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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... |
| pm2.01i 189 | Inference associated with ... |
| pm2.01d 190 | Deduction based on reducti... |
| pm2.6 191 | Theorem *2.6 of [Whitehead... |
| pm2.61 192 | Theorem *2.61 of [Whitehea... |
| pm2.65 193 | Theorem *2.65 of [Whitehea... |
| pm2.65i 194 | Inference for proof by con... |
| pm2.21dd 195 | A contradiction implies an... |
| pm2.65d 196 | Deduction for proof by con... |
| mto 197 | The rule of modus tollens.... |
| mtod 198 | Modus tollens deduction. ... |
| mtoi 199 | Modus tollens inference. ... |
| mt2 200 | A rule similar to modus to... |
| mt3 201 | A rule similar to modus to... |
| peirce 202 | Peirce's axiom. A non-int... |
| looinv 203 | The Inversion Axiom of the... |
| bijust0 204 | A self-implication (see ~ ... |
| bijust 205 | Theorem used to justify th... |
| impbi 208 | Property of the biconditio... |
| impbii 209 | Infer an equivalence from ... |
| impbidd 210 | Deduce an equivalence from... |
| impbid21d 211 | Deduce an equivalence from... |
| impbid 212 | Deduce an equivalence from... |
| dfbi1 213 | Relate the biconditional c... |
| dfbi1ALT 214 | Alternate proof of ~ dfbi1... |
| biimp 215 | Property of the biconditio... |
| biimpi 216 | Infer an implication from ... |
| sylbi 217 | A mixed syllogism inferenc... |
| sylib 218 | A mixed syllogism inferenc... |
| sylbb 219 | A mixed syllogism inferenc... |
| biimpr 220 | Property of the biconditio... |
| bicom1 221 | Commutative law for the bi... |
| bicom 222 | Commutative law for the bi... |
| bicomd 223 | Commute two sides of a bic... |
| bicomi 224 | Inference from commutative... |
| impbid1 225 | Infer an equivalence from ... |
| impbid2 226 | Infer an equivalence from ... |
| impcon4bid 227 | A variation on ~ impbid wi... |
| biimpri 228 | Infer a converse implicati... |
| biimpd 229 | Deduce an implication from... |
| mpbi 230 | An inference from a bicond... |
| mpbir 231 | An inference from a bicond... |
| mpbid 232 | A deduction from a bicondi... |
| mpbii 233 | An inference from a nested... |
| sylibr 234 | A mixed syllogism inferenc... |
| sylbir 235 | A mixed syllogism inferenc... |
| sylbbr 236 | A mixed syllogism inferenc... |
| sylbb1 237 | A mixed syllogism inferenc... |
| sylbb2 238 | A mixed syllogism inferenc... |
| sylibd 239 | A syllogism deduction. (C... |
| sylbid 240 | A syllogism deduction. (C... |
| mpbidi 241 | A deduction from a bicondi... |
| biimtrid 242 | A mixed syllogism inferenc... |
| biimtrrid 243 | A mixed syllogism inferenc... |
| imbitrid 244 | A mixed syllogism inferenc... |
| syl5ibcom 245 | A mixed syllogism inferenc... |
| imbitrrid 246 | A mixed syllogism inferenc... |
| syl5ibrcom 247 | A mixed syllogism inferenc... |
| biimprd 248 | Deduce a converse implicat... |
| biimpcd 249 | Deduce a commuted implicat... |
| biimprcd 250 | Deduce a converse commuted... |
| imbitrdi 251 | A mixed syllogism inferenc... |
| imbitrrdi 252 | A mixed syllogism inferenc... |
| biimtrdi 253 | A mixed syllogism inferenc... |
| biimtrrdi 254 | A mixed syllogism inferenc... |
| syl7bi 255 | A mixed syllogism inferenc... |
| syl8ib 256 | A syllogism rule of infere... |
| mpbird 257 | A deduction from a bicondi... |
| mpbiri 258 | An inference from a nested... |
| sylibrd 259 | A syllogism deduction. (C... |
| sylbird 260 | A syllogism deduction. (C... |
| biid 261 | Principle of identity for ... |
| biidd 262 | Principle of identity with... |
| pm5.1im 263 | Two propositions are equiv... |
| 2th 264 | Two truths are equivalent.... |
| 2thd 265 | Two truths are equivalent.... |
| monothetic 266 | Two self-implications (see... |
| ibi 267 | Inference that converts a ... |
| ibir 268 | Inference that converts a ... |
| ibd 269 | Deduction that converts a ... |
| pm5.74 270 | Distribution of implicatio... |
| pm5.74i 271 | Distribution of implicatio... |
| pm5.74ri 272 | Distribution of implicatio... |
| pm5.74d 273 | Distribution of implicatio... |
| pm5.74rd 274 | Distribution of implicatio... |
| bitri 275 | An inference from transiti... |
| bitr2i 276 | An inference from transiti... |
| bitr3i 277 | An inference from transiti... |
| bitr4i 278 | An inference from transiti... |
| bitrd 279 | Deduction form of ~ bitri ... |
| bitr2d 280 | Deduction form of ~ bitr2i... |
| bitr3d 281 | Deduction form of ~ bitr3i... |
| bitr4d 282 | Deduction form of ~ bitr4i... |
| bitrid 283 | A syllogism inference from... |
| bitr2id 284 | A syllogism inference from... |
| bitr3id 285 | A syllogism inference from... |
| bitr3di 286 | A syllogism inference from... |
| bitrdi 287 | A syllogism inference from... |
| bitr2di 288 | A syllogism inference from... |
| bitr4di 289 | A syllogism inference from... |
| bitr4id 290 | A syllogism inference from... |
| 3imtr3i 291 | A mixed syllogism inferenc... |
| 3imtr4i 292 | A mixed syllogism inferenc... |
| 3imtr3d 293 | More general version of ~ ... |
| 3imtr4d 294 | More general version of ~ ... |
| 3imtr3g 295 | More general version of ~ ... |
| 3imtr4g 296 | More general version of ~ ... |
| 3bitri 297 | A chained inference from t... |
| 3bitrri 298 | A chained inference from t... |
| 3bitr2i 299 | A chained inference from t... |
| 3bitr2ri 300 | A chained inference from t... |
| 3bitr3i 301 | A chained inference from t... |
| 3bitr3ri 302 | A chained inference from t... |
| 3bitr4i 303 | A chained inference from t... |
| 3bitr4ri 304 | A chained inference from t... |
| 3bitrd 305 | Deduction from transitivit... |
| 3bitrrd 306 | Deduction from transitivit... |
| 3bitr2d 307 | Deduction from transitivit... |
| 3bitr2rd 308 | Deduction from transitivit... |
| 3bitr3d 309 | Deduction from transitivit... |
| 3bitr3rd 310 | Deduction from transitivit... |
| 3bitr4d 311 | Deduction from transitivit... |
| 3bitr4rd 312 | Deduction from transitivit... |
| 3bitr3g 313 | More general version of ~ ... |
| 3bitr4g 314 | More general version of ~ ... |
| notnotb 315 | Double negation. Theorem ... |
| con34b 316 | A biconditional form of co... |
| con4bid 317 | A contraposition deduction... |
| notbid 318 | Deduction negating both si... |
| notbi 319 | Contraposition. Theorem *... |
| notbii 320 | Negate both sides of a log... |
| con4bii 321 | A contraposition inference... |
| mtbi 322 | An inference from a bicond... |
| mtbir 323 | An inference from a bicond... |
| mtbid 324 | A deduction from a bicondi... |
| mtbird 325 | A deduction from a bicondi... |
| mtbii 326 | An inference from a bicond... |
| mtbiri 327 | An inference from a bicond... |
| sylnib 328 | A mixed syllogism inferenc... |
| sylnibr 329 | A mixed syllogism inferenc... |
| sylnbi 330 | A mixed syllogism inferenc... |
| sylnbir 331 | A mixed syllogism inferenc... |
| xchnxbi 332 | Replacement of a subexpres... |
| xchnxbir 333 | Replacement of a subexpres... |
| xchbinx 334 | Replacement of a subexpres... |
| xchbinxr 335 | Replacement of a subexpres... |
| imbi2i 336 | Introduce an antecedent to... |
| bibi2i 337 | Inference adding a bicondi... |
| bibi1i 338 | Inference adding a bicondi... |
| bibi12i 339 | The equivalence of two equ... |
| imbi2d 340 | Deduction adding an antece... |
| imbi1d 341 | Deduction adding a consequ... |
| bibi2d 342 | Deduction adding a bicondi... |
| bibi1d 343 | Deduction adding a bicondi... |
| imbi12d 344 | Deduction joining two equi... |
| bibi12d 345 | Deduction joining two equi... |
| imbi12 346 | Closed form of ~ imbi12i .... |
| imbi1 347 | Theorem *4.84 of [Whitehea... |
| imbi2 348 | Theorem *4.85 of [Whitehea... |
| imbi1i 349 | Introduce a consequent to ... |
| imbi12i 350 | Join two logical equivalen... |
| bibi1 351 | Theorem *4.86 of [Whitehea... |
| bitr3 352 | Closed nested implication ... |
| con2bi 353 | Contraposition. Theorem *... |
| con2bid 354 | A contraposition deduction... |
| con1bid 355 | A contraposition deduction... |
| con1bii 356 | A contraposition inference... |
| con2bii 357 | A contraposition inference... |
| con1b 358 | Contraposition. Bidirecti... |
| con2b 359 | Contraposition. Bidirecti... |
| biimt 360 | A wff is equivalent to its... |
| pm5.5 361 | Theorem *5.5 of [Whitehead... |
| a1bi 362 | Inference introducing a th... |
| mt2bi 363 | A false consequent falsifi... |
| mtt 364 | Modus-tollens-like theorem... |
| imnot 365 | If a proposition is false,... |
| pm5.501 366 | Theorem *5.501 of [Whitehe... |
| ibib 367 | Implication in terms of im... |
| ibibr 368 | Implication in terms of im... |
| tbt 369 | A wff is equivalent to its... |
| nbn2 370 | The negation of a wff is e... |
| bibif 371 | Transfer negation via an e... |
| nbn 372 | The negation of a wff is e... |
| nbn3 373 | Transfer falsehood via equ... |
| pm5.21im 374 | Two propositions are equiv... |
| 2false 375 | Two falsehoods are equival... |
| 2falsed 376 | Two falsehoods are equival... |
| pm5.21ni 377 | Two propositions implying ... |
| pm5.21nii 378 | Eliminate an antecedent im... |
| pm5.21ndd 379 | Eliminate an antecedent im... |
| bija 380 | Combine antecedents into a... |
| pm5.18 381 | Theorem *5.18 of [Whitehea... |
| xor3 382 | Two ways to express "exclu... |
| nbbn 383 | Move negation outside of b... |
| biass 384 | Associative law for the bi... |
| biluk 385 | Lukasiewicz's shortest axi... |
| pm5.19 386 | Theorem *5.19 of [Whitehea... |
| bi2.04 387 | Logical equivalence of com... |
| pm5.4 388 | Antecedent absorption impl... |
| imdi 389 | Distributive law for impli... |
| pm5.41 390 | Theorem *5.41 of [Whitehea... |
| imbibi 391 | The antecedent of one side... |
| pm4.8 392 | Theorem *4.8 of [Whitehead... |
| pm4.81 393 | A formula is equivalent to... |
| imim21b 394 | Simplify an implication be... |
| pm4.63 397 | Theorem *4.63 of [Whitehea... |
| pm4.67 398 | Theorem *4.67 of [Whitehea... |
| imnan 399 | Express an implication in ... |
| imnani 400 | Infer an implication from ... |
| iman 401 | Implication in terms of co... |
| pm3.24 402 | Law of noncontradiction. ... |
| annim 403 | Express a conjunction in t... |
| pm4.61 404 | Theorem *4.61 of [Whitehea... |
| pm4.65 405 | Theorem *4.65 of [Whitehea... |
| imp 406 | Importation inference. (C... |
| impcom 407 | Importation inference with... |
| con3dimp 408 | Variant of ~ con3d with im... |
| mpnanrd 409 | Eliminate the right side o... |
| impd 410 | Importation deduction. (C... |
| impcomd 411 | Importation deduction with... |
| ex 412 | Exportation inference. (T... |
| expcom 413 | Exportation inference with... |
| expdcom 414 | Commuted form of ~ expd . ... |
| expd 415 | Exportation deduction. (C... |
| expcomd 416 | Deduction form of ~ expcom... |
| imp31 417 | An importation inference. ... |
| imp32 418 | An importation inference. ... |
| exp31 419 | An exportation inference. ... |
| exp32 420 | An exportation inference. ... |
| imp4b 421 | An importation inference. ... |
| imp4a 422 | An importation inference. ... |
| imp4c 423 | An importation inference. ... |
| imp4d 424 | An importation inference. ... |
| imp41 425 | An importation inference. ... |
| imp42 426 | An importation inference. ... |
| imp43 427 | An importation inference. ... |
| imp44 428 | An importation inference. ... |
| imp45 429 | An importation inference. ... |
| exp4b 430 | An exportation inference. ... |
| exp4a 431 | An exportation inference. ... |
| exp4c 432 | An exportation inference. ... |
| exp4d 433 | An exportation inference. ... |
| exp41 434 | An exportation inference. ... |
| exp42 435 | An exportation inference. ... |
| exp43 436 | An exportation inference. ... |
| exp44 437 | An exportation inference. ... |
| exp45 438 | An exportation inference. ... |
| imp5d 439 | An importation inference. ... |
| imp5a 440 | An importation inference. ... |
| imp5g 441 | An importation inference. ... |
| imp55 442 | An importation inference. ... |
| imp511 443 | An importation inference. ... |
| exp5c 444 | An exportation inference. ... |
| exp5j 445 | An exportation inference. ... |
| exp5l 446 | An exportation inference. ... |
| exp53 447 | An exportation inference. ... |
| pm3.3 448 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 449 | Theorem *3.31 (Imp) of [Wh... |
| impexp 450 | Import-export theorem. Pa... |
| impancom 451 | Mixed importation/commutat... |
| expdimp 452 | A deduction version of exp... |
| expimpd 453 | Exportation followed by a ... |
| impr 454 | Import a wff into a right ... |
| impl 455 | Export a wff from a left c... |
| expr 456 | Export a wff from a right ... |
| expl 457 | Export a wff from a left c... |
| ancoms 458 | Inference commuting conjun... |
| pm3.22 459 | Theorem *3.22 of [Whitehea... |
| ancom 460 | Commutative law for conjun... |
| ancomd 461 | Commutation of conjuncts i... |
| biancomi 462 | Commuting conjunction in a... |
| biancomd 463 | Commuting conjunction in a... |
| ancomst 464 | Closed form of ~ ancoms . ... |
| ancomsd 465 | Deduction commuting conjun... |
| anasss 466 | Associative law for conjun... |
| anassrs 467 | Associative law for conjun... |
| anass 468 | Associative law for conjun... |
| pm3.2 469 | Join antecedents with conj... |
| pm3.2i 470 | Infer conjunction of premi... |
| pm3.21 471 | Join antecedents with conj... |
| pm3.43i 472 | Nested conjunction of ante... |
| pm3.43 473 | Theorem *3.43 (Comp) of [W... |
| dfbi2 474 | A theorem similar to the s... |
| dfbi 475 | Definition ~ df-bi rewritt... |
| biimpa 476 | Importation inference from... |
| biimpar 477 | Importation inference from... |
| biimpac 478 | Importation inference from... |
| biimparc 479 | Importation inference from... |
| adantr 480 | Inference adding a conjunc... |
| adantl 481 | Inference adding a conjunc... |
| simpl 482 | Elimination of a conjunct.... |
| simpli 483 | Inference eliminating a co... |
| simpr 484 | Elimination of a conjunct.... |
| simpri 485 | Inference eliminating a co... |
| intnan 486 | Introduction of conjunct i... |
| intnanr 487 | Introduction of conjunct i... |
| intnand 488 | Introduction of conjunct i... |
| intnanrd 489 | Introduction of conjunct i... |
| adantld 490 | Deduction adding a conjunc... |
| adantrd 491 | Deduction adding a conjunc... |
| pm3.41 492 | Theorem *3.41 of [Whitehea... |
| pm3.42 493 | Theorem *3.42 of [Whitehea... |
| simpld 494 | Deduction eliminating a co... |
| simprd 495 | Deduction eliminating a co... |
| simplbi 496 | Deduction eliminating a co... |
| simprbi 497 | Deduction eliminating a co... |
| simprbda 498 | Deduction eliminating a co... |
| simplbda 499 | Deduction eliminating a co... |
| simplbi2 500 | Deduction eliminating a co... |
| simplbi2comt 501 | Closed form of ~ simplbi2c... |
| simplbi2com 502 | A deduction eliminating a ... |
| simpl2im 503 | Implication from an elimin... |
| simplbiim 504 | Implication from an elimin... |
| impel 505 | An inference for implicati... |
| mpan9 506 | Modus ponens conjoining di... |
| sylan9 507 | Nested syllogism inference... |
| sylan9r 508 | Nested syllogism inference... |
| sylan9bb 509 | Nested syllogism inference... |
| sylan9bbr 510 | Nested syllogism inference... |
| jca 511 | Deduce conjunction of the ... |
| jcad 512 | Deduction conjoining the c... |
| jca2 513 | Inference conjoining the c... |
| jca31 514 | Join three consequents. (... |
| jca32 515 | Join three consequents. (... |
| jcai 516 | Deduction replacing implic... |
| jcab 517 | Distributive law for impli... |
| pm4.76 518 | Theorem *4.76 of [Whitehea... |
| jctil 519 | Inference conjoining a the... |
| jctir 520 | Inference conjoining a the... |
| jccir 521 | Inference conjoining a con... |
| jccil 522 | Inference conjoining a con... |
| jctl 523 | Inference conjoining a the... |
| jctr 524 | Inference conjoining a the... |
| jctild 525 | Deduction conjoining a the... |
| jctird 526 | Deduction conjoining a the... |
| iba 527 | Introduction of antecedent... |
| ibar 528 | Introduction of antecedent... |
| biantru 529 | A wff is equivalent to its... |
| biantrur 530 | A wff is equivalent to its... |
| biantrud 531 | A wff is equivalent to its... |
| biantrurd 532 | A wff is equivalent to its... |
| bianfi 533 | A wff conjoined with false... |
| bianfd 534 | A wff conjoined with false... |
| baib 535 | Move conjunction outside o... |
| baibr 536 | Move conjunction outside o... |
| rbaibr 537 | Move conjunction outside o... |
| rbaib 538 | Move conjunction outside o... |
| baibd 539 | Move conjunction outside o... |
| rbaibd 540 | Move conjunction outside o... |
| bianabs 541 | Absorb a hypothesis into t... |
| pm5.44 542 | Theorem *5.44 of [Whitehea... |
| pm5.42 543 | Theorem *5.42 of [Whitehea... |
| ancl 544 | Conjoin antecedent to left... |
| anclb 545 | Conjoin antecedent to left... |
| ancr 546 | Conjoin antecedent to righ... |
| ancrb 547 | Conjoin antecedent to righ... |
| ancli 548 | Deduction conjoining antec... |
| ancri 549 | Deduction conjoining antec... |
| ancld 550 | Deduction conjoining antec... |
| ancrd 551 | Deduction conjoining antec... |
| impac 552 | Importation with conjuncti... |
| anc2l 553 | Conjoin antecedent to left... |
| anc2r 554 | Conjoin antecedent to righ... |
| anc2li 555 | Deduction conjoining antec... |
| anc2ri 556 | Deduction conjoining antec... |
| pm4.71 557 | Implication in terms of bi... |
| pm4.71r 558 | Implication in terms of bi... |
| pm4.71i 559 | Inference converting an im... |
| pm4.71ri 560 | Inference converting an im... |
| pm4.71d 561 | Deduction converting an im... |
| pm4.71rd 562 | Deduction converting an im... |
| pm4.24 563 | Theorem *4.24 of [Whitehea... |
| anidm 564 | Idempotent law for conjunc... |
| anidmdbi 565 | Conjunction idempotence wi... |
| anidms 566 | Inference from idempotent ... |
| imdistan 567 | Distribution of implicatio... |
| imdistani 568 | Distribution of implicatio... |
| imdistanri 569 | Distribution of implicatio... |
| imdistand 570 | Distribution of implicatio... |
| imdistanda 571 | Distribution of implicatio... |
| pm5.3 572 | Theorem *5.3 of [Whitehead... |
| pm5.32 573 | Distribution of implicatio... |
| pm5.32i 574 | Distribution of implicatio... |
| pm5.32ri 575 | Distribution of implicatio... |
| bianim 576 | Exchanging conjunction in ... |
| pm5.32d 577 | Distribution of implicatio... |
| pm5.32rd 578 | Distribution of implicatio... |
| pm5.32da 579 | Distribution of implicatio... |
| bian1d 580 | Adding a superfluous conju... |
| sylan 581 | A syllogism inference. (C... |
| sylanb 582 | A syllogism inference. (C... |
| sylanbr 583 | A syllogism inference. (C... |
| sylanbrc 584 | Syllogism inference. (Con... |
| syl2anc 585 | Syllogism inference combin... |
| syl2anc2 586 | Double syllogism inference... |
| sylancl 587 | Syllogism inference combin... |
| sylancr 588 | Syllogism inference combin... |
| sylancom 589 | Syllogism inference with c... |
| sylanblc 590 | Syllogism inference combin... |
| sylanblrc 591 | Syllogism inference combin... |
| syldan 592 | A syllogism deduction with... |
| sylbida 593 | A syllogism deduction. (C... |
| sylan2 594 | A syllogism inference. (C... |
| sylan2b 595 | A syllogism inference. (C... |
| sylan2br 596 | A syllogism inference. (C... |
| syl2an 597 | A double syllogism inferen... |
| syl2anr 598 | A double syllogism inferen... |
| syl2anb 599 | A double syllogism inferen... |
| syl2anbr 600 | A double syllogism inferen... |
| sylancb 601 | A syllogism inference comb... |
| sylancbr 602 | A syllogism inference comb... |
| syldanl 603 | A syllogism deduction with... |
| syland 604 | A syllogism deduction. (C... |
| sylani 605 | A syllogism inference. (C... |
| sylan2d 606 | A syllogism deduction. (C... |
| sylan2i 607 | A syllogism inference. (C... |
| syl2ani 608 | A syllogism inference. (C... |
| syl2and 609 | A syllogism deduction. (C... |
| anim12d 610 | Conjoin antecedents and co... |
| anim12d1 611 | Variant of ~ anim12d where... |
| anim1d 612 | Add a conjunct to right of... |
| anim2d 613 | Add a conjunct to left of ... |
| anim12i 614 | Conjoin antecedents and co... |
| anim12ci 615 | Variant of ~ anim12i with ... |
| anim1i 616 | Introduce conjunct to both... |
| anim1ci 617 | Introduce conjunct to both... |
| anim2i 618 | Introduce conjunct to both... |
| anim12ii 619 | Conjoin antecedents and co... |
| anim12dan 620 | Conjoin antecedents and co... |
| im2anan9 621 | Deduction joining nested i... |
| im2anan9r 622 | Deduction joining nested i... |
| pm3.45 623 | Theorem *3.45 (Fact) of [W... |
| anbi2i 624 | Introduce a left conjunct ... |
| anbi1i 625 | Introduce a right conjunct... |
| anbi2ci 626 | Variant of ~ anbi2i with c... |
| anbi1ci 627 | Variant of ~ anbi1i with c... |
| bianbi 628 | Exchanging conjunction in ... |
| anbi12i 629 | Conjoin both sides of two ... |
| anbi12ci 630 | Variant of ~ anbi12i with ... |
| anbi2d 631 | Deduction adding a left co... |
| anbi1d 632 | Deduction adding a right c... |
| anbi12d 633 | Deduction joining two equi... |
| anbi1 634 | Introduce a right conjunct... |
| anbi2 635 | Introduce a left conjunct ... |
| anbi1cd 636 | Introduce a proposition as... |
| an2anr 637 | Double commutation in conj... |
| pm4.38 638 | Theorem *4.38 of [Whitehea... |
| bi2anan9 639 | Deduction joining two equi... |
| bi2anan9r 640 | Deduction joining two equi... |
| bi2bian9 641 | Deduction joining two bico... |
| anbiim 642 | Adding biconditional when ... |
| bianass 643 | An inference to merge two ... |
| bianassc 644 | An inference to merge two ... |
| an21 645 | Swap two conjuncts. (Cont... |
| an12 646 | Swap two conjuncts. Note ... |
| an32 647 | A rearrangement of conjunc... |
| an13 648 | A rearrangement of conjunc... |
| an31 649 | A rearrangement of conjunc... |
| an12s 650 | Swap two conjuncts in ante... |
| ancom2s 651 | Inference commuting a nest... |
| an13s 652 | Swap two conjuncts in ante... |
| an32s 653 | Swap two conjuncts in ante... |
| ancom1s 654 | Inference commuting a nest... |
| an31s 655 | Swap two conjuncts in ante... |
| anass1rs 656 | Commutative-associative la... |
| an4 657 | Rearrangement of 4 conjunc... |
| an42 658 | Rearrangement of 4 conjunc... |
| an43 659 | Rearrangement of 4 conjunc... |
| an3 660 | A rearrangement of conjunc... |
| an4s 661 | Inference rearranging 4 co... |
| an42s 662 | Inference rearranging 4 co... |
| anabs1 663 | Absorption into embedded c... |
| anabs5 664 | Absorption into embedded c... |
| anabs7 665 | Absorption into embedded c... |
| anabsan 666 | Absorption of antecedent w... |
| anabss1 667 | Absorption of antecedent i... |
| anabss4 668 | Absorption of antecedent i... |
| anabss5 669 | Absorption of antecedent i... |
| anabsi5 670 | Absorption of antecedent i... |
| anabsi6 671 | Absorption of antecedent i... |
| anabsi7 672 | Absorption of antecedent i... |
| anabsi8 673 | Absorption of antecedent i... |
| anabss7 674 | Absorption of antecedent i... |
| anabsan2 675 | Absorption of antecedent w... |
| anabss3 676 | Absorption of antecedent i... |
| anandi 677 | Distribution of conjunctio... |
| anandir 678 | Distribution of conjunctio... |
| anandis 679 | Inference that undistribut... |
| anandirs 680 | Inference that undistribut... |
| sylanl1 681 | A syllogism inference. (C... |
| sylanl2 682 | A syllogism inference. (C... |
| sylanr1 683 | A syllogism inference. (C... |
| sylanr2 684 | A syllogism inference. (C... |
| syl6an 685 | A syllogism deduction comb... |
| syl2an2r 686 | ~ syl2anr with antecedents... |
| syl2an2 687 | ~ syl2an with antecedents ... |
| mpdan 688 | An inference based on modu... |
| mpancom 689 | An inference based on modu... |
| mpidan 690 | A deduction which "stacks"... |
| mpan 691 | An inference based on modu... |
| mpan2 692 | An inference based on modu... |
| mp2an 693 | An inference based on modu... |
| mp4an 694 | An inference based on modu... |
| mpan2d 695 | A deduction based on modus... |
| mpand 696 | A deduction based on modus... |
| mpani 697 | An inference based on modu... |
| mpan2i 698 | An inference based on modu... |
| mp2ani 699 | An inference based on modu... |
| mp2and 700 | A deduction based on modus... |
| mpanl1 701 | An inference based on modu... |
| mpanl2 702 | An inference based on modu... |
| mpanl12 703 | An inference based on modu... |
| mpanr1 704 | An inference based on modu... |
| mpanr2 705 | An inference based on modu... |
| mpanr12 706 | An inference based on modu... |
| mpanlr1 707 | An inference based on modu... |
| mpbirand 708 | Detach truth from conjunct... |
| mpbiran2d 709 | Detach truth from conjunct... |
| mpbiran 710 | Detach truth from conjunct... |
| mpbiran2 711 | Detach truth from conjunct... |
| mpbir2an 712 | Detach a conjunction of tr... |
| mpbi2and 713 | Detach a conjunction of tr... |
| mpbir2and 714 | Detach a conjunction of tr... |
| adantll 715 | Deduction adding a conjunc... |
| adantlr 716 | Deduction adding a conjunc... |
| adantrl 717 | Deduction adding a conjunc... |
| adantrr 718 | Deduction adding a conjunc... |
| adantlll 719 | Deduction adding a conjunc... |
| adantllr 720 | Deduction adding a conjunc... |
| adantlrl 721 | Deduction adding a conjunc... |
| adantlrr 722 | Deduction adding a conjunc... |
| adantrll 723 | Deduction adding a conjunc... |
| adantrlr 724 | Deduction adding a conjunc... |
| adantrrl 725 | Deduction adding a conjunc... |
| adantrrr 726 | Deduction adding a conjunc... |
| ad2antrr 727 | Deduction adding two conju... |
| ad2antlr 728 | Deduction adding two conju... |
| ad2antrl 729 | Deduction adding two conju... |
| ad2antll 730 | Deduction adding conjuncts... |
| ad3antrrr 731 | Deduction adding three con... |
| ad3antlr 732 | Deduction adding three con... |
| ad4antr 733 | Deduction adding 4 conjunc... |
| ad4antlr 734 | Deduction adding 4 conjunc... |
| ad5antr 735 | Deduction adding 5 conjunc... |
| ad5antlr 736 | Deduction adding 5 conjunc... |
| ad6antr 737 | Deduction adding 6 conjunc... |
| ad6antlr 738 | Deduction adding 6 conjunc... |
| ad7antr 739 | Deduction adding 7 conjunc... |
| ad7antlr 740 | Deduction adding 7 conjunc... |
| ad8antr 741 | Deduction adding 8 conjunc... |
| ad8antlr 742 | Deduction adding 8 conjunc... |
| ad9antr 743 | Deduction adding 9 conjunc... |
| ad9antlr 744 | Deduction adding 9 conjunc... |
| ad10antr 745 | Deduction adding 10 conjun... |
| ad10antlr 746 | Deduction adding 10 conjun... |
| ad2ant2l 747 | Deduction adding two conju... |
| ad2ant2r 748 | Deduction adding two conju... |
| ad2ant2lr 749 | Deduction adding two conju... |
| ad2ant2rl 750 | Deduction adding two conju... |
| adantl3r 751 | Deduction adding 1 conjunc... |
| ad4ant13 752 | Deduction adding conjuncts... |
| ad4ant14 753 | Deduction adding conjuncts... |
| ad4ant23 754 | Deduction adding conjuncts... |
| ad4ant24 755 | Deduction adding conjuncts... |
| adantl4r 756 | Deduction adding 1 conjunc... |
| ad5ant13 757 | Deduction adding conjuncts... |
| ad5ant14 758 | Deduction adding conjuncts... |
| ad5ant15 759 | Deduction adding conjuncts... |
| ad5ant23 760 | Deduction adding conjuncts... |
| ad5ant24 761 | Deduction adding conjuncts... |
| ad5ant25 762 | Deduction adding conjuncts... |
| adantl5r 763 | Deduction adding 1 conjunc... |
| adantl6r 764 | Deduction adding 1 conjunc... |
| pm3.33 765 | Theorem *3.33 (Syll) of [W... |
| pm3.34 766 | Theorem *3.34 (Syll) of [W... |
| simpll 767 | Simplification of a conjun... |
| simplld 768 | Deduction form of ~ simpll... |
| simplr 769 | Simplification of a conjun... |
| simplrd 770 | Deduction eliminating a do... |
| simprl 771 | Simplification of a conjun... |
| simprld 772 | Deduction eliminating a do... |
| simprr 773 | Simplification of a conjun... |
| simprrd 774 | Deduction form of ~ simprr... |
| simplll 775 | Simplification of a conjun... |
| simpllr 776 | Simplification of a conjun... |
| simplrl 777 | Simplification of a conjun... |
| simplrr 778 | Simplification of a conjun... |
| simprll 779 | Simplification of a conjun... |
| simprlr 780 | Simplification of a conjun... |
| simprrl 781 | Simplification of a conjun... |
| simprrr 782 | Simplification of a conjun... |
| simp-4l 783 | Simplification of a conjun... |
| simp-4r 784 | Simplification of a conjun... |
| simp-5l 785 | Simplification of a conjun... |
| simp-5r 786 | Simplification of a conjun... |
| simp-6l 787 | Simplification of a conjun... |
| simp-6r 788 | Simplification of a conjun... |
| simp-7l 789 | Simplification of a conjun... |
| simp-7r 790 | Simplification of a conjun... |
| simp-8l 791 | Simplification of a conjun... |
| simp-8r 792 | Simplification of a conjun... |
| simp-9l 793 | Simplification of a conjun... |
| simp-9r 794 | Simplification of a conjun... |
| simp-10l 795 | Simplification of a conjun... |
| simp-10r 796 | Simplification of a conjun... |
| simp-11l 797 | Simplification of a conjun... |
| simp-11r 798 | Simplification of a conjun... |
| pm2.01da 799 | Deduction based on reducti... |
| pm2.18da 800 | Deduction based on reducti... |
| impbida 801 | Deduce an equivalence from... |
| pm5.21nd 802 | Eliminate an antecedent im... |
| pm3.35 803 | Conjunctive detachment. T... |
| pm5.74da 804 | Distribution of implicatio... |
| bitr 805 | Theorem *4.22 of [Whitehea... |
| biantr 806 | A transitive law of equiva... |
| pm4.14 807 | Theorem *4.14 of [Whitehea... |
| pm3.37 808 | Theorem *3.37 (Transp) of ... |
| anim12 809 | Conjoin antecedents and co... |
| pm3.4 810 | Conjunction implies implic... |
| exbiri 811 | Inference form of ~ exbir ... |
| pm2.61ian 812 | Elimination of an antecede... |
| pm2.61dan 813 | Elimination of an antecede... |
| pm2.61ddan 814 | Elimination of two anteced... |
| pm2.61dda 815 | Elimination of two anteced... |
| mtand 816 | A modus tollens deduction.... |
| pm2.65da 817 | Deduction for proof by con... |
| condan 818 | Proof by contradiction. (... |
| biadan 819 | An implication is equivale... |
| biadani 820 | Inference associated with ... |
| biadaniALT 821 | Alternate proof of ~ biada... |
| biadanii 822 | Inference associated with ... |
| biadanid 823 | Deduction associated with ... |
| pm5.1 824 | Two propositions are equiv... |
| pm5.21 825 | Two propositions are equiv... |
| pm5.35 826 | Theorem *5.35 of [Whitehea... |
| abai 827 | Introduce one conjunct as ... |
| pm4.45im 828 | Conjunction with implicati... |
| impimprbi 829 | An implication and its rev... |
| nan 830 | Theorem to move a conjunct... |
| pm5.31 831 | Theorem *5.31 of [Whitehea... |
| pm5.31r 832 | Variant of ~ pm5.31 . (Co... |
| pm4.15 833 | Theorem *4.15 of [Whitehea... |
| pm5.36 834 | Theorem *5.36 of [Whitehea... |
| annotanannot 835 | A conjunction with a negat... |
| pm5.33 836 | Theorem *5.33 of [Whitehea... |
| syl12anc 837 | Syllogism combined with co... |
| syl21anc 838 | Syllogism combined with co... |
| syl22anc 839 | Syllogism combined with co... |
| bibiad 840 | Eliminate an hypothesis ` ... |
| syl1111anc 841 | Four-hypothesis eliminatio... |
| syldbl2 842 | Stacked hypotheseis implie... |
| mpsyl4anc 843 | An elimination deduction. ... |
| pm4.87 844 | Theorem *4.87 of [Whitehea... |
| bimsc1 845 | Removal of conjunct from o... |
| a2and 846 | Deduction distributing a c... |
| animpimp2impd 847 | Deduction deriving nested ... |
| pm4.64 850 | Theorem *4.64 of [Whitehea... |
| pm4.66 851 | Theorem *4.66 of [Whitehea... |
| pm2.53 852 | Theorem *2.53 of [Whitehea... |
| pm2.54 853 | Theorem *2.54 of [Whitehea... |
| imor 854 | Implication in terms of di... |
| imori 855 | Infer disjunction from imp... |
| imorri 856 | Infer implication from dis... |
| pm4.62 857 | Theorem *4.62 of [Whitehea... |
| jaoi 858 | Inference disjoining the a... |
| jao1i 859 | Add a disjunct in the ante... |
| jaod 860 | Deduction disjoining the a... |
| mpjaod 861 | Eliminate a disjunction in... |
| ori 862 | Infer implication from dis... |
| orri 863 | Infer disjunction from imp... |
| orrd 864 | Deduce disjunction from im... |
| ord 865 | Deduce implication from di... |
| orci 866 | Deduction introducing a di... |
| olci 867 | Deduction introducing a di... |
| orc 868 | Introduction of a disjunct... |
| olc 869 | Introduction of a disjunct... |
| pm1.4 870 | Axiom *1.4 of [WhiteheadRu... |
| orcom 871 | Commutative law for disjun... |
| orcomd 872 | Commutation of disjuncts i... |
| orcoms 873 | Commutation of disjuncts i... |
| orcd 874 | Deduction introducing a di... |
| olcd 875 | Deduction introducing a di... |
| orcs 876 | Deduction eliminating disj... |
| olcs 877 | Deduction eliminating disj... |
| olcnd 878 | A lemma for Conjunctive No... |
| orcnd 879 | A lemma for Conjunctive No... |
| mtord 880 | A modus tollens deduction ... |
| pm3.2ni 881 | Infer negated disjunction ... |
| pm2.45 882 | Theorem *2.45 of [Whitehea... |
| pm2.46 883 | Theorem *2.46 of [Whitehea... |
| pm2.47 884 | Theorem *2.47 of [Whitehea... |
| pm2.48 885 | Theorem *2.48 of [Whitehea... |
| pm2.49 886 | Theorem *2.49 of [Whitehea... |
| norbi 887 | If neither of two proposit... |
| nbior 888 | If two propositions are no... |
| orel1 889 | Elimination of disjunction... |
| pm2.25 890 | Theorem *2.25 of [Whitehea... |
| orel2 891 | Elimination of disjunction... |
| pm2.67-2 892 | Slight generalization of T... |
| pm2.67 893 | Theorem *2.67 of [Whitehea... |
| curryax 894 | A non-intuitionistic posit... |
| exmid 895 | Law of excluded middle, al... |
| exmidd 896 | Law of excluded middle in ... |
| pm2.1 897 | Theorem *2.1 of [Whitehead... |
| pm2.13 898 | Theorem *2.13 of [Whitehea... |
| pm2.621 899 | Theorem *2.621 of [Whitehe... |
| pm2.62 900 | Theorem *2.62 of [Whitehea... |
| pm2.68 901 | Theorem *2.68 of [Whitehea... |
| dfor2 902 | Logical 'or' expressed in ... |
| pm2.07 903 | Theorem *2.07 of [Whitehea... |
| pm1.2 904 | Axiom *1.2 of [WhiteheadRu... |
| oridm 905 | Idempotent law for disjunc... |
| pm4.25 906 | Theorem *4.25 of [Whitehea... |
| pm2.4 907 | Theorem *2.4 of [Whitehead... |
| pm2.41 908 | Theorem *2.41 of [Whitehea... |
| orim12i 909 | Disjoin antecedents and co... |
| orim1i 910 | Introduce disjunct to both... |
| orim2i 911 | Introduce disjunct to both... |
| orim12dALT 912 | Alternate proof of ~ orim1... |
| orbi2i 913 | Inference adding a left di... |
| orbi1i 914 | Inference adding a right d... |
| orbi12i 915 | Infer the disjunction of t... |
| orbi2d 916 | Deduction adding a left di... |
| orbi1d 917 | Deduction adding a right d... |
| orbi1 918 | Theorem *4.37 of [Whitehea... |
| orbi12d 919 | Deduction joining two equi... |
| pm1.5 920 | Axiom *1.5 (Assoc) of [Whi... |
| or12 921 | Swap two disjuncts. (Cont... |
| orass 922 | Associative law for disjun... |
| pm2.31 923 | Theorem *2.31 of [Whitehea... |
| pm2.32 924 | Theorem *2.32 of [Whitehea... |
| pm2.3 925 | Theorem *2.3 of [Whitehead... |
| or32 926 | A rearrangement of disjunc... |
| or4 927 | Rearrangement of 4 disjunc... |
| or42 928 | Rearrangement of 4 disjunc... |
| orordi 929 | Distribution of disjunctio... |
| orordir 930 | Distribution of disjunctio... |
| orimdi 931 | Disjunction distributes ov... |
| pm2.76 932 | Theorem *2.76 of [Whitehea... |
| pm2.85 933 | Theorem *2.85 of [Whitehea... |
| pm2.75 934 | Theorem *2.75 of [Whitehea... |
| pm4.78 935 | Implication distributes ov... |
| biort 936 | A disjunction with a true ... |
| biorf 937 | A wff is equivalent to its... |
| biortn 938 | A wff is equivalent to its... |
| biorfi 939 | The dual of ~ biorf is not... |
| biorfri 940 | A wff is equivalent to its... |
| biorfriOLD 941 | Obsolete version of ~ bior... |
| pm2.26 942 | Theorem *2.26 of [Whitehea... |
| pm2.63 943 | Theorem *2.63 of [Whitehea... |
| pm2.64 944 | Theorem *2.64 of [Whitehea... |
| pm2.42 945 | Theorem *2.42 of [Whitehea... |
| pm5.11g 946 | A general instance of Theo... |
| pm5.11 947 | Theorem *5.11 of [Whitehea... |
| pm5.12 948 | Theorem *5.12 of [Whitehea... |
| pm5.14 949 | Theorem *5.14 of [Whitehea... |
| pm5.13 950 | Theorem *5.13 of [Whitehea... |
| pm5.55 951 | Theorem *5.55 of [Whitehea... |
| pm4.72 952 | Implication in terms of bi... |
| imimorb 953 | Simplify an implication be... |
| oibabs 954 | Absorption of disjunction ... |
| orbidi 955 | Disjunction distributes ov... |
| pm5.7 956 | Disjunction distributes ov... |
| jaao 957 | Inference conjoining and d... |
| jaoa 958 | Inference disjoining and c... |
| jaoian 959 | Inference disjoining the a... |
| jaodan 960 | Deduction disjoining the a... |
| mpjaodan 961 | Eliminate a disjunction in... |
| pm3.44 962 | Theorem *3.44 of [Whitehea... |
| jao 963 | Disjunction of antecedents... |
| jaob 964 | Disjunction of antecedents... |
| pm4.77 965 | Theorem *4.77 of [Whitehea... |
| pm3.48 966 | Theorem *3.48 of [Whitehea... |
| orim12d 967 | Disjoin antecedents and co... |
| orim1d 968 | Disjoin antecedents and co... |
| orim2d 969 | Disjoin antecedents and co... |
| orim2 970 | Axiom *1.6 (Sum) of [White... |
| pm2.38 971 | Theorem *2.38 of [Whitehea... |
| pm2.36 972 | Theorem *2.36 of [Whitehea... |
| pm2.37 973 | Theorem *2.37 of [Whitehea... |
| pm2.81 974 | Theorem *2.81 of [Whitehea... |
| pm2.8 975 | Theorem *2.8 of [Whitehead... |
| pm2.73 976 | Theorem *2.73 of [Whitehea... |
| pm2.74 977 | Theorem *2.74 of [Whitehea... |
| pm2.82 978 | Theorem *2.82 of [Whitehea... |
| pm4.39 979 | Theorem *4.39 of [Whitehea... |
| animorl 980 | Conjunction implies disjun... |
| animorr 981 | Conjunction implies disjun... |
| animorlr 982 | Conjunction implies disjun... |
| animorrl 983 | Conjunction implies disjun... |
| ianor 984 | Negated conjunction in ter... |
| anor 985 | Conjunction in terms of di... |
| ioran 986 | Negated disjunction in ter... |
| pm4.52 987 | Theorem *4.52 of [Whitehea... |
| pm4.53 988 | Theorem *4.53 of [Whitehea... |
| pm4.54 989 | Theorem *4.54 of [Whitehea... |
| pm4.55 990 | Theorem *4.55 of [Whitehea... |
| pm4.56 991 | Theorem *4.56 of [Whitehea... |
| oran 992 | Disjunction in terms of co... |
| pm4.57 993 | Theorem *4.57 of [Whitehea... |
| pm3.1 994 | Theorem *3.1 of [Whitehead... |
| pm3.11 995 | Theorem *3.11 of [Whitehea... |
| pm3.12 996 | Theorem *3.12 of [Whitehea... |
| pm3.13 997 | Theorem *3.13 of [Whitehea... |
| pm3.14 998 | Theorem *3.14 of [Whitehea... |
| pm4.44 999 | Theorem *4.44 of [Whitehea... |
| pm4.45 1000 | Theorem *4.45 of [Whitehea... |
| orabs 1001 | Absorption of redundant in... |
| oranabs 1002 | Absorb a disjunct into a c... |
| pm5.61 1003 | Theorem *5.61 of [Whitehea... |
| pm5.6 1004 | Conjunction in antecedent ... |
| orcanai 1005 | Change disjunction in cons... |
| pm4.79 1006 | Theorem *4.79 of [Whitehea... |
| pm5.53 1007 | Theorem *5.53 of [Whitehea... |
| ordi 1008 | Distributive law for disju... |
| ordir 1009 | Distributive law for disju... |
| andi 1010 | Distributive law for conju... |
| andir 1011 | Distributive law for conju... |
| orddi 1012 | Double distributive law fo... |
| anddi 1013 | Double distributive law fo... |
| pm5.17 1014 | Theorem *5.17 of [Whitehea... |
| pm5.15 1015 | Theorem *5.15 of [Whitehea... |
| pm5.16 1016 | Theorem *5.16 of [Whitehea... |
| xor 1017 | Two ways to express exclus... |
| nbi2 1018 | Two ways to express "exclu... |
| xordi 1019 | Conjunction distributes ov... |
| pm5.54 1020 | Theorem *5.54 of [Whitehea... |
| pm5.62 1021 | Theorem *5.62 of [Whitehea... |
| pm5.63 1022 | Theorem *5.63 of [Whitehea... |
| niabn 1023 | Miscellaneous inference re... |
| ninba 1024 | Miscellaneous inference re... |
| pm4.43 1025 | Theorem *4.43 of [Whitehea... |
| pm4.82 1026 | Theorem *4.82 of [Whitehea... |
| pm4.83 1027 | Theorem *4.83 of [Whitehea... |
| pclem6 1028 | Negation inferred from emb... |
| bigolden 1029 | Dijkstra-Scholten's Golden... |
| pm5.71 1030 | Theorem *5.71 of [Whitehea... |
| pm5.75 1031 | Theorem *5.75 of [Whitehea... |
| ecase2d 1032 | Deduction for elimination ... |
| ecase3 1033 | Inference for elimination ... |
| ecase 1034 | Inference for elimination ... |
| ecase3d 1035 | Deduction for elimination ... |
| ecased 1036 | Deduction for elimination ... |
| ecase3ad 1037 | Deduction for elimination ... |
| ccase 1038 | Inference for combining ca... |
| ccased 1039 | Deduction for combining ca... |
| ccase2 1040 | Inference for combining ca... |
| 4cases 1041 | Inference eliminating two ... |
| 4casesdan 1042 | Deduction eliminating two ... |
| cases 1043 | Case disjunction according... |
| dedlem0a 1044 | Lemma for an alternate ver... |
| dedlem0b 1045 | Lemma for an alternate ver... |
| dedlema 1046 | Lemma for weak deduction t... |
| dedlemb 1047 | Lemma for weak deduction t... |
| cases2 1048 | Case disjunction according... |
| cases2ALT 1049 | Alternate proof of ~ cases... |
| dfbi3 1050 | An alternate definition of... |
| pm5.24 1051 | Theorem *5.24 of [Whitehea... |
| 4exmid 1052 | The disjunction of the fou... |
| consensus 1053 | The consensus theorem. Th... |
| pm4.42 1054 | Theorem *4.42 of [Whitehea... |
| prlem1 1055 | A specialized lemma for se... |
| prlem2 1056 | A specialized lemma for se... |
| oplem1 1057 | A specialized lemma for se... |
| dn1 1058 | A single axiom for Boolean... |
| bianir 1059 | A closed form of ~ mpbir ,... |
| jaoi2 1060 | Inference removing a negat... |
| jaoi3 1061 | Inference separating a dis... |
| ornld 1062 | Selecting one statement fr... |
| dfifp2 1065 | Alternate definition of th... |
| dfifp3 1066 | Alternate definition of th... |
| dfifp4 1067 | Alternate definition of th... |
| dfifp5 1068 | Alternate definition of th... |
| dfifp6 1069 | Alternate definition of th... |
| dfifp7 1070 | Alternate definition of th... |
| ifpdfbi 1071 | Define the biconditional a... |
| anifp 1072 | The conditional operator i... |
| ifpor 1073 | The conditional operator i... |
| ifpn 1074 | Conditional operator for t... |
| ifptru 1075 | Value of the conditional o... |
| ifpfal 1076 | Value of the conditional o... |
| ifpid 1077 | Value of the conditional o... |
| casesifp 1078 | Version of ~ cases express... |
| ifpbi123d 1079 | Equivalence deduction for ... |
| ifpbi23d 1080 | Equivalence deduction for ... |
| ifpimpda 1081 | Separation of the values o... |
| 1fpid3 1082 | The value of the condition... |
| elimh 1083 | Hypothesis builder for the... |
| dedt 1084 | The weak deduction theorem... |
| con3ALT 1085 | Proof of ~ con3 from its a... |
| 3orass 1090 | Associative law for triple... |
| 3orel1 1091 | Partial elimination of a t... |
| 3orrot 1092 | Rotation law for triple di... |
| 3orcoma 1093 | Commutation law for triple... |
| 3orcomb 1094 | Commutation law for triple... |
| 3anass 1095 | Associative law for triple... |
| 3anan12 1096 | Convert triple conjunction... |
| 3anan32 1097 | Convert triple conjunction... |
| 3ancoma 1098 | Commutation law for triple... |
| 3ancomb 1099 | Commutation law for triple... |
| 3anrot 1100 | Rotation law for triple co... |
| 3anrev 1101 | Reversal law for triple co... |
| anandi3 1102 | Distribution of triple con... |
| anandi3r 1103 | Distribution of triple con... |
| 3anidm 1104 | Idempotent law for conjunc... |
| 3an4anass 1105 | Associative law for four c... |
| 3ioran 1106 | Negated triple disjunction... |
| 3ianor 1107 | Negated triple conjunction... |
| 3anor 1108 | Triple conjunction express... |
| 3oran 1109 | Triple disjunction in term... |
| 3impa 1110 | Importation from double to... |
| 3imp 1111 | Importation inference. (C... |
| 3imp31 1112 | The importation inference ... |
| 3imp231 1113 | Importation inference. (C... |
| 3imp21 1114 | The importation inference ... |
| 3impb 1115 | Importation from double to... |
| bi23imp13 1116 | ~ 3imp with middle implica... |
| 3impib 1117 | Importation to triple conj... |
| 3impia 1118 | Importation to triple conj... |
| 3expa 1119 | Exportation from triple to... |
| 3exp 1120 | Exportation inference. (C... |
| 3expb 1121 | Exportation from triple to... |
| 3expia 1122 | Exportation from triple co... |
| 3expib 1123 | Exportation from triple co... |
| 3com12 1124 | Commutation in antecedent.... |
| 3com13 1125 | Commutation in antecedent.... |
| 3comr 1126 | Commutation in antecedent.... |
| 3com23 1127 | Commutation in antecedent.... |
| 3coml 1128 | Commutation in antecedent.... |
| 3jca 1129 | Join consequents with conj... |
| 3jcad 1130 | Deduction conjoining the c... |
| 3adant1 1131 | Deduction adding a conjunc... |
| 3adant2 1132 | Deduction adding a conjunc... |
| 3adant3 1133 | Deduction adding a conjunc... |
| 3ad2ant1 1134 | Deduction adding conjuncts... |
| 3ad2ant2 1135 | Deduction adding conjuncts... |
| 3ad2ant3 1136 | Deduction adding conjuncts... |
| simp1 1137 | Simplification of triple c... |
| simp2 1138 | Simplification of triple c... |
| simp3 1139 | Simplification of triple c... |
| simp1i 1140 | Infer a conjunct from a tr... |
| simp2i 1141 | Infer a conjunct from a tr... |
| simp3i 1142 | Infer a conjunct from a tr... |
| simp1d 1143 | Deduce a conjunct from a t... |
| simp2d 1144 | Deduce a conjunct from a t... |
| simp3d 1145 | Deduce a conjunct from a t... |
| simp1bi 1146 | Deduce a conjunct from a t... |
| simp2bi 1147 | Deduce a conjunct from a t... |
| simp3bi 1148 | Deduce a conjunct from a t... |
| 3simpa 1149 | Simplification of triple c... |
| 3simpb 1150 | Simplification of triple c... |
| 3simpc 1151 | Simplification of triple c... |
| 3anim123i 1152 | Join antecedents and conse... |
| 3anim1i 1153 | Add two conjuncts to antec... |
| 3anim2i 1154 | Add two conjuncts to antec... |
| 3anim3i 1155 | Add two conjuncts to antec... |
| 3anbi123i 1156 | Join 3 biconditionals with... |
| 3orbi123i 1157 | Join 3 biconditionals with... |
| 3anbi1i 1158 | Inference adding two conju... |
| 3anbi2i 1159 | Inference adding two conju... |
| 3anbi3i 1160 | Inference adding two conju... |
| syl3an 1161 | A triple syllogism inferen... |
| syl3anb 1162 | A triple syllogism inferen... |
| syl3anbr 1163 | A triple syllogism inferen... |
| syl3an1 1164 | A syllogism inference. (C... |
| syl3an2 1165 | A syllogism inference. (C... |
| syl3an3 1166 | A syllogism inference. (C... |
| syl3an132 1167 | ~ syl2an with antecedents ... |
| 3adantl1 1168 | Deduction adding a conjunc... |
| 3adantl2 1169 | Deduction adding a conjunc... |
| 3adantl3 1170 | Deduction adding a conjunc... |
| 3adantr1 1171 | Deduction adding a conjunc... |
| 3adantr2 1172 | Deduction adding a conjunc... |
| 3adantr3 1173 | Deduction adding a conjunc... |
| ad4ant123 1174 | Deduction adding conjuncts... |
| ad4ant124 1175 | Deduction adding conjuncts... |
| ad4ant134 1176 | Deduction adding conjuncts... |
| ad4ant234 1177 | Deduction adding conjuncts... |
| 3adant1l 1178 | Deduction adding a conjunc... |
| 3adant1r 1179 | Deduction adding a conjunc... |
| 3adant2l 1180 | Deduction adding a conjunc... |
| 3adant2r 1181 | Deduction adding a conjunc... |
| 3adant3l 1182 | Deduction adding a conjunc... |
| 3adant3r 1183 | Deduction adding a conjunc... |
| 3adant3r1 1184 | Deduction adding a conjunc... |
| 3adant3r2 1185 | Deduction adding a conjunc... |
| 3adant3r3 1186 | Deduction adding a conjunc... |
| 3ad2antl1 1187 | Deduction adding conjuncts... |
| 3ad2antl2 1188 | Deduction adding conjuncts... |
| 3ad2antl3 1189 | Deduction adding conjuncts... |
| 3ad2antr1 1190 | Deduction adding conjuncts... |
| 3ad2antr2 1191 | Deduction adding conjuncts... |
| 3ad2antr3 1192 | Deduction adding conjuncts... |
| simpl1 1193 | Simplification of conjunct... |
| simpl2 1194 | Simplification of conjunct... |
| simpl3 1195 | Simplification of conjunct... |
| simpr1 1196 | Simplification of conjunct... |
| simpr2 1197 | Simplification of conjunct... |
| simpr3 1198 | Simplification of conjunct... |
| simp1l 1199 | Simplification of triple c... |
| simp1r 1200 | Simplification of triple c... |
| simp2l 1201 | Simplification of triple c... |
| simp2r 1202 | Simplification of triple c... |
| simp3l 1203 | Simplification of triple c... |
| simp3r 1204 | Simplification of triple c... |
| simp11 1205 | Simplification of doubly t... |
| simp12 1206 | Simplification of doubly t... |
| simp13 1207 | Simplification of doubly t... |
| simp21 1208 | Simplification of doubly t... |
| simp22 1209 | Simplification of doubly t... |
| simp23 1210 | Simplification of doubly t... |
| simp31 1211 | Simplification of doubly t... |
| simp32 1212 | Simplification of doubly t... |
| simp33 1213 | Simplification of doubly t... |
| simpll1 1214 | Simplification of conjunct... |
| simpll2 1215 | Simplification of conjunct... |
| simpll3 1216 | Simplification of conjunct... |
| simplr1 1217 | Simplification of conjunct... |
| simplr2 1218 | Simplification of conjunct... |
| simplr3 1219 | Simplification of conjunct... |
| simprl1 1220 | Simplification of conjunct... |
| simprl2 1221 | Simplification of conjunct... |
| simprl3 1222 | Simplification of conjunct... |
| simprr1 1223 | Simplification of conjunct... |
| simprr2 1224 | Simplification of conjunct... |
| simprr3 1225 | Simplification of conjunct... |
| simpl1l 1226 | Simplification of conjunct... |
| simpl1r 1227 | Simplification of conjunct... |
| simpl2l 1228 | Simplification of conjunct... |
| simpl2r 1229 | Simplification of conjunct... |
| simpl3l 1230 | Simplification of conjunct... |
| simpl3r 1231 | Simplification of conjunct... |
| simpr1l 1232 | Simplification of conjunct... |
| simpr1r 1233 | Simplification of conjunct... |
| simpr2l 1234 | Simplification of conjunct... |
| simpr2r 1235 | Simplification of conjunct... |
| simpr3l 1236 | Simplification of conjunct... |
| simpr3r 1237 | Simplification of conjunct... |
| simp1ll 1238 | Simplification of conjunct... |
| simp1lr 1239 | Simplification of conjunct... |
| simp1rl 1240 | Simplification of conjunct... |
| simp1rr 1241 | Simplification of conjunct... |
| simp2ll 1242 | Simplification of conjunct... |
| simp2lr 1243 | Simplification of conjunct... |
| simp2rl 1244 | Simplification of conjunct... |
| simp2rr 1245 | Simplification of conjunct... |
| simp3ll 1246 | Simplification of conjunct... |
| simp3lr 1247 | Simplification of conjunct... |
| simp3rl 1248 | Simplification of conjunct... |
| simp3rr 1249 | Simplification of conjunct... |
| simpl11 1250 | Simplification of conjunct... |
| simpl12 1251 | Simplification of conjunct... |
| simpl13 1252 | Simplification of conjunct... |
| simpl21 1253 | Simplification of conjunct... |
| simpl22 1254 | Simplification of conjunct... |
| simpl23 1255 | Simplification of conjunct... |
| simpl31 1256 | Simplification of conjunct... |
| simpl32 1257 | Simplification of conjunct... |
| simpl33 1258 | Simplification of conjunct... |
| simpr11 1259 | Simplification of conjunct... |
| simpr12 1260 | Simplification of conjunct... |
| simpr13 1261 | Simplification of conjunct... |
| simpr21 1262 | Simplification of conjunct... |
| simpr22 1263 | Simplification of conjunct... |
| simpr23 1264 | Simplification of conjunct... |
| simpr31 1265 | Simplification of conjunct... |
| simpr32 1266 | Simplification of conjunct... |
| simpr33 1267 | Simplification of conjunct... |
| simp1l1 1268 | Simplification of conjunct... |
| simp1l2 1269 | Simplification of conjunct... |
| simp1l3 1270 | Simplification of conjunct... |
| simp1r1 1271 | Simplification of conjunct... |
| simp1r2 1272 | Simplification of conjunct... |
| simp1r3 1273 | Simplification of conjunct... |
| simp2l1 1274 | Simplification of conjunct... |
| simp2l2 1275 | Simplification of conjunct... |
| simp2l3 1276 | Simplification of conjunct... |
| simp2r1 1277 | Simplification of conjunct... |
| simp2r2 1278 | Simplification of conjunct... |
| simp2r3 1279 | Simplification of conjunct... |
| simp3l1 1280 | Simplification of conjunct... |
| simp3l2 1281 | Simplification of conjunct... |
| simp3l3 1282 | Simplification of conjunct... |
| simp3r1 1283 | Simplification of conjunct... |
| simp3r2 1284 | Simplification of conjunct... |
| simp3r3 1285 | Simplification of conjunct... |
| simp11l 1286 | Simplification of conjunct... |
| simp11r 1287 | Simplification of conjunct... |
| simp12l 1288 | Simplification of conjunct... |
| simp12r 1289 | Simplification of conjunct... |
| simp13l 1290 | Simplification of conjunct... |
| simp13r 1291 | Simplification of conjunct... |
| simp21l 1292 | Simplification of conjunct... |
| simp21r 1293 | Simplification of conjunct... |
| simp22l 1294 | Simplification of conjunct... |
| simp22r 1295 | Simplification of conjunct... |
| simp23l 1296 | Simplification of conjunct... |
| simp23r 1297 | Simplification of conjunct... |
| simp31l 1298 | Simplification of conjunct... |
| simp31r 1299 | Simplification of conjunct... |
| simp32l 1300 | Simplification of conjunct... |
| simp32r 1301 | Simplification of conjunct... |
| simp33l 1302 | Simplification of conjunct... |
| simp33r 1303 | Simplification of conjunct... |
| simp111 1304 | Simplification of conjunct... |
| simp112 1305 | Simplification of conjunct... |
| simp113 1306 | Simplification of conjunct... |
| simp121 1307 | Simplification of conjunct... |
| simp122 1308 | Simplification of conjunct... |
| simp123 1309 | Simplification of conjunct... |
| simp131 1310 | Simplification of conjunct... |
| simp132 1311 | Simplification of conjunct... |
| simp133 1312 | Simplification of conjunct... |
| simp211 1313 | Simplification of conjunct... |
| simp212 1314 | Simplification of conjunct... |
| simp213 1315 | Simplification of conjunct... |
| simp221 1316 | Simplification of conjunct... |
| simp222 1317 | Simplification of conjunct... |
| simp223 1318 | Simplification of conjunct... |
| simp231 1319 | Simplification of conjunct... |
| simp232 1320 | Simplification of conjunct... |
| simp233 1321 | Simplification of conjunct... |
| simp311 1322 | Simplification of conjunct... |
| simp312 1323 | Simplification of conjunct... |
| simp313 1324 | Simplification of conjunct... |
| simp321 1325 | Simplification of conjunct... |
| simp322 1326 | Simplification of conjunct... |
| simp323 1327 | Simplification of conjunct... |
| simp331 1328 | Simplification of conjunct... |
| simp332 1329 | Simplification of conjunct... |
| simp333 1330 | Simplification of conjunct... |
| 3anibar 1331 | Remove a hypothesis from t... |
| 3mix1 1332 | Introduction in triple dis... |
| 3mix2 1333 | Introduction in triple dis... |
| 3mix3 1334 | Introduction in triple dis... |
| 3mix1i 1335 | Introduction in triple dis... |
| 3mix2i 1336 | Introduction in triple dis... |
| 3mix3i 1337 | Introduction in triple dis... |
| 3mix1d 1338 | Deduction introducing trip... |
| 3mix2d 1339 | Deduction introducing trip... |
| 3mix3d 1340 | Deduction introducing trip... |
| 3pm3.2i 1341 | Infer conjunction of premi... |
| pm3.2an3 1342 | Version of ~ pm3.2 for a t... |
| mpbir3an 1343 | Detach a conjunction of tr... |
| mpbir3and 1344 | Detach a conjunction of tr... |
| syl3anbrc 1345 | Syllogism inference. (Con... |
| syl21anbrc 1346 | Syllogism inference. (Con... |
| 3imp3i2an 1347 | An elimination deduction. ... |
| ex3 1348 | Apply ~ ex to a hypothesis... |
| 3imp1 1349 | Importation to left triple... |
| 3impd 1350 | Importation deduction for ... |
| 3imp2 1351 | Importation to right tripl... |
| 3impdi 1352 | Importation inference (und... |
| 3impdir 1353 | Importation inference (und... |
| 3exp1 1354 | Exportation from left trip... |
| 3expd 1355 | Exportation deduction for ... |
| 3exp2 1356 | Exportation from right tri... |
| exp5o 1357 | A triple exportation infer... |
| exp516 1358 | A triple exportation infer... |
| exp520 1359 | A triple exportation infer... |
| 3impexp 1360 | Version of ~ impexp for a ... |
| 3an1rs 1361 | Swap conjuncts. (Contribu... |
| 3anassrs 1362 | Associative law for conjun... |
| 4anpull2 1363 | An equivalence of two four... |
| ad5ant245 1364 | Deduction adding conjuncts... |
| ad5ant234 1365 | Deduction adding conjuncts... |
| ad5ant235 1366 | Deduction adding conjuncts... |
| ad5ant123 1367 | Deduction adding conjuncts... |
| ad5ant124 1368 | Deduction adding conjuncts... |
| ad5ant125 1369 | Deduction adding conjuncts... |
| ad5ant134 1370 | Deduction adding conjuncts... |
| ad5ant135 1371 | Deduction adding conjuncts... |
| ad5ant145 1372 | Deduction adding conjuncts... |
| ad5ant2345 1373 | Deduction adding conjuncts... |
| syl3anc 1374 | Syllogism combined with co... |
| syl13anc 1375 | Syllogism combined with co... |
| syl31anc 1376 | Syllogism combined with co... |
| syl112anc 1377 | Syllogism combined with co... |
| syl121anc 1378 | Syllogism combined with co... |
| syl211anc 1379 | Syllogism combined with co... |
| syl23anc 1380 | Syllogism combined with co... |
| syl32anc 1381 | Syllogism combined with co... |
| syl122anc 1382 | Syllogism combined with co... |
| syl212anc 1383 | Syllogism combined with co... |
| syl221anc 1384 | Syllogism combined with co... |
| syl113anc 1385 | Syllogism combined with co... |
| syl131anc 1386 | Syllogism combined with co... |
| syl311anc 1387 | Syllogism combined with co... |
| syl33anc 1388 | Syllogism combined with co... |
| syl222anc 1389 | Syllogism combined with co... |
| syl123anc 1390 | Syllogism combined with co... |
| syl132anc 1391 | Syllogism combined with co... |
| syl213anc 1392 | Syllogism combined with co... |
| syl231anc 1393 | Syllogism combined with co... |
| syl312anc 1394 | Syllogism combined with co... |
| syl321anc 1395 | Syllogism combined with co... |
| syl133anc 1396 | Syllogism combined with co... |
| syl313anc 1397 | Syllogism combined with co... |
| syl331anc 1398 | Syllogism combined with co... |
| syl223anc 1399 | Syllogism combined with co... |
| syl232anc 1400 | Syllogism combined with co... |
| syl322anc 1401 | Syllogism combined with co... |
| syl233anc 1402 | Syllogism combined with co... |
| syl323anc 1403 | Syllogism combined with co... |
| syl332anc 1404 | Syllogism combined with co... |
| syl333anc 1405 | A syllogism inference comb... |
| syl3an1b 1406 | A syllogism inference. (C... |
| syl3an2b 1407 | A syllogism inference. (C... |
| syl3an3b 1408 | A syllogism inference. (C... |
| syl3an1br 1409 | A syllogism inference. (C... |
| syl3an2br 1410 | A syllogism inference. (C... |
| syl3an3br 1411 | A syllogism inference. (C... |
| syld3an3 1412 | A syllogism inference. (C... |
| syld3an1 1413 | A syllogism inference. (C... |
| syld3an2 1414 | A syllogism inference. (C... |
| syl3anl1 1415 | A syllogism inference. (C... |
| syl3anl2 1416 | A syllogism inference. (C... |
| syl3anl3 1417 | A syllogism inference. (C... |
| syl3anl 1418 | A triple syllogism inferen... |
| syl3anr1 1419 | A syllogism inference. (C... |
| syl3anr2 1420 | A syllogism inference. (C... |
| syl3anr3 1421 | A syllogism inference. (C... |
| 3anidm12 1422 | Inference from idempotent ... |
| 3anidm13 1423 | Inference from idempotent ... |
| 3anidm23 1424 | Inference from idempotent ... |
| syl2an3an 1425 | ~ syl3an with antecedents ... |
| syl2an23an 1426 | Deduction related to ~ syl... |
| 3ori 1427 | Infer implication from tri... |
| 3jao 1428 | Disjunction of three antec... |
| 3jaob 1429 | Disjunction of three antec... |
| 3jaobOLD 1430 | Obsolete version of ~ 3jao... |
| 3jaoi 1431 | Disjunction of three antec... |
| 3jaod 1432 | Disjunction of three antec... |
| 3jaoian 1433 | Disjunction of three antec... |
| 3jaodan 1434 | Disjunction of three antec... |
| mpjao3dan 1435 | Eliminate a three-way disj... |
| 3jaao 1436 | Inference conjoining and d... |
| syl3an9b 1437 | Nested syllogism inference... |
| 3orbi123d 1438 | Deduction joining 3 equiva... |
| 3anbi123d 1439 | Deduction joining 3 equiva... |
| 3anbi12d 1440 | Deduction conjoining and a... |
| 3anbi13d 1441 | Deduction conjoining and a... |
| 3anbi23d 1442 | Deduction conjoining and a... |
| 3anbi1d 1443 | Deduction adding conjuncts... |
| 3anbi2d 1444 | Deduction adding conjuncts... |
| 3anbi3d 1445 | Deduction adding conjuncts... |
| 3anim123d 1446 | Deduction joining 3 implic... |
| 3orim123d 1447 | Deduction joining 3 implic... |
| an6 1448 | Rearrangement of 6 conjunc... |
| 3an6 1449 | Analogue of ~ an4 for trip... |
| 3or6 1450 | Analogue of ~ or4 for trip... |
| mp3an1 1451 | An inference based on modu... |
| mp3an2 1452 | An inference based on modu... |
| mp3an3 1453 | An inference based on modu... |
| mp3an12 1454 | An inference based on modu... |
| mp3an13 1455 | An inference based on modu... |
| mp3an23 1456 | An inference based on modu... |
| mp3an1i 1457 | An inference based on modu... |
| mp3anl1 1458 | An inference based on modu... |
| mp3anl2 1459 | An inference based on modu... |
| mp3anl3 1460 | An inference based on modu... |
| mp3anr1 1461 | An inference based on modu... |
| mp3anr2 1462 | An inference based on modu... |
| mp3anr3 1463 | An inference based on modu... |
| mp3an 1464 | An inference based on modu... |
| mpd3an3 1465 | An inference based on modu... |
| mpd3an23 1466 | An inference based on modu... |
| mp3and 1467 | A deduction based on modus... |
| mp3an12i 1468 | ~ mp3an with antecedents i... |
| mp3an2i 1469 | ~ mp3an with antecedents i... |
| mp3an3an 1470 | ~ mp3an with antecedents i... |
| mp3an2ani 1471 | An elimination deduction. ... |
| biimp3a 1472 | Infer implication from a l... |
| biimp3ar 1473 | Infer implication from a l... |
| 3anandis 1474 | Inference that undistribut... |
| 3anandirs 1475 | Inference that undistribut... |
| ecase23d 1476 | Deduction for elimination ... |
| 3ecase 1477 | Inference for elimination ... |
| 3bior1fd 1478 | A disjunction is equivalen... |
| 3bior1fand 1479 | A disjunction is equivalen... |
| 3bior2fd 1480 | A wff is equivalent to its... |
| 3biant1d 1481 | A conjunction is equivalen... |
| intn3an1d 1482 | Introduction of a triple c... |
| intn3an2d 1483 | Introduction of a triple c... |
| intn3an3d 1484 | Introduction of a triple c... |
| an3andi 1485 | Distribution of conjunctio... |
| an33rean 1486 | Rearrange a 9-fold conjunc... |
| 3orel2 1487 | Partial elimination of a t... |
| 3orel2OLD 1488 | Obsolete version of ~ 3ore... |
| 3orel3 1489 | Partial elimination of a t... |
| 3orel13 1490 | Elimination of two disjunc... |
| 3pm3.2ni 1491 | Triple negated disjunction... |
| an42ds 1492 | Inference exchanging the l... |
| nanan 1495 | Conjunction in terms of al... |
| dfnan2 1496 | Alternative denial in term... |
| nanor 1497 | Alternative denial in term... |
| nancom 1498 | Alternative denial is comm... |
| nannan 1499 | Nested alternative denials... |
| nanim 1500 | Implication in terms of al... |
| nannot 1501 | Negation in terms of alter... |
| nanbi 1502 | Biconditional in terms of ... |
| nanbi1 1503 | Introduce a right anti-con... |
| nanbi2 1504 | Introduce a left anti-conj... |
| nanbi12 1505 | Join two logical equivalen... |
| nanbi1i 1506 | Introduce a right anti-con... |
| nanbi2i 1507 | Introduce a left anti-conj... |
| nanbi12i 1508 | Join two logical equivalen... |
| nanbi1d 1509 | Introduce a right anti-con... |
| nanbi2d 1510 | Introduce a left anti-conj... |
| nanbi12d 1511 | Join two logical equivalen... |
| nanass 1512 | A characterization of when... |
| xnor 1515 | Two ways to write XNOR (ex... |
| xorcom 1516 | The connector ` \/_ ` is c... |
| xorass 1517 | The connector ` \/_ ` is a... |
| excxor 1518 | This tautology shows that ... |
| xor2 1519 | Two ways to express "exclu... |
| xoror 1520 | Exclusive disjunction impl... |
| xornan 1521 | Exclusive disjunction impl... |
| xornan2 1522 | XOR implies NAND (written ... |
| xorneg2 1523 | The connector ` \/_ ` is n... |
| xorneg1 1524 | The connector ` \/_ ` is n... |
| xorneg 1525 | The connector ` \/_ ` is u... |
| xorbi12i 1526 | Equality property for excl... |
| xorbi12d 1527 | Equality property for excl... |
| anxordi 1528 | Conjunction distributes ov... |
| xorexmid 1529 | Exclusive-or variant of th... |
| norcom 1532 | The connector ` -\/ ` is c... |
| nornot 1533 | ` -. ` is expressible via ... |
| noran 1534 | ` /\ ` is expressible via ... |
| noror 1535 | ` \/ ` is expressible via ... |
| norasslem1 1536 | This lemma shows the equiv... |
| norasslem2 1537 | This lemma specializes ~ b... |
| norasslem3 1538 | This lemma specializes ~ b... |
| norass 1539 | A characterization of when... |
| trujust 1544 | Soundness justification th... |
| tru 1546 | The truth value ` T. ` is ... |
| dftru2 1547 | An alternate definition of... |
| trut 1548 | A proposition is equivalen... |
| mptru 1549 | Eliminate ` T. ` as an ant... |
| tbtru 1550 | A proposition is equivalen... |
| bitru 1551 | A theorem is equivalent to... |
| trud 1552 | Anything implies ` T. ` . ... |
| truan 1553 | True can be removed from a... |
| fal 1556 | The truth value ` F. ` is ... |
| nbfal 1557 | The negation of a proposit... |
| bifal 1558 | A contradiction is equival... |
| falim 1559 | The truth value ` F. ` imp... |
| falimd 1560 | The truth value ` F. ` imp... |
| dfnot 1561 | Given falsum ` F. ` , we c... |
| inegd 1562 | Negation introduction rule... |
| efald 1563 | Deduction based on reducti... |
| pm2.21fal 1564 | If a wff and its negation ... |
| truimtru 1565 | A ` -> ` identity. (Contr... |
| truimfal 1566 | A ` -> ` identity. (Contr... |
| falimtru 1567 | A ` -> ` identity. (Contr... |
| falimfal 1568 | A ` -> ` identity. (Contr... |
| nottru 1569 | A ` -. ` identity. (Contr... |
| notfal 1570 | A ` -. ` identity. (Contr... |
| trubitru 1571 | A ` <-> ` identity. (Cont... |
| falbitru 1572 | A ` <-> ` identity. (Cont... |
| trubifal 1573 | A ` <-> ` identity. (Cont... |
| falbifal 1574 | A ` <-> ` identity. (Cont... |
| truantru 1575 | A ` /\ ` identity. (Contr... |
| truanfal 1576 | A ` /\ ` identity. (Contr... |
| falantru 1577 | A ` /\ ` identity. (Contr... |
| falanfal 1578 | A ` /\ ` identity. (Contr... |
| truortru 1579 | A ` \/ ` identity. (Contr... |
| truorfal 1580 | A ` \/ ` identity. (Contr... |
| falortru 1581 | A ` \/ ` identity. (Contr... |
| falorfal 1582 | A ` \/ ` identity. (Contr... |
| trunantru 1583 | A ` -/\ ` identity. (Cont... |
| trunanfal 1584 | A ` -/\ ` identity. (Cont... |
| falnantru 1585 | A ` -/\ ` identity. (Cont... |
| falnanfal 1586 | A ` -/\ ` identity. (Cont... |
| truxortru 1587 | A ` \/_ ` identity. (Cont... |
| truxorfal 1588 | A ` \/_ ` identity. (Cont... |
| falxortru 1589 | A ` \/_ ` identity. (Cont... |
| falxorfal 1590 | A ` \/_ ` identity. (Cont... |
| trunortru 1591 | A ` -\/ ` identity. (Cont... |
| trunorfal 1592 | A ` -\/ ` identity. (Cont... |
| falnortru 1593 | A ` -\/ ` identity. (Cont... |
| falnorfal 1594 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1597 | Equality theorem for the a... |
| hadbi123i 1598 | Equality theorem for the a... |
| hadass 1599 | Associative law for the ad... |
| hadbi 1600 | The adder sum is the same ... |
| hadcoma 1601 | Commutative law for the ad... |
| hadcomb 1602 | Commutative law for the ad... |
| hadrot 1603 | Rotation law for the adder... |
| hadnot 1604 | The adder sum distributes ... |
| had1 1605 | If the first input is true... |
| had0 1606 | If the first input is fals... |
| hadifp 1607 | The value of the adder sum... |
| cador 1610 | The adder carry in disjunc... |
| cadan 1611 | The adder carry in conjunc... |
| cadbi123d 1612 | Equality theorem for the a... |
| cadbi123i 1613 | Equality theorem for the a... |
| cadcoma 1614 | Commutative law for the ad... |
| cadcomb 1615 | Commutative law for the ad... |
| cadrot 1616 | Rotation law for the adder... |
| cadnot 1617 | The adder carry distribute... |
| cad11 1618 | If (at least) two inputs a... |
| cad1 1619 | If one input is true, then... |
| cad0 1620 | If one input is false, the... |
| cadifp 1621 | The value of the carry is,... |
| cadtru 1622 | The adder carry is true as... |
| minimp 1623 | A single axiom for minimal... |
| minimp-syllsimp 1624 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1625 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1626 | Derivation of a commuted f... |
| minimp-ax2 1627 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1628 | Derivation of ~ pm2.43 (al... |
| impsingle 1629 | The shortest single axiom ... |
| impsingle-step4 1630 | Derivation of impsingle-st... |
| impsingle-step8 1631 | Derivation of impsingle-st... |
| impsingle-ax1 1632 | Derivation of impsingle-ax... |
| impsingle-step15 1633 | Derivation of impsingle-st... |
| impsingle-step18 1634 | Derivation of impsingle-st... |
| impsingle-step19 1635 | Derivation of impsingle-st... |
| impsingle-step20 1636 | Derivation of impsingle-st... |
| impsingle-step21 1637 | Derivation of impsingle-st... |
| impsingle-step22 1638 | Derivation of impsingle-st... |
| impsingle-step25 1639 | Derivation of impsingle-st... |
| impsingle-imim1 1640 | Derivation of impsingle-im... |
| impsingle-peirce 1641 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1642 | Derivation of ~ ax-2 from ... |
| meredith 1643 | Carew Meredith's sole axio... |
| merlem1 1644 | Step 3 of Meredith's proof... |
| merlem2 1645 | Step 4 of Meredith's proof... |
| merlem3 1646 | Step 7 of Meredith's proof... |
| merlem4 1647 | Step 8 of Meredith's proof... |
| merlem5 1648 | Step 11 of Meredith's proo... |
| merlem6 1649 | Step 12 of Meredith's proo... |
| merlem7 1650 | Between steps 14 and 15 of... |
| merlem8 1651 | Step 15 of Meredith's proo... |
| merlem9 1652 | Step 18 of Meredith's proo... |
| merlem10 1653 | Step 19 of Meredith's proo... |
| merlem11 1654 | Step 20 of Meredith's proo... |
| merlem12 1655 | Step 28 of Meredith's proo... |
| merlem13 1656 | Step 35 of Meredith's proo... |
| luk-1 1657 | 1 of 3 axioms for proposit... |
| luk-2 1658 | 2 of 3 axioms for proposit... |
| luk-3 1659 | 3 of 3 axioms for proposit... |
| luklem1 1660 | Used to rederive standard ... |
| luklem2 1661 | Used to rederive standard ... |
| luklem3 1662 | Used to rederive standard ... |
| luklem4 1663 | Used to rederive standard ... |
| luklem5 1664 | Used to rederive standard ... |
| luklem6 1665 | Used to rederive standard ... |
| luklem7 1666 | Used to rederive standard ... |
| luklem8 1667 | Used to rederive standard ... |
| ax1 1668 | Standard propositional axi... |
| ax2 1669 | Standard propositional axi... |
| ax3 1670 | Standard propositional axi... |
| nic-dfim 1671 | This theorem "defines" imp... |
| nic-dfneg 1672 | This theorem "defines" neg... |
| nic-mp 1673 | Derive Nicod's rule of mod... |
| nic-mpALT 1674 | A direct proof of ~ nic-mp... |
| nic-ax 1675 | Nicod's axiom derived from... |
| nic-axALT 1676 | A direct proof of ~ nic-ax... |
| nic-imp 1677 | Inference for ~ nic-mp usi... |
| nic-idlem1 1678 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1679 | Lemma for ~ nic-id . Infe... |
| nic-id 1680 | Theorem ~ id expressed wit... |
| nic-swap 1681 | The connector ` -/\ ` is s... |
| nic-isw1 1682 | Inference version of ~ nic... |
| nic-isw2 1683 | Inference for swapping nes... |
| nic-iimp1 1684 | Inference version of ~ nic... |
| nic-iimp2 1685 | Inference version of ~ nic... |
| nic-idel 1686 | Inference to remove the tr... |
| nic-ich 1687 | Chained inference. (Contr... |
| nic-idbl 1688 | Double the terms. Since d... |
| nic-bijust 1689 | Biconditional justificatio... |
| nic-bi1 1690 | Inference to extract one s... |
| nic-bi2 1691 | Inference to extract the o... |
| nic-stdmp 1692 | Derive the standard modus ... |
| nic-luk1 1693 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1694 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1695 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1696 | This alternative axiom for... |
| lukshefth1 1697 | Lemma for ~ renicax . (Co... |
| lukshefth2 1698 | Lemma for ~ renicax . (Co... |
| renicax 1699 | A rederivation of ~ nic-ax... |
| tbw-bijust 1700 | Justification for ~ tbw-ne... |
| tbw-negdf 1701 | The definition of negation... |
| tbw-ax1 1702 | The first of four axioms i... |
| tbw-ax2 1703 | The second of four axioms ... |
| tbw-ax3 1704 | The third of four axioms i... |
| tbw-ax4 1705 | The fourth of four axioms ... |
| tbwsyl 1706 | Used to rederive the Lukas... |
| tbwlem1 1707 | Used to rederive the Lukas... |
| tbwlem2 1708 | Used to rederive the Lukas... |
| tbwlem3 1709 | Used to rederive the Lukas... |
| tbwlem4 1710 | Used to rederive the Lukas... |
| tbwlem5 1711 | Used to rederive the Lukas... |
| re1luk1 1712 | ~ luk-1 derived from the T... |
| re1luk2 1713 | ~ luk-2 derived from the T... |
| re1luk3 1714 | ~ luk-3 derived from the T... |
| merco1 1715 | A single axiom for proposi... |
| merco1lem1 1716 | Used to rederive the Tarsk... |
| retbwax4 1717 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1718 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1719 | Used to rederive the Tarsk... |
| merco1lem3 1720 | Used to rederive the Tarsk... |
| merco1lem4 1721 | Used to rederive the Tarsk... |
| merco1lem5 1722 | Used to rederive the Tarsk... |
| merco1lem6 1723 | Used to rederive the Tarsk... |
| merco1lem7 1724 | Used to rederive the Tarsk... |
| retbwax3 1725 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1726 | Used to rederive the Tarsk... |
| merco1lem9 1727 | Used to rederive the Tarsk... |
| merco1lem10 1728 | Used to rederive the Tarsk... |
| merco1lem11 1729 | Used to rederive the Tarsk... |
| merco1lem12 1730 | Used to rederive the Tarsk... |
| merco1lem13 1731 | Used to rederive the Tarsk... |
| merco1lem14 1732 | Used to rederive the Tarsk... |
| merco1lem15 1733 | Used to rederive the Tarsk... |
| merco1lem16 1734 | Used to rederive the Tarsk... |
| merco1lem17 1735 | Used to rederive the Tarsk... |
| merco1lem18 1736 | Used to rederive the Tarsk... |
| retbwax1 1737 | ~ tbw-ax1 rederived from ~... |
| merco2 1738 | A single axiom for proposi... |
| mercolem1 1739 | Used to rederive the Tarsk... |
| mercolem2 1740 | Used to rederive the Tarsk... |
| mercolem3 1741 | Used to rederive the Tarsk... |
| mercolem4 1742 | Used to rederive the Tarsk... |
| mercolem5 1743 | Used to rederive the Tarsk... |
| mercolem6 1744 | Used to rederive the Tarsk... |
| mercolem7 1745 | Used to rederive the Tarsk... |
| mercolem8 1746 | Used to rederive the Tarsk... |
| re1tbw1 1747 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1748 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1749 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1750 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1751 | Justification for ~ rb-imd... |
| rb-imdf 1752 | The definition of implicat... |
| anmp 1753 | Modus ponens for ` { \/ , ... |
| rb-ax1 1754 | The first of four axioms i... |
| rb-ax2 1755 | The second of four axioms ... |
| rb-ax3 1756 | The third of four axioms i... |
| rb-ax4 1757 | The fourth of four axioms ... |
| rbsyl 1758 | Used to rederive the Lukas... |
| rblem1 1759 | Used to rederive the Lukas... |
| rblem2 1760 | Used to rederive the Lukas... |
| rblem3 1761 | Used to rederive the Lukas... |
| rblem4 1762 | Used to rederive the Lukas... |
| rblem5 1763 | Used to rederive the Lukas... |
| rblem6 1764 | Used to rederive the Lukas... |
| rblem7 1765 | Used to rederive the Lukas... |
| re1axmp 1766 | ~ ax-mp derived from Russe... |
| re2luk1 1767 | ~ luk-1 derived from Russe... |
| re2luk2 1768 | ~ luk-2 derived from Russe... |
| re2luk3 1769 | ~ luk-3 derived from Russe... |
| mptnan 1770 | Modus ponendo tollens 1, o... |
| mptxor 1771 | Modus ponendo tollens 2, o... |
| mtpor 1772 | Modus tollendo ponens (inc... |
| mtpxor 1773 | Modus tollendo ponens (ori... |
| stoic1a 1774 | Stoic logic Thema 1 (part ... |
| stoic1b 1775 | Stoic logic Thema 1 (part ... |
| stoic2a 1776 | Stoic logic Thema 2 versio... |
| stoic2b 1777 | Stoic logic Thema 2 versio... |
| stoic3 1778 | Stoic logic Thema 3. Stat... |
| stoic4a 1779 | Stoic logic Thema 4 versio... |
| stoic4b 1780 | Stoic logic Thema 4 versio... |
| alnex 1783 | Universal quantification o... |
| eximal 1784 | An equivalence between an ... |
| nf2 1787 | Alternate definition of no... |
| nf3 1788 | Alternate definition of no... |
| nf4 1789 | Alternate definition of no... |
| nfi 1790 | Deduce that ` x ` is not f... |
| nfri 1791 | Consequence of the definit... |
| nfd 1792 | Deduce that ` x ` is not f... |
| nfrd 1793 | Consequence of the definit... |
| nftht 1794 | Closed form of ~ nfth . (... |
| nfntht 1795 | Closed form of ~ nfnth . ... |
| nfntht2 1796 | Closed form of ~ nfnth . ... |
| gen2 1798 | Generalization applied twi... |
| mpg 1799 | Modus ponens combined with... |
| mpgbi 1800 | Modus ponens on biconditio... |
| mpgbir 1801 | Modus ponens on biconditio... |
| nex 1802 | Generalization rule for ne... |
| nfth 1803 | No variable is (effectivel... |
| nfnth 1804 | No variable is (effectivel... |
| hbth 1805 | No variable is (effectivel... |
| nftru 1806 | The true constant has no f... |
| nffal 1807 | The false constant has no ... |
| sptruw 1808 | Version of ~ sp when ` ph ... |
| altru 1809 | For all sets, ` T. ` is tr... |
| alfal 1810 | For all sets, ` -. F. ` is... |
| alim 1812 | Restatement of Axiom ~ ax-... |
| alimi 1813 | Inference quantifying both... |
| 2alimi 1814 | Inference doubly quantifyi... |
| ala1 1815 | Add an antecedent in a uni... |
| al2im 1816 | Closed form of ~ al2imi . ... |
| al2imi 1817 | Inference quantifying ante... |
| alanimi 1818 | Variant of ~ al2imi with c... |
| alimdh 1819 | Deduction form of Theorem ... |
| albi 1820 | Theorem 19.15 of [Margaris... |
| albii 1821 | Inference adding universal... |
| 2albii 1822 | Inference adding two unive... |
| 3albii 1823 | Inference adding three uni... |
| sylgt 1824 | Closed form of ~ sylg . (... |
| sylg 1825 | A syllogism combined with ... |
| alrimih 1826 | Inference form of Theorem ... |
| hbxfrbi 1827 | A utility lemma to transfe... |
| alex 1828 | Universal quantifier in te... |
| exnal 1829 | Existential quantification... |
| 2nalexn 1830 | Part of theorem *11.5 in [... |
| 2exnaln 1831 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1832 | Theorem *11.25 in [Whitehe... |
| alimex 1833 | An equivalence between an ... |
| aleximi 1834 | A variant of ~ al2imi : in... |
| alexbii 1835 | Biconditional form of ~ al... |
| exim 1836 | Theorem 19.22 of [Margaris... |
| eximi 1837 | Inference adding existenti... |
| 2eximi 1838 | Inference adding two exist... |
| eximii 1839 | Inference associated with ... |
| exa1 1840 | Add an antecedent in an ex... |
| 19.38 1841 | Theorem 19.38 of [Margaris... |
| 19.38a 1842 | Under a nonfreeness hypoth... |
| 19.38b 1843 | Under a nonfreeness hypoth... |
| imnang 1844 | Quantified implication in ... |
| alinexa 1845 | A transformation of quanti... |
| exnalimn 1846 | Existential quantification... |
| alexn 1847 | A relationship between two... |
| 2exnexn 1848 | Theorem *11.51 in [Whitehe... |
| exbi 1849 | Theorem 19.18 of [Margaris... |
| exbii 1850 | Inference adding existenti... |
| 2exbii 1851 | Inference adding two exist... |
| 3exbii 1852 | Inference adding three exi... |
| nfbiit 1853 | Equivalence theorem for th... |
| nfbii 1854 | Equality theorem for the n... |
| nfxfr 1855 | A utility lemma to transfe... |
| nfxfrd 1856 | A utility lemma to transfe... |
| nfnbi 1857 | A variable is nonfree in a... |
| nfnt 1858 | If a variable is nonfree i... |
| nfn 1859 | Inference associated with ... |
| nfnd 1860 | Deduction associated with ... |
| exanali 1861 | A transformation of quanti... |
| 2exanali 1862 | Theorem *11.521 in [Whiteh... |
| exancom 1863 | Commutation of conjunction... |
| exan 1864 | Place a conjunct in the sc... |
| alrimdh 1865 | Deduction form of Theorem ... |
| eximdh 1866 | Deduction from Theorem 19.... |
| nexdh 1867 | Deduction for generalizati... |
| albidh 1868 | Formula-building rule for ... |
| exbidh 1869 | Formula-building rule for ... |
| exsimpl 1870 | Simplification of an exist... |
| exsimpr 1871 | Simplification of an exist... |
| 19.26 1872 | Theorem 19.26 of [Margaris... |
| 19.26-2 1873 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1874 | Theorem ~ 19.26 with tripl... |
| 19.29 1875 | Theorem 19.29 of [Margaris... |
| 19.29r 1876 | Variation of ~ 19.29 . (C... |
| 19.29r2 1877 | Variation of ~ 19.29r with... |
| 19.29x 1878 | Variation of ~ 19.29 with ... |
| 19.35 1879 | Theorem 19.35 of [Margaris... |
| 19.35i 1880 | Inference associated with ... |
| 19.35ri 1881 | Inference associated with ... |
| 19.25 1882 | Theorem 19.25 of [Margaris... |
| 19.30 1883 | Theorem 19.30 of [Margaris... |
| 19.43 1884 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1885 | Obsolete proof of ~ 19.43 ... |
| 19.33 1886 | Theorem 19.33 of [Margaris... |
| 19.33b 1887 | The antecedent provides a ... |
| 19.40 1888 | Theorem 19.40 of [Margaris... |
| 19.40-2 1889 | Theorem *11.42 in [Whitehe... |
| 19.40b 1890 | The antecedent provides a ... |
| albiim 1891 | Split a biconditional and ... |
| 2albiim 1892 | Split a biconditional and ... |
| exintrbi 1893 | Add/remove a conjunct in t... |
| exintr 1894 | Introduce a conjunct in th... |
| alsyl 1895 | Universally quantified and... |
| nfimd 1896 | If in a context ` x ` is n... |
| nfimt 1897 | Closed form of ~ nfim and ... |
| nfim 1898 | If ` x ` is not free in ` ... |
| nfand 1899 | If in a context ` x ` is n... |
| nf3and 1900 | Deduction form of bound-va... |
| nfan 1901 | If ` x ` is not free in ` ... |
| nfnan 1902 | If ` x ` is not free in ` ... |
| nf3an 1903 | If ` x ` is not free in ` ... |
| nfbid 1904 | If in a context ` x ` is n... |
| nfbi 1905 | If ` x ` is not free in ` ... |
| nfor 1906 | If ` x ` is not free in ` ... |
| nf3or 1907 | If ` x ` is not free in ` ... |
| empty 1908 | Two characterizations of t... |
| emptyex 1909 | On the empty domain, any e... |
| emptyal 1910 | On the empty domain, any u... |
| emptynf 1911 | On the empty domain, any v... |
| ax5d 1913 | Version of ~ ax-5 with ant... |
| ax5e 1914 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1915 | If a formula holds for som... |
| nfv 1916 | If ` x ` is not present in... |
| nfvd 1917 | ~ nfv with antecedent. Us... |
| alimdv 1918 | Deduction form of Theorem ... |
| eximdv 1919 | Deduction form of Theorem ... |
| 2alimdv 1920 | Deduction form of Theorem ... |
| 2eximdv 1921 | Deduction form of Theorem ... |
| albidv 1922 | Formula-building rule for ... |
| exbidv 1923 | Formula-building rule for ... |
| nfbidv 1924 | An equality theorem for no... |
| 2albidv 1925 | Formula-building rule for ... |
| 2exbidv 1926 | Formula-building rule for ... |
| 3exbidv 1927 | Formula-building rule for ... |
| 4exbidv 1928 | Formula-building rule for ... |
| alrimiv 1929 | Inference form of Theorem ... |
| alrimivv 1930 | Inference form of Theorem ... |
| alrimdv 1931 | Deduction form of Theorem ... |
| exlimiv 1932 | Inference form of Theorem ... |
| exlimiiv 1933 | Inference (Rule C) associa... |
| exlimivv 1934 | Inference form of Theorem ... |
| exlimdv 1935 | Deduction form of Theorem ... |
| exlimdvv 1936 | Deduction form of Theorem ... |
| exlimddv 1937 | Existential elimination ru... |
| nexdv 1938 | Deduction for generalizati... |
| 2ax5 1939 | Quantification of two vari... |
| stdpc5v 1940 | Version of ~ stdpc5 with a... |
| 19.21v 1941 | Version of ~ 19.21 with a ... |
| 19.32v 1942 | Version of ~ 19.32 with a ... |
| 19.31v 1943 | Version of ~ 19.31 with a ... |
| 19.23v 1944 | Version of ~ 19.23 with a ... |
| 19.23vv 1945 | Theorem ~ 19.23v extended ... |
| pm11.53v 1946 | Version of ~ pm11.53 with ... |
| 19.36imv 1947 | One direction of ~ 19.36v ... |
| 19.36iv 1948 | Inference associated with ... |
| 19.37imv 1949 | One direction of ~ 19.37v ... |
| 19.37iv 1950 | Inference associated with ... |
| 19.41v 1951 | Version of ~ 19.41 with a ... |
| 19.41vv 1952 | Version of ~ 19.41 with tw... |
| 19.41vvv 1953 | Version of ~ 19.41 with th... |
| 19.41vvvv 1954 | Version of ~ 19.41 with fo... |
| 19.42v 1955 | Version of ~ 19.42 with a ... |
| exdistr 1956 | Distribution of existentia... |
| exdistrv 1957 | Distribute a pair of exist... |
| 4exdistrv 1958 | Distribute two pairs of ex... |
| 19.42vv 1959 | Version of ~ 19.42 with tw... |
| exdistr2 1960 | Distribution of existentia... |
| 19.42vvv 1961 | Version of ~ 19.42 with th... |
| 3exdistr 1962 | Distribution of existentia... |
| 4exdistr 1963 | Distribution of existentia... |
| weq 1964 | Extend wff definition to i... |
| speimfw 1965 | Specialization, with addit... |
| speimfwALT 1966 | Alternate proof of ~ speim... |
| spimfw 1967 | Specialization, with addit... |
| ax12i 1968 | Inference that has ~ ax-12... |
| ax6v 1970 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1971 | At least one individual ex... |
| spimw 1972 | Specialization. Lemma 8 o... |
| spimew 1973 | Existential introduction, ... |
| speiv 1974 | Inference from existential... |
| speivw 1975 | Version of ~ spei with a d... |
| exgen 1976 | Rule of existential genera... |
| extru 1977 | There exists a variable su... |
| 19.2 1978 | Theorem 19.2 of [Margaris]... |
| 19.2d 1979 | Deduction associated with ... |
| 19.8w 1980 | Weak version of ~ 19.8a an... |
| spnfw 1981 | Weak version of ~ sp . Us... |
| spfalw 1982 | Version of ~ sp when ` ph ... |
| spvw 1983 | Version of ~ sp when ` x `... |
| 19.3v 1984 | Version of ~ 19.3 with a d... |
| 19.8v 1985 | Version of ~ 19.8a with a ... |
| 19.9v 1986 | Version of ~ 19.9 with a d... |
| spimevw 1987 | Existential introduction, ... |
| spimvw 1988 | A weak form of specializat... |
| spsv 1989 | Generalization of antecede... |
| spvv 1990 | Specialization, using impl... |
| chvarvv 1991 | Implicit substitution of `... |
| 19.39 1992 | Theorem 19.39 of [Margaris... |
| 19.24 1993 | Theorem 19.24 of [Margaris... |
| 19.34 1994 | Theorem 19.34 of [Margaris... |
| 19.36v 1995 | Version of ~ 19.36 with a ... |
| 19.12vvv 1996 | Version of ~ 19.12vv with ... |
| 19.27v 1997 | Version of ~ 19.27 with a ... |
| 19.28v 1998 | Version of ~ 19.28 with a ... |
| 19.37v 1999 | Version of ~ 19.37 with a ... |
| 19.44v 2000 | Version of ~ 19.44 with a ... |
| 19.45v 2001 | Version of ~ 19.45 with a ... |
| equs4v 2002 | Version of ~ equs4 with a ... |
| alequexv 2003 | Version of ~ equs4v with i... |
| exsbim 2004 | One direction of the equiv... |
| equsv 2005 | If a formula does not cont... |
| equsalvw 2006 | Version of ~ equsalv with ... |
| equsexvw 2007 | Version of ~ equsexv with ... |
| cbvaliw 2008 | Change bound variable. Us... |
| cbvalivw 2009 | Change bound variable. Us... |
| ax7v 2011 | Weakened version of ~ ax-7... |
| ax7v1 2012 | First of two weakened vers... |
| ax7v2 2013 | Second of two weakened ver... |
| equid 2014 | Identity law for equality.... |
| nfequid 2015 | Bound-variable hypothesis ... |
| equcomiv 2016 | Weaker form of ~ equcomi w... |
| ax6evr 2017 | A commuted form of ~ ax6ev... |
| ax7 2018 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2019 | Commutative law for equali... |
| equcom 2020 | Commutative law for equali... |
| equcomd 2021 | Deduction form of ~ equcom... |
| equcoms 2022 | An inference commuting equ... |
| equtr 2023 | A transitive law for equal... |
| equtrr 2024 | A transitive law for equal... |
| equeuclr 2025 | Commuted version of ~ eque... |
| equeucl 2026 | Equality is a left-Euclide... |
| equequ1 2027 | An equivalence law for equ... |
| equequ2 2028 | An equivalence law for equ... |
| equtr2 2029 | Equality is a left-Euclide... |
| stdpc6 2030 | One of the two equality ax... |
| equvinv 2031 | A variable introduction la... |
| equvinva 2032 | A modified version of the ... |
| equvelv 2033 | A biconditional form of ~ ... |
| ax13b 2034 | An equivalence between two... |
| spfw 2035 | Weak version of ~ sp . Us... |
| spw 2036 | Weak version of the specia... |
| cbvalw 2037 | Change bound variable. Us... |
| cbvalvw 2038 | Change bound variable. Us... |
| cbvexvw 2039 | Change bound variable. Us... |
| cbvaldvaw 2040 | Rule used to change the bo... |
| cbvexdvaw 2041 | Rule used to change the bo... |
| cbval2vw 2042 | Rule used to change bound ... |
| cbvex2vw 2043 | Rule used to change bound ... |
| cbvex4vw 2044 | Rule used to change bound ... |
| alcomimw 2045 | Weak version of ~ ax-11 . ... |
| excomimw 2046 | Weak version of ~ excomim ... |
| alcomw 2047 | Weak version of ~ alcom an... |
| excomw 2048 | Weak version of ~ excom an... |
| hbn1fw 2049 | Weak version of ~ ax-10 fr... |
| hbn1w 2050 | Weak version of ~ hbn1 . ... |
| hba1w 2051 | Weak version of ~ hba1 . ... |
| hbe1w 2052 | Weak version of ~ hbe1 . ... |
| hbalw 2053 | Weak version of ~ hbal . ... |
| 19.8aw 2054 | If a formula is true, then... |
| exexw 2055 | Existential quantification... |
| spaev 2056 | A special instance of ~ sp... |
| cbvaev 2057 | Change bound variable in a... |
| aevlem0 2058 | Lemma for ~ aevlem . Inst... |
| aevlem 2059 | Lemma for ~ aev and ~ axc1... |
| aeveq 2060 | The antecedent ` A. x x = ... |
| aev 2061 | A "distinctor elimination"... |
| aev2 2062 | A version of ~ aev with tw... |
| hbaev 2063 | All variables are effectiv... |
| naev 2064 | If some set variables can ... |
| naev2 2065 | Generalization of ~ hbnaev... |
| hbnaev 2066 | Any variable is free in ` ... |
| sbjust 2067 | Justification theorem for ... |
| dfsb 2070 | Simplify definition ~ df-s... |
| sbtlem 2071 | In the case of ~ sbt , the... |
| sbt 2072 | A substitution into a theo... |
| sbtru 2073 | The result of substituting... |
| stdpc4 2074 | The specialization axiom o... |
| sbtALT 2075 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2076 | A double specialization us... |
| sbi1 2077 | Distribute substitution ov... |
| spsbim 2078 | Distribute substitution ov... |
| spsbbi 2079 | Biconditional property for... |
| sbimi 2080 | Distribute substitution ov... |
| sb2imi 2081 | Distribute substitution ov... |
| sbbii 2082 | Infer substitution into bo... |
| 2sbbii 2083 | Infer double substitution ... |
| sbimdv 2084 | Deduction substituting bot... |
| sbbidv 2085 | Deduction substituting bot... |
| sban 2086 | Conjunction inside and out... |
| sb3an 2087 | Threefold conjunction insi... |
| spsbe 2088 | Existential generalization... |
| sbequ 2089 | Equality property for subs... |
| sbequi 2090 | An equality theorem for su... |
| sb6 2091 | Alternate definition of su... |
| 2sb6 2092 | Equivalence for double sub... |
| sb1v 2093 | One direction of ~ sb5 , p... |
| sbv 2094 | Substitution for a variabl... |
| sbcom4 2095 | Commutativity law for subs... |
| pm11.07 2096 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2097 | Substitution in an implica... |
| sbbiiev 2098 | An equivalence of substitu... |
| sbievw 2099 | Conversion of implicit sub... |
| sbievwOLD 2100 | Obsolete version of ~ sbie... |
| sbiedvw 2101 | Conversion of implicit sub... |
| 2sbievw 2102 | Conversion of double impli... |
| sbcom3vv 2103 | Substituting ` y ` for ` x... |
| sbievw2 2104 | ~ sbievw applied twice, av... |
| sbco2vv 2105 | A composition law for subs... |
| cbvsbv 2106 | Change the bound variable ... |
| sbco4lem 2107 | Lemma for ~ sbco4 . It re... |
| sbco4 2108 | Two ways of exchanging two... |
| equsb3 2109 | Substitution in an equalit... |
| equsb3r 2110 | Substitution applied to th... |
| equsb1v 2111 | Substitution applied to an... |
| nsb 2112 | Any substitution in an alw... |
| sbn1 2113 | One direction of ~ sbn , u... |
| wel 2115 | Extend wff definition to i... |
| ax8v 2117 | Weakened version of ~ ax-8... |
| ax8v1 2118 | First of two weakened vers... |
| ax8v2 2119 | Second of two weakened ver... |
| ax8 2120 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2121 | An identity law for the no... |
| elsb1 2122 | Substitution for the first... |
| cleljust 2123 | When the class variables i... |
| ax9v 2125 | Weakened version of ~ ax-9... |
| ax9v1 2126 | First of two weakened vers... |
| ax9v2 2127 | Second of two weakened ver... |
| ax9 2128 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2129 | An identity law for the no... |
| elequ2g 2130 | A form of ~ elequ2 with a ... |
| elsb2 2131 | Substitution for the secon... |
| elequ12 2132 | An identity law for the no... |
| ru0 2133 | The FOL statement used in ... |
| ax6dgen 2134 | Tarski's system uses the w... |
| ax10w 2135 | Weak version of ~ ax-10 fr... |
| ax11w 2136 | Weak version of ~ ax-11 fr... |
| ax11dgen 2137 | Degenerate instance of ~ a... |
| ax12wlem 2138 | Lemma for weak version of ... |
| ax12w 2139 | Weak version of ~ ax-12 fr... |
| ax12dgen 2140 | Degenerate instance of ~ a... |
| ax12wdemo 2141 | Example of an application ... |
| ax13w 2142 | Weak version (principal in... |
| ax13dgen1 2143 | Degenerate instance of ~ a... |
| ax13dgen2 2144 | Degenerate instance of ~ a... |
| ax13dgen3 2145 | Degenerate instance of ~ a... |
| ax13dgen4 2146 | Degenerate instance of ~ a... |
| hbn1 2148 | Alias for ~ ax-10 to be us... |
| hbe1 2149 | The setvar ` x ` is not fr... |
| hbe1a 2150 | Dual statement of ~ hbe1 .... |
| nf5-1 2151 | One direction of ~ nf5 can... |
| nf5i 2152 | Deduce that ` x ` is not f... |
| nf5dh 2153 | Deduce that ` x ` is not f... |
| nf5dv 2154 | Apply the definition of no... |
| nfnaew 2155 | All variables are effectiv... |
| nfe1 2156 | The setvar ` x ` is not fr... |
| nfa1 2157 | The setvar ` x ` is not fr... |
| nfna1 2158 | A convenience theorem part... |
| nfia1 2159 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2160 | The setvar ` x ` is not fr... |
| modal5 2161 | The analogue in our predic... |
| nfs1v 2162 | The setvar ` x ` is not fr... |
| alcoms 2164 | Swap quantifiers in an ant... |
| alcom 2165 | Theorem 19.5 of [Margaris]... |
| alrot3 2166 | Theorem *11.21 in [Whitehe... |
| alrot4 2167 | Rotate four universal quan... |
| excom 2168 | Theorem 19.11 of [Margaris... |
| excomim 2169 | One direction of Theorem 1... |
| excom13 2170 | Swap 1st and 3rd existenti... |
| exrot3 2171 | Rotate existential quantif... |
| exrot4 2172 | Rotate existential quantif... |
| hbal 2173 | If ` x ` is not free in ` ... |
| hbald 2174 | Deduction form of bound-va... |
| sbal 2175 | Move universal quantifier ... |
| sbalv 2176 | Quantify with new variable... |
| hbsbw 2177 | If ` z ` is not free in ` ... |
| hbsbwOLD 2178 | Obsolete version of ~ hbsb... |
| sbcom2 2179 | Commutativity law for subs... |
| sbco4lemOLD 2180 | Obsolete version of ~ sbco... |
| sbco4OLD 2181 | Obsolete version of ~ sbco... |
| nfa2 2182 | Lemma 24 of [Monk2] p. 114... |
| nfexhe 2183 | Version of ~ nfex with the... |
| nfexa2 2184 | An inner universal quantif... |
| ax12v 2186 | This is essentially Axiom ... |
| ax12v2 2187 | It is possible to remove a... |
| ax12ev2 2188 | Version of ~ ax12v2 rewrit... |
| 19.8a 2189 | If a wff is true, it is tr... |
| 19.8ad 2190 | If a wff is true, it is tr... |
| sp 2191 | Specialization. A univers... |
| spi 2192 | Inference rule of universa... |
| sps 2193 | Generalization of antecede... |
| 2sp 2194 | A double specialization (s... |
| spsd 2195 | Deduction generalizing ant... |
| 19.2g 2196 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2197 | Inference form of ~ 19.21 ... |
| 19.21bbi 2198 | Inference removing two uni... |
| 19.23bi 2199 | Inference form of Theorem ... |
| nexr 2200 | Inference associated with ... |
| qexmid 2201 | Quantified excluded middle... |
| nf5r 2202 | Consequence of the definit... |
| nf5ri 2203 | Consequence of the definit... |
| nf5rd 2204 | Consequence of the definit... |
| spimedv 2205 | Deduction version of ~ spi... |
| spimefv 2206 | Version of ~ spime with a ... |
| nfim1 2207 | A closed form of ~ nfim . ... |
| nfan1 2208 | A closed form of ~ nfan . ... |
| 19.3t 2209 | Closed form of ~ 19.3 and ... |
| 19.3 2210 | A wff may be quantified wi... |
| 19.9d 2211 | A deduction version of one... |
| 19.9t 2212 | Closed form of ~ 19.9 and ... |
| 19.9 2213 | A wff may be existentially... |
| 19.21t 2214 | Closed form of Theorem 19.... |
| 19.21 2215 | Theorem 19.21 of [Margaris... |
| stdpc5 2216 | An axiom scheme of standar... |
| 19.21-2 2217 | Version of ~ 19.21 with tw... |
| 19.23t 2218 | Closed form of Theorem 19.... |
| 19.23 2219 | Theorem 19.23 of [Margaris... |
| alimd 2220 | Deduction form of Theorem ... |
| alrimi 2221 | Inference form of Theorem ... |
| alrimdd 2222 | Deduction form of Theorem ... |
| alrimd 2223 | Deduction form of Theorem ... |
| eximd 2224 | Deduction form of Theorem ... |
| exlimi 2225 | Inference associated with ... |
| exlimd 2226 | Deduction form of Theorem ... |
| exlimimdd 2227 | Existential elimination ru... |
| exlimdd 2228 | Existential elimination ru... |
| nexd 2229 | Deduction for generalizati... |
| albid 2230 | Formula-building rule for ... |
| exbid 2231 | Formula-building rule for ... |
| nfbidf 2232 | An equality theorem for ef... |
| 19.16 2233 | Theorem 19.16 of [Margaris... |
| 19.17 2234 | Theorem 19.17 of [Margaris... |
| 19.27 2235 | Theorem 19.27 of [Margaris... |
| 19.28 2236 | Theorem 19.28 of [Margaris... |
| 19.19 2237 | Theorem 19.19 of [Margaris... |
| 19.36 2238 | Theorem 19.36 of [Margaris... |
| 19.36i 2239 | Inference associated with ... |
| 19.37 2240 | Theorem 19.37 of [Margaris... |
| 19.32 2241 | Theorem 19.32 of [Margaris... |
| 19.31 2242 | Theorem 19.31 of [Margaris... |
| 19.41 2243 | Theorem 19.41 of [Margaris... |
| 19.42 2244 | Theorem 19.42 of [Margaris... |
| 19.44 2245 | Theorem 19.44 of [Margaris... |
| 19.45 2246 | Theorem 19.45 of [Margaris... |
| spimfv 2247 | Specialization, using impl... |
| chvarfv 2248 | Implicit substitution of `... |
| cbv3v2 2249 | Version of ~ cbv3 with two... |
| sbalex 2250 | Equivalence of two ways to... |
| sbalexOLD 2251 | Obsolete version of ~ sbal... |
| sb4av 2252 | Version of ~ sb4a with a d... |
| sbimd 2253 | Deduction substituting bot... |
| sbbid 2254 | Deduction substituting bot... |
| 2sbbid 2255 | Deduction doubly substitut... |
| sbequ1 2256 | An equality theorem for su... |
| sbequ2 2257 | An equality theorem for su... |
| stdpc7 2258 | One of the two equality ax... |
| sbequ12 2259 | An equality theorem for su... |
| sbequ12r 2260 | An equality theorem for su... |
| sbelx 2261 | Elimination of substitutio... |
| sbequ12a 2262 | An equality theorem for su... |
| sbid 2263 | An identity theorem for su... |
| sbcov 2264 | A composition law for subs... |
| sbcovOLD 2265 | Obsolete version of ~ sbco... |
| sb6a 2266 | Equivalence for substituti... |
| sbid2vw 2267 | Reverting substitution yie... |
| axc16g 2268 | Generalization of ~ axc16 ... |
| axc16 2269 | Proof of older axiom ~ ax-... |
| axc16gb 2270 | Biconditional strengthenin... |
| axc16nf 2271 | If ~ dtru is false, then t... |
| axc11v 2272 | Version of ~ axc11 with a ... |
| axc11rv 2273 | Version of ~ axc11r with a... |
| drsb2 2274 | Formula-building lemma for... |
| equsalv 2275 | An equivalence related to ... |
| equsexv 2276 | An equivalence related to ... |
| sbft 2277 | Substitution has no effect... |
| sbf 2278 | Substitution for a variabl... |
| sbf2 2279 | Substitution has no effect... |
| sbh 2280 | Substitution for a variabl... |
| hbs1 2281 | The setvar ` x ` is not fr... |
| nfs1f 2282 | If ` x ` is not free in ` ... |
| sb5 2283 | Alternate definition of su... |
| equs5av 2284 | A property related to subs... |
| 2sb5 2285 | Equivalence for double sub... |
| dfsb7 2286 | An alternate definition of... |
| sbn 2287 | Negation inside and outsid... |
| sbex 2288 | Move existential quantifie... |
| nf5 2289 | Alternate definition of ~ ... |
| nf6 2290 | An alternate definition of... |
| nf5d 2291 | Deduce that ` x ` is not f... |
| nf5di 2292 | Since the converse holds b... |
| 19.9h 2293 | A wff may be existentially... |
| 19.21h 2294 | Theorem 19.21 of [Margaris... |
| 19.23h 2295 | Theorem 19.23 of [Margaris... |
| exlimih 2296 | Inference associated with ... |
| exlimdh 2297 | Deduction form of Theorem ... |
| equsalhw 2298 | Version of ~ equsalh with ... |
| equsexhv 2299 | An equivalence related to ... |
| hba1 2300 | The setvar ` x ` is not fr... |
| hbnt 2301 | Closed theorem version of ... |
| hbn 2302 | If ` x ` is not free in ` ... |
| hbnd 2303 | Deduction form of bound-va... |
| hbim1 2304 | A closed form of ~ hbim . ... |
| hbimd 2305 | Deduction form of bound-va... |
| hbim 2306 | If ` x ` is not free in ` ... |
| hban 2307 | If ` x ` is not free in ` ... |
| hb3an 2308 | If ` x ` is not free in ` ... |
| sbi2 2309 | Introduction of implicatio... |
| sbim 2310 | Implication inside and out... |
| sbrim 2311 | Substitution in an implica... |
| sblim 2312 | Substitution in an implica... |
| sbor 2313 | Disjunction inside and out... |
| sbbi 2314 | Equivalence inside and out... |
| sblbis 2315 | Introduce left bicondition... |
| sbrbis 2316 | Introduce right biconditio... |
| sbrbif 2317 | Introduce right biconditio... |
| sbnf 2318 | Move nonfree predicate in ... |
| sbnfOLD 2319 | Obsolete version of ~ sbnf... |
| sbiev 2320 | Conversion of implicit sub... |
| sbievOLD 2321 | Obsolete version of ~ sbie... |
| sbiedw 2322 | Conversion of implicit sub... |
| axc7 2323 | Show that the original axi... |
| axc7e 2324 | Abbreviated version of ~ a... |
| modal-b 2325 | The analogue in our predic... |
| 19.9ht 2326 | A closed version of ~ 19.9... |
| axc4 2327 | Show that the original axi... |
| axc4i 2328 | Inference version of ~ axc... |
| nfal 2329 | If ` x ` is not free in ` ... |
| nfex 2330 | If ` x ` is not free in ` ... |
| hbex 2331 | If ` x ` is not free in ` ... |
| nfnf 2332 | If ` x ` is not free in ` ... |
| 19.12 2333 | Theorem 19.12 of [Margaris... |
| nfald 2334 | Deduction form of ~ nfal .... |
| nfexd 2335 | If ` x ` is not free in ` ... |
| nfsbv 2336 | If ` z ` is not free in ` ... |
| sbco2v 2337 | A composition law for subs... |
| aaan 2338 | Distribute universal quant... |
| eeor 2339 | Distribute existential qua... |
| cbv3v 2340 | Rule used to change bound ... |
| cbv1v 2341 | Rule used to change bound ... |
| cbv2w 2342 | Rule used to change bound ... |
| cbvaldw 2343 | Deduction used to change b... |
| cbvexdw 2344 | Deduction used to change b... |
| cbv3hv 2345 | Rule used to change bound ... |
| cbvalv1 2346 | Rule used to change bound ... |
| cbvexv1 2347 | Rule used to change bound ... |
| cbval2v 2348 | Rule used to change bound ... |
| cbvex2v 2349 | Rule used to change bound ... |
| dvelimhw 2350 | Proof of ~ dvelimh without... |
| pm11.53 2351 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2352 | Special case of ~ 19.12 wh... |
| eean 2353 | Distribute existential qua... |
| eeanv 2354 | Distribute a pair of exist... |
| eeeanv 2355 | Distribute three existenti... |
| ee4anv 2356 | Distribute two pairs of ex... |
| ee4anvOLD 2357 | Obsolete version of ~ ee4a... |
| sb8v 2358 | Substitution of variable i... |
| sb8f 2359 | Substitution of variable i... |
| sb8ef 2360 | Substitution of variable i... |
| 2sb8ef 2361 | An equivalent expression f... |
| sb6rfv 2362 | Reversed substitution. Ve... |
| sbnf2 2363 | Two ways of expressing " `... |
| exsb 2364 | An equivalent expression f... |
| 2exsb 2365 | An equivalent expression f... |
| sbbib 2366 | Reversal of substitution. ... |
| sbbibvv 2367 | Reversal of substitution. ... |
| cbvsbvf 2368 | Change the bound variable ... |
| cleljustALT 2369 | Alternate proof of ~ clelj... |
| cleljustALT2 2370 | Alternate proof of ~ clelj... |
| equs5aALT 2371 | Alternate proof of ~ equs5... |
| equs5eALT 2372 | Alternate proof of ~ equs5... |
| axc11r 2373 | Same as ~ axc11 but with r... |
| dral1v 2374 | Formula-building lemma for... |
| drex1v 2375 | Formula-building lemma for... |
| drnf1v 2376 | Formula-building lemma for... |
| ax13v 2378 | A weaker version of ~ ax-1... |
| ax13lem1 2379 | A version of ~ ax13v with ... |
| ax13 2380 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2381 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2382 | An equation between setvar... |
| dveeq2 2383 | Quantifier introduction wh... |
| nfeqf1 2384 | An equation between setvar... |
| dveeq1 2385 | Quantifier introduction wh... |
| nfeqf 2386 | A variable is effectively ... |
| axc9 2387 | Derive set.mm's original ~... |
| ax6e 2388 | At least one individual ex... |
| ax6 2389 | Theorem showing that ~ ax-... |
| axc10 2390 | Show that the original axi... |
| spimt 2391 | Closed theorem form of ~ s... |
| spim 2392 | Specialization, using impl... |
| spimed 2393 | Deduction version of ~ spi... |
| spime 2394 | Existential introduction, ... |
| spimv 2395 | A version of ~ spim with a... |
| spimvALT 2396 | Alternate proof of ~ spimv... |
| spimev 2397 | Distinct-variable version ... |
| spv 2398 | Specialization, using impl... |
| spei 2399 | Inference from existential... |
| chvar 2400 | Implicit substitution of `... |
| chvarv 2401 | Implicit substitution of `... |
| cbv3 2402 | Rule used to change bound ... |
| cbval 2403 | Rule used to change bound ... |
| cbvex 2404 | Rule used to change bound ... |
| cbvalv 2405 | Rule used to change bound ... |
| cbvexv 2406 | Rule used to change bound ... |
| cbv1 2407 | Rule used to change bound ... |
| cbv2 2408 | Rule used to change bound ... |
| cbv3h 2409 | Rule used to change bound ... |
| cbv1h 2410 | Rule used to change bound ... |
| cbv2h 2411 | Rule used to change bound ... |
| cbvald 2412 | Deduction used to change b... |
| cbvexd 2413 | Deduction used to change b... |
| cbvaldva 2414 | Rule used to change the bo... |
| cbvexdva 2415 | Rule used to change the bo... |
| cbval2 2416 | Rule used to change bound ... |
| cbvex2 2417 | Rule used to change bound ... |
| cbval2vv 2418 | Rule used to change bound ... |
| cbvex2vv 2419 | Rule used to change bound ... |
| cbvex4v 2420 | Rule used to change bound ... |
| equs4 2421 | Lemma used in proofs of im... |
| equsal 2422 | An equivalence related to ... |
| equsex 2423 | An equivalence related to ... |
| equsexALT 2424 | Alternate proof of ~ equse... |
| equsalh 2425 | An equivalence related to ... |
| equsexh 2426 | An equivalence related to ... |
| axc15 2427 | Derivation of set.mm's ori... |
| ax12 2428 | Rederivation of Axiom ~ ax... |
| ax12b 2429 | A bidirectional version of... |
| ax13ALT 2430 | Alternate proof of ~ ax13 ... |
| axc11n 2431 | Derive set.mm's original ~... |
| aecom 2432 | Commutation law for identi... |
| aecoms 2433 | A commutation rule for ide... |
| naecoms 2434 | A commutation rule for dis... |
| axc11 2435 | Show that ~ ax-c11 can be ... |
| hbae 2436 | All variables are effectiv... |
| hbnae 2437 | All variables are effectiv... |
| nfae 2438 | All variables are effectiv... |
| nfnae 2439 | All variables are effectiv... |
| hbnaes 2440 | Rule that applies ~ hbnae ... |
| axc16i 2441 | Inference with ~ axc16 as ... |
| axc16nfALT 2442 | Alternate proof of ~ axc16... |
| dral2 2443 | Formula-building lemma for... |
| dral1 2444 | Formula-building lemma for... |
| dral1ALT 2445 | Alternate proof of ~ dral1... |
| drex1 2446 | Formula-building lemma for... |
| drex2 2447 | Formula-building lemma for... |
| drnf1 2448 | Formula-building lemma for... |
| drnf2 2449 | Formula-building lemma for... |
| nfald2 2450 | Variation on ~ nfald which... |
| nfexd2 2451 | Variation on ~ nfexd which... |
| exdistrf 2452 | Distribution of existentia... |
| dvelimf 2453 | Version of ~ dvelimv witho... |
| dvelimdf 2454 | Deduction form of ~ dvelim... |
| dvelimh 2455 | Version of ~ dvelim withou... |
| dvelim 2456 | This theorem can be used t... |
| dvelimv 2457 | Similar to ~ dvelim with f... |
| dvelimnf 2458 | Version of ~ dvelim using ... |
| dveeq2ALT 2459 | Alternate proof of ~ dveeq... |
| equvini 2460 | A variable introduction la... |
| equvel 2461 | A variable elimination law... |
| equs5a 2462 | A property related to subs... |
| equs5e 2463 | A property related to subs... |
| equs45f 2464 | Two ways of expressing sub... |
| equs5 2465 | Lemma used in proofs of su... |
| dveel1 2466 | Quantifier introduction wh... |
| dveel2 2467 | Quantifier introduction wh... |
| axc14 2468 | Axiom ~ ax-c14 is redundan... |
| sb6x 2469 | Equivalence involving subs... |
| sbequ5 2470 | Substitution does not chan... |
| sbequ6 2471 | Substitution does not chan... |
| sb5rf 2472 | Reversed substitution. Us... |
| sb6rf 2473 | Reversed substitution. Fo... |
| ax12vALT 2474 | Alternate proof of ~ ax12v... |
| 2ax6elem 2475 | We can always find values ... |
| 2ax6e 2476 | We can always find values ... |
| 2sb5rf 2477 | Reversed double substituti... |
| 2sb6rf 2478 | Reversed double substituti... |
| sbel2x 2479 | Elimination of double subs... |
| sb4b 2480 | Simplified definition of s... |
| sb3b 2481 | Simplified definition of s... |
| sb3 2482 | One direction of a simplif... |
| sb1 2483 | One direction of a simplif... |
| sb2 2484 | One direction of a simplif... |
| sb4a 2485 | A version of one implicati... |
| dfsb1 2486 | Alternate definition of su... |
| hbsb2 2487 | Bound-variable hypothesis ... |
| nfsb2 2488 | Bound-variable hypothesis ... |
| hbsb2a 2489 | Special case of a bound-va... |
| sb4e 2490 | One direction of a simplif... |
| hbsb2e 2491 | Special case of a bound-va... |
| hbsb3 2492 | If ` y ` is not free in ` ... |
| nfs1 2493 | If ` y ` is not free in ` ... |
| axc16ALT 2494 | Alternate proof of ~ axc16... |
| axc16gALT 2495 | Alternate proof of ~ axc16... |
| equsb1 2496 | Substitution applied to an... |
| equsb2 2497 | Substitution applied to an... |
| dfsb2 2498 | An alternate definition of... |
| dfsb3 2499 | An alternate definition of... |
| drsb1 2500 | Formula-building lemma for... |
| sb2ae 2501 | In the case of two success... |
| sb6f 2502 | Equivalence for substituti... |
| sb5f 2503 | Equivalence for substituti... |
| nfsb4t 2504 | A variable not free in a p... |
| nfsb4 2505 | A variable not free in a p... |
| sbequ8 2506 | Elimination of equality fr... |
| sbie 2507 | Conversion of implicit sub... |
| sbied 2508 | Conversion of implicit sub... |
| sbiedv 2509 | Conversion of implicit sub... |
| 2sbiev 2510 | Conversion of double impli... |
| sbcom3 2511 | Substituting ` y ` for ` x... |
| sbco 2512 | A composition law for subs... |
| sbid2 2513 | An identity law for substi... |
| sbid2v 2514 | An identity law for substi... |
| sbidm 2515 | An idempotent law for subs... |
| sbco2 2516 | A composition law for subs... |
| sbco2d 2517 | A composition law for subs... |
| sbco3 2518 | A composition law for subs... |
| sbcom 2519 | A commutativity law for su... |
| sbtrt 2520 | Partially closed form of ~... |
| sbtr 2521 | A partial converse to ~ sb... |
| sb8 2522 | Substitution of variable i... |
| sb8e 2523 | Substitution of variable i... |
| sb9 2524 | Commutation of quantificat... |
| sb9i 2525 | Commutation of quantificat... |
| sbhb 2526 | Two ways of expressing " `... |
| nfsbd 2527 | Deduction version of ~ nfs... |
| nfsb 2528 | If ` z ` is not free in ` ... |
| hbsb 2529 | If ` z ` is not free in ` ... |
| sb7f 2530 | This version of ~ dfsb7 do... |
| sb7h 2531 | This version of ~ dfsb7 do... |
| sb10f 2532 | Hao Wang's identity axiom ... |
| sbal1 2533 | Check out ~ sbal for a ver... |
| sbal2 2534 | Move quantifier in and out... |
| 2sb8e 2535 | An equivalent expression f... |
| dfmoeu 2536 | An elementary proof of ~ m... |
| dfeumo 2537 | An elementary proof showin... |
| mojust 2539 | Soundness justification th... |
| dfmo 2541 | Simplify definition ~ df-m... |
| nexmo 2542 | Nonexistence implies uniqu... |
| exmo 2543 | Any proposition holds for ... |
| moabs 2544 | Absorption of existence co... |
| moim 2545 | The at-most-one quantifier... |
| moimi 2546 | The at-most-one quantifier... |
| moimdv 2547 | The at-most-one quantifier... |
| mobi 2548 | Equivalence theorem for th... |
| mobii 2549 | Formula-building rule for ... |
| mobidv 2550 | Formula-building rule for ... |
| mobid 2551 | Formula-building rule for ... |
| moa1 2552 | If an implication holds fo... |
| moan 2553 | "At most one" is still the... |
| moani 2554 | "At most one" is still tru... |
| moor 2555 | "At most one" is still the... |
| mooran1 2556 | "At most one" imports disj... |
| mooran2 2557 | "At most one" exports disj... |
| nfmo1 2558 | Bound-variable hypothesis ... |
| nfmod2 2559 | Bound-variable hypothesis ... |
| nfmodv 2560 | Bound-variable hypothesis ... |
| nfmov 2561 | Bound-variable hypothesis ... |
| nfmod 2562 | Bound-variable hypothesis ... |
| nfmo 2563 | Bound-variable hypothesis ... |
| mof 2564 | Version of ~ df-mo with di... |
| mo3 2565 | Alternate definition of th... |
| mo 2566 | Equivalent definitions of ... |
| mo4 2567 | At-most-one quantifier exp... |
| mo4f 2568 | At-most-one quantifier exp... |
| eu3v 2571 | An alternate way to expres... |
| eujust 2572 | Soundness justification th... |
| eujustALT 2573 | Alternate proof of ~ eujus... |
| eu6lem 2574 | Lemma of ~ eu6im . A diss... |
| eu6 2575 | Alternate definition of th... |
| eu6im 2576 | One direction of ~ eu6 nee... |
| euf 2577 | Version of ~ eu6 with disj... |
| euex 2578 | Existential uniqueness imp... |
| eumo 2579 | Existential uniqueness imp... |
| eumoi 2580 | Uniqueness inferred from e... |
| exmoeub 2581 | Existence implies that uni... |
| exmoeu 2582 | Existence is equivalent to... |
| moeuex 2583 | Uniqueness implies that ex... |
| moeu 2584 | Uniqueness is equivalent t... |
| eubi 2585 | Equivalence theorem for th... |
| eubii 2586 | Introduce unique existenti... |
| eubidv 2587 | Formula-building rule for ... |
| eubid 2588 | Formula-building rule for ... |
| nfeu1ALT 2589 | Alternate version of ~ nfe... |
| nfeu1 2590 | Bound-variable hypothesis ... |
| nfeud2 2591 | Bound-variable hypothesis ... |
| nfeudw 2592 | Bound-variable hypothesis ... |
| nfeud 2593 | Bound-variable hypothesis ... |
| nfeuw 2594 | Bound-variable hypothesis ... |
| nfeu 2595 | Bound-variable hypothesis ... |
| dfeu 2596 | Rederive ~ df-eu from the ... |
| dfmo2 2597 | Rederive ~ df-mo from the ... |
| euequ 2598 | There exists a unique set ... |
| sb8eulem 2599 | Lemma. Factor out the com... |
| sb8euv 2600 | Variable substitution in u... |
| sb8eu 2601 | Variable substitution in u... |
| sb8mo 2602 | Variable substitution for ... |
| cbvmovw 2603 | Change bound variable. Us... |
| cbvmow 2604 | Rule used to change bound ... |
| cbvmo 2605 | Rule used to change bound ... |
| cbveuvw 2606 | Change bound variable. Us... |
| cbveuw 2607 | Version of ~ cbveu with a ... |
| cbveu 2608 | Rule used to change bound ... |
| cbveuALT 2609 | Alternative proof of ~ cbv... |
| eu2 2610 | An alternate way of defini... |
| eu1 2611 | An alternate way to expres... |
| euor 2612 | Introduce a disjunct into ... |
| euorv 2613 | Introduce a disjunct into ... |
| euor2 2614 | Introduce or eliminate a d... |
| sbmo 2615 | Substitution into an at-mo... |
| eu4 2616 | Uniqueness using implicit ... |
| euimmo 2617 | Existential uniqueness imp... |
| euim 2618 | Add unique existential qua... |
| moanimlem 2619 | Factor out the common proo... |
| moanimv 2620 | Introduction of a conjunct... |
| moanim 2621 | Introduction of a conjunct... |
| euan 2622 | Introduction of a conjunct... |
| moanmo 2623 | Nested at-most-one quantif... |
| moaneu 2624 | Nested at-most-one and uni... |
| euanv 2625 | Introduction of a conjunct... |
| mopick 2626 | "At most one" picks a vari... |
| moexexlem 2627 | Factor out the proof skele... |
| 2moexv 2628 | Double quantification with... |
| moexexvw 2629 | "At most one" double quant... |
| 2moswapv 2630 | A condition allowing to sw... |
| 2euswapv 2631 | A condition allowing to sw... |
| 2euexv 2632 | Double quantification with... |
| 2exeuv 2633 | Double existential uniquen... |
| eupick 2634 | Existential uniqueness "pi... |
| eupicka 2635 | Version of ~ eupick with c... |
| eupickb 2636 | Existential uniqueness "pi... |
| eupickbi 2637 | Theorem *14.26 in [Whitehe... |
| mopick2 2638 | "At most one" can show the... |
| moexex 2639 | "At most one" double quant... |
| moexexv 2640 | "At most one" double quant... |
| 2moex 2641 | Double quantification with... |
| 2euex 2642 | Double quantification with... |
| 2eumo 2643 | Nested unique existential ... |
| 2eu2ex 2644 | Double existential uniquen... |
| 2moswap 2645 | A condition allowing to sw... |
| 2euswap 2646 | A condition allowing to sw... |
| 2exeu 2647 | Double existential uniquen... |
| 2mo2 2648 | Two ways of expressing "th... |
| 2mo 2649 | Two ways of expressing "th... |
| 2mos 2650 | Double "there exists at mo... |
| 2mosOLD 2651 | Obsolete version of ~ 2mos... |
| 2eu1 2652 | Double existential uniquen... |
| 2eu1v 2653 | Double existential uniquen... |
| 2eu2 2654 | Double existential uniquen... |
| 2eu3 2655 | Double existential uniquen... |
| 2eu4 2656 | This theorem provides us w... |
| 2eu5 2657 | An alternate definition of... |
| 2eu6 2658 | Two equivalent expressions... |
| 2eu7 2659 | Two equivalent expressions... |
| 2eu8 2660 | Two equivalent expressions... |
| euae 2661 | Two ways to express "exact... |
| exists1 2662 | Two ways to express "exact... |
| exists2 2663 | A condition implying that ... |
| barbara 2664 | "Barbara", one of the fund... |
| celarent 2665 | "Celarent", one of the syl... |
| darii 2666 | "Darii", one of the syllog... |
| dariiALT 2667 | Alternate proof of ~ darii... |
| ferio 2668 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2669 | Lemma for ~ barbari and th... |
| barbari 2670 | "Barbari", one of the syll... |
| barbariALT 2671 | Alternate proof of ~ barba... |
| celaront 2672 | "Celaront", one of the syl... |
| cesare 2673 | "Cesare", one of the syllo... |
| camestres 2674 | "Camestres", one of the sy... |
| festino 2675 | "Festino", one of the syll... |
| festinoALT 2676 | Alternate proof of ~ festi... |
| baroco 2677 | "Baroco", one of the syllo... |
| barocoALT 2678 | Alternate proof of ~ festi... |
| cesaro 2679 | "Cesaro", one of the syllo... |
| camestros 2680 | "Camestros", one of the sy... |
| datisi 2681 | "Datisi", one of the syllo... |
| disamis 2682 | "Disamis", one of the syll... |
| ferison 2683 | "Ferison", one of the syll... |
| bocardo 2684 | "Bocardo", one of the syll... |
