spac... |
| nmoprepnf 31733 | The norm of a Hilbert spac... |
| nmopgtmnf 31734 | The norm of a Hilbert spac... |
| nmopreltpnf 31735 | The norm of a Hilbert spac... |
| nmopre 31736 | The norm of a bounded oper... |
| elbdop2 31737 | Property defining a bounde... |
| elunop 31738 | Property defining a unitar... |
| elhmop 31739 | Property defining a Hermit... |
| hmopf 31740 | A Hermitian operator is a ... |
| hmopex 31741 | The class of Hermitian ope... |
| nmfnval 31742 | Value of the norm of a Hil... |
| nmfnsetre 31743 | The set in the supremum of... |
| nmfnsetn0 31744 | The set in the supremum of... |
| nmfnxr 31745 | The norm of any Hilbert sp... |
| nmfnrepnf 31746 | The norm of a Hilbert spac... |
| nlfnval 31747 | Value of the null space of... |
| elcnfn 31748 | Property defining a contin... |
| ellnfn 31749 | Property defining a linear... |
| lnfnf 31750 | A linear Hilbert space fun... |
| dfadj2 31751 | Alternate definition of th... |
| funadj 31752 | Functionality of the adjoi... |
| dmadjss 31753 | The domain of the adjoint ... |
| dmadjop 31754 | A member of the domain of ... |
| adjeu 31755 | Elementhood in the domain ... |
| adjval 31756 | Value of the adjoint funct... |
| adjval2 31757 | Value of the adjoint funct... |
| cnvadj 31758 | The adjoint function equal... |
| funcnvadj 31759 | The converse of the adjoin... |
| adj1o 31760 | The adjoint function maps ... |
| dmadjrn 31761 | The adjoint of an operator... |
| eigvecval 31762 | The set of eigenvectors of... |
| eigvalfval 31763 | The eigenvalues of eigenve... |
| specval 31764 | The value of the spectrum ... |
| speccl 31765 | The spectrum of an operato... |
| hhlnoi 31766 | The linear operators of Hi... |
| hhnmoi 31767 | The norm of an operator in... |
| hhbloi 31768 | A bounded linear operator ... |
| hh0oi 31769 | The zero operator in Hilbe... |
| hhcno 31770 | The continuous operators o... |
| hhcnf 31771 | The continuous functionals... |
| dmadjrnb 31772 | The adjoint of an operator... |
| nmoplb 31773 | A lower bound for an opera... |
| nmopub 31774 | An upper bound for an oper... |
| nmopub2tALT 31775 | An upper bound for an oper... |
| nmopub2tHIL 31776 | An upper bound for an oper... |
| nmopge0 31777 | The norm of any Hilbert sp... |
| nmopgt0 31778 | A linear Hilbert space ope... |
| cnopc 31779 | Basic continuity property ... |
| lnopl 31780 | Basic linearity property o... |
| unop 31781 | Basic inner product proper... |
| unopf1o 31782 | A unitary operator in Hilb... |
| unopnorm 31783 | A unitary operator is idem... |
| cnvunop 31784 | The inverse (converse) of ... |
| unopadj 31785 | The inverse (converse) of ... |
| unoplin 31786 | A unitary operator is line... |
| counop 31787 | The composition of two uni... |
| hmop 31788 | Basic inner product proper... |
| hmopre 31789 | The inner product of the v... |
| nmfnlb 31790 | A lower bound for a functi... |
| nmfnleub 31791 | An upper bound for the nor... |
| nmfnleub2 31792 | An upper bound for the nor... |
| nmfnge0 31793 | The norm of any Hilbert sp... |
| elnlfn 31794 | Membership in the null spa... |
| elnlfn2 31795 | Membership in the null spa... |
| cnfnc 31796 | Basic continuity property ... |
| lnfnl 31797 | Basic linearity property o... |
| adjcl 31798 | Closure of the adjoint of ... |
| adj1 31799 | Property of an adjoint Hil... |
| adj2 31800 | Property of an adjoint Hil... |
| adjeq 31801 | A property that determines... |
| adjadj 31802 | Double adjoint. Theorem 3... |
| adjvalval 31803 | Value of the value of the ... |
| unopadj2 31804 | The adjoint of a unitary o... |
| hmopadj 31805 | A Hermitian operator is se... |
| hmdmadj 31806 | Every Hermitian operator h... |
| hmopadj2 31807 | An operator is Hermitian i... |
| hmoplin 31808 | A Hermitian operator is li... |
| brafval 31809 | The bra of a vector, expre... |
| braval 31810 | A bra-ket juxtaposition, e... |
| braadd 31811 | Linearity property of bra ... |
| bramul 31812 | Linearity property of bra ... |
| brafn 31813 | The bra function is a func... |
| bralnfn 31814 | The Dirac bra function is ... |
| bracl 31815 | Closure of the bra functio... |
| bra0 31816 | The Dirac bra of the zero ... |
| brafnmul 31817 | Anti-linearity property of... |
| kbfval 31818 | The outer product of two v... |
| kbop 31819 | The outer product of two v... |
| kbval 31820 | The value of the operator ... |
| kbmul 31821 | Multiplication property of... |
| kbpj 31822 | If a vector ` A ` has norm... |
| eleigvec 31823 | Membership in the set of e... |
| eleigvec2 31824 | Membership in the set of e... |
| eleigveccl 31825 | Closure of an eigenvector ... |
| eigvalval 31826 | The eigenvalue of an eigen... |
| eigvalcl 31827 | An eigenvalue is a complex... |
| eigvec1 31828 | Property of an eigenvector... |
| eighmre 31829 | The eigenvalues of a Hermi... |
| eighmorth 31830 | Eigenvectors of a Hermitia... |
| nmopnegi 31831 | Value of the norm of the n... |
| lnop0 31832 | The value of a linear Hilb... |
| lnopmul 31833 | Multiplicative property of... |
| lnopli 31834 | Basic scalar product prope... |
| lnopfi 31835 | A linear Hilbert space ope... |
| lnop0i 31836 | The value of a linear Hilb... |
| lnopaddi 31837 | Additive property of a lin... |
| lnopmuli 31838 | Multiplicative property of... |
| lnopaddmuli 31839 | Sum/product property of a ... |
| lnopsubi 31840 | Subtraction property for a... |
| lnopsubmuli 31841 | Subtraction/product proper... |
| lnopmulsubi 31842 | Product/subtraction proper... |
| homco2 31843 | Move a scalar product out ... |
| idunop 31844 | The identity function (res... |
| 0cnop 31845 | The identically zero funct... |
| 0cnfn 31846 | The identically zero funct... |
| idcnop 31847 | The identity function (res... |
| idhmop 31848 | The Hilbert space identity... |
| 0hmop 31849 | The identically zero funct... |
| 0lnop 31850 | The identically zero funct... |
| 0lnfn 31851 | The identically zero funct... |
| nmop0 31852 | The norm of the zero opera... |
| nmfn0 31853 | The norm of the identicall... |
| hmopbdoptHIL 31854 | A Hermitian operator is a ... |
| hoddii 31855 | Distributive law for Hilbe... |
| hoddi 31856 | Distributive law for Hilbe... |
| nmop0h 31857 | The norm of any operator o... |
| idlnop 31858 | The identity function (res... |
| 0bdop 31859 | The identically zero opera... |
| adj0 31860 | Adjoint of the zero operat... |
| nmlnop0iALT 31861 | A linear operator with a z... |
| nmlnop0iHIL 31862 | A linear operator with a z... |
| nmlnopgt0i 31863 | A linear Hilbert space ope... |
| nmlnop0 31864 | A linear operator with a z... |
| nmlnopne0 31865 | A linear operator with a n... |
| lnopmi 31866 | The scalar product of a li... |
| lnophsi 31867 | The sum of two linear oper... |
| lnophdi 31868 | The difference of two line... |
| lnopcoi 31869 | The composition of two lin... |
| lnopco0i 31870 | The composition of a linea... |
| lnopeq0lem1 31871 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 31872 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 31873 | A condition implying that ... |
| lnopeqi 31874 | Two linear Hilbert space o... |
| lnopeq 31875 | Two linear Hilbert space o... |
| lnopunilem1 31876 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 31877 | Lemma for ~ lnopunii . (C... |
| lnopunii 31878 | If a linear operator (whos... |
| elunop2 31879 | An operator is unitary iff... |
| nmopun 31880 | Norm of a unitary Hilbert ... |
| unopbd 31881 | A unitary operator is a bo... |
| lnophmlem1 31882 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 31883 | Lemma for ~ lnophmi . (Co... |
| lnophmi 31884 | A linear operator is Hermi... |
| lnophm 31885 | A linear operator is Hermi... |
| hmops 31886 | The sum of two Hermitian o... |
| hmopm 31887 | The scalar product of a He... |
| hmopd 31888 | The difference of two Herm... |
| hmopco 31889 | The composition of two com... |
| nmbdoplbi 31890 | A lower bound for the norm... |
| nmbdoplb 31891 | A lower bound for the norm... |
| nmcexi 31892 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 31893 | The norm of a continuous l... |
| nmcoplbi 31894 | A lower bound for the norm... |
| nmcopex 31895 | The norm of a continuous l... |
| nmcoplb 31896 | A lower bound for the norm... |
| nmophmi 31897 | The norm of the scalar pro... |
| bdophmi 31898 | The scalar product of a bo... |
| lnconi 31899 | Lemma for ~ lnopconi and ~... |
| lnopconi 31900 | A condition equivalent to ... |
| lnopcon 31901 | A condition equivalent to ... |
| lnopcnbd 31902 | A linear operator is conti... |
| lncnopbd 31903 | A continuous linear operat... |
| lncnbd 31904 | A continuous linear operat... |
| lnopcnre 31905 | A linear operator is conti... |
| lnfnli 31906 | Basic property of a linear... |
| lnfnfi 31907 | A linear Hilbert space fun... |
| lnfn0i 31908 | The value of a linear Hilb... |
| lnfnaddi 31909 | Additive property of a lin... |
| lnfnmuli 31910 | Multiplicative property of... |
| lnfnaddmuli 31911 | Sum/product property of a ... |
| lnfnsubi 31912 | Subtraction property for a... |
| lnfn0 31913 | The value of a linear Hilb... |
| lnfnmul 31914 | Multiplicative property of... |
| nmbdfnlbi 31915 | A lower bound for the norm... |
| nmbdfnlb 31916 | A lower bound for the norm... |
| nmcfnexi 31917 | The norm of a continuous l... |
| nmcfnlbi 31918 | A lower bound for the norm... |
| nmcfnex 31919 | The norm of a continuous l... |
| nmcfnlb 31920 | A lower bound of the norm ... |
| lnfnconi 31921 | A condition equivalent to ... |
| lnfncon 31922 | A condition equivalent to ... |
| lnfncnbd 31923 | A linear functional is con... |
| imaelshi 31924 | The image of a subspace un... |
| rnelshi 31925 | The range of a linear oper... |
| nlelshi 31926 | The null space of a linear... |
| nlelchi 31927 | The null space of a contin... |
| riesz3i 31928 | A continuous linear functi... |
| riesz4i 31929 | A continuous linear functi... |
| riesz4 31930 | A continuous linear functi... |
| riesz1 31931 | Part 1 of the Riesz repres... |
| riesz2 31932 | Part 2 of the Riesz repres... |
| cnlnadjlem1 31933 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 31934 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 31935 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 31936 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 31937 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 31938 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 31939 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 31940 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 31941 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 31942 | Every continuous linear op... |
| cnlnadjeui 31943 | Every continuous linear op... |
| cnlnadjeu 31944 | Every continuous linear op... |
| cnlnadj 31945 | Every continuous linear op... |
| cnlnssadj 31946 | Every continuous linear Hi... |
| bdopssadj 31947 | Every bounded linear Hilbe... |
| bdopadj 31948 | Every bounded linear Hilbe... |
| adjbdln 31949 | The adjoint of a bounded l... |
| adjbdlnb 31950 | An operator is bounded and... |
| adjbd1o 31951 | The mapping of adjoints of... |
| adjlnop 31952 | The adjoint of an operator... |
| adjsslnop 31953 | Every operator with an adj... |
| nmopadjlei 31954 | Property of the norm of an... |
| nmopadjlem 31955 | Lemma for ~ nmopadji . (C... |
| nmopadji 31956 | Property of the norm of an... |
| adjeq0 31957 | An operator is zero iff it... |
| adjmul 31958 | The adjoint of the scalar ... |
| adjadd 31959 | The adjoint of the sum of ... |
| nmoptrii 31960 | Triangle inequality for th... |
| nmopcoi 31961 | Upper bound for the norm o... |
| bdophsi 31962 | The sum of two bounded lin... |
| bdophdi 31963 | The difference between two... |
| bdopcoi 31964 | The composition of two bou... |
| nmoptri2i 31965 | Triangle-type inequality f... |
| adjcoi 31966 | The adjoint of a compositi... |
| nmopcoadji 31967 | The norm of an operator co... |
| nmopcoadj2i 31968 | The norm of an operator co... |
| nmopcoadj0i 31969 | An operator composed with ... |
| unierri 31970 | If we approximate a chain ... |
| branmfn 31971 | The norm of the bra functi... |
| brabn 31972 | The bra of a vector is a b... |
| rnbra 31973 | The set of bras equals the... |
| bra11 31974 | The bra function maps vect... |
| bracnln 31975 | A bra is a continuous line... |
| cnvbraval 31976 | Value of the converse of t... |
| cnvbracl 31977 | Closure of the converse of... |
| cnvbrabra 31978 | The converse bra of the br... |
| bracnvbra 31979 | The bra of the converse br... |
| bracnlnval 31980 | The vector that a continuo... |
| cnvbramul 31981 | Multiplication property of... |
| kbass1 31982 | Dirac bra-ket associative ... |
| kbass2 31983 | Dirac bra-ket associative ... |
| kbass3 31984 | Dirac bra-ket associative ... |
| kbass4 31985 | Dirac bra-ket associative ... |
| kbass5 31986 | Dirac bra-ket associative ... |
| kbass6 31987 | Dirac bra-ket associative ... |
| leopg 31988 | Ordering relation for posi... |
| leop 31989 | Ordering relation for oper... |
| leop2 31990 | Ordering relation for oper... |
| leop3 31991 | Operator ordering in terms... |
| leoppos 31992 | Binary relation defining a... |
| leoprf2 31993 | The ordering relation for ... |
| leoprf 31994 | The ordering relation for ... |
| leopsq 31995 | The square of a Hermitian ... |
| 0leop 31996 | The zero operator is a pos... |
| idleop 31997 | The identity operator is a... |
| leopadd 31998 | The sum of two positive op... |
| leopmuli 31999 | The scalar product of a no... |
| leopmul 32000 | The scalar product of a po... |
| leopmul2i 32001 | Scalar product applied to ... |
| leoptri 32002 | The positive operator orde... |
| leoptr 32003 | The positive operator orde... |
| leopnmid 32004 | A bounded Hermitian operat... |
| nmopleid 32005 | A nonzero, bounded Hermiti... |
| opsqrlem1 32006 | Lemma for opsqri . (Contr... |
| opsqrlem2 32007 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 32008 | Lemma for opsqri . (Contr... |
| opsqrlem4 32009 | Lemma for opsqri . (Contr... |
| opsqrlem5 32010 | Lemma for opsqri . (Contr... |
| opsqrlem6 32011 | Lemma for opsqri . (Contr... |
| pjhmopi 32012 | A projector is a Hermitian... |
| pjlnopi 32013 | A projector is a linear op... |
| pjnmopi 32014 | The operator norm of a pro... |
| pjbdlni 32015 | A projector is a bounded l... |
| pjhmop 32016 | A projection is a Hermitia... |
| hmopidmchi 32017 | An idempotent Hermitian op... |
| hmopidmpji 32018 | An idempotent Hermitian op... |
| hmopidmch 32019 | An idempotent Hermitian op... |
| hmopidmpj 32020 | An idempotent Hermitian op... |
| pjsdii 32021 | Distributive law for Hilbe... |
| pjddii 32022 | Distributive law for Hilbe... |
| pjsdi2i 32023 | Chained distributive law f... |
| pjcoi 32024 | Composition of projections... |
| pjcocli 32025 | Closure of composition of ... |
| pjcohcli 32026 | Closure of composition of ... |
| pjadjcoi 32027 | Adjoint of composition of ... |
| pjcofni 32028 | Functionality of compositi... |
| pjss1coi 32029 | Subset relationship for pr... |
| pjss2coi 32030 | Subset relationship for pr... |
| pjssmi 32031 | Projection meet property. ... |
| pjssge0i 32032 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 32033 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 32034 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 32035 | Composition of projections... |
| pjscji 32036 | The projection of orthogon... |
| pjssumi 32037 | The projection on a subspa... |
| pjssposi 32038 | Projector ordering can be ... |
| pjordi 32039 | The definition of projecto... |
| pjssdif2i 32040 | The projection subspace of... |
| pjssdif1i 32041 | A necessary and sufficient... |
| pjimai 32042 | The image of a projection.... |
| pjidmcoi 32043 | A projection is idempotent... |
| pjoccoi 32044 | Composition of projections... |
| pjtoi 32045 | Subspace sum of projection... |
| pjoci 32046 | Projection of orthocomplem... |
| pjidmco 32047 | A projection operator is i... |
| dfpjop 32048 | Definition of projection o... |
| pjhmopidm 32049 | Two ways to express the se... |
| elpjidm 32050 | A projection operator is i... |
| elpjhmop 32051 | A projection operator is H... |
| 0leopj 32052 | A projector is a positive ... |
| pjadj2 32053 | A projector is self-adjoin... |
| pjadj3 32054 | A projector is self-adjoin... |
| elpjch 32055 | Reconstruction of the subs... |
| elpjrn 32056 | Reconstruction of the subs... |
| pjinvari 32057 | A closed subspace ` H ` wi... |
| pjin1i 32058 | Lemma for Theorem 1.22 of ... |
| pjin2i 32059 | Lemma for Theorem 1.22 of ... |
| pjin3i 32060 | Lemma for Theorem 1.22 of ... |
| pjclem1 32061 | Lemma for projection commu... |
| pjclem2 32062 | Lemma for projection commu... |
| pjclem3 32063 | Lemma for projection commu... |
| pjclem4a 32064 | Lemma for projection commu... |
| pjclem4 32065 | Lemma for projection commu... |
| pjci 32066 | Two subspaces commute iff ... |
| pjcmul1i 32067 | A necessary and sufficient... |
| pjcmul2i 32068 | The projection subspace of... |
| pjcohocli 32069 | Closure of composition of ... |
| pjadj2coi 32070 | Adjoint of double composit... |
| pj2cocli 32071 | Closure of double composit... |
| pj3lem1 32072 | Lemma for projection tripl... |
| pj3si 32073 | Stronger projection triple... |
| pj3i 32074 | Projection triplet theorem... |
| pj3cor1i 32075 | Projection triplet corolla... |
| pjs14i 32076 | Theorem S-14 of Watanabe, ... |
| isst 32079 | Property of a state. (Con... |
| ishst 32080 | Property of a complex Hilb... |
| sticl 32081 | ` [ 0 , 1 ] ` closure of t... |
| stcl 32082 | Real closure of the value ... |
| hstcl 32083 | Closure of the value of a ... |
| hst1a 32084 | Unit value of a Hilbert-sp... |
| hstel2 32085 | Properties of a Hilbert-sp... |
| hstorth 32086 | Orthogonality property of ... |
| hstosum 32087 | Orthogonal sum property of... |
| hstoc 32088 | Sum of a Hilbert-space-val... |
| hstnmoc 32089 | Sum of norms of a Hilbert-... |
| stge0 32090 | The value of a state is no... |
| stle1 32091 | The value of a state is le... |
| hstle1 32092 | The norm of the value of a... |
| hst1h 32093 | The norm of a Hilbert-spac... |
| hst0h 32094 | The norm of a Hilbert-spac... |
| hstpyth 32095 | Pythagorean property of a ... |
| hstle 32096 | Ordering property of a Hil... |
| hstles 32097 | Ordering property of a Hil... |
| hstoh 32098 | A Hilbert-space-valued sta... |
| hst0 32099 | A Hilbert-space-valued sta... |
| sthil 32100 | The value of a state at th... |
| stj 32101 | The value of a state on a ... |
| sto1i 32102 | The state of a subspace pl... |
| sto2i 32103 | The state of the orthocomp... |
| stge1i 32104 | If a state is greater than... |
| stle0i 32105 | If a state is less than or... |
| stlei 32106 | Ordering law for states. ... |
| stlesi 32107 | Ordering law for states. ... |
| stji1i 32108 | Join of components of Sasa... |
| stm1i 32109 | State of component of unit... |
| stm1ri 32110 | State of component of unit... |
| stm1addi 32111 | Sum of states whose meet i... |
| staddi 32112 | If the sum of 2 states is ... |
| stm1add3i 32113 | Sum of states whose meet i... |
| stadd3i 32114 | If the sum of 3 states is ... |
| st0 32115 | The state of the zero subs... |
| strlem1 32116 | Lemma for strong state the... |
| strlem2 32117 | Lemma for strong state the... |
| strlem3a 32118 | Lemma for strong state the... |
| strlem3 32119 | Lemma for strong state the... |
| strlem4 32120 | Lemma for strong state the... |
| strlem5 32121 | Lemma for strong state the... |
| strlem6 32122 | Lemma for strong state the... |
| stri 32123 | Strong state theorem. The... |
| strb 32124 | Strong state theorem (bidi... |
| hstrlem2 32125 | Lemma for strong set of CH... |
| hstrlem3a 32126 | Lemma for strong set of CH... |
| hstrlem3 32127 | Lemma for strong set of CH... |
| hstrlem4 32128 | Lemma for strong set of CH... |
| hstrlem5 32129 | Lemma for strong set of CH... |
| hstrlem6 32130 | Lemma for strong set of CH... |
| hstri 32131 | Hilbert space admits a str... |
| hstrbi 32132 | Strong CH-state theorem (b... |
| largei 32133 | A Hilbert lattice admits a... |
| jplem1 32134 | Lemma for Jauch-Piron theo... |
| jplem2 32135 | Lemma for Jauch-Piron theo... |
| jpi 32136 | The function ` S ` , that ... |
| golem1 32137 | Lemma for Godowski's equat... |
| golem2 32138 | Lemma for Godowski's equat... |
| goeqi 32139 | Godowski's equation, shown... |
| stcltr1i 32140 | Property of a strong class... |
| stcltr2i 32141 | Property of a strong class... |
| stcltrlem1 32142 | Lemma for strong classical... |
| stcltrlem2 32143 | Lemma for strong classical... |
| stcltrthi 32144 | Theorem for classically st... |
| cvbr 32148 | Binary relation expressing... |
| cvbr2 32149 | Binary relation expressing... |
| cvcon3 32150 | Contraposition law for the... |
| cvpss 32151 | The covers relation implie... |
| cvnbtwn 32152 | The covers relation implie... |
| cvnbtwn2 32153 | The covers relation implie... |
| cvnbtwn3 32154 | The covers relation implie... |
| cvnbtwn4 32155 | The covers relation implie... |
| cvnsym 32156 | The covers relation is not... |
| cvnref 32157 | The covers relation is not... |
| cvntr 32158 | The covers relation is not... |
| spansncv2 32159 | Hilbert space has the cove... |
| mdbr 32160 | Binary relation expressing... |
| mdi 32161 | Consequence of the modular... |
| mdbr2 32162 | Binary relation expressing... |
| mdbr3 32163 | Binary relation expressing... |
| mdbr4 32164 | Binary relation expressing... |
| dmdbr 32165 | Binary relation expressing... |
| dmdmd 32166 | The dual modular pair prop... |
| mddmd 32167 | The modular pair property ... |
| dmdi 32168 | Consequence of the dual mo... |
| dmdbr2 32169 | Binary relation expressing... |
| dmdi2 32170 | Consequence of the dual mo... |
| dmdbr3 32171 | Binary relation expressing... |
| dmdbr4 32172 | Binary relation expressing... |
| dmdi4 32173 | Consequence of the dual mo... |
| dmdbr5 32174 | Binary relation expressing... |
| mddmd2 32175 | Relationship between modul... |
| mdsl0 32176 | A sublattice condition tha... |
| ssmd1 32177 | Ordering implies the modul... |
| ssmd2 32178 | Ordering implies the modul... |
| ssdmd1 32179 | Ordering implies the dual ... |
| ssdmd2 32180 | Ordering implies the dual ... |
| dmdsl3 32181 | Sublattice mapping for a d... |
| mdsl3 32182 | Sublattice mapping for a m... |
| mdslle1i 32183 | Order preservation of the ... |
| mdslle2i 32184 | Order preservation of the ... |
| mdslj1i 32185 | Join preservation of the o... |
| mdslj2i 32186 | Meet preservation of the r... |
| mdsl1i 32187 | If the modular pair proper... |
| mdsl2i 32188 | If the modular pair proper... |
| mdsl2bi 32189 | If the modular pair proper... |
| cvmdi 32190 | The covering property impl... |
| mdslmd1lem1 32191 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 32192 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 32193 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 32194 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 32195 | Preservation of the modula... |
| mdslmd2i 32196 | Preservation of the modula... |
| mdsldmd1i 32197 | Preservation of the dual m... |
| mdslmd3i 32198 | Modular pair conditions th... |
| mdslmd4i 32199 | Modular pair condition tha... |
| csmdsymi 32200 | Cross-symmetry implies M-s... |
| mdexchi 32201 | An exchange lemma for modu... |
| cvmd 32202 | The covering property impl... |
| cvdmd 32203 | The covering property impl... |
| ela 32205 | Atoms in a Hilbert lattice... |
| elat2 32206 | Expanded membership relati... |
| elatcv0 32207 | A Hilbert lattice element ... |
| atcv0 32208 | An atom covers the zero su... |
| atssch 32209 | Atoms are a subset of the ... |
| atelch 32210 | An atom is a Hilbert latti... |
| atne0 32211 | An atom is not the Hilbert... |
| atss 32212 | A lattice element smaller ... |
| atsseq 32213 | Two atoms in a subset rela... |
| atcveq0 32214 | A Hilbert lattice element ... |
| h1da 32215 | A 1-dimensional subspace i... |
| spansna 32216 | The span of the singleton ... |
| sh1dle 32217 | A 1-dimensional subspace i... |
| ch1dle 32218 | A 1-dimensional subspace i... |
| atom1d 32219 | The 1-dimensional subspace... |
| superpos 32220 | Superposition Principle. ... |
| chcv1 32221 | The Hilbert lattice has th... |
| chcv2 32222 | The Hilbert lattice has th... |
| chjatom 32223 | The join of a closed subsp... |
| shatomici 32224 | The lattice of Hilbert sub... |
| hatomici 32225 | The Hilbert lattice is ato... |
| hatomic 32226 | A Hilbert lattice is atomi... |
| shatomistici 32227 | The lattice of Hilbert sub... |
| hatomistici 32228 | ` CH ` is atomistic, i.e. ... |
| chpssati 32229 | Two Hilbert lattice elemen... |
| chrelati 32230 | The Hilbert lattice is rel... |
| chrelat2i 32231 | A consequence of relative ... |
| cvati 32232 | If a Hilbert lattice eleme... |
| cvbr4i 32233 | An alternate way to expres... |
| cvexchlem 32234 | Lemma for ~ cvexchi . (Co... |
| cvexchi 32235 | The Hilbert lattice satisf... |
| chrelat2 32236 | A consequence of relative ... |
| chrelat3 32237 | A consequence of relative ... |
| chrelat3i 32238 | A consequence of the relat... |
| chrelat4i 32239 | A consequence of relative ... |
| cvexch 32240 | The Hilbert lattice satisf... |
| cvp 32241 | The Hilbert lattice satisf... |
| atnssm0 32242 | The meet of a Hilbert latt... |
| atnemeq0 32243 | The meet of distinct atoms... |
| atssma 32244 | The meet with an atom's su... |
| atcv0eq 32245 | Two atoms covering the zer... |
| atcv1 32246 | Two atoms covering the zer... |
| atexch 32247 | The Hilbert lattice satisf... |
| atomli 32248 | An assertion holding in at... |
| atoml2i 32249 | An assertion holding in at... |
| atordi 32250 | An ordering law for a Hilb... |
| atcvatlem 32251 | Lemma for ~ atcvati . (Co... |
| atcvati 32252 | A nonzero Hilbert lattice ... |
| atcvat2i 32253 | A Hilbert lattice element ... |
| atord 32254 | An ordering law for a Hilb... |
| atcvat2 32255 | A Hilbert lattice element ... |
| chirredlem1 32256 | Lemma for ~ chirredi . (C... |
| chirredlem2 32257 | Lemma for ~ chirredi . (C... |
| chirredlem3 32258 | Lemma for ~ chirredi . (C... |
| chirredlem4 32259 | Lemma for ~ chirredi . (C... |
| chirredi 32260 | The Hilbert lattice is irr... |
| chirred 32261 | The Hilbert lattice is irr... |
| atcvat3i 32262 | A condition implying that ... |
| atcvat4i 32263 | A condition implying exist... |
| atdmd 32264 | Two Hilbert lattice elemen... |
| atmd 32265 | Two Hilbert lattice elemen... |
| atmd2 32266 | Two Hilbert lattice elemen... |
| atabsi 32267 | Absorption of an incompara... |
| atabs2i 32268 | Absorption of an incompara... |
| mdsymlem1 32269 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 32270 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 32271 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 32272 | Lemma for ~ mdsymi . This... |
| mdsymlem5 32273 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 32274 | Lemma for ~ mdsymi . This... |
| mdsymlem7 32275 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 32276 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 32277 | M-symmetry of the Hilbert ... |
| mdsym 32278 | M-symmetry of the Hilbert ... |
| dmdsym 32279 | Dual M-symmetry of the Hil... |
| atdmd2 32280 | Two Hilbert lattice elemen... |
| sumdmdii 32281 | If the subspace sum of two... |
| cmmdi 32282 | Commuting subspaces form a... |
| cmdmdi 32283 | Commuting subspaces form a... |
| sumdmdlem 32284 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 32285 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 32286 | The subspace sum of two Hi... |
| dmdbr4ati 32287 | Dual modular pair property... |
| dmdbr5ati 32288 | Dual modular pair property... |
| dmdbr6ati 32289 | Dual modular pair property... |
| dmdbr7ati 32290 | Dual modular pair property... |
| mdoc1i 32291 | Orthocomplements form a mo... |
| mdoc2i 32292 | Orthocomplements form a mo... |
| dmdoc1i 32293 | Orthocomplements form a du... |
| dmdoc2i 32294 | Orthocomplements form a du... |
| mdcompli 32295 | A condition equivalent to ... |
| dmdcompli 32296 | A condition equivalent to ... |
| mddmdin0i 32297 | If dual modular implies mo... |
| cdjreui 32298 | A member of the sum of dis... |
| cdj1i 32299 | Two ways to express " ` A ... |
| cdj3lem1 32300 | A property of " ` A ` and ... |
| cdj3lem2 32301 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 32302 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 32303 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 32304 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 32305 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 32306 | Lemma for ~ cdj3i . The s... |
| cdj3i 32307 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 32308 | (_This theorem is a dummy ... |
| sa-abvi 32309 | A theorem about the univer... |
| xfree 32310 | A partial converse to ~ 19... |
| xfree2 32311 | A partial converse to ~ 19... |
| addltmulALT 32312 | A proof readability experi... |
| bian1d 32313 | Adding a superfluous conju... |
| bibiad 32314 | Eliminate an hypothesis ` ... |
| orim12da 32315 | Deduce a disjunction from ... |
| or3di 32316 | Distributive law for disju... |
| or3dir 32317 | Distributive law for disju... |
| 3o1cs 32318 | Deduction eliminating disj... |
| 3o2cs 32319 | Deduction eliminating disj... |
| 3o3cs 32320 | Deduction eliminating disj... |
| 13an22anass 32321 | Associative law for four c... |
| sbc2iedf 32322 | Conversion of implicit sub... |
| rspc2daf 32323 | Double restricted speciali... |
| ralcom4f 32324 | Commutation of restricted ... |
| rexcom4f 32325 | Commutation of restricted ... |
| 19.9d2rf 32326 | A deduction version of one... |
| 19.9d2r 32327 | A deduction version of one... |
| r19.29ffa 32328 | A commonly used pattern ba... |
| eqtrb 32329 | A transposition of equalit... |
| eqelbid 32330 | A variable elimination law... |
| opsbc2ie 32331 | Conversion of implicit sub... |
| opreu2reuALT 32332 | Correspondence between uni... |
| 2reucom 32335 | Double restricted existent... |
| 2reu2rex1 32336 | Double restricted existent... |
| 2reureurex 32337 | Double restricted existent... |
| 2reu2reu2 32338 | Double restricted existent... |
| opreu2reu1 32339 | Equivalent definition of t... |
| sq2reunnltb 32340 | There exists a unique deco... |
| addsqnot2reu 32341 | For each complex number ` ... |
| sbceqbidf 32342 | Equality theorem for class... |
| sbcies 32343 | A special version of class... |
| mo5f 32344 | Alternate definition of "a... |
| nmo 32345 | Negation of "at most one".... |
| reuxfrdf 32346 | Transfer existential uniqu... |
| rexunirn 32347 | Restricted existential qua... |
| rmoxfrd 32348 | Transfer "at most one" res... |
| rmoun 32349 | "At most one" restricted e... |
| rmounid 32350 | A case where an "at most o... |
| riotaeqbidva 32351 | Equivalent wff's yield equ... |
| dmrab 32352 | Domain of a restricted cla... |
| difrab2 32353 | Difference of two restrict... |
| rabexgfGS 32354 | Separation Scheme in terms... |
| rabsnel 32355 | Truth implied by equality ... |
| eqrrabd 32356 | Deduce equality with a res... |
| foresf1o 32357 | From a surjective function... |
| rabfodom 32358 | Domination relation for re... |
| abrexdomjm 32359 | An indexed set is dominate... |
| abrexdom2jm 32360 | An indexed set is dominate... |
| abrexexd 32361 | Existence of a class abstr... |
| elabreximd 32362 | Class substitution in an i... |
| elabreximdv 32363 | Class substitution in an i... |
| abrexss 32364 | A necessary condition for ... |
| elunsn 32365 | Elementhood to a union wit... |
| nelun 32366 | Negated membership for a u... |
| snsssng 32367 | If a singleton is a subset... |
| inin 32368 | Intersection with an inter... |
| inindif 32369 | See ~ inundif . (Contribu... |
| difininv 32370 | Condition for the intersec... |
| eldifsnd 32371 | Membership in a set with a... |
| difeq 32372 | Rewriting an equation with... |
| eqdif 32373 | If both set differences of... |
| indifbi 32374 | Two ways to express equali... |
| diffib 32375 | Case where ~ diffi is a bi... |
| difxp1ss 32376 | Difference law for Cartesi... |
| difxp2ss 32377 | Difference law for Cartesi... |
| indifundif 32378 | A remarkable equation with... |
| elpwincl1 32379 | Closure of intersection wi... |
| elpwdifcl 32380 | Closure of class differenc... |
| elpwiuncl 32381 | Closure of indexed union w... |
| eqsnd 32382 | Deduce that a set is a sin... |
| elpreq 32383 | Equality wihin a pair. (C... |
| nelpr 32384 | A set ` A ` not in a pair ... |
| inpr0 32385 | Rewrite an empty intersect... |
| neldifpr1 32386 | The first element of a pai... |
| neldifpr2 32387 | The second element of a pa... |
| unidifsnel 32388 | The other element of a pai... |
| unidifsnne 32389 | The other element of a pai... |
| ifeqeqx 32390 | An equality theorem tailor... |
| elimifd 32391 | Elimination of a condition... |
| elim2if 32392 | Elimination of two conditi... |
| elim2ifim 32393 | Elimination of two conditi... |
| ifeq3da 32394 | Given an expression ` C ` ... |
| ifnetrue 32395 | Deduce truth from a condit... |
| ifnefals 32396 | Deduce falsehood from a co... |
| ifnebib 32397 | The converse of ~ ifbi hol... |
| uniinn0 32398 | Sufficient and necessary c... |
| uniin1 32399 | Union of intersection. Ge... |
| uniin2 32400 | Union of intersection. Ge... |
| difuncomp 32401 | Express a class difference... |
| elpwunicl 32402 | Closure of a set union wit... |
| cbviunf 32403 | Rule used to change the bo... |
| iuneq12daf 32404 | Equality deduction for ind... |
| iunin1f 32405 | Indexed union of intersect... |
| ssiun3 32406 | Subset equivalence for an ... |
| ssiun2sf 32407 | Subset relationship for an... |
| iuninc 32408 | The union of an increasing... |
| iundifdifd 32409 | The intersection of a set ... |
| iundifdif 32410 | The intersection of a set ... |
| iunrdx 32411 | Re-index an indexed union.... |
| iunpreima 32412 | Preimage of an indexed uni... |
| iunrnmptss 32413 | A subset relation for an i... |
| iunxunsn 32414 | Appending a set to an inde... |
| iunxunpr 32415 | Appending two sets to an i... |
| iinabrex 32416 | Rewriting an indexed inter... |
| disjnf 32417 | In case ` x ` is not free ... |
| cbvdisjf 32418 | Change bound variables in ... |
| disjss1f 32419 | A subset of a disjoint col... |
| disjeq1f 32420 | Equality theorem for disjo... |
| disjxun0 32421 | Simplify a disjoint union.... |
| disjdifprg 32422 | A trivial partition into a... |
| disjdifprg2 32423 | A trivial partition of a s... |
| disji2f 32424 | Property of a disjoint col... |
| disjif 32425 | Property of a disjoint col... |
| disjorf 32426 | Two ways to say that a col... |
| disjorsf 32427 | Two ways to say that a col... |
| disjif2 32428 | Property of a disjoint col... |
| disjabrex 32429 | Rewriting a disjoint colle... |
| disjabrexf 32430 | Rewriting a disjoint colle... |
| disjpreima 32431 | A preimage of a disjoint s... |
| disjrnmpt 32432 | Rewriting a disjoint colle... |
| disjin 32433 | If a collection is disjoin... |
| disjin2 32434 | If a collection is disjoin... |
| disjxpin 32435 | Derive a disjunction over ... |
| iundisjf 32436 | Rewrite a countable union ... |
| iundisj2f 32437 | A disjoint union is disjoi... |
| disjrdx 32438 | Re-index a disjunct collec... |
| disjex 32439 | Two ways to say that two c... |
| disjexc 32440 | A variant of ~ disjex , ap... |
| disjunsn 32441 | Append an element to a dis... |
| disjun0 32442 | Adding the empty element p... |
| disjiunel 32443 | A set of elements B of a d... |
| disjuniel 32444 | A set of elements B of a d... |
| xpdisjres 32445 | Restriction of a constant ... |
| opeldifid 32446 | Ordered pair elementhood o... |
| difres 32447 | Case when class difference... |
| imadifxp 32448 | Image of the difference wi... |
| relfi 32449 | A relation (set) is finite... |
| 0res 32450 | Restriction of the empty f... |
| fcoinver 32451 | Build an equivalence relat... |
| fcoinvbr 32452 | Binary relation for the eq... |
| copsex2dv 32453 | Implicit substitution dedu... |
| brab2d 32454 | Expressing that two sets a... |
| brabgaf 32455 | The law of concretion for ... |
| brelg 32456 | Two things in a binary rel... |
| br8d 32457 | Substitution for an eight-... |
| opabdm 32458 | Domain of an ordered-pair ... |
| opabrn 32459 | Range of an ordered-pair c... |
| opabssi 32460 | Sufficient condition for a... |
| opabid2ss 32461 | One direction of ~ opabid2... |
| ssrelf 32462 | A subclass relationship de... |
| eqrelrd2 32463 | A version of ~ eqrelrdv2 w... |
| erbr3b 32464 | Biconditional for equivale... |
| iunsnima 32465 | Image of a singleton by an... |
| iunsnima2 32466 | Version of ~ iunsnima with... |
| ac6sf2 32467 | Alternate version of ~ ac6... |
| fnresin 32468 | Restriction of a function ... |
| f1o3d 32469 | Describe an implicit one-t... |
| eldmne0 32470 | A function of nonempty dom... |
| f1rnen 32471 | Equinumerosity of the rang... |
| rinvf1o 32472 | Sufficient conditions for ... |
| fresf1o 32473 | Conditions for a restricti... |
| nfpconfp 32474 | The set of fixed points of... |
| fmptco1f1o 32475 | The action of composing (t... |
| cofmpt2 32476 | Express composition of a m... |
| f1mptrn 32477 | Express injection for a ma... |
| dfimafnf 32478 | Alternate definition of th... |
| funimass4f 32479 | Membership relation for th... |
| elimampt 32480 | Membership in the image of... |
| suppss2f 32481 | Show that the support of a... |
| ofrn 32482 | The range of the function ... |
| ofrn2 32483 | The range of the function ... |
| off2 32484 | The function operation pro... |
| ofresid 32485 | Applying an operation rest... |
| fimarab 32486 | Expressing the image of a ... |
| unipreima 32487 | Preimage of a class union.... |
| opfv 32488 | Value of a function produc... |
| xppreima 32489 | The preimage of a Cartesia... |
| 2ndimaxp 32490 | Image of a cartesian produ... |
| djussxp2 32491 | Stronger version of ~ djus... |
| 2ndresdju 32492 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32493 | The ` 2nd ` function restr... |
| xppreima2 32494 | The preimage of a Cartesia... |
| abfmpunirn 32495 | Membership in a union of a... |
| rabfmpunirn 32496 | Membership in a union of a... |
| abfmpeld 32497 | Membership in an element o... |
| abfmpel 32498 | Membership in an element o... |
| fmptdF 32499 | Domain and codomain of the... |
| fmptcof2 32500 | Composition of two functio... |
| fcomptf 32501 | Express composition of two... |
| acunirnmpt 32502 | Axiom of choice for the un... |
| acunirnmpt2 32503 | Axiom of choice for the un... |
| acunirnmpt2f 32504 | Axiom of choice for the un... |
| aciunf1lem 32505 | Choice in an index union. ... |
| aciunf1 32506 | Choice in an index union. ... |
| ofoprabco 32507 | Function operation as a co... |
| ofpreima 32508 | Express the preimage of a ... |
| ofpreima2 32509 | Express the preimage of a ... |
| funcnvmpt 32510 | Condition for a function i... |
| funcnv5mpt 32511 | Two ways to say that a fun... |
| funcnv4mpt 32512 | Two ways to say that a fun... |
| preimane 32513 | Different elements have di... |
| fnpreimac 32514 | Choose a set ` x ` contain... |
| fgreu 32515 | Exactly one point of a fun... |
| fcnvgreu 32516 | If the converse of a relat... |
| rnmposs 32517 | The range of an operation ... |
| mptssALT 32518 | Deduce subset relation of ... |
| dfcnv2 32519 | Alternative definition of ... |
| fnimatp 32520 | The image of an unordered ... |
| mpomptxf 32521 | Express a two-argument fun... |
| suppovss 32522 | A bound for the support of... |
| suppiniseg 32523 | Relation between the suppo... |
| fsuppinisegfi 32524 | The initial segment ` ( ``... |
| fressupp 32525 | The restriction of a funct... |
| fdifsuppconst 32526 | A function is a zero const... |
| ressupprn 32527 | The range of a function re... |
| supppreima 32528 | Express the support of a f... |
| fsupprnfi 32529 | Finite support implies fin... |
| mptiffisupp 32530 | Conditions for a mapping f... |
| cosnopne 32531 | Composition of two ordered... |
| cosnop 32532 | Composition of two ordered... |
| cnvprop 32533 | Converse of a pair of orde... |
| brprop 32534 | Binary relation for a pair... |
| mptprop 32535 | Rewrite pairs of ordered p... |
| coprprop 32536 | Composition of two pairs o... |
| gtiso 32537 | Two ways to write a strict... |
| isoun 32538 | Infer an isomorphism from ... |
| disjdsct 32539 | A disjoint collection is d... |
| df1stres 32540 | Definition for a restricti... |
| df2ndres 32541 | Definition for a restricti... |
| 1stpreimas 32542 | The preimage of a singleto... |
| 1stpreima 32543 | The preimage by ` 1st ` is... |
| 2ndpreima 32544 | The preimage by ` 2nd ` is... |
| curry2ima 32545 | The image of a curried fun... |
| preiman0 32546 | The preimage of a nonempty... |
| intimafv 32547 | The intersection of an ima... |
| supssd 32548 | Inequality deduction for s... |
| infssd 32549 | Inequality deduction for i... |
| imafi2 32550 | The image by a finite set ... |
| unifi3 32551 | If a union is finite, then... |
| snct 32552 | A singleton is countable. ... |
| prct 32553 | An unordered pair is count... |
| mpocti 32554 | An operation is countable ... |
| abrexct 32555 | An image set of a countabl... |
| mptctf 32556 | A countable mapping set is... |
| abrexctf 32557 | An image set of a countabl... |
| padct 32558 | Index a countable set with... |
| cnvoprabOLD 32559 | The converse of a class ab... |
| f1od2 32560 | Sufficient condition for a... |
| fcobij 32561 | Composing functions with a... |
| fcobijfs 32562 | Composing finitely support... |
| suppss3 32563 | Deduce a function's suppor... |
| fsuppcurry1 32564 | Finite support of a currie... |
| fsuppcurry2 32565 | Finite support of a currie... |
| offinsupp1 32566 | Finite support for a funct... |
| ffs2 32567 | Rewrite a function's suppo... |
| ffsrn 32568 | The range of a finitely su... |
| resf1o 32569 | Restriction of functions t... |
| maprnin 32570 | Restricting the range of t... |
| fpwrelmapffslem 32571 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32572 | Define a canonical mapping... |
| fpwrelmapffs 32573 | Define a canonical mapping... |
| creq0 32574 | The real representation of... |
| 1nei 32575 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32576 | An integer unit times itse... |
| nnmulge 32577 | Multiplying by a positive ... |
| lt2addrd 32578 | If the right-hand side of ... |
| xrlelttric 32579 | Trichotomy law for extende... |
| xaddeq0 32580 | Two extended reals which a... |
| xrinfm 32581 | The extended real numbers ... |
| le2halvesd 32582 | A sum is less than the who... |
| xraddge02 32583 | A number is less than or e... |
| xrge0addge 32584 | A number is less than or e... |
| xlt2addrd 32585 | If the right-hand side of ... |
| xrsupssd 32586 | Inequality deduction for s... |
| xrge0infss 32587 | Any subset of nonnegative ... |
| xrge0infssd 32588 | Inequality deduction for i... |
| xrge0addcld 32589 | Nonnegative extended reals... |
| xrge0subcld 32590 | Condition for closure of n... |
| infxrge0lb 32591 | A member of a set of nonne... |
| infxrge0glb 32592 | The infimum of a set of no... |
| infxrge0gelb 32593 | The infimum of a set of no... |
| xrofsup 32594 | The supremum is preserved ... |
| supxrnemnf 32595 | The supremum of a nonempty... |
| xnn0gt0 32596 | Nonzero extended nonnegati... |
| xnn01gt 32597 | An extended nonnegative in... |
| nn0xmulclb 32598 | Finite multiplication in t... |
| joiniooico 32599 | Disjoint joining an open i... |
| ubico 32600 | A right-open interval does... |
| xeqlelt 32601 | Equality in terms of 'less... |
| eliccelico 32602 | Relate elementhood to a cl... |
| elicoelioo 32603 | Relate elementhood to a cl... |
| iocinioc2 32604 | Intersection between two o... |
| xrdifh 32605 | Class difference of a half... |
| iocinif 32606 | Relate intersection of two... |
| difioo 32607 | The difference between two... |
| difico 32608 | The difference between two... |
| uzssico 32609 | Upper integer sets are a s... |
| fz2ssnn0 32610 | A finite set of sequential... |
| nndiffz1 32611 | Upper set of the positive ... |
| ssnnssfz 32612 | For any finite subset of `... |
| fzne1 32613 | Elementhood in a finite se... |
| fzm1ne1 32614 | Elementhood of an integer ... |
| fzspl 32615 | Split the last element of ... |
| fzdif2 32616 | Split the last element of ... |
| fzodif2 32617 | Split the last element of ... |
| fzodif1 32618 | Set difference of two half... |
| fzsplit3 32619 | Split a finite interval of... |
| bcm1n 32620 | The proportion of one bino... |
| iundisjfi 32621 | Rewrite a countable union ... |
| iundisj2fi 32622 | A disjoint union is disjoi... |
| iundisjcnt 32623 | Rewrite a countable union ... |
| iundisj2cnt 32624 | A countable disjoint union... |
| fzone1 32625 | Elementhood in a half-open... |
| fzom1ne1 32626 | Elementhood in a half-open... |
| f1ocnt 32627 | Given a countable set ` A ... |
| fz1nnct 32628 | NN and integer ranges star... |
| fz1nntr 32629 | NN and integer ranges star... |
| nn0difffzod 32630 | A nonnegative integer that... |
| suppssnn0 32631 | Show that the support of a... |
| hashunif 32632 | The cardinality of a disjo... |
| hashxpe 32633 | The size of the Cartesian ... |
| hashgt1 32634 | Restate "set contains at l... |
| znumd 32635 | Numerator of an integer. ... |
| zdend 32636 | Denominator of an integer.... |
| numdenneg 32637 | Numerator and denominator ... |
| divnumden2 32638 | Calculate the reduced form... |
| nnindf 32639 | Principle of Mathematical ... |
| nn0min 32640 | Extracting the minimum pos... |
| subne0nn 32641 | A nonnegative difference i... |
| ltesubnnd 32642 | Subtracting an integer num... |
| fprodeq02 32643 | If one of the factors is z... |
| pr01ssre 32644 | The range of the indicator... |
| fprodex01 32645 | A product of factors equal... |
| prodpr 32646 | A product over a pair is t... |
| prodtp 32647 | A product over a triple is... |
| fsumub 32648 | An upper bound for a term ... |
| fsumiunle 32649 | Upper bound for a sum of n... |
| dfdec100 32650 | Split the hundreds from a ... |
| dp2eq1 32653 | Equality theorem for the d... |
| dp2eq2 32654 | Equality theorem for the d... |
| dp2eq1i 32655 | Equality theorem for the d... |
| dp2eq2i 32656 | Equality theorem for the d... |
| dp2eq12i 32657 | Equality theorem for the d... |
| dp20u 32658 | Add a zero in the tenths (... |
| dp20h 32659 | Add a zero in the unit pla... |
| dp2cl 32660 | Closure for the decimal fr... |
| dp2clq 32661 | Closure for a decimal frac... |
| rpdp2cl 32662 | Closure for a decimal frac... |
| rpdp2cl2 32663 | Closure for a decimal frac... |
| dp2lt10 32664 | Decimal fraction builds re... |
| dp2lt 32665 | Comparing two decimal frac... |
| dp2ltsuc 32666 | Comparing a decimal fracti... |
| dp2ltc 32667 | Comparing two decimal expa... |
| dpval 32670 | Define the value of the de... |
| dpcl 32671 | Prove that the closure of ... |
| dpfrac1 32672 | Prove a simple equivalence... |
| dpval2 32673 | Value of the decimal point... |
| dpval3 32674 | Value of the decimal point... |
| dpmul10 32675 | Multiply by 10 a decimal e... |
| decdiv10 32676 | Divide a decimal number by... |
| dpmul100 32677 | Multiply by 100 a decimal ... |
| dp3mul10 32678 | Multiply by 10 a decimal e... |
| dpmul1000 32679 | Multiply by 1000 a decimal... |
| dpval3rp 32680 | Value of the decimal point... |
| dp0u 32681 | Add a zero in the tenths p... |
| dp0h 32682 | Remove a zero in the units... |
| rpdpcl 32683 | Closure of the decimal poi... |
| dplt 32684 | Comparing two decimal expa... |
| dplti 32685 | Comparing a decimal expans... |
| dpgti 32686 | Comparing a decimal expans... |
| dpltc 32687 | Comparing two decimal inte... |
| dpexpp1 32688 | Add one zero to the mantis... |
| 0dp2dp 32689 | Multiply by 10 a decimal e... |
| dpadd2 32690 | Addition with one decimal,... |
| dpadd 32691 | Addition with one decimal.... |
| dpadd3 32692 | Addition with two decimals... |
| dpmul 32693 | Multiplication with one de... |
| dpmul4 32694 | An upper bound to multipli... |
| threehalves 32695 | Example theorem demonstrat... |
| 1mhdrd 32696 | Example theorem demonstrat... |
| xdivval 32699 | Value of division: the (un... |
| xrecex 32700 | Existence of reciprocal of... |
| xmulcand 32701 | Cancellation law for exten... |
| xreceu 32702 | Existential uniqueness of ... |
| xdivcld 32703 | Closure law for the extend... |
| xdivcl 32704 | Closure law for the extend... |
| xdivmul 32705 | Relationship between divis... |
| rexdiv 32706 | The extended real division... |
| xdivrec 32707 | Relationship between divis... |
| xdivid 32708 | A number divided by itself... |
| xdiv0 32709 | Division into zero is zero... |
| xdiv0rp 32710 | Division into zero is zero... |
| eliccioo 32711 | Membership in a closed int... |
| elxrge02 32712 | Elementhood in the set of ... |
| xdivpnfrp 32713 | Plus infinity divided by a... |
| rpxdivcld 32714 | Closure law for extended d... |
| xrpxdivcld 32715 | Closure law for extended d... |
| wrdfd 32716 | A word is a zero-based seq... |
| wrdres 32717 | Condition for the restrict... |
| wrdsplex 32718 | Existence of a split of a ... |
| pfx1s2 32719 | The prefix of length 1 of ... |
| pfxrn2 32720 | The range of a prefix of a... |
| pfxrn3 32721 | Express the range of a pre... |
| pfxf1 32722 | Condition for a prefix to ... |
| s1f1 32723 | Conditions for a length 1 ... |
| s2rn 32724 | Range of a length 2 string... |
| s2f1 32725 | Conditions for a length 2 ... |
| s3rn 32726 | Range of a length 3 string... |
| s3f1 32727 | Conditions for a length 3 ... |
| s3clhash 32728 | Closure of the words of le... |
| ccatf1 32729 | Conditions for a concatena... |
| pfxlsw2ccat 32730 | Reconstruct a word from it... |
| wrdt2ind 32731 | Perform an induction over ... |
| swrdrn2 32732 | The range of a subword is ... |
| swrdrn3 32733 | Express the range of a sub... |
| swrdf1 32734 | Condition for a subword to... |
| swrdrndisj 32735 | Condition for the range of... |
| splfv3 32736 | Symbols to the right of a ... |
| 1cshid 32737 | Cyclically shifting a sing... |
| cshw1s2 32738 | Cyclically shifting a leng... |
| cshwrnid 32739 | Cyclically shifting a word... |
| cshf1o 32740 | Condition for the cyclic s... |
| ressplusf 32741 | The group operation functi... |
| ressnm 32742 | The norm in a restricted s... |
| abvpropd2 32743 | Weaker version of ~ abvpro... |
| oppgle 32744 | less-than relation of an o... |
| oppgleOLD 32745 | Obsolete version of ~ oppg... |
| oppglt 32746 | less-than relation of an o... |
| ressprs 32747 | The restriction of a prose... |
| oduprs 32748 | Being a proset is a self-d... |
| posrasymb 32749 | A poset ordering is asymet... |
| resspos 32750 | The restriction of a Poset... |
| resstos 32751 | The restriction of a Toset... |
| odutos 32752 | Being a toset is a self-du... |
| tlt2 32753 | In a Toset, two elements m... |
| tlt3 32754 | In a Toset, two elements m... |
| trleile 32755 | In a Toset, two elements m... |
| toslublem 32756 | Lemma for ~ toslub and ~ x... |
| toslub 32757 | In a toset, the lowest upp... |
| tosglblem 32758 | Lemma for ~ tosglb and ~ x... |
| tosglb 32759 | Same theorem as ~ toslub ,... |
| clatp0cl 32760 | The poset zero of a comple... |
| clatp1cl 32761 | The poset one of a complet... |
| mntoval 32766 | Operation value of the mon... |
| ismnt 32767 | Express the statement " ` ... |
| ismntd 32768 | Property of being a monoto... |
| mntf 32769 | A monotone function is a f... |
| mgcoval 32770 | Operation value of the mon... |
| mgcval 32771 | Monotone Galois connection... |
| mgcf1 32772 | The lower adjoint ` F ` of... |
| mgcf2 32773 | The upper adjoint ` G ` of... |
| mgccole1 32774 | An inequality for the kern... |
| mgccole2 32775 | Inequality for the closure... |
| mgcmnt1 32776 | The lower adjoint ` F ` of... |
| mgcmnt2 32777 | The upper adjoint ` G ` of... |
| mgcmntco 32778 | A Galois connection like s... |
| dfmgc2lem 32779 | Lemma for dfmgc2, backward... |
| dfmgc2 32780 | Alternate definition of th... |
| mgcmnt1d 32781 | Galois connection implies ... |
| mgcmnt2d 32782 | Galois connection implies ... |
| mgccnv 32783 | The inverse Galois connect... |
| pwrssmgc 32784 | Given a function ` F ` , e... |
| mgcf1olem1 32785 | Property of a Galois conne... |
| mgcf1olem2 32786 | Property of a Galois conne... |
| mgcf1o 32787 | Given a Galois connection,... |
| xrs0 32790 | The zero of the extended r... |
| xrslt 32791 | The "strictly less than" r... |
| xrsinvgval 32792 | The inversion operation in... |
| xrsmulgzz 32793 | The "multiple" function in... |
| xrstos 32794 | The extended real numbers ... |
| xrsclat 32795 | The extended real numbers ... |
| xrsp0 32796 | The poset 0 of the extende... |
| xrsp1 32797 | The poset 1 of the extende... |
| xrge0base 32798 | The base of the extended n... |
| xrge00 32799 | The zero of the extended n... |
| xrge0plusg 32800 | The additive law of the ex... |
| xrge0le 32801 | The "less than or equal to... |
| xrge0mulgnn0 32802 | The group multiple functio... |
| xrge0addass 32803 | Associativity of extended ... |
| xrge0addgt0 32804 | The sum of nonnegative and... |
| xrge0adddir 32805 | Right-distributivity of ex... |
| xrge0adddi 32806 | Left-distributivity of ext... |
| xrge0npcan 32807 | Extended nonnegative real ... |
| fsumrp0cl 32808 | Closure of a finite sum of... |
| cmn4d 32809 | Commutative/associative la... |
| cmn246135 32810 | Rearrange terms in a commu... |
| cmn145236 32811 | Rearrange terms in a commu... |
| submcld 32812 | Submonoids are closed unde... |
| abliso 32813 | The image of an Abelian gr... |
| lmhmghmd 32814 | A module homomorphism is a... |
| mhmimasplusg 32815 | Value of the operation of ... |
| lmhmimasvsca 32816 | Value of the scalar produc... |
| gsumsubg 32817 | The group sum in a subgrou... |
| gsumsra 32818 | The group sum in a subring... |
| gsummpt2co 32819 | Split a finite sum into a ... |
| gsummpt2d 32820 | Express a finite sum over ... |
| lmodvslmhm 32821 | Scalar multiplication in a... |
| gsumvsmul1 32822 | Pull a scalar multiplicati... |
| gsummptres 32823 | Extend a finite group sum ... |
| gsummptres2 32824 | Extend a finite group sum ... |
| gsumzresunsn 32825 | Append an element to a fin... |
| gsumpart 32826 | Express a group sum as a d... |
| gsumhashmul 32827 | Express a group sum by gro... |
| xrge0tsmsd 32828 | Any finite or infinite sum... |
| xrge0tsmsbi 32829 | Any limit of a finite or i... |
| xrge0tsmseq 32830 | Any limit of a finite or i... |
| cntzun 32831 | The centralizer of a union... |
| cntzsnid 32832 | The centralizer of the ide... |
| cntrcrng 32833 | The center of a ring is a ... |
| isomnd 32838 | A (left) ordered monoid is... |
| isogrp 32839 | A (left-)ordered group is ... |
| ogrpgrp 32840 | A left-ordered group is a ... |
| omndmnd 32841 | A left-ordered monoid is a... |
| omndtos 32842 | A left-ordered monoid is a... |
| omndadd 32843 | In an ordered monoid, the ... |
| omndaddr 32844 | In a right ordered monoid,... |
| omndadd2d 32845 | In a commutative left orde... |
| omndadd2rd 32846 | In a left- and right- orde... |
| submomnd 32847 | A submonoid of an ordered ... |
| xrge0omnd 32848 | The nonnegative extended r... |
| omndmul2 32849 | In an ordered monoid, the ... |
| omndmul3 32850 | In an ordered monoid, the ... |
| omndmul 32851 | In a commutative ordered m... |
| ogrpinv0le 32852 | In an ordered group, the o... |
| ogrpsub 32853 | In an ordered group, the o... |
| ogrpaddlt 32854 | In an ordered group, stric... |
| ogrpaddltbi 32855 | In a right ordered group, ... |
| ogrpaddltrd 32856 | In a right ordered group, ... |
| ogrpaddltrbid 32857 | In a right ordered group, ... |
| ogrpsublt 32858 | In an ordered group, stric... |
| ogrpinv0lt 32859 | In an ordered group, the o... |
| ogrpinvlt 32860 | In an ordered group, the o... |
| gsumle 32861 | A finite sum in an ordered... |
| symgfcoeu 32862 | Uniqueness property of per... |
| symgcom 32863 | Two permutations ` X ` and... |
| symgcom2 32864 | Two permutations ` X ` and... |
| symgcntz 32865 | All elements of a (finite)... |
| odpmco 32866 | The composition of two odd... |
| symgsubg 32867 | The value of the group sub... |
| pmtrprfv2 32868 | In a transposition of two ... |
| pmtrcnel 32869 | Composing a permutation ` ... |
| pmtrcnel2 32870 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 32871 | Composing a permutation ` ... |
| pmtridf1o 32872 | Transpositions of ` X ` an... |
| pmtridfv1 32873 | Value at X of the transpos... |
| pmtridfv2 32874 | Value at Y of the transpos... |
| psgnid 32875 | Permutation sign of the id... |
| psgndmfi 32876 | For a finite base set, the... |
| pmtrto1cl 32877 | Useful lemma for the follo... |
| psgnfzto1stlem 32878 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 32879 | Value of our permutation `... |
| fzto1st1 32880 | Special case where the per... |
| fzto1st 32881 | The function moving one el... |
| fzto1stinvn 32882 | Value of the inverse of ou... |
| psgnfzto1st 32883 | The permutation sign for m... |
| tocycval 32886 | Value of the cycle builder... |
| tocycfv 32887 | Function value of a permut... |
| tocycfvres1 32888 | A cyclic permutation is a ... |
| tocycfvres2 32889 | A cyclic permutation is th... |
| cycpmfvlem 32890 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 32891 | Value of a cycle function ... |
| cycpmfv2 32892 | Value of a cycle function ... |
| cycpmfv3 32893 | Values outside of the orbi... |
| cycpmcl 32894 | Cyclic permutations are pe... |
| tocycf 32895 | The permutation cycle buil... |
| tocyc01 32896 | Permutation cycles built f... |
| cycpm2tr 32897 | A cyclic permutation of 2 ... |
| cycpm2cl 32898 | Closure for the 2-cycles. ... |
| cyc2fv1 32899 | Function value of a 2-cycl... |
| cyc2fv2 32900 | Function value of a 2-cycl... |
| trsp2cyc 32901 | Exhibit the word a transpo... |
| cycpmco2f1 32902 | The word U used in ~ cycpm... |
| cycpmco2rn 32903 | The orbit of the compositi... |
| cycpmco2lem1 32904 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 32905 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 32906 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 32907 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 32908 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 32909 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 32910 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 32911 | The composition of a cycli... |
| cyc2fvx 32912 | Function value of a 2-cycl... |
| cycpm3cl 32913 | Closure of the 3-cycles in... |
| cycpm3cl2 32914 | Closure of the 3-cycles in... |
| cyc3fv1 32915 | Function value of a 3-cycl... |
| cyc3fv2 32916 | Function value of a 3-cycl... |
| cyc3fv3 32917 | Function value of a 3-cycl... |
| cyc3co2 32918 | Represent a 3-cycle as a c... |
| cycpmconjvlem 32919 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 32920 | A formula for computing co... |
| cycpmrn 32921 | The range of the word used... |
| tocyccntz 32922 | All elements of a (finite)... |
| evpmval 32923 | Value of the set of even p... |
| cnmsgn0g 32924 | The neutral element of the... |
| evpmsubg 32925 | The alternating group is a... |
| evpmid 32926 | The identity is an even pe... |
| altgnsg 32927 | The alternating group ` ( ... |
| cyc3evpm 32928 | 3-Cycles are even permutat... |
| cyc3genpmlem 32929 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 32930 | The alternating group ` A ... |
| cycpmgcl 32931 | Cyclic permutations are pe... |
| cycpmconjslem1 32932 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 32933 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 32934 | All cycles of the same len... |
| cyc3conja 32935 | All 3-cycles are conjugate... |
| sgnsv 32938 | The sign mapping. (Contri... |
| sgnsval 32939 | The sign value. (Contribu... |
| sgnsf 32940 | The sign function. (Contr... |
| inftmrel 32945 | The infinitesimal relation... |
| isinftm 32946 | Express ` x ` is infinites... |
| isarchi 32947 | Express the predicate " ` ... |
| pnfinf 32948 | Plus infinity is an infini... |
| xrnarchi 32949 | The completed real line is... |
| isarchi2 32950 | Alternative way to express... |
| submarchi 32951 | A submonoid is archimedean... |
| isarchi3 32952 | This is the usual definiti... |
| archirng 32953 | Property of Archimedean or... |
| archirngz 32954 | Property of Archimedean le... |
| archiexdiv 32955 | In an Archimedean group, g... |
| archiabllem1a 32956 | Lemma for ~ archiabl : In... |
| archiabllem1b 32957 | Lemma for ~ archiabl . (C... |
| archiabllem1 32958 | Archimedean ordered groups... |
| archiabllem2a 32959 | Lemma for ~ archiabl , whi... |
| archiabllem2c 32960 | Lemma for ~ archiabl . (C... |
| archiabllem2b 32961 | Lemma for ~ archiabl . (C... |
| archiabllem2 32962 | Archimedean ordered groups... |
| archiabl 32963 | Archimedean left- and righ... |
| isslmd 32966 | The predicate "is a semimo... |
| slmdlema 32967 | Lemma for properties of a ... |
| lmodslmd 32968 | Left semimodules generaliz... |
| slmdcmn 32969 | A semimodule is a commutat... |
| slmdmnd 32970 | A semimodule is a monoid. ... |
| slmdsrg 32971 | The scalar component of a ... |
| slmdbn0 32972 | The base set of a semimodu... |
| slmdacl 32973 | Closure of ring addition f... |
| slmdmcl 32974 | Closure of ring multiplica... |
| slmdsn0 32975 | The set of scalars in a se... |
| slmdvacl 32976 | Closure of vector addition... |
| slmdass 32977 | Semiring left module vecto... |
| slmdvscl 32978 | Closure of scalar product ... |
| slmdvsdi 32979 | Distributive law for scala... |
| slmdvsdir 32980 | Distributive law for scala... |
| slmdvsass 32981 | Associative law for scalar... |
| slmd0cl 32982 | The ring zero in a semimod... |
| slmd1cl 32983 | The ring unity in a semiri... |
| slmdvs1 32984 | Scalar product with ring u... |
| slmd0vcl 32985 | The zero vector is a vecto... |
| slmd0vlid 32986 | Left identity law for the ... |
| slmd0vrid 32987 | Right identity law for the... |
| slmd0vs 32988 | Zero times a vector is the... |
| slmdvs0 32989 | Anything times the zero ve... |
| gsumvsca1 32990 | Scalar product of a finite... |
| gsumvsca2 32991 | Scalar product of a finite... |
| prmsimpcyc 32992 | A group of prime order is ... |
| cringmul32d 32993 | Commutative/associative la... |
| ringdid 32994 | Distributive law for the m... |
| ringdird 32995 | Distributive law for the m... |
| urpropd 32996 | Sufficient condition for r... |
| frobrhm 32997 | In a commutative ring with... |
| ress1r 32998 | ` 1r ` is unaffected by re... |
| ringinvval 32999 | The ring inverse expressed... |
| dvrcan5 33000 | Cancellation law for commo... |
| subrgchr 33001 | If ` A ` is a subring of `... |
| rmfsupp2 33002 | A mapping of a multiplicat... |
| unitnz 33003 | In a nonzero ring, a unit ... |
| irrednzr 33004 | A ring with an irreducible... |
| 0ringsubrg 33005 | A subring of a zero ring i... |
| 0ringcring 33006 | The zero ring is commutati... |
| reldmrloc 33011 | Ring localization is a pro... |
| erlval 33012 | Value of the ring localiza... |
| rlocval 33013 | Expand the value of the ri... |
| erlcl1 33014 | Closure for the ring local... |
| erlcl2 33015 | Closure for the ring local... |
| erldi 33016 | Main property of the ring ... |
| erlbrd 33017 | Deduce the ring localizati... |
| erlbr2d 33018 | Deduce the ring localizati... |
| erler 33019 | The relation used to build... |
| elrlocbasi 33020 | Membership in the basis of... |
| rlocbas 33021 | The base set of a ring loc... |
| rlocaddval 33022 | Value of the addition in t... |
| rlocmulval 33023 | Value of the addition in t... |
| rloccring 33024 | The ring localization ` L ... |
| rloc0g 33025 | The zero of a ring localiz... |
| rloc1r 33026 | The multiplicative identit... |
| rlocf1 33027 | The embedding ` F ` of a r... |
| domnlcan 33028 | Left-cancellation law for ... |
| idomrcan 33029 | Right-cancellation law for... |
| 1rrg 33030 | The multiplicative identit... |
| rrgnz 33031 | In a non-zero ring, the ze... |
| isdomn6 33032 | A ring is a domain iff non... |
| rrgsubm 33033 | The left regular elements ... |
| subrdom 33034 | A subring of a domain is a... |
| subridom 33035 | A subring of an integral d... |
| subrfld 33036 | A subring of a field is an... |
| eufndx 33039 | Index value of the Euclide... |
| eufid 33040 | Utility theorem: index-ind... |
| ringinveu 33043 | If a ring unit element ` X... |
| isdrng4 33044 | A division ring is a ring ... |
| rndrhmcl 33045 | The image of a division ri... |
| sdrgdvcl 33046 | A sub-division-ring is clo... |
| sdrginvcl 33047 | A sub-division-ring is clo... |
| primefldchr 33048 | The characteristic of a pr... |
| fracval 33051 | Value of the field of frac... |
| fracbas 33052 | The base of the field of f... |
| fracerl 33053 | Rewrite the ring localizat... |
| fracf1 33054 | The embedding of a commuta... |
| fracfld 33055 | The field of fractions of ... |
| idomsubr 33056 | Every integral domain is i... |
| fldgenval 33059 | Value of the field generat... |
| fldgenssid 33060 | The field generated by a s... |
| fldgensdrg 33061 | A generated subfield is a ... |
| fldgenssv 33062 | A generated subfield is a ... |
| fldgenss 33063 | Generated subfields preser... |
| fldgenidfld 33064 | The subfield generated by ... |
| fldgenssp 33065 | The field generated by a s... |
| fldgenid 33066 | The subfield of a field ` ... |
| fldgenfld 33067 | A generated subfield is a ... |
| primefldgen1 33068 | The prime field of a divis... |
| 1fldgenq 33069 | The field of rational numb... |
| isorng 33074 | An ordered ring is a ring ... |
| orngring 33075 | An ordered ring is a ring.... |
| orngogrp 33076 | An ordered ring is an orde... |
| isofld 33077 | An ordered field is a fiel... |
| orngmul 33078 | In an ordered ring, the or... |
| orngsqr 33079 | In an ordered ring, all sq... |
| ornglmulle 33080 | In an ordered ring, multip... |
| orngrmulle 33081 | In an ordered ring, multip... |
| ornglmullt 33082 | In an ordered ring, multip... |
| orngrmullt 33083 | In an ordered ring, multip... |
| orngmullt 33084 | In an ordered ring, the st... |
| ofldfld 33085 | An ordered field is a fiel... |
| ofldtos 33086 | An ordered field is a tota... |
| orng0le1 33087 | In an ordered ring, the ri... |
| ofldlt1 33088 | In an ordered field, the r... |
| ofldchr 33089 | The characteristic of an o... |
| suborng 33090 | Every subring of an ordere... |
| subofld 33091 | Every subfield of an order... |
| isarchiofld 33092 | Axiom of Archimedes : a ch... |
| rhmdvd 33093 | A ring homomorphism preser... |
| kerunit 33094 | If a unit element lies in ... |
| reldmresv 33097 | The scalar restriction is ... |
| resvval 33098 | Value of structure restric... |
| resvid2 33099 | General behavior of trivia... |
| resvval2 33100 | Value of nontrivial struct... |
| resvsca 33101 | Base set of a structure re... |
| resvlem 33102 | Other elements of a scalar... |
| resvlemOLD 33103 | Obsolete version of ~ resv... |
| resvbas 33104 | ` Base ` is unaffected by ... |
| resvbasOLD 33105 | Obsolete proof of ~ resvba... |
| resvplusg 33106 | ` +g ` is unaffected by sc... |
| resvplusgOLD 33107 | Obsolete proof of ~ resvpl... |
| resvvsca 33108 | ` .s ` is unaffected by sc... |
| resvvscaOLD 33109 | Obsolete proof of ~ resvvs... |
| resvmulr 33110 | ` .r ` is unaffected by sc... |
| resvmulrOLD 33111 | Obsolete proof of ~ resvmu... |
| resv0g 33112 | ` 0g ` is unaffected by sc... |
| resv1r 33113 | ` 1r ` is unaffected by sc... |
| resvcmn 33114 | Scalar restriction preserv... |
| gzcrng 33115 | The gaussian integers form... |
| reofld 33116 | The real numbers form an o... |
| nn0omnd 33117 | The nonnegative integers f... |
| rearchi 33118 | The field of the real numb... |
| nn0archi 33119 | The monoid of the nonnegat... |
| xrge0slmod 33120 | The extended nonnegative r... |
| qusker 33121 | The kernel of a quotient m... |
| eqgvscpbl 33122 | The left coset equivalence... |
| qusvscpbl 33123 | The quotient map distribut... |
| qusvsval 33124 | Value of the scalar multip... |
| imaslmod 33125 | The image structure of a l... |
| imasmhm 33126 | Given a function ` F ` wit... |
| imasghm 33127 | Given a function ` F ` wit... |
| imasrhm 33128 | Given a function ` F ` wit... |
| imaslmhm 33129 | Given a function ` F ` wit... |
| quslmod 33130 | If ` G ` is a submodule in... |
| quslmhm 33131 | If ` G ` is a submodule of... |
| quslvec 33132 | If ` S ` is a vector subsp... |
| ecxpid 33133 | The equivalence class of a... |
| qsxpid 33134 | The quotient set of a cart... |
| qusxpid 33135 | The Group quotient equival... |
| qustriv 33136 | The quotient of a group ` ... |
| qustrivr 33137 | Converse of ~ qustriv . (... |
| znfermltl 33138 | Fermat's little theorem in... |
| islinds5 33139 | A set is linearly independ... |
| ellspds 33140 | Variation on ~ ellspd . (... |
| 0ellsp 33141 | Zero is in all spans. (Co... |
| 0nellinds 33142 | The group identity cannot ... |
| rspsnel 33143 | Membership in a principal ... |
| rspsnid 33144 | A principal ideal contains... |
| elrsp 33145 | Write the elements of a ri... |
| ellpi 33146 | Elementhood in a left prin... |
| rspidlid 33147 | The ideal span of an ideal... |
| pidlnz 33148 | A principal ideal generate... |
| dvdsruassoi 33149 | If two elements ` X ` and ... |
| dvdsruasso 33150 | Two elements ` X ` and ` Y... |
| dvdsruasso2 33151 | A reformulation of ~ dvdsr... |
| dvdsrspss 33152 | In a ring, an element ` X ... |
| rspsnasso 33153 | Two elements ` X ` and ` Y... |
| lbslsp 33154 | Any element of a left modu... |
| lindssn 33155 | Any singleton of a nonzero... |
| lindflbs 33156 | Conditions for an independ... |
| islbs5 33157 | An equivalent formulation ... |
| linds2eq 33158 | Deduce equality of element... |
| lindfpropd 33159 | Property deduction for lin... |
| lindspropd 33160 | Property deduction for lin... |
| elgrplsmsn 33161 | Membership in a sumset wit... |
| lsmsnorb 33162 | The sumset of a group with... |
| lsmsnorb2 33163 | The sumset of a single ele... |
| elringlsm 33164 | Membership in a product of... |
| elringlsmd 33165 | Membership in a product of... |
| ringlsmss 33166 | Closure of the product of ... |
| ringlsmss1 33167 | The product of an ideal ` ... |
| ringlsmss2 33168 | The product with an ideal ... |
| lsmsnpridl 33169 | The product of the ring wi... |
| lsmsnidl 33170 | The product of the ring wi... |
| lsmidllsp 33171 | The sum of two ideals is t... |
| lsmidl 33172 | The sum of two ideals is a... |
| lsmssass 33173 | Group sum is associative, ... |
| grplsm0l 33174 | Sumset with the identity s... |
| grplsmid 33175 | The direct sum of an eleme... |
| qusmul 33176 | Value of the ring operatio... |
| quslsm 33177 | Express the image by the q... |
| qusbas2 33178 | Alternate definition of th... |
| qus0g 33179 | The identity element of a ... |
| qusima 33180 | The image of a subgroup by... |
| qusrn 33181 | The natural map from eleme... |
| nsgqus0 33182 | A normal subgroup ` N ` is... |
| nsgmgclem 33183 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 33184 | There is a monotone Galois... |
| nsgqusf1olem1 33185 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 33186 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 33187 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 33188 | The canonical projection h... |
| lmhmqusker 33189 | A surjective module homomo... |
| lmicqusker 33190 | The image ` H ` of a modul... |
| ghmqusnsglem1 33191 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 33192 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 33193 | The mapping ` H ` induced ... |
| intlidl 33194 | The intersection of a none... |
| rhmpreimaidl 33195 | The preimage of an ideal b... |
| kerlidl 33196 | The kernel of a ring homom... |
| lidlnsg 33197 | An ideal is a normal subgr... |
| 0ringidl 33198 | The zero ideal is the only... |
| pidlnzb 33199 | A principal ideal is nonze... |
| lidlunitel 33200 | If an ideal ` I ` contains... |
| unitpidl1 33201 | The ideal ` I ` generated ... |
| rhmquskerlem 33202 | The mapping ` J ` induced ... |
| rhmqusker 33203 | A surjective ring homomorp... |
| ricqusker 33204 | The image ` H ` of a ring ... |
| rhmqusnsg 33205 | The mapping ` J ` induced ... |
| elrspunidl 33206 | Elementhood in the span of... |
| elrspunsn 33207 | Membership to the span of ... |
| lidlincl 33208 | Ideals are closed under in... |
| idlinsubrg 33209 | The intersection between a... |
| rhmimaidl 33210 | The image of an ideal ` I ... |
| drngidl 33211 | A nonzero ring is a divisi... |
| drngidlhash 33212 | A ring is a division ring ... |
| prmidlval 33215 | The class of prime ideals ... |
| isprmidl 33216 | The predicate "is a prime ... |
| prmidlnr 33217 | A prime ideal is a proper ... |
| prmidl 33218 | The main property of a pri... |
| prmidl2 33219 | A condition that shows an ... |
| idlmulssprm 33220 | Let ` P ` be a prime ideal... |
| pridln1 33221 | A proper ideal cannot cont... |
| prmidlidl 33222 | A prime ideal is an ideal.... |
| prmidlssidl 33223 | Prime ideals as a subset o... |
| cringm4 33224 | Commutative/associative la... |
| isprmidlc 33225 | The predicate "is prime id... |
| prmidlc 33226 | Property of a prime ideal ... |
| 0ringprmidl 33227 | The trivial ring does not ... |
| prmidl0 33228 | The zero ideal of a commut... |
| rhmpreimaprmidl 33229 | The preimage of a prime id... |
| qsidomlem1 33230 | If the quotient ring of a ... |
| qsidomlem2 33231 | A quotient by a prime idea... |
| qsidom 33232 | An ideal ` I ` in the comm... |
| qsnzr 33233 | A quotient of a non-zero r... |
| mxidlval 33236 | The set of maximal ideals ... |
| ismxidl 33237 | The predicate "is a maxima... |
| mxidlidl 33238 | A maximal ideal is an idea... |
| mxidlnr 33239 | A maximal ideal is proper.... |
| mxidlmax 33240 | A maximal ideal is a maxim... |
| mxidln1 33241 | One is not contained in an... |
| mxidlnzr 33242 | A ring with a maximal idea... |
| mxidlmaxv 33243 | An ideal ` I ` strictly co... |
| crngmxidl 33244 | In a commutative ring, max... |
| mxidlprm 33245 | Every maximal ideal is pri... |
| mxidlirredi 33246 | In an integral domain, the... |
| mxidlirred 33247 | In a principal ideal domai... |
| ssmxidllem 33248 | The set ` P ` used in the ... |
| ssmxidl 33249 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 33250 | In a nonzero ring, the zer... |
| drng0mxidl 33251 | In a division ring, the ze... |
| drngmxidl 33252 | The zero ideal is the only... |
| krull 33253 | Krull's theorem: Any nonz... |
| mxidlnzrb 33254 | A ring is nonzero if and o... |
| opprabs 33255 | The opposite ring of the o... |
| oppreqg 33256 | Group coset equivalence re... |
| opprnsg 33257 | Normal subgroups of the op... |
| opprlidlabs 33258 | The ideals of the opposite... |
| oppr2idl 33259 | Two sided ideal of the opp... |
| opprmxidlabs 33260 | The maximal ideal of the o... |
| opprqusbas 33261 | The base of the quotient o... |
| opprqusplusg 33262 | The group operation of the... |
| opprqus0g 33263 | The group identity element... |
| opprqusmulr 33264 | The multiplication operati... |
| opprqus1r 33265 | The ring unity of the quot... |
| opprqusdrng 33266 | The quotient of the opposi... |
| qsdrngilem 33267 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 33268 | A quotient by a maximal le... |
| qsdrnglem2 33269 | Lemma for ~ qsdrng . (Con... |
| qsdrng 33270 | An ideal ` M ` is both lef... |
| qsfld 33271 | An ideal ` M ` in the comm... |
| mxidlprmALT 33272 | Every maximal ideal is pri... |
| idlsrgstr 33275 | A constructed semiring of ... |
| idlsrgval 33276 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 33277 | Base of the ideals of a ri... |
| idlsrgplusg 33278 | Additive operation of the ... |
| idlsrg0g 33279 | The zero ideal is the addi... |
| idlsrgmulr 33280 | Multiplicative operation o... |
| idlsrgtset 33281 | Topology component of the ... |
| idlsrgmulrval 33282 | Value of the ring multipli... |
| idlsrgmulrcl 33283 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 33284 | In a commutative ring, the... |
| idlsrgmulrss2 33285 | The product of two ideals ... |
| idlsrgmulrssin 33286 | In a commutative ring, the... |
| idlsrgmnd 33287 | The ideals of a ring form ... |
| idlsrgcmnd 33288 | The ideals of a ring form ... |
| rprmval 33289 | The prime elements of a ri... |
| isrprm 33290 | Property for ` P ` to be a... |
| rprmcl 33291 | A ring prime is an element... |
| rprmdvds 33292 | If a ring prime ` Q ` divi... |
| rprmnz 33293 | A ring prime is nonzero. ... |
| rprmnunit 33294 | A ring prime is not a unit... |
| rsprprmprmidl 33295 | In a commutative ring, ide... |
| rsprprmprmidlb 33296 | In an integral domain, an ... |
| rprmndvdsr1 33297 | A ring prime element does ... |
| rprmasso 33298 | In an integral domain, the... |
| rprmasso2 33299 | In an integral domain, if ... |
| rprmirredlem 33300 | Lemma for ~ rprmirred . (... |
| rprmirred 33301 | In an integral domain, rin... |
| rprmirredb 33302 | In a principal ideal domai... |
| rprmdvdspow 33303 | If a prime element divides... |
| rprmdvdsprod 33304 | If a prime element ` Q ` d... |
| isufd 33307 | The property of being a Un... |
| isufd2 33308 | Alternate definition of un... |
| ufdcringd 33309 | A unique factorization dom... |
| 0ringufd 33310 | A zero ring is a unique fa... |
| zringidom 33311 | The ring of integers is an... |
| zringpid 33312 | The ring of integers is a ... |
| dfprm3 33313 | The (positive) prime eleme... |
| zringfrac 33314 | The field of fractions of ... |
| 0ringmon1p 33315 | There are no monic polynom... |
| fply1 33316 | Conditions for a function ... |
| ply1lvec 33317 | In a division ring, the un... |
| evls1fn 33318 | Functionality of the subri... |
| evls1dm 33319 | The domain of the subring ... |
| evls1fvf 33320 | The subring evaluation fun... |
| ressdeg1 33321 | The degree of a univariate... |
| ressply10g 33322 | A restricted polynomial al... |
| ressply1mon1p 33323 | The monic polynomials of a... |
| ressply1invg 33324 | An element of a restricted... |
| ressply1sub 33325 | A restricted polynomial al... |
| evls1subd 33326 | Univariate polynomial eval... |
| ply1ascl1 33327 | The multiplicative identit... |
| deg1le0eq0 33328 | A polynomial with nonposit... |
| ply1asclunit 33329 | A non-zero scalar polynomi... |
| ply1unit 33330 | In a field ` F ` , a polyn... |
| m1pmeq 33331 | If two monic polynomials `... |
| ply1fermltl 33332 | Fermat's little theorem fo... |
| coe1mon 33333 | Coefficient vector of a mo... |
| ply1moneq 33334 | Two monomials are equal if... |
| ply1degltel 33335 | Characterize elementhood i... |
| ply1degleel 33336 | Characterize elementhood i... |
| ply1degltlss 33337 | The space ` S ` of the uni... |
| gsummoncoe1fzo 33338 | A coefficient of the polyn... |
| ply1gsumz 33339 | If a polynomial given as a... |
| deg1addlt 33340 | If both factors have degre... |
| ig1pnunit 33341 | The polynomial ideal gener... |
| ig1pmindeg 33342 | The polynomial ideal gener... |
| q1pdir 33343 | Distribution of univariate... |
| q1pvsca 33344 | Scalar multiplication prop... |
| r1pvsca 33345 | Scalar multiplication prop... |
| r1p0 33346 | Polynomial remainder opera... |
| r1pcyc 33347 | The polynomial remainder o... |
| r1padd1 33348 | Addition property of the p... |
| r1pid2 33349 | Identity law for polynomia... |
| r1plmhm 33350 | The univariate polynomial ... |
| r1pquslmic 33351 | The univariate polynomial ... |
| sra1r 33352 | The unity element of a sub... |
| sradrng 33353 | Condition for a subring al... |
| srasubrg 33354 | A subring of the original ... |
| sralvec 33355 | Given a sub division ring ... |
| srafldlvec 33356 | Given a subfield ` F ` of ... |
| resssra 33357 | The subring algebra of a r... |
| lsssra 33358 | A subring is a subspace of... |
| drgext0g 33359 | The additive neutral eleme... |
| drgextvsca 33360 | The scalar multiplication ... |
| drgext0gsca 33361 | The additive neutral eleme... |
| drgextsubrg 33362 | The scalar field is a subr... |
| drgextlsp 33363 | The scalar field is a subs... |
| drgextgsum 33364 | Group sum in a division ri... |
| lvecdimfi 33365 | Finite version of ~ lvecdi... |
| dimval 33368 | The dimension of a vector ... |
| dimvalfi 33369 | The dimension of a vector ... |
| dimcl 33370 | Closure of the vector spac... |
| lmimdim 33371 | Module isomorphisms preser... |
| lmicdim 33372 | Module isomorphisms preser... |
| lvecdim0i 33373 | A vector space of dimensio... |
| lvecdim0 33374 | A vector space of dimensio... |
| lssdimle 33375 | The dimension of a linear ... |
| dimpropd 33376 | If two structures have the... |
| rlmdim 33377 | The left vector space indu... |
| rgmoddimOLD 33378 | Obsolete version of ~ rlmd... |
| frlmdim 33379 | Dimension of a free left m... |
| tnglvec 33380 | Augmenting a structure wit... |
| tngdim 33381 | Dimension of a left vector... |
| rrxdim 33382 | Dimension of the generaliz... |
| matdim 33383 | Dimension of the space of ... |
| lbslsat 33384 | A nonzero vector ` X ` is ... |
| lsatdim 33385 | A line, spanned by a nonze... |
| drngdimgt0 33386 | The dimension of a vector ... |
| lmhmlvec2 33387 | A homomorphism of left vec... |
| kerlmhm 33388 | The kernel of a vector spa... |
| imlmhm 33389 | The image of a vector spac... |
| ply1degltdimlem 33390 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 33391 | The space ` S ` of the uni... |
| lindsunlem 33392 | Lemma for ~ lindsun . (Co... |
| lindsun 33393 | Condition for the union of... |
| lbsdiflsp0 33394 | The linear spans of two di... |
| dimkerim 33395 | Given a linear map ` F ` b... |
| qusdimsum 33396 | Let ` W ` be a vector spac... |
| fedgmullem1 33397 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33398 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33399 | The multiplicativity formu... |
| relfldext 33408 | The field extension is a r... |
| brfldext 33409 | The field extension relati... |
| ccfldextrr 33410 | The field of the complex n... |
| fldextfld1 33411 | A field extension is only ... |
| fldextfld2 33412 | A field extension is only ... |
| fldextsubrg 33413 | Field extension implies a ... |
| fldextress 33414 | Field extension implies a ... |
| brfinext 33415 | The finite field extension... |
| extdgval 33416 | Value of the field extensi... |
| fldextsralvec 33417 | The subring algebra associ... |
| extdgcl 33418 | Closure of the field exten... |
| extdggt0 33419 | Degrees of field extension... |
| fldexttr 33420 | Field extension is a trans... |
| fldextid 33421 | The field extension relati... |
| extdgid 33422 | A trivial field extension ... |
| extdgmul 33423 | The multiplicativity formu... |
| finexttrb 33424 | The extension ` E ` of ` K... |
| extdg1id 33425 | If the degree of the exten... |
| extdg1b 33426 | The degree of the extensio... |
| fldextchr 33427 | The characteristic of a su... |
| evls1fldgencl 33428 | Closure of the subring pol... |
| ccfldsrarelvec 33429 | The subring algebra of the... |
| ccfldextdgrr 33430 | The degree of the field ex... |
| irngval 33433 | The elements of a field ` ... |
| elirng 33434 | Property for an element ` ... |
| irngss 33435 | All elements of a subring ... |
| irngssv 33436 | An integral element is an ... |
| 0ringirng 33437 | A zero ring ` R ` has no i... |
| irngnzply1lem 33438 | In the case of a field ` E... |
| irngnzply1 33439 | In the case of a field ` E... |
| ply1annidllem 33442 | Write the set ` Q ` of pol... |
| ply1annidl 33443 | The set ` Q ` of polynomia... |
| ply1annnr 33444 | The set ` Q ` of polynomia... |
| ply1annig1p 33445 | The ideal ` Q ` of polynom... |
| minplyval 33446 | Expand the value of the mi... |
| minplycl 33447 | The minimal polynomial is ... |
| ply1annprmidl 33448 | The set ` Q ` of polynomia... |
| minplyann 33449 | The minimal polynomial for... |
| minplyirredlem 33450 | Lemma for ~ minplyirred . ... |
| minplyirred 33451 | A nonzero minimal polynomi... |
| irngnminplynz 33452 | Integral elements have non... |
| minplym1p 33453 | A minimal polynomial is mo... |
| irredminply 33454 | An irreducible, monic, ann... |
| algextdeglem1 33455 | Lemma for ~ algextdeg . (... |
| algextdeglem2 33456 | Lemma for ~ algextdeg . B... |
| algextdeglem3 33457 | Lemma for ~ algextdeg . T... |
| algextdeglem4 33458 | Lemma for ~ algextdeg . B... |
| algextdeglem5 33459 | Lemma for ~ algextdeg . T... |
| algextdeglem6 33460 | Lemma for ~ algextdeg . B... |
| algextdeglem7 33461 | Lemma for ~ algextdeg . T... |
| algextdeglem8 33462 | Lemma for ~ algextdeg . T... |
| algextdeg 33463 | The degree of an algebraic... |
| smatfval 33466 | Value of the submatrix. (... |
| smatrcl 33467 | Closure of the rectangular... |
| smatlem 33468 | Lemma for the next theorem... |
| smattl 33469 | Entries of a submatrix, to... |
| smattr 33470 | Entries of a submatrix, to... |
| smatbl 33471 | Entries of a submatrix, bo... |
| smatbr 33472 | Entries of a submatrix, bo... |
| smatcl 33473 | Closure of the square subm... |
| matmpo 33474 | Write a square matrix as a... |
| 1smat1 33475 | The submatrix of the ident... |
| submat1n 33476 | One case where the submatr... |
| submatres 33477 | Special case where the sub... |
| submateqlem1 33478 | Lemma for ~ submateq . (C... |
| submateqlem2 33479 | Lemma for ~ submateq . (C... |
| submateq 33480 | Sufficient condition for t... |
| submatminr1 33481 | If we take a submatrix by ... |
| lmatval 33484 | Value of the literal matri... |
| lmatfval 33485 | Entries of a literal matri... |
| lmatfvlem 33486 | Useful lemma to extract li... |
| lmatcl 33487 | Closure of the literal mat... |
| lmat22lem 33488 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 33489 | Entry of a 2x2 literal mat... |
| lmat22e12 33490 | Entry of a 2x2 literal mat... |
| lmat22e21 33491 | Entry of a 2x2 literal mat... |
| lmat22e22 33492 | Entry of a 2x2 literal mat... |
| lmat22det 33493 | The determinant of a liter... |
| mdetpmtr1 33494 | The determinant of a matri... |
| mdetpmtr2 33495 | The determinant of a matri... |
| mdetpmtr12 33496 | The determinant of a matri... |
| mdetlap1 33497 | A Laplace expansion of the... |
| madjusmdetlem1 33498 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 33499 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 33500 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 33501 | Lemma for ~ madjusmdet . ... |
| madjusmdet 33502 | Express the cofactor of th... |
| mdetlap 33503 | Laplace expansion of the d... |
| ist0cld 33504 | The predicate "is a T_0 sp... |
| txomap 33505 | Given two open maps ` F ` ... |
| qtopt1 33506 | If every equivalence class... |
| qtophaus 33507 | If an open map's graph in ... |
| circtopn 33508 | The topology of the unit c... |
| circcn 33509 | The function gluing the re... |
| reff 33510 | For any cover refinement, ... |
| locfinreflem 33511 | A locally finite refinemen... |
| locfinref 33512 | A locally finite refinemen... |
| iscref 33515 | The property that every op... |
| crefeq 33516 | Equality theorem for the "... |
| creftop 33517 | A space where every open c... |
| crefi 33518 | The property that every op... |
| crefdf 33519 | A formulation of ~ crefi e... |
| crefss 33520 | The "every open cover has ... |
| cmpcref 33521 | Equivalent definition of c... |
| cmpfiref 33522 | Every open cover of a Comp... |
| ldlfcntref 33525 | Every open cover of a Lind... |
| ispcmp 33528 | The predicate "is a paraco... |
| cmppcmp 33529 | Every compact space is par... |
| dispcmp 33530 | Every discrete space is pa... |
| pcmplfin 33531 | Given a paracompact topolo... |
| pcmplfinf 33532 | Given a paracompact topolo... |
| rspecval 33535 | Value of the spectrum of t... |
| rspecbas 33536 | The prime ideals form the ... |
| rspectset 33537 | Topology component of the ... |
| rspectopn 33538 | The topology component of ... |
| zarcls0 33539 | The closure of the identit... |
| zarcls1 33540 | The unit ideal ` B ` is th... |
| zarclsun 33541 | The union of two closed se... |
| zarclsiin 33542 | In a Zariski topology, the... |
| zarclsint 33543 | The intersection of a fami... |
| zarclssn 33544 | The closed points of Zaris... |
| zarcls 33545 | The open sets of the Zaris... |
| zartopn 33546 | The Zariski topology is a ... |
| zartop 33547 | The Zariski topology is a ... |
| zartopon 33548 | The points of the Zariski ... |
| zar0ring 33549 | The Zariski Topology of th... |
| zart0 33550 | The Zariski topology is T_... |
| zarmxt1 33551 | The Zariski topology restr... |
| zarcmplem 33552 | Lemma for ~ zarcmp . (Con... |
| zarcmp 33553 | The Zariski topology is co... |
| rspectps 33554 | The spectrum of a ring ` R... |
| rhmpreimacnlem 33555 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 33556 | The function mapping a pri... |
| metidval 33561 | Value of the metric identi... |
| metidss 33562 | As a relation, the metric ... |
| metidv 33563 | ` A ` and ` B ` identify b... |
| metideq 33564 | Basic property of the metr... |
| metider 33565 | The metric identification ... |
| pstmval 33566 | Value of the metric induce... |
| pstmfval 33567 | Function value of the metr... |
| pstmxmet 33568 | The metric induced by a ps... |
| hauseqcn 33569 | In a Hausdorff topology, t... |
| elunitge0 33570 | An element of the closed u... |
| unitssxrge0 33571 | The closed unit interval i... |
| unitdivcld 33572 | Necessary conditions for a... |
| iistmd 33573 | The closed unit interval f... |
| unicls 33574 | The union of the closed se... |
| tpr2tp 33575 | The usual topology on ` ( ... |
| tpr2uni 33576 | The usual topology on ` ( ... |
| xpinpreima 33577 | Rewrite the cartesian prod... |
| xpinpreima2 33578 | Rewrite the cartesian prod... |
| sqsscirc1 33579 | The complex square of side... |
| sqsscirc2 33580 | The complex square of side... |
| cnre2csqlem 33581 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 33582 | Image of a centered square... |
| tpr2rico 33583 | For any point of an open s... |
| cnvordtrestixx 33584 | The restriction of the 'gr... |
| prsdm 33585 | Domain of the relation of ... |
| prsrn 33586 | Range of the relation of a... |
| prsss 33587 | Relation of a subproset. ... |
| prsssdm 33588 | Domain of a subproset rela... |
| ordtprsval 33589 | Value of the order topolog... |
| ordtprsuni 33590 | Value of the order topolog... |
| ordtcnvNEW 33591 | The order dual generates t... |
| ordtrestNEW 33592 | The subspace topology of a... |
| ordtrest2NEWlem 33593 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 33594 | An interval-closed set ` A... |
| ordtconnlem1 33595 | Connectedness in the order... |
| ordtconn 33596 | Connectedness in the order... |
| mndpluscn 33597 | A mapping that is both a h... |
| mhmhmeotmd 33598 | Deduce a Topological Monoi... |
| rmulccn 33599 | Multiplication by a real c... |
| raddcn 33600 | Addition in the real numbe... |
| xrmulc1cn 33601 | The operation multiplying ... |
| fmcncfil 33602 | The image of a Cauchy filt... |
| xrge0hmph 33603 | The extended nonnegative r... |
| xrge0iifcnv 33604 | Define a bijection from ` ... |
| xrge0iifcv 33605 | The defined function's val... |
| xrge0iifiso 33606 | The defined bijection from... |
| xrge0iifhmeo 33607 | Expose a homeomorphism fro... |
| xrge0iifhom 33608 | The defined function from ... |
| xrge0iif1 33609 | Condition for the defined ... |
| xrge0iifmhm 33610 | The defined function from ... |
| xrge0pluscn 33611 | The addition operation of ... |
| xrge0mulc1cn 33612 | The operation multiplying ... |
| xrge0tps 33613 | The extended nonnegative r... |
| xrge0topn 33614 | The topology of the extend... |
| xrge0haus 33615 | The topology of the extend... |
| xrge0tmd 33616 | The extended nonnegative r... |
| xrge0tmdALT 33617 | Alternate proof of ~ xrge0... |
| lmlim 33618 | Relate a limit in a given ... |
| lmlimxrge0 33619 | Relate a limit in the nonn... |
| rge0scvg 33620 | Implication of convergence... |
| fsumcvg4 33621 | A serie with finite suppor... |
| pnfneige0 33622 | A neighborhood of ` +oo ` ... |
| lmxrge0 33623 | Express "sequence ` F ` co... |
| lmdvg 33624 | If a monotonic sequence of... |
| lmdvglim 33625 | If a monotonic real number... |
| pl1cn 33626 | A univariate polynomial is... |
| zringnm 33629 | The norm (function) for a ... |
| zzsnm 33630 | The norm of the ring of th... |
| zlm0 33631 | Zero of a ` ZZ ` -module. ... |
| zlm1 33632 | Unity element of a ` ZZ ` ... |
| zlmds 33633 | Distance in a ` ZZ ` -modu... |
| zlmdsOLD 33634 | Obsolete proof of ~ zlmds ... |
| zlmtset 33635 | Topology in a ` ZZ ` -modu... |
| zlmtsetOLD 33636 | Obsolete proof of ~ zlmtse... |
| zlmnm 33637 | Norm of a ` ZZ ` -module (... |
| zhmnrg 33638 | The ` ZZ ` -module built f... |
| nmmulg 33639 | The norm of a group produc... |
| zrhnm 33640 | The norm of the image by `... |
| cnzh 33641 | The ` ZZ ` -module of ` CC... |
| rezh 33642 | The ` ZZ ` -module of ` RR... |
| qqhval 33645 | Value of the canonical hom... |
| zrhf1ker 33646 | The kernel of the homomorp... |
| zrhchr 33647 | The kernel of the homomorp... |
| zrhker 33648 | The kernel of the homomorp... |
| zrhunitpreima 33649 | The preimage by ` ZRHom ` ... |
| elzrhunit 33650 | Condition for the image by... |
| elzdif0 33651 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 33652 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 33653 | Value of the canonical hom... |
| qqhvval 33654 | Value of the canonical hom... |
| qqh0 33655 | The image of ` 0 ` by the ... |
| qqh1 33656 | The image of ` 1 ` by the ... |
| qqhf 33657 | ` QQHom ` as a function. ... |
| qqhvq 33658 | The image of a quotient by... |
| qqhghm 33659 | The ` QQHom ` homomorphism... |
| qqhrhm 33660 | The ` QQHom ` homomorphism... |
| qqhnm 33661 | The norm of the image by `... |
| qqhcn 33662 | The ` QQHom ` homomorphism... |
| qqhucn 33663 | The ` QQHom ` homomorphism... |
| rrhval 33667 | Value of the canonical hom... |
| rrhcn 33668 | If the topology of ` R ` i... |
| rrhf 33669 | If the topology of ` R ` i... |
| isrrext 33671 | Express the property " ` R... |
| rrextnrg 33672 | An extension of ` RR ` is ... |
| rrextdrg 33673 | An extension of ` RR ` is ... |
| rrextnlm 33674 | The norm of an extension o... |
| rrextchr 33675 | The ring characteristic of... |
| rrextcusp 33676 | An extension of ` RR ` is ... |
| rrexttps 33677 | An extension of ` RR ` is ... |
| rrexthaus 33678 | The topology of an extensi... |
| rrextust 33679 | The uniformity of an exten... |
| rerrext 33680 | The field of the real numb... |
| cnrrext 33681 | The field of the complex n... |
| qqtopn 33682 | The topology of the field ... |
| rrhfe 33683 | If ` R ` is an extension o... |
| rrhcne 33684 | If ` R ` is an extension o... |
| rrhqima 33685 | The ` RRHom ` homomorphism... |
| rrh0 33686 | The image of ` 0 ` by the ... |
| xrhval 33689 | The value of the embedding... |
| zrhre 33690 | The ` ZRHom ` homomorphism... |
| qqhre 33691 | The ` QQHom ` homomorphism... |
| rrhre 33692 | The ` RRHom ` homomorphism... |
| relmntop 33695 | Manifold is a relation. (... |
| ismntoplly 33696 | Property of being a manifo... |
| ismntop 33697 | Property of being a manifo... |
| nexple 33698 | A lower bound for an expon... |
| indv 33701 | Value of the indicator fun... |
| indval 33702 | Value of the indicator fun... |
| indval2 33703 | Alternate value of the ind... |
| indf 33704 | An indicator function as a... |
| indfval 33705 | Value of the indicator fun... |
| ind1 33706 | Value of the indicator fun... |
| ind0 33707 | Value of the indicator fun... |
| ind1a 33708 | Value of the indicator fun... |
| indpi1 33709 | Preimage of the singleton ... |
| indsum 33710 | Finite sum of a product wi... |
| indsumin 33711 | Finite sum of a product wi... |
| prodindf 33712 | The product of indicators ... |
| indf1o 33713 | The bijection between a po... |
| indpreima 33714 | A function with range ` { ... |
| indf1ofs 33715 | The bijection between fini... |
| esumex 33718 | An extended sum is a set b... |
| esumcl 33719 | Closure for extended sum i... |
| esumeq12dvaf 33720 | Equality deduction for ext... |
| esumeq12dva 33721 | Equality deduction for ext... |
| esumeq12d 33722 | Equality deduction for ext... |
| esumeq1 33723 | Equality theorem for an ex... |
| esumeq1d 33724 | Equality theorem for an ex... |
| esumeq2 33725 | Equality theorem for exten... |
| esumeq2d 33726 | Equality deduction for ext... |
| esumeq2dv 33727 | Equality deduction for ext... |
| esumeq2sdv 33728 | Equality deduction for ext... |
| nfesum1 33729 | Bound-variable hypothesis ... |
| nfesum2 33730 | Bound-variable hypothesis ... |
| cbvesum 33731 | Change bound variable in a... |
| cbvesumv 33732 | Change bound variable in a... |
| esumid 33733 | Identify the extended sum ... |
| esumgsum 33734 | A finite extended sum is t... |
| esumval 33735 | Develop the value of the e... |
| esumel 33736 | The extended sum is a limi... |
| esumnul 33737 | Extended sum over the empt... |
| esum0 33738 | Extended sum of zero. (Co... |
| esumf1o 33739 | Re-index an extended sum u... |
| esumc 33740 | Convert from the collectio... |
| esumrnmpt 33741 | Rewrite an extended sum in... |
| esumsplit 33742 | Split an extended sum into... |
| esummono 33743 | Extended sum is monotonic.... |
| esumpad 33744 | Extend an extended sum by ... |
| esumpad2 33745 | Remove zeroes from an exte... |
| esumadd 33746 | Addition of infinite sums.... |
| esumle 33747 | If all of the terms of an ... |
| gsumesum 33748 | Relate a group sum on ` ( ... |
| esumlub 33749 | The extended sum is the lo... |
| esumaddf 33750 | Addition of infinite sums.... |
| esumlef 33751 | If all of the terms of an ... |
| esumcst 33752 | The extended sum of a cons... |
| esumsnf 33753 | The extended sum of a sing... |
| esumsn 33754 | The extended sum of a sing... |
| esumpr 33755 | Extended sum over a pair. ... |
| esumpr2 33756 | Extended sum over a pair, ... |
| esumrnmpt2 33757 | Rewrite an extended sum in... |
| esumfzf 33758 | Formulating a partial exte... |
| esumfsup 33759 | Formulating an extended su... |
| esumfsupre 33760 | Formulating an extended su... |
| esumss 33761 | Change the index set to a ... |
| esumpinfval 33762 | The value of the extended ... |
| esumpfinvallem 33763 | Lemma for ~ esumpfinval . ... |
| esumpfinval 33764 | The value of the extended ... |
| esumpfinvalf 33765 | Same as ~ esumpfinval , mi... |
| esumpinfsum 33766 | The value of the extended ... |
| esumpcvgval 33767 | The value of the extended ... |
| esumpmono 33768 | The partial sums in an ext... |
| esumcocn 33769 | Lemma for ~ esummulc2 and ... |
| esummulc1 33770 | An extended sum multiplied... |
| esummulc2 33771 | An extended sum multiplied... |
| esumdivc 33772 | An extended sum divided by... |
| hashf2 33773 | Lemma for ~ hasheuni . (C... |
| hasheuni 33774 | The cardinality of a disjo... |
| esumcvg 33775 | The sequence of partial su... |
| esumcvg2 33776 | Simpler version of ~ esumc... |
| esumcvgsum 33777 | The value of the extended ... |
| esumsup 33778 | Express an extended sum as... |
| esumgect 33779 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 33780 | All terms of a converging ... |
| esum2dlem 33781 | Lemma for ~ esum2d (finite... |
| esum2d 33782 | Write a double extended su... |
| esumiun 33783 | Sum over a nonnecessarily ... |
| ofceq 33786 | Equality theorem for funct... |
| ofcfval 33787 | Value of an operation appl... |
| ofcval 33788 | Evaluate a function/consta... |
| ofcfn 33789 | The function operation pro... |
| ofcfeqd2 33790 | Equality theorem for funct... |
| ofcfval3 33791 | General value of ` ( F oFC... |
| ofcf 33792 | The function/constant oper... |
| ofcfval2 33793 | The function operation exp... |
| ofcfval4 33794 | The function/constant oper... |
| ofcc 33795 | Left operation by a consta... |
| ofcof 33796 | Relate function operation ... |
| sigaex 33799 | Lemma for ~ issiga and ~ i... |
| sigaval 33800 | The set of sigma-algebra w... |
| issiga 33801 | An alternative definition ... |
| isrnsiga 33802 | The property of being a si... |
| 0elsiga 33803 | A sigma-algebra contains t... |
| baselsiga 33804 | A sigma-algebra contains i... |
| sigasspw 33805 | A sigma-algebra is a set o... |
| sigaclcu 33806 | A sigma-algebra is closed ... |
| sigaclcuni 33807 | A sigma-algebra is closed ... |
| sigaclfu 33808 | A sigma-algebra is closed ... |
| sigaclcu2 33809 | A sigma-algebra is closed ... |
| sigaclfu2 33810 | A sigma-algebra is closed ... |
| sigaclcu3 33811 | A sigma-algebra is closed ... |
| issgon 33812 | Property of being a sigma-... |
| sgon 33813 | A sigma-algebra is a sigma... |
| elsigass 33814 | An element of a sigma-alge... |
| elrnsiga 33815 | Dropping the base informat... |
| isrnsigau 33816 | The property of being a si... |
| unielsiga 33817 | A sigma-algebra contains i... |
| dmvlsiga 33818 | Lebesgue-measurable subset... |
| pwsiga 33819 | Any power set forms a sigm... |
| prsiga 33820 | The smallest possible sigm... |
| sigaclci 33821 | A sigma-algebra is closed ... |
| difelsiga 33822 | A sigma-algebra is closed ... |
| unelsiga 33823 | A sigma-algebra is closed ... |
| inelsiga 33824 | A sigma-algebra is closed ... |
| sigainb 33825 | Building a sigma-algebra f... |
