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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... |
| 3jaobOLD 1424 | Obsolete version of ~ 3jao... |
| 3jaoi 1425 | Disjunction of three antec... |
| 3jaod 1426 | Disjunction of three antec... |
| 3jaoian 1427 | Disjunction of three antec... |
| 3jaodan 1428 | Disjunction of three antec... |
| mpjao3dan 1429 | Eliminate a three-way disj... |
| 3jaao 1430 | Inference conjoining and d... |
| syl3an9b 1431 | Nested syllogism inference... |
| 3orbi123d 1432 | Deduction joining 3 equiva... |
| 3anbi123d 1433 | Deduction joining 3 equiva... |
| 3anbi12d 1434 | Deduction conjoining and a... |
| 3anbi13d 1435 | Deduction conjoining and a... |
| 3anbi23d 1436 | Deduction conjoining and a... |
| 3anbi1d 1437 | Deduction adding conjuncts... |
| 3anbi2d 1438 | Deduction adding conjuncts... |
| 3anbi3d 1439 | Deduction adding conjuncts... |
| 3anim123d 1440 | Deduction joining 3 implic... |
| 3orim123d 1441 | Deduction joining 3 implic... |
| an6 1442 | Rearrangement of 6 conjunc... |
| 3an6 1443 | Analogue of ~ an4 for trip... |
| 3or6 1444 | Analogue of ~ or4 for trip... |
| mp3an1 1445 | An inference based on modu... |
| mp3an2 1446 | An inference based on modu... |
| mp3an3 1447 | An inference based on modu... |
| mp3an12 1448 | An inference based on modu... |
| mp3an13 1449 | An inference based on modu... |
| mp3an23 1450 | An inference based on modu... |
| mp3an1i 1451 | An inference based on modu... |
| mp3anl1 1452 | An inference based on modu... |
| mp3anl2 1453 | An inference based on modu... |
| mp3anl3 1454 | An inference based on modu... |
| mp3anr1 1455 | An inference based on modu... |
| mp3anr2 1456 | An inference based on modu... |
| mp3anr3 1457 | An inference based on modu... |
| mp3an 1458 | An inference based on modu... |
| mpd3an3 1459 | An inference based on modu... |
| mpd3an23 1460 | An inference based on modu... |
| mp3and 1461 | A deduction based on modus... |
| mp3an12i 1462 | ~ mp3an with antecedents i... |
| mp3an2i 1463 | ~ mp3an with antecedents i... |
| mp3an3an 1464 | ~ mp3an with antecedents i... |
| mp3an2ani 1465 | An elimination deduction. ... |
| biimp3a 1466 | Infer implication from a l... |
| biimp3ar 1467 | Infer implication from a l... |
| 3anandis 1468 | Inference that undistribut... |
| 3anandirs 1469 | Inference that undistribut... |
| ecase23d 1470 | Deduction for elimination ... |
| 3ecase 1471 | Inference for elimination ... |
| 3bior1fd 1472 | A disjunction is equivalen... |
| 3bior1fand 1473 | A disjunction is equivalen... |
| 3bior2fd 1474 | A wff is equivalent to its... |
| 3biant1d 1475 | A conjunction is equivalen... |
| intn3an1d 1476 | Introduction of a triple c... |
| intn3an2d 1477 | Introduction of a triple c... |
| intn3an3d 1478 | Introduction of a triple c... |
| an3andi 1479 | Distribution of conjunctio... |
| an33rean 1480 | Rearrange a 9-fold conjunc... |
| 3orel2 1481 | Partial elimination of a t... |
| 3orel3 1482 | Partial elimination of a t... |
| 3orel13 1483 | Elimination of two disjunc... |
| 3pm3.2ni 1484 | Triple negated disjunction... |
| nanan 1487 | Conjunction in terms of al... |
| dfnan2 1488 | Alternative denial in term... |
| nanor 1489 | Alternative denial in term... |
| nancom 1490 | Alternative denial is comm... |
| nannan 1491 | Nested alternative denials... |
| nanim 1492 | Implication in terms of al... |
| nannot 1493 | Negation in terms of alter... |
| nanbi 1494 | Biconditional in terms of ... |
| nanbi1 1495 | Introduce a right anti-con... |
| nanbi2 1496 | Introduce a left anti-conj... |
| nanbi12 1497 | Join two logical equivalen... |
| nanbi1i 1498 | Introduce a right anti-con... |
| nanbi2i 1499 | Introduce a left anti-conj... |
| nanbi12i 1500 | Join two logical equivalen... |
| nanbi1d 1501 | Introduce a right anti-con... |
| nanbi2d 1502 | Introduce a left anti-conj... |
| nanbi12d 1503 | Join two logical equivalen... |
| nanass 1504 | A characterization of when... |
| xnor 1507 | Two ways to write XNOR (ex... |
| xorcom 1508 | The connector ` \/_ ` is c... |
| xorass 1509 | The connector ` \/_ ` is a... |
| excxor 1510 | This tautology shows that ... |
| xor2 1511 | Two ways to express "exclu... |
| xoror 1512 | Exclusive disjunction impl... |
| xornan 1513 | Exclusive disjunction impl... |
| xornan2 1514 | XOR implies NAND (written ... |
| xorneg2 1515 | The connector ` \/_ ` is n... |
| xorneg1 1516 | The connector ` \/_ ` is n... |
| xorneg 1517 | The connector ` \/_ ` is u... |
| xorbi12i 1518 | Equality property for excl... |
| xorbi12d 1519 | Equality property for excl... |
| anxordi 1520 | Conjunction distributes ov... |
| xorexmid 1521 | Exclusive-or variant of th... |
| norcom 1524 | The connector ` -\/ ` is c... |
| nornot 1525 | ` -. ` is expressible via ... |
| noran 1526 | ` /\ ` is expressible via ... |
| noror 1527 | ` \/ ` is expressible via ... |
| norasslem1 1528 | This lemma shows the equiv... |
| norasslem2 1529 | This lemma specializes ~ b... |
| norasslem3 1530 | This lemma specializes ~ b... |
| norass 1531 | A characterization of when... |
| trujust 1536 | Soundness justification th... |
| tru 1538 | The truth value ` T. ` is ... |
| dftru2 1539 | An alternate definition of... |
| trut 1540 | A proposition is equivalen... |
| mptru 1541 | Eliminate ` T. ` as an ant... |
| tbtru 1542 | A proposition is equivalen... |
| bitru 1543 | A theorem is equivalent to... |
| trud 1544 | Anything implies ` T. ` . ... |
| truan 1545 | True can be removed from a... |
| fal 1548 | The truth value ` F. ` is ... |
| nbfal 1549 | The negation of a proposit... |
| bifal 1550 | A contradiction is equival... |
| falim 1551 | The truth value ` F. ` imp... |
| falimd 1552 | The truth value ` F. ` imp... |
| dfnot 1553 | Given falsum ` F. ` , we c... |
| inegd 1554 | Negation introduction rule... |
| efald 1555 | Deduction based on reducti... |
| pm2.21fal 1556 | If a wff and its negation ... |
| truimtru 1557 | A ` -> ` identity. (Contr... |
| truimfal 1558 | A ` -> ` identity. (Contr... |
| falimtru 1559 | A ` -> ` identity. (Contr... |
| falimfal 1560 | A ` -> ` identity. (Contr... |
| nottru 1561 | A ` -. ` identity. (Contr... |
| notfal 1562 | A ` -. ` identity. (Contr... |
| trubitru 1563 | A ` <-> ` identity. (Cont... |
| falbitru 1564 | A ` <-> ` identity. (Cont... |
| trubifal 1565 | A ` <-> ` identity. (Cont... |
| falbifal 1566 | A ` <-> ` identity. (Cont... |
| truantru 1567 | A ` /\ ` identity. (Contr... |
| truanfal 1568 | A ` /\ ` identity. (Contr... |
| falantru 1569 | A ` /\ ` identity. (Contr... |
| falanfal 1570 | A ` /\ ` identity. (Contr... |
| truortru 1571 | A ` \/ ` identity. (Contr... |
| truorfal 1572 | A ` \/ ` identity. (Contr... |
| falortru 1573 | A ` \/ ` identity. (Contr... |
| falorfal 1574 | A ` \/ ` identity. (Contr... |
| trunantru 1575 | A ` -/\ ` identity. (Cont... |
| trunanfal 1576 | A ` -/\ ` identity. (Cont... |
| falnantru 1577 | A ` -/\ ` identity. (Cont... |
| falnanfal 1578 | A ` -/\ ` identity. (Cont... |
| truxortru 1579 | A ` \/_ ` identity. (Cont... |
| truxorfal 1580 | A ` \/_ ` identity. (Cont... |
| falxortru 1581 | A ` \/_ ` identity. (Cont... |
| falxorfal 1582 | A ` \/_ ` identity. (Cont... |
| trunortru 1583 | A ` -\/ ` identity. (Cont... |
| trunorfal 1584 | A ` -\/ ` identity. (Cont... |
| falnortru 1585 | A ` -\/ ` identity. (Cont... |
| falnorfal 1586 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1589 | Equality theorem for the a... |
| hadbi123i 1590 | Equality theorem for the a... |
| hadass 1591 | Associative law for the ad... |
| hadbi 1592 | The adder sum is the same ... |
| hadcoma 1593 | Commutative law for the ad... |
| hadcomb 1594 | Commutative law for the ad... |
| hadrot 1595 | Rotation law for the adder... |
| hadnot 1596 | The adder sum distributes ... |
| had1 1597 | If the first input is true... |
| had0 1598 | If the first input is fals... |
| hadifp 1599 | The value of the adder sum... |
| cador 1602 | The adder carry in disjunc... |
| cadan 1603 | The adder carry in conjunc... |
| cadbi123d 1604 | Equality theorem for the a... |
| cadbi123i 1605 | Equality theorem for the a... |
| cadcoma 1606 | Commutative law for the ad... |
| cadcomb 1607 | Commutative law for the ad... |
| cadrot 1608 | Rotation law for the adder... |
| cadnot 1609 | The adder carry distribute... |
| cad11 1610 | If (at least) two inputs a... |
| cad1 1611 | If one input is true, then... |
| cad0 1612 | If one input is false, the... |
| cad0OLD 1613 | Obsolete version of ~ cad0... |
| cadifp 1614 | The value of the carry is,... |
| cadtru 1615 | The adder carry is true as... |
| minimp 1616 | A single axiom for minimal... |
| minimp-syllsimp 1617 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1618 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1619 | Derivation of a commuted f... |
| minimp-ax2 1620 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1621 | Derivation of ~ pm2.43 (al... |
| impsingle 1622 | The shortest single axiom ... |
| impsingle-step4 1623 | Derivation of impsingle-st... |
| impsingle-step8 1624 | Derivation of impsingle-st... |
| impsingle-ax1 1625 | Derivation of impsingle-ax... |
| impsingle-step15 1626 | Derivation of impsingle-st... |
| impsingle-step18 1627 | Derivation of impsingle-st... |
| impsingle-step19 1628 | Derivation of impsingle-st... |
| impsingle-step20 1629 | Derivation of impsingle-st... |
| impsingle-step21 1630 | Derivation of impsingle-st... |
| impsingle-step22 1631 | Derivation of impsingle-st... |
| impsingle-step25 1632 | Derivation of impsingle-st... |
| impsingle-imim1 1633 | Derivation of impsingle-im... |
| impsingle-peirce 1634 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1635 | Derivation of ~ ax-2 from ... |
| meredith 1636 | Carew Meredith's sole axio... |
| merlem1 1637 | Step 3 of Meredith's proof... |
| merlem2 1638 | Step 4 of Meredith's proof... |
| merlem3 1639 | Step 7 of Meredith's proof... |
| merlem4 1640 | Step 8 of Meredith's proof... |
| merlem5 1641 | Step 11 of Meredith's proo... |
| merlem6 1642 | Step 12 of Meredith's proo... |
| merlem7 1643 | Between steps 14 and 15 of... |
| merlem8 1644 | Step 15 of Meredith's proo... |
| merlem9 1645 | Step 18 of Meredith's proo... |
| merlem10 1646 | Step 19 of Meredith's proo... |
| merlem11 1647 | Step 20 of Meredith's proo... |
| merlem12 1648 | Step 28 of Meredith's proo... |
| merlem13 1649 | Step 35 of Meredith's proo... |
| luk-1 1650 | 1 of 3 axioms for proposit... |
| luk-2 1651 | 2 of 3 axioms for proposit... |
| luk-3 1652 | 3 of 3 axioms for proposit... |
| luklem1 1653 | Used to rederive standard ... |
| luklem2 1654 | Used to rederive standard ... |
| luklem3 1655 | Used to rederive standard ... |
| luklem4 1656 | Used to rederive standard ... |
| luklem5 1657 | Used to rederive standard ... |
| luklem6 1658 | Used to rederive standard ... |
| luklem7 1659 | Used to rederive standard ... |
| luklem8 1660 | Used to rederive standard ... |
| ax1 1661 | Standard propositional axi... |
| ax2 1662 | Standard propositional axi... |
| ax3 1663 | Standard propositional axi... |
| nic-dfim 1664 | This theorem "defines" imp... |
| nic-dfneg 1665 | This theorem "defines" neg... |
| nic-mp 1666 | Derive Nicod's rule of mod... |
| nic-mpALT 1667 | A direct proof of ~ nic-mp... |
| nic-ax 1668 | Nicod's axiom derived from... |
| nic-axALT 1669 | A direct proof of ~ nic-ax... |
| nic-imp 1670 | Inference for ~ nic-mp usi... |
| nic-idlem1 1671 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1672 | Lemma for ~ nic-id . Infe... |
| nic-id 1673 | Theorem ~ id expressed wit... |
| nic-swap 1674 | The connector ` -/\ ` is s... |
| nic-isw1 1675 | Inference version of ~ nic... |
| nic-isw2 1676 | Inference for swapping nes... |
| nic-iimp1 1677 | Inference version of ~ nic... |
| nic-iimp2 1678 | Inference version of ~ nic... |
| nic-idel 1679 | Inference to remove the tr... |
| nic-ich 1680 | Chained inference. (Contr... |
| nic-idbl 1681 | Double the terms. Since d... |
| nic-bijust 1682 | Biconditional justificatio... |
| nic-bi1 1683 | Inference to extract one s... |
| nic-bi2 1684 | Inference to extract the o... |
| nic-stdmp 1685 | Derive the standard modus ... |
| nic-luk1 1686 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1687 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1688 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1689 | This alternative axiom for... |
| lukshefth1 1690 | Lemma for ~ renicax . (Co... |
| lukshefth2 1691 | Lemma for ~ renicax . (Co... |
| renicax 1692 | A rederivation of ~ nic-ax... |
| tbw-bijust 1693 | Justification for ~ tbw-ne... |
| tbw-negdf 1694 | The definition of negation... |
| tbw-ax1 1695 | The first of four axioms i... |
| tbw-ax2 1696 | The second of four axioms ... |
| tbw-ax3 1697 | The third of four axioms i... |
| tbw-ax4 1698 | The fourth of four axioms ... |
| tbwsyl 1699 | Used to rederive the Lukas... |
| tbwlem1 1700 | Used to rederive the Lukas... |
| tbwlem2 1701 | Used to rederive the Lukas... |
| tbwlem3 1702 | Used to rederive the Lukas... |
| tbwlem4 1703 | Used to rederive the Lukas... |
| tbwlem5 1704 | Used to rederive the Lukas... |
| re1luk1 1705 | ~ luk-1 derived from the T... |
| re1luk2 1706 | ~ luk-2 derived from the T... |
| re1luk3 1707 | ~ luk-3 derived from the T... |
| merco1 1708 | A single axiom for proposi... |
| merco1lem1 1709 | Used to rederive the Tarsk... |
| retbwax4 1710 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1711 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1712 | Used to rederive the Tarsk... |
| merco1lem3 1713 | Used to rederive the Tarsk... |
| merco1lem4 1714 | Used to rederive the Tarsk... |
| merco1lem5 1715 | Used to rederive the Tarsk... |
| merco1lem6 1716 | Used to rederive the Tarsk... |
| merco1lem7 1717 | Used to rederive the Tarsk... |
| retbwax3 1718 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1719 | Used to rederive the Tarsk... |
| merco1lem9 1720 | Used to rederive the Tarsk... |
| merco1lem10 1721 | Used to rederive the Tarsk... |
| merco1lem11 1722 | Used to rederive the Tarsk... |
| merco1lem12 1723 | Used to rederive the Tarsk... |
| merco1lem13 1724 | Used to rederive the Tarsk... |
| merco1lem14 1725 | Used to rederive the Tarsk... |
| merco1lem15 1726 | Used to rederive the Tarsk... |
| merco1lem16 1727 | Used to rederive the Tarsk... |
| merco1lem17 1728 | Used to rederive the Tarsk... |
| merco1lem18 1729 | Used to rederive the Tarsk... |
| retbwax1 1730 | ~ tbw-ax1 rederived from ~... |
| merco2 1731 | A single axiom for proposi... |
| mercolem1 1732 | Used to rederive the Tarsk... |
| mercolem2 1733 | Used to rederive the Tarsk... |
| mercolem3 1734 | Used to rederive the Tarsk... |
| mercolem4 1735 | Used to rederive the Tarsk... |
| mercolem5 1736 | Used to rederive the Tarsk... |
| mercolem6 1737 | Used to rederive the Tarsk... |
| mercolem7 1738 | Used to rederive the Tarsk... |
| mercolem8 1739 | Used to rederive the Tarsk... |
| re1tbw1 1740 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1741 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1742 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1743 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1744 | Justification for ~ rb-imd... |
| rb-imdf 1745 | The definition of implicat... |
| anmp 1746 | Modus ponens for ` { \/ , ... |
| rb-ax1 1747 | The first of four axioms i... |
| rb-ax2 1748 | The second of four axioms ... |
| rb-ax3 1749 | The third of four axioms i... |
| rb-ax4 1750 | The fourth of four axioms ... |
| rbsyl 1751 | Used to rederive the Lukas... |
| rblem1 1752 | Used to rederive the Lukas... |
| rblem2 1753 | Used to rederive the Lukas... |
| rblem3 1754 | Used to rederive the Lukas... |
| rblem4 1755 | Used to rederive the Lukas... |
| rblem5 1756 | Used to rederive the Lukas... |
| rblem6 1757 | Used to rederive the Lukas... |
| rblem7 1758 | Used to rederive the Lukas... |
| re1axmp 1759 | ~ ax-mp derived from Russe... |
| re2luk1 1760 | ~ luk-1 derived from Russe... |
| re2luk2 1761 | ~ luk-2 derived from Russe... |
| re2luk3 1762 | ~ luk-3 derived from Russe... |
| mptnan 1763 | Modus ponendo tollens 1, o... |
| mptxor 1764 | Modus ponendo tollens 2, o... |
| mtpor 1765 | Modus tollendo ponens (inc... |
| mtpxor 1766 | Modus tollendo ponens (ori... |
| stoic1a 1767 | Stoic logic Thema 1 (part ... |
| stoic1b 1768 | Stoic logic Thema 1 (part ... |
| stoic2a 1769 | Stoic logic Thema 2 versio... |
| stoic2b 1770 | Stoic logic Thema 2 versio... |
| stoic3 1771 | Stoic logic Thema 3. Stat... |
| stoic4a 1772 | Stoic logic Thema 4 versio... |
| stoic4b 1773 | Stoic logic Thema 4 versio... |
| alnex 1776 | Universal quantification o... |
| eximal 1777 | An equivalence between an ... |
| nf2 1780 | Alternate definition of no... |
| nf3 1781 | Alternate definition of no... |
| nf4 1782 | Alternate definition of no... |
| nfi 1783 | Deduce that ` x ` is not f... |
| nfri 1784 | Consequence of the definit... |
| nfd 1785 | Deduce that ` x ` is not f... |
| nfrd 1786 | Consequence of the definit... |
| nftht 1787 | Closed form of ~ nfth . (... |
| nfntht 1788 | Closed form of ~ nfnth . ... |
| nfntht2 1789 | Closed form of ~ nfnth . ... |
| gen2 1791 | Generalization applied twi... |
| mpg 1792 | Modus ponens combined with... |
| mpgbi 1793 | Modus ponens on biconditio... |
| mpgbir 1794 | Modus ponens on biconditio... |
| nex 1795 | Generalization rule for ne... |
| nfth 1796 | No variable is (effectivel... |
| nfnth 1797 | No variable is (effectivel... |
| hbth 1798 | No variable is (effectivel... |
| nftru 1799 | The true constant has no f... |
| nffal 1800 | The false constant has no ... |
| sptruw 1801 | Version of ~ sp when ` ph ... |
| altru 1802 | For all sets, ` T. ` is tr... |
| alfal 1803 | For all sets, ` -. F. ` is... |
| alim 1805 | Restatement of Axiom ~ ax-... |
| alimi 1806 | Inference quantifying both... |
| 2alimi 1807 | Inference doubly quantifyi... |
| ala1 1808 | Add an antecedent in a uni... |
| al2im 1809 | Closed form of ~ al2imi . ... |
| al2imi 1810 | Inference quantifying ante... |
| alanimi 1811 | Variant of ~ al2imi with c... |
| alimdh 1812 | Deduction form of Theorem ... |
| albi 1813 | Theorem 19.15 of [Margaris... |
| albii 1814 | Inference adding universal... |
| 2albii 1815 | Inference adding two unive... |
| 3albii 1816 | Inference adding three uni... |
| sylgt 1817 | Closed form of ~ sylg . (... |
| sylg 1818 | A syllogism combined with ... |
| alrimih 1819 | Inference form of Theorem ... |
| hbxfrbi 1820 | A utility lemma to transfe... |
| alex 1821 | Universal quantifier in te... |
| exnal 1822 | Existential quantification... |
| 2nalexn 1823 | Part of theorem *11.5 in [... |
| 2exnaln 1824 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1825 | Theorem *11.25 in [Whitehe... |
| alimex 1826 | An equivalence between an ... |
| aleximi 1827 | A variant of ~ al2imi : in... |
| alexbii 1828 | Biconditional form of ~ al... |
| exim 1829 | Theorem 19.22 of [Margaris... |
| eximi 1830 | Inference adding existenti... |
| 2eximi 1831 | Inference adding two exist... |
| eximii 1832 | Inference associated with ... |
| exa1 1833 | Add an antecedent in an ex... |
| 19.38 1834 | Theorem 19.38 of [Margaris... |
| 19.38a 1835 | Under a nonfreeness hypoth... |
| 19.38b 1836 | Under a nonfreeness hypoth... |
| imnang 1837 | Quantified implication in ... |
| alinexa 1838 | A transformation of quanti... |
| exnalimn 1839 | Existential quantification... |
| alexn 1840 | A relationship between two... |
| 2exnexn 1841 | Theorem *11.51 in [Whitehe... |
| exbi 1842 | Theorem 19.18 of [Margaris... |
| exbii 1843 | Inference adding existenti... |
| 2exbii 1844 | Inference adding two exist... |
| 3exbii 1845 | Inference adding three exi... |
| nfbiit 1846 | Equivalence theorem for th... |
| nfbii 1847 | Equality theorem for the n... |
| nfxfr 1848 | A utility lemma to transfe... |
| nfxfrd 1849 | A utility lemma to transfe... |
| nfnbi 1850 | A variable is nonfree in a... |
| nfnbiOLD 1851 | Obsolete version of ~ nfnb... |
| nfnt 1852 | If a variable is nonfree i... |
| nfn 1853 | Inference associated with ... |
| nfnd 1854 | Deduction associated with ... |
| exanali 1855 | A transformation of quanti... |
| 2exanali 1856 | Theorem *11.521 in [Whiteh... |
| exancom 1857 | Commutation of conjunction... |
| exan 1858 | Place a conjunct in the sc... |
| alrimdh 1859 | Deduction form of Theorem ... |
| eximdh 1860 | Deduction from Theorem 19.... |
| nexdh 1861 | Deduction for generalizati... |
| albidh 1862 | Formula-building rule for ... |
| exbidh 1863 | Formula-building rule for ... |
| exsimpl 1864 | Simplification of an exist... |
| exsimpr 1865 | Simplification of an exist... |
| 19.26 1866 | Theorem 19.26 of [Margaris... |
| 19.26-2 1867 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1868 | Theorem ~ 19.26 with tripl... |
| 19.29 1869 | Theorem 19.29 of [Margaris... |
| 19.29r 1870 | Variation of ~ 19.29 . (C... |
| 19.29r2 1871 | Variation of ~ 19.29r with... |
| 19.29x 1872 | Variation of ~ 19.29 with ... |
| 19.35 1873 | Theorem 19.35 of [Margaris... |
| 19.35i 1874 | Inference associated with ... |
| 19.35ri 1875 | Inference associated with ... |
| 19.25 1876 | Theorem 19.25 of [Margaris... |
| 19.30 1877 | Theorem 19.30 of [Margaris... |
| 19.43 1878 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1879 | Obsolete proof of ~ 19.43 ... |
| 19.33 1880 | Theorem 19.33 of [Margaris... |
| 19.33b 1881 | The antecedent provides a ... |
| 19.40 1882 | Theorem 19.40 of [Margaris... |
| 19.40-2 1883 | Theorem *11.42 in [Whitehe... |
| 19.40b 1884 | The antecedent provides a ... |
| albiim 1885 | Split a biconditional and ... |
| 2albiim 1886 | Split a biconditional and ... |
| exintrbi 1887 | Add/remove a conjunct in t... |
| exintr 1888 | Introduce a conjunct in th... |
| alsyl 1889 | Universally quantified and... |
| nfimd 1890 | If in a context ` x ` is n... |
| nfimt 1891 | Closed form of ~ nfim and ... |
| nfim 1892 | If ` x ` is not free in ` ... |
| nfand 1893 | If in a context ` x ` is n... |
| nf3and 1894 | Deduction form of bound-va... |
| nfan 1895 | If ` x ` is not free in ` ... |
| nfnan 1896 | If ` x ` is not free in ` ... |
| nf3an 1897 | If ` x ` is not free in ` ... |
| nfbid 1898 | If in a context ` x ` is n... |
| nfbi 1899 | If ` x ` is not free in ` ... |
| nfor 1900 | If ` x ` is not free in ` ... |
| nf3or 1901 | If ` x ` is not free in ` ... |
| empty 1902 | Two characterizations of t... |
| emptyex 1903 | On the empty domain, any e... |
| emptyal 1904 | On the empty domain, any u... |
| emptynf 1905 | On the empty domain, any v... |
| ax5d 1907 | Version of ~ ax-5 with ant... |
| ax5e 1908 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1909 | If a formula holds for som... |
| nfv 1910 | If ` x ` is not present in... |
| nfvd 1911 | ~ nfv with antecedent. Us... |
| alimdv 1912 | Deduction form of Theorem ... |
| eximdv 1913 | Deduction form of Theorem ... |
| 2alimdv 1914 | Deduction form of Theorem ... |
| 2eximdv 1915 | Deduction form of Theorem ... |
| albidv 1916 | Formula-building rule for ... |
| exbidv 1917 | Formula-building rule for ... |
| nfbidv 1918 | An equality theorem for no... |
| 2albidv 1919 | Formula-building rule for ... |
| 2exbidv 1920 | Formula-building rule for ... |
| 3exbidv 1921 | Formula-building rule for ... |
| 4exbidv 1922 | Formula-building rule for ... |
| alrimiv 1923 | Inference form of Theorem ... |
| alrimivv 1924 | Inference form of Theorem ... |
| alrimdv 1925 | Deduction form of Theorem ... |
| exlimiv 1926 | Inference form of Theorem ... |
| exlimiiv 1927 | Inference (Rule C) associa... |
| exlimivv 1928 | Inference form of Theorem ... |
| exlimdv 1929 | Deduction form of Theorem ... |
| exlimdvv 1930 | Deduction form of Theorem ... |
| exlimddv 1931 | Existential elimination ru... |
| nexdv 1932 | Deduction for generalizati... |
| 2ax5 1933 | Quantification of two vari... |
| stdpc5v 1934 | Version of ~ stdpc5 with a... |
| 19.21v 1935 | Version of ~ 19.21 with a ... |
| 19.32v 1936 | Version of ~ 19.32 with a ... |
| 19.31v 1937 | Version of ~ 19.31 with a ... |
| 19.23v 1938 | Version of ~ 19.23 with a ... |
| 19.23vv 1939 | Theorem ~ 19.23v extended ... |
| pm11.53v 1940 | Version of ~ pm11.53 with ... |
| 19.36imv 1941 | One direction of ~ 19.36v ... |
| 19.36imvOLD 1942 | Obsolete version of ~ 19.3... |
| 19.36iv 1943 | Inference associated with ... |
| 19.37imv 1944 | One direction of ~ 19.37v ... |
| 19.37iv 1945 | Inference associated with ... |
| 19.41v 1946 | Version of ~ 19.41 with a ... |
| 19.41vv 1947 | Version of ~ 19.41 with tw... |
| 19.41vvv 1948 | Version of ~ 19.41 with th... |
| 19.41vvvv 1949 | Version of ~ 19.41 with fo... |
| 19.42v 1950 | Version of ~ 19.42 with a ... |
| exdistr 1951 | Distribution of existentia... |
| exdistrv 1952 | Distribute a pair of exist... |
| 4exdistrv 1953 | Distribute two pairs of ex... |
| 19.42vv 1954 | Version of ~ 19.42 with tw... |
| exdistr2 1955 | Distribution of existentia... |
| 19.42vvv 1956 | Version of ~ 19.42 with th... |
| 3exdistr 1957 | Distribution of existentia... |
| 4exdistr 1958 | Distribution of existentia... |
| weq 1959 | Extend wff definition to i... |
| speimfw 1960 | Specialization, with addit... |
| speimfwALT 1961 | Alternate proof of ~ speim... |
| spimfw 1962 | Specialization, with addit... |
| ax12i 1963 | Inference that has ~ ax-12... |
| ax6v 1965 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1966 | At least one individual ex... |
| spimw 1967 | Specialization. Lemma 8 o... |
| spimew 1968 | Existential introduction, ... |
| speiv 1969 | Inference from existential... |
| speivw 1970 | Version of ~ spei with a d... |
| exgen 1971 | Rule of existential genera... |
| extru 1972 | There exists a variable su... |
| 19.2 1973 | Theorem 19.2 of [Margaris]... |
| 19.2d 1974 | Deduction associated with ... |
| 19.8w 1975 | Weak version of ~ 19.8a an... |
| spnfw 1976 | Weak version of ~ sp . Us... |
| spvw 1977 | Version of ~ sp when ` x `... |
| 19.3v 1978 | Version of ~ 19.3 with a d... |
| 19.8v 1979 | Version of ~ 19.8a with a ... |
| 19.9v 1980 | Version of ~ 19.9 with a d... |
| 19.39 1981 | Theorem 19.39 of [Margaris... |
| 19.24 1982 | Theorem 19.24 of [Margaris... |
| 19.34 1983 | Theorem 19.34 of [Margaris... |
| 19.36v 1984 | Version of ~ 19.36 with a ... |
| 19.12vvv 1985 | Version of ~ 19.12vv with ... |
| 19.27v 1986 | Version of ~ 19.27 with a ... |
| 19.28v 1987 | Version of ~ 19.28 with a ... |
| 19.37v 1988 | Version of ~ 19.37 with a ... |
| 19.44v 1989 | Version of ~ 19.44 with a ... |
| 19.45v 1990 | Version of ~ 19.45 with a ... |
| spimevw 1991 | Existential introduction, ... |
| spimvw 1992 | A weak form of specializat... |
| spvv 1993 | Specialization, using impl... |
| spfalw 1994 | Version of ~ sp when ` ph ... |
| chvarvv 1995 | Implicit substitution of `... |
| equs4v 1996 | Version of ~ equs4 with a ... |
| alequexv 1997 | Version of ~ equs4v with i... |
| exsbim 1998 | One direction of the equiv... |
| equsv 1999 | If a formula does not cont... |
| equsalvw 2000 | Version of ~ equsalv with ... |
| equsexvw 2001 | Version of ~ equsexv with ... |
| cbvaliw 2002 | Change bound variable. Us... |
| cbvalivw 2003 | Change bound variable. Us... |
| ax7v 2005 | Weakened version of ~ ax-7... |
| ax7v1 2006 | First of two weakened vers... |
| ax7v2 2007 | Second of two weakened ver... |
| equid 2008 | Identity law for equality.... |
| nfequid 2009 | Bound-variable hypothesis ... |
| equcomiv 2010 | Weaker form of ~ equcomi w... |
| ax6evr 2011 | A commuted form of ~ ax6ev... |
| ax7 2012 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2013 | Commutative law for equali... |
| equcom 2014 | Commutative law for equali... |
| equcomd 2015 | Deduction form of ~ equcom... |
| equcoms 2016 | An inference commuting equ... |
| equtr 2017 | A transitive law for equal... |
| equtrr 2018 | A transitive law for equal... |
| equeuclr 2019 | Commuted version of ~ eque... |
| equeucl 2020 | Equality is a left-Euclide... |
| equequ1 2021 | An equivalence law for equ... |
| equequ2 2022 | An equivalence law for equ... |
| equtr2 2023 | Equality is a left-Euclide... |
| stdpc6 2024 | One of the two equality ax... |
| equvinv 2025 | A variable introduction la... |
| equvinva 2026 | A modified version of the ... |
| equvelv 2027 | A biconditional form of ~ ... |
| ax13b 2028 | An equivalence between two... |
| spfw 2029 | Weak version of ~ sp . Us... |
| spw 2030 | Weak version of the specia... |
| cbvalw 2031 | Change bound variable. Us... |
| cbvalvw 2032 | Change bound variable. Us... |
| cbvexvw 2033 | Change bound variable. Us... |
| cbvaldvaw 2034 | Rule used to change the bo... |
| cbvexdvaw 2035 | Rule used to change the bo... |
| cbval2vw 2036 | Rule used to change bound ... |
| cbvex2vw 2037 | Rule used to change bound ... |
| cbvex4vw 2038 | Rule used to change bound ... |
| alcomimw 2039 | Weak version of ~ ax-11 . ... |
| excomimw 2040 | Weak version of ~ excomim ... |
| alcomw 2041 | Weak version of ~ alcom an... |
| hbn1fw 2042 | Weak version of ~ ax-10 fr... |
| hbn1w 2043 | Weak version of ~ hbn1 . ... |
| hba1w 2044 | Weak version of ~ hba1 . ... |
| hbe1w 2045 | Weak version of ~ hbe1 . ... |
| hbalw 2046 | Weak version of ~ hbal . ... |
| 19.8aw 2047 | If a formula is true, then... |
| exexw 2048 | Existential quantification... |
| spaev 2049 | A special instance of ~ sp... |
| cbvaev 2050 | Change bound variable in a... |
| aevlem0 2051 | Lemma for ~ aevlem . Inst... |
| aevlem 2052 | Lemma for ~ aev and ~ axc1... |
| aeveq 2053 | The antecedent ` A. x x = ... |
| aev 2054 | A "distinctor elimination"... |
| aev2 2055 | A version of ~ aev with tw... |
| hbaev 2056 | All variables are effectiv... |
| naev 2057 | If some set variables can ... |
| naev2 2058 | Generalization of ~ hbnaev... |
| hbnaev 2059 | Any variable is free in ` ... |
| sbjust 2060 | Justification theorem for ... |
| sbt 2063 | A substitution into a theo... |
| sbtru 2064 | The result of substituting... |
| stdpc4 2065 | The specialization axiom o... |
| sbtALT 2066 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2067 | A double specialization us... |
| sbi1 2068 | Distribute substitution ov... |
| spsbim 2069 | Distribute substitution ov... |
| spsbbi 2070 | Biconditional property for... |
| sbimi 2071 | Distribute substitution ov... |
| sb2imi 2072 | Distribute substitution ov... |
| sbbii 2073 | Infer substitution into bo... |
| 2sbbii 2074 | Infer double substitution ... |
| sbimdv 2075 | Deduction substituting bot... |
| sbbidv 2076 | Deduction substituting bot... |
| sban 2077 | Conjunction inside and out... |
| sb3an 2078 | Threefold conjunction insi... |
| spsbe 2079 | Existential generalization... |
| sbequ 2080 | Equality property for subs... |
| sbequi 2081 | An equality theorem for su... |
| sb6 2082 | Alternate definition of su... |
| 2sb6 2083 | Equivalence for double sub... |
| sb1v 2084 | One direction of ~ sb5 , p... |
| sbv 2085 | Substitution for a variabl... |
| sbcom4 2086 | Commutativity law for subs... |
| pm11.07 2087 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2088 | Substitution in an implica... |
| sbievw 2089 | Conversion of implicit sub... |
| sbiedvw 2090 | Conversion of implicit sub... |
| 2sbievw 2091 | Conversion of double impli... |
| sbcom3vv 2092 | Substituting ` y ` for ` x... |
| sbievw2 2093 | ~ sbievw applied twice, av... |
| sbco2vv 2094 | A composition law for subs... |
| equsb3 2095 | Substitution in an equalit... |
| equsb3r 2096 | Substitution applied to th... |
| equsb1v 2097 | Substitution applied to an... |
| nsb 2098 | Any substitution in an alw... |
| sbn1 2099 | One direction of ~ sbn , u... |
| wel 2101 | Extend wff definition to i... |
| ax8v 2103 | Weakened version of ~ ax-8... |
| ax8v1 2104 | First of two weakened vers... |
| ax8v2 2105 | Second of two weakened ver... |
| ax8 2106 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2107 | An identity law for the no... |
| elsb1 2108 | Substitution for the first... |
| cleljust 2109 | When the class variables i... |
| ax9v 2111 | Weakened version of ~ ax-9... |
| ax9v1 2112 | First of two weakened vers... |
| ax9v2 2113 | Second of two weakened ver... |
| ax9 2114 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2115 | An identity law for the no... |
| elequ2g 2116 | A form of ~ elequ2 with a ... |
| elsb2 2117 | Substitution for the secon... |
| elequ12 2118 | An identity law for the no... |
| ru0 2119 | The FOL statement used in ... |
| ax6dgen 2120 | Tarski's system uses the w... |
| ax10w 2121 | Weak version of ~ ax-10 fr... |
| ax11w 2122 | Weak version of ~ ax-11 fr... |
| ax11dgen 2123 | Degenerate instance of ~ a... |
| ax12wlem 2124 | Lemma for weak version of ... |
| ax12w 2125 | Weak version of ~ ax-12 fr... |
| ax12dgen 2126 | Degenerate instance of ~ a... |
| ax12wdemo 2127 | Example of an application ... |
| ax13w 2128 | Weak version (principal in... |
| ax13dgen1 2129 | Degenerate instance of ~ a... |
| ax13dgen2 2130 | Degenerate instance of ~ a... |
| ax13dgen3 2131 | Degenerate instance of ~ a... |
| ax13dgen4 2132 | Degenerate instance of ~ a... |
| hbn1 2134 | Alias for ~ ax-10 to be us... |
| hbe1 2135 | The setvar ` x ` is not fr... |
| hbe1a 2136 | Dual statement of ~ hbe1 .... |
| nf5-1 2137 | One direction of ~ nf5 can... |
| nf5i 2138 | Deduce that ` x ` is not f... |
| nf5dh 2139 | Deduce that ` x ` is not f... |
| nf5dv 2140 | Apply the definition of no... |
| nfnaew 2141 | All variables are effectiv... |
| nfnaewOLD 2142 | Obsolete version of ~ nfna... |
| nfe1 2143 | The setvar ` x ` is not fr... |
| nfa1 2144 | The setvar ` x ` is not fr... |
| nfna1 2145 | A convenience theorem part... |
| nfia1 2146 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2147 | The setvar ` x ` is not fr... |
| modal5 2148 | The analogue in our predic... |
| nfs1v 2149 | The setvar ` x ` is not fr... |
| alcoms 2151 | Swap quantifiers in an ant... |
| alcom 2152 | Theorem 19.5 of [Margaris]... |
| alrot3 2153 | Theorem *11.21 in [Whitehe... |
| alrot4 2154 | Rotate four universal quan... |
| excom 2155 | Theorem 19.11 of [Margaris... |
| excomim 2156 | One direction of Theorem 1... |
| excom13 2157 | Swap 1st and 3rd existenti... |
| exrot3 2158 | Rotate existential quantif... |
| exrot4 2159 | Rotate existential quantif... |
| hbal 2160 | If ` x ` is not free in ` ... |
| hbald 2161 | Deduction form of bound-va... |
| sbal 2162 | Move universal quantifier ... |
| sbalv 2163 | Quantify with new variable... |
| hbsbw 2164 | If ` z ` is not free in ` ... |
| hbsbwOLD 2165 | Obsolete version of ~ hbsb... |
| sbcom2 2166 | Commutativity law for subs... |
| sbco4lem 2167 | Lemma for ~ sbco4 . It re... |
| sbco4 2168 | Two ways of exchanging two... |
| nfa2 2169 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2171 | This is essentially Axiom ... |
| ax12v2 2172 | It is possible to remove a... |
| 19.8a 2173 | If a wff is true, it is tr... |
| 19.8ad 2174 | If a wff is true, it is tr... |
| sp 2175 | Specialization. A univers... |
| spi 2176 | Inference rule of universa... |
| sps 2177 | Generalization of antecede... |
| 2sp 2178 | A double specialization (s... |
| spsd 2179 | Deduction generalizing ant... |
| 19.2g 2180 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2181 | Inference form of ~ 19.21 ... |
| 19.21bbi 2182 | Inference removing two uni... |
| 19.23bi 2183 | Inference form of Theorem ... |
| nexr 2184 | Inference associated with ... |
| qexmid 2185 | Quantified excluded middle... |
| nf5r 2186 | Consequence of the definit... |
| nf5ri 2187 | Consequence of the definit... |
| nf5rd 2188 | Consequence of the definit... |
| spimedv 2189 | Deduction version of ~ spi... |
| spimefv 2190 | Version of ~ spime with a ... |
| nfim1 2191 | A closed form of ~ nfim . ... |
| nfan1 2192 | A closed form of ~ nfan . ... |
| 19.3t 2193 | Closed form of ~ 19.3 and ... |
| 19.3 2194 | A wff may be quantified wi... |
| 19.9d 2195 | A deduction version of one... |
| 19.9t 2196 | Closed form of ~ 19.9 and ... |
| 19.9 2197 | A wff may be existentially... |
| 19.21t 2198 | Closed form of Theorem 19.... |
| 19.21 2199 | Theorem 19.21 of [Margaris... |
| stdpc5 2200 | An axiom scheme of standar... |
| 19.21-2 2201 | Version of ~ 19.21 with tw... |
| 19.23t 2202 | Closed form of Theorem 19.... |
| 19.23 2203 | Theorem 19.23 of [Margaris... |
| alimd 2204 | Deduction form of Theorem ... |
| alrimi 2205 | Inference form of Theorem ... |
| alrimdd 2206 | Deduction form of Theorem ... |
| alrimd 2207 | Deduction form of Theorem ... |
| eximd 2208 | Deduction form of Theorem ... |
| exlimi 2209 | Inference associated with ... |
| exlimd 2210 | Deduction form of Theorem ... |
| exlimimdd 2211 | Existential elimination ru... |
| exlimdd 2212 | Existential elimination ru... |
| nexd 2213 | Deduction for generalizati... |
| albid 2214 | Formula-building rule for ... |
| exbid 2215 | Formula-building rule for ... |
| nfbidf 2216 | An equality theorem for ef... |
| 19.16 2217 | Theorem 19.16 of [Margaris... |
| 19.17 2218 | Theorem 19.17 of [Margaris... |
| 19.27 2219 | Theorem 19.27 of [Margaris... |
| 19.28 2220 | Theorem 19.28 of [Margaris... |
| 19.19 2221 | Theorem 19.19 of [Margaris... |
| 19.36 2222 | Theorem 19.36 of [Margaris... |
| 19.36i 2223 | Inference associated with ... |
| 19.37 2224 | Theorem 19.37 of [Margaris... |
| 19.32 2225 | Theorem 19.32 of [Margaris... |
| 19.31 2226 | Theorem 19.31 of [Margaris... |
| 19.41 2227 | Theorem 19.41 of [Margaris... |
| 19.42 2228 | Theorem 19.42 of [Margaris... |
| 19.44 2229 | Theorem 19.44 of [Margaris... |
| 19.45 2230 | Theorem 19.45 of [Margaris... |
| spimfv 2231 | Specialization, using impl... |
| chvarfv 2232 | Implicit substitution of `... |
| cbv3v2 2233 | Version of ~ cbv3 with two... |
| sbalex 2234 | Equivalence of two ways to... |
| sb4av 2235 | Version of ~ sb4a with a d... |
| sbimd 2236 | Deduction substituting bot... |
| sbbid 2237 | Deduction substituting bot... |
| 2sbbid 2238 | Deduction doubly substitut... |
| sbequ1 2239 | An equality theorem for su... |
| sbequ2 2240 | An equality theorem for su... |
| stdpc7 2241 | One of the two equality ax... |
| sbequ12 2242 | An equality theorem for su... |
| sbequ12r 2243 | An equality theorem for su... |
| sbelx 2244 | Elimination of substitutio... |
| sbequ12a 2245 | An equality theorem for su... |
| sbid 2246 | An identity theorem for su... |
| sbcov 2247 | A composition law for subs... |
| sb6a 2248 | Equivalence for substituti... |
| sbid2vw 2249 | Reverting substitution yie... |
| axc16g 2250 | Generalization of ~ axc16 ... |
| axc16 2251 | Proof of older axiom ~ ax-... |
| axc16gb 2252 | Biconditional strengthenin... |
| axc16nf 2253 | If ~ dtru is false, then t... |
| axc11v 2254 | Version of ~ axc11 with a ... |
| axc11rv 2255 | Version of ~ axc11r with a... |
| drsb2 2256 | Formula-building lemma for... |
| equsalv 2257 | An equivalence related to ... |
| equsexv 2258 | An equivalence related to ... |
| equsexvOLD 2259 | Obsolete version of ~ equs... |
| sbft 2260 | Substitution has no effect... |
| sbf 2261 | Substitution for a variabl... |
| sbf2 2262 | Substitution has no effect... |
| sbh 2263 | Substitution for a variabl... |
| hbs1 2264 | The setvar ` x ` is not fr... |
| nfs1f 2265 | If ` x ` is not free in ` ... |
| sb5 2266 | Alternate definition of su... |
| sb5OLD 2267 | Obsolete version of ~ sb5 ... |
| sb56OLD 2268 | Obsolete version of ~ sbal... |
| equs5av 2269 | A property related to subs... |
| 2sb5 2270 | Equivalence for double sub... |
| sbco4lemOLD 2271 | Obsolete version of ~ sbco... |
| dfsb7 2272 | An alternate definition of... |
| sbn 2273 | Negation inside and outsid... |
| sbex 2274 | Move existential quantifie... |
| nf5 2275 | Alternate definition of ~ ... |
| nf6 2276 | An alternate definition of... |
| nf5d 2277 | Deduce that ` x ` is not f... |
| nf5di 2278 | Since the converse holds b... |
| 19.9h 2279 | A wff may be existentially... |
| 19.21h 2280 | Theorem 19.21 of [Margaris... |
| 19.23h 2281 | Theorem 19.23 of [Margaris... |
| exlimih 2282 | Inference associated with ... |
| exlimdh 2283 | Deduction form of Theorem ... |
| equsalhw 2284 | Version of ~ equsalh with ... |
| equsexhv 2285 | An equivalence related to ... |
| hba1 2286 | The setvar ` x ` is not fr... |
| hbnt 2287 | Closed theorem version of ... |
| hbn 2288 | If ` x ` is not free in ` ... |
| hbnd 2289 | Deduction form of bound-va... |
| hbim1 2290 | A closed form of ~ hbim . ... |
| hbimd 2291 | Deduction form of bound-va... |
| hbim 2292 | If ` x ` is not free in ` ... |
| hban 2293 | If ` x ` is not free in ` ... |
| hb3an 2294 | If ` x ` is not free in ` ... |
| sbi2 2295 | Introduction of implicatio... |
| sbim 2296 | Implication inside and out... |
| sbrim 2297 | Substitution in an implica... |
| sbrimOLD 2298 | Obsolete version of ~ sbri... |
| sblim 2299 | Substitution in an implica... |
| sbor 2300 | Disjunction inside and out... |
| sbbi 2301 | Equivalence inside and out... |
| sblbis 2302 | Introduce left bicondition... |
| sbrbis 2303 | Introduce right biconditio... |
| sbrbif 2304 | Introduce right biconditio... |
| sbnf 2305 | Move nonfree predicate in ... |
| sbnfOLD 2306 | Obsolete version of ~ sbnf... |
| sbiev 2307 | Conversion of implicit sub... |
| sbiedw 2308 | Conversion of implicit sub... |
| axc7 2309 | Show that the original axi... |
| axc7e 2310 | Abbreviated version of ~ a... |
| modal-b 2311 | The analogue in our predic... |
| 19.9ht 2312 | A closed version of ~ 19.9... |
| axc4 2313 | Show that the original axi... |
| axc4i 2314 | Inference version of ~ axc... |
| nfal 2315 | If ` x ` is not free in ` ... |
| nfex 2316 | If ` x ` is not free in ` ... |
| hbex 2317 | If ` x ` is not free in ` ... |
| nfnf 2318 | If ` x ` is not free in ` ... |
| 19.12 2319 | Theorem 19.12 of [Margaris... |
| nfald 2320 | Deduction form of ~ nfal .... |
| nfexd 2321 | If ` x ` is not free in ` ... |
| nfsbv 2322 | If ` z ` is not free in ` ... |
| nfsbvOLD 2323 | Obsolete version of ~ nfsb... |
| sbco2v 2324 | A composition law for subs... |
| aaan 2325 | Distribute universal quant... |
| aaanOLD 2326 | Obsolete version of ~ aaan... |
| eeor 2327 | Distribute existential qua... |
| eeorOLD 2328 | Obsolete version of ~ eeor... |
| cbv3v 2329 | Rule used to change bound ... |
| cbv1v 2330 | Rule used to change bound ... |
| cbv2w 2331 | Rule used to change bound ... |
| cbvaldw 2332 | Deduction used to change b... |
| cbvexdw 2333 | Deduction used to change b... |
| cbv3hv 2334 | Rule used to change bound ... |
| cbvalv1 2335 | Rule used to change bound ... |
| cbvexv1 2336 | Rule used to change bound ... |
| cbval2v 2337 | Rule used to change bound ... |
| cbvex2v 2338 | Rule used to change bound ... |
| dvelimhw 2339 | Proof of ~ dvelimh without... |
| pm11.53 2340 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2341 | Special case of ~ 19.12 wh... |
| eean 2342 | Distribute existential qua... |
| eeanv 2343 | Distribute a pair of exist... |
| eeeanv 2344 | Distribute three existenti... |
| ee4anv 2345 | Distribute two pairs of ex... |
| sb8v 2346 | Substitution of variable i... |
| sb8f 2347 | Substitution of variable i... |
| sb8fOLD 2348 | Obsolete version of ~ sb8f... |
| sb8ef 2349 | Substitution of variable i... |
| 2sb8ef 2350 | An equivalent expression f... |
| sb6rfv 2351 | Reversed substitution. Ve... |
| sbnf2 2352 | Two ways of expressing " `... |
| exsb 2353 | An equivalent expression f... |
| 2exsb 2354 | An equivalent expression f... |
| sbbib 2355 | Reversal of substitution. ... |
| sbbibvv 2356 | Reversal of substitution. ... |
| cbvsbv 2357 | Change the bound variable ... |
| cbvsbvf 2358 | Change the bound variable ... |
| cleljustALT 2359 | Alternate proof of ~ clelj... |
| cleljustALT2 2360 | Alternate proof of ~ clelj... |
| equs5aALT 2361 | Alternate proof of ~ equs5... |
| equs5eALT 2362 | Alternate proof of ~ equs5... |
| axc11r 2363 | Same as ~ axc11 but with r... |
| dral1v 2364 | Formula-building lemma for... |
| dral1vOLD 2365 | Obsolete version of ~ dral... |
| drex1v 2366 | Formula-building lemma for... |
| drnf1v 2367 | Formula-building lemma for... |
| drnf1vOLD 2368 | Obsolete version of ~ drnf... |
| ax13v 2370 | A weaker version of ~ ax-1... |
| ax13lem1 2371 | A version of ~ ax13v with ... |
| ax13 2372 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2373 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2374 | An equation between setvar... |
| dveeq2 2375 | Quantifier introduction wh... |
| nfeqf1 2376 | An equation between setvar... |
| dveeq1 2377 | Quantifier introduction wh... |
| nfeqf 2378 | A variable is effectively ... |
| axc9 2379 | Derive set.mm's original ~... |
| ax6e 2380 | At least one individual ex... |
| ax6 2381 | Theorem showing that ~ ax-... |
| axc10 2382 | Show that the original axi... |
| spimt 2383 | Closed theorem form of ~ s... |
| spim 2384 | Specialization, using impl... |
| spimed 2385 | Deduction version of ~ spi... |
| spime 2386 | Existential introduction, ... |
| spimv 2387 | A version of ~ spim with a... |
| spimvALT 2388 | Alternate proof of ~ spimv... |
| spimev 2389 | Distinct-variable version ... |
| spv 2390 | Specialization, using impl... |
| spei 2391 | Inference from existential... |
| chvar 2392 | Implicit substitution of `... |
| chvarv 2393 | Implicit substitution of `... |
| cbv3 2394 | Rule used to change bound ... |
| cbval 2395 | Rule used to change bound ... |
| cbvex 2396 | Rule used to change bound ... |
| cbvalv 2397 | Rule used to change bound ... |
| cbvexv 2398 | Rule used to change bound ... |
| cbv1 2399 | Rule used to change bound ... |
| cbv2 2400 | Rule used to change bound ... |
| cbv3h 2401 | Rule used to change bound ... |
| cbv1h 2402 | Rule used to change bound ... |
| cbv2h 2403 | Rule used to change bound ... |
| cbvald 2404 | Deduction used to change b... |
| cbvexd 2405 | Deduction used to change b... |
| cbvaldva 2406 | Rule used to change the bo... |
| cbvexdva 2407 | Rule used to change the bo... |
| cbval2 2408 | Rule used to change bound ... |
| cbvex2 2409 | Rule used to change bound ... |
| cbval2vv 2410 | Rule used to change bound ... |
| cbvex2vv 2411 | Rule used to change bound ... |
| cbvex4v 2412 | Rule used to change bound ... |
| equs4 2413 | Lemma used in proofs of im... |
| equsal 2414 | An equivalence related to ... |
| equsex 2415 | An equivalence related to ... |
| equsexALT 2416 | Alternate proof of ~ equse... |
| equsalh 2417 | An equivalence related to ... |
| equsexh 2418 | An equivalence related to ... |
| axc15 2419 | Derivation of set.mm's ori... |
| ax12 2420 | Rederivation of Axiom ~ ax... |
| ax12b 2421 | A bidirectional version of... |
| ax13ALT 2422 | Alternate proof of ~ ax13 ... |
| axc11n 2423 | Derive set.mm's original ~... |
| aecom 2424 | Commutation law for identi... |
| aecoms 2425 | A commutation rule for ide... |
| naecoms 2426 | A commutation rule for dis... |
| axc11 2427 | Show that ~ ax-c11 can be ... |
| hbae 2428 | All variables are effectiv... |
| hbnae 2429 | All variables are effectiv... |
| nfae 2430 | All variables are effectiv... |
| nfnae 2431 | All variables are effectiv... |
| hbnaes 2432 | Rule that applies ~ hbnae ... |
| axc16i 2433 | Inference with ~ axc16 as ... |
| axc16nfALT 2434 | Alternate proof of ~ axc16... |
| dral2 2435 | Formula-building lemma for... |
| dral1 2436 | Formula-building lemma for... |
| dral1ALT 2437 | Alternate proof of ~ dral1... |
| drex1 2438 | Formula-building lemma for... |
| drex2 2439 | Formula-building lemma for... |
| drnf1 2440 | Formula-building lemma for... |
| drnf2 2441 | Formula-building lemma for... |
| nfald2 2442 | Variation on ~ nfald which... |
| nfexd2 2443 | Variation on ~ nfexd which... |
| exdistrf 2444 | Distribution of existentia... |
| dvelimf 2445 | Version of ~ dvelimv witho... |
| dvelimdf 2446 | Deduction form of ~ dvelim... |
| dvelimh 2447 | Version of ~ dvelim withou... |
| dvelim 2448 | This theorem can be used t... |
| dvelimv 2449 | Similar to ~ dvelim with f... |
| dvelimnf 2450 | Version of ~ dvelim using ... |
| dveeq2ALT 2451 | Alternate proof of ~ dveeq... |
| equvini 2452 | A variable introduction la... |
| equvel 2453 | A variable elimination law... |
| equs5a 2454 | A property related to subs... |
| equs5e 2455 | A property related to subs... |
| equs45f 2456 | Two ways of expressing sub... |
| equs5 2457 | Lemma used in proofs of su... |
| dveel1 2458 | Quantifier introduction wh... |
| dveel2 2459 | Quantifier introduction wh... |
| axc14 2460 | Axiom ~ ax-c14 is redundan... |
| sb6x 2461 | Equivalence involving subs... |
| sbequ5 2462 | Substitution does not chan... |
| sbequ6 2463 | Substitution does not chan... |
| sb5rf 2464 | Reversed substitution. Us... |
| sb6rf 2465 | Reversed substitution. Fo... |
| ax12vALT 2466 | Alternate proof of ~ ax12v... |
| 2ax6elem 2467 | We can always find values ... |
| 2ax6e 2468 | We can always find values ... |
| 2sb5rf 2469 | Reversed double substituti... |
| 2sb6rf 2470 | Reversed double substituti... |
| sbel2x 2471 | Elimination of double subs... |
| sb4b 2472 | Simplified definition of s... |
| sb3b 2473 | Simplified definition of s... |
| sb3 2474 | One direction of a simplif... |
| sb1 2475 | One direction of a simplif... |
| sb2 2476 | One direction of a simplif... |
| sb4a 2477 | A version of one implicati... |
| dfsb1 2478 | Alternate definition of su... |
| hbsb2 2479 | Bound-variable hypothesis ... |
| nfsb2 2480 | Bound-variable hypothesis ... |
| hbsb2a 2481 | Special case of a bound-va... |
| sb4e 2482 | One direction of a simplif... |
| hbsb2e 2483 | Special case of a bound-va... |
| hbsb3 2484 | If ` y ` is not free in ` ... |
| nfs1 2485 | If ` y ` is not free in ` ... |
| axc16ALT 2486 | Alternate proof of ~ axc16... |
| axc16gALT 2487 | Alternate proof of ~ axc16... |
| equsb1 2488 | Substitution applied to an... |
| equsb2 2489 | Substitution applied to an... |
| dfsb2 2490 | An alternate definition of... |
| dfsb3 2491 | An alternate definition of... |
| drsb1 2492 | Formula-building lemma for... |
| sb2ae 2493 | In the case of two success... |
| sb6f 2494 | Equivalence for substituti... |
| sb5f 2495 | Equivalence for substituti... |
| nfsb4t 2496 | A variable not free in a p... |
| nfsb4 2497 | A variable not free in a p... |
| sbequ8 2498 | Elimination of equality fr... |
| sbie 2499 | Conversion of implicit sub... |
| sbied 2500 | Conversion of implicit sub... |
| sbiedv 2501 | Conversion of implicit sub... |
| 2sbiev 2502 | Conversion of double impli... |
| sbcom3 2503 | Substituting ` y ` for ` x... |
| sbco 2504 | A composition law for subs... |
| sbid2 2505 | An identity law for substi... |
| sbid2v 2506 | An identity law for substi... |
| sbidm 2507 | An idempotent law for subs... |
| sbco2 2508 | A composition law for subs... |
| sbco2d 2509 | A composition law for subs... |
| sbco3 2510 | A composition law for subs... |
| sbcom 2511 | A commutativity law for su... |
| sbtrt 2512 | Partially closed form of ~... |
| sbtr 2513 | A partial converse to ~ sb... |
| sb8 2514 | Substitution of variable i... |
| sb8e 2515 | Substitution of variable i... |
| sb9 2516 | Commutation of quantificat... |
| sb9i 2517 | Commutation of quantificat... |
| sbhb 2518 | Two ways of expressing " `... |
| nfsbd 2519 | Deduction version of ~ nfs... |
| nfsb 2520 | If ` z ` is not free in ` ... |
| hbsb 2521 | If ` z ` is not free in ` ... |
| sb7f 2522 | This version of ~ dfsb7 do... |
| sb7h 2523 | This version of ~ dfsb7 do... |
| sb10f 2524 | Hao Wang's identity axiom ... |
| sbal1 2525 | Check out ~ sbal for a ver... |
| sbal2 2526 | Move quantifier in and out... |
| 2sb8e 2527 | An equivalent expression f... |
| dfmoeu 2528 | An elementary proof of ~ m... |
| dfeumo 2529 | An elementary proof showin... |
| mojust 2531 | Soundness justification th... |
| nexmo 2533 | Nonexistence implies uniqu... |
| exmo 2534 | Any proposition holds for ... |
| moabs 2535 | Absorption of existence co... |
| moim 2536 | The at-most-one quantifier... |
| moimi 2537 | The at-most-one quantifier... |
| moimdv 2538 | The at-most-one quantifier... |
| mobi 2539 | Equivalence theorem for th... |
| mobii 2540 | Formula-building rule for ... |
| mobidv 2541 | Formula-building rule for ... |
| mobid 2542 | Formula-building rule for ... |
| moa1 2543 | If an implication holds fo... |
| moan 2544 | "At most one" is still the... |
| moani 2545 | "At most one" is still tru... |
| moor 2546 | "At most one" is still the... |
| mooran1 2547 | "At most one" imports disj... |
| mooran2 2548 | "At most one" exports disj... |
| nfmo1 2549 | Bound-variable hypothesis ... |
| nfmod2 2550 | Bound-variable hypothesis ... |
| nfmodv 2551 | Bound-variable hypothesis ... |
| nfmov 2552 | Bound-variable hypothesis ... |
| nfmod 2553 | Bound-variable hypothesis ... |
| nfmo 2554 | Bound-variable hypothesis ... |
| mof 2555 | Version of ~ df-mo with di... |
| mo3 2556 | Alternate definition of th... |
| mo 2557 | Equivalent definitions of ... |
| mo4 2558 | At-most-one quantifier exp... |
| mo4f 2559 | At-most-one quantifier exp... |
| eu3v 2562 | An alternate way to expres... |
| eujust 2563 | Soundness justification th... |
| eujustALT 2564 | Alternate proof of ~ eujus... |
| eu6lem 2565 | Lemma of ~ eu6im . A diss... |
| eu6 2566 | Alternate definition of th... |
| eu6im 2567 | One direction of ~ eu6 nee... |
| euf 2568 | Version of ~ eu6 with disj... |
| euex 2569 | Existential uniqueness imp... |
| eumo 2570 | Existential uniqueness imp... |
| eumoi 2571 | Uniqueness inferred from e... |
| exmoeub 2572 | Existence implies that uni... |
| exmoeu 2573 | Existence is equivalent to... |
| moeuex 2574 | Uniqueness implies that ex... |
| moeu 2575 | Uniqueness is equivalent t... |
| eubi 2576 | Equivalence theorem for th... |
| eubii 2577 | Introduce unique existenti... |
| eubidv 2578 | Formula-building rule for ... |
| eubid 2579 | Formula-building rule for ... |
| nfeu1 2580 | Bound-variable hypothesis ... |
| nfeu1ALT 2581 | Alternate proof of ~ nfeu1... |
| nfeud2 2582 | Bound-variable hypothesis ... |
| nfeudw 2583 | Bound-variable hypothesis ... |
| nfeud 2584 | Bound-variable hypothesis ... |
| nfeuw 2585 | Bound-variable hypothesis ... |
| nfeu 2586 | Bound-variable hypothesis ... |
| dfeu 2587 | Rederive ~ df-eu from the ... |
| dfmo 2588 | Rederive ~ df-mo from the ... |
| euequ 2589 | There exists a unique set ... |
| sb8eulem 2590 | Lemma. Factor out the com... |
| sb8euv 2591 | Variable substitution in u... |
| sb8eu 2592 | Variable substitution in u... |
| sb8mo 2593 | Variable substitution for ... |
| cbvmovw 2594 | Change bound variable. Us... |
| cbvmow 2595 | Rule used to change bound ... |
| cbvmo 2596 | Rule used to change bound ... |
| cbveuvw 2597 | Change bound variable. Us... |
| cbveuw 2598 | Version of ~ cbveu with a ... |
| cbveu 2599 | Rule used to change bound ... |
| cbveuALT 2600 | Alternative proof of ~ cbv... |
| eu2 2601 | An alternate way of defini... |
| eu1 2602 | An alternate way to expres... |
| euor 2603 | Introduce a disjunct into ... |
| euorv 2604 | Introduce a disjunct into ... |
| euor2 2605 | Introduce or eliminate a d... |
| sbmo 2606 | Substitution into an at-mo... |
| eu4 2607 | Uniqueness using implicit ... |
| euimmo 2608 | Existential uniqueness imp... |
| euim 2609 | Add unique existential qua... |
| moanimlem 2610 | Factor out the common proo... |
| moanimv 2611 | Introduction of a conjunct... |
| moanim 2612 | Introduction of a conjunct... |
| euan 2613 | Introduction of a conjunct... |
| moanmo 2614 | Nested at-most-one quantif... |
| moaneu 2615 | Nested at-most-one and uni... |
| euanv 2616 | Introduction of a conjunct... |
| mopick 2617 | "At most one" picks a vari... |
| moexexlem 2618 | Factor out the proof skele... |
| 2moexv 2619 | Double quantification with... |
| moexexvw 2620 | "At most one" double quant... |
| 2moswapv 2621 | A condition allowing to sw... |
| 2euswapv 2622 | A condition allowing to sw... |
| 2euexv 2623 | Double quantification with... |
| 2exeuv 2624 | Double existential uniquen... |
| eupick 2625 | Existential uniqueness "pi... |
| eupicka 2626 | Version of ~ eupick with c... |
| eupickb 2627 | Existential uniqueness "pi... |
| eupickbi 2628 | Theorem *14.26 in [Whitehe... |
| mopick2 2629 | "At most one" can show the... |
| moexex 2630 | "At most one" double quant... |
| moexexv 2631 | "At most one" double quant... |
| 2moex 2632 | Double quantification with... |
| 2euex 2633 | Double quantification with... |
| 2eumo 2634 | Nested unique existential ... |
| 2eu2ex 2635 | Double existential uniquen... |
| 2moswap 2636 | A condition allowing to sw... |
| 2euswap 2637 | A condition allowing to sw... |
| 2exeu 2638 | Double existential uniquen... |
| 2mo2 2639 | Two ways of expressing "th... |
| 2mo 2640 | Two ways of expressing "th... |
| 2mos 2641 | Double "there exists at mo... |
| 2mosOLD 2642 | Obsolete version of ~ 2mos... |
| 2eu1 2643 | Double existential uniquen... |
| 2eu1v 2644 | Double existential uniquen... |
| 2eu2 2645 | Double existential uniquen... |
| 2eu3 2646 | Double existential uniquen... |
| 2eu4 2647 | This theorem provides us w... |
| 2eu5 2648 | An alternate definition of... |
| 2eu6 2649 | Two equivalent expressions... |
| 2eu7 2650 | Two equivalent expressions... |
| 2eu8 2651 | Two equivalent expressions... |
| euae 2652 | Two ways to express "exact... |
| exists1 2653 | Two ways to express "exact... |
| exists2 2654 | A condition implying that ... |
| barbara 2655 | "Barbara", one of the fund... |
| celarent 2656 | "Celarent", one of the syl... |
| darii 2657 | "Darii", one of the syllog... |
| dariiALT 2658 | Alternate proof of ~ darii... |
| ferio 2659 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2660 | Lemma for ~ barbari and th... |
| barbari 2661 | "Barbari", one of the syll... |
| barbariALT 2662 | Alternate proof of ~ barba... |
| celaront 2663 | "Celaront", one of the syl... |
| cesare 2664 | "Cesare", one of the syllo... |
| camestres 2665 | "Camestres", one of the sy... |
| festino 2666 | "Festino", one of the syll... |
| festinoALT 2667 | Alternate proof of ~ festi... |
| baroco 2668 | "Baroco", one of the syllo... |
| barocoALT 2669 | Alternate proof of ~ festi... |
| cesaro 2670 | "Cesaro", one of the syllo... |
| camestros 2671 | "Camestros", one of the sy... |
| datisi 2672 | "Datisi", one of the syllo... |
| disamis 2673 | "Disamis", one of the syll... |
| ferison 2674 | "Ferison", one of the syll... |
| bocardo 2675 | "Bocardo", one of the syll... |
| darapti 2676 | "Darapti", one of the syll... |
| daraptiALT 2677 | Alternate proof of ~ darap... |
| felapton 2678 | "Felapton", one of the syl... |
| calemes 2679 | "Calemes", one of the syll... |
| dimatis 2680 | "Dimatis", one of the syll... |
| fresison 2681 | "Fresison", one of the syl... |
| calemos 2682 | "Calemos", one of the syll... |
| fesapo 2683 | "Fesapo", one of the syllo... |
| bamalip 2684 | "Bamalip", one of the syll... |
| axia1 2685 | Left 'and' elimination (in... |
| axia2 2686 | Right 'and' elimination (i... |
| axia3 2687 | 'And' introduction (intuit... |
| axin1 2688 | 'Not' introduction (intuit... |
| axin2 2689 | 'Not' elimination (intuiti... |
| axio 2690 | Definition of 'or' (intuit... |
| axi4 2691 | Specialization (intuitioni... |
| axi5r 2692 | Converse of ~ axc4 (intuit... |
| axial 2693 | The setvar ` x ` is not fr... |
| axie1 2694 | The setvar ` x ` is not fr... |
| axie2 2695 | A key property of existent... |
| axi9 2696 | Axiom of existence (intuit... |
| axi10 2697 | Axiom of Quantifier Substi... |
| axi12 2698 | Axiom of Quantifier Introd... |
| axbnd 2699 | Axiom of Bundling (intuiti... |
| axexte 2701 | The axiom of extensionalit... |
| axextg 2702 | A generalization of the ax... |
| axextb 2703 | A bidirectional version of... |
| axextmo 2704 | There exists at most one s... |
| nulmo 2705 | There exists at most one e... |
| eleq1ab 2708 | Extension (in the sense of... |
| cleljustab 2709 | Extension of ~ cleljust fr... |
| abid 2710 | Simplification of class ab... |
| vexwt 2711 | A standard theorem of pred... |
| vexw 2712 | If ` ph ` is a theorem, th... |
| vextru 2713 | Every setvar is a member o... |
| nfsab1 2714 | Bound-variable hypothesis ... |
| hbab1 2715 | Bound-variable hypothesis ... |
| hbab1OLD 2716 | Obsolete version of ~ hbab... |
| hbab 2717 | Bound-variable hypothesis ... |
| hbabg 2718 | Bound-variable hypothesis ... |
| nfsab 2719 | Bound-variable hypothesis ... |
| nfsabg 2720 | Bound-variable hypothesis ... |
| dfcleq 2722 | The defining characterizat... |
| cvjust 2723 | Every set is a class. Pro... |
| ax9ALT 2724 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2725 | A weaker version of ~ eleq... |
| eqriv 2726 | Infer equality of classes ... |
| eqrdv 2727 | Deduce equality of classes... |
| eqrdav 2728 | Deduce equality of classes... |
| eqid 2729 | Law of identity (reflexivi... |
| eqidd 2730 | Class identity law with an... |
| eqeq1d 2731 | Deduction from equality to... |
| eqeq1dALT 2732 | Alternate proof of ~ eqeq1... |
| eqeq1 2733 | Equality implies equivalen... |
| eqeq1i 2734 | Inference from equality to... |
| eqcomd 2735 | Deduction from commutative... |
| eqcom 2736 | Commutative law for class ... |
| eqcoms 2737 | Inference applying commuta... |
| eqcomi 2738 | Inference from commutative... |
| neqcomd 2739 | Commute an inequality. (C... |
| eqeq2d 2740 | Deduction from equality to... |
| eqeq2 2741 | Equality implies equivalen... |
| eqeq2i 2742 | Inference from equality to... |
| eqeqan12d 2743 | A useful inference for sub... |
| eqeqan12rd 2744 | A useful inference for sub... |
| eqeq12d 2745 | A useful inference for sub... |
| eqeq12 2746 | Equality relationship amon... |
| eqeq12i 2747 | A useful inference for sub... |
| eqeq12OLD 2748 | Obsolete version of ~ eqeq... |
| eqeq12dOLD 2749 | Obsolete version of ~ eqeq... |
| eqeqan12dOLD 2750 | Obsolete version of ~ eqeq... |
| eqeqan12dALT 2751 | Alternate proof of ~ eqeqa... |
| eqtr 2752 | Transitive law for class e... |
| eqtr2 2753 | A transitive law for class... |
| eqtr2OLD 2754 | Obsolete version of eqtr2 ... |
| eqtr3 2755 | A transitive law for class... |
| eqtr3OLD 2756 | Obsolete version of ~ eqtr... |
| eqtri 2757 | An equality transitivity i... |
| eqtr2i 2758 | An equality transitivity i... |
| eqtr3i 2759 | An equality transitivity i... |
| eqtr4i 2760 | An equality transitivity i... |
| 3eqtri 2761 | An inference from three ch... |
| 3eqtrri 2762 | An inference from three ch... |
| 3eqtr2i 2763 | An inference from three ch... |
| 3eqtr2ri 2764 | An inference from three ch... |
| 3eqtr3i 2765 | An inference from three ch... |
| 3eqtr3ri 2766 | An inference from three ch... |
| 3eqtr4i 2767 | An inference from three ch... |
| 3eqtr4ri 2768 | An inference from three ch... |
| eqtrd 2769 | An equality transitivity d... |
| eqtr2d 2770 | An equality transitivity d... |
| eqtr3d 2771 | An equality transitivity e... |
| eqtr4d 2772 | An equality transitivity e... |
| 3eqtrd 2773 | A deduction from three cha... |
| 3eqtrrd 2774 | A deduction from three cha... |
| 3eqtr2d 2775 | A deduction from three cha... |
| 3eqtr2rd 2776 | A deduction from three cha... |
| 3eqtr3d 2777 | A deduction from three cha... |
| 3eqtr3rd 2778 | A deduction from three cha... |
| 3eqtr4d 2779 | A deduction from three cha... |
| 3eqtr4rd 2780 | A deduction from three cha... |
| eqtrid 2781 | An equality transitivity d... |
| eqtr2id 2782 | An equality transitivity d... |
| eqtr3id 2783 | An equality transitivity d... |
| eqtr3di 2784 | An equality transitivity d... |
| eqtrdi 2785 | An equality transitivity d... |
| eqtr2di 2786 | An equality transitivity d... |
| eqtr4di 2787 | An equality transitivity d... |
| eqtr4id 2788 | An equality transitivity d... |
| sylan9eq 2789 | An equality transitivity d... |
| sylan9req 2790 | An equality transitivity d... |
| sylan9eqr 2791 | An equality transitivity d... |
| 3eqtr3g 2792 | A chained equality inferen... |
| 3eqtr3a 2793 | A chained equality inferen... |
| 3eqtr4g 2794 | A chained equality inferen... |
| 3eqtr4a 2795 | A chained equality inferen... |
| eq2tri 2796 | A compound transitive infe... |
| abbi 2797 | Equivalent formulas yield ... |
| abbidv 2798 | Equivalent wff's yield equ... |
| abbii 2799 | Equivalent wff's yield equ... |
| abbid 2800 | Equivalent wff's yield equ... |
| abbib 2801 | Equal class abstractions r... |
| cbvabv 2802 | Rule used to change bound ... |
| cbvabw 2803 | Rule used to change bound ... |
| cbvab 2804 | Rule used to change bound ... |
| eqabbw 2805 | Version of ~ eqabb using i... |
| dfclel 2807 | Characterization of the el... |
| elex2 2808 | If a class contains anothe... |
| issetlem 2809 | Lemma for ~ elisset and ~ ... |
| elissetv 2810 | An element of a class exis... |
| elisset 2811 | An element of a class exis... |
| eleq1w 2812 | Weaker version of ~ eleq1 ... |
| eleq2w 2813 | Weaker version of ~ eleq2 ... |
| eleq1d 2814 | Deduction from equality to... |
| eleq2d 2815 | Deduction from equality to... |
| eleq2dALT 2816 | Alternate proof of ~ eleq2... |
| eleq1 2817 | Equality implies equivalen... |
| eleq2 2818 | Equality implies equivalen... |
| eleq12 2819 | Equality implies equivalen... |
| eleq1i 2820 | Inference from equality to... |
| eleq2i 2821 | Inference from equality to... |
| eleq12i 2822 | Inference from equality to... |
| eleq12d 2823 | Deduction from equality to... |
| eleq1a 2824 | A transitive-type law rela... |
| eqeltri 2825 | Substitution of equal clas... |
| eqeltrri 2826 | Substitution of equal clas... |
| eleqtri 2827 | Substitution of equal clas... |
| eleqtrri 2828 | Substitution of equal clas... |
| eqeltrd 2829 | Substitution of equal clas... |
| eqeltrrd 2830 | Deduction that substitutes... |
| eleqtrd 2831 | Deduction that substitutes... |
| eleqtrrd 2832 | Deduction that substitutes... |
| eqeltrid 2833 | A membership and equality ... |
| eqeltrrid 2834 | A membership and equality ... |
| eleqtrid 2835 | A membership and equality ... |
| eleqtrrid 2836 | A membership and equality ... |
| eqeltrdi 2837 | A membership and equality ... |
| eqeltrrdi 2838 | A membership and equality ... |
| eleqtrdi 2839 | A membership and equality ... |
| eleqtrrdi 2840 | A membership and equality ... |
| 3eltr3i 2841 | Substitution of equal clas... |
| 3eltr4i 2842 | Substitution of equal clas... |
| 3eltr3d 2843 | Substitution of equal clas... |
| 3eltr4d 2844 | Substitution of equal clas... |
| 3eltr3g 2845 | Substitution of equal clas... |
| 3eltr4g 2846 | Substitution of equal clas... |
| eleq2s 2847 | Substitution of equal clas... |
| eqneltri 2848 | If a class is not an eleme... |
| eqneltrd 2849 | If a class is not an eleme... |
| eqneltrrd 2850 | If a class is not an eleme... |
| neleqtrd 2851 | If a class is not an eleme... |
| neleqtrrd 2852 | If a class is not an eleme... |
| nelneq 2853 | A way of showing two class... |
| nelneq2 2854 | A way of showing two class... |
| eqsb1 2855 | Substitution for the left-... |
| clelsb1 2856 | Substitution for the first... |
| clelsb2 2857 | Substitution for the secon... |
| clelsb2OLD 2858 | Obsolete version of ~ clel... |
| cleqh 2859 | Establish equality between... |
| hbxfreq 2860 | A utility lemma to transfe... |
| hblem 2861 | Change the free variable o... |
| hblemg 2862 | Change the free variable o... |
| eqabdv 2863 | Deduction from a wff to a ... |
| eqabcdv 2864 | Deduction from a wff to a ... |
| eqabi 2865 | Equality of a class variab... |
| abid1 2866 | Every class is equal to a ... |
| abid2 2867 | A simplification of class ... |
| eqab 2868 | One direction of ~ eqabb i... |
| eqabb 2869 | Equality of a class variab... |
| eqabbOLD 2870 | Obsolete version of ~ eqab... |
| eqabcb 2871 | Equality of a class variab... |
| eqabrd 2872 | Equality of a class variab... |
| eqabri 2873 | Equality of a class variab... |
| eqabcri 2874 | Equality of a class variab... |
| clelab 2875 | Membership of a class vari... |
| clelabOLD 2876 | Obsolete version of ~ clel... |
| clabel 2877 | Membership of a class abst... |
| sbab 2878 | The right-hand side of the... |
| nfcjust 2880 | Justification theorem for ... |
| nfci 2882 | Deduce that a class ` A ` ... |
| nfcii 2883 | Deduce that a class ` A ` ... |
| nfcr 2884 | Consequence of the not-fre... |
| nfcrALT 2885 | Alternate version of ~ nfc... |
| nfcri 2886 | Consequence of the not-fre... |
| nfcd 2887 | Deduce that a class ` A ` ... |
| nfcrd 2888 | Consequence of the not-fre... |
| nfcrii 2889 | Consequence of the not-fre... |
| nfceqdf 2890 | An equality theorem for ef... |
| nfceqdfOLD 2891 | Obsolete version of ~ nfce... |
| nfceqi 2892 | Equality theorem for class... |
| nfcxfr 2893 | A utility lemma to transfe... |
| nfcxfrd 2894 | A utility lemma to transfe... |
| nfcv 2895 | If ` x ` is disjoint from ... |
| nfcvd 2896 | If ` x ` is disjoint from ... |
| nfab1 2897 | Bound-variable hypothesis ... |
| nfnfc1 2898 | The setvar ` x ` is bound ... |
| clelsb1fw 2899 | Substitution for the first... |
| clelsb1f 2900 | Substitution for the first... |
| nfab 2901 | Bound-variable hypothesis ... |
| nfabg 2902 | Bound-variable hypothesis ... |
| nfaba1 2903 | Bound-variable hypothesis ... |
| nfaba1OLD 2904 | Obsolete version of ~ nfab... |
| nfaba1g 2905 | Bound-variable hypothesis ... |
| nfeqd 2906 | Hypothesis builder for equ... |
| nfeld 2907 | Hypothesis builder for ele... |
| nfnfc 2908 | Hypothesis builder for ` F... |
| nfeq 2909 | Hypothesis builder for equ... |
| nfel 2910 | Hypothesis builder for ele... |
| nfeq1 2911 | Hypothesis builder for equ... |
| nfel1 2912 | Hypothesis builder for ele... |
| nfeq2 2913 | Hypothesis builder for equ... |
| nfel2 2914 | Hypothesis builder for ele... |
| drnfc1 2915 | Formula-building lemma for... |
| drnfc1OLD 2916 | Obsolete version of ~ drnf... |
| drnfc2 2917 | Formula-building lemma for... |
| drnfc2OLD 2918 | Obsolete version of ~ drnf... |
| nfabdw 2919 | Bound-variable hypothesis ... |
| nfabdwOLD 2920 | Obsolete version of ~ nfab... |
| nfabd 2921 | Bound-variable hypothesis ... |
| nfabd2 2922 | Bound-variable hypothesis ... |
| dvelimdc 2923 | Deduction form of ~ dvelim... |
| dvelimc 2924 | Version of ~ dvelim for cl... |
| nfcvf 2925 | If ` x ` and ` y ` are dis... |
| nfcvf2 2926 | If ` x ` and ` y ` are dis... |
| cleqf 2927 | Establish equality between... |
| eqabf 2928 | Equality of a class variab... |
| abid2f 2929 | A simplification of class ... |
| abid2fOLD 2930 | Obsolete version of ~ abid... |
| sbabel 2931 | Theorem to move a substitu... |
| sbabelOLD 2932 | Obsolete version of ~ sbab... |
| neii 2935 | Inference associated with ... |
| neir 2936 | Inference associated with ... |
| nne 2937 | Negation of inequality. (... |
| neneqd 2938 | Deduction eliminating ineq... |
| neneq 2939 | From inequality to non-equ... |
| neqned 2940 | If it is not the case that... |
| neqne 2941 | From non-equality to inequ... |
| neirr 2942 | No class is unequal to its... |
| exmidne 2943 | Excluded middle with equal... |
| eqneqall 2944 | A contradiction concerning... |
| nonconne 2945 | Law of noncontradiction wi... |
| necon3ad 2946 | Contrapositive law deducti... |
| necon3bd 2947 | Contrapositive law deducti... |
| necon2ad 2948 | Contrapositive inference f... |
| necon2bd 2949 | Contrapositive inference f... |
| necon1ad 2950 | Contrapositive deduction f... |
| necon1bd 2951 | Contrapositive deduction f... |
| necon4ad 2952 | Contrapositive inference f... |
| necon4bd 2953 | Contrapositive inference f... |
| necon3d 2954 | Contrapositive law deducti... |
| necon1d 2955 | Contrapositive law deducti... |
| necon2d 2956 | Contrapositive inference f... |
| necon4d 2957 | Contrapositive inference f... |
| necon3ai 2958 | Contrapositive inference f... |
| necon3aiOLD 2959 | Obsolete version of ~ neco... |
| necon3bi 2960 | Contrapositive inference f... |
| necon1ai 2961 | Contrapositive inference f... |
| necon1bi 2962 | Contrapositive inference f... |
| necon2ai 2963 | Contrapositive inference f... |
| necon2bi 2964 | Contrapositive inference f... |
| necon4ai 2965 | Contrapositive inference f... |
| necon3i 2966 | Contrapositive inference f... |
| necon1i 2967 | Contrapositive inference f... |
| necon2i 2968 | Contrapositive inference f... |
| necon4i 2969 | Contrapositive inference f... |
| necon3abid 2970 | Deduction from equality to... |
| necon3bbid 2971 | Deduction from equality to... |
| necon1abid 2972 | Contrapositive deduction f... |
| necon1bbid 2973 | Contrapositive inference f... |
| necon4abid 2974 | Contrapositive law deducti... |
| necon4bbid 2975 | Contrapositive law deducti... |
| necon2abid 2976 | Contrapositive deduction f... |
| necon2bbid 2977 | Contrapositive deduction f... |
| necon3bid 2978 | Deduction from equality to... |
| necon4bid 2979 | Contrapositive law deducti... |
| necon3abii 2980 | Deduction from equality to... |
| necon3bbii 2981 | Deduction from equality to... |
| necon1abii 2982 | Contrapositive inference f... |
| necon1bbii 2983 | Contrapositive inference f... |
| necon2abii 2984 | Contrapositive inference f... |
| necon2bbii 2985 | Contrapositive inference f... |
| necon3bii 2986 | Inference from equality to... |
| necom 2987 | Commutation of inequality.... |
| necomi 2988 | Inference from commutative... |
| necomd 2989 | Deduction from commutative... |
| nesym 2990 | Characterization of inequa... |
| nesymi 2991 | Inference associated with ... |
| nesymir 2992 | Inference associated with ... |
| neeq1d 2993 | Deduction for inequality. ... |
| neeq2d 2994 | Deduction for inequality. ... |
| neeq12d 2995 | Deduction for inequality. ... |
| neeq1 2996 | Equality theorem for inequ... |
| neeq2 2997 | Equality theorem for inequ... |
| neeq1i 2998 | Inference for inequality. ... |
| neeq2i 2999 | Inference for inequality. ... |
| neeq12i 3000 | Inference for inequality. ... |
| eqnetrd 3001 | Substitution of equal clas... |
| eqnetrrd 3002 | Substitution of equal clas... |
| neeqtrd 3003 | Substitution of equal clas... |
| eqnetri 3004 | Substitution of equal clas... |
| eqnetrri 3005 | Substitution of equal clas... |
| neeqtri 3006 | Substitution of equal clas... |
| neeqtrri 3007 | Substitution of equal clas... |
| neeqtrrd 3008 | Substitution of equal clas... |
| eqnetrrid 3009 | A chained equality inferen... |
| 3netr3d 3010 | Substitution of equality i... |
| 3netr4d 3011 | Substitution of equality i... |
| 3netr3g 3012 | Substitution of equality i... |
| 3netr4g 3013 | Substitution of equality i... |
| nebi 3014 | Contraposition law for ine... |
| pm13.18 3015 | Theorem *13.18 in [Whitehe... |
| pm13.181 3016 | Theorem *13.181 in [Whiteh... |
| pm13.181OLD 3017 | Obsolete version of ~ pm13... |
| pm2.61ine 3018 | Inference eliminating an i... |
| pm2.21ddne 3019 | A contradiction implies an... |
| pm2.61ne 3020 | Deduction eliminating an i... |
| pm2.61dne 3021 | Deduction eliminating an i... |
| pm2.61dane 3022 | Deduction eliminating an i... |
| pm2.61da2ne 3023 | Deduction eliminating two ... |
| pm2.61da3ne 3024 | Deduction eliminating thre... |
| pm2.61iine 3025 | Equality version of ~ pm2.... |
| mteqand 3026 | A modus tollens deduction ... |
| neor 3027 | Logical OR with an equalit... |
| neanior 3028 | A De Morgan's law for ineq... |
| ne3anior 3029 | A De Morgan's law for ineq... |
| neorian 3030 | A De Morgan's law for ineq... |
| nemtbir 3031 | An inference from an inequ... |
| nelne1 3032 | Two classes are different ... |
| nelne2 3033 | Two classes are different ... |
| nelelne 3034 | Two classes are different ... |
| neneor 3035 | If two classes are differe... |
| nfne 3036 | Bound-variable hypothesis ... |
| nfned 3037 | Bound-variable hypothesis ... |
| nabbib 3038 | Not equivalent wff's corre... |
| neli 3041 | Inference associated with ... |
| nelir 3042 | Inference associated with ... |
| nelcon3d 3043 | Contrapositive law deducti... |
| neleq12d 3044 | Equality theorem for negat... |
| neleq1 3045 | Equality theorem for negat... |
| neleq2 3046 | Equality theorem for negat... |
| nfnel 3047 | Bound-variable hypothesis ... |
| nfneld 3048 | Bound-variable hypothesis ... |
| nnel 3049 | Negation of negated member... |
| elnelne1 3050 | Two classes are different ... |
| elnelne2 3051 | Two classes are different ... |
| pm2.24nel 3052 | A contradiction concerning... |
| pm2.61danel 3053 | Deduction eliminating an e... |
| rgen 3056 | Generalization rule for re... |
| ralel 3057 | All elements of a class ar... |
| rgenw 3058 | Generalization rule for re... |
| rgen2w 3059 | Generalization rule for re... |
| mprg 3060 | Modus ponens combined with... |
| mprgbir 3061 | Modus ponens on biconditio... |
| raln 3062 | Restricted universally qua... |
| ralnex 3065 | Relationship between restr... |
| dfrex2 3066 | Relationship between restr... |
| nrex 3067 | Inference adding restricte... |
| alral 3068 | Universal quantification i... |
| rexex 3069 | Restricted existence impli... |
| rextru 3070 | Two ways of expressing tha... |
| ralimi2 3071 | Inference quantifying both... |
| reximi2 3072 | Inference quantifying both... |
| ralimia 3073 | Inference quantifying both... |
| reximia 3074 | Inference quantifying both... |
| ralimiaa 3075 | Inference quantifying both... |
| ralimi 3076 | Inference quantifying both... |
| reximi 3077 | Inference quantifying both... |
| ral2imi 3078 | Inference quantifying ante... |
| ralim 3079 | Distribution of restricted... |
| rexim 3080 | Theorem 19.22 of [Margaris... |
| reximiaOLD 3081 | Obsolete version of ~ rexi... |
| ralbii2 3082 | Inference adding different... |
| rexbii2 3083 | Inference adding different... |
| ralbiia 3084 | Inference adding restricte... |
| rexbiia 3085 | Inference adding restricte... |
| ralbii 3086 | Inference adding restricte... |
| rexbii 3087 | Inference adding restricte... |
| ralanid 3088 | Cancellation law for restr... |
| rexanid 3089 | Cancellation law for restr... |
| ralcom3 3090 | A commutation law for rest... |
| ralcom3OLD 3091 | Obsolete version of ~ ralc... |
| dfral2 3092 | Relationship between restr... |
| rexnal 3093 | Relationship between restr... |
| ralinexa 3094 | A transformation of restri... |
| rexanali 3095 | A transformation of restri... |
| ralbi 3096 | Distribute a restricted un... |
| rexbi 3097 | Distribute restricted quan... |
| rexbiOLD 3098 | Obsolete version of ~ rexb... |
| ralrexbid 3099 | Formula-building rule for ... |
| ralrexbidOLD 3100 | Obsolete version of ~ ralr... |
| r19.35 3101 | Restricted quantifier vers... |
| r19.35OLD 3102 | Obsolete version of ~ 19.3... |
| r19.26m 3103 | Version of ~ 19.26 and ~ r... |
| r19.26 3104 | Restricted quantifier vers... |
| r19.26-3 3105 | Version of ~ r19.26 with t... |
| ralbiim 3106 | Split a biconditional and ... |
| r19.29 3107 | Restricted quantifier vers... |
| r19.29OLD 3108 | Obsolete version of ~ r19.... |
| r19.29r 3109 | Restricted quantifier vers... |
| r19.29rOLD 3110 | Obsolete version of ~ r19.... |
| r19.29imd 3111 | Theorem 19.29 of [Margaris... |
| r19.40 3112 | Restricted quantifier vers... |
| r19.30 3113 | Restricted quantifier vers... |
| r19.30OLD 3114 | Obsolete version of ~ 19.3... |
| r19.43 3115 | Restricted quantifier vers... |
| 2ralimi 3116 | Inference quantifying both... |
| 3ralimi 3117 | Inference quantifying both... |
| 4ralimi 3118 | Inference quantifying both... |
| 5ralimi 3119 | Inference quantifying both... |
| 6ralimi 3120 | Inference quantifying both... |
| 2ralbii 3121 | Inference adding two restr... |
| 2rexbii 3122 | Inference adding two restr... |
| 3ralbii 3123 | Inference adding three res... |
| 4ralbii 3124 | Inference adding four rest... |
| 2ralbiim 3125 | Split a biconditional and ... |
| ralnex2 3126 | Relationship between two r... |
| ralnex3 3127 | Relationship between three... |
| rexnal2 3128 | Relationship between two r... |
| rexnal3 3129 | Relationship between three... |
| nrexralim 3130 | Negation of a complex pred... |
| r19.26-2 3131 | Restricted quantifier vers... |
| 2r19.29 3132 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3133 | Theorem 19.29 of [Margaris... |
| r19.29d2rOLD 3134 | Obsolete version of ~ r19.... |
| r2allem 3135 | Lemma factoring out common... |
| r2exlem 3136 | Lemma factoring out common... |
| hbralrimi 3137 | Inference from Theorem 19.... |
| ralrimiv 3138 | Inference from Theorem 19.... |
| ralrimiva 3139 | Inference from Theorem 19.... |
| rexlimiva 3140 | Inference from Theorem 19.... |
| rexlimiv 3141 | Inference from Theorem 19.... |
| nrexdv 3142 | Deduction adding restricte... |
| ralrimivw 3143 | Inference from Theorem 19.... |
| rexlimivw 3144 | Weaker version of ~ rexlim... |
| ralrimdv 3145 | Inference from Theorem 19.... |
| rexlimdv 3146 | Inference from Theorem 19.... |
| ralrimdva 3147 | Inference from Theorem 19.... |
| rexlimdva 3148 | Inference from Theorem 19.... |
| rexlimdvaa 3149 | Inference from Theorem 19.... |
| rexlimdva2 3150 | Inference from Theorem 19.... |
| r19.29an 3151 | A commonly used pattern in... |
| rexlimdv3a 3152 | Inference from Theorem 19.... |
| rexlimdvw 3153 | Inference from Theorem 19.... |
| rexlimddv 3154 | Restricted existential eli... |
| r19.29a 3155 | A commonly used pattern in... |
| ralimdv2 3156 | Inference quantifying both... |
| reximdv2 3157 | Deduction quantifying both... |
| reximdvai 3158 | Deduction quantifying both... |
| reximdvaiOLD 3159 | Obsolete version of ~ rexi... |
| ralimdva 3160 | Deduction quantifying both... |
| reximdva 3161 | Deduction quantifying both... |
| ralimdv 3162 | Deduction quantifying both... |
| reximdv 3163 | Deduction from Theorem 19.... |
| reximddv 3164 | Deduction from Theorem 19.... |
| reximddv3 3165 | Deduction from Theorem 19.... |
| reximssdv 3166 | Derivation of a restricted... |
| ralbidv2 3167 | Formula-building rule for ... |
| rexbidv2 3168 | Formula-building rule for ... |
| ralbidva 3169 | Formula-building rule for ... |
| rexbidva 3170 | Formula-building rule for ... |
| ralbidv 3171 | Formula-building rule for ... |
| rexbidv 3172 | Formula-building rule for ... |
| r19.21v 3173 | Restricted quantifier vers... |
| r19.21vOLD 3174 | Obsolete version of ~ r19.... |
| r19.37v 3175 | Restricted quantifier vers... |
| r19.23v 3176 | Restricted quantifier vers... |
| r19.36v 3177 | Restricted quantifier vers... |
| rexlimivOLD 3178 | Obsolete version of ~ rexl... |
| rexlimivaOLD 3179 | Obsolete version of ~ rexl... |
| rexlimivwOLD 3180 | Obsolete version of ~ rexl... |
| r19.27v 3181 | Restricted quantitifer ver... |
| r19.41v 3182 | Restricted quantifier vers... |
| r19.28v 3183 | Restricted quantifier vers... |
| r19.42v 3184 | Restricted quantifier vers... |
| r19.32v 3185 | Restricted quantifier vers... |
| r19.45v 3186 | Restricted quantifier vers... |
| r19.44v 3187 | One direction of a restric... |
| r2al 3188 | Double restricted universa... |
| r2ex 3189 | Double restricted existent... |
| r3al 3190 | Triple restricted universa... |
| r3ex 3191 | Triple existential quantif... |
| rgen2 3192 | Generalization rule for re... |
| ralrimivv 3193 | Inference from Theorem 19.... |
| rexlimivv 3194 | Inference from Theorem 19.... |
| ralrimivva 3195 | Inference from Theorem 19.... |
| ralrimdvv 3196 | Inference from Theorem 19.... |
| rgen3 3197 | Generalization rule for re... |
| ralrimivvva 3198 | Inference from Theorem 19.... |
| ralimdvva 3199 | Deduction doubly quantifyi... |
| reximdvva 3200 | Deduction doubly quantifyi... |
| ralimdvv 3201 | Deduction doubly quantifyi... |
| ralimd4v 3202 | Deduction quadrupally quan... |
| ralimd6v 3203 | Deduction sextupally quant... |
| ralrimdvva 3204 | Inference from Theorem 19.... |
| rexlimdvv 3205 | Inference from Theorem 19.... |
| rexlimdvva 3206 | Inference from Theorem 19.... |
| reximddv2 3207 | Double deduction from Theo... |
| r19.29vva 3208 | A commonly used pattern ba... |
| r19.29vvaOLD 3209 | Obsolete version of ~ r19.... |
| 2rexbiia 3210 | Inference adding two restr... |
| 2ralbidva 3211 | Formula-building rule for ... |
| 2rexbidva 3212 | Formula-building rule for ... |
| 2ralbidv 3213 | Formula-building rule for ... |
| 2rexbidv 3214 | Formula-building rule for ... |
| rexralbidv 3215 | Formula-building rule for ... |
| 3ralbidv 3216 | Formula-building rule for ... |
| 4ralbidv 3217 | Formula-building rule for ... |
| 6ralbidv 3218 | Formula-building rule for ... |
| r19.41vv 3219 | Version of ~ r19.41v with ... |
| reeanlem 3220 | Lemma factoring out common... |
| reeanv 3221 | Rearrange restricted exist... |
| 3reeanv 3222 | Rearrange three restricted... |
| 2ralor 3223 | Distribute restricted univ... |
| 2ralorOLD 3224 | Obsolete version of ~ 2ral... |
| risset 3225 | Two ways to say " ` A ` be... |
| nelb 3226 | A definition of ` -. A e. ... |
| nelbOLD 3227 | Obsolete version of ~ nelb... |
| rspw 3228 | Restricted specialization.... |
| cbvralvw 3229 | Change the bound variable ... |
| cbvrexvw 3230 | Change the bound variable ... |
| cbvraldva 3231 | Rule used to change the bo... |
| cbvrexdva 3232 | Rule used to change the bo... |
| cbvral2vw 3233 | Change bound variables of ... |
| cbvrex2vw 3234 | Change bound variables of ... |
| cbvral3vw 3235 | Change bound variables of ... |
| cbvral4vw 3236 | Change bound variables of ... |
| cbvral6vw 3237 | Change bound variables of ... |
| cbvral8vw 3238 | Change bound variables of ... |
| rsp 3239 | Restricted specialization.... |
| rspa 3240 | Restricted specialization.... |
| rspe 3241 | Restricted specialization.... |
| rspec 3242 | Specialization rule for re... |
| r19.21bi 3243 | Inference from Theorem 19.... |
| r19.21be 3244 | Inference from Theorem 19.... |
| r19.21t 3245 | Restricted quantifier vers... |
| r19.21 3246 | Restricted quantifier vers... |
| r19.23t 3247 | Closed theorem form of ~ r... |
| r19.23 3248 | Restricted quantifier vers... |
| ralrimi 3249 | Inference from Theorem 19.... |
| ralrimia 3250 | Inference from Theorem 19.... |
| rexlimi 3251 | Restricted quantifier vers... |
| ralimdaa 3252 | Deduction quantifying both... |
| reximdai 3253 | Deduction from Theorem 19.... |
| r19.37 3254 | Restricted quantifier vers... |
| r19.41 3255 | Restricted quantifier vers... |
| ralrimd 3256 | Inference from Theorem 19.... |
| rexlimd2 3257 | Version of ~ rexlimd with ... |
| rexlimd 3258 | Deduction form of ~ rexlim... |
| r19.29af2 3259 | A commonly used pattern ba... |
| r19.29af 3260 | A commonly used pattern ba... |
| reximd2a 3261 | Deduction quantifying both... |
| ralbida 3262 | Formula-building rule for ... |
| ralbidaOLD 3263 | Obsolete version of ~ ralb... |
| rexbida 3264 | Formula-building rule for ... |
| ralbid 3265 | Formula-building rule for ... |
| rexbid 3266 | Formula-building rule for ... |
| rexbidvALT 3267 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3268 | Alternate proof of ~ rexbi... |
| rsp2 3269 | Restricted specialization,... |
| rsp2e 3270 | Restricted specialization.... |
| rspec2 3271 | Specialization rule for re... |
| rspec3 3272 | Specialization rule for re... |
| r2alf 3273 | Double restricted universa... |
| r2exf 3274 | Double restricted existent... |
| 2ralbida 3275 | Formula-building rule for ... |
| nfra1 3276 | The setvar ` x ` is not fr... |
| nfre1 3277 | The setvar ` x ` is not fr... |
| ralcom4 3278 | Commutation of restricted ... |
| ralcom4OLD 3279 | Obsolete version of ~ ralc... |
| rexcom4 3280 | Commutation of restricted ... |
| ralcom 3281 | Commutation of restricted ... |
| rexcom 3282 | Commutation of restricted ... |
| rexcomOLD 3283 | Obsolete version of ~ rexc... |
| rexcom4a 3284 | Specialized existential co... |
| ralrot3 3285 | Rotate three restricted un... |
| ralcom13 3286 | Swap first and third restr... |
| ralcom13OLD 3287 | Obsolete version of ~ ralc... |
| rexcom13 3288 | Swap first and third restr... |
| rexrot4 3289 | Rotate four restricted exi... |
| 2ex2rexrot 3290 | Rotate two existential qua... |
| nfra2w 3291 | Similar to Lemma 24 of [Mo... |
| nfra2wOLD 3292 | Obsolete version of ~ nfra... |
| hbra1 3293 | The setvar ` x ` is not fr... |
| ralcomf 3294 | Commutation of restricted ... |
| rexcomf 3295 | Commutation of restricted ... |
| cbvralfw 3296 | Rule used to change bound ... |
| cbvrexfw 3297 | Rule used to change bound ... |
| cbvralw 3298 | Rule used to change bound ... |
| cbvrexw 3299 | Rule used to change bound ... |
| hbral 3300 | Bound-variable hypothesis ... |
| nfraldw 3301 | Deduction version of ~ nfr... |
| nfrexdw 3302 | Deduction version of ~ nfr... |
| nfralw 3303 | Bound-variable hypothesis ... |
| nfralwOLD 3304 | Obsolete version of ~ nfra... |
| nfrexw 3305 | Bound-variable hypothesis ... |
| r19.12 3306 | Restricted quantifier vers... |
| r19.12OLD 3307 | Obsolete version of ~ 19.1... |
| reean 3308 | Rearrange restricted exist... |
| cbvralsvw 3309 | Change bound variable by u... |
| cbvrexsvw 3310 | Change bound variable by u... |
| cbvralsvwOLD 3311 | Obsolete version of ~ cbvr... |
| cbvrexsvwOLD 3312 | Obsolete version of ~ cbvr... |
| nfraldwOLD 3313 | Obsolete version of ~ nfra... |
| nfra2wOLDOLD 3314 | Obsolete version of ~ nfra... |
| rexeq 3315 | Equality theorem for restr... |
| raleq 3316 | Equality theorem for restr... |
| raleqi 3317 | Equality inference for res... |
| rexeqi 3318 | Equality inference for res... |
| raleqdv 3319 | Equality deduction for res... |
| rexeqdv 3320 | Equality deduction for res... |
| raleqtrdv 3321 | Substitution of equal clas... |
| rexeqtrdv 3322 | Substitution of equal clas... |
| raleqtrrdv 3323 | Substitution of equal clas... |
| rexeqtrrdv 3324 | Substitution of equal clas... |
| raleqbidva 3325 | Equality deduction for res... |
| rexeqbidva 3326 | Equality deduction for res... |
| raleqbidvv 3327 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3328 | Obsolete version of ~ rale... |
| rexeqbidvv 3329 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3330 | Obsolete version of ~ rexe... |
| raleqbi1dv 3331 | Equality deduction for res... |
| rexeqbi1dv 3332 | Equality deduction for res... |
| raleqOLD 3333 | Obsolete version of ~ rale... |
| rexeqOLD 3334 | Obsolete version of ~ rale... |
| raleleq 3335 | All elements of a class ar... |
| raleleqOLD 3336 | Obsolete version of ~ rale... |
| raleqbii 3337 | Equality deduction for res... |
| rexeqbii 3338 | Equality deduction for res... |
| raleqbidv 3339 | Equality deduction for res... |
| rexeqbidv 3340 | Equality deduction for res... |
| cbvraldva2 3341 | Rule used to change the bo... |
| cbvrexdva2 3342 | Rule used to change the bo... |
| cbvrexdva2OLD 3343 | Obsolete version of ~ cbvr... |
| cbvraldvaOLD 3344 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3345 | Obsolete version of ~ cbvr... |
| raleqf 3346 | Equality theorem for restr... |
| rexeqf 3347 | Equality theorem for restr... |
| rexeqfOLD 3348 | Obsolete version of ~ rexe... |
| raleqbid 3349 | Equality deduction for res... |
| rexeqbid 3350 | Equality deduction for res... |
| sbralie 3351 | Implicit to explicit subst... |
| sbralieALT 3352 | Alternative shorter proof ... |
| cbvralf 3353 | Rule used to change bound ... |
| cbvrexf 3354 | Rule used to change bound ... |
| cbvral 3355 | Rule used to change bound ... |
| cbvrex 3356 | Rule used to change bound ... |
| cbvralv 3357 | Change the bound variable ... |
| cbvrexv 3358 | Change the bound variable ... |
| cbvralsv 3359 | Change bound variable by u... |
| cbvrexsv 3360 | Change bound variable by u... |
| cbvral2v 3361 | Change bound variables of ... |
| cbvrex2v 3362 | Change bound variables of ... |
| cbvral3v 3363 | Change bound variables of ... |
| rgen2a 3364 | Generalization rule for re... |
| nfrald 3365 | Deduction version of ~ nfr... |
| nfrexd 3366 | Deduction version of ~ nfr... |
| nfral 3367 | Bound-variable hypothesis ... |
| nfrex 3368 | Bound-variable hypothesis ... |
| nfra2 3369 | Similar to Lemma 24 of [Mo... |
| ralcom2 3370 | Commutation of restricted ... |
| reu5 3375 | Restricted uniqueness in t... |
| reurmo 3376 | Restricted existential uni... |
| reurex 3377 | Restricted unique existenc... |
| mormo 3378 | Unrestricted "at most one"... |
| rmobiia 3379 | Formula-building rule for ... |
| reubiia 3380 | Formula-building rule for ... |
| rmobii 3381 | Formula-building rule for ... |
| reubii 3382 | Formula-building rule for ... |
| rmoanid 3383 | Cancellation law for restr... |
| reuanid 3384 | Cancellation law for restr... |
| rmoanidOLD 3385 | Obsolete version of ~ rmoa... |
| reuanidOLD 3386 | Obsolete version of ~ reua... |
| 2reu2rex 3387 | Double restricted existent... |
| rmobidva 3388 | Formula-building rule for ... |
| reubidva 3389 | Formula-building rule for ... |
| rmobidv 3390 | Formula-building rule for ... |
| reubidv 3391 | Formula-building rule for ... |
| reueubd 3392 | Restricted existential uni... |
| rmo5 3393 | Restricted "at most one" i... |
| nrexrmo 3394 | Nonexistence implies restr... |
| moel 3395 | "At most one" element in a... |
| cbvrmovw 3396 | Change the bound variable ... |
| cbvreuvw 3397 | Change the bound variable ... |
| moelOLD 3398 | Obsolete version of ~ moel... |
| rmobida 3399 | Formula-building rule for ... |
| reubida 3400 | Formula-building rule for ... |
| rmobidvaOLD 3401 | Obsolete version of ~ rmob... |
| cbvrmow 3402 | Change the bound variable ... |
| cbvreuw 3403 | Change the bound variable ... |
| nfrmo1 3404 | The setvar ` x ` is not fr... |
| nfreu1 3405 | The setvar ` x ` is not fr... |
| nfrmow 3406 | Bound-variable hypothesis ... |
| nfreuw 3407 | Bound-variable hypothesis ... |
| cbvreuwOLD 3408 | Obsolete version of ~ cbvr... |
| cbvreuvwOLD 3409 | Obsolete version of ~ cbvr... |
| rmoeq1 3410 | Equality theorem for restr... |
| reueq1 3411 | Equality theorem for restr... |
| rmoeq1OLD 3412 | Obsolete version of ~ rmoe... |
| reueq1OLD 3413 | Obsolete version of ~ reue... |
| rmoeqd 3414 | Equality deduction for res... |
| reueqd 3415 | Equality deduction for res... |
| rmoeq1f 3416 | Equality theorem for restr... |
| reueq1f 3417 | Equality theorem for restr... |
| nfreuwOLD 3418 | Obsolete version of ~ nfre... |
| nfrmowOLD 3419 | Obsolete version of ~ nfrm... |
| cbvreu 3420 | Change the bound variable ... |
| cbvrmo 3421 | Change the bound variable ... |
| cbvrmov 3422 | Change the bound variable ... |
| cbvreuv 3423 | Change the bound variable ... |
| nfrmod 3424 | Deduction version of ~ nfr... |
| nfreud 3425 | Deduction version of ~ nfr... |
| nfrmo 3426 | Bound-variable hypothesis ... |
| nfreu 3427 | Bound-variable hypothesis ... |
| rabbidva2 3430 | Equivalent wff's yield equ... |
| rabbia2 3431 | Equivalent wff's yield equ... |
| rabbiia 3432 | Equivalent formulas yield ... |
| rabbiiaOLD 3433 | Obsolete version of ~ rabb... |
| rabbii 3434 | Equivalent wff's correspon... |
| rabbidva 3435 | Equivalent wff's yield equ... |
| rabbidv 3436 | Equivalent wff's yield equ... |
| rabbieq 3437 | Equivalent wff's correspon... |
| rabswap 3438 | Swap with a membership rel... |
| cbvrabv 3439 | Rule to change the bound v... |
| rabeqcda 3440 | When ` ps ` is always true... |
| rabeqc 3441 | A restricted class abstrac... |
| rabeqi 3442 | Equality theorem for restr... |
| rabeq 3443 | Equality theorem for restr... |
| rabeqdv 3444 | Equality of restricted cla... |
| rabeqbidva 3445 | Equality of restricted cla... |
| rabeqbidv 3446 | Equality of restricted cla... |
| rabrabi 3447 | Abstract builder restricte... |
| nfrab1 3448 | The abstraction variable i... |
| rabid 3449 | An "identity" law of concr... |
| rabidim1 3450 | Membership in a restricted... |
| reqabi 3451 | Inference from equality of... |
| rabrab 3452 | Abstract builder restricte... |
| rabrabiOLD 3453 | Obsolete version of ~ rabr... |
| rabbida4 3454 | Version of ~ rabbidva2 wit... |
| rabbida 3455 | Equivalent wff's yield equ... |
| rabbid 3456 | Version of ~ rabbidv with ... |
| rabeqd 3457 | Deduction form of ~ rabeq ... |
| rabeqbida 3458 | Version of ~ rabeqbidva wi... |
| rabbi 3459 | Equivalent wff's correspon... |
| rabid2f 3460 | An "identity" law for rest... |
| rabid2im 3461 | One direction of ~ rabid2 ... |
| rabid2 3462 | An "identity" law for rest... |
| rabid2OLD 3463 | Obsolete version of ~ rabi... |
| rabeqf 3464 | Equality theorem for restr... |
| cbvrabw 3465 | Rule to change the bound v... |
| cbvrabwOLD 3466 | Obsolete version of ~ cbvr... |
| nfrabw 3467 | A variable not free in a w... |
| nfrabwOLD 3468 | Obsolete version of ~ nfra... |
| rabbidaOLD 3469 | Obsolete version of ~ rabb... |
| nfrab 3470 | A variable not free in a w... |
| cbvrab 3471 | Rule to change the bound v... |
| vjust 3473 | Justification theorem for ... |
| dfv2 3475 | Alternate definition of th... |
| vex 3476 | All setvar variables are s... |
| vexOLD 3477 | Obsolete version of ~ vex ... |
| elv 3478 | If a proposition is implie... |
| elvd 3479 | If a proposition is implie... |
| el2v 3480 | If a proposition is implie... |
| el3v3 3481 | If a proposition is implie... |
| eqv 3482 | The universe contains ever... |
| eqvf 3483 | The universe contains ever... |
| abv 3484 | The class of sets verifyin... |
| abvALT 3485 | Alternate proof of ~ abv ,... |
| isset 3486 | Two ways to express that "... |
| cbvexeqsetf 3487 | The expression ` E. x x = ... |
| issetft 3488 | Closed theorem form of ~ i... |
| issetf 3489 | A version of ~ isset that ... |
| isseti 3490 | A way to say " ` A ` is a ... |
| issetri 3491 | A way to say " ` A ` is a ... |
| eqvisset 3492 | A class equal to a variabl... |
| elex 3493 | If a class is a member of ... |
| elexOLD 3494 | Obsolete version of ~ elex... |
| elexi 3495 | If a class is a member of ... |
| elexd 3496 | If a class is a member of ... |
| elex2OLD 3497 | Obsolete version of ~ elex... |
| elex22 3498 | If two classes each contai... |
| prcnel 3499 | A proper class doesn't bel... |
| ralv 3500 | A universal quantifier res... |
| rexv 3501 | An existential quantifier ... |
| reuv 3502 | A unique existential quant... |
| rmov 3503 | An at-most-one quantifier ... |
| rabab 3504 | A class abstraction restri... |
| rexcom4b 3505 | Specialized existential co... |
| ceqsal1t 3506 | One direction of ~ ceqsalt... |
| ceqsalt 3507 | Closed theorem version of ... |
| ceqsralt 3508 | Restricted quantifier vers... |
| ceqsalg 3509 | A representation of explic... |
| ceqsalgALT 3510 | Alternate proof of ~ ceqsa... |
| ceqsal 3511 | A representation of explic... |
| ceqsalALT 3512 | A representation of explic... |
| ceqsalv 3513 | A representation of explic... |
| ceqsalvOLD 3514 | Obsolete version of ~ ceqs... |
| ceqsralv 3515 | Restricted quantifier vers... |
| ceqsralvOLD 3516 | Obsolete version of ~ ceqs... |
| gencl 3517 | Implicit substitution for ... |
| 2gencl 3518 | Implicit substitution for ... |
| 3gencl 3519 | Implicit substitution for ... |
| cgsexg 3520 | Implicit substitution infe... |
| cgsex2g 3521 | Implicit substitution infe... |
| cgsex4g 3522 | An implicit substitution i... |
| cgsex4gOLD 3523 | Obsolete version of ~ cgse... |
| ceqsex 3524 | Elimination of an existent... |
| ceqsexOLD 3525 | Obsolete version of ~ ceqs... |
| ceqsexv 3526 | Elimination of an existent... |
| ceqsexvOLD 3527 | Obsolete version of ~ ceqs... |
| ceqsexvOLDOLD 3528 | Obsolete version of ~ ceqs... |
| ceqsexv2d 3529 | Elimination of an existent... |
| ceqsexv2dOLD 3530 | Obsolete version of ~ ceqs... |
| ceqsex2 3531 | Elimination of two existen... |
| ceqsex2v 3532 | Elimination of two existen... |
| ceqsex3v 3533 | Elimination of three exist... |
| ceqsex4v 3534 | Elimination of four existe... |
| ceqsex6v 3535 | Elimination of six existen... |
| ceqsex8v 3536 | Elimination of eight exist... |
| gencbvex 3537 | Change of bound variable u... |
| gencbvex2 3538 | Restatement of ~ gencbvex ... |
| gencbval 3539 | Change of bound variable u... |
| sbhypf 3540 | Introduce an explicit subs... |
| sbhypfOLD 3541 | Obsolete version of ~ sbhy... |
| spcimgft 3542 | Closed theorem form of ~ s... |
| spcimgfi1 3543 | A closed version of ~ spci... |
| spcimgfi1OLD 3544 | Obsolete version of ~ spci... |
| spcgft 3545 | A closed version of ~ spcg... |
| spcimgf 3546 | Rule of specialization, us... |
| spcimegf 3547 | Existential specialization... |
| vtoclgft 3548 | Closed theorem form of ~ v... |
| vtocleg 3549 | Implicit substitution of a... |
| vtoclg 3550 | Implicit substitution of a... |
| vtocle 3551 | Implicit substitution of a... |
| vtocleOLD 3552 | Obsolete version of ~ vtoc... |
| vtoclbg 3553 | Implicit substitution of a... |
| vtocl 3554 | Implicit substitution of a... |
| vtoclOLD 3555 | Obsolete version of ~ vtoc... |
| vtocldf 3556 | Implicit substitution of a... |
| vtocld 3557 | Implicit substitution of a... |
| vtocldOLD 3558 | Obsolete version of ~ vtoc... |
| vtocl2d 3559 | Implicit substitution of t... |
| vtoclef 3560 | Implicit substitution of a... |
| vtoclf 3561 | Implicit substitution of a... |
| vtoclfOLD 3562 | Obsolete version of ~ vtoc... |
| vtocl2 3563 | Implicit substitution of c... |
| vtocl3 3564 | Implicit substitution of c... |
| vtoclb 3565 | Implicit substitution of a... |
| vtoclgf 3566 | Implicit substitution of a... |
| vtoclg1f 3567 | Version of ~ vtoclgf with ... |
| vtoclgOLD 3568 | Obsolete version of ~ vtoc... |
| vtocl2gf 3569 | Implicit substitution of a... |
| vtocl3gf 3570 | Implicit substitution of a... |
| vtocl2g 3571 | Implicit substitution of 2... |
| vtocl3g 3572 | Implicit substitution of a... |
| vtoclgaf 3573 | Implicit substitution of a... |
| vtoclga 3574 | Implicit substitution of a... |
| vtocl2ga 3575 | Implicit substitution of 2... |
| vtocl2gaf 3576 | Implicit substitution of 2... |
| vtocl2gafOLD 3577 | Obsolete version of ~ vtoc... |
| vtocl3gaf 3578 | Implicit substitution of 3... |
| vtocl3gafOLD 3579 | Obsolete version of ~ vtoc... |
| vtocl3ga 3580 | Implicit substitution of 3... |
| vtocl3gaOLD 3581 | Obsolete version of ~ vtoc... |
| vtocl3gaOLDOLD 3582 | Obsolete version of ~ vtoc... |
| vtocl4g 3583 | Implicit substitution of 4... |
| vtocl4ga 3584 | Implicit substitution of 4... |
| vtocl4gaOLD 3585 | Obsolete version of ~ vtoc... |
| vtoclegft 3586 | Implicit substitution of a... |
| vtoclegftOLD 3587 | Obsolete version of ~ vtoc... |
| vtoclri 3588 | Implicit substitution of a... |
| spcgf 3589 | Rule of specialization, us... |
| spcegf 3590 | Existential specialization... |
| spcimdv 3591 | Restricted specialization,... |
| spcdv 3592 | Rule of specialization, us... |
| spcimedv 3593 | Restricted existential spe... |
| spcgv 3594 | Rule of specialization, us... |
| spcegv 3595 | Existential specialization... |
| spcedv 3596 | Existential specialization... |
| spc2egv 3597 | Existential specialization... |
| spc2gv 3598 | Specialization with two qu... |
| spc2ed 3599 | Existential specialization... |
| spc2d 3600 | Specialization with 2 quan... |
| spc3egv 3601 | Existential specialization... |
| spc3gv 3602 | Specialization with three ... |
| spcv 3603 | Rule of specialization, us... |
| spcev 3604 | Existential specialization... |
| spc2ev 3605 | Existential specialization... |
| rspct 3606 | A closed version of ~ rspc... |
| rspcdf 3607 | Restricted specialization,... |
| rspc 3608 | Restricted specialization,... |
| rspce 3609 | Restricted existential spe... |
| rspcimdv 3610 | Restricted specialization,... |
| rspcimedv 3611 | Restricted existential spe... |
| rspcdv 3612 | Restricted specialization,... |
| rspcedv 3613 | Restricted existential spe... |
| rspcebdv 3614 | Restricted existential spe... |
| rspcdv2 3615 | Restricted specialization,... |
| rspcv 3616 | Restricted specialization,... |
| rspccv 3617 | Restricted specialization,... |
| rspcva 3618 | Restricted specialization,... |
| rspccva 3619 | Restricted specialization,... |
| rspcev 3620 | Restricted existential spe... |
| rspcdva 3621 | Restricted specialization,... |
| rspcedvd 3622 | Restricted existential spe... |
| rspcedvdw 3623 | Version of ~ rspcedvd wher... |
| rspceb2dv 3624 | Restricted existential spe... |
| rspcime 3625 | Prove a restricted existen... |
| rspceaimv 3626 | Restricted existential spe... |
| rspcedeq1vd 3627 | Restricted existential spe... |
| rspcedeq2vd 3628 | Restricted existential spe... |
| rspc2 3629 | Restricted specialization ... |
| rspc2gv 3630 | Restricted specialization ... |
| rspc2v 3631 | 2-variable restricted spec... |
| rspc2va 3632 | 2-variable restricted spec... |
| rspc2ev 3633 | 2-variable restricted exis... |
| 2rspcedvdw 3634 | Double application of ~ rs... |
| rspc2dv 3635 | 2-variable restricted spec... |
| rspc3v 3636 | 3-variable restricted spec... |
| rspc3ev 3637 | 3-variable restricted exis... |
| rspc3dv 3638 | 3-variable restricted spec... |
| rspc4v 3639 | 4-variable restricted spec... |
| rspc6v 3640 | 6-variable restricted spec... |
| rspc8v 3641 | 8-variable restricted spec... |
| rspceeqv 3642 | Restricted existential spe... |
| ralxpxfr2d 3643 | Transfer a universal quant... |
| rexraleqim 3644 | Statement following from e... |
| eqvincg 3645 | A variable introduction la... |
| eqvinc 3646 | A variable introduction la... |
| eqvincf 3647 | A variable introduction la... |
| alexeqg 3648 | Two ways to express substi... |
| ceqex 3649 | Equality implies equivalen... |
| ceqsexg 3650 | A representation of explic... |
| ceqsexgv 3651 | Elimination of an existent... |
| ceqsrexv 3652 | Elimination of a restricte... |
| ceqsrexbv 3653 | Elimination of a restricte... |
| ceqsralbv 3654 | Elimination of a restricte... |
| ceqsrex2v 3655 | Elimination of a restricte... |
| clel2g 3656 | Alternate definition of me... |
| clel2gOLD 3657 | Obsolete version of ~ clel... |
| clel2 3658 | Alternate definition of me... |
| clel3g 3659 | Alternate definition of me... |
| clel3 3660 | Alternate definition of me... |
| clel4g 3661 | Alternate definition of me... |
| clel4 3662 | Alternate definition of me... |
| clel4OLD 3663 | Obsolete version of ~ clel... |
| clel5 3664 | Alternate definition of cl... |
| pm13.183 3665 | Compare theorem *13.183 in... |
| rr19.3v 3666 | Restricted quantifier vers... |
| rr19.28v 3667 | Restricted quantifier vers... |
| elab6g 3668 | Membership in a class abst... |
| elabd2 3669 | Membership in a class abst... |
| elabd3 3670 | Membership in a class abst... |
| elabgt 3671 | Membership in a class abst... |
| elabgtOLD 3672 | Obsolete version of ~ elab... |
| elabgtOLDOLD 3673 | Obsolete version of ~ elab... |
| elabgf 3674 | Membership in a class abst... |
| elabf 3675 | Membership in a class abst... |
| elabg 3676 | Membership in a class abst... |
| elabgOLD 3677 | Obsolete version of ~ elab... |
| elab 3678 | Membership in a class abst... |
| elabOLD 3679 | Obsolete version of ~ elab... |
| elab2g 3680 | Membership in a class abst... |
| elabd 3681 | Explicit demonstration the... |
| elab2 3682 | Membership in a class abst... |
| elab4g 3683 | Membership in a class abst... |
| elab3gf 3684 | Membership in a class abst... |
| elab3g 3685 | Membership in a class abst... |
| elab3 3686 | Membership in a class abst... |
| elrabi 3687 | Implication for the member... |
| elrabf 3688 | Membership in a restricted... |
| rabtru 3689 | Abstract builder using the... |
| rabeqcOLD 3690 | Obsolete version of ~ rabe... |
| elrab3t 3691 | Membership in a restricted... |
| elrab 3692 | Membership in a restricted... |
| elrab3 3693 | Membership in a restricted... |
| elrabd 3694 | Membership in a restricted... |
| elrab2 3695 | Membership in a restricted... |
| ralab 3696 | Universal quantification o... |
| ralabOLD 3697 | Obsolete version of ~ rala... |
| ralrab 3698 | Universal quantification o... |
| rexab 3699 | Existential quantification... |
| rexabOLD 3700 | Obsolete version of ~ rexa... |
| rexrab 3701 | Existential quantification... |
| ralab2 3702 | Universal quantification o... |
| ralrab2 3703 | Universal quantification o... |
| rexab2 3704 | Existential quantification... |
| rexrab2 3705 | Existential quantification... |
| reurab 3706 | Restricted existential uni... |
| abidnf 3707 | Identity used to create cl... |
| dedhb 3708 | A deduction theorem for co... |
| class2seteq 3709 | Writing a set as a class a... |
| nelrdva 3710 | Deduce negative membership... |
| eqeu 3711 | A condition which implies ... |
| moeq 3712 | There exists at most one s... |
| eueq 3713 | A class is a set if and on... |
| eueqi 3714 | There exists a unique set ... |
| eueq2 3715 | Equality has existential u... |
| eueq3 3716 | Equality has existential u... |
| moeq3 3717 | "At most one" property of ... |
| mosub 3718 | "At most one" remains true... |
| mo2icl 3719 | Theorem for inferring "at ... |
| mob2 3720 | Consequence of "at most on... |
| moi2 3721 | Consequence of "at most on... |
| mob 3722 | Equality implied by "at mo... |
| moi 3723 | Equality implied by "at mo... |
| morex 3724 | Derive membership from uni... |
| euxfr2w 3725 | Transfer existential uniqu... |
| euxfrw 3726 | Transfer existential uniqu... |
| euxfr2 3727 | Transfer existential uniqu... |
| euxfr 3728 | Transfer existential uniqu... |
| euind 3729 | Existential uniqueness via... |
| reu2 3730 | A way to express restricte... |
| reu6 3731 | A way to express restricte... |
| reu3 3732 | A way to express restricte... |
| reu6i 3733 | A condition which implies ... |
| eqreu 3734 | A condition which implies ... |
| rmo4 3735 | Restricted "at most one" u... |
| reu4 3736 | Restricted uniqueness usin... |
| reu7 3737 | Restricted uniqueness usin... |
| reu8 3738 | Restricted uniqueness usin... |
| rmo3f 3739 | Restricted "at most one" u... |
| rmo4f 3740 | Restricted "at most one" u... |
| reu2eqd 3741 | Deduce equality from restr... |
| reueq 3742 | Equality has existential u... |
| rmoeq 3743 | Equality's restricted exis... |
| rmoan 3744 | Restricted "at most one" s... |
| rmoim 3745 | Restricted "at most one" i... |
| rmoimia 3746 | Restricted "at most one" i... |
| rmoimi 3747 | Restricted "at most one" i... |
| rmoimi2 3748 | Restricted "at most one" i... |
| 2reu5a 3749 | Double restricted existent... |
| reuimrmo 3750 | Restricted uniqueness impl... |
| 2reuswap 3751 | A condition allowing swap ... |
| 2reuswap2 3752 | A condition allowing swap ... |
| reuxfrd 3753 | Transfer existential uniqu... |
| reuxfr 3754 | Transfer existential uniqu... |
| reuxfr1d 3755 | Transfer existential uniqu... |
| reuxfr1ds 3756 | Transfer existential uniqu... |
| reuxfr1 3757 | Transfer existential uniqu... |
| reuind 3758 | Existential uniqueness via... |
| 2rmorex 3759 | Double restricted quantifi... |
| 2reu5lem1 3760 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3761 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3762 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3763 | Double restricted existent... |
| 2reurmo 3764 | Double restricted quantifi... |
| 2reurex 3765 | Double restricted quantifi... |
| 2rmoswap 3766 | A condition allowing to sw... |
| 2rexreu 3767 | Double restricted existent... |
| cdeqi 3770 | Deduce conditional equalit... |
| cdeqri 3771 | Property of conditional eq... |
| cdeqth 3772 | Deduce conditional equalit... |
| cdeqnot 3773 | Distribute conditional equ... |
| cdeqal 3774 | Distribute conditional equ... |
| cdeqab 3775 | Distribute conditional equ... |
| cdeqal1 3776 | Distribute conditional equ... |
| cdeqab1 3777 | Distribute conditional equ... |
| cdeqim 3778 | Distribute conditional equ... |
| cdeqcv 3779 | Conditional equality for s... |
| cdeqeq 3780 | Distribute conditional equ... |
| cdeqel 3781 | Distribute conditional equ... |
| nfcdeq 3782 | If we have a conditional e... |
| nfccdeq 3783 | Variation of ~ nfcdeq for ... |
| rru 3784 | Relative version of Russel... |
| ru 3785 | Russell's Paradox. Propos... |
| ruOLD 3786 | Obsolete version of ~ ru a... |
| dfsbcq 3789 | Proper substitution of a c... |
| dfsbcq2 3790 | This theorem, which is sim... |
| sbsbc 3791 | Show that ~ df-sb and ~ df... |
| sbceq1d 3792 | Equality theorem for class... |
| sbceq1dd 3793 | Equality theorem for class... |
| sbceqbid 3794 | Equality theorem for class... |
| sbc8g 3795 | This is the closest we can... |
| sbc2or 3796 | The disjunction of two equ... |
| sbcex 3797 | By our definition of prope... |
| sbceq1a 3798 | Equality theorem for class... |
| sbceq2a 3799 | Equality theorem for class... |
| spsbc 3800 | Specialization: if a formu... |
| spsbcd 3801 | Specialization: if a formu... |
| sbcth 3802 | A substitution into a theo... |
| sbcthdv 3803 | Deduction version of ~ sbc... |
| sbcid 3804 | An identity theorem for su... |
| nfsbc1d 3805 | Deduction version of ~ nfs... |
| nfsbc1 3806 | Bound-variable hypothesis ... |
| nfsbc1v 3807 | Bound-variable hypothesis ... |
| nfsbcdw 3808 | Deduction version of ~ nfs... |
| nfsbcw 3809 | Bound-variable hypothesis ... |
| sbccow 3810 | A composition law for clas... |
| nfsbcd 3811 | Deduction version of ~ nfs... |
| nfsbc 3812 | Bound-variable hypothesis ... |
| sbcco 3813 | A composition law for clas... |
| sbcco2 3814 | A composition law for clas... |
| sbc5 3815 | An equivalence for class s... |
| sbc5ALT 3816 | Alternate proof of ~ sbc5 ... |
| sbc6g 3817 | An equivalence for class s... |
| sbc6gOLD 3818 | Obsolete version of ~ sbc6... |
| sbc6 3819 | An equivalence for class s... |
| sbc7 3820 | An equivalence for class s... |
| cbvsbcw 3821 | Change bound variables in ... |
| cbvsbcvw 3822 | Change the bound variable ... |
| cbvsbc 3823 | Change bound variables in ... |
| cbvsbcv 3824 | Change the bound variable ... |
| sbciegft 3825 | Conversion of implicit sub... |
| sbciegftOLD 3826 | Obsolete version of ~ sbci... |
| sbciegf 3827 | Conversion of implicit sub... |
| sbcieg 3828 | Conversion of implicit sub... |
| sbciegOLD 3829 | Obsolete version of ~ sbci... |
| sbcie2g 3830 | Conversion of implicit sub... |
| sbcie 3831 | Conversion of implicit sub... |
| sbciedf 3832 | Conversion of implicit sub... |
| sbcied 3833 | Conversion of implicit sub... |
| sbciedOLD 3834 | Obsolete version of ~ sbci... |
| sbcied2 3835 | Conversion of implicit sub... |
| elrabsf 3836 | Membership in a restricted... |
| eqsbc1 3837 | Substitution for the left-... |
| sbcng 3838 | Move negation in and out o... |
| sbcimg 3839 | Distribution of class subs... |
| sbcan 3840 | Distribution of class subs... |
| sbcor 3841 | Distribution of class subs... |
| sbcbig 3842 | Distribution of class subs... |
| sbcn1 3843 | Move negation in and out o... |
| sbcim1 3844 | Distribution of class subs... |
| sbcim1OLD 3845 | Obsolete version of ~ sbci... |
| sbcbid 3846 | Formula-building deduction... |
| sbcbidv 3847 | Formula-building deduction... |
| sbcbii 3848 | Formula-building inference... |
| sbcbi1 3849 | Distribution of class subs... |
| sbcbi2 3850 | Substituting into equivale... |
| sbcal 3851 | Move universal quantifier ... |
| sbcex2 3852 | Move existential quantifie... |
| sbceqal 3853 | Class version of one impli... |
| sbceqalOLD 3854 | Obsolete version of ~ sbce... |
| sbeqalb 3855 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3856 | Substitution for the right... |
| sbc3an 3857 | Distribution of class subs... |
| sbcel1v 3858 | Class substitution into a ... |
| sbcel2gv 3859 | Class substitution into a ... |
| sbcel21v 3860 | Class substitution into a ... |
| sbcimdv 3861 | Substitution analogue of T... |
| sbcimdvOLD 3862 | Obsolete version of ~ sbci... |
| sbctt 3863 | Substitution for a variabl... |
| sbcgf 3864 | Substitution for a variabl... |
| sbc19.21g 3865 | Substitution for a variabl... |
| sbcg 3866 | Substitution for a variabl... |
| sbcgOLD 3867 | Obsolete version of ~ sbcg... |
| sbcgfi 3868 | Substitution for a variabl... |
| sbc2iegf 3869 | Conversion of implicit sub... |
| sbc2ie 3870 | Conversion of implicit sub... |
| sbc2ieOLD 3871 | Obsolete version of ~ sbc2... |
| sbc2iedv 3872 | Conversion of implicit sub... |
| sbc3ie 3873 | Conversion of implicit sub... |
| sbccomlem 3874 | Lemma for ~ sbccom . (Con... |
| sbccom 3875 | Commutative law for double... |
| sbcralt 3876 | Interchange class substitu... |
| sbcrext 3877 | Interchange class substitu... |
| sbcralg 3878 | Interchange class substitu... |
| sbcrex 3879 | Interchange class substitu... |
| sbcreu 3880 | Interchange class substitu... |
| reu8nf 3881 | Restricted uniqueness usin... |
| sbcabel 3882 | Interchange class substitu... |
| rspsbc 3883 | Restricted quantifier vers... |
| rspsbca 3884 | Restricted quantifier vers... |
| rspesbca 3885 | Existence form of ~ rspsbc... |
| spesbc 3886 | Existence form of ~ spsbc ... |
| spesbcd 3887 | form of ~ spsbc . (Contri... |
| sbcth2 3888 | A substitution into a theo... |
| ra4v 3889 | Version of ~ ra4 with a di... |
| ra4 3890 | Restricted quantifier vers... |
| rmo2 3891 | Alternate definition of re... |
| rmo2i 3892 | Condition implying restric... |
| rmo3 3893 | Restricted "at most one" u... |
| rmob 3894 | Consequence of "at most on... |
| rmoi 3895 | Consequence of "at most on... |
| rmob2 3896 | Consequence of "restricted... |
| rmoi2 3897 | Consequence of "restricted... |
| rmoanim 3898 | Introduction of a conjunct... |
| rmoanimALT 3899 | Alternate proof of ~ rmoan... |
| reuan 3900 | Introduction of a conjunct... |
| 2reu1 3901 | Double restricted existent... |
| 2reu2 3902 | Double restricted existent... |
| csb2 3905 | Alternate expression for t... |
| csbeq1 3906 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3907 | Equality deduction for pro... |
| csbeq2 3908 | Substituting into equivale... |
| csbeq2d 3909 | Formula-building deduction... |
| csbeq2dv 3910 | Formula-building deduction... |
| csbeq2i 3911 | Formula-building inference... |
| csbeq12dv 3912 | Formula-building inference... |
| cbvcsbw 3913 | Change bound variables in ... |
| cbvcsb 3914 | Change bound variables in ... |
| cbvcsbv 3915 | Change the bound variable ... |
| csbid 3916 | Analogue of ~ sbid for pro... |
| csbeq1a 3917 | Equality theorem for prope... |
| csbcow 3918 | Composition law for chaine... |
| csbco 3919 | Composition law for chaine... |
| csbtt 3920 | Substitution doesn't affec... |
| csbconstgf 3921 | Substitution doesn't affec... |
| csbconstg 3922 | Substitution doesn't affec... |
| csbconstgOLD 3923 | Obsolete version of ~ csbc... |
| csbgfi 3924 | Substitution for a variabl... |
| csbconstgi 3925 | The proper substitution of... |
| nfcsb1d 3926 | Bound-variable hypothesis ... |
| nfcsb1 3927 | Bound-variable hypothesis ... |
| nfcsb1v 3928 | Bound-variable hypothesis ... |
| nfcsbd 3929 | Deduction version of ~ nfc... |
| nfcsbw 3930 | Bound-variable hypothesis ... |
| nfcsb 3931 | Bound-variable hypothesis ... |
| csbhypf 3932 | Introduce an explicit subs... |
| csbiebt 3933 | Conversion of implicit sub... |
| csbiedf 3934 | Conversion of implicit sub... |
| csbieb 3935 | Bidirectional conversion b... |
| csbiebg 3936 | Bidirectional conversion b... |
| csbiegf 3937 | Conversion of implicit sub... |
| csbief 3938 | Conversion of implicit sub... |
| csbie 3939 | Conversion of implicit sub... |
| csbieOLD 3940 | Obsolete version of ~ csbi... |
| csbied 3941 | Conversion of implicit sub... |
| csbiedOLD 3942 | Obsolete version of ~ csbi... |
| csbied2 3943 | Conversion of implicit sub... |
| csbie2t 3944 | Conversion of implicit sub... |
| csbie2 3945 | Conversion of implicit sub... |
| csbie2g 3946 | Conversion of implicit sub... |
| cbvrabcsfw 3947 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3948 | A more general version of ... |
| cbvrexcsf 3949 | A more general version of ... |
| cbvreucsf 3950 | A more general version of ... |
| cbvrabcsf 3951 | A more general version of ... |
| cbvralv2 3952 | Rule used to change the bo... |
| cbvrexv2 3953 | Rule used to change the bo... |
| rspc2vd 3954 | Deduction version of 2-var... |
| difjust 3960 | Soundness justification th... |
| unjust 3962 | Soundness justification th... |
| injust 3964 | Soundness justification th... |
| dfin5 3966 | Alternate definition for t... |
| dfdif2 3967 | Alternate definition of cl... |
| eldif 3968 | Expansion of membership in... |
| eldifd 3969 | If a class is in one class... |
| eldifad 3970 | If a class is in the diffe... |
| eldifbd 3971 | If a class is in the diffe... |
| elneeldif 3972 | The elements of a set diff... |
| velcomp 3973 | Characterization of setvar... |
| elin 3974 | Expansion of membership in... |
| dfss2 3976 | Alternate definition of th... |
| dfss 3977 | Variant of subclass defini... |
| dfss3 3979 | Alternate definition of su... |
| dfss6 3980 | Alternate definition of su... |
| dfssf 3981 | Equivalence for subclass r... |
| dfss3f 3982 | Equivalence for subclass r... |
| nfss 3983 | If ` x ` is not free in ` ... |
| ssel 3984 | Membership relationships f... |
| ssel2 3985 | Membership relationships f... |
| sseli 3986 | Membership implication fro... |
| sselii 3987 | Membership inference from ... |
| sselid 3988 | Membership inference from ... |
| sseld 3989 | Membership deduction from ... |
| sselda 3990 | Membership deduction from ... |
| sseldd 3991 | Membership inference from ... |
| ssneld 3992 | If a class is not in anoth... |
| ssneldd 3993 | If an element is not in a ... |
| ssriv 3994 | Inference based on subclas... |
| ssrd 3995 | Deduction based on subclas... |
| ssrdv 3996 | Deduction based on subclas... |
| sstr2 3997 | Transitivity of subclass r... |
| sstr2OLD 3998 | Obsolete version of ~ sstr... |
| sstr 3999 | Transitivity of subclass r... |
| sstri 4000 | Subclass transitivity infe... |
| sstrd 4001 | Subclass transitivity dedu... |
| sstrid 4002 | Subclass transitivity dedu... |
| sstrdi 4003 | Subclass transitivity dedu... |
| sylan9ss 4004 | A subclass transitivity de... |
| sylan9ssr 4005 | A subclass transitivity de... |
| eqss 4006 | The subclass relationship ... |
| eqssi 4007 | Infer equality from two su... |
| eqssd 4008 | Equality deduction from tw... |
| sssseq 4009 | If a class is a subclass o... |
| eqrd 4010 | Deduce equality of classes... |
| eqri 4011 | Infer equality of classes ... |
| eqelssd 4012 | Equality deduction from su... |
| ssid 4013 | Any class is a subclass of... |
| ssidd 4014 | Weakening of ~ ssid . (Co... |
| ssv 4015 | Any class is a subclass of... |
| sseq1 4016 | Equality theorem for subcl... |
| sseq2 4017 | Equality theorem for the s... |
| sseq12 4018 | Equality theorem for the s... |
| sseq1i 4019 | An equality inference for ... |
| sseq2i 4020 | An equality inference for ... |
| sseq12i 4021 | An equality inference for ... |
| sseq1d 4022 | An equality deduction for ... |
| sseq2d 4023 | An equality deduction for ... |
| sseq12d 4024 | An equality deduction for ... |
| eqsstri 4025 | Substitution of equality i... |
| eqsstrri 4026 | Substitution of equality i... |
| sseqtri 4027 | Substitution of equality i... |
| sseqtrri 4028 | Substitution of equality i... |
| eqsstrd 4029 | Substitution of equality i... |
| eqsstrrd 4030 | Substitution of equality i... |
| sseqtrd 4031 | Substitution of equality i... |
| sseqtrrd 4032 | Substitution of equality i... |
| 3sstr3i 4033 | Substitution of equality i... |
| 3sstr4i 4034 | Substitution of equality i... |
| 3sstr3g 4035 | Substitution of equality i... |
| 3sstr4g 4036 | Substitution of equality i... |
| 3sstr3d 4037 | Substitution of equality i... |
| 3sstr4d 4038 | Substitution of equality i... |
| eqsstrid 4039 | A chained subclass and equ... |
| eqsstrrid 4040 | A chained subclass and equ... |
| sseqtrdi 4041 | A chained subclass and equ... |
| sseqtrrdi 4042 | A chained subclass and equ... |
| sseqtrid 4043 | Subclass transitivity dedu... |
| sseqtrrid 4044 | Subclass transitivity dedu... |
| eqsstrdi 4045 | A chained subclass and equ... |
| eqsstrrdi 4046 | A chained subclass and equ... |
| eqimssd 4047 | Equality implies inclusion... |
| eqimsscd 4048 | Equality implies inclusion... |
| eqimss 4049 | Equality implies inclusion... |
| eqimss2 4050 | Equality implies inclusion... |
| eqimssi 4051 | Infer subclass relationshi... |
| eqimss2i 4052 | Infer subclass relationshi... |
| nssne1 4053 | Two classes are different ... |
| nssne2 4054 | Two classes are different ... |
| nss 4055 | Negation of subclass relat... |
| nelss 4056 | Demonstrate by witnesses t... |
| ssrexf 4057 | Restricted existential qua... |
| ssrmof 4058 | "At most one" existential ... |
| ssralv 4059 | Quantification restricted ... |
| ssrexv 4060 | Existential quantification... |
| ss2ralv 4061 | Two quantifications restri... |
| ss2rexv 4062 | Two existential quantifica... |
| ssralvOLD 4063 | Obsolete version of ~ ssra... |
| ssrexvOLD 4064 | Obsolete version of ~ ssre... |
| ralss 4065 | Restricted universal quant... |
| rexss 4066 | Restricted existential qua... |
| ss2ab 4067 | Class abstractions in a su... |
| abss 4068 | Class abstraction in a sub... |
| ssab 4069 | Subclass of a class abstra... |
| ssabral 4070 | The relation for a subclas... |
| ss2abdv 4071 | Deduction of abstraction s... |
| ss2abi 4072 | Inference of abstraction s... |
| abssdv 4073 | Deduction of abstraction s... |
| abssdvOLD 4074 | Obsolete version of ~ abss... |
| abssi 4075 | Inference of abstraction s... |
| ss2rab 4076 | Restricted abstraction cla... |
| rabss 4077 | Restricted class abstracti... |
| ssrab 4078 | Subclass of a restricted c... |
| ssrabdv 4079 | Subclass of a restricted c... |
| rabssdv 4080 | Subclass of a restricted c... |
| ss2rabdv 4081 | Deduction of restricted ab... |
| ss2rabi 4082 | Inference of restricted ab... |
| rabss2 4083 | Subclass law for restricte... |
| ssab2 4084 | Subclass relation for the ... |
| ssrab2 4085 | Subclass relation for a re... |
| rabss3d 4086 | Subclass law for restricte... |
| ssrab3 4087 | Subclass relation for a re... |
| rabssrabd 4088 | Subclass of a restricted c... |
| ssrabeq 4089 | If the restricting class o... |
| rabssab 4090 | A restricted class is a su... |
| uniiunlem 4091 | A subset relationship usef... |
| dfpss2 4092 | Alternate definition of pr... |
| dfpss3 4093 | Alternate definition of pr... |
| psseq1 4094 | Equality theorem for prope... |
| psseq2 4095 | Equality theorem for prope... |
| psseq1i 4096 | An equality inference for ... |
| psseq2i 4097 | An equality inference for ... |
| psseq12i 4098 | An equality inference for ... |
| psseq1d 4099 | An equality deduction for ... |
| psseq2d 4100 | An equality deduction for ... |
| psseq12d 4101 | An equality deduction for ... |
| pssss 4102 | A proper subclass is a sub... |
| pssne 4103 | Two classes in a proper su... |
| pssssd 4104 | Deduce subclass from prope... |
| pssned 4105 | Proper subclasses are uneq... |
| sspss 4106 | Subclass in terms of prope... |
| pssirr 4107 | Proper subclass is irrefle... |
| pssn2lp 4108 | Proper subclass has no 2-c... |
| sspsstri 4109 | Two ways of stating tricho... |
| ssnpss 4110 | Partial trichotomy law for... |
| psstr 4111 | Transitive law for proper ... |
| sspsstr 4112 | Transitive law for subclas... |
| psssstr 4113 | Transitive law for subclas... |
| psstrd 4114 | Proper subclass inclusion ... |
| sspsstrd 4115 | Transitivity involving sub... |
| psssstrd 4116 | Transitivity involving sub... |
| npss 4117 | A class is not a proper su... |
| ssnelpss 4118 | A subclass missing a membe... |
| ssnelpssd 4119 | Subclass inclusion with on... |
| ssexnelpss 4120 | If there is an element of ... |
| dfdif3 4121 | Alternate definition of cl... |
| difeq1 4122 | Equality theorem for class... |
| difeq2 4123 | Equality theorem for class... |
| difeq12 4124 | Equality theorem for class... |
| difeq1i 4125 | Inference adding differenc... |
| difeq2i 4126 | Inference adding differenc... |
| difeq12i 4127 | Equality inference for cla... |
| difeq1d 4128 | Deduction adding differenc... |
| difeq2d 4129 | Deduction adding differenc... |
| difeq12d 4130 | Equality deduction for cla... |
| difeqri 4131 | Inference from membership ... |
| nfdif 4132 | Bound-variable hypothesis ... |
| nfdifOLD 4133 | Obsolete version of ~ nfdi... |
| eldifi 4134 | Implication of membership ... |
| eldifn 4135 | Implication of membership ... |
| elndif 4136 | A set does not belong to a... |
| neldif 4137 | Implication of membership ... |
| difdif 4138 | Double class difference. ... |
| difss 4139 | Subclass relationship for ... |
| difssd 4140 | A difference of two classe... |
| difss2 4141 | If a class is contained in... |
| difss2d 4142 | If a class is contained in... |
| ssdifss 4143 | Preservation of a subclass... |
| ddif 4144 | Double complement under un... |
| ssconb 4145 | Contraposition law for sub... |
| sscon 4146 | Contraposition law for sub... |
| ssdif 4147 | Difference law for subsets... |
| ssdifd 4148 | If ` A ` is contained in `... |
| sscond 4149 | If ` A ` is contained in `... |
| ssdifssd 4150 | If ` A ` is contained in `... |
| ssdif2d 4151 | If ` A ` is contained in `... |
| raldifb 4152 | Restricted universal quant... |
| rexdifi 4153 | Restricted existential qua... |
| complss 4154 | Complementation reverses i... |
| compleq 4155 | Two classes are equal if a... |
| elun 4156 | Expansion of membership in... |
| elunnel1 4157 | A member of a union that i... |
| elunnel2 4158 | A member of a union that i... |
| uneqri 4159 | Inference from membership ... |
| unidm 4160 | Idempotent law for union o... |
| uncom 4161 | Commutative law for union ... |
| equncom 4162 | If a class equals the unio... |
| equncomi 4163 | Inference form of ~ equnco... |
| uneq1 4164 | Equality theorem for the u... |
| uneq2 4165 | Equality theorem for the u... |
| uneq12 4166 | Equality theorem for the u... |
| uneq1i 4167 | Inference adding union to ... |
| uneq2i 4168 | Inference adding union to ... |
| uneq12i 4169 | Equality inference for the... |
| uneq1d 4170 | Deduction adding union to ... |
| uneq2d 4171 | Deduction adding union to ... |
| uneq12d 4172 | Equality deduction for the... |
| nfun 4173 | Bound-variable hypothesis ... |
| nfunOLD 4174 | Obsolete version of ~ nfun... |
| unass 4175 | Associative law for union ... |
| un12 4176 | A rearrangement of union. ... |
| un23 4177 | A rearrangement of union. ... |
| un4 4178 | A rearrangement of the uni... |
| unundi 4179 | Union distributes over its... |
| unundir 4180 | Union distributes over its... |
| ssun1 4181 | Subclass relationship for ... |
| ssun2 4182 | Subclass relationship for ... |
| ssun3 4183 | Subclass law for union of ... |
| ssun4 4184 | Subclass law for union of ... |
| elun1 4185 | Membership law for union o... |
| elun2 4186 | Membership law for union o... |
| elunant 4187 | A statement is true for ev... |
| unss1 4188 | Subclass law for union of ... |
| ssequn1 4189 | A relationship between sub... |
| unss2 4190 | Subclass law for union of ... |
| unss12 4191 | Subclass law for union of ... |
| ssequn2 4192 | A relationship between sub... |
| unss 4193 | The union of two subclasse... |
| unssi 4194 | An inference showing the u... |
| unssd 4195 | A deduction showing the un... |
| unssad 4196 | If ` ( A u. B ) ` is conta... |
| unssbd 4197 | If ` ( A u. B ) ` is conta... |
| ssun 4198 | A condition that implies i... |
| rexun 4199 | Restricted existential qua... |
| ralunb 4200 | Restricted quantification ... |
| ralun 4201 | Restricted quantification ... |
| elini 4202 | Membership in an intersect... |
| elind 4203 | Deduce membership in an in... |
| elinel1 4204 | Membership in an intersect... |
| elinel2 4205 | Membership in an intersect... |
| elin2 4206 | Membership in a class defi... |
| elin1d 4207 | Elementhood in the first s... |
| elin2d 4208 | Elementhood in the first s... |
| elin3 4209 | Membership in a class defi... |
| incom 4210 | Commutative law for inters... |
| ineqcom 4211 | Two ways of expressing tha... |
| ineqcomi 4212 | Two ways of expressing tha... |
| ineqri 4213 | Inference from membership ... |
| ineq1 4214 | Equality theorem for inter... |
| ineq2 4215 | Equality theorem for inter... |
| ineq12 4216 | Equality theorem for inter... |
| ineq1i 4217 | Equality inference for int... |
| ineq2i 4218 | Equality inference for int... |
| ineq12i 4219 | Equality inference for int... |
| ineq1d 4220 | Equality deduction for int... |
| ineq2d 4221 | Equality deduction for int... |
| ineq12d 4222 | Equality deduction for int... |
| ineqan12d 4223 | Equality deduction for int... |
| sseqin2 4224 | A relationship between sub... |
| nfin 4225 | Bound-variable hypothesis ... |
| nfinOLD 4226 | Obsolete version of ~ nfin... |
| rabbi2dva 4227 | Deduction from a wff to a ... |
| inidm 4228 | Idempotent law for interse... |
| inass 4229 | Associative law for inters... |
| in12 4230 | A rearrangement of interse... |
| in32 4231 | A rearrangement of interse... |
| in13 4232 | A rearrangement of interse... |
| in31 4233 | A rearrangement of interse... |
| inrot 4234 | Rotate the intersection of... |
| in4 4235 | Rearrangement of intersect... |
| inindi 4236 | Intersection distributes o... |
| inindir 4237 | Intersection distributes o... |
| inss1 4238 | The intersection of two cl... |
| inss2 4239 | The intersection of two cl... |
| ssin 4240 | Subclass of intersection. ... |
| ssini 4241 | An inference showing that ... |
| ssind 4242 | A deduction showing that a... |
| ssrin 4243 | Add right intersection to ... |
| sslin 4244 | Add left intersection to s... |
| ssrind 4245 | Add right intersection to ... |
| ss2in 4246 | Intersection of subclasses... |
| ssinss1 4247 | Intersection preserves sub... |
| inss 4248 | Inclusion of an intersecti... |
| rexin 4249 | Restricted existential qua... |
| dfss7 4250 | Alternate definition of su... |
| symdifcom 4253 | Symmetric difference commu... |
| symdifeq1 4254 | Equality theorem for symme... |
| symdifeq2 4255 | Equality theorem for symme... |
| nfsymdif 4256 | Hypothesis builder for sym... |
| elsymdif 4257 | Membership in a symmetric ... |
| dfsymdif4 4258 | Alternate definition of th... |
| elsymdifxor 4259 | Membership in a symmetric ... |
| dfsymdif2 4260 | Alternate definition of th... |
| symdifass 4261 | Symmetric difference is as... |
| difsssymdif 4262 | The symmetric difference c... |
| difsymssdifssd 4263 | If the symmetric differenc... |
| unabs 4264 | Absorption law for union. ... |
| inabs 4265 | Absorption law for interse... |
| nssinpss 4266 | Negation of subclass expre... |
| nsspssun 4267 | Negation of subclass expre... |
| dfss4 4268 | Subclass defined in terms ... |
| dfun2 4269 | An alternate definition of... |
| dfin2 4270 | An alternate definition of... |
| difin 4271 | Difference with intersecti... |
| ssdifim 4272 | Implication of a class dif... |
| ssdifsym 4273 | Symmetric class difference... |
| dfss5 4274 | Alternate definition of su... |
| dfun3 4275 | Union defined in terms of ... |
| dfin3 4276 | Intersection defined in te... |
| dfin4 4277 | Alternate definition of th... |
| invdif 4278 | Intersection with universa... |
| indif 4279 | Intersection with class di... |
| indif2 4280 | Bring an intersection in a... |
| indif1 4281 | Bring an intersection in a... |
| indifcom 4282 | Commutation law for inters... |
| indi 4283 | Distributive law for inter... |
| undi 4284 | Distributive law for union... |
| indir 4285 | Distributive law for inter... |
| undir 4286 | Distributive law for union... |
| unineq 4287 | Infer equality from equali... |
| uneqin 4288 | Equality of union and inte... |
| difundi 4289 | Distributive law for class... |
| difundir 4290 | Distributive law for class... |
| difindi 4291 | Distributive law for class... |
| difindir 4292 | Distributive law for class... |
| indifdi 4293 | Distribute intersection ov... |
| indifdir 4294 | Distribute intersection ov... |
| indifdirOLD 4295 | Obsolete version of ~ indi... |
| difdif2 4296 | Class difference by a clas... |
| undm 4297 | De Morgan's law for union.... |
| indm 4298 | De Morgan's law for inters... |
| difun1 4299 | A relationship involving d... |
| undif3 4300 | An equality involving clas... |
| difin2 4301 | Represent a class differen... |
| dif32 4302 | Swap second and third argu... |
| difabs 4303 | Absorption-like law for cl... |
| sscon34b 4304 | Relative complementation r... |
| rcompleq 4305 | Two subclasses are equal i... |
| dfsymdif3 4306 | Alternate definition of th... |
| unabw 4307 | Union of two class abstrac... |
| unab 4308 | Union of two class abstrac... |
| inab 4309 | Intersection of two class ... |
| difab 4310 | Difference of two class ab... |
| abanssl 4311 | A class abstraction with a... |
| abanssr 4312 | A class abstraction with a... |
| notabw 4313 | A class abstraction define... |
| notab 4314 | A class abstraction define... |
| unrab 4315 | Union of two restricted cl... |
| inrab 4316 | Intersection of two restri... |
| inrab2 4317 | Intersection with a restri... |
| difrab 4318 | Difference of two restrict... |
| dfrab3 4319 | Alternate definition of re... |
| dfrab2 4320 | Alternate definition of re... |
| rabdif 4321 | Move difference in and out... |
| notrab 4322 | Complementation of restric... |
| dfrab3ss 4323 | Restricted class abstracti... |
| rabun2 4324 | Abstraction restricted to ... |
| reuun2 4325 | Transfer uniqueness to a s... |
| reuss2 4326 | Transfer uniqueness to a s... |
| reuss 4327 | Transfer uniqueness to a s... |
| reuun1 4328 | Transfer uniqueness to a s... |
| reupick 4329 | Restricted uniqueness "pic... |
| reupick3 4330 | Restricted uniqueness "pic... |
| reupick2 4331 | Restricted uniqueness "pic... |
| euelss 4332 | Transfer uniqueness of an ... |
| dfnul4 4335 | Alternate definition of th... |
| dfnul2 4336 | Alternate definition of th... |
| dfnul3 4337 | Alternate definition of th... |
| dfnul2OLD 4338 | Obsolete version of ~ dfnu... |
| dfnul3OLD 4339 | Obsolete version of ~ dfnu... |
| dfnul4OLD 4340 | Obsolete version of ~ dfnu... |
| noel 4341 | The empty set has no eleme... |
| noelOLD 4342 | Obsolete version of ~ noel... |
| nel02 4343 | The empty set has no eleme... |
| n0i 4344 | If a class has elements, t... |
| ne0i 4345 | If a class has elements, t... |
| ne0d 4346 | Deduction form of ~ ne0i .... |
| n0ii 4347 | If a class has elements, t... |
| ne0ii 4348 | If a class has elements, t... |
| vn0 4349 | The universal class is not... |
| vn0ALT 4350 | Alternate proof of ~ vn0 .... |
| eq0f 4351 | A class is equal to the em... |
| neq0f 4352 | A class is not empty if an... |
| n0f 4353 | A class is nonempty if and... |
| eq0 4354 | A class is equal to the em... |
| eq0ALT 4355 | Alternate proof of ~ eq0 .... |
| neq0 4356 | A class is not empty if an... |
| n0 4357 | A class is nonempty if and... |
| nel0 4358 | From the general negation ... |
| reximdva0 4359 | Restricted existence deduc... |
| rspn0 4360 | Specialization for restric... |
| n0rex 4361 | There is an element in a n... |
| ssn0rex 4362 | There is an element in a c... |
| n0moeu 4363 | A case of equivalence of "... |
| rex0 4364 | Vacuous restricted existen... |
| reu0 4365 | Vacuous restricted uniquen... |
| rmo0 4366 | Vacuous restricted at-most... |
| 0el 4367 | Membership of the empty se... |
| n0el 4368 | Negated membership of the ... |
| eqeuel 4369 | A condition which implies ... |
| ssdif0 4370 | Subclass expressed in term... |
| difn0 4371 | If the difference of two s... |
| pssdifn0 4372 | A proper subclass has a no... |
| pssdif 4373 | A proper subclass has a no... |
| ndisj 4374 | Express that an intersecti... |
| inn0f 4375 | A nonempty intersection. ... |
| inn0 4376 | A nonempty intersection. ... |
| difin0ss 4377 | Difference, intersection, ... |
| inssdif0 4378 | Intersection, subclass, an... |
| difid 4379 | The difference between a c... |
| difidALT 4380 | Alternate proof of ~ difid... |
| dif0 4381 | The difference between a c... |
| ab0w 4382 | The class of sets verifyin... |
| ab0 4383 | The class of sets verifyin... |
| ab0OLD 4384 | Obsolete version of ~ ab0 ... |
| ab0ALT 4385 | Alternate proof of ~ ab0 ,... |
| dfnf5 4386 | Characterization of nonfre... |
| ab0orv 4387 | The class abstraction defi... |
| ab0orvALT 4388 | Alternate proof of ~ ab0or... |
| abn0 4389 | Nonempty class abstraction... |
| abn0OLD 4390 | Obsolete version of ~ abn0... |
| rab0 4391 | Any restricted class abstr... |
| rabeq0w 4392 | Condition for a restricted... |
| rabeq0 4393 | Condition for a restricted... |
| rabn0 4394 | Nonempty restricted class ... |
| rabxm 4395 | Law of excluded middle, in... |
| rabnc 4396 | Law of noncontradiction, i... |
| elneldisj 4397 | The set of elements ` s ` ... |
| elnelun 4398 | The union of the set of el... |
| un0 4399 | The union of a class with ... |
| in0 4400 | The intersection of a clas... |
| 0un 4401 | The union of the empty set... |
| 0in 4402 | The intersection of the em... |
| inv1 4403 | The intersection of a clas... |
| unv 4404 | The union of a class with ... |
| 0ss 4405 | The null set is a subset o... |
| ss0b 4406 | Any subset of the empty se... |
| ss0 4407 | Any subset of the empty se... |
| sseq0 4408 | A subclass of an empty cla... |
| ssn0 4409 | A class with a nonempty su... |
| 0dif 4410 | The difference between the... |
| abf 4411 | A class abstraction determ... |
| eq0rdv 4412 | Deduction for equality to ... |
| eq0rdvALT 4413 | Alternate proof of ~ eq0rd... |
| csbprc 4414 | The proper substitution of... |
| csb0 4415 | The proper substitution of... |
| sbcel12 4416 | Distribute proper substitu... |
| sbceqg 4417 | Distribute proper substitu... |
| sbceqi 4418 | Distribution of class subs... |
| sbcnel12g 4419 | Distribute proper substitu... |
| sbcne12 4420 | Distribute proper substitu... |
| sbcel1g 4421 | Move proper substitution i... |
| sbceq1g 4422 | Move proper substitution t... |
| sbcel2 4423 | Move proper substitution i... |
| sbceq2g 4424 | Move proper substitution t... |
| csbcom 4425 | Commutative law for double... |
| sbcnestgfw 4426 | Nest the composition of tw... |
| csbnestgfw 4427 | Nest the composition of tw... |
| sbcnestgw 4428 | Nest the composition of tw... |
| csbnestgw 4429 | Nest the composition of tw... |
| sbcco3gw 4430 | Composition of two substit... |
| sbcnestgf 4431 | Nest the composition of tw... |
| csbnestgf 4432 | Nest the composition of tw... |
| sbcnestg 4433 | Nest the composition of tw... |
| csbnestg 4434 | Nest the composition of tw... |
| sbcco3g 4435 | Composition of two substit... |
| csbco3g 4436 | Composition of two class s... |
| csbnest1g 4437 | Nest the composition of tw... |
| csbidm 4438 | Idempotent law for class s... |
| csbvarg 4439 | The proper substitution of... |
| csbvargi 4440 | The proper substitution of... |
| sbccsb 4441 | Substitution into a wff ex... |
| sbccsb2 4442 | Substitution into a wff ex... |
| rspcsbela 4443 | Special case related to ~ ... |
| sbnfc2 4444 | Two ways of expressing " `... |
| csbab 4445 | Move substitution into a c... |
| csbun 4446 | Distribution of class subs... |
| csbin 4447 | Distribute proper substitu... |
| csbie2df 4448 | Conversion of implicit sub... |
| 2nreu 4449 | If there are two different... |
| un00 4450 | Two classes are empty iff ... |
| vss 4451 | Only the universal class h... |
| 0pss 4452 | The null set is a proper s... |
| npss0 4453 | No set is a proper subset ... |
| pssv 4454 | Any non-universal class is... |
| disj 4455 | Two ways of saying that tw... |
| disjr 4456 | Two ways of saying that tw... |
| disj1 4457 | Two ways of saying that tw... |
| reldisj 4458 | Two ways of saying that tw... |
| disj3 4459 | Two ways of saying that tw... |
| disjne 4460 | Members of disjoint sets a... |
| disjeq0 4461 | Two disjoint sets are equa... |
| disjel 4462 | A set can't belong to both... |
| disj2 4463 | Two ways of saying that tw... |
| disj4 4464 | Two ways of saying that tw... |
| ssdisj 4465 | Intersection with a subcla... |
| disjpss 4466 | A class is a proper subset... |
| undisj1 4467 | The union of disjoint clas... |
| undisj2 4468 | The union of disjoint clas... |
| ssindif0 4469 | Subclass expressed in term... |
| inelcm 4470 | The intersection of classe... |
| minel 4471 | A minimum element of a cla... |
| undif4 4472 | Distribute union over diff... |
| disjssun 4473 | Subset relation for disjoi... |
| vdif0 4474 | Universal class equality i... |
| difrab0eq 4475 | If the difference between ... |
| pssnel 4476 | A proper subclass has a me... |
| disjdif 4477 | A class and its relative c... |
| disjdifr 4478 | A class and its relative c... |
| difin0 4479 | The difference of a class ... |
| unvdif 4480 | The union of a class and i... |
| undif1 4481 | Absorption of difference b... |
| undif2 4482 | Absorption of difference b... |
| undifabs 4483 | Absorption of difference b... |
| inundif 4484 | The intersection and class... |
| disjdif2 4485 | The difference of a class ... |
| difun2 4486 | Absorption of union by dif... |
| undif 4487 | Union of complementary par... |
| undifr 4488 | Union of complementary par... |
| undifrOLD 4489 | Obsolete version of ~ undi... |
| undif5 4490 | An equality involving clas... |
| ssdifin0 4491 | A subset of a difference d... |
| ssdifeq0 4492 | A class is a subclass of i... |
| ssundif 4493 | A condition equivalent to ... |
| difcom 4494 | Swap the arguments of a cl... |
| pssdifcom1 4495 | Two ways to express overla... |
| pssdifcom2 4496 | Two ways to express non-co... |
| difdifdir 4497 | Distributive law for class... |
| uneqdifeq 4498 | Two ways to say that ` A `... |
| raldifeq 4499 | Equality theorem for restr... |
| r19.2z 4500 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4501 | A response to the notion t... |
| r19.3rz 4502 | Restricted quantification ... |
| r19.28z 4503 | Restricted quantifier vers... |
| r19.3rzv 4504 | Restricted quantification ... |
| r19.9rzv 4505 | Restricted quantification ... |
| r19.28zv 4506 | Restricted quantifier vers... |
| r19.37zv 4507 | Restricted quantifier vers... |
| r19.45zv 4508 | Restricted version of Theo... |
| r19.44zv 4509 | Restricted version of Theo... |
| r19.27z 4510 | Restricted quantifier vers... |
| r19.27zv 4511 | Restricted quantifier vers... |
| r19.36zv 4512 | Restricted quantifier vers... |
| ralidmw 4513 | Idempotent law for restric... |
| rzal 4514 | Vacuous quantification is ... |
| rzalALT 4515 | Alternate proof of ~ rzal ... |
| rexn0 4516 | Restricted existential qua... |
| ralidm 4517 | Idempotent law for restric... |
| ral0 4518 | Vacuous universal quantifi... |
| ralf0 4519 | The quantification of a fa... |
| rexn0OLD 4520 | Obsolete version of ~ rexn... |
| ralidmOLD 4521 | Obsolete version of ~ rali... |
| ral0OLD 4522 | Obsolete version of ~ ral0... |
| ralf0OLD 4523 | Obsolete version of ~ ralf... |
| ralnralall 4524 | A contradiction concerning... |
| falseral0 4525 | A false statement can only... |
| raaan 4526 | Rearrange restricted quant... |
| raaanv 4527 | Rearrange restricted quant... |
| sbss 4528 | Set substitution into the ... |
| sbcssg 4529 | Distribute proper substitu... |
| raaan2 4530 | Rearrange restricted quant... |
| 2reu4lem 4531 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4532 | Definition of double restr... |
| csbdif 4533 | Distribution of class subs... |
| dfif2 4536 | An alternate definition of... |
| dfif6 4537 | An alternate definition of... |
| ifeq1 4538 | Equality theorem for condi... |
| ifeq2 4539 | Equality theorem for condi... |
| iftrue 4540 | Value of the conditional o... |
| iftruei 4541 | Inference associated with ... |
| iftrued 4542 | Value of the conditional o... |
| iffalse 4543 | Value of the conditional o... |
| iffalsei 4544 | Inference associated with ... |
| iffalsed 4545 | Value of the conditional o... |
| ifnefalse 4546 | When values are unequal, b... |
| ifsb 4547 | Distribute a function over... |
| dfif3 4548 | Alternate definition of th... |
| dfif4 4549 | Alternate definition of th... |
| dfif5 4550 | Alternate definition of th... |
| ifssun 4551 | A conditional class is inc... |
| ifeq12 4552 | Equality theorem for condi... |
| ifeq1d 4553 | Equality deduction for con... |
| ifeq2d 4554 | Equality deduction for con... |
| ifeq12d 4555 | Equality deduction for con... |
| ifbi 4556 | Equivalence theorem for co... |
| ifbid 4557 | Equivalence deduction for ... |
| ifbieq1d 4558 | Equivalence/equality deduc... |
| ifbieq2i 4559 | Equivalence/equality infer... |
| ifbieq2d 4560 | Equivalence/equality deduc... |
| ifbieq12i 4561 | Equivalence deduction for ... |
| ifbieq12d 4562 | Equivalence deduction for ... |
| nfifd 4563 | Deduction form of ~ nfif .... |
| nfif 4564 | Bound-variable hypothesis ... |
| ifeq1da 4565 | Conditional equality. (Co... |
| ifeq2da 4566 | Conditional equality. (Co... |
| ifeq12da 4567 | Equivalence deduction for ... |
| ifbieq12d2 4568 | Equivalence deduction for ... |
| ifclda 4569 | Conditional closure. (Con... |
| ifeqda 4570 | Separation of the values o... |
| elimif 4571 | Elimination of a condition... |
| ifbothda 4572 | A wff ` th ` containing a ... |
| ifboth 4573 | A wff ` th ` containing a ... |
| ifid 4574 | Identical true and false a... |
| eqif 4575 | Expansion of an equality w... |
| ifval 4576 | Another expression of the ... |
| elif 4577 | Membership in a conditiona... |
| ifel 4578 | Membership of a conditiona... |
| ifcl 4579 | Membership (closure) of a ... |
| ifcld 4580 | Membership (closure) of a ... |
| ifcli 4581 | Inference associated with ... |
| ifexd 4582 | Existence of the condition... |
| ifexg 4583 | Existence of the condition... |
| ifex 4584 | Existence of the condition... |
| ifeqor 4585 | The possible values of a c... |
| ifnot 4586 | Negating the first argumen... |
| ifan 4587 | Rewrite a conjunction in a... |
| ifor 4588 | Rewrite a disjunction in a... |
| 2if2 4589 | Resolve two nested conditi... |
| ifcomnan 4590 | Commute the conditions in ... |
| csbif 4591 | Distribute proper substitu... |
| dedth 4592 | Weak deduction theorem tha... |
| dedth2h 4593 | Weak deduction theorem eli... |
| dedth3h 4594 | Weak deduction theorem eli... |
| dedth4h 4595 | Weak deduction theorem eli... |
| dedth2v 4596 | Weak deduction theorem for... |
| dedth3v 4597 | Weak deduction theorem for... |
| dedth4v 4598 | Weak deduction theorem for... |
| elimhyp 4599 | Eliminate a hypothesis con... |
| elimhyp2v 4600 | Eliminate a hypothesis con... |
| elimhyp3v 4601 | Eliminate a hypothesis con... |
| elimhyp4v 4602 | Eliminate a hypothesis con... |
| elimel 4603 | Eliminate a membership hyp... |
| elimdhyp 4604 | Version of ~ elimhyp where... |
| keephyp 4605 | Transform a hypothesis ` p... |
| keephyp2v 4606 | Keep a hypothesis containi... |
| keephyp3v 4607 | Keep a hypothesis containi... |
| pwjust 4609 | Soundness justification th... |
| elpwg 4611 | Membership in a power clas... |
| elpw 4612 | Membership in a power clas... |
| velpw 4613 | Setvar variable membership... |
| elpwd 4614 | Membership in a power clas... |
| elpwi 4615 | Subset relation implied by... |
| elpwb 4616 | Characterization of the el... |
| elpwid 4617 | An element of a power clas... |
| elelpwi 4618 | If ` A ` belongs to a part... |
| sspw 4619 | The powerclass preserves i... |
| sspwi 4620 | The powerclass preserves i... |
| sspwd 4621 | The powerclass preserves i... |
| pweq 4622 | Equality theorem for power... |
| pweqALT 4623 | Alternate proof of ~ pweq ... |
| pweqi 4624 | Equality inference for pow... |
| pweqd 4625 | Equality deduction for pow... |
| pwunss 4626 | The power class of the uni... |
| nfpw 4627 | Bound-variable hypothesis ... |
| pwidg 4628 | A set is an element of its... |
| pwidb 4629 | A class is an element of i... |
| pwid 4630 | A set is a member of its p... |
| pwss 4631 | Subclass relationship for ... |
| pwundif 4632 | Break up the power class o... |
| snjust 4633 | Soundness justification th... |
| sneq 4644 | Equality theorem for singl... |
| sneqi 4645 | Equality inference for sin... |
| sneqd 4646 | Equality deduction for sin... |
| dfsn2 4647 | Alternate definition of si... |
| elsng 4648 | There is exactly one eleme... |
| elsn 4649 | There is exactly one eleme... |
| velsn 4650 | There is only one element ... |
| elsni 4651 | There is at most one eleme... |
| rabsneq 4652 | Equality of class abstract... |
| absn 4653 | Condition for a class abst... |
| dfpr2 4654 | Alternate definition of a ... |
| dfsn2ALT 4655 | Alternate definition of si... |
| elprg 4656 | A member of a pair of clas... |
| elpri 4657 | If a class is an element o... |
| elpr 4658 | A member of a pair of clas... |
| elpr2g 4659 | A member of a pair of sets... |
| elpr2 4660 | A member of a pair of sets... |
| nelpr2 4661 | If a class is not an eleme... |
| nelpr1 4662 | If a class is not an eleme... |
| nelpri 4663 | If an element doesn't matc... |
| prneli 4664 | If an element doesn't matc... |
| nelprd 4665 | If an element doesn't matc... |
| eldifpr 4666 | Membership in a set with t... |
| rexdifpr 4667 | Restricted existential qua... |
| snidg 4668 | A set is a member of its s... |
| snidb 4669 | A class is a set iff it is... |
| snid 4670 | A set is a member of its s... |
| vsnid 4671 | A setvar variable is a mem... |
| elsn2g 4672 | There is exactly one eleme... |
| elsn2 4673 | There is exactly one eleme... |
| nelsn 4674 | If a class is not equal to... |
| rabeqsn 4675 | Conditions for a restricte... |
| rabsssn 4676 | Conditions for a restricte... |
| rabeqsnd 4677 | Conditions for a restricte... |
| ralsnsg 4678 | Substitution expressed in ... |
| rexsns 4679 | Restricted existential qua... |
| rexsngf 4680 | Restricted existential qua... |
| ralsngf 4681 | Restricted universal quant... |
| reusngf 4682 | Restricted existential uni... |
| ralsng 4683 | Substitution expressed in ... |
| rexsng 4684 | Restricted existential qua... |
| reusng 4685 | Restricted existential uni... |
| 2ralsng 4686 | Substitution expressed in ... |
| ralsngOLD 4687 | Obsolete version of ~ rals... |
| rexsngOLD 4688 | Obsolete version of ~ rexs... |
| rexreusng 4689 | Restricted existential uni... |
| exsnrex 4690 | There is a set being the e... |
| ralsn 4691 | Convert a universal quanti... |
| rexsn 4692 | Convert an existential qua... |
| elpwunsn 4693 | Membership in an extension... |
| eqoreldif 4694 | An element of a set is eit... |
| eltpg 4695 | Members of an unordered tr... |
| eldiftp 4696 | Membership in a set with t... |
| eltpi 4697 | A member of an unordered t... |
| eltp 4698 | A member of an unordered t... |
| dftp2 4699 | Alternate definition of un... |
| nfpr 4700 | Bound-variable hypothesis ... |
| ifpr 4701 | Membership of a conditiona... |
| ralprgf 4702 | Convert a restricted unive... |
| rexprgf 4703 | Convert a restricted exist... |
| ralprg 4704 | Convert a restricted unive... |
| ralprgOLD 4705 | Obsolete version of ~ ralp... |
| rexprg 4706 | Convert a restricted exist... |
| rexprgOLD 4707 | Obsolete version of ~ rexp... |
| raltpg 4708 | Convert a restricted unive... |
| rextpg 4709 | Convert a restricted exist... |
| ralpr 4710 | Convert a restricted unive... |
| rexpr 4711 | Convert a restricted exist... |
| reuprg0 4712 | Convert a restricted exist... |
| reuprg 4713 | Convert a restricted exist... |
| reurexprg 4714 | Convert a restricted exist... |
| raltp 4715 | Convert a universal quanti... |
| rextp 4716 | Convert an existential qua... |
| nfsn 4717 | Bound-variable hypothesis ... |
| csbsng 4718 | Distribute proper substitu... |
| csbprg 4719 | Distribute proper substitu... |
| elinsn 4720 | If the intersection of two... |
| disjsn 4721 | Intersection with the sing... |
| disjsn2 4722 | Two distinct singletons ar... |
| disjpr2 4723 | Two completely distinct un... |
| disjprsn 4724 | The disjoint intersection ... |
| disjtpsn 4725 | The disjoint intersection ... |
| disjtp2 4726 | Two completely distinct un... |
| snprc 4727 | The singleton of a proper ... |
| snnzb 4728 | A singleton is nonempty if... |
| rmosn 4729 | A restricted at-most-one q... |
| r19.12sn 4730 | Special case of ~ r19.12 w... |
| rabsn 4731 | Condition where a restrict... |
| rabsnifsb 4732 | A restricted class abstrac... |
| rabsnif 4733 | A restricted class abstrac... |
| rabrsn 4734 | A restricted class abstrac... |
| euabsn2 4735 | Another way to express exi... |
| euabsn 4736 | Another way to express exi... |
| reusn 4737 | A way to express restricte... |
| absneu 4738 | Restricted existential uni... |
| rabsneu 4739 | Restricted existential uni... |
| eusn 4740 | Two ways to express " ` A ... |
| rabsnt 4741 | Truth implied by equality ... |
| prcom 4742 | Commutative law for unorde... |
| preq1 4743 | Equality theorem for unord... |
| preq2 4744 | Equality theorem for unord... |
| preq12 4745 | Equality theorem for unord... |
| preq1i 4746 | Equality inference for uno... |
| preq2i 4747 | Equality inference for uno... |
| preq12i 4748 | Equality inference for uno... |
| preq1d 4749 | Equality deduction for uno... |
| preq2d 4750 | Equality deduction for uno... |
| preq12d 4751 | Equality deduction for uno... |
| tpeq1 4752 | Equality theorem for unord... |
| tpeq2 4753 | Equality theorem for unord... |
| tpeq3 4754 | Equality theorem for unord... |
| tpeq1d 4755 | Equality theorem for unord... |
| tpeq2d 4756 | Equality theorem for unord... |
| tpeq3d 4757 | Equality theorem for unord... |
| tpeq123d 4758 | Equality theorem for unord... |
| tprot 4759 | Rotation of the elements o... |
| tpcoma 4760 | Swap 1st and 2nd members o... |
| tpcomb 4761 | Swap 2nd and 3rd members o... |
| tpass 4762 | Split off the first elemen... |
| qdass 4763 | Two ways to write an unord... |
| qdassr 4764 | Two ways to write an unord... |
| tpidm12 4765 | Unordered triple ` { A , A... |
| tpidm13 4766 | Unordered triple ` { A , B... |
| tpidm23 4767 | Unordered triple ` { A , B... |
| tpidm 4768 | Unordered triple ` { A , A... |
| tppreq3 4769 | An unordered triple is an ... |
| prid1g 4770 | An unordered pair contains... |
| prid2g 4771 | An unordered pair contains... |
| prid1 4772 | An unordered pair contains... |
| prid2 4773 | An unordered pair contains... |
| ifpprsnss 4774 | An unordered pair is a sin... |
| prprc1 4775 | A proper class vanishes in... |
| prprc2 4776 | A proper class vanishes in... |
| prprc 4777 | An unordered pair containi... |
| tpid1 4778 | One of the three elements ... |
| tpid1g 4779 | Closed theorem form of ~ t... |
| tpid2 4780 | One of the three elements ... |
| tpid2g 4781 | Closed theorem form of ~ t... |
| tpid3g 4782 | Closed theorem form of ~ t... |
| tpid3 4783 | One of the three elements ... |
| snnzg 4784 | The singleton of a set is ... |
| snn0d 4785 | The singleton of a set is ... |
| snnz 4786 | The singleton of a set is ... |
| prnz 4787 | A pair containing a set is... |
| prnzg 4788 | A pair containing a set is... |
| tpnz 4789 | An unordered triple contai... |
| tpnzd 4790 | An unordered triple contai... |
| raltpd 4791 | Convert a universal quanti... |
| snssb 4792 | Characterization of the in... |
| snssg 4793 | The singleton formed on a ... |
| snssgOLD 4794 | Obsolete version of ~ snss... |
| snss 4795 | The singleton of an elemen... |
| eldifsn 4796 | Membership in a set with a... |
| eldifsnd 4797 | Membership in a set with a... |
| ssdifsn 4798 | Subset of a set with an el... |
| elpwdifsn 4799 | A subset of a set is an el... |
| eldifsni 4800 | Membership in a set with a... |
| eldifsnneq 4801 | An element of a difference... |
| neldifsn 4802 | The class ` A ` is not in ... |
| neldifsnd 4803 | The class ` A ` is not in ... |
| rexdifsn 4804 | Restricted existential qua... |
| raldifsni 4805 | Rearrangement of a propert... |
| raldifsnb 4806 | Restricted universal quant... |
| eldifvsn 4807 | A set is an element of the... |
| difsn 4808 | An element not in a set ca... |
| difprsnss 4809 | Removal of a singleton fro... |
| difprsn1 4810 | Removal of a singleton fro... |
| difprsn2 4811 | Removal of a singleton fro... |
| diftpsn3 4812 | Removal of a singleton fro... |
| difpr 4813 | Removing two elements as p... |
| tpprceq3 4814 | An unordered triple is an ... |
| tppreqb 4815 | An unordered triple is an ... |
| difsnb 4816 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4817 | ` ( B \ { A } ) ` is a pro... |
| snssi 4818 | The singleton of an elemen... |
| snssd 4819 | The singleton of an elemen... |
| difsnid 4820 | If we remove a single elem... |
| eldifeldifsn 4821 | An element of a difference... |
| pw0 4822 | Compute the power set of t... |
| pwpw0 4823 | Compute the power set of t... |
| snsspr1 4824 | A singleton is a subset of... |
| snsspr2 4825 | A singleton is a subset of... |
| snsstp1 4826 | A singleton is a subset of... |
| snsstp2 4827 | A singleton is a subset of... |
| snsstp3 4828 | A singleton is a subset of... |
| prssg 4829 | A pair of elements of a cl... |
| prss 4830 | A pair of elements of a cl... |
| prssi 4831 | A pair of elements of a cl... |
| prssd 4832 | Deduction version of ~ prs... |
| prsspwg 4833 | An unordered pair belongs ... |
| ssprss 4834 | A pair as subset of a pair... |
| ssprsseq 4835 | A proper pair is a subset ... |
| sssn 4836 | The subsets of a singleton... |
| ssunsn2 4837 | The property of being sand... |
| ssunsn 4838 | Possible values for a set ... |
| eqsn 4839 | Two ways to express that a... |
| eqsnd 4840 | Deduce that a set is a sin... |
| eqsndOLD 4841 | Obsolete version of ~ eqsn... |
| issn 4842 | A sufficient condition for... |
| n0snor2el 4843 | A nonempty set is either a... |
| ssunpr 4844 | Possible values for a set ... |
| sspr 4845 | The subsets of a pair. (C... |
| sstp 4846 | The subsets of an unordere... |
| tpss 4847 | An unordered triple of ele... |
| tpssi 4848 | An unordered triple of ele... |
| sneqrg 4849 | Closed form of ~ sneqr . ... |
| sneqr 4850 | If the singletons of two s... |
| snsssn 4851 | If a singleton is a subset... |
| mosneq 4852 | There exists at most one s... |
| sneqbg 4853 | Two singletons of sets are... |
| snsspw 4854 | The singleton of a class i... |
| prsspw 4855 | An unordered pair belongs ... |
| preq1b 4856 | Biconditional equality lem... |
| preq2b 4857 | Biconditional equality lem... |
| preqr1 4858 | Reverse equality lemma for... |
| preqr2 4859 | Reverse equality lemma for... |
| preq12b 4860 | Equality relationship for ... |
| opthpr 4861 | An unordered pair has the ... |
| preqr1g 4862 | Reverse equality lemma for... |
| preq12bg 4863 | Closed form of ~ preq12b .... |
| prneimg 4864 | Two pairs are not equal if... |
| prnebg 4865 | A (proper) pair is not equ... |
| pr1eqbg 4866 | A (proper) pair is equal t... |
| pr1nebg 4867 | A (proper) pair is not equ... |
| preqsnd 4868 | Equivalence for a pair equ... |
| prnesn 4869 | A proper unordered pair is... |
| prneprprc 4870 | A proper unordered pair is... |
| preqsn 4871 | Equivalence for a pair equ... |
| preq12nebg 4872 | Equality relationship for ... |
| prel12g 4873 | Equality of two unordered ... |
| opthprneg 4874 | An unordered pair has the ... |
| elpreqprlem 4875 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4876 | Equality and membership ru... |
| elpreqprb 4877 | A set is an element of an ... |
| elpr2elpr 4878 | For an element ` A ` of an... |
| dfopif 4879 | Rewrite ~ df-op using ` if... |
| dfopg 4880 | Value of the ordered pair ... |
| dfop 4881 | Value of an ordered pair w... |
| opeq1 4882 | Equality theorem for order... |
| opeq2 4883 | Equality theorem for order... |
| opeq12 4884 | Equality theorem for order... |
| opeq1i 4885 | Equality inference for ord... |
| opeq2i 4886 | Equality inference for ord... |
| opeq12i 4887 | Equality inference for ord... |
| opeq1d 4888 | Equality deduction for ord... |
| opeq2d 4889 | Equality deduction for ord... |
| opeq12d 4890 | Equality deduction for ord... |
| oteq1 4891 | Equality theorem for order... |
| oteq2 4892 | Equality theorem for order... |
| oteq3 4893 | Equality theorem for order... |
| oteq1d 4894 | Equality deduction for ord... |
| oteq2d 4895 | Equality deduction for ord... |
| oteq3d 4896 | Equality deduction for ord... |
| oteq123d 4897 | Equality deduction for ord... |
| nfop 4898 | Bound-variable hypothesis ... |
| nfopd 4899 | Deduction version of bound... |
| csbopg 4900 | Distribution of class subs... |
| opidg 4901 | The ordered pair ` <. A , ... |
| opid 4902 | The ordered pair ` <. A , ... |
| ralunsn 4903 | Restricted quantification ... |
| 2ralunsn 4904 | Double restricted quantifi... |
| opprc 4905 | Expansion of an ordered pa... |
| opprc1 4906 | Expansion of an ordered pa... |
| opprc2 4907 | Expansion of an ordered pa... |
| oprcl 4908 | If an ordered pair has an ... |
| pwsn 4909 | The power set of a singlet... |
| pwpr 4910 | The power set of an unorde... |
| pwtp 4911 | The power set of an unorde... |
| pwpwpw0 4912 | Compute the power set of t... |
| pwv 4913 | The power class of the uni... |
| prproe 4914 | For an element of a proper... |
| 3elpr2eq 4915 | If there are three element... |
| dfuni2 4918 | Alternate definition of cl... |
| eluni 4919 | Membership in class union.... |
| eluni2 4920 | Membership in class union.... |
| elunii 4921 | Membership in class union.... |
| nfunid 4922 | Deduction version of ~ nfu... |
| nfuni 4923 | Bound-variable hypothesis ... |
| uniss 4924 | Subclass relationship for ... |
| unissi 4925 | Subclass relationship for ... |
| unissd 4926 | Subclass relationship for ... |
| unieq 4927 | Equality theorem for class... |
| unieqi 4928 | Inference of equality of t... |
| unieqd 4929 | Deduction of equality of t... |
| eluniab 4930 | Membership in union of a c... |
| elunirab 4931 | Membership in union of a c... |
| uniprg 4932 | The union of a pair is the... |
| unipr 4933 | The union of a pair is the... |
| uniprOLD 4934 | Obsolete version of ~ unip... |
| uniprgOLD 4935 | Obsolete version of ~ unip... |
| unisng 4936 | A set equals the union of ... |
| unisn 4937 | A set equals the union of ... |
| unisnv 4938 | A set equals the union of ... |
| unisn3 4939 | Union of a singleton in th... |
| dfnfc2 4940 | An alternative statement o... |
| uniun 4941 | The class union of the uni... |
| uniin 4942 | The class union of the int... |
| ssuni 4943 | Subclass relationship for ... |
| uni0b 4944 | The union of a set is empt... |
| uni0c 4945 | The union of a set is empt... |
| uni0 4946 | The union of the empty set... |
| csbuni 4947 | Distribute proper substitu... |
| elssuni 4948 | An element of a class is a... |
| unissel 4949 | Condition turning a subcla... |
| unissb 4950 | Relationship involving mem... |
| unissbOLD 4951 | Obsolete version of ~ unis... |
| uniss2 4952 | A subclass condition on th... |
| unidif 4953 | If the difference ` A \ B ... |
| ssunieq 4954 | Relationship implying unio... |
| unimax 4955 | Any member of a class is t... |
| pwuni 4956 | A class is a subclass of t... |
| dfint2 4959 | Alternate definition of cl... |
| inteq 4960 | Equality law for intersect... |
| inteqi 4961 | Equality inference for cla... |
| inteqd 4962 | Equality deduction for cla... |
| elint 4963 | Membership in class inters... |
| elint2 4964 | Membership in class inters... |
| elintg 4965 | Membership in class inters... |
| elinti 4966 | Membership in class inters... |
| nfint 4967 | Bound-variable hypothesis ... |
| elintabg 4968 | Two ways of saying a set i... |
| elintab 4969 | Membership in the intersec... |
| elintabOLD 4970 | Obsolete version of ~ elin... |
| elintrab 4971 | Membership in the intersec... |
| elintrabg 4972 | Membership in the intersec... |
| int0 4973 | The intersection of the em... |
| intss1 4974 | An element of a class incl... |
| ssint 4975 | Subclass of a class inters... |
| ssintab 4976 | Subclass of the intersecti... |
| ssintub 4977 | Subclass of the least uppe... |
| ssmin 4978 | Subclass of the minimum va... |
| intmin 4979 | Any member of a class is t... |
| intss 4980 | Intersection of subclasses... |
| intssuni 4981 | The intersection of a none... |
| ssintrab 4982 | Subclass of the intersecti... |
| unissint 4983 | If the union of a class is... |
| intssuni2 4984 | Subclass relationship for ... |
| intminss 4985 | Under subset ordering, the... |
| intmin2 4986 | Any set is the smallest of... |
| intmin3 4987 | Under subset ordering, the... |
| intmin4 4988 | Elimination of a conjunct ... |
| intab 4989 | The intersection of a spec... |
| int0el 4990 | The intersection of a clas... |
| intun 4991 | The class intersection of ... |
| intprg 4992 | The intersection of a pair... |
| intpr 4993 | The intersection of a pair... |
| intprOLD 4994 | Obsolete version of ~ intp... |
| intprgOLD 4995 | Obsolete version of ~ intp... |
| intsng 4996 | Intersection of a singleto... |
| intsn 4997 | The intersection of a sing... |
| uniintsn 4998 | Two ways to express " ` A ... |
| uniintab 4999 | The union and the intersec... |
| intunsn 5000 | Theorem joining a singleto... |
| rint0 5001 | Relative intersection of a... |
| elrint 5002 | Membership in a restricted... |
| elrint2 5003 | Membership in a restricted... |
| eliun 5008 | Membership in indexed unio... |
| eliin 5009 | Membership in indexed inte... |
| eliuni 5010 | Membership in an indexed u... |
| iuncom 5011 | Commutation of indexed uni... |
| iuncom4 5012 | Commutation of union with ... |
| iunconst 5013 | Indexed union of a constan... |
| iinconst 5014 | Indexed intersection of a ... |
| iuneqconst 5015 | Indexed union of identical... |
| iuniin 5016 | Law combining indexed unio... |
| iinssiun 5017 | An indexed intersection is... |
| iunss1 5018 | Subclass theorem for index... |
| iinss1 5019 | Subclass theorem for index... |
| iuneq1 5020 | Equality theorem for index... |
| iineq1 5021 | Equality theorem for index... |
| ss2iun 5022 | Subclass theorem for index... |
| iuneq2 5023 | Equality theorem for index... |
| iineq2 5024 | Equality theorem for index... |
| iuneq2i 5025 | Equality inference for ind... |
| iineq2i 5026 | Equality inference for ind... |
| iineq2d 5027 | Equality deduction for ind... |
| iuneq2dv 5028 | Equality deduction for ind... |
| iineq2dv 5029 | Equality deduction for ind... |
| iuneq12df 5030 | Equality deduction for ind... |
| iuneq1d 5031 | Equality theorem for index... |
| iuneq12d 5032 | Equality deduction for ind... |
| iuneq2d 5033 | Equality deduction for ind... |
| nfiun 5034 | Bound-variable hypothesis ... |
| nfiin 5035 | Bound-variable hypothesis ... |
| nfiung 5036 | Bound-variable hypothesis ... |
| nfiing 5037 | Bound-variable hypothesis ... |
| nfiu1 5038 | Bound-variable hypothesis ... |
| nfiu1OLD 5039 | Obsolete version of ~ nfiu... |
| nfii1 5040 | Bound-variable hypothesis ... |
| dfiun2g 5041 | Alternate definition of in... |
| dfiun2gOLD 5042 | Obsolete version of ~ dfiu... |
| dfiin2g 5043 | Alternate definition of in... |
| dfiun2 5044 | Alternate definition of in... |
| dfiin2 5045 | Alternate definition of in... |
| dfiunv2 5046 | Define double indexed unio... |
| cbviun 5047 | Rule used to change the bo... |
| cbviin 5048 | Change bound variables in ... |
| cbviung 5049 | Rule used to change the bo... |
| cbviing 5050 | Change bound variables in ... |
| cbviunv 5051 | Rule used to change the bo... |
| cbviinv 5052 | Change bound variables in ... |
| cbviunvg 5053 | Rule used to change the bo... |
| cbviinvg 5054 | Change bound variables in ... |
| iunssf 5055 | Subset theorem for an inde... |
| iunss 5056 | Subset theorem for an inde... |
| ssiun 5057 | Subset implication for an ... |
| ssiun2 5058 | Identity law for subset of... |
| ssiun2s 5059 | Subset relationship for an... |
| iunss2 5060 | A subclass condition on th... |
| iunssd 5061 | Subset theorem for an inde... |
| iunab 5062 | The indexed union of a cla... |
| iunrab 5063 | The indexed union of a res... |
| iunxdif2 5064 | Indexed union with a class... |
| ssiinf 5065 | Subset theorem for an inde... |
| ssiin 5066 | Subset theorem for an inde... |
| iinss 5067 | Subset implication for an ... |
| iinss2 5068 | An indexed intersection is... |
| uniiun 5069 | Class union in terms of in... |
| intiin 5070 | Class intersection in term... |
| iunid 5071 | An indexed union of single... |
| iunidOLD 5072 | Obsolete version of ~ iuni... |
| iun0 5073 | An indexed union of the em... |
| 0iun 5074 | An empty indexed union is ... |
| 0iin 5075 | An empty indexed intersect... |
| viin 5076 | Indexed intersection with ... |
| iunsn 5077 | Indexed union of a singlet... |
| iunn0 5078 | There is a nonempty class ... |
| iinab 5079 | Indexed intersection of a ... |
| iinrab 5080 | Indexed intersection of a ... |
| iinrab2 5081 | Indexed intersection of a ... |
| iunin2 5082 | Indexed union of intersect... |
| iunin1 5083 | Indexed union of intersect... |
| iinun2 5084 | Indexed intersection of un... |
| iundif2 5085 | Indexed union of class dif... |
| iindif1 5086 | Indexed intersection of cl... |
| 2iunin 5087 | Rearrange indexed unions o... |
| iindif2 5088 | Indexed intersection of cl... |
| iinin2 5089 | Indexed intersection of in... |
| iinin1 5090 | Indexed intersection of in... |
| iinvdif 5091 | The indexed intersection o... |
| elriin 5092 | Elementhood in a relative ... |
| riin0 5093 | Relative intersection of a... |
| riinn0 5094 | Relative intersection of a... |
| riinrab 5095 | Relative intersection of a... |
| symdif0 5096 | Symmetric difference with ... |
| symdifv 5097 | The symmetric difference w... |
| symdifid 5098 | The symmetric difference o... |
| iinxsng 5099 | A singleton index picks ou... |
| iinxprg 5100 | Indexed intersection with ... |
| iunxsng 5101 | A singleton index picks ou... |
| iunxsn 5102 | A singleton index picks ou... |
| iunxsngf 5103 | A singleton index picks ou... |
| iunun 5104 | Separate a union in an ind... |
| iunxun 5105 | Separate a union in the in... |
| iunxdif3 5106 | An indexed union where som... |
| iunxprg 5107 | A pair index picks out two... |
| iunxiun 5108 | Separate an indexed union ... |
| iinuni 5109 | A relationship involving u... |
| iununi 5110 | A relationship involving u... |
| sspwuni 5111 | Subclass relationship for ... |
| pwssb 5112 | Two ways to express a coll... |
| elpwpw 5113 | Characterization of the el... |
| pwpwab 5114 | The double power class wri... |
| pwpwssunieq 5115 | The class of sets whose un... |
| elpwuni 5116 | Relationship for power cla... |
| iinpw 5117 | The power class of an inte... |
| iunpwss 5118 | Inclusion of an indexed un... |
| intss2 5119 | A nonempty intersection of... |
| rintn0 5120 | Relative intersection of a... |
| dfdisj2 5123 | Alternate definition for d... |
| disjss2 5124 | If each element of a colle... |
| disjeq2 5125 | Equality theorem for disjo... |
| disjeq2dv 5126 | Equality deduction for dis... |
| disjss1 5127 | A subset of a disjoint col... |
| disjeq1 5128 | Equality theorem for disjo... |
| disjeq1d 5129 | Equality theorem for disjo... |
| disjeq12d 5130 | Equality theorem for disjo... |
| cbvdisj 5131 | Change bound variables in ... |
| cbvdisjv 5132 | Change bound variables in ... |
| nfdisjw 5133 | Bound-variable hypothesis ... |
| nfdisj 5134 | Bound-variable hypothesis ... |
| nfdisj1 5135 | Bound-variable hypothesis ... |
| disjor 5136 | Two ways to say that a col... |
| disjors 5137 | Two ways to say that a col... |
| disji2 5138 | Property of a disjoint col... |
| disji 5139 | Property of a disjoint col... |
| invdisj 5140 | If there is a function ` C... |
| invdisjrab 5141 | The restricted class abstr... |
| disjiun 5142 | A disjoint collection yiel... |
| disjord 5143 | Conditions for a collectio... |
| disjiunb 5144 | Two ways to say that a col... |
| disjiund 5145 | Conditions for a collectio... |
| sndisj 5146 | Any collection of singleto... |
| 0disj 5147 | Any collection of empty se... |
| disjxsn 5148 | A singleton collection is ... |
| disjx0 5149 | An empty collection is dis... |
| disjprg 5150 | A pair collection is disjo... |
| disjxiun 5151 | An indexed union of a disj... |
| disjxun 5152 | The union of two disjoint ... |
| disjss3 5153 | Expand a disjoint collecti... |
| breq 5156 | Equality theorem for binar... |
| breq1 5157 | Equality theorem for a bin... |
| breq2 5158 | Equality theorem for a bin... |
| breq12 5159 | Equality theorem for a bin... |
| breqi 5160 | Equality inference for bin... |
| breq1i 5161 | Equality inference for a b... |
| breq2i 5162 | Equality inference for a b... |
| breq12i 5163 | Equality inference for a b... |
| breq1d 5164 | Equality deduction for a b... |
| breqd 5165 | Equality deduction for a b... |
| breq2d 5166 | Equality deduction for a b... |
| breq12d 5167 | Equality deduction for a b... |
| breq123d 5168 | Equality deduction for a b... |
| breqdi 5169 | Equality deduction for a b... |
| breqan12d 5170 | Equality deduction for a b... |
| breqan12rd 5171 | Equality deduction for a b... |
| eqnbrtrd 5172 | Substitution of equal clas... |
| nbrne1 5173 | Two classes are different ... |
| nbrne2 5174 | Two classes are different ... |
| eqbrtri 5175 | Substitution of equal clas... |
| eqbrtrd 5176 | Substitution of equal clas... |
| eqbrtrri 5177 | Substitution of equal clas... |
| eqbrtrrd 5178 | Substitution of equal clas... |
| breqtri 5179 | Substitution of equal clas... |
| breqtrd 5180 | Substitution of equal clas... |
| breqtrri 5181 | Substitution of equal clas... |
| breqtrrd 5182 | Substitution of equal clas... |
| 3brtr3i 5183 | Substitution of equality i... |
| 3brtr4i 5184 | Substitution of equality i... |
| 3brtr3d 5185 | Substitution of equality i... |
| 3brtr4d 5186 | Substitution of equality i... |
| 3brtr3g 5187 | Substitution of equality i... |
| 3brtr4g 5188 | Substitution of equality i... |
| eqbrtrid 5189 | A chained equality inferen... |
| eqbrtrrid 5190 | A chained equality inferen... |
| breqtrid 5191 | A chained equality inferen... |
| breqtrrid 5192 | A chained equality inferen... |
| eqbrtrdi 5193 | A chained equality inferen... |
| eqbrtrrdi 5194 | A chained equality inferen... |
| breqtrdi 5195 | A chained equality inferen... |
| breqtrrdi 5196 | A chained equality inferen... |
| ssbrd 5197 | Deduction from a subclass ... |
| ssbr 5198 | Implication from a subclas... |
| ssbri 5199 | Inference from a subclass ... |
| nfbrd 5200 | Deduction version of bound... |
| nfbr 5201 | Bound-variable hypothesis ... |
| brab1 5202 | Relationship between a bin... |