| darapti 2685 | "Darapti", one of the syll... |
| daraptiALT 2686 | Alternate proof of ~ darap... |
| felapton 2687 | "Felapton", one of the syl... |
| calemes 2688 | "Calemes", one of the syll... |
| dimatis 2689 | "Dimatis", one of the syll... |
| fresison 2690 | "Fresison", one of the syl... |
| calemos 2691 | "Calemos", one of the syll... |
| fesapo 2692 | "Fesapo", one of the syllo... |
| bamalip 2693 | "Bamalip", one of the syll... |
| axia1 2694 | Left 'and' elimination (in... |
| axia2 2695 | Right 'and' elimination (i... |
| axia3 2696 | 'And' introduction (intuit... |
| axin1 2697 | 'Not' introduction (intuit... |
| axin2 2698 | 'Not' elimination (intuiti... |
| axio 2699 | Definition of 'or' (intuit... |
| axi4 2700 | Specialization (intuitioni... |
| axi5r 2701 | Converse of ~ axc4 (intuit... |
| axial 2702 | The setvar ` x ` is not fr... |
| axie1 2703 | The setvar ` x ` is not fr... |
| axie2 2704 | A key property of existent... |
| axi9 2705 | Axiom of existence (intuit... |
| axi10 2706 | Axiom of Quantifier Substi... |
| axi12 2707 | Axiom of Quantifier Introd... |
| axbnd 2708 | Axiom of Bundling (intuiti... |
| axexte 2710 | The axiom of extensionalit... |
| axextg 2711 | A generalization of the ax... |
| axextb 2712 | A bidirectional version of... |
| axextmo 2713 | There exists at most one s... |
| nulmo 2714 | There exists at most one e... |
| eleq1ab 2717 | Extension (in the sense of... |
| cleljustab 2718 | Extension of ~ cleljust fr... |
| abid 2719 | Simplification of class ab... |
| vexwt 2720 | A standard theorem of pred... |
| vexw 2721 | If ` ph ` is a theorem, th... |
| vextru 2722 | Every setvar is a member o... |
| nfsab1 2723 | Bound-variable hypothesis ... |
| hbab1 2724 | Bound-variable hypothesis ... |
| hbab 2725 | Bound-variable hypothesis ... |
| hbabg 2726 | Bound-variable hypothesis ... |
| nfsab 2727 | Bound-variable hypothesis ... |
| nfsabg 2728 | Bound-variable hypothesis ... |
| dfcleq 2730 | The defining characterizat... |
| cvjust 2731 | Every set is a class. Pro... |
| ax9ALT 2732 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2733 | A weaker version of ~ eleq... |
| eqriv 2734 | Infer equality of classes ... |
| eqrdv 2735 | Deduce equality of classes... |
| eqrdav 2736 | Deduce equality of classes... |
| eqid 2737 | Law of identity (reflexivi... |
| eqidd 2738 | Class identity law with an... |
| eqeq1d 2739 | Deduction from equality to... |
| eqeq1dALT 2740 | Alternate proof of ~ eqeq1... |
| eqeq1 2741 | Equality implies equivalen... |
| eqeq1i 2742 | Inference from equality to... |
| eqcomd 2743 | Deduction from commutative... |
| eqcom 2744 | Commutative law for class ... |
| eqcoms 2745 | Inference applying commuta... |
| eqcomi 2746 | Inference from commutative... |
| neqcomd 2747 | Commute an inequality. (C... |
| eqeq2d 2748 | Deduction from equality to... |
| eqeq2 2749 | Equality implies equivalen... |
| eqeq2i 2750 | Inference from equality to... |
| eqeqan12d 2751 | A useful inference for sub... |
| eqeqan12rd 2752 | A useful inference for sub... |
| eqeq12d 2753 | A useful inference for sub... |
| eqeq12 2754 | Equality relationship amon... |
| eqeq12i 2755 | A useful inference for sub... |
| eqeqan12dALT 2756 | Alternate proof of ~ eqeqa... |
| eqtr 2757 | Transitive law for class e... |
| eqtr2 2758 | A transitive law for class... |
| eqtr3 2759 | A transitive law for class... |
| eqtri 2760 | An equality transitivity i... |
| eqtr2i 2761 | An equality transitivity i... |
| eqtr3i 2762 | An equality transitivity i... |
| eqtr4i 2763 | An equality transitivity i... |
| 3eqtri 2764 | An inference from three ch... |
| 3eqtrri 2765 | An inference from three ch... |
| 3eqtr2i 2766 | An inference from three ch... |
| 3eqtr2ri 2767 | An inference from three ch... |
| 3eqtr3i 2768 | An inference from three ch... |
| 3eqtr3ri 2769 | An inference from three ch... |
| 3eqtr4i 2770 | An inference from three ch... |
| 3eqtr4ri 2771 | An inference from three ch... |
| eqtrd 2772 | An equality transitivity d... |
| eqtr2d 2773 | An equality transitivity d... |
| eqtr3d 2774 | An equality transitivity e... |
| eqtr4d 2775 | An equality transitivity e... |
| 3eqtrd 2776 | A deduction from three cha... |
| 3eqtrrd 2777 | A deduction from three cha... |
| 3eqtr2d 2778 | A deduction from three cha... |
| 3eqtr2rd 2779 | A deduction from three cha... |
| 3eqtr3d 2780 | A deduction from three cha... |
| 3eqtr3rd 2781 | A deduction from three cha... |
| 3eqtr4d 2782 | A deduction from three cha... |
| 3eqtr4rd 2783 | A deduction from three cha... |
| eqtrid 2784 | An equality transitivity d... |
| eqtr2id 2785 | An equality transitivity d... |
| eqtr3id 2786 | An equality transitivity d... |
| eqtr3di 2787 | An equality transitivity d... |
| eqtrdi 2788 | An equality transitivity d... |
| eqtr2di 2789 | An equality transitivity d... |
| eqtr4di 2790 | An equality transitivity d... |
| eqtr4id 2791 | An equality transitivity d... |
| sylan9eq 2792 | An equality transitivity d... |
| sylan9req 2793 | An equality transitivity d... |
| sylan9eqr 2794 | An equality transitivity d... |
| 3eqtr3g 2795 | A chained equality inferen... |
| 3eqtr3a 2796 | A chained equality inferen... |
| 3eqtr4g 2797 | A chained equality inferen... |
| 3eqtr4a 2798 | A chained equality inferen... |
| eq2tri 2799 | A compound transitive infe... |
| iseqsetvlem 2800 | Lemma for ~ iseqsetv-cleq ... |
| iseqsetv-cleq 2801 | Alternate proof of ~ iseqs... |
| abbi 2802 | Equivalent formulas yield ... |
| abbidv 2803 | Equivalent wff's yield equ... |
| abbii 2804 | Equivalent wff's yield equ... |
| abbid 2805 | Equivalent wff's yield equ... |
| abbib 2806 | Equal class abstractions r... |
| cbvabv 2807 | Rule used to change bound ... |
| cbvabw 2808 | Rule used to change bound ... |
| cbvab 2809 | Rule used to change bound ... |
| eqabbw 2810 | Version of ~ eqabb using i... |
| eqabcbw 2811 | Version of ~ eqabcb using ... |
| dfclel 2813 | Characterization of the el... |
| elex2 2814 | If a class contains anothe... |
| issettru 2815 | Weak version of ~ isset . ... |
| iseqsetv-clel 2816 | Alternate proof of ~ iseqs... |
| issetlem 2817 | Lemma for ~ elisset and ~ ... |
| elissetv 2818 | An element of a class exis... |
| elisset 2819 | An element of a class exis... |
| eleq1w 2820 | Weaker version of ~ eleq1 ... |
| eleq2w 2821 | Weaker version of ~ eleq2 ... |
| eleq1d 2822 | Deduction from equality to... |
| eleq2d 2823 | Deduction from equality to... |
| eleq2dALT 2824 | Alternate proof of ~ eleq2... |
| eleq1 2825 | Equality implies equivalen... |
| eleq2 2826 | Equality implies equivalen... |
| eleq12 2827 | Equality implies equivalen... |
| eleq1i 2828 | Inference from equality to... |
| eleq2i 2829 | Inference from equality to... |
| eleq12i 2830 | Inference from equality to... |
| eleq12d 2831 | Deduction from equality to... |
| eleq1a 2832 | A transitive-type law rela... |
| eqeltri 2833 | Substitution of equal clas... |
| eqeltrri 2834 | Substitution of equal clas... |
| eleqtri 2835 | Substitution of equal clas... |
| eleqtrri 2836 | Substitution of equal clas... |
| eqeltrd 2837 | Substitution of equal clas... |
| eqeltrrd 2838 | Deduction that substitutes... |
| eleqtrd 2839 | Deduction that substitutes... |
| eleqtrrd 2840 | Deduction that substitutes... |
| eqeltrid 2841 | A membership and equality ... |
| eqeltrrid 2842 | A membership and equality ... |
| eleqtrid 2843 | A membership and equality ... |
| eleqtrrid 2844 | A membership and equality ... |
| eqeltrdi 2845 | A membership and equality ... |
| eqeltrrdi 2846 | A membership and equality ... |
| eleqtrdi 2847 | A membership and equality ... |
| eleqtrrdi 2848 | A membership and equality ... |
| 3eltr3i 2849 | Substitution of equal clas... |
| 3eltr4i 2850 | Substitution of equal clas... |
| 3eltr3d 2851 | Substitution of equal clas... |
| 3eltr4d 2852 | Substitution of equal clas... |
| 3eltr3g 2853 | Substitution of equal clas... |
| 3eltr4g 2854 | Substitution of equal clas... |
| eleq2s 2855 | Substitution of equal clas... |
| eqneltri 2856 | If a class is not an eleme... |
| eqneltrd 2857 | If a class is not an eleme... |
| eqneltrrd 2858 | If a class is not an eleme... |
| neleqtrd 2859 | If a class is not an eleme... |
| neleqtrrd 2860 | If a class is not an eleme... |
| nelneq 2861 | A way of showing two class... |
| nelneq2 2862 | A way of showing two class... |
| eqsb1 2863 | Substitution for the left-... |
| clelsb1 2864 | Substitution for the first... |
| clelsb2 2865 | Substitution for the secon... |
| cleqh 2866 | Establish equality between... |
| hbxfreq 2867 | A utility lemma to transfe... |
| hblem 2868 | Change the free variable o... |
| hblemg 2869 | Change the free variable o... |
| eqabdv 2870 | Deduction from a wff to a ... |
| eqabcdv 2871 | Deduction from a wff to a ... |
| eqabi 2872 | Equality of a class variab... |
| abid1 2873 | Every class is equal to a ... |
| abid2 2874 | A simplification of class ... |
| eqab 2875 | One direction of ~ eqabb i... |
| eqabb 2876 | Equality of a class variab... |
| eqabcb 2877 | Equality of a class variab... |
| eqabrd 2878 | Equality of a class variab... |
| eqabri 2879 | Equality of a class variab... |
| eqabcri 2880 | Equality of a class variab... |
| clelab 2881 | Membership of a class vari... |
| clabel 2882 | Membership of a class abst... |
| sbab 2883 | The right-hand side of the... |
| nfcjust 2885 | Justification theorem for ... |
| nfci 2887 | Deduce that a class ` A ` ... |
| nfcii 2888 | Deduce that a class ` A ` ... |
| nfcr 2889 | Consequence of the not-fre... |
| nfcrALT 2890 | Alternate version of ~ nfc... |
| nfcri 2891 | Consequence of the not-fre... |
| nfcd 2892 | Deduce that a class ` A ` ... |
| nfcrd 2893 | Consequence of the not-fre... |
| nfcrii 2894 | Consequence of the not-fre... |
| nfceqdf 2895 | An equality theorem for ef... |
| nfceqi 2896 | Equality theorem for class... |
| nfcxfr 2897 | A utility lemma to transfe... |
| nfcxfrd 2898 | A utility lemma to transfe... |
| nfcv 2899 | If ` x ` is disjoint from ... |
| nfcvd 2900 | If ` x ` is disjoint from ... |
| nfab1 2901 | Bound-variable hypothesis ... |
| nfnfc1 2902 | The setvar ` x ` is bound ... |
| clelsb1fw 2903 | Substitution for the first... |
| clelsb1f 2904 | Substitution for the first... |
| nfab 2905 | Bound-variable hypothesis ... |
| nfabg 2906 | Bound-variable hypothesis ... |
| nfaba1 2907 | Bound-variable hypothesis ... |
| nfaba1OLD 2908 | Obsolete version of ~ nfab... |
| nfaba1g 2909 | Bound-variable hypothesis ... |
| nfeqd 2910 | Hypothesis builder for equ... |
| nfeld 2911 | Hypothesis builder for ele... |
| nfnfc 2912 | Hypothesis builder for ` F... |
| nfeq 2913 | Hypothesis builder for equ... |
| nfel 2914 | Hypothesis builder for ele... |
| nfeq1 2915 | Hypothesis builder for equ... |
| nfel1 2916 | Hypothesis builder for ele... |
| nfeq2 2917 | Hypothesis builder for equ... |
| nfel2 2918 | Hypothesis builder for ele... |
| drnfc1 2919 | Formula-building lemma for... |
| drnfc2 2920 | Formula-building lemma for... |
| nfabdw 2921 | Bound-variable hypothesis ... |
| nfabd 2922 | Bound-variable hypothesis ... |
| nfabd2 2923 | Bound-variable hypothesis ... |
| dvelimdc 2924 | Deduction form of ~ dvelim... |
| dvelimc 2925 | Version of ~ dvelim for cl... |
| nfcvf 2926 | If ` x ` and ` y ` are dis... |
| nfcvf2 2927 | If ` x ` and ` y ` are dis... |
| cleqf 2928 | Establish equality between... |
| eqabf 2929 | Equality of a class variab... |
| abid2f 2930 | A simplification of class ... |
| abid2fOLD 2931 | Obsolete version of ~ abid... |
| sbabel 2932 | Theorem to move a substitu... |
| 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... |
| necon3bi 2959 | Contrapositive inference f... |
| necon1ai 2960 | Contrapositive inference f... |
| necon1bi 2961 | Contrapositive inference f... |
| necon2ai 2962 | Contrapositive inference f... |
| necon2bi 2963 | Contrapositive inference f... |
| necon4ai 2964 | Contrapositive inference f... |
| necon3i 2965 | Contrapositive inference f... |
| necon1i 2966 | Contrapositive inference f... |
| necon2i 2967 | Contrapositive inference f... |
| necon4i 2968 | Contrapositive inference f... |
| necon3abid 2969 | Deduction from equality to... |
| necon3bbid 2970 | Deduction from equality to... |
| necon1abid 2971 | Contrapositive deduction f... |
| necon1bbid 2972 | Contrapositive inference f... |
| necon4abid 2973 | Contrapositive law deducti... |
| necon4bbid 2974 | Contrapositive law deducti... |
| necon2abid 2975 | Contrapositive deduction f... |
| necon2bbid 2976 | Contrapositive deduction f... |
| necon3bid 2977 | Deduction from equality to... |
| necon4bid 2978 | Contrapositive law deducti... |
| necon3abii 2979 | Deduction from equality to... |
| necon3bbii 2980 | Deduction from equality to... |
| necon1abii 2981 | Contrapositive inference f... |
| necon1bbii 2982 | Contrapositive inference f... |
| necon2abii 2983 | Contrapositive inference f... |
| necon2bbii 2984 | Contrapositive inference f... |
| necon3bii 2985 | Inference from equality to... |
| necom 2986 | Commutation of inequality.... |
| necomi 2987 | Inference from commutative... |
| necomd 2988 | Deduction from commutative... |
| nesym 2989 | Characterization of inequa... |
| nesymi 2990 | Inference associated with ... |
| nesymir 2991 | Inference associated with ... |
| neeq1d 2992 | Deduction for inequality. ... |
| neeq2d 2993 | Deduction for inequality. ... |
| neeq12d 2994 | Deduction for inequality. ... |
| neeq1 2995 | Equality theorem for inequ... |
| neeq2 2996 | Equality theorem for inequ... |
| neeq1i 2997 | Inference for inequality. ... |
| neeq2i 2998 | Inference for inequality. ... |
| neeq12i 2999 | Inference for inequality. ... |
| eqnetrd 3000 | Substitution of equal clas... |
| eqnetrrd 3001 | Substitution of equal clas... |
| neeqtrd 3002 | Substitution of equal clas... |
| eqnetri 3003 | Substitution of equal clas... |
| eqnetrri 3004 | Substitution of equal clas... |
| neeqtri 3005 | Substitution of equal clas... |
| neeqtrri 3006 | Substitution of equal clas... |
| neeqtrrd 3007 | Substitution of equal clas... |
| eqnetrrid 3008 | A chained equality inferen... |
| 3netr3d 3009 | Substitution of equality i... |
| 3netr4d 3010 | Substitution of equality i... |
| 3netr3g 3011 | Substitution of equality i... |
| 3netr4g 3012 | Substitution of equality i... |
| nebi 3013 | Contraposition law for ine... |
| pm13.18 3014 | Theorem *13.18 in [Whitehe... |
| pm13.181 3015 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3016 | Inference eliminating an i... |
| pm2.21ddne 3017 | A contradiction implies an... |
| pm2.61ne 3018 | Deduction eliminating an i... |
| pm2.61dne 3019 | Deduction eliminating an i... |
| pm2.61dane 3020 | Deduction eliminating an i... |
| pm2.61da2ne 3021 | Deduction eliminating two ... |
| pm2.61da3ne 3022 | Deduction eliminating thre... |
| pm2.61iine 3023 | Equality version of ~ pm2.... |
| mteqand 3024 | A modus tollens deduction ... |
| neor 3025 | Logical OR with an equalit... |
| neanior 3026 | A De Morgan's law for ineq... |
| ne3anior 3027 | A De Morgan's law for ineq... |
| neorian 3028 | A De Morgan's law for ineq... |
| nemtbir 3029 | An inference from an inequ... |
| nelne1 3030 | Two classes are different ... |
| nelne2 3031 | Two classes are different ... |
| nelelne 3032 | Two classes are different ... |
| neneor 3033 | If two classes are differe... |
| nfne 3034 | Bound-variable hypothesis ... |
| nfned 3035 | Bound-variable hypothesis ... |
| nabbib 3036 | Not equivalent wff's corre... |
| neli 3039 | Inference associated with ... |
| nelir 3040 | Inference associated with ... |
| nelcon3d 3041 | Contrapositive law deducti... |
| neleq12d 3042 | Equality theorem for negat... |
| neleq1 3043 | Equality theorem for negat... |
| neleq2 3044 | Equality theorem for negat... |
| nfnel 3045 | Bound-variable hypothesis ... |
| nfneld 3046 | Bound-variable hypothesis ... |
| nnel 3047 | Negation of negated member... |
| elnelne1 3048 | Two classes are different ... |
| elnelne2 3049 | Two classes are different ... |
| pm2.24nel 3050 | A contradiction concerning... |
| pm2.61danel 3051 | Deduction eliminating an e... |
| rgen 3054 | Generalization rule for re... |
| ralel 3055 | All elements of a class ar... |
| rgenw 3056 | Generalization rule for re... |
| rgen2w 3057 | Generalization rule for re... |
| mprg 3058 | Modus ponens combined with... |
| mprgbir 3059 | Modus ponens on biconditio... |
| ralrid 3060 | Sufficient condition for t... |
| raln 3061 | Restricted universally qua... |
| ralnex 3064 | Relationship between restr... |
| dfrex2 3065 | Relationship between restr... |
| nrex 3066 | Inference adding restricte... |
| alral 3067 | Universal quantification i... |
| rexex 3068 | Restricted existence impli... |
| rextru 3069 | Two ways of expressing tha... |
| ralimi2 3070 | Inference quantifying both... |
| reximi2 3071 | Inference quantifying both... |
| ralimia 3072 | Inference quantifying both... |
| reximia 3073 | Inference quantifying both... |
| ralimiaa 3074 | Inference quantifying both... |
| ralimi 3075 | Inference quantifying both... |
| reximi 3076 | Inference quantifying both... |
| ral2imi 3077 | Inference quantifying ante... |
| ralim 3078 | Distribution of restricted... |
| rexim 3079 | Theorem 19.22 of [Margaris... |
| ralbii2 3080 | Inference adding different... |
| rexbii2 3081 | Inference adding different... |
| ralbiia 3082 | Inference adding restricte... |
| rexbiia 3083 | Inference adding restricte... |
| ralbii 3084 | Inference adding restricte... |
| rexbii 3085 | Inference adding restricte... |
| ralanid 3086 | Cancellation law for restr... |
| rexanid 3087 | Cancellation law for restr... |
| ralcom3 3088 | A commutation law for rest... |
| dfral2 3089 | Relationship between restr... |
| rexnal 3090 | Relationship between restr... |
| ralinexa 3091 | A transformation of restri... |
| rexanali 3092 | A transformation of restri... |
| ralbi 3093 | Distribute a restricted un... |
| rexbi 3094 | Distribute restricted quan... |
| ralrexbid 3095 | Formula-building rule for ... |
| r19.35 3096 | Restricted quantifier vers... |
| r19.26m 3097 | Version of ~ 19.26 and ~ r... |
| r19.26 3098 | Restricted quantifier vers... |
| r19.26-3 3099 | Version of ~ r19.26 with t... |
| ralbiim 3100 | Split a biconditional and ... |
| r19.29 3101 | Restricted quantifier vers... |
| r19.29r 3102 | Restricted quantifier vers... |
| r19.29imd 3103 | Theorem 19.29 of [Margaris... |
| r19.40 3104 | Restricted quantifier vers... |
| r19.30 3105 | Restricted quantifier vers... |
| r19.43 3106 | Restricted quantifier vers... |
| 3r19.43 3107 | Restricted quantifier vers... |
| 2ralimi 3108 | Inference quantifying both... |
| 3ralimi 3109 | Inference quantifying both... |
| 4ralimi 3110 | Inference quantifying both... |
| 5ralimi 3111 | Inference quantifying both... |
| 6ralimi 3112 | Inference quantifying both... |
| 2ralbii 3113 | Inference adding two restr... |
| 2rexbii 3114 | Inference adding two restr... |
| 3ralbii 3115 | Inference adding three res... |
| 4ralbii 3116 | Inference adding four rest... |
| 2ralbiim 3117 | Split a biconditional and ... |
| ralnex2 3118 | Relationship between two r... |
| ralnex3 3119 | Relationship between three... |
| rexnal2 3120 | Relationship between two r... |
| rexnal3 3121 | Relationship between three... |
| nrexralim 3122 | Negation of a complex pred... |
| r19.26-2 3123 | Restricted quantifier vers... |
| 2r19.29 3124 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3125 | Theorem 19.29 of [Margaris... |
| r2allem 3126 | Lemma factoring out common... |
| r2exlem 3127 | Lemma factoring out common... |
| hbralrimi 3128 | Inference from Theorem 19.... |
| ralrimiv 3129 | Inference from Theorem 19.... |
| ralrimiva 3130 | Inference from Theorem 19.... |
| rexlimiva 3131 | Inference from Theorem 19.... |
| rexlimiv 3132 | Inference from Theorem 19.... |
| nrexdv 3133 | Deduction adding restricte... |
| ralrimivw 3134 | Inference from Theorem 19.... |
| rexlimivw 3135 | Weaker version of ~ rexlim... |
| ralrimdv 3136 | Inference from Theorem 19.... |
| rexlimdv 3137 | Inference from Theorem 19.... |
| ralrimdva 3138 | Inference from Theorem 19.... |
| rexlimdva 3139 | Inference from Theorem 19.... |
| rexlimdvaa 3140 | Inference from Theorem 19.... |
| rexlimdva2 3141 | Inference from Theorem 19.... |
| r19.29an 3142 | A commonly used pattern in... |
| rexlimdv3a 3143 | Inference from Theorem 19.... |
| rexlimdvw 3144 | Inference from Theorem 19.... |
| rexlimddv 3145 | Restricted existential eli... |
| r19.29a 3146 | A commonly used pattern in... |
| ralimdv2 3147 | Inference quantifying both... |
| reximdv2 3148 | Deduction quantifying both... |
| reximdvai 3149 | Deduction quantifying both... |
| ralimdva 3150 | Deduction quantifying both... |
| reximdva 3151 | Deduction quantifying both... |
| ralimdv 3152 | Deduction quantifying both... |
| reximdv 3153 | Deduction from Theorem 19.... |
| reximddv 3154 | Deduction from Theorem 19.... |
| reximddv3 3155 | Deduction from Theorem 19.... |
| reximssdv 3156 | Derivation of a restricted... |
| ralbidv2 3157 | Formula-building rule for ... |
| rexbidv2 3158 | Formula-building rule for ... |
| ralbidva 3159 | Formula-building rule for ... |
| rexbidva 3160 | Formula-building rule for ... |
| ralbidv 3161 | Formula-building rule for ... |
| rexbidv 3162 | Formula-building rule for ... |
| r19.21v 3163 | Restricted quantifier vers... |
| r19.37v 3164 | Restricted quantifier vers... |
| r19.23v 3165 | Restricted quantifier vers... |
| r19.36v 3166 | Restricted quantifier vers... |
| r19.27v 3167 | Restricted quantitifer ver... |
| r19.41v 3168 | Restricted quantifier vers... |
| r19.28v 3169 | Restricted quantifier vers... |
| r19.42v 3170 | Restricted quantifier vers... |
| r19.32v 3171 | Restricted quantifier vers... |
| r19.45v 3172 | Restricted quantifier vers... |
| r19.44v 3173 | One direction of a restric... |
| r2al 3174 | Double restricted universa... |
| r2ex 3175 | Double restricted existent... |
| r3al 3176 | Triple restricted universa... |
| r3ex 3177 | Triple existential quantif... |
| rgen2 3178 | Generalization rule for re... |
| ralrimivv 3179 | Inference from Theorem 19.... |
| rexlimivv 3180 | Inference from Theorem 19.... |
| ralrimivva 3181 | Inference from Theorem 19.... |
| ralrimdvv 3182 | Inference from Theorem 19.... |
| rgen3 3183 | Generalization rule for re... |
| ralrimivvva 3184 | Inference from Theorem 19.... |
| ralimdvva 3185 | Deduction doubly quantifyi... |
| reximdvva 3186 | Deduction doubly quantifyi... |
| ralimdvv 3187 | Deduction doubly quantifyi... |
| ralimdvvOLD 3188 | Obsolete version of ~ rali... |
| ralimd4v 3189 | Deduction quadrupally quan... |
| ralimd4vOLD 3190 | Obsolete version of ~ rali... |
| ralimd6v 3191 | Deduction sextupally quant... |
| ralimd6vOLD 3192 | Obsolete version of ~ rali... |
| ralrimdvva 3193 | Inference from Theorem 19.... |
| rexlimdvv 3194 | Inference from Theorem 19.... |
| rexlimdvva 3195 | Inference from Theorem 19.... |
| rexlimdvvva 3196 | Inference from Theorem 19.... |
| reximddv2 3197 | Double deduction from Theo... |
| r19.29vva 3198 | A commonly used pattern ba... |
| 2rexbiia 3199 | Inference adding two restr... |
| 2ralbidva 3200 | Formula-building rule for ... |
| 2rexbidva 3201 | Formula-building rule for ... |
| 2ralbidv 3202 | Formula-building rule for ... |
| 2rexbidv 3203 | Formula-building rule for ... |
| rexralbidv 3204 | Formula-building rule for ... |
| 3ralbidv 3205 | Formula-building rule for ... |
| 4ralbidv 3206 | Formula-building rule for ... |
| 6ralbidv 3207 | Formula-building rule for ... |
| r19.41vv 3208 | Version of ~ r19.41v with ... |
| reeanlem 3209 | Lemma factoring out common... |
| reeanv 3210 | Rearrange restricted exist... |
| 3reeanv 3211 | Rearrange three restricted... |
| 2ralor 3212 | Distribute restricted univ... |
| risset 3213 | Two ways to say " ` A ` be... |
| nelb 3214 | A definition of ` -. A e. ... |
| rspw 3215 | Restricted specialization.... |
| cbvralvw 3216 | Change the bound variable ... |
| cbvrexvw 3217 | Change the bound variable ... |
| cbvraldva 3218 | Rule used to change the bo... |
| cbvrexdva 3219 | Rule used to change the bo... |
| cbvral2vw 3220 | Change bound variables of ... |
| cbvrex2vw 3221 | Change bound variables of ... |
| cbvral3vw 3222 | Change bound variables of ... |
| cbvral4vw 3223 | Change bound variables of ... |
| cbvral6vw 3224 | Change bound variables of ... |
| cbvral8vw 3225 | Change bound variables of ... |
| rsp 3226 | Restricted specialization.... |
| rspa 3227 | Restricted specialization.... |
| rspe 3228 | Restricted specialization.... |
| rspec 3229 | Specialization rule for re... |
| r19.21bi 3230 | Inference from Theorem 19.... |
| r19.21be 3231 | Inference from Theorem 19.... |
| r19.21t 3232 | Restricted quantifier vers... |
| r19.21 3233 | Restricted quantifier vers... |
| r19.23t 3234 | Closed theorem form of ~ r... |
| r19.23 3235 | Restricted quantifier vers... |
| ralrimi 3236 | Inference from Theorem 19.... |
| ralrimia 3237 | Inference from Theorem 19.... |
| rexlimi 3238 | Restricted quantifier vers... |
| ralimdaa 3239 | Deduction quantifying both... |
| reximdai 3240 | Deduction from Theorem 19.... |
| r19.37 3241 | Restricted quantifier vers... |
| r19.41 3242 | Restricted quantifier vers... |
| ralrimd 3243 | Inference from Theorem 19.... |
| rexlimd2 3244 | Version of ~ rexlimd with ... |
| rexlimd 3245 | Deduction form of ~ rexlim... |
| r19.29af2 3246 | A commonly used pattern ba... |
| r19.29af 3247 | A commonly used pattern ba... |
| reximd2a 3248 | Deduction quantifying both... |
| ralbida 3249 | Formula-building rule for ... |
| rexbida 3250 | Formula-building rule for ... |
| ralbid 3251 | Formula-building rule for ... |
| rexbid 3252 | Formula-building rule for ... |
| rexbidvALT 3253 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3254 | Alternate proof of ~ rexbi... |
| rsp2 3255 | Restricted specialization,... |
| rsp2e 3256 | Restricted specialization.... |
| rspec2 3257 | Specialization rule for re... |
| rspec3 3258 | Specialization rule for re... |
| r2alf 3259 | Double restricted universa... |
| r2exf 3260 | Double restricted existent... |
| 2ralbida 3261 | Formula-building rule for ... |
| nfra1 3262 | The setvar ` x ` is not fr... |
| nfre1 3263 | The setvar ` x ` is not fr... |
| ralcom4 3264 | Commutation of restricted ... |
| rexcom4 3265 | Commutation of restricted ... |
| ralcom 3266 | Commutation of restricted ... |
| rexcom 3267 | Commutation of restricted ... |
| rexcom4a 3268 | Specialized existential co... |
| ralrot3 3269 | Rotate three restricted un... |
| ralcom13 3270 | Swap first and third restr... |
| rexcom13 3271 | Swap first and third restr... |
| rexrot4 3272 | Rotate four restricted exi... |
| 2ex2rexrot 3273 | Rotate two existential qua... |
| nfra2w 3274 | Similar to Lemma 24 of [Mo... |
| hbra1 3275 | The setvar ` x ` is not fr... |
| ralcomf 3276 | Commutation of restricted ... |
| rexcomf 3277 | Commutation of restricted ... |
| cbvralfw 3278 | Rule used to change bound ... |
| cbvrexfw 3279 | Rule used to change bound ... |
| cbvralw 3280 | Rule used to change bound ... |
| cbvrexw 3281 | Rule used to change bound ... |
| hbral 3282 | Bound-variable hypothesis ... |
| nfraldw 3283 | Deduction version of ~ nfr... |
| nfrexdw 3284 | Deduction version of ~ nfr... |
| nfralw 3285 | Bound-variable hypothesis ... |
| nfrexw 3286 | Bound-variable hypothesis ... |
| r19.12 3287 | Restricted quantifier vers... |
| reean 3288 | Rearrange restricted exist... |
| cbvralsvw 3289 | Change bound variable by u... |
| cbvrexsvw 3290 | Change bound variable by u... |
| cbvralsvwOLD 3291 | Obsolete version of ~ cbvr... |
| rexeq 3292 | Equality theorem for restr... |
| raleq 3293 | Equality theorem for restr... |
| raleqi 3294 | Equality inference for res... |
| rexeqi 3295 | Equality inference for res... |
| raleqdv 3296 | Equality deduction for res... |
| rexeqdv 3297 | Equality deduction for res... |
| raleqtrdv 3298 | Substitution of equal clas... |
| rexeqtrdv 3299 | Substitution of equal clas... |
| raleqtrrdv 3300 | Substitution of equal clas... |
| rexeqtrrdv 3301 | Substitution of equal clas... |
| raleqbidva 3302 | Equality deduction for res... |
| rexeqbidva 3303 | Equality deduction for res... |
| raleqbidvv 3304 | Version of ~ raleqbidv wit... |
| rexeqbidvv 3305 | Version of ~ rexeqbidv wit... |
| raleqbi1dv 3306 | Equality deduction for res... |
| rexeqbi1dv 3307 | Equality deduction for res... |
| raleleq 3308 | All elements of a class ar... |
| raleleqOLD 3309 | Obsolete version of ~ rale... |
| raleqbii 3310 | Equality deduction for res... |
| rexeqbii 3311 | Equality deduction for res... |
| raleqbidv 3312 | Equality deduction for res... |
| rexeqbidv 3313 | Equality deduction for res... |
| cbvraldva2 3314 | Rule used to change the bo... |
| cbvrexdva2 3315 | Rule used to change the bo... |
| sbralie 3316 | Implicit to explicit subst... |
| sbralieALT 3317 | Alternative shorter proof ... |
| sbralieOLD 3318 | Obsolete version of ~ sbra... |
| raleqf 3319 | Equality theorem for restr... |
| rexeqf 3320 | Equality theorem for restr... |
| raleqbid 3321 | Equality deduction for res... |
| rexeqbid 3322 | Equality deduction for res... |
| cbvralf 3323 | Rule used to change bound ... |
| cbvrexf 3324 | Rule used to change bound ... |
| cbvral 3325 | Rule used to change bound ... |
| cbvrex 3326 | Rule used to change bound ... |
| cbvralv 3327 | Change the bound variable ... |
| cbvrexv 3328 | Change the bound variable ... |
| cbvralsv 3329 | Change bound variable by u... |
| cbvrexsv 3330 | Change bound variable by u... |
| cbvral2v 3331 | Change bound variables of ... |
| cbvrex2v 3332 | Change bound variables of ... |
| cbvral3v 3333 | Change bound variables of ... |
| rgen2a 3334 | Generalization rule for re... |
| nfrald 3335 | Deduction version of ~ nfr... |
| nfrexd 3336 | Deduction version of ~ nfr... |
| nfral 3337 | Bound-variable hypothesis ... |
| nfrex 3338 | Bound-variable hypothesis ... |
| nfra2 3339 | Similar to Lemma 24 of [Mo... |
| ralcom2 3340 | Commutation of restricted ... |
| reu5 3345 | Restricted uniqueness in t... |
| reurmo 3346 | Restricted existential uni... |
| reurex 3347 | Restricted unique existenc... |
| mormo 3348 | Unrestricted "at most one"... |
| rmobiia 3349 | Formula-building rule for ... |
| reubiia 3350 | Formula-building rule for ... |
| rmobii 3351 | Formula-building rule for ... |
| reubii 3352 | Formula-building rule for ... |
| rmoanid 3353 | Cancellation law for restr... |
| reuanid 3354 | Cancellation law for restr... |
| 2reu2rex 3355 | Double restricted existent... |
| rmobidva 3356 | Formula-building rule for ... |
| reubidva 3357 | Formula-building rule for ... |
| rmobidv 3358 | Formula-building rule for ... |
| reubidv 3359 | Formula-building rule for ... |
| reueubd 3360 | Restricted existential uni... |
| rmo5 3361 | Restricted "at most one" i... |
| nrexrmo 3362 | Nonexistence implies restr... |
| moel 3363 | "At most one" element in a... |
| cbvrmovw 3364 | Change the bound variable ... |
| cbvreuvw 3365 | Change the bound variable ... |
| rmobida 3366 | Formula-building rule for ... |
| reubida 3367 | Formula-building rule for ... |
| cbvrmow 3368 | Change the bound variable ... |
| cbvreuw 3369 | Change the bound variable ... |
| nfrmo1 3370 | The setvar ` x ` is not fr... |
| nfreu1 3371 | The setvar ` x ` is not fr... |
| nfrmow 3372 | Bound-variable hypothesis ... |
| nfreuw 3373 | Bound-variable hypothesis ... |
| rmoeq1 3374 | Equality theorem for restr... |
| reueq1 3375 | Equality theorem for restr... |
| rmoeqd 3376 | Equality deduction for res... |
| reueqd 3377 | Equality deduction for res... |
| reueqdv 3378 | Formula-building rule for ... |
| reueqbidv 3379 | Formula-building rule for ... |
| rmoeq1f 3380 | Equality theorem for restr... |
| reueq1f 3381 | Equality theorem for restr... |
| cbvreu 3382 | Change the bound variable ... |
| cbvrmo 3383 | Change the bound variable ... |
| cbvrmov 3384 | Change the bound variable ... |
| cbvreuv 3385 | Change the bound variable ... |
| nfrmod 3386 | Deduction version of ~ nfr... |
| nfreud 3387 | Deduction version of ~ nfr... |
| nfrmo 3388 | Bound-variable hypothesis ... |
| nfreu 3389 | Bound-variable hypothesis ... |
| rabbidva2 3392 | Equivalent wff's yield equ... |
| rabbia2 3393 | Equivalent wff's yield equ... |
| rabbiia 3394 | Equivalent formulas yield ... |
| rabbii 3395 | Equivalent wff's correspon... |
| rabbidva 3396 | Equivalent wff's yield equ... |
| rabbidv 3397 | Equivalent wff's yield equ... |
| rabbieq 3398 | Equivalent wff's correspon... |
| rabswap 3399 | Swap with a membership rel... |
| cbvrabv 3400 | Rule to change the bound v... |
| rabeqcda 3401 | When ` ps ` is always true... |
| rabeqc 3402 | A restricted class abstrac... |
| rabeqi 3403 | Equality theorem for restr... |
| rabeq 3404 | Equality theorem for restr... |
| rabeqdv 3405 | Equality of restricted cla... |
| rabeqbidva 3406 | Equality of restricted cla... |
| rabeqbidvaOLD 3407 | Obsolete version of ~ rabe... |
| rabeqbidv 3408 | Equality of restricted cla... |
| rabrabi 3409 | Abstract builder restricte... |
| nfrab1 3410 | The abstraction variable i... |
| rabid 3411 | An "identity" law of concr... |
| rabidim1 3412 | Membership in a restricted... |
| reqabi 3413 | Inference from equality of... |
| rabrab 3414 | Abstract builder restricte... |
| rabbida4 3415 | Version of ~ rabbidva2 wit... |
| rabbida 3416 | Equivalent wff's yield equ... |
| rabbid 3417 | Version of ~ rabbidv with ... |
| rabeqd 3418 | Deduction form of ~ rabeq ... |
| rabeqbida 3419 | Version of ~ rabeqbidva wi... |
| rabbi 3420 | Equivalent wff's correspon... |
| rabid2f 3421 | An "identity" law for rest... |
| rabid2im 3422 | One direction of ~ rabid2 ... |
| rabid2 3423 | An "identity" law for rest... |
| rabeqf 3424 | Equality theorem for restr... |
| cbvrabw 3425 | Rule to change the bound v... |
| cbvrabwOLD 3426 | Obsolete version of ~ cbvr... |
| nfrabw 3427 | A variable not free in a w... |
| nfrab 3428 | A variable not free in a w... |
| cbvrab 3429 | Rule to change the bound v... |
| vjust 3431 | Justification theorem for ... |
| dfv2 3433 | Alternate definition of th... |
| vex 3434 | All setvar variables are s... |
| elv 3435 | If a proposition is implie... |
| elvd 3436 | If a proposition is implie... |
| el2v 3437 | If a proposition is implie... |
| el3v 3438 | If a proposition is implie... |
| el3v3 3439 | If a proposition is implie... |
| eqv 3440 | The universe contains ever... |
| eqvf 3441 | The universe contains ever... |
| abv 3442 | The class of sets verifyin... |
| abvALT 3443 | Alternate proof of ~ abv ,... |
| isset 3444 | Two ways to express that "... |
| cbvexeqsetf 3445 | The expression ` E. x x = ... |
| issetft 3446 | Closed theorem form of ~ i... |
| issetf 3447 | A version of ~ isset that ... |
| isseti 3448 | A way to say " ` A ` is a ... |
| issetri 3449 | A way to say " ` A ` is a ... |
| eqvisset 3450 | A class equal to a variabl... |
| elex 3451 | If a class is a member of ... |
| elexOLD 3452 | Obsolete version of ~ elex... |
| elexi 3453 | If a class is a member of ... |
| elexd 3454 | If a class is a member of ... |
| elex22 3455 | If two classes each contai... |
| prcnel 3456 | A proper class doesn't bel... |
| ralv 3457 | A universal quantifier res... |
| rexv 3458 | An existential quantifier ... |
| reuv 3459 | A unique existential quant... |
| rmov 3460 | An at-most-one quantifier ... |
| rabab 3461 | A class abstraction restri... |
| rexcom4b 3462 | Specialized existential co... |
| ceqsal1t 3463 | One direction of ~ ceqsalt... |
| ceqsalt 3464 | Closed theorem version of ... |
| ceqsralt 3465 | Restricted quantifier vers... |
| ceqsalg 3466 | A representation of explic... |
| ceqsalgALT 3467 | Alternate proof of ~ ceqsa... |
| ceqsal 3468 | A representation of explic... |
| ceqsalALT 3469 | A representation of explic... |
| ceqsalv 3470 | A representation of explic... |
| ceqsralv 3471 | Restricted quantifier vers... |
| gencl 3472 | Implicit substitution for ... |
| 2gencl 3473 | Implicit substitution for ... |
| 3gencl 3474 | Implicit substitution for ... |
| cgsexg 3475 | Implicit substitution infe... |
| cgsex2g 3476 | Implicit substitution infe... |
| cgsex4g 3477 | An implicit substitution i... |
| ceqsex 3478 | Elimination of an existent... |
| ceqsexv 3479 | Elimination of an existent... |
| ceqsexv2d 3480 | Elimination of an existent... |
| ceqsexv2dOLD 3481 | Obsolete version of ~ ceqs... |
| ceqsex2 3482 | Elimination of two existen... |
| ceqsex2v 3483 | Elimination of two existen... |
| ceqsex3v 3484 | Elimination of three exist... |
| ceqsex4v 3485 | Elimination of four existe... |
| ceqsex6v 3486 | Elimination of six existen... |
| ceqsex8v 3487 | Elimination of eight exist... |
| gencbvex 3488 | Change of bound variable u... |
| gencbvex2 3489 | Restatement of ~ gencbvex ... |
| gencbval 3490 | Change of bound variable u... |
| sbhypf 3491 | Introduce an explicit subs... |
| spcimgft 3492 | Closed theorem form of ~ s... |
| spcimgfi1 3493 | A closed version of ~ spci... |
| spcimgfi1OLD 3494 | Obsolete version of ~ spci... |
| spcgft 3495 | A closed version of ~ spcg... |
| spcimgf 3496 | Rule of specialization, us... |
| spcimegf 3497 | Existential specialization... |
| vtoclgft 3498 | Closed theorem form of ~ v... |
| vtocleg 3499 | Implicit substitution of a... |
| vtoclg 3500 | Implicit substitution of a... |
| vtocle 3501 | Implicit substitution of a... |
| vtocleOLD 3502 | Obsolete version of ~ vtoc... |
| vtoclbg 3503 | Implicit substitution of a... |
| vtocl 3504 | Implicit substitution of a... |
| vtoclOLD 3505 | Obsolete version of ~ vtoc... |
| vtocldf 3506 | Implicit substitution of a... |
| vtocld 3507 | Implicit substitution of a... |
| vtocl2d 3508 | Implicit substitution of t... |
| vtoclef 3509 | Implicit substitution of a... |
| vtoclf 3510 | Implicit substitution of a... |
| vtocl2 3511 | Implicit substitution of c... |
| vtocl3 3512 | Implicit substitution of c... |
| vtoclb 3513 | Implicit substitution of a... |
| vtoclgf 3514 | Implicit substitution of a... |
| vtoclg1f 3515 | Version of ~ vtoclgf with ... |
| vtocl2gf 3516 | Implicit substitution of a... |
| vtocl3gf 3517 | Implicit substitution of a... |
| vtocl2g 3518 | Implicit substitution of 2... |
| vtocl3g 3519 | Implicit substitution of a... |
| vtoclgaf 3520 | Implicit substitution of a... |
| vtoclga 3521 | Implicit substitution of a... |
| vtocl2ga 3522 | Implicit substitution of 2... |
| vtocl2gaf 3523 | Implicit substitution of 2... |
| vtocl2gafOLD 3524 | Obsolete version of ~ vtoc... |
| vtocl3gaf 3525 | Implicit substitution of 3... |
| vtocl3gafOLD 3526 | Obsolete version of ~ vtoc... |
| vtocl3ga 3527 | Implicit substitution of 3... |
| vtocl3gaOLD 3528 | Obsolete version of ~ vtoc... |
| vtocl4g 3529 | Implicit substitution of 4... |
| vtocl4ga 3530 | Implicit substitution of 4... |
| vtocl4gaOLD 3531 | Obsolete version of ~ vtoc... |
| vtoclegft 3532 | Implicit substitution of a... |
| vtoclri 3533 | Implicit substitution of a... |
| spcgf 3534 | Rule of specialization, us... |
| spcegf 3535 | Existential specialization... |
| spcimdv 3536 | Restricted specialization,... |
| spcdv 3537 | Rule of specialization, us... |
| spcimedv 3538 | Restricted existential spe... |
| spcgv 3539 | Rule of specialization, us... |
| spcegv 3540 | Existential specialization... |
| spcedv 3541 | Existential specialization... |
| spc2egv 3542 | Existential specialization... |
| spc2gv 3543 | Specialization with two qu... |
| spc2ed 3544 | Existential specialization... |
| spc2d 3545 | Specialization with 2 quan... |
| spc3egv 3546 | Existential specialization... |
| spc3gv 3547 | Specialization with three ... |
| spcv 3548 | Rule of specialization, us... |
| spcev 3549 | Existential specialization... |
| spc2ev 3550 | Existential specialization... |
| rspct 3551 | A closed version of ~ rspc... |
| rspcdf 3552 | Restricted specialization,... |
| rspc 3553 | Restricted specialization,... |
| rspce 3554 | Restricted existential spe... |
| rspcimdv 3555 | Restricted specialization,... |
| rspcimedv 3556 | Restricted existential spe... |
| rspcdv 3557 | Restricted specialization,... |
| rspcedv 3558 | Restricted existential spe... |
| rspcebdv 3559 | Restricted existential spe... |
| rspcdv2 3560 | Restricted specialization,... |
| rspcv 3561 | Restricted specialization,... |
| rspccv 3562 | Restricted specialization,... |
| rspcva 3563 | Restricted specialization,... |
| rspccva 3564 | Restricted specialization,... |
| rspcev 3565 | Restricted existential spe... |
| rspcdva 3566 | Restricted specialization,... |
| rspcedvd 3567 | Restricted existential spe... |
| rspcedvdw 3568 | Version of ~ rspcedvd wher... |
| rspceb2dv 3569 | Restricted existential spe... |
| rspcime 3570 | Prove a restricted existen... |
| rspceaimv 3571 | Restricted existential spe... |
| rspcedeq1vd 3572 | Restricted existential spe... |
| rspcedeq2vd 3573 | Restricted existential spe... |
| rspc2 3574 | Restricted specialization ... |
| rspc2gv 3575 | Restricted specialization ... |
| rspc2v 3576 | 2-variable restricted spec... |
| rspc2va 3577 | 2-variable restricted spec... |
| rspc2ev 3578 | 2-variable restricted exis... |
| 2rspcedvdw 3579 | Double application of ~ rs... |
| rspc2dv 3580 | 2-variable restricted spec... |
| rspc3v 3581 | 3-variable restricted spec... |
| rspc3ev 3582 | 3-variable restricted exis... |
| 3rspcedvdw 3583 | Triple application of ~ rs... |
| rspc3dv 3584 | 3-variable restricted spec... |
| rspc4v 3585 | 4-variable restricted spec... |
| rspc6v 3586 | 6-variable restricted spec... |
| rspc8v 3587 | 8-variable restricted spec... |
| rspceeqv 3588 | Restricted existential spe... |
| ralxpxfr2d 3589 | Transfer a universal quant... |
| rexraleqim 3590 | Statement following from e... |
| eqvincg 3591 | A variable introduction la... |
| eqvinc 3592 | A variable introduction la... |
| eqvincf 3593 | A variable introduction la... |
| alexeqg 3594 | Two ways to express substi... |
| ceqex 3595 | Equality implies equivalen... |
| ceqsexg 3596 | A representation of explic... |
| ceqsexgv 3597 | Elimination of an existent... |
| ceqsrexv 3598 | Elimination of a restricte... |
| ceqsrexbv 3599 | Elimination of a restricte... |
| ceqsralbv 3600 | Elimination of a restricte... |
| ceqsrex2v 3601 | Elimination of a restricte... |
| clel2g 3602 | Alternate definition of me... |
| clel2 3603 | Alternate definition of me... |
| clel3g 3604 | Alternate definition of me... |
| clel3 3605 | Alternate definition of me... |
| clel4g 3606 | Alternate definition of me... |
| clel4 3607 | Alternate definition of me... |
| clel5 3608 | Alternate definition of cl... |
| pm13.183 3609 | Compare theorem *13.183 in... |
| rr19.3v 3610 | Restricted quantifier vers... |
| rr19.28v 3611 | Restricted quantifier vers... |
| elab6g 3612 | Membership in a class abst... |
| elabd2 3613 | Membership in a class abst... |
| elabd3 3614 | Membership in a class abst... |
| elabgt 3615 | Membership in a class abst... |
| elabgtOLD 3616 | Obsolete version of ~ elab... |
| elabgtOLDOLD 3617 | Obsolete version of ~ elab... |
| elabgf 3618 | Membership in a class abst... |
| elabf 3619 | Membership in a class abst... |
| elabg 3620 | Membership in a class abst... |
| elabgw 3621 | Membership in a class abst... |
| elab2gw 3622 | Membership in a class abst... |
| elab 3623 | Membership in a class abst... |
| elab2g 3624 | Membership in a class abst... |
| elabd 3625 | Explicit demonstration the... |
| elab2 3626 | Membership in a class abst... |
| elab4g 3627 | Membership in a class abst... |
| elab3gf 3628 | Membership in a class abst... |
| elab3g 3629 | Membership in a class abst... |
| elab3 3630 | Membership in a class abst... |
| elrabi 3631 | Implication for the member... |
| elrabf 3632 | Membership in a restricted... |
| rabtru 3633 | Abstract builder using the... |
| elrab3t 3634 | Membership in a restricted... |
| elrab 3635 | Membership in a restricted... |
| elrab3 3636 | Membership in a restricted... |
| elrabd 3637 | Membership in a restricted... |
| elrab2 3638 | Membership in a restricted... |
| elrab2w 3639 | Membership in a restricted... |
| ralab 3640 | Universal quantification o... |
| ralrab 3641 | Universal quantification o... |
| rexab 3642 | Existential quantification... |
| rexrab 3643 | Existential quantification... |
| ralab2 3644 | Universal quantification o... |
| ralrab2 3645 | Universal quantification o... |
| rexab2 3646 | Existential quantification... |
| rexrab2 3647 | Existential quantification... |
| reurab 3648 | Restricted existential uni... |
| abidnf 3649 | Identity used to create cl... |
| dedhb 3650 | A deduction theorem for co... |
| class2seteq 3651 | Writing a set as a class a... |
| nelrdva 3652 | Deduce negative membership... |
| eqeu 3653 | A condition which implies ... |
| moeq 3654 | There exists at most one s... |
| eueq 3655 | A class is a set if and on... |
| eueqi 3656 | There exists a unique set ... |
| eueq2 3657 | Equality has existential u... |
| eueq3 3658 | Equality has existential u... |
| moeq3 3659 | "At most one" property of ... |
| mosub 3660 | "At most one" remains true... |
| mo2icl 3661 | Theorem for inferring "at ... |
| mob2 3662 | Consequence of "at most on... |
| moi2 3663 | Consequence of "at most on... |
| mob 3664 | Equality implied by "at mo... |
| moi 3665 | Equality implied by "at mo... |
| morex 3666 | Derive membership from uni... |
| euxfr2w 3667 | Transfer existential uniqu... |
| euxfrw 3668 | Transfer existential uniqu... |
| euxfr2 3669 | Transfer existential uniqu... |
| euxfr 3670 | Transfer existential uniqu... |
| euind 3671 | Existential uniqueness via... |
| reu2 3672 | A way to express restricte... |
| reu6 3673 | A way to express restricte... |
| reu3 3674 | A way to express restricte... |
| reu6i 3675 | A condition which implies ... |
| eqreu 3676 | A condition which implies ... |
| rmo4 3677 | Restricted "at most one" u... |
| reu4 3678 | Restricted uniqueness usin... |
| reu7 3679 | Restricted uniqueness usin... |
| reu8 3680 | Restricted uniqueness usin... |
| rmo3f 3681 | Restricted "at most one" u... |
| rmo4f 3682 | Restricted "at most one" u... |
| reu2eqd 3683 | Deduce equality from restr... |
| reueq 3684 | Equality has existential u... |
| rmoeq 3685 | Equality's restricted exis... |
| rmoan 3686 | Restricted "at most one" s... |
| rmoim 3687 | Restricted "at most one" i... |
| rmoimia 3688 | Restricted "at most one" i... |
| rmoimi 3689 | Restricted "at most one" i... |
| rmoimi2 3690 | Restricted "at most one" i... |
| 2reu5a 3691 | Double restricted existent... |
| reuimrmo 3692 | Restricted uniqueness impl... |
| 2reuswap 3693 | A condition allowing swap ... |
| 2reuswap2 3694 | A condition allowing swap ... |
| reuxfrd 3695 | Transfer existential uniqu... |
| reuxfr 3696 | Transfer existential uniqu... |
| reuxfr1d 3697 | Transfer existential uniqu... |
| reuxfr1ds 3698 | Transfer existential uniqu... |
| reuxfr1 3699 | Transfer existential uniqu... |
| reuind 3700 | Existential uniqueness via... |
| 2rmorex 3701 | Double restricted quantifi... |
| 2reu5lem1 3702 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3703 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3704 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3705 | Double restricted existent... |
| 2reurmo 3706 | Double restricted quantifi... |
| 2reurex 3707 | Double restricted quantifi... |
| 2rmoswap 3708 | A condition allowing to sw... |
| 2rexreu 3709 | Double restricted existent... |
| cdeqi 3712 | Deduce conditional equalit... |
| cdeqri 3713 | Property of conditional eq... |
| cdeqth 3714 | Deduce conditional equalit... |
| cdeqnot 3715 | Distribute conditional equ... |
| cdeqal 3716 | Distribute conditional equ... |
| cdeqab 3717 | Distribute conditional equ... |
| cdeqal1 3718 | Distribute conditional equ... |
| cdeqab1 3719 | Distribute conditional equ... |
| cdeqim 3720 | Distribute conditional equ... |
| cdeqcv 3721 | Conditional equality for s... |
| cdeqeq 3722 | Distribute conditional equ... |
| cdeqel 3723 | Distribute conditional equ... |
| nfcdeq 3724 | If we have a conditional e... |
| nfccdeq 3725 | Variation of ~ nfcdeq for ... |
| rru 3726 | Relative version of Russel... |
| ru 3727 | Russell's Paradox. Propos... |
| ruOLD 3728 | Obsolete version of ~ ru a... |
| dfsbcq 3731 | Proper substitution of a c... |
| dfsbcq2 3732 | This theorem, which is sim... |
| sbsbc 3733 | Show that ~ df-sb and ~ df... |
| sbceq1d 3734 | Equality theorem for class... |
| sbceq1dd 3735 | Equality theorem for class... |
| sbceqbid 3736 | Equality theorem for class... |
| sbc8g 3737 | This is the closest we can... |
| sbc2or 3738 | The disjunction of two equ... |
| sbcex 3739 | By our definition of prope... |
| sbceq1a 3740 | Equality theorem for class... |
| sbceq2a 3741 | Equality theorem for class... |
| spsbc 3742 | Specialization: if a formu... |
| spsbcd 3743 | Specialization: if a formu... |
| sbcth 3744 | A substitution into a theo... |
| sbcthdv 3745 | Deduction version of ~ sbc... |
| sbcid 3746 | An identity theorem for su... |
| nfsbc1d 3747 | Deduction version of ~ nfs... |
| nfsbc1 3748 | Bound-variable hypothesis ... |
| nfsbc1v 3749 | Bound-variable hypothesis ... |
| nfsbcdw 3750 | Deduction version of ~ nfs... |
| nfsbcw 3751 | Bound-variable hypothesis ... |
| sbccow 3752 | A composition law for clas... |
| nfsbcd 3753 | Deduction version of ~ nfs... |
| nfsbc 3754 | Bound-variable hypothesis ... |
| sbcco 3755 | A composition law for clas... |
| sbcco2 3756 | A composition law for clas... |
| sbc5 3757 | An equivalence for class s... |
| sbc5ALT 3758 | Alternate proof of ~ sbc5 ... |
| sbc6g 3759 | An equivalence for class s... |
| sbc6 3760 | An equivalence for class s... |
| sbc7 3761 | An equivalence for class s... |
| cbvsbcw 3762 | Change bound variables in ... |
| cbvsbcvw 3763 | Change the bound variable ... |
| cbvsbc 3764 | Change bound variables in ... |
| cbvsbcv 3765 | Change the bound variable ... |
| sbciegft 3766 | Conversion of implicit sub... |
| sbciegftOLD 3767 | Obsolete version of ~ sbci... |
| sbciegf 3768 | Conversion of implicit sub... |
| sbcieg 3769 | Conversion of implicit sub... |
| sbcie2g 3770 | Conversion of implicit sub... |
| sbcie 3771 | Conversion of implicit sub... |
| sbciedf 3772 | Conversion of implicit sub... |
| sbcied 3773 | Conversion of implicit sub... |
| sbcied2 3774 | Conversion of implicit sub... |
| elrabsf 3775 | Membership in a restricted... |
| eqsbc1 3776 | Substitution for the left-... |
| sbcng 3777 | Move negation in and out o... |
| sbcimg 3778 | Distribution of class subs... |
| sbcan 3779 | Distribution of class subs... |
| sbcor 3780 | Distribution of class subs... |
| sbcbig 3781 | Distribution of class subs... |
| sbcn1 3782 | Move negation in and out o... |
| sbcim1 3783 | Distribution of class subs... |
| sbcbid 3784 | Formula-building deduction... |
| sbcbidv 3785 | Formula-building deduction... |
| sbcbii 3786 | Formula-building inference... |
| sbcbi1 3787 | Distribution of class subs... |
| sbcbi2 3788 | Substituting into equivale... |
| sbcal 3789 | Move universal quantifier ... |
| sbcex2 3790 | Move existential quantifie... |
| sbceqal 3791 | Class version of one impli... |
| sbeqalb 3792 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3793 | Substitution for the right... |
| sbc3an 3794 | Distribution of class subs... |
| sbcel1v 3795 | Class substitution into a ... |
| sbcel2gv 3796 | Class substitution into a ... |
| sbcel21v 3797 | Class substitution into a ... |
| sbcimdv 3798 | Substitution analogue of T... |
| sbctt 3799 | Substitution for a variabl... |
| sbcgf 3800 | Substitution for a variabl... |
| sbc19.21g 3801 | Substitution for a variabl... |
| sbcg 3802 | Substitution for a variabl... |
| sbcgfi 3803 | Substitution for a variabl... |
| sbc2iegf 3804 | Conversion of implicit sub... |
| sbc2ie 3805 | Conversion of implicit sub... |
| sbc2iedv 3806 | Conversion of implicit sub... |
| sbc3ie 3807 | Conversion of implicit sub... |
| sbccomlem 3808 | Lemma for ~ sbccom . (Con... |
| sbccomlemOLD 3809 | Obsolete version of ~ sbcc... |
| sbccom 3810 | Commutative law for double... |
| sbcralt 3811 | Interchange class substitu... |
| sbcrext 3812 | Interchange class substitu... |
| sbcralg 3813 | Interchange class substitu... |
| sbcrex 3814 | Interchange class substitu... |
| sbcreu 3815 | Interchange class substitu... |
| reu8nf 3816 | Restricted uniqueness usin... |
| sbcabel 3817 | Interchange class substitu... |
| rspsbc 3818 | Restricted quantifier vers... |
| rspsbca 3819 | Restricted quantifier vers... |
| rspesbca 3820 | Existence form of ~ rspsbc... |
| spesbc 3821 | Existence form of ~ spsbc ... |
| spesbcd 3822 | form of ~ spsbc . (Contri... |
| sbcth2 3823 | A substitution into a theo... |
| ra4v 3824 | Version of ~ ra4 with a di... |
| ra4 3825 | Restricted quantifier vers... |
| rmo2 3826 | Alternate definition of re... |
| rmo2i 3827 | Condition implying restric... |
| rmo3 3828 | Restricted "at most one" u... |
| rmob 3829 | Consequence of "at most on... |
| rmoi 3830 | Consequence of "at most on... |
| rmob2 3831 | Consequence of "restricted... |
| rmoi2 3832 | Consequence of "restricted... |
| rmoanim 3833 | Introduction of a conjunct... |
| rmoanimALT 3834 | Alternate proof of ~ rmoan... |
| reuan 3835 | Introduction of a conjunct... |
| 2reu1 3836 | Double restricted existent... |
| 2reu2 3837 | Double restricted existent... |
| csb2 3840 | Alternate expression for t... |
| csbeq1 3841 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3842 | Equality deduction for pro... |
| csbeq2 3843 | Substituting into equivale... |
| csbeq2d 3844 | Formula-building deduction... |
| csbeq2dv 3845 | Formula-building deduction... |
| csbeq2i 3846 | Formula-building inference... |
| csbeq12dv 3847 | Formula-building inference... |
| cbvcsbw 3848 | Change bound variables in ... |
| cbvcsb 3849 | Change bound variables in ... |
| cbvcsbv 3850 | Change the bound variable ... |
| csbid 3851 | Analogue of ~ sbid for pro... |
| csbeq1a 3852 | Equality theorem for prope... |
| csbcow 3853 | Composition law for chaine... |
| csbco 3854 | Composition law for chaine... |
| csbtt 3855 | Substitution doesn't affec... |
| csbconstgf 3856 | Substitution doesn't affec... |
| csbconstg 3857 | Substitution doesn't affec... |
| csbgfi 3858 | Substitution for a variabl... |
| csbconstgi 3859 | The proper substitution of... |
| nfcsb1d 3860 | Bound-variable hypothesis ... |
| nfcsb1 3861 | Bound-variable hypothesis ... |
| nfcsb1v 3862 | Bound-variable hypothesis ... |
| nfcsbd 3863 | Deduction version of ~ nfc... |
| nfcsbw 3864 | Bound-variable hypothesis ... |
| nfcsb 3865 | Bound-variable hypothesis ... |
| csbhypf 3866 | Introduce an explicit subs... |
| csbiebt 3867 | Conversion of implicit sub... |
| csbiedf 3868 | Conversion of implicit sub... |
| csbieb 3869 | Bidirectional conversion b... |
| csbiebg 3870 | Bidirectional conversion b... |
| csbiegf 3871 | Conversion of implicit sub... |
| csbief 3872 | Conversion of implicit sub... |
| csbie 3873 | Conversion of implicit sub... |
| csbied 3874 | Conversion of implicit sub... |
| csbied2 3875 | Conversion of implicit sub... |
| csbie2t 3876 | Conversion of implicit sub... |
| csbie2 3877 | Conversion of implicit sub... |
| csbie2g 3878 | Conversion of implicit sub... |
| cbvrabcsfw 3879 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3880 | A more general version of ... |
| cbvrexcsf 3881 | A more general version of ... |
| cbvreucsf 3882 | A more general version of ... |
| cbvrabcsf 3883 | A more general version of ... |
| cbvralv2 3884 | Rule used to change the bo... |
| cbvrexv2 3885 | Rule used to change the bo... |
| rspc2vd 3886 | Deduction version of 2-var... |
| difjust 3892 | Soundness justification th... |
| unjust 3894 | Soundness justification th... |
| injust 3896 | Soundness justification th... |
| dfin5 3898 | Alternate definition for t... |
| dfdif2 3899 | Alternate definition of cl... |
| eldif 3900 | Expansion of membership in... |
| eldifd 3901 | If a class is in one class... |
| eldifad 3902 | If a class is in the diffe... |
| eldifbd 3903 | If a class is in the diffe... |
| elneeldif 3904 | The elements of a set diff... |
| velcomp 3905 | Characterization of setvar... |
| elin 3906 | Expansion of membership in... |
| dfss2 3908 | Alternate definition of th... |
| dfss 3909 | Variant of subclass defini... |
| dfss3 3911 | Alternate definition of su... |
| dfss6 3912 | Alternate definition of su... |
| dfssf 3913 | Equivalence for subclass r... |
| dfss3f 3914 | Equivalence for subclass r... |
| nfss 3915 | If ` x ` is not free in ` ... |
| ssel 3916 | Membership relationships f... |
| ssel2 3917 | Membership relationships f... |
| sseli 3918 | Membership implication fro... |
| sselii 3919 | Membership inference from ... |
| sselid 3920 | Membership inference from ... |
| sseld 3921 | Membership deduction from ... |
| sselda 3922 | Membership deduction from ... |
| sseldd 3923 | Membership inference from ... |
| ssneld 3924 | If a class is not in anoth... |
| ssneldd 3925 | If an element is not in a ... |
| ssriv 3926 | Inference based on subclas... |
| ssrd 3927 | Deduction based on subclas... |
| ssrdv 3928 | Deduction based on subclas... |
| sstr2 3929 | Transitivity of subclass r... |
| sstr2OLD 3930 | Obsolete version of ~ sstr... |
| sstr 3931 | Transitivity of subclass r... |
| sstri 3932 | Subclass transitivity infe... |
| sstrd 3933 | Subclass transitivity dedu... |
| sstrid 3934 | Subclass transitivity dedu... |
| sstrdi 3935 | Subclass transitivity dedu... |
| sylan9ss 3936 | A subclass transitivity de... |
| sylan9ssr 3937 | A subclass transitivity de... |
| eqss 3938 | The subclass relationship ... |
| eqssi 3939 | Infer equality from two su... |
| eqssd 3940 | Equality deduction from tw... |
| sssseq 3941 | If a class is a subclass o... |
| eqrd 3942 | Deduce equality of classes... |
| eqri 3943 | Infer equality of classes ... |
| eqelssd 3944 | Equality deduction from su... |
| ssid 3945 | Any class is a subclass of... |
| ssidd 3946 | Weakening of ~ ssid . (Co... |
| ssv 3947 | Any class is a subclass of... |
| sseq1 3948 | Equality theorem for subcl... |
| sseq2 3949 | Equality theorem for the s... |
| sseq12 3950 | Equality theorem for the s... |
| sseq1i 3951 | An equality inference for ... |
| sseq2i 3952 | An equality inference for ... |
| sseq12i 3953 | An equality inference for ... |
| sseq1d 3954 | An equality deduction for ... |
| sseq2d 3955 | An equality deduction for ... |
| sseq12d 3956 | An equality deduction for ... |
| eqsstrd 3957 | Substitution of equality i... |
| eqsstrrd 3958 | Substitution of equality i... |
| sseqtrd 3959 | Substitution of equality i... |
| sseqtrrd 3960 | Substitution of equality i... |
| eqsstrid 3961 | A chained subclass and equ... |
| eqsstrrid 3962 | A chained subclass and equ... |
| sseqtrdi 3963 | A chained subclass and equ... |
| sseqtrrdi 3964 | A chained subclass and equ... |
| sseqtrid 3965 | Subclass transitivity dedu... |
| sseqtrrid 3966 | Subclass transitivity dedu... |
| eqsstrdi 3967 | A chained subclass and equ... |
| eqsstrrdi 3968 | A chained subclass and equ... |
| eqsstri 3969 | Substitution of equality i... |
| eqsstrri 3970 | Substitution of equality i... |
| sseqtri 3971 | Substitution of equality i... |
| sseqtrri 3972 | Substitution of equality i... |
| 3sstr3i 3973 | Substitution of equality i... |
| 3sstr4i 3974 | Substitution of equality i... |
| 3sstr3g 3975 | Substitution of equality i... |
| 3sstr4g 3976 | Substitution of equality i... |
| 3sstr3d 3977 | Substitution of equality i... |
| 3sstr4d 3978 | Substitution of equality i... |
| eqimssd 3979 | Equality implies inclusion... |
| eqimsscd 3980 | Equality implies inclusion... |
| eqimss 3981 | Equality implies inclusion... |
| eqimss2 3982 | Equality implies inclusion... |
| eqimssi 3983 | Infer subclass relationshi... |
| eqimss2i 3984 | Infer subclass relationshi... |
| nssne1 3985 | Two classes are different ... |
| nssne2 3986 | Two classes are different ... |
| nss 3987 | Negation of subclass relat... |
| nelss 3988 | Demonstrate by witnesses t... |
| ssrexf 3989 | Restricted existential qua... |
| ssrmof 3990 | "At most one" existential ... |
| ssralv 3991 | Quantification restricted ... |
| ssrexv 3992 | Existential quantification... |
| ss2ralv 3993 | Two quantifications restri... |
| ss2rexv 3994 | Two existential quantifica... |
| ssralvOLD 3995 | Obsolete version of ~ ssra... |
| ssrexvOLD 3996 | Obsolete version of ~ ssre... |
| ralss 3997 | Restricted universal quant... |
| rexss 3998 | Restricted existential qua... |
| ralssOLD 3999 | Obsolete version of ~ rals... |
| rexssOLD 4000 | Obsolete version of ~ rexs... |
| ss2abim 4001 | Class abstractions in a su... |
| ss2ab 4002 | Class abstractions in a su... |
| abss 4003 | Class abstraction in a sub... |
| ssab 4004 | Subclass of a class abstra... |
| ssabral 4005 | The relation for a subclas... |
| ss2abdv 4006 | Deduction of abstraction s... |
| ss2abi 4007 | Inference of abstraction s... |
| abssdv 4008 | Deduction of abstraction s... |
| abssi 4009 | Inference of abstraction s... |
| ss2rab 4010 | Restricted abstraction cla... |
| rabss 4011 | Restricted class abstracti... |
| ssrab 4012 | Subclass of a restricted c... |
| ss2rabd 4013 | Subclass of a restricted c... |
| ssrabdv 4014 | Subclass of a restricted c... |
| rabssdv 4015 | Subclass of a restricted c... |
| ss2rabdv 4016 | Deduction of restricted ab... |
| ss2rabi 4017 | Inference of restricted ab... |
| rabss2 4018 | Subclass law for restricte... |
| rabss2OLD 4019 | Obsolete version of ~ rabs... |
| ssab2 4020 | Subclass relation for the ... |
| ssrab2 4021 | Subclass relation for a re... |
| rabss3d 4022 | Subclass law for restricte... |
| ssrab3 4023 | Subclass relation for a re... |
| rabssrabd 4024 | Subclass of a restricted c... |
| ssrabeq 4025 | If the restricting class o... |
| rabssab 4026 | A restricted class is a su... |
| eqrrabd 4027 | Deduce equality with a res... |
| uniiunlem 4028 | A subset relationship usef... |
| dfpss2 4029 | Alternate definition of pr... |
| dfpss3 4030 | Alternate definition of pr... |
| psseq1 4031 | Equality theorem for prope... |
| psseq2 4032 | Equality theorem for prope... |
| psseq1i 4033 | An equality inference for ... |
| psseq2i 4034 | An equality inference for ... |
| psseq12i 4035 | An equality inference for ... |
| psseq1d 4036 | An equality deduction for ... |
| psseq2d 4037 | An equality deduction for ... |
| psseq12d 4038 | An equality deduction for ... |
| pssss 4039 | A proper subclass is a sub... |
| pssne 4040 | Two classes in a proper su... |
| pssssd 4041 | Deduce subclass from prope... |
| pssned 4042 | Proper subclasses are uneq... |
| sspss 4043 | Subclass in terms of prope... |
| pssirr 4044 | Proper subclass is irrefle... |
| pssn2lp 4045 | Proper subclass has no 2-c... |
| sspsstri 4046 | Two ways of stating tricho... |
| ssnpss 4047 | Partial trichotomy law for... |
| psstr 4048 | Transitive law for proper ... |
| sspsstr 4049 | Transitive law for subclas... |
| psssstr 4050 | Transitive law for subclas... |
| psstrd 4051 | Proper subclass inclusion ... |
| sspsstrd 4052 | Transitivity involving sub... |
| psssstrd 4053 | Transitivity involving sub... |
| npss 4054 | A class is not a proper su... |
| ssnelpss 4055 | A subclass missing a membe... |
| ssnelpssd 4056 | Subclass inclusion with on... |
| ssexnelpss 4057 | If there is an element of ... |
| dfdif3 4058 | Alternate definition of cl... |
| dfdif3OLD 4059 | Obsolete version of ~ dfdi... |
| difeq1 4060 | Equality theorem for class... |
| difeq2 4061 | Equality theorem for class... |
| difeq12 4062 | Equality theorem for class... |
| difeq1i 4063 | Inference adding differenc... |
| difeq2i 4064 | Inference adding differenc... |
| difeq12i 4065 | Equality inference for cla... |
| difeq1d 4066 | Deduction adding differenc... |
| difeq2d 4067 | Deduction adding differenc... |
| difeq12d 4068 | Equality deduction for cla... |
| difeqri 4069 | Inference from membership ... |
| nfdif 4070 | Bound-variable hypothesis ... |
| nfdifOLD 4071 | Obsolete version of ~ nfdi... |
| eldifi 4072 | Implication of membership ... |
| eldifn 4073 | Implication of membership ... |
| elndif 4074 | A set does not belong to a... |
| neldif 4075 | Implication of membership ... |
| difdif 4076 | Double class difference. ... |
| difss 4077 | Subclass relationship for ... |
| difssd 4078 | A difference of two classe... |
| difss2 4079 | If a class is contained in... |
| difss2d 4080 | If a class is contained in... |
| ssdifss 4081 | Preservation of a subclass... |
| ddif 4082 | Double complement under un... |
| ssconb 4083 | Contraposition law for sub... |
| sscon 4084 | Contraposition law for sub... |
| ssdif 4085 | Difference law for subsets... |
| ssdifd 4086 | If ` A ` is contained in `... |
| sscond 4087 | If ` A ` is contained in `... |
| ssdifssd 4088 | If ` A ` is contained in `... |
| ssdif2d 4089 | If ` A ` is contained in `... |
| raldifb 4090 | Restricted universal quant... |
| rexdifi 4091 | Restricted existential qua... |
| complss 4092 | Complementation reverses i... |
| compleq 4093 | Two classes are equal if a... |
| elun 4094 | Expansion of membership in... |
| elunnel1 4095 | A member of a union that i... |
| elunnel2 4096 | A member of a union that i... |
| uneqri 4097 | Inference from membership ... |
| unidm 4098 | Idempotent law for union o... |
| uncom 4099 | Commutative law for union ... |
| equncom 4100 | If a class equals the unio... |
| equncomi 4101 | Inference form of ~ equnco... |
| uneq1 4102 | Equality theorem for the u... |
| uneq2 4103 | Equality theorem for the u... |
| uneq12 4104 | Equality theorem for the u... |
| uneq1i 4105 | Inference adding union to ... |
| uneq2i 4106 | Inference adding union to ... |
| uneq12i 4107 | Equality inference for the... |
| uneq1d 4108 | Deduction adding union to ... |
| uneq2d 4109 | Deduction adding union to ... |
| uneq12d 4110 | Equality deduction for the... |
| nfun 4111 | Bound-variable hypothesis ... |
| nfunOLD 4112 | Obsolete version of ~ nfun... |
| unass 4113 | Associative law for union ... |
| un12 4114 | A rearrangement of union. ... |
| un23 4115 | A rearrangement of union. ... |
| un4 4116 | A rearrangement of the uni... |
| unundi 4117 | Union distributes over its... |
| unundir 4118 | Union distributes over its... |
| ssun1 4119 | Subclass relationship for ... |
| ssun2 4120 | Subclass relationship for ... |
| ssun3 4121 | Subclass law for union of ... |
| ssun4 4122 | Subclass law for union of ... |
| elun1 4123 | Membership law for union o... |
| elun2 4124 | Membership law for union o... |
| elunant 4125 | A statement is true for ev... |
| unss1 4126 | Subclass law for union of ... |
| ssequn1 4127 | A relationship between sub... |
| unss2 4128 | Subclass law for union of ... |
| unss12 4129 | Subclass law for union of ... |
| ssequn2 4130 | A relationship between sub... |
| unss 4131 | The union of two subclasse... |
| unssi 4132 | An inference showing the u... |
| unssd 4133 | A deduction showing the un... |
| unssad 4134 | If ` ( A u. B ) ` is conta... |
| unssbd 4135 | If ` ( A u. B ) ` is conta... |
| ssun 4136 | A condition that implies i... |
| rexun 4137 | Restricted existential qua... |
| ralunb 4138 | Restricted quantification ... |
| ralun 4139 | Restricted quantification ... |
| elini 4140 | Membership in an intersect... |
| elind 4141 | Deduce membership in an in... |
| elinel1 4142 | Membership in an intersect... |
| elinel2 4143 | Membership in an intersect... |
| elin2 4144 | Membership in a class defi... |
| elin1d 4145 | Elementhood in the first s... |
| elin2d 4146 | Elementhood in the first s... |
| elin3 4147 | Membership in a class defi... |
| nel1nelin 4148 | Membership in an intersect... |
| nel2nelin 4149 | Membership in an intersect... |
| incom 4150 | Commutative law for inters... |
| ineqcom 4151 | Two ways of expressing tha... |
| ineqcomi 4152 | Two ways of expressing tha... |
| ineqri 4153 | Inference from membership ... |
| ineq1 4154 | Equality theorem for inter... |
| ineq2 4155 | Equality theorem for inter... |
| ineq12 4156 | Equality theorem for inter... |
| ineq1i 4157 | Equality inference for int... |
| ineq2i 4158 | Equality inference for int... |
| ineq12i 4159 | Equality inference for int... |
| ineq1d 4160 | Equality deduction for int... |
| ineq2d 4161 | Equality deduction for int... |
| ineq12d 4162 | Equality deduction for int... |
| ineqan12d 4163 | Equality deduction for int... |
| sseqin2 4164 | A relationship between sub... |
| nfin 4165 | Bound-variable hypothesis ... |
| nfinOLD 4166 | Obsolete version of ~ nfin... |
| rabbi2dva 4167 | Deduction from a wff to a ... |
| inidm 4168 | Idempotent law for interse... |
| inass 4169 | Associative law for inters... |
| in12 4170 | A rearrangement of interse... |
| in32 4171 | A rearrangement of interse... |
| in13 4172 | A rearrangement of interse... |
| in31 4173 | A rearrangement of interse... |
| inrot 4174 | Rotate the intersection of... |
| in4 4175 | Rearrangement of intersect... |
| inindi 4176 | Intersection distributes o... |
| inindir 4177 | Intersection distributes o... |
| inss1 4178 | The intersection of two cl... |
| inss2 4179 | The intersection of two cl... |
| ssin 4180 | Subclass of intersection. ... |
| ssini 4181 | An inference showing that ... |
| ssind 4182 | A deduction showing that a... |
| ssrin 4183 | Add right intersection to ... |
| sslin 4184 | Add left intersection to s... |
| ssrind 4185 | Add right intersection to ... |
| ss2in 4186 | Intersection of subclasses... |
| ssinss1 4187 | Intersection preserves sub... |
| ssinss1d 4188 | Intersection preserves sub... |
| inss 4189 | Inclusion of an intersecti... |
| ralin 4190 | Restricted universal quant... |
| rexin 4191 | Restricted existential qua... |
| dfss7 4192 | Alternate definition of su... |
| symdifcom 4195 | Symmetric difference commu... |
| symdifeq1 4196 | Equality theorem for symme... |
| symdifeq2 4197 | Equality theorem for symme... |
| nfsymdif 4198 | Hypothesis builder for sym... |
| elsymdif 4199 | Membership in a symmetric ... |
| dfsymdif4 4200 | Alternate definition of th... |
| elsymdifxor 4201 | Membership in a symmetric ... |
| dfsymdif2 4202 | Alternate definition of th... |
| symdifass 4203 | Symmetric difference is as... |
| difsssymdif 4204 | The symmetric difference c... |
| difsymssdifssd 4205 | If the symmetric differenc... |
| unabs 4206 | Absorption law for union. ... |
| inabs 4207 | Absorption law for interse... |
| nssinpss 4208 | Negation of subclass expre... |
| nsspssun 4209 | Negation of subclass expre... |
| dfss4 4210 | Subclass defined in terms ... |
| dfun2 4211 | An alternate definition of... |
| dfin2 4212 | An alternate definition of... |
| difin 4213 | Difference with intersecti... |
| ssdifim 4214 | Implication of a class dif... |
| ssdifsym 4215 | Symmetric class difference... |
| dfss5 4216 | Alternate definition of su... |
| dfun3 4217 | Union defined in terms of ... |
| dfin3 4218 | Intersection defined in te... |
| dfin4 4219 | Alternate definition of th... |
| invdif 4220 | Intersection with universa... |
| indif 4221 | Intersection with class di... |
| indif2 4222 | Bring an intersection in a... |
| indif1 4223 | Bring an intersection in a... |
| indifcom 4224 | Commutation law for inters... |
| indi 4225 | Distributive law for inter... |
| undi 4226 | Distributive law for union... |
| indir 4227 | Distributive law for inter... |
| undir 4228 | Distributive law for union... |
| unineq 4229 | Infer equality from equali... |
| uneqin 4230 | Equality of union and inte... |
| difundi 4231 | Distributive law for class... |
| difundir 4232 | Distributive law for class... |
| difindi 4233 | Distributive law for class... |
| difindir 4234 | Distributive law for class... |
| indifdi 4235 | Distribute intersection ov... |
| indifdir 4236 | Distribute intersection ov... |
| difdif2 4237 | Class difference by a clas... |
| undm 4238 | De Morgan's law for union.... |
| indm 4239 | De Morgan's law for inters... |
| difun1 4240 | A relationship involving d... |
| undif3 4241 | An equality involving clas... |
| difin2 4242 | Represent a class differen... |
| dif32 4243 | Swap second and third argu... |
| difabs 4244 | Absorption-like law for cl... |
| sscon34b 4245 | Relative complementation r... |
| rcompleq 4246 | Two subclasses are equal i... |
| dfsymdif3 4247 | Alternate definition of th... |
| unabw 4248 | Union of two class abstrac... |
| unab 4249 | Union of two class abstrac... |
| inab 4250 | Intersection of two class ... |
| difab 4251 | Difference of two class ab... |
| abanssl 4252 | A class abstraction with a... |
| abanssr 4253 | A class abstraction with a... |
| notabw 4254 | A class abstraction define... |
| notab 4255 | A class abstraction define... |
| unrab 4256 | Union of two restricted cl... |
| inrab 4257 | Intersection of two restri... |
| inrab2 4258 | Intersection with a restri... |
| difrab 4259 | Difference of two restrict... |
| dfrab3 4260 | Alternate definition of re... |
| dfrab2 4261 | Alternate definition of re... |
| rabdif 4262 | Move difference in and out... |
| notrab 4263 | Complementation of restric... |
| dfrab3ss 4264 | Restricted class abstracti... |
| rabun2 4265 | Abstraction restricted to ... |
| reuun2 4266 | Transfer uniqueness to a s... |
| reuss2 4267 | Transfer uniqueness to a s... |
| reuss 4268 | Transfer uniqueness to a s... |
| reuun1 4269 | Transfer uniqueness to a s... |
| reupick 4270 | Restricted uniqueness "pic... |
| reupick3 4271 | Restricted uniqueness "pic... |
| reupick2 4272 | Restricted uniqueness "pic... |
| euelss 4273 | Transfer uniqueness of an ... |
| dfnul4 4276 | Alternate definition of th... |
| dfnul2 4277 | Alternate definition of th... |
| dfnul3 4278 | Alternate definition of th... |
| noel 4279 | The empty set has no eleme... |
| nel02 4280 | The empty set has no eleme... |
| n0i 4281 | If a class has elements, t... |
| ne0i 4282 | If a class has elements, t... |
| ne0d 4283 | Deduction form of ~ ne0i .... |
| n0ii 4284 | If a class has elements, t... |
| ne0ii 4285 | If a class has elements, t... |
| vn0 4286 | The universal class is not... |
| vn0ALT 4287 | Alternate proof of ~ vn0 .... |
| eq0f 4288 | A class is equal to the em... |
| neq0f 4289 | A class is not empty if an... |
| n0f 4290 | A class is nonempty if and... |
| eq0 4291 | A class is equal to the em... |
| eq0ALT 4292 | Alternate proof of ~ eq0 .... |
| neq0 4293 | A class is not empty if an... |
| n0 4294 | A class is nonempty if and... |
| nel0 4295 | From the general negation ... |
| reximdva0 4296 | Restricted existence deduc... |
| rspn0 4297 | Specialization for restric... |
| n0rex 4298 | There is an element in a n... |
| ssn0rex 4299 | There is an element in a c... |
| n0moeu 4300 | A case of equivalence of "... |
| rex0 4301 | Vacuous restricted existen... |
| reu0 4302 | Vacuous restricted uniquen... |
| rmo0 4303 | Vacuous restricted at-most... |
| 0el 4304 | Membership of the empty se... |
| n0el 4305 | Negated membership of the ... |
| eqeuel 4306 | A condition which implies ... |
| ssdif0 4307 | Subclass expressed in term... |
| difn0 4308 | If the difference of two s... |
| pssdifn0 4309 | A proper subclass has a no... |
| pssdif 4310 | A proper subclass has a no... |
| ndisj 4311 | Express that an intersecti... |
| inn0f 4312 | A nonempty intersection. ... |
| inn0 4313 | A nonempty intersection. ... |
| difin0ss 4314 | Difference, intersection, ... |
| inssdif0 4315 | Intersection, subclass, an... |
| inindif 4316 | The intersection and class... |
| difid 4317 | The difference between a c... |
| difidALT 4318 | Alternate proof of ~ difid... |
| dif0 4319 | The difference between a c... |
| ab0w 4320 | The class of sets verifyin... |
| ab0 4321 | The class of sets verifyin... |
| ab0ALT 4322 | Alternate proof of ~ ab0 ,... |
| dfnf5 4323 | Characterization of nonfre... |
| ab0orv 4324 | The class abstraction defi... |
| ab0orvALT 4325 | Alternate proof of ~ ab0or... |
| abn0 4326 | Nonempty class abstraction... |
| rab0 4327 | Any restricted class abstr... |
| rabeq0w 4328 | Condition for a restricted... |
| rabeq0 4329 | Condition for a restricted... |
| rabn0 4330 | Nonempty restricted class ... |
| rabxm 4331 | Law of excluded middle, in... |
| rabnc 4332 | Law of noncontradiction, i... |
| elneldisj 4333 | The set of elements ` s ` ... |
| elnelun 4334 | The union of the set of el... |
| un0 4335 | The union of a class with ... |
| in0 4336 | The intersection of a clas... |
| 0un 4337 | The union of the empty set... |
| 0in 4338 | The intersection of the em... |
| inv1 4339 | The intersection of a clas... |
| unv 4340 | The union of a class with ... |
| 0ss 4341 | The null set is a subset o... |
| ss0b 4342 | Any subset of the empty se... |
| ss0 4343 | Any subset of the empty se... |
| sseq0 4344 | A subclass of an empty cla... |
| ssn0 4345 | A class with a nonempty su... |
| 0dif 4346 | The difference between the... |
| abf 4347 | A class abstraction determ... |
| eq0rdv 4348 | Deduction for equality to ... |
| eq0rdvALT 4349 | Alternate proof of ~ eq0rd... |
| csbprc 4350 | The proper substitution of... |
| csb0 4351 | The proper substitution of... |
| sbcel12 4352 | Distribute proper substitu... |
| sbceqg 4353 | Distribute proper substitu... |
| sbceqi 4354 | Distribution of class subs... |
| sbcnel12g 4355 | Distribute proper substitu... |
| sbcne12 4356 | Distribute proper substitu... |
| sbcel1g 4357 | Move proper substitution i... |
| sbceq1g 4358 | Move proper substitution t... |
| sbcel2 4359 | Move proper substitution i... |
| sbceq2g 4360 | Move proper substitution t... |
| csbcom 4361 | Commutative law for double... |
| sbcnestgfw 4362 | Nest the composition of tw... |
| csbnestgfw 4363 | Nest the composition of tw... |
| sbcnestgw 4364 | Nest the composition of tw... |
| csbnestgw 4365 | Nest the composition of tw... |
| sbcco3gw 4366 | Composition of two substit... |
| sbcnestgf 4367 | Nest the composition of tw... |
| csbnestgf 4368 | Nest the composition of tw... |
| sbcnestg 4369 | Nest the composition of tw... |
| csbnestg 4370 | Nest the composition of tw... |
| sbcco3g 4371 | Composition of two substit... |
| csbco3g 4372 | Composition of two class s... |
| csbnest1g 4373 | Nest the composition of tw... |
| csbidm 4374 | Idempotent law for class s... |
| csbvarg 4375 | The proper substitution of... |
| csbvargi 4376 | The proper substitution of... |
| sbccsb 4377 | Substitution into a wff ex... |
| sbccsb2 4378 | Substitution into a wff ex... |
| rspcsbela 4379 | Special case related to ~ ... |
| sbnfc2 4380 | Two ways of expressing " `... |
| csbab 4381 | Move substitution into a c... |
| csbun 4382 | Distribution of class subs... |
| csbin 4383 | Distribute proper substitu... |
| csbie2df 4384 | Conversion of implicit sub... |
| 2nreu 4385 | If there are two different... |
| un00 4386 | Two classes are empty iff ... |
| vss 4387 | Only the universal class h... |
| 0pss 4388 | The null set is a proper s... |
| npss0 4389 | No set is a proper subset ... |
| pssv 4390 | Any non-universal class is... |
| disj 4391 | Two ways of saying that tw... |
| disjr 4392 | Two ways of saying that tw... |
| disj1 4393 | Two ways of saying that tw... |
| reldisj 4394 | Two ways of saying that tw... |
| disj3 4395 | Two ways of saying that tw... |
| disjne 4396 | Members of disjoint sets a... |
| disjeq0 4397 | Two disjoint sets are equa... |
| disjel 4398 | A set can't belong to both... |
| disj2 4399 | Two ways of saying that tw... |
| disj4 4400 | Two ways of saying that tw... |
| ssdisj 4401 | Intersection with a subcla... |
| disjpss 4402 | A class is a proper subset... |
| undisj1 4403 | The union of disjoint clas... |
| undisj2 4404 | The union of disjoint clas... |
| ssindif0 4405 | Subclass expressed in term... |
| inelcm 4406 | The intersection of classe... |
| minel 4407 | A minimum element of a cla... |
| undif4 4408 | Distribute union over diff... |
| disjssun 4409 | Subset relation for disjoi... |
| vdif0 4410 | Universal class equality i... |
| difrab0eq 4411 | If the difference between ... |
| pssnel 4412 | A proper subclass has a me... |
| disjdif 4413 | A class and its relative c... |
| disjdifr 4414 | A class and its relative c... |
| difin0 4415 | The difference of a class ... |
| unvdif 4416 | The union of a class and i... |
| undif1 4417 | Absorption of difference b... |
| undif2 4418 | Absorption of difference b... |
| undifabs 4419 | Absorption of difference b... |
| inundif 4420 | The intersection and class... |
| disjdif2 4421 | The difference of a class ... |
| difun2 4422 | Absorption of union by dif... |
| undif 4423 | Union of complementary par... |
| undifr 4424 | Union of complementary par... |
| undif5 4425 | An equality involving clas... |
| ssdifin0 4426 | A subset of a difference d... |
| ssdifeq0 4427 | A class is a subclass of i... |
| ssundif 4428 | A condition equivalent to ... |
| difcom 4429 | Swap the arguments of a cl... |
| pssdifcom1 4430 | Two ways to express overla... |
| pssdifcom2 4431 | Two ways to express non-co... |
| difdifdir 4432 | Distributive law for class... |
| uneqdifeq 4433 | Two ways to say that ` A `... |
| raldifeq 4434 | Equality theorem for restr... |
| rzal 4435 | Vacuous quantification is ... |
| rzalALT 4436 | Alternate proof of ~ rzal ... |
| rexn0 4437 | Restricted existential qua... |
| ralf0 4438 | The quantification of a fa... |
| ral0 4439 | Vacuous universal quantifi... |
| r19.2z 4440 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4441 | A response to the notion t... |
| r19.3rz 4442 | Restricted quantification ... |
| r19.28z 4443 | Restricted quantifier vers... |
| r19.3rzv 4444 | Restricted quantification ... |
| r19.3rzvOLD 4445 | Obsolete version of ~ r19.... |
| r19.9rzv 4446 | Restricted quantification ... |
| r19.28zv 4447 | Restricted quantifier vers... |
| r19.37zv 4448 | Restricted quantifier vers... |
| r19.45zv 4449 | Restricted version of Theo... |
| r19.44zv 4450 | Restricted version of Theo... |
| r19.27z 4451 | Restricted quantifier vers... |
| r19.27zv 4452 | Restricted quantifier vers... |
| r19.36zv 4453 | Restricted quantifier vers... |
| ralnralall 4454 | A contradiction concerning... |
| falseral0 4455 | A false statement can only... |
| falseral0OLD 4456 | Obsolete version of ~ fals... |
| ralidmw 4457 | Idempotent law for restric... |
| ralidm 4458 | Idempotent law for restric... |
| raaan 4459 | Rearrange restricted quant... |
| raaanv 4460 | Rearrange restricted quant... |
| sbss 4461 | Set substitution into the ... |
| sbcssg 4462 | Distribute proper substitu... |
| raaan2 4463 | Rearrange restricted quant... |
| 2reu4lem 4464 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4465 | Definition of double restr... |
| csbdif 4466 | Distribution of class subs... |
| dfif2 4469 | An alternate definition of... |
| dfif6 4470 | An alternate definition of... |
| ifeq1 4471 | Equality theorem for condi... |
| ifeq2 4472 | Equality theorem for condi... |
| iftrue 4473 | Value of the conditional o... |
| iftruei 4474 | Inference associated with ... |
| iftrued 4475 | Value of the conditional o... |
| iffalse 4476 | Value of the conditional o... |
| iffalsei 4477 | Inference associated with ... |
| iffalsed 4478 | Value of the conditional o... |
| ifnefalse 4479 | When values are unequal, b... |
| iftrueb 4480 | When the branches are not ... |
| ifsb 4481 | Distribute a function over... |
| dfif3 4482 | Alternate definition of th... |
| dfif4 4483 | Alternate definition of th... |
| dfif5 4484 | Alternate definition of th... |
| ifssun 4485 | A conditional class is inc... |
| ifeq12 4486 | Equality theorem for condi... |
| ifeq1d 4487 | Equality deduction for con... |
| ifeq2d 4488 | Equality deduction for con... |
| ifeq12d 4489 | Equality deduction for con... |
| ifbi 4490 | Equivalence theorem for co... |
| ifbid 4491 | Equivalence deduction for ... |
| ifbieq1d 4492 | Equivalence/equality deduc... |
| ifbieq2i 4493 | Equivalence/equality infer... |
| ifbieq2d 4494 | Equivalence/equality deduc... |
| ifbieq12i 4495 | Equivalence deduction for ... |
| ifbieq12d 4496 | Equivalence deduction for ... |
| nfifd 4497 | Deduction form of ~ nfif .... |
| nfif 4498 | Bound-variable hypothesis ... |
| ifeq1da 4499 | Conditional equality. (Co... |
| ifeq2da 4500 | Conditional equality. (Co... |
| ifeq12da 4501 | Equivalence deduction for ... |
| ifbieq12d2 4502 | Equivalence deduction for ... |
| ifclda 4503 | Conditional closure. (Con... |
| ifeqda 4504 | Separation of the values o... |
| elimif 4505 | Elimination of a condition... |
| ifbothda 4506 | A wff ` th ` containing a ... |
| ifboth 4507 | A wff ` th ` containing a ... |
| ifid 4508 | Identical true and false a... |
| eqif 4509 | Expansion of an equality w... |
| ifval 4510 | Another expression of the ... |
| elif 4511 | Membership in a conditiona... |
| ifel 4512 | Membership of a conditiona... |
| ifcl 4513 | Membership (closure) of a ... |
| ifcld 4514 | Membership (closure) of a ... |
| ifcli 4515 | Inference associated with ... |
| ifexd 4516 | Existence of the condition... |
| ifexg 4517 | Existence of the condition... |
| ifex 4518 | Existence of the condition... |
| ifeqor 4519 | The possible values of a c... |
| ifnot 4520 | Negating the first argumen... |
| ifan 4521 | Rewrite a conjunction in a... |
| ifor 4522 | Rewrite a disjunction in a... |
| 2if2 4523 | Resolve two nested conditi... |
| ifcomnan 4524 | Commute the conditions in ... |
| csbif 4525 | Distribute proper substitu... |
| dedth 4526 | Weak deduction theorem tha... |
| dedth2h 4527 | Weak deduction theorem eli... |
| dedth3h 4528 | Weak deduction theorem eli... |
| dedth4h 4529 | Weak deduction theorem eli... |
| dedth2v 4530 | Weak deduction theorem for... |
| dedth3v 4531 | Weak deduction theorem for... |
| dedth4v 4532 | Weak deduction theorem for... |
| elimhyp 4533 | Eliminate a hypothesis con... |
| elimhyp2v 4534 | Eliminate a hypothesis con... |
| elimhyp3v 4535 | Eliminate a hypothesis con... |
| elimhyp4v 4536 | Eliminate a hypothesis con... |
| elimel 4537 | Eliminate a membership hyp... |
| elimdhyp 4538 | Version of ~ elimhyp where... |
| keephyp 4539 | Transform a hypothesis ` p... |
| keephyp2v 4540 | Keep a hypothesis containi... |
| keephyp3v 4541 | Keep a hypothesis containi... |
| pwjust 4543 | Soundness justification th... |
| elpwg 4545 | Membership in a power clas... |
| elpw 4546 | Membership in a power clas... |
| velpw 4547 | Setvar variable membership... |
| elpwd 4548 | Membership in a power clas... |
| elpwi 4549 | Subset relation implied by... |
| elpwb 4550 | Characterization of the el... |
| elpwid 4551 | An element of a power clas... |
| elelpwi 4552 | If ` A ` belongs to a part... |
| sspw 4553 | The powerclass preserves i... |
| sspwi 4554 | The powerclass preserves i... |
| sspwd 4555 | The powerclass preserves i... |
| pweq 4556 | Equality theorem for power... |
| pweqALT 4557 | Alternate proof of ~ pweq ... |
| pweqi 4558 | Equality inference for pow... |
| pweqd 4559 | Equality deduction for pow... |
| pwunss 4560 | The power class of the uni... |
| nfpw 4561 | Bound-variable hypothesis ... |
| pwidg 4562 | A set is an element of its... |
| pwidb 4563 | A class is an element of i... |
| pwid 4564 | A set is a member of its p... |
| pwss 4565 | Subclass relationship for ... |
| pwundif 4566 | Break up the power class o... |
| snjust 4567 | Soundness justification th... |
| sneq 4578 | Equality theorem for singl... |
| sneqi 4579 | Equality inference for sin... |
| sneqd 4580 | Equality deduction for sin... |
| dfsn2 4581 | Alternate definition of si... |
| elsng 4582 | There is exactly one eleme... |
| elsn 4583 | There is exactly one eleme... |
| velsn 4584 | There is only one element ... |
| elsni 4585 | There is at most one eleme... |
| elsnd 4586 | There is at most one eleme... |
| rabsneq 4587 | Equality of class abstract... |
| absn 4588 | Condition for a class abst... |
| dfpr2 4589 | Alternate definition of a ... |
| dfsn2ALT 4590 | Alternate definition of si... |
| elprg 4591 | A member of a pair of clas... |
| elpri 4592 | If a class is an element o... |
| elpr 4593 | A member of a pair of clas... |
| elpr2g 4594 | A member of a pair of sets... |
| elpr2 4595 | A member of a pair of sets... |
| elprn1 4596 | A member of an unordered p... |
| elprn2 4597 | A member of an unordered p... |
| nelpr2 4598 | If a class is not an eleme... |
| nelpr1 4599 | If a class is not an eleme... |
| nelpri 4600 | If an element doesn't matc... |
| prneli 4601 | If an element doesn't matc... |
| nelprd 4602 | If an element doesn't matc... |
| eldifpr 4603 | Membership in a set with t... |
| rexdifpr 4604 | Restricted existential qua... |
| snidg 4605 | A set is a member of its s... |
| snidb 4606 | A class is a set iff it is... |
| snid 4607 | A set is a member of its s... |
| vsnid 4608 | A setvar variable is a mem... |
| elsn2g 4609 | There is exactly one eleme... |
| elsn2 4610 | There is exactly one eleme... |
| nelsn 4611 | If a class is not equal to... |
| rabeqsn 4612 | Conditions for a restricte... |
| rabsssn 4613 | Conditions for a restricte... |
| rabeqsnd 4614 | Conditions for a restricte... |
| ralsnsg 4615 | Substitution expressed in ... |
| rexsns 4616 | Restricted existential qua... |
| rexsngf 4617 | Restricted existential qua... |
| ralsngf 4618 | Restricted universal quant... |
| reusngf 4619 | Restricted existential uni... |
| ralsng 4620 | Substitution expressed in ... |
| rexsng 4621 | Restricted existential qua... |
| reusng 4622 | Restricted existential uni... |
| 2ralsng 4623 | Substitution expressed in ... |
| rexreusng 4624 | Restricted existential uni... |
| exsnrex 4625 | There is a set being the e... |
| ralsn 4626 | Convert a universal quanti... |
| rexsn 4627 | Convert an existential qua... |
| elunsn 4628 | Elementhood in a union wit... |
| elpwunsn 4629 | Membership in an extension... |
| eqoreldif 4630 | An element of a set is eit... |
| eltpg 4631 | Members of an unordered tr... |
| eldiftp 4632 | Membership in a set with t... |
| eltpi 4633 | A member of an unordered t... |
| eltp 4634 | A member of an unordered t... |
| el7g 4635 | Members of a set with seve... |
| dftp2 4636 | Alternate definition of un... |
| nfpr 4637 | Bound-variable hypothesis ... |
| ifpr 4638 | Membership of a conditiona... |
| ralprgf 4639 | Convert a restricted unive... |
| rexprgf 4640 | Convert a restricted exist... |
| ralprg 4641 | Convert a restricted unive... |
| rexprg 4642 | Convert a restricted exist... |
| raltpg 4643 | Convert a restricted unive... |
| rextpg 4644 | Convert a restricted exist... |
| ralpr 4645 | Convert a restricted unive... |
| rexpr 4646 | Convert a restricted exist... |
| reuprg0 4647 | Convert a restricted exist... |
| reuprg 4648 | Convert a restricted exist... |
| reurexprg 4649 | Convert a restricted exist... |
| raltp 4650 | Convert a universal quanti... |
| rextp 4651 | Convert an existential qua... |
| nfsn 4652 | Bound-variable hypothesis ... |
| csbsng 4653 | Distribute proper substitu... |
| csbprg 4654 | Distribute proper substitu... |
| elinsn 4655 | If the intersection of two... |
| disjsn 4656 | Intersection with the sing... |
| disjsn2 4657 | Two distinct singletons ar... |
| disjpr2 4658 | Two completely distinct un... |
| disjprsn 4659 | The disjoint intersection ... |
| disjtpsn 4660 | The disjoint intersection ... |
| disjtp2 4661 | Two completely distinct un... |
| snprc 4662 | The singleton of a proper ... |
| snnzb 4663 | A singleton is nonempty if... |
| rmosn 4664 | A restricted at-most-one q... |
| r19.12sn 4665 | Special case of ~ r19.12 w... |
| rabsn 4666 | Condition where a restrict... |
| rabsnifsb 4667 | A restricted class abstrac... |
| rabsnif 4668 | A restricted class abstrac... |
| rabrsn 4669 | A restricted class abstrac... |
| euabsn2 4670 | Another way to express exi... |
| euabsn 4671 | Another way to express exi... |
| reusn 4672 | A way to express restricte... |
| absneu 4673 | Restricted existential uni... |
| rabsneu 4674 | Restricted existential uni... |
| eusn 4675 | Two ways to express " ` A ... |
| rabsnt 4676 | Truth implied by equality ... |
| prcom 4677 | Commutative law for unorde... |
| preq1 4678 | Equality theorem for unord... |
| preq2 4679 | Equality theorem for unord... |
| preq12 4680 | Equality theorem for unord... |
| preq1i 4681 | Equality inference for uno... |
| preq2i 4682 | Equality inference for uno... |
| preq12i 4683 | Equality inference for uno... |
| preq1d 4684 | Equality deduction for uno... |
| preq2d 4685 | Equality deduction for uno... |
| preq12d 4686 | Equality deduction for uno... |
| tpeq1 4687 | Equality theorem for unord... |
| tpeq2 4688 | Equality theorem for unord... |
| tpeq3 4689 | Equality theorem for unord... |
| tpeq1d 4690 | Equality theorem for unord... |
| tpeq2d 4691 | Equality theorem for unord... |
| tpeq3d 4692 | Equality theorem for unord... |
| tpeq123d 4693 | Equality theorem for unord... |
| tprot 4694 | Rotation of the elements o... |
| tpcoma 4695 | Swap 1st and 2nd members o... |
| tpcomb 4696 | Swap 2nd and 3rd members o... |
| tpass 4697 | Split off the first elemen... |
| qdass 4698 | Two ways to write an unord... |
| qdassr 4699 | Two ways to write an unord... |
| tpidm12 4700 | Unordered triple ` { A , A... |
| tpidm13 4701 | Unordered triple ` { A , B... |
| tpidm23 4702 | Unordered triple ` { A , B... |
| tpidm 4703 | Unordered triple ` { A , A... |
| tppreq3 4704 | An unordered triple is an ... |
| prid1g 4705 | An unordered pair contains... |
| prid2g 4706 | An unordered pair contains... |
| prid1 4707 | An unordered pair contains... |
| prid2 4708 | An unordered pair contains... |
| ifpprsnss 4709 | An unordered pair is a sin... |
| prprc1 4710 | A proper class vanishes in... |
| prprc2 4711 | A proper class vanishes in... |
| prprc 4712 | An unordered pair containi... |
| tpid1 4713 | One of the three elements ... |
| tpid1g 4714 | Closed theorem form of ~ t... |
| tpid2 4715 | One of the three elements ... |
| tpid2g 4716 | Closed theorem form of ~ t... |
| tpid3g 4717 | Closed theorem form of ~ t... |
| tpid3 4718 | One of the three elements ... |
| snnzg 4719 | The singleton of a set is ... |
| snn0d 4720 | The singleton of a set is ... |
| snnz 4721 | The singleton of a set is ... |
| prnz 4722 | A pair containing a set is... |
| prnzg 4723 | A pair containing a set is... |
| tpnz 4724 | An unordered triple contai... |
| tpnzd 4725 | An unordered triple contai... |
| raltpd 4726 | Convert a universal quanti... |
| snssb 4727 | Characterization of the in... |
| snssg 4728 | The singleton formed on a ... |
| snss 4729 | The singleton of an elemen... |
| eldifsn 4730 | Membership in a set with a... |
| eldifsnd 4731 | Membership in a set with a... |
| ssdifsn 4732 | Subset of a set with an el... |
| elpwdifsn 4733 | A subset of a set is an el... |
| eldifsni 4734 | Membership in a set with a... |
| eldifsnneq 4735 | An element of a difference... |
| neldifsn 4736 | The class ` A ` is not in ... |
| neldifsnd 4737 | The class ` A ` is not in ... |
| rexdifsn 4738 | Restricted existential qua... |
| raldifsni 4739 | Rearrangement of a propert... |
| raldifsnb 4740 | Restricted universal quant... |
| eldifvsn 4741 | A set is an element of the... |
| difsn 4742 | An element not in a set ca... |
| difprsnss 4743 | Removal of a singleton fro... |
| difprsn1 4744 | Removal of a singleton fro... |
| difprsn2 4745 | Removal of a singleton fro... |
| diftpsn3 4746 | Removal of a singleton fro... |
| difpr 4747 | Removing two elements as p... |
| tpprceq3 4748 | An unordered triple is an ... |
| tppreqb 4749 | An unordered triple is an ... |
| difsnb 4750 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4751 | ` ( B \ { A } ) ` is a pro... |
| snssi 4752 | The singleton of an elemen... |
| snssd 4753 | The singleton of an elemen... |
| difsnid 4754 | If we remove a single elem... |
| eldifeldifsn 4755 | An element of a difference... |
| pw0 4756 | Compute the power set of t... |
| pwpw0 4757 | Compute the power set of t... |
| snsspr1 4758 | A singleton is a subset of... |
| snsspr2 4759 | A singleton is a subset of... |
| snsstp1 4760 | A singleton is a subset of... |
| snsstp2 4761 | A singleton is a subset of... |
| snsstp3 4762 | A singleton is a subset of... |
| prssg 4763 | A pair of elements of a cl... |
| prss 4764 | A pair of elements of a cl... |
| prssi 4765 | A pair of elements of a cl... |
| prssd 4766 | Deduction version of ~ prs... |
| prsspwg 4767 | An unordered pair belongs ... |
| ssprss 4768 | A pair as subset of a pair... |
| ssprsseq 4769 | A proper pair is a subset ... |
| sssn 4770 | The subsets of a singleton... |
| ssunsn2 4771 | The property of being sand... |
| ssunsn 4772 | Possible values for a set ... |
| eqsn 4773 | Two ways to express that a... |
| eqsnd 4774 | Deduce that a set is a sin... |
| eqsndOLD 4775 | Obsolete version of ~ eqsn... |
| issn 4776 | A sufficient condition for... |
| n0snor2el 4777 | A nonempty set is either a... |
| ssunpr 4778 | Possible values for a set ... |
| sspr 4779 | The subsets of a pair. (C... |
| sstp 4780 | The subsets of an unordere... |
| tpss 4781 | An unordered triple of ele... |
| tpssi 4782 | An unordered triple of ele... |
| sneqrg 4783 | Closed form of ~ sneqr . ... |
| sneqr 4784 | If the singletons of two s... |
| snsssn 4785 | If a singleton is a subset... |
| mosneq 4786 | There exists at most one s... |
| sneqbg 4787 | Two singletons of sets are... |
| snsspw 4788 | The singleton of a class i... |
| prsspw 4789 | An unordered pair belongs ... |
| preq1b 4790 | Biconditional equality lem... |
| preq2b 4791 | Biconditional equality lem... |
| preqr1 4792 | Reverse equality lemma for... |
| preqr2 4793 | Reverse equality lemma for... |
| preq12b 4794 | Equality relationship for ... |
| opthpr 4795 | An unordered pair has the ... |
| preqr1g 4796 | Reverse equality lemma for... |
| preq12bg 4797 | Closed form of ~ preq12b .... |
| prneimg 4798 | Two pairs are not equal if... |
| prneimg2 4799 | Two pairs are not equal if... |
| prnebg 4800 | A (proper) pair is not equ... |
| pr1eqbg 4801 | A (proper) pair is equal t... |
| pr1nebg 4802 | A (proper) pair is not equ... |
| preqsnd 4803 | Equivalence for a pair equ... |
| prnesn 4804 | A proper unordered pair is... |
| prneprprc 4805 | A proper unordered pair is... |
| preqsn 4806 | Equivalence for a pair equ... |
| preq12nebg 4807 | Equality relationship for ... |
| prel12g 4808 | Equality of two unordered ... |
| opthprneg 4809 | An unordered pair has the ... |
| elpreqprlem 4810 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4811 | Equality and membership ru... |
| elpreqprb 4812 | A set is an element of an ... |
| elpr2elpr 4813 | For an element ` A ` of an... |
| dfopif 4814 | Rewrite ~ df-op using ` if... |
| dfopg 4815 | Value of the ordered pair ... |
| dfop 4816 | Value of an ordered pair w... |
| opeq1 4817 | Equality theorem for order... |
| opeq2 4818 | Equality theorem for order... |
| opeq12 4819 | Equality theorem for order... |
| opeq1i 4820 | Equality inference for ord... |
| opeq2i 4821 | Equality inference for ord... |
| opeq12i 4822 | Equality inference for ord... |
| opeq1d 4823 | Equality deduction for ord... |
| opeq2d 4824 | Equality deduction for ord... |
| opeq12d 4825 | Equality deduction for ord... |
| oteq1 4826 | Equality theorem for order... |
| oteq2 4827 | Equality theorem for order... |
| oteq3 4828 | Equality theorem for order... |
| oteq1d 4829 | Equality deduction for ord... |
| oteq2d 4830 | Equality deduction for ord... |
| oteq3d 4831 | Equality deduction for ord... |
| oteq123d 4832 | Equality deduction for ord... |
| nfop 4833 | Bound-variable hypothesis ... |
| nfopd 4834 | Deduction version of bound... |
| csbopg 4835 | Distribution of class subs... |
| opidg 4836 | The ordered pair ` <. A , ... |
| opid 4837 | The ordered pair ` <. A , ... |
| ralunsn 4838 | Restricted quantification ... |
| 2ralunsn 4839 | Double restricted quantifi... |
| opprc 4840 | Expansion of an ordered pa... |
| opprc1 4841 | Expansion of an ordered pa... |
| opprc2 4842 | Expansion of an ordered pa... |
| oprcl 4843 | If an ordered pair has an ... |
| pwsn 4844 | The power set of a singlet... |
| pwpr 4845 | The power set of an unorde... |
| pwtp 4846 | The power set of an unorde... |
| pwpwpw0 4847 | Compute the power set of t... |
| pwv 4848 | The power class of the uni... |
| prproe 4849 | For an element of a proper... |
| 3elpr2eq 4850 | If there are three element... |
| dfuni2 4853 | Alternate definition of cl... |
| eluni 4854 | Membership in class union.... |
| eluni2 4855 | Membership in class union.... |
| elunii 4856 | Membership in class union.... |
| nfunid 4857 | Deduction version of ~ nfu... |
| nfuni 4858 | Bound-variable hypothesis ... |
| uniss 4859 | Subclass relationship for ... |
| unissi 4860 | Subclass relationship for ... |
| unissd 4861 | Subclass relationship for ... |
| unieq 4862 | Equality theorem for class... |
| unieqi 4863 | Inference of equality of t... |
| unieqd 4864 | Deduction of equality of t... |
| eluniab 4865 | Membership in union of a c... |
| elunirab 4866 | Membership in union of a c... |
| uniprg 4867 | The union of a pair is the... |
| unipr 4868 | The union of a pair is the... |
| unisng 4869 | A set equals the union of ... |
| unisn 4870 | A set equals the union of ... |
| unisnv 4871 | A set equals the union of ... |
| unisn3 4872 | Union of a singleton in th... |
| dfnfc2 4873 | An alternative statement o... |
| uniun 4874 | The class union of the uni... |
| uniin 4875 | The class union of the int... |
| ssuni 4876 | Subclass relationship for ... |
| uni0b 4877 | The union of a set is empt... |
| uni0c 4878 | The union of a set is empt... |
| uni0 4879 | The union of the empty set... |
| uni0OLD 4880 | Obsolete version of ~ uni0... |
| csbuni 4881 | Distribute proper substitu... |
| elssuni 4882 | An element of a class is a... |
| unissel 4883 | Condition turning a subcla... |
| unissb 4884 | Relationship involving mem... |
| uniss2 4885 | A subclass condition on th... |
| unidif 4886 | If the difference ` A \ B ... |
| ssunieq 4887 | Relationship implying unio... |
| unimax 4888 | Any member of a class is t... |
| pwuni 4889 | A class is a subclass of t... |
| dfint2 4892 | Alternate definition of cl... |
| inteq 4893 | Equality law for intersect... |
| inteqi 4894 | Equality inference for cla... |
| inteqd 4895 | Equality deduction for cla... |
| elint 4896 | Membership in class inters... |
| elint2 4897 | Membership in class inters... |
| elintg 4898 | Membership in class inters... |
| elinti 4899 | Membership in class inters... |
| nfint 4900 | Bound-variable hypothesis ... |
| elintabg 4901 | Two ways of saying a set i... |
| elintab 4902 | Membership in the intersec... |
| elintrab 4903 | Membership in the intersec... |
| elintrabg 4904 | Membership in the intersec... |
| int0 4905 | The intersection of the em... |
| intss1 4906 | An element of a class incl... |
| ssint 4907 | Subclass of a class inters... |
| ssintab 4908 | Subclass of the intersecti... |
| ssintub 4909 | Subclass of the least uppe... |
| ssmin 4910 | Subclass of the minimum va... |
| intmin 4911 | Any member of a class is t... |
| intss 4912 | Intersection of subclasses... |
| intssuni 4913 | The intersection of a none... |
| ssintrab 4914 | Subclass of the intersecti... |
| unissint 4915 | If the union of a class is... |
| intssuni2 4916 | Subclass relationship for ... |
| intminss 4917 | Under subset ordering, the... |
| intmin2 4918 | Any set is the smallest of... |
| intmin3 4919 | Under subset ordering, the... |
| intmin4 4920 | Elimination of a conjunct ... |
| intab 4921 | The intersection of a spec... |
| int0el 4922 | The intersection of a clas... |
| intun 4923 | The class intersection of ... |
| intprg 4924 | The intersection of a pair... |
| intpr 4925 | The intersection of a pair... |
| intsng 4926 | Intersection of a singleto... |
| intsn 4927 | The intersection of a sing... |
| uniintsn 4928 | Two ways to express " ` A ... |
| uniintab 4929 | The union and the intersec... |
| intunsn 4930 | Theorem joining a singleto... |
| rint0 4931 | Relative intersection of a... |
| elrint 4932 | Membership in a restricted... |
| elrint2 4933 | Membership in a restricted... |
| eliun 4938 | Membership in indexed unio... |
| eliin 4939 | Membership in indexed inte... |
| eliuni 4940 | Membership in an indexed u... |
| eliund 4941 | Membership in indexed unio... |
| iuncom 4942 | Commutation of indexed uni... |
| iuncom4 4943 | Commutation of union with ... |
| iunconst 4944 | Indexed union of a constan... |
| iinconst 4945 | Indexed intersection of a ... |
| iuneqconst 4946 | Indexed union of identical... |
| iuniin 4947 | Law combining indexed unio... |
| iinssiun 4948 | An indexed intersection is... |
| iunss1 4949 | Subclass theorem for index... |
| iinss1 4950 | Subclass theorem for index... |
| iuneq1 4951 | Equality theorem for index... |
| iineq1 4952 | Equality theorem for index... |
| ss2iun 4953 | Subclass theorem for index... |
| iuneq2 4954 | Equality theorem for index... |
| iineq2 4955 | Equality theorem for index... |
| iuneq2i 4956 | Equality inference for ind... |
| iineq2i 4957 | Equality inference for ind... |
| iineq2d 4958 | Equality deduction for ind... |
| iuneq2dv 4959 | Equality deduction for ind... |
| iineq2dv 4960 | Equality deduction for ind... |
| iuneq12df 4961 | Equality deduction for ind... |
| iuneq1d 4962 | Equality theorem for index... |
| iuneq12dOLD 4963 | Obsolete version of ~ iune... |
| iuneq12d 4964 | Equality deduction for ind... |
| iuneq2d 4965 | Equality deduction for ind... |
| nfiun 4966 | Bound-variable hypothesis ... |
| nfiin 4967 | Bound-variable hypothesis ... |
| nfiung 4968 | Bound-variable hypothesis ... |
| nfiing 4969 | Bound-variable hypothesis ... |
| nfiu1 4970 | Bound-variable hypothesis ... |
| nfiu1OLD 4971 | Obsolete version of ~ nfiu... |
| nfii1 4972 | Bound-variable hypothesis ... |
| dfiun2g 4973 | Alternate definition of in... |
| dfiin2g 4974 | Alternate definition of in... |
| dfiun2 4975 | Alternate definition of in... |
| dfiin2 4976 | Alternate definition of in... |
| dfiunv2 4977 | Define double indexed unio... |
| cbviun 4978 | Rule used to change the bo... |
| cbviin 4979 | Change bound variables in ... |
| cbviung 4980 | Rule used to change the bo... |
| cbviing 4981 | Change bound variables in ... |
| cbviunv 4982 | Rule used to change the bo... |
| cbviinv 4983 | Change bound variables in ... |
| cbviunvg 4984 | Rule used to change the bo... |
| cbviinvg 4985 | Change bound variables in ... |
| iunssf 4986 | Subset theorem for an inde... |
| iunssfOLD 4987 | Obsolete version of ~ iuns... |
| iunss 4988 | Subset theorem for an inde... |
| iunssOLD 4989 | Obsolete version of ~ iuns... |
| ssiun 4990 | Subset implication for an ... |
| ssiun2 4991 | Identity law for subset of... |
| ssiun2s 4992 | Subset relationship for an... |
| iunss2 4993 | A subclass condition on th... |
| iunssd 4994 | Subset theorem for an inde... |
| iunab 4995 | The indexed union of a cla... |
| iunrab 4996 | The indexed union of a res... |
| iunxdif2 4997 | Indexed union with a class... |
| ssiinf 4998 | Subset theorem for an inde... |
| ssiin 4999 | Subset theorem for an inde... |
| iinss 5000 | Subset implication for an ... |
| iinss2 5001 | An indexed intersection is... |
| uniiun 5002 | Class union in terms of in... |
| intiin 5003 | Class intersection in term... |
| iunid 5004 | An indexed union of single... |
| iun0 5005 | An indexed union of the em... |
| 0iun 5006 | An empty indexed union is ... |
| 0iin 5007 | An empty indexed intersect... |
| viin 5008 | Indexed intersection with ... |
| iunsn 5009 | Indexed union of a singlet... |
| iunn0 5010 | There is a nonempty class ... |
| iinab 5011 | Indexed intersection of a ... |
| iinrab 5012 | Indexed intersection of a ... |
| iinrab2 5013 | Indexed intersection of a ... |
| iunin2 5014 | Indexed union of intersect... |
| iunin1 5015 | Indexed union of intersect... |
| iinun2 5016 | Indexed intersection of un... |
| iundif2 5017 | Indexed union of class dif... |
| iindif1 5018 | Indexed intersection of cl... |
| 2iunin 5019 | Rearrange indexed unions o... |
| iindif2 5020 | Indexed intersection of cl... |
| iinin2 5021 | Indexed intersection of in... |
| iinin1 5022 | Indexed intersection of in... |
| iinvdif 5023 | The indexed intersection o... |
| elriin 5024 | Elementhood in a relative ... |
| riin0 5025 | Relative intersection of a... |
| riinn0 5026 | Relative intersection of a... |
| riinrab 5027 | Relative intersection of a... |
| symdif0 5028 | Symmetric difference with ... |
| symdifv 5029 | The symmetric difference w... |
| symdifid 5030 | The symmetric difference o... |
| iinxsng 5031 | A singleton index picks ou... |
| iinxprg 5032 | Indexed intersection with ... |
| iunxsng 5033 | A singleton index picks ou... |
| iunxsn 5034 | A singleton index picks ou... |
| iunxsngf 5035 | A singleton index picks ou... |
| iunun 5036 | Separate a union in an ind... |
| iunxun 5037 | Separate a union in the in... |
| iunxdif3 5038 | An indexed union where som... |
| iunxprg 5039 | A pair index picks out two... |
| iunxiun 5040 | Separate an indexed union ... |
| iinuni 5041 | A relationship involving u... |
| iununi 5042 | A relationship involving u... |
| sspwuni 5043 | Subclass relationship for ... |
| pwssb 5044 | Two ways to express a coll... |
| elpwpw 5045 | Characterization of the el... |
| pwpwab 5046 | The double power class wri... |
| pwpwssunieq 5047 | The class of sets whose un... |
| elpwuni 5048 | Relationship for power cla... |
| iinpw 5049 | The power class of an inte... |
| iunpwss 5050 | Inclusion of an indexed un... |
| intss2 5051 | A nonempty intersection of... |
| rintn0 5052 | Relative intersection of a... |
| dfdisj2 5055 | Alternate definition for d... |
| disjss2 5056 | If each element of a colle... |
| disjeq2 5057 | Equality theorem for disjo... |
| disjeq2dv 5058 | Equality deduction for dis... |
| disjss1 5059 | A subset of a disjoint col... |
| disjeq1 5060 | Equality theorem for disjo... |
| disjeq1d 5061 | Equality theorem for disjo... |
| disjeq12d 5062 | Equality theorem for disjo... |
| cbvdisj 5063 | Change bound variables in ... |
| cbvdisjv 5064 | Change bound variables in ... |
| nfdisjw 5065 | Bound-variable hypothesis ... |
| nfdisj 5066 | Bound-variable hypothesis ... |
| nfdisj1 5067 | Bound-variable hypothesis ... |
| disjor 5068 | Two ways to say that a col... |
| disjors 5069 | Two ways to say that a col... |
| disji2 5070 | Property of a disjoint col... |
| disji 5071 | Property of a disjoint col... |
| invdisj 5072 | If there is a function ` C... |
| invdisjrab 5073 | The restricted class abstr... |
| disjiun 5074 | A disjoint collection yiel... |
| disjord 5075 | Conditions for a collectio... |
| disjiunb 5076 | Two ways to say that a col... |
| disjiund 5077 | Conditions for a collectio... |
| sndisj 5078 | Any collection of singleto... |
| 0disj 5079 | Any collection of empty se... |
| disjxsn 5080 | A singleton collection is ... |
| disjx0 5081 | An empty collection is dis... |
| disjprg 5082 | A pair collection is disjo... |
| disjxiun 5083 | An indexed union of a disj... |
| disjxun 5084 | The union of two disjoint ... |
| disjss3 5085 | Expand a disjoint collecti... |
| breq 5088 | Equality theorem for binar... |
| breq1 5089 | Equality theorem for a bin... |
| breq2 5090 | Equality theorem for a bin... |
| breq12 5091 | Equality theorem for a bin... |
| breqi 5092 | Equality inference for bin... |
| breq1i 5093 | Equality inference for a b... |
| breq2i 5094 | Equality inference for a b... |
| breq12i 5095 | Equality inference for a b... |
| breq1d 5096 | Equality deduction for a b... |
| breqd 5097 | Equality deduction for a b... |
| breq2d 5098 | Equality deduction for a b... |
| breq12d 5099 | Equality deduction for a b... |
| breq123d 5100 | Equality deduction for a b... |
| breqdi 5101 | Equality deduction for a b... |
| breqan12d 5102 | Equality deduction for a b... |
| breqan12rd 5103 | Equality deduction for a b... |
| eqnbrtrd 5104 | Substitution of equal clas... |
| nbrne1 5105 | Two classes are different ... |
| nbrne2 5106 | Two classes are different ... |
| eqbrtri 5107 | Substitution of equal clas... |
| eqbrtrd 5108 | Substitution of equal clas... |
| eqbrtrri 5109 | Substitution of equal clas... |
| eqbrtrrd 5110 | Substitution of equal clas... |
| breqtri 5111 | Substitution of equal clas... |
| breqtrd 5112 | Substitution of equal clas... |
| breqtrri 5113 | Substitution of equal clas... |
| breqtrrd 5114 | Substitution of equal clas... |
| 3brtr3i 5115 | Substitution of equality i... |
| 3brtr4i 5116 | Substitution of equality i... |
| 3brtr3d 5117 | Substitution of equality i... |
| 3brtr4d 5118 | Substitution of equality i... |
| 3brtr3g 5119 | Substitution of equality i... |
| 3brtr4g 5120 | Substitution of equality i... |
| eqbrtrid 5121 | A chained equality inferen... |
| eqbrtrrid 5122 | A chained equality inferen... |
| breqtrid 5123 | A chained equality inferen... |
| breqtrrid 5124 | A chained equality inferen... |
| eqbrtrdi 5125 | A chained equality inferen... |
| eqbrtrrdi 5126 | A chained equality inferen... |
| breqtrdi 5127 | A chained equality inferen... |
| breqtrrdi 5128 | A chained equality inferen... |
| ssbrd 5129 | Deduction from a subclass ... |
| ssbr 5130 | Implication from a subclas... |
| ssbri 5131 | Inference from a subclass ... |
| nfbrd 5132 | Deduction version of bound... |
| nfbr 5133 | Bound-variable hypothesis ... |
| brab1 5134 | Relationship between a bin... |
| br0 5135 | The empty binary relation ... |
| brne0 5136 | If two sets are in a binar... |
| brun 5137 | The union of two binary re... |
| brin 5138 | The intersection of two re... |
| brdif 5139 | The difference of two bina... |
| sbcbr123 5140 | Move substitution in and o... |
| sbcbr 5141 | Move substitution in and o... |
| sbcbr12g 5142 | Move substitution in and o... |
| sbcbr1g 5143 | Move substitution in and o... |
| sbcbr2g 5144 | Move substitution in and o... |
| brsymdif 5145 | Characterization of the sy... |
| brralrspcev 5146 | Restricted existential spe... |
| brimralrspcev 5147 | Restricted existential spe... |
| opabss 5150 | The collection of ordered ... |
| opabbid 5151 | Equivalent wff's yield equ... |
| opabbidv 5152 | Equivalent wff's yield equ... |
| opabbii 5153 | Equivalent wff's yield equ... |
| nfopabd 5154 | Bound-variable hypothesis ... |
| nfopab 5155 | Bound-variable hypothesis ... |
| nfopab1 5156 | The first abstraction vari... |
| nfopab2 5157 | The second abstraction var... |
| cbvopab 5158 | Rule used to change bound ... |
| cbvopabv 5159 | Rule used to change bound ... |
| cbvopab1 5160 | Change first bound variabl... |
| cbvopab1g 5161 | Change first bound variabl... |
| cbvopab2 5162 | Change second bound variab... |
| cbvopab1s 5163 | Change first bound variabl... |
| cbvopab1v 5164 | Rule used to change the fi... |
| cbvopab2v 5165 | Rule used to change the se... |
| unopab 5166 | Union of two ordered pair ... |
| mpteq12da 5169 | An equality inference for ... |
| mpteq12df 5170 | An equality inference for ... |
| mpteq12f 5171 | An equality theorem for th... |
| mpteq12dva 5172 | An equality inference for ... |
| mpteq12dv 5173 | An equality inference for ... |
| mpteq12 5174 | An equality theorem for th... |
| mpteq1 5175 | An equality theorem for th... |
| mpteq1d 5176 | An equality theorem for th... |
| mpteq1i 5177 | An equality theorem for th... |
| mpteq2da 5178 | Slightly more general equa... |
| mpteq2dva 5179 | Slightly more general equa... |
| mpteq2dv 5180 | An equality inference for ... |
| mpteq2ia 5181 | An equality inference for ... |
| mpteq2i 5182 | An equality inference for ... |
| mpteq12i 5183 | An equality inference for ... |
| nfmpt 5184 | Bound-variable hypothesis ... |
| nfmpt1 5185 | Bound-variable hypothesis ... |
| cbvmptf 5186 | Rule to change the bound v... |
| cbvmptfg 5187 | Rule to change the bound v... |
| cbvmpt 5188 | Rule to change the bound v... |
| cbvmptg 5189 | Rule to change the bound v... |
| cbvmptv 5190 | Rule to change the bound v... |
| cbvmptvg 5191 | Rule to change the bound v... |
| mptv 5192 | Function with universal do... |
| dftr2 5195 | An alternate way of defini... |
| dftr2c 5196 | Variant of ~ dftr2 with co... |
| dftr5 5197 | An alternate way of defini... |
| dftr3 5198 | An alternate way of defini... |
| dftr4 5199 | An alternate way of defini... |
| treq 5200 | Equality theorem for the t... |
| trel 5201 | In a transitive class, the... |
| trel3 5202 | In a transitive class, the... |
| trss 5203 | An element of a transitive... |
| trin 5204 | The intersection of transi... |
| tr0 5205 | The empty set is transitiv... |