| insiga 33826 | The intersection of a coll... |
| sigagenval 33829 | Value of the generated sig... |
| sigagensiga 33830 | A generated sigma-algebra ... |
| sgsiga 33831 | A generated sigma-algebra ... |
| unisg 33832 | The sigma-algebra generate... |
| dmsigagen 33833 | A sigma-algebra can be gen... |
| sssigagen 33834 | A set is a subset of the s... |
| sssigagen2 33835 | A subset of the generating... |
| elsigagen 33836 | Any element of a set is al... |
| elsigagen2 33837 | Any countable union of ele... |
| sigagenss 33838 | The generated sigma-algebr... |
| sigagenss2 33839 | Sufficient condition for i... |
| sigagenid 33840 | The sigma-algebra generate... |
| ispisys 33841 | The property of being a pi... |
| ispisys2 33842 | The property of being a pi... |
| inelpisys 33843 | Pi-systems are closed unde... |
| sigapisys 33844 | All sigma-algebras are pi-... |
| isldsys 33845 | The property of being a la... |
| pwldsys 33846 | The power set of the unive... |
| unelldsys 33847 | Lambda-systems are closed ... |
| sigaldsys 33848 | All sigma-algebras are lam... |
| ldsysgenld 33849 | The intersection of all la... |
| sigapildsyslem 33850 | Lemma for ~ sigapildsys . ... |
| sigapildsys 33851 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 33852 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 33853 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 33854 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 33855 | The lambda system ` E ` ge... |
| dynkin 33856 | Dynkin's lambda-pi theorem... |
| isros 33857 | The property of being a ri... |
| rossspw 33858 | A ring of sets is a collec... |
| 0elros 33859 | A ring of sets contains th... |
| unelros 33860 | A ring of sets is closed u... |
| difelros 33861 | A ring of sets is closed u... |
| inelros 33862 | A ring of sets is closed u... |
| fiunelros 33863 | A ring of sets is closed u... |
| issros 33864 | The property of being a se... |
| srossspw 33865 | A semiring of sets is a co... |
| 0elsros 33866 | A semiring of sets contain... |
| inelsros 33867 | A semiring of sets is clos... |
| diffiunisros 33868 | In semiring of sets, compl... |
| rossros 33869 | Rings of sets are semiring... |
| brsiga 33872 | The Borel Algebra on real ... |
| brsigarn 33873 | The Borel Algebra is a sig... |
| brsigasspwrn 33874 | The Borel Algebra is a set... |
| unibrsiga 33875 | The union of the Borel Alg... |
| cldssbrsiga 33876 | A Borel Algebra contains a... |
| sxval 33879 | Value of the product sigma... |
| sxsiga 33880 | A product sigma-algebra is... |
| sxsigon 33881 | A product sigma-algebra is... |
| sxuni 33882 | The base set of a product ... |
| elsx 33883 | The cartesian product of t... |
| measbase 33886 | The base set of a measure ... |
| measval 33887 | The value of the ` measure... |
| ismeas 33888 | The property of being a me... |
| isrnmeas 33889 | The property of being a me... |
| dmmeas 33890 | The domain of a measure is... |
| measbasedom 33891 | The base set of a measure ... |
| measfrge0 33892 | A measure is a function ov... |
| measfn 33893 | A measure is a function on... |
| measvxrge0 33894 | The values of a measure ar... |
| measvnul 33895 | The measure of the empty s... |
| measge0 33896 | A measure is nonnegative. ... |
| measle0 33897 | If the measure of a given ... |
| measvun 33898 | The measure of a countable... |
| measxun2 33899 | The measure the union of t... |
| measun 33900 | The measure the union of t... |
| measvunilem 33901 | Lemma for ~ measvuni . (C... |
| measvunilem0 33902 | Lemma for ~ measvuni . (C... |
| measvuni 33903 | The measure of a countable... |
| measssd 33904 | A measure is monotone with... |
| measunl 33905 | A measure is sub-additive ... |
| measiuns 33906 | The measure of the union o... |
| measiun 33907 | A measure is sub-additive.... |
| meascnbl 33908 | A measure is continuous fr... |
| measinblem 33909 | Lemma for ~ measinb . (Co... |
| measinb 33910 | Building a measure restric... |
| measres 33911 | Building a measure restric... |
| measinb2 33912 | Building a measure restric... |
| measdivcst 33913 | Division of a measure by a... |
| measdivcstALTV 33914 | Alternate version of ~ mea... |
| cntmeas 33915 | The Counting measure is a ... |
| pwcntmeas 33916 | The counting measure is a ... |
| cntnevol 33917 | Counting and Lebesgue meas... |
| voliune 33918 | The Lebesgue measure funct... |
| volfiniune 33919 | The Lebesgue measure funct... |
| volmeas 33920 | The Lebesgue measure is a ... |
| ddeval1 33923 | Value of the delta measure... |
| ddeval0 33924 | Value of the delta measure... |
| ddemeas 33925 | The Dirac delta measure is... |
| relae 33929 | 'almost everywhere' is a r... |
| brae 33930 | 'almost everywhere' relati... |
| braew 33931 | 'almost everywhere' relati... |
| truae 33932 | A truth holds almost every... |
| aean 33933 | A conjunction holds almost... |
| faeval 33935 | Value of the 'almost every... |
| relfae 33936 | The 'almost everywhere' bu... |
| brfae 33937 | 'almost everywhere' relati... |
| ismbfm 33940 | The predicate " ` F ` is a... |
| elunirnmbfm 33941 | The property of being a me... |
| mbfmfun 33942 | A measurable function is a... |
| mbfmf 33943 | A measurable function as a... |
| isanmbfmOLD 33944 | Obsolete version of ~ isan... |
| mbfmcnvima 33945 | The preimage by a measurab... |
| isanmbfm 33946 | The predicate to be a meas... |
| mbfmbfmOLD 33947 | A measurable function to a... |
| mbfmbfm 33948 | A measurable function to a... |
| mbfmcst 33949 | A constant function is mea... |
| 1stmbfm 33950 | The first projection map i... |
| 2ndmbfm 33951 | The second projection map ... |
| imambfm 33952 | If the sigma-algebra in th... |
| cnmbfm 33953 | A continuous function is m... |
| mbfmco 33954 | The composition of two mea... |
| mbfmco2 33955 | The pair building of two m... |
| mbfmvolf 33956 | Measurable functions with ... |
| elmbfmvol2 33957 | Measurable functions with ... |
| mbfmcnt 33958 | All functions are measurab... |
| br2base 33959 | The base set for the gener... |
| dya2ub 33960 | An upper bound for a dyadi... |
| sxbrsigalem0 33961 | The closed half-spaces of ... |
| sxbrsigalem3 33962 | The sigma-algebra generate... |
| dya2iocival 33963 | The function ` I ` returns... |
| dya2iocress 33964 | Dyadic intervals are subse... |
| dya2iocbrsiga 33965 | Dyadic intervals are Borel... |
| dya2icobrsiga 33966 | Dyadic intervals are Borel... |
| dya2icoseg 33967 | For any point and any clos... |
| dya2icoseg2 33968 | For any point and any open... |
| dya2iocrfn 33969 | The function returning dya... |
| dya2iocct 33970 | The dyadic rectangle set i... |
| dya2iocnrect 33971 | For any point of an open r... |
| dya2iocnei 33972 | For any point of an open s... |
| dya2iocuni 33973 | Every open set of ` ( RR X... |
| dya2iocucvr 33974 | The dyadic rectangular set... |
| sxbrsigalem1 33975 | The Borel algebra on ` ( R... |
| sxbrsigalem2 33976 | The sigma-algebra generate... |
| sxbrsigalem4 33977 | The Borel algebra on ` ( R... |
| sxbrsigalem5 33978 | First direction for ~ sxbr... |
| sxbrsigalem6 33979 | First direction for ~ sxbr... |
| sxbrsiga 33980 | The product sigma-algebra ... |
| omsval 33983 | Value of the function mapp... |
| omsfval 33984 | Value of the outer measure... |
| omscl 33985 | A closure lemma for the co... |
| omsf 33986 | A constructed outer measur... |
| oms0 33987 | A constructed outer measur... |
| omsmon 33988 | A constructed outer measur... |
| omssubaddlem 33989 | For any small margin ` E `... |
| omssubadd 33990 | A constructed outer measur... |
| carsgval 33993 | Value of the Caratheodory ... |
| carsgcl 33994 | Closure of the Caratheodor... |
| elcarsg 33995 | Property of being a Carath... |
| baselcarsg 33996 | The universe set, ` O ` , ... |
| 0elcarsg 33997 | The empty set is Caratheod... |
| carsguni 33998 | The union of all Caratheod... |
| elcarsgss 33999 | Caratheodory measurable se... |
| difelcarsg 34000 | The Caratheodory measurabl... |
| inelcarsg 34001 | The Caratheodory measurabl... |
| unelcarsg 34002 | The Caratheodory-measurabl... |
| difelcarsg2 34003 | The Caratheodory-measurabl... |
| carsgmon 34004 | Utility lemma: Apply mono... |
| carsgsigalem 34005 | Lemma for the following th... |
| fiunelcarsg 34006 | The Caratheodory measurabl... |
| carsgclctunlem1 34007 | Lemma for ~ carsgclctun . ... |
| carsggect 34008 | The outer measure is count... |
| carsgclctunlem2 34009 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 34010 | Lemma for ~ carsgclctun . ... |
| carsgclctun 34011 | The Caratheodory measurabl... |
| carsgsiga 34012 | The Caratheodory measurabl... |
| omsmeas 34013 | The restriction of a const... |
| pmeasmono 34014 | This theorem's hypotheses ... |
| pmeasadd 34015 | A premeasure on a ring of ... |
| itgeq12dv 34016 | Equality theorem for an in... |
| sitgval 34022 | Value of the simple functi... |
| issibf 34023 | The predicate " ` F ` is a... |
| sibf0 34024 | The constant zero function... |
| sibfmbl 34025 | A simple function is measu... |
| sibff 34026 | A simple function is a fun... |
| sibfrn 34027 | A simple function has fini... |
| sibfima 34028 | Any preimage of a singleto... |
| sibfinima 34029 | The measure of the interse... |
| sibfof 34030 | Applying function operatio... |
| sitgfval 34031 | Value of the Bochner integ... |
| sitgclg 34032 | Closure of the Bochner int... |
| sitgclbn 34033 | Closure of the Bochner int... |
| sitgclcn 34034 | Closure of the Bochner int... |
| sitgclre 34035 | Closure of the Bochner int... |
| sitg0 34036 | The integral of the consta... |
| sitgf 34037 | The integral for simple fu... |
| sitgaddlemb 34038 | Lemma for * sitgadd . (Co... |
| sitmval 34039 | Value of the simple functi... |
| sitmfval 34040 | Value of the integral dist... |
| sitmcl 34041 | Closure of the integral di... |
| sitmf 34042 | The integral metric as a f... |
| oddpwdc 34044 | Lemma for ~ eulerpart . T... |
| oddpwdcv 34045 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 34046 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 34047 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 34048 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 34049 | Lemma for ~ eulerpart . (... |
| eulerpartlems 34050 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 34051 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 34052 | Lemma for ~ eulerpart . (... |
| eulerpartleme 34053 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 34054 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 34055 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 34056 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 34057 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 34058 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 34059 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 34060 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 34061 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 34062 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 34063 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 34064 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 34065 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 34066 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 34067 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 34068 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 34069 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 34070 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 34071 | Lemma for ~ eulerpart . (... |
| eulerpart 34072 | Euler's theorem on partiti... |
| subiwrd 34075 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 34076 | Length of a subword of an ... |
| iwrdsplit 34077 | Lemma for ~ sseqp1 . (Con... |
| sseqval 34078 | Value of the strong sequen... |
| sseqfv1 34079 | Value of the strong sequen... |
| sseqfn 34080 | A strong recursive sequenc... |
| sseqmw 34081 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 34082 | A strong recursive sequenc... |
| sseqfres 34083 | The first elements in the ... |
| sseqfv2 34084 | Value of the strong sequen... |
| sseqp1 34085 | Value of the strong sequen... |
| fiblem 34088 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 34089 | Value of the Fibonacci seq... |
| fib1 34090 | Value of the Fibonacci seq... |
| fibp1 34091 | Value of the Fibonacci seq... |
| fib2 34092 | Value of the Fibonacci seq... |
| fib3 34093 | Value of the Fibonacci seq... |
| fib4 34094 | Value of the Fibonacci seq... |
| fib5 34095 | Value of the Fibonacci seq... |
| fib6 34096 | Value of the Fibonacci seq... |
| elprob 34099 | The property of being a pr... |
| domprobmeas 34100 | A probability measure is a... |
| domprobsiga 34101 | The domain of a probabilit... |
| probtot 34102 | The probability of the uni... |
| prob01 34103 | A probability is an elemen... |
| probnul 34104 | The probability of the emp... |
| unveldomd 34105 | The universe is an element... |
| unveldom 34106 | The universe is an element... |
| nuleldmp 34107 | The empty set is an elemen... |
| probcun 34108 | The probability of the uni... |
| probun 34109 | The probability of the uni... |
| probdif 34110 | The probability of the dif... |
| probinc 34111 | A probability law is incre... |
| probdsb 34112 | The probability of the com... |
| probmeasd 34113 | A probability measure is a... |
| probvalrnd 34114 | The value of a probability... |
| probtotrnd 34115 | The probability of the uni... |
| totprobd 34116 | Law of total probability, ... |
| totprob 34117 | Law of total probability. ... |
| probfinmeasb 34118 | Build a probability measur... |
| probfinmeasbALTV 34119 | Alternate version of ~ pro... |
| probmeasb 34120 | Build a probability from a... |
| cndprobval 34123 | The value of the condition... |
| cndprobin 34124 | An identity linking condit... |
| cndprob01 34125 | The conditional probabilit... |
| cndprobtot 34126 | The conditional probabilit... |
| cndprobnul 34127 | The conditional probabilit... |
| cndprobprob 34128 | The conditional probabilit... |
| bayesth 34129 | Bayes Theorem. (Contribut... |
| rrvmbfm 34132 | A real-valued random varia... |
| isrrvv 34133 | Elementhood to the set of ... |
| rrvvf 34134 | A real-valued random varia... |
| rrvfn 34135 | A real-valued random varia... |
| rrvdm 34136 | The domain of a random var... |
| rrvrnss 34137 | The range of a random vari... |
| rrvf2 34138 | A real-valued random varia... |
| rrvdmss 34139 | The domain of a random var... |
| rrvfinvima 34140 | For a real-value random va... |
| 0rrv 34141 | The constant function equa... |
| rrvadd 34142 | The sum of two random vari... |
| rrvmulc 34143 | A random variable multipli... |
| rrvsum 34144 | An indexed sum of random v... |
| orvcval 34147 | Value of the preimage mapp... |
| orvcval2 34148 | Another way to express the... |
| elorvc 34149 | Elementhood of a preimage.... |
| orvcval4 34150 | The value of the preimage ... |
| orvcoel 34151 | If the relation produces o... |
| orvccel 34152 | If the relation produces c... |
| elorrvc 34153 | Elementhood of a preimage ... |
| orrvcval4 34154 | The value of the preimage ... |
| orrvcoel 34155 | If the relation produces o... |
| orrvccel 34156 | If the relation produces c... |
| orvcgteel 34157 | Preimage maps produced by ... |
| orvcelval 34158 | Preimage maps produced by ... |
| orvcelel 34159 | Preimage maps produced by ... |
| dstrvval 34160 | The value of the distribut... |
| dstrvprob 34161 | The distribution of a rand... |
| orvclteel 34162 | Preimage maps produced by ... |
| dstfrvel 34163 | Elementhood of preimage ma... |
| dstfrvunirn 34164 | The limit of all preimage ... |
| orvclteinc 34165 | Preimage maps produced by ... |
| dstfrvinc 34166 | A cumulative distribution ... |
| dstfrvclim1 34167 | The limit of the cumulativ... |
| coinfliplem 34168 | Division in the extended r... |
| coinflipprob 34169 | The ` P ` we defined for c... |
| coinflipspace 34170 | The space of our coin-flip... |
| coinflipuniv 34171 | The universe of our coin-f... |
| coinfliprv 34172 | The ` X ` we defined for c... |
| coinflippv 34173 | The probability of heads i... |
| coinflippvt 34174 | The probability of tails i... |
| ballotlemoex 34175 | ` O ` is a set. (Contribu... |
| ballotlem1 34176 | The size of the universe i... |
| ballotlemelo 34177 | Elementhood in ` O ` . (C... |
| ballotlem2 34178 | The probability that the f... |
| ballotlemfval 34179 | The value of ` F ` . (Con... |
| ballotlemfelz 34180 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 34181 | If the ` J ` th ballot is ... |
| ballotlemfc0 34182 | ` F ` takes value 0 betwee... |
| ballotlemfcc 34183 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 34184 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 34185 | ` ( F `` C ) ` always star... |
| ballotleme 34186 | Elements of ` E ` . (Cont... |
| ballotlemodife 34187 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 34188 | If the first pick is a vot... |
| ballotlem5 34189 | If A is not ahead througho... |
| ballotlemi 34190 | Value of ` I ` for a given... |
| ballotlemiex 34191 | Properties of ` ( I `` C )... |
| ballotlemi1 34192 | The first tie cannot be re... |
| ballotlemii 34193 | The first tie cannot be re... |
| ballotlemsup 34194 | The set of zeroes of ` F `... |
| ballotlemimin 34195 | ` ( I `` C ) ` is the firs... |
| ballotlemic 34196 | If the first vote is for B... |
| ballotlem1c 34197 | If the first vote is for A... |
| ballotlemsval 34198 | Value of ` S ` . (Contrib... |
| ballotlemsv 34199 | Value of ` S ` evaluated a... |
| ballotlemsgt1 34200 | ` S ` maps values less tha... |
| ballotlemsdom 34201 | Domain of ` S ` for a give... |
| ballotlemsel1i 34202 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 34203 | The defined ` S ` is a bij... |
| ballotlemsi 34204 | The image by ` S ` of the ... |
| ballotlemsima 34205 | The image by ` S ` of an i... |
| ballotlemieq 34206 | If two countings share the... |
| ballotlemrval 34207 | Value of ` R ` . (Contrib... |
| ballotlemscr 34208 | The image of ` ( R `` C ) ... |
| ballotlemrv 34209 | Value of ` R ` evaluated a... |
| ballotlemrv1 34210 | Value of ` R ` before the ... |
| ballotlemrv2 34211 | Value of ` R ` after the t... |
| ballotlemro 34212 | Range of ` R ` is included... |
| ballotlemgval 34213 | Expand the value of ` .^ `... |
| ballotlemgun 34214 | A property of the defined ... |
| ballotlemfg 34215 | Express the value of ` ( F... |
| ballotlemfrc 34216 | Express the value of ` ( F... |
| ballotlemfrci 34217 | Reverse counting preserves... |
| ballotlemfrceq 34218 | Value of ` F ` for a rever... |
| ballotlemfrcn0 34219 | Value of ` F ` for a rever... |
| ballotlemrc 34220 | Range of ` R ` . (Contrib... |
| ballotlemirc 34221 | Applying ` R ` does not ch... |
| ballotlemrinv0 34222 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 34223 | ` R ` is its own inverse :... |
| ballotlem1ri 34224 | When the vote on the first... |
| ballotlem7 34225 | ` R ` is a bijection betwe... |
| ballotlem8 34226 | There are as many counting... |
| ballotth 34227 | Bertrand's ballot problem ... |
| sgncl 34228 | Closure of the signum. (C... |
| sgnclre 34229 | Closure of the signum. (C... |
| sgnneg 34230 | Negation of the signum. (... |
| sgn3da 34231 | A conditional containing a... |
| sgnmul 34232 | Signum of a product. (Con... |
| sgnmulrp2 34233 | Multiplication by a positi... |
| sgnsub 34234 | Subtraction of a number of... |
| sgnnbi 34235 | Negative signum. (Contrib... |
| sgnpbi 34236 | Positive signum. (Contrib... |
| sgn0bi 34237 | Zero signum. (Contributed... |
| sgnsgn 34238 | Signum is idempotent. (Co... |
| sgnmulsgn 34239 | If two real numbers are of... |
| sgnmulsgp 34240 | If two real numbers are of... |
| fzssfzo 34241 | Condition for an integer i... |
| gsumncl 34242 | Closure of a group sum in ... |
| gsumnunsn 34243 | Closure of a group sum in ... |
| ccatmulgnn0dir 34244 | Concatenation of words fol... |
| ofcccat 34245 | Letterwise operations on w... |
| ofcs1 34246 | Letterwise operations on a... |
| ofcs2 34247 | Letterwise operations on a... |
| plymul02 34248 | Product of a polynomial wi... |
| plymulx0 34249 | Coefficients of a polynomi... |
| plymulx 34250 | Coefficients of a polynomi... |
| plyrecld 34251 | Closure of a polynomial wi... |
| signsplypnf 34252 | The quotient of a polynomi... |
| signsply0 34253 | Lemma for the rule of sign... |
| signspval 34254 | The value of the skipping ... |
| signsw0glem 34255 | Neutral element property o... |
| signswbase 34256 | The base of ` W ` is the u... |
| signswplusg 34257 | The operation of ` W ` . ... |
| signsw0g 34258 | The neutral element of ` W... |
| signswmnd 34259 | ` W ` is a monoid structur... |
| signswrid 34260 | The zero-skipping operatio... |
| signswlid 34261 | The zero-skipping operatio... |
| signswn0 34262 | The zero-skipping operatio... |
| signswch 34263 | The zero-skipping operatio... |
| signslema 34264 | Computational part of ~~? ... |
| signstfv 34265 | Value of the zero-skipping... |
| signstfval 34266 | Value of the zero-skipping... |
| signstcl 34267 | Closure of the zero skippi... |
| signstf 34268 | The zero skipping sign wor... |
| signstlen 34269 | Length of the zero skippin... |
| signstf0 34270 | Sign of a single letter wo... |
| signstfvn 34271 | Zero-skipping sign in a wo... |
| signsvtn0 34272 | If the last letter is nonz... |
| signstfvp 34273 | Zero-skipping sign in a wo... |
| signstfvneq0 34274 | In case the first letter i... |
| signstfvcl 34275 | Closure of the zero skippi... |
| signstfvc 34276 | Zero-skipping sign in a wo... |
| signstres 34277 | Restriction of a zero skip... |
| signstfveq0a 34278 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 34279 | In case the last letter is... |
| signsvvfval 34280 | The value of ` V ` , which... |
| signsvvf 34281 | ` V ` is a function. (Con... |
| signsvf0 34282 | There is no change of sign... |
| signsvf1 34283 | In a single-letter word, w... |
| signsvfn 34284 | Number of changes in a wor... |
| signsvtp 34285 | Adding a letter of the sam... |
| signsvtn 34286 | Adding a letter of a diffe... |
| signsvfpn 34287 | Adding a letter of the sam... |
| signsvfnn 34288 | Adding a letter of a diffe... |
| signlem0 34289 | Adding a zero as the highe... |
| signshf 34290 | ` H ` , corresponding to t... |
| signshwrd 34291 | ` H ` , corresponding to t... |
| signshlen 34292 | Length of ` H ` , correspo... |
| signshnz 34293 | ` H ` is not the empty wor... |
| iblidicc 34294 | The identity function is i... |
| rpsqrtcn 34295 | Continuity of the real pos... |
| divsqrtid 34296 | A real number divided by i... |
| cxpcncf1 34297 | The power function on comp... |
| efmul2picn 34298 | Multiplying by ` ( _i x. (... |
| fct2relem 34299 | Lemma for ~ ftc2re . (Con... |
| ftc2re 34300 | The Fundamental Theorem of... |
| fdvposlt 34301 | Functions with a positive ... |
| fdvneggt 34302 | Functions with a negative ... |
| fdvposle 34303 | Functions with a nonnegati... |
| fdvnegge 34304 | Functions with a nonpositi... |
| prodfzo03 34305 | A product of three factors... |
| actfunsnf1o 34306 | The action ` F ` of extend... |
| actfunsnrndisj 34307 | The action ` F ` of extend... |
| itgexpif 34308 | The basis for the circle m... |
| fsum2dsub 34309 | Lemma for ~ breprexp - Re-... |
| reprval 34312 | Value of the representatio... |
| repr0 34313 | There is exactly one repre... |
| reprf 34314 | Members of the representat... |
| reprsum 34315 | Sums of values of the memb... |
| reprle 34316 | Upper bound to the terms i... |
| reprsuc 34317 | Express the representation... |
| reprfi 34318 | Bounded representations ar... |
| reprss 34319 | Representations with terms... |
| reprinrn 34320 | Representations with term ... |
| reprlt 34321 | There are no representatio... |
| hashreprin 34322 | Express a sum of represent... |
| reprgt 34323 | There are no representatio... |
| reprinfz1 34324 | For the representation of ... |
| reprfi2 34325 | Corollary of ~ reprinfz1 .... |
| reprfz1 34326 | Corollary of ~ reprinfz1 .... |
| hashrepr 34327 | Develop the number of repr... |
| reprpmtf1o 34328 | Transposing ` 0 ` and ` X ... |
| reprdifc 34329 | Express the representation... |
| chpvalz 34330 | Value of the second Chebys... |
| chtvalz 34331 | Value of the Chebyshev fun... |
| breprexplema 34332 | Lemma for ~ breprexp (indu... |
| breprexplemb 34333 | Lemma for ~ breprexp (clos... |
| breprexplemc 34334 | Lemma for ~ breprexp (indu... |
| breprexp 34335 | Express the ` S ` th power... |
| breprexpnat 34336 | Express the ` S ` th power... |
| vtsval 34339 | Value of the Vinogradov tr... |
| vtscl 34340 | Closure of the Vinogradov ... |
| vtsprod 34341 | Express the Vinogradov tri... |
| circlemeth 34342 | The Hardy, Littlewood and ... |
| circlemethnat 34343 | The Hardy, Littlewood and ... |
| circlevma 34344 | The Circle Method, where t... |
| circlemethhgt 34345 | The circle method, where t... |
| hgt750lemc 34349 | An upper bound to the summ... |
| hgt750lemd 34350 | An upper bound to the summ... |
| hgt749d 34351 | A deduction version of ~ a... |
| logdivsqrle 34352 | Conditions for ` ( ( log `... |
| hgt750lem 34353 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 34354 | Decimal multiplication gal... |
| hgt750lemf 34355 | Lemma for the statement 7.... |
| hgt750lemg 34356 | Lemma for the statement 7.... |
| oddprm2 34357 | Two ways to write the set ... |
| hgt750lemb 34358 | An upper bound on the cont... |
| hgt750lema 34359 | An upper bound on the cont... |
| hgt750leme 34360 | An upper bound on the cont... |
| tgoldbachgnn 34361 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 34362 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 34363 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 34364 | Odd integers greater than ... |
| tgoldbachgt 34365 | Odd integers greater than ... |
| istrkg2d 34368 | Property of fulfilling dim... |
| axtglowdim2ALTV 34369 | Alternate version of ~ axt... |
| axtgupdim2ALTV 34370 | Alternate version of ~ axt... |
| afsval 34373 | Value of the AFS relation ... |
| brafs 34374 | Binary relation form of th... |
| tg5segofs 34375 | Rephrase ~ axtg5seg using ... |
| lpadval 34378 | Value of the ` leftpad ` f... |
| lpadlem1 34379 | Lemma for the ` leftpad ` ... |
| lpadlem3 34380 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 34381 | Length of a left-padded wo... |
| lpadlem2 34382 | Lemma for the ` leftpad ` ... |
| lpadlen2 34383 | Length of a left-padded wo... |
| lpadmax 34384 | Length of a left-padded wo... |
| lpadleft 34385 | The contents of prefix of ... |
| lpadright 34386 | The suffix of a left-padde... |
| bnj170 34399 | ` /\ ` -manipulation. (Co... |
| bnj240 34400 | ` /\ ` -manipulation. (Co... |
| bnj248 34401 | ` /\ ` -manipulation. (Co... |
| bnj250 34402 | ` /\ ` -manipulation. (Co... |
| bnj251 34403 | ` /\ ` -manipulation. (Co... |
| bnj252 34404 | ` /\ ` -manipulation. (Co... |
| bnj253 34405 | ` /\ ` -manipulation. (Co... |
| bnj255 34406 | ` /\ ` -manipulation. (Co... |
| bnj256 34407 | ` /\ ` -manipulation. (Co... |
| bnj257 34408 | ` /\ ` -manipulation. (Co... |
| bnj258 34409 | ` /\ ` -manipulation. (Co... |
| bnj268 34410 | ` /\ ` -manipulation. (Co... |
| bnj290 34411 | ` /\ ` -manipulation. (Co... |
| bnj291 34412 | ` /\ ` -manipulation. (Co... |
| bnj312 34413 | ` /\ ` -manipulation. (Co... |
| bnj334 34414 | ` /\ ` -manipulation. (Co... |
| bnj345 34415 | ` /\ ` -manipulation. (Co... |
| bnj422 34416 | ` /\ ` -manipulation. (Co... |
| bnj432 34417 | ` /\ ` -manipulation. (Co... |
| bnj446 34418 | ` /\ ` -manipulation. (Co... |
| bnj23 34419 | First-order logic and set ... |
| bnj31 34420 | First-order logic and set ... |
| bnj62 34421 | First-order logic and set ... |
| bnj89 34422 | First-order logic and set ... |
| bnj90 34423 | First-order logic and set ... |
| bnj101 34424 | First-order logic and set ... |
| bnj105 34425 | First-order logic and set ... |
| bnj115 34426 | First-order logic and set ... |
| bnj132 34427 | First-order logic and set ... |
| bnj133 34428 | First-order logic and set ... |
| bnj156 34429 | First-order logic and set ... |
| bnj158 34430 | First-order logic and set ... |
| bnj168 34431 | First-order logic and set ... |
| bnj206 34432 | First-order logic and set ... |
| bnj216 34433 | First-order logic and set ... |
| bnj219 34434 | First-order logic and set ... |
| bnj226 34435 | First-order logic and set ... |
| bnj228 34436 | First-order logic and set ... |
| bnj519 34437 | First-order logic and set ... |
| bnj524 34438 | First-order logic and set ... |
| bnj525 34439 | First-order logic and set ... |
| bnj534 34440 | First-order logic and set ... |
| bnj538 34441 | First-order logic and set ... |
| bnj529 34442 | First-order logic and set ... |
| bnj551 34443 | First-order logic and set ... |
| bnj563 34444 | First-order logic and set ... |
| bnj564 34445 | First-order logic and set ... |
| bnj593 34446 | First-order logic and set ... |
| bnj596 34447 | First-order logic and set ... |
| bnj610 34448 | Pass from equality ( ` x =... |
| bnj642 34449 | ` /\ ` -manipulation. (Co... |
| bnj643 34450 | ` /\ ` -manipulation. (Co... |
| bnj645 34451 | ` /\ ` -manipulation. (Co... |
| bnj658 34452 | ` /\ ` -manipulation. (Co... |
| bnj667 34453 | ` /\ ` -manipulation. (Co... |
| bnj705 34454 | ` /\ ` -manipulation. (Co... |
| bnj706 34455 | ` /\ ` -manipulation. (Co... |
| bnj707 34456 | ` /\ ` -manipulation. (Co... |
| bnj708 34457 | ` /\ ` -manipulation. (Co... |
| bnj721 34458 | ` /\ ` -manipulation. (Co... |
| bnj832 34459 | ` /\ ` -manipulation. (Co... |
| bnj835 34460 | ` /\ ` -manipulation. (Co... |
| bnj836 34461 | ` /\ ` -manipulation. (Co... |
| bnj837 34462 | ` /\ ` -manipulation. (Co... |
| bnj769 34463 | ` /\ ` -manipulation. (Co... |
| bnj770 34464 | ` /\ ` -manipulation. (Co... |
| bnj771 34465 | ` /\ ` -manipulation. (Co... |
| bnj887 34466 | ` /\ ` -manipulation. (Co... |
| bnj918 34467 | First-order logic and set ... |
| bnj919 34468 | First-order logic and set ... |
| bnj923 34469 | First-order logic and set ... |
| bnj927 34470 | First-order logic and set ... |
| bnj931 34471 | First-order logic and set ... |
| bnj937 34472 | First-order logic and set ... |
| bnj941 34473 | First-order logic and set ... |
| bnj945 34474 | Technical lemma for ~ bnj6... |
| bnj946 34475 | First-order logic and set ... |
| bnj951 34476 | ` /\ ` -manipulation. (Co... |
| bnj956 34477 | First-order logic and set ... |
| bnj976 34478 | First-order logic and set ... |
| bnj982 34479 | First-order logic and set ... |
| bnj1019 34480 | First-order logic and set ... |
| bnj1023 34481 | First-order logic and set ... |
| bnj1095 34482 | First-order logic and set ... |
| bnj1096 34483 | First-order logic and set ... |
| bnj1098 34484 | First-order logic and set ... |
| bnj1101 34485 | First-order logic and set ... |
| bnj1113 34486 | First-order logic and set ... |
| bnj1109 34487 | First-order logic and set ... |
| bnj1131 34488 | First-order logic and set ... |
| bnj1138 34489 | First-order logic and set ... |
| bnj1142 34490 | First-order logic and set ... |
| bnj1143 34491 | First-order logic and set ... |
| bnj1146 34492 | First-order logic and set ... |
| bnj1149 34493 | First-order logic and set ... |
| bnj1185 34494 | First-order logic and set ... |
| bnj1196 34495 | First-order logic and set ... |
| bnj1198 34496 | First-order logic and set ... |
| bnj1209 34497 | First-order logic and set ... |
| bnj1211 34498 | First-order logic and set ... |
| bnj1213 34499 | First-order logic and set ... |
| bnj1212 34500 | First-order logic and set ... |
| bnj1219 34501 | First-order logic and set ... |
| bnj1224 34502 | First-order logic and set ... |
| bnj1230 34503 | First-order logic and set ... |
| bnj1232 34504 | First-order logic and set ... |
| bnj1235 34505 | First-order logic and set ... |
| bnj1239 34506 | First-order logic and set ... |
| bnj1238 34507 | First-order logic and set ... |
| bnj1241 34508 | First-order logic and set ... |
| bnj1247 34509 | First-order logic and set ... |
| bnj1254 34510 | First-order logic and set ... |
| bnj1262 34511 | First-order logic and set ... |
| bnj1266 34512 | First-order logic and set ... |
| bnj1265 34513 | First-order logic and set ... |
| bnj1275 34514 | First-order logic and set ... |
| bnj1276 34515 | First-order logic and set ... |
| bnj1292 34516 | First-order logic and set ... |
| bnj1293 34517 | First-order logic and set ... |
| bnj1294 34518 | First-order logic and set ... |
| bnj1299 34519 | First-order logic and set ... |
| bnj1304 34520 | First-order logic and set ... |
| bnj1316 34521 | First-order logic and set ... |
| bnj1317 34522 | First-order logic and set ... |
| bnj1322 34523 | First-order logic and set ... |
| bnj1340 34524 | First-order logic and set ... |
| bnj1345 34525 | First-order logic and set ... |
| bnj1350 34526 | First-order logic and set ... |
| bnj1351 34527 | First-order logic and set ... |
| bnj1352 34528 | First-order logic and set ... |
| bnj1361 34529 | First-order logic and set ... |
| bnj1366 34530 | First-order logic and set ... |
| bnj1379 34531 | First-order logic and set ... |
| bnj1383 34532 | First-order logic and set ... |
| bnj1385 34533 | First-order logic and set ... |
| bnj1386 34534 | First-order logic and set ... |
| bnj1397 34535 | First-order logic and set ... |
| bnj1400 34536 | First-order logic and set ... |
| bnj1405 34537 | First-order logic and set ... |
| bnj1422 34538 | First-order logic and set ... |
| bnj1424 34539 | First-order logic and set ... |
| bnj1436 34540 | First-order logic and set ... |
| bnj1441 34541 | First-order logic and set ... |
| bnj1441g 34542 | First-order logic and set ... |
| bnj1454 34543 | First-order logic and set ... |
| bnj1459 34544 | First-order logic and set ... |
| bnj1464 34545 | Conversion of implicit sub... |
| bnj1465 34546 | First-order logic and set ... |
| bnj1468 34547 | Conversion of implicit sub... |
| bnj1476 34548 | First-order logic and set ... |
| bnj1502 34549 | First-order logic and set ... |
| bnj1503 34550 | First-order logic and set ... |
| bnj1517 34551 | First-order logic and set ... |
| bnj1521 34552 | First-order logic and set ... |
| bnj1533 34553 | First-order logic and set ... |
| bnj1534 34554 | First-order logic and set ... |
| bnj1536 34555 | First-order logic and set ... |
| bnj1538 34556 | First-order logic and set ... |
| bnj1541 34557 | First-order logic and set ... |
| bnj1542 34558 | First-order logic and set ... |
| bnj110 34559 | Well-founded induction res... |
| bnj157 34560 | Well-founded induction res... |
| bnj66 34561 | Technical lemma for ~ bnj6... |
| bnj91 34562 | First-order logic and set ... |
| bnj92 34563 | First-order logic and set ... |
| bnj93 34564 | Technical lemma for ~ bnj9... |
| bnj95 34565 | Technical lemma for ~ bnj1... |
| bnj96 34566 | Technical lemma for ~ bnj1... |
| bnj97 34567 | Technical lemma for ~ bnj1... |
| bnj98 34568 | Technical lemma for ~ bnj1... |
| bnj106 34569 | First-order logic and set ... |
| bnj118 34570 | First-order logic and set ... |
| bnj121 34571 | First-order logic and set ... |
| bnj124 34572 | Technical lemma for ~ bnj1... |
| bnj125 34573 | Technical lemma for ~ bnj1... |
| bnj126 34574 | Technical lemma for ~ bnj1... |
| bnj130 34575 | Technical lemma for ~ bnj1... |
| bnj149 34576 | Technical lemma for ~ bnj1... |
| bnj150 34577 | Technical lemma for ~ bnj1... |
| bnj151 34578 | Technical lemma for ~ bnj1... |
| bnj154 34579 | Technical lemma for ~ bnj1... |
| bnj155 34580 | Technical lemma for ~ bnj1... |
| bnj153 34581 | Technical lemma for ~ bnj8... |
| bnj207 34582 | Technical lemma for ~ bnj8... |
| bnj213 34583 | First-order logic and set ... |
| bnj222 34584 | Technical lemma for ~ bnj2... |
| bnj229 34585 | Technical lemma for ~ bnj5... |
| bnj517 34586 | Technical lemma for ~ bnj5... |
| bnj518 34587 | Technical lemma for ~ bnj8... |
| bnj523 34588 | Technical lemma for ~ bnj8... |
| bnj526 34589 | Technical lemma for ~ bnj8... |
| bnj528 34590 | Technical lemma for ~ bnj8... |
| bnj535 34591 | Technical lemma for ~ bnj8... |
| bnj539 34592 | Technical lemma for ~ bnj8... |
| bnj540 34593 | Technical lemma for ~ bnj8... |
| bnj543 34594 | Technical lemma for ~ bnj8... |
| bnj544 34595 | Technical lemma for ~ bnj8... |
| bnj545 34596 | Technical lemma for ~ bnj8... |
| bnj546 34597 | Technical lemma for ~ bnj8... |
| bnj548 34598 | Technical lemma for ~ bnj8... |
| bnj553 34599 | Technical lemma for ~ bnj8... |
| bnj554 34600 | Technical lemma for ~ bnj8... |
| bnj556 34601 | Technical lemma for ~ bnj8... |
| bnj557 34602 | Technical lemma for ~ bnj8... |
| bnj558 34603 | Technical lemma for ~ bnj8... |
| bnj561 34604 | Technical lemma for ~ bnj8... |
| bnj562 34605 | Technical lemma for ~ bnj8... |
| bnj570 34606 | Technical lemma for ~ bnj8... |
| bnj571 34607 | Technical lemma for ~ bnj8... |
| bnj605 34608 | Technical lemma. This lem... |
| bnj581 34609 | Technical lemma for ~ bnj5... |
| bnj589 34610 | Technical lemma for ~ bnj8... |
| bnj590 34611 | Technical lemma for ~ bnj8... |
| bnj591 34612 | Technical lemma for ~ bnj8... |
| bnj594 34613 | Technical lemma for ~ bnj8... |
| bnj580 34614 | Technical lemma for ~ bnj5... |
| bnj579 34615 | Technical lemma for ~ bnj8... |
| bnj602 34616 | Equality theorem for the `... |
| bnj607 34617 | Technical lemma for ~ bnj8... |
| bnj609 34618 | Technical lemma for ~ bnj8... |
| bnj611 34619 | Technical lemma for ~ bnj8... |
| bnj600 34620 | Technical lemma for ~ bnj8... |
| bnj601 34621 | Technical lemma for ~ bnj8... |
| bnj852 34622 | Technical lemma for ~ bnj6... |
| bnj864 34623 | Technical lemma for ~ bnj6... |
| bnj865 34624 | Technical lemma for ~ bnj6... |
| bnj873 34625 | Technical lemma for ~ bnj6... |
| bnj849 34626 | Technical lemma for ~ bnj6... |
| bnj882 34627 | Definition (using hypothes... |
| bnj18eq1 34628 | Equality theorem for trans... |
| bnj893 34629 | Property of ` _trCl ` . U... |
| bnj900 34630 | Technical lemma for ~ bnj6... |
| bnj906 34631 | Property of ` _trCl ` . (... |
| bnj908 34632 | Technical lemma for ~ bnj6... |
| bnj911 34633 | Technical lemma for ~ bnj6... |
| bnj916 34634 | Technical lemma for ~ bnj6... |
| bnj917 34635 | Technical lemma for ~ bnj6... |
| bnj934 34636 | Technical lemma for ~ bnj6... |
| bnj929 34637 | Technical lemma for ~ bnj6... |
| bnj938 34638 | Technical lemma for ~ bnj6... |
| bnj944 34639 | Technical lemma for ~ bnj6... |
| bnj953 34640 | Technical lemma for ~ bnj6... |
| bnj958 34641 | Technical lemma for ~ bnj6... |
| bnj1000 34642 | Technical lemma for ~ bnj8... |
| bnj965 34643 | Technical lemma for ~ bnj8... |
| bnj964 34644 | Technical lemma for ~ bnj6... |
| bnj966 34645 | Technical lemma for ~ bnj6... |
| bnj967 34646 | Technical lemma for ~ bnj6... |
| bnj969 34647 | Technical lemma for ~ bnj6... |
| bnj970 34648 | Technical lemma for ~ bnj6... |
| bnj910 34649 | Technical lemma for ~ bnj6... |
| bnj978 34650 | Technical lemma for ~ bnj6... |
| bnj981 34651 | Technical lemma for ~ bnj6... |
| bnj983 34652 | Technical lemma for ~ bnj6... |
| bnj984 34653 | Technical lemma for ~ bnj6... |
| bnj985v 34654 | Version of ~ bnj985 with a... |
| bnj985 34655 | Technical lemma for ~ bnj6... |
| bnj986 34656 | Technical lemma for ~ bnj6... |
| bnj996 34657 | Technical lemma for ~ bnj6... |
| bnj998 34658 | Technical lemma for ~ bnj6... |
| bnj999 34659 | Technical lemma for ~ bnj6... |
| bnj1001 34660 | Technical lemma for ~ bnj6... |
| bnj1006 34661 | Technical lemma for ~ bnj6... |
| bnj1014 34662 | Technical lemma for ~ bnj6... |
| bnj1015 34663 | Technical lemma for ~ bnj6... |
| bnj1018g 34664 | Version of ~ bnj1018 with ... |
| bnj1018 34665 | Technical lemma for ~ bnj6... |
| bnj1020 34666 | Technical lemma for ~ bnj6... |
| bnj1021 34667 | Technical lemma for ~ bnj6... |
| bnj907 34668 | Technical lemma for ~ bnj6... |
| bnj1029 34669 | Property of ` _trCl ` . (... |
| bnj1033 34670 | Technical lemma for ~ bnj6... |
| bnj1034 34671 | Technical lemma for ~ bnj6... |
| bnj1039 34672 | Technical lemma for ~ bnj6... |
| bnj1040 34673 | Technical lemma for ~ bnj6... |
| bnj1047 34674 | Technical lemma for ~ bnj6... |
| bnj1049 34675 | Technical lemma for ~ bnj6... |
| bnj1052 34676 | Technical lemma for ~ bnj6... |
| bnj1053 34677 | Technical lemma for ~ bnj6... |
| bnj1071 34678 | Technical lemma for ~ bnj6... |
| bnj1083 34679 | Technical lemma for ~ bnj6... |
| bnj1090 34680 | Technical lemma for ~ bnj6... |
| bnj1093 34681 | Technical lemma for ~ bnj6... |
| bnj1097 34682 | Technical lemma for ~ bnj6... |
| bnj1110 34683 | Technical lemma for ~ bnj6... |
| bnj1112 34684 | Technical lemma for ~ bnj6... |
| bnj1118 34685 | Technical lemma for ~ bnj6... |
| bnj1121 34686 | Technical lemma for ~ bnj6... |
| bnj1123 34687 | Technical lemma for ~ bnj6... |
| bnj1030 34688 | Technical lemma for ~ bnj6... |
| bnj1124 34689 | Property of ` _trCl ` . (... |
| bnj1133 34690 | Technical lemma for ~ bnj6... |
| bnj1128 34691 | Technical lemma for ~ bnj6... |
| bnj1127 34692 | Property of ` _trCl ` . (... |
| bnj1125 34693 | Property of ` _trCl ` . (... |
| bnj1145 34694 | Technical lemma for ~ bnj6... |
| bnj1147 34695 | Property of ` _trCl ` . (... |
| bnj1137 34696 | Property of ` _trCl ` . (... |
| bnj1148 34697 | Property of ` _pred ` . (... |
| bnj1136 34698 | Technical lemma for ~ bnj6... |
| bnj1152 34699 | Technical lemma for ~ bnj6... |
| bnj1154 34700 | Property of ` Fr ` . (Con... |
| bnj1171 34701 | Technical lemma for ~ bnj6... |
| bnj1172 34702 | Technical lemma for ~ bnj6... |
| bnj1173 34703 | Technical lemma for ~ bnj6... |
| bnj1174 34704 | Technical lemma for ~ bnj6... |
| bnj1175 34705 | Technical lemma for ~ bnj6... |
| bnj1176 34706 | Technical lemma for ~ bnj6... |
| bnj1177 34707 | Technical lemma for ~ bnj6... |
| bnj1186 34708 | Technical lemma for ~ bnj6... |
| bnj1190 34709 | Technical lemma for ~ bnj6... |
| bnj1189 34710 | Technical lemma for ~ bnj6... |
| bnj69 34711 | Existence of a minimal ele... |
| bnj1228 34712 | Existence of a minimal ele... |
| bnj1204 34713 | Well-founded induction. T... |
| bnj1234 34714 | Technical lemma for ~ bnj6... |
| bnj1245 34715 | Technical lemma for ~ bnj6... |
| bnj1256 34716 | Technical lemma for ~ bnj6... |
| bnj1259 34717 | Technical lemma for ~ bnj6... |
| bnj1253 34718 | Technical lemma for ~ bnj6... |
| bnj1279 34719 | Technical lemma for ~ bnj6... |
| bnj1286 34720 | Technical lemma for ~ bnj6... |
| bnj1280 34721 | Technical lemma for ~ bnj6... |
| bnj1296 34722 | Technical lemma for ~ bnj6... |
| bnj1309 34723 | Technical lemma for ~ bnj6... |
| bnj1307 34724 | Technical lemma for ~ bnj6... |
| bnj1311 34725 | Technical lemma for ~ bnj6... |
| bnj1318 34726 | Technical lemma for ~ bnj6... |
| bnj1326 34727 | Technical lemma for ~ bnj6... |
| bnj1321 34728 | Technical lemma for ~ bnj6... |
| bnj1364 34729 | Property of ` _FrSe ` . (... |
| bnj1371 34730 | Technical lemma for ~ bnj6... |
| bnj1373 34731 | Technical lemma for ~ bnj6... |
| bnj1374 34732 | Technical lemma for ~ bnj6... |
| bnj1384 34733 | Technical lemma for ~ bnj6... |
| bnj1388 34734 | Technical lemma for ~ bnj6... |
| bnj1398 34735 | Technical lemma for ~ bnj6... |
| bnj1413 34736 | Property of ` _trCl ` . (... |
| bnj1408 34737 | Technical lemma for ~ bnj1... |
| bnj1414 34738 | Property of ` _trCl ` . (... |
| bnj1415 34739 | Technical lemma for ~ bnj6... |
| bnj1416 34740 | Technical lemma for ~ bnj6... |
| bnj1418 34741 | Property of ` _pred ` . (... |
| bnj1417 34742 | Technical lemma for ~ bnj6... |
| bnj1421 34743 | Technical lemma for ~ bnj6... |
| bnj1444 34744 | Technical lemma for ~ bnj6... |
| bnj1445 34745 | Technical lemma for ~ bnj6... |
| bnj1446 34746 | Technical lemma for ~ bnj6... |
| bnj1447 34747 | Technical lemma for ~ bnj6... |
| bnj1448 34748 | Technical lemma for ~ bnj6... |
| bnj1449 34749 | Technical lemma for ~ bnj6... |
| bnj1442 34750 | Technical lemma for ~ bnj6... |
| bnj1450 34751 | Technical lemma for ~ bnj6... |
| bnj1423 34752 | Technical lemma for ~ bnj6... |
| bnj1452 34753 | Technical lemma for ~ bnj6... |
| bnj1466 34754 | Technical lemma for ~ bnj6... |
| bnj1467 34755 | Technical lemma for ~ bnj6... |
| bnj1463 34756 | Technical lemma for ~ bnj6... |
| bnj1489 34757 | Technical lemma for ~ bnj6... |
| bnj1491 34758 | Technical lemma for ~ bnj6... |
| bnj1312 34759 | Technical lemma for ~ bnj6... |
| bnj1493 34760 | Technical lemma for ~ bnj6... |
| bnj1497 34761 | Technical lemma for ~ bnj6... |
| bnj1498 34762 | Technical lemma for ~ bnj6... |
| bnj60 34763 | Well-founded recursion, pa... |
| bnj1514 34764 | Technical lemma for ~ bnj1... |
| bnj1518 34765 | Technical lemma for ~ bnj1... |
| bnj1519 34766 | Technical lemma for ~ bnj1... |
| bnj1520 34767 | Technical lemma for ~ bnj1... |
| bnj1501 34768 | Technical lemma for ~ bnj1... |
| bnj1500 34769 | Well-founded recursion, pa... |
| bnj1525 34770 | Technical lemma for ~ bnj1... |
| bnj1529 34771 | Technical lemma for ~ bnj1... |
| bnj1523 34772 | Technical lemma for ~ bnj1... |
| bnj1522 34773 | Well-founded recursion, pa... |
| exdifsn 34774 | There exists an element in... |
| srcmpltd 34775 | If a statement is true for... |
| prsrcmpltd 34776 | If a statement is true for... |
| dff15 34777 | A one-to-one function in t... |
| f1resveqaeq 34778 | If a function restricted t... |
| f1resrcmplf1dlem 34779 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 34780 | If a function's restrictio... |
| funen1cnv 34781 | If a function is equinumer... |
| fnrelpredd 34782 | A function that preserves ... |
| cardpred 34783 | The cardinality function p... |
| nummin 34784 | Every nonempty class of nu... |
| fineqvrep 34785 | If the Axiom of Infinity i... |
| fineqvpow 34786 | If the Axiom of Infinity i... |
| fineqvac 34787 | If the Axiom of Infinity i... |
| fineqvacALT 34788 | Shorter proof of ~ fineqva... |
| zltp1ne 34789 | Integer ordering relation.... |
| nnltp1ne 34790 | Positive integer ordering ... |
| nn0ltp1ne 34791 | Nonnegative integer orderi... |
| 0nn0m1nnn0 34792 | A number is zero if and on... |
| f1resfz0f1d 34793 | If a function with a seque... |
| fisshasheq 34794 | A finite set is equal to i... |
| revpfxsfxrev 34795 | The reverse of a prefix of... |
| swrdrevpfx 34796 | A subword expressed in ter... |
| lfuhgr 34797 | A hypergraph is loop-free ... |
| lfuhgr2 34798 | A hypergraph is loop-free ... |
| lfuhgr3 34799 | A hypergraph is loop-free ... |
| cplgredgex 34800 | Any two (distinct) vertice... |
| cusgredgex 34801 | Any two (distinct) vertice... |
| cusgredgex2 34802 | Any two distinct vertices ... |
| pfxwlk 34803 | A prefix of a walk is a wa... |
| revwlk 34804 | The reverse of a walk is a... |
| revwlkb 34805 | Two words represent a walk... |
| swrdwlk 34806 | Two matching subwords of a... |
| pthhashvtx 34807 | A graph containing a path ... |
| pthisspthorcycl 34808 | A path is either a simple ... |
| spthcycl 34809 | A walk is a trivial path i... |
| usgrgt2cycl 34810 | A non-trivial cycle in a s... |
| usgrcyclgt2v 34811 | A simple graph with a non-... |
| subgrwlk 34812 | If a walk exists in a subg... |
| subgrtrl 34813 | If a trail exists in a sub... |
| subgrpth 34814 | If a path exists in a subg... |
| subgrcycl 34815 | If a cycle exists in a sub... |
| cusgr3cyclex 34816 | Every complete simple grap... |
| loop1cycl 34817 | A hypergraph has a cycle o... |
| 2cycld 34818 | Construction of a 2-cycle ... |
| 2cycl2d 34819 | Construction of a 2-cycle ... |
| umgr2cycllem 34820 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 34821 | A multigraph with two dist... |
| dfacycgr1 34824 | An alternate definition of... |
| isacycgr 34825 | The property of being an a... |
| isacycgr1 34826 | The property of being an a... |
| acycgrcycl 34827 | Any cycle in an acyclic gr... |
| acycgr0v 34828 | A null graph (with no vert... |
| acycgr1v 34829 | A multigraph with one vert... |
| acycgr2v 34830 | A simple graph with two ve... |
| prclisacycgr 34831 | A proper class (representi... |
| acycgrislfgr 34832 | An acyclic hypergraph is a... |
| upgracycumgr 34833 | An acyclic pseudograph is ... |
| umgracycusgr 34834 | An acyclic multigraph is a... |
| upgracycusgr 34835 | An acyclic pseudograph is ... |
| cusgracyclt3v 34836 | A complete simple graph is... |
| pthacycspth 34837 | A path in an acyclic graph... |
| acycgrsubgr 34838 | The subgraph of an acyclic... |
| quartfull 34845 | The quartic equation, writ... |
| deranglem 34846 | Lemma for derangements. (... |
| derangval 34847 | Define the derangement fun... |
| derangf 34848 | The derangement number is ... |
| derang0 34849 | The derangement number of ... |
| derangsn 34850 | The derangement number of ... |
| derangenlem 34851 | One half of ~ derangen . ... |
| derangen 34852 | The derangement number is ... |
| subfacval 34853 | The subfactorial is define... |
| derangen2 34854 | Write the derangement numb... |
| subfacf 34855 | The subfactorial is a func... |
| subfaclefac 34856 | The subfactorial is less t... |
| subfac0 34857 | The subfactorial at zero. ... |
| subfac1 34858 | The subfactorial at one. ... |
| subfacp1lem1 34859 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 34860 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 34861 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 34862 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 34863 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 34864 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 34865 | Lemma for ~ subfacp1 . By... |
| subfacp1 34866 | A two-term recurrence for ... |
| subfacval2 34867 | A closed-form expression f... |
| subfaclim 34868 | The subfactorial converges... |
| subfacval3 34869 | Another closed form expres... |
| derangfmla 34870 | The derangements formula, ... |
| erdszelem1 34871 | Lemma for ~ erdsze . (Con... |
| erdszelem2 34872 | Lemma for ~ erdsze . (Con... |
| erdszelem3 34873 | Lemma for ~ erdsze . (Con... |
| erdszelem4 34874 | Lemma for ~ erdsze . (Con... |
| erdszelem5 34875 | Lemma for ~ erdsze . (Con... |
| erdszelem6 34876 | Lemma for ~ erdsze . (Con... |
| erdszelem7 34877 | Lemma for ~ erdsze . (Con... |
| erdszelem8 34878 | Lemma for ~ erdsze . (Con... |
| erdszelem9 34879 | Lemma for ~ erdsze . (Con... |
| erdszelem10 34880 | Lemma for ~ erdsze . (Con... |
| erdszelem11 34881 | Lemma for ~ erdsze . (Con... |
| erdsze 34882 | The Erdős-Szekeres th... |
| erdsze2lem1 34883 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 34884 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 34885 | Generalize the statement o... |
| kur14lem1 34886 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 34887 | Lemma for ~ kur14 . Write... |
| kur14lem3 34888 | Lemma for ~ kur14 . A clo... |
| kur14lem4 34889 | Lemma for ~ kur14 . Compl... |
| kur14lem5 34890 | Lemma for ~ kur14 . Closu... |
| kur14lem6 34891 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 34892 | Lemma for ~ kur14 : main p... |
| kur14lem8 34893 | Lemma for ~ kur14 . Show ... |
| kur14lem9 34894 | Lemma for ~ kur14 . Since... |
| kur14lem10 34895 | Lemma for ~ kur14 . Disch... |
| kur14 34896 | Kuratowski's closure-compl... |
| ispconn 34903 | The property of being a pa... |
| pconncn 34904 | The property of being a pa... |
| pconntop 34905 | A simply connected space i... |
| issconn 34906 | The property of being a si... |
| sconnpconn 34907 | A simply connected space i... |
| sconntop 34908 | A simply connected space i... |
| sconnpht 34909 | A closed path in a simply ... |
| cnpconn 34910 | An image of a path-connect... |
| pconnconn 34911 | A path-connected space is ... |
| txpconn 34912 | The topological product of... |
| ptpconn 34913 | The topological product of... |
| indispconn 34914 | The indiscrete topology (o... |
| connpconn 34915 | A connected and locally pa... |
| qtoppconn 34916 | A quotient of a path-conne... |
| pconnpi1 34917 | All fundamental groups in ... |
| sconnpht2 34918 | Any two paths in a simply ... |
| sconnpi1 34919 | A path-connected topologic... |
| txsconnlem 34920 | Lemma for ~ txsconn . (Co... |
| txsconn 34921 | The topological product of... |
| cvxpconn 34922 | A convex subset of the com... |
| cvxsconn 34923 | A convex subset of the com... |
| blsconn 34924 | An open ball in the comple... |
| cnllysconn 34925 | The topology of the comple... |
| resconn 34926 | A subset of ` RR ` is simp... |
| ioosconn 34927 | An open interval is simply... |
| iccsconn 34928 | A closed interval is simpl... |
| retopsconn 34929 | The real numbers are simpl... |
| iccllysconn 34930 | A closed interval is local... |
| rellysconn 34931 | The real numbers are local... |
| iisconn 34932 | The unit interval is simpl... |
| iillysconn 34933 | The unit interval is local... |
| iinllyconn 34934 | The unit interval is local... |
| fncvm 34937 | Lemma for covering maps. ... |
| cvmscbv 34938 | Change bound variables in ... |
| iscvm 34939 | The property of being a co... |
| cvmtop1 34940 | Reverse closure for a cove... |
| cvmtop2 34941 | Reverse closure for a cove... |
| cvmcn 34942 | A covering map is a contin... |
| cvmcov 34943 | Property of a covering map... |
| cvmsrcl 34944 | Reverse closure for an eve... |
| cvmsi 34945 | One direction of ~ cvmsval... |
| cvmsval 34946 | Elementhood in the set ` S... |
| cvmsss 34947 | An even covering is a subs... |
| cvmsn0 34948 | An even covering is nonemp... |
| cvmsuni 34949 | An even covering of ` U ` ... |
| cvmsdisj 34950 | An even covering of ` U ` ... |
| cvmshmeo 34951 | Every element of an even c... |
| cvmsf1o 34952 | ` F ` , localized to an el... |
| cvmscld 34953 | The sets of an even coveri... |
| cvmsss2 34954 | An open subset of an evenl... |
| cvmcov2 34955 | The covering map property ... |
| cvmseu 34956 | Every element in ` U. T ` ... |
| cvmsiota 34957 | Identify the unique elemen... |
| cvmopnlem 34958 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 34959 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 34960 | A covering map is an open ... |
| cvmliftmolem1 34961 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 34962 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 34963 | A lift of a continuous fun... |
| cvmliftmo 34964 | A lift of a continuous fun... |
| cvmliftlem1 34965 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 34966 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 34967 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 34968 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 34969 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 34970 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 34971 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 34972 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 34973 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 34974 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 34975 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 34976 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 34977 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 34978 | Lemma for ~ cvmlift . Dis... |
| cvmlift 34979 | One of the important prope... |
| cvmfo 34980 | A covering map is an onto ... |
| cvmliftiota 34981 | Write out a function ` H `... |
| cvmlift2lem1 34982 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 34983 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 34984 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 34985 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 34986 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 34987 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 34988 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 34989 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 34990 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 34991 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 34992 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 34993 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 34994 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 34995 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 34996 | A two-dimensional version ... |
| cvmliftphtlem 34997 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 34998 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 34999 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 35000 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 35001 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 35002 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 35003 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 35004 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 35005 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 35006 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 35007 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 35008 | A general version of ~ cvm... |
| snmlff 35009 | The function ` F ` from ~ ... |
| snmlfval 35010 | The function ` F ` from ~ ... |
| snmlval 35011 | The property " ` A ` is si... |
| snmlflim 35012 | If ` A ` is simply normal,... |
| goel 35027 | A "Godel-set of membership... |
| goelel3xp 35028 | A "Godel-set of membership... |
| goeleq12bg 35029 | Two "Godel-set of membersh... |
| gonafv 35030 | The "Godel-set for the She... |
| goaleq12d 35031 | Equality of the "Godel-set... |
| gonanegoal 35032 | The Godel-set for the Shef... |
| satf 35033 | The satisfaction predicate... |
| satfsucom 35034 | The satisfaction predicate... |
| satfn 35035 | The satisfaction predicate... |
| satom 35036 | The satisfaction predicate... |
| satfvsucom 35037 | The satisfaction predicate... |
| satfv0 35038 | The value of the satisfact... |
| satfvsuclem1 35039 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 35040 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 35041 | The value of the satisfact... |
| satfv1lem 35042 | Lemma for ~ satfv1 . (Con... |
| satfv1 35043 | The value of the satisfact... |
| satfsschain 35044 | The binary relation of a s... |
| satfvsucsuc 35045 | The satisfaction predicate... |
| satfbrsuc 35046 | The binary relation of a s... |
| satfrel 35047 | The value of the satisfact... |
| satfdmlem 35048 | Lemma for ~ satfdm . (Con... |
| satfdm 35049 | The domain of the satisfac... |
| satfrnmapom 35050 | The range of the satisfact... |
| satfv0fun 35051 | The value of the satisfact... |
| satf0 35052 | The satisfaction predicate... |
| satf0sucom 35053 | The satisfaction predicate... |
| satf00 35054 | The value of the satisfact... |
| satf0suclem 35055 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 35056 | The value of the satisfact... |
| satf0op 35057 | An element of a value of t... |
| satf0n0 35058 | The value of the satisfact... |
| sat1el2xp 35059 | The first component of an ... |
| fmlafv 35060 | The valid Godel formulas o... |
| fmla 35061 | The set of all valid Godel... |
| fmla0 35062 | The valid Godel formulas o... |
| fmla0xp 35063 | The valid Godel formulas o... |
| fmlasuc0 35064 | The valid Godel formulas o... |
| fmlafvel 35065 | A class is a valid Godel f... |
| fmlasuc 35066 | The valid Godel formulas o... |
| fmla1 35067 | The valid Godel formulas o... |
| isfmlasuc 35068 | The characterization of a ... |
| fmlasssuc 35069 | The Godel formulas of heig... |
| fmlaomn0 35070 | The empty set is not a God... |
| fmlan0 35071 | The empty set is not a God... |
| gonan0 35072 | The "Godel-set of NAND" is... |
| goaln0 35073 | The "Godel-set of universa... |
| gonarlem 35074 | Lemma for ~ gonar (inducti... |
| gonar 35075 | If the "Godel-set of NAND"... |
| goalrlem 35076 | Lemma for ~ goalr (inducti... |
| goalr 35077 | If the "Godel-set of unive... |
| fmla0disjsuc 35078 | The set of valid Godel for... |
| fmlasucdisj 35079 | The valid Godel formulas o... |
| satfdmfmla 35080 | The domain of the satisfac... |
| satffunlem 35081 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 35082 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 35083 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 35084 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 35085 | The domain of an ordered p... |
| satffunlem2lem2 35086 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 35087 | Lemma 1 for ~ satffun : in... |
| satffunlem2 35088 | Lemma 2 for ~ satffun : in... |
| satffun 35089 | The value of the satisfact... |
| satff 35090 | The satisfaction predicate... |
| satfun 35091 | The satisfaction predicate... |
| satfvel 35092 | An element of the value of... |
| satfv0fvfmla0 35093 | The value of the satisfact... |
| satefv 35094 | The simplified satisfactio... |
| sate0 35095 | The simplified satisfactio... |
| satef 35096 | The simplified satisfactio... |
| sate0fv0 35097 | A simplified satisfaction ... |
| satefvfmla0 35098 | The simplified satisfactio... |
| sategoelfvb 35099 | Characterization of a valu... |
| sategoelfv 35100 | Condition of a valuation `... |
| ex-sategoelel 35101 | Example of a valuation of ... |
| ex-sategoel 35102 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 35103 | The value of the satisfact... |
| 2goelgoanfmla1 35104 | Two Godel-sets of membersh... |
| satefvfmla1 35105 | The simplified satisfactio... |
| ex-sategoelelomsuc 35106 | Example of a valuation of ... |
| ex-sategoelel12 35107 | Example of a valuation of ... |
| prv 35108 | The "proves" relation on a... |
| elnanelprv 35109 | The wff ` ( A e. B -/\ B e... |
| prv0 35110 | Every wff encoded as ` U `... |
| prv1n 35111 | No wff encoded as a Godel-... |
| mvtval 35180 | The set of variable typeco... |
| mrexval 35181 | The set of "raw expression... |
| mexval 35182 | The set of expressions, wh... |
| mexval2 35183 | The set of expressions, wh... |
| mdvval 35184 | The set of disjoint variab... |
| mvrsval 35185 | The set of variables in an... |
| mvrsfpw 35186 | The set of variables in an... |
| mrsubffval 35187 | The substitution of some v... |
| mrsubfval 35188 | The substitution of some v... |
| mrsubval 35189 | The substitution of some v... |
| mrsubcv 35190 | The value of a substituted... |
| mrsubvr 35191 | The value of a substituted... |
| mrsubff 35192 | A substitution is a functi... |
| mrsubrn 35193 | Although it is defined for... |
| mrsubff1 35194 | When restricted to complet... |
| mrsubff1o 35195 | When restricted to complet... |
| mrsub0 35196 | The value of the substitut... |
| mrsubf 35197 | A substitution is a functi... |
| mrsubccat 35198 | Substitution distributes o... |
| mrsubcn 35199 | A substitution does not ch... |
| elmrsubrn 35200 | Characterization of the su... |
| mrsubco 35201 | The composition of two sub... |
| mrsubvrs 35202 | The set of variables in a ... |
| msubffval 35203 | A substitution applied to ... |
| msubfval 35204 | A substitution applied to ... |
| msubval 35205 | A substitution applied to ... |
| msubrsub 35206 | A substitution applied to ... |
| msubty 35207 | The type of a substituted ... |
| elmsubrn 35208 | Characterization of substi... |
| msubrn 35209 | Although it is defined for... |
| msubff 35210 | A substitution is a functi... |
| msubco 35211 | The composition of two sub... |
| msubf 35212 | A substitution is a functi... |
| mvhfval 35213 | Value of the function mapp... |
| mvhval 35214 | Value of the function mapp... |
| mpstval 35215 | A pre-statement is an orde... |
| elmpst 35216 | Property of being a pre-st... |
| msrfval 35217 | Value of the reduct of a p... |
| msrval 35218 | Value of the reduct of a p... |
| mpstssv 35219 | A pre-statement is an orde... |
| mpst123 35220 | Decompose a pre-statement ... |
| mpstrcl 35221 | The elements of a pre-stat... |
| msrf 35222 | The reduct of a pre-statem... |
| msrrcl 35223 | If ` X ` and ` Y ` have th... |
| mstaval 35224 | Value of the set of statem... |
| msrid 35225 | The reduct of a statement ... |
| msrfo 35226 | The reduct of a pre-statem... |
| mstapst 35227 | A statement is a pre-state... |
| elmsta 35228 | Property of being a statem... |
| ismfs 35229 | A formal system is a tuple... |
| mfsdisj 35230 | The constants and variable... |
| mtyf2 35231 | The type function maps var... |
| mtyf 35232 | The type function maps var... |
| mvtss 35233 | The set of variable typeco... |
| maxsta 35234 | An axiom is a statement. ... |
| mvtinf 35235 | Each variable typecode has... |
| msubff1 35236 | When restricted to complet... |
| msubff1o 35237 | When restricted to complet... |
| mvhf 35238 | The function mapping varia... |
| mvhf1 35239 | The function mapping varia... |
| msubvrs 35240 | The set of variables in a ... |
| mclsrcl 35241 | Reverse closure for the cl... |
| mclsssvlem 35242 | Lemma for ~ mclsssv . (Co... |
| mclsval 35243 | The function mapping varia... |
| mclsssv 35244 | The closure of a set of ex... |
| ssmclslem 35245 | Lemma for ~ ssmcls . (Con... |
| vhmcls 35246 | All variable hypotheses ar... |
| ssmcls 35247 | The original expressions a... |
| ss2mcls 35248 | The closure is monotonic u... |
| mclsax 35249 | The closure is closed unde... |
| mclsind 35250 | Induction theorem for clos... |
| mppspstlem 35251 | Lemma for ~ mppspst . (Co... |
| mppsval 35252 | Definition of a provable p... |
| elmpps 35253 | Definition of a provable p... |
| mppspst 35254 | A provable pre-statement i... |
| mthmval 35255 | A theorem is a pre-stateme... |
| elmthm 35256 | A theorem is a pre-stateme... |
| mthmi 35257 | A statement whose reduct i... |
| mthmsta 35258 | A theorem is a pre-stateme... |
| mppsthm 35259 | A provable pre-statement i... |
| mthmblem 35260 | Lemma for ~ mthmb . (Cont... |
| mthmb 35261 | If two statements have the... |
| mthmpps 35262 | Given a theorem, there is ... |
| mclsppslem 35263 | The closure is closed unde... |
| mclspps 35264 | The closure is closed unde... |
| problem1 35339 | Practice problem 1. Clues... |
| problem2 35340 | Practice problem 2. Clues... |
| problem3 35341 | Practice problem 3. Clues... |
| problem4 35342 | Practice problem 4. Clues... |
| problem5 35343 | Practice problem 5. Clues... |
| quad3 35344 | Variant of quadratic equat... |
| climuzcnv 35345 | Utility lemma to convert b... |
| sinccvglem 35346 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 35347 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 35348 | The circumference of a cir... |
| elfzm12 35349 | Membership in a curtailed ... |
| nn0seqcvg 35350 | A strictly-decreasing nonn... |
| lediv2aALT 35351 | Division of both sides of ... |
| abs2sqlei 35352 | The absolute values of two... |
| abs2sqlti 35353 | The absolute values of two... |
| abs2sqle 35354 | The absolute values of two... |
| abs2sqlt 35355 | The absolute values of two... |
| abs2difi 35356 | Difference of absolute val... |
| abs2difabsi 35357 | Absolute value of differen... |
| currybi 35358 | Biconditional version of C... |
| axextprim 35365 | ~ ax-ext without distinct ... |
| axrepprim 35366 | ~ ax-rep without distinct ... |
| axunprim 35367 | ~ ax-un without distinct v... |
| axpowprim 35368 | ~ ax-pow without distinct ... |
| axregprim 35369 | ~ ax-reg without distinct ... |
| axinfprim 35370 | ~ ax-inf without distinct ... |
| axacprim 35371 | ~ ax-ac without distinct v... |
| untelirr 35372 | We call a class "untanged"... |
| untuni 35373 | The union of a class is un... |
| untsucf 35374 | If a class is untangled, t... |
| unt0 35375 | The null set is untangled.... |
| untint 35376 | If there is an untangled e... |
| efrunt 35377 | If ` A ` is well-founded b... |
| untangtr 35378 | A transitive class is unta... |
| 3jaodd 35379 | Double deduction form of ~... |
| 3orit 35380 | Closed form of ~ 3ori . (... |
| biimpexp 35381 | A biconditional in the ant... |
| nepss 35382 | Two classes are unequal if... |
| 3ccased 35383 | Triple disjunction form of... |
| dfso3 35384 | Expansion of the definitio... |
| brtpid1 35385 | A binary relation involvin... |
| brtpid2 35386 | A binary relation involvin... |
| brtpid3 35387 | A binary relation involvin... |
| iota5f 35388 | A method for computing iot... |
| jath 35389 | Closed form of ~ ja . Pro... |
| xpab 35390 | Cartesian product of two c... |
| nnuni 35391 | The union of a finite ordi... |
| sqdivzi 35392 | Distribution of square ove... |
| supfz 35393 | The supremum of a finite s... |
| inffz 35394 | The infimum of a finite se... |
| fz0n 35395 | The sequence ` ( 0 ... ( N... |
| shftvalg 35396 | Value of a sequence shifte... |
| divcnvlin 35397 | Limit of the ratio of two ... |
| climlec3 35398 | Comparison of a constant t... |
| iexpire 35399 | ` _i ` raised to itself is... |
| bcneg1 35400 | The binomial coefficent ov... |
| bcm1nt 35401 | The proportion of one bion... |
| bcprod 35402 | A product identity for bin... |
| bccolsum 35403 | A column-sum rule for bino... |
| iprodefisumlem 35404 | Lemma for ~ iprodefisum . ... |
| iprodefisum 35405 | Applying the exponential f... |
| iprodgam 35406 | An infinite product versio... |
| faclimlem1 35407 | Lemma for ~ faclim . Clos... |
| faclimlem2 35408 | Lemma for ~ faclim . Show... |
| faclimlem3 35409 | Lemma for ~ faclim . Alge... |
| faclim 35410 | An infinite product expres... |
| iprodfac 35411 | An infinite product expres... |
| faclim2 35412 | Another factorial limit du... |
| gcd32 35413 | Swap the second and third ... |
| gcdabsorb 35414 | Absorption law for gcd. (... |
| dftr6 35415 | A potential definition of ... |
| coep 35416 | Composition with the membe... |
| coepr 35417 | Composition with the conve... |
| dffr5 35418 | A quantifier-free definiti... |
| dfso2 35419 | Quantifier-free definition... |
| br8 35420 | Substitution for an eight-... |
| br6 35421 | Substitution for a six-pla... |
| br4 35422 | Substitution for a four-pl... |
| cnvco1 35423 | Another distributive law o... |
| cnvco2 35424 | Another distributive law o... |
| eldm3 35425 | Quantifier-free definition... |
| elrn3 35426 | Quantifier-free definition... |
| pocnv 35427 | The converse of a partial ... |
| socnv 35428 | The converse of a strict o... |
| sotrd 35429 | Transitivity law for stric... |
| elintfv 35430 | Membership in an intersect... |
| funpsstri 35431 | A condition for subset tri... |
| fundmpss 35432 | If a class ` F ` is a prop... |
| funsseq 35433 | Given two functions with e... |
| fununiq 35434 | The uniqueness condition o... |
| funbreq 35435 | An equality condition for ... |
| br1steq 35436 | Uniqueness condition for t... |
| br2ndeq 35437 | Uniqueness condition for t... |
| dfdm5 35438 | Definition of domain in te... |
| dfrn5 35439 | Definition of range in ter... |
| opelco3 35440 | Alternate way of saying th... |
| elima4 35441 | Quantifier-free expression... |
| fv1stcnv 35442 | The value of the converse ... |
| fv2ndcnv 35443 | The value of the converse ... |
| setinds 35444 | Principle of set induction... |
| setinds2f 35445 | ` _E ` induction schema, u... |
| setinds2 35446 | ` _E ` induction schema, u... |
| elpotr 35447 | A class of transitive sets... |
| dford5reg 35448 | Given ~ ax-reg , an ordina... |
| dfon2lem1 35449 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 35450 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 35451 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 35452 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 35453 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 35454 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 35455 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 35456 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 35457 | Lemma for ~ dfon2 . A cla... |
| dfon2 35458 | ` On ` consists of all set... |
| rdgprc0 35459 | The value of the recursive... |
| rdgprc 35460 | The value of the recursive... |
| dfrdg2 35461 | Alternate definition of th... |
| dfrdg3 35462 | Generalization of ~ dfrdg2... |
| axextdfeq 35463 | A version of ~ ax-ext for ... |
| ax8dfeq 35464 | A version of ~ ax-8 for us... |
| axextdist 35465 | ~ ax-ext with distinctors ... |
| axextbdist 35466 | ~ axextb with distinctors ... |
| 19.12b 35467 | Version of ~ 19.12vv with ... |
| exnel 35468 | There is always a set not ... |
| distel 35469 | Distinctors in terms of me... |
| axextndbi 35470 | ~ axextnd as a bicondition... |
| hbntg 35471 | A more general form of ~ h... |
| hbimtg 35472 | A more general and closed ... |
| hbaltg 35473 | A more general and closed ... |
| hbng 35474 | A more general form of ~ h... |
| hbimg 35475 | A more general form of ~ h... |
| wsuceq123 35480 | Equality theorem for well-... |
| wsuceq1 35481 | Equality theorem for well-... |
| wsuceq2 35482 | Equality theorem for well-... |
| wsuceq3 35483 | Equality theorem for well-... |
| nfwsuc 35484 | Bound-variable hypothesis ... |
| wlimeq12 35485 | Equality theorem for the l... |
| wlimeq1 35486 | Equality theorem for the l... |
| wlimeq2 35487 | Equality theorem for the l... |
| nfwlim 35488 | Bound-variable hypothesis ... |
| elwlim 35489 | Membership in the limit cl... |
| wzel 35490 | The zero of a well-founded... |
| wsuclem 35491 | Lemma for the supremum pro... |
| wsucex 35492 | Existence theorem for well... |
| wsuccl 35493 | If ` X ` is a set with an ... |
| wsuclb 35494 | A well-founded successor i... |
| wlimss 35495 | The class of limit points ... |
| txpss3v 35544 | A tail Cartesian product i... |
| txprel 35545 | A tail Cartesian product i... |
| brtxp 35546 | Characterize a ternary rel... |
| brtxp2 35547 | The binary relation over a... |
| dfpprod2 35548 | Expanded definition of par... |
| pprodcnveq 35549 | A converse law for paralle... |
| pprodss4v 35550 | The parallel product is a ... |
| brpprod 35551 | Characterize a quaternary ... |
| brpprod3a 35552 | Condition for parallel pro... |
| brpprod3b 35553 | Condition for parallel pro... |
| relsset 35554 | The subset class is a bina... |
| brsset 35555 | For sets, the ` SSet ` bin... |
| idsset 35556 | ` _I ` is equal to the int... |
| eltrans 35557 | Membership in the class of... |
| dfon3 35558 | A quantifier-free definiti... |
| dfon4 35559 | Another quantifier-free de... |
| brtxpsd 35560 | Expansion of a common form... |
| brtxpsd2 35561 | Another common abbreviatio... |
| brtxpsd3 35562 | A third common abbreviatio... |
| relbigcup 35563 | The ` Bigcup ` relationshi... |
| brbigcup 35564 | Binary relation over ` Big... |
| dfbigcup2 35565 | ` Bigcup ` using maps-to n... |
| fobigcup 35566 | ` Bigcup ` maps the univer... |
| fnbigcup 35567 | ` Bigcup ` is a function o... |
| fvbigcup 35568 | For sets, ` Bigcup ` yield... |
| elfix 35569 | Membership in the fixpoint... |
| elfix2 35570 | Alternative membership in ... |
| dffix2 35571 | The fixpoints of a class i... |
| fixssdm 35572 | The fixpoints of a class a... |
| fixssrn 35573 | The fixpoints of a class a... |
| fixcnv 35574 | The fixpoints of a class a... |
| fixun 35575 | The fixpoint operator dist... |
| ellimits 35576 | Membership in the class of... |
| limitssson 35577 | The class of all limit ord... |
| dfom5b 35578 | A quantifier-free definiti... |
| sscoid 35579 | A condition for subset and... |
| dffun10 35580 | Another potential definiti... |
| elfuns 35581 | Membership in the class of... |
| elfunsg 35582 | Closed form of ~ elfuns . ... |
| brsingle 35583 | The binary relation form o... |
| elsingles 35584 | Membership in the class of... |
| fnsingle 35585 | The singleton relationship... |
| fvsingle 35586 | The value of the singleton... |
| dfsingles2 35587 | Alternate definition of th... |
| snelsingles 35588 | A singleton is a member of... |
| dfiota3 35589 | A definition of iota using... |
| dffv5 35590 | Another quantifier-free de... |
| unisnif 35591 | Express union of singleton... |
| brimage 35592 | Binary relation form of th... |
| brimageg 35593 | Closed form of ~ brimage .... |
| funimage 35594 | ` Image A ` is a function.... |
| fnimage 35595 | ` Image R ` is a function ... |
| imageval 35596 | The image functor in maps-... |
| fvimage 35597 | Value of the image functor... |
| brcart 35598 | Binary relation form of th... |
| brdomain 35599 | Binary relation form of th... |
| brrange 35600 | Binary relation form of th... |
| brdomaing 35601 | Closed form of ~ brdomain ... |
| brrangeg 35602 | Closed form of ~ brrange .... |
| brimg 35603 | Binary relation form of th... |
| brapply 35604 | Binary relation form of th... |
| brcup 35605 | Binary relation form of th... |
| brcap 35606 | Binary relation form of th... |
| brsuccf 35607 | Binary relation form of th... |
| funpartlem 35608 | Lemma for ~ funpartfun . ... |
| funpartfun 35609 | The functional part of ` F... |
| funpartss 35610 | The functional part of ` F... |
| funpartfv 35611 | The function value of the ... |
| fullfunfnv 35612 | The full functional part o... |
| fullfunfv 35613 | The function value of the ... |
| brfullfun 35614 | A binary relation form con... |
| brrestrict 35615 | Binary relation form of th... |
| dfrecs2 35616 | A quantifier-free definiti... |
| dfrdg4 35617 | A quantifier-free definiti... |
| dfint3 35618 | Quantifier-free definition... |
| imagesset 35619 | The Image functor applied ... |
| brub 35620 | Binary relation form of th... |
| brlb 35621 | Binary relation form of th... |
| altopex 35626 | Alternative ordered pairs ... |
| altopthsn 35627 | Two alternate ordered pair... |
| altopeq12 35628 | Equality for alternate ord... |
| altopeq1 35629 | Equality for alternate ord... |
| altopeq2 35630 | Equality for alternate ord... |
| altopth1 35631 | Equality of the first memb... |
| altopth2 35632 | Equality of the second mem... |
| altopthg 35633 | Alternate ordered pair the... |
| altopthbg 35634 | Alternate ordered pair the... |
| altopth 35635 | The alternate ordered pair... |
| altopthb 35636 | Alternate ordered pair the... |
| altopthc 35637 | Alternate ordered pair the... |
| altopthd 35638 | Alternate ordered pair the... |
| altxpeq1 35639 | Equality for alternate Car... |
| altxpeq2 35640 | Equality for alternate Car... |
| elaltxp 35641 | Membership in alternate Ca... |
| altopelaltxp 35642 | Alternate ordered pair mem... |
| altxpsspw 35643 | An inclusion rule for alte... |
| altxpexg 35644 | The alternate Cartesian pr... |
| rankaltopb 35645 | Compute the rank of an alt... |
| nfaltop 35646 | Bound-variable hypothesis ... |
| sbcaltop 35647 | Distribution of class subs... |
| cgrrflx2d 35650 | Deduction form of ~ axcgrr... |
| cgrtr4d 35651 | Deduction form of ~ axcgrt... |
| cgrtr4and 35652 | Deduction form of ~ axcgrt... |
| cgrrflx 35653 | Reflexivity law for congru... |
| cgrrflxd 35654 | Deduction form of ~ cgrrfl... |
| cgrcomim 35655 | Congruence commutes on the... |
| cgrcom 35656 | Congruence commutes betwee... |
| cgrcomand 35657 | Deduction form of ~ cgrcom... |
| cgrtr 35658 | Transitivity law for congr... |
| cgrtrand 35659 | Deduction form of ~ cgrtr ... |
| cgrtr3 35660 | Transitivity law for congr... |
| cgrtr3and 35661 | Deduction form of ~ cgrtr3... |
| cgrcoml 35662 | Congruence commutes on the... |
| cgrcomr 35663 | Congruence commutes on the... |
| cgrcomlr 35664 | Congruence commutes on bot... |
| cgrcomland 35665 | Deduction form of ~ cgrcom... |
| cgrcomrand 35666 | Deduction form of ~ cgrcom... |
| cgrcomlrand 35667 | Deduction form of ~ cgrcom... |
| cgrtriv 35668 | Degenerate segments are co... |
| cgrid2 35669 | Identity law for congruenc... |
| cgrdegen 35670 | Two congruent segments are... |
| brofs 35671 | Binary relation form of th... |
| 5segofs 35672 | Rephrase ~ ax5seg using th... |
| ofscom 35673 | The outer five segment pre... |
| cgrextend 35674 | Link congruence over a pai... |
| cgrextendand 35675 | Deduction form of ~ cgrext... |
| segconeq 35676 | Two points that satisfy th... |
| segconeu 35677 | Existential uniqueness ver... |
| btwntriv2 35678 | Betweenness always holds f... |
| btwncomim 35679 | Betweenness commutes. Imp... |
| btwncom 35680 | Betweenness commutes. (Co... |
| btwncomand 35681 | Deduction form of ~ btwnco... |
| btwntriv1 35682 | Betweenness always holds f... |
| btwnswapid 35683 | If you can swap the first ... |
| btwnswapid2 35684 | If you can swap arguments ... |
| btwnintr 35685 | Inner transitivity law for... |
| btwnexch3 35686 | Exchange the first endpoin... |
| btwnexch3and 35687 | Deduction form of ~ btwnex... |
| btwnouttr2 35688 | Outer transitivity law for... |
| btwnexch2 35689 | Exchange the outer point o... |
| btwnouttr 35690 | Outer transitivity law for... |
| btwnexch 35691 | Outer transitivity law for... |
| btwnexchand 35692 | Deduction form of ~ btwnex... |
| btwndiff 35693 | There is always a ` c ` di... |
| trisegint 35694 | A line segment between two... |
| funtransport 35697 | The ` TransportTo ` relati... |
| fvtransport 35698 | Calculate the value of the... |
| transportcl 35699 | Closure law for segment tr... |
| transportprops 35700 | Calculate the defining pro... |
| brifs 35709 | Binary relation form of th... |
| ifscgr 35710 | Inner five segment congrue... |
| cgrsub 35711 | Removing identical parts f... |
| brcgr3 35712 | Binary relation form of th... |
| cgr3permute3 35713 | Permutation law for three-... |
| cgr3permute1 35714 | Permutation law for three-... |
| cgr3permute2 35715 | Permutation law for three-... |
| cgr3permute4 35716 | Permutation law for three-... |
| cgr3permute5 35717 | Permutation law for three-... |
| cgr3tr4 35718 | Transitivity law for three... |
| cgr3com 35719 | Commutativity law for thre... |
| cgr3rflx 35720 | Identity law for three-pla... |
| cgrxfr 35721 | A line segment can be divi... |
| btwnxfr 35722 | A condition for extending ... |
| colinrel 35723 | Colinearity is a relations... |
| brcolinear2 35724 | Alternate colinearity bina... |
| brcolinear 35725 | The binary relation form o... |
| colinearex 35726 | The colinear predicate exi... |
| colineardim1 35727 | If ` A ` is colinear with ... |
| colinearperm1 35728 | Permutation law for coline... |
| colinearperm3 35729 | Permutation law for coline... |
| colinearperm2 35730 | Permutation law for coline... |
| colinearperm4 35731 | Permutation law for coline... |
| colinearperm5 35732 | Permutation law for coline... |
| colineartriv1 35733 | Trivial case of colinearit... |
| colineartriv2 35734 | Trivial case of colinearit... |
| btwncolinear1 35735 | Betweenness implies coline... |
| btwncolinear2 35736 | Betweenness implies coline... |
| btwncolinear3 35737 | Betweenness implies coline... |
| btwncolinear4 35738 | Betweenness implies coline... |
| btwncolinear5 35739 | Betweenness implies coline... |
| btwncolinear6 35740 | Betweenness implies coline... |
| colinearxfr 35741 | Transfer law for colineari... |
| lineext 35742 | Extend a line with a missi... |
| brofs2 35743 | Change some conditions for... |
| brifs2 35744 | Change some conditions for... |
| brfs 35745 | Binary relation form of th... |
| fscgr 35746 | Congruence law for the gen... |
| linecgr 35747 | Congruence rule for lines.... |
| linecgrand 35748 | Deduction form of ~ linecg... |
| lineid 35749 | Identity law for points on... |
| idinside 35750 | Law for finding a point in... |
| endofsegid 35751 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 35752 | Deduction form of ~ endofs... |
| btwnconn1lem1 35753 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 35754 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 35755 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 35756 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 35757 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 35758 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 35759 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 35760 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 35761 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 35762 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 35763 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 35764 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 35765 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 35766 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 35767 | Connectitivy law for betwe... |
| btwnconn2 35768 | Another connectivity law f... |
| btwnconn3 35769 | Inner connectivity law for... |
| midofsegid 35770 | If two points fall in the ... |
| segcon2 35771 | Generalization of ~ axsegc... |
| brsegle 35774 | Binary relation form of th... |
| brsegle2 35775 | Alternate characterization... |
| seglecgr12im 35776 | Substitution law for segme... |
| seglecgr12 35777 | Substitution law for segme... |
| seglerflx 35778 | Segment comparison is refl... |
| seglemin 35779 | Any segment is at least as... |
| segletr 35780 | Segment less than is trans... |
| segleantisym 35781 | Antisymmetry law for segme... |
| seglelin 35782 | Linearity law for segment ... |
| btwnsegle 35783 | If ` B ` falls between ` A... |
| colinbtwnle 35784 | Given three colinear point... |
| broutsideof 35787 | Binary relation form of ` ... |
| broutsideof2 35788 | Alternate form of ` Outsid... |
| outsidene1 35789 | Outsideness implies inequa... |
| outsidene2 35790 | Outsideness implies inequa... |
| btwnoutside 35791 | A principle linking outsid... |
| broutsideof3 35792 | Characterization of outsid... |
| outsideofrflx 35793 | Reflexivity of outsideness... |
| outsideofcom 35794 | Commutativity law for outs... |
| outsideoftr 35795 | Transitivity law for outsi... |
| outsideofeq 35796 | Uniqueness law for ` Outsi... |
| outsideofeu 35797 | Given a nondegenerate ray,... |
| outsidele 35798 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 35799 | Outside of implies colinea... |
| funray 35806 | Show that the ` Ray ` rela... |
| fvray 35807 | Calculate the value of the... |
| funline 35808 | Show that the ` Line ` rel... |
| linedegen 35809 | When ` Line ` is applied w... |
| fvline 35810 | Calculate the value of the... |
| liness 35811 | A line is a subset of the ... |
| fvline2 35812 | Alternate definition of a ... |
| lineunray 35813 | A line is composed of a po... |
| lineelsb2 35814 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 35815 | Reflexivity law for line m... |
| linecom 35816 | Commutativity law for line... |
| linerflx2 35817 | Reflexivity law for line m... |
| ellines 35818 | Membership in the set of a... |
| linethru 35819 | If ` A ` is a line contain... |
| hilbert1.1 35820 | There is a line through an... |
| hilbert1.2 35821 | There is at most one line ... |
| linethrueu 35822 | There is a unique line goi... |
| lineintmo 35823 | Two distinct lines interse... |
| fwddifval 35828 | Calculate the value of the... |
| fwddifnval 35829 | The value of the forward d... |
| fwddifn0 35830 | The value of the n-iterate... |
| fwddifnp1 35831 | The value of the n-iterate... |
| rankung 35832 | The rank of the union of t... |
| ranksng 35833 | The rank of a singleton. ... |
| rankelg 35834 | The membership relation is... |
| rankpwg 35835 | The rank of a power set. ... |
| rank0 35836 | The rank of the empty set ... |
| rankeq1o 35837 | The only set with rank ` 1... |
| elhf 35840 | Membership in the heredita... |
| elhf2 35841 | Alternate form of membersh... |
| elhf2g 35842 | Hereditarily finiteness vi... |
| 0hf 35843 | The empty set is a heredit... |
| hfun 35844 | The union of two HF sets i... |
| hfsn 35845 | The singleton of an HF set... |
| hfadj 35846 | Adjoining one HF element t... |
| hfelhf 35847 | Any member of an HF set is... |
| hftr 35848 | The class of all hereditar... |
| hfext 35849 | Extensionality for HF sets... |
| hfuni 35850 | The union of an HF set is ... |
| hfpw 35851 | The power class of an HF s... |
| hfninf 35852 | ` _om ` is not hereditaril... |
| mpomulnzcnf 35853 | Multiplication maps nonzer... |
| a1i14 35854 | Add two antecedents to a w... |
| a1i24 35855 | Add two antecedents to a w... |
| exp5d 35856 | An exportation inference. ... |
| exp5g 35857 | An exportation inference. ... |
| exp5k 35858 | An exportation inference. ... |
| exp56 35859 | An exportation inference. ... |
| exp58 35860 | An exportation inference. ... |
| exp510 35861 | An exportation inference. ... |
| exp511 35862 | An exportation inference. ... |
| exp512 35863 | An exportation inference. ... |
| 3com12d 35864 | Commutation in consequent.... |
| imp5p 35865 | A triple importation infer... |
| imp5q 35866 | A triple importation infer... |
| ecase13d 35867 | Deduction for elimination ... |
| subtr 35868 | Transitivity of implicit s... |
| subtr2 35869 | Transitivity of implicit s... |
| trer 35870 | A relation intersected wit... |
| elicc3 35871 | An equivalent membership c... |
| finminlem 35872 | A useful lemma about finit... |
| gtinf 35873 | Any number greater than an... |
| opnrebl 35874 | A set is open in the stand... |
| opnrebl2 35875 | A set is open in the stand... |
| nn0prpwlem 35876 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 35877 | Two nonnegative integers a... |
| topbnd 35878 | Two equivalent expressions... |
| opnbnd 35879 | A set is open iff it is di... |
| cldbnd 35880 | A set is closed iff it con... |
| ntruni 35881 | A union of interiors is a ... |
| clsun 35882 | A pairwise union of closur... |
| clsint2 35883 | The closure of an intersec... |
| opnregcld 35884 | A set is regularly closed ... |
| cldregopn 35885 | A set if regularly open if... |
| neiin 35886 | Two neighborhoods intersec... |
| hmeoclda 35887 | Homeomorphisms preserve cl... |
| hmeocldb 35888 | Homeomorphisms preserve cl... |
| ivthALT 35889 | An alternate proof of the ... |
| fnerel 35892 | Fineness is a relation. (... |
| isfne 35893 | The predicate " ` B ` is f... |
| isfne4 35894 | The predicate " ` B ` is f... |
| isfne4b 35895 | A condition for a topology... |
| isfne2 35896 | The predicate " ` B ` is f... |
| isfne3 35897 | The predicate " ` B ` is f... |
| fnebas 35898 | A finer cover covers the s... |
| fnetg 35899 | A finer cover generates a ... |
| fnessex 35900 | If ` B ` is finer than ` A... |
| fneuni 35901 | If ` B ` is finer than ` A... |
| fneint 35902 | If a cover is finer than a... |
| fness 35903 | A cover is finer than its ... |
| fneref 35904 | Reflexivity of the finenes... |
| fnetr 35905 | Transitivity of the finene... |
| fneval 35906 | Two covers are finer than ... |
| fneer 35907 | Fineness intersected with ... |
| topfne 35908 | Fineness for covers corres... |
| topfneec 35909 | A cover is equivalent to a... |
| topfneec2 35910 | A topology is precisely id... |
| fnessref 35911 | A cover is finer iff it ha... |
| refssfne 35912 | A cover is a refinement if... |
| neibastop1 35913 | A collection of neighborho... |
| neibastop2lem 35914 | Lemma for ~ neibastop2 . ... |
| neibastop2 35915 | In the topology generated ... |
| neibastop3 35916 | The topology generated by ... |
| topmtcl 35917 | The meet of a collection o... |
| topmeet 35918 | Two equivalent formulation... |
| topjoin 35919 | Two equivalent formulation... |
| fnemeet1 35920 | The meet of a collection o... |
| fnemeet2 35921 | The meet of equivalence cl... |
| fnejoin1 35922 | Join of equivalence classe... |
| fnejoin2 35923 | Join of equivalence classe... |
| fgmin 35924 | Minimality property of a g... |
| neifg 35925 | The neighborhood filter of... |
| tailfval 35926 | The tail function for a di... |
| tailval 35927 | The tail of an element in ... |
| eltail 35928 | An element of a tail. (Co... |
| tailf 35929 | The tail function of a dir... |
| tailini 35930 | A tail contains its initia... |
| tailfb 35931 | The collection of tails of... |
| filnetlem1 35932 | Lemma for ~ filnet . Chan... |
| filnetlem2 35933 | Lemma for ~ filnet . The ... |
| filnetlem3 35934 | Lemma for ~ filnet . (Con... |
| filnetlem4 35935 | Lemma for ~ filnet . (Con... |
| filnet 35936 | A filter has the same conv... |
| tb-ax1 35937 | The first of three axioms ... |
| tb-ax2 35938 | The second of three axioms... |
| tb-ax3 35939 | The third of three axioms ... |
| tbsyl 35940 | The weak syllogism from Ta... |
| re1ax2lem 35941 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 35942 | ~ ax-2 rederived from the ... |
| naim1 35943 | Constructor theorem for ` ... |
| naim2 35944 | Constructor theorem for ` ... |
| naim1i 35945 | Constructor rule for ` -/\... |
| naim2i 35946 | Constructor rule for ` -/\... |
| naim12i 35947 | Constructor rule for ` -/\... |
| nabi1i 35948 | Constructor rule for ` -/\... |
| nabi2i 35949 | Constructor rule for ` -/\... |
| nabi12i 35950 | Constructor rule for ` -/\... |
| df3nandALT1 35953 | The double nand expressed ... |
| df3nandALT2 35954 | The double nand expressed ... |
| andnand1 35955 | Double and in terms of dou... |
| imnand2 35956 | An ` -> ` nand relation. ... |
| nalfal 35957 | Not all sets hold ` F. ` a... |
| nexntru 35958 | There does not exist a set... |
| nexfal 35959 | There does not exist a set... |
| neufal 35960 | There does not exist exact... |
| neutru 35961 | There does not exist exact... |
| nmotru 35962 | There does not exist at mo... |
| mofal 35963 | There exist at most one se... |
| nrmo 35964 | "At most one" restricted e... |
| meran1 35965 | A single axiom for proposi... |
| meran2 35966 | A single axiom for proposi... |
| meran3 35967 | A single axiom for proposi... |
| waj-ax 35968 | A single axiom for proposi... |
| lukshef-ax2 35969 | A single axiom for proposi... |
| arg-ax 35970 | A single axiom for proposi... |
| negsym1 35971 | In the paper "On Variable ... |
| imsym1 35972 | A symmetry with ` -> ` . ... |
| bisym1 35973 | A symmetry with ` <-> ` . ... |
| consym1 35974 | A symmetry with ` /\ ` . ... |
| dissym1 35975 | A symmetry with ` \/ ` . ... |
| nandsym1 35976 | A symmetry with ` -/\ ` . ... |
| unisym1 35977 | A symmetry with ` A. ` . ... |
| exisym1 35978 | A symmetry with ` E. ` . ... |
| unqsym1 35979 | A symmetry with ` E! ` . ... |
| amosym1 35980 | A symmetry with ` E* ` . ... |
| subsym1 35981 | A symmetry with ` [ x / y ... |
| ontopbas 35982 | An ordinal number is a top... |
| onsstopbas 35983 | The class of ordinal numbe... |
| onpsstopbas 35984 | The class of ordinal numbe... |
| ontgval 35985 | The topology generated fro... |
| ontgsucval 35986 | The topology generated fro... |
| onsuctop 35987 | A successor ordinal number... |
| onsuctopon 35988 | One of the topologies on a... |
| ordtoplem 35989 | Membership of the class of... |
| ordtop 35990 | An ordinal is a topology i... |
| onsucconni 35991 | A successor ordinal number... |
| onsucconn 35992 | A successor ordinal number... |
| ordtopconn 35993 | An ordinal topology is con... |
| onintopssconn 35994 | An ordinal topology is con... |
| onsuct0 35995 | A successor ordinal number... |
| ordtopt0 35996 | An ordinal topology is T_0... |
| onsucsuccmpi 35997 | The successor of a success... |
| onsucsuccmp 35998 | The successor of a success... |
| limsucncmpi 35999 | The successor of a limit o... |
| limsucncmp 36000 | The successor of a limit o... |
| ordcmp 36001 | An ordinal topology is com... |
| ssoninhaus 36002 | The ordinal topologies ` 1... |
| onint1 36003 | The ordinal T_1 spaces are... |
| oninhaus 36004 | The ordinal Hausdorff spac... |
| fveleq 36005 | Please add description her... |
| findfvcl 36006 | Please add description her... |
| findreccl 36007 | Please add description her... |
| findabrcl 36008 | Please add description her... |
| nnssi2 36009 | Convert a theorem for real... |
| nnssi3 36010 | Convert a theorem for real... |
| nndivsub 36011 | Please add description her... |
| nndivlub 36012 | A factor of a positive int... |
| ee7.2aOLD 36015 | Lemma for Euclid's Element... |
| dnival 36016 | Value of the "distance to ... |
| dnicld1 36017 | Closure theorem for the "d... |
| dnicld2 36018 | Closure theorem for the "d... |
| dnif 36019 | The "distance to nearest i... |
| dnizeq0 36020 | The distance to nearest in... |
| dnizphlfeqhlf 36021 | The distance to nearest in... |
| rddif2 36022 | Variant of ~ rddif . (Con... |
| dnibndlem1 36023 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 36024 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 36025 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 36026 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 36027 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 36028 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 36029 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 36030 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 36031 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 36032 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 36033 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 36034 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 36035 | Lemma for ~ dnibnd . (Con... |
| dnibnd 36036 | The "distance to nearest i... |
| dnicn 36037 | The "distance to nearest i... |
| knoppcnlem1 36038 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 36039 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 36040 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 36041 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 36042 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 36043 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 36044 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 36045 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 36046 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 36047 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 36048 | Lemma for ~ knoppcn . (Co... |
| knoppcn 36049 | The continuous nowhere dif... |
| knoppcld 36050 | Closure theorem for Knopp'... |
| unblimceq0lem 36051 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 36052 | If ` F ` is unbounded near... |
| unbdqndv1 36053 | If the difference quotient... |
| unbdqndv2lem1 36054 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 36055 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 36056 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 36057 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 36058 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 36059 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 36060 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 36061 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 36062 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 36063 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 36064 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 36065 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 36066 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 36067 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 36068 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 36069 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 36070 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 36071 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 36072 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 36073 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 36074 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 36075 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 36076 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 36077 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 36078 | Lemma for ~ knoppndv . (C... |
| knoppndv 36079 | The continuous nowhere dif... |
| knoppf 36080 | Knopp's function is a func... |
| knoppcn2 36081 | Variant of ~ knoppcn with ... |
| cnndvlem1 36082 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 36083 | Lemma for ~ cnndv . (Cont... |
| cnndv 36084 | There exists a continuous ... |
| bj-mp2c 36085 | A double modus ponens infe... |
| bj-mp2d 36086 | A double modus ponens infe... |
| bj-0 36087 | A syntactic theorem. See ... |
| bj-1 36088 | In this proof, the use of ... |
| bj-a1k 36089 | Weakening of ~ ax-1 . As ... |
| bj-poni 36090 | Inference associated with ... |
| bj-nnclav 36091 | When ` F. ` is substituted... |
| bj-nnclavi 36092 | Inference associated with ... |
| bj-nnclavc 36093 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 36094 | Inference associated with ... |
| bj-jarrii 36095 | Inference associated with ... |
| bj-imim21 36096 | The propositional function... |
| bj-imim21i 36097 | Inference associated with ... |
| bj-peircestab 36098 | Over minimal implicational... |
| bj-stabpeirce 36099 | This minimal implicational... |
| bj-syl66ib 36100 | A mixed syllogism inferenc... |
| bj-orim2 36101 | Proof of ~ orim2 from the ... |
| bj-currypeirce 36102 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 36103 | Peirce's axiom ~ peirce im... |
| bj-animbi 36104 | Conjunction in terms of im... |
| bj-currypara 36105 | Curry's paradox. Note tha... |
| bj-con2com 36106 | A commuted form of the con... |
| bj-con2comi 36107 | Inference associated with ... |
| bj-pm2.01i 36108 | Inference associated with ... |
| bj-nimn 36109 | If a formula is true, then... |
| bj-nimni 36110 | Inference associated with ... |
| bj-peircei 36111 | Inference associated with ... |
| bj-looinvi 36112 | Inference associated with ... |
| bj-looinvii 36113 | Inference associated with ... |
| bj-mt2bi 36114 | Version of ~ mt2 where the... |
| bj-ntrufal 36115 | The negation of a theorem ... |
| bj-fal 36116 | Shortening of ~ fal using ... |