| br0 5203 | The empty binary relation ... |
| brne0 5204 | If two sets are in a binar... |
| brun 5205 | The union of two binary re... |
| brin 5206 | The intersection of two re... |
| brdif 5207 | The difference of two bina... |
| sbcbr123 5208 | Move substitution in and o... |
| sbcbr 5209 | Move substitution in and o... |
| sbcbr12g 5210 | Move substitution in and o... |
| sbcbr1g 5211 | Move substitution in and o... |
| sbcbr2g 5212 | Move substitution in and o... |
| brsymdif 5213 | Characterization of the sy... |
| brralrspcev 5214 | Restricted existential spe... |
| brimralrspcev 5215 | Restricted existential spe... |
| opabss 5218 | The collection of ordered ... |
| opabbid 5219 | Equivalent wff's yield equ... |
| opabbidv 5220 | Equivalent wff's yield equ... |
| opabbii 5221 | Equivalent wff's yield equ... |
| nfopabd 5222 | Bound-variable hypothesis ... |
| nfopab 5223 | Bound-variable hypothesis ... |
| nfopab1 5224 | The first abstraction vari... |
| nfopab2 5225 | The second abstraction var... |
| cbvopab 5226 | Rule used to change bound ... |
| cbvopabv 5227 | Rule used to change bound ... |
| cbvopabvOLD 5228 | Obsolete version of ~ cbvo... |
| cbvopab1 5229 | Change first bound variabl... |
| cbvopab1g 5230 | Change first bound variabl... |
| cbvopab2 5231 | Change second bound variab... |
| cbvopab1s 5232 | Change first bound variabl... |
| cbvopab1v 5233 | Rule used to change the fi... |
| cbvopab1vOLD 5234 | Obsolete version of ~ cbvo... |
| cbvopab2v 5235 | Rule used to change the se... |
| unopab 5236 | Union of two ordered pair ... |
| mpteq12da 5239 | An equality inference for ... |
| mpteq12df 5240 | An equality inference for ... |
| mpteq12dfOLD 5241 | Obsolete version of ~ mpte... |
| mpteq12f 5242 | An equality theorem for th... |
| mpteq12dva 5243 | An equality inference for ... |
| mpteq12dvaOLD 5244 | Obsolete version of ~ mpte... |
| mpteq12dv 5245 | An equality inference for ... |
| mpteq12 5246 | An equality theorem for th... |
| mpteq1 5247 | An equality theorem for th... |
| mpteq1OLD 5248 | Obsolete version of ~ mpte... |
| mpteq1d 5249 | An equality theorem for th... |
| mpteq1i 5250 | An equality theorem for th... |
| mpteq1iOLD 5251 | Obsolete version of ~ mpte... |
| mpteq2da 5252 | Slightly more general equa... |
| mpteq2daOLD 5253 | Obsolete version of ~ mpte... |
| mpteq2dva 5254 | Slightly more general equa... |
| mpteq2dvaOLD 5255 | Obsolete version of ~ mpte... |
| mpteq2dv 5256 | An equality inference for ... |
| mpteq2ia 5257 | An equality inference for ... |
| mpteq2iaOLD 5258 | Obsolete version of ~ mpte... |
| mpteq2i 5259 | An equality inference for ... |
| mpteq12i 5260 | An equality inference for ... |
| nfmpt 5261 | Bound-variable hypothesis ... |
| nfmpt1 5262 | Bound-variable hypothesis ... |
| cbvmptf 5263 | Rule to change the bound v... |
| cbvmptfg 5264 | Rule to change the bound v... |
| cbvmpt 5265 | Rule to change the bound v... |
| cbvmptg 5266 | Rule to change the bound v... |
| cbvmptv 5267 | Rule to change the bound v... |
| cbvmptvOLD 5268 | Obsolete version of ~ cbvm... |
| cbvmptvg 5269 | Rule to change the bound v... |
| mptv 5270 | Function with universal do... |
| dftr2 5273 | An alternate way of defini... |
| dftr2c 5274 | Variant of ~ dftr2 with co... |
| dftr5 5275 | An alternate way of defini... |
| dftr5OLD 5276 | Obsolete version of ~ dftr... |
| dftr3 5277 | An alternate way of defini... |
| dftr4 5278 | An alternate way of defini... |
| treq 5279 | Equality theorem for the t... |
| trel 5280 | In a transitive class, the... |
| trel3 5281 | In a transitive class, the... |
| trss 5282 | An element of a transitive... |
| trin 5283 | The intersection of transi... |
| tr0 5284 | The empty set is transitiv... |
| trv 5285 | The universe is transitive... |
| triun 5286 | An indexed union of a clas... |
| truni 5287 | The union of a class of tr... |
| triin 5288 | An indexed intersection of... |
| trint 5289 | The intersection of a clas... |
| trintss 5290 | Any nonempty transitive cl... |
| axrep1 5292 | The version of the Axiom o... |
| axreplem 5293 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5294 | Axiom of Replacement expre... |
| axrep3 5295 | Axiom of Replacement sligh... |
| axrep4 5296 | A more traditional version... |
| axrep5 5297 | Axiom of Replacement (simi... |
| axrep6 5298 | A condensed form of ~ ax-r... |
| axrep6g 5299 | ~ axrep6 in class notation... |
| zfrepclf 5300 | An inference based on the ... |
| zfrep3cl 5301 | An inference based on the ... |
| zfrep4 5302 | A version of Replacement u... |
| axsepgfromrep 5303 | A more general version ~ a... |
| axsep 5304 | Axiom scheme of separation... |
| axsepg 5306 | A more general version of ... |
| zfauscl 5307 | Separation Scheme (Aussond... |
| bm1.3ii 5308 | Convert implication to equ... |
| ax6vsep 5309 | Derive ~ ax6v (a weakened ... |
| axnulALT 5310 | Alternate proof of ~ axnul... |
| axnul 5311 | The Null Set Axiom of ZF s... |
| 0ex 5313 | The Null Set Axiom of ZF s... |
| al0ssb 5314 | The empty set is the uniqu... |
| sseliALT 5315 | Alternate proof of ~ sseli... |
| csbexg 5316 | The existence of proper su... |
| csbex 5317 | The existence of proper su... |
| unisn2 5318 | A version of ~ unisn witho... |
| nalset 5319 | No set contains all sets. ... |
| vnex 5320 | The universal class does n... |
| vprc 5321 | The universal class is not... |
| nvel 5322 | The universal class does n... |
| inex1 5323 | Separation Scheme (Aussond... |
| inex2 5324 | Separation Scheme (Aussond... |
| inex1g 5325 | Closed-form, generalized S... |
| inex2g 5326 | Sufficient condition for a... |
| ssex 5327 | The subset of a set is als... |
| ssexi 5328 | The subset of a set is als... |
| ssexg 5329 | The subset of a set is als... |
| ssexd 5330 | A subclass of a set is a s... |
| abexd 5331 | Conditions for a class abs... |
| abex 5332 | Conditions for a class abs... |
| prcssprc 5333 | The superclass of a proper... |
| sselpwd 5334 | Elementhood to a power set... |
| difexg 5335 | Existence of a difference.... |
| difexi 5336 | Existence of a difference,... |
| difexd 5337 | Existence of a difference.... |
| zfausab 5338 | Separation Scheme (Aussond... |
| elpw2g 5339 | Membership in a power clas... |
| elpw2 5340 | Membership in a power clas... |
| elpwi2 5341 | Membership in a power clas... |
| rabelpw 5342 | A restricted class abstrac... |
| rabexg 5343 | Separation Scheme in terms... |
| rabexgOLD 5344 | Obsolete proof of ~ rabexg... |
| rabex 5345 | Separation Scheme in terms... |
| rabexd 5346 | Separation Scheme in terms... |
| rabex2 5347 | Separation Scheme in terms... |
| rab2ex 5348 | A class abstraction based ... |
| elssabg 5349 | Membership in a class abst... |
| intex 5350 | The intersection of a none... |
| intnex 5351 | If a class intersection is... |
| intexab 5352 | The intersection of a none... |
| intexrab 5353 | The intersection of a none... |
| iinexg 5354 | The existence of a class i... |
| intabs 5355 | Absorption of a redundant ... |
| inuni 5356 | The intersection of a unio... |
| axpweq 5357 | Two equivalent ways to exp... |
| pwnss 5358 | The power set of a set is ... |
| pwne 5359 | No set equals its power se... |
| difelpw 5360 | A difference is an element... |
| class2set 5361 | The class of elements of `... |
| 0elpw 5362 | Every power class contains... |
| pwne0 5363 | A power class is never emp... |
| 0nep0 5364 | The empty set and its powe... |
| 0inp0 5365 | Something cannot be equal ... |
| unidif0 5366 | The removal of the empty s... |
| eqsnuniex 5367 | If a class is equal to the... |
| iin0 5368 | An indexed intersection of... |
| notzfaus 5369 | In the Separation Scheme ~... |
| intv 5370 | The intersection of the un... |
| zfpow 5372 | Axiom of Power Sets expres... |
| axpow2 5373 | A variant of the Axiom of ... |
| axpow3 5374 | A variant of the Axiom of ... |
| elALT2 5375 | Alternate proof of ~ el us... |
| dtruALT2 5376 | Alternate proof of ~ dtru ... |
| dtrucor 5377 | Corollary of ~ dtru . Thi... |
| dtrucor2 5378 | The theorem form of the de... |
| dvdemo1 5379 | Demonstration of a theorem... |
| dvdemo2 5380 | Demonstration of a theorem... |
| nfnid 5381 | A setvar variable is not f... |
| nfcvb 5382 | The "distinctor" expressio... |
| vpwex 5383 | Power set axiom: the power... |
| pwexg 5384 | Power set axiom expressed ... |
| pwexd 5385 | Deduction version of the p... |
| pwex 5386 | Power set axiom expressed ... |
| pwel 5387 | Quantitative version of ~ ... |
| abssexg 5388 | Existence of a class of su... |
| snexALT 5389 | Alternate proof of ~ snex ... |
| p0ex 5390 | The power set of the empty... |
| p0exALT 5391 | Alternate proof of ~ p0ex ... |
| pp0ex 5392 | The power set of the power... |
| ord3ex 5393 | The ordinal number 3 is a ... |
| dtruALT 5394 | Alternate proof of ~ dtru ... |
| axc16b 5395 | This theorem shows that Ax... |
| eunex 5396 | Existential uniqueness imp... |
| eusv1 5397 | Two ways to express single... |
| eusvnf 5398 | Even if ` x ` is free in `... |
| eusvnfb 5399 | Two ways to say that ` A (... |
| eusv2i 5400 | Two ways to express single... |
| eusv2nf 5401 | Two ways to express single... |
| eusv2 5402 | Two ways to express single... |
| reusv1 5403 | Two ways to express single... |
| reusv2lem1 5404 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5405 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5406 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5407 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5408 | Lemma for ~ reusv2 . (Con... |
| reusv2 5409 | Two ways to express single... |
| reusv3i 5410 | Two ways of expressing exi... |
| reusv3 5411 | Two ways to express single... |
| eusv4 5412 | Two ways to express single... |
| alxfr 5413 | Transfer universal quantif... |
| ralxfrd 5414 | Transfer universal quantif... |
| rexxfrd 5415 | Transfer existential quant... |
| ralxfr2d 5416 | Transfer universal quantif... |
| rexxfr2d 5417 | Transfer existential quant... |
| ralxfrd2 5418 | Transfer universal quantif... |
| rexxfrd2 5419 | Transfer existence from a ... |
| ralxfr 5420 | Transfer universal quantif... |
| ralxfrALT 5421 | Alternate proof of ~ ralxf... |
| rexxfr 5422 | Transfer existence from a ... |
| rabxfrd 5423 | Membership in a restricted... |
| rabxfr 5424 | Membership in a restricted... |
| reuhypd 5425 | A theorem useful for elimi... |
| reuhyp 5426 | A theorem useful for elimi... |
| zfpair 5427 | The Axiom of Pairing of Ze... |
| axprALT 5428 | Alternate proof of ~ axpr ... |
| axprlem1 5429 | Lemma for ~ axpr . There ... |
| axprlem2 5430 | Lemma for ~ axpr . There ... |
| axprlem3 5431 | Lemma for ~ axpr . Elimin... |
| axprlem4 5432 | Lemma for ~ axpr . The fi... |
| axprlem5 5433 | Lemma for ~ axpr . The se... |
| axpr 5434 | Unabbreviated version of t... |
| zfpair2 5436 | Derive the abbreviated ver... |
| vsnex 5437 | A singleton built on a set... |
| snexg 5438 | A singleton built on a set... |
| snex 5439 | A singleton is a set. The... |
| prex 5440 | The Axiom of Pairing using... |
| exel 5441 | There exist two sets, one ... |
| exexneq 5442 | There exist two different ... |
| exneq 5443 | Given any set (the " ` y `... |
| dtru 5444 | Given any set (the " ` y `... |
| el 5445 | Any set is an element of s... |
| sels 5446 | If a class is a set, then ... |
| selsALT 5447 | Alternate proof of ~ sels ... |
| elALT 5448 | Alternate proof of ~ el , ... |
| dtruOLD 5449 | Obsolete proof of ~ dtru a... |
| snelpwg 5450 | A singleton of a set is a ... |
| snelpwi 5451 | If a set is a member of a ... |
| snelpwiOLD 5452 | Obsolete version of ~ snel... |
| snelpw 5453 | A singleton of a set is a ... |
| prelpw 5454 | An unordered pair of two s... |
| prelpwi 5455 | If two sets are members of... |
| rext 5456 | A theorem similar to exten... |
| sspwb 5457 | The powerclass constructio... |
| unipw 5458 | A class equals the union o... |
| univ 5459 | The union of the universe ... |
| pwtr 5460 | A class is transitive iff ... |
| ssextss 5461 | An extensionality-like pri... |
| ssext 5462 | An extensionality-like pri... |
| nssss 5463 | Negation of subclass relat... |
| pweqb 5464 | Classes are equal if and o... |
| intidg 5465 | The intersection of all se... |
| intidOLD 5466 | Obsolete version of ~ inti... |
| moabex 5467 | "At most one" existence im... |
| rmorabex 5468 | Restricted "at most one" e... |
| euabex 5469 | The abstraction of a wff w... |
| nnullss 5470 | A nonempty class (even if ... |
| exss 5471 | Restricted existence in a ... |
| opex 5472 | An ordered pair of classes... |
| otex 5473 | An ordered triple of class... |
| elopg 5474 | Characterization of the el... |
| elop 5475 | Characterization of the el... |
| opi1 5476 | One of the two elements in... |
| opi2 5477 | One of the two elements of... |
| opeluu 5478 | Each member of an ordered ... |
| op1stb 5479 | Extract the first member o... |
| brv 5480 | Two classes are always in ... |
| opnz 5481 | An ordered pair is nonempt... |
| opnzi 5482 | An ordered pair is nonempt... |
| opth1 5483 | Equality of the first memb... |
| opth 5484 | The ordered pair theorem. ... |
| opthg 5485 | Ordered pair theorem. ` C ... |
| opth1g 5486 | Equality of the first memb... |
| opthg2 5487 | Ordered pair theorem. (Co... |
| opth2 5488 | Ordered pair theorem. (Co... |
| opthneg 5489 | Two ordered pairs are not ... |
| opthne 5490 | Two ordered pairs are not ... |
| otth2 5491 | Ordered triple theorem, wi... |
| otth 5492 | Ordered triple theorem. (... |
| otthg 5493 | Ordered triple theorem, cl... |
| otthne 5494 | Contrapositive of the orde... |
| eqvinop 5495 | A variable introduction la... |
| sbcop1 5496 | The proper substitution of... |
| sbcop 5497 | The proper substitution of... |
| copsexgw 5498 | Version of ~ copsexg with ... |
| copsexg 5499 | Substitution of class ` A ... |
| copsex2t 5500 | Closed theorem form of ~ c... |
| copsex2g 5501 | Implicit substitution infe... |
| copsex2gOLD 5502 | Obsolete version of ~ cops... |
| copsex4g 5503 | An implicit substitution i... |
| 0nelop 5504 | A property of ordered pair... |
| opwo0id 5505 | An ordered pair is equal t... |
| opeqex 5506 | Equivalence of existence i... |
| oteqex2 5507 | Equivalence of existence i... |
| oteqex 5508 | Equivalence of existence i... |
| opcom 5509 | An ordered pair commutes i... |
| moop2 5510 | "At most one" property of ... |
| opeqsng 5511 | Equivalence for an ordered... |
| opeqsn 5512 | Equivalence for an ordered... |
| opeqpr 5513 | Equivalence for an ordered... |
| snopeqop 5514 | Equivalence for an ordered... |
| propeqop 5515 | Equivalence for an ordered... |
| propssopi 5516 | If a pair of ordered pairs... |
| snopeqopsnid 5517 | Equivalence for an ordered... |
| mosubopt 5518 | "At most one" remains true... |
| mosubop 5519 | "At most one" remains true... |
| euop2 5520 | Transfer existential uniqu... |
| euotd 5521 | Prove existential uniquene... |
| opthwiener 5522 | Justification theorem for ... |
| uniop 5523 | The union of an ordered pa... |
| uniopel 5524 | Ordered pair membership is... |
| opthhausdorff 5525 | Justification theorem for ... |
| opthhausdorff0 5526 | Justification theorem for ... |
| otsndisj 5527 | The singletons consisting ... |
| otiunsndisj 5528 | The union of singletons co... |
| iunopeqop 5529 | Implication of an ordered ... |
| brsnop 5530 | Binary relation for an ord... |
| brtp 5531 | A necessary and sufficient... |
| opabidw 5532 | The law of concretion. Sp... |
| opabid 5533 | The law of concretion. Sp... |
| elopabw 5534 | Membership in a class abst... |
| elopab 5535 | Membership in a class abst... |
| rexopabb 5536 | Restricted existential qua... |
| vopelopabsb 5537 | The law of concretion in t... |
| opelopabsb 5538 | The law of concretion in t... |
| brabsb 5539 | The law of concretion in t... |
| opelopabt 5540 | Closed theorem form of ~ o... |
| opelopabga 5541 | The law of concretion. Th... |
| brabga 5542 | The law of concretion for ... |
| opelopab2a 5543 | Ordered pair membership in... |
| opelopaba 5544 | The law of concretion. Th... |
| braba 5545 | The law of concretion for ... |
| opelopabg 5546 | The law of concretion. Th... |
| brabg 5547 | The law of concretion for ... |
| opelopabgf 5548 | The law of concretion. Th... |
| opelopab2 5549 | Ordered pair membership in... |
| opelopab 5550 | The law of concretion. Th... |
| brab 5551 | The law of concretion for ... |
| opelopabaf 5552 | The law of concretion. Th... |
| opelopabf 5553 | The law of concretion. Th... |
| ssopab2 5554 | Equivalence of ordered pai... |
| ssopab2bw 5555 | Equivalence of ordered pai... |
| eqopab2bw 5556 | Equivalence of ordered pai... |
| ssopab2b 5557 | Equivalence of ordered pai... |
| ssopab2i 5558 | Inference of ordered pair ... |
| ssopab2dv 5559 | Inference of ordered pair ... |
| eqopab2b 5560 | Equivalence of ordered pai... |
| opabn0 5561 | Nonempty ordered pair clas... |
| opab0 5562 | Empty ordered pair class a... |
| csbopab 5563 | Move substitution into a c... |
| csbopabgALT 5564 | Move substitution into a c... |
| csbmpt12 5565 | Move substitution into a m... |
| csbmpt2 5566 | Move substitution into the... |
| iunopab 5567 | Move indexed union inside ... |
| iunopabOLD 5568 | Obsolete version of ~ iuno... |
| elopabr 5569 | Membership in an ordered-p... |
| elopabran 5570 | Membership in an ordered-p... |
| elopabrOLD 5571 | Obsolete version of ~ elop... |
| rbropapd 5572 | Properties of a pair in an... |
| rbropap 5573 | Properties of a pair in a ... |
| 2rbropap 5574 | Properties of a pair in a ... |
| 0nelopab 5575 | The empty set is never an ... |
| 0nelopabOLD 5576 | Obsolete version of ~ 0nel... |
| brabv 5577 | If two classes are in a re... |
| pwin 5578 | The power class of the int... |
| pwssun 5579 | The power class of the uni... |
| pwun 5580 | The power class of the uni... |
| dfid4 5583 | The identity function expr... |
| dfid2 5584 | Alternate definition of th... |
| dfid3 5585 | A stronger version of ~ df... |
| dfid2OLD 5586 | Obsolete version of ~ dfid... |
| epelg 5589 | The membership relation an... |
| epeli 5590 | The membership relation an... |
| epel 5591 | The membership relation an... |
| 0sn0ep 5592 | An example for the members... |
| epn0 5593 | The membership relation is... |
| poss 5598 | Subset theorem for the par... |
| poeq1 5599 | Equality theorem for parti... |
| poeq2 5600 | Equality theorem for parti... |
| nfpo 5601 | Bound-variable hypothesis ... |
| nfso 5602 | Bound-variable hypothesis ... |
| pocl 5603 | Characteristic properties ... |
| poclOLD 5604 | Obsolete version of ~ pocl... |
| ispod 5605 | Sufficient conditions for ... |
| swopolem 5606 | Perform the substitutions ... |
| swopo 5607 | A strict weak order is a p... |
| poirr 5608 | A partial order is irrefle... |
| potr 5609 | A partial order is a trans... |
| po2nr 5610 | A partial order has no 2-c... |
| po3nr 5611 | A partial order has no 3-c... |
| po2ne 5612 | Two sets related by a part... |
| po0 5613 | Any relation is a partial ... |
| pofun 5614 | The inverse image of a par... |
| sopo 5615 | A strict linear order is a... |
| soss 5616 | Subset theorem for the str... |
| soeq1 5617 | Equality theorem for the s... |
| soeq2 5618 | Equality theorem for the s... |
| sonr 5619 | A strict order relation is... |
| sotr 5620 | A strict order relation is... |
| solin 5621 | A strict order relation is... |
| so2nr 5622 | A strict order relation ha... |
| so3nr 5623 | A strict order relation ha... |
| sotric 5624 | A strict order relation sa... |
| sotrieq 5625 | Trichotomy law for strict ... |
| sotrieq2 5626 | Trichotomy law for strict ... |
| soasym 5627 | Asymmetry law for strict o... |
| sotr2 5628 | A transitivity relation. ... |
| issod 5629 | An irreflexive, transitive... |
| issoi 5630 | An irreflexive, transitive... |
| isso2i 5631 | Deduce strict ordering fro... |
| so0 5632 | Any relation is a strict o... |
| somo 5633 | A totally ordered set has ... |
| sotrine 5634 | Trichotomy law for strict ... |
| sotr3 5635 | Transitivity law for stric... |
| dffr6 5642 | Alternate definition of ~ ... |
| frd 5643 | A nonempty subset of an ` ... |
| fri 5644 | A nonempty subset of an ` ... |
| friOLD 5645 | Obsolete version of ~ fri ... |
| seex 5646 | The ` R ` -preimage of an ... |
| exse 5647 | Any relation on a set is s... |
| dffr2 5648 | Alternate definition of we... |
| dffr2ALT 5649 | Alternate proof of ~ dffr2... |
| frc 5650 | Property of well-founded r... |
| frss 5651 | Subset theorem for the wel... |
| sess1 5652 | Subset theorem for the set... |
| sess2 5653 | Subset theorem for the set... |
| freq1 5654 | Equality theorem for the w... |
| freq2 5655 | Equality theorem for the w... |
| seeq1 5656 | Equality theorem for the s... |
| seeq2 5657 | Equality theorem for the s... |
| nffr 5658 | Bound-variable hypothesis ... |
| nfse 5659 | Bound-variable hypothesis ... |
| nfwe 5660 | Bound-variable hypothesis ... |
| frirr 5661 | A well-founded relation is... |
| fr2nr 5662 | A well-founded relation ha... |
| fr0 5663 | Any relation is well-found... |
| frminex 5664 | If an element of a well-fo... |
| efrirr 5665 | A well-founded class does ... |
| efrn2lp 5666 | A well-founded class conta... |
| epse 5667 | The membership relation is... |
| tz7.2 5668 | Similar to Theorem 7.2 of ... |
| dfepfr 5669 | An alternate way of saying... |
| epfrc 5670 | A subset of a well-founded... |
| wess 5671 | Subset theorem for the wel... |
| weeq1 5672 | Equality theorem for the w... |
| weeq2 5673 | Equality theorem for the w... |
| wefr 5674 | A well-ordering is well-fo... |
| weso 5675 | A well-ordering is a stric... |
| wecmpep 5676 | The elements of a class we... |
| wetrep 5677 | On a class well-ordered by... |
| wefrc 5678 | A nonempty subclass of a c... |
| we0 5679 | Any relation is a well-ord... |
| wereu 5680 | A nonempty subset of an ` ... |
| wereu2 5681 | A nonempty subclass of an ... |
| xpeq1 5698 | Equality theorem for Carte... |
| xpss12 5699 | Subset theorem for Cartesi... |
| xpss 5700 | A Cartesian product is inc... |
| inxpssres 5701 | Intersection with a Cartes... |
| relxp 5702 | A Cartesian product is a r... |
| xpss1 5703 | Subset relation for Cartes... |
| xpss2 5704 | Subset relation for Cartes... |
| xpeq2 5705 | Equality theorem for Carte... |
| elxpi 5706 | Membership in a Cartesian ... |
| elxp 5707 | Membership in a Cartesian ... |
| elxp2 5708 | Membership in a Cartesian ... |
| xpeq12 5709 | Equality theorem for Carte... |
| xpeq1i 5710 | Equality inference for Car... |
| xpeq2i 5711 | Equality inference for Car... |
| xpeq12i 5712 | Equality inference for Car... |
| xpeq1d 5713 | Equality deduction for Car... |
| xpeq2d 5714 | Equality deduction for Car... |
| xpeq12d 5715 | Equality deduction for Car... |
| sqxpeqd 5716 | Equality deduction for a C... |
| nfxp 5717 | Bound-variable hypothesis ... |
| 0nelxp 5718 | The empty set is not a mem... |
| 0nelelxp 5719 | A member of a Cartesian pr... |
| opelxp 5720 | Ordered pair membership in... |
| opelxpi 5721 | Ordered pair membership in... |
| opelxpii 5722 | Ordered pair membership in... |
| opelxpd 5723 | Ordered pair membership in... |
| opelvv 5724 | Ordered pair membership in... |
| opelvvg 5725 | Ordered pair membership in... |
| opelxp1 5726 | The first member of an ord... |
| opelxp2 5727 | The second member of an or... |
| otelxp 5728 | Ordered triple membership ... |
| otelxp1 5729 | The first member of an ord... |
| otel3xp 5730 | An ordered triple is an el... |
| opabssxpd 5731 | An ordered-pair class abst... |
| rabxp 5732 | Class abstraction restrict... |
| brxp 5733 | Binary relation on a Carte... |
| pwvrel 5734 | A set is a binary relation... |
| pwvabrel 5735 | The powerclass of the cart... |
| brrelex12 5736 | Two classes related by a b... |
| brrelex1 5737 | If two classes are related... |
| brrelex2 5738 | If two classes are related... |
| brrelex12i 5739 | Two classes that are relat... |
| brrelex1i 5740 | The first argument of a bi... |
| brrelex2i 5741 | The second argument of a b... |
| nprrel12 5742 | Proper classes are not rel... |
| nprrel 5743 | No proper class is related... |
| 0nelrel0 5744 | A binary relation does not... |
| 0nelrel 5745 | A binary relation does not... |
| fconstmpt 5746 | Representation of a consta... |
| vtoclr 5747 | Variable to class conversi... |
| opthprc 5748 | Justification theorem for ... |
| brel 5749 | Two things in a binary rel... |
| elxp3 5750 | Membership in a Cartesian ... |
| opeliunxp 5751 | Membership in a union of C... |
| xpundi 5752 | Distributive law for Carte... |
| xpundir 5753 | Distributive law for Carte... |
| xpiundi 5754 | Distributive law for Carte... |
| xpiundir 5755 | Distributive law for Carte... |
| iunxpconst 5756 | Membership in a union of C... |
| xpun 5757 | The Cartesian product of t... |
| elvv 5758 | Membership in universal cl... |
| elvvv 5759 | Membership in universal cl... |
| elvvuni 5760 | An ordered pair contains i... |
| brinxp2 5761 | Intersection of binary rel... |
| brinxp 5762 | Intersection of binary rel... |
| opelinxp 5763 | Ordered pair element in an... |
| poinxp 5764 | Intersection of partial or... |
| soinxp 5765 | Intersection of total orde... |
| frinxp 5766 | Intersection of well-found... |
| seinxp 5767 | Intersection of set-like r... |
| weinxp 5768 | Intersection of well-order... |
| posn 5769 | Partial ordering of a sing... |
| sosn 5770 | Strict ordering on a singl... |
| frsn 5771 | Founded relation on a sing... |
| wesn 5772 | Well-ordering of a singlet... |
| elopaelxp 5773 | Membership in an ordered-p... |
| elopaelxpOLD 5774 | Obsolete version of ~ elop... |
| bropaex12 5775 | Two classes related by an ... |
| opabssxp 5776 | An abstraction relation is... |
| brab2a 5777 | The law of concretion for ... |
| optocl 5778 | Implicit substitution of c... |
| 2optocl 5779 | Implicit substitution of c... |
| 3optocl 5780 | Implicit substitution of c... |
| opbrop 5781 | Ordered pair membership in... |
| 0xp 5782 | The Cartesian product with... |
| csbxp 5783 | Distribute proper substitu... |
| releq 5784 | Equality theorem for the r... |
| releqi 5785 | Equality inference for the... |
| releqd 5786 | Equality deduction for the... |
| nfrel 5787 | Bound-variable hypothesis ... |
| sbcrel 5788 | Distribute proper substitu... |
| relss 5789 | Subclass theorem for relat... |
| ssrel 5790 | A subclass relationship de... |
| ssrelOLD 5791 | Obsolete version of ~ ssre... |
| eqrel 5792 | Extensionality principle f... |
| ssrel2 5793 | A subclass relationship de... |
| ssrel3 5794 | Subclass relation in anoth... |
| relssi 5795 | Inference from subclass pr... |
| relssdv 5796 | Deduction from subclass pr... |
| eqrelriv 5797 | Inference from extensional... |
| eqrelriiv 5798 | Inference from extensional... |
| eqbrriv 5799 | Inference from extensional... |
| eqrelrdv 5800 | Deduce equality of relatio... |
| eqbrrdv 5801 | Deduction from extensional... |
| eqbrrdiv 5802 | Deduction from extensional... |
| eqrelrdv2 5803 | A version of ~ eqrelrdv . ... |
| ssrelrel 5804 | A subclass relationship de... |
| eqrelrel 5805 | Extensionality principle f... |
| elrel 5806 | A member of a relation is ... |
| rel0 5807 | The empty set is a relatio... |
| nrelv 5808 | The universal class is not... |
| relsng 5809 | A singleton is a relation ... |
| relsnb 5810 | An at-most-singleton is a ... |
| relsnopg 5811 | A singleton of an ordered ... |
| relsn 5812 | A singleton is a relation ... |
| relsnop 5813 | A singleton of an ordered ... |
| copsex2gb 5814 | Implicit substitution infe... |
| copsex2ga 5815 | Implicit substitution infe... |
| elopaba 5816 | Membership in an ordered-p... |
| xpsspw 5817 | A Cartesian product is inc... |
| unixpss 5818 | The double class union of ... |
| relun 5819 | The union of two relations... |
| relin1 5820 | The intersection with a re... |
| relin2 5821 | The intersection with a re... |
| relinxp 5822 | Intersection with a Cartes... |
| reldif 5823 | A difference cutting down ... |
| reliun 5824 | An indexed union is a rela... |
| reliin 5825 | An indexed intersection is... |
| reluni 5826 | The union of a class is a ... |
| relint 5827 | The intersection of a clas... |
| relopabiv 5828 | A class of ordered pairs i... |
| relopabv 5829 | A class of ordered pairs i... |
| relopabi 5830 | A class of ordered pairs i... |
| relopabiALT 5831 | Alternate proof of ~ relop... |
| relopab 5832 | A class of ordered pairs i... |
| mptrel 5833 | The maps-to notation alway... |
| reli 5834 | The identity relation is a... |
| rele 5835 | The membership relation is... |
| opabid2 5836 | A relation expressed as an... |
| inopab 5837 | Intersection of two ordere... |
| difopab 5838 | Difference of two ordered-... |
| difopabOLD 5839 | Obsolete version of ~ difo... |
| inxp 5840 | Intersection of two Cartes... |
| inxpOLD 5841 | Obsolete version of ~ inxp... |
| xpindi 5842 | Distributive law for Carte... |
| xpindir 5843 | Distributive law for Carte... |
| xpiindi 5844 | Distributive law for Carte... |
| xpriindi 5845 | Distributive law for Carte... |
| eliunxp 5846 | Membership in a union of C... |
| opeliunxp2 5847 | Membership in a union of C... |
| raliunxp 5848 | Write a double restricted ... |
| rexiunxp 5849 | Write a double restricted ... |
| ralxp 5850 | Universal quantification r... |
| rexxp 5851 | Existential quantification... |
| exopxfr 5852 | Transfer ordered-pair exis... |
| exopxfr2 5853 | Transfer ordered-pair exis... |
| djussxp 5854 | Disjoint union is a subset... |
| ralxpf 5855 | Version of ~ ralxp with bo... |
| rexxpf 5856 | Version of ~ rexxp with bo... |
| iunxpf 5857 | Indexed union on a Cartesi... |
| opabbi2dv 5858 | Deduce equality of a relat... |
| relop 5859 | A necessary and sufficient... |
| ideqg 5860 | For sets, the identity rel... |
| ideq 5861 | For sets, the identity rel... |
| ididg 5862 | A set is identical to itse... |
| issetid 5863 | Two ways of expressing set... |
| coss1 5864 | Subclass theorem for compo... |
| coss2 5865 | Subclass theorem for compo... |
| coeq1 5866 | Equality theorem for compo... |
| coeq2 5867 | Equality theorem for compo... |
| coeq1i 5868 | Equality inference for com... |
| coeq2i 5869 | Equality inference for com... |
| coeq1d 5870 | Equality deduction for com... |
| coeq2d 5871 | Equality deduction for com... |
| coeq12i 5872 | Equality inference for com... |
| coeq12d 5873 | Equality deduction for com... |
| nfco 5874 | Bound-variable hypothesis ... |
| brcog 5875 | Ordered pair membership in... |
| opelco2g 5876 | Ordered pair membership in... |
| brcogw 5877 | Ordered pair membership in... |
| eqbrrdva 5878 | Deduction from extensional... |
| brco 5879 | Binary relation on a compo... |
| opelco 5880 | Ordered pair membership in... |
| cnvss 5881 | Subset theorem for convers... |
| cnveq 5882 | Equality theorem for conve... |
| cnveqi 5883 | Equality inference for con... |
| cnveqd 5884 | Equality deduction for con... |
| elcnv 5885 | Membership in a converse r... |
| elcnv2 5886 | Membership in a converse r... |
| nfcnv 5887 | Bound-variable hypothesis ... |
| brcnvg 5888 | The converse of a binary r... |
| opelcnvg 5889 | Ordered-pair membership in... |
| opelcnv 5890 | Ordered-pair membership in... |
| brcnv 5891 | The converse of a binary r... |
| csbcnv 5892 | Move class substitution in... |
| csbcnvgALT 5893 | Move class substitution in... |
| cnvco 5894 | Distributive law of conver... |
| cnvuni 5895 | The converse of a class un... |
| dfdm3 5896 | Alternate definition of do... |
| dfrn2 5897 | Alternate definition of ra... |
| dfrn3 5898 | Alternate definition of ra... |
| elrn2g 5899 | Membership in a range. (C... |
| elrng 5900 | Membership in a range. (C... |
| elrn2 5901 | Membership in a range. (C... |
| elrn 5902 | Membership in a range. (C... |
| ssrelrn 5903 | If a relation is a subset ... |
| dfdm4 5904 | Alternate definition of do... |
| dfdmf 5905 | Definition of domain, usin... |
| csbdm 5906 | Distribute proper substitu... |
| eldmg 5907 | Domain membership. Theore... |
| eldm2g 5908 | Domain membership. Theore... |
| eldm 5909 | Membership in a domain. T... |
| eldm2 5910 | Membership in a domain. T... |
| dmss 5911 | Subset theorem for domain.... |
| dmeq 5912 | Equality theorem for domai... |
| dmeqi 5913 | Equality inference for dom... |
| dmeqd 5914 | Equality deduction for dom... |
| opeldmd 5915 | Membership of first of an ... |
| opeldm 5916 | Membership of first of an ... |
| breldm 5917 | Membership of first of a b... |
| breldmg 5918 | Membership of first of a b... |
| dmun 5919 | The domain of a union is t... |
| dmin 5920 | The domain of an intersect... |
| breldmd 5921 | Membership of first of a b... |
| dmiun 5922 | The domain of an indexed u... |
| dmuni 5923 | The domain of a union. Pa... |
| dmopab 5924 | The domain of a class of o... |
| dmopabelb 5925 | A set is an element of the... |
| dmopab2rex 5926 | The domain of an ordered p... |
| dmopabss 5927 | Upper bound for the domain... |
| dmopab3 5928 | The domain of a restricted... |
| dm0 5929 | The domain of the empty se... |
| dmi 5930 | The domain of the identity... |
| dmv 5931 | The domain of the universe... |
| dmep 5932 | The domain of the membersh... |
| dm0rn0 5933 | An empty domain is equival... |
| rn0 5934 | The range of the empty set... |
| rnep 5935 | The range of the membershi... |
| reldm0 5936 | A relation is empty iff it... |
| dmxp 5937 | The domain of a Cartesian ... |
| dmxpid 5938 | The domain of a Cartesian ... |
| dmxpin 5939 | The domain of the intersec... |
| xpid11 5940 | The Cartesian square is a ... |
| dmcnvcnv 5941 | The domain of the double c... |
| rncnvcnv 5942 | The range of the double co... |
| elreldm 5943 | The first member of an ord... |
| rneq 5944 | Equality theorem for range... |
| rneqi 5945 | Equality inference for ran... |
| rneqd 5946 | Equality deduction for ran... |
| rnss 5947 | Subset theorem for range. ... |
| rnssi 5948 | Subclass inference for ran... |
| brelrng 5949 | The second argument of a b... |
| brelrn 5950 | The second argument of a b... |
| opelrn 5951 | Membership of second membe... |
| releldm 5952 | The first argument of a bi... |
| relelrn 5953 | The second argument of a b... |
| releldmb 5954 | Membership in a domain. (... |
| relelrnb 5955 | Membership in a range. (C... |
| releldmi 5956 | The first argument of a bi... |
| relelrni 5957 | The second argument of a b... |
| dfrnf 5958 | Definition of range, using... |
| nfdm 5959 | Bound-variable hypothesis ... |
| nfrn 5960 | Bound-variable hypothesis ... |
| dmiin 5961 | Domain of an intersection.... |
| rnopab 5962 | The range of a class of or... |
| rnmpt 5963 | The range of a function in... |
| elrnmpt 5964 | The range of a function in... |
| elrnmpt1s 5965 | Elementhood in an image se... |
| elrnmpt1 5966 | Elementhood in an image se... |
| elrnmptg 5967 | Membership in the range of... |
| elrnmpti 5968 | Membership in the range of... |
| elrnmptd 5969 | The range of a function in... |
| elrnmpt1d 5970 | Elementhood in an image se... |
| elrnmptdv 5971 | Elementhood in the range o... |
| elrnmpt2d 5972 | Elementhood in the range o... |
| dfiun3g 5973 | Alternate definition of in... |
| dfiin3g 5974 | Alternate definition of in... |
| dfiun3 5975 | Alternate definition of in... |
| dfiin3 5976 | Alternate definition of in... |
| riinint 5977 | Express a relative indexed... |
| relrn0 5978 | A relation is empty iff it... |
| dmrnssfld 5979 | The domain and range of a ... |
| dmcoss 5980 | Domain of a composition. ... |
| rncoss 5981 | Range of a composition. (... |
| dmcosseq 5982 | Domain of a composition. ... |
| dmcosseqOLD 5983 | Obsolete version of ~ dmco... |
| dmcoeq 5984 | Domain of a composition. ... |
| rncoeq 5985 | Range of a composition. (... |
| reseq1 5986 | Equality theorem for restr... |
| reseq2 5987 | Equality theorem for restr... |
| reseq1i 5988 | Equality inference for res... |
| reseq2i 5989 | Equality inference for res... |
| reseq12i 5990 | Equality inference for res... |
| reseq1d 5991 | Equality deduction for res... |
| reseq2d 5992 | Equality deduction for res... |
| reseq12d 5993 | Equality deduction for res... |
| nfres 5994 | Bound-variable hypothesis ... |
| csbres 5995 | Distribute proper substitu... |
| res0 5996 | A restriction to the empty... |
| dfres3 5997 | Alternate definition of re... |
| opelres 5998 | Ordered pair elementhood i... |
| brres 5999 | Binary relation on a restr... |
| opelresi 6000 | Ordered pair membership in... |
| brresi 6001 | Binary relation on a restr... |
| opres 6002 | Ordered pair membership in... |
| resieq 6003 | A restricted identity rela... |
| opelidres 6004 | ` <. A , A >. ` belongs to... |
| resres 6005 | The restriction of a restr... |
| resundi 6006 | Distributive law for restr... |
| resundir 6007 | Distributive law for restr... |
| resindi 6008 | Class restriction distribu... |
| resindir 6009 | Class restriction distribu... |
| inres 6010 | Move intersection into cla... |
| resdifcom 6011 | Commutative law for restri... |
| resiun1 6012 | Distribution of restrictio... |
| resiun2 6013 | Distribution of restrictio... |
| resss 6014 | A class includes its restr... |
| rescom 6015 | Commutative law for restri... |
| ssres 6016 | Subclass theorem for restr... |
| ssres2 6017 | Subclass theorem for restr... |
| relres 6018 | A restriction is a relatio... |
| resabs1 6019 | Absorption law for restric... |
| resabs1d 6020 | Absorption law for restric... |
| resabs2 6021 | Absorption law for restric... |
| residm 6022 | Idempotent law for restric... |
| dmresss 6023 | The domain of a restrictio... |
| dmres 6024 | The domain of a restrictio... |
| ssdmres 6025 | A domain restricted to a s... |
| dmresexg 6026 | The domain of a restrictio... |
| resima 6027 | A restriction to an image.... |
| resima2 6028 | Image under a restricted c... |
| rnresss 6029 | The range of a restriction... |
| xpssres 6030 | Restriction of a constant ... |
| elinxp 6031 | Membership in an intersect... |
| elres 6032 | Membership in a restrictio... |
| elsnres 6033 | Membership in restriction ... |
| relssres 6034 | Simplification law for res... |
| dmressnsn 6035 | The domain of a restrictio... |
| eldmressnsn 6036 | The element of the domain ... |
| eldmeldmressn 6037 | An element of the domain (... |
| resdm 6038 | A relation restricted to i... |
| resexg 6039 | The restriction of a set i... |
| resexd 6040 | The restriction of a set i... |
| resex 6041 | The restriction of a set i... |
| resindm 6042 | When restricting a relatio... |
| resdmdfsn 6043 | Restricting a relation to ... |
| reldisjun 6044 | Split a relation into two ... |
| relresdm1 6045 | Restriction of a disjoint ... |
| resopab 6046 | Restriction of a class abs... |
| iss 6047 | A subclass of the identity... |
| resopab2 6048 | Restriction of a class abs... |
| resmpt 6049 | Restriction of the mapping... |
| resmpt3 6050 | Unconditional restriction ... |
| resmptf 6051 | Restriction of the mapping... |
| resmptd 6052 | Restriction of the mapping... |
| dfres2 6053 | Alternate definition of th... |
| mptss 6054 | Sufficient condition for i... |
| elimampt 6055 | Membership in the image of... |
| elidinxp 6056 | Characterization of the el... |
| elidinxpid 6057 | Characterization of the el... |
| elrid 6058 | Characterization of the el... |
| idinxpres 6059 | The intersection of the id... |
| idinxpresid 6060 | The intersection of the id... |
| idssxp 6061 | A diagonal set as a subset... |
| opabresid 6062 | The restricted identity re... |
| mptresid 6063 | The restricted identity re... |
| dmresi 6064 | The domain of a restricted... |
| restidsing 6065 | Restriction of the identit... |
| iresn0n0 6066 | The identity function rest... |
| imaeq1 6067 | Equality theorem for image... |
| imaeq2 6068 | Equality theorem for image... |
| imaeq1i 6069 | Equality theorem for image... |
| imaeq2i 6070 | Equality theorem for image... |
| imaeq1d 6071 | Equality theorem for image... |
| imaeq2d 6072 | Equality theorem for image... |
| imaeq12d 6073 | Equality theorem for image... |
| dfima2 6074 | Alternate definition of im... |
| dfima3 6075 | Alternate definition of im... |
| elimag 6076 | Membership in an image. T... |
| elima 6077 | Membership in an image. T... |
| elima2 6078 | Membership in an image. T... |
| elima3 6079 | Membership in an image. T... |
| nfima 6080 | Bound-variable hypothesis ... |
| nfimad 6081 | Deduction version of bound... |
| imadmrn 6082 | The image of the domain of... |
| imassrn 6083 | The image of a class is a ... |
| mptima 6084 | Image of a function in map... |
| mptimass 6085 | Image of a function in map... |
| imai 6086 | Image under the identity r... |
| rnresi 6087 | The range of the restricte... |
| resiima 6088 | The image of a restriction... |
| ima0 6089 | Image of the empty set. T... |
| 0ima 6090 | Image under the empty rela... |
| csbima12 6091 | Move class substitution in... |
| imadisj 6092 | A class whose image under ... |
| imadisjlnd 6093 | Deduction form of one nega... |
| cnvimass 6094 | A preimage under any class... |
| cnvimarndm 6095 | The preimage of the range ... |
| imasng 6096 | The image of a singleton. ... |
| relimasn 6097 | The image of a singleton. ... |
| elrelimasn 6098 | Elementhood in the image o... |
| elimasng1 6099 | Membership in an image of ... |
| elimasn1 6100 | Membership in an image of ... |
| elimasng 6101 | Membership in an image of ... |
| elimasn 6102 | Membership in an image of ... |
| elimasngOLD 6103 | Obsolete version of ~ elim... |
| elimasni 6104 | Membership in an image of ... |
| args 6105 | Two ways to express the cl... |
| elinisegg 6106 | Membership in the inverse ... |
| eliniseg 6107 | Membership in the inverse ... |
| epin 6108 | Any set is equal to its pr... |
| epini 6109 | Any set is equal to its pr... |
| iniseg 6110 | An idiom that signifies an... |
| inisegn0 6111 | Nonemptiness of an initial... |
| dffr3 6112 | Alternate definition of we... |
| dfse2 6113 | Alternate definition of se... |
| imass1 6114 | Subset theorem for image. ... |
| imass2 6115 | Subset theorem for image. ... |
| ndmima 6116 | The image of a singleton o... |
| relcnv 6117 | A converse is a relation. ... |
| relbrcnvg 6118 | When ` R ` is a relation, ... |
| eliniseg2 6119 | Eliminate the class existe... |
| relbrcnv 6120 | When ` R ` is a relation, ... |
| relco 6121 | A composition is a relatio... |
| cotrg 6122 | Two ways of saying that th... |
| cotrgOLD 6123 | Obsolete version of ~ cotr... |
| cotrgOLDOLD 6124 | Obsolete version of ~ cotr... |
| cotr 6125 | Two ways of saying a relat... |
| idrefALT 6126 | Alternate proof of ~ idref... |
| cnvsym 6127 | Two ways of saying a relat... |
| cnvsymOLD 6128 | Obsolete proof of ~ cnvsym... |
| cnvsymOLDOLD 6129 | Obsolete proof of ~ cnvsym... |
| intasym 6130 | Two ways of saying a relat... |
| asymref 6131 | Two ways of saying a relat... |
| asymref2 6132 | Two ways of saying a relat... |
| intirr 6133 | Two ways of saying a relat... |
| brcodir 6134 | Two ways of saying that tw... |
| codir 6135 | Two ways of saying a relat... |
| qfto 6136 | A quantifier-free way of e... |
| xpidtr 6137 | A Cartesian square is a tr... |
| trin2 6138 | The intersection of two tr... |
| poirr2 6139 | A partial order is irrefle... |
| trinxp 6140 | The relation induced by a ... |
| soirri 6141 | A strict order relation is... |
| sotri 6142 | A strict order relation is... |
| son2lpi 6143 | A strict order relation ha... |
| sotri2 6144 | A transitivity relation. ... |
| sotri3 6145 | A transitivity relation. ... |
| poleloe 6146 | Express "less than or equa... |
| poltletr 6147 | Transitive law for general... |
| somin1 6148 | Property of a minimum in a... |
| somincom 6149 | Commutativity of minimum i... |
| somin2 6150 | Property of a minimum in a... |
| soltmin 6151 | Being less than a minimum,... |
| cnvopab 6152 | The converse of a class ab... |
| cnvopabOLD 6153 | Obsolete version of ~ cnvo... |
| mptcnv 6154 | The converse of a mapping ... |
| cnv0 6155 | The converse of the empty ... |
| cnvi 6156 | The converse of the identi... |
| cnvun 6157 | The converse of a union is... |
| cnvdif 6158 | Distributive law for conve... |
| cnvin 6159 | Distributive law for conve... |
| rnun 6160 | Distributive law for range... |
| rnin 6161 | The range of an intersecti... |
| rniun 6162 | The range of an indexed un... |
| rnuni 6163 | The range of a union. Par... |
| imaundi 6164 | Distributive law for image... |
| imaundir 6165 | The image of a union. (Co... |
| cnvimassrndm 6166 | The preimage of a superset... |
| dminss 6167 | An upper bound for interse... |
| imainss 6168 | An upper bound for interse... |
| inimass 6169 | The image of an intersecti... |
| inimasn 6170 | The intersection of the im... |
| cnvxp 6171 | The converse of a Cartesia... |
| xp0 6172 | The Cartesian product with... |
| xpnz 6173 | The Cartesian product of n... |
| xpeq0 6174 | At least one member of an ... |
| xpdisj1 6175 | Cartesian products with di... |
| xpdisj2 6176 | Cartesian products with di... |
| xpsndisj 6177 | Cartesian products with tw... |
| difxp 6178 | Difference of Cartesian pr... |
| difxp1 6179 | Difference law for Cartesi... |
| difxp2 6180 | Difference law for Cartesi... |
| djudisj 6181 | Disjoint unions with disjo... |
| xpdifid 6182 | The set of distinct couple... |
| resdisj 6183 | A double restriction to di... |
| rnxp 6184 | The range of a Cartesian p... |
| dmxpss 6185 | The domain of a Cartesian ... |
| rnxpss 6186 | The range of a Cartesian p... |
| rnxpid 6187 | The range of a Cartesian s... |
| ssxpb 6188 | A Cartesian product subcla... |
| xp11 6189 | The Cartesian product of n... |
| xpcan 6190 | Cancellation law for Carte... |
| xpcan2 6191 | Cancellation law for Carte... |
| ssrnres 6192 | Two ways to express surjec... |
| rninxp 6193 | Two ways to express surjec... |
| dminxp 6194 | Two ways to express totali... |
| imainrect 6195 | Image by a restricted and ... |
| xpima 6196 | Direct image by a Cartesia... |
| xpima1 6197 | Direct image by a Cartesia... |
| xpima2 6198 | Direct image by a Cartesia... |
| xpimasn 6199 | Direct image of a singleto... |
| sossfld 6200 | The base set of a strict o... |
| sofld 6201 | The base set of a nonempty... |
| cnvcnv3 6202 | The set of all ordered pai... |
| dfrel2 6203 | Alternate definition of re... |
| dfrel4v 6204 | A relation can be expresse... |
| dfrel4 6205 | A relation can be expresse... |
| cnvcnv 6206 | The double converse of a c... |
| cnvcnv2 6207 | The double converse of a c... |
| cnvcnvss 6208 | The double converse of a c... |
| cnvrescnv 6209 | Two ways to express the co... |
| cnveqb 6210 | Equality theorem for conve... |
| cnveq0 6211 | A relation empty iff its c... |
| dfrel3 6212 | Alternate definition of re... |
| elid 6213 | Characterization of the el... |
| dmresv 6214 | The domain of a universal ... |
| rnresv 6215 | The range of a universal r... |
| dfrn4 6216 | Range defined in terms of ... |
| csbrn 6217 | Distribute proper substitu... |
| rescnvcnv 6218 | The restriction of the dou... |
| cnvcnvres 6219 | The double converse of the... |
| imacnvcnv 6220 | The image of the double co... |
| dmsnn0 6221 | The domain of a singleton ... |
| rnsnn0 6222 | The range of a singleton i... |
| dmsn0 6223 | The domain of the singleto... |
| cnvsn0 6224 | The converse of the single... |
| dmsn0el 6225 | The domain of a singleton ... |
| relsn2 6226 | A singleton is a relation ... |
| dmsnopg 6227 | The domain of a singleton ... |
| dmsnopss 6228 | The domain of a singleton ... |
| dmpropg 6229 | The domain of an unordered... |
| dmsnop 6230 | The domain of a singleton ... |
| dmprop 6231 | The domain of an unordered... |
| dmtpop 6232 | The domain of an unordered... |
| cnvcnvsn 6233 | Double converse of a singl... |
| dmsnsnsn 6234 | The domain of the singleto... |
| rnsnopg 6235 | The range of a singleton o... |
| rnpropg 6236 | The range of a pair of ord... |
| cnvsng 6237 | Converse of a singleton of... |
| rnsnop 6238 | The range of a singleton o... |
| op1sta 6239 | Extract the first member o... |
| cnvsn 6240 | Converse of a singleton of... |
| op2ndb 6241 | Extract the second member ... |
| op2nda 6242 | Extract the second member ... |
| opswap 6243 | Swap the members of an ord... |
| cnvresima 6244 | An image under the convers... |
| resdm2 6245 | A class restricted to its ... |
| resdmres 6246 | Restriction to the domain ... |
| resresdm 6247 | A restriction by an arbitr... |
| imadmres 6248 | The image of the domain of... |
| resdmss 6249 | Subset relationship for th... |
| resdifdi 6250 | Distributive law for restr... |
| resdifdir 6251 | Distributive law for restr... |
| mptpreima 6252 | The preimage of a function... |
| mptiniseg 6253 | Converse singleton image o... |
| dmmpt 6254 | The domain of the mapping ... |
| dmmptss 6255 | The domain of a mapping is... |
| dmmptg 6256 | The domain of the mapping ... |
| rnmpt0f 6257 | The range of a function in... |
| rnmptn0 6258 | The range of a function in... |
| dfco2 6259 | Alternate definition of a ... |
| dfco2a 6260 | Generalization of ~ dfco2 ... |
| coundi 6261 | Class composition distribu... |
| coundir 6262 | Class composition distribu... |
| cores 6263 | Restricted first member of... |
| resco 6264 | Associative law for the re... |
| imaco 6265 | Image of the composition o... |
| rnco 6266 | The range of the compositi... |
| rnco2 6267 | The range of the compositi... |
| dmco 6268 | The domain of a compositio... |
| coeq0 6269 | A composition of two relat... |
| coiun 6270 | Composition with an indexe... |
| cocnvcnv1 6271 | A composition is not affec... |
| cocnvcnv2 6272 | A composition is not affec... |
| cores2 6273 | Absorption of a reverse (p... |
| co02 6274 | Composition with the empty... |
| co01 6275 | Composition with the empty... |
| coi1 6276 | Composition with the ident... |
| coi2 6277 | Composition with the ident... |
| coires1 6278 | Composition with a restric... |
| coass 6279 | Associative law for class ... |
| relcnvtrg 6280 | General form of ~ relcnvtr... |
| relcnvtr 6281 | A relation is transitive i... |
| relssdmrn 6282 | A relation is included in ... |
| relssdmrnOLD 6283 | Obsolete version of ~ rels... |
| resssxp 6284 | If the ` R ` -image of a c... |
| cnvssrndm 6285 | The converse is a subset o... |
| cossxp 6286 | Composition as a subset of... |
| relrelss 6287 | Two ways to describe the s... |
| unielrel 6288 | The membership relation fo... |
| relfld 6289 | The double union of a rela... |
| relresfld 6290 | Restriction of a relation ... |
| relcoi2 6291 | Composition with the ident... |
| relcoi1 6292 | Composition with the ident... |
| unidmrn 6293 | The double union of the co... |
| relcnvfld 6294 | if ` R ` is a relation, it... |
| dfdm2 6295 | Alternate definition of do... |
| unixp 6296 | The double class union of ... |
| unixp0 6297 | A Cartesian product is emp... |
| unixpid 6298 | Field of a Cartesian squar... |
| ressn 6299 | Restriction of a class to ... |
| cnviin 6300 | The converse of an interse... |
| cnvpo 6301 | The converse of a partial ... |
| cnvso 6302 | The converse of a strict o... |
| xpco 6303 | Composition of two Cartesi... |
| xpcoid 6304 | Composition of two Cartesi... |
| elsnxp 6305 | Membership in a Cartesian ... |
| reu3op 6306 | There is a unique ordered ... |
| reuop 6307 | There is a unique ordered ... |
| opreu2reurex 6308 | There is a unique ordered ... |
| opreu2reu 6309 | If there is a unique order... |
| dfpo2 6310 | Quantifier-free definition... |
| csbcog 6311 | Distribute proper substitu... |
| snres0 6312 | Condition for restriction ... |
| imaindm 6313 | The image is unaffected by... |
| predeq123 6316 | Equality theorem for the p... |
| predeq1 6317 | Equality theorem for the p... |
| predeq2 6318 | Equality theorem for the p... |
| predeq3 6319 | Equality theorem for the p... |
| nfpred 6320 | Bound-variable hypothesis ... |
| csbpredg 6321 | Move class substitution in... |
| predpredss 6322 | If ` A ` is a subset of ` ... |
| predss 6323 | The predecessor class of `... |
| sspred 6324 | Another subset/predecessor... |
| dfpred2 6325 | An alternate definition of... |
| dfpred3 6326 | An alternate definition of... |
| dfpred3g 6327 | An alternate definition of... |
| elpredgg 6328 | Membership in a predecesso... |
| elpredg 6329 | Membership in a predecesso... |
| elpredimg 6330 | Membership in a predecesso... |
| elpredim 6331 | Membership in a predecesso... |
| elpred 6332 | Membership in a predecesso... |
| predexg 6333 | The predecessor class exis... |
| predasetexOLD 6334 | Obsolete form of ~ predexg... |
| dffr4 6335 | Alternate definition of we... |
| predel 6336 | Membership in the predeces... |
| predtrss 6337 | If ` R ` is transitive ove... |
| predpo 6338 | Property of the predecesso... |
| predso 6339 | Property of the predecesso... |
| setlikespec 6340 | If ` R ` is set-like in ` ... |
| predidm 6341 | Idempotent law for the pre... |
| predin 6342 | Intersection law for prede... |
| predun 6343 | Union law for predecessor ... |
| preddif 6344 | Difference law for predece... |
| predep 6345 | The predecessor under the ... |
| trpred 6346 | The class of predecessors ... |
| preddowncl 6347 | A property of classes that... |
| predpoirr 6348 | Given a partial ordering, ... |
| predfrirr 6349 | Given a well-founded relat... |
| pred0 6350 | The predecessor class over... |
| dfse3 6351 | Alternate definition of se... |
| predrelss 6352 | Subset carries from relati... |
| predprc 6353 | The predecessor of a prope... |
| predres 6354 | Predecessor class is unaff... |
| frpomin 6355 | Every nonempty (possibly p... |
| frpomin2 6356 | Every nonempty (possibly p... |
| frpoind 6357 | The principle of well-foun... |
| frpoinsg 6358 | Well-Founded Induction Sch... |
| frpoins2fg 6359 | Well-Founded Induction sch... |
| frpoins2g 6360 | Well-Founded Induction sch... |
| frpoins3g 6361 | Well-Founded Induction sch... |
| tz6.26 6362 | All nonempty subclasses of... |
| tz6.26OLD 6363 | Obsolete proof of ~ tz6.26... |
| tz6.26i 6364 | All nonempty subclasses of... |
| wfi 6365 | The Principle of Well-Orde... |
| wfiOLD 6366 | Obsolete proof of ~ wfi as... |
| wfii 6367 | The Principle of Well-Orde... |
| wfisg 6368 | Well-Ordered Induction Sch... |
| wfisgOLD 6369 | Obsolete version of ~ wfis... |
| wfis 6370 | Well-Ordered Induction Sch... |
| wfis2fg 6371 | Well-Ordered Induction Sch... |
| wfis2fgOLD 6372 | Obsolete version of ~ wfis... |
| wfis2f 6373 | Well-Ordered Induction sch... |
| wfis2g 6374 | Well-Ordered Induction Sch... |
| wfis2 6375 | Well-Ordered Induction sch... |
| wfis3 6376 | Well-Ordered Induction sch... |
| ordeq 6385 | Equality theorem for the o... |
| elong 6386 | An ordinal number is an or... |
| elon 6387 | An ordinal number is an or... |
| eloni 6388 | An ordinal number has the ... |
| elon2 6389 | An ordinal number is an or... |
| limeq 6390 | Equality theorem for the l... |
| ordwe 6391 | Membership well-orders eve... |
| ordtr 6392 | An ordinal class is transi... |
| ordfr 6393 | Membership is well-founded... |
| ordelss 6394 | An element of an ordinal c... |
| trssord 6395 | A transitive subclass of a... |
| ordirr 6396 | No ordinal class is a memb... |
| nordeq 6397 | A member of an ordinal cla... |
| ordn2lp 6398 | An ordinal class cannot be... |
| tz7.5 6399 | A nonempty subclass of an ... |
| ordelord 6400 | An element of an ordinal c... |
| tron 6401 | The class of all ordinal n... |
| ordelon 6402 | An element of an ordinal c... |
| onelon 6403 | An element of an ordinal n... |
| tz7.7 6404 | A transitive class belongs... |
| ordelssne 6405 | For ordinal classes, membe... |
| ordelpss 6406 | For ordinal classes, membe... |
| ordsseleq 6407 | For ordinal classes, inclu... |
| ordin 6408 | The intersection of two or... |
| onin 6409 | The intersection of two or... |
| ordtri3or 6410 | A trichotomy law for ordin... |
| ordtri1 6411 | A trichotomy law for ordin... |
| ontri1 6412 | A trichotomy law for ordin... |
| ordtri2 6413 | A trichotomy law for ordin... |
| ordtri3 6414 | A trichotomy law for ordin... |
| ordtri4 6415 | A trichotomy law for ordin... |
| orddisj 6416 | An ordinal class and its s... |
| onfr 6417 | The ordinal class is well-... |
| onelpss 6418 | Relationship between membe... |
| onsseleq 6419 | Relationship between subse... |
| onelss 6420 | An element of an ordinal n... |
| ordtr1 6421 | Transitive law for ordinal... |
| ordtr2 6422 | Transitive law for ordinal... |
| ordtr3 6423 | Transitive law for ordinal... |
| ontr1 6424 | Transitive law for ordinal... |
| ontr2 6425 | Transitive law for ordinal... |
| onelssex 6426 | Ordinal less than is equiv... |
| ordunidif 6427 | The union of an ordinal st... |
| ordintdif 6428 | If ` B ` is smaller than `... |
| onintss 6429 | If a property is true for ... |
| oneqmini 6430 | A way to show that an ordi... |
| ord0 6431 | The empty set is an ordina... |
| 0elon 6432 | The empty set is an ordina... |
| ord0eln0 6433 | A nonempty ordinal contain... |
| on0eln0 6434 | An ordinal number contains... |
| dflim2 6435 | An alternate definition of... |
| inton 6436 | The intersection of the cl... |
| nlim0 6437 | The empty set is not a lim... |
| limord 6438 | A limit ordinal is ordinal... |
| limuni 6439 | A limit ordinal is its own... |
| limuni2 6440 | The union of a limit ordin... |
| 0ellim 6441 | A limit ordinal contains t... |
| limelon 6442 | A limit ordinal class that... |
| onn0 6443 | The class of all ordinal n... |
| suceq 6444 | Equality of successors. (... |
| elsuci 6445 | Membership in a successor.... |
| elsucg 6446 | Membership in a successor.... |
| elsuc2g 6447 | Variant of membership in a... |
| elsuc 6448 | Membership in a successor.... |
| elsuc2 6449 | Membership in a successor.... |
| nfsuc 6450 | Bound-variable hypothesis ... |
| elelsuc 6451 | Membership in a successor.... |
| sucel 6452 | Membership of a successor ... |
| suc0 6453 | The successor of the empty... |
| sucprc 6454 | A proper class is its own ... |
| unisucs 6455 | The union of the successor... |
| unisucg 6456 | A transitive class is equa... |
| unisuc 6457 | A transitive class is equa... |
| sssucid 6458 | A class is included in its... |
| sucidg 6459 | Part of Proposition 7.23 o... |
| sucid 6460 | A set belongs to its succe... |
| nsuceq0 6461 | No successor is empty. (C... |
| eqelsuc 6462 | A set belongs to the succe... |
| iunsuc 6463 | Inductive definition for t... |
| suctr 6464 | The successor of a transit... |
| trsuc 6465 | A set whose successor belo... |
| trsucss 6466 | A member of the successor ... |
| ordsssuc 6467 | An ordinal is a subset of ... |
| onsssuc 6468 | A subset of an ordinal num... |
| ordsssuc2 6469 | An ordinal subset of an or... |
| onmindif 6470 | When its successor is subt... |
| ordnbtwn 6471 | There is no set between an... |
| onnbtwn 6472 | There is no set between an... |
| sucssel 6473 | A set whose successor is a... |
| orddif 6474 | Ordinal derived from its s... |
| orduniss 6475 | An ordinal class includes ... |
| ordtri2or 6476 | A trichotomy law for ordin... |
| ordtri2or2 6477 | A trichotomy law for ordin... |
| ordtri2or3 6478 | A consequence of total ord... |
| ordelinel 6479 | The intersection of two or... |
| ordssun 6480 | Property of a subclass of ... |
| ordequn 6481 | The maximum (i.e. union) o... |
| ordun 6482 | The maximum (i.e., union) ... |
| onunel 6483 | The union of two ordinals ... |
| ordunisssuc 6484 | A subclass relationship fo... |
| suc11 6485 | The successor operation be... |
| onun2 6486 | The union of two ordinals ... |
| ontr 6487 | An ordinal number is a tra... |
| onunisuc 6488 | An ordinal number is equal... |
| onordi 6489 | An ordinal number is an or... |
| ontrciOLD 6490 | Obsolete version of ~ ontr... |
| onirri 6491 | An ordinal number is not a... |
| oneli 6492 | A member of an ordinal num... |
| onelssi 6493 | A member of an ordinal num... |
| onssneli 6494 | An ordering law for ordina... |
| onssnel2i 6495 | An ordering law for ordina... |
| onelini 6496 | An element of an ordinal n... |
| oneluni 6497 | An ordinal number equals i... |
| onunisuci 6498 | An ordinal number is equal... |
| onsseli 6499 | Subset is equivalent to me... |
| onun2i 6500 | The union of two ordinal n... |
| unizlim 6501 | An ordinal equal to its ow... |
| on0eqel 6502 | An ordinal number either e... |
| snsn0non 6503 | The singleton of the singl... |
| onxpdisj 6504 | Ordinal numbers and ordere... |
| onnev 6505 | The class of ordinal numbe... |
| iotajust 6507 | Soundness justification th... |
| dfiota2 6509 | Alternate definition for d... |
| nfiota1 6510 | Bound-variable hypothesis ... |
| nfiotadw 6511 | Deduction version of ~ nfi... |
| nfiotaw 6512 | Bound-variable hypothesis ... |
| nfiotad 6513 | Deduction version of ~ nfi... |
| nfiota 6514 | Bound-variable hypothesis ... |
| cbviotaw 6515 | Change bound variables in ... |
| cbviotavw 6516 | Change bound variables in ... |
| cbviotavwOLD 6517 | Obsolete version of ~ cbvi... |
| cbviota 6518 | Change bound variables in ... |
| cbviotav 6519 | Change bound variables in ... |
| sb8iota 6520 | Variable substitution in d... |
| iotaeq 6521 | Equality theorem for descr... |
| iotabi 6522 | Equivalence theorem for de... |
| uniabio 6523 | Part of Theorem 8.17 in [Q... |
| iotaval2 6524 | Version of ~ iotaval using... |
| iotauni2 6525 | Version of ~ iotauni using... |
| iotanul2 6526 | Version of ~ iotanul using... |
| iotaval 6527 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6528 | The ` iota ` class is a su... |
| iotaex 6529 | Theorem 8.23 in [Quine] p.... |
| iotavalOLD 6530 | Obsolete version of ~ iota... |
| iotauni 6531 | Equivalence between two di... |
| iotaint 6532 | Equivalence between two di... |
| iota1 6533 | Property of iota. (Contri... |
| iotanul 6534 | Theorem 8.22 in [Quine] p.... |
| iotassuniOLD 6535 | Obsolete version of ~ iota... |
| iotaexOLD 6536 | Obsolete version of ~ iota... |
| iota4 6537 | Theorem *14.22 in [Whitehe... |
| iota4an 6538 | Theorem *14.23 in [Whitehe... |
| iota5 6539 | A method for computing iot... |
| iotabidv 6540 | Formula-building deduction... |
| iotabii 6541 | Formula-building deduction... |
| iotacl 6542 | Membership law for descrip... |
| iota2df 6543 | A condition that allows to... |
| iota2d 6544 | A condition that allows to... |
| iota2 6545 | The unique element such th... |
| iotan0 6546 | Representation of "the uni... |
| sniota 6547 | A class abstraction with a... |
| dfiota4 6548 | The ` iota ` operation usi... |
| csbiota 6549 | Class substitution within ... |
| dffun2 6566 | Alternate definition of a ... |
| dffun2OLD 6567 | Obsolete version of ~ dffu... |
| dffun2OLDOLD 6568 | Obsolete version of ~ dffu... |
| dffun6 6569 | Alternate definition of a ... |
| dffun3 6570 | Alternate definition of fu... |
| dffun3OLD 6571 | Obsolete version of ~ dffu... |
| dffun4 6572 | Alternate definition of a ... |
| dffun5 6573 | Alternate definition of fu... |
| dffun6f 6574 | Definition of function, us... |
| dffun6OLD 6575 | Obsolete version of ~ dffu... |
| funmo 6576 | A function has at most one... |
| funmoOLD 6577 | Obsolete version of ~ funm... |
| funrel 6578 | A function is a relation. ... |
| 0nelfun 6579 | A function does not contai... |
| funss 6580 | Subclass theorem for funct... |
| funeq 6581 | Equality theorem for funct... |
| funeqi 6582 | Equality inference for the... |
| funeqd 6583 | Equality deduction for the... |
| nffun 6584 | Bound-variable hypothesis ... |
| sbcfung 6585 | Distribute proper substitu... |
| funeu 6586 | There is exactly one value... |
| funeu2 6587 | There is exactly one value... |
| dffun7 6588 | Alternate definition of a ... |
| dffun8 6589 | Alternate definition of a ... |
| dffun9 6590 | Alternate definition of a ... |
| funfn 6591 | A class is a function if a... |
| funfnd 6592 | A function is a function o... |
| funi 6593 | The identity relation is a... |
| nfunv 6594 | The universal class is not... |
| funopg 6595 | A Kuratowski ordered pair ... |
| funopab 6596 | A class of ordered pairs i... |
| funopabeq 6597 | A class of ordered pairs o... |
| funopab4 6598 | A class of ordered pairs o... |
| funmpt 6599 | A function in maps-to nota... |
| funmpt2 6600 | Functionality of a class g... |
| funco 6601 | The composition of two fun... |
| funresfunco 6602 | Composition of two functio... |
| funres 6603 | A restriction of a functio... |
| funresd 6604 | A restriction of a functio... |
| funssres 6605 | The restriction of a funct... |
| fun2ssres 6606 | Equality of restrictions o... |
| funun 6607 | The union of functions wit... |
| fununmo 6608 | If the union of classes is... |
| fununfun 6609 | If the union of classes is... |
| fundif 6610 | A function with removed el... |
| funcnvsn 6611 | The converse singleton of ... |
| funsng 6612 | A singleton of an ordered ... |
| fnsng 6613 | Functionality and domain o... |
| funsn 6614 | A singleton of an ordered ... |
| funprg 6615 | A set of two pairs is a fu... |
| funtpg 6616 | A set of three pairs is a ... |
| funpr 6617 | A function with a domain o... |
| funtp 6618 | A function with a domain o... |
| fnsn 6619 | Functionality and domain o... |
| fnprg 6620 | Function with a domain of ... |
| fntpg 6621 | Function with a domain of ... |
| fntp 6622 | A function with a domain o... |
| funcnvpr 6623 | The converse pair of order... |
| funcnvtp 6624 | The converse triple of ord... |
| funcnvqp 6625 | The converse quadruple of ... |
| fun0 6626 | The empty set is a functio... |
| funcnv0 6627 | The converse of the empty ... |
| funcnvcnv 6628 | The double converse of a f... |
| funcnv2 6629 | A simpler equivalence for ... |
| funcnv 6630 | The converse of a class is... |
| funcnv3 6631 | A condition showing a clas... |
| fun2cnv 6632 | The double converse of a c... |
| svrelfun 6633 | A single-valued relation i... |
| fncnv 6634 | Single-rootedness (see ~ f... |
| fun11 6635 | Two ways of stating that `... |
| fununi 6636 | The union of a chain (with... |
| funin 6637 | The intersection with a fu... |
| funres11 6638 | The restriction of a one-t... |
| funcnvres 6639 | The converse of a restrict... |
| cnvresid 6640 | Converse of a restricted i... |
| funcnvres2 6641 | The converse of a restrict... |
| funimacnv 6642 | The image of the preimage ... |
| funimass1 6643 | A kind of contraposition l... |
| funimass2 6644 | A kind of contraposition l... |
| imadif 6645 | The image of a difference ... |
| imain 6646 | The image of an intersecti... |
| funimaexg 6647 | Axiom of Replacement using... |
| funimaexgOLD 6648 | Obsolete version of ~ funi... |
| funimaex 6649 | The image of a set under a... |
| isarep1 6650 | Part of a study of the Axi... |
| isarep1OLD 6651 | Obsolete version of ~ isar... |
| isarep2 6652 | Part of a study of the Axi... |
| fneq1 6653 | Equality theorem for funct... |
| fneq2 6654 | Equality theorem for funct... |
| fneq1d 6655 | Equality deduction for fun... |
| fneq2d 6656 | Equality deduction for fun... |
| fneq12d 6657 | Equality deduction for fun... |
| fneq12 6658 | Equality theorem for funct... |
| fneq1i 6659 | Equality inference for fun... |
| fneq2i 6660 | Equality inference for fun... |
| nffn 6661 | Bound-variable hypothesis ... |
| fnfun 6662 | A function with domain is ... |
| fnfund 6663 | A function with domain is ... |
| fnrel 6664 | A function with domain is ... |
| fndm 6665 | The domain of a function. ... |
| fndmi 6666 | The domain of a function. ... |
| fndmd 6667 | The domain of a function. ... |
| funfni 6668 | Inference to convert a fun... |
| fndmu 6669 | A function has a unique do... |
| fnbr 6670 | The first argument of bina... |
| fnop 6671 | The first argument of an o... |
| fneu 6672 | There is exactly one value... |
| fneu2 6673 | There is exactly one value... |
| fnunres1 6674 | Restriction of a disjoint ... |
| fnunres2 6675 | Restriction of a disjoint ... |
| fnun 6676 | The union of two functions... |
| fnund 6677 | The union of two functions... |
| fnunop 6678 | Extension of a function wi... |
| fncofn 6679 | Composition of a function ... |
| fnco 6680 | Composition of two functio... |
| fncoOLD 6681 | Obsolete version of ~ fnco... |
| fnresdm 6682 | A function does not change... |
| fnresdisj 6683 | A function restricted to a... |
| 2elresin 6684 | Membership in two function... |
| fnssresb 6685 | Restriction of a function ... |
| fnssres 6686 | Restriction of a function ... |
| fnssresd 6687 | Restriction of a function ... |
| fnresin1 6688 | Restriction of a function'... |
| fnresin2 6689 | Restriction of a function'... |
| fnres 6690 | An equivalence for functio... |
| idfn 6691 | The identity relation is a... |
| fnresi 6692 | The restricted identity re... |
| fnima 6693 | The image of a function's ... |
| fn0 6694 | A function with empty doma... |
| fnimadisj 6695 | A class that is disjoint w... |
| fnimaeq0 6696 | Images under a function ne... |
| dfmpt3 6697 | Alternate definition for t... |
| mptfnf 6698 | The maps-to notation defin... |
| fnmptf 6699 | The maps-to notation defin... |
| fnopabg 6700 | Functionality and domain o... |
| fnopab 6701 | Functionality and domain o... |
| mptfng 6702 | The maps-to notation defin... |
| fnmpt 6703 | The maps-to notation defin... |
| fnmptd 6704 | The maps-to notation defin... |
| mpt0 6705 | A mapping operation with e... |
| fnmpti 6706 | Functionality and domain o... |
| dmmpti 6707 | Domain of the mapping oper... |
| dmmptd 6708 | The domain of the mapping ... |
| mptun 6709 | Union of mappings which ar... |
| partfun 6710 | Rewrite a function defined... |
| feq1 6711 | Equality theorem for funct... |
| feq2 6712 | Equality theorem for funct... |
| feq3 6713 | Equality theorem for funct... |
| feq23 6714 | Equality theorem for funct... |
| feq1d 6715 | Equality deduction for fun... |
| feq2d 6716 | Equality deduction for fun... |
| feq3d 6717 | Equality deduction for fun... |
| feq12d 6718 | Equality deduction for fun... |
| feq123d 6719 | Equality deduction for fun... |
| feq123 6720 | Equality theorem for funct... |
| feq1i 6721 | Equality inference for fun... |
| feq2i 6722 | Equality inference for fun... |
| feq12i 6723 | Equality inference for fun... |
| feq23i 6724 | Equality inference for fun... |
| feq23d 6725 | Equality deduction for fun... |
| nff 6726 | Bound-variable hypothesis ... |
| sbcfng 6727 | Distribute proper substitu... |
| sbcfg 6728 | Distribute proper substitu... |
| elimf 6729 | Eliminate a mapping hypoth... |
| ffn 6730 | A mapping is a function wi... |
| ffnd 6731 | A mapping is a function wi... |
| dffn2 6732 | Any function is a mapping ... |
| ffun 6733 | A mapping is a function. ... |
| ffund 6734 | A mapping is a function, d... |
| frel 6735 | A mapping is a relation. ... |
| freld 6736 | A mapping is a relation. ... |
| frn 6737 | The range of a mapping. (... |
| frnd 6738 | Deduction form of ~ frn . ... |
| fdm 6739 | The domain of a mapping. ... |
| fdmd 6740 | Deduction form of ~ fdm . ... |
| fdmi 6741 | Inference associated with ... |
| dffn3 6742 | A function maps to its ran... |
| ffrn 6743 | A function maps to its ran... |
| ffrnb 6744 | Characterization of a func... |
| ffrnbd 6745 | A function maps to its ran... |
| fss 6746 | Expanding the codomain of ... |
| fssd 6747 | Expanding the codomain of ... |
| fssdmd 6748 | Expressing that a class is... |
| fssdm 6749 | Expressing that a class is... |
| fimass 6750 | The image of a class under... |
| fimassd 6751 | The image of a class is a ... |
| fimacnv 6752 | The preimage of the codoma... |
| fcof 6753 | Composition of a function ... |
| fco 6754 | Composition of two functio... |
| fcoOLD 6755 | Obsolete version of ~ fco ... |
| fcod 6756 | Composition of two mapping... |
| fco2 6757 | Functionality of a composi... |
| fssxp 6758 | A mapping is a class of or... |
| funssxp 6759 | Two ways of specifying a p... |
| ffdm 6760 | A mapping is a partial fun... |
| ffdmd 6761 | The domain of a function. ... |
| fdmrn 6762 | A different way to write `... |
| funcofd 6763 | Composition of two functio... |
| fco3OLD 6764 | Obsolete version of ~ func... |
| opelf 6765 | The members of an ordered ... |
| fun 6766 | The union of two functions... |
| fun2 6767 | The union of two functions... |
| fun2d 6768 | The union of functions wit... |
| fnfco 6769 | Composition of two functio... |
| fssres 6770 | Restriction of a function ... |
| fssresd 6771 | Restriction of a function ... |
| fssres2 6772 | Restriction of a restricte... |
| fresin 6773 | An identity for the mappin... |
| resasplit 6774 | If two functions agree on ... |
| fresaun 6775 | The union of two functions... |
| fresaunres2 6776 | From the union of two func... |
| fresaunres1 6777 | From the union of two func... |
| fcoi1 6778 | Composition of a mapping a... |
| fcoi2 6779 | Composition of restricted ... |
| feu 6780 | There is exactly one value... |
| fcnvres 6781 | The converse of a restrict... |
| fimacnvdisj 6782 | The preimage of a class di... |
| fint 6783 | Function into an intersect... |
| fin 6784 | Mapping into an intersecti... |
| f0 6785 | The empty function. (Cont... |
| f00 6786 | A class is a function with... |
| f0bi 6787 | A function with empty doma... |
| f0dom0 6788 | A function is empty iff it... |
| f0rn0 6789 | If there is no element in ... |
| fconst 6790 | A Cartesian product with a... |
| fconstg 6791 | A Cartesian product with a... |
| fnconstg 6792 | A Cartesian product with a... |
| fconst6g 6793 | Constant function with loo... |
| fconst6 6794 | A constant function as a m... |
| f1eq1 6795 | Equality theorem for one-t... |
| f1eq2 6796 | Equality theorem for one-t... |
| f1eq3 6797 | Equality theorem for one-t... |
| nff1 6798 | Bound-variable hypothesis ... |
| dff12 6799 | Alternate definition of a ... |
| f1f 6800 | A one-to-one mapping is a ... |
| f1fn 6801 | A one-to-one mapping is a ... |
| f1fun 6802 | A one-to-one mapping is a ... |
| f1rel 6803 | A one-to-one onto mapping ... |
| f1dm 6804 | The domain of a one-to-one... |
| f1ss 6805 | A function that is one-to-... |
| f1ssr 6806 | A function that is one-to-... |
| f1ssres 6807 | A function that is one-to-... |
| f1resf1 6808 | The restriction of an inje... |
| f1cnvcnv 6809 | Two ways to express that a... |
| f1cof1 6810 | Composition of two one-to-... |
| f1co 6811 | Composition of one-to-one ... |
| f1coOLD 6812 | Obsolete version of ~ f1co... |
| foeq1 6813 | Equality theorem for onto ... |
| foeq2 6814 | Equality theorem for onto ... |
| foeq3 6815 | Equality theorem for onto ... |
| nffo 6816 | Bound-variable hypothesis ... |
| fof 6817 | An onto mapping is a mappi... |
| fofun 6818 | An onto mapping is a funct... |
| fofn 6819 | An onto mapping is a funct... |
| forn 6820 | The codomain of an onto fu... |
| dffo2 6821 | Alternate definition of an... |
| foima 6822 | The image of the domain of... |
| dffn4 6823 | A function maps onto its r... |
| funforn 6824 | A function maps its domain... |
| fodmrnu 6825 | An onto function has uniqu... |
| fimadmfo 6826 | A function is a function o... |
| fores 6827 | Restriction of an onto fun... |
| fimadmfoALT 6828 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6829 | The preimage of the codoma... |
| focofo 6830 | Composition of onto functi... |
| foco 6831 | Composition of onto functi... |
| foconst 6832 | A nonzero constant functio... |
| f1oeq1 6833 | Equality theorem for one-t... |
| f1oeq2 6834 | Equality theorem for one-t... |
| f1oeq3 6835 | Equality theorem for one-t... |
| f1oeq23 6836 | Equality theorem for one-t... |
| f1eq123d 6837 | Equality deduction for one... |
| foeq123d 6838 | Equality deduction for ont... |
| f1oeq123d 6839 | Equality deduction for one... |
| f1oeq1d 6840 | Equality deduction for one... |
| f1oeq2d 6841 | Equality deduction for one... |
| f1oeq3d 6842 | Equality deduction for one... |
| nff1o 6843 | Bound-variable hypothesis ... |
| f1of1 6844 | A one-to-one onto mapping ... |
| f1of 6845 | A one-to-one onto mapping ... |
| f1ofn 6846 | A one-to-one onto mapping ... |
| f1ofun 6847 | A one-to-one onto mapping ... |
| f1orel 6848 | A one-to-one onto mapping ... |
| f1odm 6849 | The domain of a one-to-one... |
| dff1o2 6850 | Alternate definition of on... |
| dff1o3 6851 | Alternate definition of on... |
| f1ofo 6852 | A one-to-one onto function... |
| dff1o4 6853 | Alternate definition of on... |
| dff1o5 6854 | Alternate definition of on... |
| f1orn 6855 | A one-to-one function maps... |
| f1f1orn 6856 | A one-to-one function maps... |
| f1ocnv 6857 | The converse of a one-to-o... |
| f1ocnvb 6858 | A relation is a one-to-one... |
| f1ores 6859 | The restriction of a one-t... |
| f1orescnv 6860 | The converse of a one-to-o... |
| f1imacnv 6861 | Preimage of an image. (Co... |
| foimacnv 6862 | A reverse version of ~ f1i... |
| foun 6863 | The union of two onto func... |
| f1oun 6864 | The union of two one-to-on... |
| f1un 6865 | The union of two one-to-on... |
| resdif 6866 | The restriction of a one-t... |
| resin 6867 | The restriction of a one-t... |
| f1oco 6868 | Composition of one-to-one ... |
| f1cnv 6869 | The converse of an injecti... |
| funcocnv2 6870 | Composition with the conve... |
| fococnv2 6871 | The composition of an onto... |
| f1ococnv2 6872 | The composition of a one-t... |
| f1cocnv2 6873 | Composition of an injectiv... |
| f1ococnv1 6874 | The composition of a one-t... |
| f1cocnv1 6875 | Composition of an injectiv... |
| funcoeqres 6876 | Express a constraint on a ... |
| f1ssf1 6877 | A subset of an injective f... |
| f10 6878 | The empty set maps one-to-... |
| f10d 6879 | The empty set maps one-to-... |
| f1o00 6880 | One-to-one onto mapping of... |
| fo00 6881 | Onto mapping of the empty ... |
| f1o0 6882 | One-to-one onto mapping of... |
| f1oi 6883 | A restriction of the ident... |
| f1ovi 6884 | The identity relation is a... |
| f1osn 6885 | A singleton of an ordered ... |
| f1osng 6886 | A singleton of an ordered ... |
| f1sng 6887 | A singleton of an ordered ... |
| fsnd 6888 | A singleton of an ordered ... |
| f1oprswap 6889 | A two-element swap is a bi... |
| f1oprg 6890 | An unordered pair of order... |
| tz6.12-2 6891 | Function value when ` F ` ... |
| fveu 6892 | The value of a function at... |
| brprcneu 6893 | If ` A ` is a proper class... |
| brprcneuALT 6894 | Alternate proof of ~ brprc... |
| fvprc 6895 | A function's value at a pr... |
| fvprcALT 6896 | Alternate proof of ~ fvprc... |
| rnfvprc 6897 | The range of a function va... |
| fv2 6898 | Alternate definition of fu... |
| dffv3 6899 | A definition of function v... |
| dffv4 6900 | The previous definition of... |
| elfv 6901 | Membership in a function v... |
| fveq1 6902 | Equality theorem for funct... |
| fveq2 6903 | Equality theorem for funct... |
| fveq1i 6904 | Equality inference for fun... |
| fveq1d 6905 | Equality deduction for fun... |
| fveq2i 6906 | Equality inference for fun... |
| fveq2d 6907 | Equality deduction for fun... |
| 2fveq3 6908 | Equality theorem for neste... |
| fveq12i 6909 | Equality deduction for fun... |
| fveq12d 6910 | Equality deduction for fun... |
| fveqeq2d 6911 | Equality deduction for fun... |
| fveqeq2 6912 | Equality deduction for fun... |
| nffv 6913 | Bound-variable hypothesis ... |
| nffvmpt1 6914 | Bound-variable hypothesis ... |
| nffvd 6915 | Deduction version of bound... |
| fvex 6916 | The value of a class exist... |
| fvexi 6917 | The value of a class exist... |
| fvexd 6918 | The value of a class exist... |
| fvif 6919 | Move a conditional outside... |
| iffv 6920 | Move a conditional outside... |
| fv3 6921 | Alternate definition of th... |
| fvres 6922 | The value of a restricted ... |
| fvresd 6923 | The value of a restricted ... |
| funssfv 6924 | The value of a member of t... |
| tz6.12c 6925 | Corollary of Theorem 6.12(... |
| tz6.12-1 6926 | Function value. Theorem 6... |
| tz6.12-1OLD 6927 | Obsolete version of ~ tz6.... |
| tz6.12 6928 | Function value. Theorem 6... |
| tz6.12f 6929 | Function value, using boun... |
| tz6.12cOLD 6930 | Obsolete version of ~ tz6.... |
| tz6.12i 6931 | Corollary of Theorem 6.12(... |
| fvbr0 6932 | Two possibilities for the ... |
| fvrn0 6933 | A function value is a memb... |
| fvn0fvelrn 6934 | If the value of a function... |
| elfvunirn 6935 | A function value is a subs... |
| fvssunirn 6936 | The result of a function v... |
| fvssunirnOLD 6937 | Obsolete version of ~ fvss... |
| ndmfv 6938 | The value of a class outsi... |
| ndmfvrcl 6939 | Reverse closure law for fu... |
| elfvdm 6940 | If a function value has a ... |
| elfvex 6941 | If a function value has a ... |
| elfvexd 6942 | If a function value has a ... |
| eliman0 6943 | A nonempty function value ... |
| nfvres 6944 | The value of a non-member ... |
| nfunsn 6945 | If the restriction of a cl... |
| fvfundmfvn0 6946 | If the "value of a class" ... |
| 0fv 6947 | Function value of the empt... |
| fv2prc 6948 | A function value of a func... |
| elfv2ex 6949 | If a function value of a f... |
| fveqres 6950 | Equal values imply equal v... |
| csbfv12 6951 | Move class substitution in... |
| csbfv2g 6952 | Move class substitution in... |
| csbfv 6953 | Substitution for a functio... |
| funbrfv 6954 | The second argument of a b... |
| funopfv 6955 | The second element in an o... |
| fnbrfvb 6956 | Equivalence of function va... |
| fnopfvb 6957 | Equivalence of function va... |
| funbrfvb 6958 | Equivalence of function va... |
| funopfvb 6959 | Equivalence of function va... |
| fnbrfvb2 6960 | Version of ~ fnbrfvb for f... |
| fdmeu 6961 | There is exactly one codom... |
| funbrfv2b 6962 | Function value in terms of... |
| dffn5 6963 | Representation of a functi... |
| fnrnfv 6964 | The range of a function ex... |
| fvelrnb 6965 | A member of a function's r... |
| foelcdmi 6966 | A member of a surjective f... |
| dfimafn 6967 | Alternate definition of th... |
| dfimafn2 6968 | Alternate definition of th... |
| funimass4 6969 | Membership relation for th... |
| fvelima 6970 | Function value in an image... |
| funimassd 6971 | Sufficient condition for t... |
| fvelimad 6972 | Function value in an image... |
| feqmptd 6973 | Deduction form of ~ dffn5 ... |
| feqresmpt 6974 | Express a restricted funct... |
| feqmptdf 6975 | Deduction form of ~ dffn5f... |
| dffn5f 6976 | Representation of a functi... |
| fvelimab 6977 | Function value in an image... |
| fvelimabd 6978 | Deduction form of ~ fvelim... |
| unima 6979 | Image of a union. (Contri... |
| fvi 6980 | The value of the identity ... |
| fviss 6981 | The value of the identity ... |
| fniinfv 6982 | The indexed intersection o... |
| fnsnfv 6983 | Singleton of function valu... |
| opabiotafun 6984 | Define a function whose va... |
| opabiotadm 6985 | Define a function whose va... |
| opabiota 6986 | Define a function whose va... |
| fnimapr 6987 | The image of a pair under ... |
| fnimatpd 6988 | The image of an unordered ... |
| ssimaex 6989 | The existence of a subimag... |
| ssimaexg 6990 | The existence of a subimag... |
| funfv 6991 | A simplified expression fo... |
| funfv2 6992 | The value of a function. ... |
| funfv2f 6993 | The value of a function. ... |
| fvun 6994 | Value of the union of two ... |
| fvun1 6995 | The value of a union when ... |
| fvun2 6996 | The value of a union when ... |
| fvun1d 6997 | The value of a union when ... |
| fvun2d 6998 | The value of a union when ... |
| dffv2 6999 | Alternate definition of fu... |
| dmfco 7000 | Domains of a function comp... |
| fvco2 7001 | Value of a function compos... |
| fvco 7002 | Value of a function compos... |
| fvco3 7003 | Value of a function compos... |
| fvco3d 7004 | Value of a function compos... |
| fvco4i 7005 | Conditions for a compositi... |
| fvopab3g 7006 | Value of a function given ... |
| fvopab3ig 7007 | Value of a function given ... |
| brfvopabrbr 7008 | The binary relation of a f... |
| fvmptg 7009 | Value of a function given ... |
| fvmpti 7010 | Value of a function given ... |
| fvmpt 7011 | Value of a function given ... |
| fvmpt2f 7012 | Value of a function given ... |
| fvtresfn 7013 | Functionality of a tuple-r... |
| fvmpts 7014 | Value of a function given ... |
| fvmpt3 7015 | Value of a function given ... |
| fvmpt3i 7016 | Value of a function given ... |
| fvmptdf 7017 | Deduction version of ~ fvm... |
| fvmptd 7018 | Deduction version of ~ fvm... |
| fvmptd2 7019 | Deduction version of ~ fvm... |
| mptrcl 7020 | Reverse closure for a mapp... |
| fvmpt2i 7021 | Value of a function given ... |
| fvmpt2 7022 | Value of a function given ... |
| fvmptss 7023 | If all the values of the m... |
| fvmpt2d 7024 | Deduction version of ~ fvm... |
| fvmptex 7025 | Express a function ` F ` w... |
| fvmptd3f 7026 | Alternate deduction versio... |
| fvmptd2f 7027 | Alternate deduction versio... |
| fvmptdv 7028 | Alternate deduction versio... |
| fvmptdv2 7029 | Alternate deduction versio... |
| mpteqb 7030 | Bidirectional equality the... |
| fvmptt 7031 | Closed theorem form of ~ f... |
| fvmptf 7032 | Value of a function given ... |
| fvmptnf 7033 | The value of a function gi... |
| fvmptd3 7034 | Deduction version of ~ fvm... |
| fvmptd4 7035 | Deduction version of ~ fvm... |
| fvmptn 7036 | This somewhat non-intuitiv... |
| fvmptss2 7037 | A mapping always evaluates... |
| elfvmptrab1w 7038 | Implications for the value... |
| elfvmptrab1 7039 | Implications for the value... |
| elfvmptrab 7040 | Implications for the value... |
| fvopab4ndm 7041 | Value of a function given ... |
| fvmptndm 7042 | Value of a function given ... |
| fvmptrabfv 7043 | Value of a function mappin... |
| fvopab5 7044 | The value of a function th... |
| fvopab6 7045 | Value of a function given ... |
| eqfnfv 7046 | Equality of functions is d... |
| eqfnfv2 7047 | Equality of functions is d... |
| eqfnfv3 7048 | Derive equality of functio... |
| eqfnfvd 7049 | Deduction for equality of ... |
| eqfnfv2f 7050 | Equality of functions is d... |
| eqfunfv 7051 | Equality of functions is d... |
| eqfnun 7052 | Two functions on ` A u. B ... |
| fvreseq0 7053 | Equality of restricted fun... |
| fvreseq1 7054 | Equality of a function res... |
| fvreseq 7055 | Equality of restricted fun... |
| fnmptfvd 7056 | A function with a given do... |
| fndmdif 7057 | Two ways to express the lo... |
| fndmdifcom 7058 | The difference set between... |
| fndmdifeq0 7059 | The difference set of two ... |
| fndmin 7060 | Two ways to express the lo... |
| fneqeql 7061 | Two functions are equal if... |
| fneqeql2 7062 | Two functions are equal if... |
| fnreseql 7063 | Two functions are equal on... |
| chfnrn 7064 | The range of a choice func... |
| funfvop 7065 | Ordered pair with function... |
| funfvbrb 7066 | Two ways to say that ` A `... |
| fvimacnvi 7067 | A member of a preimage is ... |
| fvimacnv 7068 | The argument of a function... |
| funimass3 7069 | A kind of contraposition l... |
| funimass5 7070 | A subclass of a preimage i... |
| funconstss 7071 | Two ways of specifying tha... |
| fvimacnvALT 7072 | Alternate proof of ~ fvima... |
| elpreima 7073 | Membership in the preimage... |
| elpreimad 7074 | Membership in the preimage... |
| fniniseg 7075 | Membership in the preimage... |
| fncnvima2 7076 | Inverse images under funct... |
| fniniseg2 7077 | Inverse point images under... |
| unpreima 7078 | Preimage of a union. (Con... |
| inpreima 7079 | Preimage of an intersectio... |
| difpreima 7080 | Preimage of a difference. ... |
| respreima 7081 | The preimage of a restrict... |
| cnvimainrn 7082 | The preimage of the inters... |
| sspreima 7083 | The preimage of a subset i... |
| iinpreima 7084 | Preimage of an intersectio... |
| intpreima 7085 | Preimage of an intersectio... |
| fimacnvOLD 7086 | Obsolete version of ~ fima... |
| fimacnvinrn 7087 | Taking the converse image ... |
| fimacnvinrn2 7088 | Taking the converse image ... |
| rescnvimafod 7089 | The restriction of a funct... |
| fvn0ssdmfun 7090 | If a class' function value... |
| fnopfv 7091 | Ordered pair with function... |
| fvelrn 7092 | A function's value belongs... |
| nelrnfvne 7093 | A function value cannot be... |
| fveqdmss 7094 | If the empty set is not co... |
| fveqressseq 7095 | If the empty set is not co... |
| fnfvelrn 7096 | A function's value belongs... |
| ffvelcdm 7097 | A function's value belongs... |
| fnfvelrnd 7098 | A function's value belongs... |
| ffvelcdmi 7099 | A function's value belongs... |
| ffvelcdmda 7100 | A function's value belongs... |
| ffvelcdmd 7101 | A function's value belongs... |
| feldmfvelcdm 7102 | A class is an element of t... |
| rexrn 7103 | Restricted existential qua... |
| ralrn 7104 | Restricted universal quant... |
| elrnrexdm 7105 | For any element in the ran... |
| elrnrexdmb 7106 | For any element in the ran... |
| eldmrexrn 7107 | For any element in the dom... |
| eldmrexrnb 7108 | For any element in the dom... |
| fvcofneq 7109 | The values of two function... |
| ralrnmptw 7110 | A restricted quantifier ov... |
| rexrnmptw 7111 | A restricted quantifier ov... |
| ralrnmpt 7112 | A restricted quantifier ov... |
| rexrnmpt 7113 | A restricted quantifier ov... |
| f0cli 7114 | Unconditional closure of a... |
| dff2 7115 | Alternate definition of a ... |
| dff3 7116 | Alternate definition of a ... |
| dff4 7117 | Alternate definition of a ... |
| dffo3 7118 | An onto mapping expressed ... |
| dffo4 7119 | Alternate definition of an... |
| dffo5 7120 | Alternate definition of an... |
| exfo 7121 | A relation equivalent to t... |
| dffo3f 7122 | An onto mapping expressed ... |
| foelrn 7123 | Property of a surjective f... |
| foelrnf 7124 | Property of a surjective f... |
| foco2 7125 | If a composition of two fu... |
| fmpt 7126 | Functionality of the mappi... |
| f1ompt 7127 | Express bijection for a ma... |
| fmpti 7128 | Functionality of the mappi... |
| fvmptelcdm 7129 | The value of a function at... |
| fmptd 7130 | Domain and codomain of the... |
| fmpttd 7131 | Version of ~ fmptd with in... |
| fmpt3d 7132 | Domain and codomain of the... |
| fmptdf 7133 | A version of ~ fmptd using... |
| fompt 7134 | Express being onto for a m... |
| ffnfv 7135 | A function maps to a class... |
| ffnfvf 7136 | A function maps to a class... |
| fnfvrnss 7137 | An upper bound for range d... |
| fcdmssb 7138 | A function is a function i... |
| rnmptss 7139 | The range of an operation ... |
| fmpt2d 7140 | Domain and codomain of the... |
| ffvresb 7141 | A necessary and sufficient... |
| fssrescdmd 7142 | Restriction of a function ... |
| f1oresrab 7143 | Build a bijection between ... |
| f1ossf1o 7144 | Restricting a bijection, w... |
| fmptco 7145 | Composition of two functio... |
| fmptcof 7146 | Version of ~ fmptco where ... |
| fmptcos 7147 | Composition of two functio... |
| cofmpt 7148 | Express composition of a m... |
| fcompt 7149 | Express composition of two... |
| fcoconst 7150 | Composition with a constan... |
| fsn 7151 | A function maps a singleto... |
| fsn2 7152 | A function that maps a sin... |
| fsng 7153 | A function maps a singleto... |
| fsn2g 7154 | A function that maps a sin... |
| xpsng 7155 | The Cartesian product of t... |
| xpprsng 7156 | The Cartesian product of a... |
| xpsn 7157 | The Cartesian product of t... |
| f1o2sn 7158 | A singleton consisting in ... |
| residpr 7159 | Restriction of the identit... |
| dfmpt 7160 | Alternate definition for t... |
| fnasrn 7161 | A function expressed as th... |
| idref 7162 | Two ways to state that a r... |
| funiun 7163 | A function is a union of s... |
| funopsn 7164 | If a function is an ordere... |
| funop 7165 | An ordered pair is a funct... |
| funopdmsn 7166 | The domain of a function w... |
| funsndifnop 7167 | A singleton of an ordered ... |
| funsneqopb 7168 | A singleton of an ordered ... |
| ressnop0 7169 | If ` A ` is not in ` C ` ,... |
| fpr 7170 | A function with a domain o... |
| fprg 7171 | A function with a domain o... |
| ftpg 7172 | A function with a domain o... |
| ftp 7173 | A function with a domain o... |
| fnressn 7174 | A function restricted to a... |
| funressn 7175 | A function restricted to a... |
| fressnfv 7176 | The value of a function re... |
| fvrnressn 7177 | If the value of a function... |
| fvressn 7178 | The value of a function re... |
| fvn0fvelrnOLD 7179 | Obsolete version of ~ fvn0... |
| fvconst 7180 | The value of a constant fu... |
| fnsnr 7181 | If a class belongs to a fu... |
| fnsnb 7182 | A function whose domain is... |
| fmptsn 7183 | Express a singleton functi... |
| fmptsng 7184 | Express a singleton functi... |
| fmptsnd 7185 | Express a singleton functi... |
| fmptap 7186 | Append an additional value... |
| fmptapd 7187 | Append an additional value... |
| fmptpr 7188 | Express a pair function in... |
| fvresi 7189 | The value of a restricted ... |
| fninfp 7190 | Express the class of fixed... |
| fnelfp 7191 | Property of a fixed point ... |
| fndifnfp 7192 | Express the class of non-f... |
| fnelnfp 7193 | Property of a non-fixed po... |
| fnnfpeq0 7194 | A function is the identity... |
| fvunsn 7195 | Remove an ordered pair not... |
| fvsng 7196 | The value of a singleton o... |
| fvsn 7197 | The value of a singleton o... |
| fvsnun1 7198 | The value of a function wi... |
| fvsnun2 7199 | The value of a function wi... |
| fnsnsplit 7200 | Split a function into a si... |
| fsnunf 7201 | Adjoining a point to a fun... |
| fsnunf2 7202 | Adjoining a point to a pun... |
| fsnunfv 7203 | Recover the added point fr... |
| fsnunres 7204 | Recover the original funct... |
| funresdfunsn 7205 | Restricting a function to ... |
| fvpr1g 7206 | The value of a function wi... |
| fvpr2g 7207 | The value of a function wi... |
| fvpr2gOLD 7208 | Obsolete version of ~ fvpr... |
| fvpr1 7209 | The value of a function wi... |
| fvpr1OLD 7210 | Obsolete version of ~ fvpr... |
| fvpr2 7211 | The value of a function wi... |
| fvpr2OLD 7212 | Obsolete version of ~ fvpr... |
| fprb 7213 | A condition for functionho... |
| fvtp1 7214 | The first value of a funct... |
| fvtp2 7215 | The second value of a func... |
| fvtp3 7216 | The third value of a funct... |
| fvtp1g 7217 | The value of a function wi... |
| fvtp2g 7218 | The value of a function wi... |
| fvtp3g 7219 | The value of a function wi... |
| tpres 7220 | An unordered triple of ord... |
| fvconst2g 7221 | The value of a constant fu... |
| fconst2g 7222 | A constant function expres... |
| fvconst2 7223 | The value of a constant fu... |
| fconst2 7224 | A constant function expres... |
| fconst5 7225 | Two ways to express that a... |
| rnmptc 7226 | Range of a constant functi... |
| fnprb 7227 | A function whose domain ha... |
| fntpb 7228 | A function whose domain ha... |
| fnpr2g 7229 | A function whose domain ha... |
| fpr2g 7230 | A function that maps a pai... |
| fconstfv 7231 | A constant function expres... |
| fconst3 7232 | Two ways to express a cons... |
| fconst4 7233 | Two ways to express a cons... |
| resfunexg 7234 | The restriction of a funct... |
| resiexd 7235 | The restriction of the ide... |
| fnex 7236 | If the domain of a functio... |
| fnexd 7237 | If the domain of a functio... |
| funex 7238 | If the domain of a functio... |
| opabex 7239 | Existence of a function ex... |
| mptexg 7240 | If the domain of a functio... |
| mptexgf 7241 | If the domain of a functio... |
| mptex 7242 | If the domain of a functio... |
| mptexd 7243 | If the domain of a functio... |
| mptrabex 7244 | If the domain of a functio... |
| fex 7245 | If the domain of a mapping... |
| fexd 7246 | If the domain of a mapping... |
| mptfvmpt 7247 | A function in maps-to nota... |
| eufnfv 7248 | A function is uniquely det... |
| funfvima 7249 | A function's value in a pr... |
| funfvima2 7250 | A function's value in an i... |
| funfvima2d 7251 | A function's value in a pr... |
| fnfvima 7252 | The function value of an o... |
| fnfvimad 7253 | A function's value belongs... |
| resfvresima 7254 | The value of the function ... |
| funfvima3 7255 | A class including a functi... |
| rexima 7256 | Existential quantification... |
| ralima 7257 | Universal quantification u... |
| fvclss 7258 | Upper bound for the class ... |
| elabrex 7259 | Elementhood in an image se... |
| elabrexg 7260 | Elementhood in an image se... |
| abrexco 7261 | Composition of two image m... |
| imaiun 7262 | The image of an indexed un... |
| imauni 7263 | The image of a union is th... |
| fniunfv 7264 | The indexed union of a fun... |
| funiunfv 7265 | The indexed union of a fun... |
| funiunfvf 7266 | The indexed union of a fun... |
| eluniima 7267 | Membership in the union of... |
| elunirn 7268 | Membership in the union of... |
| elunirnALT 7269 | Alternate proof of ~ eluni... |
| elunirn2OLD 7270 | Obsolete version of ~ elfv... |
| fnunirn 7271 | Membership in a union of s... |
| dff13 7272 | A one-to-one function in t... |
| dff13f 7273 | A one-to-one function in t... |
| f1veqaeq 7274 | If the values of a one-to-... |
| f1cofveqaeq 7275 | If the values of a composi... |
| f1cofveqaeqALT 7276 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7277 | If two one-to-one function... |
| f1mpt 7278 | Express injection for a ma... |
| f1fveq 7279 | Equality of function value... |
| f1elima 7280 | Membership in the image of... |
| f1imass 7281 | Taking images under a one-... |
| f1imaeq 7282 | Taking images under a one-... |
| f1imapss 7283 | Taking images under a one-... |
| fpropnf1 7284 | A function, given by an un... |
| f1dom3fv3dif 7285 | The function values for a ... |
| f1dom3el3dif 7286 | The codomain of a 1-1 func... |
| dff14a 7287 | A one-to-one function in t... |
| dff14b 7288 | A one-to-one function in t... |
| f12dfv 7289 | A one-to-one function with... |
| f13dfv 7290 | A one-to-one function with... |
| dff1o6 7291 | A one-to-one onto function... |
| f1ocnvfv1 7292 | The converse value of the ... |
| f1ocnvfv2 7293 | The value of the converse ... |
| f1ocnvfv 7294 | Relationship between the v... |
| f1ocnvfvb 7295 | Relationship between the v... |
| nvof1o 7296 | An involution is a bijecti... |
| nvocnv 7297 | The converse of an involut... |
| f1cdmsn 7298 | If a one-to-one function w... |
| fsnex 7299 | Relate a function with a s... |
| f1prex 7300 | Relate a one-to-one functi... |
| f1ocnvdm 7301 | The value of the converse ... |
| f1ocnvfvrneq 7302 | If the values of a one-to-... |
| fcof1 7303 | An application is injectiv... |
| fcofo 7304 | An application is surjecti... |
| cbvfo 7305 | Change bound variable betw... |
| cbvexfo 7306 | Change bound variable betw... |
| cocan1 7307 | An injection is left-cance... |
| cocan2 7308 | A surjection is right-canc... |
| fcof1oinvd 7309 | Show that a function is th... |
| fcof1od 7310 | A function is bijective if... |
| 2fcoidinvd 7311 | Show that a function is th... |
| fcof1o 7312 | Show that two functions ar... |
| 2fvcoidd 7313 | Show that the composition ... |
| 2fvidf1od 7314 | A function is bijective if... |
| 2fvidinvd 7315 | Show that two functions ar... |
| foeqcnvco 7316 | Condition for function equ... |
| f1eqcocnv 7317 | Condition for function equ... |
| fveqf1o 7318 | Given a bijection ` F ` , ... |
| nf1const 7319 | A constant function from a... |
| nf1oconst 7320 | A constant function from a... |
| f1ofvswap 7321 | Swapping two values in a b... |
| fvf1pr 7322 | Values of a one-to-one fun... |
| fliftrel 7323 | ` F ` , a function lift, i... |
| fliftel 7324 | Elementhood in the relatio... |
| fliftel1 7325 | Elementhood in the relatio... |
| fliftcnv 7326 | Converse of the relation `... |
| fliftfun 7327 | The function ` F ` is the ... |
| fliftfund 7328 | The function ` F ` is the ... |
| fliftfuns 7329 | The function ` F ` is the ... |
| fliftf 7330 | The domain and range of th... |
| fliftval 7331 | The value of the function ... |
| isoeq1 7332 | Equality theorem for isomo... |
| isoeq2 7333 | Equality theorem for isomo... |
| isoeq3 7334 | Equality theorem for isomo... |
| isoeq4 7335 | Equality theorem for isomo... |
| isoeq5 7336 | Equality theorem for isomo... |
| nfiso 7337 | Bound-variable hypothesis ... |
| isof1o 7338 | An isomorphism is a one-to... |
| isof1oidb 7339 | A function is a bijection ... |
| isof1oopb 7340 | A function is a bijection ... |
| isorel 7341 | An isomorphism connects bi... |
| soisores 7342 | Express the condition of i... |
| soisoi 7343 | Infer isomorphism from one... |
| isoid 7344 | Identity law for isomorphi... |
| isocnv 7345 | Converse law for isomorphi... |
| isocnv2 7346 | Converse law for isomorphi... |
| isocnv3 7347 | Complementation law for is... |
| isores2 7348 | An isomorphism from one we... |
| isores1 7349 | An isomorphism from one we... |
| isores3 7350 | Induced isomorphism on a s... |
| isotr 7351 | Composition (transitive) l... |
| isomin 7352 | Isomorphisms preserve mini... |
| isoini 7353 | Isomorphisms preserve init... |
| isoini2 7354 | Isomorphisms are isomorphi... |
| isofrlem 7355 | Lemma for ~ isofr . (Cont... |
| isoselem 7356 | Lemma for ~ isose . (Cont... |
| isofr 7357 | An isomorphism preserves w... |
| isose 7358 | An isomorphism preserves s... |
| isofr2 7359 | A weak form of ~ isofr tha... |
| isopolem 7360 | Lemma for ~ isopo . (Cont... |
| isopo 7361 | An isomorphism preserves t... |
| isosolem 7362 | Lemma for ~ isoso . (Cont... |
| isoso 7363 | An isomorphism preserves t... |
| isowe 7364 | An isomorphism preserves t... |
| isowe2 7365 | A weak form of ~ isowe tha... |
| f1oiso 7366 | Any one-to-one onto functi... |
| f1oiso2 7367 | Any one-to-one onto functi... |
| f1owe 7368 | Well-ordering of isomorphi... |
| weniso 7369 | A set-like well-ordering h... |
| weisoeq 7370 | Thus, there is at most one... |
| weisoeq2 7371 | Thus, there is at most one... |
| knatar 7372 | The Knaster-Tarski theorem... |
| fvresval 7373 | The value of a restricted ... |
| funeldmb 7374 | If ` (/) ` is not part of ... |
| eqfunresadj 7375 | Law for adjoining an eleme... |
| eqfunressuc 7376 | Law for equality of restri... |
| fnssintima 7377 | Condition for subset of an... |
| imaeqsexv 7378 | Substitute a function valu... |
| imaeqsalv 7379 | Substitute a function valu... |
| canth 7380 | No set ` A ` is equinumero... |
| ncanth 7381 | Cantor's theorem fails for... |
| riotaeqdv 7384 | Formula-building deduction... |
| riotabidv 7385 | Formula-building deduction... |
| riotaeqbidv 7386 | Equality deduction for res... |
| riotaex 7387 | Restricted iota is a set. ... |
| riotav 7388 | An iota restricted to the ... |
| riotauni 7389 | Restricted iota in terms o... |
| nfriota1 7390 | The abstraction variable i... |
| nfriotadw 7391 | Deduction version of ~ nfr... |
| cbvriotaw 7392 | Change bound variable in a... |
| cbvriotavw 7393 | Change bound variable in a... |
| cbvriotavwOLD 7394 | Obsolete version of ~ cbvr... |
| nfriotad 7395 | Deduction version of ~ nfr... |
| nfriota 7396 | A variable not free in a w... |
| cbvriota 7397 | Change bound variable in a... |
| cbvriotav 7398 | Change bound variable in a... |
| csbriota 7399 | Interchange class substitu... |
| riotacl2 7400 | Membership law for "the un... |
| riotacl 7401 | Closure of restricted iota... |
| riotasbc 7402 | Substitution law for descr... |
| riotabidva 7403 | Equivalent wff's yield equ... |
| riotabiia 7404 | Equivalent wff's yield equ... |
| riota1 7405 | Property of restricted iot... |
| riota1a 7406 | Property of iota. (Contri... |
| riota2df 7407 | A deduction version of ~ r... |
| riota2f 7408 | This theorem shows a condi... |
| riota2 7409 | This theorem shows a condi... |
| riotaeqimp 7410 | If two restricted iota des... |
| riotaprop 7411 | Properties of a restricted... |
| riota5f 7412 | A method for computing res... |
| riota5 7413 | A method for computing res... |
| riotass2 7414 | Restriction of a unique el... |
| riotass 7415 | Restriction of a unique el... |
| moriotass 7416 | Restriction of a unique el... |
| snriota 7417 | A restricted class abstrac... |
| riotaxfrd 7418 | Change the variable ` x ` ... |
| eusvobj2 7419 | Specify the same property ... |
| eusvobj1 7420 | Specify the same object in... |
| f1ofveu 7421 | There is one domain elemen... |
| f1ocnvfv3 7422 | Value of the converse of a... |
| riotaund 7423 | Restricted iota equals the... |
| riotassuni 7424 | The restricted iota class ... |
| riotaclb 7425 | Bidirectional closure of r... |
| riotarab 7426 | Restricted iota of a restr... |
| oveq 7433 | Equality theorem for opera... |
| oveq1 7434 | Equality theorem for opera... |
| oveq2 7435 | Equality theorem for opera... |
| oveq12 7436 | Equality theorem for opera... |
| oveq1i 7437 | Equality inference for ope... |
| oveq2i 7438 | Equality inference for ope... |
| oveq12i 7439 | Equality inference for ope... |
| oveqi 7440 | Equality inference for ope... |
| oveq123i 7441 | Equality inference for ope... |
| oveq1d 7442 | Equality deduction for ope... |
| oveq2d 7443 | Equality deduction for ope... |
| oveqd 7444 | Equality deduction for ope... |
| oveq12d 7445 | Equality deduction for ope... |
| oveqan12d 7446 | Equality deduction for ope... |
| oveqan12rd 7447 | Equality deduction for ope... |
| oveq123d 7448 | Equality deduction for ope... |
| fvoveq1d 7449 | Equality deduction for nes... |
| fvoveq1 7450 | Equality theorem for neste... |
| ovanraleqv 7451 | Equality theorem for a con... |
| imbrov2fvoveq 7452 | Equality theorem for neste... |
| ovrspc2v 7453 | If an operation value is e... |
| oveqrspc2v 7454 | Restricted specialization ... |
| oveqdr 7455 | Equality of two operations... |
| nfovd 7456 | Deduction version of bound... |
| nfov 7457 | Bound-variable hypothesis ... |
| oprabidw 7458 | The law of concretion. Sp... |
| oprabid 7459 | The law of concretion. Sp... |
| ovex 7460 | The result of an operation... |
| ovexi 7461 | The result of an operation... |
| ovexd 7462 | The result of an operation... |
| ovssunirn 7463 | The result of an operation... |
| 0ov 7464 | Operation value of the emp... |
| ovprc 7465 | The value of an operation ... |
| ovprc1 7466 | The value of an operation ... |
| ovprc2 7467 | The value of an operation ... |
| ovrcl 7468 | Reverse closure for an ope... |
| elfvov1 7469 | Utility theorem: reverse c... |
| csbov123 7470 | Move class substitution in... |
| csbov 7471 | Move class substitution in... |
| csbov12g 7472 | Move class substitution in... |
| csbov1g 7473 | Move class substitution in... |
| csbov2g 7474 | Move class substitution in... |
| rspceov 7475 | A frequently used special ... |
| elovimad 7476 | Elementhood of the image s... |
| fnbrovb 7477 | Value of a binary operatio... |
| fnotovb 7478 | Equivalence of operation v... |
| opabbrex 7479 | A collection of ordered pa... |
| opabresex2 7480 | Restrictions of a collecti... |
| opabresex2d 7481 | Obsolete version of ~ opab... |
| fvmptopab 7482 | The function value of a ma... |
| fvmptopabOLD 7483 | Obsolete version of ~ fvmp... |
| f1opr 7484 | Condition for an operation... |
| brfvopab 7485 | The classes involved in a ... |
| dfoprab2 7486 | Class abstraction for oper... |
| reloprab 7487 | An operation class abstrac... |
| oprabv 7488 | If a pair and a class are ... |
| nfoprab1 7489 | The abstraction variables ... |
| nfoprab2 7490 | The abstraction variables ... |
| nfoprab3 7491 | The abstraction variables ... |
| nfoprab 7492 | Bound-variable hypothesis ... |
| oprabbid 7493 | Equivalent wff's yield equ... |
| oprabbidv 7494 | Equivalent wff's yield equ... |
| oprabbii 7495 | Equivalent wff's yield equ... |
| ssoprab2 7496 | Equivalence of ordered pai... |
| ssoprab2b 7497 | Equivalence of ordered pai... |
| eqoprab2bw 7498 | Equivalence of ordered pai... |
| eqoprab2b 7499 | Equivalence of ordered pai... |
| mpoeq123 7500 | An equality theorem for th... |
| mpoeq12 7501 | An equality theorem for th... |
| mpoeq123dva 7502 | An equality deduction for ... |
| mpoeq123dv 7503 | An equality deduction for ... |
| mpoeq123i 7504 | An equality inference for ... |
| mpoeq3dva 7505 | Slightly more general equa... |
| mpoeq3ia 7506 | An equality inference for ... |
| mpoeq3dv 7507 | An equality deduction for ... |
| nfmpo1 7508 | Bound-variable hypothesis ... |
| nfmpo2 7509 | Bound-variable hypothesis ... |
| nfmpo 7510 | Bound-variable hypothesis ... |
| 0mpo0 7511 | A mapping operation with e... |
| mpo0v 7512 | A mapping operation with e... |
| mpo0 7513 | A mapping operation with e... |
| oprab4 7514 | Two ways to state the doma... |
| cbvoprab1 7515 | Rule used to change first ... |
| cbvoprab2 7516 | Change the second bound va... |
| cbvoprab12 7517 | Rule used to change first ... |
| cbvoprab12v 7518 | Rule used to change first ... |
| cbvoprab3 7519 | Rule used to change the th... |
| cbvoprab3v 7520 | Rule used to change the th... |
| cbvmpox 7521 | Rule to change the bound v... |
| cbvmpo 7522 | Rule to change the bound v... |
| cbvmpov 7523 | Rule to change the bound v... |
| elimdelov 7524 | Eliminate a hypothesis whi... |
| brif1 7525 | Move a relation inside and... |
| ovif 7526 | Move a conditional outside... |
| ovif2 7527 | Move a conditional outside... |
| ovif12 7528 | Move a conditional outside... |
| ifov 7529 | Move a conditional outside... |
| dmoprab 7530 | The domain of an operation... |
| dmoprabss 7531 | The domain of an operation... |
| rnoprab 7532 | The range of an operation ... |
| rnoprab2 7533 | The range of a restricted ... |
| reldmoprab 7534 | The domain of an operation... |
| oprabss 7535 | Structure of an operation ... |
| eloprabga 7536 | The law of concretion for ... |
| eloprabgaOLD 7537 | Obsolete version of ~ elop... |
| eloprabg 7538 | The law of concretion for ... |
| ssoprab2i 7539 | Inference of operation cla... |
| mpov 7540 | Operation with universal d... |
| mpomptx 7541 | Express a two-argument fun... |
| mpompt 7542 | Express a two-argument fun... |
| mpodifsnif 7543 | A mapping with two argumen... |
| mposnif 7544 | A mapping with two argumen... |
| fconstmpo 7545 | Representation of a consta... |
| resoprab 7546 | Restriction of an operatio... |
| resoprab2 7547 | Restriction of an operator... |
| resmpo 7548 | Restriction of the mapping... |
| funoprabg 7549 | "At most one" is a suffici... |
| funoprab 7550 | "At most one" is a suffici... |
| fnoprabg 7551 | Functionality and domain o... |
| mpofun 7552 | The maps-to notation for a... |
| fnoprab 7553 | Functionality and domain o... |
| ffnov 7554 | An operation maps to a cla... |
| fovcld 7555 | Closure law for an operati... |
| fovcl 7556 | Closure law for an operati... |
| eqfnov 7557 | Equality of two operations... |
| eqfnov2 7558 | Two operators with the sam... |
| fnov 7559 | Representation of a functi... |
| mpo2eqb 7560 | Bidirectional equality the... |
| rnmpo 7561 | The range of an operation ... |
| reldmmpo 7562 | The domain of an operation... |
| elrnmpog 7563 | Membership in the range of... |
| elrnmpo 7564 | Membership in the range of... |
| elimampo 7565 | Membership in the image of... |
| elrnmpores 7566 | Membership in the range of... |
| ralrnmpo 7567 | A restricted quantifier ov... |
| rexrnmpo 7568 | A restricted quantifier ov... |
| ovid 7569 | The value of an operation ... |
| ovidig 7570 | The value of an operation ... |
| ovidi 7571 | The value of an operation ... |
| ov 7572 | The value of an operation ... |
| ovigg 7573 | The value of an operation ... |
| ovig 7574 | The value of an operation ... |
| ovmpt4g 7575 | Value of a function given ... |
| ovmpos 7576 | Value of a function given ... |
| ov2gf 7577 | The value of an operation ... |
| ovmpodxf 7578 | Value of an operation give... |
| ovmpodx 7579 | Value of an operation give... |
| ovmpod 7580 | Value of an operation give... |
| ovmpox 7581 | The value of an operation ... |
| ovmpoga 7582 | Value of an operation give... |
| ovmpoa 7583 | Value of an operation give... |
| ovmpodf 7584 | Alternate deduction versio... |
| ovmpodv 7585 | Alternate deduction versio... |
| ovmpodv2 7586 | Alternate deduction versio... |
| ovmpog 7587 | Value of an operation give... |
| ovmpo 7588 | Value of an operation give... |
| ovmpot 7589 | The value of an operation ... |
| fvmpopr2d 7590 | Value of an operation give... |
| ov3 7591 | The value of an operation ... |
| ov6g 7592 | The value of an operation ... |
| ovg 7593 | The value of an operation ... |
| ovres 7594 | The value of a restricted ... |
| ovresd 7595 | Lemma for converting metri... |
| oprres 7596 | The restriction of an oper... |
| oprssov 7597 | The value of a member of t... |
| fovcdm 7598 | An operation's value belon... |
| fovcdmda 7599 | An operation's value belon... |
| fovcdmd 7600 | An operation's value belon... |
| fnrnov 7601 | The range of an operation ... |
| foov 7602 | An onto mapping of an oper... |
| fnovrn 7603 | An operation's value belon... |
| ovelrn 7604 | A member of an operation's... |
| funimassov 7605 | Membership relation for th... |
| ovelimab 7606 | Operation value in an imag... |
| ovima0 7607 | An operation value is a me... |
| ovconst2 7608 | The value of a constant op... |
| oprssdm 7609 | Domain of closure of an op... |
| nssdmovg 7610 | The value of an operation ... |
| ndmovg 7611 | The value of an operation ... |
| ndmov 7612 | The value of an operation ... |
| ndmovcl 7613 | The closure of an operatio... |
| ndmovrcl 7614 | Reverse closure law, when ... |
| ndmovcom 7615 | Any operation is commutati... |
| ndmovass 7616 | Any operation is associati... |
| ndmovdistr 7617 | Any operation is distribut... |
| ndmovord 7618 | Elimination of redundant a... |
| ndmovordi 7619 | Elimination of redundant a... |
| caovclg 7620 | Convert an operation closu... |
| caovcld 7621 | Convert an operation closu... |
| caovcl 7622 | Convert an operation closu... |
| caovcomg 7623 | Convert an operation commu... |
| caovcomd 7624 | Convert an operation commu... |
| caovcom 7625 | Convert an operation commu... |
| caovassg 7626 | Convert an operation assoc... |
| caovassd 7627 | Convert an operation assoc... |
| caovass 7628 | Convert an operation assoc... |
| caovcang 7629 | Convert an operation cance... |
| caovcand 7630 | Convert an operation cance... |
| caovcanrd 7631 | Commute the arguments of a... |
| caovcan 7632 | Convert an operation cance... |
| caovordig 7633 | Convert an operation order... |
| caovordid 7634 | Convert an operation order... |
| caovordg 7635 | Convert an operation order... |
| caovordd 7636 | Convert an operation order... |
| caovord2d 7637 | Operation ordering law wit... |
| caovord3d 7638 | Ordering law. (Contribute... |
| caovord 7639 | Convert an operation order... |
| caovord2 7640 | Operation ordering law wit... |
| caovord3 7641 | Ordering law. (Contribute... |
| caovdig 7642 | Convert an operation distr... |
| caovdid 7643 | Convert an operation distr... |
| caovdir2d 7644 | Convert an operation distr... |
| caovdirg 7645 | Convert an operation rever... |
| caovdird 7646 | Convert an operation distr... |
| caovdi 7647 | Convert an operation distr... |
| caov32d 7648 | Rearrange arguments in a c... |
| caov12d 7649 | Rearrange arguments in a c... |
| caov31d 7650 | Rearrange arguments in a c... |
| caov13d 7651 | Rearrange arguments in a c... |
| caov4d 7652 | Rearrange arguments in a c... |
| caov411d 7653 | Rearrange arguments in a c... |
| caov42d 7654 | Rearrange arguments in a c... |
| caov32 7655 | Rearrange arguments in a c... |
| caov12 7656 | Rearrange arguments in a c... |
| caov31 7657 | Rearrange arguments in a c... |
| caov13 7658 | Rearrange arguments in a c... |
| caov4 7659 | Rearrange arguments in a c... |
| caov411 7660 | Rearrange arguments in a c... |
| caov42 7661 | Rearrange arguments in a c... |
| caovdir 7662 | Reverse distributive law. ... |
| caovdilem 7663 | Lemma used by real number ... |
| caovlem2 7664 | Lemma used in real number ... |
| caovmo 7665 | Uniqueness of inverse elem... |
| imaeqexov 7666 | Substitute an operation va... |
| imaeqalov 7667 | Substitute an operation va... |
| mpondm0 7668 | The value of an operation ... |
| elmpocl 7669 | If a two-parameter class i... |
| elmpocl1 7670 | If a two-parameter class i... |
| elmpocl2 7671 | If a two-parameter class i... |
| elovmpod 7672 | Utility lemma for two-para... |
| elovmpo 7673 | Utility lemma for two-para... |
| elovmporab 7674 | Implications for the value... |
| elovmporab1w 7675 | Implications for the value... |
| elovmporab1 7676 | Implications for the value... |
| 2mpo0 7677 | If the operation value of ... |
| relmptopab 7678 | Any function to sets of or... |
| f1ocnvd 7679 | Describe an implicit one-t... |
| f1od 7680 | Describe an implicit one-t... |
| f1ocnv2d 7681 | Describe an implicit one-t... |
| f1o2d 7682 | Describe an implicit one-t... |
| f1opw2 7683 | A one-to-one mapping induc... |
| f1opw 7684 | A one-to-one mapping induc... |
| elovmpt3imp 7685 | If the value of a function... |
| ovmpt3rab1 7686 | The value of an operation ... |
| ovmpt3rabdm 7687 | If the value of a function... |
| elovmpt3rab1 7688 | Implications for the value... |
| elovmpt3rab 7689 | Implications for the value... |
| ofeqd 7694 | Equality theorem for funct... |
| ofeq 7695 | Equality theorem for funct... |
| ofreq 7696 | Equality theorem for funct... |
| ofexg 7697 | A function operation restr... |
| nfof 7698 | Hypothesis builder for fun... |
| nfofr 7699 | Hypothesis builder for fun... |
| ofrfvalg 7700 | Value of a relation applie... |
| offval 7701 | Value of an operation appl... |
| ofrfval 7702 | Value of a relation applie... |
| ofval 7703 | Evaluate a function operat... |
| ofrval 7704 | Exhibit a function relatio... |
| offn 7705 | The function operation pro... |
| offun 7706 | The function operation pro... |
| offval2f 7707 | The function operation exp... |
| ofmresval 7708 | Value of a restriction of ... |
| fnfvof 7709 | Function value of a pointw... |
| off 7710 | The function operation pro... |
| ofres 7711 | Restrict the operands of a... |
| offval2 7712 | The function operation exp... |
| ofrfval2 7713 | The function relation acti... |
| ofmpteq 7714 | Value of a pointwise opera... |
| coof 7715 | The composition of a _homo... |
| ofco 7716 | The composition of a funct... |
| offveq 7717 | Convert an identity of the... |
| offveqb 7718 | Equivalent expressions for... |
| ofc1 7719 | Left operation by a consta... |
| ofc2 7720 | Right operation by a const... |
| ofc12 7721 | Function operation on two ... |
| caofref 7722 | Transfer a reflexive law t... |
| caofinvl 7723 | Transfer a left inverse la... |
| caofid0l 7724 | Transfer a left identity l... |
| caofid0r 7725 | Transfer a right identity ... |
| caofid1 7726 | Transfer a right absorptio... |
| caofid2 7727 | Transfer a right absorptio... |
| caofcom 7728 | Transfer a commutative law... |
| caofrss 7729 | Transfer a relation subset... |
| caofass 7730 | Transfer an associative la... |
| caoftrn 7731 | Transfer a transitivity la... |
| caofdi 7732 | Transfer a distributive la... |
| caofdir 7733 | Transfer a reverse distrib... |
| caonncan 7734 | Transfer ~ nncan -shaped l... |
| relrpss 7737 | The proper subset relation... |
| brrpssg 7738 | The proper subset relation... |
| brrpss 7739 | The proper subset relation... |
| porpss 7740 | Every class is partially o... |
| sorpss 7741 | Express strict ordering un... |
| sorpssi 7742 | Property of a chain of set... |
| sorpssun 7743 | A chain of sets is closed ... |
| sorpssin 7744 | A chain of sets is closed ... |
| sorpssuni 7745 | In a chain of sets, a maxi... |
| sorpssint 7746 | In a chain of sets, a mini... |
| sorpsscmpl 7747 | The componentwise compleme... |
| zfun 7749 | Axiom of Union expressed w... |
| axun2 7750 | A variant of the Axiom of ... |
| uniex2 7751 | The Axiom of Union using t... |
| vuniex 7752 | The union of a setvar is a... |
| uniexg 7753 | The ZF Axiom of Union in c... |
| uniex 7754 | The Axiom of Union in clas... |
| uniexd 7755 | Deduction version of the Z... |
| unexg 7756 | The union of two sets is a... |
| unex 7757 | The union of two sets is a... |
| unexOLD 7758 | Obsolete proof of ~ unex a... |
| tpex 7759 | An unordered triple of cla... |
| unexb 7760 | Existence of union is equi... |
| unexbOLD 7761 | Obsolete proof of ~ unexb ... |
| unexgOLD 7762 | Obsolete proof of ~ unexg ... |
| xpexg 7763 | The Cartesian product of t... |
| xpexd 7764 | The Cartesian product of t... |
| 3xpexg 7765 | The Cartesian product of t... |
| xpex 7766 | The Cartesian product of t... |
| unexd 7767 | The union of two sets is a... |
| sqxpexg 7768 | The Cartesian square of a ... |
| abnexg 7769 | Sufficient condition for a... |
| abnex 7770 | Sufficient condition for a... |
| snnex 7771 | The class of all singleton... |
| pwnex 7772 | The class of all power set... |
| difex2 7773 | If the subtrahend of a cla... |
| difsnexi 7774 | If the difference of a cla... |
| uniuni 7775 | Expression for double unio... |
| uniexr 7776 | Converse of the Axiom of U... |
| uniexb 7777 | The Axiom of Union and its... |
| pwexr 7778 | Converse of the Axiom of P... |
| pwexb 7779 | The Axiom of Power Sets an... |
| elpwpwel 7780 | A class belongs to a doubl... |
| eldifpw 7781 | Membership in a power clas... |
| elpwun 7782 | Membership in the power cl... |
| pwuncl 7783 | Power classes are closed u... |
| iunpw 7784 | An indexed union of a powe... |
| fr3nr 7785 | A well-founded relation ha... |
| epne3 7786 | A well-founded class conta... |
| dfwe2 7787 | Alternate definition of we... |
| epweon 7788 | The membership relation we... |
| epweonALT 7789 | Alternate proof of ~ epweo... |
| ordon 7790 | The class of all ordinal n... |
| onprc 7791 | No set contains all ordina... |
| ssorduni 7792 | The union of a class of or... |
| ssonuni 7793 | The union of a set of ordi... |
| ssonunii 7794 | The union of a set of ordi... |
| ordeleqon 7795 | A way to express the ordin... |
| ordsson 7796 | Any ordinal class is a sub... |
| dford5 7797 | A class is ordinal iff it ... |
| onss 7798 | An ordinal number is a sub... |
| predon 7799 | The predecessor of an ordi... |
| predonOLD 7800 | Obsolete version of ~ pred... |
| ssonprc 7801 | Two ways of saying a class... |
| onuni 7802 | The union of an ordinal nu... |
| orduni 7803 | The union of an ordinal cl... |
| onint 7804 | The intersection (infimum)... |
| onint0 7805 | The intersection of a clas... |
| onssmin 7806 | A nonempty class of ordina... |
| onminesb 7807 | If a property is true for ... |
| onminsb 7808 | If a property is true for ... |
| oninton 7809 | The intersection of a none... |
| onintrab 7810 | The intersection of a clas... |
| onintrab2 7811 | An existence condition equ... |
| onnmin 7812 | No member of a set of ordi... |
| onnminsb 7813 | An ordinal number smaller ... |
| oneqmin 7814 | A way to show that an ordi... |
| uniordint 7815 | The union of a set of ordi... |
| onminex 7816 | If a wff is true for an or... |
| sucon 7817 | The class of all ordinal n... |
| sucexb 7818 | A successor exists iff its... |
| sucexg 7819 | The successor of a set is ... |
| sucex 7820 | The successor of a set is ... |
| onmindif2 7821 | The minimum of a class of ... |
| ordsuci 7822 | The successor of an ordina... |
| sucexeloni 7823 | If the successor of an ord... |
| sucexeloniOLD 7824 | Obsolete version of ~ suce... |
| onsuc 7825 | The successor of an ordina... |
| suceloniOLD 7826 | Obsolete version of ~ onsu... |
| ordsuc 7827 | A class is ordinal if and ... |
| ordsucOLD 7828 | Obsolete version of ~ ords... |
| ordpwsuc 7829 | The collection of ordinals... |
| onpwsuc 7830 | The collection of ordinal ... |
| onsucb 7831 | A class is an ordinal numb... |
| ordsucss 7832 | The successor of an elemen... |
| onpsssuc 7833 | An ordinal number is a pro... |
| ordelsuc 7834 | A set belongs to an ordina... |
| onsucmin 7835 | The successor of an ordina... |
| ordsucelsuc 7836 | Membership is inherited by... |
| ordsucsssuc 7837 | The subclass relationship ... |
| ordsucuniel 7838 | Given an element ` A ` of ... |
| ordsucun 7839 | The successor of the maxim... |
| ordunpr 7840 | The maximum of two ordinal... |
| ordunel 7841 | The maximum of two ordinal... |
| onsucuni 7842 | A class of ordinal numbers... |
| ordsucuni 7843 | An ordinal class is a subc... |
| orduniorsuc 7844 | An ordinal class is either... |
| unon 7845 | The class of all ordinal n... |
| ordunisuc 7846 | An ordinal class is equal ... |
| orduniss2 7847 | The union of the ordinal s... |
| onsucuni2 7848 | A successor ordinal is the... |
| 0elsuc 7849 | The successor of an ordina... |
| limon 7850 | The class of ordinal numbe... |
| onuniorsuc 7851 | An ordinal number is eithe... |
| onssi 7852 | An ordinal number is a sub... |
| onsuci 7853 | The successor of an ordina... |
| onuniorsuciOLD 7854 | Obsolete version of ~ onun... |
| onuninsuci 7855 | An ordinal is equal to its... |
| onsucssi 7856 | A set belongs to an ordina... |
| nlimsucg 7857 | A successor is not a limit... |
| orduninsuc 7858 | An ordinal class is equal ... |
| ordunisuc2 7859 | An ordinal equal to its un... |
| ordzsl 7860 | An ordinal is zero, a succ... |
| onzsl 7861 | An ordinal number is zero,... |
| dflim3 7862 | An alternate definition of... |
| dflim4 7863 | An alternate definition of... |
| limsuc 7864 | The successor of a member ... |
| limsssuc 7865 | A class includes a limit o... |
| nlimon 7866 | Two ways to express the cl... |
| limuni3 7867 | The union of a nonempty cl... |
| tfi 7868 | The Principle of Transfini... |
| tfisg 7869 | A closed form of ~ tfis . ... |
| tfis 7870 | Transfinite Induction Sche... |
| tfis2f 7871 | Transfinite Induction Sche... |
| tfis2 7872 | Transfinite Induction Sche... |
| tfis3 7873 | Transfinite Induction Sche... |
| tfisi 7874 | A transfinite induction sc... |
| tfinds 7875 | Principle of Transfinite I... |
| tfindsg 7876 | Transfinite Induction (inf... |
| tfindsg2 7877 | Transfinite Induction (inf... |
| tfindes 7878 | Transfinite Induction with... |
| tfinds2 7879 | Transfinite Induction (inf... |
| tfinds3 7880 | Principle of Transfinite I... |
| dfom2 7883 | An alternate definition of... |
| elom 7884 | Membership in omega. The ... |
| omsson 7885 | Omega is a subset of ` On ... |
| limomss 7886 | The class of natural numbe... |
| nnon 7887 | A natural number is an ord... |
| nnoni 7888 | A natural number is an ord... |
| nnord 7889 | A natural number is ordina... |
| trom 7890 | The class of finite ordina... |
| ordom 7891 | The class of finite ordina... |
| elnn 7892 | A member of a natural numb... |
| omon 7893 | The class of natural numbe... |
| omelon2 7894 | Omega is an ordinal number... |
| nnlim 7895 | A natural number is not a ... |
| omssnlim 7896 | The class of natural numbe... |
| limom 7897 | Omega is a limit ordinal. ... |
| peano2b 7898 | A class belongs to omega i... |
| nnsuc 7899 | A nonzero natural number i... |
| omsucne 7900 | A natural number is not th... |
| ssnlim 7901 | An ordinal subclass of non... |
| omsinds 7902 | Strong (or "total") induct... |
| omsindsOLD 7903 | Obsolete version of ~ omsi... |
| omun 7904 | The union of two finite or... |
| peano1 7905 | Zero is a natural number. ... |
| peano1OLD 7906 | Obsolete version of ~ pean... |
| peano2 7907 | The successor of any natur... |
| peano3 7908 | The successor of any natur... |
| peano4 7909 | Two natural numbers are eq... |
| peano5 7910 | The induction postulate: a... |
| peano5OLD 7911 | Obsolete version of ~ pean... |
| nn0suc 7912 | A natural number is either... |
| find 7913 | The Principle of Finite In... |
| finds 7914 | Principle of Finite Induct... |
| findsg 7915 | Principle of Finite Induct... |
| finds2 7916 | Principle of Finite Induct... |
| finds1 7917 | Principle of Finite Induct... |
| findes 7918 | Finite induction with expl... |
| dmexg 7919 | The domain of a set is a s... |
| rnexg 7920 | The range of a set is a se... |
| dmexd 7921 | The domain of a set is a s... |
| fndmexd 7922 | If a function is a set, it... |
| dmfex 7923 | If a mapping is a set, its... |
| fndmexb 7924 | The domain of a function i... |
| fdmexb 7925 | The domain of a function i... |
| dmfexALT 7926 | Alternate proof of ~ dmfex... |
| dmex 7927 | The domain of a set is a s... |
| rnex 7928 | The range of a set is a se... |
| iprc 7929 | The identity function is a... |
| resiexg 7930 | The existence of a restric... |
| imaexg 7931 | The image of a set is a se... |
| imaex 7932 | The image of a set is a se... |
| rnexd 7933 | The range of a set is a se... |
| imaexd 7934 | The image of a set is a se... |
| exse2 7935 | Any set relation is set-li... |
| xpexr 7936 | If a Cartesian product is ... |
| xpexr2 7937 | If a nonempty Cartesian pr... |
| xpexcnv 7938 | A condition where the conv... |
| soex 7939 | If the relation in a stric... |
| elxp4 7940 | Membership in a Cartesian ... |
| elxp5 7941 | Membership in a Cartesian ... |
| cnvexg 7942 | The converse of a set is a... |
| cnvex 7943 | The converse of a set is a... |
| relcnvexb 7944 | A relation is a set iff it... |
| f1oexrnex 7945 | If the range of a 1-1 onto... |
| f1oexbi 7946 | There is a one-to-one onto... |
| coexg 7947 | The composition of two set... |
| coex 7948 | The composition of two set... |
| coexd 7949 | The composition of two set... |
| funcnvuni 7950 | The union of a chain (with... |
| fun11uni 7951 | The union of a chain (with... |
| fex2 7952 | A function with bounded do... |
| fabexd 7953 | Existence of a set of func... |
| fabexg 7954 | Existence of a set of func... |
| fabexgOLD 7955 | Obsolete version of ~ fabe... |
| fabex 7956 | Existence of a set of func... |
| mapex 7957 | The class of all functions... |
| f1oabexg 7958 | The class of all 1-1-onto ... |
| f1oabexgOLD 7959 | Obsolete version of ~ f1oa... |
| fiunlem 7960 | Lemma for ~ fiun and ~ f1i... |
| fiun 7961 | The union of a chain (with... |
| f1iun 7962 | The union of a chain (with... |
| fviunfun 7963 | The function value of an i... |
| ffoss 7964 | Relationship between a map... |
| f11o 7965 | Relationship between one-t... |
| resfunexgALT 7966 | Alternate proof of ~ resfu... |
| cofunexg 7967 | Existence of a composition... |
| cofunex2g 7968 | Existence of a composition... |
| fnexALT 7969 | Alternate proof of ~ fnex ... |
| funexw 7970 | Weak version of ~ funex th... |
| mptexw 7971 | Weak version of ~ mptex th... |
| funrnex 7972 | If the domain of a functio... |
| zfrep6 7973 | A version of the Axiom of ... |
| focdmex 7974 | If the domain of an onto f... |
| f1dmex 7975 | If the codomain of a one-t... |
| f1ovv 7976 | The codomain/range of a 1-... |
| fvclex 7977 | Existence of the class of ... |
| fvresex 7978 | Existence of the class of ... |
| abrexexg 7979 | Existence of a class abstr... |
| abrexexgOLD 7980 | Obsolete version of ~ abre... |
| abrexex 7981 | Existence of a class abstr... |
| iunexg 7982 | The existence of an indexe... |
| abrexex2g 7983 | Existence of an existentia... |
| opabex3d 7984 | Existence of an ordered pa... |
| opabex3rd 7985 | Existence of an ordered pa... |
| opabex3 7986 | Existence of an ordered pa... |
| iunex 7987 | The existence of an indexe... |
| abrexex2 7988 | Existence of an existentia... |
| abexssex 7989 | Existence of a class abstr... |
| abexex 7990 | A condition where a class ... |
| f1oweALT 7991 | Alternate proof of ~ f1owe... |
| wemoiso 7992 | Thus, there is at most one... |
| wemoiso2 7993 | Thus, there is at most one... |
| oprabexd 7994 | Existence of an operator a... |
| oprabex 7995 | Existence of an operation ... |
| oprabex3 7996 | Existence of an operation ... |
| oprabrexex2 7997 | Existence of an existentia... |
| ab2rexex 7998 | Existence of a class abstr... |
| ab2rexex2 7999 | Existence of an existentia... |
| xpexgALT 8000 | Alternate proof of ~ xpexg... |
| offval3 8001 | General value of ` ( F oF ... |
| offres 8002 | Pointwise combination comm... |
| ofmres 8003 | Equivalent expressions for... |
| ofmresex 8004 | Existence of a restriction... |
| mptcnfimad 8005 | The converse of a mapping ... |
| 1stval 8010 | The value of the function ... |
| 2ndval 8011 | The value of the function ... |
| 1stnpr 8012 | Value of the first-member ... |
| 2ndnpr 8013 | Value of the second-member... |
| 1st0 8014 | The value of the first-mem... |
| 2nd0 8015 | The value of the second-me... |
| op1st 8016 | Extract the first member o... |
| op2nd 8017 | Extract the second member ... |
| op1std 8018 | Extract the first member o... |
| op2ndd 8019 | Extract the second member ... |
| op1stg 8020 | Extract the first member o... |
| op2ndg 8021 | Extract the second member ... |
| ot1stg 8022 | Extract the first member o... |
| ot2ndg 8023 | Extract the second member ... |
| ot3rdg 8024 | Extract the third member o... |
| 1stval2 8025 | Alternate value of the fun... |
| 2ndval2 8026 | Alternate value of the fun... |
| oteqimp 8027 | The components of an order... |
| fo1st 8028 | The ` 1st ` function maps ... |
| fo2nd 8029 | The ` 2nd ` function maps ... |
| br1steqg 8030 | Uniqueness condition for t... |
| br2ndeqg 8031 | Uniqueness condition for t... |
| f1stres 8032 | Mapping of a restriction o... |
| f2ndres 8033 | Mapping of a restriction o... |
| fo1stres 8034 | Onto mapping of a restrict... |
| fo2ndres 8035 | Onto mapping of a restrict... |
| 1st2val 8036 | Value of an alternate defi... |
| 2nd2val 8037 | Value of an alternate defi... |
| 1stcof 8038 | Composition of the first m... |
| 2ndcof 8039 | Composition of the second ... |
| xp1st 8040 | Location of the first elem... |
| xp2nd 8041 | Location of the second ele... |
| elxp6 8042 | Membership in a Cartesian ... |
| elxp7 8043 | Membership in a Cartesian ... |
| eqopi 8044 | Equality with an ordered p... |
| xp2 8045 | Representation of Cartesia... |
| unielxp 8046 | The membership relation fo... |
| 1st2nd2 8047 | Reconstruction of a member... |
| 1st2ndb 8048 | Reconstruction of an order... |
| xpopth 8049 | An ordered pair theorem fo... |
| eqop 8050 | Two ways to express equali... |
| eqop2 8051 | Two ways to express equali... |
| op1steq 8052 | Two ways of expressing tha... |
| opreuopreu 8053 | There is a unique ordered ... |
| el2xptp 8054 | A member of a nested Carte... |
| el2xptp0 8055 | A member of a nested Carte... |
| el2xpss 8056 | Version of ~ elrel for tri... |
| 2nd1st 8057 | Swap the members of an ord... |
| 1st2nd 8058 | Reconstruction of a member... |
| 1stdm 8059 | The first ordered pair com... |
| 2ndrn 8060 | The second ordered pair co... |
| 1st2ndbr 8061 | Express an element of a re... |
| releldm2 8062 | Two ways of expressing mem... |
| reldm 8063 | An expression for the doma... |
| releldmdifi 8064 | One way of expressing memb... |
| funfv1st2nd 8065 | The function value for the... |
| funelss 8066 | If the first component of ... |
| funeldmdif 8067 | Two ways of expressing mem... |
| sbcopeq1a 8068 | Equality theorem for subst... |
| csbopeq1a 8069 | Equality theorem for subst... |
| sbcoteq1a 8070 | Equality theorem for subst... |
| dfopab2 8071 | A way to define an ordered... |
| dfoprab3s 8072 | A way to define an operati... |
| dfoprab3 8073 | Operation class abstractio... |
| dfoprab4 8074 | Operation class abstractio... |
| dfoprab4f 8075 | Operation class abstractio... |
| opabex2 8076 | Condition for an operation... |
| opabn1stprc 8077 | An ordered-pair class abst... |
| opiota 8078 | The property of a uniquely... |
| cnvoprab 8079 | The converse of a class ab... |
| dfxp3 8080 | Define the Cartesian produ... |
| elopabi 8081 | A consequence of membershi... |
| eloprabi 8082 | A consequence of membershi... |
| mpomptsx 8083 | Express a two-argument fun... |
| mpompts 8084 | Express a two-argument fun... |
| dmmpossx 8085 | The domain of a mapping is... |
| fmpox 8086 | Functionality, domain and ... |
| fmpo 8087 | Functionality, domain and ... |
| fnmpo 8088 | Functionality and domain o... |
| fnmpoi 8089 | Functionality and domain o... |
| dmmpo 8090 | Domain of a class given by... |
| ovmpoelrn 8091 | An operation's value belon... |
| dmmpoga 8092 | Domain of an operation giv... |
| dmmpog 8093 | Domain of an operation giv... |
| mpoexxg 8094 | Existence of an operation ... |
| mpoexg 8095 | Existence of an operation ... |
| mpoexga 8096 | If the domain of an operat... |
| mpoexw 8097 | Weak version of ~ mpoex th... |
| mpoex 8098 | If the domain of an operat... |
| mptmpoopabbrd 8099 | The operation value of a f... |
| mptmpoopabbrdOLD 8100 | Obsolete version of ~ mptm... |
| mptmpoopabovd 8101 | The operation value of a f... |
| mptmpoopabbrdOLDOLD 8102 | Obsolete version of ~ mptm... |
| mptmpoopabovdOLD 8103 | Obsolete version of ~ mptm... |
| el2mpocsbcl 8104 | If the operation value of ... |
| el2mpocl 8105 | If the operation value of ... |
| fnmpoovd 8106 | A function with a Cartesia... |
| offval22 8107 | The function operation exp... |
| brovpreldm 8108 | If a binary relation holds... |
| bropopvvv 8109 | If a binary relation holds... |
| bropfvvvvlem 8110 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8111 | If a binary relation holds... |
| ovmptss 8112 | If all the values of the m... |
| relmpoopab 8113 | Any function to sets of or... |
| fmpoco 8114 | Composition of two functio... |
| oprabco 8115 | Composition of a function ... |
| oprab2co 8116 | Composition of operator ab... |
| df1st2 8117 | An alternate possible defi... |
| df2nd2 8118 | An alternate possible defi... |
| 1stconst 8119 | The mapping of a restricti... |
| 2ndconst 8120 | The mapping of a restricti... |
| dfmpo 8121 | Alternate definition for t... |
| mposn 8122 | An operation (in maps-to n... |
| curry1 8123 | Composition with ` ``' ( 2... |
| curry1val 8124 | The value of a curried fun... |
| curry1f 8125 | Functionality of a curried... |
| curry2 8126 | Composition with ` ``' ( 1... |
| curry2f 8127 | Functionality of a curried... |
| curry2val 8128 | The value of a curried fun... |
| cnvf1olem 8129 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8130 | Describe a function that m... |
| fparlem1 8131 | Lemma for ~ fpar . (Contr... |
| fparlem2 8132 | Lemma for ~ fpar . (Contr... |
| fparlem3 8133 | Lemma for ~ fpar . (Contr... |
| fparlem4 8134 | Lemma for ~ fpar . (Contr... |
| fpar 8135 | Merge two functions in par... |
| fsplit 8136 | A function that can be use... |
| fsplitfpar 8137 | Merge two functions with a... |
| offsplitfpar 8138 | Express the function opera... |
| f2ndf 8139 | The ` 2nd ` (second compon... |
| fo2ndf 8140 | The ` 2nd ` (second compon... |
| f1o2ndf1 8141 | The ` 2nd ` (second compon... |
| opco1 8142 | Value of an operation prec... |
| opco2 8143 | Value of an operation prec... |
| opco1i 8144 | Inference form of ~ opco1 ... |
| frxp 8145 | A lexicographical ordering... |
| xporderlem 8146 | Lemma for lexicographical ... |
| poxp 8147 | A lexicographical ordering... |
| soxp 8148 | A lexicographical ordering... |
| wexp 8149 | A lexicographical ordering... |
| fnwelem 8150 | Lemma for ~ fnwe . (Contr... |
| fnwe 8151 | A variant on lexicographic... |
| fnse 8152 | Condition for the well-ord... |
| fvproj 8153 | Value of a function on ord... |
| fimaproj 8154 | Image of a cartesian produ... |
| ralxpes 8155 | A version of ~ ralxp with ... |
| ralxp3f 8156 | Restricted for all over a ... |
| ralxp3 8157 | Restricted for all over a ... |
| ralxp3es 8158 | Restricted for-all over a ... |
| frpoins3xpg 8159 | Special case of founded pa... |
| frpoins3xp3g 8160 | Special case of founded pa... |
| xpord2lem 8161 | Lemma for Cartesian produc... |
| poxp2 8162 | Another way of partially o... |
| frxp2 8163 | Another way of giving a we... |
| xpord2pred 8164 | Calculate the predecessor ... |
| sexp2 8165 | Condition for the relation... |
| xpord2indlem 8166 | Induction over the Cartesi... |
| xpord2ind 8167 | Induction over the Cartesi... |
| xpord3lem 8168 | Lemma for triple ordering.... |
| poxp3 8169 | Triple Cartesian product p... |
| frxp3 8170 | Give well-foundedness over... |
| xpord3pred 8171 | Calculate the predecsessor... |
| sexp3 8172 | Show that the triple order... |
| xpord3inddlem 8173 | Induction over the triple ... |
| xpord3indd 8174 | Induction over the triple ... |
| xpord3ind 8175 | Induction over the triple ... |
| orderseqlem 8176 | Lemma for ~ poseq and ~ so... |
| poseq 8177 | A partial ordering of ordi... |
| soseq 8178 | A linear ordering of ordin... |
| suppval 8181 | The value of the operation... |
| supp0prc 8182 | The support of a class is ... |
| suppvalbr 8183 | The value of the operation... |
| supp0 8184 | The support of the empty s... |
| suppval1 8185 | The value of the operation... |
| suppvalfng 8186 | The value of the operation... |
| suppvalfn 8187 | The value of the operation... |
| elsuppfng 8188 | An element of the support ... |
| elsuppfn 8189 | An element of the support ... |
| fvdifsupp 8190 | Function value is zero out... |
| cnvimadfsn 8191 | The support of functions "... |
| suppimacnvss 8192 | The support of functions "... |
| suppimacnv 8193 | Support sets of functions ... |
| fsuppeq 8194 | Two ways of writing the su... |
| fsuppeqg 8195 | Version of ~ fsuppeq avoid... |
| suppssdm 8196 | The support of a function ... |
| suppsnop 8197 | The support of a singleton... |
| snopsuppss 8198 | The support of a singleton... |
| fvn0elsupp 8199 | If the function value for ... |
| fvn0elsuppb 8200 | The function value for a g... |
| rexsupp 8201 | Existential quantification... |
| ressuppss 8202 | The support of the restric... |
| suppun 8203 | The support of a class/fun... |
| ressuppssdif 8204 | The support of the restric... |
| mptsuppdifd 8205 | The support of a function ... |
| mptsuppd 8206 | The support of a function ... |
| extmptsuppeq 8207 | The support of an extended... |
| suppfnss 8208 | The support of a function ... |
| funsssuppss 8209 | The support of a function ... |
| fnsuppres 8210 | Two ways to express restri... |
| fnsuppeq0 8211 | The support of a function ... |
| fczsupp0 8212 | The support of a constant ... |
| suppss 8213 | Show that the support of a... |
| suppssr 8214 | A function is zero outside... |
| suppssrg 8215 | A function is zero outside... |
| suppssov1 8216 | Formula building theorem f... |
| suppssov2 8217 | Formula building theorem f... |
| suppssof1 8218 | Formula building theorem f... |
| suppss2 8219 | Show that the support of a... |
| suppsssn 8220 | Show that the support of a... |
| suppssfv 8221 | Formula building theorem f... |
| suppofssd 8222 | Condition for the support ... |
| suppofss1d 8223 | Condition for the support ... |
| suppofss2d 8224 | Condition for the support ... |
| suppco 8225 | The support of the composi... |
| suppcoss 8226 | The support of the composi... |
| supp0cosupp0 8227 | The support of the composi... |
| imacosupp 8228 | The image of the support o... |
| opeliunxp2f 8229 | Membership in a union of C... |
| mpoxeldm 8230 | If there is an element of ... |
| mpoxneldm 8231 | If the first argument of a... |
| mpoxopn0yelv 8232 | If there is an element of ... |
| mpoxopynvov0g 8233 | If the second argument of ... |
| mpoxopxnop0 8234 | If the first argument of a... |
| mpoxopx0ov0 8235 | If the first argument of a... |
| mpoxopxprcov0 8236 | If the components of the f... |
| mpoxopynvov0 8237 | If the second argument of ... |
| mpoxopoveq 8238 | Value of an operation give... |
| mpoxopovel 8239 | Element of the value of an... |
| mpoxopoveqd 8240 | Value of an operation give... |
| brovex 8241 | A binary relation of the v... |
| brovmpoex 8242 | A binary relation of the v... |
| sprmpod 8243 | The extension of a binary ... |
| tposss 8246 | Subset theorem for transpo... |
| tposeq 8247 | Equality theorem for trans... |
| tposeqd 8248 | Equality theorem for trans... |
| tposssxp 8249 | The transposition is a sub... |
| reltpos 8250 | The transposition is a rel... |
| brtpos2 8251 | Value of the transposition... |
| brtpos0 8252 | The behavior of ` tpos ` w... |
| reldmtpos 8253 | Necessary and sufficient c... |
| brtpos 8254 | The transposition swaps ar... |
| ottpos 8255 | The transposition swaps th... |
| relbrtpos 8256 | The transposition swaps ar... |
| dmtpos 8257 | The domain of ` tpos F ` w... |
| rntpos 8258 | The range of ` tpos F ` wh... |
| tposexg 8259 | The transposition of a set... |
| ovtpos 8260 | The transposition swaps th... |
| tposfun 8261 | The transposition of a fun... |
| dftpos2 8262 | Alternate definition of ` ... |
| dftpos3 8263 | Alternate definition of ` ... |
| dftpos4 8264 | Alternate definition of ` ... |
| tpostpos 8265 | Value of the double transp... |
| tpostpos2 8266 | Value of the double transp... |
| tposfn2 8267 | The domain of a transposit... |
| tposfo2 8268 | Condition for a surjective... |
| tposf2 8269 | The domain and codomain of... |
| tposf12 8270 | Condition for an injective... |
| tposf1o2 8271 | Condition of a bijective t... |
| tposfo 8272 | The domain and codomain/ra... |
| tposf 8273 | The domain and codomain of... |
| tposfn 8274 | Functionality of a transpo... |
| tpos0 8275 | Transposition of the empty... |
| tposco 8276 | Transposition of a composi... |
| tpossym 8277 | Two ways to say a function... |
| tposeqi 8278 | Equality theorem for trans... |
| tposex 8279 | A transposition is a set. ... |
| nftpos 8280 | Hypothesis builder for tra... |
| tposoprab 8281 | Transposition of a class o... |
| tposmpo 8282 | Transposition of a two-arg... |
| tposconst 8283 | The transposition of a con... |
| mpocurryd 8288 | The currying of an operati... |
| mpocurryvald 8289 | The value of a curried ope... |
| fvmpocurryd 8290 | The value of the value of ... |
| pwuninel2 8293 | Proof of ~ pwuninel under ... |
| pwuninel 8294 | The powerclass of the unio... |
| undefval 8295 | Value of the undefined val... |
| undefnel2 8296 | The undefined value genera... |
| undefnel 8297 | The undefined value genera... |
| undefne0 8298 | The undefined value genera... |
| frecseq123 8301 | Equality theorem for the w... |
| nffrecs 8302 | Bound-variable hypothesis ... |
| csbfrecsg 8303 | Move class substitution in... |
| fpr3g 8304 | Functions defined by well-... |
| frrlem1 8305 | Lemma for well-founded rec... |
| frrlem2 8306 | Lemma for well-founded rec... |
| frrlem3 8307 | Lemma for well-founded rec... |
| frrlem4 8308 | Lemma for well-founded rec... |
| frrlem5 8309 | Lemma for well-founded rec... |
| frrlem6 8310 | Lemma for well-founded rec... |
| frrlem7 8311 | Lemma for well-founded rec... |
| frrlem8 8312 | Lemma for well-founded rec... |
| frrlem9 8313 | Lemma for well-founded rec... |
| frrlem10 8314 | Lemma for well-founded rec... |
| frrlem11 8315 | Lemma for well-founded rec... |
| frrlem12 8316 | Lemma for well-founded rec... |
| frrlem13 8317 | Lemma for well-founded rec... |
| frrlem14 8318 | Lemma for well-founded rec... |
| fprlem1 8319 | Lemma for well-founded rec... |
| fprlem2 8320 | Lemma for well-founded rec... |
| fpr2a 8321 | Weak version of ~ fpr2 whi... |
| fpr1 8322 | Law of well-founded recurs... |
| fpr2 8323 | Law of well-founded recurs... |
| fpr3 8324 | Law of well-founded recurs... |
| frrrel 8325 | Show without using the axi... |
| frrdmss 8326 | Show without using the axi... |
| frrdmcl 8327 | Show without using the axi... |
| fprfung 8328 | A "function" defined by we... |
| fprresex 8329 | The restriction of a funct... |
| dfwrecsOLD 8332 | Obsolete definition of the... |
| wrecseq123 8333 | General equality theorem f... |
| wrecseq123OLD 8334 | Obsolete version of ~ wrec... |
| nfwrecs 8335 | Bound-variable hypothesis ... |
| nfwrecsOLD 8336 | Obsolete proof of ~ nfwrec... |
| wrecseq1 8337 | Equality theorem for the w... |
| wrecseq2 8338 | Equality theorem for the w... |
| wrecseq3 8339 | Equality theorem for the w... |
| csbwrecsg 8340 | Move class substitution in... |
| wfr3g 8341 | Functions defined by well-... |
| wfrlem1OLD 8342 | Lemma for well-ordered rec... |
| wfrlem2OLD 8343 | Lemma for well-ordered rec... |
| wfrlem3OLD 8344 | Lemma for well-ordered rec... |
| wfrlem3OLDa 8345 | Lemma for well-ordered rec... |
| wfrlem4OLD 8346 | Lemma for well-ordered rec... |
| wfrlem5OLD 8347 | Lemma for well-ordered rec... |
| wfrrelOLD 8348 | Obsolete proof of ~ wfrrel... |
| wfrdmssOLD 8349 | Obsolete proof of ~ wfrdms... |
| wfrlem8OLD 8350 | Lemma for well-ordered rec... |
| wfrdmclOLD 8351 | Obsolete version of ~ wfrd... |
| wfrlem10OLD 8352 | Lemma for well-ordered rec... |
| wfrfunOLD 8353 | Obsolete proof of ~ wfrfun... |
| wfrlem12OLD 8354 | Lemma for well-ordered rec... |
| wfrlem13OLD 8355 | Lemma for well-ordered rec... |
| wfrlem14OLD 8356 | Lemma for well-ordered rec... |
| wfrlem15OLD 8357 | Lemma for well-ordered rec... |
| wfrlem16OLD 8358 | Lemma for well-ordered rec... |
| wfrlem17OLD 8359 | Without using ~ ax-rep , s... |
| wfr2aOLD 8360 | Obsolete version of ~ wfr2... |
| wfr1OLD 8361 | Obsolete version of ~ wfr1... |
| wfr2OLD 8362 | Obsolete version of ~ wfr2... |
| wfrrel 8363 | The well-ordered recursion... |
| wfrdmss 8364 | The domain of the well-ord... |
| wfrdmcl 8365 | The predecessor class of a... |
| wfrfun 8366 | The "function" generated b... |
| wfrresex 8367 | Show without using the axi... |
| wfr2a 8368 | A weak version of ~ wfr2 w... |
| wfr1 8369 | The Principle of Well-Orde... |
| wfr2 8370 | The Principle of Well-Orde... |
| wfr3 8371 | The principle of Well-Orde... |
| wfr3OLD 8372 | Obsolete form of ~ wfr3 as... |
| iunon 8373 | The indexed union of a set... |
| iinon 8374 | The nonempty indexed inter... |
| onfununi 8375 | A property of functions on... |
| onovuni 8376 | A variant of ~ onfununi fo... |
| onoviun 8377 | A variant of ~ onovuni wit... |
| onnseq 8378 | There are no length ` _om ... |
| dfsmo2 8381 | Alternate definition of a ... |
| issmo 8382 | Conditions for which ` A `... |
| issmo2 8383 | Alternate definition of a ... |
| smoeq 8384 | Equality theorem for stric... |
| smodm 8385 | The domain of a strictly m... |
| smores 8386 | A strictly monotone functi... |
| smores3 8387 | A strictly monotone functi... |
| smores2 8388 | A strictly monotone ordina... |
| smodm2 8389 | The domain of a strictly m... |
| smofvon2 8390 | The function values of a s... |
| iordsmo 8391 | The identity relation rest... |
| smo0 8392 | The null set is a strictly... |
| smofvon 8393 | If ` B ` is a strictly mon... |
| smoel 8394 | If ` x ` is less than ` y ... |
| smoiun 8395 | The value of a strictly mo... |
| smoiso 8396 | If ` F ` is an isomorphism... |
| smoel2 8397 | A strictly monotone ordina... |
| smo11 8398 | A strictly monotone ordina... |
| smoord 8399 | A strictly monotone ordina... |
| smoword 8400 | A strictly monotone ordina... |
| smogt 8401 | A strictly monotone ordina... |
| smocdmdom 8402 | The codomain of a strictly... |
| smoiso2 8403 | The strictly monotone ordi... |
| dfrecs3 8406 | The old definition of tran... |
| dfrecs3OLD 8407 | Obsolete version of ~ dfre... |
| recseq 8408 | Equality theorem for ` rec... |
| nfrecs 8409 | Bound-variable hypothesis ... |
| tfrlem1 8410 | A technical lemma for tran... |
| tfrlem3a 8411 | Lemma for transfinite recu... |
| tfrlem3 8412 | Lemma for transfinite recu... |
| tfrlem4 8413 | Lemma for transfinite recu... |
| tfrlem5 8414 | Lemma for transfinite recu... |
| recsfval 8415 | Lemma for transfinite recu... |
| tfrlem6 8416 | Lemma for transfinite recu... |
| tfrlem7 8417 | Lemma for transfinite recu... |
| tfrlem8 8418 | Lemma for transfinite recu... |
| tfrlem9 8419 | Lemma for transfinite recu... |
| tfrlem9a 8420 | Lemma for transfinite recu... |
| tfrlem10 8421 | Lemma for transfinite recu... |
| tfrlem11 8422 | Lemma for transfinite recu... |
| tfrlem12 8423 | Lemma for transfinite recu... |
| tfrlem13 8424 | Lemma for transfinite recu... |
| tfrlem14 8425 | Lemma for transfinite recu... |
| tfrlem15 8426 | Lemma for transfinite recu... |
| tfrlem16 8427 | Lemma for finite recursion... |
| tfr1a 8428 | A weak version of ~ tfr1 w... |
| tfr2a 8429 | A weak version of ~ tfr2 w... |
| tfr2b 8430 | Without assuming ~ ax-rep ... |
| tfr1 8431 | Principle of Transfinite R... |
| tfr2 8432 | Principle of Transfinite R... |
| tfr3 8433 | Principle of Transfinite R... |
| tfr1ALT 8434 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8435 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8436 | Alternate proof of ~ tfr3 ... |
| recsfnon 8437 | Strong transfinite recursi... |
| recsval 8438 | Strong transfinite recursi... |
| tz7.44lem1 8439 | The ordered pair abstracti... |
| tz7.44-1 8440 | The value of ` F ` at ` (/... |
| tz7.44-2 8441 | The value of ` F ` at a su... |
| tz7.44-3 8442 | The value of ` F ` at a li... |
| rdgeq1 8445 | Equality theorem for the r... |
| rdgeq2 8446 | Equality theorem for the r... |
| rdgeq12 8447 | Equality theorem for the r... |
| nfrdg 8448 | Bound-variable hypothesis ... |
| rdglem1 8449 | Lemma used with the recurs... |
| rdgfun 8450 | The recursive definition g... |
| rdgdmlim 8451 | The domain of the recursiv... |
| rdgfnon 8452 | The recursive definition g... |
| rdgvalg 8453 | Value of the recursive def... |
| rdgval 8454 | Value of the recursive def... |
| rdg0 8455 | The initial value of the r... |
| rdgseg 8456 | The initial segments of th... |
| rdgsucg 8457 | The value of the recursive... |
| rdgsuc 8458 | The value of the recursive... |
| rdglimg 8459 | The value of the recursive... |
| rdglim 8460 | The value of the recursive... |
| rdg0g 8461 | The initial value of the r... |
| rdgsucmptf 8462 | The value of the recursive... |
| rdgsucmptnf 8463 | The value of the recursive... |
| rdgsucmpt2 8464 | This version of ~ rdgsucmp... |
| rdgsucmpt 8465 | The value of the recursive... |
| rdglim2 8466 | The value of the recursive... |
| rdglim2a 8467 | The value of the recursive... |
| rdg0n 8468 | If ` A ` is a proper class... |
| frfnom 8469 | The function generated by ... |
| fr0g 8470 | The initial value resultin... |
| frsuc 8471 | The successor value result... |
| frsucmpt 8472 | The successor value result... |
| frsucmptn 8473 | The value of the finite re... |
| frsucmpt2 8474 | The successor value result... |
| tz7.48lem 8475 | A way of showing an ordina... |
| tz7.48-2 8476 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8477 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8478 | Proposition 7.48(3) of [Ta... |
| tz7.49 8479 | Proposition 7.49 of [Takeu... |
| tz7.49c 8480 | Corollary of Proposition 7... |
| seqomlem0 8483 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8484 | Lemma for ` seqom ` . The... |
| seqomlem2 8485 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8486 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8487 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8488 | Equality theorem for ` seq... |
| fnseqom 8489 | An index-aware recursive d... |
| seqom0g 8490 | Value of an index-aware re... |
| seqomsuc 8491 | Value of an index-aware re... |
| omsucelsucb 8492 | Membership is inherited by... |
| df1o2 8507 | Expanded value of the ordi... |
| df2o3 8508 | Expanded value of the ordi... |
| df2o2 8509 | Expanded value of the ordi... |
| 1oex 8510 | Ordinal 1 is a set. (Cont... |
| 2oex 8511 | ` 2o ` is a set. (Contrib... |
| 1on 8512 | Ordinal 1 is an ordinal nu... |
| 1onOLD 8513 | Obsolete version of ~ 1on ... |
| 2on 8514 | Ordinal 2 is an ordinal nu... |
| 2onOLD 8515 | Obsolete version of ~ 2on ... |
| 2on0 8516 | Ordinal two is not zero. ... |
| ord3 8517 | Ordinal 3 is an ordinal cl... |
| 3on 8518 | Ordinal 3 is an ordinal nu... |
| 4on 8519 | Ordinal 4 is an ordinal nu... |
| 1oexOLD 8520 | Obsolete version of ~ 1oex... |
| 2oexOLD 8521 | Obsolete version of ~ 2oex... |
| 1n0 8522 | Ordinal one is not equal t... |
| nlim1 8523 | 1 is not a limit ordinal. ... |
| nlim2 8524 | 2 is not a limit ordinal. ... |
| xp01disj 8525 | Cartesian products with th... |
| xp01disjl 8526 | Cartesian products with th... |
| ordgt0ge1 8527 | Two ways to express that a... |
| ordge1n0 8528 | An ordinal greater than or... |
| el1o 8529 | Membership in ordinal one.... |
| ord1eln01 8530 | An ordinal that is not 0 o... |
| ord2eln012 8531 | An ordinal that is not 0, ... |
| 1ellim 8532 | A limit ordinal contains 1... |
| 2ellim 8533 | A limit ordinal contains 2... |
| dif1o 8534 | Two ways to say that ` A `... |
| ondif1 8535 | Two ways to say that ` A `... |
| ondif2 8536 | Two ways to say that ` A `... |
| 2oconcl 8537 | Closure of the pair swappi... |
| 0lt1o 8538 | Ordinal zero is less than ... |
| dif20el 8539 | An ordinal greater than on... |
| 0we1 8540 | The empty set is a well-or... |
| brwitnlem 8541 | Lemma for relations which ... |
| fnoa 8542 | Functionality and domain o... |
| fnom 8543 | Functionality and domain o... |
| fnoe 8544 | Functionality and domain o... |
| oav 8545 | Value of ordinal addition.... |
| omv 8546 | Value of ordinal multiplic... |
| oe0lem 8547 | A helper lemma for ~ oe0 a... |
| oev 8548 | Value of ordinal exponenti... |
| oevn0 8549 | Value of ordinal exponenti... |
| oa0 8550 | Addition with zero. Propo... |
| om0 8551 | Ordinal multiplication wit... |
| oe0m 8552 | Value of zero raised to an... |
| om0x 8553 | Ordinal multiplication wit... |
| oe0m0 8554 | Ordinal exponentiation wit... |
| oe0m1 8555 | Ordinal exponentiation wit... |
| oe0 8556 | Ordinal exponentiation wit... |
| oev2 8557 | Alternate value of ordinal... |
| oasuc 8558 | Addition with successor. ... |
| oesuclem 8559 | Lemma for ~ oesuc . (Cont... |
| omsuc 8560 | Multiplication with succes... |
| oesuc 8561 | Ordinal exponentiation wit... |
| onasuc 8562 | Addition with successor. ... |
| onmsuc 8563 | Multiplication with succes... |
| onesuc 8564 | Exponentiation with a succ... |
| oa1suc 8565 | Addition with 1 is same as... |
| oalim 8566 | Ordinal addition with a li... |
| omlim 8567 | Ordinal multiplication wit... |
| oelim 8568 | Ordinal exponentiation wit... |
| oacl 8569 | Closure law for ordinal ad... |
| omcl 8570 | Closure law for ordinal mu... |
| oecl 8571 | Closure law for ordinal ex... |
| oa0r 8572 | Ordinal addition with zero... |
| om0r 8573 | Ordinal multiplication wit... |
| o1p1e2 8574 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8575 | 2 + 2 = 4 for ordinal numb... |
| om1 8576 | Ordinal multiplication wit... |
| om1r 8577 | Ordinal multiplication wit... |
| oe1 8578 | Ordinal exponentiation wit... |
| oe1m 8579 | Ordinal exponentiation wit... |
| oaordi 8580 | Ordering property of ordin... |
| oaord 8581 | Ordering property of ordin... |
| oacan 8582 | Left cancellation law for ... |
| oaword 8583 | Weak ordering property of ... |
| oawordri 8584 | Weak ordering property of ... |
| oaord1 8585 | An ordinal is less than it... |
| oaword1 8586 | An ordinal is less than or... |
| oaword2 8587 | An ordinal is less than or... |
| oawordeulem 8588 | Lemma for ~ oawordex . (C... |
| oawordeu 8589 | Existence theorem for weak... |
| oawordexr 8590 | Existence theorem for weak... |
| oawordex 8591 | Existence theorem for weak... |
| oaordex 8592 | Existence theorem for orde... |
| oa00 8593 | An ordinal sum is zero iff... |
| oalimcl 8594 | The ordinal sum with a lim... |
| oaass 8595 | Ordinal addition is associ... |
| oarec 8596 | Recursive definition of or... |
| oaf1o 8597 | Left addition by a constan... |
| oacomf1olem 8598 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8599 | Define a bijection from ` ... |
| omordi 8600 | Ordering property of ordin... |
| omord2 8601 | Ordering property of ordin... |
| omord 8602 | Ordering property of ordin... |
| omcan 8603 | Left cancellation law for ... |
| omword 8604 | Weak ordering property of ... |
| omwordi 8605 | Weak ordering property of ... |
| omwordri 8606 | Weak ordering property of ... |
| omword1 8607 | An ordinal is less than or... |
| omword2 8608 | An ordinal is less than or... |
| om00 8609 | The product of two ordinal... |
| om00el 8610 | The product of two nonzero... |
| omordlim 8611 | Ordering involving the pro... |
| omlimcl 8612 | The product of any nonzero... |
| odi 8613 | Distributive law for ordin... |
| omass 8614 | Multiplication of ordinal ... |
| oneo 8615 | If an ordinal number is ev... |
| omeulem1 8616 | Lemma for ~ omeu : existen... |
| omeulem2 8617 | Lemma for ~ omeu : uniquen... |
| omopth2 8618 | An ordered pair-like theor... |
| omeu 8619 | The division algorithm for... |
| oen0 8620 | Ordinal exponentiation wit... |
| oeordi 8621 | Ordering law for ordinal e... |
| oeord 8622 | Ordering property of ordin... |
| oecan 8623 | Left cancellation law for ... |
| oeword 8624 | Weak ordering property of ... |
| oewordi 8625 | Weak ordering property of ... |
| oewordri 8626 | Weak ordering property of ... |
| oeworde 8627 | Ordinal exponentiation com... |
| oeordsuc 8628 | Ordering property of ordin... |
| oelim2 8629 | Ordinal exponentiation wit... |
| oeoalem 8630 | Lemma for ~ oeoa . (Contr... |
| oeoa 8631 | Sum of exponents law for o... |
| oeoelem 8632 | Lemma for ~ oeoe . (Contr... |
| oeoe 8633 | Product of exponents law f... |
| oelimcl 8634 | The ordinal exponential wi... |
| oeeulem 8635 | Lemma for ~ oeeu . (Contr... |
| oeeui 8636 | The division algorithm for... |
| oeeu 8637 | The division algorithm for... |
| nna0 8638 | Addition with zero. Theor... |
| nnm0 8639 | Multiplication with zero. ... |
| nnasuc 8640 | Addition with successor. ... |
| nnmsuc 8641 | Multiplication with succes... |
| nnesuc 8642 | Exponentiation with a succ... |
| nna0r 8643 | Addition to zero. Remark ... |
| nnm0r 8644 | Multiplication with zero. ... |
| nnacl 8645 | Closure of addition of nat... |
| nnmcl 8646 | Closure of multiplication ... |
| nnecl 8647 | Closure of exponentiation ... |
| nnacli 8648 | ` _om ` is closed under ad... |
| nnmcli 8649 | ` _om ` is closed under mu... |
| nnarcl 8650 | Reverse closure law for ad... |
| nnacom 8651 | Addition of natural number... |
| nnaordi 8652 | Ordering property of addit... |
| nnaord 8653 | Ordering property of addit... |
| nnaordr 8654 | Ordering property of addit... |
| nnawordi 8655 | Adding to both sides of an... |
| nnaass 8656 | Addition of natural number... |
| nndi 8657 | Distributive law for natur... |
| nnmass 8658 | Multiplication of natural ... |
| nnmsucr 8659 | Multiplication with succes... |
| nnmcom 8660 | Multiplication of natural ... |
| nnaword 8661 | Weak ordering property of ... |
| nnacan 8662 | Cancellation law for addit... |
| nnaword1 8663 | Weak ordering property of ... |
| nnaword2 8664 | Weak ordering property of ... |
| nnmordi 8665 | Ordering property of multi... |
| nnmord 8666 | Ordering property of multi... |
| nnmword 8667 | Weak ordering property of ... |
| nnmcan 8668 | Cancellation law for multi... |
| nnmwordi 8669 | Weak ordering property of ... |
| nnmwordri 8670 | Weak ordering property of ... |
| nnawordex 8671 | Equivalence for weak order... |
| nnaordex 8672 | Equivalence for ordering. ... |
| nnaordex2 8673 | Equivalence for ordering. ... |
| 1onn 8674 | The ordinal 1 is a natural... |
| 1onnALT 8675 | Shorter proof of ~ 1onn us... |
| 2onn 8676 | The ordinal 2 is a natural... |
| 2onnALT 8677 | Shorter proof of ~ 2onn us... |
| 3onn 8678 | The ordinal 3 is a natural... |
| 4onn 8679 | The ordinal 4 is a natural... |
| 1one2o 8680 | Ordinal one is not ordinal... |
| oaabslem 8681 | Lemma for ~ oaabs . (Cont... |
| oaabs 8682 | Ordinal addition absorbs a... |
| oaabs2 8683 | The absorption law ~ oaabs... |
| omabslem 8684 | Lemma for ~ omabs . (Cont... |
| omabs 8685 | Ordinal multiplication is ... |
| nnm1 8686 | Multiply an element of ` _... |
| nnm2 8687 | Multiply an element of ` _... |
| nn2m 8688 | Multiply an element of ` _... |
| nnneo 8689 | If a natural number is eve... |
| nneob 8690 | A natural number is even i... |
| omsmolem 8691 | Lemma for ~ omsmo . (Cont... |
| omsmo 8692 | A strictly monotonic ordin... |
| omopthlem1 8693 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8694 | Lemma for ~ omopthi . (Co... |
| omopthi 8695 | An ordered pair theorem fo... |
| omopth 8696 | An ordered pair theorem fo... |
| nnasmo 8697 | There is at most one left ... |
| eldifsucnn 8698 | Condition for membership i... |
| on2recsfn 8701 | Show that double recursion... |
| on2recsov 8702 | Calculate the value of the... |
| on2ind 8703 | Double induction over ordi... |
| on3ind 8704 | Triple induction over ordi... |
| coflton 8705 | Cofinality theorem for ord... |
| cofon1 8706 | Cofinality theorem for ord... |
| cofon2 8707 | Cofinality theorem for ord... |
| cofonr 8708 | Inverse cofinality law for... |
| naddfn 8709 | Natural addition is a func... |
| naddcllem 8710 | Lemma for ordinal addition... |
| naddcl 8711 | Closure law for natural ad... |
| naddov 8712 | The value of natural addit... |
| naddov2 8713 | Alternate expression for n... |
| naddov3 8714 | Alternate expression for n... |
| naddf 8715 | Function statement for nat... |
| naddcom 8716 | Natural addition commutes.... |
| naddrid 8717 | Ordinal zero is the additi... |
| naddlid 8718 | Ordinal zero is the additi... |
| naddssim 8719 | Ordinal less-than-or-equal... |
| naddelim 8720 | Ordinal less-than is prese... |
| naddel1 8721 | Ordinal less-than is not a... |
| naddel2 8722 | Ordinal less-than is not a... |
| naddss1 8723 | Ordinal less-than-or-equal... |
| naddss2 8724 | Ordinal less-than-or-equal... |
| naddword1 8725 | Weak-ordering principle fo... |
| naddword2 8726 | Weak-ordering principle fo... |
| naddunif 8727 | Uniformity theorem for nat... |
| naddasslem1 8728 | Lemma for ~ naddass . Exp... |
| naddasslem2 8729 | Lemma for ~ naddass . Exp... |
| naddass 8730 | Natural ordinal addition i... |
| nadd32 8731 | Commutative/associative la... |
| nadd4 8732 | Rearragement of terms in a... |
| nadd42 8733 | Rearragement of terms in a... |
| naddel12 8734 | Natural addition to both s... |
| dfer2 8739 | Alternate definition of eq... |
| dfec2 8741 | Alternate definition of ` ... |
| ecexg 8742 | An equivalence class modul... |
| ecexr 8743 | A nonempty equivalence cla... |
| ereq1 8745 | Equality theorem for equiv... |
| ereq2 8746 | Equality theorem for equiv... |
| errel 8747 | An equivalence relation is... |
| erdm 8748 | The domain of an equivalen... |
| ercl 8749 | Elementhood in the field o... |
| ersym 8750 | An equivalence relation is... |
| ercl2 8751 | Elementhood in the field o... |
| ersymb 8752 | An equivalence relation is... |
| ertr 8753 | An equivalence relation is... |
| ertrd 8754 | A transitivity relation fo... |
| ertr2d 8755 | A transitivity relation fo... |
| ertr3d 8756 | A transitivity relation fo... |
| ertr4d 8757 | A transitivity relation fo... |
| erref 8758 | An equivalence relation is... |
| ercnv 8759 | The converse of an equival... |
| errn 8760 | The range and domain of an... |
| erssxp 8761 | An equivalence relation is... |
| erex 8762 | An equivalence relation is... |
| erexb 8763 | An equivalence relation is... |
| iserd 8764 | A reflexive, symmetric, tr... |
| iseri 8765 | A reflexive, symmetric, tr... |
| iseriALT 8766 | Alternate proof of ~ iseri... |
| brinxper 8767 | Conditions for a reflexive... |
| brdifun 8768 | Evaluate the incomparabili... |
| swoer 8769 | Incomparability under a st... |
| swoord1 8770 | The incomparability equiva... |
| swoord2 8771 | The incomparability equiva... |
| swoso 8772 | If the incomparability rel... |
| eqerlem 8773 | Lemma for ~ eqer . (Contr... |
| eqer 8774 | Equivalence relation invol... |
| ider 8775 | The identity relation is a... |
| 0er 8776 | The empty set is an equiva... |
| eceq1 8777 | Equality theorem for equiv... |
| eceq1d 8778 | Equality theorem for equiv... |
| eceq2 8779 | Equality theorem for equiv... |
| eceq2i 8780 | Equality theorem for the `... |
| eceq2d 8781 | Equality theorem for the `... |
| elecg 8782 | Membership in an equivalen... |
| ecref 8783 | All elements are in their ... |
| elec 8784 | Membership in an equivalen... |
| relelec 8785 | Membership in an equivalen... |
| ecss 8786 | An equivalence class is a ... |
| ecdmn0 8787 | A representative of a none... |
| ereldm 8788 | Equality of equivalence cl... |
| erth 8789 | Basic property of equivale... |
| erth2 8790 | Basic property of equivale... |
| erthi 8791 | Basic property of equivale... |
| erdisj 8792 | Equivalence classes do not... |
| ecidsn 8793 | An equivalence class modul... |
| qseq1 8794 | Equality theorem for quoti... |
| qseq2 8795 | Equality theorem for quoti... |
| qseq2i 8796 | Equality theorem for quoti... |
| qseq1d 8797 | Equality theorem for quoti... |
| qseq2d 8798 | Equality theorem for quoti... |
| qseq12 8799 | Equality theorem for quoti... |
| 0qs 8800 | Quotient set with the empt... |
| elqsg 8801 | Closed form of ~ elqs . (... |
| elqs 8802 | Membership in a quotient s... |
| elqsi 8803 | Membership in a quotient s... |
| elqsecl 8804 | Membership in a quotient s... |
| ecelqsg 8805 | Membership of an equivalen... |
| ecelqsi 8806 | Membership of an equivalen... |
| ecopqsi 8807 | "Closure" law for equivale... |
| qsexg 8808 | A quotient set exists. (C... |
| qsex 8809 | A quotient set exists. (C... |
| uniqs 8810 | The union of a quotient se... |
| qsss 8811 | A quotient set is a set of... |
| uniqs2 8812 | The union of a quotient se... |
| snec 8813 | The singleton of an equiva... |
| ecqs 8814 | Equivalence class in terms... |
| ecid 8815 | A set is equal to its cose... |
| qsid 8816 | A set is equal to its quot... |
| ectocld 8817 | Implicit substitution of c... |
| ectocl 8818 | Implicit substitution of c... |
| elqsn0 8819 | A quotient set does not co... |
| ecelqsdm 8820 | Membership of an equivalen... |
| xpider 8821 | A Cartesian square is an e... |
| iiner 8822 | The intersection of a none... |
| riiner 8823 | The relative intersection ... |
| erinxp 8824 | A restricted equivalence r... |
| ecinxp 8825 | Restrict the relation in a... |
| qsinxp 8826 | Restrict the equivalence r... |
| qsdisj 8827 | Members of a quotient set ... |
| qsdisj2 8828 | A quotient set is a disjoi... |
| qsel 8829 | If an element of a quotien... |
| uniinqs 8830 | Class union distributes ov... |
| qliftlem 8831 | Lemma for theorems about a... |
| qliftrel 8832 | ` F ` , a function lift, i... |
| qliftel 8833 | Elementhood in the relatio... |
| qliftel1 8834 | Elementhood in the relatio... |
| qliftfun 8835 | The function ` F ` is the ... |
| qliftfund 8836 | The function ` F ` is the ... |
| qliftfuns 8837 | The function ` F ` is the ... |
| qliftf 8838 | The domain and codomain of... |
| qliftval 8839 | The value of the function ... |
| ecoptocl 8840 | Implicit substitution of c... |
| 2ecoptocl 8841 | Implicit substitution of c... |
| 3ecoptocl 8842 | Implicit substitution of c... |
| brecop 8843 | Binary relation on a quoti... |
| brecop2 8844 | Binary relation on a quoti... |
| eroveu 8845 | Lemma for ~ erov and ~ ero... |
| erovlem 8846 | Lemma for ~ erov and ~ ero... |
| erov 8847 | The value of an operation ... |
| eroprf 8848 | Functionality of an operat... |
| erov2 8849 | The value of an operation ... |
| eroprf2 8850 | Functionality of an operat... |
| ecopoveq 8851 | This is the first of sever... |
| ecopovsym 8852 | Assuming the operation ` F... |
| ecopovtrn 8853 | Assuming that operation ` ... |
| ecopover 8854 | Assuming that operation ` ... |
| eceqoveq 8855 | Equality of equivalence re... |
| ecovcom 8856 | Lemma used to transfer a c... |
| ecovass 8857 | Lemma used to transfer an ... |
| ecovdi 8858 | Lemma used to transfer a d... |
| mapprc 8863 | When ` A ` is a proper cla... |
| pmex 8864 | The class of all partial f... |
| mapexOLD 8865 | Obsolete version of ~ mape... |
| fnmap 8866 | Set exponentiation has a u... |
| fnpm 8867 | Partial function exponenti... |
| reldmmap 8868 | Set exponentiation is a we... |
| mapvalg 8869 | The value of set exponenti... |
| pmvalg 8870 | The value of the partial m... |
| mapval 8871 | The value of set exponenti... |
| elmapg 8872 | Membership relation for se... |
| elmapd 8873 | Deduction form of ~ elmapg... |
| elmapdd 8874 | Deduction associated with ... |
| mapdm0 8875 | The empty set is the only ... |
| elpmg 8876 | The predicate "is a partia... |
| elpm2g 8877 | The predicate "is a partia... |
| elpm2r 8878 | Sufficient condition for b... |
| elpmi 8879 | A partial function is a fu... |
| pmfun 8880 | A partial function is a fu... |
| elmapex 8881 | Eliminate antecedent for m... |
| elmapi 8882 | A mapping is a function, f... |
| mapfset 8883 | If ` B ` is a set, the val... |
| mapssfset 8884 | The value of the set expon... |
| mapfoss 8885 | The value of the set expon... |
| fsetsspwxp 8886 | The class of all functions... |
| fset0 8887 | The set of functions from ... |
| fsetdmprc0 8888 | The set of functions with ... |
| fsetex 8889 | The set of functions betwe... |
| f1setex 8890 | The set of injections betw... |
| fosetex 8891 | The set of surjections bet... |
| f1osetex 8892 | The set of bijections betw... |
| fsetfcdm 8893 | The class of functions wit... |
| fsetfocdm 8894 | The class of functions wit... |
| fsetprcnex 8895 | The class of all functions... |
| fsetcdmex 8896 | The class of all functions... |
| fsetexb 8897 | The class of all functions... |
| elmapfn 8898 | A mapping is a function wi... |
| elmapfun 8899 | A mapping is always a func... |
| elmapssres 8900 | A restricted mapping is a ... |
| fpmg 8901 | A total function is a part... |
| pmss12g 8902 | Subset relation for the se... |
| pmresg 8903 | Elementhood of a restricte... |
| elmap 8904 | Membership relation for se... |
| mapval2 8905 | Alternate expression for t... |
| elpm 8906 | The predicate "is a partia... |
| elpm2 8907 | The predicate "is a partia... |
| fpm 8908 | A total function is a part... |
| mapsspm 8909 | Set exponentiation is a su... |
| pmsspw 8910 | Partial maps are a subset ... |
| mapsspw 8911 | Set exponentiation is a su... |
| mapfvd 8912 | The value of a function th... |
| elmapresaun 8913 | ~ fresaun transposed to ma... |
| fvmptmap 8914 | Special case of ~ fvmpt fo... |
| map0e 8915 | Set exponentiation with an... |
| map0b 8916 | Set exponentiation with an... |
| map0g 8917 | Set exponentiation is empt... |
| 0map0sn0 8918 | The set of mappings of the... |
| mapsnd 8919 | The value of set exponenti... |
| map0 8920 | Set exponentiation is empt... |
| mapsn 8921 | The value of set exponenti... |
| mapss 8922 | Subset inheritance for set... |
| fdiagfn 8923 | Functionality of the diago... |
| fvdiagfn 8924 | Functionality of the diago... |
| mapsnconst 8925 | Every singleton map is a c... |
| mapsncnv 8926 | Expression for the inverse... |
| mapsnf1o2 8927 | Explicit bijection between... |
| mapsnf1o3 8928 | Explicit bijection in the ... |
| ralxpmap 8929 | Quantification over functi... |
| dfixp 8932 | Eliminate the expression `... |
| ixpsnval 8933 | The value of an infinite C... |
| elixp2 8934 | Membership in an infinite ... |
| fvixp 8935 | Projection of a factor of ... |
| ixpfn 8936 | A nuple is a function. (C... |
| elixp 8937 | Membership in an infinite ... |
| elixpconst 8938 | Membership in an infinite ... |
| ixpconstg 8939 | Infinite Cartesian product... |
| ixpconst 8940 | Infinite Cartesian product... |
| ixpeq1 8941 | Equality theorem for infin... |
| ixpeq1d 8942 | Equality theorem for infin... |
| ss2ixp 8943 | Subclass theorem for infin... |
| ixpeq2 8944 | Equality theorem for infin... |
| ixpeq2dva 8945 | Equality theorem for infin... |
| ixpeq2dv 8946 | Equality theorem for infin... |
| cbvixp 8947 | Change bound variable in a... |
| cbvixpv 8948 | Change bound variable in a... |
| nfixpw 8949 | Bound-variable hypothesis ... |
| nfixp 8950 | Bound-variable hypothesis ... |
| nfixp1 8951 | The index variable in an i... |
| ixpprc 8952 | A cartesian product of pro... |
| ixpf 8953 | A member of an infinite Ca... |
| uniixp 8954 | The union of an infinite C... |
| ixpexg 8955 | The existence of an infini... |
| ixpin 8956 | The intersection of two in... |
| ixpiin 8957 | The indexed intersection o... |
| ixpint 8958 | The intersection of a coll... |
| ixp0x 8959 | An infinite Cartesian prod... |
| ixpssmap2g 8960 | An infinite Cartesian prod... |
| ixpssmapg 8961 | An infinite Cartesian prod... |
| 0elixp 8962 | Membership of the empty se... |
| ixpn0 8963 | The infinite Cartesian pro... |
| ixp0 8964 | The infinite Cartesian pro... |
| ixpssmap 8965 | An infinite Cartesian prod... |
| resixp 8966 | Restriction of an element ... |
| undifixp 8967 | Union of two projections o... |
| mptelixpg 8968 | Condition for an explicit ... |
| resixpfo 8969 | Restriction of elements of... |
| elixpsn 8970 | Membership in a class of s... |
| ixpsnf1o 8971 | A bijection between a clas... |
| mapsnf1o 8972 | A bijection between a set ... |
| boxriin 8973 | A rectangular subset of a ... |
| boxcutc 8974 | The relative complement of... |
| relen 8983 | Equinumerosity is a relati... |
| reldom 8984 | Dominance is a relation. ... |
| relsdom 8985 | Strict dominance is a rela... |
| encv 8986 | If two classes are equinum... |
| breng 8987 | Equinumerosity relation. ... |
| bren 8988 | Equinumerosity relation. ... |
| brenOLD 8989 | Obsolete version of ~ bren... |
| brdom2g 8990 | Dominance relation. This ... |
| brdomg 8991 | Dominance relation. (Cont... |
| brdomgOLD 8992 | Obsolete version of ~ brdo... |
| brdomi 8993 | Dominance relation. (Cont... |
| brdomiOLD 8994 | Obsolete version of ~ brdo... |
| brdom 8995 | Dominance relation. (Cont... |
| domen 8996 | Dominance in terms of equi... |
| domeng 8997 | Dominance in terms of equi... |
| ctex 8998 | A countable set is a set. ... |
| f1oen4g 8999 | The domain and range of a ... |
| f1dom4g 9000 | The domain of a one-to-one... |
| f1oen3g 9001 | The domain and range of a ... |
| f1dom3g 9002 | The domain of a one-to-one... |
| f1oen2g 9003 | The domain and range of a ... |
| f1dom2g 9004 | The domain of a one-to-one... |
| f1dom2gOLD 9005 | Obsolete version of ~ f1do... |
| f1oeng 9006 | The domain and range of a ... |
| f1domg 9007 | The domain of a one-to-one... |
| f1oen 9008 | The domain and range of a ... |
| f1dom 9009 | The domain of a one-to-one... |
| brsdom 9010 | Strict dominance relation,... |
| isfi 9011 | Express " ` A ` is finite"... |
| enssdom 9012 | Equinumerosity implies dom... |