| trv 5206 | The universe is transitive... |
| triun 5207 | An indexed union of a clas... |
| truni 5208 | The union of a class of tr... |
| triin 5209 | An indexed intersection of... |
| trint 5210 | The intersection of a clas... |
| trintss 5211 | Any nonempty transitive cl... |
| axrep1 5213 | The version of the Axiom o... |
| axreplem 5214 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5215 | Axiom of Replacement expre... |
| axrep3 5216 | Axiom of Replacement sligh... |
| axrep4v 5217 | Version of ~ axrep4 with a... |
| axrep4 5218 | A more traditional version... |
| axrep4OLD 5219 | Obsolete version of ~ axre... |
| axrep5 5220 | Axiom of Replacement (simi... |
| axrep6 5221 | A condensed form of ~ ax-r... |
| axrep6OLD 5222 | Obsolete version of ~ axre... |
| replem 5223 | A lemma for variants of th... |
| zfrep6 5224 | A version of the Axiom of ... |
| axrep6g 5225 | ~ axrep6 in class notation... |
| zfrepclf 5226 | An inference based on the ... |
| zfrep3cl 5227 | An inference based on the ... |
| zfrep4 5228 | A version of Replacement u... |
| axsepgfromrep 5229 | A more general version ~ a... |
| axsep 5230 | Axiom scheme of separation... |
| axsepg 5232 | A more general version of ... |
| zfauscl 5233 | Separation Scheme (Aussond... |
| sepexlem 5234 | Lemma for ~ sepex . Use ~... |
| sepex 5235 | Convert implication to equ... |
| sepexi 5236 | Convert implication to equ... |
| bm1.3iiOLD 5237 | Obsolete version of ~ sepe... |
| ax6vsep 5238 | Derive ~ ax6v (a weakened ... |
| axnulALT 5239 | Alternate proof of ~ axnul... |
| axnul 5240 | The Null Set Axiom of ZF s... |
| 0ex 5242 | The Null Set Axiom of ZF s... |
| al0ssb 5243 | The empty set is the uniqu... |
| sseliALT 5244 | Alternate proof of ~ sseli... |
| csbexg 5245 | The existence of proper su... |
| csbex 5246 | The existence of proper su... |
| unisn2 5247 | A version of ~ unisn witho... |
| nalset 5248 | No set contains all sets. ... |
| vnex 5249 | The universal class does n... |
| vprc 5250 | The universal class is not... |
| nvel 5251 | The universal class does n... |
| inex1 5252 | Separation Scheme (Aussond... |
| inex2 5253 | Separation Scheme (Aussond... |
| inex1g 5254 | Closed-form, generalized S... |
| inex2g 5255 | Sufficient condition for a... |
| ssex 5256 | The subset of a set is als... |
| ssexi 5257 | The subset of a set is als... |
| ssexg 5258 | The subset of a set is als... |
| ssexd 5259 | A subclass of a set is a s... |
| abexd 5260 | Conditions for a class abs... |
| abex 5261 | Conditions for a class abs... |
| prcssprc 5262 | The superclass of a proper... |
| sselpwd 5263 | Elementhood to a power set... |
| difexg 5264 | Existence of a difference.... |
| difexi 5265 | Existence of a difference,... |
| difexd 5266 | Existence of a difference.... |
| zfausab 5267 | Separation Scheme (Aussond... |
| elpw2g 5268 | Membership in a power clas... |
| elpw2 5269 | Membership in a power clas... |
| elpwi2 5270 | Membership in a power clas... |
| rabelpw 5271 | A restricted class abstrac... |
| rabexg 5272 | Separation Scheme in terms... |
| rabexgOLD 5273 | Obsolete version of ~ rabe... |
| rabex 5274 | Separation Scheme in terms... |
| rabexd 5275 | Separation Scheme in terms... |
| rabex2 5276 | Separation Scheme in terms... |
| rab2ex 5277 | A class abstraction based ... |
| elssabg 5278 | Membership in a class abst... |
| intex 5279 | The intersection of a none... |
| intnex 5280 | If a class intersection is... |
| intexab 5281 | The intersection of a none... |
| intexrab 5282 | The intersection of a none... |
| iinexg 5283 | The existence of a class i... |
| intabs 5284 | Absorption of a redundant ... |
| inuni 5285 | The intersection of a unio... |
| axpweq 5286 | Two equivalent ways to exp... |
| pwnss 5287 | The power set of a set is ... |
| pwne 5288 | No set equals its power se... |
| difelpw 5289 | A difference is an element... |
| class2set 5290 | The class of elements of `... |
| 0elpw 5291 | Every power class contains... |
| pwne0 5292 | A power class is never emp... |
| 0nep0 5293 | The empty set and its powe... |
| 0inp0 5294 | Something cannot be equal ... |
| unidif0 5295 | The removal of the empty s... |
| eqsnuniex 5296 | If a class is equal to the... |
| iin0 5297 | An indexed intersection of... |
| notzfaus 5298 | In the Separation Scheme ~... |
| intv 5299 | The intersection of the un... |
| zfpow 5301 | Axiom of Power Sets expres... |
| axpow2 5302 | A variant of the Axiom of ... |
| axpow3 5303 | A variant of the Axiom of ... |
| elALT2 5304 | Alternate proof of ~ el us... |
| dtruALT2 5305 | Alternate proof of ~ dtru ... |
| dtrucor 5306 | Corollary of ~ dtru . Thi... |
| dtrucor2 5307 | The theorem form of the de... |
| dvdemo1 5308 | Demonstration of a theorem... |
| dvdemo2 5309 | Demonstration of a theorem... |
| nfnid 5310 | A setvar variable is not f... |
| nfcvb 5311 | The "distinctor" expressio... |
| vpwex 5312 | Power set axiom: the power... |
| pwexg 5313 | Power set axiom expressed ... |
| pwexd 5314 | Deduction version of the p... |
| pwex 5315 | Power set axiom expressed ... |
| pwel 5316 | Quantitative version of ~ ... |
| abssexg 5317 | Existence of a class of su... |
| snexALT 5318 | Alternate proof of ~ snex ... |
| p0ex 5319 | The power set of the empty... |
| p0exALT 5320 | Alternate proof of ~ p0ex ... |
| pp0ex 5321 | The power set of the power... |
| ord3ex 5322 | The ordinal number 3 is a ... |
| dtruALT 5323 | Alternate proof of ~ dtru ... |
| axc16b 5324 | This theorem shows that Ax... |
| eunex 5325 | Existential uniqueness imp... |
| eusv1 5326 | Two ways to express single... |
| eusvnf 5327 | Even if ` x ` is free in `... |
| eusvnfb 5328 | Two ways to say that ` A (... |
| eusv2i 5329 | Two ways to express single... |
| eusv2nf 5330 | Two ways to express single... |
| eusv2 5331 | Two ways to express single... |
| reusv1 5332 | Two ways to express single... |
| reusv2lem1 5333 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5334 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5335 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5336 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5337 | Lemma for ~ reusv2 . (Con... |
| reusv2 5338 | Two ways to express single... |
| reusv3i 5339 | Two ways of expressing exi... |
| reusv3 5340 | Two ways to express single... |
| eusv4 5341 | Two ways to express single... |
| alxfr 5342 | Transfer universal quantif... |
| ralxfrd 5343 | Transfer universal quantif... |
| rexxfrd 5344 | Transfer existential quant... |
| ralxfr2d 5345 | Transfer universal quantif... |
| rexxfr2d 5346 | Transfer existential quant... |
| ralxfrd2 5347 | Transfer universal quantif... |
| rexxfrd2 5348 | Transfer existence from a ... |
| ralxfr 5349 | Transfer universal quantif... |
| ralxfrALT 5350 | Alternate proof of ~ ralxf... |
| rexxfr 5351 | Transfer existence from a ... |
| rabxfrd 5352 | Membership in a restricted... |
| rabxfr 5353 | Membership in a restricted... |
| reuhypd 5354 | A theorem useful for elimi... |
| reuhyp 5355 | A theorem useful for elimi... |
| zfpair 5356 | The Axiom of Pairing of Ze... |
| axprALT 5357 | Alternate proof of ~ axpr ... |
| axprlem1 5358 | Lemma for ~ axpr . There ... |
| axprlem2 5359 | Lemma for ~ axpr . There ... |
| axprlem3 5360 | Lemma for ~ axpr . Elimin... |
| axprlem4 5361 | Lemma for ~ axpr . If an ... |
| axpr 5362 | Unabbreviated version of t... |
| axprlem1OLD 5363 | Obsolete version of ~ axpr... |
| axprlem3OLD 5364 | Obsolete version of ~ axpr... |
| axprlem4OLD 5365 | Obsolete version of ~ axpr... |
| axprlem5OLD 5366 | Obsolete version of ~ axpr... |
| axprOLD 5367 | Obsolete version of ~ axpr... |
| zfpair2 5369 | Derive the abbreviated ver... |
| vsnex 5370 | A singleton built on a set... |
| axprglem 5371 | Lemma for ~ axprg . (Cont... |
| axprg 5372 | Derive The Axiom of Pairin... |
| prex 5373 | The Axiom of Pairing using... |
| snex 5374 | A singleton is a set. The... |
| snexg 5375 | A singleton built on a set... |
| snexgALT 5376 | Alternate proof of ~ snexg... |
| snexOLD 5377 | Obsolete version of ~ snex... |
| prexOLD 5378 | Obsolete version of ~ prex... |
| exel 5379 | There exist two sets, one ... |
| exexneq 5380 | There exist two different ... |
| exneq 5381 | Given any set (the " ` y `... |
| dtru 5382 | Given any set (the " ` y `... |
| el 5383 | Any set is an element of s... |
| elOLD 5384 | Obsolete version of ~ el a... |
| sels 5385 | If a class is a set, then ... |
| selsALT 5386 | Alternate proof of ~ sels ... |
| elALT 5387 | Alternate proof of ~ el , ... |
| snelpwg 5388 | A singleton of a set is a ... |
| snelpwi 5389 | If a set is a member of a ... |
| snelpw 5390 | A singleton of a set is a ... |
| prelpw 5391 | An unordered pair of two s... |
| prelpwi 5392 | If two sets are members of... |
| rext 5393 | A theorem similar to exten... |
| sspwb 5394 | The powerclass constructio... |
| unipw 5395 | A class equals the union o... |
| univ 5396 | The union of the universe ... |
| pwtr 5397 | A class is transitive iff ... |
| ssextss 5398 | An extensionality-like pri... |
| ssext 5399 | An extensionality-like pri... |
| nssss 5400 | Negation of subclass relat... |
| pweqb 5401 | Classes are equal if and o... |
| intidg 5402 | The intersection of all se... |
| moabex 5403 | "At most one" existence im... |
| moabexOLD 5404 | Obsolete version of ~ moab... |
| rmorabex 5405 | Restricted "at most one" e... |
| euabex 5406 | The abstraction of a wff w... |
| nnullss 5407 | A nonempty class (even if ... |
| exss 5408 | Restricted existence in a ... |
| opex 5409 | An ordered pair of classes... |
| opexOLD 5410 | Obsolete version of ~ opex... |
| otex 5411 | An ordered triple of class... |
| elopg 5412 | Characterization of the el... |
| elop 5413 | Characterization of the el... |
| opi1 5414 | One of the two elements in... |
| opi2 5415 | One of the two elements of... |
| opeluu 5416 | Each member of an ordered ... |
| op1stb 5417 | Extract the first member o... |
| brv 5418 | Two classes are always in ... |
| opnz 5419 | An ordered pair is nonempt... |
| opnzi 5420 | An ordered pair is nonempt... |
| opth1 5421 | Equality of the first memb... |
| opth 5422 | The ordered pair theorem. ... |
| opthg 5423 | Ordered pair theorem. ` C ... |
| opth1g 5424 | Equality of the first memb... |
| opthg2 5425 | Ordered pair theorem. (Co... |
| opth2 5426 | Ordered pair theorem. (Co... |
| opthneg 5427 | Two ordered pairs are not ... |
| opthne 5428 | Two ordered pairs are not ... |
| otth2 5429 | Ordered triple theorem, wi... |
| otth 5430 | Ordered triple theorem. (... |
| otthg 5431 | Ordered triple theorem, cl... |
| otthne 5432 | Contrapositive of the orde... |
| eqvinop 5433 | A variable introduction la... |
| sbcop1 5434 | The proper substitution of... |
| sbcop 5435 | The proper substitution of... |
| copsexgw 5436 | Version of ~ copsexg with ... |
| copsexg 5437 | Substitution of class ` A ... |
| copsex2t 5438 | Closed theorem form of ~ c... |
| copsex2g 5439 | Implicit substitution infe... |
| copsex2dv 5440 | Implicit substitution dedu... |
| copsex4g 5441 | An implicit substitution i... |
| 0nelop 5442 | A property of ordered pair... |
| opwo0id 5443 | An ordered pair is equal t... |
| opeqex 5444 | Equivalence of existence i... |
| oteqex2 5445 | Equivalence of existence i... |
| oteqex 5446 | Equivalence of existence i... |
| opcom 5447 | An ordered pair commutes i... |
| moop2 5448 | "At most one" property of ... |
| opeqsng 5449 | Equivalence for an ordered... |
| opeqsn 5450 | Equivalence for an ordered... |
| opeqpr 5451 | Equivalence for an ordered... |
| snopeqop 5452 | Equivalence for an ordered... |
| propeqop 5453 | Equivalence for an ordered... |
| propssopi 5454 | If a pair of ordered pairs... |
| snopeqopsnid 5455 | Equivalence for an ordered... |
| mosubopt 5456 | "At most one" remains true... |
| mosubop 5457 | "At most one" remains true... |
| euop2 5458 | Transfer existential uniqu... |
| euotd 5459 | Prove existential uniquene... |
| opthwiener 5460 | Justification theorem for ... |
| uniop 5461 | The union of an ordered pa... |
| uniopel 5462 | Ordered pair membership is... |
| opthhausdorff 5463 | Justification theorem for ... |
| opthhausdorff0 5464 | Justification theorem for ... |
| otsndisj 5465 | The singletons consisting ... |
| otiunsndisj 5466 | The union of singletons co... |
| iunopeqop 5467 | Implication of an ordered ... |
| brsnop 5468 | Binary relation for an ord... |
| brtp 5469 | A necessary and sufficient... |
| opabidw 5470 | The law of concretion. Sp... |
| opabid 5471 | The law of concretion. Sp... |
| elopabw 5472 | Membership in a class abst... |
| elopab 5473 | Membership in a class abst... |
| rexopabb 5474 | Restricted existential qua... |
| vopelopabsb 5475 | The law of concretion in t... |
| opelopabsb 5476 | The law of concretion in t... |
| brabsb 5477 | The law of concretion in t... |
| opelopabt 5478 | Closed theorem form of ~ o... |
| opelopabga 5479 | The law of concretion. Th... |
| brabga 5480 | The law of concretion for ... |
| opelopab2a 5481 | Ordered pair membership in... |
| opelopaba 5482 | The law of concretion. Th... |
| braba 5483 | The law of concretion for ... |
| opelopabg 5484 | The law of concretion. Th... |
| brabg 5485 | The law of concretion for ... |
| opelopabgf 5486 | The law of concretion. Th... |
| opelopab2 5487 | Ordered pair membership in... |
| opelopab 5488 | The law of concretion. Th... |
| brab 5489 | The law of concretion for ... |
| opelopabaf 5490 | The law of concretion. Th... |
| opelopabf 5491 | The law of concretion. Th... |
| ssopab2 5492 | Equivalence of ordered pai... |
| ssopab2bw 5493 | Equivalence of ordered pai... |
| eqopab2bw 5494 | Equivalence of ordered pai... |
| ssopab2b 5495 | Equivalence of ordered pai... |
| ssopab2i 5496 | Inference of ordered pair ... |
| ssopab2dv 5497 | Inference of ordered pair ... |
| eqopab2b 5498 | Equivalence of ordered pai... |
| opabn0 5499 | Nonempty ordered pair clas... |
| opab0 5500 | Empty ordered pair class a... |
| csbopab 5501 | Move substitution into a c... |
| csbopabgALT 5502 | Move substitution into a c... |
| csbmpt12 5503 | Move substitution into a m... |
| csbmpt2 5504 | Move substitution into the... |
| iunopab 5505 | Move indexed union inside ... |
| elopabr 5506 | Membership in an ordered-p... |
| elopabran 5507 | Membership in an ordered-p... |
| rbropapd 5508 | Properties of a pair in an... |
| rbropap 5509 | Properties of a pair in a ... |
| 2rbropap 5510 | Properties of a pair in a ... |
| 0nelopab 5511 | The empty set is never an ... |
| brabv 5512 | If two classes are in a re... |
| pwin 5513 | The power class of the int... |
| pwssun 5514 | The power class of the uni... |
| pwun 5515 | The power class of the uni... |
| dfid4 5518 | The identity function expr... |
| dfid2 5519 | Alternate definition of th... |
| dfid3 5520 | A stronger version of ~ df... |
| epelg 5523 | The membership relation an... |
| epeli 5524 | The membership relation an... |
| epel 5525 | The membership relation an... |
| 0sn0ep 5526 | An example for the members... |
| epn0 5527 | The membership relation is... |
| poss 5532 | Subset theorem for the par... |
| poeq1 5533 | Equality theorem for parti... |
| poeq2 5534 | Equality theorem for parti... |
| poeq12d 5535 | Equality deduction for par... |
| nfpo 5536 | Bound-variable hypothesis ... |
| nfso 5537 | Bound-variable hypothesis ... |
| pocl 5538 | Characteristic properties ... |
| ispod 5539 | Sufficient conditions for ... |
| swopolem 5540 | Perform the substitutions ... |
| swopo 5541 | A strict weak order is a p... |
| poirr 5542 | A partial order is irrefle... |
| potr 5543 | A partial order is a trans... |
| po2nr 5544 | A partial order has no 2-c... |
| po3nr 5545 | A partial order has no 3-c... |
| po2ne 5546 | Two sets related by a part... |
| po0 5547 | Any relation is a partial ... |
| pofun 5548 | The inverse image of a par... |
| sopo 5549 | A strict linear order is a... |
| soss 5550 | Subset theorem for the str... |
| soeq1 5551 | Equality theorem for the s... |
| soeq2 5552 | Equality theorem for the s... |
| soeq12d 5553 | Equality deduction for tot... |
| sonr 5554 | A strict order relation is... |
| sotr 5555 | A strict order relation is... |
| sotrd 5556 | Transitivity law for stric... |
| solin 5557 | A strict order relation is... |
| so2nr 5558 | A strict order relation ha... |
| so3nr 5559 | A strict order relation ha... |
| sotric 5560 | A strict order relation sa... |
| sotrieq 5561 | Trichotomy law for strict ... |
| sotrieq2 5562 | Trichotomy law for strict ... |
| soasym 5563 | Asymmetry law for strict o... |
| sotr2 5564 | A transitivity relation. ... |
| issod 5565 | An irreflexive, transitive... |
| issoi 5566 | An irreflexive, transitive... |
| isso2i 5567 | Deduce strict ordering fro... |
| so0 5568 | Any relation is a strict o... |
| somo 5569 | A totally ordered set has ... |
| sotrine 5570 | Trichotomy law for strict ... |
| sotr3 5571 | Transitivity law for stric... |
| dffr6 5578 | Alternate definition of ~ ... |
| frd 5579 | A nonempty subset of an ` ... |
| fri 5580 | A nonempty subset of an ` ... |
| seex 5581 | The ` R ` -preimage of an ... |
| exse 5582 | Any relation on a set is s... |
| dffr2 5583 | Alternate definition of we... |
| dffr2ALT 5584 | Alternate proof of ~ dffr2... |
| frc 5585 | Property of well-founded r... |
| frss 5586 | Subset theorem for the wel... |
| sess1 5587 | Subset theorem for the set... |
| sess2 5588 | Subset theorem for the set... |
| freq1 5589 | Equality theorem for the w... |
| freq2 5590 | Equality theorem for the w... |
| freq12d 5591 | Equality deduction for wel... |
| seeq1 5592 | Equality theorem for the s... |
| seeq2 5593 | Equality theorem for the s... |
| seeq12d 5594 | Equality deduction for the... |
| nffr 5595 | Bound-variable hypothesis ... |
| nfse 5596 | Bound-variable hypothesis ... |
| nfwe 5597 | Bound-variable hypothesis ... |
| frirr 5598 | A well-founded relation is... |
| fr2nr 5599 | A well-founded relation ha... |
| fr0 5600 | Any relation is well-found... |
| frminex 5601 | If an element of a well-fo... |
| efrirr 5602 | A well-founded class does ... |
| efrn2lp 5603 | A well-founded class conta... |
| epse 5604 | The membership relation is... |
| tz7.2 5605 | Similar to Theorem 7.2 of ... |
| dfepfr 5606 | An alternate way of saying... |
| epfrc 5607 | A subset of a well-founded... |
| wess 5608 | Subset theorem for the wel... |
| weeq1 5609 | Equality theorem for the w... |
| weeq2 5610 | Equality theorem for the w... |
| weeq12d 5611 | Equality deduction for wel... |
| wefr 5612 | A well-ordering is well-fo... |
| weso 5613 | A well-ordering is a stric... |
| wecmpep 5614 | The elements of a class we... |
| wetrep 5615 | On a class well-ordered by... |
| wefrc 5616 | A nonempty subclass of a c... |
| we0 5617 | Any relation is a well-ord... |
| wereu 5618 | A nonempty subset of an ` ... |
| wereu2 5619 | A nonempty subclass of an ... |
| xpeq1 5636 | Equality theorem for Carte... |
| xpss12 5637 | Subset theorem for Cartesi... |
| xpss 5638 | A Cartesian product is inc... |
| inxpssres 5639 | Intersection with a Cartes... |
| relxp 5640 | A Cartesian product is a r... |
| xpss1 5641 | Subset relation for Cartes... |
| xpss2 5642 | Subset relation for Cartes... |
| xpeq2 5643 | Equality theorem for Carte... |
| elxpi 5644 | Membership in a Cartesian ... |
| elxp 5645 | Membership in a Cartesian ... |
| elxp2 5646 | Membership in a Cartesian ... |
| xpeq12 5647 | Equality theorem for Carte... |
| xpeq1i 5648 | Equality inference for Car... |
| xpeq2i 5649 | Equality inference for Car... |
| xpeq12i 5650 | Equality inference for Car... |
| xpeq1d 5651 | Equality deduction for Car... |
| xpeq2d 5652 | Equality deduction for Car... |
| xpeq12d 5653 | Equality deduction for Car... |
| sqxpeqd 5654 | Equality deduction for a C... |
| nfxp 5655 | Bound-variable hypothesis ... |
| 0nelxp 5656 | The empty set is not a mem... |
| 0nelelxp 5657 | A member of a Cartesian pr... |
| opelxp 5658 | Ordered pair membership in... |
| opelxpi 5659 | Ordered pair membership in... |
| opelxpii 5660 | Ordered pair membership in... |
| opelxpd 5661 | Ordered pair membership in... |
| opelvv 5662 | Ordered pair membership in... |
| opelvvg 5663 | Ordered pair membership in... |
| opelxp1 5664 | The first member of an ord... |
| opelxp2 5665 | The second member of an or... |
| otelxp 5666 | Ordered triple membership ... |
| otelxp1 5667 | The first member of an ord... |
| otel3xp 5668 | An ordered triple is an el... |
| opabssxpd 5669 | An ordered-pair class abst... |
| rabxp 5670 | Class abstraction restrict... |
| brxp 5671 | Binary relation on a Carte... |
| pwvrel 5672 | A set is a binary relation... |
| pwvabrel 5673 | The powerclass of the cart... |
| brrelex12 5674 | Two classes related by a b... |
| brrelex1 5675 | If two classes are related... |
| brrelex2 5676 | If two classes are related... |
| brrelex12i 5677 | Two classes that are relat... |
| brrelex1i 5678 | The first argument of a bi... |
| brrelex2i 5679 | The second argument of a b... |
| nprrel12 5680 | Proper classes are not rel... |
| nprrel 5681 | No proper class is related... |
| 0nelrel0 5682 | A binary relation does not... |
| 0nelrel 5683 | A binary relation does not... |
| fconstmpt 5684 | Representation of a consta... |
| vtoclr 5685 | Variable to class conversi... |
| opthprc 5686 | Justification theorem for ... |
| brel 5687 | Two things in a binary rel... |
| elxp3 5688 | Membership in a Cartesian ... |
| opeliunxp 5689 | Membership in a union of C... |
| opeliun2xp 5690 | Membership of an ordered p... |
| xpundi 5691 | Distributive law for Carte... |
| xpundir 5692 | Distributive law for Carte... |
| xpiundi 5693 | Distributive law for Carte... |
| xpiundir 5694 | Distributive law for Carte... |
| iunxpconst 5695 | Membership in a union of C... |
| xpun 5696 | The Cartesian product of t... |
| elvv 5697 | Membership in universal cl... |
| elvvv 5698 | Membership in universal cl... |
| elvvuni 5699 | An ordered pair contains i... |
| brinxp2 5700 | Intersection of binary rel... |
| brinxp 5701 | Intersection of binary rel... |
| opelinxp 5702 | Ordered pair element in an... |
| poinxp 5703 | Intersection of partial or... |
| soinxp 5704 | Intersection of total orde... |
| frinxp 5705 | Intersection of well-found... |
| seinxp 5706 | Intersection of set-like r... |
| weinxp 5707 | Intersection of well-order... |
| posn 5708 | Partial ordering of a sing... |
| sosn 5709 | Strict ordering on a singl... |
| frsn 5710 | Founded relation on a sing... |
| wesn 5711 | Well-ordering of a singlet... |
| elopaelxp 5712 | Membership in an ordered-p... |
| bropaex12 5713 | Two classes related by an ... |
| opabssxp 5714 | An abstraction relation is... |
| brab2a 5715 | The law of concretion for ... |
| optocl 5716 | Implicit substitution of c... |
| optoclOLD 5717 | Obsolete version of ~ opto... |
| 2optocl 5718 | Implicit substitution of c... |
| 3optocl 5719 | Implicit substitution of c... |
| opbrop 5720 | Ordered pair membership in... |
| 0xp 5721 | The Cartesian product with... |
| xp0 5722 | The Cartesian product with... |
| csbxp 5723 | Distribute proper substitu... |
| releq 5724 | Equality theorem for the r... |
| releqi 5725 | Equality inference for the... |
| releqd 5726 | Equality deduction for the... |
| nfrel 5727 | Bound-variable hypothesis ... |
| sbcrel 5728 | Distribute proper substitu... |
| relss 5729 | Subclass theorem for relat... |
| ssrel 5730 | A subclass relationship de... |
| eqrel 5731 | Extensionality principle f... |
| ssrel2 5732 | A subclass relationship de... |
| ssrel3 5733 | Subclass relation in anoth... |
| relssi 5734 | Inference from subclass pr... |
| relssdv 5735 | Deduction from subclass pr... |
| eqrelriv 5736 | Inference from extensional... |
| eqrelriiv 5737 | Inference from extensional... |
| eqbrriv 5738 | Inference from extensional... |
| eqrelrdv 5739 | Deduce equality of relatio... |
| eqbrrdv 5740 | Deduction from extensional... |
| eqbrrdiv 5741 | Deduction from extensional... |
| eqrelrdv2 5742 | A version of ~ eqrelrdv . ... |
| ssrelrel 5743 | A subclass relationship de... |
| eqrelrel 5744 | Extensionality principle f... |
| elrel 5745 | A member of a relation is ... |
| rel0 5746 | The empty set is a relatio... |
| nrelv 5747 | The universal class is not... |
| relsng 5748 | A singleton is a relation ... |
| relsnb 5749 | An at-most-singleton is a ... |
| relsnopg 5750 | A singleton of an ordered ... |
| relsn 5751 | A singleton is a relation ... |
| relsnop 5752 | A singleton of an ordered ... |
| copsex2gb 5753 | Implicit substitution infe... |
| copsex2ga 5754 | Implicit substitution infe... |
| elopaba 5755 | Membership in an ordered-p... |
| xpsspw 5756 | A Cartesian product is inc... |
| unixpss 5757 | The double class union of ... |
| relun 5758 | The union of two relations... |
| relin1 5759 | The intersection with a re... |
| relin2 5760 | The intersection with a re... |
| relinxp 5761 | Intersection with a Cartes... |
| reldif 5762 | A difference cutting down ... |
| reliun 5763 | An indexed union is a rela... |
| reliin 5764 | An indexed intersection is... |
| reluni 5765 | The union of a class is a ... |
| relint 5766 | The intersection of a clas... |
| relopabiv 5767 | A class of ordered pairs i... |
| relopabv 5768 | A class of ordered pairs i... |
| relopabi 5769 | A class of ordered pairs i... |
| relopabiALT 5770 | Alternate proof of ~ relop... |
| relopab 5771 | A class of ordered pairs i... |
| mptrel 5772 | The maps-to notation alway... |
| reli 5773 | The identity relation is a... |
| rele 5774 | The membership relation is... |
| opabid2 5775 | A relation expressed as an... |
| inopab 5776 | Intersection of two ordere... |
| difopab 5777 | Difference of two ordered-... |
| inxp 5778 | Intersection of two Cartes... |
| inxpOLD 5779 | Obsolete version of ~ inxp... |
| xpindi 5780 | Distributive law for Carte... |
| xpindir 5781 | Distributive law for Carte... |
| xpiindi 5782 | Distributive law for Carte... |
| xpriindi 5783 | Distributive law for Carte... |
| eliunxp 5784 | Membership in a union of C... |
| opeliunxp2 5785 | Membership in a union of C... |
| raliunxp 5786 | Write a double restricted ... |
| rexiunxp 5787 | Write a double restricted ... |
| ralxp 5788 | Universal quantification r... |
| rexxp 5789 | Existential quantification... |
| exopxfr 5790 | Transfer ordered-pair exis... |
| exopxfr2 5791 | Transfer ordered-pair exis... |
| djussxp 5792 | Disjoint union is a subset... |
| ralxpf 5793 | Version of ~ ralxp with bo... |
| rexxpf 5794 | Version of ~ rexxp with bo... |
| iunxpf 5795 | Indexed union on a Cartesi... |
| opabbi2dv 5796 | Deduce equality of a relat... |
| relop 5797 | A necessary and sufficient... |
| ideqg 5798 | For sets, the identity rel... |
| ideq 5799 | For sets, the identity rel... |
| ididg 5800 | A set is identical to itse... |
| issetid 5801 | Two ways of expressing set... |
| coss1 5802 | Subclass theorem for compo... |
| coss2 5803 | Subclass theorem for compo... |
| coeq1 5804 | Equality theorem for compo... |
| coeq2 5805 | Equality theorem for compo... |
| coeq1i 5806 | Equality inference for com... |
| coeq2i 5807 | Equality inference for com... |
| coeq1d 5808 | Equality deduction for com... |
| coeq2d 5809 | Equality deduction for com... |
| coeq12i 5810 | Equality inference for com... |
| coeq12d 5811 | Equality deduction for com... |
| nfco 5812 | Bound-variable hypothesis ... |
| brcog 5813 | Ordered pair membership in... |
| opelco2g 5814 | Ordered pair membership in... |
| brcogw 5815 | Ordered pair membership in... |
| eqbrrdva 5816 | Deduction from extensional... |
| brco 5817 | Binary relation on a compo... |
| opelco 5818 | Ordered pair membership in... |
| cnvss 5819 | Subset theorem for convers... |
| cnveq 5820 | Equality theorem for conve... |
| cnveqi 5821 | Equality inference for con... |
| cnveqd 5822 | Equality deduction for con... |
| elcnv 5823 | Membership in a converse r... |
| elcnv2 5824 | Membership in a converse r... |
| nfcnv 5825 | Bound-variable hypothesis ... |
| brcnvg 5826 | The converse of a binary r... |
| opelcnvg 5827 | Ordered-pair membership in... |
| opelcnv 5828 | Ordered-pair membership in... |
| brcnv 5829 | The converse of a binary r... |
| csbcnv 5830 | Move class substitution in... |
| csbcnvgALT 5831 | Move class substitution in... |
| cnvco 5832 | Distributive law of conver... |
| cnvuni 5833 | The converse of a class un... |
| dfdm3 5834 | Alternate definition of do... |
| dfrn2 5835 | Alternate definition of ra... |
| dfrn3 5836 | Alternate definition of ra... |
| elrn2g 5837 | Membership in a range. (C... |
| elrng 5838 | Membership in a range. (C... |
| elrn2 5839 | Membership in a range. (C... |
| elrn 5840 | Membership in a range. (C... |
| ssrelrn 5841 | If a relation is a subset ... |
| dfdm4 5842 | Alternate definition of do... |
| dfdmf 5843 | Definition of domain, usin... |
| csbdm 5844 | Distribute proper substitu... |
| eldmg 5845 | Domain membership. Theore... |
| eldm2g 5846 | Domain membership. Theore... |
| eldm 5847 | Membership in a domain. T... |
| eldm2 5848 | Membership in a domain. T... |
| dmss 5849 | Subset theorem for domain.... |
| dmeq 5850 | Equality theorem for domai... |
| dmeqi 5851 | Equality inference for dom... |
| dmeqd 5852 | Equality deduction for dom... |
| opeldmd 5853 | Membership of first of an ... |
| opeldm 5854 | Membership of first of an ... |
| breldm 5855 | Membership of first of a b... |
| breldmg 5856 | Membership of first of a b... |
| dmun 5857 | The domain of a union is t... |
| dmin 5858 | The domain of an intersect... |
| breldmd 5859 | Membership of first of a b... |
| dmiun 5860 | The domain of an indexed u... |
| dmuni 5861 | The domain of a union. Pa... |
| dmopab 5862 | The domain of a class of o... |
| dmopabelb 5863 | A set is an element of the... |
| dmopab2rex 5864 | The domain of an ordered p... |
| dmopabss 5865 | Upper bound for the domain... |
| dmopab3 5866 | The domain of a restricted... |
| dm0 5867 | The domain of the empty se... |
| dmi 5868 | The domain of the identity... |
| dmv 5869 | The domain of the universe... |
| dmep 5870 | The domain of the membersh... |
| dm0rn0 5871 | An empty domain is equival... |
| dm0rn0OLD 5872 | Obsolete version of ~ dm0r... |
| rn0 5873 | The range of the empty set... |
| rnep 5874 | The range of the membershi... |
| reldm0 5875 | A relation is empty iff it... |
| dmxp 5876 | The domain of a Cartesian ... |
| dmxpid 5877 | The domain of a Cartesian ... |
| dmxpin 5878 | The domain of the intersec... |
| xpid11 5879 | The Cartesian square is a ... |
| dmcnvcnv 5880 | The domain of the double c... |
| rncnvcnv 5881 | The range of the double co... |
| elreldm 5882 | The first member of an ord... |
| rneq 5883 | Equality theorem for range... |
| rneqi 5884 | Equality inference for ran... |
| rneqd 5885 | Equality deduction for ran... |
| rnss 5886 | Subset theorem for range. ... |
| rnssi 5887 | Subclass inference for ran... |
| brelrng 5888 | The second argument of a b... |
| brelrn 5889 | The second argument of a b... |
| opelrn 5890 | Membership of second membe... |
| releldm 5891 | The first argument of a bi... |
| relelrn 5892 | The second argument of a b... |
| releldmb 5893 | Membership in a domain. (... |
| relelrnb 5894 | Membership in a range. (C... |
| releldmi 5895 | The first argument of a bi... |
| relelrni 5896 | The second argument of a b... |
| dfrnf 5897 | Definition of range, using... |
| nfdm 5898 | Bound-variable hypothesis ... |
| nfrn 5899 | Bound-variable hypothesis ... |
| dmiin 5900 | Domain of an intersection.... |
| rnopab 5901 | The range of a class of or... |
| rnopabss 5902 | Upper bound for the range ... |
| rnopab3 5903 | The range of a restricted ... |
| rnmpt 5904 | The range of a function in... |
| elrnmpt 5905 | The range of a function in... |
| elrnmpt1s 5906 | Elementhood in an image se... |
| elrnmpt1 5907 | Elementhood in an image se... |
| elrnmptg 5908 | Membership in the range of... |
| elrnmpti 5909 | Membership in the range of... |
| elrnmptd 5910 | The range of a function in... |
| elrnmpt1d 5911 | Elementhood in an image se... |
| elrnmptdv 5912 | Elementhood in the range o... |
| elrnmpt2d 5913 | Elementhood in the range o... |
| nelrnmpt 5914 | Non-membership in the rang... |
| dfiun3g 5915 | Alternate definition of in... |
| dfiin3g 5916 | Alternate definition of in... |
| dfiun3 5917 | Alternate definition of in... |
| dfiin3 5918 | Alternate definition of in... |
| riinint 5919 | Express a relative indexed... |
| relrn0 5920 | A relation is empty iff it... |
| dmrnssfld 5921 | The domain and range of a ... |
| dmcoss 5922 | Domain of a composition. ... |
| dmcossOLD 5923 | Obsolete version of ~ dmco... |
| rncoss 5924 | Range of a composition. (... |
| dmcosseq 5925 | Domain of a composition. ... |
| dmcosseqOLD 5926 | Obsolete version of ~ dmco... |
| dmcosseqOLDOLD 5927 | Obsolete version of ~ dmco... |
| dmcoeq 5928 | Domain of a composition. ... |
| rncoeq 5929 | Range of a composition. (... |
| reseq1 5930 | Equality theorem for restr... |
| reseq2 5931 | Equality theorem for restr... |
| reseq1i 5932 | Equality inference for res... |
| reseq2i 5933 | Equality inference for res... |
| reseq12i 5934 | Equality inference for res... |
| reseq1d 5935 | Equality deduction for res... |
| reseq2d 5936 | Equality deduction for res... |
| reseq12d 5937 | Equality deduction for res... |
| nfres 5938 | Bound-variable hypothesis ... |
| csbres 5939 | Distribute proper substitu... |
| res0 5940 | A restriction to the empty... |
| dfres3 5941 | Alternate definition of re... |
| opelres 5942 | Ordered pair elementhood i... |
| brres 5943 | Binary relation on a restr... |
| opelresi 5944 | Ordered pair membership in... |
| brresi 5945 | Binary relation on a restr... |
| opres 5946 | Ordered pair membership in... |
| resieq 5947 | A restricted identity rela... |
| opelidres 5948 | ` <. A , A >. ` belongs to... |
| resres 5949 | The restriction of a restr... |
| resundi 5950 | Distributive law for restr... |
| resundir 5951 | Distributive law for restr... |
| resindi 5952 | Class restriction distribu... |
| resindir 5953 | Class restriction distribu... |
| inres 5954 | Move intersection into cla... |
| resdifcom 5955 | Commutative law for restri... |
| resiun1 5956 | Distribution of restrictio... |
| resiun2 5957 | Distribution of restrictio... |
| resss 5958 | A class includes its restr... |
| rescom 5959 | Commutative law for restri... |
| ssres 5960 | Subclass theorem for restr... |
| ssres2 5961 | Subclass theorem for restr... |
| relres 5962 | A restriction is a relatio... |
| resabs1 5963 | Absorption law for restric... |
| resabs1i 5964 | Absorption law for restric... |
| resabs1d 5965 | Absorption law for restric... |
| resabs2 5966 | Absorption law for restric... |
| residm 5967 | Idempotent law for restric... |
| dmresss 5968 | The domain of a restrictio... |
| dmres 5969 | The domain of a restrictio... |
| ssdmres 5970 | A domain restricted to a s... |
| dmresexg 5971 | The domain of a restrictio... |
| resima 5972 | A restriction to an image.... |
| resima2 5973 | Image under a restricted c... |
| rnresss 5974 | The range of a restriction... |
| xpssres 5975 | Restriction of a constant ... |
| elinxp 5976 | Membership in an intersect... |
| elres 5977 | Membership in a restrictio... |
| elsnres 5978 | Membership in restriction ... |
| relssres 5979 | Simplification law for res... |
| dmressnsn 5980 | The domain of a restrictio... |
| eldmressnsn 5981 | The element of the domain ... |
| eldmeldmressn 5982 | An element of the domain (... |
| resdm 5983 | A relation restricted to i... |
| resexg 5984 | The restriction of a set i... |
| resexd 5985 | The restriction of a set i... |
| resex 5986 | The restriction of a set i... |
| resindm 5987 | When restricting a relatio... |
| resdmdfsn 5988 | Restricting a relation to ... |
| reldisjun 5989 | Split a relation into two ... |
| relresdm1 5990 | Restriction of a disjoint ... |
| resopab 5991 | Restriction of a class abs... |
| iss 5992 | A subclass of the identity... |
| resopab2 5993 | Restriction of a class abs... |
| resmpt 5994 | Restriction of the mapping... |
| resmpt3 5995 | Unconditional restriction ... |
| resmptf 5996 | Restriction of the mapping... |
| resmptd 5997 | Restriction of the mapping... |
| dfres2 5998 | Alternate definition of th... |
| mptss 5999 | Sufficient condition for i... |
| elimampt 6000 | Membership in the image of... |
| elidinxp 6001 | Characterization of the el... |
| elidinxpid 6002 | Characterization of the el... |
| elrid 6003 | Characterization of the el... |
| idinxpres 6004 | The intersection of the id... |
| idinxpresid 6005 | The intersection of the id... |
| idssxp 6006 | A diagonal set as a subset... |
| opabresid 6007 | The restricted identity re... |
| mptresid 6008 | The restricted identity re... |
| dmresi 6009 | The domain of a restricted... |
| restidsing 6010 | Restriction of the identit... |
| iresn0n0 6011 | The identity function rest... |
| imaeq1 6012 | Equality theorem for image... |
| imaeq2 6013 | Equality theorem for image... |
| imaeq1i 6014 | Equality theorem for image... |
| imaeq2i 6015 | Equality theorem for image... |
| imaeq1d 6016 | Equality theorem for image... |
| imaeq2d 6017 | Equality theorem for image... |
| imaeq12d 6018 | Equality theorem for image... |
| dfima2 6019 | Alternate definition of im... |
| dfima3 6020 | Alternate definition of im... |
| elimag 6021 | Membership in an image. T... |
| elima 6022 | Membership in an image. T... |
| elima2 6023 | Membership in an image. T... |
| elima3 6024 | Membership in an image. T... |
| nfima 6025 | Bound-variable hypothesis ... |
| nfimad 6026 | Deduction version of bound... |
| imadmrn 6027 | The image of the domain of... |
| imassrn 6028 | The image of a class is a ... |
| mptima 6029 | Image of a function in map... |
| mptimass 6030 | Image of a function in map... |
| imai 6031 | Image under the identity r... |
| rnresi 6032 | The range of the restricte... |
| resiima 6033 | The image of a restriction... |
| ima0 6034 | Image of the empty set. T... |
| 0ima 6035 | Image under the empty rela... |
| csbima12 6036 | Move class substitution in... |
| imadisj 6037 | A class whose image under ... |
| imadisjlnd 6038 | Deduction form of one nega... |
| cnvimass 6039 | A preimage under any class... |
| cnvimarndm 6040 | The preimage of the range ... |
| imasng 6041 | The image of a singleton. ... |
| relimasn 6042 | The image of a singleton. ... |
| elrelimasn 6043 | Elementhood in the image o... |
| elimasng1 6044 | Membership in an image of ... |
| elimasn1 6045 | Membership in an image of ... |
| elimasng 6046 | Membership in an image of ... |
| elimasn 6047 | Membership in an image of ... |
| elimasni 6048 | Membership in an image of ... |
| args 6049 | Two ways to express the cl... |
| elinisegg 6050 | Membership in the inverse ... |
| eliniseg 6051 | Membership in the inverse ... |
| epin 6052 | Any set is equal to its pr... |
| epini 6053 | Any set is equal to its pr... |
| iniseg 6054 | An idiom that signifies an... |
| inisegn0 6055 | Nonemptiness of an initial... |
| dffr3 6056 | Alternate definition of we... |
| dfse2 6057 | Alternate definition of se... |
| imass1 6058 | Subset theorem for image. ... |
| imass2 6059 | Subset theorem for image. ... |
| ndmima 6060 | The image of a singleton o... |
| relcnv 6061 | A converse is a relation. ... |
| relbrcnvg 6062 | When ` R ` is a relation, ... |
| eliniseg2 6063 | Eliminate the class existe... |
| relbrcnv 6064 | When ` R ` is a relation, ... |
| relco 6065 | A composition is a relatio... |
| cotrg 6066 | Two ways of saying that th... |
| cotr 6067 | Two ways of saying a relat... |
| idrefALT 6068 | Alternate proof of ~ idref... |
| cnvsym 6069 | Two ways of saying a relat... |
| intasym 6070 | Two ways of saying a relat... |
| asymref 6071 | Two ways of saying a relat... |
| asymref2 6072 | Two ways of saying a relat... |
| intirr 6073 | Two ways of saying a relat... |
| brcodir 6074 | Two ways of saying that tw... |
| codir 6075 | Two ways of saying a relat... |
| qfto 6076 | A quantifier-free way of e... |
| xpidtr 6077 | A Cartesian square is a tr... |
| trin2 6078 | The intersection of two tr... |
| poirr2 6079 | A partial order is irrefle... |
| trinxp 6080 | The relation induced by a ... |
| soirri 6081 | A strict order relation is... |
| sotri 6082 | A strict order relation is... |
| son2lpi 6083 | A strict order relation ha... |
| sotri2 6084 | A transitivity relation. ... |
| sotri3 6085 | A transitivity relation. ... |
| poleloe 6086 | Express "less than or equa... |
| poltletr 6087 | Transitive law for general... |
| somin1 6088 | Property of a minimum in a... |
| somincom 6089 | Commutativity of minimum i... |
| somin2 6090 | Property of a minimum in a... |
| soltmin 6091 | Being less than a minimum,... |
| cnvopab 6092 | The converse of a class ab... |
| cnvopabOLD 6093 | Obsolete version of ~ cnvo... |
| mptcnv 6094 | The converse of a mapping ... |
| cnv0 6095 | The converse of the empty ... |
| cnv0OLD 6096 | Obsolete version of ~ cnv0... |
| cnvi 6097 | The converse of the identi... |
| cnvun 6098 | The converse of a union is... |
| cnvdif 6099 | Distributive law for conve... |
| cnvin 6100 | Distributive law for conve... |
| rnun 6101 | Distributive law for range... |
| rnin 6102 | The range of an intersecti... |
| rniun 6103 | The range of an indexed un... |
| rnuni 6104 | The range of a union. Par... |
| imaundi 6105 | Distributive law for image... |
| imaundir 6106 | The image of a union. (Co... |
| imadifssran 6107 | Condition for the range of... |
| cnvimassrndm 6108 | The preimage of a superset... |
| dminss 6109 | An upper bound for interse... |
| imainss 6110 | An upper bound for interse... |
| inimass 6111 | The image of an intersecti... |
| inimasn 6112 | The intersection of the im... |
| cnvxp 6113 | The converse of a Cartesia... |
| xp0OLD 6114 | Obsolete version of ~ xp0 ... |
| xpnz 6115 | The Cartesian product of n... |
| xpeq0 6116 | At least one member of an ... |
| xpdisj1 6117 | Cartesian products with di... |
| xpdisj2 6118 | Cartesian products with di... |
| xpsndisj 6119 | Cartesian products with tw... |
| difxp 6120 | Difference of Cartesian pr... |
| difxp1 6121 | Difference law for Cartesi... |
| difxp2 6122 | Difference law for Cartesi... |
| djudisj 6123 | Disjoint unions with disjo... |
| xpdifid 6124 | The set of distinct couple... |
| resdisj 6125 | A double restriction to di... |
| rnxp 6126 | The range of a Cartesian p... |
| dmxpss 6127 | The domain of a Cartesian ... |
| rnxpss 6128 | The range of a Cartesian p... |
| rnxpid 6129 | The range of a Cartesian s... |
| ssxpb 6130 | A Cartesian product subcla... |
| xp11 6131 | The Cartesian product of n... |
| xpcan 6132 | Cancellation law for Carte... |
| xpcan2 6133 | Cancellation law for Carte... |
| ssrnres 6134 | Two ways to express surjec... |
| rninxp 6135 | Two ways to express surjec... |
| dminxp 6136 | Two ways to express totali... |
| imainrect 6137 | Image by a restricted and ... |
| xpima 6138 | Direct image by a Cartesia... |
| xpima1 6139 | Direct image by a Cartesia... |
| xpima2 6140 | Direct image by a Cartesia... |
| xpimasn 6141 | Direct image of a singleto... |
| sossfld 6142 | The base set of a strict o... |
| sofld 6143 | The base set of a nonempty... |
| cnvcnv3 6144 | The set of all ordered pai... |
| dfrel2 6145 | Alternate definition of re... |
| dfrel4v 6146 | A relation can be expresse... |
| dfrel4 6147 | A relation can be expresse... |
| cnvcnv 6148 | The double converse of a c... |
| cnvcnv2 6149 | The double converse of a c... |
| cnvcnvss 6150 | The double converse of a c... |
| cnvrescnv 6151 | Two ways to express the co... |
| cnveqb 6152 | Equality theorem for conve... |
| cnveq0 6153 | A relation empty iff its c... |
| dfrel3 6154 | Alternate definition of re... |
| elid 6155 | Characterization of the el... |
| dmresv 6156 | The domain of a universal ... |
| rnresv 6157 | The range of a universal r... |
| dfrn4 6158 | Range defined in terms of ... |
| csbrn 6159 | Distribute proper substitu... |
| rescnvcnv 6160 | The restriction of the dou... |
| cnvcnvres 6161 | The double converse of the... |
| imacnvcnv 6162 | The image of the double co... |
| dmsnn0 6163 | The domain of a singleton ... |
| rnsnn0 6164 | The range of a singleton i... |
| dmsn0 6165 | The domain of the singleto... |
| cnvsn0 6166 | The converse of the single... |
| dmsn0el 6167 | The domain of a singleton ... |
| relsn2 6168 | A singleton is a relation ... |
| dmsnopg 6169 | The domain of a singleton ... |
| dmsnopss 6170 | The domain of a singleton ... |
| dmpropg 6171 | The domain of an unordered... |
| dmsnop 6172 | The domain of a singleton ... |
| dmprop 6173 | The domain of an unordered... |
| dmtpop 6174 | The domain of an unordered... |
| cnvcnvsn 6175 | Double converse of a singl... |
| dmsnsnsn 6176 | The domain of the singleto... |
| rnsnopg 6177 | The range of a singleton o... |
| rnpropg 6178 | The range of a pair of ord... |
| cnvsng 6179 | Converse of a singleton of... |
| rnsnop 6180 | The range of a singleton o... |
| op1sta 6181 | Extract the first member o... |
| cnvsn 6182 | Converse of a singleton of... |
| op2ndb 6183 | Extract the second member ... |
| op2nda 6184 | Extract the second member ... |
| opswap 6185 | Swap the members of an ord... |
| cnvresima 6186 | An image under the convers... |
| resdm2 6187 | A class restricted to its ... |
| resdmres 6188 | Restriction to the domain ... |
| resresdm 6189 | A restriction by an arbitr... |
| imadmres 6190 | The image of the domain of... |
| resdmss 6191 | Subset relationship for th... |
| resdifdi 6192 | Distributive law for restr... |
| resdifdir 6193 | Distributive law for restr... |
| mptpreima 6194 | The preimage of a function... |
| mptiniseg 6195 | Converse singleton image o... |
| dmmpt 6196 | The domain of the mapping ... |
| dmmptss 6197 | The domain of a mapping is... |
| dmmptg 6198 | The domain of the mapping ... |
| rnmpt0f 6199 | The range of a function in... |
| rnmptn0 6200 | The range of a function in... |
| dfco2 6201 | Alternate definition of a ... |
| dfco2a 6202 | Generalization of ~ dfco2 ... |
| coundi 6203 | Class composition distribu... |
| coundir 6204 | Class composition distribu... |
| cores 6205 | Restricted first member of... |
| resco 6206 | Associative law for the re... |
| imaco 6207 | Image of the composition o... |
| rnco 6208 | The range of the compositi... |
| rncoOLD 6209 | Obsolete version of ~ rnco... |
| rnco2 6210 | The range of the compositi... |
| dmco 6211 | The domain of a compositio... |
| coeq0 6212 | A composition of two relat... |
| coiun 6213 | Composition with an indexe... |
| cocnvcnv1 6214 | A composition is not affec... |
| cocnvcnv2 6215 | A composition is not affec... |
| cores2 6216 | Absorption of a reverse (p... |
| co02 6217 | Composition with the empty... |
| co01 6218 | Composition with the empty... |
| coi1 6219 | Composition with the ident... |
| coi2 6220 | Composition with the ident... |
| coires1 6221 | Composition with a restric... |
| coass 6222 | Associative law for class ... |
| relcnvtrg 6223 | General form of ~ relcnvtr... |
| relcnvtr 6224 | A relation is transitive i... |
| relssdmrn 6225 | A relation is included in ... |
| resssxp 6226 | If the ` R ` -image of a c... |
| cnvssrndm 6227 | The converse is a subset o... |
| cossxp 6228 | Composition as a subset of... |
| relrelss 6229 | Two ways to describe the s... |
| unielrel 6230 | The membership relation fo... |
| relfld 6231 | The double union of a rela... |
| relresfld 6232 | Restriction of a relation ... |
| relcoi2 6233 | Composition with the ident... |
| relcoi1 6234 | Composition with the ident... |
| unidmrn 6235 | The double union of the co... |
| relcnvfld 6236 | if ` R ` is a relation, it... |
| dfdm2 6237 | Alternate definition of do... |
| unixp 6238 | The double class union of ... |
| unixp0 6239 | A Cartesian product is emp... |
| unixpid 6240 | Field of a Cartesian squar... |
| ressn 6241 | Restriction of a class to ... |
| cnviin 6242 | The converse of an interse... |
| cnvpo 6243 | The converse of a partial ... |
| cnvso 6244 | The converse of a strict o... |
| xpco 6245 | Composition of two Cartesi... |
| xpcoid 6246 | Composition of two Cartesi... |
| elsnxp 6247 | Membership in a Cartesian ... |
| reu3op 6248 | There is a unique ordered ... |
| reuop 6249 | There is a unique ordered ... |
| opreu2reurex 6250 | There is a unique ordered ... |
| opreu2reu 6251 | If there is a unique order... |
| dfpo2 6252 | Quantifier-free definition... |
| csbcog 6253 | Distribute proper substitu... |
| snres0 6254 | Condition for restriction ... |
| imaindm 6255 | The image is unaffected by... |
| predeq123 6258 | Equality theorem for the p... |
| predeq1 6259 | Equality theorem for the p... |
| predeq2 6260 | Equality theorem for the p... |
| predeq3 6261 | Equality theorem for the p... |
| nfpred 6262 | Bound-variable hypothesis ... |
| csbpredg 6263 | Move class substitution in... |
| predpredss 6264 | If ` A ` is a subset of ` ... |
| predss 6265 | The predecessor class of `... |
| sspred 6266 | Another subset/predecessor... |
| dfpred2 6267 | An alternate definition of... |
| dfpred3 6268 | An alternate definition of... |
| dfpred3g 6269 | An alternate definition of... |
| elpredgg 6270 | Membership in a predecesso... |
| elpredg 6271 | Membership in a predecesso... |
| elpredimg 6272 | Membership in a predecesso... |
| elpredim 6273 | Membership in a predecesso... |
| elpred 6274 | Membership in a predecesso... |
| predexg 6275 | The predecessor class exis... |
| dffr4 6276 | Alternate definition of we... |
| predel 6277 | Membership in the predeces... |
| predtrss 6278 | If ` R ` is transitive ove... |
| predpo 6279 | Property of the predecesso... |
| predso 6280 | Property of the predecesso... |
| setlikespec 6281 | If ` R ` is set-like in ` ... |
| predidm 6282 | Idempotent law for the pre... |
| predin 6283 | Intersection law for prede... |
| predun 6284 | Union law for predecessor ... |
| preddif 6285 | Difference law for predece... |
| predep 6286 | The predecessor under the ... |
| trpred 6287 | The class of predecessors ... |
| preddowncl 6288 | A property of classes that... |
| predpoirr 6289 | Given a partial ordering, ... |
| predfrirr 6290 | Given a well-founded relat... |
| pred0 6291 | The predecessor class over... |
| dfse3 6292 | Alternate definition of se... |
| predrelss 6293 | Subset carries from relati... |
| predprc 6294 | The predecessor of a prope... |
| predres 6295 | Predecessor class is unaff... |
| frpomin 6296 | Every nonempty (possibly p... |
| frpomin2 6297 | Every nonempty (possibly p... |
| frpoind 6298 | The principle of well-foun... |
| frpoinsg 6299 | Well-Founded Induction Sch... |
| frpoins2fg 6300 | Well-Founded Induction sch... |
| frpoins2g 6301 | Well-Founded Induction sch... |
| frpoins3g 6302 | Well-Founded Induction sch... |
| tz6.26 6303 | All nonempty subclasses of... |
| tz6.26i 6304 | All nonempty subclasses of... |
| wfi 6305 | The Principle of Well-Orde... |
| wfii 6306 | The Principle of Well-Orde... |
| wfisg 6307 | Well-Ordered Induction Sch... |
| wfis 6308 | Well-Ordered Induction Sch... |
| wfis2fg 6309 | Well-Ordered Induction Sch... |
| wfis2f 6310 | Well-Ordered Induction sch... |
| wfis2g 6311 | Well-Ordered Induction Sch... |
| wfis2 6312 | Well-Ordered Induction sch... |
| wfis3 6313 | Well-Ordered Induction sch... |
| ordeq 6322 | Equality theorem for the o... |
| elong 6323 | An ordinal number is an or... |
| elon 6324 | An ordinal number is an or... |
| eloni 6325 | An ordinal number has the ... |
| elon2 6326 | An ordinal number is an or... |
| limeq 6327 | Equality theorem for the l... |
| ordwe 6328 | Membership well-orders eve... |
| ordtr 6329 | An ordinal class is transi... |
| ordfr 6330 | Membership is well-founded... |
| ordelss 6331 | An element of an ordinal c... |
| trssord 6332 | A transitive subclass of a... |
| ordirr 6333 | No ordinal class is a memb... |
| nordeq 6334 | A member of an ordinal cla... |
| ordn2lp 6335 | An ordinal class cannot be... |
| tz7.5 6336 | A nonempty subclass of an ... |
| ordelord 6337 | An element of an ordinal c... |
| tron 6338 | The class of all ordinal n... |
| ordelon 6339 | An element of an ordinal c... |
| onelon 6340 | An element of an ordinal n... |
| tz7.7 6341 | A transitive class belongs... |
| ordelssne 6342 | For ordinal classes, membe... |
| ordelpss 6343 | For ordinal classes, membe... |
| ordsseleq 6344 | For ordinal classes, inclu... |
| ordin 6345 | The intersection of two or... |
| onin 6346 | The intersection of two or... |
| ordtri3or 6347 | A trichotomy law for ordin... |
| ordtri1 6348 | A trichotomy law for ordin... |
| ontri1 6349 | A trichotomy law for ordin... |
| ordtri2 6350 | A trichotomy law for ordin... |
| ordtri3 6351 | A trichotomy law for ordin... |
| ordtri4 6352 | A trichotomy law for ordin... |
| orddisj 6353 | An ordinal class and its s... |
| onfr 6354 | The ordinal class is well-... |
| onelpss 6355 | Relationship between membe... |
| onsseleq 6356 | Relationship between subse... |
| onelss 6357 | An element of an ordinal n... |
| oneltri 6358 | The elementhood relation o... |
| ordtr1 6359 | Transitive law for ordinal... |
| ordtr2 6360 | Transitive law for ordinal... |
| ordtr3 6361 | Transitive law for ordinal... |
| ontr1 6362 | Transitive law for ordinal... |
| ontr2 6363 | Transitive law for ordinal... |
| onelssex 6364 | Ordinal less than is equiv... |
| ordunidif 6365 | The union of an ordinal st... |
| ordintdif 6366 | If ` B ` is smaller than `... |
| onintss 6367 | If a property is true for ... |
| oneqmini 6368 | A way to show that an ordi... |
| ord0 6369 | The empty set is an ordina... |
| 0elon 6370 | The empty set is an ordina... |
| ord0eln0 6371 | A nonempty ordinal contain... |
| on0eln0 6372 | An ordinal number contains... |
| dflim2 6373 | An alternate definition of... |
| inton 6374 | The intersection of the cl... |
| nlim0 6375 | The empty set is not a lim... |
| limord 6376 | A limit ordinal is ordinal... |
| limuni 6377 | A limit ordinal is its own... |
| limuni2 6378 | The union of a limit ordin... |
| 0ellim 6379 | A limit ordinal contains t... |
| limelon 6380 | A limit ordinal class that... |
| onn0 6381 | The class of all ordinal n... |
| suceqd 6382 | Deduction associated with ... |
| suceq 6383 | Equality of successors. (... |
| elsuci 6384 | Membership in a successor.... |
| elsucg 6385 | Membership in a successor.... |
| elsuc2g 6386 | Variant of membership in a... |
| elsuc 6387 | Membership in a successor.... |
| elsuc2 6388 | Membership in a successor.... |
| nfsuc 6389 | Bound-variable hypothesis ... |
| elelsuc 6390 | Membership in a successor.... |
| sucel 6391 | Membership of a successor ... |
| suc0 6392 | The successor of the empty... |
| sucprc 6393 | A proper class is its own ... |
| unisucs 6394 | The union of the successor... |
| unisucg 6395 | A transitive class is equa... |
| unisuc 6396 | A transitive class is equa... |
| sssucid 6397 | A class is included in its... |
| sucidg 6398 | Part of Proposition 7.23 o... |
| sucid 6399 | A set belongs to its succe... |
| nsuceq0 6400 | No successor is empty. (C... |
| eqelsuc 6401 | A set belongs to the succe... |
| iunsuc 6402 | Inductive definition for t... |
| suctr 6403 | The successor of a transit... |
| trsuc 6404 | A set whose successor belo... |
| trsucss 6405 | A member of the successor ... |
| ordsssuc 6406 | An ordinal is a subset of ... |
| onsssuc 6407 | A subset of an ordinal num... |
| ordsssuc2 6408 | An ordinal subset of an or... |
| onmindif 6409 | When its successor is subt... |
| ordnbtwn 6410 | There is no set between an... |
| onnbtwn 6411 | There is no set between an... |
| sucssel 6412 | A set whose successor is a... |
| orddif 6413 | Ordinal derived from its s... |
| orduniss 6414 | An ordinal class includes ... |
| ordtri2or 6415 | A trichotomy law for ordin... |
| ordtri2or2 6416 | A trichotomy law for ordin... |
| ordtri2or3 6417 | A consequence of total ord... |
| ordelinel 6418 | The intersection of two or... |
| ordssun 6419 | Property of a subclass of ... |
| ordequn 6420 | The maximum (i.e. union) o... |
| ordun 6421 | The maximum (i.e., union) ... |
| onunel 6422 | The union of two ordinals ... |
| ordunisssuc 6423 | A subclass relationship fo... |
| suc11 6424 | The successor operation be... |
| onun2 6425 | The union of two ordinals ... |
| ontr 6426 | An ordinal number is a tra... |
| onunisuc 6427 | An ordinal number is equal... |
| onordi 6428 | An ordinal number is an or... |
| onirri 6429 | An ordinal number is not a... |
| oneli 6430 | A member of an ordinal num... |
| onelssi 6431 | A member of an ordinal num... |
| onssneli 6432 | An ordering law for ordina... |
| onssnel2i 6433 | An ordering law for ordina... |
| onelini 6434 | An element of an ordinal n... |
| oneluni 6435 | An ordinal number equals i... |
| onunisuci 6436 | An ordinal number is equal... |
| onsseli 6437 | Subset is equivalent to me... |
| onun2i 6438 | The union of two ordinal n... |
| unizlim 6439 | An ordinal equal to its ow... |
| on0eqel 6440 | An ordinal number either e... |
| snsn0non 6441 | The singleton of the singl... |
| onxpdisj 6442 | Ordinal numbers and ordere... |
| onnev 6443 | The class of ordinal numbe... |
| iotajust 6445 | Soundness justification th... |
| dfiota2 6447 | Alternate definition for d... |
| nfiota1 6448 | Bound-variable hypothesis ... |
| nfiotadw 6449 | Deduction version of ~ nfi... |
| nfiotaw 6450 | Bound-variable hypothesis ... |
| nfiotad 6451 | Deduction version of ~ nfi... |
| nfiota 6452 | Bound-variable hypothesis ... |
| cbviotaw 6453 | Change bound variables in ... |
| cbviotavw 6454 | Change bound variables in ... |
| cbviota 6455 | Change bound variables in ... |
| cbviotav 6456 | Change bound variables in ... |
| sb8iota 6457 | Variable substitution in d... |
| iotaeq 6458 | Equality theorem for descr... |
| iotabi 6459 | Equivalence theorem for de... |
| uniabio 6460 | Part of Theorem 8.17 in [Q... |
| iotaval2 6461 | Version of ~ iotaval using... |
| iotauni2 6462 | Version of ~ iotauni using... |
| iotanul2 6463 | Version of ~ iotanul using... |
| iotaval 6464 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6465 | The ` iota ` class is a su... |
| iotaex 6466 | Theorem 8.23 in [Quine] p.... |
| iotauni 6467 | Equivalence between two di... |
| iotaint 6468 | Equivalence between two di... |
| iota1 6469 | Property of iota. (Contri... |
| iotanul 6470 | Theorem 8.22 in [Quine] p.... |
| iota4 6471 | Theorem *14.22 in [Whitehe... |
| iota4an 6472 | Theorem *14.23 in [Whitehe... |
| iota5 6473 | A method for computing iot... |
| iotabidv 6474 | Formula-building deduction... |
| iotabii 6475 | Formula-building deduction... |
| iotacl 6476 | Membership law for descrip... |
| iota2df 6477 | A condition that allows to... |
| iota2d 6478 | A condition that allows to... |
| iota2 6479 | The unique element such th... |
| iotan0 6480 | Representation of "the uni... |
| sniota 6481 | A class abstraction with a... |
| dfiota4 6482 | The ` iota ` operation usi... |
| csbiota 6483 | Class substitution within ... |
| dffun2 6500 | Alternate definition of a ... |
| dffun6 6501 | Alternate definition of a ... |
| dffun3 6502 | Alternate definition of fu... |
| dffun4 6503 | Alternate definition of a ... |
| dffun5 6504 | Alternate definition of fu... |
| dffun6f 6505 | Definition of function, us... |
| funmo 6506 | A function has at most one... |
| funrel 6507 | A function is a relation. ... |
| 0nelfun 6508 | A function does not contai... |
| funss 6509 | Subclass theorem for funct... |
| funeq 6510 | Equality theorem for funct... |
| funeqi 6511 | Equality inference for the... |
| funeqd 6512 | Equality deduction for the... |
| nffun 6513 | Bound-variable hypothesis ... |
| sbcfung 6514 | Distribute proper substitu... |
| funeu 6515 | There is exactly one value... |
| funeu2 6516 | There is exactly one value... |
| dffun7 6517 | Alternate definition of a ... |
| dffun8 6518 | Alternate definition of a ... |
| dffun9 6519 | Alternate definition of a ... |
| funfn 6520 | A class is a function if a... |
| funfnd 6521 | A function is a function o... |
| funi 6522 | The identity relation is a... |
| nfunv 6523 | The universal class is not... |
| funopg 6524 | A Kuratowski ordered pair ... |
| funopab 6525 | A class of ordered pairs i... |
| funopabeq 6526 | A class of ordered pairs o... |
| funopab4 6527 | A class of ordered pairs o... |
| funmpt 6528 | A function in maps-to nota... |
| funmpt2 6529 | Functionality of a class g... |
| funco 6530 | The composition of two fun... |
| funresfunco 6531 | Composition of two functio... |
| funres 6532 | A restriction of a functio... |
| funresd 6533 | A restriction of a functio... |
| funssres 6534 | The restriction of a funct... |
| fun2ssres 6535 | Equality of restrictions o... |
| funun 6536 | The union of functions wit... |
| fununmo 6537 | If the union of classes is... |
| fununfun 6538 | If the union of classes is... |
| fundif 6539 | A function with removed el... |
| funcnvsn 6540 | The converse singleton of ... |
| funsng 6541 | A singleton of an ordered ... |
| fnsng 6542 | Functionality and domain o... |
| funsn 6543 | A singleton of an ordered ... |
| funprg 6544 | A set of two pairs is a fu... |
| funtpg 6545 | A set of three pairs is a ... |
| funpr 6546 | A function with a domain o... |
| funtp 6547 | A function with a domain o... |
| fnsn 6548 | Functionality and domain o... |
| fnprg 6549 | Function with a domain of ... |
| fntpg 6550 | Function with a domain of ... |
| fntp 6551 | A function with a domain o... |
| funcnvpr 6552 | The converse pair of order... |
| funcnvtp 6553 | The converse triple of ord... |
| funcnvqp 6554 | The converse quadruple of ... |
| fun0 6555 | The empty set is a functio... |
| funcnv0 6556 | The converse of the empty ... |
| funcnvcnv 6557 | The double converse of a f... |
| funcnv2 6558 | A simpler equivalence for ... |
| funcnv 6559 | The converse of a class is... |
| funcnv3 6560 | A condition showing a clas... |
| fun2cnv 6561 | The double converse of a c... |
| svrelfun 6562 | A single-valued relation i... |
| fncnv 6563 | Single-rootedness (see ~ f... |
| fun11 6564 | Two ways of stating that `... |
| fununi 6565 | The union of a chain (with... |
| funin 6566 | The intersection with a fu... |
| funres11 6567 | The restriction of a one-t... |
| funcnvres 6568 | The converse of a restrict... |
| cnvresid 6569 | Converse of a restricted i... |
| funcnvres2 6570 | The converse of a restrict... |
| funimacnv 6571 | The image of the preimage ... |
| funimass1 6572 | A kind of contraposition l... |
| funimass2 6573 | A kind of contraposition l... |
| imadif 6574 | The image of a difference ... |
| imain 6575 | The image of an intersecti... |
| f1imadifssran 6576 | Condition for the range of... |
| funimaexg 6577 | Axiom of Replacement using... |
| funimaex 6578 | The image of a set under a... |
| isarep1 6579 | Part of a study of the Axi... |
| isarep2 6580 | Part of a study of the Axi... |
| fneq1 6581 | Equality theorem for funct... |
| fneq2 6582 | Equality theorem for funct... |
| fneq1d 6583 | Equality deduction for fun... |
| fneq2d 6584 | Equality deduction for fun... |
| fneq12d 6585 | Equality deduction for fun... |
| fneq12 6586 | Equality theorem for funct... |
| fneq1i 6587 | Equality inference for fun... |
| fneq2i 6588 | Equality inference for fun... |
| nffn 6589 | Bound-variable hypothesis ... |
| fnfun 6590 | A function with domain is ... |
| fnfund 6591 | A function with domain is ... |
| fnrel 6592 | A function with domain is ... |
| fndm 6593 | The domain of a function. ... |
| fndmi 6594 | The domain of a function. ... |
| fndmd 6595 | The domain of a function. ... |
| funfni 6596 | Inference to convert a fun... |
| fndmu 6597 | A function has a unique do... |
| fnbr 6598 | The first argument of bina... |
| fnop 6599 | The first argument of an o... |
| fneu 6600 | There is exactly one value... |
| fneu2 6601 | There is exactly one value... |
| fnunres1 6602 | Restriction of a disjoint ... |
| fnunres2 6603 | Restriction of a disjoint ... |
| fnun 6604 | The union of two functions... |
| fnund 6605 | The union of two functions... |
| fnunop 6606 | Extension of a function wi... |
| fncofn 6607 | Composition of a function ... |
| fnco 6608 | Composition of two functio... |
| fnresdm 6609 | A function does not change... |
| fnresdisj 6610 | A function restricted to a... |
| 2elresin 6611 | Membership in two function... |
| fnssresb 6612 | Restriction of a function ... |
| fnssres 6613 | Restriction of a function ... |
| fnssresd 6614 | Restriction of a function ... |
| fnresin1 6615 | Restriction of a function'... |
| fnresin2 6616 | Restriction of a function'... |
| fnres 6617 | An equivalence for functio... |
| idfn 6618 | The identity relation is a... |
| fnresi 6619 | The restricted identity re... |
| fnima 6620 | The image of a function's ... |
| fn0 6621 | A function with empty doma... |
| fnimadisj 6622 | A class that is disjoint w... |
| fnimaeq0 6623 | Images under a function ne... |
| dfmpt3 6624 | Alternate definition for t... |
| mptfnf 6625 | The maps-to notation defin... |
| fnmptf 6626 | The maps-to notation defin... |
| fnopabg 6627 | Functionality and domain o... |
| fnopab 6628 | Functionality and domain o... |
| mptfng 6629 | The maps-to notation defin... |
| fnmpt 6630 | The maps-to notation defin... |
| fnmptd 6631 | The maps-to notation defin... |
| mpt0 6632 | A mapping operation with e... |
| fnmpti 6633 | Functionality and domain o... |
| dmmpti 6634 | Domain of the mapping oper... |
| dmmptd 6635 | The domain of the mapping ... |
| mptun 6636 | Union of mappings which ar... |
| partfun 6637 | Rewrite a function defined... |
| feq1 6638 | Equality theorem for funct... |
| feq2 6639 | Equality theorem for funct... |
| feq3 6640 | Equality theorem for funct... |
| feq23 6641 | Equality theorem for funct... |
| feq1d 6642 | Equality deduction for fun... |
| feq1dd 6643 | Equality deduction for fun... |
| feq2d 6644 | Equality deduction for fun... |
| feq3d 6645 | Equality deduction for fun... |
| feq2dd 6646 | Equality deduction for fun... |
| feq3dd 6647 | Equality deduction for fun... |
| feq12d 6648 | Equality deduction for fun... |
| feq123d 6649 | Equality deduction for fun... |
| feq123 6650 | Equality theorem for funct... |
| feq1i 6651 | Equality inference for fun... |
| feq2i 6652 | Equality inference for fun... |
| feq12i 6653 | Equality inference for fun... |
| feq23i 6654 | Equality inference for fun... |
| feq23d 6655 | Equality deduction for fun... |
| nff 6656 | Bound-variable hypothesis ... |
| sbcfng 6657 | Distribute proper substitu... |
| sbcfg 6658 | Distribute proper substitu... |
| elimf 6659 | Eliminate a mapping hypoth... |
| ffn 6660 | A mapping is a function wi... |
| ffnd 6661 | A mapping is a function wi... |
| dffn2 6662 | Any function is a mapping ... |
| ffun 6663 | A mapping is a function. ... |
| ffund 6664 | A mapping is a function, d... |
| frel 6665 | A mapping is a relation. ... |
| freld 6666 | A mapping is a relation. ... |
| frn 6667 | The range of a mapping. (... |
| frnd 6668 | Deduction form of ~ frn . ... |
| fdm 6669 | The domain of a mapping. ... |
| fdmd 6670 | Deduction form of ~ fdm . ... |
| fdmi 6671 | Inference associated with ... |
| dffn3 6672 | A function maps to its ran... |
| ffrn 6673 | A function maps to its ran... |
| ffrnb 6674 | Characterization of a func... |
| ffrnbd 6675 | A function maps to its ran... |
| fss 6676 | Expanding the codomain of ... |
| fssd 6677 | Expanding the codomain of ... |
| fssdmd 6678 | Expressing that a class is... |
| fssdm 6679 | Expressing that a class is... |
| fimass 6680 | The image of a class under... |
| fimassd 6681 | The image of a class is a ... |
| fimacnv 6682 | The preimage of the codoma... |
| fcof 6683 | Composition of a function ... |
| fco 6684 | Composition of two functio... |
| fcod 6685 | Composition of two mapping... |
| fco2 6686 | Functionality of a composi... |
| fssxp 6687 | A mapping is a class of or... |
| funssxp 6688 | Two ways of specifying a p... |
| ffdm 6689 | A mapping is a partial fun... |
| ffdmd 6690 | The domain of a function. ... |
| fdmrn 6691 | A different way to write `... |
| funcofd 6692 | Composition of two functio... |
| opelf 6693 | The members of an ordered ... |
| fun 6694 | The union of two functions... |
| fun2 6695 | The union of two functions... |
| fun2d 6696 | The union of functions wit... |
| fnfco 6697 | Composition of two functio... |
| fssres 6698 | Restriction of a function ... |
| fssresd 6699 | Restriction of a function ... |
| fssres2 6700 | Restriction of a restricte... |
| fresin 6701 | An identity for the mappin... |
| resasplit 6702 | If two functions agree on ... |
| fresaun 6703 | The union of two functions... |
| fresaunres2 6704 | From the union of two func... |
| fresaunres1 6705 | From the union of two func... |
| fcoi1 6706 | Composition of a mapping a... |
| fcoi2 6707 | Composition of restricted ... |
| feu 6708 | There is exactly one value... |
| fcnvres 6709 | The converse of a restrict... |
| fimacnvdisj 6710 | The preimage of a class di... |
| fint 6711 | Function into an intersect... |
| fin 6712 | Mapping into an intersecti... |
| f0 6713 | The empty function. (Cont... |
| f00 6714 | A class is a function with... |
| f0bi 6715 | A function with empty doma... |
| f0dom0 6716 | A function is empty iff it... |
| f0rn0 6717 | If there is no element in ... |
| fconst 6718 | A Cartesian product with a... |
| fconstg 6719 | A Cartesian product with a... |
| fnconstg 6720 | A Cartesian product with a... |
| fconst6g 6721 | Constant function with loo... |
| fconst6 6722 | A constant function as a m... |
| f1eq1 6723 | Equality theorem for one-t... |
| f1eq2 6724 | Equality theorem for one-t... |
| f1eq3 6725 | Equality theorem for one-t... |
| nff1 6726 | Bound-variable hypothesis ... |
| dff12 6727 | Alternate definition of a ... |
| f1f 6728 | A one-to-one mapping is a ... |
| f1fn 6729 | A one-to-one mapping is a ... |
| f1fun 6730 | A one-to-one mapping is a ... |
| f1rel 6731 | A one-to-one onto mapping ... |
| f1dm 6732 | The domain of a one-to-one... |
| f1ss 6733 | A function that is one-to-... |
| f1ssr 6734 | A function that is one-to-... |
| f1ssres 6735 | A function that is one-to-... |
| f1resf1 6736 | The restriction of an inje... |
| f1cnvcnv 6737 | Two ways to express that a... |
| f1cof1 6738 | Composition of two one-to-... |
| f1co 6739 | Composition of one-to-one ... |
| foeq1 6740 | Equality theorem for onto ... |
| foeq2 6741 | Equality theorem for onto ... |
| foeq3 6742 | Equality theorem for onto ... |
| nffo 6743 | Bound-variable hypothesis ... |
| fof 6744 | An onto mapping is a mappi... |
| fofun 6745 | An onto mapping is a funct... |
| fofn 6746 | An onto mapping is a funct... |
| forn 6747 | The codomain of an onto fu... |
| dffo2 6748 | Alternate definition of an... |
| foima 6749 | The image of the domain of... |
| dffn4 6750 | A function maps onto its r... |
| funforn 6751 | A function maps its domain... |
| fodmrnu 6752 | An onto function has uniqu... |
| fimadmfo 6753 | A function is a function o... |
| fores 6754 | Restriction of an onto fun... |
| fimadmfoALT 6755 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6756 | The preimage of the codoma... |
| focofo 6757 | Composition of onto functi... |
| foco 6758 | Composition of onto functi... |
| foconst 6759 | A nonzero constant functio... |
| f1oeq1 6760 | Equality theorem for one-t... |
| f1oeq2 6761 | Equality theorem for one-t... |
| f1oeq3 6762 | Equality theorem for one-t... |
| f1oeq23 6763 | Equality theorem for one-t... |
| f1eq123d 6764 | Equality deduction for one... |
| foeq123d 6765 | Equality deduction for ont... |
| f1oeq123d 6766 | Equality deduction for one... |
| f1oeq1d 6767 | Equality deduction for one... |
| f1oeq2d 6768 | Equality deduction for one... |
| f1oeq3d 6769 | Equality deduction for one... |
| nff1o 6770 | Bound-variable hypothesis ... |
| f1of1 6771 | A one-to-one onto mapping ... |
| f1of 6772 | A one-to-one onto mapping ... |
| f1ofn 6773 | A one-to-one onto mapping ... |
| f1ofun 6774 | A one-to-one onto mapping ... |
| f1orel 6775 | A one-to-one onto mapping ... |
| f1odm 6776 | The domain of a one-to-one... |
| dff1o2 6777 | Alternate definition of on... |
| dff1o3 6778 | Alternate definition of on... |
| f1ofo 6779 | A one-to-one onto function... |
| dff1o4 6780 | Alternate definition of on... |
| dff1o5 6781 | Alternate definition of on... |
| f1orn 6782 | A one-to-one function maps... |
| f1f1orn 6783 | A one-to-one function maps... |
| f1ocnv 6784 | The converse of a one-to-o... |
| f1ocnvb 6785 | A relation is a one-to-one... |
| f1ores 6786 | The restriction of a one-t... |
| f1orescnv 6787 | The converse of a one-to-o... |
| f1imacnv 6788 | Preimage of an image. (Co... |
| foimacnv 6789 | A reverse version of ~ f1i... |
| foun 6790 | The union of two onto func... |
| f1oun 6791 | The union of two one-to-on... |
| f1un 6792 | The union of two one-to-on... |
| resdif 6793 | The restriction of a one-t... |
| resin 6794 | The restriction of a one-t... |
| f1oco 6795 | Composition of one-to-one ... |
| f1cnv 6796 | The converse of an injecti... |
| funcocnv2 6797 | Composition with the conve... |
| fococnv2 6798 | The composition of an onto... |
| f1ococnv2 6799 | The composition of a one-t... |
| f1cocnv2 6800 | Composition of an injectiv... |
| f1ococnv1 6801 | The composition of a one-t... |
| f1cocnv1 6802 | Composition of an injectiv... |
| funcoeqres 6803 | Express a constraint on a ... |
| f1ssf1 6804 | A subset of an injective f... |
| f10 6805 | The empty set maps one-to-... |
| f10d 6806 | The empty set maps one-to-... |
| f1o00 6807 | One-to-one onto mapping of... |
| fo00 6808 | Onto mapping of the empty ... |
| f1o0 6809 | One-to-one onto mapping of... |
| f1oi 6810 | A restriction of the ident... |
| f1oiOLD 6811 | Obsolete version of ~ f1oi... |
| f1ovi 6812 | The identity relation is a... |
| f1osn 6813 | A singleton of an ordered ... |
| f1osng 6814 | A singleton of an ordered ... |
| f1sng 6815 | A singleton of an ordered ... |
| fsnd 6816 | A singleton of an ordered ... |
| f1oprswap 6817 | A two-element swap is a bi... |
| f1oprg 6818 | An unordered pair of order... |
| tz6.12-2 6819 | Function value when ` F ` ... |
| tz6.12-2OLD 6820 | Obsolete version of ~ tz6.... |
| fveu 6821 | The value of a function at... |
| brprcneu 6822 | If ` A ` is a proper class... |
| brprcneuALT 6823 | Alternate proof of ~ brprc... |
| fvprc 6824 | A function's value at a pr... |
| fvprcALT 6825 | Alternate proof of ~ fvprc... |
| rnfvprc 6826 | The range of a function va... |
| fv2 6827 | Alternate definition of fu... |
| dffv3 6828 | A definition of function v... |
| dffv4 6829 | The previous definition of... |
| elfv 6830 | Membership in a function v... |
| fveq1 6831 | Equality theorem for funct... |
| fveq2 6832 | Equality theorem for funct... |
| fveq1i 6833 | Equality inference for fun... |
| fveq1d 6834 | Equality deduction for fun... |
| fveq2i 6835 | Equality inference for fun... |
| fveq2d 6836 | Equality deduction for fun... |
| 2fveq3 6837 | Equality theorem for neste... |
| fveq12i 6838 | Equality deduction for fun... |
| fveq12d 6839 | Equality deduction for fun... |
| fveqeq2d 6840 | Equality deduction for fun... |
| fveqeq2 6841 | Equality deduction for fun... |
| nffv 6842 | Bound-variable hypothesis ... |
| nffvmpt1 6843 | Bound-variable hypothesis ... |
| nffvd 6844 | Deduction version of bound... |
| fvex 6845 | The value of a class exist... |
| fvexi 6846 | The value of a class exist... |
| fvexd 6847 | The value of a class exist... |
| fvif 6848 | Move a conditional outside... |
| iffv 6849 | Move a conditional outside... |
| fv3 6850 | Alternate definition of th... |
| fvres 6851 | The value of a restricted ... |
| fvresd 6852 | The value of a restricted ... |
| funssfv 6853 | The value of a member of t... |
| tz6.12c 6854 | Corollary of Theorem 6.12(... |
| tz6.12-1 6855 | Function value. Theorem 6... |
| tz6.12 6856 | Function value. Theorem 6... |
| tz6.12f 6857 | Function value, using boun... |
| tz6.12i 6858 | Corollary of Theorem 6.12(... |
| fvbr0 6859 | Two possibilities for the ... |
| fvrn0 6860 | A function value is a memb... |
| fvn0fvelrn 6861 | If the value of a function... |
| elfvunirn 6862 | A function value is a subs... |
| fvssunirn 6863 | The result of a function v... |
| ndmfv 6864 | The value of a class outsi... |
| ndmfvrcl 6865 | Reverse closure law for fu... |
| elfvdm 6866 | If a function value has a ... |
| elfvex 6867 | If a function value has a ... |
| elfvexd 6868 | If a function value has a ... |
| eliman0 6869 | A nonempty function value ... |
| nfvres 6870 | The value of a non-member ... |
| nfunsn 6871 | If the restriction of a cl... |
| fvfundmfvn0 6872 | If the "value of a class" ... |
| 0fv 6873 | Function value of the empt... |
| fv2prc 6874 | A function value of a func... |
| elfv2ex 6875 | If a function value of a f... |
| fveqres 6876 | Equal values imply equal v... |
| csbfv12 6877 | Move class substitution in... |
| csbfv2g 6878 | Move class substitution in... |
| csbfv 6879 | Substitution for a functio... |
| funbrfv 6880 | The second argument of a b... |
| funopfv 6881 | The second element in an o... |
| fnbrfvb 6882 | Equivalence of function va... |
| fnopfvb 6883 | Equivalence of function va... |
| fvelima2 6884 | Function value in an image... |
| funbrfvb 6885 | Equivalence of function va... |
| funopfvb 6886 | Equivalence of function va... |
| fnbrfvb2 6887 | Version of ~ fnbrfvb for f... |
| fdmeu 6888 | There is exactly one codom... |
| funbrfv2b 6889 | Function value in terms of... |
| dffn5 6890 | Representation of a functi... |
| fnrnfv 6891 | The range of a function ex... |
| fvelrnb 6892 | A member of a function's r... |
| foelcdmi 6893 | A member of a surjective f... |
| dfimafn 6894 | Alternate definition of th... |
| dfimafn2 6895 | Alternate definition of th... |
| funimass4 6896 | Membership relation for th... |
| fvelima 6897 | Function value in an image... |
| funimassd 6898 | Sufficient condition for t... |
| fvelimad 6899 | Function value in an image... |
| feqmptd 6900 | Deduction form of ~ dffn5 ... |
| feqresmpt 6901 | Express a restricted funct... |
| feqmptdf 6902 | Deduction form of ~ dffn5f... |
| dffn5f 6903 | Representation of a functi... |
| fvelimab 6904 | Function value in an image... |
| fvelimabd 6905 | Deduction form of ~ fvelim... |
| fimarab 6906 | Expressing the image of a ... |
| unima 6907 | Image of a union. (Contri... |
| fvi 6908 | The value of the identity ... |
| fviss 6909 | The value of the identity ... |
| fniinfv 6910 | The indexed intersection o... |
| fnsnfv 6911 | Singleton of function valu... |
| opabiotafun 6912 | Define a function whose va... |
| opabiotadm 6913 | Define a function whose va... |
| opabiota 6914 | Define a function whose va... |
| fnimapr 6915 | The image of a pair under ... |
| fnimatpd 6916 | The image of an unordered ... |
| ssimaex 6917 | The existence of a subimag... |
| ssimaexg 6918 | The existence of a subimag... |
| funfv 6919 | A simplified expression fo... |
| funfv2 6920 | The value of a function. ... |
| funfv2f 6921 | The value of a function. ... |
| fvun 6922 | Value of the union of two ... |
| fvun1 6923 | The value of a union when ... |
| fvun2 6924 | The value of a union when ... |
| fvun1d 6925 | The value of a union when ... |
| fvun2d 6926 | The value of a union when ... |
| dffv2 6927 | Alternate definition of fu... |
| dmfco 6928 | Domains of a function comp... |
| fvco2 6929 | Value of a function compos... |
| fvco 6930 | Value of a function compos... |
| fvco3 6931 | Value of a function compos... |
| fvco3d 6932 | Value of a function compos... |
| fvco4i 6933 | Conditions for a compositi... |
| fvopab3g 6934 | Value of a function given ... |
| fvopab3ig 6935 | Value of a function given ... |
| brfvopabrbr 6936 | The binary relation of a f... |
| fvmptg 6937 | Value of a function given ... |
| fvmpti 6938 | Value of a function given ... |
| fvmpt 6939 | Value of a function given ... |
| fvmpt2f 6940 | Value of a function given ... |
| funcnvmpt 6941 | Condition for a function i... |
| fvtresfn 6942 | Functionality of a tuple-r... |
| fvmpts 6943 | Value of a function given ... |
| fvmpt3 6944 | Value of a function given ... |
| fvmpt3i 6945 | Value of a function given ... |
| fvmptdf 6946 | Deduction version of ~ fvm... |
| fvmptd 6947 | Deduction version of ~ fvm... |
| fvmptd2 6948 | Deduction version of ~ fvm... |
| mptrcl 6949 | Reverse closure for a mapp... |
| fvmpt2i 6950 | Value of a function given ... |
| fvmpt2 6951 | Value of a function given ... |
| fvmptss 6952 | If all the values of the m... |
| fvmpt2d 6953 | Deduction version of ~ fvm... |
| fvmptex 6954 | Express a function ` F ` w... |
| fvmptd3f 6955 | Alternate deduction versio... |
| fvmptd2f 6956 | Alternate deduction versio... |
| fvmptdv 6957 | Alternate deduction versio... |
| fvmptdv2 6958 | Alternate deduction versio... |
| mpteqb 6959 | Bidirectional equality the... |
| fvmptt 6960 | Closed theorem form of ~ f... |
| fvmptf 6961 | Value of a function given ... |
| fvmptnf 6962 | The value of a function gi... |
| fvmptd3 6963 | Deduction version of ~ fvm... |
| fvmptd4 6964 | Deduction version of ~ fvm... |
| fvmptn 6965 | This somewhat non-intuitiv... |
| fvmptss2 6966 | A mapping always evaluates... |
| elfvmptrab1w 6967 | Implications for the value... |
| elfvmptrab1 6968 | Implications for the value... |
| elfvmptrab 6969 | Implications for the value... |
| fvopab4ndm 6970 | Value of a function given ... |
| fvmptndm 6971 | Value of a function given ... |
| fvmptrabfv 6972 | Value of a function mappin... |
| fvopab5 6973 | The value of a function th... |
| fvopab6 6974 | Value of a function given ... |
| eqfnfv 6975 | Equality of functions is d... |
| eqfnfv2 6976 | Equality of functions is d... |
| eqfnfv3 6977 | Derive equality of functio... |
| eqfnfvd 6978 | Deduction for equality of ... |
| eqfnfv2f 6979 | Equality of functions is d... |
| eqfunfv 6980 | Equality of functions is d... |
| eqfnun 6981 | Two functions on ` A u. B ... |
| fvreseq0 6982 | Equality of restricted fun... |
| fvreseq1 6983 | Equality of a function res... |
| fvreseq 6984 | Equality of restricted fun... |
| fnmptfvd 6985 | A function with a given do... |
| fndmdif 6986 | Two ways to express the lo... |
| fndmdifcom 6987 | The difference set between... |
| fndmdifeq0 6988 | The difference set of two ... |
| fndmin 6989 | Two ways to express the lo... |
| fneqeql 6990 | Two functions are equal if... |
| fneqeql2 6991 | Two functions are equal if... |
| fnreseql 6992 | Two functions are equal on... |
| chfnrn 6993 | The range of a choice func... |
| funfvop 6994 | Ordered pair with function... |
| funfvbrb 6995 | Two ways to say that ` A `... |
| fvimacnvi 6996 | A member of a preimage is ... |
| fvimacnv 6997 | The argument of a function... |
| funimass3 6998 | A kind of contraposition l... |
| funimass5 6999 | A subclass of a preimage i... |
| funconstss 7000 | Two ways of specifying tha... |
| fvimacnvALT 7001 | Alternate proof of ~ fvima... |
| elpreima 7002 | Membership in the preimage... |
| elpreimad 7003 | Membership in the preimage... |
| fniniseg 7004 | Membership in the preimage... |
| fncnvima2 7005 | Inverse images under funct... |
| fniniseg2 7006 | Inverse point images under... |
| unpreima 7007 | Preimage of a union. (Con... |
| inpreima 7008 | Preimage of an intersectio... |
| difpreima 7009 | Preimage of a difference. ... |
| respreima 7010 | The preimage of a restrict... |
| cnvimainrn 7011 | The preimage of the inters... |
| sspreima 7012 | The preimage of a subset i... |
| iinpreima 7013 | Preimage of an intersectio... |
| intpreima 7014 | Preimage of an intersectio... |
| fimacnvinrn 7015 | Taking the converse image ... |
| fimacnvinrn2 7016 | Taking the converse image ... |
| rescnvimafod 7017 | The restriction of a funct... |
| fvn0ssdmfun 7018 | If a class' function value... |
| fnopfv 7019 | Ordered pair with function... |
| fvelrn 7020 | A function's value belongs... |
| nelrnfvne 7021 | A function value cannot be... |
| fveqdmss 7022 | If the empty set is not co... |
| fveqressseq 7023 | If the empty set is not co... |
| fnfvelrn 7024 | A function's value belongs... |
| ffvelcdm 7025 | A function's value belongs... |
| fnfvelrnd 7026 | A function's value belongs... |
| ffvelcdmi 7027 | A function's value belongs... |
| ffvelcdmda 7028 | A function's value belongs... |
| ffvelcdmd 7029 | A function's value belongs... |
| feldmfvelcdm 7030 | A class is an element of t... |
| rexrn 7031 | Restricted existential qua... |
| ralrn 7032 | Restricted universal quant... |
| elrnrexdm 7033 | For any element in the ran... |
| elrnrexdmb 7034 | For any element in the ran... |
| eldmrexrn 7035 | For any element in the dom... |
| eldmrexrnb 7036 | For any element in the dom... |
| fvcofneq 7037 | The values of two function... |
| ralrnmptw 7038 | A restricted quantifier ov... |
| rexrnmptw 7039 | A restricted quantifier ov... |
| ralrnmpt 7040 | A restricted quantifier ov... |
| rexrnmpt 7041 | A restricted quantifier ov... |
| f0cli 7042 | Unconditional closure of a... |
| dff2 7043 | Alternate definition of a ... |
| dff3 7044 | Alternate definition of a ... |
| dff4 7045 | Alternate definition of a ... |
| dffo3 7046 | An onto mapping expressed ... |
| dffo4 7047 | Alternate definition of an... |
| dffo5 7048 | Alternate definition of an... |
| exfo 7049 | A relation equivalent to t... |
| dffo3f 7050 | An onto mapping expressed ... |
| foelrn 7051 | Property of a surjective f... |
| foelrnf 7052 | Property of a surjective f... |
| foco2 7053 | If a composition of two fu... |
| fmpt 7054 | Functionality of the mappi... |
| f1ompt 7055 | Express bijection for a ma... |
| fmpti 7056 | Functionality of the mappi... |
| fvmptelcdm 7057 | The value of a function at... |
| fmptd 7058 | Domain and codomain of the... |
| fmpttd 7059 | Version of ~ fmptd with in... |
| fmpt3d 7060 | Domain and codomain of the... |
| fmptdf 7061 | A version of ~ fmptd using... |
| fompt 7062 | Express being onto for a m... |
| ffnfv 7063 | A function maps to a class... |
| ffnfvf 7064 | A function maps to a class... |
| fnfvrnss 7065 | An upper bound for range d... |
| fcdmssb 7066 | A function is a function i... |
| rnmptss 7067 | The range of an operation ... |
| rnmptssd 7068 | The range of a function gi... |
| fmpt2d 7069 | Domain and codomain of the... |
| ffvresb 7070 | A necessary and sufficient... |
| fssrescdmd 7071 | Restriction of a function ... |
| f1oresrab 7072 | Build a bijection between ... |
| f1ossf1o 7073 | Restricting a bijection, w... |
| fmptco 7074 | Composition of two functio... |
| fmptcof 7075 | Version of ~ fmptco where ... |
| fmptcos 7076 | Composition of two functio... |
| cofmpt 7077 | Express composition of a m... |
| fcompt 7078 | Express composition of two... |
| fcoconst 7079 | Composition with a constan... |
| fsn 7080 | A function maps a singleto... |
| fsn2 7081 | A function that maps a sin... |
| fsng 7082 | A function maps a singleto... |
| fsn2g 7083 | A function that maps a sin... |
| xpsng 7084 | The Cartesian product of t... |
| xpprsng 7085 | The Cartesian product of a... |
| xpsn 7086 | The Cartesian product of t... |
| f1o2sn 7087 | A singleton consisting in ... |
| residpr 7088 | Restriction of the identit... |
| dfmpt 7089 | Alternate definition for t... |
| fnasrn 7090 | A function expressed as th... |
| idref 7091 | Two ways to state that a r... |
| funiun 7092 | A function is a union of s... |
| funopsn 7093 | If a function is an ordere... |
| funop 7094 | An ordered pair is a funct... |
| funopdmsn 7095 | The domain of a function w... |
| funsndifnop 7096 | A singleton of an ordered ... |
| funsneqopb 7097 | A singleton of an ordered ... |
| ressnop0 7098 | If ` A ` is not in ` C ` ,... |
| fpr 7099 | A function with a domain o... |
| fprg 7100 | A function with a domain o... |
| ftpg 7101 | A function with a domain o... |
| ftp 7102 | A function with a domain o... |
| fnressn 7103 | A function restricted to a... |
| funressn 7104 | A function restricted to a... |
| fressnfv 7105 | The value of a function re... |
| fvrnressn 7106 | If the value of a function... |
| fvressn 7107 | The value of a function re... |
| fvconst 7108 | The value of a constant fu... |
| fnsnr 7109 | If a class belongs to a fu... |
| fnsnbg 7110 | A function's domain is a s... |
| fnsnb 7111 | A function whose domain is... |
| fnsnbOLD 7112 | Obsolete version of ~ fnsn... |
| fmptsn 7113 | Express a singleton functi... |
| fmptsng 7114 | Express a singleton functi... |
| fmptsnd 7115 | Express a singleton functi... |
| fmptap 7116 | Append an additional value... |
| fmptapd 7117 | Append an additional value... |
| fmptpr 7118 | Express a pair function in... |
| fvresi 7119 | The value of a restricted ... |
| fninfp 7120 | Express the class of fixed... |
| fnelfp 7121 | Property of a fixed point ... |
| fndifnfp 7122 | Express the class of non-f... |
| fnelnfp 7123 | Property of a non-fixed po... |
| fnnfpeq0 7124 | A function is the identity... |
| fvunsn 7125 | Remove an ordered pair not... |
| fvsng 7126 | The value of a singleton o... |
| fvsn 7127 | The value of a singleton o... |
| fvsnun1 7128 | The value of a function wi... |
| fvsnun2 7129 | The value of a function wi... |
| fnsnsplit 7130 | Split a function into a si... |
| fsnunf 7131 | Adjoining a point to a fun... |
| fsnunf2 7132 | Adjoining a point to a pun... |
| fsnunfv 7133 | Recover the added point fr... |
| fsnunres 7134 | Recover the original funct... |
| funresdfunsn 7135 | Restricting a function to ... |
| fvpr1g 7136 | The value of a function wi... |
| fvpr2g 7137 | The value of a function wi... |
| fvpr1 7138 | The value of a function wi... |
| fvpr2 7139 | The value of a function wi... |
| fprb 7140 | A condition for functionho... |
| fvtp1 7141 | The first value of a funct... |
| fvtp2 7142 | The second value of a func... |
| fvtp3 7143 | The third value of a funct... |
| fvtp1g 7144 | The value of a function wi... |
| fvtp2g 7145 | The value of a function wi... |
| fvtp3g 7146 | The value of a function wi... |
| tpres 7147 | An unordered triple of ord... |
| fvconst2g 7148 | The value of a constant fu... |
| fconst2g 7149 | A constant function expres... |
| fvconst2 7150 | The value of a constant fu... |
| fconst2 7151 | A constant function expres... |
| fconst5 7152 | Two ways to express that a... |
| rnmptc 7153 | Range of a constant functi... |
| fnprb 7154 | A function whose domain ha... |
| fntpb 7155 | A function whose domain ha... |
| fnpr2g 7156 | A function whose domain ha... |
| fpr2g 7157 | A function that maps a pai... |
| fconstfv 7158 | A constant function expres... |
| fconst3 7159 | Two ways to express a cons... |
| fconst4 7160 | Two ways to express a cons... |
| resfunexg 7161 | The restriction of a funct... |
| resiexd 7162 | The restriction of the ide... |
| fnex 7163 | If the domain of a functio... |
| fnexd 7164 | If the domain of a functio... |
| funex 7165 | If the domain of a functio... |
| opabex 7166 | Existence of a function ex... |
| mptexg 7167 | If the domain of a functio... |
| mptexgf 7168 | If the domain of a functio... |
| mptex 7169 | If the domain of a functio... |
| mptexd 7170 | If the domain of a functio... |
| mptrabex 7171 | If the domain of a functio... |
| fex 7172 | If the domain of a mapping... |
| fexd 7173 | If the domain of a mapping... |
| mptfvmpt 7174 | A function in maps-to nota... |
| eufnfv 7175 | A function is uniquely det... |
| funfvima 7176 | A function's value in a pr... |
| funfvima2 7177 | A function's value in an i... |
| funfvima2d 7178 | A function's value in a pr... |
| fnfvima 7179 | The function value of an o... |
| fnfvimad 7180 | A function's value belongs... |
| resfvresima 7181 | The value of the function ... |
| funfvima3 7182 | A class including a functi... |
| ralima 7183 | Universal quantification u... |
| rexima 7184 | Existential quantification... |
| reximaOLD 7185 | Obsolete version of ~ rexi... |
| ralimaOLD 7186 | Obsolete version of ~ rali... |
| fvclss 7187 | Upper bound for the class ... |
| elabrex 7188 | Elementhood in an image se... |
| elabrexg 7189 | Elementhood in an image se... |
| abrexco 7190 | Composition of two image m... |
| imaiun 7191 | The image of an indexed un... |
| imauni 7192 | The image of a union is th... |
| fniunfv 7193 | The indexed union of a fun... |
| funiunfv 7194 | The indexed union of a fun... |
| funiunfvf 7195 | The indexed union of a fun... |
| eluniima 7196 | Membership in the union of... |
| elunirn 7197 | Membership in the union of... |
| elunirnALT 7198 | Alternate proof of ~ eluni... |
| fnunirn 7199 | Membership in a union of s... |
| dff13 7200 | A one-to-one function in t... |
| dff13f 7201 | A one-to-one function in t... |
| f1veqaeq 7202 | If the values of a one-to-... |
| f1cofveqaeq 7203 | If the values of a composi... |
| f1cofveqaeqALT 7204 | Alternate proof of ~ f1cof... |
| dff14i 7205 | A one-to-one function maps... |
| 2f1fvneq 7206 | If two one-to-one function... |
| f1mpt 7207 | Express injection for a ma... |
| f1fveq 7208 | Equality of function value... |
| f1elima 7209 | Membership in the image of... |
| f1imass 7210 | Taking images under a one-... |
| f1imaeq 7211 | Taking images under a one-... |
| f1imapss 7212 | Taking images under a one-... |
| fpropnf1 7213 | A function, given by an un... |
| f1dom3fv3dif 7214 | The function values for a ... |
| f1dom3el3dif 7215 | The codomain of a 1-1 func... |
| dff14a 7216 | A one-to-one function in t... |
| dff14b 7217 | A one-to-one function in t... |
| f1ounsn 7218 | Extension of a bijection b... |
| f12dfv 7219 | A one-to-one function with... |
| f13dfv 7220 | A one-to-one function with... |
| dff1o6 7221 | A one-to-one onto function... |
| f1ocnvfv1 7222 | The converse value of the ... |
| f1ocnvfv2 7223 | The value of the converse ... |
| f1ocnvfv 7224 | Relationship between the v... |
| f1ocnvfvb 7225 | Relationship between the v... |
| nvof1o 7226 | An involution is a bijecti... |
| nvocnv 7227 | The converse of an involut... |
| f1cdmsn 7228 | If a one-to-one function w... |
| fsnex 7229 | Relate a function with a s... |
| f1prex 7230 | Relate a one-to-one functi... |
| f1ocnvdm 7231 | The value of the converse ... |
| f1ocnvfvrneq 7232 | If the values of a one-to-... |
| fcof1 7233 | An application is injectiv... |
| fcofo 7234 | An application is surjecti... |
| cbvfo 7235 | Change bound variable betw... |
| cbvexfo 7236 | Change bound variable betw... |
| cocan1 7237 | An injection is left-cance... |
| cocan2 7238 | A surjection is right-canc... |
| fcof1oinvd 7239 | Show that a function is th... |
| fcof1od 7240 | A function is bijective if... |
| 2fcoidinvd 7241 | Show that a function is th... |
| fcof1o 7242 | Show that two functions ar... |
| 2fvcoidd 7243 | Show that the composition ... |
| 2fvidf1od 7244 | A function is bijective if... |
| 2fvidinvd 7245 | Show that two functions ar... |
| foeqcnvco 7246 | Condition for function equ... |
| f1eqcocnv 7247 | Condition for function equ... |
| fveqf1o 7248 | Given a bijection ` F ` , ... |
| f1ocoima 7249 | The composition of two bij... |
| nf1const 7250 | A constant function from a... |
| nf1oconst 7251 | A constant function from a... |
| f1ofvswap 7252 | Swapping two values in a b... |
| fvf1pr 7253 | Values of a one-to-one fun... |
| fliftrel 7254 | ` F ` , a function lift, i... |
| fliftel 7255 | Elementhood in the relatio... |
| fliftel1 7256 | Elementhood in the relatio... |
| fliftcnv 7257 | Converse of the relation `... |
| fliftfun 7258 | The function ` F ` is the ... |
| fliftfund 7259 | The function ` F ` is the ... |
| fliftfuns 7260 | The function ` F ` is the ... |
| fliftf 7261 | The domain and range of th... |
| fliftval 7262 | The value of the function ... |
| isoeq1 7263 | Equality theorem for isomo... |
| isoeq2 7264 | Equality theorem for isomo... |
| isoeq3 7265 | Equality theorem for isomo... |
| isoeq4 7266 | Equality theorem for isomo... |
| isoeq5 7267 | Equality theorem for isomo... |
| nfiso 7268 | Bound-variable hypothesis ... |
| isof1o 7269 | An isomorphism is a one-to... |
| isof1oidb 7270 | A function is a bijection ... |
| isof1oopb 7271 | A function is a bijection ... |
| isorel 7272 | An isomorphism connects bi... |
| soisores 7273 | Express the condition of i... |
| soisoi 7274 | Infer isomorphism from one... |
| isoid 7275 | Identity law for isomorphi... |
| isocnv 7276 | Converse law for isomorphi... |
| isocnv2 7277 | Converse law for isomorphi... |
| isocnv3 7278 | Complementation law for is... |
| isores2 7279 | An isomorphism from one we... |
| isores1 7280 | An isomorphism from one we... |
| isores3 7281 | Induced isomorphism on a s... |
| isotr 7282 | Composition (transitive) l... |
| isomin 7283 | Isomorphisms preserve mini... |
| isoini 7284 | Isomorphisms preserve init... |
| isoini2 7285 | Isomorphisms are isomorphi... |
| isofrlem 7286 | Lemma for ~ isofr . (Cont... |
| isoselem 7287 | Lemma for ~ isose . (Cont... |
| isofr 7288 | An isomorphism preserves w... |
| isose 7289 | An isomorphism preserves s... |
| isofr2 7290 | A weak form of ~ isofr tha... |
| isopolem 7291 | Lemma for ~ isopo . (Cont... |
| isopo 7292 | An isomorphism preserves t... |
| isosolem 7293 | Lemma for ~ isoso . (Cont... |
| isoso 7294 | An isomorphism preserves t... |
| isowe 7295 | An isomorphism preserves t... |
| isowe2 7296 | A weak form of ~ isowe tha... |
| f1oiso 7297 | Any one-to-one onto functi... |
| f1oiso2 7298 | Any one-to-one onto functi... |
| f1owe 7299 | Well-ordering of isomorphi... |
| weniso 7300 | A set-like well-ordering h... |
| weisoeq 7301 | Thus, there is at most one... |
| weisoeq2 7302 | Thus, there is at most one... |
| knatar 7303 | The Knaster-Tarski theorem... |
| fvresval 7304 | The value of a restricted ... |
| funeldmb 7305 | If ` (/) ` is not part of ... |
| eqfunresadj 7306 | Law for adjoining an eleme... |
| eqfunressuc 7307 | Law for equality of restri... |
| fnssintima 7308 | Condition for subset of an... |
| imaeqsexvOLD 7309 | Obsolete version of ~ rexi... |
| imaeqsalvOLD 7310 | Obsolete version of ~ rali... |
| fnimasnd 7311 | The image of a function by... |
| canth 7312 | No set ` A ` is equinumero... |
| ncanth 7313 | Cantor's theorem fails for... |
| riotaeqdv 7316 | Formula-building deduction... |
| riotabidv 7317 | Formula-building deduction... |
| riotaeqbidv 7318 | Equality deduction for res... |
| riotaex 7319 | Restricted iota is a set. ... |
| riotav 7320 | An iota restricted to the ... |
| riotauni 7321 | Restricted iota in terms o... |
| nfriota1 7322 | The abstraction variable i... |
| nfriotadw 7323 | Deduction version of ~ nfr... |
| cbvriotaw 7324 | Change bound variable in a... |
| cbvriotavw 7325 | Change bound variable in a... |
| nfriotad 7326 | Deduction version of ~ nfr... |
| nfriota 7327 | A variable not free in a w... |
| cbvriota 7328 | Change bound variable in a... |
| cbvriotav 7329 | Change bound variable in a... |
| csbriota 7330 | Interchange class substitu... |
| riotacl2 7331 | Membership law for "the un... |
| riotacl 7332 | Closure of restricted iota... |
| riotasbc 7333 | Substitution law for descr... |
| riotabidva 7334 | Equivalent wff's yield equ... |
| riotabiia 7335 | Equivalent wff's yield equ... |
| riota1 7336 | Property of restricted iot... |
| riota1a 7337 | Property of iota. (Contri... |
| riota2df 7338 | A deduction version of ~ r... |
| riota2f 7339 | This theorem shows a condi... |
| riota2 7340 | This theorem shows a condi... |
| riotaeqimp 7341 | If two restricted iota des... |
| riotaprop 7342 | Properties of a restricted... |
| riota5f 7343 | A method for computing res... |
| riota5 7344 | A method for computing res... |
| riotass2 7345 | Restriction of a unique el... |
| riotass 7346 | Restriction of a unique el... |
| moriotass 7347 | Restriction of a unique el... |
| snriota 7348 | A restricted class abstrac... |
| riotaxfrd 7349 | Change the variable ` x ` ... |
| eusvobj2 7350 | Specify the same property ... |
| eusvobj1 7351 | Specify the same object in... |
| f1ofveu 7352 | There is one domain elemen... |
| f1ocnvfv3 7353 | Value of the converse of a... |
| riotaund 7354 | Restricted iota equals the... |
| riotassuni 7355 | The restricted iota class ... |
| riotaclb 7356 | Bidirectional closure of r... |
| riotarab 7357 | Restricted iota of a restr... |
| oveq 7364 | Equality theorem for opera... |
| oveq1 7365 | Equality theorem for opera... |
| oveq2 7366 | Equality theorem for opera... |
| oveq12 7367 | Equality theorem for opera... |
| oveq1i 7368 | Equality inference for ope... |
| oveq2i 7369 | Equality inference for ope... |
| oveq12i 7370 | Equality inference for ope... |
| oveqi 7371 | Equality inference for ope... |
| oveq123i 7372 | Equality inference for ope... |
| oveq1d 7373 | Equality deduction for ope... |
| oveq2d 7374 | Equality deduction for ope... |
| oveqd 7375 | Equality deduction for ope... |
| oveq12d 7376 | Equality deduction for ope... |
| oveqan12d 7377 | Equality deduction for ope... |
| oveqan12rd 7378 | Equality deduction for ope... |
| oveq123d 7379 | Equality deduction for ope... |
| fvoveq1d 7380 | Equality deduction for nes... |
| fvoveq1 7381 | Equality theorem for neste... |
| ovanraleqv 7382 | Equality theorem for a con... |
| imbrov2fvoveq 7383 | Equality theorem for neste... |
| ovrspc2v 7384 | If an operation value is a... |
| oveqrspc2v 7385 | Restricted specialization ... |
| oveqdr 7386 | Equality of two operations... |
| nfovd 7387 | Deduction version of bound... |
| nfov 7388 | Bound-variable hypothesis ... |
| oprabidw 7389 | The law of concretion. Sp... |
| oprabid 7390 | The law of concretion. Sp... |
| ovex 7391 | The result of an operation... |
| ovexi 7392 | The result of an operation... |
| ovexd 7393 | The result of an operation... |
| ovssunirn 7394 | The result of an operation... |
| 0ov 7395 | Operation value of the emp... |
| ovprc 7396 | The value of an operation ... |
| ovprc1 7397 | The value of an operation ... |
| ovprc2 7398 | The value of an operation ... |
| ovrcl 7399 | Reverse closure for an ope... |
| elfvov1 7400 | Utility theorem: reverse c... |
| elfvov2 7401 | Utility theorem: reverse c... |
| csbov123 7402 | Move class substitution in... |
| csbov 7403 | Move class substitution in... |
| csbov12g 7404 | Move class substitution in... |
| csbov1g 7405 | Move class substitution in... |
| csbov2g 7406 | Move class substitution in... |
| rspceov 7407 | A frequently used special ... |
| elovimad 7408 | Elementhood of the image s... |
| fnbrovb 7409 | Value of a binary operatio... |
| fnotovb 7410 | Equivalence of operation v... |
| opabbrex 7411 | A collection of ordered pa... |
| opabresex2 7412 | Restrictions of a collecti... |
| fvmptopab 7413 | The function value of a ma... |
| f1opr 7414 | Condition for an operation... |
| brfvopab 7415 | The classes involved in a ... |
| dfoprab2 7416 | Class abstraction for oper... |
| reloprab 7417 | An operation class abstrac... |
| oprabv 7418 | If a pair and a class are ... |
| nfoprab1 7419 | The abstraction variables ... |
| nfoprab2 7420 | The abstraction variables ... |
| nfoprab3 7421 | The abstraction variables ... |
| nfoprab 7422 | Bound-variable hypothesis ... |
| oprabbid 7423 | Equivalent wff's yield equ... |
| oprabbidv 7424 | Equivalent wff's yield equ... |
| oprabbii 7425 | Equivalent wff's yield equ... |
| ssoprab2 7426 | Equivalence of ordered pai... |
| ssoprab2b 7427 | Equivalence of ordered pai... |
| eqoprab2bw 7428 | Equivalence of ordered pai... |
| eqoprab2b 7429 | Equivalence of ordered pai... |
| mpoeq123 7430 | An equality theorem for th... |
| mpoeq12 7431 | An equality theorem for th... |
| mpoeq123dva 7432 | An equality deduction for ... |
| mpoeq123dv 7433 | An equality deduction for ... |
| mpoeq123i 7434 | An equality inference for ... |
| mpoeq3dva 7435 | Slightly more general equa... |
| mpoeq3ia 7436 | An equality inference for ... |
| mpoeq3dv 7437 | An equality deduction for ... |
| nfmpo1 7438 | Bound-variable hypothesis ... |
| nfmpo2 7439 | Bound-variable hypothesis ... |
| nfmpo 7440 | Bound-variable hypothesis ... |
| 0mpo0 7441 | A mapping operation with e... |
| mpo0v 7442 | A mapping operation with e... |
| mpo0 7443 | A mapping operation with e... |
| oprab4 7444 | Two ways to state the doma... |
| cbvoprab1 7445 | Rule used to change first ... |
| cbvoprab2 7446 | Change the second bound va... |
| cbvoprab12 7447 | Rule used to change first ... |
| cbvoprab12v 7448 | Rule used to change first ... |
| cbvoprab3 7449 | Rule used to change the th... |
| cbvoprab3v 7450 | Rule used to change the th... |
| cbvmpox 7451 | Rule to change the bound v... |
| cbvmpo 7452 | Rule to change the bound v... |
| cbvmpov 7453 | Rule to change the bound v... |
| elimdelov 7454 | Eliminate a hypothesis whi... |
| brif1 7455 | Move a relation inside and... |
| ovif 7456 | Move a conditional outside... |
| ovif2 7457 | Move a conditional outside... |
| ovif12 7458 | Move a conditional outside... |
| ifov 7459 | Move a conditional outside... |
| ifmpt2v 7460 | Move a conditional inside ... |
| dmoprab 7461 | The domain of an operation... |
| dmoprabss 7462 | The domain of an operation... |
| rnoprab 7463 | The range of an operation ... |
| rnoprab2 7464 | The range of a restricted ... |
| reldmoprab 7465 | The domain of an operation... |
| oprabss 7466 | Structure of an operation ... |
| eloprabga 7467 | The law of concretion for ... |
| eloprabg 7468 | The law of concretion for ... |
| ssoprab2i 7469 | Inference of operation cla... |
| mpov 7470 | Operation with universal d... |
| mpomptx 7471 | Express a two-argument fun... |
| mpompt 7472 | Express a two-argument fun... |
| mpodifsnif 7473 | A mapping with two argumen... |
| mposnif 7474 | A mapping with two argumen... |
| fconstmpo 7475 | Representation of a consta... |
| resoprab 7476 | Restriction of an operatio... |
| resoprab2 7477 | Restriction of an operator... |
| resmpo 7478 | Restriction of the mapping... |
| funoprabg 7479 | "At most one" is a suffici... |
| funoprab 7480 | "At most one" is a suffici... |
| fnoprabg 7481 | Functionality and domain o... |
| mpofun 7482 | The maps-to notation for a... |
| fnoprab 7483 | Functionality and domain o... |
| ffnov 7484 | An operation maps to a cla... |
| fovcld 7485 | Closure law for an operati... |
| fovcl 7486 | Closure law for an operati... |
| eqfnov 7487 | Equality of two operations... |
| eqfnov2 7488 | Two operators with the sam... |
| fnov 7489 | Representation of a functi... |
| mpo2eqb 7490 | Bidirectional equality the... |
| rnmpo 7491 | The range of an operation ... |
| reldmmpo 7492 | The domain of an operation... |
| elrnmpog 7493 | Membership in the range of... |
| elrnmpo 7494 | Membership in the range of... |
| elimampo 7495 | Membership in the image of... |
| elrnmpores 7496 | Membership in the range of... |
| ralrnmpo 7497 | A restricted quantifier ov... |
| rexrnmpo 7498 | A restricted quantifier ov... |
| ovid 7499 | The value of an operation ... |
| ovidig 7500 | The value of an operation ... |
| ovidi 7501 | The value of an operation ... |
| ov 7502 | The value of an operation ... |
| ovigg 7503 | The value of an operation ... |
| ovig 7504 | The value of an operation ... |
| ovmpt4g 7505 | Value of a function given ... |
| ovmpos 7506 | Value of a function given ... |
| ov2gf 7507 | The value of an operation ... |
| ovmpodxf 7508 | Value of an operation give... |
| ovmpodx 7509 | Value of an operation give... |
| ovmpod 7510 | Value of an operation give... |
| ovmpox 7511 | The value of an operation ... |
| ovmpoga 7512 | Value of an operation give... |
| ovmpoa 7513 | Value of an operation give... |
| ovmpodf 7514 | Alternate deduction versio... |
| ovmpodv 7515 | Alternate deduction versio... |
| ovmpodv2 7516 | Alternate deduction versio... |
| ovmpog 7517 | Value of an operation give... |
| ovmpo 7518 | Value of an operation give... |
| ovmpot 7519 | The value of an operation ... |
| fvmpopr2d 7520 | Value of an operation give... |
| ov3 7521 | The value of an operation ... |
| ov6g 7522 | The value of an operation ... |
| ovg 7523 | The value of an operation ... |
| ovres 7524 | The value of a restricted ... |
| ovresd 7525 | Lemma for converting metri... |
| oprres 7526 | The restriction of an oper... |
| oprssov 7527 | The value of a member of t... |
| fovcdm 7528 | An operation's value belon... |
| fovcdmda 7529 | An operation's value belon... |
| fovcdmd 7530 | An operation's value belon... |
| fnrnov 7531 | The range of an operation ... |
| foov 7532 | An onto mapping of an oper... |
| fnovrn 7533 | An operation's value belon... |
| ovelrn 7534 | A member of an operation's... |
| funimassov 7535 | Membership relation for th... |
| ovelimab 7536 | Operation value in an imag... |
| ovima0 7537 | An operation value is a me... |
| ovconst2 7538 | The value of a constant op... |
| oprssdm 7539 | Domain of closure of an op... |
| nssdmovg 7540 | The value of an operation ... |
| ndmovg 7541 | The value of an operation ... |
| ndmov 7542 | The value of an operation ... |
| ndmovcl 7543 | The closure of an operatio... |
| ndmovrcl 7544 | Reverse closure law, when ... |
| ndmovcom 7545 | Any operation is commutati... |
| ndmovass 7546 | Any operation is associati... |
| ndmovdistr 7547 | Any operation is distribut... |
| ndmovord 7548 | Elimination of redundant a... |
| ndmovordi 7549 | Elimination of redundant a... |
| caovclg 7550 | Convert an operation closu... |
| caovcld 7551 | Convert an operation closu... |
| caovcl 7552 | Convert an operation closu... |
| caovcomg 7553 | Convert an operation commu... |
| caovcomd 7554 | Convert an operation commu... |
| caovcom 7555 | Convert an operation commu... |
| caovassg 7556 | Convert an operation assoc... |
| caovassd 7557 | Convert an operation assoc... |
| caovass 7558 | Convert an operation assoc... |
| caovcang 7559 | Convert an operation cance... |
| caovcand 7560 | Convert an operation cance... |
| caovcanrd 7561 | Commute the arguments of a... |
| caovcan 7562 | Convert an operation cance... |
| caovordig 7563 | Convert an operation order... |
| caovordid 7564 | Convert an operation order... |
| caovordg 7565 | Convert an operation order... |
| caovordd 7566 | Convert an operation order... |
| caovord2d 7567 | Operation ordering law wit... |
| caovord3d 7568 | Ordering law. (Contribute... |
| caovord 7569 | Convert an operation order... |
| caovord2 7570 | Operation ordering law wit... |
| caovord3 7571 | Ordering law. (Contribute... |
| caovdig 7572 | Convert an operation distr... |
| caovdid 7573 | Convert an operation distr... |
| caovdir2d 7574 | Convert an operation distr... |
| caovdirg 7575 | Convert an operation rever... |
| caovdird 7576 | Convert an operation distr... |
| caovdi 7577 | Convert an operation distr... |
| caov32d 7578 | Rearrange arguments in a c... |
| caov12d 7579 | Rearrange arguments in a c... |
| caov31d 7580 | Rearrange arguments in a c... |
| caov13d 7581 | Rearrange arguments in a c... |
| caov4d 7582 | Rearrange arguments in a c... |
| caov411d 7583 | Rearrange arguments in a c... |
| caov42d 7584 | Rearrange arguments in a c... |
| caov32 7585 | Rearrange arguments in a c... |
| caov12 7586 | Rearrange arguments in a c... |
| caov31 7587 | Rearrange arguments in a c... |
| caov13 7588 | Rearrange arguments in a c... |
| caov4 7589 | Rearrange arguments in a c... |
| caov411 7590 | Rearrange arguments in a c... |
| caov42 7591 | Rearrange arguments in a c... |
| caovdir 7592 | Reverse distributive law. ... |
| caovdilem 7593 | Lemma used by real number ... |
| caovlem2 7594 | Lemma used in real number ... |
| caovmo 7595 | Uniqueness of inverse elem... |
| imaeqexov 7596 | Substitute an operation va... |
| imaeqalov 7597 | Substitute an operation va... |
| mpondm0 7598 | The value of an operation ... |
| elmpocl 7599 | If a two-parameter class i... |
| elmpocl1 7600 | If a two-parameter class i... |
| elmpocl2 7601 | If a two-parameter class i... |
| elovmpod 7602 | Utility lemma for two-para... |
| elovmpo 7603 | Utility lemma for two-para... |
| elovmporab 7604 | Implications for the value... |
| elovmporab1w 7605 | Implications for the value... |
| elovmporab1 7606 | Implications for the value... |
| 2mpo0 7607 | If the operation value of ... |
| relmptopab 7608 | Any function to sets of or... |
| f1ocnvd 7609 | Describe an implicit one-t... |
| f1od 7610 | Describe an implicit one-t... |
| f1ocnv2d 7611 | Describe an implicit one-t... |
| f1o2d 7612 | Describe an implicit one-t... |
| f1opw2 7613 | A one-to-one mapping induc... |
| f1opw 7614 | A one-to-one mapping induc... |
| elovmpt3imp 7615 | If the value of a function... |
| ovmpt3rab1 7616 | The value of an operation ... |
| ovmpt3rabdm 7617 | If the value of a function... |
| elovmpt3rab1 7618 | Implications for the value... |
| elovmpt3rab 7619 | Implications for the value... |
| ofeqd 7624 | Equality theorem for funct... |
| ofeq 7625 | Equality theorem for funct... |
| ofreq 7626 | Equality theorem for funct... |
| ofexg 7627 | A function operation restr... |
| nfof 7628 | Hypothesis builder for fun... |
| nfofr 7629 | Hypothesis builder for fun... |
| ofrfvalg 7630 | Value of a relation applie... |
| offval 7631 | Value of an operation appl... |
| ofrfval 7632 | Value of a relation applie... |
| ofval 7633 | Evaluate a function operat... |
| ofrval 7634 | Exhibit a function relatio... |
| offn 7635 | The function operation pro... |
| offun 7636 | The function operation pro... |
| offval2f 7637 | The function operation exp... |
| ofmresval 7638 | Value of a restriction of ... |
| fnfvof 7639 | Function value of a pointw... |
| off 7640 | The function operation pro... |
| ofres 7641 | Restrict the operands of a... |
| offval2 7642 | The function operation exp... |
| ofrfval2 7643 | The function relation acti... |
| offvalfv 7644 | The function operation exp... |
| ofmpteq 7645 | Value of a pointwise opera... |
| coof 7646 | The composition of a _homo... |
| ofco 7647 | The composition of a funct... |
| offveq 7648 | Convert an identity of the... |
| offveqb 7649 | Equivalent expressions for... |
| ofc1 7650 | Left operation by a consta... |
| ofc2 7651 | Right operation by a const... |
| ofc12 7652 | Function operation on two ... |
| caofref 7653 | Transfer a reflexive law t... |
| caofinvl 7654 | Transfer a left inverse la... |
| caofid0l 7655 | Transfer a left identity l... |
| caofid0r 7656 | Transfer a right identity ... |
| caofid1 7657 | Transfer a right absorptio... |
| caofid2 7658 | Transfer a right absorptio... |
| caofcom 7659 | Transfer a commutative law... |
| caofidlcan 7660 | Transfer a cancellation/id... |
| caofrss 7661 | Transfer a relation subset... |
| caofass 7662 | Transfer an associative la... |
| caoftrn 7663 | Transfer a transitivity la... |
| caofdi 7664 | Transfer a distributive la... |
| caofdir 7665 | Transfer a reverse distrib... |
| caonncan 7666 | Transfer ~ nncan -shaped l... |
| relrpss 7669 | The proper subset relation... |
| brrpssg 7670 | The proper subset relation... |
| brrpss 7671 | The proper subset relation... |
| porpss 7672 | Every class is partially o... |
| sorpss 7673 | Express strict ordering un... |
| sorpssi 7674 | Property of a chain of set... |
| sorpssun 7675 | A chain of sets is closed ... |
| sorpssin 7676 | A chain of sets is closed ... |
| sorpssuni 7677 | In a chain of sets, a maxi... |
| sorpssint 7678 | In a chain of sets, a mini... |
| sorpsscmpl 7679 | The componentwise compleme... |
| zfun 7681 | Axiom of Union expressed w... |
| axun2 7682 | A variant of the Axiom of ... |
| uniex2 7683 | The Axiom of Union using t... |
| vuniex 7684 | The union of a setvar is a... |
| uniexg 7685 | The ZF Axiom of Union in c... |
| uniex 7686 | The Axiom of Union in clas... |
| uniexd 7687 | Deduction version of the Z... |
| unexg 7688 | The union of two sets is a... |
| unex 7689 | The union of two sets is a... |
| unexOLD 7690 | Obsolete version of ~ unex... |
| tpex 7691 | An unordered triple of cla... |
| unexb 7692 | Existence of union is equi... |
| unexbOLD 7693 | Obsolete version of ~ unex... |
| unexgOLD 7694 | Obsolete version of ~ unex... |
| xpexg 7695 | The Cartesian product of t... |
| xpexd 7696 | The Cartesian product of t... |
| 3xpexg 7697 | The Cartesian product of t... |
| xpex 7698 | The Cartesian product of t... |
| unexd 7699 | The union of two sets is a... |
| sqxpexg 7700 | The Cartesian square of a ... |
| abnexg 7701 | Sufficient condition for a... |
| abnex 7702 | Sufficient condition for a... |
| snnex 7703 | The class of all singleton... |
| pwnex 7704 | The class of all power set... |
| difex2 7705 | If the subtrahend of a cla... |