| bj-jaoi1 36117 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 36118 | Shortens ~ consensus (110>... |
| bj-dfbi4 36119 | Alternate definition of th... |
| bj-dfbi5 36120 | Alternate definition of th... |
| bj-dfbi6 36121 | Alternate definition of th... |
| bj-bijust0ALT 36122 | Alternate proof of ~ bijus... |
| bj-bijust00 36123 | A self-implication does no... |
| bj-consensus 36124 | Version of ~ consensus exp... |
| bj-consensusALT 36125 | Alternate proof of ~ bj-co... |
| bj-df-ifc 36126 | Candidate definition for t... |
| bj-dfif 36127 | Alternate definition of th... |
| bj-ififc 36128 | A biconditional connecting... |
| bj-imbi12 36129 | Uncurried (imported) form ... |
| bj-biorfi 36130 | This should be labeled "bi... |
| bj-falor 36131 | Dual of ~ truan (which has... |
| bj-falor2 36132 | Dual of ~ truan . (Contri... |
| bj-bibibi 36133 | A property of the bicondit... |
| bj-imn3ani 36134 | Duplication of ~ bnj1224 .... |
| bj-andnotim 36135 | Two ways of expressing a c... |
| bj-bi3ant 36136 | This used to be in the mai... |
| bj-bisym 36137 | This used to be in the mai... |
| bj-bixor 36138 | Equivalence of two ternary... |
| bj-axdd2 36139 | This implication, proved u... |
| bj-axd2d 36140 | This implication, proved u... |
| bj-axtd 36141 | This implication, proved f... |
| bj-gl4 36142 | In a normal modal logic, t... |
| bj-axc4 36143 | Over minimal calculus, the... |
| prvlem1 36148 | An elementary property of ... |
| prvlem2 36149 | An elementary property of ... |
| bj-babygodel 36150 | See the section header com... |
| bj-babylob 36151 | See the section header com... |
| bj-godellob 36152 | Proof of Gödel's theo... |
| bj-genr 36153 | Generalization rule on the... |
| bj-genl 36154 | Generalization rule on the... |
| bj-genan 36155 | Generalization rule on a c... |
| bj-mpgs 36156 | From a closed form theorem... |
| bj-2alim 36157 | Closed form of ~ 2alimi . ... |
| bj-2exim 36158 | Closed form of ~ 2eximi . ... |
| bj-alanim 36159 | Closed form of ~ alanimi .... |
| bj-2albi 36160 | Closed form of ~ 2albii . ... |
| bj-notalbii 36161 | Equivalence of universal q... |
| bj-2exbi 36162 | Closed form of ~ 2exbii . ... |
| bj-3exbi 36163 | Closed form of ~ 3exbii . ... |
| bj-sylgt2 36164 | Uncurried (imported) form ... |
| bj-alrimg 36165 | The general form of the *a... |
| bj-alrimd 36166 | A slightly more general ~ ... |
| bj-sylget 36167 | Dual statement of ~ sylgt ... |
| bj-sylget2 36168 | Uncurried (imported) form ... |
| bj-exlimg 36169 | The general form of the *e... |
| bj-sylge 36170 | Dual statement of ~ sylg (... |
| bj-exlimd 36171 | A slightly more general ~ ... |
| bj-nfimexal 36172 | A weak from of nonfreeness... |
| bj-alexim 36173 | Closed form of ~ aleximi .... |
| bj-nexdh 36174 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 36175 | Uncurried (imported) form ... |
| bj-hbxfrbi 36176 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 36177 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 36178 | Distribute quantifiers ove... |
| bj-exalimi 36179 | An inference for distribut... |
| bj-exalims 36180 | Distributing quantifiers o... |
| bj-exalimsi 36181 | An inference for distribut... |
| bj-ax12ig 36182 | A lemma used to prove a we... |
| bj-ax12i 36183 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 36184 | Closed form of ~ nfim and ... |
| bj-cbvalimt 36185 | A lemma in closed form use... |
| bj-cbveximt 36186 | A lemma in closed form use... |
| bj-eximALT 36187 | Alternate proof of ~ exim ... |
| bj-aleximiALT 36188 | Alternate proof of ~ alexi... |
| bj-eximcom 36189 | A commuted form of ~ exim ... |
| bj-ax12wlem 36190 | A lemma used to prove a we... |
| bj-cbvalim 36191 | A lemma used to prove ~ bj... |
| bj-cbvexim 36192 | A lemma used to prove ~ bj... |
| bj-cbvalimi 36193 | An equality-free general i... |
| bj-cbveximi 36194 | An equality-free general i... |
| bj-cbval 36195 | Changing a bound variable ... |
| bj-cbvex 36196 | Changing a bound variable ... |
| bj-ssbeq 36199 | Substitution in an equalit... |
| bj-ssblem1 36200 | A lemma for the definiens ... |
| bj-ssblem2 36201 | An instance of ~ ax-11 pro... |
| bj-ax12v 36202 | A weaker form of ~ ax-12 a... |
| bj-ax12 36203 | Remove a DV condition from... |
| bj-ax12ssb 36204 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 36205 | Special case of ~ 19.41 pr... |
| bj-equsexval 36206 | Special case of ~ equsexv ... |
| bj-subst 36207 | Proof of ~ sbalex from cor... |
| bj-ssbid2 36208 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 36209 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 36210 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 36211 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 36212 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 36213 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 36214 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 36215 | Closed form of ~ spimvw . ... |
| bj-spnfw 36216 | Theorem close to a closed ... |
| bj-cbvexiw 36217 | Change bound variable. Th... |
| bj-cbvexivw 36218 | Change bound variable. Th... |
| bj-modald 36219 | A short form of the axiom ... |
| bj-denot 36220 | A weakening of ~ ax-6 and ... |
| bj-eqs 36221 | A lemma for substitutions,... |
| bj-cbvexw 36222 | Change bound variable. Th... |
| bj-ax12w 36223 | The general statement that... |
| bj-ax89 36224 | A theorem which could be u... |
| bj-elequ12 36225 | An identity law for the no... |
| bj-cleljusti 36226 | One direction of ~ cleljus... |
| bj-alcomexcom 36227 | Commutation of two existen... |
| bj-hbalt 36228 | Closed form of ~ hbal . W... |
| axc11n11 36229 | Proof of ~ axc11n from { ~... |
| axc11n11r 36230 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 36231 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 36232 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 36233 | Alternate proof of ~ bj-ax... |
| bj-sb 36234 | A weak variant of ~ sbid2 ... |
| bj-modalbe 36235 | The predicate-calculus ver... |
| bj-spst 36236 | Closed form of ~ sps . On... |
| bj-19.21bit 36237 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 36238 | Closed form of ~ 19.23bi .... |
| bj-nexrt 36239 | Closed form of ~ nexr . C... |
| bj-alrim 36240 | Closed form of ~ alrimi . ... |
| bj-alrim2 36241 | Uncurried (imported) form ... |
| bj-nfdt0 36242 | A theorem close to a close... |
| bj-nfdt 36243 | Closed form of ~ nf5d and ... |
| bj-nexdt 36244 | Closed form of ~ nexd . (... |
| bj-nexdvt 36245 | Closed form of ~ nexdv . ... |
| bj-alexbiex 36246 | Adding a second quantifier... |
| bj-exexbiex 36247 | Adding a second quantifier... |
| bj-alalbial 36248 | Adding a second quantifier... |
| bj-exalbial 36249 | Adding a second quantifier... |
| bj-19.9htbi 36250 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 36251 | Strengthening ~ hbnt by re... |
| bj-biexal1 36252 | A general FOL biconditiona... |
| bj-biexal2 36253 | When ` ph ` is substituted... |
| bj-biexal3 36254 | When ` ph ` is substituted... |
| bj-bialal 36255 | When ` ph ` is substituted... |
| bj-biexex 36256 | When ` ph ` is substituted... |
| bj-hbext 36257 | Closed form of ~ hbex . (... |
| bj-nfalt 36258 | Closed form of ~ nfal . (... |
| bj-nfext 36259 | Closed form of ~ nfex . (... |
| bj-eeanvw 36260 | Version of ~ exdistrv with... |
| bj-modal4 36261 | First-order logic form of ... |
| bj-modal4e 36262 | First-order logic form of ... |
| bj-modalb 36263 | A short form of the axiom ... |
| bj-wnf1 36264 | When ` ph ` is substituted... |
| bj-wnf2 36265 | When ` ph ` is substituted... |
| bj-wnfanf 36266 | When ` ph ` is substituted... |
| bj-wnfenf 36267 | When ` ph ` is substituted... |
| bj-substax12 36268 | Equivalent form of the axi... |
| bj-substw 36269 | Weak form of the LHS of ~ ... |
| bj-nnfbi 36272 | If two formulas are equiva... |
| bj-nnfbd 36273 | If two formulas are equiva... |
| bj-nnfbii 36274 | If two formulas are equiva... |
| bj-nnfa 36275 | Nonfreeness implies the eq... |
| bj-nnfad 36276 | Nonfreeness implies the eq... |
| bj-nnfai 36277 | Nonfreeness implies the eq... |
| bj-nnfe 36278 | Nonfreeness implies the eq... |
| bj-nnfed 36279 | Nonfreeness implies the eq... |
| bj-nnfei 36280 | Nonfreeness implies the eq... |
| bj-nnfea 36281 | Nonfreeness implies the eq... |
| bj-nnfead 36282 | Nonfreeness implies the eq... |
| bj-nnfeai 36283 | Nonfreeness implies the eq... |
| bj-dfnnf2 36284 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 36285 | New nonfreeness implies ol... |
| bj-wnfnf 36286 | When ` ph ` is substituted... |
| bj-nnfnt 36287 | A variable is nonfree in a... |
| bj-nnftht 36288 | A variable is nonfree in a... |
| bj-nnfth 36289 | A variable is nonfree in a... |
| bj-nnfnth 36290 | A variable is nonfree in t... |
| bj-nnfim1 36291 | A consequence of nonfreene... |
| bj-nnfim2 36292 | A consequence of nonfreene... |
| bj-nnfim 36293 | Nonfreeness in the anteced... |
| bj-nnfimd 36294 | Nonfreeness in the anteced... |
| bj-nnfan 36295 | Nonfreeness in both conjun... |
| bj-nnfand 36296 | Nonfreeness in both conjun... |
| bj-nnfor 36297 | Nonfreeness in both disjun... |
| bj-nnford 36298 | Nonfreeness in both disjun... |
| bj-nnfbit 36299 | Nonfreeness in both sides ... |
| bj-nnfbid 36300 | Nonfreeness in both sides ... |
| bj-nnfv 36301 | A non-occurring variable i... |
| bj-nnf-alrim 36302 | Proof of the closed form o... |
| bj-nnf-exlim 36303 | Proof of the closed form o... |
| bj-dfnnf3 36304 | Alternate definition of no... |
| bj-nfnnfTEMP 36305 | New nonfreeness is equival... |
| bj-nnfa1 36306 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 36307 | See ~ nfe1 . (Contributed... |
| bj-19.12 36308 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 36309 | One of four lemmas for non... |
| bj-nnflemee 36310 | One of four lemmas for non... |
| bj-nnflemae 36311 | One of four lemmas for non... |
| bj-nnflemea 36312 | One of four lemmas for non... |
| bj-nnfalt 36313 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 36314 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 36315 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 36316 | Statement ~ 19.21t proved ... |
| bj-19.23t 36317 | Statement ~ 19.23t proved ... |
| bj-19.36im 36318 | One direction of ~ 19.36 f... |
| bj-19.37im 36319 | One direction of ~ 19.37 f... |
| bj-19.42t 36320 | Closed form of ~ 19.42 fro... |
| bj-19.41t 36321 | Closed form of ~ 19.41 fro... |
| bj-sbft 36322 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 36323 | Version of ~ pm11.53v with... |
| bj-pm11.53v 36324 | Version of ~ pm11.53v with... |
| bj-pm11.53a 36325 | A variant of ~ pm11.53v . ... |
| bj-equsvt 36326 | A variant of ~ equsv . (C... |
| bj-equsalvwd 36327 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 36328 | Variant of ~ equsexvw . (... |
| bj-sbievwd 36329 | Variant of ~ sbievw . (Co... |
| bj-axc10 36330 | Alternate proof of ~ axc10... |
| bj-alequex 36331 | A fol lemma. See ~ aleque... |
| bj-spimt2 36332 | A step in the proof of ~ s... |
| bj-cbv3ta 36333 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 36334 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 36335 | A theorem close to a close... |
| bj-hbsb3 36336 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 36337 | A theorem close to a close... |
| bj-nfs1t2 36338 | A theorem close to a close... |
| bj-nfs1 36339 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 36340 | Version of ~ axc10 with a ... |
| bj-spimtv 36341 | Version of ~ spimt with a ... |
| bj-cbv3hv2 36342 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 36343 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 36344 | Version of ~ cbv2h with a ... |
| bj-cbv2v 36345 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 36346 | Version of ~ cbvald with a... |
| bj-cbvexdv 36347 | Version of ~ cbvexd with a... |
| bj-cbval2vv 36348 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 36349 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 36350 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 36351 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 36352 | Version of ~ cbvex4v with ... |
| bj-equsalhv 36353 | Version of ~ equsalh with ... |
| bj-axc11nv 36354 | Version of ~ axc11n with a... |
| bj-aecomsv 36355 | Version of ~ aecoms with a... |
| bj-axc11v 36356 | Version of ~ axc11 with a ... |
| bj-drnf2v 36357 | Version of ~ drnf2 with a ... |
| bj-equs45fv 36358 | Version of ~ equs45f with ... |
| bj-hbs1 36359 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 36360 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 36361 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 36362 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 36363 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 36364 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 36365 | Biconditional version of a... |
| bj-hbaeb 36366 | Biconditional version of ~... |
| bj-hbnaeb 36367 | Biconditional version of ~... |
| bj-dvv 36368 | A special instance of ~ bj... |
| bj-equsal1t 36369 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 36370 | Inference associated with ... |
| bj-equsal1 36371 | One direction of ~ equsal ... |
| bj-equsal2 36372 | One direction of ~ equsal ... |
| bj-equsal 36373 | Shorter proof of ~ equsal ... |
| stdpc5t 36374 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 36375 | More direct proof of ~ std... |
| 2stdpc5 36376 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 36377 | Proof of ~ 19.21t from ~ s... |
| exlimii 36378 | Inference associated with ... |
| ax11-pm 36379 | Proof of ~ ax-11 similar t... |
| ax6er 36380 | Commuted form of ~ ax6e . ... |
| exlimiieq1 36381 | Inferring a theorem when i... |
| exlimiieq2 36382 | Inferring a theorem when i... |
| ax11-pm2 36383 | Proof of ~ ax-11 from the ... |
| bj-sbsb 36384 | Biconditional showing two ... |
| bj-dfsb2 36385 | Alternate (dual) definitio... |
| bj-sbf3 36386 | Substitution has no effect... |
| bj-sbf4 36387 | Substitution has no effect... |
| bj-eu3f 36388 | Version of ~ eu3v where th... |
| bj-sblem1 36389 | Lemma for substitution. (... |
| bj-sblem2 36390 | Lemma for substitution. (... |
| bj-sblem 36391 | Lemma for substitution. (... |
| bj-sbievw1 36392 | Lemma for substitution. (... |
| bj-sbievw2 36393 | Lemma for substitution. (... |
| bj-sbievw 36394 | Lemma for substitution. C... |
| bj-sbievv 36395 | Version of ~ sbie with a s... |
| bj-moeub 36396 | Uniqueness is equivalent t... |
| bj-sbidmOLD 36397 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 36398 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 36399 | Curried (exported) form of... |
| bj-dvelimv 36400 | A version of ~ dvelim usin... |
| bj-nfeel2 36401 | Nonfreeness in a membershi... |
| bj-axc14nf 36402 | Proof of a version of ~ ax... |
| bj-axc14 36403 | Alternate proof of ~ axc14... |
| mobidvALT 36404 | Alternate proof of ~ mobid... |
| sbn1ALT 36405 | Alternate proof of ~ sbn1 ... |
| eliminable1 36406 | A theorem used to prove th... |
| eliminable2a 36407 | A theorem used to prove th... |
| eliminable2b 36408 | A theorem used to prove th... |
| eliminable2c 36409 | A theorem used to prove th... |
| eliminable3a 36410 | A theorem used to prove th... |
| eliminable3b 36411 | A theorem used to prove th... |
| eliminable-velab 36412 | A theorem used to prove th... |
| eliminable-veqab 36413 | A theorem used to prove th... |
| eliminable-abeqv 36414 | A theorem used to prove th... |
| eliminable-abeqab 36415 | A theorem used to prove th... |
| eliminable-abelv 36416 | A theorem used to prove th... |
| eliminable-abelab 36417 | A theorem used to prove th... |
| bj-denoteslem 36418 | Lemma for ~ bj-denotes . ... |
| bj-denotes 36419 | This would be the justific... |
| bj-issettru 36420 | Weak version of ~ isset wi... |
| bj-elabtru 36421 | This is as close as we can... |
| bj-issetwt 36422 | Closed form of ~ bj-issetw... |
| bj-issetw 36423 | The closest one can get to... |
| bj-elissetALT 36424 | Alternate proof of ~ eliss... |
| bj-issetiv 36425 | Version of ~ bj-isseti wit... |
| bj-isseti 36426 | Version of ~ isseti with a... |
| bj-ralvw 36427 | A weak version of ~ ralv n... |
| bj-rexvw 36428 | A weak version of ~ rexv n... |
| bj-rababw 36429 | A weak version of ~ rabab ... |
| bj-rexcom4bv 36430 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 36431 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 36432 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 36433 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 36434 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 36435 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 36436 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 36437 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 36438 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 36439 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 36440 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 36441 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 36442 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 36443 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 36444 | Remove from ~ spcimdv depe... |
| elelb 36445 | Equivalence between two co... |
| bj-pwvrelb 36446 | Characterization of the el... |
| bj-nfcsym 36447 | The nonfreeness quantifier... |
| bj-sbeqALT 36448 | Substitution in an equalit... |
| bj-sbeq 36449 | Distribute proper substitu... |
| bj-sbceqgALT 36450 | Distribute proper substitu... |
| bj-csbsnlem 36451 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 36452 | Substitution in a singleto... |
| bj-sbel1 36453 | Version of ~ sbcel1g when ... |
| bj-abv 36454 | The class of sets verifyin... |
| bj-abvALT 36455 | Alternate version of ~ bj-... |
| bj-ab0 36456 | The class of sets verifyin... |
| bj-abf 36457 | Shorter proof of ~ abf (wh... |
| bj-csbprc 36458 | More direct proof of ~ csb... |
| bj-exlimvmpi 36459 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 36460 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 36461 | Lemma for theorems of the ... |
| bj-exlimmpbir 36462 | Lemma for theorems of the ... |
| bj-vtoclf 36463 | Remove dependency on ~ ax-... |
| bj-vtocl 36464 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 36465 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 36466 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 36467 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 36468 | A version of ~ vtoclg with... |
| bj-rabeqbid 36469 | Version of ~ rabeqbidv wit... |
| bj-seex 36470 | Version of ~ seex with a d... |
| bj-nfcf 36471 | Version of ~ df-nfc with a... |
| bj-zfauscl 36472 | General version of ~ zfaus... |
| bj-elabd2ALT 36473 | Alternate proof of ~ elabd... |
| bj-unrab 36474 | Generalization of ~ unrab ... |
| bj-inrab 36475 | Generalization of ~ inrab ... |
| bj-inrab2 36476 | Shorter proof of ~ inrab .... |
| bj-inrab3 36477 | Generalization of ~ dfrab3... |
| bj-rabtr 36478 | Restricted class abstracti... |
| bj-rabtrALT 36479 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 36480 | Proof of ~ bj-rabtr found ... |
| bj-gabss 36483 | Inclusion of generalized c... |
| bj-gabssd 36484 | Inclusion of generalized c... |
| bj-gabeqd 36485 | Equality of generalized cl... |
| bj-gabeqis 36486 | Equality of generalized cl... |
| bj-elgab 36487 | Elements of a generalized ... |
| bj-gabima 36488 | Generalized class abstract... |
| bj-ru0 36491 | The FOL part of Russell's ... |
| bj-ru1 36492 | A version of Russell's par... |
| bj-ru 36493 | Remove dependency on ~ ax-... |
| currysetlem 36494 | Lemma for ~ currysetlem , ... |
| curryset 36495 | Curry's paradox in set the... |
| currysetlem1 36496 | Lemma for ~ currysetALT . ... |
| currysetlem2 36497 | Lemma for ~ currysetALT . ... |
| currysetlem3 36498 | Lemma for ~ currysetALT . ... |
| currysetALT 36499 | Alternate proof of ~ curry... |
| bj-n0i 36500 | Inference associated with ... |
| bj-disjsn01 36501 | Disjointness of the single... |
| bj-0nel1 36502 | The empty set does not bel... |
| bj-1nel0 36503 | ` 1o ` does not belong to ... |
| bj-xpimasn 36504 | The image of a singleton, ... |
| bj-xpima1sn 36505 | The image of a singleton b... |
| bj-xpima1snALT 36506 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 36507 | The image of a singleton b... |
| bj-xpnzex 36508 | If the first factor of a p... |
| bj-xpexg2 36509 | Curried (exported) form of... |
| bj-xpnzexb 36510 | If the first factor of a p... |
| bj-cleq 36511 | Substitution property for ... |
| bj-snsetex 36512 | The class of sets "whose s... |
| bj-clexab 36513 | Sethood of certain classes... |
| bj-sngleq 36516 | Substitution property for ... |
| bj-elsngl 36517 | Characterization of the el... |
| bj-snglc 36518 | Characterization of the el... |
| bj-snglss 36519 | The singletonization of a ... |
| bj-0nelsngl 36520 | The empty set is not a mem... |
| bj-snglinv 36521 | Inverse of singletonizatio... |
| bj-snglex 36522 | A class is a set if and on... |
| bj-tageq 36525 | Substitution property for ... |
| bj-eltag 36526 | Characterization of the el... |
| bj-0eltag 36527 | The empty set belongs to t... |
| bj-tagn0 36528 | The tagging of a class is ... |
| bj-tagss 36529 | The tagging of a class is ... |
| bj-snglsstag 36530 | The singletonization is in... |
| bj-sngltagi 36531 | The singletonization is in... |
| bj-sngltag 36532 | The singletonization and t... |
| bj-tagci 36533 | Characterization of the el... |
| bj-tagcg 36534 | Characterization of the el... |
| bj-taginv 36535 | Inverse of tagging. (Cont... |
| bj-tagex 36536 | A class is a set if and on... |
| bj-xtageq 36537 | The products of a given cl... |
| bj-xtagex 36538 | The product of a set and t... |
| bj-projeq 36541 | Substitution property for ... |
| bj-projeq2 36542 | Substitution property for ... |
| bj-projun 36543 | The class projection on a ... |
| bj-projex 36544 | Sethood of the class proje... |
| bj-projval 36545 | Value of the class project... |
| bj-1upleq 36548 | Substitution property for ... |
| bj-pr1eq 36551 | Substitution property for ... |
| bj-pr1un 36552 | The first projection prese... |
| bj-pr1val 36553 | Value of the first project... |
| bj-pr11val 36554 | Value of the first project... |
| bj-pr1ex 36555 | Sethood of the first proje... |
| bj-1uplth 36556 | The characteristic propert... |
| bj-1uplex 36557 | A monuple is a set if and ... |
| bj-1upln0 36558 | A monuple is nonempty. (C... |
| bj-2upleq 36561 | Substitution property for ... |
| bj-pr21val 36562 | Value of the first project... |
| bj-pr2eq 36565 | Substitution property for ... |
| bj-pr2un 36566 | The second projection pres... |
| bj-pr2val 36567 | Value of the second projec... |
| bj-pr22val 36568 | Value of the second projec... |
| bj-pr2ex 36569 | Sethood of the second proj... |
| bj-2uplth 36570 | The characteristic propert... |
| bj-2uplex 36571 | A couple is a set if and o... |
| bj-2upln0 36572 | A couple is nonempty. (Co... |
| bj-2upln1upl 36573 | A couple is never equal to... |
| bj-rcleqf 36574 | Relative version of ~ cleq... |
| bj-rcleq 36575 | Relative version of ~ dfcl... |
| bj-reabeq 36576 | Relative form of ~ eqabb .... |
| bj-disj2r 36577 | Relative version of ~ ssdi... |
| bj-sscon 36578 | Contraposition law for rel... |
| bj-abex 36579 | Two ways of stating that t... |
| bj-clex 36580 | Two ways of stating that a... |
| bj-axsn 36581 | Two ways of stating the ax... |
| bj-snexg 36583 | A singleton built on a set... |
| bj-snex 36584 | A singleton is a set. See... |
| bj-axbun 36585 | Two ways of stating the ax... |
| bj-unexg 36587 | Existence of binary unions... |
| bj-prexg 36588 | Existence of unordered pai... |
| bj-prex 36589 | Existence of unordered pai... |
| bj-axadj 36590 | Two ways of stating the ax... |
| bj-adjg1 36592 | Existence of the result of... |
| bj-snfromadj 36593 | Singleton from adjunction ... |
| bj-prfromadj 36594 | Unordered pair from adjunc... |
| bj-adjfrombun 36595 | Adjunction from singleton ... |
| eleq2w2ALT 36596 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 36597 | Alternate proof of ~ clel3... |
| bj-pw0ALT 36598 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 36599 | Quantitative version of ~ ... |
| bj-unirel 36600 | Quantitative version of ~ ... |
| bj-elpwg 36601 | If the intersection of two... |
| bj-velpwALT 36602 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 36603 | Alternate proof of ~ elpwg... |
| bj-vjust 36604 | Justification theorem for ... |
| bj-nul 36605 | Two formulations of the ax... |
| bj-nuliota 36606 | Definition of the empty se... |
| bj-nuliotaALT 36607 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 36608 | Alternate proof of ~ vtocl... |
| bj-elsn12g 36609 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 36610 | Biconditional version of ~... |
| bj-pwcfsdom 36611 | Remove hypothesis from ~ p... |
| bj-grur1 36612 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 36613 | The extension of a predica... |
| bj-dfid2ALT 36614 | Alternate version of ~ dfi... |
| bj-0nelopab 36615 | The empty set is never an ... |
| bj-brrelex12ALT 36616 | Two classes related by a b... |
| bj-epelg 36617 | The membership relation an... |
| bj-epelb 36618 | Two classes are related by... |
| bj-nsnid 36619 | A set does not contain the... |
| bj-rdg0gALT 36620 | Alternate proof of ~ rdg0g... |
| bj-evaleq 36621 | Equality theorem for the `... |
| bj-evalfun 36622 | The evaluation at a class ... |
| bj-evalfn 36623 | The evaluation at a class ... |
| bj-evalval 36624 | Value of the evaluation at... |
| bj-evalid 36625 | The evaluation at a set of... |
| bj-ndxarg 36626 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 36627 | Closed general form of ~ s... |
| bj-rest00 36630 | An elementwise intersectio... |
| bj-restsn 36631 | An elementwise intersectio... |
| bj-restsnss 36632 | Special case of ~ bj-rests... |
| bj-restsnss2 36633 | Special case of ~ bj-rests... |
| bj-restsn0 36634 | An elementwise intersectio... |
| bj-restsn10 36635 | Special case of ~ bj-rests... |
| bj-restsnid 36636 | The elementwise intersecti... |
| bj-rest10 36637 | An elementwise intersectio... |
| bj-rest10b 36638 | Alternate version of ~ bj-... |
| bj-restn0 36639 | An elementwise intersectio... |
| bj-restn0b 36640 | Alternate version of ~ bj-... |
| bj-restpw 36641 | The elementwise intersecti... |
| bj-rest0 36642 | An elementwise intersectio... |
| bj-restb 36643 | An elementwise intersectio... |
| bj-restv 36644 | An elementwise intersectio... |
| bj-resta 36645 | An elementwise intersectio... |
| bj-restuni 36646 | The union of an elementwis... |
| bj-restuni2 36647 | The union of an elementwis... |
| bj-restreg 36648 | A reformulation of the axi... |
| bj-raldifsn 36649 | All elements in a set sati... |
| bj-0int 36650 | If ` A ` is a collection o... |
| bj-mooreset 36651 | A Moore collection is a se... |
| bj-ismoore 36654 | Characterization of Moore ... |
| bj-ismoored0 36655 | Necessary condition to be ... |
| bj-ismoored 36656 | Necessary condition to be ... |
| bj-ismoored2 36657 | Necessary condition to be ... |
| bj-ismooredr 36658 | Sufficient condition to be... |
| bj-ismooredr2 36659 | Sufficient condition to be... |
| bj-discrmoore 36660 | The powerclass ` ~P A ` is... |
| bj-0nmoore 36661 | The empty set is not a Moo... |
| bj-snmoore 36662 | A singleton is a Moore col... |
| bj-snmooreb 36663 | A singleton is a Moore col... |
| bj-prmoore 36664 | A pair formed of two neste... |
| bj-0nelmpt 36665 | The empty set is not an el... |
| bj-mptval 36666 | Value of a function given ... |
| bj-dfmpoa 36667 | An equivalent definition o... |
| bj-mpomptALT 36668 | Alternate proof of ~ mpomp... |
| setsstrset 36685 | Relation between ~ df-sets... |
| bj-nfald 36686 | Variant of ~ nfald . (Con... |
| bj-nfexd 36687 | Variant of ~ nfexd . (Con... |
| copsex2d 36688 | Implicit substitution dedu... |
| copsex2b 36689 | Biconditional form of ~ co... |
| opelopabd 36690 | Membership of an ordere pa... |
| opelopabb 36691 | Membership of an ordered p... |
| opelopabbv 36692 | Membership of an ordered p... |
| bj-opelrelex 36693 | The coordinates of an orde... |
| bj-opelresdm 36694 | If an ordered pair is in a... |
| bj-brresdm 36695 | If two classes are related... |
| brabd0 36696 | Expressing that two sets a... |
| brabd 36697 | Expressing that two sets a... |
| bj-brab2a1 36698 | "Unbounded" version of ~ b... |
| bj-opabssvv 36699 | A variant of ~ relopabiv (... |
| bj-funidres 36700 | The restricted identity re... |
| bj-opelidb 36701 | Characterization of the or... |
| bj-opelidb1 36702 | Characterization of the or... |
| bj-inexeqex 36703 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 36704 | If the intersection of two... |
| bj-opelid 36705 | Characterization of the or... |
| bj-ideqg 36706 | Characterization of the cl... |
| bj-ideqgALT 36707 | Alternate proof of ~ bj-id... |
| bj-ideqb 36708 | Characterization of classe... |
| bj-idres 36709 | Alternate expression for t... |
| bj-opelidres 36710 | Characterization of the or... |
| bj-idreseq 36711 | Sufficient condition for t... |
| bj-idreseqb 36712 | Characterization for two c... |
| bj-ideqg1 36713 | For sets, the identity rel... |
| bj-ideqg1ALT 36714 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 36715 | Characterization of the co... |
| bj-elid3 36716 | Characterization of the co... |
| bj-elid4 36717 | Characterization of the el... |
| bj-elid5 36718 | Characterization of the el... |
| bj-elid6 36719 | Characterization of the el... |
| bj-elid7 36720 | Characterization of the el... |
| bj-diagval 36723 | Value of the functionalize... |
| bj-diagval2 36724 | Value of the functionalize... |
| bj-eldiag 36725 | Characterization of the el... |
| bj-eldiag2 36726 | Characterization of the el... |
| bj-imdirvallem 36729 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 36730 | Value of the functionalize... |
| bj-imdirval2lem 36731 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 36732 | Value of the functionalize... |
| bj-imdirval3 36733 | Value of the functionalize... |
| bj-imdiridlem 36734 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 36735 | Functorial property of the... |
| bj-opelopabid 36736 | Membership in an ordered-p... |
| bj-opabco 36737 | Composition of ordered-pai... |
| bj-xpcossxp 36738 | The composition of two Car... |
| bj-imdirco 36739 | Functorial property of the... |
| bj-iminvval 36742 | Value of the functionalize... |
| bj-iminvval2 36743 | Value of the functionalize... |
| bj-iminvid 36744 | Functorial property of the... |
| bj-inftyexpitaufo 36751 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 36754 | An element of the circle a... |
| bj-inftyexpiinv 36757 | Utility theorem for the in... |
| bj-inftyexpiinj 36758 | Injectivity of the paramet... |
| bj-inftyexpidisj 36759 | An element of the circle a... |
| bj-ccinftydisj 36762 | The circle at infinity is ... |
| bj-elccinfty 36763 | A lemma for infinite exten... |
| bj-ccssccbar 36766 | Complex numbers are extend... |
| bj-ccinftyssccbar 36767 | Infinite extended complex ... |
| bj-pinftyccb 36770 | The class ` pinfty ` is an... |
| bj-pinftynrr 36771 | The extended complex numbe... |
| bj-minftyccb 36774 | The class ` minfty ` is an... |
| bj-minftynrr 36775 | The extended complex numbe... |
| bj-pinftynminfty 36776 | The extended complex numbe... |
| bj-rrhatsscchat 36785 | The real projective line i... |
| bj-imafv 36800 | If the direct image of a s... |
| bj-funun 36801 | Value of a function expres... |
| bj-fununsn1 36802 | Value of a function expres... |
| bj-fununsn2 36803 | Value of a function expres... |
| bj-fvsnun1 36804 | The value of a function wi... |
| bj-fvsnun2 36805 | The value of a function wi... |
| bj-fvmptunsn1 36806 | Value of a function expres... |
| bj-fvmptunsn2 36807 | Value of a function expres... |
| bj-iomnnom 36808 | The canonical bijection fr... |
| bj-smgrpssmgm 36817 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 36818 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 36819 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 36820 | Monoids are semigroups (el... |
| bj-cmnssmnd 36821 | Commutative monoids are mo... |
| bj-cmnssmndel 36822 | Commutative monoids are mo... |
| bj-grpssmnd 36823 | Groups are monoids. (Cont... |
| bj-grpssmndel 36824 | Groups are monoids (elemen... |
| bj-ablssgrp 36825 | Abelian groups are groups.... |
| bj-ablssgrpel 36826 | Abelian groups are groups ... |
| bj-ablsscmn 36827 | Abelian groups are commuta... |
| bj-ablsscmnel 36828 | Abelian groups are commuta... |
| bj-modssabl 36829 | (The additive groups of) m... |
| bj-vecssmod 36830 | Vector spaces are modules.... |
| bj-vecssmodel 36831 | Vector spaces are modules ... |
| bj-finsumval0 36834 | Value of a finite sum. (C... |
| bj-fvimacnv0 36835 | Variant of ~ fvimacnv wher... |
| bj-isvec 36836 | The predicate "is a vector... |
| bj-fldssdrng 36837 | Fields are division rings.... |
| bj-flddrng 36838 | Fields are division rings ... |
| bj-rrdrg 36839 | The field of real numbers ... |
| bj-isclm 36840 | The predicate "is a subcom... |
| bj-isrvec 36843 | The predicate "is a real v... |
| bj-rvecmod 36844 | Real vector spaces are mod... |
| bj-rvecssmod 36845 | Real vector spaces are mod... |
| bj-rvecrr 36846 | The field of scalars of a ... |
| bj-isrvecd 36847 | The predicate "is a real v... |
| bj-rvecvec 36848 | Real vector spaces are vec... |
| bj-isrvec2 36849 | The predicate "is a real v... |
| bj-rvecssvec 36850 | Real vector spaces are vec... |
| bj-rveccmod 36851 | Real vector spaces are sub... |
| bj-rvecsscmod 36852 | Real vector spaces are sub... |
| bj-rvecsscvec 36853 | Real vector spaces are sub... |
| bj-rveccvec 36854 | Real vector spaces are sub... |
| bj-rvecssabl 36855 | (The additive groups of) r... |
| bj-rvecabl 36856 | (The additive groups of) r... |
| bj-subcom 36857 | A consequence of commutati... |
| bj-lineqi 36858 | Solution of a (scalar) lin... |
| bj-bary1lem 36859 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 36860 | Lemma for bj-bary1: comput... |
| bj-bary1 36861 | Barycentric coordinates in... |
| bj-endval 36864 | Value of the monoid of end... |
| bj-endbase 36865 | Base set of the monoid of ... |
| bj-endcomp 36866 | Composition law of the mon... |
| bj-endmnd 36867 | The monoid of endomorphism... |
| taupilem3 36868 | Lemma for tau-related theo... |
| taupilemrplb 36869 | A set of positive reals ha... |
| taupilem1 36870 | Lemma for ~ taupi . A pos... |
| taupilem2 36871 | Lemma for ~ taupi . The s... |
| taupi 36872 | Relationship between ` _ta... |
| dfgcd3 36873 | Alternate definition of th... |
| irrdifflemf 36874 | Lemma for ~ irrdiff . The... |
| irrdiff 36875 | The irrationals are exactl... |
| iccioo01 36876 | The closed unit interval i... |
| csbrecsg 36877 | Move class substitution in... |
| csbrdgg 36878 | Move class substitution in... |
| csboprabg 36879 | Move class substitution in... |
| csbmpo123 36880 | Move class substitution in... |
| con1bii2 36881 | A contraposition inference... |
| con2bii2 36882 | A contraposition inference... |
| vtoclefex 36883 | Implicit substitution of a... |
| rnmptsn 36884 | The range of a function ma... |
| f1omptsnlem 36885 | This is the core of the pr... |
| f1omptsn 36886 | A function mapping to sing... |
| mptsnunlem 36887 | This is the core of the pr... |
| mptsnun 36888 | A class ` B ` is equal to ... |
| dissneqlem 36889 | This is the core of the pr... |
| dissneq 36890 | Any topology that contains... |
| exlimim 36891 | Closed form of ~ exlimimd ... |
| exlimimd 36892 | Existential elimination ru... |
| exellim 36893 | Closed form of ~ exellimdd... |
| exellimddv 36894 | Eliminate an antecedent wh... |
| topdifinfindis 36895 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 36896 | This is the core of the pr... |
| topdifinffin 36897 | Part of Exercise 3 of [Mun... |
| topdifinf 36898 | Part of Exercise 3 of [Mun... |
| topdifinfeq 36899 | Two different ways of defi... |
| icorempo 36900 | Closed-below, open-above i... |
| icoreresf 36901 | Closed-below, open-above i... |
| icoreval 36902 | Value of the closed-below,... |
| icoreelrnab 36903 | Elementhood in the set of ... |
| isbasisrelowllem1 36904 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 36905 | Lemma for ~ isbasisrelowl ... |
| icoreclin 36906 | The set of closed-below, o... |
| isbasisrelowl 36907 | The set of all closed-belo... |
| icoreunrn 36908 | The union of all closed-be... |
| istoprelowl 36909 | The set of all closed-belo... |
| icoreelrn 36910 | A class abstraction which ... |
| iooelexlt 36911 | An element of an open inte... |
| relowlssretop 36912 | The lower limit topology o... |
| relowlpssretop 36913 | The lower limit topology o... |
| sucneqond 36914 | Inequality of an ordinal s... |
| sucneqoni 36915 | Inequality of an ordinal s... |
| onsucuni3 36916 | If an ordinal number has a... |
| 1oequni2o 36917 | The ordinal number ` 1o ` ... |
| rdgsucuni 36918 | If an ordinal number has a... |
| rdgeqoa 36919 | If a recursive function wi... |
| elxp8 36920 | Membership in a Cartesian ... |
| cbveud 36921 | Deduction used to change b... |
| cbvreud 36922 | Deduction used to change b... |
| difunieq 36923 | The difference of unions i... |
| inunissunidif 36924 | Theorem about subsets of t... |
| rdgellim 36925 | Elementhood in a recursive... |
| rdglimss 36926 | A recursive definition at ... |
| rdgssun 36927 | In a recursive definition ... |
| exrecfnlem 36928 | Lemma for ~ exrecfn . (Co... |
| exrecfn 36929 | Theorem about the existenc... |
| exrecfnpw 36930 | For any base set, a set wh... |
| finorwe 36931 | If the Axiom of Infinity i... |
| dffinxpf 36934 | This theorem is the same a... |
| finxpeq1 36935 | Equality theorem for Carte... |
| finxpeq2 36936 | Equality theorem for Carte... |
| csbfinxpg 36937 | Distribute proper substitu... |
| finxpreclem1 36938 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 36939 | Lemma for ` ^^ ` recursion... |
| finxp0 36940 | The value of Cartesian exp... |
| finxp1o 36941 | The value of Cartesian exp... |
| finxpreclem3 36942 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 36943 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 36944 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 36945 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 36946 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 36947 | The value of Cartesian exp... |
| finxp2o 36948 | The value of Cartesian exp... |
| finxp3o 36949 | The value of Cartesian exp... |
| finxpnom 36950 | Cartesian exponentiation w... |
| finxp00 36951 | Cartesian exponentiation o... |
| iunctb2 36952 | Using the axiom of countab... |
| domalom 36953 | A class which dominates ev... |
| isinf2 36954 | The converse of ~ isinf . ... |
| ctbssinf 36955 | Using the axiom of choice,... |
| ralssiun 36956 | The index set of an indexe... |
| nlpineqsn 36957 | For every point ` p ` of a... |
| nlpfvineqsn 36958 | Given a subset ` A ` of ` ... |
| fvineqsnf1 36959 | A theorem about functions ... |
| fvineqsneu 36960 | A theorem about functions ... |
| fvineqsneq 36961 | A theorem about functions ... |
| pibp16 36962 | Property P000016 of pi-bas... |
| pibp19 36963 | Property P000019 of pi-bas... |
| pibp21 36964 | Property P000021 of pi-bas... |
| pibt1 36965 | Theorem T000001 of pi-base... |
| pibt2 36966 | Theorem T000002 of pi-base... |
| wl-section-prop 36967 | Intuitionistic logic is no... |
| wl-section-boot 36971 | In this section, I provide... |
| wl-luk-imim1i 36972 | Inference adding common co... |
| wl-luk-syl 36973 | An inference version of th... |
| wl-luk-imtrid 36974 | A syllogism rule of infere... |
| wl-luk-pm2.18d 36975 | Deduction based on reducti... |
| wl-luk-con4i 36976 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 36977 | Inference rule. Copy of ~... |
| wl-luk-a1i 36978 | Inference rule. Copy of ~... |
| wl-luk-mpi 36979 | A nested modus ponens infe... |
| wl-luk-imim2i 36980 | Inference adding common an... |
| wl-luk-imtrdi 36981 | A syllogism rule of infere... |
| wl-luk-ax3 36982 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 36983 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 36984 | This theorem, called "Asse... |
| wl-luk-com12 36985 | Inference that swaps (comm... |
| wl-luk-pm2.21 36986 | From a wff and its negatio... |
| wl-luk-con1i 36987 | A contraposition inference... |
| wl-luk-ja 36988 | Inference joining the ante... |
| wl-luk-imim2 36989 | A closed form of syllogism... |
| wl-luk-a1d 36990 | Deduction introducing an e... |
| wl-luk-ax2 36991 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 36992 | Principle of identity. Th... |
| wl-luk-notnotr 36993 | Converse of double negatio... |
| wl-luk-pm2.04 36994 | Swap antecedents. Theorem... |
| wl-section-impchain 36995 | An implication like ` ( ps... |
| wl-impchain-mp-x 36996 | This series of theorems pr... |
| wl-impchain-mp-0 36997 | This theorem is the start ... |
| wl-impchain-mp-1 36998 | This theorem is in fact a ... |
| wl-impchain-mp-2 36999 | This theorem is in fact a ... |
| wl-impchain-com-1.x 37000 | It is often convenient to ... |
| wl-impchain-com-1.1 37001 | A degenerate form of antec... |
| wl-impchain-com-1.2 37002 | This theorem is in fact a ... |
| wl-impchain-com-1.3 37003 | This theorem is in fact a ... |
| wl-impchain-com-1.4 37004 | This theorem is in fact a ... |
| wl-impchain-com-n.m 37005 | This series of theorems al... |
| wl-impchain-com-2.3 37006 | This theorem is in fact a ... |
| wl-impchain-com-2.4 37007 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 37008 | This theorem is in fact a ... |
| wl-impchain-a1-x 37009 | If an implication chain is... |
| wl-impchain-a1-1 37010 | Inference rule, a copy of ... |
| wl-impchain-a1-2 37011 | Inference rule, a copy of ... |
| wl-impchain-a1-3 37012 | Inference rule, a copy of ... |
| wl-ifp-ncond1 37013 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 37014 | If one case of an ` if- ` ... |
| wl-ifpimpr 37015 | If one case of an ` if- ` ... |
| wl-ifp4impr 37016 | If one case of an ` if- ` ... |
| wl-df-3xor 37017 | Alternative definition of ... |
| wl-df3xor2 37018 | Alternative definition of ... |
| wl-df3xor3 37019 | Alternative form of ~ wl-d... |
| wl-3xortru 37020 | If the first input is true... |
| wl-3xorfal 37021 | If the first input is fals... |
| wl-3xorbi 37022 | Triple xor can be replaced... |
| wl-3xorbi2 37023 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 37024 | Equivalence theorem for tr... |
| wl-3xorbi123i 37025 | Equivalence theorem for tr... |
| wl-3xorrot 37026 | Rotation law for triple xo... |
| wl-3xorcoma 37027 | Commutative law for triple... |
| wl-3xorcomb 37028 | Commutative law for triple... |
| wl-3xornot1 37029 | Flipping the first input f... |
| wl-3xornot 37030 | Triple xor distributes ove... |
| wl-1xor 37031 | In the recursive scheme ... |
| wl-2xor 37032 | In the recursive scheme ... |
| wl-df-3mintru2 37033 | Alternative definition of ... |
| wl-df2-3mintru2 37034 | The adder carry in disjunc... |
| wl-df3-3mintru2 37035 | The adder carry in conjunc... |
| wl-df4-3mintru2 37036 | An alternative definition ... |
| wl-1mintru1 37037 | Using the recursion formul... |
| wl-1mintru2 37038 | Using the recursion formul... |
| wl-2mintru1 37039 | Using the recursion formul... |
| wl-2mintru2 37040 | Using the recursion formul... |
| wl-df3maxtru1 37041 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 37043 | A version of ~ ax-wl-13v w... |
| wl-mps 37044 | Replacing a nested consequ... |
| wl-syls1 37045 | Replacing a nested consequ... |
| wl-syls2 37046 | Replacing a nested anteced... |
| wl-embant 37047 | A true wff can always be a... |
| wl-orel12 37048 | In a conjunctive normal fo... |
| wl-cases2-dnf 37049 | A particular instance of ~... |
| wl-cbvmotv 37050 | Change bound variable. Us... |
| wl-moteq 37051 | Change bound variable. Us... |
| wl-motae 37052 | Change bound variable. Us... |
| wl-moae 37053 | Two ways to express "at mo... |
| wl-euae 37054 | Two ways to express "exact... |
| wl-nax6im 37055 | The following series of th... |
| wl-hbae1 37056 | This specialization of ~ h... |
| wl-naevhba1v 37057 | An instance of ~ hbn1w app... |
| wl-spae 37058 | Prove an instance of ~ sp ... |
| wl-speqv 37059 | Under the assumption ` -. ... |
| wl-19.8eqv 37060 | Under the assumption ` -. ... |
| wl-19.2reqv 37061 | Under the assumption ` -. ... |
| wl-nfalv 37062 | If ` x ` is not present in... |
| wl-nfimf1 37063 | An antecedent is irrelevan... |
| wl-nfae1 37064 | Unlike ~ nfae , this speci... |
| wl-nfnae1 37065 | Unlike ~ nfnae , this spec... |
| wl-aetr 37066 | A transitive law for varia... |
| wl-axc11r 37067 | Same as ~ axc11r , but usi... |
| wl-dral1d 37068 | A version of ~ dral1 with ... |
| wl-cbvalnaed 37069 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 37070 | A more general version of ... |
| wl-exeq 37071 | The semantics of ` E. x y ... |
| wl-aleq 37072 | The semantics of ` A. x y ... |
| wl-nfeqfb 37073 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 37074 | If ` y ` is not free in ` ... |
| wl-equsalvw 37075 | Version of ~ equsalv with ... |
| wl-equsald 37076 | Deduction version of ~ equ... |
| wl-equsaldv 37077 | Deduction version of ~ equ... |
| wl-equsal 37078 | A useful equivalence relat... |
| wl-equsal1t 37079 | The expression ` x = y ` i... |
| wl-equsalcom 37080 | This simple equivalence ea... |
| wl-equsal1i 37081 | The antecedent ` x = y ` i... |
| wl-sbid2ft 37082 | A more general version of ... |
| wl-cbvalsbi 37083 | Change bounded variables i... |
| wl-sbrimt 37084 | Substitution with a variab... |
| wl-sblimt 37085 | Substitution with a variab... |
| wl-sb9v 37086 | Commutation of quantificat... |
| wl-sb8ft 37087 | Substitution of variable i... |
| wl-sb8eft 37088 | Substitution of variable i... |
| wl-sb8t 37089 | Substitution of variable i... |
| wl-sb8et 37090 | Substitution of variable i... |
| wl-sbhbt 37091 | Closed form of ~ sbhb . C... |
| wl-sbnf1 37092 | Two ways expressing that `... |
| wl-equsb3 37093 | ~ equsb3 with a distinctor... |
| wl-equsb4 37094 | Substitution applied to an... |
| wl-2sb6d 37095 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 37096 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 37097 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 37098 | Version of ~ sbcom2 with a... |
| wl-sbalnae 37099 | A theorem used in eliminat... |
| wl-sbal1 37100 | A theorem used in eliminat... |
| wl-sbal2 37101 | Move quantifier in and out... |
| wl-2spsbbi 37102 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 37103 | This theorem provides a ba... |
| wl-lem-nexmo 37104 | This theorem provides a ba... |
| wl-lem-moexsb 37105 | The antecedent ` A. x ( ph... |
| wl-alanbii 37106 | This theorem extends ~ ala... |
| wl-mo2df 37107 | Version of ~ mof with a co... |
| wl-mo2tf 37108 | Closed form of ~ mof with ... |
| wl-eudf 37109 | Version of ~ eu6 with a co... |
| wl-eutf 37110 | Closed form of ~ eu6 with ... |
| wl-euequf 37111 | ~ euequ proved with a dist... |
| wl-mo2t 37112 | Closed form of ~ mof . (C... |
| wl-mo3t 37113 | Closed form of ~ mo3 . (C... |
| wl-nfsbtv 37114 | Closed form of ~ nfsbv . ... |
| wl-sb8eut 37115 | Substitution of variable i... |
| wl-sb8eutv 37116 | Substitution of variable i... |
| wl-sb8mot 37117 | Substitution of variable i... |
| wl-sb8motv 37118 | Substitution of variable i... |
| wl-issetft 37119 | A closed form of ~ issetf ... |
| wl-axc11rc11 37120 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 37122 | A transitive law for varia... |
| wl-ax11-lem2 37123 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 37124 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 37125 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 37126 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 37127 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 37128 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 37129 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 37130 | The easy part when ` x ` c... |
| wl-ax11-lem10 37131 | We now have prepared every... |
| wl-clabv 37132 | Variant of ~ df-clab , whe... |
| wl-dfclab 37133 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 37134 | Using class abstraction in... |
| wl-clabt 37135 | Using class abstraction in... |
| rabiun 37136 | Abstraction restricted to ... |
| iundif1 37137 | Indexed union of class dif... |
| imadifss 37138 | The difference of images i... |
| cureq 37139 | Equality theorem for curry... |
| unceq 37140 | Equality theorem for uncur... |
| curf 37141 | Functional property of cur... |
| uncf 37142 | Functional property of unc... |
| curfv 37143 | Value of currying. (Contr... |
| uncov 37144 | Value of uncurrying. (Con... |
| curunc 37145 | Currying of uncurrying. (... |
| unccur 37146 | Uncurrying of currying. (... |
| phpreu 37147 | Theorem related to pigeonh... |
| finixpnum 37148 | A finite Cartesian product... |
| fin2solem 37149 | Lemma for ~ fin2so . (Con... |
| fin2so 37150 | Any totally ordered Tarski... |
| ltflcei 37151 | Theorem to move the floor ... |
| leceifl 37152 | Theorem to move the floor ... |
| sin2h 37153 | Half-angle rule for sine. ... |
| cos2h 37154 | Half-angle rule for cosine... |
| tan2h 37155 | Half-angle rule for tangen... |
| lindsadd 37156 | In a vector space, the uni... |
| lindsdom 37157 | A linearly independent set... |
| lindsenlbs 37158 | A maximal linearly indepen... |
| matunitlindflem1 37159 | One direction of ~ matunit... |
| matunitlindflem2 37160 | One direction of ~ matunit... |
| matunitlindf 37161 | A matrix over a field is i... |
| ptrest 37162 | Expressing a restriction o... |
| ptrecube 37163 | Any point in an open set o... |
| poimirlem1 37164 | Lemma for ~ poimir - the v... |
| poimirlem2 37165 | Lemma for ~ poimir - conse... |
| poimirlem3 37166 | Lemma for ~ poimir to add ... |
| poimirlem4 37167 | Lemma for ~ poimir connect... |
| poimirlem5 37168 | Lemma for ~ poimir to esta... |
| poimirlem6 37169 | Lemma for ~ poimir establi... |
| poimirlem7 37170 | Lemma for ~ poimir , simil... |
| poimirlem8 37171 | Lemma for ~ poimir , estab... |
| poimirlem9 37172 | Lemma for ~ poimir , estab... |
| poimirlem10 37173 | Lemma for ~ poimir establi... |
| poimirlem11 37174 | Lemma for ~ poimir connect... |
| poimirlem12 37175 | Lemma for ~ poimir connect... |
| poimirlem13 37176 | Lemma for ~ poimir - for a... |
| poimirlem14 37177 | Lemma for ~ poimir - for a... |
| poimirlem15 37178 | Lemma for ~ poimir , that ... |
| poimirlem16 37179 | Lemma for ~ poimir establi... |
| poimirlem17 37180 | Lemma for ~ poimir establi... |
| poimirlem18 37181 | Lemma for ~ poimir stating... |
| poimirlem19 37182 | Lemma for ~ poimir establi... |
| poimirlem20 37183 | Lemma for ~ poimir establi... |
| poimirlem21 37184 | Lemma for ~ poimir stating... |
| poimirlem22 37185 | Lemma for ~ poimir , that ... |
| poimirlem23 37186 | Lemma for ~ poimir , two w... |
| poimirlem24 37187 | Lemma for ~ poimir , two w... |
| poimirlem25 37188 | Lemma for ~ poimir stating... |
| poimirlem26 37189 | Lemma for ~ poimir showing... |
| poimirlem27 37190 | Lemma for ~ poimir showing... |
| poimirlem28 37191 | Lemma for ~ poimir , a var... |
| poimirlem29 37192 | Lemma for ~ poimir connect... |
| poimirlem30 37193 | Lemma for ~ poimir combini... |
| poimirlem31 37194 | Lemma for ~ poimir , assig... |
| poimirlem32 37195 | Lemma for ~ poimir , combi... |
| poimir 37196 | Poincare-Miranda theorem. ... |
| broucube 37197 | Brouwer - or as Kulpa call... |
| heicant 37198 | Heine-Cantor theorem: a co... |
| opnmbllem0 37199 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 37200 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 37201 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 37202 | The difference between two... |
| mblfinlem4 37203 | Backward direction of ~ is... |
| ismblfin 37204 | Measurability in terms of ... |
| ovoliunnfl 37205 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 37206 | Demonstration of ~ ovoliun... |
| voliunnfl 37207 | ~ voliun is incompatible w... |
| volsupnfl 37208 | ~ volsup is incompatible w... |
| mbfresfi 37209 | Measurability of a piecewi... |
| mbfposadd 37210 | If the sum of two measurab... |
| cnambfre 37211 | A real-valued, a.e. contin... |
| dvtanlem 37212 | Lemma for ~ dvtan - the do... |
| dvtan 37213 | Derivative of tangent. (C... |
| itg2addnclem 37214 | An alternate expression fo... |
| itg2addnclem2 37215 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 37216 | Lemma incomprehensible in ... |
| itg2addnc 37217 | Alternate proof of ~ itg2a... |
| itg2gt0cn 37218 | ~ itg2gt0 holds on functio... |
| ibladdnclem 37219 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 37220 | Choice-free analogue of ~ ... |
| itgaddnclem1 37221 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 37222 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 37223 | Choice-free analogue of ~ ... |
| iblsubnc 37224 | Choice-free analogue of ~ ... |
| itgsubnc 37225 | Choice-free analogue of ~ ... |
| iblabsnclem 37226 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 37227 | Choice-free analogue of ~ ... |
| iblmulc2nc 37228 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 37229 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 37230 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 37231 | Choice-free analogue of ~ ... |
| itgabsnc 37232 | Choice-free analogue of ~ ... |
| itggt0cn 37233 | ~ itggt0 holds for continu... |
| ftc1cnnclem 37234 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 37235 | Choice-free proof of ~ ftc... |
| ftc1anclem1 37236 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 37237 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 37238 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 37239 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 37240 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 37241 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 37242 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 37243 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 37244 | ~ ftc1a holds for function... |
| ftc2nc 37245 | Choice-free proof of ~ ftc... |
| asindmre 37246 | Real part of domain of dif... |
| dvasin 37247 | Derivative of arcsine. (C... |
| dvacos 37248 | Derivative of arccosine. ... |
| dvreasin 37249 | Real derivative of arcsine... |
| dvreacos 37250 | Real derivative of arccosi... |
| areacirclem1 37251 | Antiderivative of cross-se... |
| areacirclem2 37252 | Endpoint-inclusive continu... |
| areacirclem3 37253 | Integrability of cross-sec... |
| areacirclem4 37254 | Endpoint-inclusive continu... |
| areacirclem5 37255 | Finding the cross-section ... |
| areacirc 37256 | The area of a circle of ra... |
| unirep 37257 | Define a quantity whose de... |
| cover2 37258 | Two ways of expressing the... |
| cover2g 37259 | Two ways of expressing the... |
| brabg2 37260 | Relation by a binary relat... |
| opelopab3 37261 | Ordered pair membership in... |
| cocanfo 37262 | Cancellation of a surjecti... |
| brresi2 37263 | Restriction of a binary re... |
| fnopabeqd 37264 | Equality deduction for fun... |
| fvopabf4g 37265 | Function value of an opera... |
| fnopabco 37266 | Composition of a function ... |
| opropabco 37267 | Composition of an operator... |
| cocnv 37268 | Composition with a functio... |
| f1ocan1fv 37269 | Cancel a composition by a ... |
| f1ocan2fv 37270 | Cancel a composition by th... |
| inixp 37271 | Intersection of Cartesian ... |
| upixp 37272 | Universal property of the ... |
| abrexdom 37273 | An indexed set is dominate... |
| abrexdom2 37274 | An indexed set is dominate... |
| ac6gf 37275 | Axiom of Choice. (Contrib... |
| indexa 37276 | If for every element of an... |
| indexdom 37277 | If for every element of an... |
| frinfm 37278 | A subset of a well-founded... |
| welb 37279 | A nonempty subset of a wel... |
| supex2g 37280 | Existence of supremum. (C... |
| supclt 37281 | Closure of supremum. (Con... |
| supubt 37282 | Upper bound property of su... |
| filbcmb 37283 | Combine a finite set of lo... |
| fzmul 37284 | Membership of a product in... |
| sdclem2 37285 | Lemma for ~ sdc . (Contri... |
| sdclem1 37286 | Lemma for ~ sdc . (Contri... |
| sdc 37287 | Strong dependent choice. ... |
| fdc 37288 | Finite version of dependen... |
| fdc1 37289 | Variant of ~ fdc with no s... |
| seqpo 37290 | Two ways to say that a seq... |
| incsequz 37291 | An increasing sequence of ... |
| incsequz2 37292 | An increasing sequence of ... |
| nnubfi 37293 | A bounded above set of pos... |
| nninfnub 37294 | An infinite set of positiv... |
| subspopn 37295 | An open set is open in the... |
| neificl 37296 | Neighborhoods are closed u... |
| lpss2 37297 | Limit points of a subset a... |
| metf1o 37298 | Use a bijection with a met... |
| blssp 37299 | A ball in the subspace met... |
| mettrifi 37300 | Generalized triangle inequ... |
| lmclim2 37301 | A sequence in a metric spa... |
| geomcau 37302 | If the distance between co... |
| caures 37303 | The restriction of a Cauch... |
| caushft 37304 | A shifted Cauchy sequence ... |
| constcncf 37305 | A constant function is a c... |
| cnres2 37306 | The restriction of a conti... |
| cnresima 37307 | A continuous function is c... |
| cncfres 37308 | A continuous function on c... |
| istotbnd 37312 | The predicate "is a totall... |
| istotbnd2 37313 | The predicate "is a totall... |
| istotbnd3 37314 | A metric space is totally ... |
| totbndmet 37315 | The predicate "totally bou... |
| 0totbnd 37316 | The metric (there is only ... |
| sstotbnd2 37317 | Condition for a subset of ... |
| sstotbnd 37318 | Condition for a subset of ... |
| sstotbnd3 37319 | Use a net that is not nece... |
| totbndss 37320 | A subset of a totally boun... |
| equivtotbnd 37321 | If the metric ` M ` is "st... |
| isbnd 37323 | The predicate "is a bounde... |
| bndmet 37324 | A bounded metric space is ... |
| isbndx 37325 | A "bounded extended metric... |
| isbnd2 37326 | The predicate "is a bounde... |
| isbnd3 37327 | A metric space is bounded ... |
| isbnd3b 37328 | A metric space is bounded ... |
| bndss 37329 | A subset of a bounded metr... |
| blbnd 37330 | A ball is bounded. (Contr... |
| ssbnd 37331 | A subset of a metric space... |
| totbndbnd 37332 | A totally bounded metric s... |
| equivbnd 37333 | If the metric ` M ` is "st... |
| bnd2lem 37334 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 37335 | If balls are totally bound... |
| prdsbnd 37336 | The product metric over fi... |
| prdstotbnd 37337 | The product metric over fi... |
| prdsbnd2 37338 | If balls are totally bound... |
| cntotbnd 37339 | A subset of the complex nu... |
| cnpwstotbnd 37340 | A subset of ` A ^ I ` , wh... |
| ismtyval 37343 | The set of isometries betw... |
| isismty 37344 | The condition "is an isome... |
| ismtycnv 37345 | The inverse of an isometry... |
| ismtyima 37346 | The image of a ball under ... |
| ismtyhmeolem 37347 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 37348 | An isometry is a homeomorp... |
| ismtybndlem 37349 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 37350 | Isometries preserve bounde... |
| ismtyres 37351 | A restriction of an isomet... |
| heibor1lem 37352 | Lemma for ~ heibor1 . A c... |
| heibor1 37353 | One half of ~ heibor , tha... |
| heiborlem1 37354 | Lemma for ~ heibor . We w... |
| heiborlem2 37355 | Lemma for ~ heibor . Subs... |
| heiborlem3 37356 | Lemma for ~ heibor . Usin... |
| heiborlem4 37357 | Lemma for ~ heibor . Usin... |
| heiborlem5 37358 | Lemma for ~ heibor . The ... |
| heiborlem6 37359 | Lemma for ~ heibor . Sinc... |
| heiborlem7 37360 | Lemma for ~ heibor . Sinc... |
| heiborlem8 37361 | Lemma for ~ heibor . The ... |
| heiborlem9 37362 | Lemma for ~ heibor . Disc... |
| heiborlem10 37363 | Lemma for ~ heibor . The ... |
| heibor 37364 | Generalized Heine-Borel Th... |
| bfplem1 37365 | Lemma for ~ bfp . The seq... |
| bfplem2 37366 | Lemma for ~ bfp . Using t... |
| bfp 37367 | Banach fixed point theorem... |
| rrnval 37370 | The n-dimensional Euclidea... |
| rrnmval 37371 | The value of the Euclidean... |
| rrnmet 37372 | Euclidean space is a metri... |
| rrndstprj1 37373 | The distance between two p... |
| rrndstprj2 37374 | Bound on the distance betw... |
| rrncmslem 37375 | Lemma for ~ rrncms . (Con... |
| rrncms 37376 | Euclidean space is complet... |
| repwsmet 37377 | The supremum metric on ` R... |
| rrnequiv 37378 | The supremum metric on ` R... |
| rrntotbnd 37379 | A set in Euclidean space i... |
| rrnheibor 37380 | Heine-Borel theorem for Eu... |
| ismrer1 37381 | An isometry between ` RR `... |
| reheibor 37382 | Heine-Borel theorem for re... |
| iccbnd 37383 | A closed interval in ` RR ... |
| icccmpALT 37384 | A closed interval in ` RR ... |
| isass 37389 | The predicate "is an assoc... |
| isexid 37390 | The predicate ` G ` has a ... |
| ismgmOLD 37393 | Obsolete version of ~ ismg... |
| clmgmOLD 37394 | Obsolete version of ~ mgmc... |
| opidonOLD 37395 | Obsolete version of ~ mndp... |
| rngopidOLD 37396 | Obsolete version of ~ mndp... |
| opidon2OLD 37397 | Obsolete version of ~ mndp... |
| isexid2 37398 | If ` G e. ( Magma i^i ExId... |
| exidu1 37399 | Uniqueness of the left and... |
| idrval 37400 | The value of the identity ... |
| iorlid 37401 | A magma right and left ide... |
| cmpidelt 37402 | A magma right and left ide... |
| smgrpismgmOLD 37405 | Obsolete version of ~ sgrp... |
| issmgrpOLD 37406 | Obsolete version of ~ issg... |
| smgrpmgm 37407 | A semigroup is a magma. (... |
| smgrpassOLD 37408 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 37411 | Obsolete version of ~ mnds... |
| mndoisexid 37412 | A monoid has an identity e... |
| mndoismgmOLD 37413 | Obsolete version of ~ mndm... |
| mndomgmid 37414 | A monoid is a magma with a... |
| ismndo 37415 | The predicate "is a monoid... |
| ismndo1 37416 | The predicate "is a monoid... |
| ismndo2 37417 | The predicate "is a monoid... |
| grpomndo 37418 | A group is a monoid. (Con... |
| exidcl 37419 | Closure of the binary oper... |
| exidreslem 37420 | Lemma for ~ exidres and ~ ... |
| exidres 37421 | The restriction of a binar... |
| exidresid 37422 | The restriction of a binar... |
| ablo4pnp 37423 | A commutative/associative ... |
| grpoeqdivid 37424 | Two group elements are equ... |
| grposnOLD 37425 | The group operation for th... |
| elghomlem1OLD 37428 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 37429 | Obsolete as of 15-Mar-2020... |
| elghomOLD 37430 | Obsolete version of ~ isgh... |
| ghomlinOLD 37431 | Obsolete version of ~ ghml... |
| ghomidOLD 37432 | Obsolete version of ~ ghmi... |
| ghomf 37433 | Mapping property of a grou... |
| ghomco 37434 | The composition of two gro... |
| ghomdiv 37435 | Group homomorphisms preser... |
| grpokerinj 37436 | A group homomorphism is in... |
| relrngo 37439 | The class of all unital ri... |
| isrngo 37440 | The predicate "is a (unita... |
| isrngod 37441 | Conditions that determine ... |
| rngoi 37442 | The properties of a unital... |
| rngosm 37443 | Functionality of the multi... |
| rngocl 37444 | Closure of the multiplicat... |
| rngoid 37445 | The multiplication operati... |
| rngoideu 37446 | The unity element of a rin... |
| rngodi 37447 | Distributive law for the m... |
| rngodir 37448 | Distributive law for the m... |
| rngoass 37449 | Associative law for the mu... |
| rngo2 37450 | A ring element plus itself... |
| rngoablo 37451 | A ring's addition operatio... |
| rngoablo2 37452 | In a unital ring the addit... |
| rngogrpo 37453 | A ring's addition operatio... |
| rngone0 37454 | The base set of a ring is ... |
| rngogcl 37455 | Closure law for the additi... |
| rngocom 37456 | The addition operation of ... |
| rngoaass 37457 | The addition operation of ... |
| rngoa32 37458 | The addition operation of ... |
| rngoa4 37459 | Rearrangement of 4 terms i... |
| rngorcan 37460 | Right cancellation law for... |
| rngolcan 37461 | Left cancellation law for ... |
| rngo0cl 37462 | A ring has an additive ide... |
| rngo0rid 37463 | The additive identity of a... |
| rngo0lid 37464 | The additive identity of a... |
| rngolz 37465 | The zero of a unital ring ... |
| rngorz 37466 | The zero of a unital ring ... |
| rngosn3 37467 | Obsolete as of 25-Jan-2020... |
| rngosn4 37468 | Obsolete as of 25-Jan-2020... |
| rngosn6 37469 | Obsolete as of 25-Jan-2020... |
| rngonegcl 37470 | A ring is closed under neg... |
| rngoaddneg1 37471 | Adding the negative in a r... |
| rngoaddneg2 37472 | Adding the negative in a r... |
| rngosub 37473 | Subtraction in a ring, in ... |
| rngmgmbs4 37474 | The range of an internal o... |
| rngodm1dm2 37475 | In a unital ring the domai... |
| rngorn1 37476 | In a unital ring the range... |
| rngorn1eq 37477 | In a unital ring the range... |
| rngomndo 37478 | In a unital ring the multi... |
| rngoidmlem 37479 | The unity element of a rin... |
| rngolidm 37480 | The unity element of a rin... |
| rngoridm 37481 | The unity element of a rin... |
| rngo1cl 37482 | The unity element of a rin... |
| rngoueqz 37483 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 37484 | Negation in a ring is the ... |
| rngonegmn1r 37485 | Negation in a ring is the ... |
| rngoneglmul 37486 | Negation of a product in a... |
| rngonegrmul 37487 | Negation of a product in a... |
| rngosubdi 37488 | Ring multiplication distri... |
| rngosubdir 37489 | Ring multiplication distri... |
| zerdivemp1x 37490 | In a unital ring a left in... |
| isdivrngo 37493 | The predicate "is a divisi... |
| drngoi 37494 | The properties of a divisi... |
| gidsn 37495 | Obsolete as of 23-Jan-2020... |
| zrdivrng 37496 | The zero ring is not a div... |
| dvrunz 37497 | In a division ring the rin... |
| isgrpda 37498 | Properties that determine ... |
| isdrngo1 37499 | The predicate "is a divisi... |
| divrngcl 37500 | The product of two nonzero... |
| isdrngo2 37501 | A division ring is a ring ... |
| isdrngo3 37502 | A division ring is a ring ... |
| rngohomval 37507 | The set of ring homomorphi... |
| isrngohom 37508 | The predicate "is a ring h... |
| rngohomf 37509 | A ring homomorphism is a f... |
| rngohomcl 37510 | Closure law for a ring hom... |
| rngohom1 37511 | A ring homomorphism preser... |
| rngohomadd 37512 | Ring homomorphisms preserv... |
| rngohommul 37513 | Ring homomorphisms preserv... |
| rngogrphom 37514 | A ring homomorphism is a g... |
| rngohom0 37515 | A ring homomorphism preser... |
| rngohomsub 37516 | Ring homomorphisms preserv... |
| rngohomco 37517 | The composition of two rin... |
| rngokerinj 37518 | A ring homomorphism is inj... |
| rngoisoval 37520 | The set of ring isomorphis... |
| isrngoiso 37521 | The predicate "is a ring i... |
| rngoiso1o 37522 | A ring isomorphism is a bi... |
| rngoisohom 37523 | A ring isomorphism is a ri... |
| rngoisocnv 37524 | The inverse of a ring isom... |
| rngoisoco 37525 | The composition of two rin... |
| isriscg 37527 | The ring isomorphism relat... |
| isrisc 37528 | The ring isomorphism relat... |
| risc 37529 | The ring isomorphism relat... |
| risci 37530 | Determine that two rings a... |
| riscer 37531 | Ring isomorphism is an equ... |
| iscom2 37538 | A device to add commutativ... |
| iscrngo 37539 | The predicate "is a commut... |
| iscrngo2 37540 | The predicate "is a commut... |
| iscringd 37541 | Conditions that determine ... |
| flddivrng 37542 | A field is a division ring... |
| crngorngo 37543 | A commutative ring is a ri... |
| crngocom 37544 | The multiplication operati... |
| crngm23 37545 | Commutative/associative la... |
| crngm4 37546 | Commutative/associative la... |
| fldcrngo 37547 | A field is a commutative r... |
| isfld2 37548 | The predicate "is a field"... |
| crngohomfo 37549 | The image of a homomorphis... |
| idlval 37556 | The class of ideals of a r... |
| isidl 37557 | The predicate "is an ideal... |
| isidlc 37558 | The predicate "is an ideal... |
| idlss 37559 | An ideal of ` R ` is a sub... |
| idlcl 37560 | An element of an ideal is ... |
| idl0cl 37561 | An ideal contains ` 0 ` . ... |
| idladdcl 37562 | An ideal is closed under a... |
| idllmulcl 37563 | An ideal is closed under m... |
| idlrmulcl 37564 | An ideal is closed under m... |
| idlnegcl 37565 | An ideal is closed under n... |
| idlsubcl 37566 | An ideal is closed under s... |
| rngoidl 37567 | A ring ` R ` is an ` R ` i... |
| 0idl 37568 | The set containing only ` ... |
| 1idl 37569 | Two ways of expressing the... |
| 0rngo 37570 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 37571 | The only ideals in a divis... |
| intidl 37572 | The intersection of a none... |
| inidl 37573 | The intersection of two id... |
| unichnidl 37574 | The union of a nonempty ch... |
| keridl 37575 | The kernel of a ring homom... |
| pridlval 37576 | The class of prime ideals ... |
| ispridl 37577 | The predicate "is a prime ... |
| pridlidl 37578 | A prime ideal is an ideal.... |
| pridlnr 37579 | A prime ideal is a proper ... |
| pridl 37580 | The main property of a pri... |
| ispridl2 37581 | A condition that shows an ... |
| maxidlval 37582 | The set of maximal ideals ... |
| ismaxidl 37583 | The predicate "is a maxima... |
| maxidlidl 37584 | A maximal ideal is an idea... |
| maxidlnr 37585 | A maximal ideal is proper.... |
| maxidlmax 37586 | A maximal ideal is a maxim... |
| maxidln1 37587 | One is not contained in an... |
| maxidln0 37588 | A ring with a maximal idea... |
| isprrngo 37593 | The predicate "is a prime ... |
| prrngorngo 37594 | A prime ring is a ring. (... |
| smprngopr 37595 | A simple ring (one whose o... |
| divrngpr 37596 | A division ring is a prime... |
| isdmn 37597 | The predicate "is a domain... |
| isdmn2 37598 | The predicate "is a domain... |
| dmncrng 37599 | A domain is a commutative ... |
| dmnrngo 37600 | A domain is a ring. (Cont... |
| flddmn 37601 | A field is a domain. (Con... |
| igenval 37604 | The ideal generated by a s... |
| igenss 37605 | A set is a subset of the i... |
| igenidl 37606 | The ideal generated by a s... |
| igenmin 37607 | The ideal generated by a s... |
| igenidl2 37608 | The ideal generated by an ... |
| igenval2 37609 | The ideal generated by a s... |
| prnc 37610 | A principal ideal (an idea... |
| isfldidl 37611 | Determine if a ring is a f... |
| isfldidl2 37612 | Determine if a ring is a f... |
| ispridlc 37613 | The predicate "is a prime ... |
| pridlc 37614 | Property of a prime ideal ... |
| pridlc2 37615 | Property of a prime ideal ... |
| pridlc3 37616 | Property of a prime ideal ... |
| isdmn3 37617 | The predicate "is a domain... |
| dmnnzd 37618 | A domain has no zero-divis... |
| dmncan1 37619 | Cancellation law for domai... |
| dmncan2 37620 | Cancellation law for domai... |
| efald2 37621 | A proof by contradiction. ... |
| notbinot1 37622 | Simplification rule of neg... |
| bicontr 37623 | Biconditional of its own n... |
| impor 37624 | An equivalent formula for ... |
| orfa 37625 | The falsum ` F. ` can be r... |
| notbinot2 37626 | Commutation rule between n... |
| biimpor 37627 | A rewriting rule for bicon... |
| orfa1 37628 | Add a contradicting disjun... |
| orfa2 37629 | Remove a contradicting dis... |
| bifald 37630 | Infer the equivalence to a... |
| orsild 37631 | A lemma for not-or-not eli... |
| orsird 37632 | A lemma for not-or-not eli... |
| cnf1dd 37633 | A lemma for Conjunctive No... |
| cnf2dd 37634 | A lemma for Conjunctive No... |
| cnfn1dd 37635 | A lemma for Conjunctive No... |
| cnfn2dd 37636 | A lemma for Conjunctive No... |
| or32dd 37637 | A rearrangement of disjunc... |
| notornotel1 37638 | A lemma for not-or-not eli... |
| notornotel2 37639 | A lemma for not-or-not eli... |
| contrd 37640 | A proof by contradiction, ... |
| an12i 37641 | An inference from commutin... |
| exmid2 37642 | An excluded middle law. (... |
| selconj 37643 | An inference for selecting... |
| truconj 37644 | Add true as a conjunct. (... |
| orel 37645 | An inference for disjuncti... |
| negel 37646 | An inference for negation ... |
| botel 37647 | An inference for bottom el... |
| tradd 37648 | Add top ad a conjunct. (C... |
| gm-sbtru 37649 | Substitution does not chan... |
| sbfal 37650 | Substitution does not chan... |
| sbcani 37651 | Distribution of class subs... |
| sbcori 37652 | Distribution of class subs... |
| sbcimi 37653 | Distribution of class subs... |
| sbcni 37654 | Move class substitution in... |
| sbali 37655 | Discard class substitution... |
| sbexi 37656 | Discard class substitution... |
| sbcalf 37657 | Move universal quantifier ... |
| sbcexf 37658 | Move existential quantifie... |
| sbcalfi 37659 | Move universal quantifier ... |
| sbcexfi 37660 | Move existential quantifie... |
| spsbcdi 37661 | A lemma for eliminating a ... |
| alrimii 37662 | A lemma for introducing a ... |
| spesbcdi 37663 | A lemma for introducing an... |
| exlimddvf 37664 | A lemma for eliminating an... |
| exlimddvfi 37665 | A lemma for eliminating an... |
| sbceq1ddi 37666 | A lemma for eliminating in... |
| sbccom2lem 37667 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 37668 | Commutative law for double... |
| sbccom2f 37669 | Commutative law for double... |
| sbccom2fi 37670 | Commutative law for double... |
| csbcom2fi 37671 | Commutative law for double... |
| fald 37672 | Refutation of falsity, in ... |
| tsim1 37673 | A Tseitin axiom for logica... |
| tsim2 37674 | A Tseitin axiom for logica... |
| tsim3 37675 | A Tseitin axiom for logica... |
| tsbi1 37676 | A Tseitin axiom for logica... |
| tsbi2 37677 | A Tseitin axiom for logica... |
| tsbi3 37678 | A Tseitin axiom for logica... |
| tsbi4 37679 | A Tseitin axiom for logica... |
| tsxo1 37680 | A Tseitin axiom for logica... |
| tsxo2 37681 | A Tseitin axiom for logica... |
| tsxo3 37682 | A Tseitin axiom for logica... |
| tsxo4 37683 | A Tseitin axiom for logica... |
| tsan1 37684 | A Tseitin axiom for logica... |
| tsan2 37685 | A Tseitin axiom for logica... |
| tsan3 37686 | A Tseitin axiom for logica... |
| tsna1 37687 | A Tseitin axiom for logica... |
| tsna2 37688 | A Tseitin axiom for logica... |
| tsna3 37689 | A Tseitin axiom for logica... |
| tsor1 37690 | A Tseitin axiom for logica... |
| tsor2 37691 | A Tseitin axiom for logica... |
| tsor3 37692 | A Tseitin axiom for logica... |
| ts3an1 37693 | A Tseitin axiom for triple... |
| ts3an2 37694 | A Tseitin axiom for triple... |
| ts3an3 37695 | A Tseitin axiom for triple... |
| ts3or1 37696 | A Tseitin axiom for triple... |
| ts3or2 37697 | A Tseitin axiom for triple... |
| ts3or3 37698 | A Tseitin axiom for triple... |
| iuneq2f 37699 | Equality deduction for ind... |
| rabeq12f 37700 | Equality deduction for res... |
| csbeq12 37701 | Equality deduction for sub... |
| sbeqi 37702 | Equality deduction for sub... |
| ralbi12f 37703 | Equality deduction for res... |
| oprabbi 37704 | Equality deduction for cla... |
| mpobi123f 37705 | Equality deduction for map... |
| iuneq12f 37706 | Equality deduction for ind... |
| iineq12f 37707 | Equality deduction for ind... |
| opabbi 37708 | Equality deduction for cla... |
| mptbi12f 37709 | Equality deduction for map... |
| orcomdd 37710 | Commutativity of logic dis... |
| scottexf 37711 | A version of ~ scottex wit... |
| scott0f 37712 | A version of ~ scott0 with... |
| scottn0f 37713 | A version of ~ scott0f wit... |
| ac6s3f 37714 | Generalization of the Axio... |
| ac6s6 37715 | Generalization of the Axio... |
| ac6s6f 37716 | Generalization of the Axio... |
| el2v1 37760 | New way ( ~ elv , and the ... |
| el3v 37761 | New way ( ~ elv , and the ... |
| el3v1 37762 | New way ( ~ elv , and the ... |
| el3v2 37763 | New way ( ~ elv , and the ... |
| el3v3 37764 | New way ( ~ elv , and the ... |
| el3v12 37765 | New way ( ~ elv , and the ... |
| el3v13 37766 | New way ( ~ elv , and the ... |
| el3v23 37767 | New way ( ~ elv , and the ... |
| anan 37768 | Multiple commutations in c... |
| triantru3 37769 | A wff is equivalent to its... |
| bianim 37770 | Exchanging conjunction in ... |
| biorfd 37771 | A wff is equivalent to its... |
| eqbrtr 37772 | Substitution of equal clas... |
| eqbrb 37773 | Substitution of equal clas... |
| eqeltr 37774 | Substitution of equal clas... |
| eqelb 37775 | Substitution of equal clas... |
| eqeqan2d 37776 | Implication of introducing... |
| suceqsneq 37777 | One-to-one relationship be... |
| sucdifsn2 37778 | Absorption of union with a... |
| sucdifsn 37779 | The difference between the... |
| disjresin 37780 | The restriction to a disjo... |
| disjresdisj 37781 | The intersection of restri... |
| disjresdif 37782 | The difference between res... |
| disjresundif 37783 | Lemma for ~ ressucdifsn2 .... |
| ressucdifsn2 37784 | The difference between res... |
| ressucdifsn 37785 | The difference between res... |
| inres2 37786 | Two ways of expressing the... |
| coideq 37787 | Equality theorem for compo... |
| nexmo1 37788 | If there is no case where ... |
| ralin 37789 | Restricted universal quant... |
| r2alan 37790 | Double restricted universa... |
| ssrabi 37791 | Inference of restricted ab... |
| rabimbieq 37792 | Restricted equivalent wff'... |
| abeqin 37793 | Intersection with class ab... |
| abeqinbi 37794 | Intersection with class ab... |
| rabeqel 37795 | Class element of a restric... |
| eqrelf 37796 | The equality connective be... |
| br1cnvinxp 37797 | Binary relation on the con... |
| releleccnv 37798 | Elementhood in a converse ... |
| releccnveq 37799 | Equality of converse ` R `... |
| opelvvdif 37800 | Negated elementhood of ord... |
| vvdifopab 37801 | Ordered-pair class abstrac... |
| brvdif 37802 | Binary relation with unive... |
| brvdif2 37803 | Binary relation with unive... |
| brvvdif 37804 | Binary relation with the c... |
| brvbrvvdif 37805 | Binary relation with the c... |
| brcnvep 37806 | The converse of the binary... |
| elecALTV 37807 | Elementhood in the ` R ` -... |
| brcnvepres 37808 | Restricted converse epsilo... |
| brres2 37809 | Binary relation on a restr... |
| br1cnvres 37810 | Binary relation on the con... |
| eldmres 37811 | Elementhood in the domain ... |
| elrnres 37812 | Element of the range of a ... |
| eldmressnALTV 37813 | Element of the domain of a... |
| elrnressn 37814 | Element of the range of a ... |
| eldm4 37815 | Elementhood in a domain. ... |
| eldmres2 37816 | Elementhood in the domain ... |
| eceq1i 37817 | Equality theorem for ` C `... |
| elecres 37818 | Elementhood in the restric... |
| ecres 37819 | Restricted coset of ` B ` ... |
| ecres2 37820 | The restricted coset of ` ... |
| eccnvepres 37821 | Restricted converse epsilo... |
| eleccnvep 37822 | Elementhood in the convers... |
| eccnvep 37823 | The converse epsilon coset... |
| extep 37824 | Property of epsilon relati... |
| disjeccnvep 37825 | Property of the epsilon re... |
| eccnvepres2 37826 | The restricted converse ep... |
| eccnvepres3 37827 | Condition for a restricted... |
| eldmqsres 37828 | Elementhood in a restricte... |
| eldmqsres2 37829 | Elementhood in a restricte... |
| qsss1 37830 | Subclass theorem for quoti... |
| qseq1i 37831 | Equality theorem for quoti... |
| brinxprnres 37832 | Binary relation on a restr... |
| inxprnres 37833 | Restriction of a class as ... |
| dfres4 37834 | Alternate definition of th... |
| exan3 37835 | Equivalent expressions wit... |
| exanres 37836 | Equivalent expressions wit... |
| exanres3 37837 | Equivalent expressions wit... |
| exanres2 37838 | Equivalent expressions wit... |
| cnvepres 37839 | Restricted converse epsilo... |
| eqrel2 37840 | Equality of relations. (C... |
| rncnv 37841 | Range of converse is the d... |
| dfdm6 37842 | Alternate definition of do... |
| dfrn6 37843 | Alternate definition of ra... |
| rncnvepres 37844 | The range of the restricte... |
| dmecd 37845 | Equality of the coset of `... |
| dmec2d 37846 | Equality of the coset of `... |
| brid 37847 | Property of the identity b... |
| ideq2 37848 | For sets, the identity bin... |
| idresssidinxp 37849 | Condition for the identity... |
| idreseqidinxp 37850 | Condition for the identity... |
| extid 37851 | Property of identity relat... |
| inxpss 37852 | Two ways to say that an in... |
| idinxpss 37853 | Two ways to say that an in... |
| ref5 37854 | Two ways to say that an in... |
| inxpss3 37855 | Two ways to say that an in... |
| inxpss2 37856 | Two ways to say that inter... |
| inxpssidinxp 37857 | Two ways to say that inter... |
| idinxpssinxp 37858 | Two ways to say that inter... |
| idinxpssinxp2 37859 | Identity intersection with... |
| idinxpssinxp3 37860 | Identity intersection with... |
| idinxpssinxp4 37861 | Identity intersection with... |
| relcnveq3 37862 | Two ways of saying a relat... |
| relcnveq 37863 | Two ways of saying a relat... |
| relcnveq2 37864 | Two ways of saying a relat... |
| relcnveq4 37865 | Two ways of saying a relat... |
| qsresid 37866 | Simplification of a specia... |
| n0elqs 37867 | Two ways of expressing tha... |
| n0elqs2 37868 | Two ways of expressing tha... |
| ecex2 37869 | Condition for a coset to b... |
| uniqsALTV 37870 | The union of a quotient se... |
| imaexALTV 37871 | Existence of an image of a... |
| ecexALTV 37872 | Existence of a coset, like... |
| rnresequniqs 37873 | The range of a restriction... |
| n0el2 37874 | Two ways of expressing tha... |
| cnvepresex 37875 | Sethood condition for the ... |
| eccnvepex 37876 | The converse epsilon coset... |
| cnvepimaex 37877 | The image of converse epsi... |
| cnvepima 37878 | The image of converse epsi... |
| inex3 37879 | Sufficient condition for t... |
| inxpex 37880 | Sufficient condition for a... |
| eqres 37881 | Converting a class constan... |
| brrabga 37882 | The law of concretion for ... |
| brcnvrabga 37883 | The law of concretion for ... |
| opideq 37884 | Equality conditions for or... |
| iss2 37885 | A subclass of the identity... |
| eldmcnv 37886 | Elementhood in a domain of... |
| dfrel5 37887 | Alternate definition of th... |
| dfrel6 37888 | Alternate definition of th... |
| cnvresrn 37889 | Converse restricted to ran... |
| relssinxpdmrn 37890 | Subset of restriction, spe... |
| cnvref4 37891 | Two ways to say that a rel... |
| cnvref5 37892 | Two ways to say that a rel... |
| ecin0 37893 | Two ways of saying that th... |
| ecinn0 37894 | Two ways of saying that th... |
| ineleq 37895 | Equivalence of restricted ... |
| inecmo 37896 | Equivalence of a double re... |
| inecmo2 37897 | Equivalence of a double re... |
| ineccnvmo 37898 | Equivalence of a double re... |
| alrmomorn 37899 | Equivalence of an "at most... |
| alrmomodm 37900 | Equivalence of an "at most... |
| ineccnvmo2 37901 | Equivalence of a double un... |
| inecmo3 37902 | Equivalence of a double un... |
| moeu2 37903 | Uniqueness is equivalent t... |
| mopickr 37904 | "At most one" picks a vari... |
| moantr 37905 | Sufficient condition for t... |
| brabidgaw 37906 | The law of concretion for ... |
| brabidga 37907 | The law of concretion for ... |
| inxp2 37908 | Intersection with a Cartes... |
| opabf 37909 | A class abstraction of a c... |
| ec0 37910 | The empty-coset of a class... |
| brcnvin 37911 | Intersection with a conver... |
| xrnss3v 37913 | A range Cartesian product ... |
| xrnrel 37914 | A range Cartesian product ... |
| brxrn 37915 | Characterize a ternary rel... |
| brxrn2 37916 | A characterization of the ... |
| dfxrn2 37917 | Alternate definition of th... |
| xrneq1 37918 | Equality theorem for the r... |
| xrneq1i 37919 | Equality theorem for the r... |
| xrneq1d 37920 | Equality theorem for the r... |
| xrneq2 37921 | Equality theorem for the r... |
| xrneq2i 37922 | Equality theorem for the r... |
| xrneq2d 37923 | Equality theorem for the r... |
| xrneq12 37924 | Equality theorem for the r... |
| xrneq12i 37925 | Equality theorem for the r... |
| xrneq12d 37926 | Equality theorem for the r... |
| elecxrn 37927 | Elementhood in the ` ( R |... |
| ecxrn 37928 | The ` ( R |X. S ) ` -coset... |
| disjressuc2 37929 | Double restricted quantifi... |
| disjecxrn 37930 | Two ways of saying that ` ... |
| disjecxrncnvep 37931 | Two ways of saying that co... |
| disjsuc2 37932 | Double restricted quantifi... |
| xrninxp 37933 | Intersection of a range Ca... |
| xrninxp2 37934 | Intersection of a range Ca... |
| xrninxpex 37935 | Sufficient condition for t... |
| inxpxrn 37936 | Two ways to express the in... |
| br1cnvxrn2 37937 | The converse of a binary r... |
| elec1cnvxrn2 37938 | Elementhood in the convers... |
| rnxrn 37939 | Range of the range Cartesi... |
| rnxrnres 37940 | Range of a range Cartesian... |
| rnxrncnvepres 37941 | Range of a range Cartesian... |
| rnxrnidres 37942 | Range of a range Cartesian... |
| xrnres 37943 | Two ways to express restri... |
| xrnres2 37944 | Two ways to express restri... |
| xrnres3 37945 | Two ways to express restri... |
| xrnres4 37946 | Two ways to express restri... |
| xrnresex 37947 | Sufficient condition for a... |
| xrnidresex 37948 | Sufficient condition for a... |
| xrncnvepresex 37949 | Sufficient condition for a... |
| brin2 37950 | Binary relation on an inte... |
| brin3 37951 | Binary relation on an inte... |
| dfcoss2 37954 | Alternate definition of th... |
| dfcoss3 37955 | Alternate definition of th... |
| dfcoss4 37956 | Alternate definition of th... |
| cosscnv 37957 | Class of cosets by the con... |
| coss1cnvres 37958 | Class of cosets by the con... |
| coss2cnvepres 37959 | Special case of ~ coss1cnv... |
| cossex 37960 | If ` A ` is a set then the... |
| cosscnvex 37961 | If ` A ` is a set then the... |
| 1cosscnvepresex 37962 | Sufficient condition for a... |
| 1cossxrncnvepresex 37963 | Sufficient condition for a... |
| relcoss 37964 | Cosets by ` R ` is a relat... |
| relcoels 37965 | Coelements on ` A ` is a r... |
| cossss 37966 | Subclass theorem for the c... |
| cosseq 37967 | Equality theorem for the c... |
| cosseqi 37968 | Equality theorem for the c... |
| cosseqd 37969 | Equality theorem for the c... |
| 1cossres 37970 | The class of cosets by a r... |
| dfcoels 37971 | Alternate definition of th... |
| brcoss 37972 | ` A ` and ` B ` are cosets... |
| brcoss2 37973 | Alternate form of the ` A ... |
| brcoss3 37974 | Alternate form of the ` A ... |
| brcosscnvcoss 37975 | For sets, the ` A ` and ` ... |
| brcoels 37976 | ` B ` and ` C ` are coelem... |
| cocossss 37977 | Two ways of saying that co... |
| cnvcosseq 37978 | The converse of cosets by ... |
| br2coss 37979 | Cosets by ` ,~ R ` binary ... |
| br1cossres 37980 | ` B ` and ` C ` are cosets... |
| br1cossres2 37981 | ` B ` and ` C ` are cosets... |
| brressn 37982 | Binary relation on a restr... |
| ressn2 37983 | A class ' R ' restricted t... |
| refressn 37984 | Any class ' R ' restricted... |
| antisymressn 37985 | Every class ' R ' restrict... |
| trressn 37986 | Any class ' R ' restricted... |
| relbrcoss 37987 | ` A ` and ` B ` are cosets... |
| br1cossinres 37988 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 37989 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 37990 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 37991 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 37992 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 37993 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 37994 | The domain of cosets is th... |
| dmcoss2 37995 | The domain of cosets is th... |
| rncossdmcoss 37996 | The range of cosets is the... |
| dm1cosscnvepres 37997 | The domain of cosets of th... |
| dmcoels 37998 | The domain of coelements i... |
| eldmcoss 37999 | Elementhood in the domain ... |
| eldmcoss2 38000 | Elementhood in the domain ... |
| eldm1cossres 38001 | Elementhood in the domain ... |
| eldm1cossres2 38002 | Elementhood in the domain ... |
| refrelcosslem 38003 | Lemma for the left side of... |
| refrelcoss3 38004 | The class of cosets by ` R... |
| refrelcoss2 38005 | The class of cosets by ` R... |
| symrelcoss3 38006 | The class of cosets by ` R... |
| symrelcoss2 38007 | The class of cosets by ` R... |
| cossssid 38008 | Equivalent expressions for... |
| cossssid2 38009 | Equivalent expressions for... |
| cossssid3 38010 | Equivalent expressions for... |
| cossssid4 38011 | Equivalent expressions for... |
| cossssid5 38012 | Equivalent expressions for... |
| brcosscnv 38013 | ` A ` and ` B ` are cosets... |
| brcosscnv2 38014 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 38015 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 38016 | Cosets by the converse ran... |
| cosscnvssid3 38017 | Equivalent expressions for... |
| cosscnvssid4 38018 | Equivalent expressions for... |
| cosscnvssid5 38019 | Equivalent expressions for... |
| coss0 38020 | Cosets by the empty set ar... |
| cossid 38021 | Cosets by the identity rel... |
| cosscnvid 38022 | Cosets by the converse ide... |
| trcoss 38023 | Sufficient condition for t... |
| eleccossin 38024 | Two ways of saying that th... |
| trcoss2 38025 | Equivalent expressions for... |
| elrels2 38027 | The element of the relatio... |
| elrelsrel 38028 | The element of the relatio... |
| elrelsrelim 38029 | The element of the relatio... |
| elrels5 38030 | Equivalent expressions for... |
| elrels6 38031 | Equivalent expressions for... |
| elrelscnveq3 38032 | Two ways of saying a relat... |
| elrelscnveq 38033 | Two ways of saying a relat... |
| elrelscnveq2 38034 | Two ways of saying a relat... |
| elrelscnveq4 38035 | Two ways of saying a relat... |
| cnvelrels 38036 | The converse of a set is a... |
| cosselrels 38037 | Cosets of sets are element... |
| cosscnvelrels 38038 | Cosets of converse sets ar... |
| dfssr2 38040 | Alternate definition of th... |
| relssr 38041 | The subset relation is a r... |
| brssr 38042 | The subset relation and su... |
| brssrid 38043 | Any set is a subset of its... |
| issetssr 38044 | Two ways of expressing set... |
| brssrres 38045 | Restricted subset binary r... |
| br1cnvssrres 38046 | Restricted converse subset... |
| brcnvssr 38047 | The converse of a subset r... |
| brcnvssrid 38048 | Any set is a converse subs... |
| br1cossxrncnvssrres 38049 | ` <. B , C >. ` and ` <. D... |
| extssr 38050 | Property of subset relatio... |
| dfrefrels2 38054 | Alternate definition of th... |
| dfrefrels3 38055 | Alternate definition of th... |
| dfrefrel2 38056 | Alternate definition of th... |
| dfrefrel3 38057 | Alternate definition of th... |
| dfrefrel5 38058 | Alternate definition of th... |
| elrefrels2 38059 | Element of the class of re... |
| elrefrels3 38060 | Element of the class of re... |
| elrefrelsrel 38061 | For sets, being an element... |
| refreleq 38062 | Equality theorem for refle... |
| refrelid 38063 | Identity relation is refle... |
| refrelcoss 38064 | The class of cosets by ` R... |
| refrelressn 38065 | Any class ' R ' restricted... |
| dfcnvrefrels2 38069 | Alternate definition of th... |
| dfcnvrefrels3 38070 | Alternate definition of th... |
| dfcnvrefrel2 38071 | Alternate definition of th... |
| dfcnvrefrel3 38072 | Alternate definition of th... |
| dfcnvrefrel4 38073 | Alternate definition of th... |
| dfcnvrefrel5 38074 | Alternate definition of th... |
| elcnvrefrels2 38075 | Element of the class of co... |
| elcnvrefrels3 38076 | Element of the class of co... |
| elcnvrefrelsrel 38077 | For sets, being an element... |
| cnvrefrelcoss2 38078 | Necessary and sufficient c... |
| cosselcnvrefrels2 38079 | Necessary and sufficient c... |
| cosselcnvrefrels3 38080 | Necessary and sufficient c... |
| cosselcnvrefrels4 38081 | Necessary and sufficient c... |
| cosselcnvrefrels5 38082 | Necessary and sufficient c... |
| dfsymrels2 38086 | Alternate definition of th... |
| dfsymrels3 38087 | Alternate definition of th... |
| dfsymrels4 38088 | Alternate definition of th... |
| dfsymrels5 38089 | Alternate definition of th... |
| dfsymrel2 38090 | Alternate definition of th... |
| dfsymrel3 38091 | Alternate definition of th... |
| dfsymrel4 38092 | Alternate definition of th... |
| dfsymrel5 38093 | Alternate definition of th... |
| elsymrels2 38094 | Element of the class of sy... |
| elsymrels3 38095 | Element of the class of sy... |
| elsymrels4 38096 | Element of the class of sy... |
| elsymrels5 38097 | Element of the class of sy... |
| elsymrelsrel 38098 | For sets, being an element... |
| symreleq 38099 | Equality theorem for symme... |
| symrelim 38100 | Symmetric relation implies... |
| symrelcoss 38101 | The class of cosets by ` R... |
| idsymrel 38102 | The identity relation is s... |
| epnsymrel 38103 | The membership (epsilon) r... |
| symrefref2 38104 | Symmetry is a sufficient c... |
| symrefref3 38105 | Symmetry is a sufficient c... |
| refsymrels2 38106 | Elements of the class of r... |
| refsymrels3 38107 | Elements of the class of r... |
| refsymrel2 38108 | A relation which is reflex... |
| refsymrel3 38109 | A relation which is reflex... |
| elrefsymrels2 38110 | Elements of the class of r... |
| elrefsymrels3 38111 | Elements of the class of r... |
| elrefsymrelsrel 38112 | For sets, being an element... |
| dftrrels2 38116 | Alternate definition of th... |
| dftrrels3 38117 | Alternate definition of th... |
| dftrrel2 38118 | Alternate definition of th... |
| dftrrel3 38119 | Alternate definition of th... |
| eltrrels2 38120 | Element of the class of tr... |
| eltrrels3 38121 | Element of the class of tr... |
| eltrrelsrel 38122 | For sets, being an element... |
| trreleq 38123 | Equality theorem for the t... |
| trrelressn 38124 | Any class ' R ' restricted... |
| dfeqvrels2 38129 | Alternate definition of th... |
| dfeqvrels3 38130 | Alternate definition of th... |
| dfeqvrel2 38131 | Alternate definition of th... |
| dfeqvrel3 38132 | Alternate definition of th... |
| eleqvrels2 38133 | Element of the class of eq... |
| eleqvrels3 38134 | Element of the class of eq... |
| eleqvrelsrel 38135 | For sets, being an element... |
| elcoeleqvrels 38136 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 38137 | For sets, being an element... |
| eqvrelrel 38138 | An equivalence relation is... |
| eqvrelrefrel 38139 | An equivalence relation is... |
| eqvrelsymrel 38140 | An equivalence relation is... |
| eqvreltrrel 38141 | An equivalence relation is... |
| eqvrelim 38142 | Equivalence relation impli... |
| eqvreleq 38143 | Equality theorem for equiv... |
| eqvreleqi 38144 | Equality theorem for equiv... |
| eqvreleqd 38145 | Equality theorem for equiv... |
| eqvrelsym 38146 | An equivalence relation is... |
| eqvrelsymb 38147 | An equivalence relation is... |
| eqvreltr 38148 | An equivalence relation is... |
| eqvreltrd 38149 | A transitivity relation fo... |
| eqvreltr4d 38150 | A transitivity relation fo... |
| eqvrelref 38151 | An equivalence relation is... |
| eqvrelth 38152 | Basic property of equivale... |
| eqvrelcl 38153 | Elementhood in the field o... |
| eqvrelthi 38154 | Basic property of equivale... |
| eqvreldisj 38155 | Equivalence classes do not... |
| qsdisjALTV 38156 | Elements of a quotient set... |
| eqvrelqsel 38157 | If an element of a quotien... |
| eqvrelcoss 38158 | Two ways to express equiva... |
| eqvrelcoss3 38159 | Two ways to express equiva... |
| eqvrelcoss2 38160 | Two ways to express equiva... |
| eqvrelcoss4 38161 | Two ways to express equiva... |
| dfcoeleqvrels 38162 | Alternate definition of th... |
| dfcoeleqvrel 38163 | Alternate definition of th... |
| brredunds 38167 | Binary relation on the cla... |
| brredundsredund 38168 | For sets, binary relation ... |
| redundss3 38169 | Implication of redundancy ... |
| redundeq1 38170 | Equivalence of redundancy ... |
| redundpim3 38171 | Implication of redundancy ... |
| redundpbi1 38172 | Equivalence of redundancy ... |
| refrelsredund4 38173 | The naive version of the c... |
| refrelsredund2 38174 | The naive version of the c... |
| refrelsredund3 38175 | The naive version of the c... |
| refrelredund4 38176 | The naive version of the d... |
| refrelredund2 38177 | The naive version of the d... |
| refrelredund3 38178 | The naive version of the d... |
| dmqseq 38181 | Equality theorem for domai... |
| dmqseqi 38182 | Equality theorem for domai... |
| dmqseqd 38183 | Equality theorem for domai... |
| dmqseqeq1 38184 | Equality theorem for domai... |
| dmqseqeq1i 38185 | Equality theorem for domai... |
| dmqseqeq1d 38186 | Equality theorem for domai... |
| brdmqss 38187 | The domain quotient binary... |
| brdmqssqs 38188 | If ` A ` and ` R ` are set... |
| n0eldmqs 38189 | The empty set is not an el... |
| n0eldmqseq 38190 | The empty set is not an el... |
| n0elim 38191 | Implication of that the em... |
| n0el3 38192 | Two ways of expressing tha... |
| cnvepresdmqss 38193 | The domain quotient binary... |
| cnvepresdmqs 38194 | The domain quotient predic... |
| unidmqs 38195 | The range of a relation is... |
| unidmqseq 38196 | The union of the domain qu... |
| dmqseqim 38197 | If the domain quotient of ... |
| dmqseqim2 38198 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 38199 | Elementhood in the domain ... |
| eldmqs1cossres 38200 | Elementhood in the domain ... |
| releldmqscoss 38201 | Elementhood in the domain ... |
| dmqscoelseq 38202 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 38203 | Two ways to express the eq... |
| brers 38208 | Binary equivalence relatio... |
| dferALTV2 38209 | Equivalence relation with ... |
| erALTVeq1 38210 | Equality theorem for equiv... |
| erALTVeq1i 38211 | Equality theorem for equiv... |
| erALTVeq1d 38212 | Equality theorem for equiv... |
| dfcomember 38213 | Alternate definition of th... |
| dfcomember2 38214 | Alternate definition of th... |
| dfcomember3 38215 | Alternate definition of th... |
| eqvreldmqs 38216 | Two ways to express comemb... |
| eqvreldmqs2 38217 | Two ways to express comemb... |
| brerser 38218 | Binary equivalence relatio... |
| erimeq2 38219 | Equivalence relation on it... |
| erimeq 38220 | Equivalence relation on it... |
| dffunsALTV 38224 | Alternate definition of th... |
| dffunsALTV2 38225 | Alternate definition of th... |
| dffunsALTV3 38226 | Alternate definition of th... |
| dffunsALTV4 38227 | Alternate definition of th... |
| dffunsALTV5 38228 | Alternate definition of th... |
| dffunALTV2 38229 | Alternate definition of th... |
| dffunALTV3 38230 | Alternate definition of th... |
| dffunALTV4 38231 | Alternate definition of th... |
| dffunALTV5 38232 | Alternate definition of th... |
| elfunsALTV 38233 | Elementhood in the class o... |
| elfunsALTV2 38234 | Elementhood in the class o... |
| elfunsALTV3 38235 | Elementhood in the class o... |
| elfunsALTV4 38236 | Elementhood in the class o... |
| elfunsALTV5 38237 | Elementhood in the class o... |
| elfunsALTVfunALTV 38238 | The element of the class o... |
| funALTVfun 38239 | Our definition of the func... |
| funALTVss 38240 | Subclass theorem for funct... |
| funALTVeq 38241 | Equality theorem for funct... |
| funALTVeqi 38242 | Equality inference for the... |
| funALTVeqd 38243 | Equality deduction for the... |
| dfdisjs 38249 | Alternate definition of th... |
| dfdisjs2 38250 | Alternate definition of th... |
| dfdisjs3 38251 | Alternate definition of th... |
| dfdisjs4 38252 | Alternate definition of th... |
| dfdisjs5 38253 | Alternate definition of th... |
| dfdisjALTV 38254 | Alternate definition of th... |
| dfdisjALTV2 38255 | Alternate definition of th... |
| dfdisjALTV3 38256 | Alternate definition of th... |
| dfdisjALTV4 38257 | Alternate definition of th... |
| dfdisjALTV5 38258 | Alternate definition of th... |
| dfeldisj2 38259 | Alternate definition of th... |
| dfeldisj3 38260 | Alternate definition of th... |
| dfeldisj4 38261 | Alternate definition of th... |
| dfeldisj5 38262 | Alternate definition of th... |
| eldisjs 38263 | Elementhood in the class o... |
| eldisjs2 38264 | Elementhood in the class o... |
| eldisjs3 38265 | Elementhood in the class o... |
| eldisjs4 38266 | Elementhood in the class o... |
| eldisjs5 38267 | Elementhood in the class o... |
| eldisjsdisj 38268 | The element of the class o... |
| eleldisjs 38269 | Elementhood in the disjoin... |
| eleldisjseldisj 38270 | The element of the disjoin... |
| disjrel 38271 | Disjoint relation is a rel... |
| disjss 38272 | Subclass theorem for disjo... |
| disjssi 38273 | Subclass theorem for disjo... |
| disjssd 38274 | Subclass theorem for disjo... |
| disjeq 38275 | Equality theorem for disjo... |
| disjeqi 38276 | Equality theorem for disjo... |
| disjeqd 38277 | Equality theorem for disjo... |
| disjdmqseqeq1 38278 | Lemma for the equality the... |
| eldisjss 38279 | Subclass theorem for disjo... |
| eldisjssi 38280 | Subclass theorem for disjo... |
| eldisjssd 38281 | Subclass theorem for disjo... |
| eldisjeq 38282 | Equality theorem for disjo... |
| eldisjeqi 38283 | Equality theorem for disjo... |
| eldisjeqd 38284 | Equality theorem for disjo... |
| disjres 38285 | Disjoint restriction. (Co... |
| eldisjn0elb 38286 | Two forms of disjoint elem... |