| dfdom2 9013 | Alternate definition of do... |
| endom 9014 | Equinumerosity implies dom... |
| sdomdom 9015 | Strict dominance implies d... |
| sdomnen 9016 | Strict dominance implies n... |
| brdom2 9017 | Dominance in terms of stri... |
| bren2 9018 | Equinumerosity expressed i... |
| enrefg 9019 | Equinumerosity is reflexiv... |
| enref 9020 | Equinumerosity is reflexiv... |
| eqeng 9021 | Equality implies equinumer... |
| domrefg 9022 | Dominance is reflexive. (... |
| en2d 9023 | Equinumerosity inference f... |
| en3d 9024 | Equinumerosity inference f... |
| en2i 9025 | Equinumerosity inference f... |
| en3i 9026 | Equinumerosity inference f... |
| dom2lem 9027 | A mapping (first hypothesi... |
| dom2d 9028 | A mapping (first hypothesi... |
| dom3d 9029 | A mapping (first hypothesi... |
| dom2 9030 | A mapping (first hypothesi... |
| dom3 9031 | A mapping (first hypothesi... |
| idssen 9032 | Equality implies equinumer... |
| domssl 9033 | If ` A ` is a subset of ` ... |
| domssr 9034 | If ` C ` is a superset of ... |
| ssdomg 9035 | A set dominates its subset... |
| ener 9036 | Equinumerosity is an equiv... |
| ensymb 9037 | Symmetry of equinumerosity... |
| ensym 9038 | Symmetry of equinumerosity... |
| ensymi 9039 | Symmetry of equinumerosity... |
| ensymd 9040 | Symmetry of equinumerosity... |
| entr 9041 | Transitivity of equinumero... |
| domtr 9042 | Transitivity of dominance ... |
| entri 9043 | A chained equinumerosity i... |
| entr2i 9044 | A chained equinumerosity i... |
| entr3i 9045 | A chained equinumerosity i... |
| entr4i 9046 | A chained equinumerosity i... |
| endomtr 9047 | Transitivity of equinumero... |
| domentr 9048 | Transitivity of dominance ... |
| f1imaeng 9049 | If a function is one-to-on... |
| f1imaen2g 9050 | If a function is one-to-on... |
| f1imaen3g 9051 | If a set function is one-t... |
| f1imaen 9052 | If a function is one-to-on... |
| en0 9053 | The empty set is equinumer... |
| en0OLD 9054 | Obsolete version of ~ en0 ... |
| en0ALT 9055 | Shorter proof of ~ en0 , d... |
| en0r 9056 | The empty set is equinumer... |
| ensn1 9057 | A singleton is equinumerou... |
| ensn1OLD 9058 | Obsolete version of ~ ensn... |
| ensn1g 9059 | A singleton is equinumerou... |
| enpr1g 9060 | ` { A , A } ` has only one... |
| en1 9061 | A set is equinumerous to o... |
| en1OLD 9062 | Obsolete version of ~ en1 ... |
| en1b 9063 | A set is equinumerous to o... |
| en1bOLD 9064 | Obsolete version of ~ en1b... |
| reuen1 9065 | Two ways to express "exact... |
| euen1 9066 | Two ways to express "exact... |
| euen1b 9067 | Two ways to express " ` A ... |
| en1uniel 9068 | A singleton contains its s... |
| en1unielOLD 9069 | Obsolete version of ~ en1u... |
| 2dom 9070 | A set that dominates ordin... |
| fundmen 9071 | A function is equinumerous... |
| fundmeng 9072 | A function is equinumerous... |
| cnven 9073 | A relational set is equinu... |
| cnvct 9074 | If a set is countable, so ... |
| fndmeng 9075 | A function is equinumerate... |
| mapsnend 9076 | Set exponentiation to a si... |
| mapsnen 9077 | Set exponentiation to a si... |
| snmapen 9078 | Set exponentiation: a sing... |
| snmapen1 9079 | Set exponentiation: a sing... |
| map1 9080 | Set exponentiation: ordina... |
| en2sn 9081 | Two singletons are equinum... |
| en2snOLD 9082 | Obsolete version of ~ en2s... |
| 0fi 9083 | The empty set is finite. ... |
| snfi 9084 | A singleton is finite. (C... |
| snfiOLD 9085 | Obsolete version of ~ snfi... |
| fiprc 9086 | The class of finite sets i... |
| unen 9087 | Equinumerosity of union of... |
| enrefnn 9088 | Equinumerosity is reflexiv... |
| en2prd 9089 | Two unordered pairs are eq... |
| enpr2d 9090 | A pair with distinct eleme... |
| enpr2dOLD 9091 | Obsolete version of ~ enpr... |
| ssct 9092 | Any subset of a countable ... |
| ssctOLD 9093 | Obsolete version of ~ ssct... |
| difsnen 9094 | All decrements of a set ar... |
| domdifsn 9095 | Dominance over a set with ... |
| xpsnen 9096 | A set is equinumerous to i... |
| xpsneng 9097 | A set is equinumerous to i... |
| xp1en 9098 | One times a cardinal numbe... |
| endisj 9099 | Any two sets are equinumer... |
| undom 9100 | Dominance law for union. ... |
| undomOLD 9101 | Obsolete version of ~ undo... |
| xpcomf1o 9102 | The canonical bijection fr... |
| xpcomco 9103 | Composition with the bijec... |
| xpcomen 9104 | Commutative law for equinu... |
| xpcomeng 9105 | Commutative law for equinu... |
| xpsnen2g 9106 | A set is equinumerous to i... |
| xpassen 9107 | Associative law for equinu... |
| xpdom2 9108 | Dominance law for Cartesia... |
| xpdom2g 9109 | Dominance law for Cartesia... |
| xpdom1g 9110 | Dominance law for Cartesia... |
| xpdom3 9111 | A set is dominated by its ... |
| xpdom1 9112 | Dominance law for Cartesia... |
| domunsncan 9113 | A singleton cancellation l... |
| omxpenlem 9114 | Lemma for ~ omxpen . (Con... |
| omxpen 9115 | The cardinal and ordinal p... |
| omf1o 9116 | Construct an explicit bije... |
| pw2f1olem 9117 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9118 | The power set of a set is ... |
| pw2eng 9119 | The power set of a set is ... |
| pw2en 9120 | The power set of a set is ... |
| fopwdom 9121 | Covering implies injection... |
| enfixsn 9122 | Given two equipollent sets... |
| sucdom2OLD 9123 | Obsolete version of ~ sucd... |
| sbthlem1 9124 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9125 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9126 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9127 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9128 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9129 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9130 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9131 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9132 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9133 | Lemma for ~ sbth . (Contr... |
| sbth 9134 | Schroeder-Bernstein Theore... |
| sbthb 9135 | Schroeder-Bernstein Theore... |
| sbthcl 9136 | Schroeder-Bernstein Theore... |
| dfsdom2 9137 | Alternate definition of st... |
| brsdom2 9138 | Alternate definition of st... |
| sdomnsym 9139 | Strict dominance is asymme... |
| domnsym 9140 | Theorem 22(i) of [Suppes] ... |
| 0domg 9141 | Any set dominates the empt... |
| 0domgOLD 9142 | Obsolete version of ~ 0dom... |
| dom0 9143 | A set dominated by the emp... |
| dom0OLD 9144 | Obsolete version of ~ dom0... |
| 0sdomg 9145 | A set strictly dominates t... |
| 0sdomgOLD 9146 | Obsolete version of ~ 0sdo... |
| 0dom 9147 | Any set dominates the empt... |
| 0sdom 9148 | A set strictly dominates t... |
| sdom0 9149 | The empty set does not str... |
| sdom0OLD 9150 | Obsolete version of ~ sdom... |
| sdomdomtr 9151 | Transitivity of strict dom... |
| sdomentr 9152 | Transitivity of strict dom... |
| domsdomtr 9153 | Transitivity of dominance ... |
| ensdomtr 9154 | Transitivity of equinumero... |
| sdomirr 9155 | Strict dominance is irrefl... |
| sdomtr 9156 | Strict dominance is transi... |
| sdomn2lp 9157 | Strict dominance has no 2-... |
| enen1 9158 | Equality-like theorem for ... |
| enen2 9159 | Equality-like theorem for ... |
| domen1 9160 | Equality-like theorem for ... |
| domen2 9161 | Equality-like theorem for ... |
| sdomen1 9162 | Equality-like theorem for ... |
| sdomen2 9163 | Equality-like theorem for ... |
| domtriord 9164 | Dominance is trichotomous ... |
| sdomel 9165 | For ordinals, strict domin... |
| sdomdif 9166 | The difference of a set fr... |
| onsdominel 9167 | An ordinal with more eleme... |
| domunsn 9168 | Dominance over a set with ... |
| fodomr 9169 | There exists a mapping fro... |
| pwdom 9170 | Injection of sets implies ... |
| canth2 9171 | Cantor's Theorem. No set ... |
| canth2g 9172 | Cantor's theorem with the ... |
| 2pwuninel 9173 | The power set of the power... |
| 2pwne 9174 | No set equals the power se... |
| disjen 9175 | A stronger form of ~ pwuni... |
| disjenex 9176 | Existence version of ~ dis... |
| domss2 9177 | A corollary of ~ disjenex ... |
| domssex2 9178 | A corollary of ~ disjenex ... |
| domssex 9179 | Weakening of ~ domssex2 to... |
| xpf1o 9180 | Construct a bijection on a... |
| xpen 9181 | Equinumerosity law for Car... |
| mapen 9182 | Two set exponentiations ar... |
| mapdom1 9183 | Order-preserving property ... |
| mapxpen 9184 | Equinumerosity law for dou... |
| xpmapenlem 9185 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9186 | Equinumerosity law for set... |
| mapunen 9187 | Equinumerosity law for set... |
| map2xp 9188 | A cardinal power with expo... |
| mapdom2 9189 | Order-preserving property ... |
| mapdom3 9190 | Set exponentiation dominat... |
| pwen 9191 | If two sets are equinumero... |
| ssenen 9192 | Equinumerosity of equinume... |
| limenpsi 9193 | A limit ordinal is equinum... |
| limensuci 9194 | A limit ordinal is equinum... |
| limensuc 9195 | A limit ordinal is equinum... |
| infensuc 9196 | Any infinite ordinal is eq... |
| dif1enlem 9197 | Lemma for ~ rexdif1en and ... |
| dif1enlemOLD 9198 | Obsolete version of ~ dif1... |
| rexdif1en 9199 | If a set is equinumerous t... |
| rexdif1enOLD 9200 | Obsolete version of ~ rexd... |
| dif1en 9201 | If a set ` A ` is equinume... |
| dif1ennn 9202 | If a set ` A ` is equinume... |
| dif1enOLD 9203 | Obsolete version of ~ dif1... |
| findcard 9204 | Schema for induction on th... |
| findcard2 9205 | Schema for induction on th... |
| findcard2s 9206 | Variation of ~ findcard2 r... |
| findcard2d 9207 | Deduction version of ~ fin... |
| nnfi 9208 | Natural numbers are finite... |
| pssnn 9209 | A proper subset of a natur... |
| ssnnfi 9210 | A subset of a natural numb... |
| ssnnfiOLD 9211 | Obsolete version of ~ ssnn... |
| 0finOLD 9212 | Obsolete version of ~ 0fi ... |
| unfi 9213 | The union of two finite se... |
| unfid 9214 | The union of two finite se... |
| ssfi 9215 | A subset of a finite set i... |
| ssfiALT 9216 | Shorter proof of ~ ssfi us... |
| diffi 9217 | If ` A ` is finite, ` ( A ... |
| cnvfi 9218 | If a set is finite, its co... |
| fnfi 9219 | A version of ~ fnex for fi... |
| f1oenfi 9220 | If the domain of a one-to-... |
| f1oenfirn 9221 | If the range of a one-to-o... |
| f1domfi 9222 | If the codomain of a one-t... |
| f1domfi2 9223 | If the domain of a one-to-... |
| enreffi 9224 | Equinumerosity is reflexiv... |
| ensymfib 9225 | Symmetry of equinumerosity... |
| entrfil 9226 | Transitivity of equinumero... |
| enfii 9227 | A set equinumerous to a fi... |
| enfi 9228 | Equinumerous sets have the... |
| enfiALT 9229 | Shorter proof of ~ enfi us... |
| domfi 9230 | A set dominated by a finit... |
| entrfi 9231 | Transitivity of equinumero... |
| entrfir 9232 | Transitivity of equinumero... |
| domtrfil 9233 | Transitivity of dominance ... |
| domtrfi 9234 | Transitivity of dominance ... |
| domtrfir 9235 | Transitivity of dominance ... |
| f1imaenfi 9236 | If a function is one-to-on... |
| ssdomfi 9237 | A finite set dominates its... |
| ssdomfi2 9238 | A set dominates its finite... |
| sbthfilem 9239 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9240 | Schroeder-Bernstein Theore... |
| domnsymfi 9241 | If a set dominates a finit... |
| sdomdomtrfi 9242 | Transitivity of strict dom... |
| domsdomtrfi 9243 | Transitivity of dominance ... |
| sucdom2 9244 | Strict dominance of a set ... |
| phplem1 9245 | Lemma for Pigeonhole Princ... |
| phplem2 9246 | Lemma for Pigeonhole Princ... |
| nneneq 9247 | Two equinumerous natural n... |
| php 9248 | Pigeonhole Principle. A n... |
| php2 9249 | Corollary of Pigeonhole Pr... |
| php3 9250 | Corollary of Pigeonhole Pr... |
| php4 9251 | Corollary of the Pigeonhol... |
| php5 9252 | Corollary of the Pigeonhol... |
| phpeqd 9253 | Corollary of the Pigeonhol... |
| nndomog 9254 | Cardinal ordering agrees w... |
| phplem1OLD 9255 | Obsolete lemma for ~ php a... |
| phplem2OLD 9256 | Obsolete lemma for ~ php a... |
| phplem3OLD 9257 | Obsolete version of ~ phpl... |
| phplem4OLD 9258 | Obsolete version of ~ phpl... |
| nneneqOLD 9259 | Obsolete version of ~ nnen... |
| phpOLD 9260 | Obsolete version of ~ php ... |
| php2OLD 9261 | Obsolete version of ~ php2... |
| php3OLD 9262 | Obsolete version of ~ php3... |
| phpeqdOLD 9263 | Obsolete version of ~ phpe... |
| nndomogOLD 9264 | Obsolete version of ~ nndo... |
| snnen2oOLD 9265 | Obsolete version of ~ snne... |
| onomeneq 9266 | An ordinal number equinume... |
| onomeneqOLD 9267 | Obsolete version of ~ onom... |
| onfin 9268 | An ordinal number is finit... |
| onfin2 9269 | A set is a natural number ... |
| nnfiOLD 9270 | Obsolete version of ~ nnfi... |
| nndomo 9271 | Cardinal ordering agrees w... |
| nnsdomo 9272 | Cardinal ordering agrees w... |
| sucdom 9273 | Strict dominance of a set ... |
| sucdomOLD 9274 | Obsolete version of ~ sucd... |
| snnen2o 9275 | A singleton ` { A } ` is n... |
| 0sdom1dom 9276 | Strict dominance over 0 is... |
| 0sdom1domALT 9277 | Alternate proof of ~ 0sdom... |
| 1sdom2 9278 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9279 | Alternate proof of ~ 1sdom... |
| sdom1 9280 | A set has less than one me... |
| sdom1OLD 9281 | Obsolete version of ~ sdom... |
| modom 9282 | Two ways to express "at mo... |
| modom2 9283 | Two ways to express "at mo... |
| rex2dom 9284 | A set that has at least 2 ... |
| 1sdom2dom 9285 | Strict dominance over 1 is... |
| 1sdom 9286 | A set that strictly domina... |
| 1sdomOLD 9287 | Obsolete version of ~ 1sdo... |
| unxpdomlem1 9288 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9289 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9290 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9291 | Cartesian product dominate... |
| unxpdom2 9292 | Corollary of ~ unxpdom . ... |
| sucxpdom 9293 | Cartesian product dominate... |
| pssinf 9294 | A set equinumerous to a pr... |
| fisseneq 9295 | A finite set is equal to i... |
| ominf 9296 | The set of natural numbers... |
| ominfOLD 9297 | Obsolete version of ~ omin... |
| isinf 9298 | Any set that is not finite... |
| isinfOLD 9299 | Obsolete version of ~ isin... |
| fineqvlem 9300 | Lemma for ~ fineqv . (Con... |
| fineqv 9301 | If the Axiom of Infinity i... |
| enfiiOLD 9302 | Obsolete version of ~ enfi... |
| xpfir 9303 | The components of a nonemp... |
| ssfid 9304 | A subset of a finite set i... |
| infi 9305 | The intersection of two se... |
| rabfi 9306 | A restricted class built f... |
| finresfin 9307 | The restriction of a finit... |
| f1finf1o 9308 | Any injection from one fin... |
| f1finf1oOLD 9309 | Obsolete version of ~ f1fi... |
| nfielex 9310 | If a class is not finite, ... |
| en1eqsn 9311 | A set with one element is ... |
| en1eqsnOLD 9312 | Obsolete version of ~ en1e... |
| en1eqsnbi 9313 | A set containing an elemen... |
| dif1ennnALT 9314 | Alternate proof of ~ dif1e... |
| enp1ilem 9315 | Lemma for uses of ~ enp1i ... |
| enp1i 9316 | Proof induction for ~ en2 ... |
| enp1iOLD 9317 | Obsolete version of ~ enp1... |
| en2 9318 | A set equinumerous to ordi... |
| en3 9319 | A set equinumerous to ordi... |
| en4 9320 | A set equinumerous to ordi... |
| findcard3 9321 | Schema for strong inductio... |
| findcard3OLD 9322 | Obsolete version of ~ find... |
| ac6sfi 9323 | A version of ~ ac6s for fi... |
| frfi 9324 | A partial order is well-fo... |
| fimax2g 9325 | A finite set has a maximum... |
| fimaxg 9326 | A finite set has a maximum... |
| fisupg 9327 | Lemma showing existence an... |
| wofi 9328 | A total order on a finite ... |
| ordunifi 9329 | The maximum of a finite co... |
| nnunifi 9330 | The union (supremum) of a ... |
| unblem1 9331 | Lemma for ~ unbnn . After... |
| unblem2 9332 | Lemma for ~ unbnn . The v... |
| unblem3 9333 | Lemma for ~ unbnn . The v... |
| unblem4 9334 | Lemma for ~ unbnn . The f... |
| unbnn 9335 | Any unbounded subset of na... |
| unbnn2 9336 | Version of ~ unbnn that do... |
| isfinite2 9337 | Any set strictly dominated... |
| nnsdomg 9338 | Omega strictly dominates a... |
| nnsdomgOLD 9339 | Obsolete version of ~ nnsd... |
| isfiniteg 9340 | A set is finite iff it is ... |
| infsdomnn 9341 | An infinite set strictly d... |
| infsdomnnOLD 9342 | Obsolete version of ~ infs... |
| infn0 9343 | An infinite set is not emp... |
| infn0ALT 9344 | Shorter proof of ~ infn0 u... |
| fin2inf 9345 | This (useless) theorem, wh... |
| unfilem1 9346 | Lemma for proving that the... |
| unfilem2 9347 | Lemma for proving that the... |
| unfilem3 9348 | Lemma for proving that the... |
| unfir 9349 | If a union is finite, the ... |
| unfib 9350 | A union is finite if and o... |
| unfi2 9351 | The union of two finite se... |
| difinf 9352 | An infinite set ` A ` minu... |
| fodomfi 9353 | An onto function implies d... |
| fofi 9354 | If an onto function has a ... |
| f1fi 9355 | If a 1-to-1 function has a... |
| imafi 9356 | Images of finite sets are ... |
| imafiOLD 9357 | Obsolete version of ~ imaf... |
| pwfir 9358 | If the power set of a set ... |
| pwfilem 9359 | Lemma for ~ pwfi . (Contr... |
| pwfi 9360 | The power set of a finite ... |
| xpfi 9361 | The Cartesian product of t... |
| xpfiOLD 9362 | Obsolete version of ~ xpfi... |
| 3xpfi 9363 | The Cartesian product of t... |
| domunfican 9364 | A finite set union cancell... |
| infcntss 9365 | Every infinite set has a d... |
| prfi 9366 | An unordered pair is finit... |
| prfiALT 9367 | Shorter proof of ~ prfi us... |
| tpfi 9368 | An unordered triple is fin... |
| fiint 9369 | Equivalent ways of stating... |
| fiintOLD 9370 | Obsolete version of ~ fiin... |
| fodomfir 9371 | There exists a mapping fro... |
| fodomfib 9372 | Equivalence of an onto map... |
| fodomfiOLD 9373 | Obsolete version of ~ fodo... |
| fodomfibOLD 9374 | Obsolete version of ~ fodo... |
| fofinf1o 9375 | Any surjection from one fi... |
| rneqdmfinf1o 9376 | Any function from a finite... |
| fidomdm 9377 | Any finite set dominates i... |
| dmfi 9378 | The domain of a finite set... |
| fundmfibi 9379 | A function is finite if an... |
| resfnfinfin 9380 | The restriction of a funct... |
| residfi 9381 | A restricted identity func... |
| cnvfiALT 9382 | Shorter proof of ~ cnvfi u... |
| rnfi 9383 | The range of a finite set ... |
| f1dmvrnfibi 9384 | A one-to-one function whos... |
| f1vrnfibi 9385 | A one-to-one function whic... |
| iunfi 9386 | The finite union of finite... |
| unifi 9387 | The finite union of finite... |
| unifi2 9388 | The finite union of finite... |
| infssuni 9389 | If an infinite set ` A ` i... |
| unirnffid 9390 | The union of the range of ... |
| pwfilemOLD 9391 | Obsolete version of ~ pwfi... |
| pwfiOLD 9392 | Obsolete version of ~ pwfi... |
| mapfi 9393 | Set exponentiation of fini... |
| ixpfi 9394 | A Cartesian product of fin... |
| ixpfi2 9395 | A Cartesian product of fin... |
| mptfi 9396 | A finite mapping set is fi... |
| abrexfi 9397 | An image set from a finite... |
| cnvimamptfin 9398 | A preimage of a mapping wi... |
| elfpw 9399 | Membership in a class of f... |
| unifpw 9400 | A set is the union of its ... |
| f1opwfi 9401 | A one-to-one mapping induc... |
| fissuni 9402 | A finite subset of a union... |
| fipreima 9403 | Given a finite subset ` A ... |
| finsschain 9404 | A finite subset of the uni... |
| indexfi 9405 | If for every element of a ... |
| relfsupp 9408 | The property of a function... |
| relprcnfsupp 9409 | A proper class is never fi... |
| isfsupp 9410 | The property of a class to... |
| isfsuppd 9411 | Deduction form of ~ isfsup... |
| funisfsupp 9412 | The property of a function... |
| fsuppimp 9413 | Implications of a class be... |
| fsuppimpd 9414 | A finitely supported funct... |
| fsuppfund 9415 | A finitely supported funct... |
| fisuppfi 9416 | A function on a finite set... |
| fidmfisupp 9417 | A function with a finite d... |
| fdmfisuppfi 9418 | The support of a function ... |
| fdmfifsupp 9419 | A function with a finite d... |
| fsuppmptdm 9420 | A mapping with a finite do... |
| fndmfisuppfi 9421 | The support of a function ... |
| fndmfifsupp 9422 | A function with a finite d... |
| suppeqfsuppbi 9423 | If two functions have the ... |
| suppssfifsupp 9424 | If the support of a functi... |
| fsuppsssupp 9425 | If the support of a functi... |
| fsuppsssuppgd 9426 | If the support of a functi... |
| fsuppss 9427 | A subset of a finitely sup... |
| fsuppssov1 9428 | Formula building theorem f... |
| fsuppxpfi 9429 | The cartesian product of t... |
| fczfsuppd 9430 | A constant function with v... |
| fsuppun 9431 | The union of two finitely ... |
| fsuppunfi 9432 | The union of the support o... |
| fsuppunbi 9433 | If the union of two classe... |
| 0fsupp 9434 | The empty set is a finitel... |
| snopfsupp 9435 | A singleton containing an ... |
| funsnfsupp 9436 | Finite support for a funct... |
| fsuppres 9437 | The restriction of a finit... |
| fmptssfisupp 9438 | The restriction of a mappi... |
| ressuppfi 9439 | If the support of the rest... |
| resfsupp 9440 | If the restriction of a fu... |
| resfifsupp 9441 | The restriction of a funct... |
| ffsuppbi 9442 | Two ways of saying that a ... |
| fsuppmptif 9443 | A function mapping an argu... |
| sniffsupp 9444 | A function mapping all but... |
| fsuppcolem 9445 | Lemma for ~ fsuppco . For... |
| fsuppco 9446 | The composition of a 1-1 f... |
| fsuppco2 9447 | The composition of a funct... |
| fsuppcor 9448 | The composition of a funct... |
| mapfienlem1 9449 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9450 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9451 | Lemma 3 for ~ mapfien . (... |
| mapfien 9452 | A bijection of the base se... |
| mapfien2 9453 | Equinumerousity relation f... |
| fival 9456 | The set of all the finite ... |
| elfi 9457 | Specific properties of an ... |
| elfi2 9458 | The empty intersection nee... |
| elfir 9459 | Sufficient condition for a... |
| intrnfi 9460 | Sufficient condition for t... |
| iinfi 9461 | An indexed intersection of... |
| inelfi 9462 | The intersection of two se... |
| ssfii 9463 | Any element of a set ` A `... |
| fi0 9464 | The set of finite intersec... |
| fieq0 9465 | A set is empty iff the cla... |
| fiin 9466 | The elements of ` ( fi `` ... |
| dffi2 9467 | The set of finite intersec... |
| fiss 9468 | Subset relationship for fu... |
| inficl 9469 | A set which is closed unde... |
| fipwuni 9470 | The set of finite intersec... |
| fisn 9471 | A singleton is closed unde... |
| fiuni 9472 | The union of the finite in... |
| fipwss 9473 | If a set is a family of su... |
| elfiun 9474 | A finite intersection of e... |
| dffi3 9475 | The set of finite intersec... |
| fifo 9476 | Describe a surjection from... |
| marypha1lem 9477 | Core induction for Philip ... |
| marypha1 9478 | (Philip) Hall's marriage t... |
| marypha2lem1 9479 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9480 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9481 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9482 | Lemma for ~ marypha2 . Pr... |
| marypha2 9483 | Version of ~ marypha1 usin... |
| dfsup2 9488 | Quantifier-free definition... |
| supeq1 9489 | Equality theorem for supre... |
| supeq1d 9490 | Equality deduction for sup... |
| supeq1i 9491 | Equality inference for sup... |
| supeq2 9492 | Equality theorem for supre... |
| supeq3 9493 | Equality theorem for supre... |
| supeq123d 9494 | Equality deduction for sup... |
| nfsup 9495 | Hypothesis builder for sup... |
| supmo 9496 | Any class ` B ` has at mos... |
| supexd 9497 | A supremum is a set. (Con... |
| supeu 9498 | A supremum is unique. Sim... |
| supval2 9499 | Alternate expression for t... |
| eqsup 9500 | Sufficient condition for a... |
| eqsupd 9501 | Sufficient condition for a... |
| supcl 9502 | A supremum belongs to its ... |
| supub 9503 | A supremum is an upper bou... |
| suplub 9504 | A supremum is the least up... |
| suplub2 9505 | Bidirectional form of ~ su... |
| supnub 9506 | An upper bound is not less... |
| supex 9507 | A supremum is a set. (Con... |
| sup00 9508 | The supremum under an empt... |
| sup0riota 9509 | The supremum of an empty s... |
| sup0 9510 | The supremum of an empty s... |
| supmax 9511 | The greatest element of a ... |
| fisup2g 9512 | A finite set satisfies the... |
| fisupcl 9513 | A nonempty finite set cont... |
| supgtoreq 9514 | The supremum of a finite s... |
| suppr 9515 | The supremum of a pair. (... |
| supsn 9516 | The supremum of a singleto... |
| supisolem 9517 | Lemma for ~ supiso . (Con... |
| supisoex 9518 | Lemma for ~ supiso . (Con... |
| supiso 9519 | Image of a supremum under ... |
| infeq1 9520 | Equality theorem for infim... |
| infeq1d 9521 | Equality deduction for inf... |
| infeq1i 9522 | Equality inference for inf... |
| infeq2 9523 | Equality theorem for infim... |
| infeq3 9524 | Equality theorem for infim... |
| infeq123d 9525 | Equality deduction for inf... |
| nfinf 9526 | Hypothesis builder for inf... |
| infexd 9527 | An infimum is a set. (Con... |
| eqinf 9528 | Sufficient condition for a... |
| eqinfd 9529 | Sufficient condition for a... |
| infval 9530 | Alternate expression for t... |
| infcllem 9531 | Lemma for ~ infcl , ~ infl... |
| infcl 9532 | An infimum belongs to its ... |
| inflb 9533 | An infimum is a lower boun... |
| infglb 9534 | An infimum is the greatest... |
| infglbb 9535 | Bidirectional form of ~ in... |
| infnlb 9536 | A lower bound is not great... |
| infex 9537 | An infimum is a set. (Con... |
| infmin 9538 | The smallest element of a ... |
| infmo 9539 | Any class ` B ` has at mos... |
| infeu 9540 | An infimum is unique. (Co... |
| fimin2g 9541 | A finite set has a minimum... |
| fiming 9542 | A finite set has a minimum... |
| fiinfg 9543 | Lemma showing existence an... |
| fiinf2g 9544 | A finite set satisfies the... |
| fiinfcl 9545 | A nonempty finite set cont... |
| infltoreq 9546 | The infimum of a finite se... |
| infpr 9547 | The infimum of a pair. (C... |
| infsupprpr 9548 | The infimum of a proper pa... |
| infsn 9549 | The infimum of a singleton... |
| inf00 9550 | The infimum regarding an e... |
| infempty 9551 | The infimum of an empty se... |
| infiso 9552 | Image of an infimum under ... |
| dfoi 9555 | Rewrite ~ df-oi with abbre... |
| oieq1 9556 | Equality theorem for ordin... |
| oieq2 9557 | Equality theorem for ordin... |
| nfoi 9558 | Hypothesis builder for ord... |
| ordiso2 9559 | Generalize ~ ordiso to pro... |
| ordiso 9560 | Order-isomorphic ordinal n... |
| ordtypecbv 9561 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9562 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9563 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9564 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9565 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9566 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9567 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9568 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9569 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9570 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9571 | Lemma for ~ ordtype . Usi... |
| oi0 9572 | Definition of the ordinal ... |
| oicl 9573 | The order type of the well... |
| oif 9574 | The order isomorphism of t... |
| oiiso2 9575 | The order isomorphism of t... |
| ordtype 9576 | For any set-like well-orde... |
| oiiniseg 9577 | ` ran F ` is an initial se... |
| ordtype2 9578 | For any set-like well-orde... |
| oiexg 9579 | The order isomorphism on a... |
| oion 9580 | The order type of the well... |
| oiiso 9581 | The order isomorphism of t... |
| oien 9582 | The order type of a well-o... |
| oieu 9583 | Uniqueness of the unique o... |
| oismo 9584 | When ` A ` is a subclass o... |
| oiid 9585 | The order type of an ordin... |
| hartogslem1 9586 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9587 | Lemma for ~ hartogs . (Co... |
| hartogs 9588 | The class of ordinals domi... |
| wofib 9589 | The only sets which are we... |
| wemaplem1 9590 | Value of the lexicographic... |
| wemaplem2 9591 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9592 | Lemma for ~ wemapso . Tra... |
| wemappo 9593 | Construct lexicographic or... |
| wemapsolem 9594 | Lemma for ~ wemapso . (Co... |
| wemapso 9595 | Construct lexicographic or... |
| wemapso2lem 9596 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9597 | An alternative to having a... |
| card2on 9598 | The alternate definition o... |
| card2inf 9599 | The alternate definition o... |
| harf 9602 | Functionality of the Harto... |
| harcl 9603 | Values of the Hartogs func... |
| harval 9604 | Function value of the Hart... |
| elharval 9605 | The Hartogs number of a se... |
| harndom 9606 | The Hartogs number of a se... |
| harword 9607 | Weak ordering property of ... |
| relwdom 9610 | Weak dominance is a relati... |
| brwdom 9611 | Property of weak dominance... |
| brwdomi 9612 | Property of weak dominance... |
| brwdomn0 9613 | Weak dominance over nonemp... |
| 0wdom 9614 | Any set weakly dominates t... |
| fowdom 9615 | An onto function implies w... |
| wdomref 9616 | Reflexivity of weak domina... |
| brwdom2 9617 | Alternate characterization... |
| domwdom 9618 | Weak dominance is implied ... |
| wdomtr 9619 | Transitivity of weak domin... |
| wdomen1 9620 | Equality-like theorem for ... |
| wdomen2 9621 | Equality-like theorem for ... |
| wdompwdom 9622 | Weak dominance strengthens... |
| canthwdom 9623 | Cantor's Theorem, stated u... |
| wdom2d 9624 | Deduce weak dominance from... |
| wdomd 9625 | Deduce weak dominance from... |
| brwdom3 9626 | Condition for weak dominan... |
| brwdom3i 9627 | Weak dominance implies exi... |
| unwdomg 9628 | Weak dominance of a (disjo... |
| xpwdomg 9629 | Weak dominance of a Cartes... |
| wdomima2g 9630 | A set is weakly dominant o... |
| wdomimag 9631 | A set is weakly dominant o... |
| unxpwdom2 9632 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9633 | If a Cartesian product is ... |
| ixpiunwdom 9634 | Describe an onto function ... |
| harwdom 9635 | The value of the Hartogs f... |
| axreg2 9637 | Axiom of Regularity expres... |
| zfregcl 9638 | The Axiom of Regularity wi... |
| zfreg 9639 | The Axiom of Regularity us... |
| elirrv 9640 | The membership relation is... |
| elirr 9641 | No class is a member of it... |
| elneq 9642 | A class is not equal to an... |
| nelaneq 9643 | A class is not an element ... |
| epinid0 9644 | The membership relation an... |
| sucprcreg 9645 | A class is equal to its su... |
| ruv 9646 | The Russell class is equal... |
| ruALT 9647 | Alternate proof of ~ ru , ... |
| disjcsn 9648 | A class is disjoint from i... |
| zfregfr 9649 | The membership relation is... |
| en2lp 9650 | No class has 2-cycle membe... |
| elnanel 9651 | Two classes are not elemen... |
| cnvepnep 9652 | The membership (epsilon) r... |
| epnsym 9653 | The membership (epsilon) r... |
| elnotel 9654 | A class cannot be an eleme... |
| elnel 9655 | A class cannot be an eleme... |
| en3lplem1 9656 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9657 | Lemma for ~ en3lp . (Cont... |
| en3lp 9658 | No class has 3-cycle membe... |
| preleqg 9659 | Equality of two unordered ... |
| preleq 9660 | Equality of two unordered ... |
| preleqALT 9661 | Alternate proof of ~ prele... |
| opthreg 9662 | Theorem for alternate repr... |
| suc11reg 9663 | The successor operation be... |
| dford2 9664 | Assuming ~ ax-reg , an ord... |
| inf0 9665 | Existence of ` _om ` impli... |
| inf1 9666 | Variation of Axiom of Infi... |
| inf2 9667 | Variation of Axiom of Infi... |
| inf3lema 9668 | Lemma for our Axiom of Inf... |
| inf3lemb 9669 | Lemma for our Axiom of Inf... |
| inf3lemc 9670 | Lemma for our Axiom of Inf... |
| inf3lemd 9671 | Lemma for our Axiom of Inf... |
| inf3lem1 9672 | Lemma for our Axiom of Inf... |
| inf3lem2 9673 | Lemma for our Axiom of Inf... |
| inf3lem3 9674 | Lemma for our Axiom of Inf... |
| inf3lem4 9675 | Lemma for our Axiom of Inf... |
| inf3lem5 9676 | Lemma for our Axiom of Inf... |
| inf3lem6 9677 | Lemma for our Axiom of Inf... |
| inf3lem7 9678 | Lemma for our Axiom of Inf... |
| inf3 9679 | Our Axiom of Infinity ~ ax... |
| infeq5i 9680 | Half of ~ infeq5 . (Contr... |
| infeq5 9681 | The statement "there exist... |
| zfinf 9683 | Axiom of Infinity expresse... |
| axinf2 9684 | A standard version of Axio... |
| zfinf2 9686 | A standard version of the ... |
| omex 9687 | The existence of omega (th... |
| axinf 9688 | The first version of the A... |
| inf5 9689 | The statement "there exist... |
| omelon 9690 | Omega is an ordinal number... |
| dfom3 9691 | The class of natural numbe... |
| elom3 9692 | A simplification of ~ elom... |
| dfom4 9693 | A simplification of ~ df-o... |
| dfom5 9694 | ` _om ` is the smallest li... |
| oancom 9695 | Ordinal addition is not co... |
| isfinite 9696 | A set is finite iff it is ... |
| fict 9697 | A finite set is countable ... |
| nnsdom 9698 | A natural number is strict... |
| omenps 9699 | Omega is equinumerous to a... |
| omensuc 9700 | The set of natural numbers... |
| infdifsn 9701 | Removing a singleton from ... |
| infdiffi 9702 | Removing a finite set from... |
| unbnn3 9703 | Any unbounded subset of na... |
| noinfep 9704 | Using the Axiom of Regular... |
| cantnffval 9707 | The value of the Cantor no... |
| cantnfdm 9708 | The domain of the Cantor n... |
| cantnfvalf 9709 | Lemma for ~ cantnf . The ... |
| cantnfs 9710 | Elementhood in the set of ... |
| cantnfcl 9711 | Basic properties of the or... |
| cantnfval 9712 | The value of the Cantor no... |
| cantnfval2 9713 | Alternate expression for t... |
| cantnfsuc 9714 | The value of the recursive... |
| cantnfle 9715 | A lower bound on the ` CNF... |
| cantnflt 9716 | An upper bound on the part... |
| cantnflt2 9717 | An upper bound on the ` CN... |
| cantnff 9718 | The ` CNF ` function is a ... |
| cantnf0 9719 | The value of the zero func... |
| cantnfrescl 9720 | A function is finitely sup... |
| cantnfres 9721 | The ` CNF ` function respe... |
| cantnfp1lem1 9722 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9723 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9724 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9725 | If ` F ` is created by add... |
| oemapso 9726 | The relation ` T ` is a st... |
| oemapval 9727 | Value of the relation ` T ... |
| oemapvali 9728 | If ` F < G ` , then there ... |
| cantnflem1a 9729 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9730 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9731 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9732 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9733 | Lemma for ~ cantnf . This... |
| cantnflem2 9734 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9735 | Lemma for ~ cantnf . Here... |
| cantnflem4 9736 | Lemma for ~ cantnf . Comp... |
| cantnf 9737 | The Cantor Normal Form the... |
| oemapwe 9738 | The lexicographic order on... |
| cantnffval2 9739 | An alternate definition of... |
| cantnff1o 9740 | Simplify the isomorphism o... |
| wemapwe 9741 | Construct lexicographic or... |
| oef1o 9742 | A bijection of the base se... |
| cnfcomlem 9743 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9744 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9745 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9746 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9747 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9748 | Any infinite ordinal ` B `... |
| cnfcom3clem 9749 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9750 | Wrap the construction of ~... |
| ttrcleq 9753 | Equality theorem for trans... |
| nfttrcld 9754 | Bound variable hypothesis ... |
| nfttrcl 9755 | Bound variable hypothesis ... |
| relttrcl 9756 | The transitive closure of ... |
| brttrcl 9757 | Characterization of elemen... |
| brttrcl2 9758 | Characterization of elemen... |
| ssttrcl 9759 | If ` R ` is a relation, th... |
| ttrcltr 9760 | The transitive closure of ... |
| ttrclresv 9761 | The transitive closure of ... |
| ttrclco 9762 | Composition law for the tr... |
| cottrcl 9763 | Composition law for the tr... |
| ttrclss 9764 | If ` R ` is a subclass of ... |
| dmttrcl 9765 | The domain of a transitive... |
| rnttrcl 9766 | The range of a transitive ... |
| ttrclexg 9767 | If ` R ` is a set, then so... |
| dfttrcl2 9768 | When ` R ` is a set and a ... |
| ttrclselem1 9769 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9770 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9771 | If ` R ` is set-like over ... |
| trcl 9772 | For any set ` A ` , show t... |
| tz9.1 9773 | Every set has a transitive... |
| tz9.1c 9774 | Alternate expression for t... |
| epfrs 9775 | The strong form of the Axi... |
| zfregs 9776 | The strong form of the Axi... |
| zfregs2 9777 | Alternate strong form of t... |
| setind 9778 | Set (epsilon) induction. ... |
| setind2 9779 | Set (epsilon) induction, s... |
| tcvalg 9782 | Value of the transitive cl... |
| tcid 9783 | Defining property of the t... |
| tctr 9784 | Defining property of the t... |
| tcmin 9785 | Defining property of the t... |
| tc2 9786 | A variant of the definitio... |
| tcsni 9787 | The transitive closure of ... |
| tcss 9788 | The transitive closure fun... |
| tcel 9789 | The transitive closure fun... |
| tcidm 9790 | The transitive closure fun... |
| tc0 9791 | The transitive closure of ... |
| tc00 9792 | The transitive closure is ... |
| frmin 9793 | Every (possibly proper) su... |
| frind 9794 | A subclass of a well-found... |
| frinsg 9795 | Well-Founded Induction Sch... |
| frins 9796 | Well-Founded Induction Sch... |
| frins2f 9797 | Well-Founded Induction sch... |
| frins2 9798 | Well-Founded Induction sch... |
| frins3 9799 | Well-Founded Induction sch... |
| frr3g 9800 | Functions defined by well-... |
| frrlem15 9801 | Lemma for general well-fou... |
| frrlem16 9802 | Lemma for general well-fou... |
| frr1 9803 | Law of general well-founde... |
| frr2 9804 | Law of general well-founde... |
| frr3 9805 | Law of general well-founde... |
| r1funlim 9810 | The cumulative hierarchy o... |
| r1fnon 9811 | The cumulative hierarchy o... |
| r10 9812 | Value of the cumulative hi... |
| r1sucg 9813 | Value of the cumulative hi... |
| r1suc 9814 | Value of the cumulative hi... |
| r1limg 9815 | Value of the cumulative hi... |
| r1lim 9816 | Value of the cumulative hi... |
| r1fin 9817 | The first ` _om ` levels o... |
| r1sdom 9818 | Each stage in the cumulati... |
| r111 9819 | The cumulative hierarchy i... |
| r1tr 9820 | The cumulative hierarchy o... |
| r1tr2 9821 | The union of a cumulative ... |
| r1ordg 9822 | Ordering relation for the ... |
| r1ord3g 9823 | Ordering relation for the ... |
| r1ord 9824 | Ordering relation for the ... |
| r1ord2 9825 | Ordering relation for the ... |
| r1ord3 9826 | Ordering relation for the ... |
| r1sssuc 9827 | The value of the cumulativ... |
| r1pwss 9828 | Each set of the cumulative... |
| r1sscl 9829 | Each set of the cumulative... |
| r1val1 9830 | The value of the cumulativ... |
| tz9.12lem1 9831 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9832 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9833 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9834 | A set is well-founded if a... |
| tz9.13 9835 | Every set is well-founded,... |
| tz9.13g 9836 | Every set is well-founded,... |
| rankwflemb 9837 | Two ways of saying a set i... |
| rankf 9838 | The domain and codomain of... |
| rankon 9839 | The rank of a set is an or... |
| r1elwf 9840 | Any member of the cumulati... |
| rankvalb 9841 | Value of the rank function... |
| rankr1ai 9842 | One direction of ~ rankr1a... |
| rankvaln 9843 | Value of the rank function... |
| rankidb 9844 | Identity law for the rank ... |
| rankdmr1 9845 | A rank is a member of the ... |
| rankr1ag 9846 | A version of ~ rankr1a tha... |
| rankr1bg 9847 | A relationship between ran... |
| r1rankidb 9848 | Any set is a subset of the... |
| r1elssi 9849 | The range of the ` R1 ` fu... |
| r1elss 9850 | The range of the ` R1 ` fu... |
| pwwf 9851 | A power set is well-founde... |
| sswf 9852 | A subset of a well-founded... |
| snwf 9853 | A singleton is well-founde... |
| unwf 9854 | A binary union is well-fou... |
| prwf 9855 | An unordered pair is well-... |
| opwf 9856 | An ordered pair is well-fo... |
| unir1 9857 | The cumulative hierarchy o... |
| jech9.3 9858 | Every set belongs to some ... |
| rankwflem 9859 | Every set is well-founded,... |
| rankval 9860 | Value of the rank function... |
| rankvalg 9861 | Value of the rank function... |
| rankval2 9862 | Value of an alternate defi... |
| uniwf 9863 | A union is well-founded if... |
| rankr1clem 9864 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9865 | A relationship between the... |
| rankidn 9866 | A relationship between the... |
| rankpwi 9867 | The rank of a power set. ... |
| rankelb 9868 | The membership relation is... |
| wfelirr 9869 | A well-founded set is not ... |
| rankval3b 9870 | The value of the rank func... |
| ranksnb 9871 | The rank of a singleton. ... |
| rankonidlem 9872 | Lemma for ~ rankonid . (C... |
| rankonid 9873 | The rank of an ordinal num... |
| onwf 9874 | The ordinals are all well-... |
| onssr1 9875 | Initial segments of the or... |
| rankr1g 9876 | A relationship between the... |
| rankid 9877 | Identity law for the rank ... |
| rankr1 9878 | A relationship between the... |
| ssrankr1 9879 | A relationship between an ... |
| rankr1a 9880 | A relationship between ran... |
| r1val2 9881 | The value of the cumulativ... |
| r1val3 9882 | The value of the cumulativ... |
| rankel 9883 | The membership relation is... |
| rankval3 9884 | The value of the rank func... |
| bndrank 9885 | Any class whose elements h... |
| unbndrank 9886 | The elements of a proper c... |
| rankpw 9887 | The rank of a power set. ... |
| ranklim 9888 | The rank of a set belongs ... |
| r1pw 9889 | A stronger property of ` R... |
| r1pwALT 9890 | Alternate shorter proof of... |
| r1pwcl 9891 | The cumulative hierarchy o... |
| rankssb 9892 | The subset relation is inh... |
| rankss 9893 | The subset relation is inh... |
| rankunb 9894 | The rank of the union of t... |
| rankprb 9895 | The rank of an unordered p... |
| rankopb 9896 | The rank of an ordered pai... |
| rankuni2b 9897 | The value of the rank func... |
| ranksn 9898 | The rank of a singleton. ... |
| rankuni2 9899 | The rank of a union. Part... |
| rankun 9900 | The rank of the union of t... |
| rankpr 9901 | The rank of an unordered p... |
| rankop 9902 | The rank of an ordered pai... |
| r1rankid 9903 | Any set is a subset of the... |
| rankeq0b 9904 | A set is empty iff its ran... |
| rankeq0 9905 | A set is empty iff its ran... |
| rankr1id 9906 | The rank of the hierarchy ... |
| rankuni 9907 | The rank of a union. Part... |
| rankr1b 9908 | A relationship between ran... |
| ranksuc 9909 | The rank of a successor. ... |
| rankuniss 9910 | Upper bound of the rank of... |
| rankval4 9911 | The rank of a set is the s... |
| rankbnd 9912 | The rank of a set is bound... |
| rankbnd2 9913 | The rank of a set is bound... |
| rankc1 9914 | A relationship that can be... |
| rankc2 9915 | A relationship that can be... |
| rankelun 9916 | Rank membership is inherit... |
| rankelpr 9917 | Rank membership is inherit... |
| rankelop 9918 | Rank membership is inherit... |
| rankxpl 9919 | A lower bound on the rank ... |
| rankxpu 9920 | An upper bound on the rank... |
| rankfu 9921 | An upper bound on the rank... |
| rankmapu 9922 | An upper bound on the rank... |
| rankxplim 9923 | The rank of a Cartesian pr... |
| rankxplim2 9924 | If the rank of a Cartesian... |
| rankxplim3 9925 | The rank of a Cartesian pr... |
| rankxpsuc 9926 | The rank of a Cartesian pr... |
| tcwf 9927 | The transitive closure fun... |
| tcrank 9928 | This theorem expresses two... |
| scottex 9929 | Scott's trick collects all... |
| scott0 9930 | Scott's trick collects all... |
| scottexs 9931 | Theorem scheme version of ... |
| scott0s 9932 | Theorem scheme version of ... |
| cplem1 9933 | Lemma for the Collection P... |
| cplem2 9934 | Lemma for the Collection P... |
| cp 9935 | Collection Principle. Thi... |
| bnd 9936 | A very strong generalizati... |
| bnd2 9937 | A variant of the Boundedne... |
| kardex 9938 | The collection of all sets... |
| karden 9939 | If we allow the Axiom of R... |
| htalem 9940 | Lemma for defining an emul... |
| hta 9941 | A ZFC emulation of Hilbert... |
| djueq12 9948 | Equality theorem for disjo... |
| djueq1 9949 | Equality theorem for disjo... |
| djueq2 9950 | Equality theorem for disjo... |
| nfdju 9951 | Bound-variable hypothesis ... |
| djuex 9952 | The disjoint union of sets... |
| djuexb 9953 | The disjoint union of two ... |
| djulcl 9954 | Left closure of disjoint u... |
| djurcl 9955 | Right closure of disjoint ... |
| djulf1o 9956 | The left injection functio... |
| djurf1o 9957 | The right injection functi... |
| inlresf 9958 | The left injection restric... |
| inlresf1 9959 | The left injection restric... |
| inrresf 9960 | The right injection restri... |
| inrresf1 9961 | The right injection restri... |
| djuin 9962 | The images of any classes ... |
| djur 9963 | A member of a disjoint uni... |
| djuss 9964 | A disjoint union is a subc... |
| djuunxp 9965 | The union of a disjoint un... |
| djuexALT 9966 | Alternate proof of ~ djuex... |
| eldju1st 9967 | The first component of an ... |
| eldju2ndl 9968 | The second component of an... |
| eldju2ndr 9969 | The second component of an... |
| djuun 9970 | The disjoint union of two ... |
| 1stinl 9971 | The first component of the... |
| 2ndinl 9972 | The second component of th... |
| 1stinr 9973 | The first component of the... |
| 2ndinr 9974 | The second component of th... |
| updjudhf 9975 | The mapping of an element ... |
| updjudhcoinlf 9976 | The composition of the map... |
| updjudhcoinrg 9977 | The composition of the map... |
| updjud 9978 | Universal property of the ... |
| cardf2 9987 | The cardinality function i... |
| cardon 9988 | The cardinal number of a s... |
| isnum2 9989 | A way to express well-orde... |
| isnumi 9990 | A set equinumerous to an o... |
| ennum 9991 | Equinumerous sets are equi... |
| finnum 9992 | Every finite set is numera... |
| onenon 9993 | Every ordinal number is nu... |
| tskwe 9994 | A Tarski set is well-order... |
| xpnum 9995 | The cartesian product of n... |
| cardval3 9996 | An alternate definition of... |
| cardid2 9997 | Any numerable set is equin... |
| isnum3 9998 | A set is numerable iff it ... |
| oncardval 9999 | The value of the cardinal ... |
| oncardid 10000 | Any ordinal number is equi... |
| cardonle 10001 | The cardinal of an ordinal... |
| card0 10002 | The cardinality of the emp... |
| cardidm 10003 | The cardinality function i... |
| oncard 10004 | A set is a cardinal number... |
| ficardom 10005 | The cardinal number of a f... |
| ficardid 10006 | A finite set is equinumero... |
| cardnn 10007 | The cardinality of a natur... |
| cardnueq0 10008 | The empty set is the only ... |
| cardne 10009 | No member of a cardinal nu... |
| carden2a 10010 | If two sets have equal non... |
| carden2b 10011 | If two sets are equinumero... |
| card1 10012 | A set has cardinality one ... |
| cardsn 10013 | A singleton has cardinalit... |
| carddomi2 10014 | Two sets have the dominanc... |
| sdomsdomcardi 10015 | A set strictly dominates i... |
| cardlim 10016 | An infinite cardinal is a ... |
| cardsdomelir 10017 | A cardinal strictly domina... |
| cardsdomel 10018 | A cardinal strictly domina... |
| iscard 10019 | Two ways to express the pr... |
| iscard2 10020 | Two ways to express the pr... |
| carddom2 10021 | Two numerable sets have th... |
| harcard 10022 | The class of ordinal numbe... |
| cardprclem 10023 | Lemma for ~ cardprc . (Co... |
| cardprc 10024 | The class of all cardinal ... |
| carduni 10025 | The union of a set of card... |
| cardiun 10026 | The indexed union of a set... |
| cardennn 10027 | If ` A ` is equinumerous t... |
| cardsucinf 10028 | The cardinality of the suc... |
| cardsucnn 10029 | The cardinality of the suc... |
| cardom 10030 | The set of natural numbers... |
| carden2 10031 | Two numerable sets are equ... |
| cardsdom2 10032 | A numerable set is strictl... |
| domtri2 10033 | Trichotomy of dominance fo... |
| nnsdomel 10034 | Strict dominance and eleme... |
| cardval2 10035 | An alternate version of th... |
| isinffi 10036 | An infinite set contains s... |
| fidomtri 10037 | Trichotomy of dominance wi... |
| fidomtri2 10038 | Trichotomy of dominance wi... |
| harsdom 10039 | The Hartogs number of a we... |
| onsdom 10040 | Any well-orderable set is ... |
| harval2 10041 | An alternate expression fo... |
| harsucnn 10042 | The next cardinal after a ... |
| cardmin2 10043 | The smallest ordinal that ... |
| pm54.43lem 10044 | In Theorem *54.43 of [Whit... |
| pm54.43 10045 | Theorem *54.43 of [Whitehe... |
| enpr2 10046 | An unordered pair with dis... |
| pr2nelemOLD 10047 | Obsolete version of ~ enpr... |
| pr2ne 10048 | If an unordered pair has t... |
| pr2neOLD 10049 | Obsolete version of ~ pr2n... |
| prdom2 10050 | An unordered pair has at m... |
| en2eqpr 10051 | Building a set with two el... |
| en2eleq 10052 | Express a set of pair card... |
| en2other2 10053 | Taking the other element t... |
| dif1card 10054 | The cardinality of a nonem... |
| leweon 10055 | Lexicographical order is a... |
| r0weon 10056 | A set-like well-ordering o... |
| infxpenlem 10057 | Lemma for ~ infxpen . (Co... |
| infxpen 10058 | Every infinite ordinal is ... |
| xpomen 10059 | The Cartesian product of o... |
| xpct 10060 | The cartesian product of t... |
| infxpidm2 10061 | Every infinite well-ordera... |
| infxpenc 10062 | A canonical version of ~ i... |
| infxpenc2lem1 10063 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10064 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10065 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10066 | Existence form of ~ infxpe... |
| iunmapdisj 10067 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10068 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10069 | Lemma for ~ fseqen . (Con... |
| fseqdom 10070 | One half of ~ fseqen . (C... |
| fseqen 10071 | A set that is equinumerous... |
| infpwfidom 10072 | The collection of finite s... |
| dfac8alem 10073 | Lemma for ~ dfac8a . If t... |
| dfac8a 10074 | Numeration theorem: every ... |
| dfac8b 10075 | The well-ordering theorem:... |
| dfac8clem 10076 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10077 | If the union of a set is w... |
| ac10ct 10078 | A proof of the well-orderi... |
| ween 10079 | A set is numerable iff it ... |
| ac5num 10080 | A version of ~ ac5b with t... |
| ondomen 10081 | If a set is dominated by a... |
| numdom 10082 | A set dominated by a numer... |
| ssnum 10083 | A subset of a numerable se... |
| onssnum 10084 | All subsets of the ordinal... |
| indcardi 10085 | Indirect strong induction ... |
| acnrcl 10086 | Reverse closure for the ch... |
| acneq 10087 | Equality theorem for the c... |
| isacn 10088 | The property of being a ch... |
| acni 10089 | The property of being a ch... |
| acni2 10090 | The property of being a ch... |
| acni3 10091 | The property of being a ch... |
| acnlem 10092 | Construct a mapping satisf... |
| numacn 10093 | A well-orderable set has c... |
| finacn 10094 | Every set has finite choic... |
| acndom 10095 | A set with long choice seq... |
| acnnum 10096 | A set ` X ` which has choi... |
| acnen 10097 | The class of choice sets o... |
| acndom2 10098 | A set smaller than one wit... |
| acnen2 10099 | The class of sets with cho... |
| fodomacn 10100 | A version of ~ fodom that ... |
| fodomnum 10101 | A version of ~ fodom that ... |
| fonum 10102 | A surjection maps numerabl... |
| numwdom 10103 | A surjection maps numerabl... |
| fodomfi2 10104 | Onto functions define domi... |
| wdomfil 10105 | Weak dominance agrees with... |
| infpwfien 10106 | Any infinite well-orderabl... |
| inffien 10107 | The set of finite intersec... |
| wdomnumr 10108 | Weak dominance agrees with... |
| alephfnon 10109 | The aleph function is a fu... |
| aleph0 10110 | The first infinite cardina... |
| alephlim 10111 | Value of the aleph functio... |
| alephsuc 10112 | Value of the aleph functio... |
| alephon 10113 | An aleph is an ordinal num... |
| alephcard 10114 | Every aleph is a cardinal ... |
| alephnbtwn 10115 | No cardinal can be sandwic... |
| alephnbtwn2 10116 | No set has equinumerosity ... |
| alephordilem1 10117 | Lemma for ~ alephordi . (... |
| alephordi 10118 | Strict ordering property o... |
| alephord 10119 | Ordering property of the a... |
| alephord2 10120 | Ordering property of the a... |
| alephord2i 10121 | Ordering property of the a... |
| alephord3 10122 | Ordering property of the a... |
| alephsucdom 10123 | A set dominated by an alep... |
| alephsuc2 10124 | An alternate representatio... |
| alephdom 10125 | Relationship between inclu... |
| alephgeom 10126 | Every aleph is greater tha... |
| alephislim 10127 | Every aleph is a limit ord... |
| aleph11 10128 | The aleph function is one-... |
| alephf1 10129 | The aleph function is a on... |
| alephsdom 10130 | If an ordinal is smaller t... |
| alephdom2 10131 | A dominated initial ordina... |
| alephle 10132 | The argument of the aleph ... |
| cardaleph 10133 | Given any transfinite card... |
| cardalephex 10134 | Every transfinite cardinal... |
| infenaleph 10135 | An infinite numerable set ... |
| isinfcard 10136 | Two ways to express the pr... |
| iscard3 10137 | Two ways to express the pr... |
| cardnum 10138 | Two ways to express the cl... |
| alephinit 10139 | An infinite initial ordina... |
| carduniima 10140 | The union of the image of ... |
| cardinfima 10141 | If a mapping to cardinals ... |
| alephiso 10142 | Aleph is an order isomorph... |
| alephprc 10143 | The class of all transfini... |
| alephsson 10144 | The class of transfinite c... |
| unialeph 10145 | The union of the class of ... |
| alephsmo 10146 | The aleph function is stri... |
| alephf1ALT 10147 | Alternate proof of ~ aleph... |
| alephfplem1 10148 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10149 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10150 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10151 | Lemma for ~ alephfp . (Co... |
| alephfp 10152 | The aleph function has a f... |
| alephfp2 10153 | The aleph function has at ... |
| alephval3 10154 | An alternate way to expres... |
| alephsucpw2 10155 | The power set of an aleph ... |
| mappwen 10156 | Power rule for cardinal ar... |
| finnisoeu 10157 | A finite totally ordered s... |
| iunfictbso 10158 | Countability of a countabl... |
| aceq1 10161 | Equivalence of two version... |
| aceq0 10162 | Equivalence of two version... |
| aceq2 10163 | Equivalence of two version... |
| aceq3lem 10164 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10165 | Equivalence of two version... |
| dfac4 10166 | Equivalence of two version... |
| dfac5lem1 10167 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10168 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10169 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10170 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10171 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4OLD 10172 | Obsolete version of ~ dfac... |
| dfac5 10173 | Equivalence of two version... |
| dfac2a 10174 | Our Axiom of Choice (in th... |
| dfac2b 10175 | Axiom of Choice (first for... |
| dfac2 10176 | Axiom of Choice (first for... |
| dfac7 10177 | Equivalence of the Axiom o... |
| dfac0 10178 | Equivalence of two version... |
| dfac1 10179 | Equivalence of two version... |
| dfac8 10180 | A proof of the equivalency... |
| dfac9 10181 | Equivalence of the axiom o... |
| dfac10 10182 | Axiom of Choice equivalent... |
| dfac10c 10183 | Axiom of Choice equivalent... |
| dfac10b 10184 | Axiom of Choice equivalent... |
| acacni 10185 | A choice equivalent: every... |
| dfacacn 10186 | A choice equivalent: every... |
| dfac13 10187 | The axiom of choice holds ... |
| dfac12lem1 10188 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10189 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10190 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10191 | The axiom of choice holds ... |
| dfac12k 10192 | Equivalence of ~ dfac12 an... |
| dfac12a 10193 | The axiom of choice holds ... |
| dfac12 10194 | The axiom of choice holds ... |
| kmlem1 10195 | Lemma for 5-quantifier AC ... |
| kmlem2 10196 | Lemma for 5-quantifier AC ... |
| kmlem3 10197 | Lemma for 5-quantifier AC ... |
| kmlem4 10198 | Lemma for 5-quantifier AC ... |
| kmlem5 10199 | Lemma for 5-quantifier AC ... |
| kmlem6 10200 | Lemma for 5-quantifier AC ... |
| kmlem7 10201 | Lemma for 5-quantifier AC ... |
| kmlem8 10202 | Lemma for 5-quantifier AC ... |
| kmlem9 10203 | Lemma for 5-quantifier AC ... |
| kmlem10 10204 | Lemma for 5-quantifier AC ... |
| kmlem11 10205 | Lemma for 5-quantifier AC ... |
| kmlem12 10206 | Lemma for 5-quantifier AC ... |
| kmlem13 10207 | Lemma for 5-quantifier AC ... |
| kmlem14 10208 | Lemma for 5-quantifier AC ... |
| kmlem15 10209 | Lemma for 5-quantifier AC ... |
| kmlem16 10210 | Lemma for 5-quantifier AC ... |
| dfackm 10211 | Equivalence of the Axiom o... |
| undjudom 10212 | Cardinal addition dominate... |
| endjudisj 10213 | Equinumerosity of a disjoi... |
| djuen 10214 | Disjoint unions of equinum... |
| djuenun 10215 | Disjoint union is equinume... |
| dju1en 10216 | Cardinal addition with car... |
| dju1dif 10217 | Adding and subtracting one... |
| dju1p1e2 10218 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10219 | Alternate proof of ~ dju1p... |
| dju0en 10220 | Cardinal addition with car... |
| xp2dju 10221 | Two times a cardinal numbe... |
| djucomen 10222 | Commutative law for cardin... |
| djuassen 10223 | Associative law for cardin... |
| xpdjuen 10224 | Cardinal multiplication di... |
| mapdjuen 10225 | Sum of exponents law for c... |
| pwdjuen 10226 | Sum of exponents law for c... |
| djudom1 10227 | Ordering law for cardinal ... |
| djudom2 10228 | Ordering law for cardinal ... |
| djudoml 10229 | A set is dominated by its ... |
| djuxpdom 10230 | Cartesian product dominate... |
| djufi 10231 | The disjoint union of two ... |
| cdainflem 10232 | Any partition of omega int... |
| djuinf 10233 | A set is infinite iff the ... |
| infdju1 10234 | An infinite set is equinum... |
| pwdju1 10235 | The sum of a powerset with... |
| pwdjuidm 10236 | If the natural numbers inj... |
| djulepw 10237 | If ` A ` is idempotent und... |
| onadju 10238 | The cardinal and ordinal s... |
| cardadju 10239 | The cardinal sum is equinu... |
| djunum 10240 | The disjoint union of two ... |
| unnum 10241 | The union of two numerable... |
| nnadju 10242 | The cardinal and ordinal s... |
| nnadjuALT 10243 | Shorter proof of ~ nnadju ... |
| ficardadju 10244 | The disjoint union of fini... |
| ficardun 10245 | The cardinality of the uni... |
| ficardun2 10246 | The cardinality of the uni... |
| pwsdompw 10247 | Lemma for ~ domtriom . Th... |
| unctb 10248 | The union of two countable... |
| infdjuabs 10249 | Absorption law for additio... |
| infunabs 10250 | An infinite set is equinum... |
| infdju 10251 | The sum of two cardinal nu... |
| infdif 10252 | The cardinality of an infi... |
| infdif2 10253 | Cardinality ordering for a... |
| infxpdom 10254 | Dominance law for multipli... |
| infxpabs 10255 | Absorption law for multipl... |
| infunsdom1 10256 | The union of two sets that... |
| infunsdom 10257 | The union of two sets that... |
| infxp 10258 | Absorption law for multipl... |
| pwdjudom 10259 | A property of dominance ov... |
| infpss 10260 | Every infinite set has an ... |
| infmap2 10261 | An exponentiation law for ... |
| ackbij2lem1 10262 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10263 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10264 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10265 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10266 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10267 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10268 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10269 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10270 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10271 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10272 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10273 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10274 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10275 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10276 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10277 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10278 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10279 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10280 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10281 | The Ackermann bijection, p... |
| ackbij1b 10282 | The Ackermann bijection, p... |
| ackbij2lem2 10283 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10284 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10285 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10286 | The Ackermann bijection, p... |
| r1om 10287 | The set of hereditarily fi... |
| fictb 10288 | A set is countable iff its... |
| cflem 10289 | A lemma used to simplify c... |
| cflemOLD 10290 | Obsolete version of ~ cfle... |
| cfval 10291 | Value of the cofinality fu... |
| cff 10292 | Cofinality is a function o... |
| cfub 10293 | An upper bound on cofinali... |
| cflm 10294 | Value of the cofinality fu... |
| cf0 10295 | Value of the cofinality fu... |
| cardcf 10296 | Cofinality is a cardinal n... |
| cflecard 10297 | Cofinality is bounded by t... |
| cfle 10298 | Cofinality is bounded by i... |
| cfon 10299 | The cofinality of any set ... |
| cfeq0 10300 | Only the ordinal zero has ... |
| cfsuc 10301 | Value of the cofinality fu... |
| cff1 10302 | There is always a map from... |
| cfflb 10303 | If there is a cofinal map ... |
| cfval2 10304 | Another expression for the... |
| coflim 10305 | A simpler expression for t... |
| cflim3 10306 | Another expression for the... |
| cflim2 10307 | The cofinality function is... |
| cfom 10308 | Value of the cofinality fu... |
| cfss 10309 | There is a cofinal subset ... |
| cfslb 10310 | Any cofinal subset of ` A ... |
| cfslbn 10311 | Any subset of ` A ` smalle... |
| cfslb2n 10312 | Any small collection of sm... |
| cofsmo 10313 | Any cofinal map implies th... |
| cfsmolem 10314 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10315 | The map in ~ cff1 can be a... |
| cfcoflem 10316 | Lemma for ~ cfcof , showin... |
| coftr 10317 | If there is a cofinal map ... |
| cfcof 10318 | If there is a cofinal map ... |
| cfidm 10319 | The cofinality function is... |
| alephsing 10320 | The cofinality of a limit ... |
| sornom 10321 | The range of a single-step... |
| isfin1a 10336 | Definition of a Ia-finite ... |
| fin1ai 10337 | Property of a Ia-finite se... |
| isfin2 10338 | Definition of a II-finite ... |
| fin2i 10339 | Property of a II-finite se... |
| isfin3 10340 | Definition of a III-finite... |
| isfin4 10341 | Definition of a IV-finite ... |
| fin4i 10342 | Infer that a set is IV-inf... |
| isfin5 10343 | Definition of a V-finite s... |
| isfin6 10344 | Definition of a VI-finite ... |
| isfin7 10345 | Definition of a VII-finite... |
| sdom2en01 10346 | A set with less than two e... |
| infpssrlem1 10347 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10348 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10349 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10350 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10351 | Lemma for ~ infpssr . (Co... |
| infpssr 10352 | Dedekind infinity implies ... |
| fin4en1 10353 | Dedekind finite is a cardi... |
| ssfin4 10354 | Dedekind finite sets have ... |
| domfin4 10355 | A set dominated by a Dedek... |
| ominf4 10356 | ` _om ` is Dedekind infini... |
| infpssALT 10357 | Alternate proof of ~ infps... |
| isfin4-2 10358 | Alternate definition of IV... |
| isfin4p1 10359 | Alternate definition of IV... |
| fin23lem7 10360 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10361 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10362 | A II-finite set contains m... |
| isfin2-2 10363 | ` Fin2 ` expressed in term... |
| ssfin2 10364 | A subset of a II-finite se... |
| enfin2i 10365 | II-finiteness is a cardina... |
| fin23lem24 10366 | Lemma for ~ fin23 . In a ... |
| fincssdom 10367 | In a chain of finite sets,... |
| fin23lem25 10368 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10369 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10370 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10371 | Lemma for ~ fin23 but coul... |
| fin23lem27 10372 | The mapping constructed in... |
| isfin3ds 10373 | Property of a III-finite s... |
| ssfin3ds 10374 | A subset of a III-finite s... |
| fin23lem12 10375 | The beginning of the proof... |
| fin23lem13 10376 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10377 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10378 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10379 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10380 | Lemma for ~ fin23 . The f... |
| fin23lem20 10381 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10382 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10383 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10384 | Lemma for ~ fin23 . The r... |
| fin23lem29 10385 | Lemma for ~ fin23 . The r... |
| fin23lem30 10386 | Lemma for ~ fin23 . The r... |
| fin23lem31 10387 | Lemma for ~ fin23 . The r... |
| fin23lem32 10388 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10389 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10390 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10391 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10392 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10393 | Lemma for ~ fin23 . The c... |
| fin23lem39 10394 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10395 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10396 | Lemma for ~ fin23 . A set... |
| isf32lem1 10397 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10398 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10399 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10400 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10401 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10402 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10403 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10404 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10405 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10406 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10407 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10408 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10409 | One half of ~ isfin3-2 . ... |
| isf33lem 10410 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10411 | Weakly Dedekind-infinite s... |
| isfin3-3 10412 | Weakly Dedekind-infinite s... |
| fin33i 10413 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10414 | Lemma for ~ compsscnv . (... |
| compsscnv 10415 | Complementation on a power... |
| isf34lem1 10416 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10417 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10418 | Complementation is an anti... |
| isf34lem3 10419 | Lemma for ~ isfin3-4 . (C... |
| compss 10420 | Express image under of the... |
| isf34lem4 10421 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10422 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10423 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10424 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10425 | Inference from ~ isfin3-4 ... |
| isfin3-4 10426 | Weakly Dedekind-infinite s... |
| fin11a 10427 | Every I-finite set is Ia-f... |
| enfin1ai 10428 | Ia-finiteness is a cardina... |
| isfin1-2 10429 | A set is finite in the usu... |
| isfin1-3 10430 | A set is I-finite iff ever... |
| isfin1-4 10431 | A set is I-finite iff ever... |
| dffin1-5 10432 | Compact quantifier-free ve... |
| fin23 10433 | Every II-finite set (every... |
| fin34 10434 | Every III-finite set is IV... |
| isfin5-2 10435 | Alternate definition of V-... |
| fin45 10436 | Every IV-finite set is V-f... |
| fin56 10437 | Every V-finite set is VI-f... |
| fin17 10438 | Every I-finite set is VII-... |
| fin67 10439 | Every VI-finite set is VII... |
| isfin7-2 10440 | A set is VII-finite iff it... |
| fin71num 10441 | A well-orderable set is VI... |
| dffin7-2 10442 | Class form of ~ isfin7-2 .... |
| dfacfin7 10443 | Axiom of Choice equivalent... |
| fin1a2lem1 10444 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10445 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10446 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10447 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10448 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10449 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10450 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10451 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10452 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10453 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10454 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10455 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10456 | Lemma for ~ fin1a2 . (Con... |
| fin12 10457 | Weak theorem which skips I... |
| fin1a2s 10458 | An II-infinite set can hav... |
| fin1a2 10459 | Every Ia-finite set is II-... |
| itunifval 10460 | Function value of iterated... |
| itunifn 10461 | Functionality of the itera... |
| ituni0 10462 | A zero-fold iterated union... |
| itunisuc 10463 | Successor iterated union. ... |
| itunitc1 10464 | Each union iterate is a me... |
| itunitc 10465 | The union of all union ite... |
| ituniiun 10466 | Unwrap an iterated union f... |
| hsmexlem7 10467 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10468 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10469 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10470 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10471 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10472 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10473 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10474 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10475 | Lemma for ~ hsmex . (Cont... |
| hsmex 10476 | The collection of heredita... |
| hsmex2 10477 | The set of hereditary size... |
| hsmex3 10478 | The set of hereditary size... |
| axcc2lem 10480 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10481 | A possibly more useful ver... |
| axcc3 10482 | A possibly more useful ver... |
| axcc4 10483 | A version of ~ axcc3 that ... |
| acncc 10484 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10485 | Relax the constraint on ~ ... |
| domtriomlem 10486 | Lemma for ~ domtriom . (C... |
| domtriom 10487 | Trichotomy of equinumerosi... |
| fin41 10488 | Under countable choice, th... |
| dominf 10489 | A nonempty set that is a s... |
| dcomex 10491 | The Axiom of Dependent Cho... |
| axdc2lem 10492 | Lemma for ~ axdc2 . We co... |
| axdc2 10493 | An apparent strengthening ... |
| axdc3lem 10494 | The class ` S ` of finite ... |
| axdc3lem2 10495 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10496 | Simple substitution lemma ... |
| axdc3lem4 10497 | Lemma for ~ axdc3 . We ha... |
| axdc3 10498 | Dependent Choice. Axiom D... |
| axdc4lem 10499 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10500 | A more general version of ... |
| axcclem 10501 | Lemma for ~ axcc . (Contr... |
| axcc 10502 | Although CC can be proven ... |
| zfac 10504 | Axiom of Choice expressed ... |
| ac2 10505 | Axiom of Choice equivalent... |
| ac3 10506 | Axiom of Choice using abbr... |
| axac3 10508 | This theorem asserts that ... |
| ackm 10509 | A remarkable equivalent to... |
| axac2 10510 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10511 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10512 | Apply a choice equivalent.... |
| cardeqv 10513 | All sets are well-orderabl... |
| numth3 10514 | All sets are well-orderabl... |
| numth2 10515 | Numeration theorem: any se... |
| numth 10516 | Numeration theorem: every ... |
| ac7 10517 | An Axiom of Choice equival... |
| ac7g 10518 | An Axiom of Choice equival... |
| ac4 10519 | Equivalent of Axiom of Cho... |
| ac4c 10520 | Equivalent of Axiom of Cho... |
| ac5 10521 | An Axiom of Choice equival... |
| ac5b 10522 | Equivalent of Axiom of Cho... |
| ac6num 10523 | A version of ~ ac6 which t... |
| ac6 10524 | Equivalent of Axiom of Cho... |
| ac6c4 10525 | Equivalent of Axiom of Cho... |
| ac6c5 10526 | Equivalent of Axiom of Cho... |
| ac9 10527 | An Axiom of Choice equival... |
| ac6s 10528 | Equivalent of Axiom of Cho... |
| ac6n 10529 | Equivalent of Axiom of Cho... |
| ac6s2 10530 | Generalization of the Axio... |
| ac6s3 10531 | Generalization of the Axio... |
| ac6sg 10532 | ~ ac6s with sethood as ant... |
| ac6sf 10533 | Version of ~ ac6 with boun... |
| ac6s4 10534 | Generalization of the Axio... |
| ac6s5 10535 | Generalization of the Axio... |
| ac8 10536 | An Axiom of Choice equival... |
| ac9s 10537 | An Axiom of Choice equival... |
| numthcor 10538 | Any set is strictly domina... |
| weth 10539 | Well-ordering theorem: any... |
| zorn2lem1 10540 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10541 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10542 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10543 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10544 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10545 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10546 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10547 | Zorn's Lemma of [Monk1] p.... |
| zorng 10548 | Zorn's Lemma. If the unio... |
| zornn0g 10549 | Variant of Zorn's lemma ~ ... |
| zorn2 10550 | Zorn's Lemma of [Monk1] p.... |
| zorn 10551 | Zorn's Lemma. If the unio... |
| zornn0 10552 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10553 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10554 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10555 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10556 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10557 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10558 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10559 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10560 | The Teichmüller-Tukey... |
| ttukeyg 10561 | The Teichmüller-Tukey... |
| ttukey 10562 | The Teichmüller-Tukey... |
| axdclem 10563 | Lemma for ~ axdc . (Contr... |
| axdclem2 10564 | Lemma for ~ axdc . Using ... |
| axdc 10565 | This theorem derives ~ ax-... |
| fodomg 10566 | An onto function implies d... |
| fodom 10567 | An onto function implies d... |
| dmct 10568 | The domain of a countable ... |
| rnct 10569 | The range of a countable s... |
| fodomb 10570 | Equivalence of an onto map... |
| wdomac 10571 | When assuming AC, weak and... |
| brdom3 10572 | Equivalence to a dominance... |
| brdom5 10573 | An equivalence to a domina... |
| brdom4 10574 | An equivalence to a domina... |
| brdom7disj 10575 | An equivalence to a domina... |
| brdom6disj 10576 | An equivalence to a domina... |
| fin71ac 10577 | Once we allow AC, the "str... |
| imadomg 10578 | An image of a function und... |
| fimact 10579 | The image by a function of... |
| fnrndomg 10580 | The range of a function is... |
| fnct 10581 | If the domain of a functio... |
| mptct 10582 | A countable mapping set is... |
| iunfo 10583 | Existence of an onto funct... |
| iundom2g 10584 | An upper bound for the car... |
| iundomg 10585 | An upper bound for the car... |
| iundom 10586 | An upper bound for the car... |
| unidom 10587 | An upper bound for the car... |
| uniimadom 10588 | An upper bound for the car... |
| uniimadomf 10589 | An upper bound for the car... |
| cardval 10590 | The value of the cardinal ... |
| cardid 10591 | Any set is equinumerous to... |
| cardidg 10592 | Any set is equinumerous to... |
| cardidd 10593 | Any set is equinumerous to... |
| cardf 10594 | The cardinality function i... |
| carden 10595 | Two sets are equinumerous ... |
| cardeq0 10596 | Only the empty set has car... |
| unsnen 10597 | Equinumerosity of a set wi... |
| carddom 10598 | Two sets have the dominanc... |
| cardsdom 10599 | Two sets have the strict d... |
| domtri 10600 | Trichotomy law for dominan... |
| entric 10601 | Trichotomy of equinumerosi... |
| entri2 10602 | Trichotomy of dominance an... |
| entri3 10603 | Trichotomy of dominance. ... |
| sdomsdomcard 10604 | A set strictly dominates i... |
| canth3 10605 | Cantor's theorem in terms ... |
| infxpidm 10606 | Every infinite class is eq... |
| ondomon 10607 | The class of ordinals domi... |
| cardmin 10608 | The smallest ordinal that ... |
| ficard 10609 | A set is finite iff its ca... |
| infinf 10610 | Equivalence between two in... |
| unirnfdomd 10611 | The union of the range of ... |
| konigthlem 10612 | Lemma for ~ konigth . (Co... |
| konigth 10613 | Konig's Theorem. If ` m (... |
| alephsucpw 10614 | The power set of an aleph ... |
| aleph1 10615 | The set exponentiation of ... |
| alephval2 10616 | An alternate way to expres... |
| dominfac 10617 | A nonempty set that is a s... |
| iunctb 10618 | The countable union of cou... |
| unictb 10619 | The countable union of cou... |
| infmap 10620 | An exponentiation law for ... |
| alephadd 10621 | The sum of two alephs is t... |
| alephmul 10622 | The product of two alephs ... |
| alephexp1 10623 | An exponentiation law for ... |
| alephsuc3 10624 | An alternate representatio... |
| alephexp2 10625 | An expression equinumerous... |
| alephreg 10626 | A successor aleph is regul... |
| pwcfsdom 10627 | A corollary of Konig's The... |
| cfpwsdom 10628 | A corollary of Konig's The... |
| alephom 10629 | From ~ canth2 , we know th... |
| smobeth 10630 | The beth function is stric... |
| nd1 10631 | A lemma for proving condit... |
| nd2 10632 | A lemma for proving condit... |
| nd3 10633 | A lemma for proving condit... |
| nd4 10634 | A lemma for proving condit... |
| axextnd 10635 | A version of the Axiom of ... |
| axrepndlem1 10636 | Lemma for the Axiom of Rep... |
| axrepndlem2 10637 | Lemma for the Axiom of Rep... |
| axrepnd 10638 | A version of the Axiom of ... |
| axunndlem1 10639 | Lemma for the Axiom of Uni... |
| axunnd 10640 | A version of the Axiom of ... |
| axpowndlem1 10641 | Lemma for the Axiom of Pow... |
| axpowndlem2 10642 | Lemma for the Axiom of Pow... |
| axpowndlem3 10643 | Lemma for the Axiom of Pow... |
| axpowndlem4 10644 | Lemma for the Axiom of Pow... |
| axpownd 10645 | A version of the Axiom of ... |
| axregndlem1 10646 | Lemma for the Axiom of Reg... |
| axregndlem2 10647 | Lemma for the Axiom of Reg... |
| axregnd 10648 | A version of the Axiom of ... |
| axinfndlem1 10649 | Lemma for the Axiom of Inf... |
| axinfnd 10650 | A version of the Axiom of ... |
| axacndlem1 10651 | Lemma for the Axiom of Cho... |
| axacndlem2 10652 | Lemma for the Axiom of Cho... |
| axacndlem3 10653 | Lemma for the Axiom of Cho... |
| axacndlem4 10654 | Lemma for the Axiom of Cho... |
| axacndlem5 10655 | Lemma for the Axiom of Cho... |
| axacnd 10656 | A version of the Axiom of ... |
| zfcndext 10657 | Axiom of Extensionality ~ ... |
| zfcndrep 10658 | Axiom of Replacement ~ ax-... |
| zfcndun 10659 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10660 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10661 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10662 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10663 | Axiom of Choice ~ ax-ac , ... |
| elgch 10666 | Elementhood in the collect... |
| fingch 10667 | A finite set is a GCH-set.... |
| gchi 10668 | The only GCH-sets which ha... |
| gchen1 10669 | If ` A <_ B < ~P A ` , and... |
| gchen2 10670 | If ` A < B <_ ~P A ` , and... |
| gchor 10671 | If ` A <_ B <_ ~P A ` , an... |
| engch 10672 | The property of being a GC... |
| gchdomtri 10673 | Under certain conditions, ... |
| fpwwe2cbv 10674 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10675 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10676 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10677 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10678 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10679 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10680 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10681 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10682 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10683 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10684 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10685 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10686 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10687 | Given any function ` F ` f... |
| fpwwecbv 10688 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10689 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10690 | Given any function ` F ` f... |
| canth4 10691 | An "effective" form of Can... |
| canthnumlem 10692 | Lemma for ~ canthnum . (C... |
| canthnum 10693 | The set of well-orderable ... |
| canthwelem 10694 | Lemma for ~ canthwe . (Co... |
| canthwe 10695 | The set of well-orders of ... |
| canthp1lem1 10696 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10697 | Lemma for ~ canthp1 . (Co... |
| canthp1 10698 | A slightly stronger form o... |
| finngch 10699 | The exclusion of finite se... |
| gchdju1 10700 | An infinite GCH-set is ide... |
| gchinf 10701 | An infinite GCH-set is Ded... |
| pwfseqlem1 10702 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10703 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10704 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10705 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10706 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10707 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10708 | The powerset of a Dedekind... |
| pwxpndom2 10709 | The powerset of a Dedekind... |
| pwxpndom 10710 | The powerset of a Dedekind... |
| pwdjundom 10711 | The powerset of a Dedekind... |
| gchdjuidm 10712 | An infinite GCH-set is ide... |
| gchxpidm 10713 | An infinite GCH-set is ide... |
| gchpwdom 10714 | A relationship between dom... |
| gchaleph 10715 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10716 | If ` ( aleph `` A ) ` and ... |
| hargch 10717 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10718 | If ` ( aleph `` suc A ) ` ... |
| gch2 10719 | It is sufficient to requir... |
| gch3 10720 | An equivalent formulation ... |
| gch-kn 10721 | The equivalence of two ver... |
| gchaclem 10722 | Lemma for ~ gchac (obsolet... |
| gchhar 10723 | A "local" form of ~ gchac ... |
| gchacg 10724 | A "local" form of ~ gchac ... |
| gchac 10725 | The Generalized Continuum ... |
| elwina 10730 | Conditions of weak inacces... |
| elina 10731 | Conditions of strong inacc... |
| winaon 10732 | A weakly inaccessible card... |
| inawinalem 10733 | Lemma for ~ inawina . (Co... |
| inawina 10734 | Every strongly inaccessibl... |
| omina 10735 | ` _om ` is a strongly inac... |
| winacard 10736 | A weakly inaccessible card... |
| winainflem 10737 | A weakly inaccessible card... |
| winainf 10738 | A weakly inaccessible card... |
| winalim 10739 | A weakly inaccessible card... |
| winalim2 10740 | A nontrivial weakly inacce... |
| winafp 10741 | A nontrivial weakly inacce... |
| winafpi 10742 | This theorem, which states... |
| gchina 10743 | Assuming the GCH, weakly a... |
| iswun 10748 | Properties of a weak unive... |
| wuntr 10749 | A weak universe is transit... |
| wununi 10750 | A weak universe is closed ... |
| wunpw 10751 | A weak universe is closed ... |
| wunelss 10752 | The elements of a weak uni... |
| wunpr 10753 | A weak universe is closed ... |
| wunun 10754 | A weak universe is closed ... |
| wuntp 10755 | A weak universe is closed ... |
| wunss 10756 | A weak universe is closed ... |
| wunin 10757 | A weak universe is closed ... |
| wundif 10758 | A weak universe is closed ... |
| wunint 10759 | A weak universe is closed ... |
| wunsn 10760 | A weak universe is closed ... |
| wunsuc 10761 | A weak universe is closed ... |
| wun0 10762 | A weak universe contains t... |
| wunr1om 10763 | A weak universe is infinit... |
| wunom 10764 | A weak universe contains a... |
| wunfi 10765 | A weak universe contains a... |
| wunop 10766 | A weak universe is closed ... |
| wunot 10767 | A weak universe is closed ... |
| wunxp 10768 | A weak universe is closed ... |
| wunpm 10769 | A weak universe is closed ... |
| wunmap 10770 | A weak universe is closed ... |
| wunf 10771 | A weak universe is closed ... |
| wundm 10772 | A weak universe is closed ... |
| wunrn 10773 | A weak universe is closed ... |
| wuncnv 10774 | A weak universe is closed ... |
| wunres 10775 | A weak universe is closed ... |
| wunfv 10776 | A weak universe is closed ... |
| wunco 10777 | A weak universe is closed ... |
| wuntpos 10778 | A weak universe is closed ... |
| intwun 10779 | The intersection of a coll... |
| r1limwun 10780 | Each limit stage in the cu... |
| r1wunlim 10781 | The weak universes in the ... |
| wunex2 10782 | Construct a weak universe ... |
| wunex 10783 | Construct a weak universe ... |
| uniwun 10784 | Every set is contained in ... |
| wunex3 10785 | Construct a weak universe ... |
| wuncval 10786 | Value of the weak universe... |
| wuncid 10787 | The weak universe closure ... |
| wunccl 10788 | The weak universe closure ... |
| wuncss 10789 | The weak universe closure ... |
| wuncidm 10790 | The weak universe closure ... |
| wuncval2 10791 | Our earlier expression for... |
| eltskg 10794 | Properties of a Tarski cla... |
| eltsk2g 10795 | Properties of a Tarski cla... |
| tskpwss 10796 | First axiom of a Tarski cl... |
| tskpw 10797 | Second axiom of a Tarski c... |
| tsken 10798 | Third axiom of a Tarski cl... |
| 0tsk 10799 | The empty set is a (transi... |
| tsksdom 10800 | An element of a Tarski cla... |
| tskssel 10801 | A part of a Tarski class s... |
| tskss 10802 | The subsets of an element ... |
| tskin 10803 | The intersection of two el... |
| tsksn 10804 | A singleton of an element ... |
| tsktrss 10805 | A transitive element of a ... |
| tsksuc 10806 | If an element of a Tarski ... |
| tsk0 10807 | A nonempty Tarski class co... |
| tsk1 10808 | One is an element of a non... |
| tsk2 10809 | Two is an element of a non... |
| 2domtsk 10810 | If a Tarski class is not e... |
| tskr1om 10811 | A nonempty Tarski class is... |
| tskr1om2 10812 | A nonempty Tarski class co... |
| tskinf 10813 | A nonempty Tarski class is... |
| tskpr 10814 | If ` A ` and ` B ` are mem... |
| tskop 10815 | If ` A ` and ` B ` are mem... |
| tskxpss 10816 | A Cartesian product of two... |
| tskwe2 10817 | A Tarski class is well-ord... |
| inttsk 10818 | The intersection of a coll... |
| inar1 10819 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10820 | Alternate proof of ~ r1om ... |
| rankcf 10821 | Any set must be at least a... |
| inatsk 10822 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10823 | The set of hereditarily fi... |
| tskord 10824 | A Tarski class contains al... |
| tskcard 10825 | An even more direct relati... |
| r1tskina 10826 | There is a direct relation... |
| tskuni 10827 | The union of an element of... |
| tskwun 10828 | A nonempty transitive Tars... |
| tskint 10829 | The intersection of an ele... |
| tskun 10830 | The union of two elements ... |
| tskxp 10831 | The Cartesian product of t... |
| tskmap 10832 | Set exponentiation is an e... |
| tskurn 10833 | A transitive Tarski class ... |
| elgrug 10836 | Properties of a Grothendie... |
| grutr 10837 | A Grothendieck universe is... |
| gruelss 10838 | A Grothendieck universe is... |
| grupw 10839 | A Grothendieck universe co... |
| gruss 10840 | Any subset of an element o... |
| grupr 10841 | A Grothendieck universe co... |
| gruurn 10842 | A Grothendieck universe co... |
| gruiun 10843 | If ` B ( x ) ` is a family... |
| gruuni 10844 | A Grothendieck universe co... |
| grurn 10845 | A Grothendieck universe co... |
| gruima 10846 | A Grothendieck universe co... |
| gruel 10847 | Any element of an element ... |
| grusn 10848 | A Grothendieck universe co... |
| gruop 10849 | A Grothendieck universe co... |
| gruun 10850 | A Grothendieck universe co... |
| gruxp 10851 | A Grothendieck universe co... |
| grumap 10852 | A Grothendieck universe co... |
| gruixp 10853 | A Grothendieck universe co... |
| gruiin 10854 | A Grothendieck universe co... |
| gruf 10855 | A Grothendieck universe co... |
| gruen 10856 | A Grothendieck universe co... |
| gruwun 10857 | A nonempty Grothendieck un... |
| intgru 10858 | The intersection of a fami... |
| ingru 10859 | The intersection of a univ... |
| wfgru 10860 | The wellfounded part of a ... |
| grudomon 10861 | Each ordinal that is compa... |
| gruina 10862 | If a Grothendieck universe... |
| grur1a 10863 | A characterization of Grot... |
| grur1 10864 | A characterization of Grot... |
| grutsk1 10865 | Grothendieck universes are... |
| grutsk 10866 | Grothendieck universes are... |
| axgroth5 10868 | The Tarski-Grothendieck ax... |
| axgroth2 10869 | Alternate version of the T... |
| grothpw 10870 | Derive the Axiom of Power ... |
| grothpwex 10871 | Derive the Axiom of Power ... |
| axgroth6 10872 | The Tarski-Grothendieck ax... |
| grothomex 10873 | The Tarski-Grothendieck Ax... |
| grothac 10874 | The Tarski-Grothendieck Ax... |
| axgroth3 10875 | Alternate version of the T... |
| axgroth4 10876 | Alternate version of the T... |
| grothprimlem 10877 | Lemma for ~ grothprim . E... |
| grothprim 10878 | The Tarski-Grothendieck Ax... |
| grothtsk 10879 | The Tarski-Grothendieck Ax... |
| inaprc 10880 | An equivalent to the Tarsk... |
| tskmval 10883 | Value of our tarski map. ... |
| tskmid 10884 | The set ` A ` is an elemen... |
| tskmcl 10885 | A Tarski class that contai... |
| sstskm 10886 | Being a part of ` ( tarski... |
| eltskm 10887 | Belonging to ` ( tarskiMap... |
| elni 10920 | Membership in the class of... |
| elni2 10921 | Membership in the class of... |
| pinn 10922 | A positive integer is a na... |
| pion 10923 | A positive integer is an o... |
| piord 10924 | A positive integer is ordi... |
| niex 10925 | The class of positive inte... |
| 0npi 10926 | The empty set is not a pos... |
| 1pi 10927 | Ordinal 'one' is a positiv... |
| addpiord 10928 | Positive integer addition ... |
| mulpiord 10929 | Positive integer multiplic... |
| mulidpi 10930 | 1 is an identity element f... |
| ltpiord 10931 | Positive integer 'less tha... |
| ltsopi 10932 | Positive integer 'less tha... |
| ltrelpi 10933 | Positive integer 'less tha... |
| dmaddpi 10934 | Domain of addition on posi... |
| dmmulpi 10935 | Domain of multiplication o... |
| addclpi 10936 | Closure of addition of pos... |
| mulclpi 10937 | Closure of multiplication ... |
| addcompi 10938 | Addition of positive integ... |
| addasspi 10939 | Addition of positive integ... |
| mulcompi 10940 | Multiplication of positive... |
| mulasspi 10941 | Multiplication of positive... |
| distrpi 10942 | Multiplication of positive... |
| addcanpi 10943 | Addition cancellation law ... |
| mulcanpi 10944 | Multiplication cancellatio... |
| addnidpi 10945 | There is no identity eleme... |
| ltexpi 10946 | Ordering on positive integ... |
| ltapi 10947 | Ordering property of addit... |
| ltmpi 10948 | Ordering property of multi... |
| 1lt2pi 10949 | One is less than two (one ... |
| nlt1pi 10950 | No positive integer is les... |
| indpi 10951 | Principle of Finite Induct... |
| enqbreq 10963 | Equivalence relation for p... |
| enqbreq2 10964 | Equivalence relation for p... |
| enqer 10965 | The equivalence relation f... |
| enqex 10966 | The equivalence relation f... |
| nqex 10967 | The class of positive frac... |
| 0nnq 10968 | The empty set is not a pos... |
| elpqn 10969 | Each positive fraction is ... |
| ltrelnq 10970 | Positive fraction 'less th... |
| pinq 10971 | The representatives of pos... |
| 1nq 10972 | The positive fraction 'one... |
| nqereu 10973 | There is a unique element ... |
| nqerf 10974 | Corollary of ~ nqereu : th... |
| nqercl 10975 | Corollary of ~ nqereu : cl... |
| nqerrel 10976 | Any member of ` ( N. X. N.... |
| nqerid 10977 | Corollary of ~ nqereu : th... |
| enqeq 10978 | Corollary of ~ nqereu : if... |
| nqereq 10979 | The function ` /Q ` acts a... |
| addpipq2 10980 | Addition of positive fract... |
| addpipq 10981 | Addition of positive fract... |
| addpqnq 10982 | Addition of positive fract... |
| mulpipq2 10983 | Multiplication of positive... |
| mulpipq 10984 | Multiplication of positive... |
| mulpqnq 10985 | Multiplication of positive... |
| ordpipq 10986 | Ordering of positive fract... |
| ordpinq 10987 | Ordering of positive fract... |
| addpqf 10988 | Closure of addition on pos... |
| addclnq 10989 | Closure of addition on pos... |
| mulpqf 10990 | Closure of multiplication ... |
| mulclnq 10991 | Closure of multiplication ... |
| addnqf 10992 | Domain of addition on posi... |
| mulnqf 10993 | Domain of multiplication o... |
| addcompq 10994 | Addition of positive fract... |
| addcomnq 10995 | Addition of positive fract... |
| mulcompq 10996 | Multiplication of positive... |
| mulcomnq 10997 | Multiplication of positive... |
| adderpqlem 10998 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10999 | Lemma for ~ mulerpq . (Co... |
| adderpq 11000 | Addition is compatible wit... |
| mulerpq 11001 | Multiplication is compatib... |
| addassnq 11002 | Addition of positive fract... |
| mulassnq 11003 | Multiplication of positive... |
| mulcanenq 11004 | Lemma for distributive law... |
| distrnq 11005 | Multiplication of positive... |
| 1nqenq 11006 | The equivalence class of r... |
| mulidnq 11007 | Multiplication identity el... |
| recmulnq 11008 | Relationship between recip... |
| recidnq 11009 | A positive fraction times ... |
| recclnq 11010 | Closure law for positive f... |
| recrecnq 11011 | Reciprocal of reciprocal o... |
| dmrecnq 11012 | Domain of reciprocal on po... |
| ltsonq 11013 | 'Less than' is a strict or... |
| lterpq 11014 | Compatibility of ordering ... |
| ltanq 11015 | Ordering property of addit... |
| ltmnq 11016 | Ordering property of multi... |
| 1lt2nq 11017 | One is less than two (one ... |
| ltaddnq 11018 | The sum of two fractions i... |
| ltexnq 11019 | Ordering on positive fract... |
| halfnq 11020 | One-half of any positive f... |
| nsmallnq 11021 | The is no smallest positiv... |
| ltbtwnnq 11022 | There exists a number betw... |
| ltrnq 11023 | Ordering property of recip... |
| archnq 11024 | For any fraction, there is... |
| npex 11030 | The class of positive real... |
| elnp 11031 | Membership in positive rea... |
| elnpi 11032 | Membership in positive rea... |
| prn0 11033 | A positive real is not emp... |
| prpssnq 11034 | A positive real is a subse... |
| elprnq 11035 | A positive real is a set o... |
| 0npr 11036 | The empty set is not a pos... |
| prcdnq 11037 | A positive real is closed ... |
| prub 11038 | A positive fraction not in... |
| prnmax 11039 | A positive real has no lar... |
| npomex 11040 | A simplifying observation,... |
| prnmadd 11041 | A positive real has no lar... |
| ltrelpr 11042 | Positive real 'less than' ... |
| genpv 11043 | Value of general operation... |
| genpelv 11044 | Membership in value of gen... |
| genpprecl 11045 | Pre-closure law for genera... |
| genpdm 11046 | Domain of general operatio... |
| genpn0 11047 | The result of an operation... |
| genpss 11048 | The result of an operation... |
| genpnnp 11049 | The result of an operation... |
| genpcd 11050 | Downward closure of an ope... |
| genpnmax 11051 | An operation on positive r... |
| genpcl 11052 | Closure of an operation on... |
| genpass 11053 | Associativity of an operat... |
| plpv 11054 | Value of addition on posit... |
| mpv 11055 | Value of multiplication on... |
| dmplp 11056 | Domain of addition on posi... |
| dmmp 11057 | Domain of multiplication o... |
| nqpr 11058 | The canonical embedding of... |
| 1pr 11059 | The positive real number '... |
| addclprlem1 11060 | Lemma to prove downward cl... |
| addclprlem2 11061 | Lemma to prove downward cl... |
| addclpr 11062 | Closure of addition on pos... |
| mulclprlem 11063 | Lemma to prove downward cl... |
| mulclpr 11064 | Closure of multiplication ... |
| addcompr 11065 | Addition of positive reals... |
| addasspr 11066 | Addition of positive reals... |
| mulcompr 11067 | Multiplication of positive... |
| mulasspr 11068 | Multiplication of positive... |
| distrlem1pr 11069 | Lemma for distributive law... |
| distrlem4pr 11070 | Lemma for distributive law... |
| distrlem5pr 11071 | Lemma for distributive law... |
| distrpr 11072 | Multiplication of positive... |
| 1idpr 11073 | 1 is an identity element f... |
| ltprord 11074 | Positive real 'less than' ... |
| psslinpr 11075 | Proper subset is a linear ... |
| ltsopr 11076 | Positive real 'less than' ... |
| prlem934 11077 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11078 | The sum of two positive re... |
| ltaddpr2 11079 | The sum of two positive re... |
| ltexprlem1 11080 | Lemma for Proposition 9-3.... |
| ltexprlem2 11081 | Lemma for Proposition 9-3.... |
| ltexprlem3 11082 | Lemma for Proposition 9-3.... |
| ltexprlem4 11083 | Lemma for Proposition 9-3.... |
| ltexprlem5 11084 | Lemma for Proposition 9-3.... |
| ltexprlem6 11085 | Lemma for Proposition 9-3.... |
| ltexprlem7 11086 | Lemma for Proposition 9-3.... |
| ltexpri 11087 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11088 | Lemma for Proposition 9-3.... |
| ltapr 11089 | Ordering property of addit... |
| addcanpr 11090 | Addition cancellation law ... |
| prlem936 11091 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11092 | Lemma for Proposition 9-3.... |
| reclem3pr 11093 | Lemma for Proposition 9-3.... |
| reclem4pr 11094 | Lemma for Proposition 9-3.... |
| recexpr 11095 | The reciprocal of a positi... |
| suplem1pr 11096 | The union of a nonempty, b... |
| suplem2pr 11097 | The union of a set of posi... |
| supexpr 11098 | The union of a nonempty, b... |
| enrer 11107 | The equivalence relation f... |
| nrex1 11108 | The class of signed reals ... |
| enrbreq 11109 | Equivalence relation for s... |
| enreceq 11110 | Equivalence class equality... |
| enrex 11111 | The equivalence relation f... |
| ltrelsr 11112 | Signed real 'less than' is... |
| addcmpblnr 11113 | Lemma showing compatibilit... |
| mulcmpblnrlem 11114 | Lemma used in lemma showin... |
| mulcmpblnr 11115 | Lemma showing compatibilit... |
| prsrlem1 11116 | Decomposing signed reals i... |
| addsrmo 11117 | There is at most one resul... |
| mulsrmo 11118 | There is at most one resul... |
| addsrpr 11119 | Addition of signed reals i... |
| mulsrpr 11120 | Multiplication of signed r... |
| ltsrpr 11121 | Ordering of signed reals i... |
| gt0srpr 11122 | Greater than zero in terms... |
| 0nsr 11123 | The empty set is not a sig... |
| 0r 11124 | The constant ` 0R ` is a s... |
| 1sr 11125 | The constant ` 1R ` is a s... |
| m1r 11126 | The constant ` -1R ` is a ... |
| addclsr 11127 | Closure of addition on sig... |
| mulclsr 11128 | Closure of multiplication ... |
| dmaddsr 11129 | Domain of addition on sign... |
| dmmulsr 11130 | Domain of multiplication o... |
| addcomsr 11131 | Addition of signed reals i... |
| addasssr 11132 | Addition of signed reals i... |
| mulcomsr 11133 | Multiplication of signed r... |
| mulasssr 11134 | Multiplication of signed r... |
| distrsr 11135 | Multiplication of signed r... |
| m1p1sr 11136 | Minus one plus one is zero... |
| m1m1sr 11137 | Minus one times minus one ... |
| ltsosr 11138 | Signed real 'less than' is... |
| 0lt1sr 11139 | 0 is less than 1 for signe... |
| 1ne0sr 11140 | 1 and 0 are distinct for s... |
| 0idsr 11141 | The signed real number 0 i... |
| 1idsr 11142 | 1 is an identity element f... |
| 00sr 11143 | A signed real times 0 is 0... |
| ltasr 11144 | Ordering property of addit... |
| pn0sr 11145 | A signed real plus its neg... |
| negexsr 11146 | Existence of negative sign... |
| recexsrlem 11147 | The reciprocal of a positi... |
| addgt0sr 11148 | The sum of two positive si... |
| mulgt0sr 11149 | The product of two positiv... |
| sqgt0sr 11150 | The square of a nonzero si... |
| recexsr 11151 | The reciprocal of a nonzer... |
| mappsrpr 11152 | Mapping from positive sign... |
| ltpsrpr 11153 | Mapping of order from posi... |
| map2psrpr 11154 | Equivalence for positive s... |
| supsrlem 11155 | Lemma for supremum theorem... |
| supsr 11156 | A nonempty, bounded set of... |
| opelcn 11173 | Ordered pair membership in... |
| opelreal 11174 | Ordered pair membership in... |
| elreal 11175 | Membership in class of rea... |
| elreal2 11176 | Ordered pair membership in... |
| 0ncn 11177 | The empty set is not a com... |
| ltrelre 11178 | 'Less than' is a relation ... |
| addcnsr 11179 | Addition of complex number... |
| mulcnsr 11180 | Multiplication of complex ... |
| eqresr 11181 | Equality of real numbers i... |
| addresr 11182 | Addition of real numbers i... |
| mulresr 11183 | Multiplication of real num... |
| ltresr 11184 | Ordering of real subset of... |
| ltresr2 11185 | Ordering of real subset of... |
| dfcnqs 11186 | Technical trick to permit ... |
| addcnsrec 11187 | Technical trick to permit ... |
| mulcnsrec 11188 | Technical trick to permit ... |
| axaddf 11189 | Addition is an operation o... |
| axmulf 11190 | Multiplication is an opera... |
| axcnex 11191 | The complex numbers form a... |
| axresscn 11192 | The real numbers are a sub... |
| ax1cn 11193 | 1 is a complex number. Ax... |
| axicn 11194 | ` _i ` is a complex number... |
| axaddcl 11195 | Closure law for addition o... |
| axaddrcl 11196 | Closure law for addition i... |
| axmulcl 11197 | Closure law for multiplica... |
| axmulrcl 11198 | Closure law for multiplica... |
| axmulcom 11199 | Multiplication of complex ... |
| axaddass 11200 | Addition of complex number... |
| axmulass 11201 | Multiplication of complex ... |
| axdistr 11202 | Distributive law for compl... |
| axi2m1 11203 | i-squared equals -1 (expre... |
| ax1ne0 11204 | 1 and 0 are distinct. Axi... |
| ax1rid 11205 | ` 1 ` is an identity eleme... |
| axrnegex 11206 | Existence of negative of r... |
| axrrecex 11207 | Existence of reciprocal of... |
| axcnre 11208 | A complex number can be ex... |
| axpre-lttri 11209 | Ordering on reals satisfie... |
| axpre-lttrn 11210 | Ordering on reals is trans... |
| axpre-ltadd 11211 | Ordering property of addit... |
| axpre-mulgt0 11212 | The product of two positiv... |
| axpre-sup 11213 | A nonempty, bounded-above ... |
| wuncn 11214 | A weak universe containing... |
| cnex 11240 | Alias for ~ ax-cnex . See... |
| addcl 11241 | Alias for ~ ax-addcl , for... |
| readdcl 11242 | Alias for ~ ax-addrcl , fo... |
| mulcl 11243 | Alias for ~ ax-mulcl , for... |
| remulcl 11244 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11245 | Alias for ~ ax-mulcom , fo... |
| addass 11246 | Alias for ~ ax-addass , fo... |
| mulass 11247 | Alias for ~ ax-mulass , fo... |
| adddi 11248 | Alias for ~ ax-distr , for... |
| recn 11249 | A real number is a complex... |
| reex 11250 | The real numbers form a se... |
| reelprrecn 11251 | Reals are a subset of the ... |
| cnelprrecn 11252 | Complex numbers are a subs... |
| mpoaddf 11253 | Addition is an operation o... |
| mpomulf 11254 | Multiplication is an opera... |
| elimne0 11255 | Hypothesis for weak deduct... |
| adddir 11256 | Distributive law for compl... |
| 0cn 11257 | Zero is a complex number. ... |
| 0cnd 11258 | Zero is a complex number, ... |
| c0ex 11259 | Zero is a set. (Contribut... |
| 1cnd 11260 | One is a complex number, d... |
| 1ex 11261 | One is a set. (Contribute... |
| cnre 11262 | Alias for ~ ax-cnre , for ... |
| mulrid 11263 | The number 1 is an identit... |
| mullid 11264 | Identity law for multiplic... |
| 1re 11265 | The number 1 is real. Thi... |
| 1red 11266 | The number 1 is real, dedu... |
| 0re 11267 | The number 0 is real. Rem... |
| 0red 11268 | The number 0 is real, dedu... |
| mulridi 11269 | Identity law for multiplic... |
| mullidi 11270 | Identity law for multiplic... |
| addcli 11271 | Closure law for addition. ... |
| mulcli 11272 | Closure law for multiplica... |
| mulcomi 11273 | Commutative law for multip... |
| mulcomli 11274 | Commutative law for multip... |
| addassi 11275 | Associative law for additi... |
| mulassi 11276 | Associative law for multip... |
| adddii 11277 | Distributive law (left-dis... |
| adddiri 11278 | Distributive law (right-di... |
| recni 11279 | A real number is a complex... |
| readdcli 11280 | Closure law for addition o... |
| remulcli 11281 | Closure law for multiplica... |
| mulridd 11282 | Identity law for multiplic... |
| mullidd 11283 | Identity law for multiplic... |
| addcld 11284 | Closure law for addition. ... |
| mulcld 11285 | Closure law for multiplica... |
| mulcomd 11286 | Commutative law for multip... |
| addassd 11287 | Associative law for additi... |
| mulassd 11288 | Associative law for multip... |
| adddid 11289 | Distributive law (left-dis... |
| adddird 11290 | Distributive law (right-di... |
| adddirp1d 11291 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11292 | Join AB+CB into (A+C) on L... |
| recnd 11293 | Deduction from real number... |
| readdcld 11294 | Closure law for addition o... |
| remulcld 11295 | Closure law for multiplica... |
| pnfnre 11306 | Plus infinity is not a rea... |
| pnfnre2 11307 | Plus infinity is not a rea... |
| mnfnre 11308 | Minus infinity is not a re... |
| ressxr 11309 | The standard reals are a s... |
| rexpssxrxp 11310 | The Cartesian product of s... |
| rexr 11311 | A standard real is an exte... |
| 0xr 11312 | Zero is an extended real. ... |
| renepnf 11313 | No (finite) real equals pl... |
| renemnf 11314 | No real equals minus infin... |
| rexrd 11315 | A standard real is an exte... |
| renepnfd 11316 | No (finite) real equals pl... |
| renemnfd 11317 | No real equals minus infin... |
| pnfex 11318 | Plus infinity exists. (Co... |
| pnfxr 11319 | Plus infinity belongs to t... |
| pnfnemnf 11320 | Plus and minus infinity ar... |
| mnfnepnf 11321 | Minus and plus infinity ar... |
| mnfxr 11322 | Minus infinity belongs to ... |
| rexri 11323 | A standard real is an exte... |
| 1xr 11324 | ` 1 ` is an extended real ... |
| renfdisj 11325 | The reals and the infiniti... |
| ltrelxr 11326 | "Less than" is a relation ... |
| ltrel 11327 | "Less than" is a relation.... |
| lerelxr 11328 | "Less than or equal to" is... |
| lerel 11329 | "Less than or equal to" is... |
| xrlenlt 11330 | "Less than or equal to" ex... |
| xrlenltd 11331 | "Less than or equal to" ex... |
| xrltnle 11332 | "Less than" expressed in t... |
| xrnltled 11333 | "Not less than" implies "l... |
| ssxr 11334 | The three (non-exclusive) ... |
| ltxrlt 11335 | The standard less-than ` <... |
| axlttri 11336 | Ordering on reals satisfie... |
| axlttrn 11337 | Ordering on reals is trans... |
| axltadd 11338 | Ordering property of addit... |
| axmulgt0 11339 | The product of two positiv... |
| axsup 11340 | A nonempty, bounded-above ... |
| lttr 11341 | Alias for ~ axlttrn , for ... |
| mulgt0 11342 | The product of two positiv... |
| lenlt 11343 | 'Less than or equal to' ex... |
| ltnle 11344 | 'Less than' expressed in t... |
| ltso 11345 | 'Less than' is a strict or... |
| gtso 11346 | 'Greater than' is a strict... |
| lttri2 11347 | Consequence of trichotomy.... |
| lttri3 11348 | Trichotomy law for 'less t... |
| lttri4 11349 | Trichotomy law for 'less t... |
| letri3 11350 | Trichotomy law. (Contribu... |
| leloe 11351 | 'Less than or equal to' ex... |
| eqlelt 11352 | Equality in terms of 'less... |
| ltle 11353 | 'Less than' implies 'less ... |
| leltne 11354 | 'Less than or equal to' im... |
| lelttr 11355 | Transitive law. (Contribu... |
| leltletr 11356 | Transitive law, weaker for... |
| ltletr 11357 | Transitive law. (Contribu... |
| ltleletr 11358 | Transitive law, weaker for... |
| letr 11359 | Transitive law. (Contribu... |
| ltnr 11360 | 'Less than' is irreflexive... |
| leid 11361 | 'Less than or equal to' is... |
| ltne 11362 | 'Less than' implies not eq... |
| ltnsym 11363 | 'Less than' is not symmetr... |
| ltnsym2 11364 | 'Less than' is antisymmetr... |
| letric 11365 | Trichotomy law. (Contribu... |
| ltlen 11366 | 'Less than' expressed in t... |
| eqle 11367 | Equality implies 'less tha... |
| eqled 11368 | Equality implies 'less tha... |
| ltadd2 11369 | Addition to both sides of ... |
| ne0gt0 11370 | A nonzero nonnegative numb... |
| lecasei 11371 | Ordering elimination by ca... |
| lelttric 11372 | Trichotomy law. (Contribu... |
| ltlecasei 11373 | Ordering elimination by ca... |
| ltnri 11374 | 'Less than' is irreflexive... |
| eqlei 11375 | Equality implies 'less tha... |
| eqlei2 11376 | Equality implies 'less tha... |
| gtneii 11377 | 'Less than' implies not eq... |
| ltneii 11378 | 'Greater than' implies not... |
| lttri2i 11379 | Consequence of trichotomy.... |
| lttri3i 11380 | Consequence of trichotomy.... |
| letri3i 11381 | Consequence of trichotomy.... |
| leloei 11382 | 'Less than or equal to' in... |
| ltleni 11383 | 'Less than' expressed in t... |
| ltnsymi 11384 | 'Less than' is not symmetr... |
| lenlti 11385 | 'Less than or equal to' in... |
| ltnlei 11386 | 'Less than' in terms of 'l... |
| ltlei 11387 | 'Less than' implies 'less ... |
| ltleii 11388 | 'Less than' implies 'less ... |
| ltnei 11389 | 'Less than' implies not eq... |
| letrii 11390 | Trichotomy law for 'less t... |
| lttri 11391 | 'Less than' is transitive.... |
| lelttri 11392 | 'Less than or equal to', '... |
| ltletri 11393 | 'Less than', 'less than or... |
| letri 11394 | 'Less than or equal to' is... |
| le2tri3i 11395 | Extended trichotomy law fo... |
| ltadd2i 11396 | Addition to both sides of ... |
| mulgt0i 11397 | The product of two positiv... |
| mulgt0ii 11398 | The product of two positiv... |
| ltnrd 11399 | 'Less than' is irreflexive... |
| gtned 11400 | 'Less than' implies not eq... |
| ltned 11401 | 'Greater than' implies not... |
| ne0gt0d 11402 | A nonzero nonnegative numb... |
| lttrid 11403 | Ordering on reals satisfie... |
| lttri2d 11404 | Consequence of trichotomy.... |
| lttri3d 11405 | Consequence of trichotomy.... |
| lttri4d 11406 | Trichotomy law for 'less t... |
| letri3d 11407 | Consequence of trichotomy.... |
| leloed 11408 | 'Less than or equal to' in... |
| eqleltd 11409 | Equality in terms of 'less... |
| ltlend 11410 | 'Less than' expressed in t... |
| lenltd 11411 | 'Less than or equal to' in... |
| ltnled 11412 | 'Less than' in terms of 'l... |
| ltled 11413 | 'Less than' implies 'less ... |
| ltnsymd 11414 | 'Less than' implies 'less ... |
| nltled 11415 | 'Not less than ' implies '... |
| lensymd 11416 | 'Less than or equal to' im... |
| letrid 11417 | Trichotomy law for 'less t... |
| leltned 11418 | 'Less than or equal to' im... |
| leneltd 11419 | 'Less than or equal to' an... |
| mulgt0d 11420 | The product of two positiv... |
| ltadd2d 11421 | Addition to both sides of ... |
| letrd 11422 | Transitive law deduction f... |
| lelttrd 11423 | Transitive law deduction f... |
| ltadd2dd 11424 | Addition to both sides of ... |
| ltletrd 11425 | Transitive law deduction f... |
| lttrd 11426 | Transitive law deduction f... |
| lelttrdi 11427 | If a number is less than a... |
| dedekind 11428 | The Dedekind cut theorem. ... |
| dedekindle 11429 | The Dedekind cut theorem, ... |
| mul12 11430 | Commutative/associative la... |
| mul32 11431 | Commutative/associative la... |
| mul31 11432 | Commutative/associative la... |
| mul4 11433 | Rearrangement of 4 factors... |
| mul4r 11434 | Rearrangement of 4 factors... |
| muladd11 11435 | A simple product of sums e... |
| 1p1times 11436 | Two times a number. (Cont... |
| peano2cn 11437 | A theorem for complex numb... |
| peano2re 11438 | A theorem for reals analog... |
| readdcan 11439 | Cancellation law for addit... |
| 00id 11440 | ` 0 ` is its own additive ... |
| mul02lem1 11441 | Lemma for ~ mul02 . If an... |
| mul02lem2 11442 | Lemma for ~ mul02 . Zero ... |
| mul02 11443 | Multiplication by ` 0 ` . ... |
| mul01 11444 | Multiplication by ` 0 ` . ... |
| addrid 11445 | ` 0 ` is an additive ident... |
| cnegex 11446 | Existence of the negative ... |
| cnegex2 11447 | Existence of a left invers... |
| addlid 11448 | ` 0 ` is a left identity f... |
| addcan 11449 | Cancellation law for addit... |
| addcan2 11450 | Cancellation law for addit... |
| addcom 11451 | Addition commutes. This u... |
| addridi 11452 | ` 0 ` is an additive ident... |
| addlidi 11453 | ` 0 ` is a left identity f... |
| mul02i 11454 | Multiplication by 0. Theo... |
| mul01i 11455 | Multiplication by ` 0 ` . ... |
| addcomi 11456 | Addition commutes. Based ... |
| addcomli 11457 | Addition commutes. (Contr... |
| addcani 11458 | Cancellation law for addit... |
| addcan2i 11459 | Cancellation law for addit... |
| mul12i 11460 | Commutative/associative la... |
| mul32i 11461 | Commutative/associative la... |
| mul4i 11462 | Rearrangement of 4 factors... |
| mul02d 11463 | Multiplication by 0. Theo... |
| mul01d 11464 | Multiplication by ` 0 ` . ... |
| addridd 11465 | ` 0 ` is an additive ident... |
| addlidd 11466 | ` 0 ` is a left identity f... |
| addcomd 11467 | Addition commutes. Based ... |
| addcand 11468 | Cancellation law for addit... |
| addcan2d 11469 | Cancellation law for addit... |
| addcanad 11470 | Cancelling a term on the l... |
| addcan2ad 11471 | Cancelling a term on the r... |
| addneintrd 11472 | Introducing a term on the ... |
| addneintr2d 11473 | Introducing a term on the ... |
| mul12d 11474 | Commutative/associative la... |
| mul32d 11475 | Commutative/associative la... |
| mul31d 11476 | Commutative/associative la... |
| mul4d 11477 | Rearrangement of 4 factors... |
| muladd11r 11478 | A simple product of sums e... |
| comraddd 11479 | Commute RHS addition, in d... |
| ltaddneg 11480 | Adding a negative number t... |
| ltaddnegr 11481 | Adding a negative number t... |
| add12 11482 | Commutative/associative la... |
| add32 11483 | Commutative/associative la... |
| add32r 11484 | Commutative/associative la... |
| add4 11485 | Rearrangement of 4 terms i... |
| add42 11486 | Rearrangement of 4 terms i... |
| add12i 11487 | Commutative/associative la... |
| add32i 11488 | Commutative/associative la... |
| add4i 11489 | Rearrangement of 4 terms i... |
| add42i 11490 | Rearrangement of 4 terms i... |
| add12d 11491 | Commutative/associative la... |
| add32d 11492 | Commutative/associative la... |
| add4d 11493 | Rearrangement of 4 terms i... |
| add42d 11494 | Rearrangement of 4 terms i... |
| 0cnALT 11499 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11500 | Alternate proof of ~ 0cnAL... |
| negeu 11501 | Existential uniqueness of ... |
| subval 11502 | Value of subtraction, whic... |
| negeq 11503 | Equality theorem for negat... |
| negeqi 11504 | Equality inference for neg... |
| negeqd 11505 | Equality deduction for neg... |
| nfnegd 11506 | Deduction version of ~ nfn... |
| nfneg 11507 | Bound-variable hypothesis ... |
| csbnegg 11508 | Move class substitution in... |
| negex 11509 | A negative is a set. (Con... |
| subcl 11510 | Closure law for subtractio... |
| negcl 11511 | Closure law for negative. ... |
| negicn 11512 | ` -u _i ` is a complex num... |
| subf 11513 | Subtraction is an operatio... |
| subadd 11514 | Relationship between subtr... |
| subadd2 11515 | Relationship between subtr... |
| subsub23 11516 | Swap subtrahend and result... |
| pncan 11517 | Cancellation law for subtr... |
| pncan2 11518 | Cancellation law for subtr... |
| pncan3 11519 | Subtraction and addition o... |
| npcan 11520 | Cancellation law for subtr... |
| addsubass 11521 | Associative-type law for a... |
| addsub 11522 | Law for addition and subtr... |
| subadd23 11523 | Commutative/associative la... |
| addsub12 11524 | Commutative/associative la... |
| 2addsub 11525 | Law for subtraction and ad... |
| addsubeq4 11526 | Relation between sums and ... |
| pncan3oi 11527 | Subtraction and addition o... |
| mvrraddi 11528 | Move the right term in a s... |
| mvlladdi 11529 | Move the left term in a su... |
| subid 11530 | Subtraction of a number fr... |
| subid1 11531 | Identity law for subtracti... |
| npncan 11532 | Cancellation law for subtr... |
| nppcan 11533 | Cancellation law for subtr... |
| nnpcan 11534 | Cancellation law for subtr... |
| nppcan3 11535 | Cancellation law for subtr... |
| subcan2 11536 | Cancellation law for subtr... |
| subeq0 11537 | If the difference between ... |
| npncan2 11538 | Cancellation law for subtr... |
| subsub2 11539 | Law for double subtraction... |
| nncan 11540 | Cancellation law for subtr... |
| subsub 11541 | Law for double subtraction... |
| nppcan2 11542 | Cancellation law for subtr... |
| subsub3 11543 | Law for double subtraction... |
| subsub4 11544 | Law for double subtraction... |
| sub32 11545 | Swap the second and third ... |
| nnncan 11546 | Cancellation law for subtr... |
| nnncan1 11547 | Cancellation law for subtr... |
| nnncan2 11548 | Cancellation law for subtr... |
| npncan3 11549 | Cancellation law for subtr... |
| pnpcan 11550 | Cancellation law for mixed... |
| pnpcan2 11551 | Cancellation law for mixed... |
| pnncan 11552 | Cancellation law for mixed... |
| ppncan 11553 | Cancellation law for mixed... |
| addsub4 11554 | Rearrangement of 4 terms i... |
| subadd4 11555 | Rearrangement of 4 terms i... |
| sub4 11556 | Rearrangement of 4 terms i... |
| neg0 11557 | Minus 0 equals 0. (Contri... |
| negid 11558 | Addition of a number and i... |
| negsub 11559 | Relationship between subtr... |
| subneg 11560 | Relationship between subtr... |
| negneg 11561 | A number is equal to the n... |
| neg11 11562 | Negative is one-to-one. (... |
| negcon1 11563 | Negative contraposition la... |
| negcon2 11564 | Negative contraposition la... |
| negeq0 11565 | A number is zero iff its n... |
| subcan 11566 | Cancellation law for subtr... |
| negsubdi 11567 | Distribution of negative o... |
| negdi 11568 | Distribution of negative o... |
| negdi2 11569 | Distribution of negative o... |
| negsubdi2 11570 | Distribution of negative o... |
| neg2sub 11571 | Relationship between subtr... |
| renegcli 11572 | Closure law for negative o... |
| resubcli 11573 | Closure law for subtractio... |
| renegcl 11574 | Closure law for negative o... |
| resubcl 11575 | Closure law for subtractio... |
| negreb 11576 | The negative of a real is ... |
| peano2cnm 11577 | "Reverse" second Peano pos... |
| peano2rem 11578 | "Reverse" second Peano pos... |
| negcli 11579 | Closure law for negative. ... |
| negidi 11580 | Addition of a number and i... |
| negnegi 11581 | A number is equal to the n... |
| subidi 11582 | Subtraction of a number fr... |
| subid1i 11583 | Identity law for subtracti... |
| negne0bi 11584 | A number is nonzero iff it... |
| negrebi 11585 | The negative of a real is ... |
| negne0i 11586 | The negative of a nonzero ... |
| subcli 11587 | Closure law for subtractio... |
| pncan3i 11588 | Subtraction and addition o... |
| negsubi 11589 | Relationship between subtr... |
| subnegi 11590 | Relationship between subtr... |
| subeq0i 11591 | If the difference between ... |
| neg11i 11592 | Negative is one-to-one. (... |
| negcon1i 11593 | Negative contraposition la... |
| negcon2i 11594 | Negative contraposition la... |
| negdii 11595 | Distribution of negative o... |
| negsubdii 11596 | Distribution of negative o... |
| negsubdi2i 11597 | Distribution of negative o... |
| subaddi 11598 | Relationship between subtr... |
| subadd2i 11599 | Relationship between subtr... |
| subaddrii 11600 | Relationship between subtr... |
| subsub23i 11601 | Swap subtrahend and result... |
| addsubassi 11602 | Associative-type law for s... |
| addsubi 11603 | Law for subtraction and ad... |
| subcani 11604 | Cancellation law for subtr... |
| subcan2i 11605 | Cancellation law for subtr... |
| pnncani 11606 | Cancellation law for mixed... |
| addsub4i 11607 | Rearrangement of 4 terms i... |
| 0reALT 11608 | Alternate proof of ~ 0re .... |
| negcld 11609 | Closure law for negative. ... |
| subidd 11610 | Subtraction of a number fr... |
| subid1d 11611 | Identity law for subtracti... |
| negidd 11612 | Addition of a number and i... |
| negnegd 11613 | A number is equal to the n... |
| negeq0d 11614 | A number is zero iff its n... |
| negne0bd 11615 | A number is nonzero iff it... |
| negcon1d 11616 | Contraposition law for una... |
| negcon1ad 11617 | Contraposition law for una... |
| neg11ad 11618 | The negatives of two compl... |
| negned 11619 | If two complex numbers are... |
| negne0d 11620 | The negative of a nonzero ... |
| negrebd 11621 | The negative of a real is ... |
| subcld 11622 | Closure law for subtractio... |
| pncand 11623 | Cancellation law for subtr... |
| pncan2d 11624 | Cancellation law for subtr... |
| pncan3d 11625 | Subtraction and addition o... |
| npcand 11626 | Cancellation law for subtr... |
| nncand 11627 | Cancellation law for subtr... |
| negsubd 11628 | Relationship between subtr... |
| subnegd 11629 | Relationship between subtr... |
| subeq0d 11630 | If the difference between ... |
| subne0d 11631 | Two unequal numbers have n... |
| subeq0ad 11632 | The difference of two comp... |
| subne0ad 11633 | If the difference of two c... |
| neg11d 11634 | If the difference between ... |
| negdid 11635 | Distribution of negative o... |
| negdi2d 11636 | Distribution of negative o... |
| negsubdid 11637 | Distribution of negative o... |
| negsubdi2d 11638 | Distribution of negative o... |
| neg2subd 11639 | Relationship between subtr... |
| subaddd 11640 | Relationship between subtr... |
| subadd2d 11641 | Relationship between subtr... |
| addsubassd 11642 | Associative-type law for s... |
| addsubd 11643 | Law for subtraction and ad... |
| subadd23d 11644 | Commutative/associative la... |
| addsub12d 11645 | Commutative/associative la... |
| npncand 11646 | Cancellation law for subtr... |
| nppcand 11647 | Cancellation law for subtr... |
| nppcan2d 11648 | Cancellation law for subtr... |
| nppcan3d 11649 | Cancellation law for subtr... |
| subsubd 11650 | Law for double subtraction... |
| subsub2d 11651 | Law for double subtraction... |
| subsub3d 11652 | Law for double subtraction... |
| subsub4d 11653 | Law for double subtraction... |
| sub32d 11654 | Swap the second and third ... |
| nnncand 11655 | Cancellation law for subtr... |
| nnncan1d 11656 | Cancellation law for subtr... |
| nnncan2d 11657 | Cancellation law for subtr... |
| npncan3d 11658 | Cancellation law for subtr... |
| pnpcand 11659 | Cancellation law for mixed... |
| pnpcan2d 11660 | Cancellation law for mixed... |
| pnncand 11661 | Cancellation law for mixed... |
| ppncand 11662 | Cancellation law for mixed... |
| subcand 11663 | Cancellation law for subtr... |
| subcan2d 11664 | Cancellation law for subtr... |
| subcanad 11665 | Cancellation law for subtr... |
| subneintrd 11666 | Introducing subtraction on... |
| subcan2ad 11667 | Cancellation law for subtr... |
| subneintr2d 11668 | Introducing subtraction on... |
| addsub4d 11669 | Rearrangement of 4 terms i... |
| subadd4d 11670 | Rearrangement of 4 terms i... |
| sub4d 11671 | Rearrangement of 4 terms i... |
| 2addsubd 11672 | Law for subtraction and ad... |
| addsubeq4d 11673 | Relation between sums and ... |
| subeqxfrd 11674 | Transfer two terms of a su... |
| mvlraddd 11675 | Move the right term in a s... |
| mvlladdd 11676 | Move the left term in a su... |
| mvrraddd 11677 | Move the right term in a s... |
| mvrladdd 11678 | Move the left term in a su... |
| assraddsubd 11679 | Associate RHS addition-sub... |
| subaddeqd 11680 | Transfer two terms of a su... |
| addlsub 11681 | Left-subtraction: Subtrac... |
| addrsub 11682 | Right-subtraction: Subtra... |
| subexsub 11683 | A subtraction law: Exchan... |
| addid0 11684 | If adding a number to a an... |
| addn0nid 11685 | Adding a nonzero number to... |
| pnpncand 11686 | Addition/subtraction cance... |
| subeqrev 11687 | Reverse the order of subtr... |
| addeq0 11688 | Two complex numbers add up... |
| pncan1 11689 | Cancellation law for addit... |
| npcan1 11690 | Cancellation law for subtr... |
| subeq0bd 11691 | If two complex numbers are... |
| renegcld 11692 | Closure law for negative o... |
| resubcld 11693 | Closure law for subtractio... |
| negn0 11694 | The image under negation o... |
| negf1o 11695 | Negation is an isomorphism... |
| kcnktkm1cn 11696 | k times k minus 1 is a com... |
| muladd 11697 | Product of two sums. (Con... |
| subdi 11698 | Distribution of multiplica... |
| subdir 11699 | Distribution of multiplica... |
| ine0 11700 | The imaginary unit ` _i ` ... |
| mulneg1 11701 | Product with negative is n... |
| mulneg2 11702 | The product with a negativ... |
| mulneg12 11703 | Swap the negative sign in ... |
| mul2neg 11704 | Product of two negatives. ... |
| submul2 11705 | Convert a subtraction to a... |
| mulm1 11706 | Product with minus one is ... |
| addneg1mul 11707 | Addition with product with... |
| mulsub 11708 | Product of two differences... |
| mulsub2 11709 | Swap the order of subtract... |
| mulm1i 11710 | Product with minus one is ... |
| mulneg1i 11711 | Product with negative is n... |
| mulneg2i 11712 | Product with negative is n... |
| mul2negi 11713 | Product of two negatives. ... |
| subdii 11714 | Distribution of multiplica... |
| subdiri 11715 | Distribution of multiplica... |
| muladdi 11716 | Product of two sums. (Con... |
| mulm1d 11717 | Product with minus one is ... |
| mulneg1d 11718 | Product with negative is n... |
| mulneg2d 11719 | Product with negative is n... |
| mul2negd 11720 | Product of two negatives. ... |
| subdid 11721 | Distribution of multiplica... |
| subdird 11722 | Distribution of multiplica... |
| muladdd 11723 | Product of two sums. (Con... |
| mulsubd 11724 | Product of two differences... |
| muls1d 11725 | Multiplication by one minu... |
| mulsubfacd 11726 | Multiplication followed by... |
| addmulsub 11727 | The product of a sum and a... |
| subaddmulsub 11728 | The difference with a prod... |
| mulsubaddmulsub 11729 | A special difference of a ... |
| gt0ne0 11730 | Positive implies nonzero. ... |
| lt0ne0 11731 | A number which is less tha... |
| ltadd1 11732 | Addition to both sides of ... |
| leadd1 11733 | Addition to both sides of ... |
| leadd2 11734 | Addition to both sides of ... |
| ltsubadd 11735 | 'Less than' relationship b... |
| ltsubadd2 11736 | 'Less than' relationship b... |
| lesubadd 11737 | 'Less than or equal to' re... |
| lesubadd2 11738 | 'Less than or equal to' re... |
| ltaddsub 11739 | 'Less than' relationship b... |
| ltaddsub2 11740 | 'Less than' relationship b... |
| leaddsub 11741 | 'Less than or equal to' re... |
| leaddsub2 11742 | 'Less than or equal to' re... |
| suble 11743 | Swap subtrahends in an ine... |
| lesub 11744 | Swap subtrahends in an ine... |
| ltsub23 11745 | 'Less than' relationship b... |
| ltsub13 11746 | 'Less than' relationship b... |
| le2add 11747 | Adding both sides of two '... |
| ltleadd 11748 | Adding both sides of two o... |
| leltadd 11749 | Adding both sides of two o... |
| lt2add 11750 | Adding both sides of two '... |
| addgt0 11751 | The sum of 2 positive numb... |
| addgegt0 11752 | The sum of nonnegative and... |
| addgtge0 11753 | The sum of nonnegative and... |
| addge0 11754 | The sum of 2 nonnegative n... |
| ltaddpos 11755 | Adding a positive number t... |
| ltaddpos2 11756 | Adding a positive number t... |
| ltsubpos 11757 | Subtracting a positive num... |
| posdif 11758 | Comparison of two numbers ... |
| lesub1 11759 | Subtraction from both side... |
| lesub2 11760 | Subtraction of both sides ... |
| ltsub1 11761 | Subtraction from both side... |
| ltsub2 11762 | Subtraction of both sides ... |
| lt2sub 11763 | Subtracting both sides of ... |
| le2sub 11764 | Subtracting both sides of ... |
| ltneg 11765 | Negative of both sides of ... |
| ltnegcon1 11766 | Contraposition of negative... |
| ltnegcon2 11767 | Contraposition of negative... |
| leneg 11768 | Negative of both sides of ... |
| lenegcon1 11769 | Contraposition of negative... |
| lenegcon2 11770 | Contraposition of negative... |
| lt0neg1 11771 | Comparison of a number and... |
| lt0neg2 11772 | Comparison of a number and... |
| le0neg1 11773 | Comparison of a number and... |
| le0neg2 11774 | Comparison of a number and... |
| addge01 11775 | A number is less than or e... |
| addge02 11776 | A number is less than or e... |
| add20 11777 | Two nonnegative numbers ar... |
| subge0 11778 | Nonnegative subtraction. ... |
| suble0 11779 | Nonpositive subtraction. ... |
| leaddle0 11780 | The sum of a real number a... |
| subge02 11781 | Nonnegative subtraction. ... |
| lesub0 11782 | Lemma to show a nonnegativ... |
| mulge0 11783 | The product of two nonnega... |
| mullt0 11784 | The product of two negativ... |
| msqgt0 11785 | A nonzero square is positi... |
| msqge0 11786 | A square is nonnegative. ... |
| 0lt1 11787 | 0 is less than 1. Theorem... |
| 0le1 11788 | 0 is less than or equal to... |
| relin01 11789 | An interval law for less t... |
| ltordlem 11790 | Lemma for ~ ltord1 . (Con... |
| ltord1 11791 | Infer an ordering relation... |
| leord1 11792 | Infer an ordering relation... |
| eqord1 11793 | A strictly increasing real... |
| ltord2 11794 | Infer an ordering relation... |
| leord2 11795 | Infer an ordering relation... |
| eqord2 11796 | A strictly decreasing real... |
| wloglei 11797 | Form of ~ wlogle where bot... |
| wlogle 11798 | If the predicate ` ch ( x ... |
| leidi 11799 | 'Less than or equal to' is... |
| gt0ne0i 11800 | Positive means nonzero (us... |
| gt0ne0ii 11801 | Positive implies nonzero. ... |
| msqgt0i 11802 | A nonzero square is positi... |
| msqge0i 11803 | A square is nonnegative. ... |
| addgt0i 11804 | Addition of 2 positive num... |
| addge0i 11805 | Addition of 2 nonnegative ... |
| addgegt0i 11806 | Addition of nonnegative an... |
| addgt0ii 11807 | Addition of 2 positive num... |
| add20i 11808 | Two nonnegative numbers ar... |
| ltnegi 11809 | Negative of both sides of ... |
| lenegi 11810 | Negative of both sides of ... |
| ltnegcon2i 11811 | Contraposition of negative... |
| mulge0i 11812 | The product of two nonnega... |
| lesub0i 11813 | Lemma to show a nonnegativ... |
| ltaddposi 11814 | Adding a positive number t... |
| posdifi 11815 | Comparison of two numbers ... |
| ltnegcon1i 11816 | Contraposition of negative... |
| lenegcon1i 11817 | Contraposition of negative... |
| subge0i 11818 | Nonnegative subtraction. ... |
| ltadd1i 11819 | Addition to both sides of ... |
| leadd1i 11820 | Addition to both sides of ... |
| leadd2i 11821 | Addition to both sides of ... |
| ltsubaddi 11822 | 'Less than' relationship b... |
| lesubaddi 11823 | 'Less than or equal to' re... |
| ltsubadd2i 11824 | 'Less than' relationship b... |
| lesubadd2i 11825 | 'Less than or equal to' re... |
| ltaddsubi 11826 | 'Less than' relationship b... |
| lt2addi 11827 | Adding both side of two in... |
| le2addi 11828 | Adding both side of two in... |
| gt0ne0d 11829 | Positive implies nonzero. ... |
| lt0ne0d 11830 | Something less than zero i... |
| leidd 11831 | 'Less than or equal to' is... |
| msqgt0d 11832 | A nonzero square is positi... |
| msqge0d 11833 | A square is nonnegative. ... |
| lt0neg1d 11834 | Comparison of a number and... |
| lt0neg2d 11835 | Comparison of a number and... |
| le0neg1d 11836 | Comparison of a number and... |
| le0neg2d 11837 | Comparison of a number and... |
| addgegt0d 11838 | Addition of nonnegative an... |
| addgtge0d 11839 | Addition of positive and n... |
| addgt0d 11840 | Addition of 2 positive num... |
| addge0d 11841 | Addition of 2 nonnegative ... |
| mulge0d 11842 | The product of two nonnega... |
| ltnegd 11843 | Negative of both sides of ... |
| lenegd 11844 | Negative of both sides of ... |
| ltnegcon1d 11845 | Contraposition of negative... |
| ltnegcon2d 11846 | Contraposition of negative... |
| lenegcon1d 11847 | Contraposition of negative... |
| lenegcon2d 11848 | Contraposition of negative... |
| ltaddposd 11849 | Adding a positive number t... |
| ltaddpos2d 11850 | Adding a positive number t... |
| ltsubposd 11851 | Subtracting a positive num... |
| posdifd 11852 | Comparison of two numbers ... |
| addge01d 11853 | A number is less than or e... |
| addge02d 11854 | A number is less than or e... |
| subge0d 11855 | Nonnegative subtraction. ... |
| suble0d 11856 | Nonpositive subtraction. ... |
| subge02d 11857 | Nonnegative subtraction. ... |
| ltadd1d 11858 | Addition to both sides of ... |
| leadd1d 11859 | Addition to both sides of ... |
| leadd2d 11860 | Addition to both sides of ... |
| ltsubaddd 11861 | 'Less than' relationship b... |
| lesubaddd 11862 | 'Less than or equal to' re... |
| ltsubadd2d 11863 | 'Less than' relationship b... |
| lesubadd2d 11864 | 'Less than or equal to' re... |
| ltaddsubd 11865 | 'Less than' relationship b... |
| ltaddsub2d 11866 | 'Less than' relationship b... |
| leaddsub2d 11867 | 'Less than or equal to' re... |
| subled 11868 | Swap subtrahends in an ine... |
| lesubd 11869 | Swap subtrahends in an ine... |
| ltsub23d 11870 | 'Less than' relationship b... |
| ltsub13d 11871 | 'Less than' relationship b... |
| lesub1d 11872 | Subtraction from both side... |
| lesub2d 11873 | Subtraction of both sides ... |
| ltsub1d 11874 | Subtraction from both side... |
| ltsub2d 11875 | Subtraction of both sides ... |
| ltadd1dd 11876 | Addition to both sides of ... |
| ltsub1dd 11877 | Subtraction from both side... |
| ltsub2dd 11878 | Subtraction of both sides ... |
| leadd1dd 11879 | Addition to both sides of ... |
| leadd2dd 11880 | Addition to both sides of ... |
| lesub1dd 11881 | Subtraction from both side... |
| lesub2dd 11882 | Subtraction of both sides ... |
| lesub3d 11883 | The result of subtracting ... |
| le2addd 11884 | Adding both side of two in... |
| le2subd 11885 | Subtracting both sides of ... |
| ltleaddd 11886 | Adding both sides of two o... |
| leltaddd 11887 | Adding both sides of two o... |
| lt2addd 11888 | Adding both side of two in... |
| lt2subd 11889 | Subtracting both sides of ... |
| possumd 11890 | Condition for a positive s... |
| sublt0d 11891 | When a subtraction gives a... |
| ltaddsublt 11892 | Addition and subtraction o... |
| 1le1 11893 | One is less than or equal ... |
| ixi 11894 | ` _i ` times itself is min... |
| recextlem1 11895 | Lemma for ~ recex . (Cont... |
| recextlem2 11896 | Lemma for ~ recex . (Cont... |
| recex 11897 | Existence of reciprocal of... |
| mulcand 11898 | Cancellation law for multi... |
| mulcan2d 11899 | Cancellation law for multi... |
| mulcanad 11900 | Cancellation of a nonzero ... |
| mulcan2ad 11901 | Cancellation of a nonzero ... |
| mulcan 11902 | Cancellation law for multi... |
| mulcan2 11903 | Cancellation law for multi... |
| mulcani 11904 | Cancellation law for multi... |
| mul0or 11905 | If a product is zero, one ... |
| mulne0b 11906 | The product of two nonzero... |
| mulne0 11907 | The product of two nonzero... |
| mulne0i 11908 | The product of two nonzero... |
| muleqadd 11909 | Property of numbers whose ... |
| receu 11910 | Existential uniqueness of ... |
| mulnzcnf 11911 | Multiplication maps nonzer... |
| msq0i 11912 | A number is zero iff its s... |
| mul0ori 11913 | If a product is zero, one ... |
| msq0d 11914 | A number is zero iff its s... |
| mul0ord 11915 | If a product is zero, one ... |
| mulne0bd 11916 | The product of two nonzero... |
| mulne0d 11917 | The product of two nonzero... |
| mulcan1g 11918 | A generalized form of the ... |
| mulcan2g 11919 | A generalized form of the ... |
| mulne0bad 11920 | A factor of a nonzero comp... |
| mulne0bbd 11921 | A factor of a nonzero comp... |
| 1div0 11924 | You can't divide by zero, ... |
| 1div0OLD 11925 | Obsolete version of ~ 1div... |
| divval 11926 | Value of division: if ` A ... |
| divmul 11927 | Relationship between divis... |
| divmul2 11928 | Relationship between divis... |
| divmul3 11929 | Relationship between divis... |
| divcl 11930 | Closure law for division. ... |
| reccl 11931 | Closure law for reciprocal... |
| divcan2 11932 | A cancellation law for div... |
| divcan1 11933 | A cancellation law for div... |
| diveq0 11934 | A ratio is zero iff the nu... |
| divne0b 11935 | The ratio of nonzero numbe... |
| divne0 11936 | The ratio of nonzero numbe... |
| recne0 11937 | The reciprocal of a nonzer... |
| recid 11938 | Multiplication of a number... |
| recid2 11939 | Multiplication of a number... |
| divrec 11940 | Relationship between divis... |
| divrec2 11941 | Relationship between divis... |
| divass 11942 | An associative law for div... |
| div23 11943 | A commutative/associative ... |
| div32 11944 | A commutative/associative ... |
| div13 11945 | A commutative/associative ... |
| div12 11946 | A commutative/associative ... |
| divmulass 11947 | An associative law for div... |
| divmulasscom 11948 | An associative/commutative... |
| divdir 11949 | Distribution of division o... |
| divcan3 11950 | A cancellation law for div... |
| divcan4 11951 | A cancellation law for div... |
| div11 11952 | One-to-one relationship fo... |
| div11OLD 11953 | Obsolete version of ~ div1... |
| diveq1 11954 | Equality in terms of unit ... |
| divid 11955 | A number divided by itself... |
| dividOLD 11956 | Obsolete version of ~ divi... |
| div0 11957 | Division into zero is zero... |
| div0OLD 11958 | Obsolete version of ~ div0... |
| div1 11959 | A number divided by 1 is i... |
| 1div1e1 11960 | 1 divided by 1 is 1. (Con... |
| divneg 11961 | Move negative sign inside ... |
| muldivdir 11962 | Distribution of division o... |
| divsubdir 11963 | Distribution of division o... |
| subdivcomb1 11964 | Bring a term in a subtract... |
| subdivcomb2 11965 | Bring a term in a subtract... |
| recrec 11966 | A number is equal to the r... |
| rec11 11967 | Reciprocal is one-to-one. ... |
| rec11r 11968 | Mutual reciprocals. (Cont... |
| divmuldiv 11969 | Multiplication of two rati... |
| divdivdiv 11970 | Division of two ratios. T... |
| divcan5 11971 | Cancellation of common fac... |
| divmul13 11972 | Swap the denominators in t... |
| divmul24 11973 | Swap the numerators in the... |
| divmuleq 11974 | Cross-multiply in an equal... |
| recdiv 11975 | The reciprocal of a ratio.... |
| divcan6 11976 | Cancellation of inverted f... |
| divdiv32 11977 | Swap denominators in a div... |
| divcan7 11978 | Cancel equal divisors in a... |
| dmdcan 11979 | Cancellation law for divis... |
| divdiv1 11980 | Division into a fraction. ... |
| divdiv2 11981 | Division by a fraction. (... |
| recdiv2 11982 | Division into a reciprocal... |
| ddcan 11983 | Cancellation in a double d... |
| divadddiv 11984 | Addition of two ratios. T... |
| divsubdiv 11985 | Subtraction of two ratios.... |
| conjmul 11986 | Two numbers whose reciproc... |
| rereccl 11987 | Closure law for reciprocal... |
| redivcl 11988 | Closure law for division o... |
| eqneg 11989 | A number equal to its nega... |
| eqnegd 11990 | A complex number equals it... |
| eqnegad 11991 | If a complex number equals... |
| div2neg 11992 | Quotient of two negatives.... |
| divneg2 11993 | Move negative sign inside ... |
| recclzi 11994 | Closure law for reciprocal... |
| recne0zi 11995 | The reciprocal of a nonzer... |
| recidzi 11996 | Multiplication of a number... |
| div1i 11997 | A number divided by 1 is i... |
| eqnegi 11998 | A number equal to its nega... |
| reccli 11999 | Closure law for reciprocal... |
| recidi 12000 | Multiplication of a number... |
| recreci 12001 | A number is equal to the r... |
| dividi 12002 | A number divided by itself... |
| div0i 12003 | Division into zero is zero... |
| divclzi 12004 | Closure law for division. ... |
| divcan1zi 12005 | A cancellation law for div... |
| divcan2zi 12006 | A cancellation law for div... |
| divreczi 12007 | Relationship between divis... |
| divcan3zi 12008 | A cancellation law for div... |
| divcan4zi 12009 | A cancellation law for div... |
| rec11i 12010 | Reciprocal is one-to-one. ... |
| divcli 12011 | Closure law for division. ... |
| divcan2i 12012 | A cancellation law for div... |
| divcan1i 12013 | A cancellation law for div... |
| divreci 12014 | Relationship between divis... |
| divcan3i 12015 | A cancellation law for div... |
| divcan4i 12016 | A cancellation law for div... |
| divne0i 12017 | The ratio of nonzero numbe... |
| rec11ii 12018 | Reciprocal is one-to-one. ... |
| divasszi 12019 | An associative law for div... |
| divmulzi 12020 | Relationship between divis... |
| divdirzi 12021 | Distribution of division o... |
| divdiv23zi 12022 | Swap denominators in a div... |
| divmuli 12023 | Relationship between divis... |
| divdiv32i 12024 | Swap denominators in a div... |
| divassi 12025 | An associative law for div... |
| divdiri 12026 | Distribution of division o... |
| div23i 12027 | A commutative/associative ... |
| div11i 12028 | One-to-one relationship fo... |
| divmuldivi 12029 | Multiplication of two rati... |
| divmul13i 12030 | Swap denominators of two r... |
| divadddivi 12031 | Addition of two ratios. T... |
| divdivdivi 12032 | Division of two ratios. T... |
| rerecclzi 12033 | Closure law for reciprocal... |
| rereccli 12034 | Closure law for reciprocal... |
| redivclzi 12035 | Closure law for division o... |
| redivcli 12036 | Closure law for division o... |
| div1d 12037 | A number divided by 1 is i... |
| reccld 12038 | Closure law for reciprocal... |
| recne0d 12039 | The reciprocal of a nonzer... |
| recidd 12040 | Multiplication of a number... |
| recid2d 12041 | Multiplication of a number... |
| recrecd 12042 | A number is equal to the r... |
| dividd 12043 | A number divided by itself... |
| div0d 12044 | Division into zero is zero... |
| divcld 12045 | Closure law for division. ... |
| divcan1d 12046 | A cancellation law for div... |
| divcan2d 12047 | A cancellation law for div... |
| divrecd 12048 | Relationship between divis... |
| divrec2d 12049 | Relationship between divis... |
| divcan3d 12050 | A cancellation law for div... |
| divcan4d 12051 | A cancellation law for div... |
| diveq0d 12052 | A ratio is zero iff the nu... |
| diveq1d 12053 | Equality in terms of unit ... |
| diveq1ad 12054 | The quotient of two comple... |
| diveq0ad 12055 | A fraction of complex numb... |
| divne1d 12056 | If two complex numbers are... |
| divne0bd 12057 | A ratio is zero iff the nu... |
| divnegd 12058 | Move negative sign inside ... |
| divneg2d 12059 | Move negative sign inside ... |
| div2negd 12060 | Quotient of two negatives.... |
| divne0d 12061 | The ratio of nonzero numbe... |
| recdivd 12062 | The reciprocal of a ratio.... |
| recdiv2d 12063 | Division into a reciprocal... |
| divcan6d 12064 | Cancellation of inverted f... |
| ddcand 12065 | Cancellation in a double d... |
| rec11d 12066 | Reciprocal is one-to-one. ... |
| divmuld 12067 | Relationship between divis... |
| div32d 12068 | A commutative/associative ... |
| div13d 12069 | A commutative/associative ... |
| divdiv32d 12070 | Swap denominators in a div... |
| divcan5d 12071 | Cancellation of common fac... |
| divcan5rd 12072 | Cancellation of common fac... |
| divcan7d 12073 | Cancel equal divisors in a... |
| dmdcand 12074 | Cancellation law for divis... |
| dmdcan2d 12075 | Cancellation law for divis... |
| divdiv1d 12076 | Division into a fraction. ... |
| divdiv2d 12077 | Division by a fraction. (... |
| divmul2d 12078 | Relationship between divis... |
| divmul3d 12079 | Relationship between divis... |
| divassd 12080 | An associative law for div... |
| div12d 12081 | A commutative/associative ... |
| div23d 12082 | A commutative/associative ... |
| divdird 12083 | Distribution of division o... |
| divsubdird 12084 | Distribution of division o... |
| div11d 12085 | One-to-one relationship fo... |
| divmuldivd 12086 | Multiplication of two rati... |
| divmul13d 12087 | Swap denominators of two r... |
| divmul24d 12088 | Swap the numerators in the... |
| divadddivd 12089 | Addition of two ratios. T... |
| divsubdivd 12090 | Subtraction of two ratios.... |
| divmuleqd 12091 | Cross-multiply in an equal... |
| divdivdivd 12092 | Division of two ratios. T... |
| diveq1bd 12093 | If two complex numbers are... |
| div2sub 12094 | Swap the order of subtract... |
| div2subd 12095 | Swap subtrahend and minuen... |
| rereccld 12096 | Closure law for reciprocal... |
| redivcld 12097 | Closure law for division o... |
| subrec 12098 | Subtraction of reciprocals... |
| subreci 12099 | Subtraction of reciprocals... |
| subrecd 12100 | Subtraction of reciprocals... |
| mvllmuld 12101 | Move the left term in a pr... |
| mvllmuli 12102 | Move the left term in a pr... |
| ldiv 12103 | Left-division. (Contribut... |
| rdiv 12104 | Right-division. (Contribu... |
| mdiv 12105 | A division law. (Contribu... |
| lineq 12106 | Solution of a (scalar) lin... |
| elimgt0 12107 | Hypothesis for weak deduct... |
| elimge0 12108 | Hypothesis for weak deduct... |
| ltp1 12109 | A number is less than itse... |
| lep1 12110 | A number is less than or e... |
| ltm1 12111 | A number minus 1 is less t... |
| lem1 12112 | A number minus 1 is less t... |
| letrp1 12113 | A transitive property of '... |
| p1le 12114 | A transitive property of p... |
| recgt0 12115 | The reciprocal of a positi... |
| prodgt0 12116 | Infer that a multiplicand ... |
| prodgt02 12117 | Infer that a multiplier is... |
| ltmul1a 12118 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12119 | Multiplication of both sid... |
| ltmul2 12120 | Multiplication of both sid... |
| lemul1 12121 | Multiplication of both sid... |
| lemul2 12122 | Multiplication of both sid... |
| lemul1a 12123 | Multiplication of both sid... |
| lemul2a 12124 | Multiplication of both sid... |
| ltmul12a 12125 | Comparison of product of t... |
| lemul12b 12126 | Comparison of product of t... |
| lemul12a 12127 | Comparison of product of t... |
| mulgt1OLD 12128 | Obsolete version of ~ mulg... |
| ltmulgt11 12129 | Multiplication by a number... |
| ltmulgt12 12130 | Multiplication by a number... |
| mulgt1 12131 | The product of two numbers... |
| lemulge11 12132 | Multiplication by a number... |
| lemulge12 12133 | Multiplication by a number... |
| ltdiv1 12134 | Division of both sides of ... |
| lediv1 12135 | Division of both sides of ... |
| gt0div 12136 | Division of a positive num... |
| ge0div 12137 | Division of a nonnegative ... |
| divgt0 12138 | The ratio of two positive ... |
| divge0 12139 | The ratio of nonnegative a... |
| mulge0b 12140 | A condition for multiplica... |
| mulle0b 12141 | A condition for multiplica... |
| mulsuble0b 12142 | A condition for multiplica... |
| ltmuldiv 12143 | 'Less than' relationship b... |
| ltmuldiv2 12144 | 'Less than' relationship b... |
| ltdivmul 12145 | 'Less than' relationship b... |
| ledivmul 12146 | 'Less than or equal to' re... |
| ltdivmul2 12147 | 'Less than' relationship b... |
| lt2mul2div 12148 | 'Less than' relationship b... |
| ledivmul2 12149 | 'Less than or equal to' re... |
| lemuldiv 12150 | 'Less than or equal' relat... |
| lemuldiv2 12151 | 'Less than or equal' relat... |
| ltrec 12152 | The reciprocal of both sid... |
| lerec 12153 | The reciprocal of both sid... |
| lt2msq1 12154 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12155 | Two nonnegative numbers co... |
| ltdiv2 12156 | Division of a positive num... |
| ltrec1 12157 | Reciprocal swap in a 'less... |
| lerec2 12158 | Reciprocal swap in a 'less... |
| ledivdiv 12159 | Invert ratios of positive ... |
| lediv2 12160 | Division of a positive num... |
| ltdiv23 12161 | Swap denominator with othe... |
| lediv23 12162 | Swap denominator with othe... |
| lediv12a 12163 | Comparison of ratio of two... |
| lediv2a 12164 | Division of both sides of ... |
| reclt1 12165 | The reciprocal of a positi... |
| recgt1 12166 | The reciprocal of a positi... |
| recgt1i 12167 | The reciprocal of a number... |
| recp1lt1 12168 | Construct a number less th... |
| recreclt 12169 | Given a positive number ` ... |
| le2msq 12170 | The square function on non... |
| msq11 12171 | The square of a nonnegativ... |
| ledivp1 12172 | "Less than or equal to" an... |
| squeeze0 12173 | If a nonnegative number is... |
| ltp1i 12174 | A number is less than itse... |
| recgt0i 12175 | The reciprocal of a positi... |
| recgt0ii 12176 | The reciprocal of a positi... |
| prodgt0i 12177 | Infer that a multiplicand ... |
| divgt0i 12178 | The ratio of two positive ... |
| divge0i 12179 | The ratio of nonnegative a... |
| ltreci 12180 | The reciprocal of both sid... |
| lereci 12181 | The reciprocal of both sid... |
| lt2msqi 12182 | The square function on non... |
| le2msqi 12183 | The square function on non... |
| msq11i 12184 | The square of a nonnegativ... |
| divgt0i2i 12185 | The ratio of two positive ... |
| ltrecii 12186 | The reciprocal of both sid... |
| divgt0ii 12187 | The ratio of two positive ... |
| ltmul1i 12188 | Multiplication of both sid... |
| ltdiv1i 12189 | Division of both sides of ... |
| ltmuldivi 12190 | 'Less than' relationship b... |
| ltmul2i 12191 | Multiplication of both sid... |
| lemul1i 12192 | Multiplication of both sid... |
| lemul2i 12193 | Multiplication of both sid... |
| ltdiv23i 12194 | Swap denominator with othe... |
| ledivp1i 12195 | "Less than or equal to" an... |
| ltdivp1i 12196 | Less-than and division rel... |
| ltdiv23ii 12197 | Swap denominator with othe... |
| ltmul1ii 12198 | Multiplication of both sid... |
| ltdiv1ii 12199 | Division of both sides of ... |
| ltp1d 12200 | A number is less than itse... |
| lep1d 12201 | A number is less than or e... |
| ltm1d 12202 | A number minus 1 is less t... |
| lem1d 12203 | A number minus 1 is less t... |
| recgt0d 12204 | The reciprocal of a positi... |
| divgt0d 12205 | The ratio of two positive ... |
| mulgt1d 12206 | The product of two numbers... |
| lemulge11d 12207 | Multiplication by a number... |
| lemulge12d 12208 | Multiplication by a number... |
| lemul1ad 12209 | Multiplication of both sid... |
| lemul2ad 12210 | Multiplication of both sid... |
| ltmul12ad 12211 | Comparison of product of t... |
| lemul12ad 12212 | Comparison of product of t... |
| lemul12bd 12213 | Comparison of product of t... |
| fimaxre 12214 | A finite set of real numbe... |
| fimaxre2 12215 | A nonempty finite set of r... |
| fimaxre3 12216 | A nonempty finite set of r... |
| fiminre 12217 | A nonempty finite set of r... |
| fiminre2 12218 | A nonempty finite set of r... |
| negfi 12219 | The negation of a finite s... |
| lbreu 12220 | If a set of reals contains... |
| lbcl 12221 | If a set of reals contains... |
| lble 12222 | If a set of reals contains... |
| lbinf 12223 | If a set of reals contains... |
| lbinfcl 12224 | If a set of reals contains... |
| lbinfle 12225 | If a set of reals contains... |
| sup2 12226 | A nonempty, bounded-above ... |
| sup3 12227 | A version of the completen... |
| infm3lem 12228 | Lemma for ~ infm3 . (Cont... |
| infm3 12229 | The completeness axiom for... |
| suprcl 12230 | Closure of supremum of a n... |
| suprub 12231 | A member of a nonempty bou... |
| suprubd 12232 | Natural deduction form of ... |
| suprcld 12233 | Natural deduction form of ... |
| suprlub 12234 | The supremum of a nonempty... |
| suprnub 12235 | An upper bound is not less... |
| suprleub 12236 | The supremum of a nonempty... |
| supaddc 12237 | The supremum function dist... |
| supadd 12238 | The supremum function dist... |
| supmul1 12239 | The supremum function dist... |
| supmullem1 12240 | Lemma for ~ supmul . (Con... |
| supmullem2 12241 | Lemma for ~ supmul . (Con... |
| supmul 12242 | The supremum function dist... |
| sup3ii 12243 | A version of the completen... |
| suprclii 12244 | Closure of supremum of a n... |
| suprubii 12245 | A member of a nonempty bou... |
| suprlubii 12246 | The supremum of a nonempty... |
| suprnubii 12247 | An upper bound is not less... |
| suprleubii 12248 | The supremum of a nonempty... |
| riotaneg 12249 | The negative of the unique... |
| negiso 12250 | Negation is an order anti-... |
| dfinfre 12251 | The infimum of a set of re... |
| infrecl 12252 | Closure of infimum of a no... |
| infrenegsup 12253 | The infimum of a set of re... |
| infregelb 12254 | Any lower bound of a nonem... |
| infrelb 12255 | If a nonempty set of real ... |
| infrefilb 12256 | The infimum of a finite se... |
| supfirege 12257 | The supremum of a finite s... |
| inelr 12258 | The imaginary unit ` _i ` ... |
| rimul 12259 | A real number times the im... |
| cru 12260 | The representation of comp... |
| crne0 12261 | The real representation of... |
| creur 12262 | The real part of a complex... |
| creui 12263 | The imaginary part of a co... |
| cju 12264 | The complex conjugate of a... |
| ofsubeq0 12265 | Function analogue of ~ sub... |
| ofnegsub 12266 | Function analogue of ~ neg... |
| ofsubge0 12267 | Function analogue of ~ sub... |
| nnexALT 12270 | Alternate proof of ~ nnex ... |
| peano5nni 12271 | Peano's inductive postulat... |
| nnssre 12272 | The positive integers are ... |
| nnsscn 12273 | The positive integers are ... |
| nnex 12274 | The set of positive intege... |
| nnre 12275 | A positive integer is a re... |
| nncn 12276 | A positive integer is a co... |
| nnrei 12277 | A positive integer is a re... |
| nncni 12278 | A positive integer is a co... |
| 1nn 12279 | Peano postulate: 1 is a po... |
| peano2nn 12280 | Peano postulate: a success... |
| dfnn2 12281 | Alternate definition of th... |
| dfnn3 12282 | Alternate definition of th... |
| nnred 12283 | A positive integer is a re... |
| nncnd 12284 | A positive integer is a co... |
| peano2nnd 12285 | Peano postulate: a success... |
| nnind 12286 | Principle of Mathematical ... |
| nnindALT 12287 | Principle of Mathematical ... |
| nnindd 12288 | Principle of Mathematical ... |
| nn1m1nn 12289 | Every positive integer is ... |
| nn1suc 12290 | If a statement holds for 1... |
| nnaddcl 12291 | Closure of addition of pos... |
| nnmulcl 12292 | Closure of multiplication ... |
| nnmulcli 12293 | Closure of multiplication ... |
| nnmtmip 12294 | "Minus times minus is plus... |
| nn2ge 12295 | There exists a positive in... |
| nnge1 12296 | A positive integer is one ... |
| nngt1ne1 12297 | A positive integer is grea... |
| nnle1eq1 12298 | A positive integer is less... |
| nngt0 12299 | A positive integer is posi... |
| nnnlt1 12300 | A positive integer is not ... |
| nnnle0 12301 | A positive integer is not ... |
| nnne0 12302 | A positive integer is nonz... |
| nnneneg 12303 | No positive integer is equ... |
| 0nnn 12304 | Zero is not a positive int... |
| 0nnnALT 12305 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12306 | Alternate version of ~ nnn... |
| nngt0i 12307 | A positive integer is posi... |
| nnne0i 12308 | A positive integer is nonz... |
| nndivre 12309 | The quotient of a real and... |
| nnrecre 12310 | The reciprocal of a positi... |
| nnrecgt0 12311 | The reciprocal of a positi... |
| nnsub 12312 | Subtraction of positive in... |
| nnsubi 12313 | Subtraction of positive in... |
| nndiv 12314 | Two ways to express " ` A ... |
| nndivtr 12315 | Transitive property of div... |
| nnge1d 12316 | A positive integer is one ... |
| nngt0d 12317 | A positive integer is posi... |
| nnne0d 12318 | A positive integer is nonz... |
| nnrecred 12319 | The reciprocal of a positi... |
| nnaddcld 12320 | Closure of addition of pos... |
| nnmulcld 12321 | Closure of multiplication ... |
| nndivred 12322 | A positive integer is one ... |
| 0ne1 12339 | Zero is different from one... |
| 1m1e0 12340 | One minus one equals zero.... |
| 2nn 12341 | 2 is a positive integer. ... |
| 2re 12342 | The number 2 is real. (Co... |
| 2cn 12343 | The number 2 is a complex ... |
| 2cnALT 12344 | Alternate proof of ~ 2cn .... |
| 2ex 12345 | The number 2 is a set. (C... |
| 2cnd 12346 | The number 2 is a complex ... |
| 3nn 12347 | 3 is a positive integer. ... |
| 3re 12348 | The number 3 is real. (Co... |
| 3cn 12349 | The number 3 is a complex ... |
| 3ex 12350 | The number 3 is a set. (C... |
| 4nn 12351 | 4 is a positive integer. ... |
| 4re 12352 | The number 4 is real. (Co... |
| 4cn 12353 | The number 4 is a complex ... |
| 5nn 12354 | 5 is a positive integer. ... |
| 5re 12355 | The number 5 is real. (Co... |
| 5cn 12356 | The number 5 is a complex ... |
| 6nn 12357 | 6 is a positive integer. ... |
| 6re 12358 | The number 6 is real. (Co... |
| 6cn 12359 | The number 6 is a complex ... |
| 7nn 12360 | 7 is a positive integer. ... |
| 7re 12361 | The number 7 is real. (Co... |
| 7cn 12362 | The number 7 is a complex ... |
| 8nn 12363 | 8 is a positive integer. ... |
| 8re 12364 | The number 8 is real. (Co... |
| 8cn 12365 | The number 8 is a complex ... |
| 9nn 12366 | 9 is a positive integer. ... |
| 9re 12367 | The number 9 is real. (Co... |
| 9cn 12368 | The number 9 is a complex ... |
| 0le0 12369 | Zero is nonnegative. (Con... |
| 0le2 12370 | The number 0 is less than ... |
| 2pos 12371 | The number 2 is positive. ... |
| 2ne0 12372 | The number 2 is nonzero. ... |
| 3pos 12373 | The number 3 is positive. ... |
| 3ne0 12374 | The number 3 is nonzero. ... |
| 4pos 12375 | The number 4 is positive. ... |
| 4ne0 12376 | The number 4 is nonzero. ... |
| 5pos 12377 | The number 5 is positive. ... |
| 6pos 12378 | The number 6 is positive. ... |
| 7pos 12379 | The number 7 is positive. ... |
| 8pos 12380 | The number 8 is positive. ... |
| 9pos 12381 | The number 9 is positive. ... |
| neg1cn 12382 | -1 is a complex number. (... |
| neg1rr 12383 | -1 is a real number. (Con... |
| neg1ne0 12384 | -1 is nonzero. (Contribut... |
| neg1lt0 12385 | -1 is less than 0. (Contr... |
| negneg1e1 12386 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 12387 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12388 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12389 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12390 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12391 | Function value at ` N + 1 ... |
| 1p0e1 12392 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12393 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12394 | 2 - 1 = 1. The result is ... |
| 1e2m1 12395 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12396 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12397 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12398 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12399 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12400 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12401 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12402 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12403 | Two plus two equals four. ... |
| 2times 12404 | Two times a number. (Cont... |
| times2 12405 | A number times 2. (Contri... |
| 2timesi 12406 | Two times a number. (Cont... |
| times2i 12407 | A number times 2. (Contri... |
| 2txmxeqx 12408 | Two times a complex number... |
| 2div2e1 12409 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12410 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12411 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12412 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12413 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12414 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12415 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12416 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12417 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12418 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12419 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12420 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12421 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12422 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12423 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12424 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12425 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12426 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12427 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12428 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12429 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12430 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12431 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12432 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12433 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12434 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12435 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12436 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12437 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12438 | One half of four is two. ... |
| 1lt2 12439 | 1 is less than 2. (Contri... |
| 2lt3 12440 | 2 is less than 3. (Contri... |
| 1lt3 12441 | 1 is less than 3. (Contri... |
| 3lt4 12442 | 3 is less than 4. (Contri... |
| 2lt4 12443 | 2 is less than 4. (Contri... |
| 1lt4 12444 | 1 is less than 4. (Contri... |
| 4lt5 12445 | 4 is less than 5. (Contri... |
| 3lt5 12446 | 3 is less than 5. (Contri... |
| 2lt5 12447 | 2 is less than 5. (Contri... |
| 1lt5 12448 | 1 is less than 5. (Contri... |
| 5lt6 12449 | 5 is less than 6. (Contri... |
| 4lt6 12450 | 4 is less than 6. (Contri... |
| 3lt6 12451 | 3 is less than 6. (Contri... |
| 2lt6 12452 | 2 is less than 6. (Contri... |
| 1lt6 12453 | 1 is less than 6. (Contri... |
| 6lt7 12454 | 6 is less than 7. (Contri... |
| 5lt7 12455 | 5 is less than 7. (Contri... |
| 4lt7 12456 | 4 is less than 7. (Contri... |
| 3lt7 12457 | 3 is less than 7. (Contri... |
| 2lt7 12458 | 2 is less than 7. (Contri... |
| 1lt7 12459 | 1 is less than 7. (Contri... |
| 7lt8 12460 | 7 is less than 8. (Contri... |
| 6lt8 12461 | 6 is less than 8. (Contri... |
| 5lt8 12462 | 5 is less than 8. (Contri... |
| 4lt8 12463 | 4 is less than 8. (Contri... |
| 3lt8 12464 | 3 is less than 8. (Contri... |
| 2lt8 12465 | 2 is less than 8. (Contri... |
| 1lt8 12466 | 1 is less than 8. (Contri... |
| 8lt9 12467 | 8 is less than 9. (Contri... |
| 7lt9 12468 | 7 is less than 9. (Contri... |
| 6lt9 12469 | 6 is less than 9. (Contri... |
| 5lt9 12470 | 5 is less than 9. (Contri... |
| 4lt9 12471 | 4 is less than 9. (Contri... |
| 3lt9 12472 | 3 is less than 9. (Contri... |
| 2lt9 12473 | 2 is less than 9. (Contri... |
| 1lt9 12474 | 1 is less than 9. (Contri... |
| 0ne2 12475 | 0 is not equal to 2. (Con... |
| 1ne2 12476 | 1 is not equal to 2. (Con... |
| 1le2 12477 | 1 is less than or equal to... |
| 2cnne0 12478 | 2 is a nonzero complex num... |
| 2rene0 12479 | 2 is a nonzero real number... |
| 1le3 12480 | 1 is less than or equal to... |
| neg1mulneg1e1 12481 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12482 | One-half is real. (Contri... |
| halfcn 12483 | One-half is a complex numb... |
| halfgt0 12484 | One-half is greater than z... |
| halfge0 12485 | One-half is not negative. ... |
| halflt1 12486 | One-half is less than one.... |
| 1mhlfehlf 12487 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12488 | An eighth of four thirds i... |
| halfpm6th 12489 | One half plus or minus one... |
| it0e0 12490 | i times 0 equals 0. (Cont... |
| 2mulicn 12491 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12492 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12493 | Closure of half of a numbe... |
| rehalfcl 12494 | Real closure of half. (Co... |
| half0 12495 | Half of a number is zero i... |
| 2halves 12496 | Two halves make a whole. ... |
| halfpos2 12497 | A number is positive iff i... |
| halfpos 12498 | A positive number is great... |
| halfnneg2 12499 | A number is nonnegative if... |
| halfaddsubcl 12500 | Closure of half-sum and ha... |
| halfaddsub 12501 | Sum and difference of half... |
| subhalfhalf 12502 | Subtracting the half of a ... |
| lt2halves 12503 | A sum is less than the who... |
| addltmul 12504 | Sum is less than product f... |
| nominpos 12505 | There is no smallest posit... |
| avglt1 12506 | Ordering property for aver... |
| avglt2 12507 | Ordering property for aver... |
| avgle1 12508 | Ordering property for aver... |
| avgle2 12509 | Ordering property for aver... |
| avgle 12510 | The average of two numbers... |
| 2timesd 12511 | Two times a number. (Cont... |
| times2d 12512 | A number times 2. (Contri... |
| halfcld 12513 | Closure of half of a numbe... |
| 2halvesd 12514 | Two halves make a whole. ... |
| rehalfcld 12515 | Real closure of half. (Co... |
| lt2halvesd 12516 | A sum is less than the who... |
| rehalfcli 12517 | Half a real number is real... |
| lt2addmuld 12518 | If two real numbers are le... |
| add1p1 12519 | Adding two times 1 to a nu... |
| sub1m1 12520 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12521 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12522 | A complex number increased... |
| div4p1lem1div2 12523 | An integer greater than 5,... |
| nnunb 12524 | The set of positive intege... |
| arch 12525 | Archimedean property of re... |
| nnrecl 12526 | There exists a positive in... |
| bndndx 12527 | A bounded real sequence ` ... |
| elnn0 12530 | Nonnegative integers expre... |
| nnssnn0 12531 | Positive naturals are a su... |
| nn0ssre 12532 | Nonnegative integers are a... |
| nn0sscn 12533 | Nonnegative integers are a... |
| nn0ex 12534 | The set of nonnegative int... |
| nnnn0 12535 | A positive integer is a no... |
| nnnn0i 12536 | A positive integer is a no... |
| nn0re 12537 | A nonnegative integer is a... |
| nn0cn 12538 | A nonnegative integer is a... |
| nn0rei 12539 | A nonnegative integer is a... |
| nn0cni 12540 | A nonnegative integer is a... |
| dfn2 12541 | The set of positive intege... |
| elnnne0 12542 | The positive integer prope... |
| 0nn0 12543 | 0 is a nonnegative integer... |
| 1nn0 12544 | 1 is a nonnegative integer... |
| 2nn0 12545 | 2 is a nonnegative integer... |
| 3nn0 12546 | 3 is a nonnegative integer... |
| 4nn0 12547 | 4 is a nonnegative integer... |
| 5nn0 12548 | 5 is a nonnegative integer... |
| 6nn0 12549 | 6 is a nonnegative integer... |
| 7nn0 12550 | 7 is a nonnegative integer... |
| 8nn0 12551 | 8 is a nonnegative integer... |
| 9nn0 12552 | 9 is a nonnegative integer... |
| nn0ge0 12553 | A nonnegative integer is g... |
| nn0nlt0 12554 | A nonnegative integer is n... |
| nn0ge0i 12555 | Nonnegative integers are n... |
| nn0le0eq0 12556 | A nonnegative integer is l... |
| nn0p1gt0 12557 | A nonnegative integer incr... |
| nnnn0addcl 12558 | A positive integer plus a ... |
| nn0nnaddcl 12559 | A nonnegative integer plus... |
| 0mnnnnn0 12560 | The result of subtracting ... |
| un0addcl 12561 | If ` S ` is closed under a... |
| un0mulcl 12562 | If ` S ` is closed under m... |
| nn0addcl 12563 | Closure of addition of non... |
| nn0mulcl 12564 | Closure of multiplication ... |
| nn0addcli 12565 | Closure of addition of non... |
| nn0mulcli 12566 | Closure of multiplication ... |
| nn0p1nn 12567 | A nonnegative integer plus... |
| peano2nn0 12568 | Second Peano postulate for... |
| nnm1nn0 12569 | A positive integer minus 1... |
| elnn0nn 12570 | The nonnegative integer pr... |
| elnnnn0 12571 | The positive integer prope... |
| elnnnn0b 12572 | The positive integer prope... |
| elnnnn0c 12573 | The positive integer prope... |
| nn0addge1 12574 | A number is less than or e... |
| nn0addge2 12575 | A number is less than or e... |
| nn0addge1i 12576 | A number is less than or e... |
| nn0addge2i 12577 | A number is less than or e... |
| nn0sub 12578 | Subtraction of nonnegative... |
| ltsubnn0 12579 | Subtracting a nonnegative ... |
| nn0negleid 12580 | A nonnegative integer is g... |
| difgtsumgt 12581 | If the difference of a rea... |
| nn0le2xi 12582 | A nonnegative integer is l... |
| nn0lele2xi 12583 | 'Less than or equal to' im... |
| fcdmnn0supp 12584 | Two ways to write the supp... |
| fcdmnn0fsupp 12585 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12586 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12587 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12588 | A positive integer is a no... |
| nn0red 12589 | A nonnegative integer is a... |
| nn0cnd 12590 | A nonnegative integer is a... |
| nn0ge0d 12591 | A nonnegative integer is g... |
| nn0addcld 12592 | Closure of addition of non... |
| nn0mulcld 12593 | Closure of multiplication ... |
| nn0readdcl 12594 | Closure law for addition o... |
| nn0n0n1ge2 12595 | A nonnegative integer whic... |
| nn0n0n1ge2b 12596 | A nonnegative integer is n... |
| nn0ge2m1nn 12597 | If a nonnegative integer i... |
| nn0ge2m1nn0 12598 | If a nonnegative integer i... |
| nn0nndivcl 12599 | Closure law for dividing o... |
| elxnn0 12602 | An extended nonnegative in... |
| nn0ssxnn0 12603 | The standard nonnegative i... |
| nn0xnn0 12604 | A standard nonnegative int... |
| xnn0xr 12605 | An extended nonnegative in... |
| 0xnn0 12606 | Zero is an extended nonneg... |
| pnf0xnn0 12607 | Positive infinity is an ex... |
| nn0nepnf 12608 | No standard nonnegative in... |
| nn0xnn0d 12609 | A standard nonnegative int... |
| nn0nepnfd 12610 | No standard nonnegative in... |
| xnn0nemnf 12611 | No extended nonnegative in... |
| xnn0xrnemnf 12612 | The extended nonnegative i... |
| xnn0nnn0pnf 12613 | An extended nonnegative in... |
| elz 12616 | Membership in the set of i... |
| nnnegz 12617 | The negative of a positive... |
| zre 12618 | An integer is a real. (Co... |
| zcn 12619 | An integer is a complex nu... |
| zrei 12620 | An integer is a real numbe... |
| zssre 12621 | The integers are a subset ... |
| zsscn 12622 | The integers are a subset ... |
| zex 12623 | The set of integers exists... |
| elnnz 12624 | Positive integer property ... |
| 0z 12625 | Zero is an integer. (Cont... |
| 0zd 12626 | Zero is an integer, deduct... |
| elnn0z 12627 | Nonnegative integer proper... |
| elznn0nn 12628 | Integer property expressed... |
| elznn0 12629 | Integer property expressed... |
| elznn 12630 | Integer property expressed... |
| zle0orge1 12631 | There is no integer in the... |
| elz2 12632 | Membership in the set of i... |
| dfz2 12633 | Alternative definition of ... |
| zexALT 12634 | Alternate proof of ~ zex .... |
| nnz 12635 | A positive integer is an i... |
| nnssz 12636 | Positive integers are a su... |
| nn0ssz 12637 | Nonnegative integers are a... |
| nnzOLD 12638 | Obsolete version of ~ nnz ... |
| nn0z 12639 | A nonnegative integer is a... |
| nn0zd 12640 | A nonnegative integer is a... |
| nnzd 12641 | A positive integer is an i... |
| nnzi 12642 | A positive integer is an i... |
| nn0zi 12643 | A nonnegative integer is a... |
| elnnz1 12644 | Positive integer property ... |
| znnnlt1 12645 | An integer is not a positi... |
| nnzrab 12646 | Positive integers expresse... |
| nn0zrab 12647 | Nonnegative integers expre... |
| 1z 12648 | One is an integer. (Contr... |
| 1zzd 12649 | One is an integer, deducti... |
| 2z 12650 | 2 is an integer. (Contrib... |
| 3z 12651 | 3 is an integer. (Contrib... |
| 4z 12652 | 4 is an integer. (Contrib... |
| znegcl 12653 | Closure law for negative i... |
| neg1z 12654 | -1 is an integer. (Contri... |
| znegclb 12655 | A complex number is an int... |
| nn0negz 12656 | The negative of a nonnegat... |
| nn0negzi 12657 | The negative of a nonnegat... |
| zaddcl 12658 | Closure of addition of int... |
| peano2z 12659 | Second Peano postulate gen... |
| zsubcl 12660 | Closure of subtraction of ... |
| peano2zm 12661 | "Reverse" second Peano pos... |
| zletr 12662 | Transitive law of ordering... |
| zrevaddcl 12663 | Reverse closure law for ad... |
| znnsub 12664 | The positive difference of... |
| znn0sub 12665 | The nonnegative difference... |
| nzadd 12666 | The sum of a real number n... |
| zmulcl 12667 | Closure of multiplication ... |
| zltp1le 12668 | Integer ordering relation.... |
| zleltp1 12669 | Integer ordering relation.... |
| zlem1lt 12670 | Integer ordering relation.... |
| zltlem1 12671 | Integer ordering relation.... |
| zgt0ge1 12672 | An integer greater than ` ... |
| nnleltp1 12673 | Positive integer ordering ... |
| nnltp1le 12674 | Positive integer ordering ... |
| nnaddm1cl 12675 | Closure of addition of pos... |
| nn0ltp1le 12676 | Nonnegative integer orderi... |
| nn0leltp1 12677 | Nonnegative integer orderi... |
| nn0ltlem1 12678 | Nonnegative integer orderi... |
| nn0sub2 12679 | Subtraction of nonnegative... |
| nn0lt10b 12680 | A nonnegative integer less... |
| nn0lt2 12681 | A nonnegative integer less... |
| nn0le2is012 12682 | A nonnegative integer whic... |
| nn0lem1lt 12683 | Nonnegative integer orderi... |
| nnlem1lt 12684 | Positive integer ordering ... |
| nnltlem1 12685 | Positive integer ordering ... |
| nnm1ge0 12686 | A positive integer decreas... |
| nn0ge0div 12687 | Division of a nonnegative ... |
| zdiv 12688 | Two ways to express " ` M ... |
| zdivadd 12689 | Property of divisibility: ... |
| zdivmul 12690 | Property of divisibility: ... |
| zextle 12691 | An extensionality-like pro... |
| zextlt 12692 | An extensionality-like pro... |
| recnz 12693 | The reciprocal of a number... |
| btwnnz 12694 | A number between an intege... |
| gtndiv 12695 | A larger number does not d... |
| halfnz 12696 | One-half is not an integer... |
| 3halfnz 12697 | Three halves is not an int... |
| suprzcl 12698 | The supremum of a bounded-... |
| prime 12699 | Two ways to express " ` A ... |
| msqznn 12700 | The square of a nonzero in... |
| zneo 12701 | No even integer equals an ... |
| nneo 12702 | A positive integer is even... |
| nneoi 12703 | A positive integer is even... |
| zeo 12704 | An integer is even or odd.... |
| zeo2 12705 | An integer is even or odd ... |
| peano2uz2 12706 | Second Peano postulate for... |
| peano5uzi 12707 | Peano's inductive postulat... |
| peano5uzti 12708 | Peano's inductive postulat... |
| dfuzi 12709 | An expression for the uppe... |
| uzind 12710 | Induction on the upper int... |
| uzind2 12711 | Induction on the upper int... |
| uzind3 12712 | Induction on the upper int... |
| nn0ind 12713 | Principle of Mathematical ... |
| nn0indALT 12714 | Principle of Mathematical ... |
| nn0indd 12715 | Principle of Mathematical ... |
| fzind 12716 | Induction on the integers ... |
| fnn0ind 12717 | Induction on the integers ... |
| nn0ind-raph 12718 | Principle of Mathematical ... |
| zindd 12719 | Principle of Mathematical ... |
| fzindd 12720 | Induction on the integers ... |
| btwnz 12721 | Any real number can be san... |
| zred 12722 | An integer is a real numbe... |
| zcnd 12723 | An integer is a complex nu... |
| znegcld 12724 | Closure law for negative i... |
| peano2zd 12725 | Deduction from second Pean... |
| zaddcld 12726 | Closure of addition of int... |
| zsubcld 12727 | Closure of subtraction of ... |
| zmulcld 12728 | Closure of multiplication ... |
| znnn0nn 12729 | The negative of a negative... |
| zadd2cl 12730 | Increasing an integer by 2... |
| zriotaneg 12731 | The negative of the unique... |
| suprfinzcl 12732 | The supremum of a nonempty... |
| 9p1e10 12735 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12736 | Version of the definition ... |
| decex 12737 | A decimal number is a set.... |
| deceq1 12738 | Equality theorem for the d... |
| deceq2 12739 | Equality theorem for the d... |
| deceq1i 12740 | Equality theorem for the d... |
| deceq2i 12741 | Equality theorem for the d... |
| deceq12i 12742 | Equality theorem for the d... |
| numnncl 12743 | Closure for a numeral (wit... |
| num0u 12744 | Add a zero in the units pl... |
| num0h 12745 | Add a zero in the higher p... |
| numcl 12746 | Closure for a decimal inte... |
| numsuc 12747 | The successor of a decimal... |
| deccl 12748 | Closure for a numeral. (C... |
| 10nn 12749 | 10 is a positive integer. ... |
| 10pos 12750 | The number 10 is positive.... |
| 10nn0 12751 | 10 is a nonnegative intege... |
| 10re 12752 | The number 10 is real. (C... |
| decnncl 12753 | Closure for a numeral. (C... |
| dec0u 12754 | Add a zero in the units pl... |
| dec0h 12755 | Add a zero in the higher p... |
| numnncl2 12756 | Closure for a decimal inte... |
| decnncl2 12757 | Closure for a decimal inte... |
| numlt 12758 | Comparing two decimal inte... |
| numltc 12759 | Comparing two decimal inte... |
| le9lt10 12760 | A "decimal digit" (i.e. a ... |
| declt 12761 | Comparing two decimal inte... |
| decltc 12762 | Comparing two decimal inte... |
| declth 12763 | Comparing two decimal inte... |
| decsuc 12764 | The successor of a decimal... |
| 3declth 12765 | Comparing two decimal inte... |
| 3decltc 12766 | Comparing two decimal inte... |
| decle 12767 | Comparing two decimal inte... |
| decleh 12768 | Comparing two decimal inte... |
| declei 12769 | Comparing a digit to a dec... |
| numlti 12770 | Comparing a digit to a dec... |
| declti 12771 | Comparing a digit to a dec... |
| decltdi 12772 | Comparing a digit to a dec... |
| numsucc 12773 | The successor of a decimal... |
| decsucc 12774 | The successor of a decimal... |
| 1e0p1 12775 | The successor of zero. (C... |
| dec10p 12776 | Ten plus an integer. (Con... |
| numma 12777 | Perform a multiply-add of ... |
| nummac 12778 | Perform a multiply-add of ... |
| numma2c 12779 | Perform a multiply-add of ... |
| numadd 12780 | Add two decimal integers `... |
| numaddc 12781 | Add two decimal integers `... |
| nummul1c 12782 | The product of a decimal i... |
| nummul2c 12783 | The product of a decimal i... |
| decma 12784 | Perform a multiply-add of ... |
| decmac 12785 | Perform a multiply-add of ... |
| decma2c 12786 | Perform a multiply-add of ... |