| difsnexi 7706 | If the difference of a cla... |
| uniuni 7707 | Expression for double unio... |
| uniexr 7708 | Converse of the Axiom of U... |
| uniexb 7709 | The Axiom of Union and its... |
| pwexr 7710 | Converse of the Axiom of P... |
| pwexb 7711 | The Axiom of Power Sets an... |
| elpwpwel 7712 | A class belongs to a doubl... |
| eldifpw 7713 | Membership in a power clas... |
| elpwun 7714 | Membership in the power cl... |
| pwuncl 7715 | Power classes are closed u... |
| iunpw 7716 | An indexed union of a powe... |
| fr3nr 7717 | A well-founded relation ha... |
| epne3 7718 | A well-founded class conta... |
| dfwe2 7719 | Alternate definition of we... |
| epweon 7720 | The membership relation we... |
| epweonALT 7721 | Alternate proof of ~ epweo... |
| ordon 7722 | The class of all ordinal n... |
| onprc 7723 | No set contains all ordina... |
| ssorduni 7724 | The union of a class of or... |
| ssonuni 7725 | The union of a set of ordi... |
| ssonunii 7726 | The union of a set of ordi... |
| ordeleqon 7727 | A way to express the ordin... |
| ordsson 7728 | Any ordinal class is a sub... |
| dford5 7729 | A class is ordinal iff it ... |
| onss 7730 | An ordinal number is a sub... |
| predon 7731 | The predecessor of an ordi... |
| ssonprc 7732 | Two ways of saying a class... |
| onuni 7733 | The union of an ordinal nu... |
| orduni 7734 | The union of an ordinal cl... |
| onint 7735 | The intersection (infimum)... |
| onint0 7736 | The intersection of a clas... |
| onssmin 7737 | A nonempty class of ordina... |
| onminesb 7738 | If a property is true for ... |
| onminsb 7739 | If a property is true for ... |
| oninton 7740 | The intersection of a none... |
| onintrab 7741 | The intersection of a clas... |
| onintrab2 7742 | An existence condition equ... |
| onnmin 7743 | No member of a set of ordi... |
| onnminsb 7744 | An ordinal number smaller ... |
| oneqmin 7745 | A way to show that an ordi... |
| uniordint 7746 | The union of a set of ordi... |
| onminex 7747 | If a wff is true for an or... |
| sucon 7748 | The class of all ordinal n... |
| sucexb 7749 | A successor exists iff its... |
| sucexg 7750 | The successor of a set is ... |
| sucex 7751 | The successor of a set is ... |
| onmindif2 7752 | The minimum of a class of ... |
| ordsuci 7753 | The successor of an ordina... |
| sucexeloni 7754 | If the successor of an ord... |
| onsuc 7755 | The successor of an ordina... |
| ordsuc 7756 | A class is ordinal if and ... |
| ordpwsuc 7757 | The collection of ordinals... |
| onpwsuc 7758 | The collection of ordinal ... |
| onsucb 7759 | A class is an ordinal numb... |
| ordsucss 7760 | The successor of an elemen... |
| onpsssuc 7761 | An ordinal number is a pro... |
| ordelsuc 7762 | A set belongs to an ordina... |
| onsucmin 7763 | The successor of an ordina... |
| ordsucelsuc 7764 | Membership is inherited by... |
| ordsucsssuc 7765 | The subclass relationship ... |
| ordsucuniel 7766 | Given an element ` A ` of ... |
| ordsucun 7767 | The successor of the maxim... |
| ordunpr 7768 | The maximum of two ordinal... |
| ordunel 7769 | The maximum of two ordinal... |
| onsucuni 7770 | A class of ordinal numbers... |
| ordsucuni 7771 | An ordinal class is a subc... |
| orduniorsuc 7772 | An ordinal class is either... |
| unon 7773 | The class of all ordinal n... |
| ordunisuc 7774 | An ordinal class is equal ... |
| orduniss2 7775 | The union of the ordinal s... |
| onsucuni2 7776 | A successor ordinal is the... |
| 0elsuc 7777 | The successor of an ordina... |
| limon 7778 | The class of ordinal numbe... |
| onuniorsuc 7779 | An ordinal number is eithe... |
| onssi 7780 | An ordinal number is a sub... |
| onsuci 7781 | The successor of an ordina... |
| onuninsuci 7782 | An ordinal is equal to its... |
| onsucssi 7783 | A set belongs to an ordina... |
| nlimsucg 7784 | A successor is not a limit... |
| orduninsuc 7785 | An ordinal class is equal ... |
| ordunisuc2 7786 | An ordinal equal to its un... |
| ordzsl 7787 | An ordinal is zero, a succ... |
| onzsl 7788 | An ordinal number is zero,... |
| dflim3 7789 | An alternate definition of... |
| dflim4 7790 | An alternate definition of... |
| limsuc 7791 | The successor of a member ... |
| limsssuc 7792 | A class includes a limit o... |
| nlimon 7793 | Two ways to express the cl... |
| limuni3 7794 | The union of a nonempty cl... |
| tfi 7795 | The Principle of Transfini... |
| tfisg 7796 | A closed form of ~ tfis . ... |
| tfis 7797 | Transfinite Induction Sche... |
| tfis2f 7798 | Transfinite Induction Sche... |
| tfis2 7799 | Transfinite Induction Sche... |
| tfis3 7800 | Transfinite Induction Sche... |
| tfisi 7801 | A transfinite induction sc... |
| tfinds 7802 | Principle of Transfinite I... |
| tfindsg 7803 | Transfinite Induction (inf... |
| tfindsg2 7804 | Transfinite Induction (inf... |
| tfindes 7805 | Transfinite Induction with... |
| tfinds2 7806 | Transfinite Induction (inf... |
| tfinds3 7807 | Principle of Transfinite I... |
| dfom2 7810 | An alternate definition of... |
| elom 7811 | Membership in omega. The ... |
| omsson 7812 | Omega is a subset of ` On ... |
| limomss 7813 | The class of natural numbe... |
| nnon 7814 | A natural number is an ord... |
| nnoni 7815 | A natural number is an ord... |
| nnord 7816 | A natural number is ordina... |
| trom 7817 | The class of finite ordina... |
| ordom 7818 | The class of finite ordina... |
| elnn 7819 | A member of a natural numb... |
| omon 7820 | The class of natural numbe... |
| omelon2 7821 | Omega is an ordinal number... |
| nnlim 7822 | A natural number is not a ... |
| omssnlim 7823 | The class of natural numbe... |
| limom 7824 | Omega is a limit ordinal. ... |
| peano2b 7825 | A class belongs to omega i... |
| nnsuc 7826 | A nonzero natural number i... |
| omsucne 7827 | A natural number is not th... |
| ssnlim 7828 | An ordinal subclass of non... |
| omsinds 7829 | Strong (or "total") induct... |
| omun 7830 | The union of two finite or... |
| peano1 7831 | Zero is a natural number. ... |
| peano2 7832 | The successor of any natur... |
| peano3 7833 | The successor of any natur... |
| peano4 7834 | Two natural numbers are eq... |
| peano5 7835 | The induction postulate: a... |
| nn0suc 7836 | A natural number is either... |
| find 7837 | The Principle of Finite In... |
| finds 7838 | Principle of Finite Induct... |
| findsg 7839 | Principle of Finite Induct... |
| finds2 7840 | Principle of Finite Induct... |
| finds1 7841 | Principle of Finite Induct... |
| findes 7842 | Finite induction with expl... |
| dmexg 7843 | The domain of a set is a s... |
| rnexg 7844 | The range of a set is a se... |
| dmexd 7845 | The domain of a set is a s... |
| fndmexd 7846 | If a function is a set, it... |
| dmfex 7847 | If a mapping is a set, its... |
| fndmexb 7848 | The domain of a function i... |
| fdmexb 7849 | The domain of a function i... |
| dmfexALT 7850 | Alternate proof of ~ dmfex... |
| dmex 7851 | The domain of a set is a s... |
| rnex 7852 | The range of a set is a se... |
| iprc 7853 | The identity function is a... |
| resiexg 7854 | The existence of a restric... |
| imaexg 7855 | The image of a set is a se... |
| imaex 7856 | The image of a set is a se... |
| rnexd 7857 | The range of a set is a se... |
| imaexd 7858 | The image of a set is a se... |
| exse2 7859 | Any set relation is set-li... |
| xpexr 7860 | If a Cartesian product is ... |
| xpexr2 7861 | If a nonempty Cartesian pr... |
| xpexcnv 7862 | A condition where the conv... |
| soex 7863 | If the relation in a stric... |
| elxp4 7864 | Membership in a Cartesian ... |
| elxp5 7865 | Membership in a Cartesian ... |
| cnvexg 7866 | The converse of a set is a... |
| cnvex 7867 | The converse of a set is a... |
| relcnvexb 7868 | A relation is a set iff it... |
| f1oexrnex 7869 | If the range of a 1-1 onto... |
| f1oexbi 7870 | There is a one-to-one onto... |
| coexg 7871 | The composition of two set... |
| coex 7872 | The composition of two set... |
| coexd 7873 | The composition of two set... |
| funcnvuni 7874 | The union of a chain (with... |
| fun11uni 7875 | The union of a chain (with... |
| resf1extb 7876 | Extension of an injection ... |
| resf1ext2b 7877 | Extension of an injection ... |
| fex2 7878 | A function with bounded do... |
| fabexd 7879 | Existence of a set of func... |
| fabexg 7880 | Existence of a set of func... |
| fabexgOLD 7881 | Obsolete version of ~ fabe... |
| fabex 7882 | Existence of a set of func... |
| mapex 7883 | The class of all functions... |
| f1oabexg 7884 | The class of all 1-1-onto ... |
| f1oabexgOLD 7885 | Obsolete version of ~ f1oa... |
| fiunlem 7886 | Lemma for ~ fiun and ~ f1i... |
| fiun 7887 | The union of a chain (with... |
| f1iun 7888 | The union of a chain (with... |
| fviunfun 7889 | The function value of an i... |
| ffoss 7890 | Relationship between a map... |
| f11o 7891 | Relationship between one-t... |
| resfunexgALT 7892 | Alternate proof of ~ resfu... |
| cofunexg 7893 | Existence of a composition... |
| cofunex2g 7894 | Existence of a composition... |
| fnexALT 7895 | Alternate proof of ~ fnex ... |
| funexw 7896 | Weak version of ~ funex th... |
| mptexw 7897 | Weak version of ~ mptex th... |
| funrnex 7898 | If the domain of a functio... |
| zfrep6OLD 7899 | Obsolete proof of ~ zfrep6... |
| focdmex 7900 | If the domain of an onto f... |
| f1dmex 7901 | If the codomain of a one-t... |
| f1ovv 7902 | The codomain/range of a 1-... |
| fvclex 7903 | Existence of the class of ... |
| fvresex 7904 | Existence of the class of ... |
| abrexexg 7905 | Existence of a class abstr... |
| abrexex 7906 | Existence of a class abstr... |
| iunexg 7907 | The existence of an indexe... |
| abrexex2g 7908 | Existence of an existentia... |
| opabex3d 7909 | Existence of an ordered pa... |
| opabex3rd 7910 | Existence of an ordered pa... |
| opabex3 7911 | Existence of an ordered pa... |
| iunex 7912 | The existence of an indexe... |
| abrexex2 7913 | Existence of an existentia... |
| abexssex 7914 | Existence of a class abstr... |
| abexex 7915 | A condition where a class ... |
| f1oweALT 7916 | Alternate proof of ~ f1owe... |
| wemoiso 7917 | Thus, there is at most one... |
| wemoiso2 7918 | Thus, there is at most one... |
| oprabexd 7919 | Existence of an operator a... |
| oprabex 7920 | Existence of an operation ... |
| oprabex3 7921 | Existence of an operation ... |
| oprabrexex2 7922 | Existence of an existentia... |
| ab2rexex 7923 | Existence of a class abstr... |
| ab2rexex2 7924 | Existence of an existentia... |
| xpexgALT 7925 | Alternate proof of ~ xpexg... |
| offval3 7926 | General value of ` ( F oF ... |
| offres 7927 | Pointwise combination comm... |
| ofmres 7928 | Equivalent expressions for... |
| ofmresex 7929 | Existence of a restriction... |
| mptcnfimad 7930 | The converse of a mapping ... |
| 1stval 7935 | The value of the function ... |
| 2ndval 7936 | The value of the function ... |
| 1stnpr 7937 | Value of the first-member ... |
| 2ndnpr 7938 | Value of the second-member... |
| 1st0 7939 | The value of the first-mem... |
| 2nd0 7940 | The value of the second-me... |
| op1st 7941 | Extract the first member o... |
| op2nd 7942 | Extract the second member ... |
| op1std 7943 | Extract the first member o... |
| op2ndd 7944 | Extract the second member ... |
| op1stg 7945 | Extract the first member o... |
| op2ndg 7946 | Extract the second member ... |
| ot1stg 7947 | Extract the first member o... |
| ot2ndg 7948 | Extract the second member ... |
| ot3rdg 7949 | Extract the third member o... |
| 1stval2 7950 | Alternate value of the fun... |
| 2ndval2 7951 | Alternate value of the fun... |
| oteqimp 7952 | The components of an order... |
| fo1st 7953 | The ` 1st ` function maps ... |
| fo2nd 7954 | The ` 2nd ` function maps ... |
| br1steqg 7955 | Uniqueness condition for t... |
| br2ndeqg 7956 | Uniqueness condition for t... |
| f1stres 7957 | Mapping of a restriction o... |
| f2ndres 7958 | Mapping of a restriction o... |
| fo1stres 7959 | Onto mapping of a restrict... |
| fo2ndres 7960 | Onto mapping of a restrict... |
| 1st2val 7961 | Value of an alternate defi... |
| 2nd2val 7962 | Value of an alternate defi... |
| 1stcof 7963 | Composition of the first m... |
| 2ndcof 7964 | Composition of the second ... |
| xp1st 7965 | Location of the first elem... |
| xp2nd 7966 | Location of the second ele... |
| elxp6 7967 | Membership in a Cartesian ... |
| elxp7 7968 | Membership in a Cartesian ... |
| eqopi 7969 | Equality with an ordered p... |
| xp2 7970 | Representation of Cartesia... |
| unielxp 7971 | The membership relation fo... |
| 1st2nd2 7972 | Reconstruction of a member... |
| 1st2ndb 7973 | Reconstruction of an order... |
| xpopth 7974 | An ordered pair theorem fo... |
| eqop 7975 | Two ways to express equali... |
| eqop2 7976 | Two ways to express equali... |
| op1steq 7977 | Two ways of expressing tha... |
| opreuopreu 7978 | There is a unique ordered ... |
| el2xptp 7979 | A member of a nested Carte... |
| el2xptp0 7980 | A member of a nested Carte... |
| el2xpss 7981 | Version of ~ elrel for tri... |
| 2nd1st 7982 | Swap the members of an ord... |
| 1st2nd 7983 | Reconstruction of a member... |
| 1stdm 7984 | The first ordered pair com... |
| 2ndrn 7985 | The second ordered pair co... |
| 1st2ndbr 7986 | Express an element of a re... |
| releldm2 7987 | Two ways of expressing mem... |
| reldm 7988 | An expression for the doma... |
| releldmdifi 7989 | One way of expressing memb... |
| funfv1st2nd 7990 | The function value for the... |
| funelss 7991 | If the first component of ... |
| funeldmdif 7992 | Two ways of expressing mem... |
| sbcopeq1a 7993 | Equality theorem for subst... |
| csbopeq1a 7994 | Equality theorem for subst... |
| sbcoteq1a 7995 | Equality theorem for subst... |
| dfopab2 7996 | A way to define an ordered... |
| dfoprab3s 7997 | A way to define an operati... |
| dfoprab3 7998 | Operation class abstractio... |
| dfoprab4 7999 | Operation class abstractio... |
| dfoprab4f 8000 | Operation class abstractio... |
| opabex2 8001 | Condition for an operation... |
| opabn1stprc 8002 | An ordered-pair class abst... |
| opiota 8003 | The property of a uniquely... |
| cnvoprab 8004 | The converse of a class ab... |
| dfxp3 8005 | Define the Cartesian produ... |
| elopabi 8006 | A consequence of membershi... |
| eloprabi 8007 | A consequence of membershi... |
| mpomptsx 8008 | Express a two-argument fun... |
| mpompts 8009 | Express a two-argument fun... |
| dmmpossx 8010 | The domain of a mapping is... |
| fmpox 8011 | Functionality, domain and ... |
| fmpo 8012 | Functionality, domain and ... |
| fnmpo 8013 | Functionality and domain o... |
| fnmpoi 8014 | Functionality and domain o... |
| dmmpo 8015 | Domain of a class given by... |
| ovmpoelrn 8016 | An operation's value belon... |
| dmmpoga 8017 | Domain of an operation giv... |
| dmmpog 8018 | Domain of an operation giv... |
| mpoexxg 8019 | Existence of an operation ... |
| mpoexg 8020 | Existence of an operation ... |
| mpoexga 8021 | If the domain of an operat... |
| mpoexw 8022 | Weak version of ~ mpoex th... |
| mpoex 8023 | If the domain of an operat... |
| mptmpoopabbrd 8024 | The operation value of a f... |
| mptmpoopabovd 8025 | The operation value of a f... |
| el2mpocsbcl 8026 | If the operation value of ... |
| el2mpocl 8027 | If the operation value of ... |
| fnmpoovd 8028 | A function with a Cartesia... |
| offval22 8029 | The function operation exp... |
| brovpreldm 8030 | If a binary relation holds... |
| bropopvvv 8031 | If a binary relation holds... |
| bropfvvvvlem 8032 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8033 | If a binary relation holds... |
| ovmptss 8034 | If all the values of the m... |
| relmpoopab 8035 | Any function to sets of or... |
| fmpoco 8036 | Composition of two functio... |
| oprabco 8037 | Composition of a function ... |
| oprab2co 8038 | Composition of operator ab... |
| df1st2 8039 | An alternate possible defi... |
| df2nd2 8040 | An alternate possible defi... |
| 1stconst 8041 | The mapping of a restricti... |
| 2ndconst 8042 | The mapping of a restricti... |
| dfmpo 8043 | Alternate definition for t... |
| mposn 8044 | An operation (in maps-to n... |
| curry1 8045 | Composition with ` ``' ( 2... |
| curry1val 8046 | The value of a curried fun... |
| curry1f 8047 | Functionality of a curried... |
| curry2 8048 | Composition with ` ``' ( 1... |
| curry2f 8049 | Functionality of a curried... |
| curry2val 8050 | The value of a curried fun... |
| cnvf1olem 8051 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8052 | Describe a function that m... |
| fparlem1 8053 | Lemma for ~ fpar . (Contr... |
| fparlem2 8054 | Lemma for ~ fpar . (Contr... |
| fparlem3 8055 | Lemma for ~ fpar . (Contr... |
| fparlem4 8056 | Lemma for ~ fpar . (Contr... |
| fpar 8057 | Merge two functions in par... |
| fsplit 8058 | A function that can be use... |
| fsplitfpar 8059 | Merge two functions with a... |
| offsplitfpar 8060 | Express the function opera... |
| f2ndf 8061 | The ` 2nd ` (second compon... |
| fo2ndf 8062 | The ` 2nd ` (second compon... |
| f1o2ndf1 8063 | The ` 2nd ` (second compon... |
| opco1 8064 | Value of an operation prec... |
| opco2 8065 | Value of an operation prec... |
| opco1i 8066 | Inference form of ~ opco1 ... |
| frxp 8067 | A lexicographical ordering... |
| xporderlem 8068 | Lemma for lexicographical ... |
| poxp 8069 | A lexicographical ordering... |
| soxp 8070 | A lexicographical ordering... |
| wexp 8071 | A lexicographical ordering... |
| fnwelem 8072 | Lemma for ~ fnwe . (Contr... |
| fnwe 8073 | A variant on lexicographic... |
| fnse 8074 | Condition for the well-ord... |
| fvproj 8075 | Value of a function on ord... |
| fimaproj 8076 | Image of a cartesian produ... |
| ralxpes 8077 | A version of ~ ralxp with ... |
| ralxp3f 8078 | Restricted for all over a ... |
| ralxp3 8079 | Restricted for all over a ... |
| ralxp3es 8080 | Restricted for-all over a ... |
| frpoins3xpg 8081 | Special case of founded pa... |
| frpoins3xp3g 8082 | Special case of founded pa... |
| xpord2lem 8083 | Lemma for Cartesian produc... |
| poxp2 8084 | Another way of partially o... |
| frxp2 8085 | Another way of giving a we... |
| xpord2pred 8086 | Calculate the predecessor ... |
| sexp2 8087 | Condition for the relation... |
| xpord2indlem 8088 | Induction over the Cartesi... |
| xpord2ind 8089 | Induction over the Cartesi... |
| xpord3lem 8090 | Lemma for triple ordering.... |
| poxp3 8091 | Triple Cartesian product p... |
| frxp3 8092 | Give well-foundedness over... |
| xpord3pred 8093 | Calculate the predecsessor... |
| sexp3 8094 | Show that the triple order... |
| xpord3inddlem 8095 | Induction over the triple ... |
| xpord3indd 8096 | Induction over the triple ... |
| xpord3ind 8097 | Induction over the triple ... |
| orderseqlem 8098 | Lemma for ~ poseq and ~ so... |
| poseq 8099 | A partial ordering of ordi... |
| soseq 8100 | A linear ordering of ordin... |
| suppval 8103 | The value of the operation... |
| supp0prc 8104 | The support of a class is ... |
| suppvalbr 8105 | The value of the operation... |
| supp0 8106 | The support of the empty s... |
| suppval1 8107 | The value of the operation... |
| suppvalfng 8108 | The value of the operation... |
| suppvalfn 8109 | The value of the operation... |
| elsuppfng 8110 | An element of the support ... |
| elsuppfn 8111 | An element of the support ... |
| fvdifsupp 8112 | Function value is zero out... |
| cnvimadfsn 8113 | The support of functions "... |
| suppimacnvss 8114 | The support of functions "... |
| suppimacnv 8115 | Support sets of functions ... |
| fsuppeq 8116 | Two ways of writing the su... |
| fsuppeqg 8117 | Version of ~ fsuppeq avoid... |
| suppssdm 8118 | The support of a function ... |
| suppsnop 8119 | The support of a singleton... |
| snopsuppss 8120 | The support of a singleton... |
| fvn0elsupp 8121 | If the function value for ... |
| fvn0elsuppb 8122 | The function value for a g... |
| rexsupp 8123 | Existential quantification... |
| ressuppss 8124 | The support of the restric... |
| suppun 8125 | The support of a class/fun... |
| ressuppssdif 8126 | The support of the restric... |
| mptsuppdifd 8127 | The support of a function ... |
| mptsuppd 8128 | The support of a function ... |
| extmptsuppeq 8129 | The support of an extended... |
| suppfnss 8130 | The support of a function ... |
| funsssuppss 8131 | The support of a function ... |
| fnsuppres 8132 | Two ways to express restri... |
| fnsuppeq0 8133 | The support of a function ... |
| fczsupp0 8134 | The support of a constant ... |
| suppss 8135 | Show that the support of a... |
| suppssr 8136 | A function is zero outside... |
| suppssrg 8137 | A function is zero outside... |
| suppssov1 8138 | Formula building theorem f... |
| suppssov2 8139 | Formula building theorem f... |
| suppssof1 8140 | Formula building theorem f... |
| suppss2 8141 | Show that the support of a... |
| suppsssn 8142 | Show that the support of a... |
| suppssfv 8143 | Formula building theorem f... |
| suppofssd 8144 | Condition for the support ... |
| suppofss1d 8145 | Condition for the support ... |
| suppofss2d 8146 | Condition for the support ... |
| suppco 8147 | The support of the composi... |
| suppcoss 8148 | The support of the composi... |
| supp0cosupp0 8149 | The support of the composi... |
| imacosupp 8150 | The image of the support o... |
| opeliunxp2f 8151 | Membership in a union of C... |
| mpoxeldm 8152 | If there is an element of ... |
| mpoxneldm 8153 | If the first argument of a... |
| mpoxopn0yelv 8154 | If there is an element of ... |
| mpoxopynvov0g 8155 | If the second argument of ... |
| mpoxopxnop0 8156 | If the first argument of a... |
| mpoxopx0ov0 8157 | If the first argument of a... |
| mpoxopxprcov0 8158 | If the components of the f... |
| mpoxopynvov0 8159 | If the second argument of ... |
| mpoxopoveq 8160 | Value of an operation give... |
| mpoxopovel 8161 | Element of the value of an... |
| mpoxopoveqd 8162 | Value of an operation give... |
| brovex 8163 | A binary relation of the v... |
| brovmpoex 8164 | A binary relation of the v... |
| sprmpod 8165 | The extension of a binary ... |
| tposss 8168 | Subset theorem for transpo... |
| tposeq 8169 | Equality theorem for trans... |
| tposeqd 8170 | Equality theorem for trans... |
| tposssxp 8171 | The transposition is a sub... |
| reltpos 8172 | The transposition is a rel... |
| brtpos2 8173 | Value of the transposition... |
| brtpos0 8174 | The behavior of ` tpos ` w... |
| reldmtpos 8175 | Necessary and sufficient c... |
| brtpos 8176 | The transposition swaps ar... |
| ottpos 8177 | The transposition swaps th... |
| relbrtpos 8178 | The transposition swaps ar... |
| dmtpos 8179 | The domain of ` tpos F ` w... |
| rntpos 8180 | The range of ` tpos F ` wh... |
| tposexg 8181 | The transposition of a set... |
| ovtpos 8182 | The transposition swaps th... |
| tposfun 8183 | The transposition of a fun... |
| dftpos2 8184 | Alternate definition of ` ... |
| dftpos3 8185 | Alternate definition of ` ... |
| dftpos4 8186 | Alternate definition of ` ... |
| tpostpos 8187 | Value of the double transp... |
| tpostpos2 8188 | Value of the double transp... |
| tposfn2 8189 | The domain of a transposit... |
| tposfo2 8190 | Condition for a surjective... |
| tposf2 8191 | The domain and codomain of... |
| tposf12 8192 | Condition for an injective... |
| tposf1o2 8193 | Condition of a bijective t... |
| tposfo 8194 | The domain and codomain/ra... |
| tposf 8195 | The domain and codomain of... |
| tposfn 8196 | Functionality of a transpo... |
| tpos0 8197 | Transposition of the empty... |
| tposco 8198 | Transposition of a composi... |
| tpossym 8199 | Two ways to say a function... |
| tposeqi 8200 | Equality theorem for trans... |
| tposex 8201 | A transposition is a set. ... |
| nftpos 8202 | Hypothesis builder for tra... |
| tposoprab 8203 | Transposition of a class o... |
| tposmpo 8204 | Transposition of a two-arg... |
| tposconst 8205 | The transposition of a con... |
| mpocurryd 8210 | The currying of an operati... |
| mpocurryvald 8211 | The value of a curried ope... |
| fvmpocurryd 8212 | The value of the value of ... |
| pwuninel2 8215 | Proof of ~ pwuninel under ... |
| pwuninel 8216 | The powerclass of the unio... |
| undefval 8217 | Value of the undefined val... |
| undefnel2 8218 | The undefined value genera... |
| undefnel 8219 | The undefined value genera... |
| undefne0 8220 | The undefined value genera... |
| frecseq123 8223 | Equality theorem for the w... |
| nffrecs 8224 | Bound-variable hypothesis ... |
| csbfrecsg 8225 | Move class substitution in... |
| fpr3g 8226 | Functions defined by well-... |
| frrlem1 8227 | Lemma for well-founded rec... |
| frrlem2 8228 | Lemma for well-founded rec... |
| frrlem3 8229 | Lemma for well-founded rec... |
| frrlem4 8230 | Lemma for well-founded rec... |
| frrlem5 8231 | Lemma for well-founded rec... |
| frrlem6 8232 | Lemma for well-founded rec... |
| frrlem7 8233 | Lemma for well-founded rec... |
| frrlem8 8234 | Lemma for well-founded rec... |
| frrlem9 8235 | Lemma for well-founded rec... |
| frrlem10 8236 | Lemma for well-founded rec... |
| frrlem11 8237 | Lemma for well-founded rec... |
| frrlem12 8238 | Lemma for well-founded rec... |
| frrlem13 8239 | Lemma for well-founded rec... |
| frrlem14 8240 | Lemma for well-founded rec... |
| fprlem1 8241 | Lemma for well-founded rec... |
| fprlem2 8242 | Lemma for well-founded rec... |
| fpr2a 8243 | Weak version of ~ fpr2 whi... |
| fpr1 8244 | Law of well-founded recurs... |
| fpr2 8245 | Law of well-founded recurs... |
| fpr3 8246 | Law of well-founded recurs... |
| frrrel 8247 | Show without using the axi... |
| frrdmss 8248 | Show without using the axi... |
| frrdmcl 8249 | Show without using the axi... |
| fprfung 8250 | A "function" defined by we... |
| fprresex 8251 | The restriction of a funct... |
| wrecseq123 8254 | General equality theorem f... |
| nfwrecs 8255 | Bound-variable hypothesis ... |
| wrecseq1 8256 | Equality theorem for the w... |
| wrecseq2 8257 | Equality theorem for the w... |
| wrecseq3 8258 | Equality theorem for the w... |
| csbwrecsg 8259 | Move class substitution in... |
| wfr3g 8260 | Functions defined by well-... |
| wfrrel 8261 | The well-ordered recursion... |
| wfrdmss 8262 | The domain of the well-ord... |
| wfrdmcl 8263 | The predecessor class of a... |
| wfrfun 8264 | The "function" generated b... |
| wfrresex 8265 | Show without using the axi... |
| wfr2a 8266 | A weak version of ~ wfr2 w... |
| wfr1 8267 | The Principle of Well-Orde... |
| wfr2 8268 | The Principle of Well-Orde... |
| wfr3 8269 | The principle of Well-Orde... |
| iunon 8270 | The indexed union of a set... |
| iinon 8271 | The nonempty indexed inter... |
| onfununi 8272 | A property of functions on... |
| onovuni 8273 | A variant of ~ onfununi fo... |
| onoviun 8274 | A variant of ~ onovuni wit... |
| onnseq 8275 | There are no length ` _om ... |
| dfsmo2 8278 | Alternate definition of a ... |
| issmo 8279 | Conditions for which ` A `... |
| issmo2 8280 | Alternate definition of a ... |
| smoeq 8281 | Equality theorem for stric... |
| smodm 8282 | The domain of a strictly m... |
| smores 8283 | A strictly monotone functi... |
| smores3 8284 | A strictly monotone functi... |
| smores2 8285 | A strictly monotone ordina... |
| smodm2 8286 | The domain of a strictly m... |
| smofvon2 8287 | The function values of a s... |
| iordsmo 8288 | The identity relation rest... |
| smo0 8289 | The null set is a strictly... |
| smofvon 8290 | If ` B ` is a strictly mon... |
| smoel 8291 | If ` x ` is less than ` y ... |
| smoiun 8292 | The value of a strictly mo... |
| smoiso 8293 | If ` F ` is an isomorphism... |
| smoel2 8294 | A strictly monotone ordina... |
| smo11 8295 | A strictly monotone ordina... |
| smoord 8296 | A strictly monotone ordina... |
| smoword 8297 | A strictly monotone ordina... |
| smogt 8298 | A strictly monotone ordina... |
| smocdmdom 8299 | The codomain of a strictly... |
| smoiso2 8300 | The strictly monotone ordi... |
| dfrecs3 8303 | The old definition of tran... |
| recseq 8304 | Equality theorem for ` rec... |
| nfrecs 8305 | Bound-variable hypothesis ... |
| tfrlem1 8306 | A technical lemma for tran... |
| tfrlem3a 8307 | Lemma for transfinite recu... |
| tfrlem3 8308 | Lemma for transfinite recu... |
| tfrlem4 8309 | Lemma for transfinite recu... |
| tfrlem5 8310 | Lemma for transfinite recu... |
| recsfval 8311 | Lemma for transfinite recu... |
| tfrlem6 8312 | Lemma for transfinite recu... |
| tfrlem7 8313 | Lemma for transfinite recu... |
| tfrlem8 8314 | Lemma for transfinite recu... |
| tfrlem9 8315 | Lemma for transfinite recu... |
| tfrlem9a 8316 | Lemma for transfinite recu... |
| tfrlem10 8317 | Lemma for transfinite recu... |
| tfrlem11 8318 | Lemma for transfinite recu... |
| tfrlem12 8319 | Lemma for transfinite recu... |
| tfrlem13 8320 | Lemma for transfinite recu... |
| tfrlem14 8321 | Lemma for transfinite recu... |
| tfrlem15 8322 | Lemma for transfinite recu... |
| tfrlem16 8323 | Lemma for finite recursion... |
| tfr1a 8324 | A weak version of ~ tfr1 w... |
| tfr2a 8325 | A weak version of ~ tfr2 w... |
| tfr2b 8326 | Without assuming ~ ax-rep ... |
| tfr1 8327 | Principle of Transfinite R... |
| tfr2 8328 | Principle of Transfinite R... |
| tfr3 8329 | Principle of Transfinite R... |
| tfr1ALT 8330 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8331 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8332 | Alternate proof of ~ tfr3 ... |
| recsfnon 8333 | Strong transfinite recursi... |
| recsval 8334 | Strong transfinite recursi... |
| tz7.44lem1 8335 | The ordered pair abstracti... |
| tz7.44-1 8336 | The value of ` F ` at ` (/... |
| tz7.44-2 8337 | The value of ` F ` at a su... |
| tz7.44-3 8338 | The value of ` F ` at a li... |
| rdgeq1 8341 | Equality theorem for the r... |
| rdgeq2 8342 | Equality theorem for the r... |
| rdgeq12 8343 | Equality theorem for the r... |
| nfrdg 8344 | Bound-variable hypothesis ... |
| rdglem1 8345 | Lemma used with the recurs... |
| rdgfun 8346 | The recursive definition g... |
| rdgdmlim 8347 | The domain of the recursiv... |
| rdgfnon 8348 | The recursive definition g... |
| rdgvalg 8349 | Value of the recursive def... |
| rdgval 8350 | Value of the recursive def... |
| rdg0 8351 | The initial value of the r... |
| rdgseg 8352 | The initial segments of th... |
| rdgsucg 8353 | The value of the recursive... |
| rdgsuc 8354 | The value of the recursive... |
| rdglimg 8355 | The value of the recursive... |
| rdglim 8356 | The value of the recursive... |
| rdg0g 8357 | The initial value of the r... |
| rdgsucmptf 8358 | The value of the recursive... |
| rdgsucmptnf 8359 | The value of the recursive... |
| rdgsucmpt2 8360 | This version of ~ rdgsucmp... |
| rdgsucmpt 8361 | The value of the recursive... |
| rdglim2 8362 | The value of the recursive... |
| rdglim2a 8363 | The value of the recursive... |
| rdg0n 8364 | If ` A ` is a proper class... |
| frfnom 8365 | The function generated by ... |
| fr0g 8366 | The initial value resultin... |
| frsuc 8367 | The successor value result... |
| frsucmpt 8368 | The successor value result... |
| frsucmptn 8369 | The value of the finite re... |
| frsucmpt2 8370 | The successor value result... |
| tz7.48lem 8371 | A way of showing an ordina... |
| tz7.48-2 8372 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8373 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8374 | Proposition 7.48(3) of [Ta... |
| tz7.49 8375 | Proposition 7.49 of [Takeu... |
| tz7.49c 8376 | Corollary of Proposition 7... |
| seqomlem0 8379 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8380 | Lemma for ` seqom ` . The... |
| seqomlem2 8381 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8382 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8383 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8384 | Equality theorem for ` seq... |
| fnseqom 8385 | An index-aware recursive d... |
| seqom0g 8386 | Value of an index-aware re... |
| seqomsuc 8387 | Value of an index-aware re... |
| omsucelsucb 8388 | Membership is inherited by... |
| df1o2 8403 | Expanded value of the ordi... |
| df2o3 8404 | Expanded value of the ordi... |
| df2o2 8405 | Expanded value of the ordi... |
| 1oex 8406 | Ordinal 1 is a set. (Cont... |
| 2oex 8407 | ` 2o ` is a set. (Contrib... |
| 1on 8408 | Ordinal 1 is an ordinal nu... |
| 2on 8409 | Ordinal 2 is an ordinal nu... |
| 2on0 8410 | Ordinal two is not zero. ... |
| ord3 8411 | Ordinal 3 is an ordinal cl... |
| 3on 8412 | Ordinal 3 is an ordinal nu... |
| 4on 8413 | Ordinal 4 is an ordinal nu... |
| 1n0 8414 | Ordinal one is not equal t... |
| nlim1 8415 | 1 is not a limit ordinal. ... |
| nlim2 8416 | 2 is not a limit ordinal. ... |
| xp01disj 8417 | Cartesian products with th... |
| xp01disjl 8418 | Cartesian products with th... |
| ordgt0ge1 8419 | Two ways to express that a... |
| ordge1n0 8420 | An ordinal greater than or... |
| el1o 8421 | Membership in ordinal one.... |
| ord1eln01 8422 | An ordinal that is not 0 o... |
| ord2eln012 8423 | An ordinal that is not 0, ... |
| 1ellim 8424 | A limit ordinal contains 1... |
| 2ellim 8425 | A limit ordinal contains 2... |
| dif1o 8426 | Two ways to say that ` A `... |
| ondif1 8427 | Two ways to say that ` A `... |
| ondif2 8428 | Two ways to say that ` A `... |
| 2oconcl 8429 | Closure of the pair swappi... |
| 0lt1o 8430 | Ordinal zero is less than ... |
| dif20el 8431 | An ordinal greater than on... |
| 0we1 8432 | The empty set is a well-or... |
| brwitnlem 8433 | Lemma for relations which ... |
| fnoa 8434 | Functionality and domain o... |
| fnom 8435 | Functionality and domain o... |
| fnoe 8436 | Functionality and domain o... |
| oav 8437 | Value of ordinal addition.... |
| omv 8438 | Value of ordinal multiplic... |
| oe0lem 8439 | A helper lemma for ~ oe0 a... |
| oev 8440 | Value of ordinal exponenti... |
| oevn0 8441 | Value of ordinal exponenti... |
| oa0 8442 | Addition with zero. Propo... |
| om0 8443 | Ordinal multiplication wit... |
| oe0m 8444 | Value of zero raised to an... |
| om0x 8445 | Ordinal multiplication wit... |
| oe0m0 8446 | Ordinal exponentiation wit... |
| oe0m1 8447 | Ordinal exponentiation wit... |
| oe0 8448 | Ordinal exponentiation wit... |
| oev2 8449 | Alternate value of ordinal... |
| oasuc 8450 | Addition with successor. ... |
| oesuclem 8451 | Lemma for ~ oesuc . (Cont... |
| omsuc 8452 | Multiplication with succes... |
| oesuc 8453 | Ordinal exponentiation wit... |
| onasuc 8454 | Addition with successor. ... |
| onmsuc 8455 | Multiplication with succes... |
| onesuc 8456 | Exponentiation with a succ... |
| oa1suc 8457 | Addition with 1 is same as... |
| oalim 8458 | Ordinal addition with a li... |
| omlim 8459 | Ordinal multiplication wit... |
| oelim 8460 | Ordinal exponentiation wit... |
| oacl 8461 | Closure law for ordinal ad... |
| omcl 8462 | Closure law for ordinal mu... |
| oecl 8463 | Closure law for ordinal ex... |
| oa0r 8464 | Ordinal addition with zero... |
| om0r 8465 | Ordinal multiplication wit... |
| o1p1e2 8466 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8467 | 2 + 2 = 4 for ordinal numb... |
| om1 8468 | Ordinal multiplication wit... |
| om1r 8469 | Ordinal multiplication wit... |
| oe1 8470 | Ordinal exponentiation wit... |
| oe1m 8471 | Ordinal exponentiation wit... |
| oaordi 8472 | Ordering property of ordin... |
| oaord 8473 | Ordering property of ordin... |
| oacan 8474 | Left cancellation law for ... |
| oaword 8475 | Weak ordering property of ... |
| oawordri 8476 | Weak ordering property of ... |
| oaord1 8477 | An ordinal is less than it... |
| oaword1 8478 | An ordinal is less than or... |
| oaword2 8479 | An ordinal is less than or... |
| oawordeulem 8480 | Lemma for ~ oawordex . (C... |
| oawordeu 8481 | Existence theorem for weak... |
| oawordexr 8482 | Existence theorem for weak... |
| oawordex 8483 | Existence theorem for weak... |
| oaordex 8484 | Existence theorem for orde... |
| oa00 8485 | An ordinal sum is zero iff... |
| oalimcl 8486 | The ordinal sum with a lim... |
| oaass 8487 | Ordinal addition is associ... |
| oarec 8488 | Recursive definition of or... |
| oaf1o 8489 | Left addition by a constan... |
| oacomf1olem 8490 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8491 | Define a bijection from ` ... |
| omordi 8492 | Ordering property of ordin... |
| omord2 8493 | Ordering property of ordin... |
| omord 8494 | Ordering property of ordin... |
| omcan 8495 | Left cancellation law for ... |
| omword 8496 | Weak ordering property of ... |
| omwordi 8497 | Weak ordering property of ... |
| omwordri 8498 | Weak ordering property of ... |
| omword1 8499 | An ordinal is less than or... |
| omword2 8500 | An ordinal is less than or... |
| om00 8501 | The product of two ordinal... |
| om00el 8502 | The product of two nonzero... |
| omordlim 8503 | Ordering involving the pro... |
| omlimcl 8504 | The product of any nonzero... |
| odi 8505 | Distributive law for ordin... |
| omass 8506 | Multiplication of ordinal ... |
| oneo 8507 | If an ordinal number is ev... |
| omeulem1 8508 | Lemma for ~ omeu : existen... |
| omeulem2 8509 | Lemma for ~ omeu : uniquen... |
| omopth2 8510 | An ordered pair-like theor... |
| omeu 8511 | The division algorithm for... |
| om2 8512 | Two ways to double an ordi... |
| oen0 8513 | Ordinal exponentiation wit... |
| oeordi 8514 | Ordering law for ordinal e... |
| oeord 8515 | Ordering property of ordin... |
| oecan 8516 | Left cancellation law for ... |
| oeword 8517 | Weak ordering property of ... |
| oewordi 8518 | Weak ordering property of ... |
| oewordri 8519 | Weak ordering property of ... |
| oeworde 8520 | Ordinal exponentiation com... |
| oeordsuc 8521 | Ordering property of ordin... |
| oelim2 8522 | Ordinal exponentiation wit... |
| oeoalem 8523 | Lemma for ~ oeoa . (Contr... |
| oeoa 8524 | Sum of exponents law for o... |
| oeoelem 8525 | Lemma for ~ oeoe . (Contr... |
| oeoe 8526 | Product of exponents law f... |
| oelimcl 8527 | The ordinal exponential wi... |
| oeeulem 8528 | Lemma for ~ oeeu . (Contr... |
| oeeui 8529 | The division algorithm for... |
| oeeu 8530 | The division algorithm for... |
| nna0 8531 | Addition with zero. Theor... |
| nnm0 8532 | Multiplication with zero. ... |
| nnasuc 8533 | Addition with successor. ... |
| nnmsuc 8534 | Multiplication with succes... |
| nnesuc 8535 | Exponentiation with a succ... |
| nna0r 8536 | Addition to zero. Remark ... |
| nnm0r 8537 | Multiplication with zero. ... |
| nnacl 8538 | Closure of addition of nat... |
| nnmcl 8539 | Closure of multiplication ... |
| nnecl 8540 | Closure of exponentiation ... |
| nnacli 8541 | ` _om ` is closed under ad... |
| nnmcli 8542 | ` _om ` is closed under mu... |
| nnarcl 8543 | Reverse closure law for ad... |
| nnacom 8544 | Addition of natural number... |
| nnaordi 8545 | Ordering property of addit... |
| nnaord 8546 | Ordering property of addit... |
| nnaordr 8547 | Ordering property of addit... |
| nnawordi 8548 | Adding to both sides of an... |
| nnaass 8549 | Addition of natural number... |
| nndi 8550 | Distributive law for natur... |
| nnmass 8551 | Multiplication of natural ... |
| nnmsucr 8552 | Multiplication with succes... |
| nnmcom 8553 | Multiplication of natural ... |
| nnaword 8554 | Weak ordering property of ... |
| nnacan 8555 | Cancellation law for addit... |
| nnaword1 8556 | Weak ordering property of ... |
| nnaword2 8557 | Weak ordering property of ... |
| nnmordi 8558 | Ordering property of multi... |
| nnmord 8559 | Ordering property of multi... |
| nnmword 8560 | Weak ordering property of ... |
| nnmcan 8561 | Cancellation law for multi... |
| nnmwordi 8562 | Weak ordering property of ... |
| nnmwordri 8563 | Weak ordering property of ... |
| nnawordex 8564 | Equivalence for weak order... |
| nnaordex 8565 | Equivalence for ordering. ... |
| nnaordex2 8566 | Equivalence for ordering. ... |
| 1onn 8567 | The ordinal 1 is a natural... |
| 1onnALT 8568 | Shorter proof of ~ 1onn us... |
| 2onn 8569 | The ordinal 2 is a natural... |
| 2onnALT 8570 | Shorter proof of ~ 2onn us... |
| 3onn 8571 | The ordinal 3 is a natural... |
| 4onn 8572 | The ordinal 4 is a natural... |
| 1one2o 8573 | Ordinal one is not ordinal... |
| oaabslem 8574 | Lemma for ~ oaabs . (Cont... |
| oaabs 8575 | Ordinal addition absorbs a... |
| oaabs2 8576 | The absorption law ~ oaabs... |
| omabslem 8577 | Lemma for ~ omabs . (Cont... |
| omabs 8578 | Ordinal multiplication is ... |
| nnm1 8579 | Multiply an element of ` _... |
| nnm2 8580 | Multiply an element of ` _... |
| nn2m 8581 | Multiply an element of ` _... |
| nnneo 8582 | If a natural number is eve... |
| nneob 8583 | A natural number is even i... |
| omsmolem 8584 | Lemma for ~ omsmo . (Cont... |
| omsmo 8585 | A strictly monotonic ordin... |
| omopthlem1 8586 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8587 | Lemma for ~ omopthi . (Co... |
| omopthi 8588 | An ordered pair theorem fo... |
| omopth 8589 | An ordered pair theorem fo... |
| nnasmo 8590 | There is at most one left ... |
| eldifsucnn 8591 | Condition for membership i... |
| on2recsfn 8594 | Show that double recursion... |
| on2recsov 8595 | Calculate the value of the... |
| on2ind 8596 | Double induction over ordi... |
| on3ind 8597 | Triple induction over ordi... |
| coflton 8598 | Cofinality theorem for ord... |
| cofon1 8599 | Cofinality theorem for ord... |
| cofon2 8600 | Cofinality theorem for ord... |
| cofonr 8601 | Inverse cofinality law for... |
| naddfn 8602 | Natural addition is a func... |
| naddcllem 8603 | Lemma for ordinal addition... |
| naddcl 8604 | Closure law for natural ad... |
| naddov 8605 | The value of natural addit... |
| naddov2 8606 | Alternate expression for n... |
| naddov3 8607 | Alternate expression for n... |
| naddf 8608 | Function statement for nat... |
| naddcom 8609 | Natural addition commutes.... |
| naddrid 8610 | Ordinal zero is the additi... |
| naddlid 8611 | Ordinal zero is the additi... |
| naddssim 8612 | Ordinal less-than-or-equal... |
| naddelim 8613 | Ordinal less-than is prese... |
| naddel1 8614 | Ordinal less-than is not a... |
| naddel2 8615 | Ordinal less-than is not a... |
| naddss1 8616 | Ordinal less-than-or-equal... |
| naddss2 8617 | Ordinal less-than-or-equal... |
| naddword1 8618 | Weak-ordering principle fo... |
| naddword2 8619 | Weak-ordering principle fo... |
| naddunif 8620 | Uniformity theorem for nat... |
| naddasslem1 8621 | Lemma for ~ naddass . Exp... |
| naddasslem2 8622 | Lemma for ~ naddass . Exp... |
| naddass 8623 | Natural ordinal addition i... |
| nadd32 8624 | Commutative/associative la... |
| nadd4 8625 | Rearragement of terms in a... |
| nadd42 8626 | Rearragement of terms in a... |
| naddel12 8627 | Natural addition to both s... |
| naddsuc2 8628 | Natural addition with succ... |
| naddoa 8629 | Natural addition of a natu... |
| omnaddcl 8630 | The naturals are closed un... |
| dfer2 8635 | Alternate definition of eq... |
| dfec2 8637 | Alternate definition of ` ... |
| ecexg 8638 | An equivalence class modul... |
| ecexr 8639 | A nonempty equivalence cla... |
| dfqs2 8641 | Alternate definition of qu... |
| ereq1 8642 | Equality theorem for equiv... |
| ereq2 8643 | Equality theorem for equiv... |
| errel 8644 | An equivalence relation is... |
| erdm 8645 | The domain of an equivalen... |
| ercl 8646 | Elementhood in the field o... |
| ersym 8647 | An equivalence relation is... |
| ercl2 8648 | Elementhood in the field o... |
| ersymb 8649 | An equivalence relation is... |
| ertr 8650 | An equivalence relation is... |
| ertrd 8651 | A transitivity relation fo... |
| ertr2d 8652 | A transitivity relation fo... |
| ertr3d 8653 | A transitivity relation fo... |
| ertr4d 8654 | A transitivity relation fo... |
| erref 8655 | An equivalence relation is... |
| ercnv 8656 | The converse of an equival... |
| errn 8657 | The range and domain of an... |
| erssxp 8658 | An equivalence relation is... |
| erex 8659 | An equivalence relation is... |
| erexb 8660 | An equivalence relation is... |
| iserd 8661 | A reflexive, symmetric, tr... |
| iseri 8662 | A reflexive, symmetric, tr... |
| iseriALT 8663 | Alternate proof of ~ iseri... |
| brinxper 8664 | Conditions for a reflexive... |
| brdifun 8665 | Evaluate the incomparabili... |
| swoer 8666 | Incomparability under a st... |
| swoord1 8667 | The incomparability equiva... |
| swoord2 8668 | The incomparability equiva... |
| swoso 8669 | If the incomparability rel... |
| eqerlem 8670 | Lemma for ~ eqer . (Contr... |
| eqer 8671 | Equivalence relation invol... |
| ider 8672 | The identity relation is a... |
| 0er 8673 | The empty set is an equiva... |
| eceq1 8674 | Equality theorem for equiv... |
| eceq1d 8675 | Equality theorem for equiv... |
| eceq2 8676 | Equality theorem for equiv... |
| eceq2i 8677 | Equality theorem for the `... |
| eceq2d 8678 | Equality theorem for the `... |
| elecg 8679 | Membership in an equivalen... |
| ecref 8680 | All elements are in their ... |
| elec 8681 | Membership in an equivalen... |
| relelec 8682 | Membership in an equivalen... |
| elecres 8683 | Elementhood in the restric... |
| elecreseq 8684 | The restricted coset of ` ... |
| elecex 8685 | Condition for a coset to b... |
| ecss 8686 | An equivalence class is a ... |
| ecdmn0 8687 | A representative of a none... |
| ereldm 8688 | Equality of equivalence cl... |
| erth 8689 | Basic property of equivale... |
| erth2 8690 | Basic property of equivale... |
| erthi 8691 | Basic property of equivale... |
| erdisj 8692 | Equivalence classes do not... |
| ecidsn 8693 | An equivalence class modul... |
| qseq1 8694 | Equality theorem for quoti... |
| qseq2 8695 | Equality theorem for quoti... |
| qseq2i 8696 | Equality theorem for quoti... |
| qseq1d 8697 | Equality theorem for quoti... |
| qseq2d 8698 | Equality theorem for quoti... |
| qseq12 8699 | Equality theorem for quoti... |
| 0qs 8700 | Quotient set with the empt... |
| elqsg 8701 | Closed form of ~ elqs . (... |
| elqs 8702 | Membership in a quotient s... |
| elqsi 8703 | Membership in a quotient s... |
| elqsecl 8704 | Membership in a quotient s... |
| ecelqs 8705 | Membership of an equivalen... |
| ecelqsw 8706 | Membership of an equivalen... |
| ecelqsi 8707 | Membership of an equivalen... |
| ecopqsi 8708 | "Closure" law for equivale... |
| qsexg 8709 | A quotient set exists. (C... |
| qsex 8710 | A quotient set exists. (C... |
| uniqs 8711 | The union of a quotient se... |
| uniqsw 8712 | The union of a quotient se... |
| qsss 8713 | A quotient set is a set of... |
| uniqs2 8714 | The union of a quotient se... |
| snecg 8715 | The singleton of a coset i... |
| snec 8716 | The singleton of an equiva... |
| ecqs 8717 | Equivalence class in terms... |
| ecid 8718 | A set is equal to its cose... |
| qsid 8719 | A set is equal to its quot... |
| ectocld 8720 | Implicit substitution of c... |
| ectocl 8721 | Implicit substitution of c... |
| elqsn0 8722 | A quotient set does not co... |
| ecelqsdm 8723 | Membership of an equivalen... |
| ecelqsdmb 8724 | ` R ` -coset of ` B ` in a... |
| eceldmqs 8725 | ` R ` -coset in its domain... |
| xpider 8726 | A Cartesian square is an e... |
| iiner 8727 | The intersection of a none... |
| riiner 8728 | The relative intersection ... |
| erinxp 8729 | A restricted equivalence r... |
| ecinxp 8730 | Restrict the relation in a... |
| qsinxp 8731 | Restrict the equivalence r... |
| qsdisj 8732 | Members of a quotient set ... |
| qsdisj2 8733 | A quotient set is a disjoi... |
| qsel 8734 | If an element of a quotien... |
| uniinqs 8735 | Class union distributes ov... |
| qliftlem 8736 | Lemma for theorems about a... |
| qliftrel 8737 | ` F ` , a function lift, i... |
| qliftel 8738 | Elementhood in the relatio... |
| qliftel1 8739 | Elementhood in the relatio... |
| qliftfun 8740 | The function ` F ` is the ... |
| qliftfund 8741 | The function ` F ` is the ... |
| qliftfuns 8742 | The function ` F ` is the ... |
| qliftf 8743 | The domain and codomain of... |
| qliftval 8744 | The value of the function ... |
| ecoptocl 8745 | Implicit substitution of c... |
| 2ecoptocl 8746 | Implicit substitution of c... |
| 3ecoptocl 8747 | Implicit substitution of c... |
| brecop 8748 | Binary relation on a quoti... |
| brecop2 8749 | Binary relation on a quoti... |
| eroveu 8750 | Lemma for ~ erov and ~ ero... |
| erovlem 8751 | Lemma for ~ erov and ~ ero... |
| erov 8752 | The value of an operation ... |
| eroprf 8753 | Functionality of an operat... |
| erov2 8754 | The value of an operation ... |
| eroprf2 8755 | Functionality of an operat... |
| ecopoveq 8756 | This is the first of sever... |
| ecopovsym 8757 | Assuming the operation ` F... |
| ecopovtrn 8758 | Assuming that operation ` ... |
| ecopover 8759 | Assuming that operation ` ... |
| eceqoveq 8760 | Equality of equivalence re... |
| ecovcom 8761 | Lemma used to transfer a c... |
| ecovass 8762 | Lemma used to transfer an ... |
| ecovdi 8763 | Lemma used to transfer a d... |
| mapprc 8768 | When ` A ` is a proper cla... |
| pmex 8769 | The class of all partial f... |
| mapexOLD 8770 | Obsolete version of ~ mape... |
| fnmap 8771 | Set exponentiation has a u... |
| fnpm 8772 | Partial function exponenti... |
| reldmmap 8773 | Set exponentiation is a we... |
| mapvalg 8774 | The value of set exponenti... |
| pmvalg 8775 | The value of the partial m... |
| mapval 8776 | The value of set exponenti... |
| elmapg 8777 | Membership relation for se... |
| elmapd 8778 | Deduction form of ~ elmapg... |
| elmapdd 8779 | Deduction associated with ... |
| mapdm0 8780 | The empty set is the only ... |
| elpmg 8781 | The predicate "is a partia... |
| elpm2g 8782 | The predicate "is a partia... |
| elpm2r 8783 | Sufficient condition for b... |
| elpmi 8784 | A partial function is a fu... |
| pmfun 8785 | A partial function is a fu... |
| elmapex 8786 | Eliminate antecedent for m... |
| elmapi 8787 | A mapping is a function, f... |
| mapfset 8788 | If ` B ` is a set, the val... |
| mapssfset 8789 | The value of the set expon... |
| mapfoss 8790 | The value of the set expon... |
| fsetsspwxp 8791 | The class of all functions... |
| fset0 8792 | The set of functions from ... |
| fsetdmprc0 8793 | The set of functions with ... |
| fsetex 8794 | The set of functions betwe... |
| f1setex 8795 | The set of injections betw... |
| fosetex 8796 | The set of surjections bet... |
| f1osetex 8797 | The set of bijections betw... |
| fsetfcdm 8798 | The class of functions wit... |
| fsetfocdm 8799 | The class of functions wit... |
| fsetprcnex 8800 | The class of all functions... |
| fsetcdmex 8801 | The class of all functions... |
| fsetexb 8802 | The class of all functions... |
| elmapfn 8803 | A mapping is a function wi... |
| elmapfun 8804 | A mapping is always a func... |
| elmapssres 8805 | A restricted mapping is a ... |
| elmapssresd 8806 | A restricted mapping is a ... |
| fpmg 8807 | A total function is a part... |
| pmss12g 8808 | Subset relation for the se... |
| pmresg 8809 | Elementhood of a restricte... |
| elmap 8810 | Membership relation for se... |
| mapval2 8811 | Alternate expression for t... |
| elpm 8812 | The predicate "is a partia... |
| elpm2 8813 | The predicate "is a partia... |
| fpm 8814 | A total function is a part... |
| mapsspm 8815 | Set exponentiation is a su... |
| pmsspw 8816 | Partial maps are a subset ... |
| mapsspw 8817 | Set exponentiation is a su... |
| mapfvd 8818 | The value of a function th... |
| elmapresaun 8819 | ~ fresaun transposed to ma... |
| fvmptmap 8820 | Special case of ~ fvmpt fo... |
| map0e 8821 | Set exponentiation with an... |
| map0b 8822 | Set exponentiation with an... |
| map0g 8823 | Set exponentiation is empt... |
| 0map0sn0 8824 | The set of mappings of the... |
| mapsnd 8825 | The value of set exponenti... |
| map0 8826 | Set exponentiation is empt... |
| mapsn 8827 | The value of set exponenti... |
| mapss 8828 | Subset inheritance for set... |
| fdiagfn 8829 | Functionality of the diago... |
| fvdiagfn 8830 | Functionality of the diago... |
| mapsnconst 8831 | Every singleton map is a c... |
| mapsncnv 8832 | Expression for the inverse... |
| mapsnf1o2 8833 | Explicit bijection between... |
| mapsnf1o3 8834 | Explicit bijection in the ... |
| ralxpmap 8835 | Quantification over functi... |
| dfixp 8838 | Eliminate the expression `... |
| ixpsnval 8839 | The value of an infinite C... |
| elixp2 8840 | Membership in an infinite ... |
| fvixp 8841 | Projection of a factor of ... |
| ixpfn 8842 | A nuple is a function. (C... |
| elixp 8843 | Membership in an infinite ... |
| elixpconst 8844 | Membership in an infinite ... |
| ixpconstg 8845 | Infinite Cartesian product... |
| ixpconst 8846 | Infinite Cartesian product... |
| ixpeq1 8847 | Equality theorem for infin... |
| ixpeq1d 8848 | Equality theorem for infin... |
| ss2ixp 8849 | Subclass theorem for infin... |
| ixpeq2 8850 | Equality theorem for infin... |
| ixpeq2dva 8851 | Equality theorem for infin... |
| ixpeq2dv 8852 | Equality theorem for infin... |
| cbvixp 8853 | Change bound variable in a... |
| cbvixpv 8854 | Change bound variable in a... |
| nfixpw 8855 | Bound-variable hypothesis ... |
| nfixp 8856 | Bound-variable hypothesis ... |
| nfixp1 8857 | The index variable in an i... |
| ixpprc 8858 | A cartesian product of pro... |
| ixpf 8859 | A member of an infinite Ca... |
| uniixp 8860 | The union of an infinite C... |
| ixpexg 8861 | The existence of an infini... |
| ixpin 8862 | The intersection of two in... |
| ixpiin 8863 | The indexed intersection o... |
| ixpint 8864 | The intersection of a coll... |
| ixp0x 8865 | An infinite Cartesian prod... |
| ixpssmap2g 8866 | An infinite Cartesian prod... |
| ixpssmapg 8867 | An infinite Cartesian prod... |
| 0elixp 8868 | Membership of the empty se... |
| ixpn0 8869 | The infinite Cartesian pro... |
| ixp0 8870 | The infinite Cartesian pro... |
| ixpssmap 8871 | An infinite Cartesian prod... |
| resixp 8872 | Restriction of an element ... |
| undifixp 8873 | Union of two projections o... |
| mptelixpg 8874 | Condition for an explicit ... |
| resixpfo 8875 | Restriction of elements of... |
| elixpsn 8876 | Membership in a class of s... |
| ixpsnf1o 8877 | A bijection between a clas... |
| mapsnf1o 8878 | A bijection between a set ... |
| boxriin 8879 | A rectangular subset of a ... |
| boxcutc 8880 | The relative complement of... |
| relen 8889 | Equinumerosity is a relati... |
| reldom 8890 | Dominance is a relation. ... |
| relsdom 8891 | Strict dominance is a rela... |
| encv 8892 | If two classes are equinum... |
| breng 8893 | Equinumerosity relation. ... |
| bren 8894 | Equinumerosity relation. ... |
| brdom2g 8895 | Dominance relation. This ... |
| brdomg 8896 | Dominance relation. (Cont... |
| brdomi 8897 | Dominance relation. (Cont... |
| brdom 8898 | Dominance relation. (Cont... |
| domen 8899 | Dominance in terms of equi... |
| domeng 8900 | Dominance in terms of equi... |
| ctex 8901 | A countable set is a set. ... |
| f1oen4g 8902 | The domain and range of a ... |
| f1dom4g 8903 | The domain of a one-to-one... |
| f1oen3g 8904 | The domain and range of a ... |
| f1dom3g 8905 | The domain of a one-to-one... |
| f1oen2g 8906 | The domain and range of a ... |
| f1dom2g 8907 | The domain of a one-to-one... |
| f1oeng 8908 | The domain and range of a ... |
| f1domg 8909 | The domain of a one-to-one... |
| f1oen 8910 | The domain and range of a ... |
| f1dom 8911 | The domain of a one-to-one... |
| brsdom 8912 | Strict dominance relation,... |
| isfi 8913 | Express " ` A ` is finite"... |
| enssdom 8914 | Equinumerosity implies dom... |
| enssdomOLD 8915 | Obsolete version of ~ enss... |
| dfdom2 8916 | Alternate definition of do... |
| endom 8917 | Equinumerosity implies dom... |
| sdomdom 8918 | Strict dominance implies d... |
| sdomnen 8919 | Strict dominance implies n... |
| brdom2 8920 | Dominance in terms of stri... |
| bren2 8921 | Equinumerosity expressed i... |
| enrefg 8922 | Equinumerosity is reflexiv... |
| enref 8923 | Equinumerosity is reflexiv... |
| eqeng 8924 | Equality implies equinumer... |
| domrefg 8925 | Dominance is reflexive. (... |
| en2d 8926 | Equinumerosity inference f... |
| en3d 8927 | Equinumerosity inference f... |
| en2i 8928 | Equinumerosity inference f... |
| en3i 8929 | Equinumerosity inference f... |
| dom2lem 8930 | A mapping (first hypothesi... |
| dom2d 8931 | A mapping (first hypothesi... |
| dom3d 8932 | A mapping (first hypothesi... |
| dom2 8933 | A mapping (first hypothesi... |
| dom3 8934 | A mapping (first hypothesi... |
| idssen 8935 | Equality implies equinumer... |
| domssl 8936 | If ` A ` is a subset of ` ... |
| domssr 8937 | If ` C ` is a superset of ... |
| ssdomg 8938 | A set dominates its subset... |
| ener 8939 | Equinumerosity is an equiv... |
| ensymb 8940 | Symmetry of equinumerosity... |
| ensym 8941 | Symmetry of equinumerosity... |
| ensymi 8942 | Symmetry of equinumerosity... |
| ensymd 8943 | Symmetry of equinumerosity... |
| entr 8944 | Transitivity of equinumero... |
| domtr 8945 | Transitivity of dominance ... |
| entri 8946 | A chained equinumerosity i... |
| entr2i 8947 | A chained equinumerosity i... |
| entr3i 8948 | A chained equinumerosity i... |
| entr4i 8949 | A chained equinumerosity i... |
| endomtr 8950 | Transitivity of equinumero... |
| domentr 8951 | Transitivity of dominance ... |
| f1imaeng 8952 | If a function is one-to-on... |
| f1imaen2g 8953 | If a function is one-to-on... |
| f1imaen3g 8954 | If a set function is one-t... |
| f1imaen 8955 | If a function is one-to-on... |
| en0 8956 | The empty set is equinumer... |
| en0ALT 8957 | Shorter proof of ~ en0 , d... |
| en0r 8958 | The empty set is equinumer... |
| ensn1 8959 | A singleton is equinumerou... |
| ensn1g 8960 | A singleton is equinumerou... |
| enpr1g 8961 | ` { A , A } ` has only one... |
| en1 8962 | A set is equinumerous to o... |
| en1b 8963 | A set is equinumerous to o... |
| reuen1 8964 | Two ways to express "exact... |
| euen1 8965 | Two ways to express "exact... |
| euen1b 8966 | Two ways to express " ` A ... |
| en1uniel 8967 | A singleton contains its s... |
| 2dom 8968 | A set that dominates ordin... |
| fundmen 8969 | A function is equinumerous... |
| fundmeng 8970 | A function is equinumerous... |
| cnven 8971 | A relational set is equinu... |
| cnvct 8972 | If a set is countable, so ... |
| fndmeng 8973 | A function is equinumerate... |
| mapsnend 8974 | Set exponentiation to a si... |
| mapsnen 8975 | Set exponentiation to a si... |
| snmapen 8976 | Set exponentiation: a sing... |
| snmapen1 8977 | Set exponentiation: a sing... |
| map1 8978 | Set exponentiation: ordina... |
| en2sn 8979 | Two singletons are equinum... |
| 0fi 8980 | The empty set is finite. ... |
| snfi 8981 | A singleton is finite. (C... |
| fiprc 8982 | The class of finite sets i... |
| unen 8983 | Equinumerosity of union of... |
| enrefnn 8984 | Equinumerosity is reflexiv... |
| en2prd 8985 | Two proper unordered pairs... |
| enpr2d 8986 | A pair with distinct eleme... |
| ssct 8987 | Any subset of a countable ... |
| difsnen 8988 | All decrements of a set ar... |
| domdifsn 8989 | Dominance over a set with ... |
| xpsnen 8990 | A set is equinumerous to i... |
| xpsneng 8991 | A set is equinumerous to i... |
| xp1en 8992 | One times a cardinal numbe... |
| endisj 8993 | Any two sets are equinumer... |
| undom 8994 | Dominance law for union. ... |
| xpcomf1o 8995 | The canonical bijection fr... |
| xpcomco 8996 | Composition with the bijec... |
| xpcomen 8997 | Commutative law for equinu... |
| xpcomeng 8998 | Commutative law for equinu... |
| xpsnen2g 8999 | A set is equinumerous to i... |
| xpassen 9000 | Associative law for equinu... |
| xpdom2 9001 | Dominance law for Cartesia... |
| xpdom2g 9002 | Dominance law for Cartesia... |
| xpdom1g 9003 | Dominance law for Cartesia... |
| xpdom3 9004 | A set is dominated by its ... |
| xpdom1 9005 | Dominance law for Cartesia... |
| domunsncan 9006 | A singleton cancellation l... |
| omxpenlem 9007 | Lemma for ~ omxpen . (Con... |
| omxpen 9008 | The cardinal and ordinal p... |
| omf1o 9009 | Construct an explicit bije... |
| pw2f1olem 9010 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9011 | The power set of a set is ... |
| pw2eng 9012 | The power set of a set is ... |
| pw2en 9013 | The power set of a set is ... |
| fopwdom 9014 | Covering implies injection... |
| enfixsn 9015 | Given two equipollent sets... |
| sbthlem1 9016 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9017 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9018 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9019 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9020 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9021 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9022 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9023 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9024 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9025 | Lemma for ~ sbth . (Contr... |
| sbth 9026 | Schroeder-Bernstein Theore... |
| sbthb 9027 | Schroeder-Bernstein Theore... |
| sbthcl 9028 | Schroeder-Bernstein Theore... |
| dfsdom2 9029 | Alternate definition of st... |
| brsdom2 9030 | Alternate definition of st... |
| sdomnsym 9031 | Strict dominance is asymme... |
| domnsym 9032 | Theorem 22(i) of [Suppes] ... |
| 0domg 9033 | Any set dominates the empt... |
| dom0 9034 | A set dominated by the emp... |
| 0sdomg 9035 | A set strictly dominates t... |
| 0dom 9036 | Any set dominates the empt... |
| 0sdom 9037 | A set strictly dominates t... |
| sdom0 9038 | The empty set does not str... |
| sdomdomtr 9039 | Transitivity of strict dom... |
| sdomentr 9040 | Transitivity of strict dom... |
| domsdomtr 9041 | Transitivity of dominance ... |
| ensdomtr 9042 | Transitivity of equinumero... |
| sdomirr 9043 | Strict dominance is irrefl... |
| sdomtr 9044 | Strict dominance is transi... |
| sdomn2lp 9045 | Strict dominance has no 2-... |
| enen1 9046 | Equality-like theorem for ... |
| enen2 9047 | Equality-like theorem for ... |
| domen1 9048 | Equality-like theorem for ... |
| domen2 9049 | Equality-like theorem for ... |
| sdomen1 9050 | Equality-like theorem for ... |
| sdomen2 9051 | Equality-like theorem for ... |
| domtriord 9052 | Dominance is trichotomous ... |
| sdomel 9053 | For ordinals, strict domin... |
| sdomdif 9054 | The difference of a set fr... |
| onsdominel 9055 | An ordinal with more eleme... |
| domunsn 9056 | Dominance over a set with ... |
| fodomr 9057 | There exists a mapping fro... |
| pwdom 9058 | Injection of sets implies ... |
| canth2 9059 | Cantor's Theorem. No set ... |
| canth2g 9060 | Cantor's theorem with the ... |
| 2pwuninel 9061 | The power set of the power... |
| 2pwne 9062 | No set equals the power se... |
| disjen 9063 | A stronger form of ~ pwuni... |
| disjenex 9064 | Existence version of ~ dis... |
| domss2 9065 | A corollary of ~ disjenex ... |
| domssex2 9066 | A corollary of ~ disjenex ... |
| domssex 9067 | Weakening of ~ domssex2 to... |
| xpf1o 9068 | Construct a bijection on a... |
| xpen 9069 | Equinumerosity law for Car... |
| mapen 9070 | Two set exponentiations ar... |
| mapdom1 9071 | Order-preserving property ... |
| mapxpen 9072 | Equinumerosity law for dou... |
| xpmapenlem 9073 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9074 | Equinumerosity law for set... |
| mapunen 9075 | Equinumerosity law for set... |
| map2xp 9076 | A cardinal power with expo... |
| mapdom2 9077 | Order-preserving property ... |
| mapdom3 9078 | Set exponentiation dominat... |
| pwen 9079 | If two sets are equinumero... |
| ssenen 9080 | Equinumerosity of equinume... |
| limenpsi 9081 | A limit ordinal is equinum... |
| limensuci 9082 | A limit ordinal is equinum... |
| limensuc 9083 | A limit ordinal is equinum... |
| infensuc 9084 | Any infinite ordinal is eq... |
| dif1enlem 9085 | Lemma for ~ rexdif1en and ... |
| rexdif1en 9086 | If a set is equinumerous t... |
| dif1en 9087 | If a set ` A ` is equinume... |
| dif1ennn 9088 | If a set ` A ` is equinume... |
| findcard 9089 | Schema for induction on th... |
| findcard2 9090 | Schema for induction on th... |
| findcard2s 9091 | Variation of ~ findcard2 r... |
| findcard2d 9092 | Deduction version of ~ fin... |
| nnfi 9093 | Natural numbers are finite... |
| pssnn 9094 | A proper subset of a natur... |
| ssnnfi 9095 | A subset of a natural numb... |
| unfi 9096 | The union of two finite se... |
| unfid 9097 | The union of two finite se... |
| ssfi 9098 | A subset of a finite set i... |
| ssfiALT 9099 | Shorter proof of ~ ssfi us... |
| diffi 9100 | If ` A ` is finite, ` ( A ... |
| cnvfi 9101 | If a set is finite, its co... |
| pwssfi 9102 | Every element of the power... |
| fnfi 9103 | A version of ~ fnex for fi... |
| f1oenfi 9104 | If the domain of a one-to-... |
| f1oenfirn 9105 | If the range of a one-to-o... |
| f1domfi 9106 | If the codomain of a one-t... |
| f1domfi2 9107 | If the domain of a one-to-... |
| enreffi 9108 | Equinumerosity is reflexiv... |
| ensymfib 9109 | Symmetry of equinumerosity... |
| entrfil 9110 | Transitivity of equinumero... |
| enfii 9111 | A set equinumerous to a fi... |
| enfi 9112 | Equinumerous sets have the... |
| enfiALT 9113 | Shorter proof of ~ enfi us... |
| domfi 9114 | A set dominated by a finit... |
| entrfi 9115 | Transitivity of equinumero... |
| entrfir 9116 | Transitivity of equinumero... |
| domtrfil 9117 | Transitivity of dominance ... |
| domtrfi 9118 | Transitivity of dominance ... |
| domtrfir 9119 | Transitivity of dominance ... |
| f1imaenfi 9120 | If a function is one-to-on... |
| ssdomfi 9121 | A finite set dominates its... |
| ssdomfi2 9122 | A set dominates its finite... |
| sbthfilem 9123 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9124 | Schroeder-Bernstein Theore... |
| domnsymfi 9125 | If a set dominates a finit... |
| sdomdomtrfi 9126 | Transitivity of strict dom... |
| domsdomtrfi 9127 | Transitivity of dominance ... |
| sucdom2 9128 | Strict dominance of a set ... |
| phplem1 9129 | Lemma for Pigeonhole Princ... |
| phplem2 9130 | Lemma for Pigeonhole Princ... |
| nneneq 9131 | Two equinumerous natural n... |
| php 9132 | Pigeonhole Principle. A n... |
| php2 9133 | Corollary of Pigeonhole Pr... |
| php3 9134 | Corollary of Pigeonhole Pr... |
| php4 9135 | Corollary of the Pigeonhol... |
| php5 9136 | Corollary of the Pigeonhol... |
| phpeqd 9137 | Corollary of the Pigeonhol... |
| nndomog 9138 | Cardinal ordering agrees w... |
| onomeneq 9139 | An ordinal number equinume... |
| onfin 9140 | An ordinal number is finit... |
| ordfin 9141 | A generalization of ~ onfi... |
| onfin2 9142 | A set is a natural number ... |
| nndomo 9143 | Cardinal ordering agrees w... |
| nnsdomo 9144 | Cardinal ordering agrees w... |
| sucdom 9145 | Strict dominance of a set ... |
| snnen2o 9146 | A singleton ` { A } ` is n... |
| 0sdom1dom 9147 | Strict dominance over 0 is... |
| 0sdom1domALT 9148 | Alternate proof of ~ 0sdom... |
| 1sdom2 9149 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9150 | Alternate proof of ~ 1sdom... |
| sdom1 9151 | A set has less than one me... |
| modom 9152 | Two ways to express "at mo... |
| modom2 9153 | Two ways to express "at mo... |
| rex2dom 9154 | A set that has at least 2 ... |
| 1sdom2dom 9155 | Strict dominance over 1 is... |
| 1sdom 9156 | A set that strictly domina... |
| unxpdomlem1 9157 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9158 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9159 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9160 | Cartesian product dominate... |
| unxpdom2 9161 | Corollary of ~ unxpdom . ... |
| sucxpdom 9162 | Cartesian product dominate... |
| pssinf 9163 | A set equinumerous to a pr... |
| fisseneq 9164 | A finite set is equal to i... |
| ominf 9165 | The set of natural numbers... |
| isinf 9166 | Any set that is not finite... |
| fineqvlem 9167 | Lemma for ~ fineqv . (Con... |
| fineqv 9168 | If the Axiom of Infinity i... |
| xpfir 9169 | The components of a nonemp... |
| ssfid 9170 | A subset of a finite set i... |
| infi 9171 | The intersection of two se... |
| rabfi 9172 | A restricted class built f... |
| finresfin 9173 | The restriction of a finit... |
| f1finf1o 9174 | Any injection from one fin... |
| nfielex 9175 | If a class is not finite, ... |
| en1eqsn 9176 | A set with one element is ... |
| en1eqsnbi 9177 | A set containing an elemen... |
| dif1ennnALT 9178 | Alternate proof of ~ dif1e... |
| enp1ilem 9179 | Lemma for uses of ~ enp1i ... |
| enp1i 9180 | Proof induction for ~ en2 ... |
| en2 9181 | A set equinumerous to ordi... |
| en3 9182 | A set equinumerous to ordi... |
| en4 9183 | A set equinumerous to ordi... |
| findcard3 9184 | Schema for strong inductio... |
| ac6sfi 9185 | A version of ~ ac6s for fi... |
| frfi 9186 | A partial order is well-fo... |
| fimax2g 9187 | A finite set has a maximum... |
| fimaxg 9188 | A finite set has a maximum... |
| fisupg 9189 | Lemma showing existence an... |
| wofi 9190 | A total order on a finite ... |
| ordunifi 9191 | The maximum of a finite co... |
| nnunifi 9192 | The union (supremum) of a ... |
| unblem1 9193 | Lemma for ~ unbnn . After... |
| unblem2 9194 | Lemma for ~ unbnn . The v... |
| unblem3 9195 | Lemma for ~ unbnn . The v... |
| unblem4 9196 | Lemma for ~ unbnn . The f... |
| unbnn 9197 | Any unbounded subset of na... |
| unbnn2 9198 | Version of ~ unbnn that do... |
| isfinite2 9199 | Any set strictly dominated... |
| nnsdomg 9200 | Omega strictly dominates a... |
| isfiniteg 9201 | A set is finite iff it is ... |
| infsdomnn 9202 | An infinite set strictly d... |
| infn0 9203 | An infinite set is not emp... |
| infn0ALT 9204 | Shorter proof of ~ infn0 u... |
| fin2inf 9205 | This (useless) theorem, wh... |
| unfilem1 9206 | Lemma for proving that the... |
| unfilem2 9207 | Lemma for proving that the... |
| unfilem3 9208 | Lemma for proving that the... |
| unfir 9209 | If a union is finite, the ... |
| unfib 9210 | A union is finite if and o... |
| unfi2 9211 | The union of two finite se... |
| difinf 9212 | An infinite set ` A ` minu... |
| fodomfi 9213 | An onto function implies d... |
| fofi 9214 | If an onto function has a ... |
| f1fi 9215 | If a 1-to-1 function has a... |
| imafi 9216 | Images of finite sets are ... |
| imafiOLD 9217 | Obsolete version of ~ imaf... |
| pwfir 9218 | If the power set of a set ... |
| pwfilem 9219 | Lemma for ~ pwfi . (Contr... |
| pwfi 9220 | The power set of a finite ... |
| xpfi 9221 | The Cartesian product of t... |
| 3xpfi 9222 | The Cartesian product of t... |
| domunfican 9223 | A finite set union cancell... |
| infcntss 9224 | Every infinite set has a d... |
| prfi 9225 | An unordered pair is finit... |
| prfiALT 9226 | Shorter proof of ~ prfi us... |
| tpfi 9227 | An unordered triple is fin... |
| fiint 9228 | Equivalent ways of stating... |
| fodomfir 9229 | There exists a mapping fro... |
| fodomfib 9230 | Equivalence of an onto map... |
| fodomfiOLD 9231 | Obsolete version of ~ fodo... |
| fodomfibOLD 9232 | Obsolete version of ~ fodo... |
| fofinf1o 9233 | Any surjection from one fi... |
| rneqdmfinf1o 9234 | Any function from a finite... |
| fidomdm 9235 | Any finite set dominates i... |
| dmfi 9236 | The domain of a finite set... |
| fundmfibi 9237 | A function is finite if an... |
| resfnfinfin 9238 | The restriction of a funct... |
| residfi 9239 | A restricted identity func... |
| cnvfiALT 9240 | Shorter proof of ~ cnvfi u... |
| rnfi 9241 | The range of a finite set ... |
| f1dmvrnfibi 9242 | A one-to-one function whos... |
| f1vrnfibi 9243 | A one-to-one function whic... |
| iunfi 9244 | The finite union of finite... |
| unifi 9245 | The finite union of finite... |
| unifi2 9246 | The finite union of finite... |
| infssuni 9247 | If an infinite set ` A ` i... |
| unirnffid 9248 | The union of the range of ... |
| mapfi 9249 | Set exponentiation of fini... |
| ixpfi 9250 | A Cartesian product of fin... |
| ixpfi2 9251 | A Cartesian product of fin... |
| mptfi 9252 | A finite mapping set is fi... |
| abrexfi 9253 | An image set from a finite... |
| cnvimamptfin 9254 | A preimage of a mapping wi... |
| elfpw 9255 | Membership in a class of f... |
| unifpw 9256 | A set is the union of its ... |
| f1opwfi 9257 | A one-to-one mapping induc... |
| fissuni 9258 | A finite subset of a union... |
| fipreima 9259 | Given a finite subset ` A ... |
| finsschain 9260 | A finite subset of the uni... |
| indexfi 9261 | If for every element of a ... |
| imafi2 9262 | The image by a finite set ... |
| unifi3 9263 | If a union is finite, then... |
| tfsnfin2 9264 | A transfinite sequence is ... |
| relfsupp 9267 | The property of a function... |
| relprcnfsupp 9268 | A proper class is never fi... |
| isfsupp 9269 | The property of a class to... |
| isfsuppd 9270 | Deduction form of ~ isfsup... |
| funisfsupp 9271 | The property of a function... |
| fsuppimp 9272 | Implications of a class be... |
| fsuppimpd 9273 | A finitely supported funct... |
| fsuppfund 9274 | A finitely supported funct... |
| fisuppfi 9275 | A function on a finite set... |
| fidmfisupp 9276 | A function with a finite d... |
| finnzfsuppd 9277 | If a function is zero outs... |
| fdmfisuppfi 9278 | The support of a function ... |
| fdmfifsupp 9279 | A function with a finite d... |
| fsuppmptdm 9280 | A mapping with a finite do... |
| fndmfisuppfi 9281 | The support of a function ... |
| fndmfifsupp 9282 | A function with a finite d... |
| suppeqfsuppbi 9283 | If two functions have the ... |
| suppssfifsupp 9284 | If the support of a functi... |
| fsuppsssupp 9285 | If the support of a functi... |
| fsuppsssuppgd 9286 | If the support of a functi... |
| fsuppss 9287 | A subset of a finitely sup... |
| fsuppssov1 9288 | Formula building theorem f... |
| fsuppxpfi 9289 | The cartesian product of t... |
| fczfsuppd 9290 | A constant function with v... |
| fsuppun 9291 | The union of two finitely ... |
| fsuppunfi 9292 | The union of the support o... |
| fsuppunbi 9293 | If the union of two classe... |
| 0fsupp 9294 | The empty set is a finitel... |
| snopfsupp 9295 | A singleton containing an ... |
| funsnfsupp 9296 | Finite support for a funct... |
| fsuppres 9297 | The restriction of a finit... |
| fmptssfisupp 9298 | The restriction of a mappi... |
| ressuppfi 9299 | If the support of the rest... |
| resfsupp 9300 | If the restriction of a fu... |
| resfifsupp 9301 | The restriction of a funct... |
| ffsuppbi 9302 | Two ways of saying that a ... |
| fsuppmptif 9303 | A function mapping an argu... |
| sniffsupp 9304 | A function mapping all but... |
| fsuppcolem 9305 | Lemma for ~ fsuppco . For... |
| fsuppco 9306 | The composition of a 1-1 f... |
| fsuppco2 9307 | The composition of a funct... |
| fsuppcor 9308 | The composition of a funct... |
| mapfienlem1 9309 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9310 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9311 | Lemma 3 for ~ mapfien . (... |
| mapfien 9312 | A bijection of the base se... |
| mapfien2 9313 | Equinumerousity relation f... |
| fival 9316 | The set of all the finite ... |
| elfi 9317 | Specific properties of an ... |
| elfi2 9318 | The empty intersection nee... |
| elfir 9319 | Sufficient condition for a... |
| intrnfi 9320 | Sufficient condition for t... |
| iinfi 9321 | An indexed intersection of... |
| inelfi 9322 | The intersection of two se... |
| ssfii 9323 | Any element of a set ` A `... |
| fi0 9324 | The set of finite intersec... |
| fieq0 9325 | A set is empty iff the cla... |
| fiin 9326 | The elements of ` ( fi `` ... |
| dffi2 9327 | The set of finite intersec... |
| fiss 9328 | Subset relationship for fu... |
| inficl 9329 | A set which is closed unde... |
| fipwuni 9330 | The set of finite intersec... |
| fisn 9331 | A singleton is closed unde... |
| fiuni 9332 | The union of the finite in... |
| fipwss 9333 | If a set is a family of su... |
| elfiun 9334 | A finite intersection of e... |
| dffi3 9335 | The set of finite intersec... |
| fifo 9336 | Describe a surjection from... |
| marypha1lem 9337 | Core induction for Philip ... |
| marypha1 9338 | (Philip) Hall's marriage t... |
| marypha2lem1 9339 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9340 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9341 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9342 | Lemma for ~ marypha2 . Pr... |
| marypha2 9343 | Version of ~ marypha1 usin... |
| dfsup2 9348 | Quantifier-free definition... |
| supeq1 9349 | Equality theorem for supre... |
| supeq1d 9350 | Equality deduction for sup... |
| supeq1i 9351 | Equality inference for sup... |
| supeq2 9352 | Equality theorem for supre... |
| supeq3 9353 | Equality theorem for supre... |
| supeq123d 9354 | Equality deduction for sup... |
| nfsup 9355 | Hypothesis builder for sup... |
| supmo 9356 | Any class ` B ` has at mos... |
| supexd 9357 | A supremum is a set. (Con... |
| supeu 9358 | A supremum is unique. Sim... |
| supval2 9359 | Alternate expression for t... |
| eqsup 9360 | Sufficient condition for a... |
| eqsupd 9361 | Sufficient condition for a... |
| supcl 9362 | A supremum belongs to its ... |
| supub 9363 | A supremum is an upper bou... |
| suplub 9364 | A supremum is the least up... |
| suplub2 9365 | Bidirectional form of ~ su... |
| supnub 9366 | An upper bound is not less... |
| supssd 9367 | Inequality deduction for s... |
| supex 9368 | A supremum is a set. (Con... |
| sup00 9369 | The supremum under an empt... |
| sup0riota 9370 | The supremum of an empty s... |
| sup0 9371 | The supremum of an empty s... |
| supmax 9372 | The greatest element of a ... |
| fisup2g 9373 | A finite set satisfies the... |
| fisupcl 9374 | A nonempty finite set cont... |
| supgtoreq 9375 | The supremum of a finite s... |
| suppr 9376 | The supremum of a pair. (... |
| supsn 9377 | The supremum of a singleto... |
| supisolem 9378 | Lemma for ~ supiso . (Con... |
| supisoex 9379 | Lemma for ~ supiso . (Con... |
| supiso 9380 | Image of a supremum under ... |
| infeq1 9381 | Equality theorem for infim... |
| infeq1d 9382 | Equality deduction for inf... |
| infeq1i 9383 | Equality inference for inf... |
| infeq2 9384 | Equality theorem for infim... |
| infeq3 9385 | Equality theorem for infim... |
| infeq123d 9386 | Equality deduction for inf... |
| nfinf 9387 | Hypothesis builder for inf... |
| infexd 9388 | An infimum is a set. (Con... |
| eqinf 9389 | Sufficient condition for a... |
| eqinfd 9390 | Sufficient condition for a... |
| infval 9391 | Alternate expression for t... |
| infcllem 9392 | Lemma for ~ infcl , ~ infl... |
| infcl 9393 | An infimum belongs to its ... |
| inflb 9394 | An infimum is a lower boun... |
| infglb 9395 | An infimum is the greatest... |
| infglbb 9396 | Bidirectional form of ~ in... |
| infnlb 9397 | A lower bound is not great... |
| infssd 9398 | Inequality deduction for i... |
| infex 9399 | An infimum is a set. (Con... |
| infmin 9400 | The smallest element of a ... |
| infmo 9401 | Any class ` B ` has at mos... |
| infeu 9402 | An infimum is unique. (Co... |
| fimin2g 9403 | A finite set has a minimum... |
| fiming 9404 | A finite set has a minimum... |
| fiinfg 9405 | Lemma showing existence an... |
| fiinf2g 9406 | A finite set satisfies the... |
| fiinfcl 9407 | A nonempty finite set cont... |
| infltoreq 9408 | The infimum of a finite se... |
| infpr 9409 | The infimum of a pair. (C... |
| infsupprpr 9410 | The infimum of a proper pa... |
| infsn 9411 | The infimum of a singleton... |
| inf00 9412 | The infimum regarding an e... |
| infempty 9413 | The infimum of an empty se... |
| infiso 9414 | Image of an infimum under ... |
| dfoi 9417 | Rewrite ~ df-oi with abbre... |
| oieq1 9418 | Equality theorem for ordin... |
| oieq2 9419 | Equality theorem for ordin... |
| nfoi 9420 | Hypothesis builder for ord... |
| ordiso2 9421 | Generalize ~ ordiso to pro... |
| ordiso 9422 | Order-isomorphic ordinal n... |
| ordtypecbv 9423 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9424 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9425 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9426 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9427 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9428 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9429 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9430 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9431 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9432 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9433 | Lemma for ~ ordtype . Usi... |
| oi0 9434 | Definition of the ordinal ... |
| oicl 9435 | The order type of the well... |
| oif 9436 | The order isomorphism of t... |
| oiiso2 9437 | The order isomorphism of t... |
| ordtype 9438 | For any set-like well-orde... |
| oiiniseg 9439 | ` ran F ` is an initial se... |
| ordtype2 9440 | For any set-like well-orde... |
| oiexg 9441 | The order isomorphism on a... |
| oion 9442 | The order type of the well... |
| oiiso 9443 | The order isomorphism of t... |
| oien 9444 | The order type of a well-o... |
| oieu 9445 | Uniqueness of the unique o... |
| oismo 9446 | When ` A ` is a subclass o... |
| oiid 9447 | The order type of an ordin... |
| hartogslem1 9448 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9449 | Lemma for ~ hartogs . (Co... |
| hartogs 9450 | The class of ordinals domi... |
| wofib 9451 | The only sets which are we... |
| wemaplem1 9452 | Value of the lexicographic... |
| wemaplem2 9453 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9454 | Lemma for ~ wemapso . Tra... |
| wemappo 9455 | Construct lexicographic or... |
| wemapsolem 9456 | Lemma for ~ wemapso . (Co... |
| wemapso 9457 | Construct lexicographic or... |
| wemapso2lem 9458 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9459 | An alternative to having a... |
| card2on 9460 | The alternate definition o... |
| card2inf 9461 | The alternate definition o... |
| harf 9464 | Functionality of the Harto... |
| harcl 9465 | Values of the Hartogs func... |
| harval 9466 | Function value of the Hart... |
| elharval 9467 | The Hartogs number of a se... |
| harndom 9468 | The Hartogs number of a se... |
| harword 9469 | Weak ordering property of ... |
| relwdom 9472 | Weak dominance is a relati... |
| brwdom 9473 | Property of weak dominance... |
| brwdomi 9474 | Property of weak dominance... |
| brwdomn0 9475 | Weak dominance over nonemp... |
| 0wdom 9476 | Any set weakly dominates t... |
| fowdom 9477 | An onto function implies w... |
| wdomref 9478 | Reflexivity of weak domina... |
| brwdom2 9479 | Alternate characterization... |
| domwdom 9480 | Weak dominance is implied ... |
| wdomtr 9481 | Transitivity of weak domin... |
| wdomen1 9482 | Equality-like theorem for ... |
| wdomen2 9483 | Equality-like theorem for ... |
| wdompwdom 9484 | Weak dominance strengthens... |
| canthwdom 9485 | Cantor's Theorem, stated u... |
| wdom2d 9486 | Deduce weak dominance from... |
| wdomd 9487 | Deduce weak dominance from... |
| brwdom3 9488 | Condition for weak dominan... |
| brwdom3i 9489 | Weak dominance implies exi... |
| unwdomg 9490 | Weak dominance of a (disjo... |
| xpwdomg 9491 | Weak dominance of a Cartes... |
| wdomima2g 9492 | A set is weakly dominant o... |
| wdomimag 9493 | A set is weakly dominant o... |
| unxpwdom2 9494 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9495 | If a Cartesian product is ... |
| ixpiunwdom 9496 | Describe an onto function ... |
| harwdom 9497 | The value of the Hartogs f... |
| axreg2 9499 | Axiom of Regularity expres... |
| zfregcl 9500 | The Axiom of Regularity wi... |
| zfregclOLD 9501 | Obsolete version of ~ zfre... |
| zfreg 9502 | The Axiom of Regularity us... |
| elirrv 9503 | The membership relation is... |
| elirrvOLD 9504 | Obsolete version of ~ elir... |
| elirr 9505 | No class is a member of it... |
| elneq 9506 | A class is not equal to an... |
| nelaneq 9507 | A class is not an element ... |
| nelaneqOLD 9508 | Obsolete version of ~ nela... |
| epinid0 9509 | The membership relation an... |
| sucprcreg 9510 | A class is equal to its su... |
| ruv 9511 | The Russell class is equal... |
| ruALT 9512 | Alternate proof of ~ ru , ... |
| disjcsn 9513 | A class is disjoint from i... |
| zfregfr 9514 | The membership relation is... |
| elirrvALT 9515 | Alternate proof of ~ elirr... |
| en2lp 9516 | No class has 2-cycle membe... |
| elnanel 9517 | Two classes are not elemen... |
| cnvepnep 9518 | The membership (epsilon) r... |
| epnsym 9519 | The membership (epsilon) r... |
| elnotel 9520 | A class cannot be an eleme... |
| elnel 9521 | A class cannot be an eleme... |
| en3lplem1 9522 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9523 | Lemma for ~ en3lp . (Cont... |
| en3lp 9524 | No class has 3-cycle membe... |
| preleqg 9525 | Equality of two unordered ... |
| preleq 9526 | Equality of two unordered ... |
| preleqALT 9527 | Alternate proof of ~ prele... |
| opthreg 9528 | Theorem for alternate repr... |
| suc11reg 9529 | The successor operation be... |
| dford2 9530 | Assuming ~ ax-reg , an ord... |
| inf0 9531 | Existence of ` _om ` impli... |
| inf1 9532 | Variation of Axiom of Infi... |
| inf2 9533 | Variation of Axiom of Infi... |
| inf3lema 9534 | Lemma for our Axiom of Inf... |
| inf3lemb 9535 | Lemma for our Axiom of Inf... |
| inf3lemc 9536 | Lemma for our Axiom of Inf... |
| inf3lemd 9537 | Lemma for our Axiom of Inf... |
| inf3lem1 9538 | Lemma for our Axiom of Inf... |
| inf3lem2 9539 | Lemma for our Axiom of Inf... |
| inf3lem3 9540 | Lemma for our Axiom of Inf... |
| inf3lem4 9541 | Lemma for our Axiom of Inf... |
| inf3lem5 9542 | Lemma for our Axiom of Inf... |
| inf3lem6 9543 | Lemma for our Axiom of Inf... |
| inf3lem7 9544 | Lemma for our Axiom of Inf... |
| inf3 9545 | Our Axiom of Infinity ~ ax... |
| infeq5i 9546 | Half of ~ infeq5 . (Contr... |
| infeq5 9547 | The statement "there exist... |
| zfinf 9549 | Axiom of Infinity expresse... |
| axinf2 9550 | A standard version of Axio... |
| zfinf2 9552 | A standard version of the ... |
| omex 9553 | The existence of omega (th... |
| axinf 9554 | The first version of the A... |
| inf5 9555 | The statement "there exist... |
| omelon 9556 | Omega is an ordinal number... |
| dfom3 9557 | The class of natural numbe... |
| elom3 9558 | A simplification of ~ elom... |
| dfom4 9559 | A simplification of ~ df-o... |
| dfom5 9560 | ` _om ` is the smallest li... |
| oancom 9561 | Ordinal addition is not co... |
| isfinite 9562 | A set is finite iff it is ... |
| fict 9563 | A finite set is countable ... |
| nnsdom 9564 | A natural number is strict... |
| omenps 9565 | Omega is equinumerous to a... |
| omensuc 9566 | The set of natural numbers... |
| infdifsn 9567 | Removing a singleton from ... |
| infdiffi 9568 | Removing a finite set from... |
| unbnn3 9569 | Any unbounded subset of na... |
| noinfep 9570 | Using the Axiom of Regular... |
| cantnffval 9573 | The value of the Cantor no... |
| cantnfdm 9574 | The domain of the Cantor n... |
| cantnfvalf 9575 | Lemma for ~ cantnf . The ... |
| cantnfs 9576 | Elementhood in the set of ... |
| cantnfcl 9577 | Basic properties of the or... |
| cantnfval 9578 | The value of the Cantor no... |
| cantnfval2 9579 | Alternate expression for t... |
| cantnfsuc 9580 | The value of the recursive... |
| cantnfle 9581 | A lower bound on the ` CNF... |
| cantnflt 9582 | An upper bound on the part... |
| cantnflt2 9583 | An upper bound on the ` CN... |
| cantnff 9584 | The ` CNF ` function is a ... |
| cantnf0 9585 | The value of the zero func... |
| cantnfrescl 9586 | A function is finitely sup... |
| cantnfres 9587 | The ` CNF ` function respe... |
| cantnfp1lem1 9588 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9589 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9590 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9591 | If ` F ` is created by add... |
| oemapso 9592 | The relation ` T ` is a st... |
| oemapval 9593 | Value of the relation ` T ... |
| oemapvali 9594 | If ` F < G ` , then there ... |
| cantnflem1a 9595 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9596 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9597 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9598 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9599 | Lemma for ~ cantnf . This... |
| cantnflem2 9600 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9601 | Lemma for ~ cantnf . Here... |
| cantnflem4 9602 | Lemma for ~ cantnf . Comp... |
| cantnf 9603 | The Cantor Normal Form the... |
| oemapwe 9604 | The lexicographic order on... |
| cantnffval2 9605 | An alternate definition of... |
| cantnff1o 9606 | Simplify the isomorphism o... |
| wemapwe 9607 | Construct lexicographic or... |
| oef1o 9608 | A bijection of the base se... |
| cnfcomlem 9609 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9610 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9611 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9612 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9613 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9614 | Any infinite ordinal ` B `... |
| cnfcom3clem 9615 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9616 | Wrap the construction of ~... |
| ttrcleq 9619 | Equality theorem for trans... |
| nfttrcld 9620 | Bound variable hypothesis ... |
| nfttrcl 9621 | Bound variable hypothesis ... |
| relttrcl 9622 | The transitive closure of ... |
| brttrcl 9623 | Characterization of elemen... |
| brttrcl2 9624 | Characterization of elemen... |
| ssttrcl 9625 | If ` R ` is a relation, th... |
| ttrcltr 9626 | The transitive closure of ... |
| ttrclresv 9627 | The transitive closure of ... |
| ttrclco 9628 | Composition law for the tr... |
| cottrcl 9629 | Composition law for the tr... |
| ttrclss 9630 | If ` R ` is a subclass of ... |
| dmttrcl 9631 | The domain of a transitive... |
| rnttrcl 9632 | The range of a transitive ... |
| ttrclexg 9633 | If ` R ` is a set, then so... |
| dfttrcl2 9634 | When ` R ` is a set and a ... |
| ttrclselem1 9635 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9636 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9637 | If ` R ` is set-like over ... |
| trcl 9638 | For any set ` A ` , show t... |
| tz9.1 9639 | Every set has a transitive... |
| tz9.1c 9640 | Alternate expression for t... |
| epfrs 9641 | The strong form of the Axi... |
| zfregs 9642 | The strong form of the Axi... |
| zfregs2 9643 | Alternate strong form of t... |
| tcvalg 9646 | Value of the transitive cl... |
| tcid 9647 | Defining property of the t... |
| tctr 9648 | Defining property of the t... |
| tcmin 9649 | Defining property of the t... |
| tc2 9650 | A variant of the definitio... |
| tcsni 9651 | The transitive closure of ... |
| tcss 9652 | The transitive closure fun... |
| tcel 9653 | The transitive closure fun... |
| tcidm 9654 | The transitive closure fun... |
| tc0 9655 | The transitive closure of ... |
| tc00 9656 | The transitive closure is ... |
| setind 9657 | Set (epsilon) induction. ... |
| setind2 9658 | Set (epsilon) induction, s... |
| setinds 9659 | Principle of set induction... |
| setinds2f 9660 | ` _E ` induction schema, u... |
| setinds2 9661 | ` _E ` induction schema, u... |
| frmin 9662 | Every (possibly proper) su... |
| frind 9663 | A subclass of a well-found... |
| frinsg 9664 | Well-Founded Induction Sch... |
| frins 9665 | Well-Founded Induction Sch... |
| frins2f 9666 | Well-Founded Induction sch... |
| frins2 9667 | Well-Founded Induction sch... |
| frins3 9668 | Well-Founded Induction sch... |
| frr3g 9669 | Functions defined by well-... |
| frrlem15 9670 | Lemma for general well-fou... |
| frrlem16 9671 | Lemma for general well-fou... |
| frr1 9672 | Law of general well-founde... |
| frr2 9673 | Law of general well-founde... |
| frr3 9674 | Law of general well-founde... |
| r1funlim 9679 | The cumulative hierarchy o... |
| r1fnon 9680 | The cumulative hierarchy o... |
| r10 9681 | Value of the cumulative hi... |
| r1sucg 9682 | Value of the cumulative hi... |
| r1suc 9683 | Value of the cumulative hi... |
| r1limg 9684 | Value of the cumulative hi... |
| r1lim 9685 | Value of the cumulative hi... |
| r1fin 9686 | The first ` _om ` levels o... |
| r1sdom 9687 | Each stage in the cumulati... |
| r111 9688 | The cumulative hierarchy i... |
| r1tr 9689 | The cumulative hierarchy o... |
| r1tr2 9690 | The union of a cumulative ... |
| r1ordg 9691 | Ordering relation for the ... |
| r1ord3g 9692 | Ordering relation for the ... |
| r1ord 9693 | Ordering relation for the ... |
| r1ord2 9694 | Ordering relation for the ... |
| r1ord3 9695 | Ordering relation for the ... |
| r1sssuc 9696 | The value of the cumulativ... |
| r1pwss 9697 | Each set of the cumulative... |
| r1sscl 9698 | Each set of the cumulative... |
| r1val1 9699 | The value of the cumulativ... |
| tz9.12lem1 9700 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9701 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9702 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9703 | A set is well-founded if a... |
| tz9.13 9704 | Every set is well-founded,... |
| tz9.13g 9705 | Every set is well-founded,... |
| rankwflemb 9706 | Two ways of saying a set i... |
| rankf 9707 | The domain and codomain of... |
| rankon 9708 | The rank of a set is an or... |
| r1elwf 9709 | Any member of the cumulati... |
| rankvalb 9710 | Value of the rank function... |
| rankr1ai 9711 | One direction of ~ rankr1a... |
| rankvaln 9712 | Value of the rank function... |
| rankidb 9713 | Identity law for the rank ... |
| rankdmr1 9714 | A rank is a member of the ... |
| rankr1ag 9715 | A version of ~ rankr1a tha... |
| rankr1bg 9716 | A relationship between ran... |
| r1rankidb 9717 | Any set is a subset of the... |
| r1elssi 9718 | The range of the ` R1 ` fu... |
| r1elss 9719 | The range of the ` R1 ` fu... |
| pwwf 9720 | A power set is well-founde... |
| sswf 9721 | A subset of a well-founded... |
| snwf 9722 | A singleton is well-founde... |
| unwf 9723 | A binary union is well-fou... |
| prwf 9724 | An unordered pair is well-... |
| opwf 9725 | An ordered pair is well-fo... |
| unir1 9726 | The cumulative hierarchy o... |
| jech9.3 9727 | Every set belongs to some ... |
| rankwflem 9728 | Every set is well-founded,... |
| rankval 9729 | Value of the rank function... |
| rankvalg 9730 | Value of the rank function... |
| rankval2 9731 | Value of an alternate defi... |
| uniwf 9732 | A union is well-founded if... |
| rankr1clem 9733 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9734 | A relationship between the... |
| rankidn 9735 | A relationship between the... |
| rankpwi 9736 | The rank of a power set. ... |
| rankelb 9737 | The membership relation is... |
| wfelirr 9738 | A well-founded set is not ... |
| rankval3b 9739 | The value of the rank func... |
| ranksnb 9740 | The rank of a singleton. ... |
| rankonidlem 9741 | Lemma for ~ rankonid . (C... |
| rankonid 9742 | The rank of an ordinal num... |
| onwf 9743 | The ordinals are all well-... |
| onssr1 9744 | Initial segments of the or... |
| rankr1g 9745 | A relationship between the... |
| rankid 9746 | Identity law for the rank ... |
| rankr1 9747 | A relationship between the... |
| ssrankr1 9748 | A relationship between an ... |
| rankr1a 9749 | A relationship between ran... |
| r1val2 9750 | The value of the cumulativ... |
| r1val3 9751 | The value of the cumulativ... |
| rankel 9752 | The membership relation is... |
| rankval3 9753 | The value of the rank func... |
| bndrank 9754 | Any class whose elements h... |
| unbndrank 9755 | The elements of a proper c... |
| rankpw 9756 | The rank of a power set. ... |
| ranklim 9757 | The rank of a set belongs ... |
| r1pw 9758 | A stronger property of ` R... |
| r1pwALT 9759 | Alternate shorter proof of... |
| r1pwcl 9760 | The cumulative hierarchy o... |
| rankssb 9761 | The subset relation is inh... |
| rankss 9762 | The subset relation is inh... |
| rankunb 9763 | The rank of the union of t... |
| rankprb 9764 | The rank of an unordered p... |
| rankopb 9765 | The rank of an ordered pai... |
| rankuni2b 9766 | The value of the rank func... |
| ranksn 9767 | The rank of a singleton. ... |
| rankuni2 9768 | The rank of a union. Part... |
| rankun 9769 | The rank of the union of t... |
| rankpr 9770 | The rank of an unordered p... |
| rankop 9771 | The rank of an ordered pai... |
| r1rankid 9772 | Any set is a subset of the... |
| rankeq0b 9773 | A set is empty iff its ran... |
| rankeq0 9774 | A set is empty iff its ran... |
| rankr1id 9775 | The rank of the hierarchy ... |
| rankuni 9776 | The rank of a union. Part... |
| rankr1b 9777 | A relationship between ran... |
| ranksuc 9778 | The rank of a successor. ... |
| rankuniss 9779 | Upper bound of the rank of... |
| rankval4 9780 | The rank of a set is the s... |
| rankbnd 9781 | The rank of a set is bound... |
| rankbnd2 9782 | The rank of a set is bound... |
| rankc1 9783 | A relationship that can be... |
| rankc2 9784 | A relationship that can be... |
| rankelun 9785 | Rank membership is inherit... |
| rankelpr 9786 | Rank membership is inherit... |
| rankelop 9787 | Rank membership is inherit... |
| rankxpl 9788 | A lower bound on the rank ... |
| rankxpu 9789 | An upper bound on the rank... |
| rankfu 9790 | An upper bound on the rank... |
| rankmapu 9791 | An upper bound on the rank... |
| rankxplim 9792 | The rank of a Cartesian pr... |
| rankxplim2 9793 | If the rank of a Cartesian... |
| rankxplim3 9794 | The rank of a Cartesian pr... |
| rankxpsuc 9795 | The rank of a Cartesian pr... |
| tcwf 9796 | The transitive closure fun... |
| tcrank 9797 | This theorem expresses two... |
| scottex 9798 | Scott's trick collects all... |
| scott0 9799 | Scott's trick collects all... |
| scottexs 9800 | Theorem scheme version of ... |
| scott0s 9801 | Theorem scheme version of ... |
| cplem1 9802 | Lemma for the Collection P... |
| cplem2 9803 | Lemma for the Collection P... |
| cp 9804 | Collection Principle. Thi... |
| bnd 9805 | A very strong generalizati... |
| bnd2 9806 | A variant of the Boundedne... |
| kardex 9807 | The collection of all sets... |
| karden 9808 | If we allow the Axiom of R... |
| htalem 9809 | Lemma for defining an emul... |
| hta 9810 | A ZFC emulation of Hilbert... |
| djueq12 9817 | Equality theorem for disjo... |
| djueq1 9818 | Equality theorem for disjo... |
| djueq2 9819 | Equality theorem for disjo... |
| nfdju 9820 | Bound-variable hypothesis ... |
| djuex 9821 | The disjoint union of sets... |
| djuexb 9822 | The disjoint union of two ... |
| djulcl 9823 | Left closure of disjoint u... |
| djurcl 9824 | Right closure of disjoint ... |
| djulf1o 9825 | The left injection functio... |
| djurf1o 9826 | The right injection functi... |
| inlresf 9827 | The left injection restric... |
| inlresf1 9828 | The left injection restric... |
| inrresf 9829 | The right injection restri... |
| inrresf1 9830 | The right injection restri... |
| djuin 9831 | The images of any classes ... |
| djur 9832 | A member of a disjoint uni... |
| djuss 9833 | A disjoint union is a subc... |
| djuunxp 9834 | The union of a disjoint un... |
| djuexALT 9835 | Alternate proof of ~ djuex... |
| eldju1st 9836 | The first component of an ... |
| eldju2ndl 9837 | The second component of an... |
| eldju2ndr 9838 | The second component of an... |
| djuun 9839 | The disjoint union of two ... |
| 1stinl 9840 | The first component of the... |
| 2ndinl 9841 | The second component of th... |
| 1stinr 9842 | The first component of the... |
| 2ndinr 9843 | The second component of th... |
| updjudhf 9844 | The mapping of an element ... |
| updjudhcoinlf 9845 | The composition of the map... |
| updjudhcoinrg 9846 | The composition of the map... |
| updjud 9847 | Universal property of the ... |
| cardf2 9856 | The cardinality function i... |
| cardon 9857 | The cardinal number of a s... |
| isnum2 9858 | A way to express well-orde... |
| isnumi 9859 | A set equinumerous to an o... |
| ennum 9860 | Equinumerous sets are equi... |
| finnum 9861 | Every finite set is numera... |
| onenon 9862 | Every ordinal number is nu... |
| tskwe 9863 | A Tarski set is well-order... |
| xpnum 9864 | The cartesian product of n... |
| cardval3 9865 | An alternate definition of... |
| cardid2 9866 | Any numerable set is equin... |
| isnum3 9867 | A set is numerable iff it ... |
| oncardval 9868 | The value of the cardinal ... |
| oncardid 9869 | Any ordinal number is equi... |
| cardonle 9870 | The cardinal of an ordinal... |
| card0 9871 | The cardinality of the emp... |
| cardidm 9872 | The cardinality function i... |
| oncard 9873 | A set is a cardinal number... |
| ficardom 9874 | The cardinal number of a f... |
| ficardid 9875 | A finite set is equinumero... |
| cardnn 9876 | The cardinality of a natur... |
| cardnueq0 9877 | The empty set is the only ... |
| cardne 9878 | No member of a cardinal nu... |
| carden2a 9879 | If two sets have equal non... |
| carden2b 9880 | If two sets are equinumero... |
| card1 9881 | A set has cardinality one ... |
| cardsn 9882 | A singleton has cardinalit... |
| carddomi2 9883 | Two sets have the dominanc... |
| sdomsdomcardi 9884 | A set strictly dominates i... |
| cardlim 9885 | An infinite cardinal is a ... |
| cardsdomelir 9886 | A cardinal strictly domina... |
| cardsdomel 9887 | A cardinal strictly domina... |
| iscard 9888 | Two ways to express the pr... |
| iscard2 9889 | Two ways to express the pr... |
| carddom2 9890 | Two numerable sets have th... |
| harcard 9891 | The class of ordinal numbe... |
| cardprclem 9892 | Lemma for ~ cardprc . (Co... |
| cardprc 9893 | The class of all cardinal ... |
| carduni 9894 | The union of a set of card... |
| cardiun 9895 | The indexed union of a set... |
| cardennn 9896 | If ` A ` is equinumerous t... |
| cardsucinf 9897 | The cardinality of the suc... |
| cardsucnn 9898 | The cardinality of the suc... |
| cardom 9899 | The set of natural numbers... |
| carden2 9900 | Two numerable sets are equ... |
| cardsdom2 9901 | A numerable set is strictl... |
| domtri2 9902 | Trichotomy of dominance fo... |
| nnsdomel 9903 | Strict dominance and eleme... |
| cardval2 9904 | An alternate version of th... |
| isinffi 9905 | An infinite set contains s... |
| fidomtri 9906 | Trichotomy of dominance wi... |
| fidomtri2 9907 | Trichotomy of dominance wi... |
| harsdom 9908 | The Hartogs number of a we... |
| onsdom 9909 | Any well-orderable set is ... |
| harval2 9910 | An alternate expression fo... |
| harsucnn 9911 | The next cardinal after a ... |
| cardmin2 9912 | The smallest ordinal that ... |
| pm54.43lem 9913 | In Theorem *54.43 of [Whit... |
| pm54.43 9914 | Theorem *54.43 of [Whitehe... |
| enpr2 9915 | An unordered pair with dis... |
| pr2ne 9916 | If an unordered pair has t... |
| prdom2 9917 | An unordered pair has at m... |
| en2eqpr 9918 | Building a set with two el... |
| en2eleq 9919 | Express a set of pair card... |
| en2other2 9920 | Taking the other element t... |
| dif1card 9921 | The cardinality of a nonem... |
| leweon 9922 | Lexicographical order is a... |
| r0weon 9923 | A set-like well-ordering o... |
| infxpenlem 9924 | Lemma for ~ infxpen . (Co... |
| infxpen 9925 | Every infinite ordinal is ... |
| xpomen 9926 | The Cartesian product of o... |
| xpct 9927 | The cartesian product of t... |
| infxpidm2 9928 | Every infinite well-ordera... |
| infxpenc 9929 | A canonical version of ~ i... |
| infxpenc2lem1 9930 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9931 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9932 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9933 | Existence form of ~ infxpe... |
| iunmapdisj 9934 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9935 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9936 | Lemma for ~ fseqen . (Con... |
| fseqdom 9937 | One half of ~ fseqen . (C... |
| fseqen 9938 | A set that is equinumerous... |
| infpwfidom 9939 | The collection of finite s... |
| dfac8alem 9940 | Lemma for ~ dfac8a . If t... |
| dfac8a 9941 | Numeration theorem: every ... |
| dfac8b 9942 | The well-ordering theorem:... |
| dfac8clem 9943 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9944 | If the union of a set is w... |
| ac10ct 9945 | A proof of the well-orderi... |
| ween 9946 | A set is numerable iff it ... |
| ac5num 9947 | A version of ~ ac5b with t... |
| ondomen 9948 | If a set is dominated by a... |
| numdom 9949 | A set dominated by a numer... |
| ssnum 9950 | A subset of a numerable se... |
| onssnum 9951 | All subsets of the ordinal... |
| indcardi 9952 | Indirect strong induction ... |
| acnrcl 9953 | Reverse closure for the ch... |
| acneq 9954 | Equality theorem for the c... |
| isacn 9955 | The property of being a ch... |
| acni 9956 | The property of being a ch... |
| acni2 9957 | The property of being a ch... |
| acni3 9958 | The property of being a ch... |
| acnlem 9959 | Construct a mapping satisf... |
| numacn 9960 | A well-orderable set has c... |
| finacn 9961 | Every set has finite choic... |
| acndom 9962 | A set with long choice seq... |
| acnnum 9963 | A set ` X ` which has choi... |
| acnen 9964 | The class of choice sets o... |
| acndom2 9965 | A set smaller than one wit... |
| acnen2 9966 | The class of sets with cho... |
| fodomacn 9967 | A version of ~ fodom that ... |
| fodomnum 9968 | A version of ~ fodom that ... |
| fonum 9969 | A surjection maps numerabl... |
| numwdom 9970 | A surjection maps numerabl... |
| fodomfi2 9971 | Onto functions define domi... |
| wdomfil 9972 | Weak dominance agrees with... |
| infpwfien 9973 | Any infinite well-orderabl... |
| inffien 9974 | The set of finite intersec... |
| wdomnumr 9975 | Weak dominance agrees with... |
| alephfnon 9976 | The aleph function is a fu... |
| aleph0 9977 | The first infinite cardina... |
| alephlim 9978 | Value of the aleph functio... |
| alephsuc 9979 | Value of the aleph functio... |
| alephon 9980 | An aleph is an ordinal num... |
| alephcard 9981 | Every aleph is a cardinal ... |
| alephnbtwn 9982 | No cardinal can be sandwic... |
| alephnbtwn2 9983 | No set has equinumerosity ... |
| alephordilem1 9984 | Lemma for ~ alephordi . (... |
| alephordi 9985 | Strict ordering property o... |
| alephord 9986 | Ordering property of the a... |
| alephord2 9987 | Ordering property of the a... |
| alephord2i 9988 | Ordering property of the a... |
| alephord3 9989 | Ordering property of the a... |
| alephsucdom 9990 | A set dominated by an alep... |
| alephsuc2 9991 | An alternate representatio... |
| alephdom 9992 | Relationship between inclu... |
| alephgeom 9993 | Every aleph is greater tha... |
| alephislim 9994 | Every aleph is a limit ord... |
| aleph11 9995 | The aleph function is one-... |
| alephf1 9996 | The aleph function is a on... |
| alephsdom 9997 | If an ordinal is smaller t... |
| alephdom2 9998 | A dominated initial ordina... |
| alephle 9999 | The argument of the aleph ... |
| cardaleph 10000 | Given any transfinite card... |
| cardalephex 10001 | Every transfinite cardinal... |
| infenaleph 10002 | An infinite numerable set ... |
| isinfcard 10003 | Two ways to express the pr... |
| iscard3 10004 | Two ways to express the pr... |
| cardnum 10005 | Two ways to express the cl... |
| alephinit 10006 | An infinite initial ordina... |
| carduniima 10007 | The union of the image of ... |
| cardinfima 10008 | If a mapping to cardinals ... |
| alephiso 10009 | Aleph is an order isomorph... |
| alephprc 10010 | The class of all transfini... |
| alephsson 10011 | The class of transfinite c... |
| unialeph 10012 | The union of the class of ... |
| alephsmo 10013 | The aleph function is stri... |
| alephf1ALT 10014 | Alternate proof of ~ aleph... |
| alephfplem1 10015 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10016 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10017 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10018 | Lemma for ~ alephfp . (Co... |
| alephfp 10019 | The aleph function has a f... |
| alephfp2 10020 | The aleph function has at ... |
| alephval3 10021 | An alternate way to expres... |
| alephsucpw2 10022 | The power set of an aleph ... |
| mappwen 10023 | Power rule for cardinal ar... |
| finnisoeu 10024 | A finite totally ordered s... |
| iunfictbso 10025 | Countability of a countabl... |
| aceq1 10028 | Equivalence of two version... |
| aceq0 10029 | Equivalence of two version... |
| aceq2 10030 | Equivalence of two version... |
| aceq3lem 10031 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10032 | Equivalence of two version... |
| dfac4 10033 | Equivalence of two version... |
| dfac5lem1 10034 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10035 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10036 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10037 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10038 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4OLD 10039 | Obsolete version of ~ dfac... |
| dfac5 10040 | Equivalence of two version... |
| dfac2a 10041 | Our Axiom of Choice (in th... |
| dfac2b 10042 | Axiom of Choice (first for... |
| dfac2 10043 | Axiom of Choice (first for... |
| dfac7 10044 | Equivalence of the Axiom o... |
| dfac0 10045 | Equivalence of two version... |
| dfac1 10046 | Equivalence of two version... |
| dfac8 10047 | A proof of the equivalency... |
| dfac9 10048 | Equivalence of the axiom o... |
| dfac10 10049 | Axiom of Choice equivalent... |
| dfac10c 10050 | Axiom of Choice equivalent... |
| dfac10b 10051 | Axiom of Choice equivalent... |
| acacni 10052 | A choice equivalent: every... |
| dfacacn 10053 | A choice equivalent: every... |
| dfac13 10054 | The axiom of choice holds ... |
| dfac12lem1 10055 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10056 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10057 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10058 | The axiom of choice holds ... |
| dfac12k 10059 | Equivalence of ~ dfac12 an... |
| dfac12a 10060 | The axiom of choice holds ... |
| dfac12 10061 | The axiom of choice holds ... |
| kmlem1 10062 | Lemma for 5-quantifier AC ... |
| kmlem2 10063 | Lemma for 5-quantifier AC ... |
| kmlem3 10064 | Lemma for 5-quantifier AC ... |
| kmlem4 10065 | Lemma for 5-quantifier AC ... |
| kmlem5 10066 | Lemma for 5-quantifier AC ... |
| kmlem6 10067 | Lemma for 5-quantifier AC ... |
| kmlem7 10068 | Lemma for 5-quantifier AC ... |
| kmlem8 10069 | Lemma for 5-quantifier AC ... |
| kmlem9 10070 | Lemma for 5-quantifier AC ... |
| kmlem10 10071 | Lemma for 5-quantifier AC ... |
| kmlem11 10072 | Lemma for 5-quantifier AC ... |
| kmlem12 10073 | Lemma for 5-quantifier AC ... |
| kmlem13 10074 | Lemma for 5-quantifier AC ... |
| kmlem14 10075 | Lemma for 5-quantifier AC ... |
| kmlem15 10076 | Lemma for 5-quantifier AC ... |
| kmlem16 10077 | Lemma for 5-quantifier AC ... |
| dfackm 10078 | Equivalence of the Axiom o... |
| undjudom 10079 | Cardinal addition dominate... |
| endjudisj 10080 | Equinumerosity of a disjoi... |
| djuen 10081 | Disjoint unions of equinum... |
| djuenun 10082 | Disjoint union is equinume... |
| dju1en 10083 | Cardinal addition with car... |
| dju1dif 10084 | Adding and subtracting one... |
| dju1p1e2 10085 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10086 | Alternate proof of ~ dju1p... |
| dju0en 10087 | Cardinal addition with car... |
| xp2dju 10088 | Two times a cardinal numbe... |
| djucomen 10089 | Commutative law for cardin... |
| djuassen 10090 | Associative law for cardin... |
| xpdjuen 10091 | Cardinal multiplication di... |
| mapdjuen 10092 | Sum of exponents law for c... |
| pwdjuen 10093 | Sum of exponents law for c... |
| djudom1 10094 | Ordering law for cardinal ... |
| djudom2 10095 | Ordering law for cardinal ... |
| djudoml 10096 | A set is dominated by its ... |
| djuxpdom 10097 | Cartesian product dominate... |
| djufi 10098 | The disjoint union of two ... |
| cdainflem 10099 | Any partition of omega int... |
| djuinf 10100 | A set is infinite iff the ... |
| infdju1 10101 | An infinite set is equinum... |
| pwdju1 10102 | The sum of a powerset with... |
| pwdjuidm 10103 | If the natural numbers inj... |
| djulepw 10104 | If ` A ` is idempotent und... |
| onadju 10105 | The cardinal and ordinal s... |
| cardadju 10106 | The cardinal sum is equinu... |
| djunum 10107 | The disjoint union of two ... |
| unnum 10108 | The union of two numerable... |
| nnadju 10109 | The cardinal and ordinal s... |
| nnadjuALT 10110 | Shorter proof of ~ nnadju ... |
| ficardadju 10111 | The disjoint union of fini... |
| ficardun 10112 | The cardinality of the uni... |
| ficardun2 10113 | The cardinality of the uni... |
| pwsdompw 10114 | Lemma for ~ domtriom . Th... |
| unctb 10115 | The union of two countable... |
| infdjuabs 10116 | Absorption law for additio... |
| infunabs 10117 | An infinite set is equinum... |
| infdju 10118 | The sum of two cardinal nu... |
| infdif 10119 | The cardinality of an infi... |
| infdif2 10120 | Cardinality ordering for a... |
| infxpdom 10121 | Dominance law for multipli... |
| infxpabs 10122 | Absorption law for multipl... |
| infunsdom1 10123 | The union of two sets that... |
| infunsdom 10124 | The union of two sets that... |
| infxp 10125 | Absorption law for multipl... |
| pwdjudom 10126 | A property of dominance ov... |
| infpss 10127 | Every infinite set has an ... |
| infmap2 10128 | An exponentiation law for ... |
| ackbij2lem1 10129 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10130 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10131 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10132 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10133 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10134 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10135 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10136 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10137 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10138 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10139 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10140 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10141 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10142 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10143 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10144 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10145 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10146 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10147 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10148 | The Ackermann bijection, p... |
| ackbij1b 10149 | The Ackermann bijection, p... |
| ackbij2lem2 10150 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10151 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10152 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10153 | The Ackermann bijection, p... |
| r1om 10154 | The set of hereditarily fi... |
| fictb 10155 | A set is countable iff its... |
| cflem 10156 | A lemma used to simplify c... |
| cflemOLD 10157 | Obsolete version of ~ cfle... |
| cfval 10158 | Value of the cofinality fu... |
| cff 10159 | Cofinality is a function o... |
| cfub 10160 | An upper bound on cofinali... |
| cflm 10161 | Value of the cofinality fu... |
| cf0 10162 | Value of the cofinality fu... |
| cardcf 10163 | Cofinality is a cardinal n... |
| cflecard 10164 | Cofinality is bounded by t... |
| cfle 10165 | Cofinality is bounded by i... |
| cfon 10166 | The cofinality of any set ... |
| cfeq0 10167 | Only the ordinal zero has ... |
| cfsuc 10168 | Value of the cofinality fu... |
| cff1 10169 | There is always a map from... |
| cfflb 10170 | If there is a cofinal map ... |
| cfval2 10171 | Another expression for the... |
| coflim 10172 | A simpler expression for t... |
| cflim3 10173 | Another expression for the... |
| cflim2 10174 | The cofinality function is... |
| cfom 10175 | Value of the cofinality fu... |
| cfss 10176 | There is a cofinal subset ... |
| cfslb 10177 | Any cofinal subset of ` A ... |
| cfslbn 10178 | Any subset of ` A ` smalle... |
| cfslb2n 10179 | Any small collection of sm... |
| cofsmo 10180 | Any cofinal map implies th... |
| cfsmolem 10181 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10182 | The map in ~ cff1 can be a... |
| cfcoflem 10183 | Lemma for ~ cfcof , showin... |
| coftr 10184 | If there is a cofinal map ... |
| cfcof 10185 | If there is a cofinal map ... |
| cfidm 10186 | The cofinality function is... |
| alephsing 10187 | The cofinality of a limit ... |
| sornom 10188 | The range of a single-step... |
| isfin1a 10203 | Definition of a Ia-finite ... |
| fin1ai 10204 | Property of a Ia-finite se... |
| isfin2 10205 | Definition of a II-finite ... |
| fin2i 10206 | Property of a II-finite se... |
| isfin3 10207 | Definition of a III-finite... |
| isfin4 10208 | Definition of a IV-finite ... |
| fin4i 10209 | Infer that a set is IV-inf... |
| isfin5 10210 | Definition of a V-finite s... |
| isfin6 10211 | Definition of a VI-finite ... |
| isfin7 10212 | Definition of a VII-finite... |
| sdom2en01 10213 | A set with less than two e... |