| disjxrn 38287 | Two ways of saying that a ... |
| disjxrnres5 38288 | Disjoint range Cartesian p... |
| disjorimxrn 38289 | Disjointness condition for... |
| disjimxrn 38290 | Disjointness condition for... |
| disjimres 38291 | Disjointness condition for... |
| disjimin 38292 | Disjointness condition for... |
| disjiminres 38293 | Disjointness condition for... |
| disjimxrnres 38294 | Disjointness condition for... |
| disjALTV0 38295 | The null class is disjoint... |
| disjALTVid 38296 | The class of identity rela... |
| disjALTVidres 38297 | The class of identity rela... |
| disjALTVinidres 38298 | The intersection with rest... |
| disjALTVxrnidres 38299 | The class of range Cartesi... |
| disjsuc 38300 | Disjoint range Cartesian p... |
| dfantisymrel4 38302 | Alternate definition of th... |
| dfantisymrel5 38303 | Alternate definition of th... |
| antisymrelres 38304 | (Contributed by Peter Mazs... |
| antisymrelressn 38305 | (Contributed by Peter Mazs... |
| dfpart2 38310 | Alternate definition of th... |
| dfmembpart2 38311 | Alternate definition of th... |
| brparts 38312 | Binary partitions relation... |
| brparts2 38313 | Binary partitions relation... |
| brpartspart 38314 | Binary partition and the p... |
| parteq1 38315 | Equality theorem for parti... |
| parteq2 38316 | Equality theorem for parti... |
| parteq12 38317 | Equality theorem for parti... |
| parteq1i 38318 | Equality theorem for parti... |
| parteq1d 38319 | Equality theorem for parti... |
| partsuc2 38320 | Property of the partition.... |
| partsuc 38321 | Property of the partition.... |
| disjim 38322 | The "Divide et Aequivalere... |
| disjimi 38323 | Every disjoint relation ge... |
| detlem 38324 | If a relation is disjoint,... |
| eldisjim 38325 | If the elements of ` A ` a... |
| eldisjim2 38326 | Alternate form of ~ eldisj... |
| eqvrel0 38327 | The null class is an equiv... |
| det0 38328 | The cosets by the null cla... |
| eqvrelcoss0 38329 | The cosets by the null cla... |
| eqvrelid 38330 | The identity relation is a... |
| eqvrel1cossidres 38331 | The cosets by a restricted... |
| eqvrel1cossinidres 38332 | The cosets by an intersect... |
| eqvrel1cossxrnidres 38333 | The cosets by a range Cart... |
| detid 38334 | The cosets by the identity... |
| eqvrelcossid 38335 | The cosets by the identity... |
| detidres 38336 | The cosets by the restrict... |
| detinidres 38337 | The cosets by the intersec... |
| detxrnidres 38338 | The cosets by the range Ca... |
| disjlem14 38339 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 38340 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 38341 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 38342 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 38343 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 38344 | Lemma for ~ disjdmqseq via... |
| disjdmqs 38345 | If a relation is disjoint,... |
| disjdmqseq 38346 | If a relation is disjoint,... |
| eldisjn0el 38347 | Special case of ~ disjdmqs... |
| partim2 38348 | Disjoint relation on its n... |
| partim 38349 | Partition implies equivale... |
| partimeq 38350 | Partition implies that the... |
| eldisjlem19 38351 | Special case of ~ disjlem1... |
| membpartlem19 38352 | Together with ~ disjlem19 ... |
| petlem 38353 | If you can prove that the ... |
| petlemi 38354 | If you can prove disjointn... |
| pet02 38355 | Class ` A ` is a partition... |
| pet0 38356 | Class ` A ` is a partition... |
| petid2 38357 | Class ` A ` is a partition... |
| petid 38358 | A class is a partition by ... |
| petidres2 38359 | Class ` A ` is a partition... |
| petidres 38360 | A class is a partition by ... |
| petinidres2 38361 | Class ` A ` is a partition... |
| petinidres 38362 | A class is a partition by ... |
| petxrnidres2 38363 | Class ` A ` is a partition... |
| petxrnidres 38364 | A class is a partition by ... |
| eqvreldisj1 38365 | The elements of the quotie... |
| eqvreldisj2 38366 | The elements of the quotie... |
| eqvreldisj3 38367 | The elements of the quotie... |
| eqvreldisj4 38368 | Intersection with the conv... |
| eqvreldisj5 38369 | Range Cartesian product wi... |
| eqvrelqseqdisj2 38370 | Implication of ~ eqvreldis... |
| fences3 38371 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 38372 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 38373 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 38374 | Lemma for the Partition-Eq... |
| mainer 38375 | The Main Theorem of Equiva... |
| partimcomember 38376 | Partition with general ` R... |
| mpet3 38377 | Member Partition-Equivalen... |
| cpet2 38378 | The conventional form of t... |
| cpet 38379 | The conventional form of M... |
| mpet 38380 | Member Partition-Equivalen... |
| mpet2 38381 | Member Partition-Equivalen... |
| mpets2 38382 | Member Partition-Equivalen... |
| mpets 38383 | Member Partition-Equivalen... |
| mainpart 38384 | Partition with general ` R... |
| fences 38385 | The Theorem of Fences by E... |
| fences2 38386 | The Theorem of Fences by E... |
| mainer2 38387 | The Main Theorem of Equiva... |
| mainerim 38388 | Every equivalence relation... |
| petincnvepres2 38389 | A partition-equivalence th... |
| petincnvepres 38390 | The shortest form of a par... |
| pet2 38391 | Partition-Equivalence Theo... |
| pet 38392 | Partition-Equivalence Theo... |
| pets 38393 | Partition-Equivalence Theo... |
| prtlem60 38394 | Lemma for ~ prter3 . (Con... |
| bicomdd 38395 | Commute two sides of a bic... |
| jca2r 38396 | Inference conjoining the c... |
| jca3 38397 | Inference conjoining the c... |
| prtlem70 38398 | Lemma for ~ prter3 : a rea... |
| ibdr 38399 | Reverse of ~ ibd . (Contr... |
| prtlem100 38400 | Lemma for ~ prter3 . (Con... |
| prtlem5 38401 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 38402 | Lemma for ~ prter2 . (Con... |
| brabsb2 38403 | A closed form of ~ brabsb ... |
| eqbrrdv2 38404 | Other version of ~ eqbrrdi... |
| prtlem9 38405 | Lemma for ~ prter3 . (Con... |
| prtlem10 38406 | Lemma for ~ prter3 . (Con... |
| prtlem11 38407 | Lemma for ~ prter2 . (Con... |
| prtlem12 38408 | Lemma for ~ prtex and ~ pr... |
| prtlem13 38409 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 38410 | Lemma for ~ prtex , ~ prte... |
| prtlem400 38411 | Lemma for ~ prter2 and als... |
| erprt 38414 | The quotient set of an equ... |
| prtlem14 38415 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 38416 | Lemma for ~ prter1 and ~ p... |
| prtlem17 38417 | Lemma for ~ prter2 . (Con... |
| prtlem18 38418 | Lemma for ~ prter2 . (Con... |
| prtlem19 38419 | Lemma for ~ prter2 . (Con... |
| prter1 38420 | Every partition generates ... |
| prtex 38421 | The equivalence relation g... |
| prter2 38422 | The quotient set of the eq... |
| prter3 38423 | For every partition there ... |
| axc5 38434 | This theorem repeats ~ sp ... |
| ax4fromc4 38435 | Rederivation of Axiom ~ ax... |
| ax10fromc7 38436 | Rederivation of Axiom ~ ax... |
| ax6fromc10 38437 | Rederivation of Axiom ~ ax... |
| hba1-o 38438 | The setvar ` x ` is not fr... |
| axc4i-o 38439 | Inference version of ~ ax-... |
| equid1 38440 | Proof of ~ equid from our ... |
| equcomi1 38441 | Proof of ~ equcomi from ~ ... |
| aecom-o 38442 | Commutation law for identi... |
| aecoms-o 38443 | A commutation rule for ide... |
| hbae-o 38444 | All variables are effectiv... |
| dral1-o 38445 | Formula-building lemma for... |
| ax12fromc15 38446 | Rederivation of Axiom ~ ax... |
| ax13fromc9 38447 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 38448 | Axiom to quantify a variab... |
| sps-o 38449 | Generalization of antecede... |
| hbequid 38450 | Bound-variable hypothesis ... |
| nfequid-o 38451 | Bound-variable hypothesis ... |
| axc5c7 38452 | Proof of a single axiom th... |
| axc5c7toc5 38453 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 38454 | Rederivation of ~ ax-c7 fr... |
| axc711 38455 | Proof of a single axiom th... |
| nfa1-o 38456 | ` x ` is not free in ` A. ... |
| axc711toc7 38457 | Rederivation of ~ ax-c7 fr... |
| axc711to11 38458 | Rederivation of ~ ax-11 fr... |
| axc5c711 38459 | Proof of a single axiom th... |
| axc5c711toc5 38460 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 38461 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 38462 | Rederivation of ~ ax-11 fr... |
| equidqe 38463 | ~ equid with existential q... |
| axc5sp1 38464 | A special case of ~ ax-c5 ... |
| equidq 38465 | ~ equid with universal qua... |
| equid1ALT 38466 | Alternate proof of ~ equid... |
| axc11nfromc11 38467 | Rederivation of ~ ax-c11n ... |
| naecoms-o 38468 | A commutation rule for dis... |
| hbnae-o 38469 | All variables are effectiv... |
| dvelimf-o 38470 | Proof of ~ dvelimh that us... |
| dral2-o 38471 | Formula-building lemma for... |
| aev-o 38472 | A "distinctor elimination"... |
| ax5eq 38473 | Theorem to add distinct qu... |
| dveeq2-o 38474 | Quantifier introduction wh... |
| axc16g-o 38475 | A generalization of Axiom ... |
| dveeq1-o 38476 | Quantifier introduction wh... |
| dveeq1-o16 38477 | Version of ~ dveeq1 using ... |
| ax5el 38478 | Theorem to add distinct qu... |
| axc11n-16 38479 | This theorem shows that, g... |
| dveel2ALT 38480 | Alternate proof of ~ dveel... |
| ax12f 38481 | Basis step for constructin... |
| ax12eq 38482 | Basis step for constructin... |
| ax12el 38483 | Basis step for constructin... |
| ax12indn 38484 | Induction step for constru... |
| ax12indi 38485 | Induction step for constru... |
| ax12indalem 38486 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 38487 | Alternate proof of ~ ax12i... |
| ax12inda2 38488 | Induction step for constru... |
| ax12inda 38489 | Induction step for constru... |
| ax12v2-o 38490 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 38491 | Derive ~ ax-c15 from a hyp... |
| axc11-o 38492 | Show that ~ ax-c11 can be ... |
| fsumshftd 38493 | Index shift of a finite su... |
| riotaclbgBAD 38495 | Closure of restricted iota... |
| riotaclbBAD 38496 | Closure of restricted iota... |
| riotasvd 38497 | Deduction version of ~ rio... |
| riotasv2d 38498 | Value of description binde... |
| riotasv2s 38499 | The value of description b... |
| riotasv 38500 | Value of description binde... |
| riotasv3d 38501 | A property ` ch ` holding ... |
| elimhyps 38502 | A version of ~ elimhyp usi... |
| dedths 38503 | A version of weak deductio... |
| renegclALT 38504 | Closure law for negative o... |
| elimhyps2 38505 | Generalization of ~ elimhy... |
| dedths2 38506 | Generalization of ~ dedths... |
| nfcxfrdf 38507 | A utility lemma to transfe... |
| nfded 38508 | A deduction theorem that c... |
| nfded2 38509 | A deduction theorem that c... |
| nfunidALT2 38510 | Deduction version of ~ nfu... |
| nfunidALT 38511 | Deduction version of ~ nfu... |
| nfopdALT 38512 | Deduction version of bound... |
| cnaddcom 38513 | Recover the commutative la... |
| toycom 38514 | Show the commutative law f... |
| lshpset 38519 | The set of all hyperplanes... |
| islshp 38520 | The predicate "is a hyperp... |
| islshpsm 38521 | Hyperplane properties expr... |
| lshplss 38522 | A hyperplane is a subspace... |
| lshpne 38523 | A hyperplane is not equal ... |
| lshpnel 38524 | A hyperplane's generating ... |
| lshpnelb 38525 | The subspace sum of a hype... |
| lshpnel2N 38526 | Condition that determines ... |
| lshpne0 38527 | The member of the span in ... |
| lshpdisj 38528 | A hyperplane and the span ... |
| lshpcmp 38529 | If two hyperplanes are com... |
| lshpinN 38530 | The intersection of two di... |
| lsatset 38531 | The set of all 1-dim subsp... |
| islsat 38532 | The predicate "is a 1-dim ... |
| lsatlspsn2 38533 | The span of a nonzero sing... |
| lsatlspsn 38534 | The span of a nonzero sing... |
| islsati 38535 | A 1-dim subspace (atom) (o... |
| lsateln0 38536 | A 1-dim subspace (atom) (o... |
| lsatlss 38537 | The set of 1-dim subspaces... |
| lsatlssel 38538 | An atom is a subspace. (C... |
| lsatssv 38539 | An atom is a set of vector... |
| lsatn0 38540 | A 1-dim subspace (atom) of... |
| lsatspn0 38541 | The span of a vector is an... |
| lsator0sp 38542 | The span of a vector is ei... |
| lsatssn0 38543 | A subspace (or any class) ... |
| lsatcmp 38544 | If two atoms are comparabl... |
| lsatcmp2 38545 | If an atom is included in ... |
| lsatel 38546 | A nonzero vector in an ato... |
| lsatelbN 38547 | A nonzero vector in an ato... |
| lsat2el 38548 | Two atoms sharing a nonzer... |
| lsmsat 38549 | Convert comparison of atom... |
| lsatfixedN 38550 | Show equality with the spa... |
| lsmsatcv 38551 | Subspace sum has the cover... |
| lssatomic 38552 | The lattice of subspaces i... |
| lssats 38553 | The lattice of subspaces i... |
| lpssat 38554 | Two subspaces in a proper ... |
| lrelat 38555 | Subspaces are relatively a... |
| lssatle 38556 | The ordering of two subspa... |
| lssat 38557 | Two subspaces in a proper ... |
| islshpat 38558 | Hyperplane properties expr... |
| lcvfbr 38561 | The covers relation for a ... |
| lcvbr 38562 | The covers relation for a ... |
| lcvbr2 38563 | The covers relation for a ... |
| lcvbr3 38564 | The covers relation for a ... |
| lcvpss 38565 | The covers relation implie... |
| lcvnbtwn 38566 | The covers relation implie... |
| lcvntr 38567 | The covers relation is not... |
| lcvnbtwn2 38568 | The covers relation implie... |
| lcvnbtwn3 38569 | The covers relation implie... |
| lsmcv2 38570 | Subspace sum has the cover... |
| lcvat 38571 | If a subspace covers anoth... |
| lsatcv0 38572 | An atom covers the zero su... |
| lsatcveq0 38573 | A subspace covered by an a... |
| lsat0cv 38574 | A subspace is an atom iff ... |
| lcvexchlem1 38575 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 38576 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 38577 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 38578 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 38579 | Lemma for ~ lcvexch . (Co... |
| lcvexch 38580 | Subspaces satisfy the exch... |
| lcvp 38581 | Covering property of Defin... |
| lcv1 38582 | Covering property of a sub... |
| lcv2 38583 | Covering property of a sub... |
| lsatexch 38584 | The atom exchange property... |
| lsatnle 38585 | The meet of a subspace and... |
| lsatnem0 38586 | The meet of distinct atoms... |
| lsatexch1 38587 | The atom exch1ange propert... |
| lsatcv0eq 38588 | If the sum of two atoms co... |
| lsatcv1 38589 | Two atoms covering the zer... |
| lsatcvatlem 38590 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 38591 | A nonzero subspace less th... |
| lsatcvat2 38592 | A subspace covered by the ... |
| lsatcvat3 38593 | A condition implying that ... |
| islshpcv 38594 | Hyperplane properties expr... |
| l1cvpat 38595 | A subspace covered by the ... |
| l1cvat 38596 | Create an atom under an el... |
| lshpat 38597 | Create an atom under a hyp... |
| lflset 38600 | The set of linear function... |
| islfl 38601 | The predicate "is a linear... |
| lfli 38602 | Property of a linear funct... |
| islfld 38603 | Properties that determine ... |
| lflf 38604 | A linear functional is a f... |
| lflcl 38605 | A linear functional value ... |
| lfl0 38606 | A linear functional is zer... |
| lfladd 38607 | Property of a linear funct... |
| lflsub 38608 | Property of a linear funct... |
| lflmul 38609 | Property of a linear funct... |
| lfl0f 38610 | The zero function is a fun... |
| lfl1 38611 | A nonzero functional has a... |
| lfladdcl 38612 | Closure of addition of two... |
| lfladdcom 38613 | Commutativity of functiona... |
| lfladdass 38614 | Associativity of functiona... |
| lfladd0l 38615 | Functional addition with t... |
| lflnegcl 38616 | Closure of the negative of... |
| lflnegl 38617 | A functional plus its nega... |
| lflvscl 38618 | Closure of a scalar produc... |
| lflvsdi1 38619 | Distributive law for (righ... |
| lflvsdi2 38620 | Reverse distributive law f... |
| lflvsdi2a 38621 | Reverse distributive law f... |
| lflvsass 38622 | Associative law for (right... |
| lfl0sc 38623 | The (right vector space) s... |
| lflsc0N 38624 | The scalar product with th... |
| lfl1sc 38625 | The (right vector space) s... |
| lkrfval 38628 | The kernel of a functional... |
| lkrval 38629 | Value of the kernel of a f... |
| ellkr 38630 | Membership in the kernel o... |
| lkrval2 38631 | Value of the kernel of a f... |
| ellkr2 38632 | Membership in the kernel o... |
| lkrcl 38633 | A member of the kernel of ... |
| lkrf0 38634 | The value of a functional ... |
| lkr0f 38635 | The kernel of the zero fun... |
| lkrlss 38636 | The kernel of a linear fun... |
| lkrssv 38637 | The kernel of a linear fun... |
| lkrsc 38638 | The kernel of a nonzero sc... |
| lkrscss 38639 | The kernel of a scalar pro... |
| eqlkr 38640 | Two functionals with the s... |
| eqlkr2 38641 | Two functionals with the s... |
| eqlkr3 38642 | Two functionals with the s... |
| lkrlsp 38643 | The subspace sum of a kern... |
| lkrlsp2 38644 | The subspace sum of a kern... |
| lkrlsp3 38645 | The subspace sum of a kern... |
| lkrshp 38646 | The kernel of a nonzero fu... |
| lkrshp3 38647 | The kernels of nonzero fun... |
| lkrshpor 38648 | The kernel of a functional... |
| lkrshp4 38649 | A kernel is a hyperplane i... |
| lshpsmreu 38650 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 38651 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 38652 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 38653 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 38654 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 38655 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 38656 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 38657 | The set ` G ` defined by h... |
| lshpkr 38658 | The kernel of functional `... |
| lshpkrex 38659 | There exists a functional ... |
| lshpset2N 38660 | The set of all hyperplanes... |
| islshpkrN 38661 | The predicate "is a hyperp... |
| lfl1dim 38662 | Equivalent expressions for... |
| lfl1dim2N 38663 | Equivalent expressions for... |
| ldualset 38666 | Define the (left) dual of ... |
| ldualvbase 38667 | The vectors of a dual spac... |
| ldualelvbase 38668 | Utility theorem for conver... |
| ldualfvadd 38669 | Vector addition in the dua... |
| ldualvadd 38670 | Vector addition in the dua... |
| ldualvaddcl 38671 | The value of vector additi... |
| ldualvaddval 38672 | The value of the value of ... |
| ldualsca 38673 | The ring of scalars of the... |
| ldualsbase 38674 | Base set of scalar ring fo... |
| ldualsaddN 38675 | Scalar addition for the du... |
| ldualsmul 38676 | Scalar multiplication for ... |
| ldualfvs 38677 | Scalar product operation f... |
| ldualvs 38678 | Scalar product operation v... |
| ldualvsval 38679 | Value of scalar product op... |
| ldualvscl 38680 | The scalar product operati... |
| ldualvaddcom 38681 | Commutative law for vector... |
| ldualvsass 38682 | Associative law for scalar... |
| ldualvsass2 38683 | Associative law for scalar... |
| ldualvsdi1 38684 | Distributive law for scala... |
| ldualvsdi2 38685 | Reverse distributive law f... |
| ldualgrplem 38686 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 38687 | The dual of a vector space... |
| ldual0 38688 | The zero scalar of the dua... |
| ldual1 38689 | The unit scalar of the dua... |
| ldualneg 38690 | The negative of a scalar o... |
| ldual0v 38691 | The zero vector of the dua... |
| ldual0vcl 38692 | The dual zero vector is a ... |
| lduallmodlem 38693 | Lemma for ~ lduallmod . (... |
| lduallmod 38694 | The dual of a left module ... |
| lduallvec 38695 | The dual of a left vector ... |
| ldualvsub 38696 | The value of vector subtra... |
| ldualvsubcl 38697 | Closure of vector subtract... |
| ldualvsubval 38698 | The value of the value of ... |
| ldualssvscl 38699 | Closure of scalar product ... |
| ldualssvsubcl 38700 | Closure of vector subtract... |
| ldual0vs 38701 | Scalar zero times a functi... |
| lkr0f2 38702 | The kernel of the zero fun... |
| lduallkr3 38703 | The kernels of nonzero fun... |
| lkrpssN 38704 | Proper subset relation bet... |
| lkrin 38705 | Intersection of the kernel... |
| eqlkr4 38706 | Two functionals with the s... |
| ldual1dim 38707 | Equivalent expressions for... |
| ldualkrsc 38708 | The kernel of a nonzero sc... |
| lkrss 38709 | The kernel of a scalar pro... |
| lkrss2N 38710 | Two functionals with kerne... |
| lkreqN 38711 | Proportional functionals h... |
| lkrlspeqN 38712 | Condition for colinear fun... |
| isopos 38721 | The predicate "is an ortho... |
| opposet 38722 | Every orthoposet is a pose... |
| oposlem 38723 | Lemma for orthoposet prope... |
| op01dm 38724 | Conditions necessary for z... |
| op0cl 38725 | An orthoposet has a zero e... |
| op1cl 38726 | An orthoposet has a unity ... |
| op0le 38727 | Orthoposet zero is less th... |
| ople0 38728 | An element less than or eq... |
| opnlen0 38729 | An element not less than a... |
| lub0N 38730 | The least upper bound of t... |
| opltn0 38731 | A lattice element greater ... |
| ople1 38732 | Any element is less than t... |
| op1le 38733 | If the orthoposet unity is... |
| glb0N 38734 | The greatest lower bound o... |
| opoccl 38735 | Closure of orthocomplement... |
| opococ 38736 | Double negative law for or... |
| opcon3b 38737 | Contraposition law for ort... |
| opcon2b 38738 | Orthocomplement contraposi... |
| opcon1b 38739 | Orthocomplement contraposi... |
| oplecon3 38740 | Contraposition law for ort... |
| oplecon3b 38741 | Contraposition law for ort... |
| oplecon1b 38742 | Contraposition law for str... |
| opoc1 38743 | Orthocomplement of orthopo... |
| opoc0 38744 | Orthocomplement of orthopo... |
| opltcon3b 38745 | Contraposition law for str... |
| opltcon1b 38746 | Contraposition law for str... |
| opltcon2b 38747 | Contraposition law for str... |
| opexmid 38748 | Law of excluded middle for... |
| opnoncon 38749 | Law of contradiction for o... |
| riotaocN 38750 | The orthocomplement of the... |
| cmtfvalN 38751 | Value of commutes relation... |
| cmtvalN 38752 | Equivalence for commutes r... |
| isolat 38753 | The predicate "is an ortho... |
| ollat 38754 | An ortholattice is a latti... |
| olop 38755 | An ortholattice is an orth... |
| olposN 38756 | An ortholattice is a poset... |
| isolatiN 38757 | Properties that determine ... |
| oldmm1 38758 | De Morgan's law for meet i... |
| oldmm2 38759 | De Morgan's law for meet i... |
| oldmm3N 38760 | De Morgan's law for meet i... |
| oldmm4 38761 | De Morgan's law for meet i... |
| oldmj1 38762 | De Morgan's law for join i... |
| oldmj2 38763 | De Morgan's law for join i... |
| oldmj3 38764 | De Morgan's law for join i... |
| oldmj4 38765 | De Morgan's law for join i... |
| olj01 38766 | An ortholattice element jo... |
| olj02 38767 | An ortholattice element jo... |
| olm11 38768 | The meet of an ortholattic... |
| olm12 38769 | The meet of an ortholattic... |
| latmassOLD 38770 | Ortholattice meet is assoc... |
| latm12 38771 | A rearrangement of lattice... |
| latm32 38772 | A rearrangement of lattice... |
| latmrot 38773 | Rotate lattice meet of 3 c... |
| latm4 38774 | Rearrangement of lattice m... |
| latmmdiN 38775 | Lattice meet distributes o... |
| latmmdir 38776 | Lattice meet distributes o... |
| olm01 38777 | Meet with lattice zero is ... |
| olm02 38778 | Meet with lattice zero is ... |
| isoml 38779 | The predicate "is an ortho... |
| isomliN 38780 | Properties that determine ... |
| omlol 38781 | An orthomodular lattice is... |
| omlop 38782 | An orthomodular lattice is... |
| omllat 38783 | An orthomodular lattice is... |
| omllaw 38784 | The orthomodular law. (Co... |
| omllaw2N 38785 | Variation of orthomodular ... |
| omllaw3 38786 | Orthomodular law equivalen... |
| omllaw4 38787 | Orthomodular law equivalen... |
| omllaw5N 38788 | The orthomodular law. Rem... |
| cmtcomlemN 38789 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 38790 | Commutation is symmetric. ... |
| cmt2N 38791 | Commutation with orthocomp... |
| cmt3N 38792 | Commutation with orthocomp... |
| cmt4N 38793 | Commutation with orthocomp... |
| cmtbr2N 38794 | Alternate definition of th... |
| cmtbr3N 38795 | Alternate definition for t... |
| cmtbr4N 38796 | Alternate definition for t... |
| lecmtN 38797 | Ordered elements commute. ... |
| cmtidN 38798 | Any element commutes with ... |
| omlfh1N 38799 | Foulis-Holland Theorem, pa... |
| omlfh3N 38800 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 38801 | Analogue of modular law ~ ... |
| omlspjN 38802 | Contraction of a Sasaki pr... |
| cvrfval 38809 | Value of covers relation "... |
| cvrval 38810 | Binary relation expressing... |
| cvrlt 38811 | The covers relation implie... |
| cvrnbtwn 38812 | There is no element betwee... |
| ncvr1 38813 | No element covers the latt... |
| cvrletrN 38814 | Property of an element abo... |
| cvrval2 38815 | Binary relation expressing... |
| cvrnbtwn2 38816 | The covers relation implie... |
| cvrnbtwn3 38817 | The covers relation implie... |
| cvrcon3b 38818 | Contraposition law for the... |
| cvrle 38819 | The covers relation implie... |
| cvrnbtwn4 38820 | The covers relation implie... |
| cvrnle 38821 | The covers relation implie... |
| cvrne 38822 | The covers relation implie... |
| cvrnrefN 38823 | The covers relation is not... |
| cvrcmp 38824 | If two lattice elements th... |
| cvrcmp2 38825 | If two lattice elements co... |
| pats 38826 | The set of atoms in a pose... |
| isat 38827 | The predicate "is an atom"... |
| isat2 38828 | The predicate "is an atom"... |
| atcvr0 38829 | An atom covers zero. ( ~ ... |
| atbase 38830 | An atom is a member of the... |
| atssbase 38831 | The set of atoms is a subs... |
| 0ltat 38832 | An atom is greater than ze... |
| leatb 38833 | A poset element less than ... |
| leat 38834 | A poset element less than ... |
| leat2 38835 | A nonzero poset element le... |
| leat3 38836 | A poset element less than ... |
| meetat 38837 | The meet of any element wi... |
| meetat2 38838 | The meet of any element wi... |
| isatl 38840 | The predicate "is an atomi... |
| atllat 38841 | An atomic lattice is a lat... |
| atlpos 38842 | An atomic lattice is a pos... |
| atl0dm 38843 | Condition necessary for ze... |
| atl0cl 38844 | An atomic lattice has a ze... |
| atl0le 38845 | Orthoposet zero is less th... |
| atlle0 38846 | An element less than or eq... |
| atlltn0 38847 | A lattice element greater ... |
| isat3 38848 | The predicate "is an atom"... |
| atn0 38849 | An atom is not zero. ( ~ ... |
| atnle0 38850 | An atom is not less than o... |
| atlen0 38851 | A lattice element is nonze... |
| atcmp 38852 | If two atoms are comparabl... |
| atncmp 38853 | Frequently-used variation ... |
| atnlt 38854 | Two atoms cannot satisfy t... |
| atcvreq0 38855 | An element covered by an a... |
| atncvrN 38856 | Two atoms cannot satisfy t... |
| atlex 38857 | Every nonzero element of a... |
| atnle 38858 | Two ways of expressing "an... |
| atnem0 38859 | The meet of distinct atoms... |
| atlatmstc 38860 | An atomic, complete, ortho... |
| atlatle 38861 | The ordering of two Hilber... |
| atlrelat1 38862 | An atomistic lattice with ... |
| iscvlat 38864 | The predicate "is an atomi... |
| iscvlat2N 38865 | The predicate "is an atomi... |
| cvlatl 38866 | An atomic lattice with the... |
| cvllat 38867 | An atomic lattice with the... |
| cvlposN 38868 | An atomic lattice with the... |
| cvlexch1 38869 | An atomic covering lattice... |
| cvlexch2 38870 | An atomic covering lattice... |
| cvlexchb1 38871 | An atomic covering lattice... |
| cvlexchb2 38872 | An atomic covering lattice... |
| cvlexch3 38873 | An atomic covering lattice... |
| cvlexch4N 38874 | An atomic covering lattice... |
| cvlatexchb1 38875 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 38876 | A version of ~ cvlexchb2 f... |
| cvlatexch1 38877 | Atom exchange property. (... |
| cvlatexch2 38878 | Atom exchange property. (... |
| cvlatexch3 38879 | Atom exchange property. (... |
| cvlcvr1 38880 | The covering property. Pr... |
| cvlcvrp 38881 | A Hilbert lattice satisfie... |
| cvlatcvr1 38882 | An atom is covered by its ... |
| cvlatcvr2 38883 | An atom is covered by its ... |
| cvlsupr2 38884 | Two equivalent ways of exp... |
| cvlsupr3 38885 | Two equivalent ways of exp... |
| cvlsupr4 38886 | Consequence of superpositi... |
| cvlsupr5 38887 | Consequence of superpositi... |
| cvlsupr6 38888 | Consequence of superpositi... |
| cvlsupr7 38889 | Consequence of superpositi... |
| cvlsupr8 38890 | Consequence of superpositi... |
| ishlat1 38893 | The predicate "is a Hilber... |
| ishlat2 38894 | The predicate "is a Hilber... |
| ishlat3N 38895 | The predicate "is a Hilber... |
| ishlatiN 38896 | Properties that determine ... |
| hlomcmcv 38897 | A Hilbert lattice is ortho... |
| hloml 38898 | A Hilbert lattice is ortho... |
| hlclat 38899 | A Hilbert lattice is compl... |
| hlcvl 38900 | A Hilbert lattice is an at... |
| hlatl 38901 | A Hilbert lattice is atomi... |
| hlol 38902 | A Hilbert lattice is an or... |
| hlop 38903 | A Hilbert lattice is an or... |
| hllat 38904 | A Hilbert lattice is a lat... |
| hllatd 38905 | Deduction form of ~ hllat ... |
| hlomcmat 38906 | A Hilbert lattice is ortho... |
| hlpos 38907 | A Hilbert lattice is a pos... |
| hlatjcl 38908 | Closure of join operation.... |
| hlatjcom 38909 | Commutatitivity of join op... |
| hlatjidm 38910 | Idempotence of join operat... |
| hlatjass 38911 | Lattice join is associativ... |
| hlatj12 38912 | Swap 1st and 2nd members o... |
| hlatj32 38913 | Swap 2nd and 3rd members o... |
| hlatjrot 38914 | Rotate lattice join of 3 c... |
| hlatj4 38915 | Rearrangement of lattice j... |
| hlatlej1 38916 | A join's first argument is... |
| hlatlej2 38917 | A join's second argument i... |
| glbconN 38918 | De Morgan's law for GLB an... |
| glbconNOLD 38919 | Obsolete version of ~ glbc... |
| glbconxN 38920 | De Morgan's law for GLB an... |
| atnlej1 38921 | If an atom is not less tha... |
| atnlej2 38922 | If an atom is not less tha... |
| hlsuprexch 38923 | A Hilbert lattice has the ... |
| hlexch1 38924 | A Hilbert lattice has the ... |
| hlexch2 38925 | A Hilbert lattice has the ... |
| hlexchb1 38926 | A Hilbert lattice has the ... |
| hlexchb2 38927 | A Hilbert lattice has the ... |
| hlsupr 38928 | A Hilbert lattice has the ... |
| hlsupr2 38929 | A Hilbert lattice has the ... |
| hlhgt4 38930 | A Hilbert lattice has a he... |
| hlhgt2 38931 | A Hilbert lattice has a he... |
| hl0lt1N 38932 | Lattice 0 is less than lat... |
| hlexch3 38933 | A Hilbert lattice has the ... |
| hlexch4N 38934 | A Hilbert lattice has the ... |
| hlatexchb1 38935 | A version of ~ hlexchb1 fo... |
| hlatexchb2 38936 | A version of ~ hlexchb2 fo... |
| hlatexch1 38937 | Atom exchange property. (... |
| hlatexch2 38938 | Atom exchange property. (... |
| hlatmstcOLDN 38939 | An atomic, complete, ortho... |
| hlatle 38940 | The ordering of two Hilber... |
| hlateq 38941 | The equality of two Hilber... |
| hlrelat1 38942 | An atomistic lattice with ... |
| hlrelat5N 38943 | An atomistic lattice with ... |
| hlrelat 38944 | A Hilbert lattice is relat... |
| hlrelat2 38945 | A consequence of relative ... |
| exatleN 38946 | A condition for an atom to... |
| hl2at 38947 | A Hilbert lattice has at l... |
| atex 38948 | At least one atom exists. ... |
| intnatN 38949 | If the intersection with a... |
| 2llnne2N 38950 | Condition implying that tw... |
| 2llnneN 38951 | Condition implying that tw... |
| cvr1 38952 | A Hilbert lattice has the ... |
| cvr2N 38953 | Less-than and covers equiv... |
| hlrelat3 38954 | The Hilbert lattice is rel... |
| cvrval3 38955 | Binary relation expressing... |
| cvrval4N 38956 | Binary relation expressing... |
| cvrval5 38957 | Binary relation expressing... |
| cvrp 38958 | A Hilbert lattice satisfie... |
| atcvr1 38959 | An atom is covered by its ... |
| atcvr2 38960 | An atom is covered by its ... |
| cvrexchlem 38961 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 38962 | A Hilbert lattice satisfie... |
| cvratlem 38963 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 38964 | A nonzero Hilbert lattice ... |
| ltltncvr 38965 | A chained strong ordering ... |
| ltcvrntr 38966 | Non-transitive condition f... |
| cvrntr 38967 | The covers relation is not... |
| atcvr0eq 38968 | The covers relation is not... |
| lnnat 38969 | A line (the join of two di... |
| atcvrj0 38970 | Two atoms covering the zer... |
| cvrat2 38971 | A Hilbert lattice element ... |
| atcvrneN 38972 | Inequality derived from at... |
| atcvrj1 38973 | Condition for an atom to b... |
| atcvrj2b 38974 | Condition for an atom to b... |
| atcvrj2 38975 | Condition for an atom to b... |
| atleneN 38976 | Inequality derived from at... |
| atltcvr 38977 | An equivalence of less-tha... |
| atle 38978 | Any nonzero element has an... |
| atlt 38979 | Two atoms are unequal iff ... |
| atlelt 38980 | Transfer less-than relatio... |
| 2atlt 38981 | Given an atom less than an... |
| atexchcvrN 38982 | Atom exchange property. V... |
| atexchltN 38983 | Atom exchange property. V... |
| cvrat3 38984 | A condition implying that ... |
| cvrat4 38985 | A condition implying exist... |
| cvrat42 38986 | Commuted version of ~ cvra... |
| 2atjm 38987 | The meet of a line (expres... |
| atbtwn 38988 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 38989 | There exists a 3rd atom ` ... |
| atbtwnex 38990 | Given atoms ` P ` in ` X `... |
| 3noncolr2 38991 | Two ways to express 3 non-... |
| 3noncolr1N 38992 | Two ways to express 3 non-... |
| hlatcon3 38993 | Atom exchange combined wit... |
| hlatcon2 38994 | Atom exchange combined wit... |
| 4noncolr3 38995 | A way to express 4 non-col... |
| 4noncolr2 38996 | A way to express 4 non-col... |
| 4noncolr1 38997 | A way to express 4 non-col... |
| athgt 38998 | A Hilbert lattice, whose h... |
| 3dim0 38999 | There exists a 3-dimension... |
| 3dimlem1 39000 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 39001 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 39002 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 39003 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 39004 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 39005 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 39006 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 39007 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 39008 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 39009 | Construct a 3-dimensional ... |
| 3dim2 39010 | Construct 2 new layers on ... |
| 3dim3 39011 | Construct a new layer on t... |
| 2dim 39012 | Generate a height-3 elemen... |
| 1dimN 39013 | An atom is covered by a he... |
| 1cvrco 39014 | The orthocomplement of an ... |
| 1cvratex 39015 | There exists an atom less ... |
| 1cvratlt 39016 | An atom less than or equal... |
| 1cvrjat 39017 | An element covered by the ... |
| 1cvrat 39018 | Create an atom under an el... |
| ps-1 39019 | The join of two atoms ` R ... |
| ps-2 39020 | Lattice analogue for the p... |
| 2atjlej 39021 | Two atoms are different if... |
| hlatexch3N 39022 | Rearrange join of atoms in... |
| hlatexch4 39023 | Exchange 2 atoms. (Contri... |
| ps-2b 39024 | Variation of projective ge... |
| 3atlem1 39025 | Lemma for ~ 3at . (Contri... |
| 3atlem2 39026 | Lemma for ~ 3at . (Contri... |
| 3atlem3 39027 | Lemma for ~ 3at . (Contri... |
| 3atlem4 39028 | Lemma for ~ 3at . (Contri... |
| 3atlem5 39029 | Lemma for ~ 3at . (Contri... |
| 3atlem6 39030 | Lemma for ~ 3at . (Contri... |
| 3atlem7 39031 | Lemma for ~ 3at . (Contri... |
| 3at 39032 | Any three non-colinear ato... |
| llnset 39047 | The set of lattice lines i... |
| islln 39048 | The predicate "is a lattic... |
| islln4 39049 | The predicate "is a lattic... |
| llni 39050 | Condition implying a latti... |
| llnbase 39051 | A lattice line is a lattic... |
| islln3 39052 | The predicate "is a lattic... |
| islln2 39053 | The predicate "is a lattic... |
| llni2 39054 | The join of two different ... |
| llnnleat 39055 | An atom cannot majorize a ... |
| llnneat 39056 | A lattice line is not an a... |
| 2atneat 39057 | The join of two distinct a... |
| llnn0 39058 | A lattice line is nonzero.... |
| islln2a 39059 | The predicate "is a lattic... |
| llnle 39060 | Any element greater than 0... |
| atcvrlln2 39061 | An atom under a line is co... |
| atcvrlln 39062 | An element covering an ato... |
| llnexatN 39063 | Given an atom on a line, t... |
| llncmp 39064 | If two lattice lines are c... |
| llnnlt 39065 | Two lattice lines cannot s... |
| 2llnmat 39066 | Two intersecting lines int... |
| 2at0mat0 39067 | Special case of ~ 2atmat0 ... |
| 2atmat0 39068 | The meet of two unequal li... |
| 2atm 39069 | An atom majorized by two d... |
| ps-2c 39070 | Variation of projective ge... |
| lplnset 39071 | The set of lattice planes ... |
| islpln 39072 | The predicate "is a lattic... |
| islpln4 39073 | The predicate "is a lattic... |
| lplni 39074 | Condition implying a latti... |
| islpln3 39075 | The predicate "is a lattic... |
| lplnbase 39076 | A lattice plane is a latti... |
| islpln5 39077 | The predicate "is a lattic... |
| islpln2 39078 | The predicate "is a lattic... |
| lplni2 39079 | The join of 3 different at... |
| lvolex3N 39080 | There is an atom outside o... |
| llnmlplnN 39081 | The intersection of a line... |
| lplnle 39082 | Any element greater than 0... |
| lplnnle2at 39083 | A lattice line (or atom) c... |
| lplnnleat 39084 | A lattice plane cannot maj... |
| lplnnlelln 39085 | A lattice plane is not les... |
| 2atnelpln 39086 | The join of two atoms is n... |
| lplnneat 39087 | No lattice plane is an ato... |
| lplnnelln 39088 | No lattice plane is a latt... |
| lplnn0N 39089 | A lattice plane is nonzero... |
| islpln2a 39090 | The predicate "is a lattic... |
| islpln2ah 39091 | The predicate "is a lattic... |
| lplnriaN 39092 | Property of a lattice plan... |
| lplnribN 39093 | Property of a lattice plan... |
| lplnric 39094 | Property of a lattice plan... |
| lplnri1 39095 | Property of a lattice plan... |
| lplnri2N 39096 | Property of a lattice plan... |
| lplnri3N 39097 | Property of a lattice plan... |
| lplnllnneN 39098 | Two lattice lines defined ... |
| llncvrlpln2 39099 | A lattice line under a lat... |
| llncvrlpln 39100 | An element covering a latt... |
| 2lplnmN 39101 | If the join of two lattice... |
| 2llnmj 39102 | The meet of two lattice li... |
| 2atmat 39103 | The meet of two intersecti... |
| lplncmp 39104 | If two lattice planes are ... |
| lplnexatN 39105 | Given a lattice line on a ... |
| lplnexllnN 39106 | Given an atom on a lattice... |
| lplnnlt 39107 | Two lattice planes cannot ... |
| 2llnjaN 39108 | The join of two different ... |
| 2llnjN 39109 | The join of two different ... |
| 2llnm2N 39110 | The meet of two different ... |
| 2llnm3N 39111 | Two lattice lines in a lat... |
| 2llnm4 39112 | Two lattice lines that maj... |
| 2llnmeqat 39113 | An atom equals the interse... |
| lvolset 39114 | The set of 3-dim lattice v... |
| islvol 39115 | The predicate "is a 3-dim ... |
| islvol4 39116 | The predicate "is a 3-dim ... |
| lvoli 39117 | Condition implying a 3-dim... |
| islvol3 39118 | The predicate "is a 3-dim ... |
| lvoli3 39119 | Condition implying a 3-dim... |
| lvolbase 39120 | A 3-dim lattice volume is ... |
| islvol5 39121 | The predicate "is a 3-dim ... |
| islvol2 39122 | The predicate "is a 3-dim ... |
| lvoli2 39123 | The join of 4 different at... |
| lvolnle3at 39124 | A lattice plane (or lattic... |
| lvolnleat 39125 | An atom cannot majorize a ... |
| lvolnlelln 39126 | A lattice line cannot majo... |
| lvolnlelpln 39127 | A lattice plane cannot maj... |
| 3atnelvolN 39128 | The join of 3 atoms is not... |
| 2atnelvolN 39129 | The join of two atoms is n... |
| lvolneatN 39130 | No lattice volume is an at... |
| lvolnelln 39131 | No lattice volume is a lat... |
| lvolnelpln 39132 | No lattice volume is a lat... |
| lvoln0N 39133 | A lattice volume is nonzer... |
| islvol2aN 39134 | The predicate "is a lattic... |
| 4atlem0a 39135 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 39136 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 39137 | Lemma for ~ 4at . (Contri... |
| 4atlem3 39138 | Lemma for ~ 4at . Break i... |
| 4atlem3a 39139 | Lemma for ~ 4at . Break i... |
| 4atlem3b 39140 | Lemma for ~ 4at . Break i... |
| 4atlem4a 39141 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 39142 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 39143 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 39144 | Lemma for ~ 4at . Frequen... |
| 4atlem9 39145 | Lemma for ~ 4at . Substit... |
| 4atlem10a 39146 | Lemma for ~ 4at . Substit... |
| 4atlem10b 39147 | Lemma for ~ 4at . Substit... |
| 4atlem10 39148 | Lemma for ~ 4at . Combine... |
| 4atlem11a 39149 | Lemma for ~ 4at . Substit... |
| 4atlem11b 39150 | Lemma for ~ 4at . Substit... |
| 4atlem11 39151 | Lemma for ~ 4at . Combine... |
| 4atlem12a 39152 | Lemma for ~ 4at . Substit... |
| 4atlem12b 39153 | Lemma for ~ 4at . Substit... |
| 4atlem12 39154 | Lemma for ~ 4at . Combine... |
| 4at 39155 | Four atoms determine a lat... |
| 4at2 39156 | Four atoms determine a lat... |
| lplncvrlvol2 39157 | A lattice line under a lat... |
| lplncvrlvol 39158 | An element covering a latt... |
| lvolcmp 39159 | If two lattice planes are ... |
| lvolnltN 39160 | Two lattice volumes cannot... |
| 2lplnja 39161 | The join of two different ... |
| 2lplnj 39162 | The join of two different ... |
| 2lplnm2N 39163 | The meet of two different ... |
| 2lplnmj 39164 | The meet of two lattice pl... |
| dalemkehl 39165 | Lemma for ~ dath . Freque... |
| dalemkelat 39166 | Lemma for ~ dath . Freque... |
| dalemkeop 39167 | Lemma for ~ dath . Freque... |
| dalempea 39168 | Lemma for ~ dath . Freque... |
| dalemqea 39169 | Lemma for ~ dath . Freque... |
| dalemrea 39170 | Lemma for ~ dath . Freque... |
| dalemsea 39171 | Lemma for ~ dath . Freque... |
| dalemtea 39172 | Lemma for ~ dath . Freque... |
| dalemuea 39173 | Lemma for ~ dath . Freque... |
| dalemyeo 39174 | Lemma for ~ dath . Freque... |
| dalemzeo 39175 | Lemma for ~ dath . Freque... |
| dalemclpjs 39176 | Lemma for ~ dath . Freque... |
| dalemclqjt 39177 | Lemma for ~ dath . Freque... |
| dalemclrju 39178 | Lemma for ~ dath . Freque... |
| dalem-clpjq 39179 | Lemma for ~ dath . Freque... |
| dalemceb 39180 | Lemma for ~ dath . Freque... |
| dalempeb 39181 | Lemma for ~ dath . Freque... |
| dalemqeb 39182 | Lemma for ~ dath . Freque... |
| dalemreb 39183 | Lemma for ~ dath . Freque... |
| dalemseb 39184 | Lemma for ~ dath . Freque... |
| dalemteb 39185 | Lemma for ~ dath . Freque... |
| dalemueb 39186 | Lemma for ~ dath . Freque... |
| dalempjqeb 39187 | Lemma for ~ dath . Freque... |
| dalemsjteb 39188 | Lemma for ~ dath . Freque... |
| dalemtjueb 39189 | Lemma for ~ dath . Freque... |
| dalemqrprot 39190 | Lemma for ~ dath . Freque... |
| dalemyeb 39191 | Lemma for ~ dath . Freque... |
| dalemcnes 39192 | Lemma for ~ dath . Freque... |
| dalempnes 39193 | Lemma for ~ dath . Freque... |
| dalemqnet 39194 | Lemma for ~ dath . Freque... |
| dalempjsen 39195 | Lemma for ~ dath . Freque... |
| dalemply 39196 | Lemma for ~ dath . Freque... |
| dalemsly 39197 | Lemma for ~ dath . Freque... |
| dalemswapyz 39198 | Lemma for ~ dath . Swap t... |
| dalemrot 39199 | Lemma for ~ dath . Rotate... |
| dalemrotyz 39200 | Lemma for ~ dath . Rotate... |
| dalem1 39201 | Lemma for ~ dath . Show t... |
| dalemcea 39202 | Lemma for ~ dath . Freque... |
| dalem2 39203 | Lemma for ~ dath . Show t... |
| dalemdea 39204 | Lemma for ~ dath . Freque... |
| dalemeea 39205 | Lemma for ~ dath . Freque... |
| dalem3 39206 | Lemma for ~ dalemdnee . (... |
| dalem4 39207 | Lemma for ~ dalemdnee . (... |
| dalemdnee 39208 | Lemma for ~ dath . Axis o... |
| dalem5 39209 | Lemma for ~ dath . Atom `... |
| dalem6 39210 | Lemma for ~ dath . Analog... |
| dalem7 39211 | Lemma for ~ dath . Analog... |
| dalem8 39212 | Lemma for ~ dath . Plane ... |
| dalem-cly 39213 | Lemma for ~ dalem9 . Cent... |
| dalem9 39214 | Lemma for ~ dath . Since ... |
| dalem10 39215 | Lemma for ~ dath . Atom `... |
| dalem11 39216 | Lemma for ~ dath . Analog... |
| dalem12 39217 | Lemma for ~ dath . Analog... |
| dalem13 39218 | Lemma for ~ dalem14 . (Co... |
| dalem14 39219 | Lemma for ~ dath . Planes... |
| dalem15 39220 | Lemma for ~ dath . The ax... |
| dalem16 39221 | Lemma for ~ dath . The at... |
| dalem17 39222 | Lemma for ~ dath . When p... |
| dalem18 39223 | Lemma for ~ dath . Show t... |
| dalem19 39224 | Lemma for ~ dath . Show t... |
| dalemccea 39225 | Lemma for ~ dath . Freque... |
| dalemddea 39226 | Lemma for ~ dath . Freque... |
| dalem-ccly 39227 | Lemma for ~ dath . Freque... |
| dalem-ddly 39228 | Lemma for ~ dath . Freque... |
| dalemccnedd 39229 | Lemma for ~ dath . Freque... |
| dalemclccjdd 39230 | Lemma for ~ dath . Freque... |
| dalemcceb 39231 | Lemma for ~ dath . Freque... |
| dalemswapyzps 39232 | Lemma for ~ dath . Swap t... |
| dalemrotps 39233 | Lemma for ~ dath . Rotate... |
| dalemcjden 39234 | Lemma for ~ dath . Show t... |
| dalem20 39235 | Lemma for ~ dath . Show t... |
| dalem21 39236 | Lemma for ~ dath . Show t... |
| dalem22 39237 | Lemma for ~ dath . Show t... |
| dalem23 39238 | Lemma for ~ dath . Show t... |
| dalem24 39239 | Lemma for ~ dath . Show t... |
| dalem25 39240 | Lemma for ~ dath . Show t... |
| dalem27 39241 | Lemma for ~ dath . Show t... |
| dalem28 39242 | Lemma for ~ dath . Lemma ... |
| dalem29 39243 | Lemma for ~ dath . Analog... |
| dalem30 39244 | Lemma for ~ dath . Analog... |
| dalem31N 39245 | Lemma for ~ dath . Analog... |
| dalem32 39246 | Lemma for ~ dath . Analog... |
| dalem33 39247 | Lemma for ~ dath . Analog... |
| dalem34 39248 | Lemma for ~ dath . Analog... |
| dalem35 39249 | Lemma for ~ dath . Analog... |
| dalem36 39250 | Lemma for ~ dath . Analog... |
| dalem37 39251 | Lemma for ~ dath . Analog... |
| dalem38 39252 | Lemma for ~ dath . Plane ... |
| dalem39 39253 | Lemma for ~ dath . Auxili... |
| dalem40 39254 | Lemma for ~ dath . Analog... |
| dalem41 39255 | Lemma for ~ dath . (Contr... |
| dalem42 39256 | Lemma for ~ dath . Auxili... |
| dalem43 39257 | Lemma for ~ dath . Planes... |
| dalem44 39258 | Lemma for ~ dath . Dummy ... |
| dalem45 39259 | Lemma for ~ dath . Dummy ... |
| dalem46 39260 | Lemma for ~ dath . Analog... |
| dalem47 39261 | Lemma for ~ dath . Analog... |
| dalem48 39262 | Lemma for ~ dath . Analog... |
| dalem49 39263 | Lemma for ~ dath . Analog... |
| dalem50 39264 | Lemma for ~ dath . Analog... |
| dalem51 39265 | Lemma for ~ dath . Constr... |
| dalem52 39266 | Lemma for ~ dath . Lines ... |
| dalem53 39267 | Lemma for ~ dath . The au... |
| dalem54 39268 | Lemma for ~ dath . Line `... |
| dalem55 39269 | Lemma for ~ dath . Lines ... |
| dalem56 39270 | Lemma for ~ dath . Analog... |
| dalem57 39271 | Lemma for ~ dath . Axis o... |
| dalem58 39272 | Lemma for ~ dath . Analog... |
| dalem59 39273 | Lemma for ~ dath . Analog... |
| dalem60 39274 | Lemma for ~ dath . ` B ` i... |
| dalem61 39275 | Lemma for ~ dath . Show t... |
| dalem62 39276 | Lemma for ~ dath . Elimin... |
| dalem63 39277 | Lemma for ~ dath . Combin... |
| dath 39278 | Desargues's theorem of pro... |
| dath2 39279 | Version of Desargues's the... |
| lineset 39280 | The set of lines in a Hilb... |
| isline 39281 | The predicate "is a line".... |
| islinei 39282 | Condition implying "is a l... |
| pointsetN 39283 | The set of points in a Hil... |
| ispointN 39284 | The predicate "is a point"... |
| atpointN 39285 | The singleton of an atom i... |
| psubspset 39286 | The set of projective subs... |
| ispsubsp 39287 | The predicate "is a projec... |
| ispsubsp2 39288 | The predicate "is a projec... |
| psubspi 39289 | Property of a projective s... |
| psubspi2N 39290 | Property of a projective s... |
| 0psubN 39291 | The empty set is a project... |
| snatpsubN 39292 | The singleton of an atom i... |
| pointpsubN 39293 | A point (singleton of an a... |
| linepsubN 39294 | A line is a projective sub... |
| atpsubN 39295 | The set of all atoms is a ... |
| psubssat 39296 | A projective subspace cons... |
| psubatN 39297 | A member of a projective s... |
| pmapfval 39298 | The projective map of a Hi... |
| pmapval 39299 | Value of the projective ma... |
| elpmap 39300 | Member of a projective map... |
| pmapssat 39301 | The projective map of a Hi... |
| pmapssbaN 39302 | A weakening of ~ pmapssat ... |
| pmaple 39303 | The projective map of a Hi... |
| pmap11 39304 | The projective map of a Hi... |
| pmapat 39305 | The projective map of an a... |
| elpmapat 39306 | Member of the projective m... |
| pmap0 39307 | Value of the projective ma... |
| pmapeq0 39308 | A projective map value is ... |
| pmap1N 39309 | Value of the projective ma... |
| pmapsub 39310 | The projective map of a Hi... |
| pmapglbx 39311 | The projective map of the ... |
| pmapglb 39312 | The projective map of the ... |
| pmapglb2N 39313 | The projective map of the ... |
| pmapglb2xN 39314 | The projective map of the ... |
| pmapmeet 39315 | The projective map of a me... |
| isline2 39316 | Definition of line in term... |
| linepmap 39317 | A line described with a pr... |
| isline3 39318 | Definition of line in term... |
| isline4N 39319 | Definition of line in term... |
| lneq2at 39320 | A line equals the join of ... |
| lnatexN 39321 | There is an atom in a line... |
| lnjatN 39322 | Given an atom in a line, t... |
| lncvrelatN 39323 | A lattice element covered ... |
| lncvrat 39324 | A line covers the atoms it... |
| lncmp 39325 | If two lines are comparabl... |
| 2lnat 39326 | Two intersecting lines int... |
| 2atm2atN 39327 | Two joins with a common at... |
| 2llnma1b 39328 | Generalization of ~ 2llnma... |
| 2llnma1 39329 | Two different intersecting... |
| 2llnma3r 39330 | Two different intersecting... |
| 2llnma2 39331 | Two different intersecting... |
| 2llnma2rN 39332 | Two different intersecting... |
| cdlema1N 39333 | A condition for required f... |
| cdlema2N 39334 | A condition for required f... |
| cdlemblem 39335 | Lemma for ~ cdlemb . (Con... |
| cdlemb 39336 | Given two atoms not less t... |
| paddfval 39339 | Projective subspace sum op... |
| paddval 39340 | Projective subspace sum op... |
| elpadd 39341 | Member of a projective sub... |
| elpaddn0 39342 | Member of projective subsp... |
| paddvaln0N 39343 | Projective subspace sum op... |
| elpaddri 39344 | Condition implying members... |
| elpaddatriN 39345 | Condition implying members... |
| elpaddat 39346 | Membership in a projective... |
| elpaddatiN 39347 | Consequence of membership ... |
| elpadd2at 39348 | Membership in a projective... |
| elpadd2at2 39349 | Membership in a projective... |
| paddunssN 39350 | Projective subspace sum in... |
| elpadd0 39351 | Member of projective subsp... |
| paddval0 39352 | Projective subspace sum wi... |
| padd01 39353 | Projective subspace sum wi... |
| padd02 39354 | Projective subspace sum wi... |
| paddcom 39355 | Projective subspace sum co... |
| paddssat 39356 | A projective subspace sum ... |
| sspadd1 39357 | A projective subspace sum ... |
| sspadd2 39358 | A projective subspace sum ... |
| paddss1 39359 | Subset law for projective ... |
| paddss2 39360 | Subset law for projective ... |
| paddss12 39361 | Subset law for projective ... |
| paddasslem1 39362 | Lemma for ~ paddass . (Co... |
| paddasslem2 39363 | Lemma for ~ paddass . (Co... |
| paddasslem3 39364 | Lemma for ~ paddass . Res... |
| paddasslem4 39365 | Lemma for ~ paddass . Com... |
| paddasslem5 39366 | Lemma for ~ paddass . Sho... |
| paddasslem6 39367 | Lemma for ~ paddass . (Co... |
| paddasslem7 39368 | Lemma for ~ paddass . Com... |
| paddasslem8 39369 | Lemma for ~ paddass . (Co... |
| paddasslem9 39370 | Lemma for ~ paddass . Com... |
| paddasslem10 39371 | Lemma for ~ paddass . Use... |
| paddasslem11 39372 | Lemma for ~ paddass . The... |
| paddasslem12 39373 | Lemma for ~ paddass . The... |
| paddasslem13 39374 | Lemma for ~ paddass . The... |
| paddasslem14 39375 | Lemma for ~ paddass . Rem... |
| paddasslem15 39376 | Lemma for ~ paddass . Use... |
| paddasslem16 39377 | Lemma for ~ paddass . Use... |
| paddasslem17 39378 | Lemma for ~ paddass . The... |
| paddasslem18 39379 | Lemma for ~ paddass . Com... |
| paddass 39380 | Projective subspace sum is... |
| padd12N 39381 | Commutative/associative la... |
| padd4N 39382 | Rearrangement of 4 terms i... |
| paddidm 39383 | Projective subspace sum is... |
| paddclN 39384 | The projective sum of two ... |
| paddssw1 39385 | Subset law for projective ... |
| paddssw2 39386 | Subset law for projective ... |
| paddss 39387 | Subset law for projective ... |
| pmodlem1 39388 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 39389 | Lemma for ~ pmod1i . (Con... |
| pmod1i 39390 | The modular law holds in a... |
| pmod2iN 39391 | Dual of the modular law. ... |
| pmodN 39392 | The modular law for projec... |
| pmodl42N 39393 | Lemma derived from modular... |
| pmapjoin 39394 | The projective map of the ... |
| pmapjat1 39395 | The projective map of the ... |
| pmapjat2 39396 | The projective map of the ... |
| pmapjlln1 39397 | The projective map of the ... |
| hlmod1i 39398 | A version of the modular l... |
| atmod1i1 39399 | Version of modular law ~ p... |
| atmod1i1m 39400 | Version of modular law ~ p... |
| atmod1i2 39401 | Version of modular law ~ p... |
| llnmod1i2 39402 | Version of modular law ~ p... |
| atmod2i1 39403 | Version of modular law ~ p... |
| atmod2i2 39404 | Version of modular law ~ p... |
| llnmod2i2 39405 | Version of modular law ~ p... |
| atmod3i1 39406 | Version of modular law tha... |
| atmod3i2 39407 | Version of modular law tha... |
| atmod4i1 39408 | Version of modular law tha... |
| atmod4i2 39409 | Version of modular law tha... |
| llnexchb2lem 39410 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 39411 | Line exchange property (co... |
| llnexch2N 39412 | Line exchange property (co... |
| dalawlem1 39413 | Lemma for ~ dalaw . Speci... |
| dalawlem2 39414 | Lemma for ~ dalaw . Utili... |
| dalawlem3 39415 | Lemma for ~ dalaw . First... |
| dalawlem4 39416 | Lemma for ~ dalaw . Secon... |
| dalawlem5 39417 | Lemma for ~ dalaw . Speci... |
| dalawlem6 39418 | Lemma for ~ dalaw . First... |
| dalawlem7 39419 | Lemma for ~ dalaw . Secon... |
| dalawlem8 39420 | Lemma for ~ dalaw . Speci... |
| dalawlem9 39421 | Lemma for ~ dalaw . Speci... |
| dalawlem10 39422 | Lemma for ~ dalaw . Combi... |
| dalawlem11 39423 | Lemma for ~ dalaw . First... |
| dalawlem12 39424 | Lemma for ~ dalaw . Secon... |
| dalawlem13 39425 | Lemma for ~ dalaw . Speci... |
| dalawlem14 39426 | Lemma for ~ dalaw . Combi... |
| dalawlem15 39427 | Lemma for ~ dalaw . Swap ... |
| dalaw 39428 | Desargues's law, derived f... |
| pclfvalN 39431 | The projective subspace cl... |
| pclvalN 39432 | Value of the projective su... |
| pclclN 39433 | Closure of the projective ... |
| elpclN 39434 | Membership in the projecti... |
| elpcliN 39435 | Implication of membership ... |
| pclssN 39436 | Ordering is preserved by s... |
| pclssidN 39437 | A set of atoms is included... |
| pclidN 39438 | The projective subspace cl... |
| pclbtwnN 39439 | A projective subspace sand... |
| pclunN 39440 | The projective subspace cl... |
| pclun2N 39441 | The projective subspace cl... |
| pclfinN 39442 | The projective subspace cl... |
| pclcmpatN 39443 | The set of projective subs... |
| polfvalN 39446 | The projective subspace po... |
| polvalN 39447 | Value of the projective su... |
| polval2N 39448 | Alternate expression for v... |
| polsubN 39449 | The polarity of a set of a... |
| polssatN 39450 | The polarity of a set of a... |
| pol0N 39451 | The polarity of the empty ... |
| pol1N 39452 | The polarity of the whole ... |
| 2pol0N 39453 | The closed subspace closur... |
| polpmapN 39454 | The polarity of a projecti... |
| 2polpmapN 39455 | Double polarity of a proje... |
| 2polvalN 39456 | Value of double polarity. ... |
| 2polssN 39457 | A set of atoms is a subset... |
| 3polN 39458 | Triple polarity cancels to... |
| polcon3N 39459 | Contraposition law for pol... |
| 2polcon4bN 39460 | Contraposition law for pol... |
| polcon2N 39461 | Contraposition law for pol... |
| polcon2bN 39462 | Contraposition law for pol... |
| pclss2polN 39463 | The projective subspace cl... |
| pcl0N 39464 | The projective subspace cl... |
| pcl0bN 39465 | The projective subspace cl... |
| pmaplubN 39466 | The LUB of a projective ma... |
| sspmaplubN 39467 | A set of atoms is a subset... |
| 2pmaplubN 39468 | Double projective map of a... |
| paddunN 39469 | The closure of the project... |
| poldmj1N 39470 | De Morgan's law for polari... |
| pmapj2N 39471 | The projective map of the ... |
| pmapocjN 39472 | The projective map of the ... |
| polatN 39473 | The polarity of the single... |
| 2polatN 39474 | Double polarity of the sin... |
| pnonsingN 39475 | The intersection of a set ... |
| psubclsetN 39478 | The set of closed projecti... |
| ispsubclN 39479 | The predicate "is a closed... |
| psubcliN 39480 | Property of a closed proje... |
| psubcli2N 39481 | Property of a closed proje... |
| psubclsubN 39482 | A closed projective subspa... |
| psubclssatN 39483 | A closed projective subspa... |
| pmapidclN 39484 | Projective map of the LUB ... |
| 0psubclN 39485 | The empty set is a closed ... |
| 1psubclN 39486 | The set of all atoms is a ... |
| atpsubclN 39487 | A point (singleton of an a... |
| pmapsubclN 39488 | A projective map value is ... |
| ispsubcl2N 39489 | Alternate predicate for "i... |
| psubclinN 39490 | The intersection of two cl... |
| paddatclN 39491 | The projective sum of a cl... |
| pclfinclN 39492 | The projective subspace cl... |
| linepsubclN 39493 | A line is a closed project... |
| polsubclN 39494 | A polarity is a closed pro... |
| poml4N 39495 | Orthomodular law for proje... |
| poml5N 39496 | Orthomodular law for proje... |
| poml6N 39497 | Orthomodular law for proje... |
| osumcllem1N 39498 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 39499 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 39500 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 39501 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 39502 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 39503 | Lemma for ~ osumclN . Use... |
| osumcllem7N 39504 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 39505 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 39506 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 39507 | Lemma for ~ osumclN . Con... |
| osumcllem11N 39508 | Lemma for ~ osumclN . (Co... |
| osumclN 39509 | Closure of orthogonal sum.... |
| pmapojoinN 39510 | For orthogonal elements, p... |
| pexmidN 39511 | Excluded middle law for cl... |
| pexmidlem1N 39512 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 39513 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 39514 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 39515 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 39516 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 39517 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 39518 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 39519 | Lemma for ~ pexmidN . The... |
| pexmidALTN 39520 | Excluded middle law for cl... |
| pl42lem1N 39521 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 39522 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 39523 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 39524 | Lemma for ~ pl42N . (Cont... |
| pl42N 39525 | Law holding in a Hilbert l... |
| watfvalN 39534 | The W atoms function. (Co... |
| watvalN 39535 | Value of the W atoms funct... |
| iswatN 39536 | The predicate "is a W atom... |
| lhpset 39537 | The set of co-atoms (latti... |
| islhp 39538 | The predicate "is a co-ato... |
| islhp2 39539 | The predicate "is a co-ato... |
| lhpbase 39540 | A co-atom is a member of t... |
| lhp1cvr 39541 | The lattice unity covers a... |
| lhplt 39542 | An atom under a co-atom is... |
| lhp2lt 39543 | The join of two atoms unde... |
| lhpexlt 39544 | There exists an atom less ... |
| lhp0lt 39545 | A co-atom is greater than ... |
| lhpn0 39546 | A co-atom is nonzero. TOD... |
| lhpexle 39547 | There exists an atom under... |
| lhpexnle 39548 | There exists an atom not u... |
| lhpexle1lem 39549 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 39550 | There exists an atom under... |
| lhpexle2lem 39551 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 39552 | There exists atom under a ... |
| lhpexle3lem 39553 | There exists atom under a ... |
| lhpexle3 39554 | There exists atom under a ... |
| lhpex2leN 39555 | There exist at least two d... |
| lhpoc 39556 | The orthocomplement of a c... |
| lhpoc2N 39557 | The orthocomplement of an ... |
| lhpocnle 39558 | The orthocomplement of a c... |
| lhpocat 39559 | The orthocomplement of a c... |
| lhpocnel 39560 | The orthocomplement of a c... |
| lhpocnel2 39561 | The orthocomplement of a c... |
| lhpjat1 39562 | The join of a co-atom (hyp... |
| lhpjat2 39563 | The join of a co-atom (hyp... |
| lhpj1 39564 | The join of a co-atom (hyp... |
| lhpmcvr 39565 | The meet of a lattice hype... |
| lhpmcvr2 39566 | Alternate way to express t... |
| lhpmcvr3 39567 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 39568 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 39569 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 39570 | Specialization of ~ lhpmcv... |
| lhpm0atN 39571 | If the meet of a lattice h... |
| lhpmat 39572 | An element covered by the ... |
| lhpmatb 39573 | An element covered by the ... |
| lhp2at0 39574 | Join and meet with differe... |
| lhp2atnle 39575 | Inequality for 2 different... |
| lhp2atne 39576 | Inequality for joins with ... |
| lhp2at0nle 39577 | Inequality for 2 different... |
| lhp2at0ne 39578 | Inequality for joins with ... |
| lhpelim 39579 | Eliminate an atom not unde... |
| lhpmod2i2 39580 | Modular law for hyperplane... |
| lhpmod6i1 39581 | Modular law for hyperplane... |
| lhprelat3N 39582 | The Hilbert lattice is rel... |
| cdlemb2 39583 | Given two atoms not under ... |
| lhple 39584 | Property of a lattice elem... |
| lhpat 39585 | Create an atom under a co-... |
| lhpat4N 39586 | Property of an atom under ... |
| lhpat2 39587 | Create an atom under a co-... |
| lhpat3 39588 | There is only one atom und... |
| 4atexlemk 39589 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 39590 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 39591 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 39592 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 39593 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 39594 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 39595 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 39596 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 39597 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 39598 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 39599 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 39600 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 39601 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 39602 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 39603 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 39604 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 39605 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 39606 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 39607 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 39608 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 39609 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 39610 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 39611 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 39612 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 39613 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 39614 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 39615 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 39616 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 39617 | Whenever there are at leas... |
| 4atex 39618 | Whenever there are at leas... |
| 4atex2 39619 | More general version of ~ ... |
| 4atex2-0aOLDN 39620 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 39621 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 39622 | Same as ~ 4atex2 except th... |
| 4atex3 39623 | More general version of ~ ... |
| lautset 39624 | The set of lattice automor... |
| islaut 39625 | The predicate "is a lattic... |
| lautle 39626 | Less-than or equal propert... |
| laut1o 39627 | A lattice automorphism is ... |
| laut11 39628 | One-to-one property of a l... |
| lautcl 39629 | A lattice automorphism val... |
| lautcnvclN 39630 | Reverse closure of a latti... |
| lautcnvle 39631 | Less-than or equal propert... |
| lautcnv 39632 | The converse of a lattice ... |
| lautlt 39633 | Less-than property of a la... |
| lautcvr 39634 | Covering property of a lat... |
| lautj 39635 | Meet property of a lattice... |
| lautm 39636 | Meet property of a lattice... |
| lauteq 39637 | A lattice automorphism arg... |
| idlaut 39638 | The identity function is a... |
| lautco 39639 | The composition of two lat... |
| pautsetN 39640 | The set of projective auto... |
| ispautN 39641 | The predicate "is a projec... |
| ldilfset 39650 | The mapping from fiducial ... |
| ldilset 39651 | The set of lattice dilatio... |
| isldil 39652 | The predicate "is a lattic... |
| ldillaut 39653 | A lattice dilation is an a... |
| ldil1o 39654 | A lattice dilation is a on... |
| ldilval 39655 | Value of a lattice dilatio... |
| idldil 39656 | The identity function is a... |
| ldilcnv 39657 | The converse of a lattice ... |
| ldilco 39658 | The composition of two lat... |
| ltrnfset 39659 | The set of all lattice tra... |
| ltrnset 39660 | The set of lattice transla... |
| isltrn 39661 | The predicate "is a lattic... |
| isltrn2N 39662 | The predicate "is a lattic... |
| ltrnu 39663 | Uniqueness property of a l... |
| ltrnldil 39664 | A lattice translation is a... |
| ltrnlaut 39665 | A lattice translation is a... |
| ltrn1o 39666 | A lattice translation is a... |
| ltrncl 39667 | Closure of a lattice trans... |
| ltrn11 39668 | One-to-one property of a l... |
| ltrncnvnid 39669 | If a translation is differ... |
| ltrncoidN 39670 | Two translations are equal... |
| ltrnle 39671 | Less-than or equal propert... |
| ltrncnvleN 39672 | Less-than or equal propert... |
| ltrnm 39673 | Lattice translation of a m... |
| ltrnj 39674 | Lattice translation of a m... |
| ltrncvr 39675 | Covering property of a lat... |
| ltrnval1 39676 | Value of a lattice transla... |
| ltrnid 39677 | A lattice translation is t... |
| ltrnnid 39678 | If a lattice translation i... |
| ltrnatb 39679 | The lattice translation of... |
| ltrncnvatb 39680 | The converse of the lattic... |
| ltrnel 39681 | The lattice translation of... |
| ltrnat 39682 | The lattice translation of... |
| ltrncnvat 39683 | The converse of the lattic... |
| ltrncnvel 39684 | The converse of the lattic... |
| ltrncoelN 39685 | Composition of lattice tra... |
| ltrncoat 39686 | Composition of lattice tra... |
| ltrncoval 39687 | Two ways to express value ... |
| ltrncnv 39688 | The converse of a lattice ... |
| ltrn11at 39689 | Frequently used one-to-one... |
| ltrneq2 39690 | The equality of two transl... |
| ltrneq 39691 | The equality of two transl... |
| idltrn 39692 | The identity function is a... |
| ltrnmw 39693 | Property of lattice transl... |
| dilfsetN 39694 | The mapping from fiducial ... |
| dilsetN 39695 | The set of dilations for a... |
| isdilN 39696 | The predicate "is a dilati... |
| trnfsetN 39697 | The mapping from fiducial ... |
| trnsetN 39698 | The set of translations fo... |
| istrnN 39699 | The predicate "is a transl... |
| trlfset 39702 | The set of all traces of l... |
| trlset 39703 | The set of traces of latti... |
| trlval 39704 | The value of the trace of ... |
| trlval2 39705 | The value of the trace of ... |
| trlcl 39706 | Closure of the trace of a ... |
| trlcnv 39707 | The trace of the converse ... |
| trljat1 39708 | The value of a translation... |
| trljat2 39709 | The value of a translation... |
| trljat3 39710 | The value of a translation... |
| trlat 39711 | If an atom differs from it... |
| trl0 39712 | If an atom not under the f... |
| trlator0 39713 | The trace of a lattice tra... |
| trlatn0 39714 | The trace of a lattice tra... |
| trlnidat 39715 | The trace of a lattice tra... |
| ltrnnidn 39716 | If a lattice translation i... |
| ltrnideq 39717 | Property of the identity l... |
| trlid0 39718 | The trace of the identity ... |
| trlnidatb 39719 | A lattice translation is n... |
| trlid0b 39720 | A lattice translation is t... |
| trlnid 39721 | Different translations wit... |
| ltrn2ateq 39722 | Property of the equality o... |
| ltrnateq 39723 | If any atom (under ` W ` )... |
| ltrnatneq 39724 | If any atom (under ` W ` )... |
| ltrnatlw 39725 | If the value of an atom eq... |
| trlle 39726 | The trace of a lattice tra... |
| trlne 39727 | The trace of a lattice tra... |
| trlnle 39728 | The atom not under the fid... |
| trlval3 39729 | The value of the trace of ... |
| trlval4 39730 | The value of the trace of ... |
| trlval5 39731 | The value of the trace of ... |
| arglem1N 39732 | Lemma for Desargues's law.... |
| cdlemc1 39733 | Part of proof of Lemma C i... |
| cdlemc2 39734 | Part of proof of Lemma C i... |
| cdlemc3 39735 | Part of proof of Lemma C i... |
| cdlemc4 39736 | Part of proof of Lemma C i... |
| cdlemc5 39737 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 39738 | Lemma for ~ cdlemc . (Con... |
| cdlemc 39739 | Lemma C in [Crawley] p. 11... |
| cdlemd1 39740 | Part of proof of Lemma D i... |
| cdlemd2 39741 | Part of proof of Lemma D i... |
| cdlemd3 39742 | Part of proof of Lemma D i... |
| cdlemd4 39743 | Part of proof of Lemma D i... |
| cdlemd5 39744 | Part of proof of Lemma D i... |
| cdlemd6 39745 | Part of proof of Lemma D i... |
| cdlemd7 39746 | Part of proof of Lemma D i... |
| cdlemd8 39747 | Part of proof of Lemma D i... |
| cdlemd9 39748 | Part of proof of Lemma D i... |
| cdlemd 39749 | If two translations agree ... |
| ltrneq3 39750 | Two translations agree at ... |
| cdleme00a 39751 | Part of proof of Lemma E i... |
| cdleme0aa 39752 | Part of proof of Lemma E i... |
| cdleme0a 39753 | Part of proof of Lemma E i... |
| cdleme0b 39754 | Part of proof of Lemma E i... |
| cdleme0c 39755 | Part of proof of Lemma E i... |
| cdleme0cp 39756 | Part of proof of Lemma E i... |
| cdleme0cq 39757 | Part of proof of Lemma E i... |
| cdleme0dN 39758 | Part of proof of Lemma E i... |
| cdleme0e 39759 | Part of proof of Lemma E i... |
| cdleme0fN 39760 | Part of proof of Lemma E i... |
| cdleme0gN 39761 | Part of proof of Lemma E i... |
| cdlemeulpq 39762 | Part of proof of Lemma E i... |
| cdleme01N 39763 | Part of proof of Lemma E i... |
| cdleme02N 39764 | Part of proof of Lemma E i... |
| cdleme0ex1N 39765 | Part of proof of Lemma E i... |
| cdleme0ex2N 39766 | Part of proof of Lemma E i... |
| cdleme0moN 39767 | Part of proof of Lemma E i... |
| cdleme1b 39768 | Part of proof of Lemma E i... |
| cdleme1 39769 | Part of proof of Lemma E i... |
| cdleme2 39770 | Part of proof of Lemma E i... |
| cdleme3b 39771 | Part of proof of Lemma E i... |
| cdleme3c 39772 | Part of proof of Lemma E i... |
| cdleme3d 39773 | Part of proof of Lemma E i... |
| cdleme3e 39774 | Part of proof of Lemma E i... |
| cdleme3fN 39775 | Part of proof of Lemma E i... |
| cdleme3g 39776 | Part of proof of Lemma E i... |
| cdleme3h 39777 | Part of proof of Lemma E i... |
| cdleme3fa 39778 | Part of proof of Lemma E i... |
| cdleme3 39779 | Part of proof of Lemma E i... |
| cdleme4 39780 | Part of proof of Lemma E i... |
| cdleme4a 39781 | Part of proof of Lemma E i... |
| cdleme5 39782 | Part of proof of Lemma E i... |
| cdleme6 39783 | Part of proof of Lemma E i... |
| cdleme7aa 39784 | Part of proof of Lemma E i... |
| cdleme7a 39785 | Part of proof of Lemma E i... |
| cdleme7b 39786 | Part of proof of Lemma E i... |
| cdleme7c 39787 | Part of proof of Lemma E i... |
| cdleme7d 39788 | Part of proof of Lemma E i... |
| cdleme7e 39789 | Part of proof of Lemma E i... |
| cdleme7ga 39790 | Part of proof of Lemma E i... |
| cdleme7 39791 | Part of proof of Lemma E i... |
| cdleme8 39792 | Part of proof of Lemma E i... |
| cdleme9a 39793 | Part of proof of Lemma E i... |
| cdleme9b 39794 | Utility lemma for Lemma E ... |
| cdleme9 39795 | Part of proof of Lemma E i... |
| cdleme10 39796 | Part of proof of Lemma E i... |
| cdleme8tN 39797 | Part of proof of Lemma E i... |
| cdleme9taN 39798 | Part of proof of Lemma E i... |
| cdleme9tN 39799 | Part of proof of Lemma E i... |
| cdleme10tN 39800 | Part of proof of Lemma E i... |
| cdleme16aN 39801 | Part of proof of Lemma E i... |
| cdleme11a 39802 | Part of proof of Lemma E i... |
| cdleme11c 39803 | Part of proof of Lemma E i... |
| cdleme11dN 39804 | Part of proof of Lemma E i... |
| cdleme11e 39805 | Part of proof of Lemma E i... |
| cdleme11fN 39806 | Part of proof of Lemma E i... |
| cdleme11g 39807 | Part of proof of Lemma E i... |
| cdleme11h 39808 | Part of proof of Lemma E i... |
| cdleme11j 39809 | Part of proof of Lemma E i... |
| cdleme11k 39810 | Part of proof of Lemma E i... |
| cdleme11l 39811 | Part of proof of Lemma E i... |
| cdleme11 39812 | Part of proof of Lemma E i... |
| cdleme12 39813 | Part of proof of Lemma E i... |
| cdleme13 39814 | Part of proof of Lemma E i... |
| cdleme14 39815 | Part of proof of Lemma E i... |
| cdleme15a 39816 | Part of proof of Lemma E i... |
| cdleme15b 39817 | Part of proof of Lemma E i... |
| cdleme15c 39818 | Part of proof of Lemma E i... |
| cdleme15d 39819 | Part of proof of Lemma E i... |
| cdleme15 39820 | Part of proof of Lemma E i... |
| cdleme16b 39821 | Part of proof of Lemma E i... |
| cdleme16c 39822 | Part of proof of Lemma E i... |
| cdleme16d 39823 | Part of proof of Lemma E i... |
| cdleme16e 39824 | Part of proof of Lemma E i... |
| cdleme16f 39825 | Part of proof of Lemma E i... |
| cdleme16g 39826 | Part of proof of Lemma E i... |
| cdleme16 39827 | Part of proof of Lemma E i... |
| cdleme17a 39828 | Part of proof of Lemma E i... |
| cdleme17b 39829 | Lemma leading to ~ cdleme1... |
| cdleme17c 39830 | Part of proof of Lemma E i... |
| cdleme17d1 39831 | Part of proof of Lemma E i... |
| cdleme0nex 39832 | Part of proof of Lemma E i... |
| cdleme18a 39833 | Part of proof of Lemma E i... |
| cdleme18b 39834 | Part of proof of Lemma E i... |
| cdleme18c 39835 | Part of proof of Lemma E i... |
| cdleme22gb 39836 | Utility lemma for Lemma E ... |
| cdleme18d 39837 | Part of proof of Lemma E i... |
| cdlemesner 39838 | Part of proof of Lemma E i... |
| cdlemedb 39839 | Part of proof of Lemma E i... |
| cdlemeda 39840 | Part of proof of Lemma E i... |
| cdlemednpq 39841 | Part of proof of Lemma E i... |
| cdlemednuN 39842 | Part of proof of Lemma E i... |
| cdleme20zN 39843 | Part of proof of Lemma E i... |
| cdleme20y 39844 | Part of proof of Lemma E i... |
| cdleme19a 39845 | Part of proof of Lemma E i... |
| cdleme19b 39846 | Part of proof of Lemma E i... |
| cdleme19c 39847 | Part of proof of Lemma E i... |
| cdleme19d 39848 | Part of proof of Lemma E i... |
| cdleme19e 39849 | Part of proof of Lemma E i... |
| cdleme19f 39850 | Part of proof of Lemma E i... |
| cdleme20aN 39851 | Part of proof of Lemma E i... |
| cdleme20bN 39852 | Part of proof of Lemma E i... |
| cdleme20c 39853 | Part of proof of Lemma E i... |
| cdleme20d 39854 | Part of proof of Lemma E i... |
| cdleme20e 39855 | Part of proof of Lemma E i... |
| cdleme20f 39856 | Part of proof of Lemma E i... |
| cdleme20g 39857 | Part of proof of Lemma E i... |
| cdleme20h 39858 | Part of proof of Lemma E i... |
| cdleme20i 39859 | Part of proof of Lemma E i... |
| cdleme20j 39860 | Part of proof of Lemma E i... |
| cdleme20k 39861 | Part of proof of Lemma E i... |
| cdleme20l1 39862 | Part of proof of Lemma E i... |
| cdleme20l2 39863 | Part of proof of Lemma E i... |
| cdleme20l 39864 | Part of proof of Lemma E i... |
| cdleme20m 39865 | Part of proof of Lemma E i... |
| cdleme20 39866 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 39867 | Part of proof of Lemma E i... |
| cdleme21b 39868 | Part of proof of Lemma E i... |
| cdleme21c 39869 | Part of proof of Lemma E i... |
| cdleme21at 39870 | Part of proof of Lemma E i... |
| cdleme21ct 39871 | Part of proof of Lemma E i... |
| cdleme21d 39872 | Part of proof of Lemma E i... |
| cdleme21e 39873 | Part of proof of Lemma E i... |
| cdleme21f 39874 | Part of proof of Lemma E i... |
| cdleme21g 39875 | Part of proof of Lemma E i... |
| cdleme21h 39876 | Part of proof of Lemma E i... |
| cdleme21i 39877 | Part of proof of Lemma E i... |
| cdleme21j 39878 | Combine ~ cdleme20 and ~ c... |
| cdleme21 39879 | Part of proof of Lemma E i... |
| cdleme21k 39880 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 39881 | Part of proof of Lemma E i... |
| cdleme22a 39882 | Part of proof of Lemma E i... |
| cdleme22b 39883 | Part of proof of Lemma E i... |
| cdleme22cN 39884 | Part of proof of Lemma E i... |
| cdleme22d 39885 | Part of proof of Lemma E i... |
| cdleme22e 39886 | Part of proof of Lemma E i... |
| cdleme22eALTN 39887 | Part of proof of Lemma E i... |
| cdleme22f 39888 | Part of proof of Lemma E i... |
| cdleme22f2 39889 | Part of proof of Lemma E i... |
| cdleme22g 39890 | Part of proof of Lemma E i... |
| cdleme23a 39891 | Part of proof of Lemma E i... |
| cdleme23b 39892 | Part of proof of Lemma E i... |
| cdleme23c 39893 | Part of proof of Lemma E i... |
| cdleme24 39894 | Quantified version of ~ cd... |
| cdleme25a 39895 | Lemma for ~ cdleme25b . (... |
| cdleme25b 39896 | Transform ~ cdleme24 . TO... |
| cdleme25c 39897 | Transform ~ cdleme25b . (... |
| cdleme25dN 39898 | Transform ~ cdleme25c . (... |
| cdleme25cl 39899 | Show closure of the unique... |
| cdleme25cv 39900 | Change bound variables in ... |
| cdleme26e 39901 | Part of proof of Lemma E i... |
| cdleme26ee 39902 | Part of proof of Lemma E i... |
| cdleme26eALTN 39903 | Part of proof of Lemma E i... |
| cdleme26fALTN 39904 | Part of proof of Lemma E i... |
| cdleme26f 39905 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 39906 | Part of proof of Lemma E i... |
| cdleme26f2 39907 | Part of proof of Lemma E i... |
| cdleme27cl 39908 | Part of proof of Lemma E i... |
| cdleme27a 39909 | Part of proof of Lemma E i... |
| cdleme27b 39910 | Lemma for ~ cdleme27N . (... |
| cdleme27N 39911 | Part of proof of Lemma E i... |
| cdleme28a 39912 | Lemma for ~ cdleme25b . T... |
| cdleme28b 39913 | Lemma for ~ cdleme25b . T... |
| cdleme28c 39914 | Part of proof of Lemma E i... |
| cdleme28 39915 | Quantified version of ~ cd... |
| cdleme29ex 39916 | Lemma for ~ cdleme29b . (... |
| cdleme29b 39917 | Transform ~ cdleme28 . (C... |
| cdleme29c 39918 | Transform ~ cdleme28b . (... |
| cdleme29cl 39919 | Show closure of the unique... |
| cdleme30a 39920 | Part of proof of Lemma E i... |
| cdleme31so 39921 | Part of proof of Lemma E i... |
| cdleme31sn 39922 | Part of proof of Lemma E i... |
| cdleme31sn1 39923 | Part of proof of Lemma E i... |
| cdleme31se 39924 | Part of proof of Lemma D i... |
| cdleme31se2 39925 | Part of proof of Lemma D i... |
| cdleme31sc 39926 | Part of proof of Lemma E i... |
| cdleme31sde 39927 | Part of proof of Lemma D i... |
| cdleme31snd 39928 | Part of proof of Lemma D i... |
| cdleme31sdnN 39929 | Part of proof of Lemma E i... |
| cdleme31sn1c 39930 | Part of proof of Lemma E i... |
| cdleme31sn2 39931 | Part of proof of Lemma E i... |
| cdleme31fv 39932 | Part of proof of Lemma E i... |
| cdleme31fv1 39933 | Part of proof of Lemma E i... |
| cdleme31fv1s 39934 | Part of proof of Lemma E i... |
| cdleme31fv2 39935 | Part of proof of Lemma E i... |
| cdleme31id 39936 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 39937 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 39938 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 39939 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 39940 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 39941 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 39942 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 39943 | Part of proof of Lemma E i... |
| cdlemefr29exN 39944 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 39945 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 39946 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 39947 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 39948 | TODO fix comment. (Contri... |
| cdlemefr29clN 39949 | Show closure of the unique... |
| cdleme43frv1snN 39950 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 39951 | Part of proof of Lemma E i... |
| cdlemefr32fva1 39952 | Part of proof of Lemma E i... |
| cdlemefr31fv1 39953 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 39954 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 39955 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 39956 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 39957 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 39958 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 39959 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 39960 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 39961 | Show closure of the unique... |
| cdleme43fsv1snlem 39962 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 39963 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 39964 | Part of proof of Lemma E i... |
| cdlemefs32fva1 39965 | Part of proof of Lemma E i... |
| cdlemefs31fv1 39966 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 39967 | Value of f(r) when r is an... |
| cdlemefs44 39968 | Value of f_s(r) when r is ... |
| cdlemefr45 39969 | Value of f(r) when r is an... |
| cdlemefr45e 39970 | Explicit expansion of ~ cd... |
| cdlemefs45 39971 | Value of f_s(r) when r is ... |
| cdlemefs45ee 39972 | Explicit expansion of ~ cd... |
| cdlemefs45eN 39973 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 39974 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 39975 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 39976 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 39977 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 39978 | Show closure of ` [_ R / s... |
| cdleme32fva 39979 | Part of proof of Lemma D i... |
| cdleme32fva1 39980 | Part of proof of Lemma D i... |
| cdleme32fvaw 39981 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 39982 | Part of proof of Lemma D i... |
| cdleme32a 39983 | Part of proof of Lemma D i... |
| cdleme32b 39984 | Part of proof of Lemma D i... |
| cdleme32c 39985 | Part of proof of Lemma D i... |
| cdleme32d 39986 | Part of proof of Lemma D i... |
| cdleme32e 39987 | Part of proof of Lemma D i... |
| cdleme32f 39988 | Part of proof of Lemma D i... |
| cdleme32le 39989 | Part of proof of Lemma D i... |
| cdleme35a 39990 | Part of proof of Lemma E i... |
| cdleme35fnpq 39991 | Part of proof of Lemma E i... |
| cdleme35b 39992 | Part of proof of Lemma E i... |
| cdleme35c 39993 | Part of proof of Lemma E i... |
| cdleme35d 39994 | Part of proof of Lemma E i... |
| cdleme35e 39995 | Part of proof of Lemma E i... |
| cdleme35f 39996 | Part of proof of Lemma E i... |
| cdleme35g 39997 | Part of proof of Lemma E i... |
| cdleme35h 39998 | Part of proof of Lemma E i... |
| cdleme35h2 39999 | Part of proof of Lemma E i... |
| cdleme35sn2aw 40000 | Part of proof of Lemma E i... |
| cdleme35sn3a 40001 | Part of proof of Lemma E i... |
| cdleme36a 40002 | Part of proof of Lemma E i... |
| cdleme36m 40003 | Part of proof of Lemma E i... |
| cdleme37m 40004 | Part of proof of Lemma E i... |
| cdleme38m 40005 | Part of proof of Lemma E i... |
| cdleme38n 40006 | Part of proof of Lemma E i... |
| cdleme39a 40007 | Part of proof of Lemma E i... |
| cdleme39n 40008 | Part of proof of Lemma E i... |
| cdleme40m 40009 | Part of proof of Lemma E i... |
| cdleme40n 40010 | Part of proof of Lemma E i... |
| cdleme40v 40011 | Part of proof of Lemma E i... |
| cdleme40w 40012 | Part of proof of Lemma E i... |
| cdleme42a 40013 | Part of proof of Lemma E i... |
| cdleme42c 40014 | Part of proof of Lemma E i... |
| cdleme42d 40015 | Part of proof of Lemma E i... |
| cdleme41sn3aw 40016 | Part of proof of Lemma E i... |
| cdleme41sn4aw 40017 | Part of proof of Lemma E i... |
| cdleme41snaw 40018 | Part of proof of Lemma E i... |
| cdleme41fva11 40019 | Part of proof of Lemma E i... |
| cdleme42b 40020 | Part of proof of Lemma E i... |
| cdleme42e 40021 | Part of proof of Lemma E i... |
| cdleme42f 40022 | Part of proof of Lemma E i... |
| cdleme42g 40023 | Part of proof of Lemma E i... |
| cdleme42h 40024 | Part of proof of Lemma E i... |
| cdleme42i 40025 | Part of proof of Lemma E i... |
| cdleme42k 40026 | Part of proof of Lemma E i... |
| cdleme42ke 40027 | Part of proof of Lemma E i... |
| cdleme42keg 40028 | Part of proof of Lemma E i... |
| cdleme42mN 40029 | Part of proof of Lemma E i... |
| cdleme42mgN 40030 | Part of proof of Lemma E i... |
| cdleme43aN 40031 | Part of proof of Lemma E i... |
| cdleme43bN 40032 | Lemma for Lemma E in [Craw... |
| cdleme43cN 40033 | Part of proof of Lemma E i... |
| cdleme43dN 40034 | Part of proof of Lemma E i... |
| cdleme46f2g2 40035 | Conversion for ` G ` to re... |
| cdleme46f2g1 40036 | Conversion for ` G ` to re... |
| cdleme17d2 40037 | Part of proof of Lemma E i... |
| cdleme17d3 40038 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 40039 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 40040 | Part of proof of Lemma E i... |
| cdleme48fv 40041 | Part of proof of Lemma D i... |
| cdleme48fvg 40042 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 40043 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 40044 | TODO: fix comment. TODO: ... |
| cdleme48b 40045 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 40046 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 40047 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 40048 | TODO: fix comment. (Contr... |
| cdleme4gfv 40049 | Part of proof of Lemma D i... |
| cdlemeg47b 40050 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 40051 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 40052 | Value of g_s(r) when r is ... |
| cdlemeg49le 40053 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 40054 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 40055 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 40056 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 40057 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 40058 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 40059 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 40060 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 40061 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 40062 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 40063 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 40064 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 40065 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 40066 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 40067 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 40068 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 40069 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 40070 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 40071 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 40072 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 40073 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 40074 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 40075 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 40076 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 40077 | TODO: fix comment. (Contr... |
| cdleme48gfv1 40078 | TODO: fix comment. (Contr... |
| cdleme48gfv 40079 | TODO: fix comment. (Contr... |
| cdleme48fgv 40080 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 40081 | Part of proof of Lemma D i... |
| cdleme50lebi 40082 | Part of proof of Lemma D i... |
| cdleme50eq 40083 | Part of proof of Lemma D i... |
| cdleme50f 40084 | Part of proof of Lemma D i... |
| cdleme50f1 40085 | Part of proof of Lemma D i... |
| cdleme50rnlem 40086 | Part of proof of Lemma D i... |
| cdleme50rn 40087 | Part of proof of Lemma D i... |
| cdleme50f1o 40088 | Part of proof of Lemma D i... |
| cdleme50laut 40089 | Part of proof of Lemma D i... |
| cdleme50ldil 40090 | Part of proof of Lemma D i... |
| cdleme50trn1 40091 | Part of proof that ` F ` i... |
| cdleme50trn2a 40092 | Part of proof that ` F ` i... |
| cdleme50trn2 40093 | Part of proof that ` F ` i... |
| cdleme50trn12 40094 | Part of proof that ` F ` i... |
| cdleme50trn3 40095 | Part of proof that ` F ` i... |
| cdleme50trn123 40096 | Part of proof that ` F ` i... |
| cdleme51finvfvN 40097 | Part of proof of Lemma E i... |
| cdleme51finvN 40098 | Part of proof of Lemma E i... |
| cdleme50ltrn 40099 | Part of proof of Lemma E i... |
| cdleme51finvtrN 40100 | Part of proof of Lemma E i... |
| cdleme50ex 40101 | Part of Lemma E in [Crawle... |
| cdleme 40102 | Lemma E in [Crawley] p. 11... |
| cdlemf1 40103 | Part of Lemma F in [Crawle... |
| cdlemf2 40104 | Part of Lemma F in [Crawle... |
| cdlemf 40105 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 40106 | ~ cdlemf with additional c... |
| cdlemftr3 40107 | Special case of ~ cdlemf s... |
| cdlemftr2 40108 | Special case of ~ cdlemf s... |
| cdlemftr1 40109 | Part of proof of Lemma G o... |
| cdlemftr0 40110 | Special case of ~ cdlemf s... |
| trlord 40111 | The ordering of two Hilber... |
| cdlemg1a 40112 | Shorter expression for ` G... |
| cdlemg1b2 40113 | This theorem can be used t... |
| cdlemg1idlemN 40114 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 40115 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 40116 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 40117 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 40118 | This theorem can be used t... |
| cdlemg1idN 40119 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 40120 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 40121 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 40122 | Version of ~ cdleme51finvt... |
| ltrniotaval 40123 | Value of the unique transl... |
| ltrniotacnvval 40124 | Converse value of the uniq... |
| ltrniotaidvalN 40125 | Value of the unique transl... |
| ltrniotavalbN 40126 | Value of the unique transl... |
| cdlemeiota 40127 | A translation is uniquely ... |
| cdlemg1ci2 40128 | Any function of the form o... |
| cdlemg1cN 40129 | Any translation belongs to... |
| cdlemg1cex 40130 | Any translation is one of ... |
| cdlemg2cN 40131 | Any translation belongs to... |
| cdlemg2dN 40132 | This theorem can be used t... |
| cdlemg2cex 40133 | Any translation is one of ... |
| cdlemg2ce 40134 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 40135 | Part of proof of Lemma E i... |
| cdlemg2fvlem 40136 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 40137 | ~ cdleme42keg with simpler... |
| cdlemg2idN 40138 | Version of ~ cdleme31id wi... |
| cdlemg3a 40139 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 40140 | TODO: Replace this with ~... |
| cdlemg2fv 40141 | Value of a translation in ... |
| cdlemg2fv2 40142 | Value of a translation in ... |
| cdlemg2k 40143 | ~ cdleme42keg with simpler... |
| cdlemg2kq 40144 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 40145 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 40146 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 40147 | TODO: Is there a simpler ... |
| cdlemb3 40148 | Given two atoms not under ... |
| cdlemg7fvbwN 40149 | Properties of a translatio... |
| cdlemg4a 40150 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 40151 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 40152 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 40153 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 40154 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 40155 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 40156 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 40157 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 40158 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 40159 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 40160 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 40161 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 40162 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 40163 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 40164 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 40165 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 40166 | Value of a translation com... |
| cdlemg7aN 40167 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 40168 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 40169 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 40170 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 40171 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 40172 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 40173 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 40174 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 40175 | The triples ` <. P , ( F `... |
| cdlemg9 40176 | The triples ` <. P , ( F `... |
| cdlemg10b 40177 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 40178 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 40179 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 40180 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 40181 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 40182 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 40183 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 40184 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 40185 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 40186 | The triples ` <. P , ( F `... |
| cdlemg12c 40187 | The triples ` <. P , ( F `... |
| cdlemg12d 40188 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 40189 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 40190 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 40191 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 40192 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 40193 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 40194 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 40195 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 40196 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 40197 | Eliminate the ` ( F `` P )... |
| cdlemg15 40198 | Eliminate the ` ( (... |
| cdlemg16 40199 | Part of proof of Lemma G o... |
| cdlemg16ALTN 40200 | This version of ~ cdlemg16... |
| cdlemg16z 40201 | Eliminate ` ( ( F `... |
| cdlemg16zz 40202 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 40203 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 40204 | Part of proof of Lemma G i... |
| cdlemg17dN 40205 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 40206 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 40207 | TODO: fix comment. (Contr... |
| cdlemg17f 40208 | TODO: fix comment. (Contr... |
| cdlemg17g 40209 | TODO: fix comment. (Contr... |
| cdlemg17h 40210 | TODO: fix comment. (Contr... |
| cdlemg17i 40211 | TODO: fix comment. (Contr... |
| cdlemg17ir 40212 | TODO: fix comment. (Contr... |
| cdlemg17j 40213 | TODO: fix comment. (Contr... |
| cdlemg17pq 40214 | Utility theorem for swappi... |
| cdlemg17bq 40215 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 40216 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 40217 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 40218 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 40219 | Part of Lemma G of [Crawle... |
| cdlemg18a 40220 | Show two lines are differe... |
| cdlemg18b 40221 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 40222 | Show two lines intersect a... |
| cdlemg18d 40223 | Show two lines intersect a... |
| cdlemg18 40224 | Show two lines intersect a... |
| cdlemg19a 40225 | Show two lines intersect a... |
| cdlemg19 40226 | Show two lines intersect a... |
| cdlemg20 40227 | Show two lines intersect a... |
| cdlemg21 40228 | Version of cdlemg19 with `... |
| cdlemg22 40229 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 40230 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 40231 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 40232 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 40233 | ~ cdlemg16zz restated for ... |
| cdlemg27a 40234 | For use with case when ` (... |
| cdlemg28a 40235 | Part of proof of Lemma G o... |
| cdlemg31b0N 40236 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 40237 | TODO: Fix comment. (Cont... |
| cdlemg27b 40238 | TODO: Fix comment. (Cont... |
| cdlemg31a 40239 | TODO: fix comment. (Contr... |
| cdlemg31b 40240 | TODO: fix comment. (Contr... |
| cdlemg31c 40241 | Show that when ` N ` is an... |
| cdlemg31d 40242 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 40243 | TODO: Fix comment. (Cont... |
| cdlemg33c0 40244 | TODO: Fix comment. (Cont... |
| cdlemg28b 40245 | Part of proof of Lemma G o... |
| cdlemg28 40246 | Part of proof of Lemma G o... |
| cdlemg29 40247 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 40248 | TODO: Fix comment. (Cont... |
| cdlemg33b 40249 | TODO: Fix comment. (Cont... |
| cdlemg33c 40250 | TODO: Fix comment. (Cont... |
| cdlemg33d 40251 | TODO: Fix comment. (Cont... |
| cdlemg33e 40252 | TODO: Fix comment. (Cont... |
| cdlemg33 40253 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 40254 | Use cdlemg33 to eliminate ... |
| cdlemg35 40255 | TODO: Fix comment. TODO:... |
| cdlemg36 40256 | Use cdlemg35 to eliminate ... |
| cdlemg38 40257 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 40258 | Eliminate ` =/= ` conditio... |
| cdlemg40 40259 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 40260 | Convert ~ cdlemg40 to func... |
| ltrnco 40261 | The composition of two tra... |
| trlcocnv 40262 | Swap the arguments of the ... |
| trlcoabs 40263 | Absorption into a composit... |
| trlcoabs2N 40264 | Absorption of the trace of... |
| trlcoat 40265 | The trace of a composition... |
| trlcocnvat 40266 | Commonly used special case... |
| trlconid 40267 | The composition of two dif... |
| trlcolem 40268 | Lemma for ~ trlco . (Cont... |
| trlco 40269 | The trace of a composition... |
| trlcone 40270 | If two translations have d... |
| cdlemg42 40271 | Part of proof of Lemma G o... |
| cdlemg43 40272 | Part of proof of Lemma G o... |
| cdlemg44a 40273 | Part of proof of Lemma G o... |
| cdlemg44b 40274 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 40275 | Part of proof of Lemma G o... |
| cdlemg47a 40276 | TODO: fix comment. TODO: ... |
| cdlemg46 40277 | Part of proof of Lemma G o... |
| cdlemg47 40278 | Part of proof of Lemma G o... |
| cdlemg48 40279 | Eliminate ` h ` from ~ cdl... |
| ltrncom 40280 | Composition is commutative... |
| ltrnco4 40281 | Rearrange a composition of... |
| trljco 40282 | Trace joined with trace of... |
| trljco2 40283 | Trace joined with trace of... |
| tgrpfset 40286 | The translation group maps... |
| tgrpset 40287 | The translation group for ... |
| tgrpbase 40288 | The base set of the transl... |
| tgrpopr 40289 | The group operation of the... |
| tgrpov 40290 | The group operation value ... |
| tgrpgrplem 40291 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 40292 | The translation group is a... |
| tgrpabl 40293 | The translation group is a... |
| tendofset 40300 | The set of all trace-prese... |
| tendoset 40301 | The set of trace-preservin... |
| istendo 40302 | The predicate "is a trace-... |
| tendotp 40303 | Trace-preserving property ... |
| istendod 40304 | Deduce the predicate "is a... |
| tendof 40305 | Functionality of a trace-p... |
| tendoeq1 40306 | Condition determining equa... |
| tendovalco 40307 | Value of composition of tr... |
| tendocoval 40308 | Value of composition of en... |
| tendocl 40309 | Closure of a trace-preserv... |
| tendoco2 40310 | Distribution of compositio... |
| tendoidcl 40311 | The identity is a trace-pr... |
| tendo1mul 40312 | Multiplicative identity mu... |
| tendo1mulr 40313 | Multiplicative identity mu... |
| tendococl 40314 | The composition of two tra... |
| tendoid 40315 | The identity value of a tr... |
| tendoeq2 40316 | Condition determining equa... |
| tendoplcbv 40317 | Define sum operation for t... |
| tendopl 40318 | Value of endomorphism sum ... |
| tendopl2 40319 | Value of result of endomor... |
| tendoplcl2 40320 | Value of result of endomor... |
| tendoplco2 40321 | Value of result of endomor... |
| tendopltp 40322 | Trace-preserving property ... |
| tendoplcl 40323 | Endomorphism sum is a trac... |
| tendoplcom 40324 | The endomorphism sum opera... |
| tendoplass 40325 | The endomorphism sum opera... |