| decadd 12787 | Add two numerals ` M ` and... |
| decaddc 12788 | Add two numerals ` M ` and... |
| decaddc2 12789 | Add two numerals ` M ` and... |
| decrmanc 12790 | Perform a multiply-add of ... |
| decrmac 12791 | Perform a multiply-add of ... |
| decaddm10 12792 | The sum of two multiples o... |
| decaddi 12793 | Add two numerals ` M ` and... |
| decaddci 12794 | Add two numerals ` M ` and... |
| decaddci2 12795 | Add two numerals ` M ` and... |
| decsubi 12796 | Difference between a numer... |
| decmul1 12797 | The product of a numeral w... |
| decmul1c 12798 | The product of a numeral w... |
| decmul2c 12799 | The product of a numeral w... |
| decmulnc 12800 | The product of a numeral w... |
| 11multnc 12801 | The product of 11 (as nume... |
| decmul10add 12802 | A multiplication of a numb... |
| 6p5lem 12803 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12804 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12805 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12806 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12807 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12808 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12809 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12810 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12811 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12812 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12813 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12814 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12815 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12816 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12817 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12818 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12819 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12820 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12821 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12822 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12823 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12824 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12825 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12826 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12827 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12828 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12829 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12830 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12831 | 4 times 3 equals 12. (Con... |
| 4t4e16 12832 | 4 times 4 equals 16. (Con... |
| 5t2e10 12833 | 5 times 2 equals 10. (Con... |
| 5t3e15 12834 | 5 times 3 equals 15. (Con... |
| 5t4e20 12835 | 5 times 4 equals 20. (Con... |
| 5t5e25 12836 | 5 times 5 equals 25. (Con... |
| 6t2e12 12837 | 6 times 2 equals 12. (Con... |
| 6t3e18 12838 | 6 times 3 equals 18. (Con... |
| 6t4e24 12839 | 6 times 4 equals 24. (Con... |
| 6t5e30 12840 | 6 times 5 equals 30. (Con... |
| 6t6e36 12841 | 6 times 6 equals 36. (Con... |
| 7t2e14 12842 | 7 times 2 equals 14. (Con... |
| 7t3e21 12843 | 7 times 3 equals 21. (Con... |
| 7t4e28 12844 | 7 times 4 equals 28. (Con... |
| 7t5e35 12845 | 7 times 5 equals 35. (Con... |
| 7t6e42 12846 | 7 times 6 equals 42. (Con... |
| 7t7e49 12847 | 7 times 7 equals 49. (Con... |
| 8t2e16 12848 | 8 times 2 equals 16. (Con... |
| 8t3e24 12849 | 8 times 3 equals 24. (Con... |
| 8t4e32 12850 | 8 times 4 equals 32. (Con... |
| 8t5e40 12851 | 8 times 5 equals 40. (Con... |
| 8t6e48 12852 | 8 times 6 equals 48. (Con... |
| 8t7e56 12853 | 8 times 7 equals 56. (Con... |
| 8t8e64 12854 | 8 times 8 equals 64. (Con... |
| 9t2e18 12855 | 9 times 2 equals 18. (Con... |
| 9t3e27 12856 | 9 times 3 equals 27. (Con... |
| 9t4e36 12857 | 9 times 4 equals 36. (Con... |
| 9t5e45 12858 | 9 times 5 equals 45. (Con... |
| 9t6e54 12859 | 9 times 6 equals 54. (Con... |
| 9t7e63 12860 | 9 times 7 equals 63. (Con... |
| 9t8e72 12861 | 9 times 8 equals 72. (Con... |
| 9t9e81 12862 | 9 times 9 equals 81. (Con... |
| 9t11e99 12863 | 9 times 11 equals 99. (Co... |
| 9lt10 12864 | 9 is less than 10. (Contr... |
| 8lt10 12865 | 8 is less than 10. (Contr... |
| 7lt10 12866 | 7 is less than 10. (Contr... |
| 6lt10 12867 | 6 is less than 10. (Contr... |
| 5lt10 12868 | 5 is less than 10. (Contr... |
| 4lt10 12869 | 4 is less than 10. (Contr... |
| 3lt10 12870 | 3 is less than 10. (Contr... |
| 2lt10 12871 | 2 is less than 10. (Contr... |
| 1lt10 12872 | 1 is less than 10. (Contr... |
| decbin0 12873 | Decompose base 4 into base... |
| decbin2 12874 | Decompose base 4 into base... |
| decbin3 12875 | Decompose base 4 into base... |
| halfthird 12876 | Half minus a third. (Cont... |
| 5recm6rec 12877 | One fifth minus one sixth.... |
| uzval 12880 | The value of the upper int... |
| uzf 12881 | The domain and codomain of... |
| eluz1 12882 | Membership in the upper se... |
| eluzel2 12883 | Implication of membership ... |
| eluz2 12884 | Membership in an upper set... |
| eluzmn 12885 | Membership in an earlier u... |
| eluz1i 12886 | Membership in an upper set... |
| eluzuzle 12887 | An integer in an upper set... |
| eluzelz 12888 | A member of an upper set o... |
| eluzelre 12889 | A member of an upper set o... |
| eluzelcn 12890 | A member of an upper set o... |
| eluzle 12891 | Implication of membership ... |
| eluz 12892 | Membership in an upper set... |
| uzid 12893 | Membership of the least me... |
| uzidd 12894 | Membership of the least me... |
| uzn0 12895 | The upper integers are all... |
| uztrn 12896 | Transitive law for sets of... |
| uztrn2 12897 | Transitive law for sets of... |
| uzneg 12898 | Contraposition law for upp... |
| uzssz 12899 | An upper set of integers i... |
| uzssre 12900 | An upper set of integers i... |
| uzss 12901 | Subset relationship for tw... |
| uztric 12902 | Totality of the ordering r... |
| uz11 12903 | The upper integers functio... |
| eluzp1m1 12904 | Membership in the next upp... |
| eluzp1l 12905 | Strict ordering implied by... |
| eluzp1p1 12906 | Membership in the next upp... |
| eluzadd 12907 | Membership in a later uppe... |
| eluzsub 12908 | Membership in an earlier u... |
| eluzaddi 12909 | Membership in a later uppe... |
| eluzaddiOLD 12910 | Obsolete version of ~ eluz... |
| eluzsubi 12911 | Membership in an earlier u... |
| eluzsubiOLD 12912 | Obsolete version of ~ eluz... |
| eluzaddOLD 12913 | Obsolete version of ~ eluz... |
| eluzsubOLD 12914 | Obsolete version of ~ eluz... |
| subeluzsub 12915 | Membership of a difference... |
| uzm1 12916 | Choices for an element of ... |
| uznn0sub 12917 | The nonnegative difference... |
| uzin 12918 | Intersection of two upper ... |
| uzp1 12919 | Choices for an element of ... |
| nn0uz 12920 | Nonnegative integers expre... |
| nnuz 12921 | Positive integers expresse... |
| elnnuz 12922 | A positive integer express... |
| elnn0uz 12923 | A nonnegative integer expr... |
| eluz2nn 12924 | An integer greater than or... |
| eluz4eluz2 12925 | An integer greater than or... |
| eluz4nn 12926 | An integer greater than or... |
| eluzge2nn0 12927 | If an integer is greater t... |
| eluz2n0 12928 | An integer greater than or... |
| uzuzle23 12929 | An integer in the upper se... |
| eluzge3nn 12930 | If an integer is greater t... |
| uz3m2nn 12931 | An integer greater than or... |
| 1eluzge0 12932 | 1 is an integer greater th... |
| 2eluzge0 12933 | 2 is an integer greater th... |
| 2eluzge1 12934 | 2 is an integer greater th... |
| uznnssnn 12935 | The upper integers startin... |
| raluz 12936 | Restricted universal quant... |
| raluz2 12937 | Restricted universal quant... |
| rexuz 12938 | Restricted existential qua... |
| rexuz2 12939 | Restricted existential qua... |
| 2rexuz 12940 | Double existential quantif... |
| peano2uz 12941 | Second Peano postulate for... |
| peano2uzs 12942 | Second Peano postulate for... |
| peano2uzr 12943 | Reversed second Peano axio... |
| uzaddcl 12944 | Addition closure law for a... |
| nn0pzuz 12945 | The sum of a nonnegative i... |
| uzind4 12946 | Induction on the upper set... |
| uzind4ALT 12947 | Induction on the upper set... |
| uzind4s 12948 | Induction on the upper set... |
| uzind4s2 12949 | Induction on the upper set... |
| uzind4i 12950 | Induction on the upper int... |
| uzwo 12951 | Well-ordering principle: a... |
| uzwo2 12952 | Well-ordering principle: a... |
| nnwo 12953 | Well-ordering principle: a... |
| nnwof 12954 | Well-ordering principle: a... |
| nnwos 12955 | Well-ordering principle: a... |
| indstr 12956 | Strong Mathematical Induct... |
| eluznn0 12957 | Membership in a nonnegativ... |
| eluznn 12958 | Membership in a positive u... |
| eluz2b1 12959 | Two ways to say "an intege... |
| eluz2gt1 12960 | An integer greater than or... |
| eluz2b2 12961 | Two ways to say "an intege... |
| eluz2b3 12962 | Two ways to say "an intege... |
| uz2m1nn 12963 | One less than an integer g... |
| 1nuz2 12964 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12965 | A positive integer is eith... |
| uz2mulcl 12966 | Closure of multiplication ... |
| indstr2 12967 | Strong Mathematical Induct... |
| uzinfi 12968 | Extract the lower bound of... |
| nninf 12969 | The infimum of the set of ... |
| nn0inf 12970 | The infimum of the set of ... |
| infssuzle 12971 | The infimum of a subset of... |
| infssuzcl 12972 | The infimum of a subset of... |
| ublbneg 12973 | The image under negation o... |
| eqreznegel 12974 | Two ways to express the im... |
| supminf 12975 | The supremum of a bounded-... |
| lbzbi 12976 | If a set of reals is bound... |
| zsupss 12977 | Any nonempty bounded subse... |
| suprzcl2 12978 | The supremum of a bounded-... |
| suprzub 12979 | The supremum of a bounded-... |
| uzsupss 12980 | Any bounded subset of an u... |
| nn01to3 12981 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12982 | Alternate proof of ~ nn0ge... |
| uzwo3 12983 | Well-ordering principle: a... |
| zmin 12984 | There is a unique smallest... |
| zmax 12985 | There is a unique largest ... |
| zbtwnre 12986 | There is a unique integer ... |
| rebtwnz 12987 | There is a unique greatest... |
| elq 12990 | Membership in the set of r... |
| qmulz 12991 | If ` A ` is rational, then... |
| znq 12992 | The ratio of an integer an... |
| qre 12993 | A rational number is a rea... |
| zq 12994 | An integer is a rational n... |
| qred 12995 | A rational number is a rea... |
| zssq 12996 | The integers are a subset ... |
| nn0ssq 12997 | The nonnegative integers a... |
| nnssq 12998 | The positive integers are ... |
| qssre 12999 | The rationals are a subset... |
| qsscn 13000 | The rationals are a subset... |
| qex 13001 | The set of rational number... |
| nnq 13002 | A positive integer is rati... |
| qcn 13003 | A rational number is a com... |
| qexALT 13004 | Alternate proof of ~ qex .... |
| qaddcl 13005 | Closure of addition of rat... |
| qnegcl 13006 | Closure law for the negati... |
| qmulcl 13007 | Closure of multiplication ... |
| qsubcl 13008 | Closure of subtraction of ... |
| qreccl 13009 | Closure of reciprocal of r... |
| qdivcl 13010 | Closure of division of rat... |
| qrevaddcl 13011 | Reverse closure law for ad... |
| nnrecq 13012 | The reciprocal of a positi... |
| irradd 13013 | The sum of an irrational n... |
| irrmul 13014 | The product of an irration... |
| elpq 13015 | A positive rational is the... |
| elpqb 13016 | A class is a positive rati... |
| rpnnen1lem2 13017 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 13018 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 13019 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 13020 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 13021 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 13022 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 13023 | One half of ~ rpnnen , whe... |
| reexALT 13024 | Alternate proof of ~ reex ... |
| cnref1o 13025 | There is a natural one-to-... |
| cnexALT 13026 | The set of complex numbers... |
| xrex 13027 | The set of extended reals ... |
| mpoaddex 13028 | The addition operation is ... |
| addex 13029 | The addition operation is ... |
| mpomulex 13030 | The multiplication operati... |
| mulex 13031 | The multiplication operati... |
| elrp 13034 | Membership in the set of p... |
| elrpii 13035 | Membership in the set of p... |
| 1rp 13036 | 1 is a positive real. (Co... |
| 2rp 13037 | 2 is a positive real. (Co... |
| 3rp 13038 | 3 is a positive real. (Co... |
| rpssre 13039 | The positive reals are a s... |
| rpre 13040 | A positive real is a real.... |
| rpxr 13041 | A positive real is an exte... |
| rpcn 13042 | A positive real is a compl... |
| nnrp 13043 | A positive integer is a po... |
| rpgt0 13044 | A positive real is greater... |
| rpge0 13045 | A positive real is greater... |
| rpregt0 13046 | A positive real is a posit... |
| rprege0 13047 | A positive real is a nonne... |
| rpne0 13048 | A positive real is nonzero... |
| rprene0 13049 | A positive real is a nonze... |
| rpcnne0 13050 | A positive real is a nonze... |
| rpcndif0 13051 | A positive real number is ... |
| ralrp 13052 | Quantification over positi... |
| rexrp 13053 | Quantification over positi... |
| rpaddcl 13054 | Closure law for addition o... |
| rpmulcl 13055 | Closure law for multiplica... |
| rpmtmip 13056 | "Minus times minus is plus... |
| rpdivcl 13057 | Closure law for division o... |
| rpreccl 13058 | Closure law for reciprocat... |
| rphalfcl 13059 | Closure law for half of a ... |
| rpgecl 13060 | A number greater than or e... |
| rphalflt 13061 | Half of a positive real is... |
| rerpdivcl 13062 | Closure law for division o... |
| ge0p1rp 13063 | A nonnegative number plus ... |
| rpneg 13064 | Either a nonzero real or i... |
| negelrp 13065 | Elementhood of a negation ... |
| negelrpd 13066 | The negation of a negative... |
| 0nrp 13067 | Zero is not a positive rea... |
| ltsubrp 13068 | Subtracting a positive rea... |
| ltaddrp 13069 | Adding a positive number t... |
| difrp 13070 | Two ways to say one number... |
| elrpd 13071 | Membership in the set of p... |
| nnrpd 13072 | A positive integer is a po... |
| zgt1rpn0n1 13073 | An integer greater than 1 ... |
| rpred 13074 | A positive real is a real.... |
| rpxrd 13075 | A positive real is an exte... |
| rpcnd 13076 | A positive real is a compl... |
| rpgt0d 13077 | A positive real is greater... |
| rpge0d 13078 | A positive real is greater... |
| rpne0d 13079 | A positive real is nonzero... |
| rpregt0d 13080 | A positive real is real an... |
| rprege0d 13081 | A positive real is real an... |
| rprene0d 13082 | A positive real is a nonze... |
| rpcnne0d 13083 | A positive real is a nonze... |
| rpreccld 13084 | Closure law for reciprocat... |
| rprecred 13085 | Closure law for reciprocat... |
| rphalfcld 13086 | Closure law for half of a ... |
| reclt1d 13087 | The reciprocal of a positi... |
| recgt1d 13088 | The reciprocal of a positi... |
| rpaddcld 13089 | Closure law for addition o... |
| rpmulcld 13090 | Closure law for multiplica... |
| rpdivcld 13091 | Closure law for division o... |
| ltrecd 13092 | The reciprocal of both sid... |
| lerecd 13093 | The reciprocal of both sid... |
| ltrec1d 13094 | Reciprocal swap in a 'less... |
| lerec2d 13095 | Reciprocal swap in a 'less... |
| lediv2ad 13096 | Division of both sides of ... |
| ltdiv2d 13097 | Division of a positive num... |
| lediv2d 13098 | Division of a positive num... |
| ledivdivd 13099 | Invert ratios of positive ... |
| divge1 13100 | The ratio of a number over... |
| divlt1lt 13101 | A real number divided by a... |
| divle1le 13102 | A real number divided by a... |
| ledivge1le 13103 | If a number is less than o... |
| ge0p1rpd 13104 | A nonnegative number plus ... |
| rerpdivcld 13105 | Closure law for division o... |
| ltsubrpd 13106 | Subtracting a positive rea... |
| ltaddrpd 13107 | Adding a positive number t... |
| ltaddrp2d 13108 | Adding a positive number t... |
| ltmulgt11d 13109 | Multiplication by a number... |
| ltmulgt12d 13110 | Multiplication by a number... |
| gt0divd 13111 | Division of a positive num... |
| ge0divd 13112 | Division of a nonnegative ... |
| rpgecld 13113 | A number greater than or e... |
| divge0d 13114 | The ratio of nonnegative a... |
| ltmul1d 13115 | The ratio of nonnegative a... |
| ltmul2d 13116 | Multiplication of both sid... |
| lemul1d 13117 | Multiplication of both sid... |
| lemul2d 13118 | Multiplication of both sid... |
| ltdiv1d 13119 | Division of both sides of ... |
| lediv1d 13120 | Division of both sides of ... |
| ltmuldivd 13121 | 'Less than' relationship b... |
| ltmuldiv2d 13122 | 'Less than' relationship b... |
| lemuldivd 13123 | 'Less than or equal to' re... |
| lemuldiv2d 13124 | 'Less than or equal to' re... |
| ltdivmuld 13125 | 'Less than' relationship b... |
| ltdivmul2d 13126 | 'Less than' relationship b... |
| ledivmuld 13127 | 'Less than or equal to' re... |
| ledivmul2d 13128 | 'Less than or equal to' re... |
| ltmul1dd 13129 | The ratio of nonnegative a... |
| ltmul2dd 13130 | Multiplication of both sid... |
| ltdiv1dd 13131 | Division of both sides of ... |
| lediv1dd 13132 | Division of both sides of ... |
| lediv12ad 13133 | Comparison of ratio of two... |
| mul2lt0rlt0 13134 | If the result of a multipl... |
| mul2lt0rgt0 13135 | If the result of a multipl... |
| mul2lt0llt0 13136 | If the result of a multipl... |
| mul2lt0lgt0 13137 | If the result of a multipl... |
| mul2lt0bi 13138 | If the result of a multipl... |
| prodge0rd 13139 | Infer that a multiplicand ... |
| prodge0ld 13140 | Infer that a multiplier is... |
| ltdiv23d 13141 | Swap denominator with othe... |
| lediv23d 13142 | Swap denominator with othe... |
| lt2mul2divd 13143 | The ratio of nonnegative a... |
| nnledivrp 13144 | Division of a positive int... |
| nn0ledivnn 13145 | Division of a nonnegative ... |
| addlelt 13146 | If the sum of a real numbe... |
| ltxr 13153 | The 'less than' binary rel... |
| elxr 13154 | Membership in the set of e... |
| xrnemnf 13155 | An extended real other tha... |
| xrnepnf 13156 | An extended real other tha... |
| xrltnr 13157 | The extended real 'less th... |
| ltpnf 13158 | Any (finite) real is less ... |
| ltpnfd 13159 | Any (finite) real is less ... |
| 0ltpnf 13160 | Zero is less than plus inf... |
| mnflt 13161 | Minus infinity is less tha... |
| mnfltd 13162 | Minus infinity is less tha... |
| mnflt0 13163 | Minus infinity is less tha... |
| mnfltpnf 13164 | Minus infinity is less tha... |
| mnfltxr 13165 | Minus infinity is less tha... |
| pnfnlt 13166 | No extended real is greate... |
| nltmnf 13167 | No extended real is less t... |
| pnfge 13168 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13169 | An extended nonnegative in... |
| 0lepnf 13170 | 0 less than or equal to po... |
| xnn0ge0 13171 | An extended nonnegative in... |
| mnfle 13172 | Minus infinity is less tha... |
| mnfled 13173 | Minus infinity is less tha... |
| xrltnsym 13174 | Ordering on the extended r... |
| xrltnsym2 13175 | 'Less than' is antisymmetr... |
| xrlttri 13176 | Ordering on the extended r... |
| xrlttr 13177 | Ordering on the extended r... |
| xrltso 13178 | 'Less than' is a strict or... |
| xrlttri2 13179 | Trichotomy law for 'less t... |
| xrlttri3 13180 | Trichotomy law for 'less t... |
| xrleloe 13181 | 'Less than or equal' expre... |
| xrleltne 13182 | 'Less than or equal to' im... |
| xrltlen 13183 | 'Less than' expressed in t... |
| dfle2 13184 | Alternative definition of ... |
| dflt2 13185 | Alternative definition of ... |
| xrltle 13186 | 'Less than' implies 'less ... |
| xrltled 13187 | 'Less than' implies 'less ... |
| xrleid 13188 | 'Less than or equal to' is... |
| xrleidd 13189 | 'Less than or equal to' is... |
| xrletri 13190 | Trichotomy law for extende... |
| xrletri3 13191 | Trichotomy law for extende... |
| xrletrid 13192 | Trichotomy law for extende... |
| xrlelttr 13193 | Transitive law for orderin... |
| xrltletr 13194 | Transitive law for orderin... |
| xrletr 13195 | Transitive law for orderin... |
| xrlttrd 13196 | Transitive law for orderin... |
| xrlelttrd 13197 | Transitive law for orderin... |
| xrltletrd 13198 | Transitive law for orderin... |
| xrletrd 13199 | Transitive law for orderin... |
| xrltne 13200 | 'Less than' implies not eq... |
| nltpnft 13201 | An extended real is not le... |
| xgepnf 13202 | An extended real which is ... |
| ngtmnft 13203 | An extended real is not gr... |
| xlemnf 13204 | An extended real which is ... |
| xrrebnd 13205 | An extended real is real i... |
| xrre 13206 | A way of proving that an e... |
| xrre2 13207 | An extended real between t... |
| xrre3 13208 | A way of proving that an e... |
| ge0gtmnf 13209 | A nonnegative extended rea... |
| ge0nemnf 13210 | A nonnegative extended rea... |
| xrrege0 13211 | A nonnegative extended rea... |
| xrmax1 13212 | An extended real is less t... |
| xrmax2 13213 | An extended real is less t... |
| xrmin1 13214 | The minimum of two extende... |
| xrmin2 13215 | The minimum of two extende... |
| xrmaxeq 13216 | The maximum of two extende... |
| xrmineq 13217 | The minimum of two extende... |
| xrmaxlt 13218 | Two ways of saying the max... |
| xrltmin 13219 | Two ways of saying an exte... |
| xrmaxle 13220 | Two ways of saying the max... |
| xrlemin 13221 | Two ways of saying a numbe... |
| max1 13222 | A number is less than or e... |
| max1ALT 13223 | A number is less than or e... |
| max2 13224 | A number is less than or e... |
| 2resupmax 13225 | The supremum of two real n... |
| min1 13226 | The minimum of two numbers... |
| min2 13227 | The minimum of two numbers... |
| maxle 13228 | Two ways of saying the max... |
| lemin 13229 | Two ways of saying a numbe... |
| maxlt 13230 | Two ways of saying the max... |
| ltmin 13231 | Two ways of saying a numbe... |
| lemaxle 13232 | A real number which is les... |
| max0sub 13233 | Decompose a real number in... |
| ifle 13234 | An if statement transforms... |
| z2ge 13235 | There exists an integer gr... |
| qbtwnre 13236 | The rational numbers are d... |
| qbtwnxr 13237 | The rational numbers are d... |
| qsqueeze 13238 | If a nonnegative real is l... |
| qextltlem 13239 | Lemma for ~ qextlt and qex... |
| qextlt 13240 | An extensionality-like pro... |
| qextle 13241 | An extensionality-like pro... |
| xralrple 13242 | Show that ` A ` is less th... |
| alrple 13243 | Show that ` A ` is less th... |
| xnegeq 13244 | Equality of two extended n... |
| xnegex 13245 | A negative extended real e... |
| xnegpnf 13246 | Minus ` +oo ` . Remark of... |
| xnegmnf 13247 | Minus ` -oo ` . Remark of... |
| rexneg 13248 | Minus a real number. Rema... |
| xneg0 13249 | The negative of zero. (Co... |
| xnegcl 13250 | Closure of extended real n... |
| xnegneg 13251 | Extended real version of ~... |
| xneg11 13252 | Extended real version of ~... |
| xltnegi 13253 | Forward direction of ~ xlt... |
| xltneg 13254 | Extended real version of ~... |
| xleneg 13255 | Extended real version of ~... |
| xlt0neg1 13256 | Extended real version of ~... |
| xlt0neg2 13257 | Extended real version of ~... |
| xle0neg1 13258 | Extended real version of ~... |
| xle0neg2 13259 | Extended real version of ~... |
| xaddval 13260 | Value of the extended real... |
| xaddf 13261 | The extended real addition... |
| xmulval 13262 | Value of the extended real... |
| xaddpnf1 13263 | Addition of positive infin... |
| xaddpnf2 13264 | Addition of positive infin... |
| xaddmnf1 13265 | Addition of negative infin... |
| xaddmnf2 13266 | Addition of negative infin... |
| pnfaddmnf 13267 | Addition of positive and n... |
| mnfaddpnf 13268 | Addition of negative and p... |
| rexadd 13269 | The extended real addition... |
| rexsub 13270 | Extended real subtraction ... |
| rexaddd 13271 | The extended real addition... |
| xnn0xaddcl 13272 | The extended nonnegative i... |
| xaddnemnf 13273 | Closure of extended real a... |
| xaddnepnf 13274 | Closure of extended real a... |
| xnegid 13275 | Extended real version of ~... |
| xaddcl 13276 | The extended real addition... |
| xaddcom 13277 | The extended real addition... |
| xaddrid 13278 | Extended real version of ~... |
| xaddlid 13279 | Extended real version of ~... |
| xaddridd 13280 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13281 | Extended nonnegative integ... |
| xnn0lenn0nn0 13282 | An extended nonnegative in... |
| xnn0le2is012 13283 | An extended nonnegative in... |
| xnn0xadd0 13284 | The sum of two extended no... |
| xnegdi 13285 | Extended real version of ~... |
| xaddass 13286 | Associativity of extended ... |
| xaddass2 13287 | Associativity of extended ... |
| xpncan 13288 | Extended real version of ~... |
| xnpcan 13289 | Extended real version of ~... |
| xleadd1a 13290 | Extended real version of ~... |
| xleadd2a 13291 | Commuted form of ~ xleadd1... |
| xleadd1 13292 | Weakened version of ~ xlea... |
| xltadd1 13293 | Extended real version of ~... |
| xltadd2 13294 | Extended real version of ~... |
| xaddge0 13295 | The sum of nonnegative ext... |
| xle2add 13296 | Extended real version of ~... |
| xlt2add 13297 | Extended real version of ~... |
| xsubge0 13298 | Extended real version of ~... |
| xposdif 13299 | Extended real version of ~... |
| xlesubadd 13300 | Under certain conditions, ... |
| xmullem 13301 | Lemma for ~ rexmul . (Con... |
| xmullem2 13302 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13303 | Extended real multiplicati... |
| xmul01 13304 | Extended real version of ~... |
| xmul02 13305 | Extended real version of ~... |
| xmulneg1 13306 | Extended real version of ~... |
| xmulneg2 13307 | Extended real version of ~... |
| rexmul 13308 | The extended real multipli... |
| xmulf 13309 | The extended real multipli... |
| xmulcl 13310 | Closure of extended real m... |
| xmulpnf1 13311 | Multiplication by plus inf... |
| xmulpnf2 13312 | Multiplication by plus inf... |
| xmulmnf1 13313 | Multiplication by minus in... |
| xmulmnf2 13314 | Multiplication by minus in... |
| xmulpnf1n 13315 | Multiplication by plus inf... |
| xmulrid 13316 | Extended real version of ~... |
| xmullid 13317 | Extended real version of ~... |
| xmulm1 13318 | Extended real version of ~... |
| xmulasslem2 13319 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13320 | Extended real version of ~... |
| xmulge0 13321 | Extended real version of ~... |
| xmulasslem 13322 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13323 | Lemma for ~ xmulass . (Co... |
| xmulass 13324 | Associativity of the exten... |
| xlemul1a 13325 | Extended real version of ~... |
| xlemul2a 13326 | Extended real version of ~... |
| xlemul1 13327 | Extended real version of ~... |
| xlemul2 13328 | Extended real version of ~... |
| xltmul1 13329 | Extended real version of ~... |
| xltmul2 13330 | Extended real version of ~... |
| xadddilem 13331 | Lemma for ~ xadddi . (Con... |
| xadddi 13332 | Distributive property for ... |
| xadddir 13333 | Commuted version of ~ xadd... |
| xadddi2 13334 | The assumption that the mu... |
| xadddi2r 13335 | Commuted version of ~ xadd... |
| x2times 13336 | Extended real version of ~... |
| xnegcld 13337 | Closure of extended real n... |
| xaddcld 13338 | The extended real addition... |
| xmulcld 13339 | Closure of extended real m... |
| xadd4d 13340 | Rearrangement of 4 terms i... |
| xnn0add4d 13341 | Rearrangement of 4 terms i... |
| xrsupexmnf 13342 | Adding minus infinity to a... |
| xrinfmexpnf 13343 | Adding plus infinity to a ... |
| xrsupsslem 13344 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13345 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13346 | Any subset of extended rea... |
| xrinfmss 13347 | Any subset of extended rea... |
| xrinfmss2 13348 | Any subset of extended rea... |
| xrub 13349 | By quantifying only over r... |
| supxr 13350 | The supremum of a set of e... |
| supxr2 13351 | The supremum of a set of e... |
| supxrcl 13352 | The supremum of an arbitra... |
| supxrun 13353 | The supremum of the union ... |
| supxrmnf 13354 | Adding minus infinity to a... |
| supxrpnf 13355 | The supremum of a set of e... |
| supxrunb1 13356 | The supremum of an unbound... |
| supxrunb2 13357 | The supremum of an unbound... |
| supxrbnd1 13358 | The supremum of a bounded-... |
| supxrbnd2 13359 | The supremum of a bounded-... |
| xrsup0 13360 | The supremum of an empty s... |
| supxrub 13361 | A member of a set of exten... |
| supxrlub 13362 | The supremum of a set of e... |
| supxrleub 13363 | The supremum of a set of e... |
| supxrre 13364 | The real and extended real... |
| supxrbnd 13365 | The supremum of a bounded-... |
| supxrgtmnf 13366 | The supremum of a nonempty... |
| supxrre1 13367 | The supremum of a nonempty... |
| supxrre2 13368 | The supremum of a nonempty... |
| supxrss 13369 | Smaller sets of extended r... |
| infxrcl 13370 | The infimum of an arbitrar... |
| infxrlb 13371 | A member of a set of exten... |
| infxrgelb 13372 | The infimum of a set of ex... |
| infxrre 13373 | The real and extended real... |
| infxrmnf 13374 | The infinimum of a set of ... |
| xrinf0 13375 | The infimum of the empty s... |
| infxrss 13376 | Larger sets of extended re... |
| reltre 13377 | For all real numbers there... |
| rpltrp 13378 | For all positive real numb... |
| reltxrnmnf 13379 | For all extended real numb... |
| infmremnf 13380 | The infimum of the reals i... |
| infmrp1 13381 | The infimum of the positiv... |
| ixxval 13390 | Value of the interval func... |
| elixx1 13391 | Membership in an interval ... |
| ixxf 13392 | The set of intervals of ex... |
| ixxex 13393 | The set of intervals of ex... |
| ixxssxr 13394 | The set of intervals of ex... |
| elixx3g 13395 | Membership in a set of ope... |
| ixxssixx 13396 | An interval is a subset of... |
| ixxdisj 13397 | Split an interval into dis... |
| ixxun 13398 | Split an interval into two... |
| ixxin 13399 | Intersection of two interv... |
| ixxss1 13400 | Subset relationship for in... |
| ixxss2 13401 | Subset relationship for in... |
| ixxss12 13402 | Subset relationship for in... |
| ixxub 13403 | Extract the upper bound of... |
| ixxlb 13404 | Extract the lower bound of... |
| iooex 13405 | The set of open intervals ... |
| iooval 13406 | Value of the open interval... |
| ioo0 13407 | An empty open interval of ... |
| ioon0 13408 | An open interval of extend... |
| ndmioo 13409 | The open interval function... |
| iooid 13410 | An open interval with iden... |
| elioo3g 13411 | Membership in a set of ope... |
| elioore 13412 | A member of an open interv... |
| lbioo 13413 | An open interval does not ... |
| ubioo 13414 | An open interval does not ... |
| iooval2 13415 | Value of the open interval... |
| iooin 13416 | Intersection of two open i... |
| iooss1 13417 | Subset relationship for op... |
| iooss2 13418 | Subset relationship for op... |
| iocval 13419 | Value of the open-below, c... |
| icoval 13420 | Value of the closed-below,... |
| iccval 13421 | Value of the closed interv... |
| elioo1 13422 | Membership in an open inte... |
| elioo2 13423 | Membership in an open inte... |
| elioc1 13424 | Membership in an open-belo... |
| elico1 13425 | Membership in a closed-bel... |
| elicc1 13426 | Membership in a closed int... |
| iccid 13427 | A closed interval with ide... |
| ico0 13428 | An empty open interval of ... |
| ioc0 13429 | An empty open interval of ... |
| icc0 13430 | An empty closed interval o... |
| dfrp2 13431 | Alternate definition of th... |
| elicod 13432 | Membership in a left-close... |
| icogelb 13433 | An element of a left-close... |
| elicore 13434 | A member of a left-closed ... |
| ubioc1 13435 | The upper bound belongs to... |
| lbico1 13436 | The lower bound belongs to... |
| iccleub 13437 | An element of a closed int... |
| iccgelb 13438 | An element of a closed int... |
| elioo5 13439 | Membership in an open inte... |
| eliooxr 13440 | A nonempty open interval s... |
| eliooord 13441 | Ordering implied by a memb... |
| elioo4g 13442 | Membership in an open inte... |
| ioossre 13443 | An open interval is a set ... |
| ioosscn 13444 | An open interval is a set ... |
| elioc2 13445 | Membership in an open-belo... |
| elico2 13446 | Membership in a closed-bel... |
| elicc2 13447 | Membership in a closed rea... |
| elicc2i 13448 | Inference for membership i... |
| elicc4 13449 | Membership in a closed rea... |
| iccss 13450 | Condition for a closed int... |
| iccssioo 13451 | Condition for a closed int... |
| icossico 13452 | Condition for a closed-bel... |
| iccss2 13453 | Condition for a closed int... |
| iccssico 13454 | Condition for a closed int... |
| iccssioo2 13455 | Condition for a closed int... |
| iccssico2 13456 | Condition for a closed int... |
| ioomax 13457 | The open interval from min... |
| iccmax 13458 | The closed interval from m... |
| ioopos 13459 | The set of positive reals ... |
| ioorp 13460 | The set of positive reals ... |
| iooshf 13461 | Shift the arguments of the... |
| iocssre 13462 | A closed-above interval wi... |
| icossre 13463 | A closed-below interval wi... |
| iccssre 13464 | A closed real interval is ... |
| iccssxr 13465 | A closed interval is a set... |
| iocssxr 13466 | An open-below, closed-abov... |
| icossxr 13467 | A closed-below, open-above... |
| ioossicc 13468 | An open interval is a subs... |
| iccssred 13469 | A closed real interval is ... |
| eliccxr 13470 | A member of a closed inter... |
| icossicc 13471 | A closed-below, open-above... |
| iocssicc 13472 | A closed-above, open-below... |
| ioossico 13473 | An open interval is a subs... |
| iocssioo 13474 | Condition for a closed int... |
| icossioo 13475 | Condition for a closed int... |
| ioossioo 13476 | Condition for an open inte... |
| iccsupr 13477 | A nonempty subset of a clo... |
| elioopnf 13478 | Membership in an unbounded... |
| elioomnf 13479 | Membership in an unbounded... |
| elicopnf 13480 | Membership in a closed unb... |
| repos 13481 | Two ways of saying that a ... |
| ioof 13482 | The set of open intervals ... |
| iccf 13483 | The set of closed interval... |
| unirnioo 13484 | The union of the range of ... |
| dfioo2 13485 | Alternate definition of th... |
| ioorebas 13486 | Open intervals are element... |
| xrge0neqmnf 13487 | A nonnegative extended rea... |
| xrge0nre 13488 | An extended real which is ... |
| elrege0 13489 | The predicate "is a nonneg... |
| nn0rp0 13490 | A nonnegative integer is a... |
| rge0ssre 13491 | Nonnegative real numbers a... |
| elxrge0 13492 | Elementhood in the set of ... |
| 0e0icopnf 13493 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13494 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13495 | The nonnegative reals are ... |
| ge0mulcl 13496 | The nonnegative reals are ... |
| ge0xaddcl 13497 | The nonnegative reals are ... |
| ge0xmulcl 13498 | The nonnegative extended r... |
| lbicc2 13499 | The lower bound of a close... |
| ubicc2 13500 | The upper bound of a close... |
| elicc01 13501 | Membership in the closed r... |
| elunitrn 13502 | The closed unit interval i... |
| elunitcn 13503 | The closed unit interval i... |
| 0elunit 13504 | Zero is an element of the ... |
| 1elunit 13505 | One is an element of the c... |
| iooneg 13506 | Membership in a negated op... |
| iccneg 13507 | Membership in a negated cl... |
| icoshft 13508 | A shifted real is a member... |
| icoshftf1o 13509 | Shifting a closed-below, o... |
| icoun 13510 | The union of two adjacent ... |
| icodisj 13511 | Adjacent left-closed right... |
| ioounsn 13512 | The union of an open inter... |
| snunioo 13513 | The closure of one end of ... |
| snunico 13514 | The closure of the open en... |
| snunioc 13515 | The closure of the open en... |
| prunioo 13516 | The closure of an open rea... |
| ioodisj 13517 | If the upper bound of one ... |
| ioojoin 13518 | Join two open intervals to... |
| difreicc 13519 | The class difference of ` ... |
| iccsplit 13520 | Split a closed interval in... |
| iccshftr 13521 | Membership in a shifted in... |
| iccshftri 13522 | Membership in a shifted in... |
| iccshftl 13523 | Membership in a shifted in... |
| iccshftli 13524 | Membership in a shifted in... |
| iccdil 13525 | Membership in a dilated in... |
| iccdili 13526 | Membership in a dilated in... |
| icccntr 13527 | Membership in a contracted... |
| icccntri 13528 | Membership in a contracted... |
| divelunit 13529 | A condition for a ratio to... |
| lincmb01cmp 13530 | A linear combination of tw... |
| iccf1o 13531 | Describe a bijection from ... |
| iccen 13532 | Any nontrivial closed inte... |
| xov1plusxeqvd 13533 | A complex number ` X ` is ... |
| unitssre 13534 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13535 | The closed unit interval i... |
| supicc 13536 | Supremum of a bounded set ... |
| supiccub 13537 | The supremum of a bounded ... |
| supicclub 13538 | The supremum of a bounded ... |
| supicclub2 13539 | The supremum of a bounded ... |
| zltaddlt1le 13540 | The sum of an integer and ... |
| xnn0xrge0 13541 | An extended nonnegative in... |
| fzval 13544 | The value of a finite set ... |
| fzval2 13545 | An alternative way of expr... |
| fzf 13546 | Establish the domain and c... |
| elfz1 13547 | Membership in a finite set... |
| elfz 13548 | Membership in a finite set... |
| elfz2 13549 | Membership in a finite set... |
| elfzd 13550 | Membership in a finite set... |
| elfz5 13551 | Membership in a finite set... |
| elfz4 13552 | Membership in a finite set... |
| elfzuzb 13553 | Membership in a finite set... |
| eluzfz 13554 | Membership in a finite set... |
| elfzuz 13555 | A member of a finite set o... |
| elfzuz3 13556 | Membership in a finite set... |
| elfzel2 13557 | Membership in a finite set... |
| elfzel1 13558 | Membership in a finite set... |
| elfzelz 13559 | A member of a finite set o... |
| elfzelzd 13560 | A member of a finite set o... |
| fzssz 13561 | A finite sequence of integ... |
| elfzle1 13562 | A member of a finite set o... |
| elfzle2 13563 | A member of a finite set o... |
| elfzuz2 13564 | Implication of membership ... |
| elfzle3 13565 | Membership in a finite set... |
| eluzfz1 13566 | Membership in a finite set... |
| eluzfz2 13567 | Membership in a finite set... |
| eluzfz2b 13568 | Membership in a finite set... |
| elfz3 13569 | Membership in a finite set... |
| elfz1eq 13570 | Membership in a finite set... |
| elfzubelfz 13571 | If there is a member in a ... |
| peano2fzr 13572 | A Peano-postulate-like the... |
| fzn0 13573 | Properties of a finite int... |
| fz0 13574 | A finite set of sequential... |
| fzn 13575 | A finite set of sequential... |
| fzen 13576 | A shifted finite set of se... |
| fz1n 13577 | A 1-based finite set of se... |
| 0nelfz1 13578 | 0 is not an element of a f... |
| 0fz1 13579 | Two ways to say a finite 1... |
| fz10 13580 | There are no integers betw... |
| uzsubsubfz 13581 | Membership of an integer g... |
| uzsubsubfz1 13582 | Membership of an integer g... |
| ige3m2fz 13583 | Membership of an integer g... |
| fzsplit2 13584 | Split a finite interval of... |
| fzsplit 13585 | Split a finite interval of... |
| fzdisj 13586 | Condition for two finite i... |
| fz01en 13587 | 0-based and 1-based finite... |
| elfznn 13588 | A member of a finite set o... |
| elfz1end 13589 | A nonempty finite range of... |
| fz1ssnn 13590 | A finite set of positive i... |
| fznn0sub 13591 | Subtraction closure for a ... |
| fzmmmeqm 13592 | Subtracting the difference... |
| fzaddel 13593 | Membership of a sum in a f... |
| fzadd2 13594 | Membership of a sum in a f... |
| fzsubel 13595 | Membership of a difference... |
| fzopth 13596 | A finite set of sequential... |
| fzass4 13597 | Two ways to express a nond... |
| fzss1 13598 | Subset relationship for fi... |
| fzss2 13599 | Subset relationship for fi... |
| fzssuz 13600 | A finite set of sequential... |
| fzsn 13601 | A finite interval of integ... |
| fzssp1 13602 | Subset relationship for fi... |
| fzssnn 13603 | Finite sets of sequential ... |
| ssfzunsnext 13604 | A subset of a finite seque... |
| ssfzunsn 13605 | A subset of a finite seque... |
| fzsuc 13606 | Join a successor to the en... |
| fzpred 13607 | Join a predecessor to the ... |
| fzpreddisj 13608 | A finite set of sequential... |
| elfzp1 13609 | Append an element to a fin... |
| fzp1ss 13610 | Subset relationship for fi... |
| fzelp1 13611 | Membership in a set of seq... |
| fzp1elp1 13612 | Add one to an element of a... |
| fznatpl1 13613 | Shift membership in a fini... |
| fzpr 13614 | A finite interval of integ... |
| fztp 13615 | A finite interval of integ... |
| fz12pr 13616 | An integer range between 1... |
| fzsuc2 13617 | Join a successor to the en... |
| fzp1disj 13618 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13619 | Remove a successor from th... |
| fzprval 13620 | Two ways of defining the f... |
| fztpval 13621 | Two ways of defining the f... |
| fzrev 13622 | Reversal of start and end ... |
| fzrev2 13623 | Reversal of start and end ... |
| fzrev2i 13624 | Reversal of start and end ... |
| fzrev3 13625 | The "complement" of a memb... |
| fzrev3i 13626 | The "complement" of a memb... |
| fznn 13627 | Finite set of sequential i... |
| elfz1b 13628 | Membership in a 1-based fi... |
| elfz1uz 13629 | Membership in a 1-based fi... |
| elfzm11 13630 | Membership in a finite set... |
| uzsplit 13631 | Express an upper integer s... |
| uzdisj 13632 | The first ` N ` elements o... |
| fseq1p1m1 13633 | Add/remove an item to/from... |
| fseq1m1p1 13634 | Add/remove an item to/from... |
| fz1sbc 13635 | Quantification over a one-... |
| elfzp1b 13636 | An integer is a member of ... |
| elfzm1b 13637 | An integer is a member of ... |
| elfzp12 13638 | Options for membership in ... |
| fzm1 13639 | Choices for an element of ... |
| fzneuz 13640 | No finite set of sequentia... |
| fznuz 13641 | Disjointness of the upper ... |
| uznfz 13642 | Disjointness of the upper ... |
| fzp1nel 13643 | One plus the upper bound o... |
| fzrevral 13644 | Reversal of scanning order... |
| fzrevral2 13645 | Reversal of scanning order... |
| fzrevral3 13646 | Reversal of scanning order... |
| fzshftral 13647 | Shift the scanning order i... |
| ige2m1fz1 13648 | Membership of an integer g... |
| ige2m1fz 13649 | Membership in a 0-based fi... |
| elfz2nn0 13650 | Membership in a finite set... |
| fznn0 13651 | Characterization of a fini... |
| elfznn0 13652 | A member of a finite set o... |
| elfz3nn0 13653 | The upper bound of a nonem... |
| fz0ssnn0 13654 | Finite sets of sequential ... |
| fz1ssfz0 13655 | Subset relationship for fi... |
| 0elfz 13656 | 0 is an element of a finit... |
| nn0fz0 13657 | A nonnegative integer is a... |
| elfz0add 13658 | An element of a finite set... |
| fz0sn 13659 | An integer range from 0 to... |
| fz0tp 13660 | An integer range from 0 to... |
| fz0to3un2pr 13661 | An integer range from 0 to... |
| fz0to4untppr 13662 | An integer range from 0 to... |
| elfz0ubfz0 13663 | An element of a finite set... |
| elfz0fzfz0 13664 | A member of a finite set o... |
| fz0fzelfz0 13665 | If a member of a finite se... |
| fznn0sub2 13666 | Subtraction closure for a ... |
| uzsubfz0 13667 | Membership of an integer g... |
| fz0fzdiffz0 13668 | The difference of an integ... |
| elfzmlbm 13669 | Subtracting the lower boun... |
| elfzmlbp 13670 | Subtracting the lower boun... |
| fzctr 13671 | Lemma for theorems about t... |
| difelfzle 13672 | The difference of two inte... |
| difelfznle 13673 | The difference of two inte... |
| nn0split 13674 | Express the set of nonnega... |
| nn0disj 13675 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13676 | A finite set of sequential... |
| fvffz0 13677 | The function value of a fu... |
| 1fv 13678 | A function on a singleton.... |
| 4fvwrd4 13679 | The first four function va... |
| 2ffzeq 13680 | Two functions over 0-based... |
| preduz 13681 | The value of the predecess... |
| prednn 13682 | The value of the predecess... |
| prednn0 13683 | The value of the predecess... |
| predfz 13684 | Calculate the predecessor ... |
| fzof 13687 | Functionality of the half-... |
| elfzoel1 13688 | Reverse closure for half-o... |
| elfzoel2 13689 | Reverse closure for half-o... |
| elfzoelz 13690 | Reverse closure for half-o... |
| fzoval 13691 | Value of the half-open int... |
| elfzo 13692 | Membership in a half-open ... |
| elfzo2 13693 | Membership in a half-open ... |
| elfzouz 13694 | Membership in a half-open ... |
| nelfzo 13695 | An integer not being a mem... |
| fzolb 13696 | The left endpoint of a hal... |
| fzolb2 13697 | The left endpoint of a hal... |
| elfzole1 13698 | A member in a half-open in... |
| elfzolt2 13699 | A member in a half-open in... |
| elfzolt3 13700 | Membership in a half-open ... |
| elfzolt2b 13701 | A member in a half-open in... |
| elfzolt3b 13702 | Membership in a half-open ... |
| elfzop1le2 13703 | A member in a half-open in... |
| fzonel 13704 | A half-open range does not... |
| elfzouz2 13705 | The upper bound of a half-... |
| elfzofz 13706 | A half-open range is conta... |
| elfzo3 13707 | Express membership in a ha... |
| fzon0 13708 | A half-open integer interv... |
| fzossfz 13709 | A half-open range is conta... |
| fzossz 13710 | A half-open integer interv... |
| fzon 13711 | A half-open set of sequent... |
| fzo0n 13712 | A half-open range of nonne... |
| fzonlt0 13713 | A half-open integer range ... |
| fzo0 13714 | Half-open sets with equal ... |
| fzonnsub 13715 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13716 | If ` M < N ` then ` N - M ... |
| fzoss1 13717 | Subset relationship for ha... |
| fzoss2 13718 | Subset relationship for ha... |
| fzossrbm1 13719 | Subset of a half-open rang... |
| fzo0ss1 13720 | Subset relationship for ha... |
| fzossnn0 13721 | A half-open integer range ... |
| fzospliti 13722 | One direction of splitting... |
| fzosplit 13723 | Split a half-open integer ... |
| fzodisj 13724 | Abutting half-open integer... |
| fzouzsplit 13725 | Split an upper integer set... |
| fzouzdisj 13726 | A half-open integer range ... |
| fzoun 13727 | A half-open integer range ... |
| fzodisjsn 13728 | A half-open integer range ... |
| prinfzo0 13729 | The intersection of a half... |
| lbfzo0 13730 | An integer is strictly gre... |
| elfzo0 13731 | Membership in a half-open ... |
| elfzo0z 13732 | Membership in a half-open ... |
| nn0p1elfzo 13733 | A nonnegative integer incr... |
| elfzo0le 13734 | A member in a half-open ra... |
| elfzonn0 13735 | A member of a half-open ra... |
| fzonmapblen 13736 | The result of subtracting ... |
| fzofzim 13737 | If a nonnegative integer i... |
| fz1fzo0m1 13738 | Translation of one between... |
| fzossnn 13739 | Half-open integer ranges s... |
| elfzo1 13740 | Membership in a half-open ... |
| fzo1fzo0n0 13741 | An integer between 1 and a... |
| fzo0n0 13742 | A half-open integer range ... |
| fzoaddel 13743 | Translate membership in a ... |
| fzo0addel 13744 | Translate membership in a ... |
| fzo0addelr 13745 | Translate membership in a ... |
| fzoaddel2 13746 | Translate membership in a ... |
| elfzoext 13747 | Membership of an integer i... |
| elincfzoext 13748 | Membership of an increased... |
| fzosubel 13749 | Translate membership in a ... |
| fzosubel2 13750 | Membership in a translated... |
| fzosubel3 13751 | Membership in a translated... |
| eluzgtdifelfzo 13752 | Membership of the differen... |
| ige2m2fzo 13753 | Membership of an integer g... |
| fzocatel 13754 | Translate membership in a ... |
| ubmelfzo 13755 | If an integer in a 1-based... |
| elfzodifsumelfzo 13756 | If an integer is in a half... |
| elfzom1elp1fzo 13757 | Membership of an integer i... |
| elfzom1elfzo 13758 | Membership in a half-open ... |
| fzval3 13759 | Expressing a closed intege... |
| fz0add1fz1 13760 | Translate membership in a ... |
| fzosn 13761 | Expressing a singleton as ... |
| elfzomin 13762 | Membership of an integer i... |
| zpnn0elfzo 13763 | Membership of an integer i... |
| zpnn0elfzo1 13764 | Membership of an integer i... |
| fzosplitsnm1 13765 | Removing a singleton from ... |
| elfzonlteqm1 13766 | If an element of a half-op... |
| fzonn0p1 13767 | A nonnegative integer is e... |
| fzossfzop1 13768 | A half-open range of nonne... |
| fzonn0p1p1 13769 | If a nonnegative integer i... |
| elfzom1p1elfzo 13770 | Increasing an element of a... |
| fzo0ssnn0 13771 | Half-open integer ranges s... |
| fzo01 13772 | Expressing the singleton o... |
| fzo12sn 13773 | A 1-based half-open intege... |
| fzo13pr 13774 | A 1-based half-open intege... |
| fzo0to2pr 13775 | A half-open integer range ... |
| fzo0to3tp 13776 | A half-open integer range ... |
| fzo0to42pr 13777 | A half-open integer range ... |
| fzo1to4tp 13778 | A half-open integer range ... |
| fzo0sn0fzo1 13779 | A half-open range of nonne... |
| elfzo0l 13780 | A member of a half-open ra... |
| fzoend 13781 | The endpoint of a half-ope... |
| fzo0end 13782 | The endpoint of a zero-bas... |
| ssfzo12 13783 | Subset relationship for ha... |
| ssfzoulel 13784 | If a half-open integer ran... |
| ssfzo12bi 13785 | Subset relationship for ha... |
| fzoopth 13786 | A half-open integer range ... |
| ubmelm1fzo 13787 | The result of subtracting ... |
| fzofzp1 13788 | If a point is in a half-op... |
| fzofzp1b 13789 | If a point is in a half-op... |
| elfzom1b 13790 | An integer is a member of ... |
| elfzom1elp1fzo1 13791 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13792 | Membership of a positive i... |
| elfzonelfzo 13793 | If an element of a half-op... |
| fzonfzoufzol 13794 | If an element of a half-op... |
| elfzomelpfzo 13795 | An integer increased by an... |
| elfznelfzo 13796 | A value in a finite set of... |
| elfznelfzob 13797 | A value in a finite set of... |
| peano2fzor 13798 | A Peano-postulate-like the... |
| fzosplitsn 13799 | Extending a half-open rang... |
| fzosplitpr 13800 | Extending a half-open inte... |
| fzosplitprm1 13801 | Extending a half-open inte... |
| fzosplitsni 13802 | Membership in a half-open ... |
| fzisfzounsn 13803 | A finite interval of integ... |
| elfzr 13804 | A member of a finite inter... |
| elfzlmr 13805 | A member of a finite inter... |
| elfz0lmr 13806 | A member of a finite inter... |
| fzostep1 13807 | Two possibilities for a nu... |
| fzoshftral 13808 | Shift the scanning order i... |
| fzind2 13809 | Induction on the integers ... |
| fvinim0ffz 13810 | The function values for th... |
| injresinjlem 13811 | Lemma for ~ injresinj . (... |
| injresinj 13812 | A function whose restricti... |
| subfzo0 13813 | The difference between two... |
| fvf1tp 13814 | Values of a one-to-one fun... |
| flval 13819 | Value of the floor (greate... |
| flcl 13820 | The floor (greatest intege... |
| reflcl 13821 | The floor (greatest intege... |
| fllelt 13822 | A basic property of the fl... |
| flcld 13823 | The floor (greatest intege... |
| flle 13824 | A basic property of the fl... |
| flltp1 13825 | A basic property of the fl... |
| fllep1 13826 | A basic property of the fl... |
| fraclt1 13827 | The fractional part of a r... |
| fracle1 13828 | The fractional part of a r... |
| fracge0 13829 | The fractional part of a r... |
| flge 13830 | The floor function value i... |
| fllt 13831 | The floor function value i... |
| flflp1 13832 | Move floor function betwee... |
| flid 13833 | An integer is its own floo... |
| flidm 13834 | The floor function is idem... |
| flidz 13835 | A real number equals its f... |
| flltnz 13836 | The floor of a non-integer... |
| flwordi 13837 | Ordering relation for the ... |
| flword2 13838 | Ordering relation for the ... |
| flval2 13839 | An alternate way to define... |
| flval3 13840 | An alternate way to define... |
| flbi 13841 | A condition equivalent to ... |
| flbi2 13842 | A condition equivalent to ... |
| adddivflid 13843 | The floor of a sum of an i... |
| ico01fl0 13844 | The floor of a real number... |
| flge0nn0 13845 | The floor of a number grea... |
| flge1nn 13846 | The floor of a number grea... |
| fldivnn0 13847 | The floor function of a di... |
| refldivcl 13848 | The floor function of a di... |
| divfl0 13849 | The floor of a fraction is... |
| fladdz 13850 | An integer can be moved in... |
| flzadd 13851 | An integer can be moved in... |
| flmulnn0 13852 | Move a nonnegative integer... |
| btwnzge0 13853 | A real bounded between an ... |
| 2tnp1ge0ge0 13854 | Two times an integer plus ... |
| flhalf 13855 | Ordering relation for the ... |
| fldivle 13856 | The floor function of a di... |
| fldivnn0le 13857 | The floor function of a di... |
| flltdivnn0lt 13858 | The floor function of a di... |
| ltdifltdiv 13859 | If the dividend of a divis... |
| fldiv4p1lem1div2 13860 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13861 | The floor of an integer gr... |
| fldiv4lem1div2 13862 | The floor of a positive in... |
| ceilval 13863 | The value of the ceiling f... |
| dfceil2 13864 | Alternative definition of ... |
| ceilval2 13865 | The value of the ceiling f... |
| ceicl 13866 | The ceiling function retur... |
| ceilcl 13867 | Closure of the ceiling fun... |
| ceilcld 13868 | Closure of the ceiling fun... |
| ceige 13869 | The ceiling of a real numb... |
| ceilge 13870 | The ceiling of a real numb... |
| ceilged 13871 | The ceiling of a real numb... |
| ceim1l 13872 | One less than the ceiling ... |
| ceilm1lt 13873 | One less than the ceiling ... |
| ceile 13874 | The ceiling of a real numb... |
| ceille 13875 | The ceiling of a real numb... |
| ceilid 13876 | An integer is its own ceil... |
| ceilidz 13877 | A real number equals its c... |
| flleceil 13878 | The floor of a real number... |
| fleqceilz 13879 | A real number is an intege... |
| quoremz 13880 | Quotient and remainder of ... |
| quoremnn0 13881 | Quotient and remainder of ... |
| quoremnn0ALT 13882 | Alternate proof of ~ quore... |
| intfrac2 13883 | Decompose a real into inte... |
| intfracq 13884 | Decompose a rational numbe... |
| fldiv 13885 | Cancellation of the embedd... |
| fldiv2 13886 | Cancellation of an embedde... |
| fznnfl 13887 | Finite set of sequential i... |
| uzsup 13888 | An upper set of integers i... |
| ioopnfsup 13889 | An upper set of reals is u... |
| icopnfsup 13890 | An upper set of reals is u... |
| rpsup 13891 | The positive reals are unb... |
| resup 13892 | The real numbers are unbou... |
| xrsup 13893 | The extended real numbers ... |
| modval 13896 | The value of the modulo op... |
| modvalr 13897 | The value of the modulo op... |
| modcl 13898 | Closure law for the modulo... |
| flpmodeq 13899 | Partition of a division in... |
| modcld 13900 | Closure law for the modulo... |
| mod0 13901 | ` A mod B ` is zero iff ` ... |
| mulmod0 13902 | The product of an integer ... |
| negmod0 13903 | ` A ` is divisible by ` B ... |
| modge0 13904 | The modulo operation is no... |
| modlt 13905 | The modulo operation is le... |
| modelico 13906 | Modular reduction produces... |
| moddiffl 13907 | Value of the modulo operat... |
| moddifz 13908 | The modulo operation diffe... |
| modfrac 13909 | The fractional part of a n... |
| flmod 13910 | The floor function express... |
| intfrac 13911 | Break a number into its in... |
| zmod10 13912 | An integer modulo 1 is 0. ... |
| zmod1congr 13913 | Two arbitrary integers are... |
| modmulnn 13914 | Move a positive integer in... |
| modvalp1 13915 | The value of the modulo op... |
| zmodcl 13916 | Closure law for the modulo... |
| zmodcld 13917 | Closure law for the modulo... |
| zmodfz 13918 | An integer mod ` B ` lies ... |
| zmodfzo 13919 | An integer mod ` B ` lies ... |
| zmodfzp1 13920 | An integer mod ` B ` lies ... |
| modid 13921 | Identity law for modulo. ... |
| modid0 13922 | A positive real number mod... |
| modid2 13923 | Identity law for modulo. ... |
| zmodid2 13924 | Identity law for modulo re... |
| zmodidfzo 13925 | Identity law for modulo re... |
| zmodidfzoimp 13926 | Identity law for modulo re... |
| 0mod 13927 | Special case: 0 modulo a p... |
| 1mod 13928 | Special case: 1 modulo a r... |
| modabs 13929 | Absorption law for modulo.... |
| modabs2 13930 | Absorption law for modulo.... |
| modcyc 13931 | The modulo operation is pe... |
| modcyc2 13932 | The modulo operation is pe... |
| modadd1 13933 | Addition property of the m... |
| modaddabs 13934 | Absorption law for modulo.... |
| modaddmod 13935 | The sum of a real number m... |
| muladdmodid 13936 | The sum of a positive real... |
| mulp1mod1 13937 | The product of an integer ... |
| modmuladd 13938 | Decomposition of an intege... |
| modmuladdim 13939 | Implication of a decomposi... |
| modmuladdnn0 13940 | Implication of a decomposi... |
| negmod 13941 | The negation of a number m... |
| m1modnnsub1 13942 | Minus one modulo a positiv... |
| m1modge3gt1 13943 | Minus one modulo an intege... |
| addmodid 13944 | The sum of a positive inte... |
| addmodidr 13945 | The sum of a positive inte... |
| modadd2mod 13946 | The sum of a real number m... |
| modm1p1mod0 13947 | If a real number modulo a ... |
| modltm1p1mod 13948 | If a real number modulo a ... |
| modmul1 13949 | Multiplication property of... |
| modmul12d 13950 | Multiplication property of... |
| modnegd 13951 | Negation property of the m... |
| modadd12d 13952 | Additive property of the m... |
| modsub12d 13953 | Subtraction property of th... |
| modsubmod 13954 | The difference of a real n... |
| modsubmodmod 13955 | The difference of a real n... |
| 2txmodxeq0 13956 | Two times a positive real ... |
| 2submod 13957 | If a real number is betwee... |
| modifeq2int 13958 | If a nonnegative integer i... |
| modaddmodup 13959 | The sum of an integer modu... |
| modaddmodlo 13960 | The sum of an integer modu... |
| modmulmod 13961 | The product of a real numb... |
| modmulmodr 13962 | The product of an integer ... |
| modaddmulmod 13963 | The sum of a real number a... |
| moddi 13964 | Distribute multiplication ... |
| modsubdir 13965 | Distribute the modulo oper... |
| modeqmodmin 13966 | A real number equals the d... |
| modirr 13967 | A number modulo an irratio... |
| modfzo0difsn 13968 | For a number within a half... |
| modsumfzodifsn 13969 | The sum of a number within... |
| modlteq 13970 | Two nonnegative integers l... |
| addmodlteq 13971 | Two nonnegative integers l... |
| om2uz0i 13972 | The mapping ` G ` is a one... |
| om2uzsuci 13973 | The value of ` G ` (see ~ ... |
| om2uzuzi 13974 | The value ` G ` (see ~ om2... |
| om2uzlti 13975 | Less-than relation for ` G... |
| om2uzlt2i 13976 | The mapping ` G ` (see ~ o... |
| om2uzrani 13977 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13978 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13979 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13980 | An alternative definition ... |
| om2uzrdg 13981 | A helper lemma for the val... |
| uzrdglem 13982 | A helper lemma for the val... |
| uzrdgfni 13983 | The recursive definition g... |
| uzrdg0i 13984 | Initial value of a recursi... |
| uzrdgsuci 13985 | Successor value of a recur... |
| ltweuz 13986 | ` < ` is a well-founded re... |
| ltwenn 13987 | Less than well-orders the ... |
| ltwefz 13988 | Less than well-orders a se... |
| uzenom 13989 | An upper integer set is de... |
| uzinf 13990 | An upper integer set is in... |
| nnnfi 13991 | The set of positive intege... |
| uzrdgxfr 13992 | Transfer the value of the ... |
| fzennn 13993 | The cardinality of a finit... |
| fzen2 13994 | The cardinality of a finit... |
| cardfz 13995 | The cardinality of a finit... |
| hashgf1o 13996 | ` G ` maps ` _om ` one-to-... |
| fzfi 13997 | A finite interval of integ... |
| fzfid 13998 | Commonly used special case... |
| fzofi 13999 | Half-open integer sets are... |
| fsequb 14000 | The values of a finite rea... |
| fsequb2 14001 | The values of a finite rea... |
| fseqsupcl 14002 | The values of a finite rea... |
| fseqsupubi 14003 | The values of a finite rea... |
| nn0ennn 14004 | The nonnegative integers a... |
| nnenom 14005 | The set of positive intege... |
| nnct 14006 | ` NN ` is countable. (Con... |
| uzindi 14007 | Indirect strong induction ... |
| axdc4uzlem 14008 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 14009 | A version of ~ axdc4 that ... |
| ssnn0fi 14010 | A subset of the nonnegativ... |
| rabssnn0fi 14011 | A subset of the nonnegativ... |
| uzsinds 14012 | Strong (or "total") induct... |
| nnsinds 14013 | Strong (or "total") induct... |
| nn0sinds 14014 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 14015 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 14016 | If all functions of a fini... |
| fsuppmapnn0fiubex 14017 | If all functions of a fini... |
| fsuppmapnn0fiub0 14018 | If all functions of a fini... |
| suppssfz 14019 | Condition for a function o... |
| fsuppmapnn0ub 14020 | If a function over the non... |
| fsuppmapnn0fz 14021 | If a function over the non... |
| mptnn0fsupp 14022 | A mapping from the nonnega... |
| mptnn0fsuppd 14023 | A mapping from the nonnega... |
| mptnn0fsuppr 14024 | A finitely supported mappi... |
| f13idfv 14025 | A one-to-one function with... |
| seqex 14028 | Existence of the sequence ... |
| seqeq1 14029 | Equality theorem for the s... |
| seqeq2 14030 | Equality theorem for the s... |
| seqeq3 14031 | Equality theorem for the s... |
| seqeq1d 14032 | Equality deduction for the... |
| seqeq2d 14033 | Equality deduction for the... |
| seqeq3d 14034 | Equality deduction for the... |
| seqeq123d 14035 | Equality deduction for the... |
| nfseq 14036 | Hypothesis builder for the... |
| seqval 14037 | Value of the sequence buil... |
| seqfn 14038 | The sequence builder funct... |
| seq1 14039 | Value of the sequence buil... |
| seq1i 14040 | Value of the sequence buil... |
| seqp1 14041 | Value of the sequence buil... |
| seqexw 14042 | Weak version of ~ seqex th... |
| seqp1d 14043 | Value of the sequence buil... |
| seqm1 14044 | Value of the sequence buil... |
| seqcl2 14045 | Closure properties of the ... |
| seqf2 14046 | Range of the recursive seq... |
| seqcl 14047 | Closure properties of the ... |
| seqf 14048 | Range of the recursive seq... |
| seqfveq2 14049 | Equality of sequences. (C... |
| seqfeq2 14050 | Equality of sequences. (C... |
| seqfveq 14051 | Equality of sequences. (C... |
| seqfeq 14052 | Equality of sequences. (C... |
| seqshft2 14053 | Shifting the index set of ... |
| seqres 14054 | Restricting its characteri... |
| serf 14055 | An infinite series of comp... |
| serfre 14056 | An infinite series of real... |
| monoord 14057 | Ordering relation for a mo... |
| monoord2 14058 | Ordering relation for a mo... |
| sermono 14059 | The partial sums in an inf... |
| seqsplit 14060 | Split a sequence into two ... |
| seq1p 14061 | Removing the first term fr... |
| seqcaopr3 14062 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14063 | The sum of two infinite se... |
| seqcaopr 14064 | The sum of two infinite se... |
| seqf1olem2a 14065 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14066 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14067 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14068 | Rearrange a sum via an arb... |
| seradd 14069 | The sum of two infinite se... |
| sersub 14070 | The difference of two infi... |
| seqid3 14071 | A sequence that consists e... |
| seqid 14072 | Discarding the first few t... |
| seqid2 14073 | The last few partial sums ... |
| seqhomo 14074 | Apply a homomorphism to a ... |
| seqz 14075 | If the operation ` .+ ` ha... |
| seqfeq4 14076 | Equality of series under d... |
| seqfeq3 14077 | Equality of series under d... |
| seqdistr 14078 | The distributive property ... |
| ser0 14079 | The value of the partial s... |
| ser0f 14080 | A zero-valued infinite ser... |
| serge0 14081 | A finite sum of nonnegativ... |
| serle 14082 | Comparison of partial sums... |
| ser1const 14083 | Value of the partial serie... |
| seqof 14084 | Distribute function operat... |
| seqof2 14085 | Distribute function operat... |
| expval 14088 | Value of exponentiation to... |
| expnnval 14089 | Value of exponentiation to... |
| exp0 14090 | Value of a complex number ... |
| 0exp0e1 14091 | The zeroth power of zero e... |
| exp1 14092 | Value of a complex number ... |
| expp1 14093 | Value of a complex number ... |
| expneg 14094 | Value of a complex number ... |
| expneg2 14095 | Value of a complex number ... |
| expn1 14096 | A complex number raised to... |
| expcllem 14097 | Lemma for proving nonnegat... |
| expcl2lem 14098 | Lemma for proving integer ... |
| nnexpcl 14099 | Closure of exponentiation ... |
| nn0expcl 14100 | Closure of exponentiation ... |
| zexpcl 14101 | Closure of exponentiation ... |
| qexpcl 14102 | Closure of exponentiation ... |
| reexpcl 14103 | Closure of exponentiation ... |
| expcl 14104 | Closure law for nonnegativ... |
| rpexpcl 14105 | Closure law for integer ex... |
| qexpclz 14106 | Closure of integer exponen... |
| reexpclz 14107 | Closure of integer exponen... |
| expclzlem 14108 | Lemma for ~ expclz . (Con... |
| expclz 14109 | Closure law for integer ex... |
| m1expcl2 14110 | Closure of integer exponen... |
| m1expcl 14111 | Closure of exponentiation ... |
| zexpcld 14112 | Closure of exponentiation ... |
| nn0expcli 14113 | Closure of exponentiation ... |
| nn0sqcl 14114 | The square of a nonnegativ... |
| expm1t 14115 | Exponentiation in terms of... |
| 1exp 14116 | Value of 1 raised to an in... |
| expeq0 14117 | A positive integer power i... |
| expne0 14118 | A positive integer power i... |
| expne0i 14119 | An integer power is nonzer... |
| expgt0 14120 | A positive real raised to ... |
| expnegz 14121 | Value of a nonzero complex... |
| 0exp 14122 | Value of zero raised to a ... |
| expge0 14123 | A nonnegative real raised ... |
| expge1 14124 | A real greater than or equ... |
| expgt1 14125 | A real greater than 1 rais... |
| mulexp 14126 | Nonnegative integer expone... |
| mulexpz 14127 | Integer exponentiation of ... |
| exprec 14128 | Integer exponentiation of ... |
| expadd 14129 | Sum of exponents law for n... |
| expaddzlem 14130 | Lemma for ~ expaddz . (Co... |
| expaddz 14131 | Sum of exponents law for i... |
| expmul 14132 | Product of exponents law f... |
| expmulz 14133 | Product of exponents law f... |
| m1expeven 14134 | Exponentiation of negative... |
| expsub 14135 | Exponent subtraction law f... |
| expp1z 14136 | Value of a nonzero complex... |
| expm1 14137 | Value of a nonzero complex... |
| expdiv 14138 | Nonnegative integer expone... |
| sqval 14139 | Value of the square of a c... |
| sqneg 14140 | The square of the negative... |
| sqsubswap 14141 | Swap the order of subtract... |
| sqcl 14142 | Closure of square. (Contr... |
| sqmul 14143 | Distribution of squaring o... |
| sqeq0 14144 | A complex number is zero i... |
| sqdiv 14145 | Distribution of squaring o... |
| sqdivid 14146 | The square of a nonzero co... |
| sqne0 14147 | A complex number is nonzer... |
| resqcl 14148 | Closure of squaring in rea... |
| resqcld 14149 | Closure of squaring in rea... |
| sqgt0 14150 | The square of a nonzero re... |
| sqn0rp 14151 | The square of a nonzero re... |
| nnsqcl 14152 | The positive naturals are ... |
| zsqcl 14153 | Integers are closed under ... |
| qsqcl 14154 | The square of a rational i... |
| sq11 14155 | The square function is one... |
| nn0sq11 14156 | The square function is one... |
| lt2sq 14157 | The square function is inc... |
| le2sq 14158 | The square function is non... |
| le2sq2 14159 | The square function is non... |
| sqge0 14160 | The square of a real is no... |
| sqge0d 14161 | The square of a real is no... |
| zsqcl2 14162 | The square of an integer i... |
| 0expd 14163 | Value of zero raised to a ... |
| exp0d 14164 | Value of a complex number ... |
| exp1d 14165 | Value of a complex number ... |
| expeq0d 14166 | If a positive integer powe... |
| sqvald 14167 | Value of square. Inferenc... |
| sqcld 14168 | Closure of square. (Contr... |
| sqeq0d 14169 | A number is zero iff its s... |
| expcld 14170 | Closure law for nonnegativ... |
| expp1d 14171 | Value of a complex number ... |
| expaddd 14172 | Sum of exponents law for n... |
| expmuld 14173 | Product of exponents law f... |
| sqrecd 14174 | Square of reciprocal is re... |
| expclzd 14175 | Closure law for integer ex... |
| expne0d 14176 | A nonnegative integer powe... |
| expnegd 14177 | Value of a nonzero complex... |
| exprecd 14178 | An integer power of a reci... |
| expp1zd 14179 | Value of a nonzero complex... |
| expm1d 14180 | Value of a nonzero complex... |
| expsubd 14181 | Exponent subtraction law f... |
| sqmuld 14182 | Distribution of squaring o... |
| sqdivd 14183 | Distribution of squaring o... |
| expdivd 14184 | Nonnegative integer expone... |
| mulexpd 14185 | Nonnegative integer expone... |
| znsqcld 14186 | The square of a nonzero in... |
| reexpcld 14187 | Closure of exponentiation ... |
| expge0d 14188 | A nonnegative real raised ... |
| expge1d 14189 | A real greater than or equ... |
| ltexp2a 14190 | Exponent ordering relation... |
| expmordi 14191 | Base ordering relationship... |
| rpexpmord 14192 | Base ordering relationship... |
| expcan 14193 | Cancellation law for integ... |
| ltexp2 14194 | Strict ordering law for ex... |
| leexp2 14195 | Ordering law for exponenti... |
| leexp2a 14196 | Weak ordering relationship... |
| ltexp2r 14197 | The integer powers of a fi... |
| leexp2r 14198 | Weak ordering relationship... |
| leexp1a 14199 | Weak base ordering relatio... |
| exple1 14200 | A real between 0 and 1 inc... |
| expubnd 14201 | An upper bound on ` A ^ N ... |
| sumsqeq0 14202 | The sum of two squres of r... |
| sqvali 14203 | Value of square. Inferenc... |
| sqcli 14204 | Closure of square. (Contr... |
| sqeq0i 14205 | A complex number is zero i... |
| sqrecii 14206 | The square of a reciprocal... |
| sqmuli 14207 | Distribution of squaring o... |
| sqdivi 14208 | Distribution of squaring o... |
| resqcli 14209 | Closure of square in reals... |
| sqgt0i 14210 | The square of a nonzero re... |
| sqge0i 14211 | The square of a real is no... |
| lt2sqi 14212 | The square function on non... |
| le2sqi 14213 | The square function on non... |
| sq11i 14214 | The square function is one... |
| sq0 14215 | The square of 0 is 0. (Co... |
| sq0i 14216 | If a number is zero, then ... |
| sq0id 14217 | If a number is zero, then ... |
| sq1 14218 | The square of 1 is 1. (Co... |
| neg1sqe1 14219 | The square of ` -u 1 ` is ... |
| sq2 14220 | The square of 2 is 4. (Co... |
| sq3 14221 | The square of 3 is 9. (Co... |
| sq4e2t8 14222 | The square of 4 is 2 times... |
| cu2 14223 | The cube of 2 is 8. (Cont... |
| irec 14224 | The reciprocal of ` _i ` .... |
| i2 14225 | ` _i ` squared. (Contribu... |
| i3 14226 | ` _i ` cubed. (Contribute... |
| i4 14227 | ` _i ` to the fourth power... |
| nnlesq 14228 | A positive integer is less... |
| zzlesq 14229 | An integer is less than or... |
| iexpcyc 14230 | Taking ` _i ` to the ` K `... |
| expnass 14231 | A counterexample showing t... |
| sqlecan 14232 | Cancel one factor of a squ... |
| subsq 14233 | Factor the difference of t... |
| subsq2 14234 | Express the difference of ... |
| binom2i 14235 | The square of a binomial. ... |
| subsqi 14236 | Factor the difference of t... |
| sqeqori 14237 | The squares of two complex... |
| subsq0i 14238 | The two solutions to the d... |
| sqeqor 14239 | The squares of two complex... |
| binom2 14240 | The square of a binomial. ... |
| binom2d 14241 | Deduction form of ~ binom2... |
| binom21 14242 | Special case of ~ binom2 w... |
| binom2sub 14243 | Expand the square of a sub... |
| binom2sub1 14244 | Special case of ~ binom2su... |
| binom2subi 14245 | Expand the square of a sub... |
| mulbinom2 14246 | The square of a binomial w... |
| binom3 14247 | The cube of a binomial. (... |
| sq01 14248 | If a complex number equals... |
| zesq 14249 | An integer is even iff its... |
| nnesq 14250 | A positive integer is even... |
| crreczi 14251 | Reciprocal of a complex nu... |
| bernneq 14252 | Bernoulli's inequality, du... |
| bernneq2 14253 | Variation of Bernoulli's i... |
| bernneq3 14254 | A corollary of ~ bernneq .... |
| expnbnd 14255 | Exponentiation with a base... |
| expnlbnd 14256 | The reciprocal of exponent... |
| expnlbnd2 14257 | The reciprocal of exponent... |
| expmulnbnd 14258 | Exponentiation with a base... |
| digit2 14259 | Two ways to express the ` ... |
| digit1 14260 | Two ways to express the ` ... |
| modexp 14261 | Exponentiation property of... |
| discr1 14262 | A nonnegative quadratic fo... |
| discr 14263 | If a quadratic polynomial ... |
| expnngt1 14264 | If an integer power with a... |
| expnngt1b 14265 | An integer power with an i... |
| sqoddm1div8 14266 | A squared odd number minus... |
| nnsqcld 14267 | The naturals are closed un... |
| nnexpcld 14268 | Closure of exponentiation ... |
| nn0expcld 14269 | Closure of exponentiation ... |
| rpexpcld 14270 | Closure law for exponentia... |
| ltexp2rd 14271 | The power of a positive nu... |
| reexpclzd 14272 | Closure of exponentiation ... |
| sqgt0d 14273 | The square of a nonzero re... |
| ltexp2d 14274 | Ordering relationship for ... |
| leexp2d 14275 | Ordering law for exponenti... |
| expcand 14276 | Ordering relationship for ... |
| leexp2ad 14277 | Ordering relationship for ... |
| leexp2rd 14278 | Ordering relationship for ... |
| lt2sqd 14279 | The square function on non... |
| le2sqd 14280 | The square function on non... |
| sq11d 14281 | The square function is one... |
| ltexp1d 14282 | Elevating to a positive po... |
| ltexp1dd 14283 | Raising both sides of 'les... |
| exp11nnd 14284 | The function elevating non... |
| mulsubdivbinom2 14285 | The square of a binomial w... |
| muldivbinom2 14286 | The square of a binomial w... |
| sq10 14287 | The square of 10 is 100. ... |
| sq10e99m1 14288 | The square of 10 is 99 plu... |
| 3dec 14289 | A "decimal constructor" wh... |
| nn0le2msqi 14290 | The square function on non... |
| nn0opthlem1 14291 | A rather pretty lemma for ... |
| nn0opthlem2 14292 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14293 | An ordered pair theorem fo... |
| nn0opth2i 14294 | An ordered pair theorem fo... |
| nn0opth2 14295 | An ordered pair theorem fo... |
| facnn 14298 | Value of the factorial fun... |
| fac0 14299 | The factorial of 0. (Cont... |
| fac1 14300 | The factorial of 1. (Cont... |
| facp1 14301 | The factorial of a success... |
| fac2 14302 | The factorial of 2. (Cont... |
| fac3 14303 | The factorial of 3. (Cont... |
| fac4 14304 | The factorial of 4. (Cont... |
| facnn2 14305 | Value of the factorial fun... |
| faccl 14306 | Closure of the factorial f... |
| faccld 14307 | Closure of the factorial f... |
| facmapnn 14308 | The factorial function res... |
| facne0 14309 | The factorial function is ... |
| facdiv 14310 | A positive integer divides... |
| facndiv 14311 | No positive integer (great... |
| facwordi 14312 | Ordering property of facto... |
| faclbnd 14313 | A lower bound for the fact... |
| faclbnd2 14314 | A lower bound for the fact... |
| faclbnd3 14315 | A lower bound for the fact... |
| faclbnd4lem1 14316 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14317 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14318 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14319 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14320 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14321 | The factorial function gro... |
| faclbnd6 14322 | Geometric lower bound for ... |
| facubnd 14323 | An upper bound for the fac... |
| facavg 14324 | The product of two factori... |
| bcval 14327 | Value of the binomial coef... |
| bcval2 14328 | Value of the binomial coef... |
| bcval3 14329 | Value of the binomial coef... |
| bcval4 14330 | Value of the binomial coef... |
| bcrpcl 14331 | Closure of the binomial co... |
| bccmpl 14332 | "Complementing" its second... |
| bcn0 14333 | ` N ` choose 0 is 1. Rema... |
| bc0k 14334 | The binomial coefficient "... |
| bcnn 14335 | ` N ` choose ` N ` is 1. ... |
| bcn1 14336 | Binomial coefficient: ` N ... |
| bcnp1n 14337 | Binomial coefficient: ` N ... |
| bcm1k 14338 | The proportion of one bino... |
| bcp1n 14339 | The proportion of one bino... |
| bcp1nk 14340 | The proportion of one bino... |
| bcval5 14341 | Write out the top and bott... |
| bcn2 14342 | Binomial coefficient: ` N ... |
| bcp1m1 14343 | Compute the binomial coeff... |
| bcpasc 14344 | Pascal's rule for the bino... |
| bccl 14345 | A binomial coefficient, in... |
| bccl2 14346 | A binomial coefficient, in... |
| bcn2m1 14347 | Compute the binomial coeff... |
| bcn2p1 14348 | Compute the binomial coeff... |
| permnn 14349 | The number of permutations... |
| bcnm1 14350 | The binomial coefficient o... |
| 4bc3eq4 14351 | The value of four choose t... |
| 4bc2eq6 14352 | The value of four choose t... |
| hashkf 14355 | The finite part of the siz... |
| hashgval 14356 | The value of the ` # ` fun... |
| hashginv 14357 | The converse of ` G ` maps... |
| hashinf 14358 | The value of the ` # ` fun... |
| hashbnd 14359 | If ` A ` has size bounded ... |
| hashfxnn0 14360 | The size function is a fun... |
| hashf 14361 | The size function maps all... |
| hashxnn0 14362 | The value of the hash func... |
| hashresfn 14363 | Restriction of the domain ... |
| dmhashres 14364 | Restriction of the domain ... |
| hashnn0pnf 14365 | The value of the hash func... |
| hashnnn0genn0 14366 | If the size of a set is no... |
| hashnemnf 14367 | The size of a set is never... |
| hashv01gt1 14368 | The size of a set is eithe... |
| hashfz1 14369 | The set ` ( 1 ... N ) ` ha... |
| hashen 14370 | Two finite sets have the s... |
| hasheni 14371 | Equinumerous sets have the... |
| hasheqf1o 14372 | The size of two finite set... |
| fiinfnf1o 14373 | There is no bijection betw... |
| hasheqf1oi 14374 | The size of two sets is eq... |
| hashf1rn 14375 | The size of a finite set w... |
| hasheqf1od 14376 | The size of two sets is eq... |
| fz1eqb 14377 | Two possibly-empty 1-based... |
| hashcard 14378 | The size function of the c... |
| hashcl 14379 | Closure of the ` # ` funct... |
| hashxrcl 14380 | Extended real closure of t... |
| hashclb 14381 | Reverse closure of the ` #... |
| nfile 14382 | The size of any infinite s... |
| hashvnfin 14383 | A set of finite size is a ... |
| hashnfinnn0 14384 | The size of an infinite se... |
| isfinite4 14385 | A finite set is equinumero... |
| hasheq0 14386 | Two ways of saying a set i... |
| hashneq0 14387 | Two ways of saying a set i... |
| hashgt0n0 14388 | If the size of a set is gr... |
| hashnncl 14389 | Positive natural closure o... |
| hash0 14390 | The empty set has size zer... |
| hashelne0d 14391 | A set with an element has ... |
| hashsng 14392 | The size of a singleton. ... |
| hashen1 14393 | A set has size 1 if and on... |
| hash1elsn 14394 | A set of size 1 with a kno... |
| hashrabrsn 14395 | The size of a restricted c... |
| hashrabsn01 14396 | The size of a restricted c... |
| hashrabsn1 14397 | If the size of a restricte... |
| hashfn 14398 | A function is equinumerous... |
| fseq1hash 14399 | The value of the size func... |
| hashgadd 14400 | ` G ` maps ordinal additio... |
| hashgval2 14401 | A short expression for the... |
| hashdom 14402 | Dominance relation for the... |
| hashdomi 14403 | Non-strict order relation ... |
| hashsdom 14404 | Strict dominance relation ... |
| hashun 14405 | The size of the union of d... |
| hashun2 14406 | The size of the union of f... |
| hashun3 14407 | The size of the union of f... |
| hashinfxadd 14408 | The extended real addition... |
| hashunx 14409 | The size of the union of d... |
| hashge0 14410 | The cardinality of a set i... |
| hashgt0 14411 | The cardinality of a nonem... |
| hashge1 14412 | The cardinality of a nonem... |
| 1elfz0hash 14413 | 1 is an element of the fin... |
| hashnn0n0nn 14414 | If a nonnegative integer i... |
| hashunsng 14415 | The size of the union of a... |
| hashunsngx 14416 | The size of the union of a... |
| hashunsnggt 14417 | The size of a set is great... |
| hashprg 14418 | The size of an unordered p... |
| elprchashprn2 14419 | If one element of an unord... |
| hashprb 14420 | The size of an unordered p... |
| hashprdifel 14421 | The elements of an unorder... |
| prhash2ex 14422 | There is (at least) one se... |
| hashle00 14423 | If the size of a set is le... |
| hashgt0elex 14424 | If the size of a set is gr... |
| hashgt0elexb 14425 | The size of a set is great... |
| hashp1i 14426 | Size of a finite ordinal. ... |
| hash1 14427 | Size of a finite ordinal. ... |
| hash2 14428 | Size of a finite ordinal. ... |
| hash3 14429 | Size of a finite ordinal. ... |
| hash4 14430 | Size of a finite ordinal. ... |
| pr0hash2ex 14431 | There is (at least) one se... |
| hashss 14432 | The size of a subset is le... |
| prsshashgt1 14433 | The size of a superset of ... |
| hashin 14434 | The size of the intersecti... |
| hashssdif 14435 | The size of the difference... |
| hashdif 14436 | The size of the difference... |
| hashdifsn 14437 | The size of the difference... |
| hashdifpr 14438 | The size of the difference... |
| hashsn01 14439 | The size of a singleton is... |
| hashsnle1 14440 | The size of a singleton is... |
| hashsnlei 14441 | Get an upper bound on a co... |
| hash1snb 14442 | The size of a set is 1 if ... |
| euhash1 14443 | The size of a set is 1 in ... |
| hash1n0 14444 | If the size of a set is 1 ... |
| hashgt12el 14445 | In a set with more than on... |
| hashgt12el2 14446 | In a set with more than on... |
| hashgt23el 14447 | A set with more than two e... |
| hashunlei 14448 | Get an upper bound on a co... |
| hashsslei 14449 | Get an upper bound on a co... |
| hashfz 14450 | Value of the numeric cardi... |
| fzsdom2 14451 | Condition for finite range... |
| hashfzo 14452 | Cardinality of a half-open... |
| hashfzo0 14453 | Cardinality of a half-open... |
| hashfzp1 14454 | Value of the numeric cardi... |
| hashfz0 14455 | Value of the numeric cardi... |
| hashxplem 14456 | Lemma for ~ hashxp . (Con... |
| hashxp 14457 | The size of the Cartesian ... |
| hashmap 14458 | The size of the set expone... |
| hashpw 14459 | The size of the power set ... |
| hashfun 14460 | A finite set is a function... |
| hashres 14461 | The number of elements of ... |
| hashreshashfun 14462 | The number of elements of ... |
| hashimarn 14463 | The size of the image of a... |
| hashimarni 14464 | If the size of the image o... |
| hashfundm 14465 | The size of a set function... |
| hashf1dmrn 14466 | The size of the domain of ... |
| hashf1dmcdm 14467 | The size of the domain of ... |
| resunimafz0 14468 | TODO-AV: Revise using ` F... |
| fnfz0hash 14469 | The size of a function on ... |
| ffz0hash 14470 | The size of a function on ... |
| fnfz0hashnn0 14471 | The size of a function on ... |
| ffzo0hash 14472 | The size of a function on ... |
| fnfzo0hash 14473 | The size of a function on ... |
| fnfzo0hashnn0 14474 | The value of the size func... |
| hashbclem 14475 | Lemma for ~ hashbc : induc... |
| hashbc 14476 | The binomial coefficient c... |
| hashfacen 14477 | The number of bijections b... |
| hashf1lem1 14478 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14479 | Lemma for ~ hashf1 . (Con... |
| hashf1 14480 | The permutation number ` |... |
| hashfac 14481 | A factorial counts the num... |
| leiso 14482 | Two ways to write a strict... |
| leisorel 14483 | Version of ~ isorel for st... |
| fz1isolem 14484 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14485 | Any finite ordered set has... |
| ishashinf 14486 | Any set that is not finite... |
| seqcoll 14487 | The function ` F ` contain... |
| seqcoll2 14488 | The function ` F ` contain... |
| phphashd 14489 | Corollary of the Pigeonhol... |
| phphashrd 14490 | Corollary of the Pigeonhol... |
| hashprlei 14491 | An unordered pair has at m... |
| hash2pr 14492 | A set of size two is an un... |
| hash2prde 14493 | A set of size two is an un... |
| hash2exprb 14494 | A set of size two is an un... |
| hash2prb 14495 | A set of size two is a pro... |
| prprrab 14496 | The set of proper pairs of... |
| nehash2 14497 | The cardinality of a set w... |
| hash2prd 14498 | A set of size two is an un... |
| hash2pwpr 14499 | If the size of a subset of... |
| hashle2pr 14500 | A nonempty set of size les... |
| hashle2prv 14501 | A nonempty subset of a pow... |
| pr2pwpr 14502 | The set of subsets of a pa... |
| hashge2el2dif 14503 | A set with size at least 2... |
| hashge2el2difr 14504 | A set with at least 2 diff... |
| hashge2el2difb 14505 | A set has size at least 2 ... |
| hashdmpropge2 14506 | The size of the domain of ... |
| hashtplei 14507 | An unordered triple has at... |
| hashtpg 14508 | The size of an unordered t... |
| hashge3el3dif 14509 | A set with size at least 3... |
| elss2prb 14510 | An element of the set of s... |
| hash2sspr 14511 | A subset of size two is an... |
| exprelprel 14512 | If there is an element of ... |
| hash3tr 14513 | A set of size three is an ... |
| hash1to3 14514 | If the size of a set is be... |
| hash3tpde 14515 | A set of size three is an ... |
| hash3tpexb 14516 | A set of size three is an ... |
| hash3tpb 14517 | A set of size three is a p... |
| tpf1ofv0 14518 | The value of a one-to-one ... |
| tpf1ofv1 14519 | The value of a one-to-one ... |
| tpf1ofv2 14520 | The value of a one-to-one ... |
| tpf 14521 | A function into a (proper)... |
| tpfo 14522 | A function onto a (proper)... |
| tpf1o 14523 | A bijection onto a (proper... |
| fundmge2nop0 14524 | A function with a domain c... |
| fundmge2nop 14525 | A function with a domain c... |
| fun2dmnop0 14526 | A function with a domain c... |
| fun2dmnop 14527 | A function with a domain c... |
| hashdifsnp1 14528 | If the size of a set is a ... |
| fi1uzind 14529 | Properties of an ordered p... |
| brfi1uzind 14530 | Properties of a binary rel... |
| brfi1ind 14531 | Properties of a binary rel... |
| brfi1indALT 14532 | Alternate proof of ~ brfi1... |
| opfi1uzind 14533 | Properties of an ordered p... |
| opfi1ind 14534 | Properties of an ordered p... |
| iswrd 14537 | Property of being a word o... |
| wrdval 14538 | Value of the set of words ... |
| iswrdi 14539 | A zero-based sequence is a... |
| wrdf 14540 | A word is a zero-based seq... |
| iswrdb 14541 | A word over an alphabet is... |
| wrddm 14542 | The indices of a word (i.e... |
| sswrd 14543 | The set of words respects ... |
| snopiswrd 14544 | A singleton of an ordered ... |
| wrdexg 14545 | The set of words over a se... |
| wrdexb 14546 | The set of words over a se... |
| wrdexi 14547 | The set of words over a se... |
| wrdsymbcl 14548 | A symbol within a word ove... |
| wrdfn 14549 | A word is a function with ... |
| wrdv 14550 | A word over an alphabet is... |
| wrdlndm 14551 | The length of a word is no... |
| iswrdsymb 14552 | An arbitrary word is a wor... |
| wrdfin 14553 | A word is a finite set. (... |
| lencl 14554 | The length of a word is a ... |
| lennncl 14555 | The length of a nonempty w... |
| wrdffz 14556 | A word is a function from ... |
| wrdeq 14557 | Equality theorem for the s... |
| wrdeqi 14558 | Equality theorem for the s... |
| iswrddm0 14559 | A function with empty doma... |
| wrd0 14560 | The empty set is a word (t... |
| 0wrd0 14561 | The empty word is the only... |
| ffz0iswrd 14562 | A sequence with zero-based... |
| wrdsymb 14563 | A word is a word over the ... |
| nfwrd 14564 | Hypothesis builder for ` W... |
| csbwrdg 14565 | Class substitution for the... |
| wrdnval 14566 | Words of a fixed length ar... |
| wrdmap 14567 | Words as a mapping. (Cont... |
| hashwrdn 14568 | If there is only a finite ... |
| wrdnfi 14569 | If there is only a finite ... |
| wrdsymb0 14570 | A symbol at a position "ou... |
| wrdlenge1n0 14571 | A word with length at leas... |
| len0nnbi 14572 | The length of a word is a ... |
| wrdlenge2n0 14573 | A word with length at leas... |
| wrdsymb1 14574 | The first symbol of a none... |
| wrdlen1 14575 | A word of length 1 starts ... |
| fstwrdne 14576 | The first symbol of a none... |
| fstwrdne0 14577 | The first symbol of a none... |
| eqwrd 14578 | Two words are equal iff th... |
| elovmpowrd 14579 | Implications for the value... |
| elovmptnn0wrd 14580 | Implications for the value... |
| wrdred1 14581 | A word truncated by a symb... |
| wrdred1hash 14582 | The length of a word trunc... |
| lsw 14585 | Extract the last symbol of... |
| lsw0 14586 | The last symbol of an empt... |
| lsw0g 14587 | The last symbol of an empt... |
| lsw1 14588 | The last symbol of a word ... |
| lswcl 14589 | Closure of the last symbol... |
| lswlgt0cl 14590 | The last symbol of a nonem... |
| ccatfn 14593 | The concatenation operator... |
| ccatfval 14594 | Value of the concatenation... |
| ccatcl 14595 | The concatenation of two w... |
| ccatlen 14596 | The length of a concatenat... |
| ccat0 14597 | The concatenation of two w... |
| ccatval1 14598 | Value of a symbol in the l... |
| ccatval2 14599 | Value of a symbol in the r... |
| ccatval3 14600 | Value of a symbol in the r... |
| elfzelfzccat 14601 | An element of a finite set... |
| ccatvalfn 14602 | The concatenation of two w... |
| ccatsymb 14603 | The symbol at a given posi... |
| ccatfv0 14604 | The first symbol of a conc... |
| ccatval1lsw 14605 | The last symbol of the lef... |
| ccatval21sw 14606 | The first symbol of the ri... |
| ccatlid 14607 | Concatenation of a word by... |
| ccatrid 14608 | Concatenation of a word by... |
| ccatass 14609 | Associative law for concat... |
| ccatrn 14610 | The range of a concatenate... |
| ccatidid 14611 | Concatenation of the empty... |
| lswccatn0lsw 14612 | The last symbol of a word ... |
| lswccat0lsw 14613 | The last symbol of a word ... |
| ccatalpha 14614 | A concatenation of two arb... |
| ccatrcl1 14615 | Reverse closure of a conca... |
| ids1 14618 | Identity function protecti... |
| s1val 14619 | Value of a singleton word.... |
| s1rn 14620 | The range of a singleton w... |
| s1eq 14621 | Equality theorem for a sin... |
| s1eqd 14622 | Equality theorem for a sin... |
| s1cl 14623 | A singleton word is a word... |
| s1cld 14624 | A singleton word is a word... |
| s1prc 14625 | Value of a singleton word ... |
| s1cli 14626 | A singleton word is a word... |
| s1len 14627 | Length of a singleton word... |
| s1nz 14628 | A singleton word is not th... |
| s1dm 14629 | The domain of a singleton ... |
| s1dmALT 14630 | Alternate version of ~ s1d... |
| s1fv 14631 | Sole symbol of a singleton... |
| lsws1 14632 | The last symbol of a singl... |
| eqs1 14633 | A word of length 1 is a si... |
| wrdl1exs1 14634 | A word of length 1 is a si... |
| wrdl1s1 14635 | A word of length 1 is a si... |
| s111 14636 | The singleton word functio... |
| ccatws1cl 14637 | The concatenation of a wor... |
| ccatws1clv 14638 | The concatenation of a wor... |
| ccat2s1cl 14639 | The concatenation of two s... |
| ccats1alpha 14640 | A concatenation of a word ... |
| ccatws1len 14641 | The length of the concaten... |
| ccatws1lenp1b 14642 | The length of a word is ` ... |
| wrdlenccats1lenm1 14643 | The length of a word is th... |
| ccat2s1len 14644 | The length of the concaten... |
| ccatw2s1cl 14645 | The concatenation of a wor... |
| ccatw2s1len 14646 | The length of the concaten... |
| ccats1val1 14647 | Value of a symbol in the l... |
| ccats1val2 14648 | Value of the symbol concat... |
| ccat1st1st 14649 | The first symbol of a word... |
| ccat2s1p1 14650 | Extract the first of two c... |
| ccat2s1p2 14651 | Extract the second of two ... |
| ccatw2s1ass 14652 | Associative law for a conc... |
| ccatws1n0 14653 | The concatenation of a wor... |
| ccatws1ls 14654 | The last symbol of the con... |
| lswccats1 14655 | The last symbol of a word ... |
| lswccats1fst 14656 | The last symbol of a nonem... |
| ccatw2s1p1 14657 | Extract the symbol of the ... |
| ccatw2s1p2 14658 | Extract the second of two ... |
| ccat2s1fvw 14659 | Extract a symbol of a word... |
| ccat2s1fst 14660 | The first symbol of the co... |
| swrdnznd 14663 | The value of a subword ope... |
| swrdval 14664 | Value of a subword. (Cont... |
| swrd00 14665 | A zero length substring. ... |
| swrdcl 14666 | Closure of the subword ext... |
| swrdval2 14667 | Value of the subword extra... |
| swrdlen 14668 | Length of an extracted sub... |
| swrdfv 14669 | A symbol in an extracted s... |
| swrdfv0 14670 | The first symbol in an ext... |
| swrdf 14671 | A subword of a word is a f... |
| swrdvalfn 14672 | Value of the subword extra... |
| swrdrn 14673 | The range of a subword of ... |
| swrdlend 14674 | The value of the subword e... |
| swrdnd 14675 | The value of the subword e... |
| swrdnd2 14676 | Value of the subword extra... |
| swrdnnn0nd 14677 | The value of a subword ope... |
| swrdnd0 14678 | The value of a subword ope... |
| swrd0 14679 | A subword of an empty set ... |
| swrdrlen 14680 | Length of a right-anchored... |
| swrdlen2 14681 | Length of an extracted sub... |
| swrdfv2 14682 | A symbol in an extracted s... |
| swrdwrdsymb 14683 | A subword is a word over t... |
| swrdsb0eq 14684 | Two subwords with the same... |
| swrdsbslen 14685 | Two subwords with the same... |
| swrdspsleq 14686 | Two words have a common su... |
| swrds1 14687 | Extract a single symbol fr... |
| swrdlsw 14688 | Extract the last single sy... |
| ccatswrd 14689 | Joining two adjacent subwo... |
| swrdccat2 14690 | Recover the right half of ... |
| pfxnndmnd 14693 | The value of a prefix oper... |
| pfxval 14694 | Value of a prefix operatio... |
| pfx00 14695 | The zero length prefix is ... |
| pfx0 14696 | A prefix of an empty set i... |
| pfxval0 14697 | Value of a prefix operatio... |
| pfxcl 14698 | Closure of the prefix extr... |
| pfxmpt 14699 | Value of the prefix extrac... |
| pfxres 14700 | Value of the subword extra... |
| pfxf 14701 | A prefix of a word is a fu... |
| pfxfn 14702 | Value of the prefix extrac... |
| pfxfv 14703 | A symbol in a prefix of a ... |
| pfxlen 14704 | Length of a prefix. (Cont... |
| pfxid 14705 | A word is a prefix of itse... |
| pfxrn 14706 | The range of a prefix of a... |
| pfxn0 14707 | A prefix consisting of at ... |
| pfxnd 14708 | The value of a prefix oper... |
| pfxnd0 14709 | The value of a prefix oper... |
| pfxwrdsymb 14710 | A prefix of a word is a wo... |
| addlenrevpfx 14711 | The sum of the lengths of ... |
| addlenpfx 14712 | The sum of the lengths of ... |
| pfxfv0 14713 | The first symbol of a pref... |
| pfxtrcfv 14714 | A symbol in a word truncat... |
| pfxtrcfv0 14715 | The first symbol in a word... |
| pfxfvlsw 14716 | The last symbol in a nonem... |
| pfxeq 14717 | The prefixes of two words ... |
| pfxtrcfvl 14718 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14719 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14720 | Two (nonempty) words are e... |
| disjwrdpfx 14721 | Sets of words are disjoint... |
| ccatpfx 14722 | Concatenating a prefix wit... |
| pfxccat1 14723 | Recover the left half of a... |
| pfx1 14724 | The prefix of length one o... |
| swrdswrdlem 14725 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14726 | A subword of a subword is ... |
| pfxswrd 14727 | A prefix of a subword is a... |
| swrdpfx 14728 | A subword of a prefix is a... |
| pfxpfx 14729 | A prefix of a prefix is a ... |
| pfxpfxid 14730 | A prefix of a prefix with ... |
| pfxcctswrd 14731 | The concatenation of the p... |
| lenpfxcctswrd 14732 | The length of the concaten... |
| lenrevpfxcctswrd 14733 | The length of the concaten... |
| pfxlswccat 14734 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14735 | The last symbol of a word ... |
| ccats1pfxeqrex 14736 | There exists a symbol such... |
| ccatopth 14737 | An ~ opth -like theorem fo... |
| ccatopth2 14738 | An ~ opth -like theorem fo... |
| ccatlcan 14739 | Concatenation of words is ... |
| ccatrcan 14740 | Concatenation of words is ... |
| wrdeqs1cat 14741 | Decompose a nonempty word ... |
| cats1un 14742 | Express a word with an ext... |
| wrdind 14743 | Perform induction over the... |
| wrd2ind 14744 | Perform induction over the... |
| swrdccatfn 14745 | The subword of a concatena... |
| swrdccatin1 14746 | The subword of a concatena... |
| pfxccatin12lem4 14747 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14748 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14749 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14750 | The subword of a concatena... |
| pfxccatin12lem2c 14751 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14752 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14753 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14754 | The subword of a concatena... |
| pfxccat3 14755 | The subword of a concatena... |
| swrdccat 14756 | The subword of a concatena... |
| pfxccatpfx1 14757 | A prefix of a concatenatio... |
| pfxccatpfx2 14758 | A prefix of a concatenatio... |
| pfxccat3a 14759 | A prefix of a concatenatio... |
| swrdccat3blem 14760 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14761 | A suffix of a concatenatio... |
| pfxccatid 14762 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14763 | A word is a prefix of a wo... |
| swrdccatin1d 14764 | The subword of a concatena... |
| swrdccatin2d 14765 | The subword of a concatena... |
| pfxccatin12d 14766 | The subword of a concatena... |
| reuccatpfxs1lem 14767 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14768 | There is a unique word hav... |
| reuccatpfxs1v 14769 | There is a unique word hav... |
| splval 14772 | Value of the substring rep... |
| splcl 14773 | Closure of the substring r... |
| splid 14774 | Splicing a subword for the... |
| spllen 14775 | The length of a splice. (... |
| splfv1 14776 | Symbols to the left of a s... |
| splfv2a 14777 | Symbols within the replace... |
| splval2 14778 | Value of a splice, assumin... |
| revval 14781 | Value of the word reversin... |
| revcl 14782 | The reverse of a word is a... |
| revlen 14783 | The reverse of a word has ... |
| revfv 14784 | Reverse of a word at a poi... |
| rev0 14785 | The empty word is its own ... |
| revs1 14786 | Singleton words are their ... |
| revccat 14787 | Antiautomorphic property o... |
| revrev 14788 | Reversal is an involution ... |
| reps 14791 | Construct a function mappi... |
| repsundef 14792 | A function mapping a half-... |
| repsconst 14793 | Construct a function mappi... |
| repsf 14794 | The constructed function m... |
| repswsymb 14795 | The symbols of a "repeated... |
| repsw 14796 | A function mapping a half-... |
| repswlen 14797 | The length of a "repeated ... |
| repsw0 14798 | The "repeated symbol word"... |
| repsdf2 14799 | Alternative definition of ... |
| repswsymball 14800 | All the symbols of a "repe... |
| repswsymballbi 14801 | A word is a "repeated symb... |
| repswfsts 14802 | The first symbol of a none... |
| repswlsw 14803 | The last symbol of a nonem... |
| repsw1 14804 | The "repeated symbol word"... |
| repswswrd 14805 | A subword of a "repeated s... |
| repswpfx 14806 | A prefix of a repeated sym... |
| repswccat 14807 | The concatenation of two "... |
| repswrevw 14808 | The reverse of a "repeated... |
| cshfn 14811 | Perform a cyclical shift f... |
| cshword 14812 | Perform a cyclical shift f... |
| cshnz 14813 | A cyclical shift is the em... |
| 0csh0 14814 | Cyclically shifting an emp... |
| cshw0 14815 | A word cyclically shifted ... |
| cshwmodn 14816 | Cyclically shifting a word... |
| cshwsublen 14817 | Cyclically shifting a word... |
| cshwn 14818 | A word cyclically shifted ... |
| cshwcl 14819 | A cyclically shifted word ... |
| cshwlen 14820 | The length of a cyclically... |
| cshwf 14821 | A cyclically shifted word ... |
| cshwfn 14822 | A cyclically shifted word ... |
| cshwrn 14823 | The range of a cyclically ... |
| cshwidxmod 14824 | The symbol at a given inde... |
| cshwidxmodr 14825 | The symbol at a given inde... |
| cshwidx0mod 14826 | The symbol at index 0 of a... |
| cshwidx0 14827 | The symbol at index 0 of a... |
| cshwidxm1 14828 | The symbol at index ((n-N)... |
| cshwidxm 14829 | The symbol at index (n-N) ... |
| cshwidxn 14830 | The symbol at index (n-1) ... |
| cshf1 14831 | Cyclically shifting a word... |
| cshinj 14832 | If a word is injectiv (reg... |
| repswcshw 14833 | A cyclically shifted "repe... |
| 2cshw 14834 | Cyclically shifting a word... |
| 2cshwid 14835 | Cyclically shifting a word... |
| lswcshw 14836 | The last symbol of a word ... |
| 2cshwcom 14837 | Cyclically shifting a word... |
| cshwleneq 14838 | If the results of cyclical... |
| 3cshw 14839 | Cyclically shifting a word... |
| cshweqdif2 14840 | If cyclically shifting two... |
| cshweqdifid 14841 | If cyclically shifting a w... |
| cshweqrep 14842 | If cyclically shifting a w... |
| cshw1 14843 | If cyclically shifting a w... |
| cshw1repsw 14844 | If cyclically shifting a w... |
| cshwsexa 14845 | The class of (different!) ... |
| cshwsexaOLD 14846 | Obsolete version of ~ cshw... |
| 2cshwcshw 14847 | If a word is a cyclically ... |
| scshwfzeqfzo 14848 | For a nonempty word the se... |
| cshwcshid 14849 | A cyclically shifted word ... |
| cshwcsh2id 14850 | A cyclically shifted word ... |
| cshimadifsn 14851 | The image of a cyclically ... |
| cshimadifsn0 14852 | The image of a cyclically ... |
| wrdco 14853 | Mapping a word by a functi... |
| lenco 14854 | Length of a mapped word is... |
| s1co 14855 | Mapping of a singleton wor... |
| revco 14856 | Mapping of words (i.e., a ... |
| ccatco 14857 | Mapping of words commutes ... |
| cshco 14858 | Mapping of words commutes ... |
| swrdco 14859 | Mapping of words commutes ... |
| pfxco 14860 | Mapping of words commutes ... |
| lswco 14861 | Mapping of (nonempty) word... |
| repsco 14862 | Mapping of words commutes ... |
| cats1cld 14877 | Closure of concatenation w... |
| cats1co 14878 | Closure of concatenation w... |
| cats1cli 14879 | Closure of concatenation w... |
| cats1fvn 14880 | The last symbol of a conca... |
| cats1fv 14881 | A symbol other than the la... |
| cats1len 14882 | The length of concatenatio... |
| cats1cat 14883 | Closure of concatenation w... |
| cats2cat 14884 | Closure of concatenation o... |
| s2eqd 14885 | Equality theorem for a dou... |
| s3eqd 14886 | Equality theorem for a len... |
| s4eqd 14887 | Equality theorem for a len... |
| s5eqd 14888 | Equality theorem for a len... |
| s6eqd 14889 | Equality theorem for a len... |
| s7eqd 14890 | Equality theorem for a len... |
| s8eqd 14891 | Equality theorem for a len... |
| s3eq2 14892 | Equality theorem for a len... |
| s2cld 14893 | A doubleton word is a word... |
| s3cld 14894 | A length 3 string is a wor... |
| s4cld 14895 | A length 4 string is a wor... |
| s5cld 14896 | A length 5 string is a wor... |
| s6cld 14897 | A length 6 string is a wor... |
| s7cld 14898 | A length 7 string is a wor... |
| s8cld 14899 | A length 7 string is a wor... |
| s2cl 14900 | A doubleton word is a word... |
| s3cl 14901 | A length 3 string is a wor... |
| s2cli 14902 | A doubleton word is a word... |
| s3cli 14903 | A length 3 string is a wor... |
| s4cli 14904 | A length 4 string is a wor... |
| s5cli 14905 | A length 5 string is a wor... |
| s6cli 14906 | A length 6 string is a wor... |
| s7cli 14907 | A length 7 string is a wor... |
| s8cli 14908 | A length 8 string is a wor... |
| s2fv0 14909 | Extract the first symbol f... |
| s2fv1 14910 | Extract the second symbol ... |
| s2len 14911 | The length of a doubleton ... |
| s2dm 14912 | The domain of a doubleton ... |
| s3fv0 14913 | Extract the first symbol f... |
| s3fv1 14914 | Extract the second symbol ... |
| s3fv2 14915 | Extract the third symbol f... |
| s3len 14916 | The length of a length 3 s... |
| s4fv0 14917 | Extract the first symbol f... |
| s4fv1 14918 | Extract the second symbol ... |
| s4fv2 14919 | Extract the third symbol f... |
| s4fv3 14920 | Extract the fourth symbol ... |
| s4len 14921 | The length of a length 4 s... |
| s5len 14922 | The length of a length 5 s... |
| s6len 14923 | The length of a length 6 s... |
| s7len 14924 | The length of a length 7 s... |
| s8len 14925 | The length of a length 8 s... |
| lsws2 14926 | The last symbol of a doubl... |
| lsws3 14927 | The last symbol of a 3 let... |
| lsws4 14928 | The last symbol of a 4 let... |
| s2prop 14929 | A length 2 word is an unor... |
| s2dmALT 14930 | Alternate version of ~ s2d... |
| s3tpop 14931 | A length 3 word is an unor... |
| s4prop 14932 | A length 4 word is a union... |
| s3fn 14933 | A length 3 word is a funct... |
| funcnvs1 14934 | The converse of a singleto... |
| funcnvs2 14935 | The converse of a length 2... |
| funcnvs3 14936 | The converse of a length 3... |
| funcnvs4 14937 | The converse of a length 4... |
| s2f1o 14938 | A length 2 word with mutua... |
| f1oun2prg 14939 | A union of unordered pairs... |
| s4f1o 14940 | A length 4 word with mutua... |
| s4dom 14941 | The domain of a length 4 w... |
| s2co 14942 | Mapping a doubleton word b... |
| s3co 14943 | Mapping a length 3 string ... |
| s0s1 14944 | Concatenation of fixed len... |
| s1s2 14945 | Concatenation of fixed len... |
| s1s3 14946 | Concatenation of fixed len... |
| s1s4 14947 | Concatenation of fixed len... |
| s1s5 14948 | Concatenation of fixed len... |
| s1s6 14949 | Concatenation of fixed len... |
| s1s7 14950 | Concatenation of fixed len... |
| s2s2 14951 | Concatenation of fixed len... |
| s4s2 14952 | Concatenation of fixed len... |
| s4s3 14953 | Concatenation of fixed len... |
| s4s4 14954 | Concatenation of fixed len... |
| s3s4 14955 | Concatenation of fixed len... |
| s2s5 14956 | Concatenation of fixed len... |
| s5s2 14957 | Concatenation of fixed len... |
| s2eq2s1eq 14958 | Two length 2 words are equ... |
| s2eq2seq 14959 | Two length 2 words are equ... |
| s3eqs2s1eq 14960 | Two length 3 words are equ... |
| s3eq3seq 14961 | Two length 3 words are equ... |
| swrds2 14962 | Extract two adjacent symbo... |
| swrds2m 14963 | Extract two adjacent symbo... |
| wrdlen2i 14964 | Implications of a word of ... |
| wrd2pr2op 14965 | A word of length two repre... |
| wrdlen2 14966 | A word of length two. (Co... |
| wrdlen2s2 14967 | A word of length two as do... |
| wrdl2exs2 14968 | A word of length two is a ... |
| pfx2 14969 | A prefix of length two. (... |
| wrd3tpop 14970 | A word of length three rep... |
| wrdlen3s3 14971 | A word of length three as ... |
| repsw2 14972 | The "repeated symbol word"... |
| repsw3 14973 | The "repeated symbol word"... |
| swrd2lsw 14974 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14975 | Two words of length at lea... |
| ccatw2s1ccatws2 14976 | The concatenation of a wor... |
| ccat2s1fvwALT 14977 | Alternate proof of ~ ccat2... |
| wwlktovf 14978 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14979 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14980 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14981 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14982 | There is a bijection betwe... |
| eqwrds3 14983 | A word is equal with a len... |
| wrdl3s3 14984 | A word of length 3 is a le... |
| s3sndisj 14985 | The singletons consisting ... |
| s3iunsndisj 14986 | The union of singletons co... |
| ofccat 14987 | Letterwise operations on w... |
| ofs1 14988 | Letterwise operations on a... |
| ofs2 14989 | Letterwise operations on a... |
| coss12d 14990 | Subset deduction for compo... |
| trrelssd 14991 | The composition of subclas... |
| xpcogend 14992 | The most interesting case ... |
| xpcoidgend 14993 | If two classes are not dis... |
| cotr2g 14994 | Two ways of saying that th... |
| cotr2 14995 | Two ways of saying a relat... |
| cotr3 14996 | Two ways of saying a relat... |
| coemptyd 14997 | Deduction about compositio... |
| xptrrel 14998 | The cross product is alway... |
| 0trrel 14999 | The empty class is a trans... |
| cleq1lem 15000 | Equality implies bijection... |
| cleq1 15001 | Equality of relations impl... |
| clsslem 15002 | The closure of a subclass ... |
| trcleq1 15007 | Equality of relations impl... |
| trclsslem 15008 | The transitive closure (as... |
| trcleq2lem 15009 | Equality implies bijection... |
| cvbtrcl 15010 | Change of bound variable i... |
| trcleq12lem 15011 | Equality implies bijection... |
| trclexlem 15012 | Existence of relation impl... |
| trclublem 15013 | If a relation exists then ... |
| trclubi 15014 | The Cartesian product of t... |
| trclubgi 15015 | The union with the Cartesi... |
| trclub 15016 | The Cartesian product of t... |
| trclubg 15017 | The union with the Cartesi... |
| trclfv 15018 | The transitive closure of ... |
| brintclab 15019 | Two ways to express a bina... |
| brtrclfv 15020 | Two ways of expressing the... |
| brcnvtrclfv 15021 | Two ways of expressing the... |
| brtrclfvcnv 15022 | Two ways of expressing the... |
| brcnvtrclfvcnv 15023 | Two ways of expressing the... |
| trclfvss 15024 | The transitive closure (as... |
| trclfvub 15025 | The transitive closure of ... |
| trclfvlb 15026 | The transitive closure of ... |
| trclfvcotr 15027 | The transitive closure of ... |
| trclfvlb2 15028 | The transitive closure of ... |
| trclfvlb3 15029 | The transitive closure of ... |
| cotrtrclfv 15030 | The transitive closure of ... |
| trclidm 15031 | The transitive closure of ... |
| trclun 15032 | Transitive closure of a un... |
| trclfvg 15033 | The value of the transitiv... |
| trclfvcotrg 15034 | The value of the transitiv... |
| reltrclfv 15035 | The transitive closure of ... |
| dmtrclfv 15036 | The domain of the transiti... |
| reldmrelexp 15039 | The domain of the repeated... |
| relexp0g 15040 | A relation composed zero t... |
| relexp0 15041 | A relation composed zero t... |
| relexp0d 15042 | A relation composed zero t... |
| relexpsucnnr 15043 | A reduction for relation e... |
| relexp1g 15044 | A relation composed once i... |
| dfid5 15045 | Identity relation is equal... |
| dfid6 15046 | Identity relation expresse... |
| relexp1d 15047 | A relation composed once i... |
| relexpsucnnl 15048 | A reduction for relation e... |
| relexpsucl 15049 | A reduction for relation e... |
| relexpsucr 15050 | A reduction for relation e... |
| relexpsucrd 15051 | A reduction for relation e... |
| relexpsucld 15052 | A reduction for relation e... |
| relexpcnv 15053 | Commutation of converse an... |
| relexpcnvd 15054 | Commutation of converse an... |
| relexp0rel 15055 | The exponentiation of a cl... |
| relexprelg 15056 | The exponentiation of a cl... |
| relexprel 15057 | The exponentiation of a re... |
| relexpreld 15058 | The exponentiation of a re... |
| relexpnndm 15059 | The domain of an exponenti... |
| relexpdmg 15060 | The domain of an exponenti... |
| relexpdm 15061 | The domain of an exponenti... |
| relexpdmd 15062 | The domain of an exponenti... |
| relexpnnrn 15063 | The range of an exponentia... |
| relexprng 15064 | The range of an exponentia... |
| relexprn 15065 | The range of an exponentia... |
| relexprnd 15066 | The range of an exponentia... |
| relexpfld 15067 | The field of an exponentia... |
| relexpfldd 15068 | The field of an exponentia... |
| relexpaddnn 15069 | Relation composition becom... |
| relexpuzrel 15070 | The exponentiation of a cl... |
| relexpaddg 15071 | Relation composition becom... |
| relexpaddd 15072 | Relation composition becom... |
| rtrclreclem1 15075 | The reflexive, transitive ... |
| dfrtrclrec2 15076 | If two elements are connec... |
| rtrclreclem2 15077 | The reflexive, transitive ... |
| rtrclreclem3 15078 | The reflexive, transitive ... |
| rtrclreclem4 15079 | The reflexive, transitive ... |
| dfrtrcl2 15080 | The two definitions ` t* `... |
| relexpindlem 15081 | Principle of transitive in... |
| relexpind 15082 | Principle of transitive in... |
| rtrclind 15083 | Principle of transitive in... |
| shftlem 15086 | Two ways to write a shifte... |
| shftuz 15087 | A shift of the upper integ... |
| shftfval 15088 | The value of the sequence ... |
| shftdm 15089 | Domain of a relation shift... |
| shftfib 15090 | Value of a fiber of the re... |
| shftfn 15091 | Functionality and domain o... |
| shftval 15092 | Value of a sequence shifte... |
| shftval2 15093 | Value of a sequence shifte... |
| shftval3 15094 | Value of a sequence shifte... |
| shftval4 15095 | Value of a sequence shifte... |
| shftval5 15096 | Value of a shifted sequenc... |
| shftf 15097 | Functionality of a shifted... |
| 2shfti 15098 | Composite shift operations... |
| shftidt2 15099 | Identity law for the shift... |
| shftidt 15100 | Identity law for the shift... |
| shftcan1 15101 | Cancellation law for the s... |
| shftcan2 15102 | Cancellation law for the s... |
| seqshft 15103 | Shifting the index set of ... |
| sgnval 15106 | Value of the signum functi... |
| sgn0 15107 | The signum of 0 is 0. (Co... |
| sgnp 15108 | The signum of a positive e... |
| sgnrrp 15109 | The signum of a positive r... |
| sgn1 15110 | The signum of 1 is 1. (Co... |
| sgnpnf 15111 | The signum of ` +oo ` is 1... |
| sgnn 15112 | The signum of a negative e... |
| sgnmnf 15113 | The signum of ` -oo ` is -... |
| cjval 15120 | The value of the conjugate... |
| cjth 15121 | The defining property of t... |
| cjf 15122 | Domain and codomain of the... |
| cjcl 15123 | The conjugate of a complex... |
| reval 15124 | The value of the real part... |
| imval 15125 | The value of the imaginary... |
| imre 15126 | The imaginary part of a co... |
| reim 15127 | The real part of a complex... |
| recl 15128 | The real part of a complex... |
| imcl 15129 | The imaginary part of a co... |
| ref 15130 | Domain and codomain of the... |
| imf 15131 | Domain and codomain of the... |
| crre 15132 | The real part of a complex... |
| crim 15133 | The real part of a complex... |
| replim 15134 | Reconstruct a complex numb... |
| remim 15135 | Value of the conjugate of ... |
| reim0 15136 | The imaginary part of a re... |
| reim0b 15137 | A number is real iff its i... |
| rereb 15138 | A number is real iff it eq... |
| mulre 15139 | A product with a nonzero r... |
| rere 15140 | A real number equals its r... |
| cjreb 15141 | A number is real iff it eq... |
| recj 15142 | Real part of a complex con... |
| reneg 15143 | Real part of negative. (C... |
| readd 15144 | Real part distributes over... |
| resub 15145 | Real part distributes over... |
| remullem 15146 | Lemma for ~ remul , ~ immu... |
| remul 15147 | Real part of a product. (... |
| remul2 15148 | Real part of a product. (... |
| rediv 15149 | Real part of a division. ... |
| imcj 15150 | Imaginary part of a comple... |
| imneg 15151 | The imaginary part of a ne... |
| imadd 15152 | Imaginary part distributes... |
| imsub 15153 | Imaginary part distributes... |
| immul 15154 | Imaginary part of a produc... |
| immul2 15155 | Imaginary part of a produc... |
| imdiv 15156 | Imaginary part of a divisi... |
| cjre 15157 | A real number equals its c... |
| cjcj 15158 | The conjugate of the conju... |
| cjadd 15159 | Complex conjugate distribu... |
| cjmul 15160 | Complex conjugate distribu... |
| ipcnval 15161 | Standard inner product on ... |
| cjmulrcl 15162 | A complex number times its... |
| cjmulval 15163 | A complex number times its... |
| cjmulge0 15164 | A complex number times its... |
| cjneg 15165 | Complex conjugate of negat... |
| addcj 15166 | A number plus its conjugat... |
| cjsub 15167 | Complex conjugate distribu... |
| cjexp 15168 | Complex conjugate of posit... |
| imval2 15169 | The imaginary part of a nu... |
| re0 15170 | The real part of zero. (C... |
| im0 15171 | The imaginary part of zero... |
| re1 15172 | The real part of one. (Co... |
| im1 15173 | The imaginary part of one.... |
| rei 15174 | The real part of ` _i ` . ... |
| imi 15175 | The imaginary part of ` _i... |
| cj0 15176 | The conjugate of zero. (C... |
| cji 15177 | The complex conjugate of t... |
| cjreim 15178 | The conjugate of a represe... |
| cjreim2 15179 | The conjugate of the repre... |
| cj11 15180 | Complex conjugate is a one... |
| cjne0 15181 | A number is nonzero iff it... |
| cjdiv 15182 | Complex conjugate distribu... |
| cnrecnv 15183 | The inverse to the canonic... |
| sqeqd 15184 | A deduction for showing tw... |
| recli 15185 | The real part of a complex... |
| imcli 15186 | The imaginary part of a co... |
| cjcli 15187 | Closure law for complex co... |
| replimi 15188 | Construct a complex number... |
| cjcji 15189 | The conjugate of the conju... |
| reim0bi 15190 | A number is real iff its i... |
| rerebi 15191 | A real number equals its r... |
| cjrebi 15192 | A number is real iff it eq... |
| recji 15193 | Real part of a complex con... |
| imcji 15194 | Imaginary part of a comple... |
| cjmulrcli 15195 | A complex number times its... |
| cjmulvali 15196 | A complex number times its... |
| cjmulge0i 15197 | A complex number times its... |
| renegi 15198 | Real part of negative. (C... |
| imnegi 15199 | Imaginary part of negative... |
| cjnegi 15200 | Complex conjugate of negat... |
| addcji 15201 | A number plus its conjugat... |
| readdi 15202 | Real part distributes over... |
| imaddi 15203 | Imaginary part distributes... |
| remuli 15204 | Real part of a product. (... |
| immuli 15205 | Imaginary part of a produc... |
| cjaddi 15206 | Complex conjugate distribu... |
| cjmuli 15207 | Complex conjugate distribu... |
| ipcni 15208 | Standard inner product on ... |
| cjdivi 15209 | Complex conjugate distribu... |
| crrei 15210 | The real part of a complex... |
| crimi 15211 | The imaginary part of a co... |
| recld 15212 | The real part of a complex... |
| imcld 15213 | The imaginary part of a co... |
| cjcld 15214 | Closure law for complex co... |
| replimd 15215 | Construct a complex number... |
| remimd 15216 | Value of the conjugate of ... |
| cjcjd 15217 | The conjugate of the conju... |
| reim0bd 15218 | A number is real iff its i... |
| rerebd 15219 | A real number equals its r... |
| cjrebd 15220 | A number is real iff it eq... |
| cjne0d 15221 | A number is nonzero iff it... |
| recjd 15222 | Real part of a complex con... |
| imcjd 15223 | Imaginary part of a comple... |
| cjmulrcld 15224 | A complex number times its... |
| cjmulvald 15225 | A complex number times its... |
| cjmulge0d 15226 | A complex number times its... |
| renegd 15227 | Real part of negative. (C... |
| imnegd 15228 | Imaginary part of negative... |
| cjnegd 15229 | Complex conjugate of negat... |
| addcjd 15230 | A number plus its conjugat... |
| cjexpd 15231 | Complex conjugate of posit... |
| readdd 15232 | Real part distributes over... |
| imaddd 15233 | Imaginary part distributes... |
| resubd 15234 | Real part distributes over... |
| imsubd 15235 | Imaginary part distributes... |
| remuld 15236 | Real part of a product. (... |
| immuld 15237 | Imaginary part of a produc... |
| cjaddd 15238 | Complex conjugate distribu... |
| cjmuld 15239 | Complex conjugate distribu... |
| ipcnd 15240 | Standard inner product on ... |
| cjdivd 15241 | Complex conjugate distribu... |
| rered 15242 | A real number equals its r... |
| reim0d 15243 | The imaginary part of a re... |
| cjred 15244 | A real number equals its c... |
| remul2d 15245 | Real part of a product. (... |
| immul2d 15246 | Imaginary part of a produc... |
| redivd 15247 | Real part of a division. ... |
| imdivd 15248 | Imaginary part of a divisi... |
| crred 15249 | The real part of a complex... |
| crimd 15250 | The imaginary part of a co... |
| sqrtval 15255 | Value of square root funct... |
| absval 15256 | The absolute value (modulu... |
| rennim 15257 | A real number does not lie... |
| cnpart 15258 | The specification of restr... |
| sqrt0 15259 | The square root of zero is... |
| 01sqrexlem1 15260 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15261 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15262 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15263 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15264 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15265 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15266 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15267 | Existence of a square root... |
| resqrex 15268 | Existence of a square root... |
| sqrmo 15269 | Uniqueness for the square ... |
| resqreu 15270 | Existence and uniqueness f... |
| resqrtcl 15271 | Closure of the square root... |
| resqrtthlem 15272 | Lemma for ~ resqrtth . (C... |
| resqrtth 15273 | Square root theorem over t... |
| remsqsqrt 15274 | Square of square root. (C... |
| sqrtge0 15275 | The square root function i... |
| sqrtgt0 15276 | The square root function i... |
| sqrtmul 15277 | Square root distributes ov... |
| sqrtle 15278 | Square root is monotonic. ... |
| sqrtlt 15279 | Square root is strictly mo... |
| sqrt11 15280 | The square root function i... |
| sqrt00 15281 | A square root is zero iff ... |
| rpsqrtcl 15282 | The square root of a posit... |
| sqrtdiv 15283 | Square root distributes ov... |
| sqrtneglem 15284 | The square root of a negat... |
| sqrtneg 15285 | The square root of a negat... |
| sqrtsq2 15286 | Relationship between squar... |
| sqrtsq 15287 | Square root of square. (C... |
| sqrtmsq 15288 | Square root of square. (C... |
| sqrt1 15289 | The square root of 1 is 1.... |
| sqrt4 15290 | The square root of 4 is 2.... |
| sqrt9 15291 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15292 | The square root of 2 is bo... |
| sqrtm1 15293 | The imaginary unit is the ... |
| nn0sqeq1 15294 | A natural number with squa... |
| absneg 15295 | Absolute value of the nega... |
| abscl 15296 | Real closure of absolute v... |
| abscj 15297 | The absolute value of a nu... |
| absvalsq 15298 | Square of value of absolut... |
| absvalsq2 15299 | Square of value of absolut... |
| sqabsadd 15300 | Square of absolute value o... |
| sqabssub 15301 | Square of absolute value o... |
| absval2 15302 | Value of absolute value fu... |
| abs0 15303 | The absolute value of 0. ... |
| absi 15304 | The absolute value of the ... |
| absge0 15305 | Absolute value is nonnegat... |
| absrpcl 15306 | The absolute value of a no... |
| abs00 15307 | The absolute value of a nu... |
| abs00ad 15308 | A complex number is zero i... |
| abs00bd 15309 | If a complex number is zer... |
| absreimsq 15310 | Square of the absolute val... |
| absreim 15311 | Absolute value of a number... |
| absmul 15312 | Absolute value distributes... |
| absdiv 15313 | Absolute value distributes... |
| absid 15314 | A nonnegative number is it... |
| abs1 15315 | The absolute value of one ... |
| absnid 15316 | For a negative number, its... |
| leabs 15317 | A real number is less than... |
| absor 15318 | The absolute value of a re... |
| absre 15319 | Absolute value of a real n... |
| absresq 15320 | Square of the absolute val... |
| absmod0 15321 | ` A ` is divisible by ` B ... |
| absexp 15322 | Absolute value of positive... |
| absexpz 15323 | Absolute value of integer ... |
| abssq 15324 | Square can be moved in and... |
| sqabs 15325 | The squares of two reals a... |
| absrele 15326 | The absolute value of a co... |
| absimle 15327 | The absolute value of a co... |
| max0add 15328 | The sum of the positive an... |
| absz 15329 | A real number is an intege... |
| nn0abscl 15330 | The absolute value of an i... |
| zabscl 15331 | The absolute value of an i... |
| abslt 15332 | Absolute value and 'less t... |
| absle 15333 | Absolute value and 'less t... |
| abssubne0 15334 | If the absolute value of a... |
| absdiflt 15335 | The absolute value of a di... |
| absdifle 15336 | The absolute value of a di... |
| elicc4abs 15337 | Membership in a symmetric ... |
| lenegsq 15338 | Comparison to a nonnegativ... |
| releabs 15339 | The real part of a number ... |
| recval 15340 | Reciprocal expressed with ... |
| absidm 15341 | The absolute value functio... |
| absgt0 15342 | The absolute value of a no... |
| nnabscl 15343 | The absolute value of a no... |
| abssub 15344 | Swapping order of subtract... |
| abssubge0 15345 | Absolute value of a nonneg... |
| abssuble0 15346 | Absolute value of a nonpos... |
| absmax 15347 | The maximum of two numbers... |
| abstri 15348 | Triangle inequality for ab... |
| abs3dif 15349 | Absolute value of differen... |
| abs2dif 15350 | Difference of absolute val... |
| abs2dif2 15351 | Difference of absolute val... |
| abs2difabs 15352 | Absolute value of differen... |
| abs1m 15353 | For any complex number, th... |
| recan 15354 | Cancellation law involving... |
| absf 15355 | Mapping domain and codomai... |
| abs3lem 15356 | Lemma involving absolute v... |
| abslem2 15357 | Lemma involving absolute v... |
| rddif 15358 | The difference between a r... |
| absrdbnd 15359 | Bound on the absolute valu... |
| fzomaxdiflem 15360 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15361 | A bound on the separation ... |
| uzin2 15362 | The upper integers are clo... |
| rexanuz 15363 | Combine two different uppe... |
| rexanre 15364 | Combine two different uppe... |
| rexfiuz 15365 | Combine finitely many diff... |
| rexuz3 15366 | Restrict the base of the u... |
| rexanuz2 15367 | Combine two different uppe... |
| r19.29uz 15368 | A version of ~ 19.29 for u... |
| r19.2uz 15369 | A version of ~ r19.2z for ... |
| rexuzre 15370 | Convert an upper real quan... |
| rexico 15371 | Restrict the base of an up... |
| cau3lem 15372 | Lemma for ~ cau3 . (Contr... |
| cau3 15373 | Convert between three-quan... |
| cau4 15374 | Change the base of a Cauch... |
| caubnd2 15375 | A Cauchy sequence of compl... |
| caubnd 15376 | A Cauchy sequence of compl... |
| sqreulem 15377 | Lemma for ~ sqreu : write ... |
| sqreu 15378 | Existence and uniqueness f... |
| sqrtcl 15379 | Closure of the square root... |
| sqrtthlem 15380 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15381 | Mapping domain and codomai... |
| sqrtth 15382 | Square root theorem over t... |
| sqrtrege0 15383 | The square root function m... |
| eqsqrtor 15384 | Solve an equation containi... |
| eqsqrtd 15385 | A deduction for showing th... |
| eqsqrt2d 15386 | A deduction for showing th... |
| amgm2 15387 | Arithmetic-geometric mean ... |
| sqrtthi 15388 | Square root theorem. Theo... |
| sqrtcli 15389 | The square root of a nonne... |
| sqrtgt0i 15390 | The square root of a posit... |
| sqrtmsqi 15391 | Square root of square. (C... |
| sqrtsqi 15392 | Square root of square. (C... |
| sqsqrti 15393 | Square of square root. (C... |
| sqrtge0i 15394 | The square root of a nonne... |
| absidi 15395 | A nonnegative number is it... |
| absnidi 15396 | A negative number is the n... |
| leabsi 15397 | A real number is less than... |
| absori 15398 | The absolute value of a re... |
| absrei 15399 | Absolute value of a real n... |
| sqrtpclii 15400 | The square root of a posit... |
| sqrtgt0ii 15401 | The square root of a posit... |
| sqrt11i 15402 | The square root function i... |
| sqrtmuli 15403 | Square root distributes ov... |
| sqrtmulii 15404 | Square root distributes ov... |
| sqrtmsq2i 15405 | Relationship between squar... |
| sqrtlei 15406 | Square root is monotonic. ... |
| sqrtlti 15407 | Square root is strictly mo... |
| abslti 15408 | Absolute value and 'less t... |
| abslei 15409 | Absolute value and 'less t... |
| cnsqrt00 15410 | A square root of a complex... |
| absvalsqi 15411 | Square of value of absolut... |
| absvalsq2i 15412 | Square of value of absolut... |
| abscli 15413 | Real closure of absolute v... |
| absge0i 15414 | Absolute value is nonnegat... |
| absval2i 15415 | Value of absolute value fu... |
| abs00i 15416 | The absolute value of a nu... |
| absgt0i 15417 | The absolute value of a no... |
| absnegi 15418 | Absolute value of negative... |
| abscji 15419 | The absolute value of a nu... |
| releabsi 15420 | The real part of a number ... |
| abssubi 15421 | Swapping order of subtract... |
| absmuli 15422 | Absolute value distributes... |
| sqabsaddi 15423 | Square of absolute value o... |
| sqabssubi 15424 | Square of absolute value o... |
| absdivzi 15425 | Absolute value distributes... |
| abstrii 15426 | Triangle inequality for ab... |
| abs3difi 15427 | Absolute value of differen... |
| abs3lemi 15428 | Lemma involving absolute v... |
| rpsqrtcld 15429 | The square root of a posit... |
| sqrtgt0d 15430 | The square root of a posit... |
| absnidd 15431 | A negative number is the n... |
| leabsd 15432 | A real number is less than... |
| absord 15433 | The absolute value of a re... |
| absred 15434 | Absolute value of a real n... |
| resqrtcld 15435 | The square root of a nonne... |
| sqrtmsqd 15436 | Square root of square. (C... |
| sqrtsqd 15437 | Square root of square. (C... |
| sqrtge0d 15438 | The square root of a nonne... |
| sqrtnegd 15439 | The square root of a negat... |
| absidd 15440 | A nonnegative number is it... |
| sqrtdivd 15441 | Square root distributes ov... |
| sqrtmuld 15442 | Square root distributes ov... |
| sqrtsq2d 15443 | Relationship between squar... |
| sqrtled 15444 | Square root is monotonic. ... |
| sqrtltd 15445 | Square root is strictly mo... |
| sqr11d 15446 | The square root function i... |
| absltd 15447 | Absolute value and 'less t... |
| absled 15448 | Absolute value and 'less t... |
| abssubge0d 15449 | Absolute value of a nonneg... |
| abssuble0d 15450 | Absolute value of a nonpos... |
| absdifltd 15451 | The absolute value of a di... |
| absdifled 15452 | The absolute value of a di... |
| icodiamlt 15453 | Two elements in a half-ope... |
| abscld 15454 | Real closure of absolute v... |
| sqrtcld 15455 | Closure of the square root... |
| sqrtrege0d 15456 | The real part of the squar... |
| sqsqrtd 15457 | Square root theorem. Theo... |
| msqsqrtd 15458 | Square root theorem. Theo... |
| sqr00d 15459 | A square root is zero iff ... |
| absvalsqd 15460 | Square of value of absolut... |
| absvalsq2d 15461 | Square of value of absolut... |
| absge0d 15462 | Absolute value is nonnegat... |
| absval2d 15463 | Value of absolute value fu... |
| abs00d 15464 | The absolute value of a nu... |
| absne0d 15465 | The absolute value of a nu... |
| absrpcld 15466 | The absolute value of a no... |
| absnegd 15467 | Absolute value of negative... |
| abscjd 15468 | The absolute value of a nu... |
| releabsd 15469 | The real part of a number ... |
| absexpd 15470 | Absolute value of positive... |
| abssubd 15471 | Swapping order of subtract... |
| absmuld 15472 | Absolute value distributes... |
| absdivd 15473 | Absolute value distributes... |
| abstrid 15474 | Triangle inequality for ab... |
| abs2difd 15475 | Difference of absolute val... |
| abs2dif2d 15476 | Difference of absolute val... |
| abs2difabsd 15477 | Absolute value of differen... |
| abs3difd 15478 | Absolute value of differen... |
| abs3lemd 15479 | Lemma involving absolute v... |
| reusq0 15480 | A complex number is the sq... |
| bhmafibid1cn 15481 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15482 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15483 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15484 | The Brahmagupta-Fibonacci ... |
| limsupgord 15487 | Ordering property of the s... |
| limsupcl 15488 | Closure of the superior li... |
| limsupval 15489 | The superior limit of an i... |
| limsupgf 15490 | Closure of the superior li... |
| limsupgval 15491 | Value of the superior limi... |
| limsupgle 15492 | The defining property of t... |
| limsuple 15493 | The defining property of t... |
| limsuplt 15494 | The defining property of t... |
| limsupval2 15495 | The superior limit, relati... |
| limsupgre 15496 | If a sequence of real numb... |
| limsupbnd1 15497 | If a sequence is eventuall... |
| limsupbnd2 15498 | If a sequence is eventuall... |
| climrel 15507 | The limit relation is a re... |
| rlimrel 15508 | The limit relation is a re... |
| clim 15509 | Express the predicate: Th... |
| rlim 15510 | Express the predicate: Th... |
| rlim2 15511 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15512 | Use strictly less-than in ... |
| rlim3 15513 | Restrict the range of the ... |
| climcl 15514 | Closure of the limit of a ... |
| rlimpm 15515 | Closure of a function with... |
| rlimf 15516 | Closure of a function with... |
| rlimss 15517 | Domain closure of a functi... |
| rlimcl 15518 | Closure of the limit of a ... |
| clim2 15519 | Express the predicate: Th... |
| clim2c 15520 | Express the predicate ` F ... |
| clim0 15521 | Express the predicate ` F ... |
| clim0c 15522 | Express the predicate ` F ... |
| rlim0 15523 | Express the predicate ` B ... |
| rlim0lt 15524 | Use strictly less-than in ... |
| climi 15525 | Convergence of a sequence ... |
| climi2 15526 | Convergence of a sequence ... |
| climi0 15527 | Convergence of a sequence ... |
| rlimi 15528 | Convergence at infinity of... |
| rlimi2 15529 | Convergence at infinity of... |
| ello1 15530 | Elementhood in the set of ... |
| ello12 15531 | Elementhood in the set of ... |
| ello12r 15532 | Sufficient condition for e... |
| lo1f 15533 | An eventually upper bounde... |
| lo1dm 15534 | An eventually upper bounde... |
| lo1bdd 15535 | The defining property of a... |
| ello1mpt 15536 | Elementhood in the set of ... |
| ello1mpt2 15537 | Elementhood in the set of ... |
| ello1d 15538 | Sufficient condition for e... |
| lo1bdd2 15539 | If an eventually bounded f... |
| lo1bddrp 15540 | Refine ~ o1bdd2 to give a ... |
| elo1 15541 | Elementhood in the set of ... |
| elo12 15542 | Elementhood in the set of ... |
| elo12r 15543 | Sufficient condition for e... |
| o1f 15544 | An eventually bounded func... |
| o1dm 15545 | An eventually bounded func... |
| o1bdd 15546 | The defining property of a... |
| lo1o1 15547 | A function is eventually b... |
| lo1o12 15548 | A function is eventually b... |
| elo1mpt 15549 | Elementhood in the set of ... |
| elo1mpt2 15550 | Elementhood in the set of ... |
| elo1d 15551 | Sufficient condition for e... |
| o1lo1 15552 | A real function is eventua... |
| o1lo12 15553 | A lower bounded real funct... |
| o1lo1d 15554 | A real eventually bounded ... |
| icco1 15555 | Derive eventual boundednes... |
| o1bdd2 15556 | If an eventually bounded f... |
| o1bddrp 15557 | Refine ~ o1bdd2 to give a ... |
| climconst 15558 | An (eventually) constant s... |
| rlimconst 15559 | A constant sequence conver... |
| rlimclim1 15560 | Forward direction of ~ rli... |
| rlimclim 15561 | A sequence on an upper int... |
| climrlim2 15562 | Produce a real limit from ... |
| climconst2 15563 | A constant sequence conver... |
| climz 15564 | The zero sequence converge... |
| rlimuni 15565 | A real function whose doma... |
| rlimdm 15566 | Two ways to express that a... |
| climuni 15567 | An infinite sequence of co... |
| fclim 15568 | The limit relation is func... |
| climdm 15569 | Two ways to express that a... |
| climeu 15570 | An infinite sequence of co... |
| climreu 15571 | An infinite sequence of co... |
| climmo 15572 | An infinite sequence of co... |
| rlimres 15573 | The restriction of a funct... |
| lo1res 15574 | The restriction of an even... |
| o1res 15575 | The restriction of an even... |
| rlimres2 15576 | The restriction of a funct... |
| lo1res2 15577 | The restriction of a funct... |
| o1res2 15578 | The restriction of a funct... |
| lo1resb 15579 | The restriction of a funct... |
| rlimresb 15580 | The restriction of a funct... |
| o1resb 15581 | The restriction of a funct... |
| climeq 15582 | Two functions that are eve... |
| lo1eq 15583 | Two functions that are eve... |
| rlimeq 15584 | Two functions that are eve... |
| o1eq 15585 | Two functions that are eve... |
| climmpt 15586 | Exhibit a function ` G ` w... |
| 2clim 15587 | If two sequences converge ... |
| climmpt2 15588 | Relate an integer limit on... |
| climshftlem 15589 | A shifted function converg... |
| climres 15590 | A function restricted to u... |
| climshft 15591 | A shifted function converg... |
| serclim0 15592 | The zero series converges ... |
| rlimcld2 15593 | If ` D ` is a closed set i... |
| rlimrege0 15594 | The limit of a sequence of... |
| rlimrecl 15595 | The limit of a real sequen... |
| rlimge0 15596 | The limit of a sequence of... |
| climshft2 15597 | A shifted function converg... |
| climrecl 15598 | The limit of a convergent ... |
| climge0 15599 | A nonnegative sequence con... |
| climabs0 15600 | Convergence to zero of the... |
| o1co 15601 | Sufficient condition for t... |
| o1compt 15602 | Sufficient condition for t... |
| rlimcn1 15603 | Image of a limit under a c... |
| rlimcn1b 15604 | Image of a limit under a c... |
| rlimcn3 15605 | Image of a limit under a c... |
| rlimcn2 15606 | Image of a limit under a c... |
| climcn1 15607 | Image of a limit under a c... |
| climcn2 15608 | Image of a limit under a c... |
| addcn2 15609 | Complex number addition is... |
| subcn2 15610 | Complex number subtraction... |
| mulcn2 15611 | Complex number multiplicat... |
| reccn2 15612 | The reciprocal function is... |
| cn1lem 15613 | A sufficient condition for... |
| abscn2 15614 | The absolute value functio... |
| cjcn2 15615 | The complex conjugate func... |
| recn2 15616 | The real part function is ... |
| imcn2 15617 | The imaginary part functio... |
| climcn1lem 15618 | The limit of a continuous ... |
| climabs 15619 | Limit of the absolute valu... |
| climcj 15620 | Limit of the complex conju... |
| climre 15621 | Limit of the real part of ... |
| climim 15622 | Limit of the imaginary par... |
| rlimmptrcl 15623 | Reverse closure for a real... |
| rlimabs 15624 | Limit of the absolute valu... |
| rlimcj 15625 | Limit of the complex conju... |
| rlimre 15626 | Limit of the real part of ... |
| rlimim 15627 | Limit of the imaginary par... |
| o1of2 15628 | Show that a binary operati... |
| o1add 15629 | The sum of two eventually ... |
| o1mul 15630 | The product of two eventua... |
| o1sub 15631 | The difference of two even... |
| rlimo1 15632 | Any function with a finite... |
| rlimdmo1 15633 | A convergent function is e... |
| o1rlimmul 15634 | The product of an eventual... |
| o1const 15635 | A constant function is eve... |
| lo1const 15636 | A constant function is eve... |
| lo1mptrcl 15637 | Reverse closure for an eve... |
| o1mptrcl 15638 | Reverse closure for an eve... |
| o1add2 15639 | The sum of two eventually ... |
| o1mul2 15640 | The product of two eventua... |
| o1sub2 15641 | The product of two eventua... |
| lo1add 15642 | The sum of two eventually ... |
| lo1mul 15643 | The product of an eventual... |
| lo1mul2 15644 | The product of an eventual... |
| o1dif 15645 | If the difference of two f... |
| lo1sub 15646 | The difference of an event... |
| climadd 15647 | Limit of the sum of two co... |
| climmul 15648 | Limit of the product of tw... |
| climsub 15649 | Limit of the difference of... |
| climaddc1 15650 | Limit of a constant ` C ` ... |
| climaddc2 15651 | Limit of a constant ` C ` ... |
| climmulc2 15652 | Limit of a sequence multip... |
| climsubc1 15653 | Limit of a constant ` C ` ... |
| climsubc2 15654 | Limit of a constant ` C ` ... |
| climle 15655 | Comparison of the limits o... |
| climsqz 15656 | Convergence of a sequence ... |
| climsqz2 15657 | Convergence of a sequence ... |
| rlimadd 15658 | Limit of the sum of two co... |
| rlimaddOLD 15659 | Obsolete version of ~ rlim... |
| rlimsub 15660 | Limit of the difference of... |
| rlimmul 15661 | Limit of the product of tw... |
| rlimmulOLD 15662 | Obsolete version of ~ rlim... |
| rlimdiv 15663 | Limit of the quotient of t... |
| rlimneg 15664 | Limit of the negative of a... |
| rlimle 15665 | Comparison of the limits o... |
| rlimsqzlem 15666 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15667 | Convergence of a sequence ... |
| rlimsqz2 15668 | Convergence of a sequence ... |
| lo1le 15669 | Transfer eventual upper bo... |
| o1le 15670 | Transfer eventual boundedn... |
| rlimno1 15671 | A function whose inverse c... |
| clim2ser 15672 | The limit of an infinite s... |
| clim2ser2 15673 | The limit of an infinite s... |
| iserex 15674 | An infinite series converg... |
| isermulc2 15675 | Multiplication of an infin... |
| climlec2 15676 | Comparison of a constant t... |
| iserle 15677 | Comparison of the limits o... |
| iserge0 15678 | The limit of an infinite s... |
| climub 15679 | The limit of a monotonic s... |
| climserle 15680 | The partial sums of a conv... |
| isershft 15681 | Index shift of the limit o... |
| isercolllem1 15682 | Lemma for ~ isercoll . (C... |
| isercolllem2 15683 | Lemma for ~ isercoll . (C... |
| isercolllem3 15684 | Lemma for ~ isercoll . (C... |
| isercoll 15685 | Rearrange an infinite seri... |
| isercoll2 15686 | Generalize ~ isercoll so t... |
| climsup 15687 | A bounded monotonic sequen... |
| climcau 15688 | A converging sequence of c... |
| climbdd 15689 | A converging sequence of c... |
| caucvgrlem 15690 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15691 | A Cauchy sequence of real ... |
| caucvgrlem2 15692 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15693 | A Cauchy sequence of compl... |
| caurcvg 15694 | A Cauchy sequence of real ... |
| caurcvg2 15695 | A Cauchy sequence of real ... |
| caucvg 15696 | A Cauchy sequence of compl... |
| caucvgb 15697 | A function is convergent i... |
| serf0 15698 | If an infinite series conv... |
| iseraltlem1 15699 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15700 | Lemma for ~ iseralt . The... |
| iseraltlem3 15701 | Lemma for ~ iseralt . Fro... |
| iseralt 15702 | The alternating series tes... |
| sumex 15705 | A sum is a set. (Contribu... |
| sumeq1 15706 | Equality theorem for a sum... |
| nfsum1 15707 | Bound-variable hypothesis ... |
| nfsum 15708 | Bound-variable hypothesis ... |
| sumeq2w 15709 | Equality theorem for sum, ... |
| sumeq2ii 15710 | Equality theorem for sum, ... |
| sumeq2 15711 | Equality theorem for sum. ... |
| cbvsum 15712 | Change bound variable in a... |
| cbvsumv 15713 | Change bound variable in a... |
| cbvsumi 15714 | Change bound variable in a... |
| sumeq1i 15715 | Equality inference for sum... |
| sumeq2i 15716 | Equality inference for sum... |
| sumeq12i 15717 | Equality inference for sum... |
| sumeq1d 15718 | Equality deduction for sum... |
| sumeq2d 15719 | Equality deduction for sum... |
| sumeq2dv 15720 | Equality deduction for sum... |
| sumeq2sdv 15721 | Equality deduction for sum... |
| 2sumeq2dv 15722 | Equality deduction for dou... |
| sumeq12dv 15723 | Equality deduction for sum... |
| sumeq12rdv 15724 | Equality deduction for sum... |
| sum2id 15725 | The second class argument ... |
| sumfc 15726 | A lemma to facilitate conv... |
| fz1f1o 15727 | A lemma for working with f... |
| sumrblem 15728 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15729 | The sequence of partial su... |
| sumrb 15730 | Rebase the starting point ... |
| summolem3 15731 | Lemma for ~ summo . (Cont... |
| summolem2a 15732 | Lemma for ~ summo . (Cont... |
| summolem2 15733 | Lemma for ~ summo . (Cont... |
| summo 15734 | A sum has at most one limi... |
| zsum 15735 | Series sum with index set ... |
| isum 15736 | Series sum with an upper i... |
| fsum 15737 | The value of a sum over a ... |
| sum0 15738 | Any sum over the empty set... |
| sumz 15739 | Any sum of zero over a sum... |
| fsumf1o 15740 | Re-index a finite sum usin... |
| sumss 15741 | Change the index set to a ... |
| fsumss 15742 | Change the index set to a ... |
| sumss2 15743 | Change the index set of a ... |
| fsumcvg2 15744 | The sequence of partial su... |
| fsumsers 15745 | Special case of series sum... |
| fsumcvg3 15746 | A finite sum is convergent... |
| fsumser 15747 | A finite sum expressed in ... |
| fsumcl2lem 15748 | - Lemma for finite sum clo... |
| fsumcllem 15749 | - Lemma for finite sum clo... |
| fsumcl 15750 | Closure of a finite sum of... |
| fsumrecl 15751 | Closure of a finite sum of... |
| fsumzcl 15752 | Closure of a finite sum of... |
| fsumnn0cl 15753 | Closure of a finite sum of... |
| fsumrpcl 15754 | Closure of a finite sum of... |
| fsumclf 15755 | Closure of a finite sum of... |
| fsumzcl2 15756 | A finite sum with integer ... |
| fsumadd 15757 | The sum of two finite sums... |
| fsumsplit 15758 | Split a sum into two parts... |
| fsumsplitf 15759 | Split a sum into two parts... |
| sumsnf 15760 | A sum of a singleton is th... |
| fsumsplitsn 15761 | Separate out a term in a f... |
| fsumsplit1 15762 | Separate out a term in a f... |
| sumsn 15763 | A sum of a singleton is th... |
| fsum1 15764 | The finite sum of ` A ( k ... |
| sumpr 15765 | A sum over a pair is the s... |
| sumtp 15766 | A sum over a triple is the... |
| sumsns 15767 | A sum of a singleton is th... |
| fsumm1 15768 | Separate out the last term... |
| fzosump1 15769 | Separate out the last term... |
| fsum1p 15770 | Separate out the first ter... |
| fsummsnunz 15771 | A finite sum all of whose ... |
| fsumsplitsnun 15772 | Separate out a term in a f... |
| fsump1 15773 | The addition of the next t... |
| isumclim 15774 | An infinite sum equals the... |
| isumclim2 15775 | A converging series conver... |
| isumclim3 15776 | The sequence of partial fi... |
| sumnul 15777 | The sum of a non-convergen... |
| isumcl 15778 | The sum of a converging in... |
| isummulc2 15779 | An infinite sum multiplied... |
| isummulc1 15780 | An infinite sum multiplied... |
| isumdivc 15781 | An infinite sum divided by... |
| isumrecl 15782 | The sum of a converging in... |
| isumge0 15783 | An infinite sum of nonnega... |
| isumadd 15784 | Addition of infinite sums.... |
| sumsplit 15785 | Split a sum into two parts... |
| fsump1i 15786 | Optimized version of ~ fsu... |
| fsum2dlem 15787 | Lemma for ~ fsum2d - induc... |
| fsum2d 15788 | Write a double sum as a su... |
| fsumxp 15789 | Combine two sums into a si... |
| fsumcnv 15790 | Transform a region of summ... |
| fsumcom2 15791 | Interchange order of summa... |
| fsumcom 15792 | Interchange order of summa... |
| fsum0diaglem 15793 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15794 | Two ways to express "the s... |
| mptfzshft 15795 | 1-1 onto function in maps-... |
| fsumrev 15796 | Reversal of a finite sum. ... |
| fsumshft 15797 | Index shift of a finite su... |
| fsumshftm 15798 | Negative index shift of a ... |
| fsumrev2 15799 | Reversal of a finite sum. ... |
| fsum0diag2 15800 | Two ways to express "the s... |
| fsummulc2 15801 | A finite sum multiplied by... |
| fsummulc1 15802 | A finite sum multiplied by... |
| fsumdivc 15803 | A finite sum divided by a ... |
| fsumneg 15804 | Negation of a finite sum. ... |
| fsumsub 15805 | Split a finite sum over a ... |
| fsum2mul 15806 | Separate the nested sum of... |
| fsumconst 15807 | The sum of constant terms ... |
| fsumdifsnconst 15808 | The sum of constant terms ... |
| modfsummodslem1 15809 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15810 | Induction step for ~ modfs... |
| modfsummod 15811 | A finite sum modulo a posi... |
| fsumge0 15812 | If all of the terms of a f... |
| fsumless 15813 | A shorter sum of nonnegati... |
| fsumge1 15814 | A sum of nonnegative numbe... |
| fsum00 15815 | A sum of nonnegative numbe... |
| fsumle 15816 | If all of the terms of fin... |
| fsumlt 15817 | If every term in one finit... |
| fsumabs 15818 | Generalized triangle inequ... |
| telfsumo 15819 | Sum of a telescoping serie... |
| telfsumo2 15820 | Sum of a telescoping serie... |
| telfsum 15821 | Sum of a telescoping serie... |
| telfsum2 15822 | Sum of a telescoping serie... |
| fsumparts 15823 | Summation by parts. (Cont... |
| fsumrelem 15824 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15825 | The real part of a sum. (... |
| fsumim 15826 | The imaginary part of a su... |
| fsumcj 15827 | The complex conjugate of a... |
| fsumrlim 15828 | Limit of a finite sum of c... |
| fsumo1 15829 | The finite sum of eventual... |
| o1fsum 15830 | If ` A ( k ) ` is O(1), th... |
| seqabs 15831 | Generalized triangle inequ... |
| iserabs 15832 | Generalized triangle inequ... |
| cvgcmp 15833 | A comparison test for conv... |
| cvgcmpub 15834 | An upper bound for the lim... |
| cvgcmpce 15835 | A comparison test for conv... |
| abscvgcvg 15836 | An absolutely convergent s... |
| climfsum 15837 | Limit of a finite sum of c... |
| fsumiun 15838 | Sum over a disjoint indexe... |
| hashiun 15839 | The cardinality of a disjo... |
| hash2iun 15840 | The cardinality of a neste... |
| hash2iun1dif1 15841 | The cardinality of a neste... |
| hashrabrex 15842 | The number of elements in ... |
| hashuni 15843 | The cardinality of a disjo... |
| qshash 15844 | The cardinality of a set w... |
| ackbijnn 15845 | Translate the Ackermann bi... |
| binomlem 15846 | Lemma for ~ binom (binomia... |
| binom 15847 | The binomial theorem: ` ( ... |
| binom1p 15848 | Special case of the binomi... |
| binom11 15849 | Special case of the binomi... |
| binom1dif 15850 | A summation for the differ... |
| bcxmaslem1 15851 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15852 | Parallel summation (Christ... |
| incexclem 15853 | Lemma for ~ incexc . (Con... |
| incexc 15854 | The inclusion/exclusion pr... |
| incexc2 15855 | The inclusion/exclusion pr... |
| isumshft 15856 | Index shift of an infinite... |
| isumsplit 15857 | Split off the first ` N ` ... |
| isum1p 15858 | The infinite sum of a conv... |
| isumnn0nn 15859 | Sum from 0 to infinity in ... |
| isumrpcl 15860 | The infinite sum of positi... |
| isumle 15861 | Comparison of two infinite... |
| isumless 15862 | A finite sum of nonnegativ... |
| isumsup2 15863 | An infinite sum of nonnega... |
| isumsup 15864 | An infinite sum of nonnega... |
| isumltss 15865 | A partial sum of a series ... |
| climcndslem1 15866 | Lemma for ~ climcnds : bou... |
| climcndslem2 15867 | Lemma for ~ climcnds : bou... |
| climcnds 15868 | The Cauchy condensation te... |
| divrcnv 15869 | The sequence of reciprocal... |
| divcnv 15870 | The sequence of reciprocal... |
| flo1 15871 | The floor function satisfi... |
| divcnvshft 15872 | Limit of a ratio function.... |
| supcvg 15873 | Extract a sequence ` f ` i... |
| infcvgaux1i 15874 | Auxiliary theorem for appl... |
| infcvgaux2i 15875 | Auxiliary theorem for appl... |
| harmonic 15876 | The harmonic series ` H ` ... |
| arisum 15877 | Arithmetic series sum of t... |
| arisum2 15878 | Arithmetic series sum of t... |
| trireciplem 15879 | Lemma for ~ trirecip . Sh... |
| trirecip 15880 | The sum of the reciprocals... |
| expcnv 15881 | A sequence of powers of a ... |
| explecnv 15882 | A sequence of terms conver... |
| geoserg 15883 | The value of the finite ge... |
| geoser 15884 | The value of the finite ge... |
| pwdif 15885 | The difference of two numb... |
| pwm1geoser 15886 | The n-th power of a number... |
| geolim 15887 | The partial sums in the in... |
| geolim2 15888 | The partial sums in the ge... |
| georeclim 15889 | The limit of a geometric s... |
| geo2sum 15890 | The value of the finite ge... |
| geo2sum2 15891 | The value of the finite ge... |
| geo2lim 15892 | The value of the infinite ... |
| geomulcvg 15893 | The geometric series conve... |
| geoisum 15894 | The infinite sum of ` 1 + ... |
| geoisumr 15895 | The infinite sum of recipr... |
| geoisum1 15896 | The infinite sum of ` A ^ ... |
| geoisum1c 15897 | The infinite sum of ` A x.... |
| 0.999... 15898 | The recurring decimal 0.99... |
| geoihalfsum 15899 | Prove that the infinite ge... |
| cvgrat 15900 | Ratio test for convergence... |
| mertenslem1 15901 | Lemma for ~ mertens . (Co... |
| mertenslem2 15902 | Lemma for ~ mertens . (Co... |
| mertens 15903 | Mertens' theorem. If ` A ... |
| prodf 15904 | An infinite product of com... |
| clim2prod 15905 | The limit of an infinite p... |
| clim2div 15906 | The limit of an infinite p... |
| prodfmul 15907 | The product of two infinit... |
| prodf1 15908 | The value of the partial p... |
| prodf1f 15909 | A one-valued infinite prod... |
| prodfclim1 15910 | The constant one product c... |
| prodfn0 15911 | No term of a nonzero infin... |
| prodfrec 15912 | The reciprocal of an infin... |
| prodfdiv 15913 | The quotient of two infini... |
| ntrivcvg 15914 | A non-trivially converging... |
| ntrivcvgn0 15915 | A product that converges t... |
| ntrivcvgfvn0 15916 | Any value of a product seq... |
| ntrivcvgtail 15917 | A tail of a non-trivially ... |
| ntrivcvgmullem 15918 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15919 | The product of two non-tri... |
| prodex 15922 | A product is a set. (Cont... |
| prodeq1f 15923 | Equality theorem for a pro... |
| prodeq1 15924 | Equality theorem for a pro... |
| nfcprod1 15925 | Bound-variable hypothesis ... |
| nfcprod 15926 | Bound-variable hypothesis ... |
| prodeq2w 15927 | Equality theorem for produ... |
| prodeq2ii 15928 | Equality theorem for produ... |
| prodeq2 15929 | Equality theorem for produ... |
| cbvprod 15930 | Change bound variable in a... |
| cbvprodv 15931 | Change bound variable in a... |
| cbvprodi 15932 | Change bound variable in a... |
| prodeq1i 15933 | Equality inference for pro... |
| prodeq2i 15934 | Equality inference for pro... |
| prodeq12i 15935 | Equality inference for pro... |
| prodeq1d 15936 | Equality deduction for pro... |
| prodeq2d 15937 | Equality deduction for pro... |
| prodeq2dv 15938 | Equality deduction for pro... |
| prodeq2sdv 15939 | Equality deduction for pro... |
| 2cprodeq2dv 15940 | Equality deduction for dou... |
| prodeq12dv 15941 | Equality deduction for pro... |
| prodeq12rdv 15942 | Equality deduction for pro... |
| prod2id 15943 | The second class argument ... |
| prodrblem 15944 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15945 | The sequence of partial pr... |
| prodrblem2 15946 | Lemma for ~ prodrb . (Con... |
| prodrb 15947 | Rebase the starting point ... |
| prodmolem3 15948 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15949 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15950 | Lemma for ~ prodmo . (Con... |
| prodmo 15951 | A product has at most one ... |
| zprod 15952 | Series product with index ... |
| iprod 15953 | Series product with an upp... |
| zprodn0 15954 | Nonzero series product wit... |
| iprodn0 15955 | Nonzero series product wit... |
| fprod 15956 | The value of a product ove... |
| fprodntriv 15957 | A non-triviality lemma for... |
| prod0 15958 | A product over the empty s... |
| prod1 15959 | Any product of one over a ... |
| prodfc 15960 | A lemma to facilitate conv... |
| fprodf1o 15961 | Re-index a finite product ... |
| prodss 15962 | Change the index set to a ... |
| fprodss 15963 | Change the index set to a ... |
| fprodser 15964 | A finite product expressed... |
| fprodcl2lem 15965 | Finite product closure lem... |
| fprodcllem 15966 | Finite product closure lem... |
| fprodcl 15967 | Closure of a finite produc... |
| fprodrecl 15968 | Closure of a finite produc... |
| fprodzcl 15969 | Closure of a finite produc... |
| fprodnncl 15970 | Closure of a finite produc... |
| fprodrpcl 15971 | Closure of a finite produc... |
| fprodnn0cl 15972 | Closure of a finite produc... |
| fprodcllemf 15973 | Finite product closure lem... |
| fprodreclf 15974 | Closure of a finite produc... |
| fprodmul 15975 | The product of two finite ... |
| fproddiv 15976 | The quotient of two finite... |
| prodsn 15977 | A product of a singleton i... |
| fprod1 15978 | A finite product of only o... |
| prodsnf 15979 | A product of a singleton i... |
| climprod1 15980 | The limit of a product ove... |
| fprodsplit 15981 | Split a finite product int... |
| fprodm1 15982 | Separate out the last term... |
| fprod1p 15983 | Separate out the first ter... |
| fprodp1 15984 | Multiply in the last term ... |
| fprodm1s 15985 | Separate out the last term... |
| fprodp1s 15986 | Multiply in the last term ... |
| prodsns 15987 | A product of the singleton... |
| fprodfac 15988 | Factorial using product no... |
| fprodabs 15989 | The absolute value of a fi... |
| fprodeq0 15990 | Any finite product contain... |
| fprodshft 15991 | Shift the index of a finit... |
| fprodrev 15992 | Reversal of a finite produ... |
| fprodconst 15993 | The product of constant te... |
| fprodn0 15994 | A finite product of nonzer... |
| fprod2dlem 15995 | Lemma for ~ fprod2d - indu... |
| fprod2d 15996 | Write a double product as ... |
| fprodxp 15997 | Combine two products into ... |
| fprodcnv 15998 | Transform a product region... |
| fprodcom2 15999 | Interchange order of multi... |
| fprodcom 16000 | Interchange product order.... |
| fprod0diag 16001 | Two ways to express "the p... |
| fproddivf 16002 | The quotient of two finite... |
| fprodsplitf 16003 | Split a finite product int... |
| fprodsplitsn 16004 | Separate out a term in a f... |
| fprodsplit1f 16005 | Separate out a term in a f... |
| fprodn0f 16006 | A finite product of nonzer... |
| fprodclf 16007 | Closure of a finite produc... |
| fprodge0 16008 | If all the terms of a fini... |
| fprodeq0g 16009 | Any finite product contain... |
| fprodge1 16010 | If all of the terms of a f... |
| fprodle 16011 | If all the terms of two fi... |
| fprodmodd 16012 | If all factors of two fini... |
| iprodclim 16013 | An infinite product equals... |
| iprodclim2 16014 | A converging product conve... |
| iprodclim3 16015 | The sequence of partial fi... |
| iprodcl 16016 | The product of a non-trivi... |
| iprodrecl 16017 | The product of a non-trivi... |
| iprodmul 16018 | Multiplication of infinite... |
| risefacval 16023 | The value of the rising fa... |
| fallfacval 16024 | The value of the falling f... |
| risefacval2 16025 | One-based value of rising ... |
| fallfacval2 16026 | One-based value of falling... |
| fallfacval3 16027 | A product representation o... |
| risefaccllem 16028 | Lemma for rising factorial... |
| fallfaccllem 16029 | Lemma for falling factoria... |
| risefaccl 16030 | Closure law for rising fac... |
| fallfaccl 16031 | Closure law for falling fa... |
| rerisefaccl 16032 | Closure law for rising fac... |
| refallfaccl 16033 | Closure law for falling fa... |
| nnrisefaccl 16034 | Closure law for rising fac... |
| zrisefaccl 16035 | Closure law for rising fac... |
| zfallfaccl 16036 | Closure law for falling fa... |
| nn0risefaccl 16037 | Closure law for rising fac... |
| rprisefaccl 16038 | Closure law for rising fac... |
| risefallfac 16039 | A relationship between ris... |
| fallrisefac 16040 | A relationship between fal... |
| risefall0lem 16041 | Lemma for ~ risefac0 and ~... |
| risefac0 16042 | The value of the rising fa... |
| fallfac0 16043 | The value of the falling f... |
| risefacp1 16044 | The value of the rising fa... |
| fallfacp1 16045 | The value of the falling f... |
| risefacp1d 16046 | The value of the rising fa... |
| fallfacp1d 16047 | The value of the falling f... |
| risefac1 16048 | The value of rising factor... |
| fallfac1 16049 | The value of falling facto... |
| risefacfac 16050 | Relate rising factorial to... |
| fallfacfwd 16051 | The forward difference of ... |
| 0fallfac 16052 | The value of the zero fall... |
| 0risefac 16053 | The value of the zero risi... |
| binomfallfaclem1 16054 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 16055 | Lemma for ~ binomfallfac .... |
| binomfallfac 16056 | A version of the binomial ... |
| binomrisefac 16057 | A version of the binomial ... |
| fallfacval4 16058 | Represent the falling fact... |
| bcfallfac 16059 | Binomial coefficient in te... |
| fallfacfac 16060 | Relate falling factorial t... |
| bpolylem 16063 | Lemma for ~ bpolyval . (C... |
| bpolyval 16064 | The value of the Bernoulli... |
| bpoly0 16065 | The value of the Bernoulli... |
| bpoly1 16066 | The value of the Bernoulli... |
| bpolycl 16067 | Closure law for Bernoulli ... |
| bpolysum 16068 | A sum for Bernoulli polyno... |
| bpolydiflem 16069 | Lemma for ~ bpolydif . (C... |
| bpolydif 16070 | Calculate the difference b... |
| fsumkthpow 16071 | A closed-form expression f... |
| bpoly2 16072 | The Bernoulli polynomials ... |
| bpoly3 16073 | The Bernoulli polynomials ... |
| bpoly4 16074 | The Bernoulli polynomials ... |
| fsumcube 16075 | Express the sum of cubes i... |
| eftcl 16088 | Closure of a term in the s... |
| reeftcl 16089 | The terms of the series ex... |
| eftabs 16090 | The absolute value of a te... |
| eftval 16091 | The value of a term in the... |
| efcllem 16092 | Lemma for ~ efcl . The se... |
| ef0lem 16093 | The series defining the ex... |
| efval 16094 | Value of the exponential f... |
| esum 16095 | Value of Euler's constant ... |
| eff 16096 | Domain and codomain of the... |
| efcl 16097 | Closure law for the expone... |
| efcld 16098 | Closure law for the expone... |
| efval2 16099 | Value of the exponential f... |
| efcvg 16100 | The series that defines th... |
| efcvgfsum 16101 | Exponential function conve... |
| reefcl 16102 | The exponential function i... |
| reefcld 16103 | The exponential function i... |
| ere 16104 | Euler's constant ` _e ` = ... |
| ege2le3 16105 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16106 | Value of the exponential f... |
| efcj 16107 | The exponential of a compl... |
| efaddlem 16108 | Lemma for ~ efadd (exponen... |
| efadd 16109 | Sum of exponents law for e... |
| fprodefsum 16110 | Move the exponential funct... |
| efcan 16111 | Cancellation law for expon... |
| efne0 16112 | The exponential of a compl... |
| efneg 16113 | The exponential of the opp... |
| eff2 16114 | The exponential function m... |
| efsub 16115 | Difference of exponents la... |
| efexp 16116 | The exponential of an inte... |
| efzval 16117 | Value of the exponential f... |
| efgt0 16118 | The exponential of a real ... |
| rpefcl 16119 | The exponential of a real ... |
| rpefcld 16120 | The exponential of a real ... |
| eftlcvg 16121 | The tail series of the exp... |
| eftlcl 16122 | Closure of the sum of an i... |
| reeftlcl 16123 | Closure of the sum of an i... |
| eftlub 16124 | An upper bound on the abso... |
| efsep 16125 | Separate out the next term... |
| effsumlt 16126 | The partial sums of the se... |
| eft0val 16127 | The value of the first ter... |
| ef4p 16128 | Separate out the first fou... |
| efgt1p2 16129 | The exponential of a posit... |
| efgt1p 16130 | The exponential of a posit... |
| efgt1 16131 | The exponential of a posit... |
| eflt 16132 | The exponential function o... |
| efle 16133 | The exponential function o... |
| reef11 16134 | The exponential function o... |
| reeff1 16135 | The exponential function m... |
| eflegeo 16136 | The exponential function o... |
| sinval 16137 | Value of the sine function... |
| cosval 16138 | Value of the cosine functi... |
| sinf 16139 | Domain and codomain of the... |
| cosf 16140 | Domain and codomain of the... |
| sincl 16141 | Closure of the sine functi... |
| coscl 16142 | Closure of the cosine func... |
| tanval 16143 | Value of the tangent funct... |
| tancl 16144 | The closure of the tangent... |
| sincld 16145 | Closure of the sine functi... |
| coscld 16146 | Closure of the cosine func... |
| tancld 16147 | Closure of the tangent fun... |
| tanval2 16148 | Express the tangent functi... |
| tanval3 16149 | Express the tangent functi... |
| resinval 16150 | The sine of a real number ... |
| recosval 16151 | The cosine of a real numbe... |
| efi4p 16152 | Separate out the first fou... |
| resin4p 16153 | Separate out the first fou... |
| recos4p 16154 | Separate out the first fou... |
| resincl 16155 | The sine of a real number ... |
| recoscl 16156 | The cosine of a real numbe... |
| retancl 16157 | The closure of the tangent... |
| resincld 16158 | Closure of the sine functi... |
| recoscld 16159 | Closure of the cosine func... |
| retancld 16160 | Closure of the tangent fun... |
| sinneg 16161 | The sine of a negative is ... |
| cosneg 16162 | The cosines of a number an... |
| tanneg 16163 | The tangent of a negative ... |
| sin0 16164 | Value of the sine function... |
| cos0 16165 | Value of the cosine functi... |
| tan0 16166 | The value of the tangent f... |
| efival 16167 | The exponential function i... |
| efmival 16168 | The exponential function i... |
| sinhval 16169 | Value of the hyperbolic si... |
| coshval 16170 | Value of the hyperbolic co... |
| resinhcl 16171 | The hyperbolic sine of a r... |
| rpcoshcl 16172 | The hyperbolic cosine of a... |
| recoshcl 16173 | The hyperbolic cosine of a... |
| retanhcl 16174 | The hyperbolic tangent of ... |
| tanhlt1 16175 | The hyperbolic tangent of ... |
| tanhbnd 16176 | The hyperbolic tangent of ... |
| efeul 16177 | Eulerian representation of... |
| efieq 16178 | The exponentials of two im... |
| sinadd 16179 | Addition formula for sine.... |
| cosadd 16180 | Addition formula for cosin... |
| tanaddlem 16181 | A useful intermediate step... |
| tanadd 16182 | Addition formula for tange... |
| sinsub 16183 | Sine of difference. (Cont... |
| cossub 16184 | Cosine of difference. (Co... |
| addsin 16185 | Sum of sines. (Contribute... |
| subsin 16186 | Difference of sines. (Con... |
| sinmul 16187 | Product of sines can be re... |
| cosmul 16188 | Product of cosines can be ... |
| addcos 16189 | Sum of cosines. (Contribu... |
| subcos 16190 | Difference of cosines. (C... |
| sincossq 16191 | Sine squared plus cosine s... |
| sin2t 16192 | Double-angle formula for s... |
| cos2t 16193 | Double-angle formula for c... |
| cos2tsin 16194 | Double-angle formula for c... |
| sinbnd 16195 | The sine of a real number ... |
| cosbnd 16196 | The cosine of a real numbe... |
| sinbnd2 16197 | The sine of a real number ... |
| cosbnd2 16198 | The cosine of a real numbe... |
| ef01bndlem 16199 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16200 | Bounds on the sine of a po... |
| cos01bnd 16201 | Bounds on the cosine of a ... |
| cos1bnd 16202 | Bounds on the cosine of 1.... |
| cos2bnd 16203 | Bounds on the cosine of 2.... |
| sinltx 16204 | The sine of a positive rea... |
| sin01gt0 16205 | The sine of a positive rea... |
| cos01gt0 16206 | The cosine of a positive r... |
| sin02gt0 16207 | The sine of a positive rea... |
| sincos1sgn 16208 | The signs of the sine and ... |
| sincos2sgn 16209 | The signs of the sine and ... |
| sin4lt0 16210 | The sine of 4 is negative.... |
| absefi 16211 | The absolute value of the ... |
| absef 16212 | The absolute value of the ... |
| absefib 16213 | A complex number is real i... |
| efieq1re 16214 | A number whose imaginary e... |
| demoivre 16215 | De Moivre's Formula. Proo... |
| demoivreALT 16216 | Alternate proof of ~ demoi... |
| eirrlem 16219 | Lemma for ~ eirr . (Contr... |
| eirr 16220 | ` _e ` is irrational. (Co... |
| egt2lt3 16221 | Euler's constant ` _e ` = ... |
| epos 16222 | Euler's constant ` _e ` is... |
| epr 16223 | Euler's constant ` _e ` is... |
| ene0 16224 | ` _e ` is not 0. (Contrib... |
| ene1 16225 | ` _e ` is not 1. (Contrib... |
| xpnnen 16226 | The Cartesian product of t... |
| znnen 16227 | The set of integers and th... |
| qnnen 16228 | The rational numbers are c... |
| rpnnen2lem1 16229 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16230 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16231 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16232 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16233 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16234 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16235 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16236 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16237 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16238 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16239 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16240 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16241 | The other half of ~ rpnnen... |
| rpnnen 16242 | The cardinality of the con... |
| rexpen 16243 | The real numbers are equin... |
| cpnnen 16244 | The complex numbers are eq... |
| rucALT 16245 | Alternate proof of ~ ruc .... |
| ruclem1 16246 | Lemma for ~ ruc (the reals... |
| ruclem2 16247 | Lemma for ~ ruc . Orderin... |
| ruclem3 16248 | Lemma for ~ ruc . The con... |
| ruclem4 16249 | Lemma for ~ ruc . Initial... |
| ruclem6 16250 | Lemma for ~ ruc . Domain ... |
| ruclem7 16251 | Lemma for ~ ruc . Success... |
| ruclem8 16252 | Lemma for ~ ruc . The int... |
| ruclem9 16253 | Lemma for ~ ruc . The fir... |
| ruclem10 16254 | Lemma for ~ ruc . Every f... |
| ruclem11 16255 | Lemma for ~ ruc . Closure... |
| ruclem12 16256 | Lemma for ~ ruc . The sup... |
| ruclem13 16257 | Lemma for ~ ruc . There i... |
| ruc 16258 | The set of positive intege... |
| resdomq 16259 | The set of rationals is st... |
| aleph1re 16260 | There are at least aleph-o... |
| aleph1irr 16261 | There are at least aleph-o... |
| cnso 16262 | The complex numbers can be... |
| sqrt2irrlem 16263 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16264 | The square root of 2 is ir... |
| sqrt2re 16265 | The square root of 2 exist... |
| sqrt2irr0 16266 | The square root of 2 is an... |
| nthruc 16267 | The sequence ` NN ` , ` ZZ... |
| nthruz 16268 | The sequence ` NN ` , ` NN... |
| divides 16271 | Define the divides relatio... |
| dvdsval2 16272 | One nonzero integer divide... |
| dvdsval3 16273 | One nonzero integer divide... |
| dvdszrcl 16274 | Reverse closure for the di... |
| dvdsmod0 16275 | If a positive integer divi... |
| p1modz1 16276 | If a number greater than 1... |
| dvdsmodexp 16277 | If a positive integer divi... |
| nndivdvds 16278 | Strong form of ~ dvdsval2 ... |
| nndivides 16279 | Definition of the divides ... |
| moddvds 16280 | Two ways to say ` A == B `... |
| modm1div 16281 | An integer greater than on... |
| dvds0lem 16282 | A lemma to assist theorems... |
| dvds1lem 16283 | A lemma to assist theorems... |
| dvds2lem 16284 | A lemma to assist theorems... |
| iddvds 16285 | An integer divides itself.... |
| 1dvds 16286 | 1 divides any integer. Th... |
| dvds0 16287 | Any integer divides 0. Th... |
| negdvdsb 16288 | An integer divides another... |
| dvdsnegb 16289 | An integer divides another... |
| absdvdsb 16290 | An integer divides another... |
| dvdsabsb 16291 | An integer divides another... |
| 0dvds 16292 | Only 0 is divisible by 0. ... |
| dvdsmul1 16293 | An integer divides a multi... |
| dvdsmul2 16294 | An integer divides a multi... |
| iddvdsexp 16295 | An integer divides a posit... |
| muldvds1 16296 | If a product divides an in... |
| muldvds2 16297 | If a product divides an in... |
| dvdscmul 16298 | Multiplication by a consta... |
| dvdsmulc 16299 | Multiplication by a consta... |
| dvdscmulr 16300 | Cancellation law for the d... |
| dvdsmulcr 16301 | Cancellation law for the d... |
| summodnegmod 16302 | The sum of two integers mo... |
| modmulconst 16303 | Constant multiplication in... |
| dvds2ln 16304 | If an integer divides each... |
| dvds2add 16305 | If an integer divides each... |
| dvds2sub 16306 | If an integer divides each... |
| dvds2addd 16307 | Deduction form of ~ dvds2a... |
| dvds2subd 16308 | Deduction form of ~ dvds2s... |
| dvdstr 16309 | The divides relation is tr... |
| dvdstrd 16310 | The divides relation is tr... |
| dvdsmultr1 16311 | If an integer divides anot... |
| dvdsmultr1d 16312 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16313 | If an integer divides anot... |
| dvdsmultr2d 16314 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16315 | If an integer divides eith... |
| dvdssub2 16316 | If an integer divides a di... |
| dvdsadd 16317 | An integer divides another... |
| dvdsaddr 16318 | An integer divides another... |
| dvdssub 16319 | An integer divides another... |
| dvdssubr 16320 | An integer divides another... |
| dvdsadd2b 16321 | Adding a multiple of the b... |
| dvdsaddre2b 16322 | Adding a multiple of the b... |
| fsumdvds 16323 | If every term in a sum is ... |
| dvdslelem 16324 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16325 | The divisors of a positive... |
| dvdsleabs 16326 | The divisors of a nonzero ... |
| dvdsleabs2 16327 | Transfer divisibility to a... |
| dvdsabseq 16328 | If two integers divide eac... |
| dvdseq 16329 | If two nonnegative integer... |
| divconjdvds 16330 | If a nonzero integer ` M `... |
| dvdsdivcl 16331 | The complement of a diviso... |
| dvdsflip 16332 | An involution of the divis... |
| dvdsssfz1 16333 | The set of divisors of a n... |
| dvds1 16334 | The only nonnegative integ... |
| alzdvds 16335 | Only 0 is divisible by all... |
| dvdsext 16336 | Poset extensionality for d... |
| fzm1ndvds 16337 | No number between ` 1 ` an... |
| fzo0dvdseq 16338 | Zero is the only one of th... |
| fzocongeq 16339 | Two different elements of ... |
| addmodlteqALT 16340 | Two nonnegative integers l... |
| dvdsfac 16341 | A positive integer divides... |
| dvdsexp2im 16342 | If an integer divides anot... |
| dvdsexp 16343 | A power divides a power wi... |
| dvdsmod 16344 | Any number ` K ` whose mod... |
| mulmoddvds 16345 | If an integer is divisible... |
| 3dvds 16346 | A rule for divisibility by... |
| 3dvdsdec 16347 | A decimal number is divisi... |
| 3dvds2dec 16348 | A decimal number is divisi... |
| fprodfvdvdsd 16349 | A finite product of intege... |
| fproddvdsd 16350 | A finite product of intege... |
| evenelz 16351 | An even number is an integ... |
| zeo3 16352 | An integer is even or odd.... |
| zeo4 16353 | An integer is even or odd ... |
| zeneo 16354 | No even integer equals an ... |
| odd2np1lem 16355 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16356 | An integer is odd iff it i... |
| even2n 16357 | An integer is even iff it ... |
| oddm1even 16358 | An integer is odd iff its ... |
| oddp1even 16359 | An integer is odd iff its ... |
| oexpneg 16360 | The exponential of the neg... |
| mod2eq0even 16361 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16362 | An integer is 1 modulo 2 i... |
| oddnn02np1 16363 | A nonnegative integer is o... |
| oddge22np1 16364 | An integer greater than on... |
| evennn02n 16365 | A nonnegative integer is e... |
| evennn2n 16366 | A positive integer is even... |
| 2tp1odd 16367 | A number which is twice an... |
| mulsucdiv2z 16368 | An integer multiplied with... |
| sqoddm1div8z 16369 | A squared odd number minus... |
| 2teven 16370 | A number which is twice an... |
| zeo5 16371 | An integer is either even ... |
| evend2 16372 | An integer is even iff its... |
| oddp1d2 16373 | An integer is odd iff its ... |
| zob 16374 | Alternate characterization... |
| oddm1d2 16375 | An integer is odd iff its ... |
| ltoddhalfle 16376 | An integer is less than ha... |
| halfleoddlt 16377 | An integer is greater than... |
| opoe 16378 | The sum of two odds is eve... |
| omoe 16379 | The difference of two odds... |
| opeo 16380 | The sum of an odd and an e... |
| omeo 16381 | The difference of an odd a... |
| z0even 16382 | 2 divides 0. That means 0... |
| n2dvds1 16383 | 2 does not divide 1. That... |
| n2dvdsm1 16384 | 2 does not divide -1. Tha... |
| z2even 16385 | 2 divides 2. That means 2... |
| n2dvds3 16386 | 2 does not divide 3. That... |
| z4even 16387 | 2 divides 4. That means 4... |
| 4dvdseven 16388 | An integer which is divisi... |
| m1expe 16389 | Exponentiation of -1 by an... |
| m1expo 16390 | Exponentiation of -1 by an... |
| m1exp1 16391 | Exponentiation of negative... |
| nn0enne 16392 | A positive integer is an e... |
| nn0ehalf 16393 | The half of an even nonneg... |
| nnehalf 16394 | The half of an even positi... |
| nn0onn 16395 | An odd nonnegative integer... |
| nn0o1gt2 16396 | An odd nonnegative integer... |
| nno 16397 | An alternate characterizat... |
| nn0o 16398 | An alternate characterizat... |
| nn0ob 16399 | Alternate characterization... |
| nn0oddm1d2 16400 | A positive integer is odd ... |
| nnoddm1d2 16401 | A positive integer is odd ... |
| sumeven 16402 | If every term in a sum is ... |
| sumodd 16403 | If every term in a sum is ... |
| evensumodd 16404 | If every term in a sum wit... |
| oddsumodd 16405 | If every term in a sum wit... |
| pwp1fsum 16406 | The n-th power of a number... |
| oddpwp1fsum 16407 | An odd power of a number i... |
| divalglem0 16408 | Lemma for ~ divalg . (Con... |
| divalglem1 16409 | Lemma for ~ divalg . (Con... |
| divalglem2 16410 | Lemma for ~ divalg . (Con... |
| divalglem4 16411 | Lemma for ~ divalg . (Con... |
| divalglem5 16412 | Lemma for ~ divalg . (Con... |
| divalglem6 16413 | Lemma for ~ divalg . (Con... |
| divalglem7 16414 | Lemma for ~ divalg . (Con... |
| divalglem8 16415 | Lemma for ~ divalg . (Con... |
| divalglem9 16416 | Lemma for ~ divalg . (Con... |
| divalglem10 16417 | Lemma for ~ divalg . (Con... |
| divalg 16418 | The division algorithm (th... |
| divalgb 16419 | Express the division algor... |
| divalg2 16420 | The division algorithm (th... |
| divalgmod 16421 | The result of the ` mod ` ... |
| divalgmodcl 16422 | The result of the ` mod ` ... |
| modremain 16423 | The result of the modulo o... |
| ndvdssub 16424 | Corollary of the division ... |
| ndvdsadd 16425 | Corollary of the division ... |
| ndvdsp1 16426 | Special case of ~ ndvdsadd... |
| ndvdsi 16427 | A quick test for non-divis... |
| flodddiv4 16428 | The floor of an odd intege... |
| fldivndvdslt 16429 | The floor of an integer di... |
| flodddiv4lt 16430 | The floor of an odd number... |
| flodddiv4t2lthalf 16431 | The floor of an odd number... |
| bitsfval 16436 | Expand the definition of t... |
| bitsval 16437 | Expand the definition of t... |
| bitsval2 16438 | Expand the definition of t... |
| bitsss 16439 | The set of bits of an inte... |
| bitsf 16440 | The ` bits ` function is a... |
| bits0 16441 | Value of the zeroth bit. ... |
| bits0e 16442 | The zeroth bit of an even ... |
| bits0o 16443 | The zeroth bit of an odd n... |
| bitsp1 16444 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16445 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16446 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16447 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16448 | The bits of a number are a... |
| bitsmod 16449 | Truncating the bit sequenc... |
| bitsfi 16450 | Every number is associated... |
| bitscmp 16451 | The bit complement of ` N ... |
| 0bits 16452 | The bits of zero. (Contri... |
| m1bits 16453 | The bits of negative one. ... |