| infpssrlem1 10214 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10215 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10216 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10217 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10218 | Lemma for ~ infpssr . (Co... |
| infpssr 10219 | Dedekind infinity implies ... |
| fin4en1 10220 | Dedekind finite is a cardi... |
| ssfin4 10221 | Dedekind finite sets have ... |
| domfin4 10222 | A set dominated by a Dedek... |
| ominf4 10223 | ` _om ` is Dedekind infini... |
| infpssALT 10224 | Alternate proof of ~ infps... |
| isfin4-2 10225 | Alternate definition of IV... |
| isfin4p1 10226 | Alternate definition of IV... |
| fin23lem7 10227 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10228 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10229 | A II-finite set contains m... |
| isfin2-2 10230 | ` Fin2 ` expressed in term... |
| ssfin2 10231 | A subset of a II-finite se... |
| enfin2i 10232 | II-finiteness is a cardina... |
| fin23lem24 10233 | Lemma for ~ fin23 . In a ... |
| fincssdom 10234 | In a chain of finite sets,... |
| fin23lem25 10235 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10236 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10237 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10238 | Lemma for ~ fin23 but coul... |
| fin23lem27 10239 | The mapping constructed in... |
| isfin3ds 10240 | Property of a III-finite s... |
| ssfin3ds 10241 | A subset of a III-finite s... |
| fin23lem12 10242 | The beginning of the proof... |
| fin23lem13 10243 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10244 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10245 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10246 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10247 | Lemma for ~ fin23 . The f... |
| fin23lem20 10248 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10249 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10250 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10251 | Lemma for ~ fin23 . The r... |
| fin23lem29 10252 | Lemma for ~ fin23 . The r... |
| fin23lem30 10253 | Lemma for ~ fin23 . The r... |
| fin23lem31 10254 | Lemma for ~ fin23 . The r... |
| fin23lem32 10255 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10256 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10257 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10258 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10259 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10260 | Lemma for ~ fin23 . The c... |
| fin23lem39 10261 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10262 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10263 | Lemma for ~ fin23 . A set... |
| isf32lem1 10264 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10265 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10266 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10267 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10268 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10269 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10270 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10271 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10272 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10273 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10274 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10275 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10276 | One half of ~ isfin3-2 . ... |
| isf33lem 10277 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10278 | Weakly Dedekind-infinite s... |
| isfin3-3 10279 | Weakly Dedekind-infinite s... |
| fin33i 10280 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10281 | Lemma for ~ compsscnv . (... |
| compsscnv 10282 | Complementation on a power... |
| isf34lem1 10283 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10284 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10285 | Complementation is an anti... |
| isf34lem3 10286 | Lemma for ~ isfin3-4 . (C... |
| compss 10287 | Express image under of the... |
| isf34lem4 10288 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10289 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10290 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10291 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10292 | Inference from ~ isfin3-4 ... |
| isfin3-4 10293 | Weakly Dedekind-infinite s... |
| fin11a 10294 | Every I-finite set is Ia-f... |
| enfin1ai 10295 | Ia-finiteness is a cardina... |
| isfin1-2 10296 | A set is finite in the usu... |
| isfin1-3 10297 | A set is I-finite iff ever... |
| isfin1-4 10298 | A set is I-finite iff ever... |
| dffin1-5 10299 | Compact quantifier-free ve... |
| fin23 10300 | Every II-finite set (every... |
| fin34 10301 | Every III-finite set is IV... |
| isfin5-2 10302 | Alternate definition of V-... |
| fin45 10303 | Every IV-finite set is V-f... |
| fin56 10304 | Every V-finite set is VI-f... |
| fin17 10305 | Every I-finite set is VII-... |
| fin67 10306 | Every VI-finite set is VII... |
| isfin7-2 10307 | A set is VII-finite iff it... |
| fin71num 10308 | A well-orderable set is VI... |
| dffin7-2 10309 | Class form of ~ isfin7-2 .... |
| dfacfin7 10310 | Axiom of Choice equivalent... |
| fin1a2lem1 10311 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10312 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10313 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10314 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10315 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10316 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10317 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10318 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10319 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10320 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10321 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10322 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10323 | Lemma for ~ fin1a2 . (Con... |
| fin12 10324 | Weak theorem which skips I... |
| fin1a2s 10325 | An II-infinite set can hav... |
| fin1a2 10326 | Every Ia-finite set is II-... |
| itunifval 10327 | Function value of iterated... |
| itunifn 10328 | Functionality of the itera... |
| ituni0 10329 | A zero-fold iterated union... |
| itunisuc 10330 | Successor iterated union. ... |
| itunitc1 10331 | Each union iterate is a me... |
| itunitc 10332 | The union of all union ite... |
| ituniiun 10333 | Unwrap an iterated union f... |
| hsmexlem7 10334 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10335 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10336 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10337 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10338 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10339 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10340 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10341 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10342 | Lemma for ~ hsmex . (Cont... |
| hsmex 10343 | The collection of heredita... |
| hsmex2 10344 | The set of hereditary size... |
| hsmex3 10345 | The set of hereditary size... |
| axcc2lem 10347 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10348 | A possibly more useful ver... |
| axcc3 10349 | A possibly more useful ver... |
| axcc4 10350 | A version of ~ axcc3 that ... |
| acncc 10351 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10352 | Relax the constraint on ~ ... |
| domtriomlem 10353 | Lemma for ~ domtriom . (C... |
| domtriom 10354 | Trichotomy of equinumerosi... |
| fin41 10355 | Under countable choice, th... |
| dominf 10356 | A nonempty set that is a s... |
| dcomex 10358 | The Axiom of Dependent Cho... |
| axdc2lem 10359 | Lemma for ~ axdc2 . We co... |
| axdc2 10360 | An apparent strengthening ... |
| axdc3lem 10361 | The class ` S ` of finite ... |
| axdc3lem2 10362 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10363 | Simple substitution lemma ... |
| axdc3lem4 10364 | Lemma for ~ axdc3 . We ha... |
| axdc3 10365 | Dependent Choice. Axiom D... |
| axdc4lem 10366 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10367 | A more general version of ... |
| axcclem 10368 | Lemma for ~ axcc . (Contr... |
| axcc 10369 | Although CC can be proven ... |
| zfac 10371 | Axiom of Choice expressed ... |
| ac2 10372 | Axiom of Choice equivalent... |
| ac3 10373 | Axiom of Choice using abbr... |
| axac3 10375 | This theorem asserts that ... |
| ackm 10376 | A remarkable equivalent to... |
| axac2 10377 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10378 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10379 | Apply a choice equivalent.... |
| cardeqv 10380 | All sets are well-orderabl... |
| numth3 10381 | All sets are well-orderabl... |
| numth2 10382 | Numeration theorem: any se... |
| numth 10383 | Numeration theorem: every ... |
| ac7 10384 | An Axiom of Choice equival... |
| ac7g 10385 | An Axiom of Choice equival... |
| ac4 10386 | Equivalent of Axiom of Cho... |
| ac4c 10387 | Equivalent of Axiom of Cho... |
| ac5 10388 | An Axiom of Choice equival... |
| ac5b 10389 | Equivalent of Axiom of Cho... |
| ac6num 10390 | A version of ~ ac6 which t... |
| ac6 10391 | Equivalent of Axiom of Cho... |
| ac6c4 10392 | Equivalent of Axiom of Cho... |
| ac6c5 10393 | Equivalent of Axiom of Cho... |
| ac9 10394 | An Axiom of Choice equival... |
| ac6s 10395 | Equivalent of Axiom of Cho... |
| ac6n 10396 | Equivalent of Axiom of Cho... |
| ac6s2 10397 | Generalization of the Axio... |
| ac6s3 10398 | Generalization of the Axio... |
| ac6sg 10399 | ~ ac6s with sethood as ant... |
| ac6sf 10400 | Version of ~ ac6 with boun... |
| ac6s4 10401 | Generalization of the Axio... |
| ac6s5 10402 | Generalization of the Axio... |
| ac8 10403 | An Axiom of Choice equival... |
| ac9s 10404 | An Axiom of Choice equival... |
| numthcor 10405 | Any set is strictly domina... |
| weth 10406 | Well-ordering theorem: any... |
| zorn2lem1 10407 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10408 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10409 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10410 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10411 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10412 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10413 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10414 | Zorn's Lemma of [Monk1] p.... |
| zorng 10415 | Zorn's Lemma. If the unio... |
| zornn0g 10416 | Variant of Zorn's lemma ~ ... |
| zorn2 10417 | Zorn's Lemma of [Monk1] p.... |
| zorn 10418 | Zorn's Lemma. If the unio... |
| zornn0 10419 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10420 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10421 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10422 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10423 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10424 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10425 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10426 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10427 | The Teichmüller-Tukey... |
| ttukeyg 10428 | The Teichmüller-Tukey... |
| ttukey 10429 | The Teichmüller-Tukey... |
| axdclem 10430 | Lemma for ~ axdc . (Contr... |
| axdclem2 10431 | Lemma for ~ axdc . Using ... |
| axdc 10432 | This theorem derives ~ ax-... |
| fodomg 10433 | An onto function implies d... |
| fodom 10434 | An onto function implies d... |
| dmct 10435 | The domain of a countable ... |
| rnct 10436 | The range of a countable s... |
| fodomb 10437 | Equivalence of an onto map... |
| wdomac 10438 | When assuming AC, weak and... |
| brdom3 10439 | Equivalence to a dominance... |
| brdom5 10440 | An equivalence to a domina... |
| brdom4 10441 | An equivalence to a domina... |
| brdom7disj 10442 | An equivalence to a domina... |
| brdom6disj 10443 | An equivalence to a domina... |
| fin71ac 10444 | Once we allow AC, the "str... |
| imadomg 10445 | An image of a function und... |
| fimact 10446 | The image by a function of... |
| fnrndomg 10447 | The range of a function is... |
| fnct 10448 | If the domain of a functio... |
| mptct 10449 | A countable mapping set is... |
| iunfo 10450 | Existence of an onto funct... |
| iundom2g 10451 | An upper bound for the car... |
| iundomg 10452 | An upper bound for the car... |
| iundom 10453 | An upper bound for the car... |
| unidom 10454 | An upper bound for the car... |
| uniimadom 10455 | An upper bound for the car... |
| uniimadomf 10456 | An upper bound for the car... |
| cardval 10457 | The value of the cardinal ... |
| cardid 10458 | Any set is equinumerous to... |
| cardidg 10459 | Any set is equinumerous to... |
| cardidd 10460 | Any set is equinumerous to... |
| cardf 10461 | The cardinality function i... |
| carden 10462 | Two sets are equinumerous ... |
| cardeq0 10463 | Only the empty set has car... |
| unsnen 10464 | Equinumerosity of a set wi... |
| carddom 10465 | Two sets have the dominanc... |
| cardsdom 10466 | Two sets have the strict d... |
| domtri 10467 | Trichotomy law for dominan... |
| entric 10468 | Trichotomy of equinumerosi... |
| entri2 10469 | Trichotomy of dominance an... |
| entri3 10470 | Trichotomy of dominance. ... |
| sdomsdomcard 10471 | A set strictly dominates i... |
| canth3 10472 | Cantor's theorem in terms ... |
| infxpidm 10473 | Every infinite class is eq... |
| ondomon 10474 | The class of ordinals domi... |
| cardmin 10475 | The smallest ordinal that ... |
| ficard 10476 | A set is finite iff its ca... |
| infinfg 10477 | Equivalence between two in... |
| infinf 10478 | Equivalence between two in... |
| unirnfdomd 10479 | The union of the range of ... |
| konigthlem 10480 | Lemma for ~ konigth . (Co... |
| konigth 10481 | Konig's Theorem. If ` m (... |
| alephsucpw 10482 | The power set of an aleph ... |
| aleph1 10483 | The set exponentiation of ... |
| alephval2 10484 | An alternate way to expres... |
| dominfac 10485 | A nonempty set that is a s... |
| iunctb 10486 | The countable union of cou... |
| unictb 10487 | The countable union of cou... |
| infmap 10488 | An exponentiation law for ... |
| alephadd 10489 | The sum of two alephs is t... |
| alephmul 10490 | The product of two alephs ... |
| alephexp1 10491 | An exponentiation law for ... |
| alephsuc3 10492 | An alternate representatio... |
| alephexp2 10493 | An expression equinumerous... |
| alephreg 10494 | A successor aleph is regul... |
| pwcfsdom 10495 | A corollary of Konig's The... |
| cfpwsdom 10496 | A corollary of Konig's The... |
| alephom 10497 | From ~ canth2 , we know th... |
| smobeth 10498 | The beth function is stric... |
| nd1 10499 | A lemma for proving condit... |
| nd2 10500 | A lemma for proving condit... |
| nd3 10501 | A lemma for proving condit... |
| nd4 10502 | A lemma for proving condit... |
| axextnd 10503 | A version of the Axiom of ... |
| axrepndlem1 10504 | Lemma for the Axiom of Rep... |
| axrepndlem2 10505 | Lemma for the Axiom of Rep... |
| axrepnd 10506 | A version of the Axiom of ... |
| axunndlem1 10507 | Lemma for the Axiom of Uni... |
| axunnd 10508 | A version of the Axiom of ... |
| axpowndlem1 10509 | Lemma for the Axiom of Pow... |
| axpowndlem2 10510 | Lemma for the Axiom of Pow... |
| axpowndlem3 10511 | Lemma for the Axiom of Pow... |
| axpowndlem4 10512 | Lemma for the Axiom of Pow... |
| axpownd 10513 | A version of the Axiom of ... |
| axregndlem1 10514 | Lemma for the Axiom of Reg... |
| axregndlem2 10515 | Lemma for the Axiom of Reg... |
| axregnd 10516 | A version of the Axiom of ... |
| axinfndlem1 10517 | Lemma for the Axiom of Inf... |
| axinfnd 10518 | A version of the Axiom of ... |
| axacndlem1 10519 | Lemma for the Axiom of Cho... |
| axacndlem2 10520 | Lemma for the Axiom of Cho... |
| axacndlem3 10521 | Lemma for the Axiom of Cho... |
| axacndlem4 10522 | Lemma for the Axiom of Cho... |
| axacndlem5 10523 | Lemma for the Axiom of Cho... |
| axacnd 10524 | A version of the Axiom of ... |
| zfcndext 10525 | Axiom of Extensionality ~ ... |
| zfcndrep 10526 | Axiom of Replacement ~ ax-... |
| zfcndun 10527 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10528 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10529 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10530 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10531 | Axiom of Choice ~ ax-ac , ... |
| elgch 10534 | Elementhood in the collect... |
| fingch 10535 | A finite set is a GCH-set.... |
| gchi 10536 | The only GCH-sets which ha... |
| gchen1 10537 | If ` A <_ B < ~P A ` , and... |
| gchen2 10538 | If ` A < B <_ ~P A ` , and... |
| gchor 10539 | If ` A <_ B <_ ~P A ` , an... |
| engch 10540 | The property of being a GC... |
| gchdomtri 10541 | Under certain conditions, ... |
| fpwwe2cbv 10542 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10543 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10544 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10545 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10546 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10547 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10548 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10549 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10550 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10551 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10552 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10553 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10554 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10555 | Given any function ` F ` f... |
| fpwwecbv 10556 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10557 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10558 | Given any function ` F ` f... |
| canth4 10559 | An "effective" form of Can... |
| canthnumlem 10560 | Lemma for ~ canthnum . (C... |
| canthnum 10561 | The set of well-orderable ... |
| canthwelem 10562 | Lemma for ~ canthwe . (Co... |
| canthwe 10563 | The set of well-orders of ... |
| canthp1lem1 10564 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10565 | Lemma for ~ canthp1 . (Co... |
| canthp1 10566 | A slightly stronger form o... |
| finngch 10567 | The exclusion of finite se... |
| gchdju1 10568 | An infinite GCH-set is ide... |
| gchinf 10569 | An infinite GCH-set is Ded... |
| pwfseqlem1 10570 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10571 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10572 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10573 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10574 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10575 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10576 | The powerset of a Dedekind... |
| pwxpndom2 10577 | The powerset of a Dedekind... |
| pwxpndom 10578 | The powerset of a Dedekind... |
| pwdjundom 10579 | The powerset of a Dedekind... |
| gchdjuidm 10580 | An infinite GCH-set is ide... |
| gchxpidm 10581 | An infinite GCH-set is ide... |
| gchpwdom 10582 | A relationship between dom... |
| gchaleph 10583 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10584 | If ` ( aleph `` A ) ` and ... |
| hargch 10585 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10586 | If ` ( aleph `` suc A ) ` ... |
| gch2 10587 | It is sufficient to requir... |
| gch3 10588 | An equivalent formulation ... |
| gch-kn 10589 | The equivalence of two ver... |
| gchaclem 10590 | Lemma for ~ gchac (obsolet... |
| gchhar 10591 | A "local" form of ~ gchac ... |
| gchacg 10592 | A "local" form of ~ gchac ... |
| gchac 10593 | The Generalized Continuum ... |
| elwina 10598 | Conditions of weak inacces... |
| elina 10599 | Conditions of strong inacc... |
| winaon 10600 | A weakly inaccessible card... |
| inawinalem 10601 | Lemma for ~ inawina . (Co... |
| inawina 10602 | Every strongly inaccessibl... |
| omina 10603 | ` _om ` is a strongly inac... |
| winacard 10604 | A weakly inaccessible card... |
| winainflem 10605 | A weakly inaccessible card... |
| winainf 10606 | A weakly inaccessible card... |
| winalim 10607 | A weakly inaccessible card... |
| winalim2 10608 | A nontrivial weakly inacce... |
| winafp 10609 | A nontrivial weakly inacce... |
| winafpi 10610 | This theorem, which states... |
| gchina 10611 | Assuming the GCH, weakly a... |
| iswun 10616 | Properties of a weak unive... |
| wuntr 10617 | A weak universe is transit... |
| wununi 10618 | A weak universe is closed ... |
| wunpw 10619 | A weak universe is closed ... |
| wunelss 10620 | The elements of a weak uni... |
| wunpr 10621 | A weak universe is closed ... |
| wunun 10622 | A weak universe is closed ... |
| wuntp 10623 | A weak universe is closed ... |
| wunss 10624 | A weak universe is closed ... |
| wunin 10625 | A weak universe is closed ... |
| wundif 10626 | A weak universe is closed ... |
| wunint 10627 | A weak universe is closed ... |
| wunsn 10628 | A weak universe is closed ... |
| wunsuc 10629 | A weak universe is closed ... |
| wun0 10630 | A weak universe contains t... |
| wunr1om 10631 | A weak universe is infinit... |
| wunom 10632 | A weak universe contains a... |
| wunfi 10633 | A weak universe contains a... |
| wunop 10634 | A weak universe is closed ... |
| wunot 10635 | A weak universe is closed ... |
| wunxp 10636 | A weak universe is closed ... |
| wunpm 10637 | A weak universe is closed ... |
| wunmap 10638 | A weak universe is closed ... |
| wunf 10639 | A weak universe is closed ... |
| wundm 10640 | A weak universe is closed ... |
| wunrn 10641 | A weak universe is closed ... |
| wuncnv 10642 | A weak universe is closed ... |
| wunres 10643 | A weak universe is closed ... |
| wunfv 10644 | A weak universe is closed ... |
| wunco 10645 | A weak universe is closed ... |
| wuntpos 10646 | A weak universe is closed ... |
| intwun 10647 | The intersection of a coll... |
| r1limwun 10648 | Each limit stage in the cu... |
| r1wunlim 10649 | The weak universes in the ... |
| wunex2 10650 | Construct a weak universe ... |
| wunex 10651 | Construct a weak universe ... |
| uniwun 10652 | Every set is contained in ... |
| wunex3 10653 | Construct a weak universe ... |
| wuncval 10654 | Value of the weak universe... |
| wuncid 10655 | The weak universe closure ... |
| wunccl 10656 | The weak universe closure ... |
| wuncss 10657 | The weak universe closure ... |
| wuncidm 10658 | The weak universe closure ... |
| wuncval2 10659 | Our earlier expression for... |
| eltskg 10662 | Properties of a Tarski cla... |
| eltsk2g 10663 | Properties of a Tarski cla... |
| tskpwss 10664 | First axiom of a Tarski cl... |
| tskpw 10665 | Second axiom of a Tarski c... |
| tsken 10666 | Third axiom of a Tarski cl... |
| 0tsk 10667 | The empty set is a (transi... |
| tsksdom 10668 | An element of a Tarski cla... |
| tskssel 10669 | A part of a Tarski class s... |
| tskss 10670 | The subsets of an element ... |
| tskin 10671 | The intersection of two el... |
| tsksn 10672 | A singleton of an element ... |
| tsktrss 10673 | A transitive element of a ... |
| tsksuc 10674 | If an element of a Tarski ... |
| tsk0 10675 | A nonempty Tarski class co... |
| tsk1 10676 | One is an element of a non... |
| tsk2 10677 | Two is an element of a non... |
| 2domtsk 10678 | If a Tarski class is not e... |
| tskr1om 10679 | A nonempty Tarski class is... |
| tskr1om2 10680 | A nonempty Tarski class co... |
| tskinf 10681 | A nonempty Tarski class is... |
| tskpr 10682 | If ` A ` and ` B ` are mem... |
| tskop 10683 | If ` A ` and ` B ` are mem... |
| tskxpss 10684 | A Cartesian product of two... |
| tskwe2 10685 | A Tarski class is well-ord... |
| inttsk 10686 | The intersection of a coll... |
| inar1 10687 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10688 | Alternate proof of ~ r1om ... |
| rankcf 10689 | Any set must be at least a... |
| inatsk 10690 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10691 | The set of hereditarily fi... |
| tskord 10692 | A Tarski class contains al... |
| tskcard 10693 | An even more direct relati... |
| r1tskina 10694 | There is a direct relation... |
| tskuni 10695 | The union of an element of... |
| tskwun 10696 | A nonempty transitive Tars... |
| tskint 10697 | The intersection of an ele... |
| tskun 10698 | The union of two elements ... |
| tskxp 10699 | The Cartesian product of t... |
| tskmap 10700 | Set exponentiation is an e... |
| tskurn 10701 | A transitive Tarski class ... |
| elgrug 10704 | Properties of a Grothendie... |
| grutr 10705 | A Grothendieck universe is... |
| gruelss 10706 | A Grothendieck universe is... |
| grupw 10707 | A Grothendieck universe co... |
| gruss 10708 | Any subset of an element o... |
| grupr 10709 | A Grothendieck universe co... |
| gruurn 10710 | A Grothendieck universe co... |
| gruiun 10711 | If ` B ( x ) ` is a family... |
| gruuni 10712 | A Grothendieck universe co... |
| grurn 10713 | A Grothendieck universe co... |
| gruima 10714 | A Grothendieck universe co... |
| gruel 10715 | Any element of an element ... |
| grusn 10716 | A Grothendieck universe co... |
| gruop 10717 | A Grothendieck universe co... |
| gruun 10718 | A Grothendieck universe co... |
| gruxp 10719 | A Grothendieck universe co... |
| grumap 10720 | A Grothendieck universe co... |
| gruixp 10721 | A Grothendieck universe co... |
| gruiin 10722 | A Grothendieck universe co... |
| gruf 10723 | A Grothendieck universe co... |
| gruen 10724 | A Grothendieck universe co... |
| gruwun 10725 | A nonempty Grothendieck un... |
| intgru 10726 | The intersection of a fami... |
| ingru 10727 | The intersection of a univ... |
| wfgru 10728 | The wellfounded part of a ... |
| grudomon 10729 | Each ordinal that is compa... |
| gruina 10730 | If a Grothendieck universe... |
| grur1a 10731 | A characterization of Grot... |
| grur1 10732 | A characterization of Grot... |
| grutsk1 10733 | Grothendieck universes are... |
| grutsk 10734 | Grothendieck universes are... |
| axgroth5 10736 | The Tarski-Grothendieck ax... |
| axgroth2 10737 | Alternate version of the T... |
| grothpw 10738 | Derive the Axiom of Power ... |
| grothpwex 10739 | Derive the Axiom of Power ... |
| axgroth6 10740 | The Tarski-Grothendieck ax... |
| grothomex 10741 | The Tarski-Grothendieck Ax... |
| grothac 10742 | The Tarski-Grothendieck Ax... |
| axgroth3 10743 | Alternate version of the T... |
| axgroth4 10744 | Alternate version of the T... |
| grothprimlem 10745 | Lemma for ~ grothprim . E... |
| grothprim 10746 | The Tarski-Grothendieck Ax... |
| grothtsk 10747 | The Tarski-Grothendieck Ax... |
| inaprc 10748 | An equivalent to the Tarsk... |
| tskmval 10751 | Value of our tarski map. ... |
| tskmid 10752 | The set ` A ` is an elemen... |
| tskmcl 10753 | A Tarski class that contai... |
| sstskm 10754 | Being a part of ` ( tarski... |
| eltskm 10755 | Belonging to ` ( tarskiMap... |
| elni 10788 | Membership in the class of... |
| elni2 10789 | Membership in the class of... |
| pinn 10790 | A positive integer is a na... |
| pion 10791 | A positive integer is an o... |
| piord 10792 | A positive integer is ordi... |
| niex 10793 | The class of positive inte... |
| 0npi 10794 | The empty set is not a pos... |
| 1pi 10795 | Ordinal 'one' is a positiv... |
| addpiord 10796 | Positive integer addition ... |
| mulpiord 10797 | Positive integer multiplic... |
| mulidpi 10798 | 1 is an identity element f... |
| ltpiord 10799 | Positive integer 'less tha... |
| ltsopi 10800 | Positive integer 'less tha... |
| ltrelpi 10801 | Positive integer 'less tha... |
| dmaddpi 10802 | Domain of addition on posi... |
| dmmulpi 10803 | Domain of multiplication o... |
| addclpi 10804 | Closure of addition of pos... |
| mulclpi 10805 | Closure of multiplication ... |
| addcompi 10806 | Addition of positive integ... |
| addasspi 10807 | Addition of positive integ... |
| mulcompi 10808 | Multiplication of positive... |
| mulasspi 10809 | Multiplication of positive... |
| distrpi 10810 | Multiplication of positive... |
| addcanpi 10811 | Addition cancellation law ... |
| mulcanpi 10812 | Multiplication cancellatio... |
| addnidpi 10813 | There is no identity eleme... |
| ltexpi 10814 | Ordering on positive integ... |
| ltapi 10815 | Ordering property of addit... |
| ltmpi 10816 | Ordering property of multi... |
| 1lt2pi 10817 | One is less than two (one ... |
| nlt1pi 10818 | No positive integer is les... |
| indpi 10819 | Principle of Finite Induct... |
| enqbreq 10831 | Equivalence relation for p... |
| enqbreq2 10832 | Equivalence relation for p... |
| enqer 10833 | The equivalence relation f... |
| enqex 10834 | The equivalence relation f... |
| nqex 10835 | The class of positive frac... |
| 0nnq 10836 | The empty set is not a pos... |
| elpqn 10837 | Each positive fraction is ... |
| ltrelnq 10838 | Positive fraction 'less th... |
| pinq 10839 | The representatives of pos... |
| 1nq 10840 | The positive fraction 'one... |
| nqereu 10841 | There is a unique element ... |
| nqerf 10842 | Corollary of ~ nqereu : th... |
| nqercl 10843 | Corollary of ~ nqereu : cl... |
| nqerrel 10844 | Any member of ` ( N. X. N.... |
| nqerid 10845 | Corollary of ~ nqereu : th... |
| enqeq 10846 | Corollary of ~ nqereu : if... |
| nqereq 10847 | The function ` /Q ` acts a... |
| addpipq2 10848 | Addition of positive fract... |
| addpipq 10849 | Addition of positive fract... |
| addpqnq 10850 | Addition of positive fract... |
| mulpipq2 10851 | Multiplication of positive... |
| mulpipq 10852 | Multiplication of positive... |
| mulpqnq 10853 | Multiplication of positive... |
| ordpipq 10854 | Ordering of positive fract... |
| ordpinq 10855 | Ordering of positive fract... |
| addpqf 10856 | Closure of addition on pos... |
| addclnq 10857 | Closure of addition on pos... |
| mulpqf 10858 | Closure of multiplication ... |
| mulclnq 10859 | Closure of multiplication ... |
| addnqf 10860 | Domain of addition on posi... |
| mulnqf 10861 | Domain of multiplication o... |
| addcompq 10862 | Addition of positive fract... |
| addcomnq 10863 | Addition of positive fract... |
| mulcompq 10864 | Multiplication of positive... |
| mulcomnq 10865 | Multiplication of positive... |
| adderpqlem 10866 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10867 | Lemma for ~ mulerpq . (Co... |
| adderpq 10868 | Addition is compatible wit... |
| mulerpq 10869 | Multiplication is compatib... |
| addassnq 10870 | Addition of positive fract... |
| mulassnq 10871 | Multiplication of positive... |
| mulcanenq 10872 | Lemma for distributive law... |
| distrnq 10873 | Multiplication of positive... |
| 1nqenq 10874 | The equivalence class of r... |
| mulidnq 10875 | Multiplication identity el... |
| recmulnq 10876 | Relationship between recip... |
| recidnq 10877 | A positive fraction times ... |
| recclnq 10878 | Closure law for positive f... |
| recrecnq 10879 | Reciprocal of reciprocal o... |
| dmrecnq 10880 | Domain of reciprocal on po... |
| ltsonq 10881 | 'Less than' is a strict or... |
| lterpq 10882 | Compatibility of ordering ... |
| ltanq 10883 | Ordering property of addit... |
| ltmnq 10884 | Ordering property of multi... |
| 1lt2nq 10885 | One is less than two (one ... |
| ltaddnq 10886 | The sum of two fractions i... |
| ltexnq 10887 | Ordering on positive fract... |
| halfnq 10888 | One-half of any positive f... |
| nsmallnq 10889 | The is no smallest positiv... |
| ltbtwnnq 10890 | There exists a number betw... |
| ltrnq 10891 | Ordering property of recip... |
| archnq 10892 | For any fraction, there is... |
| npex 10898 | The class of positive real... |
| elnp 10899 | Membership in positive rea... |
| elnpi 10900 | Membership in positive rea... |
| prn0 10901 | A positive real is not emp... |
| prpssnq 10902 | A positive real is a subse... |
| elprnq 10903 | A positive real is a set o... |
| 0npr 10904 | The empty set is not a pos... |
| prcdnq 10905 | A positive real is closed ... |
| prub 10906 | A positive fraction not in... |
| prnmax 10907 | A positive real has no lar... |
| npomex 10908 | A simplifying observation,... |
| prnmadd 10909 | A positive real has no lar... |
| ltrelpr 10910 | Positive real 'less than' ... |
| genpv 10911 | Value of general operation... |
| genpelv 10912 | Membership in value of gen... |
| genpprecl 10913 | Pre-closure law for genera... |
| genpdm 10914 | Domain of general operatio... |
| genpn0 10915 | The result of an operation... |
| genpss 10916 | The result of an operation... |
| genpnnp 10917 | The result of an operation... |
| genpcd 10918 | Downward closure of an ope... |
| genpnmax 10919 | An operation on positive r... |
| genpcl 10920 | Closure of an operation on... |
| genpass 10921 | Associativity of an operat... |
| plpv 10922 | Value of addition on posit... |
| mpv 10923 | Value of multiplication on... |
| dmplp 10924 | Domain of addition on posi... |
| dmmp 10925 | Domain of multiplication o... |
| nqpr 10926 | The canonical embedding of... |
| 1pr 10927 | The positive real number '... |
| addclprlem1 10928 | Lemma to prove downward cl... |
| addclprlem2 10929 | Lemma to prove downward cl... |
| addclpr 10930 | Closure of addition on pos... |
| mulclprlem 10931 | Lemma to prove downward cl... |
| mulclpr 10932 | Closure of multiplication ... |
| addcompr 10933 | Addition of positive reals... |
| addasspr 10934 | Addition of positive reals... |
| mulcompr 10935 | Multiplication of positive... |
| mulasspr 10936 | Multiplication of positive... |
| distrlem1pr 10937 | Lemma for distributive law... |
| distrlem4pr 10938 | Lemma for distributive law... |
| distrlem5pr 10939 | Lemma for distributive law... |
| distrpr 10940 | Multiplication of positive... |
| 1idpr 10941 | 1 is an identity element f... |
| ltprord 10942 | Positive real 'less than' ... |
| psslinpr 10943 | Proper subset is a linear ... |
| ltsopr 10944 | Positive real 'less than' ... |
| prlem934 10945 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10946 | The sum of two positive re... |
| ltaddpr2 10947 | The sum of two positive re... |
| ltexprlem1 10948 | Lemma for Proposition 9-3.... |
| ltexprlem2 10949 | Lemma for Proposition 9-3.... |
| ltexprlem3 10950 | Lemma for Proposition 9-3.... |
| ltexprlem4 10951 | Lemma for Proposition 9-3.... |
| ltexprlem5 10952 | Lemma for Proposition 9-3.... |
| ltexprlem6 10953 | Lemma for Proposition 9-3.... |
| ltexprlem7 10954 | Lemma for Proposition 9-3.... |
| ltexpri 10955 | Proposition 9-3.5(iv) of [... |
| ltaprlem 10956 | Lemma for Proposition 9-3.... |
| ltapr 10957 | Ordering property of addit... |
| addcanpr 10958 | Addition cancellation law ... |
| prlem936 10959 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 10960 | Lemma for Proposition 9-3.... |
| reclem3pr 10961 | Lemma for Proposition 9-3.... |
| reclem4pr 10962 | Lemma for Proposition 9-3.... |
| recexpr 10963 | The reciprocal of a positi... |
| suplem1pr 10964 | The union of a nonempty, b... |
| suplem2pr 10965 | The union of a set of posi... |
| supexpr 10966 | The union of a nonempty, b... |
| enrer 10975 | The equivalence relation f... |
| nrex1 10976 | The class of signed reals ... |
| enrbreq 10977 | Equivalence relation for s... |
| enreceq 10978 | Equivalence class equality... |
| enrex 10979 | The equivalence relation f... |
| ltrelsr 10980 | Signed real 'less than' is... |
| addcmpblnr 10981 | Lemma showing compatibilit... |
| mulcmpblnrlem 10982 | Lemma used in lemma showin... |
| mulcmpblnr 10983 | Lemma showing compatibilit... |
| prsrlem1 10984 | Decomposing signed reals i... |
| addsrmo 10985 | There is at most one resul... |
| mulsrmo 10986 | There is at most one resul... |
| addsrpr 10987 | Addition of signed reals i... |
| mulsrpr 10988 | Multiplication of signed r... |
| ltsrpr 10989 | Ordering of signed reals i... |
| gt0srpr 10990 | Greater than zero in terms... |
| 0nsr 10991 | The empty set is not a sig... |
| 0r 10992 | The constant ` 0R ` is a s... |
| 1sr 10993 | The constant ` 1R ` is a s... |
| m1r 10994 | The constant ` -1R ` is a ... |
| addclsr 10995 | Closure of addition on sig... |
| mulclsr 10996 | Closure of multiplication ... |
| dmaddsr 10997 | Domain of addition on sign... |
| dmmulsr 10998 | Domain of multiplication o... |
| addcomsr 10999 | Addition of signed reals i... |
| addasssr 11000 | Addition of signed reals i... |
| mulcomsr 11001 | Multiplication of signed r... |
| mulasssr 11002 | Multiplication of signed r... |
| distrsr 11003 | Multiplication of signed r... |
| m1p1sr 11004 | Minus one plus one is zero... |
| m1m1sr 11005 | Minus one times minus one ... |
| ltsosr 11006 | Signed real 'less than' is... |
| 0lt1sr 11007 | 0 is less than 1 for signe... |
| 1ne0sr 11008 | 1 and 0 are distinct for s... |
| 0idsr 11009 | The signed real number 0 i... |
| 1idsr 11010 | 1 is an identity element f... |
| 00sr 11011 | A signed real times 0 is 0... |
| ltasr 11012 | Ordering property of addit... |
| pn0sr 11013 | A signed real plus its neg... |
| negexsr 11014 | Existence of negative sign... |
| recexsrlem 11015 | The reciprocal of a positi... |
| addgt0sr 11016 | The sum of two positive si... |
| mulgt0sr 11017 | The product of two positiv... |
| sqgt0sr 11018 | The square of a nonzero si... |
| recexsr 11019 | The reciprocal of a nonzer... |
| mappsrpr 11020 | Mapping from positive sign... |
| ltpsrpr 11021 | Mapping of order from posi... |
| map2psrpr 11022 | Equivalence for positive s... |
| supsrlem 11023 | Lemma for supremum theorem... |
| supsr 11024 | A nonempty, bounded set of... |
| opelcn 11041 | Ordered pair membership in... |
| opelreal 11042 | Ordered pair membership in... |
| elreal 11043 | Membership in class of rea... |
| elreal2 11044 | Ordered pair membership in... |
| 0ncn 11045 | The empty set is not a com... |
| ltrelre 11046 | 'Less than' is a relation ... |
| addcnsr 11047 | Addition of complex number... |
| mulcnsr 11048 | Multiplication of complex ... |
| eqresr 11049 | Equality of real numbers i... |
| addresr 11050 | Addition of real numbers i... |
| mulresr 11051 | Multiplication of real num... |
| ltresr 11052 | Ordering of real subset of... |
| ltresr2 11053 | Ordering of real subset of... |
| dfcnqs 11054 | Technical trick to permit ... |
| addcnsrec 11055 | Technical trick to permit ... |
| mulcnsrec 11056 | Technical trick to permit ... |
| axaddf 11057 | Addition is an operation o... |
| axmulf 11058 | Multiplication is an opera... |
| axcnex 11059 | The complex numbers form a... |
| axresscn 11060 | The real numbers are a sub... |
| ax1cn 11061 | 1 is a complex number. Ax... |
| axicn 11062 | ` _i ` is a complex number... |
| axaddcl 11063 | Closure law for addition o... |
| axaddrcl 11064 | Closure law for addition i... |
| axmulcl 11065 | Closure law for multiplica... |
| axmulrcl 11066 | Closure law for multiplica... |
| axmulcom 11067 | Multiplication of complex ... |
| axaddass 11068 | Addition of complex number... |
| axmulass 11069 | Multiplication of complex ... |
| axdistr 11070 | Distributive law for compl... |
| axi2m1 11071 | i-squared equals -1 (expre... |
| ax1ne0 11072 | 1 and 0 are distinct. Axi... |
| ax1rid 11073 | ` 1 ` is an identity eleme... |
| axrnegex 11074 | Existence of negative of r... |
| axrrecex 11075 | Existence of reciprocal of... |
| axcnre 11076 | A complex number can be ex... |
| axpre-lttri 11077 | Ordering on reals satisfie... |
| axpre-lttrn 11078 | Ordering on reals is trans... |
| axpre-ltadd 11079 | Ordering property of addit... |
| axpre-mulgt0 11080 | The product of two positiv... |
| axpre-sup 11081 | A nonempty, bounded-above ... |
| wuncn 11082 | A weak universe containing... |
| cnex 11108 | Alias for ~ ax-cnex . See... |
| addcl 11109 | Alias for ~ ax-addcl , for... |
| readdcl 11110 | Alias for ~ ax-addrcl , fo... |
| mulcl 11111 | Alias for ~ ax-mulcl , for... |
| remulcl 11112 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11113 | Alias for ~ ax-mulcom , fo... |
| addass 11114 | Alias for ~ ax-addass , fo... |
| mulass 11115 | Alias for ~ ax-mulass , fo... |
| adddi 11116 | Alias for ~ ax-distr , for... |
| recn 11117 | A real number is a complex... |
| reex 11118 | The real numbers form a se... |
| reelprrecn 11119 | Reals are a subset of the ... |
| cnelprrecn 11120 | Complex numbers are a subs... |
| mpoaddf 11121 | Addition is an operation o... |
| mpomulf 11122 | Multiplication is an opera... |
| elimne0 11123 | Hypothesis for weak deduct... |
| adddir 11124 | Distributive law for compl... |
| 0cn 11125 | Zero is a complex number. ... |
| 0cnd 11126 | Zero is a complex number, ... |
| c0ex 11127 | Zero is a set. (Contribut... |
| 1cnd 11128 | One is a complex number, d... |
| 1ex 11129 | One is a set. (Contribute... |
| cnre 11130 | Alias for ~ ax-cnre , for ... |
| mulrid 11131 | The number 1 is an identit... |
| mullid 11132 | Identity law for multiplic... |
| 1re 11133 | The number 1 is real. Thi... |
| 1red 11134 | The number 1 is real, dedu... |
| 0re 11135 | The number 0 is real. Rem... |
| 0red 11136 | The number 0 is real, dedu... |
| mulridi 11137 | Identity law for multiplic... |
| mullidi 11138 | Identity law for multiplic... |
| addcli 11139 | Closure law for addition. ... |
| mulcli 11140 | Closure law for multiplica... |
| mulcomi 11141 | Commutative law for multip... |
| mulcomli 11142 | Commutative law for multip... |
| addassi 11143 | Associative law for additi... |
| mulassi 11144 | Associative law for multip... |
| adddii 11145 | Distributive law (left-dis... |
| adddiri 11146 | Distributive law (right-di... |
| recni 11147 | A real number is a complex... |
| readdcli 11148 | Closure law for addition o... |
| remulcli 11149 | Closure law for multiplica... |
| mulridd 11150 | Identity law for multiplic... |
| mullidd 11151 | Identity law for multiplic... |
| addcld 11152 | Closure law for addition. ... |
| mulcld 11153 | Closure law for multiplica... |
| mulcomd 11154 | Commutative law for multip... |
| addassd 11155 | Associative law for additi... |
| mulassd 11156 | Associative law for multip... |
| adddid 11157 | Distributive law (left-dis... |
| adddird 11158 | Distributive law (right-di... |
| adddirp1d 11159 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11160 | Join AB+CB into (A+C) on L... |
| recnd 11161 | Deduction from real number... |
| readdcld 11162 | Closure law for addition o... |
| remulcld 11163 | Closure law for multiplica... |
| pnfnre 11174 | Plus infinity is not a rea... |
| pnfnre2 11175 | Plus infinity is not a rea... |
| mnfnre 11176 | Minus infinity is not a re... |
| ressxr 11177 | The standard reals are a s... |
| rexpssxrxp 11178 | The Cartesian product of s... |
| rexr 11179 | A standard real is an exte... |
| 0xr 11180 | Zero is an extended real. ... |
| renepnf 11181 | No (finite) real equals pl... |
| renemnf 11182 | No real equals minus infin... |
| rexrd 11183 | A standard real is an exte... |
| renepnfd 11184 | No (finite) real equals pl... |
| renemnfd 11185 | No real equals minus infin... |
| pnfex 11186 | Plus infinity exists. (Co... |
| pnfxr 11187 | Plus infinity belongs to t... |
| pnfnemnf 11188 | Plus and minus infinity ar... |
| mnfnepnf 11189 | Minus and plus infinity ar... |
| mnfxr 11190 | Minus infinity belongs to ... |
| rexri 11191 | A standard real is an exte... |
| 1xr 11192 | ` 1 ` is an extended real ... |
| renfdisj 11193 | The reals and the infiniti... |
| ltrelxr 11194 | "Less than" is a relation ... |
| ltrel 11195 | "Less than" is a relation.... |
| lerelxr 11196 | "Less than or equal to" is... |
| lerel 11197 | "Less than or equal to" is... |
| xrlenlt 11198 | "Less than or equal to" ex... |
| xrlenltd 11199 | "Less than or equal to" ex... |
| xrltnle 11200 | "Less than" expressed in t... |
| xrltnled 11201 | 'Less than' in terms of 'l... |
| xrnltled 11202 | "Not less than" implies "l... |
| ssxr 11203 | The three (non-exclusive) ... |
| ltxrlt 11204 | The standard less-than ` <... |
| axlttri 11205 | Ordering on reals satisfie... |
| axlttrn 11206 | Ordering on reals is trans... |
| axltadd 11207 | Ordering property of addit... |
| axmulgt0 11208 | The product of two positiv... |
| axsup 11209 | A nonempty, bounded-above ... |
| lttr 11210 | Alias for ~ axlttrn , for ... |
| mulgt0 11211 | The product of two positiv... |
| lenlt 11212 | 'Less than or equal to' ex... |
| ltnle 11213 | 'Less than' expressed in t... |
| ltso 11214 | 'Less than' is a strict or... |
| gtso 11215 | 'Greater than' is a strict... |
| lttri2 11216 | Consequence of trichotomy.... |
| lttri3 11217 | Trichotomy law for 'less t... |
| lttri4 11218 | Trichotomy law for 'less t... |
| letri3 11219 | Trichotomy law. (Contribu... |
| leloe 11220 | 'Less than or equal to' ex... |
| eqlelt 11221 | Equality in terms of 'less... |
| ltle 11222 | 'Less than' implies 'less ... |
| leltne 11223 | 'Less than or equal to' im... |
| lelttr 11224 | Transitive law. (Contribu... |
| leltletr 11225 | Transitive law, weaker for... |
| ltletr 11226 | Transitive law. (Contribu... |
| ltleletr 11227 | Transitive law, weaker for... |
| letr 11228 | Transitive law. (Contribu... |
| ltnr 11229 | 'Less than' is irreflexive... |
| leid 11230 | 'Less than or equal to' is... |
| ltne 11231 | 'Less than' implies not eq... |
| ltnsym 11232 | 'Less than' is not symmetr... |
| ltnsym2 11233 | 'Less than' is antisymmetr... |
| letric 11234 | Trichotomy law. (Contribu... |
| ltlen 11235 | 'Less than' expressed in t... |
| eqle 11236 | Equality implies 'less tha... |
| eqled 11237 | Equality implies 'less tha... |
| ltadd2 11238 | Addition to both sides of ... |
| ne0gt0 11239 | A nonzero nonnegative numb... |
| lecasei 11240 | Ordering elimination by ca... |
| lelttric 11241 | Trichotomy law. (Contribu... |
| ltlecasei 11242 | Ordering elimination by ca... |
| ltnri 11243 | 'Less than' is irreflexive... |
| eqlei 11244 | Equality implies 'less tha... |
| eqlei2 11245 | Equality implies 'less tha... |
| gtneii 11246 | 'Less than' implies not eq... |
| ltneii 11247 | 'Greater than' implies not... |
| lttri2i 11248 | Consequence of trichotomy.... |
| lttri3i 11249 | Consequence of trichotomy.... |
| letri3i 11250 | Consequence of trichotomy.... |
| leloei 11251 | 'Less than or equal to' in... |
| ltleni 11252 | 'Less than' expressed in t... |
| ltnsymi 11253 | 'Less than' is not symmetr... |
| lenlti 11254 | 'Less than or equal to' in... |
| ltnlei 11255 | 'Less than' in terms of 'l... |
| ltlei 11256 | 'Less than' implies 'less ... |
| ltleii 11257 | 'Less than' implies 'less ... |
| ltnei 11258 | 'Less than' implies not eq... |
| letrii 11259 | Trichotomy law for 'less t... |
| lttri 11260 | 'Less than' is transitive.... |
| lelttri 11261 | 'Less than or equal to', '... |
| ltletri 11262 | 'Less than', 'less than or... |
| letri 11263 | 'Less than or equal to' is... |
| le2tri3i 11264 | Extended trichotomy law fo... |
| ltadd2i 11265 | Addition to both sides of ... |
| mulgt0i 11266 | The product of two positiv... |
| mulgt0ii 11267 | The product of two positiv... |
| ltnrd 11268 | 'Less than' is irreflexive... |
| gtned 11269 | 'Less than' implies not eq... |
| ltned 11270 | 'Greater than' implies not... |
| ne0gt0d 11271 | A nonzero nonnegative numb... |
| lttrid 11272 | Ordering on reals satisfie... |
| lttri2d 11273 | Consequence of trichotomy.... |
| lttri3d 11274 | Consequence of trichotomy.... |
| lttri4d 11275 | Trichotomy law for 'less t... |
| letri3d 11276 | Consequence of trichotomy.... |
| leloed 11277 | 'Less than or equal to' in... |
| eqleltd 11278 | Equality in terms of 'less... |
| ltlend 11279 | 'Less than' expressed in t... |
| lenltd 11280 | 'Less than or equal to' in... |
| ltnled 11281 | 'Less than' in terms of 'l... |
| ltled 11282 | 'Less than' implies 'less ... |
| ltnsymd 11283 | 'Less than' implies 'less ... |
| nltled 11284 | 'Not less than ' implies '... |
| lensymd 11285 | 'Less than or equal to' im... |
| letrid 11286 | Trichotomy law for 'less t... |
| leltned 11287 | 'Less than or equal to' im... |
| leneltd 11288 | 'Less than or equal to' an... |
| mulgt0d 11289 | The product of two positiv... |
| ltadd2d 11290 | Addition to both sides of ... |
| letrd 11291 | Transitive law deduction f... |
| lelttrd 11292 | Transitive law deduction f... |
| ltadd2dd 11293 | Addition to both sides of ... |
| ltletrd 11294 | Transitive law deduction f... |
| lttrd 11295 | Transitive law deduction f... |
| lelttrdi 11296 | If a number is less than a... |
| dedekind 11297 | The Dedekind cut theorem. ... |
| dedekindle 11298 | The Dedekind cut theorem, ... |
| mul12 11299 | Commutative/associative la... |
| mul32 11300 | Commutative/associative la... |
| mul31 11301 | Commutative/associative la... |
| mul4 11302 | Rearrangement of 4 factors... |
| mul4r 11303 | Rearrangement of 4 factors... |
| muladd11 11304 | A simple product of sums e... |
| 1p1times 11305 | Two times a number. (Cont... |
| peano2cn 11306 | A theorem for complex numb... |
| peano2re 11307 | A theorem for reals analog... |
| readdcan 11308 | Cancellation law for addit... |
| 00id 11309 | ` 0 ` is its own additive ... |
| mul02lem1 11310 | Lemma for ~ mul02 . If an... |
| mul02lem2 11311 | Lemma for ~ mul02 . Zero ... |
| mul02 11312 | Multiplication by ` 0 ` . ... |
| mul01 11313 | Multiplication by ` 0 ` . ... |
| addrid 11314 | ` 0 ` is an additive ident... |
| cnegex 11315 | Existence of the negative ... |
| cnegex2 11316 | Existence of a left invers... |
| addlid 11317 | ` 0 ` is a left identity f... |
| addcan 11318 | Cancellation law for addit... |
| addcan2 11319 | Cancellation law for addit... |
| addcom 11320 | Addition commutes. This u... |
| addridi 11321 | ` 0 ` is an additive ident... |
| addlidi 11322 | ` 0 ` is a left identity f... |
| mul02i 11323 | Multiplication by 0. Theo... |
| mul01i 11324 | Multiplication by ` 0 ` . ... |
| addcomi 11325 | Addition commutes. Based ... |
| addcomli 11326 | Addition commutes. (Contr... |
| addcani 11327 | Cancellation law for addit... |
| addcan2i 11328 | Cancellation law for addit... |
| mul12i 11329 | Commutative/associative la... |
| mul32i 11330 | Commutative/associative la... |
| mul4i 11331 | Rearrangement of 4 factors... |
| mul02d 11332 | Multiplication by 0. Theo... |
| mul01d 11333 | Multiplication by ` 0 ` . ... |
| addridd 11334 | ` 0 ` is an additive ident... |
| addlidd 11335 | ` 0 ` is a left identity f... |
| addcomd 11336 | Addition commutes. Based ... |
| addcand 11337 | Cancellation law for addit... |
| addcan2d 11338 | Cancellation law for addit... |
| addcanad 11339 | Cancelling a term on the l... |
| addcan2ad 11340 | Cancelling a term on the r... |
| addneintrd 11341 | Introducing a term on the ... |
| addneintr2d 11342 | Introducing a term on the ... |
| mul12d 11343 | Commutative/associative la... |
| mul32d 11344 | Commutative/associative la... |
| mul31d 11345 | Commutative/associative la... |
| mul4d 11346 | Rearrangement of 4 factors... |
| muladd11r 11347 | A simple product of sums e... |
| comraddd 11348 | Commute RHS addition, in d... |
| comraddi 11349 | Commute RHS addition. See... |
| ltaddneg 11350 | Adding a negative number t... |
| ltaddnegr 11351 | Adding a negative number t... |
| add12 11352 | Commutative/associative la... |
| add32 11353 | Commutative/associative la... |
| add32r 11354 | Commutative/associative la... |
| add4 11355 | Rearrangement of 4 terms i... |
| add42 11356 | Rearrangement of 4 terms i... |
| add12i 11357 | Commutative/associative la... |
| add32i 11358 | Commutative/associative la... |
| add4i 11359 | Rearrangement of 4 terms i... |
| add42i 11360 | Rearrangement of 4 terms i... |
| add12d 11361 | Commutative/associative la... |
| add32d 11362 | Commutative/associative la... |
| add4d 11363 | Rearrangement of 4 terms i... |
| add42d 11364 | Rearrangement of 4 terms i... |
| 0cnALT 11369 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11370 | Alternate proof of ~ 0cnAL... |
| negeu 11371 | Existential uniqueness of ... |
| subval 11372 | Value of subtraction, whic... |
| negeq 11373 | Equality theorem for negat... |
| negeqi 11374 | Equality inference for neg... |
| negeqd 11375 | Equality deduction for neg... |
| nfnegd 11376 | Deduction version of ~ nfn... |
| nfneg 11377 | Bound-variable hypothesis ... |
| csbnegg 11378 | Move class substitution in... |
| negex 11379 | A negative is a set. (Con... |
| subcl 11380 | Closure law for subtractio... |
| negcl 11381 | Closure law for negative. ... |
| negicn 11382 | ` -u _i ` is a complex num... |
| subf 11383 | Subtraction is an operatio... |
| subadd 11384 | Relationship between subtr... |
| subadd2 11385 | Relationship between subtr... |
| subsub23 11386 | Swap subtrahend and result... |
| pncan 11387 | Cancellation law for subtr... |
| pncan2 11388 | Cancellation law for subtr... |
| pncan3 11389 | Subtraction and addition o... |
| npcan 11390 | Cancellation law for subtr... |
| addsubass 11391 | Associative-type law for a... |
| addsub 11392 | Law for addition and subtr... |
| subadd23 11393 | Commutative/associative la... |
| addsub12 11394 | Commutative/associative la... |
| 2addsub 11395 | Law for subtraction and ad... |
| addsubeq4 11396 | Relation between sums and ... |
| pncan3oi 11397 | Subtraction and addition o... |
| mvrraddi 11398 | Move the right term in a s... |
| mvrladdi 11399 | Move the left term in a su... |
| mvlladdi 11400 | Move the left term in a su... |
| subid 11401 | Subtraction of a number fr... |
| subid1 11402 | Identity law for subtracti... |
| npncan 11403 | Cancellation law for subtr... |
| nppcan 11404 | Cancellation law for subtr... |
| nnpcan 11405 | Cancellation law for subtr... |
| nppcan3 11406 | Cancellation law for subtr... |
| subcan2 11407 | Cancellation law for subtr... |
| subeq0 11408 | If the difference between ... |
| npncan2 11409 | Cancellation law for subtr... |
| subsub2 11410 | Law for double subtraction... |
| nncan 11411 | Cancellation law for subtr... |
| subsub 11412 | Law for double subtraction... |
| nppcan2 11413 | Cancellation law for subtr... |
| subsub3 11414 | Law for double subtraction... |
| subsub4 11415 | Law for double subtraction... |
| sub32 11416 | Swap the second and third ... |
| nnncan 11417 | Cancellation law for subtr... |
| nnncan1 11418 | Cancellation law for subtr... |
| nnncan2 11419 | Cancellation law for subtr... |
| npncan3 11420 | Cancellation law for subtr... |
| pnpcan 11421 | Cancellation law for mixed... |
| pnpcan2 11422 | Cancellation law for mixed... |
| pnncan 11423 | Cancellation law for mixed... |
| ppncan 11424 | Cancellation law for mixed... |
| addsub4 11425 | Rearrangement of 4 terms i... |
| subadd4 11426 | Rearrangement of 4 terms i... |
| sub4 11427 | Rearrangement of 4 terms i... |
| neg0 11428 | Minus 0 equals 0. (Contri... |
| negid 11429 | Addition of a number and i... |
| negsub 11430 | Relationship between subtr... |
| subneg 11431 | Relationship between subtr... |
| negneg 11432 | A number is equal to the n... |
| neg11 11433 | Negative is one-to-one. (... |
| negcon1 11434 | Negative contraposition la... |
| negcon2 11435 | Negative contraposition la... |
| negeq0 11436 | A number is zero iff its n... |
| subcan 11437 | Cancellation law for subtr... |
| negsubdi 11438 | Distribution of negative o... |
| negdi 11439 | Distribution of negative o... |
| negdi2 11440 | Distribution of negative o... |
| negsubdi2 11441 | Distribution of negative o... |
| neg2sub 11442 | Relationship between subtr... |
| renegcli 11443 | Closure law for negative o... |
| resubcli 11444 | Closure law for subtractio... |
| renegcl 11445 | Closure law for negative o... |
| resubcl 11446 | Closure law for subtractio... |
| negreb 11447 | The negative of a real is ... |
| peano2cnm 11448 | "Reverse" second Peano pos... |
| peano2rem 11449 | "Reverse" second Peano pos... |
| negcli 11450 | Closure law for negative. ... |
| negidi 11451 | Addition of a number and i... |
| negnegi 11452 | A number is equal to the n... |
| subidi 11453 | Subtraction of a number fr... |
| subid1i 11454 | Identity law for subtracti... |
| negne0bi 11455 | A number is nonzero iff it... |
| negrebi 11456 | The negative of a real is ... |
| negne0i 11457 | The negative of a nonzero ... |
| subcli 11458 | Closure law for subtractio... |
| pncan3i 11459 | Subtraction and addition o... |
| negsubi 11460 | Relationship between subtr... |
| subnegi 11461 | Relationship between subtr... |
| subeq0i 11462 | If the difference between ... |
| neg11i 11463 | Negative is one-to-one. (... |
| negcon1i 11464 | Negative contraposition la... |
| negcon2i 11465 | Negative contraposition la... |
| negdii 11466 | Distribution of negative o... |
| negsubdii 11467 | Distribution of negative o... |
| negsubdi2i 11468 | Distribution of negative o... |
| subaddi 11469 | Relationship between subtr... |
| subadd2i 11470 | Relationship between subtr... |
| subaddrii 11471 | Relationship between subtr... |
| subsub23i 11472 | Swap subtrahend and result... |
| addsubassi 11473 | Associative-type law for s... |
| addsubi 11474 | Law for subtraction and ad... |
| subcani 11475 | Cancellation law for subtr... |
| subcan2i 11476 | Cancellation law for subtr... |
| pnncani 11477 | Cancellation law for mixed... |
| addsub4i 11478 | Rearrangement of 4 terms i... |
| 0reALT 11479 | Alternate proof of ~ 0re .... |
| negcld 11480 | Closure law for negative. ... |
| subidd 11481 | Subtraction of a number fr... |
| subid1d 11482 | Identity law for subtracti... |
| negidd 11483 | Addition of a number and i... |
| negnegd 11484 | A number is equal to the n... |
| negeq0d 11485 | A number is zero iff its n... |
| negne0bd 11486 | A number is nonzero iff it... |
| negcon1d 11487 | Contraposition law for una... |
| negcon1ad 11488 | Contraposition law for una... |
| neg11ad 11489 | The negatives of two compl... |
| negned 11490 | If two complex numbers are... |
| negne0d 11491 | The negative of a nonzero ... |
| negrebd 11492 | The negative of a real is ... |
| subcld 11493 | Closure law for subtractio... |
| pncand 11494 | Cancellation law for subtr... |
| pncan2d 11495 | Cancellation law for subtr... |
| pncan3d 11496 | Subtraction and addition o... |
| npcand 11497 | Cancellation law for subtr... |
| nncand 11498 | Cancellation law for subtr... |
| negsubd 11499 | Relationship between subtr... |
| subnegd 11500 | Relationship between subtr... |
| subeq0d 11501 | If the difference between ... |
| subne0d 11502 | Two unequal numbers have n... |
| subeq0ad 11503 | The difference of two comp... |
| subne0ad 11504 | If the difference of two c... |
| neg11d 11505 | If the difference between ... |
| negdid 11506 | Distribution of negative o... |
| negdi2d 11507 | Distribution of negative o... |
| negsubdid 11508 | Distribution of negative o... |
| negsubdi2d 11509 | Distribution of negative o... |
| neg2subd 11510 | Relationship between subtr... |
| subaddd 11511 | Relationship between subtr... |
| subadd2d 11512 | Relationship between subtr... |
| addsubassd 11513 | Associative-type law for s... |
| addsubd 11514 | Law for subtraction and ad... |
| subadd23d 11515 | Commutative/associative la... |
| addsub12d 11516 | Commutative/associative la... |
| npncand 11517 | Cancellation law for subtr... |
| nppcand 11518 | Cancellation law for subtr... |
| nppcan2d 11519 | Cancellation law for subtr... |
| nppcan3d 11520 | Cancellation law for subtr... |
| subsubd 11521 | Law for double subtraction... |
| subsub2d 11522 | Law for double subtraction... |
| subsub3d 11523 | Law for double subtraction... |
| subsub4d 11524 | Law for double subtraction... |
| sub32d 11525 | Swap the second and third ... |
| nnncand 11526 | Cancellation law for subtr... |
| nnncan1d 11527 | Cancellation law for subtr... |
| nnncan2d 11528 | Cancellation law for subtr... |
| npncan3d 11529 | Cancellation law for subtr... |
| pnpcand 11530 | Cancellation law for mixed... |
| pnpcan2d 11531 | Cancellation law for mixed... |
| pnncand 11532 | Cancellation law for mixed... |
| ppncand 11533 | Cancellation law for mixed... |
| subcand 11534 | Cancellation law for subtr... |
| subcan2d 11535 | Cancellation law for subtr... |
| subcanad 11536 | Cancellation law for subtr... |
| subneintrd 11537 | Introducing subtraction on... |
| subcan2ad 11538 | Cancellation law for subtr... |
| subneintr2d 11539 | Introducing subtraction on... |
| addsub4d 11540 | Rearrangement of 4 terms i... |
| subadd4d 11541 | Rearrangement of 4 terms i... |
| sub4d 11542 | Rearrangement of 4 terms i... |
| 2addsubd 11543 | Law for subtraction and ad... |
| addsubeq4d 11544 | Relation between sums and ... |
| subsubadd23 11545 | Swap the second and the th... |
| addsubsub23 11546 | Swap the second and the th... |
| subeqxfrd 11547 | Transfer two terms of a su... |
| mvlraddd 11548 | Move the right term in a s... |
| mvlladdd 11549 | Move the left term in a su... |
| mvrraddd 11550 | Move the right term in a s... |
| mvrladdd 11551 | Move the left term in a su... |
| assraddsubd 11552 | Associate RHS addition-sub... |
| subaddeqd 11553 | Transfer two terms of a su... |
| addlsub 11554 | Left-subtraction: Subtrac... |
| addrsub 11555 | Right-subtraction: Subtra... |
| subexsub 11556 | A subtraction law: Exchan... |
| addid0 11557 | If adding a number to a an... |
| addn0nid 11558 | Adding a nonzero number to... |
| pnpncand 11559 | Addition/subtraction cance... |
| subeqrev 11560 | Reverse the order of subtr... |
| addeq0 11561 | Two complex numbers add up... |
| pncan1 11562 | Cancellation law for addit... |
| npcan1 11563 | Cancellation law for subtr... |
| subeq0bd 11564 | If two complex numbers are... |
| renegcld 11565 | Closure law for negative o... |
| resubcld 11566 | Closure law for subtractio... |
| negn0 11567 | The image under negation o... |
| negf1o 11568 | Negation is an isomorphism... |
| kcnktkm1cn 11569 | k times k minus 1 is a com... |
| muladd 11570 | Product of two sums. (Con... |
| subdi 11571 | Distribution of multiplica... |
| subdir 11572 | Distribution of multiplica... |
| ine0 11573 | The imaginary unit ` _i ` ... |
| mulneg1 11574 | Product with negative is n... |
| mulneg2 11575 | The product with a negativ... |
| mulneg12 11576 | Swap the negative sign in ... |
| mul2neg 11577 | Product of two negatives. ... |
| submul2 11578 | Convert a subtraction to a... |
| mulm1 11579 | Product with minus one is ... |
| addneg1mul 11580 | Addition with product with... |
| mulsub 11581 | Product of two differences... |
| mulsub2 11582 | Swap the order of subtract... |
| mulm1i 11583 | Product with minus one is ... |
| mulneg1i 11584 | Product with negative is n... |
| mulneg2i 11585 | Product with negative is n... |
| mul2negi 11586 | Product of two negatives. ... |
| subdii 11587 | Distribution of multiplica... |
| subdiri 11588 | Distribution of multiplica... |
| muladdi 11589 | Product of two sums. (Con... |
| mulm1d 11590 | Product with minus one is ... |
| mulneg1d 11591 | Product with negative is n... |
| mulneg2d 11592 | Product with negative is n... |
| mul2negd 11593 | Product of two negatives. ... |
| subdid 11594 | Distribution of multiplica... |
| subdird 11595 | Distribution of multiplica... |
| muladdd 11596 | Product of two sums. (Con... |
| mulsubd 11597 | Product of two differences... |
| muls1d 11598 | Multiplication by one minu... |
| mulsubfacd 11599 | Multiplication followed by... |
| addmulsub 11600 | The product of a sum and a... |
| subaddmulsub 11601 | The difference with a prod... |
| mulsubaddmulsub 11602 | A special difference of a ... |
| gt0ne0 11603 | Positive implies nonzero. ... |
| lt0ne0 11604 | A number which is less tha... |
| ltadd1 11605 | Addition to both sides of ... |
| leadd1 11606 | Addition to both sides of ... |
| leadd2 11607 | Addition to both sides of ... |
| ltsubadd 11608 | 'Less than' relationship b... |
| ltsubadd2 11609 | 'Less than' relationship b... |
| lesubadd 11610 | 'Less than or equal to' re... |
| lesubadd2 11611 | 'Less than or equal to' re... |
| ltaddsub 11612 | 'Less than' relationship b... |
| ltaddsub2 11613 | 'Less than' relationship b... |
| leaddsub 11614 | 'Less than or equal to' re... |
| leaddsub2 11615 | 'Less than or equal to' re... |
| suble 11616 | Swap subtrahends in an ine... |
| lesub 11617 | Swap subtrahends in an ine... |
| ltsub23 11618 | 'Less than' relationship b... |
| ltsub13 11619 | 'Less than' relationship b... |
| le2add 11620 | Adding both sides of two '... |
| ltleadd 11621 | Adding both sides of two o... |
| leltadd 11622 | Adding both sides of two o... |
| lt2add 11623 | Adding both sides of two '... |
| addgt0 11624 | The sum of 2 positive numb... |
| addgegt0 11625 | The sum of nonnegative and... |
| addgtge0 11626 | The sum of nonnegative and... |
| addge0 11627 | The sum of 2 nonnegative n... |
| ltaddpos 11628 | Adding a positive number t... |
| ltaddpos2 11629 | Adding a positive number t... |
| ltsubpos 11630 | Subtracting a positive num... |
| posdif 11631 | Comparison of two numbers ... |
| lesub1 11632 | Subtraction from both side... |
| lesub2 11633 | Subtraction of both sides ... |
| ltsub1 11634 | Subtraction from both side... |
| ltsub2 11635 | Subtraction of both sides ... |
| lt2sub 11636 | Subtracting both sides of ... |
| le2sub 11637 | Subtracting both sides of ... |
| ltneg 11638 | Negative of both sides of ... |
| ltnegcon1 11639 | Contraposition of negative... |
| ltnegcon2 11640 | Contraposition of negative... |
| leneg 11641 | Negative of both sides of ... |
| lenegcon1 11642 | Contraposition of negative... |
| lenegcon2 11643 | Contraposition of negative... |
| lt0neg1 11644 | Comparison of a number and... |
| lt0neg2 11645 | Comparison of a number and... |
| le0neg1 11646 | Comparison of a number and... |
| le0neg2 11647 | Comparison of a number and... |
| addge01 11648 | A number is less than or e... |
| addge02 11649 | A number is less than or e... |
| add20 11650 | Two nonnegative numbers ar... |
| subge0 11651 | Nonnegative subtraction. ... |
| suble0 11652 | Nonpositive subtraction. ... |
| leaddle0 11653 | The sum of a real number a... |
| subge02 11654 | Nonnegative subtraction. ... |
| lesub0 11655 | Lemma to show a nonnegativ... |
| mulge0 11656 | The product of two nonnega... |
| mullt0 11657 | The product of two negativ... |
| msqgt0 11658 | A nonzero square is positi... |
| msqge0 11659 | A square is nonnegative. ... |
| 0lt1 11660 | 0 is less than 1. Theorem... |
| 0le1 11661 | 0 is less than or equal to... |
| relin01 11662 | An interval law for less t... |
| ltordlem 11663 | Lemma for ~ ltord1 . (Con... |
| ltord1 11664 | Infer an ordering relation... |
| leord1 11665 | Infer an ordering relation... |
| eqord1 11666 | A strictly increasing real... |
| ltord2 11667 | Infer an ordering relation... |
| leord2 11668 | Infer an ordering relation... |
| eqord2 11669 | A strictly decreasing real... |
| wloglei 11670 | Form of ~ wlogle where bot... |
| wlogle 11671 | If the predicate ` ch ( x ... |
| leidi 11672 | 'Less than or equal to' is... |
| gt0ne0i 11673 | Positive means nonzero (us... |
| gt0ne0ii 11674 | Positive implies nonzero. ... |
| msqgt0i 11675 | A nonzero square is positi... |
| msqge0i 11676 | A square is nonnegative. ... |
| addgt0i 11677 | Addition of 2 positive num... |
| addge0i 11678 | Addition of 2 nonnegative ... |
| addgegt0i 11679 | Addition of nonnegative an... |
| addgt0ii 11680 | Addition of 2 positive num... |
| add20i 11681 | Two nonnegative numbers ar... |
| ltnegi 11682 | Negative of both sides of ... |
| lenegi 11683 | Negative of both sides of ... |
| ltnegcon2i 11684 | Contraposition of negative... |
| mulge0i 11685 | The product of two nonnega... |
| lesub0i 11686 | Lemma to show a nonnegativ... |
| ltaddposi 11687 | Adding a positive number t... |
| posdifi 11688 | Comparison of two numbers ... |
| ltnegcon1i 11689 | Contraposition of negative... |
| lenegcon1i 11690 | Contraposition of negative... |
| subge0i 11691 | Nonnegative subtraction. ... |
| ltadd1i 11692 | Addition to both sides of ... |
| leadd1i 11693 | Addition to both sides of ... |
| leadd2i 11694 | Addition to both sides of ... |
| ltsubaddi 11695 | 'Less than' relationship b... |
| lesubaddi 11696 | 'Less than or equal to' re... |
| ltsubadd2i 11697 | 'Less than' relationship b... |
| lesubadd2i 11698 | 'Less than or equal to' re... |
| ltaddsubi 11699 | 'Less than' relationship b... |
| lt2addi 11700 | Adding both side of two in... |
| le2addi 11701 | Adding both side of two in... |
| gt0ne0d 11702 | Positive implies nonzero. ... |
| lt0ne0d 11703 | Something less than zero i... |
| leidd 11704 | 'Less than or equal to' is... |
| msqgt0d 11705 | A nonzero square is positi... |
| msqge0d 11706 | A square is nonnegative. ... |
| lt0neg1d 11707 | Comparison of a number and... |
| lt0neg2d 11708 | Comparison of a number and... |
| le0neg1d 11709 | Comparison of a number and... |
| le0neg2d 11710 | Comparison of a number and... |
| addgegt0d 11711 | Addition of nonnegative an... |
| addgtge0d 11712 | Addition of positive and n... |
| addgt0d 11713 | Addition of 2 positive num... |
| addge0d 11714 | Addition of 2 nonnegative ... |
| mulge0d 11715 | The product of two nonnega... |
| ltnegd 11716 | Negative of both sides of ... |
| lenegd 11717 | Negative of both sides of ... |
| ltnegcon1d 11718 | Contraposition of negative... |
| ltnegcon2d 11719 | Contraposition of negative... |
| lenegcon1d 11720 | Contraposition of negative... |
| lenegcon2d 11721 | Contraposition of negative... |
| ltaddposd 11722 | Adding a positive number t... |
| ltaddpos2d 11723 | Adding a positive number t... |
| ltsubposd 11724 | Subtracting a positive num... |
| posdifd 11725 | Comparison of two numbers ... |
| addge01d 11726 | A number is less than or e... |
| addge02d 11727 | A number is less than or e... |
| subge0d 11728 | Nonnegative subtraction. ... |
| suble0d 11729 | Nonpositive subtraction. ... |
| subge02d 11730 | Nonnegative subtraction. ... |
| ltadd1d 11731 | Addition to both sides of ... |
| leadd1d 11732 | Addition to both sides of ... |
| leadd2d 11733 | Addition to both sides of ... |
| ltsubaddd 11734 | 'Less than' relationship b... |
| lesubaddd 11735 | 'Less than or equal to' re... |
| ltsubadd2d 11736 | 'Less than' relationship b... |
| lesubadd2d 11737 | 'Less than or equal to' re... |
| ltaddsubd 11738 | 'Less than' relationship b... |
| ltaddsub2d 11739 | 'Less than' relationship b... |
| leaddsub2d 11740 | 'Less than or equal to' re... |
| subled 11741 | Swap subtrahends in an ine... |
| lesubd 11742 | Swap subtrahends in an ine... |
| ltsub23d 11743 | 'Less than' relationship b... |
| ltsub13d 11744 | 'Less than' relationship b... |
| lesub1d 11745 | Subtraction from both side... |
| lesub2d 11746 | Subtraction of both sides ... |
| ltsub1d 11747 | Subtraction from both side... |
| ltsub2d 11748 | Subtraction of both sides ... |
| ltadd1dd 11749 | Addition to both sides of ... |
| ltsub1dd 11750 | Subtraction from both side... |
| ltsub2dd 11751 | Subtraction of both sides ... |
| leadd1dd 11752 | Addition to both sides of ... |
| leadd2dd 11753 | Addition to both sides of ... |
| lesub1dd 11754 | Subtraction from both side... |
| lesub2dd 11755 | Subtraction of both sides ... |
| lesub3d 11756 | The result of subtracting ... |
| le2addd 11757 | Adding both side of two in... |
| le2subd 11758 | Subtracting both sides of ... |
| ltleaddd 11759 | Adding both sides of two o... |
| leltaddd 11760 | Adding both sides of two o... |
| lt2addd 11761 | Adding both side of two in... |
| lt2subd 11762 | Subtracting both sides of ... |
| possumd 11763 | Condition for a positive s... |
| sublt0d 11764 | When a subtraction gives a... |
| ltaddsublt 11765 | Addition and subtraction o... |
| 1le1 11766 | One is less than or equal ... |
| ixi 11767 | ` _i ` times itself is min... |
| recextlem1 11768 | Lemma for ~ recex . (Cont... |
| recextlem2 11769 | Lemma for ~ recex . (Cont... |
| recex 11770 | Existence of reciprocal of... |
| mulcand 11771 | Cancellation law for multi... |
| mulcan2d 11772 | Cancellation law for multi... |
| mulcanad 11773 | Cancellation of a nonzero ... |
| mulcan2ad 11774 | Cancellation of a nonzero ... |
| mulcan 11775 | Cancellation law for multi... |
| mulcan2 11776 | Cancellation law for multi... |
| mulcani 11777 | Cancellation law for multi... |
| mul0or 11778 | If a product is zero, one ... |
| mulne0b 11779 | The product of two nonzero... |
| mulne0 11780 | The product of two nonzero... |
| mulne0i 11781 | The product of two nonzero... |
| muleqadd 11782 | Property of numbers whose ... |
| receu 11783 | Existential uniqueness of ... |
| mulnzcnf 11784 | Multiplication maps nonzer... |
| mul0ori 11785 | If a product is zero, one ... |
| mul0ord 11786 | If a product is zero, one ... |
| msq0i 11787 | A number is zero iff its s... |
| msq0d 11788 | A number is zero iff its s... |
| mulne0bd 11789 | The product of two nonzero... |
| mulne0d 11790 | The product of two nonzero... |
| mulcan1g 11791 | A generalized form of the ... |
| mulcan2g 11792 | A generalized form of the ... |
| mulne0bad 11793 | A factor of a nonzero comp... |
| mulne0bbd 11794 | A factor of a nonzero comp... |
| 1div0 11797 | You can't divide by zero, ... |
| 1div0OLD 11798 | Obsolete version of ~ 1div... |
| divval 11799 | Value of division: if ` A ... |
| divmul 11800 | Relationship between divis... |
| divmul2 11801 | Relationship between divis... |
| divmul3 11802 | Relationship between divis... |
| divcl 11803 | Closure law for division. ... |
| reccl 11804 | Closure law for reciprocal... |
| divcan2 11805 | A cancellation law for div... |
| divcan1 11806 | A cancellation law for div... |
| diveq0 11807 | A ratio is zero iff the nu... |
| divne0b 11808 | The ratio of nonzero numbe... |
| divne0 11809 | The ratio of nonzero numbe... |
| recne0 11810 | The reciprocal of a nonzer... |
| recid 11811 | Multiplication of a number... |
| recid2 11812 | Multiplication of a number... |
| divrec 11813 | Relationship between divis... |
| divrec2 11814 | Relationship between divis... |
| divass 11815 | An associative law for div... |
| div23 11816 | A commutative/associative ... |
| div32 11817 | A commutative/associative ... |
| div13 11818 | A commutative/associative ... |
| div12 11819 | A commutative/associative ... |
| divmulass 11820 | An associative law for div... |
| divmulasscom 11821 | An associative/commutative... |
| divdir 11822 | Distribution of division o... |
| divcan3 11823 | A cancellation law for div... |
| divcan4 11824 | A cancellation law for div... |
| div11 11825 | One-to-one relationship fo... |
| div11OLD 11826 | Obsolete version of ~ div1... |
| diveq1 11827 | Equality in terms of unit ... |
| divid 11828 | A number divided by itself... |
| dividOLD 11829 | Obsolete version of ~ divi... |
| div0 11830 | Division into zero is zero... |
| div0OLD 11831 | Obsolete version of ~ div0... |
| div1 11832 | A number divided by 1 is i... |
| 1div1e1 11833 | 1 divided by 1 is 1. (Con... |
| divneg 11834 | Move negative sign inside ... |
| muldivdir 11835 | Distribution of division o... |
| divsubdir 11836 | Distribution of division o... |
| subdivcomb1 11837 | Bring a term in a subtract... |
| subdivcomb2 11838 | Bring a term in a subtract... |
| recrec 11839 | A number is equal to the r... |
| rec11 11840 | Reciprocal is one-to-one. ... |
| rec11r 11841 | Mutual reciprocals. (Cont... |
| divmuldiv 11842 | Multiplication of two rati... |
| divdivdiv 11843 | Division of two ratios. T... |
| divcan5 11844 | Cancellation of common fac... |
| divmul13 11845 | Swap the denominators in t... |
| divmul24 11846 | Swap the numerators in the... |
| divmuleq 11847 | Cross-multiply in an equal... |
| recdiv 11848 | The reciprocal of a ratio.... |
| divcan6 11849 | Cancellation of inverted f... |
| divdiv32 11850 | Swap denominators in a div... |
| divcan7 11851 | Cancel equal divisors in a... |
| dmdcan 11852 | Cancellation law for divis... |
| divdiv1 11853 | Division into a fraction. ... |
| divdiv2 11854 | Division by a fraction. (... |
| recdiv2 11855 | Division into a reciprocal... |
| ddcan 11856 | Cancellation in a double d... |
| divadddiv 11857 | Addition of two ratios. T... |
| divsubdiv 11858 | Subtraction of two ratios.... |
| conjmul 11859 | Two numbers whose reciproc... |
| rereccl 11860 | Closure law for reciprocal... |
| redivcl 11861 | Closure law for division o... |
| eqneg 11862 | A number equal to its nega... |
| eqnegd 11863 | A complex number equals it... |
| eqnegad 11864 | If a complex number equals... |
| div2neg 11865 | Quotient of two negatives.... |
| divneg2 11866 | Move negative sign inside ... |
| recclzi 11867 | Closure law for reciprocal... |
| recne0zi 11868 | The reciprocal of a nonzer... |
| recidzi 11869 | Multiplication of a number... |
| div1i 11870 | A number divided by 1 is i... |
| eqnegi 11871 | A number equal to its nega... |
| reccli 11872 | Closure law for reciprocal... |
| recidi 11873 | Multiplication of a number... |
| recreci 11874 | A number is equal to the r... |
| dividi 11875 | A number divided by itself... |
| div0i 11876 | Division into zero is zero... |
| divclzi 11877 | Closure law for division. ... |
| divcan1zi 11878 | A cancellation law for div... |
| divcan2zi 11879 | A cancellation law for div... |
| divreczi 11880 | Relationship between divis... |
| divcan3zi 11881 | A cancellation law for div... |
| divcan4zi 11882 | A cancellation law for div... |
| rec11i 11883 | Reciprocal is one-to-one. ... |
| divcli 11884 | Closure law for division. ... |
| divcan2i 11885 | A cancellation law for div... |
| divcan1i 11886 | A cancellation law for div... |
| divreci 11887 | Relationship between divis... |
| divcan3i 11888 | A cancellation law for div... |
| divcan4i 11889 | A cancellation law for div... |
| divne0i 11890 | The ratio of nonzero numbe... |
| rec11ii 11891 | Reciprocal is one-to-one. ... |
| divasszi 11892 | An associative law for div... |
| divmulzi 11893 | Relationship between divis... |
| divdirzi 11894 | Distribution of division o... |
| divdiv23zi 11895 | Swap denominators in a div... |
| divmuli 11896 | Relationship between divis... |
| divdiv32i 11897 | Swap denominators in a div... |
| divassi 11898 | An associative law for div... |
| divdiri 11899 | Distribution of division o... |
| div23i 11900 | A commutative/associative ... |
| div11i 11901 | One-to-one relationship fo... |
| divmuldivi 11902 | Multiplication of two rati... |
| divmul13i 11903 | Swap denominators of two r... |
| divadddivi 11904 | Addition of two ratios. T... |
| divdivdivi 11905 | Division of two ratios. T... |
| rerecclzi 11906 | Closure law for reciprocal... |
| rereccli 11907 | Closure law for reciprocal... |
| redivclzi 11908 | Closure law for division o... |
| redivcli 11909 | Closure law for division o... |
| div1d 11910 | A number divided by 1 is i... |
| reccld 11911 | Closure law for reciprocal... |
| recne0d 11912 | The reciprocal of a nonzer... |
| recidd 11913 | Multiplication of a number... |
| recid2d 11914 | Multiplication of a number... |
| recrecd 11915 | A number is equal to the r... |
| dividd 11916 | A number divided by itself... |
| div0d 11917 | Division into zero is zero... |
| divcld 11918 | Closure law for division. ... |
| divcan1d 11919 | A cancellation law for div... |
| divcan2d 11920 | A cancellation law for div... |
| divrecd 11921 | Relationship between divis... |
| divrec2d 11922 | Relationship between divis... |
| divcan3d 11923 | A cancellation law for div... |
| divcan4d 11924 | A cancellation law for div... |
| diveq0d 11925 | A ratio is zero iff the nu... |
| diveq1d 11926 | Equality in terms of unit ... |
| diveq1ad 11927 | The quotient of two comple... |
| diveq0ad 11928 | A fraction of complex numb... |
| divne1d 11929 | If two complex numbers are... |
| divne0bd 11930 | A ratio is zero iff the nu... |
| divnegd 11931 | Move negative sign inside ... |
| divneg2d 11932 | Move negative sign inside ... |
| div2negd 11933 | Quotient of two negatives.... |
| divne0d 11934 | The ratio of nonzero numbe... |
| recdivd 11935 | The reciprocal of a ratio.... |
| recdiv2d 11936 | Division into a reciprocal... |
| divcan6d 11937 | Cancellation of inverted f... |
| ddcand 11938 | Cancellation in a double d... |
| rec11d 11939 | Reciprocal is one-to-one. ... |
| divmuld 11940 | Relationship between divis... |
| div32d 11941 | A commutative/associative ... |
| div13d 11942 | A commutative/associative ... |
| divdiv32d 11943 | Swap denominators in a div... |
| divcan5d 11944 | Cancellation of common fac... |
| divcan5rd 11945 | Cancellation of common fac... |
| divcan7d 11946 | Cancel equal divisors in a... |
| dmdcand 11947 | Cancellation law for divis... |
| dmdcan2d 11948 | Cancellation law for divis... |
| divdiv1d 11949 | Division into a fraction. ... |
| divdiv2d 11950 | Division by a fraction. (... |
| divmul2d 11951 | Relationship between divis... |
| divmul3d 11952 | Relationship between divis... |
| divassd 11953 | An associative law for div... |
| div12d 11954 | A commutative/associative ... |
| div23d 11955 | A commutative/associative ... |
| divdird 11956 | Distribution of division o... |
| divsubdird 11957 | Distribution of division o... |
| div11d 11958 | One-to-one relationship fo... |
| divmuldivd 11959 | Multiplication of two rati... |
| divmul13d 11960 | Swap denominators of two r... |
| divmul24d 11961 | Swap the numerators in the... |
| divadddivd 11962 | Addition of two ratios. T... |
| divsubdivd 11963 | Subtraction of two ratios.... |
| divmuleqd 11964 | Cross-multiply in an equal... |
| divdivdivd 11965 | Division of two ratios. T... |
| diveq1bd 11966 | If two complex numbers are... |
| div2sub 11967 | Swap the order of subtract... |
| div2subd 11968 | Swap subtrahend and minuen... |
| rereccld 11969 | Closure law for reciprocal... |
| redivcld 11970 | Closure law for division o... |
| subrecd 11971 | Subtraction of reciprocals... |
| subrec 11972 | Subtraction of reciprocals... |
| subreci 11973 | Subtraction of reciprocals... |
| mvllmuld 11974 | Move the left term in a pr... |
| mvllmuli 11975 | Move the left term in a pr... |
| ldiv 11976 | Left-division. (Contribut... |
| rdiv 11977 | Right-division. (Contribu... |
| mdiv 11978 | A division law. (Contribu... |
| lineq 11979 | Solution of a (scalar) lin... |
| elimgt0 11980 | Hypothesis for weak deduct... |
| elimge0 11981 | Hypothesis for weak deduct... |
| ltp1 11982 | A number is less than itse... |
| lep1 11983 | A number is less than or e... |
| ltm1 11984 | A number minus 1 is less t... |
| lem1 11985 | A number minus 1 is less t... |
| letrp1 11986 | A transitive property of '... |
| p1le 11987 | A transitive property of p... |
| recgt0 11988 | The reciprocal of a positi... |
| prodgt0 11989 | Infer that a multiplicand ... |
| prodgt02 11990 | Infer that a multiplier is... |
| ltmul1a 11991 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 11992 | Multiplication of both sid... |
| ltmul2 11993 | Multiplication of both sid... |
| lemul1 11994 | Multiplication of both sid... |
| lemul2 11995 | Multiplication of both sid... |
| lemul1a 11996 | Multiplication of both sid... |
| lemul2a 11997 | Multiplication of both sid... |
| ltmul12a 11998 | Comparison of product of t... |
| lemul12b 11999 | Comparison of product of t... |
| lemul12a 12000 | Comparison of product of t... |
| mulgt1OLD 12001 | Obsolete version of ~ mulg... |
| ltmulgt11 12002 | Multiplication by a number... |
| ltmulgt12 12003 | Multiplication by a number... |
| mulgt1 12004 | The product of two numbers... |
| lemulge11 12005 | Multiplication by a number... |
| lemulge12 12006 | Multiplication by a number... |
| ltdiv1 12007 | Division of both sides of ... |
| lediv1 12008 | Division of both sides of ... |
| gt0div 12009 | Division of a positive num... |
| ge0div 12010 | Division of a nonnegative ... |
| divgt0 12011 | The ratio of two positive ... |
| divge0 12012 | The ratio of nonnegative a... |
| mulge0b 12013 | A condition for multiplica... |
| mulle0b 12014 | A condition for multiplica... |
| mulsuble0b 12015 | A condition for multiplica... |
| ltmuldiv 12016 | 'Less than' relationship b... |
| ltmuldiv2 12017 | 'Less than' relationship b... |
| ltdivmul 12018 | 'Less than' relationship b... |
| ledivmul 12019 | 'Less than or equal to' re... |
| ltdivmul2 12020 | 'Less than' relationship b... |
| lt2mul2div 12021 | 'Less than' relationship b... |
| ledivmul2 12022 | 'Less than or equal to' re... |
| lemuldiv 12023 | 'Less than or equal' relat... |
| lemuldiv2 12024 | 'Less than or equal' relat... |
| ltrec 12025 | The reciprocal of both sid... |
| lerec 12026 | The reciprocal of both sid... |
| lt2msq1 12027 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12028 | Two nonnegative numbers co... |
| ltdiv2 12029 | Division of a positive num... |
| ltrec1 12030 | Reciprocal swap in a 'less... |
| lerec2 12031 | Reciprocal swap in a 'less... |
| ledivdiv 12032 | Invert ratios of positive ... |
| lediv2 12033 | Division of a positive num... |
| ltdiv23 12034 | Swap denominator with othe... |
| lediv23 12035 | Swap denominator with othe... |
| lediv12a 12036 | Comparison of ratio of two... |
| lediv2a 12037 | Division of both sides of ... |
| reclt1 12038 | The reciprocal of a positi... |
| recgt1 12039 | The reciprocal of a positi... |
| recgt1i 12040 | The reciprocal of a number... |
| recp1lt1 12041 | Construct a number less th... |
| recreclt 12042 | Given a positive number ` ... |
| le2msq 12043 | The square function on non... |
| msq11 12044 | The square of a nonnegativ... |
| ledivp1 12045 | "Less than or equal to" an... |
| squeeze0 12046 | If a nonnegative number is... |
| ltp1i 12047 | A number is less than itse... |
| recgt0i 12048 | The reciprocal of a positi... |
| recgt0ii 12049 | The reciprocal of a positi... |
| prodgt0i 12050 | Infer that a multiplicand ... |
| divgt0i 12051 | The ratio of two positive ... |
| divge0i 12052 | The ratio of nonnegative a... |
| ltreci 12053 | The reciprocal of both sid... |
| lereci 12054 | The reciprocal of both sid... |
| lt2msqi 12055 | The square function on non... |
| le2msqi 12056 | The square function on non... |
| msq11i 12057 | The square of a nonnegativ... |
| divgt0i2i 12058 | The ratio of two positive ... |
| ltrecii 12059 | The reciprocal of both sid... |
| divgt0ii 12060 | The ratio of two positive ... |
| ltmul1i 12061 | Multiplication of both sid... |
| ltdiv1i 12062 | Division of both sides of ... |
| ltmuldivi 12063 | 'Less than' relationship b... |
| ltmul2i 12064 | Multiplication of both sid... |
| lemul1i 12065 | Multiplication of both sid... |
| lemul2i 12066 | Multiplication of both sid... |
| ltdiv23i 12067 | Swap denominator with othe... |
| ledivp1i 12068 | "Less than or equal to" an... |
| ltdivp1i 12069 | Less-than and division rel... |
| ltdiv23ii 12070 | Swap denominator with othe... |
| ltmul1ii 12071 | Multiplication of both sid... |
| ltdiv1ii 12072 | Division of both sides of ... |
| ltp1d 12073 | A number is less than itse... |
| lep1d 12074 | A number is less than or e... |
| ltm1d 12075 | A number minus 1 is less t... |
| lem1d 12076 | A number minus 1 is less t... |
| recgt0d 12077 | The reciprocal of a positi... |
| divgt0d 12078 | The ratio of two positive ... |
| mulgt1d 12079 | The product of two numbers... |
| lemulge11d 12080 | Multiplication by a number... |
| lemulge12d 12081 | Multiplication by a number... |
| lemul1ad 12082 | Multiplication of both sid... |
| lemul2ad 12083 | Multiplication of both sid... |
| ltmul12ad 12084 | Comparison of product of t... |
| lemul12ad 12085 | Comparison of product of t... |
| lemul12bd 12086 | Comparison of product of t... |
| fimaxre 12087 | A finite set of real numbe... |
| fimaxre2 12088 | A nonempty finite set of r... |
| fimaxre3 12089 | A nonempty finite set of r... |
| fiminre 12090 | A nonempty finite set of r... |
| fiminre2 12091 | A nonempty finite set of r... |
| negfi 12092 | The negation of a finite s... |
| lbreu 12093 | If a set of reals contains... |
| lbcl 12094 | If a set of reals contains... |
| lble 12095 | If a set of reals contains... |
| lbinf 12096 | If a set of reals contains... |
| lbinfcl 12097 | If a set of reals contains... |
| lbinfle 12098 | If a set of reals contains... |
| sup2 12099 | A nonempty, bounded-above ... |
| sup3 12100 | A version of the completen... |
| infm3lem 12101 | Lemma for ~ infm3 . (Cont... |
| infm3 12102 | The completeness axiom for... |
| suprcl 12103 | Closure of supremum of a n... |
| suprub 12104 | A member of a nonempty bou... |
| suprubd 12105 | Natural deduction form of ... |
| suprcld 12106 | Natural deduction form of ... |
| suprlub 12107 | The supremum of a nonempty... |
| suprnub 12108 | An upper bound is not less... |
| suprleub 12109 | The supremum of a nonempty... |
| supaddc 12110 | The supremum function dist... |
| supadd 12111 | The supremum function dist... |
| supmul1 12112 | The supremum function dist... |
| supmullem1 12113 | Lemma for ~ supmul . (Con... |
| supmullem2 12114 | Lemma for ~ supmul . (Con... |
| supmul 12115 | The supremum function dist... |
| sup3ii 12116 | A version of the completen... |
| suprclii 12117 | Closure of supremum of a n... |
| suprubii 12118 | A member of a nonempty bou... |
| suprlubii 12119 | The supremum of a nonempty... |
| suprnubii 12120 | An upper bound is not less... |
| suprleubii 12121 | The supremum of a nonempty... |
| riotaneg 12122 | The negative of the unique... |
| negiso 12123 | Negation is an order anti-... |
| dfinfre 12124 | The infimum of a set of re... |
| infrecl 12125 | Closure of infimum of a no... |
| infrenegsup 12126 | The infimum of a set of re... |
| infregelb 12127 | Any lower bound of a nonem... |
| infrelb 12128 | If a nonempty set of real ... |
| infrefilb 12129 | The infimum of a finite se... |
| supfirege 12130 | The supremum of a finite s... |
| neg1cn 12131 | -1 is a complex number. (... |
| neg1rr 12132 | -1 is a real number. (Con... |
| neg1ne0 12133 | -1 is nonzero. (Contribut... |
| neg1lt0 12134 | -1 is less than 0. (Contr... |
| negneg1e1 12135 | ` -u -u 1 ` is 1. (Contri... |
| inelr 12136 | The imaginary unit ` _i ` ... |
| rimul 12137 | A real number times the im... |
| cru 12138 | The representation of comp... |
| crne0 12139 | The real representation of... |
| creur 12140 | The real part of a complex... |
| creui 12141 | The imaginary part of a co... |
| cju 12142 | The complex conjugate of a... |
| ofsubeq0 12143 | Function analogue of ~ sub... |
| ofnegsub 12144 | Function analogue of ~ neg... |
| ofsubge0 12145 | Function analogue of ~ sub... |
| nnexALT 12148 | Alternate proof of ~ nnex ... |
| peano5nni 12149 | Peano's inductive postulat... |
| nnssre 12150 | The positive integers are ... |
| nnsscn 12151 | The positive integers are ... |
| nnex 12152 | The set of positive intege... |
| nnre 12153 | A positive integer is a re... |
| nncn 12154 | A positive integer is a co... |
| nnrei 12155 | A positive integer is a re... |
| nncni 12156 | A positive integer is a co... |
| 1nn 12157 | Peano postulate: 1 is a po... |
| peano2nn 12158 | Peano postulate: a success... |
| dfnn2 12159 | Alternate definition of th... |
| dfnn3 12160 | Alternate definition of th... |
| nnred 12161 | A positive integer is a re... |
| nncnd 12162 | A positive integer is a co... |
| peano2nnd 12163 | Peano postulate: a success... |
| nnind 12164 | Principle of Mathematical ... |
| nnindALT 12165 | Principle of Mathematical ... |
| nnindd 12166 | Principle of Mathematical ... |
| nn1m1nn 12167 | Every positive integer is ... |
| nn1suc 12168 | If a statement holds for 1... |
| nnaddcl 12169 | Closure of addition of pos... |
| nnmulcl 12170 | Closure of multiplication ... |
| nnmulcli 12171 | Closure of multiplication ... |
| nnmtmip 12172 | "Minus times minus is plus... |
| nn2ge 12173 | There exists a positive in... |
| nnge1 12174 | A positive integer is one ... |
| nngt1ne1 12175 | A positive integer is grea... |
| nnle1eq1 12176 | A positive integer is less... |
| nngt0 12177 | A positive integer is posi... |
| nnnlt1 12178 | A positive integer is not ... |
| nnnle0 12179 | A positive integer is not ... |
| nnne0 12180 | A positive integer is nonz... |
| nnneneg 12181 | No positive integer is equ... |
| 0nnn 12182 | Zero is not a positive int... |
| 0nnnALT 12183 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12184 | Alternate version of ~ nnn... |
| nngt0i 12185 | A positive integer is posi... |
| nnne0i 12186 | A positive integer is nonz... |
| nndivre 12187 | The quotient of a real and... |
| nnrecre 12188 | The reciprocal of a positi... |
| nnrecgt0 12189 | The reciprocal of a positi... |
| nnsub 12190 | Subtraction of positive in... |
| nnsubi 12191 | Subtraction of positive in... |
| nndiv 12192 | Two ways to express " ` A ... |
| nndivtr 12193 | Transitive property of div... |
| nnge1d 12194 | A positive integer is one ... |
| nngt0d 12195 | A positive integer is posi... |
| nnne0d 12196 | A positive integer is nonz... |
| nnrecred 12197 | The reciprocal of a positi... |
| nnaddcld 12198 | Closure of addition of pos... |
| nnmulcld 12199 | Closure of multiplication ... |
| nndivred 12200 | A positive integer is one ... |
| 0ne1 12217 | Zero is different from one... |
| 1m1e0 12218 | One minus one equals zero.... |
| 2nn 12219 | 2 is a positive integer. ... |
| 2re 12220 | The number 2 is real. (Co... |
| 2cn 12221 | The number 2 is a complex ... |
| 2cnALT 12222 | Alternate proof of ~ 2cn .... |
| 2ex 12223 | The number 2 is a set. (C... |
| 2cnd 12224 | The number 2 is a complex ... |
| 3nn 12225 | 3 is a positive integer. ... |
| 3re 12226 | The number 3 is real. (Co... |
| 3cn 12227 | The number 3 is a complex ... |
| 3ex 12228 | The number 3 is a set. (C... |
| 4nn 12229 | 4 is a positive integer. ... |
| 4re 12230 | The number 4 is real. (Co... |
| 4cn 12231 | The number 4 is a complex ... |
| 5nn 12232 | 5 is a positive integer. ... |
| 5re 12233 | The number 5 is real. (Co... |
| 5cn 12234 | The number 5 is a complex ... |
| 6nn 12235 | 6 is a positive integer. ... |
| 6re 12236 | The number 6 is real. (Co... |
| 6cn 12237 | The number 6 is a complex ... |
| 7nn 12238 | 7 is a positive integer. ... |
| 7re 12239 | The number 7 is real. (Co... |
| 7cn 12240 | The number 7 is a complex ... |
| 8nn 12241 | 8 is a positive integer. ... |
| 8re 12242 | The number 8 is real. (Co... |
| 8cn 12243 | The number 8 is a complex ... |
| 9nn 12244 | 9 is a positive integer. ... |
| 9re 12245 | The number 9 is real. (Co... |
| 9cn 12246 | The number 9 is a complex ... |
| 0le0 12247 | Zero is nonnegative. (Con... |
| 0le2 12248 | The number 0 is less than ... |
| 2pos 12249 | The number 2 is positive. ... |
| 2ne0 12250 | The number 2 is nonzero. ... |
| 3pos 12251 | The number 3 is positive. ... |
| 3ne0 12252 | The number 3 is nonzero. ... |
| 4pos 12253 | The number 4 is positive. ... |
| 4ne0 12254 | The number 4 is nonzero. ... |
| 5pos 12255 | The number 5 is positive. ... |
| 6pos 12256 | The number 6 is positive. ... |
| 7pos 12257 | The number 7 is positive. ... |
| 8pos 12258 | The number 8 is positive. ... |
| 9pos 12259 | The number 9 is positive. ... |
| 1pneg1e0 12260 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12261 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12262 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12263 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12264 | Function value at ` N + 1 ... |
| 1p0e1 12265 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12266 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12267 | 2 - 1 = 1. The result is ... |
| 1e2m1 12268 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12269 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12270 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12271 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12272 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12273 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12274 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12275 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12276 | Two plus two equals four. ... |
| 2times 12277 | Two times a number. (Cont... |
| times2 12278 | A number times 2. (Contri... |
| 2timesi 12279 | Two times a number. (Cont... |
| times2i 12280 | A number times 2. (Contri... |
| 2txmxeqx 12281 | Two times a complex number... |
| 2div2e1 12282 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12283 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12284 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12285 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12286 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12287 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12288 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12289 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12290 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12291 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12292 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12293 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12294 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12295 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12296 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12297 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12298 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12299 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12300 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12301 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12302 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12303 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12304 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12305 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12306 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12307 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12308 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12309 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12310 | 2 times 0 equals 0. (Cont... |
| 4div2e2 12311 | One half of four is two. ... |
| 1lt2 12312 | 1 is less than 2. (Contri... |
| 2lt3 12313 | 2 is less than 3. (Contri... |
| 1lt3 12314 | 1 is less than 3. (Contri... |
| 3lt4 12315 | 3 is less than 4. (Contri... |
| 2lt4 12316 | 2 is less than 4. (Contri... |
| 1lt4 12317 | 1 is less than 4. (Contri... |
| 4lt5 12318 | 4 is less than 5. (Contri... |
| 3lt5 12319 | 3 is less than 5. (Contri... |
| 2lt5 12320 | 2 is less than 5. (Contri... |
| 1lt5 12321 | 1 is less than 5. (Contri... |
| 5lt6 12322 | 5 is less than 6. (Contri... |
| 4lt6 12323 | 4 is less than 6. (Contri... |
| 3lt6 12324 | 3 is less than 6. (Contri... |
| 2lt6 12325 | 2 is less than 6. (Contri... |
| 1lt6 12326 | 1 is less than 6. (Contri... |
| 6lt7 12327 | 6 is less than 7. (Contri... |
| 5lt7 12328 | 5 is less than 7. (Contri... |
| 4lt7 12329 | 4 is less than 7. (Contri... |
| 3lt7 12330 | 3 is less than 7. (Contri... |
| 2lt7 12331 | 2 is less than 7. (Contri... |
| 1lt7 12332 | 1 is less than 7. (Contri... |
| 7lt8 12333 | 7 is less than 8. (Contri... |
| 6lt8 12334 | 6 is less than 8. (Contri... |
| 5lt8 12335 | 5 is less than 8. (Contri... |
| 4lt8 12336 | 4 is less than 8. (Contri... |
| 3lt8 12337 | 3 is less than 8. (Contri... |
| 2lt8 12338 | 2 is less than 8. (Contri... |
| 1lt8 12339 | 1 is less than 8. (Contri... |
| 8lt9 12340 | 8 is less than 9. (Contri... |
| 7lt9 12341 | 7 is less than 9. (Contri... |
| 6lt9 12342 | 6 is less than 9. (Contri... |
| 5lt9 12343 | 5 is less than 9. (Contri... |
| 4lt9 12344 | 4 is less than 9. (Contri... |
| 3lt9 12345 | 3 is less than 9. (Contri... |
| 2lt9 12346 | 2 is less than 9. (Contri... |
| 1lt9 12347 | 1 is less than 9. (Contri... |
| 0ne2 12348 | 0 is not equal to 2. (Con... |
| 1ne2 12349 | 1 is not equal to 2. (Con... |
| 1le2 12350 | 1 is less than or equal to... |
| 2cnne0 12351 | 2 is a nonzero complex num... |
| 2rene0 12352 | 2 is a nonzero real number... |
| 1le3 12353 | 1 is less than or equal to... |
| neg1mulneg1e1 12354 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12355 | One-half is real. (Contri... |
| halfcn 12356 | One-half is a complex numb... |
| halfgt0 12357 | One-half is greater than z... |
| halfge0 12358 | One-half is not negative. ... |
| halflt1 12359 | One-half is less than one.... |
| 2halves 12360 | Two halves make a whole. ... |
| 1mhlfehlf 12361 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12362 | An eighth of four thirds i... |
| halfthird 12363 | Half minus a third. (Cont... |
| halfpm6th 12364 | One half plus or minus one... |
| it0e0 12365 | i times 0 equals 0. (Cont... |
| 2mulicn 12366 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12367 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12368 | Closure of half of a numbe... |
| rehalfcl 12369 | Real closure of half. (Co... |
| half0 12370 | Half of a number is zero i... |
| halfpos2 12371 | A number is positive iff i... |
| halfpos 12372 | A positive number is great... |
| halfnneg2 12373 | A number is nonnegative if... |
| halfaddsubcl 12374 | Closure of half-sum and ha... |
| halfaddsub 12375 | Sum and difference of half... |
| subhalfhalf 12376 | Subtracting the half of a ... |
| lt2halves 12377 | A sum is less than the who... |
| addltmul 12378 | Sum is less than product f... |
| nominpos 12379 | There is no smallest posit... |
| avglt1 12380 | Ordering property for aver... |
| avglt2 12381 | Ordering property for aver... |
| avgle1 12382 | Ordering property for aver... |
| avgle2 12383 | Ordering property for aver... |
| avgle 12384 | The average of two numbers... |
| 2timesd 12385 | Two times a number. (Cont... |
| times2d 12386 | A number times 2. (Contri... |
| halfcld 12387 | Closure of half of a numbe... |
| 2halvesd 12388 | Two halves make a whole. ... |
| rehalfcld 12389 | Real closure of half. (Co... |
| lt2halvesd 12390 | A sum is less than the who... |
| rehalfcli 12391 | Half a real number is real... |
| lt2addmuld 12392 | If two real numbers are le... |
| add1p1 12393 | Adding two times 1 to a nu... |
| sub1m1 12394 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12395 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12396 | A complex number increased... |
| div4p1lem1div2 12397 | An integer greater than 5,... |
| nnunb 12398 | The set of positive intege... |
| arch 12399 | Archimedean property of re... |
| nnrecl 12400 | There exists a positive in... |
| bndndx 12401 | A bounded real sequence ` ... |
| elnn0 12404 | Nonnegative integers expre... |
| nnssnn0 12405 | Positive naturals are a su... |
| nn0ssre 12406 | Nonnegative integers are a... |
| nn0sscn 12407 | Nonnegative integers are a... |
| nn0ex 12408 | The set of nonnegative int... |
| nnnn0 12409 | A positive integer is a no... |
| nnnn0i 12410 | A positive integer is a no... |
| nn0re 12411 | A nonnegative integer is a... |
| nn0cn 12412 | A nonnegative integer is a... |
| nn0rei 12413 | A nonnegative integer is a... |
| nn0cni 12414 | A nonnegative integer is a... |
| dfn2 12415 | The set of positive intege... |
| elnnne0 12416 | The positive integer prope... |
| 0nn0 12417 | 0 is a nonnegative integer... |
| 1nn0 12418 | 1 is a nonnegative integer... |
| 2nn0 12419 | 2 is a nonnegative integer... |
| 3nn0 12420 | 3 is a nonnegative integer... |
| 4nn0 12421 | 4 is a nonnegative integer... |
| 5nn0 12422 | 5 is a nonnegative integer... |
| 6nn0 12423 | 6 is a nonnegative integer... |
| 7nn0 12424 | 7 is a nonnegative integer... |
| 8nn0 12425 | 8 is a nonnegative integer... |
| 9nn0 12426 | 9 is a nonnegative integer... |
| nn0ge0 12427 | A nonnegative integer is g... |
| nn0nlt0 12428 | A nonnegative integer is n... |
| nn0ge0i 12429 | Nonnegative integers are n... |
| nn0le0eq0 12430 | A nonnegative integer is l... |
| nn0p1gt0 12431 | A nonnegative integer incr... |
| nnnn0addcl 12432 | A positive integer plus a ... |
| nn0nnaddcl 12433 | A nonnegative integer plus... |
| 0mnnnnn0 12434 | The result of subtracting ... |
| un0addcl 12435 | If ` S ` is closed under a... |
| un0mulcl 12436 | If ` S ` is closed under m... |
| nn0addcl 12437 | Closure of addition of non... |
| nn0mulcl 12438 | Closure of multiplication ... |
| nn0addcli 12439 | Closure of addition of non... |
| nn0mulcli 12440 | Closure of multiplication ... |
| nn0p1nn 12441 | A nonnegative integer plus... |
| peano2nn0 12442 | Second Peano postulate for... |
| nnm1nn0 12443 | A positive integer minus 1... |
| elnn0nn 12444 | The nonnegative integer pr... |
| elnnnn0 12445 | The positive integer prope... |
| elnnnn0b 12446 | The positive integer prope... |
| elnnnn0c 12447 | The positive integer prope... |
| nn0addge1 12448 | A number is less than or e... |
| nn0addge2 12449 | A number is less than or e... |
| nn0addge1i 12450 | A number is less than or e... |
| nn0addge2i 12451 | A number is less than or e... |
| nn0sub 12452 | Subtraction of nonnegative... |
| ltsubnn0 12453 | Subtracting a nonnegative ... |
| nn0negleid 12454 | A nonnegative integer is g... |
| difgtsumgt 12455 | If the difference of a rea... |
| nn0le2x 12456 | A nonnegative integer is l... |
| nn0le2xi 12457 | A nonnegative integer is l... |
| nn0lele2xi 12458 | 'Less than or equal to' im... |
| fcdmnn0supp 12459 | Two ways to write the supp... |
| fcdmnn0fsupp 12460 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12461 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12462 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12463 | A positive integer is a no... |
| nn0red 12464 | A nonnegative integer is a... |
| nn0cnd 12465 | A nonnegative integer is a... |
| nn0ge0d 12466 | A nonnegative integer is g... |
| nn0addcld 12467 | Closure of addition of non... |
| nn0mulcld 12468 | Closure of multiplication ... |
| nn0readdcl 12469 | Closure law for addition o... |
| nn0n0n1ge2 12470 | A nonnegative integer whic... |
| nn0n0n1ge2b 12471 | A nonnegative integer is n... |
| nn0ge2m1nn 12472 | If a nonnegative integer i... |
| nn0ge2m1nn0 12473 | If a nonnegative integer i... |
| nn0nndivcl 12474 | Closure law for dividing o... |
| elxnn0 12477 | An extended nonnegative in... |
| nn0ssxnn0 12478 | The standard nonnegative i... |
| nn0xnn0 12479 | A standard nonnegative int... |
| xnn0xr 12480 | An extended nonnegative in... |
| 0xnn0 12481 | Zero is an extended nonneg... |
| pnf0xnn0 12482 | Positive infinity is an ex... |
| nn0nepnf 12483 | No standard nonnegative in... |
| nn0xnn0d 12484 | A standard nonnegative int... |
| nn0nepnfd 12485 | No standard nonnegative in... |
| xnn0nemnf 12486 | No extended nonnegative in... |
| xnn0xrnemnf 12487 | The extended nonnegative i... |
| xnn0nnn0pnf 12488 | An extended nonnegative in... |
| elz 12491 | Membership in the set of i... |
| nnnegz 12492 | The negative of a positive... |
| zre 12493 | An integer is a real. (Co... |
| zcn 12494 | An integer is a complex nu... |
| zrei 12495 | An integer is a real numbe... |
| zssre 12496 | The integers are a subset ... |
| zsscn 12497 | The integers are a subset ... |
| zex 12498 | The set of integers exists... |
| elnnz 12499 | Positive integer property ... |
| 0z 12500 | Zero is an integer. (Cont... |
| 0zd 12501 | Zero is an integer, deduct... |
| elnn0z 12502 | Nonnegative integer proper... |
| elznn0nn 12503 | Integer property expressed... |
| elznn0 12504 | Integer property expressed... |
| elznn 12505 | Integer property expressed... |
| zle0orge1 12506 | There is no integer in the... |
| elz2 12507 | Membership in the set of i... |
| dfz2 12508 | Alternative definition of ... |
| zexALT 12509 | Alternate proof of ~ zex .... |
| nnz 12510 | A positive integer is an i... |
| nnssz 12511 | Positive integers are a su... |
| nn0ssz 12512 | Nonnegative integers are a... |
| nn0z 12513 | A nonnegative integer is a... |
| nn0zd 12514 | A nonnegative integer is a... |
| nnzd 12515 | A positive integer is an i... |
| nnzi 12516 | A positive integer is an i... |
| nn0zi 12517 | A nonnegative integer is a... |
| elnnz1 12518 | Positive integer property ... |
| znnnlt1 12519 | An integer is not a positi... |
| nnzrab 12520 | Positive integers expresse... |
| nn0zrab 12521 | Nonnegative integers expre... |
| 1z 12522 | One is an integer. (Contr... |
| 1zzd 12523 | One is an integer, deducti... |
| 2z 12524 | 2 is an integer. (Contrib... |
| 3z 12525 | 3 is an integer. (Contrib... |
| 4z 12526 | 4 is an integer. (Contrib... |
| znegcl 12527 | Closure law for negative i... |
| neg1z 12528 | -1 is an integer. (Contri... |
| znegclb 12529 | A complex number is an int... |
| nn0negz 12530 | The negative of a nonnegat... |
| nn0negzi 12531 | The negative of a nonnegat... |
| zaddcl 12532 | Closure of addition of int... |
| peano2z 12533 | Second Peano postulate gen... |
| zsubcl 12534 | Closure of subtraction of ... |
| peano2zm 12535 | "Reverse" second Peano pos... |
| zletr 12536 | Transitive law of ordering... |
| zrevaddcl 12537 | Reverse closure law for ad... |
| znnsub 12538 | The positive difference of... |
| znn0sub 12539 | The nonnegative difference... |
| nzadd 12540 | The sum of a real number n... |
| zmulcl 12541 | Closure of multiplication ... |
| zltp1le 12542 | Integer ordering relation.... |
| zleltp1 12543 | Integer ordering relation.... |
| zlem1lt 12544 | Integer ordering relation.... |
| zltlem1 12545 | Integer ordering relation.... |
| zltlem1d 12546 | Integer ordering relation,... |
| zgt0ge1 12547 | An integer greater than ` ... |
| nnleltp1 12548 | Positive integer ordering ... |
| nnltp1le 12549 | Positive integer ordering ... |
| nnaddm1cl 12550 | Closure of addition of pos... |
| nn0ltp1le 12551 | Nonnegative integer orderi... |
| nn0leltp1 12552 | Nonnegative integer orderi... |
| nn0ltlem1 12553 | Nonnegative integer orderi... |
| nn0sub2 12554 | Subtraction of nonnegative... |
| nn0lt10b 12555 | A nonnegative integer less... |
| nn0lt2 12556 | A nonnegative integer less... |
| nn0le2is012 12557 | A nonnegative integer whic... |
| nn0lem1lt 12558 | Nonnegative integer orderi... |
| nnlem1lt 12559 | Positive integer ordering ... |
| nnltlem1 12560 | Positive integer ordering ... |
| nnm1ge0 12561 | A positive integer decreas... |
| nn0ge0div 12562 | Division of a nonnegative ... |
| zdiv 12563 | Two ways to express " ` M ... |
| zdivadd 12564 | Property of divisibility: ... |
| zdivmul 12565 | Property of divisibility: ... |
| zextle 12566 | An extensionality-like pro... |
| zextlt 12567 | An extensionality-like pro... |
| recnz 12568 | The reciprocal of a number... |
| btwnnz 12569 | A number between an intege... |
| gtndiv 12570 | A larger number does not d... |
| halfnz 12571 | One-half is not an integer... |
| 3halfnz 12572 | Three halves is not an int... |
| suprzcl 12573 | The supremum of a bounded-... |
| prime 12574 | Two ways to express " ` A ... |
| msqznn 12575 | The square of a nonzero in... |
| zneo 12576 | No even integer equals an ... |
| nneo 12577 | A positive integer is even... |
| nneoi 12578 | A positive integer is even... |
| zeo 12579 | An integer is even or odd.... |
| zeo2 12580 | An integer is even or odd ... |
| peano2uz2 12581 | Second Peano postulate for... |
| peano5uzi 12582 | Peano's inductive postulat... |
| peano5uzti 12583 | Peano's inductive postulat... |
| dfuzi 12584 | An expression for the uppe... |
| uzind 12585 | Induction on the upper int... |
| uzind2 12586 | Induction on the upper int... |
| uzind3 12587 | Induction on the upper int... |
| nn0ind 12588 | Principle of Mathematical ... |
| nn0indALT 12589 | Principle of Mathematical ... |
| nn0indd 12590 | Principle of Mathematical ... |
| fzind 12591 | Induction on the integers ... |
| fnn0ind 12592 | Induction on the integers ... |
| nn0ind-raph 12593 | Principle of Mathematical ... |
| zindd 12594 | Principle of Mathematical ... |
| fzindd 12595 | Induction on the integers ... |
| btwnz 12596 | Any real number can be san... |
| zred 12597 | An integer is a real numbe... |
| zcnd 12598 | An integer is a complex nu... |
| znegcld 12599 | Closure law for negative i... |
| peano2zd 12600 | Deduction from second Pean... |
| zaddcld 12601 | Closure of addition of int... |
| zsubcld 12602 | Closure of subtraction of ... |
| zmulcld 12603 | Closure of multiplication ... |
| znnn0nn 12604 | The negative of a negative... |
| zadd2cl 12605 | Increasing an integer by 2... |
| zriotaneg 12606 | The negative of the unique... |
| suprfinzcl 12607 | The supremum of a nonempty... |
| 9p1e10 12610 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12611 | Version of the definition ... |
| decex 12612 | A decimal number is a set.... |
| deceq1 12613 | Equality theorem for the d... |
| deceq2 12614 | Equality theorem for the d... |
| deceq1i 12615 | Equality theorem for the d... |
| deceq2i 12616 | Equality theorem for the d... |
| deceq12i 12617 | Equality theorem for the d... |
| numnncl 12618 | Closure for a numeral (wit... |
| num0u 12619 | Add a zero in the units pl... |
| num0h 12620 | Add a zero in the higher p... |
| numcl 12621 | Closure for a decimal inte... |
| numsuc 12622 | The successor of a decimal... |
| deccl 12623 | Closure for a numeral. (C... |
| 10nn 12624 | 10 is a positive integer. ... |
| 10pos 12625 | The number 10 is positive.... |
| 10nn0 12626 | 10 is a nonnegative intege... |
| 10re 12627 | The number 10 is real. (C... |
| decnncl 12628 | Closure for a numeral. (C... |
| dec0u 12629 | Add a zero in the units pl... |
| dec0h 12630 | Add a zero in the higher p... |
| numnncl2 12631 | Closure for a decimal inte... |
| decnncl2 12632 | Closure for a decimal inte... |
| numlt 12633 | Comparing two decimal inte... |
| numltc 12634 | Comparing two decimal inte... |
| le9lt10 12635 | A "decimal digit" (i.e. a ... |
| declt 12636 | Comparing two decimal inte... |
| decltc 12637 | Comparing two decimal inte... |
| declth 12638 | Comparing two decimal inte... |
| decsuc 12639 | The successor of a decimal... |
| 3declth 12640 | Comparing two decimal inte... |
| 3decltc 12641 | Comparing two decimal inte... |
| decle 12642 | Comparing two decimal inte... |
| decleh 12643 | Comparing two decimal inte... |
| declei 12644 | Comparing a digit to a dec... |
| numlti 12645 | Comparing a digit to a dec... |
| declti 12646 | Comparing a digit to a dec... |
| decltdi 12647 | Comparing a digit to a dec... |
| numsucc 12648 | The successor of a decimal... |
| decsucc 12649 | The successor of a decimal... |
| 1e0p1 12650 | The successor of zero. (C... |
| dec10p 12651 | Ten plus an integer. (Con... |
| numma 12652 | Perform a multiply-add of ... |
| nummac 12653 | Perform a multiply-add of ... |
| numma2c 12654 | Perform a multiply-add of ... |
| numadd 12655 | Add two decimal integers `... |
| numaddc 12656 | Add two decimal integers `... |
| nummul1c 12657 | The product of a decimal i... |
| nummul2c 12658 | The product of a decimal i... |
| decma 12659 | Perform a multiply-add of ... |
| decmac 12660 | Perform a multiply-add of ... |
| decma2c 12661 | Perform a multiply-add of ... |
| decadd 12662 | Add two numerals ` M ` and... |
| decaddc 12663 | Add two numerals ` M ` and... |
| decaddc2 12664 | Add two numerals ` M ` and... |
| decrmanc 12665 | Perform a multiply-add of ... |
| decrmac 12666 | Perform a multiply-add of ... |
| decaddm10 12667 | The sum of two multiples o... |
| decaddi 12668 | Add two numerals ` M ` and... |
| decaddci 12669 | Add two numerals ` M ` and... |
| decaddci2 12670 | Add two numerals ` M ` and... |
| decsubi 12671 | Difference between a numer... |
| decmul1 12672 | The product of a numeral w... |
| decmul1c 12673 | The product of a numeral w... |
| decmul2c 12674 | The product of a numeral w... |
| decmulnc 12675 | The product of a numeral w... |
| 11multnc 12676 | The product of 11 (as nume... |
| decmul10add 12677 | A multiplication of a numb... |
| 6p5lem 12678 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12679 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12680 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12681 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12682 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12683 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12684 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12685 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12686 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12687 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12688 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12689 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12690 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12691 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12692 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12693 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12694 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12695 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12696 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12697 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12698 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12699 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12700 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12701 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12702 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12703 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12704 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12705 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12706 | 4 times 3 equals 12. (Con... |
| 4t4e16 12707 | 4 times 4 equals 16. (Con... |