| tendodi1 40326 | Endomorphism composition d... |
| tendodi2 40327 | Endomorphism composition d... |
| tendo0cbv 40328 | Define additive identity f... |
| tendo02 40329 | Value of additive identity... |
| tendo0co2 40330 | The additive identity trac... |
| tendo0tp 40331 | Trace-preserving property ... |
| tendo0cl 40332 | The additive identity is a... |
| tendo0pl 40333 | Property of the additive i... |
| tendo0plr 40334 | Property of the additive i... |
| tendoicbv 40335 | Define inverse function fo... |
| tendoi 40336 | Value of inverse endomorph... |
| tendoi2 40337 | Value of additive inverse ... |
| tendoicl 40338 | Closure of the additive in... |
| tendoipl 40339 | Property of the additive i... |
| tendoipl2 40340 | Property of the additive i... |
| erngfset 40341 | The division rings on trac... |
| erngset 40342 | The division ring on trace... |
| erngbase 40343 | The base set of the divisi... |
| erngfplus 40344 | Ring addition operation. ... |
| erngplus 40345 | Ring addition operation. ... |
| erngplus2 40346 | Ring addition operation. ... |
| erngfmul 40347 | Ring multiplication operat... |
| erngmul 40348 | Ring addition operation. ... |
| erngfset-rN 40349 | The division rings on trac... |
| erngset-rN 40350 | The division ring on trace... |
| erngbase-rN 40351 | The base set of the divisi... |
| erngfplus-rN 40352 | Ring addition operation. ... |
| erngplus-rN 40353 | Ring addition operation. ... |
| erngplus2-rN 40354 | Ring addition operation. ... |
| erngfmul-rN 40355 | Ring multiplication operat... |
| erngmul-rN 40356 | Ring addition operation. ... |
| cdlemh1 40357 | Part of proof of Lemma H o... |
| cdlemh2 40358 | Part of proof of Lemma H o... |
| cdlemh 40359 | Lemma H of [Crawley] p. 11... |
| cdlemi1 40360 | Part of proof of Lemma I o... |
| cdlemi2 40361 | Part of proof of Lemma I o... |
| cdlemi 40362 | Lemma I of [Crawley] p. 11... |
| cdlemj1 40363 | Part of proof of Lemma J o... |
| cdlemj2 40364 | Part of proof of Lemma J o... |
| cdlemj3 40365 | Part of proof of Lemma J o... |
| tendocan 40366 | Cancellation law: if the v... |
| tendoid0 40367 | A trace-preserving endomor... |
| tendo0mul 40368 | Additive identity multipli... |
| tendo0mulr 40369 | Additive identity multipli... |
| tendo1ne0 40370 | The identity (unity) is no... |
| tendoconid 40371 | The composition (product) ... |
| tendotr 40372 | The trace of the value of ... |
| cdlemk1 40373 | Part of proof of Lemma K o... |
| cdlemk2 40374 | Part of proof of Lemma K o... |
| cdlemk3 40375 | Part of proof of Lemma K o... |
| cdlemk4 40376 | Part of proof of Lemma K o... |
| cdlemk5a 40377 | Part of proof of Lemma K o... |
| cdlemk5 40378 | Part of proof of Lemma K o... |
| cdlemk6 40379 | Part of proof of Lemma K o... |
| cdlemk8 40380 | Part of proof of Lemma K o... |
| cdlemk9 40381 | Part of proof of Lemma K o... |
| cdlemk9bN 40382 | Part of proof of Lemma K o... |
| cdlemki 40383 | Part of proof of Lemma K o... |
| cdlemkvcl 40384 | Part of proof of Lemma K o... |
| cdlemk10 40385 | Part of proof of Lemma K o... |
| cdlemksv 40386 | Part of proof of Lemma K o... |
| cdlemksel 40387 | Part of proof of Lemma K o... |
| cdlemksat 40388 | Part of proof of Lemma K o... |
| cdlemksv2 40389 | Part of proof of Lemma K o... |
| cdlemk7 40390 | Part of proof of Lemma K o... |
| cdlemk11 40391 | Part of proof of Lemma K o... |
| cdlemk12 40392 | Part of proof of Lemma K o... |
| cdlemkoatnle 40393 | Utility lemma. (Contribut... |
| cdlemk13 40394 | Part of proof of Lemma K o... |
| cdlemkole 40395 | Utility lemma. (Contribut... |
| cdlemk14 40396 | Part of proof of Lemma K o... |
| cdlemk15 40397 | Part of proof of Lemma K o... |
| cdlemk16a 40398 | Part of proof of Lemma K o... |
| cdlemk16 40399 | Part of proof of Lemma K o... |
| cdlemk17 40400 | Part of proof of Lemma K o... |
| cdlemk1u 40401 | Part of proof of Lemma K o... |
| cdlemk5auN 40402 | Part of proof of Lemma K o... |
| cdlemk5u 40403 | Part of proof of Lemma K o... |
| cdlemk6u 40404 | Part of proof of Lemma K o... |
| cdlemkj 40405 | Part of proof of Lemma K o... |
| cdlemkuvN 40406 | Part of proof of Lemma K o... |
| cdlemkuel 40407 | Part of proof of Lemma K o... |
| cdlemkuat 40408 | Part of proof of Lemma K o... |
| cdlemkuv2 40409 | Part of proof of Lemma K o... |
| cdlemk18 40410 | Part of proof of Lemma K o... |
| cdlemk19 40411 | Part of proof of Lemma K o... |
| cdlemk7u 40412 | Part of proof of Lemma K o... |
| cdlemk11u 40413 | Part of proof of Lemma K o... |
| cdlemk12u 40414 | Part of proof of Lemma K o... |
| cdlemk21N 40415 | Part of proof of Lemma K o... |
| cdlemk20 40416 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 40417 | Utility lemma. (Contribut... |
| cdlemk13-2N 40418 | Part of proof of Lemma K o... |
| cdlemkole-2N 40419 | Utility lemma. (Contribut... |
| cdlemk14-2N 40420 | Part of proof of Lemma K o... |
| cdlemk15-2N 40421 | Part of proof of Lemma K o... |
| cdlemk16-2N 40422 | Part of proof of Lemma K o... |
| cdlemk17-2N 40423 | Part of proof of Lemma K o... |
| cdlemkj-2N 40424 | Part of proof of Lemma K o... |
| cdlemkuv-2N 40425 | Part of proof of Lemma K o... |
| cdlemkuel-2N 40426 | Part of proof of Lemma K o... |
| cdlemkuv2-2 40427 | Part of proof of Lemma K o... |
| cdlemk18-2N 40428 | Part of proof of Lemma K o... |
| cdlemk19-2N 40429 | Part of proof of Lemma K o... |
| cdlemk7u-2N 40430 | Part of proof of Lemma K o... |
| cdlemk11u-2N 40431 | Part of proof of Lemma K o... |
| cdlemk12u-2N 40432 | Part of proof of Lemma K o... |
| cdlemk21-2N 40433 | Part of proof of Lemma K o... |
| cdlemk20-2N 40434 | Part of proof of Lemma K o... |
| cdlemk22 40435 | Part of proof of Lemma K o... |
| cdlemk30 40436 | Part of proof of Lemma K o... |
| cdlemkuu 40437 | Convert between function a... |
| cdlemk31 40438 | Part of proof of Lemma K o... |
| cdlemk32 40439 | Part of proof of Lemma K o... |
| cdlemkuel-3 40440 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 40441 | Part of proof of Lemma K o... |
| cdlemk18-3N 40442 | Part of proof of Lemma K o... |
| cdlemk22-3 40443 | Part of proof of Lemma K o... |
| cdlemk23-3 40444 | Part of proof of Lemma K o... |
| cdlemk24-3 40445 | Part of proof of Lemma K o... |
| cdlemk25-3 40446 | Part of proof of Lemma K o... |
| cdlemk26b-3 40447 | Part of proof of Lemma K o... |
| cdlemk26-3 40448 | Part of proof of Lemma K o... |
| cdlemk27-3 40449 | Part of proof of Lemma K o... |
| cdlemk28-3 40450 | Part of proof of Lemma K o... |
| cdlemk33N 40451 | Part of proof of Lemma K o... |
| cdlemk34 40452 | Part of proof of Lemma K o... |
| cdlemk29-3 40453 | Part of proof of Lemma K o... |
| cdlemk35 40454 | Part of proof of Lemma K o... |
| cdlemk36 40455 | Part of proof of Lemma K o... |
| cdlemk37 40456 | Part of proof of Lemma K o... |
| cdlemk38 40457 | Part of proof of Lemma K o... |
| cdlemk39 40458 | Part of proof of Lemma K o... |
| cdlemk40 40459 | TODO: fix comment. (Contr... |
| cdlemk40t 40460 | TODO: fix comment. (Contr... |
| cdlemk40f 40461 | TODO: fix comment. (Contr... |
| cdlemk41 40462 | Part of proof of Lemma K o... |
| cdlemkfid1N 40463 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 40464 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 40465 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 40466 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 40467 | TODO: is this useful or sh... |
| cdlemky 40468 | Part of proof of Lemma K o... |
| cdlemkyu 40469 | Convert between function a... |
| cdlemkyuu 40470 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 40471 | Part of proof of Lemma K o... |
| cdlemk19ylem 40472 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 40473 | Part of proof of Lemma K o... |
| cdlemk19y 40474 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 40475 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 40476 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 40477 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 40478 | The value of the tau funct... |
| cdlemk35s 40479 | Substitution version of ~ ... |
| cdlemk35s-id 40480 | Substitution version of ~ ... |
| cdlemk39s 40481 | Substitution version of ~ ... |
| cdlemk39s-id 40482 | Substitution version of ~ ... |
| cdlemk42 40483 | Part of proof of Lemma K o... |
| cdlemk19xlem 40484 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 40485 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 40486 | Part of proof of Lemma K o... |
| cdlemk11tc 40487 | Part of proof of Lemma K o... |
| cdlemk11t 40488 | Part of proof of Lemma K o... |
| cdlemk45 40489 | Part of proof of Lemma K o... |
| cdlemk46 40490 | Part of proof of Lemma K o... |
| cdlemk47 40491 | Part of proof of Lemma K o... |
| cdlemk48 40492 | Part of proof of Lemma K o... |
| cdlemk49 40493 | Part of proof of Lemma K o... |
| cdlemk50 40494 | Part of proof of Lemma K o... |
| cdlemk51 40495 | Part of proof of Lemma K o... |
| cdlemk52 40496 | Part of proof of Lemma K o... |
| cdlemk53a 40497 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 40498 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 40499 | Part of proof of Lemma K o... |
| cdlemk54 40500 | Part of proof of Lemma K o... |
| cdlemk55a 40501 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 40502 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 40503 | Part of proof of Lemma K o... |
| cdlemkyyN 40504 | Part of proof of Lemma K o... |
| cdlemk43N 40505 | Part of proof of Lemma K o... |
| cdlemk35u 40506 | Substitution version of ~ ... |
| cdlemk55u1 40507 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 40508 | Part of proof of Lemma K o... |
| cdlemk39u1 40509 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 40510 | Part of proof of Lemma K o... |
| cdlemk19u1 40511 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 40512 | Part of Lemma K of [Crawle... |
| cdlemk56 40513 | Part of Lemma K of [Crawle... |
| cdlemk19w 40514 | Use a fixed element to eli... |
| cdlemk56w 40515 | Use a fixed element to eli... |
| cdlemk 40516 | Lemma K of [Crawley] p. 11... |
| tendoex 40517 | Generalization of Lemma K ... |
| cdleml1N 40518 | Part of proof of Lemma L o... |
| cdleml2N 40519 | Part of proof of Lemma L o... |
| cdleml3N 40520 | Part of proof of Lemma L o... |
| cdleml4N 40521 | Part of proof of Lemma L o... |
| cdleml5N 40522 | Part of proof of Lemma L o... |
| cdleml6 40523 | Part of proof of Lemma L o... |
| cdleml7 40524 | Part of proof of Lemma L o... |
| cdleml8 40525 | Part of proof of Lemma L o... |
| cdleml9 40526 | Part of proof of Lemma L o... |
| dva1dim 40527 | Two expressions for the 1-... |
| dvhb1dimN 40528 | Two expressions for the 1-... |
| erng1lem 40529 | Value of the endomorphism ... |
| erngdvlem1 40530 | Lemma for ~ eringring . (... |
| erngdvlem2N 40531 | Lemma for ~ eringring . (... |
| erngdvlem3 40532 | Lemma for ~ eringring . (... |
| erngdvlem4 40533 | Lemma for ~ erngdv . (Con... |
| eringring 40534 | An endomorphism ring is a ... |
| erngdv 40535 | An endomorphism ring is a ... |
| erng0g 40536 | The division ring zero of ... |
| erng1r 40537 | The division ring unity of... |
| erngdvlem1-rN 40538 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 40539 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 40540 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 40541 | Lemma for ~ erngdv . (Con... |
| erngring-rN 40542 | An endomorphism ring is a ... |
| erngdv-rN 40543 | An endomorphism ring is a ... |
| dvafset 40546 | The constructed partial ve... |
| dvaset 40547 | The constructed partial ve... |
| dvasca 40548 | The ring base set of the c... |
| dvabase 40549 | The ring base set of the c... |
| dvafplusg 40550 | Ring addition operation fo... |
| dvaplusg 40551 | Ring addition operation fo... |
| dvaplusgv 40552 | Ring addition operation fo... |
| dvafmulr 40553 | Ring multiplication operat... |
| dvamulr 40554 | Ring multiplication operat... |
| dvavbase 40555 | The vectors (vector base s... |
| dvafvadd 40556 | The vector sum operation f... |
| dvavadd 40557 | Ring addition operation fo... |
| dvafvsca 40558 | Ring addition operation fo... |
| dvavsca 40559 | Ring addition operation fo... |
| tendospcl 40560 | Closure of endomorphism sc... |
| tendospass 40561 | Associative law for endomo... |
| tendospdi1 40562 | Forward distributive law f... |
| tendocnv 40563 | Converse of a trace-preser... |
| tendospdi2 40564 | Reverse distributive law f... |
| tendospcanN 40565 | Cancellation law for trace... |
| dvaabl 40566 | The constructed partial ve... |
| dvalveclem 40567 | Lemma for ~ dvalvec . (Co... |
| dvalvec 40568 | The constructed partial ve... |
| dva0g 40569 | The zero vector of partial... |
| diaffval 40572 | The partial isomorphism A ... |
| diafval 40573 | The partial isomorphism A ... |
| diaval 40574 | The partial isomorphism A ... |
| diaelval 40575 | Member of the partial isom... |
| diafn 40576 | Functionality and domain o... |
| diadm 40577 | Domain of the partial isom... |
| diaeldm 40578 | Member of domain of the pa... |
| diadmclN 40579 | A member of domain of the ... |
| diadmleN 40580 | A member of domain of the ... |
| dian0 40581 | The value of the partial i... |
| dia0eldmN 40582 | The lattice zero belongs t... |
| dia1eldmN 40583 | The fiducial hyperplane (t... |
| diass 40584 | The value of the partial i... |
| diael 40585 | A member of the value of t... |
| diatrl 40586 | Trace of a member of the p... |
| diaelrnN 40587 | Any value of the partial i... |
| dialss 40588 | The value of partial isomo... |
| diaord 40589 | The partial isomorphism A ... |
| dia11N 40590 | The partial isomorphism A ... |
| diaf11N 40591 | The partial isomorphism A ... |
| diaclN 40592 | Closure of partial isomorp... |
| diacnvclN 40593 | Closure of partial isomorp... |
| dia0 40594 | The value of the partial i... |
| dia1N 40595 | The value of the partial i... |
| dia1elN 40596 | The largest subspace in th... |
| diaglbN 40597 | Partial isomorphism A of a... |
| diameetN 40598 | Partial isomorphism A of a... |
| diainN 40599 | Inverse partial isomorphis... |
| diaintclN 40600 | The intersection of partia... |
| diasslssN 40601 | The partial isomorphism A ... |
| diassdvaN 40602 | The partial isomorphism A ... |
| dia1dim 40603 | Two expressions for the 1-... |
| dia1dim2 40604 | Two expressions for a 1-di... |
| dia1dimid 40605 | A vector (translation) bel... |
| dia2dimlem1 40606 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 40607 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 40608 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 40609 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 40610 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 40611 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 40612 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 40613 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 40614 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 40615 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 40616 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 40617 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 40618 | Lemma for ~ dia2dim . Eli... |
| dia2dim 40619 | A two-dimensional subspace... |
| dvhfset 40622 | The constructed full vecto... |
| dvhset 40623 | The constructed full vecto... |
| dvhsca 40624 | The ring of scalars of the... |
| dvhbase 40625 | The ring base set of the c... |
| dvhfplusr 40626 | Ring addition operation fo... |
| dvhfmulr 40627 | Ring multiplication operat... |
| dvhmulr 40628 | Ring multiplication operat... |
| dvhvbase 40629 | The vectors (vector base s... |
| dvhelvbasei 40630 | Vector membership in the c... |
| dvhvaddcbv 40631 | Change bound variables to ... |
| dvhvaddval 40632 | The vector sum operation f... |
| dvhfvadd 40633 | The vector sum operation f... |
| dvhvadd 40634 | The vector sum operation f... |
| dvhopvadd 40635 | The vector sum operation f... |
| dvhopvadd2 40636 | The vector sum operation f... |
| dvhvaddcl 40637 | Closure of the vector sum ... |
| dvhvaddcomN 40638 | Commutativity of vector su... |
| dvhvaddass 40639 | Associativity of vector su... |
| dvhvscacbv 40640 | Change bound variables to ... |
| dvhvscaval 40641 | The scalar product operati... |
| dvhfvsca 40642 | Scalar product operation f... |
| dvhvsca 40643 | Scalar product operation f... |
| dvhopvsca 40644 | Scalar product operation f... |
| dvhvscacl 40645 | Closure of the scalar prod... |
| tendoinvcl 40646 | Closure of multiplicative ... |
| tendolinv 40647 | Left multiplicative invers... |
| tendorinv 40648 | Right multiplicative inver... |
| dvhgrp 40649 | The full vector space ` U ... |
| dvhlveclem 40650 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 40651 | The full vector space ` U ... |
| dvhlmod 40652 | The full vector space ` U ... |
| dvh0g 40653 | The zero vector of vector ... |
| dvheveccl 40654 | Properties of a unit vecto... |
| dvhopclN 40655 | Closure of a ` DVecH ` vec... |
| dvhopaddN 40656 | Sum of ` DVecH ` vectors e... |
| dvhopspN 40657 | Scalar product of ` DVecH ... |
| dvhopN 40658 | Decompose a ` DVecH ` vect... |
| dvhopellsm 40659 | Ordered pair membership in... |
| cdlemm10N 40660 | The image of the map ` G `... |
| docaffvalN 40663 | Subspace orthocomplement f... |
| docafvalN 40664 | Subspace orthocomplement f... |
| docavalN 40665 | Subspace orthocomplement f... |
| docaclN 40666 | Closure of subspace orthoc... |
| diaocN 40667 | Value of partial isomorphi... |
| doca2N 40668 | Double orthocomplement of ... |
| doca3N 40669 | Double orthocomplement of ... |
| dvadiaN 40670 | Any closed subspace is a m... |
| diarnN 40671 | Partial isomorphism A maps... |
| diaf1oN 40672 | The partial isomorphism A ... |
| djaffvalN 40675 | Subspace join for ` DVecA ... |
| djafvalN 40676 | Subspace join for ` DVecA ... |
| djavalN 40677 | Subspace join for ` DVecA ... |
| djaclN 40678 | Closure of subspace join f... |
| djajN 40679 | Transfer lattice join to `... |
| dibffval 40682 | The partial isomorphism B ... |
| dibfval 40683 | The partial isomorphism B ... |
| dibval 40684 | The partial isomorphism B ... |
| dibopelvalN 40685 | Member of the partial isom... |
| dibval2 40686 | Value of the partial isomo... |
| dibopelval2 40687 | Member of the partial isom... |
| dibval3N 40688 | Value of the partial isomo... |
| dibelval3 40689 | Member of the partial isom... |
| dibopelval3 40690 | Member of the partial isom... |
| dibelval1st 40691 | Membership in value of the... |
| dibelval1st1 40692 | Membership in value of the... |
| dibelval1st2N 40693 | Membership in value of the... |
| dibelval2nd 40694 | Membership in value of the... |
| dibn0 40695 | The value of the partial i... |
| dibfna 40696 | Functionality and domain o... |
| dibdiadm 40697 | Domain of the partial isom... |
| dibfnN 40698 | Functionality and domain o... |
| dibdmN 40699 | Domain of the partial isom... |
| dibeldmN 40700 | Member of domain of the pa... |
| dibord 40701 | The isomorphism B for a la... |
| dib11N 40702 | The isomorphism B for a la... |
| dibf11N 40703 | The partial isomorphism A ... |
| dibclN 40704 | Closure of partial isomorp... |
| dibvalrel 40705 | The value of partial isomo... |
| dib0 40706 | The value of partial isomo... |
| dib1dim 40707 | Two expressions for the 1-... |
| dibglbN 40708 | Partial isomorphism B of a... |
| dibintclN 40709 | The intersection of partia... |
| dib1dim2 40710 | Two expressions for a 1-di... |
| dibss 40711 | The partial isomorphism B ... |
| diblss 40712 | The value of partial isomo... |
| diblsmopel 40713 | Membership in subspace sum... |
| dicffval 40716 | The partial isomorphism C ... |
| dicfval 40717 | The partial isomorphism C ... |
| dicval 40718 | The partial isomorphism C ... |
| dicopelval 40719 | Membership in value of the... |
| dicelvalN 40720 | Membership in value of the... |
| dicval2 40721 | The partial isomorphism C ... |
| dicelval3 40722 | Member of the partial isom... |
| dicopelval2 40723 | Membership in value of the... |
| dicelval2N 40724 | Membership in value of the... |
| dicfnN 40725 | Functionality and domain o... |
| dicdmN 40726 | Domain of the partial isom... |
| dicvalrelN 40727 | The value of partial isomo... |
| dicssdvh 40728 | The partial isomorphism C ... |
| dicelval1sta 40729 | Membership in value of the... |
| dicelval1stN 40730 | Membership in value of the... |
| dicelval2nd 40731 | Membership in value of the... |
| dicvaddcl 40732 | Membership in value of the... |
| dicvscacl 40733 | Membership in value of the... |
| dicn0 40734 | The value of the partial i... |
| diclss 40735 | The value of partial isomo... |
| diclspsn 40736 | The value of isomorphism C... |
| cdlemn2 40737 | Part of proof of Lemma N o... |
| cdlemn2a 40738 | Part of proof of Lemma N o... |
| cdlemn3 40739 | Part of proof of Lemma N o... |
| cdlemn4 40740 | Part of proof of Lemma N o... |
| cdlemn4a 40741 | Part of proof of Lemma N o... |
| cdlemn5pre 40742 | Part of proof of Lemma N o... |
| cdlemn5 40743 | Part of proof of Lemma N o... |
| cdlemn6 40744 | Part of proof of Lemma N o... |
| cdlemn7 40745 | Part of proof of Lemma N o... |
| cdlemn8 40746 | Part of proof of Lemma N o... |
| cdlemn9 40747 | Part of proof of Lemma N o... |
| cdlemn10 40748 | Part of proof of Lemma N o... |
| cdlemn11a 40749 | Part of proof of Lemma N o... |
| cdlemn11b 40750 | Part of proof of Lemma N o... |
| cdlemn11c 40751 | Part of proof of Lemma N o... |
| cdlemn11pre 40752 | Part of proof of Lemma N o... |
| cdlemn11 40753 | Part of proof of Lemma N o... |
| cdlemn 40754 | Lemma N of [Crawley] p. 12... |
| dihordlem6 40755 | Part of proof of Lemma N o... |
| dihordlem7 40756 | Part of proof of Lemma N o... |
| dihordlem7b 40757 | Part of proof of Lemma N o... |
| dihjustlem 40758 | Part of proof after Lemma ... |
| dihjust 40759 | Part of proof after Lemma ... |
| dihord1 40760 | Part of proof after Lemma ... |
| dihord2a 40761 | Part of proof after Lemma ... |
| dihord2b 40762 | Part of proof after Lemma ... |
| dihord2cN 40763 | Part of proof after Lemma ... |
| dihord11b 40764 | Part of proof after Lemma ... |
| dihord10 40765 | Part of proof after Lemma ... |
| dihord11c 40766 | Part of proof after Lemma ... |
| dihord2pre 40767 | Part of proof after Lemma ... |
| dihord2pre2 40768 | Part of proof after Lemma ... |
| dihord2 40769 | Part of proof after Lemma ... |
| dihffval 40772 | The isomorphism H for a la... |
| dihfval 40773 | Isomorphism H for a lattic... |
| dihval 40774 | Value of isomorphism H for... |
| dihvalc 40775 | Value of isomorphism H for... |
| dihlsscpre 40776 | Closure of isomorphism H f... |
| dihvalcqpre 40777 | Value of isomorphism H for... |
| dihvalcq 40778 | Value of isomorphism H for... |
| dihvalb 40779 | Value of isomorphism H for... |
| dihopelvalbN 40780 | Ordered pair member of the... |
| dihvalcqat 40781 | Value of isomorphism H for... |
| dih1dimb 40782 | Two expressions for a 1-di... |
| dih1dimb2 40783 | Isomorphism H at an atom u... |
| dih1dimc 40784 | Isomorphism H at an atom n... |
| dib2dim 40785 | Extend ~ dia2dim to partia... |
| dih2dimb 40786 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 40787 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 40788 | Ordered pair member of the... |
| dihvalcq2 40789 | Value of isomorphism H for... |
| dihopelvalcpre 40790 | Member of value of isomorp... |
| dihopelvalc 40791 | Member of value of isomorp... |
| dihlss 40792 | The value of isomorphism H... |
| dihss 40793 | The value of isomorphism H... |
| dihssxp 40794 | An isomorphism H value is ... |
| dihopcl 40795 | Closure of an ordered pair... |
| xihopellsmN 40796 | Ordered pair membership in... |
| dihopellsm 40797 | Ordered pair membership in... |
| dihord6apre 40798 | Part of proof that isomorp... |
| dihord3 40799 | The isomorphism H for a la... |
| dihord4 40800 | The isomorphism H for a la... |
| dihord5b 40801 | Part of proof that isomorp... |
| dihord6b 40802 | Part of proof that isomorp... |
| dihord6a 40803 | Part of proof that isomorp... |
| dihord5apre 40804 | Part of proof that isomorp... |
| dihord5a 40805 | Part of proof that isomorp... |
| dihord 40806 | The isomorphism H is order... |
| dih11 40807 | The isomorphism H is one-t... |
| dihf11lem 40808 | Functionality of the isomo... |
| dihf11 40809 | The isomorphism H for a la... |
| dihfn 40810 | Functionality and domain o... |
| dihdm 40811 | Domain of isomorphism H. (... |
| dihcl 40812 | Closure of isomorphism H. ... |
| dihcnvcl 40813 | Closure of isomorphism H c... |
| dihcnvid1 40814 | The converse isomorphism o... |
| dihcnvid2 40815 | The isomorphism of a conve... |
| dihcnvord 40816 | Ordering property for conv... |
| dihcnv11 40817 | The converse of isomorphis... |
| dihsslss 40818 | The isomorphism H maps to ... |
| dihrnlss 40819 | The isomorphism H maps to ... |
| dihrnss 40820 | The isomorphism H maps to ... |
| dihvalrel 40821 | The value of isomorphism H... |
| dih0 40822 | The value of isomorphism H... |
| dih0bN 40823 | A lattice element is zero ... |
| dih0vbN 40824 | A vector is zero iff its s... |
| dih0cnv 40825 | The isomorphism H converse... |
| dih0rn 40826 | The zero subspace belongs ... |
| dih0sb 40827 | A subspace is zero iff the... |
| dih1 40828 | The value of isomorphism H... |
| dih1rn 40829 | The full vector space belo... |
| dih1cnv 40830 | The isomorphism H converse... |
| dihwN 40831 | Value of isomorphism H at ... |
| dihmeetlem1N 40832 | Isomorphism H of a conjunc... |
| dihglblem5apreN 40833 | A conjunction property of ... |
| dihglblem5aN 40834 | A conjunction property of ... |
| dihglblem2aN 40835 | Lemma for isomorphism H of... |
| dihglblem2N 40836 | The GLB of a set of lattic... |
| dihglblem3N 40837 | Isomorphism H of a lattice... |
| dihglblem3aN 40838 | Isomorphism H of a lattice... |
| dihglblem4 40839 | Isomorphism H of a lattice... |
| dihglblem5 40840 | Isomorphism H of a lattice... |
| dihmeetlem2N 40841 | Isomorphism H of a conjunc... |
| dihglbcpreN 40842 | Isomorphism H of a lattice... |
| dihglbcN 40843 | Isomorphism H of a lattice... |
| dihmeetcN 40844 | Isomorphism H of a lattice... |
| dihmeetbN 40845 | Isomorphism H of a lattice... |
| dihmeetbclemN 40846 | Lemma for isomorphism H of... |
| dihmeetlem3N 40847 | Lemma for isomorphism H of... |
| dihmeetlem4preN 40848 | Lemma for isomorphism H of... |
| dihmeetlem4N 40849 | Lemma for isomorphism H of... |
| dihmeetlem5 40850 | Part of proof that isomorp... |
| dihmeetlem6 40851 | Lemma for isomorphism H of... |
| dihmeetlem7N 40852 | Lemma for isomorphism H of... |
| dihjatc1 40853 | Lemma for isomorphism H of... |
| dihjatc2N 40854 | Isomorphism H of join with... |
| dihjatc3 40855 | Isomorphism H of join with... |
| dihmeetlem8N 40856 | Lemma for isomorphism H of... |
| dihmeetlem9N 40857 | Lemma for isomorphism H of... |
| dihmeetlem10N 40858 | Lemma for isomorphism H of... |
| dihmeetlem11N 40859 | Lemma for isomorphism H of... |
| dihmeetlem12N 40860 | Lemma for isomorphism H of... |
| dihmeetlem13N 40861 | Lemma for isomorphism H of... |
| dihmeetlem14N 40862 | Lemma for isomorphism H of... |
| dihmeetlem15N 40863 | Lemma for isomorphism H of... |
| dihmeetlem16N 40864 | Lemma for isomorphism H of... |
| dihmeetlem17N 40865 | Lemma for isomorphism H of... |
| dihmeetlem18N 40866 | Lemma for isomorphism H of... |
| dihmeetlem19N 40867 | Lemma for isomorphism H of... |
| dihmeetlem20N 40868 | Lemma for isomorphism H of... |
| dihmeetALTN 40869 | Isomorphism H of a lattice... |
| dih1dimatlem0 40870 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 40871 | Lemma for ~ dih1dimat . (... |
| dih1dimat 40872 | Any 1-dimensional subspace... |
| dihlsprn 40873 | The span of a vector belon... |
| dihlspsnssN 40874 | A subspace included in a 1... |
| dihlspsnat 40875 | The inverse isomorphism H ... |
| dihatlat 40876 | The isomorphism H of an at... |
| dihat 40877 | There exists at least one ... |
| dihpN 40878 | The value of isomorphism H... |
| dihlatat 40879 | The reverse isomorphism H ... |
| dihatexv 40880 | There is a nonzero vector ... |
| dihatexv2 40881 | There is a nonzero vector ... |
| dihglblem6 40882 | Isomorphism H of a lattice... |
| dihglb 40883 | Isomorphism H of a lattice... |
| dihglb2 40884 | Isomorphism H of a lattice... |
| dihmeet 40885 | Isomorphism H of a lattice... |
| dihintcl 40886 | The intersection of closed... |
| dihmeetcl 40887 | Closure of closed subspace... |
| dihmeet2 40888 | Reverse isomorphism H of a... |
| dochffval 40891 | Subspace orthocomplement f... |
| dochfval 40892 | Subspace orthocomplement f... |
| dochval 40893 | Subspace orthocomplement f... |
| dochval2 40894 | Subspace orthocomplement f... |
| dochcl 40895 | Closure of subspace orthoc... |
| dochlss 40896 | A subspace orthocomplement... |
| dochssv 40897 | A subspace orthocomplement... |
| dochfN 40898 | Domain and codomain of the... |
| dochvalr 40899 | Orthocomplement of a close... |
| doch0 40900 | Orthocomplement of the zer... |
| doch1 40901 | Orthocomplement of the uni... |
| dochoc0 40902 | The zero subspace is close... |
| dochoc1 40903 | The unit subspace (all vec... |
| dochvalr2 40904 | Orthocomplement of a close... |
| dochvalr3 40905 | Orthocomplement of a close... |
| doch2val2 40906 | Double orthocomplement for... |
| dochss 40907 | Subset law for orthocomple... |
| dochocss 40908 | Double negative law for or... |
| dochoc 40909 | Double negative law for or... |
| dochsscl 40910 | If a set of vectors is inc... |
| dochoccl 40911 | A set of vectors is closed... |
| dochord 40912 | Ordering law for orthocomp... |
| dochord2N 40913 | Ordering law for orthocomp... |
| dochord3 40914 | Ordering law for orthocomp... |
| doch11 40915 | Orthocomplement is one-to-... |
| dochsordN 40916 | Strict ordering law for or... |
| dochn0nv 40917 | An orthocomplement is nonz... |
| dihoml4c 40918 | Version of ~ dihoml4 with ... |
| dihoml4 40919 | Orthomodular law for const... |
| dochspss 40920 | The span of a set of vecto... |
| dochocsp 40921 | The span of an orthocomple... |
| dochspocN 40922 | The span of an orthocomple... |
| dochocsn 40923 | The double orthocomplement... |
| dochsncom 40924 | Swap vectors in an orthoco... |
| dochsat 40925 | The double orthocomplement... |
| dochshpncl 40926 | If a hyperplane is not clo... |
| dochlkr 40927 | Equivalent conditions for ... |
| dochkrshp 40928 | The closure of a kernel is... |
| dochkrshp2 40929 | Properties of the closure ... |
| dochkrshp3 40930 | Properties of the closure ... |
| dochkrshp4 40931 | Properties of the closure ... |
| dochdmj1 40932 | De Morgan-like law for sub... |
| dochnoncon 40933 | Law of noncontradiction. ... |
| dochnel2 40934 | A nonzero member of a subs... |
| dochnel 40935 | A nonzero vector doesn't b... |
| djhffval 40938 | Subspace join for ` DVecH ... |
| djhfval 40939 | Subspace join for ` DVecH ... |
| djhval 40940 | Subspace join for ` DVecH ... |
| djhval2 40941 | Value of subspace join for... |
| djhcl 40942 | Closure of subspace join f... |
| djhlj 40943 | Transfer lattice join to `... |
| djhljjN 40944 | Lattice join in terms of `... |
| djhjlj 40945 | ` DVecH ` vector space clo... |
| djhj 40946 | ` DVecH ` vector space clo... |
| djhcom 40947 | Subspace join commutes. (... |
| djhspss 40948 | Subspace span of union is ... |
| djhsumss 40949 | Subspace sum is a subset o... |
| dihsumssj 40950 | The subspace sum of two is... |
| djhunssN 40951 | Subspace union is a subset... |
| dochdmm1 40952 | De Morgan-like law for clo... |
| djhexmid 40953 | Excluded middle property o... |
| djh01 40954 | Closed subspace join with ... |
| djh02 40955 | Closed subspace join with ... |
| djhlsmcl 40956 | A closed subspace sum equa... |
| djhcvat42 40957 | A covering property. ( ~ ... |
| dihjatb 40958 | Isomorphism H of lattice j... |
| dihjatc 40959 | Isomorphism H of lattice j... |
| dihjatcclem1 40960 | Lemma for isomorphism H of... |
| dihjatcclem2 40961 | Lemma for isomorphism H of... |
| dihjatcclem3 40962 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 40963 | Lemma for isomorphism H of... |
| dihjatcc 40964 | Isomorphism H of lattice j... |
| dihjat 40965 | Isomorphism H of lattice j... |
| dihprrnlem1N 40966 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 40967 | Lemma for ~ dihprrn . (Co... |
| dihprrn 40968 | The span of a vector pair ... |
| djhlsmat 40969 | The sum of two subspace at... |
| dihjat1lem 40970 | Subspace sum of a closed s... |
| dihjat1 40971 | Subspace sum of a closed s... |
| dihsmsprn 40972 | Subspace sum of a closed s... |
| dihjat2 40973 | The subspace sum of a clos... |
| dihjat3 40974 | Isomorphism H of lattice j... |
| dihjat4 40975 | Transfer the subspace sum ... |
| dihjat6 40976 | Transfer the subspace sum ... |
| dihsmsnrn 40977 | The subspace sum of two si... |
| dihsmatrn 40978 | The subspace sum of a clos... |
| dihjat5N 40979 | Transfer lattice join with... |
| dvh4dimat 40980 | There is an atom that is o... |
| dvh3dimatN 40981 | There is an atom that is o... |
| dvh2dimatN 40982 | Given an atom, there exist... |
| dvh1dimat 40983 | There exists an atom. (Co... |
| dvh1dim 40984 | There exists a nonzero vec... |
| dvh4dimlem 40985 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 40986 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 40987 | There is a vector that is ... |
| dvh3dim 40988 | There is a vector that is ... |
| dvh4dimN 40989 | There is a vector that is ... |
| dvh3dim2 40990 | There is a vector that is ... |
| dvh3dim3N 40991 | There is a vector that is ... |
| dochsnnz 40992 | The orthocomplement of a s... |
| dochsatshp 40993 | The orthocomplement of a s... |
| dochsatshpb 40994 | The orthocomplement of a s... |
| dochsnshp 40995 | The orthocomplement of a n... |
| dochshpsat 40996 | A hyperplane is closed iff... |
| dochkrsat 40997 | The orthocomplement of a k... |
| dochkrsat2 40998 | The orthocomplement of a k... |
| dochsat0 40999 | The orthocomplement of a k... |
| dochkrsm 41000 | The subspace sum of a clos... |
| dochexmidat 41001 | Special case of excluded m... |
| dochexmidlem1 41002 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 41003 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 41004 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 41005 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 41006 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 41007 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 41008 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 41009 | Lemma for ~ dochexmid . T... |
| dochexmid 41010 | Excluded middle law for cl... |
| dochsnkrlem1 41011 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 41012 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 41013 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 41014 | A (closed) kernel expresse... |
| dochsnkr2 41015 | Kernel of the explicit fun... |
| dochsnkr2cl 41016 | The ` X ` determining func... |
| dochflcl 41017 | Closure of the explicit fu... |
| dochfl1 41018 | The value of the explicit ... |
| dochfln0 41019 | The value of a functional ... |
| dochkr1 41020 | A nonzero functional has a... |
| dochkr1OLDN 41021 | A nonzero functional has a... |
| lpolsetN 41024 | The set of polarities of a... |
| islpolN 41025 | The predicate "is a polari... |
| islpoldN 41026 | Properties that determine ... |
| lpolfN 41027 | Functionality of a polarit... |
| lpolvN 41028 | The polarity of the whole ... |
| lpolconN 41029 | Contraposition property of... |
| lpolsatN 41030 | The polarity of an atomic ... |
| lpolpolsatN 41031 | Property of a polarity. (... |
| dochpolN 41032 | The subspace orthocompleme... |
| lcfl1lem 41033 | Property of a functional w... |
| lcfl1 41034 | Property of a functional w... |
| lcfl2 41035 | Property of a functional w... |
| lcfl3 41036 | Property of a functional w... |
| lcfl4N 41037 | Property of a functional w... |
| lcfl5 41038 | Property of a functional w... |
| lcfl5a 41039 | Property of a functional w... |
| lcfl6lem 41040 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 41041 | Lemma for ~ lcfl7N . If t... |
| lcfl6 41042 | Property of a functional w... |
| lcfl7N 41043 | Property of a functional w... |
| lcfl8 41044 | Property of a functional w... |
| lcfl8a 41045 | Property of a functional w... |
| lcfl8b 41046 | Property of a nonzero func... |
| lcfl9a 41047 | Property implying that a f... |
| lclkrlem1 41048 | The set of functionals hav... |
| lclkrlem2a 41049 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 41050 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 41051 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 41052 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 41053 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 41054 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 41055 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 41056 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 41057 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 41058 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 41059 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 41060 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 41061 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 41062 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 41063 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 41064 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 41065 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 41066 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 41067 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 41068 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 41069 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 41070 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 41071 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 41072 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 41073 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 41074 | The set of functionals hav... |
| lclkr 41075 | The set of functionals wit... |
| lcfls1lem 41076 | Property of a functional w... |
| lcfls1N 41077 | Property of a functional w... |
| lcfls1c 41078 | Property of a functional w... |
| lclkrslem1 41079 | The set of functionals hav... |
| lclkrslem2 41080 | The set of functionals hav... |
| lclkrs 41081 | The set of functionals hav... |
| lclkrs2 41082 | The set of functionals wit... |
| lcfrvalsnN 41083 | Reconstruction from the du... |
| lcfrlem1 41084 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 41085 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 41086 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 41087 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 41088 | Lemma for ~ lcfr . The se... |
| lcfrlem6 41089 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 41090 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 41091 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 41092 | Lemma for ~ lcf1o . (This... |
| lcf1o 41093 | Define a function ` J ` th... |
| lcfrlem10 41094 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 41095 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 41096 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 41097 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 41098 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 41099 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 41100 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 41101 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 41102 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 41103 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 41104 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 41105 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 41106 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 41107 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 41108 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 41109 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 41110 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 41111 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 41112 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 41113 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 41114 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 41115 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 41116 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 41117 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 41118 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 41119 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 41120 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 41121 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 41122 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 41123 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 41124 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 41125 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 41126 | Lemma for ~ lcfr . Elimin... |
| lcfr 41127 | Reconstruction of a subspa... |
| lcdfval 41130 | Dual vector space of funct... |
| lcdval 41131 | Dual vector space of funct... |
| lcdval2 41132 | Dual vector space of funct... |
| lcdlvec 41133 | The dual vector space of f... |
| lcdlmod 41134 | The dual vector space of f... |
| lcdvbase 41135 | Vector base set of a dual ... |
| lcdvbasess 41136 | The vector base set of the... |
| lcdvbaselfl 41137 | A vector in the base set o... |
| lcdvbasecl 41138 | Closure of the value of a ... |
| lcdvadd 41139 | Vector addition for the cl... |
| lcdvaddval 41140 | The value of the value of ... |
| lcdsca 41141 | The ring of scalars of the... |
| lcdsbase 41142 | Base set of scalar ring fo... |
| lcdsadd 41143 | Scalar addition for the cl... |
| lcdsmul 41144 | Scalar multiplication for ... |
| lcdvs 41145 | Scalar product for the clo... |
| lcdvsval 41146 | Value of scalar product op... |
| lcdvscl 41147 | The scalar product operati... |
| lcdlssvscl 41148 | Closure of scalar product ... |
| lcdvsass 41149 | Associative law for scalar... |
| lcd0 41150 | The zero scalar of the clo... |
| lcd1 41151 | The unit scalar of the clo... |
| lcdneg 41152 | The unit scalar of the clo... |
| lcd0v 41153 | The zero functional in the... |
| lcd0v2 41154 | The zero functional in the... |
| lcd0vvalN 41155 | Value of the zero function... |
| lcd0vcl 41156 | Closure of the zero functi... |
| lcd0vs 41157 | A scalar zero times a func... |
| lcdvs0N 41158 | A scalar times the zero fu... |
| lcdvsub 41159 | The value of vector subtra... |
| lcdvsubval 41160 | The value of the value of ... |
| lcdlss 41161 | Subspaces of a dual vector... |
| lcdlss2N 41162 | Subspaces of a dual vector... |
| lcdlsp 41163 | Span in the set of functio... |
| lcdlkreqN 41164 | Colinear functionals have ... |
| lcdlkreq2N 41165 | Colinear functionals have ... |
| mapdffval 41168 | Projectivity from vector s... |
| mapdfval 41169 | Projectivity from vector s... |
| mapdval 41170 | Value of projectivity from... |
| mapdvalc 41171 | Value of projectivity from... |
| mapdval2N 41172 | Value of projectivity from... |
| mapdval3N 41173 | Value of projectivity from... |
| mapdval4N 41174 | Value of projectivity from... |
| mapdval5N 41175 | Value of projectivity from... |
| mapdordlem1a 41176 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 41177 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 41178 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 41179 | Lemma for ~ mapdord . Ord... |
| mapdord 41180 | Ordering property of the m... |
| mapd11 41181 | The map defined by ~ df-ma... |
| mapddlssN 41182 | The mapping of a subspace ... |
| mapdsn 41183 | Value of the map defined b... |
| mapdsn2 41184 | Value of the map defined b... |
| mapdsn3 41185 | Value of the map defined b... |
| mapd1dim2lem1N 41186 | Value of the map defined b... |
| mapdrvallem2 41187 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 41188 | Lemma for ~ mapdrval . (C... |
| mapdrval 41189 | Given a dual subspace ` R ... |
| mapd1o 41190 | The map defined by ~ df-ma... |
| mapdrn 41191 | Range of the map defined b... |
| mapdunirnN 41192 | Union of the range of the ... |
| mapdrn2 41193 | Range of the map defined b... |
| mapdcnvcl 41194 | Closure of the converse of... |
| mapdcl 41195 | Closure the value of the m... |
| mapdcnvid1N 41196 | Converse of the value of t... |
| mapdsord 41197 | Strong ordering property o... |
| mapdcl2 41198 | The mapping of a subspace ... |
| mapdcnvid2 41199 | Value of the converse of t... |
| mapdcnvordN 41200 | Ordering property of the c... |
| mapdcnv11N 41201 | The converse of the map de... |
| mapdcv 41202 | Covering property of the c... |
| mapdincl 41203 | Closure of dual subspace i... |
| mapdin 41204 | Subspace intersection is p... |
| mapdlsmcl 41205 | Closure of dual subspace s... |
| mapdlsm 41206 | Subspace sum is preserved ... |
| mapd0 41207 | Projectivity map of the ze... |
| mapdcnvatN 41208 | Atoms are preserved by the... |
| mapdat 41209 | Atoms are preserved by the... |
| mapdspex 41210 | The map of a span equals t... |
| mapdn0 41211 | Transfer nonzero property ... |
| mapdncol 41212 | Transfer non-colinearity f... |
| mapdindp 41213 | Transfer (part of) vector ... |
| mapdpglem1 41214 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 41215 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 41216 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 41217 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 41218 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 41219 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 41220 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 41221 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 41222 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 41223 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 41224 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 41225 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 41226 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 41227 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 41228 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 41229 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 41230 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 41231 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 41232 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 41233 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 41234 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 41235 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 41236 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 41237 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 41238 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 41239 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 41240 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 41241 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 41242 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 41243 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 41244 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 41245 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 41246 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 41247 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 41248 | Part 1 of proof of the fir... |
| baerlem3lem1 41249 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 41250 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 41251 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 41252 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 41253 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 41254 | Lemma for ~ baerlem5b . (... |
| baerlem3 41255 | An equality that holds whe... |
| baerlem5a 41256 | An equality that holds whe... |
| baerlem5b 41257 | An equality that holds whe... |
| baerlem5amN 41258 | An equality that holds whe... |
| baerlem5bmN 41259 | An equality that holds whe... |
| baerlem5abmN 41260 | An equality that holds whe... |
| mapdindp0 41261 | Vector independence lemma.... |
| mapdindp1 41262 | Vector independence lemma.... |
| mapdindp2 41263 | Vector independence lemma.... |
| mapdindp3 41264 | Vector independence lemma.... |
| mapdindp4 41265 | Vector independence lemma.... |
| mapdhval 41266 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 41267 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 41268 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 41269 | Lemmma for ~~? mapdh . (C... |
| mapdheq 41270 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 41271 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 41272 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 41273 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 41274 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 41275 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 41276 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 41277 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 41278 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 41279 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 41280 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 41281 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 41282 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 41283 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 41284 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 41285 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 41286 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 41287 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 41288 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 41289 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 41290 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 41291 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 41292 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 41293 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 41294 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 41295 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 41296 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 41297 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 41300 | Map from nonzero vectors t... |
| hvmapfval 41301 | Map from nonzero vectors t... |
| hvmapval 41302 | Value of map from nonzero ... |
| hvmapvalvalN 41303 | Value of value of map (i.e... |
| hvmapidN 41304 | The value of the vector to... |
| hvmap1o 41305 | The vector to functional m... |
| hvmapclN 41306 | Closure of the vector to f... |
| hvmap1o2 41307 | The vector to functional m... |
| hvmapcl2 41308 | Closure of the vector to f... |
| hvmaplfl 41309 | The vector to functional m... |
| hvmaplkr 41310 | Kernel of the vector to fu... |
| mapdhvmap 41311 | Relationship between ` map... |
| lspindp5 41312 | Obtain an independent vect... |
| hdmaplem1 41313 | Lemma to convert a frequen... |
| hdmaplem2N 41314 | Lemma to convert a frequen... |
| hdmaplem3 41315 | Lemma to convert a frequen... |
| hdmaplem4 41316 | Lemma to convert a frequen... |
| mapdh8a 41317 | Part of Part (8) in [Baer]... |
| mapdh8aa 41318 | Part of Part (8) in [Baer]... |
| mapdh8ab 41319 | Part of Part (8) in [Baer]... |
| mapdh8ac 41320 | Part of Part (8) in [Baer]... |
| mapdh8ad 41321 | Part of Part (8) in [Baer]... |
| mapdh8b 41322 | Part of Part (8) in [Baer]... |
| mapdh8c 41323 | Part of Part (8) in [Baer]... |
| mapdh8d0N 41324 | Part of Part (8) in [Baer]... |
| mapdh8d 41325 | Part of Part (8) in [Baer]... |
| mapdh8e 41326 | Part of Part (8) in [Baer]... |
| mapdh8g 41327 | Part of Part (8) in [Baer]... |
| mapdh8i 41328 | Part of Part (8) in [Baer]... |
| mapdh8j 41329 | Part of Part (8) in [Baer]... |
| mapdh8 41330 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 41331 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 41332 | Lemma for part (9) in [Bae... |
| hdmap1ffval 41337 | Preliminary map from vecto... |
| hdmap1fval 41338 | Preliminary map from vecto... |
| hdmap1vallem 41339 | Value of preliminary map f... |
| hdmap1val 41340 | Value of preliminary map f... |
| hdmap1val0 41341 | Value of preliminary map f... |
| hdmap1val2 41342 | Value of preliminary map f... |
| hdmap1eq 41343 | The defining equation for ... |
| hdmap1cbv 41344 | Frequently used lemma to c... |
| hdmap1valc 41345 | Connect the value of the p... |
| hdmap1cl 41346 | Convert closure theorem ~ ... |
| hdmap1eq2 41347 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 41348 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 41349 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 41350 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 41351 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 41352 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 41353 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 41354 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 41355 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 41356 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 41357 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 41358 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 41359 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 41360 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 41361 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 41362 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 41363 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 41364 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 41365 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 41366 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 41367 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 41368 | Map from vectors to functi... |
| hdmapfval 41369 | Map from vectors to functi... |
| hdmapval 41370 | Value of map from vectors ... |
| hdmapfnN 41371 | Functionality of map from ... |
| hdmapcl 41372 | Closure of map from vector... |
| hdmapval2lem 41373 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 41374 | Value of map from vectors ... |
| hdmapval0 41375 | Value of map from vectors ... |
| hdmapeveclem 41376 | Lemma for ~ hdmapevec . T... |
| hdmapevec 41377 | Value of map from vectors ... |
| hdmapevec2 41378 | The inner product of the r... |
| hdmapval3lemN 41379 | Value of map from vectors ... |
| hdmapval3N 41380 | Value of map from vectors ... |
| hdmap10lem 41381 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 41382 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 41383 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 41384 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 41385 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 41386 | Part of proof of part 12 i... |
| hdmapnzcl 41387 | Nonzero vector closure of ... |
| hdmapneg 41388 | Part of proof of part 12 i... |
| hdmapsub 41389 | Part of proof of part 12 i... |
| hdmap11 41390 | Part of proof of part 12 i... |
| hdmaprnlem1N 41391 | Part of proof of part 12 i... |
| hdmaprnlem3N 41392 | Part of proof of part 12 i... |
| hdmaprnlem3uN 41393 | Part of proof of part 12 i... |
| hdmaprnlem4tN 41394 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 41395 | Part of proof of part 12 i... |
| hdmaprnlem6N 41396 | Part of proof of part 12 i... |
| hdmaprnlem7N 41397 | Part of proof of part 12 i... |
| hdmaprnlem8N 41398 | Part of proof of part 12 i... |
| hdmaprnlem9N 41399 | Part of proof of part 12 i... |
| hdmaprnlem3eN 41400 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 41401 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 41402 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 41403 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 41404 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 41405 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 41406 | Part of proof of part 12 i... |
| hdmapf1oN 41407 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 41408 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 41409 | Prior to part 14 in [Baer]... |
| hdmap14lem1 41410 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 41411 | Prior to part 14 in [Baer]... |
| hdmap14lem3 41412 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 41413 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 41414 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 41415 | Case where ` F ` is zero. ... |
| hdmap14lem7 41416 | Combine cases of ` F ` . ... |
| hdmap14lem8 41417 | Part of proof of part 14 i... |
| hdmap14lem9 41418 | Part of proof of part 14 i... |
| hdmap14lem10 41419 | Part of proof of part 14 i... |
| hdmap14lem11 41420 | Part of proof of part 14 i... |
| hdmap14lem12 41421 | Lemma for proof of part 14... |
| hdmap14lem13 41422 | Lemma for proof of part 14... |
| hdmap14lem14 41423 | Part of proof of part 14 i... |
| hdmap14lem15 41424 | Part of proof of part 14 i... |
| hgmapffval 41427 | Map from the scalar divisi... |
| hgmapfval 41428 | Map from the scalar divisi... |
| hgmapval 41429 | Value of map from the scal... |
| hgmapfnN 41430 | Functionality of scalar si... |
| hgmapcl 41431 | Closure of scalar sigma ma... |
| hgmapdcl 41432 | Closure of the vector spac... |
| hgmapvs 41433 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 41434 | Value of the scalar sigma ... |
| hgmapval1 41435 | Value of the scalar sigma ... |
| hgmapadd 41436 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 41437 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 41438 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 41439 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 41440 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 41441 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 41442 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 41443 | Part of proof of part 16 i... |
| hgmap11 41444 | The scalar sigma map is on... |
| hgmapf1oN 41445 | The scalar sigma map is a ... |
| hgmapeq0 41446 | The scalar sigma map is ze... |
| hdmapipcl 41447 | The inner product (Hermiti... |
| hdmapln1 41448 | Linearity property that wi... |
| hdmaplna1 41449 | Additive property of first... |
| hdmaplns1 41450 | Subtraction property of fi... |
| hdmaplnm1 41451 | Multiplicative property of... |
| hdmaplna2 41452 | Additive property of secon... |
| hdmapglnm2 41453 | g-linear property of secon... |
| hdmapgln2 41454 | g-linear property that wil... |
| hdmaplkr 41455 | Kernel of the vector to du... |
| hdmapellkr 41456 | Membership in the kernel (... |
| hdmapip0 41457 | Zero property that will be... |
| hdmapip1 41458 | Construct a proportional v... |
| hdmapip0com 41459 | Commutation property of Ba... |
| hdmapinvlem1 41460 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 41461 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 41462 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 41463 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 41464 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 41465 | Involution property of sca... |
| hgmapvvlem2 41466 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 41467 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 41468 | Value of a double involuti... |
| hdmapglem7a 41469 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 41470 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 41471 | Lemma for ~ hdmapg . Line... |
| hdmapg 41472 | Apply the scalar sigma fun... |
| hdmapoc 41473 | Express our constructed or... |
| hlhilset 41476 | The final Hilbert space co... |
| hlhilsca 41477 | The scalar of the final co... |
| hlhilbase 41478 | The base set of the final ... |
| hlhilplus 41479 | The vector addition for th... |
| hlhilslem 41480 | Lemma for ~ hlhilsbase etc... |
| hlhilslemOLD 41481 | Obsolete version of ~ hlhi... |
| hlhilsbase 41482 | The scalar base set of the... |
| hlhilsbaseOLD 41483 | Obsolete version of ~ hlhi... |
| hlhilsplus 41484 | Scalar addition for the fi... |
| hlhilsplusOLD 41485 | Obsolete version of ~ hlhi... |
| hlhilsmul 41486 | Scalar multiplication for ... |
| hlhilsmulOLD 41487 | Obsolete version of ~ hlhi... |
| hlhilsbase2 41488 | The scalar base set of the... |
| hlhilsplus2 41489 | Scalar addition for the fi... |
| hlhilsmul2 41490 | Scalar multiplication for ... |
| hlhils0 41491 | The scalar ring zero for t... |
| hlhils1N 41492 | The scalar ring unity for ... |
| hlhilvsca 41493 | The scalar product for the... |
| hlhilip 41494 | Inner product operation fo... |
| hlhilipval 41495 | Value of inner product ope... |
| hlhilnvl 41496 | The involution operation o... |
| hlhillvec 41497 | The final constructed Hilb... |
| hlhildrng 41498 | The star division ring for... |
| hlhilsrnglem 41499 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 41500 | The star division ring for... |
| hlhil0 41501 | The zero vector for the fi... |
| hlhillsm 41502 | The vector sum operation f... |
| hlhilocv 41503 | The orthocomplement for th... |
| hlhillcs 41504 | The closed subspaces of th... |
| hlhilphllem 41505 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 41506 | Lemma for ~ hlhil . (Cont... |
| hlathil 41507 | Construction of a Hilbert ... |
| iscsrg 41510 | A commutative semiring is ... |
| leexp1ad 41511 | Weak base ordering relatio... |
| relogbcld 41512 | Closure of the general log... |
| relogbexpd 41513 | Identity law for general l... |
| relogbzexpd 41514 | Power law for the general ... |
| logblebd 41515 | The general logarithm is m... |
| uzindd 41516 | Induction on the upper int... |
| fzadd2d 41517 | Membership of a sum in a f... |
| zltlem1d 41518 | Integer ordering relation,... |
| zltp1led 41519 | Integer ordering relation,... |
| fzne2d 41520 | Elementhood in a finite se... |
| eqfnfv2d2 41521 | Equality of functions is d... |
| fzsplitnd 41522 | Split a finite interval of... |
| fzsplitnr 41523 | Split a finite interval of... |
| addassnni 41524 | Associative law for additi... |
| addcomnni 41525 | Commutative law for additi... |
| mulassnni 41526 | Associative law for multip... |
| mulcomnni 41527 | Commutative law for multip... |
| gcdcomnni 41528 | Commutative law for gcd. ... |
| gcdnegnni 41529 | Negation invariance for gc... |
| neggcdnni 41530 | Negation invariance for gc... |
| bccl2d 41531 | Closure of the binomial co... |
| recbothd 41532 | Take reciprocal on both si... |
| gcdmultiplei 41533 | The GCD of a multiple of a... |
| gcdaddmzz2nni 41534 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 41535 | Adding a multiple of one o... |
| gcdnncli 41536 | Closure of the gcd operato... |
| muldvds1d 41537 | If a product divides an in... |
| muldvds2d 41538 | If a product divides an in... |
| nndivdvdsd 41539 | A positive integer divides... |
| nnproddivdvdsd 41540 | A product of natural numbe... |
| coprmdvds2d 41541 | If an integer is divisible... |
| imadomfi 41542 | An image of a function und... |
| 12gcd5e1 41543 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 41544 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 41545 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 41546 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 41547 | Calculate the least common... |
| 12lcm5e60 41548 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 41549 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 41550 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 41551 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 41552 | Useful equation to calcula... |
| lcm1un 41553 | Least common multiple of n... |
| lcm2un 41554 | Least common multiple of n... |
| lcm3un 41555 | Least common multiple of n... |
| lcm4un 41556 | Least common multiple of n... |
| lcm5un 41557 | Least common multiple of n... |
| lcm6un 41558 | Least common multiple of n... |
| lcm7un 41559 | Least common multiple of n... |
| lcm8un 41560 | Least common multiple of n... |
| 3factsumint1 41561 | Move constants out of inte... |
| 3factsumint2 41562 | Move constants out of inte... |
| 3factsumint3 41563 | Move constants out of inte... |
| 3factsumint4 41564 | Move constants out of inte... |
| 3factsumint 41565 | Helpful equation for lcm i... |
| resopunitintvd 41566 | Restrict continuous functi... |
| resclunitintvd 41567 | Restrict continuous functi... |
| resdvopclptsd 41568 | Restrict derivative on uni... |
| lcmineqlem1 41569 | Part of lcm inequality lem... |
| lcmineqlem2 41570 | Part of lcm inequality lem... |
| lcmineqlem3 41571 | Part of lcm inequality lem... |
| lcmineqlem4 41572 | Part of lcm inequality lem... |
| lcmineqlem5 41573 | Technical lemma for recipr... |
| lcmineqlem6 41574 | Part of lcm inequality lem... |
| lcmineqlem7 41575 | Derivative of 1-x for chai... |
| lcmineqlem8 41576 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 41577 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 41578 | Induction step of ~ lcmine... |
| lcmineqlem11 41579 | Induction step, continuati... |
| lcmineqlem12 41580 | Base case for induction. ... |
| lcmineqlem13 41581 | Induction proof for lcm in... |
| lcmineqlem14 41582 | Technical lemma for inequa... |
| lcmineqlem15 41583 | F times the least common m... |
| lcmineqlem16 41584 | Technical divisibility lem... |
| lcmineqlem17 41585 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 41586 | Technical lemma to shift f... |
| lcmineqlem19 41587 | Dividing implies inequalit... |
| lcmineqlem20 41588 | Inequality for lcm lemma. ... |
| lcmineqlem21 41589 | The lcm inequality lemma w... |
| lcmineqlem22 41590 | The lcm inequality lemma w... |
| lcmineqlem23 41591 | Penultimate step to the lc... |
| lcmineqlem 41592 | The least common multiple ... |
| 3exp7 41593 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 41594 | First inequality in inequa... |
| 3lexlogpow5ineq2 41595 | Second inequality in inequ... |
| 3lexlogpow5ineq4 41596 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 41597 | Combined inequality chain ... |
| 3lexlogpow2ineq1 41598 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 41599 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 41600 | Result for bound in AKS in... |
| intlewftc 41601 | Inequality inference by in... |
| aks4d1lem1 41602 | Technical lemma to reduce ... |
| aks4d1p1p1 41603 | Exponential law for finite... |
| dvrelog2 41604 | The derivative of the loga... |
| dvrelog3 41605 | The derivative of the loga... |
| dvrelog2b 41606 | Derivative of the binary l... |
| 0nonelalab 41607 | Technical lemma for open i... |
| dvrelogpow2b 41608 | Derivative of the power of... |
| aks4d1p1p3 41609 | Bound of a ceiling of the ... |
| aks4d1p1p2 41610 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 41611 | Technical step for inequal... |
| dvle2 41612 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 41613 | Inequality lift to differe... |
| aks4d1p1p7 41614 | Bound of intermediary of i... |
| aks4d1p1p5 41615 | Show inequality for existe... |
| aks4d1p1 41616 | Show inequality for existe... |
| aks4d1p2 41617 | Technical lemma for existe... |
| aks4d1p3 41618 | There exists a small enoug... |
| aks4d1p4 41619 | There exists a small enoug... |
| aks4d1p5 41620 | Show that ` N ` and ` R ` ... |
| aks4d1p6 41621 | The maximal prime power ex... |
| aks4d1p7d1 41622 | Technical step in AKS lemm... |
| aks4d1p7 41623 | Technical step in AKS lemm... |
| aks4d1p8d1 41624 | If a prime divides one num... |
| aks4d1p8d2 41625 | Any prime power dividing a... |
| aks4d1p8d3 41626 | The remainder of a divisio... |
| aks4d1p8 41627 | Show that ` N ` and ` R ` ... |
| aks4d1p9 41628 | Show that the order is bou... |
| aks4d1 41629 | Lemma 4.1 from ~ https://w... |
| fldhmf1 41630 | A field homomorphism is in... |
| isprimroot 41633 | The value of a primitive r... |
| mndmolinv 41634 | An element of a monoid tha... |
| linvh 41635 | If an element has a unique... |
| primrootsunit1 41636 | Primitive roots have left ... |
| primrootsunit 41637 | Primitive roots have left ... |
| ressmulgnnd 41638 | Values for the group multi... |
| primrootscoprmpow 41639 | Coprime powers of primitiv... |
| posbezout 41640 | Bezout's identity restrict... |
| primrootscoprf 41641 | Coprime powers of primitiv... |
| primrootscoprbij 41642 | A bijection between coprim... |
| primrootscoprbij2 41643 | A bijection between coprim... |
| remexz 41644 | Division with rest. (Cont... |
| primrootlekpowne0 41645 | There is no smaller power ... |
| primrootspoweq0 41646 | The power of a ` R ` -th p... |
| aks6d1c1p1 41647 | Definition of the introspe... |
| aks6d1c1p1rcl 41648 | Reverse closure of the int... |
| aks6d1c1p2 41649 | ` P ` and linear factors a... |
| aks6d1c1p3 41650 | In a field with a Frobeniu... |
| aks6d1c1p4 41651 | The product of polynomials... |
| aks6d1c1p5 41652 | The product of exponents i... |
| aks6d1c1p7 41653 | ` X ` is introspective to ... |
| aks6d1c1p6 41654 | If a polynomials ` F ` is ... |
| aks6d1c1p8 41655 | If a number ` E ` is intro... |
| aks6d1c1 41656 | Claim 1 of Theorem 6.1 ~ h... |
| evl1gprodd 41657 | Polynomial evaluation buil... |
| aks6d1c2p1 41658 | In the AKS-theorem the sub... |
| aks6d1c2p2 41659 | Injective condition for co... |
| hashscontpowcl 41660 | Closure of E for ~ https:/... |
| hashscontpow1 41661 | Helper lemma for to prove ... |
| hashscontpow 41662 | If a set contains all ` N ... |
| aks6d1c3 41663 | Claim 3 of Theorem 6.1 of ... |
| aks6d1c4 41664 | Claim 4 of Theorem 6.1 of ... |
| aks6d1c1rh 41665 | Claim 1 of AKS primality p... |
| aks6d1c2lem3 41666 | Lemma for ~ aks6d1c2 to si... |
| aks6d1c2lem4 41667 | Claim 2 of Theorem 6.1 AKS... |
| hashnexinj 41668 | If the number of elements ... |
| hashnexinjle 41669 | If the number of elements ... |
| aks6d1c2 41670 | Claim 2 of Theorem 6.1 of ... |
| rspcsbnea 41671 | Special case related to ~ ... |
| idomnnzpownz 41672 | A non-zero power in an int... |
| idomnnzgmulnz 41673 | A finite product of non-ze... |
| ringexp0nn 41674 | Zero to the power of a pos... |
| aks6d1c5lem0 41675 | Lemma for Claim 5 of Theor... |
| aks6d1c5lem1 41676 | Lemma for claim 5, evaluat... |
| aks6d1c5lem3 41677 | Lemma for Claim 5, polynom... |
| aks6d1c5lem2 41678 | Lemma for Claim 5, contrad... |
| aks6d1c5 41679 | Claim 5 of Theorem 6.1 ~ h... |
| deg1mul 41680 | Degree of multiplication o... |
| deg1gprod 41681 | Degree multiplication is a... |
| deg1pow 41682 | Exact degree of a power of... |
| 5bc2eq10 41683 | The value of 5 choose 2. ... |
| facp2 41684 | The factorial of a success... |
| 2np3bcnp1 41685 | Part of induction step for... |
| 2ap1caineq 41686 | Inequality for Theorem 6.6... |
| sticksstones1 41687 | Different strictly monoton... |
| sticksstones2 41688 | The range function on stri... |
| sticksstones3 41689 | The range function on stri... |
| sticksstones4 41690 | Equinumerosity lemma for s... |
| sticksstones5 41691 | Count the number of strict... |
| sticksstones6 41692 | Function induces an order ... |
| sticksstones7 41693 | Closure property of sticks... |
| sticksstones8 41694 | Establish mapping between ... |
| sticksstones9 41695 | Establish mapping between ... |
| sticksstones10 41696 | Establish mapping between ... |
| sticksstones11 41697 | Establish bijective mappin... |
| sticksstones12a 41698 | Establish bijective mappin... |
| sticksstones12 41699 | Establish bijective mappin... |
| sticksstones13 41700 | Establish bijective mappin... |
| sticksstones14 41701 | Sticks and stones with def... |
| sticksstones15 41702 | Sticks and stones with alm... |
| sticksstones16 41703 | Sticks and stones with col... |
| sticksstones17 41704 | Extend sticks and stones t... |
| sticksstones18 41705 | Extend sticks and stones t... |
| sticksstones19 41706 | Extend sticks and stones t... |
| sticksstones20 41707 | Lift sticks and stones to ... |
| sticksstones21 41708 | Lift sticks and stones to ... |
| sticksstones22 41709 | Non-exhaustive sticks and ... |
| sticksstones23 41710 | Non-exhaustive sticks and ... |
| aks6d1c6lem1 41711 | Lemma for claim 6, deduce ... |
| aks6d1c6lem2 41712 | Every primitive root is ro... |
| aks6d1c6lem3 41713 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6lem4 41714 | Claim 6 of Theorem 6.1 of ... |
| aks6d1c6isolem1 41715 | Lemma to construct the map... |
| aks6d1c6isolem2 41716 | Lemma to construct the gro... |
| aks6d1c6isolem3 41717 | The preimage of a map send... |
| aks6d1c6lem5 41718 | Eliminate the size hypothe... |
| bcled 41719 | Inequality for binomial co... |
| bcle2d 41720 | Inequality for binomial co... |
| aks6d1c7lem1 41721 | The last set of inequaliti... |
| aks6d1c7lem2 41722 | Contradiction to Claim 2 a... |
| aks6d1c7lem3 41723 | Remove lots of hypotheses ... |
| aks6d1c7lem4 41724 | In the AKS algorithm there... |
| aks6d1c7 41725 | ` N ` is a prime power if ... |
| metakunt1 41726 | A is an endomapping. (Con... |
| metakunt2 41727 | A is an endomapping. (Con... |
| metakunt3 41728 | Value of A. (Contributed b... |
| metakunt4 41729 | Value of A. (Contributed b... |
| metakunt5 41730 | C is the left inverse for ... |
| metakunt6 41731 | C is the left inverse for ... |
| metakunt7 41732 | C is the left inverse for ... |
| metakunt8 41733 | C is the left inverse for ... |
| metakunt9 41734 | C is the left inverse for ... |
| metakunt10 41735 | C is the right inverse for... |
| metakunt11 41736 | C is the right inverse for... |
| metakunt12 41737 | C is the right inverse for... |
| metakunt13 41738 | C is the right inverse for... |
| metakunt14 41739 | A is a primitive permutati... |
| metakunt15 41740 | Construction of another pe... |
| metakunt16 41741 | Construction of another pe... |
| metakunt17 41742 | The union of three disjoin... |
| metakunt18 41743 | Disjoint domains and codom... |
| metakunt19 41744 | Domains on restrictions of... |
| metakunt20 41745 | Show that B coincides on t... |
| metakunt21 41746 | Show that B coincides on t... |
| metakunt22 41747 | Show that B coincides on t... |
| metakunt23 41748 | B coincides on the union o... |
| metakunt24 41749 | Technical condition such t... |
| metakunt25 41750 | B is a permutation. (Cont... |
| metakunt26 41751 | Construction of one soluti... |
| metakunt27 41752 | Construction of one soluti... |
| metakunt28 41753 | Construction of one soluti... |
| metakunt29 41754 | Construction of one soluti... |
| metakunt30 41755 | Construction of one soluti... |
| metakunt31 41756 | Construction of one soluti... |
| metakunt32 41757 | Construction of one soluti... |
| metakunt33 41758 | Construction of one soluti... |
| metakunt34 41759 | ` D ` is a permutation. (... |
| fac2xp3 41760 | Factorial of 2x+3, sublemm... |
| prodsplit 41761 | Product split into two fac... |
| 2xp3dxp2ge1d 41762 | 2x+3 is greater than or eq... |
| factwoffsmonot 41763 | A factorial with offset is... |
| intnanrt 41764 | Introduction of conjunct i... |
| ioin9i8 41765 | Miscellaneous inference cr... |
| jaodd 41766 | Double deduction form of ~... |
| syl3an12 41767 | A double syllogism inferen... |
| sbtd 41768 | A true statement is true u... |
| sbor2 41769 | One direction of ~ sbor , ... |
| 19.9dev 41770 | ~ 19.9d in the case of an ... |
| 3rspcedvdw 41771 | Triple application of ~ rs... |
| 3rspcedvd 41772 | Triple application of ~ rs... |
| sn-axrep5v 41773 | A condensed form of ~ axre... |
| sn-axprlem3 41774 | ~ axprlem3 using only Tars... |
| sn-exelALT 41775 | Alternate proof of ~ exel ... |
| ss2ab1 41776 | Class abstractions in a su... |
| ssabdv 41777 | Deduction of abstraction s... |
| sn-iotalem 41778 | An unused lemma showing th... |
| sn-iotalemcor 41779 | Corollary of ~ sn-iotalem ... |
| abbi1sn 41780 | Originally part of ~ uniab... |
| brif2 41781 | Move a relation inside and... |
| brif12 41782 | Move a relation inside and... |
| pssexg 41783 | The proper subset of a set... |
| pssn0 41784 | A proper superset is nonem... |
| psspwb 41785 | Classes are proper subclas... |
| xppss12 41786 | Proper subset theorem for ... |
| elpwbi 41787 | Membership in a power set,... |
| imaopab 41788 | The image of a class of or... |
| fnsnbt 41789 | A function's domain is a s... |
| fnimasnd 41790 | The image of a function by... |
| eqresfnbd 41791 | Property of being the rest... |
| f1o2d2 41792 | Sufficient condition for a... |
| fmpocos 41793 | Composition of two functio... |
| ovmpogad 41794 | Value of an operation give... |
| ofun 41795 | A function operation of un... |
| dfqs2 41796 | Alternate definition of qu... |
| dfqs3 41797 | Alternate definition of qu... |
| qseq12d 41798 | Equality theorem for quoti... |
| qsalrel 41799 | The quotient set is equal ... |
| elmapssresd 41800 | A restricted mapping is a ... |
| mapcod 41801 | Compose two mappings. (Co... |
| fzosumm1 41802 | Separate out the last term... |
| ccatcan2d 41803 | Cancellation law for conca... |
| nelsubginvcld 41804 | The inverse of a non-subgr... |
| nelsubgcld 41805 | A non-subgroup-member plus... |
| nelsubgsubcld 41806 | A non-subgroup-member minu... |
| rnasclg 41807 | The set of injected scalar... |
| frlmfielbas 41808 | The vectors of a finite fr... |
| frlmfzwrd 41809 | A vector of a module with ... |
| frlmfzowrd 41810 | A vector of a module with ... |
| frlmfzolen 41811 | The dimension of a vector ... |
| frlmfzowrdb 41812 | The vectors of a module wi... |
| frlmfzoccat 41813 | The concatenation of two v... |
| frlmvscadiccat 41814 | Scalar multiplication dist... |
| grpasscan2d 41815 | An associative cancellatio... |
| grpcominv1 41816 | If two elements commute, t... |
| grpcominv2 41817 | If two elements commute, t... |
| finsubmsubg 41818 | A submonoid of a finite gr... |
| imacrhmcl 41819 | The image of a commutative... |
| rimrcl1 41820 | Reverse closure of a ring ... |
| rimrcl2 41821 | Reverse closure of a ring ... |
| rimcnv 41822 | The converse of a ring iso... |
| rimco 41823 | The composition of ring is... |
| ricsym 41824 | Ring isomorphism is symmet... |
| rictr 41825 | Ring isomorphism is transi... |
| riccrng1 41826 | Ring isomorphism preserves... |
| riccrng 41827 | A ring is commutative if a... |
| drnginvrn0d 41828 | A multiplicative inverse i... |
| drngmulcanad 41829 | Cancellation of a nonzero ... |
| drngmulcan2ad 41830 | Cancellation of a nonzero ... |
| drnginvmuld 41831 | Inverse of a nonzero produ... |
| ricdrng1 41832 | A ring isomorphism maps a ... |
| ricdrng 41833 | A ring is a division ring ... |
| ricfld 41834 | A ring is a field if and o... |
| lvecgrp 41835 | A vector space is a group.... |
| lvecring 41836 | The scalar component of a ... |
| frlm0vald 41837 | All coordinates of the zer... |
| frlmsnic 41838 | Given a free module with a... |
| uvccl 41839 | A unit vector is a vector.... |
| uvcn0 41840 | A unit vector is nonzero. ... |
| pwselbasr 41841 | The reverse direction of ~... |
| pwsgprod 41842 | Finite products in a power... |
| psrmnd 41843 | The ring of power series i... |
| psrbagres 41844 | Restrict a bag of variable... |
| mplcrngd 41845 | The polynomial ring is a c... |
| mplsubrgcl 41846 | An element of a polynomial... |
| mhmcopsr 41847 | The composition of a monoi... |
| mhmcoaddpsr 41848 | Show that the ring homomor... |
| rhmcomulpsr 41849 | Show that the ring homomor... |
| rhmpsr 41850 | Provide a ring homomorphis... |
| rhmpsr1 41851 | Provide a ring homomorphis... |
| mplascl0 41852 | The zero scalar as a polyn... |
| mplascl1 41853 | The one scalar as a polyno... |
| mplmapghm 41854 | The function ` H ` mapping... |
| evl0 41855 | The zero polynomial evalua... |
| evlscl 41856 | A polynomial over the ring... |
| evlsval3 41857 | Give a formula for the pol... |
| evlsvval 41858 | Give a formula for the eva... |
| evlsvvvallem 41859 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 41860 | Lemma for theorems using ~... |
| evlsvvval 41861 | Give a formula for the eva... |
| evlsscaval 41862 | Polynomial evaluation buil... |
| evlsvarval 41863 | Polynomial evaluation buil... |
| evlsbagval 41864 | Polynomial evaluation buil... |
| evlsexpval 41865 | Polynomial evaluation buil... |
| evlsaddval 41866 | Polynomial evaluation buil... |
| evlsmulval 41867 | Polynomial evaluation buil... |
| evlsmaprhm 41868 | The function ` F ` mapping... |
| evlsevl 41869 | Evaluation in a subring is... |
| evlcl 41870 | A polynomial over the ring... |
| evlvvval 41871 | Give a formula for the eva... |
| evlvvvallem 41872 | Lemma for theorems using ~... |
| evladdval 41873 | Polynomial evaluation buil... |
| evlmulval 41874 | Polynomial evaluation buil... |
| selvcllem1 41875 | ` T ` is an associative al... |
| selvcllem2 41876 | ` D ` is a ring homomorphi... |
| selvcllem3 41877 | The third argument passed ... |
| selvcllemh 41878 | Apply the third argument (... |
| selvcllem4 41879 | The fourth argument passed... |
| selvcllem5 41880 | The fifth argument passed ... |
| selvcl 41881 | Closure of the "variable s... |
| selvval2 41882 | Value of the "variable sel... |
| selvvvval 41883 | Recover the original polyn... |
| evlselvlem 41884 | Lemma for ~ evlselv . Use... |
| evlselv 41885 | Evaluating a selection of ... |
| selvadd 41886 | The "variable selection" f... |
| selvmul 41887 | The "variable selection" f... |
| fsuppind 41888 | Induction on functions ` F... |
| fsuppssindlem1 41889 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 41890 | Lemma for ~ fsuppssind . ... |
| fsuppssind 41891 | Induction on functions ` F... |
| mhpind 41892 | The homogeneous polynomial... |
| evlsmhpvvval 41893 | Give a formula for the eva... |
| mhphflem 41894 | Lemma for ~ mhphf . Add s... |
| mhphf 41895 | A homogeneous polynomial d... |
| mhphf2 41896 | A homogeneous polynomial d... |
| mhphf3 41897 | A homogeneous polynomial d... |
| mhphf4 41898 | A homogeneous polynomial d... |
| c0exALT 41899 | Alternate proof of ~ c0ex ... |
| 0cnALT3 41900 | Alternate proof of ~ 0cn u... |
| elre0re 41901 | Specialized version of ~ 0... |
| 1t1e1ALT 41902 | Alternate proof of ~ 1t1e1... |
| remulcan2d 41903 | ~ mulcan2d for real number... |
| readdridaddlidd 41904 | Given some real number ` B... |
| sn-1ne2 41905 | A proof of ~ 1ne2 without ... |
| nnn1suc 41906 | A positive integer that is... |
| nnadd1com 41907 | Addition with 1 is commuta... |
| nnaddcom 41908 | Addition is commutative fo... |
| nnaddcomli 41909 | Version of ~ addcomli for ... |
| nnadddir 41910 | Right-distributivity for n... |
| nnmul1com 41911 | Multiplication with 1 is c... |
| nnmulcom 41912 | Multiplication is commutat... |
| readdrcl2d 41913 | Reverse closure for additi... |
| mvrrsubd 41914 | Move a subtraction in the ... |
| laddrotrd 41915 | Rotate the variables right... |
| raddcom12d 41916 | Swap the first two variabl... |
| lsubrotld 41917 | Rotate the variables left ... |
| lsubcom23d 41918 | Swap the second and third ... |
| addsubeq4com 41919 | Relation between sums and ... |
| sqsumi 41920 | A sum squared. (Contribut... |
| negn0nposznnd 41921 | Lemma for ~ dffltz . (Con... |
| sqmid3api 41922 | Value of the square of the... |
| decaddcom 41923 | Commute ones place in addi... |
| sqn5i 41924 | The square of a number end... |
| sqn5ii 41925 | The square of a number end... |
| decpmulnc 41926 | Partial products algorithm... |
| decpmul 41927 | Partial products algorithm... |
| sqdeccom12 41928 | The square of a number in ... |
| sq3deccom12 41929 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 41930 | 4 times 5 equals 20. (Con... |
| sq9 41931 | The square of 9 is 81. (C... |
| 235t711 41932 | Calculate a product by lon... |
| ex-decpmul 41933 | Example usage of ~ decpmul... |
| fz1sumconst 41934 | The sum of ` N ` constant ... |
| fz1sump1 41935 | Add one more term to a sum... |
| oddnumth 41936 | The Odd Number Theorem. T... |
| nicomachus 41937 | Nicomachus's Theorem. The... |
| sumcubes 41938 | The sum of the first ` N `... |
| pine0 41939 | ` _pi ` is nonzero. (Cont... |
| ine1 41940 | ` _i ` is not 1. (Contrib... |
| 0tie0 41941 | 0 times ` _i ` equals 0. ... |
| it1ei 41942 | ` _i ` times 1 equals ` _i... |
| 1tiei 41943 | 1 times ` _i ` equals ` _i... |
| itrere 41944 | ` _i ` times a real is rea... |
| retire 41945 | A real times ` _i ` is rea... |
| oexpreposd 41946 | Lemma for ~ dffltz . TODO... |
| ltexp1d 41947 | ~ ltmul1d for exponentiati... |
| ltexp1dd 41948 | Raising both sides of 'les... |
| exp11nnd 41949 | ~ sq11d for positive real ... |
| exp11d 41950 | ~ exp11nnd for nonzero int... |
| 0dvds0 41951 | 0 divides 0. (Contributed... |
| absdvdsabsb 41952 | Divisibility is invariant ... |
| dvdsexpim 41953 | ~ dvdssqim generalized to ... |
| gcdnn0id 41954 | The ` gcd ` of a nonnegati... |
| gcdle1d 41955 | The greatest common diviso... |
| gcdle2d 41956 | The greatest common diviso... |
| dvdsexpad 41957 | Deduction associated with ... |
| nn0rppwr 41958 | If ` A ` and ` B ` are rel... |
| expgcd 41959 | Exponentiation distributes... |
| nn0expgcd 41960 | Exponentiation distributes... |
| zexpgcd 41961 | Exponentiation distributes... |
| numdenexp 41962 | ~ numdensq extended to non... |
| numexp 41963 | ~ numsq extended to nonneg... |
| denexp 41964 | ~ densq extended to nonneg... |
| dvdsexpnn 41965 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 41966 | ~ dvdsexpnn generalized to... |
| dvdsexpb 41967 | ~ dvdssq generalized to po... |
| posqsqznn 41968 | When a positive rational s... |
| zrtelqelz 41969 | ~ zsqrtelqelz generalized ... |
| zrtdvds 41970 | A positive integer root di... |
| rtprmirr 41971 | The root of a prime number... |
| zdivgd 41972 | Two ways to express " ` N ... |
| efne0d 41973 | The exponential of a compl... |
| efsubd 41974 | Difference of exponents la... |
| ef11d 41975 | General condition for the ... |
| logccne0d 41976 | The logarithm isn't 0 if i... |
| cxp112d 41977 | General condition for comp... |
| cxp111d 41978 | General condition for comp... |
| cxpi11d 41979 | ` _i ` to the powers of ` ... |
| logne0d 41980 | Deduction form of ~ logne0... |
| rxp112d 41981 | Real exponentiation is one... |
| log11d 41982 | The natural logarithm is o... |
| rplog11d 41983 | The natural logarithm is o... |
| rxp11d 41984 | Real exponentiation is one... |
| resubval 41987 | Value of real subtraction,... |
| renegeulemv 41988 | Lemma for ~ renegeu and si... |
| renegeulem 41989 | Lemma for ~ renegeu and si... |
| renegeu 41990 | Existential uniqueness of ... |
| rernegcl 41991 | Closure law for negative r... |
| renegadd 41992 | Relationship between real ... |
| renegid 41993 | Addition of a real number ... |
| reneg0addlid 41994 | Negative zero is a left ad... |
| resubeulem1 41995 | Lemma for ~ resubeu . A v... |
| resubeulem2 41996 | Lemma for ~ resubeu . A v... |
| resubeu 41997 | Existential uniqueness of ... |
| rersubcl 41998 | Closure for real subtracti... |
| resubadd 41999 | Relation between real subt... |
| resubaddd 42000 | Relationship between subtr... |
| resubf 42001 | Real subtraction is an ope... |
| repncan2 42002 | Addition and subtraction o... |
| repncan3 42003 | Addition and subtraction o... |
| readdsub 42004 | Law for addition and subtr... |
| reladdrsub 42005 | Move LHS of a sum into RHS... |
| reltsub1 42006 | Subtraction from both side... |
| reltsubadd2 42007 | 'Less than' relationship b... |
| resubcan2 42008 | Cancellation law for real ... |
| resubsub4 42009 | Law for double subtraction... |
| rennncan2 42010 | Cancellation law for real ... |
| renpncan3 42011 | Cancellation law for real ... |
| repnpcan 42012 | Cancellation law for addit... |
| reppncan 42013 | Cancellation law for mixed... |
| resubidaddlidlem 42014 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 42015 | Any real number subtracted... |
| resubdi 42016 | Distribution of multiplica... |
| re1m1e0m0 42017 | Equality of two left-addit... |
| sn-00idlem1 42018 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 42019 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 42020 | Lemma for ~ sn-00id . (Co... |
| sn-00id 42021 | ~ 00id proven without ~ ax... |
| re0m0e0 42022 | Real number version of ~ 0... |
| readdlid 42023 | Real number version of ~ a... |
| sn-addlid 42024 | ~ addlid without ~ ax-mulc... |
| remul02 42025 | Real number version of ~ m... |
| sn-0ne2 42026 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 42027 | Real number version of ~ m... |
| resubid 42028 | Subtraction of a real numb... |
| readdrid 42029 | Real number version of ~ a... |
| resubid1 42030 | Real number version of ~ s... |
| renegneg 42031 | A real number is equal to ... |
| readdcan2 42032 | Commuted version of ~ read... |
| renegid2 42033 | Commuted version of ~ rene... |
| remulneg2d 42034 | Product with negative is n... |
| sn-it0e0 42035 | Proof of ~ it0e0 without ~... |
| sn-negex12 42036 | A combination of ~ cnegex ... |
| sn-negex 42037 | Proof of ~ cnegex without ... |
| sn-negex2 42038 | Proof of ~ cnegex2 without... |
| sn-addcand 42039 | ~ addcand without ~ ax-mul... |
| sn-addrid 42040 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 42041 | ~ addcan2d without ~ ax-mu... |
| reixi 42042 | ~ ixi without ~ ax-mulcom ... |
| rei4 42043 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 42044 | A number that sums to itse... |
| sn-mul01 42045 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 42046 | ~ negeu without ~ ax-mulco... |
| sn-subcl 42047 | ~ subcl without ~ ax-mulco... |
| sn-subf 42048 | ~ subf without ~ ax-mulcom... |
| resubeqsub 42049 | Equivalence between real s... |
| subresre 42050 | Subtraction restricted to ... |
| addinvcom 42051 | A number commutes with its... |
| remulinvcom 42052 | A left multiplicative inve... |
| remullid 42053 | Commuted version of ~ ax-1... |
| sn-1ticom 42054 | Lemma for ~ sn-mullid and ... |
| sn-mullid 42055 | ~ mullid without ~ ax-mulc... |
| sn-it1ei 42056 | ~ it1ei without ~ ax-mulco... |
| ipiiie0 42057 | The multiplicative inverse... |
| remulcand 42058 | Commuted version of ~ remu... |
| sn-0tie0 42059 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 42060 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 42061 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 42062 | ~ ltaddneg without ~ ax-mu... |
| reposdif 42063 | Comparison of two numbers ... |
| relt0neg1 42064 | Comparison of a real and i... |
| relt0neg2 42065 | Comparison of a real and i... |
| sn-addlt0d 42066 | The sum of negative number... |
| sn-addgt0d 42067 | The sum of positive number... |
| sn-nnne0 42068 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 42069 | ~ elznn0nn restated using ... |
| nn0addcom 42070 | Addition is commutative fo... |
| zaddcomlem 42071 | Lemma for ~ zaddcom . (Co... |
| zaddcom 42072 | Addition is commutative fo... |
| renegmulnnass 42073 | Move multiplication by a n... |
| nn0mulcom 42074 | Multiplication is commutat... |
| zmulcomlem 42075 | Lemma for ~ zmulcom . (Co... |
| zmulcom 42076 | Multiplication is commutat... |
| mulgt0con1dlem 42077 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 42078 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 42079 | Lemma for ~ mulgt0b2d and ... |
| mulgt0b2d 42080 | Biconditional, deductive f... |
| sn-ltmul2d 42081 | ~ ltmul2d without ~ ax-mul... |
| sn-0lt1 42082 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 42083 | ~ ltp1 without ~ ax-mulcom... |
| reneg1lt0 42084 | Lemma for ~ sn-inelr . (C... |
| sn-inelr 42085 | ~ inelr without ~ ax-mulco... |
| sn-itrere 42086 | ` _i ` times a real is rea... |
| sn-retire 42087 | Commuted version of ~ sn-i... |
| cnreeu 42088 | The reals in the expressio... |
| sn-sup2 42089 | ~ sup2 with exactly the sa... |
| prjspval 42092 | Value of the projective sp... |
| prjsprel 42093 | Utility theorem regarding ... |
| prjspertr 42094 | The relation in ` PrjSp ` ... |
| prjsperref 42095 | The relation in ` PrjSp ` ... |
| prjspersym 42096 | The relation in ` PrjSp ` ... |
| prjsper 42097 | The relation used to defin... |
| prjspreln0 42098 | Two nonzero vectors are eq... |
| prjspvs 42099 | A nonzero multiple of a ve... |
| prjsprellsp 42100 | Two vectors are equivalent... |
| prjspeclsp 42101 | The vectors equivalent to ... |
| prjspval2 42102 | Alternate definition of pr... |
| prjspnval 42105 | Value of the n-dimensional... |
| prjspnerlem 42106 | A lemma showing that the e... |
| prjspnval2 42107 | Value of the n-dimensional... |
| prjspner 42108 | The relation used to defin... |
| prjspnvs 42109 | A nonzero multiple of a ve... |
| prjspnssbas 42110 | A projective point spans a... |
| prjspnn0 42111 | A projective point is none... |
| 0prjspnlem 42112 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 42113 | Any vector is equivalent t... |
| prjspner01 42114 | Any vector is equivalent t... |
| prjspner1 42115 | Two vectors whose zeroth c... |
| 0prjspnrel 42116 | In the zero-dimensional pr... |
| 0prjspn 42117 | A zero-dimensional project... |
| prjcrvfval 42120 | Value of the projective cu... |
| prjcrvval 42121 | Value of the projective cu... |
| prjcrv0 42122 | The "curve" (zero set) cor... |
| dffltz 42123 | Fermat's Last Theorem (FLT... |
| fltmul 42124 | A counterexample to FLT st... |
| fltdiv 42125 | A counterexample to FLT st... |
| flt0 42126 | A counterexample for FLT d... |
| fltdvdsabdvdsc 42127 | Any factor of both ` A ` a... |
| fltabcoprmex 42128 | A counterexample to FLT im... |
| fltaccoprm 42129 | A counterexample to FLT wi... |
| fltbccoprm 42130 | A counterexample to FLT wi... |
| fltabcoprm 42131 | A counterexample to FLT wi... |
| infdesc 42132 | Infinite descent. The hyp... |
| fltne 42133 | If a counterexample to FLT... |
| flt4lem 42134 | Raising a number to the fo... |
| flt4lem1 42135 | Satisfy the antecedent use... |
| flt4lem2 42136 | If ` A ` is even, ` B ` is... |
| flt4lem3 42137 | Equivalent to ~ pythagtrip... |
| flt4lem4 42138 | If the product of two copr... |
| flt4lem5 42139 | In the context of the lemm... |
| flt4lem5elem 42140 | Version of ~ fltaccoprm an... |
| flt4lem5a 42141 | Part 1 of Equation 1 of ... |
| flt4lem5b 42142 | Part 2 of Equation 1 of ... |
| flt4lem5c 42143 | Part 2 of Equation 2 of ... |
| flt4lem5d 42144 | Part 3 of Equation 2 of ... |
| flt4lem5e 42145 | Satisfy the hypotheses of ... |
| flt4lem5f 42146 | Final equation of ~... |
| flt4lem6 42147 | Remove shared factors in a... |
| flt4lem7 42148 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 42149 | Strengthening of Fermat's ... |
| fltltc 42150 | ` ( C ^ N ) ` is the large... |
| fltnltalem 42151 | Lemma for ~ fltnlta . A l... |
| fltnlta 42152 | In a Fermat counterexample... |
| iddii 42153 | Version of ~ a1ii with the... |
| bicomdALT 42154 | Alternate proof of ~ bicom... |
| elabgw 42155 | Membership in a class abst... |
| elab2gw 42156 | Membership in a class abst... |
| elrab2w 42157 | Membership in a restricted... |
| ruvALT 42158 | Alternate proof of ~ ruv w... |
| sn-wcdeq 42159 | Alternative to ~ wcdeq and... |
| sq45 42160 | 45 squared is 2025. (Cont... |
| sum9cubes 42161 | The sum of the first nine ... |
| acos1half 42162 | The arccosine of ` 1 / 2 `... |
| aprilfools2025 42163 | An abuse of notation. (Co... |
| binom2d 42164 | Deduction form of binom2. ... |
| cu3addd 42165 | Cube of sum of three numbe... |
| sqnegd 42166 | The square of the negative... |
| negexpidd 42167 | The sum of a real number t... |
| rexlimdv3d 42168 | An extended version of ~ r... |
| 3cubeslem1 42169 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 42170 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 42171 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 42172 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 42173 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 42174 | Lemma for ~ 3cubes . This... |
| 3cubes 42175 | Every rational number is a... |
| rntrclfvOAI 42176 | The range of the transitiv... |
| moxfr 42177 | Transfer at-most-one betwe... |
| imaiinfv 42178 | Indexed intersection of an... |
| elrfi 42179 | Elementhood in a set of re... |
| elrfirn 42180 | Elementhood in a set of re... |
| elrfirn2 42181 | Elementhood in a set of re... |
| cmpfiiin 42182 | In a compact topology, a s... |
| ismrcd1 42183 | Any function from the subs... |
| ismrcd2 42184 | Second half of ~ ismrcd1 .... |
| istopclsd 42185 | A closure function which s... |
| ismrc 42186 | A function is a Moore clos... |
| isnacs 42189 | Expand definition of Noeth... |
| nacsfg 42190 | In a Noetherian-type closu... |
| isnacs2 42191 | Express Noetherian-type cl... |
| mrefg2 42192 | Slight variation on finite... |
| mrefg3 42193 | Slight variation on finite... |
| nacsacs 42194 | A closure system of Noethe... |
| isnacs3 42195 | A choice-free order equiva... |
| incssnn0 42196 | Transitivity induction of ... |
| nacsfix 42197 | An increasing sequence of ... |
| constmap 42198 | A constant (represented wi... |
| mapco2g 42199 | Renaming indices in a tupl... |
| mapco2 42200 | Post-composition (renaming... |
| mapfzcons 42201 | Extending a one-based mapp... |
| mapfzcons1 42202 | Recover prefix mapping fro... |
| mapfzcons1cl 42203 | A nonempty mapping has a p... |
| mapfzcons2 42204 | Recover added element from... |
| mptfcl 42205 | Interpret range of a maps-... |
| mzpclval 42210 | Substitution lemma for ` m... |
| elmzpcl 42211 | Double substitution lemma ... |
| mzpclall 42212 | The set of all functions w... |
| mzpcln0 42213 | Corollary of ~ mzpclall : ... |
| mzpcl1 42214 | Defining property 1 of a p... |
| mzpcl2 42215 | Defining property 2 of a p... |
| mzpcl34 42216 | Defining properties 3 and ... |
| mzpval 42217 | Value of the ` mzPoly ` fu... |
| dmmzp 42218 | ` mzPoly ` is defined for ... |
| mzpincl 42219 | Polynomial closedness is a... |
| mzpconst 42220 | Constant functions are pol... |
| mzpf 42221 | A polynomial function is a... |
| mzpproj 42222 | A projection function is p... |
| mzpadd 42223 | The pointwise sum of two p... |
| mzpmul 42224 | The pointwise product of t... |
| mzpconstmpt 42225 | A constant function expres... |
| mzpaddmpt 42226 | Sum of polynomial function... |
| mzpmulmpt 42227 | Product of polynomial func... |
| mzpsubmpt 42228 | The difference of two poly... |
| mzpnegmpt 42229 | Negation of a polynomial f... |
| mzpexpmpt 42230 | Raise a polynomial functio... |
| mzpindd 42231 | "Structural" induction to ... |
| mzpmfp 42232 | Relationship between multi... |
| mzpsubst 42233 | Substituting polynomials f... |
| mzprename 42234 | Simplified version of ~ mz... |
| mzpresrename 42235 | A polynomial is a polynomi... |
| mzpcompact2lem 42236 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 42237 | Polynomials are finitary o... |
| coeq0i 42238 | ~ coeq0 but without explic... |
| fzsplit1nn0 42239 | Split a finite 1-based set... |
| eldiophb 42242 | Initial expression of Diop... |
| eldioph 42243 | Condition for a set to be ... |
| diophrw 42244 | Renaming and adding unused... |
| eldioph2lem1 42245 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 42246 | Lemma for ~ eldioph2 . Co... |
| eldioph2 42247 | Construct a Diophantine se... |
| eldioph2b 42248 | While Diophantine sets wer... |
| eldiophelnn0 42249 | Remove antecedent on ` B `... |
| eldioph3b 42250 | Define Diophantine sets in... |
| eldioph3 42251 | Inference version of ~ eld... |
| ellz1 42252 | Membership in a lower set ... |
| lzunuz 42253 | The union of a lower set o... |
| fz1eqin 42254 | Express a one-based finite... |
| lzenom 42255 | Lower integers are countab... |
| elmapresaunres2 42256 | ~ fresaunres2 transposed t... |
| diophin 42257 | If two sets are Diophantin... |
| diophun 42258 | If two sets are Diophantin... |
| eldiophss 42259 | Diophantine sets are sets ... |
| diophrex 42260 | Projecting a Diophantine s... |
| eq0rabdioph 42261 | This is the first of a num... |
| eqrabdioph 42262 | Diophantine set builder fo... |
| 0dioph 42263 | The null set is Diophantin... |
| vdioph 42264 | The "universal" set (as la... |
| anrabdioph 42265 | Diophantine set builder fo... |
| orrabdioph 42266 | Diophantine set builder fo... |
| 3anrabdioph 42267 | Diophantine set builder fo... |
| 3orrabdioph 42268 | Diophantine set builder fo... |
| 2sbcrex 42269 | Exchange an existential qu... |
| sbcrexgOLD 42270 | Interchange class substitu... |
| 2sbcrexOLD 42271 | Exchange an existential qu... |
| sbc2rex 42272 | Exchange a substitution wi... |
| sbc2rexgOLD 42273 | Exchange a substitution wi... |
| sbc4rex 42274 | Exchange a substitution wi... |
| sbc4rexgOLD 42275 | Exchange a substitution wi... |
| sbcrot3 42276 | Rotate a sequence of three... |
| sbcrot5 42277 | Rotate a sequence of five ... |
| sbccomieg 42278 | Commute two explicit subst... |
| rexrabdioph 42279 | Diophantine set builder fo... |
| rexfrabdioph 42280 | Diophantine set builder fo... |
| 2rexfrabdioph 42281 | Diophantine set builder fo... |
| 3rexfrabdioph 42282 | Diophantine set builder fo... |
| 4rexfrabdioph 42283 | Diophantine set builder fo... |
| 6rexfrabdioph 42284 | Diophantine set builder fo... |
| 7rexfrabdioph 42285 | Diophantine set builder fo... |
| rabdiophlem1 42286 | Lemma for arithmetic dioph... |
| rabdiophlem2 42287 | Lemma for arithmetic dioph... |
| elnn0rabdioph 42288 | Diophantine set builder fo... |
| rexzrexnn0 42289 | Rewrite an existential qua... |