| bitsinv1lem 16454 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16455 | There is an explicit inver... |
| bitsinv2 16456 | There is an explicit inver... |
| bitsf1ocnv 16457 | The ` bits ` function rest... |
| bitsf1o 16458 | The ` bits ` function rest... |
| bitsf1 16459 | The ` bits ` function is a... |
| 2ebits 16460 | The bits of a power of two... |
| bitsinv 16461 | The inverse of the ` bits ... |
| bitsinvp1 16462 | Recursive definition of th... |
| sadadd2lem2 16463 | The core of the proof of ~... |
| sadfval 16465 | Define the addition of two... |
| sadcf 16466 | The carry sequence is a se... |
| sadc0 16467 | The initial element of the... |
| sadcp1 16468 | The carry sequence (which ... |
| sadval 16469 | The full adder sequence is... |
| sadcaddlem 16470 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16471 | Non-recursive definition o... |
| sadadd2lem 16472 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16473 | Sum of initial segments of... |
| sadadd3 16474 | Sum of initial segments of... |
| sadcl 16475 | The sum of two sequences i... |
| sadcom 16476 | The adder sequence functio... |
| saddisjlem 16477 | Lemma for ~ sadadd . (Con... |
| saddisj 16478 | The sum of disjoint sequen... |
| sadaddlem 16479 | Lemma for ~ sadadd . (Con... |
| sadadd 16480 | For sequences that corresp... |
| sadid1 16481 | The adder sequence functio... |
| sadid2 16482 | The adder sequence functio... |
| sadasslem 16483 | Lemma for ~ sadass . (Con... |
| sadass 16484 | Sequence addition is assoc... |
| sadeq 16485 | Any element of a sequence ... |
| bitsres 16486 | Restrict the bits of a num... |
| bitsuz 16487 | The bits of a number are a... |
| bitsshft 16488 | Shifting a bit sequence to... |
| smufval 16490 | The multiplication of two ... |
| smupf 16491 | The sequence of partial su... |
| smup0 16492 | The initial element of the... |
| smupp1 16493 | The initial element of the... |
| smuval 16494 | Define the addition of two... |
| smuval2 16495 | The partial sum sequence s... |
| smupvallem 16496 | If ` A ` only has elements... |
| smucl 16497 | The product of two sequenc... |
| smu01lem 16498 | Lemma for ~ smu01 and ~ sm... |
| smu01 16499 | Multiplication of a sequen... |
| smu02 16500 | Multiplication of a sequen... |
| smupval 16501 | Rewrite the elements of th... |
| smup1 16502 | Rewrite ~ smupp1 using onl... |
| smueqlem 16503 | Any element of a sequence ... |
| smueq 16504 | Any element of a sequence ... |
| smumullem 16505 | Lemma for ~ smumul . (Con... |
| smumul 16506 | For sequences that corresp... |
| gcdval 16509 | The value of the ` gcd ` o... |
| gcd0val 16510 | The value, by convention, ... |
| gcdn0val 16511 | The value of the ` gcd ` o... |
| gcdcllem1 16512 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16513 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16514 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16515 | Closure of the ` gcd ` ope... |
| gcddvds 16516 | The gcd of two integers di... |
| dvdslegcd 16517 | An integer which divides b... |
| nndvdslegcd 16518 | A positive integer which d... |
| gcdcl 16519 | Closure of the ` gcd ` ope... |
| gcdnncl 16520 | Closure of the ` gcd ` ope... |
| gcdcld 16521 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16522 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16523 | An integer is the product ... |
| divgcdz 16524 | An integer divided by the ... |
| gcdf 16525 | Domain and codomain of the... |
| gcdcom 16526 | The ` gcd ` operator is co... |
| gcdcomd 16527 | The ` gcd ` operator is co... |
| divgcdnn 16528 | A positive integer divided... |
| divgcdnnr 16529 | A positive integer divided... |
| gcdeq0 16530 | The gcd of two integers is... |
| gcdn0gt0 16531 | The gcd of two integers is... |
| gcd0id 16532 | The gcd of 0 and an intege... |
| gcdid0 16533 | The gcd of an integer and ... |
| nn0gcdid0 16534 | The gcd of a nonnegative i... |
| gcdneg 16535 | Negating one operand of th... |
| neggcd 16536 | Negating one operand of th... |
| gcdaddmlem 16537 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16538 | Adding a multiple of one o... |
| gcdadd 16539 | The GCD of two numbers is ... |
| gcdid 16540 | The gcd of a number and it... |
| gcd1 16541 | The gcd of a number with 1... |
| gcdabs1 16542 | ` gcd ` of the absolute va... |
| gcdabs2 16543 | ` gcd ` of the absolute va... |
| gcdabs 16544 | The gcd of two integers is... |
| gcdabsOLD 16545 | Obsolete version of ~ gcda... |
| modgcd 16546 | The gcd remains unchanged ... |
| 1gcd 16547 | The GCD of one and an inte... |
| gcdmultipled 16548 | The greatest common diviso... |
| gcdmultiplez 16549 | The GCD of a multiple of a... |
| gcdmultiple 16550 | The GCD of a multiple of a... |
| dvdsgcdidd 16551 | The greatest common diviso... |
| 6gcd4e2 16552 | The greatest common diviso... |
| bezoutlem1 16553 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16554 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16555 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16556 | Lemma for ~ bezout . (Con... |
| bezout 16557 | Bézout's identity: ... |
| dvdsgcd 16558 | An integer which divides e... |
| dvdsgcdb 16559 | Biconditional form of ~ dv... |
| dfgcd2 16560 | Alternate definition of th... |
| gcdass 16561 | Associative law for ` gcd ... |
| mulgcd 16562 | Distribute multiplication ... |
| absmulgcd 16563 | Distribute absolute value ... |
| mulgcdr 16564 | Reverse distribution law f... |
| gcddiv 16565 | Division law for GCD. (Con... |
| gcdzeq 16566 | A positive integer ` A ` i... |
| gcdeq 16567 | ` A ` is equal to its gcd ... |
| dvdssqim 16568 | Unidirectional form of ~ d... |
| dvdsexpim 16569 | If two numbers are divisib... |
| dvdsmulgcd 16570 | A divisibility equivalent ... |
| rpmulgcd 16571 | If ` K ` and ` M ` are rel... |
| rplpwr 16572 | If ` A ` and ` B ` are rel... |
| rprpwr 16573 | If ` A ` and ` B ` are rel... |
| rppwr 16574 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16575 | If ` A ` and ` B ` are rel... |
| sqgcd 16576 | Square distributes over gc... |
| expgcd 16577 | Exponentiation distributes... |
| nn0expgcd 16578 | Exponentiation distributes... |
| zexpgcd 16579 | Exponentiation distributes... |
| dvdssqlem 16580 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16581 | Two numbers are divisible ... |
| bezoutr 16582 | Partial converse to ~ bezo... |
| bezoutr1 16583 | Converse of ~ bezout for w... |
| nn0seqcvgd 16584 | A strictly-decreasing nonn... |
| seq1st 16585 | A sequence whose iteration... |
| algr0 16586 | The value of the algorithm... |
| algrf 16587 | An algorithm is a step fun... |
| algrp1 16588 | The value of the algorithm... |
| alginv 16589 | If ` I ` is an invariant o... |
| algcvg 16590 | One way to prove that an a... |
| algcvgblem 16591 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16592 | Two ways of expressing tha... |
| algcvga 16593 | The countdown function ` C... |
| algfx 16594 | If ` F ` reaches a fixed p... |
| eucalgval2 16595 | The value of the step func... |
| eucalgval 16596 | Euclid's Algorithm ~ eucal... |
| eucalgf 16597 | Domain and codomain of the... |
| eucalginv 16598 | The invariant of the step ... |
| eucalglt 16599 | The second member of the s... |
| eucalgcvga 16600 | Once Euclid's Algorithm ha... |
| eucalg 16601 | Euclid's Algorithm compute... |
| lcmval 16606 | Value of the ` lcm ` opera... |
| lcmcom 16607 | The ` lcm ` operator is co... |
| lcm0val 16608 | The value, by convention, ... |
| lcmn0val 16609 | The value of the ` lcm ` o... |
| lcmcllem 16610 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16611 | Closure of the ` lcm ` ope... |
| dvdslcm 16612 | The lcm of two integers is... |
| lcmledvds 16613 | A positive integer which b... |
| lcmeq0 16614 | The lcm of two integers is... |
| lcmcl 16615 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16616 | The greatest common diviso... |
| lcmneg 16617 | Negating one operand of th... |
| neglcm 16618 | Negating one operand of th... |
| lcmabs 16619 | The lcm of two integers is... |
| lcmgcdlem 16620 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16621 | The product of two numbers... |
| lcmdvds 16622 | The lcm of two integers di... |
| lcmid 16623 | The lcm of an integer and ... |
| lcm1 16624 | The lcm of an integer and ... |
| lcmgcdnn 16625 | The product of two positiv... |
| lcmgcdeq 16626 | Two integers' absolute val... |
| lcmdvdsb 16627 | Biconditional form of ~ lc... |
| lcmass 16628 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16629 | The least common multiple ... |
| 6lcm4e12 16630 | The least common multiple ... |
| absproddvds 16631 | The absolute value of the ... |
| absprodnn 16632 | The absolute value of the ... |
| fissn0dvds 16633 | For each finite subset of ... |
| fissn0dvdsn0 16634 | For each finite subset of ... |
| lcmfval 16635 | Value of the ` _lcm ` func... |
| lcmf0val 16636 | The value, by convention, ... |
| lcmfn0val 16637 | The value of the ` _lcm ` ... |
| lcmfnnval 16638 | The value of the ` _lcm ` ... |
| lcmfcllem 16639 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16640 | Closure of the ` _lcm ` fu... |
| lcmfpr 16641 | The value of the ` _lcm ` ... |
| lcmfcl 16642 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16643 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16644 | The least common multiple ... |
| dvdslcmf 16645 | The least common multiple ... |
| lcmfledvds 16646 | A positive integer which i... |
| lcmf 16647 | Characterization of the le... |
| lcmf0 16648 | The least common multiple ... |
| lcmfsn 16649 | The least common multiple ... |
| lcmftp 16650 | The least common multiple ... |
| lcmfunsnlem1 16651 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16652 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16653 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16654 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16655 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16656 | The least common multiple ... |
| lcmfdvdsb 16657 | Biconditional form of ~ lc... |
| lcmfunsn 16658 | The ` _lcm ` function for ... |
| lcmfun 16659 | The ` _lcm ` function for ... |
| lcmfass 16660 | Associative law for the ` ... |
| lcmf2a3a4e12 16661 | The least common multiple ... |
| lcmflefac 16662 | The least common multiple ... |
| coprmgcdb 16663 | Two positive integers are ... |
| ncoprmgcdne1b 16664 | Two positive integers are ... |
| ncoprmgcdgt1b 16665 | Two positive integers are ... |
| coprmdvds1 16666 | If two positive integers a... |
| coprmdvds 16667 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16668 | If an integer is divisible... |
| mulgcddvds 16669 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16670 | If ` M ` is relatively pri... |
| qredeq 16671 | Two equal reduced fraction... |
| qredeu 16672 | Every rational number has ... |
| rpmul 16673 | If ` K ` is relatively pri... |
| rpdvds 16674 | If ` K ` is relatively pri... |
| coprmprod 16675 | The product of the element... |
| coprmproddvdslem 16676 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16677 | If a positive integer is d... |
| congr 16678 | Definition of congruence b... |
| divgcdcoprm0 16679 | Integers divided by gcd ar... |
| divgcdcoprmex 16680 | Integers divided by gcd ar... |
| cncongr1 16681 | One direction of the bicon... |
| cncongr2 16682 | The other direction of the... |
| cncongr 16683 | Cancellability of Congruen... |
| cncongrcoprm 16684 | Corollary 1 of Cancellabil... |
| isprm 16687 | The predicate "is a prime ... |
| prmnn 16688 | A prime number is a positi... |
| prmz 16689 | A prime number is an integ... |
| prmssnn 16690 | The prime numbers are a su... |
| prmex 16691 | The set of prime numbers e... |
| 0nprm 16692 | 0 is not a prime number. ... |
| 1nprm 16693 | 1 is not a prime number. ... |
| 1idssfct 16694 | The positive divisors of a... |
| isprm2lem 16695 | Lemma for ~ isprm2 . (Con... |
| isprm2 16696 | The predicate "is a prime ... |
| isprm3 16697 | The predicate "is a prime ... |
| isprm4 16698 | The predicate "is a prime ... |
| prmind2 16699 | A variation on ~ prmind as... |
| prmind 16700 | Perform induction over the... |
| dvdsprime 16701 | If ` M ` divides a prime, ... |
| nprm 16702 | A product of two integers ... |
| nprmi 16703 | An inference for composite... |
| dvdsnprmd 16704 | If a number is divisible b... |
| prm2orodd 16705 | A prime number is either 2... |
| 2prm 16706 | 2 is a prime number. (Con... |
| 2mulprm 16707 | A multiple of two is prime... |
| 3prm 16708 | 3 is a prime number. (Con... |
| 4nprm 16709 | 4 is not a prime number. ... |
| prmuz2 16710 | A prime number is an integ... |
| prmgt1 16711 | A prime number is an integ... |
| prmm2nn0 16712 | Subtracting 2 from a prime... |
| oddprmgt2 16713 | An odd prime is greater th... |
| oddprmge3 16714 | An odd prime is greater th... |
| ge2nprmge4 16715 | A composite integer greate... |
| sqnprm 16716 | A square is never prime. ... |
| dvdsprm 16717 | An integer greater than or... |
| exprmfct 16718 | Every integer greater than... |
| prmdvdsfz 16719 | Each integer greater than ... |
| nprmdvds1 16720 | No prime number divides 1.... |
| isprm5 16721 | One need only check prime ... |
| isprm7 16722 | One need only check prime ... |
| maxprmfct 16723 | The set of prime factors o... |
| divgcdodd 16724 | Either ` A / ( A gcd B ) `... |
| coprm 16725 | A prime number either divi... |
| prmrp 16726 | Unequal prime numbers are ... |
| euclemma 16727 | Euclid's lemma. A prime n... |
| isprm6 16728 | A number is prime iff it s... |
| prmdvdsexp 16729 | A prime divides a positive... |
| prmdvdsexpb 16730 | A prime divides a positive... |
| prmdvdsexpr 16731 | If a prime divides a nonne... |
| prmdvdssq 16732 | Condition for a prime divi... |
| prmdvdssqOLD 16733 | Obsolete version of ~ prmd... |
| prmexpb 16734 | Two positive prime powers ... |
| prmfac1 16735 | The factorial of a number ... |
| dvdszzq 16736 | Divisibility for an intege... |
| rpexp 16737 | If two numbers ` A ` and `... |
| rpexp1i 16738 | Relative primality passes ... |
| rpexp12i 16739 | Relative primality passes ... |
| prmndvdsfaclt 16740 | A prime number does not di... |
| prmdvdsbc 16741 | Condition for a prime numb... |
| prmdvdsncoprmbd 16742 | Two positive integers are ... |
| ncoprmlnprm 16743 | If two positive integers a... |
| cncongrprm 16744 | Corollary 2 of Cancellabil... |
| isevengcd2 16745 | The predicate "is an even ... |
| isoddgcd1 16746 | The predicate "is an odd n... |
| 3lcm2e6 16747 | The least common multiple ... |
| qnumval 16752 | Value of the canonical num... |
| qdenval 16753 | Value of the canonical den... |
| qnumdencl 16754 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16755 | The canonical numerator of... |
| qdencl 16756 | The canonical denominator ... |
| fnum 16757 | Canonical numerator define... |
| fden 16758 | Canonical denominator defi... |
| qnumdenbi 16759 | Two numbers are the canoni... |
| qnumdencoprm 16760 | The canonical representati... |
| qeqnumdivden 16761 | Recover a rational number ... |
| qmuldeneqnum 16762 | Multiplying a rational by ... |
| divnumden 16763 | Calculate the reduced form... |
| divdenle 16764 | Reducing a quotient never ... |
| qnumgt0 16765 | A rational is positive iff... |
| qgt0numnn 16766 | A rational is positive iff... |
| nn0gcdsq 16767 | Squaring commutes with GCD... |
| zgcdsq 16768 | ~ nn0gcdsq extended to int... |
| numdensq 16769 | Squaring a rational square... |
| numsq 16770 | Square commutes with canon... |
| densq 16771 | Square commutes with canon... |
| qden1elz 16772 | A rational is an integer i... |
| zsqrtelqelz 16773 | If an integer has a ration... |
| nonsq 16774 | Any integer strictly betwe... |
| numdenexp 16775 | Elevating a rational numbe... |
| numexp 16776 | Elevating to a nonnegative... |
| denexp 16777 | Elevating to a nonnegative... |
| phival 16782 | Value of the Euler ` phi `... |
| phicl2 16783 | Bounds and closure for the... |
| phicl 16784 | Closure for the value of t... |
| phibndlem 16785 | Lemma for ~ phibnd . (Con... |
| phibnd 16786 | A slightly tighter bound o... |
| phicld 16787 | Closure for the value of t... |
| phi1 16788 | Value of the Euler ` phi `... |
| dfphi2 16789 | Alternate definition of th... |
| hashdvds 16790 | The number of numbers in a... |
| phiprmpw 16791 | Value of the Euler ` phi `... |
| phiprm 16792 | Value of the Euler ` phi `... |
| crth 16793 | The Chinese Remainder Theo... |
| phimullem 16794 | Lemma for ~ phimul . (Con... |
| phimul 16795 | The Euler ` phi ` function... |
| eulerthlem1 16796 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16797 | Lemma for ~ eulerth . (Co... |
| eulerth 16798 | Euler's theorem, a general... |
| fermltl 16799 | Fermat's little theorem. ... |
| prmdiv 16800 | Show an explicit expressio... |
| prmdiveq 16801 | The modular inverse of ` A... |
| prmdivdiv 16802 | The (modular) inverse of t... |
| hashgcdlem 16803 | A correspondence between e... |
| hashgcdeq 16804 | Number of initial positive... |
| phisum 16805 | The divisor sum identity o... |
| odzval 16806 | Value of the order functio... |
| odzcllem 16807 | - Lemma for ~ odzcl , show... |
| odzcl 16808 | The order of a group eleme... |
| odzid 16809 | Any element raised to the ... |
| odzdvds 16810 | The only powers of ` A ` t... |
| odzphi 16811 | The order of any group ele... |
| modprm1div 16812 | A prime number divides an ... |
| m1dvdsndvds 16813 | If an integer minus 1 is d... |
| modprminv 16814 | Show an explicit expressio... |
| modprminveq 16815 | The modular inverse of ` A... |
| vfermltl 16816 | Variant of Fermat's little... |
| vfermltlALT 16817 | Alternate proof of ~ vferm... |
| powm2modprm 16818 | If an integer minus 1 is d... |
| reumodprminv 16819 | For any prime number and f... |
| modprm0 16820 | For two positive integers ... |
| nnnn0modprm0 16821 | For a positive integer and... |
| modprmn0modprm0 16822 | For an integer not being 0... |
| coprimeprodsq 16823 | If three numbers are copri... |
| coprimeprodsq2 16824 | If three numbers are copri... |
| oddprm 16825 | A prime not equal to ` 2 `... |
| nnoddn2prm 16826 | A prime not equal to ` 2 `... |
| oddn2prm 16827 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16828 | A number is a prime number... |
| prm23lt5 16829 | A prime less than 5 is eit... |
| prm23ge5 16830 | A prime is either 2 or 3 o... |
| pythagtriplem1 16831 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16832 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16833 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16834 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16835 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16836 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16837 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16838 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16839 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16840 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16841 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16842 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16843 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16844 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16845 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16846 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16847 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16848 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16849 | Parameterize the Pythagore... |
| iserodd 16850 | Collect the odd terms in a... |
| pclem 16853 | - Lemma for the prime powe... |
| pcprecl 16854 | Closure of the prime power... |
| pcprendvds 16855 | Non-divisibility property ... |
| pcprendvds2 16856 | Non-divisibility property ... |
| pcpre1 16857 | Value of the prime power p... |
| pcpremul 16858 | Multiplicative property of... |
| pcval 16859 | The value of the prime pow... |
| pceulem 16860 | Lemma for ~ pceu . (Contr... |
| pceu 16861 | Uniqueness for the prime p... |
| pczpre 16862 | Connect the prime count pr... |
| pczcl 16863 | Closure of the prime power... |
| pccl 16864 | Closure of the prime power... |
| pccld 16865 | Closure of the prime power... |
| pcmul 16866 | Multiplication property of... |
| pcdiv 16867 | Division property of the p... |
| pcqmul 16868 | Multiplication property of... |
| pc0 16869 | The value of the prime pow... |
| pc1 16870 | Value of the prime count f... |
| pcqcl 16871 | Closure of the general pri... |
| pcqdiv 16872 | Division property of the p... |
| pcrec 16873 | Prime power of a reciproca... |
| pcexp 16874 | Prime power of an exponent... |
| pcxnn0cl 16875 | Extended nonnegative integ... |
| pcxcl 16876 | Extended real closure of t... |
| pcge0 16877 | The prime count of an inte... |
| pczdvds 16878 | Defining property of the p... |
| pcdvds 16879 | Defining property of the p... |
| pczndvds 16880 | Defining property of the p... |
| pcndvds 16881 | Defining property of the p... |
| pczndvds2 16882 | The remainder after dividi... |
| pcndvds2 16883 | The remainder after dividi... |
| pcdvdsb 16884 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16885 | There are a positive numbe... |
| pceq0 16886 | There are zero powers of a... |
| pcidlem 16887 | The prime count of a prime... |
| pcid 16888 | The prime count of a prime... |
| pcneg 16889 | The prime count of a negat... |
| pcabs 16890 | The prime count of an abso... |
| pcdvdstr 16891 | The prime count increases ... |
| pcgcd1 16892 | The prime count of a GCD i... |
| pcgcd 16893 | The prime count of a GCD i... |
| pc2dvds 16894 | A characterization of divi... |
| pc11 16895 | The prime count function, ... |
| pcz 16896 | The prime count function c... |
| pcprmpw2 16897 | Self-referential expressio... |
| pcprmpw 16898 | Self-referential expressio... |
| dvdsprmpweq 16899 | If a positive integer divi... |
| dvdsprmpweqnn 16900 | If an integer greater than... |
| dvdsprmpweqle 16901 | If a positive integer divi... |
| difsqpwdvds 16902 | If the difference of two s... |
| pcaddlem 16903 | Lemma for ~ pcadd . The o... |
| pcadd 16904 | An inequality for the prim... |
| pcadd2 16905 | The inequality of ~ pcadd ... |
| pcmptcl 16906 | Closure for the prime powe... |
| pcmpt 16907 | Construct a function with ... |
| pcmpt2 16908 | Dividing two prime count m... |
| pcmptdvds 16909 | The partial products of th... |
| pcprod 16910 | The product of the primes ... |
| sumhash 16911 | The sum of 1 over a set is... |
| fldivp1 16912 | The difference between the... |
| pcfaclem 16913 | Lemma for ~ pcfac . (Cont... |
| pcfac 16914 | Calculate the prime count ... |
| pcbc 16915 | Calculate the prime count ... |
| qexpz 16916 | If a power of a rational n... |
| expnprm 16917 | A second or higher power o... |
| oddprmdvds 16918 | Every positive integer whi... |
| prmpwdvds 16919 | A relation involving divis... |
| pockthlem 16920 | Lemma for ~ pockthg . (Co... |
| pockthg 16921 | The generalized Pocklingto... |
| pockthi 16922 | Pocklington's theorem, whi... |
| unbenlem 16923 | Lemma for ~ unben . (Cont... |
| unben 16924 | An unbounded set of positi... |
| infpnlem1 16925 | Lemma for ~ infpn . The s... |
| infpnlem2 16926 | Lemma for ~ infpn . For a... |
| infpn 16927 | There exist infinitely man... |
| infpn2 16928 | There exist infinitely man... |
| prmunb 16929 | The primes are unbounded. ... |
| prminf 16930 | There are an infinite numb... |
| prmreclem1 16931 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16932 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16933 | Lemma for ~ prmrec . The ... |
| prmreclem4 16934 | Lemma for ~ prmrec . Show... |
| prmreclem5 16935 | Lemma for ~ prmrec . Here... |
| prmreclem6 16936 | Lemma for ~ prmrec . If t... |
| prmrec 16937 | The sum of the reciprocals... |
| 1arithlem1 16938 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16939 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16940 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16941 | Lemma for ~ 1arith . (Con... |
| 1arith 16942 | Fundamental theorem of ari... |
| 1arith2 16943 | Fundamental theorem of ari... |
| elgz 16946 | Elementhood in the gaussia... |
| gzcn 16947 | A gaussian integer is a co... |
| zgz 16948 | An integer is a gaussian i... |
| igz 16949 | ` _i ` is a gaussian integ... |
| gznegcl 16950 | The gaussian integers are ... |
| gzcjcl 16951 | The gaussian integers are ... |
| gzaddcl 16952 | The gaussian integers are ... |
| gzmulcl 16953 | The gaussian integers are ... |
| gzreim 16954 | Construct a gaussian integ... |
| gzsubcl 16955 | The gaussian integers are ... |
| gzabssqcl 16956 | The squared norm of a gaus... |
| 4sqlem5 16957 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16958 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16959 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16960 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16961 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16962 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16963 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16964 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16965 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16966 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16967 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16968 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16969 | Euler's four-square identi... |
| 4sqlem11 16970 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16971 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16972 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16973 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16974 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16975 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16976 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16977 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16978 | Lemma for ~ 4sq . The pro... |
| 4sq 16979 | Lagrange's four-square the... |
| vdwapfval 16986 | Define the arithmetic prog... |
| vdwapf 16987 | The arithmetic progression... |
| vdwapval 16988 | Value of the arithmetic pr... |
| vdwapun 16989 | Remove the first element o... |
| vdwapid1 16990 | The first element of an ar... |
| vdwap0 16991 | Value of a length-1 arithm... |
| vdwap1 16992 | Value of a length-1 arithm... |
| vdwmc 16993 | The predicate " The ` <. R... |
| vdwmc2 16994 | Expand out the definition ... |
| vdwpc 16995 | The predicate " The colori... |
| vdwlem1 16996 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16997 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16998 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16999 | Lemma for ~ vdw . (Contri... |
| vdwlem5 17000 | Lemma for ~ vdw . (Contri... |
| vdwlem6 17001 | Lemma for ~ vdw . (Contri... |
| vdwlem7 17002 | Lemma for ~ vdw . (Contri... |
| vdwlem8 17003 | Lemma for ~ vdw . (Contri... |
| vdwlem9 17004 | Lemma for ~ vdw . (Contri... |
| vdwlem10 17005 | Lemma for ~ vdw . Set up ... |
| vdwlem11 17006 | Lemma for ~ vdw . (Contri... |
| vdwlem12 17007 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 17008 | Lemma for ~ vdw . Main in... |
| vdw 17009 | Van der Waerden's theorem.... |
| vdwnnlem1 17010 | Corollary of ~ vdw , and l... |
| vdwnnlem2 17011 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 17012 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 17013 | Van der Waerden's theorem,... |
| ramtlecl 17015 | The set ` T ` of numbers w... |
| hashbcval 17017 | Value of the "binomial set... |
| hashbccl 17018 | The binomial set is a fini... |
| hashbcss 17019 | Subset relation for the bi... |
| hashbc0 17020 | The set of subsets of size... |
| hashbc2 17021 | The size of the binomial s... |
| 0hashbc 17022 | There are no subsets of th... |
| ramval 17023 | The value of the Ramsey nu... |
| ramcl2lem 17024 | Lemma for extended real cl... |
| ramtcl 17025 | The Ramsey number has the ... |
| ramtcl2 17026 | The Ramsey number is an in... |
| ramtub 17027 | The Ramsey number is a low... |
| ramub 17028 | The Ramsey number is a low... |
| ramub2 17029 | It is sufficient to check ... |
| rami 17030 | The defining property of a... |
| ramcl2 17031 | The Ramsey number is eithe... |
| ramxrcl 17032 | The Ramsey number is an ex... |
| ramubcl 17033 | If the Ramsey number is up... |
| ramlb 17034 | Establish a lower bound on... |
| 0ram 17035 | The Ramsey number when ` M... |
| 0ram2 17036 | The Ramsey number when ` M... |
| ram0 17037 | The Ramsey number when ` R... |
| 0ramcl 17038 | Lemma for ~ ramcl : Exist... |
| ramz2 17039 | The Ramsey number when ` F... |
| ramz 17040 | The Ramsey number when ` F... |
| ramub1lem1 17041 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 17042 | Lemma for ~ ramub1 . (Con... |
| ramub1 17043 | Inductive step for Ramsey'... |
| ramcl 17044 | Ramsey's theorem: the Rams... |
| ramsey 17045 | Ramsey's theorem with the ... |
| prmoval 17048 | Value of the primorial fun... |
| prmocl 17049 | Closure of the primorial f... |
| prmone0 17050 | The primorial function is ... |
| prmo0 17051 | The primorial of 0. (Cont... |
| prmo1 17052 | The primorial of 1. (Cont... |
| prmop1 17053 | The primorial of a success... |
| prmonn2 17054 | Value of the primorial fun... |
| prmo2 17055 | The primorial of 2. (Cont... |
| prmo3 17056 | The primorial of 3. (Cont... |
| prmdvdsprmo 17057 | The primorial of a number ... |
| prmdvdsprmop 17058 | The primorial of a number ... |
| fvprmselelfz 17059 | The value of the prime sel... |
| fvprmselgcd1 17060 | The greatest common diviso... |
| prmolefac 17061 | The primorial of a positiv... |
| prmodvdslcmf 17062 | The primorial of a nonnega... |
| prmolelcmf 17063 | The primorial of a positiv... |
| prmgaplem1 17064 | Lemma for ~ prmgap : The ... |
| prmgaplem2 17065 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 17066 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 17067 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 17068 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 17069 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 17070 | Lemma for ~ prmgap : for e... |
| prmgaplem6 17071 | Lemma for ~ prmgap : for e... |
| prmgaplem7 17072 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 17073 | Lemma for ~ prmgap . (Con... |
| prmgap 17074 | The prime gap theorem: for... |
| prmgaplcm 17075 | Alternate proof of ~ prmga... |
| prmgapprmolem 17076 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17077 | Alternate proof of ~ prmga... |
| dec2dvds 17078 | Divisibility by two is obv... |
| dec5dvds 17079 | Divisibility by five is ob... |
| dec5dvds2 17080 | Divisibility by five is ob... |
| dec5nprm 17081 | Divisibility by five is ob... |
| dec2nprm 17082 | Divisibility by two is obv... |
| modxai 17083 | Add exponents in a power m... |
| mod2xi 17084 | Double exponents in a powe... |
| modxp1i 17085 | Add one to an exponent in ... |
| mod2xnegi 17086 | Version of ~ mod2xi with a... |
| modsubi 17087 | Subtract from within a mod... |
| gcdi 17088 | Calculate a GCD via Euclid... |
| gcdmodi 17089 | Calculate a GCD via Euclid... |
| decexp2 17090 | Calculate a power of two. ... |
| numexp0 17091 | Calculate an integer power... |
| numexp1 17092 | Calculate an integer power... |
| numexpp1 17093 | Calculate an integer power... |
| numexp2x 17094 | Double an integer power. ... |
| decsplit0b 17095 | Split a decimal number int... |
| decsplit0 17096 | Split a decimal number int... |
| decsplit1 17097 | Split a decimal number int... |
| decsplit 17098 | Split a decimal number int... |
| karatsuba 17099 | The Karatsuba multiplicati... |
| 2exp4 17100 | Two to the fourth power is... |
| 2exp5 17101 | Two to the fifth power is ... |
| 2exp6 17102 | Two to the sixth power is ... |
| 2exp7 17103 | Two to the seventh power i... |
| 2exp8 17104 | Two to the eighth power is... |
| 2exp11 17105 | Two to the eleventh power ... |
| 2exp16 17106 | Two to the sixteenth power... |
| 3exp3 17107 | Three to the third power i... |
| 2expltfac 17108 | The factorial grows faster... |
| cshwsidrepsw 17109 | If cyclically shifting a w... |
| cshwsidrepswmod0 17110 | If cyclically shifting a w... |
| cshwshashlem1 17111 | If cyclically shifting a w... |
| cshwshashlem2 17112 | If cyclically shifting a w... |
| cshwshashlem3 17113 | If cyclically shifting a w... |
| cshwsdisj 17114 | The singletons resulting b... |
| cshwsiun 17115 | The set of (different!) wo... |
| cshwsex 17116 | The class of (different!) ... |
| cshws0 17117 | The size of the set of (di... |
| cshwrepswhash1 17118 | The size of the set of (di... |
| cshwshashnsame 17119 | If a word (not consisting ... |
| cshwshash 17120 | If a word has a length bei... |
| prmlem0 17121 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17122 | A quick proof skeleton to ... |
| prmlem1 17123 | A quick proof skeleton to ... |
| 5prm 17124 | 5 is a prime number. (Con... |
| 6nprm 17125 | 6 is not a prime number. ... |
| 7prm 17126 | 7 is a prime number. (Con... |
| 8nprm 17127 | 8 is not a prime number. ... |
| 9nprm 17128 | 9 is not a prime number. ... |
| 10nprm 17129 | 10 is not a prime number. ... |
| 11prm 17130 | 11 is a prime number. (Co... |
| 13prm 17131 | 13 is a prime number. (Co... |
| 17prm 17132 | 17 is a prime number. (Co... |
| 19prm 17133 | 19 is a prime number. (Co... |
| 23prm 17134 | 23 is a prime number. (Co... |
| prmlem2 17135 | Our last proving session g... |
| 37prm 17136 | 37 is a prime number. (Co... |
| 43prm 17137 | 43 is a prime number. (Co... |
| 83prm 17138 | 83 is a prime number. (Co... |
| 139prm 17139 | 139 is a prime number. (C... |
| 163prm 17140 | 163 is a prime number. (C... |
| 317prm 17141 | 317 is a prime number. (C... |
| 631prm 17142 | 631 is a prime number. (C... |
| prmo4 17143 | The primorial of 4. (Cont... |
| prmo5 17144 | The primorial of 5. (Cont... |
| prmo6 17145 | The primorial of 6. (Cont... |
| 1259lem1 17146 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17147 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17148 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17149 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17150 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17151 | 1259 is a prime number. (... |
| 2503lem1 17152 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17153 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17154 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17155 | 2503 is a prime number. (... |
| 4001lem1 17156 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17157 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17158 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17159 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17160 | 4001 is a prime number. (... |
| brstruct 17163 | The structure relation is ... |
| isstruct2 17164 | The property of being a st... |
| structex 17165 | A structure is a set. (Co... |
| structn0fun 17166 | A structure without the em... |
| isstruct 17167 | The property of being a st... |
| structcnvcnv 17168 | Two ways to express the re... |
| structfung 17169 | The converse of the conver... |
| structfun 17170 | Convert between two kinds ... |
| structfn 17171 | Convert between two kinds ... |
| strleun 17172 | Combine two structures int... |
| strle1 17173 | Make a structure from a si... |
| strle2 17174 | Make a structure from a pa... |
| strle3 17175 | Make a structure from a tr... |
| sbcie2s 17176 | A special version of class... |
| sbcie3s 17177 | A special version of class... |
| reldmsets 17180 | The structure override ope... |
| setsvalg 17181 | Value of the structure rep... |
| setsval 17182 | Value of the structure rep... |
| fvsetsid 17183 | The value of the structure... |
| fsets 17184 | The structure replacement ... |
| setsdm 17185 | The domain of a structure ... |
| setsfun 17186 | A structure with replaceme... |
| setsfun0 17187 | A structure with replaceme... |
| setsn0fun 17188 | The value of the structure... |
| setsstruct2 17189 | An extensible structure wi... |
| setsexstruct2 17190 | An extensible structure wi... |
| setsstruct 17191 | An extensible structure wi... |
| wunsets 17192 | Closure of structure repla... |
| setsres 17193 | The structure replacement ... |
| setsabs 17194 | Replacing the same compone... |
| setscom 17195 | Different components can b... |
| sloteq 17198 | Equality theorem for the `... |
| slotfn 17199 | A slot is a function on se... |
| strfvnd 17200 | Deduction version of ~ str... |
| strfvn 17201 | Value of a structure compo... |
| strfvss 17202 | A structure component extr... |
| wunstr 17203 | Closure of a structure ind... |
| str0 17204 | All components of the empt... |
| strfvi 17205 | Structure slot extractors ... |
| fveqprc 17206 | Lemma for showing the equa... |
| oveqprc 17207 | Lemma for showing the equa... |
| wunndx 17210 | Closure of the index extra... |
| ndxarg 17211 | Get the numeric argument f... |
| ndxid 17212 | A structure component extr... |
| strndxid 17213 | The value of a structure c... |
| setsidvald 17214 | Value of the structure rep... |
| setsidvaldOLD 17215 | Obsolete version of ~ sets... |
| strfvd 17216 | Deduction version of ~ str... |
| strfv2d 17217 | Deduction version of ~ str... |
| strfv2 17218 | A variation on ~ strfv to ... |
| strfv 17219 | Extract a structure compon... |
| strfv3 17220 | Variant on ~ strfv for lar... |
| strssd 17221 | Deduction version of ~ str... |
| strss 17222 | Propagate component extrac... |
| setsid 17223 | Value of the structure rep... |
| setsnid 17224 | Value of the structure rep... |
| setsnidOLD 17225 | Obsolete proof of ~ setsni... |
| baseval 17228 | Value of the base set extr... |
| baseid 17229 | Utility theorem: index-ind... |
| basfn 17230 | The base set extractor is ... |
| base0 17231 | The base set of the empty ... |
| elbasfv 17232 | Utility theorem: reverse c... |
| elbasov 17233 | Utility theorem: reverse c... |
| strov2rcl 17234 | Partial reverse closure fo... |
| basendx 17235 | Index value of the base se... |
| basendxnn 17236 | The index value of the bas... |
| basendxnnOLD 17237 | Obsolete proof of ~ basend... |
| basndxelwund 17238 | The index of the base set ... |
| basprssdmsets 17239 | The pair of the base index... |
| opelstrbas 17240 | The base set of a structur... |
| 1strstr 17241 | A constructed one-slot str... |
| 1strstr1 17242 | A constructed one-slot str... |
| 1strbas 17243 | The base set of a construc... |
| 1strbasOLD 17244 | Obsolete proof of ~ 1strba... |
| 1strwunbndx 17245 | A constructed one-slot str... |
| 1strwun 17246 | A constructed one-slot str... |
| 1strwunOLD 17247 | Obsolete version of ~ 1str... |
| 2strstr 17248 | A constructed two-slot str... |
| 2strbas 17249 | The base set of a construc... |
| 2strop 17250 | The other slot of a constr... |
| 2strstr1 17251 | A constructed two-slot str... |
| 2strstr1OLD 17252 | Obsolete version of ~ 2str... |
| 2strbas1 17253 | The base set of a construc... |
| 2strop1 17254 | The other slot of a constr... |
| reldmress 17257 | The structure restriction ... |
| ressval 17258 | Value of structure restric... |
| ressid2 17259 | General behavior of trivia... |
| ressval2 17260 | Value of nontrivial struct... |
| ressbas 17261 | Base set of a structure re... |
| ressbasOLD 17262 | Obsolete proof of ~ ressba... |
| ressbasssg 17263 | The base set of a restrict... |
| ressbas2 17264 | Base set of a structure re... |
| ressbasss 17265 | The base set of a restrict... |
| ressbasssOLD 17266 | Obsolete proof of ~ ressba... |
| ressbasss2 17267 | The base set of a restrict... |
| resseqnbas 17268 | The components of an exten... |
| resslemOLD 17269 | Obsolete version of ~ ress... |
| ress0 17270 | All restrictions of the nu... |
| ressid 17271 | Behavior of trivial restri... |
| ressinbas 17272 | Restriction only cares abo... |
| ressval3d 17273 | Value of structure restric... |
| ressval3dOLD 17274 | Obsolete version of ~ ress... |
| ressress 17275 | Restriction composition la... |
| ressabs 17276 | Restriction absorption law... |
| wunress 17277 | Closure of structure restr... |
| wunressOLD 17278 | Obsolete proof of ~ wunres... |
| plusgndx 17305 | Index value of the ~ df-pl... |
| plusgid 17306 | Utility theorem: index-ind... |
| plusgndxnn 17307 | The index of the slot for ... |
| basendxltplusgndx 17308 | The index of the slot for ... |
| basendxnplusgndx 17309 | The slot for the base set ... |
| basendxnplusgndxOLD 17310 | Obsolete version of ~ base... |
| grpstr 17311 | A constructed group is a s... |
| grpstrndx 17312 | A constructed group is a s... |
| grpbase 17313 | The base set of a construc... |
| grpbaseOLD 17314 | Obsolete version of ~ grpb... |
| grpplusg 17315 | The operation of a constru... |
| grpplusgOLD 17316 | Obsolete version of ~ grpp... |
| ressplusg 17317 | ` +g ` is unaffected by re... |
| grpbasex 17318 | The base of an explicitly ... |
| grpplusgx 17319 | The operation of an explic... |
| mulrndx 17320 | Index value of the ~ df-mu... |
| mulridx 17321 | Utility theorem: index-ind... |
| basendxnmulrndx 17322 | The slot for the base set ... |
| basendxnmulrndxOLD 17323 | Obsolete proof of ~ basend... |
| plusgndxnmulrndx 17324 | The slot for the group (ad... |
| rngstr 17325 | A constructed ring is a st... |
| rngbase 17326 | The base set of a construc... |
| rngplusg 17327 | The additive operation of ... |
| rngmulr 17328 | The multiplicative operati... |
| starvndx 17329 | Index value of the ~ df-st... |
| starvid 17330 | Utility theorem: index-ind... |
| starvndxnbasendx 17331 | The slot for the involutio... |
| starvndxnplusgndx 17332 | The slot for the involutio... |
| starvndxnmulrndx 17333 | The slot for the involutio... |
| ressmulr 17334 | ` .r ` is unaffected by re... |
| ressstarv 17335 | ` *r ` is unaffected by re... |
| srngstr 17336 | A constructed star ring is... |
| srngbase 17337 | The base set of a construc... |
| srngplusg 17338 | The addition operation of ... |
| srngmulr 17339 | The multiplication operati... |
| srnginvl 17340 | The involution function of... |
| scandx 17341 | Index value of the ~ df-sc... |
| scaid 17342 | Utility theorem: index-ind... |
| scandxnbasendx 17343 | The slot for the scalar is... |
| scandxnplusgndx 17344 | The slot for the scalar fi... |
| scandxnmulrndx 17345 | The slot for the scalar fi... |
| vscandx 17346 | Index value of the ~ df-vs... |
| vscaid 17347 | Utility theorem: index-ind... |
| vscandxnbasendx 17348 | The slot for the scalar pr... |
| vscandxnplusgndx 17349 | The slot for the scalar pr... |
| vscandxnmulrndx 17350 | The slot for the scalar pr... |
| vscandxnscandx 17351 | The slot for the scalar pr... |
| lmodstr 17352 | A constructed left module ... |
| lmodbase 17353 | The base set of a construc... |
| lmodplusg 17354 | The additive operation of ... |
| lmodsca 17355 | The set of scalars of a co... |
| lmodvsca 17356 | The scalar product operati... |
| ipndx 17357 | Index value of the ~ df-ip... |
| ipid 17358 | Utility theorem: index-ind... |
| ipndxnbasendx 17359 | The slot for the inner pro... |
| ipndxnplusgndx 17360 | The slot for the inner pro... |
| ipndxnmulrndx 17361 | The slot for the inner pro... |
| slotsdifipndx 17362 | The slot for the scalar is... |
| ipsstr 17363 | Lemma to shorten proofs of... |
| ipsbase 17364 | The base set of a construc... |
| ipsaddg 17365 | The additive operation of ... |
| ipsmulr 17366 | The multiplicative operati... |
| ipssca 17367 | The set of scalars of a co... |
| ipsvsca 17368 | The scalar product operati... |
| ipsip 17369 | The multiplicative operati... |
| resssca 17370 | ` Scalar ` is unaffected b... |
| ressvsca 17371 | ` .s ` is unaffected by re... |
| ressip 17372 | The inner product is unaff... |
| phlstr 17373 | A constructed pre-Hilbert ... |
| phlbase 17374 | The base set of a construc... |
| phlplusg 17375 | The additive operation of ... |
| phlsca 17376 | The ring of scalars of a c... |
| phlvsca 17377 | The scalar product operati... |
| phlip 17378 | The inner product (Hermiti... |
| tsetndx 17379 | Index value of the ~ df-ts... |
| tsetid 17380 | Utility theorem: index-ind... |
| tsetndxnn 17381 | The index of the slot for ... |
| basendxlttsetndx 17382 | The index of the slot for ... |
| tsetndxnbasendx 17383 | The slot for the topology ... |
| tsetndxnplusgndx 17384 | The slot for the topology ... |
| tsetndxnmulrndx 17385 | The slot for the topology ... |
| tsetndxnstarvndx 17386 | The slot for the topology ... |
| slotstnscsi 17387 | The slots ` Scalar ` , ` .... |
| topgrpstr 17388 | A constructed topological ... |
| topgrpbas 17389 | The base set of a construc... |
| topgrpplusg 17390 | The additive operation of ... |
| topgrptset 17391 | The topology of a construc... |
| resstset 17392 | ` TopSet ` is unaffected b... |
| plendx 17393 | Index value of the ~ df-pl... |
| pleid 17394 | Utility theorem: self-refe... |
| plendxnn 17395 | The index value of the ord... |
| basendxltplendx 17396 | The index value of the ` B... |
| plendxnbasendx 17397 | The slot for the order is ... |
| plendxnplusgndx 17398 | The slot for the "less tha... |
| plendxnmulrndx 17399 | The slot for the "less tha... |
| plendxnscandx 17400 | The slot for the "less tha... |
| plendxnvscandx 17401 | The slot for the "less tha... |
| slotsdifplendx 17402 | The index of the slot for ... |
| otpsstr 17403 | Functionality of a topolog... |
| otpsbas 17404 | The base set of a topologi... |
| otpstset 17405 | The open sets of a topolog... |
| otpsle 17406 | The order of a topological... |
| ressle 17407 | ` le ` is unaffected by re... |
| ocndx 17408 | Index value of the ~ df-oc... |
| ocid 17409 | Utility theorem: index-ind... |
| basendxnocndx 17410 | The slot for the orthocomp... |
| plendxnocndx 17411 | The slot for the orthocomp... |
| dsndx 17412 | Index value of the ~ df-ds... |
| dsid 17413 | Utility theorem: index-ind... |
| dsndxnn 17414 | The index of the slot for ... |
| basendxltdsndx 17415 | The index of the slot for ... |
| dsndxnbasendx 17416 | The slot for the distance ... |
| dsndxnplusgndx 17417 | The slot for the distance ... |
| dsndxnmulrndx 17418 | The slot for the distance ... |
| slotsdnscsi 17419 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17420 | The slot for the distance ... |
| slotsdifdsndx 17421 | The index of the slot for ... |
| unifndx 17422 | Index value of the ~ df-un... |
| unifid 17423 | Utility theorem: index-ind... |
| unifndxnn 17424 | The index of the slot for ... |
| basendxltunifndx 17425 | The index of the slot for ... |
| unifndxnbasendx 17426 | The slot for the uniform s... |
| unifndxntsetndx 17427 | The slot for the uniform s... |
| slotsdifunifndx 17428 | The index of the slot for ... |
| ressunif 17429 | ` UnifSet ` is unaffected ... |
| odrngstr 17430 | Functionality of an ordere... |
| odrngbas 17431 | The base set of an ordered... |
| odrngplusg 17432 | The addition operation of ... |
| odrngmulr 17433 | The multiplication operati... |
| odrngtset 17434 | The open sets of an ordere... |
| odrngle 17435 | The order of an ordered me... |
| odrngds 17436 | The metric of an ordered m... |
| ressds 17437 | ` dist ` is unaffected by ... |
| homndx 17438 | Index value of the ~ df-ho... |
| homid 17439 | Utility theorem: index-ind... |
| ccondx 17440 | Index value of the ~ df-cc... |
| ccoid 17441 | Utility theorem: index-ind... |
| slotsbhcdif 17442 | The slots ` Base ` , ` Hom... |
| slotsbhcdifOLD 17443 | Obsolete proof of ~ slotsb... |
| slotsdifplendx2 17444 | The index of the slot for ... |
| slotsdifocndx 17445 | The index of the slot for ... |
| resshom 17446 | ` Hom ` is unaffected by r... |
| ressco 17447 | ` comp ` is unaffected by ... |
| restfn 17452 | The subspace topology oper... |
| topnfn 17453 | The topology extractor fun... |
| restval 17454 | The subspace topology indu... |
| elrest 17455 | The predicate "is an open ... |
| elrestr 17456 | Sufficient condition for b... |
| 0rest 17457 | Value of the structure res... |
| restid2 17458 | The subspace topology over... |
| restsspw 17459 | The subspace topology is a... |
| firest 17460 | The finite intersections o... |
| restid 17461 | The subspace topology of t... |
| topnval 17462 | Value of the topology extr... |
| topnid 17463 | Value of the topology extr... |
| topnpropd 17464 | The topology extractor fun... |
| reldmprds 17476 | The structure product is a... |
| prdsbasex 17478 | Lemma for structure produc... |
| imasvalstr 17479 | An image structure value i... |
| prdsvalstr 17480 | Structure product value is... |
| prdsbaslem 17481 | Lemma for ~ prdsbas and si... |
| prdsvallem 17482 | Lemma for ~ prdsval . (Co... |
| prdsval 17483 | Value of the structure pro... |
| prdssca 17484 | Scalar ring of a structure... |
| prdsbas 17485 | Base set of a structure pr... |
| prdsplusg 17486 | Addition in a structure pr... |
| prdsmulr 17487 | Multiplication in a struct... |
| prdsvsca 17488 | Scalar multiplication in a... |
| prdsip 17489 | Inner product in a structu... |
| prdsle 17490 | Structure product weak ord... |
| prdsless 17491 | Closure of the order relat... |
| prdsds 17492 | Structure product distance... |
| prdsdsfn 17493 | Structure product distance... |
| prdstset 17494 | Structure product topology... |
| prdshom 17495 | Structure product hom-sets... |
| prdsco 17496 | Structure product composit... |
| prdsbas2 17497 | The base set of a structur... |
| prdsbasmpt 17498 | A constructed tuple is a p... |
| prdsbasfn 17499 | Points in the structure pr... |
| prdsbasprj 17500 | Each point in a structure ... |
| prdsplusgval 17501 | Value of a componentwise s... |
| prdsplusgfval 17502 | Value of a structure produ... |
| prdsmulrval 17503 | Value of a componentwise r... |
| prdsmulrfval 17504 | Value of a structure produ... |
| prdsleval 17505 | Value of the product order... |
| prdsdsval 17506 | Value of the metric in a s... |
| prdsvscaval 17507 | Scalar multiplication in a... |
| prdsvscafval 17508 | Scalar multiplication of a... |
| prdsbas3 17509 | The base set of an indexed... |
| prdsbasmpt2 17510 | A constructed tuple is a p... |
| prdsbascl 17511 | An element of the base has... |
| prdsdsval2 17512 | Value of the metric in a s... |
| prdsdsval3 17513 | Value of the metric in a s... |
| pwsval 17514 | Value of a structure power... |
| pwsbas 17515 | Base set of a structure po... |
| pwselbasb 17516 | Membership in the base set... |
| pwselbas 17517 | An element of a structure ... |
| pwsplusgval 17518 | Value of addition in a str... |
| pwsmulrval 17519 | Value of multiplication in... |
| pwsle 17520 | Ordering in a structure po... |
| pwsleval 17521 | Ordering in a structure po... |
| pwsvscafval 17522 | Scalar multiplication in a... |
| pwsvscaval 17523 | Scalar multiplication of a... |
| pwssca 17524 | The ring of scalars of a s... |
| pwsdiagel 17525 | Membership of diagonal ele... |
| pwssnf1o 17526 | Triviality of singleton po... |
| imasval 17539 | Value of an image structur... |
| imasbas 17540 | The base set of an image s... |
| imasds 17541 | The distance function of a... |
| imasdsfn 17542 | The distance function is a... |
| imasdsval 17543 | The distance function of a... |
| imasdsval2 17544 | The distance function of a... |
| imasplusg 17545 | The group operation in an ... |
| imasmulr 17546 | The ring multiplication in... |
| imassca 17547 | The scalar field of an ima... |
| imasvsca 17548 | The scalar multiplication ... |
| imasip 17549 | The inner product of an im... |
| imastset 17550 | The topology of an image s... |
| imasle 17551 | The ordering of an image s... |
| f1ocpbllem 17552 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17553 | An injection is compatible... |
| f1ovscpbl 17554 | An injection is compatible... |
| f1olecpbl 17555 | An injection is compatible... |
| imasaddfnlem 17556 | The image structure operat... |
| imasaddvallem 17557 | The operation of an image ... |
| imasaddflem 17558 | The image set operations a... |
| imasaddfn 17559 | The image structure's grou... |
| imasaddval 17560 | The value of an image stru... |
| imasaddf 17561 | The image structure's grou... |
| imasmulfn 17562 | The image structure's ring... |
| imasmulval 17563 | The value of an image stru... |
| imasmulf 17564 | The image structure's ring... |
| imasvscafn 17565 | The image structure's scal... |
| imasvscaval 17566 | The value of an image stru... |
| imasvscaf 17567 | The image structure's scal... |
| imasless 17568 | The order relation defined... |
| imasleval 17569 | The value of the image str... |
| qusval 17570 | Value of a quotient struct... |
| quslem 17571 | The function in ~ qusval i... |
| qusin 17572 | Restrict the equivalence r... |
| qusbas 17573 | Base set of a quotient str... |
| quss 17574 | The scalar field of a quot... |
| divsfval 17575 | Value of the function in ~... |
| ercpbllem 17576 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17577 | Translate the function com... |
| erlecpbl 17578 | Translate the relation com... |
| qusaddvallem 17579 | Value of an operation defi... |
| qusaddflem 17580 | The operation of a quotien... |
| qusaddval 17581 | The addition in a quotient... |
| qusaddf 17582 | The addition in a quotient... |
| qusmulval 17583 | The multiplication in a qu... |
| qusmulf 17584 | The multiplication in a qu... |
| fnpr2o 17585 | Function with a domain of ... |
| fnpr2ob 17586 | Biconditional version of ~... |
| fvpr0o 17587 | The value of a function wi... |
| fvpr1o 17588 | The value of a function wi... |
| fvprif 17589 | The value of the pair func... |
| xpsfrnel 17590 | Elementhood in the target ... |
| xpsfeq 17591 | A function on ` 2o ` is de... |
| xpsfrnel2 17592 | Elementhood in the target ... |
| xpscf 17593 | Equivalent condition for t... |
| xpsfval 17594 | The value of the function ... |
| xpsff1o 17595 | The function appearing in ... |
| xpsfrn 17596 | A short expression for the... |
| xpsff1o2 17597 | The function appearing in ... |
| xpsval 17598 | Value of the binary struct... |
| xpsrnbas 17599 | The indexed structure prod... |
| xpsbas 17600 | The base set of the binary... |
| xpsaddlem 17601 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17602 | Value of the addition oper... |
| xpsmul 17603 | Value of the multiplicatio... |
| xpssca 17604 | Value of the scalar field ... |
| xpsvsca 17605 | Value of the scalar multip... |
| xpsless 17606 | Closure of the ordering in... |
| xpsle 17607 | Value of the ordering in a... |
| ismre 17616 | Property of being a Moore ... |
| fnmre 17617 | The Moore collection gener... |
| mresspw 17618 | A Moore collection is a su... |
| mress 17619 | A Moore-closed subset is a... |
| mre1cl 17620 | In any Moore collection th... |
| mreintcl 17621 | A nonempty collection of c... |
| mreiincl 17622 | A nonempty indexed interse... |
| mrerintcl 17623 | The relative intersection ... |
| mreriincl 17624 | The relative intersection ... |
| mreincl 17625 | Two closed sets have a clo... |
| mreuni 17626 | Since the entire base set ... |
| mreunirn 17627 | Two ways to express the no... |
| ismred 17628 | Properties that determine ... |
| ismred2 17629 | Properties that determine ... |
| mremre 17630 | The Moore collections of s... |
| submre 17631 | The subcollection of a clo... |
| mrcflem 17632 | The domain and codomain of... |
| fnmrc 17633 | Moore-closure is a well-be... |
| mrcfval 17634 | Value of the function expr... |
| mrcf 17635 | The Moore closure is a fun... |
| mrcval 17636 | Evaluation of the Moore cl... |
| mrccl 17637 | The Moore closure of a set... |
| mrcsncl 17638 | The Moore closure of a sin... |
| mrcid 17639 | The closure of a closed se... |
| mrcssv 17640 | The closure of a set is a ... |
| mrcidb 17641 | A set is closed iff it is ... |
| mrcss 17642 | Closure preserves subset o... |
| mrcssid 17643 | The closure of a set is a ... |
| mrcidb2 17644 | A set is closed iff it con... |
| mrcidm 17645 | The closure operation is i... |
| mrcsscl 17646 | The closure is the minimal... |
| mrcuni 17647 | Idempotence of closure und... |
| mrcun 17648 | Idempotence of closure und... |
| mrcssvd 17649 | The Moore closure of a set... |
| mrcssd 17650 | Moore closure preserves su... |
| mrcssidd 17651 | A set is contained in its ... |
| mrcidmd 17652 | Moore closure is idempoten... |
| mressmrcd 17653 | In a Moore system, if a se... |
| submrc 17654 | In a closure system which ... |
| mrieqvlemd 17655 | In a Moore system, if ` Y ... |
| mrisval 17656 | Value of the set of indepe... |
| ismri 17657 | Criterion for a set to be ... |
| ismri2 17658 | Criterion for a subset of ... |
| ismri2d 17659 | Criterion for a subset of ... |
| ismri2dd 17660 | Definition of independence... |
| mriss 17661 | An independent set of a Mo... |
| mrissd 17662 | An independent set of a Mo... |
| ismri2dad 17663 | Consequence of a set in a ... |
| mrieqvd 17664 | In a Moore system, a set i... |
| mrieqv2d 17665 | In a Moore system, a set i... |
| mrissmrcd 17666 | In a Moore system, if an i... |
| mrissmrid 17667 | In a Moore system, subsets... |
| mreexd 17668 | In a Moore system, the clo... |
| mreexmrid 17669 | In a Moore system whose cl... |
| mreexexlemd 17670 | This lemma is used to gene... |
| mreexexlem2d 17671 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17672 | Base case of the induction... |
| mreexexlem4d 17673 | Induction step of the indu... |
| mreexexd 17674 | Exchange-type theorem. In... |
| mreexdomd 17675 | In a Moore system whose cl... |
| mreexfidimd 17676 | In a Moore system whose cl... |
| isacs 17677 | A set is an algebraic clos... |
| acsmre 17678 | Algebraic closure systems ... |
| isacs2 17679 | In the definition of an al... |
| acsfiel 17680 | A set is closed in an alge... |
| acsfiel2 17681 | A set is closed in an alge... |
| acsmred 17682 | An algebraic closure syste... |
| isacs1i 17683 | A closure system determine... |
| mreacs 17684 | Algebraicity is a composab... |
| acsfn 17685 | Algebraicity of a conditio... |
| acsfn0 17686 | Algebraicity of a point cl... |
| acsfn1 17687 | Algebraicity of a one-argu... |
| acsfn1c 17688 | Algebraicity of a one-argu... |
| acsfn2 17689 | Algebraicity of a two-argu... |
| iscat 17698 | The predicate "is a catego... |
| iscatd 17699 | Properties that determine ... |
| catidex 17700 | Each object in a category ... |
| catideu 17701 | Each object in a category ... |
| cidfval 17702 | Each object in a category ... |
| cidval 17703 | Each object in a category ... |
| cidffn 17704 | The identity arrow constru... |
| cidfn 17705 | The identity arrow operato... |
| catidd 17706 | Deduce the identity arrow ... |
| iscatd2 17707 | Version of ~ iscatd with a... |
| catidcl 17708 | Each object in a category ... |
| catlid 17709 | Left identity property of ... |
| catrid 17710 | Right identity property of... |
| catcocl 17711 | Closure of a composition a... |
| catass 17712 | Associativity of compositi... |
| catcone0 17713 | Composition of non-empty h... |
| 0catg 17714 | Any structure with an empt... |
| 0cat 17715 | The empty set is a categor... |
| homffval 17716 | Value of the functionalize... |
| fnhomeqhomf 17717 | If the Hom-set operation i... |
| homfval 17718 | Value of the functionalize... |
| homffn 17719 | The functionalized Hom-set... |
| homfeq 17720 | Condition for two categori... |
| homfeqd 17721 | If two structures have the... |
| homfeqbas 17722 | Deduce equality of base se... |
| homfeqval 17723 | Value of the functionalize... |
| comfffval 17724 | Value of the functionalize... |
| comffval 17725 | Value of the functionalize... |
| comfval 17726 | Value of the functionalize... |
| comfffval2 17727 | Value of the functionalize... |
| comffval2 17728 | Value of the functionalize... |
| comfval2 17729 | Value of the functionalize... |
| comfffn 17730 | The functionalized composi... |
| comffn 17731 | The functionalized composi... |
| comfeq 17732 | Condition for two categori... |
| comfeqd 17733 | Condition for two categori... |
| comfeqval 17734 | Equality of two compositio... |
| catpropd 17735 | Two structures with the sa... |
| cidpropd 17736 | Two structures with the sa... |
| oppcval 17739 | Value of the opposite cate... |
| oppchomfval 17740 | Hom-sets of the opposite c... |
| oppchomfvalOLD 17741 | Obsolete proof of ~ oppcho... |
| oppchom 17742 | Hom-sets of the opposite c... |
| oppccofval 17743 | Composition in the opposit... |
| oppcco 17744 | Composition in the opposit... |
| oppcbas 17745 | Base set of an opposite ca... |
| oppcbasOLD 17746 | Obsolete version of ~ oppc... |
| oppccatid 17747 | Lemma for ~ oppccat . (Co... |
| oppchomf 17748 | Hom-sets of the opposite c... |
| oppcid 17749 | Identity function of an op... |
| oppccat 17750 | An opposite category is a ... |
| 2oppcbas 17751 | The double opposite catego... |
| 2oppchomf 17752 | The double opposite catego... |
| 2oppccomf 17753 | The double opposite catego... |
| oppchomfpropd 17754 | If two categories have the... |
| oppccomfpropd 17755 | If two categories have the... |
| oppccatf 17756 | ` oppCat ` restricted to `... |
| monfval 17761 | Definition of a monomorphi... |
| ismon 17762 | Definition of a monomorphi... |
| ismon2 17763 | Write out the monomorphism... |
| monhom 17764 | A monomorphism is a morphi... |
| moni 17765 | Property of a monomorphism... |
| monpropd 17766 | If two categories have the... |
| oppcmon 17767 | A monomorphism in the oppo... |
| oppcepi 17768 | An epimorphism in the oppo... |
| isepi 17769 | Definition of an epimorphi... |
| isepi2 17770 | Write out the epimorphism ... |
| epihom 17771 | An epimorphism is a morphi... |
| epii 17772 | Property of an epimorphism... |
| sectffval 17779 | Value of the section opera... |
| sectfval 17780 | Value of the section relat... |
| sectss 17781 | The section relation is a ... |
| issect 17782 | The property " ` F ` is a ... |
| issect2 17783 | Property of being a sectio... |
| sectcan 17784 | If ` G ` is a section of `... |
| sectco 17785 | Composition of two section... |
| isofval 17786 | Function value of the func... |
| invffval 17787 | Value of the inverse relat... |
| invfval 17788 | Value of the inverse relat... |
| isinv 17789 | Value of the inverse relat... |
| invss 17790 | The inverse relation is a ... |
| invsym 17791 | The inverse relation is sy... |
| invsym2 17792 | The inverse relation is sy... |
| invfun 17793 | The inverse relation is a ... |
| isoval 17794 | The isomorphisms are the d... |
| inviso1 17795 | If ` G ` is an inverse to ... |
| inviso2 17796 | If ` G ` is an inverse to ... |
| invf 17797 | The inverse relation is a ... |
| invf1o 17798 | The inverse relation is a ... |
| invinv 17799 | The inverse of the inverse... |
| invco 17800 | The composition of two iso... |
| dfiso2 17801 | Alternate definition of an... |
| dfiso3 17802 | Alternate definition of an... |
| inveq 17803 | If there are two inverses ... |
| isofn 17804 | The function value of the ... |
| isohom 17805 | An isomorphism is a homomo... |
| isoco 17806 | The composition of two iso... |
| oppcsect 17807 | A section in the opposite ... |
| oppcsect2 17808 | A section in the opposite ... |
| oppcinv 17809 | An inverse in the opposite... |
| oppciso 17810 | An isomorphism in the oppo... |
| sectmon 17811 | If ` F ` is a section of `... |
| monsect 17812 | If ` F ` is a monomorphism... |
| sectepi 17813 | If ` F ` is a section of `... |
| episect 17814 | If ` F ` is an epimorphism... |
| sectid 17815 | The identity is a section ... |
| invid 17816 | The inverse of the identit... |
| idiso 17817 | The identity is an isomorp... |
| idinv 17818 | The inverse of the identit... |
| invisoinvl 17819 | The inverse of an isomorph... |
| invisoinvr 17820 | The inverse of an isomorph... |
| invcoisoid 17821 | The inverse of an isomorph... |
| isocoinvid 17822 | The inverse of an isomorph... |
| rcaninv 17823 | Right cancellation of an i... |
| cicfval 17826 | The set of isomorphic obje... |
| brcic 17827 | The relation "is isomorphi... |
| cic 17828 | Objects ` X ` and ` Y ` in... |
| brcici 17829 | Prove that two objects are... |
| cicref 17830 | Isomorphism is reflexive. ... |
| ciclcl 17831 | Isomorphism implies the le... |
| cicrcl 17832 | Isomorphism implies the ri... |
| cicsym 17833 | Isomorphism is symmetric. ... |
| cictr 17834 | Isomorphism is transitive.... |
| cicer 17835 | Isomorphism is an equivale... |
| sscrel 17842 | The subcategory subset rel... |
| brssc 17843 | The subcategory subset rel... |
| sscpwex 17844 | An analogue of ~ pwex for ... |
| subcrcl 17845 | Reverse closure for the su... |
| sscfn1 17846 | The subcategory subset rel... |
| sscfn2 17847 | The subcategory subset rel... |
| ssclem 17848 | Lemma for ~ ssc1 and simil... |
| isssc 17849 | Value of the subcategory s... |
| ssc1 17850 | Infer subset relation on o... |
| ssc2 17851 | Infer subset relation on m... |
| sscres 17852 | Any function restricted to... |
| sscid 17853 | The subcategory subset rel... |
| ssctr 17854 | The subcategory subset rel... |
| ssceq 17855 | The subcategory subset rel... |
| rescval 17856 | Value of the category rest... |
| rescval2 17857 | Value of the category rest... |
| rescbas 17858 | Base set of the category r... |
| rescbasOLD 17859 | Obsolete version of ~ resc... |
| reschom 17860 | Hom-sets of the category r... |
| reschomf 17861 | Hom-sets of the category r... |
| rescco 17862 | Composition in the categor... |
| resccoOLD 17863 | Obsolete proof of ~ rescco... |
| rescabs 17864 | Restriction absorption law... |
| rescabsOLD 17865 | Obsolete proof of ~ seqp1d... |
| rescabs2 17866 | Restriction absorption law... |
| issubc 17867 | Elementhood in the set of ... |
| issubc2 17868 | Elementhood in the set of ... |
| 0ssc 17869 | For any category ` C ` , t... |
| 0subcat 17870 | For any category ` C ` , t... |
| catsubcat 17871 | For any category ` C ` , `... |
| subcssc 17872 | An element in the set of s... |
| subcfn 17873 | An element in the set of s... |
| subcss1 17874 | The objects of a subcatego... |
| subcss2 17875 | The morphisms of a subcate... |
| subcidcl 17876 | The identity of the origin... |
| subccocl 17877 | A subcategory is closed un... |
| subccatid 17878 | A subcategory is a categor... |
| subcid 17879 | The identity in a subcateg... |
| subccat 17880 | A subcategory is a categor... |
| issubc3 17881 | Alternate definition of a ... |
| fullsubc 17882 | The full subcategory gener... |
| fullresc 17883 | The category formed by str... |
| resscat 17884 | A category restricted to a... |
| subsubc 17885 | A subcategory of a subcate... |
| relfunc 17894 | The set of functors is a r... |
| funcrcl 17895 | Reverse closure for a func... |
| isfunc 17896 | Value of the set of functo... |
| isfuncd 17897 | Deduce that an operation i... |
| funcf1 17898 | The object part of a funct... |
| funcixp 17899 | The morphism part of a fun... |
| funcf2 17900 | The morphism part of a fun... |
| funcfn2 17901 | The morphism part of a fun... |
| funcid 17902 | A functor maps each identi... |
| funcco 17903 | A functor maps composition... |
| funcsect 17904 | The image of a section und... |
| funcinv 17905 | The image of an inverse un... |
| funciso 17906 | The image of an isomorphis... |
| funcoppc 17907 | A functor on categories yi... |
| idfuval 17908 | Value of the identity func... |
| idfu2nd 17909 | Value of the morphism part... |
| idfu2 17910 | Value of the morphism part... |
| idfu1st 17911 | Value of the object part o... |
| idfu1 17912 | Value of the object part o... |
| idfucl 17913 | The identity functor is a ... |
| cofuval 17914 | Value of the composition o... |
| cofu1st 17915 | Value of the object part o... |
| cofu1 17916 | Value of the object part o... |
| cofu2nd 17917 | Value of the morphism part... |
| cofu2 17918 | Value of the morphism part... |
| cofuval2 17919 | Value of the composition o... |
| cofucl 17920 | The composition of two fun... |
| cofuass 17921 | Functor composition is ass... |
| cofulid 17922 | The identity functor is a ... |
| cofurid 17923 | The identity functor is a ... |
| resfval 17924 | Value of the functor restr... |
| resfval2 17925 | Value of the functor restr... |
| resf1st 17926 | Value of the functor restr... |
| resf2nd 17927 | Value of the functor restr... |
| funcres 17928 | A functor restricted to a ... |
| funcres2b 17929 | Condition for a functor to... |
| funcres2 17930 | A functor into a restricte... |
| idfusubc0 17931 | The identity functor for a... |
| idfusubc 17932 | The identity functor for a... |
| wunfunc 17933 | A weak universe is closed ... |
| wunfuncOLD 17934 | Obsolete proof of ~ wunfun... |
| funcpropd 17935 | If two categories have the... |
| funcres2c 17936 | Condition for a functor to... |
| fullfunc 17941 | A full functor is a functo... |
| fthfunc 17942 | A faithful functor is a fu... |
| relfull 17943 | The set of full functors i... |
| relfth 17944 | The set of faithful functo... |
| isfull 17945 | Value of the set of full f... |
| isfull2 17946 | Equivalent condition for a... |
| fullfo 17947 | The morphism map of a full... |
| fulli 17948 | The morphism map of a full... |
| isfth 17949 | Value of the set of faithf... |
| isfth2 17950 | Equivalent condition for a... |
| isffth2 17951 | A fully faithful functor i... |
| fthf1 17952 | The morphism map of a fait... |
| fthi 17953 | The morphism map of a fait... |
| ffthf1o 17954 | The morphism map of a full... |
| fullpropd 17955 | If two categories have the... |
| fthpropd 17956 | If two categories have the... |
| fulloppc 17957 | The opposite functor of a ... |
| fthoppc 17958 | The opposite functor of a ... |
| ffthoppc 17959 | The opposite functor of a ... |
| fthsect 17960 | A faithful functor reflect... |
| fthinv 17961 | A faithful functor reflect... |
| fthmon 17962 | A faithful functor reflect... |
| fthepi 17963 | A faithful functor reflect... |
| ffthiso 17964 | A fully faithful functor r... |
| fthres2b 17965 | Condition for a faithful f... |
| fthres2c 17966 | Condition for a faithful f... |
| fthres2 17967 | A faithful functor into a ... |
| idffth 17968 | The identity functor is a ... |
| cofull 17969 | The composition of two ful... |
| cofth 17970 | The composition of two fai... |
| coffth 17971 | The composition of two ful... |
| rescfth 17972 | The inclusion functor from... |
| ressffth 17973 | The inclusion functor from... |
| fullres2c 17974 | Condition for a full funct... |
| ffthres2c 17975 | Condition for a fully fait... |
| inclfusubc 17976 | The "inclusion functor" fr... |
| fnfuc 17981 | The ` FuncCat ` operation ... |
| natfval 17982 | Value of the function givi... |
| isnat 17983 | Property of being a natura... |
| isnat2 17984 | Property of being a natura... |
| natffn 17985 | The natural transformation... |
| natrcl 17986 | Reverse closure for a natu... |
| nat1st2nd 17987 | Rewrite the natural transf... |
| natixp 17988 | A natural transformation i... |
| natcl 17989 | A component of a natural t... |
| natfn 17990 | A natural transformation i... |
| nati 17991 | Naturality property of a n... |
| wunnat 17992 | A weak universe is closed ... |
| wunnatOLD 17993 | Obsolete proof of ~ wunnat... |
| catstr 17994 | A category structure is a ... |
| fucval 17995 | Value of the functor categ... |
| fuccofval 17996 | Value of the functor categ... |
| fucbas 17997 | The objects of the functor... |
| fuchom 17998 | The morphisms in the funct... |
| fuchomOLD 17999 | Obsolete proof of ~ fuchom... |
| fucco 18000 | Value of the composition o... |
| fuccoval 18001 | Value of the functor categ... |
| fuccocl 18002 | The composition of two nat... |
| fucidcl 18003 | The identity natural trans... |
| fuclid 18004 | Left identity of natural t... |
| fucrid 18005 | Right identity of natural ... |
| fucass 18006 | Associativity of natural t... |
| fuccatid 18007 | The functor category is a ... |
| fuccat 18008 | The functor category is a ... |
| fucid 18009 | The identity morphism in t... |
| fucsect 18010 | Two natural transformation... |
| fucinv 18011 | Two natural transformation... |
| invfuc 18012 | If ` V ( x ) ` is an inver... |
| fuciso 18013 | A natural transformation i... |
| natpropd 18014 | If two categories have the... |
| fucpropd 18015 | If two categories have the... |
| initofn 18022 | ` InitO ` is a function on... |
| termofn 18023 | ` TermO ` is a function on... |
| zeroofn 18024 | ` ZeroO ` is a function on... |
| initorcl 18025 | Reverse closure for an ini... |
| termorcl 18026 | Reverse closure for a term... |
| zeroorcl 18027 | Reverse closure for a zero... |
| initoval 18028 | The value of the initial o... |
| termoval 18029 | The value of the terminal ... |
| zerooval 18030 | The value of the zero obje... |
| isinito 18031 | The predicate "is an initi... |
| istermo 18032 | The predicate "is a termin... |
| iszeroo 18033 | The predicate "is a zero o... |
| isinitoi 18034 | Implication of a class bei... |
| istermoi 18035 | Implication of a class bei... |
| initoid 18036 | For an initial object, the... |
| termoid 18037 | For a terminal object, the... |
| dfinito2 18038 | An initial object is a ter... |
| dftermo2 18039 | A terminal object is an in... |
| dfinito3 18040 | An alternate definition of... |
| dftermo3 18041 | An alternate definition of... |
| initoo 18042 | An initial object is an ob... |
| termoo 18043 | A terminal object is an ob... |
| iszeroi 18044 | Implication of a class bei... |
| 2initoinv 18045 | Morphisms between two init... |
| initoeu1 18046 | Initial objects are essent... |
| initoeu1w 18047 | Initial objects are essent... |
| initoeu2lem0 18048 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 18049 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 18050 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 18051 | Initial objects are essent... |
| 2termoinv 18052 | Morphisms between two term... |
| termoeu1 18053 | Terminal objects are essen... |
| termoeu1w 18054 | Terminal objects are essen... |
| homarcl 18063 | Reverse closure for an arr... |
| homafval 18064 | Value of the disjointified... |
| homaf 18065 | Functionality of the disjo... |
| homaval 18066 | Value of the disjointified... |
| elhoma 18067 | Value of the disjointified... |
| elhomai 18068 | Produce an arrow from a mo... |
| elhomai2 18069 | Produce an arrow from a mo... |
| homarcl2 18070 | Reverse closure for the do... |
| homarel 18071 | An arrow is an ordered pai... |
| homa1 18072 | The first component of an ... |
| homahom2 18073 | The second component of an... |
| homahom 18074 | The second component of an... |
| homadm 18075 | The domain of an arrow wit... |
| homacd 18076 | The codomain of an arrow w... |
| homadmcd 18077 | Decompose an arrow into do... |
| arwval 18078 | The set of arrows is the u... |
| arwrcl 18079 | The first component of an ... |
| arwhoma 18080 | An arrow is contained in t... |
| homarw 18081 | A hom-set is a subset of t... |
| arwdm 18082 | The domain of an arrow is ... |
| arwcd 18083 | The codomain of an arrow i... |
| dmaf 18084 | The domain function is a f... |
| cdaf 18085 | The codomain function is a... |
| arwhom 18086 | The second component of an... |
| arwdmcd 18087 | Decompose an arrow into do... |
| idafval 18092 | Value of the identity arro... |
| idaval 18093 | Value of the identity arro... |
| ida2 18094 | Morphism part of the ident... |
| idahom 18095 | Domain and codomain of the... |
| idadm 18096 | Domain of the identity arr... |
| idacd 18097 | Codomain of the identity a... |
| idaf 18098 | The identity arrow functio... |
| coafval 18099 | The value of the compositi... |
| eldmcoa 18100 | A pair ` <. G , F >. ` is ... |
| dmcoass 18101 | The domain of composition ... |
| homdmcoa 18102 | If ` F : X --> Y ` and ` G... |
| coaval 18103 | Value of composition for c... |
| coa2 18104 | The morphism part of arrow... |
| coahom 18105 | The composition of two com... |
| coapm 18106 | Composition of arrows is a... |
| arwlid 18107 | Left identity of a categor... |
| arwrid 18108 | Right identity of a catego... |
| arwass 18109 | Associativity of compositi... |
| setcval 18112 | Value of the category of s... |
| setcbas 18113 | Set of objects of the cate... |
| setchomfval 18114 | Set of arrows of the categ... |
| setchom 18115 | Set of arrows of the categ... |
| elsetchom 18116 | A morphism of sets is a fu... |
| setccofval 18117 | Composition in the categor... |
| setcco 18118 | Composition in the categor... |
| setccatid 18119 | Lemma for ~ setccat . (Co... |
| setccat 18120 | The category of sets is a ... |
| setcid 18121 | The identity arrow in the ... |
| setcmon 18122 | A monomorphism of sets is ... |
| setcepi 18123 | An epimorphism of sets is ... |
| setcsect 18124 | A section in the category ... |
| setcinv 18125 | An inverse in the category... |
| setciso 18126 | An isomorphism in the cate... |
| resssetc 18127 | The restriction of the cat... |
| funcsetcres2 18128 | A functor into a smaller c... |
| setc2obas 18129 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18130 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18131 | The category of sets in a ... |
| cat1 18132 | The definition of category... |
| catcval 18135 | Value of the category of c... |
| catcbas 18136 | Set of objects of the cate... |
| catchomfval 18137 | Set of arrows of the categ... |
| catchom 18138 | Set of arrows of the categ... |
| catccofval 18139 | Composition in the categor... |
| catcco 18140 | Composition in the categor... |
| catccatid 18141 | Lemma for ~ catccat . (Co... |
| catcid 18142 | The identity arrow in the ... |
| catccat 18143 | The category of categories... |
| resscatc 18144 | The restriction of the cat... |
| catcisolem 18145 | Lemma for ~ catciso . (Co... |
| catciso 18146 | A functor is an isomorphis... |
| catcbascl 18147 | An element of the base set... |
| catcslotelcl 18148 | A slot entry of an element... |
| catcbaselcl 18149 | The base set of an element... |
| catchomcl 18150 | The Hom-set of an element ... |
| catcccocl 18151 | The composition operation ... |
| catcoppccl 18152 | The category of categories... |
| catcoppcclOLD 18153 | Obsolete proof of ~ catcop... |
| catcfuccl 18154 | The category of categories... |
| catcfucclOLD 18155 | Obsolete proof of ~ catcfu... |
| fncnvimaeqv 18156 | The inverse images of the ... |
| bascnvimaeqv 18157 | The inverse image of the u... |
| estrcval 18160 | Value of the category of e... |
| estrcbas 18161 | Set of objects of the cate... |
| estrchomfval 18162 | Set of morphisms ("arrows"... |
| estrchom 18163 | The morphisms between exte... |
| elestrchom 18164 | A morphism between extensi... |
| estrccofval 18165 | Composition in the categor... |
| estrcco 18166 | Composition in the categor... |
| estrcbasbas 18167 | An element of the base set... |
| estrccatid 18168 | Lemma for ~ estrccat . (C... |
| estrccat 18169 | The category of extensible... |
| estrcid 18170 | The identity arrow in the ... |
| estrchomfn 18171 | The Hom-set operation in t... |
| estrchomfeqhom 18172 | The functionalized Hom-set... |
| estrreslem1 18173 | Lemma 1 for ~ estrres . (... |
| estrreslem1OLD 18174 | Obsolete version of ~ estr... |
| estrreslem2 18175 | Lemma 2 for ~ estrres . (... |
| estrres 18176 | Any restriction of a categ... |
| funcestrcsetclem1 18177 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18178 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18179 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18180 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18181 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18182 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18183 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18184 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18185 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18186 | The "natural forgetful fun... |
| fthestrcsetc 18187 | The "natural forgetful fun... |
| fullestrcsetc 18188 | The "natural forgetful fun... |
| equivestrcsetc 18189 | The "natural forgetful fun... |
| setc1strwun 18190 | A constructed one-slot str... |
| funcsetcestrclem1 18191 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18192 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18193 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18194 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18195 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18196 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18197 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18198 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18199 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18200 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18201 | The "embedding functor" fr... |
| fthsetcestrc 18202 | The "embedding functor" fr... |
| fullsetcestrc 18203 | The "embedding functor" fr... |
| embedsetcestrc 18204 | The "embedding functor" fr... |
| fnxpc 18213 | The binary product of cate... |
| xpcval 18214 | Value of the binary produc... |
| xpcbas 18215 | Set of objects of the bina... |
| xpchomfval 18216 | Set of morphisms of the bi... |
| xpchom 18217 | Set of morphisms of the bi... |
| relxpchom 18218 | A hom-set in the binary pr... |
| xpccofval 18219 | Value of composition in th... |
| xpcco 18220 | Value of composition in th... |
| xpcco1st 18221 | Value of composition in th... |
| xpcco2nd 18222 | Value of composition in th... |
| xpchom2 18223 | Value of the set of morphi... |
| xpcco2 18224 | Value of composition in th... |
| xpccatid 18225 | The product of two categor... |
| xpcid 18226 | The identity morphism in t... |
| xpccat 18227 | The product of two categor... |
| 1stfval 18228 | Value of the first project... |
| 1stf1 18229 | Value of the first project... |
| 1stf2 18230 | Value of the first project... |
| 2ndfval 18231 | Value of the first project... |
| 2ndf1 18232 | Value of the first project... |
| 2ndf2 18233 | Value of the first project... |
| 1stfcl 18234 | The first projection funct... |
| 2ndfcl 18235 | The second projection func... |
| prfval 18236 | Value of the pairing funct... |
| prf1 18237 | Value of the pairing funct... |
| prf2fval 18238 | Value of the pairing funct... |
| prf2 18239 | Value of the pairing funct... |
| prfcl 18240 | The pairing of functors ` ... |
| prf1st 18241 | Cancellation of pairing wi... |
| prf2nd 18242 | Cancellation of pairing wi... |
| 1st2ndprf 18243 | Break a functor into a pro... |
| catcxpccl 18244 | The category of categories... |
| catcxpcclOLD 18245 | Obsolete proof of ~ catcxp... |
| xpcpropd 18246 | If two categories have the... |
| evlfval 18255 | Value of the evaluation fu... |
| evlf2 18256 | Value of the evaluation fu... |
| evlf2val 18257 | Value of the evaluation na... |
| evlf1 18258 | Value of the evaluation fu... |
| evlfcllem 18259 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18260 | The evaluation functor is ... |
| curfval 18261 | Value of the curry functor... |
| curf1fval 18262 | Value of the object part o... |
| curf1 18263 | Value of the object part o... |
| curf11 18264 | Value of the double evalua... |
| curf12 18265 | The partially evaluated cu... |
| curf1cl 18266 | The partially evaluated cu... |
| curf2 18267 | Value of the curry functor... |
| curf2val 18268 | Value of a component of th... |
| curf2cl 18269 | The curry functor at a mor... |
| curfcl 18270 | The curry functor of a fun... |
| curfpropd 18271 | If two categories have the... |
| uncfval 18272 | Value of the uncurry funct... |
| uncfcl 18273 | The uncurry operation take... |
| uncf1 18274 | Value of the uncurry funct... |
| uncf2 18275 | Value of the uncurry funct... |
| curfuncf 18276 | Cancellation of curry with... |
| uncfcurf 18277 | Cancellation of uncurry wi... |
| diagval 18278 | Define the diagonal functo... |
| diagcl 18279 | The diagonal functor is a ... |
| diag1cl 18280 | The constant functor of ` ... |
| diag11 18281 | Value of the constant func... |
| diag12 18282 | Value of the constant func... |
| diag2 18283 | Value of the diagonal func... |
| diag2cl 18284 | The diagonal functor at a ... |
| curf2ndf 18285 | As shown in ~ diagval , th... |
| hofval 18290 | Value of the Hom functor, ... |
| hof1fval 18291 | The object part of the Hom... |
| hof1 18292 | The object part of the Hom... |
| hof2fval 18293 | The morphism part of the H... |
| hof2val 18294 | The morphism part of the H... |
| hof2 18295 | The morphism part of the H... |
| hofcllem 18296 | Lemma for ~ hofcl . (Cont... |
| hofcl 18297 | Closure of the Hom functor... |
| oppchofcl 18298 | Closure of the opposite Ho... |
| yonval 18299 | Value of the Yoneda embedd... |
| yoncl 18300 | The Yoneda embedding is a ... |
| yon1cl 18301 | The Yoneda embedding at an... |
| yon11 18302 | Value of the Yoneda embedd... |
| yon12 18303 | Value of the Yoneda embedd... |
| yon2 18304 | Value of the Yoneda embedd... |
| hofpropd 18305 | If two categories have the... |
| yonpropd 18306 | If two categories have the... |
| oppcyon 18307 | Value of the opposite Yone... |
| oyoncl 18308 | The opposite Yoneda embedd... |
| oyon1cl 18309 | The opposite Yoneda embedd... |
| yonedalem1 18310 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18311 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18312 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18313 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18314 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18315 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18316 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18317 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18318 | Lemma for ~ yoneda . (Con... |
| yonedainv 18319 | The Yoneda Lemma with expl... |
| yonffthlem 18320 | Lemma for ~ yonffth . (Co... |
| yoneda 18321 | The Yoneda Lemma. There i... |
| yonffth 18322 | The Yoneda Lemma. The Yon... |
| yoniso 18323 | If the codomain is recover... |
| oduval 18326 | Value of an order dual str... |
| oduleval 18327 | Value of the less-equal re... |
| oduleg 18328 | Truth of the less-equal re... |
| odubas 18329 | Base set of an order dual ... |
| odubasOLD 18330 | Obsolete proof of ~ odubas... |
| isprs 18335 | Property of being a preord... |
| prslem 18336 | Lemma for ~ prsref and ~ p... |
| prsref 18337 | "Less than or equal to" is... |
| prstr 18338 | "Less than or equal to" is... |
| isdrs 18339 | Property of being a direct... |
| drsdir 18340 | Direction of a directed se... |
| drsprs 18341 | A directed set is a proset... |
| drsbn0 18342 | The base of a directed set... |
| drsdirfi 18343 | Any _finite_ number of ele... |
| isdrs2 18344 | Directed sets may be defin... |
| ispos 18352 | The predicate "is a poset"... |
| ispos2 18353 | A poset is an antisymmetri... |
| posprs 18354 | A poset is a proset. (Con... |
| posi 18355 | Lemma for poset properties... |
| posref 18356 | A poset ordering is reflex... |
| posasymb 18357 | A poset ordering is asymme... |
| postr 18358 | A poset ordering is transi... |
| 0pos 18359 | Technical lemma to simplif... |
| 0posOLD 18360 | Obsolete proof of ~ 0pos a... |
| isposd 18361 | Properties that determine ... |
| isposi 18362 | Properties that determine ... |
| isposix 18363 | Properties that determine ... |
| isposixOLD 18364 | Obsolete proof of ~ isposi... |
| pospropd 18365 | Posethood is determined on... |
| odupos 18366 | Being a poset is a self-du... |
| oduposb 18367 | Being a poset is a self-du... |
| pltfval 18369 | Value of the less-than rel... |
| pltval 18370 | Less-than relation. ( ~ d... |
| pltle 18371 | "Less than" implies "less ... |
| pltne 18372 | The "less than" relation i... |
| pltirr 18373 | The "less than" relation i... |
| pleval2i 18374 | One direction of ~ pleval2... |
| pleval2 18375 | "Less than or equal to" in... |
| pltnle 18376 | "Less than" implies not co... |
| pltval3 18377 | Alternate expression for t... |
| pltnlt 18378 | The less-than relation imp... |
| pltn2lp 18379 | The less-than relation has... |
| plttr 18380 | The less-than relation is ... |
| pltletr 18381 | Transitive law for chained... |
| plelttr 18382 | Transitive law for chained... |
| pospo 18383 | Write a poset structure in... |
| lubfval 18388 | Value of the least upper b... |
| lubdm 18389 | Domain of the least upper ... |
| lubfun 18390 | The LUB is a function. (C... |
| lubeldm 18391 | Member of the domain of th... |
| lubelss 18392 | A member of the domain of ... |
| lubeu 18393 | Unique existence proper of... |
| lubval 18394 | Value of the least upper b... |
| lubcl 18395 | The least upper bound func... |
| lubprop 18396 | Properties of greatest low... |
| luble 18397 | The greatest lower bound i... |
| lublecllem 18398 | Lemma for ~ lublecl and ~ ... |
| lublecl 18399 | The set of all elements le... |
| lubid 18400 | The LUB of elements less t... |
| glbfval 18401 | Value of the greatest lowe... |
| glbdm 18402 | Domain of the greatest low... |
| glbfun 18403 | The GLB is a function. (C... |
| glbeldm 18404 | Member of the domain of th... |
| glbelss 18405 | A member of the domain of ... |
| glbeu 18406 | Unique existence proper of... |
| glbval 18407 | Value of the greatest lowe... |
| glbcl 18408 | The least upper bound func... |
| glbprop 18409 | Properties of greatest low... |
| glble 18410 | The greatest lower bound i... |
| joinfval 18411 | Value of join function for... |
| joinfval2 18412 | Value of join function for... |
| joindm 18413 | Domain of join function fo... |
| joindef 18414 | Two ways to say that a joi... |
| joinval 18415 | Join value. Since both si... |
| joincl 18416 | Closure of join of element... |
| joindmss 18417 | Subset property of domain ... |
| joinval2lem 18418 | Lemma for ~ joinval2 and ~... |
| joinval2 18419 | Value of join for a poset ... |
| joineu 18420 | Uniqueness of join of elem... |
| joinlem 18421 | Lemma for join properties.... |
| lejoin1 18422 | A join's first argument is... |
| lejoin2 18423 | A join's second argument i... |
| joinle 18424 | A join is less than or equ... |
| meetfval 18425 | Value of meet function for... |
| meetfval2 18426 | Value of meet function for... |
| meetdm 18427 | Domain of meet function fo... |
| meetdef 18428 | Two ways to say that a mee... |
| meetval 18429 | Meet value. Since both si... |
| meetcl 18430 | Closure of meet of element... |
| meetdmss 18431 | Subset property of domain ... |
| meetval2lem 18432 | Lemma for ~ meetval2 and ~... |
| meetval2 18433 | Value of meet for a poset ... |
| meeteu 18434 | Uniqueness of meet of elem... |
| meetlem 18435 | Lemma for meet properties.... |
| lemeet1 18436 | A meet's first argument is... |
| lemeet2 18437 | A meet's second argument i... |
| meetle 18438 | A meet is less than or equ... |
| joincomALT 18439 | The join of a poset is com... |
| joincom 18440 | The join of a poset is com... |
| meetcomALT 18441 | The meet of a poset is com... |
| meetcom 18442 | The meet of a poset is com... |
| join0 18443 | Lemma for ~ odumeet . (Co... |
| meet0 18444 | Lemma for ~ odujoin . (Co... |
| odulub 18445 | Least upper bounds in a du... |
| odujoin 18446 | Joins in a dual order are ... |
| oduglb 18447 | Greatest lower bounds in a... |
| odumeet 18448 | Meets in a dual order are ... |
| poslubmo 18449 | Least upper bounds in a po... |
| posglbmo 18450 | Greatest lower bounds in a... |
| poslubd 18451 | Properties which determine... |
| poslubdg 18452 | Properties which determine... |
| posglbdg 18453 | Properties which determine... |
| istos 18456 | The predicate "is a toset"... |
| tosso 18457 | Write the totally ordered ... |
| tospos 18458 | A Toset is a Poset. (Cont... |
| tleile 18459 | In a Toset, any two elemen... |
| tltnle 18460 | In a Toset, "less than" is... |
| p0val 18465 | Value of poset zero. (Con... |
| p1val 18466 | Value of poset zero. (Con... |
| p0le 18467 | Any element is less than o... |
| ple1 18468 | Any element is less than o... |
| islat 18471 | The predicate "is a lattic... |
| odulatb 18472 | Being a lattice is self-du... |
| odulat 18473 | Being a lattice is self-du... |
| latcl2 18474 | The join and meet of any t... |
| latlem 18475 | Lemma for lattice properti... |
| latpos 18476 | A lattice is a poset. (Co... |
| latjcl 18477 | Closure of join operation ... |
| latmcl 18478 | Closure of meet operation ... |
| latref 18479 | A lattice ordering is refl... |
| latasymb 18480 | A lattice ordering is asym... |
| latasym 18481 | A lattice ordering is asym... |
| lattr 18482 | A lattice ordering is tran... |
| latasymd 18483 | Deduce equality from latti... |
| lattrd 18484 | A lattice ordering is tran... |
| latjcom 18485 | The join of a lattice comm... |
| latlej1 18486 | A join's first argument is... |
| latlej2 18487 | A join's second argument i... |
| latjle12 18488 | A join is less than or equ... |
| latleeqj1 18489 | "Less than or equal to" in... |
| latleeqj2 18490 | "Less than or equal to" in... |
| latjlej1 18491 | Add join to both sides of ... |
| latjlej2 18492 | Add join to both sides of ... |
| latjlej12 18493 | Add join to both sides of ... |
| latnlej 18494 | An idiom to express that a... |
| latnlej1l 18495 | An idiom to express that a... |
| latnlej1r 18496 | An idiom to express that a... |
| latnlej2 18497 | An idiom to express that a... |
| latnlej2l 18498 | An idiom to express that a... |
| latnlej2r 18499 | An idiom to express that a... |
| latjidm 18500 | Lattice join is idempotent... |
| latmcom 18501 | The join of a lattice comm... |
| latmle1 18502 | A meet is less than or equ... |
| latmle2 18503 | A meet is less than or equ... |
| latlem12 18504 | An element is less than or... |
| latleeqm1 18505 | "Less than or equal to" in... |
| latleeqm2 18506 | "Less than or equal to" in... |
| latmlem1 18507 | Add meet to both sides of ... |
| latmlem2 18508 | Add meet to both sides of ... |
| latmlem12 18509 | Add join to both sides of ... |
| latnlemlt 18510 | Negation of "less than or ... |
| latnle 18511 | Equivalent expressions for... |
| latmidm 18512 | Lattice meet is idempotent... |
| latabs1 18513 | Lattice absorption law. F... |
| latabs2 18514 | Lattice absorption law. F... |
| latledi 18515 | An ortholattice is distrib... |
| latmlej11 18516 | Ordering of a meet and joi... |
| latmlej12 18517 | Ordering of a meet and joi... |
| latmlej21 18518 | Ordering of a meet and joi... |
| latmlej22 18519 | Ordering of a meet and joi... |
| lubsn 18520 | The least upper bound of a... |
| latjass 18521 | Lattice join is associativ... |
| latj12 18522 | Swap 1st and 2nd members o... |
| latj32 18523 | Swap 2nd and 3rd members o... |
| latj13 18524 | Swap 1st and 3rd members o... |
| latj31 18525 | Swap 2nd and 3rd members o... |
| latjrot 18526 | Rotate lattice join of 3 c... |
| latj4 18527 | Rearrangement of lattice j... |
| latj4rot 18528 | Rotate lattice join of 4 c... |
| latjjdi 18529 | Lattice join distributes o... |
| latjjdir 18530 | Lattice join distributes o... |
| mod1ile 18531 | The weak direction of the ... |
| mod2ile 18532 | The weak direction of the ... |
| latmass 18533 | Lattice meet is associativ... |
| latdisdlem 18534 | Lemma for ~ latdisd . (Co... |
| latdisd 18535 | In a lattice, joins distri... |
| isclat 18538 | The predicate "is a comple... |
| clatpos 18539 | A complete lattice is a po... |
| clatlem 18540 | Lemma for properties of a ... |
| clatlubcl 18541 | Any subset of the base set... |
| clatlubcl2 18542 | Any subset of the base set... |
| clatglbcl 18543 | Any subset of the base set... |
| clatglbcl2 18544 | Any subset of the base set... |
| oduclatb 18545 | Being a complete lattice i... |
| clatl 18546 | A complete lattice is a la... |
| isglbd 18547 | Properties that determine ... |
| lublem 18548 | Lemma for the least upper ... |
| lubub 18549 | The LUB of a complete latt... |
| lubl 18550 | The LUB of a complete latt... |
| lubss 18551 | Subset law for least upper... |
| lubel 18552 | An element of a set is les... |
| lubun 18553 | The LUB of a union. (Cont... |
| clatglb 18554 | Properties of greatest low... |
| clatglble 18555 | The greatest lower bound i... |
| clatleglb 18556 | Two ways of expressing "le... |
| clatglbss 18557 | Subset law for greatest lo... |
| isdlat 18560 | Property of being a distri... |
| dlatmjdi 18561 | In a distributive lattice,... |
| dlatl 18562 | A distributive lattice is ... |
| odudlatb 18563 | The dual of a distributive... |
| dlatjmdi 18564 | In a distributive lattice,... |
| ipostr 18567 | The structure of ~ df-ipo ... |
| ipoval 18568 | Value of the inclusion pos... |
| ipobas 18569 | Base set of the inclusion ... |
| ipolerval 18570 | Relation of the inclusion ... |
| ipotset 18571 | Topology of the inclusion ... |
| ipole 18572 | Weak order condition of th... |
| ipolt 18573 | Strict order condition of ... |
| ipopos 18574 | The inclusion poset on a f... |
| isipodrs 18575 | Condition for a family of ... |
| ipodrscl 18576 | Direction by inclusion as ... |
| ipodrsfi 18577 | Finite upper bound propert... |
| fpwipodrs 18578 | The finite subsets of any ... |
| ipodrsima 18579 | The monotone image of a di... |
| isacs3lem 18580 | An algebraic closure syste... |
| acsdrsel 18581 | An algebraic closure syste... |
| isacs4lem 18582 | In a closure system in whi... |
| isacs5lem 18583 | If closure commutes with d... |
| acsdrscl 18584 | In an algebraic closure sy... |
| acsficl 18585 | A closure in an algebraic ... |
| isacs5 18586 | A closure system is algebr... |
| isacs4 18587 | A closure system is algebr... |
| isacs3 18588 | A closure system is algebr... |
| acsficld 18589 | In an algebraic closure sy... |
| acsficl2d 18590 | In an algebraic closure sy... |
| acsfiindd 18591 | In an algebraic closure sy... |
| acsmapd 18592 | In an algebraic closure sy... |
| acsmap2d 18593 | In an algebraic closure sy... |
| acsinfd 18594 | In an algebraic closure sy... |
| acsdomd 18595 | In an algebraic closure sy... |
| acsinfdimd 18596 | In an algebraic closure sy... |
| acsexdimd 18597 | In an algebraic closure sy... |
| mrelatglb 18598 | Greatest lower bounds in a... |
| mrelatglb0 18599 | The empty intersection in ... |
| mrelatlub 18600 | Least upper bounds in a Mo... |
| mreclatBAD 18601 | A Moore space is a complet... |
| isps 18606 | The predicate "is a poset"... |
| psrel 18607 | A poset is a relation. (C... |
| psref2 18608 | A poset is antisymmetric a... |
| pstr2 18609 | A poset is transitive. (C... |
| pslem 18610 | Lemma for ~ psref and othe... |
| psdmrn 18611 | The domain and range of a ... |
| psref 18612 | A poset is reflexive. (Co... |
| psrn 18613 | The range of a poset equal... |
| psasym 18614 | A poset is antisymmetric. ... |
| pstr 18615 | A poset is transitive. (C... |
| cnvps 18616 | The converse of a poset is... |
| cnvpsb 18617 | The converse of a poset is... |
| psss 18618 | Any subset of a partially ... |
| psssdm2 18619 | Field of a subposet. (Con... |
| psssdm 18620 | Field of a subposet. (Con... |
| istsr 18621 | The predicate is a toset. ... |
| istsr2 18622 | The predicate is a toset. ... |
| tsrlin 18623 | A toset is a linear order.... |
| tsrlemax 18624 | Two ways of saying a numbe... |
| tsrps 18625 | A toset is a poset. (Cont... |
| cnvtsr 18626 | The converse of a toset is... |
| tsrss 18627 | Any subset of a totally or... |
| ledm 18628 | The domain of ` <_ ` is ` ... |
| lern 18629 | The range of ` <_ ` is ` R... |
| lefld 18630 | The field of the 'less or ... |
| letsr 18631 | The "less than or equal to... |
| isdir 18636 | A condition for a relation... |
| reldir 18637 | A direction is a relation.... |
| dirdm 18638 | A direction's domain is eq... |
| dirref 18639 | A direction is reflexive. ... |
| dirtr 18640 | A direction is transitive.... |
| dirge 18641 | For any two elements of a ... |
| tsrdir 18642 | A totally ordered set is a... |
| ismgm 18647 | The predicate "is a magma"... |
| ismgmn0 18648 | The predicate "is a magma"... |
| mgmcl 18649 | Closure of the operation o... |
| isnmgm 18650 | A condition for a structur... |
| mgmsscl 18651 | If the base set of a magma... |
| plusffval 18652 | The group addition operati... |
| plusfval 18653 | The group addition operati... |
| plusfeq 18654 | If the addition operation ... |
| plusffn 18655 | The group addition operati... |
| mgmplusf 18656 | The group addition functio... |
| mgmpropd 18657 | If two structures have the... |
| ismgmd 18658 | Deduce a magma from its pr... |
| issstrmgm 18659 | Characterize a substructur... |
| intopsn 18660 | The internal operation for... |
| mgmb1mgm1 18661 | The only magma with a base... |
| mgm0 18662 | Any set with an empty base... |
| mgm0b 18663 | The structure with an empt... |
| mgm1 18664 | The structure with one ele... |
| opifismgm 18665 | A structure with a group a... |
| mgmidmo 18666 | A two-sided identity eleme... |
| grpidval 18667 | The value of the identity ... |
| grpidpropd 18668 | If two structures have the... |
| fn0g 18669 | The group zero extractor i... |
| 0g0 18670 | The identity element funct... |
| ismgmid 18671 | The identity element of a ... |
| mgmidcl 18672 | The identity element of a ... |
| mgmlrid 18673 | The identity element of a ... |
| ismgmid2 18674 | Show that a given element ... |
| lidrideqd 18675 | If there is a left and rig... |
| lidrididd 18676 | If there is a left and rig... |
| grpidd 18677 | Deduce the identity elemen... |
| mgmidsssn0 18678 | Property of the set of ide... |
| grpinvalem 18679 | Lemma for ~ grpinva . (Co... |
| grpinva 18680 | Deduce right inverse from ... |
| grprida 18681 | Deduce right identity from... |
| gsumvalx 18682 | Expand out the substitutio... |
| gsumval 18683 | Expand out the substitutio... |
| gsumpropd 18684 | The group sum depends only... |
| gsumpropd2lem 18685 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18686 | A stronger version of ~ gs... |
| gsummgmpropd 18687 | A stronger version of ~ gs... |
| gsumress 18688 | The group sum in a substru... |
| gsumval1 18689 | Value of the group sum ope... |
| gsum0 18690 | Value of the empty group s... |
| gsumval2a 18691 | Value of the group sum ope... |
| gsumval2 18692 | Value of the group sum ope... |
| gsumsplit1r 18693 | Splitting off the rightmos... |
| gsumprval 18694 | Value of the group sum ope... |
| gsumpr12val 18695 | Value of the group sum ope... |
| mgmhmrcl 18700 | Reverse closure of a magma... |
| submgmrcl 18701 | Reverse closure for submag... |
| ismgmhm 18702 | Property of a magma homomo... |
| mgmhmf 18703 | A magma homomorphism is a ... |
| mgmhmpropd 18704 | Magma homomorphism depends... |
| mgmhmlin 18705 | A magma homomorphism prese... |
| mgmhmf1o 18706 | A magma homomorphism is bi... |
| idmgmhm 18707 | The identity homomorphism ... |
| issubmgm 18708 | Expand definition of a sub... |
| issubmgm2 18709 | Submagmas are subsets that... |
| rabsubmgmd 18710 | Deduction for proving that... |
| submgmss 18711 | Submagmas are subsets of t... |
| submgmid 18712 | Every magma is trivially a... |
| submgmcl 18713 | Submagmas are closed under... |
| submgmmgm 18714 | Submagmas are themselves m... |
| submgmbas 18715 | The base set of a submagma... |
| subsubmgm 18716 | A submagma of a submagma i... |
| resmgmhm 18717 | Restriction of a magma hom... |
| resmgmhm2 18718 | One direction of ~ resmgmh... |
| resmgmhm2b 18719 | Restriction of the codomai... |
| mgmhmco 18720 | The composition of magma h... |
| mgmhmima 18721 | The homomorphic image of a... |
| mgmhmeql 18722 | The equalizer of two magma... |
| submgmacs 18723 | Submagmas are an algebraic... |
| issgrp 18726 | The predicate "is a semigr... |
| issgrpv 18727 | The predicate "is a semigr... |
| issgrpn0 18728 | The predicate "is a semigr... |
| isnsgrp 18729 | A condition for a structur... |
| sgrpmgm 18730 | A semigroup is a magma. (... |
| sgrpass 18731 | A semigroup operation is a... |
| sgrpcl 18732 | Closure of the operation o... |
| sgrp0 18733 | Any set with an empty base... |
| sgrp0b 18734 | The structure with an empt... |
| sgrp1 18735 | The structure with one ele... |
| issgrpd 18736 | Deduce a semigroup from it... |
| sgrppropd 18737 | If two structures are sets... |
| prdsplusgsgrpcl 18738 | Structure product pointwis... |
| prdssgrpd 18739 | The product of a family of... |
| ismnddef 18742 | The predicate "is a monoid... |
| ismnd 18743 | The predicate "is a monoid... |
| isnmnd 18744 | A condition for a structur... |
| sgrpidmnd 18745 | A semigroup with an identi... |
| mndsgrp 18746 | A monoid is a semigroup. ... |
| mndmgm 18747 | A monoid is a magma. (Con... |
| mndcl 18748 | Closure of the operation o... |
| mndass 18749 | A monoid operation is asso... |
| mndid 18750 | A monoid has a two-sided i... |
| mndideu 18751 | The two-sided identity ele... |
| mnd32g 18752 | Commutative/associative la... |
| mnd12g 18753 | Commutative/associative la... |
| mnd4g 18754 | Commutative/associative la... |
| mndidcl 18755 | The identity element of a ... |
| mndbn0 18756 | The base set of a monoid i... |
| hashfinmndnn 18757 | A finite monoid has positi... |
| mndplusf 18758 | The group addition operati... |
| mndlrid 18759 | A monoid's identity elemen... |
| mndlid 18760 | The identity element of a ... |
| mndrid 18761 | The identity element of a ... |
| ismndd 18762 | Deduce a monoid from its p... |
| mndpfo 18763 | The addition operation of ... |
| mndfo 18764 | The addition operation of ... |
| mndpropd 18765 | If two structures have the... |
| mndprop 18766 | If two structures have the... |
| issubmnd 18767 | Characterize a submonoid b... |
| ress0g 18768 | ` 0g ` is unaffected by re... |
| submnd0 18769 | The zero of a submonoid is... |
| mndinvmod 18770 | Uniqueness of an inverse e... |
| prdsplusgcl 18771 | Structure product pointwis... |
| prdsidlem 18772 | Characterization of identi... |
| prdsmndd 18773 | The product of a family of... |
| prds0g 18774 | Zero in a product of monoi... |
| pwsmnd 18775 | The structure power of a m... |
| pws0g 18776 | Zero in a structure power ... |
| imasmnd2 18777 | The image structure of a m... |
| imasmnd 18778 | The image structure of a m... |
| imasmndf1 18779 | The image of a monoid unde... |
| xpsmnd 18780 | The binary product of mono... |
| xpsmnd0 18781 | The identity element of a ... |
| mnd1 18782 | The (smallest) structure r... |
| mnd1id 18783 | The singleton element of a... |
| ismhm 18788 | Property of a monoid homom... |
| ismhmd 18789 | Deduction version of ~ ism... |
| mhmrcl1 18790 | Reverse closure of a monoi... |
| mhmrcl2 18791 | Reverse closure of a monoi... |
| mhmf 18792 | A monoid homomorphism is a... |
| ismhm0 18793 | Property of a monoid homom... |
| mhmismgmhm 18794 | Each monoid homomorphism i... |
| mhmpropd 18795 | Monoid homomorphism depend... |
| mhmlin 18796 | A monoid homomorphism comm... |
| mhm0 18797 | A monoid homomorphism pres... |
| idmhm 18798 | The identity homomorphism ... |
| mhmf1o 18799 | A monoid homomorphism is b... |
| mndvcl 18800 | Tuple-wise additive closur... |
| mndvass 18801 | Tuple-wise associativity i... |
| mndvlid 18802 | Tuple-wise left identity i... |
| mndvrid 18803 | Tuple-wise right identity ... |
| mhmvlin 18804 | Tuple extension of monoid ... |
| submrcl 18805 | Reverse closure for submon... |
| issubm 18806 | Expand definition of a sub... |
| issubm2 18807 | Submonoids are subsets tha... |
| issubmndb 18808 | The submonoid predicate. ... |
| issubmd 18809 | Deduction for proving a su... |
| mndissubm 18810 | If the base set of a monoi... |
| resmndismnd 18811 | If the base set of a monoi... |
| submss 18812 | Submonoids are subsets of ... |
| submid 18813 | Every monoid is trivially ... |
| subm0cl 18814 | Submonoids contain zero. ... |
| submcl 18815 | Submonoids are closed unde... |
| submmnd 18816 | Submonoids are themselves ... |
| submbas 18817 | The base set of a submonoi... |
| subm0 18818 | Submonoids have the same i... |
| subsubm 18819 | A submonoid of a submonoid... |
| 0subm 18820 | The zero submonoid of an a... |
| insubm 18821 | The intersection of two su... |
| 0mhm 18822 | The constant zero linear f... |
| resmhm 18823 | Restriction of a monoid ho... |
| resmhm2 18824 | One direction of ~ resmhm2... |
| resmhm2b 18825 | Restriction of the codomai... |
| mhmco 18826 | The composition of monoid ... |
| mhmimalem 18827 | Lemma for ~ mhmima and sim... |
| mhmima 18828 | The homomorphic image of a... |
| mhmeql 18829 | The equalizer of two monoi... |
| submacs 18830 | Submonoids are an algebrai... |
| mndind 18831 | Induction in a monoid. In... |
| prdspjmhm 18832 | A projection from a produc... |
| pwspjmhm 18833 | A projection from a struct... |
| pwsdiagmhm 18834 | Diagonal monoid homomorphi... |
| pwsco1mhm 18835 | Right composition with a f... |
| pwsco2mhm 18836 | Left composition with a mo... |
| gsumvallem2 18837 | Lemma for properties of th... |
| gsumsubm 18838 | Evaluate a group sum in a ... |
| gsumz 18839 | Value of a group sum over ... |
| gsumwsubmcl 18840 | Closure of the composite i... |
| gsumws1 18841 | A singleton composite reco... |
| gsumwcl 18842 | Closure of the composite o... |
| gsumsgrpccat 18843 | Homomorphic property of no... |
| gsumccat 18844 | Homomorphic property of co... |
| gsumws2 18845 | Valuation of a pair in a m... |
| gsumccatsn 18846 | Homomorphic property of co... |
| gsumspl 18847 | The primary purpose of the... |
| gsumwmhm 18848 | Behavior of homomorphisms ... |
| gsumwspan 18849 | The submonoid generated by... |
| frmdval 18854 | Value of the free monoid c... |
| frmdbas 18855 | The base set of a free mon... |
| frmdelbas 18856 | An element of the base set... |
| frmdplusg 18857 | The monoid operation of a ... |
| frmdadd 18858 | Value of the monoid operat... |
| vrmdfval 18859 | The canonical injection fr... |
| vrmdval 18860 | The value of the generatin... |
| vrmdf 18861 | The mapping from the index... |
| frmdmnd 18862 | A free monoid is a monoid.... |
| frmd0 18863 | The identity of the free m... |
| frmdsssubm 18864 | The set of words taking va... |
| frmdgsum 18865 | Any word in a free monoid ... |
| frmdss2 18866 | A subset of generators is ... |
| frmdup1 18867 | Any assignment of the gene... |
| frmdup2 18868 | The evaluation map has the... |
| frmdup3lem 18869 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18870 | Universal property of the ... |
| efmnd 18873 | The monoid of endofunction... |
| efmndbas 18874 | The base set of the monoid... |
| efmndbasabf 18875 | The base set of the monoid... |
| elefmndbas 18876 | Two ways of saying a funct... |
| elefmndbas2 18877 | Two ways of saying a funct... |
| efmndbasf 18878 | Elements in the monoid of ... |
| efmndhash 18879 | The monoid of endofunction... |
| efmndbasfi 18880 | The monoid of endofunction... |
| efmndfv 18881 | The function value of an e... |
| efmndtset 18882 | The topology of the monoid... |
| efmndplusg 18883 | The group operation of a m... |
| efmndov 18884 | The value of the group ope... |
| efmndcl 18885 | The group operation of the... |
| efmndtopn 18886 | The topology of the monoid... |
| symggrplem 18887 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18888 | The monoid of endofunction... |
| efmndsgrp 18889 | The monoid of endofunction... |
| ielefmnd 18890 | The identity function rest... |
| efmndid 18891 | The identity function rest... |
| efmndmnd 18892 | The monoid of endofunction... |
| efmnd0nmnd 18893 | Even the monoid of endofun... |
| efmndbas0 18894 | The base set of the monoid... |
| efmnd1hash 18895 | The monoid of endofunction... |
| efmnd1bas 18896 | The monoid of endofunction... |
| efmnd2hash 18897 | The monoid of endofunction... |
| submefmnd 18898 | If the base set of a monoi... |
| sursubmefmnd 18899 | The set of surjective endo... |
| injsubmefmnd 18900 | The set of injective endof... |
| idressubmefmnd 18901 | The singleton containing o... |
| idresefmnd 18902 | The structure with the sin... |
| smndex1ibas 18903 | The modulo function ` I ` ... |
| smndex1iidm 18904 | The modulo function ` I ` ... |
| smndex1gbas 18905 | The constant functions ` (... |
| smndex1gid 18906 | The composition of a const... |
| smndex1igid 18907 | The composition of the mod... |
| smndex1basss 18908 | The modulo function ` I ` ... |
| smndex1bas 18909 | The base set of the monoid... |
| smndex1mgm 18910 | The monoid of endofunction... |
| smndex1sgrp 18911 | The monoid of endofunction... |
| smndex1mndlem 18912 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18913 | The monoid of endofunction... |
| smndex1id 18914 | The modulo function ` I ` ... |
| smndex1n0mnd 18915 | The identity of the monoid... |
| nsmndex1 18916 | The base set ` B ` of the ... |
| smndex2dbas 18917 | The doubling function ` D ... |
| smndex2dnrinv 18918 | The doubling function ` D ... |
| smndex2hbas 18919 | The halving functions ` H ... |
| smndex2dlinvh 18920 | The halving functions ` H ... |
| mgm2nsgrplem1 18921 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18922 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18923 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18924 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18925 | A small magma (with two el... |
| sgrp2nmndlem1 18926 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18927 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18928 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18929 | A small semigroup (with tw... |
| sgrp2rid2ex 18930 | A small semigroup (with tw... |
| sgrp2nmndlem4 18931 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18932 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18933 | A small semigroup (with tw... |
| mgmnsgrpex 18934 | There is a magma which is ... |
| sgrpnmndex 18935 | There is a semigroup which... |
| sgrpssmgm 18936 | The class of all semigroup... |
| mndsssgrp 18937 | The class of all monoids i... |
| pwmndgplus 18938 | The operation of the monoi... |
| pwmndid 18939 | The identity of the monoid... |
| pwmnd 18940 | The power set of a class `... |
| isgrp 18947 | The predicate "is a group"... |
| grpmnd 18948 | A group is a monoid. (Con... |
| grpcl 18949 | Closure of the operation o... |
| grpass 18950 | A group operation is assoc... |
| grpinvex 18951 | Every member of a group ha... |
| grpideu 18952 | The two-sided identity ele... |
| grpassd 18953 | A group operation is assoc... |
| grpmndd 18954 | A group is a monoid. (Con... |
| grpcld 18955 | Closure of the operation o... |
| grpplusf 18956 | The group addition operati... |
| grpplusfo 18957 | The group addition operati... |
| resgrpplusfrn 18958 | The underlying set of a gr... |
| grppropd 18959 | If two structures have the... |
| grpprop 18960 | If two structures have the... |
| grppropstr 18961 | Generalize a specific 2-el... |
| grpss 18962 | Show that a structure exte... |
| isgrpd2e 18963 | Deduce a group from its pr... |
| isgrpd2 18964 | Deduce a group from its pr... |
| isgrpde 18965 | Deduce a group from its pr... |
| isgrpd 18966 | Deduce a group from its pr... |
| isgrpi 18967 | Properties that determine ... |
| grpsgrp 18968 | A group is a semigroup. (... |
| grpmgmd 18969 | A group is a magma, deduct... |
| dfgrp2 18970 | Alternate definition of a ... |
| dfgrp2e 18971 | Alternate definition of a ... |
| isgrpix 18972 | Properties that determine ... |
| grpidcl 18973 | The identity element of a ... |
| grpbn0 18974 | The base set of a group is... |
| grplid 18975 | The identity element of a ... |
| grprid 18976 | The identity element of a ... |
| grplidd 18977 | The identity element of a ... |
| grpridd 18978 | The identity element of a ... |
| grpn0 18979 | A group is not empty. (Co... |
| hashfingrpnn 18980 | A finite group has positiv... |
| grprcan 18981 | Right cancellation law for... |
| grpinveu 18982 | The left inverse element o... |
| grpid 18983 | Two ways of saying that an... |
| isgrpid2 18984 | Properties showing that an... |
| grpidd2 18985 | Deduce the identity elemen... |
| grpinvfval 18986 | The inverse function of a ... |
| grpinvfvalALT 18987 | Shorter proof of ~ grpinvf... |
| grpinvval 18988 | The inverse of a group ele... |
| grpinvfn 18989 | Functionality of the group... |
| grpinvfvi 18990 | The group inverse function... |
| grpsubfval 18991 | Group subtraction (divisio... |
| grpsubfvalALT 18992 | Shorter proof of ~ grpsubf... |
| grpsubval 18993 | Group subtraction (divisio... |
| grpinvf 18994 | The group inversion operat... |
| grpinvcl 18995 | A group element's inverse ... |
| grpinvcld 18996 | A group element's inverse ... |
| grplinv 18997 | The left inverse of a grou... |
| grprinv 18998 | The right inverse of a gro... |
| grpinvid1 18999 | The inverse of a group ele... |
| grpinvid2 19000 | The inverse of a group ele... |
| isgrpinv 19001 | Properties showing that a ... |
| grplinvd 19002 | The left inverse of a grou... |
| grprinvd 19003 | The right inverse of a gro... |
| grplrinv 19004 | In a group, every member h... |
| grpidinv2 19005 | A group's properties using... |
| grpidinv 19006 | A group has a left and rig... |
| grpinvid 19007 | The inverse of the identit... |
| grplcan 19008 | Left cancellation law for ... |
| grpasscan1 19009 | An associative cancellatio... |
| grpasscan2 19010 | An associative cancellatio... |
| grpidrcan 19011 | If right adding an element... |
| grpidlcan 19012 | If left adding an element ... |
| grpinvinv 19013 | Double inverse law for gro... |
| grpinvcnv 19014 | The group inverse is its o... |
| grpinv11 19015 | The group inverse is one-t... |
| grpinv11OLD 19016 | Obsolete version of ~ grpi... |
| grpinvf1o 19017 | The group inverse is a one... |
| grpinvnz 19018 | The inverse of a nonzero g... |
| grpinvnzcl 19019 | The inverse of a nonzero g... |
| grpsubinv 19020 | Subtraction of an inverse.... |
| grplmulf1o 19021 | Left multiplication by a g... |
| grpraddf1o 19022 | Right addition by a group ... |
| grpinvpropd 19023 | If two structures have the... |
| grpidssd 19024 | If the base set of a group... |
| grpinvssd 19025 | If the base set of a group... |
| grpinvadd 19026 | The inverse of the group o... |
| grpsubf 19027 | Functionality of group sub... |
| grpsubcl 19028 | Closure of group subtracti... |
| grpsubrcan 19029 | Right cancellation law for... |
| grpinvsub 19030 | Inverse of a group subtrac... |
| grpinvval2 19031 | A ~ df-neg -like equation ... |
| grpsubid 19032 | Subtraction of a group ele... |
| grpsubid1 19033 | Subtraction of the identit... |
| grpsubeq0 19034 | If the difference between ... |
| grpsubadd0sub 19035 | Subtraction expressed as a... |
| grpsubadd 19036 | Relationship between group... |
| grpsubsub 19037 | Double group subtraction. ... |
| grpaddsubass 19038 | Associative-type law for g... |
| grppncan 19039 | Cancellation law for subtr... |
| grpnpcan 19040 | Cancellation law for subtr... |
| grpsubsub4 19041 | Double group subtraction (... |
| grppnpcan2 19042 | Cancellation law for mixed... |
| grpnpncan 19043 | Cancellation law for group... |
| grpnpncan0 19044 | Cancellation law for group... |
| grpnnncan2 19045 | Cancellation law for group... |
| dfgrp3lem 19046 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 19047 | Alternate definition of a ... |
| dfgrp3e 19048 | Alternate definition of a ... |
| grplactfval 19049 | The left group action of e... |
| grplactval 19050 | The value of the left grou... |
| grplactcnv 19051 | The left group action of e... |
| grplactf1o 19052 | The left group action of e... |
| grpsubpropd 19053 | Weak property deduction fo... |
| grpsubpropd2 19054 | Strong property deduction ... |
| grp1 19055 | The (smallest) structure r... |
| grp1inv 19056 | The inverse function of th... |
| prdsinvlem 19057 | Characterization of invers... |
| prdsgrpd 19058 | The product of a family of... |
| prdsinvgd 19059 | Negation in a product of g... |
| pwsgrp 19060 | A structure power of a gro... |
| pwsinvg 19061 | Negation in a group power.... |
| pwssub 19062 | Subtraction in a group pow... |
| imasgrp2 19063 | The image structure of a g... |
| imasgrp 19064 | The image structure of a g... |
| imasgrpf1 19065 | The image of a group under... |
| qusgrp2 19066 | Prove that a quotient stru... |
| xpsgrp 19067 | The binary product of grou... |
| xpsinv 19068 | Value of the negation oper... |
| xpsgrpsub 19069 | Value of the subtraction o... |
| mhmlem 19070 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 19071 | A surjective monoid morphi... |
| mhmmnd 19072 | The image of a monoid ` G ... |
| mhmfmhm 19073 | The function fulfilling th... |
| ghmgrp 19074 | The image of a group ` G `... |
| mulgfval 19077 | Group multiple (exponentia... |
| mulgfvalALT 19078 | Shorter proof of ~ mulgfva... |
| mulgval 19079 | Value of the group multipl... |
| mulgfn 19080 | Functionality of the group... |
| mulgfvi 19081 | The group multiple operati... |
| mulg0 19082 | Group multiple (exponentia... |
| mulgnn 19083 | Group multiple (exponentia... |
| ressmulgnn 19084 | Values for the group multi... |
| ressmulgnn0 19085 | Values for the group multi... |
| ressmulgnnd 19086 | Values for the group multi... |
| mulgnngsum 19087 | Group multiple (exponentia... |
| mulgnn0gsum 19088 | Group multiple (exponentia... |
| mulg1 19089 | Group multiple (exponentia... |
| mulgnnp1 19090 | Group multiple (exponentia... |
| mulg2 19091 | Group multiple (exponentia... |
| mulgnegnn 19092 | Group multiple (exponentia... |
| mulgnn0p1 19093 | Group multiple (exponentia... |
| mulgnnsubcl 19094 | Closure of the group multi... |
| mulgnn0subcl 19095 | Closure of the group multi... |
| mulgsubcl 19096 | Closure of the group multi... |
| mulgnncl 19097 | Closure of the group multi... |
| mulgnn0cl 19098 | Closure of the group multi... |
| mulgcl 19099 | Closure of the group multi... |
| mulgneg 19100 | Group multiple (exponentia... |
| mulgnegneg 19101 | The inverse of a negative ... |
| mulgm1 19102 | Group multiple (exponentia... |
| mulgnn0cld 19103 | Closure of the group multi... |
| mulgcld 19104 | Deduction associated with ... |
| mulgaddcomlem 19105 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19106 | The group multiple operato... |
| mulginvcom 19107 | The group multiple operato... |
| mulginvinv 19108 | The group multiple operato... |
| mulgnn0z 19109 | A group multiple of the id... |
| mulgz 19110 | A group multiple of the id... |
| mulgnndir 19111 | Sum of group multiples, fo... |
| mulgnn0dir 19112 | Sum of group multiples, ge... |
| mulgdirlem 19113 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19114 | Sum of group multiples, ge... |
| mulgp1 19115 | Group multiple (exponentia... |
| mulgneg2 19116 | Group multiple (exponentia... |
| mulgnnass 19117 | Product of group multiples... |
| mulgnn0ass 19118 | Product of group multiples... |
| mulgass 19119 | Product of group multiples... |
| mulgassr 19120 | Reversed product of group ... |
| mulgmodid 19121 | Casting out multiples of t... |
| mulgsubdir 19122 | Distribution of group mult... |
| mhmmulg 19123 | A homomorphism of monoids ... |
| mulgpropd 19124 | Two structures with the sa... |
| submmulgcl 19125 | Closure of the group multi... |
| submmulg 19126 | A group multiple is the sa... |
| pwsmulg 19127 | Value of a group multiple ... |
| issubg 19134 | The subgroup predicate. (... |
| subgss 19135 | A subgroup is a subset. (... |
| subgid 19136 | A group is a subgroup of i... |
| subggrp 19137 | A subgroup is a group. (C... |
| subgbas 19138 | The base of the restricted... |
| subgrcl 19139 | Reverse closure for the su... |
| subg0 19140 | A subgroup of a group must... |
| subginv 19141 | The inverse of an element ... |
| subg0cl 19142 | The group identity is an e... |
| subginvcl 19143 | The inverse of an element ... |
| subgcl 19144 | A subgroup is closed under... |
| subgsubcl 19145 | A subgroup is closed under... |
| subgsub 19146 | The subtraction of element... |
| subgmulgcl 19147 | Closure of the group multi... |
| subgmulg 19148 | A group multiple is the sa... |
| issubg2 19149 | Characterize the subgroups... |
| issubgrpd2 19150 | Prove a subgroup by closur... |
| issubgrpd 19151 | Prove a subgroup by closur... |
| issubg3 19152 | A subgroup is a symmetric ... |
| issubg4 19153 | A subgroup is a nonempty s... |
| grpissubg 19154 | If the base set of a group... |
| resgrpisgrp 19155 | If the base set of a group... |
| subgsubm 19156 | A subgroup is a submonoid.... |
| subsubg 19157 | A subgroup of a subgroup i... |
| subgint 19158 | The intersection of a none... |
| 0subg 19159 | The zero subgroup of an ar... |
| 0subgOLD 19160 | Obsolete version of ~ 0sub... |
| trivsubgd 19161 | The only subgroup of a tri... |
| trivsubgsnd 19162 | The only subgroup of a tri... |
| isnsg 19163 | Property of being a normal... |
| isnsg2 19164 | Weaken the condition of ~ ... |
| nsgbi 19165 | Defining property of a nor... |
| nsgsubg 19166 | A normal subgroup is a sub... |
| nsgconj 19167 | The conjugation of an elem... |
| isnsg3 19168 | A subgroup is normal iff t... |
| subgacs 19169 | Subgroups are an algebraic... |
| nsgacs 19170 | Normal subgroups form an a... |
| elnmz 19171 | Elementhood in the normali... |
| nmzbi 19172 | Defining property of the n... |
| nmzsubg 19173 | The normalizer N_G(S) of a... |
| ssnmz 19174 | A subgroup is a subset of ... |
| isnsg4 19175 | A subgroup is normal iff i... |
| nmznsg 19176 | Any subgroup is a normal s... |
| 0nsg 19177 | The zero subgroup is norma... |
| nsgid 19178 | The whole group is a norma... |
| 0idnsgd 19179 | The whole group and the ze... |
| trivnsgd 19180 | The only normal subgroup o... |
| triv1nsgd 19181 | A trivial group has exactl... |
| 1nsgtrivd 19182 | A group with exactly one n... |
| releqg 19183 | The left coset equivalence... |
| eqgfval 19184 | Value of the subgroup left... |
| eqgval 19185 | Value of the subgroup left... |
| eqger 19186 | The subgroup coset equival... |
| eqglact 19187 | A left coset can be expres... |
| eqgid 19188 | The left coset containing ... |
| eqgen 19189 | Each coset is equipotent t... |
| eqgcpbl 19190 | The subgroup coset equival... |
| eqg0el 19191 | Equivalence class of a quo... |
| quselbas 19192 | Membership in the base set... |
| quseccl0 19193 | Closure of the quotient ma... |
| qusgrp 19194 | If ` Y ` is a normal subgr... |
| quseccl 19195 | Closure of the quotient ma... |
| qusadd 19196 | Value of the group operati... |
| qus0 19197 | Value of the group identit... |
| qusinv 19198 | Value of the group inverse... |
| qussub 19199 | Value of the group subtrac... |
| ecqusaddd 19200 | Addition of equivalence cl... |
| ecqusaddcl 19201 | Closure of the addition in... |
| lagsubg2 19202 | Lagrange's theorem for fin... |
| lagsubg 19203 | Lagrange's theorem for Gro... |
| eqg0subg 19204 | The coset equivalence rela... |
| eqg0subgecsn 19205 | The equivalence classes mo... |
| qus0subgbas 19206 | The base set of a quotient... |
| qus0subgadd 19207 | The addition in a quotient... |
| cycsubmel 19208 | Characterization of an ele... |
| cycsubmcl 19209 | The set of nonnegative int... |
| cycsubm 19210 | The set of nonnegative int... |
| cyccom 19211 | Condition for an operation... |
| cycsubmcom 19212 | The operation of a monoid ... |
| cycsubggend 19213 | The cyclic subgroup genera... |
| cycsubgcl 19214 | The set of integer powers ... |
| cycsubgss 19215 | The cyclic subgroup genera... |
| cycsubg 19216 | The cyclic group generated... |
| cycsubgcld 19217 | The cyclic subgroup genera... |
| cycsubg2 19218 | The subgroup generated by ... |
| cycsubg2cl 19219 | Any multiple of an element... |
| reldmghm 19222 | Lemma for group homomorphi... |
| isghm 19223 | Property of being a homomo... |
| isghmOLD 19224 | Obsolete version of ~ isgh... |
| isghm3 19225 | Property of a group homomo... |
| ghmgrp1 19226 | A group homomorphism is on... |
| ghmgrp2 19227 | A group homomorphism is on... |
| ghmf 19228 | A group homomorphism is a ... |
| ghmlin 19229 | A homomorphism of groups i... |
| ghmid 19230 | A homomorphism of groups p... |
| ghminv 19231 | A homomorphism of groups p... |
| ghmsub 19232 | Linearity of subtraction t... |
| isghmd 19233 | Deduction for a group homo... |
| ghmmhm 19234 | A group homomorphism is a ... |
| ghmmhmb 19235 | Group homomorphisms and mo... |
| ghmmulg 19236 | A group homomorphism prese... |
| ghmrn 19237 | The range of a homomorphis... |
| 0ghm 19238 | The constant zero linear f... |
| idghm 19239 | The identity homomorphism ... |
| resghm 19240 | Restriction of a homomorph... |
| resghm2 19241 | One direction of ~ resghm2... |
| resghm2b 19242 | Restriction of the codomai... |
| ghmghmrn 19243 | A group homomorphism from ... |
| ghmco 19244 | The composition of group h... |
| ghmima 19245 | The image of a subgroup un... |
| ghmpreima 19246 | The inverse image of a sub... |
| ghmeql 19247 | The equalizer of two group... |
| ghmnsgima 19248 | The image of a normal subg... |
| ghmnsgpreima 19249 | The inverse image of a nor... |
| ghmker 19250 | The kernel of a homomorphi... |
| ghmeqker 19251 | Two source points map to t... |
| pwsdiagghm 19252 | Diagonal homomorphism into... |
| f1ghm0to0 19253 | If a group homomorphism ` ... |
| ghmf1 19254 | Two ways of saying a group... |
| kerf1ghm 19255 | A group homomorphism ` F `... |
| ghmf1o 19256 | A bijective group homomorp... |
| conjghm 19257 | Conjugation is an automorp... |
| conjsubg 19258 | A conjugated subgroup is a... |
| conjsubgen 19259 | A conjugated subgroup is e... |
| conjnmz 19260 | A subgroup is unchanged un... |
| conjnmzb 19261 | Alternative condition for ... |
| conjnsg 19262 | A normal subgroup is uncha... |
| qusghm 19263 | If ` Y ` is a normal subgr... |
| ghmpropd 19264 | Group homomorphism depends... |
| gimfn 19269 | The group isomorphism func... |
| isgim 19270 | An isomorphism of groups i... |
| gimf1o 19271 | An isomorphism of groups i... |
| gimghm 19272 | An isomorphism of groups i... |
| isgim2 19273 | A group isomorphism is a h... |
| subggim 19274 | Behavior of subgroups unde... |
| gimcnv 19275 | The converse of a group is... |
| gimco 19276 | The composition of group i... |
| gim0to0 19277 | A group isomorphism maps t... |
| brgic 19278 | The relation "is isomorphi... |
| brgici 19279 | Prove isomorphic by an exp... |
| gicref 19280 | Isomorphism is reflexive. ... |
| giclcl 19281 | Isomorphism implies the le... |
| gicrcl 19282 | Isomorphism implies the ri... |
| gicsym 19283 | Isomorphism is symmetric. ... |
| gictr 19284 | Isomorphism is transitive.... |
| gicer 19285 | Isomorphism is an equivale... |
| gicen 19286 | Isomorphic groups have equ... |
| gicsubgen 19287 | A less trivial example of ... |
| ghmqusnsglem1 19288 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19289 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19290 | The mapping ` H ` induced ... |
| ghmquskerlem1 19291 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19292 | In the case of theorem ~ g... |
| ghmquskerlem2 19293 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19294 | The mapping ` H ` induced ... |
| ghmqusker 19295 | A surjective group homomor... |
| gicqusker 19296 | The image ` H ` of a group... |
| isga 19299 | The predicate "is a (left)... |
| gagrp 19300 | The left argument of a gro... |
| gaset 19301 | The right argument of a gr... |
| gagrpid 19302 | The identity of the group ... |
| gaf 19303 | The mapping of the group a... |
| gafo 19304 | A group action is onto its... |
| gaass 19305 | An "associative" property ... |
| ga0 19306 | The action of a group on t... |
| gaid 19307 | The trivial action of a gr... |
| subgga 19308 | A subgroup acts on its par... |
| gass 19309 | A subset of a group action... |
| gasubg 19310 | The restriction of a group... |
| gaid2 19311 | A group operation is a lef... |
| galcan 19312 | The action of a particular... |
| gacan 19313 | Group inverses cancel in a... |
| gapm 19314 | The action of a particular... |
| gaorb 19315 | The orbit equivalence rela... |
| gaorber 19316 | The orbit equivalence rela... |
| gastacl 19317 | The stabilizer subgroup in... |
| gastacos 19318 | Write the coset relation f... |
| orbstafun 19319 | Existence and uniqueness f... |
| orbstaval 19320 | Value of the function at a... |
| orbsta 19321 | The Orbit-Stabilizer theor... |
| orbsta2 19322 | Relation between the size ... |
| cntrval 19327 | Substitute definition of t... |
| cntzfval 19328 | First level substitution f... |
| cntzval 19329 | Definition substitution fo... |
| elcntz 19330 | Elementhood in the central... |
| cntzel 19331 | Membership in a centralize... |
| cntzsnval 19332 | Special substitution for t... |
| elcntzsn 19333 | Value of the centralizer o... |
| sscntz 19334 | A centralizer expression f... |
| cntzrcl 19335 | Reverse closure for elemen... |
| cntzssv 19336 | The centralizer is uncondi... |
| cntzi 19337 | Membership in a centralize... |
| elcntr 19338 | Elementhood in the center ... |
| cntrss 19339 | The center is a subset of ... |
| cntri 19340 | Defining property of the c... |
| resscntz 19341 | Centralizer in a substruct... |
| cntzsgrpcl 19342 | Centralizers are closed un... |
| cntz2ss 19343 | Centralizers reverse the s... |
| cntzrec 19344 | Reciprocity relationship f... |
| cntziinsn 19345 | Express any centralizer as... |
| cntzsubm 19346 | Centralizers in a monoid a... |
| cntzsubg 19347 | Centralizers in a group ar... |
| cntzidss 19348 | If the elements of ` S ` c... |
| cntzmhm 19349 | Centralizers in a monoid a... |
| cntzmhm2 19350 | Centralizers in a monoid a... |
| cntrsubgnsg 19351 | A central subgroup is norm... |
| cntrnsg 19352 | The center of a group is a... |
| oppgval 19355 | Value of the opposite grou... |
| oppgplusfval 19356 | Value of the addition oper... |
| oppgplus 19357 | Value of the addition oper... |
| setsplusg 19358 | The other components of an... |
| oppglemOLD 19359 | Obsolete version of ~ sets... |
| oppgbas 19360 | Base set of an opposite gr... |
| oppgbasOLD 19361 | Obsolete version of ~ oppg... |
| oppgtset 19362 | Topology of an opposite gr... |
| oppgtsetOLD 19363 | Obsolete version of ~ oppg... |
| oppgtopn 19364 | Topology of an opposite gr... |
| oppgmnd 19365 | The opposite of a monoid i... |
| oppgmndb 19366 | Bidirectional form of ~ op... |
| oppgid 19367 | Zero in a monoid is a symm... |
| oppggrp 19368 | The opposite of a group is... |
| oppggrpb 19369 | Bidirectional form of ~ op... |
| oppginv 19370 | Inverses in a group are a ... |
| invoppggim 19371 | The inverse is an antiauto... |
| oppggic 19372 | Every group is (naturally)... |
| oppgsubm 19373 | Being a submonoid is a sym... |
| oppgsubg 19374 | Being a subgroup is a symm... |
| oppgcntz 19375 | A centralizer in a group i... |
| oppgcntr 19376 | The center of a group is t... |
| gsumwrev 19377 | A sum in an opposite monoi... |
| symgval 19380 | The value of the symmetric... |
| symgbas 19381 | The base set of the symmet... |
| elsymgbas2 19382 | Two ways of saying a funct... |
| elsymgbas 19383 | Two ways of saying a funct... |
| symgbasf1o 19384 | Elements in the symmetric ... |
| symgbasf 19385 | A permutation (element of ... |
| symgbasmap 19386 | A permutation (element of ... |
| symghash 19387 | The symmetric group on ` n... |
| symgbasfi 19388 | The symmetric group on a f... |
| symgfv 19389 | The function value of a pe... |
| symgfvne 19390 | The function values of a p... |
| symgressbas 19391 | The symmetric group on ` A... |
| symgplusg 19392 | The group operation of a s... |
| symgov 19393 | The value of the group ope... |
| symgcl 19394 | The group operation of the... |
| idresperm 19395 | The identity function rest... |
| symgmov1 19396 | For a permutation of a set... |
| symgmov2 19397 | For a permutation of a set... |
| symgbas0 19398 | The base set of the symmet... |
| symg1hash 19399 | The symmetric group on a s... |
| symg1bas 19400 | The symmetric group on a s... |
| symg2hash 19401 | The symmetric group on a (... |
| symg2bas 19402 | The symmetric group on a p... |
| 0symgefmndeq 19403 | The symmetric group on the... |
| snsymgefmndeq 19404 | The symmetric group on a s... |
| symgpssefmnd 19405 | For a set ` A ` with more ... |
| symgvalstruct 19406 | The value of the symmetric... |
| symgvalstructOLD 19407 | Obsolete proof of ~ symgva... |
| symgsubmefmnd 19408 | The symmetric group on a s... |
| symgtset 19409 | The topology of the symmet... |
| symggrp 19410 | The symmetric group on a s... |
| symgid 19411 | The group identity element... |
| symginv 19412 | The group inverse in the s... |
| symgsubmefmndALT 19413 | The symmetric group on a s... |
| galactghm 19414 | The currying of a group ac... |
| lactghmga 19415 | The converse of ~ galactgh... |
| symgtopn 19416 | The topology of the symmet... |
| symgga 19417 | The symmetric group induce... |
| pgrpsubgsymgbi 19418 | Every permutation group is... |
| pgrpsubgsymg 19419 | Every permutation group is... |
| idressubgsymg 19420 | The singleton containing o... |
| idrespermg 19421 | The structure with the sin... |
| cayleylem1 19422 | Lemma for ~ cayley . (Con... |
| cayleylem2 19423 | Lemma for ~ cayley . (Con... |
| cayley 19424 | Cayley's Theorem (construc... |
| cayleyth 19425 | Cayley's Theorem (existenc... |
| symgfix2 19426 | If a permutation does not ... |
| symgextf 19427 | The extension of a permuta... |
| symgextfv 19428 | The function value of the ... |
| symgextfve 19429 | The function value of the ... |
| symgextf1lem 19430 | Lemma for ~ symgextf1 . (... |
| symgextf1 19431 | The extension of a permuta... |
| symgextfo 19432 | The extension of a permuta... |
| symgextf1o 19433 | The extension of a permuta... |
| symgextsymg 19434 | The extension of a permuta... |
| symgextres 19435 | The restriction of the ext... |
| gsumccatsymgsn 19436 | Homomorphic property of co... |
| gsmsymgrfixlem1 19437 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19438 | The composition of permuta... |
| fvcosymgeq 19439 | The values of two composit... |
| gsmsymgreqlem1 19440 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19441 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19442 | Two combination of permuta... |
| symgfixelq 19443 | A permutation of a set fix... |
| symgfixels 19444 | The restriction of a permu... |
| symgfixelsi 19445 | The restriction of a permu... |
| symgfixf 19446 | The mapping of a permutati... |
| symgfixf1 19447 | The mapping of a permutati... |
| symgfixfolem1 19448 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19449 | The mapping of a permutati... |
| symgfixf1o 19450 | The mapping of a permutati... |
| f1omvdmvd 19453 | A permutation of any class... |
| f1omvdcnv 19454 | A permutation and its inve... |
| mvdco 19455 | Composing two permutations... |
| f1omvdconj 19456 | Conjugation of a permutati... |
| f1otrspeq 19457 | A transposition is charact... |
| f1omvdco2 19458 | If exactly one of two perm... |
| f1omvdco3 19459 | If a point is moved by exa... |
| pmtrfval 19460 | The function generating tr... |
| pmtrval 19461 | A generated transposition,... |
| pmtrfv 19462 | General value of mapping a... |
| pmtrprfv 19463 | In a transposition of two ... |
| pmtrprfv3 19464 | In a transposition of two ... |
| pmtrf 19465 | Functionality of a transpo... |
| pmtrmvd 19466 | A transposition moves prec... |
| pmtrrn 19467 | Transposing two points giv... |
| pmtrfrn 19468 | A transposition (as a kind... |
| pmtrffv 19469 | Mapping of a point under a... |
| pmtrrn2 19470 | For any transposition ther... |
| pmtrfinv 19471 | A transposition function i... |
| pmtrfmvdn0 19472 | A transposition moves at l... |
| pmtrff1o 19473 | A transposition function i... |
| pmtrfcnv 19474 | A transposition function i... |
| pmtrfb 19475 | An intrinsic characterizat... |
| pmtrfconj 19476 | Any conjugate of a transpo... |
| symgsssg 19477 | The symmetric group has su... |
| symgfisg 19478 | The symmetric group has a ... |
| symgtrf 19479 | Transpositions are element... |
| symggen 19480 | The span of the transposit... |
| symggen2 19481 | A finite permutation group... |
| symgtrinv 19482 | To invert a permutation re... |
| pmtr3ncomlem1 19483 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19484 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19485 | Transpositions over sets w... |
| pmtrdifellem1 19486 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19487 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19488 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19489 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19490 | A transposition of element... |
| pmtrdifwrdellem1 19491 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19492 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19493 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19494 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19495 | A sequence of transpositio... |
| pmtrdifwrdel2 19496 | A sequence of transpositio... |
| pmtrprfval 19497 | The transpositions on a pa... |
| pmtrprfvalrn 19498 | The range of the transposi... |
| psgnunilem1 19503 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19504 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19505 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19506 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19507 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19508 | Addition and subtraction o... |
| psgnuni 19509 | If the same permutation ca... |
| psgnfval 19510 | Function definition of the... |
| psgnfn 19511 | Functionality and domain o... |
| psgndmsubg 19512 | The finitary permutations ... |
| psgneldm 19513 | Property of being a finita... |
| psgneldm2 19514 | The finitary permutations ... |
| psgneldm2i 19515 | A sequence of transpositio... |
| psgneu 19516 | A finitary permutation has... |
| psgnval 19517 | Value of the permutation s... |
| psgnvali 19518 | A finitary permutation has... |
| psgnvalii 19519 | Any representation of a pe... |
| psgnpmtr 19520 | All transpositions are odd... |
| psgn0fv0 19521 | The permutation sign funct... |
| sygbasnfpfi 19522 | The class of non-fixed poi... |
| psgnfvalfi 19523 | Function definition of the... |
| psgnvalfi 19524 | Value of the permutation s... |
| psgnran 19525 | The range of the permutati... |
| gsmtrcl 19526 | The group sum of transposi... |
| psgnfitr 19527 | A permutation of a finite ... |
| psgnfieu 19528 | A permutation of a finite ... |
| pmtrsn 19529 | The value of the transposi... |
| psgnsn 19530 | The permutation sign funct... |
| psgnprfval 19531 | The permutation sign funct... |
| psgnprfval1 19532 | The permutation sign of th... |
| psgnprfval2 19533 | The permutation sign of th... |
| odfval 19542 | Value of the order functio... |
| odfvalALT 19543 | Shorter proof of ~ odfval ... |
| odval 19544 | Second substitution for th... |
| odlem1 19545 | The group element order is... |
| odcl 19546 | The order of a group eleme... |
| odf 19547 | Functionality of the group... |
| odid 19548 | Any element to the power o... |
| odlem2 19549 | Any positive annihilator o... |
| odmodnn0 19550 | Reduce the argument of a g... |
| mndodconglem 19551 | Lemma for ~ mndodcong . (... |
| mndodcong 19552 | If two multipliers are con... |
| mndodcongi 19553 | If two multipliers are con... |
| oddvdsnn0 19554 | The only multiples of ` A ... |
| odnncl 19555 | If a nonzero multiple of a... |
| odmod 19556 | Reduce the argument of a g... |
| oddvds 19557 | The only multiples of ` A ... |
| oddvdsi 19558 | Any group element is annih... |
| odcong 19559 | If two multipliers are con... |
| odeq 19560 | The ~ oddvds property uniq... |
| odval2 19561 | A non-conditional definiti... |
| odcld 19562 | The order of a group eleme... |
| odm1inv 19563 | The (order-1)th multiple o... |
| odmulgid 19564 | A relationship between the... |
| odmulg2 19565 | The order of a multiple di... |
| odmulg 19566 | Relationship between the o... |
| odmulgeq 19567 | A multiple of a point of f... |
| odbezout 19568 | If ` N ` is coprime to the... |
| od1 19569 | The order of the group ide... |
| odeq1 19570 | The group identity is the ... |
| odinv 19571 | The order of the inverse o... |
| odf1 19572 | The multiples of an elemen... |
| odinf 19573 | The multiples of an elemen... |
| dfod2 19574 | An alternative definition ... |
| odcl2 19575 | The order of an element of... |
| oddvds2 19576 | The order of an element of... |
| finodsubmsubg 19577 | A submonoid whose elements... |
| 0subgALT 19578 | A shorter proof of ~ 0subg... |
| submod 19579 | The order of an element is... |
| subgod 19580 | The order of an element is... |
| odsubdvds 19581 | The order of an element of... |
| odf1o1 19582 | An element with zero order... |
| odf1o2 19583 | An element with nonzero or... |
| odhash 19584 | An element of zero order g... |
| odhash2 19585 | If an element has nonzero ... |
| odhash3 19586 | An element which generates... |
| odngen 19587 | A cyclic subgroup of size ... |
| gexval 19588 | Value of the exponent of a... |
| gexlem1 19589 | The group element order is... |
| gexcl 19590 | The exponent of a group is... |
| gexid 19591 | Any element to the power o... |
| gexlem2 19592 | Any positive annihilator o... |
| gexdvdsi 19593 | Any group element is annih... |
| gexdvds 19594 | The only ` N ` that annihi... |
| gexdvds2 19595 | An integer divides the gro... |
| gexod 19596 | Any group element is annih... |
| gexcl3 19597 | If the order of every grou... |
| gexnnod 19598 | Every group element has fi... |
| gexcl2 19599 | The exponent of a finite g... |
| gexdvds3 19600 | The exponent of a finite g... |
| gex1 19601 | A group or monoid has expo... |
| ispgp 19602 | A group is a ` P ` -group ... |
| pgpprm 19603 | Reverse closure for the fi... |
| pgpgrp 19604 | Reverse closure for the se... |
| pgpfi1 19605 | A finite group with order ... |
| pgp0 19606 | The identity subgroup is a... |
| subgpgp 19607 | A subgroup of a p-group is... |
| sylow1lem1 19608 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19609 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19610 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19611 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19612 | Lemma for ~ sylow1 . Usin... |
| sylow1 19613 | Sylow's first theorem. If... |
| odcau 19614 | Cauchy's theorem for the o... |
| pgpfi 19615 | The converse to ~ pgpfi1 .... |
| pgpfi2 19616 | Alternate version of ~ pgp... |
| pgphash 19617 | The order of a p-group. (... |
| isslw 19618 | The property of being a Sy... |
| slwprm 19619 | Reverse closure for the fi... |
| slwsubg 19620 | A Sylow ` P ` -subgroup is... |
| slwispgp 19621 | Defining property of a Syl... |
| slwpss 19622 | A proper superset of a Syl... |
| slwpgp 19623 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19624 | Every ` P ` -subgroup is c... |
| slwn0 19625 | Every finite group contain... |
| subgslw 19626 | A Sylow subgroup that is c... |
| sylow2alem1 19627 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19628 | Lemma for ~ sylow2a . All... |
| sylow2a 19629 | A named lemma of Sylow's s... |
| sylow2blem1 19630 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19631 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19632 | Sylow's second theorem. P... |
| sylow2b 19633 | Sylow's second theorem. A... |
| slwhash 19634 | A sylow subgroup has cardi... |
| fislw 19635 | The sylow subgroups of a f... |
| sylow2 19636 | Sylow's second theorem. S... |
| sylow3lem1 19637 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19638 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19639 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19640 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19641 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19642 | Lemma for ~ sylow3 , secon... |
| sylow3 19643 | Sylow's third theorem. Th... |
| lsmfval 19648 | The subgroup sum function ... |
| lsmvalx 19649 | Subspace sum value (for a ... |
| lsmelvalx 19650 | Subspace sum membership (f... |
| lsmelvalix 19651 | Subspace sum membership (f... |
| oppglsm 19652 | The subspace sum operation... |
| lsmssv 19653 | Subgroup sum is a subset o... |
| lsmless1x 19654 | Subset implies subgroup su... |
| lsmless2x 19655 | Subset implies subgroup su... |
| lsmub1x 19656 | Subgroup sum is an upper b... |
| lsmub2x 19657 | Subgroup sum is an upper b... |
| lsmval 19658 | Subgroup sum value (for a ... |
| lsmelval 19659 | Subgroup sum membership (f... |
| lsmelvali 19660 | Subgroup sum membership (f... |
| lsmelvalm 19661 | Subgroup sum membership an... |
| lsmelvalmi 19662 | Membership of vector subtr... |
| lsmsubm 19663 | The sum of two commuting s... |
| lsmsubg 19664 | The sum of two commuting s... |
| lsmcom2 19665 | Subgroup sum commutes. (C... |
| smndlsmidm 19666 | The direct product is idem... |
| lsmub1 19667 | Subgroup sum is an upper b... |
| lsmub2 19668 | Subgroup sum is an upper b... |
| lsmunss 19669 | Union of subgroups is a su... |
| lsmless1 19670 | Subset implies subgroup su... |
| lsmless2 19671 | Subset implies subgroup su... |
| lsmless12 19672 | Subset implies subgroup su... |
| lsmidm 19673 | Subgroup sum is idempotent... |
| lsmlub 19674 | The least upper bound prop... |
| lsmss1 19675 | Subgroup sum with a subset... |
| lsmss1b 19676 | Subgroup sum with a subset... |
| lsmss2 19677 | Subgroup sum with a subset... |
| lsmss2b 19678 | Subgroup sum with a subset... |
| lsmass 19679 | Subgroup sum is associativ... |
| mndlsmidm 19680 | Subgroup sum is idempotent... |
| lsm01 19681 | Subgroup sum with the zero... |
| lsm02 19682 | Subgroup sum with the zero... |
| subglsm 19683 | The subgroup sum evaluated... |
| lssnle 19684 | Equivalent expressions for... |
| lsmmod 19685 | The modular law holds for ... |
| lsmmod2 19686 | Modular law dual for subgr... |
| lsmpropd 19687 | If two structures have the... |
| cntzrecd 19688 | Commute the "subgroups com... |
| lsmcntz 19689 | The "subgroups commute" pr... |
| lsmcntzr 19690 | The "subgroups commute" pr... |
| lsmdisj 19691 | Disjointness from a subgro... |
| lsmdisj2 19692 | Association of the disjoin... |
| lsmdisj3 19693 | Association of the disjoin... |
| lsmdisjr 19694 | Disjointness from a subgro... |
| lsmdisj2r 19695 | Association of the disjoin... |
| lsmdisj3r 19696 | Association of the disjoin... |
| lsmdisj2a 19697 | Association of the disjoin... |
| lsmdisj2b 19698 | Association of the disjoin... |
| lsmdisj3a 19699 | Association of the disjoin... |
| lsmdisj3b 19700 | Association of the disjoin... |
| subgdisj1 19701 | Vectors belonging to disjo... |
| subgdisj2 19702 | Vectors belonging to disjo... |
| subgdisjb 19703 | Vectors belonging to disjo... |
| pj1fval 19704 | The left projection functi... |
| pj1val 19705 | The left projection functi... |
| pj1eu 19706 | Uniqueness of a left proje... |
| pj1f 19707 | The left projection functi... |
| pj2f 19708 | The right projection funct... |
| pj1id 19709 | Any element of a direct su... |
| pj1eq 19710 | Any element of a direct su... |
| pj1lid 19711 | The left projection functi... |
| pj1rid 19712 | The left projection functi... |
| pj1ghm 19713 | The left projection functi... |
| pj1ghm2 19714 | The left projection functi... |
| lsmhash 19715 | The order of the direct pr... |
| efgmval 19722 | Value of the formal invers... |
| efgmf 19723 | The formal inverse operati... |
| efgmnvl 19724 | The inversion function on ... |
| efgrcl 19725 | Lemma for ~ efgval . (Con... |
| efglem 19726 | Lemma for ~ efgval . (Con... |
| efgval 19727 | Value of the free group co... |
| efger 19728 | Value of the free group co... |
| efgi 19729 | Value of the free group co... |
| efgi0 19730 | Value of the free group co... |
| efgi1 19731 | Value of the free group co... |
| efgtf 19732 | Value of the free group co... |
| efgtval 19733 | Value of the extension fun... |
| efgval2 19734 | Value of the free group co... |
| efgi2 19735 | Value of the free group co... |
| efgtlen 19736 | Value of the free group co... |
| efginvrel2 19737 | The inverse of the reverse... |
| efginvrel1 19738 | The inverse of the reverse... |
| efgsf 19739 | Value of the auxiliary fun... |
| efgsdm 19740 | Elementhood in the domain ... |
| efgsval 19741 | Value of the auxiliary fun... |
| efgsdmi 19742 | Property of the last link ... |
| efgsval2 19743 | Value of the auxiliary fun... |
| efgsrel 19744 | The start and end of any e... |
| efgs1 19745 | A singleton of an irreduci... |
| efgs1b 19746 | Every extension sequence e... |
| efgsp1 19747 | If ` F ` is an extension s... |
| efgsres 19748 | An initial segment of an e... |
| efgsfo 19749 | For any word, there is a s... |
| efgredlema 19750 | The reduced word that form... |
| efgredlemf 19751 | Lemma for ~ efgredleme . ... |
| efgredlemg 19752 | Lemma for ~ efgred . (Con... |
| efgredleme 19753 | Lemma for ~ efgred . (Con... |
| efgredlemd 19754 | The reduced word that form... |
| efgredlemc 19755 | The reduced word that form... |
| efgredlemb 19756 | The reduced word that form... |
| efgredlem 19757 | The reduced word that form... |
| efgred 19758 | The reduced word that form... |
| efgrelexlema 19759 | If two words ` A , B ` are... |
| efgrelexlemb 19760 | If two words ` A , B ` are... |
| efgrelex 19761 | If two words ` A , B ` are... |
| efgredeu 19762 | There is a unique reduced ... |
| efgred2 19763 | Two extension sequences ha... |
| efgcpbllema 19764 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19765 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19766 | Two extension sequences ha... |
| efgcpbl2 19767 | Two extension sequences ha... |
| frgpval 19768 | Value of the free group co... |
| frgpcpbl 19769 | Compatibility of the group... |
| frgp0 19770 | The free group is a group.... |
| frgpeccl 19771 | Closure of the quotient ma... |
| frgpgrp 19772 | The free group is a group.... |
| frgpadd 19773 | Addition in the free group... |
| frgpinv 19774 | The inverse of an element ... |
| frgpmhm 19775 | The "natural map" from wor... |
| vrgpfval 19776 | The canonical injection fr... |
| vrgpval 19777 | The value of the generatin... |
| vrgpf 19778 | The mapping from the index... |
| vrgpinv 19779 | The inverse of a generatin... |
| frgpuptf 19780 | Any assignment of the gene... |
| frgpuptinv 19781 | Any assignment of the gene... |
| frgpuplem 19782 | Any assignment of the gene... |
| frgpupf 19783 | Any assignment of the gene... |
| frgpupval 19784 | Any assignment of the gene... |
| frgpup1 19785 | Any assignment of the gene... |
| frgpup2 19786 | The evaluation map has the... |
| frgpup3lem 19787 | The evaluation map has the... |
| frgpup3 19788 | Universal property of the ... |
| 0frgp 19789 | The free group on zero gen... |
| isabl 19794 | The predicate "is an Abeli... |
| ablgrp 19795 | An Abelian group is a grou... |
| ablgrpd 19796 | An Abelian group is a grou... |
| ablcmn 19797 | An Abelian group is a comm... |
| ablcmnd 19798 | An Abelian group is a comm... |
| iscmn 19799 | The predicate "is a commut... |
| isabl2 19800 | The predicate "is an Abeli... |
| cmnpropd 19801 | If two structures have the... |
| ablpropd 19802 | If two structures have the... |
| ablprop 19803 | If two structures have the... |
| iscmnd 19804 | Properties that determine ... |
| isabld 19805 | Properties that determine ... |
| isabli 19806 | Properties that determine ... |
| cmnmnd 19807 | A commutative monoid is a ... |
| cmncom 19808 | A commutative monoid is co... |
| ablcom 19809 | An Abelian group operation... |
| cmn32 19810 | Commutative/associative la... |
| cmn4 19811 | Commutative/associative la... |
| cmn12 19812 | Commutative/associative la... |
| abl32 19813 | Commutative/associative la... |
| cmnmndd 19814 | A commutative monoid is a ... |
| cmnbascntr 19815 | The base set of a commutat... |
| rinvmod 19816 | Uniqueness of a right inve... |
| ablinvadd 19817 | The inverse of an Abelian ... |
| ablsub2inv 19818 | Abelian group subtraction ... |
| ablsubadd 19819 | Relationship between Abeli... |
| ablsub4 19820 | Commutative/associative su... |
| abladdsub4 19821 | Abelian group addition/sub... |
| abladdsub 19822 | Associative-type law for g... |
| ablsubadd23 19823 | Commutative/associative la... |
| ablsubaddsub 19824 | Double subtraction and add... |
| ablpncan2 19825 | Cancellation law for subtr... |
| ablpncan3 19826 | A cancellation law for Abe... |
| ablsubsub 19827 | Law for double subtraction... |
| ablsubsub4 19828 | Law for double subtraction... |
| ablpnpcan 19829 | Cancellation law for mixed... |
| ablnncan 19830 | Cancellation law for group... |
| ablsub32 19831 | Swap the second and third ... |
| ablnnncan 19832 | Cancellation law for group... |
| ablnnncan1 19833 | Cancellation law for group... |
| ablsubsub23 19834 | Swap subtrahend and result... |
| mulgnn0di 19835 | Group multiple of a sum, f... |
| mulgdi 19836 | Group multiple of a sum. ... |
| mulgmhm 19837 | The map from ` x ` to ` n ... |
| mulgghm 19838 | The map from ` x ` to ` n ... |
| mulgsubdi 19839 | Group multiple of a differ... |
| ghmfghm 19840 | The function fulfilling th... |
| ghmcmn 19841 | The image of a commutative... |
| ghmabl 19842 | The image of an abelian gr... |
| invghm 19843 | The inversion map is a gro... |
| eqgabl 19844 | Value of the subgroup cose... |
| qusecsub 19845 | Two subgroup cosets are eq... |
| subgabl 19846 | A subgroup of an abelian g... |
| subcmn 19847 | A submonoid of a commutati... |
| submcmn 19848 | A submonoid of a commutati... |
| submcmn2 19849 | A submonoid is commutative... |
| cntzcmn 19850 | The centralizer of any sub... |
| cntzcmnss 19851 | Any subset in a commutativ... |
| cntrcmnd 19852 | The center of a monoid is ... |
| cntrabl 19853 | The center of a group is a... |
| cntzspan 19854 | If the generators commute,... |
| cntzcmnf 19855 | Discharge the centralizer ... |
| ghmplusg 19856 | The pointwise sum of two l... |
| ablnsg 19857 | Every subgroup of an abeli... |
| odadd1 19858 | The order of a product in ... |
| odadd2 19859 | The order of a product in ... |
| odadd 19860 | The order of a product is ... |
| gex2abl 19861 | A group with exponent 2 (o... |
| gexexlem 19862 | Lemma for ~ gexex . (Cont... |
| gexex 19863 | In an abelian group with f... |
| torsubg 19864 | The set of all elements of... |
| oddvdssubg 19865 | The set of all elements wh... |
| lsmcomx 19866 | Subgroup sum commutes (ext... |
| ablcntzd 19867 | All subgroups in an abelia... |
| lsmcom 19868 | Subgroup sum commutes. (C... |
| lsmsubg2 19869 | The sum of two subgroups i... |
| lsm4 19870 | Commutative/associative la... |
| prdscmnd 19871 | The product of a family of... |
| prdsabld 19872 | The product of a family of... |
| pwscmn 19873 | The structure power on a c... |
| pwsabl 19874 | The structure power on an ... |
| qusabl 19875 | If ` Y ` is a subgroup of ... |
| abl1 19876 | The (smallest) structure r... |
| abln0 19877 | Abelian groups (and theref... |
| cnaddablx 19878 | The complex numbers are an... |
| cnaddabl 19879 | The complex numbers are an... |
| cnaddid 19880 | The group identity element... |
| cnaddinv 19881 | Value of the group inverse... |
| zaddablx 19882 | The integers are an Abelia... |
| frgpnabllem1 19883 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19884 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19885 | The free group on two or m... |
| imasabl 19886 | The image structure of an ... |
| iscyg 19889 | Definition of a cyclic gro... |
| iscyggen 19890 | The property of being a cy... |
| iscyggen2 19891 | The property of being a cy... |
| iscyg2 19892 | A cyclic group is a group ... |
| cyggeninv 19893 | The inverse of a cyclic ge... |
| cyggenod 19894 | An element is the generato... |
| cyggenod2 19895 | In an infinite cyclic grou... |
| iscyg3 19896 | Definition of a cyclic gro... |
| iscygd 19897 | Definition of a cyclic gro... |
| iscygodd 19898 | Show that a group with an ... |
| cycsubmcmn 19899 | The set of nonnegative int... |
| cyggrp 19900 | A cyclic group is a group.... |
| cygabl 19901 | A cyclic group is abelian.... |
| cygctb 19902 | A cyclic group is countabl... |
| 0cyg 19903 | The trivial group is cycli... |
| prmcyg 19904 | A group with prime order i... |
| lt6abl 19905 | A group with fewer than ` ... |
| ghmcyg 19906 | The image of a cyclic grou... |
| cyggex2 19907 | The exponent of a cyclic g... |
| cyggex 19908 | The exponent of a finite c... |
| cyggexb 19909 | A finite abelian group is ... |
| giccyg 19910 | Cyclicity is a group prope... |
| cycsubgcyg 19911 | The cyclic subgroup genera... |
| cycsubgcyg2 19912 | The cyclic subgroup genera... |
| gsumval3a 19913 | Value of the group sum ope... |
| gsumval3eu 19914 | The group sum as defined i... |
| gsumval3lem1 19915 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19916 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19917 | Value of the group sum ope... |
| gsumcllem 19918 | Lemma for ~ gsumcl and rel... |
| gsumzres 19919 | Extend a finite group sum ... |
| gsumzcl2 19920 | Closure of a finite group ... |
| gsumzcl 19921 | Closure of a finite group ... |
| gsumzf1o 19922 | Re-index a finite group su... |
| gsumres 19923 | Extend a finite group sum ... |
| gsumcl2 19924 | Closure of a finite group ... |
| gsumcl 19925 | Closure of a finite group ... |
| gsumf1o 19926 | Re-index a finite group su... |
| gsumreidx 19927 | Re-index a finite group su... |
| gsumzsubmcl 19928 | Closure of a group sum in ... |
| gsumsubmcl 19929 | Closure of a group sum in ... |
| gsumsubgcl 19930 | Closure of a group sum in ... |
| gsumzaddlem 19931 | The sum of two group sums.... |
| gsumzadd 19932 | The sum of two group sums.... |
| gsumadd 19933 | The sum of two group sums.... |
| gsummptfsadd 19934 | The sum of two group sums ... |
| gsummptfidmadd 19935 | The sum of two group sums ... |
| gsummptfidmadd2 19936 | The sum of two group sums ... |
| gsumzsplit 19937 | Split a group sum into two... |
| gsumsplit 19938 | Split a group sum into two... |
| gsumsplit2 19939 | Split a group sum into two... |
| gsummptfidmsplit 19940 | Split a group sum expresse... |
| gsummptfidmsplitres 19941 | Split a group sum expresse... |
| gsummptfzsplit 19942 | Split a group sum expresse... |
| gsummptfzsplitl 19943 | Split a group sum expresse... |
| gsumconst 19944 | Sum of a constant series. ... |
| gsumconstf 19945 | Sum of a constant series. ... |
| gsummptshft 19946 | Index shift of a finite gr... |
| gsumzmhm 19947 | Apply a group homomorphism... |
| gsummhm 19948 | Apply a group homomorphism... |
| gsummhm2 19949 | Apply a group homomorphism... |
| gsummptmhm 19950 | Apply a group homomorphism... |
| gsummulglem 19951 | Lemma for ~ gsummulg and ~... |
| gsummulg 19952 | Nonnegative multiple of a ... |
| gsummulgz 19953 | Integer multiple of a grou... |
| gsumzoppg 19954 | The opposite of a group su... |
| gsumzinv 19955 | Inverse of a group sum. (... |
| gsuminv 19956 | Inverse of a group sum. (... |
| gsummptfidminv 19957 | Inverse of a group sum exp... |
| gsumsub 19958 | The difference of two grou... |
| gsummptfssub 19959 | The difference of two grou... |
| gsummptfidmsub 19960 | The difference of two grou... |
| gsumsnfd 19961 | Group sum of a singleton, ... |
| gsumsnd 19962 | Group sum of a singleton, ... |
| gsumsnf 19963 | Group sum of a singleton, ... |
| gsumsn 19964 | Group sum of a singleton. ... |
| gsumpr 19965 | Group sum of a pair. (Con... |
| gsumzunsnd 19966 | Append an element to a fin... |
| gsumunsnfd 19967 | Append an element to a fin... |
| gsumunsnd 19968 | Append an element to a fin... |
| gsumunsnf 19969 | Append an element to a fin... |
| gsumunsn 19970 | Append an element to a fin... |
| gsumdifsnd 19971 | Extract a summand from a f... |
| gsumpt 19972 | Sum of a family that is no... |
| gsummptf1o 19973 | Re-index a finite group su... |
| gsummptun 19974 | Group sum of a disjoint un... |
| gsummpt1n0 19975 | If only one summand in a f... |
| gsummptif1n0 19976 | If only one summand in a f... |
| gsummptcl 19977 | Closure of a finite group ... |
| gsummptfif1o 19978 | Re-index a finite group su... |
| gsummptfzcl 19979 | Closure of a finite group ... |
| gsum2dlem1 19980 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19981 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19982 | Write a sum over a two-dim... |
| gsum2d2lem 19983 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19984 | Write a group sum over a t... |
| gsumcom2 19985 | Two-dimensional commutatio... |
| gsumxp 19986 | Write a group sum over a c... |
| gsumcom 19987 | Commute the arguments of a... |
| gsumcom3 19988 | A commutative law for fini... |
| gsumcom3fi 19989 | A commutative law for fini... |
| gsumxp2 19990 | Write a group sum over a c... |
| prdsgsum 19991 | Finite commutative sums in... |
| pwsgsum 19992 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19993 | Replacing a finitely suppo... |
| nn0gsumfz 19994 | Replacing a finitely suppo... |
| nn0gsumfz0 19995 | Replacing a finitely suppo... |
| gsummptnn0fz 19996 | A final group sum over a f... |
| gsummptnn0fzfv 19997 | A final group sum over a f... |
| telgsumfzslem 19998 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19999 | Telescoping group sum rang... |
| telgsumfz 20000 | Telescoping group sum rang... |
| telgsumfz0s 20001 | Telescoping finite group s... |
| telgsumfz0 20002 | Telescoping finite group s... |
| telgsums 20003 | Telescoping finitely suppo... |
| telgsum 20004 | Telescoping finitely suppo... |
| reldmdprd 20009 | The domain of the internal... |
| dmdprd 20010 | The domain of definition o... |
| dmdprdd 20011 | Show that a given family i... |
| dprddomprc 20012 | A family of subgroups inde... |
| dprddomcld 20013 | If a family of subgroups i... |
| dprdval0prc 20014 | The internal direct produc... |
| dprdval 20015 | The value of the internal ... |
| eldprd 20016 | A class ` A ` is an intern... |
| dprdgrp 20017 | Reverse closure for the in... |
| dprdf 20018 | The function ` S ` is a fa... |
| dprdf2 20019 | The function ` S ` is a fa... |
| dprdcntz 20020 | The function ` S ` is a fa... |
| dprddisj 20021 | The function ` S ` is a fa... |
| dprdw 20022 | The property of being a fi... |
| dprdwd 20023 | A mapping being a finitely... |
| dprdff 20024 | A finitely supported funct... |
| dprdfcl 20025 | A finitely supported funct... |
| dprdffsupp 20026 | A finitely supported funct... |
| dprdfcntz 20027 | A function on the elements... |
| dprdssv 20028 | The internal direct produc... |
| dprdfid 20029 | A function mapping all but... |
| eldprdi 20030 | The domain of definition o... |
| dprdfinv 20031 | Take the inverse of a grou... |
| dprdfadd 20032 | Take the sum of group sums... |
| dprdfsub 20033 | Take the difference of gro... |
| dprdfeq0 20034 | The zero function is the o... |
| dprdf11 20035 | Two group sums over a dire... |
| dprdsubg 20036 | The internal direct produc... |
| dprdub 20037 | Each factor is a subset of... |
| dprdlub 20038 | The direct product is smal... |
| dprdspan 20039 | The direct product is the ... |
| dprdres 20040 | Restriction of a direct pr... |
| dprdss 20041 | Create a direct product by... |
| dprdz 20042 | A family consisting entire... |
| dprd0 20043 | The empty family is an int... |
| dprdf1o 20044 | Rearrange the index set of... |
| dprdf1 20045 | Rearrange the index set of... |
| subgdmdprd 20046 | A direct product in a subg... |
| subgdprd 20047 | A direct product in a subg... |
| dprdsn 20048 | A singleton family is an i... |
| dmdprdsplitlem 20049 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 20050 | The function ` S ` is a fa... |
| dprddisj2 20051 | The function ` S ` is a fa... |
| dprd2dlem2 20052 | The direct product of a co... |
| dprd2dlem1 20053 | The direct product of a co... |
| dprd2da 20054 | The direct product of a co... |
| dprd2db 20055 | The direct product of a co... |
| dprd2d2 20056 | The direct product of a co... |
| dmdprdsplit2lem 20057 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 20058 | The direct product splits ... |
| dmdprdsplit 20059 | The direct product splits ... |
| dprdsplit 20060 | The direct product is the ... |
| dmdprdpr 20061 | A singleton family is an i... |
| dprdpr 20062 | A singleton family is an i... |
| dpjlem 20063 | Lemma for theorems about d... |
| dpjcntz 20064 | The two subgroups that app... |
| dpjdisj 20065 | The two subgroups that app... |
| dpjlsm 20066 | The two subgroups that app... |
| dpjfval 20067 | Value of the direct produc... |
| dpjval 20068 | Value of the direct produc... |
| dpjf 20069 | The ` X ` -th index projec... |
| dpjidcl 20070 | The key property of projec... |
| dpjeq 20071 | Decompose a group sum into... |
| dpjid 20072 | The key property of projec... |
| dpjlid 20073 | The ` X ` -th index projec... |
| dpjrid 20074 | The ` Y ` -th index projec... |
| dpjghm 20075 | The direct product is the ... |
| dpjghm2 20076 | The direct product is the ... |
| ablfacrplem 20077 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20078 | A finite abelian group who... |
| ablfacrp2 20079 | The factors ` K , L ` of ~... |
| ablfac1lem 20080 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20081 | The factors of ~ ablfac1b ... |
| ablfac1b 20082 | Any abelian group is the d... |
| ablfac1c 20083 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20084 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20085 | The factorization of ~ abl... |
| pgpfac1lem1 20086 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20087 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20088 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20089 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20090 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20091 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20092 | Factorization of a finite ... |
| pgpfaclem1 20093 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20094 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20095 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20096 | Full factorization of a fi... |
| ablfaclem1 20097 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20098 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20099 | Lemma for ~ ablfac . (Con... |
| ablfac 20100 | The Fundamental Theorem of... |
| ablfac2 20101 | Choose generators for each... |
| issimpg 20104 | The predicate "is a simple... |
| issimpgd 20105 | Deduce a simple group from... |
| simpggrp 20106 | A simple group is a group.... |
| simpggrpd 20107 | A simple group is a group.... |
| simpg2nsg 20108 | A simple group has two nor... |
| trivnsimpgd 20109 | Trivial groups are not sim... |
| simpgntrivd 20110 | Simple groups are nontrivi... |
| simpgnideld 20111 | A simple group contains a ... |
| simpgnsgd 20112 | The only normal subgroups ... |
| simpgnsgeqd 20113 | A normal subgroup of a sim... |
| 2nsgsimpgd 20114 | If any normal subgroup of ... |
| simpgnsgbid 20115 | A nontrivial group is simp... |
| ablsimpnosubgd 20116 | A subgroup of an abelian s... |
| ablsimpg1gend 20117 | An abelian simple group is... |
| ablsimpgcygd 20118 | An abelian simple group is... |
| ablsimpgfindlem1 20119 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20120 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20121 | Relationship between the o... |
| ablsimpgfind 20122 | An abelian simple group is... |
| fincygsubgd 20123 | The subgroup referenced in... |
| fincygsubgodd 20124 | Calculate the order of a s... |
| fincygsubgodexd 20125 | A finite cyclic group has ... |
| prmgrpsimpgd 20126 | A group of prime order is ... |
| ablsimpgprmd 20127 | An abelian simple group ha... |
| ablsimpgd 20128 | An abelian group is simple... |
| fnmgp 20131 | The multiplicative group o... |
| mgpval 20132 | Value of the multiplicatio... |
| mgpplusg 20133 | Value of the group operati... |
| mgplemOLD 20134 | Obsolete version of ~ sets... |
| mgpbas 20135 | Base set of the multiplica... |
| mgpbasOLD 20136 | Obsolete version of ~ mgpb... |
| mgpsca 20137 | The multiplication monoid ... |
| mgpscaOLD 20138 | Obsolete version of ~ mgps... |
| mgptset 20139 | Topology component of the ... |
| mgptsetOLD 20140 | Obsolete version of ~ mgpt... |
| mgptopn 20141 | Topology of the multiplica... |
| mgpds 20142 | Distance function of the m... |
| mgpdsOLD 20143 | Obsolete version of ~ mgpd... |
| mgpress 20144 | Subgroup commutes with the... |
| mgpressOLD 20145 | Obsolete version of ~ mgpr... |
| prdsmgp 20146 | The multiplicative monoid ... |
| isrng 20149 | The predicate "is a non-un... |
| rngabl 20150 | A non-unital ring is an (a... |
| rngmgp 20151 | A non-unital ring is a sem... |
| rngmgpf 20152 | Restricted functionality o... |
| rnggrp 20153 | A non-unital ring is a (ad... |
| rngass 20154 | Associative law for the mu... |
| rngdi 20155 | Distributive law for the m... |
| rngdir 20156 | Distributive law for the m... |
| rngacl 20157 | Closure of the addition op... |
| rng0cl 20158 | The zero element of a non-... |
| rngcl 20159 | Closure of the multiplicat... |
| rnglz 20160 | The zero of a non-unital r... |
| rngrz 20161 | The zero of a non-unital r... |
| rngmneg1 20162 | Negation of a product in a... |
| rngmneg2 20163 | Negation of a product in a... |
| rngm2neg 20164 | Double negation of a produ... |
| rngansg 20165 | Every additive subgroup of... |
| rngsubdi 20166 | Ring multiplication distri... |
| rngsubdir 20167 | Ring multiplication distri... |
| isrngd 20168 | Properties that determine ... |
| rngpropd 20169 | If two structures have the... |
| prdsmulrngcl 20170 | Closure of the multiplicat... |
| prdsrngd 20171 | A product of non-unital ri... |
| imasrng 20172 | The image structure of a n... |
| imasrngf1 20173 | The image of a non-unital ... |
| xpsrngd 20174 | A product of two non-unita... |
| qusrng 20175 | The quotient structure of ... |
| ringidval 20178 | The value of the unity ele... |
| dfur2 20179 | The multiplicative identit... |
| ringurd 20180 | Deduce the unity element o... |
| issrg 20183 | The predicate "is a semiri... |
| srgcmn 20184 | A semiring is a commutativ... |
| srgmnd 20185 | A semiring is a monoid. (... |
| srgmgp 20186 | A semiring is a monoid und... |
| srgdilem 20187 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20188 | Closure of the multiplicat... |
| srgass 20189 | Associative law for the mu... |
| srgideu 20190 | The unity element of a sem... |
| srgfcl 20191 | Functionality of the multi... |
| srgdi 20192 | Distributive law for the m... |
| srgdir 20193 | Distributive law for the m... |
| srgidcl 20194 | The unity element of a sem... |
| srg0cl 20195 | The zero element of a semi... |
| srgidmlem 20196 | Lemma for ~ srglidm and ~ ... |
| srglidm 20197 | The unity element of a sem... |
| srgridm 20198 | The unity element of a sem... |
| issrgid 20199 | Properties showing that an... |
| srgacl 20200 | Closure of the addition op... |
| srgcom 20201 | Commutativity of the addit... |
| srgrz 20202 | The zero of a semiring is ... |
| srglz 20203 | The zero of a semiring is ... |
| srgisid 20204 | In a semiring, the only le... |
| o2timesd 20205 | An element of a ring-like ... |
| rglcom4d 20206 | Restricted commutativity o... |
| srgo2times 20207 | A semiring element plus it... |
| srgcom4lem 20208 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20209 | Restricted commutativity o... |
| srg1zr 20210 | The only semiring with a b... |
| srgen1zr 20211 | The only semiring with one... |
| srgmulgass 20212 | An associative property be... |
| srgpcomp 20213 | If two elements of a semir... |
| srgpcompp 20214 | If two elements of a semir... |
| srgpcomppsc 20215 | If two elements of a semir... |
| srglmhm 20216 | Left-multiplication in a s... |
| srgrmhm 20217 | Right-multiplication in a ... |
| srgsummulcr 20218 | A finite semiring sum mult... |
| sgsummulcl 20219 | A finite semiring sum mult... |
| srg1expzeq1 20220 | The exponentiation (by a n... |
| srgbinomlem1 20221 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20222 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20223 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20224 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20225 | Lemma for ~ srgbinom . In... |
| srgbinom 20226 | The binomial theorem for c... |
| csrgbinom 20227 | The binomial theorem for c... |
| isring 20232 | The predicate "is a (unita... |
| ringgrp 20233 | A ring is a group. (Contr... |
| ringmgp 20234 | A ring is a monoid under m... |
| iscrng 20235 | A commutative ring is a ri... |
| crngmgp 20236 | A commutative ring's multi... |
| ringgrpd 20237 | A ring is a group. (Contr... |
| ringmnd 20238 | A ring is a monoid under a... |
| ringmgm 20239 | A ring is a magma. (Contr... |
| crngring 20240 | A commutative ring is a ri... |
| crngringd 20241 | A commutative ring is a ri... |
| crnggrpd 20242 | A commutative ring is a gr... |
| mgpf 20243 | Restricted functionality o... |
| ringdilem 20244 | Properties of a unital rin... |
| ringcl 20245 | Closure of the multiplicat... |
| crngcom 20246 | A commutative ring's multi... |
| iscrng2 20247 | A commutative ring is a ri... |
| ringass 20248 | Associative law for multip... |
| ringideu 20249 | The unity element of a rin... |
| crngcomd 20250 | Multiplication is commutat... |
| crngbascntr 20251 | The base set of a commutat... |
| ringassd 20252 | Associative law for multip... |
| crng12d 20253 | Commutative/associative la... |
| ringcld 20254 | Closure of the multiplicat... |
| ringdi 20255 | Distributive law for the m... |
| ringdir 20256 | Distributive law for the m... |
| ringidcl 20257 | The unity element of a rin... |
| ring0cl 20258 | The zero element of a ring... |
| ringidmlem 20259 | Lemma for ~ ringlidm and ~... |
| ringlidm 20260 | The unity element of a rin... |
| ringridm 20261 | The unity element of a rin... |
| isringid 20262 | Properties showing that an... |
| ringlidmd 20263 | The unity element of a rin... |
| ringridmd 20264 | The unity element of a rin... |
| ringid 20265 | The multiplication operati... |
| ringo2times 20266 | A ring element plus itself... |
| ringadd2 20267 | A ring element plus itself... |
| ringidss 20268 | A subset of the multiplica... |
| ringacl 20269 | Closure of the addition op... |
| ringcomlem 20270 | Lemma for ~ ringcom . Thi... |
| ringcom 20271 | Commutativity of the addit... |
| ringabl 20272 | A ring is an Abelian group... |
| ringcmn 20273 | A ring is a commutative mo... |
| ringabld 20274 | A ring is an Abelian group... |
| ringcmnd 20275 | A ring is a commutative mo... |