| 5t2e10 12708 | 5 times 2 equals 10. (Con... |
| 5t3e15 12709 | 5 times 3 equals 15. (Con... |
| 5t4e20 12710 | 5 times 4 equals 20. (Con... |
| 5t5e25 12711 | 5 times 5 equals 25. (Con... |
| 6t2e12 12712 | 6 times 2 equals 12. (Con... |
| 6t3e18 12713 | 6 times 3 equals 18. (Con... |
| 6t4e24 12714 | 6 times 4 equals 24. (Con... |
| 6t5e30 12715 | 6 times 5 equals 30. (Con... |
| 6t6e36 12716 | 6 times 6 equals 36. (Con... |
| 7t2e14 12717 | 7 times 2 equals 14. (Con... |
| 7t3e21 12718 | 7 times 3 equals 21. (Con... |
| 7t4e28 12719 | 7 times 4 equals 28. (Con... |
| 7t5e35 12720 | 7 times 5 equals 35. (Con... |
| 7t6e42 12721 | 7 times 6 equals 42. (Con... |
| 7t7e49 12722 | 7 times 7 equals 49. (Con... |
| 8t2e16 12723 | 8 times 2 equals 16. (Con... |
| 8t3e24 12724 | 8 times 3 equals 24. (Con... |
| 8t4e32 12725 | 8 times 4 equals 32. (Con... |
| 8t5e40 12726 | 8 times 5 equals 40. (Con... |
| 8t6e48 12727 | 8 times 6 equals 48. (Con... |
| 8t7e56 12728 | 8 times 7 equals 56. (Con... |
| 8t8e64 12729 | 8 times 8 equals 64. (Con... |
| 9t2e18 12730 | 9 times 2 equals 18. (Con... |
| 9t3e27 12731 | 9 times 3 equals 27. (Con... |
| 9t4e36 12732 | 9 times 4 equals 36. (Con... |
| 9t5e45 12733 | 9 times 5 equals 45. (Con... |
| 9t6e54 12734 | 9 times 6 equals 54. (Con... |
| 9t7e63 12735 | 9 times 7 equals 63. (Con... |
| 9t8e72 12736 | 9 times 8 equals 72. (Con... |
| 9t9e81 12737 | 9 times 9 equals 81. (Con... |
| 9t11e99 12738 | 9 times 11 equals 99. (Co... |
| 9lt10 12739 | 9 is less than 10. (Contr... |
| 8lt10 12740 | 8 is less than 10. (Contr... |
| 7lt10 12741 | 7 is less than 10. (Contr... |
| 6lt10 12742 | 6 is less than 10. (Contr... |
| 5lt10 12743 | 5 is less than 10. (Contr... |
| 4lt10 12744 | 4 is less than 10. (Contr... |
| 3lt10 12745 | 3 is less than 10. (Contr... |
| 2lt10 12746 | 2 is less than 10. (Contr... |
| 1lt10 12747 | 1 is less than 10. (Contr... |
| decbin0 12748 | Decompose base 4 into base... |
| decbin2 12749 | Decompose base 4 into base... |
| decbin3 12750 | Decompose base 4 into base... |
| 5recm6rec 12751 | One fifth minus one sixth.... |
| uzval 12754 | The value of the upper int... |
| uzf 12755 | The domain and codomain of... |
| eluz1 12756 | Membership in the upper se... |
| eluzel2 12757 | Implication of membership ... |
| eluz2 12758 | Membership in an upper set... |
| eluzmn 12759 | Membership in an earlier u... |
| eluz1i 12760 | Membership in an upper set... |
| eluzuzle 12761 | An integer in an upper set... |
| eluzelz 12762 | A member of an upper set o... |
| eluzelre 12763 | A member of an upper set o... |
| eluzelcn 12764 | A member of an upper set o... |
| eluzle 12765 | Implication of membership ... |
| eluz 12766 | Membership in an upper set... |
| uzid 12767 | Membership of the least me... |
| uzidd 12768 | Membership of the least me... |
| uzn0 12769 | The upper integers are all... |
| uztrn 12770 | Transitive law for sets of... |
| uztrn2 12771 | Transitive law for sets of... |
| uzneg 12772 | Contraposition law for upp... |
| uzssz 12773 | An upper set of integers i... |
| uzssre 12774 | An upper set of integers i... |
| uzss 12775 | Subset relationship for tw... |
| uztric 12776 | Totality of the ordering r... |
| uz11 12777 | The upper integers functio... |
| eluzp1m1 12778 | Membership in the next upp... |
| eluzp1l 12779 | Strict ordering implied by... |
| eluzp1p1 12780 | Membership in the next upp... |
| eluzadd 12781 | Membership in a later uppe... |
| eluzsub 12782 | Membership in an earlier u... |
| eluzaddi 12783 | Membership in a later uppe... |
| eluzsubi 12784 | Membership in an earlier u... |
| subeluzsub 12785 | Membership of a difference... |
| uzm1 12786 | Choices for an element of ... |
| uznn0sub 12787 | The nonnegative difference... |
| uzin 12788 | Intersection of two upper ... |
| uzp1 12789 | Choices for an element of ... |
| nn0uz 12790 | Nonnegative integers expre... |
| nnuz 12791 | Positive integers expresse... |
| elnnuz 12792 | A positive integer express... |
| elnn0uz 12793 | A nonnegative integer expr... |
| 1eluzge0 12794 | 1 is an integer greater th... |
| 2eluzge0 12795 | 2 is an integer greater th... |
| 2eluzge1 12796 | 2 is an integer greater th... |
| 5eluz3 12797 | 5 is an integer greater th... |
| uzuzle23 12798 | An integer greater than or... |
| uzuzle24 12799 | An integer greater than or... |
| uzuzle34 12800 | An integer greater than or... |
| uzuzle35 12801 | An integer greater than or... |
| eluz2nn 12802 | An integer greater than or... |
| eluz3nn 12803 | An integer greater than or... |
| eluz4nn 12804 | An integer greater than or... |
| eluz5nn 12805 | An integer greater than or... |
| eluzge2nn0 12806 | If an integer is greater t... |
| eluz2n0 12807 | An integer greater than or... |
| uz3m2nn 12808 | An integer greater than or... |
| uznnssnn 12809 | The upper integers startin... |
| raluz 12810 | Restricted universal quant... |
| raluz2 12811 | Restricted universal quant... |
| rexuz 12812 | Restricted existential qua... |
| rexuz2 12813 | Restricted existential qua... |
| 2rexuz 12814 | Double existential quantif... |
| peano2uz 12815 | Second Peano postulate for... |
| peano2uzs 12816 | Second Peano postulate for... |
| peano2uzr 12817 | Reversed second Peano axio... |
| uzaddcl 12818 | Addition closure law for a... |
| nn0pzuz 12819 | The sum of a nonnegative i... |
| uzind4 12820 | Induction on the upper set... |
| uzind4ALT 12821 | Induction on the upper set... |
| uzind4s 12822 | Induction on the upper set... |
| uzind4s2 12823 | Induction on the upper set... |
| uzind4i 12824 | Induction on the upper int... |
| uzwo 12825 | Well-ordering principle: a... |
| uzwo2 12826 | Well-ordering principle: a... |
| nnwo 12827 | Well-ordering principle: a... |
| nnwof 12828 | Well-ordering principle: a... |
| nnwos 12829 | Well-ordering principle: a... |
| indstr 12830 | Strong Mathematical Induct... |
| eluznn0 12831 | Membership in a nonnegativ... |
| eluznn 12832 | Membership in a positive u... |
| eluz2b1 12833 | Two ways to say "an intege... |
| eluz2gt1 12834 | An integer greater than or... |
| eluz2b2 12835 | Two ways to say "an intege... |
| eluz2b3 12836 | Two ways to say "an intege... |
| uz2m1nn 12837 | One less than an integer g... |
| 1nuz2 12838 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12839 | A positive integer is eith... |
| uz2mulcl 12840 | Closure of multiplication ... |
| indstr2 12841 | Strong Mathematical Induct... |
| uzinfi 12842 | Extract the lower bound of... |
| nninf 12843 | The infimum of the set of ... |
| nn0inf 12844 | The infimum of the set of ... |
| infssuzle 12845 | The infimum of a subset of... |
| infssuzcl 12846 | The infimum of a subset of... |
| ublbneg 12847 | The image under negation o... |
| eqreznegel 12848 | Two ways to express the im... |
| supminf 12849 | The supremum of a bounded-... |
| lbzbi 12850 | If a set of reals is bound... |
| zsupss 12851 | Any nonempty bounded subse... |
| suprzcl2 12852 | The supremum of a bounded-... |
| suprzub 12853 | The supremum of a bounded-... |
| uzsupss 12854 | Any bounded subset of an u... |
| nn01to3 12855 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12856 | Alternate proof of ~ nn0ge... |
| uzwo3 12857 | Well-ordering principle: a... |
| zmin 12858 | There is a unique smallest... |
| zmax 12859 | There is a unique largest ... |
| zbtwnre 12860 | There is a unique integer ... |
| rebtwnz 12861 | There is a unique greatest... |
| elq 12864 | Membership in the set of r... |
| qmulz 12865 | If ` A ` is rational, then... |
| znq 12866 | The ratio of an integer an... |
| qre 12867 | A rational number is a rea... |
| zq 12868 | An integer is a rational n... |
| qred 12869 | A rational number is a rea... |
| zssq 12870 | The integers are a subset ... |
| nn0ssq 12871 | The nonnegative integers a... |
| nnssq 12872 | The positive integers are ... |
| qssre 12873 | The rationals are a subset... |
| qsscn 12874 | The rationals are a subset... |
| qex 12875 | The set of rational number... |
| nnq 12876 | A positive integer is rati... |
| qcn 12877 | A rational number is a com... |
| qexALT 12878 | Alternate proof of ~ qex .... |
| qaddcl 12879 | Closure of addition of rat... |
| qnegcl 12880 | Closure law for the negati... |
| qmulcl 12881 | Closure of multiplication ... |
| qsubcl 12882 | Closure of subtraction of ... |
| qreccl 12883 | Closure of reciprocal of r... |
| qdivcl 12884 | Closure of division of rat... |
| qrevaddcl 12885 | Reverse closure law for ad... |
| nnrecq 12886 | The reciprocal of a positi... |
| irradd 12887 | The sum of an irrational n... |
| irrmul 12888 | The product of an irration... |
| elpq 12889 | A positive rational is the... |
| elpqb 12890 | A class is a positive rati... |
| rpnnen1lem2 12891 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12892 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12893 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12894 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12895 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12896 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12897 | One half of ~ rpnnen , whe... |
| reexALT 12898 | Alternate proof of ~ reex ... |
| cnref1o 12899 | There is a natural one-to-... |
| cnexALT 12900 | The set of complex numbers... |
| xrex 12901 | The set of extended reals ... |
| mpoaddex 12902 | The addition operation is ... |
| addex 12903 | The addition operation is ... |
| mpomulex 12904 | The multiplication operati... |
| mulex 12905 | The multiplication operati... |
| elrp 12908 | Membership in the set of p... |
| elrpii 12909 | Membership in the set of p... |
| 1rp 12910 | 1 is a positive real. (Co... |
| 2rp 12911 | 2 is a positive real. (Co... |
| 3rp 12912 | 3 is a positive real. (Co... |
| 5rp 12913 | 5 is a positive real. (Co... |
| rpssre 12914 | The positive reals are a s... |
| rpre 12915 | A positive real is a real.... |
| rpxr 12916 | A positive real is an exte... |
| rpcn 12917 | A positive real is a compl... |
| nnrp 12918 | A positive integer is a po... |
| rpgt0 12919 | A positive real is greater... |
| rpge0 12920 | A positive real is greater... |
| rpregt0 12921 | A positive real is a posit... |
| rprege0 12922 | A positive real is a nonne... |
| rpne0 12923 | A positive real is nonzero... |
| rprene0 12924 | A positive real is a nonze... |
| rpcnne0 12925 | A positive real is a nonze... |
| neglt 12926 | The negative of a positive... |
| rpcndif0 12927 | A positive real number is ... |
| ralrp 12928 | Quantification over positi... |
| rexrp 12929 | Quantification over positi... |
| rpaddcl 12930 | Closure law for addition o... |
| rpmulcl 12931 | Closure law for multiplica... |
| rpmtmip 12932 | "Minus times minus is plus... |
| rpdivcl 12933 | Closure law for division o... |
| rpreccl 12934 | Closure law for reciprocat... |
| rphalfcl 12935 | Closure law for half of a ... |
| rpgecl 12936 | A number greater than or e... |
| rphalflt 12937 | Half of a positive real is... |
| rerpdivcl 12938 | Closure law for division o... |
| ge0p1rp 12939 | A nonnegative number plus ... |
| rpneg 12940 | Either a nonzero real or i... |
| negelrp 12941 | Elementhood of a negation ... |
| negelrpd 12942 | The negation of a negative... |
| 0nrp 12943 | Zero is not a positive rea... |
| ltsubrp 12944 | Subtracting a positive rea... |
| ltaddrp 12945 | Adding a positive number t... |
| difrp 12946 | Two ways to say one number... |
| elrpd 12947 | Membership in the set of p... |
| nnrpd 12948 | A positive integer is a po... |
| zgt1rpn0n1 12949 | An integer greater than 1 ... |
| rpred 12950 | A positive real is a real.... |
| rpxrd 12951 | A positive real is an exte... |
| rpcnd 12952 | A positive real is a compl... |
| rpgt0d 12953 | A positive real is greater... |
| rpge0d 12954 | A positive real is greater... |
| rpne0d 12955 | A positive real is nonzero... |
| rpregt0d 12956 | A positive real is real an... |
| rprege0d 12957 | A positive real is real an... |
| rprene0d 12958 | A positive real is a nonze... |
| rpcnne0d 12959 | A positive real is a nonze... |
| rpreccld 12960 | Closure law for reciprocat... |
| rprecred 12961 | Closure law for reciprocat... |
| rphalfcld 12962 | Closure law for half of a ... |
| reclt1d 12963 | The reciprocal of a positi... |
| recgt1d 12964 | The reciprocal of a positi... |
| rpaddcld 12965 | Closure law for addition o... |
| rpmulcld 12966 | Closure law for multiplica... |
| rpdivcld 12967 | Closure law for division o... |
| ltrecd 12968 | The reciprocal of both sid... |
| lerecd 12969 | The reciprocal of both sid... |
| ltrec1d 12970 | Reciprocal swap in a 'less... |
| lerec2d 12971 | Reciprocal swap in a 'less... |
| lediv2ad 12972 | Division of both sides of ... |
| ltdiv2d 12973 | Division of a positive num... |
| lediv2d 12974 | Division of a positive num... |
| ledivdivd 12975 | Invert ratios of positive ... |
| divge1 12976 | The ratio of a number over... |
| divlt1lt 12977 | A real number divided by a... |
| divle1le 12978 | A real number divided by a... |
| ledivge1le 12979 | If a number is less than o... |
| ge0p1rpd 12980 | A nonnegative number plus ... |
| rerpdivcld 12981 | Closure law for division o... |
| ltsubrpd 12982 | Subtracting a positive rea... |
| ltaddrpd 12983 | Adding a positive number t... |
| ltaddrp2d 12984 | Adding a positive number t... |
| ltmulgt11d 12985 | Multiplication by a number... |
| ltmulgt12d 12986 | Multiplication by a number... |
| gt0divd 12987 | Division of a positive num... |
| ge0divd 12988 | Division of a nonnegative ... |
| rpgecld 12989 | A number greater than or e... |
| divge0d 12990 | The ratio of nonnegative a... |
| ltmul1d 12991 | The ratio of nonnegative a... |
| ltmul2d 12992 | Multiplication of both sid... |
| lemul1d 12993 | Multiplication of both sid... |
| lemul2d 12994 | Multiplication of both sid... |
| ltdiv1d 12995 | Division of both sides of ... |
| lediv1d 12996 | Division of both sides of ... |
| ltmuldivd 12997 | 'Less than' relationship b... |
| ltmuldiv2d 12998 | 'Less than' relationship b... |
| lemuldivd 12999 | 'Less than or equal to' re... |
| lemuldiv2d 13000 | 'Less than or equal to' re... |
| ltdivmuld 13001 | 'Less than' relationship b... |
| ltdivmul2d 13002 | 'Less than' relationship b... |
| ledivmuld 13003 | 'Less than or equal to' re... |
| ledivmul2d 13004 | 'Less than or equal to' re... |
| ltmul1dd 13005 | The ratio of nonnegative a... |
| ltmul2dd 13006 | Multiplication of both sid... |
| ltdiv1dd 13007 | Division of both sides of ... |
| lediv1dd 13008 | Division of both sides of ... |
| lediv12ad 13009 | Comparison of ratio of two... |
| mul2lt0rlt0 13010 | If the result of a multipl... |
| mul2lt0rgt0 13011 | If the result of a multipl... |
| mul2lt0llt0 13012 | If the result of a multipl... |
| mul2lt0lgt0 13013 | If the result of a multipl... |
| mul2lt0bi 13014 | If the result of a multipl... |
| prodge0rd 13015 | Infer that a multiplicand ... |
| prodge0ld 13016 | Infer that a multiplier is... |
| ltdiv23d 13017 | Swap denominator with othe... |
| lediv23d 13018 | Swap denominator with othe... |
| lt2mul2divd 13019 | The ratio of nonnegative a... |
| nnledivrp 13020 | Division of a positive int... |
| nn0ledivnn 13021 | Division of a nonnegative ... |
| addlelt 13022 | If the sum of a real numbe... |
| ge2halflem1 13023 | Half of an integer greater... |
| ltxr 13030 | The 'less than' binary rel... |
| elxr 13031 | Membership in the set of e... |
| xrnemnf 13032 | An extended real other tha... |
| xrnepnf 13033 | An extended real other tha... |
| xrltnr 13034 | The extended real 'less th... |
| ltpnf 13035 | Any (finite) real is less ... |
| ltpnfd 13036 | Any (finite) real is less ... |
| 0ltpnf 13037 | Zero is less than plus inf... |
| mnflt 13038 | Minus infinity is less tha... |
| mnfltd 13039 | Minus infinity is less tha... |
| mnflt0 13040 | Minus infinity is less tha... |
| mnfltpnf 13041 | Minus infinity is less tha... |
| mnfltxr 13042 | Minus infinity is less tha... |
| pnfnlt 13043 | No extended real is greate... |
| nltmnf 13044 | No extended real is less t... |
| pnfge 13045 | Plus infinity is an upper ... |
| pnfged 13046 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13047 | An extended nonnegative in... |
| 0lepnf 13048 | 0 less than or equal to po... |
| xnn0ge0 13049 | An extended nonnegative in... |
| mnfle 13050 | Minus infinity is less tha... |
| mnfled 13051 | Minus infinity is less tha... |
| xrltnsym 13052 | Ordering on the extended r... |
| xrltnsym2 13053 | 'Less than' is antisymmetr... |
| xrlttri 13054 | Ordering on the extended r... |
| xrlttr 13055 | Ordering on the extended r... |
| xrltso 13056 | 'Less than' is a strict or... |
| xrlttri2 13057 | Trichotomy law for 'less t... |
| xrlttri3 13058 | Trichotomy law for 'less t... |
| xrleloe 13059 | 'Less than or equal' expre... |
| xrleltne 13060 | 'Less than or equal to' im... |
| xrltlen 13061 | 'Less than' expressed in t... |
| dfle2 13062 | Alternative definition of ... |
| dflt2 13063 | Alternative definition of ... |
| xrltle 13064 | 'Less than' implies 'less ... |
| xrltled 13065 | 'Less than' implies 'less ... |
| xrleid 13066 | 'Less than or equal to' is... |
| xrleidd 13067 | 'Less than or equal to' is... |
| xrletri 13068 | Trichotomy law for extende... |
| xrletri3 13069 | Trichotomy law for extende... |
| xrletrid 13070 | Trichotomy law for extende... |
| xrlelttr 13071 | Transitive law for orderin... |
| xrltletr 13072 | Transitive law for orderin... |
| xrletr 13073 | Transitive law for orderin... |
| xrlttrd 13074 | Transitive law for orderin... |
| xrlelttrd 13075 | Transitive law for orderin... |
| xrltletrd 13076 | Transitive law for orderin... |
| xrletrd 13077 | Transitive law for orderin... |
| xrltne 13078 | 'Less than' implies not eq... |
| xrgtned 13079 | 'Greater than' implies not... |
| nltpnft 13080 | An extended real is not le... |
| xgepnf 13081 | An extended real which is ... |
| ngtmnft 13082 | An extended real is not gr... |
| xlemnf 13083 | An extended real which is ... |
| xrrebnd 13084 | An extended real is real i... |
| xrre 13085 | A way of proving that an e... |
| xrre2 13086 | An extended real between t... |
| xrre3 13087 | A way of proving that an e... |
| ge0gtmnf 13088 | A nonnegative extended rea... |
| ge0nemnf 13089 | A nonnegative extended rea... |
| xrrege0 13090 | A nonnegative extended rea... |
| xrmax1 13091 | An extended real is less t... |
| xrmax2 13092 | An extended real is less t... |
| xrmin1 13093 | The minimum of two extende... |
| xrmin2 13094 | The minimum of two extende... |
| xrmaxeq 13095 | The maximum of two extende... |
| xrmineq 13096 | The minimum of two extende... |
| xrmaxlt 13097 | Two ways of saying the max... |
| xrltmin 13098 | Two ways of saying an exte... |
| xrmaxle 13099 | Two ways of saying the max... |
| xrlemin 13100 | Two ways of saying a numbe... |
| max1 13101 | A number is less than or e... |
| max1ALT 13102 | A number is less than or e... |
| max2 13103 | A number is less than or e... |
| 2resupmax 13104 | The supremum of two real n... |
| min1 13105 | The minimum of two numbers... |
| min2 13106 | The minimum of two numbers... |
| maxle 13107 | Two ways of saying the max... |
| lemin 13108 | Two ways of saying a numbe... |
| maxlt 13109 | Two ways of saying the max... |
| ltmin 13110 | Two ways of saying a numbe... |
| lemaxle 13111 | A real number which is les... |
| max0sub 13112 | Decompose a real number in... |
| ifle 13113 | An if statement transforms... |
| z2ge 13114 | There exists an integer gr... |
| qbtwnre 13115 | The rational numbers are d... |
| qbtwnxr 13116 | The rational numbers are d... |
| qsqueeze 13117 | If a nonnegative real is l... |
| qextltlem 13118 | Lemma for ~ qextlt and qex... |
| qextlt 13119 | An extensionality-like pro... |
| qextle 13120 | An extensionality-like pro... |
| xralrple 13121 | Show that ` A ` is less th... |
| alrple 13122 | Show that ` A ` is less th... |
| xnegeq 13123 | Equality of two extended n... |
| xnegex 13124 | A negative extended real e... |
| xnegpnf 13125 | Minus ` +oo ` . Remark of... |
| xnegmnf 13126 | Minus ` -oo ` . Remark of... |
| rexneg 13127 | Minus a real number. Rema... |
| xneg0 13128 | The negative of zero. (Co... |
| xnegcl 13129 | Closure of extended real n... |
| xnegneg 13130 | Extended real version of ~... |
| xneg11 13131 | Extended real version of ~... |
| xltnegi 13132 | Forward direction of ~ xlt... |
| xltneg 13133 | Extended real version of ~... |
| xleneg 13134 | Extended real version of ~... |
| xlt0neg1 13135 | Extended real version of ~... |
| xlt0neg2 13136 | Extended real version of ~... |
| xle0neg1 13137 | Extended real version of ~... |
| xle0neg2 13138 | Extended real version of ~... |
| xaddval 13139 | Value of the extended real... |
| xaddf 13140 | The extended real addition... |
| xmulval 13141 | Value of the extended real... |
| xaddpnf1 13142 | Addition of positive infin... |
| xaddpnf2 13143 | Addition of positive infin... |
| xaddmnf1 13144 | Addition of negative infin... |
| xaddmnf2 13145 | Addition of negative infin... |
| pnfaddmnf 13146 | Addition of positive and n... |
| mnfaddpnf 13147 | Addition of negative and p... |
| rexadd 13148 | The extended real addition... |
| rexsub 13149 | Extended real subtraction ... |
| rexaddd 13150 | The extended real addition... |
| xnn0xaddcl 13151 | The extended nonnegative i... |
| xaddnemnf 13152 | Closure of extended real a... |
| xaddnepnf 13153 | Closure of extended real a... |
| xnegid 13154 | Extended real version of ~... |
| xaddcl 13155 | The extended real addition... |
| xaddcom 13156 | The extended real addition... |
| xaddrid 13157 | Extended real version of ~... |
| xaddlid 13158 | Extended real version of ~... |
| xaddridd 13159 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13160 | Extended nonnegative integ... |
| xnn0lenn0nn0 13161 | An extended nonnegative in... |
| xnn0le2is012 13162 | An extended nonnegative in... |
| xnn0xadd0 13163 | The sum of two extended no... |
| xnegdi 13164 | Extended real version of ~... |
| xaddass 13165 | Associativity of extended ... |
| xaddass2 13166 | Associativity of extended ... |
| xpncan 13167 | Extended real version of ~... |
| xnpcan 13168 | Extended real version of ~... |
| xleadd1a 13169 | Extended real version of ~... |
| xleadd2a 13170 | Commuted form of ~ xleadd1... |
| xleadd1 13171 | Weakened version of ~ xlea... |
| xltadd1 13172 | Extended real version of ~... |
| xltadd2 13173 | Extended real version of ~... |
| xaddge0 13174 | The sum of nonnegative ext... |
| xle2add 13175 | Extended real version of ~... |
| xlt2add 13176 | Extended real version of ~... |
| xsubge0 13177 | Extended real version of ~... |
| xposdif 13178 | Extended real version of ~... |
| xlesubadd 13179 | Under certain conditions, ... |
| xmullem 13180 | Lemma for ~ rexmul . (Con... |
| xmullem2 13181 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13182 | Extended real multiplicati... |
| xmul01 13183 | Extended real version of ~... |
| xmul02 13184 | Extended real version of ~... |
| xmulneg1 13185 | Extended real version of ~... |
| xmulneg2 13186 | Extended real version of ~... |
| rexmul 13187 | The extended real multipli... |
| xmulf 13188 | The extended real multipli... |
| xmulcl 13189 | Closure of extended real m... |
| xmulpnf1 13190 | Multiplication by plus inf... |
| xmulpnf2 13191 | Multiplication by plus inf... |
| xmulmnf1 13192 | Multiplication by minus in... |
| xmulmnf2 13193 | Multiplication by minus in... |
| xmulpnf1n 13194 | Multiplication by plus inf... |
| xmulrid 13195 | Extended real version of ~... |
| xmullid 13196 | Extended real version of ~... |
| xmulm1 13197 | Extended real version of ~... |
| xmulasslem2 13198 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13199 | Extended real version of ~... |
| xmulge0 13200 | Extended real version of ~... |
| xmulasslem 13201 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13202 | Lemma for ~ xmulass . (Co... |
| xmulass 13203 | Associativity of the exten... |
| xlemul1a 13204 | Extended real version of ~... |
| xlemul2a 13205 | Extended real version of ~... |
| xlemul1 13206 | Extended real version of ~... |
| xlemul2 13207 | Extended real version of ~... |
| xltmul1 13208 | Extended real version of ~... |
| xltmul2 13209 | Extended real version of ~... |
| xadddilem 13210 | Lemma for ~ xadddi . (Con... |
| xadddi 13211 | Distributive property for ... |
| xadddir 13212 | Commuted version of ~ xadd... |
| xadddi2 13213 | The assumption that the mu... |
| xadddi2r 13214 | Commuted version of ~ xadd... |
| x2times 13215 | Extended real version of ~... |
| xnegcld 13216 | Closure of extended real n... |
| xaddcld 13217 | The extended real addition... |
| xmulcld 13218 | Closure of extended real m... |
| xadd4d 13219 | Rearrangement of 4 terms i... |
| xnn0add4d 13220 | Rearrangement of 4 terms i... |
| xrsupexmnf 13221 | Adding minus infinity to a... |
| xrinfmexpnf 13222 | Adding plus infinity to a ... |
| xrsupsslem 13223 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13224 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13225 | Any subset of extended rea... |
| xrinfmss 13226 | Any subset of extended rea... |
| xrinfmss2 13227 | Any subset of extended rea... |
| xrub 13228 | By quantifying only over r... |
| supxr 13229 | The supremum of a set of e... |
| supxr2 13230 | The supremum of a set of e... |
| supxrcl 13231 | The supremum of an arbitra... |
| supxrun 13232 | The supremum of the union ... |
| supxrmnf 13233 | Adding minus infinity to a... |
| supxrpnf 13234 | The supremum of a set of e... |
| supxrunb1 13235 | The supremum of an unbound... |
| supxrunb2 13236 | The supremum of an unbound... |
| supxrbnd1 13237 | The supremum of a bounded-... |
| supxrbnd2 13238 | The supremum of a bounded-... |
| xrsup0 13239 | The supremum of an empty s... |
| supxrub 13240 | A member of a set of exten... |
| supxrlub 13241 | The supremum of a set of e... |
| supxrleub 13242 | The supremum of a set of e... |
| supxrre 13243 | The real and extended real... |
| supxrbnd 13244 | The supremum of a bounded-... |
| supxrgtmnf 13245 | The supremum of a nonempty... |
| supxrre1 13246 | The supremum of a nonempty... |
| supxrre2 13247 | The supremum of a nonempty... |
| supxrss 13248 | Smaller sets of extended r... |
| xrsupssd 13249 | Inequality deduction for s... |
| infxrcl 13250 | The infimum of an arbitrar... |
| infxrlb 13251 | A member of a set of exten... |
| infxrgelb 13252 | The infimum of a set of ex... |
| infxrre 13253 | The real and extended real... |
| infxrmnf 13254 | The infinimum of a set of ... |
| xrinf0 13255 | The infimum of the empty s... |
| infxrss 13256 | Larger sets of extended re... |
| reltre 13257 | For all real numbers there... |
| rpltrp 13258 | For all positive real numb... |
| reltxrnmnf 13259 | For all extended real numb... |
| infmremnf 13260 | The infimum of the reals i... |
| infmrp1 13261 | The infimum of the positiv... |
| ixxval 13270 | Value of the interval func... |
| elixx1 13271 | Membership in an interval ... |
| ixxf 13272 | The set of intervals of ex... |
| ixxex 13273 | The set of intervals of ex... |
| ixxssxr 13274 | The set of intervals of ex... |
| elixx3g 13275 | Membership in a set of ope... |
| ixxssixx 13276 | An interval is a subset of... |
| ixxdisj 13277 | Split an interval into dis... |
| ixxun 13278 | Split an interval into two... |
| ixxin 13279 | Intersection of two interv... |
| ixxss1 13280 | Subset relationship for in... |
| ixxss2 13281 | Subset relationship for in... |
| ixxss12 13282 | Subset relationship for in... |
| ixxub 13283 | Extract the upper bound of... |
| ixxlb 13284 | Extract the lower bound of... |
| iooex 13285 | The set of open intervals ... |
| iooval 13286 | Value of the open interval... |
| ioo0 13287 | An empty open interval of ... |
| ioon0 13288 | An open interval of extend... |
| ndmioo 13289 | The open interval function... |
| iooid 13290 | An open interval with iden... |
| elioo3g 13291 | Membership in a set of ope... |
| elioore 13292 | A member of an open interv... |
| lbioo 13293 | An open interval does not ... |
| ubioo 13294 | An open interval does not ... |
| iooval2 13295 | Value of the open interval... |
| iooin 13296 | Intersection of two open i... |
| iooss1 13297 | Subset relationship for op... |
| iooss2 13298 | Subset relationship for op... |
| iocval 13299 | Value of the open-below, c... |
| icoval 13300 | Value of the closed-below,... |
| iccval 13301 | Value of the closed interv... |
| elioo1 13302 | Membership in an open inte... |
| elioo2 13303 | Membership in an open inte... |
| elioc1 13304 | Membership in an open-belo... |
| elico1 13305 | Membership in a closed-bel... |
| elicc1 13306 | Membership in a closed int... |
| iccid 13307 | A closed interval with ide... |
| ico0 13308 | An empty open interval of ... |
| ioc0 13309 | An empty open interval of ... |
| icc0 13310 | An empty closed interval o... |
| dfrp2 13311 | Alternate definition of th... |
| elicod 13312 | Membership in a left-close... |
| icogelb 13313 | An element of a left-close... |
| icogelbd 13314 | An element of a left-close... |
| elicore 13315 | A member of a left-closed ... |
| ubioc1 13316 | The upper bound belongs to... |
| lbico1 13317 | The lower bound belongs to... |
| iccleub 13318 | An element of a closed int... |
| iccgelb 13319 | An element of a closed int... |
| elioo5 13320 | Membership in an open inte... |
| eliooxr 13321 | A nonempty open interval s... |
| eliooord 13322 | Ordering implied by a memb... |
| elioo4g 13323 | Membership in an open inte... |
| ioossre 13324 | An open interval is a set ... |
| ioosscn 13325 | An open interval is a set ... |
| elioc2 13326 | Membership in an open-belo... |
| elico2 13327 | Membership in a closed-bel... |
| elicc2 13328 | Membership in a closed rea... |
| elicc2i 13329 | Inference for membership i... |
| elicc4 13330 | Membership in a closed rea... |
| iccss 13331 | Condition for a closed int... |
| iccssioo 13332 | Condition for a closed int... |
| icossico 13333 | Condition for a closed-bel... |
| iccss2 13334 | Condition for a closed int... |
| iccssico 13335 | Condition for a closed int... |
| iccssioo2 13336 | Condition for a closed int... |
| iccssico2 13337 | Condition for a closed int... |
| icossico2d 13338 | Condition for a closed-bel... |
| ioomax 13339 | The open interval from min... |
| iccmax 13340 | The closed interval from m... |
| ioopos 13341 | The set of positive reals ... |
| ioorp 13342 | The set of positive reals ... |
| iooshf 13343 | Shift the arguments of the... |
| iocssre 13344 | A closed-above interval wi... |
| icossre 13345 | A closed-below interval wi... |
| iccssre 13346 | A closed real interval is ... |
| iccssxr 13347 | A closed interval is a set... |
| iocssxr 13348 | An open-below, closed-abov... |
| icossxr 13349 | A closed-below, open-above... |
| ioossicc 13350 | An open interval is a subs... |
| iccssred 13351 | A closed real interval is ... |
| eliccxr 13352 | A member of a closed inter... |
| icossicc 13353 | A closed-below, open-above... |
| iocssicc 13354 | A closed-above, open-below... |
| ioossico 13355 | An open interval is a subs... |
| iocssioo 13356 | Condition for a closed int... |
| icossioo 13357 | Condition for a closed int... |
| ioossioo 13358 | Condition for an open inte... |
| iccsupr 13359 | A nonempty subset of a clo... |
| elioopnf 13360 | Membership in an unbounded... |
| elioomnf 13361 | Membership in an unbounded... |
| elicopnf 13362 | Membership in a closed unb... |
| repos 13363 | Two ways of saying that a ... |
| ioof 13364 | The set of open intervals ... |
| iccf 13365 | The set of closed interval... |
| unirnioo 13366 | The union of the range of ... |
| dfioo2 13367 | Alternate definition of th... |
| ioorebas 13368 | Open intervals are element... |
| xrge0neqmnf 13369 | A nonnegative extended rea... |
| xrge0nre 13370 | An extended real which is ... |
| elrege0 13371 | The predicate "is a nonneg... |
| nn0rp0 13372 | A nonnegative integer is a... |
| rge0ssre 13373 | Nonnegative real numbers a... |
| elxrge0 13374 | Elementhood in the set of ... |
| 0e0icopnf 13375 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13376 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13377 | The nonnegative reals are ... |
| ge0mulcl 13378 | The nonnegative reals are ... |
| ge0xaddcl 13379 | The nonnegative reals are ... |
| ge0xmulcl 13380 | The nonnegative extended r... |
| lbicc2 13381 | The lower bound of a close... |
| ubicc2 13382 | The upper bound of a close... |
| elicc01 13383 | Membership in the closed r... |
| elunitrn 13384 | The closed unit interval i... |
| elunitcn 13385 | The closed unit interval i... |
| 0elunit 13386 | Zero is an element of the ... |
| 1elunit 13387 | One is an element of the c... |
| iooneg 13388 | Membership in a negated op... |
| iccneg 13389 | Membership in a negated cl... |
| icoshft 13390 | A shifted real is a member... |
| icoshftf1o 13391 | Shifting a closed-below, o... |
| icoun 13392 | The union of two adjacent ... |
| icodisj 13393 | Adjacent left-closed right... |
| ioounsn 13394 | The union of an open inter... |
| snunioo 13395 | The closure of one end of ... |
| snunico 13396 | The closure of the open en... |
| snunioc 13397 | The closure of the open en... |
| prunioo 13398 | The closure of an open rea... |
| ioodisj 13399 | If the upper bound of one ... |
| ioojoin 13400 | Join two open intervals to... |
| difreicc 13401 | The class difference of ` ... |
| iccsplit 13402 | Split a closed interval in... |
| iccshftr 13403 | Membership in a shifted in... |
| iccshftri 13404 | Membership in a shifted in... |
| iccshftl 13405 | Membership in a shifted in... |
| iccshftli 13406 | Membership in a shifted in... |
| iccdil 13407 | Membership in a dilated in... |
| iccdili 13408 | Membership in a dilated in... |
| icccntr 13409 | Membership in a contracted... |
| icccntri 13410 | Membership in a contracted... |
| divelunit 13411 | A condition for a ratio to... |
| lincmb01cmp 13412 | A linear combination of tw... |
| iccf1o 13413 | Describe a bijection from ... |
| iccen 13414 | Any nontrivial closed inte... |
| xov1plusxeqvd 13415 | A complex number ` X ` is ... |
| unitssre 13416 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13417 | The closed unit interval i... |
| supicc 13418 | Supremum of a bounded set ... |
| supiccub 13419 | The supremum of a bounded ... |
| supicclub 13420 | The supremum of a bounded ... |
| supicclub2 13421 | The supremum of a bounded ... |
| zltaddlt1le 13422 | The sum of an integer and ... |
| xnn0xrge0 13423 | An extended nonnegative in... |
| fzval 13426 | The value of a finite set ... |
| fzval2 13427 | An alternative way of expr... |
| fzf 13428 | Establish the domain and c... |
| elfz1 13429 | Membership in a finite set... |
| elfz 13430 | Membership in a finite set... |
| elfz2 13431 | Membership in a finite set... |
| elfzd 13432 | Membership in a finite set... |
| elfz5 13433 | Membership in a finite set... |
| elfz4 13434 | Membership in a finite set... |
| elfzuzb 13435 | Membership in a finite set... |
| eluzfz 13436 | Membership in a finite set... |
| elfzuz 13437 | A member of a finite set o... |
| elfzuz3 13438 | Membership in a finite set... |
| elfzel2 13439 | Membership in a finite set... |
| elfzel1 13440 | Membership in a finite set... |
| elfzelz 13441 | A member of a finite set o... |
| elfzelzd 13442 | A member of a finite set o... |
| fzssz 13443 | A finite sequence of integ... |
| elfzle1 13444 | A member of a finite set o... |
| elfzle2 13445 | A member of a finite set o... |
| elfzuz2 13446 | Implication of membership ... |
| elfzle3 13447 | Membership in a finite set... |
| eluzfz1 13448 | Membership in a finite set... |
| eluzfz2 13449 | Membership in a finite set... |
| eluzfz2b 13450 | Membership in a finite set... |
| elfz3 13451 | Membership in a finite set... |
| elfz1eq 13452 | Membership in a finite set... |
| elfzubelfz 13453 | If there is a member in a ... |
| peano2fzr 13454 | A Peano-postulate-like the... |
| fzn0 13455 | Properties of a finite int... |
| fz0 13456 | A finite set of sequential... |
| fzn 13457 | A finite set of sequential... |
| fzen 13458 | A shifted finite set of se... |
| fz1n 13459 | A 1-based finite set of se... |
| 0nelfz1 13460 | 0 is not an element of a f... |
| 0fz1 13461 | Two ways to say a finite 1... |
| fz10 13462 | There are no integers betw... |
| uzsubsubfz 13463 | Membership of an integer g... |
| uzsubsubfz1 13464 | Membership of an integer g... |
| ige3m2fz 13465 | Membership of an integer g... |
| fzsplit2 13466 | Split a finite interval of... |
| fzsplit 13467 | Split a finite interval of... |
| fzdisj 13468 | Condition for two finite i... |
| fz01en 13469 | 0-based and 1-based finite... |
| elfznn 13470 | A member of a finite set o... |
| elfz1end 13471 | A nonempty finite range of... |
| fz1ssnn 13472 | A finite set of positive i... |
| fznn0sub 13473 | Subtraction closure for a ... |
| fzmmmeqm 13474 | Subtracting the difference... |
| fzaddel 13475 | Membership of a sum in a f... |
| fzadd2 13476 | Membership of a sum in a f... |
| fzsubel 13477 | Membership of a difference... |
| fzopth 13478 | A finite set of sequential... |
| fzass4 13479 | Two ways to express a nond... |
| fzss1 13480 | Subset relationship for fi... |
| fzss2 13481 | Subset relationship for fi... |
| fzssuz 13482 | A finite set of sequential... |
| fzsn 13483 | A finite interval of integ... |
| fzssp1 13484 | Subset relationship for fi... |
| fzssnn 13485 | Finite sets of sequential ... |
| ssfzunsnext 13486 | A subset of a finite seque... |
| ssfzunsn 13487 | A subset of a finite seque... |
| fzsuc 13488 | Join a successor to the en... |
| fzpred 13489 | Join a predecessor to the ... |
| fzpreddisj 13490 | A finite set of sequential... |
| elfzp1 13491 | Append an element to a fin... |
| fzp1ss 13492 | Subset relationship for fi... |
| fzelp1 13493 | Membership in a set of seq... |
| fzp1elp1 13494 | Add one to an element of a... |
| fznatpl1 13495 | Shift membership in a fini... |
| fzpr 13496 | A finite interval of integ... |
| fztp 13497 | A finite interval of integ... |
| fz12pr 13498 | An integer range between 1... |
| fzsuc2 13499 | Join a successor to the en... |
| fzp1disj 13500 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13501 | Remove a successor from th... |
| fzprval 13502 | Two ways of defining the f... |
| fztpval 13503 | Two ways of defining the f... |
| fzrev 13504 | Reversal of start and end ... |
| fzrev2 13505 | Reversal of start and end ... |
| fzrev2i 13506 | Reversal of start and end ... |
| fzrev3 13507 | The "complement" of a memb... |
| fzrev3i 13508 | The "complement" of a memb... |
| fznn 13509 | Finite set of sequential i... |
| elfz1b 13510 | Membership in a 1-based fi... |
| elfz1uz 13511 | Membership in a 1-based fi... |
| elfzm11 13512 | Membership in a finite set... |
| uzsplit 13513 | Express an upper integer s... |
| uzdisj 13514 | The first ` N ` elements o... |
| fseq1p1m1 13515 | Add/remove an item to/from... |
| fseq1m1p1 13516 | Add/remove an item to/from... |
| fz1sbc 13517 | Quantification over a one-... |
| elfzp1b 13518 | An integer is a member of ... |
| elfzm1b 13519 | An integer is a member of ... |
| elfzp12 13520 | Options for membership in ... |
| fzne1 13521 | Elementhood in a finite se... |
| fzdif1 13522 | Split the first element of... |
| fz0dif1 13523 | Split the first element of... |
| fzm1 13524 | Choices for an element of ... |
| fzneuz 13525 | No finite set of sequentia... |
| fznuz 13526 | Disjointness of the upper ... |
| uznfz 13527 | Disjointness of the upper ... |
| fzp1nel 13528 | One plus the upper bound o... |
| fzrevral 13529 | Reversal of scanning order... |
| fzrevral2 13530 | Reversal of scanning order... |
| fzrevral3 13531 | Reversal of scanning order... |
| fzshftral 13532 | Shift the scanning order i... |
| ige2m1fz1 13533 | Membership of an integer g... |
| ige2m1fz 13534 | Membership in a 0-based fi... |
| elfz2nn0 13535 | Membership in a finite set... |
| fznn0 13536 | Characterization of a fini... |
| elfznn0 13537 | A member of a finite set o... |
| elfz3nn0 13538 | The upper bound of a nonem... |
| fz0ssnn0 13539 | Finite sets of sequential ... |
| fz1ssfz0 13540 | Subset relationship for fi... |
| 0elfz 13541 | 0 is an element of a finit... |
| nn0fz0 13542 | A nonnegative integer is a... |
| elfz0add 13543 | An element of a finite set... |
| fz0sn 13544 | An integer range from 0 to... |
| fz0tp 13545 | An integer range from 0 to... |
| fz0to3un2pr 13546 | An integer range from 0 to... |
| fz0to4untppr 13547 | An integer range from 0 to... |
| fz0to5un2tp 13548 | An integer range from 0 to... |
| elfz0ubfz0 13549 | An element of a finite set... |
| elfz0fzfz0 13550 | A member of a finite set o... |
| fz0fzelfz0 13551 | If a member of a finite se... |
| fznn0sub2 13552 | Subtraction closure for a ... |
| uzsubfz0 13553 | Membership of an integer g... |
| fz0fzdiffz0 13554 | The difference of an integ... |
| elfzmlbm 13555 | Subtracting the lower boun... |
| elfzmlbp 13556 | Subtracting the lower boun... |
| fzctr 13557 | Lemma for theorems about t... |
| difelfzle 13558 | The difference of two inte... |
| difelfznle 13559 | The difference of two inte... |
| nn0split 13560 | Express the set of nonnega... |
| nn0disj 13561 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13562 | A finite set of sequential... |
| fvffz0 13563 | The function value of a fu... |
| 1fv 13564 | A function on a singleton.... |
| 4fvwrd4 13565 | The first four function va... |
| 2ffzeq 13566 | Two functions over 0-based... |
| preduz 13567 | The value of the predecess... |
| prednn 13568 | The value of the predecess... |
| prednn0 13569 | The value of the predecess... |
| predfz 13570 | Calculate the predecessor ... |
| fzof 13573 | Functionality of the half-... |
| elfzoel1 13574 | Reverse closure for half-o... |
| elfzoel2 13575 | Reverse closure for half-o... |
| elfzoelz 13576 | Reverse closure for half-o... |
| fzoval 13577 | Value of the half-open int... |
| elfzo 13578 | Membership in a half-open ... |
| elfzo2 13579 | Membership in a half-open ... |
| elfzouz 13580 | Membership in a half-open ... |
| nelfzo 13581 | An integer not being a mem... |
| fzolb 13582 | The left endpoint of a hal... |
| fzolb2 13583 | The left endpoint of a hal... |
| elfzole1 13584 | A member in a half-open in... |
| elfzolt2 13585 | A member in a half-open in... |
| elfzolt3 13586 | Membership in a half-open ... |
| elfzolt2b 13587 | A member in a half-open in... |
| elfzolt3b 13588 | Membership in a half-open ... |
| elfzop1le2 13589 | A member in a half-open in... |
| fzonel 13590 | A half-open range does not... |
| elfzouz2 13591 | The upper bound of a half-... |
| elfzofz 13592 | A half-open range is conta... |
| elfzo3 13593 | Express membership in a ha... |
| fzon0 13594 | A half-open integer interv... |
| fzossfz 13595 | A half-open range is conta... |
| fzossz 13596 | A half-open integer interv... |
| fzon 13597 | A half-open set of sequent... |
| fzo0n 13598 | A half-open range of nonne... |
| fzonlt0 13599 | A half-open integer range ... |
| fzo0 13600 | Half-open sets with equal ... |
| fzonnsub 13601 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13602 | If ` M < N ` then ` N - M ... |
| fzoss1 13603 | Subset relationship for ha... |
| fzoss2 13604 | Subset relationship for ha... |
| fzossrbm1 13605 | Subset of a half-open rang... |
| fzo0ss1 13606 | Subset relationship for ha... |
| fzossnn0 13607 | A half-open integer range ... |
| fzospliti 13608 | One direction of splitting... |
| fzosplit 13609 | Split a half-open integer ... |
| fzodisj 13610 | Abutting half-open integer... |
| fzouzsplit 13611 | Split an upper integer set... |
| fzouzdisj 13612 | A half-open integer range ... |
| fzoun 13613 | A half-open integer range ... |
| fzodisjsn 13614 | A half-open integer range ... |
| prinfzo0 13615 | The intersection of a half... |
| lbfzo0 13616 | An integer is strictly gre... |
| elfzo0 13617 | Membership in a half-open ... |
| elfzo0z 13618 | Membership in a half-open ... |
| nn0p1elfzo 13619 | A nonnegative integer incr... |
| elfzo0le 13620 | A member in a half-open ra... |
| elfzolem1 13621 | A member in a half-open in... |
| elfzo0subge1 13622 | The difference of the uppe... |
| elfzo0suble 13623 | The difference of the uppe... |
| elfzonn0 13624 | A member of a half-open ra... |
| fzonmapblen 13625 | The result of subtracting ... |
| fzofzim 13626 | If a nonnegative integer i... |
| fz1fzo0m1 13627 | Translation of one between... |
| fzossnn 13628 | Half-open integer ranges s... |
| elfzo1 13629 | Membership in a half-open ... |
| fzo1lb 13630 | 1 is the left endpoint of ... |
| 1elfzo1 13631 | 1 is in a half-open range ... |
| fzo1fzo0n0 13632 | An integer between 1 and a... |
| fzo0n0 13633 | A half-open integer range ... |
| fzoaddel 13634 | Translate membership in a ... |
| fzo0addel 13635 | Translate membership in a ... |
| fzo0addelr 13636 | Translate membership in a ... |
| fzoaddel2 13637 | Translate membership in a ... |
| elfzoextl 13638 | Membership of an integer i... |
| elfzoext 13639 | Membership of an integer i... |
| elincfzoext 13640 | Membership of an increased... |
| fzosubel 13641 | Translate membership in a ... |
| fzosubel2 13642 | Membership in a translated... |
| fzosubel3 13643 | Membership in a translated... |
| eluzgtdifelfzo 13644 | Membership of the differen... |
| ige2m2fzo 13645 | Membership of an integer g... |
| fzocatel 13646 | Translate membership in a ... |
| ubmelfzo 13647 | If an integer in a 1-based... |
| elfzodifsumelfzo 13648 | If an integer is in a half... |
| elfzom1elp1fzo 13649 | Membership of an integer i... |
| elfzom1elfzo 13650 | Membership in a half-open ... |
| fzval3 13651 | Expressing a closed intege... |
| fz0add1fz1 13652 | Translate membership in a ... |
| fzosn 13653 | Expressing a singleton as ... |
| elfzomin 13654 | Membership of an integer i... |
| zpnn0elfzo 13655 | Membership of an integer i... |
| zpnn0elfzo1 13656 | Membership of an integer i... |
| fzosplitsnm1 13657 | Removing a singleton from ... |
| elfzonlteqm1 13658 | If an element of a half-op... |
| fzonn0p1 13659 | A nonnegative integer is a... |
| fzossfzop1 13660 | A half-open range of nonne... |
| fzonn0p1p1 13661 | If a nonnegative integer i... |
| elfzom1p1elfzo 13662 | Increasing an element of a... |
| fzo0ssnn0 13663 | Half-open integer ranges s... |
| fzo01 13664 | Expressing the singleton o... |
| fzo12sn 13665 | A 1-based half-open intege... |
| fzo13pr 13666 | A 1-based half-open intege... |
| fzo0to2pr 13667 | A half-open integer range ... |
| fz01pr 13668 | An integer range between 0... |
| fzo0to3tp 13669 | A half-open integer range ... |
| fzo0to42pr 13670 | A half-open integer range ... |
| fzo1to4tp 13671 | A half-open integer range ... |
| fzo0sn0fzo1 13672 | A half-open range of nonne... |
| elfzo0l 13673 | A member of a half-open ra... |
| fzoend 13674 | The endpoint of a half-ope... |
| fzo0end 13675 | The endpoint of a zero-bas... |
| ssfzo12 13676 | Subset relationship for ha... |
| ssfzoulel 13677 | If a half-open integer ran... |
| ssfzo12bi 13678 | Subset relationship for ha... |
| fzoopth 13679 | A half-open integer range ... |
| ubmelm1fzo 13680 | The result of subtracting ... |
| fzofzp1 13681 | If a point is in a half-op... |
| fzofzp1b 13682 | If a point is in a half-op... |
| elfzom1b 13683 | An integer is a member of ... |
| elfzom1elp1fzo1 13684 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13685 | Membership of a positive i... |
| elfzonelfzo 13686 | If an element of a half-op... |
| elfzodif0 13687 | If an integer ` M ` is in ... |
| fzonfzoufzol 13688 | If an element of a half-op... |
| elfzomelpfzo 13689 | An integer increased by an... |
| elfznelfzo 13690 | A value in a finite set of... |
| elfznelfzob 13691 | A value in a finite set of... |
| peano2fzor 13692 | A Peano-postulate-like the... |
| fzosplitsn 13693 | Extending a half-open rang... |
| fzosplitpr 13694 | Extending a half-open inte... |
| fzosplitprm1 13695 | Extending a half-open inte... |
| fzosplitsni 13696 | Membership in a half-open ... |
| fzisfzounsn 13697 | A finite interval of integ... |
| elfzr 13698 | A member of a finite inter... |
| elfzlmr 13699 | A member of a finite inter... |
| elfz0lmr 13700 | A member of a finite inter... |
| fzone1 13701 | Elementhood in a half-open... |
| fzom1ne1 13702 | Elementhood in a half-open... |
| fzostep1 13703 | Two possibilities for a nu... |
| fzoshftral 13704 | Shift the scanning order i... |
| fzind2 13705 | Induction on the integers ... |
| fvinim0ffz 13706 | The function values for th... |
| injresinjlem 13707 | Lemma for ~ injresinj . (... |
| injresinj 13708 | A function whose restricti... |
| subfzo0 13709 | The difference between two... |
| fvf1tp 13710 | Values of a one-to-one fun... |
| flval 13715 | Value of the floor (greate... |
| flcl 13716 | The floor (greatest intege... |
| reflcl 13717 | The floor (greatest intege... |
| fllelt 13718 | A basic property of the fl... |
| flcld 13719 | The floor (greatest intege... |
| flle 13720 | A basic property of the fl... |
| flltp1 13721 | A basic property of the fl... |
| fllep1 13722 | A basic property of the fl... |
| fraclt1 13723 | The fractional part of a r... |
| fracle1 13724 | The fractional part of a r... |
| fracge0 13725 | The fractional part of a r... |
| flge 13726 | The floor function value i... |
| fllt 13727 | The floor function value i... |
| flflp1 13728 | Move floor function betwee... |
| flid 13729 | An integer is its own floo... |
| flidm 13730 | The floor function is idem... |
| flidz 13731 | A real number equals its f... |
| flltnz 13732 | The floor of a non-integer... |
| flwordi 13733 | Ordering relation for the ... |
| flword2 13734 | Ordering relation for the ... |
| flval2 13735 | An alternate way to define... |
| flval3 13736 | An alternate way to define... |
| flbi 13737 | A condition equivalent to ... |
| flbi2 13738 | A condition equivalent to ... |
| adddivflid 13739 | The floor of a sum of an i... |
| ico01fl0 13740 | The floor of a real number... |
| flge0nn0 13741 | The floor of a number grea... |
| flge1nn 13742 | The floor of a number grea... |
| fldivnn0 13743 | The floor function of a di... |
| refldivcl 13744 | The floor function of a di... |
| divfl0 13745 | The floor of a fraction is... |
| fladdz 13746 | An integer can be moved in... |
| flzadd 13747 | An integer can be moved in... |
| flmulnn0 13748 | Move a nonnegative integer... |
| btwnzge0 13749 | A real bounded between an ... |
| 2tnp1ge0ge0 13750 | Two times an integer plus ... |
| flhalf 13751 | Ordering relation for the ... |
| fldivle 13752 | The floor function of a di... |
| fldivnn0le 13753 | The floor function of a di... |
| flltdivnn0lt 13754 | The floor function of a di... |
| ltdifltdiv 13755 | If the dividend of a divis... |
| fldiv4p1lem1div2 13756 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13757 | The floor of an integer gr... |
| fldiv4lem1div2 13758 | The floor of a positive in... |
| ceilval 13759 | The value of the ceiling f... |
| dfceil2 13760 | Alternative definition of ... |
| ceilval2 13761 | The value of the ceiling f... |
| ceicl 13762 | The ceiling function retur... |
| ceilcl 13763 | Closure of the ceiling fun... |
| ceilcld 13764 | Closure of the ceiling fun... |
| ceige 13765 | The ceiling of a real numb... |
| ceilge 13766 | The ceiling of a real numb... |
| ceilged 13767 | The ceiling of a real numb... |
| ceim1l 13768 | One less than the ceiling ... |
| ceilm1lt 13769 | One less than the ceiling ... |
| ceile 13770 | The ceiling of a real numb... |
| ceille 13771 | The ceiling of a real numb... |
| ceilid 13772 | An integer is its own ceil... |
| ceilidz 13773 | A real number equals its c... |
| flleceil 13774 | The floor of a real number... |
| fleqceilz 13775 | A real number is an intege... |
| quoremz 13776 | Quotient and remainder of ... |
| quoremnn0 13777 | Quotient and remainder of ... |
| quoremnn0ALT 13778 | Alternate proof of ~ quore... |
| intfrac2 13779 | Decompose a real into inte... |
| intfracq 13780 | Decompose a rational numbe... |
| fldiv 13781 | Cancellation of the embedd... |
| fldiv2 13782 | Cancellation of an embedde... |
| fznnfl 13783 | Finite set of sequential i... |
| uzsup 13784 | An upper set of integers i... |
| ioopnfsup 13785 | An upper set of reals is u... |
| icopnfsup 13786 | An upper set of reals is u... |
| rpsup 13787 | The positive reals are unb... |
| resup 13788 | The real numbers are unbou... |
| xrsup 13789 | The extended real numbers ... |
| modval 13792 | The value of the modulo op... |
| modvalr 13793 | The value of the modulo op... |
| modcl 13794 | Closure law for the modulo... |
| flpmodeq 13795 | Partition of a division in... |
| modcld 13796 | Closure law for the modulo... |
| mod0 13797 | ` A mod B ` is zero iff ` ... |
| mulmod0 13798 | The product of an integer ... |
| negmod0 13799 | ` A ` is divisible by ` B ... |
| modge0 13800 | The modulo operation is no... |
| modlt 13801 | The modulo operation is le... |
| modelico 13802 | Modular reduction produces... |
| moddiffl 13803 | Value of the modulo operat... |
| moddifz 13804 | The modulo operation diffe... |
| modfrac 13805 | The fractional part of a n... |
| flmod 13806 | The floor function express... |
| intfrac 13807 | Break a number into its in... |
| zmod10 13808 | An integer modulo 1 is 0. ... |
| zmod1congr 13809 | Two arbitrary integers are... |
| modmulnn 13810 | Move a positive integer in... |
| modvalp1 13811 | The value of the modulo op... |
| zmodcl 13812 | Closure law for the modulo... |
| zmodcld 13813 | Closure law for the modulo... |
| zmodfz 13814 | An integer mod ` B ` lies ... |
| zmodfzo 13815 | An integer mod ` B ` lies ... |
| zmodfzp1 13816 | An integer mod ` B ` lies ... |
| modid 13817 | Identity law for modulo. ... |
| modid0 13818 | A positive real number mod... |
| modid2 13819 | Identity law for modulo. ... |
| zmodid2 13820 | Identity law for modulo re... |
| zmodidfzo 13821 | Identity law for modulo re... |
| zmodidfzoimp 13822 | Identity law for modulo re... |
| 0mod 13823 | Special case: 0 modulo a p... |
| 1mod 13824 | Special case: 1 modulo a r... |
| modabs 13825 | Absorption law for modulo.... |
| modabs2 13826 | Absorption law for modulo.... |
| modcyc 13827 | The modulo operation is pe... |
| modcyc2 13828 | The modulo operation is pe... |
| modadd1 13829 | Addition property of the m... |
| modaddb 13830 | Addition property of the m... |
| modaddid 13831 | The sums of two nonnegativ... |
| modaddabs 13832 | Absorption law for modulo.... |
| modaddmod 13833 | The sum of a real number m... |
| muladdmodid 13834 | The sum of a positive real... |
| mulp1mod1 13835 | The product of an integer ... |
| muladdmod 13836 | A real number is the sum o... |
| modmuladd 13837 | Decomposition of an intege... |
| modmuladdim 13838 | Implication of a decomposi... |
| modmuladdnn0 13839 | Implication of a decomposi... |
| negmod 13840 | The negation of a number m... |
| m1modnnsub1 13841 | Minus one modulo a positiv... |
| m1modge3gt1 13842 | Minus one modulo an intege... |
| addmodid 13843 | The sum of a positive inte... |
| addmodidr 13844 | The sum of a positive inte... |
| modadd2mod 13845 | The sum of a real number m... |
| modm1p1mod0 13846 | If a real number modulo a ... |
| modltm1p1mod 13847 | If a real number modulo a ... |
| modmul1 13848 | Multiplication property of... |
| modmul12d 13849 | Multiplication property of... |
| modnegd 13850 | Negation property of the m... |
| modadd12d 13851 | Additive property of the m... |
| modsub12d 13852 | Subtraction property of th... |
| modsubmod 13853 | The difference of a real n... |
| modsubmodmod 13854 | The difference of a real n... |
| 2txmodxeq0 13855 | Two times a positive real ... |
| 2submod 13856 | If a real number is betwee... |
| modifeq2int 13857 | If a nonnegative integer i... |
| modaddmodup 13858 | The sum of an integer modu... |
| modaddmodlo 13859 | The sum of an integer modu... |
| modmulmod 13860 | The product of a real numb... |
| modmulmodr 13861 | The product of an integer ... |
| modaddmulmod 13862 | The sum of a real number a... |
| moddi 13863 | Distribute multiplication ... |
| modsubdir 13864 | Distribute the modulo oper... |
| modeqmodmin 13865 | A real number equals the d... |
| modirr 13866 | A number modulo an irratio... |
| modfzo0difsn 13867 | For a number within a half... |
| modsumfzodifsn 13868 | The sum of a number within... |
| modlteq 13869 | Two nonnegative integers l... |
| addmodlteq 13870 | Two nonnegative integers l... |
| om2uz0i 13871 | The mapping ` G ` is a one... |
| om2uzsuci 13872 | The value of ` G ` (see ~ ... |
| om2uzuzi 13873 | The value ` G ` (see ~ om2... |
| om2uzlti 13874 | Less-than relation for ` G... |
| om2uzlt2i 13875 | The mapping ` G ` (see ~ o... |
| om2uzrani 13876 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13877 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13878 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13879 | An alternative definition ... |
| om2uzrdg 13880 | A helper lemma for the val... |
| uzrdglem 13881 | A helper lemma for the val... |
| uzrdgfni 13882 | The recursive definition g... |
| uzrdg0i 13883 | Initial value of a recursi... |
| uzrdgsuci 13884 | Successor value of a recur... |
| ltweuz 13885 | ` < ` is a well-founded re... |
| ltwenn 13886 | Less than well-orders the ... |
| ltwefz 13887 | Less than well-orders a se... |
| uzenom 13888 | An upper integer set is de... |
| uzinf 13889 | An upper integer set is in... |
| nnnfi 13890 | The set of positive intege... |
| uzrdgxfr 13891 | Transfer the value of the ... |
| fzennn 13892 | The cardinality of a finit... |
| fzen2 13893 | The cardinality of a finit... |
| cardfz 13894 | The cardinality of a finit... |
| hashgf1o 13895 | ` G ` maps ` _om ` one-to-... |
| fzfi 13896 | A finite interval of integ... |
| fzfid 13897 | Commonly used special case... |
| fzofi 13898 | Half-open integer sets are... |
| fsequb 13899 | The values of a finite rea... |
| fsequb2 13900 | The values of a finite rea... |
| fseqsupcl 13901 | The values of a finite rea... |
| fseqsupubi 13902 | The values of a finite rea... |
| nn0ennn 13903 | The nonnegative integers a... |
| nnenom 13904 | The set of positive intege... |
| nnct 13905 | ` NN ` is countable. (Con... |
| uzindi 13906 | Indirect strong induction ... |
| axdc4uzlem 13907 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13908 | A version of ~ axdc4 that ... |
| ssnn0fi 13909 | A subset of the nonnegativ... |
| rabssnn0fi 13910 | A subset of the nonnegativ... |
| uzsinds 13911 | Strong (or "total") induct... |
| nnsinds 13912 | Strong (or "total") induct... |
| nn0sinds 13913 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13914 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13915 | If all functions of a fini... |
| fsuppmapnn0fiubex 13916 | If all functions of a fini... |
| fsuppmapnn0fiub0 13917 | If all functions of a fini... |
| suppssfz 13918 | Condition for a function o... |
| fsuppmapnn0ub 13919 | If a function over the non... |
| fsuppmapnn0fz 13920 | If a function over the non... |
| mptnn0fsupp 13921 | A mapping from the nonnega... |
| mptnn0fsuppd 13922 | A mapping from the nonnega... |
| mptnn0fsuppr 13923 | A finitely supported mappi... |
| f13idfv 13924 | A one-to-one function with... |
| seqex 13927 | Existence of the sequence ... |
| seqeq1 13928 | Equality theorem for the s... |
| seqeq2 13929 | Equality theorem for the s... |
| seqeq3 13930 | Equality theorem for the s... |
| seqeq1d 13931 | Equality deduction for the... |
| seqeq2d 13932 | Equality deduction for the... |
| seqeq3d 13933 | Equality deduction for the... |
| seqeq123d 13934 | Equality deduction for the... |
| nfseq 13935 | Hypothesis builder for the... |
| seqval 13936 | Value of the sequence buil... |
| seqfn 13937 | The sequence builder funct... |
| seq1 13938 | Value of the sequence buil... |
| seq1i 13939 | Value of the sequence buil... |
| seqp1 13940 | Value of the sequence buil... |
| seqexw 13941 | Weak version of ~ seqex th... |
| seqp1d 13942 | Value of the sequence buil... |
| seqm1 13943 | Value of the sequence buil... |
| seqcl2 13944 | Closure properties of the ... |
| seqf2 13945 | Range of the recursive seq... |
| seqcl 13946 | Closure properties of the ... |
| seqf 13947 | Range of the recursive seq... |
| seqfveq2 13948 | Equality of sequences. (C... |
| seqfeq2 13949 | Equality of sequences. (C... |
| seqfveq 13950 | Equality of sequences. (C... |
| seqfeq 13951 | Equality of sequences. (C... |
| seqshft2 13952 | Shifting the index set of ... |
| seqres 13953 | Restricting its characteri... |
| serf 13954 | An infinite series of comp... |
| serfre 13955 | An infinite series of real... |
| monoord 13956 | Ordering relation for a mo... |
| monoord2 13957 | Ordering relation for a mo... |
| sermono 13958 | The partial sums in an inf... |
| seqsplit 13959 | Split a sequence into two ... |
| seq1p 13960 | Removing the first term fr... |
| seqcaopr3 13961 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 13962 | The sum of two infinite se... |
| seqcaopr 13963 | The sum of two infinite se... |
| seqf1olem2a 13964 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 13965 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 13966 | Lemma for ~ seqf1o . (Con... |
| seqf1o 13967 | Rearrange a sum via an arb... |
| seradd 13968 | The sum of two infinite se... |
| sersub 13969 | The difference of two infi... |
| seqid3 13970 | A sequence that consists e... |
| seqid 13971 | Discarding the first few t... |
| seqid2 13972 | The last few partial sums ... |
| seqhomo 13973 | Apply a homomorphism to a ... |
| seqz 13974 | If the operation ` .+ ` ha... |
| seqfeq4 13975 | Equality of series under d... |
| seqfeq3 13976 | Equality of series under d... |
| seqdistr 13977 | The distributive property ... |
| ser0 13978 | The value of the partial s... |
| ser0f 13979 | A zero-valued infinite ser... |
| serge0 13980 | A finite sum of nonnegativ... |
| serle 13981 | Comparison of partial sums... |
| ser1const 13982 | Value of the partial serie... |
| seqof 13983 | Distribute function operat... |
| seqof2 13984 | Distribute function operat... |
| expval 13987 | Value of exponentiation to... |
| expnnval 13988 | Value of exponentiation to... |
| exp0 13989 | Value of a complex number ... |
| 0exp0e1 13990 | The zeroth power of zero e... |
| exp1 13991 | Value of a complex number ... |
| expp1 13992 | Value of a complex number ... |
| expneg 13993 | Value of a complex number ... |
| expneg2 13994 | Value of a complex number ... |
| expn1 13995 | A complex number raised to... |
| expcllem 13996 | Lemma for proving nonnegat... |
| expcl2lem 13997 | Lemma for proving integer ... |
| nnexpcl 13998 | Closure of exponentiation ... |
| nn0expcl 13999 | Closure of exponentiation ... |
| zexpcl 14000 | Closure of exponentiation ... |
| qexpcl 14001 | Closure of exponentiation ... |
| reexpcl 14002 | Closure of exponentiation ... |
| expcl 14003 | Closure law for nonnegativ... |
| rpexpcl 14004 | Closure law for integer ex... |
| qexpclz 14005 | Closure of integer exponen... |
| reexpclz 14006 | Closure of integer exponen... |
| expclzlem 14007 | Lemma for ~ expclz . (Con... |
| expclz 14008 | Closure law for integer ex... |
| m1expcl2 14009 | Closure of integer exponen... |
| m1expcl 14010 | Closure of exponentiation ... |
| zexpcld 14011 | Closure of exponentiation ... |
| nn0expcli 14012 | Closure of exponentiation ... |
| nn0sqcl 14013 | The square of a nonnegativ... |
| expm1t 14014 | Exponentiation in terms of... |
| 1exp 14015 | Value of 1 raised to an in... |
| expeq0 14016 | A positive integer power i... |
| expne0 14017 | A positive integer power i... |
| expne0i 14018 | An integer power is nonzer... |
| expgt0 14019 | A positive real raised to ... |
| expnegz 14020 | Value of a nonzero complex... |
| 0exp 14021 | Value of zero raised to a ... |
| expge0 14022 | A nonnegative real raised ... |
| expge1 14023 | A real greater than or equ... |
| expgt1 14024 | A real greater than 1 rais... |
| mulexp 14025 | Nonnegative integer expone... |
| mulexpz 14026 | Integer exponentiation of ... |
| exprec 14027 | Integer exponentiation of ... |
| expadd 14028 | Sum of exponents law for n... |
| expaddzlem 14029 | Lemma for ~ expaddz . (Co... |
| expaddz 14030 | Sum of exponents law for i... |
| expmul 14031 | Product of exponents law f... |
| expmulz 14032 | Product of exponents law f... |
| m1expeven 14033 | Exponentiation of negative... |
| expsub 14034 | Exponent subtraction law f... |
| expp1z 14035 | Value of a nonzero complex... |
| expm1 14036 | Value of a nonzero complex... |
| expdiv 14037 | Nonnegative integer expone... |
| sqval 14038 | Value of the square of a c... |
| sqneg 14039 | The square of the negative... |
| sqnegd 14040 | The square of the negative... |
| sqsubswap 14041 | Swap the order of subtract... |
| sqcl 14042 | Closure of square. (Contr... |
| sqmul 14043 | Distribution of squaring o... |
| sqeq0 14044 | A complex number is zero i... |
| sqdiv 14045 | Distribution of squaring o... |
| sqdivid 14046 | The square of a nonzero co... |
| sqne0 14047 | A complex number is nonzer... |
| resqcl 14048 | Closure of squaring in rea... |
| resqcld 14049 | Closure of squaring in rea... |
| sqgt0 14050 | The square of a nonzero re... |
| sqn0rp 14051 | The square of a nonzero re... |
| nnsqcl 14052 | The positive naturals are ... |
| zsqcl 14053 | Integers are closed under ... |
| qsqcl 14054 | The square of a rational i... |
| sq11 14055 | The square function is one... |
| nn0sq11 14056 | The square function is one... |
| lt2sq 14057 | The square function is inc... |
| le2sq 14058 | The square function is non... |
| le2sq2 14059 | The square function is non... |
| sqge0 14060 | The square of a real is no... |
| sqge0d 14061 | The square of a real is no... |
| zsqcl2 14062 | The square of an integer i... |
| 0expd 14063 | Value of zero raised to a ... |
| exp0d 14064 | Value of a complex number ... |
| exp1d 14065 | Value of a complex number ... |
| expeq0d 14066 | If a positive integer powe... |
| sqvald 14067 | Value of square. Inferenc... |
| sqcld 14068 | Closure of square. (Contr... |
| sqeq0d 14069 | A number is zero iff its s... |
| expcld 14070 | Closure law for nonnegativ... |
| expp1d 14071 | Value of a complex number ... |
| expaddd 14072 | Sum of exponents law for n... |
| expmuld 14073 | Product of exponents law f... |
| sqrecd 14074 | Square of reciprocal is re... |
| expclzd 14075 | Closure law for integer ex... |
| expne0d 14076 | A nonnegative integer powe... |
| expnegd 14077 | Value of a nonzero complex... |
| exprecd 14078 | An integer power of a reci... |
| expp1zd 14079 | Value of a nonzero complex... |
| expm1d 14080 | Value of a nonzero complex... |
| expsubd 14081 | Exponent subtraction law f... |
| sqmuld 14082 | Distribution of squaring o... |
| sqdivd 14083 | Distribution of squaring o... |
| expdivd 14084 | Nonnegative integer expone... |
| mulexpd 14085 | Nonnegative integer expone... |
| znsqcld 14086 | The square of a nonzero in... |
| reexpcld 14087 | Closure of exponentiation ... |
| expge0d 14088 | A nonnegative real raised ... |
| expge1d 14089 | A real greater than or equ... |
| ltexp2a 14090 | Exponent ordering relation... |
| expmordi 14091 | Base ordering relationship... |
| rpexpmord 14092 | Base ordering relationship... |
| expcan 14093 | Cancellation law for integ... |
| ltexp2 14094 | Strict ordering law for ex... |
| leexp2 14095 | Ordering law for exponenti... |
| leexp2a 14096 | Weak ordering relationship... |
| ltexp2r 14097 | The integer powers of a fi... |
| leexp2r 14098 | Weak ordering relationship... |
| leexp1a 14099 | Weak base ordering relatio... |
| leexp1ad 14100 | Weak base ordering relatio... |
| exple1 14101 | A real between 0 and 1 inc... |
| expubnd 14102 | An upper bound on ` A ^ N ... |
| sumsqeq0 14103 | The sum of two squres of r... |
| sqvali 14104 | Value of square. Inferenc... |
| sqcli 14105 | Closure of square. (Contr... |
| sqeq0i 14106 | A complex number is zero i... |
| sqrecii 14107 | The square of a reciprocal... |
| sqmuli 14108 | Distribution of squaring o... |
| sqdivi 14109 | Distribution of squaring o... |
| resqcli 14110 | Closure of square in reals... |
| sqgt0i 14111 | The square of a nonzero re... |
| sqge0i 14112 | The square of a real is no... |
| lt2sqi 14113 | The square function on non... |
| le2sqi 14114 | The square function on non... |
| sq11i 14115 | The square function is one... |
| sq0 14116 | The square of 0 is 0. (Co... |
| sq0i 14117 | If a number is zero, then ... |
| sq0id 14118 | If a number is zero, then ... |
| sq1 14119 | The square of 1 is 1. (Co... |
| neg1sqe1 14120 | The square of ` -u 1 ` is ... |
| sq2 14121 | The square of 2 is 4. (Co... |
| sq3 14122 | The square of 3 is 9. (Co... |
| sq4e2t8 14123 | The square of 4 is 2 times... |
| cu2 14124 | The cube of 2 is 8. (Cont... |
| irec 14125 | The reciprocal of ` _i ` .... |
| i2 14126 | ` _i ` squared. (Contribu... |
| i3 14127 | ` _i ` cubed. (Contribute... |
| i4 14128 | ` _i ` to the fourth power... |
| nnlesq 14129 | A positive integer is less... |
| zzlesq 14130 | An integer is less than or... |
| iexpcyc 14131 | Taking ` _i ` to the ` K `... |
| expnass 14132 | A counterexample showing t... |
| sqlecan 14133 | Cancel one factor of a squ... |
| subsq 14134 | Factor the difference of t... |
| subsq2 14135 | Express the difference of ... |
| binom2i 14136 | The square of a binomial. ... |
| subsqi 14137 | Factor the difference of t... |
| sqeqori 14138 | The squares of two complex... |
| subsq0i 14139 | The two solutions to the d... |
| sqeqor 14140 | The squares of two complex... |
| binom2 14141 | The square of a binomial. ... |
| binom2d 14142 | Deduction form of ~ binom2... |
| binom21 14143 | Special case of ~ binom2 w... |
| binom2sub 14144 | Expand the square of a sub... |
| binom2sub1 14145 | Special case of ~ binom2su... |
| binom2subi 14146 | Expand the square of a sub... |
| mulbinom2 14147 | The square of a binomial w... |
| binom3 14148 | The cube of a binomial. (... |
| sq01 14149 | If a complex number equals... |
| zesq 14150 | An integer is even iff its... |
| nnesq 14151 | A positive integer is even... |
| crreczi 14152 | Reciprocal of a complex nu... |
| bernneq 14153 | Bernoulli's inequality, du... |
| bernneq2 14154 | Variation of Bernoulli's i... |
| bernneq3 14155 | A corollary of ~ bernneq .... |
| expnbnd 14156 | Exponentiation with a base... |
| expnlbnd 14157 | The reciprocal of exponent... |
| expnlbnd2 14158 | The reciprocal of exponent... |
| expmulnbnd 14159 | Exponentiation with a base... |
| digit2 14160 | Two ways to express the ` ... |
| digit1 14161 | Two ways to express the ` ... |
| modexp 14162 | Exponentiation property of... |
| discr1 14163 | A nonnegative quadratic fo... |
| discr 14164 | If a quadratic polynomial ... |
| expnngt1 14165 | If an integer power with a... |
| expnngt1b 14166 | An integer power with an i... |
| sqoddm1div8 14167 | A squared odd number minus... |
| nnsqcld 14168 | The naturals are closed un... |
| nnexpcld 14169 | Closure of exponentiation ... |
| nn0expcld 14170 | Closure of exponentiation ... |
| rpexpcld 14171 | Closure law for exponentia... |
| ltexp2rd 14172 | The power of a positive nu... |
| reexpclzd 14173 | Closure of exponentiation ... |
| sqgt0d 14174 | The square of a nonzero re... |
| ltexp2d 14175 | Ordering relationship for ... |
| leexp2d 14176 | Ordering law for exponenti... |
| expcand 14177 | Ordering relationship for ... |
| leexp2ad 14178 | Ordering relationship for ... |
| leexp2rd 14179 | Ordering relationship for ... |
| lt2sqd 14180 | The square function on non... |
| le2sqd 14181 | The square function on non... |
| sq11d 14182 | The square function is one... |
| ltexp1d 14183 | Elevating to a positive po... |
| ltexp1dd 14184 | Raising both sides of 'les... |
| exp11nnd 14185 | The function elevating non... |
| mulsubdivbinom2 14186 | The square of a binomial w... |
| muldivbinom2 14187 | The square of a binomial w... |
| sq10 14188 | The square of 10 is 100. ... |
| sq10e99m1 14189 | The square of 10 is 99 plu... |
| 3dec 14190 | A "decimal constructor" wh... |
| nn0le2msqi 14191 | The square function on non... |
| nn0opthlem1 14192 | A rather pretty lemma for ... |
| nn0opthlem2 14193 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14194 | An ordered pair theorem fo... |
| nn0opth2i 14195 | An ordered pair theorem fo... |
| nn0opth2 14196 | An ordered pair theorem fo... |
| facnn 14199 | Value of the factorial fun... |
| fac0 14200 | The factorial of 0. (Cont... |
| fac1 14201 | The factorial of 1. (Cont... |
| facp1 14202 | The factorial of a success... |
| fac2 14203 | The factorial of 2. (Cont... |
| fac3 14204 | The factorial of 3. (Cont... |
| fac4 14205 | The factorial of 4. (Cont... |
| facnn2 14206 | Value of the factorial fun... |
| faccl 14207 | Closure of the factorial f... |
| faccld 14208 | Closure of the factorial f... |
| facmapnn 14209 | The factorial function res... |
| facne0 14210 | The factorial function is ... |
| facdiv 14211 | A positive integer divides... |
| facndiv 14212 | No positive integer (great... |
| facwordi 14213 | Ordering property of facto... |
| faclbnd 14214 | A lower bound for the fact... |
| faclbnd2 14215 | A lower bound for the fact... |
| faclbnd3 14216 | A lower bound for the fact... |
| faclbnd4lem1 14217 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14218 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14219 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14220 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14221 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14222 | The factorial function gro... |
| faclbnd6 14223 | Geometric lower bound for ... |
| facubnd 14224 | An upper bound for the fac... |
| facavg 14225 | The product of two factori... |
| bcval 14228 | Value of the binomial coef... |
| bcval2 14229 | Value of the binomial coef... |
| bcval3 14230 | Value of the binomial coef... |
| bcval4 14231 | Value of the binomial coef... |
| bcrpcl 14232 | Closure of the binomial co... |
| bccmpl 14233 | "Complementing" its second... |
| bcn0 14234 | ` N ` choose 0 is 1. Rema... |
| bc0k 14235 | The binomial coefficient "... |
| bcnn 14236 | ` N ` choose ` N ` is 1. ... |
| bcn1 14237 | Binomial coefficient: ` N ... |
| bcnp1n 14238 | Binomial coefficient: ` N ... |
| bcm1k 14239 | The proportion of one bino... |
| bcp1n 14240 | The proportion of one bino... |
| bcp1nk 14241 | The proportion of one bino... |
| bcval5 14242 | Write out the top and bott... |
| bcn2 14243 | Binomial coefficient: ` N ... |
| bcp1m1 14244 | Compute the binomial coeff... |
| bcpasc 14245 | Pascal's rule for the bino... |
| bccl 14246 | A binomial coefficient, in... |
| bccl2 14247 | A binomial coefficient, in... |
| bcn2m1 14248 | Compute the binomial coeff... |
| bcn2p1 14249 | Compute the binomial coeff... |
| permnn 14250 | The number of permutations... |
| bcnm1 14251 | The binomial coefficient o... |
| 4bc3eq4 14252 | The value of four choose t... |
| 4bc2eq6 14253 | The value of four choose t... |
| hashkf 14256 | The finite part of the siz... |
| hashgval 14257 | The value of the ` # ` fun... |
| hashginv 14258 | The converse of ` G ` maps... |
| hashinf 14259 | The value of the ` # ` fun... |
| hashbnd 14260 | If ` A ` has size bounded ... |
| hashfxnn0 14261 | The size function is a fun... |
| hashf 14262 | The size function maps all... |
| hashxnn0 14263 | The value of the hash func... |
| hashresfn 14264 | Restriction of the domain ... |
| dmhashres 14265 | Restriction of the domain ... |
| hashnn0pnf 14266 | The value of the hash func... |
| hashnnn0genn0 14267 | If the size of a set is no... |
| hashnemnf 14268 | The size of a set is never... |
| hashv01gt1 14269 | The size of a set is eithe... |
| hashfz1 14270 | The set ` ( 1 ... N ) ` ha... |
| hashen 14271 | Two finite sets have the s... |
| hasheni 14272 | Equinumerous sets have the... |
| hasheqf1o 14273 | The size of two finite set... |
| fiinfnf1o 14274 | There is no bijection betw... |
| hasheqf1oi 14275 | The size of two sets is eq... |
| hashf1rn 14276 | The size of a finite set w... |
| hasheqf1od 14277 | The size of two sets is eq... |
| fz1eqb 14278 | Two possibly-empty 1-based... |
| hashcard 14279 | The size function of the c... |
| hashcl 14280 | Closure of the ` # ` funct... |
| hashxrcl 14281 | Extended real closure of t... |
| hashclb 14282 | Reverse closure of the ` #... |
| nfile 14283 | The size of any infinite s... |
| hashvnfin 14284 | A set of finite size is a ... |
| hashnfinnn0 14285 | The size of an infinite se... |
| isfinite4 14286 | A finite set is equinumero... |
| hasheq0 14287 | Two ways of saying a set i... |
| hashneq0 14288 | Two ways of saying a set i... |
| hashgt0n0 14289 | If the size of a set is gr... |
| hashnncl 14290 | Positive natural closure o... |
| hash0 14291 | The empty set has size zer... |
| hashelne0d 14292 | A set with an element has ... |
| hashsng 14293 | The size of a singleton. ... |
| hashen1 14294 | A set has size 1 if and on... |
| hash1elsn 14295 | A set of size 1 with a kno... |
| hashrabrsn 14296 | The size of a restricted c... |
| hashrabsn01 14297 | The size of a restricted c... |
| hashrabsn1 14298 | If the size of a restricte... |
| hashfn 14299 | A function is equinumerous... |
| fseq1hash 14300 | The value of the size func... |
| hashgadd 14301 | ` G ` maps ordinal additio... |
| hashgval2 14302 | A short expression for the... |
| hashdom 14303 | Dominance relation for the... |
| hashdomi 14304 | Non-strict order relation ... |
| hashsdom 14305 | Strict dominance relation ... |
| hashun 14306 | The size of the union of d... |
| hashun2 14307 | The size of the union of f... |
| hashun3 14308 | The size of the union of f... |
| hashinfxadd 14309 | The extended real addition... |
| hashunx 14310 | The size of the union of d... |
| hashge0 14311 | The cardinality of a set i... |
| hashgt0 14312 | The cardinality of a nonem... |
| hashge1 14313 | The cardinality of a nonem... |
| 1elfz0hash 14314 | 1 is an element of the fin... |
| hashnn0n0nn 14315 | If a nonnegative integer i... |
| hashunsng 14316 | The size of the union of a... |
| hashunsngx 14317 | The size of the union of a... |
| hashunsnggt 14318 | The size of a set is great... |
| hashprg 14319 | The size of an unordered p... |
| elprchashprn2 14320 | If one element of an unord... |
| hashprb 14321 | The size of an unordered p... |
| hashprdifel 14322 | The elements of an unorder... |
| prhash2ex 14323 | There is (at least) one se... |
| hashle00 14324 | If the size of a set is le... |
| hashgt0elex 14325 | If the size of a set is gr... |
| hashgt0elexb 14326 | The size of a set is great... |
| hashp1i 14327 | Size of a finite ordinal. ... |
| hash1 14328 | Size of a finite ordinal. ... |
| hash2 14329 | Size of a finite ordinal. ... |
| hash3 14330 | Size of a finite ordinal. ... |
| hash4 14331 | Size of a finite ordinal. ... |
| pr0hash2ex 14332 | There is (at least) one se... |
| hashss 14333 | The size of a subset is le... |
| prsshashgt1 14334 | The size of a superset of ... |
| hashin 14335 | The size of the intersecti... |
| hashssdif 14336 | The size of the difference... |
| hashdif 14337 | The size of the difference... |
| hashdifsn 14338 | The size of the difference... |
| hashdifpr 14339 | The size of the difference... |
| hashsn01 14340 | The size of a singleton is... |
| hashsnle1 14341 | The size of a singleton is... |
| hashsnlei 14342 | Get an upper bound on a co... |
| hash1snb 14343 | The size of a set is 1 if ... |
| euhash1 14344 | The size of a set is 1 in ... |
| hash1n0 14345 | If the size of a set is 1 ... |
| hashgt12el 14346 | In a set with more than on... |
| hashgt12el2 14347 | In a set with more than on... |
| hashgt23el 14348 | A set with more than two e... |
| hashunlei 14349 | Get an upper bound on a co... |
| hashsslei 14350 | Get an upper bound on a co... |
| hashfz 14351 | Value of the numeric cardi... |
| fzsdom2 14352 | Condition for finite range... |
| hashfzo 14353 | Cardinality of a half-open... |
| hashfzo0 14354 | Cardinality of a half-open... |
| hashfzp1 14355 | Value of the numeric cardi... |
| hashfz0 14356 | Value of the numeric cardi... |
| hashxplem 14357 | Lemma for ~ hashxp . (Con... |
| hashxp 14358 | The size of the Cartesian ... |
| hashmap 14359 | The size of the set expone... |
| hashpw 14360 | The size of the power set ... |
| hashfun 14361 | A finite set is a function... |
| hashres 14362 | The number of elements of ... |
| hashreshashfun 14363 | The number of elements of ... |
| hashimarn 14364 | The size of the image of a... |
| hashimarni 14365 | If the size of the image o... |
| hashfundm 14366 | The size of a set function... |
| hashf1dmrn 14367 | The size of the domain of ... |
| hashf1dmcdm 14368 | The size of the domain of ... |
| resunimafz0 14369 | TODO-AV: Revise using ` F... |
| fnfz0hash 14370 | The size of a function on ... |
| ffz0hash 14371 | The size of a function on ... |
| fnfz0hashnn0 14372 | The size of a function on ... |
| ffzo0hash 14373 | The size of a function on ... |
| fnfzo0hash 14374 | The size of a function on ... |
| fnfzo0hashnn0 14375 | The value of the size func... |
| hashbclem 14376 | Lemma for ~ hashbc : induc... |
| hashbc 14377 | The binomial coefficient c... |
| hashfacen 14378 | The number of bijections b... |
| hashf1lem1 14379 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 14380 | Lemma for ~ hashf1 . (Con... |
| hashf1 14381 | The permutation number ` |... |
| hashfac 14382 | A factorial counts the num... |
| leiso 14383 | Two ways to write a strict... |
| leisorel 14384 | Version of ~ isorel for st... |
| fz1isolem 14385 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14386 | Any finite ordered set has... |
| ishashinf 14387 | Any set that is not finite... |
| seqcoll 14388 | The function ` F ` contain... |
| seqcoll2 14389 | The function ` F ` contain... |
| phphashd 14390 | Corollary of the Pigeonhol... |
| phphashrd 14391 | Corollary of the Pigeonhol... |
| hashprlei 14392 | An unordered pair has at m... |
| hash2pr 14393 | A set of size two is an un... |
| hash2prde 14394 | A set of size two is an un... |
| hash2exprb 14395 | A set of size two is an un... |
| hash2prb 14396 | A set of size two is a pro... |
| prprrab 14397 | The set of proper pairs of... |
| nehash2 14398 | The cardinality of a set w... |
| hash2prd 14399 | A set of size two is an un... |
| hash2pwpr 14400 | If the size of a subset of... |
| hashle2pr 14401 | A nonempty set of size les... |
| hashle2prv 14402 | A nonempty subset of a pow... |
| pr2pwpr 14403 | The set of subsets of a pa... |
| hashge2el2dif 14404 | A set with size at least 2... |
| hashge2el2difr 14405 | A set with at least 2 diff... |
| hashge2el2difb 14406 | A set has size at least 2 ... |
| hashdmpropge2 14407 | The size of the domain of ... |
| hashtplei 14408 | An unordered triple has at... |
| hashtpg 14409 | The size of an unordered t... |
| hash7g 14410 | The size of an unordered s... |
| hashge3el3dif 14411 | A set with size at least 3... |
| elss2prb 14412 | An element of the set of s... |
| hash2sspr 14413 | A subset of size two is an... |
| exprelprel 14414 | If there is an element of ... |
| hash3tr 14415 | A set of size three is an ... |
| hash1to3 14416 | If the size of a set is be... |
| hash3tpde 14417 | A set of size three is an ... |
| hash3tpexb 14418 | A set of size three is an ... |
| hash3tpb 14419 | A set of size three is a p... |
| tpf1ofv0 14420 | The value of a one-to-one ... |
| tpf1ofv1 14421 | The value of a one-to-one ... |
| tpf1ofv2 14422 | The value of a one-to-one ... |
| tpf 14423 | A function into a (proper)... |
| tpfo 14424 | A function onto a (proper)... |
| tpf1o 14425 | A bijection onto a (proper... |
| fundmge2nop0 14426 | A function with a domain c... |
| fundmge2nop 14427 | A function with a domain c... |
| fun2dmnop0 14428 | A function with a domain c... |
| fun2dmnop 14429 | A function with a domain c... |
| hashdifsnp1 14430 | If the size of a set is a ... |
| fi1uzind 14431 | Properties of an ordered p... |
| brfi1uzind 14432 | Properties of a binary rel... |
| brfi1ind 14433 | Properties of a binary rel... |
| brfi1indALT 14434 | Alternate proof of ~ brfi1... |
| opfi1uzind 14435 | Properties of an ordered p... |
| opfi1ind 14436 | Properties of an ordered p... |
| iswrd 14439 | Property of being a word o... |
| wrdval 14440 | Value of the set of words ... |
| iswrdi 14441 | A zero-based sequence is a... |
| wrdf 14442 | A word is a zero-based seq... |
| wrdfd 14443 | A word is a zero-based seq... |
| iswrdb 14444 | A word over an alphabet is... |
| wrddm 14445 | The indices of a word (i.e... |
| sswrd 14446 | The set of words respects ... |
| snopiswrd 14447 | A singleton of an ordered ... |
| wrdexg 14448 | The set of words over a se... |
| wrdexb 14449 | The set of words over a se... |
| wrdexi 14450 | The set of words over a se... |
| wrdsymbcl 14451 | A symbol within a word ove... |
| wrdfn 14452 | A word is a function with ... |
| wrdv 14453 | A word over an alphabet is... |
| wrdlndm 14454 | The length of a word is no... |
| iswrdsymb 14455 | An arbitrary word is a wor... |
| wrdfin 14456 | A word is a finite set. (... |
| lencl 14457 | The length of a word is a ... |
| lennncl 14458 | The length of a nonempty w... |
| wrdffz 14459 | A word is a function from ... |
| wrdeq 14460 | Equality theorem for the s... |
| wrdeqi 14461 | Equality theorem for the s... |
| iswrddm0 14462 | A function with empty doma... |
| wrd0 14463 | The empty set is a word (t... |
| 0wrd0 14464 | The empty word is the only... |
| ffz0iswrd 14465 | A sequence with zero-based... |
| wrdsymb 14466 | A word is a word over the ... |
| nfwrd 14467 | Hypothesis builder for ` W... |
| csbwrdg 14468 | Class substitution for the... |
| wrdnval 14469 | Words of a fixed length ar... |
| wrdmap 14470 | Words as a mapping. (Cont... |
| hashwrdn 14471 | If there is only a finite ... |
| wrdnfi 14472 | If there is only a finite ... |
| wrdsymb0 14473 | A symbol at a position "ou... |
| wrdlenge1n0 14474 | A word with length at leas... |
| len0nnbi 14475 | The length of a word is a ... |
| wrdlenge2n0 14476 | A word with length at leas... |
| wrdsymb1 14477 | The first symbol of a none... |
| wrdlen1 14478 | A word of length 1 starts ... |
| fstwrdne 14479 | The first symbol of a none... |
| fstwrdne0 14480 | The first symbol of a none... |
| eqwrd 14481 | Two words are equal iff th... |
| elovmpowrd 14482 | Implications for the value... |
| elovmptnn0wrd 14483 | Implications for the value... |
| wrdred1 14484 | A word truncated by a symb... |
| wrdred1hash 14485 | The length of a word trunc... |
| lsw 14488 | Extract the last symbol of... |
| lsw0 14489 | The last symbol of an empt... |
| lsw0g 14490 | The last symbol of an empt... |
| lsw1 14491 | The last symbol of a word ... |
| lswcl 14492 | Closure of the last symbol... |
| lswlgt0cl 14493 | The last symbol of a nonem... |
| ccatfn 14496 | The concatenation operator... |
| ccatfval 14497 | Value of the concatenation... |
| ccatcl 14498 | The concatenation of two w... |
| ccatlen 14499 | The length of a concatenat... |
| ccat0 14500 | The concatenation of two w... |
| ccatval1 14501 | Value of a symbol in the l... |
| ccatval2 14502 | Value of a symbol in the r... |
| ccatval3 14503 | Value of a symbol in the r... |
| elfzelfzccat 14504 | An element of a finite set... |
| ccatvalfn 14505 | The concatenation of two w... |
| ccatdmss 14506 | The domain of a concatenat... |
| ccatsymb 14507 | The symbol at a given posi... |
| ccatfv0 14508 | The first symbol of a conc... |
| ccatval1lsw 14509 | The last symbol of the lef... |
| ccatval21sw 14510 | The first symbol of the ri... |
| ccatlid 14511 | Concatenation of a word by... |
| ccatrid 14512 | Concatenation of a word by... |
| ccatass 14513 | Associative law for concat... |
| ccatrn 14514 | The range of a concatenate... |
| ccatidid 14515 | Concatenation of the empty... |
| lswccatn0lsw 14516 | The last symbol of a word ... |
| lswccat0lsw 14517 | The last symbol of a word ... |
| ccatalpha 14518 | A concatenation of two arb... |
| ccatrcl1 14519 | Reverse closure of a conca... |
| ids1 14522 | Identity function protecti... |
| s1val 14523 | Value of a singleton word.... |
| s1rn 14524 | The range of a singleton w... |
| s1eq 14525 | Equality theorem for a sin... |
| s1eqd 14526 | Equality theorem for a sin... |
| s1cl 14527 | A singleton word is a word... |
| s1cld 14528 | A singleton word is a word... |
| s1prc 14529 | Value of a singleton word ... |
| s1cli 14530 | A singleton word is a word... |
| s1len 14531 | Length of a singleton word... |
| s1nz 14532 | A singleton word is not th... |
| s1dm 14533 | The domain of a singleton ... |
| s1dmALT 14534 | Alternate version of ~ s1d... |
| s1fv 14535 | Sole symbol of a singleton... |
| lsws1 14536 | The last symbol of a singl... |
| eqs1 14537 | A word of length 1 is a si... |
| wrdl1exs1 14538 | A word of length 1 is a si... |
| wrdl1s1 14539 | A word of length 1 is a si... |
| s111 14540 | The singleton word functio... |
| ccatws1cl 14541 | The concatenation of a wor... |
| ccatws1clv 14542 | The concatenation of a wor... |
| ccat2s1cl 14543 | The concatenation of two s... |
| ccats1alpha 14544 | A concatenation of a word ... |
| ccatws1len 14545 | The length of the concaten... |
| ccatws1lenp1b 14546 | The length of a word is ` ... |
| wrdlenccats1lenm1 14547 | The length of a word is th... |
| ccat2s1len 14548 | The length of the concaten... |
| ccatw2s1cl 14549 | The concatenation of a wor... |
| ccatw2s1len 14550 | The length of the concaten... |
| ccats1val1 14551 | Value of a symbol in the l... |
| ccats1val2 14552 | Value of the symbol concat... |
| ccat1st1st 14553 | The first symbol of a word... |
| ccat2s1p1 14554 | Extract the first of two c... |
| ccat2s1p2 14555 | Extract the second of two ... |
| ccatw2s1ass 14556 | Associative law for a conc... |
| ccatws1n0 14557 | The concatenation of a wor... |
| ccatws1ls 14558 | The last symbol of the con... |
| lswccats1 14559 | The last symbol of a word ... |
| lswccats1fst 14560 | The last symbol of a nonem... |
| ccatw2s1p1 14561 | Extract the symbol of the ... |
| ccatw2s1p2 14562 | Extract the second of two ... |
| ccat2s1fvw 14563 | Extract a symbol of a word... |
| ccat2s1fst 14564 | The first symbol of the co... |
| swrdnznd 14567 | The value of a subword ope... |
| swrdval 14568 | Value of a subword. (Cont... |
| swrd00 14569 | A zero length substring. ... |
| swrdcl 14570 | Closure of the subword ext... |
| swrdval2 14571 | Value of the subword extra... |
| swrdlen 14572 | Length of an extracted sub... |
| swrdfv 14573 | A symbol in an extracted s... |
| swrdfv0 14574 | The first symbol in an ext... |
| swrdf 14575 | A subword of a word is a f... |
| swrdvalfn 14576 | Value of the subword extra... |
| swrdrn 14577 | The range of a subword of ... |
| swrdlend 14578 | The value of the subword e... |
| swrdnd 14579 | The value of the subword e... |
| swrdnd2 14580 | Value of the subword extra... |
| swrdnnn0nd 14581 | The value of a subword ope... |
| swrdnd0 14582 | The value of a subword ope... |
| swrd0 14583 | A subword of an empty set ... |
| swrdrlen 14584 | Length of a right-anchored... |
| swrdlen2 14585 | Length of an extracted sub... |
| swrdfv2 14586 | A symbol in an extracted s... |
| swrdwrdsymb 14587 | A subword is a word over t... |
| swrdsb0eq 14588 | Two subwords with the same... |
| swrdsbslen 14589 | Two subwords with the same... |
| swrdspsleq 14590 | Two words have a common su... |
| swrds1 14591 | Extract a single symbol fr... |
| swrdlsw 14592 | Extract the last single sy... |
| ccatswrd 14593 | Joining two adjacent subwo... |
| swrdccat2 14594 | Recover the right half of ... |
| pfxnndmnd 14597 | The value of a prefix oper... |
| pfxval 14598 | Value of a prefix operatio... |
| pfx00 14599 | The zero length prefix is ... |
| pfx0 14600 | A prefix of an empty set i... |
| pfxval0 14601 | Value of a prefix operatio... |
| pfxcl 14602 | Closure of the prefix extr... |
| pfxmpt 14603 | Value of the prefix extrac... |
| pfxres 14604 | Value of the prefix extrac... |
| pfxf 14605 | A prefix of a word is a fu... |
| pfxfn 14606 | Value of the prefix extrac... |
| pfxfv 14607 | A symbol in a prefix of a ... |
| pfxlen 14608 | Length of a prefix. (Cont... |
| pfxid 14609 | A word is a prefix of itse... |
| pfxrn 14610 | The range of a prefix of a... |
| pfxn0 14611 | A prefix consisting of at ... |
| pfxnd 14612 | The value of a prefix oper... |
| pfxnd0 14613 | The value of a prefix oper... |
| pfxwrdsymb 14614 | A prefix of a word is a wo... |
| addlenpfx 14615 | The sum of the lengths of ... |
| pfxfv0 14616 | The first symbol of a pref... |
| pfxtrcfv 14617 | A symbol in a word truncat... |
| pfxtrcfv0 14618 | The first symbol in a word... |
| pfxfvlsw 14619 | The last symbol in a nonem... |
| pfxeq 14620 | The prefixes of two words ... |
| pfxtrcfvl 14621 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14622 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14623 | Two (nonempty) words are e... |
| disjwrdpfx 14624 | Sets of words are disjoint... |
| ccatpfx 14625 | Concatenating a prefix wit... |
| pfxccat1 14626 | Recover the left half of a... |
| pfx1 14627 | The prefix of length one o... |
| swrdswrdlem 14628 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14629 | A subword of a subword is ... |
| pfxswrd 14630 | A prefix of a subword is a... |
| swrdpfx 14631 | A subword of a prefix is a... |
| pfxpfx 14632 | A prefix of a prefix is a ... |
| pfxpfxid 14633 | A prefix of a prefix with ... |
| pfxcctswrd 14634 | The concatenation of the p... |
| lenpfxcctswrd 14635 | The length of the concaten... |
| lenrevpfxcctswrd 14636 | The length of the concaten... |
| pfxlswccat 14637 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14638 | The last symbol of a word ... |
| ccats1pfxeqrex 14639 | There exists a symbol such... |
| ccatopth 14640 | An ~ opth -like theorem fo... |
| ccatopth2 14641 | An ~ opth -like theorem fo... |
| ccatlcan 14642 | Concatenation of words is ... |
| ccatrcan 14643 | Concatenation of words is ... |
| wrdeqs1cat 14644 | Decompose a nonempty word ... |
| cats1un 14645 | Express a word with an ext... |
| wrdind 14646 | Perform induction over the... |
| wrd2ind 14647 | Perform induction over the... |
| swrdccatfn 14648 | The subword of a concatena... |
| swrdccatin1 14649 | The subword of a concatena... |
| pfxccatin12lem4 14650 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14651 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14652 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14653 | The subword of a concatena... |
| pfxccatin12lem2c 14654 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14655 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14656 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14657 | The subword of a concatena... |
| pfxccat3 14658 | The subword of a concatena... |
| swrdccat 14659 | The subword of a concatena... |
| pfxccatpfx1 14660 | A prefix of a concatenatio... |
| pfxccatpfx2 14661 | A prefix of a concatenatio... |
| pfxccat3a 14662 | A prefix of a concatenatio... |
| swrdccat3blem 14663 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14664 | A suffix of a concatenatio... |
| pfxccatid 14665 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14666 | A word is a prefix of a wo... |
| swrdccatin1d 14667 | The subword of a concatena... |
| swrdccatin2d 14668 | The subword of a concatena... |
| pfxccatin12d 14669 | The subword of a concatena... |
| reuccatpfxs1lem 14670 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14671 | There is a unique word hav... |
| reuccatpfxs1v 14672 | There is a unique word hav... |
| splval 14675 | Value of the substring rep... |
| splcl 14676 | Closure of the substring r... |
| splid 14677 | Splicing a subword for the... |
| spllen 14678 | The length of a splice. (... |
| splfv1 14679 | Symbols to the left of a s... |
| splfv2a 14680 | Symbols within the replace... |
| splval2 14681 | Value of a splice, assumin... |
| revval 14684 | Value of the word reversin... |
| revcl 14685 | The reverse of a word is a... |
| revlen 14686 | The reverse of a word has ... |
| revfv 14687 | Reverse of a word at a poi... |
| rev0 14688 | The empty word is its own ... |
| revs1 14689 | Singleton words are their ... |
| revccat 14690 | Antiautomorphic property o... |
| revrev 14691 | Reversal is an involution ... |
| reps 14694 | Construct a function mappi... |
| repsundef 14695 | A function mapping a half-... |
| repsconst 14696 | Construct a function mappi... |
| repsf 14697 | The constructed function m... |
| repswsymb 14698 | The symbols of a "repeated... |
| repsw 14699 | A function mapping a half-... |
| repswlen 14700 | The length of a "repeated ... |
| repsw0 14701 | The "repeated symbol word"... |
| repsdf2 14702 | Alternative definition of ... |
| repswsymball 14703 | All the symbols of a "repe... |
| repswsymballbi 14704 | A word is a "repeated symb... |
| repswfsts 14705 | The first symbol of a none... |
| repswlsw 14706 | The last symbol of a nonem... |
| repsw1 14707 | The "repeated symbol word"... |
| repswswrd 14708 | A subword of a "repeated s... |
| repswpfx 14709 | A prefix of a repeated sym... |
| repswccat 14710 | The concatenation of two "... |
| repswrevw 14711 | The reverse of a "repeated... |
| cshfn 14714 | Perform a cyclical shift f... |
| cshword 14715 | Perform a cyclical shift f... |
| cshnz 14716 | A cyclical shift is the em... |
| 0csh0 14717 | Cyclically shifting an emp... |
| cshw0 14718 | A word cyclically shifted ... |
| cshwmodn 14719 | Cyclically shifting a word... |
| cshwsublen 14720 | Cyclically shifting a word... |
| cshwn 14721 | A word cyclically shifted ... |
| cshwcl 14722 | A cyclically shifted word ... |
| cshwlen 14723 | The length of a cyclically... |
| cshwf 14724 | A cyclically shifted word ... |
| cshwfn 14725 | A cyclically shifted word ... |
| cshwrn 14726 | The range of a cyclically ... |
| cshwidxmod 14727 | The symbol at a given inde... |
| cshwidxmodr 14728 | The symbol at a given inde... |
| cshwidx0mod 14729 | The symbol at index 0 of a... |
| cshwidx0 14730 | The symbol at index 0 of a... |
| cshwidxm1 14731 | The symbol at index ((n-N)... |
| cshwidxm 14732 | The symbol at index (n-N) ... |
| cshwidxn 14733 | The symbol at index (n-1) ... |
| cshf1 14734 | Cyclically shifting a word... |
| cshinj 14735 | If a word is injectiv (reg... |
| repswcshw 14736 | A cyclically shifted "repe... |
| 2cshw 14737 | Cyclically shifting a word... |
| 2cshwid 14738 | Cyclically shifting a word... |
| lswcshw 14739 | The last symbol of a word ... |
| 2cshwcom 14740 | Cyclically shifting a word... |
| cshwleneq 14741 | If the results of cyclical... |
| 3cshw 14742 | Cyclically shifting a word... |
| cshweqdif2 14743 | If cyclically shifting two... |
| cshweqdifid 14744 | If cyclically shifting a w... |
| cshweqrep 14745 | If cyclically shifting a w... |
| cshw1 14746 | If cyclically shifting a w... |
| cshw1repsw 14747 | If cyclically shifting a w... |
| cshwsexa 14748 | The class of (different!) ... |
| 2cshwcshw 14749 | If a word is a cyclically ... |
| scshwfzeqfzo 14750 | For a nonempty word the se... |
| cshwcshid 14751 | A cyclically shifted word ... |
| cshwcsh2id 14752 | A cyclically shifted word ... |
| cshimadifsn 14753 | The image of a cyclically ... |
| cshimadifsn0 14754 | The image of a cyclically ... |
| wrdco 14755 | Mapping a word by a functi... |
| lenco 14756 | Length of a mapped word is... |
| s1co 14757 | Mapping of a singleton wor... |
| revco 14758 | Mapping of words (i.e., a ... |
| ccatco 14759 | Mapping of words commutes ... |
| cshco 14760 | Mapping of words commutes ... |
| swrdco 14761 | Mapping of words commutes ... |
| pfxco 14762 | Mapping of words commutes ... |
| lswco 14763 | Mapping of (nonempty) word... |
| repsco 14764 | Mapping of words commutes ... |
| cats1cld 14779 | Closure of concatenation w... |
| cats1co 14780 | Closure of concatenation w... |
| cats1cli 14781 | Closure of concatenation w... |
| cats1fvn 14782 | The last symbol of a conca... |
| cats1fv 14783 | A symbol other than the la... |
| cats1len 14784 | The length of concatenatio... |
| cats1cat 14785 | Closure of concatenation w... |
| cats2cat 14786 | Closure of concatenation o... |
| s2eqd 14787 | Equality theorem for a dou... |
| s3eqd 14788 | Equality theorem for a len... |
| s4eqd 14789 | Equality theorem for a len... |
| s5eqd 14790 | Equality theorem for a len... |
| s6eqd 14791 | Equality theorem for a len... |
| s7eqd 14792 | Equality theorem for a len... |
| s8eqd 14793 | Equality theorem for a len... |
| s3eq2 14794 | Equality theorem for a len... |
| s2cld 14795 | A doubleton word is a word... |
| s3cld 14796 | A length 3 string is a wor... |
| s4cld 14797 | A length 4 string is a wor... |
| s5cld 14798 | A length 5 string is a wor... |
| s6cld 14799 | A length 6 string is a wor... |
| s7cld 14800 | A length 7 string is a wor... |
| s8cld 14801 | A length 8 string is a wor... |
| s2cl 14802 | A doubleton word is a word... |
| s3cl 14803 | A length 3 string is a wor... |
| s2cli 14804 | A doubleton word is a word... |
| s3cli 14805 | A length 3 string is a wor... |
| s4cli 14806 | A length 4 string is a wor... |
| s5cli 14807 | A length 5 string is a wor... |
| s6cli 14808 | A length 6 string is a wor... |
| s7cli 14809 | A length 7 string is a wor... |
| s8cli 14810 | A length 8 string is a wor... |
| s2fv0 14811 | Extract the first symbol f... |
| s2fv1 14812 | Extract the second symbol ... |
| s2len 14813 | The length of a doubleton ... |
| s2dm 14814 | The domain of a doubleton ... |
| s3fv0 14815 | Extract the first symbol f... |
| s3fv1 14816 | Extract the second symbol ... |
| s3fv2 14817 | Extract the third symbol f... |
| s3len 14818 | The length of a length 3 s... |
| s4fv0 14819 | Extract the first symbol f... |
| s4fv1 14820 | Extract the second symbol ... |
| s4fv2 14821 | Extract the third symbol f... |
| s4fv3 14822 | Extract the fourth symbol ... |
| s4len 14823 | The length of a length 4 s... |
| s5len 14824 | The length of a length 5 s... |
| s6len 14825 | The length of a length 6 s... |
| s7len 14826 | The length of a length 7 s... |
| s8len 14827 | The length of a length 8 s... |
| lsws2 14828 | The last symbol of a doubl... |
| lsws3 14829 | The last symbol of a 3 let... |
| lsws4 14830 | The last symbol of a 4 let... |
| s2prop 14831 | A length 2 word is an unor... |
| s2dmALT 14832 | Alternate version of ~ s2d... |
| s3tpop 14833 | A length 3 word is an unor... |
| s4prop 14834 | A length 4 word is a union... |
| s3fn 14835 | A length 3 word is a funct... |
| funcnvs1 14836 | The converse of a singleto... |
| funcnvs2 14837 | The converse of a length 2... |
| funcnvs3 14838 | The converse of a length 3... |
| funcnvs4 14839 | The converse of a length 4... |
| s2f1o 14840 | A length 2 word with mutua... |
| f1oun2prg 14841 | A union of unordered pairs... |
| s4f1o 14842 | A length 4 word with mutua... |
| s4dom 14843 | The domain of a length 4 w... |
| s2co 14844 | Mapping a doubleton word b... |
| s3co 14845 | Mapping a length 3 string ... |
| s0s1 14846 | Concatenation of fixed len... |
| s1s2 14847 | Concatenation of fixed len... |
| s1s3 14848 | Concatenation of fixed len... |
| s1s4 14849 | Concatenation of fixed len... |
| s1s5 14850 | Concatenation of fixed len... |
| s1s6 14851 | Concatenation of fixed len... |
| s1s7 14852 | Concatenation of fixed len... |
| s2s2 14853 | Concatenation of fixed len... |
| s4s2 14854 | Concatenation of fixed len... |
| s4s3 14855 | Concatenation of fixed len... |
| s4s4 14856 | Concatenation of fixed len... |
| s3s4 14857 | Concatenation of fixed len... |
| s2s5 14858 | Concatenation of fixed len... |
| s5s2 14859 | Concatenation of fixed len... |
| s2eq2s1eq 14860 | Two length 2 words are equ... |
| s2eq2seq 14861 | Two length 2 words are equ... |
| s3eqs2s1eq 14862 | Two length 3 words are equ... |
| s3eq3seq 14863 | Two length 3 words are equ... |
| swrds2 14864 | Extract two adjacent symbo... |
| swrds2m 14865 | Extract two adjacent symbo... |
| wrdlen2i 14866 | Implications of a word of ... |
| wrd2pr2op 14867 | A word of length two repre... |
| wrdlen2 14868 | A word of length two. (Co... |
| wrdlen2s2 14869 | A word of length two as do... |
| wrdl2exs2 14870 | A word of length two is a ... |
| pfx2 14871 | A prefix of length two. (... |
| wrd3tpop 14872 | A word of length three rep... |
| wrdlen3s3 14873 | A word of length three as ... |
| repsw2 14874 | The "repeated symbol word"... |
| repsw3 14875 | The "repeated symbol word"... |
| swrd2lsw 14876 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14877 | Two words of length at lea... |
| ccatw2s1ccatws2 14878 | The concatenation of a wor... |
| ccat2s1fvwALT 14879 | Alternate proof of ~ ccat2... |
| wwlktovf 14880 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14881 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14882 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14883 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14884 | There is a bijection betwe... |
| eqwrds3 14885 | A word is equal with a len... |
| wrdl3s3 14886 | A word of length 3 is a le... |
| s2rn 14887 | Range of a length 2 string... |
| s3rn 14888 | Range of a length 3 string... |
| s7rn 14889 | Range of a length 7 string... |
| s7f1o 14890 | A length 7 word with mutua... |
| s3sndisj 14891 | The singletons consisting ... |
| s3iunsndisj 14892 | The union of singletons co... |
| ofccat 14893 | Letterwise operations on w... |
| ofs1 14894 | Letterwise operations on a... |
| ofs2 14895 | Letterwise operations on a... |
| coss12d 14896 | Subset deduction for compo... |
| trrelssd 14897 | The composition of subclas... |
| xpcogend 14898 | The most interesting case ... |
| xpcoidgend 14899 | If two classes are not dis... |
| cotr2g 14900 | Two ways of saying that th... |
| cotr2 14901 | Two ways of saying a relat... |
| cotr3 14902 | Two ways of saying a relat... |
| coemptyd 14903 | Deduction about compositio... |
| xptrrel 14904 | The cross product is alway... |
| 0trrel 14905 | The empty class is a trans... |
| cleq1lem 14906 | Equality implies bijection... |
| cleq1 14907 | Equality of relations impl... |
| clsslem 14908 | The closure of a subclass ... |
| trcleq1 14913 | Equality of relations impl... |
| trclsslem 14914 | The transitive closure (as... |
| trcleq2lem 14915 | Equality implies bijection... |
| cvbtrcl 14916 | Change of bound variable i... |
| trcleq12lem 14917 | Equality implies bijection... |
| trclexlem 14918 | Existence of relation impl... |
| trclublem 14919 | If a relation exists then ... |
| trclubi 14920 | The Cartesian product of t... |
| trclubgi 14921 | The union with the Cartesi... |
| trclub 14922 | The Cartesian product of t... |
| trclubg 14923 | The union with the Cartesi... |
| trclfv 14924 | The transitive closure of ... |
| brintclab 14925 | Two ways to express a bina... |
| brtrclfv 14926 | Two ways of expressing the... |
| brcnvtrclfv 14927 | Two ways of expressing the... |
| brtrclfvcnv 14928 | Two ways of expressing the... |
| brcnvtrclfvcnv 14929 | Two ways of expressing the... |
| trclfvss 14930 | The transitive closure (as... |
| trclfvub 14931 | The transitive closure of ... |
| trclfvlb 14932 | The transitive closure of ... |
| trclfvcotr 14933 | The transitive closure of ... |
| trclfvlb2 14934 | The transitive closure of ... |
| trclfvlb3 14935 | The transitive closure of ... |
| cotrtrclfv 14936 | The transitive closure of ... |
| trclidm 14937 | The transitive closure of ... |
| trclun 14938 | Transitive closure of a un... |
| trclfvg 14939 | The value of the transitiv... |
| trclfvcotrg 14940 | The value of the transitiv... |
| reltrclfv 14941 | The transitive closure of ... |
| dmtrclfv 14942 | The domain of the transiti... |
| reldmrelexp 14945 | The domain of the repeated... |
| relexp0g 14946 | A relation composed zero t... |
| relexp0 14947 | A relation composed zero t... |
| relexp0d 14948 | A relation composed zero t... |
| relexpsucnnr 14949 | A reduction for relation e... |
| relexp1g 14950 | A relation composed once i... |
| dfid5 14951 | Identity relation is equal... |
| dfid6 14952 | Identity relation expresse... |
| relexp1d 14953 | A relation composed once i... |
| relexpsucnnl 14954 | A reduction for relation e... |
| relexpsucl 14955 | A reduction for relation e... |
| relexpsucr 14956 | A reduction for relation e... |
| relexpsucrd 14957 | A reduction for relation e... |
| relexpsucld 14958 | A reduction for relation e... |
| relexpcnv 14959 | Commutation of converse an... |
| relexpcnvd 14960 | Commutation of converse an... |
| relexp0rel 14961 | The exponentiation of a cl... |
| relexprelg 14962 | The exponentiation of a cl... |
| relexprel 14963 | The exponentiation of a re... |
| relexpreld 14964 | The exponentiation of a re... |
| relexpnndm 14965 | The domain of an exponenti... |
| relexpdmg 14966 | The domain of an exponenti... |
| relexpdm 14967 | The domain of an exponenti... |
| relexpdmd 14968 | The domain of an exponenti... |
| relexpnnrn 14969 | The range of an exponentia... |
| relexprng 14970 | The range of an exponentia... |
| relexprn 14971 | The range of an exponentia... |
| relexprnd 14972 | The range of an exponentia... |
| relexpfld 14973 | The field of an exponentia... |
| relexpfldd 14974 | The field of an exponentia... |
| relexpaddnn 14975 | Relation composition becom... |
| relexpuzrel 14976 | The exponentiation of a cl... |
| relexpaddg 14977 | Relation composition becom... |
| relexpaddd 14978 | Relation composition becom... |
| rtrclreclem1 14981 | The reflexive, transitive ... |
| dfrtrclrec2 14982 | If two elements are connec... |
| rtrclreclem2 14983 | The reflexive, transitive ... |
| rtrclreclem3 14984 | The reflexive, transitive ... |
| rtrclreclem4 14985 | The reflexive, transitive ... |
| dfrtrcl2 14986 | The two definitions ` t* `... |
| relexpindlem 14987 | Principle of transitive in... |
| relexpind 14988 | Principle of transitive in... |
| rtrclind 14989 | Principle of transitive in... |
| shftlem 14992 | Two ways to write a shifte... |
| shftuz 14993 | A shift of the upper integ... |
| shftfval 14994 | The value of the sequence ... |
| shftdm 14995 | Domain of a relation shift... |
| shftfib 14996 | Value of a fiber of the re... |
| shftfn 14997 | Functionality and domain o... |
| shftval 14998 | Value of a sequence shifte... |
| shftval2 14999 | Value of a sequence shifte... |
| shftval3 15000 | Value of a sequence shifte... |
| shftval4 15001 | Value of a sequence shifte... |
| shftval5 15002 | Value of a shifted sequenc... |
| shftf 15003 | Functionality of a shifted... |
| 2shfti 15004 | Composite shift operations... |
| shftidt2 15005 | Identity law for the shift... |
| shftidt 15006 | Identity law for the shift... |
| shftcan1 15007 | Cancellation law for the s... |
| shftcan2 15008 | Cancellation law for the s... |
| seqshft 15009 | Shifting the index set of ... |
| sgnval 15012 | Value of the signum functi... |
| sgn0 15013 | The signum of 0 is 0. (Co... |
| sgnp 15014 | The signum of a positive e... |
| sgnrrp 15015 | The signum of a positive r... |
| sgn1 15016 | The signum of 1 is 1. (Co... |
| sgnpnf 15017 | The signum of ` +oo ` is 1... |
| sgnn 15018 | The signum of a negative e... |
| sgnmnf 15019 | The signum of ` -oo ` is -... |
| cjval 15026 | The value of the conjugate... |
| cjth 15027 | The defining property of t... |
| cjf 15028 | Domain and codomain of the... |
| cjcl 15029 | The conjugate of a complex... |
| reval 15030 | The value of the real part... |
| imval 15031 | The value of the imaginary... |
| imre 15032 | The imaginary part of a co... |
| reim 15033 | The real part of a complex... |
| recl 15034 | The real part of a complex... |
| imcl 15035 | The imaginary part of a co... |
| ref 15036 | Domain and codomain of the... |
| imf 15037 | Domain and codomain of the... |
| crre 15038 | The real part of a complex... |
| crim 15039 | The real part of a complex... |
| replim 15040 | Reconstruct a complex numb... |
| remim 15041 | Value of the conjugate of ... |
| reim0 15042 | The imaginary part of a re... |
| reim0b 15043 | A number is real iff its i... |
| rereb 15044 | A number is real iff it eq... |
| mulre 15045 | A product with a nonzero r... |
| rere 15046 | A real number equals its r... |
| cjreb 15047 | A number is real iff it eq... |
| recj 15048 | Real part of a complex con... |
| reneg 15049 | Real part of negative. (C... |
| readd 15050 | Real part distributes over... |
| resub 15051 | Real part distributes over... |
| remullem 15052 | Lemma for ~ remul , ~ immu... |
| remul 15053 | Real part of a product. (... |
| remul2 15054 | Real part of a product. (... |
| rediv 15055 | Real part of a division. ... |
| imcj 15056 | Imaginary part of a comple... |
| imneg 15057 | The imaginary part of a ne... |
| imadd 15058 | Imaginary part distributes... |
| imsub 15059 | Imaginary part distributes... |
| immul 15060 | Imaginary part of a produc... |
| immul2 15061 | Imaginary part of a produc... |
| imdiv 15062 | Imaginary part of a divisi... |
| cjre 15063 | A real number equals its c... |
| cjcj 15064 | The conjugate of the conju... |
| cjadd 15065 | Complex conjugate distribu... |
| cjmul 15066 | Complex conjugate distribu... |
| ipcnval 15067 | Standard inner product on ... |
| cjmulrcl 15068 | A complex number times its... |
| cjmulval 15069 | A complex number times its... |
| cjmulge0 15070 | A complex number times its... |
| cjneg 15071 | Complex conjugate of negat... |
| addcj 15072 | A number plus its conjugat... |
| cjsub 15073 | Complex conjugate distribu... |
| cjexp 15074 | Complex conjugate of posit... |
| imval2 15075 | The imaginary part of a nu... |
| re0 15076 | The real part of zero. (C... |
| im0 15077 | The imaginary part of zero... |
| re1 15078 | The real part of one. (Co... |
| im1 15079 | The imaginary part of one.... |
| rei 15080 | The real part of ` _i ` . ... |
| imi 15081 | The imaginary part of ` _i... |
| cj0 15082 | The conjugate of zero. (C... |
| cji 15083 | The complex conjugate of t... |
| cjreim 15084 | The conjugate of a represe... |
| cjreim2 15085 | The conjugate of the repre... |
| cj11 15086 | Complex conjugate is a one... |
| cjne0 15087 | A number is nonzero iff it... |
| cjdiv 15088 | Complex conjugate distribu... |
| cnrecnv 15089 | The inverse to the canonic... |
| sqeqd 15090 | A deduction for showing tw... |
| recli 15091 | The real part of a complex... |
| imcli 15092 | The imaginary part of a co... |
| cjcli 15093 | Closure law for complex co... |
| replimi 15094 | Construct a complex number... |
| cjcji 15095 | The conjugate of the conju... |
| reim0bi 15096 | A number is real iff its i... |
| rerebi 15097 | A real number equals its r... |
| cjrebi 15098 | A number is real iff it eq... |
| recji 15099 | Real part of a complex con... |
| imcji 15100 | Imaginary part of a comple... |
| cjmulrcli 15101 | A complex number times its... |
| cjmulvali 15102 | A complex number times its... |
| cjmulge0i 15103 | A complex number times its... |
| renegi 15104 | Real part of negative. (C... |
| imnegi 15105 | Imaginary part of negative... |
| cjnegi 15106 | Complex conjugate of negat... |
| addcji 15107 | A number plus its conjugat... |
| readdi 15108 | Real part distributes over... |
| imaddi 15109 | Imaginary part distributes... |
| remuli 15110 | Real part of a product. (... |
| immuli 15111 | Imaginary part of a produc... |
| cjaddi 15112 | Complex conjugate distribu... |
| cjmuli 15113 | Complex conjugate distribu... |
| ipcni 15114 | Standard inner product on ... |
| cjdivi 15115 | Complex conjugate distribu... |
| crrei 15116 | The real part of a complex... |
| crimi 15117 | The imaginary part of a co... |
| recld 15118 | The real part of a complex... |
| imcld 15119 | The imaginary part of a co... |
| cjcld 15120 | Closure law for complex co... |
| replimd 15121 | Construct a complex number... |
| remimd 15122 | Value of the conjugate of ... |
| cjcjd 15123 | The conjugate of the conju... |
| reim0bd 15124 | A number is real iff its i... |
| rerebd 15125 | A real number equals its r... |
| cjrebd 15126 | A number is real iff it eq... |
| cjne0d 15127 | A number is nonzero iff it... |
| recjd 15128 | Real part of a complex con... |
| imcjd 15129 | Imaginary part of a comple... |
| cjmulrcld 15130 | A complex number times its... |
| cjmulvald 15131 | A complex number times its... |
| cjmulge0d 15132 | A complex number times its... |
| renegd 15133 | Real part of negative. (C... |
| imnegd 15134 | Imaginary part of negative... |
| cjnegd 15135 | Complex conjugate of negat... |
| addcjd 15136 | A number plus its conjugat... |
| cjexpd 15137 | Complex conjugate of posit... |
| readdd 15138 | Real part distributes over... |
| imaddd 15139 | Imaginary part distributes... |