| lerabdioph 42290 | Diophantine set builder fo... |
| eluzrabdioph 42291 | Diophantine set builder fo... |
| elnnrabdioph 42292 | Diophantine set builder fo... |
| ltrabdioph 42293 | Diophantine set builder fo... |
| nerabdioph 42294 | Diophantine set builder fo... |
| dvdsrabdioph 42295 | Divisibility is a Diophant... |
| eldioph4b 42296 | Membership in ` Dioph ` ex... |
| eldioph4i 42297 | Forward-only version of ~ ... |
| diophren 42298 | Change variables in a Diop... |
| rabrenfdioph 42299 | Change variable numbers in... |
| rabren3dioph 42300 | Change variable numbers in... |
| fphpd 42301 | Pigeonhole principle expre... |
| fphpdo 42302 | Pigeonhole principle for s... |
| ctbnfien 42303 | An infinite subset of a co... |
| fiphp3d 42304 | Infinite pigeonhole princi... |
| rencldnfilem 42305 | Lemma for ~ rencldnfi . (... |
| rencldnfi 42306 | A set of real numbers whic... |
| irrapxlem1 42307 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 42308 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 42309 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 42310 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 42311 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 42312 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 42313 | Dirichlet's approximation ... |
| pellexlem1 42314 | Lemma for ~ pellex . Arit... |
| pellexlem2 42315 | Lemma for ~ pellex . Arit... |
| pellexlem3 42316 | Lemma for ~ pellex . To e... |
| pellexlem4 42317 | Lemma for ~ pellex . Invo... |
| pellexlem5 42318 | Lemma for ~ pellex . Invo... |
| pellexlem6 42319 | Lemma for ~ pellex . Doin... |
| pellex 42320 | Every Pell equation has a ... |
| pell1qrval 42331 | Value of the set of first-... |
| elpell1qr 42332 | Membership in a first-quad... |
| pell14qrval 42333 | Value of the set of positi... |
| elpell14qr 42334 | Membership in the set of p... |
| pell1234qrval 42335 | Value of the set of genera... |
| elpell1234qr 42336 | Membership in the set of g... |
| pell1234qrre 42337 | General Pell solutions are... |
| pell1234qrne0 42338 | No solution to a Pell equa... |
| pell1234qrreccl 42339 | General solutions of the P... |
| pell1234qrmulcl 42340 | General solutions of the P... |
| pell14qrss1234 42341 | A positive Pell solution i... |
| pell14qrre 42342 | A positive Pell solution i... |
| pell14qrne0 42343 | A positive Pell solution i... |
| pell14qrgt0 42344 | A positive Pell solution i... |
| pell14qrrp 42345 | A positive Pell solution i... |
| pell1234qrdich 42346 | A general Pell solution is... |
| elpell14qr2 42347 | A number is a positive Pel... |
| pell14qrmulcl 42348 | Positive Pell solutions ar... |
| pell14qrreccl 42349 | Positive Pell solutions ar... |
| pell14qrdivcl 42350 | Positive Pell solutions ar... |
| pell14qrexpclnn0 42351 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 42352 | Positive Pell solutions ar... |
| pell1qrss14 42353 | First-quadrant Pell soluti... |
| pell14qrdich 42354 | A positive Pell solution i... |
| pell1qrge1 42355 | A Pell solution in the fir... |
| pell1qr1 42356 | 1 is a Pell solution and i... |
| elpell1qr2 42357 | The first quadrant solutio... |
| pell1qrgaplem 42358 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 42359 | First-quadrant Pell soluti... |
| pell14qrgap 42360 | Positive Pell solutions ar... |
| pell14qrgapw 42361 | Positive Pell solutions ar... |
| pellqrexplicit 42362 | Condition for a calculated... |
| infmrgelbi 42363 | Any lower bound of a nonem... |
| pellqrex 42364 | There is a nontrivial solu... |
| pellfundval 42365 | Value of the fundamental s... |
| pellfundre 42366 | The fundamental solution o... |
| pellfundge 42367 | Lower bound on the fundame... |
| pellfundgt1 42368 | Weak lower bound on the Pe... |
| pellfundlb 42369 | A nontrivial first quadran... |
| pellfundglb 42370 | If a real is larger than t... |
| pellfundex 42371 | The fundamental solution a... |
| pellfund14gap 42372 | There are no solutions bet... |
| pellfundrp 42373 | The fundamental Pell solut... |
| pellfundne1 42374 | The fundamental Pell solut... |
| reglogcl 42375 | General logarithm is a rea... |
| reglogltb 42376 | General logarithm preserve... |
| reglogleb 42377 | General logarithm preserve... |
| reglogmul 42378 | Multiplication law for gen... |
| reglogexp 42379 | Power law for general log.... |
| reglogbas 42380 | General log of the base is... |
| reglog1 42381 | General log of 1 is 0. (C... |
| reglogexpbas 42382 | General log of a power of ... |
| pellfund14 42383 | Every positive Pell soluti... |
| pellfund14b 42384 | The positive Pell solution... |
| rmxfval 42389 | Value of the X sequence. ... |
| rmyfval 42390 | Value of the Y sequence. ... |
| rmspecsqrtnq 42391 | The discriminant used to d... |
| rmspecnonsq 42392 | The discriminant used to d... |
| qirropth 42393 | This lemma implements the ... |
| rmspecfund 42394 | The base of exponent used ... |
| rmxyelqirr 42395 | The solutions used to cons... |
| rmxyelqirrOLD 42396 | Obsolete version of ~ rmxy... |
| rmxypairf1o 42397 | The function used to extra... |
| rmxyelxp 42398 | Lemma for ~ frmx and ~ frm... |
| frmx 42399 | The X sequence is a nonneg... |
| frmy 42400 | The Y sequence is an integ... |
| rmxyval 42401 | Main definition of the X a... |
| rmspecpos 42402 | The discriminant used to d... |
| rmxycomplete 42403 | The X and Y sequences take... |
| rmxynorm 42404 | The X and Y sequences defi... |
| rmbaserp 42405 | The base of exponentiation... |
| rmxyneg 42406 | Negation law for X and Y s... |
| rmxyadd 42407 | Addition formula for X and... |
| rmxy1 42408 | Value of the X and Y seque... |
| rmxy0 42409 | Value of the X and Y seque... |
| rmxneg 42410 | Negation law (even functio... |
| rmx0 42411 | Value of X sequence at 0. ... |
| rmx1 42412 | Value of X sequence at 1. ... |
| rmxadd 42413 | Addition formula for X seq... |
| rmyneg 42414 | Negation formula for Y seq... |
| rmy0 42415 | Value of Y sequence at 0. ... |
| rmy1 42416 | Value of Y sequence at 1. ... |
| rmyadd 42417 | Addition formula for Y seq... |
| rmxp1 42418 | Special addition-of-1 form... |
| rmyp1 42419 | Special addition of 1 form... |
| rmxm1 42420 | Subtraction of 1 formula f... |
| rmym1 42421 | Subtraction of 1 formula f... |
| rmxluc 42422 | The X sequence is a Lucas ... |
| rmyluc 42423 | The Y sequence is a Lucas ... |
| rmyluc2 42424 | Lucas sequence property of... |
| rmxdbl 42425 | "Double-angle formula" for... |
| rmydbl 42426 | "Double-angle formula" for... |
| monotuz 42427 | A function defined on an u... |
| monotoddzzfi 42428 | A function which is odd an... |
| monotoddzz 42429 | A function (given implicit... |
| oddcomabszz 42430 | An odd function which take... |
| 2nn0ind 42431 | Induction on nonnegative i... |
| zindbi 42432 | Inductively transfer a pro... |
| rmxypos 42433 | For all nonnegative indice... |
| ltrmynn0 42434 | The Y-sequence is strictly... |
| ltrmxnn0 42435 | The X-sequence is strictly... |
| lermxnn0 42436 | The X-sequence is monotoni... |
| rmxnn 42437 | The X-sequence is defined ... |
| ltrmy 42438 | The Y-sequence is strictly... |
| rmyeq0 42439 | Y is zero only at zero. (... |
| rmyeq 42440 | Y is one-to-one. (Contrib... |
| lermy 42441 | Y is monotonic (non-strict... |
| rmynn 42442 | ` rmY ` is positive for po... |
| rmynn0 42443 | ` rmY ` is nonnegative for... |
| rmyabs 42444 | ` rmY ` commutes with ` ab... |
| jm2.24nn 42445 | X(n) is strictly greater t... |
| jm2.17a 42446 | First half of lemma 2.17 o... |
| jm2.17b 42447 | Weak form of the second ha... |
| jm2.17c 42448 | Second half of lemma 2.17 ... |
| jm2.24 42449 | Lemma 2.24 of [JonesMatija... |
| rmygeid 42450 | Y(n) increases faster than... |
| congtr 42451 | A wff of the form ` A || (... |
| congadd 42452 | If two pairs of numbers ar... |
| congmul 42453 | If two pairs of numbers ar... |
| congsym 42454 | Congruence mod ` A ` is a ... |
| congneg 42455 | If two integers are congru... |
| congsub 42456 | If two pairs of numbers ar... |
| congid 42457 | Every integer is congruent... |
| mzpcong 42458 | Polynomials commute with c... |
| congrep 42459 | Every integer is congruent... |
| congabseq 42460 | If two integers are congru... |
| acongid 42461 | A wff like that in this th... |
| acongsym 42462 | Symmetry of alternating co... |
| acongneg2 42463 | Negate right side of alter... |
| acongtr 42464 | Transitivity of alternatin... |
| acongeq12d 42465 | Substitution deduction for... |
| acongrep 42466 | Every integer is alternati... |
| fzmaxdif 42467 | Bound on the difference be... |
| fzneg 42468 | Reflection of a finite ran... |
| acongeq 42469 | Two numbers in the fundame... |
| dvdsacongtr 42470 | Alternating congruence pas... |
| coprmdvdsb 42471 | Multiplication by a coprim... |
| modabsdifz 42472 | Divisibility in terms of m... |
| dvdsabsmod0 42473 | Divisibility in terms of m... |
| jm2.18 42474 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 42475 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 42476 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 42477 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 42478 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 42479 | Lemma 2.19 of [JonesMatija... |
| jm2.21 42480 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 42481 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 42482 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 42483 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 42484 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 42485 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 42486 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 42487 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 42488 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 42489 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 42490 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 42491 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 42492 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 42493 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 42494 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 42495 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 42496 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 42497 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 42498 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 42499 | Lemma for ~ rmydioph . Us... |
| rmydioph 42500 | ~ jm2.27 restated in terms... |
| rmxdiophlem 42501 | X can be expressed in term... |
| rmxdioph 42502 | X is a Diophantine functio... |
| jm3.1lem1 42503 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 42504 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 42505 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 42506 | Diophantine expression for... |
| expdiophlem1 42507 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 42508 | Lemma for ~ expdioph . Ex... |
| expdioph 42509 | The exponential function i... |
| setindtr 42510 | Set induction for sets con... |
| setindtrs 42511 | Set induction scheme witho... |
| dford3lem1 42512 | Lemma for ~ dford3 . (Con... |
| dford3lem2 42513 | Lemma for ~ dford3 . (Con... |
| dford3 42514 | Ordinals are precisely the... |
| dford4 42515 | ~ dford3 expressed in prim... |
| wopprc 42516 | Unrelated: Wiener pairs t... |
| rpnnen3lem 42517 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 42518 | Dedekind cut injection of ... |
| axac10 42519 | Characterization of choice... |
| harinf 42520 | The Hartogs number of an i... |
| wdom2d2 42521 | Deduction for weak dominan... |
| ttac 42522 | Tarski's theorem about cho... |
| pw2f1ocnv 42523 | Define a bijection between... |
| pw2f1o2 42524 | Define a bijection between... |
| pw2f1o2val 42525 | Function value of the ~ pw... |
| pw2f1o2val2 42526 | Membership in a mapped set... |
| soeq12d 42527 | Equality deduction for tot... |
| freq12d 42528 | Equality deduction for fou... |
| weeq12d 42529 | Equality deduction for wel... |
| limsuc2 42530 | Limit ordinals in the sens... |
| wepwsolem 42531 | Transfer an ordering on ch... |
| wepwso 42532 | A well-ordering induces a ... |
| dnnumch1 42533 | Define an enumeration of a... |
| dnnumch2 42534 | Define an enumeration (wea... |
| dnnumch3lem 42535 | Value of the ordinal injec... |
| dnnumch3 42536 | Define an injection from a... |
| dnwech 42537 | Define a well-ordering fro... |
| fnwe2val 42538 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 42539 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 42540 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 42541 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 42542 | A well-ordering can be con... |
| aomclem1 42543 | Lemma for ~ dfac11 . This... |
| aomclem2 42544 | Lemma for ~ dfac11 . Succ... |
| aomclem3 42545 | Lemma for ~ dfac11 . Succ... |
| aomclem4 42546 | Lemma for ~ dfac11 . Limi... |
| aomclem5 42547 | Lemma for ~ dfac11 . Comb... |
| aomclem6 42548 | Lemma for ~ dfac11 . Tran... |
| aomclem7 42549 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 42550 | Lemma for ~ dfac11 . Perf... |
| dfac11 42551 | The right-hand side of thi... |
| kelac1 42552 | Kelley's choice, basic for... |
| kelac2lem 42553 | Lemma for ~ kelac2 and ~ d... |
| kelac2 42554 | Kelley's choice, most comm... |
| dfac21 42555 | Tychonoff's theorem is a c... |
| islmodfg 42558 | Property of a finitely gen... |
| islssfg 42559 | Property of a finitely gen... |
| islssfg2 42560 | Property of a finitely gen... |
| islssfgi 42561 | Finitely spanned subspaces... |
| fglmod 42562 | Finitely generated left mo... |
| lsmfgcl 42563 | The sum of two finitely ge... |
| islnm 42566 | Property of being a Noethe... |
| islnm2 42567 | Property of being a Noethe... |
| lnmlmod 42568 | A Noetherian left module i... |
| lnmlssfg 42569 | A submodule of Noetherian ... |
| lnmlsslnm 42570 | All submodules of a Noethe... |
| lnmfg 42571 | A Noetherian left module i... |
| kercvrlsm 42572 | The domain of a linear fun... |
| lmhmfgima 42573 | A homomorphism maps finite... |
| lnmepi 42574 | Epimorphic images of Noeth... |
| lmhmfgsplit 42575 | If the kernel and range of... |
| lmhmlnmsplit 42576 | If the kernel and range of... |
| lnmlmic 42577 | Noetherian is an invariant... |
| pwssplit4 42578 | Splitting for structure po... |
| filnm 42579 | Finite left modules are No... |
| pwslnmlem0 42580 | Zeroeth powers are Noether... |
| pwslnmlem1 42581 | First powers are Noetheria... |
| pwslnmlem2 42582 | A sum of powers is Noether... |
| pwslnm 42583 | Finite powers of Noetheria... |
| unxpwdom3 42584 | Weaker version of ~ unxpwd... |
| pwfi2f1o 42585 | The ~ pw2f1o bijection rel... |
| pwfi2en 42586 | Finitely supported indicat... |
| frlmpwfi 42587 | Formal linear combinations... |
| gicabl 42588 | Being Abelian is a group i... |
| imasgim 42589 | A relabeling of the elemen... |
| isnumbasgrplem1 42590 | A set which is equipollent... |
| harn0 42591 | The Hartogs number of a se... |
| numinfctb 42592 | A numerable infinite set c... |
| isnumbasgrplem2 42593 | If the (to be thought of a... |
| isnumbasgrplem3 42594 | Every nonempty numerable s... |
| isnumbasabl 42595 | A set is numerable iff it ... |
| isnumbasgrp 42596 | A set is numerable iff it ... |
| dfacbasgrp 42597 | A choice equivalent in abs... |
| islnr 42600 | Property of a left-Noether... |
| lnrring 42601 | Left-Noetherian rings are ... |
| lnrlnm 42602 | Left-Noetherian rings have... |
| islnr2 42603 | Property of being a left-N... |
| islnr3 42604 | Relate left-Noetherian rin... |
| lnr2i 42605 | Given an ideal in a left-N... |
| lpirlnr 42606 | Left principal ideal rings... |
| lnrfrlm 42607 | Finite-dimensional free mo... |
| lnrfg 42608 | Finitely-generated modules... |
| lnrfgtr 42609 | A submodule of a finitely ... |
| hbtlem1 42612 | Value of the leading coeff... |
| hbtlem2 42613 | Leading coefficient ideals... |
| hbtlem7 42614 | Functionality of leading c... |
| hbtlem4 42615 | The leading ideal function... |
| hbtlem3 42616 | The leading ideal function... |
| hbtlem5 42617 | The leading ideal function... |
| hbtlem6 42618 | There is a finite set of p... |
| hbt 42619 | The Hilbert Basis Theorem ... |
| dgrsub2 42624 | Subtracting two polynomial... |
| elmnc 42625 | Property of a monic polyno... |
| mncply 42626 | A monic polynomial is a po... |
| mnccoe 42627 | A monic polynomial has lea... |
| mncn0 42628 | A monic polynomial is not ... |
| dgraaval 42633 | Value of the degree functi... |
| dgraalem 42634 | Properties of the degree o... |
| dgraacl 42635 | Closure of the degree func... |
| dgraaf 42636 | Degree function on algebra... |
| dgraaub 42637 | Upper bound on degree of a... |
| dgraa0p 42638 | A rational polynomial of d... |
| mpaaeu 42639 | An algebraic number has ex... |
| mpaaval 42640 | Value of the minimal polyn... |
| mpaalem 42641 | Properties of the minimal ... |
| mpaacl 42642 | Minimal polynomial is a po... |
| mpaadgr 42643 | Minimal polynomial has deg... |
| mpaaroot 42644 | The minimal polynomial of ... |
| mpaamn 42645 | Minimal polynomial is moni... |
| itgoval 42650 | Value of the integral-over... |
| aaitgo 42651 | The standard algebraic num... |
| itgoss 42652 | An integral element is int... |
| itgocn 42653 | All integral elements are ... |
| cnsrexpcl 42654 | Exponentiation is closed i... |
| fsumcnsrcl 42655 | Finite sums are closed in ... |
| cnsrplycl 42656 | Polynomials are closed in ... |
| rgspnval 42657 | Value of the ring-span of ... |
| rgspncl 42658 | The ring-span of a set is ... |
| rgspnssid 42659 | The ring-span of a set con... |
| rgspnmin 42660 | The ring-span is contained... |
| rgspnid 42661 | The span of a subring is i... |
| rngunsnply 42662 | Adjoining one element to a... |
| flcidc 42663 | Finite linear combinations... |
| algstr 42666 | Lemma to shorten proofs of... |
| algbase 42667 | The base set of a construc... |
| algaddg 42668 | The additive operation of ... |
| algmulr 42669 | The multiplicative operati... |
| algsca 42670 | The set of scalars of a co... |
| algvsca 42671 | The scalar product operati... |
| mendval 42672 | Value of the module endomo... |
| mendbas 42673 | Base set of the module end... |
| mendplusgfval 42674 | Addition in the module end... |
| mendplusg 42675 | A specific addition in the... |
| mendmulrfval 42676 | Multiplication in the modu... |
| mendmulr 42677 | A specific multiplication ... |
| mendsca 42678 | The module endomorphism al... |
| mendvscafval 42679 | Scalar multiplication in t... |
| mendvsca 42680 | A specific scalar multipli... |
| mendring 42681 | The module endomorphism al... |
| mendlmod 42682 | The module endomorphism al... |
| mendassa 42683 | The module endomorphism al... |
| idomodle 42684 | Limit on the number of ` N... |
| fiuneneq 42685 | Two finite sets of equal s... |
| idomsubgmo 42686 | The units of an integral d... |
| proot1mul 42687 | Any primitive ` N ` -th ro... |
| proot1hash 42688 | If an integral domain has ... |
| proot1ex 42689 | The complex field has prim... |
| mon1psubm 42692 | Monic polynomials are a mu... |
| deg1mhm 42693 | Homomorphic property of th... |
| cytpfn 42694 | Functionality of the cyclo... |
| cytpval 42695 | Substitutions for the Nth ... |
| fgraphopab 42696 | Express a function as a su... |
| fgraphxp 42697 | Express a function as a su... |
| hausgraph 42698 | The graph of a continuous ... |
| r1sssucd 42703 | Deductive form of ~ r1sssu... |
| iocunico 42704 | Split an open interval int... |
| iocinico 42705 | The intersection of two se... |
| iocmbl 42706 | An open-below, closed-abov... |
| cnioobibld 42707 | A bounded, continuous func... |
| arearect 42708 | The area of a rectangle wh... |
| areaquad 42709 | The area of a quadrilatera... |
| uniel 42710 | Two ways to say a union is... |
| unielss 42711 | Two ways to say the union ... |
| unielid 42712 | Two ways to say the union ... |
| ssunib 42713 | Two ways to say a class is... |
| rp-intrabeq 42714 | Equality theorem for supre... |
| rp-unirabeq 42715 | Equality theorem for infim... |
| onmaxnelsup 42716 | Two ways to say the maximu... |
| onsupneqmaxlim0 42717 | If the supremum of a class... |
| onsupcl2 42718 | The supremum of a set of o... |
| onuniintrab 42719 | The union of a set of ordi... |
| onintunirab 42720 | The intersection of a non-... |
| onsupnmax 42721 | If the union of a class of... |
| onsupuni 42722 | The supremum of a set of o... |
| onsupuni2 42723 | The supremum of a set of o... |
| onsupintrab 42724 | The supremum of a set of o... |
| onsupintrab2 42725 | The supremum of a set of o... |
| onsupcl3 42726 | The supremum of a set of o... |
| onsupex3 42727 | The supremum of a set of o... |
| onuniintrab2 42728 | The union of a set of ordi... |
| oninfint 42729 | The infimum of a non-empty... |
| oninfunirab 42730 | The infimum of a non-empty... |
| oninfcl2 42731 | The infimum of a non-empty... |
| onsupmaxb 42732 | The union of a class of or... |
| onexgt 42733 | For any ordinal, there is ... |
| onexomgt 42734 | For any ordinal, there is ... |
| omlimcl2 42735 | The product of a limit ord... |
| onexlimgt 42736 | For any ordinal, there is ... |
| onexoegt 42737 | For any ordinal, there is ... |
| oninfex2 42738 | The infimum of a non-empty... |
| onsupeqmax 42739 | Condition when the supremu... |
| onsupeqnmax 42740 | Condition when the supremu... |
| onsuplub 42741 | The supremum of a set of o... |
| onsupnub 42742 | An upper bound of a set of... |
| onfisupcl 42743 | Sufficient condition when ... |
| onelord 42744 | Every element of a ordinal... |
| onepsuc 42745 | Every ordinal is less than... |
| epsoon 42746 | The ordinals are strictly ... |
| epirron 42747 | The strict order on the or... |
| oneptr 42748 | The strict order on the or... |
| oneltr 42749 | The elementhood relation o... |
| oneptri 42750 | The strict, complete (line... |
| oneltri 42751 | The elementhood relation o... |
| ordeldif 42752 | Membership in the differen... |
| ordeldifsucon 42753 | Membership in the differen... |
| ordeldif1o 42754 | Membership in the differen... |
| ordne0gt0 42755 | Ordinal zero is less than ... |
| ondif1i 42756 | Ordinal zero is less than ... |
| onsucelab 42757 | The successor of every ord... |
| dflim6 42758 | A limit ordinal is a non-z... |
| limnsuc 42759 | A limit ordinal is not an ... |
| onsucss 42760 | If one ordinal is less tha... |
| ordnexbtwnsuc 42761 | For any distinct pair of o... |
| orddif0suc 42762 | For any distinct pair of o... |
| onsucf1lem 42763 | For ordinals, the successo... |
| onsucf1olem 42764 | The successor operation is... |
| onsucrn 42765 | The successor operation is... |
| onsucf1o 42766 | The successor operation is... |
| dflim7 42767 | A limit ordinal is a non-z... |
| onov0suclim 42768 | Compactly express rules fo... |
| oa0suclim 42769 | Closed form expression of ... |
| om0suclim 42770 | Closed form expression of ... |
| oe0suclim 42771 | Closed form expression of ... |
| oaomoecl 42772 | The operations of addition... |
| onsupsucismax 42773 | If the union of a set of o... |
| onsssupeqcond 42774 | If for every element of a ... |
| limexissup 42775 | An ordinal which is a limi... |
| limiun 42776 | A limit ordinal is the uni... |
| limexissupab 42777 | An ordinal which is a limi... |
| om1om1r 42778 | Ordinal one is both a left... |
| oe0rif 42779 | Ordinal zero raised to any... |
| oasubex 42780 | While subtraction can't be... |
| nnamecl 42781 | Natural numbers are closed... |
| onsucwordi 42782 | The successor operation pr... |
| oalim2cl 42783 | The ordinal sum of any ord... |
| oaltublim 42784 | Given ` C ` is a limit ord... |
| oaordi3 42785 | Ordinal addition of the sa... |
| oaord3 42786 | When the same ordinal is a... |
| 1oaomeqom 42787 | Ordinal one plus omega is ... |
| oaabsb 42788 | The right addend absorbs t... |
| oaordnrex 42789 | When omega is added on the... |
| oaordnr 42790 | When the same ordinal is a... |
| omge1 42791 | Any non-zero ordinal produ... |
| omge2 42792 | Any non-zero ordinal produ... |
| omlim2 42793 | The non-zero product with ... |
| omord2lim 42794 | Given a limit ordinal, the... |
| omord2i 42795 | Ordinal multiplication of ... |
| omord2com 42796 | When the same non-zero ord... |
| 2omomeqom 42797 | Ordinal two times omega is... |
| omnord1ex 42798 | When omega is multiplied o... |
| omnord1 42799 | When the same non-zero ord... |
| oege1 42800 | Any non-zero ordinal power... |
| oege2 42801 | Any power of an ordinal at... |
| rp-oelim2 42802 | The power of an ordinal at... |
| oeord2lim 42803 | Given a limit ordinal, the... |
| oeord2i 42804 | Ordinal exponentiation of ... |
| oeord2com 42805 | When the same base at leas... |
| nnoeomeqom 42806 | Any natural number at leas... |
| df3o2 42807 | Ordinal 3 is the unordered... |
| df3o3 42808 | Ordinal 3, fully expanded.... |
| oenord1ex 42809 | When ordinals two and thre... |
| oenord1 42810 | When two ordinals (both at... |
| oaomoencom 42811 | Ordinal addition, multipli... |
| oenassex 42812 | Ordinal two raised to two ... |
| oenass 42813 | Ordinal exponentiation is ... |
| cantnftermord 42814 | For terms of the form of a... |
| cantnfub 42815 | Given a finite number of t... |
| cantnfub2 42816 | Given a finite number of t... |
| bropabg 42817 | Equivalence for two classe... |
| cantnfresb 42818 | A Cantor normal form which... |
| cantnf2 42819 | For every ordinal, ` A ` ,... |
| oawordex2 42820 | If ` C ` is between ` A ` ... |
| nnawordexg 42821 | If an ordinal, ` B ` , is ... |
| succlg 42822 | Closure law for ordinal su... |
| dflim5 42823 | A limit ordinal is either ... |
| oacl2g 42824 | Closure law for ordinal ad... |
| onmcl 42825 | If an ordinal is less than... |
| omabs2 42826 | Ordinal multiplication by ... |
| omcl2 42827 | Closure law for ordinal mu... |
| omcl3g 42828 | Closure law for ordinal mu... |
| ordsssucb 42829 | An ordinal number is less ... |
| tfsconcatlem 42830 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 42831 | The concatenation of two t... |
| tfsconcatfn 42832 | The concatenation of two t... |
| tfsconcatfv1 42833 | An early value of the conc... |
| tfsconcatfv2 42834 | A latter value of the conc... |
| tfsconcatfv 42835 | The value of the concatena... |
| tfsconcatrn 42836 | The range of the concatena... |
| tfsconcatfo 42837 | The concatenation of two t... |
| tfsconcatb0 42838 | The concatentation with th... |
| tfsconcat0i 42839 | The concatentation with th... |
| tfsconcat0b 42840 | The concatentation with th... |
| tfsconcat00 42841 | The concatentation of two ... |
| tfsconcatrev 42842 | If the domain of a transfi... |
| tfsconcatrnss12 42843 | The range of the concatena... |
| tfsconcatrnss 42844 | The concatenation of trans... |
| tfsconcatrnsson 42845 | The concatenation of trans... |
| tfsnfin 42846 | A transfinite sequence is ... |
| rp-tfslim 42847 | The limit of a sequence of... |
| ofoafg 42848 | Addition operator for func... |
| ofoaf 42849 | Addition operator for func... |
| ofoafo 42850 | Addition operator for func... |
| ofoacl 42851 | Closure law for component ... |
| ofoaid1 42852 | Identity law for component... |
| ofoaid2 42853 | Identity law for component... |
| ofoaass 42854 | Component-wise addition of... |
| ofoacom 42855 | Component-wise addition of... |
| naddcnff 42856 | Addition operator for Cant... |
| naddcnffn 42857 | Addition operator for Cant... |
| naddcnffo 42858 | Addition of Cantor normal ... |
| naddcnfcl 42859 | Closure law for component-... |
| naddcnfcom 42860 | Component-wise ordinal add... |
| naddcnfid1 42861 | Identity law for component... |
| naddcnfid2 42862 | Identity law for component... |
| naddcnfass 42863 | Component-wise addition of... |
| onsucunifi 42864 | The successor to the union... |
| sucunisn 42865 | The successor to the union... |
| onsucunipr 42866 | The successor to the union... |
| onsucunitp 42867 | The successor to the union... |
| oaun3lem1 42868 | The class of all ordinal s... |
| oaun3lem2 42869 | The class of all ordinal s... |
| oaun3lem3 42870 | The class of all ordinal s... |
| oaun3lem4 42871 | The class of all ordinal s... |
| rp-abid 42872 | Two ways to express a clas... |
| oadif1lem 42873 | Express the set difference... |
| oadif1 42874 | Express the set difference... |
| oaun2 42875 | Ordinal addition as a unio... |
| oaun3 42876 | Ordinal addition as a unio... |
| naddov4 42877 | Alternate expression for n... |
| nadd2rabtr 42878 | The set of ordinals which ... |
| nadd2rabord 42879 | The set of ordinals which ... |
| nadd2rabex 42880 | The class of ordinals whic... |
| nadd2rabon 42881 | The set of ordinals which ... |
| nadd1rabtr 42882 | The set of ordinals which ... |
| nadd1rabord 42883 | The set of ordinals which ... |
| nadd1rabex 42884 | The class of ordinals whic... |
| nadd1rabon 42885 | The set of ordinals which ... |
| nadd1suc 42886 | Natural addition with 1 is... |
| naddsuc2 42887 | Natural addition with succ... |
| naddass1 42888 | Natural addition of ordina... |
| naddgeoa 42889 | Natural addition results i... |
| naddonnn 42890 | Natural addition with a na... |
| naddwordnexlem0 42891 | When ` A ` is the sum of a... |
| naddwordnexlem1 42892 | When ` A ` is the sum of a... |
| naddwordnexlem2 42893 | When ` A ` is the sum of a... |
| naddwordnexlem3 42894 | When ` A ` is the sum of a... |
| oawordex3 42895 | When ` A ` is the sum of a... |
| naddwordnexlem4 42896 | When ` A ` is the sum of a... |
| ordsssucim 42897 | If an ordinal is less than... |
| insucid 42898 | The intersection of a clas... |
| om2 42899 | Two ways to double an ordi... |
| oaltom 42900 | Multiplication eventually ... |
| oe2 42901 | Two ways to square an ordi... |
| omltoe 42902 | Exponentiation eventually ... |
| abeqabi 42903 | Generalized condition for ... |
| abpr 42904 | Condition for a class abst... |
| abtp 42905 | Condition for a class abst... |
| ralopabb 42906 | Restricted universal quant... |
| fpwfvss 42907 | Functions into a powerset ... |
| sdomne0 42908 | A class that strictly domi... |
| sdomne0d 42909 | A class that strictly domi... |
| safesnsupfiss 42910 | If ` B ` is a finite subse... |
| safesnsupfiub 42911 | If ` B ` is a finite subse... |
| safesnsupfidom1o 42912 | If ` B ` is a finite subse... |
| safesnsupfilb 42913 | If ` B ` is a finite subse... |
| isoeq145d 42914 | Equality deduction for iso... |
| resisoeq45d 42915 | Equality deduction for equ... |
| negslem1 42916 | An equivalence between ide... |
| nvocnvb 42917 | Equivalence to saying the ... |
| rp-brsslt 42918 | Binary relation form of a ... |
| nla0002 42919 | Extending a linear order t... |
| nla0003 42920 | Extending a linear order t... |
| nla0001 42921 | Extending a linear order t... |
| faosnf0.11b 42922 | ` B ` is called a non-limi... |
| dfno2 42923 | A surreal number, in the f... |
| onnog 42924 | Every ordinal maps to a su... |
| onnobdayg 42925 | Every ordinal maps to a su... |
| bdaybndex 42926 | Bounds formed from the bir... |
| bdaybndbday 42927 | Bounds formed from the bir... |
| onno 42928 | Every ordinal maps to a su... |
| onnoi 42929 | Every ordinal maps to a su... |
| 0no 42930 | Ordinal zero maps to a sur... |
| 1no 42931 | Ordinal one maps to a surr... |
| 2no 42932 | Ordinal two maps to a surr... |
| 3no 42933 | Ordinal three maps to a su... |
| 4no 42934 | Ordinal four maps to a sur... |
| fnimafnex 42935 | The functional image of a ... |
| nlimsuc 42936 | A successor is not a limit... |
| nlim1NEW 42937 | 1 is not a limit ordinal. ... |
| nlim2NEW 42938 | 2 is not a limit ordinal. ... |
| nlim3 42939 | 3 is not a limit ordinal. ... |
| nlim4 42940 | 4 is not a limit ordinal. ... |
| oa1un 42941 | Given ` A e. On ` , let ` ... |
| oa1cl 42942 | ` A +o 1o ` is in ` On ` .... |
| 0finon 42943 | 0 is a finite ordinal. Se... |
| 1finon 42944 | 1 is a finite ordinal. Se... |
| 2finon 42945 | 2 is a finite ordinal. Se... |
| 3finon 42946 | 3 is a finite ordinal. Se... |
| 4finon 42947 | 4 is a finite ordinal. Se... |
| finona1cl 42948 | The finite ordinals are cl... |
| finonex 42949 | The finite ordinals are a ... |
| fzunt 42950 | Union of two adjacent fini... |
| fzuntd 42951 | Union of two adjacent fini... |
| fzunt1d 42952 | Union of two overlapping f... |
| fzuntgd 42953 | Union of two adjacent or o... |
| ifpan123g 42954 | Conjunction of conditional... |
| ifpan23 42955 | Conjunction of conditional... |
| ifpdfor2 42956 | Define or in terms of cond... |
| ifporcor 42957 | Corollary of commutation o... |
| ifpdfan2 42958 | Define and with conditiona... |
| ifpancor 42959 | Corollary of commutation o... |
| ifpdfor 42960 | Define or in terms of cond... |
| ifpdfan 42961 | Define and with conditiona... |
| ifpbi2 42962 | Equivalence theorem for co... |
| ifpbi3 42963 | Equivalence theorem for co... |
| ifpim1 42964 | Restate implication as con... |
| ifpnot 42965 | Restate negated wff as con... |
| ifpid2 42966 | Restate wff as conditional... |
| ifpim2 42967 | Restate implication as con... |
| ifpbi23 42968 | Equivalence theorem for co... |
| ifpbiidcor 42969 | Restatement of ~ biid . (... |
| ifpbicor 42970 | Corollary of commutation o... |
| ifpxorcor 42971 | Corollary of commutation o... |
| ifpbi1 42972 | Equivalence theorem for co... |
| ifpnot23 42973 | Negation of conditional lo... |
| ifpnotnotb 42974 | Factor conditional logic o... |
| ifpnorcor 42975 | Corollary of commutation o... |
| ifpnancor 42976 | Corollary of commutation o... |
| ifpnot23b 42977 | Negation of conditional lo... |
| ifpbiidcor2 42978 | Restatement of ~ biid . (... |
| ifpnot23c 42979 | Negation of conditional lo... |
| ifpnot23d 42980 | Negation of conditional lo... |
| ifpdfnan 42981 | Define nand as conditional... |
| ifpdfxor 42982 | Define xor as conditional ... |
| ifpbi12 42983 | Equivalence theorem for co... |
| ifpbi13 42984 | Equivalence theorem for co... |
| ifpbi123 42985 | Equivalence theorem for co... |
| ifpidg 42986 | Restate wff as conditional... |
| ifpid3g 42987 | Restate wff as conditional... |
| ifpid2g 42988 | Restate wff as conditional... |
| ifpid1g 42989 | Restate wff as conditional... |
| ifpim23g 42990 | Restate implication as con... |
| ifpim3 42991 | Restate implication as con... |
| ifpnim1 42992 | Restate negated implicatio... |
| ifpim4 42993 | Restate implication as con... |
| ifpnim2 42994 | Restate negated implicatio... |
| ifpim123g 42995 | Implication of conditional... |
| ifpim1g 42996 | Implication of conditional... |
| ifp1bi 42997 | Substitute the first eleme... |
| ifpbi1b 42998 | When the first variable is... |
| ifpimimb 42999 | Factor conditional logic o... |
| ifpororb 43000 | Factor conditional logic o... |
| ifpananb 43001 | Factor conditional logic o... |
| ifpnannanb 43002 | Factor conditional logic o... |
| ifpor123g 43003 | Disjunction of conditional... |
| ifpimim 43004 | Consequnce of implication.... |
| ifpbibib 43005 | Factor conditional logic o... |
| ifpxorxorb 43006 | Factor conditional logic o... |
| rp-fakeimass 43007 | A special case where impli... |
| rp-fakeanorass 43008 | A special case where a mix... |
| rp-fakeoranass 43009 | A special case where a mix... |
| rp-fakeinunass 43010 | A special case where a mix... |
| rp-fakeuninass 43011 | A special case where a mix... |
| rp-isfinite5 43012 | A set is said to be finite... |
| rp-isfinite6 43013 | A set is said to be finite... |
| intabssd 43014 | When for each element ` y ... |
| eu0 43015 | There is only one empty se... |
| epelon2 43016 | Over the ordinal numbers, ... |
| ontric3g 43017 | For all ` x , y e. On ` , ... |
| dfsucon 43018 | ` A ` is called a successo... |
| snen1g 43019 | A singleton is equinumerou... |
| snen1el 43020 | A singleton is equinumerou... |
| sn1dom 43021 | A singleton is dominated b... |
| pr2dom 43022 | An unordered pair is domin... |
| tr3dom 43023 | An unordered triple is dom... |
| ensucne0 43024 | A class equinumerous to a ... |
| ensucne0OLD 43025 | A class equinumerous to a ... |
| dfom6 43026 | Let ` _om ` be defined to ... |
| infordmin 43027 | ` _om ` is the smallest in... |
| iscard4 43028 | Two ways to express the pr... |
| minregex 43029 | Given any cardinal number ... |
| minregex2 43030 | Given any cardinal number ... |
| iscard5 43031 | Two ways to express the pr... |
| elrncard 43032 | Let us define a cardinal n... |
| harval3 43033 | ` ( har `` A ) ` is the le... |
| harval3on 43034 | For any ordinal number ` A... |
| omssrncard 43035 | All natural numbers are ca... |
| 0iscard 43036 | 0 is a cardinal number. (... |
| 1iscard 43037 | 1 is a cardinal number. (... |
| omiscard 43038 | ` _om ` is a cardinal numb... |
| sucomisnotcard 43039 | ` _om +o 1o ` is not a car... |
| nna1iscard 43040 | For any natural number, th... |
| har2o 43041 | The least cardinal greater... |
| en2pr 43042 | A class is equinumerous to... |
| pr2cv 43043 | If an unordered pair is eq... |
| pr2el1 43044 | If an unordered pair is eq... |
| pr2cv1 43045 | If an unordered pair is eq... |
| pr2el2 43046 | If an unordered pair is eq... |
| pr2cv2 43047 | If an unordered pair is eq... |
| pren2 43048 | An unordered pair is equin... |
| pr2eldif1 43049 | If an unordered pair is eq... |
| pr2eldif2 43050 | If an unordered pair is eq... |
| pren2d 43051 | A pair of two distinct set... |
| aleph1min 43052 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 43053 | ` aleph ` is a strictly or... |
| alephiso3 43054 | ` aleph ` is a strictly or... |
| pwelg 43055 | The powerclass is an eleme... |
| pwinfig 43056 | The powerclass of an infin... |
| pwinfi2 43057 | The powerclass of an infin... |
| pwinfi3 43058 | The powerclass of an infin... |
| pwinfi 43059 | The powerclass of an infin... |
| fipjust 43060 | A definition of the finite... |
| cllem0 43061 | The class of all sets with... |
| superficl 43062 | The class of all supersets... |
| superuncl 43063 | The class of all supersets... |
| ssficl 43064 | The class of all subsets o... |
| ssuncl 43065 | The class of all subsets o... |
| ssdifcl 43066 | The class of all subsets o... |
| sssymdifcl 43067 | The class of all subsets o... |
| fiinfi 43068 | If two classes have the fi... |
| rababg 43069 | Condition when restricted ... |
| elinintab 43070 | Two ways of saying a set i... |
| elmapintrab 43071 | Two ways to say a set is a... |
| elinintrab 43072 | Two ways of saying a set i... |
| inintabss 43073 | Upper bound on intersectio... |
| inintabd 43074 | Value of the intersection ... |
| xpinintabd 43075 | Value of the intersection ... |
| relintabex 43076 | If the intersection of a c... |
| elcnvcnvintab 43077 | Two ways of saying a set i... |
| relintab 43078 | Value of the intersection ... |
| nonrel 43079 | A non-relation is equal to... |
| elnonrel 43080 | Only an ordered pair where... |
| cnvssb 43081 | Subclass theorem for conve... |
| relnonrel 43082 | The non-relation part of a... |
| cnvnonrel 43083 | The converse of the non-re... |
| brnonrel 43084 | A non-relation cannot rela... |
| dmnonrel 43085 | The domain of the non-rela... |
| rnnonrel 43086 | The range of the non-relat... |
| resnonrel 43087 | A restriction of the non-r... |
| imanonrel 43088 | An image under the non-rel... |
| cononrel1 43089 | Composition with the non-r... |
| cononrel2 43090 | Composition with the non-r... |
| elmapintab 43091 | Two ways to say a set is a... |
| fvnonrel 43092 | The function value of any ... |
| elinlem 43093 | Two ways to say a set is a... |
| elcnvcnvlem 43094 | Two ways to say a set is a... |
| cnvcnvintabd 43095 | Value of the relationship ... |
| elcnvlem 43096 | Two ways to say a set is a... |
| elcnvintab 43097 | Two ways of saying a set i... |
| cnvintabd 43098 | Value of the converse of t... |
| undmrnresiss 43099 | Two ways of saying the ide... |
| reflexg 43100 | Two ways of saying a relat... |
| cnvssco 43101 | A condition weaker than re... |
| refimssco 43102 | Reflexive relations are su... |
| cleq2lem 43103 | Equality implies bijection... |
| cbvcllem 43104 | Change of bound variable i... |
| clublem 43105 | If a superset ` Y ` of ` X... |
| clss2lem 43106 | The closure of a property ... |
| dfid7 43107 | Definition of identity rel... |
| mptrcllem 43108 | Show two versions of a clo... |
| cotrintab 43109 | The intersection of a clas... |
| rclexi 43110 | The reflexive closure of a... |
| rtrclexlem 43111 | Existence of relation impl... |
| rtrclex 43112 | The reflexive-transitive c... |
| trclubgNEW 43113 | If a relation exists then ... |
| trclubNEW 43114 | If a relation exists then ... |
| trclexi 43115 | The transitive closure of ... |
| rtrclexi 43116 | The reflexive-transitive c... |
| clrellem 43117 | When the property ` ps ` h... |
| clcnvlem 43118 | When ` A ` , an upper boun... |
| cnvtrucl0 43119 | The converse of the trivia... |
| cnvrcl0 43120 | The converse of the reflex... |
| cnvtrcl0 43121 | The converse of the transi... |
| dmtrcl 43122 | The domain of the transiti... |
| rntrcl 43123 | The range of the transitiv... |
| dfrtrcl5 43124 | Definition of reflexive-tr... |
| trcleq2lemRP 43125 | Equality implies bijection... |
| sqrtcvallem1 43126 | Two ways of saying a compl... |
| reabsifneg 43127 | Alternate expression for t... |
| reabsifnpos 43128 | Alternate expression for t... |
| reabsifpos 43129 | Alternate expression for t... |
| reabsifnneg 43130 | Alternate expression for t... |
| reabssgn 43131 | Alternate expression for t... |
| sqrtcvallem2 43132 | Equivalent to saying that ... |
| sqrtcvallem3 43133 | Equivalent to saying that ... |
| sqrtcvallem4 43134 | Equivalent to saying that ... |
| sqrtcvallem5 43135 | Equivalent to saying that ... |
| sqrtcval 43136 | Explicit formula for the c... |
| sqrtcval2 43137 | Explicit formula for the c... |
| resqrtval 43138 | Real part of the complex s... |
| imsqrtval 43139 | Imaginary part of the comp... |
| resqrtvalex 43140 | Example for ~ resqrtval . ... |
| imsqrtvalex 43141 | Example for ~ imsqrtval . ... |
| al3im 43142 | Version of ~ ax-4 for a ne... |
| intima0 43143 | Two ways of expressing the... |
| elimaint 43144 | Element of image of inters... |
| cnviun 43145 | Converse of indexed union.... |
| imaiun1 43146 | The image of an indexed un... |
| coiun1 43147 | Composition with an indexe... |
| elintima 43148 | Element of intersection of... |
| intimass 43149 | The image under the inters... |
| intimass2 43150 | The image under the inters... |
| intimag 43151 | Requirement for the image ... |
| intimasn 43152 | Two ways to express the im... |
| intimasn2 43153 | Two ways to express the im... |
| ss2iundf 43154 | Subclass theorem for index... |
| ss2iundv 43155 | Subclass theorem for index... |
| cbviuneq12df 43156 | Rule used to change the bo... |
| cbviuneq12dv 43157 | Rule used to change the bo... |
| conrel1d 43158 | Deduction about compositio... |
| conrel2d 43159 | Deduction about compositio... |
| trrelind 43160 | The intersection of transi... |
| xpintrreld 43161 | The intersection of a tran... |
| restrreld 43162 | The restriction of a trans... |
| trrelsuperreldg 43163 | Concrete construction of a... |
| trficl 43164 | The class of all transitiv... |
| cnvtrrel 43165 | The converse of a transiti... |
| trrelsuperrel2dg 43166 | Concrete construction of a... |
| dfrcl2 43169 | Reflexive closure of a rel... |
| dfrcl3 43170 | Reflexive closure of a rel... |
| dfrcl4 43171 | Reflexive closure of a rel... |
| relexp2 43172 | A set operated on by the r... |
| relexpnul 43173 | If the domain and range of... |
| eliunov2 43174 | Membership in the indexed ... |
| eltrclrec 43175 | Membership in the indexed ... |
| elrtrclrec 43176 | Membership in the indexed ... |
| briunov2 43177 | Two classes related by the... |
| brmptiunrelexpd 43178 | If two elements are connec... |
| fvmptiunrelexplb0d 43179 | If the indexed union range... |
| fvmptiunrelexplb0da 43180 | If the indexed union range... |
| fvmptiunrelexplb1d 43181 | If the indexed union range... |
| brfvid 43182 | If two elements are connec... |
| brfvidRP 43183 | If two elements are connec... |
| fvilbd 43184 | A set is a subset of its i... |
| fvilbdRP 43185 | A set is a subset of its i... |
| brfvrcld 43186 | If two elements are connec... |
| brfvrcld2 43187 | If two elements are connec... |
| fvrcllb0d 43188 | A restriction of the ident... |
| fvrcllb0da 43189 | A restriction of the ident... |
| fvrcllb1d 43190 | A set is a subset of its i... |
| brtrclrec 43191 | Two classes related by the... |
| brrtrclrec 43192 | Two classes related by the... |
| briunov2uz 43193 | Two classes related by the... |
| eliunov2uz 43194 | Membership in the indexed ... |
| ov2ssiunov2 43195 | Any particular operator va... |
| relexp0eq 43196 | The zeroth power of relati... |
| iunrelexp0 43197 | Simplification of zeroth p... |
| relexpxpnnidm 43198 | Any positive power of a Ca... |
| relexpiidm 43199 | Any power of any restricti... |
| relexpss1d 43200 | The relational power of a ... |
| comptiunov2i 43201 | The composition two indexe... |
| corclrcl 43202 | The reflexive closure is i... |
| iunrelexpmin1 43203 | The indexed union of relat... |
| relexpmulnn 43204 | With exponents limited to ... |
| relexpmulg 43205 | With ordered exponents, th... |
| trclrelexplem 43206 | The union of relational po... |
| iunrelexpmin2 43207 | The indexed union of relat... |
| relexp01min 43208 | With exponents limited to ... |
| relexp1idm 43209 | Repeated raising a relatio... |
| relexp0idm 43210 | Repeated raising a relatio... |
| relexp0a 43211 | Absorption law for zeroth ... |
| relexpxpmin 43212 | The composition of powers ... |
| relexpaddss 43213 | The composition of two pow... |
| iunrelexpuztr 43214 | The indexed union of relat... |
| dftrcl3 43215 | Transitive closure of a re... |
| brfvtrcld 43216 | If two elements are connec... |
| fvtrcllb1d 43217 | A set is a subset of its i... |
| trclfvcom 43218 | The transitive closure of ... |
| cnvtrclfv 43219 | The converse of the transi... |
| cotrcltrcl 43220 | The transitive closure is ... |
| trclimalb2 43221 | Lower bound for image unde... |
| brtrclfv2 43222 | Two ways to indicate two e... |
| trclfvdecomr 43223 | The transitive closure of ... |
| trclfvdecoml 43224 | The transitive closure of ... |
| dmtrclfvRP 43225 | The domain of the transiti... |
| rntrclfvRP 43226 | The range of the transitiv... |
| rntrclfv 43227 | The range of the transitiv... |
| dfrtrcl3 43228 | Reflexive-transitive closu... |
| brfvrtrcld 43229 | If two elements are connec... |
| fvrtrcllb0d 43230 | A restriction of the ident... |
| fvrtrcllb0da 43231 | A restriction of the ident... |
| fvrtrcllb1d 43232 | A set is a subset of its i... |
| dfrtrcl4 43233 | Reflexive-transitive closu... |
| corcltrcl 43234 | The composition of the ref... |
| cortrcltrcl 43235 | Composition with the refle... |
| corclrtrcl 43236 | Composition with the refle... |
| cotrclrcl 43237 | The composition of the ref... |
| cortrclrcl 43238 | Composition with the refle... |
| cotrclrtrcl 43239 | Composition with the refle... |
| cortrclrtrcl 43240 | The reflexive-transitive c... |
| frege77d 43241 | If the images of both ` { ... |
| frege81d 43242 | If the image of ` U ` is a... |
| frege83d 43243 | If the image of the union ... |
| frege96d 43244 | If ` C ` follows ` A ` in ... |
| frege87d 43245 | If the images of both ` { ... |
| frege91d 43246 | If ` B ` follows ` A ` in ... |
| frege97d 43247 | If ` A ` contains all elem... |
| frege98d 43248 | If ` C ` follows ` A ` and... |
| frege102d 43249 | If either ` A ` and ` C ` ... |
| frege106d 43250 | If ` B ` follows ` A ` in ... |
| frege108d 43251 | If either ` A ` and ` C ` ... |
| frege109d 43252 | If ` A ` contains all elem... |
| frege114d 43253 | If either ` R ` relates ` ... |
| frege111d 43254 | If either ` A ` and ` C ` ... |
| frege122d 43255 | If ` F ` is a function, ` ... |
| frege124d 43256 | If ` F ` is a function, ` ... |
| frege126d 43257 | If ` F ` is a function, ` ... |
| frege129d 43258 | If ` F ` is a function and... |
| frege131d 43259 | If ` F ` is a function and... |
| frege133d 43260 | If ` F ` is a function and... |
| dfxor4 43261 | Express exclusive-or in te... |
| dfxor5 43262 | Express exclusive-or in te... |
| df3or2 43263 | Express triple-or in terms... |
| df3an2 43264 | Express triple-and in term... |
| nev 43265 | Express that not every set... |
| 0pssin 43266 | Express that an intersecti... |
| dfhe2 43269 | The property of relation `... |
| dfhe3 43270 | The property of relation `... |
| heeq12 43271 | Equality law for relations... |
| heeq1 43272 | Equality law for relations... |
| heeq2 43273 | Equality law for relations... |
| sbcheg 43274 | Distribute proper substitu... |
| hess 43275 | Subclass law for relations... |
| xphe 43276 | Any Cartesian product is h... |
| 0he 43277 | The empty relation is here... |
| 0heALT 43278 | The empty relation is here... |
| he0 43279 | Any relation is hereditary... |
| unhe1 43280 | The union of two relations... |
| snhesn 43281 | Any singleton is hereditar... |
| idhe 43282 | The identity relation is h... |
| psshepw 43283 | The relation between sets ... |
| sshepw 43284 | The relation between sets ... |
| rp-simp2-frege 43287 | Simplification of triple c... |
| rp-simp2 43288 | Simplification of triple c... |
| rp-frege3g 43289 | Add antecedent to ~ ax-fre... |
| frege3 43290 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 43291 | Double-use of ~ ax-frege2 ... |
| rp-frege24 43292 | Introducing an embedded an... |
| rp-frege4g 43293 | Deduction related to distr... |
| frege4 43294 | Special case of closed for... |
| frege5 43295 | A closed form of ~ syl . ... |
| rp-7frege 43296 | Distribute antecedent and ... |
| rp-4frege 43297 | Elimination of a nested an... |
| rp-6frege 43298 | Elimination of a nested an... |
| rp-8frege 43299 | Eliminate antecedent when ... |
| rp-frege25 43300 | Closed form for ~ a1dd . ... |
| frege6 43301 | A closed form of ~ imim2d ... |
| axfrege8 43302 | Swap antecedents. Identic... |
| frege7 43303 | A closed form of ~ syl6 . ... |
| frege26 43305 | Identical to ~ idd . Prop... |
| frege27 43306 | We cannot (at the same tim... |
| frege9 43307 | Closed form of ~ syl with ... |
| frege12 43308 | A closed form of ~ com23 .... |
| frege11 43309 | Elimination of a nested an... |
| frege24 43310 | Closed form for ~ a1d . D... |
| frege16 43311 | A closed form of ~ com34 .... |
| frege25 43312 | Closed form for ~ a1dd . ... |
| frege18 43313 | Closed form of a syllogism... |
| frege22 43314 | A closed form of ~ com45 .... |
| frege10 43315 | Result commuting anteceden... |
| frege17 43316 | A closed form of ~ com3l .... |
| frege13 43317 | A closed form of ~ com3r .... |
| frege14 43318 | Closed form of a deduction... |
| frege19 43319 | A closed form of ~ syl6 . ... |
| frege23 43320 | Syllogism followed by rota... |
| frege15 43321 | A closed form of ~ com4r .... |
| frege21 43322 | Replace antecedent in ante... |
| frege20 43323 | A closed form of ~ syl8 . ... |
| axfrege28 43324 | Contraposition. Identical... |
| frege29 43326 | Closed form of ~ con3d . ... |
| frege30 43327 | Commuted, closed form of ~... |
| axfrege31 43328 | Identical to ~ notnotr . ... |
| frege32 43330 | Deduce ~ con1 from ~ con3 ... |
| frege33 43331 | If ` ph ` or ` ps ` takes ... |
| frege34 43332 | If as a consequence of the... |
| frege35 43333 | Commuted, closed form of ~... |
| frege36 43334 | The case in which ` ps ` i... |
| frege37 43335 | If ` ch ` is a necessary c... |
| frege38 43336 | Identical to ~ pm2.21 . P... |
| frege39 43337 | Syllogism between ~ pm2.18... |
| frege40 43338 | Anything implies ~ pm2.18 ... |
| axfrege41 43339 | Identical to ~ notnot . A... |
| frege42 43341 | Not not ~ id . Propositio... |
| frege43 43342 | If there is a choice only ... |
| frege44 43343 | Similar to a commuted ~ pm... |
| frege45 43344 | Deduce ~ pm2.6 from ~ con1... |
| frege46 43345 | If ` ps ` holds when ` ph ... |
| frege47 43346 | Deduce consequence follows... |
| frege48 43347 | Closed form of syllogism w... |
| frege49 43348 | Closed form of deduction w... |
| frege50 43349 | Closed form of ~ jaoi . P... |
| frege51 43350 | Compare with ~ jaod . Pro... |
| axfrege52a 43351 | Justification for ~ ax-fre... |
| frege52aid 43353 | The case when the content ... |
| frege53aid 43354 | Specialization of ~ frege5... |
| frege53a 43355 | Lemma for ~ frege55a . Pr... |
| axfrege54a 43356 | Justification for ~ ax-fre... |
| frege54cor0a 43358 | Synonym for logical equiva... |
| frege54cor1a 43359 | Reflexive equality. (Cont... |
| frege55aid 43360 | Lemma for ~ frege57aid . ... |
| frege55lem1a 43361 | Necessary deduction regard... |
| frege55lem2a 43362 | Core proof of Proposition ... |
| frege55a 43363 | Proposition 55 of [Frege18... |
| frege55cor1a 43364 | Proposition 55 of [Frege18... |
| frege56aid 43365 | Lemma for ~ frege57aid . ... |
| frege56a 43366 | Proposition 56 of [Frege18... |
| frege57aid 43367 | This is the all imporant f... |
| frege57a 43368 | Analogue of ~ frege57aid .... |
| axfrege58a 43369 | Identical to ~ anifp . Ju... |
| frege58acor 43371 | Lemma for ~ frege59a . (C... |
| frege59a 43372 | A kind of Aristotelian inf... |
| frege60a 43373 | Swap antecedents of ~ ax-f... |
| frege61a 43374 | Lemma for ~ frege65a . Pr... |
| frege62a 43375 | A kind of Aristotelian inf... |
| frege63a 43376 | Proposition 63 of [Frege18... |
| frege64a 43377 | Lemma for ~ frege65a . Pr... |
| frege65a 43378 | A kind of Aristotelian inf... |
| frege66a 43379 | Swap antecedents of ~ freg... |
| frege67a 43380 | Lemma for ~ frege68a . Pr... |
| frege68a 43381 | Combination of applying a ... |
| axfrege52c 43382 | Justification for ~ ax-fre... |
| frege52b 43384 | The case when the content ... |
| frege53b 43385 | Lemma for frege102 (via ~ ... |
| axfrege54c 43386 | Reflexive equality of clas... |
| frege54b 43388 | Reflexive equality of sets... |
| frege54cor1b 43389 | Reflexive equality. (Cont... |
| frege55lem1b 43390 | Necessary deduction regard... |
| frege55lem2b 43391 | Lemma for ~ frege55b . Co... |
| frege55b 43392 | Lemma for ~ frege57b . Pr... |
| frege56b 43393 | Lemma for ~ frege57b . Pr... |
| frege57b 43394 | Analogue of ~ frege57aid .... |
| axfrege58b 43395 | If ` A. x ph ` is affirmed... |
| frege58bid 43397 | If ` A. x ph ` is affirmed... |
| frege58bcor 43398 | Lemma for ~ frege59b . (C... |
| frege59b 43399 | A kind of Aristotelian inf... |
| frege60b 43400 | Swap antecedents of ~ ax-f... |
| frege61b 43401 | Lemma for ~ frege65b . Pr... |
| frege62b 43402 | A kind of Aristotelian inf... |
| frege63b 43403 | Lemma for ~ frege91 . Pro... |
| frege64b 43404 | Lemma for ~ frege65b . Pr... |
| frege65b 43405 | A kind of Aristotelian inf... |
| frege66b 43406 | Swap antecedents of ~ freg... |
| frege67b 43407 | Lemma for ~ frege68b . Pr... |
| frege68b 43408 | Combination of applying a ... |
| frege53c 43409 | Proposition 53 of [Frege18... |
| frege54cor1c 43410 | Reflexive equality. (Cont... |
| frege55lem1c 43411 | Necessary deduction regard... |
| frege55lem2c 43412 | Core proof of Proposition ... |
| frege55c 43413 | Proposition 55 of [Frege18... |
| frege56c 43414 | Lemma for ~ frege57c . Pr... |
| frege57c 43415 | Swap order of implication ... |
| frege58c 43416 | Principle related to ~ sp ... |
| frege59c 43417 | A kind of Aristotelian inf... |
| frege60c 43418 | Swap antecedents of ~ freg... |
| frege61c 43419 | Lemma for ~ frege65c . Pr... |
| frege62c 43420 | A kind of Aristotelian inf... |
| frege63c 43421 | Analogue of ~ frege63b . ... |
| frege64c 43422 | Lemma for ~ frege65c . Pr... |
| frege65c 43423 | A kind of Aristotelian inf... |
| frege66c 43424 | Swap antecedents of ~ freg... |
| frege67c 43425 | Lemma for ~ frege68c . Pr... |
| frege68c 43426 | Combination of applying a ... |
| dffrege69 43427 | If from the proposition th... |
| frege70 43428 | Lemma for ~ frege72 . Pro... |
| frege71 43429 | Lemma for ~ frege72 . Pro... |
| frege72 43430 | If property ` A ` is hered... |
| frege73 43431 | Lemma for ~ frege87 . Pro... |
| frege74 43432 | If ` X ` has a property ` ... |
| frege75 43433 | If from the proposition th... |
| dffrege76 43434 | If from the two propositio... |
| frege77 43435 | If ` Y ` follows ` X ` in ... |
| frege78 43436 | Commuted form of ~ frege77... |
| frege79 43437 | Distributed form of ~ freg... |
| frege80 43438 | Add additional condition t... |
| frege81 43439 | If ` X ` has a property ` ... |
| frege82 43440 | Closed-form deduction base... |
| frege83 43441 | Apply commuted form of ~ f... |
| frege84 43442 | Commuted form of ~ frege81... |
| frege85 43443 | Commuted form of ~ frege77... |
| frege86 43444 | Conclusion about element o... |
| frege87 43445 | If ` Z ` is a result of an... |
| frege88 43446 | Commuted form of ~ frege87... |
| frege89 43447 | One direction of ~ dffrege... |
| frege90 43448 | Add antecedent to ~ frege8... |
| frege91 43449 | Every result of an applica... |
| frege92 43450 | Inference from ~ frege91 .... |
| frege93 43451 | Necessary condition for tw... |
| frege94 43452 | Looking one past a pair re... |
| frege95 43453 | Looking one past a pair re... |
| frege96 43454 | Every result of an applica... |
| frege97 43455 | The property of following ... |
| frege98 43456 | If ` Y ` follows ` X ` and... |
| dffrege99 43457 | If ` Z ` is identical with... |
| frege100 43458 | One direction of ~ dffrege... |
| frege101 43459 | Lemma for ~ frege102 . Pr... |
| frege102 43460 | If ` Z ` belongs to the ` ... |
| frege103 43461 | Proposition 103 of [Frege1... |
| frege104 43462 | Proposition 104 of [Frege1... |
| frege105 43463 | Proposition 105 of [Frege1... |
| frege106 43464 | Whatever follows ` X ` in ... |
| frege107 43465 | Proposition 107 of [Frege1... |
| frege108 43466 | If ` Y ` belongs to the ` ... |
| frege109 43467 | The property of belonging ... |
| frege110 43468 | Proposition 110 of [Frege1... |
| frege111 43469 | If ` Y ` belongs to the ` ... |
| frege112 43470 | Identity implies belonging... |
| frege113 43471 | Proposition 113 of [Frege1... |
| frege114 43472 | If ` X ` belongs to the ` ... |
| dffrege115 43473 | If from the circumstance t... |
| frege116 43474 | One direction of ~ dffrege... |
| frege117 43475 | Lemma for ~ frege118 . Pr... |
| frege118 43476 | Simplified application of ... |
| frege119 43477 | Lemma for ~ frege120 . Pr... |
| frege120 43478 | Simplified application of ... |
| frege121 43479 | Lemma for ~ frege122 . Pr... |
| frege122 43480 | If ` X ` is a result of an... |
| frege123 43481 | Lemma for ~ frege124 . Pr... |
| frege124 43482 | If ` X ` is a result of an... |
| frege125 43483 | Lemma for ~ frege126 . Pr... |
| frege126 43484 | If ` M ` follows ` Y ` in ... |
| frege127 43485 | Communte antecedents of ~ ... |
| frege128 43486 | Lemma for ~ frege129 . Pr... |
| frege129 43487 | If the procedure ` R ` is ... |
| frege130 43488 | Lemma for ~ frege131 . Pr... |
| frege131 43489 | If the procedure ` R ` is ... |
| frege132 43490 | Lemma for ~ frege133 . Pr... |
| frege133 43491 | If the procedure ` R ` is ... |
| enrelmap 43492 | The set of all possible re... |
| enrelmapr 43493 | The set of all possible re... |
| enmappw 43494 | The set of all mappings fr... |
| enmappwid 43495 | The set of all mappings fr... |
| rfovd 43496 | Value of the operator, ` (... |
| rfovfvd 43497 | Value of the operator, ` (... |
| rfovfvfvd 43498 | Value of the operator, ` (... |
| rfovcnvf1od 43499 | Properties of the operator... |
| rfovcnvd 43500 | Value of the converse of t... |
| rfovf1od 43501 | The value of the operator,... |
| rfovcnvfvd 43502 | Value of the converse of t... |
| fsovd 43503 | Value of the operator, ` (... |
| fsovrfovd 43504 | The operator which gives a... |
| fsovfvd 43505 | Value of the operator, ` (... |
| fsovfvfvd 43506 | Value of the operator, ` (... |
| fsovfd 43507 | The operator, ` ( A O B ) ... |
| fsovcnvlem 43508 | The ` O ` operator, which ... |
| fsovcnvd 43509 | The value of the converse ... |
| fsovcnvfvd 43510 | The value of the converse ... |
| fsovf1od 43511 | The value of ` ( A O B ) `... |
| dssmapfvd 43512 | Value of the duality opera... |
| dssmapfv2d 43513 | Value of the duality opera... |
| dssmapfv3d 43514 | Value of the duality opera... |
| dssmapnvod 43515 | For any base set ` B ` the... |
| dssmapf1od 43516 | For any base set ` B ` the... |
| dssmap2d 43517 | For any base set ` B ` the... |
| or3or 43518 | Decompose disjunction into... |
| andi3or 43519 | Distribute over triple dis... |
| uneqsn 43520 | If a union of classes is e... |
| brfvimex 43521 | If a binary relation holds... |
| brovmptimex 43522 | If a binary relation holds... |
| brovmptimex1 43523 | If a binary relation holds... |
| brovmptimex2 43524 | If a binary relation holds... |
| brcoffn 43525 | Conditions allowing the de... |
| brcofffn 43526 | Conditions allowing the de... |
| brco2f1o 43527 | Conditions allowing the de... |
| brco3f1o 43528 | Conditions allowing the de... |
| ntrclsbex 43529 | If (pseudo-)interior and (... |
| ntrclsrcomplex 43530 | The relative complement of... |
| neik0imk0p 43531 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 43532 | If an interior function is... |
| ntrkbimka 43533 | If the interiors of disjoi... |
| ntrk0kbimka 43534 | If the interiors of disjoi... |
| clsk3nimkb 43535 | If the base set is not emp... |
| clsk1indlem0 43536 | The ansatz closure functio... |
| clsk1indlem2 43537 | The ansatz closure functio... |
| clsk1indlem3 43538 | The ansatz closure functio... |
| clsk1indlem4 43539 | The ansatz closure functio... |
| clsk1indlem1 43540 | The ansatz closure functio... |
| clsk1independent 43541 | For generalized closure fu... |
| neik0pk1imk0 43542 | Kuratowski's K0' and K1 ax... |
| isotone1 43543 | Two different ways to say ... |
| isotone2 43544 | Two different ways to say ... |
| ntrk1k3eqk13 43545 | An interior function is bo... |
| ntrclsf1o 43546 | If (pseudo-)interior and (... |
| ntrclsnvobr 43547 | If (pseudo-)interior and (... |
| ntrclsiex 43548 | If (pseudo-)interior and (... |
| ntrclskex 43549 | If (pseudo-)interior and (... |
| ntrclsfv1 43550 | If (pseudo-)interior and (... |
| ntrclsfv2 43551 | If (pseudo-)interior and (... |
| ntrclselnel1 43552 | If (pseudo-)interior and (... |
| ntrclselnel2 43553 | If (pseudo-)interior and (... |
| ntrclsfv 43554 | The value of the interior ... |
| ntrclsfveq1 43555 | If interior and closure fu... |
| ntrclsfveq2 43556 | If interior and closure fu... |
| ntrclsfveq 43557 | If interior and closure fu... |
| ntrclsss 43558 | If interior and closure fu... |
| ntrclsneine0lem 43559 | If (pseudo-)interior and (... |
| ntrclsneine0 43560 | If (pseudo-)interior and (... |
| ntrclscls00 43561 | If (pseudo-)interior and (... |
| ntrclsiso 43562 | If (pseudo-)interior and (... |
| ntrclsk2 43563 | An interior function is co... |
| ntrclskb 43564 | The interiors of disjoint ... |
| ntrclsk3 43565 | The intersection of interi... |
| ntrclsk13 43566 | The interior of the inters... |
| ntrclsk4 43567 | Idempotence of the interio... |
| ntrneibex 43568 | If (pseudo-)interior and (... |
| ntrneircomplex 43569 | The relative complement of... |
| ntrneif1o 43570 | If (pseudo-)interior and (... |
| ntrneiiex 43571 | If (pseudo-)interior and (... |
| ntrneinex 43572 | If (pseudo-)interior and (... |
| ntrneicnv 43573 | If (pseudo-)interior and (... |
| ntrneifv1 43574 | If (pseudo-)interior and (... |
| ntrneifv2 43575 | If (pseudo-)interior and (... |
| ntrneiel 43576 | If (pseudo-)interior and (... |
| ntrneifv3 43577 | The value of the neighbors... |
| ntrneineine0lem 43578 | If (pseudo-)interior and (... |
| ntrneineine1lem 43579 | If (pseudo-)interior and (... |
| ntrneifv4 43580 | The value of the interior ... |
| ntrneiel2 43581 | Membership in iterated int... |
| ntrneineine0 43582 | If (pseudo-)interior and (... |
| ntrneineine1 43583 | If (pseudo-)interior and (... |
| ntrneicls00 43584 | If (pseudo-)interior and (... |
| ntrneicls11 43585 | If (pseudo-)interior and (... |
| ntrneiiso 43586 | If (pseudo-)interior and (... |
| ntrneik2 43587 | An interior function is co... |
| ntrneix2 43588 | An interior (closure) func... |
| ntrneikb 43589 | The interiors of disjoint ... |
| ntrneixb 43590 | The interiors (closures) o... |
| ntrneik3 43591 | The intersection of interi... |
| ntrneix3 43592 | The closure of the union o... |
| ntrneik13 43593 | The interior of the inters... |
| ntrneix13 43594 | The closure of the union o... |
| ntrneik4w 43595 | Idempotence of the interio... |
| ntrneik4 43596 | Idempotence of the interio... |
| clsneibex 43597 | If (pseudo-)closure and (p... |
| clsneircomplex 43598 | The relative complement of... |
| clsneif1o 43599 | If a (pseudo-)closure func... |
| clsneicnv 43600 | If a (pseudo-)closure func... |
| clsneikex 43601 | If closure and neighborhoo... |
| clsneinex 43602 | If closure and neighborhoo... |
| clsneiel1 43603 | If a (pseudo-)closure func... |
| clsneiel2 43604 | If a (pseudo-)closure func... |
| clsneifv3 43605 | Value of the neighborhoods... |
| clsneifv4 43606 | Value of the closure (inte... |
| neicvgbex 43607 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 43608 | The relative complement of... |
| neicvgf1o 43609 | If neighborhood and conver... |
| neicvgnvo 43610 | If neighborhood and conver... |
| neicvgnvor 43611 | If neighborhood and conver... |
| neicvgmex 43612 | If the neighborhoods and c... |
| neicvgnex 43613 | If the neighborhoods and c... |
| neicvgel1 43614 | A subset being an element ... |
| neicvgel2 43615 | The complement of a subset... |
| neicvgfv 43616 | The value of the neighborh... |
| ntrrn 43617 | The range of the interior ... |
| ntrf 43618 | The interior function of a... |
| ntrf2 43619 | The interior function is a... |
| ntrelmap 43620 | The interior function is a... |
| clsf2 43621 | The closure function is a ... |
| clselmap 43622 | The closure function is a ... |
| dssmapntrcls 43623 | The interior and closure o... |
| dssmapclsntr 43624 | The closure and interior o... |
| gneispa 43625 | Each point ` p ` of the ne... |
| gneispb 43626 | Given a neighborhood ` N `... |
| gneispace2 43627 | The predicate that ` F ` i... |
| gneispace3 43628 | The predicate that ` F ` i... |
| gneispace 43629 | The predicate that ` F ` i... |
| gneispacef 43630 | A generic neighborhood spa... |
| gneispacef2 43631 | A generic neighborhood spa... |
| gneispacefun 43632 | A generic neighborhood spa... |
| gneispacern 43633 | A generic neighborhood spa... |
| gneispacern2 43634 | A generic neighborhood spa... |
| gneispace0nelrn 43635 | A generic neighborhood spa... |
| gneispace0nelrn2 43636 | A generic neighborhood spa... |
| gneispace0nelrn3 43637 | A generic neighborhood spa... |
| gneispaceel 43638 | Every neighborhood of a po... |
| gneispaceel2 43639 | Every neighborhood of a po... |
| gneispacess 43640 | All supersets of a neighbo... |
| gneispacess2 43641 | All supersets of a neighbo... |
| k0004lem1 43642 | Application of ~ ssin to r... |
| k0004lem2 43643 | A mapping with a particula... |
| k0004lem3 43644 | When the value of a mappin... |
| k0004val 43645 | The topological simplex of... |
| k0004ss1 43646 | The topological simplex of... |
| k0004ss2 43647 | The topological simplex of... |
| k0004ss3 43648 | The topological simplex of... |
| k0004val0 43649 | The topological simplex of... |
| inductionexd 43650 | Simple induction example. ... |
| wwlemuld 43651 | Natural deduction form of ... |
| leeq1d 43652 | Specialization of ~ breq1d... |
| leeq2d 43653 | Specialization of ~ breq2d... |
| absmulrposd 43654 | Specialization of absmuld ... |
| imadisjld 43655 | Natural dduction form of o... |
| wnefimgd 43656 | The image of a mapping fro... |
| fco2d 43657 | Natural deduction form of ... |
| wfximgfd 43658 | The value of a function on... |
| extoimad 43659 | If |f(x)| <= C for all x t... |
| imo72b2lem0 43660 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 43661 | Natural deduction form of ... |
| imo72b2lem2 43662 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 43663 | Natural deduction form of ... |
| imo72b2lem1 43664 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 43665 | 'Less than or equal to' re... |
| lemuldiv4d 43666 | 'Less than or equal to' re... |
| imo72b2 43667 | IMO 1972 B2. (14th Intern... |
| int-addcomd 43668 | AdditionCommutativity gene... |
| int-addassocd 43669 | AdditionAssociativity gene... |
| int-addsimpd 43670 | AdditionSimplification gen... |
| int-mulcomd 43671 | MultiplicationCommutativit... |
| int-mulassocd 43672 | MultiplicationAssociativit... |
| int-mulsimpd 43673 | MultiplicationSimplificati... |
| int-leftdistd 43674 | AdditionMultiplicationLeft... |
| int-rightdistd 43675 | AdditionMultiplicationRigh... |
| int-sqdefd 43676 | SquareDefinition generator... |
| int-mul11d 43677 | First MultiplicationOne ge... |
| int-mul12d 43678 | Second MultiplicationOne g... |
| int-add01d 43679 | First AdditionZero generat... |
| int-add02d 43680 | Second AdditionZero genera... |
| int-sqgeq0d 43681 | SquareGEQZero generator ru... |
| int-eqprincd 43682 | PrincipleOfEquality genera... |
| int-eqtransd 43683 | EqualityTransitivity gener... |
| int-eqmvtd 43684 | EquMoveTerm generator rule... |
| int-eqineqd 43685 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 43686 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 43687 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 43688 | SecondPrincipleOfInequalit... |
| int-ineqtransd 43689 | InequalityTransitivity gen... |
| unitadd 43690 | Theorem used in conjunctio... |
| gsumws3 43691 | Valuation of a length 3 wo... |
| gsumws4 43692 | Valuation of a length 4 wo... |
| amgm2d 43693 | Arithmetic-geometric mean ... |
| amgm3d 43694 | Arithmetic-geometric mean ... |
| amgm4d 43695 | Arithmetic-geometric mean ... |
| spALT 43696 | ~ sp can be proven from th... |
| elnelneqd 43697 | Two classes are not equal ... |
| elnelneq2d 43698 | Two classes are not equal ... |
| rr-spce 43699 | Prove an existential. (Co... |
| rexlimdvaacbv 43700 | Unpack a restricted existe... |
| rexlimddvcbvw 43701 | Unpack a restricted existe... |
| rexlimddvcbv 43702 | Unpack a restricted existe... |
| rr-elrnmpt3d 43703 | Elementhood in an image se... |
| finnzfsuppd 43704 | If a function is zero outs... |
| rr-phpd 43705 | Equivalent of ~ php withou... |
| suceqd 43706 | Deduction associated with ... |
| tfindsd 43707 | Deduction associated with ... |
| mnringvald 43710 | Value of the monoid ring f... |
| mnringnmulrd 43711 | Components of a monoid rin... |
| mnringnmulrdOLD 43712 | Obsolete version of ~ mnri... |
| mnringbased 43713 | The base set of a monoid r... |
| mnringbasedOLD 43714 | Obsolete version of ~ mnri... |
| mnringbaserd 43715 | The base set of a monoid r... |
| mnringelbased 43716 | Membership in the base set... |
| mnringbasefd 43717 | Elements of a monoid ring ... |
| mnringbasefsuppd 43718 | Elements of a monoid ring ... |
| mnringaddgd 43719 | The additive operation of ... |
| mnringaddgdOLD 43720 | Obsolete version of ~ mnri... |
| mnring0gd 43721 | The additive identity of a... |
| mnring0g2d 43722 | The additive identity of a... |
| mnringmulrd 43723 | The ring product of a mono... |
| mnringscad 43724 | The scalar ring of a monoi... |
| mnringscadOLD 43725 | Obsolete version of ~ mnri... |
| mnringvscad 43726 | The scalar product of a mo... |
| mnringvscadOLD 43727 | Obsolete version of ~ mnri... |
| mnringlmodd 43728 | Monoid rings are left modu... |
| mnringmulrvald 43729 | Value of multiplication in... |
| mnringmulrcld 43730 | Monoid rings are closed un... |
| gru0eld 43731 | A nonempty Grothendieck un... |
| grusucd 43732 | Grothendieck universes are... |
| r1rankcld 43733 | Any rank of the cumulative... |
| grur1cld 43734 | Grothendieck universes are... |
| grurankcld 43735 | Grothendieck universes are... |
| grurankrcld 43736 | If a Grothendieck universe... |
| scotteqd 43739 | Equality theorem for the S... |
| scotteq 43740 | Closed form of ~ scotteqd ... |
| nfscott 43741 | Bound-variable hypothesis ... |
| scottabf 43742 | Value of the Scott operati... |
| scottab 43743 | Value of the Scott operati... |
| scottabes 43744 | Value of the Scott operati... |
| scottss 43745 | Scott's trick produces a s... |
| elscottab 43746 | An element of the output o... |
| scottex2 43747 | ~ scottex expressed using ... |
| scotteld 43748 | The Scott operation sends ... |
| scottelrankd 43749 | Property of a Scott's tric... |
| scottrankd 43750 | Rank of a nonempty Scott's... |
| gruscottcld 43751 | If a Grothendieck universe... |
| dfcoll2 43754 | Alternate definition of th... |
| colleq12d 43755 | Equality theorem for the c... |
| colleq1 43756 | Equality theorem for the c... |
| colleq2 43757 | Equality theorem for the c... |
| nfcoll 43758 | Bound-variable hypothesis ... |
| collexd 43759 | The output of the collecti... |
| cpcolld 43760 | Property of the collection... |
| cpcoll2d 43761 | ~ cpcolld with an extra ex... |
| grucollcld 43762 | A Grothendieck universe co... |
| ismnu 43763 | The hypothesis of this the... |
| mnuop123d 43764 | Operations of a minimal un... |
| mnussd 43765 | Minimal universes are clos... |
| mnuss2d 43766 | ~ mnussd with arguments pr... |
| mnu0eld 43767 | A nonempty minimal univers... |
| mnuop23d 43768 | Second and third operation... |
| mnupwd 43769 | Minimal universes are clos... |
| mnusnd 43770 | Minimal universes are clos... |
| mnuprssd 43771 | A minimal universe contain... |
| mnuprss2d 43772 | Special case of ~ mnuprssd... |
| mnuop3d 43773 | Third operation of a minim... |
| mnuprdlem1 43774 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 43775 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 43776 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 43777 | Lemma for ~ mnuprd . Gene... |
| mnuprd 43778 | Minimal universes are clos... |
| mnuunid 43779 | Minimal universes are clos... |
| mnuund 43780 | Minimal universes are clos... |
| mnutrcld 43781 | Minimal universes contain ... |
| mnutrd 43782 | Minimal universes are tran... |
| mnurndlem1 43783 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 43784 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 43785 | Minimal universes contain ... |
| mnugrud 43786 | Minimal universes are Grot... |
| grumnudlem 43787 | Lemma for ~ grumnud . (Co... |
| grumnud 43788 | Grothendieck universes are... |
| grumnueq 43789 | The class of Grothendieck ... |
| expandan 43790 | Expand conjunction to prim... |
| expandexn 43791 | Expand an existential quan... |
| expandral 43792 | Expand a restricted univer... |
| expandrexn 43793 | Expand a restricted existe... |
| expandrex 43794 | Expand a restricted existe... |
| expanduniss 43795 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 43796 | Express the predicate on `... |
| rr-grothprimbi 43797 | Express "every set is cont... |
| inagrud 43798 | Inaccessible levels of the... |
| inaex 43799 | Assuming the Tarski-Grothe... |
| gruex 43800 | Assuming the Tarski-Grothe... |
| rr-groth 43801 | An equivalent of ~ ax-grot... |
| rr-grothprim 43802 | An equivalent of ~ ax-grot... |
| ismnushort 43803 | Express the predicate on `... |
| dfuniv2 43804 | Alternative definition of ... |
| rr-grothshortbi 43805 | Express "every set is cont... |
| rr-grothshort 43806 | A shorter equivalent of ~ ... |
| nanorxor 43807 | 'nand' is equivalent to th... |
| undisjrab 43808 | Union of two disjoint rest... |
| iso0 43809 | The empty set is an ` R , ... |
| ssrecnpr 43810 | ` RR ` is a subset of both... |
| seff 43811 | Let set ` S ` be the real ... |
| sblpnf 43812 | The infinity ball in the a... |
| prmunb2 43813 | The primes are unbounded. ... |
| dvgrat 43814 | Ratio test for divergence ... |
| cvgdvgrat 43815 | Ratio test for convergence... |
| radcnvrat 43816 | Let ` L ` be the limit, if... |
| reldvds 43817 | The divides relation is in... |
| nznngen 43818 | All positive integers in t... |
| nzss 43819 | The set of multiples of _m... |
| nzin 43820 | The intersection of the se... |
| nzprmdif 43821 | Subtract one prime's multi... |
| hashnzfz 43822 | Special case of ~ hashdvds... |
| hashnzfz2 43823 | Special case of ~ hashnzfz... |
| hashnzfzclim 43824 | As the upper bound ` K ` o... |
| caofcan 43825 | Transfer a cancellation la... |
| ofsubid 43826 | Function analogue of ~ sub... |
| ofmul12 43827 | Function analogue of ~ mul... |
| ofdivrec 43828 | Function analogue of ~ div... |
| ofdivcan4 43829 | Function analogue of ~ div... |
| ofdivdiv2 43830 | Function analogue of ~ div... |
| lhe4.4ex1a 43831 | Example of the Fundamental... |
| dvsconst 43832 | Derivative of a constant f... |
| dvsid 43833 | Derivative of the identity... |
| dvsef 43834 | Derivative of the exponent... |
| expgrowthi 43835 | Exponential growth and dec... |
| dvconstbi 43836 | The derivative of a functi... |
| expgrowth 43837 | Exponential growth and dec... |
| bccval 43840 | Value of the generalized b... |
| bcccl 43841 | Closure of the generalized... |
| bcc0 43842 | The generalized binomial c... |
| bccp1k 43843 | Generalized binomial coeff... |
| bccm1k 43844 | Generalized binomial coeff... |
| bccn0 43845 | Generalized binomial coeff... |
| bccn1 43846 | Generalized binomial coeff... |
| bccbc 43847 | The binomial coefficient a... |
| uzmptshftfval 43848 | When ` F ` is a maps-to fu... |
| dvradcnv2 43849 | The radius of convergence ... |
| binomcxplemwb 43850 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 43851 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 43852 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 43853 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 43854 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 43855 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 43856 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 43857 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 43858 | Lemma for ~ binomcxp . Wh... |
| binomcxp 43859 | Generalize the binomial th... |
| pm10.12 43860 | Theorem *10.12 in [Whitehe... |
| pm10.14 43861 | Theorem *10.14 in [Whitehe... |
| pm10.251 43862 | Theorem *10.251 in [Whiteh... |
| pm10.252 43863 | Theorem *10.252 in [Whiteh... |
| pm10.253 43864 | Theorem *10.253 in [Whiteh... |
| albitr 43865 | Theorem *10.301 in [Whiteh... |
| pm10.42 43866 | Theorem *10.42 in [Whitehe... |
| pm10.52 43867 | Theorem *10.52 in [Whitehe... |
| pm10.53 43868 | Theorem *10.53 in [Whitehe... |
| pm10.541 43869 | Theorem *10.541 in [Whiteh... |
| pm10.542 43870 | Theorem *10.542 in [Whiteh... |
| pm10.55 43871 | Theorem *10.55 in [Whitehe... |
| pm10.56 43872 | Theorem *10.56 in [Whitehe... |
| pm10.57 43873 | Theorem *10.57 in [Whitehe... |
| 2alanimi 43874 | Removes two universal quan... |
| 2al2imi 43875 | Removes two universal quan... |
| pm11.11 43876 | Theorem *11.11 in [Whitehe... |
| pm11.12 43877 | Theorem *11.12 in [Whitehe... |
| 19.21vv 43878 | Compare Theorem *11.3 in [... |
| 2alim 43879 | Theorem *11.32 in [Whitehe... |
| 2albi 43880 | Theorem *11.33 in [Whitehe... |
| 2exim 43881 | Theorem *11.34 in [Whitehe... |
| 2exbi 43882 | Theorem *11.341 in [Whiteh... |
| spsbce-2 43883 | Theorem *11.36 in [Whitehe... |
| 19.33-2 43884 | Theorem *11.421 in [Whiteh... |
| 19.36vv 43885 | Theorem *11.43 in [Whitehe... |
| 19.31vv 43886 | Theorem *11.44 in [Whitehe... |
| 19.37vv 43887 | Theorem *11.46 in [Whitehe... |
| 19.28vv 43888 | Theorem *11.47 in [Whitehe... |
| pm11.52 43889 | Theorem *11.52 in [Whitehe... |
| aaanv 43890 | Theorem *11.56 in [Whitehe... |
| pm11.57 43891 | Theorem *11.57 in [Whitehe... |
| pm11.58 43892 | Theorem *11.58 in [Whitehe... |
| pm11.59 43893 | Theorem *11.59 in [Whitehe... |
| pm11.6 43894 | Theorem *11.6 in [Whitehea... |
| pm11.61 43895 | Theorem *11.61 in [Whitehe... |
| pm11.62 43896 | Theorem *11.62 in [Whitehe... |
| pm11.63 43897 | Theorem *11.63 in [Whitehe... |
| pm11.7 43898 | Theorem *11.7 in [Whitehea... |
| pm11.71 43899 | Theorem *11.71 in [Whitehe... |
| sbeqal1 43900 | If ` x = y ` always implie... |
| sbeqal1i 43901 | Suppose you know ` x = y `... |
| sbeqal2i 43902 | If ` x = y ` implies ` x =... |
| axc5c4c711 43903 | Proof of a theorem that ca... |
| axc5c4c711toc5 43904 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 43905 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 43906 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 43907 | Rederivation of ~ ax-11 fr... |
| axc11next 43908 | This theorem shows that, g... |
| pm13.13a 43909 | One result of theorem *13.... |
| pm13.13b 43910 | Theorem *13.13 in [Whitehe... |
| pm13.14 43911 | Theorem *13.14 in [Whitehe... |
| pm13.192 43912 | Theorem *13.192 in [Whiteh... |
| pm13.193 43913 | Theorem *13.193 in [Whiteh... |
| pm13.194 43914 | Theorem *13.194 in [Whiteh... |
| pm13.195 43915 | Theorem *13.195 in [Whiteh... |
| pm13.196a 43916 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 43917 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 43918 | Theorem *13.22 in [Whitehe... |
| iotain 43919 | Equivalence between two di... |
| iotaexeu 43920 | The iota class exists. Th... |
| iotasbc 43921 | Definition *14.01 in [Whit... |
| iotasbc2 43922 | Theorem *14.111 in [Whiteh... |
| pm14.12 43923 | Theorem *14.12 in [Whitehe... |
| pm14.122a 43924 | Theorem *14.122 in [Whiteh... |
| pm14.122b 43925 | Theorem *14.122 in [Whiteh... |
| pm14.122c 43926 | Theorem *14.122 in [Whiteh... |
| pm14.123a 43927 | Theorem *14.123 in [Whiteh... |
| pm14.123b 43928 | Theorem *14.123 in [Whiteh... |
| pm14.123c 43929 | Theorem *14.123 in [Whiteh... |
| pm14.18 43930 | Theorem *14.18 in [Whitehe... |
| iotaequ 43931 | Theorem *14.2 in [Whitehea... |
| iotavalb 43932 | Theorem *14.202 in [Whiteh... |
| iotasbc5 43933 | Theorem *14.205 in [Whiteh... |
| pm14.24 43934 | Theorem *14.24 in [Whitehe... |
| iotavalsb 43935 | Theorem *14.242 in [Whiteh... |
| sbiota1 43936 | Theorem *14.25 in [Whitehe... |
| sbaniota 43937 | Theorem *14.26 in [Whitehe... |
| eubiOLD 43938 | Obsolete proof of ~ eubi a... |
| iotasbcq 43939 | Theorem *14.272 in [Whiteh... |
| elnev 43940 | Any set that contains one ... |
| rusbcALT 43941 | A version of Russell's par... |
| compeq 43942 | Equality between two ways ... |
| compne 43943 | The complement of ` A ` is... |
| compab 43944 | Two ways of saying "the co... |
| conss2 43945 | Contrapositive law for sub... |
| conss1 43946 | Contrapositive law for sub... |
| ralbidar 43947 | More general form of ~ ral... |
| rexbidar 43948 | More general form of ~ rex... |
| dropab1 43949 | Theorem to aid use of the ... |
| dropab2 43950 | Theorem to aid use of the ... |
| ipo0 43951 | If the identity relation p... |
| ifr0 43952 | A class that is founded by... |
| ordpss 43953 | ~ ordelpss with an anteced... |
| fvsb 43954 | Explicit substitution of a... |
| fveqsb 43955 | Implicit substitution of a... |
| xpexb 43956 | A Cartesian product exists... |
| trelpss 43957 | An element of a transitive... |
| addcomgi 43958 | Generalization of commutat... |
| addrval 43968 | Value of the operation of ... |
| subrval 43969 | Value of the operation of ... |
| mulvval 43970 | Value of the operation of ... |
| addrfv 43971 | Vector addition at a value... |
| subrfv 43972 | Vector subtraction at a va... |
| mulvfv 43973 | Scalar multiplication at a... |
| addrfn 43974 | Vector addition produces a... |
| subrfn 43975 | Vector subtraction produce... |
| mulvfn 43976 | Scalar multiplication prod... |
| addrcom 43977 | Vector addition is commuta... |
| idiALT 43981 | Placeholder for ~ idi . T... |
| exbir 43982 | Exportation implication al... |
| 3impexpbicom 43983 | Version of ~ 3impexp where... |
| 3impexpbicomi 43984 | Inference associated with ... |
| bi1imp 43985 | Importation inference simi... |
| bi2imp 43986 | Importation inference simi... |
| bi3impb 43987 | Similar to ~ 3impb with im... |
| bi3impa 43988 | Similar to ~ 3impa with im... |
| bi23impib 43989 | ~ 3impib with the inner im... |
| bi13impib 43990 | ~ 3impib with the outer im... |
| bi123impib 43991 | ~ 3impib with the implicat... |
| bi13impia 43992 | ~ 3impia with the outer im... |
| bi123impia 43993 | ~ 3impia with the implicat... |
| bi33imp12 43994 | ~ 3imp with innermost impl... |
| bi23imp13 43995 | ~ 3imp with middle implica... |
| bi13imp23 43996 | ~ 3imp with outermost impl... |
| bi13imp2 43997 | Similar to ~ 3imp except t... |
| bi12imp3 43998 | Similar to ~ 3imp except a... |
| bi23imp1 43999 | Similar to ~ 3imp except a... |
| bi123imp0 44000 | Similar to ~ 3imp except a... |
| 4animp1 44001 | A single hypothesis unific... |
| 4an31 44002 | A rearrangement of conjunc... |
| 4an4132 44003 | A rearrangement of conjunc... |
| expcomdg 44004 | Biconditional form of ~ ex... |
| iidn3 44005 | ~ idn3 without virtual ded... |
| ee222 44006 | ~ e222 without virtual ded... |
| ee3bir 44007 | Right-biconditional form o... |
| ee13 44008 | ~ e13 without virtual dedu... |
| ee121 44009 | ~ e121 without virtual ded... |
| ee122 44010 | ~ e122 without virtual ded... |
| ee333 44011 | ~ e333 without virtual ded... |
| ee323 44012 | ~ e323 without virtual ded... |
| 3ornot23 44013 | If the second and third di... |
| orbi1r 44014 | ~ orbi1 with order of disj... |
| 3orbi123 44015 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 44016 | Closed form of ~ syl5 . D... |
| impexpd 44017 | The following User's Proof... |
| com3rgbi 44018 | The following User's Proof... |
| impexpdcom 44019 | The following User's Proof... |
| ee1111 44020 | Non-virtual deduction form... |
| pm2.43bgbi 44021 | Logical equivalence of a 2... |
| pm2.43cbi 44022 | Logical equivalence of a 3... |
| ee233 44023 | Non-virtual deduction form... |
| imbi13 44024 | Join three logical equival... |
| ee33 44025 | Non-virtual deduction form... |
| con5 44026 | Biconditional contrapositi... |
| con5i 44027 | Inference form of ~ con5 .... |
| exlimexi 44028 | Inference similar to Theor... |
| sb5ALT 44029 | Equivalence for substituti... |
| eexinst01 44030 | ~ exinst01 without virtual... |
| eexinst11 44031 | ~ exinst11 without virtual... |
| vk15.4j 44032 | Excercise 4j of Unit 15 of... |
| notnotrALT 44033 | Converse of double negatio... |
| con3ALT2 44034 | Contraposition. Alternate... |
| ssralv2 44035 | Quantification restricted ... |
| sbc3or 44036 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 44037 | Closed form of ~ alrimi wi... |
| rspsbc2 44038 | ~ rspsbc with two quantify... |
| sbcoreleleq 44039 | Substitution of a setvar v... |
| tratrb 44040 | If a class is transitive a... |
| ordelordALT 44041 | An element of an ordinal c... |
| sbcim2g 44042 | Distribution of class subs... |
| sbcbi 44043 | Implication form of ~ sbcb... |
| trsbc 44044 | Formula-building inference... |
| truniALT 44045 | The union of a class of tr... |
| onfrALTlem5 44046 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 44047 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 44048 | Lemma for ~ onfrALT . (Co... |
| ggen31 44049 | ~ gen31 without virtual de... |
| onfrALTlem2 44050 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 44051 | A theorem pertaining to th... |
| onfrALTlem1 44052 | Lemma for ~ onfrALT . (Co... |
| onfrALT 44053 | The membership relation is... |
| 19.41rg 44054 | Closed form of right-to-le... |
| opelopab4 44055 | Ordered pair membership in... |
| 2pm13.193 44056 | ~ pm13.193 for two variabl... |
| hbntal 44057 | A closed form of ~ hbn . ~... |
| hbimpg 44058 | A closed form of ~ hbim . ... |
| hbalg 44059 | Closed form of ~ hbal . D... |
| hbexg 44060 | Closed form of ~ nfex . D... |
| ax6e2eq 44061 | Alternate form of ~ ax6e f... |
| ax6e2nd 44062 | If at least two sets exist... |
| ax6e2ndeq 44063 | "At least two sets exist" ... |
| 2sb5nd 44064 | Equivalence for double sub... |
| 2uasbanh 44065 | Distribute the unabbreviat... |
| 2uasban 44066 | Distribute the unabbreviat... |
| e2ebind 44067 | Absorption of an existenti... |
| elpwgded 44068 | ~ elpwgdedVD in convention... |
| trelded 44069 | Deduction form of ~ trel .... |
| jaoded 44070 | Deduction form of ~ jao . ... |
| sbtT 44071 | A substitution into a theo... |
| not12an2impnot1 44072 | If a double conjunction is... |
| in1 44075 | Inference form of ~ df-vd1... |
| iin1 44076 | ~ in1 without virtual dedu... |
| dfvd1ir 44077 | Inference form of ~ df-vd1... |
| idn1 44078 | Virtual deduction identity... |
| dfvd1imp 44079 | Left-to-right part of defi... |
| dfvd1impr 44080 | Right-to-left part of defi... |
| dfvd2 44083 | Definition of a 2-hypothes... |
| dfvd2an 44086 | Definition of a 2-hypothes... |
| dfvd2ani 44087 | Inference form of ~ dfvd2a... |
| dfvd2anir 44088 | Right-to-left inference fo... |
| dfvd2i 44089 | Inference form of ~ dfvd2 ... |
| dfvd2ir 44090 | Right-to-left inference fo... |
| dfvd3 44095 | Definition of a 3-hypothes... |
| dfvd3i 44096 | Inference form of ~ dfvd3 ... |
| dfvd3ir 44097 | Right-to-left inference fo... |
| dfvd3an 44098 | Definition of a 3-hypothes... |
| dfvd3ani 44099 | Inference form of ~ dfvd3a... |
| dfvd3anir 44100 | Right-to-left inference fo... |
| vd01 44101 | A virtual hypothesis virtu... |
| vd02 44102 | Two virtual hypotheses vir... |
| vd03 44103 | A theorem is virtually inf... |
| vd12 44104 | A virtual deduction with 1... |
| vd13 44105 | A virtual deduction with 1... |
| vd23 44106 | A virtual deduction with 2... |
| dfvd2imp 44107 | The virtual deduction form... |
| dfvd2impr 44108 | A 2-antecedent nested impl... |
| in2 44109 | The virtual deduction intr... |
| int2 44110 | The virtual deduction intr... |
| iin2 44111 | ~ in2 without virtual dedu... |
| in2an 44112 | The virtual deduction intr... |
| in3 44113 | The virtual deduction intr... |
| iin3 44114 | ~ in3 without virtual dedu... |
| in3an 44115 | The virtual deduction intr... |
| int3 44116 | The virtual deduction intr... |
| idn2 44117 | Virtual deduction identity... |
| iden2 44118 | Virtual deduction identity... |
| idn3 44119 | Virtual deduction identity... |
| gen11 44120 | Virtual deduction generali... |
| gen11nv 44121 | Virtual deduction generali... |
| gen12 44122 | Virtual deduction generali... |
| gen21 44123 | Virtual deduction generali... |
| gen21nv 44124 | Virtual deduction form of ... |
| gen31 44125 | Virtual deduction generali... |
| gen22 44126 | Virtual deduction generali... |
| ggen22 44127 | ~ gen22 without virtual de... |
| exinst 44128 | Existential Instantiation.... |
| exinst01 44129 | Existential Instantiation.... |
| exinst11 44130 | Existential Instantiation.... |
| e1a 44131 | A Virtual deduction elimin... |
| el1 44132 | A Virtual deduction elimin... |
| e1bi 44133 | Biconditional form of ~ e1... |
| e1bir 44134 | Right biconditional form o... |
| e2 44135 | A virtual deduction elimin... |
| e2bi 44136 | Biconditional form of ~ e2... |
| e2bir 44137 | Right biconditional form o... |
| ee223 44138 | ~ e223 without virtual ded... |
| e223 44139 | A virtual deduction elimin... |
| e222 44140 | A virtual deduction elimin... |
| e220 44141 | A virtual deduction elimin... |
| ee220 44142 | ~ e220 without virtual ded... |
| e202 44143 | A virtual deduction elimin... |
| ee202 44144 | ~ e202 without virtual ded... |
| e022 44145 | A virtual deduction elimin... |
| ee022 44146 | ~ e022 without virtual ded... |
| e002 44147 | A virtual deduction elimin... |
| ee002 44148 | ~ e002 without virtual ded... |
| e020 44149 | A virtual deduction elimin... |
| ee020 44150 | ~ e020 without virtual ded... |
| e200 44151 | A virtual deduction elimin... |
| ee200 44152 | ~ e200 without virtual ded... |
| e221 44153 | A virtual deduction elimin... |
| ee221 44154 | ~ e221 without virtual ded... |
| e212 44155 | A virtual deduction elimin... |
| ee212 44156 | ~ e212 without virtual ded... |
| e122 44157 | A virtual deduction elimin... |
| e112 44158 | A virtual deduction elimin... |
| ee112 44159 | ~ e112 without virtual ded... |
| e121 44160 | A virtual deduction elimin... |
| e211 44161 | A virtual deduction elimin... |
| ee211 44162 | ~ e211 without virtual ded... |
| e210 44163 | A virtual deduction elimin... |
| ee210 44164 | ~ e210 without virtual ded... |
| e201 44165 | A virtual deduction elimin... |
| ee201 44166 | ~ e201 without virtual ded... |
| e120 44167 | A virtual deduction elimin... |
| ee120 44168 | Virtual deduction rule ~ e... |
| e021 44169 | A virtual deduction elimin... |
| ee021 44170 | ~ e021 without virtual ded... |
| e012 44171 | A virtual deduction elimin... |
| ee012 44172 | ~ e012 without virtual ded... |
| e102 44173 | A virtual deduction elimin... |
| ee102 44174 | ~ e102 without virtual ded... |
| e22 44175 | A virtual deduction elimin... |
| e22an 44176 | Conjunction form of ~ e22 ... |
| ee22an 44177 | ~ e22an without virtual de... |
| e111 44178 | A virtual deduction elimin... |
| e1111 44179 | A virtual deduction elimin... |
| e110 44180 | A virtual deduction elimin... |
| ee110 44181 | ~ e110 without virtual ded... |
| e101 44182 | A virtual deduction elimin... |
| ee101 44183 | ~ e101 without virtual ded... |
| e011 44184 | A virtual deduction elimin... |
| ee011 44185 | ~ e011 without virtual ded... |
| e100 44186 | A virtual deduction elimin... |
| ee100 44187 | ~ e100 without virtual ded... |
| e010 44188 | A virtual deduction elimin... |
| ee010 44189 | ~ e010 without virtual ded... |
| e001 44190 | A virtual deduction elimin... |
| ee001 44191 | ~ e001 without virtual ded... |
| e11 44192 | A virtual deduction elimin... |
| e11an 44193 | Conjunction form of ~ e11 ... |
| ee11an 44194 | ~ e11an without virtual de... |
| e01 44195 | A virtual deduction elimin... |
| e01an 44196 | Conjunction form of ~ e01 ... |
| ee01an 44197 | ~ e01an without virtual de... |
| e10 44198 | A virtual deduction elimin... |
| e10an 44199 | Conjunction form of ~ e10 ... |
| ee10an 44200 | ~ e10an without virtual de... |
| e02 44201 | A virtual deduction elimin... |
| e02an 44202 | Conjunction form of ~ e02 ... |
| ee02an 44203 | ~ e02an without virtual de... |
| eel021old 44204 | ~ el021old without virtual... |
| el021old 44205 | A virtual deduction elimin... |
| eel132 44206 | ~ syl2an with antecedents ... |
| eel000cT 44207 | An elimination deduction. ... |
| eel0TT 44208 | An elimination deduction. ... |
| eelT00 44209 | An elimination deduction. ... |
| eelTTT 44210 | An elimination deduction. ... |
| eelT11 44211 | An elimination deduction. ... |
| eelT1 44212 | Syllogism inference combin... |
| eelT12 44213 | An elimination deduction. ... |
| eelTT1 44214 | An elimination deduction. ... |
| eelT01 44215 | An elimination deduction. ... |
| eel0T1 44216 | An elimination deduction. ... |
| eel12131 44217 | An elimination deduction. ... |
| eel2131 44218 | ~ syl2an with antecedents ... |
| eel3132 44219 | ~ syl2an with antecedents ... |
| eel0321old 44220 | ~ el0321old without virtua... |
| el0321old 44221 | A virtual deduction elimin... |
| eel2122old 44222 | ~ el2122old without virtua... |
| el2122old 44223 | A virtual deduction elimin... |
| eel0000 44224 | Elimination rule similar t... |
| eel00001 44225 | An elimination deduction. ... |
| eel00000 44226 | Elimination rule similar ~... |
| eel11111 44227 | Five-hypothesis eliminatio... |
| e12 44228 | A virtual deduction elimin... |
| e12an 44229 | Conjunction form of ~ e12 ... |
| el12 44230 | Virtual deduction form of ... |
| e20 44231 | A virtual deduction elimin... |
| e20an 44232 | Conjunction form of ~ e20 ... |
| ee20an 44233 | ~ e20an without virtual de... |
| e21 44234 | A virtual deduction elimin... |
| e21an 44235 | Conjunction form of ~ e21 ... |
| ee21an 44236 | ~ e21an without virtual de... |
| e333 44237 | A virtual deduction elimin... |
| e33 44238 | A virtual deduction elimin... |
| e33an 44239 | Conjunction form of ~ e33 ... |
| ee33an 44240 | ~ e33an without virtual de... |
| e3 44241 | Meta-connective form of ~ ... |
| e3bi 44242 | Biconditional form of ~ e3... |
| e3bir 44243 | Right biconditional form o... |
| e03 44244 | A virtual deduction elimin... |
| ee03 44245 | ~ e03 without virtual dedu... |
| e03an 44246 | Conjunction form of ~ e03 ... |
| ee03an 44247 | Conjunction form of ~ ee03... |
| e30 44248 | A virtual deduction elimin... |
| ee30 44249 | ~ e30 without virtual dedu... |
| e30an 44250 | A virtual deduction elimin... |
| ee30an 44251 | Conjunction form of ~ ee30... |
| e13 44252 | A virtual deduction elimin... |
| e13an 44253 | A virtual deduction elimin... |
| ee13an 44254 | ~ e13an without virtual de... |
| e31 44255 | A virtual deduction elimin... |
| ee31 44256 | ~ e31 without virtual dedu... |
| e31an 44257 | A virtual deduction elimin... |
| ee31an 44258 | ~ e31an without virtual de... |
| e23 44259 | A virtual deduction elimin... |
| e23an 44260 | A virtual deduction elimin... |
| ee23an 44261 | ~ e23an without virtual de... |
| e32 44262 | A virtual deduction elimin... |
| ee32 44263 | ~ e32 without virtual dedu... |
| e32an 44264 | A virtual deduction elimin... |
| ee32an 44265 | ~ e33an without virtual de... |
| e123 44266 | A virtual deduction elimin... |
| ee123 44267 | ~ e123 without virtual ded... |
| el123 44268 | A virtual deduction elimin... |
| e233 44269 | A virtual deduction elimin... |
| e323 44270 | A virtual deduction elimin... |
| e000 44271 | A virtual deduction elimin... |
| e00 44272 | Elimination rule identical... |
| e00an 44273 | Elimination rule identical... |
| eel00cT 44274 | An elimination deduction. ... |
| eelTT 44275 | An elimination deduction. ... |
| e0a 44276 | Elimination rule identical... |
| eelT 44277 | An elimination deduction. ... |
| eel0cT 44278 | An elimination deduction. ... |
| eelT0 44279 | An elimination deduction. ... |
| e0bi 44280 | Elimination rule identical... |
| e0bir 44281 | Elimination rule identical... |
| uun0.1 44282 | Convention notation form o... |
| un0.1 44283 | ` T. ` is the constant tru... |
| uunT1 44284 | A deduction unionizing a n... |
| uunT1p1 44285 | A deduction unionizing a n... |
| uunT21 44286 | A deduction unionizing a n... |
| uun121 44287 | A deduction unionizing a n... |
| uun121p1 44288 | A deduction unionizing a n... |
| uun132 44289 | A deduction unionizing a n... |
| uun132p1 44290 | A deduction unionizing a n... |
| anabss7p1 44291 | A deduction unionizing a n... |
| un10 44292 | A unionizing deduction. (... |
| un01 44293 | A unionizing deduction. (... |
| un2122 44294 | A deduction unionizing a n... |
| uun2131 44295 | A deduction unionizing a n... |
| uun2131p1 44296 | A deduction unionizing a n... |
| uunTT1 44297 | A deduction unionizing a n... |
| uunTT1p1 44298 | A deduction unionizing a n... |
| uunTT1p2 44299 | A deduction unionizing a n... |
| uunT11 44300 | A deduction unionizing a n... |
| uunT11p1 44301 | A deduction unionizing a n... |
| uunT11p2 44302 | A deduction unionizing a n... |
| uunT12 44303 | A deduction unionizing a n... |
| uunT12p1 44304 | A deduction unionizing a n... |
| uunT12p2 44305 | A deduction unionizing a n... |
| uunT12p3 44306 | A deduction unionizing a n... |
| uunT12p4 44307 | A deduction unionizing a n... |
| uunT12p5 44308 | A deduction unionizing a n... |
| uun111 44309 | A deduction unionizing a n... |
| 3anidm12p1 44310 | A deduction unionizing a n... |
| 3anidm12p2 44311 | A deduction unionizing a n... |
| uun123 44312 | A deduction unionizing a n... |
| uun123p1 44313 | A deduction unionizing a n... |
| uun123p2 44314 | A deduction unionizing a n... |
| uun123p3 44315 | A deduction unionizing a n... |
| uun123p4 44316 | A deduction unionizing a n... |
| uun2221 44317 | A deduction unionizing a n... |
| uun2221p1 44318 | A deduction unionizing a n... |
| uun2221p2 44319 | A deduction unionizing a n... |
| 3impdirp1 44320 | A deduction unionizing a n... |
| 3impcombi 44321 | A 1-hypothesis proposition... |
| trsspwALT 44322 | Virtual deduction proof of... |
| trsspwALT2 44323 | Virtual deduction proof of... |
| trsspwALT3 44324 | Short predicate calculus p... |
| sspwtr 44325 | Virtual deduction proof of... |
| sspwtrALT 44326 | Virtual deduction proof of... |
| sspwtrALT2 44327 | Short predicate calculus p... |
| pwtrVD 44328 | Virtual deduction proof of... |
| pwtrrVD 44329 | Virtual deduction proof of... |
| suctrALT 44330 | The successor of a transit... |
| snssiALTVD 44331 | Virtual deduction proof of... |
| snssiALT 44332 | If a class is an element o... |
| snsslVD 44333 | Virtual deduction proof of... |
| snssl 44334 | If a singleton is a subcla... |
| snelpwrVD 44335 | Virtual deduction proof of... |
| unipwrVD 44336 | Virtual deduction proof of... |
| unipwr 44337 | A class is a subclass of t... |
| sstrALT2VD 44338 | Virtual deduction proof of... |
| sstrALT2 44339 | Virtual deduction proof of... |
| suctrALT2VD 44340 | Virtual deduction proof of... |
| suctrALT2 44341 | Virtual deduction proof of... |
| elex2VD 44342 | Virtual deduction proof of... |
| elex22VD 44343 | Virtual deduction proof of... |
| eqsbc2VD 44344 | Virtual deduction proof of... |
| zfregs2VD 44345 | Virtual deduction proof of... |
| tpid3gVD 44346 | Virtual deduction proof of... |
| en3lplem1VD 44347 | Virtual deduction proof of... |