| ringrng 20276 | A unital ring is a non-uni... |
| ringssrng 20277 | The unital rings are non-u... |
| isringrng 20278 | The predicate "is a unital... |
| ringpropd 20279 | If two structures have the... |
| crngpropd 20280 | If two structures have the... |
| ringprop 20281 | If two structures have the... |
| isringd 20282 | Properties that determine ... |
| iscrngd 20283 | Properties that determine ... |
| ringlz 20284 | The zero of a unital ring ... |
| ringrz 20285 | The zero of a unital ring ... |
| ringlzd 20286 | The zero of a unital ring ... |
| ringrzd 20287 | The zero of a unital ring ... |
| ringsrg 20288 | Any ring is also a semirin... |
| ring1eq0 20289 | If one and zero are equal,... |
| ring1ne0 20290 | If a ring has at least two... |
| ringinvnz1ne0 20291 | In a unital ring, a left i... |
| ringinvnzdiv 20292 | In a unital ring, a left i... |
| ringnegl 20293 | Negation in a ring is the ... |
| ringnegr 20294 | Negation in a ring is the ... |
| ringmneg1 20295 | Negation of a product in a... |
| ringmneg2 20296 | Negation of a product in a... |
| ringm2neg 20297 | Double negation of a produ... |
| ringsubdi 20298 | Ring multiplication distri... |
| ringsubdir 20299 | Ring multiplication distri... |
| mulgass2 20300 | An associative property be... |
| ring1 20301 | The (smallest) structure r... |
| ringn0 20302 | Rings exist. (Contributed... |
| ringlghm 20303 | Left-multiplication in a r... |
| ringrghm 20304 | Right-multiplication in a ... |
| gsummulc1OLD 20305 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20306 | Obsolete version of ~ gsum... |
| gsummulc1 20307 | A finite ring sum multipli... |
| gsummulc2 20308 | A finite ring sum multipli... |
| gsummgp0 20309 | If one factor in a finite ... |
| gsumdixp 20310 | Distribute a binary produc... |
| prdsmulrcl 20311 | A structure product of rin... |
| prdsringd 20312 | A product of rings is a ri... |
| prdscrngd 20313 | A product of commutative r... |
| prds1 20314 | Value of the ring unity in... |
| pwsring 20315 | A structure power of a rin... |
| pws1 20316 | Value of the ring unity in... |
| pwscrng 20317 | A structure power of a com... |
| pwsmgp 20318 | The multiplicative group o... |
| pwspjmhmmgpd 20319 | The projection given by ~ ... |
| pwsexpg 20320 | Value of a group exponenti... |
| imasring 20321 | The image structure of a r... |
| imasringf1 20322 | The image of a ring under ... |
| xpsringd 20323 | A product of two rings is ... |
| xpsring1d 20324 | The multiplicative identit... |
| qusring2 20325 | The quotient structure of ... |
| crngbinom 20326 | The binomial theorem for c... |
| opprval 20329 | Value of the opposite ring... |
| opprmulfval 20330 | Value of the multiplicatio... |
| opprmul 20331 | Value of the multiplicatio... |
| crngoppr 20332 | In a commutative ring, the... |
| opprlem 20333 | Lemma for ~ opprbas and ~ ... |
| opprlemOLD 20334 | Obsolete version of ~ oppr... |
| opprbas 20335 | Base set of an opposite ri... |
| opprbasOLD 20336 | Obsolete proof of ~ opprba... |
| oppradd 20337 | Addition operation of an o... |
| oppraddOLD 20338 | Obsolete proof of ~ opprba... |
| opprrng 20339 | An opposite non-unital rin... |
| opprrngb 20340 | A class is a non-unital ri... |
| opprring 20341 | An opposite ring is a ring... |
| opprringb 20342 | Bidirectional form of ~ op... |
| oppr0 20343 | Additive identity of an op... |
| oppr1 20344 | Multiplicative identity of... |
| opprneg 20345 | The negative function in a... |
| opprsubg 20346 | Being a subgroup is a symm... |
| mulgass3 20347 | An associative property be... |
| reldvdsr 20354 | The divides relation is a ... |
| dvdsrval 20355 | Value of the divides relat... |
| dvdsr 20356 | Value of the divides relat... |
| dvdsr2 20357 | Value of the divides relat... |
| dvdsrmul 20358 | A left-multiple of ` X ` i... |
| dvdsrcl 20359 | Closure of a dividing elem... |
| dvdsrcl2 20360 | Closure of a dividing elem... |
| dvdsrid 20361 | An element in a (unital) r... |
| dvdsrtr 20362 | Divisibility is transitive... |
| dvdsrmul1 20363 | The divisibility relation ... |
| dvdsrneg 20364 | An element divides its neg... |
| dvdsr01 20365 | In a ring, zero is divisib... |
| dvdsr02 20366 | Only zero is divisible by ... |
| isunit 20367 | Property of being a unit o... |
| 1unit 20368 | The multiplicative identit... |
| unitcl 20369 | A unit is an element of th... |
| unitss 20370 | The set of units is contai... |
| opprunit 20371 | Being a unit is a symmetri... |
| crngunit 20372 | Property of being a unit i... |
| dvdsunit 20373 | A divisor of a unit is a u... |
| unitmulcl 20374 | The product of units is a ... |
| unitmulclb 20375 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20376 | The base set of the group ... |
| unitgrp 20377 | The group of units is a gr... |
| unitabl 20378 | The group of units of a co... |
| unitgrpid 20379 | The identity of the group ... |
| unitsubm 20380 | The group of units is a su... |
| invrfval 20383 | Multiplicative inverse fun... |
| unitinvcl 20384 | The inverse of a unit exis... |
| unitinvinv 20385 | The inverse of the inverse... |
| ringinvcl 20386 | The inverse of a unit is a... |
| unitlinv 20387 | A unit times its inverse i... |
| unitrinv 20388 | A unit times its inverse i... |
| 1rinv 20389 | The inverse of the ring un... |
| 0unit 20390 | The additive identity is a... |
| unitnegcl 20391 | The negative of a unit is ... |
| ringunitnzdiv 20392 | In a unitary ring, a unit ... |
| ring1nzdiv 20393 | In a unitary ring, the rin... |
| dvrfval 20396 | Division operation in a ri... |
| dvrval 20397 | Division operation in a ri... |
| dvrcl 20398 | Closure of division operat... |
| unitdvcl 20399 | The units are closed under... |
| dvrid 20400 | A ring element divided by ... |
| dvr1 20401 | A ring element divided by ... |
| dvrass 20402 | An associative law for div... |
| dvrcan1 20403 | A cancellation law for div... |
| dvrcan3 20404 | A cancellation law for div... |
| dvreq1 20405 | Equality in terms of ratio... |
| dvrdir 20406 | Distributive law for the d... |
| rdivmuldivd 20407 | Multiplication of two rati... |
| ringinvdv 20408 | Write the inverse function... |
| rngidpropd 20409 | The ring unity depends onl... |
| dvdsrpropd 20410 | The divisibility relation ... |
| unitpropd 20411 | The set of units depends o... |
| invrpropd 20412 | The ring inverse function ... |
| isirred 20413 | An irreducible element of ... |
| isnirred 20414 | The property of being a no... |
| isirred2 20415 | Expand out the class diffe... |
| opprirred 20416 | Irreducibility is symmetri... |
| irredn0 20417 | The additive identity is n... |
| irredcl 20418 | An irreducible element is ... |
| irrednu 20419 | An irreducible element is ... |
| irredn1 20420 | The multiplicative identit... |
| irredrmul 20421 | The product of an irreduci... |
| irredlmul 20422 | The product of a unit and ... |
| irredmul 20423 | If product of two elements... |
| irredneg 20424 | The negative of an irreduc... |
| irrednegb 20425 | An element is irreducible ... |
| rnghmrcl 20432 | Reverse closure of a non-u... |
| rnghmfn 20433 | The mapping of two non-uni... |
| rnghmval 20434 | The set of the non-unital ... |
| isrnghm 20435 | A function is a non-unital... |
| isrnghmmul 20436 | A function is a non-unital... |
| rnghmmgmhm 20437 | A non-unital ring homomorp... |
| rnghmval2 20438 | The non-unital ring homomo... |
| isrngim 20439 | An isomorphism of non-unit... |
| rngimrcl 20440 | Reverse closure for an iso... |
| rnghmghm 20441 | A non-unital ring homomorp... |
| rnghmf 20442 | A ring homomorphism is a f... |
| rnghmmul 20443 | A homomorphism of non-unit... |
| isrnghm2d 20444 | Demonstration of non-unita... |
| isrnghmd 20445 | Demonstration of non-unita... |
| rnghmf1o 20446 | A non-unital ring homomorp... |
| isrngim2 20447 | An isomorphism of non-unit... |
| rngimf1o 20448 | An isomorphism of non-unit... |
| rngimrnghm 20449 | An isomorphism of non-unit... |
| rngimcnv 20450 | The converse of an isomorp... |
| rnghmco 20451 | The composition of non-uni... |
| idrnghm 20452 | The identity homomorphism ... |
| c0mgm 20453 | The constant mapping to ze... |
| c0mhm 20454 | The constant mapping to ze... |
| c0ghm 20455 | The constant mapping to ze... |
| c0snmgmhm 20456 | The constant mapping to ze... |
| c0snmhm 20457 | The constant mapping to ze... |
| c0snghm 20458 | The constant mapping to ze... |
| rngisomfv1 20459 | If there is a non-unital r... |
| rngisom1 20460 | If there is a non-unital r... |
| rngisomring 20461 | If there is a non-unital r... |
| rngisomring1 20462 | If there is a non-unital r... |
| dfrhm2 20468 | The property of a ring hom... |
| rhmrcl1 20470 | Reverse closure of a ring ... |
| rhmrcl2 20471 | Reverse closure of a ring ... |
| isrhm 20472 | A function is a ring homom... |
| rhmmhm 20473 | A ring homomorphism is a h... |
| rhmisrnghm 20474 | Each unital ring homomorph... |
| isrim0OLD 20475 | Obsolete version of ~ isri... |
| rimrcl 20476 | Reverse closure for an iso... |
| isrim0 20477 | A ring isomorphism is a ho... |
| rhmghm 20478 | A ring homomorphism is an ... |
| rhmf 20479 | A ring homomorphism is a f... |
| rhmmul 20480 | A homomorphism of rings pr... |
| isrhm2d 20481 | Demonstration of ring homo... |
| isrhmd 20482 | Demonstration of ring homo... |
| rhm1 20483 | Ring homomorphisms are req... |
| idrhm 20484 | The identity homomorphism ... |
| rhmf1o 20485 | A ring homomorphism is bij... |
| isrim 20486 | An isomorphism of rings is... |
| isrimOLD 20487 | Obsolete version of ~ isri... |
| rimf1o 20488 | An isomorphism of rings is... |
| rimrhmOLD 20489 | Obsolete version of ~ rimr... |
| rimrhm 20490 | A ring isomorphism is a ho... |
| rimgim 20491 | An isomorphism of rings is... |
| rimisrngim 20492 | Each unital ring isomorphi... |
| rhmfn 20493 | The mapping of two rings t... |
| rhmval 20494 | The ring homomorphisms bet... |
| rhmco 20495 | The composition of ring ho... |
| pwsco1rhm 20496 | Right composition with a f... |
| pwsco2rhm 20497 | Left composition with a ri... |
| brric 20498 | The relation "is isomorphi... |
| brrici 20499 | Prove isomorphic by an exp... |
| brric2 20500 | The relation "is isomorphi... |
| ricgic 20501 | If two rings are (ring) is... |
| rhmdvdsr 20502 | A ring homomorphism preser... |
| rhmopp 20503 | A ring homomorphism is als... |
| elrhmunit 20504 | Ring homomorphisms preserv... |
| rhmunitinv 20505 | Ring homomorphisms preserv... |
| isnzr 20508 | Property of a nonzero ring... |
| nzrnz 20509 | One and zero are different... |
| nzrring 20510 | A nonzero ring is a ring. ... |
| nzrringOLD 20511 | Obsolete version of ~ nzrr... |
| isnzr2 20512 | Equivalent characterizatio... |
| isnzr2hash 20513 | Equivalent characterizatio... |
| nzrpropd 20514 | If two structures have the... |
| opprnzrb 20515 | The opposite of a nonzero ... |
| opprnzr 20516 | The opposite of a nonzero ... |
| ringelnzr 20517 | A ring is nonzero if it ha... |
| nzrunit 20518 | A unit is nonzero in any n... |
| 0ringnnzr 20519 | A ring is a zero ring iff ... |
| 0ring 20520 | If a ring has only one ele... |
| 0ringdif 20521 | A zero ring is a ring whic... |
| 0ringbas 20522 | The base set of a zero rin... |
| 0ring01eq 20523 | In a ring with only one el... |
| 01eq0ring 20524 | If the zero and the identi... |
| 01eq0ringOLD 20525 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20526 | In a unital ring the zero ... |
| 0ring1eq0 20527 | In a zero ring, a ring whi... |
| c0rhm 20528 | The constant mapping to ze... |
| c0rnghm 20529 | The constant mapping to ze... |
| zrrnghm 20530 | The constant mapping to ze... |
| nrhmzr 20531 | There is no ring homomorph... |
| islring 20534 | The predicate "is a local ... |
| lringnzr 20535 | A local ring is a nonzero ... |
| lringring 20536 | A local ring is a ring. (... |
| lringnz 20537 | A local ring is a nonzero ... |
| lringuplu 20538 | If the sum of two elements... |
| issubrng 20541 | The subring of non-unital ... |
| subrngss 20542 | A subring is a subset. (C... |
| subrngid 20543 | Every non-unital ring is a... |
| subrngrng 20544 | A subring is a non-unital ... |
| subrngrcl 20545 | Reverse closure for a subr... |
| subrngsubg 20546 | A subring is a subgroup. ... |
| subrngringnsg 20547 | A subring is a normal subg... |
| subrngbas 20548 | Base set of a subring stru... |
| subrng0 20549 | A subring always has the s... |
| subrngacl 20550 | A subring is closed under ... |
| subrngmcl 20551 | A subring is closed under ... |
| issubrng2 20552 | Characterize the subrings ... |
| opprsubrng 20553 | Being a subring is a symme... |
| subrngint 20554 | The intersection of a none... |
| subrngin 20555 | The intersection of two su... |
| subrngmre 20556 | The subrings of a non-unit... |
| subsubrng 20557 | A subring of a subring is ... |
| subsubrng2 20558 | The set of subrings of a s... |
| rhmimasubrnglem 20559 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20560 | The homomorphic image of a... |
| cntzsubrng 20561 | Centralizers in a non-unit... |
| subrngpropd 20562 | If two structures have the... |
| issubrg 20567 | The subring predicate. (C... |
| subrgss 20568 | A subring is a subset. (C... |
| subrgid 20569 | Every ring is a subring of... |
| subrgring 20570 | A subring is a ring. (Con... |
| subrgcrng 20571 | A subring of a commutative... |
| subrgrcl 20572 | Reverse closure for a subr... |
| subrgsubg 20573 | A subring is a subgroup. ... |
| subrgsubrng 20574 | A subring of a unital ring... |
| subrg0 20575 | A subring always has the s... |
| subrg1cl 20576 | A subring contains the mul... |
| subrgbas 20577 | Base set of a subring stru... |
| subrg1 20578 | A subring always has the s... |
| subrgacl 20579 | A subring is closed under ... |
| subrgmcl 20580 | A subring is closed under ... |
| subrgsubm 20581 | A subring is a submonoid o... |
| subrgdvds 20582 | If an element divides anot... |
| subrguss 20583 | A unit of a subring is a u... |
| subrginv 20584 | A subring always has the s... |
| subrgdv 20585 | A subring always has the s... |
| subrgunit 20586 | An element of a ring is a ... |
| subrgugrp 20587 | The units of a subring for... |
| issubrg2 20588 | Characterize the subrings ... |
| opprsubrg 20589 | Being a subring is a symme... |
| subrgnzr 20590 | A subring of a nonzero rin... |
| subrgint 20591 | The intersection of a none... |
| subrgin 20592 | The intersection of two su... |
| subrgmre 20593 | The subrings of a ring are... |
| subsubrg 20594 | A subring of a subring is ... |
| subsubrg2 20595 | The set of subrings of a s... |
| issubrg3 20596 | A subring is an additive s... |
| resrhm 20597 | Restriction of a ring homo... |
| resrhm2b 20598 | Restriction of the codomai... |
| rhmeql 20599 | The equalizer of two ring ... |
| rhmima 20600 | The homomorphic image of a... |
| rnrhmsubrg 20601 | The range of a ring homomo... |
| cntzsubr 20602 | Centralizers in a ring are... |
| pwsdiagrhm 20603 | Diagonal homomorphism into... |
| subrgpropd 20604 | If two structures have the... |
| rhmpropd 20605 | Ring homomorphism depends ... |
| rngcval 20608 | Value of the category of n... |
| rnghmresfn 20609 | The class of non-unital ri... |
| rnghmresel 20610 | An element of the non-unit... |
| rngcbas 20611 | Set of objects of the cate... |
| rngchomfval 20612 | Set of arrows of the categ... |
| rngchom 20613 | Set of arrows of the categ... |
| elrngchom 20614 | A morphism of non-unital r... |
| rngchomfeqhom 20615 | The functionalized Hom-set... |
| rngccofval 20616 | Composition in the categor... |
| rngcco 20617 | Composition in the categor... |
| dfrngc2 20618 | Alternate definition of th... |
| rnghmsscmap2 20619 | The non-unital ring homomo... |
| rnghmsscmap 20620 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20621 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20622 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20623 | The non-unital ring homomo... |
| rngccat 20624 | The category of non-unital... |
| rngcid 20625 | The identity arrow in the ... |
| rngcsect 20626 | A section in the category ... |
| rngcinv 20627 | An inverse in the category... |
| rngciso 20628 | An isomorphism in the cate... |
| rngcifuestrc 20629 | The "inclusion functor" fr... |
| funcrngcsetc 20630 | The "natural forgetful fun... |
| funcrngcsetcALT 20631 | Alternate proof of ~ funcr... |
| zrinitorngc 20632 | The zero ring is an initia... |
| zrtermorngc 20633 | The zero ring is a termina... |
| zrzeroorngc 20634 | The zero ring is a zero ob... |
| ringcval 20637 | Value of the category of u... |
| rhmresfn 20638 | The class of unital ring h... |
| rhmresel 20639 | An element of the unital r... |
| ringcbas 20640 | Set of objects of the cate... |
| ringchomfval 20641 | Set of arrows of the categ... |
| ringchom 20642 | Set of arrows of the categ... |
| elringchom 20643 | A morphism of unital rings... |
| ringchomfeqhom 20644 | The functionalized Hom-set... |
| ringccofval 20645 | Composition in the categor... |
| ringcco 20646 | Composition in the categor... |
| dfringc2 20647 | Alternate definition of th... |
| rhmsscmap2 20648 | The unital ring homomorphi... |
| rhmsscmap 20649 | The unital ring homomorphi... |
| rhmsubcsetclem1 20650 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20651 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20652 | The unital ring homomorphi... |
| ringccat 20653 | The category of unital rin... |
| ringcid 20654 | The identity arrow in the ... |
| rhmsscrnghm 20655 | The unital ring homomorphi... |
| rhmsubcrngclem1 20656 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20657 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20658 | The unital ring homomorphi... |
| rngcresringcat 20659 | The restriction of the cat... |
| ringcsect 20660 | A section in the category ... |
| ringcinv 20661 | An inverse in the category... |
| ringciso 20662 | An isomorphism in the cate... |
| ringcbasbas 20663 | An element of the base set... |
| funcringcsetc 20664 | The "natural forgetful fun... |
| zrtermoringc 20665 | The zero ring is a termina... |
| zrninitoringc 20666 | The zero ring is not an in... |
| srhmsubclem1 20667 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20668 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20669 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20670 | According to ~ df-subc , t... |
| sringcat 20671 | The restriction of the cat... |
| crhmsubc 20672 | According to ~ df-subc , t... |
| cringcat 20673 | The restriction of the cat... |
| rngcrescrhm 20674 | The category of non-unital... |
| rhmsubclem1 20675 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20676 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20677 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20678 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20679 | According to ~ df-subc , t... |
| rhmsubccat 20680 | The restriction of the cat... |
| rrgval 20687 | Value of the set or left-r... |
| isrrg 20688 | Membership in the set of l... |
| rrgeq0i 20689 | Property of a left-regular... |
| rrgeq0 20690 | Left-multiplication by a l... |
| rrgsupp 20691 | Left multiplication by a l... |
| rrgss 20692 | Left-regular elements are ... |
| unitrrg 20693 | Units are regular elements... |
| rrgnz 20694 | In a nonzero ring, the zer... |
| isdomn 20695 | Expand definition of a dom... |
| domnnzr 20696 | A domain is a nonzero ring... |
| domnring 20697 | A domain is a ring. (Cont... |
| domneq0 20698 | In a domain, a product is ... |
| domnmuln0 20699 | In a domain, a product of ... |
| isdomn5 20700 | The equivalence between th... |
| isdomn2 20701 | A ring is a domain iff all... |
| isdomn2OLD 20702 | Obsolete version of ~ isdo... |
| domnrrg 20703 | In a domain, a nonzero ele... |
| isdomn6 20704 | A ring is a domain iff the... |
| isdomn3 20705 | Nonzero elements form a mu... |
| isdomn4 20706 | A ring is a domain iff it ... |
| opprdomnb 20707 | A class is a domain if and... |
| opprdomn 20708 | The opposite of a domain i... |
| isdomn4r 20709 | A ring is a domain iff it ... |
| domnlcanb 20710 | Left-cancellation law for ... |
| domnlcan 20711 | Left-cancellation law for ... |
| domnrcanb 20712 | Right-cancellation law for... |
| domnrcan 20713 | Right-cancellation law for... |
| domneq0r 20714 | Right multiplication by a ... |
| isidom 20715 | An integral domain is a co... |
| idomdomd 20716 | An integral domain is a do... |
| idomcringd 20717 | An integral domain is a co... |
| idomringd 20718 | An integral domain is a ri... |
| isdrng 20723 | The predicate "is a divisi... |
| drngunit 20724 | Elementhood in the set of ... |
| drngui 20725 | The set of units of a divi... |
| drngring 20726 | A division ring is a ring.... |
| drngringd 20727 | A division ring is a ring.... |
| drnggrpd 20728 | A division ring is a group... |
| drnggrp 20729 | A division ring is a group... |
| isfld 20730 | A field is a commutative d... |
| flddrngd 20731 | A field is a division ring... |
| fldcrngd 20732 | A field is a commutative r... |
| isdrng2 20733 | A division ring can equiva... |
| drngprop 20734 | If two structures have the... |
| drngmgp 20735 | A division ring contains a... |
| drngid 20736 | A division ring's unity is... |
| drngunz 20737 | A division ring's unity is... |
| drngnzr 20738 | A division ring is a nonze... |
| drngdomn 20739 | A division ring is a domai... |
| drngmcl 20740 | The product of two nonzero... |
| drngmclOLD 20741 | Obsolete version of ~ drng... |
| drngid2 20742 | Properties showing that an... |
| drnginvrcl 20743 | Closure of the multiplicat... |
| drnginvrn0 20744 | The multiplicative inverse... |
| drnginvrcld 20745 | Closure of the multiplicat... |
| drnginvrl 20746 | Property of the multiplica... |
| drnginvrr 20747 | Property of the multiplica... |
| drnginvrld 20748 | Property of the multiplica... |
| drnginvrrd 20749 | Property of the multiplica... |
| drngmul0or 20750 | A product is zero iff one ... |
| drngmul0orOLD 20751 | Obsolete version of ~ drng... |
| drngmulne0 20752 | A product is nonzero iff b... |
| drngmuleq0 20753 | An element is zero iff its... |
| opprdrng 20754 | The opposite of a division... |
| isdrngd 20755 | Properties that characteri... |
| isdrngrd 20756 | Properties that characteri... |
| isdrngdOLD 20757 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20758 | Obsolete version of ~ isdr... |
| drngpropd 20759 | If two structures have the... |
| fldpropd 20760 | If two structures have the... |
| fldidom 20761 | A field is an integral dom... |
| fldidomOLD 20762 | Obsolete version of ~ fldi... |
| fidomndrnglem 20763 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20764 | A finite domain is a divis... |
| fiidomfld 20765 | A finite integral domain i... |
| rng1nnzr 20766 | The (smallest) structure r... |
| ring1zr 20767 | The only (unital) ring wit... |
| rngen1zr 20768 | The only (unital) ring wit... |
| ringen1zr 20769 | The only unital ring with ... |
| rng1nfld 20770 | The zero ring is not a fie... |
| issubdrg 20771 | Characterize the subfields... |
| drhmsubc 20772 | According to ~ df-subc , t... |
| drngcat 20773 | The restriction of the cat... |
| fldcat 20774 | The restriction of the cat... |
| fldc 20775 | The restriction of the cat... |
| fldhmsubc 20776 | According to ~ df-subc , t... |
| issdrg 20779 | Property of a division sub... |
| sdrgrcl 20780 | Reverse closure for a sub-... |
| sdrgdrng 20781 | A sub-division-ring is a d... |
| sdrgsubrg 20782 | A sub-division-ring is a s... |
| sdrgid 20783 | Every division ring is a d... |
| sdrgss 20784 | A division subring is a su... |
| sdrgbas 20785 | Base set of a sub-division... |
| issdrg2 20786 | Property of a division sub... |
| sdrgunit 20787 | A unit of a sub-division-r... |
| imadrhmcl 20788 | The image of a (nontrivial... |
| fldsdrgfld 20789 | A sub-division-ring of a f... |
| acsfn1p 20790 | Construction of a closure ... |
| subrgacs 20791 | Closure property of subrin... |
| sdrgacs 20792 | Closure property of divisi... |
| cntzsdrg 20793 | Centralizers in division r... |
| subdrgint 20794 | The intersection of a none... |
| sdrgint 20795 | The intersection of a none... |
| primefld 20796 | The smallest sub division ... |
| primefld0cl 20797 | The prime field contains t... |
| primefld1cl 20798 | The prime field contains t... |
| abvfval 20801 | Value of the set of absolu... |
| isabv 20802 | Elementhood in the set of ... |
| isabvd 20803 | Properties that determine ... |
| abvrcl 20804 | Reverse closure for the ab... |
| abvfge0 20805 | An absolute value is a fun... |
| abvf 20806 | An absolute value is a fun... |
| abvcl 20807 | An absolute value is a fun... |
| abvge0 20808 | The absolute value of a nu... |
| abveq0 20809 | The value of an absolute v... |
| abvne0 20810 | The absolute value of a no... |
| abvgt0 20811 | The absolute value of a no... |
| abvmul 20812 | An absolute value distribu... |
| abvtri 20813 | An absolute value satisfie... |
| abv0 20814 | The absolute value of zero... |
| abv1z 20815 | The absolute value of one ... |
| abv1 20816 | The absolute value of one ... |
| abvneg 20817 | The absolute value of a ne... |
| abvsubtri 20818 | An absolute value satisfie... |
| abvrec 20819 | The absolute value distrib... |
| abvdiv 20820 | The absolute value distrib... |
| abvdom 20821 | Any ring with an absolute ... |
| abvres 20822 | The restriction of an abso... |
| abvtrivd 20823 | The trivial absolute value... |
| abvtrivg 20824 | The trivial absolute value... |
| abvtriv 20825 | The trivial absolute value... |
| abvpropd 20826 | If two structures have the... |
| abvn0b 20827 | Another characterization o... |
| staffval 20832 | The functionalization of t... |
| stafval 20833 | The functionalization of t... |
| staffn 20834 | The functionalization is e... |
| issrng 20835 | The predicate "is a star r... |
| srngrhm 20836 | The involution function in... |
| srngring 20837 | A star ring is a ring. (C... |
| srngcnv 20838 | The involution function in... |
| srngf1o 20839 | The involution function in... |
| srngcl 20840 | The involution function in... |
| srngnvl 20841 | The involution function in... |
| srngadd 20842 | The involution function in... |
| srngmul 20843 | The involution function in... |
| srng1 20844 | The conjugate of the ring ... |
| srng0 20845 | The conjugate of the ring ... |
| issrngd 20846 | Properties that determine ... |
| idsrngd 20847 | A commutative ring is a st... |
| islmod 20852 | The predicate "is a left m... |
| lmodlema 20853 | Lemma for properties of a ... |
| islmodd 20854 | Properties that determine ... |
| lmodgrp 20855 | A left module is a group. ... |
| lmodring 20856 | The scalar component of a ... |
| lmodfgrp 20857 | The scalar component of a ... |
| lmodgrpd 20858 | A left module is a group. ... |
| lmodbn0 20859 | The base set of a left mod... |
| lmodacl 20860 | Closure of ring addition f... |
| lmodmcl 20861 | Closure of ring multiplica... |
| lmodsn0 20862 | The set of scalars in a le... |
| lmodvacl 20863 | Closure of vector addition... |
| lmodass 20864 | Left module vector sum is ... |
| lmodlcan 20865 | Left cancellation law for ... |
| lmodvscl 20866 | Closure of scalar product ... |
| lmodvscld 20867 | Closure of scalar product ... |
| scaffval 20868 | The scalar multiplication ... |
| scafval 20869 | The scalar multiplication ... |
| scafeq 20870 | If the scalar multiplicati... |
| scaffn 20871 | The scalar multiplication ... |
| lmodscaf 20872 | The scalar multiplication ... |
| lmodvsdi 20873 | Distributive law for scala... |
| lmodvsdir 20874 | Distributive law for scala... |
| lmodvsass 20875 | Associative law for scalar... |
| lmod0cl 20876 | The ring zero in a left mo... |
| lmod1cl 20877 | The ring unity in a left m... |
| lmodvs1 20878 | Scalar product with the ri... |
| lmod0vcl 20879 | The zero vector is a vecto... |
| lmod0vlid 20880 | Left identity law for the ... |
| lmod0vrid 20881 | Right identity law for the... |
| lmod0vid 20882 | Identity equivalent to the... |
| lmod0vs 20883 | Zero times a vector is the... |
| lmodvs0 20884 | Anything times the zero ve... |
| lmodvsmmulgdi 20885 | Distributive law for a gro... |
| lmodfopnelem1 20886 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20887 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20888 | The (functionalized) opera... |
| lcomf 20889 | A linear-combination sum i... |
| lcomfsupp 20890 | A linear-combination sum i... |
| lmodvnegcl 20891 | Closure of vector negative... |
| lmodvnegid 20892 | Addition of a vector with ... |
| lmodvneg1 20893 | Minus 1 times a vector is ... |
| lmodvsneg 20894 | Multiplication of a vector... |
| lmodvsubcl 20895 | Closure of vector subtract... |
| lmodcom 20896 | Left module vector sum is ... |
| lmodabl 20897 | A left module is an abelia... |
| lmodcmn 20898 | A left module is a commuta... |
| lmodnegadd 20899 | Distribute negation throug... |
| lmod4 20900 | Commutative/associative la... |
| lmodvsubadd 20901 | Relationship between vecto... |
| lmodvaddsub4 20902 | Vector addition/subtractio... |
| lmodvpncan 20903 | Addition/subtraction cance... |
| lmodvnpcan 20904 | Cancellation law for vecto... |
| lmodvsubval2 20905 | Value of vector subtractio... |
| lmodsubvs 20906 | Subtraction of a scalar pr... |
| lmodsubdi 20907 | Scalar multiplication dist... |
| lmodsubdir 20908 | Scalar multiplication dist... |
| lmodsubeq0 20909 | If the difference between ... |
| lmodsubid 20910 | Subtraction of a vector fr... |
| lmodvsghm 20911 | Scalar multiplication of t... |
| lmodprop2d 20912 | If two structures have the... |
| lmodpropd 20913 | If two structures have the... |
| gsumvsmul 20914 | Pull a scalar multiplicati... |
| mptscmfsupp0 20915 | A mapping to a scalar prod... |
| mptscmfsuppd 20916 | A function mapping to a sc... |
| rmodislmodlem 20917 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20918 | The right module ` R ` ind... |
| rmodislmodOLD 20919 | Obsolete version of ~ rmod... |
| lssset 20922 | The set of all (not necess... |
| islss 20923 | The predicate "is a subspa... |
| islssd 20924 | Properties that determine ... |
| lssss 20925 | A subspace is a set of vec... |
| lssel 20926 | A subspace member is a vec... |
| lss1 20927 | The set of vectors in a le... |
| lssuni 20928 | The union of all subspaces... |
| lssn0 20929 | A subspace is not empty. ... |
| 00lss 20930 | The empty structure has no... |
| lsscl 20931 | Closure property of a subs... |
| lssvacl 20932 | Closure of vector addition... |
| lssvsubcl 20933 | Closure of vector subtract... |
| lssvancl1 20934 | Non-closure: if one vector... |
| lssvancl2 20935 | Non-closure: if one vector... |
| lss0cl 20936 | The zero vector belongs to... |
| lsssn0 20937 | The singleton of the zero ... |
| lss0ss 20938 | The zero subspace is inclu... |
| lssle0 20939 | No subspace is smaller tha... |
| lssne0 20940 | A nonzero subspace has a n... |
| lssvneln0 20941 | A vector ` X ` which doesn... |
| lssneln0 20942 | A vector ` X ` which doesn... |
| lssssr 20943 | Conclude subspace ordering... |
| lssvscl 20944 | Closure of scalar product ... |
| lssvnegcl 20945 | Closure of negative vector... |
| lsssubg 20946 | All subspaces are subgroup... |
| lsssssubg 20947 | All subspaces are subgroup... |
| islss3 20948 | A linear subspace of a mod... |
| lsslmod 20949 | A submodule is a module. ... |
| lsslss 20950 | The subspaces of a subspac... |
| islss4 20951 | A linear subspace is a sub... |
| lss1d 20952 | One-dimensional subspace (... |
| lssintcl 20953 | The intersection of a none... |
| lssincl 20954 | The intersection of two su... |
| lssmre 20955 | The subspaces of a module ... |
| lssacs 20956 | Submodules are an algebrai... |
| prdsvscacl 20957 | Pointwise scalar multiplic... |
| prdslmodd 20958 | The product of a family of... |
| pwslmod 20959 | A structure power of a lef... |
| lspfval 20962 | The span function for a le... |
| lspf 20963 | The span function on a lef... |
| lspval 20964 | The span of a set of vecto... |
| lspcl 20965 | The span of a set of vecto... |
| lspsncl 20966 | The span of a singleton is... |
| lspprcl 20967 | The span of a pair is a su... |
| lsptpcl 20968 | The span of an unordered t... |
| lspsnsubg 20969 | The span of a singleton is... |
| 00lsp 20970 | ~ fvco4i lemma for linear ... |
| lspid 20971 | The span of a subspace is ... |
| lspssv 20972 | A span is a set of vectors... |
| lspss 20973 | Span preserves subset orde... |
| lspssid 20974 | A set of vectors is a subs... |
| lspidm 20975 | The span of a set of vecto... |
| lspun 20976 | The span of union is the s... |
| lspssp 20977 | If a set of vectors is a s... |
| mrclsp 20978 | Moore closure generalizes ... |
| lspsnss 20979 | The span of the singleton ... |
| ellspsn3 20980 | A member of the span of th... |
| lspprss 20981 | The span of a pair of vect... |
| lspsnid 20982 | A vector belongs to the sp... |
| ellspsn6 20983 | Relationship between a vec... |
| ellspsn5b 20984 | Relationship between a vec... |
| ellspsn5 20985 | Relationship between a vec... |
| lspprid1 20986 | A member of a pair of vect... |
| lspprid2 20987 | A member of a pair of vect... |
| lspprvacl 20988 | The sum of two vectors bel... |
| lssats2 20989 | A way to express atomistic... |
| ellspsni 20990 | A scalar product with a ve... |
| lspsn 20991 | Span of the singleton of a... |
| ellspsn 20992 | Member of span of the sing... |
| lspsnvsi 20993 | Span of a scalar product o... |
| lspsnss2 20994 | Comparable spans of single... |
| lspsnneg 20995 | Negation does not change t... |
| lspsnsub 20996 | Swapping subtraction order... |
| lspsn0 20997 | Span of the singleton of t... |
| lsp0 20998 | Span of the empty set. (C... |
| lspuni0 20999 | Union of the span of the e... |
| lspun0 21000 | The span of a union with t... |
| lspsneq0 21001 | Span of the singleton is t... |
| lspsneq0b 21002 | Equal singleton spans impl... |
| lmodindp1 21003 | Two independent (non-colin... |
| lsslsp 21004 | Spans in submodules corres... |
| lsslspOLD 21005 | Obsolete version of ~ lssl... |
| lss0v 21006 | The zero vector in a submo... |
| lsspropd 21007 | If two structures have the... |
| lsppropd 21008 | If two structures have the... |
| reldmlmhm 21015 | Lemma for module homomorph... |
| lmimfn 21016 | Lemma for module isomorphi... |
| islmhm 21017 | Property of being a homomo... |
| islmhm3 21018 | Property of a module homom... |
| lmhmlem 21019 | Non-quantified consequence... |
| lmhmsca 21020 | A homomorphism of left mod... |
| lmghm 21021 | A homomorphism of left mod... |
| lmhmlmod2 21022 | A homomorphism of left mod... |
| lmhmlmod1 21023 | A homomorphism of left mod... |
| lmhmf 21024 | A homomorphism of left mod... |
| lmhmlin 21025 | A homomorphism of left mod... |
| lmodvsinv 21026 | Multiplication of a vector... |
| lmodvsinv2 21027 | Multiplying a negated vect... |
| islmhm2 21028 | A one-equation proof of li... |
| islmhmd 21029 | Deduction for a module hom... |
| 0lmhm 21030 | The constant zero linear f... |
| idlmhm 21031 | The identity function on a... |
| invlmhm 21032 | The negative function on a... |
| lmhmco 21033 | The composition of two mod... |
| lmhmplusg 21034 | The pointwise sum of two l... |
| lmhmvsca 21035 | The pointwise scalar produ... |
| lmhmf1o 21036 | A bijective module homomor... |
| lmhmima 21037 | The image of a subspace un... |
| lmhmpreima 21038 | The inverse image of a sub... |
| lmhmlsp 21039 | Homomorphisms preserve spa... |
| lmhmrnlss 21040 | The range of a homomorphis... |
| lmhmkerlss 21041 | The kernel of a homomorphi... |
| reslmhm 21042 | Restriction of a homomorph... |
| reslmhm2 21043 | Expansion of the codomain ... |
| reslmhm2b 21044 | Expansion of the codomain ... |
| lmhmeql 21045 | The equalizer of two modul... |
| lspextmo 21046 | A linear function is compl... |
| pwsdiaglmhm 21047 | Diagonal homomorphism into... |
| pwssplit0 21048 | Splitting for structure po... |
| pwssplit1 21049 | Splitting for structure po... |
| pwssplit2 21050 | Splitting for structure po... |
| pwssplit3 21051 | Splitting for structure po... |
| islmim 21052 | An isomorphism of left mod... |
| lmimf1o 21053 | An isomorphism of left mod... |
| lmimlmhm 21054 | An isomorphism of modules ... |
| lmimgim 21055 | An isomorphism of modules ... |
| islmim2 21056 | An isomorphism of left mod... |
| lmimcnv 21057 | The converse of a bijectiv... |
| brlmic 21058 | The relation "is isomorphi... |
| brlmici 21059 | Prove isomorphic by an exp... |
| lmiclcl 21060 | Isomorphism implies the le... |
| lmicrcl 21061 | Isomorphism implies the ri... |
| lmicsym 21062 | Module isomorphism is symm... |
| lmhmpropd 21063 | Module homomorphism depend... |
| islbs 21066 | The predicate " ` B ` is a... |
| lbsss 21067 | A basis is a set of vector... |
| lbsel 21068 | An element of a basis is a... |
| lbssp 21069 | The span of a basis is the... |
| lbsind 21070 | A basis is linearly indepe... |
| lbsind2 21071 | A basis is linearly indepe... |
| lbspss 21072 | No proper subset of a basi... |
| lsmcl 21073 | The sum of two subspaces i... |
| lsmspsn 21074 | Member of subspace sum of ... |
| lsmelval2 21075 | Subspace sum membership in... |
| lsmsp 21076 | Subspace sum in terms of s... |
| lsmsp2 21077 | Subspace sum of spans of s... |
| lsmssspx 21078 | Subspace sum (in its exten... |
| lsmpr 21079 | The span of a pair of vect... |
| lsppreli 21080 | A vector expressed as a su... |
| lsmelpr 21081 | Two ways to say that a vec... |
| lsppr0 21082 | The span of a vector paire... |
| lsppr 21083 | Span of a pair of vectors.... |
| lspprel 21084 | Member of the span of a pa... |
| lspprabs 21085 | Absorption of vector sum i... |
| lspvadd 21086 | The span of a vector sum i... |
| lspsntri 21087 | Triangle-type inequality f... |
| lspsntrim 21088 | Triangle-type inequality f... |
| lbspropd 21089 | If two structures have the... |
| pj1lmhm 21090 | The left projection functi... |
| pj1lmhm2 21091 | The left projection functi... |
| islvec 21094 | The predicate "is a left v... |
| lvecdrng 21095 | The set of scalars of a le... |
| lveclmod 21096 | A left vector space is a l... |
| lveclmodd 21097 | A vector space is a left m... |
| lvecgrpd 21098 | A vector space is a group.... |
| lsslvec 21099 | A vector subspace is a vec... |
| lmhmlvec 21100 | The property for modules t... |
| lvecvs0or 21101 | If a scalar product is zer... |
| lvecvsn0 21102 | A scalar product is nonzer... |
| lssvs0or 21103 | If a scalar product belong... |
| lvecvscan 21104 | Cancellation law for scala... |
| lvecvscan2 21105 | Cancellation law for scala... |
| lvecinv 21106 | Invert coefficient of scal... |
| lspsnvs 21107 | A nonzero scalar product d... |
| lspsneleq 21108 | Membership relation that i... |
| lspsncmp 21109 | Comparable spans of nonzer... |
| lspsnne1 21110 | Two ways to express that v... |
| lspsnne2 21111 | Two ways to express that v... |
| lspsnnecom 21112 | Swap two vectors with diff... |
| lspabs2 21113 | Absorption law for span of... |
| lspabs3 21114 | Absorption law for span of... |
| lspsneq 21115 | Equal spans of singletons ... |
| lspsneu 21116 | Nonzero vectors with equal... |
| ellspsn4 21117 | A member of the span of th... |
| lspdisj 21118 | The span of a vector not i... |
| lspdisjb 21119 | A nonzero vector is not in... |
| lspdisj2 21120 | Unequal spans are disjoint... |
| lspfixed 21121 | Show membership in the spa... |
| lspexch 21122 | Exchange property for span... |
| lspexchn1 21123 | Exchange property for span... |
| lspexchn2 21124 | Exchange property for span... |
| lspindpi 21125 | Partial independence prope... |
| lspindp1 21126 | Alternate way to say 3 vec... |
| lspindp2l 21127 | Alternate way to say 3 vec... |
| lspindp2 21128 | Alternate way to say 3 vec... |
| lspindp3 21129 | Independence of 2 vectors ... |
| lspindp4 21130 | (Partial) independence of ... |
| lvecindp 21131 | Compute the ` X ` coeffici... |
| lvecindp2 21132 | Sums of independent vector... |
| lspsnsubn0 21133 | Unequal singleton spans im... |
| lsmcv 21134 | Subspace sum has the cover... |
| lspsolvlem 21135 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21136 | If ` X ` is in the span of... |
| lssacsex 21137 | In a vector space, subspac... |
| lspsnat 21138 | There is no subspace stric... |
| lspsncv0 21139 | The span of a singleton co... |
| lsppratlem1 21140 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21141 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21142 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21143 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21144 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21145 | Lemma for ~ lspprat . Neg... |
| lspprat 21146 | A proper subspace of the s... |
| islbs2 21147 | An equivalent formulation ... |
| islbs3 21148 | An equivalent formulation ... |
| lbsacsbs 21149 | Being a basis in a vector ... |
| lvecdim 21150 | The dimension theorem for ... |
| lbsextlem1 21151 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21152 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21153 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21154 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21155 | For any linearly independe... |
| lbsext 21156 | For any linearly independe... |
| lbsexg 21157 | Every vector space has a b... |
| lbsex 21158 | Every vector space has a b... |
| lvecprop2d 21159 | If two structures have the... |
| lvecpropd 21160 | If two structures have the... |
| sraval 21165 | Lemma for ~ srabase throug... |
| sralem 21166 | Lemma for ~ srabase and si... |
| sralemOLD 21167 | Obsolete version of ~ sral... |
| srabase 21168 | Base set of a subring alge... |
| srabaseOLD 21169 | Obsolete proof of ~ srabas... |
| sraaddg 21170 | Additive operation of a su... |
| sraaddgOLD 21171 | Obsolete proof of ~ sraadd... |
| sramulr 21172 | Multiplicative operation o... |
| sramulrOLD 21173 | Obsolete proof of ~ sramul... |
| srasca 21174 | The set of scalars of a su... |
| srascaOLD 21175 | Obsolete proof of ~ srasca... |
| sravsca 21176 | The scalar product operati... |
| sravscaOLD 21177 | Obsolete proof of ~ sravsc... |
| sraip 21178 | The inner product operatio... |
| sratset 21179 | Topology component of a su... |
| sratsetOLD 21180 | Obsolete proof of ~ sratse... |
| sratopn 21181 | Topology component of a su... |
| srads 21182 | Distance function of a sub... |
| sradsOLD 21183 | Obsolete proof of ~ srads ... |
| sraring 21184 | Condition for a subring al... |
| sralmod 21185 | The subring algebra is a l... |
| sralmod0 21186 | The subring module inherit... |
| issubrgd 21187 | Prove a subring by closure... |
| rlmfn 21188 | ` ringLMod ` is a function... |
| rlmval 21189 | Value of the ring module. ... |
| rlmval2 21190 | Value of the ring module e... |
| rlmbas 21191 | Base set of the ring modul... |
| rlmplusg 21192 | Vector addition in the rin... |
| rlm0 21193 | Zero vector in the ring mo... |
| rlmsub 21194 | Subtraction in the ring mo... |
| rlmmulr 21195 | Ring multiplication in the... |
| rlmsca 21196 | Scalars in the ring module... |
| rlmsca2 21197 | Scalars in the ring module... |
| rlmvsca 21198 | Scalar multiplication in t... |
| rlmtopn 21199 | Topology component of the ... |
| rlmds 21200 | Metric component of the ri... |
| rlmlmod 21201 | The ring module is a modul... |
| rlmlvec 21202 | The ring module over a div... |
| rlmlsm 21203 | Subgroup sum of the ring m... |
| rlmvneg 21204 | Vector negation in the rin... |
| rlmscaf 21205 | Functionalized scalar mult... |
| ixpsnbasval 21206 | The value of an infinite C... |
| lidlval 21211 | Value of the set of ring i... |
| rspval 21212 | Value of the ring span fun... |
| lidlss 21213 | An ideal is a subset of th... |
| lidlssbas 21214 | The base set of the restri... |
| lidlbas 21215 | A (left) ideal of a ring i... |
| islidl 21216 | Predicate of being a (left... |
| rnglidlmcl 21217 | A (left) ideal containing ... |
| rngridlmcl 21218 | A right ideal (which is a ... |
| dflidl2rng 21219 | Alternate (the usual textb... |
| isridlrng 21220 | A right ideal is a left id... |
| lidl0cl 21221 | An ideal contains 0. (Con... |
| lidlacl 21222 | An ideal is closed under a... |
| lidlnegcl 21223 | An ideal contains negative... |
| lidlsubg 21224 | An ideal is a subgroup of ... |
| lidlsubcl 21225 | An ideal is closed under s... |
| lidlmcl 21226 | An ideal is closed under l... |
| lidl1el 21227 | An ideal contains 1 iff it... |
| dflidl2 21228 | Alternate (the usual textb... |
| lidl0ALT 21229 | Alternate proof for ~ lidl... |
| rnglidl0 21230 | Every non-unital ring cont... |
| lidl0 21231 | Every ring contains a zero... |
| lidl1ALT 21232 | Alternate proof for ~ lidl... |
| rnglidl1 21233 | The base set of every non-... |
| lidl1 21234 | Every ring contains a unit... |
| lidlacs 21235 | The ideal system is an alg... |
| rspcl 21236 | The span of a set of ring ... |
| rspssid 21237 | The span of a set of ring ... |
| rsp1 21238 | The span of the identity e... |
| rsp0 21239 | The span of the zero eleme... |
| rspssp 21240 | The ideal span of a set of... |
| elrspsn 21241 | Membership in a principal ... |
| mrcrsp 21242 | Moore closure generalizes ... |
| lidlnz 21243 | A nonzero ideal contains a... |
| drngnidl 21244 | A division ring has only t... |
| lidlrsppropd 21245 | The left ideals and ring s... |
| rnglidlmmgm 21246 | The multiplicative group o... |
| rnglidlmsgrp 21247 | The multiplicative group o... |
| rnglidlrng 21248 | A (left) ideal of a non-un... |
| lidlnsg 21249 | An ideal is a normal subgr... |
| 2idlval 21252 | Definition of a two-sided ... |
| isridl 21253 | A right ideal is a left id... |
| 2idlelb 21254 | Membership in a two-sided ... |
| 2idllidld 21255 | A two-sided ideal is a lef... |
| 2idlridld 21256 | A two-sided ideal is a rig... |
| df2idl2rng 21257 | Alternate (the usual textb... |
| df2idl2 21258 | Alternate (the usual textb... |
| ridl0 21259 | Every ring contains a zero... |
| ridl1 21260 | Every ring contains a unit... |
| 2idl0 21261 | Every ring contains a zero... |
| 2idl1 21262 | Every ring contains a unit... |
| 2idlss 21263 | A two-sided ideal is a sub... |
| 2idlbas 21264 | The base set of a two-side... |
| 2idlelbas 21265 | The base set of a two-side... |
| rng2idlsubrng 21266 | A two-sided ideal of a non... |
| rng2idlnsg 21267 | A two-sided ideal of a non... |
| rng2idl0 21268 | The zero (additive identit... |
| rng2idlsubgsubrng 21269 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21270 | A two-sided ideal of a non... |
| rng2idlsubg0 21271 | The zero (additive identit... |
| 2idlcpblrng 21272 | The coset equivalence rela... |
| 2idlcpbl 21273 | The coset equivalence rela... |
| qus2idrng 21274 | The quotient of a non-unit... |
| qus1 21275 | The multiplicative identit... |
| qusring 21276 | If ` S ` is a two-sided id... |
| qusrhm 21277 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21278 | The preimage of an ideal b... |
| kerlidl 21279 | The kernel of a ring homom... |
| qusmul2idl 21280 | Value of the ring operatio... |
| crngridl 21281 | In a commutative ring, the... |
| crng2idl 21282 | In a commutative ring, a t... |
| qusmulrng 21283 | Value of the multiplicatio... |
| quscrng 21284 | The quotient of a commutat... |
| qusmulcrng 21285 | Value of the ring operatio... |
| rhmqusnsg 21286 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21287 | The ring unity of a two-si... |
| rngqiprngghmlem1 21288 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21289 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21290 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21291 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21292 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21293 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21294 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21295 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21296 | The base set of the produc... |
| rngqiprng 21297 | The product of the quotien... |
| rngqiprngimf 21298 | ` F ` is a function from (... |
| rngqiprngimfv 21299 | The value of the function ... |
| rngqiprngghm 21300 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21301 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21302 | ` F ` is a function from (... |
| rngqiprnglin 21303 | ` F ` is linear with respe... |
| rngqiprngho 21304 | ` F ` is a homomorphism of... |
| rngqiprngim 21305 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21306 | The unity of a two-sided i... |
| rngringbdlem1 21307 | In a unital ring, the quot... |
| rngringbdlem2 21308 | A non-unital ring is unita... |
| rngringbd 21309 | A non-unital ring is unita... |
| ring2idlqus 21310 | For every unital ring ther... |
| ring2idlqusb 21311 | A non-unital ring is unita... |
| rngqiprngfulem1 21312 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21313 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21314 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21315 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21316 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21317 | The ring unity of the prod... |
| rngqiprngfu 21318 | The function value of ` F ... |
| rngqiprngu 21319 | If a non-unital ring has a... |
| ring2idlqus1 21320 | If a non-unital ring has a... |
| lpival 21325 | Value of the set of princi... |
| islpidl 21326 | Property of being a princi... |
| lpi0 21327 | The zero ideal is always p... |
| lpi1 21328 | The unit ideal is always p... |
| islpir 21329 | Principal ideal rings are ... |
| lpiss 21330 | Principal ideals are a sub... |
| islpir2 21331 | Principal ideal rings are ... |
| lpirring 21332 | Principal ideal rings are ... |
| drnglpir 21333 | Division rings are princip... |
| rspsn 21334 | Membership in principal id... |
| lidldvgen 21335 | An element generates an id... |
| lpigen 21336 | An ideal is principal iff ... |
| cnfldstr 21357 | The field of complex numbe... |
| cnfldex 21358 | The field of complex numbe... |
| cnfldbas 21359 | The base set of the field ... |
| mpocnfldadd 21360 | The addition operation of ... |
| cnfldadd 21361 | The addition operation of ... |
| mpocnfldmul 21362 | The multiplication operati... |
| cnfldmul 21363 | The multiplication operati... |
| cnfldcj 21364 | The conjugation operation ... |
| cnfldtset 21365 | The topology component of ... |
| cnfldle 21366 | The ordering of the field ... |
| cnfldds 21367 | The metric of the field of... |
| cnfldunif 21368 | The uniform structure comp... |
| cnfldfun 21369 | The field of complex numbe... |
| cnfldfunALT 21370 | The field of complex numbe... |
| dfcnfldOLD 21371 | Obsolete version of ~ df-c... |
| cnfldstrOLD 21372 | Obsolete version of ~ cnfl... |
| cnfldexOLD 21373 | Obsolete version of ~ cnfl... |
| cnfldbasOLD 21374 | Obsolete version of ~ cnfl... |
| cnfldaddOLD 21375 | Obsolete version of ~ cnfl... |
| cnfldmulOLD 21376 | Obsolete version of ~ cnfl... |
| cnfldcjOLD 21377 | Obsolete version of ~ cnfl... |
| cnfldtsetOLD 21378 | Obsolete version of ~ cnfl... |
| cnfldleOLD 21379 | Obsolete version of ~ cnfl... |
| cnflddsOLD 21380 | Obsolete version of ~ cnfl... |
| cnfldunifOLD 21381 | Obsolete version of ~ cnfl... |
| cnfldfunOLD 21382 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLD 21383 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLDOLD 21384 | Obsolete proof of ~ cnfldf... |
| xrsstr 21385 | The extended real structur... |
| xrsex 21386 | The extended real structur... |
| xrsbas 21387 | The base set of the extend... |
| xrsadd 21388 | The addition operation of ... |
| xrsmul 21389 | The multiplication operati... |
| xrstset 21390 | The topology component of ... |
| xrsle 21391 | The ordering of the extend... |
| cncrng 21392 | The complex numbers form a... |
| cncrngOLD 21393 | Obsolete version of ~ cncr... |
| cnring 21394 | The complex numbers form a... |
| xrsmcmn 21395 | The "multiplicative group"... |
| cnfld0 21396 | Zero is the zero element o... |
| cnfld1 21397 | One is the unity element o... |
| cnfld1OLD 21398 | Obsolete version of ~ cnfl... |
| cnfldneg 21399 | The additive inverse in th... |
| cnfldplusf 21400 | The functionalized additio... |
| cnfldsub 21401 | The subtraction operator i... |
| cndrng 21402 | The complex numbers form a... |
| cndrngOLD 21403 | Obsolete version of ~ cndr... |
| cnflddiv 21404 | The division operation in ... |
| cnflddivOLD 21405 | Obsolete version of ~ cnfl... |
| cnfldinv 21406 | The multiplicative inverse... |
| cnfldmulg 21407 | The group multiple functio... |
| cnfldexp 21408 | The exponentiation operato... |
| cnsrng 21409 | The complex numbers form a... |
| xrsmgm 21410 | The "additive group" of th... |
| xrsnsgrp 21411 | The "additive group" of th... |
| xrsmgmdifsgrp 21412 | The "additive group" of th... |
| xrs1mnd 21413 | The extended real numbers,... |
| xrs10 21414 | The zero of the extended r... |
| xrs1cmn 21415 | The extended real numbers ... |
| xrge0subm 21416 | The nonnegative extended r... |
| xrge0cmn 21417 | The nonnegative extended r... |
| xrsds 21418 | The metric of the extended... |
| xrsdsval 21419 | The metric of the extended... |
| xrsdsreval 21420 | The metric of the extended... |
| xrsdsreclblem 21421 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21422 | The metric of the extended... |
| cnsubmlem 21423 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21424 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21425 | Lemma for ~ resubdrg and f... |
| cnsubrglemOLD 21426 | Obsolete version of ~ cnsu... |
| cnsubdrglem 21427 | Lemma for ~ resubdrg and f... |
| qsubdrg 21428 | The rational numbers form ... |
| zsubrg 21429 | The integers form a subrin... |
| gzsubrg 21430 | The gaussian integers form... |
| nn0subm 21431 | The nonnegative integers f... |
| rege0subm 21432 | The nonnegative reals form... |
| absabv 21433 | The regular absolute value... |
| zsssubrg 21434 | The integers are a subset ... |
| qsssubdrg 21435 | The rational numbers are a... |
| cnsubrg 21436 | There are no subrings of t... |
| cnmgpabl 21437 | The unit group of the comp... |
| cnmgpid 21438 | The group identity element... |
| cnmsubglem 21439 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21440 | The positive reals form a ... |
| gzrngunitlem 21441 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21442 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21443 | Relate a group sum on ` CC... |
| regsumfsum 21444 | Relate a group sum on ` ( ... |
| expmhm 21445 | Exponentiation is a monoid... |
| nn0srg 21446 | The nonnegative integers f... |
| rge0srg 21447 | The nonnegative real numbe... |
| zringcrng 21450 | The ring of integers is a ... |
| zringring 21451 | The ring of integers is a ... |
| zringrng 21452 | The ring of integers is a ... |
| zringabl 21453 | The ring of integers is an... |
| zringgrp 21454 | The ring of integers is an... |
| zringbas 21455 | The integers are the base ... |
| zringplusg 21456 | The addition operation of ... |
| zringsub 21457 | The subtraction of element... |
| zringmulg 21458 | The multiplication (group ... |
| zringmulr 21459 | The multiplication operati... |
| zring0 21460 | The zero element of the ri... |
| zring1 21461 | The unity element of the r... |
| zringnzr 21462 | The ring of integers is a ... |
| dvdsrzring 21463 | Ring divisibility in the r... |
| zringlpirlem1 21464 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21465 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21466 | Lemma for ~ zringlpir . A... |
| zringinvg 21467 | The additive inverse of an... |
| zringunit 21468 | The units of ` ZZ ` are th... |
| zringlpir 21469 | The integers are a princip... |
| zringndrg 21470 | The integers are not a div... |
| zringcyg 21471 | The integers are a cyclic ... |
| zringsubgval 21472 | Subtraction in the ring of... |
| zringmpg 21473 | The multiplicative group o... |
| prmirredlem 21474 | A positive integer is irre... |
| dfprm2 21475 | The positive irreducible e... |
| prmirred 21476 | The irreducible elements o... |
| expghm 21477 | Exponentiation is a group ... |
| mulgghm2 21478 | The powers of a group elem... |
| mulgrhm 21479 | The powers of the element ... |
| mulgrhm2 21480 | The powers of the element ... |
| irinitoringc 21481 | The ring of integers is an... |
| nzerooringczr 21482 | There is no zero object in... |
| pzriprnglem1 21483 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21484 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21485 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21486 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21487 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21488 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21489 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21490 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21491 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21492 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21493 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21494 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21495 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21496 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21497 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21498 | The ring unity of the ring... |
| pzriprng 21499 | The non-unital ring ` ( ZZ... |
| pzriprng1 21500 | The ring unity of the ring... |
| zrhval 21509 | Define the unique homomorp... |
| zrhval2 21510 | Alternate value of the ` Z... |
| zrhmulg 21511 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21512 | The ` ZRHom ` homomorphism... |
| zrhrhm 21513 | The ` ZRHom ` homomorphism... |
| zrh1 21514 | Interpretation of 1 in a r... |
| zrh0 21515 | Interpretation of 0 in a r... |
| zrhpropd 21516 | The ` ZZ ` ring homomorphi... |
| zlmval 21517 | Augment an abelian group w... |
| zlmlem 21518 | Lemma for ~ zlmbas and ~ z... |
| zlmlemOLD 21519 | Obsolete version of ~ zlml... |
| zlmbas 21520 | Base set of a ` ZZ ` -modu... |
| zlmbasOLD 21521 | Obsolete version of ~ zlmb... |
| zlmplusg 21522 | Group operation of a ` ZZ ... |
| zlmplusgOLD 21523 | Obsolete version of ~ zlmb... |
| zlmmulr 21524 | Ring operation of a ` ZZ `... |
| zlmmulrOLD 21525 | Obsolete version of ~ zlmb... |
| zlmsca 21526 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21527 | Scalar multiplication oper... |
| zlmlmod 21528 | The ` ZZ ` -module operati... |
| chrval 21529 | Definition substitution of... |
| chrcl 21530 | Closure of the characteris... |
| chrid 21531 | The canonical ` ZZ ` ring ... |
| chrdvds 21532 | The ` ZZ ` ring homomorphi... |
| chrcong 21533 | If two integers are congru... |
| dvdschrmulg 21534 | In a ring, any multiple of... |
| fermltlchr 21535 | A generalization of Fermat... |
| chrnzr 21536 | Nonzero rings are precisel... |
| chrrhm 21537 | The characteristic restric... |
| domnchr 21538 | The characteristic of a do... |
| znlidl 21539 | The set ` n ZZ ` is an ide... |
| zncrng2 21540 | Making a commutative ring ... |
| znval 21541 | The value of the ` Z/nZ ` ... |
| znle 21542 | The value of the ` Z/nZ ` ... |
| znval2 21543 | Self-referential expressio... |
| znbaslem 21544 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 21545 | Obsolete version of ~ znba... |
| znbas2 21546 | The base set of ` Z/nZ ` i... |
| znbas2OLD 21547 | Obsolete version of ~ znba... |
| znadd 21548 | The additive structure of ... |
| znaddOLD 21549 | Obsolete version of ~ znad... |
| znmul 21550 | The multiplicative structu... |
| znmulOLD 21551 | Obsolete version of ~ znad... |
| znzrh 21552 | The ` ZZ ` ring homomorphi... |
| znbas 21553 | The base set of ` Z/nZ ` s... |
| zncrng 21554 | ` Z/nZ ` is a commutative ... |
| znzrh2 21555 | The ` ZZ ` ring homomorphi... |
| znzrhval 21556 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21557 | The ` ZZ ` ring homomorphi... |
| zncyg 21558 | The group ` ZZ / n ZZ ` is... |
| zndvds 21559 | Express equality of equiva... |
| zndvds0 21560 | Special case of ~ zndvds w... |
| znf1o 21561 | The function ` F ` enumera... |
| zzngim 21562 | The ` ZZ ` ring homomorphi... |
| znle2 21563 | The ordering of the ` Z/nZ... |
| znleval 21564 | The ordering of the ` Z/nZ... |
| znleval2 21565 | The ordering of the ` Z/nZ... |
| zntoslem 21566 | Lemma for ~ zntos . (Cont... |
| zntos 21567 | The ` Z/nZ ` structure is ... |
| znhash 21568 | The ` Z/nZ ` structure has... |
| znfi 21569 | The ` Z/nZ ` structure is ... |
| znfld 21570 | The ` Z/nZ ` structure is ... |
| znidomb 21571 | The ` Z/nZ ` structure is ... |
| znchr 21572 | Cyclic rings are defined b... |
| znunit 21573 | The units of ` Z/nZ ` are ... |
| znunithash 21574 | The size of the unit group... |
| znrrg 21575 | The regular elements of ` ... |
| cygznlem1 21576 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21577 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21578 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21579 | A cyclic group with ` n ` ... |
| cygzn 21580 | A cyclic group with ` n ` ... |
| cygth 21581 | The "fundamental theorem o... |
| cyggic 21582 | Cyclic groups are isomorph... |
| frgpcyg 21583 | A free group is cyclic iff... |
| freshmansdream 21584 | For a prime number ` P ` ,... |
| frobrhm 21585 | In a commutative ring with... |
| cnmsgnsubg 21586 | The signs form a multiplic... |
| cnmsgnbas 21587 | The base set of the sign s... |
| cnmsgngrp 21588 | The group of signs under m... |
| psgnghm 21589 | The sign is a homomorphism... |
| psgnghm2 21590 | The sign is a homomorphism... |
| psgninv 21591 | The sign of a permutation ... |
| psgnco 21592 | Multiplicativity of the pe... |
| zrhpsgnmhm 21593 | Embedding of permutation s... |
| zrhpsgninv 21594 | The embedded sign of a per... |
| evpmss 21595 | Even permutations are perm... |
| psgnevpmb 21596 | A class is an even permuta... |
| psgnodpm 21597 | A permutation which is odd... |
| psgnevpm 21598 | A permutation which is eve... |
| psgnodpmr 21599 | If a permutation has sign ... |
| zrhpsgnevpm 21600 | The sign of an even permut... |
| zrhpsgnodpm 21601 | The sign of an odd permuta... |
| cofipsgn 21602 | Composition of any class `... |
| zrhpsgnelbas 21603 | Embedding of permutation s... |
| zrhcopsgnelbas 21604 | Embedding of permutation s... |
| evpmodpmf1o 21605 | The function for performin... |
| pmtrodpm 21606 | A transposition is an odd ... |
| psgnfix1 21607 | A permutation of a finite ... |
| psgnfix2 21608 | A permutation of a finite ... |
| psgndiflemB 21609 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21610 | Lemma 2 for ~ psgndif . (... |
| psgndif 21611 | Embedding of permutation s... |
| copsgndif 21612 | Embedding of permutation s... |
| rebase 21615 | The base of the field of r... |
| remulg 21616 | The multiplication (group ... |
| resubdrg 21617 | The real numbers form a di... |
| resubgval 21618 | Subtraction in the field o... |
| replusg 21619 | The addition operation of ... |
| remulr 21620 | The multiplication operati... |
| re0g 21621 | The zero element of the fi... |
| re1r 21622 | The unity element of the f... |
| rele2 21623 | The ordering relation of t... |
| relt 21624 | The ordering relation of t... |
| reds 21625 | The distance of the field ... |
| redvr 21626 | The division operation of ... |
| retos 21627 | The real numbers are a tot... |
| refld 21628 | The real numbers form a fi... |
| refldcj 21629 | The conjugation operation ... |
| resrng 21630 | The real numbers form a st... |
| regsumsupp 21631 | The group sum over the rea... |
| rzgrp 21632 | The quotient group ` RR / ... |
| isphl 21637 | The predicate "is a genera... |
| phllvec 21638 | A pre-Hilbert space is a l... |
| phllmod 21639 | A pre-Hilbert space is a l... |
| phlsrng 21640 | The scalar ring of a pre-H... |
| phllmhm 21641 | The inner product of a pre... |
| ipcl 21642 | Closure of the inner produ... |
| ipcj 21643 | Conjugate of an inner prod... |
| iporthcom 21644 | Orthogonality (meaning inn... |
| ip0l 21645 | Inner product with a zero ... |
| ip0r 21646 | Inner product with a zero ... |
| ipeq0 21647 | The inner product of a vec... |
| ipdir 21648 | Distributive law for inner... |
| ipdi 21649 | Distributive law for inner... |
| ip2di 21650 | Distributive law for inner... |
| ipsubdir 21651 | Distributive law for inner... |
| ipsubdi 21652 | Distributive law for inner... |
| ip2subdi 21653 | Distributive law for inner... |
| ipass 21654 | Associative law for inner ... |
| ipassr 21655 | "Associative" law for seco... |
| ipassr2 21656 | "Associative" law for inne... |
| ipffval 21657 | The inner product operatio... |
| ipfval 21658 | The inner product operatio... |
| ipfeq 21659 | If the inner product opera... |
| ipffn 21660 | The inner product operatio... |
| phlipf 21661 | The inner product operatio... |
| ip2eq 21662 | Two vectors are equal iff ... |
| isphld 21663 | Properties that determine ... |
| phlpropd 21664 | If two structures have the... |
| ssipeq 21665 | The inner product on a sub... |
| phssipval 21666 | The inner product on a sub... |
| phssip 21667 | The inner product (as a fu... |
| phlssphl 21668 | A subspace of an inner pro... |
| ocvfval 21675 | The orthocomplement operat... |
| ocvval 21676 | Value of the orthocompleme... |
| elocv 21677 | Elementhood in the orthoco... |
| ocvi 21678 | Property of a member of th... |
| ocvss 21679 | The orthocomplement of a s... |
| ocvocv 21680 | A set is contained in its ... |
| ocvlss 21681 | The orthocomplement of a s... |
| ocv2ss 21682 | Orthocomplements reverse s... |
| ocvin 21683 | An orthocomplement has tri... |
| ocvsscon 21684 | Two ways to say that ` S `... |
| ocvlsp 21685 | The orthocomplement of a l... |
| ocv0 21686 | The orthocomplement of the... |
| ocvz 21687 | The orthocomplement of the... |
| ocv1 21688 | The orthocomplement of the... |
| unocv 21689 | The orthocomplement of a u... |
| iunocv 21690 | The orthocomplement of an ... |
| cssval 21691 | The set of closed subspace... |
| iscss 21692 | The predicate "is a closed... |
| cssi 21693 | Property of a closed subsp... |
| cssss 21694 | A closed subspace is a sub... |
| iscss2 21695 | It is sufficient to prove ... |
| ocvcss 21696 | The orthocomplement of any... |
| cssincl 21697 | The zero subspace is a clo... |
| css0 21698 | The zero subspace is a clo... |
| css1 21699 | The whole space is a close... |
| csslss 21700 | A closed subspace of a pre... |
| lsmcss 21701 | A subset of a pre-Hilbert ... |
| cssmre 21702 | The closed subspaces of a ... |
| mrccss 21703 | The Moore closure correspo... |
| thlval 21704 | Value of the Hilbert latti... |
| thlbas 21705 | Base set of the Hilbert la... |
| thlbasOLD 21706 | Obsolete proof of ~ thlbas... |
| thlle 21707 | Ordering on the Hilbert la... |
| thlleOLD 21708 | Obsolete proof of ~ thlle ... |
| thlleval 21709 | Ordering on the Hilbert la... |
| thloc 21710 | Orthocomplement on the Hil... |
| pjfval 21717 | The value of the projectio... |
| pjdm 21718 | A subspace is in the domai... |
| pjpm 21719 | The projection map is a pa... |
| pjfval2 21720 | Value of the projection ma... |
| pjval 21721 | Value of the projection ma... |
| pjdm2 21722 | A subspace is in the domai... |
| pjff 21723 | A projection is a linear o... |
| pjf 21724 | A projection is a function... |
| pjf2 21725 | A projection is a function... |
| pjfo 21726 | A projection is a surjecti... |
| pjcss 21727 | A projection subspace is a... |
| ocvpj 21728 | The orthocomplement of a p... |
| ishil 21729 | The predicate "is a Hilber... |
| ishil2 21730 | The predicate "is a Hilber... |
| isobs 21731 | The predicate "is an ortho... |
| obsip 21732 | The inner product of two e... |
| obsipid 21733 | A basis element has length... |
| obsrcl 21734 | Reverse closure for an ort... |
| obsss 21735 | An orthonormal basis is a ... |
| obsne0 21736 | A basis element is nonzero... |
| obsocv 21737 | An orthonormal basis has t... |
| obs2ocv 21738 | The double orthocomplement... |
| obselocv 21739 | A basis element is in the ... |
| obs2ss 21740 | A basis has no proper subs... |
| obslbs 21741 | An orthogonal basis is a l... |
| reldmdsmm 21744 | The direct sum is a well-b... |
| dsmmval 21745 | Value of the module direct... |
| dsmmbase 21746 | Base set of the module dir... |
| dsmmval2 21747 | Self-referential definitio... |
| dsmmbas2 21748 | Base set of the direct sum... |
| dsmmfi 21749 | For finite products, the d... |
| dsmmelbas 21750 | Membership in the finitely... |
| dsmm0cl 21751 | The all-zero vector is con... |
| dsmmacl 21752 | The finite hull is closed ... |
| prdsinvgd2 21753 | Negation of a single coord... |
| dsmmsubg 21754 | The finite hull of a produ... |
| dsmmlss 21755 | The finite hull of a produ... |
| dsmmlmod 21756 | The direct sum of a family... |
| frlmval 21759 | Value of the "free module"... |
| frlmlmod 21760 | The free module is a modul... |
| frlmpws 21761 | The free module as a restr... |
| frlmlss 21762 | The base set of the free m... |
| frlmpwsfi 21763 | The finite free module is ... |
| frlmsca 21764 | The ring of scalars of a f... |
| frlm0 21765 | Zero in a free module (rin... |
| frlmbas 21766 | Base set of the free modul... |
| frlmelbas 21767 | Membership in the base set... |
| frlmrcl 21768 | If a free module is inhabi... |
| frlmbasfsupp 21769 | Elements of the free modul... |
| frlmbasmap 21770 | Elements of the free modul... |
| frlmbasf 21771 | Elements of the free modul... |
| frlmlvec 21772 | The free module over a div... |
| frlmfibas 21773 | The base set of the finite... |
| elfrlmbasn0 21774 | If the dimension of a free... |
| frlmplusgval 21775 | Addition in a free module.... |
| frlmsubgval 21776 | Subtraction in a free modu... |
| frlmvscafval 21777 | Scalar multiplication in a... |
| frlmvplusgvalc 21778 | Coordinates of a sum with ... |
| frlmvscaval 21779 | Coordinates of a scalar mu... |
| frlmplusgvalb 21780 | Addition in a free module ... |
| frlmvscavalb 21781 | Scalar multiplication in a... |
| frlmvplusgscavalb 21782 | Addition combined with sca... |
| frlmgsum 21783 | Finite commutative sums in... |
| frlmsplit2 21784 | Restriction is homomorphic... |
| frlmsslss 21785 | A subset of a free module ... |
| frlmsslss2 21786 | A subset of a free module ... |
| frlmbas3 21787 | An element of the base set... |
| mpofrlmd 21788 | Elements of the free modul... |
| frlmip 21789 | The inner product of a fre... |
| frlmipval 21790 | The inner product of a fre... |
| frlmphllem 21791 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21792 | Conditions for a free modu... |
| uvcfval 21795 | Value of the unit-vector g... |
| uvcval 21796 | Value of a single unit vec... |
| uvcvval 21797 | Value of a unit vector coo... |
| uvcvvcl 21798 | A coordinate of a unit vec... |
| uvcvvcl2 21799 | A unit vector coordinate i... |
| uvcvv1 21800 | The unit vector is one at ... |
| uvcvv0 21801 | The unit vector is zero at... |
| uvcff 21802 | Domain and codomain of the... |
| uvcf1 21803 | In a nonzero ring, each un... |
| uvcresum 21804 | Any element of a free modu... |
| frlmssuvc1 21805 | A scalar multiple of a uni... |
| frlmssuvc2 21806 | A nonzero scalar multiple ... |
| frlmsslsp 21807 | A subset of a free module ... |
| frlmlbs 21808 | The unit vectors comprise ... |
| frlmup1 21809 | Any assignment of unit vec... |
| frlmup2 21810 | The evaluation map has the... |
| frlmup3 21811 | The range of such an evalu... |
| frlmup4 21812 | Universal property of the ... |
| ellspd 21813 | The elements of the span o... |
| elfilspd 21814 | Simplified version of ~ el... |
| rellindf 21819 | The independent-family pre... |
| islinds 21820 | Property of an independent... |
| linds1 21821 | An independent set of vect... |
| linds2 21822 | An independent set of vect... |
| islindf 21823 | Property of an independent... |
| islinds2 21824 | Expanded property of an in... |
| islindf2 21825 | Property of an independent... |
| lindff 21826 | Functional property of a l... |
| lindfind 21827 | A linearly independent fam... |
| lindsind 21828 | A linearly independent set... |
| lindfind2 21829 | In a linearly independent ... |
| lindsind2 21830 | In a linearly independent ... |
| lindff1 21831 | A linearly independent fam... |
| lindfrn 21832 | The range of an independen... |
| f1lindf 21833 | Rearranging and deleting e... |
| lindfres 21834 | Any restriction of an inde... |
| lindsss 21835 | Any subset of an independe... |
| f1linds 21836 | A family constructed from ... |
| islindf3 21837 | In a nonzero ring, indepen... |
| lindfmm 21838 | Linear independence of a f... |
| lindsmm 21839 | Linear independence of a s... |
| lindsmm2 21840 | The monomorphic image of a... |
| lsslindf 21841 | Linear independence is unc... |
| lsslinds 21842 | Linear independence is unc... |
| islbs4 21843 | A basis is an independent ... |
| lbslinds 21844 | A basis is independent. (... |
| islinds3 21845 | A subset is linearly indep... |
| islinds4 21846 | A set is independent in a ... |
| lmimlbs 21847 | The isomorphic image of a ... |
| lmiclbs 21848 | Having a basis is an isomo... |
| islindf4 21849 | A family is independent if... |
| islindf5 21850 | A family is independent if... |
| indlcim 21851 | An independent, spanning f... |
| lbslcic 21852 | A module with a basis is i... |
| lmisfree 21853 | A module has a basis iff i... |
| lvecisfrlm 21854 | Every vector space is isom... |
| lmimco 21855 | The composition of two iso... |
| lmictra 21856 | Module isomorphism is tran... |
| uvcf1o 21857 | In a nonzero ring, the map... |
| uvcendim 21858 | In a nonzero ring, the num... |
| frlmisfrlm 21859 | A free module is isomorphi... |
| frlmiscvec 21860 | Every free module is isomo... |
| isassa 21867 | The properties of an assoc... |
| assalem 21868 | The properties of an assoc... |
| assaass 21869 | Left-associative property ... |
| assaassr 21870 | Right-associative property... |
| assalmod 21871 | An associative algebra is ... |
| assaring 21872 | An associative algebra is ... |
| assasca 21873 | The scalars of an associat... |
| assa2ass 21874 | Left- and right-associativ... |
| isassad 21875 | Sufficient condition for b... |
| issubassa3 21876 | A subring that is also a s... |
| issubassa 21877 | The subalgebras of an asso... |
| sraassab 21878 | A subring algebra is an as... |
| sraassa 21879 | The subring algebra over a... |
| sraassaOLD 21880 | Obsolete version of ~ sraa... |
| rlmassa 21881 | The ring module over a com... |
| assapropd 21882 | If two structures have the... |
| aspval 21883 | Value of the algebraic clo... |
| asplss 21884 | The algebraic span of a se... |
| aspid 21885 | The algebraic span of a su... |
| aspsubrg 21886 | The algebraic span of a se... |
| aspss 21887 | Span preserves subset orde... |
| aspssid 21888 | A set of vectors is a subs... |
| asclfval 21889 | Function value of the alge... |
| asclval 21890 | Value of a mapped algebra ... |
| asclfn 21891 | Unconditional functionalit... |
| asclf 21892 | The algebra scalars functi... |
| asclghm 21893 | The algebra scalars functi... |
| ascl0 21894 | The scalar 0 embedded into... |
| ascl1 21895 | The scalar 1 embedded into... |
| asclmul1 21896 | Left multiplication by a l... |
| asclmul2 21897 | Right multiplication by a ... |
| ascldimul 21898 | The algebra scalars functi... |
| asclinvg 21899 | The group inverse (negatio... |
| asclrhm 21900 | The scalar injection is a ... |
| rnascl 21901 | The set of injected scalar... |
| issubassa2 21902 | A subring of a unital alge... |
| rnasclsubrg 21903 | The scalar multiples of th... |
| rnasclmulcl 21904 | (Vector) multiplication is... |
| rnasclassa 21905 | The scalar multiples of th... |
| ressascl 21906 | The injection of scalars i... |
| asclpropd 21907 | If two structures have the... |
| aspval2 21908 | The algebraic closure is t... |
| assamulgscmlem1 21909 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21910 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21911 | Exponentiation of a scalar... |
| asclmulg 21912 | Apply group multiplication... |
| zlmassa 21913 | The ` ZZ ` -module operati... |
| reldmpsr 21924 | The multivariate power ser... |
| psrval 21925 | Value of the multivariate ... |
| psrvalstr 21926 | The multivariate power ser... |
| psrbag 21927 | Elementhood in the set of ... |
| psrbagf 21928 | A finite bag is a function... |
| psrbagfsupp 21929 | Finite bags have finite su... |
| snifpsrbag 21930 | A bag containing one eleme... |
| fczpsrbag 21931 | The constant function equa... |
| psrbaglesupp 21932 | The support of a dominated... |
| psrbaglecl 21933 | The set of finite bags is ... |
| psrbagaddcl 21934 | The sum of two finite bags... |
| psrbagcon 21935 | The analogue of the statem... |
| psrbaglefi 21936 | There are finitely many ba... |
| psrbagconcl 21937 | The complement of a bag is... |
| psrbagleadd1 21938 | The analogue of " ` X <_ F... |
| psrbagconf1o 21939 | Bag complementation is a b... |
| gsumbagdiaglem 21940 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21941 | Two-dimensional commutatio... |
| psrass1lem 21942 | A group sum commutation us... |
| psrbas 21943 | The base set of the multiv... |
| psrelbas 21944 | An element of the set of p... |
| psrelbasfun 21945 | An element of the set of p... |
| psrplusg 21946 | The addition operation of ... |
| psradd 21947 | The addition operation of ... |
| psraddcl 21948 | Closure of the power serie... |
| psraddclOLD 21949 | Obsolete version of ~ psra... |
| rhmpsrlem1 21950 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21951 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21952 | The multiplication operati... |
| psrmulfval 21953 | The multiplication operati... |
| psrmulval 21954 | The multiplication operati... |
| psrmulcllem 21955 | Closure of the power serie... |
| psrmulcl 21956 | Closure of the power serie... |
| psrsca 21957 | The scalar field of the mu... |
| psrvscafval 21958 | The scalar multiplication ... |
| psrvsca 21959 | The scalar multiplication ... |
| psrvscaval 21960 | The scalar multiplication ... |
| psrvscacl 21961 | Closure of the power serie... |
| psr0cl 21962 | The zero element of the ri... |
| psr0lid 21963 | The zero element of the ri... |
| psrnegcl 21964 | The negative function in t... |
| psrlinv 21965 | The negative function in t... |
| psrgrp 21966 | The ring of power series i... |
| psrgrpOLD 21967 | Obsolete proof of ~ psrgrp... |
| psr0 21968 | The zero element of the ri... |
| psrneg 21969 | The negative function of t... |
| psrlmod 21970 | The ring of power series i... |
| psr1cl 21971 | The identity element of th... |
| psrlidm 21972 | The identity element of th... |
| psrridm 21973 | The identity element of th... |
| psrass1 21974 | Associative identity for t... |
| psrdi 21975 | Distributive law for the r... |
| psrdir 21976 | Distributive law for the r... |
| psrass23l 21977 | Associative identity for t... |
| psrcom 21978 | Commutative law for the ri... |
| psrass23 21979 | Associative identities for... |
| psrring 21980 | The ring of power series i... |
| psr1 21981 | The identity element of th... |
| psrcrng 21982 | The ring of power series i... |
| psrassa 21983 | The ring of power series i... |
| resspsrbas 21984 | A restricted power series ... |
| resspsradd 21985 | A restricted power series ... |
| resspsrmul 21986 | A restricted power series ... |
| resspsrvsca 21987 | A restricted power series ... |
| subrgpsr 21988 | A subring of the base ring... |
| psrascl 21989 | Value of the scalar inject... |
| psrasclcl 21990 | A scalar is lifted into a ... |
| mvrfval 21991 | Value of the generating el... |
| mvrval 21992 | Value of the generating el... |
| mvrval2 21993 | Value of the generating el... |
| mvrid 21994 | The ` X i ` -th coefficien... |
| mvrf 21995 | The power series variable ... |
| mvrf1 21996 | The power series variable ... |
| mvrcl2 21997 | A power series variable is... |
| reldmmpl 21998 | The multivariate polynomia... |
| mplval 21999 | Value of the set of multiv... |
| mplbas 22000 | Base set of the set of mul... |
| mplelbas 22001 | Property of being a polyno... |
| mvrcl 22002 | A power series variable is... |
| mvrf2 22003 | The power series/polynomia... |
| mplrcl 22004 | Reverse closure for the po... |
| mplelsfi 22005 | A polynomial treated as a ... |
| mplval2 22006 | Self-referential expressio... |
| mplbasss 22007 | The set of polynomials is ... |
| mplelf 22008 | A polynomial is defined as... |
| mplsubglem 22009 | If ` A ` is an ideal of se... |
| mpllsslem 22010 | If ` A ` is an ideal of su... |
| mplsubglem2 22011 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 22012 | The set of polynomials is ... |
| mpllss 22013 | The set of polynomials is ... |
| mplsubrglem 22014 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 22015 | The set of polynomials is ... |
| mpl0 22016 | The zero polynomial. (Con... |
| mplplusg 22017 | Value of addition in a pol... |
| mplmulr 22018 | Value of multiplication in... |
| mpladd 22019 | The addition operation on ... |
| mplneg 22020 | The negative function on m... |
| mplmul 22021 | The multiplication operati... |
| mpl1 22022 | The identity element of th... |
| mplsca 22023 | The scalar field of a mult... |
| mplvsca2 22024 | The scalar multiplication ... |
| mplvsca 22025 | The scalar multiplication ... |
| mplvscaval 22026 | The scalar multiplication ... |
| mplgrp 22027 | The polynomial ring is a g... |
| mpllmod 22028 | The polynomial ring is a l... |
| mplring 22029 | The polynomial ring is a r... |
| mpllvec 22030 | The polynomial ring is a v... |
| mplcrng 22031 | The polynomial ring is a c... |
| mplassa 22032 | The polynomial ring is an ... |
| mplringd 22033 | The polynomial ring is a r... |
| mpllmodd 22034 | The polynomial ring is a l... |
| ressmplbas2 22035 | The base set of a restrict... |
| ressmplbas 22036 | A restricted polynomial al... |
| ressmpladd 22037 | A restricted polynomial al... |
| ressmplmul 22038 | A restricted polynomial al... |
| ressmplvsca 22039 | A restricted power series ... |
| subrgmpl 22040 | A subring of the base ring... |
| subrgmvr 22041 | The variables in a subring... |
| subrgmvrf 22042 | The variables in a polynom... |
| mplmon 22043 | A monomial is a polynomial... |
| mplmonmul 22044 | The product of two monomia... |
| mplcoe1 22045 | Decompose a polynomial int... |
| mplcoe3 22046 | Decompose a monomial in on... |
| mplcoe5lem 22047 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 22048 | Decompose a monomial into ... |
| mplcoe2 22049 | Decompose a monomial into ... |
| mplbas2 22050 | An alternative expression ... |
| ltbval 22051 | Value of the well-order on... |
| ltbwe 22052 | The finite bag order is a ... |
| reldmopsr 22053 | Lemma for ordered power se... |
| opsrval 22054 | The value of the "ordered ... |
| opsrle 22055 | An alternative expression ... |
| opsrval2 22056 | Self-referential expressio... |
| opsrbaslem 22057 | Get a component of the ord... |
| opsrbaslemOLD 22058 | Obsolete version of ~ opsr... |
| opsrbas 22059 | The base set of the ordere... |
| opsrbasOLD 22060 | Obsolete version of ~ opsr... |
| opsrplusg 22061 | The addition operation of ... |
| opsrplusgOLD 22062 | Obsolete version of ~ opsr... |
| opsrmulr 22063 | The multiplication operati... |
| opsrmulrOLD 22064 | Obsolete version of ~ opsr... |
| opsrvsca 22065 | The scalar product operati... |
| opsrvscaOLD 22066 | Obsolete version of ~ opsr... |
| opsrsca 22067 | The scalar ring of the ord... |
| opsrscaOLD 22068 | Obsolete version of ~ opsr... |
| opsrtoslem1 22069 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22070 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22071 | The ordered power series s... |
| opsrso 22072 | The ordered power series s... |
| opsrcrng 22073 | The ring of ordered power ... |
| opsrassa 22074 | The ring of ordered power ... |
| mplmon2 22075 | Express a scaled monomial.... |
| psrbag0 22076 | The empty bag is a bag. (... |
| psrbagsn 22077 | A singleton bag is a bag. ... |
| mplascl 22078 | Value of the scalar inject... |
| mplasclf 22079 | The scalar injection is a ... |
| subrgascl 22080 | The scalar injection funct... |
| subrgasclcl 22081 | The scalars in a polynomia... |
| mplmon2cl 22082 | A scaled monomial is a pol... |
| mplmon2mul 22083 | Product of scaled monomial... |
| mplind 22084 | Prove a property of polyno... |
| mplcoe4 22085 | Decompose a polynomial int... |
| evlslem4 22090 | The support of a tensor pr... |
| psrbagev1 22091 | A bag of multipliers provi... |
| psrbagev2 22092 | Closure of a sum using a b... |
| evlslem2 22093 | A linear function on the p... |
| evlslem3 22094 | Lemma for ~ evlseu . Poly... |
| evlslem6 22095 | Lemma for ~ evlseu . Fini... |
| evlslem1 22096 | Lemma for ~ evlseu , give ... |
| evlseu 22097 | For a given interpretation... |
| reldmevls 22098 | Well-behaved binary operat... |
| mpfrcl 22099 | Reverse closure for the se... |
| evlsval 22100 | Value of the polynomial ev... |
| evlsval2 22101 | Characterizing properties ... |
| evlsrhm 22102 | Polynomial evaluation is a... |
| evlssca 22103 | Polynomial evaluation maps... |
| evlsvar 22104 | Polynomial evaluation maps... |
| evlsgsumadd 22105 | Polynomial evaluation maps... |
| evlsgsummul 22106 | Polynomial evaluation maps... |
| evlspw 22107 | Polynomial evaluation for ... |
| evlsvarpw 22108 | Polynomial evaluation for ... |
| evlval 22109 | Value of the simple/same r... |
| evlrhm 22110 | The simple evaluation map ... |
| evlsscasrng 22111 | The evaluation of a scalar... |
| evlsca 22112 | Simple polynomial evaluati... |
| evlsvarsrng 22113 | The evaluation of the vari... |
| evlvar 22114 | Simple polynomial evaluati... |
| mpfconst 22115 | Constants are multivariate... |
| mpfproj 22116 | Projections are multivaria... |
| mpfsubrg 22117 | Polynomial functions are a... |
| mpff 22118 | Polynomial functions are f... |
| mpfaddcl 22119 | The sum of multivariate po... |
| mpfmulcl 22120 | The product of multivariat... |
| mpfind 22121 | Prove a property of polyno... |
| selvffval 22127 | Value of the "variable sel... |
| selvfval 22128 | Value of the "variable sel... |
| selvval 22129 | Value of the "variable sel... |
| reldmmhp 22131 | The domain of the homogene... |
| mhpfval 22132 | Value of the "homogeneous ... |
| mhpval 22133 | Value of the "homogeneous ... |
| ismhp 22134 | Property of being a homoge... |
| ismhp2 22135 | Deduce a homogeneous polyn... |
| ismhp3 22136 | A polynomial is homogeneou... |
| mhpmpl 22137 | A homogeneous polynomial i... |
| mhpdeg 22138 | All nonzero terms of a hom... |
| mhp0cl 22139 | The zero polynomial is hom... |
| mhpsclcl 22140 | A scalar (or constant) pol... |
| mhpvarcl 22141 | A power series variable is... |
| mhpmulcl 22142 | A product of homogeneous p... |
| mhppwdeg 22143 | Degree of a homogeneous po... |
| mhpaddcl 22144 | Homogeneous polynomials ar... |
| mhpinvcl 22145 | Homogeneous polynomials ar... |
| mhpsubg 22146 | Homogeneous polynomials fo... |
| mhpvscacl 22147 | Homogeneous polynomials ar... |
| mhplss 22148 | Homogeneous polynomials fo... |
| psdffval 22150 | Value of the power series ... |
| psdfval 22151 | Give a map between power s... |
| psdval 22152 | Evaluate the partial deriv... |
| psdcoef 22153 | Coefficient of a term of t... |
| psdcl 22154 | The derivative of a power ... |
| psdmplcl 22155 | The derivative of a polyno... |
| psdadd 22156 | The derivative of a sum is... |
| psdvsca 22157 | The derivative of a scaled... |
| psdmullem 22158 | Lemma for ~ psdmul . Tran... |
| psdmul 22159 | Product rule for power ser... |
| psd1 22160 | The derivative of one is z... |
| psdascl 22161 | The derivative of a consta... |
| psr1baslem 22173 | The set of finite bags on ... |
| psr1val 22174 | Value of the ring of univa... |
| psr1crng 22175 | The ring of univariate pow... |
| psr1assa 22176 | The ring of univariate pow... |
| psr1tos 22177 | The ordered power series s... |
| psr1bas2 22178 | The base set of the ring o... |
| psr1bas 22179 | The base set of the ring o... |
| vr1val 22180 | The value of the generator... |
| vr1cl2 22181 | The variable ` X ` is a me... |
| ply1val 22182 | The value of the set of un... |
| ply1bas 22183 | The value of the base set ... |
| ply1basOLD 22184 | Obsolete version of ~ ply1... |
| ply1lss 22185 | Univariate polynomials for... |
| ply1subrg 22186 | Univariate polynomials for... |
| ply1crng 22187 | The ring of univariate pol... |
| ply1assa 22188 | The ring of univariate pol... |
| psr1bascl 22189 | A univariate power series ... |
| psr1basf 22190 | Univariate power series ba... |
| ply1basf 22191 | Univariate polynomial base... |
| ply1bascl 22192 | A univariate polynomial is... |
| ply1bascl2 22193 | A univariate polynomial is... |
| coe1fval 22194 | Value of the univariate po... |
| coe1fv 22195 | Value of an evaluated coef... |
| fvcoe1 22196 | Value of a multivariate co... |
| coe1fval3 22197 | Univariate power series co... |
| coe1f2 22198 | Functionality of univariat... |
| coe1fval2 22199 | Univariate polynomial coef... |
| coe1f 22200 | Functionality of univariat... |
| coe1fvalcl 22201 | A coefficient of a univari... |
| coe1sfi 22202 | Finite support of univaria... |
| coe1fsupp 22203 | The coefficient vector of ... |
| mptcoe1fsupp 22204 | A mapping involving coeffi... |
| coe1ae0 22205 | The coefficient vector of ... |
| vr1cl 22206 | The generator of a univari... |
| opsr0 22207 | Zero in the ordered power ... |
| opsr1 22208 | One in the ordered power s... |
| psr1plusg 22209 | Value of addition in a uni... |
| psr1vsca 22210 | Value of scalar multiplica... |
| psr1mulr 22211 | Value of multiplication in... |
| ply1plusg 22212 | Value of addition in a uni... |
| ply1vsca 22213 | Value of scalar multiplica... |
| ply1mulr 22214 | Value of multiplication in... |
| ply1ass23l 22215 | Associative identity with ... |
| ressply1bas2 22216 | The base set of a restrict... |
| ressply1bas 22217 | A restricted polynomial al... |
| ressply1add 22218 | A restricted polynomial al... |
| ressply1mul 22219 | A restricted polynomial al... |
| ressply1vsca 22220 | A restricted power series ... |
| subrgply1 22221 | A subring of the base ring... |
| gsumply1subr 22222 | Evaluate a group sum in a ... |
| psrbaspropd 22223 | Property deduction for pow... |
| psrplusgpropd 22224 | Property deduction for pow... |
| mplbaspropd 22225 | Property deduction for pol... |
| psropprmul 22226 | Reversing multiplication i... |
| ply1opprmul 22227 | Reversing multiplication i... |
| 00ply1bas 22228 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22229 | Protection compatibility o... |
| ply1plusgfvi 22230 | Protection compatibility o... |
| ply1baspropd 22231 | Property deduction for uni... |
| ply1plusgpropd 22232 | Property deduction for uni... |
| opsrring 22233 | Ordered power series form ... |
| opsrlmod 22234 | Ordered power series form ... |
| psr1ring 22235 | Univariate power series fo... |
| ply1ring 22236 | Univariate polynomials for... |
| psr1lmod 22237 | Univariate power series fo... |
| psr1sca 22238 | Scalars of a univariate po... |
| psr1sca2 22239 | Scalars of a univariate po... |
| ply1lmod 22240 | Univariate polynomials for... |
| ply1sca 22241 | Scalars of a univariate po... |
| ply1sca2 22242 | Scalars of a univariate po... |
| ply1ascl0 22243 | The zero scalar as a polyn... |
| ply1ascl1 22244 | The multiplicative identit... |
| ply1mpl0 22245 | The univariate polynomial ... |
| ply10s0 22246 | Zero times a univariate po... |
| ply1mpl1 22247 | The univariate polynomial ... |
| ply1ascl 22248 | The univariate polynomial ... |
| subrg1ascl 22249 | The scalar injection funct... |
| subrg1asclcl 22250 | The scalars in a polynomia... |
| subrgvr1 22251 | The variables in a subring... |
| subrgvr1cl 22252 | The variables in a polynom... |
| coe1z 22253 | The coefficient vector of ... |
| coe1add 22254 | The coefficient vector of ... |
| coe1addfv 22255 | A particular coefficient o... |
| coe1subfv 22256 | A particular coefficient o... |
| coe1mul2lem1 22257 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22258 | An equivalence for ~ coe1m... |
| coe1mul2 22259 | The coefficient vector of ... |
| coe1mul 22260 | The coefficient vector of ... |
| ply1moncl 22261 | Closure of the expression ... |
| ply1tmcl 22262 | Closure of the expression ... |
| coe1tm 22263 | Coefficient vector of a po... |
| coe1tmfv1 22264 | Nonzero coefficient of a p... |
| coe1tmfv2 22265 | Zero coefficient of a poly... |
| coe1tmmul2 22266 | Coefficient vector of a po... |
| coe1tmmul 22267 | Coefficient vector of a po... |
| coe1tmmul2fv 22268 | Function value of a right-... |
| coe1pwmul 22269 | Coefficient vector of a po... |
| coe1pwmulfv 22270 | Function value of a right-... |
| ply1scltm 22271 | A scalar is a term with ze... |
| coe1sclmul 22272 | Coefficient vector of a po... |
| coe1sclmulfv 22273 | A single coefficient of a ... |
| coe1sclmul2 22274 | Coefficient vector of a po... |
| ply1sclf 22275 | A scalar polynomial is a p... |
| ply1sclcl 22276 | The value of the algebra s... |
| coe1scl 22277 | Coefficient vector of a sc... |
| ply1sclid 22278 | Recover the base scalar fr... |
| ply1sclf1 22279 | The polynomial scalar func... |
| ply1scl0 22280 | The zero scalar is zero. ... |
| ply1scl0OLD 22281 | Obsolete version of ~ ply1... |
| ply1scln0 22282 | Nonzero scalars create non... |
| ply1scl1 22283 | The one scalar is the unit... |
| ply1scl1OLD 22284 | Obsolete version of ~ ply1... |
| ply1idvr1 22285 | The identity of a polynomi... |
| ply1idvr1OLD 22286 | Obsolete version of ~ ply1... |
| cply1mul 22287 | The product of two constan... |
| ply1coefsupp 22288 | The decomposition of a uni... |
| ply1coe 22289 | Decompose a univariate pol... |
| eqcoe1ply1eq 22290 | Two polynomials over the s... |
| ply1coe1eq 22291 | Two polynomials over the s... |
| cply1coe0 22292 | All but the first coeffici... |
| cply1coe0bi 22293 | A polynomial is constant (... |
| coe1fzgsumdlem 22294 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22295 | Value of an evaluated coef... |
| ply1scleq 22296 | Equality of a constant pol... |
| ply1chr 22297 | The characteristic of a po... |
| gsumsmonply1 22298 | A finite group sum of scal... |
| gsummoncoe1 22299 | A coefficient of the polyn... |
| gsumply1eq 22300 | Two univariate polynomials... |
| lply1binom 22301 | The binomial theorem for l... |
| lply1binomsc 22302 | The binomial theorem for l... |
| ply1fermltlchr 22303 | Fermat's little theorem fo... |
| reldmevls1 22308 | Well-behaved binary operat... |
| ply1frcl 22309 | Reverse closure for the se... |
| evls1fval 22310 | Value of the univariate po... |
| evls1val 22311 | Value of the univariate po... |
| evls1rhmlem 22312 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22313 | Polynomial evaluation is a... |
| evls1sca 22314 | Univariate polynomial eval... |
| evls1gsumadd 22315 | Univariate polynomial eval... |
| evls1gsummul 22316 | Univariate polynomial eval... |
| evls1pw 22317 | Univariate polynomial eval... |
| evls1varpw 22318 | Univariate polynomial eval... |
| evl1fval 22319 | Value of the simple/same r... |
| evl1val 22320 | Value of the simple/same r... |
| evl1fval1lem 22321 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22322 | Value of the simple/same r... |
| evl1rhm 22323 | Polynomial evaluation is a... |
| fveval1fvcl 22324 | The function value of the ... |
| evl1sca 22325 | Polynomial evaluation maps... |
| evl1scad 22326 | Polynomial evaluation buil... |
| evl1var 22327 | Polynomial evaluation maps... |
| evl1vard 22328 | Polynomial evaluation buil... |
| evls1var 22329 | Univariate polynomial eval... |
| evls1scasrng 22330 | The evaluation of a scalar... |
| evls1varsrng 22331 | The evaluation of the vari... |
| evl1addd 22332 | Polynomial evaluation buil... |
| evl1subd 22333 | Polynomial evaluation buil... |
| evl1muld 22334 | Polynomial evaluation buil... |
| evl1vsd 22335 | Polynomial evaluation buil... |
| evl1expd 22336 | Polynomial evaluation buil... |
| pf1const 22337 | Constants are polynomial f... |
| pf1id 22338 | The identity is a polynomi... |
| pf1subrg 22339 | Polynomial functions are a... |
| pf1rcl 22340 | Reverse closure for the se... |
| pf1f 22341 | Polynomial functions are f... |
| mpfpf1 22342 | Convert a multivariate pol... |
| pf1mpf 22343 | Convert a univariate polyn... |
| pf1addcl 22344 | The sum of multivariate po... |
| pf1mulcl 22345 | The product of multivariat... |
| pf1ind 22346 | Prove a property of polyno... |
| evl1gsumdlem 22347 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22348 | Polynomial evaluation buil... |
| evl1gsumadd 22349 | Univariate polynomial eval... |
| evl1gsumaddval 22350 | Value of a univariate poly... |
| evl1gsummul 22351 | Univariate polynomial eval... |
| evl1varpw 22352 | Univariate polynomial eval... |
| evl1varpwval 22353 | Value of a univariate poly... |
| evl1scvarpw 22354 | Univariate polynomial eval... |
| evl1scvarpwval 22355 | Value of a univariate poly... |
| evl1gsummon 22356 | Value of a univariate poly... |
| evls1scafv 22357 | Value of the univariate po... |
| evls1expd 22358 | Univariate polynomial eval... |
| evls1varpwval 22359 | Univariate polynomial eval... |
| evls1fpws 22360 | Evaluation of a univariate... |
| ressply1evl 22361 | Evaluation of a univariate... |
| evls1addd 22362 | Univariate polynomial eval... |
| evls1muld 22363 | Univariate polynomial eval... |
| evls1vsca 22364 | Univariate polynomial eval... |
| asclply1subcl 22365 | Closure of the algebra sca... |
| evls1fvcl 22366 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22367 | The function ` F ` mapping... |
| evls1maplmhm 22368 | The function ` F ` mapping... |
| evls1maprnss 22369 | The function ` F ` mapping... |
| evl1maprhm 22370 | The function ` F ` mapping... |
| mhmcompl 22371 | The composition of a monoi... |
| mhmcoaddmpl 22372 | Show that the ring homomor... |
| rhmcomulmpl 22373 | Show that the ring homomor... |
| rhmmpl 22374 | Provide a ring homomorphis... |
| ply1vscl 22375 | Closure of scalar multipli... |
| mhmcoply1 22376 | The composition of a monoi... |
| rhmply1 22377 | Provide a ring homomorphis... |
| rhmply1vr1 22378 | A ring homomorphism betwee... |
| rhmply1vsca 22379 | Apply a ring homomorphism ... |
| rhmply1mon 22380 | Apply a ring homomorphism ... |
| mamufval 22383 | Functional value of the ma... |
| mamuval 22384 | Multiplication of two matr... |
| mamufv 22385 | A cell in the multiplicati... |
| mamudm 22386 | The domain of the matrix m... |
| mamufacex 22387 | Every solution of the equa... |
| mamures 22388 | Rows in a matrix product a... |
| grpvlinv 22389 | Tuple-wise left inverse in... |
| grpvrinv 22390 | Tuple-wise right inverse i... |
| ringvcl 22391 | Tuple-wise multiplication ... |
| mamucl 22392 | Operation closure of matri... |
| mamuass 22393 | Matrix multiplication is a... |
| mamudi 22394 | Matrix multiplication dist... |
| mamudir 22395 | Matrix multiplication dist... |
| mamuvs1 22396 | Matrix multiplication dist... |
| mamuvs2 22397 | Matrix multiplication dist... |
| matbas0pc 22400 | There is no matrix with a ... |
| matbas0 22401 | There is no matrix for a n... |
| matval 22402 | Value of the matrix algebr... |
| matrcl 22403 | Reverse closure for the ma... |
| matbas 22404 | The matrix ring has the sa... |
| matplusg 22405 | The matrix ring has the sa... |
| matsca 22406 | The matrix ring has the sa... |
| matscaOLD 22407 | Obsolete proof of ~ matsca... |
| matvsca 22408 | The matrix ring has the sa... |
| matvscaOLD 22409 | Obsolete proof of ~ matvsc... |
| mat0 22410 | The matrix ring has the sa... |
| matinvg 22411 | The matrix ring has the sa... |
| mat0op 22412 | Value of a zero matrix as ... |
| matsca2 22413 | The scalars of the matrix ... |
| matbas2 22414 | The base set of the matrix... |
| matbas2i 22415 | A matrix is a function. (... |
| matbas2d 22416 | The base set of the matrix... |
| eqmat 22417 | Two square matrices of the... |
| matecl 22418 | Each entry (according to W... |
| matecld 22419 | Each entry (according to W... |
| matplusg2 22420 | Addition in the matrix rin... |
| matvsca2 22421 | Scalar multiplication in t... |
| matlmod 22422 | The matrix ring is a linea... |
| matgrp 22423 | The matrix ring is a group... |
| matvscl 22424 | Closure of the scalar mult... |
| matsubg 22425 | The matrix ring has the sa... |
| matplusgcell 22426 | Addition in the matrix rin... |
| matsubgcell 22427 | Subtraction in the matrix ... |
| matinvgcell 22428 | Additive inversion in the ... |
| matvscacell 22429 | Scalar multiplication in t... |
| matgsum 22430 | Finite commutative sums in... |
| matmulr 22431 | Multiplication in the matr... |
| mamumat1cl 22432 | The identity matrix (as op... |
| mat1comp 22433 | The components of the iden... |
| mamulid 22434 | The identity matrix (as op... |
| mamurid 22435 | The identity matrix (as op... |
| matring 22436 | Existence of the matrix ri... |
| matassa 22437 | Existence of the matrix al... |
| matmulcell 22438 | Multiplication in the matr... |
| mpomatmul 22439 | Multiplication of two N x ... |
| mat1 22440 | Value of an identity matri... |
| mat1ov 22441 | Entries of an identity mat... |
| mat1bas 22442 | The identity matrix is a m... |
| matsc 22443 | The identity matrix multip... |
| ofco2 22444 | Distribution law for the f... |
| oftpos 22445 | The transposition of the v... |
| mattposcl 22446 | The transpose of a square ... |
| mattpostpos 22447 | The transpose of the trans... |
| mattposvs 22448 | The transposition of a mat... |
| mattpos1 22449 | The transposition of the i... |
| tposmap 22450 | The transposition of an I ... |
| mamutpos 22451 | Behavior of transposes in ... |
| mattposm 22452 | Multiplying two transposed... |
| matgsumcl 22453 | Closure of a group sum ove... |
| madetsumid 22454 | The identity summand in th... |
| matepmcl 22455 | Each entry of a matrix wit... |
| matepm2cl 22456 | Each entry of a matrix wit... |
| madetsmelbas 22457 | A summand of the determina... |
| madetsmelbas2 22458 | A summand of the determina... |
| mat0dimbas0 22459 | The empty set is the one a... |
| mat0dim0 22460 | The zero of the algebra of... |
| mat0dimid 22461 | The identity of the algebr... |
| mat0dimscm 22462 | The scalar multiplication ... |
| mat0dimcrng 22463 | The algebra of matrices wi... |
| mat1dimelbas 22464 | A matrix with dimension 1 ... |
| mat1dimbas 22465 | A matrix with dimension 1 ... |
| mat1dim0 22466 | The zero of the algebra of... |
| mat1dimid 22467 | The identity of the algebr... |
| mat1dimscm 22468 | The scalar multiplication ... |
| mat1dimmul 22469 | The ring multiplication in... |
| mat1dimcrng 22470 | The algebra of matrices wi... |
| mat1f1o 22471 | There is a 1-1 function fr... |
| mat1rhmval 22472 | The value of the ring homo... |
| mat1rhmelval 22473 | The value of the ring homo... |
| mat1rhmcl 22474 | The value of the ring homo... |
| mat1f 22475 | There is a function from a... |
| mat1ghm 22476 | There is a group homomorph... |
| mat1mhm 22477 | There is a monoid homomorp... |
| mat1rhm 22478 | There is a ring homomorphi... |
| mat1rngiso 22479 | There is a ring isomorphis... |
| mat1ric 22480 | A ring is isomorphic to th... |
| dmatval 22485 | The set of ` N ` x ` N ` d... |
| dmatel 22486 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22487 | An ` N ` x ` N ` diagonal ... |
| dmatid 22488 | The identity matrix is a d... |
| dmatelnd 22489 | An extradiagonal entry of ... |
| dmatmul 22490 | The product of two diagona... |
| dmatsubcl 22491 | The difference of two diag... |
| dmatsgrp 22492 | The set of diagonal matric... |
| dmatmulcl 22493 | The product of two diagona... |
| dmatsrng 22494 | The set of diagonal matric... |
| dmatcrng 22495 | The subring of diagonal ma... |
| dmatscmcl 22496 | The multiplication of a di... |
| scmatval 22497 | The set of ` N ` x ` N ` s... |
| scmatel 22498 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22499 | A scalar matrix can be exp... |
| scmatscmide 22500 | An entry of a scalar matri... |
| scmatscmiddistr 22501 | Distributive law for scala... |
| scmatmat 22502 | An ` N ` x ` N ` scalar ma... |
| scmate 22503 | An entry of an ` N ` x ` N... |
| scmatmats 22504 | The set of an ` N ` x ` N ... |
| scmateALT 22505 | Alternate proof of ~ scmat... |
| scmatscm 22506 | The multiplication of a ma... |
| scmatid 22507 | The identity matrix is a s... |
| scmatdmat 22508 | A scalar matrix is a diago... |
| scmataddcl 22509 | The sum of two scalar matr... |
| scmatsubcl 22510 | The difference of two scal... |
| scmatmulcl 22511 | The product of two scalar ... |
| scmatsgrp 22512 | The set of scalar matrices... |
| scmatsrng 22513 | The set of scalar matrices... |
| scmatcrng 22514 | The subring of scalar matr... |
| scmatsgrp1 22515 | The set of scalar matrices... |
| scmatsrng1 22516 | The set of scalar matrices... |
| smatvscl 22517 | Closure of the scalar mult... |
| scmatlss 22518 | The set of scalar matrices... |
| scmatstrbas 22519 | The set of scalar matrices... |
| scmatrhmval 22520 | The value of the ring homo... |
| scmatrhmcl 22521 | The value of the ring homo... |
| scmatf 22522 | There is a function from a... |
| scmatfo 22523 | There is a function from a... |
| scmatf1 22524 | There is a 1-1 function fr... |
| scmatf1o 22525 | There is a bijection betwe... |
| scmatghm 22526 | There is a group homomorph... |
| scmatmhm 22527 | There is a monoid homomorp... |
| scmatrhm 22528 | There is a ring homomorphi... |
| scmatrngiso 22529 | There is a ring isomorphis... |
| scmatric 22530 | A ring is isomorphic to ev... |
| mat0scmat 22531 | The empty matrix over a ri... |
| mat1scmat 22532 | A 1-dimensional matrix ove... |
| mvmulfval 22535 | Functional value of the ma... |
| mvmulval 22536 | Multiplication of a vector... |
| mvmulfv 22537 | A cell/element in the vect... |
| mavmulval 22538 | Multiplication of a vector... |
| mavmulfv 22539 | A cell/element in the vect... |
| mavmulcl 22540 | Multiplication of an NxN m... |
| 1mavmul 22541 | Multiplication of the iden... |
| mavmulass 22542 | Associativity of the multi... |
| mavmuldm 22543 | The domain of the matrix v... |
| mavmulsolcl 22544 | Every solution of the equa... |
| mavmul0 22545 | Multiplication of a 0-dime... |
| mavmul0g 22546 | The result of the 0-dimens... |
| mvmumamul1 22547 | The multiplication of an M... |
| mavmumamul1 22548 | The multiplication of an N... |
| marrepfval 22553 | First substitution for the... |
| marrepval0 22554 | Second substitution for th... |
| marrepval 22555 | Third substitution for the... |
| marrepeval 22556 | An entry of a matrix with ... |
| marrepcl 22557 | Closure of the row replace... |
| marepvfval 22558 | First substitution for the... |
| marepvval0 22559 | Second substitution for th... |
| marepvval 22560 | Third substitution for the... |
| marepveval 22561 | An entry of a matrix with ... |
| marepvcl 22562 | Closure of the column repl... |
| ma1repvcl 22563 | Closure of the column repl... |
| ma1repveval 22564 | An entry of an identity ma... |
| mulmarep1el 22565 | Element by element multipl... |
| mulmarep1gsum1 22566 | The sum of element by elem... |
| mulmarep1gsum2 22567 | The sum of element by elem... |
| 1marepvmarrepid 22568 | Replacing the ith row by 0... |
| submabas 22571 | Any subset of the index se... |
| submafval 22572 | First substitution for a s... |
| submaval0 22573 | Second substitution for a ... |
| submaval 22574 | Third substitution for a s... |
| submaeval 22575 | An entry of a submatrix of... |
| 1marepvsma1 22576 | The submatrix of the ident... |
| mdetfval 22579 | First substitution for the... |
| mdetleib 22580 | Full substitution of our d... |
| mdetleib2 22581 | Leibniz' formula can also ... |
| nfimdetndef 22582 | The determinant is not def... |
| mdetfval1 22583 | First substitution of an a... |
| mdetleib1 22584 | Full substitution of an al... |
| mdet0pr 22585 | The determinant function f... |
| mdet0f1o 22586 | The determinant function f... |
| mdet0fv0 22587 | The determinant of the emp... |
| mdetf 22588 | Functionality of the deter... |
| mdetcl 22589 | The determinant evaluates ... |
| m1detdiag 22590 | The determinant of a 1-dim... |
| mdetdiaglem 22591 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22592 | The determinant of a diago... |
| mdetdiagid 22593 | The determinant of a diago... |
| mdet1 22594 | The determinant of the ide... |
| mdetrlin 22595 | The determinant function i... |
| mdetrsca 22596 | The determinant function i... |
| mdetrsca2 22597 | The determinant function i... |
| mdetr0 22598 | The determinant of a matri... |
| mdet0 22599 | The determinant of the zer... |
| mdetrlin2 22600 | The determinant function i... |
| mdetralt 22601 | The determinant function i... |
| mdetralt2 22602 | The determinant function i... |
| mdetero 22603 | The determinant function i... |
| mdettpos 22604 | Determinant is invariant u... |
| mdetunilem1 22605 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22606 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22607 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22608 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22609 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22610 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22611 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22612 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22613 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22614 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22615 | According to the definitio... |
| mdetmul 22616 | Multiplicativity of the de... |
| m2detleiblem1 22617 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22618 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22619 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22620 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22621 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22622 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22623 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22624 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22629 | Lemma for ~ maducoeval2 . ... |
| madufval 22630 | First substitution for the... |
| maduval 22631 | Second substitution for th... |
| maducoeval 22632 | An entry of the adjunct (c... |
| maducoeval2 22633 | An entry of the adjunct (c... |
| maduf 22634 | Creating the adjunct of ma... |
| madutpos 22635 | The adjuct of a transposed... |
| madugsum 22636 | The determinant of a matri... |
| madurid 22637 | Multiplying a matrix with ... |
| madulid 22638 | Multiplying the adjunct of... |
| minmar1fval 22639 | First substitution for the... |
| minmar1val0 22640 | Second substitution for th... |
| minmar1val 22641 | Third substitution for the... |
| minmar1eval 22642 | An entry of a matrix for a... |
| minmar1marrep 22643 | The minor matrix is a spec... |
| minmar1cl 22644 | Closure of the row replace... |
| maducoevalmin1 22645 | The coefficients of an adj... |
| symgmatr01lem 22646 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22647 | Applying a permutation tha... |
| gsummatr01lem1 22648 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22649 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22650 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22651 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22652 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22653 | Replacing a row of a squar... |
| smadiadetlem0 22654 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22655 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22656 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22657 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22658 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22659 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22660 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22661 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22662 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22663 | The determinant of a subma... |
| smadiadetglem1 22664 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22665 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22666 | The determinant of a squar... |
| smadiadetg0 22667 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22668 | The determinant of a squar... |
| invrvald 22669 | If a matrix multiplied wit... |
| matinv 22670 | The inverse of a matrix is... |
| matunit 22671 | A matrix is a unit in the ... |
| slesolvec 22672 | Every solution of a system... |
| slesolinv 22673 | The solution of a system o... |
| slesolinvbi 22674 | The solution of a system o... |
| slesolex 22675 | Every system of linear equ... |
| cramerimplem1 22676 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22677 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22678 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22679 | One direction of Cramer's ... |
| cramerlem1 22680 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22681 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22682 | Lemma 3 for ~ cramer . (C... |
| cramer0 22683 | Special case of Cramer's r... |
| cramer 22684 | Cramer's rule. According ... |
| pmatring 22685 | The set of polynomial matr... |
| pmatlmod 22686 | The set of polynomial matr... |
| pmatassa 22687 | The set of polynomial matr... |
| pmat0op 22688 | The zero polynomial matrix... |
| pmat1op 22689 | The identity polynomial ma... |
| pmat1ovd 22690 | Entries of the identity po... |
| pmat0opsc 22691 | The zero polynomial matrix... |
| pmat1opsc 22692 | The identity polynomial ma... |
| pmat1ovscd 22693 | Entries of the identity po... |
| pmatcoe1fsupp 22694 | For a polynomial matrix th... |
| 1pmatscmul 22695 | The scalar product of the ... |
| cpmat 22702 | Value of the constructor o... |
| cpmatpmat 22703 | A constant polynomial matr... |
| cpmatel 22704 | Property of a constant pol... |
| cpmatelimp 22705 | Implication of a set being... |
| cpmatel2 22706 | Another property of a cons... |
| cpmatelimp2 22707 | Another implication of a s... |
| 1elcpmat 22708 | The identity of the ring o... |
| cpmatacl 22709 | The set of all constant po... |
| cpmatinvcl 22710 | The set of all constant po... |
| cpmatmcllem 22711 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22712 | The set of all constant po... |
| cpmatsubgpmat 22713 | The set of all constant po... |
| cpmatsrgpmat 22714 | The set of all constant po... |
| 0elcpmat 22715 | The zero of the ring of al... |
| mat2pmatfval 22716 | Value of the matrix transf... |
| mat2pmatval 22717 | The result of a matrix tra... |
| mat2pmatvalel 22718 | A (matrix) element of the ... |
| mat2pmatbas 22719 | The result of a matrix tra... |
| mat2pmatbas0 22720 | The result of a matrix tra... |
| mat2pmatf 22721 | The matrix transformation ... |
| mat2pmatf1 22722 | The matrix transformation ... |
| mat2pmatghm 22723 | The transformation of matr... |
| mat2pmatmul 22724 | The transformation of matr... |
| mat2pmat1 22725 | The transformation of the ... |
| mat2pmatmhm 22726 | The transformation of matr... |
| mat2pmatrhm 22727 | The transformation of matr... |
| mat2pmatlin 22728 | The transformation of matr... |
| 0mat2pmat 22729 | The transformed zero matri... |
| idmatidpmat 22730 | The transformed identity m... |
| d0mat2pmat 22731 | The transformed empty set ... |
| d1mat2pmat 22732 | The transformation of a ma... |
| mat2pmatscmxcl 22733 | A transformed matrix multi... |
| m2cpm 22734 | The result of a matrix tra... |
| m2cpmf 22735 | The matrix transformation ... |
| m2cpmf1 22736 | The matrix transformation ... |
| m2cpmghm 22737 | The transformation of matr... |
| m2cpmmhm 22738 | The transformation of matr... |
| m2cpmrhm 22739 | The transformation of matr... |
| m2pmfzmap 22740 | The transformed values of ... |
| m2pmfzgsumcl 22741 | Closure of the sum of scal... |
| cpm2mfval 22742 | Value of the inverse matri... |
| cpm2mval 22743 | The result of an inverse m... |
| cpm2mvalel 22744 | A (matrix) element of the ... |
| cpm2mf 22745 | The inverse matrix transfo... |
| m2cpminvid 22746 | The inverse transformation... |
| m2cpminvid2lem 22747 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22748 | The transformation applied... |
| m2cpmfo 22749 | The matrix transformation ... |
| m2cpmf1o 22750 | The matrix transformation ... |
| m2cpmrngiso 22751 | The transformation of matr... |
| matcpmric 22752 | The ring of matrices over ... |
| m2cpminv 22753 | The inverse matrix transfo... |
| m2cpminv0 22754 | The inverse matrix transfo... |
| decpmatval0 22757 | The matrix consisting of t... |
| decpmatval 22758 | The matrix consisting of t... |
| decpmate 22759 | An entry of the matrix con... |
| decpmatcl 22760 | Closure of the decompositi... |
| decpmataa0 22761 | The matrix consisting of t... |
| decpmatfsupp 22762 | The mapping to the matrice... |
| decpmatid 22763 | The matrix consisting of t... |
| decpmatmullem 22764 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22765 | The matrix consisting of t... |
| decpmatmulsumfsupp 22766 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22767 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22768 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22769 | Write a polynomial matrix ... |
| pmatcollpw2lem 22770 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22771 | Write a polynomial matrix ... |
| monmatcollpw 22772 | The matrix consisting of t... |
| pmatcollpwlem 22773 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22774 | Write a polynomial matrix ... |
| pmatcollpwfi 22775 | Write a polynomial matrix ... |
| pmatcollpw3lem 22776 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22777 | Write a polynomial matrix ... |
| pmatcollpw3fi 22778 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22779 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22780 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22781 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22782 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22783 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22784 | Write a scalar matrix over... |
| pm2mpf1lem 22787 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22788 | Value of the transformatio... |
| pm2mpfval 22789 | A polynomial matrix transf... |
| pm2mpcl 22790 | The transformation of poly... |
| pm2mpf 22791 | The transformation of poly... |
| pm2mpf1 22792 | The transformation of poly... |
| pm2mpcoe1 22793 | A coefficient of the polyn... |
| idpm2idmp 22794 | The transformation of the ... |
| mptcoe1matfsupp 22795 | The mapping extracting the... |
| mply1topmatcllem 22796 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22797 | A polynomial over matrices... |
| mply1topmatcl 22798 | A polynomial over matrices... |
| mp2pm2mplem1 22799 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22800 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22801 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22802 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22803 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22804 | A polynomial over matrices... |
| pm2mpghmlem2 22805 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22806 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22807 | The transformation of poly... |
| pm2mpf1o 22808 | The transformation of poly... |
| pm2mpghm 22809 | The transformation of poly... |
| pm2mpgrpiso 22810 | The transformation of poly... |
| pm2mpmhmlem1 22811 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22812 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22813 | The transformation of poly... |
| pm2mprhm 22814 | The transformation of poly... |
| pm2mprngiso 22815 | The transformation of poly... |
| pmmpric 22816 | The ring of polynomial mat... |
| monmat2matmon 22817 | The transformation of a po... |
| pm2mp 22818 | The transformation of a su... |
| chmatcl 22821 | Closure of the characteris... |
| chmatval 22822 | The entries of the charact... |
| chpmatfval 22823 | Value of the characteristi... |
| chpmatval 22824 | The characteristic polynom... |
| chpmatply1 22825 | The characteristic polynom... |
| chpmatval2 22826 | The characteristic polynom... |
| chpmat0d 22827 | The characteristic polynom... |
| chpmat1dlem 22828 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22829 | The characteristic polynom... |
| chpdmatlem0 22830 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22831 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22832 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22833 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22834 | The characteristic polynom... |
| chpscmat 22835 | The characteristic polynom... |
| chpscmat0 22836 | The characteristic polynom... |
| chpscmatgsumbin 22837 | The characteristic polynom... |
| chpscmatgsummon 22838 | The characteristic polynom... |
| chp0mat 22839 | The characteristic polynom... |
| chpidmat 22840 | The characteristic polynom... |
| chmaidscmat 22841 | The characteristic polynom... |
| fvmptnn04if 22842 | The function values of a m... |
| fvmptnn04ifa 22843 | The function value of a ma... |
| fvmptnn04ifb 22844 | The function value of a ma... |
| fvmptnn04ifc 22845 | The function value of a ma... |
| fvmptnn04ifd 22846 | The function value of a ma... |
| chfacfisf 22847 | The "characteristic factor... |
| chfacfisfcpmat 22848 | The "characteristic factor... |
| chfacffsupp 22849 | The "characteristic factor... |
| chfacfscmulcl 22850 | Closure of a scaled value ... |
| chfacfscmul0 22851 | A scaled value of the "cha... |
| chfacfscmulfsupp 22852 | A mapping of scaled values... |
| chfacfscmulgsum 22853 | Breaking up a sum of value... |
| chfacfpmmulcl 22854 | Closure of the value of th... |
| chfacfpmmul0 22855 | The value of the "characte... |
| chfacfpmmulfsupp 22856 | A mapping of values of the... |
| chfacfpmmulgsum 22857 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22858 | Breaking up a sum of value... |
| cayhamlem1 22859 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22860 | The right-hand fundamental... |
| cpmidgsum 22861 | Representation of the iden... |
| cpmidgsumm2pm 22862 | Representation of the iden... |
| cpmidpmatlem1 22863 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22864 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22865 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22866 | Representation of the iden... |
| cpmadugsumlemB 22867 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22868 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22869 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22870 | The product of the charact... |
| cpmadugsum 22871 | The product of the charact... |
| cpmidgsum2 22872 | Representation of the iden... |
| cpmidg2sum 22873 | Equality of two sums repre... |
| cpmadumatpolylem1 22874 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22875 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22876 | The product of the charact... |
| cayhamlem2 22877 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22878 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22879 | The coefficients of the ch... |
| cayhamlem3 22880 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22881 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22882 | The Cayley-Hamilton theore... |
| cayleyhamilton 22883 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22884 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22885 | The Cayley-Hamilton theore... |
| istopg 22888 | Express the predicate " ` ... |
| istop2g 22889 | Express the predicate " ` ... |
| uniopn 22890 | The union of a subset of a... |
| iunopn 22891 | The indexed union of a sub... |
| inopn 22892 | The intersection of two op... |
| fitop 22893 | A topology is closed under... |
| fiinopn 22894 | The intersection of a none... |
| iinopn 22895 | The intersection of a none... |
| unopn 22896 | The union of two open sets... |
| 0opn 22897 | The empty set is an open s... |
| 0ntop 22898 | The empty set is not a top... |
| topopn 22899 | The underlying set of a to... |
| eltopss 22900 | A member of a topology is ... |
| riinopn 22901 | A finite indexed relative ... |
| rintopn 22902 | A finite relative intersec... |
| istopon 22905 | Property of being a topolo... |
| topontop 22906 | A topology on a given base... |
| toponuni 22907 | The base set of a topology... |
| topontopi 22908 | A topology on a given base... |
| toponunii 22909 | The base set of a topology... |
| toptopon 22910 | Alternative definition of ... |
| toptopon2 22911 | A topology is the same thi... |
| topontopon 22912 | A topology on a set is a t... |
| funtopon 22913 | The class ` TopOn ` is a f... |
| toponrestid 22914 | Given a topology on a set,... |
| toponsspwpw 22915 | The set of topologies on a... |
| dmtopon 22916 | The domain of ` TopOn ` is... |
| fntopon 22917 | The class ` TopOn ` is a f... |
| toprntopon 22918 | A topology is the same thi... |
| toponmax 22919 | The base set of a topology... |
| toponss 22920 | A member of a topology is ... |
| toponcom 22921 | If ` K ` is a topology on ... |
| toponcomb 22922 | Biconditional form of ~ to... |
| topgele 22923 | The topologies over the sa... |
| topsn 22924 | The only topology on a sin... |
| istps 22927 | Express the predicate "is ... |
| istps2 22928 | Express the predicate "is ... |
| tpsuni 22929 | The base set of a topologi... |
| tpstop 22930 | The topology extractor on ... |
| tpspropd 22931 | A topological space depend... |
| tpsprop2d 22932 | A topological space depend... |
| topontopn 22933 | Express the predicate "is ... |
| tsettps 22934 | If the topology component ... |
| istpsi 22935 | Properties that determine ... |
| eltpsg 22936 | Properties that determine ... |
| eltpsgOLD 22937 | Obsolete version of ~ eltp... |
| eltpsi 22938 | Properties that determine ... |
| isbasisg 22941 | Express the predicate "the... |
| isbasis2g 22942 | Express the predicate "the... |
| isbasis3g 22943 | Express the predicate "the... |
| basis1 22944 | Property of a basis. (Con... |
| basis2 22945 | Property of a basis. (Con... |
| fiinbas 22946 | If a set is closed under f... |
| basdif0 22947 | A basis is not affected by... |
| baspartn 22948 | A disjoint system of sets ... |
| tgval 22949 | The topology generated by ... |
| tgval2 22950 | Definition of a topology g... |
| eltg 22951 | Membership in a topology g... |
| eltg2 22952 | Membership in a topology g... |
| eltg2b 22953 | Membership in a topology g... |
| eltg4i 22954 | An open set in a topology ... |
| eltg3i 22955 | The union of a set of basi... |
| eltg3 22956 | Membership in a topology g... |
| tgval3 22957 | Alternate expression for t... |
| tg1 22958 | Property of a member of a ... |
| tg2 22959 | Property of a member of a ... |
| bastg 22960 | A member of a basis is a s... |
| unitg 22961 | The topology generated by ... |
| tgss 22962 | Subset relation for genera... |
| tgcl 22963 | Show that a basis generate... |
| tgclb 22964 | The property ~ tgcl can be... |
| tgtopon 22965 | A basis generates a topolo... |
| topbas 22966 | A topology is its own basi... |
| tgtop 22967 | A topology is its own basi... |
| eltop 22968 | Membership in a topology, ... |
| eltop2 22969 | Membership in a topology. ... |
| eltop3 22970 | Membership in a topology. ... |
| fibas 22971 | A collection of finite int... |
| tgdom 22972 | A space has no more open s... |
| tgiun 22973 | The indexed union of a set... |
| tgidm 22974 | The topology generator fun... |
| bastop 22975 | Two ways to express that a... |
| tgtop11 22976 | The topology generation fu... |
| 0top 22977 | The singleton of the empty... |
| en1top 22978 | ` { (/) } ` is the only to... |
| en2top 22979 | If a topology has two elem... |
| tgss3 22980 | A criterion for determinin... |
| tgss2 22981 | A criterion for determinin... |
| basgen 22982 | Given a topology ` J ` , s... |
| basgen2 22983 | Given a topology ` J ` , s... |
| 2basgen 22984 | Conditions that determine ... |
| tgfiss 22985 | If a subbase is included i... |
| tgdif0 22986 | A generated topology is no... |
| bastop1 22987 | A subset of a topology is ... |
| bastop2 22988 | A version of ~ bastop1 tha... |
| distop 22989 | The discrete topology on a... |
| topnex 22990 | The class of all topologie... |
| distopon 22991 | The discrete topology on a... |
| sn0topon 22992 | The singleton of the empty... |
| sn0top 22993 | The singleton of the empty... |
| indislem 22994 | A lemma to eliminate some ... |
| indistopon 22995 | The indiscrete topology on... |
| indistop 22996 | The indiscrete topology on... |
| indisuni 22997 | The base set of the indisc... |
| fctop 22998 | The finite complement topo... |
| fctop2 22999 | The finite complement topo... |
| cctop 23000 | The countable complement t... |
| ppttop 23001 | The particular point topol... |
| pptbas 23002 | The particular point topol... |
| epttop 23003 | The excluded point topolog... |
| indistpsx 23004 | The indiscrete topology on... |
| indistps 23005 | The indiscrete topology on... |
| indistps2 23006 | The indiscrete topology on... |
| indistpsALT 23007 | The indiscrete topology on... |
| indistpsALTOLD 23008 | Obsolete version of ~ indi... |
| indistps2ALT 23009 | The indiscrete topology on... |
| distps 23010 | The discrete topology on a... |
| fncld 23017 | The closed-set generator i... |
| cldval 23018 | The set of closed sets of ... |
| ntrfval 23019 | The interior function on t... |
| clsfval 23020 | The closure function on th... |
| cldrcl 23021 | Reverse closure of the clo... |
| iscld 23022 | The predicate "the class `... |
| iscld2 23023 | A subset of the underlying... |
| cldss 23024 | A closed set is a subset o... |
| cldss2 23025 | The set of closed sets is ... |
| cldopn 23026 | The complement of a closed... |
| isopn2 23027 | A subset of the underlying... |
| opncld 23028 | The complement of an open ... |
| difopn 23029 | The difference of a closed... |
| topcld 23030 | The underlying set of a to... |
| ntrval 23031 | The interior of a subset o... |
| clsval 23032 | The closure of a subset of... |
| 0cld 23033 | The empty set is closed. ... |
| iincld 23034 | The indexed intersection o... |
| intcld 23035 | The intersection of a set ... |
| uncld 23036 | The union of two closed se... |
| cldcls 23037 | A closed subset equals its... |
| incld 23038 | The intersection of two cl... |
| riincld 23039 | An indexed relative inters... |
| iuncld 23040 | A finite indexed union of ... |
| unicld 23041 | A finite union of closed s... |
| clscld 23042 | The closure of a subset of... |
| clsf 23043 | The closure function is a ... |
| ntropn 23044 | The interior of a subset o... |
| clsval2 23045 | Express closure in terms o... |
| ntrval2 23046 | Interior expressed in term... |
| ntrdif 23047 | An interior of a complemen... |
| clsdif 23048 | A closure of a complement ... |
| clsss 23049 | Subset relationship for cl... |
| ntrss 23050 | Subset relationship for in... |
| sscls 23051 | A subset of a topology's u... |
| ntrss2 23052 | A subset includes its inte... |
| ssntr 23053 | An open subset of a set is... |
| clsss3 23054 | The closure of a subset of... |
| ntrss3 23055 | The interior of a subset o... |
| ntrin 23056 | A pairwise intersection of... |
| cmclsopn 23057 | The complement of a closur... |
| cmntrcld 23058 | The complement of an inter... |
| iscld3 23059 | A subset is closed iff it ... |
| iscld4 23060 | A subset is closed iff it ... |
| isopn3 23061 | A subset is open iff it eq... |
| clsidm 23062 | The closure operation is i... |
| ntridm 23063 | The interior operation is ... |
| clstop 23064 | The closure of a topology'... |
| ntrtop 23065 | The interior of a topology... |
| 0ntr 23066 | A subset with an empty int... |
| clsss2 23067 | If a subset is included in... |
| elcls 23068 | Membership in a closure. ... |
| elcls2 23069 | Membership in a closure. ... |
| clsndisj 23070 | Any open set containing a ... |
| ntrcls0 23071 | A subset whose closure has... |
| ntreq0 23072 | Two ways to say that a sub... |
| cldmre 23073 | The closed sets of a topol... |
| mrccls 23074 | Moore closure generalizes ... |
| cls0 23075 | The closure of the empty s... |
| ntr0 23076 | The interior of the empty ... |
| isopn3i 23077 | An open subset equals its ... |
| elcls3 23078 | Membership in a closure in... |
| opncldf1 23079 | A bijection useful for con... |
| opncldf2 23080 | The values of the open-clo... |
| opncldf3 23081 | The values of the converse... |
| isclo 23082 | A set ` A ` is clopen iff ... |
| isclo2 23083 | A set ` A ` is clopen iff ... |
| discld 23084 | The open sets of a discret... |
| sn0cld 23085 | The closed sets of the top... |
| indiscld 23086 | The closed sets of an indi... |
| mretopd 23087 | A Moore collection which i... |
| toponmre 23088 | The topologies over a give... |
| cldmreon 23089 | The closed sets of a topol... |
| iscldtop 23090 | A family is the closed set... |
| mreclatdemoBAD 23091 | The closed subspaces of a ... |
| neifval 23094 | Value of the neighborhood ... |
| neif 23095 | The neighborhood function ... |
| neiss2 23096 | A set with a neighborhood ... |
| neival 23097 | Value of the set of neighb... |
| isnei 23098 | The predicate "the class `... |
| neiint 23099 | An intuitive definition of... |
| isneip 23100 | The predicate "the class `... |
| neii1 23101 | A neighborhood is included... |
| neisspw 23102 | The neighborhoods of any s... |
| neii2 23103 | Property of a neighborhood... |
| neiss 23104 | Any neighborhood of a set ... |
| ssnei 23105 | A set is included in any o... |
| elnei 23106 | A point belongs to any of ... |
| 0nnei 23107 | The empty set is not a nei... |
| neips 23108 | A neighborhood of a set is... |
| opnneissb 23109 | An open set is a neighborh... |
| opnssneib 23110 | Any superset of an open se... |
| ssnei2 23111 | Any subset ` M ` of ` X ` ... |
| neindisj 23112 | Any neighborhood of an ele... |
| opnneiss 23113 | An open set is a neighborh... |
| opnneip 23114 | An open set is a neighborh... |
| opnnei 23115 | A set is open iff it is a ... |
| tpnei 23116 | The underlying set of a to... |
| neiuni 23117 | The union of the neighborh... |
| neindisj2 23118 | A point ` P ` belongs to t... |
| topssnei 23119 | A finer topology has more ... |
| innei 23120 | The intersection of two ne... |
| opnneiid 23121 | Only an open set is a neig... |
| neissex 23122 | For any neighborhood ` N `... |
| 0nei 23123 | The empty set is a neighbo... |
| neipeltop 23124 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23125 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23126 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23127 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23128 | If, to each element ` P ` ... |
| lpfval 23133 | The limit point function o... |
| lpval 23134 | The set of limit points of... |
| islp 23135 | The predicate "the class `... |
| lpsscls 23136 | The limit points of a subs... |
| lpss 23137 | The limit points of a subs... |
| lpdifsn 23138 | ` P ` is a limit point of ... |
| lpss3 23139 | Subset relationship for li... |
| islp2 23140 | The predicate " ` P ` is a... |
| islp3 23141 | The predicate " ` P ` is a... |
| maxlp 23142 | A point is a limit point o... |
| clslp 23143 | The closure of a subset of... |
| islpi 23144 | A point belonging to a set... |
| cldlp 23145 | A subset of a topological ... |
| isperf 23146 | Definition of a perfect sp... |
| isperf2 23147 | Definition of a perfect sp... |
| isperf3 23148 | A perfect space is a topol... |
| perflp 23149 | The limit points of a perf... |
| perfi 23150 | Property of a perfect spac... |
| perftop 23151 | A perfect space is a topol... |
| restrcl 23152 | Reverse closure for the su... |
| restbas 23153 | A subspace topology basis ... |
| tgrest 23154 | A subspace can be generate... |
| resttop 23155 | A subspace topology is a t... |
| resttopon 23156 | A subspace topology is a t... |
| restuni 23157 | The underlying set of a su... |
| stoig 23158 | The topological space buil... |
| restco 23159 | Composition of subspaces. ... |
| restabs 23160 | Equivalence of being a sub... |
| restin 23161 | When the subspace region i... |
| restuni2 23162 | The underlying set of a su... |
| resttopon2 23163 | The underlying set of a su... |
| rest0 23164 | The subspace topology indu... |
| restsn 23165 | The only subspace topology... |
| restsn2 23166 | The subspace topology indu... |
| restcld 23167 | A closed set of a subspace... |
| restcldi 23168 | A closed set is closed in ... |
| restcldr 23169 | A set which is closed in t... |
| restopnb 23170 | If ` B ` is an open subset... |
| ssrest 23171 | If ` K ` is a finer topolo... |
| restopn2 23172 | If ` A ` is open, then ` B... |
| restdis 23173 | A subspace of a discrete t... |
| restfpw 23174 | The restriction of the set... |
| neitr 23175 | The neighborhood of a trac... |
| restcls 23176 | A closure in a subspace to... |
| restntr 23177 | An interior in a subspace ... |
| restlp 23178 | The limit points of a subs... |
| restperf 23179 | Perfection of a subspace. ... |
| perfopn 23180 | An open subset of a perfec... |
| resstopn 23181 | The topology of a restrict... |
| resstps 23182 | A restricted topological s... |
| ordtbaslem 23183 | Lemma for ~ ordtbas . In ... |
| ordtval 23184 | Value of the order topolog... |
| ordtuni 23185 | Value of the order topolog... |
| ordtbas2 23186 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23187 | In a total order, the fini... |
| ordttopon 23188 | Value of the order topolog... |
| ordtopn1 23189 | An upward ray ` ( P , +oo ... |
| ordtopn2 23190 | A downward ray ` ( -oo , P... |
| ordtopn3 23191 | An open interval ` ( A , B... |
| ordtcld1 23192 | A downward ray ` ( -oo , P... |
| ordtcld2 23193 | An upward ray ` [ P , +oo ... |
| ordtcld3 23194 | A closed interval ` [ A , ... |
| ordttop 23195 | The order topology is a to... |
| ordtcnv 23196 | The order dual generates t... |
| ordtrest 23197 | The subspace topology of a... |
| ordtrest2lem 23198 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23199 | An interval-closed set ` A... |
| letopon 23200 | The topology of the extend... |
| letop 23201 | The topology of the extend... |
| letopuni 23202 | The topology of the extend... |
| xrstopn 23203 | The topology component of ... |
| xrstps 23204 | The extended real number s... |
| leordtvallem1 23205 | Lemma for ~ leordtval . (... |
| leordtvallem2 23206 | Lemma for ~ leordtval . (... |
| leordtval2 23207 | The topology of the extend... |
| leordtval 23208 | The topology of the extend... |
| iccordt 23209 | A closed interval is close... |
| iocpnfordt 23210 | An unbounded above open in... |
| icomnfordt 23211 | An unbounded above open in... |
| iooordt 23212 | An open interval is open i... |
| reordt 23213 | The real numbers are an op... |
| lecldbas 23214 | The set of closed interval... |
| pnfnei 23215 | A neighborhood of ` +oo ` ... |
| mnfnei 23216 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23217 | The restriction of the les... |
| ordtresticc 23218 | The restriction of the les... |
| lmrel 23225 | The topological space conv... |
| lmrcl 23226 | Reverse closure for the co... |
| lmfval 23227 | The relation "sequence ` f... |
| cnfval 23228 | The set of all continuous ... |
| cnpfval 23229 | The function mapping the p... |
| iscn 23230 | The predicate "the class `... |
| cnpval 23231 | The set of all functions f... |
| iscnp 23232 | The predicate "the class `... |
| iscn2 23233 | The predicate "the class `... |
| iscnp2 23234 | The predicate "the class `... |
| cntop1 23235 | Reverse closure for a cont... |
| cntop2 23236 | Reverse closure for a cont... |
| cnptop1 23237 | Reverse closure for a func... |
| cnptop2 23238 | Reverse closure for a func... |
| iscnp3 23239 | The predicate "the class `... |
| cnprcl 23240 | Reverse closure for a func... |
| cnf 23241 | A continuous function is a... |
| cnpf 23242 | A continuous function at p... |
| cnpcl 23243 | The value of a continuous ... |
| cnf2 23244 | A continuous function is a... |
| cnpf2 23245 | A continuous function at p... |
| cnprcl2 23246 | Reverse closure for a func... |
| tgcn 23247 | The continuity predicate w... |
| tgcnp 23248 | The "continuous at a point... |
| subbascn 23249 | The continuity predicate w... |
| ssidcn 23250 | The identity function is a... |
| cnpimaex 23251 | Property of a function con... |
| idcn 23252 | A restricted identity func... |
| lmbr 23253 | Express the binary relatio... |
| lmbr2 23254 | Express the binary relatio... |
| lmbrf 23255 | Express the binary relatio... |
| lmconst 23256 | A constant sequence conver... |
| lmcvg 23257 | Convergence property of a ... |
| iscnp4 23258 | The predicate "the class `... |
| cnpnei 23259 | A condition for continuity... |
| cnima 23260 | An open subset of the codo... |
| cnco 23261 | The composition of two con... |
| cnpco 23262 | The composition of a funct... |
| cnclima 23263 | A closed subset of the cod... |
| iscncl 23264 | A characterization of a co... |
| cncls2i 23265 | Property of the preimage o... |
| cnntri 23266 | Property of the preimage o... |
| cnclsi 23267 | Property of the image of a... |
| cncls2 23268 | Continuity in terms of clo... |
| cncls 23269 | Continuity in terms of clo... |
| cnntr 23270 | Continuity in terms of int... |
| cnss1 23271 | If the topology ` K ` is f... |
| cnss2 23272 | If the topology ` K ` is f... |
| cncnpi 23273 | A continuous function is c... |
| cnsscnp 23274 | The set of continuous func... |
| cncnp 23275 | A continuous function is c... |
| cncnp2 23276 | A continuous function is c... |
| cnnei 23277 | Continuity in terms of nei... |
| cnconst2 23278 | A constant function is con... |
| cnconst 23279 | A constant function is con... |
| cnrest 23280 | Continuity of a restrictio... |
| cnrest2 23281 | Equivalence of continuity ... |
| cnrest2r 23282 | Equivalence of continuity ... |
| cnpresti 23283 | One direction of ~ cnprest... |
| cnprest 23284 | Equivalence of continuity ... |
| cnprest2 23285 | Equivalence of point-conti... |
| cndis 23286 | Every function is continuo... |
| cnindis 23287 | Every function is continuo... |
| cnpdis 23288 | If ` A ` is an isolated po... |
| paste 23289 | Pasting lemma. If ` A ` a... |
| lmfpm 23290 | If ` F ` converges, then `... |
| lmfss 23291 | Inclusion of a function ha... |
| lmcl 23292 | Closure of a limit. (Cont... |
| lmss 23293 | Limit on a subspace. (Con... |
| sslm 23294 | A finer topology has fewer... |
| lmres 23295 | A function converges iff i... |
| lmff 23296 | If ` F ` converges, there ... |
| lmcls 23297 | Any convergent sequence of... |
| lmcld 23298 | Any convergent sequence of... |
| lmcnp 23299 | The image of a convergent ... |
| lmcn 23300 | The image of a convergent ... |
| ist0 23315 | The predicate "is a T_0 sp... |
| ist1 23316 | The predicate "is a T_1 sp... |
| ishaus 23317 | The predicate "is a Hausdo... |
| iscnrm 23318 | The property of being comp... |
| t0sep 23319 | Any two topologically indi... |
| t0dist 23320 | Any two distinct points in... |
| t1sncld 23321 | In a T_1 space, singletons... |
| t1ficld 23322 | In a T_1 space, finite set... |
| hausnei 23323 | Neighborhood property of a... |
| t0top 23324 | A T_0 space is a topologic... |
| t1top 23325 | A T_1 space is a topologic... |
| haustop 23326 | A Hausdorff space is a top... |
| isreg 23327 | The predicate "is a regula... |
| regtop 23328 | A regular space is a topol... |
| regsep 23329 | In a regular space, every ... |
| isnrm 23330 | The predicate "is a normal... |
| nrmtop 23331 | A normal space is a topolo... |
| cnrmtop 23332 | A completely normal space ... |
| iscnrm2 23333 | The property of being comp... |
| ispnrm 23334 | The property of being perf... |
| pnrmnrm 23335 | A perfectly normal space i... |
| pnrmtop 23336 | A perfectly normal space i... |
| pnrmcld 23337 | A closed set in a perfectl... |
| pnrmopn 23338 | An open set in a perfectly... |
| ist0-2 23339 | The predicate "is a T_0 sp... |
| ist0-3 23340 | The predicate "is a T_0 sp... |
| cnt0 23341 | The preimage of a T_0 topo... |
| ist1-2 23342 | An alternate characterizat... |
| t1t0 23343 | A T_1 space is a T_0 space... |
| ist1-3 23344 | A space is T_1 iff every p... |
| cnt1 23345 | The preimage of a T_1 topo... |
| ishaus2 23346 | Express the predicate " ` ... |
| haust1 23347 | A Hausdorff space is a T_1... |
| hausnei2 23348 | The Hausdorff condition st... |
| cnhaus 23349 | The preimage of a Hausdorf... |
| nrmsep3 23350 | In a normal space, given a... |
| nrmsep2 23351 | In a normal space, any two... |
| nrmsep 23352 | In a normal space, disjoin... |
| isnrm2 23353 | An alternate characterizat... |
| isnrm3 23354 | A topological space is nor... |
| cnrmi 23355 | A subspace of a completely... |
| cnrmnrm 23356 | A completely normal space ... |
| restcnrm 23357 | A subspace of a completely... |
| resthauslem 23358 | Lemma for ~ resthaus and s... |
| lpcls 23359 | The limit points of the cl... |
| perfcls 23360 | A subset of a perfect spac... |
| restt0 23361 | A subspace of a T_0 topolo... |
| restt1 23362 | A subspace of a T_1 topolo... |
| resthaus 23363 | A subspace of a Hausdorff ... |
| t1sep2 23364 | Any two points in a T_1 sp... |
| t1sep 23365 | Any two distinct points in... |
| sncld 23366 | A singleton is closed in a... |
| sshauslem 23367 | Lemma for ~ sshaus and sim... |
| sst0 23368 | A topology finer than a T_... |
| sst1 23369 | A topology finer than a T_... |
| sshaus 23370 | A topology finer than a Ha... |
| regsep2 23371 | In a regular space, a clos... |
| isreg2 23372 | A topological space is reg... |
| dnsconst 23373 | If a continuous mapping to... |
| ordtt1 23374 | The order topology is T_1 ... |
| lmmo 23375 | A sequence in a Hausdorff ... |
| lmfun 23376 | The convergence relation i... |
| dishaus 23377 | A discrete topology is Hau... |
| ordthauslem 23378 | Lemma for ~ ordthaus . (C... |
| ordthaus 23379 | The order topology of a to... |
| xrhaus 23380 | The topology of the extend... |
| iscmp 23383 | The predicate "is a compac... |
| cmpcov 23384 | An open cover of a compact... |
| cmpcov2 23385 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23386 | Combine ~ cmpcov with ~ ac... |
| cncmp 23387 | Compactness is respected b... |
| fincmp 23388 | A finite topology is compa... |
| 0cmp 23389 | The singleton of the empty... |
| cmptop 23390 | A compact topology is a to... |
| rncmp 23391 | The image of a compact set... |
| imacmp 23392 | The image of a compact set... |
| discmp 23393 | A discrete topology is com... |
| cmpsublem 23394 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23395 | Two equivalent ways of des... |
| tgcmp 23396 | A topology generated by a ... |
| cmpcld 23397 | A closed subset of a compa... |
| uncmp 23398 | The union of two compact s... |
| fiuncmp 23399 | A finite union of compact ... |
| sscmp 23400 | A subset of a compact topo... |
| hauscmplem 23401 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23402 | A compact subspace of a T2... |
| cmpfi 23403 | If a topology is compact a... |
| cmpfii 23404 | In a compact topology, a s... |
| bwth 23405 | The glorious Bolzano-Weier... |
| isconn 23408 | The predicate ` J ` is a c... |
| isconn2 23409 | The predicate ` J ` is a c... |
| connclo 23410 | The only nonempty clopen s... |
| conndisj 23411 | If a topology is connected... |
| conntop 23412 | A connected topology is a ... |
| indisconn 23413 | The indiscrete topology (o... |
| dfconn2 23414 | An alternate definition of... |
| connsuba 23415 | Connectedness for a subspa... |
| connsub 23416 | Two equivalent ways of say... |
| cnconn 23417 | Connectedness is respected... |
| nconnsubb 23418 | Disconnectedness for a sub... |
| connsubclo 23419 | If a clopen set meets a co... |
| connima 23420 | The image of a connected s... |
| conncn 23421 | A continuous function from... |
| iunconnlem 23422 | Lemma for ~ iunconn . (Co... |
| iunconn 23423 | The indexed union of conne... |
| unconn 23424 | The union of two connected... |
| clsconn 23425 | The closure of a connected... |
| conncompid 23426 | The connected component co... |
| conncompconn 23427 | The connected component co... |
| conncompss 23428 | The connected component co... |
| conncompcld 23429 | The connected component co... |
| conncompclo 23430 | The connected component co... |
| t1connperf 23431 | A connected T_1 space is p... |
| is1stc 23436 | The predicate "is a first-... |
| is1stc2 23437 | An equivalent way of sayin... |
| 1stctop 23438 | A first-countable topology... |
| 1stcclb 23439 | A property of points in a ... |
| 1stcfb 23440 | For any point ` A ` in a f... |
| is2ndc 23441 | The property of being seco... |
| 2ndctop 23442 | A second-countable topolog... |
| 2ndci 23443 | A countable basis generate... |
| 2ndcsb 23444 | Having a countable subbase... |
| 2ndcredom 23445 | A second-countable space h... |
| 2ndc1stc 23446 | A second-countable space i... |
| 1stcrestlem 23447 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23448 | A subspace of a first-coun... |
| 2ndcrest 23449 | A subspace of a second-cou... |
| 2ndcctbss 23450 | If a topology is second-co... |
| 2ndcdisj 23451 | Any disjoint family of ope... |
| 2ndcdisj2 23452 | Any disjoint collection of... |
| 2ndcomap 23453 | A surjective continuous op... |
| 2ndcsep 23454 | A second-countable topolog... |
| dis2ndc 23455 | A discrete space is second... |
| 1stcelcls 23456 | A point belongs to the clo... |
| 1stccnp 23457 | A mapping is continuous at... |
| 1stccn 23458 | A mapping ` X --> Y ` , wh... |
| islly 23463 | The property of being a lo... |
| isnlly 23464 | The property of being an n... |
| llyeq 23465 | Equality theorem for the `... |
| nllyeq 23466 | Equality theorem for the `... |
| llytop 23467 | A locally ` A ` space is a... |
| nllytop 23468 | A locally ` A ` space is a... |
| llyi 23469 | The property of a locally ... |
| nllyi 23470 | The property of an n-local... |
| nlly2i 23471 | Eliminate the neighborhood... |
| llynlly 23472 | A locally ` A ` space is n... |
| llyssnlly 23473 | A locally ` A ` space is n... |
| llyss 23474 | The "locally" predicate re... |
| nllyss 23475 | The "n-locally" predicate ... |
| subislly 23476 | The property of a subspace... |
| restnlly 23477 | If the property ` A ` pass... |
| restlly 23478 | If the property ` A ` pass... |
| islly2 23479 | An alternative expression ... |
| llyrest 23480 | An open subspace of a loca... |
| nllyrest 23481 | An open subspace of an n-l... |
| loclly 23482 | If ` A ` is a local proper... |
| llyidm 23483 | Idempotence of the "locall... |
| nllyidm 23484 | Idempotence of the "n-loca... |
| toplly 23485 | A topology is locally a to... |
| topnlly 23486 | A topology is n-locally a ... |
| hauslly 23487 | A Hausdorff space is local... |
| hausnlly 23488 | A Hausdorff space is n-loc... |
| hausllycmp 23489 | A compact Hausdorff space ... |
| cldllycmp 23490 | A closed subspace of a loc... |
| lly1stc 23491 | First-countability is a lo... |
| dislly 23492 | The discrete space ` ~P X ... |
| disllycmp 23493 | A discrete space is locall... |
| dis1stc 23494 | A discrete space is first-... |
| hausmapdom 23495 | If ` X ` is a first-counta... |
| hauspwdom 23496 | Simplify the cardinal ` A ... |
| refrel 23503 | Refinement is a relation. ... |
| isref 23504 | The property of being a re... |
| refbas 23505 | A refinement covers the sa... |
| refssex 23506 | Every set in a refinement ... |
| ssref 23507 | A subcover is a refinement... |
| refref 23508 | Reflexivity of refinement.... |
| reftr 23509 | Refinement is transitive. ... |
| refun0 23510 | Adding the empty set prese... |
| isptfin 23511 | The statement "is a point-... |
| islocfin 23512 | The statement "is a locall... |
| finptfin 23513 | A finite cover is a point-... |
| ptfinfin 23514 | A point covered by a point... |
| finlocfin 23515 | A finite cover of a topolo... |
| locfintop 23516 | A locally finite cover cov... |
| locfinbas 23517 | A locally finite cover mus... |
| locfinnei 23518 | A point covered by a local... |
| lfinpfin 23519 | A locally finite cover is ... |
| lfinun 23520 | Adding a finite set preser... |
| locfincmp 23521 | For a compact space, the l... |
| unisngl 23522 | Taking the union of the se... |
| dissnref 23523 | The set of singletons is a... |
| dissnlocfin 23524 | The set of singletons is l... |
| locfindis 23525 | The locally finite covers ... |
| locfincf 23526 | A locally finite cover in ... |
| comppfsc 23527 | A space where every open c... |
| kgenval 23530 | Value of the compact gener... |
| elkgen 23531 | Value of the compact gener... |
| kgeni 23532 | Property of the open sets ... |
| kgentopon 23533 | The compact generator gene... |
| kgenuni 23534 | The base set of the compac... |
| kgenftop 23535 | The compact generator gene... |
| kgenf 23536 | The compact generator is a... |
| kgentop 23537 | A compactly generated spac... |
| kgenss 23538 | The compact generator gene... |
| kgenhaus 23539 | The compact generator gene... |
| kgencmp 23540 | The compact generator topo... |
| kgencmp2 23541 | The compact generator topo... |
| kgenidm 23542 | The compact generator is i... |
| iskgen2 23543 | A space is compactly gener... |
| iskgen3 23544 | Derive the usual definitio... |
| llycmpkgen2 23545 | A locally compact space is... |
| cmpkgen 23546 | A compact space is compact... |
| llycmpkgen 23547 | A locally compact space is... |
| 1stckgenlem 23548 | The one-point compactifica... |
| 1stckgen 23549 | A first-countable space is... |
| kgen2ss 23550 | The compact generator pres... |
| kgencn 23551 | A function from a compactl... |
| kgencn2 23552 | A function ` F : J --> K `... |
| kgencn3 23553 | The set of continuous func... |
| kgen2cn 23554 | A continuous function is a... |
| txval 23559 | Value of the binary topolo... |
| txuni2 23560 | The underlying set of the ... |
| txbasex 23561 | The basis for the product ... |
| txbas 23562 | The set of Cartesian produ... |
| eltx 23563 | A set in a product is open... |
| txtop 23564 | The product of two topolog... |
| ptval 23565 | The value of the product t... |
| ptpjpre1 23566 | The preimage of a projecti... |
| elpt 23567 | Elementhood in the bases o... |
| elptr 23568 | A basic open set in the pr... |
| elptr2 23569 | A basic open set in the pr... |
| ptbasid 23570 | The base set of the produc... |
| ptuni2 23571 | The base set for the produ... |
| ptbasin 23572 | The basis for a product to... |
| ptbasin2 23573 | The basis for a product to... |
| ptbas 23574 | The basis for a product to... |
| ptpjpre2 23575 | The basis for a product to... |
| ptbasfi 23576 | The basis for the product ... |
| pttop 23577 | The product topology is a ... |
| ptopn 23578 | A basic open set in the pr... |
| ptopn2 23579 | A sub-basic open set in th... |
| xkotf 23580 | Functionality of function ... |
| xkobval 23581 | Alternative expression for... |
| xkoval 23582 | Value of the compact-open ... |
| xkotop 23583 | The compact-open topology ... |
| xkoopn 23584 | A basic open set of the co... |
| txtopi 23585 | The product of two topolog... |
| txtopon 23586 | The underlying set of the ... |
| txuni 23587 | The underlying set of the ... |
| txunii 23588 | The underlying set of the ... |
| ptuni 23589 | The base set for the produ... |
| ptunimpt 23590 | Base set of a product topo... |
| pttopon 23591 | The base set for the produ... |
| pttoponconst 23592 | The base set for a product... |
| ptuniconst 23593 | The base set for a product... |
| xkouni 23594 | The base set of the compac... |
| xkotopon 23595 | The base set of the compac... |
| ptval2 23596 | The value of the product t... |
| txopn 23597 | The product of two open se... |
| txcld 23598 | The product of two closed ... |
| txcls 23599 | Closure of a rectangle in ... |
| txss12 23600 | Subset property of the top... |
| txbasval 23601 | It is sufficient to consid... |
| neitx 23602 | The Cartesian product of t... |
| txcnpi 23603 | Continuity of a two-argume... |
| tx1cn 23604 | Continuity of the first pr... |
| tx2cn 23605 | Continuity of the second p... |
| ptpjcn 23606 | Continuity of a projection... |
| ptpjopn 23607 | The projection map is an o... |
| ptcld 23608 | A closed box in the produc... |
| ptcldmpt 23609 | A closed box in the produc... |
| ptclsg 23610 | The closure of a box in th... |
| ptcls 23611 | The closure of a box in th... |
| dfac14lem 23612 | Lemma for ~ dfac14 . By e... |
| dfac14 23613 | Theorem ~ ptcls is an equi... |
| xkoccn 23614 | The "constant function" fu... |
| txcnp 23615 | If two functions are conti... |
| ptcnplem 23616 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23617 | If every projection of a f... |
| upxp 23618 | Universal property of the ... |
| txcnmpt 23619 | A map into the product of ... |
| uptx 23620 | Universal property of the ... |
| txcn 23621 | A map into the product of ... |
| ptcn 23622 | If every projection of a f... |
| prdstopn 23623 | Topology of a structure pr... |
| prdstps 23624 | A structure product of top... |
| pwstps 23625 | A structure power of a top... |
| txrest 23626 | The subspace of a topologi... |
| txdis 23627 | The topological product of... |
| txindislem 23628 | Lemma for ~ txindis . (Co... |
| txindis 23629 | The topological product of... |
| txdis1cn 23630 | A function is jointly cont... |
| txlly 23631 | If the property ` A ` is p... |
| txnlly 23632 | If the property ` A ` is p... |
| pthaus 23633 | The product of a collectio... |
| ptrescn 23634 | Restriction is a continuou... |
| txtube 23635 | The "tube lemma". If ` X ... |
| txcmplem1 23636 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23637 | Lemma for ~ txcmp . (Cont... |
| txcmp 23638 | The topological product of... |
| txcmpb 23639 | The topological product of... |
| hausdiag 23640 | A topology is Hausdorff if... |
| hauseqlcld 23641 | In a Hausdorff topology, t... |
| txhaus 23642 | The topological product of... |
| txlm 23643 | Two sequences converge iff... |
| lmcn2 23644 | The image of a convergent ... |
| tx1stc 23645 | The topological product of... |
| tx2ndc 23646 | The topological product of... |
| txkgen 23647 | The topological product of... |
| xkohaus 23648 | If the codomain space is H... |
| xkoptsub 23649 | The compact-open topology ... |
| xkopt 23650 | The compact-open topology ... |
| xkopjcn 23651 | Continuity of a projection... |
| xkoco1cn 23652 | If ` F ` is a continuous f... |
| xkoco2cn 23653 | If ` F ` is a continuous f... |
| xkococnlem 23654 | Continuity of the composit... |
| xkococn 23655 | Continuity of the composit... |
| cnmptid 23656 | The identity function is c... |
| cnmptc 23657 | A constant function is con... |
| cnmpt11 23658 | The composition of continu... |
| cnmpt11f 23659 | The composition of continu... |
| cnmpt1t 23660 | The composition of continu... |
| cnmpt12f 23661 | The composition of continu... |
| cnmpt12 23662 | The composition of continu... |
| cnmpt1st 23663 | The projection onto the fi... |
| cnmpt2nd 23664 | The projection onto the se... |
| cnmpt2c 23665 | A constant function is con... |
| cnmpt21 23666 | The composition of continu... |
| cnmpt21f 23667 | The composition of continu... |
| cnmpt2t 23668 | The composition of continu... |
| cnmpt22 23669 | The composition of continu... |
| cnmpt22f 23670 | The composition of continu... |
| cnmpt1res 23671 | The restriction of a conti... |
| cnmpt2res 23672 | The restriction of a conti... |
| cnmptcom 23673 | The argument converse of a... |
| cnmptkc 23674 | The curried first projecti... |
| cnmptkp 23675 | The evaluation of the inne... |
| cnmptk1 23676 | The composition of a curri... |
| cnmpt1k 23677 | The composition of a one-a... |
| cnmptkk 23678 | The composition of two cur... |
| xkofvcn 23679 | Joint continuity of the fu... |
| cnmptk1p 23680 | The evaluation of a currie... |
| cnmptk2 23681 | The uncurrying of a currie... |
| xkoinjcn 23682 | Continuity of "injection",... |
| cnmpt2k 23683 | The currying of a two-argu... |
| txconn 23684 | The topological product of... |
| imasnopn 23685 | If a relation graph is ope... |
| imasncld 23686 | If a relation graph is clo... |
| imasncls 23687 | If a relation graph is clo... |
| qtopval 23690 | Value of the quotient topo... |
| qtopval2 23691 | Value of the quotient topo... |
| elqtop 23692 | Value of the quotient topo... |
| qtopres 23693 | The quotient topology is u... |
| qtoptop2 23694 | The quotient topology is a... |
| qtoptop 23695 | The quotient topology is a... |
| elqtop2 23696 | Value of the quotient topo... |
| qtopuni 23697 | The base set of the quotie... |
| elqtop3 23698 | Value of the quotient topo... |
| qtoptopon 23699 | The base set of the quotie... |
| qtopid 23700 | A quotient map is a contin... |
| idqtop 23701 | The quotient topology indu... |
| qtopcmplem 23702 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23703 | A quotient of a compact sp... |
| qtopconn 23704 | A quotient of a connected ... |
| qtopkgen 23705 | A quotient of a compactly ... |
| basqtop 23706 | An injection maps bases to... |
| tgqtop 23707 | An injection maps generate... |
| qtopcld 23708 | The property of being a cl... |
| qtopcn 23709 | Universal property of a qu... |
| qtopss 23710 | A surjective continuous fu... |
| qtopeu 23711 | Universal property of the ... |
| qtoprest 23712 | If ` A ` is a saturated op... |
| qtopomap 23713 | If ` F ` is a surjective c... |
| qtopcmap 23714 | If ` F ` is a surjective c... |
| imastopn 23715 | The topology of an image s... |
| imastps 23716 | The image of a topological... |
| qustps 23717 | A quotient structure is a ... |
| kqfval 23718 | Value of the function appe... |
| kqfeq 23719 | Two points in the Kolmogor... |
| kqffn 23720 | The topological indistingu... |
| kqval 23721 | Value of the quotient topo... |
| kqtopon 23722 | The Kolmogorov quotient is... |
| kqid 23723 | The topological indistingu... |
| ist0-4 23724 | The topological indistingu... |
| kqfvima 23725 | When the image set is open... |
| kqsat 23726 | Any open set is saturated ... |
| kqdisj 23727 | A version of ~ imain for t... |
| kqcldsat 23728 | Any closed set is saturate... |
| kqopn 23729 | The topological indistingu... |
| kqcld 23730 | The topological indistingu... |
| kqt0lem 23731 | Lemma for ~ kqt0 . (Contr... |
| isr0 23732 | The property " ` J ` is an... |
| r0cld 23733 | The analogue of the T_1 ax... |
| regr1lem 23734 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23735 | A Kolmogorov quotient of a... |
| kqreglem1 23736 | A Kolmogorov quotient of a... |
| kqreglem2 23737 | If the Kolmogorov quotient... |
| kqnrmlem1 23738 | A Kolmogorov quotient of a... |
| kqnrmlem2 23739 | If the Kolmogorov quotient... |
| kqtop 23740 | The Kolmogorov quotient is... |
| kqt0 23741 | The Kolmogorov quotient is... |
| kqf 23742 | The Kolmogorov quotient is... |
| r0sep 23743 | The separation property of... |
| nrmr0reg 23744 | A normal R_0 space is also... |
| regr1 23745 | A regular space is R_1, wh... |
| kqreg 23746 | The Kolmogorov quotient of... |
| kqnrm 23747 | The Kolmogorov quotient of... |
| hmeofn 23752 | The set of homeomorphisms ... |
| hmeofval 23753 | The set of all the homeomo... |
| ishmeo 23754 | The predicate F is a homeo... |
| hmeocn 23755 | A homeomorphism is continu... |
| hmeocnvcn 23756 | The converse of a homeomor... |
| hmeocnv 23757 | The converse of a homeomor... |
| hmeof1o2 23758 | A homeomorphism is a 1-1-o... |
| hmeof1o 23759 | A homeomorphism is a 1-1-o... |
| hmeoima 23760 | The image of an open set b... |
| hmeoopn 23761 | Homeomorphisms preserve op... |
| hmeocld 23762 | Homeomorphisms preserve cl... |
| hmeocls 23763 | Homeomorphisms preserve cl... |
| hmeontr 23764 | Homeomorphisms preserve in... |
| hmeoimaf1o 23765 | The function mapping open ... |
| hmeores 23766 | The restriction of a homeo... |
| hmeoco 23767 | The composite of two homeo... |
| idhmeo 23768 | The identity function is a... |
| hmeocnvb 23769 | The converse of a homeomor... |
| hmeoqtop 23770 | A homeomorphism is a quoti... |
| hmph 23771 | Express the predicate ` J ... |
| hmphi 23772 | If there is a homeomorphis... |
| hmphtop 23773 | Reverse closure for the ho... |
| hmphtop1 23774 | The relation "being homeom... |
| hmphtop2 23775 | The relation "being homeom... |
| hmphref 23776 | "Is homeomorphic to" is re... |
| hmphsym 23777 | "Is homeomorphic to" is sy... |
| hmphtr 23778 | "Is homeomorphic to" is tr... |
| hmpher 23779 | "Is homeomorphic to" is an... |
| hmphen 23780 | Homeomorphisms preserve th... |
| hmphsymb 23781 | "Is homeomorphic to" is sy... |
| haushmphlem 23782 | Lemma for ~ haushmph and s... |
| cmphmph 23783 | Compactness is a topologic... |
| connhmph 23784 | Connectedness is a topolog... |
| t0hmph 23785 | T_0 is a topological prope... |
| t1hmph 23786 | T_1 is a topological prope... |
| haushmph 23787 | Hausdorff-ness is a topolo... |
| reghmph 23788 | Regularity is a topologica... |
| nrmhmph 23789 | Normality is a topological... |
| hmph0 23790 | A topology homeomorphic to... |
| hmphdis 23791 | Homeomorphisms preserve to... |
| hmphindis 23792 | Homeomorphisms preserve to... |
| indishmph 23793 | Equinumerous sets equipped... |
| hmphen2 23794 | Homeomorphisms preserve th... |
| cmphaushmeo 23795 | A continuous bijection fro... |
| ordthmeolem 23796 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23797 | An order isomorphism is a ... |
| txhmeo 23798 | Lift a pair of homeomorphi... |
| txswaphmeolem 23799 | Show inverse for the "swap... |
| txswaphmeo 23800 | There is a homeomorphism f... |
| pt1hmeo 23801 | The canonical homeomorphis... |
| ptuncnv 23802 | Exhibit the converse funct... |
| ptunhmeo 23803 | Define a homeomorphism fro... |
| xpstopnlem1 23804 | The function ` F ` used in... |
| xpstps 23805 | A binary product of topolo... |
| xpstopnlem2 23806 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23807 | The topology on a binary p... |
| ptcmpfi 23808 | A topological product of f... |
| xkocnv 23809 | The inverse of the "curryi... |
| xkohmeo 23810 | The Exponential Law for to... |
| qtopf1 23811 | If a quotient map is injec... |
| qtophmeo 23812 | If two functions on a base... |
| t0kq 23813 | A topological space is T_0... |
| kqhmph 23814 | A topological space is T_0... |
| ist1-5lem 23815 | Lemma for ~ ist1-5 and sim... |
| t1r0 23816 | A T_1 space is R_0. That ... |
| ist1-5 23817 | A topological space is T_1... |
| ishaus3 23818 | A topological space is Hau... |
| nrmreg 23819 | A normal T_1 space is regu... |
| reghaus 23820 | A regular T_0 space is Hau... |
| nrmhaus 23821 | A T_1 normal space is Haus... |
| elmptrab 23822 | Membership in a one-parame... |
| elmptrab2 23823 | Membership in a one-parame... |
| isfbas 23824 | The predicate " ` F ` is a... |
| fbasne0 23825 | There are no empty filter ... |
| 0nelfb 23826 | No filter base contains th... |
| fbsspw 23827 | A filter base on a set is ... |
| fbelss 23828 | An element of the filter b... |
| fbdmn0 23829 | The domain of a filter bas... |
| isfbas2 23830 | The predicate " ` F ` is a... |
| fbasssin 23831 | A filter base contains sub... |
| fbssfi 23832 | A filter base contains sub... |
| fbssint 23833 | A filter base contains sub... |
| fbncp 23834 | A filter base does not con... |
| fbun 23835 | A necessary and sufficient... |
| fbfinnfr 23836 | No filter base containing ... |
| opnfbas 23837 | The collection of open sup... |
| trfbas2 23838 | Conditions for the trace o... |
| trfbas 23839 | Conditions for the trace o... |
| isfil 23842 | The predicate "is a filter... |
| filfbas 23843 | A filter is a filter base.... |
| 0nelfil 23844 | The empty set doesn't belo... |
| fileln0 23845 | An element of a filter is ... |
| filsspw 23846 | A filter is a subset of th... |
| filelss 23847 | An element of a filter is ... |
| filss 23848 | A filter is closed under t... |
| filin 23849 | A filter is closed under t... |
| filtop 23850 | The underlying set belongs... |
| isfil2 23851 | Derive the standard axioms... |
| isfildlem 23852 | Lemma for ~ isfild . (Con... |
| isfild 23853 | Sufficient condition for a... |
| filfi 23854 | A filter is closed under t... |
| filinn0 23855 | The intersection of two el... |
| filintn0 23856 | A filter has the finite in... |
| filn0 23857 | The empty set is not a fil... |
| infil 23858 | The intersection of two fi... |
| snfil 23859 | A singleton is a filter. ... |
| fbasweak 23860 | A filter base on any set i... |
| snfbas 23861 | Condition for a singleton ... |
| fsubbas 23862 | A condition for a set to g... |
| fbasfip 23863 | A filter base has the fini... |
| fbunfip 23864 | A helpful lemma for showin... |
| fgval 23865 | The filter generating clas... |
| elfg 23866 | A condition for elements o... |
| ssfg 23867 | A filter base is a subset ... |
| fgss 23868 | A bigger base generates a ... |
| fgss2 23869 | A condition for a filter t... |
| fgfil 23870 | A filter generates itself.... |
| elfilss 23871 | An element belongs to a fi... |
| filfinnfr 23872 | No filter containing a fin... |
| fgcl 23873 | A generated filter is a fi... |
| fgabs 23874 | Absorption law for filter ... |
| neifil 23875 | The neighborhoods of a non... |
| filunibas 23876 | Recover the base set from ... |
| filunirn 23877 | Two ways to express a filt... |
| filconn 23878 | A filter gives rise to a c... |
| fbasrn 23879 | Given a filter on a domain... |
| filuni 23880 | The union of a nonempty se... |
| trfil1 23881 | Conditions for the trace o... |
| trfil2 23882 | Conditions for the trace o... |
| trfil3 23883 | Conditions for the trace o... |
| trfilss 23884 | If ` A ` is a member of th... |
| fgtr 23885 | If ` A ` is a member of th... |
| trfg 23886 | The trace operation and th... |
| trnei 23887 | The trace, over a set ` A ... |
| cfinfil 23888 | Relative complements of th... |
| csdfil 23889 | The set of all elements wh... |
| supfil 23890 | The supersets of a nonempt... |
| zfbas 23891 | The set of upper sets of i... |
| uzrest 23892 | The restriction of the set... |
| uzfbas 23893 | The set of upper sets of i... |
| isufil 23898 | The property of being an u... |
| ufilfil 23899 | An ultrafilter is a filter... |
| ufilss 23900 | For any subset of the base... |
| ufilb 23901 | The complement is in an ul... |
| ufilmax 23902 | Any filter finer than an u... |
| isufil2 23903 | The maximal property of an... |
| ufprim 23904 | An ultrafilter is a prime ... |
| trufil 23905 | Conditions for the trace o... |
| filssufilg 23906 | A filter is contained in s... |
| filssufil 23907 | A filter is contained in s... |
| isufl 23908 | Define the (strong) ultraf... |
| ufli 23909 | Property of a set that sat... |
| numufl 23910 | Consequence of ~ filssufil... |
| fiufl 23911 | A finite set satisfies the... |
| acufl 23912 | The axiom of choice implie... |
| ssufl 23913 | If ` Y ` is a subset of ` ... |
| ufileu 23914 | If the ultrafilter contain... |
| filufint 23915 | A filter is equal to the i... |
| uffix 23916 | Lemma for ~ fixufil and ~ ... |
| fixufil 23917 | The condition describing a... |
| uffixfr 23918 | An ultrafilter is either f... |
| uffix2 23919 | A classification of fixed ... |
| uffixsn 23920 | The singleton of the gener... |
| ufildom1 23921 | An ultrafilter is generate... |
| uffinfix 23922 | An ultrafilter containing ... |
| cfinufil 23923 | An ultrafilter is free iff... |
| ufinffr 23924 | An infinite subset is cont... |
| ufilen 23925 | Any infinite set has an ul... |
| ufildr 23926 | An ultrafilter gives rise ... |
| fin1aufil 23927 | There are no definable fre... |
| fmval 23938 | Introduce a function that ... |
| fmfil 23939 | A mapping filter is a filt... |
| fmf 23940 | Pushing-forward via a func... |
| fmss 23941 | A finer filter produces a ... |
| elfm 23942 | An element of a mapping fi... |
| elfm2 23943 | An element of a mapping fi... |
| fmfg 23944 | The image filter of a filt... |
| elfm3 23945 | An alternate formulation o... |
| imaelfm 23946 | An image of a filter eleme... |
| rnelfmlem 23947 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23948 | A condition for a filter t... |
| fmfnfmlem1 23949 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23950 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23951 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23952 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23953 | A filter finer than an ima... |
| fmufil 23954 | An image filter of an ultr... |
| fmid 23955 | The filter map applied to ... |
| fmco 23956 | Composition of image filte... |
| ufldom 23957 | The ultrafilter lemma prop... |
| flimval 23958 | The set of limit points of... |
| elflim2 23959 | The predicate "is a limit ... |
| flimtop 23960 | Reverse closure for the li... |
| flimneiss 23961 | A filter contains the neig... |
| flimnei 23962 | A filter contains all of t... |
| flimelbas 23963 | A limit point of a filter ... |
| flimfil 23964 | Reverse closure for the li... |
| flimtopon 23965 | Reverse closure for the li... |
| elflim 23966 | The predicate "is a limit ... |
| flimss2 23967 | A limit point of a filter ... |
| flimss1 23968 | A limit point of a filter ... |
| neiflim 23969 | A point is a limit point o... |
| flimopn 23970 | The condition for being a ... |
| fbflim 23971 | A condition for a filter t... |
| fbflim2 23972 | A condition for a filter b... |
| flimclsi 23973 | The convergent points of a... |
| hausflimlem 23974 | If ` A ` and ` B ` are bot... |
| hausflimi 23975 | One direction of ~ hausfli... |
| hausflim 23976 | A condition for a topology... |
| flimcf 23977 | Fineness is properly chara... |
| flimrest 23978 | The set of limit points in... |
| flimclslem 23979 | Lemma for ~ flimcls . (Co... |
| flimcls 23980 | Closure in terms of filter... |
| flimsncls 23981 | If ` A ` is a limit point ... |
| hauspwpwf1 23982 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23983 | If ` X ` is a Hausdorff sp... |
| flffval 23984 | Given a topology and a fil... |
| flfval 23985 | Given a function from a fi... |
| flfnei 23986 | The property of being a li... |
| flfneii 23987 | A neighborhood of a limit ... |
| isflf 23988 | The property of being a li... |
| flfelbas 23989 | A limit point of a functio... |
| flffbas 23990 | Limit points of a function... |
| flftg 23991 | Limit points of a function... |
| hausflf 23992 | If a function has its valu... |
| hausflf2 23993 | If a convergent function h... |
| cnpflfi 23994 | Forward direction of ~ cnp... |
| cnpflf2 23995 | ` F ` is continuous at poi... |
| cnpflf 23996 | Continuity of a function a... |
| cnflf 23997 | A function is continuous i... |
| cnflf2 23998 | A function is continuous i... |
| flfcnp 23999 | A continuous function pres... |
| lmflf 24000 | The topological limit rela... |
| txflf 24001 | Two sequences converge in ... |
| flfcnp2 24002 | The image of a convergent ... |
| fclsval 24003 | The set of all cluster poi... |
| isfcls 24004 | A cluster point of a filte... |
| fclsfil 24005 | Reverse closure for the cl... |
| fclstop 24006 | Reverse closure for the cl... |
| fclstopon 24007 | Reverse closure for the cl... |
| isfcls2 24008 | A cluster point of a filte... |
| fclsopn 24009 | Write the cluster point co... |
| fclsopni 24010 | An open neighborhood of a ... |
| fclselbas 24011 | A cluster point is in the ... |
| fclsneii 24012 | A neighborhood of a cluste... |
| fclssscls 24013 | The set of cluster points ... |
| fclsnei 24014 | Cluster points in terms of... |
| supnfcls 24015 | The filter of supersets of... |
| fclsbas 24016 | Cluster points in terms of... |
| fclsss1 24017 | A finer topology has fewer... |
| fclsss2 24018 | A finer filter has fewer c... |
| fclsrest 24019 | The set of cluster points ... |
| fclscf 24020 | Characterization of finene... |
| flimfcls 24021 | A limit point is a cluster... |
| fclsfnflim 24022 | A filter clusters at a poi... |
| flimfnfcls 24023 | A filter converges to a po... |
| fclscmpi 24024 | Forward direction of ~ fcl... |
| fclscmp 24025 | A space is compact iff eve... |
| uffclsflim 24026 | The cluster points of an u... |
| ufilcmp 24027 | A space is compact iff eve... |
| fcfval 24028 | The set of cluster points ... |
| isfcf 24029 | The property of being a cl... |
| fcfnei 24030 | The property of being a cl... |
| fcfelbas 24031 | A cluster point of a funct... |
| fcfneii 24032 | A neighborhood of a cluste... |
| flfssfcf 24033 | A limit point of a functio... |
| uffcfflf 24034 | If the domain filter is an... |
| cnpfcfi 24035 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 24036 | A function ` F ` is contin... |
| cnfcf 24037 | Continuity of a function i... |
| flfcntr 24038 | A continuous function's va... |
| alexsublem 24039 | Lemma for ~ alexsub . (Co... |
| alexsub 24040 | The Alexander Subbase Theo... |
| alexsubb 24041 | Biconditional form of the ... |
| alexsubALTlem1 24042 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 24043 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 24044 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 24045 | Lemma for ~ alexsubALT . ... |
| alexsubALT 24046 | The Alexander Subbase Theo... |
| ptcmplem1 24047 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 24048 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 24049 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 24050 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 24051 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24052 | Tychonoff's theorem: The ... |
| ptcmp 24053 | Tychonoff's theorem: The ... |
| cnextval 24056 | The function applying cont... |
| cnextfval 24057 | The continuous extension o... |
| cnextrel 24058 | In the general case, a con... |
| cnextfun 24059 | If the target space is Hau... |
| cnextfvval 24060 | The value of the continuou... |
| cnextf 24061 | Extension by continuity. ... |
| cnextcn 24062 | Extension by continuity. ... |
| cnextfres1 24063 | ` F ` and its extension by... |
| cnextfres 24064 | ` F ` and its extension by... |
| istmd 24069 | The predicate "is a topolo... |
| tmdmnd 24070 | A topological monoid is a ... |
| tmdtps 24071 | A topological monoid is a ... |
| istgp 24072 | The predicate "is a topolo... |
| tgpgrp 24073 | A topological group is a g... |
| tgptmd 24074 | A topological group is a t... |