| resubd 15140 | Real part distributes over... |
| imsubd 15141 | Imaginary part distributes... |
| remuld 15142 | Real part of a product. (... |
| immuld 15143 | Imaginary part of a produc... |
| cjaddd 15144 | Complex conjugate distribu... |
| cjmuld 15145 | Complex conjugate distribu... |
| ipcnd 15146 | Standard inner product on ... |
| cjdivd 15147 | Complex conjugate distribu... |
| rered 15148 | A real number equals its r... |
| reim0d 15149 | The imaginary part of a re... |
| cjred 15150 | A real number equals its c... |
| remul2d 15151 | Real part of a product. (... |
| immul2d 15152 | Imaginary part of a produc... |
| redivd 15153 | Real part of a division. ... |
| imdivd 15154 | Imaginary part of a divisi... |
| crred 15155 | The real part of a complex... |
| crimd 15156 | The imaginary part of a co... |
| sqrtval 15161 | Value of square root funct... |
| absval 15162 | The absolute value (modulu... |
| rennim 15163 | A real number does not lie... |
| cnpart 15164 | The specification of restr... |
| sqrt0 15165 | The square root of zero is... |
| 01sqrexlem1 15166 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15167 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15168 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15169 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15170 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15171 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15172 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15173 | Existence of a square root... |
| resqrex 15174 | Existence of a square root... |
| sqrmo 15175 | Uniqueness for the square ... |
| resqreu 15176 | Existence and uniqueness f... |
| resqrtcl 15177 | Closure of the square root... |
| resqrtthlem 15178 | Lemma for ~ resqrtth . (C... |
| resqrtth 15179 | Square root theorem over t... |
| remsqsqrt 15180 | Square of square root. (C... |
| sqrtge0 15181 | The square root function i... |
| sqrtgt0 15182 | The square root function i... |
| sqrtmul 15183 | Square root distributes ov... |
| sqrtle 15184 | Square root is monotonic. ... |
| sqrtlt 15185 | Square root is strictly mo... |
| sqrt11 15186 | The square root function i... |
| sqrt00 15187 | A square root is zero iff ... |
| rpsqrtcl 15188 | The square root of a posit... |
| sqrtdiv 15189 | Square root distributes ov... |
| sqrtneglem 15190 | The square root of a negat... |
| sqrtneg 15191 | The square root of a negat... |
| sqrtsq2 15192 | Relationship between squar... |
| sqrtsq 15193 | Square root of square. (C... |
| sqrtmsq 15194 | Square root of square. (C... |
| sqrt1 15195 | The square root of 1 is 1.... |
| sqrt4 15196 | The square root of 4 is 2.... |
| sqrt9 15197 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15198 | The square root of 2 is bo... |
| sqrtm1 15199 | The imaginary unit is the ... |
| nn0sqeq1 15200 | A natural number with squa... |
| absneg 15201 | Absolute value of the nega... |
| abscl 15202 | Real closure of absolute v... |
| abscj 15203 | The absolute value of a nu... |
| absvalsq 15204 | Square of value of absolut... |
| absvalsq2 15205 | Square of value of absolut... |
| sqabsadd 15206 | Square of absolute value o... |
| sqabssub 15207 | Square of absolute value o... |
| absval2 15208 | Value of absolute value fu... |
| abs0 15209 | The absolute value of 0. ... |
| absi 15210 | The absolute value of the ... |
| absge0 15211 | Absolute value is nonnegat... |
| absrpcl 15212 | The absolute value of a no... |
| abs00 15213 | The absolute value of a nu... |
| abs00ad 15214 | A complex number is zero i... |
| abs00bd 15215 | If a complex number is zer... |
| absreimsq 15216 | Square of the absolute val... |
| absreim 15217 | Absolute value of a number... |
| absmul 15218 | Absolute value distributes... |
| absdiv 15219 | Absolute value distributes... |
| absid 15220 | A nonnegative number is it... |
| abs1 15221 | The absolute value of one ... |
| absnid 15222 | For a negative number, its... |
| leabs 15223 | A real number is less than... |
| absor 15224 | The absolute value of a re... |
| absre 15225 | Absolute value of a real n... |
| absresq 15226 | Square of the absolute val... |
| absmod0 15227 | ` A ` is divisible by ` B ... |
| absexp 15228 | Absolute value of positive... |
| absexpz 15229 | Absolute value of integer ... |
| abssq 15230 | Square can be moved in and... |
| sqabs 15231 | The squares of two reals a... |
| absrele 15232 | The absolute value of a co... |
| absimle 15233 | The absolute value of a co... |
| max0add 15234 | The sum of the positive an... |
| absz 15235 | A real number is an intege... |
| nn0abscl 15236 | The absolute value of an i... |
| zabscl 15237 | The absolute value of an i... |
| zabs0b 15238 | An integer has an absolute... |
| abslt 15239 | Absolute value and 'less t... |
| absle 15240 | Absolute value and 'less t... |
| abssubne0 15241 | If the absolute value of a... |
| absdiflt 15242 | The absolute value of a di... |
| absdifle 15243 | The absolute value of a di... |
| elicc4abs 15244 | Membership in a symmetric ... |
| lenegsq 15245 | Comparison to a nonnegativ... |
| releabs 15246 | The real part of a number ... |
| recval 15247 | Reciprocal expressed with ... |
| absidm 15248 | The absolute value functio... |
| absgt0 15249 | The absolute value of a no... |
| nnabscl 15250 | The absolute value of a no... |
| abssub 15251 | Swapping order of subtract... |
| abssubge0 15252 | Absolute value of a nonneg... |
| abssuble0 15253 | Absolute value of a nonpos... |
| absmax 15254 | The maximum of two numbers... |
| abstri 15255 | Triangle inequality for ab... |
| abs3dif 15256 | Absolute value of differen... |
| abs2dif 15257 | Difference of absolute val... |
| abs2dif2 15258 | Difference of absolute val... |
| abs2difabs 15259 | Absolute value of differen... |
| abs1m 15260 | For any complex number, th... |
| recan 15261 | Cancellation law involving... |
| absf 15262 | Mapping domain and codomai... |
| abs3lem 15263 | Lemma involving absolute v... |
| abslem2 15264 | Lemma involving absolute v... |
| rddif 15265 | The difference between a r... |
| absrdbnd 15266 | Bound on the absolute valu... |
| fzomaxdiflem 15267 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15268 | A bound on the separation ... |
| uzin2 15269 | The upper integers are clo... |
| rexanuz 15270 | Combine two different uppe... |
| rexanre 15271 | Combine two different uppe... |
| rexfiuz 15272 | Combine finitely many diff... |
| rexuz3 15273 | Restrict the base of the u... |
| rexanuz2 15274 | Combine two different uppe... |
| r19.29uz 15275 | A version of ~ 19.29 for u... |
| r19.2uz 15276 | A version of ~ r19.2z for ... |
| rexuzre 15277 | Convert an upper real quan... |
| rexico 15278 | Restrict the base of an up... |
| cau3lem 15279 | Lemma for ~ cau3 . (Contr... |
| cau3 15280 | Convert between three-quan... |
| cau4 15281 | Change the base of a Cauch... |
| caubnd2 15282 | A Cauchy sequence of compl... |
| caubnd 15283 | A Cauchy sequence of compl... |
| sqreulem 15284 | Lemma for ~ sqreu : write ... |
| sqreu 15285 | Existence and uniqueness f... |
| sqrtcl 15286 | Closure of the square root... |
| sqrtthlem 15287 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15288 | Mapping domain and codomai... |
| sqrtth 15289 | Square root theorem over t... |
| sqrtrege0 15290 | The square root function m... |
| eqsqrtor 15291 | Solve an equation containi... |
| eqsqrtd 15292 | A deduction for showing th... |
| eqsqrt2d 15293 | A deduction for showing th... |
| amgm2 15294 | Arithmetic-geometric mean ... |
| sqrtthi 15295 | Square root theorem. Theo... |
| sqrtcli 15296 | The square root of a nonne... |
| sqrtgt0i 15297 | The square root of a posit... |
| sqrtmsqi 15298 | Square root of square. (C... |
| sqrtsqi 15299 | Square root of square. (C... |
| sqsqrti 15300 | Square of square root. (C... |
| sqrtge0i 15301 | The square root of a nonne... |
| absidi 15302 | A nonnegative number is it... |
| absnidi 15303 | A negative number is the n... |
| leabsi 15304 | A real number is less than... |
| absori 15305 | The absolute value of a re... |
| absrei 15306 | Absolute value of a real n... |
| sqrtpclii 15307 | The square root of a posit... |
| sqrtgt0ii 15308 | The square root of a posit... |
| sqrt11i 15309 | The square root function i... |
| sqrtmuli 15310 | Square root distributes ov... |
| sqrtmulii 15311 | Square root distributes ov... |
| sqrtmsq2i 15312 | Relationship between squar... |
| sqrtlei 15313 | Square root is monotonic. ... |
| sqrtlti 15314 | Square root is strictly mo... |
| abslti 15315 | Absolute value and 'less t... |
| abslei 15316 | Absolute value and 'less t... |
| cnsqrt00 15317 | A square root of a complex... |
| absvalsqi 15318 | Square of value of absolut... |
| absvalsq2i 15319 | Square of value of absolut... |
| abscli 15320 | Real closure of absolute v... |
| absge0i 15321 | Absolute value is nonnegat... |
| absval2i 15322 | Value of absolute value fu... |
| abs00i 15323 | The absolute value of a nu... |
| absgt0i 15324 | The absolute value of a no... |
| absnegi 15325 | Absolute value of negative... |
| abscji 15326 | The absolute value of a nu... |
| releabsi 15327 | The real part of a number ... |
| abssubi 15328 | Swapping order of subtract... |
| absmuli 15329 | Absolute value distributes... |
| sqabsaddi 15330 | Square of absolute value o... |
| sqabssubi 15331 | Square of absolute value o... |
| absdivzi 15332 | Absolute value distributes... |
| abstrii 15333 | Triangle inequality for ab... |
| abs3difi 15334 | Absolute value of differen... |
| abs3lemi 15335 | Lemma involving absolute v... |
| rpsqrtcld 15336 | The square root of a posit... |
| sqrtgt0d 15337 | The square root of a posit... |
| absnidd 15338 | A negative number is the n... |
| leabsd 15339 | A real number is less than... |
| absord 15340 | The absolute value of a re... |
| absred 15341 | Absolute value of a real n... |
| resqrtcld 15342 | The square root of a nonne... |
| sqrtmsqd 15343 | Square root of square. (C... |
| sqrtsqd 15344 | Square root of square. (C... |
| sqrtge0d 15345 | The square root of a nonne... |
| sqrtnegd 15346 | The square root of a negat... |
| absidd 15347 | A nonnegative number is it... |
| sqrtdivd 15348 | Square root distributes ov... |
| sqrtmuld 15349 | Square root distributes ov... |
| sqrtsq2d 15350 | Relationship between squar... |
| sqrtled 15351 | Square root is monotonic. ... |
| sqrtltd 15352 | Square root is strictly mo... |
| sqr11d 15353 | The square root function i... |
| nn0absid 15354 | A nonnegative integer is i... |
| nn0absidi 15355 | A nonnegative integer is i... |
| absltd 15356 | Absolute value and 'less t... |
| absled 15357 | Absolute value and 'less t... |
| abssubge0d 15358 | Absolute value of a nonneg... |
| abssuble0d 15359 | Absolute value of a nonpos... |
| absdifltd 15360 | The absolute value of a di... |
| absdifled 15361 | The absolute value of a di... |
| icodiamlt 15362 | Two elements in a half-ope... |
| abscld 15363 | Real closure of absolute v... |
| sqrtcld 15364 | Closure of the square root... |
| sqrtrege0d 15365 | The real part of the squar... |
| sqsqrtd 15366 | Square root theorem. Theo... |
| msqsqrtd 15367 | Square root theorem. Theo... |
| sqr00d 15368 | A square root is zero iff ... |
| absvalsqd 15369 | Square of value of absolut... |
| absvalsq2d 15370 | Square of value of absolut... |
| absge0d 15371 | Absolute value is nonnegat... |
| absval2d 15372 | Value of absolute value fu... |
| abs00d 15373 | The absolute value of a nu... |
| absne0d 15374 | The absolute value of a nu... |
| absrpcld 15375 | The absolute value of a no... |
| absnegd 15376 | Absolute value of negative... |
| abscjd 15377 | The absolute value of a nu... |
| releabsd 15378 | The real part of a number ... |
| absexpd 15379 | Absolute value of positive... |
| abssubd 15380 | Swapping order of subtract... |
| absmuld 15381 | Absolute value distributes... |
| absdivd 15382 | Absolute value distributes... |
| abstrid 15383 | Triangle inequality for ab... |
| abs2difd 15384 | Difference of absolute val... |
| abs2dif2d 15385 | Difference of absolute val... |
| abs2difabsd 15386 | Absolute value of differen... |
| abs3difd 15387 | Absolute value of differen... |
| abs3lemd 15388 | Lemma involving absolute v... |
| reusq0 15389 | A complex number is the sq... |
| bhmafibid1cn 15390 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15391 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15392 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15393 | The Brahmagupta-Fibonacci ... |
| limsupgord 15396 | Ordering property of the s... |
| limsupcl 15397 | Closure of the superior li... |
| limsupval 15398 | The superior limit of an i... |
| limsupgf 15399 | Closure of the superior li... |
| limsupgval 15400 | Value of the superior limi... |
| limsupgle 15401 | The defining property of t... |
| limsuple 15402 | The defining property of t... |
| limsuplt 15403 | The defining property of t... |
| limsupval2 15404 | The superior limit, relati... |
| limsupgre 15405 | If a sequence of real numb... |
| limsupbnd1 15406 | If a sequence is eventuall... |
| limsupbnd2 15407 | If a sequence is eventuall... |
| climrel 15416 | The limit relation is a re... |
| rlimrel 15417 | The limit relation is a re... |
| clim 15418 | Express the predicate: Th... |
| rlim 15419 | Express the predicate: Th... |
| rlim2 15420 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15421 | Use strictly less-than in ... |
| rlim3 15422 | Restrict the range of the ... |
| climcl 15423 | Closure of the limit of a ... |
| rlimpm 15424 | Closure of a function with... |
| rlimf 15425 | Closure of a function with... |
| rlimss 15426 | Domain closure of a functi... |
| rlimcl 15427 | Closure of the limit of a ... |
| clim2 15428 | Express the predicate: Th... |
| clim2c 15429 | Express the predicate ` F ... |
| clim0 15430 | Express the predicate ` F ... |
| clim0c 15431 | Express the predicate ` F ... |
| rlim0 15432 | Express the predicate ` B ... |
| rlim0lt 15433 | Use strictly less-than in ... |
| climi 15434 | Convergence of a sequence ... |
| climi2 15435 | Convergence of a sequence ... |
| climi0 15436 | Convergence of a sequence ... |
| rlimi 15437 | Convergence at infinity of... |
| rlimi2 15438 | Convergence at infinity of... |
| ello1 15439 | Elementhood in the set of ... |
| ello12 15440 | Elementhood in the set of ... |
| ello12r 15441 | Sufficient condition for e... |
| lo1f 15442 | An eventually upper bounde... |
| lo1dm 15443 | An eventually upper bounde... |
| lo1bdd 15444 | The defining property of a... |
| ello1mpt 15445 | Elementhood in the set of ... |
| ello1mpt2 15446 | Elementhood in the set of ... |
| ello1d 15447 | Sufficient condition for e... |
| lo1bdd2 15448 | If an eventually bounded f... |
| lo1bddrp 15449 | Refine ~ o1bdd2 to give a ... |
| elo1 15450 | Elementhood in the set of ... |
| elo12 15451 | Elementhood in the set of ... |
| elo12r 15452 | Sufficient condition for e... |
| o1f 15453 | An eventually bounded func... |
| o1dm 15454 | An eventually bounded func... |
| o1bdd 15455 | The defining property of a... |
| lo1o1 15456 | A function is eventually b... |
| lo1o12 15457 | A function is eventually b... |
| elo1mpt 15458 | Elementhood in the set of ... |
| elo1mpt2 15459 | Elementhood in the set of ... |
| elo1d 15460 | Sufficient condition for e... |
| o1lo1 15461 | A real function is eventua... |
| o1lo12 15462 | A lower bounded real funct... |
| o1lo1d 15463 | A real eventually bounded ... |
| icco1 15464 | Derive eventual boundednes... |
| o1bdd2 15465 | If an eventually bounded f... |
| o1bddrp 15466 | Refine ~ o1bdd2 to give a ... |
| climconst 15467 | An (eventually) constant s... |
| rlimconst 15468 | A constant sequence conver... |
| rlimclim1 15469 | Forward direction of ~ rli... |
| rlimclim 15470 | A sequence on an upper int... |
| climrlim2 15471 | Produce a real limit from ... |
| climconst2 15472 | A constant sequence conver... |
| climz 15473 | The zero sequence converge... |
| rlimuni 15474 | A real function whose doma... |
| rlimdm 15475 | Two ways to express that a... |
| climuni 15476 | An infinite sequence of co... |
| fclim 15477 | The limit relation is func... |
| climdm 15478 | Two ways to express that a... |
| climeu 15479 | An infinite sequence of co... |
| climreu 15480 | An infinite sequence of co... |
| climmo 15481 | An infinite sequence of co... |
| rlimres 15482 | The restriction of a funct... |
| lo1res 15483 | The restriction of an even... |
| o1res 15484 | The restriction of an even... |
| rlimres2 15485 | The restriction of a funct... |
| lo1res2 15486 | The restriction of a funct... |
| o1res2 15487 | The restriction of a funct... |
| lo1resb 15488 | The restriction of a funct... |
| rlimresb 15489 | The restriction of a funct... |
| o1resb 15490 | The restriction of a funct... |
| climeq 15491 | Two functions that are eve... |
| lo1eq 15492 | Two functions that are eve... |
| rlimeq 15493 | Two functions that are eve... |
| o1eq 15494 | Two functions that are eve... |
| climmpt 15495 | Exhibit a function ` G ` w... |
| 2clim 15496 | If two sequences converge ... |
| climmpt2 15497 | Relate an integer limit on... |
| climshftlem 15498 | A shifted function converg... |
| climres 15499 | A function restricted to u... |
| climshft 15500 | A shifted function converg... |
| serclim0 15501 | The zero series converges ... |
| rlimcld2 15502 | If ` D ` is a closed set i... |
| rlimrege0 15503 | The limit of a sequence of... |
| rlimrecl 15504 | The limit of a real sequen... |
| rlimge0 15505 | The limit of a sequence of... |
| climshft2 15506 | A shifted function converg... |
| climrecl 15507 | The limit of a convergent ... |
| climge0 15508 | A nonnegative sequence con... |
| climabs0 15509 | Convergence to zero of the... |
| o1co 15510 | Sufficient condition for t... |
| o1compt 15511 | Sufficient condition for t... |
| rlimcn1 15512 | Image of a limit under a c... |
| rlimcn1b 15513 | Image of a limit under a c... |
| rlimcn3 15514 | Image of a limit under a c... |
| rlimcn2 15515 | Image of a limit under a c... |
| climcn1 15516 | Image of a limit under a c... |
| climcn2 15517 | Image of a limit under a c... |
| addcn2 15518 | Complex number addition is... |
| subcn2 15519 | Complex number subtraction... |
| mulcn2 15520 | Complex number multiplicat... |
| reccn2 15521 | The reciprocal function is... |
| cn1lem 15522 | A sufficient condition for... |
| abscn2 15523 | The absolute value functio... |
| cjcn2 15524 | The complex conjugate func... |
| recn2 15525 | The real part function is ... |
| imcn2 15526 | The imaginary part functio... |
| climcn1lem 15527 | The limit of a continuous ... |
| climabs 15528 | Limit of the absolute valu... |
| climcj 15529 | Limit of the complex conju... |
| climre 15530 | Limit of the real part of ... |
| climim 15531 | Limit of the imaginary par... |
| rlimmptrcl 15532 | Reverse closure for a real... |
| rlimabs 15533 | Limit of the absolute valu... |
| rlimcj 15534 | Limit of the complex conju... |
| rlimre 15535 | Limit of the real part of ... |
| rlimim 15536 | Limit of the imaginary par... |
| o1of2 15537 | Show that a binary operati... |
| o1add 15538 | The sum of two eventually ... |
| o1mul 15539 | The product of two eventua... |
| o1sub 15540 | The difference of two even... |
| rlimo1 15541 | Any function with a finite... |
| rlimdmo1 15542 | A convergent function is e... |
| o1rlimmul 15543 | The product of an eventual... |
| o1const 15544 | A constant function is eve... |
| lo1const 15545 | A constant function is eve... |
| lo1mptrcl 15546 | Reverse closure for an eve... |
| o1mptrcl 15547 | Reverse closure for an eve... |
| o1add2 15548 | The sum of two eventually ... |
| o1mul2 15549 | The product of two eventua... |
| o1sub2 15550 | The product of two eventua... |
| lo1add 15551 | The sum of two eventually ... |
| lo1mul 15552 | The product of an eventual... |
| lo1mul2 15553 | The product of an eventual... |
| o1dif 15554 | If the difference of two f... |
| lo1sub 15555 | The difference of an event... |
| climadd 15556 | Limit of the sum of two co... |
| climmul 15557 | Limit of the product of tw... |
| climsub 15558 | Limit of the difference of... |
| climaddc1 15559 | Limit of a constant ` C ` ... |
| climaddc2 15560 | Limit of a constant ` C ` ... |
| climmulc2 15561 | Limit of a sequence multip... |
| climsubc1 15562 | Limit of a constant ` C ` ... |
| climsubc2 15563 | Limit of a constant ` C ` ... |
| climle 15564 | Comparison of the limits o... |
| climsqz 15565 | Convergence of a sequence ... |
| climsqz2 15566 | Convergence of a sequence ... |
| rlimadd 15567 | Limit of the sum of two co... |
| rlimsub 15568 | Limit of the difference of... |
| rlimmul 15569 | Limit of the product of tw... |
| rlimdiv 15570 | Limit of the quotient of t... |
| rlimneg 15571 | Limit of the negative of a... |
| rlimle 15572 | Comparison of the limits o... |
| rlimsqzlem 15573 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15574 | Convergence of a sequence ... |
| rlimsqz2 15575 | Convergence of a sequence ... |
| lo1le 15576 | Transfer eventual upper bo... |
| o1le 15577 | Transfer eventual boundedn... |
| rlimno1 15578 | A function whose inverse c... |
| clim2ser 15579 | The limit of an infinite s... |
| clim2ser2 15580 | The limit of an infinite s... |
| iserex 15581 | An infinite series converg... |
| isermulc2 15582 | Multiplication of an infin... |
| climlec2 15583 | Comparison of a constant t... |
| iserle 15584 | Comparison of the limits o... |
| iserge0 15585 | The limit of an infinite s... |
| climub 15586 | The limit of a monotonic s... |
| climserle 15587 | The partial sums of a conv... |
| isershft 15588 | Index shift of the limit o... |
| isercolllem1 15589 | Lemma for ~ isercoll . (C... |
| isercolllem2 15590 | Lemma for ~ isercoll . (C... |
| isercolllem3 15591 | Lemma for ~ isercoll . (C... |
| isercoll 15592 | Rearrange an infinite seri... |
| isercoll2 15593 | Generalize ~ isercoll so t... |
| climsup 15594 | A bounded monotonic sequen... |
| climcau 15595 | A converging sequence of c... |
| climbdd 15596 | A converging sequence of c... |
| caucvgrlem 15597 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15598 | A Cauchy sequence of real ... |
| caucvgrlem2 15599 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15600 | A Cauchy sequence of compl... |
| caurcvg 15601 | A Cauchy sequence of real ... |
| caurcvg2 15602 | A Cauchy sequence of real ... |
| caucvg 15603 | A Cauchy sequence of compl... |
| caucvgb 15604 | A function is convergent i... |
| serf0 15605 | If an infinite series conv... |
| iseraltlem1 15606 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15607 | Lemma for ~ iseralt . The... |
| iseraltlem3 15608 | Lemma for ~ iseralt . Fro... |
| iseralt 15609 | The alternating series tes... |
| sumex 15612 | A sum is a set. (Contribu... |
| sumeq1 15613 | Equality theorem for a sum... |
| nfsum1 15614 | Bound-variable hypothesis ... |
| nfsum 15615 | Bound-variable hypothesis ... |
| sumeq2w 15616 | Equality theorem for sum, ... |
| sumeq2ii 15617 | Equality theorem for sum, ... |
| sumeq2 15618 | Equality theorem for sum. ... |
| cbvsum 15619 | Change bound variable in a... |
| cbvsumv 15620 | Change bound variable in a... |
| sumeq1i 15621 | Equality inference for sum... |
| sumeq2i 15622 | Equality inference for sum... |
| sumeq12i 15623 | Equality inference for sum... |
| sumeq1d 15624 | Equality deduction for sum... |
| sumeq2d 15625 | Equality deduction for sum... |
| sumeq2dv 15626 | Equality deduction for sum... |
| sumeq2sdv 15627 | Equality deduction for sum... |
| sumeq2sdvOLD 15628 | Obsolete version of ~ sume... |
| 2sumeq2dv 15629 | Equality deduction for dou... |
| sumeq12dv 15630 | Equality deduction for sum... |
| sumeq12rdv 15631 | Equality deduction for sum... |
| sum2id 15632 | The second class argument ... |
| sumfc 15633 | A lemma to facilitate conv... |
| fz1f1o 15634 | A lemma for working with f... |
| sumrblem 15635 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15636 | The sequence of partial su... |
| sumrb 15637 | Rebase the starting point ... |
| summolem3 15638 | Lemma for ~ summo . (Cont... |
| summolem2a 15639 | Lemma for ~ summo . (Cont... |
| summolem2 15640 | Lemma for ~ summo . (Cont... |
| summo 15641 | A sum has at most one limi... |
| zsum 15642 | Series sum with index set ... |
| isum 15643 | Series sum with an upper i... |
| fsum 15644 | The value of a sum over a ... |
| sum0 15645 | Any sum over the empty set... |
| sumz 15646 | Any sum of zero over a sum... |
| fsumf1o 15647 | Re-index a finite sum usin... |
| sumss 15648 | Change the index set to a ... |
| fsumss 15649 | Change the index set to a ... |
| sumss2 15650 | Change the index set of a ... |
| fsumcvg2 15651 | The sequence of partial su... |
| fsumsers 15652 | Special case of series sum... |
| fsumcvg3 15653 | A finite sum is convergent... |
| fsumser 15654 | A finite sum expressed in ... |
| fsumcl2lem 15655 | - Lemma for finite sum clo... |
| fsumcllem 15656 | - Lemma for finite sum clo... |
| fsumcl 15657 | Closure of a finite sum of... |
| fsumrecl 15658 | Closure of a finite sum of... |
| fsumzcl 15659 | Closure of a finite sum of... |
| fsumnn0cl 15660 | Closure of a finite sum of... |
| fsumrpcl 15661 | Closure of a finite sum of... |
| fsumclf 15662 | Closure of a finite sum of... |
| fsumzcl2 15663 | A finite sum with integer ... |
| fsumadd 15664 | The sum of two finite sums... |
| fsumsplit 15665 | Split a sum into two parts... |
| fsumsplitf 15666 | Split a sum into two parts... |
| sumsnf 15667 | A sum of a singleton is th... |
| fsumsplitsn 15668 | Separate out a term in a f... |
| fsumsplit1 15669 | Separate out a term in a f... |
| sumsn 15670 | A sum of a singleton is th... |
| fsum1 15671 | The finite sum of ` A ( k ... |
| sumpr 15672 | A sum over a pair is the s... |
| sumtp 15673 | A sum over a triple is the... |
| sumsns 15674 | A sum of a singleton is th... |
| fsumm1 15675 | Separate out the last term... |
| fzosump1 15676 | Separate out the last term... |
| fsum1p 15677 | Separate out the first ter... |
| fsummsnunz 15678 | A finite sum all of whose ... |
| fsumsplitsnun 15679 | Separate out a term in a f... |
| fsump1 15680 | The addition of the next t... |
| isumclim 15681 | An infinite sum equals the... |
| isumclim2 15682 | A converging series conver... |
| isumclim3 15683 | The sequence of partial fi... |
| sumnul 15684 | The sum of a non-convergen... |
| isumcl 15685 | The sum of a converging in... |
| isummulc2 15686 | An infinite sum multiplied... |
| isummulc1 15687 | An infinite sum multiplied... |
| isumdivc 15688 | An infinite sum divided by... |
| isumrecl 15689 | The sum of a converging in... |
| isumge0 15690 | An infinite sum of nonnega... |
| isumadd 15691 | Addition of infinite sums.... |
| sumsplit 15692 | Split a sum into two parts... |
| fsump1i 15693 | Optimized version of ~ fsu... |
| fsum2dlem 15694 | Lemma for ~ fsum2d - induc... |
| fsum2d 15695 | Write a double sum as a su... |
| fsumxp 15696 | Combine two sums into a si... |
| fsumcnv 15697 | Transform a region of summ... |
| fsumcom2 15698 | Interchange order of summa... |
| fsumcom 15699 | Interchange order of summa... |
| fsum0diaglem 15700 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15701 | Two ways to express "the s... |
| mptfzshft 15702 | 1-1 onto function in maps-... |
| fsumrev 15703 | Reversal of a finite sum. ... |
| fsumshft 15704 | Index shift of a finite su... |
| fsumshftm 15705 | Negative index shift of a ... |
| fsumrev2 15706 | Reversal of a finite sum. ... |
| fsum0diag2 15707 | Two ways to express "the s... |
| fsummulc2 15708 | A finite sum multiplied by... |
| fsummulc1 15709 | A finite sum multiplied by... |
| fsumdivc 15710 | A finite sum divided by a ... |
| fsumneg 15711 | Negation of a finite sum. ... |
| fsumsub 15712 | Split a finite sum over a ... |
| fsum2mul 15713 | Separate the nested sum of... |
| fsumconst 15714 | The sum of constant terms ... |
| fsumdifsnconst 15715 | The sum of constant terms ... |
| modfsummodslem1 15716 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15717 | Induction step for ~ modfs... |
| modfsummod 15718 | A finite sum modulo a posi... |
| fsumge0 15719 | If all of the terms of a f... |
| fsumless 15720 | A shorter sum of nonnegati... |
| fsumge1 15721 | A sum of nonnegative numbe... |
| fsum00 15722 | A sum of nonnegative numbe... |
| fsumle 15723 | If all of the terms of fin... |
| fsumlt 15724 | If every term in one finit... |
| fsumabs 15725 | Generalized triangle inequ... |
| telfsumo 15726 | Sum of a telescoping serie... |
| telfsumo2 15727 | Sum of a telescoping serie... |
| telfsum 15728 | Sum of a telescoping serie... |
| telfsum2 15729 | Sum of a telescoping serie... |
| fsumparts 15730 | Summation by parts. (Cont... |
| fsumrelem 15731 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15732 | The real part of a sum. (... |
| fsumim 15733 | The imaginary part of a su... |
| fsumcj 15734 | The complex conjugate of a... |
| fsumrlim 15735 | Limit of a finite sum of c... |
| fsumo1 15736 | The finite sum of eventual... |
| o1fsum 15737 | If ` A ( k ) ` is O(1), th... |
| seqabs 15738 | Generalized triangle inequ... |
| iserabs 15739 | Generalized triangle inequ... |
| cvgcmp 15740 | A comparison test for conv... |
| cvgcmpub 15741 | An upper bound for the lim... |
| cvgcmpce 15742 | A comparison test for conv... |
| abscvgcvg 15743 | An absolutely convergent s... |
| climfsum 15744 | Limit of a finite sum of c... |
| fsumiun 15745 | Sum over a disjoint indexe... |
| hashiun 15746 | The cardinality of a disjo... |
| hash2iun 15747 | The cardinality of a neste... |
| hash2iun1dif1 15748 | The cardinality of a neste... |
| hashrabrex 15749 | The number of elements in ... |
| hashuni 15750 | The cardinality of a disjo... |
| qshash 15751 | The cardinality of a set w... |
| ackbijnn 15752 | Translate the Ackermann bi... |
| binomlem 15753 | Lemma for ~ binom (binomia... |
| binom 15754 | The binomial theorem: ` ( ... |
| binom1p 15755 | Special case of the binomi... |
| binom11 15756 | Special case of the binomi... |
| binom1dif 15757 | A summation for the differ... |
| bcxmaslem1 15758 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15759 | Parallel summation (Christ... |
| incexclem 15760 | Lemma for ~ incexc . (Con... |
| incexc 15761 | The inclusion/exclusion pr... |
| incexc2 15762 | The inclusion/exclusion pr... |
| isumshft 15763 | Index shift of an infinite... |
| isumsplit 15764 | Split off the first ` N ` ... |
| isum1p 15765 | The infinite sum of a conv... |
| isumnn0nn 15766 | Sum from 0 to infinity in ... |
| isumrpcl 15767 | The infinite sum of positi... |
| isumle 15768 | Comparison of two infinite... |
| isumless 15769 | A finite sum of nonnegativ... |
| isumsup2 15770 | An infinite sum of nonnega... |
| isumsup 15771 | An infinite sum of nonnega... |
| isumltss 15772 | A partial sum of a series ... |
| climcndslem1 15773 | Lemma for ~ climcnds : bou... |
| climcndslem2 15774 | Lemma for ~ climcnds : bou... |
| climcnds 15775 | The Cauchy condensation te... |
| divrcnv 15776 | The sequence of reciprocal... |
| divcnv 15777 | The sequence of reciprocal... |
| flo1 15778 | The floor function satisfi... |
| divcnvshft 15779 | Limit of a ratio function.... |
| supcvg 15780 | Extract a sequence ` f ` i... |
| infcvgaux1i 15781 | Auxiliary theorem for appl... |
| infcvgaux2i 15782 | Auxiliary theorem for appl... |
| harmonic 15783 | The harmonic series ` H ` ... |
| arisum 15784 | Arithmetic series sum of t... |
| arisum2 15785 | Arithmetic series sum of t... |
| trireciplem 15786 | Lemma for ~ trirecip . Sh... |
| trirecip 15787 | The sum of the reciprocals... |
| expcnv 15788 | A sequence of powers of a ... |
| explecnv 15789 | A sequence of terms conver... |
| geoserg 15790 | The value of the finite ge... |
| geoser 15791 | The value of the finite ge... |
| pwdif 15792 | The difference of two numb... |
| pwm1geoser 15793 | The n-th power of a number... |
| geolim 15794 | The partial sums in the in... |
| geolim2 15795 | The partial sums in the ge... |
| georeclim 15796 | The limit of a geometric s... |
| geo2sum 15797 | The value of the finite ge... |
| geo2sum2 15798 | The value of the finite ge... |
| geo2lim 15799 | The value of the infinite ... |
| geomulcvg 15800 | The geometric series conve... |
| geoisum 15801 | The infinite sum of ` 1 + ... |
| geoisumr 15802 | The infinite sum of recipr... |
| geoisum1 15803 | The infinite sum of ` A ^ ... |
| geoisum1c 15804 | The infinite sum of ` A x.... |
| 0.999... 15805 | The recurring decimal 0.99... |
| geoihalfsum 15806 | Prove that the infinite ge... |
| cvgrat 15807 | Ratio test for convergence... |
| mertenslem1 15808 | Lemma for ~ mertens . (Co... |
| mertenslem2 15809 | Lemma for ~ mertens . (Co... |
| mertens 15810 | Mertens' theorem. If ` A ... |
| prodf 15811 | An infinite product of com... |
| clim2prod 15812 | The limit of an infinite p... |
| clim2div 15813 | The limit of an infinite p... |
| prodfmul 15814 | The product of two infinit... |
| prodf1 15815 | The value of the partial p... |
| prodf1f 15816 | A one-valued infinite prod... |
| prodfclim1 15817 | The constant one product c... |
| prodfn0 15818 | No term of a nonzero infin... |
| prodfrec 15819 | The reciprocal of an infin... |
| prodfdiv 15820 | The quotient of two infini... |
| ntrivcvg 15821 | A non-trivially converging... |
| ntrivcvgn0 15822 | A product that converges t... |
| ntrivcvgfvn0 15823 | Any value of a product seq... |
| ntrivcvgtail 15824 | A tail of a non-trivially ... |
| ntrivcvgmullem 15825 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15826 | The product of two non-tri... |
| prodex 15829 | A product is a set. (Cont... |
| prodeq1f 15830 | Equality theorem for a pro... |
| prodeq1 15831 | Equality theorem for a pro... |
| nfcprod1 15832 | Bound-variable hypothesis ... |
| nfcprod 15833 | Bound-variable hypothesis ... |
| prodeq2w 15834 | Equality theorem for produ... |
| prodeq2ii 15835 | Equality theorem for produ... |
| prodeq2 15836 | Equality theorem for produ... |
| cbvprod 15837 | Change bound variable in a... |
| cbvprodv 15838 | Change bound variable in a... |
| cbvprodi 15839 | Change bound variable in a... |
| prodeq1i 15840 | Equality inference for pro... |
| prodeq1iOLD 15841 | Obsolete version of ~ prod... |
| prodeq2i 15842 | Equality inference for pro... |
| prodeq12i 15843 | Equality inference for pro... |
| prodeq1d 15844 | Equality deduction for pro... |
| prodeq2d 15845 | Equality deduction for pro... |
| prodeq2dv 15846 | Equality deduction for pro... |
| prodeq2sdv 15847 | Equality deduction for pro... |
| prodeq2sdvOLD 15848 | Obsolete version of ~ prod... |
| 2cprodeq2dv 15849 | Equality deduction for dou... |
| prodeq12dv 15850 | Equality deduction for pro... |
| prodeq12rdv 15851 | Equality deduction for pro... |
| prod2id 15852 | The second class argument ... |
| prodrblem 15853 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15854 | The sequence of partial pr... |
| prodrblem2 15855 | Lemma for ~ prodrb . (Con... |
| prodrb 15856 | Rebase the starting point ... |
| prodmolem3 15857 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15858 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15859 | Lemma for ~ prodmo . (Con... |
| prodmo 15860 | A product has at most one ... |
| zprod 15861 | Series product with index ... |
| iprod 15862 | Series product with an upp... |
| zprodn0 15863 | Nonzero series product wit... |
| iprodn0 15864 | Nonzero series product wit... |
| fprod 15865 | The value of a product ove... |
| fprodntriv 15866 | A non-triviality lemma for... |
| prod0 15867 | A product over the empty s... |
| prod1 15868 | Any product of one over a ... |
| prodfc 15869 | A lemma to facilitate conv... |
| fprodf1o 15870 | Re-index a finite product ... |
| prodss 15871 | Change the index set to a ... |
| fprodss 15872 | Change the index set to a ... |
| fprodser 15873 | A finite product expressed... |
| fprodcl2lem 15874 | Finite product closure lem... |
| fprodcllem 15875 | Finite product closure lem... |
| fprodcl 15876 | Closure of a finite produc... |
| fprodrecl 15877 | Closure of a finite produc... |
| fprodzcl 15878 | Closure of a finite produc... |
| fprodnncl 15879 | Closure of a finite produc... |
| fprodrpcl 15880 | Closure of a finite produc... |
| fprodnn0cl 15881 | Closure of a finite produc... |
| fprodcllemf 15882 | Finite product closure lem... |
| fprodreclf 15883 | Closure of a finite produc... |
| fprodmul 15884 | The product of two finite ... |
| fproddiv 15885 | The quotient of two finite... |
| prodsn 15886 | A product of a singleton i... |
| fprod1 15887 | A finite product of only o... |
| prodsnf 15888 | A product of a singleton i... |
| climprod1 15889 | The limit of a product ove... |
| fprodsplit 15890 | Split a finite product int... |
| fprodm1 15891 | Separate out the last term... |
| fprod1p 15892 | Separate out the first ter... |
| fprodp1 15893 | Multiply in the last term ... |
| fprodm1s 15894 | Separate out the last term... |
| fprodp1s 15895 | Multiply in the last term ... |
| prodsns 15896 | A product of the singleton... |
| fprodfac 15897 | Factorial using product no... |
| fprodabs 15898 | The absolute value of a fi... |
| fprodeq0 15899 | Any finite product contain... |
| fprodshft 15900 | Shift the index of a finit... |
| fprodrev 15901 | Reversal of a finite produ... |
| fprodconst 15902 | The product of constant te... |
| fprodn0 15903 | A finite product of nonzer... |
| fprod2dlem 15904 | Lemma for ~ fprod2d - indu... |
| fprod2d 15905 | Write a double product as ... |
| fprodxp 15906 | Combine two products into ... |
| fprodcnv 15907 | Transform a product region... |
| fprodcom2 15908 | Interchange order of multi... |
| fprodcom 15909 | Interchange product order.... |
| fprod0diag 15910 | Two ways to express "the p... |
| fproddivf 15911 | The quotient of two finite... |
| fprodsplitf 15912 | Split a finite product int... |
| fprodsplitsn 15913 | Separate out a term in a f... |
| fprodsplit1f 15914 | Separate out a term in a f... |
| fprodn0f 15915 | A finite product of nonzer... |
| fprodclf 15916 | Closure of a finite produc... |
| fprodge0 15917 | If all the terms of a fini... |
| fprodeq0g 15918 | Any finite product contain... |
| fprodge1 15919 | If all of the terms of a f... |
| fprodle 15920 | If all the terms of two fi... |
| fprodmodd 15921 | If all factors of two fini... |
| iprodclim 15922 | An infinite product equals... |
| iprodclim2 15923 | A converging product conve... |
| iprodclim3 15924 | The sequence of partial fi... |
| iprodcl 15925 | The product of a non-trivi... |
| iprodrecl 15926 | The product of a non-trivi... |
| iprodmul 15927 | Multiplication of infinite... |
| risefacval 15932 | The value of the rising fa... |
| fallfacval 15933 | The value of the falling f... |
| risefacval2 15934 | One-based value of rising ... |
| fallfacval2 15935 | One-based value of falling... |
| fallfacval3 15936 | A product representation o... |
| risefaccllem 15937 | Lemma for rising factorial... |
| fallfaccllem 15938 | Lemma for falling factoria... |
| risefaccl 15939 | Closure law for rising fac... |
| fallfaccl 15940 | Closure law for falling fa... |
| rerisefaccl 15941 | Closure law for rising fac... |
| refallfaccl 15942 | Closure law for falling fa... |
| nnrisefaccl 15943 | Closure law for rising fac... |
| zrisefaccl 15944 | Closure law for rising fac... |
| zfallfaccl 15945 | Closure law for falling fa... |
| nn0risefaccl 15946 | Closure law for rising fac... |
| rprisefaccl 15947 | Closure law for rising fac... |
| risefallfac 15948 | A relationship between ris... |
| fallrisefac 15949 | A relationship between fal... |
| risefall0lem 15950 | Lemma for ~ risefac0 and ~... |
| risefac0 15951 | The value of the rising fa... |
| fallfac0 15952 | The value of the falling f... |
| risefacp1 15953 | The value of the rising fa... |
| fallfacp1 15954 | The value of the falling f... |
| risefacp1d 15955 | The value of the rising fa... |
| fallfacp1d 15956 | The value of the falling f... |
| risefac1 15957 | The value of rising factor... |
| fallfac1 15958 | The value of falling facto... |
| risefacfac 15959 | Relate rising factorial to... |
| fallfacfwd 15960 | The forward difference of ... |
| 0fallfac 15961 | The value of the zero fall... |
| 0risefac 15962 | The value of the zero risi... |
| binomfallfaclem1 15963 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 15964 | Lemma for ~ binomfallfac .... |
| binomfallfac 15965 | A version of the binomial ... |
| binomrisefac 15966 | A version of the binomial ... |
| fallfacval4 15967 | Represent the falling fact... |
| bcfallfac 15968 | Binomial coefficient in te... |
| fallfacfac 15969 | Relate falling factorial t... |
| bpolylem 15972 | Lemma for ~ bpolyval . (C... |
| bpolyval 15973 | The value of the Bernoulli... |
| bpoly0 15974 | The value of the Bernoulli... |
| bpoly1 15975 | The value of the Bernoulli... |
| bpolycl 15976 | Closure law for Bernoulli ... |
| bpolysum 15977 | A sum for Bernoulli polyno... |
| bpolydiflem 15978 | Lemma for ~ bpolydif . (C... |
| bpolydif 15979 | Calculate the difference b... |
| fsumkthpow 15980 | A closed-form expression f... |
| bpoly2 15981 | The Bernoulli polynomials ... |
| bpoly3 15982 | The Bernoulli polynomials ... |
| bpoly4 15983 | The Bernoulli polynomials ... |
| fsumcube 15984 | Express the sum of cubes i... |
| eftcl 15997 | Closure of a term in the s... |
| reeftcl 15998 | The terms of the series ex... |
| eftabs 15999 | The absolute value of a te... |
| eftval 16000 | The value of a term in the... |
| efcllem 16001 | Lemma for ~ efcl . The se... |
| ef0lem 16002 | The series defining the ex... |
| efval 16003 | Value of the exponential f... |
| esum 16004 | Value of Euler's constant ... |
| eff 16005 | Domain and codomain of the... |
| efcl 16006 | Closure law for the expone... |
| efcld 16007 | Closure law for the expone... |
| efval2 16008 | Value of the exponential f... |
| efcvg 16009 | The series that defines th... |
| efcvgfsum 16010 | Exponential function conve... |
| reefcl 16011 | The exponential function i... |
| reefcld 16012 | The exponential function i... |
| ere 16013 | Euler's constant ` _e ` = ... |
| ege2le3 16014 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16015 | Value of the exponential f... |
| efcj 16016 | The exponential of a compl... |
| efaddlem 16017 | Lemma for ~ efadd (exponen... |
| efadd 16018 | Sum of exponents law for e... |
| fprodefsum 16019 | Move the exponential funct... |
| efcan 16020 | Cancellation law for expon... |
| efne0d 16021 | The exponential of a compl... |
| efne0 16022 | The exponential of a compl... |
| efne0OLD 16023 | Obsolete version of ~ efne... |
| efneg 16024 | The exponential of the opp... |
| eff2 16025 | The exponential function m... |
| efsub 16026 | Difference of exponents la... |
| efexp 16027 | The exponential of an inte... |
| efzval 16028 | Value of the exponential f... |
| efgt0 16029 | The exponential of a real ... |
| rpefcl 16030 | The exponential of a real ... |
| rpefcld 16031 | The exponential of a real ... |
| eftlcvg 16032 | The tail series of the exp... |
| eftlcl 16033 | Closure of the sum of an i... |
| reeftlcl 16034 | Closure of the sum of an i... |
| eftlub 16035 | An upper bound on the abso... |
| efsep 16036 | Separate out the next term... |
| effsumlt 16037 | The partial sums of the se... |
| eft0val 16038 | The value of the first ter... |
| ef4p 16039 | Separate out the first fou... |
| efgt1p2 16040 | The exponential of a posit... |
| efgt1p 16041 | The exponential of a posit... |
| efgt1 16042 | The exponential of a posit... |
| eflt 16043 | The exponential function o... |
| efle 16044 | The exponential function o... |
| reef11 16045 | The exponential function o... |
| reeff1 16046 | The exponential function m... |
| eflegeo 16047 | The exponential function o... |
| sinval 16048 | Value of the sine function... |
| cosval 16049 | Value of the cosine functi... |
| sinf 16050 | Domain and codomain of the... |
| cosf 16051 | Domain and codomain of the... |
| sincl 16052 | Closure of the sine functi... |
| coscl 16053 | Closure of the cosine func... |
| tanval 16054 | Value of the tangent funct... |
| tancl 16055 | The closure of the tangent... |
| sincld 16056 | Closure of the sine functi... |
| coscld 16057 | Closure of the cosine func... |
| tancld 16058 | Closure of the tangent fun... |
| tanval2 16059 | Express the tangent functi... |
| tanval3 16060 | Express the tangent functi... |
| resinval 16061 | The sine of a real number ... |
| recosval 16062 | The cosine of a real numbe... |
| efi4p 16063 | Separate out the first fou... |
| resin4p 16064 | Separate out the first fou... |
| recos4p 16065 | Separate out the first fou... |
| resincl 16066 | The sine of a real number ... |
| recoscl 16067 | The cosine of a real numbe... |
| retancl 16068 | The closure of the tangent... |
| resincld 16069 | Closure of the sine functi... |
| recoscld 16070 | Closure of the cosine func... |
| retancld 16071 | Closure of the tangent fun... |
| sinneg 16072 | The sine of a negative is ... |
| cosneg 16073 | The cosines of a number an... |
| tanneg 16074 | The tangent of a negative ... |
| sin0 16075 | Value of the sine function... |
| cos0 16076 | Value of the cosine functi... |
| tan0 16077 | The value of the tangent f... |
| efival 16078 | The exponential function i... |
| efmival 16079 | The exponential function i... |
| sinhval 16080 | Value of the hyperbolic si... |
| coshval 16081 | Value of the hyperbolic co... |
| resinhcl 16082 | The hyperbolic sine of a r... |
| rpcoshcl 16083 | The hyperbolic cosine of a... |
| recoshcl 16084 | The hyperbolic cosine of a... |
| retanhcl 16085 | The hyperbolic tangent of ... |
| tanhlt1 16086 | The hyperbolic tangent of ... |
| tanhbnd 16087 | The hyperbolic tangent of ... |
| efeul 16088 | Eulerian representation of... |
| efieq 16089 | The exponentials of two im... |
| sinadd 16090 | Addition formula for sine.... |
| cosadd 16091 | Addition formula for cosin... |
| tanaddlem 16092 | A useful intermediate step... |
| tanadd 16093 | Addition formula for tange... |
| sinsub 16094 | Sine of difference. (Cont... |
| cossub 16095 | Cosine of difference. (Co... |
| addsin 16096 | Sum of sines. (Contribute... |
| subsin 16097 | Difference of sines. (Con... |
| sinmul 16098 | Product of sines can be re... |
| cosmul 16099 | Product of cosines can be ... |
| addcos 16100 | Sum of cosines. (Contribu... |
| subcos 16101 | Difference of cosines. (C... |
| sincossq 16102 | Sine squared plus cosine s... |
| sin2t 16103 | Double-angle formula for s... |
| cos2t 16104 | Double-angle formula for c... |
| cos2tsin 16105 | Double-angle formula for c... |
| sinbnd 16106 | The sine of a real number ... |
| cosbnd 16107 | The cosine of a real numbe... |
| sinbnd2 16108 | The sine of a real number ... |
| cosbnd2 16109 | The cosine of a real numbe... |
| ef01bndlem 16110 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16111 | Bounds on the sine of a po... |
| cos01bnd 16112 | Bounds on the cosine of a ... |
| cos1bnd 16113 | Bounds on the cosine of 1.... |
| cos2bnd 16114 | Bounds on the cosine of 2.... |
| sinltx 16115 | The sine of a positive rea... |
| sin01gt0 16116 | The sine of a positive rea... |
| cos01gt0 16117 | The cosine of a positive r... |
| sin02gt0 16118 | The sine of a positive rea... |
| sincos1sgn 16119 | The signs of the sine and ... |
| sincos2sgn 16120 | The signs of the sine and ... |
| sin4lt0 16121 | The sine of 4 is negative.... |
| absefi 16122 | The absolute value of the ... |
| absef 16123 | The absolute value of the ... |
| absefib 16124 | A complex number is real i... |
| efieq1re 16125 | A number whose imaginary e... |
| demoivre 16126 | De Moivre's Formula. Proo... |
| demoivreALT 16127 | Alternate proof of ~ demoi... |
| eirrlem 16130 | Lemma for ~ eirr . (Contr... |
| eirr 16131 | ` _e ` is irrational. (Co... |
| egt2lt3 16132 | Euler's constant ` _e ` = ... |
| epos 16133 | Euler's constant ` _e ` is... |
| epr 16134 | Euler's constant ` _e ` is... |
| ene0 16135 | ` _e ` is not 0. (Contrib... |
| ene1 16136 | ` _e ` is not 1. (Contrib... |
| xpnnen 16137 | The Cartesian product of t... |
| znnen 16138 | The set of integers and th... |
| qnnen 16139 | The rational numbers are c... |
| rpnnen2lem1 16140 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16141 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16142 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16143 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16144 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16145 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16146 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16147 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16148 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16149 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16150 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16151 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16152 | The other half of ~ rpnnen... |
| rpnnen 16153 | The cardinality of the con... |
| rexpen 16154 | The real numbers are equin... |
| cpnnen 16155 | The complex numbers are eq... |
| rucALT 16156 | Alternate proof of ~ ruc .... |
| ruclem1 16157 | Lemma for ~ ruc (the reals... |
| ruclem2 16158 | Lemma for ~ ruc . Orderin... |
| ruclem3 16159 | Lemma for ~ ruc . The con... |
| ruclem4 16160 | Lemma for ~ ruc . Initial... |
| ruclem6 16161 | Lemma for ~ ruc . Domain ... |
| ruclem7 16162 | Lemma for ~ ruc . Success... |
| ruclem8 16163 | Lemma for ~ ruc . The int... |
| ruclem9 16164 | Lemma for ~ ruc . The fir... |
| ruclem10 16165 | Lemma for ~ ruc . Every f... |
| ruclem11 16166 | Lemma for ~ ruc . Closure... |
| ruclem12 16167 | Lemma for ~ ruc . The sup... |
| ruclem13 16168 | Lemma for ~ ruc . There i... |
| ruc 16169 | The set of positive intege... |
| resdomq 16170 | The set of rationals is st... |
| aleph1re 16171 | There are at least aleph-o... |
| aleph1irr 16172 | There are at least aleph-o... |
| cnso 16173 | The complex numbers can be... |
| sqrt2irrlem 16174 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16175 | The square root of 2 is ir... |
| sqrt2re 16176 | The square root of 2 exist... |
| sqrt2irr0 16177 | The square root of 2 is an... |
| nthruc 16178 | The sequence ` NN ` , ` ZZ... |
| nthruz 16179 | The sequence ` NN ` , ` NN... |
| divides 16182 | Define the divides relatio... |
| dvdsval2 16183 | One nonzero integer divide... |
| dvdsval3 16184 | One nonzero integer divide... |
| dvdszrcl 16185 | Reverse closure for the di... |
| dvdsmod0 16186 | If a positive integer divi... |
| p1modz1 16187 | If a number greater than 1... |
| dvdsmodexp 16188 | If a positive integer divi... |
| nndivdvds 16189 | Strong form of ~ dvdsval2 ... |
| nndivides 16190 | Definition of the divides ... |
| moddvds 16191 | Two ways to say ` A == B `... |
| modm1div 16192 | An integer greater than on... |
| addmulmodb 16193 | An integer plus a product ... |
| dvds0lem 16194 | A lemma to assist theorems... |
| dvds1lem 16195 | A lemma to assist theorems... |
| dvds2lem 16196 | A lemma to assist theorems... |
| iddvds 16197 | An integer divides itself.... |
| 1dvds 16198 | 1 divides any integer. Th... |
| dvds0 16199 | Any integer divides 0. Th... |
| negdvdsb 16200 | An integer divides another... |
| dvdsnegb 16201 | An integer divides another... |
| absdvdsb 16202 | An integer divides another... |
| dvdsabsb 16203 | An integer divides another... |
| 0dvds 16204 | Only 0 is divisible by 0. ... |
| dvdsmul1 16205 | An integer divides a multi... |
| dvdsmul2 16206 | An integer divides a multi... |
| iddvdsexp 16207 | An integer divides a posit... |
| muldvds1 16208 | If a product divides an in... |
| muldvds2 16209 | If a product divides an in... |
| dvdscmul 16210 | Multiplication by a consta... |
| dvdsmulc 16211 | Multiplication by a consta... |
| dvdscmulr 16212 | Cancellation law for the d... |
| dvdsmulcr 16213 | Cancellation law for the d... |
| summodnegmod 16214 | The sum of two integers mo... |
| difmod0 16215 | The difference of two inte... |
| modmulconst 16216 | Constant multiplication in... |
| dvds2ln 16217 | If an integer divides each... |
| dvds2add 16218 | If an integer divides each... |
| dvds2sub 16219 | If an integer divides each... |
| dvds2addd 16220 | Deduction form of ~ dvds2a... |
| dvds2subd 16221 | Deduction form of ~ dvds2s... |
| dvdstr 16222 | The divides relation is tr... |
| dvdstrd 16223 | The divides relation is tr... |
| dvdsmultr1 16224 | If an integer divides anot... |
| dvdsmultr1d 16225 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16226 | If an integer divides anot... |
| dvdsmultr2d 16227 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16228 | If an integer divides eith... |
| dvdssub2 16229 | If an integer divides a di... |
| dvdsadd 16230 | An integer divides another... |
| dvdsaddr 16231 | An integer divides another... |
| dvdssub 16232 | An integer divides another... |
| dvdssubr 16233 | An integer divides another... |
| dvdsadd2b 16234 | Adding a multiple of the b... |
| dvdsaddre2b 16235 | Adding a multiple of the b... |
| fsumdvds 16236 | If every term in a sum is ... |
| dvdslelem 16237 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16238 | The divisors of a positive... |
| dvdsleabs 16239 | The divisors of a nonzero ... |
| dvdsleabs2 16240 | Transfer divisibility to a... |
| dvdsabseq 16241 | If two integers divide eac... |
| dvdseq 16242 | If two nonnegative integer... |
| divconjdvds 16243 | If a nonzero integer ` M `... |
| dvdsdivcl 16244 | The complement of a diviso... |
| dvdsflip 16245 | An involution of the divis... |
| dvdsssfz1 16246 | The set of divisors of a n... |
| dvds1 16247 | The only nonnegative integ... |
| alzdvds 16248 | Only 0 is divisible by all... |
| dvdsext 16249 | Poset extensionality for d... |
| fzm1ndvds 16250 | No number between ` 1 ` an... |
| fzo0dvdseq 16251 | Zero is the only one of th... |
| fzocongeq 16252 | Two different elements of ... |
| addmodlteqALT 16253 | Two nonnegative integers l... |
| dvdsfac 16254 | A positive integer divides... |
| dvdsexp2im 16255 | If an integer divides anot... |
| dvdsexp 16256 | A power divides a power wi... |
| dvdsmod 16257 | Any number ` K ` whose mod... |
| mulmoddvds 16258 | If an integer is divisible... |
| 3dvds 16259 | A rule for divisibility by... |
| 3dvdsdec 16260 | A decimal number is divisi... |
| 3dvds2dec 16261 | A decimal number is divisi... |
| fprodfvdvdsd 16262 | A finite product of intege... |
| fproddvdsd 16263 | A finite product of intege... |
| evenelz 16264 | An even number is an integ... |
| zeo3 16265 | An integer is even or odd.... |
| zeo4 16266 | An integer is even or odd ... |
| zeneo 16267 | No even integer equals an ... |
| odd2np1lem 16268 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16269 | An integer is odd iff it i... |
| even2n 16270 | An integer is even iff it ... |
| oddm1even 16271 | An integer is odd iff its ... |
| oddp1even 16272 | An integer is odd iff its ... |
| oexpneg 16273 | The exponential of the neg... |
| mod2eq0even 16274 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16275 | An integer is 1 modulo 2 i... |
| oddnn02np1 16276 | A nonnegative integer is o... |
| oddge22np1 16277 | An integer greater than on... |
| evennn02n 16278 | A nonnegative integer is e... |
| evennn2n 16279 | A positive integer is even... |
| 2tp1odd 16280 | A number which is twice an... |
| mulsucdiv2z 16281 | An integer multiplied with... |
| sqoddm1div8z 16282 | A squared odd number minus... |
| 2teven 16283 | A number which is twice an... |
| zeo5 16284 | An integer is either even ... |
| evend2 16285 | An integer is even iff its... |
| oddp1d2 16286 | An integer is odd iff its ... |
| zob 16287 | Alternate characterization... |
| oddm1d2 16288 | An integer is odd iff its ... |
| ltoddhalfle 16289 | An integer is less than ha... |
| halfleoddlt 16290 | An integer is greater than... |
| opoe 16291 | The sum of two odds is eve... |
| omoe 16292 | The difference of two odds... |
| opeo 16293 | The sum of an odd and an e... |
| omeo 16294 | The difference of an odd a... |
| z0even 16295 | 2 divides 0. That means 0... |
| n2dvds1 16296 | 2 does not divide 1. That... |
| n2dvdsm1 16297 | 2 does not divide -1. Tha... |
| z2even 16298 | 2 divides 2. That means 2... |
| n2dvds3 16299 | 2 does not divide 3. That... |
| z4even 16300 | 2 divides 4. That means 4... |
| 4dvdseven 16301 | An integer which is divisi... |
| m1expe 16302 | Exponentiation of -1 by an... |
| m1expo 16303 | Exponentiation of -1 by an... |
| m1exp1 16304 | Exponentiation of negative... |
| nn0enne 16305 | A positive integer is an e... |
| nn0ehalf 16306 | The half of an even nonneg... |
| nnehalf 16307 | The half of an even positi... |
| nn0onn 16308 | An odd nonnegative integer... |
| nn0o1gt2 16309 | An odd nonnegative integer... |
| nno 16310 | An alternate characterizat... |
| nn0o 16311 | An alternate characterizat... |
| nn0ob 16312 | Alternate characterization... |
| nn0oddm1d2 16313 | A positive integer is odd ... |
| nnoddm1d2 16314 | A positive integer is odd ... |
| sumeven 16315 | If every term in a sum is ... |
| sumodd 16316 | If every term in a sum is ... |
| evensumodd 16317 | If every term in a sum wit... |
| oddsumodd 16318 | If every term in a sum wit... |
| pwp1fsum 16319 | The n-th power of a number... |
| oddpwp1fsum 16320 | An odd power of a number i... |
| divalglem0 16321 | Lemma for ~ divalg . (Con... |
| divalglem1 16322 | Lemma for ~ divalg . (Con... |
| divalglem2 16323 | Lemma for ~ divalg . (Con... |
| divalglem4 16324 | Lemma for ~ divalg . (Con... |
| divalglem5 16325 | Lemma for ~ divalg . (Con... |
| divalglem6 16326 | Lemma for ~ divalg . (Con... |
| divalglem7 16327 | Lemma for ~ divalg . (Con... |
| divalglem8 16328 | Lemma for ~ divalg . (Con... |
| divalglem9 16329 | Lemma for ~ divalg . (Con... |
| divalglem10 16330 | Lemma for ~ divalg . (Con... |
| divalg 16331 | The division algorithm (th... |
| divalgb 16332 | Express the division algor... |
| divalg2 16333 | The division algorithm (th... |
| divalgmod 16334 | The result of the ` mod ` ... |
| divalgmodcl 16335 | The result of the ` mod ` ... |
| modremain 16336 | The result of the modulo o... |
| ndvdssub 16337 | Corollary of the division ... |
| ndvdsadd 16338 | Corollary of the division ... |
| ndvdsp1 16339 | Special case of ~ ndvdsadd... |
| ndvdsi 16340 | A quick test for non-divis... |
| 5ndvds3 16341 | 5 does not divide 3. (Con... |
| 5ndvds6 16342 | 5 does not divide 6. (Con... |
| flodddiv4 16343 | The floor of an odd intege... |
| fldivndvdslt 16344 | The floor of an integer di... |
| flodddiv4lt 16345 | The floor of an odd number... |
| flodddiv4t2lthalf 16346 | The floor of an odd number... |
| bitsfval 16351 | Expand the definition of t... |
| bitsval 16352 | Expand the definition of t... |
| bitsval2 16353 | Expand the definition of t... |
| bitsss 16354 | The set of bits of an inte... |
| bitsf 16355 | The ` bits ` function is a... |
| bits0 16356 | Value of the zeroth bit. ... |
| bits0e 16357 | The zeroth bit of an even ... |
| bits0o 16358 | The zeroth bit of an odd n... |
| bitsp1 16359 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16360 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16361 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16362 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16363 | The bits of a number are a... |
| bitsmod 16364 | Truncating the bit sequenc... |
| bitsfi 16365 | Every number is associated... |
| bitscmp 16366 | The bit complement of ` N ... |
| 0bits 16367 | The bits of zero. (Contri... |
| m1bits 16368 | The bits of negative one. ... |
| bitsinv1lem 16369 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16370 | There is an explicit inver... |
| bitsinv2 16371 | There is an explicit inver... |
| bitsf1ocnv 16372 | The ` bits ` function rest... |
| bitsf1o 16373 | The ` bits ` function rest... |
| bitsf1 16374 | The ` bits ` function is a... |
| 2ebits 16375 | The bits of a power of two... |
| bitsinv 16376 | The inverse of the ` bits ... |
| bitsinvp1 16377 | Recursive definition of th... |
| sadadd2lem2 16378 | The core of the proof of ~... |
| sadfval 16380 | Define the addition of two... |
| sadcf 16381 | The carry sequence is a se... |
| sadc0 16382 | The initial element of the... |
| sadcp1 16383 | The carry sequence (which ... |
| sadval 16384 | The full adder sequence is... |
| sadcaddlem 16385 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16386 | Non-recursive definition o... |
| sadadd2lem 16387 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16388 | Sum of initial segments of... |
| sadadd3 16389 | Sum of initial segments of... |
| sadcl 16390 | The sum of two sequences i... |
| sadcom 16391 | The adder sequence functio... |
| saddisjlem 16392 | Lemma for ~ sadadd . (Con... |
| saddisj 16393 | The sum of disjoint sequen... |
| sadaddlem 16394 | Lemma for ~ sadadd . (Con... |
| sadadd 16395 | For sequences that corresp... |
| sadid1 16396 | The adder sequence functio... |
| sadid2 16397 | The adder sequence functio... |
| sadasslem 16398 | Lemma for ~ sadass . (Con... |
| sadass 16399 | Sequence addition is assoc... |
| sadeq 16400 | Any element of a sequence ... |
| bitsres 16401 | Restrict the bits of a num... |
| bitsuz 16402 | The bits of a number are a... |
| bitsshft 16403 | Shifting a bit sequence to... |
| smufval 16405 | The multiplication of two ... |
| smupf 16406 | The sequence of partial su... |
| smup0 16407 | The initial element of the... |
| smupp1 16408 | The initial element of the... |
| smuval 16409 | Define the addition of two... |
| smuval2 16410 | The partial sum sequence s... |
| smupvallem 16411 | If ` A ` only has elements... |
| smucl 16412 | The product of two sequenc... |
| smu01lem 16413 | Lemma for ~ smu01 and ~ sm... |
| smu01 16414 | Multiplication of a sequen... |
| smu02 16415 | Multiplication of a sequen... |
| smupval 16416 | Rewrite the elements of th... |
| smup1 16417 | Rewrite ~ smupp1 using onl... |
| smueqlem 16418 | Any element of a sequence ... |
| smueq 16419 | Any element of a sequence ... |
| smumullem 16420 | Lemma for ~ smumul . (Con... |
| smumul 16421 | For sequences that corresp... |
| gcdval 16424 | The value of the ` gcd ` o... |
| gcd0val 16425 | The value, by convention, ... |
| gcdn0val 16426 | The value of the ` gcd ` o... |
| gcdcllem1 16427 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16428 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16429 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16430 | Closure of the ` gcd ` ope... |
| gcddvds 16431 | The gcd of two integers di... |
| dvdslegcd 16432 | An integer which divides b... |
| nndvdslegcd 16433 | A positive integer which d... |
| gcdcl 16434 | Closure of the ` gcd ` ope... |
| gcdnncl 16435 | Closure of the ` gcd ` ope... |
| gcdcld 16436 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16437 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16438 | An integer is the product ... |
| divgcdz 16439 | An integer divided by the ... |
| gcdf 16440 | Domain and codomain of the... |
| gcdcom 16441 | The ` gcd ` operator is co... |
| gcdcomd 16442 | The ` gcd ` operator is co... |
| divgcdnn 16443 | A positive integer divided... |
| divgcdnnr 16444 | A positive integer divided... |
| gcdeq0 16445 | The gcd of two integers is... |
| gcdn0gt0 16446 | The gcd of two integers is... |
| gcd0id 16447 | The gcd of 0 and an intege... |
| gcdid0 16448 | The gcd of an integer and ... |
| nn0gcdid0 16449 | The gcd of a nonnegative i... |
| gcdneg 16450 | Negating one operand of th... |
| neggcd 16451 | Negating one operand of th... |
| gcdaddmlem 16452 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16453 | Adding a multiple of one o... |
| gcdadd 16454 | The GCD of two numbers is ... |
| gcdid 16455 | The gcd of a number and it... |
| gcd1 16456 | The gcd of a number with 1... |
| gcdabs1 16457 | ` gcd ` of the absolute va... |
| gcdabs2 16458 | ` gcd ` of the absolute va... |
| gcdabs 16459 | The gcd of two integers is... |
| modgcd 16460 | The gcd remains unchanged ... |
| 1gcd 16461 | The GCD of one and an inte... |
| gcdmultipled 16462 | The greatest common diviso... |
| gcdmultiplez 16463 | The GCD of a multiple of a... |
| gcdmultiple 16464 | The GCD of a multiple of a... |
| dvdsgcdidd 16465 | The greatest common diviso... |
| 6gcd4e2 16466 | The greatest common diviso... |
| bezoutlem1 16467 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16468 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16469 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16470 | Lemma for ~ bezout . (Con... |
| bezout 16471 | Bézout's identity: ... |
| dvdsgcd 16472 | An integer which divides e... |
| dvdsgcdb 16473 | Biconditional form of ~ dv... |
| dfgcd2 16474 | Alternate definition of th... |
| gcdass 16475 | Associative law for ` gcd ... |
| mulgcd 16476 | Distribute multiplication ... |
| absmulgcd 16477 | Distribute absolute value ... |
| mulgcdr 16478 | Reverse distribution law f... |
| gcddiv 16479 | Division law for GCD. (Con... |
| gcdzeq 16480 | A positive integer ` A ` i... |
| gcdeq 16481 | ` A ` is equal to its gcd ... |
| dvdssqim 16482 | Unidirectional form of ~ d... |
| dvdsexpim 16483 | If two numbers are divisib... |
| dvdsmulgcd 16484 | A divisibility equivalent ... |
| rpmulgcd 16485 | If ` K ` and ` M ` are rel... |
| rplpwr 16486 | If ` A ` and ` B ` are rel... |
| rprpwr 16487 | If ` A ` and ` B ` are rel... |
| rppwr 16488 | If ` A ` and ` B ` are rel... |
| nn0rppwr 16489 | If ` A ` and ` B ` are rel... |
| sqgcd 16490 | Square distributes over gc... |
| expgcd 16491 | Exponentiation distributes... |
| nn0expgcd 16492 | Exponentiation distributes... |
| zexpgcd 16493 | Exponentiation distributes... |
| dvdssqlem 16494 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16495 | Two numbers are divisible ... |
| bezoutr 16496 | Partial converse to ~ bezo... |
| bezoutr1 16497 | Converse of ~ bezout for w... |
| nn0seqcvgd 16498 | A strictly-decreasing nonn... |
| seq1st 16499 | A sequence whose iteration... |
| algr0 16500 | The value of the algorithm... |
| algrf 16501 | An algorithm is a step fun... |
| algrp1 16502 | The value of the algorithm... |
| alginv 16503 | If ` I ` is an invariant o... |
| algcvg 16504 | One way to prove that an a... |
| algcvgblem 16505 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16506 | Two ways of expressing tha... |
| algcvga 16507 | The countdown function ` C... |
| algfx 16508 | If ` F ` reaches a fixed p... |
| eucalgval2 16509 | The value of the step func... |
| eucalgval 16510 | Euclid's Algorithm ~ eucal... |
| eucalgf 16511 | Domain and codomain of the... |
| eucalginv 16512 | The invariant of the step ... |
| eucalglt 16513 | The second member of the s... |
| eucalgcvga 16514 | Once Euclid's Algorithm ha... |
| eucalg 16515 | Euclid's Algorithm compute... |
| lcmval 16520 | Value of the ` lcm ` opera... |
| lcmcom 16521 | The ` lcm ` operator is co... |
| lcm0val 16522 | The value, by convention, ... |
| lcmn0val 16523 | The value of the ` lcm ` o... |
| lcmcllem 16524 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16525 | Closure of the ` lcm ` ope... |
| dvdslcm 16526 | The lcm of two integers is... |
| lcmledvds 16527 | A positive integer which b... |
| lcmeq0 16528 | The lcm of two integers is... |
| lcmcl 16529 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16530 | The greatest common diviso... |
| lcmneg 16531 | Negating one operand of th... |
| neglcm 16532 | Negating one operand of th... |
| lcmabs 16533 | The lcm of two integers is... |
| lcmgcdlem 16534 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16535 | The product of two numbers... |
| lcmdvds 16536 | The lcm of two integers di... |
| lcmid 16537 | The lcm of an integer and ... |
| lcm1 16538 | The lcm of an integer and ... |
| lcmgcdnn 16539 | The product of two positiv... |
| lcmgcdeq 16540 | Two integers' absolute val... |
| lcmdvdsb 16541 | Biconditional form of ~ lc... |
| lcmass 16542 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16543 | The least common multiple ... |
| 6lcm4e12 16544 | The least common multiple ... |
| absproddvds 16545 | The absolute value of the ... |
| absprodnn 16546 | The absolute value of the ... |
| fissn0dvds 16547 | For each finite subset of ... |
| fissn0dvdsn0 16548 | For each finite subset of ... |
| lcmfval 16549 | Value of the ` _lcm ` func... |
| lcmf0val 16550 | The value, by convention, ... |
| lcmfn0val 16551 | The value of the ` _lcm ` ... |
| lcmfnnval 16552 | The value of the ` _lcm ` ... |
| lcmfcllem 16553 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16554 | Closure of the ` _lcm ` fu... |
| lcmfpr 16555 | The value of the ` _lcm ` ... |
| lcmfcl 16556 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16557 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16558 | The least common multiple ... |
| dvdslcmf 16559 | The least common multiple ... |
| lcmfledvds 16560 | A positive integer which i... |
| lcmf 16561 | Characterization of the le... |
| lcmf0 16562 | The least common multiple ... |
| lcmfsn 16563 | The least common multiple ... |
| lcmftp 16564 | The least common multiple ... |
| lcmfunsnlem1 16565 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16566 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16567 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16568 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16569 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16570 | The least common multiple ... |
| lcmfdvdsb 16571 | Biconditional form of ~ lc... |
| lcmfunsn 16572 | The ` _lcm ` function for ... |
| lcmfun 16573 | The ` _lcm ` function for ... |
| lcmfass 16574 | Associative law for the ` ... |
| lcmf2a3a4e12 16575 | The least common multiple ... |
| lcmflefac 16576 | The least common multiple ... |
| coprmgcdb 16577 | Two positive integers are ... |
| ncoprmgcdne1b 16578 | Two positive integers are ... |
| ncoprmgcdgt1b 16579 | Two positive integers are ... |
| coprmdvds1 16580 | If two positive integers a... |
| coprmdvds 16581 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16582 | If an integer is divisible... |
| mulgcddvds 16583 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16584 | If ` M ` is relatively pri... |
| qredeq 16585 | Two equal reduced fraction... |
| qredeu 16586 | Every rational number has ... |
| rpmul 16587 | If ` K ` is relatively pri... |
| rpdvds 16588 | If ` K ` is relatively pri... |
| coprmprod 16589 | The product of the element... |
| coprmproddvdslem 16590 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16591 | If a positive integer is d... |
| congr 16592 | Definition of congruence b... |
| divgcdcoprm0 16593 | Integers divided by gcd ar... |
| divgcdcoprmex 16594 | Integers divided by gcd ar... |
| cncongr1 16595 | One direction of the bicon... |
| cncongr2 16596 | The other direction of the... |
| cncongr 16597 | Cancellability of Congruen... |
| cncongrcoprm 16598 | Corollary 1 of Cancellabil... |
| isprm 16601 | The predicate "is a prime ... |
| prmnn 16602 | A prime number is a positi... |
| prmz 16603 | A prime number is an integ... |
| prmssnn 16604 | The prime numbers are a su... |
| prmex 16605 | The set of prime numbers e... |
| 0nprm 16606 | 0 is not a prime number. ... |
| 1nprm 16607 | 1 is not a prime number. ... |
| 1idssfct 16608 | The positive divisors of a... |
| isprm2lem 16609 | Lemma for ~ isprm2 . (Con... |
| isprm2 16610 | The predicate "is a prime ... |
| isprm3 16611 | The predicate "is a prime ... |
| isprm4 16612 | The predicate "is a prime ... |
| prmind2 16613 | A variation on ~ prmind as... |
| prmind 16614 | Perform induction over the... |
| dvdsprime 16615 | If ` M ` divides a prime, ... |
| nprm 16616 | A product of two integers ... |
| nprmi 16617 | An inference for composite... |
| dvdsnprmd 16618 | If a number is divisible b... |
| prm2orodd 16619 | A prime number is either 2... |
| 2prm 16620 | 2 is a prime number. (Con... |
| 2mulprm 16621 | A multiple of two is prime... |
| 3prm 16622 | 3 is a prime number. (Con... |
| 4nprm 16623 | 4 is not a prime number. ... |
| prmuz2 16624 | A prime number is an integ... |
| prmgt1 16625 | A prime number is an integ... |
| prmm2nn0 16626 | Subtracting 2 from a prime... |
| oddprmgt2 16627 | An odd prime is greater th... |
| oddprmge3 16628 | An odd prime is greater th... |
| ge2nprmge4 16629 | A composite integer greate... |
| sqnprm 16630 | A square is never prime. ... |
| dvdsprm 16631 | An integer greater than or... |
| exprmfct 16632 | Every integer greater than... |
| prmdvdsfz 16633 | Each integer greater than ... |
| nprmdvds1 16634 | No prime number divides 1.... |
| isprm5 16635 | One need only check prime ... |
| isprm7 16636 | One need only check prime ... |
| maxprmfct 16637 | The set of prime factors o... |
| divgcdodd 16638 | Either ` A / ( A gcd B ) `... |
| coprm 16639 | A prime number either divi... |
| prmrp 16640 | Unequal prime numbers are ... |
| euclemma 16641 | Euclid's lemma. A prime n... |
| isprm6 16642 | A number is prime iff it s... |
| prmdvdsexp 16643 | A prime divides a positive... |
| prmdvdsexpb 16644 | A prime divides a positive... |
| prmdvdsexpr 16645 | If a prime divides a nonne... |
| prmdvdssq 16646 | Condition for a prime divi... |
| prmexpb 16647 | Two positive prime powers ... |
| prmfac1 16648 | The factorial of a number ... |
| dvdszzq 16649 | Divisibility for an intege... |
| rpexp 16650 | If two numbers ` A ` and `... |
| rpexp1i 16651 | Relative primality passes ... |
| rpexp12i 16652 | Relative primality passes ... |
| prmndvdsfaclt 16653 | A prime number does not di... |
| prmdvdsbc 16654 | Condition for a prime numb... |
| prmdvdsncoprmbd 16655 | Two positive integers are ... |
| ncoprmlnprm 16656 | If two positive integers a... |
| cncongrprm 16657 | Corollary 2 of Cancellabil... |
| isevengcd2 16658 | The predicate "is an even ... |
| isoddgcd1 16659 | The predicate "is an odd n... |
| 3lcm2e6 16660 | The least common multiple ... |
| qnumval 16665 | Value of the canonical num... |
| qdenval 16666 | Value of the canonical den... |
| qnumdencl 16667 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16668 | The canonical numerator of... |
| qdencl 16669 | The canonical denominator ... |
| fnum 16670 | Canonical numerator define... |
| fden 16671 | Canonical denominator defi... |
| qnumdenbi 16672 | Two numbers are the canoni... |
| qnumdencoprm 16673 | The canonical representati... |
| qeqnumdivden 16674 | Recover a rational number ... |
| qmuldeneqnum 16675 | Multiplying a rational by ... |
| divnumden 16676 | Calculate the reduced form... |
| divdenle 16677 | Reducing a quotient never ... |
| qnumgt0 16678 | A rational is positive iff... |
| qgt0numnn 16679 | A rational is positive iff... |
| nn0gcdsq 16680 | Squaring commutes with GCD... |
| zgcdsq 16681 | ~ nn0gcdsq extended to int... |
| numdensq 16682 | Squaring a rational square... |
| numsq 16683 | Square commutes with canon... |
| densq 16684 | Square commutes with canon... |
| qden1elz 16685 | A rational is an integer i... |
| zsqrtelqelz 16686 | If an integer has a ration... |
| nonsq 16687 | Any integer strictly betwe... |
| numdenexp 16688 | Elevating a rational numbe... |
| numexp 16689 | Elevating to a nonnegative... |
| denexp 16690 | Elevating to a nonnegative... |
| phival 16695 | Value of the Euler ` phi `... |
| phicl2 16696 | Bounds and closure for the... |
| phicl 16697 | Closure for the value of t... |
| phibndlem 16698 | Lemma for ~ phibnd . (Con... |
| phibnd 16699 | A slightly tighter bound o... |
| phicld 16700 | Closure for the value of t... |
| phi1 16701 | Value of the Euler ` phi `... |
| dfphi2 16702 | Alternate definition of th... |
| hashdvds 16703 | The number of numbers in a... |
| phiprmpw 16704 | Value of the Euler ` phi `... |
| phiprm 16705 | Value of the Euler ` phi `... |
| crth 16706 | The Chinese Remainder Theo... |
| phimullem 16707 | Lemma for ~ phimul . (Con... |
| phimul 16708 | The Euler ` phi ` function... |
| eulerthlem1 16709 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16710 | Lemma for ~ eulerth . (Co... |
| eulerth 16711 | Euler's theorem, a general... |
| fermltl 16712 | Fermat's little theorem. ... |
| prmdiv 16713 | Show an explicit expressio... |
| prmdiveq 16714 | The modular inverse of ` A... |
| prmdivdiv 16715 | The (modular) inverse of t... |
| hashgcdlem 16716 | A correspondence between e... |
| dvdsfi 16717 | A natural number has finit... |
| hashgcdeq 16718 | Number of initial positive... |
| phisum 16719 | The divisor sum identity o... |
| odzval 16720 | Value of the order functio... |
| odzcllem 16721 | - Lemma for ~ odzcl , show... |
| odzcl 16722 | The order of a group eleme... |
| odzid 16723 | Any element raised to the ... |
| odzdvds 16724 | The only powers of ` A ` t... |
| odzphi 16725 | The order of any group ele... |
| modprm1div 16726 | A prime number divides an ... |
| m1dvdsndvds 16727 | If an integer minus 1 is d... |
| modprminv 16728 | Show an explicit expressio... |
| modprminveq 16729 | The modular inverse of ` A... |
| vfermltl 16730 | Variant of Fermat's little... |
| vfermltlALT 16731 | Alternate proof of ~ vferm... |
| powm2modprm 16732 | If an integer minus 1 is d... |
| reumodprminv 16733 | For any prime number and f... |
| modprm0 16734 | For two positive integers ... |
| nnnn0modprm0 16735 | For a positive integer and... |
| modprmn0modprm0 16736 | For an integer not being 0... |
| coprimeprodsq 16737 | If three numbers are copri... |
| coprimeprodsq2 16738 | If three numbers are copri... |
| oddprm 16739 | A prime not equal to ` 2 `... |
| nnoddn2prm 16740 | A prime not equal to ` 2 `... |
| oddn2prm 16741 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16742 | A number is a prime number... |
| prm23lt5 16743 | A prime less than 5 is eit... |
| prm23ge5 16744 | A prime is either 2 or 3 o... |
| pythagtriplem1 16745 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16746 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16747 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16748 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16749 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16750 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16751 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16752 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16753 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16754 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16755 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16756 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16757 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16758 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16759 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16760 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16761 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16762 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16763 | Parameterize the Pythagore... |
| iserodd 16764 | Collect the odd terms in a... |
| pclem 16767 | - Lemma for the prime powe... |
| pcprecl 16768 | Closure of the prime power... |
| pcprendvds 16769 | Non-divisibility property ... |
| pcprendvds2 16770 | Non-divisibility property ... |
| pcpre1 16771 | Value of the prime power p... |
| pcpremul 16772 | Multiplicative property of... |
| pcval 16773 | The value of the prime pow... |
| pceulem 16774 | Lemma for ~ pceu . (Contr... |
| pceu 16775 | Uniqueness for the prime p... |
| pczpre 16776 | Connect the prime count pr... |
| pczcl 16777 | Closure of the prime power... |
| pccl 16778 | Closure of the prime power... |
| pccld 16779 | Closure of the prime power... |
| pcmul 16780 | Multiplication property of... |
| pcdiv 16781 | Division property of the p... |
| pcqmul 16782 | Multiplication property of... |
| pc0 16783 | The value of the prime pow... |
| pc1 16784 | Value of the prime count f... |
| pcqcl 16785 | Closure of the general pri... |
| pcqdiv 16786 | Division property of the p... |
| pcrec 16787 | Prime power of a reciproca... |
| pcexp 16788 | Prime power of an exponent... |
| pcxnn0cl 16789 | Extended nonnegative integ... |
| pcxcl 16790 | Extended real closure of t... |
| pcge0 16791 | The prime count of an inte... |
| pczdvds 16792 | Defining property of the p... |
| pcdvds 16793 | Defining property of the p... |
| pczndvds 16794 | Defining property of the p... |
| pcndvds 16795 | Defining property of the p... |
| pczndvds2 16796 | The remainder after dividi... |
| pcndvds2 16797 | The remainder after dividi... |
| pcdvdsb 16798 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16799 | There are a positive numbe... |
| pceq0 16800 | There are zero powers of a... |
| pcidlem 16801 | The prime count of a prime... |
| pcid 16802 | The prime count of a prime... |
| pcneg 16803 | The prime count of a negat... |
| pcabs 16804 | The prime count of an abso... |
| pcdvdstr 16805 | The prime count increases ... |
| pcgcd1 16806 | The prime count of a GCD i... |
| pcgcd 16807 | The prime count of a GCD i... |
| pc2dvds 16808 | A characterization of divi... |
| pc11 16809 | The prime count function, ... |
| pcz 16810 | The prime count function c... |
| pcprmpw2 16811 | Self-referential expressio... |
| pcprmpw 16812 | Self-referential expressio... |
| dvdsprmpweq 16813 | If a positive integer divi... |
| dvdsprmpweqnn 16814 | If an integer greater than... |
| dvdsprmpweqle 16815 | If a positive integer divi... |
| difsqpwdvds 16816 | If the difference of two s... |
| pcaddlem 16817 | Lemma for ~ pcadd . The o... |
| pcadd 16818 | An inequality for the prim... |
| pcadd2 16819 | The inequality of ~ pcadd ... |
| pcmptcl 16820 | Closure for the prime powe... |
| pcmpt 16821 | Construct a function with ... |
| pcmpt2 16822 | Dividing two prime count m... |
| pcmptdvds 16823 | The partial products of th... |
| pcprod 16824 | The product of the primes ... |
| sumhash 16825 | The sum of 1 over a set is... |
| fldivp1 16826 | The difference between the... |
| pcfaclem 16827 | Lemma for ~ pcfac . (Cont... |
| pcfac 16828 | Calculate the prime count ... |
| pcbc 16829 | Calculate the prime count ... |
| qexpz 16830 | If a power of a rational n... |
| expnprm 16831 | A second or higher power o... |
| oddprmdvds 16832 | Every positive integer whi... |
| prmpwdvds 16833 | A relation involving divis... |
| pockthlem 16834 | Lemma for ~ pockthg . (Co... |
| pockthg 16835 | The generalized Pocklingto... |
| pockthi 16836 | Pocklington's theorem, whi... |
| unbenlem 16837 | Lemma for ~ unben . (Cont... |
| unben 16838 | An unbounded set of positi... |
| infpnlem1 16839 | Lemma for ~ infpn . The s... |
| infpnlem2 16840 | Lemma for ~ infpn . For a... |
| infpn 16841 | There exist infinitely man... |
| infpn2 16842 | There exist infinitely man... |
| prmunb 16843 | The primes are unbounded. ... |
| prminf 16844 | There are an infinite numb... |
| prmreclem1 16845 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16846 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16847 | Lemma for ~ prmrec . The ... |
| prmreclem4 16848 | Lemma for ~ prmrec . Show... |
| prmreclem5 16849 | Lemma for ~ prmrec . Here... |
| prmreclem6 16850 | Lemma for ~ prmrec . If t... |
| prmrec 16851 | The sum of the reciprocals... |
| 1arithlem1 16852 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16853 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16854 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16855 | Lemma for ~ 1arith . (Con... |
| 1arith 16856 | Fundamental theorem of ari... |
| 1arith2 16857 | Fundamental theorem of ari... |
| elgz 16860 | Elementhood in the gaussia... |
| gzcn 16861 | A gaussian integer is a co... |
| zgz 16862 | An integer is a gaussian i... |
| igz 16863 | ` _i ` is a gaussian integ... |
| gznegcl 16864 | The gaussian integers are ... |
| gzcjcl 16865 | The gaussian integers are ... |
| gzaddcl 16866 | The gaussian integers are ... |
| gzmulcl 16867 | The gaussian integers are ... |
| gzreim 16868 | Construct a gaussian integ... |
| gzsubcl 16869 | The gaussian integers are ... |
| gzabssqcl 16870 | The squared norm of a gaus... |
| 4sqlem5 16871 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16872 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16873 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16874 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16875 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16876 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16877 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16878 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16879 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16880 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16881 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16882 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16883 | Euler's four-square identi... |
| 4sqlem11 16884 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16885 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16886 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16887 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16888 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16889 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16890 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16891 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16892 | Lemma for ~ 4sq . The pro... |
| 4sq 16893 | Lagrange's four-square the... |
| vdwapfval 16900 | Define the arithmetic prog... |
| vdwapf 16901 | The arithmetic progression... |
| vdwapval 16902 | Value of the arithmetic pr... |
| vdwapun 16903 | Remove the first element o... |
| vdwapid1 16904 | The first element of an ar... |
| vdwap0 16905 | Value of a length-1 arithm... |
| vdwap1 16906 | Value of a length-1 arithm... |
| vdwmc 16907 | The predicate " The ` <. R... |
| vdwmc2 16908 | Expand out the definition ... |
| vdwpc 16909 | The predicate " The colori... |
| vdwlem1 16910 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16911 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16912 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16913 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16914 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16915 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16916 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16917 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16918 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16919 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16920 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16921 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16922 | Lemma for ~ vdw . Main in... |
| vdw 16923 | Van der Waerden's theorem.... |
| vdwnnlem1 16924 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16925 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16926 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16927 | Van der Waerden's theorem,... |
| ramtlecl 16929 | The set ` T ` of numbers w... |
| hashbcval 16931 | Value of the "binomial set... |
| hashbccl 16932 | The binomial set is a fini... |
| hashbcss 16933 | Subset relation for the bi... |
| hashbc0 16934 | The set of subsets of size... |
| hashbc2 16935 | The size of the binomial s... |
| 0hashbc 16936 | There are no subsets of th... |
| ramval 16937 | The value of the Ramsey nu... |
| ramcl2lem 16938 | Lemma for extended real cl... |
| ramtcl 16939 | The Ramsey number has the ... |
| ramtcl2 16940 | The Ramsey number is an in... |
| ramtub 16941 | The Ramsey number is a low... |
| ramub 16942 | The Ramsey number is a low... |
| ramub2 16943 | It is sufficient to check ... |
| rami 16944 | The defining property of a... |
| ramcl2 16945 | The Ramsey number is eithe... |
| ramxrcl 16946 | The Ramsey number is an ex... |
| ramubcl 16947 | If the Ramsey number is up... |
| ramlb 16948 | Establish a lower bound on... |
| 0ram 16949 | The Ramsey number when ` M... |
| 0ram2 16950 | The Ramsey number when ` M... |
| ram0 16951 | The Ramsey number when ` R... |
| 0ramcl 16952 | Lemma for ~ ramcl : Exist... |
| ramz2 16953 | The Ramsey number when ` F... |
| ramz 16954 | The Ramsey number when ` F... |
| ramub1lem1 16955 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16956 | Lemma for ~ ramub1 . (Con... |
| ramub1 16957 | Inductive step for Ramsey'... |
| ramcl 16958 | Ramsey's theorem: the Rams... |
| ramsey 16959 | Ramsey's theorem with the ... |
| prmoval 16962 | Value of the primorial fun... |
| prmocl 16963 | Closure of the primorial f... |
| prmone0 16964 | The primorial function is ... |
| prmo0 16965 | The primorial of 0. (Cont... |
| prmo1 16966 | The primorial of 1. (Cont... |
| prmop1 16967 | The primorial of a success... |
| prmonn2 16968 | Value of the primorial fun... |
| prmo2 16969 | The primorial of 2. (Cont... |
| prmo3 16970 | The primorial of 3. (Cont... |
| prmdvdsprmo 16971 | The primorial of a number ... |
| prmdvdsprmop 16972 | The primorial of a number ... |
| fvprmselelfz 16973 | The value of the prime sel... |
| fvprmselgcd1 16974 | The greatest common diviso... |
| prmolefac 16975 | The primorial of a positiv... |
| prmodvdslcmf 16976 | The primorial of a nonnega... |
| prmolelcmf 16977 | The primorial of a positiv... |
| prmgaplem1 16978 | Lemma for ~ prmgap : The ... |
| prmgaplem2 16979 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 16980 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 16981 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 16982 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 16983 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 16984 | Lemma for ~ prmgap : for e... |
| prmgaplem6 16985 | Lemma for ~ prmgap : for e... |
| prmgaplem7 16986 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 16987 | Lemma for ~ prmgap . (Con... |
| prmgap 16988 | The prime gap theorem: for... |
| prmgaplcm 16989 | Alternate proof of ~ prmga... |
| prmgapprmolem 16990 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 16991 | Alternate proof of ~ prmga... |
| dec2dvds 16992 | Divisibility by two is obv... |
| dec5dvds 16993 | Divisibility by five is ob... |
| dec5dvds2 16994 | Divisibility by five is ob... |
| dec5nprm 16995 | A decimal number greater t... |
| dec2nprm 16996 | A decimal number greater t... |
| modxai 16997 | Add exponents in a power m... |
| mod2xi 16998 | Double exponents in a powe... |
| modxp1i 16999 | Add one to an exponent in ... |
| mod2xnegi 17000 | Version of ~ mod2xi with a... |
| modsubi 17001 | Subtract from within a mod... |
| gcdi 17002 | Calculate a GCD via Euclid... |
| gcdmodi 17003 | Calculate a GCD via Euclid... |
| numexp0 17004 | Calculate an integer power... |
| numexp1 17005 | Calculate an integer power... |
| numexpp1 17006 | Calculate an integer power... |
| numexp2x 17007 | Double an integer power. ... |
| decsplit0b 17008 | Split a decimal number int... |
| decsplit0 17009 | Split a decimal number int... |
| decsplit1 17010 | Split a decimal number int... |
| decsplit 17011 | Split a decimal number int... |
| karatsuba 17012 | The Karatsuba multiplicati... |
| 2exp4 17013 | Two to the fourth power is... |
| 2exp5 17014 | Two to the fifth power is ... |
| 2exp6 17015 | Two to the sixth power is ... |
| 2exp7 17016 | Two to the seventh power i... |
| 2exp8 17017 | Two to the eighth power is... |
| 2exp11 17018 | Two to the eleventh power ... |
| 2exp16 17019 | Two to the sixteenth power... |
| 3exp3 17020 | Three to the third power i... |
| 2expltfac 17021 | The factorial grows faster... |
| cshwsidrepsw 17022 | If cyclically shifting a w... |
| cshwsidrepswmod0 17023 | If cyclically shifting a w... |
| cshwshashlem1 17024 | If cyclically shifting a w... |
| cshwshashlem2 17025 | If cyclically shifting a w... |
| cshwshashlem3 17026 | If cyclically shifting a w... |
| cshwsdisj 17027 | The singletons resulting b... |
| cshwsiun 17028 | The set of (different!) wo... |
| cshwsex 17029 | The class of (different!) ... |
| cshws0 17030 | The size of the set of (di... |
| cshwrepswhash1 17031 | The size of the set of (di... |
| cshwshashnsame 17032 | If a word (not consisting ... |
| cshwshash 17033 | If a word has a length bei... |
| prmlem0 17034 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17035 | A quick proof skeleton to ... |
| prmlem1 17036 | A quick proof skeleton to ... |
| 5prm 17037 | 5 is a prime number. (Con... |
| 6nprm 17038 | 6 is not a prime number. ... |
| 7prm 17039 | 7 is a prime number. (Con... |
| 8nprm 17040 | 8 is not a prime number. ... |
| 9nprm 17041 | 9 is not a prime number. ... |
| 10nprm 17042 | 10 is not a prime number. ... |
| 11prm 17043 | 11 is a prime number. (Co... |
| 13prm 17044 | 13 is a prime number. (Co... |
| 17prm 17045 | 17 is a prime number. (Co... |
| 19prm 17046 | 19 is a prime number. (Co... |
| 23prm 17047 | 23 is a prime number. (Co... |
| prmlem2 17048 | Our last proving session g... |
| 37prm 17049 | 37 is a prime number. (Co... |
| 43prm 17050 | 43 is a prime number. (Co... |
| 83prm 17051 | 83 is a prime number. (Co... |
| 139prm 17052 | 139 is a prime number. (C... |
| 163prm 17053 | 163 is a prime number. (C... |
| 317prm 17054 | 317 is a prime number. (C... |
| 631prm 17055 | 631 is a prime number. (C... |
| prmo4 17056 | The primorial of 4. (Cont... |
| prmo5 17057 | The primorial of 5. (Cont... |
| prmo6 17058 | The primorial of 6. (Cont... |
| 1259lem1 17059 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17060 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17061 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17062 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17063 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17064 | 1259 is a prime number. (... |
| 2503lem1 17065 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17066 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17067 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17068 | 2503 is a prime number. (... |
| 4001lem1 17069 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17070 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17071 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17072 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17073 | 4001 is a prime number. (... |
| brstruct 17076 | The structure relation is ... |
| isstruct2 17077 | The property of being a st... |
| structex 17078 | A structure is a set. (Co... |
| structn0fun 17079 | A structure without the em... |
| isstruct 17080 | The property of being a st... |
| structcnvcnv 17081 | Two ways to express the re... |
| structfung 17082 | The converse of the conver... |
| structfun 17083 | Convert between two kinds ... |
| structfn 17084 | Convert between two kinds ... |
| strleun 17085 | Combine two structures int... |
| strle1 17086 | Make a structure from a si... |
| strle2 17087 | Make a structure from a pa... |
| strle3 17088 | Make a structure from a tr... |
| sbcie2s 17089 | A special version of class... |
| sbcie3s 17090 | A special version of class... |
| reldmsets 17093 | The structure override ope... |
| setsvalg 17094 | Value of the structure rep... |
| setsval 17095 | Value of the structure rep... |
| fvsetsid 17096 | The value of the structure... |
| fsets 17097 | The structure replacement ... |
| setsdm 17098 | The domain of a structure ... |
| setsfun 17099 | A structure with replaceme... |
| setsfun0 17100 | A structure with replaceme... |
| setsn0fun 17101 | The value of the structure... |
| setsstruct2 17102 | An extensible structure wi... |
| setsexstruct2 17103 | An extensible structure wi... |
| setsstruct 17104 | An extensible structure wi... |
| wunsets 17105 | Closure of structure repla... |
| setsres 17106 | The structure replacement ... |
| setsabs 17107 | Replacing the same compone... |
| setscom 17108 | Different components can b... |
| sloteq 17111 | Equality theorem for the `... |
| slotfn 17112 | A slot is a function on se... |
| strfvnd 17113 | Deduction version of ~ str... |
| strfvn 17114 | Value of a structure compo... |
| strfvss 17115 | A structure component extr... |
| wunstr 17116 | Closure of a structure ind... |
| str0 17117 | All components of the empt... |
| strfvi 17118 | Structure slot extractors ... |
| fveqprc 17119 | Lemma for showing the equa... |
| oveqprc 17120 | Lemma for showing the equa... |
| wunndx 17123 | Closure of the index extra... |
| ndxarg 17124 | Get the numeric argument f... |
| ndxid 17125 | A structure component extr... |
| strndxid 17126 | The value of a structure c... |
| setsidvald 17127 | Value of the structure rep... |
| strfvd 17128 | Deduction version of ~ str... |
| strfv2d 17129 | Deduction version of ~ str... |
| strfv2 17130 | A variation on ~ strfv to ... |
| strfv 17131 | Extract a structure compon... |
| strfv3 17132 | Variant on ~ strfv for lar... |
| strssd 17133 | Deduction version of ~ str... |
| strss 17134 | Propagate component extrac... |
| setsid 17135 | Value of the structure rep... |
| setsnid 17136 | Value of the structure rep... |
| baseval 17139 | Value of the base set extr... |
| baseid 17140 | Utility theorem: index-ind... |
| basfn 17141 | The base set extractor is ... |
| base0 17142 | The base set of the empty ... |
| elbasfv 17143 | Utility theorem: reverse c... |
| elbasov 17144 | Utility theorem: reverse c... |
| strov2rcl 17145 | Partial reverse closure fo... |
| basendx 17146 | Index value of the base se... |
| basendxnn 17147 | The index value of the bas... |
| basndxelwund 17148 | The index of the base set ... |
| basprssdmsets 17149 | The pair of the base index... |
| opelstrbas 17150 | The base set of a structur... |
| 1strstr 17151 | A constructed one-slot str... |
| 1strbas 17152 | The base set of a construc... |
| 1strwunbndx 17153 | A constructed one-slot str... |
| 1strwun 17154 | A constructed one-slot str... |
| 2strstr 17155 | A constructed two-slot str... |
| 2strbas 17156 | The base set of a construc... |
| 2strop 17157 | The other slot of a constr... |
| reldmress 17160 | The structure restriction ... |
| ressval 17161 | Value of structure restric... |
| ressid2 17162 | General behavior of trivia... |
| ressval2 17163 | Value of nontrivial struct... |
| ressbas 17164 | Base set of a structure re... |
| ressbasssg 17165 | The base set of a restrict... |
| ressbas2 17166 | Base set of a structure re... |
| ressbasss 17167 | The base set of a restrict... |
| ressbasssOLD 17168 | Obsolete version of ~ ress... |
| ressbasss2 17169 | The base set of a restrict... |
| resseqnbas 17170 | The components of an exten... |
| ress0 17171 | All restrictions of the nu... |
| ressid 17172 | Behavior of trivial restri... |
| ressinbas 17173 | Restriction only cares abo... |
| ressval3d 17174 | Value of structure restric... |
| ressress 17175 | Restriction composition la... |
| ressabs 17176 | Restriction absorption law... |
| wunress 17177 | Closure of structure restr... |
| plusgndx 17204 | Index value of the ~ df-pl... |
| plusgid 17205 | Utility theorem: index-ind... |
| plusgndxnn 17206 | The index of the slot for ... |
| basendxltplusgndx 17207 | The index of the slot for ... |
| basendxnplusgndx 17208 | The slot for the base set ... |
| grpstr 17209 | A constructed group is a s... |
| grpbase 17210 | The base set of a construc... |
| grpplusg 17211 | The operation of a constru... |
| ressplusg 17212 | ` +g ` is unaffected by re... |
| grpbasex 17213 | The base of an explicitly ... |
| grpplusgx 17214 | The operation of an explic... |
| mulrndx 17215 | Index value of the ~ df-mu... |
| mulridx 17216 | Utility theorem: index-ind... |
| basendxnmulrndx 17217 | The slot for the base set ... |
| plusgndxnmulrndx 17218 | The slot for the group (ad... |
| rngstr 17219 | A constructed ring is a st... |
| rngbase 17220 | The base set of a construc... |
| rngplusg 17221 | The additive operation of ... |
| rngmulr 17222 | The multiplicative operati... |
| starvndx 17223 | Index value of the ~ df-st... |
| starvid 17224 | Utility theorem: index-ind... |
| starvndxnbasendx 17225 | The slot for the involutio... |
| starvndxnplusgndx 17226 | The slot for the involutio... |
| starvndxnmulrndx 17227 | The slot for the involutio... |
| ressmulr 17228 | ` .r ` is unaffected by re... |
| ressstarv 17229 | ` *r ` is unaffected by re... |
| srngstr 17230 | A constructed star ring is... |
| srngbase 17231 | The base set of a construc... |
| srngplusg 17232 | The addition operation of ... |
| srngmulr 17233 | The multiplication operati... |
| srnginvl 17234 | The involution function of... |
| scandx 17235 | Index value of the ~ df-sc... |
| scaid 17236 | Utility theorem: index-ind... |
| scandxnbasendx 17237 | The slot for the scalar is... |
| scandxnplusgndx 17238 | The slot for the scalar fi... |
| scandxnmulrndx 17239 | The slot for the scalar fi... |
| vscandx 17240 | Index value of the ~ df-vs... |
| vscaid 17241 | Utility theorem: index-ind... |
| vscandxnbasendx 17242 | The slot for the scalar pr... |
| vscandxnplusgndx 17243 | The slot for the scalar pr... |
| vscandxnmulrndx 17244 | The slot for the scalar pr... |
| vscandxnscandx 17245 | The slot for the scalar pr... |
| lmodstr 17246 | A constructed left module ... |
| lmodbase 17247 | The base set of a construc... |
| lmodplusg 17248 | The additive operation of ... |
| lmodsca 17249 | The set of scalars of a co... |
| lmodvsca 17250 | The scalar product operati... |
| ipndx 17251 | Index value of the ~ df-ip... |
| ipid 17252 | Utility theorem: index-ind... |
| ipndxnbasendx 17253 | The slot for the inner pro... |
| ipndxnplusgndx 17254 | The slot for the inner pro... |
| ipndxnmulrndx 17255 | The slot for the inner pro... |
| slotsdifipndx 17256 | The slot for the scalar is... |
| ipsstr 17257 | Lemma to shorten proofs of... |
| ipsbase 17258 | The base set of a construc... |
| ipsaddg 17259 | The additive operation of ... |
| ipsmulr 17260 | The multiplicative operati... |
| ipssca 17261 | The set of scalars of a co... |
| ipsvsca 17262 | The scalar product operati... |
| ipsip 17263 | The multiplicative operati... |
| resssca 17264 | ` Scalar ` is unaffected b... |
| ressvsca 17265 | ` .s ` is unaffected by re... |
| ressip 17266 | The inner product is unaff... |
| phlstr 17267 | A constructed pre-Hilbert ... |
| phlbase 17268 | The base set of a construc... |
| phlplusg 17269 | The additive operation of ... |
| phlsca 17270 | The ring of scalars of a c... |
| phlvsca 17271 | The scalar product operati... |
| phlip 17272 | The inner product (Hermiti... |
| tsetndx 17273 | Index value of the ~ df-ts... |
| tsetid 17274 | Utility theorem: index-ind... |
| tsetndxnn 17275 | The index of the slot for ... |
| basendxlttsetndx 17276 | The index of the slot for ... |
| tsetndxnbasendx 17277 | The slot for the topology ... |
| tsetndxnplusgndx 17278 | The slot for the topology ... |
| tsetndxnmulrndx 17279 | The slot for the topology ... |
| tsetndxnstarvndx 17280 | The slot for the topology ... |
| slotstnscsi 17281 | The slots ` Scalar ` , ` .... |
| topgrpstr 17282 | A constructed topological ... |
| topgrpbas 17283 | The base set of a construc... |
| topgrpplusg 17284 | The additive operation of ... |
| topgrptset 17285 | The topology of a construc... |
| resstset 17286 | ` TopSet ` is unaffected b... |
| plendx 17287 | Index value of the ~ df-pl... |
| pleid 17288 | Utility theorem: self-refe... |
| plendxnn 17289 | The index value of the ord... |
| basendxltplendx 17290 | The index value of the ` B... |
| plendxnbasendx 17291 | The slot for the order is ... |
| plendxnplusgndx 17292 | The slot for the "less tha... |
| plendxnmulrndx 17293 | The slot for the "less tha... |
| plendxnscandx 17294 | The slot for the "less tha... |
| plendxnvscandx 17295 | The slot for the "less tha... |
| slotsdifplendx 17296 | The index of the slot for ... |
| otpsstr 17297 | Functionality of a topolog... |
| otpsbas 17298 | The base set of a topologi... |
| otpstset 17299 | The open sets of a topolog... |
| otpsle 17300 | The order of a topological... |
| ressle 17301 | ` le ` is unaffected by re... |
| ocndx 17302 | Index value of the ~ df-oc... |
| ocid 17303 | Utility theorem: index-ind... |
| basendxnocndx 17304 | The slot for the orthocomp... |
| plendxnocndx 17305 | The slot for the orthocomp... |
| dsndx 17306 | Index value of the ~ df-ds... |
| dsid 17307 | Utility theorem: index-ind... |
| dsndxnn 17308 | The index of the slot for ... |
| basendxltdsndx 17309 | The index of the slot for ... |
| dsndxnbasendx 17310 | The slot for the distance ... |
| dsndxnplusgndx 17311 | The slot for the distance ... |
| dsndxnmulrndx 17312 | The slot for the distance ... |
| slotsdnscsi 17313 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17314 | The slot for the distance ... |
| slotsdifdsndx 17315 | The index of the slot for ... |
| unifndx 17316 | Index value of the ~ df-un... |
| unifid 17317 | Utility theorem: index-ind... |
| unifndxnn 17318 | The index of the slot for ... |
| basendxltunifndx 17319 | The index of the slot for ... |
| unifndxnbasendx 17320 | The slot for the uniform s... |
| unifndxntsetndx 17321 | The slot for the uniform s... |
| slotsdifunifndx 17322 | The index of the slot for ... |
| ressunif 17323 | ` UnifSet ` is unaffected ... |
| odrngstr 17324 | Functionality of an ordere... |
| odrngbas 17325 | The base set of an ordered... |
| odrngplusg 17326 | The addition operation of ... |
| odrngmulr 17327 | The multiplication operati... |
| odrngtset 17328 | The open sets of an ordere... |
| odrngle 17329 | The order of an ordered me... |
| odrngds 17330 | The metric of an ordered m... |
| ressds 17331 | ` dist ` is unaffected by ... |
| homndx 17332 | Index value of the ~ df-ho... |
| homid 17333 | Utility theorem: index-ind... |
| ccondx 17334 | Index value of the ~ df-cc... |
| ccoid 17335 | Utility theorem: index-ind... |
| slotsbhcdif 17336 | The slots ` Base ` , ` Hom... |
| slotsdifplendx2 17337 | The index of the slot for ... |
| slotsdifocndx 17338 | The index of the slot for ... |
| resshom 17339 | ` Hom ` is unaffected by r... |
| ressco 17340 | ` comp ` is unaffected by ... |
| restfn 17345 | The subspace topology oper... |
| topnfn 17346 | The topology extractor fun... |
| restval 17347 | The subspace topology indu... |
| elrest 17348 | The predicate "is an open ... |
| elrestr 17349 | Sufficient condition for b... |
| 0rest 17350 | Value of the structure res... |
| restid2 17351 | The subspace topology over... |
| restsspw 17352 | The subspace topology is a... |
| firest 17353 | The finite intersections o... |
| restid 17354 | The subspace topology of t... |
| topnval 17355 | Value of the topology extr... |
| topnid 17356 | Value of the topology extr... |
| topnpropd 17357 | The topology extractor fun... |
| reldmprds 17369 | The structure product is a... |
| prdsbasex 17371 | Lemma for structure produc... |
| imasvalstr 17372 | An image structure value i... |
| prdsvalstr 17373 | Structure product value is... |
| prdsbaslem 17374 | Lemma for ~ prdsbas and si... |
| prdsvallem 17375 | Lemma for ~ prdsval . (Co... |
| prdsval 17376 | Value of the structure pro... |
| prdssca 17377 | Scalar ring of a structure... |
| prdsbas 17378 | Base set of a structure pr... |
| prdsplusg 17379 | Addition in a structure pr... |
| prdsmulr 17380 | Multiplication in a struct... |
| prdsvsca 17381 | Scalar multiplication in a... |
| prdsip 17382 | Inner product in a structu... |
| prdsle 17383 | Structure product weak ord... |
| prdsless 17384 | Closure of the order relat... |
| prdsds 17385 | Structure product distance... |
| prdsdsfn 17386 | Structure product distance... |
| prdstset 17387 | Structure product topology... |
| prdshom 17388 | Structure product hom-sets... |
| prdsco 17389 | Structure product composit... |
| prdsbas2 17390 | The base set of a structur... |
| prdsbasmpt 17391 | A constructed tuple is a p... |
| prdsbasfn 17392 | Points in the structure pr... |
| prdsbasprj 17393 | Each point in a structure ... |
| prdsplusgval 17394 | Value of a componentwise s... |
| prdsplusgfval 17395 | Value of a structure produ... |
| prdsmulrval 17396 | Value of a componentwise r... |
| prdsmulrfval 17397 | Value of a structure produ... |
| prdsleval 17398 | Value of the product order... |
| prdsdsval 17399 | Value of the metric in a s... |
| prdsvscaval 17400 | Scalar multiplication in a... |
| prdsvscafval 17401 | Scalar multiplication of a... |
| prdsbas3 17402 | The base set of an indexed... |
| prdsbasmpt2 17403 | A constructed tuple is a p... |
| prdsbascl 17404 | An element of the base has... |
| prdsdsval2 17405 | Value of the metric in a s... |
| prdsdsval3 17406 | Value of the metric in a s... |
| pwsval 17407 | Value of a structure power... |
| pwsbas 17408 | Base set of a structure po... |
| pwselbasb 17409 | Membership in the base set... |
| pwselbas 17410 | An element of a structure ... |
| pwselbasr 17411 | The reverse direction of ~... |
| pwsplusgval 17412 | Value of addition in a str... |
| pwsmulrval 17413 | Value of multiplication in... |
| pwsle 17414 | Ordering in a structure po... |
| pwsleval 17415 | Ordering in a structure po... |
| pwsvscafval 17416 | Scalar multiplication in a... |
| pwsvscaval 17417 | Scalar multiplication of a... |
| pwssca 17418 | The ring of scalars of a s... |
| pwsdiagel 17419 | Membership of diagonal ele... |
| pwssnf1o 17420 | Triviality of singleton po... |
| imasval 17433 | Value of an image structur... |
| imasbas 17434 | The base set of an image s... |
| imasds 17435 | The distance function of a... |
| imasdsfn 17436 | The distance function is a... |
| imasdsval 17437 | The distance function of a... |
| imasdsval2 17438 | The distance function of a... |
| imasplusg 17439 | The group operation in an ... |
| imasmulr 17440 | The ring multiplication in... |
| imassca 17441 | The scalar field of an ima... |
| imasvsca 17442 | The scalar multiplication ... |
| imasip 17443 | The inner product of an im... |
| imastset 17444 | The topology of an image s... |
| imasle 17445 | The ordering of an image s... |
| f1ocpbllem 17446 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17447 | An injection is compatible... |
| f1ovscpbl 17448 | An injection is compatible... |
| f1olecpbl 17449 | An injection is compatible... |
| imasaddfnlem 17450 | The image structure operat... |
| imasaddvallem 17451 | The operation of an image ... |
| imasaddflem 17452 | The image set operations a... |
| imasaddfn 17453 | The image structure's grou... |
| imasaddval 17454 | The value of an image stru... |
| imasaddf 17455 | The image structure's grou... |
| imasmulfn 17456 | The image structure's ring... |
| imasmulval 17457 | The value of an image stru... |
| imasmulf 17458 | The image structure's ring... |
| imasvscafn 17459 | The image structure's scal... |
| imasvscaval 17460 | The value of an image stru... |
| imasvscaf 17461 | The image structure's scal... |
| imasless 17462 | The order relation defined... |
| imasleval 17463 | The value of the image str... |
| qusval 17464 | Value of a quotient struct... |
| quslem 17465 | The function in ~ qusval i... |
| qusin 17466 | Restrict the equivalence r... |
| qusbas 17467 | Base set of a quotient str... |
| quss 17468 | The scalar field of a quot... |
| divsfval 17469 | Value of the function in ~... |
| ercpbllem 17470 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17471 | Translate the function com... |
| erlecpbl 17472 | Translate the relation com... |
| qusaddvallem 17473 | Value of an operation defi... |
| qusaddflem 17474 | The operation of a quotien... |
| qusaddval 17475 | The addition in a quotient... |
| qusaddf 17476 | The addition in a quotient... |
| qusmulval 17477 | The multiplication in a qu... |
| qusmulf 17478 | The multiplication in a qu... |
| fnpr2o 17479 | Function with a domain of ... |
| fnpr2ob 17480 | Biconditional version of ~... |
| fvpr0o 17481 | The value of a function wi... |
| fvpr1o 17482 | The value of a function wi... |
| fvprif 17483 | The value of the pair func... |
| xpsfrnel 17484 | Elementhood in the target ... |
| xpsfeq 17485 | A function on ` 2o ` is de... |
| xpsfrnel2 17486 | Elementhood in the target ... |
| xpscf 17487 | Equivalent condition for t... |
| xpsfval 17488 | The value of the function ... |
| xpsff1o 17489 | The function appearing in ... |
| xpsfrn 17490 | A short expression for the... |
| xpsff1o2 17491 | The function appearing in ... |
| xpsval 17492 | Value of the binary struct... |
| xpsrnbas 17493 | The indexed structure prod... |
| xpsbas 17494 | The base set of the binary... |
| xpsaddlem 17495 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17496 | Value of the addition oper... |
| xpsmul 17497 | Value of the multiplicatio... |
| xpssca 17498 | Value of the scalar field ... |
| xpsvsca 17499 | Value of the scalar multip... |
| xpsless 17500 | Closure of the ordering in... |
| xpsle 17501 | Value of the ordering in a... |
| ismre 17510 | Property of being a Moore ... |
| fnmre 17511 | The Moore collection gener... |
| mresspw 17512 | A Moore collection is a su... |
| mress 17513 | A Moore-closed subset is a... |
| mre1cl 17514 | In any Moore collection th... |
| mreintcl 17515 | A nonempty collection of c... |
| mreiincl 17516 | A nonempty indexed interse... |
| mrerintcl 17517 | The relative intersection ... |
| mreriincl 17518 | The relative intersection ... |
| mreincl 17519 | Two closed sets have a clo... |
| mreuni 17520 | Since the entire base set ... |
| mreunirn 17521 | Two ways to express the no... |
| ismred 17522 | Properties that determine ... |
| ismred2 17523 | Properties that determine ... |
| mremre 17524 | The Moore collections of s... |
| submre 17525 | The subcollection of a clo... |
| xrsle 17526 | The ordering of the extend... |
| xrge0le 17527 | The "less than or equal to... |
| xrsbas 17528 | The base set of the extend... |
| xrge0base 17529 | The base of the extended n... |
| mrcflem 17530 | The domain and codomain of... |
| fnmrc 17531 | Moore-closure is a well-be... |
| mrcfval 17532 | Value of the function expr... |
| mrcf 17533 | The Moore closure is a fun... |
| mrcval 17534 | Evaluation of the Moore cl... |
| mrccl 17535 | The Moore closure of a set... |
| mrcsncl 17536 | The Moore closure of a sin... |
| mrcid 17537 | The closure of a closed se... |
| mrcssv 17538 | The closure of a set is a ... |
| mrcidb 17539 | A set is closed iff it is ... |
| mrcss 17540 | Closure preserves subset o... |
| mrcssid 17541 | The closure of a set is a ... |
| mrcidb2 17542 | A set is closed iff it con... |
| mrcidm 17543 | The closure operation is i... |
| mrcsscl 17544 | The closure is the minimal... |
| mrcuni 17545 | Idempotence of closure und... |
| mrcun 17546 | Idempotence of closure und... |
| mrcssvd 17547 | The Moore closure of a set... |
| mrcssd 17548 | Moore closure preserves su... |
| mrcssidd 17549 | A set is contained in its ... |
| mrcidmd 17550 | Moore closure is idempoten... |
| mressmrcd 17551 | In a Moore system, if a se... |
| submrc 17552 | In a closure system which ... |
| mrieqvlemd 17553 | In a Moore system, if ` Y ... |
| mrisval 17554 | Value of the set of indepe... |
| ismri 17555 | Criterion for a set to be ... |
| ismri2 17556 | Criterion for a subset of ... |
| ismri2d 17557 | Criterion for a subset of ... |
| ismri2dd 17558 | Definition of independence... |
| mriss 17559 | An independent set of a Mo... |
| mrissd 17560 | An independent set of a Mo... |
| ismri2dad 17561 | Consequence of a set in a ... |
| mrieqvd 17562 | In a Moore system, a set i... |
| mrieqv2d 17563 | In a Moore system, a set i... |
| mrissmrcd 17564 | In a Moore system, if an i... |
| mrissmrid 17565 | In a Moore system, subsets... |
| mreexd 17566 | In a Moore system, the clo... |
| mreexmrid 17567 | In a Moore system whose cl... |
| mreexexlemd 17568 | This lemma is used to gene... |
| mreexexlem2d 17569 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17570 | Base case of the induction... |
| mreexexlem4d 17571 | Induction step of the indu... |
| mreexexd 17572 | Exchange-type theorem. In... |
| mreexdomd 17573 | In a Moore system whose cl... |
| mreexfidimd 17574 | In a Moore system whose cl... |
| isacs 17575 | A set is an algebraic clos... |
| acsmre 17576 | Algebraic closure systems ... |
| isacs2 17577 | In the definition of an al... |
| acsfiel 17578 | A set is closed in an alge... |
| acsfiel2 17579 | A set is closed in an alge... |
| acsmred 17580 | An algebraic closure syste... |
| isacs1i 17581 | A closure system determine... |
| mreacs 17582 | Algebraicity is a composab... |
| acsfn 17583 | Algebraicity of a conditio... |
| acsfn0 17584 | Algebraicity of a point cl... |
| acsfn1 17585 | Algebraicity of a one-argu... |
| acsfn1c 17586 | Algebraicity of a one-argu... |
| acsfn2 17587 | Algebraicity of a two-argu... |
| iscat 17596 | The predicate "is a catego... |
| iscatd 17597 | Properties that determine ... |
| catidex 17598 | Each object in a category ... |