| en3lplem2VD 44348 | Virtual deduction proof of... |
| en3lpVD 44349 | Virtual deduction proof of... |
| simplbi2VD 44350 | Virtual deduction proof of... |
| 3ornot23VD 44351 | Virtual deduction proof of... |
| orbi1rVD 44352 | Virtual deduction proof of... |
| bitr3VD 44353 | Virtual deduction proof of... |
| 3orbi123VD 44354 | Virtual deduction proof of... |
| sbc3orgVD 44355 | Virtual deduction proof of... |
| 19.21a3con13vVD 44356 | Virtual deduction proof of... |
| exbirVD 44357 | Virtual deduction proof of... |
| exbiriVD 44358 | Virtual deduction proof of... |
| rspsbc2VD 44359 | Virtual deduction proof of... |
| 3impexpVD 44360 | Virtual deduction proof of... |
| 3impexpbicomVD 44361 | Virtual deduction proof of... |
| 3impexpbicomiVD 44362 | Virtual deduction proof of... |
| sbcoreleleqVD 44363 | Virtual deduction proof of... |
| hbra2VD 44364 | Virtual deduction proof of... |
| tratrbVD 44365 | Virtual deduction proof of... |
| al2imVD 44366 | Virtual deduction proof of... |
| syl5impVD 44367 | Virtual deduction proof of... |
| idiVD 44368 | Virtual deduction proof of... |
| ancomstVD 44369 | Closed form of ~ ancoms . ... |
| ssralv2VD 44370 | Quantification restricted ... |
| ordelordALTVD 44371 | An element of an ordinal c... |
| equncomVD 44372 | If a class equals the unio... |
| equncomiVD 44373 | Inference form of ~ equnco... |
| sucidALTVD 44374 | A set belongs to its succe... |
| sucidALT 44375 | A set belongs to its succe... |
| sucidVD 44376 | A set belongs to its succe... |
| imbi12VD 44377 | Implication form of ~ imbi... |
| imbi13VD 44378 | Join three logical equival... |
| sbcim2gVD 44379 | Distribution of class subs... |
| sbcbiVD 44380 | Implication form of ~ sbcb... |
| trsbcVD 44381 | Formula-building inference... |
| truniALTVD 44382 | The union of a class of tr... |
| ee33VD 44383 | Non-virtual deduction form... |
| trintALTVD 44384 | The intersection of a clas... |
| trintALT 44385 | The intersection of a clas... |
| undif3VD 44386 | The first equality of Exer... |
| sbcssgVD 44387 | Virtual deduction proof of... |
| csbingVD 44388 | Virtual deduction proof of... |
| onfrALTlem5VD 44389 | Virtual deduction proof of... |
| onfrALTlem4VD 44390 | Virtual deduction proof of... |
| onfrALTlem3VD 44391 | Virtual deduction proof of... |
| simplbi2comtVD 44392 | Virtual deduction proof of... |
| onfrALTlem2VD 44393 | Virtual deduction proof of... |
| onfrALTlem1VD 44394 | Virtual deduction proof of... |
| onfrALTVD 44395 | Virtual deduction proof of... |
| csbeq2gVD 44396 | Virtual deduction proof of... |
| csbsngVD 44397 | Virtual deduction proof of... |
| csbxpgVD 44398 | Virtual deduction proof of... |
| csbresgVD 44399 | Virtual deduction proof of... |
| csbrngVD 44400 | Virtual deduction proof of... |
| csbima12gALTVD 44401 | Virtual deduction proof of... |
| csbunigVD 44402 | Virtual deduction proof of... |
| csbfv12gALTVD 44403 | Virtual deduction proof of... |
| con5VD 44404 | Virtual deduction proof of... |
| relopabVD 44405 | Virtual deduction proof of... |
| 19.41rgVD 44406 | Virtual deduction proof of... |
| 2pm13.193VD 44407 | Virtual deduction proof of... |
| hbimpgVD 44408 | Virtual deduction proof of... |
| hbalgVD 44409 | Virtual deduction proof of... |
| hbexgVD 44410 | Virtual deduction proof of... |
| ax6e2eqVD 44411 | The following User's Proof... |
| ax6e2ndVD 44412 | The following User's Proof... |
| ax6e2ndeqVD 44413 | The following User's Proof... |
| 2sb5ndVD 44414 | The following User's Proof... |
| 2uasbanhVD 44415 | The following User's Proof... |
| e2ebindVD 44416 | The following User's Proof... |
| sb5ALTVD 44417 | The following User's Proof... |
| vk15.4jVD 44418 | The following User's Proof... |
| notnotrALTVD 44419 | The following User's Proof... |
| con3ALTVD 44420 | The following User's Proof... |
| elpwgdedVD 44421 | Membership in a power clas... |
| sspwimp 44422 | If a class is a subclass o... |
| sspwimpVD 44423 | The following User's Proof... |
| sspwimpcf 44424 | If a class is a subclass o... |
| sspwimpcfVD 44425 | The following User's Proof... |
| suctrALTcf 44426 | The sucessor of a transiti... |
| suctrALTcfVD 44427 | The following User's Proof... |
| suctrALT3 44428 | The successor of a transit... |
| sspwimpALT 44429 | If a class is a subclass o... |
| unisnALT 44430 | A set equals the union of ... |
| notnotrALT2 44431 | Converse of double negatio... |
| sspwimpALT2 44432 | If a class is a subclass o... |
| e2ebindALT 44433 | Absorption of an existenti... |
| ax6e2ndALT 44434 | If at least two sets exist... |
| ax6e2ndeqALT 44435 | "At least two sets exist" ... |
| 2sb5ndALT 44436 | Equivalence for double sub... |
| chordthmALT 44437 | The intersecting chords th... |
| isosctrlem1ALT 44438 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 44439 | The indexed union of conne... |
| iunconnALT 44440 | The indexed union of conne... |
| sineq0ALT 44441 | A complex number whose sin... |
| evth2f 44442 | A version of ~ evth2 using... |
| elunif 44443 | A version of ~ eluni using... |
| rzalf 44444 | A version of ~ rzal using ... |
| fvelrnbf 44445 | A version of ~ fvelrnb usi... |
| rfcnpre1 44446 | If F is a continuous funct... |
| ubelsupr 44447 | If U belongs to A and U is... |
| fsumcnf 44448 | A finite sum of functions ... |
| mulltgt0 44449 | The product of a negative ... |
| rspcegf 44450 | A version of ~ rspcev usin... |
| rabexgf 44451 | A version of ~ rabexg usin... |
| fcnre 44452 | A function continuous with... |
| sumsnd 44453 | A sum of a singleton is th... |
| evthf 44454 | A version of ~ evth using ... |
| cnfex 44455 | The class of continuous fu... |
| fnchoice 44456 | For a finite set, a choice... |
| refsumcn 44457 | A finite sum of continuous... |
| rfcnpre2 44458 | If ` F ` is a continuous f... |
| cncmpmax 44459 | When the hypothesis for th... |
| rfcnpre3 44460 | If F is a continuous funct... |
| rfcnpre4 44461 | If F is a continuous funct... |
| sumpair 44462 | Sum of two distinct comple... |
| rfcnnnub 44463 | Given a real continuous fu... |
| refsum2cnlem1 44464 | This is the core Lemma for... |
| refsum2cn 44465 | The sum of two continuus r... |
| adantlllr 44466 | Deduction adding a conjunc... |
| 3adantlr3 44467 | Deduction adding a conjunc... |
| 3adantll2 44468 | Deduction adding a conjunc... |
| 3adantll3 44469 | Deduction adding a conjunc... |
| ssnel 44470 | If not element of a set, t... |
| sncldre 44471 | A singleton is closed w.r.... |
| n0p 44472 | A polynomial with a nonzer... |
| pm2.65ni 44473 | Inference rule for proof b... |
| pwssfi 44474 | Every element of the power... |
| iuneq2df 44475 | Equality deduction for ind... |
| nnfoctb 44476 | There exists a mapping fro... |
| ssinss1d 44477 | Intersection preserves sub... |
| elpwinss 44478 | An element of the powerset... |
| unidmex 44479 | If ` F ` is a set, then ` ... |
| ndisj2 44480 | A non-disjointness conditi... |
| zenom 44481 | The set of integer numbers... |
| uzwo4 44482 | Well-ordering principle: a... |
| unisn0 44483 | The union of the singleton... |
| ssin0 44484 | If two classes are disjoin... |
| inabs3 44485 | Absorption law for interse... |
| pwpwuni 44486 | Relationship between power... |
| disjiun2 44487 | In a disjoint collection, ... |
| 0pwfi 44488 | The empty set is in any po... |
| ssinss2d 44489 | Intersection preserves sub... |
| zct 44490 | The set of integer numbers... |
| pwfin0 44491 | A finite set always belong... |
| uzct 44492 | An upper integer set is co... |
| iunxsnf 44493 | A singleton index picks ou... |
| fiiuncl 44494 | If a set is closed under t... |
| iunp1 44495 | The addition of the next s... |
| fiunicl 44496 | If a set is closed under t... |
| ixpeq2d 44497 | Equality theorem for infin... |
| disjxp1 44498 | The sets of a cartesian pr... |
| disjsnxp 44499 | The sets in the cartesian ... |
| eliind 44500 | Membership in indexed inte... |
| rspcef 44501 | Restricted existential spe... |
| inn0f 44502 | A nonempty intersection. ... |
| ixpssmapc 44503 | An infinite Cartesian prod... |
| inn0 44504 | A nonempty intersection. ... |
| elintd 44505 | Membership in class inters... |
| ssdf 44506 | A sufficient condition for... |
| brneqtrd 44507 | Substitution of equal clas... |
| ssnct 44508 | A set containing an uncoun... |
| ssuniint 44509 | Sufficient condition for b... |
| elintdv 44510 | Membership in class inters... |
| ssd 44511 | A sufficient condition for... |
| ralimralim 44512 | Introducing any antecedent... |
| snelmap 44513 | Membership of the element ... |
| xrnmnfpnf 44514 | An extended real that is n... |
| nelrnmpt 44515 | Non-membership in the rang... |
| iuneq1i 44516 | Equality theorem for index... |
| nssrex 44517 | Negation of subclass relat... |
| ssinc 44518 | Inclusion relation for a m... |
| ssdec 44519 | Inclusion relation for a m... |
| elixpconstg 44520 | Membership in an infinite ... |
| iineq1d 44521 | Equality theorem for index... |
| metpsmet 44522 | A metric is a pseudometric... |
| ixpssixp 44523 | Subclass theorem for infin... |
| ballss3 44524 | A sufficient condition for... |
| iunincfi 44525 | Given a sequence of increa... |
| nsstr 44526 | If it's not a subclass, it... |
| rexanuz3 44527 | Combine two different uppe... |
| cbvmpo2 44528 | Rule to change the second ... |
| cbvmpo1 44529 | Rule to change the first b... |
| eliuniin 44530 | Indexed union of indexed i... |
| ssabf 44531 | Subclass of a class abstra... |
| pssnssi 44532 | A proper subclass does not... |
| rabidim2 44533 | Membership in a restricted... |
| eluni2f 44534 | Membership in class union.... |
| eliin2f 44535 | Membership in indexed inte... |
| nssd 44536 | Negation of subclass relat... |
| iineq12dv 44537 | Equality deduction for ind... |
| supxrcld 44538 | The supremum of an arbitra... |
| elrestd 44539 | A sufficient condition for... |
| eliuniincex 44540 | Counterexample to show tha... |
| eliincex 44541 | Counterexample to show tha... |
| eliinid 44542 | Membership in an indexed i... |
| abssf 44543 | Class abstraction in a sub... |
| supxrubd 44544 | A member of a set of exten... |
| ssrabf 44545 | Subclass of a restricted c... |
| ssrabdf 44546 | Subclass of a restricted c... |
| eliin2 44547 | Membership in indexed inte... |
| ssrab2f 44548 | Subclass relation for a re... |
| restuni3 44549 | The underlying set of a su... |
| rabssf 44550 | Restricted class abstracti... |
| eliuniin2 44551 | Indexed union of indexed i... |
| restuni4 44552 | The underlying set of a su... |
| restuni6 44553 | The underlying set of a su... |
| restuni5 44554 | The underlying set of a su... |
| unirestss 44555 | The union of an elementwis... |
| iniin1 44556 | Indexed intersection of in... |
| iniin2 44557 | Indexed intersection of in... |
| cbvrabv2 44558 | A more general version of ... |
| cbvrabv2w 44559 | A more general version of ... |
| iinssiin 44560 | Subset implication for an ... |
| eliind2 44561 | Membership in indexed inte... |
| iinssd 44562 | Subset implication for an ... |
| rabbida2 44563 | Equivalent wff's yield equ... |
| iinexd 44564 | The existence of an indexe... |
| rabexf 44565 | Separation Scheme in terms... |
| rabbida3 44566 | Equivalent wff's yield equ... |
| r19.36vf 44567 | Restricted quantifier vers... |
| raleqd 44568 | Equality deduction for res... |
| iinssf 44569 | Subset implication for an ... |
| iinssdf 44570 | Subset implication for an ... |
| resabs2i 44571 | Absorption law for restric... |
| ssdf2 44572 | A sufficient condition for... |
| rabssd 44573 | Restricted class abstracti... |
| rexnegd 44574 | Minus a real number. (Con... |
| rexlimd3 44575 | * Inference from Theorem 1... |
| resabs1i 44576 | Absorption law for restric... |
| nel1nelin 44577 | Membership in an intersect... |
| nel2nelin 44578 | Membership in an intersect... |
| nel1nelini 44579 | Membership in an intersect... |
| nel2nelini 44580 | Membership in an intersect... |
| eliunid 44581 | Membership in indexed unio... |
| reximdd 44582 | Deduction from Theorem 19.... |
| unfid 44583 | The union of two finite se... |
| inopnd 44584 | The intersection of two op... |
| ss2rabdf 44585 | Deduction of restricted ab... |
| restopn3 44586 | If ` A ` is open, then ` A... |
| restopnssd 44587 | A topology restricted to a... |
| restsubel 44588 | A subset belongs in the sp... |
| toprestsubel 44589 | A subset is open in the to... |
| rabidd 44590 | An "identity" law of concr... |
| iunssdf 44591 | Subset theorem for an inde... |
| iinss2d 44592 | Subset implication for an ... |
| r19.3rzf 44593 | Restricted quantification ... |
| r19.28zf 44594 | Restricted quantifier vers... |
| iindif2f 44595 | Indexed intersection of cl... |
| ralfal 44596 | Two ways of expressing emp... |
| archd 44597 | Archimedean property of re... |
| eliund 44598 | Membership in indexed unio... |
| nimnbi 44599 | If an implication is false... |
| nimnbi2 44600 | If an implication is false... |
| notbicom 44601 | Commutative law for the ne... |
| rexeqif 44602 | Equality inference for res... |
| rspced 44603 | Restricted existential spe... |
| feq1dd 44604 | Equality deduction for fun... |
| fnresdmss 44605 | A function does not change... |
| fmptsnxp 44606 | Maps-to notation and Carte... |
| fvmpt2bd 44607 | Value of a function given ... |
| rnmptfi 44608 | The range of a function wi... |
| fresin2 44609 | Restriction of a function ... |
| ffi 44610 | A function with finite dom... |
| suprnmpt 44611 | An explicit bound for the ... |
| rnffi 44612 | The range of a function wi... |
| mptelpm 44613 | A function in maps-to nota... |
| rnmptpr 44614 | Range of a function define... |
| resmpti 44615 | Restriction of the mapping... |
| founiiun 44616 | Union expressed as an inde... |
| rnresun 44617 | Distribution law for range... |
| elrnmptf 44618 | The range of a function in... |
| rnmptssrn 44619 | Inclusion relation for two... |
| disjf1 44620 | A 1 to 1 mapping built fro... |
| rnsnf 44621 | The range of a function wh... |
| wessf1ornlem 44622 | Given a function ` F ` on ... |
| wessf1orn 44623 | Given a function ` F ` on ... |
| nelrnres 44624 | If ` A ` is not in the ran... |
| disjrnmpt2 44625 | Disjointness of the range ... |
| elrnmpt1sf 44626 | Elementhood in an image se... |
| founiiun0 44627 | Union expressed as an inde... |
| disjf1o 44628 | A bijection built from dis... |
| disjinfi 44629 | Only a finite number of di... |
| fvovco 44630 | Value of the composition o... |
| ssnnf1octb 44631 | There exists a bijection b... |
| nnf1oxpnn 44632 | There is a bijection betwe... |
| rnmptssd 44633 | The range of a function gi... |
| projf1o 44634 | A biijection from a set to... |
| fvmap 44635 | Function value for a membe... |
| fvixp2 44636 | Projection of a factor of ... |
| choicefi 44637 | For a finite set, a choice... |
| mpct 44638 | The exponentiation of a co... |
| cnmetcoval 44639 | Value of the distance func... |
| fcomptss 44640 | Express composition of two... |
| elmapsnd 44641 | Membership in a set expone... |
| mapss2 44642 | Subset inheritance for set... |
| fsneq 44643 | Equality condition for two... |
| difmap 44644 | Difference of two sets exp... |
| unirnmap 44645 | Given a subset of a set ex... |
| inmap 44646 | Intersection of two sets e... |
| fcoss 44647 | Composition of two mapping... |
| fsneqrn 44648 | Equality condition for two... |
| difmapsn 44649 | Difference of two sets exp... |
| mapssbi 44650 | Subset inheritance for set... |
| unirnmapsn 44651 | Equality theorem for a sub... |
| iunmapss 44652 | The indexed union of set e... |
| ssmapsn 44653 | A subset ` C ` of a set ex... |
| iunmapsn 44654 | The indexed union of set e... |
| absfico 44655 | Mapping domain and codomai... |
| icof 44656 | The set of left-closed rig... |
| elpmrn 44657 | The range of a partial fun... |
| imaexi 44658 | The image of a set is a se... |
| axccdom 44659 | Relax the constraint on ax... |
| dmmptdff 44660 | The domain of the mapping ... |
| dmmptdf 44661 | The domain of the mapping ... |
| elpmi2 44662 | The domain of a partial fu... |
| dmrelrnrel 44663 | A relation preserving func... |
| fvcod 44664 | Value of a function compos... |
| elrnmpoid 44665 | Membership in the range of... |
| axccd 44666 | An alternative version of ... |
| axccd2 44667 | An alternative version of ... |
| feqresmptf 44668 | Express a restricted funct... |
| dmmptssf 44669 | The domain of a mapping is... |
| dmmptdf2 44670 | The domain of the mapping ... |
| dmuz 44671 | Domain of the upper intege... |
| fmptd2f 44672 | Domain and codomain of the... |
| mpteq1df 44673 | An equality theorem for th... |
| mpteq1dfOLD 44674 | Obsolete version of ~ mpte... |
| mptexf 44675 | If the domain of a functio... |
| fvmpt4 44676 | Value of a function given ... |
| fmptf 44677 | Functionality of the mappi... |
| resimass 44678 | The image of a restriction... |
| mptssid 44679 | The mapping operation expr... |
| mptfnd 44680 | The maps-to notation defin... |
| mpteq12daOLD 44681 | Obsolete version of ~ mpte... |
| rnmptlb 44682 | Boundness below of the ran... |
| rnmptbddlem 44683 | Boundness of the range of ... |
| rnmptbdd 44684 | Boundness of the range of ... |
| funimaeq 44685 | Membership relation for th... |
| rnmptssf 44686 | The range of a function gi... |
| rnmptbd2lem 44687 | Boundness below of the ran... |
| rnmptbd2 44688 | Boundness below of the ran... |
| infnsuprnmpt 44689 | The indexed infimum of rea... |
| suprclrnmpt 44690 | Closure of the indexed sup... |
| suprubrnmpt2 44691 | A member of a nonempty ind... |
| suprubrnmpt 44692 | A member of a nonempty ind... |
| rnmptssdf 44693 | The range of a function gi... |
| rnmptbdlem 44694 | Boundness above of the ran... |
| rnmptbd 44695 | Boundness above of the ran... |
| rnmptss2 44696 | The range of a function gi... |
| elmptima 44697 | The image of a function in... |
| ralrnmpt3 44698 | A restricted quantifier ov... |
| fvelima2 44699 | Function value in an image... |
| rnmptssbi 44700 | The range of a function gi... |
| imass2d 44701 | Subset theorem for image. ... |
| imassmpt 44702 | Membership relation for th... |
| fpmd 44703 | A total function is a part... |
| fconst7 44704 | An alternative way to expr... |
| fnmptif 44705 | Functionality and domain o... |
| dmmptif 44706 | Domain of the mapping oper... |
| mpteq2dfa 44707 | Slightly more general equa... |
| dmmpt1 44708 | The domain of the mapping ... |
| fmptff 44709 | Functionality of the mappi... |
| fvmptelcdmf 44710 | The value of a function at... |
| fmptdff 44711 | A version of ~ fmptd using... |
| fvmpt2df 44712 | Deduction version of ~ fvm... |
| rn1st 44713 | The range of a function wi... |
| rnmptssff 44714 | The range of a function gi... |
| rnmptssdff 44715 | The range of a function gi... |
| fvmpt4d 44716 | Value of a function given ... |
| sub2times 44717 | Subtracting from a number,... |
| nnxrd 44718 | A natural number is an ext... |
| nnxr 44719 | A natural number is an ext... |
| abssubrp 44720 | The distance of two distin... |
| elfzfzo 44721 | Relationship between membe... |
| oddfl 44722 | Odd number representation ... |
| abscosbd 44723 | Bound for the absolute val... |
| mul13d 44724 | Commutative/associative la... |
| negpilt0 44725 | Negative ` _pi ` is negati... |
| dstregt0 44726 | A complex number ` A ` tha... |
| subadd4b 44727 | Rearrangement of 4 terms i... |
| xrlttri5d 44728 | Not equal and not larger i... |
| neglt 44729 | The negative of a positive... |
| zltlesub 44730 | If an integer ` N ` is les... |
| divlt0gt0d 44731 | The ratio of a negative nu... |
| subsub23d 44732 | Swap subtrahend and result... |
| 2timesgt 44733 | Double of a positive real ... |
| reopn 44734 | The reals are open with re... |
| sub31 44735 | Swap the first and third t... |
| nnne1ge2 44736 | A positive integer which i... |
| lefldiveq 44737 | A closed enough, smaller r... |
| negsubdi3d 44738 | Distribution of negative o... |
| ltdiv2dd 44739 | Division of a positive num... |
| abssinbd 44740 | Bound for the absolute val... |
| halffl 44741 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 44742 | Ordering relation for a st... |
| hashssle 44743 | The size of a subset of a ... |
| lttri5d 44744 | Not equal and not larger i... |
| fzisoeu 44745 | A finite ordered set has a... |
| lt3addmuld 44746 | If three real numbers are ... |
| absnpncan2d 44747 | Triangular inequality, com... |
| fperiodmullem 44748 | A function with period ` T... |
| fperiodmul 44749 | A function with period T i... |
| upbdrech 44750 | Choice of an upper bound f... |
| lt4addmuld 44751 | If four real numbers are l... |
| absnpncan3d 44752 | Triangular inequality, com... |
| upbdrech2 44753 | Choice of an upper bound f... |
| ssfiunibd 44754 | A finite union of bounded ... |
| fzdifsuc2 44755 | Remove a successor from th... |
| fzsscn 44756 | A finite sequence of integ... |
| divcan8d 44757 | A cancellation law for div... |
| dmmcand 44758 | Cancellation law for divis... |
| fzssre 44759 | A finite sequence of integ... |
| bccld 44760 | A binomial coefficient, in... |
| leadd12dd 44761 | Addition to both sides of ... |
| fzssnn0 44762 | A finite set of sequential... |
| xreqle 44763 | Equality implies 'less tha... |
| xaddlidd 44764 | ` 0 ` is a left identity f... |
| xadd0ge 44765 | A number is less than or e... |
| elfzolem1 44766 | A member in a half-open in... |
| xrgtned 44767 | 'Greater than' implies not... |
| xrleneltd 44768 | 'Less than or equal to' an... |
| xaddcomd 44769 | The extended real addition... |
| supxrre3 44770 | The supremum of a nonempty... |
| uzfissfz 44771 | For any finite subset of t... |
| xleadd2d 44772 | Addition of extended reals... |
| suprltrp 44773 | The supremum of a nonempty... |
| xleadd1d 44774 | Addition of extended reals... |
| xreqled 44775 | Equality implies 'less tha... |
| xrgepnfd 44776 | An extended real greater t... |
| xrge0nemnfd 44777 | A nonnegative extended rea... |
| supxrgere 44778 | If a real number can be ap... |
| iuneqfzuzlem 44779 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 44780 | If two unions indexed by u... |
| xle2addd 44781 | Adding both side of two in... |
| supxrgelem 44782 | If an extended real number... |
| supxrge 44783 | If an extended real number... |
| suplesup 44784 | If any element of ` A ` ca... |
| infxrglb 44785 | The infimum of a set of ex... |
| xadd0ge2 44786 | A number is less than or e... |
| nepnfltpnf 44787 | An extended real that is n... |
| ltadd12dd 44788 | Addition to both sides of ... |
| nemnftgtmnft 44789 | An extended real that is n... |
| xrgtso 44790 | 'Greater than' is a strict... |
| rpex 44791 | The positive reals form a ... |
| xrge0ge0 44792 | A nonnegative extended rea... |
| xrssre 44793 | A subset of extended reals... |
| ssuzfz 44794 | A finite subset of the upp... |
| absfun 44795 | The absolute value is a fu... |
| infrpge 44796 | The infimum of a nonempty,... |
| xrlexaddrp 44797 | If an extended real number... |
| supsubc 44798 | The supremum function dist... |
| xralrple2 44799 | Show that ` A ` is less th... |
| nnuzdisj 44800 | The first ` N ` elements o... |
| ltdivgt1 44801 | Divsion by a number greate... |
| xrltned 44802 | 'Less than' implies not eq... |
| nnsplit 44803 | Express the set of positiv... |
| divdiv3d 44804 | Division into a fraction. ... |
| abslt2sqd 44805 | Comparison of the square o... |
| qenom 44806 | The set of rational number... |
| qct 44807 | The set of rational number... |
| xrltnled 44808 | 'Less than' in terms of 'l... |
| lenlteq 44809 | 'less than or equal to' bu... |
| xrred 44810 | An extended real that is n... |
| rr2sscn2 44811 | The cartesian square of ` ... |
| infxr 44812 | The infimum of a set of ex... |
| infxrunb2 44813 | The infimum of an unbounde... |
| infxrbnd2 44814 | The infimum of a bounded-b... |
| infleinflem1 44815 | Lemma for ~ infleinf , cas... |
| infleinflem2 44816 | Lemma for ~ infleinf , whe... |
| infleinf 44817 | If any element of ` B ` ca... |
| xralrple4 44818 | Show that ` A ` is less th... |
| xralrple3 44819 | Show that ` A ` is less th... |
| eluzelzd 44820 | A member of an upper set o... |
| suplesup2 44821 | If any element of ` A ` is... |
| recnnltrp 44822 | ` N ` is a natural number ... |
| nnn0 44823 | The set of positive intege... |
| fzct 44824 | A finite set of sequential... |
| rpgtrecnn 44825 | Any positive real number i... |
| fzossuz 44826 | A half-open integer interv... |
| infxrrefi 44827 | The real and extended real... |
| xrralrecnnle 44828 | Show that ` A ` is less th... |
| fzoct 44829 | A finite set of sequential... |
| frexr 44830 | A function taking real val... |
| nnrecrp 44831 | The reciprocal of a positi... |
| reclt0d 44832 | The reciprocal of a negati... |
| lt0neg1dd 44833 | If a number is negative, i... |
| infxrcld 44834 | The infimum of an arbitrar... |
| xrralrecnnge 44835 | Show that ` A ` is less th... |
| reclt0 44836 | The reciprocal of a negati... |
| ltmulneg 44837 | Multiplying by a negative ... |
| allbutfi 44838 | For all but finitely many.... |
| ltdiv23neg 44839 | Swap denominator with othe... |
| xreqnltd 44840 | A consequence of trichotom... |
| mnfnre2 44841 | Minus infinity is not a re... |
| zssxr 44842 | The integers are a subset ... |
| fisupclrnmpt 44843 | A nonempty finite indexed ... |
| supxrunb3 44844 | The supremum of an unbound... |
| elfzod 44845 | Membership in a half-open ... |
| fimaxre4 44846 | A nonempty finite set of r... |
| ren0 44847 | The set of reals is nonemp... |
| eluzelz2 44848 | A member of an upper set o... |
| resabs2d 44849 | Absorption law for restric... |
| uzid2 44850 | Membership of the least me... |
| supxrleubrnmpt 44851 | The supremum of a nonempty... |
| uzssre2 44852 | An upper set of integers i... |
| uzssd 44853 | Subset relationship for tw... |
| eluzd 44854 | Membership in an upper set... |
| infxrlbrnmpt2 44855 | A member of a nonempty ind... |
| xrre4 44856 | An extended real is real i... |
| uz0 44857 | The upper integers functio... |
| eluzelz2d 44858 | A member of an upper set o... |
| infleinf2 44859 | If any element in ` B ` is... |
| unb2ltle 44860 | "Unbounded below" expresse... |
| uzidd2 44861 | Membership of the least me... |
| uzssd2 44862 | Subset relationship for tw... |
| rexabslelem 44863 | An indexed set of absolute... |
| rexabsle 44864 | An indexed set of absolute... |
| allbutfiinf 44865 | Given a "for all but finit... |
| supxrrernmpt 44866 | The real and extended real... |
| suprleubrnmpt 44867 | The supremum of a nonempty... |
| infrnmptle 44868 | An indexed infimum of exte... |
| infxrunb3 44869 | The infimum of an unbounde... |
| uzn0d 44870 | The upper integers are all... |
| uzssd3 44871 | Subset relationship for tw... |
| rexabsle2 44872 | An indexed set of absolute... |
| infxrunb3rnmpt 44873 | The infimum of an unbounde... |
| supxrre3rnmpt 44874 | The indexed supremum of a ... |
| uzublem 44875 | A set of reals, indexed by... |
| uzub 44876 | A set of reals, indexed by... |
| ssrexr 44877 | A subset of the reals is a... |
| supxrmnf2 44878 | Removing minus infinity fr... |
| supxrcli 44879 | The supremum of an arbitra... |
| uzid3 44880 | Membership of the least me... |
| infxrlesupxr 44881 | The supremum of a nonempty... |
| xnegeqd 44882 | Equality of two extended n... |
| xnegrecl 44883 | The extended real negative... |
| xnegnegi 44884 | Extended real version of ~... |
| xnegeqi 44885 | Equality of two extended n... |
| nfxnegd 44886 | Deduction version of ~ nfx... |
| xnegnegd 44887 | Extended real version of ~... |
| uzred 44888 | An upper integer is a real... |
| xnegcli 44889 | Closure of extended real n... |
| supminfrnmpt 44890 | The indexed supremum of a ... |
| infxrpnf 44891 | Adding plus infinity to a ... |
| infxrrnmptcl 44892 | The infimum of an arbitrar... |
| leneg2d 44893 | Negative of one side of 'l... |
| supxrltinfxr 44894 | The supremum of the empty ... |
| max1d 44895 | A number is less than or e... |
| supxrleubrnmptf 44896 | The supremum of a nonempty... |
| nleltd 44897 | 'Not less than or equal to... |
| zxrd 44898 | An integer is an extended ... |
| infxrgelbrnmpt 44899 | The infimum of an indexed ... |
| rphalfltd 44900 | Half of a positive real is... |
| uzssz2 44901 | An upper set of integers i... |
| leneg3d 44902 | Negative of one side of 'l... |
| max2d 44903 | A number is less than or e... |
| uzn0bi 44904 | The upper integers functio... |
| xnegrecl2 44905 | If the extended real negat... |
| nfxneg 44906 | Bound-variable hypothesis ... |
| uzxrd 44907 | An upper integer is an ext... |
| infxrpnf2 44908 | Removing plus infinity fro... |
| supminfxr 44909 | The extended real suprema ... |
| infrpgernmpt 44910 | The infimum of a nonempty,... |
| xnegre 44911 | An extended real is real i... |
| xnegrecl2d 44912 | If the extended real negat... |
| uzxr 44913 | An upper integer is an ext... |
| supminfxr2 44914 | The extended real suprema ... |
| xnegred 44915 | An extended real is real i... |
| supminfxrrnmpt 44916 | The indexed supremum of a ... |
| min1d 44917 | The minimum of two numbers... |
| min2d 44918 | The minimum of two numbers... |
| pnfged 44919 | Plus infinity is an upper ... |
| xrnpnfmnf 44920 | An extended real that is n... |
| uzsscn 44921 | An upper set of integers i... |
| absimnre 44922 | The absolute value of the ... |
| uzsscn2 44923 | An upper set of integers i... |
| xrtgcntopre 44924 | The standard topologies on... |
| absimlere 44925 | The absolute value of the ... |
| rpssxr 44926 | The positive reals are a s... |
| monoordxrv 44927 | Ordering relation for a mo... |
| monoordxr 44928 | Ordering relation for a mo... |
| monoord2xrv 44929 | Ordering relation for a mo... |
| monoord2xr 44930 | Ordering relation for a mo... |
| xrpnf 44931 | An extended real is plus i... |
| xlenegcon1 44932 | Extended real version of ~... |
| xlenegcon2 44933 | Extended real version of ~... |
| pimxrneun 44934 | The preimage of a set of e... |
| caucvgbf 44935 | A function is convergent i... |
| cvgcau 44936 | A convergent function is C... |
| cvgcaule 44937 | A convergent function is C... |
| rexanuz2nf 44938 | A simple counterexample re... |
| gtnelioc 44939 | A real number larger than ... |
| ioossioc 44940 | An open interval is a subs... |
| ioondisj2 44941 | A condition for two open i... |
| ioondisj1 44942 | A condition for two open i... |
| ioogtlb 44943 | An element of a closed int... |
| evthiccabs 44944 | Extreme Value Theorem on y... |
| ltnelicc 44945 | A real number smaller than... |
| eliood 44946 | Membership in an open real... |
| iooabslt 44947 | An upper bound for the dis... |
| gtnelicc 44948 | A real number greater than... |
| iooinlbub 44949 | An open interval has empty... |
| iocgtlb 44950 | An element of a left-open ... |
| iocleub 44951 | An element of a left-open ... |
| eliccd 44952 | Membership in a closed rea... |
| eliccre 44953 | A member of a closed inter... |
| eliooshift 44954 | Element of an open interva... |
| eliocd 44955 | Membership in a left-open ... |
| icoltub 44956 | An element of a left-close... |
| eliocre 44957 | A member of a left-open ri... |
| iooltub 44958 | An element of an open inte... |
| ioontr 44959 | The interior of an interva... |
| snunioo1 44960 | The closure of one end of ... |
| lbioc 44961 | A left-open right-closed i... |
| ioomidp 44962 | The midpoint is an element... |
| iccdifioo 44963 | If the open inverval is re... |
| iccdifprioo 44964 | An open interval is the cl... |
| ioossioobi 44965 | Biconditional form of ~ io... |
| iccshift 44966 | A closed interval shifted ... |
| iccsuble 44967 | An upper bound to the dist... |
| iocopn 44968 | A left-open right-closed i... |
| eliccelioc 44969 | Membership in a closed int... |
| iooshift 44970 | An open interval shifted b... |
| iccintsng 44971 | Intersection of two adiace... |
| icoiccdif 44972 | Left-closed right-open int... |
| icoopn 44973 | A left-closed right-open i... |
| icoub 44974 | A left-closed, right-open ... |
| eliccxrd 44975 | Membership in a closed rea... |
| pnfel0pnf 44976 | ` +oo ` is a nonnegative e... |
| eliccnelico 44977 | An element of a closed int... |
| eliccelicod 44978 | A member of a closed inter... |
| ge0xrre 44979 | A nonnegative extended rea... |
| ge0lere 44980 | A nonnegative extended Rea... |
| elicores 44981 | Membership in a left-close... |
| inficc 44982 | The infimum of a nonempty ... |
| qinioo 44983 | The rational numbers are d... |
| lenelioc 44984 | A real number smaller than... |
| ioonct 44985 | A nonempty open interval i... |
| xrgtnelicc 44986 | A real number greater than... |
| iccdificc 44987 | The difference of two clos... |
| iocnct 44988 | A nonempty left-open, righ... |
| iccnct 44989 | A closed interval, with mo... |
| iooiinicc 44990 | A closed interval expresse... |
| iccgelbd 44991 | An element of a closed int... |
| iooltubd 44992 | An element of an open inte... |
| icoltubd 44993 | An element of a left-close... |
| qelioo 44994 | The rational numbers are d... |
| tgqioo2 44995 | Every open set of reals is... |
| iccleubd 44996 | An element of a closed int... |
| elioored 44997 | A member of an open interv... |
| ioogtlbd 44998 | An element of a closed int... |
| ioofun 44999 | ` (,) ` is a function. (C... |
| icomnfinre 45000 | A left-closed, right-open,... |
| sqrlearg 45001 | The square compared with i... |
| ressiocsup 45002 | If the supremum belongs to... |
| ressioosup 45003 | If the supremum does not b... |
| iooiinioc 45004 | A left-open, right-closed ... |
| ressiooinf 45005 | If the infimum does not be... |
| icogelbd 45006 | An element of a left-close... |
| iocleubd 45007 | An element of a left-open ... |
| uzinico 45008 | An upper interval of integ... |
| preimaiocmnf 45009 | Preimage of a right-closed... |
| uzinico2 45010 | An upper interval of integ... |
| uzinico3 45011 | An upper interval of integ... |
| icossico2 45012 | Condition for a closed-bel... |
| dmico 45013 | The domain of the closed-b... |
| ndmico 45014 | The closed-below, open-abo... |
| uzubioo 45015 | The upper integers are unb... |
| uzubico 45016 | The upper integers are unb... |
| uzubioo2 45017 | The upper integers are unb... |
| uzubico2 45018 | The upper integers are unb... |
| iocgtlbd 45019 | An element of a left-open ... |
| xrtgioo2 45020 | The topology on the extend... |
| tgioo4 45021 | The standard topology on t... |
| fsummulc1f 45022 | Closure of a finite sum of... |
| fsumnncl 45023 | Closure of a nonempty, fin... |
| fsumge0cl 45024 | The finite sum of nonnegat... |
| fsumf1of 45025 | Re-index a finite sum usin... |
| fsumiunss 45026 | Sum over a disjoint indexe... |
| fsumreclf 45027 | Closure of a finite sum of... |
| fsumlessf 45028 | A shorter sum of nonnegati... |
| fsumsupp0 45029 | Finite sum of function val... |
| fsumsermpt 45030 | A finite sum expressed in ... |
| fmul01 45031 | Multiplying a finite numbe... |
| fmulcl 45032 | If ' Y ' is closed under t... |
| fmuldfeqlem1 45033 | induction step for the pro... |
| fmuldfeq 45034 | X and Z are two equivalent... |
| fmul01lt1lem1 45035 | Given a finite multiplicat... |
| fmul01lt1lem2 45036 | Given a finite multiplicat... |
| fmul01lt1 45037 | Given a finite multiplicat... |
| cncfmptss 45038 | A continuous complex funct... |
| rrpsscn 45039 | The positive reals are a s... |
| mulc1cncfg 45040 | A version of ~ mulc1cncf u... |
| infrglb 45041 | The infimum of a nonempty ... |
| expcnfg 45042 | If ` F ` is a complex cont... |
| prodeq2ad 45043 | Equality deduction for pro... |
| fprodsplit1 45044 | Separate out a term in a f... |
| fprodexp 45045 | Positive integer exponenti... |
| fprodabs2 45046 | The absolute value of a fi... |
| fprod0 45047 | A finite product with a ze... |
| mccllem 45048 | * Induction step for ~ mcc... |
| mccl 45049 | A multinomial coefficient,... |
| fprodcnlem 45050 | A finite product of functi... |
| fprodcn 45051 | A finite product of functi... |
| clim1fr1 45052 | A class of sequences of fr... |
| isumneg 45053 | Negation of a converging s... |
| climrec 45054 | Limit of the reciprocal of... |
| climmulf 45055 | A version of ~ climmul usi... |
| climexp 45056 | The limit of natural power... |
| climinf 45057 | A bounded monotonic noninc... |
| climsuselem1 45058 | The subsequence index ` I ... |
| climsuse 45059 | A subsequence ` G ` of a c... |
| climrecf 45060 | A version of ~ climrec usi... |
| climneg 45061 | Complex limit of the negat... |
| climinff 45062 | A version of ~ climinf usi... |
| climdivf 45063 | Limit of the ratio of two ... |
| climreeq 45064 | If ` F ` is a real functio... |
| ellimciota 45065 | An explicit value for the ... |
| climaddf 45066 | A version of ~ climadd usi... |
| mullimc 45067 | Limit of the product of tw... |
| ellimcabssub0 45068 | An equivalent condition fo... |
| limcdm0 45069 | If a function has empty do... |
| islptre 45070 | An equivalence condition f... |
| limccog 45071 | Limit of the composition o... |
| limciccioolb 45072 | The limit of a function at... |
| climf 45073 | Express the predicate: Th... |
| mullimcf 45074 | Limit of the multiplicatio... |
| constlimc 45075 | Limit of constant function... |
| rexlim2d 45076 | Inference removing two res... |
| idlimc 45077 | Limit of the identity func... |
| divcnvg 45078 | The sequence of reciprocal... |
| limcperiod 45079 | If ` F ` is a periodic fun... |
| limcrecl 45080 | If ` F ` is a real-valued ... |
| sumnnodd 45081 | A series indexed by ` NN `... |
| lptioo2 45082 | The upper bound of an open... |
| lptioo1 45083 | The lower bound of an open... |
| elprn1 45084 | A member of an unordered p... |
| elprn2 45085 | A member of an unordered p... |
| limcmptdm 45086 | The domain of a maps-to fu... |
| clim2f 45087 | Express the predicate: Th... |
| limcicciooub 45088 | The limit of a function at... |
| ltmod 45089 | A sufficient condition for... |
| islpcn 45090 | A characterization for a l... |
| lptre2pt 45091 | If a set in the real line ... |
| limsupre 45092 | If a sequence is bounded, ... |
| limcresiooub 45093 | The left limit doesn't cha... |
| limcresioolb 45094 | The right limit doesn't ch... |
| limcleqr 45095 | If the left and the right ... |
| lptioo2cn 45096 | The upper bound of an open... |
| lptioo1cn 45097 | The lower bound of an open... |
| neglimc 45098 | Limit of the negative func... |
| addlimc 45099 | Sum of two limits. (Contr... |
| 0ellimcdiv 45100 | If the numerator converges... |
| clim2cf 45101 | Express the predicate ` F ... |
| limclner 45102 | For a limit point, both fr... |
| sublimc 45103 | Subtraction of two limits.... |
| reclimc 45104 | Limit of the reciprocal of... |
| clim0cf 45105 | Express the predicate ` F ... |
| limclr 45106 | For a limit point, both fr... |
| divlimc 45107 | Limit of the quotient of t... |
| expfac 45108 | Factorial grows faster tha... |
| climconstmpt 45109 | A constant sequence conver... |
| climresmpt 45110 | A function restricted to u... |
| climsubmpt 45111 | Limit of the difference of... |
| climsubc2mpt 45112 | Limit of the difference of... |
| climsubc1mpt 45113 | Limit of the difference of... |
| fnlimfv 45114 | The value of the limit fun... |
| climreclf 45115 | The limit of a convergent ... |
| climeldmeq 45116 | Two functions that are eve... |
| climf2 45117 | Express the predicate: Th... |
| fnlimcnv 45118 | The sequence of function v... |
| climeldmeqmpt 45119 | Two functions that are eve... |
| climfveq 45120 | Two functions that are eve... |
| clim2f2 45121 | Express the predicate: Th... |
| climfveqmpt 45122 | Two functions that are eve... |
| climd 45123 | Express the predicate: Th... |
| clim2d 45124 | The limit of complex numbe... |
| fnlimfvre 45125 | The limit function of real... |
| allbutfifvre 45126 | Given a sequence of real-v... |
| climleltrp 45127 | The limit of complex numbe... |
| fnlimfvre2 45128 | The limit function of real... |
| fnlimf 45129 | The limit function of real... |
| fnlimabslt 45130 | A sequence of function val... |
| climfveqf 45131 | Two functions that are eve... |
| climmptf 45132 | Exhibit a function ` G ` w... |
| climfveqmpt3 45133 | Two functions that are eve... |
| climeldmeqf 45134 | Two functions that are eve... |
| climreclmpt 45135 | The limit of B convergent ... |
| limsupref 45136 | If a sequence is bounded, ... |
| limsupbnd1f 45137 | If a sequence is eventuall... |
| climbddf 45138 | A converging sequence of c... |
| climeqf 45139 | Two functions that are eve... |
| climeldmeqmpt3 45140 | Two functions that are eve... |
| limsupcld 45141 | Closure of the superior li... |
| climfv 45142 | The limit of a convergent ... |
| limsupval3 45143 | The superior limit of an i... |
| climfveqmpt2 45144 | Two functions that are eve... |
| limsup0 45145 | The superior limit of the ... |
| climeldmeqmpt2 45146 | Two functions that are eve... |
| limsupresre 45147 | The supremum limit of a fu... |
| climeqmpt 45148 | Two functions that are eve... |
| climfvd 45149 | The limit of a convergent ... |
| limsuplesup 45150 | An upper bound for the sup... |
| limsupresico 45151 | The superior limit doesn't... |
| limsuppnfdlem 45152 | If the restriction of a fu... |
| limsuppnfd 45153 | If the restriction of a fu... |
| limsupresuz 45154 | If the real part of the do... |
| limsupub 45155 | If the limsup is not ` +oo... |
| limsupres 45156 | The superior limit of a re... |
| climinf2lem 45157 | A convergent, nonincreasin... |
| climinf2 45158 | A convergent, nonincreasin... |
| limsupvaluz 45159 | The superior limit, when t... |
| limsupresuz2 45160 | If the domain of a functio... |
| limsuppnflem 45161 | If the restriction of a fu... |
| limsuppnf 45162 | If the restriction of a fu... |
| limsupubuzlem 45163 | If the limsup is not ` +oo... |
| limsupubuz 45164 | For a real-valued function... |
| climinf2mpt 45165 | A bounded below, monotonic... |
| climinfmpt 45166 | A bounded below, monotonic... |
| climinf3 45167 | A convergent, nonincreasin... |
| limsupvaluzmpt 45168 | The superior limit, when t... |
| limsupequzmpt2 45169 | Two functions that are eve... |
| limsupubuzmpt 45170 | If the limsup is not ` +oo... |
| limsupmnflem 45171 | The superior limit of a fu... |
| limsupmnf 45172 | The superior limit of a fu... |
| limsupequzlem 45173 | Two functions that are eve... |
| limsupequz 45174 | Two functions that are eve... |
| limsupre2lem 45175 | Given a function on the ex... |
| limsupre2 45176 | Given a function on the ex... |
| limsupmnfuzlem 45177 | The superior limit of a fu... |
| limsupmnfuz 45178 | The superior limit of a fu... |
| limsupequzmptlem 45179 | Two functions that are eve... |
| limsupequzmpt 45180 | Two functions that are eve... |
| limsupre2mpt 45181 | Given a function on the ex... |
| limsupequzmptf 45182 | Two functions that are eve... |
| limsupre3lem 45183 | Given a function on the ex... |
| limsupre3 45184 | Given a function on the ex... |
| limsupre3mpt 45185 | Given a function on the ex... |
| limsupre3uzlem 45186 | Given a function on the ex... |
| limsupre3uz 45187 | Given a function on the ex... |
| limsupreuz 45188 | Given a function on the re... |
| limsupvaluz2 45189 | The superior limit, when t... |
| limsupreuzmpt 45190 | Given a function on the re... |
| supcnvlimsup 45191 | If a function on a set of ... |
| supcnvlimsupmpt 45192 | If a function on a set of ... |
| 0cnv 45193 | If ` (/) ` is a complex nu... |
| climuzlem 45194 | Express the predicate: Th... |
| climuz 45195 | Express the predicate: Th... |
| lmbr3v 45196 | Express the binary relatio... |
| climisp 45197 | If a sequence converges to... |
| lmbr3 45198 | Express the binary relatio... |
| climrescn 45199 | A sequence converging w.r.... |
| climxrrelem 45200 | If a sequence ranging over... |
| climxrre 45201 | If a sequence ranging over... |
| limsuplt2 45204 | The defining property of t... |
| liminfgord 45205 | Ordering property of the i... |
| limsupvald 45206 | The superior limit of a se... |
| limsupresicompt 45207 | The superior limit doesn't... |
| limsupcli 45208 | Closure of the superior li... |
| liminfgf 45209 | Closure of the inferior li... |
| liminfval 45210 | The inferior limit of a se... |
| climlimsup 45211 | A sequence of real numbers... |
| limsupge 45212 | The defining property of t... |
| liminfgval 45213 | Value of the inferior limi... |
| liminfcl 45214 | Closure of the inferior li... |
| liminfvald 45215 | The inferior limit of a se... |
| liminfval5 45216 | The inferior limit of an i... |
| limsupresxr 45217 | The superior limit of a fu... |
| liminfresxr 45218 | The inferior limit of a fu... |
| liminfval2 45219 | The superior limit, relati... |
| climlimsupcex 45220 | Counterexample for ~ climl... |
| liminfcld 45221 | Closure of the inferior li... |
| liminfresico 45222 | The inferior limit doesn't... |
| limsup10exlem 45223 | The range of the given fun... |
| limsup10ex 45224 | The superior limit of a fu... |
| liminf10ex 45225 | The inferior limit of a fu... |
| liminflelimsuplem 45226 | The superior limit is grea... |
| liminflelimsup 45227 | The superior limit is grea... |
| limsupgtlem 45228 | For any positive real, the... |
| limsupgt 45229 | Given a sequence of real n... |
| liminfresre 45230 | The inferior limit of a fu... |
| liminfresicompt 45231 | The inferior limit doesn't... |
| liminfltlimsupex 45232 | An example where the ` lim... |
| liminfgelimsup 45233 | The inferior limit is grea... |
| liminfvalxr 45234 | Alternate definition of ` ... |
| liminfresuz 45235 | If the real part of the do... |
| liminflelimsupuz 45236 | The superior limit is grea... |
| liminfvalxrmpt 45237 | Alternate definition of ` ... |
| liminfresuz2 45238 | If the domain of a functio... |
| liminfgelimsupuz 45239 | The inferior limit is grea... |
| liminfval4 45240 | Alternate definition of ` ... |
| liminfval3 45241 | Alternate definition of ` ... |
| liminfequzmpt2 45242 | Two functions that are eve... |
| liminfvaluz 45243 | Alternate definition of ` ... |
| liminf0 45244 | The inferior limit of the ... |
| limsupval4 45245 | Alternate definition of ` ... |
| liminfvaluz2 45246 | Alternate definition of ` ... |
| liminfvaluz3 45247 | Alternate definition of ` ... |
| liminflelimsupcex 45248 | A counterexample for ~ lim... |
| limsupvaluz3 45249 | Alternate definition of ` ... |
| liminfvaluz4 45250 | Alternate definition of ` ... |
| limsupvaluz4 45251 | Alternate definition of ` ... |
| climliminflimsupd 45252 | If a sequence of real numb... |
| liminfreuzlem 45253 | Given a function on the re... |
| liminfreuz 45254 | Given a function on the re... |
| liminfltlem 45255 | Given a sequence of real n... |
| liminflt 45256 | Given a sequence of real n... |
| climliminf 45257 | A sequence of real numbers... |
| liminflimsupclim 45258 | A sequence of real numbers... |
| climliminflimsup 45259 | A sequence of real numbers... |
| climliminflimsup2 45260 | A sequence of real numbers... |
| climliminflimsup3 45261 | A sequence of real numbers... |
| climliminflimsup4 45262 | A sequence of real numbers... |
| limsupub2 45263 | A extended real valued fun... |
| limsupubuz2 45264 | A sequence with values in ... |
| xlimpnfxnegmnf 45265 | A sequence converges to ` ... |
| liminflbuz2 45266 | A sequence with values in ... |
| liminfpnfuz 45267 | The inferior limit of a fu... |
| liminflimsupxrre 45268 | A sequence with values in ... |
| xlimrel 45271 | The limit on extended real... |
| xlimres 45272 | A function converges iff i... |
| xlimcl 45273 | The limit of a sequence of... |
| rexlimddv2 45274 | Restricted existential eli... |
| xlimclim 45275 | Given a sequence of reals,... |
| xlimconst 45276 | A constant sequence conver... |
| climxlim 45277 | A converging sequence in t... |
| xlimbr 45278 | Express the binary relatio... |
| fuzxrpmcn 45279 | A function mapping from an... |
| cnrefiisplem 45280 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 45281 | A non-real, complex number... |
| xlimxrre 45282 | If a sequence ranging over... |
| xlimmnfvlem1 45283 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 45284 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 45285 | A function converges to mi... |
| xlimconst2 45286 | A sequence that eventually... |
| xlimpnfvlem1 45287 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 45288 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 45289 | A function converges to pl... |
| xlimclim2lem 45290 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 45291 | Given a sequence of extend... |
| xlimmnf 45292 | A function converges to mi... |
| xlimpnf 45293 | A function converges to pl... |
| xlimmnfmpt 45294 | A function converges to pl... |
| xlimpnfmpt 45295 | A function converges to pl... |
| climxlim2lem 45296 | In this lemma for ~ climxl... |
| climxlim2 45297 | A sequence of extended rea... |
| dfxlim2v 45298 | An alternative definition ... |
| dfxlim2 45299 | An alternative definition ... |
| climresd 45300 | A function restricted to u... |
| climresdm 45301 | A real function converges ... |
| dmclimxlim 45302 | A real valued sequence tha... |
| xlimmnflimsup2 45303 | A sequence of extended rea... |
| xlimuni 45304 | An infinite sequence conve... |
| xlimclimdm 45305 | A sequence of extended rea... |
| xlimfun 45306 | The convergence relation o... |
| xlimmnflimsup 45307 | If a sequence of extended ... |
| xlimdm 45308 | Two ways to express that a... |
| xlimpnfxnegmnf2 45309 | A sequence converges to ` ... |
| xlimresdm 45310 | A function converges in th... |
| xlimpnfliminf 45311 | If a sequence of extended ... |
| xlimpnfliminf2 45312 | A sequence of extended rea... |
| xlimliminflimsup 45313 | A sequence of extended rea... |
| xlimlimsupleliminf 45314 | A sequence of extended rea... |
| coseq0 45315 | A complex number whose cos... |
| sinmulcos 45316 | Multiplication formula for... |
| coskpi2 45317 | The cosine of an integer m... |
| cosnegpi 45318 | The cosine of negative ` _... |
| sinaover2ne0 45319 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 45320 | The cosine of an integer m... |
| mulcncff 45321 | The multiplication of two ... |
| cncfmptssg 45322 | A continuous complex funct... |
| constcncfg 45323 | A constant function is a c... |
| idcncfg 45324 | The identity function is a... |
| cncfshift 45325 | A periodic continuous func... |
| resincncf 45326 | ` sin ` restricted to real... |
| addccncf2 45327 | Adding a constant is a con... |
| 0cnf 45328 | The empty set is a continu... |
| fsumcncf 45329 | The finite sum of continuo... |
| cncfperiod 45330 | A periodic continuous func... |
| subcncff 45331 | The subtraction of two con... |
| negcncfg 45332 | The opposite of a continuo... |
| cnfdmsn 45333 | A function with a singleto... |
| cncfcompt 45334 | Composition of continuous ... |
| addcncff 45335 | The sum of two continuous ... |
| ioccncflimc 45336 | Limit at the upper bound o... |
| cncfuni 45337 | A complex function on a su... |
| icccncfext 45338 | A continuous function on a... |
| cncficcgt0 45339 | A the absolute value of a ... |
| icocncflimc 45340 | Limit at the lower bound, ... |
| cncfdmsn 45341 | A complex function with a ... |
| divcncff 45342 | The quotient of two contin... |
| cncfshiftioo 45343 | A periodic continuous func... |
| cncfiooicclem1 45344 | A continuous function ` F ... |
| cncfiooicc 45345 | A continuous function ` F ... |
| cncfiooiccre 45346 | A continuous function ` F ... |
| cncfioobdlem 45347 | ` G ` actually extends ` F... |
| cncfioobd 45348 | A continuous function ` F ... |
| jumpncnp 45349 | Jump discontinuity or disc... |
| cxpcncf2 45350 | The complex power function... |
| fprodcncf 45351 | The finite product of cont... |
| add1cncf 45352 | Addition to a constant is ... |
| add2cncf 45353 | Addition to a constant is ... |
| sub1cncfd 45354 | Subtracting a constant is ... |
| sub2cncfd 45355 | Subtraction from a constan... |
| fprodsub2cncf 45356 | ` F ` is continuous. (Con... |
| fprodadd2cncf 45357 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 45358 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 45359 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 45360 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 45361 | The sequence ` S ` of fini... |
| dvsinexp 45362 | The derivative of sin^N . ... |
| dvcosre 45363 | The real derivative of the... |
| dvsinax 45364 | Derivative exercise: the d... |
| dvsubf 45365 | The subtraction rule for e... |
| dvmptconst 45366 | Function-builder for deriv... |
| dvcnre 45367 | From complex differentiati... |
| dvmptidg 45368 | Function-builder for deriv... |
| dvresntr 45369 | Function-builder for deriv... |
| fperdvper 45370 | The derivative of a period... |
| dvasinbx 45371 | Derivative exercise: the d... |
| dvresioo 45372 | Restriction of a derivativ... |
| dvdivf 45373 | The quotient rule for ever... |
| dvdivbd 45374 | A sufficient condition for... |
| dvsubcncf 45375 | A sufficient condition for... |
| dvmulcncf 45376 | A sufficient condition for... |
| dvcosax 45377 | Derivative exercise: the d... |
| dvdivcncf 45378 | A sufficient condition for... |
| dvbdfbdioolem1 45379 | Given a function with boun... |
| dvbdfbdioolem2 45380 | A function on an open inte... |
| dvbdfbdioo 45381 | A function on an open inte... |
| ioodvbdlimc1lem1 45382 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 45383 | Limit at the lower bound o... |
| ioodvbdlimc1 45384 | A real function with bound... |
| ioodvbdlimc2lem 45385 | Limit at the upper bound o... |
| ioodvbdlimc2 45386 | A real function with bound... |
| dvdmsscn 45387 | ` X ` is a subset of ` CC ... |
| dvmptmulf 45388 | Function-builder for deriv... |
| dvnmptdivc 45389 | Function-builder for itera... |
| dvdsn1add 45390 | If ` K ` divides ` N ` but... |
| dvxpaek 45391 | Derivative of the polynomi... |
| dvnmptconst 45392 | The ` N ` -th derivative o... |
| dvnxpaek 45393 | The ` n ` -th derivative o... |
| dvnmul 45394 | Function-builder for the `... |
| dvmptfprodlem 45395 | Induction step for ~ dvmpt... |
| dvmptfprod 45396 | Function-builder for deriv... |
| dvnprodlem1 45397 | ` D ` is bijective. (Cont... |
| dvnprodlem2 45398 | Induction step for ~ dvnpr... |
| dvnprodlem3 45399 | The multinomial formula fo... |
| dvnprod 45400 | The multinomial formula fo... |
| itgsin0pilem1 45401 | Calculation of the integra... |
| ibliccsinexp 45402 | sin^n on a closed interval... |
| itgsin0pi 45403 | Calculation of the integra... |
| iblioosinexp 45404 | sin^n on an open integral ... |
| itgsinexplem1 45405 | Integration by parts is ap... |
| itgsinexp 45406 | A recursive formula for th... |
| iblconstmpt 45407 | A constant function is int... |
| itgeq1d 45408 | Equality theorem for an in... |
| mbfres2cn 45409 | Measurability of a piecewi... |
| vol0 45410 | The measure of the empty s... |
| ditgeqiooicc 45411 | A function ` F ` on an ope... |
| volge0 45412 | The volume of a set is alw... |
| cnbdibl 45413 | A continuous bounded funct... |
| snmbl 45414 | A singleton is measurable.... |
| ditgeq3d 45415 | Equality theorem for the d... |
| iblempty 45416 | The empty function is inte... |
| iblsplit 45417 | The union of two integrabl... |
| volsn 45418 | A singleton has 0 Lebesgue... |
| itgvol0 45419 | If the domani is negligibl... |
| itgcoscmulx 45420 | Exercise: the integral of ... |
| iblsplitf 45421 | A version of ~ iblsplit us... |
| ibliooicc 45422 | If a function is integrabl... |
| volioc 45423 | The measure of a left-open... |
| iblspltprt 45424 | If a function is integrabl... |
| itgsincmulx 45425 | Exercise: the integral of ... |
| itgsubsticclem 45426 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 45427 | Integration by u-substitut... |
| itgioocnicc 45428 | The integral of a piecewis... |
| iblcncfioo 45429 | A continuous function ` F ... |
| itgspltprt 45430 | The ` S. ` integral splits... |
| itgiccshift 45431 | The integral of a function... |
| itgperiod 45432 | The integral of a periodic... |
| itgsbtaddcnst 45433 | Integral substitution, add... |
| volico 45434 | The measure of left-closed... |
| sublevolico 45435 | The Lebesgue measure of a ... |
| dmvolss 45436 | Lebesgue measurable sets a... |
| ismbl3 45437 | The predicate " ` A ` is L... |
| volioof 45438 | The function that assigns ... |
| ovolsplit 45439 | The Lebesgue outer measure... |
| fvvolioof 45440 | The function value of the ... |
| volioore 45441 | The measure of an open int... |
| fvvolicof 45442 | The function value of the ... |
| voliooico 45443 | An open interval and a lef... |
| ismbl4 45444 | The predicate " ` A ` is L... |
| volioofmpt 45445 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 45446 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 45447 | The Lebesgue measure of op... |
| volicofmpt 45448 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 45449 | The Lebesgue measure of a ... |
| voliccico 45450 | A closed interval and a le... |
| mbfdmssre 45451 | The domain of a measurable... |
| stoweidlem1 45452 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 45453 | lemma for ~ stoweid : here... |
| stoweidlem3 45454 | Lemma for ~ stoweid : if `... |
| stoweidlem4 45455 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 45456 | There exists a δ as ... |
| stoweidlem6 45457 | Lemma for ~ stoweid : two ... |
| stoweidlem7 45458 | This lemma is used to prov... |
| stoweidlem8 45459 | Lemma for ~ stoweid : two ... |
| stoweidlem9 45460 | Lemma for ~ stoweid : here... |
| stoweidlem10 45461 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 45462 | This lemma is used to prov... |
| stoweidlem12 45463 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 45464 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 45465 | There exists a ` k ` as in... |
| stoweidlem15 45466 | This lemma is used to prov... |
| stoweidlem16 45467 | Lemma for ~ stoweid . The... |
| stoweidlem17 45468 | This lemma proves that the... |
| stoweidlem18 45469 | This theorem proves Lemma ... |
| stoweidlem19 45470 | If a set of real functions... |
| stoweidlem20 45471 | If a set A of real functio... |
| stoweidlem21 45472 | Once the Stone Weierstrass... |
| stoweidlem22 45473 | If a set of real functions... |
| stoweidlem23 45474 | This lemma is used to prov... |
| stoweidlem24 45475 | This lemma proves that for... |
| stoweidlem25 45476 | This lemma proves that for... |
| stoweidlem26 45477 | This lemma is used to prov... |
| stoweidlem27 45478 | This lemma is used to prov... |
| stoweidlem28 45479 | There exists a δ as ... |
| stoweidlem29 45480 | When the hypothesis for th... |
| stoweidlem30 45481 | This lemma is used to prov... |
| stoweidlem31 45482 | This lemma is used to prov... |
| stoweidlem32 45483 | If a set A of real functio... |
| stoweidlem33 45484 | If a set of real functions... |
| stoweidlem34 45485 | This lemma proves that for... |
| stoweidlem35 45486 | This lemma is used to prov... |
| stoweidlem36 45487 | This lemma is used to prov... |
| stoweidlem37 45488 | This lemma is used to prov... |
| stoweidlem38 45489 | This lemma is used to prov... |
| stoweidlem39 45490 | This lemma is used to prov... |
| stoweidlem40 45491 | This lemma proves that q_n... |
| stoweidlem41 45492 | This lemma is used to prov... |
| stoweidlem42 45493 | This lemma is used to prov... |
| stoweidlem43 45494 | This lemma is used to prov... |
| stoweidlem44 45495 | This lemma is used to prov... |
| stoweidlem45 45496 | This lemma proves that, gi... |
| stoweidlem46 45497 | This lemma proves that set... |
| stoweidlem47 45498 | Subtracting a constant fro... |
| stoweidlem48 45499 | This lemma is used to prov... |
| stoweidlem49 45500 | There exists a function q_... |
| stoweidlem50 45501 | This lemma proves that set... |
| stoweidlem51 45502 | There exists a function x ... |
| stoweidlem52 45503 | There exists a neighborhoo... |
| stoweidlem53 45504 | This lemma is used to prov... |
| stoweidlem54 45505 | There exists a function ` ... |
| stoweidlem55 45506 | This lemma proves the exis... |
| stoweidlem56 45507 | This theorem proves Lemma ... |
| stoweidlem57 45508 | There exists a function x ... |
| stoweidlem58 45509 | This theorem proves Lemma ... |
| stoweidlem59 45510 | This lemma proves that the... |
| stoweidlem60 45511 | This lemma proves that the... |
| stoweidlem61 45512 | This lemma proves that the... |
| stoweidlem62 45513 | This theorem proves the St... |
| stoweid 45514 | This theorem proves the St... |
| stowei 45515 | This theorem proves the St... |
| wallispilem1 45516 | ` I ` is monotone: increas... |
| wallispilem2 45517 | A first set of properties ... |
| wallispilem3 45518 | I maps to real values. (C... |
| wallispilem4 45519 | ` F ` maps to explicit exp... |
| wallispilem5 45520 | The sequence ` H ` converg... |
| wallispi 45521 | Wallis' formula for π :... |
| wallispi2lem1 45522 | An intermediate step betwe... |
| wallispi2lem2 45523 | Two expressions are proven... |
| wallispi2 45524 | An alternative version of ... |
| stirlinglem1 45525 | A simple limit of fraction... |
| stirlinglem2 45526 | ` A ` maps to positive rea... |
| stirlinglem3 45527 | Long but simple algebraic ... |
| stirlinglem4 45528 | Algebraic manipulation of ... |
| stirlinglem5 45529 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 45530 | A series that converges to... |
| stirlinglem7 45531 | Algebraic manipulation of ... |
| stirlinglem8 45532 | If ` A ` converges to ` C ... |
| stirlinglem9 45533 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 45534 | A bound for any B(N)-B(N +... |
| stirlinglem11 45535 | ` B ` is decreasing. (Con... |
| stirlinglem12 45536 | The sequence ` B ` is boun... |
| stirlinglem13 45537 | ` B ` is decreasing and ha... |
| stirlinglem14 45538 | The sequence ` A ` converg... |
| stirlinglem15 45539 | The Stirling's formula is ... |
| stirling 45540 | Stirling's approximation f... |
| stirlingr 45541 | Stirling's approximation f... |
| dirkerval 45542 | The N_th Dirichlet Kernel.... |
| dirker2re 45543 | The Dirichlet Kernel value... |
| dirkerdenne0 45544 | The Dirichlet Kernel denom... |
| dirkerval2 45545 | The N_th Dirichlet Kernel ... |
| dirkerre 45546 | The Dirichlet Kernel at an... |
| dirkerper 45547 | the Dirichlet Kernel has p... |
| dirkerf 45548 | For any natural number ` N... |
| dirkertrigeqlem1 45549 | Sum of an even number of a... |
| dirkertrigeqlem2 45550 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 45551 | Trigonometric equality lem... |
| dirkertrigeq 45552 | Trigonometric equality for... |
| dirkeritg 45553 | The definite integral of t... |
| dirkercncflem1 45554 | If ` Y ` is a multiple of ... |
| dirkercncflem2 45555 | Lemma used to prove that t... |
| dirkercncflem3 45556 | The Dirichlet Kernel is co... |
| dirkercncflem4 45557 | The Dirichlet Kernel is co... |
| dirkercncf 45558 | For any natural number ` N... |
| fourierdlem1 45559 | A partition interval is a ... |
| fourierdlem2 45560 | Membership in a partition.... |
| fourierdlem3 45561 | Membership in a partition.... |
| fourierdlem4 45562 | ` E ` is a function that m... |
| fourierdlem5 45563 | ` S ` is a function. (Con... |
| fourierdlem6 45564 | ` X ` is in the periodic p... |
| fourierdlem7 45565 | The difference between the... |
| fourierdlem8 45566 | A partition interval is a ... |
| fourierdlem9 45567 | ` H ` is a complex functio... |
| fourierdlem10 45568 | Condition on the bounds of... |
| fourierdlem11 45569 | If there is a partition, t... |
| fourierdlem12 45570 | A point of a partition is ... |
| fourierdlem13 45571 | Value of ` V ` in terms of... |
| fourierdlem14 45572 | Given the partition ` V ` ... |
| fourierdlem15 45573 | The range of the partition... |
| fourierdlem16 45574 | The coefficients of the fo... |
| fourierdlem17 45575 | The defined ` L ` is actua... |
| fourierdlem18 45576 | The function ` S ` is cont... |
| fourierdlem19 45577 | If two elements of ` D ` h... |
| fourierdlem20 45578 | Every interval in the part... |
| fourierdlem21 45579 | The coefficients of the fo... |
| fourierdlem22 45580 | The coefficients of the fo... |
| fourierdlem23 45581 | If ` F ` is continuous and... |
| fourierdlem24 45582 | A sufficient condition for... |
| fourierdlem25 45583 | If ` C ` is not in the ran... |
| fourierdlem26 45584 | Periodic image of a point ... |
| fourierdlem27 45585 | A partition open interval ... |
| fourierdlem28 45586 | Derivative of ` ( F `` ( X... |
| fourierdlem29 45587 | Explicit function value fo... |
| fourierdlem30 45588 | Sum of three small pieces ... |
| fourierdlem31 45589 | If ` A ` is finite and for... |
| fourierdlem32 45590 | Limit of a continuous func... |
| fourierdlem33 45591 | Limit of a continuous func... |
| fourierdlem34 45592 | A partition is one to one.... |
| fourierdlem35 45593 | There is a single point in... |
| fourierdlem36 45594 | ` F ` is an isomorphism. ... |
| fourierdlem37 45595 | ` I ` is a function that m... |
| fourierdlem38 45596 | The function ` F ` is cont... |
| fourierdlem39 45597 | Integration by parts of ... |
| fourierdlem40 45598 | ` H ` is a continuous func... |
| fourierdlem41 45599 | Lemma used to prove that e... |
| fourierdlem42 45600 | The set of points in a mov... |
| fourierdlem43 45601 | ` K ` is a real function. ... |
| fourierdlem44 45602 | A condition for having ` (... |
| fourierdlem46 45603 | The function ` F ` has a l... |
| fourierdlem47 45604 | For ` r ` large enough, th... |
| fourierdlem48 45605 | The given periodic functio... |
| fourierdlem49 45606 | The given periodic functio... |
| fourierdlem50 45607 | Continuity of ` O ` and it... |
| fourierdlem51 45608 | ` X ` is in the periodic p... |
| fourierdlem52 45609 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 45610 | The limit of ` F ( s ) ` a... |
| fourierdlem54 45611 | Given a partition ` Q ` an... |
| fourierdlem55 45612 | ` U ` is a real function. ... |
| fourierdlem56 45613 | Derivative of the ` K ` fu... |
| fourierdlem57 45614 | The derivative of ` O ` . ... |
| fourierdlem58 45615 | The derivative of ` K ` is... |
| fourierdlem59 45616 | The derivative of ` H ` is... |
| fourierdlem60 45617 | Given a differentiable fun... |
| fourierdlem61 45618 | Given a differentiable fun... |
| fourierdlem62 45619 | The function ` K ` is cont... |
| fourierdlem63 45620 | The upper bound of interva... |
| fourierdlem64 45621 | The partition ` V ` is fin... |
| fourierdlem65 45622 | The distance of two adjace... |
| fourierdlem66 45623 | Value of the ` G ` functio... |
| fourierdlem67 45624 | ` G ` is a function. (Con... |
| fourierdlem68 45625 | The derivative of ` O ` is... |
| fourierdlem69 45626 | A piecewise continuous fun... |
| fourierdlem70 45627 | A piecewise continuous fun... |
| fourierdlem71 45628 | A periodic piecewise conti... |
| fourierdlem72 45629 | The derivative of ` O ` is... |
| fourierdlem73 45630 | A version of the Riemann L... |
| fourierdlem74 45631 | Given a piecewise smooth f... |
| fourierdlem75 45632 | Given a piecewise smooth f... |
| fourierdlem76 45633 | Continuity of ` O ` and it... |
| fourierdlem77 45634 | If ` H ` is bounded, then ... |
| fourierdlem78 45635 | ` G ` is continuous when r... |
| fourierdlem79 45636 | ` E ` projects every inter... |
| fourierdlem80 45637 | The derivative of ` O ` is... |
| fourierdlem81 45638 | The integral of a piecewis... |
| fourierdlem82 45639 | Integral by substitution, ... |
| fourierdlem83 45640 | The fourier partial sum fo... |
| fourierdlem84 45641 | If ` F ` is piecewise coni... |
| fourierdlem85 45642 | Limit of the function ` G ... |
| fourierdlem86 45643 | Continuity of ` O ` and it... |
| fourierdlem87 45644 | The integral of ` G ` goes... |
| fourierdlem88 45645 | Given a piecewise continuo... |
| fourierdlem89 45646 | Given a piecewise continuo... |
| fourierdlem90 45647 | Given a piecewise continuo... |
| fourierdlem91 45648 | Given a piecewise continuo... |
| fourierdlem92 45649 | The integral of a piecewis... |
| fourierdlem93 45650 | Integral by substitution (... |
| fourierdlem94 45651 | For a piecewise smooth fun... |
| fourierdlem95 45652 | Algebraic manipulation of ... |
| fourierdlem96 45653 | limit for ` F ` at the low... |
| fourierdlem97 45654 | ` F ` is continuous on the... |
| fourierdlem98 45655 | ` F ` is continuous on the... |
| fourierdlem99 45656 | limit for ` F ` at the upp... |
| fourierdlem100 45657 | A piecewise continuous fun... |
| fourierdlem101 45658 | Integral by substitution f... |
| fourierdlem102 45659 | For a piecewise smooth fun... |
| fourierdlem103 45660 | The half lower part of the... |
| fourierdlem104 45661 | The half upper part of the... |
| fourierdlem105 45662 | A piecewise continuous fun... |
| fourierdlem106 45663 | For a piecewise smooth fun... |
| fourierdlem107 45664 | The integral of a piecewis... |
| fourierdlem108 45665 | The integral of a piecewis... |
| fourierdlem109 45666 | The integral of a piecewis... |
| fourierdlem110 45667 | The integral of a piecewis... |
| fourierdlem111 45668 | The fourier partial sum fo... |
| fourierdlem112 45669 | Here abbreviations (local ... |
| fourierdlem113 45670 | Fourier series convergence... |
| fourierdlem114 45671 | Fourier series convergence... |
| fourierdlem115 45672 | Fourier serier convergence... |
| fourierd 45673 | Fourier series convergence... |
| fourierclimd 45674 | Fourier series convergence... |
| fourierclim 45675 | Fourier series convergence... |
| fourier 45676 | Fourier series convergence... |
| fouriercnp 45677 | If ` F ` is continuous at ... |
| fourier2 45678 | Fourier series convergence... |
| sqwvfoura 45679 | Fourier coefficients for t... |
| sqwvfourb 45680 | Fourier series ` B ` coeff... |
| fourierswlem 45681 | The Fourier series for the... |
| fouriersw 45682 | Fourier series convergence... |
| fouriercn 45683 | If the derivative of ` F `... |
| elaa2lem 45684 | Elementhood in the set of ... |
| elaa2 45685 | Elementhood in the set of ... |
| etransclem1 45686 | ` H ` is a function. (Con... |
| etransclem2 45687 | Derivative of ` G ` . (Co... |
| etransclem3 45688 | The given ` if ` term is a... |
| etransclem4 45689 | ` F ` expressed as a finit... |
| etransclem5 45690 | A change of bound variable... |
| etransclem6 45691 | A change of bound variable... |
| etransclem7 45692 | The given product is an in... |
| etransclem8 45693 | ` F ` is a function. (Con... |
| etransclem9 45694 | If ` K ` divides ` N ` but... |
| etransclem10 45695 | The given ` if ` term is a... |
| etransclem11 45696 | A change of bound variable... |
| etransclem12 45697 | ` C ` applied to ` N ` . ... |
| etransclem13 45698 | ` F ` applied to ` Y ` . ... |
| etransclem14 45699 | Value of the term ` T ` , ... |
| etransclem15 45700 | Value of the term ` T ` , ... |
| etransclem16 45701 | Every element in the range... |
| etransclem17 45702 | The ` N ` -th derivative o... |
| etransclem18 45703 | The given function is inte... |
| etransclem19 45704 | The ` N ` -th derivative o... |
| etransclem20 45705 | ` H ` is smooth. (Contrib... |
| etransclem21 45706 | The ` N ` -th derivative o... |
| etransclem22 45707 | The ` N ` -th derivative o... |
| etransclem23 45708 | This is the claim proof in... |
| etransclem24 45709 | ` P ` divides the I -th de... |
| etransclem25 45710 | ` P ` factorial divides th... |
| etransclem26 45711 | Every term in the sum of t... |
| etransclem27 45712 | The ` N ` -th derivative o... |
| etransclem28 45713 | ` ( P - 1 ) ` factorial di... |
| etransclem29 45714 | The ` N ` -th derivative o... |
| etransclem30 45715 | The ` N ` -th derivative o... |
| etransclem31 45716 | The ` N ` -th derivative o... |
| etransclem32 45717 | This is the proof for the ... |
| etransclem33 45718 | ` F ` is smooth. (Contrib... |
| etransclem34 45719 | The ` N ` -th derivative o... |
| etransclem35 45720 | ` P ` does not divide the ... |
| etransclem36 45721 | The ` N ` -th derivative o... |
| etransclem37 45722 | ` ( P - 1 ) ` factorial di... |
| etransclem38 45723 | ` P ` divides the I -th de... |
| etransclem39 45724 | ` G ` is a function. (Con... |
| etransclem40 45725 | The ` N ` -th derivative o... |
| etransclem41 45726 | ` P ` does not divide the ... |
| etransclem42 45727 | The ` N ` -th derivative o... |
| etransclem43 45728 | ` G ` is a continuous func... |
| etransclem44 45729 | The given finite sum is no... |
| etransclem45 45730 | ` K ` is an integer. (Con... |
| etransclem46 45731 | This is the proof for equa... |
| etransclem47 45732 | ` _e ` is transcendental. ... |
| etransclem48 45733 | ` _e ` is transcendental. ... |
| etransc 45734 | ` _e ` is transcendental. ... |
| rrxtopn 45735 | The topology of the genera... |
| rrxngp 45736 | Generalized Euclidean real... |
| rrxtps 45737 | Generalized Euclidean real... |
| rrxtopnfi 45738 | The topology of the n-dime... |
| rrxtopon 45739 | The topology on generalize... |
| rrxtop 45740 | The topology on generalize... |
| rrndistlt 45741 | Given two points in the sp... |
| rrxtoponfi 45742 | The topology on n-dimensio... |
| rrxunitopnfi 45743 | The base set of the standa... |
| rrxtopn0 45744 | The topology of the zero-d... |
| qndenserrnbllem 45745 | n-dimensional rational num... |
| qndenserrnbl 45746 | n-dimensional rational num... |
| rrxtopn0b 45747 | The topology of the zero-d... |
| qndenserrnopnlem 45748 | n-dimensional rational num... |
| qndenserrnopn 45749 | n-dimensional rational num... |
| qndenserrn 45750 | n-dimensional rational num... |
| rrxsnicc 45751 | A multidimensional singlet... |
| rrnprjdstle 45752 | The distance between two p... |
| rrndsmet 45753 | ` D ` is a metric for the ... |
| rrndsxmet 45754 | ` D ` is an extended metri... |
| ioorrnopnlem 45755 | The a point in an indexed ... |
| ioorrnopn 45756 | The indexed product of ope... |
| ioorrnopnxrlem 45757 | Given a point ` F ` that b... |
| ioorrnopnxr 45758 | The indexed product of ope... |
| issal 45765 | Express the predicate " ` ... |
| pwsal 45766 | The power set of a given s... |
| salunicl 45767 | SAlg sigma-algebra is clos... |
| saluncl 45768 | The union of two sets in a... |
| prsal 45769 | The pair of the empty set ... |
| saldifcl 45770 | The complement of an eleme... |
| 0sal 45771 | The empty set belongs to e... |
| salgenval 45772 | The sigma-algebra generate... |
| saliunclf 45773 | SAlg sigma-algebra is clos... |
| saliuncl 45774 | SAlg sigma-algebra is clos... |
| salincl 45775 | The intersection of two se... |
| saluni 45776 | A set is an element of any... |
| saliinclf 45777 | SAlg sigma-algebra is clos... |
| saliincl 45778 | SAlg sigma-algebra is clos... |
| saldifcl2 45779 | The difference of two elem... |
| intsaluni 45780 | The union of an arbitrary ... |
| intsal 45781 | The arbitrary intersection... |
| salgenn0 45782 | The set used in the defini... |
| salgencl 45783 | ` SalGen ` actually genera... |
| issald 45784 | Sufficient condition to pr... |
| salexct 45785 | An example of nontrivial s... |
| sssalgen 45786 | A set is a subset of the s... |
| salgenss 45787 | The sigma-algebra generate... |
| salgenuni 45788 | The base set of the sigma-... |
| issalgend 45789 | One side of ~ dfsalgen2 . ... |
| salexct2 45790 | An example of a subset tha... |
| unisalgen 45791 | The union of a set belongs... |
| dfsalgen2 45792 | Alternate characterization... |
| salexct3 45793 | An example of a sigma-alge... |
| salgencntex 45794 | This counterexample shows ... |
| salgensscntex 45795 | This counterexample shows ... |
| issalnnd 45796 | Sufficient condition to pr... |
| dmvolsal 45797 | Lebesgue measurable sets f... |
| saldifcld 45798 | The complement of an eleme... |
| saluncld 45799 | The union of two sets in a... |
| salgencld 45800 | ` SalGen ` actually genera... |
| 0sald 45801 | The empty set belongs to e... |
| iooborel 45802 | An open interval is a Bore... |
| salincld 45803 | The intersection of two se... |
| salunid 45804 | A set is an element of any... |
| unisalgen2 45805 | The union of a set belongs... |
| bor1sal 45806 | The Borel sigma-algebra on... |
| iocborel 45807 | A left-open, right-closed ... |
| subsaliuncllem 45808 | A subspace sigma-algebra i... |
| subsaliuncl 45809 | A subspace sigma-algebra i... |
| subsalsal 45810 | A subspace sigma-algebra i... |
| subsaluni 45811 | A set belongs to the subsp... |
| salrestss 45812 | A sigma-algebra restricted... |
| sge0rnre 45815 | When ` sum^ ` is applied t... |
| fge0icoicc 45816 | If ` F ` maps to nonnegati... |
| sge0val 45817 | The value of the sum of no... |
| fge0npnf 45818 | If ` F ` maps to nonnegati... |
| sge0rnn0 45819 | The range used in the defi... |
| sge0vald 45820 | The value of the sum of no... |
| fge0iccico 45821 | A range of nonnegative ext... |
| gsumge0cl 45822 | Closure of group sum, for ... |
| sge0reval 45823 | Value of the sum of nonneg... |
| sge0pnfval 45824 | If a term in the sum of no... |
| fge0iccre 45825 | A range of nonnegative ext... |
| sge0z 45826 | Any nonnegative extended s... |
| sge00 45827 | The sum of nonnegative ext... |
| fsumlesge0 45828 | Every finite subsum of non... |
| sge0revalmpt 45829 | Value of the sum of nonneg... |
| sge0sn 45830 | A sum of a nonnegative ext... |
| sge0tsms 45831 | ` sum^ ` applied to a nonn... |
| sge0cl 45832 | The arbitrary sum of nonne... |
| sge0f1o 45833 | Re-index a nonnegative ext... |
| sge0snmpt 45834 | A sum of a nonnegative ext... |
| sge0ge0 45835 | The sum of nonnegative ext... |
| sge0xrcl 45836 | The arbitrary sum of nonne... |
| sge0repnf 45837 | The of nonnegative extende... |
| sge0fsum 45838 | The arbitrary sum of a fin... |
| sge0rern 45839 | If the sum of nonnegative ... |
| sge0supre 45840 | If the arbitrary sum of no... |
| sge0fsummpt 45841 | The arbitrary sum of a fin... |
| sge0sup 45842 | The arbitrary sum of nonne... |
| sge0less 45843 | A shorter sum of nonnegati... |
| sge0rnbnd 45844 | The range used in the defi... |
| sge0pr 45845 | Sum of a pair of nonnegati... |
| sge0gerp 45846 | The arbitrary sum of nonne... |
| sge0pnffigt 45847 | If the sum of nonnegative ... |
| sge0ssre 45848 | If a sum of nonnegative ex... |
| sge0lefi 45849 | A sum of nonnegative exten... |
| sge0lessmpt 45850 | A shorter sum of nonnegati... |
| sge0ltfirp 45851 | If the sum of nonnegative ... |
| sge0prle 45852 | The sum of a pair of nonne... |
| sge0gerpmpt 45853 | The arbitrary sum of nonne... |
| sge0resrnlem 45854 | The sum of nonnegative ext... |
| sge0resrn 45855 | The sum of nonnegative ext... |
| sge0ssrempt 45856 | If a sum of nonnegative ex... |
| sge0resplit 45857 | ` sum^ ` splits into two p... |
| sge0le 45858 | If all of the terms of sum... |
| sge0ltfirpmpt 45859 | If the extended sum of non... |
| sge0split 45860 | Split a sum of nonnegative... |
| sge0lempt 45861 | If all of the terms of sum... |
| sge0splitmpt 45862 | Split a sum of nonnegative... |
| sge0ss 45863 | Change the index set to a ... |
| sge0iunmptlemfi 45864 | Sum of nonnegative extende... |
| sge0p1 45865 | The addition of the next t... |
| sge0iunmptlemre 45866 | Sum of nonnegative extende... |
| sge0fodjrnlem 45867 | Re-index a nonnegative ext... |
| sge0fodjrn 45868 | Re-index a nonnegative ext... |
| sge0iunmpt 45869 | Sum of nonnegative extende... |
| sge0iun 45870 | Sum of nonnegative extende... |
| sge0nemnf 45871 | The generalized sum of non... |
| sge0rpcpnf 45872 | The sum of an infinite num... |
| sge0rernmpt 45873 | If the sum of nonnegative ... |
| sge0lefimpt 45874 | A sum of nonnegative exten... |
| nn0ssge0 45875 | Nonnegative integers are n... |
| sge0clmpt 45876 | The generalized sum of non... |
| sge0ltfirpmpt2 45877 | If the extended sum of non... |
| sge0isum 45878 | If a series of nonnegative... |
| sge0xrclmpt 45879 | The generalized sum of non... |
| sge0xp 45880 | Combine two generalized su... |
| sge0isummpt 45881 | If a series of nonnegative... |
| sge0ad2en 45882 | The value of the infinite ... |
| sge0isummpt2 45883 | If a series of nonnegative... |
| sge0xaddlem1 45884 | The extended addition of t... |
| sge0xaddlem2 45885 | The extended addition of t... |
| sge0xadd 45886 | The extended addition of t... |
| sge0fsummptf 45887 | The generalized sum of a f... |
| sge0snmptf 45888 | A sum of a nonnegative ext... |
| sge0ge0mpt 45889 | The sum of nonnegative ext... |
| sge0repnfmpt 45890 | The of nonnegative extende... |
| sge0pnffigtmpt 45891 | If the generalized sum of ... |
| sge0splitsn 45892 | Separate out a term in a g... |
| sge0pnffsumgt 45893 | If the sum of nonnegative ... |
| sge0gtfsumgt 45894 | If the generalized sum of ... |
| sge0uzfsumgt 45895 | If a real number is smalle... |
| sge0pnfmpt 45896 | If a term in the sum of no... |
| sge0seq 45897 | A series of nonnegative re... |
| sge0reuz 45898 | Value of the generalized s... |
| sge0reuzb 45899 | Value of the generalized s... |
| ismea 45902 | Express the predicate " ` ... |
| dmmeasal 45903 | The domain of a measure is... |
| meaf 45904 | A measure is a function th... |
| mea0 45905 | The measure of the empty s... |
| nnfoctbdjlem 45906 | There exists a mapping fro... |
| nnfoctbdj 45907 | There exists a mapping fro... |
| meadjuni 45908 | The measure of the disjoin... |
| meacl 45909 | The measure of a set is a ... |
| iundjiunlem 45910 | The sets in the sequence `... |
| iundjiun 45911 | Given a sequence ` E ` of ... |
| meaxrcl 45912 | The measure of a set is an... |
| meadjun 45913 | The measure of the union o... |
| meassle 45914 | The measure of a set is gr... |
| meaunle 45915 | The measure of the union o... |
| meadjiunlem 45916 | The sum of nonnegative ext... |
| meadjiun 45917 | The measure of the disjoin... |
| ismeannd 45918 | Sufficient condition to pr... |
| meaiunlelem 45919 | The measure of the union o... |
| meaiunle 45920 | The measure of the union o... |
| psmeasurelem 45921 | ` M ` applied to a disjoin... |
| psmeasure 45922 | Point supported measure, R... |
| voliunsge0lem 45923 | The Lebesgue measure funct... |
| voliunsge0 45924 | The Lebesgue measure funct... |
| volmea 45925 | The Lebesgue measure on th... |
| meage0 45926 | If the measure of a measur... |
| meadjunre 45927 | The measure of the union o... |
| meassre 45928 | If the measure of a measur... |
| meale0eq0 45929 | A measure that is less tha... |
| meadif 45930 | The measure of the differe... |
| meaiuninclem 45931 | Measures are continuous fr... |
| meaiuninc 45932 | Measures are continuous fr... |
| meaiuninc2 45933 | Measures are continuous fr... |
| meaiunincf 45934 | Measures are continuous fr... |
| meaiuninc3v 45935 | Measures are continuous fr... |
| meaiuninc3 45936 | Measures are continuous fr... |
| meaiininclem 45937 | Measures are continuous fr... |
| meaiininc 45938 | Measures are continuous fr... |
| meaiininc2 45939 | Measures are continuous fr... |
| caragenval 45944 | The sigma-algebra generate... |
| isome 45945 | Express the predicate " ` ... |
| caragenel 45946 | Membership in the Caratheo... |
| omef 45947 | An outer measure is a func... |
| ome0 45948 | The outer measure of the e... |
| omessle 45949 | The outer measure of a set... |
| omedm 45950 | The domain of an outer mea... |
| caragensplit 45951 | If ` E ` is in the set gen... |
| caragenelss 45952 | An element of the Caratheo... |
| carageneld 45953 | Membership in the Caratheo... |
| omecl 45954 | The outer measure of a set... |
| caragenss 45955 | The sigma-algebra generate... |
| omeunile 45956 | The outer measure of the u... |
| caragen0 45957 | The empty set belongs to a... |
| omexrcl 45958 | The outer measure of a set... |
| caragenunidm 45959 | The base set of an outer m... |
| caragensspw 45960 | The sigma-algebra generate... |
| omessre 45961 | If the outer measure of a ... |
| caragenuni 45962 | The base set of the sigma-... |
| caragenuncllem 45963 | The Caratheodory's constru... |
| caragenuncl 45964 | The Caratheodory's constru... |
| caragendifcl 45965 | The Caratheodory's constru... |
| caragenfiiuncl 45966 | The Caratheodory's constru... |
| omeunle 45967 | The outer measure of the u... |
| omeiunle 45968 | The outer measure of the i... |
| omelesplit 45969 | The outer measure of a set... |
| omeiunltfirp 45970 | If the outer measure of a ... |
| omeiunlempt 45971 | The outer measure of the i... |
| carageniuncllem1 45972 | The outer measure of ` A i... |
| carageniuncllem2 45973 | The Caratheodory's constru... |
| carageniuncl 45974 | The Caratheodory's constru... |
| caragenunicl 45975 | The Caratheodory's constru... |
| caragensal 45976 | Caratheodory's method gene... |
| caratheodorylem1 45977 | Lemma used to prove that C... |
| caratheodorylem2 45978 | Caratheodory's constructio... |
| caratheodory 45979 | Caratheodory's constructio... |
| 0ome 45980 | The map that assigns 0 to ... |
| isomenndlem 45981 | ` O ` is sub-additive w.r.... |
| isomennd 45982 | Sufficient condition to pr... |
| caragenel2d 45983 | Membership in the Caratheo... |
| omege0 45984 | If the outer measure of a ... |
| omess0 45985 | If the outer measure of a ... |
| caragencmpl 45986 | A measure built with the C... |
| vonval 45991 | Value of the Lebesgue meas... |
| ovnval 45992 | Value of the Lebesgue oute... |
| elhoi 45993 | Membership in a multidimen... |
| icoresmbl 45994 | A closed-below, open-above... |
| hoissre 45995 | The projection of a half-o... |
| ovnval2 45996 | Value of the Lebesgue oute... |
| volicorecl 45997 | The Lebesgue measure of a ... |
| hoiprodcl 45998 | The pre-measure of half-op... |
| hoicvr 45999 | ` I ` is a countable set o... |
| hoissrrn 46000 | A half-open interval is a ... |
| ovn0val 46001 | The Lebesgue outer measure... |
| ovnn0val 46002 | The value of a (multidimen... |
| ovnval2b 46003 | Value of the Lebesgue oute... |
| volicorescl 46004 | The Lebesgue measure of a ... |
| ovnprodcl 46005 | The product used in the de... |
| hoiprodcl2 46006 | The pre-measure of half-op... |
| hoicvrrex 46007 | Any subset of the multidim... |
| ovnsupge0 46008 | The set used in the defini... |
| ovnlecvr 46009 | Given a subset of multidim... |
| ovnpnfelsup 46010 | ` +oo ` is an element of t... |
| ovnsslelem 46011 | The (multidimensional, non... |
| ovnssle 46012 | The (multidimensional) Leb... |
| ovnlerp 46013 | The Lebesgue outer measure... |
| ovnf 46014 | The Lebesgue outer measure... |
| ovncvrrp 46015 | The Lebesgue outer measure... |
| ovn0lem 46016 | For any finite dimension, ... |
| ovn0 46017 | For any finite dimension, ... |
| ovncl 46018 | The Lebesgue outer measure... |
| ovn02 46019 | For the zero-dimensional s... |
| ovnxrcl 46020 | The Lebesgue outer measure... |
| ovnsubaddlem1 46021 | The Lebesgue outer measure... |
| ovnsubaddlem2 46022 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 46023 | ` ( voln* `` X ) ` is suba... |
| ovnome 46024 | ` ( voln* `` X ) ` is an o... |
| vonmea 46025 | ` ( voln `` X ) ` is a mea... |
| volicon0 46026 | The measure of a nonempty ... |
| hsphoif 46027 | ` H ` is a function (that ... |
| hoidmvval 46028 | The dimensional volume of ... |
| hoissrrn2 46029 | A half-open interval is a ... |
| hsphoival 46030 | ` H ` is a function (that ... |
| hoiprodcl3 46031 | The pre-measure of half-op... |
| volicore 46032 | The Lebesgue measure of a ... |
| hoidmvcl 46033 | The dimensional volume of ... |
| hoidmv0val 46034 | The dimensional volume of ... |
| hoidmvn0val 46035 | The dimensional volume of ... |
| hsphoidmvle2 46036 | The dimensional volume of ... |
| hsphoidmvle 46037 | The dimensional volume of ... |
| hoidmvval0 46038 | The dimensional volume of ... |
| hoiprodp1 46039 | The dimensional volume of ... |
| sge0hsphoire 46040 | If the generalized sum of ... |
| hoidmvval0b 46041 | The dimensional volume of ... |
| hoidmv1lelem1 46042 | The supremum of ` U ` belo... |
| hoidmv1lelem2 46043 | This is the contradiction ... |
| hoidmv1lelem3 46044 | The dimensional volume of ... |
| hoidmv1le 46045 | The dimensional volume of ... |
| hoidmvlelem1 46046 | The supremum of ` U ` belo... |
| hoidmvlelem2 46047 | This is the contradiction ... |
| hoidmvlelem3 46048 | This is the contradiction ... |
| hoidmvlelem4 46049 | The dimensional volume of ... |
| hoidmvlelem5 46050 | The dimensional volume of ... |
| hoidmvle 46051 | The dimensional volume of ... |
| ovnhoilem1 46052 | The Lebesgue outer measure... |
| ovnhoilem2 46053 | The Lebesgue outer measure... |
| ovnhoi 46054 | The Lebesgue outer measure... |
| dmovn 46055 | The domain of the Lebesgue... |
| hoicoto2 46056 | The half-open interval exp... |
| dmvon 46057 | Lebesgue measurable n-dime... |
| hoi2toco 46058 | The half-open interval exp... |
| hoidifhspval 46059 | ` D ` is a function that r... |
| hspval 46060 | The value of the half-spac... |
| ovnlecvr2 46061 | Given a subset of multidim... |
| ovncvr2 46062 | ` B ` and ` T ` are the le... |
| dmovnsal 46063 | The domain of the Lebesgue... |
| unidmovn 46064 | Base set of the n-dimensio... |
| rrnmbl 46065 | The set of n-dimensional R... |
| hoidifhspval2 46066 | ` D ` is a function that r... |
| hspdifhsp 46067 | A n-dimensional half-open ... |
| unidmvon 46068 | Base set of the n-dimensio... |
| hoidifhspf 46069 | ` D ` is a function that r... |
| hoidifhspval3 46070 | ` D ` is a function that r... |
| hoidifhspdmvle 46071 | The dimensional volume of ... |
| voncmpl 46072 | The Lebesgue measure is co... |
| hoiqssbllem1 46073 | The center of the n-dimens... |
| hoiqssbllem2 46074 | The center of the n-dimens... |
| hoiqssbllem3 46075 | A n-dimensional ball conta... |
| hoiqssbl 46076 | A n-dimensional ball conta... |
| hspmbllem1 46077 | Any half-space of the n-di... |
| hspmbllem2 46078 | Any half-space of the n-di... |
| hspmbllem3 46079 | Any half-space of the n-di... |
| hspmbl 46080 | Any half-space of the n-di... |
| hoimbllem 46081 | Any n-dimensional half-ope... |
| hoimbl 46082 | Any n-dimensional half-ope... |
| opnvonmbllem1 46083 | The half-open interval exp... |
| opnvonmbllem2 46084 | An open subset of the n-di... |
| opnvonmbl 46085 | An open subset of the n-di... |
| opnssborel 46086 | Open sets of a generalized... |
| borelmbl 46087 | All Borel subsets of the n... |
| volicorege0 46088 | The Lebesgue measure of a ... |
| isvonmbl 46089 | The predicate " ` A ` is m... |
| mblvon 46090 | The n-dimensional Lebesgue... |
| vonmblss 46091 | n-dimensional Lebesgue mea... |
| volico2 46092 | The measure of left-closed... |
| vonmblss2 46093 | n-dimensional Lebesgue mea... |
| ovolval2lem 46094 | The value of the Lebesgue ... |
| ovolval2 46095 | The value of the Lebesgue ... |
| ovnsubadd2lem 46096 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 46097 | ` ( voln* `` X ) ` is suba... |
| ovolval3 46098 | The value of the Lebesgue ... |
| ovnsplit 46099 | The n-dimensional Lebesgue... |
| ovolval4lem1 46100 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 46101 | The value of the Lebesgue ... |
| ovolval4 46102 | The value of the Lebesgue ... |
| ovolval5lem1 46103 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 46104 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 46105 | The value of the Lebesgue ... |
| ovolval5 46106 | The value of the Lebesgue ... |
| ovnovollem1 46107 | if ` F ` is a cover of ` B... |
| ovnovollem2 46108 | if ` I ` is a cover of ` (... |
| ovnovollem3 46109 | The 1-dimensional Lebesgue... |
| ovnovol 46110 | The 1-dimensional Lebesgue... |
| vonvolmbllem 46111 | If a subset ` B ` of real ... |
| vonvolmbl 46112 | A subset of Real numbers i... |
| vonvol 46113 | The 1-dimensional Lebesgue... |
| vonvolmbl2 46114 | A subset ` X ` of the spac... |
| vonvol2 46115 | The 1-dimensional Lebesgue... |
| hoimbl2 46116 | Any n-dimensional half-ope... |
| voncl 46117 | The Lebesgue measure of a ... |
| vonhoi 46118 | The Lebesgue outer measure... |
| vonxrcl 46119 | The Lebesgue measure of a ... |
| ioosshoi 46120 | A n-dimensional open inter... |
| vonn0hoi 46121 | The Lebesgue outer measure... |
| von0val 46122 | The Lebesgue measure (for ... |
| vonhoire 46123 | The Lebesgue measure of a ... |
| iinhoiicclem 46124 | A n-dimensional closed int... |
| iinhoiicc 46125 | A n-dimensional closed int... |
| iunhoiioolem 46126 | A n-dimensional open inter... |
| iunhoiioo 46127 | A n-dimensional open inter... |
| ioovonmbl 46128 | Any n-dimensional open int... |
| iccvonmbllem 46129 | Any n-dimensional closed i... |
| iccvonmbl 46130 | Any n-dimensional closed i... |
| vonioolem1 46131 | The sequence of the measur... |
| vonioolem2 46132 | The n-dimensional Lebesgue... |
| vonioo 46133 | The n-dimensional Lebesgue... |
| vonicclem1 46134 | The sequence of the measur... |
| vonicclem2 46135 | The n-dimensional Lebesgue... |
| vonicc 46136 | The n-dimensional Lebesgue... |
| snvonmbl 46137 | A n-dimensional singleton ... |
| vonn0ioo 46138 | The n-dimensional Lebesgue... |
| vonn0icc 46139 | The n-dimensional Lebesgue... |
| ctvonmbl 46140 | Any n-dimensional countabl... |
| vonn0ioo2 46141 | The n-dimensional Lebesgue... |
| vonsn 46142 | The n-dimensional Lebesgue... |
| vonn0icc2 46143 | The n-dimensional Lebesgue... |
| vonct 46144 | The n-dimensional Lebesgue... |
| vitali2 46145 | There are non-measurable s... |
| pimltmnf2f 46148 | Given a real-valued functi... |
| pimltmnf2 46149 | Given a real-valued functi... |
| preimagelt 46150 | The preimage of a right-op... |
| preimalegt 46151 | The preimage of a left-ope... |
| pimconstlt0 46152 | Given a constant function,... |
| pimconstlt1 46153 | Given a constant function,... |
| pimltpnff 46154 | Given a real-valued functi... |
| pimltpnf 46155 | Given a real-valued functi... |
| pimgtpnf2f 46156 | Given a real-valued functi... |
| pimgtpnf2 46157 | Given a real-valued functi... |
| salpreimagelt 46158 | If all the preimages of le... |
| pimrecltpos 46159 | The preimage of an unbound... |
| salpreimalegt 46160 | If all the preimages of ri... |
| pimiooltgt 46161 | The preimage of an open in... |
| preimaicomnf 46162 | Preimage of an open interv... |
| pimltpnf2f 46163 | Given a real-valued functi... |
| pimltpnf2 46164 | Given a real-valued functi... |
| pimgtmnf2 46165 | Given a real-valued functi... |
| pimdecfgtioc 46166 | Given a nonincreasing func... |
| pimincfltioc 46167 | Given a nondecreasing func... |
| pimdecfgtioo 46168 | Given a nondecreasing func... |
| pimincfltioo 46169 | Given a nondecreasing func... |
| preimaioomnf 46170 | Preimage of an open interv... |
| preimageiingt 46171 | A preimage of a left-close... |
| preimaleiinlt 46172 | A preimage of a left-open,... |
| pimgtmnff 46173 | Given a real-valued functi... |
| pimgtmnf 46174 | Given a real-valued functi... |
| pimrecltneg 46175 | The preimage of an unbound... |
| salpreimagtge 46176 | If all the preimages of le... |
| salpreimaltle 46177 | If all the preimages of ri... |
| issmflem 46178 | The predicate " ` F ` is a... |
| issmf 46179 | The predicate " ` F ` is a... |
| salpreimalelt 46180 | If all the preimages of ri... |
| salpreimagtlt 46181 | If all the preimages of le... |
| smfpreimalt 46182 | Given a function measurabl... |
| smff 46183 | A function measurable w.r.... |
| smfdmss 46184 | The domain of a function m... |
| issmff 46185 | The predicate " ` F ` is a... |
| issmfd 46186 | A sufficient condition for... |
| smfpreimaltf 46187 | Given a function measurabl... |
| issmfdf 46188 | A sufficient condition for... |
| sssmf 46189 | The restriction of a sigma... |
| mbfresmf 46190 | A real-valued measurable f... |
| cnfsmf 46191 | A continuous function is m... |
| incsmflem 46192 | A nondecreasing function i... |
| incsmf 46193 | A real-valued, nondecreasi... |
| smfsssmf 46194 | If a function is measurabl... |
| issmflelem 46195 | The predicate " ` F ` is a... |
| issmfle 46196 | The predicate " ` F ` is a... |
| smfpimltmpt 46197 | Given a function measurabl... |
| smfpimltxr 46198 | Given a function measurabl... |
| issmfdmpt 46199 | A sufficient condition for... |
| smfconst 46200 | Given a sigma-algebra over... |
| sssmfmpt 46201 | The restriction of a sigma... |
| cnfrrnsmf 46202 | A function, continuous fro... |
| smfid 46203 | The identity function is B... |
| bormflebmf 46204 | A Borel measurable functio... |
| smfpreimale 46205 | Given a function measurabl... |
| issmfgtlem 46206 | The predicate " ` F ` is a... |
| issmfgt 46207 | The predicate " ` F ` is a... |
| issmfled 46208 | A sufficient condition for... |
| smfpimltxrmptf 46209 | Given a function measurabl... |
| smfpimltxrmpt 46210 | Given a function measurabl... |
| smfmbfcex 46211 | A constant function, with ... |
| issmfgtd 46212 | A sufficient condition for... |
| smfpreimagt 46213 | Given a function measurabl... |
| smfaddlem1 46214 | Given the sum of two funct... |
| smfaddlem2 46215 | The sum of two sigma-measu... |
| smfadd 46216 | The sum of two sigma-measu... |
| decsmflem 46217 | A nonincreasing function i... |
| decsmf 46218 | A real-valued, nonincreasi... |
| smfpreimagtf 46219 | Given a function measurabl... |
| issmfgelem 46220 | The predicate " ` F ` is a... |
| issmfge 46221 | The predicate " ` F ` is a... |
| smflimlem1 46222 | Lemma for the proof that t... |
| smflimlem2 46223 | Lemma for the proof that t... |
| smflimlem3 46224 | The limit of sigma-measura... |
| smflimlem4 46225 | Lemma for the proof that t... |
| smflimlem5 46226 | Lemma for the proof that t... |
| smflimlem6 46227 | Lemma for the proof that t... |
| smflim 46228 | The limit of sigma-measura... |
| nsssmfmbflem 46229 | The sigma-measurable funct... |
| nsssmfmbf 46230 | The sigma-measurable funct... |
| smfpimgtxr 46231 | Given a function measurabl... |
| smfpimgtmpt 46232 | Given a function measurabl... |
| smfpreimage 46233 | Given a function measurabl... |
| mbfpsssmf 46234 | Real-valued measurable fun... |
| smfpimgtxrmptf 46235 | Given a function measurabl... |
| smfpimgtxrmpt 46236 | Given a function measurabl... |
| smfpimioompt 46237 | Given a function measurabl... |
| smfpimioo 46238 | Given a function measurabl... |
| smfresal 46239 | Given a sigma-measurable f... |
| smfrec 46240 | The reciprocal of a sigma-... |
| smfres 46241 | The restriction of sigma-m... |
| smfmullem1 46242 | The multiplication of two ... |
| smfmullem2 46243 | The multiplication of two ... |
| smfmullem3 46244 | The multiplication of two ... |
| smfmullem4 46245 | The multiplication of two ... |
| smfmul 46246 | The multiplication of two ... |
| smfmulc1 46247 | A sigma-measurable functio... |
| smfdiv 46248 | The fraction of two sigma-... |
| smfpimbor1lem1 46249 | Every open set belongs to ... |
| smfpimbor1lem2 46250 | Given a sigma-measurable f... |
| smfpimbor1 46251 | Given a sigma-measurable f... |
| smf2id 46252 | Twice the identity functio... |
| smfco 46253 | The composition of a Borel... |
| smfneg 46254 | The negative of a sigma-me... |
| smffmptf 46255 | A function measurable w.r.... |
| smffmpt 46256 | A function measurable w.r.... |
| smflim2 46257 | The limit of a sequence of... |
| smfpimcclem 46258 | Lemma for ~ smfpimcc given... |
| smfpimcc 46259 | Given a countable set of s... |
| issmfle2d 46260 | A sufficient condition for... |
| smflimmpt 46261 | The limit of a sequence of... |
| smfsuplem1 46262 | The supremum of a countabl... |
| smfsuplem2 46263 | The supremum of a countabl... |
| smfsuplem3 46264 | The supremum of a countabl... |
| smfsup 46265 | The supremum of a countabl... |
| smfsupmpt 46266 | The supremum of a countabl... |
| smfsupxr 46267 | The supremum of a countabl... |
| smfinflem 46268 | The infimum of a countable... |
| smfinf 46269 | The infimum of a countable... |
| smfinfmpt 46270 | The infimum of a countable... |
| smflimsuplem1 46271 | If ` H ` converges, the ` ... |
| smflimsuplem2 46272 | The superior limit of a se... |
| smflimsuplem3 46273 | The limit of the ` ( H `` ... |
| smflimsuplem4 46274 | If ` H ` converges, the ` ... |
| smflimsuplem5 46275 | ` H ` converges to the sup... |
| smflimsuplem6 46276 | The superior limit of a se... |
| smflimsuplem7 46277 | The superior limit of a se... |
| smflimsuplem8 46278 | The superior limit of a se... |
| smflimsup 46279 | The superior limit of a se... |
| smflimsupmpt 46280 | The superior limit of a se... |
| smfliminflem 46281 | The inferior limit of a co... |
| smfliminf 46282 | The inferior limit of a co... |
| smfliminfmpt 46283 | The inferior limit of a co... |
| adddmmbl 46284 | If two functions have doma... |
| adddmmbl2 46285 | If two functions have doma... |
| muldmmbl 46286 | If two functions have doma... |
| muldmmbl2 46287 | If two functions have doma... |
| smfdmmblpimne 46288 | If a measurable function w... |
| smfdivdmmbl 46289 | If a functions and a sigma... |
| smfpimne 46290 | Given a function measurabl... |
| smfpimne2 46291 | Given a function measurabl... |
| smfdivdmmbl2 46292 | If a functions and a sigma... |
| fsupdm 46293 | The domain of the sup func... |
| fsupdm2 46294 | The domain of the sup func... |
| smfsupdmmbllem 46295 | If a countable set of sigm... |
| smfsupdmmbl 46296 | If a countable set of sigm... |
| finfdm 46297 | The domain of the inf func... |
| finfdm2 46298 | The domain of the inf func... |
| smfinfdmmbllem 46299 | If a countable set of sigm... |
| smfinfdmmbl 46300 | If a countable set of sigm... |
| sigarval 46301 | Define the signed area by ... |