| tgptps 24075 | A topological group is a t... |
| tmdtopon 24076 | The topology of a topologi... |
| tgptopon 24077 | The topology of a topologi... |
| tmdcn 24078 | In a topological monoid, t... |
| tgpcn 24079 | In a topological group, th... |
| tgpinv 24080 | In a topological group, th... |
| grpinvhmeo 24081 | The inverse function in a ... |
| cnmpt1plusg 24082 | Continuity of the group su... |
| cnmpt2plusg 24083 | Continuity of the group su... |
| tmdcn2 24084 | Write out the definition o... |
| tgpsubcn 24085 | In a topological group, th... |
| istgp2 24086 | A group with a topology is... |
| tmdmulg 24087 | In a topological monoid, t... |
| tgpmulg 24088 | In a topological group, th... |
| tgpmulg2 24089 | In a topological monoid, t... |
| tmdgsum 24090 | In a topological monoid, t... |
| tmdgsum2 24091 | For any neighborhood ` U `... |
| oppgtmd 24092 | The opposite of a topologi... |
| oppgtgp 24093 | The opposite of a topologi... |
| distgp 24094 | Any group equipped with th... |
| indistgp 24095 | Any group equipped with th... |
| efmndtmd 24096 | The monoid of endofunction... |
| tmdlactcn 24097 | The left group action of e... |
| tgplacthmeo 24098 | The left group action of e... |
| submtmd 24099 | A submonoid of a topologic... |
| subgtgp 24100 | A subgroup of a topologica... |
| symgtgp 24101 | The symmetric group is a t... |
| subgntr 24102 | A subgroup of a topologica... |
| opnsubg 24103 | An open subgroup of a topo... |
| clssubg 24104 | The closure of a subgroup ... |
| clsnsg 24105 | The closure of a normal su... |
| cldsubg 24106 | A subgroup of finite index... |
| tgpconncompeqg 24107 | The connected component co... |
| tgpconncomp 24108 | The identity component, th... |
| tgpconncompss 24109 | The identity component is ... |
| ghmcnp 24110 | A group homomorphism on to... |
| snclseqg 24111 | The coset of the closure o... |
| tgphaus 24112 | A topological group is Hau... |
| tgpt1 24113 | Hausdorff and T1 are equiv... |
| tgpt0 24114 | Hausdorff and T0 are equiv... |
| qustgpopn 24115 | A quotient map in a topolo... |
| qustgplem 24116 | Lemma for ~ qustgp . (Con... |
| qustgp 24117 | The quotient of a topologi... |
| qustgphaus 24118 | The quotient of a topologi... |
| prdstmdd 24119 | The product of a family of... |
| prdstgpd 24120 | The product of a family of... |
| tsmsfbas 24123 | The collection of all sets... |
| tsmslem1 24124 | The finite partial sums of... |
| tsmsval2 24125 | Definition of the topologi... |
| tsmsval 24126 | Definition of the topologi... |
| tsmspropd 24127 | The group sum depends only... |
| eltsms 24128 | The property of being a su... |
| tsmsi 24129 | The property of being a su... |
| tsmscl 24130 | A sum in a topological gro... |
| haustsms 24131 | In a Hausdorff topological... |
| haustsms2 24132 | In a Hausdorff topological... |
| tsmscls 24133 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24134 | The convergent points of a... |
| tsmsid 24135 | If a sum is finite, the us... |
| haustsmsid 24136 | In a Hausdorff topological... |
| tsms0 24137 | The sum of zero is zero. ... |
| tsmssubm 24138 | Evaluate an infinite group... |
| tsmsres 24139 | Extend an infinite group s... |
| tsmsf1o 24140 | Re-index an infinite group... |
| tsmsmhm 24141 | Apply a continuous group h... |
| tsmsadd 24142 | The sum of two infinite gr... |
| tsmsinv 24143 | Inverse of an infinite gro... |
| tsmssub 24144 | The difference of two infi... |
| tgptsmscls 24145 | A sum in a topological gro... |
| tgptsmscld 24146 | The set of limit points to... |
| tsmssplit 24147 | Split a topological group ... |
| tsmsxplem1 24148 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24149 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24150 | Write a sum over a two-dim... |
| istrg 24159 | Express the predicate " ` ... |
| trgtmd 24160 | The multiplicative monoid ... |
| istdrg 24161 | Express the predicate " ` ... |
| tdrgunit 24162 | The unit group of a topolo... |
| trgtgp 24163 | A topological ring is a to... |
| trgtmd2 24164 | A topological ring is a to... |
| trgtps 24165 | A topological ring is a to... |
| trgring 24166 | A topological ring is a ri... |
| trggrp 24167 | A topological ring is a gr... |
| tdrgtrg 24168 | A topological division rin... |
| tdrgdrng 24169 | A topological division rin... |
| tdrgring 24170 | A topological division rin... |
| tdrgtmd 24171 | A topological division rin... |
| tdrgtps 24172 | A topological division rin... |
| istdrg2 24173 | A topological-ring divisio... |
| mulrcn 24174 | The functionalization of t... |
| invrcn2 24175 | The multiplicative inverse... |
| invrcn 24176 | The multiplicative inverse... |
| cnmpt1mulr 24177 | Continuity of ring multipl... |
| cnmpt2mulr 24178 | Continuity of ring multipl... |
| dvrcn 24179 | The division function is c... |
| istlm 24180 | The predicate " ` W ` is a... |
| vscacn 24181 | The scalar multiplication ... |
| tlmtmd 24182 | A topological module is a ... |
| tlmtps 24183 | A topological module is a ... |
| tlmlmod 24184 | A topological module is a ... |
| tlmtrg 24185 | The scalar ring of a topol... |
| tlmscatps 24186 | The scalar ring of a topol... |
| istvc 24187 | A topological vector space... |
| tvctdrg 24188 | The scalar field of a topo... |
| cnmpt1vsca 24189 | Continuity of scalar multi... |
| cnmpt2vsca 24190 | Continuity of scalar multi... |
| tlmtgp 24191 | A topological vector space... |
| tvctlm 24192 | A topological vector space... |
| tvclmod 24193 | A topological vector space... |
| tvclvec 24194 | A topological vector space... |
| ustfn 24197 | The defined uniform struct... |
| ustval 24198 | The class of all uniform s... |
| isust 24199 | The predicate " ` U ` is a... |
| ustssxp 24200 | Entourages are subsets of ... |
| ustssel 24201 | A uniform structure is upw... |
| ustbasel 24202 | The full set is always an ... |
| ustincl 24203 | A uniform structure is clo... |
| ustdiag 24204 | The diagonal set is includ... |
| ustinvel 24205 | If ` V ` is an entourage, ... |
| ustexhalf 24206 | For each entourage ` V ` t... |
| ustrel 24207 | The elements of uniform st... |
| ustfilxp 24208 | A uniform structure on a n... |
| ustne0 24209 | A uniform structure cannot... |
| ustssco 24210 | In an uniform structure, a... |
| ustexsym 24211 | In an uniform structure, f... |
| ustex2sym 24212 | In an uniform structure, f... |
| ustex3sym 24213 | In an uniform structure, f... |
| ustref 24214 | Any element of the base se... |
| ust0 24215 | The unique uniform structu... |
| ustn0 24216 | The empty set is not an un... |
| ustund 24217 | If two intersecting sets `... |
| ustelimasn 24218 | Any point ` A ` is near en... |
| ustneism 24219 | For a point ` A ` in ` X `... |
| elrnustOLD 24220 | Obsolete version of ~ elfv... |
| ustbas2 24221 | Second direction for ~ ust... |
| ustuni 24222 | The set union of a uniform... |
| ustbas 24223 | Recover the base of an uni... |
| ustimasn 24224 | Lemma for ~ ustuqtop . (C... |
| trust 24225 | The trace of a uniform str... |
| utopval 24228 | The topology induced by a ... |
| elutop 24229 | Open sets in the topology ... |
| utoptop 24230 | The topology induced by a ... |
| utopbas 24231 | The base of the topology i... |
| utoptopon 24232 | Topology induced by a unif... |
| restutop 24233 | Restriction of a topology ... |
| restutopopn 24234 | The restriction of the top... |
| ustuqtoplem 24235 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24236 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24237 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24238 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24239 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24240 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24241 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24242 | For a given uniform struct... |
| utopsnneiplem 24243 | The neighborhoods of a poi... |
| utopsnneip 24244 | The neighborhoods of a poi... |
| utopsnnei 24245 | Images of singletons by en... |
| utop2nei 24246 | For any symmetrical entour... |
| utop3cls 24247 | Relation between a topolog... |
| utopreg 24248 | All Hausdorff uniform spac... |
| ussval 24255 | The uniform structure on u... |
| ussid 24256 | In case the base of the ` ... |
| isusp 24257 | The predicate ` W ` is a u... |
| ressuss 24258 | Value of the uniform struc... |
| ressust 24259 | The uniform structure of a... |
| ressusp 24260 | The restriction of a unifo... |
| tusval 24261 | The value of the uniform s... |
| tuslem 24262 | Lemma for ~ tusbas , ~ tus... |
| tuslemOLD 24263 | Obsolete proof of ~ tuslem... |
| tusbas 24264 | The base set of a construc... |
| tusunif 24265 | The uniform structure of a... |
| tususs 24266 | The uniform structure of a... |
| tustopn 24267 | The topology induced by a ... |
| tususp 24268 | A constructed uniform spac... |
| tustps 24269 | A constructed uniform spac... |
| uspreg 24270 | If a uniform space is Haus... |
| ucnval 24273 | The set of all uniformly c... |
| isucn 24274 | The predicate " ` F ` is a... |
| isucn2 24275 | The predicate " ` F ` is a... |
| ucnimalem 24276 | Reformulate the ` G ` func... |
| ucnima 24277 | An equivalent statement of... |
| ucnprima 24278 | The preimage by a uniforml... |
| iducn 24279 | The identity is uniformly ... |
| cstucnd 24280 | A constant function is uni... |
| ucncn 24281 | Uniform continuity implies... |
| iscfilu 24284 | The predicate " ` F ` is a... |
| cfilufbas 24285 | A Cauchy filter base is a ... |
| cfiluexsm 24286 | For a Cauchy filter base a... |
| fmucndlem 24287 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24288 | The image of a Cauchy filt... |
| cfilufg 24289 | The filter generated by a ... |
| trcfilu 24290 | Condition for the trace of... |
| cfiluweak 24291 | A Cauchy filter base is al... |
| neipcfilu 24292 | In an uniform space, a nei... |
| iscusp 24295 | The predicate " ` W ` is a... |
| cuspusp 24296 | A complete uniform space i... |
| cuspcvg 24297 | In a complete uniform spac... |
| iscusp2 24298 | The predicate " ` W ` is a... |
| cnextucn 24299 | Extension by continuity. ... |
| ucnextcn 24300 | Extension by continuity. ... |
| ispsmet 24301 | Express the predicate " ` ... |
| psmetdmdm 24302 | Recover the base set from ... |
| psmetf 24303 | The distance function of a... |
| psmetcl 24304 | Closure of the distance fu... |
| psmet0 24305 | The distance function of a... |
| psmettri2 24306 | Triangle inequality for th... |
| psmetsym 24307 | The distance function of a... |
| psmettri 24308 | Triangle inequality for th... |
| psmetge0 24309 | The distance function of a... |
| psmetxrge0 24310 | The distance function of a... |
| psmetres2 24311 | Restriction of a pseudomet... |
| psmetlecl 24312 | Real closure of an extende... |
| distspace 24313 | A set ` X ` together with ... |
| ismet 24320 | Express the predicate " ` ... |
| isxmet 24321 | Express the predicate " ` ... |
| ismeti 24322 | Properties that determine ... |
| isxmetd 24323 | Properties that determine ... |
| isxmet2d 24324 | It is safe to only require... |
| metflem 24325 | Lemma for ~ metf and other... |
| xmetf 24326 | Mapping of the distance fu... |
| metf 24327 | Mapping of the distance fu... |
| xmetcl 24328 | Closure of the distance fu... |
| metcl 24329 | Closure of the distance fu... |
| ismet2 24330 | An extended metric is a me... |
| metxmet 24331 | A metric is an extended me... |
| xmetdmdm 24332 | Recover the base set from ... |
| metdmdm 24333 | Recover the base set from ... |
| xmetunirn 24334 | Two ways to express an ext... |
| xmeteq0 24335 | The value of an extended m... |
| meteq0 24336 | The value of a metric is z... |
| xmettri2 24337 | Triangle inequality for th... |
| mettri2 24338 | Triangle inequality for th... |
| xmet0 24339 | The distance function of a... |
| met0 24340 | The distance function of a... |
| xmetge0 24341 | The distance function of a... |
| metge0 24342 | The distance function of a... |
| xmetlecl 24343 | Real closure of an extende... |
| xmetsym 24344 | The distance function of a... |
| xmetpsmet 24345 | An extended metric is a ps... |
| xmettpos 24346 | The distance function of a... |
| metsym 24347 | The distance function of a... |
| xmettri 24348 | Triangle inequality for th... |
| mettri 24349 | Triangle inequality for th... |
| xmettri3 24350 | Triangle inequality for th... |
| mettri3 24351 | Triangle inequality for th... |
| xmetrtri 24352 | One half of the reverse tr... |
| xmetrtri2 24353 | The reverse triangle inequ... |
| metrtri 24354 | Reverse triangle inequalit... |
| xmetgt0 24355 | The distance function of a... |
| metgt0 24356 | The distance function of a... |
| metn0 24357 | A metric space is nonempty... |
| xmetres2 24358 | Restriction of an extended... |
| metreslem 24359 | Lemma for ~ metres . (Con... |
| metres2 24360 | Lemma for ~ metres . (Con... |
| xmetres 24361 | A restriction of an extend... |
| metres 24362 | A restriction of a metric ... |
| 0met 24363 | The empty metric. (Contri... |
| prdsdsf 24364 | The product metric is a fu... |
| prdsxmetlem 24365 | The product metric is an e... |
| prdsxmet 24366 | The product metric is an e... |
| prdsmet 24367 | The product metric is a me... |
| ressprdsds 24368 | Restriction of a product m... |
| resspwsds 24369 | Restriction of a power met... |
| imasdsf1olem 24370 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24371 | The distance function is t... |
| imasf1oxmet 24372 | The image of an extended m... |
| imasf1omet 24373 | The image of a metric is a... |
| xpsdsfn 24374 | Closure of the metric in a... |
| xpsdsfn2 24375 | Closure of the metric in a... |
| xpsxmetlem 24376 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24377 | A product metric of extend... |
| xpsdsval 24378 | Value of the metric in a b... |
| xpsmet 24379 | The direct product of two ... |
| blfvalps 24380 | The value of the ball func... |
| blfval 24381 | The value of the ball func... |
| blvalps 24382 | The ball around a point ` ... |
| blval 24383 | The ball around a point ` ... |
| elblps 24384 | Membership in a ball. (Co... |
| elbl 24385 | Membership in a ball. (Co... |
| elbl2ps 24386 | Membership in a ball. (Co... |
| elbl2 24387 | Membership in a ball. (Co... |
| elbl3ps 24388 | Membership in a ball, with... |
| elbl3 24389 | Membership in a ball, with... |
| blcomps 24390 | Commute the arguments to t... |
| blcom 24391 | Commute the arguments to t... |
| xblpnfps 24392 | The infinity ball in an ex... |
| xblpnf 24393 | The infinity ball in an ex... |
| blpnf 24394 | The infinity ball in a sta... |
| bldisj 24395 | Two balls are disjoint if ... |
| blgt0 24396 | A nonempty ball implies th... |
| bl2in 24397 | Two balls are disjoint if ... |
| xblss2ps 24398 | One ball is contained in a... |
| xblss2 24399 | One ball is contained in a... |
| blss2ps 24400 | One ball is contained in a... |
| blss2 24401 | One ball is contained in a... |
| blhalf 24402 | A ball of radius ` R / 2 `... |
| blfps 24403 | Mapping of a ball. (Contr... |
| blf 24404 | Mapping of a ball. (Contr... |
| blrnps 24405 | Membership in the range of... |
| blrn 24406 | Membership in the range of... |
| xblcntrps 24407 | A ball contains its center... |
| xblcntr 24408 | A ball contains its center... |
| blcntrps 24409 | A ball contains its center... |
| blcntr 24410 | A ball contains its center... |
| xbln0 24411 | A ball is nonempty iff the... |
| bln0 24412 | A ball is not empty. (Con... |
| blelrnps 24413 | A ball belongs to the set ... |
| blelrn 24414 | A ball belongs to the set ... |
| blssm 24415 | A ball is a subset of the ... |
| unirnblps 24416 | The union of the set of ba... |
| unirnbl 24417 | The union of the set of ba... |
| blin 24418 | The intersection of two ba... |
| ssblps 24419 | The size of a ball increas... |
| ssbl 24420 | The size of a ball increas... |
| blssps 24421 | Any point ` P ` in a ball ... |
| blss 24422 | Any point ` P ` in a ball ... |
| blssexps 24423 | Two ways to express the ex... |
| blssex 24424 | Two ways to express the ex... |
| ssblex 24425 | A nested ball exists whose... |
| blin2 24426 | Given any two balls and a ... |
| blbas 24427 | The balls of a metric spac... |
| blres 24428 | A ball in a restricted met... |
| xmeterval 24429 | Value of the "finitely sep... |
| xmeter 24430 | The "finitely separated" r... |
| xmetec 24431 | The equivalence classes un... |
| blssec 24432 | A ball centered at ` P ` i... |
| blpnfctr 24433 | The infinity ball in an ex... |
| xmetresbl 24434 | An extended metric restric... |
| mopnval 24435 | An open set is a subset of... |
| mopntopon 24436 | The set of open sets of a ... |
| mopntop 24437 | The set of open sets of a ... |
| mopnuni 24438 | The union of all open sets... |
| elmopn 24439 | The defining property of a... |
| mopnfss 24440 | The family of open sets of... |
| mopnm 24441 | The base set of a metric s... |
| elmopn2 24442 | A defining property of an ... |
| mopnss 24443 | An open set of a metric sp... |
| isxms 24444 | Express the predicate " ` ... |
| isxms2 24445 | Express the predicate " ` ... |
| isms 24446 | Express the predicate " ` ... |
| isms2 24447 | Express the predicate " ` ... |
| xmstopn 24448 | The topology component of ... |
| mstopn 24449 | The topology component of ... |
| xmstps 24450 | An extended metric space i... |
| msxms 24451 | A metric space is an exten... |
| mstps 24452 | A metric space is a topolo... |
| xmsxmet 24453 | The distance function, sui... |
| msmet 24454 | The distance function, sui... |
| msf 24455 | The distance function of a... |
| xmsxmet2 24456 | The distance function, sui... |
| msmet2 24457 | The distance function, sui... |
| mscl 24458 | Closure of the distance fu... |
| xmscl 24459 | Closure of the distance fu... |
| xmsge0 24460 | The distance function in a... |
| xmseq0 24461 | The distance between two p... |
| xmssym 24462 | The distance function in a... |
| xmstri2 24463 | Triangle inequality for th... |
| mstri2 24464 | Triangle inequality for th... |
| xmstri 24465 | Triangle inequality for th... |
| mstri 24466 | Triangle inequality for th... |
| xmstri3 24467 | Triangle inequality for th... |
| mstri3 24468 | Triangle inequality for th... |
| msrtri 24469 | Reverse triangle inequalit... |
| xmspropd 24470 | Property deduction for an ... |
| mspropd 24471 | Property deduction for a m... |
| setsmsbas 24472 | The base set of a construc... |
| setsmsbasOLD 24473 | Obsolete proof of ~ setsms... |
| setsmsds 24474 | The distance function of a... |
| setsmsdsOLD 24475 | Obsolete proof of ~ setsms... |
| setsmstset 24476 | The topology of a construc... |
| setsmstopn 24477 | The topology of a construc... |
| setsxms 24478 | The constructed metric spa... |
| setsms 24479 | The constructed metric spa... |
| tmsval 24480 | For any metric there is an... |
| tmslem 24481 | Lemma for ~ tmsbas , ~ tms... |
| tmslemOLD 24482 | Obsolete version of ~ tmsl... |
| tmsbas 24483 | The base set of a construc... |
| tmsds 24484 | The metric of a constructe... |
| tmstopn 24485 | The topology of a construc... |
| tmsxms 24486 | The constructed metric spa... |
| tmsms 24487 | The constructed metric spa... |
| imasf1obl 24488 | The image of a metric spac... |
| imasf1oxms 24489 | The image of a metric spac... |
| imasf1oms 24490 | The image of a metric spac... |
| prdsbl 24491 | A ball in the product metr... |
| mopni 24492 | An open set of a metric sp... |
| mopni2 24493 | An open set of a metric sp... |
| mopni3 24494 | An open set of a metric sp... |
| blssopn 24495 | The balls of a metric spac... |
| unimopn 24496 | The union of a collection ... |
| mopnin 24497 | The intersection of two op... |
| mopn0 24498 | The empty set is an open s... |
| rnblopn 24499 | A ball of a metric space i... |
| blopn 24500 | A ball of a metric space i... |
| neibl 24501 | The neighborhoods around a... |
| blnei 24502 | A ball around a point is a... |
| lpbl 24503 | Every ball around a limit ... |
| blsscls2 24504 | A smaller closed ball is c... |
| blcld 24505 | A "closed ball" in a metri... |
| blcls 24506 | The closure of an open bal... |
| blsscls 24507 | If two concentric balls ha... |
| metss 24508 | Two ways of saying that me... |
| metequiv 24509 | Two ways of saying that tw... |
| metequiv2 24510 | If there is a sequence of ... |
| metss2lem 24511 | Lemma for ~ metss2 . (Con... |
| metss2 24512 | If the metric ` D ` is "st... |
| comet 24513 | The composition of an exte... |
| stdbdmetval 24514 | Value of the standard boun... |
| stdbdxmet 24515 | The standard bounded metri... |
| stdbdmet 24516 | The standard bounded metri... |
| stdbdbl 24517 | The standard bounded metri... |
| stdbdmopn 24518 | The standard bounded metri... |
| mopnex 24519 | The topology generated by ... |
| methaus 24520 | The topology generated by ... |
| met1stc 24521 | The topology generated by ... |
| met2ndci 24522 | A separable metric space (... |
| met2ndc 24523 | A metric space is second-c... |
| metrest 24524 | Two alternate formulations... |
| ressxms 24525 | The restriction of a metri... |
| ressms 24526 | The restriction of a metri... |
| prdsmslem1 24527 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24528 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24529 | Lemma for ~ prdsxms . The... |
| prdsxms 24530 | The indexed product struct... |
| prdsms 24531 | The indexed product struct... |
| pwsxms 24532 | A power of an extended met... |
| pwsms 24533 | A power of a metric space ... |
| xpsxms 24534 | A binary product of metric... |
| xpsms 24535 | A binary product of metric... |
| tmsxps 24536 | Express the product of two... |
| tmsxpsmopn 24537 | Express the product of two... |
| tmsxpsval 24538 | Value of the product of tw... |
| tmsxpsval2 24539 | Value of the product of tw... |
| metcnp3 24540 | Two ways to express that `... |
| metcnp 24541 | Two ways to say a mapping ... |
| metcnp2 24542 | Two ways to say a mapping ... |
| metcn 24543 | Two ways to say a mapping ... |
| metcnpi 24544 | Epsilon-delta property of ... |
| metcnpi2 24545 | Epsilon-delta property of ... |
| metcnpi3 24546 | Epsilon-delta property of ... |
| txmetcnp 24547 | Continuity of a binary ope... |
| txmetcn 24548 | Continuity of a binary ope... |
| metuval 24549 | Value of the uniform struc... |
| metustel 24550 | Define a filter base ` F `... |
| metustss 24551 | Range of the elements of t... |
| metustrel 24552 | Elements of the filter bas... |
| metustto 24553 | Any two elements of the fi... |
| metustid 24554 | The identity diagonal is i... |
| metustsym 24555 | Elements of the filter bas... |
| metustexhalf 24556 | For any element ` A ` of t... |
| metustfbas 24557 | The filter base generated ... |
| metust 24558 | The uniform structure gene... |
| cfilucfil 24559 | Given a metric ` D ` and a... |
| metuust 24560 | The uniform structure gene... |
| cfilucfil2 24561 | Given a metric ` D ` and a... |
| blval2 24562 | The ball around a point ` ... |
| elbl4 24563 | Membership in a ball, alte... |
| metuel 24564 | Elementhood in the uniform... |
| metuel2 24565 | Elementhood in the uniform... |
| metustbl 24566 | The "section" image of an ... |
| psmetutop 24567 | The topology induced by a ... |
| xmetutop 24568 | The topology induced by a ... |
| xmsusp 24569 | If the uniform set of a me... |
| restmetu 24570 | The uniform structure gene... |
| metucn 24571 | Uniform continuity in metr... |
| dscmet 24572 | The discrete metric on any... |
| dscopn 24573 | The discrete metric genera... |
| nrmmetd 24574 | Show that a group norm gen... |
| abvmet 24575 | An absolute value ` F ` ge... |
| nmfval 24588 | The value of the norm func... |
| nmval 24589 | The value of the norm as t... |
| nmfval0 24590 | The value of the norm func... |
| nmfval2 24591 | The value of the norm func... |
| nmval2 24592 | The value of the norm on a... |
| nmf2 24593 | The norm on a metric group... |
| nmpropd 24594 | Weak property deduction fo... |
| nmpropd2 24595 | Strong property deduction ... |
| isngp 24596 | The property of being a no... |
| isngp2 24597 | The property of being a no... |
| isngp3 24598 | The property of being a no... |
| ngpgrp 24599 | A normed group is a group.... |
| ngpms 24600 | A normed group is a metric... |
| ngpxms 24601 | A normed group is an exten... |
| ngptps 24602 | A normed group is a topolo... |
| ngpmet 24603 | The (induced) metric of a ... |
| ngpds 24604 | Value of the distance func... |
| ngpdsr 24605 | Value of the distance func... |
| ngpds2 24606 | Write the distance between... |
| ngpds2r 24607 | Write the distance between... |
| ngpds3 24608 | Write the distance between... |
| ngpds3r 24609 | Write the distance between... |
| ngprcan 24610 | Cancel right addition insi... |
| ngplcan 24611 | Cancel left addition insid... |
| isngp4 24612 | Express the property of be... |
| ngpinvds 24613 | Two elements are the same ... |
| ngpsubcan 24614 | Cancel right subtraction i... |
| nmf 24615 | The norm on a normed group... |
| nmcl 24616 | The norm of a normed group... |
| nmge0 24617 | The norm of a normed group... |
| nmeq0 24618 | The identity is the only e... |
| nmne0 24619 | The norm of a nonzero elem... |
| nmrpcl 24620 | The norm of a nonzero elem... |
| nminv 24621 | The norm of a negated elem... |
| nmmtri 24622 | The triangle inequality fo... |
| nmsub 24623 | The norm of the difference... |
| nmrtri 24624 | Reverse triangle inequalit... |
| nm2dif 24625 | Inequality for the differe... |
| nmtri 24626 | The triangle inequality fo... |
| nmtri2 24627 | Triangle inequality for th... |
| ngpi 24628 | The properties of a normed... |
| nm0 24629 | Norm of the identity eleme... |
| nmgt0 24630 | The norm of a nonzero elem... |
| sgrim 24631 | The induced metric on a su... |
| sgrimval 24632 | The induced metric on a su... |
| subgnm 24633 | The norm in a subgroup. (... |
| subgnm2 24634 | A substructure assigns the... |
| subgngp 24635 | A normed group restricted ... |
| ngptgp 24636 | A normed abelian group is ... |
| ngppropd 24637 | Property deduction for a n... |
| reldmtng 24638 | The function ` toNrmGrp ` ... |
| tngval 24639 | Value of the function whic... |
| tnglem 24640 | Lemma for ~ tngbas and sim... |
| tnglemOLD 24641 | Obsolete version of ~ tngl... |
| tngbas 24642 | The base set of a structur... |
| tngbasOLD 24643 | Obsolete proof of ~ tngbas... |
| tngplusg 24644 | The group addition of a st... |
| tngplusgOLD 24645 | Obsolete proof of ~ tngplu... |
| tng0 24646 | The group identity of a st... |
| tngmulr 24647 | The ring multiplication of... |
| tngmulrOLD 24648 | Obsolete proof of ~ tngmul... |
| tngsca 24649 | The scalar ring of a struc... |
| tngscaOLD 24650 | Obsolete proof of ~ tngsca... |
| tngvsca 24651 | The scalar multiplication ... |
| tngvscaOLD 24652 | Obsolete proof of ~ tngvsc... |
| tngip 24653 | The inner product operatio... |
| tngipOLD 24654 | Obsolete proof of ~ tngip ... |
| tngds 24655 | The metric function of a s... |
| tngdsOLD 24656 | Obsolete proof of ~ tngds ... |
| tngtset 24657 | The topology generated by ... |
| tngtopn 24658 | The topology generated by ... |
| tngnm 24659 | The topology generated by ... |
| tngngp2 24660 | A norm turns a group into ... |
| tngngpd 24661 | Derive the axioms for a no... |
| tngngp 24662 | Derive the axioms for a no... |
| tnggrpr 24663 | If a structure equipped wi... |
| tngngp3 24664 | Alternate definition of a ... |
| nrmtngdist 24665 | The augmentation of a norm... |
| nrmtngnrm 24666 | The augmentation of a norm... |
| tngngpim 24667 | The induced metric of a no... |
| isnrg 24668 | A normed ring is a ring wi... |
| nrgabv 24669 | The norm of a normed ring ... |
| nrgngp 24670 | A normed ring is a normed ... |
| nrgring 24671 | A normed ring is a ring. ... |
| nmmul 24672 | The norm of a product in a... |
| nrgdsdi 24673 | Distribute a distance calc... |
| nrgdsdir 24674 | Distribute a distance calc... |
| nm1 24675 | The norm of one in a nonze... |
| unitnmn0 24676 | The norm of a unit is nonz... |
| nminvr 24677 | The norm of an inverse in ... |
| nmdvr 24678 | The norm of a division in ... |
| nrgdomn 24679 | A nonzero normed ring is a... |
| nrgtgp 24680 | A normed ring is a topolog... |
| subrgnrg 24681 | A normed ring restricted t... |
| tngnrg 24682 | Given any absolute value o... |
| isnlm 24683 | A normed (left) module is ... |
| nmvs 24684 | Defining property of a nor... |
| nlmngp 24685 | A normed module is a norme... |
| nlmlmod 24686 | A normed module is a left ... |
| nlmnrg 24687 | The scalar component of a ... |
| nlmngp2 24688 | The scalar component of a ... |
| nlmdsdi 24689 | Distribute a distance calc... |
| nlmdsdir 24690 | Distribute a distance calc... |
| nlmmul0or 24691 | If a scalar product is zer... |
| sranlm 24692 | The subring algebra over a... |
| nlmvscnlem2 24693 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24694 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24695 | The scalar multiplication ... |
| rlmnlm 24696 | The ring module over a nor... |
| rlmnm 24697 | The norm function in the r... |
| nrgtrg 24698 | A normed ring is a topolog... |
| nrginvrcnlem 24699 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24700 | The ring inverse function ... |
| nrgtdrg 24701 | A normed division ring is ... |
| nlmtlm 24702 | A normed module is a topol... |
| isnvc 24703 | A normed vector space is j... |
| nvcnlm 24704 | A normed vector space is a... |
| nvclvec 24705 | A normed vector space is a... |
| nvclmod 24706 | A normed vector space is a... |
| isnvc2 24707 | A normed vector space is j... |
| nvctvc 24708 | A normed vector space is a... |
| lssnlm 24709 | A subspace of a normed mod... |
| lssnvc 24710 | A subspace of a normed vec... |
| rlmnvc 24711 | The ring module over a nor... |
| ngpocelbl 24712 | Membership of an off-cente... |
| nmoffn 24719 | The function producing ope... |
| reldmnghm 24720 | Lemma for normed group hom... |
| reldmnmhm 24721 | Lemma for module homomorph... |
| nmofval 24722 | Value of the operator norm... |
| nmoval 24723 | Value of the operator norm... |
| nmogelb 24724 | Property of the operator n... |
| nmolb 24725 | Any upper bound on the val... |
| nmolb2d 24726 | Any upper bound on the val... |
| nmof 24727 | The operator norm is a fun... |
| nmocl 24728 | The operator norm of an op... |
| nmoge0 24729 | The operator norm of an op... |
| nghmfval 24730 | A normed group homomorphis... |
| isnghm 24731 | A normed group homomorphis... |
| isnghm2 24732 | A normed group homomorphis... |
| isnghm3 24733 | A normed group homomorphis... |
| bddnghm 24734 | A bounded group homomorphi... |
| nghmcl 24735 | A normed group homomorphis... |
| nmoi 24736 | The operator norm achieves... |
| nmoix 24737 | The operator norm is a bou... |
| nmoi2 24738 | The operator norm is a bou... |
| nmoleub 24739 | The operator norm, defined... |
| nghmrcl1 24740 | Reverse closure for a norm... |
| nghmrcl2 24741 | Reverse closure for a norm... |
| nghmghm 24742 | A normed group homomorphis... |
| nmo0 24743 | The operator norm of the z... |
| nmoeq0 24744 | The operator norm is zero ... |
| nmoco 24745 | An upper bound on the oper... |
| nghmco 24746 | The composition of normed ... |
| nmotri 24747 | Triangle inequality for th... |
| nghmplusg 24748 | The sum of two bounded lin... |
| 0nghm 24749 | The zero operator is a nor... |
| nmoid 24750 | The operator norm of the i... |
| idnghm 24751 | The identity operator is a... |
| nmods 24752 | Upper bound for the distan... |
| nghmcn 24753 | A normed group homomorphis... |
| isnmhm 24754 | A normed module homomorphi... |
| nmhmrcl1 24755 | Reverse closure for a norm... |
| nmhmrcl2 24756 | Reverse closure for a norm... |
| nmhmlmhm 24757 | A normed module homomorphi... |
| nmhmnghm 24758 | A normed module homomorphi... |
| nmhmghm 24759 | A normed module homomorphi... |
| isnmhm2 24760 | A normed module homomorphi... |
| nmhmcl 24761 | A normed module homomorphi... |
| idnmhm 24762 | The identity operator is a... |
| 0nmhm 24763 | The zero operator is a bou... |
| nmhmco 24764 | The composition of bounded... |
| nmhmplusg 24765 | The sum of two bounded lin... |
| qtopbaslem 24766 | The set of open intervals ... |
| qtopbas 24767 | The set of open intervals ... |
| retopbas 24768 | A basis for the standard t... |
| retop 24769 | The standard topology on t... |
| uniretop 24770 | The underlying set of the ... |
| retopon 24771 | The standard topology on t... |
| retps 24772 | The standard topological s... |
| iooretop 24773 | Open intervals are open se... |
| icccld 24774 | Closed intervals are close... |
| icopnfcld 24775 | Right-unbounded closed int... |
| iocmnfcld 24776 | Left-unbounded closed inte... |
| qdensere 24777 | ` QQ ` is dense in the sta... |
| cnmetdval 24778 | Value of the distance func... |
| cnmet 24779 | The absolute value metric ... |
| cnxmet 24780 | The absolute value metric ... |
| cnbl0 24781 | Two ways to write the open... |
| cnblcld 24782 | Two ways to write the clos... |
| cnfldms 24783 | The complex number field i... |
| cnfldxms 24784 | The complex number field i... |
| cnfldtps 24785 | The complex number field i... |
| cnfldnm 24786 | The norm of the field of c... |
| cnngp 24787 | The complex numbers form a... |
| cnnrg 24788 | The complex numbers form a... |
| cnfldtopn 24789 | The topology of the comple... |
| cnfldtopon 24790 | The topology of the comple... |
| cnfldtop 24791 | The topology of the comple... |
| cnfldhaus 24792 | The topology of the comple... |
| unicntop 24793 | The underlying set of the ... |
| cnopn 24794 | The set of complex numbers... |
| zringnrg 24795 | The ring of integers is a ... |
| remetdval 24796 | Value of the distance func... |
| remet 24797 | The absolute value metric ... |
| rexmet 24798 | The absolute value metric ... |
| bl2ioo 24799 | A ball in terms of an open... |
| ioo2bl 24800 | An open interval of reals ... |
| ioo2blex 24801 | An open interval of reals ... |
| blssioo 24802 | The balls of the standard ... |
| tgioo 24803 | The topology generated by ... |
| qdensere2 24804 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24805 | An open ball in the comple... |
| rehaus 24806 | The standard topology on t... |
| tgqioo 24807 | The topology generated by ... |
| re2ndc 24808 | The standard topology on t... |
| resubmet 24809 | The subspace topology indu... |
| tgioo2 24810 | The standard topology on t... |
| rerest 24811 | The subspace topology indu... |
| tgioo3 24812 | The standard topology on t... |
| xrtgioo 24813 | The topology on the extend... |
| xrrest 24814 | The subspace topology indu... |
| xrrest2 24815 | The subspace topology indu... |
| xrsxmet 24816 | The metric on the extended... |
| xrsdsre 24817 | The metric on the extended... |
| xrsblre 24818 | Any ball of the metric of ... |
| xrsmopn 24819 | The metric on the extended... |
| zcld 24820 | The integers are a closed ... |
| recld2 24821 | The real numbers are a clo... |
| zcld2 24822 | The integers are a closed ... |
| zdis 24823 | The integers are a discret... |
| sszcld 24824 | Every subset of the intege... |
| reperflem 24825 | A subset of the real numbe... |
| reperf 24826 | The real numbers are a per... |
| cnperf 24827 | The complex numbers are a ... |
| iccntr 24828 | The interior of a closed i... |
| icccmplem1 24829 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24830 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24831 | Lemma for ~ icccmp . (Con... |
| icccmp 24832 | A closed interval in ` RR ... |
| reconnlem1 24833 | Lemma for ~ reconn . Conn... |
| reconnlem2 24834 | Lemma for ~ reconn . (Con... |
| reconn 24835 | A subset of the reals is c... |
| retopconn 24836 | Corollary of ~ reconn . T... |
| iccconn 24837 | A closed interval is conne... |
| opnreen 24838 | Every nonempty open set is... |
| rectbntr0 24839 | A countable subset of the ... |
| xrge0gsumle 24840 | A finite sum in the nonneg... |
| xrge0tsms 24841 | Any finite or infinite sum... |
| xrge0tsms2 24842 | Any finite or infinite sum... |
| metdcnlem 24843 | The metric function of a m... |
| xmetdcn2 24844 | The metric function of an ... |
| xmetdcn 24845 | The metric function of an ... |
| metdcn2 24846 | The metric function of a m... |
| metdcn 24847 | The metric function of a m... |
| msdcn 24848 | The metric function of a m... |
| cnmpt1ds 24849 | Continuity of the metric f... |
| cnmpt2ds 24850 | Continuity of the metric f... |
| nmcn 24851 | The norm of a normed group... |
| ngnmcncn 24852 | The norm of a normed group... |
| abscn 24853 | The absolute value functio... |
| metdsval 24854 | Value of the "distance to ... |
| metdsf 24855 | The distance from a point ... |
| metdsge 24856 | The distance from the poin... |
| metds0 24857 | If a point is in a set, it... |
| metdstri 24858 | A generalization of the tr... |
| metdsle 24859 | The distance from a point ... |
| metdsre 24860 | The distance from a point ... |
| metdseq0 24861 | The distance from a point ... |
| metdscnlem 24862 | Lemma for ~ metdscn . (Co... |
| metdscn 24863 | The function ` F ` which g... |
| metdscn2 24864 | The function ` F ` which g... |
| metnrmlem1a 24865 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24866 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24867 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24868 | Lemma for ~ metnrm . (Con... |
| metnrm 24869 | A metric space is normal. ... |
| metreg 24870 | A metric space is regular.... |
| addcnlem 24871 | Lemma for ~ addcn , ~ subc... |
| addcn 24872 | Complex number addition is... |
| subcn 24873 | Complex number subtraction... |
| mulcn 24874 | Complex number multiplicat... |
| divcnOLD 24875 | Obsolete version of ~ divc... |
| mpomulcn 24876 | Complex number multiplicat... |
| divcn 24877 | Complex number division is... |
| cnfldtgp 24878 | The complex numbers form a... |
| fsumcn 24879 | A finite sum of functions ... |
| fsum2cn 24880 | Version of ~ fsumcn for tw... |
| expcn 24881 | The power function on comp... |
| divccn 24882 | Division by a nonzero cons... |
| expcnOLD 24883 | Obsolete version of ~ expc... |
| divccnOLD 24884 | Obsolete version of ~ divc... |
| sqcn 24885 | The square function on com... |
| iitopon 24890 | The unit interval is a top... |
| iitop 24891 | The unit interval is a top... |
| iiuni 24892 | The base set of the unit i... |
| dfii2 24893 | Alternate definition of th... |
| dfii3 24894 | Alternate definition of th... |
| dfii4 24895 | Alternate definition of th... |
| dfii5 24896 | The unit interval expresse... |
| iicmp 24897 | The unit interval is compa... |
| iiconn 24898 | The unit interval is conne... |
| cncfval 24899 | The value of the continuou... |
| elcncf 24900 | Membership in the set of c... |
| elcncf2 24901 | Version of ~ elcncf with a... |
| cncfrss 24902 | Reverse closure of the con... |
| cncfrss2 24903 | Reverse closure of the con... |
| cncff 24904 | A continuous complex funct... |
| cncfi 24905 | Defining property of a con... |
| elcncf1di 24906 | Membership in the set of c... |
| elcncf1ii 24907 | Membership in the set of c... |
| rescncf 24908 | A continuous complex funct... |
| cncfcdm 24909 | Change the codomain of a c... |
| cncfss 24910 | The set of continuous func... |
| climcncf 24911 | Image of a limit under a c... |
| abscncf 24912 | Absolute value is continuo... |
| recncf 24913 | Real part is continuous. ... |
| imcncf 24914 | Imaginary part is continuo... |
| cjcncf 24915 | Complex conjugate is conti... |
| mulc1cncf 24916 | Multiplication by a consta... |
| divccncf 24917 | Division by a constant is ... |
| cncfco 24918 | The composition of two con... |
| cncfcompt2 24919 | Composition of continuous ... |
| cncfmet 24920 | Relate complex function co... |
| cncfcn 24921 | Relate complex function co... |
| cncfcn1 24922 | Relate complex function co... |
| cncfmptc 24923 | A constant function is a c... |
| cncfmptid 24924 | The identity function is a... |
| cncfmpt1f 24925 | Composition of continuous ... |
| cncfmpt2f 24926 | Composition of continuous ... |
| cncfmpt2ss 24927 | Composition of continuous ... |
| addccncf 24928 | Adding a constant is a con... |
| idcncf 24929 | The identity function is a... |
| sub1cncf 24930 | Subtracting a constant is ... |
| sub2cncf 24931 | Subtraction from a constan... |
| cdivcncf 24932 | Division with a constant n... |
| negcncf 24933 | The negative function is c... |
| negcncfOLD 24934 | Obsolete version of ~ negc... |
| negfcncf 24935 | The negative of a continuo... |
| abscncfALT 24936 | Absolute value is continuo... |
| cncfcnvcn 24937 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24938 | The power function on comp... |
| cnmptre 24939 | Lemma for ~ iirevcn and re... |
| cnmpopc 24940 | Piecewise definition of a ... |
| iirev 24941 | Reverse the unit interval.... |
| iirevcn 24942 | The reversion function is ... |
| iihalf1 24943 | Map the first half of ` II... |
| iihalf1cn 24944 | The first half function is... |
| iihalf1cnOLD 24945 | Obsolete version of ~ iiha... |
| iihalf2 24946 | Map the second half of ` I... |
| iihalf2cn 24947 | The second half function i... |
| iihalf2cnOLD 24948 | Obsolete version of ~ iiha... |
| elii1 24949 | Divide the unit interval i... |
| elii2 24950 | Divide the unit interval i... |
| iimulcl 24951 | The unit interval is close... |
| iimulcn 24952 | Multiplication is a contin... |
| iimulcnOLD 24953 | Obsolete version of ~ iimu... |
| icoopnst 24954 | A half-open interval start... |
| iocopnst 24955 | A half-open interval endin... |
| icchmeo 24956 | The natural bijection from... |
| icchmeoOLD 24957 | Obsolete version of ~ icch... |
| icopnfcnv 24958 | Define a bijection from ` ... |
| icopnfhmeo 24959 | The defined bijection from... |
| iccpnfcnv 24960 | Define a bijection from ` ... |
| iccpnfhmeo 24961 | The defined bijection from... |
| xrhmeo 24962 | The bijection from ` [ -u ... |
| xrhmph 24963 | The extended reals are hom... |
| xrcmp 24964 | The topology of the extend... |
| xrconn 24965 | The topology of the extend... |
| icccvx 24966 | A linear combination of tw... |
| oprpiece1res1 24967 | Restriction to the first p... |
| oprpiece1res2 24968 | Restriction to the second ... |
| cnrehmeo 24969 | The canonical bijection fr... |
| cnrehmeoOLD 24970 | Obsolete version of ~ cnre... |
| cnheiborlem 24971 | Lemma for ~ cnheibor . (C... |
| cnheibor 24972 | Heine-Borel theorem for co... |
| cnllycmp 24973 | The topology on the comple... |
| rellycmp 24974 | The topology on the reals ... |
| bndth 24975 | The Boundedness Theorem. ... |
| evth 24976 | The Extreme Value Theorem.... |
| evth2 24977 | The Extreme Value Theorem,... |
| lebnumlem1 24978 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24979 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24980 | Lemma for ~ lebnum . By t... |
| lebnum 24981 | The Lebesgue number lemma,... |
| xlebnum 24982 | Generalize ~ lebnum to ext... |
| lebnumii 24983 | Specialize the Lebesgue nu... |
| ishtpy 24989 | Membership in the class of... |
| htpycn 24990 | A homotopy is a continuous... |
| htpyi 24991 | A homotopy evaluated at it... |
| ishtpyd 24992 | Deduction for membership i... |
| htpycom 24993 | Given a homotopy from ` F ... |
| htpyid 24994 | A homotopy from a function... |
| htpyco1 24995 | Compose a homotopy with a ... |
| htpyco2 24996 | Compose a homotopy with a ... |
| htpycc 24997 | Concatenate two homotopies... |
| isphtpy 24998 | Membership in the class of... |
| phtpyhtpy 24999 | A path homotopy is a homot... |
| phtpycn 25000 | A path homotopy is a conti... |
| phtpyi 25001 | Membership in the class of... |
| phtpy01 25002 | Two path-homotopic paths h... |
| isphtpyd 25003 | Deduction for membership i... |
| isphtpy2d 25004 | Deduction for membership i... |
| phtpycom 25005 | Given a homotopy from ` F ... |
| phtpyid 25006 | A homotopy from a path to ... |
| phtpyco2 25007 | Compose a path homotopy wi... |
| phtpycc 25008 | Concatenate two path homot... |
| phtpcrel 25010 | The path homotopy relation... |
| isphtpc 25011 | The relation "is path homo... |
| phtpcer 25012 | Path homotopy is an equiva... |
| phtpc01 25013 | Path homotopic paths have ... |
| reparphti 25014 | Lemma for ~ reparpht . (C... |
| reparphtiOLD 25015 | Obsolete version of ~ repa... |
| reparpht 25016 | Reparametrization lemma. ... |
| phtpcco2 25017 | Compose a path homotopy wi... |
| pcofval 25028 | The value of the path conc... |
| pcoval 25029 | The concatenation of two p... |
| pcovalg 25030 | Evaluate the concatenation... |
| pcoval1 25031 | Evaluate the concatenation... |
| pco0 25032 | The starting point of a pa... |
| pco1 25033 | The ending point of a path... |
| pcoval2 25034 | Evaluate the concatenation... |
| pcocn 25035 | The concatenation of two p... |
| copco 25036 | The composition of a conca... |
| pcohtpylem 25037 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 25038 | Homotopy invariance of pat... |
| pcoptcl 25039 | A constant function is a p... |
| pcopt 25040 | Concatenation with a point... |
| pcopt2 25041 | Concatenation with a point... |
| pcoass 25042 | Order of concatenation doe... |
| pcorevcl 25043 | Closure for a reversed pat... |
| pcorevlem 25044 | Lemma for ~ pcorev . Prov... |
| pcorev 25045 | Concatenation with the rev... |
| pcorev2 25046 | Concatenation with the rev... |
| pcophtb 25047 | The path homotopy equivale... |
| om1val 25048 | The definition of the loop... |
| om1bas 25049 | The base set of the loop s... |
| om1elbas 25050 | Elementhood in the base se... |
| om1addcl 25051 | Closure of the group opera... |
| om1plusg 25052 | The group operation (which... |
| om1tset 25053 | The topology of the loop s... |
| om1opn 25054 | The topology of the loop s... |
| pi1val 25055 | The definition of the fund... |
| pi1bas 25056 | The base set of the fundam... |
| pi1blem 25057 | Lemma for ~ pi1buni . (Co... |
| pi1buni 25058 | Another way to write the l... |
| pi1bas2 25059 | The base set of the fundam... |
| pi1eluni 25060 | Elementhood in the base se... |
| pi1bas3 25061 | The base set of the fundam... |
| pi1cpbl 25062 | The group operation, loop ... |
| elpi1 25063 | The elements of the fundam... |
| elpi1i 25064 | The elements of the fundam... |
| pi1addf 25065 | The group operation of ` p... |
| pi1addval 25066 | The concatenation of two p... |
| pi1grplem 25067 | Lemma for ~ pi1grp . (Con... |
| pi1grp 25068 | The fundamental group is a... |
| pi1id 25069 | The identity element of th... |
| pi1inv 25070 | An inverse in the fundamen... |
| pi1xfrf 25071 | Functionality of the loop ... |
| pi1xfrval 25072 | The value of the loop tran... |
| pi1xfr 25073 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25074 | Given a path ` F ` between... |
| pi1xfrcnv 25075 | Given a path ` F ` between... |
| pi1xfrgim 25076 | The mapping ` G ` between ... |
| pi1cof 25077 | Functionality of the loop ... |
| pi1coval 25078 | The value of the loop tran... |
| pi1coghm 25079 | The mapping ` G ` between ... |
| isclm 25082 | A subcomplex module is a l... |
| clmsca 25083 | The ring of scalars ` F ` ... |
| clmsubrg 25084 | The base set of the ring o... |
| clmlmod 25085 | A subcomplex module is a l... |
| clmgrp 25086 | A subcomplex module is an ... |
| clmabl 25087 | A subcomplex module is an ... |
| clmring 25088 | The scalar ring of a subco... |
| clmfgrp 25089 | The scalar ring of a subco... |
| clm0 25090 | The zero of the scalar rin... |
| clm1 25091 | The identity of the scalar... |
| clmadd 25092 | The addition of the scalar... |
| clmmul 25093 | The multiplication of the ... |
| clmcj 25094 | The conjugation of the sca... |
| isclmi 25095 | Reverse direction of ~ isc... |
| clmzss 25096 | The scalar ring of a subco... |
| clmsscn 25097 | The scalar ring of a subco... |
| clmsub 25098 | Subtraction in the scalar ... |
| clmneg 25099 | Negation in the scalar rin... |
| clmneg1 25100 | Minus one is in the scalar... |
| clmabs 25101 | Norm in the scalar ring of... |
| clmacl 25102 | Closure of ring addition f... |
| clmmcl 25103 | Closure of ring multiplica... |
| clmsubcl 25104 | Closure of ring subtractio... |
| lmhmclm 25105 | The domain of a linear ope... |
| clmvscl 25106 | Closure of scalar product ... |
| clmvsass 25107 | Associative law for scalar... |
| clmvscom 25108 | Commutative law for the sc... |
| clmvsdir 25109 | Distributive law for scala... |
| clmvsdi 25110 | Distributive law for scala... |
| clmvs1 25111 | Scalar product with ring u... |
| clmvs2 25112 | A vector plus itself is tw... |
| clm0vs 25113 | Zero times a vector is the... |
| clmopfne 25114 | The (functionalized) opera... |
| isclmp 25115 | The predicate "is a subcom... |
| isclmi0 25116 | Properties that determine ... |
| clmvneg1 25117 | Minus 1 times a vector is ... |
| clmvsneg 25118 | Multiplication of a vector... |
| clmmulg 25119 | The group multiple functio... |
| clmsubdir 25120 | Scalar multiplication dist... |
| clmpm1dir 25121 | Subtractive distributive l... |
| clmnegneg 25122 | Double negative of a vecto... |
| clmnegsubdi2 25123 | Distribution of negative o... |
| clmsub4 25124 | Rearrangement of 4 terms i... |
| clmvsrinv 25125 | A vector minus itself. (C... |
| clmvslinv 25126 | Minus a vector plus itself... |
| clmvsubval 25127 | Value of vector subtractio... |
| clmvsubval2 25128 | Value of vector subtractio... |
| clmvz 25129 | Two ways to express the ne... |
| zlmclm 25130 | The ` ZZ ` -module operati... |
| clmzlmvsca 25131 | The scalar product of a su... |
| nmoleub2lem 25132 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25133 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25134 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25135 | The operator norm is the s... |
| nmoleub2b 25136 | The operator norm is the s... |
| nmoleub3 25137 | The operator norm is the s... |
| nmhmcn 25138 | A linear operator over a n... |
| cmodscexp 25139 | The powers of ` _i ` belon... |
| cmodscmulexp 25140 | The scalar product of a ve... |
| cvslvec 25143 | A subcomplex vector space ... |
| cvsclm 25144 | A subcomplex vector space ... |
| iscvs 25145 | A subcomplex vector space ... |
| iscvsp 25146 | The predicate "is a subcom... |
| iscvsi 25147 | Properties that determine ... |
| cvsi 25148 | The properties of a subcom... |
| cvsunit 25149 | Unit group of the scalar r... |
| cvsdiv 25150 | Division of the scalar rin... |
| cvsdivcl 25151 | The scalar field of a subc... |
| cvsmuleqdivd 25152 | An equality involving rati... |
| cvsdiveqd 25153 | An equality involving rati... |
| cnlmodlem1 25154 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25155 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25156 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25157 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25158 | The set of complex numbers... |
| cnstrcvs 25159 | The set of complex numbers... |
| cnrbas 25160 | The set of complex numbers... |
| cnrlmod 25161 | The complex left module of... |
| cnrlvec 25162 | The complex left module of... |
| cncvs 25163 | The complex left module of... |
| recvs 25164 | The field of the real numb... |
| recvsOLD 25165 | Obsolete version of ~ recv... |
| qcvs 25166 | The field of rational numb... |
| zclmncvs 25167 | The ring of integers as le... |
| isncvsngp 25168 | A normed subcomplex vector... |
| isncvsngpd 25169 | Properties that determine ... |
| ncvsi 25170 | The properties of a normed... |
| ncvsprp 25171 | Proportionality property o... |
| ncvsge0 25172 | The norm of a scalar produ... |
| ncvsm1 25173 | The norm of the opposite o... |
| ncvsdif 25174 | The norm of the difference... |
| ncvspi 25175 | The norm of a vector plus ... |
| ncvs1 25176 | From any nonzero vector of... |
| cnrnvc 25177 | The module of complex numb... |
| cnncvs 25178 | The module of complex numb... |
| cnnm 25179 | The norm of the normed sub... |
| ncvspds 25180 | Value of the distance func... |
| cnindmet 25181 | The metric induced on the ... |
| cnncvsaddassdemo 25182 | Derive the associative law... |
| cnncvsmulassdemo 25183 | Derive the associative law... |
| cnncvsabsnegdemo 25184 | Derive the absolute value ... |
| iscph 25189 | A subcomplex pre-Hilbert s... |
| cphphl 25190 | A subcomplex pre-Hilbert s... |
| cphnlm 25191 | A subcomplex pre-Hilbert s... |
| cphngp 25192 | A subcomplex pre-Hilbert s... |
| cphlmod 25193 | A subcomplex pre-Hilbert s... |
| cphlvec 25194 | A subcomplex pre-Hilbert s... |
| cphnvc 25195 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25196 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25197 | Lemma for ~ cphreccl . (C... |
| cphsca 25198 | A subcomplex pre-Hilbert s... |
| cphsubrg 25199 | The scalar field of a subc... |
| cphreccl 25200 | The scalar field of a subc... |
| cphdivcl 25201 | The scalar field of a subc... |
| cphcjcl 25202 | The scalar field of a subc... |
| cphsqrtcl 25203 | The scalar field of a subc... |
| cphabscl 25204 | The scalar field of a subc... |
| cphsqrtcl2 25205 | The scalar field of a subc... |
| cphsqrtcl3 25206 | If the scalar field of a s... |
| cphqss 25207 | The scalar field of a subc... |
| cphclm 25208 | A subcomplex pre-Hilbert s... |
| cphnmvs 25209 | Norm of a scalar product. ... |
| cphipcl 25210 | An inner product is a memb... |
| cphnmfval 25211 | The value of the norm in a... |
| cphnm 25212 | The square of the norm is ... |
| nmsq 25213 | The square of the norm is ... |
| cphnmf 25214 | The norm of a vector is a ... |
| cphnmcl 25215 | The norm of a vector is a ... |
| reipcl 25216 | An inner product of an ele... |
| ipge0 25217 | The inner product in a sub... |
| cphipcj 25218 | Conjugate of an inner prod... |
| cphipipcj 25219 | An inner product times its... |
| cphorthcom 25220 | Orthogonality (meaning inn... |
| cphip0l 25221 | Inner product with a zero ... |
| cphip0r 25222 | Inner product with a zero ... |
| cphipeq0 25223 | The inner product of a vec... |
| cphdir 25224 | Distributive law for inner... |
| cphdi 25225 | Distributive law for inner... |
| cph2di 25226 | Distributive law for inner... |
| cphsubdir 25227 | Distributive law for inner... |
| cphsubdi 25228 | Distributive law for inner... |
| cph2subdi 25229 | Distributive law for inner... |
| cphass 25230 | Associative law for inner ... |
| cphassr 25231 | "Associative" law for seco... |
| cph2ass 25232 | Move scalar multiplication... |
| cphassi 25233 | Associative law for the fi... |
| cphassir 25234 | "Associative" law for the ... |
| cphpyth 25235 | The pythagorean theorem fo... |
| tcphex 25236 | Lemma for ~ tcphbas and si... |
| tcphval 25237 | Define a function to augme... |
| tcphbas 25238 | The base set of a subcompl... |
| tchplusg 25239 | The addition operation of ... |
| tcphsub 25240 | The subtraction operation ... |
| tcphmulr 25241 | The ring operation of a su... |
| tcphsca 25242 | The scalar field of a subc... |
| tcphvsca 25243 | The scalar multiplication ... |
| tcphip 25244 | The inner product of a sub... |
| tcphtopn 25245 | The topology of a subcompl... |
| tcphphl 25246 | Augmentation of a subcompl... |
| tchnmfval 25247 | The norm of a subcomplex p... |
| tcphnmval 25248 | The norm of a subcomplex p... |
| cphtcphnm 25249 | The norm of a norm-augment... |
| tcphds 25250 | The distance of a pre-Hilb... |
| phclm 25251 | A pre-Hilbert space whose ... |
| tcphcphlem3 25252 | Lemma for ~ tcphcph : real... |
| ipcau2 25253 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25254 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25255 | Lemma for ~ tcphcph : homo... |
| tcphcph 25256 | The standard definition of... |
| ipcau 25257 | The Cauchy-Schwarz inequal... |
| nmparlem 25258 | Lemma for ~ nmpar . (Cont... |
| nmpar 25259 | A subcomplex pre-Hilbert s... |
| cphipval2 25260 | Value of the inner product... |
| 4cphipval2 25261 | Four times the inner produ... |
| cphipval 25262 | Value of the inner product... |
| ipcnlem2 25263 | The inner product operatio... |
| ipcnlem1 25264 | The inner product operatio... |
| ipcn 25265 | The inner product operatio... |
| cnmpt1ip 25266 | Continuity of inner produc... |
| cnmpt2ip 25267 | Continuity of inner produc... |
| csscld 25268 | A "closed subspace" in a s... |
| clsocv 25269 | The orthogonal complement ... |
| cphsscph 25270 | A subspace of a subcomplex... |
| lmmbr 25277 | Express the binary relatio... |
| lmmbr2 25278 | Express the binary relatio... |
| lmmbr3 25279 | Express the binary relatio... |
| lmmcvg 25280 | Convergence property of a ... |
| lmmbrf 25281 | Express the binary relatio... |
| lmnn 25282 | A condition that implies c... |
| cfilfval 25283 | The set of Cauchy filters ... |
| iscfil 25284 | The property of being a Ca... |
| iscfil2 25285 | The property of being a Ca... |
| cfilfil 25286 | A Cauchy filter is a filte... |
| cfili 25287 | Property of a Cauchy filte... |
| cfil3i 25288 | A Cauchy filter contains b... |
| cfilss 25289 | A filter finer than a Cauc... |
| fgcfil 25290 | The Cauchy filter conditio... |
| fmcfil 25291 | The Cauchy filter conditio... |
| iscfil3 25292 | A filter is Cauchy iff it ... |
| cfilfcls 25293 | Similar to ultrafilters ( ... |
| caufval 25294 | The set of Cauchy sequence... |
| iscau 25295 | Express the property " ` F... |
| iscau2 25296 | Express the property " ` F... |
| iscau3 25297 | Express the Cauchy sequenc... |
| iscau4 25298 | Express the property " ` F... |
| iscauf 25299 | Express the property " ` F... |
| caun0 25300 | A metric with a Cauchy seq... |
| caufpm 25301 | Inclusion of a Cauchy sequ... |
| caucfil 25302 | A Cauchy sequence predicat... |
| iscmet 25303 | The property " ` D ` is a ... |
| cmetcvg 25304 | The convergence of a Cauch... |
| cmetmet 25305 | A complete metric space is... |
| cmetmeti 25306 | A complete metric space is... |
| cmetcaulem 25307 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25308 | The convergence of a Cauch... |
| iscmet3lem3 25309 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25310 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25311 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25312 | The property " ` D ` is a ... |
| iscmet2 25313 | A metric ` D ` is complete... |
| cfilresi 25314 | A Cauchy filter on a metri... |
| cfilres 25315 | Cauchy filter on a metric ... |
| caussi 25316 | Cauchy sequence on a metri... |
| causs 25317 | Cauchy sequence on a metri... |
| equivcfil 25318 | If the metric ` D ` is "st... |
| equivcau 25319 | If the metric ` D ` is "st... |
| lmle 25320 | If the distance from each ... |
| nglmle 25321 | If the norm of each member... |
| lmclim 25322 | Relate a limit on the metr... |
| lmclimf 25323 | Relate a limit on the metr... |
| metelcls 25324 | A point belongs to the clo... |
| metcld 25325 | A subset of a metric space... |
| metcld2 25326 | A subset of a metric space... |
| caubl 25327 | Sufficient condition to en... |
| caublcls 25328 | The convergent point of a ... |
| metcnp4 25329 | Two ways to say a mapping ... |
| metcn4 25330 | Two ways to say a mapping ... |
| iscmet3i 25331 | Properties that determine ... |
| lmcau 25332 | Every convergent sequence ... |
| flimcfil 25333 | Every convergent filter in... |
| metsscmetcld 25334 | A complete subspace of a m... |
| cmetss 25335 | A subspace of a complete m... |
| equivcmet 25336 | If two metrics are strongl... |
| relcmpcmet 25337 | If ` D ` is a metric space... |
| cmpcmet 25338 | A compact metric space is ... |
| cfilucfil3 25339 | Given a metric ` D ` and a... |
| cfilucfil4 25340 | Given a metric ` D ` and a... |
| cncmet 25341 | The set of complex numbers... |
| recmet 25342 | The real numbers are a com... |
| bcthlem1 25343 | Lemma for ~ bcth . Substi... |
| bcthlem2 25344 | Lemma for ~ bcth . The ba... |
| bcthlem3 25345 | Lemma for ~ bcth . The li... |
| bcthlem4 25346 | Lemma for ~ bcth . Given ... |
| bcthlem5 25347 | Lemma for ~ bcth . The pr... |
| bcth 25348 | Baire's Category Theorem. ... |
| bcth2 25349 | Baire's Category Theorem, ... |
| bcth3 25350 | Baire's Category Theorem, ... |
| isbn 25357 | A Banach space is a normed... |
| bnsca 25358 | The scalar field of a Bana... |
| bnnvc 25359 | A Banach space is a normed... |
| bnnlm 25360 | A Banach space is a normed... |
| bnngp 25361 | A Banach space is a normed... |
| bnlmod 25362 | A Banach space is a left m... |
| bncms 25363 | A Banach space is a comple... |
| iscms 25364 | A complete metric space is... |
| cmscmet 25365 | The induced metric on a co... |
| bncmet 25366 | The induced metric on Bana... |
| cmsms 25367 | A complete metric space is... |
| cmspropd 25368 | Property deduction for a c... |
| cmssmscld 25369 | The restriction of a metri... |
| cmsss 25370 | The restriction of a compl... |
| lssbn 25371 | A subspace of a Banach spa... |
| cmetcusp1 25372 | If the uniform set of a co... |
| cmetcusp 25373 | The uniform space generate... |
| cncms 25374 | The field of complex numbe... |
| cnflduss 25375 | The uniform structure of t... |
| cnfldcusp 25376 | The field of complex numbe... |
| resscdrg 25377 | The real numbers are a sub... |
| cncdrg 25378 | The only complete subfield... |
| srabn 25379 | The subring algebra over a... |
| rlmbn 25380 | The ring module over a com... |
| ishl 25381 | The predicate "is a subcom... |
| hlbn 25382 | Every subcomplex Hilbert s... |
| hlcph 25383 | Every subcomplex Hilbert s... |
| hlphl 25384 | Every subcomplex Hilbert s... |
| hlcms 25385 | Every subcomplex Hilbert s... |
| hlprlem 25386 | Lemma for ~ hlpr . (Contr... |
| hlress 25387 | The scalar field of a subc... |
| hlpr 25388 | The scalar field of a subc... |
| ishl2 25389 | A Hilbert space is a compl... |
| cphssphl 25390 | A Banach subspace of a sub... |
| cmslssbn 25391 | A complete linear subspace... |
| cmscsscms 25392 | A closed subspace of a com... |
| bncssbn 25393 | A closed subspace of a Ban... |
| cssbn 25394 | A complete subspace of a n... |
| csschl 25395 | A complete subspace of a c... |
| cmslsschl 25396 | A complete linear subspace... |
| chlcsschl 25397 | A closed subspace of a sub... |
| retopn 25398 | The topology of the real n... |
| recms 25399 | The real numbers form a co... |
| reust 25400 | The Uniform structure of t... |
| recusp 25401 | The real numbers form a co... |
| rrxval 25406 | Value of the generalized E... |
| rrxbase 25407 | The base of the generalize... |
| rrxprds 25408 | Expand the definition of t... |
| rrxip 25409 | The inner product of the g... |
| rrxnm 25410 | The norm of the generalize... |
| rrxcph 25411 | Generalized Euclidean real... |
| rrxds 25412 | The distance over generali... |
| rrxvsca 25413 | The scalar product over ge... |
| rrxplusgvscavalb 25414 | The result of the addition... |
| rrxsca 25415 | The field of real numbers ... |
| rrx0 25416 | The zero ("origin") in a g... |
| rrx0el 25417 | The zero ("origin") in a g... |
| csbren 25418 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25419 | Triangle inequality in R^n... |
| rrxf 25420 | Euclidean vectors as funct... |
| rrxfsupp 25421 | Euclidean vectors are of f... |
| rrxsuppss 25422 | Support of Euclidean vecto... |
| rrxmvallem 25423 | Support of the function us... |
| rrxmval 25424 | The value of the Euclidean... |
| rrxmfval 25425 | The value of the Euclidean... |
| rrxmetlem 25426 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25427 | Euclidean space is a metri... |
| rrxdstprj1 25428 | The distance between two p... |
| rrxbasefi 25429 | The base of the generalize... |
| rrxdsfi 25430 | The distance over generali... |
| rrxmetfi 25431 | Euclidean space is a metri... |
| rrxdsfival 25432 | The value of the Euclidean... |
| ehlval 25433 | Value of the Euclidean spa... |
| ehlbase 25434 | The base of the Euclidean ... |
| ehl0base 25435 | The base of the Euclidean ... |
| ehl0 25436 | The Euclidean space of dim... |
| ehleudis 25437 | The Euclidean distance fun... |
| ehleudisval 25438 | The value of the Euclidean... |
| ehl1eudis 25439 | The Euclidean distance fun... |
| ehl1eudisval 25440 | The value of the Euclidean... |
| ehl2eudis 25441 | The Euclidean distance fun... |
| ehl2eudisval 25442 | The value of the Euclidean... |
| minveclem1 25443 | Lemma for ~ minvec . The ... |
| minveclem4c 25444 | Lemma for ~ minvec . The ... |
| minveclem2 25445 | Lemma for ~ minvec . Any ... |
| minveclem3a 25446 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25447 | Lemma for ~ minvec . The ... |
| minveclem3 25448 | Lemma for ~ minvec . The ... |
| minveclem4a 25449 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25450 | Lemma for ~ minvec . The ... |
| minveclem4 25451 | Lemma for ~ minvec . The ... |
| minveclem5 25452 | Lemma for ~ minvec . Disc... |
| minveclem6 25453 | Lemma for ~ minvec . Any ... |
| minveclem7 25454 | Lemma for ~ minvec . Sinc... |
| minvec 25455 | Minimizing vector theorem,... |
| pjthlem1 25456 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25457 | Lemma for ~ pjth . (Contr... |
| pjth 25458 | Projection Theorem: Any H... |
| pjth2 25459 | Projection Theorem with ab... |
| cldcss 25460 | Corollary of the Projectio... |
| cldcss2 25461 | Corollary of the Projectio... |
| hlhil 25462 | Corollary of the Projectio... |
| addcncf 25463 | The addition of two contin... |
| subcncf 25464 | The subtraction of two con... |
| mulcncf 25465 | The multiplication of two ... |
| mulcncfOLD 25466 | Obsolete version of ~ mulc... |
| divcncf 25467 | The quotient of two contin... |
| pmltpclem1 25468 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25469 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25470 | Any function on the reals ... |
| ivthlem1 25471 | Lemma for ~ ivth . The se... |
| ivthlem2 25472 | Lemma for ~ ivth . Show t... |
| ivthlem3 25473 | Lemma for ~ ivth , the int... |
| ivth 25474 | The intermediate value the... |
| ivth2 25475 | The intermediate value the... |
| ivthle 25476 | The intermediate value the... |
| ivthle2 25477 | The intermediate value the... |
| ivthicc 25478 | The interval between any t... |
| evthicc 25479 | Specialization of the Extr... |
| evthicc2 25480 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25481 | A continuous function on a... |
| ovolfcl 25486 | Closure for the interval e... |
| ovolfioo 25487 | Unpack the interval coveri... |
| ovolficc 25488 | Unpack the interval coveri... |
| ovolficcss 25489 | Any (closed) interval cove... |
| ovolfsval 25490 | The value of the interval ... |
| ovolfsf 25491 | Closure for the interval l... |
| ovolsf 25492 | Closure for the partial su... |
| ovolval 25493 | The value of the outer mea... |
| elovolmlem 25494 | Lemma for ~ elovolm and re... |
| elovolm 25495 | Elementhood in the set ` M... |
| elovolmr 25496 | Sufficient condition for e... |
| ovolmge0 25497 | The set ` M ` is composed ... |
| ovolcl 25498 | The volume of a set is an ... |
| ovollb 25499 | The outer volume is a lowe... |
| ovolgelb 25500 | The outer volume is the gr... |
| ovolge0 25501 | The volume of a set is alw... |
| ovolf 25502 | The domain and codomain of... |
| ovollecl 25503 | If an outer volume is boun... |
| ovolsslem 25504 | Lemma for ~ ovolss . (Con... |
| ovolss 25505 | The volume of a set is mon... |
| ovolsscl 25506 | If a set is contained in a... |
| ovolssnul 25507 | A subset of a nullset is n... |
| ovollb2lem 25508 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25509 | It is often more convenien... |
| ovolctb 25510 | The volume of a denumerabl... |
| ovolq 25511 | The rational numbers have ... |
| ovolctb2 25512 | The volume of a countable ... |
| ovol0 25513 | The empty set has 0 outer ... |
| ovolfi 25514 | A finite set has 0 outer L... |
| ovolsn 25515 | A singleton has 0 outer Le... |
| ovolunlem1a 25516 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25517 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25518 | Lemma for ~ ovolun . (Con... |
| ovolun 25519 | The Lebesgue outer measure... |
| ovolunnul 25520 | Adding a nullset does not ... |
| ovolfiniun 25521 | The Lebesgue outer measure... |
| ovoliunlem1 25522 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25523 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25524 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25525 | The Lebesgue outer measure... |
| ovoliun2 25526 | The Lebesgue outer measure... |
| ovoliunnul 25527 | A countable union of nulls... |
| shft2rab 25528 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25529 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25530 | Lemma for ~ ovolshft . (C... |
| ovolshft 25531 | The Lebesgue outer measure... |
| sca2rab 25532 | If ` B ` is a scale of ` A... |
| ovolscalem1 25533 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25534 | Lemma for ~ ovolshft . (C... |
| ovolsca 25535 | The Lebesgue outer measure... |
| ovolicc1 25536 | The measure of a closed in... |
| ovolicc2lem1 25537 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25538 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25539 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25540 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25541 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25542 | The measure of a closed in... |
| ovolicc 25543 | The measure of a closed in... |
| ovolicopnf 25544 | The measure of a right-unb... |
| ovolre 25545 | The measure of the real nu... |
| ismbl 25546 | The predicate " ` A ` is L... |
| ismbl2 25547 | From ~ ovolun , it suffice... |
| volres 25548 | A self-referencing abbrevi... |
| volf 25549 | The domain and codomain of... |
| mblvol 25550 | The volume of a measurable... |
| mblss 25551 | A measurable set is a subs... |
| mblsplit 25552 | The defining property of m... |
| volss 25553 | The Lebesgue measure is mo... |
| cmmbl 25554 | The complement of a measur... |
| nulmbl 25555 | A nullset is measurable. ... |
| nulmbl2 25556 | A set of outer measure zer... |
| unmbl 25557 | A union of measurable sets... |
| shftmbl 25558 | A shift of a measurable se... |
| 0mbl 25559 | The empty set is measurabl... |
| rembl 25560 | The set of all real number... |
| unidmvol 25561 | The union of the Lebesgue ... |
| inmbl 25562 | An intersection of measura... |
| difmbl 25563 | A difference of measurable... |
| finiunmbl 25564 | A finite union of measurab... |
| volun 25565 | The Lebesgue measure funct... |
| volinun 25566 | Addition of non-disjoint s... |
| volfiniun 25567 | The volume of a disjoint f... |
| iundisj 25568 | Rewrite a countable union ... |
| iundisj2 25569 | A disjoint union is disjoi... |
| voliunlem1 25570 | Lemma for ~ voliun . (Con... |
| voliunlem2 25571 | Lemma for ~ voliun . (Con... |
| voliunlem3 25572 | Lemma for ~ voliun . (Con... |
| iunmbl 25573 | The measurable sets are cl... |
| voliun 25574 | The Lebesgue measure funct... |
| volsuplem 25575 | Lemma for ~ volsup . (Con... |
| volsup 25576 | The volume of the limit of... |
| iunmbl2 25577 | The measurable sets are cl... |
| ioombl1lem1 25578 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25579 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25580 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25581 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25582 | An open right-unbounded in... |
| icombl1 25583 | A closed unbounded-above i... |
| icombl 25584 | A closed-below, open-above... |
| ioombl 25585 | An open real interval is m... |
| iccmbl 25586 | A closed real interval is ... |
| iccvolcl 25587 | A closed real interval has... |
| ovolioo 25588 | The measure of an open int... |
| volioo 25589 | The measure of an open int... |
| ioovolcl 25590 | An open real interval has ... |
| ovolfs2 25591 | Alternative expression for... |
| ioorcl2 25592 | An open interval with fini... |
| ioorf 25593 | Define a function from ope... |
| ioorval 25594 | Define a function from ope... |
| ioorinv2 25595 | The function ` F ` is an "... |
| ioorinv 25596 | The function ` F ` is an "... |
| ioorcl 25597 | The function ` F ` does no... |
| uniiccdif 25598 | A union of closed interval... |
| uniioovol 25599 | A disjoint union of open i... |
| uniiccvol 25600 | An almost-disjoint union o... |
| uniioombllem1 25601 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25602 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25603 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25604 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25605 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25606 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25607 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25608 | Lemma for ~ uniioombl . (... |
| uniioombl 25609 | A disjoint union of open i... |
| uniiccmbl 25610 | An almost-disjoint union o... |
| dyadf 25611 | The function ` F ` returns... |
| dyadval 25612 | Value of the dyadic ration... |
| dyadovol 25613 | Volume of a dyadic rationa... |
| dyadss 25614 | Two closed dyadic rational... |
| dyaddisjlem 25615 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25616 | Two closed dyadic rational... |
| dyadmaxlem 25617 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25618 | Any nonempty set of dyadic... |
| dyadmbllem 25619 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25620 | Any union of dyadic ration... |
| opnmbllem 25621 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25622 | All open sets are measurab... |
| opnmblALT 25623 | All open sets are measurab... |
| subopnmbl 25624 | Sets which are open in a m... |
| volsup2 25625 | The volume of ` A ` is the... |
| volcn 25626 | The function formed by res... |
| volivth 25627 | The Intermediate Value The... |
| vitalilem1 25628 | Lemma for ~ vitali . (Con... |
| vitalilem2 25629 | Lemma for ~ vitali . (Con... |
| vitalilem3 25630 | Lemma for ~ vitali . (Con... |
| vitalilem4 25631 | Lemma for ~ vitali . (Con... |
| vitalilem5 25632 | Lemma for ~ vitali . (Con... |
| vitali 25633 | If the reals can be well-o... |
| ismbf1 25644 | The predicate " ` F ` is a... |
| mbff 25645 | A measurable function is a... |
| mbfdm 25646 | The domain of a measurable... |
| mbfconstlem 25647 | Lemma for ~ mbfconst and r... |
| ismbf 25648 | The predicate " ` F ` is a... |
| ismbfcn 25649 | A complex function is meas... |
| mbfima 25650 | Definitional property of a... |
| mbfimaicc 25651 | The preimage of any closed... |
| mbfimasn 25652 | The preimage of a point un... |
| mbfconst 25653 | A constant function is mea... |
| mbf0 25654 | The empty function is meas... |
| mbfid 25655 | The identity function is m... |
| mbfmptcl 25656 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25657 | The domain of a measurable... |
| ismbfcn2 25658 | A complex function is meas... |
| ismbfd 25659 | Deduction to prove measura... |
| ismbf2d 25660 | Deduction to prove measura... |
| mbfeqalem1 25661 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25662 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25663 | If two functions are equal... |
| mbfres 25664 | The restriction of a measu... |
| mbfres2 25665 | Measurability of a piecewi... |
| mbfss 25666 | Change the domain of a mea... |
| mbfmulc2lem 25667 | Multiplication by a consta... |
| mbfmulc2re 25668 | Multiplication by a consta... |
| mbfmax 25669 | The maximum of two functio... |
| mbfneg 25670 | The negative of a measurab... |
| mbfpos 25671 | The positive part of a mea... |
| mbfposr 25672 | Converse to ~ mbfpos . (C... |
| mbfposb 25673 | A function is measurable i... |
| ismbf3d 25674 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25675 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25676 | The preimage of any open s... |
| mbfimaopn2 25677 | The preimage of any set op... |
| cncombf 25678 | The composition of a conti... |
| cnmbf 25679 | A continuous function is m... |
| mbfaddlem 25680 | The sum of two measurable ... |
| mbfadd 25681 | The sum of two measurable ... |
| mbfsub 25682 | The difference of two meas... |
| mbfmulc2 25683 | A complex constant times a... |
| mbfsup 25684 | The supremum of a sequence... |
| mbfinf 25685 | The infimum of a sequence ... |
| mbflimsup 25686 | The limit supremum of a se... |
| mbflimlem 25687 | The pointwise limit of a s... |
| mbflim 25688 | The pointwise limit of a s... |
| 0pval 25691 | The zero function evaluate... |
| 0plef 25692 | Two ways to say that the f... |
| 0pledm 25693 | Adjust the domain of the l... |
| isi1f 25694 | The predicate " ` F ` is a... |
| i1fmbf 25695 | Simple functions are measu... |
| i1ff 25696 | A simple function is a fun... |
| i1frn 25697 | A simple function has fini... |
| i1fima 25698 | Any preimage of a simple f... |
| i1fima2 25699 | Any preimage of a simple f... |
| i1fima2sn 25700 | Preimage of a singleton. ... |
| i1fd 25701 | A simplified set of assump... |
| i1f0rn 25702 | Any simple function takes ... |
| itg1val 25703 | The value of the integral ... |
| itg1val2 25704 | The value of the integral ... |
| itg1cl 25705 | Closure of the integral on... |
| itg1ge0 25706 | Closure of the integral on... |
| i1f0 25707 | The zero function is simpl... |
| itg10 25708 | The zero function has zero... |
| i1f1lem 25709 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25710 | Base case simple functions... |
| itg11 25711 | The integral of an indicat... |
| itg1addlem1 25712 | Decompose a preimage, whic... |
| i1faddlem 25713 | Decompose the preimage of ... |
| i1fmullem 25714 | Decompose the preimage of ... |
| i1fadd 25715 | The sum of two simple func... |
| i1fmul 25716 | The pointwise product of t... |
| itg1addlem2 25717 | Lemma for ~ itg1add . The... |
| itg1addlem3 25718 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25719 | Lemma for ~ itg1add . (Co... |
| itg1addlem4OLD 25720 | Obsolete version of ~ itg1... |
| itg1addlem5 25721 | Lemma for ~ itg1add . (Co... |
| itg1add 25722 | The integral of a sum of s... |
| i1fmulclem 25723 | Decompose the preimage of ... |
| i1fmulc 25724 | A nonnegative constant tim... |
| itg1mulc 25725 | The integral of a constant... |
| i1fres 25726 | The "restriction" of a sim... |
| i1fpos 25727 | The positive part of a sim... |
| i1fposd 25728 | Deduction form of ~ i1fpos... |
| i1fsub 25729 | The difference of two simp... |
| itg1sub 25730 | The integral of a differen... |
| itg10a 25731 | The integral of a simple f... |
| itg1ge0a 25732 | The integral of an almost ... |
| itg1lea 25733 | Approximate version of ~ i... |
| itg1le 25734 | If one simple function dom... |
| itg1climres 25735 | Restricting the simple fun... |
| mbfi1fseqlem1 25736 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25737 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25738 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25739 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25740 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25741 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25742 | A characterization of meas... |
| mbfi1flimlem 25743 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25744 | Any real measurable functi... |
| mbfmullem2 25745 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25746 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25747 | The product of two measura... |
| itg2lcl 25748 | The set of lower sums is a... |
| itg2val 25749 | Value of the integral on n... |
| itg2l 25750 | Elementhood in the set ` L... |
| itg2lr 25751 | Sufficient condition for e... |
| xrge0f 25752 | A real function is a nonne... |
| itg2cl 25753 | The integral of a nonnegat... |
| itg2ub 25754 | The integral of a nonnegat... |
| itg2leub 25755 | Any upper bound on the int... |
| itg2ge0 25756 | The integral of a nonnegat... |
| itg2itg1 25757 | The integral of a nonnegat... |
| itg20 25758 | The integral of the zero f... |
| itg2lecl 25759 | If an ` S.2 ` integral is ... |
| itg2le 25760 | If one function dominates ... |
| itg2const 25761 | Integral of a constant fun... |
| itg2const2 25762 | When the base set of a con... |
| itg2seq 25763 | Definitional property of t... |
| itg2uba 25764 | Approximate version of ~ i... |
| itg2lea 25765 | Approximate version of ~ i... |
| itg2eqa 25766 | Approximate equality of in... |
| itg2mulclem 25767 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25768 | The integral of a nonnegat... |
| itg2splitlem 25769 | Lemma for ~ itg2split . (... |
| itg2split 25770 | The ` S.2 ` integral split... |
| itg2monolem1 25771 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25772 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25773 | Lemma for ~ itg2mono . (C... |
| itg2mono 25774 | The Monotone Convergence T... |
| itg2i1fseqle 25775 | Subject to the conditions ... |
| itg2i1fseq 25776 | Subject to the conditions ... |
| itg2i1fseq2 25777 | In an extension to the res... |
| itg2i1fseq3 25778 | Special case of ~ itg2i1fs... |
| itg2addlem 25779 | Lemma for ~ itg2add . (Co... |
| itg2add 25780 | The ` S.2 ` integral is li... |
| itg2gt0 25781 | If the function ` F ` is s... |
| itg2cnlem1 25782 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25783 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25784 | A sort of absolute continu... |
| ibllem 25785 | Conditioned equality theor... |
| isibl 25786 | The predicate " ` F ` is i... |
| isibl2 25787 | The predicate " ` F ` is i... |
| iblmbf 25788 | An integrable function is ... |
| iblitg 25789 | If a function is integrabl... |
| dfitg 25790 | Evaluate the class substit... |
| itgex 25791 | An integral is a set. (Co... |
| itgeq1f 25792 | Equality theorem for an in... |
| itgeq1 25793 | Equality theorem for an in... |
| nfitg1 25794 | Bound-variable hypothesis ... |
| nfitg 25795 | Bound-variable hypothesis ... |
| cbvitg 25796 | Change bound variable in a... |
| cbvitgv 25797 | Change bound variable in a... |
| itgeq2 25798 | Equality theorem for an in... |
| itgresr 25799 | The domain of an integral ... |
| itg0 25800 | The integral of anything o... |
| itgz 25801 | The integral of zero on an... |
| itgeq2dv 25802 | Equality theorem for an in... |
| itgmpt 25803 | Change bound variable in a... |
| itgcl 25804 | The integral of an integra... |
| itgvallem 25805 | Substitution lemma. (Cont... |
| itgvallem3 25806 | Lemma for ~ itgposval and ... |
| ibl0 25807 | The zero function is integ... |
| iblcnlem1 25808 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25809 | Expand out the universal q... |
| itgcnlem 25810 | Expand out the sum in ~ df... |
| iblrelem 25811 | Integrability of a real fu... |
| iblposlem 25812 | Lemma for ~ iblpos . (Con... |
| iblpos 25813 | Integrability of a nonnega... |
| iblre 25814 | Integrability of a real fu... |
| itgrevallem1 25815 | Lemma for ~ itgposval and ... |
| itgposval 25816 | The integral of a nonnegat... |
| itgreval 25817 | Decompose the integral of ... |
| itgrecl 25818 | Real closure of an integra... |
| iblcn 25819 | Integrability of a complex... |
| itgcnval 25820 | Decompose the integral of ... |
| itgre 25821 | Real part of an integral. ... |
| itgim 25822 | Imaginary part of an integ... |
| iblneg 25823 | The negative of an integra... |
| itgneg 25824 | Negation of an integral. ... |
| iblss 25825 | A subset of an integrable ... |
| iblss2 25826 | Change the domain of an in... |
| itgitg2 25827 | Transfer an integral using... |
| i1fibl 25828 | A simple function is integ... |
| itgitg1 25829 | Transfer an integral using... |
| itgle 25830 | Monotonicity of an integra... |
| itgge0 25831 | The integral of a positive... |
| itgss 25832 | Expand the set of an integ... |
| itgss2 25833 | Expand the set of an integ... |
| itgeqa 25834 | Approximate equality of in... |
| itgss3 25835 | Expand the set of an integ... |
| itgioo 25836 | Equality of integrals on o... |
| itgless 25837 | Expand the integral of a n... |
| iblconst 25838 | A constant function is int... |
| itgconst 25839 | Integral of a constant fun... |
| ibladdlem 25840 | Lemma for ~ ibladd . (Con... |
| ibladd 25841 | Add two integrals over the... |
| iblsub 25842 | Subtract two integrals ove... |
| itgaddlem1 25843 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25844 | Lemma for ~ itgadd . (Con... |
| itgadd 25845 | Add two integrals over the... |
| itgsub 25846 | Subtract two integrals ove... |
| itgfsum 25847 | Take a finite sum of integ... |
| iblabslem 25848 | Lemma for ~ iblabs . (Con... |
| iblabs 25849 | The absolute value of an i... |
| iblabsr 25850 | A measurable function is i... |
| iblmulc2 25851 | Multiply an integral by a ... |
| itgmulc2lem1 25852 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25853 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25854 | Multiply an integral by a ... |
| itgabs 25855 | The triangle inequality fo... |
| itgsplit 25856 | The ` S. ` integral splits... |
| itgspliticc 25857 | The ` S. ` integral splits... |
| itgsplitioo 25858 | The ` S. ` integral splits... |
| bddmulibl 25859 | A bounded function times a... |
| bddibl 25860 | A bounded function is inte... |
| cniccibl 25861 | A continuous function on a... |
| bddiblnc 25862 | Choice-free proof of ~ bdd... |
| cnicciblnc 25863 | Choice-free proof of ~ cni... |
| itggt0 25864 | The integral of a strictly... |
| itgcn 25865 | Transfer ~ itg2cn to the f... |
| ditgeq1 25868 | Equality theorem for the d... |
| ditgeq2 25869 | Equality theorem for the d... |
| ditgeq3 25870 | Equality theorem for the d... |
| ditgeq3dv 25871 | Equality theorem for the d... |
| ditgex 25872 | A directed integral is a s... |
| ditg0 25873 | Value of the directed inte... |
| cbvditg 25874 | Change bound variable in a... |
| cbvditgv 25875 | Change bound variable in a... |
| ditgpos 25876 | Value of the directed inte... |
| ditgneg 25877 | Value of the directed inte... |
| ditgcl 25878 | Closure of a directed inte... |
| ditgswap 25879 | Reverse a directed integra... |
| ditgsplitlem 25880 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25881 | This theorem is the raison... |
| reldv 25890 | The derivative function is... |
| limcvallem 25891 | Lemma for ~ ellimc . (Con... |
| limcfval 25892 | Value and set bounds on th... |
| ellimc 25893 | Value of the limit predica... |
| limcrcl 25894 | Reverse closure for the li... |
| limccl 25895 | Closure of the limit opera... |
| limcdif 25896 | It suffices to consider fu... |
| ellimc2 25897 | Write the definition of a ... |
| limcnlp 25898 | If ` B ` is not a limit po... |
| ellimc3 25899 | Write the epsilon-delta de... |
| limcflflem 25900 | Lemma for ~ limcflf . (Co... |
| limcflf 25901 | The limit operator can be ... |
| limcmo 25902 | If ` B ` is a limit point ... |
| limcmpt 25903 | Express the limit operator... |
| limcmpt2 25904 | Express the limit operator... |
| limcresi 25905 | Any limit of ` F ` is also... |
| limcres 25906 | If ` B ` is an interior po... |
| cnplimc 25907 | A function is continuous a... |
| cnlimc 25908 | ` F ` is a continuous func... |
| cnlimci 25909 | If ` F ` is a continuous f... |
| cnmptlimc 25910 | If ` F ` is a continuous f... |
| limccnp 25911 | If the limit of ` F ` at `... |
| limccnp2 25912 | The image of a convergent ... |
| limcco 25913 | Composition of two limits.... |
| limciun 25914 | A point is a limit of ` F ... |
| limcun 25915 | A point is a limit of ` F ... |
| dvlem 25916 | Closure for a difference q... |
| dvfval 25917 | Value and set bounds on th... |
| eldv 25918 | The differentiable predica... |
| dvcl 25919 | The derivative function ta... |
| dvbssntr 25920 | The set of differentiable ... |
| dvbss 25921 | The set of differentiable ... |
| dvbsss 25922 | The set of differentiable ... |
| perfdvf 25923 | The derivative is a functi... |
| recnprss 25924 | Both ` RR ` and ` CC ` are... |
| recnperf 25925 | Both ` RR ` and ` CC ` are... |
| dvfg 25926 | Explicitly write out the f... |
| dvf 25927 | The derivative is a functi... |
| dvfcn 25928 | The derivative is a functi... |
| dvreslem 25929 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25930 | Lemma for ~ dvres2 . (Con... |
| dvres 25931 | Restriction of a derivativ... |
| dvres2 25932 | Restriction of the base se... |
| dvres3 25933 | Restriction of a complex d... |
| dvres3a 25934 | Restriction of a complex d... |
| dvidlem 25935 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25936 | Derivative of a function r... |
| dvconst 25937 | Derivative of a constant f... |
| dvid 25938 | Derivative of the identity... |
| dvcnp 25939 | The difference quotient is... |
| dvcnp2 25940 | A function is continuous a... |
| dvcnp2OLD 25941 | Obsolete version of ~ dvcn... |
| dvcn 25942 | A differentiable function ... |
| dvnfval 25943 | Value of the iterated deri... |
| dvnff 25944 | The iterated derivative is... |
| dvn0 25945 | Zero times iterated deriva... |
| dvnp1 25946 | Successor iterated derivat... |
| dvn1 25947 | One times iterated derivat... |
| dvnf 25948 | The N-times derivative is ... |
| dvnbss 25949 | The set of N-times differe... |
| dvnadd 25950 | The ` N ` -th derivative o... |
| dvn2bss 25951 | An N-times differentiable ... |
| dvnres 25952 | Multiple derivative versio... |
| cpnfval 25953 | Condition for n-times cont... |
| fncpn 25954 | The ` C^n ` object is a fu... |
| elcpn 25955 | Condition for n-times cont... |
| cpnord 25956 | ` C^n ` conditions are ord... |
| cpncn 25957 | A ` C^n ` function is cont... |
| cpnres 25958 | The restriction of a ` C^n... |
| dvaddbr 25959 | The sum rule for derivativ... |
| dvmulbr 25960 | The product rule for deriv... |
| dvmulbrOLD 25961 | Obsolete version of ~ dvmu... |
| dvadd 25962 | The sum rule for derivativ... |
| dvmul 25963 | The product rule for deriv... |
| dvaddf 25964 | The sum rule for everywher... |
| dvmulf 25965 | The product rule for every... |
| dvcmul 25966 | The product rule when one ... |
| dvcmulf 25967 | The product rule when one ... |
| dvcobr 25968 | The chain rule for derivat... |
| dvcobrOLD 25969 | Obsolete version of ~ dvco... |
| dvco 25970 | The chain rule for derivat... |
| dvcof 25971 | The chain rule for everywh... |
| dvcjbr 25972 | The derivative of the conj... |
| dvcj 25973 | The derivative of the conj... |
| dvfre 25974 | The derivative of a real f... |
| dvnfre 25975 | The ` N ` -th derivative o... |
| dvexp 25976 | Derivative of a power func... |
| dvexp2 25977 | Derivative of an exponenti... |
| dvrec 25978 | Derivative of the reciproc... |
| dvmptres3 25979 | Function-builder for deriv... |
| dvmptid 25980 | Function-builder for deriv... |
| dvmptc 25981 | Function-builder for deriv... |
| dvmptcl 25982 | Closure lemma for ~ dvmptc... |
| dvmptadd 25983 | Function-builder for deriv... |
| dvmptmul 25984 | Function-builder for deriv... |
| dvmptres2 25985 | Function-builder for deriv... |
| dvmptres 25986 | Function-builder for deriv... |
| dvmptcmul 25987 | Function-builder for deriv... |
| dvmptdivc 25988 | Function-builder for deriv... |
| dvmptneg 25989 | Function-builder for deriv... |
| dvmptsub 25990 | Function-builder for deriv... |
| dvmptcj 25991 | Function-builder for deriv... |
| dvmptre 25992 | Function-builder for deriv... |
| dvmptim 25993 | Function-builder for deriv... |
| dvmptntr 25994 | Function-builder for deriv... |
| dvmptco 25995 | Function-builder for deriv... |
| dvrecg 25996 | Derivative of the reciproc... |
| dvmptdiv 25997 | Function-builder for deriv... |
| dvmptfsum 25998 | Function-builder for deriv... |
| dvcnvlem 25999 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 26000 | A weak version of ~ dvcnvr... |
| dvexp3 26001 | Derivative of an exponenti... |
| dveflem 26002 | Derivative of the exponent... |
| dvef 26003 | Derivative of the exponent... |
| dvsincos 26004 | Derivative of the sine and... |
| dvsin 26005 | Derivative of the sine fun... |
| dvcos 26006 | Derivative of the cosine f... |
| dvferm1lem 26007 | Lemma for ~ dvferm . (Con... |
| dvferm1 26008 | One-sided version of ~ dvf... |
| dvferm2lem 26009 | Lemma for ~ dvferm . (Con... |
| dvferm2 26010 | One-sided version of ~ dvf... |
| dvferm 26011 | Fermat's theorem on statio... |
| rollelem 26012 | Lemma for ~ rolle . (Cont... |
| rolle 26013 | Rolle's theorem. If ` F `... |
| cmvth 26014 | Cauchy's Mean Value Theore... |
| cmvthOLD 26015 | Obsolete version of ~ cmvt... |
| mvth 26016 | The Mean Value Theorem. I... |
| dvlip 26017 | A function with derivative... |
| dvlipcn 26018 | A complex function with de... |
| dvlip2 26019 | Combine the results of ~ d... |
| c1liplem1 26020 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 26021 | C^1 functions are Lipschit... |
| c1lip2 26022 | C^1 functions are Lipschit... |
| c1lip3 26023 | C^1 functions are Lipschit... |
| dveq0 26024 | If a continuous function h... |
| dv11cn 26025 | Two functions defined on a... |
| dvgt0lem1 26026 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 26027 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 26028 | A function on a closed int... |
| dvlt0 26029 | A function on a closed int... |
| dvge0 26030 | A function on a closed int... |
| dvle 26031 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 26032 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 26033 | Lemma for ~ dvivth . (Con... |
| dvivth 26034 | Darboux' theorem, or the i... |
| dvne0 26035 | A function on a closed int... |
| dvne0f1 26036 | A function on a closed int... |
| lhop1lem 26037 | Lemma for ~ lhop1 . (Cont... |
| lhop1 26038 | L'Hôpital's Rule for... |
| lhop2 26039 | L'Hôpital's Rule for... |
| lhop 26040 | L'Hôpital's Rule. I... |
| dvcnvrelem1 26041 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 26042 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 26043 | The derivative rule for in... |
| dvcvx 26044 | A real function with stric... |
| dvfsumle 26045 | Compare a finite sum to an... |
| dvfsumleOLD 26046 | Obsolete version of ~ dvfs... |
| dvfsumge 26047 | Compare a finite sum to an... |
| dvfsumabs 26048 | Compare a finite sum to an... |
| dvmptrecl 26049 | Real closure of a derivati... |
| dvfsumrlimf 26050 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 26051 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 26052 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2OLD 26053 | Obsolete version of ~ dvfs... |
| dvfsumlem3 26054 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 26055 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 26056 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 26057 | Compare a finite sum to an... |
| dvfsumrlim2 26058 | Compare a finite sum to an... |
| dvfsumrlim3 26059 | Conjoin the statements of ... |
| dvfsum2 26060 | The reverse of ~ dvfsumrli... |
| ftc1lem1 26061 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 26062 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 26063 | The Fundamental Theorem of... |
| ftc1lem3 26064 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 26065 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 26066 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 26067 | Lemma for ~ ftc1 . (Contr... |
| ftc1 26068 | The Fundamental Theorem of... |
| ftc1cn 26069 | Strengthen the assumptions... |
| ftc2 26070 | The Fundamental Theorem of... |
| ftc2ditglem 26071 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 26072 | Directed integral analogue... |
| itgparts 26073 | Integration by parts. If ... |
| itgsubstlem 26074 | Lemma for ~ itgsubst . (C... |
| itgsubst 26075 | Integration by ` u ` -subs... |
| itgpowd 26076 | The integral of a monomial... |
| reldmmdeg 26081 | Multivariate degree is a b... |
| tdeglem1 26082 | Functionality of the total... |
| tdeglem3 26083 | Additivity of the total de... |
| tdeglem4 26084 | There is only one multi-in... |
| tdeglem2 26085 | Simplification of total de... |
| mdegfval 26086 | Value of the multivariate ... |
| mdegval 26087 | Value of the multivariate ... |
| mdegleb 26088 | Property of being of limit... |
| mdeglt 26089 | If there is an upper limit... |
| mdegldg 26090 | A nonzero polynomial has s... |
| mdegxrcl 26091 | Closure of polynomial degr... |
| mdegxrf 26092 | Functionality of polynomia... |
| mdegcl 26093 | Sharp closure for multivar... |
| mdeg0 26094 | Degree of the zero polynom... |
| mdegnn0cl 26095 | Degree of a nonzero polyno... |
| degltlem1 26096 | Theorem on arithmetic of e... |
| degltp1le 26097 | Theorem on arithmetic of e... |
| mdegaddle 26098 | The degree of a sum is at ... |
| mdegvscale 26099 | The degree of a scalar mul... |
| mdegvsca 26100 | The degree of a scalar mul... |
| mdegle0 26101 | A polynomial has nonpositi... |
| mdegmullem 26102 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26103 | The multivariate degree of... |
| deg1fval 26104 | Relate univariate polynomi... |
| deg1xrf 26105 | Functionality of univariat... |
| deg1xrcl 26106 | Closure of univariate poly... |
| deg1cl 26107 | Sharp closure of univariat... |
| mdegpropd 26108 | Property deduction for pol... |
| deg1fvi 26109 | Univariate polynomial degr... |
| deg1propd 26110 | Property deduction for pol... |
| deg1z 26111 | Degree of the zero univari... |
| deg1nn0cl 26112 | Degree of a nonzero univar... |
| deg1n0ima 26113 | Degree image of a set of p... |
| deg1nn0clb 26114 | A polynomial is nonzero if... |
| deg1lt0 26115 | A polynomial is zero iff i... |
| deg1ldg 26116 | A nonzero univariate polyn... |
| deg1ldgn 26117 | An index at which a polyno... |
| deg1ldgdomn 26118 | A nonzero univariate polyn... |
| deg1leb 26119 | Property of being of limit... |
| deg1val 26120 | Value of the univariate de... |
| deg1lt 26121 | If the degree of a univari... |
| deg1ge 26122 | Conversely, a nonzero coef... |
| coe1mul3 26123 | The coefficient vector of ... |
| coe1mul4 26124 | Value of the "leading" coe... |
| deg1addle 26125 | The degree of a sum is at ... |
| deg1addle2 26126 | If both factors have degre... |
| deg1add 26127 | Exact degree of a sum of t... |
| deg1vscale 26128 | The degree of a scalar tim... |
| deg1vsca 26129 | The degree of a scalar tim... |
| deg1invg 26130 | The degree of the negated ... |
| deg1suble 26131 | The degree of a difference... |
| deg1sub 26132 | Exact degree of a differen... |
| deg1mulle2 26133 | Produce a bound on the pro... |
| deg1sublt 26134 | Subtraction of two polynom... |
| deg1le0 26135 | A polynomial has nonpositi... |
| deg1sclle 26136 | A scalar polynomial has no... |
| deg1scl 26137 | A nonzero scalar polynomia... |
| deg1mul2 26138 | Degree of multiplication o... |
| deg1mul 26139 | Degree of multiplication o... |
| deg1mul3 26140 | Degree of multiplication o... |
| deg1mul3le 26141 | Degree of multiplication o... |
| deg1tmle 26142 | Limiting degree of a polyn... |
| deg1tm 26143 | Exact degree of a polynomi... |
| deg1pwle 26144 | Limiting degree of a varia... |
| deg1pw 26145 | Exact degree of a variable... |
| ply1nz 26146 | Univariate polynomials ove... |
| ply1nzb 26147 | Univariate polynomials are... |
| ply1domn 26148 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26149 | The ring of univariate pol... |
| ply1divmo 26160 | Uniqueness of a quotient i... |
| ply1divex 26161 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26162 | The division algorithm for... |
| ply1divalg2 26163 | Reverse the order of multi... |
| uc1pval 26164 | Value of the set of unitic... |
| isuc1p 26165 | Being a unitic polynomial.... |
| mon1pval 26166 | Value of the set of monic ... |
| ismon1p 26167 | Being a monic polynomial. ... |
| uc1pcl 26168 | Unitic polynomials are pol... |
| mon1pcl 26169 | Monic polynomials are poly... |
| uc1pn0 26170 | Unitic polynomials are not... |
| mon1pn0 26171 | Monic polynomials are not ... |
| uc1pdeg 26172 | Unitic polynomials have no... |
| uc1pldg 26173 | Unitic polynomials have un... |
| mon1pldg 26174 | Unitic polynomials have on... |
| mon1puc1p 26175 | Monic polynomials are unit... |
| uc1pmon1p 26176 | Make a unitic polynomial m... |
| deg1submon1p 26177 | The difference of two moni... |
| mon1pid 26178 | Monicity and degree of the... |
| q1pval 26179 | Value of the univariate po... |
| q1peqb 26180 | Characterizing property of... |
| q1pcl 26181 | Closure of the quotient by... |
| r1pval 26182 | Value of the polynomial re... |
| r1pcl 26183 | Closure of remainder follo... |
| r1pdeglt 26184 | The remainder has a degree... |
| r1pid 26185 | Express the original polyn... |
| r1pid2 26186 | Identity law for polynomia... |
| dvdsq1p 26187 | Divisibility in a polynomi... |
| dvdsr1p 26188 | Divisibility in a polynomi... |
| ply1remlem 26189 | A term of the form ` x - N... |
| ply1rem 26190 | The polynomial remainder t... |
| facth1 26191 | The factor theorem and its... |
| fta1glem1 26192 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26193 | Lemma for ~ fta1g . (Cont... |
| fta1g 26194 | The one-sided fundamental ... |
| fta1blem 26195 | Lemma for ~ fta1b . (Cont... |
| fta1b 26196 | The assumption that ` R ` ... |
| idomrootle 26197 | No element of an integral ... |
| drnguc1p 26198 | Over a division ring, all ... |
| ig1peu 26199 | There is a unique monic po... |
| ig1pval 26200 | Substitutions for the poly... |
| ig1pval2 26201 | Generator of the zero idea... |
| ig1pval3 26202 | Characterizing properties ... |
| ig1pcl 26203 | The monic generator of an ... |
| ig1pdvds 26204 | The monic generator of an ... |
| ig1prsp 26205 | Any ideal of polynomials o... |
| ply1lpir 26206 | The ring of polynomials ov... |
| ply1pid 26207 | The polynomials over a fie... |
| plyco0 26216 | Two ways to say that a fun... |
| plyval 26217 | Value of the polynomial se... |
| plybss 26218 | Reverse closure of the par... |
| elply 26219 | Definition of a polynomial... |
| elply2 26220 | The coefficient function c... |
| plyun0 26221 | The set of polynomials is ... |
| plyf 26222 | The polynomial is a functi... |
| plyss 26223 | The polynomial set functio... |
| plyssc 26224 | Every polynomial ring is c... |
| elplyr 26225 | Sufficient condition for e... |
| elplyd 26226 | Sufficient condition for e... |
| ply1termlem 26227 | Lemma for ~ ply1term . (C... |
| ply1term 26228 | A one-term polynomial. (C... |
| plypow 26229 | A power is a polynomial. ... |
| plyconst 26230 | A constant function is a p... |
| ne0p 26231 | A test to show that a poly... |
| ply0 26232 | The zero function is a pol... |
| plyid 26233 | The identity function is a... |
| plyeq0lem 26234 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26235 | If a polynomial is zero at... |
| plypf1 26236 | Write the set of complex p... |
| plyaddlem1 26237 | Derive the coefficient fun... |
| plymullem1 26238 | Derive the coefficient fun... |
| plyaddlem 26239 | Lemma for ~ plyadd . (Con... |
| plymullem 26240 | Lemma for ~ plymul . (Con... |
| plyadd 26241 | The sum of two polynomials... |
| plymul 26242 | The product of two polynom... |
| plysub 26243 | The difference of two poly... |
| plyaddcl 26244 | The sum of two polynomials... |
| plymulcl 26245 | The product of two polynom... |
| plysubcl 26246 | The difference of two poly... |
| coeval 26247 | Value of the coefficient f... |
| coeeulem 26248 | Lemma for ~ coeeu . (Cont... |
| coeeu 26249 | Uniqueness of the coeffici... |
| coelem 26250 | Lemma for properties of th... |
| coeeq 26251 | If ` A ` satisfies the pro... |
| dgrval 26252 | Value of the degree functi... |
| dgrlem 26253 | Lemma for ~ dgrcl and simi... |
| coef 26254 | The domain and codomain of... |
| coef2 26255 | The domain and codomain of... |
| coef3 26256 | The domain and codomain of... |
| dgrcl 26257 | The degree of any polynomi... |
| dgrub 26258 | If the ` M ` -th coefficie... |
| dgrub2 26259 | All the coefficients above... |
| dgrlb 26260 | If all the coefficients ab... |
| coeidlem 26261 | Lemma for ~ coeid . (Cont... |
| coeid 26262 | Reconstruct a polynomial a... |
| coeid2 26263 | Reconstruct a polynomial a... |
| coeid3 26264 | Reconstruct a polynomial a... |
| plyco 26265 | The composition of two pol... |
| coeeq2 26266 | Compute the coefficient fu... |
| dgrle 26267 | Given an explicit expressi... |
| dgreq 26268 | If the highest term in a p... |
| 0dgr 26269 | A constant function has de... |
| 0dgrb 26270 | A function has degree zero... |
| dgrnznn 26271 | A nonzero polynomial with ... |
| coefv0 26272 | The result of evaluating a... |
| coeaddlem 26273 | Lemma for ~ coeadd and ~ d... |
| coemullem 26274 | Lemma for ~ coemul and ~ d... |
| coeadd 26275 | The coefficient function o... |
| coemul 26276 | A coefficient of a product... |
| coe11 26277 | The coefficient function i... |
| coemulhi 26278 | The leading coefficient of... |
| coemulc 26279 | The coefficient function i... |
| coe0 26280 | The coefficients of the ze... |
| coesub 26281 | The coefficient function o... |
| coe1termlem 26282 | The coefficient function o... |
| coe1term 26283 | The coefficient function o... |
| dgr1term 26284 | The degree of a monomial. ... |
| plycn 26285 | A polynomial is a continuo... |
| plycnOLD 26286 | Obsolete version of ~ plyc... |
| dgr0 26287 | The degree of the zero pol... |
| coeidp 26288 | The coefficients of the id... |
| dgrid 26289 | The degree of the identity... |
| dgreq0 26290 | The leading coefficient of... |
| dgrlt 26291 | Two ways to say that the d... |
| dgradd 26292 | The degree of a sum of pol... |
| dgradd2 26293 | The degree of a sum of pol... |
| dgrmul2 26294 | The degree of a product of... |
| dgrmul 26295 | The degree of a product of... |
| dgrmulc 26296 | Scalar multiplication by a... |
| dgrsub 26297 | The degree of a difference... |
| dgrcolem1 26298 | The degree of a compositio... |
| dgrcolem2 26299 | Lemma for ~ dgrco . (Cont... |
| dgrco 26300 | The degree of a compositio... |
| plycjlem 26301 | Lemma for ~ plycj and ~ co... |
| plycj 26302 | The double conjugation of ... |
| coecj 26303 | Double conjugation of a po... |
| plyrecj 26304 | A polynomial with real coe... |
| plymul0or 26305 | Polynomial multiplication ... |
| ofmulrt 26306 | The set of roots of a prod... |
| plyreres 26307 | Real-coefficient polynomia... |
| dvply1 26308 | Derivative of a polynomial... |
| dvply2g 26309 | The derivative of a polyno... |
| dvply2gOLD 26310 | Obsolete version of ~ dvpl... |
| dvply2 26311 | The derivative of a polyno... |
| dvnply2 26312 | Polynomials have polynomia... |
| dvnply 26313 | Polynomials have polynomia... |
| plycpn 26314 | Polynomials are smooth. (... |
| quotval 26317 | Value of the quotient func... |
| plydivlem1 26318 | Lemma for ~ plydivalg . (... |
| plydivlem2 26319 | Lemma for ~ plydivalg . (... |
| plydivlem3 26320 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26321 | Lemma for ~ plydivex . In... |
| plydivex 26322 | Lemma for ~ plydivalg . (... |
| plydiveu 26323 | Lemma for ~ plydivalg . (... |
| plydivalg 26324 | The division algorithm on ... |
| quotlem 26325 | Lemma for properties of th... |
| quotcl 26326 | The quotient of two polyno... |
| quotcl2 26327 | Closure of the quotient fu... |
| quotdgr 26328 | Remainder property of the ... |
| plyremlem 26329 | Closure of a linear factor... |
| plyrem 26330 | The polynomial remainder t... |
| facth 26331 | The factor theorem. If a ... |
| fta1lem 26332 | Lemma for ~ fta1 . (Contr... |
| fta1 26333 | The easy direction of the ... |
| quotcan 26334 | Exact division with a mult... |
| vieta1lem1 26335 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26336 | Lemma for ~ vieta1 : induc... |
| vieta1 26337 | The first-order Vieta's fo... |
| plyexmo 26338 | An infinite set of values ... |
| elaa 26341 | Elementhood in the set of ... |
| aacn 26342 | An algebraic number is a c... |
| aasscn 26343 | The algebraic numbers are ... |
| elqaalem1 26344 | Lemma for ~ elqaa . The f... |
| elqaalem2 26345 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26346 | Lemma for ~ elqaa . (Cont... |
| elqaa 26347 | The set of numbers generat... |
| qaa 26348 | Every rational number is a... |
| qssaa 26349 | The rational numbers are c... |
| iaa 26350 | The imaginary unit is alge... |
| aareccl 26351 | The reciprocal of an algeb... |
| aacjcl 26352 | The conjugate of an algebr... |
| aannenlem1 26353 | Lemma for ~ aannen . (Con... |
| aannenlem2 26354 | Lemma for ~ aannen . (Con... |
| aannenlem3 26355 | The algebraic numbers are ... |
| aannen 26356 | The algebraic numbers are ... |
| aalioulem1 26357 | Lemma for ~ aaliou . An i... |
| aalioulem2 26358 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26359 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26360 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26361 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26362 | Lemma for ~ aaliou . (Con... |
| aaliou 26363 | Liouville's theorem on dio... |
| geolim3 26364 | Geometric series convergen... |
| aaliou2 26365 | Liouville's approximation ... |
| aaliou2b 26366 | Liouville's approximation ... |
| aaliou3lem1 26367 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26368 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26369 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26370 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26371 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26372 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26373 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26374 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26375 | Example of a "Liouville nu... |
| aaliou3 26376 | Example of a "Liouville nu... |
| taylfvallem1 26381 | Lemma for ~ taylfval . (C... |
| taylfvallem 26382 | Lemma for ~ taylfval . (C... |
| taylfval 26383 | Define the Taylor polynomi... |
| eltayl 26384 | Value of the Taylor series... |
| taylf 26385 | The Taylor series defines ... |
| tayl0 26386 | The Taylor series is alway... |
| taylplem1 26387 | Lemma for ~ taylpfval and ... |
| taylplem2 26388 | Lemma for ~ taylpfval and ... |
| taylpfval 26389 | Define the Taylor polynomi... |
| taylpf 26390 | The Taylor polynomial is a... |
| taylpval 26391 | Value of the Taylor polyno... |
| taylply2 26392 | The Taylor polynomial is a... |
| taylply2OLD 26393 | Obsolete version of ~ tayl... |
| taylply 26394 | The Taylor polynomial is a... |
| dvtaylp 26395 | The derivative of the Tayl... |
| dvntaylp 26396 | The ` M ` -th derivative o... |
| dvntaylp0 26397 | The first ` N ` derivative... |
| taylthlem1 26398 | Lemma for ~ taylth . This... |
| taylthlem2 26399 | Lemma for ~ taylth . (Con... |
| taylthlem2OLD 26400 | Obsolete version of ~ tayl... |
| taylth 26401 | Taylor's theorem. The Tay... |
| ulmrel 26404 | The uniform limit relation... |
| ulmscl 26405 | Closure of the base set in... |
| ulmval 26406 | Express the predicate: Th... |
| ulmcl 26407 | Closure of a uniform limit... |
| ulmf 26408 | Closure of a uniform limit... |
| ulmpm 26409 | Closure of a uniform limit... |
| ulmf2 26410 | Closure of a uniform limit... |
| ulm2 26411 | Simplify ~ ulmval when ` F... |
| ulmi 26412 | The uniform limit property... |
| ulmclm 26413 | A uniform limit of functio... |
| ulmres 26414 | A sequence of functions co... |
| ulmshftlem 26415 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26416 | A sequence of functions co... |
| ulm0 26417 | Every function converges u... |
| ulmuni 26418 | A sequence of functions un... |
| ulmdm 26419 | Two ways to express that a... |
| ulmcaulem 26420 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26421 | A sequence of functions co... |
| ulmcau2 26422 | A sequence of functions co... |
| ulmss 26423 | A uniform limit of functio... |
| ulmbdd 26424 | A uniform limit of bounded... |
| ulmcn 26425 | A uniform limit of continu... |
| ulmdvlem1 26426 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26427 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26428 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26429 | If ` F ` is a sequence of ... |
| mtest 26430 | The Weierstrass M-test. I... |
| mtestbdd 26431 | Given the hypotheses of th... |
| mbfulm 26432 | A uniform limit of measura... |
| iblulm 26433 | A uniform limit of integra... |
| itgulm 26434 | A uniform limit of integra... |
| itgulm2 26435 | A uniform limit of integra... |
| pserval 26436 | Value of the function ` G ... |
| pserval2 26437 | Value of the function ` G ... |
| psergf 26438 | The sequence of terms in t... |
| radcnvlem1 26439 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26440 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26441 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26442 | Zero is always a convergen... |
| radcnvcl 26443 | The radius of convergence ... |
| radcnvlt1 26444 | If ` X ` is within the ope... |
| radcnvlt2 26445 | If ` X ` is within the ope... |
| radcnvle 26446 | If ` X ` is a convergent p... |
| dvradcnv 26447 | The radius of convergence ... |
| pserulm 26448 | If ` S ` is a region conta... |
| psercn2 26449 | Since by ~ pserulm the ser... |
| psercn2OLD 26450 | Obsolete version of ~ pser... |
| psercnlem2 26451 | Lemma for ~ psercn . (Con... |
| psercnlem1 26452 | Lemma for ~ psercn . (Con... |
| psercn 26453 | An infinite series converg... |
| pserdvlem1 26454 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26455 | Lemma for ~ pserdv . (Con... |
| pserdv 26456 | The derivative of a power ... |
| pserdv2 26457 | The derivative of a power ... |
| abelthlem1 26458 | Lemma for ~ abelth . (Con... |
| abelthlem2 26459 | Lemma for ~ abelth . The ... |
| abelthlem3 26460 | Lemma for ~ abelth . (Con... |
| abelthlem4 26461 | Lemma for ~ abelth . (Con... |
| abelthlem5 26462 | Lemma for ~ abelth . (Con... |
| abelthlem6 26463 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26464 | Lemma for ~ abelth . (Con... |
| abelthlem7 26465 | Lemma for ~ abelth . (Con... |
| abelthlem8 26466 | Lemma for ~ abelth . (Con... |
| abelthlem9 26467 | Lemma for ~ abelth . By a... |
| abelth 26468 | Abel's theorem. If the po... |
| abelth2 26469 | Abel's theorem, restricted... |
| efcn 26470 | The exponential function i... |
| sincn 26471 | Sine is continuous. (Cont... |
| coscn 26472 | Cosine is continuous. (Co... |
| reeff1olem 26473 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26474 | The real exponential funct... |
| reefiso 26475 | The exponential function o... |
| efcvx 26476 | The exponential function o... |
| reefgim 26477 | The exponential function i... |
| pilem1 26478 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26479 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26480 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26481 | ` _pi ` is between 2 and 4... |
| sinpi 26482 | The sine of ` _pi ` is 0. ... |
| pire 26483 | ` _pi ` is a real number. ... |
| picn 26484 | ` _pi ` is a complex numbe... |
| pipos 26485 | ` _pi ` is positive. (Con... |
| pirp 26486 | ` _pi ` is a positive real... |
| negpicn 26487 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26488 | Lemma for ~ sinhalfpi and ... |
| halfpire 26489 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26490 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26491 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26492 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26493 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26494 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26495 | The cosine of ` -u _pi / 2... |
| efhalfpi 26496 | The exponential of ` _i _p... |
| cospi 26497 | The cosine of ` _pi ` is `... |
| efipi 26498 | The exponential of ` _i x.... |
| eulerid 26499 | Euler's identity. (Contri... |
| sin2pi 26500 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26501 | The cosine of ` 2 _pi ` is... |
| ef2pi 26502 | The exponential of ` 2 _pi... |
| ef2kpi 26503 | If ` K ` is an integer, th... |
| efper 26504 | The exponential function i... |
| sinperlem 26505 | Lemma for ~ sinper and ~ c... |
| sinper 26506 | The sine function is perio... |
| cosper 26507 | The cosine function is per... |
| sin2kpi 26508 | If ` K ` is an integer, th... |
| cos2kpi 26509 | If ` K ` is an integer, th... |
| sin2pim 26510 | Sine of a number subtracte... |
| cos2pim 26511 | Cosine of a number subtrac... |
| sinmpi 26512 | Sine of a number less ` _p... |
| cosmpi 26513 | Cosine of a number less ` ... |
| sinppi 26514 | Sine of a number plus ` _p... |
| cosppi 26515 | Cosine of a number plus ` ... |
| efimpi 26516 | The exponential function a... |
| sinhalfpip 26517 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26518 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26519 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26520 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26521 | Ptolemy's Theorem. This t... |
| sincosq1lem 26522 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26523 | The signs of the sine and ... |
| sincosq2sgn 26524 | The signs of the sine and ... |
| sincosq3sgn 26525 | The signs of the sine and ... |
| sincosq4sgn 26526 | The signs of the sine and ... |
| coseq00topi 26527 | Location of the zeroes of ... |
| coseq0negpitopi 26528 | Location of the zeroes of ... |
| tanrpcl 26529 | Positive real closure of t... |
| tangtx 26530 | The tangent function is gr... |
| tanabsge 26531 | The tangent function is gr... |
| sinq12gt0 26532 | The sine of a number stric... |
| sinq12ge0 26533 | The sine of a number betwe... |
| sinq34lt0t 26534 | The sine of a number stric... |
| cosq14gt0 26535 | The cosine of a number str... |
| cosq14ge0 26536 | The cosine of a number bet... |
| sincosq1eq 26537 | Complementarity of the sin... |
| sincos4thpi 26538 | The sine and cosine of ` _... |
| tan4thpi 26539 | The tangent of ` _pi / 4 `... |
| sincos6thpi 26540 | The sine and cosine of ` _... |
| sincos3rdpi 26541 | The sine and cosine of ` _... |
| pigt3 26542 | ` _pi ` is greater than 3.... |
| pige3 26543 | ` _pi ` is greater than or... |
| pige3ALT 26544 | Alternate proof of ~ pige3... |
| abssinper 26545 | The absolute value of sine... |
| sinkpi 26546 | The sine of an integer mul... |
| coskpi 26547 | The absolute value of the ... |
| sineq0 26548 | A complex number whose sin... |
| coseq1 26549 | A complex number whose cos... |
| cos02pilt1 26550 | Cosine is less than one be... |
| cosq34lt1 26551 | Cosine is less than one in... |
| efeq1 26552 | A complex number whose exp... |
| cosne0 26553 | The cosine function has no... |
| cosordlem 26554 | Lemma for ~ cosord . (Con... |
| cosord 26555 | Cosine is decreasing over ... |
| cos0pilt1 26556 | Cosine is between minus on... |
| cos11 26557 | Cosine is one-to-one over ... |
| sinord 26558 | Sine is increasing over th... |
| recosf1o 26559 | The cosine function is a b... |
| resinf1o 26560 | The sine function is a bij... |
| tanord1 26561 | The tangent function is st... |
| tanord 26562 | The tangent function is st... |
| tanregt0 26563 | The real part of the tange... |
| negpitopissre 26564 | The interval ` ( -u _pi (,... |
| efgh 26565 | The exponential function o... |
| efif1olem1 26566 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26567 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26568 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26569 | The exponential function o... |
| efif1o 26570 | The exponential function o... |
| efifo 26571 | The exponential function o... |
| eff1olem 26572 | The exponential function m... |
| eff1o 26573 | The exponential function m... |
| efabl 26574 | The image of a subgroup of... |
| efsubm 26575 | The image of a subgroup of... |
| circgrp 26576 | The circle group ` T ` is ... |
| circsubm 26577 | The circle group ` T ` is ... |
| logrn 26582 | The range of the natural l... |
| ellogrn 26583 | Write out the property ` A... |
| dflog2 26584 | The natural logarithm func... |
| relogrn 26585 | The range of the natural l... |
| logrncn 26586 | The range of the natural l... |
| eff1o2 26587 | The exponential function r... |
| logf1o 26588 | The natural logarithm func... |
| dfrelog 26589 | The natural logarithm func... |
| relogf1o 26590 | The natural logarithm func... |
| logrncl 26591 | Closure of the natural log... |
| logcl 26592 | Closure of the natural log... |
| logimcl 26593 | Closure of the imaginary p... |
| logcld 26594 | The logarithm of a nonzero... |
| logimcld 26595 | The imaginary part of the ... |
| logimclad 26596 | The imaginary part of the ... |
| abslogimle 26597 | The imaginary part of the ... |
| logrnaddcl 26598 | The range of the natural l... |
| relogcl 26599 | Closure of the natural log... |
| eflog 26600 | Relationship between the n... |
| logeq0im1 26601 | If the logarithm of a numb... |
| logccne0 26602 | The logarithm isn't 0 if i... |
| logne0 26603 | Logarithm of a non-1 posit... |
| reeflog 26604 | Relationship between the n... |
| logef 26605 | Relationship between the n... |
| relogef 26606 | Relationship between the n... |
| logeftb 26607 | Relationship between the n... |
| relogeftb 26608 | Relationship between the n... |
| log1 26609 | The natural logarithm of `... |
| loge 26610 | The natural logarithm of `... |
| logi 26611 | The natural logarithm of `... |
| logneg 26612 | The natural logarithm of a... |
| logm1 26613 | The natural logarithm of n... |
| lognegb 26614 | If a number has imaginary ... |
| relogoprlem 26615 | Lemma for ~ relogmul and ~... |
| relogmul 26616 | The natural logarithm of t... |
| relogdiv 26617 | The natural logarithm of t... |
| explog 26618 | Exponentiation of a nonzer... |
| reexplog 26619 | Exponentiation of a positi... |
| relogexp 26620 | The natural logarithm of p... |
| relog 26621 | Real part of a logarithm. ... |
| relogiso 26622 | The natural logarithm func... |
| reloggim 26623 | The natural logarithm is a... |
| logltb 26624 | The natural logarithm func... |
| logfac 26625 | The logarithm of a factori... |
| eflogeq 26626 | Solve an equation involvin... |
| logleb 26627 | Natural logarithm preserve... |
| rplogcl 26628 | Closure of the logarithm f... |
| logge0 26629 | The logarithm of a number ... |
| logcj 26630 | The natural logarithm dist... |
| efiarg 26631 | The exponential of the "ar... |
| cosargd 26632 | The cosine of the argument... |
| cosarg0d 26633 | The cosine of the argument... |
| argregt0 26634 | Closure of the argument of... |
| argrege0 26635 | Closure of the argument of... |
| argimgt0 26636 | Closure of the argument of... |
| argimlt0 26637 | Closure of the argument of... |
| logimul 26638 | Multiplying a number by ` ... |
| logneg2 26639 | The logarithm of the negat... |
| logmul2 26640 | Generalization of ~ relogm... |
| logdiv2 26641 | Generalization of ~ relogd... |
| abslogle 26642 | Bound on the magnitude of ... |
| tanarg 26643 | The basic relation between... |
| logdivlti 26644 | The ` log x / x ` function... |
| logdivlt 26645 | The ` log x / x ` function... |
| logdivle 26646 | The ` log x / x ` function... |
| relogcld 26647 | Closure of the natural log... |
| reeflogd 26648 | Relationship between the n... |
| relogmuld 26649 | The natural logarithm of t... |
| relogdivd 26650 | The natural logarithm of t... |
| logled 26651 | Natural logarithm preserve... |
| relogefd 26652 | Relationship between the n... |
| rplogcld 26653 | Closure of the logarithm f... |
| logge0d 26654 | The logarithm of a number ... |
| logge0b 26655 | The logarithm of a number ... |
| loggt0b 26656 | The logarithm of a number ... |
| logle1b 26657 | The logarithm of a number ... |
| loglt1b 26658 | The logarithm of a number ... |
| divlogrlim 26659 | The inverse logarithm func... |
| logno1 26660 | The logarithm function is ... |
| dvrelog 26661 | The derivative of the real... |
| relogcn 26662 | The real logarithm functio... |
| ellogdm 26663 | Elementhood in the "contin... |
| logdmn0 26664 | A number in the continuous... |
| logdmnrp 26665 | A number in the continuous... |
| logdmss 26666 | The continuity domain of `... |
| logcnlem2 26667 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26668 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26669 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26670 | Lemma for ~ logcn . (Cont... |
| logcn 26671 | The logarithm function is ... |
| dvloglem 26672 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26673 | The "continuous domain" of... |
| logf1o2 26674 | The logarithm maps its con... |
| dvlog 26675 | The derivative of the comp... |
| dvlog2lem 26676 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26677 | The derivative of the comp... |
| advlog 26678 | The antiderivative of the ... |
| advlogexp 26679 | The antiderivative of a po... |
| efopnlem1 26680 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26681 | Lemma for ~ efopn . (Cont... |
| efopn 26682 | The exponential map is an ... |
| logtayllem 26683 | Lemma for ~ logtayl . (Co... |
| logtayl 26684 | The Taylor series for ` -u... |
| logtaylsum 26685 | The Taylor series for ` -u... |
| logtayl2 26686 | Power series expression fo... |
| logccv 26687 | The natural logarithm func... |
| cxpval 26688 | Value of the complex power... |
| cxpef 26689 | Value of the complex power... |
| 0cxp 26690 | Value of the complex power... |
| cxpexpz 26691 | Relate the complex power f... |
| cxpexp 26692 | Relate the complex power f... |
| logcxp 26693 | Logarithm of a complex pow... |
| cxp0 26694 | Value of the complex power... |
| cxp1 26695 | Value of the complex power... |
| 1cxp 26696 | Value of the complex power... |
| ecxp 26697 | Write the exponential func... |
| cxpcl 26698 | Closure of the complex pow... |
| recxpcl 26699 | Real closure of the comple... |
| rpcxpcl 26700 | Positive real closure of t... |
| cxpne0 26701 | Complex exponentiation is ... |
| cxpeq0 26702 | Complex exponentiation is ... |
| cxpadd 26703 | Sum of exponents law for c... |
| cxpp1 26704 | Value of a nonzero complex... |
| cxpneg 26705 | Value of a complex number ... |
| cxpsub 26706 | Exponent subtraction law f... |
| cxpge0 26707 | Nonnegative exponentiation... |
| mulcxplem 26708 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26709 | Complex exponentiation of ... |
| cxprec 26710 | Complex exponentiation of ... |
| divcxp 26711 | Complex exponentiation of ... |
| cxpmul 26712 | Product of exponents law f... |
| cxpmul2 26713 | Product of exponents law f... |
| cxproot 26714 | The complex power function... |
| cxpmul2z 26715 | Generalize ~ cxpmul2 to ne... |
| abscxp 26716 | Absolute value of a power,... |
| abscxp2 26717 | Absolute value of a power,... |
| cxplt 26718 | Ordering property for comp... |
| cxple 26719 | Ordering property for comp... |
| cxplea 26720 | Ordering property for comp... |
| cxple2 26721 | Ordering property for comp... |
| cxplt2 26722 | Ordering property for comp... |
| cxple2a 26723 | Ordering property for comp... |
| cxplt3 26724 | Ordering property for comp... |
| cxple3 26725 | Ordering property for comp... |
| cxpsqrtlem 26726 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26727 | The complex exponential fu... |
| logsqrt 26728 | Logarithm of a square root... |
| cxp0d 26729 | Value of the complex power... |
| cxp1d 26730 | Value of the complex power... |
| 1cxpd 26731 | Value of the complex power... |
| cxpcld 26732 | Closure of the complex pow... |
| cxpmul2d 26733 | Product of exponents law f... |
| 0cxpd 26734 | Value of the complex power... |
| cxpexpzd 26735 | Relate the complex power f... |
| cxpefd 26736 | Value of the complex power... |
| cxpne0d 26737 | Complex exponentiation is ... |
| cxpp1d 26738 | Value of a nonzero complex... |
| cxpnegd 26739 | Value of a complex number ... |
| cxpmul2zd 26740 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26741 | Sum of exponents law for c... |
| cxpsubd 26742 | Exponent subtraction law f... |
| cxpltd 26743 | Ordering property for comp... |
| cxpled 26744 | Ordering property for comp... |
| cxplead 26745 | Ordering property for comp... |
| divcxpd 26746 | Complex exponentiation of ... |
| recxpcld 26747 | Positive real closure of t... |
| cxpge0d 26748 | Nonnegative exponentiation... |
| cxple2ad 26749 | Ordering property for comp... |
| cxplt2d 26750 | Ordering property for comp... |
| cxple2d 26751 | Ordering property for comp... |
| mulcxpd 26752 | Complex exponentiation of ... |
| recxpf1lem 26753 | Complex exponentiation on ... |
| cxpsqrtth 26754 | Square root theorem over t... |
| 2irrexpq 26755 | There exist irrational num... |
| cxprecd 26756 | Complex exponentiation of ... |
| rpcxpcld 26757 | Positive real closure of t... |
| logcxpd 26758 | Logarithm of a complex pow... |
| cxplt3d 26759 | Ordering property for comp... |
| cxple3d 26760 | Ordering property for comp... |
| cxpmuld 26761 | Product of exponents law f... |
| cxpgt0d 26762 | A positive real raised to ... |
| cxpcom 26763 | Commutative law for real e... |
| dvcxp1 26764 | The derivative of a comple... |
| dvcxp2 26765 | The derivative of a comple... |
| dvsqrt 26766 | The derivative of the real... |
| dvcncxp1 26767 | Derivative of complex powe... |
| dvcnsqrt 26768 | Derivative of square root ... |
| cxpcn 26769 | Domain of continuity of th... |
| cxpcnOLD 26770 | Obsolete version of ~ cxpc... |
| cxpcn2 26771 | Continuity of the complex ... |
| cxpcn3lem 26772 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26773 | Extend continuity of the c... |
| resqrtcn 26774 | Continuity of the real squ... |
| sqrtcn 26775 | Continuity of the square r... |
| cxpaddlelem 26776 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26777 | Ordering property for comp... |
| abscxpbnd 26778 | Bound on the absolute valu... |
| root1id 26779 | Property of an ` N ` -th r... |
| root1eq1 26780 | The only powers of an ` N ... |
| root1cj 26781 | Within the ` N ` -th roots... |
| cxpeq 26782 | Solve an equation involvin... |
| zrtelqelz 26783 | If the ` N ` -th root of a... |
| zrtdvds 26784 | A positive integer root di... |
| rtprmirr 26785 | The root of a prime number... |
| loglesqrt 26786 | An upper bound on the loga... |
| logreclem 26787 | Symmetry of the natural lo... |
| logrec 26788 | Logarithm of a reciprocal ... |
| logbval 26791 | Define the value of the ` ... |
| logbcl 26792 | General logarithm closure.... |
| logbid1 26793 | General logarithm is 1 whe... |
| logb1 26794 | The logarithm of ` 1 ` to ... |
| elogb 26795 | The general logarithm of a... |
| logbchbase 26796 | Change of base for logarit... |
| relogbval 26797 | Value of the general logar... |
| relogbcl 26798 | Closure of the general log... |
| relogbzcl 26799 | Closure of the general log... |
| relogbreexp 26800 | Power law for the general ... |
| relogbzexp 26801 | Power law for the general ... |
| relogbmul 26802 | The logarithm of the produ... |
| relogbmulexp 26803 | The logarithm of the produ... |
| relogbdiv 26804 | The logarithm of the quoti... |
| relogbexp 26805 | Identity law for general l... |
| nnlogbexp 26806 | Identity law for general l... |
| logbrec 26807 | Logarithm of a reciprocal ... |
| logbleb 26808 | The general logarithm func... |
| logblt 26809 | The general logarithm func... |
| relogbcxp 26810 | Identity law for the gener... |
| cxplogb 26811 | Identity law for the gener... |
| relogbcxpb 26812 | The logarithm is the inver... |
| logbmpt 26813 | The general logarithm to a... |
| logbf 26814 | The general logarithm to a... |
| logbfval 26815 | The general logarithm of a... |
| relogbf 26816 | The general logarithm to a... |
| logblog 26817 | The general logarithm to t... |
| logbgt0b 26818 | The logarithm of a positiv... |
| logbgcd1irr 26819 | The logarithm of an intege... |
| 2logb9irr 26820 | Example for ~ logbgcd1irr ... |
| logbprmirr 26821 | The logarithm of a prime t... |
| 2logb3irr 26822 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26823 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26824 | The square root of two to ... |
| 2irrexpqALT 26825 | Alternate proof of ~ 2irre... |
| angval 26826 | Define the angle function,... |
| angcan 26827 | Cancel a constant multipli... |
| angneg 26828 | Cancel a negative sign in ... |
| angvald 26829 | The (signed) angle between... |
| angcld 26830 | The (signed) angle between... |
| angrteqvd 26831 | Two vectors are at a right... |
| cosangneg2d 26832 | The cosine of the angle be... |
| angrtmuld 26833 | Perpendicularity of two ve... |
| ang180lem1 26834 | Lemma for ~ ang180 . Show... |
| ang180lem2 26835 | Lemma for ~ ang180 . Show... |
| ang180lem3 26836 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26837 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26838 | Lemma for ~ ang180 : Redu... |
| ang180 26839 | The sum of angles ` m A B ... |
| lawcoslem1 26840 | Lemma for ~ lawcos . Here... |
| lawcos 26841 | Law of cosines (also known... |
| pythag 26842 | Pythagorean theorem. Give... |
| isosctrlem1 26843 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26844 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26845 | Lemma for ~ isosctr . Cor... |
| isosctr 26846 | Isosceles triangle theorem... |
| ssscongptld 26847 | If two triangles have equa... |
| affineequiv 26848 | Equivalence between two wa... |
| affineequiv2 26849 | Equivalence between two wa... |
| affineequiv3 26850 | Equivalence between two wa... |
| affineequiv4 26851 | Equivalence between two wa... |
| affineequivne 26852 | Equivalence between two wa... |
| angpieqvdlem 26853 | Equivalence used in the pr... |
| angpieqvdlem2 26854 | Equivalence used in ~ angp... |
| angpined 26855 | If the angle at ABC is ` _... |
| angpieqvd 26856 | The angle ABC is ` _pi ` i... |
| chordthmlem 26857 | If ` M ` is the midpoint o... |
| chordthmlem2 26858 | If M is the midpoint of AB... |
| chordthmlem3 26859 | If M is the midpoint of AB... |
| chordthmlem4 26860 | If P is on the segment AB ... |
| chordthmlem5 26861 | If P is on the segment AB ... |
| chordthm 26862 | The intersecting chords th... |
| heron 26863 | Heron's formula gives the ... |
| quad2 26864 | The quadratic equation, wi... |
| quad 26865 | The quadratic equation. (... |
| 1cubrlem 26866 | The cube roots of unity. ... |
| 1cubr 26867 | The cube roots of unity. ... |
| dcubic1lem 26868 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26869 | Reverse direction of ~ dcu... |
| dcubic1 26870 | Forward direction of ~ dcu... |
| dcubic 26871 | Solutions to the depressed... |
| mcubic 26872 | Solutions to a monic cubic... |
| cubic2 26873 | The solution to the genera... |
| cubic 26874 | The cubic equation, which ... |
| binom4 26875 | Work out a quartic binomia... |
| dquartlem1 26876 | Lemma for ~ dquart . (Con... |
| dquartlem2 26877 | Lemma for ~ dquart . (Con... |
| dquart 26878 | Solve a depressed quartic ... |
| quart1cl 26879 | Closure lemmas for ~ quart... |
| quart1lem 26880 | Lemma for ~ quart1 . (Con... |
| quart1 26881 | Depress a quartic equation... |
| quartlem1 26882 | Lemma for ~ quart . (Cont... |
| quartlem2 26883 | Closure lemmas for ~ quart... |
| quartlem3 26884 | Closure lemmas for ~ quart... |
| quartlem4 26885 | Closure lemmas for ~ quart... |
| quart 26886 | The quartic equation, writ... |
| asinlem 26893 | The argument to the logari... |
| asinlem2 26894 | The argument to the logari... |
| asinlem3a 26895 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26896 | The argument to the logari... |
| asinf 26897 | Domain and codomain of the... |
| asincl 26898 | Closure for the arcsin fun... |
| acosf 26899 | Domain and codoamin of the... |
| acoscl 26900 | Closure for the arccos fun... |
| atandm 26901 | Since the property is a li... |
| atandm2 26902 | This form of ~ atandm is a... |
| atandm3 26903 | A compact form of ~ atandm... |
| atandm4 26904 | A compact form of ~ atandm... |
| atanf 26905 | Domain and codoamin of the... |
| atancl 26906 | Closure for the arctan fun... |
| asinval 26907 | Value of the arcsin functi... |
| acosval 26908 | Value of the arccos functi... |
| atanval 26909 | Value of the arctan functi... |
| atanre 26910 | A real number is in the do... |
| asinneg 26911 | The arcsine function is od... |
| acosneg 26912 | The negative symmetry rela... |
| efiasin 26913 | The exponential of the arc... |
| sinasin 26914 | The arcsine function is an... |
| cosacos 26915 | The arccosine function is ... |
| asinsinlem 26916 | Lemma for ~ asinsin . (Co... |
| asinsin 26917 | The arcsine function compo... |
| acoscos 26918 | The arccosine function is ... |
| asin1 26919 | The arcsine of ` 1 ` is ` ... |
| acos1 26920 | The arccosine of ` 1 ` is ... |
| reasinsin 26921 | The arcsine function compo... |
| asinsinb 26922 | Relationship between sine ... |
| acoscosb 26923 | Relationship between cosin... |
| asinbnd 26924 | The arcsine function has r... |
| acosbnd 26925 | The arccosine function has... |
| asinrebnd 26926 | Bounds on the arcsine func... |
| asinrecl 26927 | The arcsine function is re... |
| acosrecl 26928 | The arccosine function is ... |
| cosasin 26929 | The cosine of the arcsine ... |
| sinacos 26930 | The sine of the arccosine ... |
| atandmneg 26931 | The domain of the arctange... |
| atanneg 26932 | The arctangent function is... |
| atan0 26933 | The arctangent of zero is ... |
| atandmcj 26934 | The arctangent function di... |
| atancj 26935 | The arctangent function di... |
| atanrecl 26936 | The arctangent function is... |
| efiatan 26937 | Value of the exponential o... |
| atanlogaddlem 26938 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26939 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26940 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26941 | A variation on ~ atanlogad... |
| efiatan2 26942 | Value of the exponential o... |
| 2efiatan 26943 | Value of the exponential o... |
| tanatan 26944 | The arctangent function is... |
| atandmtan 26945 | The tangent function has r... |
| cosatan 26946 | The cosine of an arctangen... |
| cosatanne0 26947 | The arctangent function ha... |
| atantan 26948 | The arctangent function is... |
| atantanb 26949 | Relationship between tange... |
| atanbndlem 26950 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26951 | The arctangent function is... |
| atanord 26952 | The arctangent function is... |
| atan1 26953 | The arctangent of ` 1 ` is... |
| bndatandm 26954 | A point in the open unit d... |
| atans 26955 | The "domain of continuity"... |
| atans2 26956 | It suffices to show that `... |
| atansopn 26957 | The domain of continuity o... |
| atansssdm 26958 | The domain of continuity o... |
| ressatans 26959 | The real number line is a ... |
| dvatan 26960 | The derivative of the arct... |
| atancn 26961 | The arctangent is a contin... |
| atantayl 26962 | The Taylor series for ` ar... |
| atantayl2 26963 | The Taylor series for ` ar... |
| atantayl3 26964 | The Taylor series for ` ar... |
| leibpilem1 26965 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26966 | The Leibniz formula for ` ... |
| leibpi 26967 | The Leibniz formula for ` ... |
| leibpisum 26968 | The Leibniz formula for ` ... |
| log2cnv 26969 | Using the Taylor series fo... |
| log2tlbnd 26970 | Bound the error term in th... |
| log2ublem1 26971 | Lemma for ~ log2ub . The ... |
| log2ublem2 26972 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26973 | Lemma for ~ log2ub . In d... |
| log2ub 26974 | ` log 2 ` is less than ` 2... |
| log2le1 26975 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26976 | Lemma for ~ birthday . (C... |
| birthdaylem2 26977 | For general ` N ` and ` K ... |
| birthdaylem3 26978 | For general ` N ` and ` K ... |
| birthday 26979 | The Birthday Problem. The... |
| dmarea 26982 | The domain of the area fun... |
| areambl 26983 | The fibers of a measurable... |
| areass 26984 | A measurable region is a s... |
| dfarea 26985 | Rewrite ~ df-area self-ref... |
| areaf 26986 | Area measurement is a func... |
| areacl 26987 | The area of a measurable r... |
| areage0 26988 | The area of a measurable r... |
| areaval 26989 | The area of a measurable r... |
| rlimcnp 26990 | Relate a limit of a real-v... |
| rlimcnp2 26991 | Relate a limit of a real-v... |
| rlimcnp3 26992 | Relate a limit of a real-v... |
| xrlimcnp 26993 | Relate a limit of a real-v... |
| efrlim 26994 | The limit of the sequence ... |
| efrlimOLD 26995 | Obsolete version of ~ efrl... |
| dfef2 26996 | The limit of the sequence ... |
| cxplim 26997 | A power to a negative expo... |
| sqrtlim 26998 | The inverse square root fu... |
| rlimcxp 26999 | Any power to a positive ex... |
| o1cxp 27000 | An eventually bounded func... |
| cxp2limlem 27001 | A linear factor grows slow... |
| cxp2lim 27002 | Any power grows slower tha... |
| cxploglim 27003 | The logarithm grows slower... |
| cxploglim2 27004 | Every power of the logarit... |
| divsqrtsumlem 27005 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 27006 | The function ` F ` used in... |
| divsqrsum 27007 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 27008 | A bound on the distance of... |
| divsqrtsumo1 27009 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 27010 | Closure of a 0-1 linear co... |
| scvxcvx 27011 | A strictly convex function... |
| jensenlem1 27012 | Lemma for ~ jensen . (Con... |
| jensenlem2 27013 | Lemma for ~ jensen . (Con... |
| jensen 27014 | Jensen's inequality, a fin... |
| amgmlem 27015 | Lemma for ~ amgm . (Contr... |
| amgm 27016 | Inequality of arithmetic a... |
| logdifbnd 27019 | Bound on the difference of... |
| logdiflbnd 27020 | Lower bound on the differe... |
| emcllem1 27021 | Lemma for ~ emcl . The se... |
| emcllem2 27022 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 27023 | Lemma for ~ emcl . The fu... |
| emcllem4 27024 | Lemma for ~ emcl . The di... |
| emcllem5 27025 | Lemma for ~ emcl . The pa... |
| emcllem6 27026 | Lemma for ~ emcl . By the... |
| emcllem7 27027 | Lemma for ~ emcl and ~ har... |
| emcl 27028 | Closure and bounds for the... |
| harmonicbnd 27029 | A bound on the harmonic se... |
| harmonicbnd2 27030 | A bound on the harmonic se... |
| emre 27031 | The Euler-Mascheroni const... |
| emgt0 27032 | The Euler-Mascheroni const... |
| harmonicbnd3 27033 | A bound on the harmonic se... |
| harmoniclbnd 27034 | A bound on the harmonic se... |
| harmonicubnd 27035 | A bound on the harmonic se... |
| harmonicbnd4 27036 | The asymptotic behavior of... |
| fsumharmonic 27037 | Bound a finite sum based o... |
| zetacvg 27040 | The zeta series is converg... |
| eldmgm 27047 | Elementhood in the set of ... |
| dmgmaddn0 27048 | If ` A ` is not a nonposit... |
| dmlogdmgm 27049 | If ` A ` is in the continu... |
| rpdmgm 27050 | A positive real number is ... |
| dmgmn0 27051 | If ` A ` is not a nonposit... |
| dmgmaddnn0 27052 | If ` A ` is not a nonposit... |
| dmgmdivn0 27053 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 27054 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 27055 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 27056 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 27057 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 27058 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 27059 | The series ` G ` is unifor... |
| lgamgulm 27060 | The series ` G ` is unifor... |
| lgamgulm2 27061 | Rewrite the limit of the s... |
| lgambdd 27062 | The log-Gamma function is ... |
| lgamucov 27063 | The ` U ` regions used in ... |
| lgamucov2 27064 | The ` U ` regions used in ... |
| lgamcvglem 27065 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 27066 | The log-Gamma function is ... |
| lgamf 27067 | The log-Gamma function is ... |
| gamf 27068 | The Gamma function is a co... |
| gamcl 27069 | The exponential of the log... |
| eflgam 27070 | The exponential of the log... |
| gamne0 27071 | The Gamma function is neve... |
| igamval 27072 | Value of the inverse Gamma... |
| igamz 27073 | Value of the inverse Gamma... |
| igamgam 27074 | Value of the inverse Gamma... |
| igamlgam 27075 | Value of the inverse Gamma... |
| igamf 27076 | Closure of the inverse Gam... |
| igamcl 27077 | Closure of the inverse Gam... |
| gamigam 27078 | The Gamma function is the ... |
| lgamcvg 27079 | The series ` G ` converges... |
| lgamcvg2 27080 | The series ` G ` converges... |
| gamcvg 27081 | The pointwise exponential ... |
| lgamp1 27082 | The functional equation of... |
| gamp1 27083 | The functional equation of... |
| gamcvg2lem 27084 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27085 | An infinite product expres... |
| regamcl 27086 | The Gamma function is real... |
| relgamcl 27087 | The log-Gamma function is ... |
| rpgamcl 27088 | The log-Gamma function is ... |
| lgam1 27089 | The log-Gamma function at ... |
| gam1 27090 | The log-Gamma function at ... |
| facgam 27091 | The Gamma function general... |
| gamfac 27092 | The Gamma function general... |
| wilthlem1 27093 | The only elements that are... |
| wilthlem2 27094 | Lemma for ~ wilth : induct... |
| wilthlem3 27095 | Lemma for ~ wilth . Here ... |
| wilth 27096 | Wilson's theorem. A numbe... |
| wilthimp 27097 | The forward implication of... |
| ftalem1 27098 | Lemma for ~ fta : "growth... |
| ftalem2 27099 | Lemma for ~ fta . There e... |
| ftalem3 27100 | Lemma for ~ fta . There e... |
| ftalem4 27101 | Lemma for ~ fta : Closure... |
| ftalem5 27102 | Lemma for ~ fta : Main pr... |
| ftalem6 27103 | Lemma for ~ fta : Dischar... |
| ftalem7 27104 | Lemma for ~ fta . Shift t... |
| fta 27105 | The Fundamental Theorem of... |
| basellem1 27106 | Lemma for ~ basel . Closu... |
| basellem2 27107 | Lemma for ~ basel . Show ... |
| basellem3 27108 | Lemma for ~ basel . Using... |
| basellem4 27109 | Lemma for ~ basel . By ~ ... |
| basellem5 27110 | Lemma for ~ basel . Using... |
| basellem6 27111 | Lemma for ~ basel . The f... |
| basellem7 27112 | Lemma for ~ basel . The f... |
| basellem8 27113 | Lemma for ~ basel . The f... |
| basellem9 27114 | Lemma for ~ basel . Since... |
| basel 27115 | The sum of the inverse squ... |
| efnnfsumcl 27128 | Finite sum closure in the ... |
| ppisval 27129 | The set of primes less tha... |
| ppisval2 27130 | The set of primes less tha... |
| ppifi 27131 | The set of primes less tha... |
| prmdvdsfi 27132 | The set of prime divisors ... |
| chtf 27133 | Domain and codoamin of the... |
| chtcl 27134 | Real closure of the Chebys... |
| chtval 27135 | Value of the Chebyshev fun... |
| efchtcl 27136 | The Chebyshev function is ... |
| chtge0 27137 | The Chebyshev function is ... |
| vmaval 27138 | Value of the von Mangoldt ... |
| isppw 27139 | Two ways to say that ` A `... |
| isppw2 27140 | Two ways to say that ` A `... |
| vmappw 27141 | Value of the von Mangoldt ... |
| vmaprm 27142 | Value of the von Mangoldt ... |
| vmacl 27143 | Closure for the von Mangol... |
| vmaf 27144 | Functionality of the von M... |
| efvmacl 27145 | The von Mangoldt is closed... |
| vmage0 27146 | The von Mangoldt function ... |
| chpval 27147 | Value of the second Chebys... |
| chpf 27148 | Functionality of the secon... |
| chpcl 27149 | Closure for the second Che... |
| efchpcl 27150 | The second Chebyshev funct... |
| chpge0 27151 | The second Chebyshev funct... |
| ppival 27152 | Value of the prime-countin... |
| ppival2 27153 | Value of the prime-countin... |
| ppival2g 27154 | Value of the prime-countin... |
| ppif 27155 | Domain and codomain of the... |
| ppicl 27156 | Real closure of the prime-... |
| muval 27157 | The value of the Möbi... |
| muval1 27158 | The value of the Möbi... |
| muval2 27159 | The value of the Möbi... |
| isnsqf 27160 | Two ways to say that a num... |
| issqf 27161 | Two ways to say that a num... |
| sqfpc 27162 | The prime count of a squar... |
| dvdssqf 27163 | A divisor of a squarefree ... |
| sqf11 27164 | A squarefree number is com... |
| muf 27165 | The Möbius function i... |
| mucl 27166 | Closure of the Möbius... |
| sgmval 27167 | The value of the divisor f... |
| sgmval2 27168 | The value of the divisor f... |
| 0sgm 27169 | The value of the sum-of-di... |
| sgmf 27170 | The divisor function is a ... |
| sgmcl 27171 | Closure of the divisor fun... |
| sgmnncl 27172 | Closure of the divisor fun... |
| mule1 27173 | The Möbius function t... |
| chtfl 27174 | The Chebyshev function doe... |
| chpfl 27175 | The second Chebyshev funct... |
| ppiprm 27176 | The prime-counting functio... |
| ppinprm 27177 | The prime-counting functio... |
| chtprm 27178 | The Chebyshev function at ... |
| chtnprm 27179 | The Chebyshev function at ... |
| chpp1 27180 | The second Chebyshev funct... |
| chtwordi 27181 | The Chebyshev function is ... |
| chpwordi 27182 | The second Chebyshev funct... |
| chtdif 27183 | The difference of the Cheb... |
| efchtdvds 27184 | The exponentiated Chebyshe... |
| ppifl 27185 | The prime-counting functio... |
| ppip1le 27186 | The prime-counting functio... |
| ppiwordi 27187 | The prime-counting functio... |
| ppidif 27188 | The difference of the prim... |
| ppi1 27189 | The prime-counting functio... |
| cht1 27190 | The Chebyshev function at ... |
| vma1 27191 | The von Mangoldt function ... |
| chp1 27192 | The second Chebyshev funct... |
| ppi1i 27193 | Inference form of ~ ppiprm... |
| ppi2i 27194 | Inference form of ~ ppinpr... |
| ppi2 27195 | The prime-counting functio... |
| ppi3 27196 | The prime-counting functio... |
| cht2 27197 | The Chebyshev function at ... |
| cht3 27198 | The Chebyshev function at ... |
| ppinncl 27199 | Closure of the prime-count... |
| chtrpcl 27200 | Closure of the Chebyshev f... |
| ppieq0 27201 | The prime-counting functio... |
| ppiltx 27202 | The prime-counting functio... |
| prmorcht 27203 | Relate the primorial (prod... |
| mumullem1 27204 | Lemma for ~ mumul . A mul... |
| mumullem2 27205 | Lemma for ~ mumul . The p... |
| mumul 27206 | The Möbius function i... |
| sqff1o 27207 | There is a bijection from ... |
| fsumdvdsdiaglem 27208 | A "diagonal commutation" o... |
| fsumdvdsdiag 27209 | A "diagonal commutation" o... |
| fsumdvdscom 27210 | A double commutation of di... |
| dvdsppwf1o 27211 | A bijection from the divis... |
| dvdsflf1o 27212 | A bijection from the numbe... |
| dvdsflsumcom 27213 | A sum commutation from ` s... |
| fsumfldivdiaglem 27214 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27215 | The right-hand side of ~ d... |
| musum 27216 | The sum of the Möbius... |
| musumsum 27217 | Evaluate a collapsing sum ... |
| muinv 27218 | The Möbius inversion ... |
| mpodvdsmulf1o 27219 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27220 | Product of two divisor sum... |
| dvdsmulf1o 27221 | If ` M ` and ` N ` are two... |
| fsumdvdsmulOLD 27222 | Obsolete version of ~ fsum... |
| sgmppw 27223 | The value of the divisor f... |
| 0sgmppw 27224 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27225 | The sum of divisors for a ... |
| 1sgm2ppw 27226 | The sum of the divisors of... |
| sgmmul 27227 | The divisor function for f... |
| ppiublem1 27228 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27229 | A prime greater than ` 3 `... |
| ppiub 27230 | An upper bound on the prim... |
| vmalelog 27231 | The von Mangoldt function ... |
| chtlepsi 27232 | The first Chebyshev functi... |
| chprpcl 27233 | Closure of the second Cheb... |
| chpeq0 27234 | The second Chebyshev funct... |
| chteq0 27235 | The first Chebyshev functi... |
| chtleppi 27236 | Upper bound on the ` theta... |
| chtublem 27237 | Lemma for ~ chtub . (Cont... |
| chtub 27238 | An upper bound on the Cheb... |
| fsumvma 27239 | Rewrite a sum over the von... |
| fsumvma2 27240 | Apply ~ fsumvma for the co... |
| pclogsum 27241 | The logarithmic analogue o... |
| vmasum 27242 | The sum of the von Mangold... |
| logfac2 27243 | Another expression for the... |
| chpval2 27244 | Express the second Chebysh... |
| chpchtsum 27245 | The second Chebyshev funct... |
| chpub 27246 | An upper bound on the seco... |
| logfacubnd 27247 | A simple upper bound on th... |
| logfaclbnd 27248 | A lower bound on the logar... |
| logfacbnd3 27249 | Show the stronger statemen... |
| logfacrlim 27250 | Combine the estimates ~ lo... |
| logexprlim 27251 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27252 | Write out ~ logfacrlim as ... |
| mersenne 27253 | A Mersenne prime is a prim... |
| perfect1 27254 | Euclid's contribution to t... |
| perfectlem1 27255 | Lemma for ~ perfect . (Co... |
| perfectlem2 27256 | Lemma for ~ perfect . (Co... |
| perfect 27257 | The Euclid-Euler theorem, ... |
| dchrval 27260 | Value of the group of Diri... |
| dchrbas 27261 | Base set of the group of D... |
| dchrelbas 27262 | A Dirichlet character is a... |
| dchrelbas2 27263 | A Dirichlet character is a... |
| dchrelbas3 27264 | A Dirichlet character is a... |
| dchrelbasd 27265 | A Dirichlet character is a... |
| dchrrcl 27266 | Reverse closure for a Diri... |
| dchrmhm 27267 | A Dirichlet character is a... |
| dchrf 27268 | A Dirichlet character is a... |
| dchrelbas4 27269 | A Dirichlet character is a... |
| dchrzrh1 27270 | Value of a Dirichlet chara... |
| dchrzrhcl 27271 | A Dirichlet character take... |
| dchrzrhmul 27272 | A Dirichlet character is c... |
| dchrplusg 27273 | Group operation on the gro... |
| dchrmul 27274 | Group operation on the gro... |
| dchrmulcl 27275 | Closure of the group opera... |
| dchrn0 27276 | A Dirichlet character is n... |
| dchr1cl 27277 | Closure of the principal D... |
| dchrmullid 27278 | Left identity for the prin... |
| dchrinvcl 27279 | Closure of the group inver... |
| dchrabl 27280 | The set of Dirichlet chara... |
| dchrfi 27281 | The group of Dirichlet cha... |
| dchrghm 27282 | A Dirichlet character rest... |
| dchr1 27283 | Value of the principal Dir... |
| dchreq 27284 | A Dirichlet character is d... |
| dchrresb 27285 | A Dirichlet character is d... |
| dchrabs 27286 | A Dirichlet character take... |
| dchrinv 27287 | The inverse of a Dirichlet... |
| dchrabs2 27288 | A Dirichlet character take... |
| dchr1re 27289 | The principal Dirichlet ch... |
| dchrptlem1 27290 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27291 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27292 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27293 | For any element other than... |
| dchrsum2 27294 | An orthogonality relation ... |
| dchrsum 27295 | An orthogonality relation ... |
| sumdchr2 27296 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27297 | There are exactly ` phi ( ... |
| sumdchr 27298 | An orthogonality relation ... |
| dchr2sum 27299 | An orthogonality relation ... |
| sum2dchr 27300 | An orthogonality relation ... |
| bcctr 27301 | Value of the central binom... |
| pcbcctr 27302 | Prime count of a central b... |
| bcmono 27303 | The binomial coefficient i... |
| bcmax 27304 | The binomial coefficient t... |
| bcp1ctr 27305 | Ratio of two central binom... |
| bclbnd 27306 | A bound on the binomial co... |
| efexple 27307 | Convert a bound on a power... |
| bpos1lem 27308 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27309 | Bertrand's postulate, chec... |
| bposlem1 27310 | An upper bound on the prim... |
| bposlem2 27311 | There are no odd primes in... |
| bposlem3 27312 | Lemma for ~ bpos . Since ... |
| bposlem4 27313 | Lemma for ~ bpos . (Contr... |
| bposlem5 27314 | Lemma for ~ bpos . Bound ... |
| bposlem6 27315 | Lemma for ~ bpos . By usi... |
| bposlem7 27316 | Lemma for ~ bpos . The fu... |
| bposlem8 27317 | Lemma for ~ bpos . Evalua... |
| bposlem9 27318 | Lemma for ~ bpos . Derive... |
| bpos 27319 | Bertrand's postulate: ther... |
| zabsle1 27322 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27323 | When ` a ` is coprime to t... |
| lgslem2 27324 | The set ` Z ` of all integ... |
| lgslem3 27325 | The set ` Z ` of all integ... |
| lgslem4 27326 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27327 | Value of the Legendre symb... |
| lgsfval 27328 | Value of the function ` F ... |
| lgsfcl2 27329 | The function ` F ` is clos... |
| lgscllem 27330 | The Legendre symbol is an ... |
| lgsfcl 27331 | Closure of the function ` ... |
| lgsfle1 27332 | The function ` F ` has mag... |
| lgsval2lem 27333 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27334 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27335 | The Legendre symbol is an ... |
| lgs0 27336 | The Legendre symbol when t... |
| lgscl 27337 | The Legendre symbol is an ... |
| lgsle1 27338 | The Legendre symbol has ab... |
| lgsval2 27339 | The Legendre symbol at a p... |
| lgs2 27340 | The Legendre symbol at ` 2... |
| lgsval3 27341 | The Legendre symbol at an ... |
| lgsvalmod 27342 | The Legendre symbol is equ... |
| lgsval4 27343 | Restate ~ lgsval for nonze... |
| lgsfcl3 27344 | Closure of the function ` ... |
| lgsval4a 27345 | Same as ~ lgsval4 for posi... |
| lgscl1 27346 | The value of the Legendre ... |
| lgsneg 27347 | The Legendre symbol is eit... |
| lgsneg1 27348 | The Legendre symbol for no... |
| lgsmod 27349 | The Legendre (Jacobi) symb... |
| lgsdilem 27350 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27351 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27352 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27353 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27354 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27355 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27356 | The Legendre symbol is com... |
| lgsdirprm 27357 | The Legendre symbol is com... |
| lgsdir 27358 | The Legendre symbol is com... |
| lgsdilem2 27359 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27360 | The Legendre symbol is com... |
| lgsne0 27361 | The Legendre symbol is non... |
| lgsabs1 27362 | The Legendre symbol is non... |
| lgssq 27363 | The Legendre symbol at a s... |
| lgssq2 27364 | The Legendre symbol at a s... |
| lgsprme0 27365 | The Legendre symbol at any... |
| 1lgs 27366 | The Legendre symbol at ` 1... |
| lgs1 27367 | The Legendre symbol at ` 1... |
| lgsmodeq 27368 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27369 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27370 | Variation on ~ lgsdir vali... |
| lgsdinn0 27371 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27372 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27373 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27374 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27375 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27376 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27377 | The Legendre symbol for od... |
| lgsqrmod 27378 | If the Legendre symbol of ... |
| lgsqrmodndvds 27379 | If the Legendre symbol of ... |
| lgsdchrval 27380 | The Legendre symbol functi... |
| lgsdchr 27381 | The Legendre symbol functi... |
| gausslemma2dlem0a 27382 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27383 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27384 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27385 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27386 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27387 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27388 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27389 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27390 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27391 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27392 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27393 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27394 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27395 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27396 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27397 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27398 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27399 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27400 | Gauss' Lemma (see also the... |
| lgseisenlem1 27401 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27402 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27403 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27404 | Lemma for ~ lgseisen . Th... |
| lgseisen 27405 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27406 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27407 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27408 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27409 | The Law of Quadratic Recip... |
| lgsquad2lem1 27410 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27411 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27412 | Extend ~ lgsquad to coprim... |
| lgsquad3 27413 | Extend ~ lgsquad2 to integ... |
| m1lgs 27414 | The first supplement to th... |
| 2lgslem1a1 27415 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27416 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27417 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27418 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27419 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27420 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27421 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27422 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27423 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27424 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27425 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27426 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27427 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27428 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27429 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27430 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27431 | The Legendre symbol for ` ... |
| 2lgslem4 27432 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27433 | The second supplement to t... |
| 2lgsoddprmlem1 27434 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27435 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27436 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27437 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27438 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27439 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27440 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27441 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27442 | The second supplement to t... |
| 2sqlem1 27443 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27444 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27445 | Fibonacci's identity (actu... |
| 2sqlem3 27446 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27447 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27448 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27449 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27450 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27451 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27452 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27453 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27454 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27455 | Lemma for ~ 2sq . (Contri... |
| 2sq 27456 | All primes of the form ` 4... |
| 2sqblem 27457 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27458 | The converse to ~ 2sq . (... |
| 2sq2 27459 | ` 2 ` is the sum of square... |
| 2sqn0 27460 | If the sum of two squares ... |
| 2sqcoprm 27461 | If the sum of two squares ... |
| 2sqmod 27462 | Given two decompositions o... |
| 2sqmo 27463 | There exists at most one d... |
| 2sqnn0 27464 | All primes of the form ` 4... |
| 2sqnn 27465 | All primes of the form ` 4... |
| addsq2reu 27466 | For each complex number ` ... |
| addsqn2reu 27467 | For each complex number ` ... |
| addsqrexnreu 27468 | For each complex number, t... |
| addsqnreup 27469 | There is no unique decompo... |
| addsq2nreurex 27470 | For each complex number ` ... |
| addsqn2reurex2 27471 | For each complex number ` ... |
| 2sqreulem1 27472 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27473 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27474 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27475 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27476 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27477 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27478 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27479 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27480 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27481 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27482 | There exists a unique deco... |
| 2sqreunn 27483 | There exists a unique deco... |
| 2sqreult 27484 | There exists a unique deco... |
| 2sqreultb 27485 | There exists a unique deco... |
| 2sqreunnlt 27486 | There exists a unique deco... |
| 2sqreunnltb 27487 | There exists a unique deco... |
| 2sqreuop 27488 | There exists a unique deco... |
| 2sqreuopnn 27489 | There exists a unique deco... |
| 2sqreuoplt 27490 | There exists a unique deco... |
| 2sqreuopltb 27491 | There exists a unique deco... |
| 2sqreuopnnlt 27492 | There exists a unique deco... |
| 2sqreuopnnltb 27493 | There exists a unique deco... |
| 2sqreuopb 27494 | There exists a unique deco... |
| chebbnd1lem1 27495 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27496 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27497 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27498 | The Chebyshev bound: The ... |
| chtppilimlem1 27499 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27500 | Lemma for ~ chtppilim . (... |
| chtppilim 27501 | The ` theta ` function is ... |
| chto1ub 27502 | The ` theta ` function is ... |
| chebbnd2 27503 | The Chebyshev bound, part ... |
| chto1lb 27504 | The ` theta ` function is ... |
| chpchtlim 27505 | The ` psi ` and ` theta ` ... |
| chpo1ub 27506 | The ` psi ` function is up... |
| chpo1ubb 27507 | The ` psi ` function is up... |
| vmadivsum 27508 | The sum of the von Mangold... |
| vmadivsumb 27509 | Give a total bound on the ... |
| rplogsumlem1 27510 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27511 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27512 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27513 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27514 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27515 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27516 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27517 | Lemma for ~ dchrisum . Le... |
| dchrisum 27518 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27519 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27520 | The sum of the Möbius... |
| dchrvmasumlem1 27521 | An alternative expression ... |
| dchrvmasum2lem 27522 | Give an expression for ` l... |
| dchrvmasum2if 27523 | Combine the results of ~ d... |
| dchrvmasumlem2 27524 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27525 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27526 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27527 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27528 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27529 | An asymptotic approximatio... |
| dchrvmaeq0 27530 | The set ` W ` is the colle... |
| dchrisum0fval 27531 | Value of the function ` F ... |
| dchrisum0fmul 27532 | The function ` F ` , the d... |
| dchrisum0ff 27533 | The function ` F ` is a re... |
| dchrisum0flblem1 27534 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27535 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27536 | The divisor sum of a real ... |
| dchrisum0fno1 27537 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27538 | A partial result along the... |
| dchrisum0re 27539 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27540 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27541 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27542 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27543 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27544 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27545 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27546 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27547 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27548 | The sum of the Möbius... |
| dchrvmasumlem 27549 | The sum of the Möbius... |
| dchrmusum 27550 | The sum of the Möbius... |
| dchrvmasum 27551 | The sum of the von Mangold... |
| rpvmasum 27552 | The sum of the von Mangold... |
| rplogsum 27553 | The sum of ` log p / p ` o... |
| dirith2 27554 | Dirichlet's theorem: there... |
| dirith 27555 | Dirichlet's theorem: there... |
| mudivsum 27556 | Asymptotic formula for ` s... |
| mulogsumlem 27557 | Lemma for ~ mulogsum . (C... |
| mulogsum 27558 | Asymptotic formula for ... |
| logdivsum 27559 | Asymptotic analysis of ... |
| mulog2sumlem1 27560 | Asymptotic formula for ... |
| mulog2sumlem2 27561 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27562 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27563 | Asymptotic formula for ... |
| vmalogdivsum2 27564 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27565 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27566 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27567 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27568 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27569 | The Möbius inverse of... |
| log2sumbnd 27570 | Bound on the difference be... |
| selberglem1 27571 | Lemma for ~ selberg . Est... |
| selberglem2 27572 | Lemma for ~ selberg . (Co... |
| selberglem3 27573 | Lemma for ~ selberg . Est... |
| selberg 27574 | Selberg's symmetry formula... |
| selbergb 27575 | Convert eventual boundedne... |
| selberg2lem 27576 | Lemma for ~ selberg2 . Eq... |
| selberg2 27577 | Selberg's symmetry formula... |
| selberg2b 27578 | Convert eventual boundedne... |
| chpdifbndlem1 27579 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27580 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27581 | A bound on the difference ... |
| logdivbnd 27582 | A bound on a sum of logs, ... |
| selberg3lem1 27583 | Introduce a log weighting ... |
| selberg3lem2 27584 | Lemma for ~ selberg3 . Eq... |
| selberg3 27585 | Introduce a log weighting ... |
| selberg4lem1 27586 | Lemma for ~ selberg4 . Eq... |
| selberg4 27587 | The Selberg symmetry formu... |
| pntrval 27588 | Define the residual of the... |
| pntrf 27589 | Functionality of the resid... |
| pntrmax 27590 | There is a bound on the re... |
| pntrsumo1 27591 | A bound on a sum over ` R ... |
| pntrsumbnd 27592 | A bound on a sum over ` R ... |
| pntrsumbnd2 27593 | A bound on a sum over ` R ... |
| selbergr 27594 | Selberg's symmetry formula... |
| selberg3r 27595 | Selberg's symmetry formula... |
| selberg4r 27596 | Selberg's symmetry formula... |
| selberg34r 27597 | The sum of ~ selberg3r and... |
| pntsval 27598 | Define the "Selberg functi... |
| pntsf 27599 | Functionality of the Selbe... |
| selbergs 27600 | Selberg's symmetry formula... |
| selbergsb 27601 | Selberg's symmetry formula... |
| pntsval2 27602 | The Selberg function can b... |
| pntrlog2bndlem1 27603 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27604 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27605 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27606 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27607 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27608 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27609 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27610 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27611 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27612 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27613 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27614 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27615 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27616 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27617 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27618 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27619 | Lemma for ~ pnt . Establi... |
| pntlemd 27620 | Lemma for ~ pnt . Closure... |
| pntlemc 27621 | Lemma for ~ pnt . Closure... |
| pntlema 27622 | Lemma for ~ pnt . Closure... |
| pntlemb 27623 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27624 | Lemma for ~ pnt . Closure... |
| pntlemh 27625 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27626 | Lemma for ~ pnt . The "na... |
| pntlemq 27627 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27628 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27629 | Lemma for ~ pnt . The ind... |
| pntlemi 27630 | Lemma for ~ pnt . Elimina... |
| pntlemf 27631 | Lemma for ~ pnt . Add up ... |
| pntlemk 27632 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27633 | Lemma for ~ pnt . Combine... |
| pntleme 27634 | Lemma for ~ pnt . Package... |
| pntlem3 27635 | Lemma for ~ pnt . Equatio... |
| pntlemp 27636 | Lemma for ~ pnt . Wrappin... |
| pntleml 27637 | Lemma for ~ pnt . Equatio... |
| pnt3 27638 | The Prime Number Theorem, ... |
| pnt2 27639 | The Prime Number Theorem, ... |
| pnt 27640 | The Prime Number Theorem: ... |
| abvcxp 27641 | Raising an absolute value ... |
| padicfval 27642 | Value of the p-adic absolu... |
| padicval 27643 | Value of the p-adic absolu... |
| ostth2lem1 27644 | Lemma for ~ ostth2 , altho... |
| qrngbas 27645 | The base set of the field ... |
| qdrng 27646 | The rationals form a divis... |
| qrng0 27647 | The zero element of the fi... |
| qrng1 27648 | The unity element of the f... |
| qrngneg 27649 | The additive inverse in th... |
| qrngdiv 27650 | The division operation in ... |
| qabvle 27651 | By using induction on ` N ... |
| qabvexp 27652 | Induct the product rule ~ ... |
| ostthlem1 27653 | Lemma for ~ ostth . If tw... |
| ostthlem2 27654 | Lemma for ~ ostth . Refin... |
| qabsabv 27655 | The regular absolute value... |
| padicabv 27656 | The p-adic absolute value ... |
| padicabvf 27657 | The p-adic absolute value ... |
| padicabvcxp 27658 | All positive powers of the... |
| ostth1 27659 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27660 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27661 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27662 | Lemma for ~ ostth2 . (Con... |
| ostth2 27663 | - Lemma for ~ ostth : regu... |
| ostth3 27664 | - Lemma for ~ ostth : p-ad... |
| ostth 27665 | Ostrowski's theorem, which... |
| elno 27672 | Membership in the surreals... |
| elnoOLD 27673 | Obsolete version of ~ elno... |
| sltval 27674 | The value of the surreal l... |
| bdayval 27675 | The value of the birthday ... |
| nofun 27676 | A surreal is a function. ... |
| nodmon 27677 | The domain of a surreal is... |
| norn 27678 | The range of a surreal is ... |
| nofnbday 27679 | A surreal is a function ov... |
| nodmord 27680 | The domain of a surreal ha... |
| elno2 27681 | An alternative condition f... |
| elno3 27682 | Another condition for memb... |
| sltval2 27683 | Alternate expression for s... |
| nofv 27684 | The function value of a su... |
| nosgnn0 27685 | ` (/) ` is not a surreal s... |
| nosgnn0i 27686 | If ` X ` is a surreal sign... |
| noreson 27687 | The restriction of a surre... |
| sltintdifex 27688 |
If ` A |
| sltres 27689 | If the restrictions of two... |
| noxp1o 27690 | The Cartesian product of a... |
| noseponlem 27691 | Lemma for ~ nosepon . Con... |
| nosepon 27692 | Given two unequal surreals... |
| noextend 27693 | Extending a surreal by one... |
| noextendseq 27694 | Extend a surreal by a sequ... |
| noextenddif 27695 | Calculate the place where ... |
| noextendlt 27696 | Extending a surreal with a... |
| noextendgt 27697 | Extending a surreal with a... |
| nolesgn2o 27698 | Given ` A ` less-than or e... |
| nolesgn2ores 27699 | Given ` A ` less-than or e... |
| nogesgn1o 27700 | Given ` A ` greater than o... |
| nogesgn1ores 27701 | Given ` A ` greater than o... |
| sltsolem1 27702 | Lemma for ~ sltso . The "... |
| sltso 27703 | Less-than totally orders t... |
| bdayfo 27704 | The birthday function maps... |
| fvnobday 27705 | The value of a surreal at ... |
| nosepnelem 27706 | Lemma for ~ nosepne . (Co... |
| nosepne 27707 | The value of two non-equal... |
| nosep1o 27708 | If the value of a surreal ... |
| nosep2o 27709 | If the value of a surreal ... |
| nosepdmlem 27710 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27711 | The first place two surrea... |
| nosepeq 27712 | The values of two surreals... |
| nosepssdm 27713 | Given two non-equal surrea... |
| nodenselem4 27714 | Lemma for ~ nodense . Sho... |
| nodenselem5 27715 | Lemma for ~ nodense . If ... |
| nodenselem6 27716 | The restriction of a surre... |
| nodenselem7 27717 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27718 | Lemma for ~ nodense . Giv... |
| nodense 27719 | Given two distinct surreal... |
| bdayimaon 27720 | Lemma for full-eta propert... |
| nolt02olem 27721 | Lemma for ~ nolt02o . If ... |
| nolt02o 27722 | Given ` A ` less-than ` B ... |
| nogt01o 27723 | Given ` A ` greater than `... |
| noresle 27724 | Restriction law for surrea... |
| nomaxmo 27725 | A class of surreals has at... |
| nominmo 27726 | A class of surreals has at... |
| nosupprefixmo 27727 | In any class of surreals, ... |
| noinfprefixmo 27728 | In any class of surreals, ... |
| nosupcbv 27729 | Lemma to change bound vari... |
| nosupno 27730 | The next several theorems ... |
| nosupdm 27731 | The domain of the surreal ... |
| nosupbday 27732 | Birthday bounding law for ... |
| nosupfv 27733 | The value of surreal supre... |
| nosupres 27734 | A restriction law for surr... |
| nosupbnd1lem1 27735 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27736 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27737 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27738 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27739 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27740 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27741 | Bounding law from below fo... |
| nosupbnd2lem1 27742 | Bounding law from above wh... |
| nosupbnd2 27743 | Bounding law from above fo... |
| noinfcbv 27744 | Change bound variables for... |
| noinfno 27745 | The next several theorems ... |
| noinfdm 27746 | Next, we calculate the dom... |
| noinfbday 27747 | Birthday bounding law for ... |
| noinffv 27748 | The value of surreal infim... |
| noinfres 27749 | The restriction of surreal... |
| noinfbnd1lem1 27750 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27751 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27752 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27753 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27754 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27755 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27756 | Bounding law from above fo... |
| noinfbnd2lem1 27757 | Bounding law from below wh... |
| noinfbnd2 27758 | Bounding law from below fo... |
| nosupinfsep 27759 | Given two sets of surreals... |
| noetasuplem1 27760 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27761 | Lemma for ~ noeta . The r... |
| noetasuplem3 27762 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27763 | Lemma for ~ noeta . When ... |
| noetainflem1 27764 | Lemma for ~ noeta . Estab... |
| noetainflem2 27765 | Lemma for ~ noeta . The r... |
| noetainflem3 27766 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27767 | Lemma for ~ noeta . If ` ... |
| noetalem1 27768 | Lemma for ~ noeta . Eithe... |
| noetalem2 27769 | Lemma for ~ noeta . The f... |
| noeta 27770 | The full-eta axiom for the... |
| sltirr 27773 | Surreal less-than is irref... |
| slttr 27774 | Surreal less-than is trans... |
| sltasym 27775 | Surreal less-than is asymm... |
| sltlin 27776 | Surreal less-than obeys tr... |
| slttrieq2 27777 | Trichotomy law for surreal... |
| slttrine 27778 | Trichotomy law for surreal... |
| slenlt 27779 | Surreal less-than or equal... |
| sltnle 27780 | Surreal less-than in terms... |
| sleloe 27781 | Surreal less-than or equal... |
| sletri3 27782 | Trichotomy law for surreal... |
| sltletr 27783 | Surreal transitive law. (... |
| slelttr 27784 | Surreal transitive law. (... |
| sletr 27785 | Surreal transitive law. (... |
| slttrd 27786 | Surreal less-than is trans... |
| sltletrd 27787 | Surreal less-than is trans... |
| slelttrd 27788 | Surreal less-than is trans... |
| sletrd 27789 | Surreal less-than or equal... |
| slerflex 27790 | Surreal less-than or equal... |
| sletric 27791 | Surreal trichotomy law. (... |
| maxs1 27792 | A surreal is less than or ... |
| maxs2 27793 | A surreal is less than or ... |
| mins1 27794 | The minimum of two surreal... |
| mins2 27795 | The minimum of two surreal... |
| sltled 27796 | Surreal less-than implies ... |
| sltne 27797 | Surreal less-than implies ... |
| sltlend 27798 | Surreal less-than in terms... |
| bdayfun 27799 | The birthday function is a... |
| bdayfn 27800 | The birthday function is a... |
| bdaydm 27801 | The birthday function's do... |
| bdayrn 27802 | The birthday function's ra... |
| bdayelon 27803 | The value of the birthday ... |
| nocvxminlem 27804 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27805 | Given a nonempty convex cl... |
| noprc 27806 | The surreal numbers are a ... |
| noeta2 27811 | A version of ~ noeta with ... |
| brsslt 27812 | Binary relation form of th... |
| ssltex1 27813 | The first argument of surr... |
| ssltex2 27814 | The second argument of sur... |
| ssltss1 27815 | The first argument of surr... |
| ssltss2 27816 | The second argument of sur... |
| ssltsep 27817 | The separation property of... |
| ssltd 27818 | Deduce surreal set less-th... |
| ssltsn 27819 | Surreal set less-than of t... |
| ssltsepc 27820 | Two elements of separated ... |
| ssltsepcd 27821 | Two elements of separated ... |
| sssslt1 27822 | Relation between surreal s... |
| sssslt2 27823 | Relation between surreal s... |
| nulsslt 27824 | The empty set is less-than... |
| nulssgt 27825 | The empty set is greater t... |
| conway 27826 | Conway's Simplicity Theore... |
| scutval 27827 | The value of the surreal c... |
| scutcut 27828 | Cut properties of the surr... |
| scutcl 27829 | Closure law for surreal cu... |
| scutcld 27830 | Closure law for surreal cu... |
| scutbday 27831 | The birthday of the surrea... |
| eqscut 27832 | Condition for equality to ... |
| eqscut2 27833 | Condition for equality to ... |
| sslttr 27834 | Transitive law for surreal... |
| ssltun1 27835 | Union law for surreal set ... |
| ssltun2 27836 | Union law for surreal set ... |
| scutun12 27837 | Union law for surreal cuts... |
| dmscut 27838 | The domain of the surreal ... |
| scutf 27839 | Functionality statement fo... |
| etasslt 27840 | A restatement of ~ noeta u... |
| etasslt2 27841 | A version of ~ etasslt wit... |
| scutbdaybnd 27842 | An upper bound on the birt... |
| scutbdaybnd2 27843 | An upper bound on the birt... |
| scutbdaybnd2lim 27844 | An upper bound on the birt... |
| scutbdaylt 27845 | If a surreal lies in a gap... |
| slerec 27846 | A comparison law for surre... |
| sltrec 27847 | A comparison law for surre... |
| ssltdisj 27848 | If ` A ` preceeds ` B ` , ... |
| 0sno 27853 | Surreal zero is a surreal.... |
| 1sno 27854 | Surreal one is a surreal. ... |
| bday0s 27855 | Calculate the birthday of ... |
| 0slt1s 27856 | Surreal zero is less than ... |
| bday0b 27857 | The only surreal with birt... |
| bday1s 27858 | The birthday of surreal on... |
| cuteq0 27859 | Condition for a surreal cu... |
| cuteq1 27860 | Condition for a surreal cu... |
| sgt0ne0 27861 | A positive surreal is not ... |
| sgt0ne0d 27862 | A positive surreal is not ... |
| madeval 27873 | The value of the made by f... |
| madeval2 27874 | Alternative characterizati... |
| oldval 27875 | The value of the old optio... |
| newval 27876 | The value of the new optio... |
| madef 27877 | The made function is a fun... |
| oldf 27878 | The older function is a fu... |
| newf 27879 | The new function is a func... |
| old0 27880 | No surreal is older than `... |
| madessno 27881 | Made sets are surreals. (... |
| oldssno 27882 | Old sets are surreals. (C... |
| newssno 27883 | New sets are surreals. (C... |
| leftval 27884 | The value of the left opti... |
| rightval 27885 | The value of the right opt... |
| leftf 27886 | The functionality of the l... |
| rightf 27887 | The functionality of the r... |
| elmade 27888 | Membership in the made fun... |
| elmade2 27889 | Membership in the made fun... |
| elold 27890 | Membership in an old set. ... |
| ssltleft 27891 | A surreal is greater than ... |
| ssltright 27892 | A surreal is less than its... |
| lltropt 27893 | The left options of a surr... |
| made0 27894 | The only surreal made on d... |
| new0 27895 | The only surreal new on da... |
| old1 27896 | The only surreal older tha... |
| madess 27897 | If ` A ` is less than or e... |
| oldssmade 27898 | The older-than set is a su... |
| leftssold 27899 | The left options are a sub... |
| rightssold 27900 | The right options are a su... |
| leftssno 27901 | The left set of a surreal ... |
| rightssno 27902 | The right set of a surreal... |
| madecut 27903 | Given a section that is a ... |
| madeun 27904 | The made set is the union ... |
| madeoldsuc 27905 | The made set is the old se... |
| oldsuc 27906 | The value of the old set a... |
| oldlim 27907 | The value of the old set a... |
| madebdayim 27908 | If a surreal is a member o... |
| oldbdayim 27909 | If ` X ` is in the old set... |
| oldirr 27910 | No surreal is a member of ... |
| leftirr 27911 | No surreal is a member of ... |
| rightirr 27912 | No surreal is a member of ... |
| left0s 27913 | The left set of ` 0s ` is ... |
| right0s 27914 | The right set of ` 0s ` is... |
| left1s 27915 | The left set of ` 1s ` is ... |
| right1s 27916 | The right set of ` 1s ` is... |
| lrold 27917 | The union of the left and ... |
| madebdaylemold 27918 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27919 | Lemma for ~ madebday . If... |
| madebday 27920 | A surreal is part of the s... |
| oldbday 27921 | A surreal is part of the s... |
| newbday 27922 | A surreal is an element of... |
| lrcut 27923 | A surreal is equal to the ... |
| scutfo 27924 | The surreal cut function i... |
| sltn0 27925 | If ` X ` is less than ` Y ... |
| lruneq 27926 | If two surreals share a bi... |
| sltlpss 27927 | If two surreals share a bi... |
| slelss 27928 | If two surreals ` A ` and ... |
| 0elold 27929 | Zero is in the old set of ... |
| 0elleft 27930 | Zero is in the left set of... |
| 0elright 27931 | Zero is in the right set o... |
| cofsslt 27932 | If every element of ` A ` ... |
| coinitsslt 27933 | If ` B ` is coinitial with... |
| cofcut1 27934 | If ` C ` is cofinal with `... |
| cofcut1d 27935 | If ` C ` is cofinal with `... |
| cofcut2 27936 | If ` A ` and ` C ` are mut... |
| cofcut2d 27937 | If ` A ` and ` C ` are mut... |
| cofcutr 27938 | If ` X ` is the cut of ` A... |
| cofcutr1d 27939 | If ` X ` is the cut of ` A... |
| cofcutr2d 27940 | If ` X ` is the cut of ` A... |
| cofcutrtime 27941 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27942 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27943 | If ` X ` is a timely cut o... |
| cofss 27944 | Cofinality for a subset. ... |
| coiniss 27945 | Coinitiality for a subset.... |
| cutlt 27946 | Eliminating all elements b... |
| cutpos 27947 | Reduce the elements of a c... |
| lrrecval 27950 | The next step in the devel... |
| lrrecval2 27951 | Next, we establish an alte... |
| lrrecpo 27952 | Now, we establish that ` R... |
| lrrecse 27953 | Next, we show that ` R ` i... |
| lrrecfr 27954 | Now we show that ` R ` is ... |
| lrrecpred 27955 | Finally, we calculate the ... |
| noinds 27956 | Induction principle for a ... |
| norecfn 27957 | Surreal recursion over one... |
| norecov 27958 | Calculate the value of the... |
| noxpordpo 27961 | To get through most of the... |
| noxpordfr 27962 | Next we establish the foun... |
| noxpordse 27963 | Next we establish the set-... |
| noxpordpred 27964 | Next we calculate the pred... |
| no2indslem 27965 | Double induction on surrea... |
| no2inds 27966 | Double induction on surrea... |
| norec2fn 27967 | The double-recursion opera... |
| norec2ov 27968 | The value of the double-re... |
| no3inds 27969 | Triple induction over surr... |
| addsfn 27972 | Surreal addition is a func... |
| addsval 27973 | The value of surreal addit... |
| addsval2 27974 | The value of surreal addit... |
| addsrid 27975 | Surreal addition to zero i... |
| addsridd 27976 | Surreal addition to zero i... |
| addscom 27977 | Surreal addition commutes.... |
| addscomd 27978 | Surreal addition commutes.... |
| addslid 27979 | Surreal addition to zero i... |
| addsproplem1 27980 | Lemma for surreal addition... |
| addsproplem2 27981 | Lemma for surreal addition... |
| addsproplem3 27982 | Lemma for surreal addition... |
| addsproplem4 27983 | Lemma for surreal addition... |
| addsproplem5 27984 | Lemma for surreal addition... |
| addsproplem6 27985 | Lemma for surreal addition... |
| addsproplem7 27986 | Lemma for surreal addition... |
| addsprop 27987 | Inductively show that surr... |
| addscutlem 27988 | Lemma for ~ addscut . Sho... |
| addscut 27989 | Demonstrate the cut proper... |
| addscut2 27990 | Show that the cut involved... |
| addscld 27991 | Surreal numbers are closed... |
| addscl 27992 | Surreal numbers are closed... |
| addsf 27993 | Function statement for sur... |
| addsfo 27994 | Surreal addition is onto. ... |
| peano2no 27995 | A theorem for surreals tha... |
| sltadd1im 27996 | Surreal less-than is prese... |
| sltadd2im 27997 | Surreal less-than is prese... |
| sleadd1im 27998 | Surreal less-than or equal... |
| sleadd2im 27999 | Surreal less-than or equal... |
| sleadd1 28000 | Addition to both sides of ... |
| sleadd2 28001 | Addition to both sides of ... |
| sltadd2 28002 | Addition to both sides of ... |
| sltadd1 28003 | Addition to both sides of ... |
| addscan2 28004 | Cancellation law for surre... |
| addscan1 28005 | Cancellation law for surre... |
| sleadd1d 28006 | Addition to both sides of ... |
| sleadd2d 28007 | Addition to both sides of ... |
| sltadd2d 28008 | Addition to both sides of ... |
| sltadd1d 28009 | Addition to both sides of ... |
| addscan2d 28010 | Cancellation law for surre... |
| addscan1d 28011 | Cancellation law for surre... |
| addsuniflem 28012 | Lemma for ~ addsunif . St... |
| addsunif 28013 | Uniformity theorem for sur... |
| addsasslem1 28014 | Lemma for addition associa... |
| addsasslem2 28015 | Lemma for addition associa... |
| addsass 28016 | Surreal addition is associ... |
| addsassd 28017 | Surreal addition is associ... |
| adds32d 28018 | Commutative/associative la... |
| adds12d 28019 | Commutative/associative la... |
| adds4d 28020 | Rearrangement of four term... |
| adds42d 28021 | Rearrangement of four term... |
| sltaddpos1d 28022 | Addition of a positive num... |
| sltaddpos2d 28023 | Addition of a positive num... |
| slt2addd 28024 | Adding both sides of two s... |
| addsgt0d 28025 | The sum of two positive su... |
| negsfn 28030 | Surreal negation is a func... |
| subsfn 28031 | Surreal subtraction is a f... |
| negsval 28032 | The value of the surreal n... |
| negs0s 28033 | Negative surreal zero is s... |
| negs1s 28034 | An expression for negative... |
| negsproplem1 28035 | Lemma for surreal negation... |
| negsproplem2 28036 | Lemma for surreal negation... |
| negsproplem3 28037 | Lemma for surreal negation... |
| negsproplem4 28038 | Lemma for surreal negation... |
| negsproplem5 28039 | Lemma for surreal negation... |
| negsproplem6 28040 | Lemma for surreal negation... |
| negsproplem7 28041 | Lemma for surreal negation... |
| negsprop 28042 | Show closure and ordering ... |
| negscl 28043 | The surreals are closed un... |
| negscld 28044 | The surreals are closed un... |
| sltnegim 28045 | The forward direction of t... |
| negscut 28046 | The cut properties of surr... |
| negscut2 28047 | The cut that defines surre... |
| negsid 28048 | Surreal addition of a numb... |
| negsidd 28049 | Surreal addition of a numb... |
| negsex 28050 | Every surreal has a negati... |
| negnegs 28051 | A surreal is equal to the ... |
| sltneg 28052 | Negative of both sides of ... |
| sleneg 28053 | Negative of both sides of ... |
| sltnegd 28054 | Negative of both sides of ... |
| slenegd 28055 | Negative of both sides of ... |
| negs11 28056 | Surreal negation is one-to... |
| negsdi 28057 | Distribution of surreal ne... |
| slt0neg2d 28058 | Comparison of a surreal an... |
| negsf 28059 | Function statement for sur... |
| negsfo 28060 | Function statement for sur... |
| negsf1o 28061 | Surreal negation is a bije... |
| negsunif 28062 | Uniformity property for su... |
| negsbdaylem 28063 | Lemma for ~ negsbday . Bo... |
| negsbday 28064 | Negation of a surreal numb... |
| subsval 28065 | The value of surreal subtr... |
| subsvald 28066 | The value of surreal subtr... |
| subscl 28067 | Closure law for surreal su... |
| subscld 28068 | Closure law for surreal su... |
| subsf 28069 | Function statement for sur... |
| subsfo 28070 | Surreal subtraction is an ... |
| negsval2 28071 | Surreal negation in terms ... |
| negsval2d 28072 | Surreal negation in terms ... |
| subsid1 28073 | Identity law for subtracti... |
| subsid 28074 | Subtraction of a surreal f... |
| subadds 28075 | Relationship between addit... |
| subaddsd 28076 | Relationship between addit... |
| pncans 28077 | Cancellation law for surre... |
| pncan3s 28078 | Subtraction and addition o... |
| pncan2s 28079 | Cancellation law for surre... |
| npcans 28080 | Cancellation law for surre... |
| sltsub1 28081 | Subtraction from both side... |
| sltsub2 28082 | Subtraction from both side... |
| sltsub1d 28083 | Subtraction from both side... |
| sltsub2d 28084 | Subtraction from both side... |
| negsubsdi2d 28085 | Distribution of negative o... |
| addsubsassd 28086 | Associative-type law for s... |
| addsubsd 28087 | Law for surreal addition a... |
| sltsubsubbd 28088 | Equivalence for the surrea... |
| sltsubsub2bd 28089 | Equivalence for the surrea... |
| sltsubsub3bd 28090 | Equivalence for the surrea... |
| slesubsubbd 28091 | Equivalence for the surrea... |
| slesubsub2bd 28092 | Equivalence for the surrea... |
| slesubsub3bd 28093 | Equivalence for the surrea... |
| sltsubaddd 28094 | Surreal less-than relation... |
| sltsubadd2d 28095 | Surreal less-than relation... |
| sltaddsubd 28096 | Surreal less-than relation... |
| sltaddsub2d 28097 | Surreal less-than relation... |
| slesubaddd 28098 | Surreal less-than or equal... |
| subsubs4d 28099 | Law for double surreal sub... |
| subsubs2d 28100 | Law for double surreal sub... |
| nncansd 28101 | Cancellation law for surre... |
| posdifsd 28102 | Comparison of two surreals... |
| sltsubposd 28103 | Subtraction of a positive ... |
| subsge0d 28104 | Non-negative subtraction. ... |
| addsubs4d 28105 | Rearrangement of four term... |
| mulsfn 28108 | Surreal multiplication is ... |
| mulsval 28109 | The value of surreal multi... |
| mulsval2lem 28110 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28111 | The value of surreal multi... |
| muls01 28112 | Surreal multiplication by ... |
| mulsrid 28113 | Surreal one is a right ide... |
| mulsridd 28114 | Surreal one is a right ide... |
| mulsproplemcbv 28115 | Lemma for surreal multipli... |
| mulsproplem1 28116 | Lemma for surreal multipli... |
| mulsproplem2 28117 | Lemma for surreal multipli... |
| mulsproplem3 28118 | Lemma for surreal multipli... |
| mulsproplem4 28119 | Lemma for surreal multipli... |
| mulsproplem5 28120 | Lemma for surreal multipli... |
| mulsproplem6 28121 | Lemma for surreal multipli... |
| mulsproplem7 28122 | Lemma for surreal multipli... |
| mulsproplem8 28123 | Lemma for surreal multipli... |
| mulsproplem9 28124 | Lemma for surreal multipli... |
| mulsproplem10 28125 | Lemma for surreal multipli... |
| mulsproplem11 28126 | Lemma for surreal multipli... |
| mulsproplem12 28127 | Lemma for surreal multipli... |
| mulsproplem13 28128 | Lemma for surreal multipli... |
| mulsproplem14 28129 | Lemma for surreal multipli... |
| mulsprop 28130 | Surreals are closed under ... |
| mulscutlem 28131 | Lemma for ~ mulscut . Sta... |
| mulscut 28132 | Show the cut properties of... |
| mulscut2 28133 | Show that the cut involved... |
| mulscl 28134 | The surreals are closed un... |
| mulscld 28135 | The surreals are closed un... |
| sltmul 28136 | An ordering relationship f... |
| sltmuld 28137 | An ordering relationship f... |
| slemuld 28138 | An ordering relationship f... |
| mulscom 28139 | Surreal multiplication com... |
| mulscomd 28140 | Surreal multiplication com... |
| muls02 28141 | Surreal multiplication by ... |
| mulslid 28142 | Surreal one is a left iden... |
| mulslidd 28143 | Surreal one is a left iden... |
| mulsgt0 28144 | The product of two positiv... |
| mulsgt0d 28145 | The product of two positiv... |
| mulsge0d 28146 | The product of two non-neg... |
| ssltmul1 28147 | One surreal set less-than ... |
| ssltmul2 28148 | One surreal set less-than ... |
| mulsuniflem 28149 | Lemma for ~ mulsunif . St... |
| mulsunif 28150 | Surreal multiplication has... |
| addsdilem1 28151 | Lemma for surreal distribu... |
| addsdilem2 28152 | Lemma for surreal distribu... |
| addsdilem3 28153 | Lemma for ~ addsdi . Show... |
| addsdilem4 28154 | Lemma for ~ addsdi . Show... |
| addsdi 28155 | Distributive law for surre... |
| addsdid 28156 | Distributive law for surre... |
| addsdird 28157 | Distributive law for surre... |
| subsdid 28158 | Distribution of surreal mu... |
| subsdird 28159 | Distribution of surreal mu... |
| mulnegs1d 28160 | Product with negative is n... |
| mulnegs2d 28161 | Product with negative is n... |
| mul2negsd 28162 | Surreal product of two neg... |
| mulsasslem1 28163 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28164 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28165 | Lemma for ~ mulsass . Dem... |
| mulsass 28166 | Associative law for surrea... |
| mulsassd 28167 | Associative law for surrea... |
| muls4d 28168 | Rearrangement of four surr... |
| mulsunif2lem 28169 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28170 | Alternate expression for s... |
| sltmul2 28171 | Multiplication of both sid... |
| sltmul2d 28172 | Multiplication of both sid... |
| sltmul1d 28173 | Multiplication of both sid... |
| slemul2d 28174 | Multiplication of both sid... |
| slemul1d 28175 | Multiplication of both sid... |
| sltmulneg1d 28176 | Multiplication of both sid... |
| sltmulneg2d 28177 | Multiplication of both sid... |
| mulscan2dlem 28178 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28179 | Cancellation of surreal mu... |
| mulscan1d 28180 | Cancellation of surreal mu... |
| muls12d 28181 | Commutative/associative la... |
| slemul1ad 28182 | Multiplication of both sid... |
| sltmul12ad 28183 | Comparison of the product ... |
| divsmo 28184 | Uniqueness of surreal inve... |
| muls0ord 28185 | If a surreal product is ze... |
| mulsne0bd 28186 | The product of two non-zer... |
| divsval 28189 | The value of surreal divis... |
| norecdiv 28190 | If a surreal has a recipro... |
| noreceuw 28191 | If a surreal has a recipro... |
| divsmulw 28192 | Relationship between surre... |
| divsmulwd 28193 | Relationship between surre... |
| divsclw 28194 | Weak division closure law.... |
| divsclwd 28195 | Weak division closure law.... |
| divscan2wd 28196 | A weak cancellation law fo... |
| divscan1wd 28197 | A weak cancellation law fo... |
| sltdivmulwd 28198 | Surreal less-than relation... |
| sltdivmul2wd 28199 | Surreal less-than relation... |
| sltmuldivwd 28200 | Surreal less-than relation... |
| sltmuldiv2wd 28201 | Surreal less-than relation... |
| divsasswd 28202 | An associative law for sur... |
| divs1 28203 | A surreal divided by one i... |
| precsexlemcbv 28204 | Lemma for surreal reciproc... |
| precsexlem1 28205 | Lemma for surreal reciproc... |
| precsexlem2 28206 | Lemma for surreal reciproc... |
| precsexlem3 28207 | Lemma for surreal reciproc... |
| precsexlem4 28208 | Lemma for surreal reciproc... |
| precsexlem5 28209 | Lemma for surreal reciproc... |
| precsexlem6 28210 | Lemma for surreal reciproc... |
| precsexlem7 28211 | Lemma for surreal reciproc... |
| precsexlem8 28212 | Lemma for surreal reciproc... |
| precsexlem9 28213 | Lemma for surreal reciproc... |
| precsexlem10 28214 | Lemma for surreal reciproc... |
| precsexlem11 28215 | Lemma for surreal reciproc... |
| precsex 28216 | Every positive surreal has... |
| recsex 28217 | A non-zero surreal has a r... |
| recsexd 28218 | A non-zero surreal has a r... |
| divsmul 28219 | Relationship between surre... |
| divsmuld 28220 | Relationship between surre... |
| divscl 28221 | Surreal division closure l... |
| divscld 28222 | Surreal division closure l... |
| divscan2d 28223 | A cancellation law for sur... |
| divscan1d 28224 | A cancellation law for sur... |
| sltdivmuld 28225 | Surreal less-than relation... |
| sltdivmul2d 28226 | Surreal less-than relation... |
| sltmuldivd 28227 | Surreal less-than relation... |
| sltmuldiv2d 28228 | Surreal less-than relation... |
| divsassd 28229 | An associative law for sur... |
| divmuldivsd 28230 | Multiplication of two surr... |
| divdivs1d 28231 | Surreal division into a fr... |
| abssval 28234 | The value of surreal absol... |
| absscl 28235 | Closure law for surreal ab... |
| abssid 28236 | The absolute value of a no... |
| abs0s 28237 | The absolute value of surr... |
| abssnid 28238 | For a negative surreal, it... |
| absmuls 28239 | Surreal absolute value dis... |
| abssge0 28240 | The absolute value of a su... |
| abssor 28241 | The absolute value of a su... |
| abssneg 28242 | Surreal absolute value of ... |
| sleabs 28243 | A surreal is less than or ... |
| absslt 28244 | Surreal absolute value and... |
| elons 28247 | Membership in the class of... |
| onssno 28248 | The surreal ordinals are a... |
| onsno 28249 | A surreal ordinal is a sur... |
| 0ons 28250 | Surreal zero is a surreal ... |
| 1ons 28251 | Surreal one is a surreal o... |
| elons2 28252 | A surreal is ordinal iff i... |
| elons2d 28253 | The cut of any set of surr... |
| sltonold 28254 | The class of ordinals less... |
| sltonex 28255 | The class of ordinals less... |
| onscutleft 28256 | A surreal ordinal is equal... |
| seqsex 28259 | Existence of the surreal s... |
| seqseq123d 28260 | Equality deduction for the... |
| nfseqs 28261 | Hypothesis builder for the... |
| seqsval 28262 | The value of the surreal s... |
| noseqex 28263 | The next several theorems ... |
| noseq0 28264 | The surreal ` A ` is a mem... |
| noseqp1 28265 | One plus an element of ` Z... |
| noseqind 28266 | Peano's inductive postulat... |
| noseqinds 28267 | Induction schema for surre... |
| noseqssno 28268 | A surreal sequence is a su... |
| noseqno 28269 | An element of a surreal se... |
| om2noseq0 28270 | The mapping ` G ` is a one... |
| om2noseqsuc 28271 | The value of ` G ` at a su... |
| om2noseqfo 28272 | Function statement for ` G... |
| om2noseqlt 28273 | Surreal less-than relation... |
| om2noseqlt2 28274 | The mapping ` G ` preserve... |
| om2noseqf1o 28275 | ` G ` is a bijection. (Co... |
| om2noseqiso 28276 | ` G ` is an isomorphism fr... |
| om2noseqoi 28277 | An alternative definition ... |
| om2noseqrdg 28278 | A helper lemma for the val... |
| noseqrdglem 28279 | A helper lemma for the val... |
| noseqrdgfn 28280 | The recursive definition g... |
| noseqrdg0 28281 | Initial value of a recursi... |
| noseqrdgsuc 28282 | Successor value of a recur... |
| seqsfn 28283 | The surreal sequence build... |
| seqs1 28284 | The value of the surreal s... |
| seqsp1 28285 | The value of the surreal s... |
| n0sex 28290 | The set of all non-negativ... |
| nnsex 28291 | The set of all positive su... |
| peano5n0s 28292 | Peano's inductive postulat... |
| n0ssno 28293 | The non-negative surreal i... |
| nnssn0s 28294 | The positive surreal integ... |
| nnssno 28295 | The positive surreal integ... |
| n0sno 28296 | A non-negative surreal int... |
| nnsno 28297 | A positive surreal integer... |
| n0snod 28298 | A non-negative surreal int... |
| nnsnod 28299 | A positive surreal integer... |
| nnn0s 28300 | A positive surreal integer... |
| nnn0sd 28301 | A positive surreal integer... |
| 0n0s 28302 | Peano postulate: ` 0s ` is... |
| peano2n0s 28303 | Peano postulate: the succe... |
| dfn0s2 28304 | Alternate definition of th... |
| n0sind 28305 | Principle of Mathematical ... |
| n0scut 28306 | A cut form for surreal nat... |
| n0ons 28307 | A surreal natural is a sur... |
| nnne0s 28308 | A surreal positive integer... |
| n0sge0 28309 | A non-negative integer is ... |
| nnsgt0 28310 | A positive integer is grea... |
| elnns 28311 | Membership in the positive... |
| elnns2 28312 | A positive surreal integer... |
| n0s0suc 28313 | A non-negative surreal int... |
| nnsge1 28314 | A positive surreal integer... |
| n0addscl 28315 | The non-negative surreal i... |
| n0mulscl 28316 | The non-negative surreal i... |
| nnaddscl 28317 | The positive surreal integ... |
| nnmulscl 28318 | The positive surreal integ... |
| 1n0s 28319 | Surreal one is a non-negat... |
| 1nns 28320 | Surreal one is a positive ... |
| peano2nns 28321 | Peano postulate for positi... |
| n0sbday 28322 | A non-negative surreal int... |
| n0ssold 28323 | The non-negative surreal i... |
| nnsrecgt0d 28324 | The reciprocal of a positi... |
| seqn0sfn 28325 | The surreal sequence build... |
| eln0s 28326 | A non-negative surreal int... |
| n0s0m1 28327 | Every non-negative surreal... |
| n0subs 28328 | Subtraction of non-negativ... |
| n0p1nns 28329 | One plus a non-negative su... |
| dfnns2 28330 | Alternate definition of th... |
| nnsind 28331 | Principle of Mathematical ... |
| zsex 28334 | The surreal integers form ... |
| zssno 28335 | The surreal integers are a... |
| zno 28336 | A surreal integer is a sur... |
| znod 28337 | A surreal integer is a sur... |
| elzs 28338 | Membership in the set of s... |
| nnzsubs 28339 | The difference of two surr... |
| nnzs 28340 | A positive surreal integer... |
| nnzsd 28341 | A positive surreal integer... |
| 0zs 28342 | Zero is a surreal integer.... |
| n0zs 28343 | A non-negative surreal int... |
| n0zsd 28344 | A non-negative surreal int... |
| 1zs 28345 | One is a surreal integer. ... |
| znegscl 28346 | The surreal integers are c... |
| znegscld 28347 | The surreal integers are c... |
| zaddscl 28348 | The surreal integers are c... |
| zaddscld 28349 | The surreal integers are c... |
| zsubscld 28350 | The surreal integers are c... |
| elzn0s 28351 | A surreal integer is a sur... |
| elzs2 28352 | A surreal integer is eithe... |
| eln0zs 28353 | Non-negative surreal integ... |
| elnnzs 28354 | Positive surreal integer p... |
| elznns 28355 | Surreal integer property e... |
| zn0subs 28356 | The non-negative differenc... |
| peano5uzs 28357 | Peano's inductive postulat... |
| uzsind 28358 | Induction on the upper sur... |
| zsbday 28359 | A surreal integer has a fi... |
| 1p1e2s 28366 | One plus one is two. Surr... |
| no2times 28367 | Version of ~ 2times for su... |
| 2nns 28368 | Surreal two is a surreal n... |
| 2sno 28369 | Surreal two is a surreal n... |
| 2ne0s 28370 | Surreal two is non-zero. ... |
| nohalf 28371 | An explicit expression for... |
| expsval 28372 | The value of surreal expon... |
| expsnnval 28373 | Value of surreal exponenti... |
| exps0 28374 | Surreal exponentiation to ... |
| exps1 28375 | Surreal exponentiation to ... |
| expsp1 28376 | Value of a surreal number ... |
| expscl 28377 | Closure law for surreal ex... |
| expsne0 28378 | A non-negative surreal int... |
| expsgt0 28379 | A non-negative surreal int... |
| halfcut 28380 | Relate the cut of twice of... |
| cutpw2 28381 | A cut expression for inver... |
| pw2bday 28382 | The inverses of powers of ... |
| elzs12 28383 | Membership in the dyadic f... |
| zs12ex 28384 | The class of dyadic fracti... |
| zzs12 28385 | A surreal integer is a dya... |
| elreno 28388 | Membership in the set of s... |
| recut 28389 | The cut involved in defini... |
| 0reno 28390 | Surreal zero is a surreal ... |
| renegscl 28391 | The surreal reals are clos... |
| readdscl 28392 | The surreal reals are clos... |
| remulscllem1 28393 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28394 | Lemma for ~ remulscl . Bo... |
| remulscl 28395 | The surreal reals are clos... |
| itvndx 28406 | Index value of the Interva... |
| lngndx 28407 | Index value of the "line" ... |
| itvid 28408 | Utility theorem: index-ind... |
| lngid 28409 | Utility theorem: index-ind... |
| slotsinbpsd 28410 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28411 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28412 | The slot for the line is n... |
| trkgstr 28413 | Functionality of a Tarski ... |
| trkgbas 28414 | The base set of a Tarski g... |
| trkgdist 28415 | The measure of a distance ... |
| trkgitv 28416 | The congruence relation in... |
| istrkgc 28423 | Property of being a Tarski... |
| istrkgb 28424 | Property of being a Tarski... |
| istrkgcb 28425 | Property of being a Tarski... |
| istrkge 28426 | Property of fulfilling Euc... |
| istrkgl 28427 | Building lines from the se... |
| istrkgld 28428 | Property of fulfilling the... |
| istrkg2ld 28429 | Property of fulfilling the... |
| istrkg3ld 28430 | Property of fulfilling the... |
| axtgcgrrflx 28431 | Axiom of reflexivity of co... |
| axtgcgrid 28432 | Axiom of identity of congr... |
| axtgsegcon 28433 | Axiom of segment construct... |
| axtg5seg 28434 | Five segments axiom, Axiom... |
| axtgbtwnid 28435 | Identity of Betweenness. ... |
| axtgpasch 28436 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28437 | Axiom of Continuity. Axio... |
| axtgcont 28438 | Axiom of Continuity. Axio... |
| axtglowdim2 28439 | Lower dimension axiom for ... |
| axtgupdim2 28440 | Upper dimension axiom for ... |
| axtgeucl 28441 | Euclid's Axiom. Axiom A10... |
| tgjustf 28442 | Given any function ` F ` ,... |
| tgjustr 28443 | Given any equivalence rela... |
| tgjustc1 28444 | A justification for using ... |
| tgjustc2 28445 | A justification for using ... |
| tgcgrcomimp 28446 | Congruence commutes on the... |
| tgcgrcomr 28447 | Congruence commutes on the... |
| tgcgrcoml 28448 | Congruence commutes on the... |
| tgcgrcomlr 28449 | Congruence commutes on bot... |
| tgcgreqb 28450 | Congruence and equality. ... |
| tgcgreq 28451 | Congruence and equality. ... |
| tgcgrneq 28452 | Congruence and equality. ... |
| tgcgrtriv 28453 | Degenerate segments are co... |
| tgcgrextend 28454 | Link congruence over a pai... |
| tgsegconeq 28455 | Two points that satisfy th... |
| tgbtwntriv2 28456 | Betweenness always holds f... |
| tgbtwncom 28457 | Betweenness commutes. The... |
| tgbtwncomb 28458 | Betweenness commutes, bico... |
| tgbtwnne 28459 | Betweenness and inequality... |
| tgbtwntriv1 28460 | Betweenness always holds f... |
| tgbtwnswapid 28461 | If you can swap the first ... |
| tgbtwnintr 28462 | Inner transitivity law for... |
| tgbtwnexch3 28463 | Exchange the first endpoin... |
| tgbtwnouttr2 28464 | Outer transitivity law for... |
| tgbtwnexch2 28465 | Exchange the outer point o... |
| tgbtwnouttr 28466 | Outer transitivity law for... |
| tgbtwnexch 28467 | Outer transitivity law for... |
| tgtrisegint 28468 | A line segment between two... |
| tglowdim1 28469 | Lower dimension axiom for ... |
| tglowdim1i 28470 | Lower dimension axiom for ... |
| tgldimor 28471 | Excluded-middle like state... |
| tgldim0eq 28472 | In dimension zero, any two... |
| tgldim0itv 28473 | In dimension zero, any two... |
| tgldim0cgr 28474 | In dimension zero, any two... |
| tgbtwndiff 28475 | There is always a ` c ` di... |
| tgdim01 28476 | In geometries of dimension... |
| tgifscgr 28477 | Inner five segment congrue... |
| tgcgrsub 28478 | Removing identical parts f... |
| iscgrg 28481 | The congruence property fo... |
| iscgrgd 28482 | The property for two seque... |
| iscgrglt 28483 | The property for two seque... |
| trgcgrg 28484 | The property for two trian... |
| trgcgr 28485 | Triangle congruence. (Con... |
| ercgrg 28486 | The shape congruence relat... |
| tgcgrxfr 28487 | A line segment can be divi... |
| cgr3id 28488 | Reflexivity law for three-... |
| cgr3simp1 28489 | Deduce segment congruence ... |
| cgr3simp2 28490 | Deduce segment congruence ... |
| cgr3simp3 28491 | Deduce segment congruence ... |
| cgr3swap12 28492 | Permutation law for three-... |
| cgr3swap23 28493 | Permutation law for three-... |
| cgr3swap13 28494 | Permutation law for three-... |
| cgr3rotr 28495 | Permutation law for three-... |
| cgr3rotl 28496 | Permutation law for three-... |
| trgcgrcom 28497 | Commutative law for three-... |
| cgr3tr 28498 | Transitivity law for three... |
| tgbtwnxfr 28499 | A condition for extending ... |
| tgcgr4 28500 | Two quadrilaterals to be c... |
| isismt 28503 | Property of being an isome... |
| ismot 28504 | Property of being an isome... |
| motcgr 28505 | Property of a motion: dist... |
| idmot 28506 | The identity is a motion. ... |
| motf1o 28507 | Motions are bijections. (... |
| motcl 28508 | Closure of motions. (Cont... |
| motco 28509 | The composition of two mot... |
| cnvmot 28510 | The converse of a motion i... |
| motplusg 28511 | The operation for motions ... |
| motgrp 28512 | The motions of a geometry ... |
| motcgrg 28513 | Property of a motion: dist... |
| motcgr3 28514 | Property of a motion: dist... |
| tglng 28515 | Lines of a Tarski Geometry... |
| tglnfn 28516 | Lines as functions. (Cont... |
| tglnunirn 28517 | Lines are sets of points. ... |
| tglnpt 28518 | Lines are sets of points. ... |
| tglngne 28519 | It takes two different poi... |
| tglngval 28520 | The line going through poi... |
| tglnssp 28521 | Lines are subset of the ge... |
| tgellng 28522 | Property of lying on the l... |
| tgcolg 28523 | We choose the notation ` (... |
| btwncolg1 28524 | Betweenness implies coline... |
| btwncolg2 28525 | Betweenness implies coline... |
| btwncolg3 28526 | Betweenness implies coline... |
| colcom 28527 | Swapping the points defini... |
| colrot1 28528 | Rotating the points defini... |
| colrot2 28529 | Rotating the points defini... |
| ncolcom 28530 | Swapping non-colinear poin... |
| ncolrot1 28531 | Rotating non-colinear poin... |
| ncolrot2 28532 | Rotating non-colinear poin... |
| tgdim01ln 28533 | In geometries of dimension... |
| ncoltgdim2 28534 | If there are three non-col... |
| lnxfr 28535 | Transfer law for colineari... |
| lnext 28536 | Extend a line with a missi... |
| tgfscgr 28537 | Congruence law for the gen... |
| lncgr 28538 | Congruence rule for lines.... |
| lnid 28539 | Identity law for points on... |
| tgidinside 28540 | Law for finding a point in... |
| tgbtwnconn1lem1 28541 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28542 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28543 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28544 | Connectivity law for betwe... |
| tgbtwnconn2 28545 | Another connectivity law f... |
| tgbtwnconn3 28546 | Inner connectivity law for... |
| tgbtwnconnln3 28547 | Derive colinearity from be... |
| tgbtwnconn22 28548 | Double connectivity law fo... |
| tgbtwnconnln1 28549 | Derive colinearity from be... |
| tgbtwnconnln2 28550 | Derive colinearity from be... |
| legval 28553 | Value of the less-than rel... |
| legov 28554 | Value of the less-than rel... |
| legov2 28555 | An equivalent definition o... |
| legid 28556 | Reflexivity of the less-th... |
| btwnleg 28557 | Betweenness implies less-t... |
| legtrd 28558 | Transitivity of the less-t... |
| legtri3 28559 | Equality from the less-tha... |
| legtrid 28560 | Trichotomy law for the les... |
| leg0 28561 | Degenerated (zero-length) ... |
| legeq 28562 | Deduce equality from "less... |
| legbtwn 28563 | Deduce betweenness from "l... |
| tgcgrsub2 28564 | Removing identical parts f... |
| ltgseg 28565 | The set ` E ` denotes the ... |
| ltgov 28566 | Strict "shorter than" geom... |
| legov3 28567 | An equivalent definition o... |
| legso 28568 | The "shorter than" relatio... |
| ishlg 28571 | Rays : Definition 6.1 of ... |
| hlcomb 28572 | The half-line relation com... |
| hlcomd 28573 | The half-line relation com... |
| hlne1 28574 | The half-line relation imp... |
| hlne2 28575 | The half-line relation imp... |
| hlln 28576 | The half-line relation imp... |
| hleqnid 28577 | The endpoint does not belo... |
| hlid 28578 | The half-line relation is ... |
| hltr 28579 | The half-line relation is ... |
| hlbtwn 28580 | Betweenness is a sufficien... |
| btwnhl1 28581 | Deduce half-line from betw... |
| btwnhl2 28582 | Deduce half-line from betw... |
| btwnhl 28583 | Swap betweenness for a hal... |
| lnhl 28584 | Either a point ` C ` on th... |
| hlcgrex 28585 | Construct a point on a hal... |
| hlcgreulem 28586 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28587 | The point constructed in ~... |
| btwnlng1 28588 | Betweenness implies coline... |
| btwnlng2 28589 | Betweenness implies coline... |
| btwnlng3 28590 | Betweenness implies coline... |
| lncom 28591 | Swapping the points defini... |
| lnrot1 28592 | Rotating the points defini... |
| lnrot2 28593 | Rotating the points defini... |
| ncolne1 28594 | Non-colinear points are di... |
| ncolne2 28595 | Non-colinear points are di... |
| tgisline 28596 | The property of being a pr... |
| tglnne 28597 | It takes two different poi... |
| tglndim0 28598 | There are no lines in dime... |
| tgelrnln 28599 | The property of being a pr... |
| tglineeltr 28600 | Transitivity law for lines... |
| tglineelsb2 28601 | If ` S ` lies on PQ , then... |
| tglinerflx1 28602 | Reflexivity law for line m... |
| tglinerflx2 28603 | Reflexivity law for line m... |
| tglinecom 28604 | Commutativity law for line... |
| tglinethru 28605 | If ` A ` is a line contain... |
| tghilberti1 28606 | There is a line through an... |
| tghilberti2 28607 | There is at most one line ... |
| tglinethrueu 28608 | There is a unique line goi... |
| tglnne0 28609 | A line ` A ` has at least ... |
| tglnpt2 28610 | Find a second point on a l... |
| tglineintmo 28611 | Two distinct lines interse... |
| tglineineq 28612 | Two distinct lines interse... |
| tglineneq 28613 | Given three non-colinear p... |
| tglineinteq 28614 | Two distinct lines interse... |
| ncolncol 28615 | Deduce non-colinearity fro... |
| coltr 28616 | A transitivity law for col... |
| coltr3 28617 | A transitivity law for col... |
| colline 28618 | Three points are colinear ... |
| tglowdim2l 28619 | Reformulation of the lower... |
| tglowdim2ln 28620 | There is always one point ... |
| mirreu3 28623 | Existential uniqueness of ... |
| mirval 28624 | Value of the point inversi... |
| mirfv 28625 | Value of the point inversi... |
| mircgr 28626 | Property of the image by t... |
| mirbtwn 28627 | Property of the image by t... |
| ismir 28628 | Property of the image by t... |
| mirf 28629 | Point inversion as functio... |
| mircl 28630 | Closure of the point inver... |
| mirmir 28631 | The point inversion functi... |
| mircom 28632 | Variation on ~ mirmir . (... |
| mirreu 28633 | Any point has a unique ant... |
| mireq 28634 | Equality deduction for poi... |
| mirinv 28635 | The only invariant point o... |
| mirne 28636 | Mirror of non-center point... |
| mircinv 28637 | The center point is invari... |
| mirf1o 28638 | The point inversion functi... |
| miriso 28639 | The point inversion functi... |
| mirbtwni 28640 | Point inversion preserves ... |
| mirbtwnb 28641 | Point inversion preserves ... |
| mircgrs 28642 | Point inversion preserves ... |
| mirmir2 28643 | Point inversion of a point... |
| mirmot 28644 | Point investion is a motio... |
| mirln 28645 | If two points are on the s... |
| mirln2 28646 | If a point and its mirror ... |
| mirconn 28647 | Point inversion of connect... |
| mirhl 28648 | If two points ` X ` and ` ... |
| mirbtwnhl 28649 | If the center of the point... |
| mirhl2 28650 | Deduce half-line relation ... |
| mircgrextend 28651 | Link congruence over a pai... |
| mirtrcgr 28652 | Point inversion of one poi... |
| mirauto 28653 | Point inversion preserves ... |
| miduniq 28654 | Uniqueness of the middle p... |
| miduniq1 28655 | Uniqueness of the middle p... |
| miduniq2 28656 | If two point inversions co... |
| colmid 28657 | Colinearity and equidistan... |
| symquadlem 28658 | Lemma of the symetrial qua... |
| krippenlem 28659 | Lemma for ~ krippen . We ... |
| krippen 28660 | Krippenlemma (German for c... |
| midexlem 28661 | Lemma for the existence of... |
| israg 28666 | Property for 3 points A, B... |
| ragcom 28667 | Commutative rule for right... |
| ragcol 28668 | The right angle property i... |
| ragmir 28669 | Right angle property is pr... |
| mirrag 28670 | Right angle is conserved b... |
| ragtrivb 28671 | Trivial right angle. Theo... |
| ragflat2 28672 | Deduce equality from two r... |
| ragflat 28673 | Deduce equality from two r... |
| ragtriva 28674 | Trivial right angle. Theo... |
| ragflat3 28675 | Right angle and colinearit... |
| ragcgr 28676 | Right angle and colinearit... |
| motrag 28677 | Right angles are preserved... |
| ragncol 28678 | Right angle implies non-co... |
| perpln1 28679 | Derive a line from perpend... |
| perpln2 28680 | Derive a line from perpend... |
| isperp 28681 | Property for 2 lines A, B ... |
| perpcom 28682 | The "perpendicular" relati... |
| perpneq 28683 | Two perpendicular lines ar... |
| isperp2 28684 | Property for 2 lines A, B,... |
| isperp2d 28685 | One direction of ~ isperp2... |
| ragperp 28686 | Deduce that two lines are ... |
| footexALT 28687 | Alternative version of ~ f... |
| footexlem1 28688 | Lemma for ~ footex . (Con... |
| footexlem2 28689 | Lemma for ~ footex . (Con... |
| footex 28690 | From a point ` C ` outside... |
| foot 28691 | From a point ` C ` outside... |
| footne 28692 | Uniqueness of the foot poi... |
| footeq 28693 | Uniqueness of the foot poi... |
| hlperpnel 28694 | A point on a half-line whi... |
| perprag 28695 | Deduce a right angle from ... |
| perpdragALT 28696 | Deduce a right angle from ... |
| perpdrag 28697 | Deduce a right angle from ... |
| colperp 28698 | Deduce a perpendicularity ... |
| colperpexlem1 28699 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28700 | Lemma for ~ colperpex . S... |
| colperpexlem3 28701 | Lemma for ~ colperpex . C... |
| colperpex 28702 | In dimension 2 and above, ... |
| mideulem2 28703 | Lemma for ~ opphllem , whi... |
| opphllem 28704 | Lemma 8.24 of [Schwabhause... |
| mideulem 28705 | Lemma for ~ mideu . We ca... |
| midex 28706 | Existence of the midpoint,... |
| mideu 28707 | Existence and uniqueness o... |
| islnopp 28708 | The property for two point... |
| islnoppd 28709 | Deduce that ` A ` and ` B ... |
| oppne1 28710 | Points lying on opposite s... |
| oppne2 28711 | Points lying on opposite s... |
| oppne3 28712 | Points lying on opposite s... |
| oppcom 28713 | Commutativity rule for "op... |
| opptgdim2 28714 | If two points opposite to ... |
| oppnid 28715 | The "opposite to a line" r... |
| opphllem1 28716 | Lemma for ~ opphl . (Cont... |
| opphllem2 28717 | Lemma for ~ opphl . Lemma... |
| opphllem3 28718 | Lemma for ~ opphl : We as... |
| opphllem4 28719 | Lemma for ~ opphl . (Cont... |
| opphllem5 28720 | Second part of Lemma 9.4 o... |
| opphllem6 28721 | First part of Lemma 9.4 of... |
| oppperpex 28722 | Restating ~ colperpex usin... |
| opphl 28723 | If two points ` A ` and ` ... |
| outpasch 28724 | Axiom of Pasch, outer form... |
| hlpasch 28725 | An application of the axio... |
| ishpg 28728 | Value of the half-plane re... |
| hpgbr 28729 | Half-planes : property for... |
| hpgne1 28730 | Points on the open half pl... |
| hpgne2 28731 | Points on the open half pl... |
| lnopp2hpgb 28732 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28733 | If two points lie on the o... |
| hpgerlem 28734 | Lemma for the proof that t... |
| hpgid 28735 | The half-plane relation is... |
| hpgcom 28736 | The half-plane relation co... |
| hpgtr 28737 | The half-plane relation is... |
| colopp 28738 | Opposite sides of a line f... |
| colhp 28739 | Half-plane relation for co... |
| hphl 28740 | If two points are on the s... |
| midf 28745 | Midpoint as a function. (... |
| midcl 28746 | Closure of the midpoint. ... |
| ismidb 28747 | Property of the midpoint. ... |
| midbtwn 28748 | Betweenness of midpoint. ... |
| midcgr 28749 | Congruence of midpoint. (... |
| midid 28750 | Midpoint of a null segment... |
| midcom 28751 | Commutativity rule for the... |
| mirmid 28752 | Point inversion preserves ... |
| lmieu 28753 | Uniqueness of the line mir... |
| lmif 28754 | Line mirror as a function.... |
| lmicl 28755 | Closure of the line mirror... |
| islmib 28756 | Property of the line mirro... |
| lmicom 28757 | The line mirroring functio... |
| lmilmi 28758 | Line mirroring is an invol... |
| lmireu 28759 | Any point has a unique ant... |
| lmieq 28760 | Equality deduction for lin... |
| lmiinv 28761 | The invariants of the line... |
| lmicinv 28762 | The mirroring line is an i... |
| lmimid 28763 | If we have a right angle, ... |
| lmif1o 28764 | The line mirroring functio... |
| lmiisolem 28765 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28766 | The line mirroring functio... |
| lmimot 28767 | Line mirroring is a motion... |
| hypcgrlem1 28768 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28769 | Lemma for ~ hypcgr , case ... |
| hypcgr 28770 | If the catheti of two righ... |
| lmiopp 28771 | Line mirroring produces po... |
| lnperpex 28772 | Existence of a perpendicul... |
| trgcopy 28773 | Triangle construction: a c... |
| trgcopyeulem 28774 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28775 | Triangle construction: a c... |
| iscgra 28778 | Property for two angles AB... |
| iscgra1 28779 | A special version of ~ isc... |
| iscgrad 28780 | Sufficient conditions for ... |
| cgrane1 28781 | Angles imply inequality. ... |
| cgrane2 28782 | Angles imply inequality. ... |
| cgrane3 28783 | Angles imply inequality. ... |
| cgrane4 28784 | Angles imply inequality. ... |
| cgrahl1 28785 | Angle congruence is indepe... |
| cgrahl2 28786 | Angle congruence is indepe... |
| cgracgr 28787 | First direction of proposi... |
| cgraid 28788 | Angle congruence is reflex... |
| cgraswap 28789 | Swap rays in a congruence ... |
| cgrcgra 28790 | Triangle congruence implie... |
| cgracom 28791 | Angle congruence commutes.... |
| cgratr 28792 | Angle congruence is transi... |
| flatcgra 28793 | Flat angles are congruent.... |
| cgraswaplr 28794 | Swap both side of angle co... |
| cgrabtwn 28795 | Angle congruence preserves... |
| cgrahl 28796 | Angle congruence preserves... |
| cgracol 28797 | Angle congruence preserves... |
| cgrancol 28798 | Angle congruence preserves... |
| dfcgra2 28799 | This is the full statement... |
| sacgr 28800 | Supplementary angles of co... |
| oacgr 28801 | Vertical angle theorem. V... |
| acopy 28802 | Angle construction. Theor... |
| acopyeu 28803 | Angle construction. Theor... |
| isinag 28807 | Property for point ` X ` t... |
| isinagd 28808 | Sufficient conditions for ... |
| inagflat 28809 | Any point lies in a flat a... |
| inagswap 28810 | Swap the order of the half... |
| inagne1 28811 | Deduce inequality from the... |
| inagne2 28812 | Deduce inequality from the... |
| inagne3 28813 | Deduce inequality from the... |
| inaghl 28814 | The "point lie in angle" r... |
| isleag 28816 | Geometrical "less than" pr... |
| isleagd 28817 | Sufficient condition for "... |
| leagne1 28818 | Deduce inequality from the... |
| leagne2 28819 | Deduce inequality from the... |
| leagne3 28820 | Deduce inequality from the... |
| leagne4 28821 | Deduce inequality from the... |
| cgrg3col4 28822 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28823 | First congruence theorem: ... |
| tgsas 28824 | First congruence theorem: ... |
| tgsas2 28825 | First congruence theorem: ... |
| tgsas3 28826 | First congruence theorem: ... |
| tgasa1 28827 | Second congruence theorem:... |
| tgasa 28828 | Second congruence theorem:... |
| tgsss1 28829 | Third congruence theorem: ... |
| tgsss2 28830 | Third congruence theorem: ... |
| tgsss3 28831 | Third congruence theorem: ... |
| dfcgrg2 28832 | Congruence for two triangl... |
| isoas 28833 | Congruence theorem for iso... |
| iseqlg 28836 | Property of a triangle bei... |
| iseqlgd 28837 | Condition for a triangle t... |
| f1otrgds 28838 | Convenient lemma for ~ f1o... |
| f1otrgitv 28839 | Convenient lemma for ~ f1o... |
| f1otrg 28840 | A bijection between bases ... |
| f1otrge 28841 | A bijection between bases ... |
| ttgval 28844 | Define a function to augme... |
| ttgvalOLD 28845 | Obsolete proof of ~ ttgval... |
| ttglem 28846 | Lemma for ~ ttgbas , ~ ttg... |
| ttglemOLD 28847 | Obsolete version of ~ ttgl... |
| ttgbas 28848 | The base set of a subcompl... |
| ttgbasOLD 28849 | Obsolete proof of ~ ttgbas... |
| ttgplusg 28850 | The addition operation of ... |
| ttgplusgOLD 28851 | Obsolete proof of ~ ttgplu... |
| ttgsub 28852 | The subtraction operation ... |
| ttgvsca 28853 | The scalar product of a su... |
| ttgvscaOLD 28854 | Obsolete proof of ~ ttgvsc... |
| ttgds 28855 | The metric of a subcomplex... |
| ttgdsOLD 28856 | Obsolete proof of ~ ttgds ... |
| ttgitvval 28857 | Betweenness for a subcompl... |
| ttgelitv 28858 | Betweenness for a subcompl... |
| ttgbtwnid 28859 | Any subcomplex module equi... |
| ttgcontlem1 28860 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28861 | Any metric space fulfills ... |
| cchhllem 28862 | Lemma for chlbas and chlvs... |
| cchhllemOLD 28863 | Obsolete version of ~ cchh... |
| elee 28870 | Membership in a Euclidean ... |
| mptelee 28871 | A condition for a mapping ... |
| eleenn 28872 | If ` A ` is in ` ( EE `` N... |
| eleei 28873 | The forward direction of ~... |
| eedimeq 28874 | A point belongs to at most... |
| brbtwn 28875 | The binary relation form o... |
| brcgr 28876 | The binary relation form o... |
| fveere 28877 | The function value of a po... |
| fveecn 28878 | The function value of a po... |
| eqeefv 28879 | Two points are equal iff t... |
| eqeelen 28880 | Two points are equal iff t... |
| brbtwn2 28881 | Alternate characterization... |
| colinearalglem1 28882 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28883 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28884 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28885 | Lemma for ~ colinearalg . ... |
| colinearalg 28886 | An algebraic characterizat... |
| eleesub 28887 | Membership of a subtractio... |
| eleesubd 28888 | Membership of a subtractio... |
| axdimuniq 28889 | The unique dimension axiom... |
| axcgrrflx 28890 | ` A ` is as far from ` B `... |
| axcgrtr 28891 | Congruence is transitive. ... |
| axcgrid 28892 | If there is no distance be... |
| axsegconlem1 28893 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28894 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28895 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28896 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28897 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28898 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28899 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28900 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28901 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28902 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28903 | Any segment ` A B ` can be... |
| ax5seglem1 28904 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28905 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28906 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28907 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28908 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28909 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28910 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28911 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28912 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28913 | Lemma for ~ ax5seg . Take... |
| ax5seg 28914 | The five segment axiom. T... |
| axbtwnid 28915 | Points are indivisible. T... |
| axpaschlem 28916 | Lemma for ~ axpasch . Set... |
| axpasch 28917 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28918 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 28919 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 28920 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 28921 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 28922 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 28923 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 28924 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 28925 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 28926 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 28927 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 28928 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 28929 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 28930 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 28931 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 28932 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 28933 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 28934 | Lemma for ~ axlowdim . Es... |
| axlowdim1 28935 | The lower dimension axiom ... |
| axlowdim2 28936 | The lower two-dimensional ... |
| axlowdim 28937 | The general lower dimensio... |
| axeuclidlem 28938 | Lemma for ~ axeuclid . Ha... |
| axeuclid 28939 | Euclid's axiom. Take an a... |
| axcontlem1 28940 | Lemma for ~ axcont . Chan... |
| axcontlem2 28941 | Lemma for ~ axcont . The ... |
| axcontlem3 28942 | Lemma for ~ axcont . Give... |
| axcontlem4 28943 | Lemma for ~ axcont . Give... |
| axcontlem5 28944 | Lemma for ~ axcont . Comp... |
| axcontlem6 28945 | Lemma for ~ axcont . Stat... |
| axcontlem7 28946 | Lemma for ~ axcont . Give... |
| axcontlem8 28947 | Lemma for ~ axcont . A po... |
| axcontlem9 28948 | Lemma for ~ axcont . Give... |
| axcontlem10 28949 | Lemma for ~ axcont . Give... |
| axcontlem11 28950 | Lemma for ~ axcont . Elim... |
| axcontlem12 28951 | Lemma for ~ axcont . Elim... |
| axcont 28952 | The axiom of continuity. ... |
| eengv 28955 | The value of the Euclidean... |
| eengstr 28956 | The Euclidean geometry as ... |
| eengbas 28957 | The Base of the Euclidean ... |
| ebtwntg 28958 | The betweenness relation u... |
| ecgrtg 28959 | The congruence relation us... |
| elntg 28960 | The line definition in the... |
| elntg2 28961 | The line definition in the... |
| eengtrkg 28962 | The geometry structure for... |
| eengtrkge 28963 | The geometry structure for... |
| edgfid 28966 | Utility theorem: index-ind... |
| edgfndx 28967 | Index value of the ~ df-ed... |
| edgfndxnn 28968 | The index value of the edg... |
| edgfndxid 28969 | The value of the edge func... |
| edgfndxidOLD 28970 | Obsolete version of ~ edgf... |
| basendxltedgfndx 28971 | The index value of the ` B... |
| baseltedgfOLD 28972 | Obsolete proof of ~ basend... |
| basendxnedgfndx 28973 | The slots ` Base ` and ` .... |
| vtxval 28978 | The set of vertices of a g... |
| iedgval 28979 | The set of indexed edges o... |
| 1vgrex 28980 | A graph with at least one ... |
| opvtxval 28981 | The set of vertices of a g... |
| opvtxfv 28982 | The set of vertices of a g... |
| opvtxov 28983 | The set of vertices of a g... |
| opiedgval 28984 | The set of indexed edges o... |
| opiedgfv 28985 | The set of indexed edges o... |
| opiedgov 28986 | The set of indexed edges o... |
| opvtxfvi 28987 | The set of vertices of a g... |
| opiedgfvi 28988 | The set of indexed edges o... |
| funvtxdmge2val 28989 | The set of vertices of an ... |
| funiedgdmge2val 28990 | The set of indexed edges o... |
| funvtxdm2val 28991 | The set of vertices of an ... |
| funiedgdm2val 28992 | The set of indexed edges o... |
| funvtxval0 28993 | The set of vertices of an ... |
| basvtxval 28994 | The set of vertices of a g... |
| edgfiedgval 28995 | The set of indexed edges o... |
| funvtxval 28996 | The set of vertices of a g... |
| funiedgval 28997 | The set of indexed edges o... |
| structvtxvallem 28998 | Lemma for ~ structvtxval a... |
| structvtxval 28999 | The set of vertices of an ... |
| structiedg0val 29000 | The set of indexed edges o... |
| structgrssvtxlem 29001 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29002 | The set of vertices of a g... |
| structgrssiedg 29003 | The set of indexed edges o... |
| struct2grstr 29004 | A graph represented as an ... |
| struct2grvtx 29005 | The set of vertices of a g... |
| struct2griedg 29006 | The set of indexed edges o... |
| graop 29007 | Any representation of a gr... |
| grastruct 29008 | Any representation of a gr... |
| gropd 29009 | If any representation of a... |
| grstructd 29010 | If any representation of a... |
| gropeld 29011 | If any representation of a... |
| grstructeld 29012 | If any representation of a... |
| setsvtx 29013 | The vertices of a structur... |
| setsiedg 29014 | The (indexed) edges of a s... |
| snstrvtxval 29015 | The set of vertices of a g... |
| snstriedgval 29016 | The set of indexed edges o... |
| vtxval0 29017 | Degenerated case 1 for ver... |
| iedgval0 29018 | Degenerated case 1 for edg... |
| vtxvalsnop 29019 | Degenerated case 2 for ver... |
| iedgvalsnop 29020 | Degenerated case 2 for edg... |
| vtxval3sn 29021 | Degenerated case 3 for ver... |
| iedgval3sn 29022 | Degenerated case 3 for edg... |
| vtxvalprc 29023 | Degenerated case 4 for ver... |
| iedgvalprc 29024 | Degenerated case 4 for edg... |
| edgval 29027 | The edges of a graph. (Co... |
| iedgedg 29028 | An indexed edge is an edge... |
| edgopval 29029 | The edges of a graph repre... |
| edgov 29030 | The edges of a graph repre... |
| edgstruct 29031 | The edges of a graph repre... |
| edgiedgb 29032 | A set is an edge iff it is... |
| edg0iedg0 29033 | There is no edge in a grap... |
| isuhgr 29038 | The predicate "is an undir... |
| isushgr 29039 | The predicate "is an undir... |
| uhgrf 29040 | The edge function of an un... |
| ushgrf 29041 | The edge function of an un... |
| uhgrss 29042 | An edge is a subset of ver... |
| uhgreq12g 29043 | If two sets have the same ... |
| uhgrfun 29044 | The edge function of an un... |
| uhgrn0 29045 | An edge is a nonempty subs... |
| lpvtx 29046 | The endpoints of a loop (w... |
| ushgruhgr 29047 | An undirected simple hyper... |
| isuhgrop 29048 | The property of being an u... |
| uhgr0e 29049 | The empty graph, with vert... |
| uhgr0vb 29050 | The null graph, with no ve... |
| uhgr0 29051 | The null graph represented... |
| uhgrun 29052 | The union ` U ` of two (un... |
| uhgrunop 29053 | The union of two (undirect... |
| ushgrun 29054 | The union ` U ` of two (un... |
| ushgrunop 29055 | The union of two (undirect... |
| uhgrstrrepe 29056 | Replacing (or adding) the ... |
| incistruhgr 29057 | An _incidence structure_ `... |
| isupgr 29062 | The property of being an u... |
| wrdupgr 29063 | The property of being an u... |
| upgrf 29064 | The edge function of an un... |
| upgrfn 29065 | The edge function of an un... |
| upgrss 29066 | An edge is a subset of ver... |
| upgrn0 29067 | An edge is a nonempty subs... |
| upgrle 29068 | An edge of an undirected p... |
| upgrfi 29069 | An edge is a finite subset... |
| upgrex 29070 | An edge is an unordered pa... |
| upgrbi 29071 | Show that an unordered pai... |
| upgrop 29072 | A pseudograph represented ... |
| isumgr 29073 | The property of being an u... |
| isumgrs 29074 | The simplified property of... |
| wrdumgr 29075 | The property of being an u... |
| umgrf 29076 | The edge function of an un... |
| umgrfn 29077 | The edge function of an un... |
| umgredg2 29078 | An edge of a multigraph ha... |
| umgrbi 29079 | Show that an unordered pai... |
| upgruhgr 29080 | An undirected pseudograph ... |
| umgrupgr 29081 | An undirected multigraph i... |
| umgruhgr 29082 | An undirected multigraph i... |
| upgrle2 29083 | An edge of an undirected p... |
| umgrnloopv 29084 | In a multigraph, there is ... |
| umgredgprv 29085 | In a multigraph, an edge i... |
| umgrnloop 29086 | In a multigraph, there is ... |
| umgrnloop0 29087 | A multigraph has no loops.... |
| umgr0e 29088 | The empty graph, with vert... |
| upgr0e 29089 | The empty graph, with vert... |
| upgr1elem 29090 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29091 | A pseudograph with one edg... |
| upgr0eop 29092 | The empty graph, with vert... |
| upgr1eop 29093 | A pseudograph with one edg... |
| upgr0eopALT 29094 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29095 | Alternate proof of ~ upgr1... |
| upgrun 29096 | The union ` U ` of two pse... |
| upgrunop 29097 | The union of two pseudogra... |
| umgrun 29098 | The union ` U ` of two mul... |
| umgrunop 29099 | The union of two multigrap... |
| umgrislfupgrlem 29100 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29101 | A multigraph is a loop-fre... |
| lfgredgge2 29102 | An edge of a loop-free gra... |
| lfgrnloop 29103 | A loop-free graph has no l... |
| uhgredgiedgb 29104 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29105 | A hypergraph has no edges ... |
| uhgredgn0 29106 | An edge of a hypergraph is... |
| edguhgr 29107 | An edge of a hypergraph is... |
| uhgredgrnv 29108 | An edge of a hypergraph co... |
| uhgredgss 29109 | The set of edges of a hype... |
| upgredgss 29110 | The set of edges of a pseu... |
| umgredgss 29111 | The set of edges of a mult... |
| edgupgr 29112 | Properties of an edge of a... |
| edgumgr 29113 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29114 | In a hypergraph, a vertex ... |
| upgredg 29115 | For each edge in a pseudog... |
| umgredg 29116 | For each edge in a multigr... |
| upgrpredgv 29117 | An edge of a pseudograph a... |
| umgrpredgv 29118 | An edge of a multigraph al... |
| upgredg2vtx 29119 | For a vertex incident to a... |
| upgredgpr 29120 | If a proper pair (of verti... |
| edglnl 29121 | The edges incident with a ... |
| numedglnl 29122 | The number of edges incide... |
| umgredgne 29123 | An edge of a multigraph al... |
| umgrnloop2 29124 | A multigraph has no loops.... |
| umgredgnlp 29125 | An edge of a multigraph is... |
| isuspgr 29130 | The property of being a si... |
| isusgr 29131 | The property of being a si... |
| uspgrf 29132 | The edge function of a sim... |
| usgrf 29133 | The edge function of a sim... |
| isusgrs 29134 | The property of being a si... |
| usgrfs 29135 | The edge function of a sim... |
| usgrfun 29136 | The edge function of a sim... |
| usgredgss 29137 | The set of edges of a simp... |
| edgusgr 29138 | An edge of a simple graph ... |
| isuspgrop 29139 | The property of being an u... |
| isusgrop 29140 | The property of being an u... |
| usgrop 29141 | A simple graph represented... |
| isausgr 29142 | The property of an unorder... |
| ausgrusgrb 29143 | The equivalence of the def... |
| usgrausgri 29144 | A simple graph represented... |
| ausgrumgri 29145 | If an alternatively define... |
| ausgrusgri 29146 | The equivalence of the def... |
| usgrausgrb 29147 | The equivalence of the def... |
| usgredgop 29148 | An edge of a simple graph ... |
| usgrf1o 29149 | The edge function of a sim... |
| usgrf1 29150 | The edge function of a sim... |
| uspgrf1oedg 29151 | The edge function of a sim... |
| usgrss 29152 | An edge is a subset of ver... |
| uspgredgiedg 29153 | In a simple pseudograph, f... |
| uspgriedgedg 29154 | In a simple pseudograph, f... |
| uspgrushgr 29155 | A simple pseudograph is an... |
| uspgrupgr 29156 | A simple pseudograph is an... |
| uspgrupgrushgr 29157 | A graph is a simple pseudo... |
| usgruspgr 29158 | A simple graph is a simple... |
| usgrumgr 29159 | A simple graph is an undir... |
| usgrumgruspgr 29160 | A graph is a simple graph ... |
| usgruspgrb 29161 | A class is a simple graph ... |
| uspgruhgr 29162 | An undirected simple pseud... |
| usgrupgr 29163 | A simple graph is an undir... |
| usgruhgr 29164 | A simple graph is an undir... |
| usgrislfuspgr 29165 | A simple graph is a loop-f... |
| uspgrun 29166 | The union ` U ` of two sim... |
| uspgrunop 29167 | The union of two simple ps... |
| usgrun 29168 | The union ` U ` of two sim... |
| usgrunop 29169 | The union of two simple gr... |
| usgredg2 29170 | The value of the "edge fun... |
| usgredg2ALT 29171 | Alternate proof of ~ usgre... |
| usgredgprv 29172 | In a simple graph, an edge... |
| usgredgprvALT 29173 | Alternate proof of ~ usgre... |
| usgredgppr 29174 | An edge of a simple graph ... |
| usgrpredgv 29175 | An edge of a simple graph ... |
| edgssv2 29176 | An edge of a simple graph ... |
| usgredg 29177 | For each edge in a simple ... |
| usgrnloopv 29178 | In a simple graph, there i... |
| usgrnloopvALT 29179 | Alternate proof of ~ usgrn... |
| usgrnloop 29180 | In a simple graph, there i... |
| usgrnloopALT 29181 | Alternate proof of ~ usgrn... |
| usgrnloop0 29182 | A simple graph has no loop... |
| usgrnloop0ALT 29183 | Alternate proof of ~ usgrn... |
| usgredgne 29184 | An edge of a simple graph ... |
| usgrf1oedg 29185 | The edge function of a sim... |
| uhgr2edg 29186 | If a vertex is adjacent to... |
| umgr2edg 29187 | If a vertex is adjacent to... |
| usgr2edg 29188 | If a vertex is adjacent to... |
| umgr2edg1 29189 | If a vertex is adjacent to... |
| usgr2edg1 29190 | If a vertex is adjacent to... |
| umgrvad2edg 29191 | If a vertex is adjacent to... |
| umgr2edgneu 29192 | If a vertex is adjacent to... |
| usgrsizedg 29193 | In a simple graph, the siz... |
| usgredg3 29194 | The value of the "edge fun... |
| usgredg4 29195 | For a vertex incident to a... |
| usgredgreu 29196 | For a vertex incident to a... |
| usgredg2vtx 29197 | For a vertex incident to a... |
| uspgredg2vtxeu 29198 | For a vertex incident to a... |
| usgredg2vtxeu 29199 | For a vertex incident to a... |
| usgredg2vtxeuALT 29200 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29201 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29202 | In a simple pseudograph, t... |
| usgredg2vlem1 29203 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29204 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29205 | In a simple graph, the map... |
| usgriedgleord 29206 | Alternate version of ~ usg... |
| ushgredgedg 29207 | In a simple hypergraph the... |
| usgredgedg 29208 | In a simple graph there is... |
| ushgredgedgloop 29209 | In a simple hypergraph the... |
| uspgredgleord 29210 | In a simple pseudograph th... |
| usgredgleord 29211 | In a simple graph the numb... |
| usgredgleordALT 29212 | Alternate proof for ~ usgr... |
| usgrstrrepe 29213 | Replacing (or adding) the ... |
| usgr0e 29214 | The empty graph, with vert... |
| usgr0vb 29215 | The null graph, with no ve... |
| uhgr0v0e 29216 | The null graph, with no ve... |
| uhgr0vsize0 29217 | The size of a hypergraph w... |
| uhgr0edgfi 29218 | A graph of order 0 (i.e. w... |
| usgr0v 29219 | The null graph, with no ve... |
| uhgr0vusgr 29220 | The null graph, with no ve... |
| usgr0 29221 | The null graph represented... |
| uspgr1e 29222 | A simple pseudograph with ... |
| usgr1e 29223 | A simple graph with one ed... |
| usgr0eop 29224 | The empty graph, with vert... |
| uspgr1eop 29225 | A simple pseudograph with ... |
| uspgr1ewop 29226 | A simple pseudograph with ... |
| uspgr1v1eop 29227 | A simple pseudograph with ... |
| usgr1eop 29228 | A simple graph with (at le... |
| uspgr2v1e2w 29229 | A simple pseudograph with ... |
| usgr2v1e2w 29230 | A simple graph with two ve... |
| edg0usgr 29231 | A class without edges is a... |
| lfuhgr1v0e 29232 | A loop-free hypergraph wit... |
| usgr1vr 29233 | A simple graph with one ve... |
| usgr1v 29234 | A class with one (or no) v... |
| usgr1v0edg 29235 | A class with one (or no) v... |
| usgrexmpldifpr 29236 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29237 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29238 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29239 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29240 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29241 | ` G ` is a simple graph of... |
| griedg0prc 29242 | The class of empty graphs ... |
| griedg0ssusgr 29243 | The class of all simple gr... |
| usgrprc 29244 | The class of simple graphs... |
| relsubgr 29247 | The class of the subgraph ... |
| subgrv 29248 | If a class is a subgraph o... |
| issubgr 29249 | The property of a set to b... |
| issubgr2 29250 | The property of a set to b... |
| subgrprop 29251 | The properties of a subgra... |
| subgrprop2 29252 | The properties of a subgra... |
| uhgrissubgr 29253 | The property of a hypergra... |
| subgrprop3 29254 | The properties of a subgra... |
| egrsubgr 29255 | An empty graph consisting ... |
| 0grsubgr 29256 | The null graph (represente... |
| 0uhgrsubgr 29257 | The null graph (as hypergr... |
| uhgrsubgrself 29258 | A hypergraph is a subgraph... |
| subgrfun 29259 | The edge function of a sub... |
| subgruhgrfun 29260 | The edge function of a sub... |
| subgreldmiedg 29261 | An element of the domain o... |
| subgruhgredgd 29262 | An edge of a subgraph of a... |
| subumgredg2 29263 | An edge of a subgraph of a... |
| subuhgr 29264 | A subgraph of a hypergraph... |
| subupgr 29265 | A subgraph of a pseudograp... |
| subumgr 29266 | A subgraph of a multigraph... |
| subusgr 29267 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29268 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29269 | A spanning subgraph ` S ` ... |
| uhgrspan 29270 | A spanning subgraph ` S ` ... |
| upgrspan 29271 | A spanning subgraph ` S ` ... |
| umgrspan 29272 | A spanning subgraph ` S ` ... |
| usgrspan 29273 | A spanning subgraph ` S ` ... |
| uhgrspanop 29274 | A spanning subgraph of a h... |
| upgrspanop 29275 | A spanning subgraph of a p... |
| umgrspanop 29276 | A spanning subgraph of a m... |
| usgrspanop 29277 | A spanning subgraph of a s... |
| uhgrspan1lem1 29278 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29279 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29280 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29281 | The induced subgraph ` S `... |
| upgrreslem 29282 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29283 | Lemma for ~ umgrres and ~ ... |
| upgrres 29284 | A subgraph obtained by rem... |
| umgrres 29285 | A subgraph obtained by rem... |
| usgrres 29286 | A subgraph obtained by rem... |
| upgrres1lem1 29287 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29288 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29289 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29290 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29291 | A pseudograph obtained by ... |
| umgrres1 29292 | A multigraph obtained by r... |
| usgrres1 29293 | Restricting a simple graph... |
| isfusgr 29296 | The property of being a fi... |
| fusgrvtxfi 29297 | A finite simple graph has ... |
| isfusgrf1 29298 | The property of being a fi... |
| isfusgrcl 29299 | The property of being a fi... |
| fusgrusgr 29300 | A finite simple graph is a... |
| opfusgr 29301 | A finite simple graph repr... |
| usgredgffibi 29302 | The number of edges in a s... |
| fusgredgfi 29303 | In a finite simple graph t... |
| usgr1v0e 29304 | The size of a (finite) sim... |
| usgrfilem 29305 | In a finite simple graph, ... |
| fusgrfisbase 29306 | Induction base for ~ fusgr... |
| fusgrfisstep 29307 | Induction step in ~ fusgrf... |
| fusgrfis 29308 | A finite simple graph is o... |
| fusgrfupgrfs 29309 | A finite simple graph is a... |
| nbgrprc0 29312 | The set of neighbors is em... |
| nbgrcl 29313 | If a class ` X ` has at le... |
| nbgrval 29314 | The set of neighbors of a ... |
| dfnbgr2 29315 | Alternate definition of th... |
| dfnbgr3 29316 | Alternate definition of th... |
| nbgrnvtx0 29317 | If a class ` X ` is not a ... |
| nbgrel 29318 | Characterization of a neig... |
| nbgrisvtx 29319 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29320 | The neighbors of a vertex ... |
| nbuhgr 29321 | The set of neighbors of a ... |
| nbupgr 29322 | The set of neighbors of a ... |
| nbupgrel 29323 | A neighbor of a vertex in ... |
| nbumgrvtx 29324 | The set of neighbors of a ... |
| nbumgr 29325 | The set of neighbors of an... |
| nbusgrvtx 29326 | The set of neighbors of a ... |
| nbusgr 29327 | The set of neighbors of an... |
| nbgr2vtx1edg 29328 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29329 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29330 | If a hypergraph has two ve... |
| nbusgreledg 29331 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29332 | A vertex which is not endp... |
| nbgr0vtx 29333 | In a null graph (with no v... |
| nbgr0edglem 29334 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29335 | In an empty graph (with no... |
| nbgr1vtx 29336 | In a graph with one vertex... |
| nbgrnself 29337 | A vertex in a graph is not... |
| nbgrnself2 29338 | A class ` X ` is not a nei... |
| nbgrssovtx 29339 | The neighbors of a vertex ... |
| nbgrssvwo2 29340 | The neighbors of a vertex ... |
| nbgrsym 29341 | In a graph, the neighborho... |
| nbupgrres 29342 | The neighborhood of a vert... |
| usgrnbcnvfv 29343 | Applying the edge function... |
| nbusgredgeu 29344 | For each neighbor of a ver... |
| edgnbusgreu 29345 | For each edge incident to ... |
| nbusgredgeu0 29346 | For each neighbor of a ver... |
| nbusgrf1o0 29347 | The mapping of neighbors o... |
| nbusgrf1o1 29348 | The set of neighbors of a ... |
| nbusgrf1o 29349 | The set of neighbors of a ... |
| nbedgusgr 29350 | The number of neighbors of... |
| edgusgrnbfin 29351 | The number of neighbors of... |
| nbusgrfi 29352 | The class of neighbors of ... |
| nbfiusgrfi 29353 | The class of neighbors of ... |
| hashnbusgrnn0 29354 | The number of neighbors of... |
| nbfusgrlevtxm1 29355 | The number of neighbors of... |
| nbfusgrlevtxm2 29356 | If there is a vertex which... |
| nbusgrvtxm1 29357 | If the number of neighbors... |
| nb3grprlem1 29358 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29359 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29360 | The neighbors of a vertex ... |
| nb3grpr2 29361 | The neighbors of a vertex ... |
| nb3gr2nb 29362 | If the neighbors of two ve... |
| uvtxval 29365 | The set of all universal v... |
| uvtxel 29366 | A universal vertex, i.e. a... |
| uvtxisvtx 29367 | A universal vertex is a ve... |
| uvtxssvtx 29368 | The set of the universal v... |
| vtxnbuvtx 29369 | A universal vertex has all... |
| uvtxnbgrss 29370 | A universal vertex has all... |
| uvtxnbgrvtx 29371 | A universal vertex is neig... |
| uvtx0 29372 | There is no universal vert... |
| isuvtx 29373 | The set of all universal v... |
| uvtxel1 29374 | Characterization of a univ... |
| uvtx01vtx 29375 | If a graph/class has no ed... |
| uvtx2vtx1edg 29376 | If a graph has two vertice... |
| uvtx2vtx1edgb 29377 | If a hypergraph has two ve... |
| uvtxnbgr 29378 | A universal vertex has all... |
| uvtxnbgrb 29379 | A vertex is universal iff ... |
| uvtxusgr 29380 | The set of all universal v... |
| uvtxusgrel 29381 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29382 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29383 | If the number of neighbors... |
| uvtxnbvtxm1 29384 | A universal vertex has ` n... |
| nbupgruvtxres 29385 | The neighborhood of a univ... |
| uvtxupgrres 29386 | A universal vertex is univ... |
| cplgruvtxb 29391 | A graph ` G ` is complete ... |
| prcliscplgr 29392 | A proper class (representi... |
| iscplgr 29393 | The property of being a co... |
| iscplgrnb 29394 | A graph is complete iff al... |
| iscplgredg 29395 | A graph ` G ` is complete ... |
| iscusgr 29396 | The property of being a co... |
| cusgrusgr 29397 | A complete simple graph is... |
| cusgrcplgr 29398 | A complete simple graph is... |
| iscusgrvtx 29399 | A simple graph is complete... |
| cusgruvtxb 29400 | A simple graph is complete... |
| iscusgredg 29401 | A simple graph is complete... |
| cusgredg 29402 | In a complete simple graph... |
| cplgr0 29403 | The null graph (with no ve... |
| cusgr0 29404 | The null graph (with no ve... |
| cplgr0v 29405 | A null graph (with no vert... |
| cusgr0v 29406 | A graph with no vertices a... |
| cplgr1vlem 29407 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29408 | A graph with one vertex is... |
| cusgr1v 29409 | A graph with one vertex an... |
| cplgr2v 29410 | An undirected hypergraph w... |
| cplgr2vpr 29411 | An undirected hypergraph w... |
| nbcplgr 29412 | In a complete graph, each ... |
| cplgr3v 29413 | A pseudograph with three (... |
| cusgr3vnbpr 29414 | The neighbors of a vertex ... |
| cplgrop 29415 | A complete graph represent... |
| cusgrop 29416 | A complete simple graph re... |
| cusgrexilem1 29417 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29418 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29419 | An arbitrary set regarded ... |
| cusgrexilem2 29420 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29421 | An arbitrary set ` V ` reg... |
| cusgrexg 29422 | For each set there is a se... |
| structtousgr 29423 | Any (extensible) structure... |
| structtocusgr 29424 | Any (extensible) structure... |
| cffldtocusgr 29425 | The field of complex numbe... |
| cffldtocusgrOLD 29426 | Obsolete version of ~ cffl... |
| cusgrres 29427 | Restricting a complete sim... |
| cusgrsizeindb0 29428 | Base case of the induction... |
| cusgrsizeindb1 29429 | Base case of the induction... |
| cusgrsizeindslem 29430 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29431 | Part 1 of induction step i... |
| cusgrsize2inds 29432 | Induction step in ~ cusgrs... |
| cusgrsize 29433 | The size of a finite compl... |
| cusgrfilem1 29434 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29435 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29436 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29437 | If the size of a complete ... |
| usgredgsscusgredg 29438 | A simple graph is a subgra... |
| usgrsscusgr 29439 | A simple graph is a subgra... |
| sizusglecusglem1 29440 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29441 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29442 | The size of a simple graph... |
| fusgrmaxsize 29443 | The maximum size of a fini... |
| vtxdgfval 29446 | The value of the vertex de... |
| vtxdgval 29447 | The degree of a vertex. (... |
| vtxdgfival 29448 | The degree of a vertex for... |
| vtxdgop 29449 | The vertex degree expresse... |
| vtxdgf 29450 | The vertex degree function... |
| vtxdgelxnn0 29451 | The degree of a vertex is ... |
| vtxdg0v 29452 | The degree of a vertex in ... |
| vtxdg0e 29453 | The degree of a vertex in ... |
| vtxdgfisnn0 29454 | The degree of a vertex in ... |
| vtxdgfisf 29455 | The vertex degree function... |
| vtxdeqd 29456 | Equality theorem for the v... |
| vtxduhgr0e 29457 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29458 | The degree of the vertex i... |
| vdumgr0 29459 | A vertex in a multigraph h... |
| vtxdun 29460 | The degree of a vertex in ... |
| vtxdfiun 29461 | The degree of a vertex in ... |
| vtxduhgrun 29462 | The degree of a vertex in ... |
| vtxduhgrfiun 29463 | The degree of a vertex in ... |
| vtxdlfgrval 29464 | The value of the vertex de... |
| vtxdumgrval 29465 | The value of the vertex de... |
| vtxdusgrval 29466 | The value of the vertex de... |
| vtxd0nedgb 29467 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29468 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29469 | The value of the vertex de... |
| vtxdusgrfvedg 29470 | The value of the vertex de... |
| vtxduhgr0nedg 29471 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29472 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29473 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29474 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29475 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29476 | The vertex degree function... |
| vtxdgfusgr 29477 | In a finite simple graph, ... |
| fusgrn0degnn0 29478 | In a nonempty, finite grap... |
| 1loopgruspgr 29479 | A graph with one edge whic... |
| 1loopgredg 29480 | The set of edges in a grap... |
| 1loopgrnb0 29481 | In a graph (simple pseudog... |
| 1loopgrvd2 29482 | The vertex degree of a one... |
| 1loopgrvd0 29483 | The vertex degree of a one... |
| 1hevtxdg0 29484 | The vertex degree of verte... |
| 1hevtxdg1 29485 | The vertex degree of verte... |
| 1hegrvtxdg1 29486 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29487 | The vertex degree of a gra... |
| 1egrvtxdg1 29488 | The vertex degree of a one... |
| 1egrvtxdg1r 29489 | The vertex degree of a one... |
| 1egrvtxdg0 29490 | The vertex degree of a one... |
| p1evtxdeqlem 29491 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29492 | If an edge ` E ` which doe... |
| p1evtxdp1 29493 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29494 | The set of vertices in a g... |
| uspgrloopvtxel 29495 | A vertex in a graph (simpl... |
| uspgrloopiedg 29496 | The set of edges in a grap... |
| uspgrloopedg 29497 | The set of edges in a grap... |
| uspgrloopnb0 29498 | In a graph (simple pseudog... |
| uspgrloopvd2 29499 | The vertex degree of a one... |
| umgr2v2evtx 29500 | The set of vertices in a m... |
| umgr2v2evtxel 29501 | A vertex in a multigraph w... |
| umgr2v2eiedg 29502 | The edge function in a mul... |
| umgr2v2eedg 29503 | The set of edges in a mult... |
| umgr2v2e 29504 | A multigraph with two edge... |
| umgr2v2enb1 29505 | In a multigraph with two e... |
| umgr2v2evd2 29506 | In a multigraph with two e... |
| hashnbusgrvd 29507 | In a simple graph, the num... |
| usgruvtxvdb 29508 | In a finite simple graph w... |
| vdiscusgrb 29509 | A finite simple graph with... |
| vdiscusgr 29510 | In a finite complete simpl... |
| vtxdusgradjvtx 29511 | The degree of a vertex in ... |
| usgrvd0nedg 29512 | If a vertex in a simple gr... |
| uhgrvd00 29513 | If every vertex in a hyper... |
| usgrvd00 29514 | If every vertex in a simpl... |
| vdegp1ai 29515 | The induction step for a v... |
| vdegp1bi 29516 | The induction step for a v... |
| vdegp1ci 29517 | The induction step for a v... |
| vtxdginducedm1lem1 29518 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29519 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29520 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29521 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29522 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29523 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29524 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29525 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29526 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29527 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29528 | Induction step of ~ finsum... |
| finsumvtxdg2size 29529 | The sum of the degrees of ... |
| fusgr1th 29530 | The sum of the degrees of ... |
| finsumvtxdgeven 29531 | The sum of the degrees of ... |
| vtxdgoddnumeven 29532 | The number of vertices of ... |
| fusgrvtxdgonume 29533 | The number of vertices of ... |
| isrgr 29538 | The property of a class be... |
| rgrprop 29539 | The properties of a k-regu... |
| isrusgr 29540 | The property of being a k-... |
| rusgrprop 29541 | The properties of a k-regu... |
| rusgrrgr 29542 | A k-regular simple graph i... |
| rusgrusgr 29543 | A k-regular simple graph i... |
| finrusgrfusgr 29544 | A finite regular simple gr... |
| isrusgr0 29545 | The property of being a k-... |
| rusgrprop0 29546 | The properties of a k-regu... |
| usgreqdrusgr 29547 | If all vertices in a simpl... |
| fusgrregdegfi 29548 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29549 | If all vertices in a nonem... |
| frusgrnn0 29550 | In a nonempty finite k-reg... |
| 0edg0rgr 29551 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29552 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29553 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29554 | A simple graph is 0-regula... |
| 0vtxrgr 29555 | A null graph (with no vert... |
| 0vtxrusgr 29556 | A graph with no vertices a... |
| 0uhgrrusgr 29557 | The null graph as hypergra... |
| 0grrusgr 29558 | The null graph represented... |
| 0grrgr 29559 | The null graph represented... |
| cusgrrusgr 29560 | A complete simple graph wi... |
| cusgrm1rusgr 29561 | A finite simple graph with... |
| rusgrpropnb 29562 | The properties of a k-regu... |
| rusgrpropedg 29563 | The properties of a k-regu... |
| rusgrpropadjvtx 29564 | The properties of a k-regu... |
| rusgrnumwrdl2 29565 | In a k-regular simple grap... |
| rusgr1vtxlem 29566 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29567 | If a k-regular simple grap... |
| rgrusgrprc 29568 | The class of 0-regular sim... |
| rusgrprc 29569 | The class of 0-regular sim... |
| rgrprc 29570 | The class of 0-regular gra... |
| rgrprcx 29571 | The class of 0-regular gra... |
| rgrx0ndm 29572 | 0 is not in the domain of ... |
| rgrx0nd 29573 | The potentially alternativ... |
| ewlksfval 29580 | The set of s-walks of edge... |
| isewlk 29581 | Conditions for a function ... |
| ewlkprop 29582 | Properties of an s-walk of... |
| ewlkinedg 29583 | The intersection (common v... |
| ewlkle 29584 | An s-walk of edges is also... |
| upgrewlkle2 29585 | In a pseudograph, there is... |
| wkslem1 29586 | Lemma 1 for walks to subst... |
| wkslem2 29587 | Lemma 2 for walks to subst... |
| wksfval 29588 | The set of walks (in an un... |
| iswlk 29589 | Properties of a pair of fu... |
| wlkprop 29590 | Properties of a walk. (Co... |
| wlkv 29591 | The classes involved in a ... |
| iswlkg 29592 | Generalization of ~ iswlk ... |
| wlkf 29593 | The mapping enumerating th... |
| wlkcl 29594 | A walk has length ` # ( F ... |
| wlkp 29595 | The mapping enumerating th... |
| wlkpwrd 29596 | The sequence of vertices o... |
| wlklenvp1 29597 | The number of vertices of ... |
| wksv 29598 | The class of walks is a se... |
| wksvOLD 29599 | Obsolete version of ~ wksv... |
| wlkn0 29600 | The sequence of vertices o... |
| wlklenvm1 29601 | The number of edges of a w... |
| ifpsnprss 29602 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29603 | Each pair of adjacent vert... |
| wlkvtxiedg 29604 | The vertices of a walk are... |
| relwlk 29605 | The set ` ( Walks `` G ) `... |
| wlkvv 29606 | If there is at least one w... |
| wlkop 29607 | A walk is an ordered pair.... |
| wlkcpr 29608 | A walk as class with two c... |
| wlk2f 29609 | If there is a walk ` W ` t... |
| wlkcomp 29610 | A walk expressed by proper... |
| wlkcompim 29611 | Implications for the prope... |
| wlkelwrd 29612 | The components of a walk a... |
| wlkeq 29613 | Conditions for two walks (... |
| edginwlk 29614 | The value of the edge func... |
| upgredginwlk 29615 | The value of the edge func... |
| iedginwlk 29616 | The value of the edge func... |
| wlkl1loop 29617 | A walk of length 1 from a ... |
| wlk1walk 29618 | A walk is a 1-walk "on the... |
| wlk1ewlk 29619 | A walk is an s-walk "on th... |
| upgriswlk 29620 | Properties of a pair of fu... |
| upgrwlkedg 29621 | The edges of a walk in a p... |
| upgrwlkcompim 29622 | Implications for the prope... |
| wlkvtxedg 29623 | The vertices of a walk are... |
| upgrwlkvtxedg 29624 | The pairs of connected ver... |
| uspgr2wlkeq 29625 | Conditions for two walks w... |
| uspgr2wlkeq2 29626 | Conditions for two walks w... |
| uspgr2wlkeqi 29627 | Conditions for two walks w... |
| umgrwlknloop 29628 | In a multigraph, each walk... |
| wlkResOLD 29629 | Obsolete version of ~ opab... |
| wlkv0 29630 | If there is a walk in the ... |
| g0wlk0 29631 | There is no walk in a null... |
| 0wlk0 29632 | There is no walk for the e... |
| wlk0prc 29633 | There is no walk in a null... |
| wlklenvclwlk 29634 | The number of vertices in ... |
| wlkson 29635 | The set of walks between t... |
| iswlkon 29636 | Properties of a pair of fu... |
| wlkonprop 29637 | Properties of a walk betwe... |
| wlkpvtx 29638 | A walk connects vertices. ... |
| wlkepvtx 29639 | The endpoints of a walk ar... |
| wlkoniswlk 29640 | A walk between two vertice... |
| wlkonwlk 29641 | A walk is a walk between i... |
| wlkonwlk1l 29642 | A walk is a walk from its ... |
| wlksoneq1eq2 29643 | Two walks with identical s... |
| wlkonl1iedg 29644 | If there is a walk between... |
| wlkon2n0 29645 | The length of a walk betwe... |
| 2wlklem 29646 | Lemma for theorems for wal... |
| upgr2wlk 29647 | Properties of a pair of fu... |
| wlkreslem 29648 | Lemma for ~ wlkres . (Con... |
| wlkres 29649 | The restriction ` <. H , Q... |
| redwlklem 29650 | Lemma for ~ redwlk . (Con... |
| redwlk 29651 | A walk ending at the last ... |
| wlkp1lem1 29652 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29653 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29654 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29655 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29656 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29657 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29658 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29659 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29660 | Append one path segment (e... |
| wlkdlem1 29661 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29662 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29663 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29664 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29665 | Two words representing a w... |
| lfgrwlkprop 29666 | Two adjacent vertices in a... |
| lfgriswlk 29667 | Conditions for a pair of f... |
| lfgrwlknloop 29668 | In a loop-free graph, each... |
| reltrls 29673 | The set ` ( Trails `` G ) ... |
| trlsfval 29674 | The set of trails (in an u... |
| istrl 29675 | Conditions for a pair of c... |
| trliswlk 29676 | A trail is a walk. (Contr... |
| trlf1 29677 | The enumeration ` F ` of a... |
| trlreslem 29678 | Lemma for ~ trlres . Form... |
| trlres 29679 | The restriction ` <. H , Q... |
| upgrtrls 29680 | The set of trails in a pse... |
| upgristrl 29681 | Properties of a pair of fu... |
| upgrf1istrl 29682 | Properties of a pair of a ... |
| wksonproplem 29683 | Lemma for theorems for pro... |
| wksonproplemOLD 29684 | Obsolete version of ~ wkso... |
| trlsonfval 29685 | The set of trails between ... |
| istrlson 29686 | Properties of a pair of fu... |
| trlsonprop 29687 | Properties of a trail betw... |
| trlsonistrl 29688 | A trail between two vertic... |
| trlsonwlkon 29689 | A trail between two vertic... |
| trlontrl 29690 | A trail is a trail between... |
| relpths 29699 | The set ` ( Paths `` G ) `... |
| pthsfval 29700 | The set of paths (in an un... |
| spthsfval 29701 | The set of simple paths (i... |
| ispth 29702 | Conditions for a pair of c... |
| isspth 29703 | Conditions for a pair of c... |
| pthistrl 29704 | A path is a trail (in an u... |
| spthispth 29705 | A simple path is a path (i... |
| pthiswlk 29706 | A path is a walk (in an un... |
| spthiswlk 29707 | A simple path is a walk (i... |
| pthdivtx 29708 | The inner vertices of a pa... |
| pthdadjvtx 29709 | The adjacent vertices of a... |
| 2pthnloop 29710 | A path of length at least ... |
| upgr2pthnlp 29711 | A path of length at least ... |
| spthdifv 29712 | The vertices of a simple p... |
| spthdep 29713 | A simple path (at least of... |
| pthdepisspth 29714 | A path with different star... |
| upgrwlkdvdelem 29715 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29716 | In a pseudograph, all edge... |
| upgrspthswlk 29717 | The set of simple paths in... |
| upgrwlkdvspth 29718 | A walk consisting of diffe... |
| pthsonfval 29719 | The set of paths between t... |
| spthson 29720 | The set of simple paths be... |
| ispthson 29721 | Properties of a pair of fu... |
| isspthson 29722 | Properties of a pair of fu... |
| pthsonprop 29723 | Properties of a path betwe... |
| spthonprop 29724 | Properties of a simple pat... |
| pthonispth 29725 | A path between two vertice... |
| pthontrlon 29726 | A path between two vertice... |
| pthonpth 29727 | A path is a path between i... |
| isspthonpth 29728 | A pair of functions is a s... |
| spthonisspth 29729 | A simple path between to v... |
| spthonpthon 29730 | A simple path between two ... |
| spthonepeq 29731 | The endpoints of a simple ... |
| uhgrwkspthlem1 29732 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29733 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29734 | Any walk of length 1 betwe... |
| usgr2wlkneq 29735 | The vertices and edges are... |
| usgr2wlkspthlem1 29736 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29737 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29738 | In a simple graph, any wal... |
| usgr2trlncl 29739 | In a simple graph, any tra... |
| usgr2trlspth 29740 | In a simple graph, any tra... |
| usgr2pthspth 29741 | In a simple graph, any pat... |
| usgr2pthlem 29742 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29743 | In a simple graph, there i... |
| usgr2pth0 29744 | In a simply graph, there i... |
| pthdlem1 29745 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29746 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29747 | Lemma 2 for ~ pthd . (Con... |
| pthd 29748 | Two words representing a t... |
| clwlks 29751 | The set of closed walks (i... |
| isclwlk 29752 | A pair of functions repres... |
| clwlkiswlk 29753 | A closed walk is a walk (i... |
| clwlkwlk 29754 | Closed walks are walks (in... |
| clwlkswks 29755 | Closed walks are walks (in... |
| isclwlke 29756 | Properties of a pair of fu... |
| isclwlkupgr 29757 | Properties of a pair of fu... |
| clwlkcomp 29758 | A closed walk expressed by... |
| clwlkcompim 29759 | Implications for the prope... |
| upgrclwlkcompim 29760 | Implications for the prope... |
| clwlkcompbp 29761 | Basic properties of the co... |
| clwlkl1loop 29762 | A closed walk of length 1 ... |
| crcts 29767 | The set of circuits (in an... |
| cycls 29768 | The set of cycles (in an u... |
| iscrct 29769 | Sufficient and necessary c... |
| iscycl 29770 | Sufficient and necessary c... |
| crctprop 29771 | The properties of a circui... |
| cyclprop 29772 | The properties of a cycle:... |
| crctisclwlk 29773 | A circuit is a closed walk... |
| crctistrl 29774 | A circuit is a trail. (Co... |
| crctiswlk 29775 | A circuit is a walk. (Con... |
| cyclispth 29776 | A cycle is a path. (Contr... |
| cycliswlk 29777 | A cycle is a walk. (Contr... |
| cycliscrct 29778 | A cycle is a circuit. (Co... |
| cyclnspth 29779 | A (non-trivial) cycle is n... |
| cyclispthon 29780 | A cycle is a path starting... |
| lfgrn1cycl 29781 | In a loop-free graph there... |
| usgr2trlncrct 29782 | In a simple graph, any tra... |
| umgrn1cycl 29783 | In a multigraph graph (wit... |
| uspgrn2crct 29784 | In a simple pseudograph th... |
| usgrn2cycl 29785 | In a simple graph there ar... |
| crctcshwlkn0lem1 29786 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29787 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29788 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29789 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29790 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29791 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29792 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29793 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29794 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29795 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29796 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29797 | Cyclically shifting the in... |
| crctcshwlk 29798 | Cyclically shifting the in... |
| crctcshtrl 29799 | Cyclically shifting the in... |
| crctcsh 29800 | Cyclically shifting the in... |
| wwlks 29811 | The set of walks (in an un... |
| iswwlks 29812 | A word over the set of ver... |
| wwlksn 29813 | The set of walks (in an un... |
| iswwlksn 29814 | A word over the set of ver... |
| wwlksnprcl 29815 | Derivation of the length o... |
| iswwlksnx 29816 | Properties of a word to re... |
| wwlkbp 29817 | Basic properties of a walk... |
| wwlknbp 29818 | Basic properties of a walk... |
| wwlknp 29819 | Properties of a set being ... |
| wwlknbp1 29820 | Other basic properties of ... |
| wwlknvtx 29821 | The symbols of a word ` W ... |
| wwlknllvtx 29822 | If a word ` W ` represents... |
| wwlknlsw 29823 | If a word represents a wal... |
| wspthsn 29824 | The set of simple paths of... |
| iswspthn 29825 | An element of the set of s... |
| wspthnp 29826 | Properties of a set being ... |
| wwlksnon 29827 | The set of walks of a fixe... |
| wspthsnon 29828 | The set of simple paths of... |
| iswwlksnon 29829 | The set of walks of a fixe... |
| wwlksnon0 29830 | Sufficient conditions for ... |
| wwlksonvtx 29831 | If a word ` W ` represents... |
| iswspthsnon 29832 | The set of simple paths of... |
| wwlknon 29833 | An element of the set of w... |
| wspthnon 29834 | An element of the set of s... |
| wspthnonp 29835 | Properties of a set being ... |
| wspthneq1eq2 29836 | Two simple paths with iden... |
| wwlksn0s 29837 | The set of all walks as wo... |
| wwlkssswrd 29838 | Walks (represented by word... |
| wwlksn0 29839 | A walk of length 0 is repr... |
| 0enwwlksnge1 29840 | In graphs without edges, t... |
| wwlkswwlksn 29841 | A walk of a fixed length a... |
| wwlkssswwlksn 29842 | The walks of a fixed lengt... |
| wlkiswwlks1 29843 | The sequence of vertices i... |
| wlklnwwlkln1 29844 | The sequence of vertices i... |
| wlkiswwlks2lem1 29845 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29846 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29847 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29848 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29849 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29850 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29851 | A walk as word corresponds... |
| wlkiswwlks 29852 | A walk as word corresponds... |
| wlkiswwlksupgr2 29853 | A walk as word corresponds... |
| wlkiswwlkupgr 29854 | A walk as word corresponds... |
| wlkswwlksf1o 29855 | The mapping of (ordinary) ... |
| wlkswwlksen 29856 | The set of walks as words ... |
| wwlksm1edg 29857 | Removing the trailing edge... |
| wlklnwwlkln2lem 29858 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29859 | A walk of length ` N ` as ... |
| wlklnwwlkn 29860 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29861 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29862 | A walk of length ` N ` as ... |
| wlknewwlksn 29863 | If a walk in a pseudograph... |
| wlknwwlksnbij 29864 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29865 | In a simple pseudograph, t... |
| wlknwwlksneqs 29866 | The set of walks of a fixe... |
| wwlkseq 29867 | Equality of two walks (as ... |
| wwlksnred 29868 | Reduction of a walk (as wo... |
| wwlksnext 29869 | Extension of a walk (as wo... |
| wwlksnextbi 29870 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29871 | For each walk (as word) of... |
| wwlksnredwwlkn0 29872 | For each walk (as word) of... |
| wwlksnextwrd 29873 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29874 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29875 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29876 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29877 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29878 | There is a bijection betwe... |
| wwlksnexthasheq 29879 | The number of the extensio... |
| disjxwwlksn 29880 | Sets of walks (as words) e... |
| wwlksnndef 29881 | Conditions for ` WWalksN `... |
| wwlksnfi 29882 | The number of walks repres... |
| wlksnfi 29883 | The number of walks of fix... |
| wlksnwwlknvbij 29884 | There is a bijection betwe... |
| wwlksnextproplem1 29885 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29886 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29887 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29888 | Adding additional properti... |
| disjxwwlkn 29889 | Sets of walks (as words) e... |
| hashwwlksnext 29890 | Number of walks (as words)... |
| wwlksnwwlksnon 29891 | A walk of fixed length is ... |
| wspthsnwspthsnon 29892 | A simple path of fixed len... |
| wspthsnonn0vne 29893 | If the set of simple paths... |
| wspthsswwlkn 29894 | The set of simple paths of... |
| wspthnfi 29895 | In a finite graph, the set... |
| wwlksnonfi 29896 | In a finite graph, the set... |
| wspthsswwlknon 29897 | The set of simple paths of... |
| wspthnonfi 29898 | In a finite graph, the set... |
| wspniunwspnon 29899 | The set of nonempty simple... |
| wspn0 29900 | If there are no vertices, ... |
| 2wlkdlem1 29901 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29902 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29903 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29904 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29905 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29906 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29907 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29908 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29909 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29910 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29911 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29912 | Construction of a walk fro... |
| 2wlkond 29913 | A walk of length 2 from on... |
| 2trld 29914 | Construction of a trail fr... |
| 2trlond 29915 | A trail of length 2 from o... |
| 2pthd 29916 | A path of length 2 from on... |
| 2spthd 29917 | A simple path of length 2 ... |
| 2pthond 29918 | A simple path of length 2 ... |
| 2pthon3v 29919 | For a vertex adjacent to t... |
| umgr2adedgwlklem 29920 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 29921 | In a multigraph, two adjac... |
| umgr2adedgwlkon 29922 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 29923 | Alternate proof for ~ umgr... |
| umgr2adedgspth 29924 | In a multigraph, two adjac... |
| umgr2wlk 29925 | In a multigraph, there is ... |
| umgr2wlkon 29926 | For each pair of adjacent ... |
| elwwlks2s3 29927 | A walk of length 2 as word... |
| midwwlks2s3 29928 | There is a vertex between ... |
| wwlks2onv 29929 | If a length 3 string repre... |
| elwwlks2ons3im 29930 | A walk as word of length 2... |
| elwwlks2ons3 29931 | For each walk of length 2 ... |
| s3wwlks2on 29932 | A length 3 string which re... |
| umgrwwlks2on 29933 | A walk of length 2 between... |
| wwlks2onsym 29934 | There is a walk of length ... |
| elwwlks2on 29935 | A walk of length 2 between... |
| elwspths2on 29936 | A simple path of length 2 ... |
| wpthswwlks2on 29937 | For two different vertices... |
| 2wspdisj 29938 | All simple paths of length... |
| 2wspiundisj 29939 | All simple paths of length... |
| usgr2wspthons3 29940 | A simple path of length 2 ... |
| usgr2wspthon 29941 | A simple path of length 2 ... |
| elwwlks2 29942 | A walk of length 2 between... |
| elwspths2spth 29943 | A simple path of length 2 ... |
| rusgrnumwwlkl1 29944 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 29945 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 29946 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 29947 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 29948 | Induction base 1 for ~ rus... |
| rusgr0edg 29949 | Special case for graphs wi... |
| rusgrnumwwlks 29950 | Induction step for ~ rusgr... |
| rusgrnumwwlk 29951 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 29952 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 29953 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 29954 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 29955 | In a ` K `-regular graph, ... |
| clwwlk 29958 | The set of closed walks (i... |
| isclwwlk 29959 | Properties of a word to re... |
| clwwlkbp 29960 | Basic properties of a clos... |
| clwwlkgt0 29961 | There is no empty closed w... |
| clwwlksswrd 29962 | Closed walks (represented ... |
| clwwlk1loop 29963 | A closed walk of length 1 ... |
| clwwlkccatlem 29964 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 29965 | The concatenation of two w... |
| umgrclwwlkge2 29966 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 29967 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 29968 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 29969 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 29970 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 29971 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 29972 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 29973 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 29974 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 29975 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 29976 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 29977 | A closed walk as word of l... |
| clwlkclwwlk2 29978 | A closed walk corresponds ... |
| clwlkclwwlkflem 29979 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 29980 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 29981 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 29982 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 29983 | ` F ` is a function from t... |
| clwlkclwwlkfo 29984 | ` F ` is a function from t... |
| clwlkclwwlkf1 29985 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 29986 | ` F ` is a bijection betwe... |
| clwlkclwwlken 29987 | The set of the nonempty cl... |
| clwwisshclwwslemlem 29988 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 29989 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 29990 | Cyclically shifting a clos... |
| clwwisshclwwsn 29991 | Cyclically shifting a clos... |
| erclwwlkrel 29992 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 29993 | Two classes are equivalent... |
| erclwwlkeqlen 29994 | If two classes are equival... |
| erclwwlkref 29995 | ` .~ ` is a reflexive rela... |
| erclwwlksym 29996 | ` .~ ` is a symmetric rela... |
| erclwwlktr 29997 | ` .~ ` is a transitive rel... |
| erclwwlk 29998 | ` .~ ` is an equivalence r... |
| clwwlkn 30001 | The set of closed walks of... |
| isclwwlkn 30002 | A word over the set of ver... |
| clwwlkn0 30003 | There is no closed walk of... |
| clwwlkneq0 30004 | Sufficient conditions for ... |
| clwwlkclwwlkn 30005 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30006 | The closed walks of a fixe... |
| clwwlknlen 30007 | The length of a word repre... |
| clwwlknnn 30008 | The length of a closed wal... |
| clwwlknwrd 30009 | A closed walk of a fixed l... |
| clwwlknbp 30010 | Basic properties of a clos... |
| isclwwlknx 30011 | Characterization of a word... |
| clwwlknp 30012 | Properties of a set being ... |
| clwwlknwwlksn 30013 | A word representing a clos... |
| clwwlknlbonbgr1 30014 | The last but one vertex in... |
| clwwlkinwwlk 30015 | If the initial vertex of a... |
| clwwlkn1 30016 | A closed walk of length 1 ... |
| loopclwwlkn1b 30017 | The singleton word consist... |
| clwwlkn1loopb 30018 | A word represents a closed... |
| clwwlkn2 30019 | A closed walk of length 2 ... |
| clwwlknfi 30020 | If there is only a finite ... |
| clwwlkel 30021 | Obtaining a closed walk (a... |
| clwwlkf 30022 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30023 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30024 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30025 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30026 | F is a 1-1 onto function, ... |
| clwwlken 30027 | The set of closed walks of... |
| clwwlknwwlkncl 30028 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30029 | A nonempty word over verti... |
| clwwlknwwlksnb 30030 | A word over vertices repre... |
| clwwlkext2edg 30031 | If a word concatenated wit... |
| wwlksext2clwwlk 30032 | If a word represents a wal... |
| wwlksubclwwlk 30033 | Any prefix of a word repre... |
| clwwnisshclwwsn 30034 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30035 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30036 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30037 | The set of cyclical shifts... |
| clwwlknccat 30038 | The concatenation of two w... |
| umgr2cwwk2dif 30039 | If a word represents a clo... |
| umgr2cwwkdifex 30040 | If a word represents a clo... |
| erclwwlknrel 30041 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30042 | Two classes are equivalent... |
| erclwwlkneqlen 30043 | If two classes are equival... |
| erclwwlknref 30044 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30045 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30046 | ` .~ ` is a transitive rel... |
| erclwwlkn 30047 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30048 | The quotient set of the se... |
| hashclwwlkn0 30049 | The number of closed walks... |
| eclclwwlkn1 30050 | An equivalence class accor... |
| eleclclwwlkn 30051 | A member of an equivalence... |
| hashecclwwlkn1 30052 | The size of every equivale... |
| umgrhashecclwwlk 30053 | The size of every equivale... |
| fusgrhashclwwlkn 30054 | The size of the set of clo... |
| clwwlkndivn 30055 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30056 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30057 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30058 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30059 | There is a one-to-one onto... |
| clwlkssizeeq 30060 | The size of the set of clo... |
| clwlksndivn 30061 | The size of the set of clo... |
| clwwlknonmpo 30064 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30065 | The set of closed walks on... |
| isclwwlknon 30066 | A word over the set of ver... |
| clwwlk0on0 30067 | There is no word over the ... |
| clwwlknon0 30068 | Sufficient conditions for ... |
| clwwlknonfin 30069 | In a finite graph ` G ` , ... |
| clwwlknonel 30070 | Characterization of a word... |
| clwwlknonccat 30071 | The concatenation of two w... |
| clwwlknon1 30072 | The set of closed walks on... |
| clwwlknon1loop 30073 | If there is a loop at vert... |
| clwwlknon1nloop 30074 | If there is no loop at ver... |
| clwwlknon1sn 30075 | The set of (closed) walks ... |
| clwwlknon1le1 30076 | There is at most one (clos... |
| clwwlknon2 30077 | The set of closed walks on... |
| clwwlknon2x 30078 | The set of closed walks on... |
| s2elclwwlknon2 30079 | Sufficient conditions of a... |
| clwwlknon2num 30080 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30081 | A word over vertices repre... |
| clwwlknonex2lem1 30082 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30083 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30084 | Extending a closed walk ` ... |
| clwwlknonex2e 30085 | Extending a closed walk ` ... |
| clwwlknondisj 30086 | The sets of closed walks o... |
| clwwlknun 30087 | The set of closed walks of... |
| clwwlkvbij 30088 | There is a bijection betwe... |
| 0ewlk 30089 | The empty set (empty seque... |
| 1ewlk 30090 | A sequence of 1 edge is an... |
| 0wlk 30091 | A pair of an empty set (of... |
| is0wlk 30092 | A pair of an empty set (of... |
| 0wlkonlem1 30093 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30094 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30095 | A walk of length 0 from a ... |
| 0wlkons1 30096 | A walk of length 0 from a ... |
| 0trl 30097 | A pair of an empty set (of... |
| is0trl 30098 | A pair of an empty set (of... |
| 0trlon 30099 | A trail of length 0 from a... |
| 0pth 30100 | A pair of an empty set (of... |
| 0spth 30101 | A pair of an empty set (of... |
| 0pthon 30102 | A path of length 0 from a ... |
| 0pthon1 30103 | A path of length 0 from a ... |
| 0pthonv 30104 | For each vertex there is a... |
| 0clwlk 30105 | A pair of an empty set (of... |
| 0clwlkv 30106 | Any vertex (more precisely... |
| 0clwlk0 30107 | There is no closed walk in... |
| 0crct 30108 | A pair of an empty set (of... |
| 0cycl 30109 | A pair of an empty set (of... |
| 1pthdlem1 30110 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30111 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30112 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30113 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30114 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30115 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30116 | In a graph with two vertic... |
| 1trld 30117 | In a graph with two vertic... |
| 1pthd 30118 | In a graph with two vertic... |
| 1pthond 30119 | In a graph with two vertic... |
| upgr1wlkdlem1 30120 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30121 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30122 | In a pseudograph with two ... |
| upgr1trld 30123 | In a pseudograph with two ... |
| upgr1pthd 30124 | In a pseudograph with two ... |
| upgr1pthond 30125 | In a pseudograph with two ... |
| lppthon 30126 | A loop (which is an edge a... |
| lp1cycl 30127 | A loop (which is an edge a... |
| 1pthon2v 30128 | For each pair of adjacent ... |
| 1pthon2ve 30129 | For each pair of adjacent ... |
| wlk2v2elem1 30130 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30131 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30132 | In a graph with two vertic... |
| ntrl2v2e 30133 | A walk which is not a trai... |
| 3wlkdlem1 30134 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30135 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30136 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30137 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30138 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30139 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30140 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30141 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30142 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30143 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30144 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30145 | Construction of a walk fro... |
| 3wlkond 30146 | A walk of length 3 from on... |
| 3trld 30147 | Construction of a trail fr... |
| 3trlond 30148 | A trail of length 3 from o... |
| 3pthd 30149 | A path of length 3 from on... |
| 3pthond 30150 | A path of length 3 from on... |
| 3spthd 30151 | A simple path of length 3 ... |
| 3spthond 30152 | A simple path of length 3 ... |
| 3cycld 30153 | Construction of a 3-cycle ... |
| 3cyclpd 30154 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30155 | If there is a cycle of len... |
| uhgr3cyclexlem 30156 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30157 | If there are three differe... |
| umgr3cyclex 30158 | If there are three (differ... |
| umgr3v3e3cycl 30159 | If and only if there is a ... |
| upgr4cycl4dv4e 30160 | If there is a cycle of len... |
| dfconngr1 30163 | Alternative definition of ... |
| isconngr 30164 | The property of being a co... |
| isconngr1 30165 | The property of being a co... |
| cusconngr 30166 | A complete hypergraph is c... |
| 0conngr 30167 | A graph without vertices i... |
| 0vconngr 30168 | A graph without vertices i... |
| 1conngr 30169 | A graph with (at most) one... |
| conngrv2edg 30170 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30171 | A vertex in a connected mu... |
| releupth 30174 | The set ` ( EulerPaths `` ... |
| eupths 30175 | The Eulerian paths on the ... |
| iseupth 30176 | The property " ` <. F , P ... |
| iseupthf1o 30177 | The property " ` <. F , P ... |
| eupthi 30178 | Properties of an Eulerian ... |
| eupthf1o 30179 | The ` F ` function in an E... |
| eupthfi 30180 | Any graph with an Eulerian... |
| eupthseg 30181 | The ` N ` -th edge in an e... |
| upgriseupth 30182 | The property " ` <. F , P ... |
| upgreupthi 30183 | Properties of an Eulerian ... |
| upgreupthseg 30184 | The ` N ` -th edge in an e... |
| eupthcl 30185 | An Eulerian path has lengt... |
| eupthistrl 30186 | An Eulerian path is a trai... |
| eupthiswlk 30187 | An Eulerian path is a walk... |
| eupthpf 30188 | The ` P ` function in an E... |
| eupth0 30189 | There is an Eulerian path ... |
| eupthres 30190 | The restriction ` <. H , Q... |
| eupthp1 30191 | Append one path segment to... |
| eupth2eucrct 30192 | Append one path segment to... |
| eupth2lem1 30193 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30194 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30195 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30196 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30197 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30198 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30199 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30200 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30201 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30202 | Formerly part of proof of ... |
| eupth2lem3lem1 30203 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30204 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30205 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30206 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30207 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30208 | Formerly part of proof of ... |
| eupth2lem3lem7 30209 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30210 | Formerly part of proof of ... |
| eupth2lem3 30211 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30212 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30213 | Lemma for ~ eupth2 (induct... |
| eupth2 30214 | The only vertices of odd d... |
| eulerpathpr 30215 | A graph with an Eulerian p... |
| eulerpath 30216 | A pseudograph with an Eule... |
| eulercrct 30217 | A pseudograph with an Eule... |
| eucrctshift 30218 | Cyclically shifting the in... |
| eucrct2eupth1 30219 | Removing one edge ` ( I ``... |
| eucrct2eupth 30220 | Removing one edge ` ( I ``... |
| konigsbergvtx 30221 | The set of vertices of the... |
| konigsbergiedg 30222 | The indexed edges of the K... |
| konigsbergiedgw 30223 | The indexed edges of the K... |
| konigsbergssiedgwpr 30224 | Each subset of the indexed... |
| konigsbergssiedgw 30225 | Each subset of the indexed... |
| konigsbergumgr 30226 | The Königsberg graph ... |
| konigsberglem1 30227 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30228 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30229 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30230 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30231 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30232 | The Königsberg Bridge... |
| isfrgr 30235 | The property of being a fr... |
| frgrusgr 30236 | A friendship graph is a si... |
| frgr0v 30237 | Any null graph (set with n... |
| frgr0vb 30238 | Any null graph (without ve... |
| frgruhgr0v 30239 | Any null graph (without ve... |
| frgr0 30240 | The null graph (graph with... |
| frcond1 30241 | The friendship condition: ... |
| frcond2 30242 | The friendship condition: ... |
| frgreu 30243 | Variant of ~ frcond2 : An... |
| frcond3 30244 | The friendship condition, ... |
| frcond4 30245 | The friendship condition, ... |
| frgr1v 30246 | Any graph with (at most) o... |
| nfrgr2v 30247 | Any graph with two (differ... |
| frgr3vlem1 30248 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30249 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30250 | Any graph with three verti... |
| 1vwmgr 30251 | Every graph with one verte... |
| 3vfriswmgrlem 30252 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30253 | Every friendship graph wit... |
| 1to2vfriswmgr 30254 | Every friendship graph wit... |
| 1to3vfriswmgr 30255 | Every friendship graph wit... |
| 1to3vfriendship 30256 | The friendship theorem for... |
| 2pthfrgrrn 30257 | Between any two (different... |
| 2pthfrgrrn2 30258 | Between any two (different... |
| 2pthfrgr 30259 | Between any two (different... |
| 3cyclfrgrrn1 30260 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30261 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30262 | Every vertex in a friendsh... |
| 3cyclfrgr 30263 | Every vertex in a friendsh... |
| 4cycl2v2nb 30264 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30265 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30266 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30267 | A graph with a 4-cycle is ... |
| frgrnbnb 30268 | If two neighbors ` U ` and... |
| frgrconngr 30269 | A friendship graph is conn... |
| vdgn0frgrv2 30270 | A vertex in a friendship g... |
| vdgn1frgrv2 30271 | Any vertex in a friendship... |
| vdgn1frgrv3 30272 | Any vertex in a friendship... |
| vdgfrgrgt2 30273 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30274 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30275 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30276 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30277 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30278 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30279 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30280 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30281 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30282 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30283 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30284 | In a friendship graph, two... |
| frgrwopreglem4a 30285 | In a friendship graph any ... |
| frgrwopreglem5a 30286 | If a friendship graph has ... |
| frgrwopreglem1 30287 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30288 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30289 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30290 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30291 | According to statement 5 i... |
| frgrwopregbsn 30292 | According to statement 5 i... |
| frgrwopreg1 30293 | According to statement 5 i... |
| frgrwopreg2 30294 | According to statement 5 i... |
| frgrwopreglem5lem 30295 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30296 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30297 | Alternate direct proof of ... |
| frgrwopreg 30298 | In a friendship graph ther... |
| frgrregorufr0 30299 | In a friendship graph ther... |
| frgrregorufr 30300 | If there is a vertex havin... |
| frgrregorufrg 30301 | If there is a vertex havin... |
| frgr2wwlkeu 30302 | For two different vertices... |
| frgr2wwlkn0 30303 | In a friendship graph, the... |
| frgr2wwlk1 30304 | In a friendship graph, the... |
| frgr2wsp1 30305 | In a friendship graph, the... |
| frgr2wwlkeqm 30306 | If there is a (simple) pat... |
| frgrhash2wsp 30307 | The number of simple paths... |
| fusgreg2wsplem 30308 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30309 | The set of paths of length... |
| fusgreghash2wspv 30310 | According to statement 7 i... |
| fusgreg2wsp 30311 | In a finite simple graph, ... |
| 2wspmdisj 30312 | The sets of paths of lengt... |
| fusgreghash2wsp 30313 | In a finite k-regular grap... |
| frrusgrord0lem 30314 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30315 | If a nonempty finite frien... |
| frrusgrord 30316 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30317 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30318 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30319 | If the initial vertex of a... |
| clwwnonrepclwwnon 30320 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30321 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30322 | Value of operation ` C ` ,... |
| 2clwwlk2 30323 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30324 | Characterization of an ele... |
| 2clwwlk2clwwlk 30325 | An element of the value of... |
| numclwwlk1lem2foalem 30326 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30327 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30328 | Characterization of an ele... |
| numclwwlk1lem2foa 30329 | Going forth and back from ... |
| numclwwlk1lem2f 30330 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30331 | Value of the function ` T ... |
| numclwwlk1lem2f1 30332 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30333 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30334 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30335 | The set of double loops of... |
| numclwwlk1 30336 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30337 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30338 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30339 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30340 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30341 | The sets of the two repres... |
| wlkl0 30342 | There is exactly one walk ... |
| clwlknon2num 30343 | There are k walks of lengt... |
| numclwlk1lem1 30344 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30345 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30346 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30347 | Value of operation ` H ` ,... |
| numclwwlkovh 30348 | Value of operation ` H ` ,... |
| numclwwlkovq 30349 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30350 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30351 | In a friendship graph, for... |
| numclwlk2lem2f 30352 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30353 | Value of the function ` R ... |
| numclwlk2lem2f1o 30354 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30355 | In a friendship graph, the... |
| numclwwlk2 30356 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30357 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30358 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30359 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30360 | Statement 12 in [Huneke] p... |
| numclwwlk4 30361 | The total number of closed... |
| numclwwlk5lem 30362 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30363 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30364 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30365 | For a prime divisor ` P ` ... |
| numclwwlk7 30366 | Statement 14 in [Huneke] p... |
| numclwwlk8 30367 | The size of the set of clo... |
| frgrreggt1 30368 | If a finite nonempty frien... |
| frgrreg 30369 | If a finite nonempty frien... |
| frgrregord013 30370 | If a finite friendship gra... |
| frgrregord13 30371 | If a nonempty finite frien... |
| frgrogt3nreg 30372 | If a finite friendship gra... |
| friendshipgt3 30373 | The friendship theorem for... |
| friendship 30374 | The friendship theorem: I... |
| conventions 30375 |
H... |
| conventions-labels 30376 |
... |
| conventions-comments 30377 |
... |
| natded 30378 | Here are typical n... |
| ex-natded5.2 30379 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30380 | A more efficient proof of ... |
| ex-natded5.2i 30381 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30382 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30383 | A more efficient proof of ... |
| ex-natded5.3i 30384 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30385 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30386 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30387 | A more efficient proof of ... |
| ex-natded5.8 30388 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30389 | A more efficient proof of ... |
| ex-natded5.13 30390 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30391 | A more efficient proof of ... |
| ex-natded9.20 30392 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30393 | A more efficient proof of ... |
| ex-natded9.26 30394 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30395 | A more efficient proof of ... |
| ex-or 30396 | Example for ~ df-or . Exa... |
| ex-an 30397 | Example for ~ df-an . Exa... |
| ex-dif 30398 | Example for ~ df-dif . Ex... |
| ex-un 30399 | Example for ~ df-un . Exa... |
| ex-in 30400 | Example for ~ df-in . Exa... |
| ex-uni 30401 | Example for ~ df-uni . Ex... |
| ex-ss 30402 | Example for ~ df-ss . Exa... |
| ex-pss 30403 | Example for ~ df-pss . Ex... |
| ex-pw 30404 | Example for ~ df-pw . Exa... |
| ex-pr 30405 | Example for ~ df-pr . (Co... |
| ex-br 30406 | Example for ~ df-br . Exa... |
| ex-opab 30407 | Example for ~ df-opab . E... |
| ex-eprel 30408 | Example for ~ df-eprel . ... |
| ex-id 30409 | Example for ~ df-id . Exa... |
| ex-po 30410 | Example for ~ df-po . Exa... |
| ex-xp 30411 | Example for ~ df-xp . Exa... |
| ex-cnv 30412 | Example for ~ df-cnv . Ex... |
| ex-co 30413 | Example for ~ df-co . Exa... |
| ex-dm 30414 | Example for ~ df-dm . Exa... |
| ex-rn 30415 | Example for ~ df-rn . Exa... |
| ex-res 30416 | Example for ~ df-res . Ex... |
| ex-ima 30417 | Example for ~ df-ima . Ex... |
| ex-fv 30418 | Example for ~ df-fv . Exa... |
| ex-1st 30419 | Example for ~ df-1st . Ex... |
| ex-2nd 30420 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30421 | Example for ~ df-dec , 100... |
| ex-fl 30422 | Example for ~ df-fl . Exa... |
| ex-ceil 30423 | Example for ~ df-ceil . (... |
| ex-mod 30424 | Example for ~ df-mod . (C... |
| ex-exp 30425 | Example for ~ df-exp . (C... |
| ex-fac 30426 | Example for ~ df-fac . (C... |
| ex-bc 30427 | Example for ~ df-bc . (Co... |
| ex-hash 30428 | Example for ~ df-hash . (... |
| ex-sqrt 30429 | Example for ~ df-sqrt . (... |
| ex-abs 30430 | Example for ~ df-abs . (C... |
| ex-dvds 30431 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30432 | Example for ~ df-gcd . (C... |
| ex-lcm 30433 | Example for ~ df-lcm . (C... |
| ex-prmo 30434 | Example for ~ df-prmo : ` ... |
| aevdemo 30435 | Proof illustrating the com... |
| ex-ind-dvds 30436 | Example of a proof by indu... |
| ex-fpar 30437 | Formalized example provide... |
| avril1 30438 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30439 | The law of excluded middle... |
| helloworld 30440 | The classic "Hello world" ... |
| 1p1e2apr1 30441 | One plus one equals two. ... |
| eqid1 30442 | Law of identity (reflexivi... |
| 1div0apr 30443 | Division by zero is forbid... |
| topnfbey 30444 | Nothing seems to be imposs... |
| 9p10ne21 30445 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30446 | 9 + 10 equals 21. This as... |
| nrt2irr 30448 | The ` N ` -th root of 2 is... |
| isplig 30451 | The predicate "is a planar... |
| ispligb 30452 | The predicate "is a planar... |
| tncp 30453 | In any planar incidence ge... |
| l2p 30454 | For any line in a planar i... |
| lpni 30455 | For any line in a planar i... |
| nsnlplig 30456 | There is no "one-point lin... |
| nsnlpligALT 30457 | Alternate version of ~ nsn... |
| n0lplig 30458 | There is no "empty line" i... |
| n0lpligALT 30459 | Alternate version of ~ n0l... |
| eulplig 30460 | Through two distinct point... |
| pliguhgr 30461 | Any planar incidence geome... |
| dummylink 30462 | Alias for ~ a1ii that may ... |
| id1 30463 | Alias for ~ idALT that may... |
| isgrpo 30472 | The predicate "is a group ... |
| isgrpoi 30473 | Properties that determine ... |
| grpofo 30474 | A group operation maps ont... |
| grpocl 30475 | Closure law for a group op... |
| grpolidinv 30476 | A group has a left identit... |
| grpon0 30477 | The base set of a group is... |
| grpoass 30478 | A group operation is assoc... |
| grpoidinvlem1 30479 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30480 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30481 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30482 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30483 | A group has a left and rig... |
| grpoideu 30484 | The left identity element ... |
| grporndm 30485 | A group's range in terms o... |
| 0ngrp 30486 | The empty set is not a gro... |
| gidval 30487 | The value of the identity ... |
| grpoidval 30488 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30489 | The identity element of a ... |
| grpoidinv2 30490 | A group's properties using... |
| grpolid 30491 | The identity element of a ... |
| grporid 30492 | The identity element of a ... |
| grporcan 30493 | Right cancellation law for... |
| grpoinveu 30494 | The left inverse element o... |
| grpoid 30495 | Two ways of saying that an... |
| grporn 30496 | The range of a group opera... |
| grpoinvfval 30497 | The inverse function of a ... |
| grpoinvval 30498 | The inverse of a group ele... |
| grpoinvcl 30499 | A group element's inverse ... |
| grpoinv 30500 | The properties of a group ... |
| grpolinv 30501 | The left inverse of a grou... |
| grporinv 30502 | The right inverse of a gro... |
| grpoinvid1 30503 | The inverse of a group ele... |
| grpoinvid2 30504 | The inverse of a group ele... |
| grpolcan 30505 | Left cancellation law for ... |
| grpo2inv 30506 | Double inverse law for gro... |
| grpoinvf 30507 | Mapping of the inverse fun... |
| grpoinvop 30508 | The inverse of the group o... |
| grpodivfval 30509 | Group division (or subtrac... |
| grpodivval 30510 | Group division (or subtrac... |
| grpodivinv 30511 | Group division by an inver... |
| grpoinvdiv 30512 | Inverse of a group divisio... |
| grpodivf 30513 | Mapping for group division... |
| grpodivcl 30514 | Closure of group division ... |
| grpodivdiv 30515 | Double group division. (C... |
| grpomuldivass 30516 | Associative-type law for m... |
| grpodivid 30517 | Division of a group member... |
| grponpcan 30518 | Cancellation law for group... |
| isablo 30521 | The predicate "is an Abeli... |
| ablogrpo 30522 | An Abelian group operation... |
| ablocom 30523 | An Abelian group operation... |
| ablo32 30524 | Commutative/associative la... |
| ablo4 30525 | Commutative/associative la... |
| isabloi 30526 | Properties that determine ... |
| ablomuldiv 30527 | Law for group multiplicati... |
| ablodivdiv 30528 | Law for double group divis... |
| ablodivdiv4 30529 | Law for double group divis... |
| ablodiv32 30530 | Swap the second and third ... |
| ablonncan 30531 | Cancellation law for group... |
| ablonnncan1 30532 | Cancellation law for group... |
| vcrel 30535 | The class of all complex v... |
| vciOLD 30536 | Obsolete version of ~ cvsi... |
| vcsm 30537 | Functionality of th scalar... |
| vccl 30538 | Closure of the scalar prod... |
| vcidOLD 30539 | Identity element for the s... |
| vcdi 30540 | Distributive law for the s... |
| vcdir 30541 | Distributive law for the s... |
| vcass 30542 | Associative law for the sc... |
| vc2OLD 30543 | A vector plus itself is tw... |
| vcablo 30544 | Vector addition is an Abel... |
| vcgrp 30545 | Vector addition is a group... |
| vclcan 30546 | Left cancellation law for ... |
| vczcl 30547 | The zero vector is a vecto... |
| vc0rid 30548 | The zero vector is a right... |
| vc0 30549 | Zero times a vector is the... |
| vcz 30550 | Anything times the zero ve... |
| vcm 30551 | Minus 1 times a vector is ... |
| isvclem 30552 | Lemma for ~ isvcOLD . (Co... |
| vcex 30553 | The components of a comple... |
| isvcOLD 30554 | The predicate "is a comple... |
| isvciOLD 30555 | Properties that determine ... |
| cnaddabloOLD 30556 | Obsolete version of ~ cnad... |
| cnidOLD 30557 | Obsolete version of ~ cnad... |
| cncvcOLD 30558 | Obsolete version of ~ cncv... |
| nvss 30568 | Structure of the class of ... |
| nvvcop 30569 | A normed complex vector sp... |
| nvrel 30577 | The class of all normed co... |
| vafval 30578 | Value of the function for ... |
| bafval 30579 | Value of the function for ... |
| smfval 30580 | Value of the function for ... |
| 0vfval 30581 | Value of the function for ... |
| nmcvfval 30582 | Value of the norm function... |
| nvop2 30583 | A normed complex vector sp... |
| nvvop 30584 | The vector space component... |
| isnvlem 30585 | Lemma for ~ isnv . (Contr... |
| nvex 30586 | The components of a normed... |
| isnv 30587 | The predicate "is a normed... |
| isnvi 30588 | Properties that determine ... |
| nvi 30589 | The properties of a normed... |
| nvvc 30590 | The vector space component... |
| nvablo 30591 | The vector addition operat... |
| nvgrp 30592 | The vector addition operat... |
| nvgf 30593 | Mapping for the vector add... |
| nvsf 30594 | Mapping for the scalar mul... |
| nvgcl 30595 | Closure law for the vector... |
| nvcom 30596 | The vector addition (group... |
| nvass 30597 | The vector addition (group... |
| nvadd32 30598 | Commutative/associative la... |
| nvrcan 30599 | Right cancellation law for... |
| nvadd4 30600 | Rearrangement of 4 terms i... |
| nvscl 30601 | Closure law for the scalar... |
| nvsid 30602 | Identity element for the s... |
| nvsass 30603 | Associative law for the sc... |
| nvscom 30604 | Commutative law for the sc... |
| nvdi 30605 | Distributive law for the s... |
| nvdir 30606 | Distributive law for the s... |
| nv2 30607 | A vector plus itself is tw... |
| vsfval 30608 | Value of the function for ... |
| nvzcl 30609 | Closure law for the zero v... |
| nv0rid 30610 | The zero vector is a right... |
| nv0lid 30611 | The zero vector is a left ... |
| nv0 30612 | Zero times a vector is the... |
| nvsz 30613 | Anything times the zero ve... |
| nvinv 30614 | Minus 1 times a vector is ... |
| nvinvfval 30615 | Function for the negative ... |
| nvm 30616 | Vector subtraction in term... |
| nvmval 30617 | Value of vector subtractio... |
| nvmval2 30618 | Value of vector subtractio... |
| nvmfval 30619 | Value of the function for ... |
| nvmf 30620 | Mapping for the vector sub... |
| nvmcl 30621 | Closure law for the vector... |
| nvnnncan1 30622 | Cancellation law for vecto... |
| nvmdi 30623 | Distributive law for scala... |
| nvnegneg 30624 | Double negative of a vecto... |
| nvmul0or 30625 | If a scalar product is zer... |
| nvrinv 30626 | A vector minus itself. (C... |
| nvlinv 30627 | Minus a vector plus itself... |
| nvpncan2 30628 | Cancellation law for vecto... |
| nvpncan 30629 | Cancellation law for vecto... |
| nvaddsub 30630 | Commutative/associative la... |
| nvnpcan 30631 | Cancellation law for a nor... |
| nvaddsub4 30632 | Rearrangement of 4 terms i... |
| nvmeq0 30633 | The difference between two... |
| nvmid 30634 | A vector minus itself is t... |
| nvf 30635 | Mapping for the norm funct... |
| nvcl 30636 | The norm of a normed compl... |
| nvcli 30637 | The norm of a normed compl... |
| nvs 30638 | Proportionality property o... |
| nvsge0 30639 | The norm of a scalar produ... |
| nvm1 30640 | The norm of the negative o... |
| nvdif 30641 | The norm of the difference... |
| nvpi 30642 | The norm of a vector plus ... |
| nvz0 30643 | The norm of a zero vector ... |
| nvz 30644 | The norm of a vector is ze... |
| nvtri 30645 | Triangle inequality for th... |
| nvmtri 30646 | Triangle inequality for th... |
| nvabs 30647 | Norm difference property o... |
| nvge0 30648 | The norm of a normed compl... |
| nvgt0 30649 | A nonzero norm is positive... |
| nv1 30650 | From any nonzero vector, c... |
| nvop 30651 | A complex inner product sp... |
| cnnv 30652 | The set of complex numbers... |
| cnnvg 30653 | The vector addition (group... |
| cnnvba 30654 | The base set of the normed... |
| cnnvs 30655 | The scalar product operati... |
| cnnvnm 30656 | The norm operation of the ... |
| cnnvm 30657 | The vector subtraction ope... |
| elimnv 30658 | Hypothesis elimination lem... |
| elimnvu 30659 | Hypothesis elimination lem... |
| imsval 30660 | Value of the induced metri... |
| imsdval 30661 | Value of the induced metri... |
| imsdval2 30662 | Value of the distance func... |
| nvnd 30663 | The norm of a normed compl... |
| imsdf 30664 | Mapping for the induced me... |
| imsmetlem 30665 | Lemma for ~ imsmet . (Con... |
| imsmet 30666 | The induced metric of a no... |
| imsxmet 30667 | The induced metric of a no... |
| cnims 30668 | The metric induced on the ... |
| vacn 30669 | Vector addition is jointly... |
| nmcvcn 30670 | The norm of a normed compl... |
| nmcnc 30671 | The norm of a normed compl... |
| smcnlem 30672 | Lemma for ~ smcn . (Contr... |
| smcn 30673 | Scalar multiplication is j... |
| vmcn 30674 | Vector subtraction is join... |
| dipfval 30677 | The inner product function... |
| ipval 30678 | Value of the inner product... |
| ipval2lem2 30679 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30680 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30681 | Lemma for ~ ipval3 . (Con... |
| ipval2 30682 | Expansion of the inner pro... |
| 4ipval2 30683 | Four times the inner produ... |
| ipval3 30684 | Expansion of the inner pro... |
| ipidsq 30685 | The inner product of a vec... |
| ipnm 30686 | Norm expressed in terms of... |
| dipcl 30687 | An inner product is a comp... |
| ipf 30688 | Mapping for the inner prod... |
| dipcj 30689 | The complex conjugate of a... |
| ipipcj 30690 | An inner product times its... |
| diporthcom 30691 | Orthogonality (meaning inn... |
| dip0r 30692 | Inner product with a zero ... |
| dip0l 30693 | Inner product with a zero ... |
| ipz 30694 | The inner product of a vec... |
| dipcn 30695 | Inner product is jointly c... |
| sspval 30698 | The set of all subspaces o... |
| isssp 30699 | The predicate "is a subspa... |
| sspid 30700 | A normed complex vector sp... |
| sspnv 30701 | A subspace is a normed com... |
| sspba 30702 | The base set of a subspace... |
| sspg 30703 | Vector addition on a subsp... |
| sspgval 30704 | Vector addition on a subsp... |
| ssps 30705 | Scalar multiplication on a... |
| sspsval 30706 | Scalar multiplication on a... |
| sspmlem 30707 | Lemma for ~ sspm and other... |
| sspmval 30708 | Vector addition on a subsp... |
| sspm 30709 | Vector subtraction on a su... |
| sspz 30710 | The zero vector of a subsp... |
| sspn 30711 | The norm on a subspace is ... |
| sspnval 30712 | The norm on a subspace in ... |
| sspimsval 30713 | The induced metric on a su... |
| sspims 30714 | The induced metric on a su... |
| lnoval 30727 | The set of linear operator... |
| islno 30728 | The predicate "is a linear... |
| lnolin 30729 | Basic linearity property o... |
| lnof 30730 | A linear operator is a map... |
| lno0 30731 | The value of a linear oper... |
| lnocoi 30732 | The composition of two lin... |
| lnoadd 30733 | Addition property of a lin... |
| lnosub 30734 | Subtraction property of a ... |
| lnomul 30735 | Scalar multiplication prop... |
| nvo00 30736 | Two ways to express a zero... |
| nmoofval 30737 | The operator norm function... |
| nmooval 30738 | The operator norm function... |
| nmosetre 30739 | The set in the supremum of... |
| nmosetn0 30740 | The set in the supremum of... |
| nmoxr 30741 | The norm of an operator is... |
| nmooge0 30742 | The norm of an operator is... |
| nmorepnf 30743 | The norm of an operator is... |
| nmoreltpnf 30744 | The norm of any operator i... |
| nmogtmnf 30745 | The norm of an operator is... |
| nmoolb 30746 | A lower bound for an opera... |
| nmoubi 30747 | An upper bound for an oper... |
| nmoub3i 30748 | An upper bound for an oper... |
| nmoub2i 30749 | An upper bound for an oper... |
| nmobndi 30750 | Two ways to express that a... |
| nmounbi 30751 | Two ways two express that ... |
| nmounbseqi 30752 | An unbounded operator dete... |
| nmounbseqiALT 30753 | Alternate shorter proof of... |
| nmobndseqi 30754 | A bounded sequence determi... |
| nmobndseqiALT 30755 | Alternate shorter proof of... |
| bloval 30756 | The class of bounded linea... |
| isblo 30757 | The predicate "is a bounde... |
| isblo2 30758 | The predicate "is a bounde... |
| bloln 30759 | A bounded operator is a li... |
| blof 30760 | A bounded operator is an o... |
| nmblore 30761 | The norm of a bounded oper... |
| 0ofval 30762 | The zero operator between ... |
| 0oval 30763 | Value of the zero operator... |
| 0oo 30764 | The zero operator is an op... |
| 0lno 30765 | The zero operator is linea... |
| nmoo0 30766 | The operator norm of the z... |
| 0blo 30767 | The zero operator is a bou... |
| nmlno0lem 30768 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30769 | The norm of a linear opera... |
| nmlno0 30770 | The norm of a linear opera... |
| nmlnoubi 30771 | An upper bound for the ope... |
| nmlnogt0 30772 | The norm of a nonzero line... |
| lnon0 30773 | The domain of a nonzero li... |
| nmblolbii 30774 | A lower bound for the norm... |
| nmblolbi 30775 | A lower bound for the norm... |
| isblo3i 30776 | The predicate "is a bounde... |
| blo3i 30777 | Properties that determine ... |
| blometi 30778 | Upper bound for the distan... |
| blocnilem 30779 | Lemma for ~ blocni and ~ l... |
| blocni 30780 | A linear operator is conti... |
| lnocni 30781 | If a linear operator is co... |
| blocn 30782 | A linear operator is conti... |
| blocn2 30783 | A bounded linear operator ... |
| ajfval 30784 | The adjoint function. (Co... |
| hmoval 30785 | The set of Hermitian (self... |
| ishmo 30786 | The predicate "is a hermit... |
| phnv 30789 | Every complex inner produc... |
| phrel 30790 | The class of all complex i... |
| phnvi 30791 | Every complex inner produc... |
| isphg 30792 | The predicate "is a comple... |
| phop 30793 | A complex inner product sp... |
| cncph 30794 | The set of complex numbers... |
| elimph 30795 | Hypothesis elimination lem... |
| elimphu 30796 | Hypothesis elimination lem... |
| isph 30797 | The predicate "is an inner... |
| phpar2 30798 | The parallelogram law for ... |
| phpar 30799 | The parallelogram law for ... |
| ip0i 30800 | A slight variant of Equati... |
| ip1ilem 30801 | Lemma for ~ ip1i . (Contr... |
| ip1i 30802 | Equation 6.47 of [Ponnusam... |
| ip2i 30803 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30804 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30805 | Distributive law for inner... |
| ipasslem1 30806 | Lemma for ~ ipassi . Show... |
| ipasslem2 30807 | Lemma for ~ ipassi . Show... |
| ipasslem3 30808 | Lemma for ~ ipassi . Show... |
| ipasslem4 30809 | Lemma for ~ ipassi . Show... |
| ipasslem5 30810 | Lemma for ~ ipassi . Show... |
| ipasslem7 30811 | Lemma for ~ ipassi . Show... |
| ipasslem8 30812 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30813 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30814 | Lemma for ~ ipassi . Show... |
| ipasslem11 30815 | Lemma for ~ ipassi . Show... |
| ipassi 30816 | Associative law for inner ... |
| dipdir 30817 | Distributive law for inner... |
| dipdi 30818 | Distributive law for inner... |
| ip2dii 30819 | Inner product of two sums.... |
| dipass 30820 | Associative law for inner ... |
| dipassr 30821 | "Associative" law for seco... |
| dipassr2 30822 | "Associative" law for inne... |
| dipsubdir 30823 | Distributive law for inner... |
| dipsubdi 30824 | Distributive law for inner... |
| pythi 30825 | The Pythagorean theorem fo... |
| siilem1 30826 | Lemma for ~ sii . (Contri... |
| siilem2 30827 | Lemma for ~ sii . (Contri... |
| siii 30828 | Inference from ~ sii . (C... |
| sii 30829 | Obsolete version of ~ ipca... |
| ipblnfi 30830 | A function ` F ` generated... |
| ip2eqi 30831 | Two vectors are equal iff ... |
| phoeqi 30832 | A condition implying that ... |
| ajmoi 30833 | Every operator has at most... |
| ajfuni 30834 | The adjoint function is a ... |
| ajfun 30835 | The adjoint function is a ... |
| ajval 30836 | Value of the adjoint funct... |
| iscbn 30839 | A complex Banach space is ... |
| cbncms 30840 | The induced metric on comp... |
| bnnv 30841 | Every complex Banach space... |
| bnrel 30842 | The class of all complex B... |
| bnsscmcl 30843 | A subspace of a Banach spa... |
| cnbn 30844 | The set of complex numbers... |
| ubthlem1 30845 | Lemma for ~ ubth . The fu... |
| ubthlem2 30846 | Lemma for ~ ubth . Given ... |
| ubthlem3 30847 | Lemma for ~ ubth . Prove ... |
| ubth 30848 | Uniform Boundedness Theore... |
| minvecolem1 30849 | Lemma for ~ minveco . The... |
| minvecolem2 30850 | Lemma for ~ minveco . Any... |
| minvecolem3 30851 | Lemma for ~ minveco . The... |
| minvecolem4a 30852 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30853 | Lemma for ~ minveco . The... |
| minvecolem4c 30854 | Lemma for ~ minveco . The... |
| minvecolem4 30855 | Lemma for ~ minveco . The... |
| minvecolem5 30856 | Lemma for ~ minveco . Dis... |
| minvecolem6 30857 | Lemma for ~ minveco . Any... |
| minvecolem7 30858 | Lemma for ~ minveco . Sin... |
| minveco 30859 | Minimizing vector theorem,... |
| ishlo 30862 | The predicate "is a comple... |
| hlobn 30863 | Every complex Hilbert spac... |
| hlph 30864 | Every complex Hilbert spac... |
| hlrel 30865 | The class of all complex H... |
| hlnv 30866 | Every complex Hilbert spac... |
| hlnvi 30867 | Every complex Hilbert spac... |
| hlvc 30868 | Every complex Hilbert spac... |
| hlcmet 30869 | The induced metric on a co... |
| hlmet 30870 | The induced metric on a co... |
| hlpar2 30871 | The parallelogram law sati... |
| hlpar 30872 | The parallelogram law sati... |
| hlex 30873 | The base set of a Hilbert ... |
| hladdf 30874 | Mapping for Hilbert space ... |
| hlcom 30875 | Hilbert space vector addit... |
| hlass 30876 | Hilbert space vector addit... |
| hl0cl 30877 | The Hilbert space zero vec... |
| hladdid 30878 | Hilbert space addition wit... |
| hlmulf 30879 | Mapping for Hilbert space ... |
| hlmulid 30880 | Hilbert space scalar multi... |
| hlmulass 30881 | Hilbert space scalar multi... |
| hldi 30882 | Hilbert space scalar multi... |
| hldir 30883 | Hilbert space scalar multi... |
| hlmul0 30884 | Hilbert space scalar multi... |
| hlipf 30885 | Mapping for Hilbert space ... |
| hlipcj 30886 | Conjugate law for Hilbert ... |
| hlipdir 30887 | Distributive law for Hilbe... |
| hlipass 30888 | Associative law for Hilber... |
| hlipgt0 30889 | The inner product of a Hil... |
| hlcompl 30890 | Completeness of a Hilbert ... |
| cnchl 30891 | The set of complex numbers... |
| htthlem 30892 | Lemma for ~ htth . The co... |
| htth 30893 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 30949 | The group (addition) opera... |
| h2hsm 30950 | The scalar product operati... |
| h2hnm 30951 | The norm function of Hilbe... |
| h2hvs 30952 | The vector subtraction ope... |
| h2hmetdval 30953 | Value of the distance func... |
| h2hcau 30954 | The Cauchy sequences of Hi... |
| h2hlm 30955 | The limit sequences of Hil... |
| axhilex-zf 30956 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 30957 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 30958 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 30959 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 30960 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 30961 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 30962 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 30963 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 30964 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 30965 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 30966 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 30967 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 30968 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 30969 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 30970 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 30971 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 30972 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 30973 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 30986 | The Hilbert space scalar p... |
| hvaddcl 30987 | Closure of vector addition... |
| hvmulcl 30988 | Closure of scalar multipli... |
| hvmulcli 30989 | Closure inference for scal... |
| hvsubf 30990 | Mapping domain and codomai... |
| hvsubval 30991 | Value of vector subtractio... |
| hvsubcl 30992 | Closure of vector subtract... |
| hvaddcli 30993 | Closure of vector addition... |
| hvcomi 30994 | Commutation of vector addi... |
| hvsubvali 30995 | Value of vector subtractio... |
| hvsubcli 30996 | Closure of vector subtract... |
| ifhvhv0 30997 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 30998 | Addition with the zero vec... |
| hvmul0 30999 | Scalar multiplication with... |
| hvmul0or 31000 | If a scalar product is zer... |
| hvsubid 31001 | Subtraction of a vector fr... |
| hvnegid 31002 | Addition of negative of a ... |
| hv2neg 31003 | Two ways to express the ne... |
| hvaddlidi 31004 | Addition with the zero vec... |
| hvnegidi 31005 | Addition of negative of a ... |
| hv2negi 31006 | Two ways to express the ne... |
| hvm1neg 31007 | Convert minus one times a ... |
| hvaddsubval 31008 | Value of vector addition i... |
| hvadd32 31009 | Commutative/associative la... |
| hvadd12 31010 | Commutative/associative la... |
| hvadd4 31011 | Hilbert vector space addit... |
| hvsub4 31012 | Hilbert vector space addit... |
| hvaddsub12 31013 | Commutative/associative la... |
| hvpncan 31014 | Addition/subtraction cance... |
| hvpncan2 31015 | Addition/subtraction cance... |
| hvaddsubass 31016 | Associativity of sum and d... |
| hvpncan3 31017 | Subtraction and addition o... |
| hvmulcom 31018 | Scalar multiplication comm... |
| hvsubass 31019 | Hilbert vector space assoc... |
| hvsub32 31020 | Hilbert vector space commu... |
| hvmulassi 31021 | Scalar multiplication asso... |
| hvmulcomi 31022 | Scalar multiplication comm... |
| hvmul2negi 31023 | Double negative in scalar ... |
| hvsubdistr1 31024 | Scalar multiplication dist... |
| hvsubdistr2 31025 | Scalar multiplication dist... |
| hvdistr1i 31026 | Scalar multiplication dist... |
| hvsubdistr1i 31027 | Scalar multiplication dist... |
| hvassi 31028 | Hilbert vector space assoc... |
| hvadd32i 31029 | Hilbert vector space commu... |
| hvsubassi 31030 | Hilbert vector space assoc... |
| hvsub32i 31031 | Hilbert vector space commu... |
| hvadd12i 31032 | Hilbert vector space commu... |
| hvadd4i 31033 | Hilbert vector space addit... |
| hvsubsub4i 31034 | Hilbert vector space addit... |
| hvsubsub4 31035 | Hilbert vector space addit... |
| hv2times 31036 | Two times a vector. (Cont... |
| hvnegdii 31037 | Distribution of negative o... |
| hvsubeq0i 31038 | If the difference between ... |
| hvsubcan2i 31039 | Vector cancellation law. ... |
| hvaddcani 31040 | Cancellation law for vecto... |
| hvsubaddi 31041 | Relationship between vecto... |
| hvnegdi 31042 | Distribution of negative o... |
| hvsubeq0 31043 | If the difference between ... |
| hvaddeq0 31044 | If the sum of two vectors ... |
| hvaddcan 31045 | Cancellation law for vecto... |
| hvaddcan2 31046 | Cancellation law for vecto... |
| hvmulcan 31047 | Cancellation law for scala... |
| hvmulcan2 31048 | Cancellation law for scala... |
| hvsubcan 31049 | Cancellation law for vecto... |
| hvsubcan2 31050 | Cancellation law for vecto... |
| hvsub0 31051 | Subtraction of a zero vect... |
| hvsubadd 31052 | Relationship between vecto... |
| hvaddsub4 31053 | Hilbert vector space addit... |
| hicl 31055 | Closure of inner product. ... |
| hicli 31056 | Closure inference for inne... |
| his5 31061 | Associative law for inner ... |
| his52 31062 | Associative law for inner ... |
| his35 31063 | Move scalar multiplication... |
| his35i 31064 | Move scalar multiplication... |
| his7 31065 | Distributive law for inner... |
| hiassdi 31066 | Distributive/associative l... |
| his2sub 31067 | Distributive law for inner... |
| his2sub2 31068 | Distributive law for inner... |
| hire 31069 | A necessary and sufficient... |
| hiidrcl 31070 | Real closure of inner prod... |
| hi01 31071 | Inner product with the 0 v... |
| hi02 31072 | Inner product with the 0 v... |
| hiidge0 31073 | Inner product with self is... |
| his6 31074 | Zero inner product with se... |
| his1i 31075 | Conjugate law for inner pr... |
| abshicom 31076 | Commuted inner products ha... |
| hial0 31077 | A vector whose inner produ... |
| hial02 31078 | A vector whose inner produ... |
| hisubcomi 31079 | Two vector subtractions si... |
| hi2eq 31080 | Lemma used to prove equali... |
| hial2eq 31081 | Two vectors whose inner pr... |
| hial2eq2 31082 | Two vectors whose inner pr... |
| orthcom 31083 | Orthogonality commutes. (... |
| normlem0 31084 | Lemma used to derive prope... |
| normlem1 31085 | Lemma used to derive prope... |
| normlem2 31086 | Lemma used to derive prope... |
| normlem3 31087 | Lemma used to derive prope... |
| normlem4 31088 | Lemma used to derive prope... |
| normlem5 31089 | Lemma used to derive prope... |
| normlem6 31090 | Lemma used to derive prope... |
| normlem7 31091 | Lemma used to derive prope... |
| normlem8 31092 | Lemma used to derive prope... |
| normlem9 31093 | Lemma used to derive prope... |
| normlem7tALT 31094 | Lemma used to derive prope... |
| bcseqi 31095 | Equality case of Bunjakova... |
| normlem9at 31096 | Lemma used to derive prope... |
| dfhnorm2 31097 | Alternate definition of th... |
| normf 31098 | The norm function maps fro... |
| normval 31099 | The value of the norm of a... |
| normcl 31100 | Real closure of the norm o... |
| normge0 31101 | The norm of a vector is no... |
| normgt0 31102 | The norm of nonzero vector... |
| norm0 31103 | The norm of a zero vector.... |
| norm-i 31104 | Theorem 3.3(i) of [Beran] ... |
| normne0 31105 | A norm is nonzero iff its ... |
| normcli 31106 | Real closure of the norm o... |
| normsqi 31107 | The square of a norm. (Co... |
| norm-i-i 31108 | Theorem 3.3(i) of [Beran] ... |
| normsq 31109 | The square of a norm. (Co... |
| normsub0i 31110 | Two vectors are equal iff ... |
| normsub0 31111 | Two vectors are equal iff ... |
| norm-ii-i 31112 | Triangle inequality for no... |
| norm-ii 31113 | Triangle inequality for no... |
| norm-iii-i 31114 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31115 | Theorem 3.3(iii) of [Beran... |
| normsubi 31116 | Negative doesn't change th... |
| normpythi 31117 | Analogy to Pythagorean the... |
| normsub 31118 | Swapping order of subtract... |
| normneg 31119 | The norm of a vector equal... |
| normpyth 31120 | Analogy to Pythagorean the... |
| normpyc 31121 | Corollary to Pythagorean t... |
| norm3difi 31122 | Norm of differences around... |
| norm3adifii 31123 | Norm of differences around... |
| norm3lem 31124 | Lemma involving norm of di... |
| norm3dif 31125 | Norm of differences around... |
| norm3dif2 31126 | Norm of differences around... |
| norm3lemt 31127 | Lemma involving norm of di... |
| norm3adifi 31128 | Norm of differences around... |
| normpari 31129 | Parallelogram law for norm... |
| normpar 31130 | Parallelogram law for norm... |
| normpar2i 31131 | Corollary of parallelogram... |
| polid2i 31132 | Generalized polarization i... |
| polidi 31133 | Polarization identity. Re... |
| polid 31134 | Polarization identity. Re... |
| hilablo 31135 | Hilbert space vector addit... |
| hilid 31136 | The group identity element... |
| hilvc 31137 | Hilbert space is a complex... |
| hilnormi 31138 | Hilbert space norm in term... |
| hilhhi 31139 | Deduce the structure of Hi... |
| hhnv 31140 | Hilbert space is a normed ... |
| hhva 31141 | The group (addition) opera... |
| hhba 31142 | The base set of Hilbert sp... |
| hh0v 31143 | The zero vector of Hilbert... |
| hhsm 31144 | The scalar product operati... |
| hhvs 31145 | The vector subtraction ope... |
| hhnm 31146 | The norm function of Hilbe... |
| hhims 31147 | The induced metric of Hilb... |
| hhims2 31148 | Hilbert space distance met... |
| hhmet 31149 | The induced metric of Hilb... |
| hhxmet 31150 | The induced metric of Hilb... |
| hhmetdval 31151 | Value of the distance func... |
| hhip 31152 | The inner product operatio... |
| hhph 31153 | The Hilbert space of the H... |
| bcsiALT 31154 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31155 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31156 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31157 | Corollary of the Bunjakova... |
| bcs3 31158 | Corollary of the Bunjakova... |
| hcau 31159 | Member of the set of Cauch... |
| hcauseq 31160 | A Cauchy sequences on a Hi... |
| hcaucvg 31161 | A Cauchy sequence on a Hil... |
| seq1hcau 31162 | A sequence on a Hilbert sp... |
| hlimi 31163 | Express the predicate: Th... |
| hlimseqi 31164 | A sequence with a limit on... |
| hlimveci 31165 | Closure of the limit of a ... |
| hlimconvi 31166 | Convergence of a sequence ... |
| hlim2 31167 | The limit of a sequence on... |
| hlimadd 31168 | Limit of the sum of two se... |
| hilmet 31169 | The Hilbert space norm det... |
| hilxmet 31170 | The Hilbert space norm det... |
| hilmetdval 31171 | Value of the distance func... |
| hilims 31172 | Hilbert space distance met... |
| hhcau 31173 | The Cauchy sequences of Hi... |
| hhlm 31174 | The limit sequences of Hil... |
| hhcmpl 31175 | Lemma used for derivation ... |
| hilcompl 31176 | Lemma used for derivation ... |
| hhcms 31178 | The Hilbert space induced ... |
| hhhl 31179 | The Hilbert space structur... |
| hilcms 31180 | The Hilbert space norm det... |
| hilhl 31181 | The Hilbert space of the H... |
| issh 31183 | Subspace ` H ` of a Hilber... |
| issh2 31184 | Subspace ` H ` of a Hilber... |
| shss 31185 | A subspace is a subset of ... |
| shel 31186 | A member of a subspace of ... |
| shex 31187 | The set of subspaces of a ... |
| shssii 31188 | A closed subspace of a Hil... |
| sheli 31189 | A member of a subspace of ... |
| shelii 31190 | A member of a subspace of ... |
| sh0 31191 | The zero vector belongs to... |
| shaddcl 31192 | Closure of vector addition... |
| shmulcl 31193 | Closure of vector scalar m... |
| issh3 31194 | Subspace ` H ` of a Hilber... |
| shsubcl 31195 | Closure of vector subtract... |
| isch 31197 | Closed subspace ` H ` of a... |
| isch2 31198 | Closed subspace ` H ` of a... |
| chsh 31199 | A closed subspace is a sub... |
| chsssh 31200 | Closed subspaces are subsp... |
| chex 31201 | The set of closed subspace... |
| chshii 31202 | A closed subspace is a sub... |
| ch0 31203 | The zero vector belongs to... |
| chss 31204 | A closed subspace of a Hil... |
| chel 31205 | A member of a closed subsp... |
| chssii 31206 | A closed subspace of a Hil... |
| cheli 31207 | A member of a closed subsp... |
| chelii 31208 | A member of a closed subsp... |
| chlimi 31209 | The limit property of a cl... |
| hlim0 31210 | The zero sequence in Hilbe... |
| hlimcaui 31211 | If a sequence in Hilbert s... |
| hlimf 31212 | Function-like behavior of ... |
| hlimuni 31213 | A Hilbert space sequence c... |
| hlimreui 31214 | The limit of a Hilbert spa... |
| hlimeui 31215 | The limit of a Hilbert spa... |
| isch3 31216 | A Hilbert subspace is clos... |
| chcompl 31217 | Completeness of a closed s... |
| helch 31218 | The Hilbert lattice one (w... |
| ifchhv 31219 | Prove ` if ( A e. CH , A ,... |
| helsh 31220 | Hilbert space is a subspac... |
| shsspwh 31221 | Subspaces are subsets of H... |
| chsspwh 31222 | Closed subspaces are subse... |
| hsn0elch 31223 | The zero subspace belongs ... |
| norm1 31224 | From any nonzero Hilbert s... |
| norm1exi 31225 | A normalized vector exists... |
| norm1hex 31226 | A normalized vector can ex... |
| elch0 31229 | Membership in zero for clo... |
| h0elch 31230 | The zero subspace is a clo... |
| h0elsh 31231 | The zero subspace is a sub... |
| hhssva 31232 | The vector addition operat... |
| hhsssm 31233 | The scalar multiplication ... |
| hhssnm 31234 | The norm operation on a su... |
| issubgoilem 31235 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31236 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31237 | Abelian group property of ... |
| hhssablo 31238 | Abelian group property of ... |
| hhssnv 31239 | Normed complex vector spac... |
| hhssnvt 31240 | Normed complex vector spac... |
| hhsst 31241 | A member of ` SH ` is a su... |
| hhshsslem1 31242 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31243 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31244 | The predicate " ` H ` is a... |
| hhsssh2 31245 | The predicate " ` H ` is a... |
| hhssba 31246 | The base set of a subspace... |
| hhssvs 31247 | The vector subtraction ope... |
| hhssvsf 31248 | Mapping of the vector subt... |
| hhssims 31249 | Induced metric of a subspa... |
| hhssims2 31250 | Induced metric of a subspa... |
| hhssmet 31251 | Induced metric of a subspa... |
| hhssmetdval 31252 | Value of the distance func... |
| hhsscms 31253 | The induced metric of a cl... |
| hhssbnOLD 31254 | Obsolete version of ~ cssb... |
| ocval 31255 | Value of orthogonal comple... |
| ocel 31256 | Membership in orthogonal c... |
| shocel 31257 | Membership in orthogonal c... |
| ocsh 31258 | The orthogonal complement ... |
| shocsh 31259 | The orthogonal complement ... |
| ocss 31260 | An orthogonal complement i... |
| shocss 31261 | An orthogonal complement i... |
| occon 31262 | Contraposition law for ort... |
| occon2 31263 | Double contraposition for ... |
| occon2i 31264 | Double contraposition for ... |
| oc0 31265 | The zero vector belongs to... |
| ocorth 31266 | Members of a subset and it... |
| shocorth 31267 | Members of a subspace and ... |
| ococss 31268 | Inclusion in complement of... |
| shococss 31269 | Inclusion in complement of... |
| shorth 31270 | Members of orthogonal subs... |
| ocin 31271 | Intersection of a Hilbert ... |
| occon3 31272 | Hilbert lattice contraposi... |
| ocnel 31273 | A nonzero vector in the co... |
| chocvali 31274 | Value of the orthogonal co... |
| shuni 31275 | Two subspaces with trivial... |
| chocunii 31276 | Lemma for uniqueness part ... |
| pjhthmo 31277 | Projection Theorem, unique... |
| occllem 31278 | Lemma for ~ occl . (Contr... |
| occl 31279 | Closure of complement of H... |
| shoccl 31280 | Closure of complement of H... |
| choccl 31281 | Closure of complement of H... |
| choccli 31282 | Closure of ` CH ` orthocom... |
| shsval 31287 | Value of subspace sum of t... |
| shsss 31288 | The subspace sum is a subs... |
| shsel 31289 | Membership in the subspace... |
| shsel3 31290 | Membership in the subspace... |
| shseli 31291 | Membership in subspace sum... |
| shscli 31292 | Closure of subspace sum. ... |
| shscl 31293 | Closure of subspace sum. ... |
| shscom 31294 | Commutative law for subspa... |
| shsva 31295 | Vector sum belongs to subs... |
| shsel1 31296 | A subspace sum contains a ... |
| shsel2 31297 | A subspace sum contains a ... |
| shsvs 31298 | Vector subtraction belongs... |
| shsub1 31299 | Subspace sum is an upper b... |
| shsub2 31300 | Subspace sum is an upper b... |
| choc0 31301 | The orthocomplement of the... |
| choc1 31302 | The orthocomplement of the... |
| chocnul 31303 | Orthogonal complement of t... |
| shintcli 31304 | Closure of intersection of... |
| shintcl 31305 | The intersection of a none... |
| chintcli 31306 | The intersection of a none... |
| chintcl 31307 | The intersection (infimum)... |
| spanval 31308 | Value of the linear span o... |
| hsupval 31309 | Value of supremum of set o... |
| chsupval 31310 | The value of the supremum ... |
| spancl 31311 | The span of a subset of Hi... |
| elspancl 31312 | A member of a span is a ve... |
| shsupcl 31313 | Closure of the subspace su... |
| hsupcl 31314 | Closure of supremum of set... |
| chsupcl 31315 | Closure of supremum of sub... |
| hsupss 31316 | Subset relation for suprem... |
| chsupss 31317 | Subset relation for suprem... |
| hsupunss 31318 | The union of a set of Hilb... |
| chsupunss 31319 | The union of a set of clos... |
| spanss2 31320 | A subset of Hilbert space ... |
| shsupunss 31321 | The union of a set of subs... |
| spanid 31322 | A subspace of Hilbert spac... |
| spanss 31323 | Ordering relationship for ... |
| spanssoc 31324 | The span of a subset of Hi... |
| sshjval 31325 | Value of join for subsets ... |
| shjval 31326 | Value of join in ` SH ` . ... |
| chjval 31327 | Value of join in ` CH ` . ... |
| chjvali 31328 | Value of join in ` CH ` . ... |
| sshjval3 31329 | Value of join for subsets ... |
| sshjcl 31330 | Closure of join for subset... |
| shjcl 31331 | Closure of join in ` SH ` ... |
| chjcl 31332 | Closure of join in ` CH ` ... |
| shjcom 31333 | Commutative law for Hilber... |
| shless 31334 | Subset implies subset of s... |
| shlej1 31335 | Add disjunct to both sides... |
| shlej2 31336 | Add disjunct to both sides... |
| shincli 31337 | Closure of intersection of... |
| shscomi 31338 | Commutative law for subspa... |
| shsvai 31339 | Vector sum belongs to subs... |
| shsel1i 31340 | A subspace sum contains a ... |
| shsel2i 31341 | A subspace sum contains a ... |
| shsvsi 31342 | Vector subtraction belongs... |
| shunssi 31343 | Union is smaller than subs... |
| shunssji 31344 | Union is smaller than Hilb... |
| shsleji 31345 | Subspace sum is smaller th... |
| shjcomi 31346 | Commutative law for join i... |
| shsub1i 31347 | Subspace sum is an upper b... |
| shsub2i 31348 | Subspace sum is an upper b... |
| shub1i 31349 | Hilbert lattice join is an... |
| shjcli 31350 | Closure of ` CH ` join. (... |
| shjshcli 31351 | ` SH ` closure of join. (... |
| shlessi 31352 | Subset implies subset of s... |
| shlej1i 31353 | Add disjunct to both sides... |
| shlej2i 31354 | Add disjunct to both sides... |
| shslej 31355 | Subspace sum is smaller th... |
| shincl 31356 | Closure of intersection of... |
| shub1 31357 | Hilbert lattice join is an... |
| shub2 31358 | A subspace is a subset of ... |
| shsidmi 31359 | Idempotent law for Hilbert... |
| shslubi 31360 | The least upper bound law ... |
| shlesb1i 31361 | Hilbert lattice ordering i... |
| shsval2i 31362 | An alternate way to expres... |
| shsval3i 31363 | An alternate way to expres... |
| shmodsi 31364 | The modular law holds for ... |
| shmodi 31365 | The modular law is implied... |
| pjhthlem1 31366 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31367 | Lemma for ~ pjhth . (Cont... |
| pjhth 31368 | Projection Theorem: Any H... |
| pjhtheu 31369 | Projection Theorem: Any H... |
| pjhfval 31371 | The value of the projectio... |
| pjhval 31372 | Value of a projection. (C... |
| pjpreeq 31373 | Equality with a projection... |
| pjeq 31374 | Equality with a projection... |
| axpjcl 31375 | Closure of a projection in... |
| pjhcl 31376 | Closure of a projection in... |
| omlsilem 31377 | Lemma for orthomodular law... |
| omlsii 31378 | Subspace inference form of... |
| omlsi 31379 | Subspace form of orthomodu... |
| ococi 31380 | Complement of complement o... |
| ococ 31381 | Complement of complement o... |
| dfch2 31382 | Alternate definition of th... |
| ococin 31383 | The double complement is t... |
| hsupval2 31384 | Alternate definition of su... |
| chsupval2 31385 | The value of the supremum ... |
| sshjval2 31386 | Value of join in the set o... |
| chsupid 31387 | A subspace is the supremum... |
| chsupsn 31388 | Value of supremum of subse... |
| shlub 31389 | Hilbert lattice join is th... |
| shlubi 31390 | Hilbert lattice join is th... |
| pjhtheu2 31391 | Uniqueness of ` y ` for th... |
| pjcli 31392 | Closure of a projection in... |
| pjhcli 31393 | Closure of a projection in... |
| pjpjpre 31394 | Decomposition of a vector ... |
| axpjpj 31395 | Decomposition of a vector ... |
| pjclii 31396 | Closure of a projection in... |
| pjhclii 31397 | Closure of a projection in... |
| pjpj0i 31398 | Decomposition of a vector ... |
| pjpji 31399 | Decomposition of a vector ... |
| pjpjhth 31400 | Projection Theorem: Any H... |
| pjpjhthi 31401 | Projection Theorem: Any H... |
| pjop 31402 | Orthocomplement projection... |
| pjpo 31403 | Projection in terms of ort... |
| pjopi 31404 | Orthocomplement projection... |
| pjpoi 31405 | Projection in terms of ort... |
| pjoc1i 31406 | Projection of a vector in ... |
| pjchi 31407 | Projection of a vector in ... |
| pjoccl 31408 | The part of a vector that ... |
| pjoc1 31409 | Projection of a vector in ... |
| pjomli 31410 | Subspace form of orthomodu... |
| pjoml 31411 | Subspace form of orthomodu... |
| pjococi 31412 | Proof of orthocomplement t... |
| pjoc2i 31413 | Projection of a vector in ... |
| pjoc2 31414 | Projection of a vector in ... |
| sh0le 31415 | The zero subspace is the s... |
| ch0le 31416 | The zero subspace is the s... |
| shle0 31417 | No subspace is smaller tha... |
| chle0 31418 | No Hilbert lattice element... |
| chnlen0 31419 | A Hilbert lattice element ... |
| ch0pss 31420 | The zero subspace is a pro... |
| orthin 31421 | The intersection of orthog... |
| ssjo 31422 | The lattice join of a subs... |
| shne0i 31423 | A nonzero subspace has a n... |
| shs0i 31424 | Hilbert subspace sum with ... |
| shs00i 31425 | Two subspaces are zero iff... |
| ch0lei 31426 | The closed subspace zero i... |
| chle0i 31427 | No Hilbert closed subspace... |
| chne0i 31428 | A nonzero closed subspace ... |
| chocini 31429 | Intersection of a closed s... |
| chj0i 31430 | Join with lattice zero in ... |
| chm1i 31431 | Meet with lattice one in `... |
| chjcli 31432 | Closure of ` CH ` join. (... |
| chsleji 31433 | Subspace sum is smaller th... |
| chseli 31434 | Membership in subspace sum... |
| chincli 31435 | Closure of Hilbert lattice... |
| chsscon3i 31436 | Hilbert lattice contraposi... |
| chsscon1i 31437 | Hilbert lattice contraposi... |
| chsscon2i 31438 | Hilbert lattice contraposi... |
| chcon2i 31439 | Hilbert lattice contraposi... |
| chcon1i 31440 | Hilbert lattice contraposi... |
| chcon3i 31441 | Hilbert lattice contraposi... |
| chunssji 31442 | Union is smaller than ` CH... |
| chjcomi 31443 | Commutative law for join i... |
| chub1i 31444 | ` CH ` join is an upper bo... |
| chub2i 31445 | ` CH ` join is an upper bo... |
| chlubi 31446 | Hilbert lattice join is th... |
| chlubii 31447 | Hilbert lattice join is th... |
| chlej1i 31448 | Add join to both sides of ... |
| chlej2i 31449 | Add join to both sides of ... |
| chlej12i 31450 | Add join to both sides of ... |
| chlejb1i 31451 | Hilbert lattice ordering i... |
| chdmm1i 31452 | De Morgan's law for meet i... |
| chdmm2i 31453 | De Morgan's law for meet i... |
| chdmm3i 31454 | De Morgan's law for meet i... |
| chdmm4i 31455 | De Morgan's law for meet i... |
| chdmj1i 31456 | De Morgan's law for join i... |
| chdmj2i 31457 | De Morgan's law for join i... |
| chdmj3i 31458 | De Morgan's law for join i... |
| chdmj4i 31459 | De Morgan's law for join i... |
| chnlei 31460 | Equivalent expressions for... |
| chjassi 31461 | Associative law for Hilber... |
| chj00i 31462 | Two Hilbert lattice elemen... |
| chjoi 31463 | The join of a closed subsp... |
| chj1i 31464 | Join with Hilbert lattice ... |
| chm0i 31465 | Meet with Hilbert lattice ... |
| chm0 31466 | Meet with Hilbert lattice ... |
| shjshsi 31467 | Hilbert lattice join equal... |
| shjshseli 31468 | A closed subspace sum equa... |
| chne0 31469 | A nonzero closed subspace ... |
| chocin 31470 | Intersection of a closed s... |
| chssoc 31471 | A closed subspace less tha... |
| chj0 31472 | Join with Hilbert lattice ... |
| chslej 31473 | Subspace sum is smaller th... |
| chincl 31474 | Closure of Hilbert lattice... |
| chsscon3 31475 | Hilbert lattice contraposi... |
| chsscon1 31476 | Hilbert lattice contraposi... |
| chsscon2 31477 | Hilbert lattice contraposi... |
| chpsscon3 31478 | Hilbert lattice contraposi... |
| chpsscon1 31479 | Hilbert lattice contraposi... |
| chpsscon2 31480 | Hilbert lattice contraposi... |
| chjcom 31481 | Commutative law for Hilber... |
| chub1 31482 | Hilbert lattice join is gr... |
| chub2 31483 | Hilbert lattice join is gr... |
| chlub 31484 | Hilbert lattice join is th... |
| chlej1 31485 | Add join to both sides of ... |
| chlej2 31486 | Add join to both sides of ... |
| chlejb1 31487 | Hilbert lattice ordering i... |
| chlejb2 31488 | Hilbert lattice ordering i... |
| chnle 31489 | Equivalent expressions for... |
| chjo 31490 | The join of a closed subsp... |
| chabs1 31491 | Hilbert lattice absorption... |
| chabs2 31492 | Hilbert lattice absorption... |
| chabs1i 31493 | Hilbert lattice absorption... |
| chabs2i 31494 | Hilbert lattice absorption... |
| chjidm 31495 | Idempotent law for Hilbert... |
| chjidmi 31496 | Idempotent law for Hilbert... |
| chj12i 31497 | A rearrangement of Hilbert... |
| chj4i 31498 | Rearrangement of the join ... |
| chjjdiri 31499 | Hilbert lattice join distr... |
| chdmm1 31500 | De Morgan's law for meet i... |
| chdmm2 31501 | De Morgan's law for meet i... |
| chdmm3 31502 | De Morgan's law for meet i... |
| chdmm4 31503 | De Morgan's law for meet i... |
| chdmj1 31504 | De Morgan's law for join i... |
| chdmj2 31505 | De Morgan's law for join i... |
| chdmj3 31506 | De Morgan's law for join i... |
| chdmj4 31507 | De Morgan's law for join i... |
| chjass 31508 | Associative law for Hilber... |
| chj12 31509 | A rearrangement of Hilbert... |
| chj4 31510 | Rearrangement of the join ... |
| ledii 31511 | An ortholattice is distrib... |
| lediri 31512 | An ortholattice is distrib... |
| lejdii 31513 | An ortholattice is distrib... |
| lejdiri 31514 | An ortholattice is distrib... |
| ledi 31515 | An ortholattice is distrib... |
| spansn0 31516 | The span of the singleton ... |
| span0 31517 | The span of the empty set ... |
| elspani 31518 | Membership in the span of ... |
| spanuni 31519 | The span of a union is the... |
| spanun 31520 | The span of a union is the... |
| sshhococi 31521 | The join of two Hilbert sp... |
| hne0 31522 | Hilbert space has a nonzer... |
| chsup0 31523 | The supremum of the empty ... |
| h1deoi | |