| sigarim 46302 | Signed area takes value in... |
| sigarac 46303 | Signed area is anticommuta... |
| sigaraf 46304 | Signed area is additive by... |
| sigarmf 46305 | Signed area is additive (w... |
| sigaras 46306 | Signed area is additive by... |
| sigarms 46307 | Signed area is additive (w... |
| sigarls 46308 | Signed area is linear by t... |
| sigarid 46309 | Signed area of a flat para... |
| sigarexp 46310 | Expand the signed area for... |
| sigarperm 46311 | Signed area ` ( A - C ) G ... |
| sigardiv 46312 | If signed area between vec... |
| sigarimcd 46313 | Signed area takes value in... |
| sigariz 46314 | If signed area is zero, th... |
| sigarcol 46315 | Given three points ` A ` ,... |
| sharhght 46316 | Let ` A B C ` be a triangl... |
| sigaradd 46317 | Subtracting (double) area ... |
| cevathlem1 46318 | Ceva's theorem first lemma... |
| cevathlem2 46319 | Ceva's theorem second lemm... |
| cevath 46320 | Ceva's theorem. Let ` A B... |
| simpcntrab 46321 | The center of a simple gro... |
| et-ltneverrefl 46322 | Less-than class is never r... |
| et-equeucl 46323 | Alternative proof that equ... |
| et-sqrtnegnre 46324 | The square root of a negat... |
| natlocalincr 46325 | Global monotonicity on hal... |
| natglobalincr 46326 | Local monotonicity on half... |
| upwordnul 46329 | Empty set is an increasing... |
| upwordisword 46330 | Any increasing sequence is... |
| singoutnword 46331 | Singleton with character o... |
| singoutnupword 46332 | Singleton with character o... |
| upwordsing 46333 | Singleton is an increasing... |
| upwordsseti 46334 | Strictly increasing sequen... |
| tworepnotupword 46335 | Concatenation of identical... |
| upwrdfi 46336 | There is a finite number o... |
| hirstL-ax3 46337 | The third axiom of a syste... |
| ax3h 46338 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 46339 | A closed form showing (a i... |
| aibandbiaiaiffb 46340 | A closed form showing (a i... |
| notatnand 46341 | Do not use. Use intnanr i... |
| aistia 46342 | Given a is equivalent to `... |
| aisfina 46343 | Given a is equivalent to `... |
| bothtbothsame 46344 | Given both a, b are equiva... |
| bothfbothsame 46345 | Given both a, b are equiva... |
| aiffbbtat 46346 | Given a is equivalent to b... |
| aisbbisfaisf 46347 | Given a is equivalent to b... |
| axorbtnotaiffb 46348 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 46349 | Given a is equivalent to (... |
| axorbciffatcxorb 46350 | Given a is equivalent to (... |
| aibnbna 46351 | Given a implies b, (not b)... |
| aibnbaif 46352 | Given a implies b, not b, ... |
| aiffbtbat 46353 | Given a is equivalent to b... |
| astbstanbst 46354 | Given a is equivalent to T... |
| aistbistaandb 46355 | Given a is equivalent to T... |
| aisbnaxb 46356 | Given a is equivalent to b... |
| atbiffatnnb 46357 | If a implies b, then a imp... |
| bisaiaisb 46358 | Application of bicom1 with... |
| atbiffatnnbalt 46359 | If a implies b, then a imp... |
| abnotbtaxb 46360 | Assuming a, not b, there e... |
| abnotataxb 46361 | Assuming not a, b, there e... |
| conimpf 46362 | Assuming a, not b, and a i... |
| conimpfalt 46363 | Assuming a, not b, and a i... |
| aistbisfiaxb 46364 | Given a is equivalent to T... |
| aisfbistiaxb 46365 | Given a is equivalent to F... |
| aifftbifffaibif 46366 | Given a is equivalent to T... |
| aifftbifffaibifff 46367 | Given a is equivalent to T... |
| atnaiana 46368 | Given a, it is not the cas... |
| ainaiaandna 46369 | Given a, a implies it is n... |
| abcdta 46370 | Given (((a and b) and c) a... |
| abcdtb 46371 | Given (((a and b) and c) a... |
| abcdtc 46372 | Given (((a and b) and c) a... |
| abcdtd 46373 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 46374 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 46375 | Operands in a biconditiona... |
| nabctnabc 46376 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 46377 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 46378 | From one negated implicati... |
| twonotinotbothi 46379 | From these two negated imp... |
| clifte 46380 | show d is the same as an i... |
| cliftet 46381 | show d is the same as an i... |
| clifteta 46382 | show d is the same as an i... |
| cliftetb 46383 | show d is the same as an i... |
| confun 46384 | Given the hypotheses there... |
| confun2 46385 | Confun simplified to two p... |
| confun3 46386 | Confun's more complex form... |
| confun4 46387 | An attempt at derivative. ... |
| confun5 46388 | An attempt at derivative. ... |
| plcofph 46389 | Given, a,b and a "definiti... |
| pldofph 46390 | Given, a,b c, d, "definiti... |
| plvcofph 46391 | Given, a,b,d, and "definit... |
| plvcofphax 46392 | Given, a,b,d, and "definit... |
| plvofpos 46393 | rh is derivable because ON... |
| mdandyv0 46394 | Given the equivalences set... |
| mdandyv1 46395 | Given the equivalences set... |
| mdandyv2 46396 | Given the equivalences set... |
| mdandyv3 46397 | Given the equivalences set... |
| mdandyv4 46398 | Given the equivalences set... |
| mdandyv5 46399 | Given the equivalences set... |
| mdandyv6 46400 | Given the equivalences set... |
| mdandyv7 46401 | Given the equivalences set... |
| mdandyv8 46402 | Given the equivalences set... |
| mdandyv9 46403 | Given the equivalences set... |
| mdandyv10 46404 | Given the equivalences set... |
| mdandyv11 46405 | Given the equivalences set... |
| mdandyv12 46406 | Given the equivalences set... |
| mdandyv13 46407 | Given the equivalences set... |
| mdandyv14 46408 | Given the equivalences set... |
| mdandyv15 46409 | Given the equivalences set... |
| mdandyvr0 46410 | Given the equivalences set... |
| mdandyvr1 46411 | Given the equivalences set... |
| mdandyvr2 46412 | Given the equivalences set... |
| mdandyvr3 46413 | Given the equivalences set... |
| mdandyvr4 46414 | Given the equivalences set... |
| mdandyvr5 46415 | Given the equivalences set... |
| mdandyvr6 46416 | Given the equivalences set... |
| mdandyvr7 46417 | Given the equivalences set... |
| mdandyvr8 46418 | Given the equivalences set... |
| mdandyvr9 46419 | Given the equivalences set... |
| mdandyvr10 46420 | Given the equivalences set... |
| mdandyvr11 46421 | Given the equivalences set... |
| mdandyvr12 46422 | Given the equivalences set... |
| mdandyvr13 46423 | Given the equivalences set... |
| mdandyvr14 46424 | Given the equivalences set... |
| mdandyvr15 46425 | Given the equivalences set... |
| mdandyvrx0 46426 | Given the exclusivities se... |
| mdandyvrx1 46427 | Given the exclusivities se... |
| mdandyvrx2 46428 | Given the exclusivities se... |
| mdandyvrx3 46429 | Given the exclusivities se... |
| mdandyvrx4 46430 | Given the exclusivities se... |
| mdandyvrx5 46431 | Given the exclusivities se... |
| mdandyvrx6 46432 | Given the exclusivities se... |
| mdandyvrx7 46433 | Given the exclusivities se... |
| mdandyvrx8 46434 | Given the exclusivities se... |
| mdandyvrx9 46435 | Given the exclusivities se... |
| mdandyvrx10 46436 | Given the exclusivities se... |
| mdandyvrx11 46437 | Given the exclusivities se... |
| mdandyvrx12 46438 | Given the exclusivities se... |
| mdandyvrx13 46439 | Given the exclusivities se... |
| mdandyvrx14 46440 | Given the exclusivities se... |
| mdandyvrx15 46441 | Given the exclusivities se... |
| H15NH16TH15IH16 46442 | Given 15 hypotheses and a ... |
| dandysum2p2e4 46443 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 46444 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 46445 | Replacement of a nested an... |
| adh-minim 46446 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 46447 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 46448 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 46449 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 46450 | Fourth lemma for the deriv... |
| adh-minim-ax1 46451 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 46452 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 46453 | Sixth lemma for the deriva... |
| adh-minim-ax2c 46454 | Derivation of a commuted f... |
| adh-minim-ax2 46455 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 46456 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 46457 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 46458 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 46459 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 46460 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 46461 | Third lemma for the deriva... |
| adh-minimp-sylsimp 46462 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 46463 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 46464 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 46465 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 46466 | Fourth lemma for the deriv... |
| adh-minimp-ax2 46467 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 46468 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 46469 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 46470 | If a class with one elemen... |
| eusnsn 46471 | There is a unique element ... |
| absnsb 46472 | If the class abstraction `... |
| euabsneu 46473 | Another way to express exi... |
| elprneb 46474 | An element of a proper uno... |
| oppr 46475 | Equality for ordered pairs... |
| opprb 46476 | Equality for unordered pai... |
| or2expropbilem1 46477 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 46478 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 46479 | If two classes are strictl... |
| eubrv 46480 | If there is a unique set w... |
| eubrdm 46481 | If there is a unique set w... |
| eldmressn 46482 | Element of the domain of a... |
| iota0def 46483 | Example for a defined iota... |
| iota0ndef 46484 | Example for an undefined i... |
| fveqvfvv 46485 | If a function's value at a... |
| fnresfnco 46486 | Composition of two functio... |
| funcoressn 46487 | A composition restricted t... |
| funressnfv 46488 | A restriction to a singlet... |
| funressndmfvrn 46489 | The value of a function ` ... |
| funressnvmo 46490 | A function restricted to a... |
| funressnmo 46491 | A function restricted to a... |
| funressneu 46492 | There is exactly one value... |
| fresfo 46493 | Conditions for a restricti... |
| fsetsniunop 46494 | The class of all functions... |
| fsetabsnop 46495 | The class of all functions... |
| fsetsnf 46496 | The mapping of an element ... |
| fsetsnf1 46497 | The mapping of an element ... |
| fsetsnfo 46498 | The mapping of an element ... |
| fsetsnf1o 46499 | The mapping of an element ... |
| fsetsnprcnex 46500 | The class of all functions... |
| cfsetssfset 46501 | The class of constant func... |
| cfsetsnfsetfv 46502 | The function value of the ... |
| cfsetsnfsetf 46503 | The mapping of the class o... |
| cfsetsnfsetf1 46504 | The mapping of the class o... |
| cfsetsnfsetfo 46505 | The mapping of the class o... |
| cfsetsnfsetf1o 46506 | The mapping of the class o... |
| fsetprcnexALT 46507 | First version of proof for... |
| fcoreslem1 46508 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 46509 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 46510 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 46511 | Lemma 4 for ~ fcores . (C... |
| fcores 46512 | Every composite function `... |
| fcoresf1lem 46513 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 46514 | If a composition is inject... |
| fcoresf1b 46515 | A composition is injective... |
| fcoresfo 46516 | If a composition is surjec... |
| fcoresfob 46517 | A composition is surjectiv... |
| fcoresf1ob 46518 | A composition is bijective... |
| f1cof1blem 46519 | Lemma for ~ f1cof1b and ~ ... |
| f1cof1b 46520 | If the range of ` F ` equa... |
| funfocofob 46521 | If the domain of a functio... |
| fnfocofob 46522 | If the domain of a functio... |
| focofob 46523 | If the domain of a functio... |
| f1ocof1ob 46524 | If the range of ` F ` equa... |
| f1ocof1ob2 46525 | If the range of ` F ` equa... |
| aiotajust 46527 | Soundness justification th... |
| dfaiota2 46529 | Alternate definition of th... |
| reuabaiotaiota 46530 | The iota and the alternate... |
| reuaiotaiota 46531 | The iota and the alternate... |
| aiotaexb 46532 | The alternate iota over a ... |
| aiotavb 46533 | The alternate iota over a ... |
| aiotaint 46534 | This is to ~ df-aiota what... |
| dfaiota3 46535 | Alternate definition of ` ... |
| iotan0aiotaex 46536 | If the iota over a wff ` p... |
| aiotaexaiotaiota 46537 | The alternate iota over a ... |
| aiotaval 46538 | Theorem 8.19 in [Quine] p.... |
| aiota0def 46539 | Example for a defined alte... |
| aiota0ndef 46540 | Example for an undefined a... |
| r19.32 46541 | Theorem 19.32 of [Margaris... |
| rexsb 46542 | An equivalent expression f... |
| rexrsb 46543 | An equivalent expression f... |
| 2rexsb 46544 | An equivalent expression f... |
| 2rexrsb 46545 | An equivalent expression f... |
| cbvral2 46546 | Change bound variables of ... |
| cbvrex2 46547 | Change bound variables of ... |
| ralndv1 46548 | Example for a theorem abou... |
| ralndv2 46549 | Second example for a theor... |
| reuf1odnf 46550 | There is exactly one eleme... |
| reuf1od 46551 | There is exactly one eleme... |
| euoreqb 46552 | There is a set which is eq... |
| 2reu3 46553 | Double restricted existent... |
| 2reu7 46554 | Two equivalent expressions... |
| 2reu8 46555 | Two equivalent expressions... |
| 2reu8i 46556 | Implication of a double re... |
| 2reuimp0 46557 | Implication of a double re... |
| 2reuimp 46558 | Implication of a double re... |
| ralbinrald 46565 | Elemination of a restricte... |
| nvelim 46566 | If a class is the universa... |
| alneu 46567 | If a statement holds for a... |
| eu2ndop1stv 46568 | If there is a unique secon... |
| dfateq12d 46569 | Equality deduction for "de... |
| nfdfat 46570 | Bound-variable hypothesis ... |
| dfdfat2 46571 | Alternate definition of th... |
| fundmdfat 46572 | A function is defined at a... |
| dfatprc 46573 | A function is not defined ... |
| dfatelrn 46574 | The value of a function ` ... |
| dfafv2 46575 | Alternative definition of ... |
| afveq12d 46576 | Equality deduction for fun... |
| afveq1 46577 | Equality theorem for funct... |
| afveq2 46578 | Equality theorem for funct... |
| nfafv 46579 | Bound-variable hypothesis ... |
| csbafv12g 46580 | Move class substitution in... |
| afvfundmfveq 46581 | If a class is a function r... |
| afvnfundmuv 46582 | If a set is not in the dom... |
| ndmafv 46583 | The value of a class outsi... |
| afvvdm 46584 | If the function value of a... |
| nfunsnafv 46585 | If the restriction of a cl... |
| afvvfunressn 46586 | If the function value of a... |
| afvprc 46587 | A function's value at a pr... |
| afvvv 46588 | If a function's value at a... |
| afvpcfv0 46589 | If the value of the altern... |
| afvnufveq 46590 | The value of the alternati... |
| afvvfveq 46591 | The value of the alternati... |
| afv0fv0 46592 | If the value of the altern... |
| afvfvn0fveq 46593 | If the function's value at... |
| afv0nbfvbi 46594 | The function's value at an... |
| afvfv0bi 46595 | The function's value at an... |
| afveu 46596 | The value of a function at... |
| fnbrafvb 46597 | Equivalence of function va... |
| fnopafvb 46598 | Equivalence of function va... |
| funbrafvb 46599 | Equivalence of function va... |
| funopafvb 46600 | Equivalence of function va... |
| funbrafv 46601 | The second argument of a b... |
| funbrafv2b 46602 | Function value in terms of... |
| dfafn5a 46603 | Representation of a functi... |
| dfafn5b 46604 | Representation of a functi... |
| fnrnafv 46605 | The range of a function ex... |
| afvelrnb 46606 | A member of a function's r... |
| afvelrnb0 46607 | A member of a function's r... |
| dfaimafn 46608 | Alternate definition of th... |
| dfaimafn2 46609 | Alternate definition of th... |
| afvelima 46610 | Function value in an image... |
| afvelrn 46611 | A function's value belongs... |
| fnafvelrn 46612 | A function's value belongs... |
| fafvelcdm 46613 | A function's value belongs... |
| ffnafv 46614 | A function maps to a class... |
| afvres 46615 | The value of a restricted ... |
| tz6.12-afv 46616 | Function value. Theorem 6... |
| tz6.12-1-afv 46617 | Function value (Theorem 6.... |
| dmfcoafv 46618 | Domains of a function comp... |
| afvco2 46619 | Value of a function compos... |
| rlimdmafv 46620 | Two ways to express that a... |
| aoveq123d 46621 | Equality deduction for ope... |
| nfaov 46622 | Bound-variable hypothesis ... |
| csbaovg 46623 | Move class substitution in... |
| aovfundmoveq 46624 | If a class is a function r... |
| aovnfundmuv 46625 | If an ordered pair is not ... |
| ndmaov 46626 | The value of an operation ... |
| ndmaovg 46627 | The value of an operation ... |
| aovvdm 46628 | If the operation value of ... |
| nfunsnaov 46629 | If the restriction of a cl... |
| aovvfunressn 46630 | If the operation value of ... |
| aovprc 46631 | The value of an operation ... |
| aovrcl 46632 | Reverse closure for an ope... |
| aovpcov0 46633 | If the alternative value o... |
| aovnuoveq 46634 | The alternative value of t... |
| aovvoveq 46635 | The alternative value of t... |
| aov0ov0 46636 | If the alternative value o... |
| aovovn0oveq 46637 | If the operation's value a... |
| aov0nbovbi 46638 | The operation's value on a... |
| aovov0bi 46639 | The operation's value on a... |
| rspceaov 46640 | A frequently used special ... |
| fnotaovb 46641 | Equivalence of operation v... |
| ffnaov 46642 | An operation maps to a cla... |
| faovcl 46643 | Closure law for an operati... |
| aovmpt4g 46644 | Value of a function given ... |
| aoprssdm 46645 | Domain of closure of an op... |
| ndmaovcl 46646 | The "closure" of an operat... |
| ndmaovrcl 46647 | Reverse closure law, in co... |
| ndmaovcom 46648 | Any operation is commutati... |
| ndmaovass 46649 | Any operation is associati... |
| ndmaovdistr 46650 | Any operation is distribut... |
| dfatafv2iota 46653 | If a function is defined a... |
| ndfatafv2 46654 | The alternate function val... |
| ndfatafv2undef 46655 | The alternate function val... |
| dfatafv2ex 46656 | The alternate function val... |
| afv2ex 46657 | The alternate function val... |
| afv2eq12d 46658 | Equality deduction for fun... |
| afv2eq1 46659 | Equality theorem for funct... |
| afv2eq2 46660 | Equality theorem for funct... |
| nfafv2 46661 | Bound-variable hypothesis ... |
| csbafv212g 46662 | Move class substitution in... |
| fexafv2ex 46663 | The alternate function val... |
| ndfatafv2nrn 46664 | The alternate function val... |
| ndmafv2nrn 46665 | The value of a class outsi... |
| funressndmafv2rn 46666 | The alternate function val... |
| afv2ndefb 46667 | Two ways to say that an al... |
| nfunsnafv2 46668 | If the restriction of a cl... |
| afv2prc 46669 | A function's value at a pr... |
| dfatafv2rnb 46670 | The alternate function val... |
| afv2orxorb 46671 | If a set is in the range o... |
| dmafv2rnb 46672 | The alternate function val... |
| fundmafv2rnb 46673 | The alternate function val... |
| afv2elrn 46674 | An alternate function valu... |
| afv20defat 46675 | If the alternate function ... |
| fnafv2elrn 46676 | An alternate function valu... |
| fafv2elcdm 46677 | An alternate function valu... |
| fafv2elrnb 46678 | An alternate function valu... |
| fcdmvafv2v 46679 | If the codomain of a funct... |
| tz6.12-2-afv2 46680 | Function value when ` F ` ... |
| afv2eu 46681 | The value of a function at... |
| afv2res 46682 | The value of a restricted ... |
| tz6.12-afv2 46683 | Function value (Theorem 6.... |
| tz6.12-1-afv2 46684 | Function value (Theorem 6.... |
| tz6.12c-afv2 46685 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 46686 | Corollary of Theorem 6.12(... |
| funressnbrafv2 46687 | The second argument of a b... |
| dfatbrafv2b 46688 | Equivalence of function va... |
| dfatopafv2b 46689 | Equivalence of function va... |
| funbrafv2 46690 | The second argument of a b... |
| fnbrafv2b 46691 | Equivalence of function va... |
| fnopafv2b 46692 | Equivalence of function va... |
| funbrafv22b 46693 | Equivalence of function va... |
| funopafv2b 46694 | Equivalence of function va... |
| dfatsnafv2 46695 | Singleton of function valu... |
| dfafv23 46696 | A definition of function v... |
| dfatdmfcoafv2 46697 | Domain of a function compo... |
| dfatcolem 46698 | Lemma for ~ dfatco . (Con... |
| dfatco 46699 | The predicate "defined at"... |
| afv2co2 46700 | Value of a function compos... |
| rlimdmafv2 46701 | Two ways to express that a... |
| dfafv22 46702 | Alternate definition of ` ... |
| afv2ndeffv0 46703 | If the alternate function ... |
| dfatafv2eqfv 46704 | If a function is defined a... |
| afv2rnfveq 46705 | If the alternate function ... |
| afv20fv0 46706 | If the alternate function ... |
| afv2fvn0fveq 46707 | If the function's value at... |
| afv2fv0 46708 | If the function's value at... |
| afv2fv0b 46709 | The function's value at an... |
| afv2fv0xorb 46710 | If a set is in the range o... |
| an4com24 46711 | Rearrangement of 4 conjunc... |
| 3an4ancom24 46712 | Commutative law for a conj... |
| 4an21 46713 | Rearrangement of 4 conjunc... |
| dfnelbr2 46716 | Alternate definition of th... |
| nelbr 46717 | The binary relation of a s... |
| nelbrim 46718 | If a set is related to ano... |
| nelbrnel 46719 | A set is related to anothe... |
| nelbrnelim 46720 | If a set is related to ano... |
| ralralimp 46721 | Selecting one of two alter... |
| otiunsndisjX 46722 | The union of singletons co... |
| fvifeq 46723 | Equality of function value... |
| rnfdmpr 46724 | The range of a one-to-one ... |
| imarnf1pr 46725 | The image of the range of ... |
| funop1 46726 | A function is an ordered p... |
| fun2dmnopgexmpl 46727 | A function with a domain c... |
| opabresex0d 46728 | A collection of ordered pa... |
| opabbrfex0d 46729 | A collection of ordered pa... |
| opabresexd 46730 | A collection of ordered pa... |
| opabbrfexd 46731 | A collection of ordered pa... |
| f1oresf1orab 46732 | Build a bijection by restr... |
| f1oresf1o 46733 | Build a bijection by restr... |
| f1oresf1o2 46734 | Build a bijection by restr... |
| fvmptrab 46735 | Value of a function mappin... |
| fvmptrabdm 46736 | Value of a function mappin... |
| cnambpcma 46737 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 46738 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 46739 | The sum of two complex num... |
| leaddsuble 46740 | Addition and subtraction o... |
| 2leaddle2 46741 | If two real numbers are le... |
| ltnltne 46742 | Variant of trichotomy law ... |
| p1lep2 46743 | A real number increasd by ... |
| ltsubsubaddltsub 46744 | If the result of subtracti... |
| zm1nn 46745 | An integer minus 1 is posi... |
| readdcnnred 46746 | The sum of a real number a... |
| resubcnnred 46747 | The difference of a real n... |
| recnmulnred 46748 | The product of a real numb... |
| cndivrenred 46749 | The quotient of an imagina... |
| sqrtnegnre 46750 | The square root of a negat... |
| nn0resubcl 46751 | Closure law for subtractio... |
| zgeltp1eq 46752 | If an integer is between a... |
| 1t10e1p1e11 46753 | 11 is 1 times 10 to the po... |
| deccarry 46754 | Add 1 to a 2 digit number ... |
| eluzge0nn0 46755 | If an integer is greater t... |
| nltle2tri 46756 | Negated extended trichotom... |
| ssfz12 46757 | Subset relationship for fi... |
| elfz2z 46758 | Membership of an integer i... |
| 2elfz3nn0 46759 | If there are two elements ... |
| fz0addcom 46760 | The addition of two member... |
| 2elfz2melfz 46761 | If the sum of two integers... |
| fz0addge0 46762 | The sum of two integers in... |
| elfzlble 46763 | Membership of an integer i... |
| elfzelfzlble 46764 | Membership of an element o... |
| fzopred 46765 | Join a predecessor to the ... |
| fzopredsuc 46766 | Join a predecessor and a s... |
| 1fzopredsuc 46767 | Join 0 and a successor to ... |
| el1fzopredsuc 46768 | An element of an open inte... |
| subsubelfzo0 46769 | Subtracting a difference f... |
| fzoopth 46770 | A half-open integer range ... |
| 2ffzoeq 46771 | Two functions over a half-... |
| m1mod0mod1 46772 | An integer decreased by 1 ... |
| elmod2 46773 | An integer modulo 2 is eit... |
| smonoord 46774 | Ordering relation for a st... |
| fsummsndifre 46775 | A finite sum with one of i... |
| fsumsplitsndif 46776 | Separate out a term in a f... |
| fsummmodsndifre 46777 | A finite sum of summands m... |
| fsummmodsnunz 46778 | A finite sum of summands m... |
| setsidel 46779 | The injected slot is an el... |
| setsnidel 46780 | The injected slot is an el... |
| setsv 46781 | The value of the structure... |
| preimafvsnel 46782 | The preimage of a function... |
| preimafvn0 46783 | The preimage of a function... |
| uniimafveqt 46784 | The union of the image of ... |
| uniimaprimaeqfv 46785 | The union of the image of ... |
| setpreimafvex 46786 | The class ` P ` of all pre... |
| elsetpreimafvb 46787 | The characterization of an... |
| elsetpreimafv 46788 | An element of the class ` ... |
| elsetpreimafvssdm 46789 | An element of the class ` ... |
| fvelsetpreimafv 46790 | There is an element in a p... |
| preimafvelsetpreimafv 46791 | The preimage of a function... |
| preimafvsspwdm 46792 | The class ` P ` of all pre... |
| 0nelsetpreimafv 46793 | The empty set is not an el... |
| elsetpreimafvbi 46794 | An element of the preimage... |
| elsetpreimafveqfv 46795 | The elements of the preima... |
| eqfvelsetpreimafv 46796 | If an element of the domai... |
| elsetpreimafvrab 46797 | An element of the preimage... |
| imaelsetpreimafv 46798 | The image of an element of... |
| uniimaelsetpreimafv 46799 | The union of the image of ... |
| elsetpreimafveq 46800 | If two preimages of functi... |
| fundcmpsurinjlem1 46801 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 46802 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 46803 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 46804 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 46805 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 46806 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 46807 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 46808 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 46809 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 46810 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 46811 | Every function ` F : A -->... |
| fundcmpsurinj 46812 | Every function ` F : A -->... |
| fundcmpsurbijinj 46813 | Every function ` F : A -->... |
| fundcmpsurinjimaid 46814 | Every function ` F : A -->... |
| fundcmpsurinjALT 46815 | Alternate proof of ~ fundc... |
| iccpval 46818 | Partition consisting of a ... |
| iccpart 46819 | A special partition. Corr... |
| iccpartimp 46820 | Implications for a class b... |
| iccpartres 46821 | The restriction of a parti... |
| iccpartxr 46822 | If there is a partition, t... |
| iccpartgtprec 46823 | If there is a partition, t... |
| iccpartipre 46824 | If there is a partition, t... |
| iccpartiltu 46825 | If there is a partition, t... |
| iccpartigtl 46826 | If there is a partition, t... |
| iccpartlt 46827 | If there is a partition, t... |
| iccpartltu 46828 | If there is a partition, t... |
| iccpartgtl 46829 | If there is a partition, t... |
| iccpartgt 46830 | If there is a partition, t... |
| iccpartleu 46831 | If there is a partition, t... |
| iccpartgel 46832 | If there is a partition, t... |
| iccpartrn 46833 | If there is a partition, t... |
| iccpartf 46834 | The range of the partition... |
| iccpartel 46835 | If there is a partition, t... |
| iccelpart 46836 | An element of any partitio... |
| iccpartiun 46837 | A half-open interval of ex... |
| icceuelpartlem 46838 | Lemma for ~ icceuelpart . ... |
| icceuelpart 46839 | An element of a partitione... |
| iccpartdisj 46840 | The segments of a partitio... |
| iccpartnel 46841 | A point of a partition is ... |
| fargshiftfv 46842 | If a class is a function, ... |
| fargshiftf 46843 | If a class is a function, ... |
| fargshiftf1 46844 | If a function is 1-1, then... |
| fargshiftfo 46845 | If a function is onto, the... |
| fargshiftfva 46846 | The values of a shifted fu... |
| lswn0 46847 | The last symbol of a not e... |
| nfich1 46850 | The first interchangeable ... |
| nfich2 46851 | The second interchangeable... |
| ichv 46852 | Setvar variables are inter... |
| ichf 46853 | Setvar variables are inter... |
| ichid 46854 | A setvar variable is alway... |
| icht 46855 | A theorem is interchangeab... |
| ichbidv 46856 | Formula building rule for ... |
| ichcircshi 46857 | The setvar variables are i... |
| ichan 46858 | If two setvar variables ar... |
| ichn 46859 | Negation does not affect i... |
| ichim 46860 | Formula building rule for ... |
| dfich2 46861 | Alternate definition of th... |
| ichcom 46862 | The interchangeability of ... |
| ichbi12i 46863 | Equivalence for interchang... |
| icheqid 46864 | In an equality for the sam... |
| icheq 46865 | In an equality of setvar v... |
| ichnfimlem 46866 | Lemma for ~ ichnfim : A s... |
| ichnfim 46867 | If in an interchangeabilit... |
| ichnfb 46868 | If ` x ` and ` y ` are int... |
| ichal 46869 | Move a universal quantifie... |
| ich2al 46870 | Two setvar variables are a... |
| ich2ex 46871 | Two setvar variables are a... |
| ichexmpl1 46872 | Example for interchangeabl... |
| ichexmpl2 46873 | Example for interchangeabl... |
| ich2exprop 46874 | If the setvar variables ar... |
| ichnreuop 46875 | If the setvar variables ar... |
| ichreuopeq 46876 | If the setvar variables ar... |
| sprid 46877 | Two identical representati... |
| elsprel 46878 | An unordered pair is an el... |
| spr0nelg 46879 | The empty set is not an el... |
| sprval 46882 | The set of all unordered p... |
| sprvalpw 46883 | The set of all unordered p... |
| sprssspr 46884 | The set of all unordered p... |
| spr0el 46885 | The empty set is not an un... |
| sprvalpwn0 46886 | The set of all unordered p... |
| sprel 46887 | An element of the set of a... |
| prssspr 46888 | An element of a subset of ... |
| prelspr 46889 | An unordered pair of eleme... |
| prsprel 46890 | The elements of a pair fro... |
| prsssprel 46891 | The elements of a pair fro... |
| sprvalpwle2 46892 | The set of all unordered p... |
| sprsymrelfvlem 46893 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 46894 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 46895 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 46896 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 46897 | The value of the function ... |
| sprsymrelf 46898 | The mapping ` F ` is a fun... |
| sprsymrelf1 46899 | The mapping ` F ` is a one... |
| sprsymrelfo 46900 | The mapping ` F ` is a fun... |
| sprsymrelf1o 46901 | The mapping ` F ` is a bij... |
| sprbisymrel 46902 | There is a bijection betwe... |
| sprsymrelen 46903 | The class ` P ` of subsets... |
| prpair 46904 | Characterization of a prop... |
| prproropf1olem0 46905 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 46906 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 46907 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 46908 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 46909 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 46910 | There is a bijection betwe... |
| prproropen 46911 | The set of proper pairs an... |
| prproropreud 46912 | There is exactly one order... |
| pairreueq 46913 | Two equivalent representat... |
| paireqne 46914 | Two sets are not equal iff... |
| prprval 46917 | The set of all proper unor... |
| prprvalpw 46918 | The set of all proper unor... |
| prprelb 46919 | An element of the set of a... |
| prprelprb 46920 | A set is an element of the... |
| prprspr2 46921 | The set of all proper unor... |
| prprsprreu 46922 | There is a unique proper u... |
| prprreueq 46923 | There is a unique proper u... |
| sbcpr 46924 | The proper substitution of... |
| reupr 46925 | There is a unique unordere... |
| reuprpr 46926 | There is a unique proper u... |
| poprelb 46927 | Equality for unordered pai... |
| 2exopprim 46928 | The existence of an ordere... |
| reuopreuprim 46929 | There is a unique unordere... |
| fmtno 46932 | The ` N ` th Fermat number... |
| fmtnoge3 46933 | Each Fermat number is grea... |
| fmtnonn 46934 | Each Fermat number is a po... |
| fmtnom1nn 46935 | A Fermat number minus one ... |
| fmtnoodd 46936 | Each Fermat number is odd.... |
| fmtnorn 46937 | A Fermat number is a funct... |
| fmtnof1 46938 | The enumeration of the Fer... |
| fmtnoinf 46939 | The set of Fermat numbers ... |
| fmtnorec1 46940 | The first recurrence relat... |
| sqrtpwpw2p 46941 | The floor of the square ro... |
| fmtnosqrt 46942 | The floor of the square ro... |
| fmtno0 46943 | The ` 0 ` th Fermat number... |
| fmtno1 46944 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 46945 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 46946 | The second recurrence rela... |
| fmtnodvds 46947 | Any Fermat number divides ... |
| goldbachthlem1 46948 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 46949 | Lemma 2 for ~ goldbachth .... |
| goldbachth 46950 | Goldbach's theorem: Two d... |
| fmtnorec3 46951 | The third recurrence relat... |
| fmtnorec4 46952 | The fourth recurrence rela... |
| fmtno2 46953 | The ` 2 ` nd Fermat number... |
| fmtno3 46954 | The ` 3 ` rd Fermat number... |
| fmtno4 46955 | The ` 4 ` th Fermat number... |
| fmtno5lem1 46956 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 46957 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 46958 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 46959 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 46960 | The ` 5 ` th Fermat number... |
| fmtno0prm 46961 | The ` 0 ` th Fermat number... |
| fmtno1prm 46962 | The ` 1 ` st Fermat number... |
| fmtno2prm 46963 | The ` 2 ` nd Fermat number... |
| 257prm 46964 | 257 is a prime number (the... |
| fmtno3prm 46965 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 46966 | Any power of two is coprim... |
| fmtnoprmfac1lem 46967 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 46968 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 46969 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 46970 | Divisor of Fermat number (... |
| fmtnofac2lem 46971 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 46972 | Divisor of Fermat number (... |
| fmtnofac1 46973 | Divisor of Fermat number (... |
| fmtno4sqrt 46974 | The floor of the square ro... |
| fmtno4prmfac 46975 | If P was a (prime) factor ... |
| fmtno4prmfac193 46976 | If P was a (prime) factor ... |
| fmtno4nprmfac193 46977 | 193 is not a (prime) facto... |
| fmtno4prm 46978 | The ` 4 `-th Fermat number... |
| 65537prm 46979 | 65537 is a prime number (t... |
| fmtnofz04prm 46980 | The first five Fermat numb... |
| fmtnole4prm 46981 | The first five Fermat numb... |
| fmtno5faclem1 46982 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 46983 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 46984 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 46985 | The factorisation of the `... |
| fmtno5nprm 46986 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 46987 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 46988 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 46989 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 46990 | The mapping of a Fermat nu... |
| prminf2 46991 | The set of prime numbers i... |
| 2pwp1prm 46992 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 46993 | Every prime number of the ... |
| m2prm 46994 | The second Mersenne number... |
| m3prm 46995 | The third Mersenne number ... |
| flsqrt 46996 | A condition equivalent to ... |
| flsqrt5 46997 | The floor of the square ro... |
| 3ndvds4 46998 | 3 does not divide 4. (Con... |
| 139prmALT 46999 | 139 is a prime number. In... |
| 31prm 47000 | 31 is a prime number. In ... |
| m5prm 47001 | The fifth Mersenne number ... |
| 127prm 47002 | 127 is a prime number. (C... |
| m7prm 47003 | The seventh Mersenne numbe... |
| m11nprm 47004 | The eleventh Mersenne numb... |
| mod42tp1mod8 47005 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 47006 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 47007 | If ` P ` is a Sophie Germa... |
| lighneallem1 47008 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 47009 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 47010 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 47011 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 47012 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 47013 | Lemma 3 for ~ lighneal . ... |
| lighneal 47014 | If a power of a prime ` P ... |
| modexp2m1d 47015 | The square of an integer w... |
| proththdlem 47016 | Lemma for ~ proththd . (C... |
| proththd 47017 | Proth's theorem (1878). I... |
| 5tcu2e40 47018 | 5 times the cube of 2 is 4... |
| 3exp4mod41 47019 | 3 to the fourth power is -... |
| 41prothprmlem1 47020 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 47021 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 47022 | 41 is a _Proth prime_. (C... |
| quad1 47023 | A condition for a quadrati... |
| requad01 47024 | A condition for a quadrati... |
| requad1 47025 | A condition for a quadrati... |
| requad2 47026 | A condition for a quadrati... |
| iseven 47031 | The predicate "is an even ... |
| isodd 47032 | The predicate "is an odd n... |
| evenz 47033 | An even number is an integ... |
| oddz 47034 | An odd number is an intege... |
| evendiv2z 47035 | The result of dividing an ... |
| oddp1div2z 47036 | The result of dividing an ... |
| oddm1div2z 47037 | The result of dividing an ... |
| isodd2 47038 | The predicate "is an odd n... |
| dfodd2 47039 | Alternate definition for o... |
| dfodd6 47040 | Alternate definition for o... |
| dfeven4 47041 | Alternate definition for e... |
| evenm1odd 47042 | The predecessor of an even... |
| evenp1odd 47043 | The successor of an even n... |
| oddp1eveni 47044 | The successor of an odd nu... |
| oddm1eveni 47045 | The predecessor of an odd ... |
| evennodd 47046 | An even number is not an o... |
| oddneven 47047 | An odd number is not an ev... |
| enege 47048 | The negative of an even nu... |
| onego 47049 | The negative of an odd num... |
| m1expevenALTV 47050 | Exponentiation of -1 by an... |
| m1expoddALTV 47051 | Exponentiation of -1 by an... |
| dfeven2 47052 | Alternate definition for e... |
| dfodd3 47053 | Alternate definition for o... |
| iseven2 47054 | The predicate "is an even ... |
| isodd3 47055 | The predicate "is an odd n... |
| 2dvdseven 47056 | 2 divides an even number. ... |
| m2even 47057 | A multiple of 2 is an even... |
| 2ndvdsodd 47058 | 2 does not divide an odd n... |
| 2dvdsoddp1 47059 | 2 divides an odd number in... |
| 2dvdsoddm1 47060 | 2 divides an odd number de... |
| dfeven3 47061 | Alternate definition for e... |
| dfodd4 47062 | Alternate definition for o... |
| dfodd5 47063 | Alternate definition for o... |
| zefldiv2ALTV 47064 | The floor of an even numbe... |
| zofldiv2ALTV 47065 | The floor of an odd numer ... |
| oddflALTV 47066 | Odd number representation ... |
| iseven5 47067 | The predicate "is an even ... |
| isodd7 47068 | The predicate "is an odd n... |
| dfeven5 47069 | Alternate definition for e... |
| dfodd7 47070 | Alternate definition for o... |
| gcd2odd1 47071 | The greatest common diviso... |
| zneoALTV 47072 | No even integer equals an ... |
| zeoALTV 47073 | An integer is even or odd.... |
| zeo2ALTV 47074 | An integer is even or odd ... |
| nneoALTV 47075 | A positive integer is even... |
| nneoiALTV 47076 | A positive integer is even... |
| odd2np1ALTV 47077 | An integer is odd iff it i... |
| oddm1evenALTV 47078 | An integer is odd iff its ... |
| oddp1evenALTV 47079 | An integer is odd iff its ... |
| oexpnegALTV 47080 | The exponential of the neg... |
| oexpnegnz 47081 | The exponential of the neg... |
| bits0ALTV 47082 | Value of the zeroth bit. ... |
| bits0eALTV 47083 | The zeroth bit of an even ... |
| bits0oALTV 47084 | The zeroth bit of an odd n... |
| divgcdoddALTV 47085 | Either ` A / ( A gcd B ) `... |
| opoeALTV 47086 | The sum of two odds is eve... |
| opeoALTV 47087 | The sum of an odd and an e... |
| omoeALTV 47088 | The difference of two odds... |
| omeoALTV 47089 | The difference of an odd a... |
| oddprmALTV 47090 | A prime not equal to ` 2 `... |
| 0evenALTV 47091 | 0 is an even number. (Con... |
| 0noddALTV 47092 | 0 is not an odd number. (... |
| 1oddALTV 47093 | 1 is an odd number. (Cont... |
| 1nevenALTV 47094 | 1 is not an even number. ... |
| 2evenALTV 47095 | 2 is an even number. (Con... |
| 2noddALTV 47096 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 47097 | An odd nonnegative integer... |
| nnoALTV 47098 | An alternate characterizat... |
| nn0oALTV 47099 | An alternate characterizat... |
| nn0e 47100 | An alternate characterizat... |
| nneven 47101 | An alternate characterizat... |
| nn0onn0exALTV 47102 | For each odd nonnegative i... |
| nn0enn0exALTV 47103 | For each even nonnegative ... |
| nnennexALTV 47104 | For each even positive int... |
| nnpw2evenALTV 47105 | 2 to the power of a positi... |
| epoo 47106 | The sum of an even and an ... |
| emoo 47107 | The difference of an even ... |
| epee 47108 | The sum of two even number... |
| emee 47109 | The difference of two even... |
| evensumeven 47110 | If a summand is even, the ... |
| 3odd 47111 | 3 is an odd number. (Cont... |
| 4even 47112 | 4 is an even number. (Con... |
| 5odd 47113 | 5 is an odd number. (Cont... |
| 6even 47114 | 6 is an even number. (Con... |
| 7odd 47115 | 7 is an odd number. (Cont... |
| 8even 47116 | 8 is an even number. (Con... |
| evenprm2 47117 | A prime number is even iff... |
| oddprmne2 47118 | Every prime number not bei... |
| oddprmuzge3 47119 | A prime number which is od... |
| evenltle 47120 | If an even number is great... |
| odd2prm2 47121 | If an odd number is the su... |
| even3prm2 47122 | If an even number is the s... |
| mogoldbblem 47123 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 47124 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 47125 | Lemma for ~ perfectALTV . ... |
| perfectALTV 47126 | The Euclid-Euler theorem, ... |
| fppr 47129 | The set of Fermat pseudopr... |
| fpprmod 47130 | The set of Fermat pseudopr... |
| fpprel 47131 | A Fermat pseudoprime to th... |
| fpprbasnn 47132 | The base of a Fermat pseud... |
| fpprnn 47133 | A Fermat pseudoprime to th... |
| fppr2odd 47134 | A Fermat pseudoprime to th... |
| 11t31e341 47135 | 341 is the product of 11 a... |
| 2exp340mod341 47136 | Eight to the eighth power ... |
| 341fppr2 47137 | 341 is the (smallest) _Pou... |
| 4fppr1 47138 | 4 is the (smallest) Fermat... |
| 8exp8mod9 47139 | Eight to the eighth power ... |
| 9fppr8 47140 | 9 is the (smallest) Fermat... |
| dfwppr 47141 | Alternate definition of a ... |
| fpprwppr 47142 | A Fermat pseudoprime to th... |
| fpprwpprb 47143 | An integer ` X ` which is ... |
| fpprel2 47144 | An alternate definition fo... |
| nfermltl8rev 47145 | Fermat's little theorem wi... |
| nfermltl2rev 47146 | Fermat's little theorem wi... |
| nfermltlrev 47147 | Fermat's little theorem re... |
| isgbe 47154 | The predicate "is an even ... |
| isgbow 47155 | The predicate "is a weak o... |
| isgbo 47156 | The predicate "is an odd G... |
| gbeeven 47157 | An even Goldbach number is... |
| gbowodd 47158 | A weak odd Goldbach number... |
| gbogbow 47159 | A (strong) odd Goldbach nu... |
| gboodd 47160 | An odd Goldbach number is ... |
| gbepos 47161 | Any even Goldbach number i... |
| gbowpos 47162 | Any weak odd Goldbach numb... |
| gbopos 47163 | Any odd Goldbach number is... |
| gbegt5 47164 | Any even Goldbach number i... |
| gbowgt5 47165 | Any weak odd Goldbach numb... |
| gbowge7 47166 | Any weak odd Goldbach numb... |
| gboge9 47167 | Any odd Goldbach number is... |
| gbege6 47168 | Any even Goldbach number i... |
| gbpart6 47169 | The Goldbach partition of ... |
| gbpart7 47170 | The (weak) Goldbach partit... |
| gbpart8 47171 | The Goldbach partition of ... |
| gbpart9 47172 | The (strong) Goldbach part... |
| gbpart11 47173 | The (strong) Goldbach part... |
| 6gbe 47174 | 6 is an even Goldbach numb... |
| 7gbow 47175 | 7 is a weak odd Goldbach n... |
| 8gbe 47176 | 8 is an even Goldbach numb... |
| 9gbo 47177 | 9 is an odd Goldbach numbe... |
| 11gbo 47178 | 11 is an odd Goldbach numb... |
| stgoldbwt 47179 | If the strong ternary Gold... |
| sbgoldbwt 47180 | If the strong binary Goldb... |
| sbgoldbst 47181 | If the strong binary Goldb... |
| sbgoldbaltlem1 47182 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 47183 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 47184 | An alternate (related to t... |
| sbgoldbb 47185 | If the strong binary Goldb... |
| sgoldbeven3prm 47186 | If the binary Goldbach con... |
| sbgoldbm 47187 | If the strong binary Goldb... |
| mogoldbb 47188 | If the modern version of t... |
| sbgoldbmb 47189 | The strong binary Goldbach... |
| sbgoldbo 47190 | If the strong binary Goldb... |
| nnsum3primes4 47191 | 4 is the sum of at most 3 ... |
| nnsum4primes4 47192 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 47193 | Every prime is "the sum of... |
| nnsum4primesprm 47194 | Every prime is "the sum of... |
| nnsum3primesgbe 47195 | Any even Goldbach number i... |
| nnsum4primesgbe 47196 | Any even Goldbach number i... |
| nnsum3primesle9 47197 | Every integer greater than... |
| nnsum4primesle9 47198 | Every integer greater than... |
| nnsum4primesodd 47199 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 47200 | If the (strong) ternary Go... |
| evengpop3 47201 | If the (weak) ternary Gold... |
| evengpoap3 47202 | If the (strong) ternary Go... |
| nnsum4primeseven 47203 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 47204 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 47205 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 47206 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 47207 | If the binary Goldbach con... |
| bgoldbtbndlem1 47208 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 47209 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 47210 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 47211 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 47212 | If the binary Goldbach con... |
| tgoldbachgtALTV 47215 | Variant of Thierry Arnoux'... |
| bgoldbachlt 47216 | The binary Goldbach conjec... |
| tgblthelfgott 47218 | The ternary Goldbach conje... |
| tgoldbachlt 47219 | The ternary Goldbach conje... |
| tgoldbach 47220 | The ternary Goldbach conje... |
| clnbgrprc0 47223 | The closed neighborhood is... |
| clnbgrcl 47224 | If a class ` X ` has at le... |
| clnbgrval 47225 | The closed neighborhood of... |
| dfclnbgr2 47226 | Alternate definition of th... |
| dfclnbgr4 47227 | Alternate definition of th... |
| dfclnbgr3 47228 | Alternate definition of th... |
| clnbgrnvtx0 47229 | If a class ` X ` is not a ... |
| clnbgrel 47230 | Characterization of a memb... |
| clnbgrvtxel 47231 | Every vertex ` K ` is a me... |
| clnbgrisvtx 47232 | Every member ` N ` of the ... |
| clnbgrssvtx 47233 | The closed neighborhood of... |
| clnbgrn0 47234 | The closed neighborhood of... |
| clnbupgr 47235 | The closed neighborhood of... |
| clnbupgrel 47236 | A member of the closed nei... |
| clnbgr0vtx 47237 | In a null graph (with no v... |
| clnbgr0edg 47238 | In an empty graph (with no... |
| clnbgrsym 47239 | In a graph, the closed nei... |
| edgusgrclnbfin 47240 | The size of the closed nei... |
| clnbusgrfi 47241 | The closed neighborhood of... |
| clnbfiusgrfi 47242 | The closed neighborhood of... |
| clnbgrlevtx 47243 | The size of the closed nei... |
| dfsclnbgr2 47244 | Alternate definition of th... |
| sclnbgrel 47245 | Characterization of a memb... |
| sclnbgrelself 47246 | A vertex ` N ` is a member... |
| sclnbgrisvtx 47247 | Every member ` X ` of the ... |
| dfclnbgr5 47248 | Alternate definition of th... |
| dfnbgr5 47249 | Alternate definition of th... |
| dfnbgrss 47250 | Subset chain for different... |
| dfvopnbgr2 47251 | Alternate definition of th... |
| vopnbgrel 47252 | Characterization of a memb... |
| vopnbgrelself 47253 | A vertex ` N ` is a member... |
| dfclnbgr6 47254 | Alternate definition of th... |
| dfnbgr6 47255 | Alternate definition of th... |
| dfsclnbgr6 47256 | Alternate definition of a ... |
| dfnbgrss2 47257 | Subset chain for different... |
| isisubgr 47260 | The subgraph induced by a ... |
| isubgriedg 47261 | The edges of an induced su... |
| isubgrvtxuhgr 47262 | The subgraph induced by th... |
| isubgrvtx 47263 | The vertices of an induced... |
| isubgruhgr 47264 | An induced subgraph of a h... |
| isubgrsubgr 47265 | An induced subgraph of a h... |
| isubgrupgr 47266 | An induced subgraph of a p... |
| isubgrumgr 47267 | An induced subgraph of a m... |
| isubgrusgr 47268 | An induced subgraph of a s... |
| isubgr0uhgr 47269 | The subgraph induced by an... |
| grimfn 47275 | The graph isomorphism func... |
| grimdmrel 47276 | The domain of the graph is... |
| isgrim 47278 | An isomorphism of graphs i... |
| grimprop 47279 | Properties of an isomorphi... |
| grimf1o 47280 | An isomorphism of graphs i... |
| isuspgrim0lem 47281 | An isomorphism of simple p... |
| isuspgrim0 47282 | An isomorphism of simple p... |
| uspgrimprop 47283 | An isomorphism of simple p... |
| isuspgrimlem 47284 | Lemma for ~ isuspgrim . (... |
| isuspgrim 47285 | A class is an isomorphism ... |
| grimidvtxedg 47286 | The identity relation rest... |
| grimid 47287 | The identity relation rest... |
| grimuhgr 47288 | If there is a graph isomor... |
| grimcnv 47289 | The converse of a graph is... |
| grimco 47290 | The composition of graph i... |
| brgric 47291 | The relation "is isomorphi... |
| brgrici 47292 | Prove that two graphs are ... |
| dfgric2 47293 | Alternate, explicit defini... |
| gricbri 47294 | Implications of two graphs... |
| gricushgr 47295 | The "is isomorphic to" rel... |
| gricuspgr 47296 | The "is isomorphic to" rel... |
| gricrel 47297 | The "is isomorphic to" rel... |
| gricref 47298 | Graph isomorphism is refle... |
| gricsym 47299 | Graph isomorphism is symme... |
| gricsymb 47300 | Graph isomorphism is symme... |
| grictr 47301 | Graph isomorphism is trans... |
| gricer 47302 | Isomorphism is an equivale... |
| gricen 47303 | Isomorphic graphs have equ... |
| opstrgric 47304 | A graph represented as an ... |
| ushggricedg 47305 | A simple hypergraph (with ... |
| 1hegrlfgr 47306 | A graph ` G ` with one hyp... |
| upwlksfval 47309 | The set of simple walks (i... |
| isupwlk 47310 | Properties of a pair of fu... |
| isupwlkg 47311 | Generalization of ~ isupwl... |
| upwlkbprop 47312 | Basic properties of a simp... |
| upwlkwlk 47313 | A simple walk is a walk. ... |
| upgrwlkupwlk 47314 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 47315 | In a pseudograph, the defi... |
| upgrisupwlkALT 47316 | Alternate proof of ~ upgri... |
| upgredgssspr 47317 | The set of edges of a pseu... |
| uspgropssxp 47318 | The set ` G ` of "simple p... |
| uspgrsprfv 47319 | The value of the function ... |
| uspgrsprf 47320 | The mapping ` F ` is a fun... |
| uspgrsprf1 47321 | The mapping ` F ` is a one... |
| uspgrsprfo 47322 | The mapping ` F ` is a fun... |
| uspgrsprf1o 47323 | The mapping ` F ` is a bij... |
| uspgrex 47324 | The class ` G ` of all "si... |
| uspgrbispr 47325 | There is a bijection betwe... |
| uspgrspren 47326 | The set ` G ` of the "simp... |
| uspgrymrelen 47327 | The set ` G ` of the "simp... |
| uspgrbisymrel 47328 | There is a bijection betwe... |
| uspgrbisymrelALT 47329 | Alternate proof of ~ uspgr... |
| ovn0dmfun 47330 | If a class operation value... |
| xpsnopab 47331 | A Cartesian product with a... |
| xpiun 47332 | A Cartesian product expres... |
| ovn0ssdmfun 47333 | If a class' operation valu... |
| fnxpdmdm 47334 | The domain of the domain o... |
| cnfldsrngbas 47335 | The base set of a subring ... |
| cnfldsrngadd 47336 | The group addition operati... |
| cnfldsrngmul 47337 | The ring multiplication op... |
| plusfreseq 47338 | If the empty set is not co... |
| mgmplusfreseq 47339 | If the empty set is not co... |
| 0mgm 47340 | A set with an empty base s... |
| opmpoismgm 47341 | A structure with a group a... |
| copissgrp 47342 | A structure with a constan... |
| copisnmnd 47343 | A structure with a constan... |
| 0nodd 47344 | 0 is not an odd integer. ... |
| 1odd 47345 | 1 is an odd integer. (Con... |
| 2nodd 47346 | 2 is not an odd integer. ... |
| oddibas 47347 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 47348 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 47349 | The structure of all odd i... |
| nnsgrpmgm 47350 | The structure of positive ... |
| nnsgrp 47351 | The structure of positive ... |
| nnsgrpnmnd 47352 | The structure of positive ... |
| nn0mnd 47353 | The set of nonnegative int... |
| gsumsplit2f 47354 | Split a group sum into two... |
| gsumdifsndf 47355 | Extract a summand from a f... |
| gsumfsupp 47356 | A group sum of a family ca... |
| iscllaw 47363 | The predicate "is a closed... |
| iscomlaw 47364 | The predicate "is a commut... |
| clcllaw 47365 | Closure of a closed operat... |
| isasslaw 47366 | The predicate "is an assoc... |
| asslawass 47367 | Associativity of an associ... |
| mgmplusgiopALT 47368 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 47369 | Slot 2 (group operation) o... |
| intopval 47376 | The internal (binary) oper... |
| intop 47377 | An internal (binary) opera... |
| clintopval 47378 | The closed (internal binar... |
| assintopval 47379 | The associative (closed in... |
| assintopmap 47380 | The associative (closed in... |
| isclintop 47381 | The predicate "is a closed... |
| clintop 47382 | A closed (internal binary)... |
| assintop 47383 | An associative (closed int... |
| isassintop 47384 | The predicate "is an assoc... |
| clintopcllaw 47385 | The closure law holds for ... |
| assintopcllaw 47386 | The closure low holds for ... |
| assintopasslaw 47387 | The associative low holds ... |
| assintopass 47388 | An associative (closed int... |
| ismgmALT 47397 | The predicate "is a magma"... |
| iscmgmALT 47398 | The predicate "is a commut... |
| issgrpALT 47399 | The predicate "is a semigr... |
| iscsgrpALT 47400 | The predicate "is a commut... |
| mgm2mgm 47401 | Equivalence of the two def... |
| sgrp2sgrp 47402 | Equivalence of the two def... |
| lmod0rng 47403 | If the scalar ring of a mo... |
| nzrneg1ne0 47404 | The additive inverse of th... |
| lidldomn1 47405 | If a (left) ideal (which i... |
| lidlabl 47406 | A (left) ideal of a ring i... |
| lidlrng 47407 | A (left) ideal of a ring i... |
| zlidlring 47408 | The zero (left) ideal of a... |
| uzlidlring 47409 | Only the zero (left) ideal... |
| lidldomnnring 47410 | A (left) ideal of a domain... |
| 0even 47411 | 0 is an even integer. (Co... |
| 1neven 47412 | 1 is not an even integer. ... |
| 2even 47413 | 2 is an even integer. (Co... |
| 2zlidl 47414 | The even integers are a (l... |
| 2zrng 47415 | The ring of integers restr... |
| 2zrngbas 47416 | The base set of R is the s... |
| 2zrngadd 47417 | The group addition operati... |
| 2zrng0 47418 | The additive identity of R... |
| 2zrngamgm 47419 | R is an (additive) magma. ... |
| 2zrngasgrp 47420 | R is an (additive) semigro... |
| 2zrngamnd 47421 | R is an (additive) monoid.... |
| 2zrngacmnd 47422 | R is a commutative (additi... |
| 2zrngagrp 47423 | R is an (additive) group. ... |
| 2zrngaabl 47424 | R is an (additive) abelian... |
| 2zrngmul 47425 | The ring multiplication op... |
| 2zrngmmgm 47426 | R is a (multiplicative) ma... |
| 2zrngmsgrp 47427 | R is a (multiplicative) se... |
| 2zrngALT 47428 | The ring of integers restr... |
| 2zrngnmlid 47429 | R has no multiplicative (l... |
| 2zrngnmrid 47430 | R has no multiplicative (r... |
| 2zrngnmlid2 47431 | R has no multiplicative (l... |
| 2zrngnring 47432 | R is not a unital ring. (... |
| cznrnglem 47433 | Lemma for ~ cznrng : The ... |
| cznabel 47434 | The ring constructed from ... |
| cznrng 47435 | The ring constructed from ... |
| cznnring 47436 | The ring constructed from ... |
| rngcvalALTV 47439 | Value of the category of n... |
| rngcbasALTV 47440 | Set of objects of the cate... |
| rngchomfvalALTV 47441 | Set of arrows of the categ... |
| rngchomALTV 47442 | Set of arrows of the categ... |
| elrngchomALTV 47443 | A morphism of non-unital r... |
| rngccofvalALTV 47444 | Composition in the categor... |
| rngccoALTV 47445 | Composition in the categor... |
| rngccatidALTV 47446 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 47447 | The category of non-unital... |
| rngcidALTV 47448 | The identity arrow in the ... |
| rngcsectALTV 47449 | A section in the category ... |
| rngcinvALTV 47450 | An inverse in the category... |
| rngcisoALTV 47451 | An isomorphism in the cate... |
| rngchomffvalALTV 47452 | The value of the functiona... |
| rngchomrnghmresALTV 47453 | The value of the functiona... |
| rngcrescrhmALTV 47454 | The category of non-unital... |
| rhmsubcALTVlem1 47455 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 47456 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 47457 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 47458 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 47459 | According to ~ df-subc , t... |
| rhmsubcALTVcat 47460 | The restriction of the cat... |
| ringcvalALTV 47463 | Value of the category of r... |
| funcringcsetcALTV2lem1 47464 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 47465 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 47466 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 47467 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 47468 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 47469 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 47470 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 47471 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 47472 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 47473 | The "natural forgetful fun... |
| ringcbasALTV 47474 | Set of objects of the cate... |
| ringchomfvalALTV 47475 | Set of arrows of the categ... |
| ringchomALTV 47476 | Set of arrows of the categ... |
| elringchomALTV 47477 | A morphism of rings is a f... |
| ringccofvalALTV 47478 | Composition in the categor... |
| ringccoALTV 47479 | Composition in the categor... |
| ringccatidALTV 47480 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 47481 | The category of rings is a... |
| ringcidALTV 47482 | The identity arrow in the ... |
| ringcsectALTV 47483 | A section in the category ... |
| ringcinvALTV 47484 | An inverse in the category... |
| ringcisoALTV 47485 | An isomorphism in the cate... |
| ringcbasbasALTV 47486 | An element of the base set... |
| funcringcsetclem1ALTV 47487 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 47488 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 47489 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 47490 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 47491 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 47492 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 47493 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 47494 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 47495 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 47496 | The "natural forgetful fun... |
| srhmsubcALTVlem1 47497 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 47498 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 47499 | According to ~ df-subc , t... |
| sringcatALTV 47500 | The restriction of the cat... |
| crhmsubcALTV 47501 | According to ~ df-subc , t... |
| cringcatALTV 47502 | The restriction of the cat... |
| drhmsubcALTV 47503 | According to ~ df-subc , t... |
| drngcatALTV 47504 | The restriction of the cat... |
| fldcatALTV 47505 | The restriction of the cat... |
| fldcALTV 47506 | The restriction of the cat... |
| fldhmsubcALTV 47507 | According to ~ df-subc , t... |
| opeliun2xp 47508 | Membership of an ordered p... |
| eliunxp2 47509 | Membership in a union of C... |
| mpomptx2 47510 | Express a two-argument fun... |
| cbvmpox2 47511 | Rule to change the bound v... |
| dmmpossx2 47512 | The domain of a mapping is... |
| mpoexxg2 47513 | Existence of an operation ... |
| ovmpordxf 47514 | Value of an operation give... |
| ovmpordx 47515 | Value of an operation give... |
| ovmpox2 47516 | The value of an operation ... |
| fdmdifeqresdif 47517 | The restriction of a condi... |
| offvalfv 47518 | The function operation exp... |
| ofaddmndmap 47519 | The function operation app... |
| mapsnop 47520 | A singleton of an ordered ... |
| fprmappr 47521 | A function with a domain o... |
| mapprop 47522 | An unordered pair containi... |
| ztprmneprm 47523 | A prime is not an integer ... |
| 2t6m3t4e0 47524 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 47525 | For any finite subset of `... |
| nn0sumltlt 47526 | If the sum of two nonnegat... |
| bcpascm1 47527 | Pascal's rule for the bino... |
| altgsumbc 47528 | The sum of binomial coeffi... |
| altgsumbcALT 47529 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 47530 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 47531 | An element of the (base se... |
| zlmodzxz0 47532 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 47533 | The scalar multiplication ... |
| zlmodzxzadd 47534 | The addition of the ` ZZ `... |
| zlmodzxzsubm 47535 | The subtraction of the ` Z... |
| zlmodzxzsub 47536 | The subtraction of the ` Z... |
| mgpsumunsn 47537 | Extract a summand/factor f... |
| mgpsumz 47538 | If the group sum for the m... |
| mgpsumn 47539 | If the group sum for the m... |
| exple2lt6 47540 | A nonnegative integer to t... |
| pgrple2abl 47541 | Every symmetric group on a... |
| pgrpgt2nabl 47542 | Every symmetric group on a... |
| invginvrid 47543 | Identity for a multiplicat... |
| rmsupp0 47544 | The support of a mapping o... |
| domnmsuppn0 47545 | The support of a mapping o... |
| rmsuppss 47546 | The support of a mapping o... |
| mndpsuppss 47547 | The support of a mapping o... |
| scmsuppss 47548 | The support of a mapping o... |
| rmsuppfi 47549 | The support of a mapping o... |
| rmfsupp 47550 | A mapping of a multiplicat... |
| mndpsuppfi 47551 | The support of a mapping o... |
| mndpfsupp 47552 | A mapping of a scalar mult... |
| scmsuppfi 47553 | The support of a mapping o... |
| scmfsupp 47554 | A mapping of a scalar mult... |
| suppmptcfin 47555 | The support of a mapping w... |
| mptcfsupp 47556 | A mapping with value 0 exc... |
| fsuppmptdmf 47557 | A mapping with a finite do... |
| lmodvsmdi 47558 | Multiple distributive law ... |
| gsumlsscl 47559 | Closure of a group sum in ... |
| assaascl0 47560 | The scalar 0 embedded into... |
| assaascl1 47561 | The scalar 1 embedded into... |
| ply1vr1smo 47562 | The variable in a polynomi... |
| ply1sclrmsm 47563 | The ring multiplication of... |
| coe1id 47564 | Coefficient vector of the ... |
| coe1sclmulval 47565 | The value of the coefficie... |
| ply1mulgsumlem1 47566 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 47567 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 47568 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 47569 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 47570 | The product of two polynom... |
| evl1at0 47571 | Polynomial evaluation for ... |
| evl1at1 47572 | Polynomial evaluation for ... |
| linply1 47573 | A term of the form ` x - C... |
| lineval 47574 | A term of the form ` x - C... |
| linevalexample 47575 | The polynomial ` x - 3 ` o... |
| dmatALTval 47580 | The algebra of ` N ` x ` N... |
| dmatALTbas 47581 | The base set of the algebr... |
| dmatALTbasel 47582 | An element of the base set... |
| dmatbas 47583 | The set of all ` N ` x ` N... |
| lincop 47588 | A linear combination as op... |
| lincval 47589 | The value of a linear comb... |
| dflinc2 47590 | Alternative definition of ... |
| lcoop 47591 | A linear combination as op... |
| lcoval 47592 | The value of a linear comb... |
| lincfsuppcl 47593 | A linear combination of ve... |
| linccl 47594 | A linear combination of ve... |
| lincval0 47595 | The value of an empty line... |
| lincvalsng 47596 | The linear combination ove... |
| lincvalsn 47597 | The linear combination ove... |
| lincvalpr 47598 | The linear combination ove... |
| lincval1 47599 | The linear combination ove... |
| lcosn0 47600 | Properties of a linear com... |
| lincvalsc0 47601 | The linear combination whe... |
| lcoc0 47602 | Properties of a linear com... |
| linc0scn0 47603 | If a set contains the zero... |
| lincdifsn 47604 | A vector is a linear combi... |
| linc1 47605 | A vector is a linear combi... |
| lincellss 47606 | A linear combination of a ... |
| lco0 47607 | The set of empty linear co... |
| lcoel0 47608 | The zero vector is always ... |
| lincsum 47609 | The sum of two linear comb... |
| lincscm 47610 | A linear combinations mult... |
| lincsumcl 47611 | The sum of two linear comb... |
| lincscmcl 47612 | The multiplication of a li... |
| lincsumscmcl 47613 | The sum of a linear combin... |
| lincolss 47614 | According to the statement... |
| ellcoellss 47615 | Every linear combination o... |
| lcoss 47616 | A set of vectors of a modu... |
| lspsslco 47617 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 47618 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 47619 | Equivalence of a _span_ of... |
| rellininds 47623 | The class defining the rel... |
| linindsv 47625 | The classes of the module ... |
| islininds 47626 | The property of being a li... |
| linindsi 47627 | The implications of being ... |
| linindslinci 47628 | The implications of being ... |
| islinindfis 47629 | The property of being a li... |
| islinindfiss 47630 | The property of being a li... |
| linindscl 47631 | A linearly independent set... |
| lindepsnlininds 47632 | A linearly dependent subse... |
| islindeps 47633 | The property of being a li... |
| lincext1 47634 | Property 1 of an extension... |
| lincext2 47635 | Property 2 of an extension... |
| lincext3 47636 | Property 3 of an extension... |
| lindslinindsimp1 47637 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 47638 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 47639 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 47640 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 47641 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 47642 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 47643 | Implication 2 for ~ lindsl... |
| lindslininds 47644 | Equivalence of definitions... |
| linds0 47645 | The empty set is always a ... |
| el0ldep 47646 | A set containing the zero ... |
| el0ldepsnzr 47647 | A set containing the zero ... |
| lindsrng01 47648 | Any subset of a module is ... |
| lindszr 47649 | Any subset of a module ove... |
| snlindsntorlem 47650 | Lemma for ~ snlindsntor . ... |
| snlindsntor 47651 | A singleton is linearly in... |
| ldepsprlem 47652 | Lemma for ~ ldepspr . (Co... |
| ldepspr 47653 | If a vector is a scalar mu... |
| lincresunit3lem3 47654 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 47655 | Lemma 1 for properties of ... |
| lincresunitlem2 47656 | Lemma for properties of a ... |
| lincresunit1 47657 | Property 1 of a specially ... |
| lincresunit2 47658 | Property 2 of a specially ... |
| lincresunit3lem1 47659 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 47660 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 47661 | Property 3 of a specially ... |
| lincreslvec3 47662 | Property 3 of a specially ... |
| islindeps2 47663 | Conditions for being a lin... |
| islininds2 47664 | Implication of being a lin... |
| isldepslvec2 47665 | Alternative definition of ... |
| lindssnlvec 47666 | A singleton not containing... |
| lmod1lem1 47667 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 47668 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 47669 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 47670 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 47671 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 47672 | The (smallest) structure r... |
| lmod1zr 47673 | The (smallest) structure r... |
| lmod1zrnlvec 47674 | There is a (left) module (... |
| lmodn0 47675 | Left modules exist. (Cont... |
| zlmodzxzequa 47676 | Example of an equation wit... |
| zlmodzxznm 47677 | Example of a linearly depe... |
| zlmodzxzldeplem 47678 | A and B are not equal. (C... |
| zlmodzxzequap 47679 | Example of an equation wit... |
| zlmodzxzldeplem1 47680 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 47681 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 47682 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 47683 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 47684 | { A , B } is a linearly de... |
| ldepsnlinclem1 47685 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 47686 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 47687 | The class of all (left) ve... |
| ldepsnlinc 47688 | The reverse implication of... |
| ldepslinc 47689 | For (left) vector spaces, ... |
| suppdm 47690 | If the range of a function... |
| eluz2cnn0n1 47691 | An integer greater than 1 ... |
| divge1b 47692 | The ratio of a real number... |
| divgt1b 47693 | The ratio of a real number... |
| ltsubaddb 47694 | Equivalence for the "less ... |
| ltsubsubb 47695 | Equivalence for the "less ... |
| ltsubadd2b 47696 | Equivalence for the "less ... |
| divsub1dir 47697 | Distribution of division o... |
| expnegico01 47698 | An integer greater than 1 ... |
| elfzolborelfzop1 47699 | An element of a half-open ... |
| pw2m1lepw2m1 47700 | 2 to the power of a positi... |
| zgtp1leeq 47701 | If an integer is between a... |
| flsubz 47702 | An integer can be moved in... |
| fldivmod 47703 | Expressing the floor of a ... |
| mod0mul 47704 | If an integer is 0 modulo ... |
| modn0mul 47705 | If an integer is not 0 mod... |
| m1modmmod 47706 | An integer decreased by 1 ... |
| difmodm1lt 47707 | The difference between an ... |
| nn0onn0ex 47708 | For each odd nonnegative i... |
| nn0enn0ex 47709 | For each even nonnegative ... |
| nnennex 47710 | For each even positive int... |
| nneop 47711 | A positive integer is even... |
| nneom 47712 | A positive integer is even... |
| nn0eo 47713 | A nonnegative integer is e... |
| nnpw2even 47714 | 2 to the power of a positi... |
| zefldiv2 47715 | The floor of an even integ... |
| zofldiv2 47716 | The floor of an odd intege... |
| nn0ofldiv2 47717 | The floor of an odd nonneg... |
| flnn0div2ge 47718 | The floor of a positive in... |
| flnn0ohalf 47719 | The floor of the half of a... |
| logcxp0 47720 | Logarithm of a complex pow... |
| regt1loggt0 47721 | The natural logarithm for ... |
| fdivval 47724 | The quotient of two functi... |
| fdivmpt 47725 | The quotient of two functi... |
| fdivmptf 47726 | The quotient of two functi... |
| refdivmptf 47727 | The quotient of two functi... |
| fdivpm 47728 | The quotient of two functi... |
| refdivpm 47729 | The quotient of two functi... |
| fdivmptfv 47730 | The function value of a qu... |
| refdivmptfv 47731 | The function value of a qu... |
| bigoval 47734 | Set of functions of order ... |
| elbigofrcl 47735 | Reverse closure of the "bi... |
| elbigo 47736 | Properties of a function o... |
| elbigo2 47737 | Properties of a function o... |
| elbigo2r 47738 | Sufficient condition for a... |
| elbigof 47739 | A function of order G(x) i... |
| elbigodm 47740 | The domain of a function o... |
| elbigoimp 47741 | The defining property of a... |
| elbigolo1 47742 | A function (into the posit... |
| rege1logbrege0 47743 | The general logarithm, wit... |
| rege1logbzge0 47744 | The general logarithm, wit... |
| fllogbd 47745 | A real number is between t... |
| relogbmulbexp 47746 | The logarithm of the produ... |
| relogbdivb 47747 | The logarithm of the quoti... |
| logbge0b 47748 | The logarithm of a number ... |
| logblt1b 47749 | The logarithm of a number ... |
| fldivexpfllog2 47750 | The floor of a positive re... |
| nnlog2ge0lt1 47751 | A positive integer is 1 if... |
| logbpw2m1 47752 | The floor of the binary lo... |
| fllog2 47753 | The floor of the binary lo... |
| blenval 47756 | The binary length of an in... |
| blen0 47757 | The binary length of 0. (... |
| blenn0 47758 | The binary length of a "nu... |
| blenre 47759 | The binary length of a pos... |
| blennn 47760 | The binary length of a pos... |
| blennnelnn 47761 | The binary length of a pos... |
| blennn0elnn 47762 | The binary length of a non... |
| blenpw2 47763 | The binary length of a pow... |
| blenpw2m1 47764 | The binary length of a pow... |
| nnpw2blen 47765 | A positive integer is betw... |
| nnpw2blenfzo 47766 | A positive integer is betw... |
| nnpw2blenfzo2 47767 | A positive integer is eith... |
| nnpw2pmod 47768 | Every positive integer can... |
| blen1 47769 | The binary length of 1. (... |
| blen2 47770 | The binary length of 2. (... |
| nnpw2p 47771 | Every positive integer can... |
| nnpw2pb 47772 | A number is a positive int... |
| blen1b 47773 | The binary length of a non... |
| blennnt2 47774 | The binary length of a pos... |
| nnolog2flm1 47775 | The floor of the binary lo... |
| blennn0em1 47776 | The binary length of the h... |
| blennngt2o2 47777 | The binary length of an od... |
| blengt1fldiv2p1 47778 | The binary length of an in... |
| blennn0e2 47779 | The binary length of an ev... |
| digfval 47782 | Operation to obtain the ` ... |
| digval 47783 | The ` K ` th digit of a no... |
| digvalnn0 47784 | The ` K ` th digit of a no... |
| nn0digval 47785 | The ` K ` th digit of a no... |
| dignn0fr 47786 | The digits of the fraction... |
| dignn0ldlem 47787 | Lemma for ~ dignnld . (Co... |
| dignnld 47788 | The leading digits of a po... |
| dig2nn0ld 47789 | The leading digits of a po... |
| dig2nn1st 47790 | The first (relevant) digit... |
| dig0 47791 | All digits of 0 are 0. (C... |
| digexp 47792 | The ` K ` th digit of a po... |
| dig1 47793 | All but one digits of 1 ar... |
| 0dig1 47794 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 47795 | The integers 0 and 1 corre... |
| dig2nn0 47796 | A digit of a nonnegative i... |
| 0dig2nn0e 47797 | The last bit of an even in... |
| 0dig2nn0o 47798 | The last bit of an odd int... |
| dig2bits 47799 | The ` K ` th digit of a no... |
| dignn0flhalflem1 47800 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 47801 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 47802 | The digits of the half of ... |
| dignn0flhalf 47803 | The digits of the rounded ... |
| nn0sumshdiglemA 47804 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 47805 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 47806 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 47807 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 47808 | A nonnegative integer can ... |
| nn0mulfsum 47809 | Trivial algorithm to calcu... |
| nn0mullong 47810 | Standard algorithm (also k... |
| naryfval 47813 | The set of the n-ary (endo... |
| naryfvalixp 47814 | The set of the n-ary (endo... |
| naryfvalel 47815 | An n-ary (endo)function on... |
| naryrcl 47816 | Reverse closure for n-ary ... |
| naryfvalelfv 47817 | The value of an n-ary (end... |
| naryfvalelwrdf 47818 | An n-ary (endo)function on... |
| 0aryfvalel 47819 | A nullary (endo)function o... |
| 0aryfvalelfv 47820 | The value of a nullary (en... |
| 1aryfvalel 47821 | A unary (endo)function on ... |
| fv1arycl 47822 | Closure of a unary (endo)f... |
| 1arympt1 47823 | A unary (endo)function in ... |
| 1arympt1fv 47824 | The value of a unary (endo... |
| 1arymaptfv 47825 | The value of the mapping o... |
| 1arymaptf 47826 | The mapping of unary (endo... |
| 1arymaptf1 47827 | The mapping of unary (endo... |
| 1arymaptfo 47828 | The mapping of unary (endo... |
| 1arymaptf1o 47829 | The mapping of unary (endo... |
| 1aryenef 47830 | The set of unary (endo)fun... |
| 1aryenefmnd 47831 | The set of unary (endo)fun... |
| 2aryfvalel 47832 | A binary (endo)function on... |
| fv2arycl 47833 | Closure of a binary (endo)... |
| 2arympt 47834 | A binary (endo)function in... |
| 2arymptfv 47835 | The value of a binary (end... |
| 2arymaptfv 47836 | The value of the mapping o... |
| 2arymaptf 47837 | The mapping of binary (end... |
| 2arymaptf1 47838 | The mapping of binary (end... |
| 2arymaptfo 47839 | The mapping of binary (end... |
| 2arymaptf1o 47840 | The mapping of binary (end... |
| 2aryenef 47841 | The set of binary (endo)fu... |
| itcoval 47846 | The value of the function ... |
| itcoval0 47847 | A function iterated zero t... |
| itcoval1 47848 | A function iterated once. ... |
| itcoval2 47849 | A function iterated twice.... |
| itcoval3 47850 | A function iterated three ... |
| itcoval0mpt 47851 | A mapping iterated zero ti... |
| itcovalsuc 47852 | The value of the function ... |
| itcovalsucov 47853 | The value of the function ... |
| itcovalendof 47854 | The n-th iterate of an end... |
| itcovalpclem1 47855 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 47856 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 47857 | The value of the function ... |
| itcovalt2lem2lem1 47858 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 47859 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 47860 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 47861 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 47862 | The value of the function ... |
| ackvalsuc1mpt 47863 | The Ackermann function at ... |
| ackvalsuc1 47864 | The Ackermann function at ... |
| ackval0 47865 | The Ackermann function at ... |
| ackval1 47866 | The Ackermann function at ... |
| ackval2 47867 | The Ackermann function at ... |
| ackval3 47868 | The Ackermann function at ... |
| ackendofnn0 47869 | The Ackermann function at ... |
| ackfnnn0 47870 | The Ackermann function at ... |
| ackval0val 47871 | The Ackermann function at ... |
| ackvalsuc0val 47872 | The Ackermann function at ... |
| ackvalsucsucval 47873 | The Ackermann function at ... |
| ackval0012 47874 | The Ackermann function at ... |
| ackval1012 47875 | The Ackermann function at ... |
| ackval2012 47876 | The Ackermann function at ... |
| ackval3012 47877 | The Ackermann function at ... |
| ackval40 47878 | The Ackermann function at ... |
| ackval41a 47879 | The Ackermann function at ... |
| ackval41 47880 | The Ackermann function at ... |
| ackval42 47881 | The Ackermann function at ... |
| ackval42a 47882 | The Ackermann function at ... |
| ackval50 47883 | The Ackermann function at ... |
| fv1prop 47884 | The function value of unor... |
| fv2prop 47885 | The function value of unor... |
| submuladdmuld 47886 | Transformation of a sum of... |
| affinecomb1 47887 | Combination of two real af... |
| affinecomb2 47888 | Combination of two real af... |
| affineid 47889 | Identity of an affine comb... |
| 1subrec1sub 47890 | Subtract the reciprocal of... |
| resum2sqcl 47891 | The sum of two squares of ... |
| resum2sqgt0 47892 | The sum of the square of a... |
| resum2sqrp 47893 | The sum of the square of a... |
| resum2sqorgt0 47894 | The sum of the square of t... |
| reorelicc 47895 | Membership in and outside ... |
| rrx2pxel 47896 | The x-coordinate of a poin... |
| rrx2pyel 47897 | The y-coordinate of a poin... |
| prelrrx2 47898 | An unordered pair of order... |
| prelrrx2b 47899 | An unordered pair of order... |
| rrx2pnecoorneor 47900 | If two different points ` ... |
| rrx2pnedifcoorneor 47901 | If two different points ` ... |
| rrx2pnedifcoorneorr 47902 | If two different points ` ... |
| rrx2xpref1o 47903 | There is a bijection betwe... |
| rrx2xpreen 47904 | The set of points in the t... |
| rrx2plord 47905 | The lexicographical orderi... |
| rrx2plord1 47906 | The lexicographical orderi... |
| rrx2plord2 47907 | The lexicographical orderi... |
| rrx2plordisom 47908 | The set of points in the t... |
| rrx2plordso 47909 | The lexicographical orderi... |
| ehl2eudisval0 47910 | The Euclidean distance of ... |
| ehl2eudis0lt 47911 | An upper bound of the Eucl... |
| lines 47916 | The lines passing through ... |
| line 47917 | The line passing through t... |
| rrxlines 47918 | Definition of lines passin... |
| rrxline 47919 | The line passing through t... |
| rrxlinesc 47920 | Definition of lines passin... |
| rrxlinec 47921 | The line passing through t... |
| eenglngeehlnmlem1 47922 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 47923 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 47924 | The line definition in the... |
| rrx2line 47925 | The line passing through t... |
| rrx2vlinest 47926 | The vertical line passing ... |
| rrx2linest 47927 | The line passing through t... |
| rrx2linesl 47928 | The line passing through t... |
| rrx2linest2 47929 | The line passing through t... |
| elrrx2linest2 47930 | The line passing through t... |
| spheres 47931 | The spheres for given cent... |
| sphere 47932 | A sphere with center ` X `... |
| rrxsphere 47933 | The sphere with center ` M... |
| 2sphere 47934 | The sphere with center ` M... |
| 2sphere0 47935 | The sphere around the orig... |
| line2ylem 47936 | Lemma for ~ line2y . This... |
| line2 47937 | Example for a line ` G ` p... |
| line2xlem 47938 | Lemma for ~ line2x . This... |
| line2x 47939 | Example for a horizontal l... |
| line2y 47940 | Example for a vertical lin... |
| itsclc0lem1 47941 | Lemma for theorems about i... |
| itsclc0lem2 47942 | Lemma for theorems about i... |
| itsclc0lem3 47943 | Lemma for theorems about i... |
| itscnhlc0yqe 47944 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 47945 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 47946 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 47947 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 47948 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 47949 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 47950 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 47951 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 47952 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 47953 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 47954 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 47955 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 47956 | The intersection points of... |
| itsclc0b 47957 | The intersection points of... |
| itsclinecirc0 47958 | The intersection points of... |
| itsclinecirc0b 47959 | The intersection points of... |
| itsclinecirc0in 47960 | The intersection points of... |
| itsclquadb 47961 | Quadratic equation for the... |
| itsclquadeu 47962 | Quadratic equation for the... |
| 2itscplem1 47963 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 47964 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 47965 | Lemma D for ~ 2itscp . (C... |
| 2itscp 47966 | A condition for a quadrati... |
| itscnhlinecirc02plem1 47967 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 47968 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 47969 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 47970 | Intersection of a nonhoriz... |
| inlinecirc02plem 47971 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 47972 | Intersection of a line wit... |
| inlinecirc02preu 47973 | Intersection of a line wit... |
| pm4.71da 47974 | Deduction converting a bic... |
| logic1 47975 | Distribution of implicatio... |
| logic1a 47976 | Variant of ~ logic1 . (Co... |
| logic2 47977 | Variant of ~ logic1 . (Co... |
| pm5.32dav 47978 | Distribution of implicatio... |
| pm5.32dra 47979 | Reverse distribution of im... |
| exp12bd 47980 | The import-export theorem ... |
| mpbiran3d 47981 | Equivalence with a conjunc... |
| mpbiran4d 47982 | Equivalence with a conjunc... |
| dtrucor3 47983 | An example of how ~ ax-5 w... |
| ralbidb 47984 | Formula-building rule for ... |
| ralbidc 47985 | Formula-building rule for ... |
| r19.41dv 47986 | A complex deduction form o... |
| rmotru 47987 | Two ways of expressing "at... |
| reutru 47988 | Two ways of expressing "ex... |
| reutruALT 47989 | Alternate proof for ~ reut... |
| ssdisjd 47990 | Subset preserves disjointn... |
| ssdisjdr 47991 | Subset preserves disjointn... |
| disjdifb 47992 | Relative complement is ant... |
| predisj 47993 | Preimages of disjoint sets... |
| vsn 47994 | The singleton of the unive... |
| mosn 47995 | "At most one" element in a... |
| mo0 47996 | "At most one" element in a... |
| mosssn 47997 | "At most one" element in a... |
| mo0sn 47998 | Two ways of expressing "at... |
| mosssn2 47999 | Two ways of expressing "at... |
| unilbss 48000 | Superclass of the greatest... |
| inpw 48001 | Two ways of expressing a c... |
| mof0 48002 | There is at most one funct... |
| mof02 48003 | A variant of ~ mof0 . (Co... |
| mof0ALT 48004 | Alternate proof for ~ mof0... |
| eufsnlem 48005 | There is exactly one funct... |
| eufsn 48006 | There is exactly one funct... |
| eufsn2 48007 | There is exactly one funct... |
| mofsn 48008 | There is at most one funct... |
| mofsn2 48009 | There is at most one funct... |
| mofsssn 48010 | There is at most one funct... |
| mofmo 48011 | There is at most one funct... |
| mofeu 48012 | The uniqueness of a functi... |
| elfvne0 48013 | If a function value has a ... |
| fdomne0 48014 | A function with non-empty ... |
| f1sn2g 48015 | A function that maps a sin... |
| f102g 48016 | A function that maps the e... |
| f1mo 48017 | A function that maps a set... |
| f002 48018 | A function with an empty c... |
| map0cor 48019 | A function exists iff an e... |
| fvconstr 48020 | Two ways of expressing ` A... |
| fvconstrn0 48021 | Two ways of expressing ` A... |
| fvconstr2 48022 | Two ways of expressing ` A... |
| fvconst0ci 48023 | A constant function's valu... |
| fvconstdomi 48024 | A constant function's valu... |
| f1omo 48025 | There is at most one eleme... |
| f1omoALT 48026 | There is at most one eleme... |
| iccin 48027 | Intersection of two closed... |
| iccdisj2 48028 | If the upper bound of one ... |
| iccdisj 48029 | If the upper bound of one ... |
| mreuniss 48030 | The union of a collection ... |
| clduni 48031 | The union of closed sets i... |
| opncldeqv 48032 | Conditions on open sets ar... |
| opndisj 48033 | Two ways of saying that tw... |
| clddisj 48034 | Two ways of saying that tw... |
| neircl 48035 | Reverse closure of the nei... |
| opnneilem 48036 | Lemma factoring out common... |
| opnneir 48037 | If something is true for a... |
| opnneirv 48038 | A variant of ~ opnneir wit... |
| opnneilv 48039 | The converse of ~ opnneir ... |
| opnneil 48040 | A variant of ~ opnneilv . ... |
| opnneieqv 48041 | The equivalence between ne... |
| opnneieqvv 48042 | The equivalence between ne... |
| restcls2lem 48043 | A closed set in a subspace... |
| restcls2 48044 | A closed set in a subspace... |
| restclsseplem 48045 | Lemma for ~ restclssep . ... |
| restclssep 48046 | Two disjoint closed sets i... |
| cnneiima 48047 | Given a continuous functio... |
| iooii 48048 | Open intervals are open se... |
| icccldii 48049 | Closed intervals are close... |
| i0oii 48050 | ` ( 0 [,) A ) ` is open in... |
| io1ii 48051 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 48052 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 48053 | Open neighborhood and neig... |
| sepnsepo 48054 | Open neighborhood and neig... |
| sepdisj 48055 | Separated sets are disjoin... |
| seposep 48056 | If two sets are separated ... |
| sepcsepo 48057 | If two sets are separated ... |
| sepfsepc 48058 | If two sets are separated ... |
| seppsepf 48059 | If two sets are precisely ... |
| seppcld 48060 | If two sets are precisely ... |
| isnrm4 48061 | A topological space is nor... |
| dfnrm2 48062 | A topological space is nor... |
| dfnrm3 48063 | A topological space is nor... |
| iscnrm3lem1 48064 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 48065 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem3 48066 | Lemma for ~ iscnrm3lem4 . ... |
| iscnrm3lem4 48067 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 48068 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 48069 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 48070 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 48071 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 48072 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 48073 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 48074 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 48075 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 48076 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 48077 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 48078 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 48079 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 48080 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 48081 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 48082 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 48083 | A completely normal topolo... |
| iscnrm3v 48084 | A topology is completely n... |
| iscnrm4 48085 | A completely normal topolo... |
| isprsd 48086 | Property of being a preord... |
| lubeldm2 48087 | Member of the domain of th... |
| glbeldm2 48088 | Member of the domain of th... |
| lubeldm2d 48089 | Member of the domain of th... |
| glbeldm2d 48090 | Member of the domain of th... |
| lubsscl 48091 | If a subset of ` S ` conta... |
| glbsscl 48092 | If a subset of ` S ` conta... |
| lubprlem 48093 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 48094 | The set of two comparable ... |
| lubpr 48095 | The LUB of the set of two ... |
| glbprlem 48096 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 48097 | The set of two comparable ... |
| glbpr 48098 | The GLB of the set of two ... |
| joindm2 48099 | The join of any two elemen... |
| joindm3 48100 | The join of any two elemen... |
| meetdm2 48101 | The meet of any two elemen... |
| meetdm3 48102 | The meet of any two elemen... |
| posjidm 48103 | Poset join is idempotent. ... |
| posmidm 48104 | Poset meet is idempotent. ... |
| toslat 48105 | A toset is a lattice. (Co... |
| isclatd 48106 | The predicate "is a comple... |
| intubeu 48107 | Existential uniqueness of ... |
| unilbeu 48108 | Existential uniqueness of ... |
| ipolublem 48109 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 48110 | The domain of the LUB of t... |
| ipolub 48111 | The LUB of the inclusion p... |
| ipoglblem 48112 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 48113 | The domain of the GLB of t... |
| ipoglb 48114 | The GLB of the inclusion p... |
| ipolub0 48115 | The LUB of the empty set i... |
| ipolub00 48116 | The LUB of the empty set i... |
| ipoglb0 48117 | The GLB of the empty set i... |
| mrelatlubALT 48118 | Least upper bounds in a Mo... |
| mrelatglbALT 48119 | Greatest lower bounds in a... |
| mreclat 48120 | A Moore space is a complet... |
| topclat 48121 | A topology is a complete l... |
| toplatglb0 48122 | The empty intersection in ... |
| toplatlub 48123 | Least upper bounds in a to... |
| toplatglb 48124 | Greatest lower bounds in a... |
| toplatjoin 48125 | Joins in a topology are re... |
| toplatmeet 48126 | Meets in a topology are re... |
| topdlat 48127 | A topology is a distributi... |
| catprslem 48128 | Lemma for ~ catprs . (Con... |
| catprs 48129 | A preorder can be extracte... |
| catprs2 48130 | A category equipped with t... |
| catprsc 48131 | A construction of the preo... |
| catprsc2 48132 | An alternate construction ... |
| endmndlem 48133 | A diagonal hom-set in a ca... |
| idmon 48134 | An identity arrow, or an i... |
| idepi 48135 | An identity arrow, or an i... |
| funcf2lem 48136 | A utility theorem for prov... |
| isthinc 48139 | The predicate "is a thin c... |
| isthinc2 48140 | A thin category is a categ... |
| isthinc3 48141 | A thin category is a categ... |
| thincc 48142 | A thin category is a categ... |
| thinccd 48143 | A thin category is a categ... |
| thincssc 48144 | A thin category is a categ... |
| isthincd2lem1 48145 | Lemma for ~ isthincd2 and ... |
| thincmo2 48146 | Morphisms in the same hom-... |
| thincmo 48147 | There is at most one morph... |
| thincmoALT 48148 | Alternate proof for ~ thin... |
| thincmod 48149 | At most one morphism in ea... |
| thincn0eu 48150 | In a thin category, a hom-... |
| thincid 48151 | In a thin category, a morp... |
| thincmon 48152 | In a thin category, all mo... |
| thincepi 48153 | In a thin category, all mo... |
| isthincd2lem2 48154 | Lemma for ~ isthincd2 . (... |
| isthincd 48155 | The predicate "is a thin c... |
| isthincd2 48156 | The predicate " ` C ` is a... |
| oppcthin 48157 | The opposite category of a... |
| subthinc 48158 | A subcategory of a thin ca... |
| functhinclem1 48159 | Lemma for ~ functhinc . G... |
| functhinclem2 48160 | Lemma for ~ functhinc . (... |
| functhinclem3 48161 | Lemma for ~ functhinc . T... |
| functhinclem4 48162 | Lemma for ~ functhinc . O... |
| functhinc 48163 | A functor to a thin catego... |
| fullthinc 48164 | A functor to a thin catego... |
| fullthinc2 48165 | A full functor to a thin c... |
| thincfth 48166 | A functor from a thin cate... |
| thincciso 48167 | Two thin categories are is... |
| 0thincg 48168 | Any structure with an empt... |
| 0thinc 48169 | The empty category (see ~ ... |
| indthinc 48170 | An indiscrete category in ... |
| indthincALT 48171 | An alternate proof for ~ i... |
| prsthinc 48172 | Preordered sets as categor... |
| setcthin 48173 | A category of sets all of ... |
| setc2othin 48174 | The category ` ( SetCat ``... |
| thincsect 48175 | In a thin category, one mo... |
| thincsect2 48176 | In a thin category, ` F ` ... |
| thincinv 48177 | In a thin category, ` F ` ... |
| thinciso 48178 | In a thin category, ` F : ... |
| thinccic 48179 | In a thin category, two ob... |
| prstcval 48182 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 48183 | Lemma for ~ prstcnid and ~... |
| prstcnid 48184 | Components other than ` Ho... |
| prstcbas 48185 | The base set is unchanged.... |
| prstcleval 48186 | Value of the less-than-or-... |
| prstclevalOLD 48187 | Obsolete proof of ~ prstcl... |
| prstcle 48188 | Value of the less-than-or-... |
| prstcocval 48189 | Orthocomplementation is un... |
| prstcocvalOLD 48190 | Obsolete proof of ~ prstco... |
| prstcoc 48191 | Orthocomplementation is un... |
| prstchomval 48192 | Hom-sets of the constructe... |
| prstcprs 48193 | The category is a preorder... |
| prstcthin 48194 | The preordered set is equi... |
| prstchom 48195 | Hom-sets of the constructe... |
| prstchom2 48196 | Hom-sets of the constructe... |
| prstchom2ALT 48197 | Hom-sets of the constructe... |
| postcpos 48198 | The converted category is ... |
| postcposALT 48199 | Alternate proof for ~ post... |
| postc 48200 | The converted category is ... |
| mndtcval 48203 | Value of the category buil... |
| mndtcbasval 48204 | The base set of the catego... |
| mndtcbas 48205 | The category built from a ... |
| mndtcob 48206 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 48207 | Two objects in a category ... |
| mndtchom 48208 | The only hom-set of the ca... |
| mndtcco 48209 | The composition of the cat... |
| mndtcco2 48210 | The composition of the cat... |
| mndtccatid 48211 | Lemma for ~ mndtccat and ~... |
| mndtccat 48212 | The function value is a ca... |
| mndtcid 48213 | The identity morphism, or ... |
| grptcmon 48214 | All morphisms in a categor... |
| grptcepi 48215 | All morphisms in a categor... |
| nfintd 48216 | Bound-variable hypothesis ... |
| nfiund 48217 | Bound-variable hypothesis ... |
| nfiundg 48218 | Bound-variable hypothesis ... |
| iunord 48219 | The indexed union of a col... |
| iunordi 48220 | The indexed union of a col... |
| spd 48221 | Specialization deduction, ... |
| spcdvw 48222 | A version of ~ spcdv where... |
| tfis2d 48223 | Transfinite Induction Sche... |
| bnd2d 48224 | Deduction form of ~ bnd2 .... |
| dffun3f 48225 | Alternate definition of fu... |
| setrecseq 48228 | Equality theorem for set r... |
| nfsetrecs 48229 | Bound-variable hypothesis ... |
| setrec1lem1 48230 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 48231 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 48232 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 48233 | Lemma for ~ setrec1 . If ... |
| setrec1 48234 | This is the first of two f... |
| setrec2fun 48235 | This is the second of two ... |
| setrec2lem1 48236 | Lemma for ~ setrec2 . The... |
| setrec2lem2 48237 | Lemma for ~ setrec2 . The... |
| setrec2 48238 | This is the second of two ... |
| setrec2v 48239 | Version of ~ setrec2 with ... |
| setrec2mpt 48240 | Version of ~ setrec2 where... |
| setis 48241 | Version of ~ setrec2 expre... |
| elsetrecslem 48242 | Lemma for ~ elsetrecs . A... |
| elsetrecs 48243 | A set ` A ` is an element ... |
| setrecsss 48244 | The ` setrecs ` operator r... |
| setrecsres 48245 | A recursively generated cl... |
| vsetrec 48246 | Construct ` _V ` using set... |
| 0setrec 48247 | If a function sends the em... |
| onsetreclem1 48248 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 48249 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 48250 | Lemma for ~ onsetrec . (C... |
| onsetrec 48251 | Construct ` On ` using set... |
| elpglem1 48254 | Lemma for ~ elpg . (Contr... |
| elpglem2 48255 | Lemma for ~ elpg . (Contr... |
| elpglem3 48256 | Lemma for ~ elpg . (Contr... |
| elpg 48257 | Membership in the class of... |
| pgindlem 48258 | Lemma for ~ pgind . (Cont... |
| pgindnf 48259 | Version of ~ pgind with ex... |
| pgind 48260 | Induction on partizan game... |
| sbidd 48261 | An identity theorem for su... |
| sbidd-misc 48262 | An identity theorem for su... |
| gte-lte 48267 | Simple relationship betwee... |
| gt-lt 48268 | Simple relationship betwee... |
| gte-lteh 48269 | Relationship between ` <_ ... |
| gt-lth 48270 | Relationship between ` < `... |
| ex-gt 48271 | Simple example of ` > ` , ... |
| ex-gte 48272 | Simple example of ` >_ ` ,... |
| sinhval-named 48279 | Value of the named sinh fu... |
| coshval-named 48280 | Value of the named cosh fu... |
| tanhval-named 48281 | Value of the named tanh fu... |
| sinh-conventional 48282 | Conventional definition of... |
| sinhpcosh 48283 | Prove that ` ( sinh `` A )... |
| secval 48290 | Value of the secant functi... |
| cscval 48291 | Value of the cosecant func... |
| cotval 48292 | Value of the cotangent fun... |
| seccl 48293 | The closure of the secant ... |
| csccl 48294 | The closure of the cosecan... |
| cotcl 48295 | The closure of the cotange... |
| reseccl 48296 | The closure of the secant ... |
| recsccl 48297 | The closure of the cosecan... |
| recotcl 48298 | The closure of the cotange... |
| recsec 48299 | The reciprocal of secant i... |
| reccsc 48300 | The reciprocal of cosecant... |
| reccot 48301 | The reciprocal of cotangen... |
| rectan 48302 | The reciprocal of tangent ... |
| sec0 48303 | The value of the secant fu... |
| onetansqsecsq 48304 | Prove the tangent squared ... |
| cotsqcscsq 48305 | Prove the tangent squared ... |
| ifnmfalse 48306 | If A is not a member of B,... |
| logb2aval 48307 | Define the value of the ` ... |
| comraddi 48314 | Commute RHS addition. See... |
| mvlraddi 48315 | Move the right term in a s... |
| mvrladdi 48316 | Move the left term in a su... |
| assraddsubi 48317 | Associate RHS addition-sub... |
| joinlmuladdmuli 48318 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 48319 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 48320 | Join AB-CB into (A-C) on L... |
| mvlrmuld 48321 | Move the right term in a p... |
| mvlrmuli 48322 | Move the right term in a p... |
| i2linesi 48323 | Solve for the intersection... |
| i2linesd 48324 | Solve for the intersection... |
| alimp-surprise 48325 | Demonstrate that when usin... |
| alimp-no-surprise 48326 | There is no "surprise" in ... |
| empty-surprise 48327 | Demonstrate that when usin... |
| empty-surprise2 48328 | "Prove" that false is true... |
| eximp-surprise 48329 | Show what implication insi... |
| eximp-surprise2 48330 | Show that "there exists" w... |
| alsconv 48335 | There is an equivalence be... |
| alsi1d 48336 | Deduction rule: Given "al... |
| alsi2d 48337 | Deduction rule: Given "al... |
| alsc1d 48338 | Deduction rule: Given "al... |
| alsc2d 48339 | Deduction rule: Given "al... |
| alscn0d 48340 | Deduction rule: Given "al... |
| alsi-no-surprise 48341 | Demonstrate that there is ... |
| 5m4e1 48342 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 48343 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 48344 | Resolution rule. This is ... |
| testable 48345 | In classical logic all wff... |
| aacllem 48346 | Lemma for other theorems a... |
| amgmwlem 48347 | Weighted version of ~ amgm... |
| amgmlemALT 48348 | Alternate proof of ~ amgml... |
| amgmw2d 48349 | Weighted arithmetic-geomet... |
| young2d 48350 | Young's inequality for ` n... |
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