| catideu 17599 | Each object in a category ... |
| cidfval 17600 | Each object in a category ... |
| cidval 17601 | Each object in a category ... |
| cidffn 17602 | The identity arrow constru... |
| cidfn 17603 | The identity arrow operato... |
| catidd 17604 | Deduce the identity arrow ... |
| iscatd2 17605 | Version of ~ iscatd with a... |
| catidcl 17606 | Each object in a category ... |
| catlid 17607 | Left identity property of ... |
| catrid 17608 | Right identity property of... |
| catcocl 17609 | Closure of a composition a... |
| catass 17610 | Associativity of compositi... |
| catcone0 17611 | Composition of non-empty h... |
| 0catg 17612 | Any structure with an empt... |
| 0cat 17613 | The empty set is a categor... |
| homffval 17614 | Value of the functionalize... |
| fnhomeqhomf 17615 | If the Hom-set operation i... |
| homfval 17616 | Value of the functionalize... |
| homffn 17617 | The functionalized Hom-set... |
| homfeq 17618 | Condition for two categori... |
| homfeqd 17619 | If two structures have the... |
| homfeqbas 17620 | Deduce equality of base se... |
| homfeqval 17621 | Value of the functionalize... |
| comfffval 17622 | Value of the functionalize... |
| comffval 17623 | Value of the functionalize... |
| comfval 17624 | Value of the functionalize... |
| comfffval2 17625 | Value of the functionalize... |
| comffval2 17626 | Value of the functionalize... |
| comfval2 17627 | Value of the functionalize... |
| comfffn 17628 | The functionalized composi... |
| comffn 17629 | The functionalized composi... |
| comfeq 17630 | Condition for two categori... |
| comfeqd 17631 | Condition for two categori... |
| comfeqval 17632 | Equality of two compositio... |
| catpropd 17633 | Two structures with the sa... |
| cidpropd 17634 | Two structures with the sa... |
| oppcval 17637 | Value of the opposite cate... |
| oppchomfval 17638 | Hom-sets of the opposite c... |
| oppchom 17639 | Hom-sets of the opposite c... |
| oppccofval 17640 | Composition in the opposit... |
| oppcco 17641 | Composition in the opposit... |
| oppcbas 17642 | Base set of an opposite ca... |
| oppccatid 17643 | Lemma for ~ oppccat . (Co... |
| oppchomf 17644 | Hom-sets of the opposite c... |
| oppcid 17645 | Identity function of an op... |
| oppccat 17646 | An opposite category is a ... |
| 2oppcbas 17647 | The double opposite catego... |
| 2oppchomf 17648 | The double opposite catego... |
| 2oppccomf 17649 | The double opposite catego... |
| oppchomfpropd 17650 | If two categories have the... |
| oppccomfpropd 17651 | If two categories have the... |
| oppccatf 17652 | ` oppCat ` restricted to `... |
| monfval 17657 | Definition of a monomorphi... |
| ismon 17658 | Definition of a monomorphi... |
| ismon2 17659 | Write out the monomorphism... |
| monhom 17660 | A monomorphism is a morphi... |
| moni 17661 | Property of a monomorphism... |
| monpropd 17662 | If two categories have the... |
| oppcmon 17663 | A monomorphism in the oppo... |
| oppcepi 17664 | An epimorphism in the oppo... |
| isepi 17665 | Definition of an epimorphi... |
| isepi2 17666 | Write out the epimorphism ... |
| epihom 17667 | An epimorphism is a morphi... |
| epii 17668 | Property of an epimorphism... |
| sectffval 17675 | Value of the section opera... |
| sectfval 17676 | Value of the section relat... |
| sectss 17677 | The section relation is a ... |
| issect 17678 | The property " ` F ` is a ... |
| issect2 17679 | Property of being a sectio... |
| sectcan 17680 | If ` G ` is a section of `... |
| sectco 17681 | Composition of two section... |
| isofval 17682 | Function value of the func... |
| invffval 17683 | Value of the inverse relat... |
| invfval 17684 | Value of the inverse relat... |
| isinv 17685 | Value of the inverse relat... |
| invss 17686 | The inverse relation is a ... |
| invsym 17687 | The inverse relation is sy... |
| invsym2 17688 | The inverse relation is sy... |
| invfun 17689 | The inverse relation is a ... |
| isoval 17690 | The isomorphisms are the d... |
| inviso1 17691 | If ` G ` is an inverse to ... |
| inviso2 17692 | If ` G ` is an inverse to ... |
| invf 17693 | The inverse relation is a ... |
| invf1o 17694 | The inverse relation is a ... |
| invinv 17695 | The inverse of the inverse... |
| invco 17696 | The composition of two iso... |
| dfiso2 17697 | Alternate definition of an... |
| dfiso3 17698 | Alternate definition of an... |
| inveq 17699 | If there are two inverses ... |
| isofn 17700 | The function value of the ... |
| isohom 17701 | An isomorphism is a homomo... |
| isoco 17702 | The composition of two iso... |
| oppcsect 17703 | A section in the opposite ... |
| oppcsect2 17704 | A section in the opposite ... |
| oppcinv 17705 | An inverse in the opposite... |
| oppciso 17706 | An isomorphism in the oppo... |
| sectmon 17707 | If ` F ` is a section of `... |
| monsect 17708 | If ` F ` is a monomorphism... |
| sectepi 17709 | If ` F ` is a section of `... |
| episect 17710 | If ` F ` is an epimorphism... |
| sectid 17711 | The identity is a section ... |
| invid 17712 | The inverse of the identit... |
| idiso 17713 | The identity is an isomorp... |
| idinv 17714 | The inverse of the identit... |
| invisoinvl 17715 | The inverse of an isomorph... |
| invisoinvr 17716 | The inverse of an isomorph... |
| invcoisoid 17717 | The inverse of an isomorph... |
| isocoinvid 17718 | The inverse of an isomorph... |
| rcaninv 17719 | Right cancellation of an i... |
| cicfval 17722 | The set of isomorphic obje... |
| brcic 17723 | The relation "is isomorphi... |
| cic 17724 | Objects ` X ` and ` Y ` in... |
| brcici 17725 | Prove that two objects are... |
| cicref 17726 | Isomorphism is reflexive. ... |
| ciclcl 17727 | Isomorphism implies the le... |
| cicrcl 17728 | Isomorphism implies the ri... |
| cicsym 17729 | Isomorphism is symmetric. ... |
| cictr 17730 | Isomorphism is transitive.... |
| cicer 17731 | Isomorphism is an equivale... |
| sscrel 17738 | The subcategory subset rel... |
| brssc 17739 | The subcategory subset rel... |
| sscpwex 17740 | An analogue of ~ pwex for ... |
| subcrcl 17741 | Reverse closure for the su... |
| sscfn1 17742 | The subcategory subset rel... |
| sscfn2 17743 | The subcategory subset rel... |
| ssclem 17744 | Lemma for ~ ssc1 and simil... |
| isssc 17745 | Value of the subcategory s... |
| ssc1 17746 | Infer subset relation on o... |
| ssc2 17747 | Infer subset relation on m... |
| sscres 17748 | Any function restricted to... |
| sscid 17749 | The subcategory subset rel... |
| ssctr 17750 | The subcategory subset rel... |
| ssceq 17751 | The subcategory subset rel... |
| rescval 17752 | Value of the category rest... |
| rescval2 17753 | Value of the category rest... |
| rescbas 17754 | Base set of the category r... |
| reschom 17755 | Hom-sets of the category r... |
| reschomf 17756 | Hom-sets of the category r... |
| rescco 17757 | Composition in the categor... |
| rescabs 17758 | Restriction absorption law... |
| rescabs2 17759 | Restriction absorption law... |
| issubc 17760 | Elementhood in the set of ... |
| issubc2 17761 | Elementhood in the set of ... |
| 0ssc 17762 | For any category ` C ` , t... |
| 0subcat 17763 | For any category ` C ` , t... |
| catsubcat 17764 | For any category ` C ` , `... |
| subcssc 17765 | An element in the set of s... |
| subcfn 17766 | An element in the set of s... |
| subcss1 17767 | The objects of a subcatego... |
| subcss2 17768 | The morphisms of a subcate... |
| subcidcl 17769 | The identity of the origin... |
| subccocl 17770 | A subcategory is closed un... |
| subccatid 17771 | A subcategory is a categor... |
| subcid 17772 | The identity in a subcateg... |
| subccat 17773 | A subcategory is a categor... |
| issubc3 17774 | Alternate definition of a ... |
| fullsubc 17775 | The full subcategory gener... |
| fullresc 17776 | The category formed by str... |
| resscat 17777 | A category restricted to a... |
| subsubc 17778 | A subcategory of a subcate... |
| relfunc 17787 | The set of functors is a r... |
| funcrcl 17788 | Reverse closure for a func... |
| isfunc 17789 | Value of the set of functo... |
| isfuncd 17790 | Deduce that an operation i... |
| funcf1 17791 | The object part of a funct... |
| funcixp 17792 | The morphism part of a fun... |
| funcf2 17793 | The morphism part of a fun... |
| funcfn2 17794 | The morphism part of a fun... |
| funcid 17795 | A functor maps each identi... |
| funcco 17796 | A functor maps composition... |
| funcsect 17797 | The image of a section und... |
| funcinv 17798 | The image of an inverse un... |
| funciso 17799 | The image of an isomorphis... |
| funcoppc 17800 | A functor on categories yi... |
| idfuval 17801 | Value of the identity func... |
| idfu2nd 17802 | Value of the morphism part... |
| idfu2 17803 | Value of the morphism part... |
| idfu1st 17804 | Value of the object part o... |
| idfu1 17805 | Value of the object part o... |
| idfucl 17806 | The identity functor is a ... |
| cofuval 17807 | Value of the composition o... |
| cofu1st 17808 | Value of the object part o... |
| cofu1 17809 | Value of the object part o... |
| cofu2nd 17810 | Value of the morphism part... |
| cofu2 17811 | Value of the morphism part... |
| cofuval2 17812 | Value of the composition o... |
| cofucl 17813 | The composition of two fun... |
| cofuass 17814 | Functor composition is ass... |
| cofulid 17815 | The identity functor is a ... |
| cofurid 17816 | The identity functor is a ... |
| resfval 17817 | Value of the functor restr... |
| resfval2 17818 | Value of the functor restr... |
| resf1st 17819 | Value of the functor restr... |
| resf2nd 17820 | Value of the functor restr... |
| funcres 17821 | A functor restricted to a ... |
| funcres2b 17822 | Condition for a functor to... |
| funcres2 17823 | A functor into a restricte... |
| idfusubc0 17824 | The identity functor for a... |
| idfusubc 17825 | The identity functor for a... |
| wunfunc 17826 | A weak universe is closed ... |
| funcpropd 17827 | If two categories have the... |
| funcres2c 17828 | Condition for a functor to... |
| fullfunc 17833 | A full functor is a functo... |
| fthfunc 17834 | A faithful functor is a fu... |
| relfull 17835 | The set of full functors i... |
| relfth 17836 | The set of faithful functo... |
| isfull 17837 | Value of the set of full f... |
| isfull2 17838 | Equivalent condition for a... |
| fullfo 17839 | The morphism map of a full... |
| fulli 17840 | The morphism map of a full... |
| isfth 17841 | Value of the set of faithf... |
| isfth2 17842 | Equivalent condition for a... |
| isffth2 17843 | A fully faithful functor i... |
| fthf1 17844 | The morphism map of a fait... |
| fthi 17845 | The morphism map of a fait... |
| ffthf1o 17846 | The morphism map of a full... |
| fullpropd 17847 | If two categories have the... |
| fthpropd 17848 | If two categories have the... |
| fulloppc 17849 | The opposite functor of a ... |
| fthoppc 17850 | The opposite functor of a ... |
| ffthoppc 17851 | The opposite functor of a ... |
| fthsect 17852 | A faithful functor reflect... |
| fthinv 17853 | A faithful functor reflect... |
| fthmon 17854 | A faithful functor reflect... |
| fthepi 17855 | A faithful functor reflect... |
| ffthiso 17856 | A fully faithful functor r... |
| fthres2b 17857 | Condition for a faithful f... |
| fthres2c 17858 | Condition for a faithful f... |
| fthres2 17859 | A faithful functor into a ... |
| idffth 17860 | The identity functor is a ... |
| cofull 17861 | The composition of two ful... |
| cofth 17862 | The composition of two fai... |
| coffth 17863 | The composition of two ful... |
| rescfth 17864 | The inclusion functor from... |
| ressffth 17865 | The inclusion functor from... |
| fullres2c 17866 | Condition for a full funct... |
| ffthres2c 17867 | Condition for a fully fait... |
| inclfusubc 17868 | The "inclusion functor" fr... |
| fnfuc 17873 | The ` FuncCat ` operation ... |
| natfval 17874 | Value of the function givi... |
| isnat 17875 | Property of being a natura... |
| isnat2 17876 | Property of being a natura... |
| natffn 17877 | The natural transformation... |
| natrcl 17878 | Reverse closure for a natu... |
| nat1st2nd 17879 | Rewrite the natural transf... |
| natixp 17880 | A natural transformation i... |
| natcl 17881 | A component of a natural t... |
| natfn 17882 | A natural transformation i... |
| nati 17883 | Naturality property of a n... |
| wunnat 17884 | A weak universe is closed ... |
| catstr 17885 | A category structure is a ... |
| fucval 17886 | Value of the functor categ... |
| fuccofval 17887 | Value of the functor categ... |
| fucbas 17888 | The objects of the functor... |
| fuchom 17889 | The morphisms in the funct... |
| fucco 17890 | Value of the composition o... |
| fuccoval 17891 | Value of the functor categ... |
| fuccocl 17892 | The composition of two nat... |
| fucidcl 17893 | The identity natural trans... |
| fuclid 17894 | Left identity of natural t... |
| fucrid 17895 | Right identity of natural ... |
| fucass 17896 | Associativity of natural t... |
| fuccatid 17897 | The functor category is a ... |
| fuccat 17898 | The functor category is a ... |
| fucid 17899 | The identity morphism in t... |
| fucsect 17900 | Two natural transformation... |
| fucinv 17901 | Two natural transformation... |
| invfuc 17902 | If ` V ( x ) ` is an inver... |
| fuciso 17903 | A natural transformation i... |
| natpropd 17904 | If two categories have the... |
| fucpropd 17905 | If two categories have the... |
| initofn 17912 | ` InitO ` is a function on... |
| termofn 17913 | ` TermO ` is a function on... |
| zeroofn 17914 | ` ZeroO ` is a function on... |
| initorcl 17915 | Reverse closure for an ini... |
| termorcl 17916 | Reverse closure for a term... |
| zeroorcl 17917 | Reverse closure for a zero... |
| initoval 17918 | The value of the initial o... |
| termoval 17919 | The value of the terminal ... |
| zerooval 17920 | The value of the zero obje... |
| isinito 17921 | The predicate "is an initi... |
| istermo 17922 | The predicate "is a termin... |
| iszeroo 17923 | The predicate "is a zero o... |
| isinitoi 17924 | Implication of a class bei... |
| istermoi 17925 | Implication of a class bei... |
| initoid 17926 | For an initial object, the... |
| termoid 17927 | For a terminal object, the... |
| dfinito2 17928 | An initial object is a ter... |
| dftermo2 17929 | A terminal object is an in... |
| dfinito3 17930 | An alternate definition of... |
| dftermo3 17931 | An alternate definition of... |
| initoo 17932 | An initial object is an ob... |
| termoo 17933 | A terminal object is an ob... |
| iszeroi 17934 | Implication of a class bei... |
| 2initoinv 17935 | Morphisms between two init... |
| initoeu1 17936 | Initial objects are essent... |
| initoeu1w 17937 | Initial objects are essent... |
| initoeu2lem0 17938 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17939 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17940 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17941 | Initial objects are essent... |
| 2termoinv 17942 | Morphisms between two term... |
| termoeu1 17943 | Terminal objects are essen... |
| termoeu1w 17944 | Terminal objects are essen... |
| homarcl 17953 | Reverse closure for an arr... |
| homafval 17954 | Value of the disjointified... |
| homaf 17955 | Functionality of the disjo... |
| homaval 17956 | Value of the disjointified... |
| elhoma 17957 | Value of the disjointified... |
| elhomai 17958 | Produce an arrow from a mo... |
| elhomai2 17959 | Produce an arrow from a mo... |
| homarcl2 17960 | Reverse closure for the do... |
| homarel 17961 | An arrow is an ordered pai... |
| homa1 17962 | The first component of an ... |
| homahom2 17963 | The second component of an... |
| homahom 17964 | The second component of an... |
| homadm 17965 | The domain of an arrow wit... |
| homacd 17966 | The codomain of an arrow w... |
| homadmcd 17967 | Decompose an arrow into do... |
| arwval 17968 | The set of arrows is the u... |
| arwrcl 17969 | The first component of an ... |
| arwhoma 17970 | An arrow is contained in t... |
| homarw 17971 | A hom-set is a subset of t... |
| arwdm 17972 | The domain of an arrow is ... |
| arwcd 17973 | The codomain of an arrow i... |
| dmaf 17974 | The domain function is a f... |
| cdaf 17975 | The codomain function is a... |
| arwhom 17976 | The second component of an... |
| arwdmcd 17977 | Decompose an arrow into do... |
| idafval 17982 | Value of the identity arro... |
| idaval 17983 | Value of the identity arro... |
| ida2 17984 | Morphism part of the ident... |
| idahom 17985 | Domain and codomain of the... |
| idadm 17986 | Domain of the identity arr... |
| idacd 17987 | Codomain of the identity a... |
| idaf 17988 | The identity arrow functio... |
| coafval 17989 | The value of the compositi... |
| eldmcoa 17990 | A pair ` <. G , F >. ` is ... |
| dmcoass 17991 | The domain of composition ... |
| homdmcoa 17992 | If ` F : X --> Y ` and ` G... |
| coaval 17993 | Value of composition for c... |
| coa2 17994 | The morphism part of arrow... |
| coahom 17995 | The composition of two com... |
| coapm 17996 | Composition of arrows is a... |
| arwlid 17997 | Left identity of a categor... |
| arwrid 17998 | Right identity of a catego... |
| arwass 17999 | Associativity of compositi... |
| setcval 18002 | Value of the category of s... |
| setcbas 18003 | Set of objects of the cate... |
| setchomfval 18004 | Set of arrows of the categ... |
| setchom 18005 | Set of arrows of the categ... |
| elsetchom 18006 | A morphism of sets is a fu... |
| setccofval 18007 | Composition in the categor... |
| setcco 18008 | Composition in the categor... |
| setccatid 18009 | Lemma for ~ setccat . (Co... |
| setccat 18010 | The category of sets is a ... |
| setcid 18011 | The identity arrow in the ... |
| setcmon 18012 | A monomorphism of sets is ... |
| setcepi 18013 | An epimorphism of sets is ... |
| setcsect 18014 | A section in the category ... |
| setcinv 18015 | An inverse in the category... |
| setciso 18016 | An isomorphism in the cate... |
| resssetc 18017 | The restriction of the cat... |
| funcsetcres2 18018 | A functor into a smaller c... |
| setc2obas 18019 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18020 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18021 | The category of sets in a ... |
| cat1 18022 | The definition of category... |
| catcval 18025 | Value of the category of c... |
| catcbas 18026 | Set of objects of the cate... |
| catchomfval 18027 | Set of arrows of the categ... |
| catchom 18028 | Set of arrows of the categ... |
| catccofval 18029 | Composition in the categor... |
| catcco 18030 | Composition in the categor... |
| catccatid 18031 | Lemma for ~ catccat . (Co... |
| catcid 18032 | The identity arrow in the ... |
| catccat 18033 | The category of categories... |
| resscatc 18034 | The restriction of the cat... |
| catcisolem 18035 | Lemma for ~ catciso . (Co... |
| catciso 18036 | A functor is an isomorphis... |
| catcbascl 18037 | An element of the base set... |
| catcslotelcl 18038 | A slot entry of an element... |
| catcbaselcl 18039 | The base set of an element... |
| catchomcl 18040 | The Hom-set of an element ... |
| catcccocl 18041 | The composition operation ... |
| catcoppccl 18042 | The category of categories... |
| catcfuccl 18043 | The category of categories... |
| fncnvimaeqv 18044 | The inverse images of the ... |
| bascnvimaeqv 18045 | The inverse image of the u... |
| estrcval 18048 | Value of the category of e... |
| estrcbas 18049 | Set of objects of the cate... |
| estrchomfval 18050 | Set of morphisms ("arrows"... |
| estrchom 18051 | The morphisms between exte... |
| elestrchom 18052 | A morphism between extensi... |
| estrccofval 18053 | Composition in the categor... |
| estrcco 18054 | Composition in the categor... |
| estrcbasbas 18055 | An element of the base set... |
| estrccatid 18056 | Lemma for ~ estrccat . (C... |
| estrccat 18057 | The category of extensible... |
| estrcid 18058 | The identity arrow in the ... |
| estrchomfn 18059 | The Hom-set operation in t... |
| estrchomfeqhom 18060 | The functionalized Hom-set... |
| estrreslem1 18061 | Lemma 1 for ~ estrres . (... |
| estrreslem2 18062 | Lemma 2 for ~ estrres . (... |
| estrres 18063 | Any restriction of a categ... |
| funcestrcsetclem1 18064 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18065 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18066 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18067 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18068 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18069 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18070 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18071 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18072 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18073 | The "natural forgetful fun... |
| fthestrcsetc 18074 | The "natural forgetful fun... |
| fullestrcsetc 18075 | The "natural forgetful fun... |
| equivestrcsetc 18076 | The "natural forgetful fun... |
| setc1strwun 18077 | A constructed one-slot str... |
| funcsetcestrclem1 18078 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18079 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18080 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18081 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18082 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18083 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18084 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18085 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18086 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18087 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18088 | The "embedding functor" fr... |
| fthsetcestrc 18089 | The "embedding functor" fr... |
| fullsetcestrc 18090 | The "embedding functor" fr... |
| embedsetcestrc 18091 | The "embedding functor" fr... |
| fnxpc 18100 | The binary product of cate... |
| xpcval 18101 | Value of the binary produc... |
| xpcbas 18102 | Set of objects of the bina... |
| xpchomfval 18103 | Set of morphisms of the bi... |
| xpchom 18104 | Set of morphisms of the bi... |
| relxpchom 18105 | A hom-set in the binary pr... |
| xpccofval 18106 | Value of composition in th... |
| xpcco 18107 | Value of composition in th... |
| xpcco1st 18108 | Value of composition in th... |
| xpcco2nd 18109 | Value of composition in th... |
| xpchom2 18110 | Value of the set of morphi... |
| xpcco2 18111 | Value of composition in th... |
| xpccatid 18112 | The product of two categor... |
| xpcid 18113 | The identity morphism in t... |
| xpccat 18114 | The product of two categor... |
| 1stfval 18115 | Value of the first project... |
| 1stf1 18116 | Value of the first project... |
| 1stf2 18117 | Value of the first project... |
| 2ndfval 18118 | Value of the first project... |
| 2ndf1 18119 | Value of the first project... |
| 2ndf2 18120 | Value of the first project... |
| 1stfcl 18121 | The first projection funct... |
| 2ndfcl 18122 | The second projection func... |
| prfval 18123 | Value of the pairing funct... |
| prf1 18124 | Value of the pairing funct... |
| prf2fval 18125 | Value of the pairing funct... |
| prf2 18126 | Value of the pairing funct... |
| prfcl 18127 | The pairing of functors ` ... |
| prf1st 18128 | Cancellation of pairing wi... |
| prf2nd 18129 | Cancellation of pairing wi... |
| 1st2ndprf 18130 | Break a functor into a pro... |
| catcxpccl 18131 | The category of categories... |
| xpcpropd 18132 | If two categories have the... |
| evlfval 18141 | Value of the evaluation fu... |
| evlf2 18142 | Value of the evaluation fu... |
| evlf2val 18143 | Value of the evaluation na... |
| evlf1 18144 | Value of the evaluation fu... |
| evlfcllem 18145 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18146 | The evaluation functor is ... |
| curfval 18147 | Value of the curry functor... |
| curf1fval 18148 | Value of the object part o... |
| curf1 18149 | Value of the object part o... |
| curf11 18150 | Value of the double evalua... |
| curf12 18151 | The partially evaluated cu... |
| curf1cl 18152 | The partially evaluated cu... |
| curf2 18153 | Value of the curry functor... |
| curf2val 18154 | Value of a component of th... |
| curf2cl 18155 | The curry functor at a mor... |
| curfcl 18156 | The curry functor of a fun... |
| curfpropd 18157 | If two categories have the... |
| uncfval 18158 | Value of the uncurry funct... |
| uncfcl 18159 | The uncurry operation take... |
| uncf1 18160 | Value of the uncurry funct... |
| uncf2 18161 | Value of the uncurry funct... |
| curfuncf 18162 | Cancellation of curry with... |
| uncfcurf 18163 | Cancellation of uncurry wi... |
| diagval 18164 | Define the diagonal functo... |
| diagcl 18165 | The diagonal functor is a ... |
| diag1cl 18166 | The constant functor of ` ... |
| diag11 18167 | Value of the constant func... |
| diag12 18168 | Value of the constant func... |
| diag2 18169 | Value of the diagonal func... |
| diag2cl 18170 | The diagonal functor at a ... |
| curf2ndf 18171 | As shown in ~ diagval , th... |
| hofval 18176 | Value of the Hom functor, ... |
| hof1fval 18177 | The object part of the Hom... |
| hof1 18178 | The object part of the Hom... |
| hof2fval 18179 | The morphism part of the H... |
| hof2val 18180 | The morphism part of the H... |
| hof2 18181 | The morphism part of the H... |
| hofcllem 18182 | Lemma for ~ hofcl . (Cont... |
| hofcl 18183 | Closure of the Hom functor... |
| oppchofcl 18184 | Closure of the opposite Ho... |
| yonval 18185 | Value of the Yoneda embedd... |
| yoncl 18186 | The Yoneda embedding is a ... |
| yon1cl 18187 | The Yoneda embedding at an... |
| yon11 18188 | Value of the Yoneda embedd... |
| yon12 18189 | Value of the Yoneda embedd... |
| yon2 18190 | Value of the Yoneda embedd... |
| hofpropd 18191 | If two categories have the... |
| yonpropd 18192 | If two categories have the... |
| oppcyon 18193 | Value of the opposite Yone... |
| oyoncl 18194 | The opposite Yoneda embedd... |
| oyon1cl 18195 | The opposite Yoneda embedd... |
| yonedalem1 18196 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18197 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18198 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18199 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18200 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18201 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18202 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18203 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18204 | Lemma for ~ yoneda . (Con... |
| yonedainv 18205 | The Yoneda Lemma with expl... |
| yonffthlem 18206 | Lemma for ~ yonffth . (Co... |
| yoneda 18207 | The Yoneda Lemma. There i... |
| yonffth 18208 | The Yoneda Lemma. The Yon... |
| yoniso 18209 | If the codomain is recover... |
| oduval 18212 | Value of an order dual str... |
| oduleval 18213 | Value of the less-equal re... |
| oduleg 18214 | Truth of the less-equal re... |
| odubas 18215 | Base set of an order dual ... |
| isprs 18220 | Property of being a preord... |
| prslem 18221 | Lemma for ~ prsref and ~ p... |
| prsref 18222 | "Less than or equal to" is... |
| prstr 18223 | "Less than or equal to" is... |
| oduprs 18224 | Being a proset is a self-d... |
| isdrs 18225 | Property of being a direct... |
| drsdir 18226 | Direction of a directed se... |
| drsprs 18227 | A directed set is a proset... |
| drsbn0 18228 | The base of a directed set... |
| drsdirfi 18229 | Any _finite_ number of ele... |
| isdrs2 18230 | Directed sets may be defin... |
| ispos 18238 | The predicate "is a poset"... |
| ispos2 18239 | A poset is an antisymmetri... |
| posprs 18240 | A poset is a proset. (Con... |
| posi 18241 | Lemma for poset properties... |
| posref 18242 | A poset ordering is reflex... |
| posasymb 18243 | A poset ordering is asymme... |
| postr 18244 | A poset ordering is transi... |
| 0pos 18245 | Technical lemma to simplif... |
| isposd 18246 | Properties that determine ... |
| isposi 18247 | Properties that determine ... |
| isposix 18248 | Properties that determine ... |
| pospropd 18249 | Posethood is determined on... |
| odupos 18250 | Being a poset is a self-du... |
| oduposb 18251 | Being a poset is a self-du... |
| pltfval 18253 | Value of the less-than rel... |
| pltval 18254 | Less-than relation. ( ~ d... |
| pltle 18255 | "Less than" implies "less ... |
| pltne 18256 | The "less than" relation i... |
| pltirr 18257 | The "less than" relation i... |
| pleval2i 18258 | One direction of ~ pleval2... |
| pleval2 18259 | "Less than or equal to" in... |
| pltnle 18260 | "Less than" implies not co... |
| pltval3 18261 | Alternate expression for t... |
| pltnlt 18262 | The less-than relation imp... |
| pltn2lp 18263 | The less-than relation has... |
| plttr 18264 | The less-than relation is ... |
| pltletr 18265 | Transitive law for chained... |
| plelttr 18266 | Transitive law for chained... |
| pospo 18267 | Write a poset structure in... |
| lubfval 18272 | Value of the least upper b... |
| lubdm 18273 | Domain of the least upper ... |
| lubfun 18274 | The LUB is a function. (C... |
| lubeldm 18275 | Member of the domain of th... |
| lubelss 18276 | A member of the domain of ... |
| lubeu 18277 | Unique existence proper of... |
| lubval 18278 | Value of the least upper b... |
| lubcl 18279 | The least upper bound func... |
| lubprop 18280 | Properties of greatest low... |
| luble 18281 | The greatest lower bound i... |
| lublecllem 18282 | Lemma for ~ lublecl and ~ ... |
| lublecl 18283 | The set of all elements le... |
| lubid 18284 | The LUB of elements less t... |
| glbfval 18285 | Value of the greatest lowe... |
| glbdm 18286 | Domain of the greatest low... |
| glbfun 18287 | The GLB is a function. (C... |
| glbeldm 18288 | Member of the domain of th... |
| glbelss 18289 | A member of the domain of ... |
| glbeu 18290 | Unique existence proper of... |
| glbval 18291 | Value of the greatest lowe... |
| glbcl 18292 | The least upper bound func... |
| glbprop 18293 | Properties of greatest low... |
| glble 18294 | The greatest lower bound i... |
| joinfval 18295 | Value of join function for... |
| joinfval2 18296 | Value of join function for... |
| joindm 18297 | Domain of join function fo... |
| joindef 18298 | Two ways to say that a joi... |
| joinval 18299 | Join value. Since both si... |
| joincl 18300 | Closure of join of element... |
| joindmss 18301 | Subset property of domain ... |
| joinval2lem 18302 | Lemma for ~ joinval2 and ~... |
| joinval2 18303 | Value of join for a poset ... |
| joineu 18304 | Uniqueness of join of elem... |
| joinlem 18305 | Lemma for join properties.... |
| lejoin1 18306 | A join's first argument is... |
| lejoin2 18307 | A join's second argument i... |
| joinle 18308 | A join is less than or equ... |
| meetfval 18309 | Value of meet function for... |
| meetfval2 18310 | Value of meet function for... |
| meetdm 18311 | Domain of meet function fo... |
| meetdef 18312 | Two ways to say that a mee... |
| meetval 18313 | Meet value. Since both si... |
| meetcl 18314 | Closure of meet of element... |
| meetdmss 18315 | Subset property of domain ... |
| meetval2lem 18316 | Lemma for ~ meetval2 and ~... |
| meetval2 18317 | Value of meet for a poset ... |
| meeteu 18318 | Uniqueness of meet of elem... |
| meetlem 18319 | Lemma for meet properties.... |
| lemeet1 18320 | A meet's first argument is... |
| lemeet2 18321 | A meet's second argument i... |
| meetle 18322 | A meet is less than or equ... |
| joincomALT 18323 | The join of a poset is com... |
| joincom 18324 | The join of a poset is com... |
| meetcomALT 18325 | The meet of a poset is com... |
| meetcom 18326 | The meet of a poset is com... |
| join0 18327 | Lemma for ~ odumeet . (Co... |
| meet0 18328 | Lemma for ~ odujoin . (Co... |
| odulub 18329 | Least upper bounds in a du... |
| odujoin 18330 | Joins in a dual order are ... |
| oduglb 18331 | Greatest lower bounds in a... |
| odumeet 18332 | Meets in a dual order are ... |
| poslubmo 18333 | Least upper bounds in a po... |
| posglbmo 18334 | Greatest lower bounds in a... |
| poslubd 18335 | Properties which determine... |
| poslubdg 18336 | Properties which determine... |
| posglbdg 18337 | Properties which determine... |
| istos 18340 | The predicate "is a toset"... |
| tosso 18341 | Write the totally ordered ... |
| tospos 18342 | A Toset is a Poset. (Cont... |
| tleile 18343 | In a Toset, any two elemen... |
| tltnle 18344 | In a Toset, "less than" is... |
| p0val 18349 | Value of poset zero. (Con... |
| p1val 18350 | Value of poset zero. (Con... |
| p0le 18351 | Any element is less than o... |
| ple1 18352 | Any element is less than o... |
| resspos 18353 | The restriction of a Poset... |
| resstos 18354 | The restriction of a Toset... |
| islat 18357 | The predicate "is a lattic... |
| odulatb 18358 | Being a lattice is self-du... |
| odulat 18359 | Being a lattice is self-du... |
| latcl2 18360 | The join and meet of any t... |
| latlem 18361 | Lemma for lattice properti... |
| latpos 18362 | A lattice is a poset. (Co... |
| latjcl 18363 | Closure of join operation ... |
| latmcl 18364 | Closure of meet operation ... |
| latref 18365 | A lattice ordering is refl... |
| latasymb 18366 | A lattice ordering is asym... |
| latasym 18367 | A lattice ordering is asym... |
| lattr 18368 | A lattice ordering is tran... |
| latasymd 18369 | Deduce equality from latti... |
| lattrd 18370 | A lattice ordering is tran... |
| latjcom 18371 | The join of a lattice comm... |
| latlej1 18372 | A join's first argument is... |
| latlej2 18373 | A join's second argument i... |
| latjle12 18374 | A join is less than or equ... |
| latleeqj1 18375 | "Less than or equal to" in... |
| latleeqj2 18376 | "Less than or equal to" in... |
| latjlej1 18377 | Add join to both sides of ... |
| latjlej2 18378 | Add join to both sides of ... |
| latjlej12 18379 | Add join to both sides of ... |
| latnlej 18380 | An idiom to express that a... |
| latnlej1l 18381 | An idiom to express that a... |
| latnlej1r 18382 | An idiom to express that a... |
| latnlej2 18383 | An idiom to express that a... |
| latnlej2l 18384 | An idiom to express that a... |
| latnlej2r 18385 | An idiom to express that a... |
| latjidm 18386 | Lattice join is idempotent... |
| latmcom 18387 | The join of a lattice comm... |
| latmle1 18388 | A meet is less than or equ... |
| latmle2 18389 | A meet is less than or equ... |
| latlem12 18390 | An element is less than or... |
| latleeqm1 18391 | "Less than or equal to" in... |
| latleeqm2 18392 | "Less than or equal to" in... |
| latmlem1 18393 | Add meet to both sides of ... |
| latmlem2 18394 | Add meet to both sides of ... |
| latmlem12 18395 | Add join to both sides of ... |
| latnlemlt 18396 | Negation of "less than or ... |
| latnle 18397 | Equivalent expressions for... |
| latmidm 18398 | Lattice meet is idempotent... |
| latabs1 18399 | Lattice absorption law. F... |
| latabs2 18400 | Lattice absorption law. F... |
| latledi 18401 | An ortholattice is distrib... |
| latmlej11 18402 | Ordering of a meet and joi... |
| latmlej12 18403 | Ordering of a meet and joi... |
| latmlej21 18404 | Ordering of a meet and joi... |
| latmlej22 18405 | Ordering of a meet and joi... |
| lubsn 18406 | The least upper bound of a... |
| latjass 18407 | Lattice join is associativ... |
| latj12 18408 | Swap 1st and 2nd members o... |
| latj32 18409 | Swap 2nd and 3rd members o... |
| latj13 18410 | Swap 1st and 3rd members o... |
| latj31 18411 | Swap 2nd and 3rd members o... |
| latjrot 18412 | Rotate lattice join of 3 c... |
| latj4 18413 | Rearrangement of lattice j... |
| latj4rot 18414 | Rotate lattice join of 4 c... |
| latjjdi 18415 | Lattice join distributes o... |
| latjjdir 18416 | Lattice join distributes o... |
| mod1ile 18417 | The weak direction of the ... |
| mod2ile 18418 | The weak direction of the ... |
| latmass 18419 | Lattice meet is associativ... |
| latdisdlem 18420 | Lemma for ~ latdisd . (Co... |
| latdisd 18421 | In a lattice, joins distri... |
| isclat 18424 | The predicate "is a comple... |
| clatpos 18425 | A complete lattice is a po... |
| clatlem 18426 | Lemma for properties of a ... |
| clatlubcl 18427 | Any subset of the base set... |
| clatlubcl2 18428 | Any subset of the base set... |
| clatglbcl 18429 | Any subset of the base set... |
| clatglbcl2 18430 | Any subset of the base set... |
| oduclatb 18431 | Being a complete lattice i... |
| clatl 18432 | A complete lattice is a la... |
| isglbd 18433 | Properties that determine ... |
| lublem 18434 | Lemma for the least upper ... |
| lubub 18435 | The LUB of a complete latt... |
| lubl 18436 | The LUB of a complete latt... |
| lubss 18437 | Subset law for least upper... |
| lubel 18438 | An element of a set is les... |
| lubun 18439 | The LUB of a union. (Cont... |
| clatglb 18440 | Properties of greatest low... |
| clatglble 18441 | The greatest lower bound i... |
| clatleglb 18442 | Two ways of expressing "le... |
| clatglbss 18443 | Subset law for greatest lo... |
| isdlat 18446 | Property of being a distri... |
| dlatmjdi 18447 | In a distributive lattice,... |
| dlatl 18448 | A distributive lattice is ... |
| odudlatb 18449 | The dual of a distributive... |
| dlatjmdi 18450 | In a distributive lattice,... |
| ipostr 18453 | The structure of ~ df-ipo ... |
| ipoval 18454 | Value of the inclusion pos... |
| ipobas 18455 | Base set of the inclusion ... |
| ipolerval 18456 | Relation of the inclusion ... |
| ipotset 18457 | Topology of the inclusion ... |
| ipole 18458 | Weak order condition of th... |
| ipolt 18459 | Strict order condition of ... |
| ipopos 18460 | The inclusion poset on a f... |
| isipodrs 18461 | Condition for a family of ... |
| ipodrscl 18462 | Direction by inclusion as ... |
| ipodrsfi 18463 | Finite upper bound propert... |
| fpwipodrs 18464 | The finite subsets of any ... |
| ipodrsima 18465 | The monotone image of a di... |
| isacs3lem 18466 | An algebraic closure syste... |
| acsdrsel 18467 | An algebraic closure syste... |
| isacs4lem 18468 | In a closure system in whi... |
| isacs5lem 18469 | If closure commutes with d... |
| acsdrscl 18470 | In an algebraic closure sy... |
| acsficl 18471 | A closure in an algebraic ... |
| isacs5 18472 | A closure system is algebr... |
| isacs4 18473 | A closure system is algebr... |
| isacs3 18474 | A closure system is algebr... |
| acsficld 18475 | In an algebraic closure sy... |
| acsficl2d 18476 | In an algebraic closure sy... |
| acsfiindd 18477 | In an algebraic closure sy... |
| acsmapd 18478 | In an algebraic closure sy... |
| acsmap2d 18479 | In an algebraic closure sy... |
| acsinfd 18480 | In an algebraic closure sy... |
| acsdomd 18481 | In an algebraic closure sy... |
| acsinfdimd 18482 | In an algebraic closure sy... |
| acsexdimd 18483 | In an algebraic closure sy... |
| mrelatglb 18484 | Greatest lower bounds in a... |
| mrelatglb0 18485 | The empty intersection in ... |
| mrelatlub 18486 | Least upper bounds in a Mo... |
| mreclatBAD 18487 | A Moore space is a complet... |
| isps 18492 | The predicate "is a poset"... |
| psrel 18493 | A poset is a relation. (C... |
| psref2 18494 | A poset is antisymmetric a... |
| pstr2 18495 | A poset is transitive. (C... |
| pslem 18496 | Lemma for ~ psref and othe... |
| psdmrn 18497 | The domain and range of a ... |
| psref 18498 | A poset is reflexive. (Co... |
| psrn 18499 | The range of a poset equal... |
| psasym 18500 | A poset is antisymmetric. ... |
| pstr 18501 | A poset is transitive. (C... |
| cnvps 18502 | The converse of a poset is... |
| cnvpsb 18503 | The converse of a poset is... |
| psss 18504 | Any subset of a partially ... |
| psssdm2 18505 | Field of a subposet. (Con... |
| psssdm 18506 | Field of a subposet. (Con... |
| istsr 18507 | The predicate is a toset. ... |
| istsr2 18508 | The predicate is a toset. ... |
| tsrlin 18509 | A toset is a linear order.... |
| tsrlemax 18510 | Two ways of saying a numbe... |
| tsrps 18511 | A toset is a poset. (Cont... |
| cnvtsr 18512 | The converse of a toset is... |
| tsrss 18513 | Any subset of a totally or... |
| ledm 18514 | The domain of ` <_ ` is ` ... |
| lern 18515 | The range of ` <_ ` is ` R... |
| lefld 18516 | The field of the 'less or ... |
| letsr 18517 | The "less than or equal to... |
| isdir 18522 | A condition for a relation... |
| reldir 18523 | A direction is a relation.... |
| dirdm 18524 | A direction's domain is eq... |
| dirref 18525 | A direction is reflexive. ... |
| dirtr 18526 | A direction is transitive.... |
| dirge 18527 | For any two elements of a ... |
| tsrdir 18528 | A totally ordered set is a... |
| ischn 18531 | Property of being a chain.... |
| chnwrd 18532 | A chain is an ordered sequ... |
| chnltm1 18533 | Basic property of a chain.... |
| pfxchn 18534 | A prefix of a chain is sti... |
| nfchnd 18535 | Bound-variable hypothesis ... |
| chneq1 18536 | Equality theorem for chain... |
| chneq2 18537 | Equality theorem for chain... |
| chneq12 18538 | Equality theorem for chain... |
| chnrss 18539 | Chains under a relation ar... |
| chndss 18540 | Chains with an alphabet ar... |
| chnrdss 18541 | Subset theorem for chains.... |
| chnexg 18542 | Chains with a set given fo... |
| nulchn 18543 | Empty set is an increasing... |
| s1chn 18544 | A singleton word is always... |
| chnind 18545 | Induction over a chain. S... |
| chnub 18546 | In a chain, the last eleme... |
| chnlt 18547 | Compare any two elements i... |
| chnso 18548 | A chain induces a total or... |
| chnccats1 18549 | Extend a chain with a sing... |
| chnccat 18550 | Concatenate two chains. (... |
| chnrev 18551 | Reverse of a chain is chai... |
| chnflenfi 18552 | There is a finite number o... |
| chnf 18553 | A chain is a zero-based fi... |
| chnpof1 18554 | A chain under relation whi... |
| chnpoadomd 18555 | A chain under relation whi... |
| chnpolleha 18556 | A chain under relation whi... |
| chnpolfz 18557 | Provided that chain's rela... |
| chnfi 18558 | There is a finite number o... |
| chninf 18559 | There is an infinite numbe... |
| chnfibg 18560 | Given a partial order, the... |
| ex-chn1 18561 | Example: a doubleton of tw... |
| ex-chn2 18562 | Example: sequence <" ZZ NN... |
| ismgm 18567 | The predicate "is a magma"... |
| ismgmn0 18568 | The predicate "is a magma"... |
| mgmcl 18569 | Closure of the operation o... |
| isnmgm 18570 | A condition for a structur... |
| mgmsscl 18571 | If the base set of a magma... |
| plusffval 18572 | The group addition operati... |
| plusfval 18573 | The group addition operati... |
| plusfeq 18574 | If the addition operation ... |
| plusffn 18575 | The group addition operati... |
| mgmplusf 18576 | The group addition functio... |
| mgmpropd 18577 | If two structures have the... |
| ismgmd 18578 | Deduce a magma from its pr... |
| issstrmgm 18579 | Characterize a substructur... |
| intopsn 18580 | The internal operation for... |
| mgmb1mgm1 18581 | The only magma with a base... |
| mgm0 18582 | Any set with an empty base... |
| mgm0b 18583 | The structure with an empt... |
| mgm1 18584 | The structure with one ele... |
| opifismgm 18585 | A structure with a group a... |
| mgmidmo 18586 | A two-sided identity eleme... |
| grpidval 18587 | The value of the identity ... |
| grpidpropd 18588 | If two structures have the... |
| fn0g 18589 | The group zero extractor i... |
| 0g0 18590 | The identity element funct... |
| ismgmid 18591 | The identity element of a ... |
| mgmidcl 18592 | The identity element of a ... |
| mgmlrid 18593 | The identity element of a ... |
| ismgmid2 18594 | Show that a given element ... |
| lidrideqd 18595 | If there is a left and rig... |
| lidrididd 18596 | If there is a left and rig... |
| grpidd 18597 | Deduce the identity elemen... |
| mgmidsssn0 18598 | Property of the set of ide... |
| grpinvalem 18599 | Lemma for ~ grpinva . (Co... |
| grpinva 18600 | Deduce right inverse from ... |
| grprida 18601 | Deduce right identity from... |
| gsumvalx 18602 | Expand out the substitutio... |
| gsumval 18603 | Expand out the substitutio... |
| gsumpropd 18604 | The group sum depends only... |
| gsumpropd2lem 18605 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18606 | A stronger version of ~ gs... |
| gsummgmpropd 18607 | A stronger version of ~ gs... |
| gsumress 18608 | The group sum in a substru... |
| gsumval1 18609 | Value of the group sum ope... |
| gsum0 18610 | Value of the empty group s... |
| gsumval2a 18611 | Value of the group sum ope... |
| gsumval2 18612 | Value of the group sum ope... |
| gsumsplit1r 18613 | Splitting off the rightmos... |
| gsumprval 18614 | Value of the group sum ope... |
| gsumpr12val 18615 | Value of the group sum ope... |
| mgmhmrcl 18620 | Reverse closure of a magma... |
| submgmrcl 18621 | Reverse closure for submag... |
| ismgmhm 18622 | Property of a magma homomo... |
| mgmhmf 18623 | A magma homomorphism is a ... |
| mgmhmpropd 18624 | Magma homomorphism depends... |
| mgmhmlin 18625 | A magma homomorphism prese... |
| mgmhmf1o 18626 | A magma homomorphism is bi... |
| idmgmhm 18627 | The identity homomorphism ... |
| issubmgm 18628 | Expand definition of a sub... |
| issubmgm2 18629 | Submagmas are subsets that... |
| rabsubmgmd 18630 | Deduction for proving that... |
| submgmss 18631 | Submagmas are subsets of t... |
| submgmid 18632 | Every magma is trivially a... |
| submgmcl 18633 | Submagmas are closed under... |
| submgmmgm 18634 | Submagmas are themselves m... |
| submgmbas 18635 | The base set of a submagma... |
| subsubmgm 18636 | A submagma of a submagma i... |
| resmgmhm 18637 | Restriction of a magma hom... |
| resmgmhm2 18638 | One direction of ~ resmgmh... |
| resmgmhm2b 18639 | Restriction of the codomai... |
| mgmhmco 18640 | The composition of magma h... |
| mgmhmima 18641 | The homomorphic image of a... |
| mgmhmeql 18642 | The equalizer of two magma... |
| submgmacs 18643 | Submagmas are an algebraic... |
| issgrp 18646 | The predicate "is a semigr... |
| issgrpv 18647 | The predicate "is a semigr... |
| issgrpn0 18648 | The predicate "is a semigr... |
| isnsgrp 18649 | A condition for a structur... |
| sgrpmgm 18650 | A semigroup is a magma. (... |
| sgrpass 18651 | A semigroup operation is a... |
| sgrpcl 18652 | Closure of the operation o... |
| sgrp0 18653 | Any set with an empty base... |
| sgrp0b 18654 | The structure with an empt... |
| sgrp1 18655 | The structure with one ele... |
| issgrpd 18656 | Deduce a semigroup from it... |
| sgrppropd 18657 | If two structures are sets... |
| prdsplusgsgrpcl 18658 | Structure product pointwis... |
| prdssgrpd 18659 | The product of a family of... |
| ismnddef 18662 | The predicate "is a monoid... |
| ismnd 18663 | The predicate "is a monoid... |
| isnmnd 18664 | A condition for a structur... |
| sgrpidmnd 18665 | A semigroup with an identi... |
| mndsgrp 18666 | A monoid is a semigroup. ... |
| mndmgm 18667 | A monoid is a magma. (Con... |
| mndcl 18668 | Closure of the operation o... |
| mndass 18669 | A monoid operation is asso... |
| mndid 18670 | A monoid has a two-sided i... |
| mndideu 18671 | The two-sided identity ele... |
| mnd32g 18672 | Commutative/associative la... |
| mnd12g 18673 | Commutative/associative la... |
| mnd4g 18674 | Commutative/associative la... |
| mndidcl 18675 | The identity element of a ... |
| mndbn0 18676 | The base set of a monoid i... |
| hashfinmndnn 18677 | A finite monoid has positi... |
| mndplusf 18678 | The group addition operati... |
| mndlrid 18679 | A monoid's identity elemen... |
| mndlid 18680 | The identity element of a ... |
| mndrid 18681 | The identity element of a ... |
| ismndd 18682 | Deduce a monoid from its p... |
| mndpfo 18683 | The addition operation of ... |
| mndfo 18684 | The addition operation of ... |
| mndpropd 18685 | If two structures have the... |
| mndprop 18686 | If two structures have the... |
| issubmnd 18687 | Characterize a submonoid b... |
| ress0g 18688 | ` 0g ` is unaffected by re... |
| submnd0 18689 | The zero of a submonoid is... |
| mndinvmod 18690 | Uniqueness of an inverse e... |
| mndpsuppss 18691 | The support of a mapping o... |
| mndpsuppfi 18692 | The support of a mapping o... |
| mndpfsupp 18693 | A mapping of a scalar mult... |
| prdsplusgcl 18694 | Structure product pointwis... |
| prdsidlem 18695 | Characterization of identi... |
| prdsmndd 18696 | The product of a family of... |
| prds0g 18697 | The identity in a product ... |
| pwsmnd 18698 | The structure power of a m... |
| pws0g 18699 | The identity in a structur... |
| imasmnd2 18700 | The image structure of a m... |
| imasmnd 18701 | The image structure of a m... |
| imasmndf1 18702 | The image of a monoid unde... |
| xpsmnd 18703 | The binary product of mono... |
| xpsmnd0 18704 | The identity element of a ... |
| mnd1 18705 | The (smallest) structure r... |
| mnd1id 18706 | The singleton element of a... |
| ismhm 18711 | Property of a monoid homom... |
| ismhmd 18712 | Deduction version of ~ ism... |
| mhmrcl1 18713 | Reverse closure of a monoi... |
| mhmrcl2 18714 | Reverse closure of a monoi... |
| mhmf 18715 | A monoid homomorphism is a... |
| ismhm0 18716 | Property of a monoid homom... |
| mhmismgmhm 18717 | Each monoid homomorphism i... |
| mhmpropd 18718 | Monoid homomorphism depend... |
| mhmlin 18719 | A monoid homomorphism comm... |
| mhm0 18720 | A monoid homomorphism pres... |
| idmhm 18721 | The identity homomorphism ... |
| mhmf1o 18722 | A monoid homomorphism is b... |
| mndvcl 18723 | Tuple-wise additive closur... |
| mndvass 18724 | Tuple-wise associativity i... |
| mndvlid 18725 | Tuple-wise left identity i... |
| mndvrid 18726 | Tuple-wise right identity ... |
| mhmvlin 18727 | Tuple extension of monoid ... |
| submrcl 18728 | Reverse closure for submon... |
| issubm 18729 | Expand definition of a sub... |
| issubm2 18730 | Submonoids are subsets tha... |
| issubmndb 18731 | The submonoid predicate. ... |
| issubmd 18732 | Deduction for proving a su... |
| mndissubm 18733 | If the base set of a monoi... |
| resmndismnd 18734 | If the base set of a monoi... |
| submss 18735 | Submonoids are subsets of ... |
| submid 18736 | Every monoid is trivially ... |
| subm0cl 18737 | Submonoids contain zero. ... |
| submcl 18738 | Submonoids are closed unde... |
| submmnd 18739 | Submonoids are themselves ... |
| submbas 18740 | The base set of a submonoi... |
| subm0 18741 | Submonoids have the same i... |
| subsubm 18742 | A submonoid of a submonoid... |
| 0subm 18743 | The zero submonoid of an a... |
| insubm 18744 | The intersection of two su... |
| 0mhm 18745 | The constant zero linear f... |
| resmhm 18746 | Restriction of a monoid ho... |
| resmhm2 18747 | One direction of ~ resmhm2... |
| resmhm2b 18748 | Restriction of the codomai... |
| mhmco 18749 | The composition of monoid ... |
| mhmimalem 18750 | Lemma for ~ mhmima and sim... |
| mhmima 18751 | The homomorphic image of a... |
| mhmeql 18752 | The equalizer of two monoi... |
| submacs 18753 | Submonoids are an algebrai... |
| mndind 18754 | Induction in a monoid. In... |
| prdspjmhm 18755 | A projection from a produc... |
| pwspjmhm 18756 | A projection from a struct... |
| pwsdiagmhm 18757 | Diagonal monoid homomorphi... |
| pwsco1mhm 18758 | Right composition with a f... |
| pwsco2mhm 18759 | Left composition with a mo... |
| gsumvallem2 18760 | Lemma for properties of th... |
| gsumsubm 18761 | Evaluate a group sum in a ... |
| gsumz 18762 | Value of a group sum over ... |
| gsumwsubmcl 18763 | Closure of the composite i... |
| gsumws1 18764 | A singleton composite reco... |
| gsumwcl 18765 | Closure of the composite o... |
| gsumsgrpccat 18766 | Homomorphic property of no... |
| gsumccat 18767 | Homomorphic property of co... |
| gsumws2 18768 | Valuation of a pair in a m... |
| gsumccatsn 18769 | Homomorphic property of co... |
| gsumspl 18770 | The primary purpose of the... |
| gsumwmhm 18771 | Behavior of homomorphisms ... |
| gsumwspan 18772 | The submonoid generated by... |
| frmdval 18777 | Value of the free monoid c... |
| frmdbas 18778 | The base set of a free mon... |
| frmdelbas 18779 | An element of the base set... |
| frmdplusg 18780 | The monoid operation of a ... |
| frmdadd 18781 | Value of the monoid operat... |
| vrmdfval 18782 | The canonical injection fr... |
| vrmdval 18783 | The value of the generatin... |
| vrmdf 18784 | The mapping from the index... |
| frmdmnd 18785 | A free monoid is a monoid.... |
| frmd0 18786 | The identity of the free m... |
| frmdsssubm 18787 | The set of words taking va... |
| frmdgsum 18788 | Any word in a free monoid ... |
| frmdss2 18789 | A subset of generators is ... |
| frmdup1 18790 | Any assignment of the gene... |
| frmdup2 18791 | The evaluation map has the... |
| frmdup3lem 18792 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18793 | Universal property of the ... |
| efmnd 18796 | The monoid of endofunction... |
| efmndbas 18797 | The base set of the monoid... |
| efmndbasabf 18798 | The base set of the monoid... |
| elefmndbas 18799 | Two ways of saying a funct... |
| elefmndbas2 18800 | Two ways of saying a funct... |
| efmndbasf 18801 | Elements in the monoid of ... |
| efmndhash 18802 | The monoid of endofunction... |
| efmndbasfi 18803 | The monoid of endofunction... |
| efmndfv 18804 | The function value of an e... |
| efmndtset 18805 | The topology of the monoid... |
| efmndplusg 18806 | The group operation of a m... |
| efmndov 18807 | The value of the group ope... |
| efmndcl 18808 | The group operation of the... |
| efmndtopn 18809 | The topology of the monoid... |
| symggrplem 18810 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18811 | The monoid of endofunction... |
| efmndsgrp 18812 | The monoid of endofunction... |
| ielefmnd 18813 | The identity function rest... |
| efmndid 18814 | The identity function rest... |
| efmndmnd 18815 | The monoid of endofunction... |
| efmnd0nmnd 18816 | Even the monoid of endofun... |
| efmndbas0 18817 | The base set of the monoid... |
| efmnd1hash 18818 | The monoid of endofunction... |
| efmnd1bas 18819 | The monoid of endofunction... |
| efmnd2hash 18820 | The monoid of endofunction... |
| submefmnd 18821 | If the base set of a monoi... |
| sursubmefmnd 18822 | The set of surjective endo... |
| injsubmefmnd 18823 | The set of injective endof... |
| idressubmefmnd 18824 | The singleton containing o... |
| idresefmnd 18825 | The structure with the sin... |
| smndex1ibas 18826 | The modulo function ` I ` ... |
| smndex1iidm 18827 | The modulo function ` I ` ... |
| smndex1gbas 18828 | The constant functions ` (... |
| smndex1gbasOLD 18829 | Obsolete version of ~ smnd... |
| smndex1gid 18830 | The composition of a const... |
| smndex1gidOLD 18831 | Obsolete version of ~ smnd... |
| smndex1igid 18832 | The composition of the mod... |
| smndex1igidOLD 18833 | Obsolete version of ~ smnd... |
| smndex1basss 18834 | The modulo function ` I ` ... |
| smndex1bas 18835 | The base set of the monoid... |
| smndex1mgm 18836 | The monoid of endofunction... |
| smndex1sgrp 18837 | The monoid of endofunction... |
| smndex1mndlem 18838 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18839 | The monoid of endofunction... |
| smndex1id 18840 | The modulo function ` I ` ... |
| smndex1n0mnd 18841 | The identity of the monoid... |
| nsmndex1 18842 | The base set ` B ` of the ... |
| smndex2dbas 18843 | The doubling function ` D ... |
| smndex2dnrinv 18844 | The doubling function ` D ... |
| smndex2hbas 18845 | The halving functions ` H ... |
| smndex2dlinvh 18846 | The halving functions ` H ... |
| mgm2nsgrplem1 18847 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18848 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18849 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18850 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18851 | A small magma (with two el... |
| sgrp2nmndlem1 18852 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18853 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18854 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18855 | A small semigroup (with tw... |
| sgrp2rid2ex 18856 | A small semigroup (with tw... |
| sgrp2nmndlem4 18857 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18858 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18859 | A small semigroup (with tw... |
| mgmnsgrpex 18860 | There is a magma which is ... |
| sgrpnmndex 18861 | There is a semigroup which... |
| sgrpssmgm 18862 | The class of all semigroup... |
| mndsssgrp 18863 | The class of all monoids i... |
| pwmndgplus 18864 | The operation of the monoi... |
| pwmndid 18865 | The identity of the monoid... |
| pwmnd 18866 | The power set of a class `... |
| isgrp 18873 | The predicate "is a group"... |
| grpmnd 18874 | A group is a monoid. (Con... |
| grpcl 18875 | Closure of the operation o... |
| grpass 18876 | A group operation is assoc... |
| grpinvex 18877 | Every member of a group ha... |
| grpideu 18878 | The two-sided identity ele... |
| grpassd 18879 | A group operation is assoc... |
| grpmndd 18880 | A group is a monoid. (Con... |
| grpcld 18881 | Closure of the operation o... |
| grpplusf 18882 | The group addition operati... |
| grpplusfo 18883 | The group addition operati... |
| resgrpplusfrn 18884 | The underlying set of a gr... |
| grppropd 18885 | If two structures have the... |
| grpprop 18886 | If two structures have the... |
| grppropstr 18887 | Generalize a specific 2-el... |
| grpss 18888 | Show that a structure exte... |
| isgrpd2e 18889 | Deduce a group from its pr... |
| isgrpd2 18890 | Deduce a group from its pr... |
| isgrpde 18891 | Deduce a group from its pr... |
| isgrpd 18892 | Deduce a group from its pr... |
| isgrpi 18893 | Properties that determine ... |
| grpsgrp 18894 | A group is a semigroup. (... |
| grpmgmd 18895 | A group is a magma, deduct... |
| dfgrp2 18896 | Alternate definition of a ... |
| dfgrp2e 18897 | Alternate definition of a ... |
| isgrpix 18898 | Properties that determine ... |
| grpidcl 18899 | The identity element of a ... |
| grpbn0 18900 | The base set of a group is... |
| grplid 18901 | The identity element of a ... |
| grprid 18902 | The identity element of a ... |
| grplidd 18903 | The identity element of a ... |
| grpridd 18904 | The identity element of a ... |
| grpn0 18905 | A group is not empty. (Co... |
| hashfingrpnn 18906 | A finite group has positiv... |
| grprcan 18907 | Right cancellation law for... |
| grpinveu 18908 | The left inverse element o... |
| grpid 18909 | Two ways of saying that an... |
| isgrpid2 18910 | Properties showing that an... |
| grpidd2 18911 | Deduce the identity elemen... |
| grpinvfval 18912 | The inverse function of a ... |
| grpinvfvalALT 18913 | Shorter proof of ~ grpinvf... |
| grpinvval 18914 | The inverse of a group ele... |
| grpinvfn 18915 | Functionality of the group... |
| grpinvfvi 18916 | The group inverse function... |
| grpsubfval 18917 | Group subtraction (divisio... |
| grpsubfvalALT 18918 | Shorter proof of ~ grpsubf... |
| grpsubval 18919 | Group subtraction (divisio... |
| grpinvf 18920 | The group inversion operat... |
| grpinvcl 18921 | A group element's inverse ... |
| grpinvcld 18922 | A group element's inverse ... |
| grplinv 18923 | The left inverse of a grou... |
| grprinv 18924 | The right inverse of a gro... |
| grpinvid1 18925 | The inverse of a group ele... |
| grpinvid2 18926 | The inverse of a group ele... |
| isgrpinv 18927 | Properties showing that a ... |
| grplinvd 18928 | The left inverse of a grou... |
| grprinvd 18929 | The right inverse of a gro... |
| grplrinv 18930 | In a group, every member h... |
| grpidinv2 18931 | A group's properties using... |
| grpidinv 18932 | A group has a left and rig... |
| grpinvid 18933 | The inverse of the identit... |
| grplcan 18934 | Left cancellation law for ... |
| grpasscan1 18935 | An associative cancellatio... |
| grpasscan2 18936 | An associative cancellatio... |
| grpidrcan 18937 | If right adding an element... |
| grpidlcan 18938 | If left adding an element ... |
| grpinvinv 18939 | Double inverse law for gro... |
| grpinvcnv 18940 | The group inverse is its o... |
| grpinv11 18941 | The group inverse is one-t... |
| grpinv11OLD 18942 | Obsolete version of ~ grpi... |
| grpinvf1o 18943 | The group inverse is a one... |
| grpinvnz 18944 | The inverse of a nonzero g... |
| grpinvnzcl 18945 | The inverse of a nonzero g... |
| grpsubinv 18946 | Subtraction of an inverse.... |
| grplmulf1o 18947 | Left multiplication by a g... |
| grpraddf1o 18948 | Right addition by a group ... |
| grpinvpropd 18949 | If two structures have the... |
| grpidssd 18950 | If the base set of a group... |
| grpinvssd 18951 | If the base set of a group... |
| grpinvadd 18952 | The inverse of the group o... |
| grpsubf 18953 | Functionality of group sub... |
| grpsubcl 18954 | Closure of group subtracti... |
| grpsubrcan 18955 | Right cancellation law for... |
| grpinvsub 18956 | Inverse of a group subtrac... |
| grpinvval2 18957 | A ~ df-neg -like equation ... |
| grpsubid 18958 | Subtraction of a group ele... |
| grpsubid1 18959 | Subtraction of the identit... |
| grpsubeq0 18960 | If the difference between ... |
| grpsubadd0sub 18961 | Subtraction expressed as a... |
| grpsubadd 18962 | Relationship between group... |
| grpsubsub 18963 | Double group subtraction. ... |
| grpaddsubass 18964 | Associative-type law for g... |
| grppncan 18965 | Cancellation law for subtr... |
| grpnpcan 18966 | Cancellation law for subtr... |
| grpsubsub4 18967 | Double group subtraction (... |
| grppnpcan2 18968 | Cancellation law for mixed... |
| grpnpncan 18969 | Cancellation law for group... |
| grpnpncan0 18970 | Cancellation law for group... |
| grpnnncan2 18971 | Cancellation law for group... |
| dfgrp3lem 18972 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18973 | Alternate definition of a ... |
| dfgrp3e 18974 | Alternate definition of a ... |
| grplactfval 18975 | The left group action of e... |
| grplactval 18976 | The value of the left grou... |
| grplactcnv 18977 | The left group action of e... |
| grplactf1o 18978 | The left group action of e... |
| grpsubpropd 18979 | Weak property deduction fo... |
| grpsubpropd2 18980 | Strong property deduction ... |
| grp1 18981 | The (smallest) structure r... |
| grp1inv 18982 | The inverse function of th... |
| prdsinvlem 18983 | Characterization of invers... |
| prdsgrpd 18984 | The product of a family of... |
| prdsinvgd 18985 | Negation in a product of g... |
| pwsgrp 18986 | A structure power of a gro... |
| pwsinvg 18987 | Negation in a group power.... |
| pwssub 18988 | Subtraction in a group pow... |
| imasgrp2 18989 | The image structure of a g... |
| imasgrp 18990 | The image structure of a g... |
| imasgrpf1 18991 | The image of a group under... |
| qusgrp2 18992 | Prove that a quotient stru... |
| xpsgrp 18993 | The binary product of grou... |
| xpsinv 18994 | Value of the negation oper... |
| xpsgrpsub 18995 | Value of the subtraction o... |
| mhmlem 18996 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18997 | A surjective monoid morphi... |
| mhmmnd 18998 | The image of a monoid ` G ... |
| mhmfmhm 18999 | The function fulfilling th... |
| ghmgrp 19000 | The image of a group ` G `... |
| mulgfval 19003 | Group multiple (exponentia... |
| mulgfvalALT 19004 | Shorter proof of ~ mulgfva... |
| mulgval 19005 | Value of the group multipl... |
| mulgfn 19006 | Functionality of the group... |
| mulgfvi 19007 | The group multiple operati... |
| mulg0 19008 | Group multiple (exponentia... |
| mulgnn 19009 | Group multiple (exponentia... |
| ressmulgnn 19010 | Values for the group multi... |
| ressmulgnn0 19011 | Values for the group multi... |
| ressmulgnnd 19012 | Values for the group multi... |
| mulgnngsum 19013 | Group multiple (exponentia... |
| mulgnn0gsum 19014 | Group multiple (exponentia... |
| mulg1 19015 | Group multiple (exponentia... |
| mulgnnp1 19016 | Group multiple (exponentia... |
| mulg2 19017 | Group multiple (exponentia... |
| mulgnegnn 19018 | Group multiple (exponentia... |
| mulgnn0p1 19019 | Group multiple (exponentia... |
| mulgnnsubcl 19020 | Closure of the group multi... |
| mulgnn0subcl 19021 | Closure of the group multi... |
| mulgsubcl 19022 | Closure of the group multi... |
| mulgnncl 19023 | Closure of the group multi... |
| mulgnn0cl 19024 | Closure of the group multi... |
| mulgcl 19025 | Closure of the group multi... |
| mulgneg 19026 | Group multiple (exponentia... |
| mulgnegneg 19027 | The inverse of a negative ... |
| mulgm1 19028 | Group multiple (exponentia... |
| mulgnn0cld 19029 | Closure of the group multi... |
| mulgcld 19030 | Deduction associated with ... |
| mulgaddcomlem 19031 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19032 | The group multiple operato... |
| mulginvcom 19033 | The group multiple operato... |
| mulginvinv 19034 | The group multiple operato... |
| mulgnn0z 19035 | A group multiple of the id... |
| mulgz 19036 | A group multiple of the id... |
| mulgnndir 19037 | Sum of group multiples, fo... |
| mulgnn0dir 19038 | Sum of group multiples, ge... |
| mulgdirlem 19039 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19040 | Sum of group multiples, ge... |
| mulgp1 19041 | Group multiple (exponentia... |
| mulgneg2 19042 | Group multiple (exponentia... |
| mulgnnass 19043 | Product of group multiples... |
| mulgnn0ass 19044 | Product of group multiples... |
| mulgass 19045 | Product of group multiples... |
| mulgassr 19046 | Reversed product of group ... |
| mulgmodid 19047 | Casting out multiples of t... |
| mulgsubdir 19048 | Distribution of group mult... |
| mhmmulg 19049 | A homomorphism of monoids ... |
| mulgpropd 19050 | Two structures with the sa... |
| submmulgcl 19051 | Closure of the group multi... |
| submmulg 19052 | A group multiple is the sa... |
| pwsmulg 19053 | Value of a group multiple ... |
| issubg 19060 | The subgroup predicate. (... |
| subgss 19061 | A subgroup is a subset. (... |
| subgid 19062 | A group is a subgroup of i... |
| subggrp 19063 | A subgroup is a group. (C... |
| subgbas 19064 | The base of the restricted... |
| subgrcl 19065 | Reverse closure for the su... |
| subg0 19066 | A subgroup of a group must... |
| subginv 19067 | The inverse of an element ... |
| subg0cl 19068 | The group identity is an e... |
| subginvcl 19069 | The inverse of an element ... |
| subgcl 19070 | A subgroup is closed under... |
| subgsubcl 19071 | A subgroup is closed under... |
| subgsub 19072 | The subtraction of element... |
| subgmulgcl 19073 | Closure of the group multi... |
| subgmulg 19074 | A group multiple is the sa... |
| issubg2 19075 | Characterize the subgroups... |
| issubgrpd2 19076 | Prove a subgroup by closur... |
| issubgrpd 19077 | Prove a subgroup by closur... |
| issubg3 19078 | A subgroup is a symmetric ... |
| issubg4 19079 | A subgroup is a nonempty s... |
| grpissubg 19080 | If the base set of a group... |
| resgrpisgrp 19081 | If the base set of a group... |
| subgsubm 19082 | A subgroup is a submonoid.... |
| subsubg 19083 | A subgroup of a subgroup i... |
| subgint 19084 | The intersection of a none... |
| 0subg 19085 | The zero subgroup of an ar... |
| trivsubgd 19086 | The only subgroup of a tri... |
| trivsubgsnd 19087 | The only subgroup of a tri... |
| isnsg 19088 | Property of being a normal... |
| isnsg2 19089 | Weaken the condition of ~ ... |
| nsgbi 19090 | Defining property of a nor... |
| nsgsubg 19091 | A normal subgroup is a sub... |
| nsgconj 19092 | The conjugation of an elem... |
| isnsg3 19093 | A subgroup is normal iff t... |
| subgacs 19094 | Subgroups are an algebraic... |
| nsgacs 19095 | Normal subgroups form an a... |
| elnmz 19096 | Elementhood in the normali... |
| nmzbi 19097 | Defining property of the n... |
| nmzsubg 19098 | The normalizer N_G(S) of a... |
| ssnmz 19099 | A subgroup is a subset of ... |
| isnsg4 19100 | A subgroup is normal iff i... |
| nmznsg 19101 | Any subgroup is a normal s... |
| 0nsg 19102 | The zero subgroup is norma... |
| nsgid 19103 | The whole group is a norma... |
| 0idnsgd 19104 | The whole group and the ze... |
| trivnsgd 19105 | The only normal subgroup o... |
| triv1nsgd 19106 | A trivial group has exactl... |
| 1nsgtrivd 19107 | A group with exactly one n... |
| releqg 19108 | The left coset equivalence... |
| eqgfval 19109 | Value of the subgroup left... |
| eqgval 19110 | Value of the subgroup left... |
| eqger 19111 | The subgroup coset equival... |
| eqglact 19112 | A left coset can be expres... |
| eqgid 19113 | The left coset containing ... |
| eqgen 19114 | Each coset is equipotent t... |
| eqgcpbl 19115 | The subgroup coset equival... |
| eqg0el 19116 | Equivalence class of a quo... |
| quselbas 19117 | Membership in the base set... |
| quseccl0 19118 | Closure of the quotient ma... |
| qusgrp 19119 | If ` Y ` is a normal subgr... |
| quseccl 19120 | Closure of the quotient ma... |
| qusadd 19121 | Value of the group operati... |
| qus0 19122 | Value of the group identit... |
| qusinv 19123 | Value of the group inverse... |
| qussub 19124 | Value of the group subtrac... |
| ecqusaddd 19125 | Addition of equivalence cl... |
| ecqusaddcl 19126 | Closure of the addition in... |
| lagsubg2 19127 | Lagrange's theorem for fin... |
| lagsubg 19128 | Lagrange's theorem for Gro... |
| eqg0subg 19129 | The coset equivalence rela... |
| eqg0subgecsn 19130 | The equivalence classes mo... |
| qus0subgbas 19131 | The base set of a quotient... |
| qus0subgadd 19132 | The addition in a quotient... |
| cycsubmel 19133 | Characterization of an ele... |
| cycsubmcl 19134 | The set of nonnegative int... |
| cycsubm 19135 | The set of nonnegative int... |
| cyccom 19136 | Condition for an operation... |
| cycsubmcom 19137 | The operation of a monoid ... |
| cycsubggend 19138 | The cyclic subgroup genera... |
| cycsubgcl 19139 | The set of integer powers ... |
| cycsubgss 19140 | The cyclic subgroup genera... |
| cycsubg 19141 | The cyclic group generated... |
| cycsubgcld 19142 | The cyclic subgroup genera... |
| cycsubg2 19143 | The subgroup generated by ... |
| cycsubg2cl 19144 | Any multiple of an element... |
| reldmghm 19147 | Lemma for group homomorphi... |
| isghm 19148 | Property of being a homomo... |
| isghmOLD 19149 | Obsolete version of ~ isgh... |
| isghm3 19150 | Property of a group homomo... |
| ghmgrp1 19151 | A group homomorphism is on... |
| ghmgrp2 19152 | A group homomorphism is on... |
| ghmf 19153 | A group homomorphism is a ... |
| ghmlin 19154 | A homomorphism of groups i... |
| ghmid 19155 | A homomorphism of groups p... |
| ghminv 19156 | A homomorphism of groups p... |
| ghmsub 19157 | Linearity of subtraction t... |
| isghmd 19158 | Deduction for a group homo... |
| ghmmhm 19159 | A group homomorphism is a ... |
| ghmmhmb 19160 | Group homomorphisms and mo... |
| ghmmulg 19161 | A group homomorphism prese... |
| ghmrn 19162 | The range of a homomorphis... |
| 0ghm 19163 | The constant zero linear f... |
| idghm 19164 | The identity homomorphism ... |
| resghm 19165 | Restriction of a homomorph... |
| resghm2 19166 | One direction of ~ resghm2... |
| resghm2b 19167 | Restriction of the codomai... |
| ghmghmrn 19168 | A group homomorphism from ... |
| ghmco 19169 | The composition of group h... |
| ghmima 19170 | The image of a subgroup un... |
| ghmpreima 19171 | The inverse image of a sub... |
| ghmeql 19172 | The equalizer of two group... |
| ghmnsgima 19173 | The image of a normal subg... |
| ghmnsgpreima 19174 | The inverse image of a nor... |
| ghmker 19175 | The kernel of a homomorphi... |
| ghmeqker 19176 | Two source points map to t... |
| pwsdiagghm 19177 | Diagonal homomorphism into... |
| f1ghm0to0 19178 | If a group homomorphism ` ... |
| ghmf1 19179 | Two ways of saying a group... |
| kerf1ghm 19180 | A group homomorphism ` F `... |
| ghmf1o 19181 | A bijective group homomorp... |
| conjghm 19182 | Conjugation is an automorp... |
| conjsubg 19183 | A conjugated subgroup is a... |
| conjsubgen 19184 | A conjugated subgroup is e... |
| conjnmz 19185 | A subgroup is unchanged un... |
| conjnmzb 19186 | Alternative condition for ... |
| conjnsg 19187 | A normal subgroup is uncha... |
| qusghm 19188 | If ` Y ` is a normal subgr... |
| ghmpropd 19189 | Group homomorphism depends... |
| gimfn 19194 | The group isomorphism func... |
| isgim 19195 | An isomorphism of groups i... |
| gimf1o 19196 | An isomorphism of groups i... |
| gimghm 19197 | An isomorphism of groups i... |
| isgim2 19198 | A group isomorphism is a h... |
| subggim 19199 | Behavior of subgroups unde... |
| gimcnv 19200 | The converse of a group is... |
| gimco 19201 | The composition of group i... |
| gim0to0 19202 | A group isomorphism maps t... |
| brgic 19203 | The relation "is isomorphi... |
| brgici 19204 | Prove isomorphic by an exp... |
| gicref 19205 | Isomorphism is reflexive. ... |
| giclcl 19206 | Isomorphism implies the le... |
| gicrcl 19207 | Isomorphism implies the ri... |
| gicsym 19208 | Isomorphism is symmetric. ... |
| gictr 19209 | Isomorphism is transitive.... |
| gicer 19210 | Isomorphism is an equivale... |
| gicen 19211 | Isomorphic groups have equ... |
| gicsubgen 19212 | A less trivial example of ... |
| ghmqusnsglem1 19213 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsglem2 19214 | Lemma for ~ ghmqusnsg . (... |
| ghmqusnsg 19215 | The mapping ` H ` induced ... |
| ghmquskerlem1 19216 | Lemma for ~ ghmqusker . (... |
| ghmquskerco 19217 | In the case of theorem ~ g... |
| ghmquskerlem2 19218 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 19219 | The mapping ` H ` induced ... |
| ghmqusker 19220 | A surjective group homomor... |
| gicqusker 19221 | The image ` H ` of a group... |
| isga 19224 | The predicate "is a (left)... |
| gagrp 19225 | The left argument of a gro... |
| gaset 19226 | The right argument of a gr... |
| gagrpid 19227 | The identity of the group ... |
| gaf 19228 | The mapping of the group a... |
| gafo 19229 | A group action is onto its... |
| gaass 19230 | An "associative" property ... |
| ga0 19231 | The action of a group on t... |
| gaid 19232 | The trivial action of a gr... |
| subgga 19233 | A subgroup acts on its par... |
| gass 19234 | A subset of a group action... |
| gasubg 19235 | The restriction of a group... |
| gaid2 19236 | A group operation is a lef... |
| galcan 19237 | The action of a particular... |
| gacan 19238 | Group inverses cancel in a... |
| gapm 19239 | The action of a particular... |
| gaorb 19240 | The orbit equivalence rela... |
| gaorber 19241 | The orbit equivalence rela... |
| gastacl 19242 | The stabilizer subgroup in... |
| gastacos 19243 | Write the coset relation f... |
| orbstafun 19244 | Existence and uniqueness f... |
| orbstaval 19245 | Value of the function at a... |
| orbsta 19246 | The Orbit-Stabilizer theor... |
| orbsta2 19247 | Relation between the size ... |
| cntrval 19252 | Substitute definition of t... |
| cntzfval 19253 | First level substitution f... |
| cntzval 19254 | Definition substitution fo... |
| elcntz 19255 | Elementhood in the central... |
| cntzel 19256 | Membership in a centralize... |
| cntzsnval 19257 | Special substitution for t... |
| elcntzsn 19258 | Value of the centralizer o... |
| sscntz 19259 | A centralizer expression f... |
| cntzrcl 19260 | Reverse closure for elemen... |
| cntzssv 19261 | The centralizer is uncondi... |
| cntzi 19262 | Membership in a centralize... |
| elcntr 19263 | Elementhood in the center ... |
| cntrss 19264 | The center is a subset of ... |
| cntri 19265 | Defining property of the c... |
| resscntz 19266 | Centralizer in a substruct... |
| cntzsgrpcl 19267 | Centralizers are closed un... |
| cntz2ss 19268 | Centralizers reverse the s... |
| cntzrec 19269 | Reciprocity relationship f... |
| cntziinsn 19270 | Express any centralizer as... |
| cntzsubm 19271 | Centralizers in a monoid a... |
| cntzsubg 19272 | Centralizers in a group ar... |
| cntzidss 19273 | If the elements of ` S ` c... |
| cntzmhm 19274 | Centralizers in a monoid a... |
| cntzmhm2 19275 | Centralizers in a monoid a... |
| cntrsubgnsg 19276 | A central subgroup is norm... |
| cntrnsg 19277 | The center of a group is a... |
| oppgval 19280 | Value of the opposite grou... |
| oppgplusfval 19281 | Value of the addition oper... |
| oppgplus 19282 | Value of the addition oper... |
| setsplusg 19283 | The other components of an... |
| oppgbas 19284 | Base set of an opposite gr... |
| oppgtset 19285 | Topology of an opposite gr... |
| oppgtopn 19286 | Topology of an opposite gr... |
| oppgmnd 19287 | The opposite of a monoid i... |
| oppgmndb 19288 | Bidirectional form of ~ op... |
| oppgid 19289 | Zero in a monoid is a symm... |
| oppggrp 19290 | The opposite of a group is... |
| oppggrpb 19291 | Bidirectional form of ~ op... |
| oppginv 19292 | Inverses in a group are a ... |
| invoppggim 19293 | The inverse is an antiauto... |
| oppggic 19294 | Every group is (naturally)... |
| oppgsubm 19295 | Being a submonoid is a sym... |
| oppgsubg 19296 | Being a subgroup is a symm... |
| oppgcntz 19297 | A centralizer in a group i... |
| oppgcntr 19298 | The center of a group is t... |
| gsumwrev 19299 | A sum in an opposite monoi... |
| oppgle 19300 | less-than relation of an o... |
| oppglt 19301 | less-than relation of an o... |
| symgval 19304 | The value of the symmetric... |
| symgbas 19305 | The base set of the symmet... |
| elsymgbas2 19306 | Two ways of saying a funct... |
| elsymgbas 19307 | Two ways of saying a funct... |
| symgbasf1o 19308 | Elements in the symmetric ... |
| symgbasf 19309 | A permutation (element of ... |
| symgbasmap 19310 | A permutation (element of ... |
| symghash 19311 | The symmetric group on ` n... |
| symgbasfi 19312 | The symmetric group on a f... |
| symgfv 19313 | The function value of a pe... |
| symgfvne 19314 | The function values of a p... |
| symgressbas 19315 | The symmetric group on ` A... |
| symgplusg 19316 | The group operation of a s... |
| symgov 19317 | The value of the group ope... |
| symgcl 19318 | The group operation of the... |
| idresperm 19319 | The identity function rest... |
| symgmov1 19320 | For a permutation of a set... |
| symgmov2 19321 | For a permutation of a set... |
| symgbas0 19322 | The base set of the symmet... |
| symg1hash 19323 | The symmetric group on a s... |
| symg1bas 19324 | The symmetric group on a s... |
| symg2hash 19325 | The symmetric group on a (... |
| symg2bas 19326 | The symmetric group on a p... |
| 0symgefmndeq 19327 | The symmetric group on the... |
| snsymgefmndeq 19328 | The symmetric group on a s... |
| symgpssefmnd 19329 | For a set ` A ` with more ... |
| symgvalstruct 19330 | The value of the symmetric... |
| symgsubmefmnd 19331 | The symmetric group on a s... |
| symgtset 19332 | The topology of the symmet... |
| symggrp 19333 | The symmetric group on a s... |
| symgid 19334 | The group identity element... |
| symginv 19335 | The group inverse in the s... |
| symgsubmefmndALT 19336 | The symmetric group on a s... |
| galactghm 19337 | The currying of a group ac... |
| lactghmga 19338 | The converse of ~ galactgh... |
| symgtopn 19339 | The topology of the symmet... |
| symgga 19340 | The symmetric group induce... |
| pgrpsubgsymgbi 19341 | Every permutation group is... |
| pgrpsubgsymg 19342 | Every permutation group is... |
| idressubgsymg 19343 | The singleton containing o... |
| idrespermg 19344 | The structure with the sin... |
| cayleylem1 19345 | Lemma for ~ cayley . (Con... |
| cayleylem2 19346 | Lemma for ~ cayley . (Con... |
| cayley 19347 | Cayley's Theorem (construc... |
| cayleyth 19348 | Cayley's Theorem (existenc... |
| symgfix2 19349 | If a permutation does not ... |
| symgextf 19350 | The extension of a permuta... |
| symgextfv 19351 | The function value of the ... |
| symgextfve 19352 | The function value of the ... |
| symgextf1lem 19353 | Lemma for ~ symgextf1 . (... |
| symgextf1 19354 | The extension of a permuta... |
| symgextfo 19355 | The extension of a permuta... |
| symgextf1o 19356 | The extension of a permuta... |
| symgextsymg 19357 | The extension of a permuta... |
| symgextres 19358 | The restriction of the ext... |
| gsumccatsymgsn 19359 | Homomorphic property of co... |
| gsmsymgrfixlem1 19360 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19361 | The composition of permuta... |
| fvcosymgeq 19362 | The values of two composit... |
| gsmsymgreqlem1 19363 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19364 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19365 | Two combination of permuta... |
| symgfixelq 19366 | A permutation of a set fix... |
| symgfixels 19367 | The restriction of a permu... |
| symgfixelsi 19368 | The restriction of a permu... |
| symgfixf 19369 | The mapping of a permutati... |
| symgfixf1 19370 | The mapping of a permutati... |
| symgfixfolem1 19371 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19372 | The mapping of a permutati... |
| symgfixf1o 19373 | The mapping of a permutati... |
| f1omvdmvd 19376 | A permutation of any class... |
| f1omvdcnv 19377 | A permutation and its inve... |
| mvdco 19378 | Composing two permutations... |
| f1omvdconj 19379 | Conjugation of a permutati... |
| f1otrspeq 19380 | A transposition is charact... |
| f1omvdco2 19381 | If exactly one of two perm... |
| f1omvdco3 19382 | If a point is moved by exa... |
| pmtrfval 19383 | The function generating tr... |
| pmtrval 19384 | A generated transposition,... |
| pmtrfv 19385 | General value of mapping a... |
| pmtrprfv 19386 | In a transposition of two ... |
| pmtrprfv3 19387 | In a transposition of two ... |
| pmtrf 19388 | Functionality of a transpo... |
| pmtrmvd 19389 | A transposition moves prec... |
| pmtrrn 19390 | Transposing two points giv... |
| pmtrfrn 19391 | A transposition (as a kind... |
| pmtrffv 19392 | Mapping of a point under a... |
| pmtrrn2 19393 | For any transposition ther... |
| pmtrfinv 19394 | A transposition function i... |
| pmtrfmvdn0 19395 | A transposition moves at l... |
| pmtrff1o 19396 | A transposition function i... |
| pmtrfcnv 19397 | A transposition function i... |
| pmtrfb 19398 | An intrinsic characterizat... |
| pmtrfconj 19399 | Any conjugate of a transpo... |
| symgsssg 19400 | The symmetric group has su... |
| symgfisg 19401 | The symmetric group has a ... |
| symgtrf 19402 | Transpositions are element... |
| symggen 19403 | The span of the transposit... |
| symggen2 19404 | A finite permutation group... |
| symgtrinv 19405 | To invert a permutation re... |
| pmtr3ncomlem1 19406 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19407 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19408 | Transpositions over sets w... |
| pmtrdifellem1 19409 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19410 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19411 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19412 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19413 | A transposition of element... |
| pmtrdifwrdellem1 19414 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19415 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19416 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19417 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19418 | A sequence of transpositio... |
| pmtrdifwrdel2 19419 | A sequence of transpositio... |
| pmtrprfval 19420 | The transpositions on a pa... |
| pmtrprfvalrn 19421 | The range of the transposi... |
| psgnunilem1 19426 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19427 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19428 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19429 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19430 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19431 | Addition and subtraction o... |
| psgnuni 19432 | If the same permutation ca... |
| psgnfval 19433 | Function definition of the... |
| psgnfn 19434 | Functionality and domain o... |
| psgndmsubg 19435 | The finitary permutations ... |
| psgneldm 19436 | Property of being a finita... |
| psgneldm2 19437 | The finitary permutations ... |
| psgneldm2i 19438 | A sequence of transpositio... |
| psgneu 19439 | A finitary permutation has... |
| psgnval 19440 | Value of the permutation s... |
| psgnvali 19441 | A finitary permutation has... |
| psgnvalii 19442 | Any representation of a pe... |
| psgnpmtr 19443 | All transpositions are odd... |
| psgn0fv0 19444 | The permutation sign funct... |
| sygbasnfpfi 19445 | The class of non-fixed poi... |
| psgnfvalfi 19446 | Function definition of the... |
| psgnvalfi 19447 | Value of the permutation s... |
| psgnran 19448 | The range of the permutati... |
| gsmtrcl 19449 | The group sum of transposi... |
| psgnfitr 19450 | A permutation of a finite ... |
| psgnfieu 19451 | A permutation of a finite ... |
| pmtrsn 19452 | The value of the transposi... |
| psgnsn 19453 | The permutation sign funct... |
| psgnprfval 19454 | The permutation sign funct... |
| psgnprfval1 19455 | The permutation sign of th... |
| psgnprfval2 19456 | The permutation sign of th... |
| odfval 19465 | Value of the order functio... |
| odfvalALT 19466 | Shorter proof of ~ odfval ... |
| odval 19467 | Second substitution for th... |
| odlem1 19468 | The group element order is... |
| odcl 19469 | The order of a group eleme... |
| odf 19470 | Functionality of the group... |
| odid 19471 | Any element to the power o... |
| odlem2 19472 | Any positive annihilator o... |
| odmodnn0 19473 | Reduce the argument of a g... |
| mndodconglem 19474 | Lemma for ~ mndodcong . (... |
| mndodcong 19475 | If two multipliers are con... |
| mndodcongi 19476 | If two multipliers are con... |
| oddvdsnn0 19477 | The only multiples of ` A ... |
| odnncl 19478 | If a nonzero multiple of a... |
| odmod 19479 | Reduce the argument of a g... |
| oddvds 19480 | The only multiples of ` A ... |
| oddvdsi 19481 | Any group element is annih... |
| odcong 19482 | If two multipliers are con... |
| odeq 19483 | The ~ oddvds property uniq... |
| odval2 19484 | A non-conditional definiti... |
| odcld 19485 | The order of a group eleme... |
| odm1inv 19486 | The (order-1)th multiple o... |
| odmulgid 19487 | A relationship between the... |
| odmulg2 19488 | The order of a multiple di... |
| odmulg 19489 | Relationship between the o... |
| odmulgeq 19490 | A multiple of a point of f... |
| odbezout 19491 | If ` N ` is coprime to the... |
| od1 19492 | The order of the group ide... |
| odeq1 19493 | The group identity is the ... |
| odinv 19494 | The order of the inverse o... |
| odf1 19495 | The multiples of an elemen... |
| odinf 19496 | The multiples of an elemen... |
| dfod2 19497 | An alternative definition ... |
| odcl2 19498 | The order of an element of... |
| oddvds2 19499 | The order of an element of... |
| finodsubmsubg 19500 | A submonoid whose elements... |
| 0subgALT 19501 | A shorter proof of ~ 0subg... |
| submod 19502 | The order of an element is... |
| subgod 19503 | The order of an element is... |
| odsubdvds 19504 | The order of an element of... |
| odf1o1 19505 | An element with zero order... |
| odf1o2 19506 | An element with nonzero or... |
| odhash 19507 | An element of zero order g... |
| odhash2 19508 | If an element has nonzero ... |
| odhash3 19509 | An element which generates... |
| odngen 19510 | A cyclic subgroup of size ... |
| gexval 19511 | Value of the exponent of a... |
| gexlem1 19512 | The group element order is... |
| gexcl 19513 | The exponent of a group is... |
| gexid 19514 | Any element to the power o... |
| gexlem2 19515 | Any positive annihilator o... |
| gexdvdsi 19516 | Any group element is annih... |
| gexdvds 19517 | The only ` N ` that annihi... |
| gexdvds2 19518 | An integer divides the gro... |
| gexod 19519 | Any group element is annih... |
| gexcl3 19520 | If the order of every grou... |
| gexnnod 19521 | Every group element has fi... |
| gexcl2 19522 | The exponent of a finite g... |
| gexdvds3 19523 | The exponent of a finite g... |
| gex1 19524 | A group or monoid has expo... |
| ispgp 19525 | A group is a ` P ` -group ... |
| pgpprm 19526 | Reverse closure for the fi... |
| pgpgrp 19527 | Reverse closure for the se... |
| pgpfi1 19528 | A finite group with order ... |
| pgp0 19529 | The identity subgroup is a... |
| subgpgp 19530 | A subgroup of a p-group is... |
| sylow1lem1 19531 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19532 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19533 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19534 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19535 | Lemma for ~ sylow1 . Usin... |
| sylow1 19536 | Sylow's first theorem. If... |
| odcau 19537 | Cauchy's theorem for the o... |
| pgpfi 19538 | The converse to ~ pgpfi1 .... |
| pgpfi2 19539 | Alternate version of ~ pgp... |
| pgphash 19540 | The order of a p-group. (... |
| isslw 19541 | The property of being a Sy... |
| slwprm 19542 | Reverse closure for the fi... |
| slwsubg 19543 | A Sylow ` P ` -subgroup is... |
| slwispgp 19544 | Defining property of a Syl... |
| slwpss 19545 | A proper superset of a Syl... |
| slwpgp 19546 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19547 | Every ` P ` -subgroup is c... |
| slwn0 19548 | Every finite group contain... |
| subgslw 19549 | A Sylow subgroup that is c... |
| sylow2alem1 19550 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19551 | Lemma for ~ sylow2a . All... |
| sylow2a 19552 | A named lemma of Sylow's s... |
| sylow2blem1 19553 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19554 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19555 | Sylow's second theorem. P... |
| sylow2b 19556 | Sylow's second theorem. A... |
| slwhash 19557 | A sylow subgroup has cardi... |
| fislw 19558 | The sylow subgroups of a f... |
| sylow2 19559 | Sylow's second theorem. S... |
| sylow3lem1 19560 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19561 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19562 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19563 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19564 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19565 | Lemma for ~ sylow3 , secon... |
| sylow3 19566 | Sylow's third theorem. Th... |
| lsmfval 19571 | The subgroup sum function ... |
| lsmvalx 19572 | Subspace sum value (for a ... |
| lsmelvalx 19573 | Subspace sum membership (f... |
| lsmelvalix 19574 | Subspace sum membership (f... |
| oppglsm 19575 | The subspace sum operation... |
| lsmssv 19576 | Subgroup sum is a subset o... |
| lsmless1x 19577 | Subset implies subgroup su... |
| lsmless2x 19578 | Subset implies subgroup su... |
| lsmub1x 19579 | Subgroup sum is an upper b... |
| lsmub2x 19580 | Subgroup sum is an upper b... |
| lsmval 19581 | Subgroup sum value (for a ... |
| lsmelval 19582 | Subgroup sum membership (f... |
| lsmelvali 19583 | Subgroup sum membership (f... |
| lsmelvalm 19584 | Subgroup sum membership an... |
| lsmelvalmi 19585 | Membership of vector subtr... |
| lsmsubm 19586 | The sum of two commuting s... |
| lsmsubg 19587 | The sum of two commuting s... |
| lsmcom2 19588 | Subgroup sum commutes. (C... |
| smndlsmidm 19589 | The direct product is idem... |
| lsmub1 19590 | Subgroup sum is an upper b... |
| lsmub2 19591 | Subgroup sum is an upper b... |
| lsmunss 19592 | Union of subgroups is a su... |
| lsmless1 19593 | Subset implies subgroup su... |
| lsmless2 19594 | Subset implies subgroup su... |
| lsmless12 19595 | Subset implies subgroup su... |
| lsmidm 19596 | Subgroup sum is idempotent... |
| lsmlub 19597 | The least upper bound prop... |
| lsmss1 19598 | Subgroup sum with a subset... |
| lsmss1b 19599 | Subgroup sum with a subset... |
| lsmss2 19600 | Subgroup sum with a subset... |
| lsmss2b 19601 | Subgroup sum with a subset... |
| lsmass 19602 | Subgroup sum is associativ... |
| mndlsmidm 19603 | Subgroup sum is idempotent... |
| lsm01 19604 | Subgroup sum with the zero... |
| lsm02 19605 | Subgroup sum with the zero... |
| subglsm 19606 | The subgroup sum evaluated... |
| lssnle 19607 | Equivalent expressions for... |
| lsmmod 19608 | The modular law holds for ... |
| lsmmod2 19609 | Modular law dual for subgr... |
| lsmpropd 19610 | If two structures have the... |
| cntzrecd 19611 | Commute the "subgroups com... |
| lsmcntz 19612 | The "subgroups commute" pr... |
| lsmcntzr 19613 | The "subgroups commute" pr... |
| lsmdisj 19614 | Disjointness from a subgro... |
| lsmdisj2 19615 | Association of the disjoin... |
| lsmdisj3 19616 | Association of the disjoin... |
| lsmdisjr 19617 | Disjointness from a subgro... |
| lsmdisj2r 19618 | Association of the disjoin... |
| lsmdisj3r 19619 | Association of the disjoin... |
| lsmdisj2a 19620 | Association of the disjoin... |
| lsmdisj2b 19621 | Association of the disjoin... |
| lsmdisj3a 19622 | Association of the disjoin... |
| lsmdisj3b 19623 | Association of the disjoin... |
| subgdisj1 19624 | Vectors belonging to disjo... |
| subgdisj2 19625 | Vectors belonging to disjo... |
| subgdisjb 19626 | Vectors belonging to disjo... |
| pj1fval 19627 | The left projection functi... |
| pj1val 19628 | The left projection functi... |
| pj1eu 19629 | Uniqueness of a left proje... |
| pj1f 19630 | The left projection functi... |
| pj2f 19631 | The right projection funct... |
| pj1id 19632 | Any element of a direct su... |
| pj1eq 19633 | Any element of a direct su... |
| pj1lid 19634 | The left projection functi... |
| pj1rid 19635 | The left projection functi... |
| pj1ghm 19636 | The left projection functi... |
| pj1ghm2 19637 | The left projection functi... |
| lsmhash 19638 | The order of the direct pr... |
| efgmval 19645 | Value of the formal invers... |
| efgmf 19646 | The formal inverse operati... |
| efgmnvl 19647 | The inversion function on ... |
| efgrcl 19648 | Lemma for ~ efgval . (Con... |
| efglem 19649 | Lemma for ~ efgval . (Con... |
| efgval 19650 | Value of the free group co... |
| efger 19651 | Value of the free group co... |
| efgi 19652 | Value of the free group co... |
| efgi0 19653 | Value of the free group co... |
| efgi1 19654 | Value of the free group co... |
| efgtf 19655 | Value of the free group co... |
| efgtval 19656 | Value of the extension fun... |
| efgval2 19657 | Value of the free group co... |
| efgi2 19658 | Value of the free group co... |
| efgtlen 19659 | Value of the free group co... |
| efginvrel2 19660 | The inverse of the reverse... |
| efginvrel1 19661 | The inverse of the reverse... |
| efgsf 19662 | Value of the auxiliary fun... |
| efgsdm 19663 | Elementhood in the domain ... |
| efgsval 19664 | Value of the auxiliary fun... |
| efgsdmi 19665 | Property of the last link ... |
| efgsval2 19666 | Value of the auxiliary fun... |
| efgsrel 19667 | The start and end of any e... |
| efgs1 19668 | A singleton of an irreduci... |
| efgs1b 19669 | Every extension sequence e... |
| efgsp1 19670 | If ` F ` is an extension s... |
| efgsres 19671 | An initial segment of an e... |
| efgsfo 19672 | For any word, there is a s... |
| efgredlema 19673 | The reduced word that form... |
| efgredlemf 19674 | Lemma for ~ efgredleme . ... |
| efgredlemg 19675 | Lemma for ~ efgred . (Con... |
| efgredleme 19676 | Lemma for ~ efgred . (Con... |
| efgredlemd 19677 | The reduced word that form... |
| efgredlemc 19678 | The reduced word that form... |
| efgredlemb 19679 | The reduced word that form... |
| efgredlem 19680 | The reduced word that form... |
| efgred 19681 | The reduced word that form... |
| efgrelexlema 19682 | If two words ` A , B ` are... |
| efgrelexlemb 19683 | If two words ` A , B ` are... |
| efgrelex 19684 | If two words ` A , B ` are... |
| efgredeu 19685 | There is a unique reduced ... |
| efgred2 19686 | Two extension sequences ha... |
| efgcpbllema 19687 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19688 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19689 | Two extension sequences ha... |
| efgcpbl2 19690 | Two extension sequences ha... |
| frgpval 19691 | Value of the free group co... |
| frgpcpbl 19692 | Compatibility of the group... |
| frgp0 19693 | The free group is a group.... |
| frgpeccl 19694 | Closure of the quotient ma... |
| frgpgrp 19695 | The free group is a group.... |
| frgpadd 19696 | Addition in the free group... |
| frgpinv 19697 | The inverse of an element ... |
| frgpmhm 19698 | The "natural map" from wor... |
| vrgpfval 19699 | The canonical injection fr... |
| vrgpval 19700 | The value of the generatin... |
| vrgpf 19701 | The mapping from the index... |
| vrgpinv 19702 | The inverse of a generatin... |
| frgpuptf 19703 | Any assignment of the gene... |
| frgpuptinv 19704 | Any assignment of the gene... |
| frgpuplem 19705 | Any assignment of the gene... |
| frgpupf 19706 | Any assignment of the gene... |
| frgpupval 19707 | Any assignment of the gene... |
| frgpup1 19708 | Any assignment of the gene... |
| frgpup2 19709 | The evaluation map has the... |
| frgpup3lem 19710 | The evaluation map has the... |
| frgpup3 19711 | Universal property of the ... |
| 0frgp 19712 | The free group on zero gen... |
| isabl 19717 | The predicate "is an Abeli... |
| ablgrp 19718 | An Abelian group is a grou... |
| ablgrpd 19719 | An Abelian group is a grou... |
| ablcmn 19720 | An Abelian group is a comm... |
| ablcmnd 19721 | An Abelian group is a comm... |
| iscmn 19722 | The predicate "is a commut... |
| isabl2 19723 | The predicate "is an Abeli... |
| cmnpropd 19724 | If two structures have the... |
| ablpropd 19725 | If two structures have the... |
| ablprop 19726 | If two structures have the... |
| iscmnd 19727 | Properties that determine ... |
| isabld 19728 | Properties that determine ... |
| isabli 19729 | Properties that determine ... |
| cmnmnd 19730 | A commutative monoid is a ... |
| cmncom 19731 | A commutative monoid is co... |
| ablcom 19732 | An Abelian group operation... |
| cmn32 19733 | Commutative/associative la... |
| cmn4 19734 | Commutative/associative la... |
| cmn12 19735 | Commutative/associative la... |
| abl32 19736 | Commutative/associative la... |
| cmnmndd 19737 | A commutative monoid is a ... |
| cmnbascntr 19738 | The base set of a commutat... |
| rinvmod 19739 | Uniqueness of a right inve... |
| ablinvadd 19740 | The inverse of an Abelian ... |
| ablsub2inv 19741 | Abelian group subtraction ... |
| ablsubadd 19742 | Relationship between Abeli... |
| ablsub4 19743 | Commutative/associative su... |
| abladdsub4 19744 | Abelian group addition/sub... |
| abladdsub 19745 | Associative-type law for g... |
| ablsubadd23 19746 | Commutative/associative la... |
| ablsubaddsub 19747 | Double subtraction and add... |
| ablpncan2 19748 | Cancellation law for subtr... |
| ablpncan3 19749 | A cancellation law for Abe... |
| ablsubsub 19750 | Law for double subtraction... |
| ablsubsub4 19751 | Law for double subtraction... |
| ablpnpcan 19752 | Cancellation law for mixed... |
| ablnncan 19753 | Cancellation law for group... |
| ablsub32 19754 | Swap the second and third ... |
| ablnnncan 19755 | Cancellation law for group... |
| ablnnncan1 19756 | Cancellation law for group... |
| ablsubsub23 19757 | Swap subtrahend and result... |
| mulgnn0di 19758 | Group multiple of a sum, f... |
| mulgdi 19759 | Group multiple of a sum. ... |
| mulgmhm 19760 | The map from ` x ` to ` n ... |
| mulgghm 19761 | The map from ` x ` to ` n ... |
| mulgsubdi 19762 | Group multiple of a differ... |
| ghmfghm 19763 | The function fulfilling th... |
| ghmcmn 19764 | The image of a commutative... |
| ghmabl 19765 | The image of an abelian gr... |
| invghm 19766 | The inversion map is a gro... |
| eqgabl 19767 | Value of the subgroup cose... |
| qusecsub 19768 | Two subgroup cosets are eq... |
| subgabl 19769 | A subgroup of an abelian g... |
| subcmn 19770 | A submonoid of a commutati... |
| submcmn 19771 | A submonoid of a commutati... |
| submcmn2 19772 | A submonoid is commutative... |
| cntzcmn 19773 | The centralizer of any sub... |
| cntzcmnss 19774 | Any subset in a commutativ... |
| cntrcmnd 19775 | The center of a monoid is ... |
| cntrabl 19776 | The center of a group is a... |
| cntzspan 19777 | If the generators commute,... |
| cntzcmnf 19778 | Discharge the centralizer ... |
| ghmplusg 19779 | The pointwise sum of two l... |
| ablnsg 19780 | Every subgroup of an abeli... |
| odadd1 19781 | The order of a product in ... |
| odadd2 19782 | The order of a product in ... |
| odadd 19783 | The order of a product is ... |
| gex2abl 19784 | A group with exponent 2 (o... |
| gexexlem 19785 | Lemma for ~ gexex . (Cont... |
| gexex 19786 | In an abelian group with f... |
| torsubg 19787 | The set of all elements of... |
| oddvdssubg 19788 | The set of all elements wh... |
| lsmcomx 19789 | Subgroup sum commutes (ext... |
| ablcntzd 19790 | All subgroups in an abelia... |
| lsmcom 19791 | Subgroup sum commutes. (C... |
| lsmsubg2 19792 | The sum of two subgroups i... |
| lsm4 19793 | Commutative/associative la... |
| prdscmnd 19794 | The product of a family of... |
| prdsabld 19795 | The product of a family of... |
| pwscmn 19796 | The structure power on a c... |
| pwsabl 19797 | The structure power on an ... |
| qusabl 19798 | If ` Y ` is a subgroup of ... |
| abl1 19799 | The (smallest) structure r... |
| abln0 19800 | Abelian groups (and theref... |
| cnaddablx 19801 | The complex numbers are an... |
| cnaddabl 19802 | The complex numbers are an... |
| cnaddid 19803 | The group identity element... |
| cnaddinv 19804 | Value of the group inverse... |
| zaddablx 19805 | The integers are an Abelia... |
| frgpnabllem1 19806 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19807 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19808 | The free group on two or m... |
| imasabl 19809 | The image structure of an ... |
| iscyg 19812 | Definition of a cyclic gro... |
| iscyggen 19813 | The property of being a cy... |
| iscyggen2 19814 | The property of being a cy... |
| iscyg2 19815 | A cyclic group is a group ... |
| cyggeninv 19816 | The inverse of a cyclic ge... |
| cyggenod 19817 | An element is the generato... |
| cyggenod2 19818 | In an infinite cyclic grou... |
| iscyg3 19819 | Definition of a cyclic gro... |
| iscygd 19820 | Definition of a cyclic gro... |
| iscygodd 19821 | Show that a group with an ... |
| cycsubmcmn 19822 | The set of nonnegative int... |
| cyggrp 19823 | A cyclic group is a group.... |
| cygabl 19824 | A cyclic group is abelian.... |
| cygctb 19825 | A cyclic group is countabl... |
| 0cyg 19826 | The trivial group is cycli... |
| prmcyg 19827 | A group with prime order i... |
| lt6abl 19828 | A group with fewer than ` ... |
| ghmcyg 19829 | The image of a cyclic grou... |
| cyggex2 19830 | The exponent of a cyclic g... |
| cyggex 19831 | The exponent of a finite c... |
| cyggexb 19832 | A finite abelian group is ... |
| giccyg 19833 | Cyclicity is a group prope... |
| cycsubgcyg 19834 | The cyclic subgroup genera... |
| cycsubgcyg2 19835 | The cyclic subgroup genera... |
| gsumval3a 19836 | Value of the group sum ope... |
| gsumval3eu 19837 | The group sum as defined i... |
| gsumval3lem1 19838 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19839 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19840 | Value of the group sum ope... |
| gsumcllem 19841 | Lemma for ~ gsumcl and rel... |
| gsumzres 19842 | Extend a finite group sum ... |
| gsumzcl2 19843 | Closure of a finite group ... |
| gsumzcl 19844 | Closure of a finite group ... |
| gsumzf1o 19845 | Re-index a finite group su... |
| gsumres 19846 | Extend a finite group sum ... |
| gsumcl2 19847 | Closure of a finite group ... |
| gsumcl 19848 | Closure of a finite group ... |
| gsumf1o 19849 | Re-index a finite group su... |
| gsumreidx 19850 | Re-index a finite group su... |
| gsumzsubmcl 19851 | Closure of a group sum in ... |
| gsumsubmcl 19852 | Closure of a group sum in ... |
| gsumsubgcl 19853 | Closure of a group sum in ... |
| gsumzaddlem 19854 | The sum of two group sums.... |
| gsumzadd 19855 | The sum of two group sums.... |
| gsumadd 19856 | The sum of two group sums.... |
| gsummptfsadd 19857 | The sum of two group sums ... |
| gsummptfidmadd 19858 | The sum of two group sums ... |
| gsummptfidmadd2 19859 | The sum of two group sums ... |
| gsumzsplit 19860 | Split a group sum into two... |
| gsumsplit 19861 | Split a group sum into two... |
| gsumsplit2 19862 | Split a group sum into two... |
| gsummptfidmsplit 19863 | Split a group sum expresse... |
| gsummptfidmsplitres 19864 | Split a group sum expresse... |
| gsummptfzsplit 19865 | Split a group sum expresse... |
| gsummptfzsplitl 19866 | Split a group sum expresse... |
| gsumconst 19867 | Sum of a constant series. ... |
| gsumconstf 19868 | Sum of a constant series. ... |
| gsummptshft 19869 | Index shift of a finite gr... |
| gsumzmhm 19870 | Apply a group homomorphism... |
| gsummhm 19871 | Apply a group homomorphism... |
| gsummhm2 19872 | Apply a group homomorphism... |
| gsummptmhm 19873 | Apply a group homomorphism... |
| gsummulglem 19874 | Lemma for ~ gsummulg and ~... |
| gsummulg 19875 | Nonnegative multiple of a ... |
| gsummulgz 19876 | Integer multiple of a grou... |
| gsumzoppg 19877 | The opposite of a group su... |
| gsumzinv 19878 | Inverse of a group sum. (... |
| gsuminv 19879 | Inverse of a group sum. (... |
| gsummptfidminv 19880 | Inverse of a group sum exp... |
| gsumsub 19881 | The difference of two grou... |
| gsummptfssub 19882 | The difference of two grou... |
| gsummptfidmsub 19883 | The difference of two grou... |
| gsumsnfd 19884 | Group sum of a singleton, ... |
| gsumsnd 19885 | Group sum of a singleton, ... |
| gsumsnf 19886 | Group sum of a singleton, ... |
| gsumsn 19887 | Group sum of a singleton. ... |
| gsumpr 19888 | Group sum of a pair. (Con... |
| gsumzunsnd 19889 | Append an element to a fin... |
| gsumunsnfd 19890 | Append an element to a fin... |
| gsumunsnd 19891 | Append an element to a fin... |
| gsumunsnf 19892 | Append an element to a fin... |
| gsumunsn 19893 | Append an element to a fin... |
| gsumdifsnd 19894 | Extract a summand from a f... |
| gsumpt 19895 | Sum of a family that is no... |
| gsummptf1o 19896 | Re-index a finite group su... |
| gsummptun 19897 | Group sum of a disjoint un... |
| gsummpt1n0 19898 | If only one summand in a f... |
| gsummptif1n0 19899 | If only one summand in a f... |
| gsummptcl 19900 | Closure of a finite group ... |
| gsummptfif1o 19901 | Re-index a finite group su... |
| gsummptfzcl 19902 | Closure of a finite group ... |
| gsum2dlem1 19903 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19904 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19905 | Write a sum over a two-dim... |
| gsum2d2lem 19906 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19907 | Write a group sum over a t... |
| gsumcom2 19908 | Two-dimensional commutatio... |
| gsumxp 19909 | Write a group sum over a c... |
| gsumcom 19910 | Commute the arguments of a... |
| gsumcom3 19911 | A commutative law for fini... |
| gsumcom3fi 19912 | A commutative law for fini... |
| gsumxp2 19913 | Write a group sum over a c... |
| prdsgsum 19914 | Finite commutative sums in... |
| pwsgsum 19915 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19916 | Replacing a finitely suppo... |
| nn0gsumfz 19917 | Replacing a finitely suppo... |
| nn0gsumfz0 19918 | Replacing a finitely suppo... |
| gsummptnn0fz 19919 | A final group sum over a f... |
| gsummptnn0fzfv 19920 | A final group sum over a f... |
| telgsumfzslem 19921 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19922 | Telescoping group sum rang... |
| telgsumfz 19923 | Telescoping group sum rang... |
| telgsumfz0s 19924 | Telescoping finite group s... |
| telgsumfz0 19925 | Telescoping finite group s... |
| telgsums 19926 | Telescoping finitely suppo... |
| telgsum 19927 | Telescoping finitely suppo... |
| reldmdprd 19932 | The domain of the internal... |
| dmdprd 19933 | The domain of definition o... |
| dmdprdd 19934 | Show that a given family i... |
| dprddomprc 19935 | A family of subgroups inde... |
| dprddomcld 19936 | If a family of subgroups i... |
| dprdval0prc 19937 | The internal direct produc... |
| dprdval 19938 | The value of the internal ... |
| eldprd 19939 | A class ` A ` is an intern... |
| dprdgrp 19940 | Reverse closure for the in... |
| dprdf 19941 | The function ` S ` is a fa... |
| dprdf2 19942 | The function ` S ` is a fa... |
| dprdcntz 19943 | The function ` S ` is a fa... |
| dprddisj 19944 | The function ` S ` is a fa... |
| dprdw 19945 | The property of being a fi... |
| dprdwd 19946 | A mapping being a finitely... |
| dprdff 19947 | A finitely supported funct... |
| dprdfcl 19948 | A finitely supported funct... |
| dprdffsupp 19949 | A finitely supported funct... |
| dprdfcntz 19950 | A function on the elements... |
| dprdssv 19951 | The internal direct produc... |
| dprdfid 19952 | A function mapping all but... |
| eldprdi 19953 | The domain of definition o... |
| dprdfinv 19954 | Take the inverse of a grou... |
| dprdfadd 19955 | Take the sum of group sums... |
| dprdfsub 19956 | Take the difference of gro... |
| dprdfeq0 19957 | The zero function is the o... |
| dprdf11 19958 | Two group sums over a dire... |
| dprdsubg 19959 | The internal direct produc... |
| dprdub 19960 | Each factor is a subset of... |
| dprdlub 19961 | The direct product is smal... |
| dprdspan 19962 | The direct product is the ... |
| dprdres 19963 | Restriction of a direct pr... |
| dprdss 19964 | Create a direct product by... |
| dprdz 19965 | A family consisting entire... |
| dprd0 19966 | The empty family is an int... |
| dprdf1o 19967 | Rearrange the index set of... |
| dprdf1 19968 | Rearrange the index set of... |
| subgdmdprd 19969 | A direct product in a subg... |
| subgdprd 19970 | A direct product in a subg... |
| dprdsn 19971 | A singleton family is an i... |
| dmdprdsplitlem 19972 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19973 | The function ` S ` is a fa... |
| dprddisj2 19974 | The function ` S ` is a fa... |
| dprd2dlem2 19975 | The direct product of a co... |
| dprd2dlem1 19976 | The direct product of a co... |
| dprd2da 19977 | The direct product of a co... |
| dprd2db 19978 | The direct product of a co... |
| dprd2d2 19979 | The direct product of a co... |
| dmdprdsplit2lem 19980 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19981 | The direct product splits ... |
| dmdprdsplit 19982 | The direct product splits ... |
| dprdsplit 19983 | The direct product is the ... |
| dmdprdpr 19984 | A singleton family is an i... |
| dprdpr 19985 | A singleton family is an i... |
| dpjlem 19986 | Lemma for theorems about d... |
| dpjcntz 19987 | The two subgroups that app... |
| dpjdisj 19988 | The two subgroups that app... |
| dpjlsm 19989 | The two subgroups that app... |
| dpjfval 19990 | Value of the direct produc... |
| dpjval 19991 | Value of the direct produc... |
| dpjf 19992 | The ` X ` -th index projec... |
| dpjidcl 19993 | The key property of projec... |
| dpjeq 19994 | Decompose a group sum into... |
| dpjid 19995 | The key property of projec... |
| dpjlid 19996 | The ` X ` -th index projec... |
| dpjrid 19997 | The ` Y ` -th index projec... |
| dpjghm 19998 | The direct product is the ... |
| dpjghm2 19999 | The direct product is the ... |
| ablfacrplem 20000 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 20001 | A finite abelian group who... |
| ablfacrp2 20002 | The factors ` K , L ` of ~... |
| ablfac1lem 20003 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 20004 | The factors of ~ ablfac1b ... |
| ablfac1b 20005 | Any abelian group is the d... |
| ablfac1c 20006 | The factors of ~ ablfac1b ... |
| ablfac1eulem 20007 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 20008 | The factorization of ~ abl... |
| pgpfac1lem1 20009 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 20010 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 20011 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 20012 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 20013 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 20014 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 20015 | Factorization of a finite ... |
| pgpfaclem1 20016 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 20017 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 20018 | Lemma for ~ pgpfac . (Con... |
| pgpfac 20019 | Full factorization of a fi... |
| ablfaclem1 20020 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 20021 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 20022 | Lemma for ~ ablfac . (Con... |
| ablfac 20023 | The Fundamental Theorem of... |
| ablfac2 20024 | Choose generators for each... |
| issimpg 20027 | The predicate "is a simple... |
| issimpgd 20028 | Deduce a simple group from... |
| simpggrp 20029 | A simple group is a group.... |
| simpggrpd 20030 | A simple group is a group.... |
| simpg2nsg 20031 | A simple group has two nor... |
| trivnsimpgd 20032 | Trivial groups are not sim... |
| simpgntrivd 20033 | Simple groups are nontrivi... |
| simpgnideld 20034 | A simple group contains a ... |
| simpgnsgd 20035 | The only normal subgroups ... |
| simpgnsgeqd 20036 | A normal subgroup of a sim... |
| 2nsgsimpgd 20037 | If any normal subgroup of ... |
| simpgnsgbid 20038 | A nontrivial group is simp... |
| ablsimpnosubgd 20039 | A subgroup of an abelian s... |
| ablsimpg1gend 20040 | An abelian simple group is... |
| ablsimpgcygd 20041 | An abelian simple group is... |
| ablsimpgfindlem1 20042 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20043 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20044 | Relationship between the o... |
| ablsimpgfind 20045 | An abelian simple group is... |
| fincygsubgd 20046 | The subgroup referenced in... |
| fincygsubgodd 20047 | Calculate the order of a s... |
| fincygsubgodexd 20048 | A finite cyclic group has ... |
| prmgrpsimpgd 20049 | A group of prime order is ... |
| ablsimpgprmd 20050 | An abelian simple group ha... |
| ablsimpgd 20051 | An abelian group is simple... |
| isomnd 20056 | A (left) ordered monoid is... |
| isogrp 20057 | A (left-)ordered group is ... |
| ogrpgrp 20058 | A left-ordered group is a ... |
| omndmnd 20059 | A left-ordered monoid is a... |
| omndtos 20060 | A left-ordered monoid is a... |
| omndadd 20061 | In an ordered monoid, the ... |
| omndaddr 20062 | In a right ordered monoid,... |
| omndadd2d 20063 | In a commutative left orde... |
| omndadd2rd 20064 | In a left- and right- orde... |
| submomnd 20065 | A submonoid of an ordered ... |
| omndmul2 20066 | In an ordered monoid, the ... |
| omndmul3 20067 | In an ordered monoid, the ... |
| omndmul 20068 | In a commutative ordered m... |
| ogrpinv0le 20069 | In an ordered group, the o... |
| ogrpsub 20070 | In an ordered group, the o... |
| ogrpaddlt 20071 | In an ordered group, stric... |
| ogrpaddltbi 20072 | In a right ordered group, ... |
| ogrpaddltrd 20073 | In a right ordered group, ... |
| ogrpaddltrbid 20074 | In a right ordered group, ... |
| ogrpsublt 20075 | In an ordered group, stric... |
| ogrpinv0lt 20076 | In an ordered group, the o... |
| ogrpinvlt 20077 | In an ordered group, the o... |
| gsumle 20078 | A finite sum in an ordered... |
| fnmgp 20081 | The multiplicative group o... |
| mgpval 20082 | Value of the multiplicatio... |
| mgpplusg 20083 | Value of the group operati... |
| mgpbas 20084 | Base set of the multiplica... |
| mgpsca 20085 | The multiplication monoid ... |
| mgptset 20086 | Topology component of the ... |
| mgptopn 20087 | Topology of the multiplica... |
| mgpds 20088 | Distance function of the m... |
| mgpress 20089 | Subgroup commutes with the... |
| prdsmgp 20090 | The multiplicative monoid ... |
| isrng 20093 | The predicate "is a non-un... |
| rngabl 20094 | A non-unital ring is an (a... |
| rngmgp 20095 | A non-unital ring is a sem... |
| rngmgpf 20096 | Restricted functionality o... |
| rnggrp 20097 | A non-unital ring is a (ad... |
| rngass 20098 | Associative law for the mu... |
| rngdi 20099 | Distributive law for the m... |
| rngdir 20100 | Distributive law for the m... |
| rngacl 20101 | Closure of the addition op... |
| rng0cl 20102 | The zero element of a non-... |
| rngcl 20103 | Closure of the multiplicat... |
| rnglz 20104 | The zero of a non-unital r... |
| rngrz 20105 | The zero of a non-unital r... |
| rngmneg1 20106 | Negation of a product in a... |
| rngmneg2 20107 | Negation of a product in a... |
| rngm2neg 20108 | Double negation of a produ... |
| rngansg 20109 | Every additive subgroup of... |
| rngsubdi 20110 | Ring multiplication distri... |
| rngsubdir 20111 | Ring multiplication distri... |
| isrngd 20112 | Properties that determine ... |
| rngpropd 20113 | If two structures have the... |
| prdsmulrngcl 20114 | Closure of the multiplicat... |
| prdsrngd 20115 | A product of non-unital ri... |
| imasrng 20116 | The image structure of a n... |
| imasrngf1 20117 | The image of a non-unital ... |
| xpsrngd 20118 | A product of two non-unita... |
| qusrng 20119 | The quotient structure of ... |
| ringidval 20122 | The value of the unity ele... |
| dfur2 20123 | The multiplicative identit... |
| ringurd 20124 | Deduce the unity element o... |
| issrg 20127 | The predicate "is a semiri... |
| srgcmn 20128 | A semiring is a commutativ... |
| srgmnd 20129 | A semiring is a monoid. (... |
| srgmgp 20130 | A semiring is a monoid und... |
| srgdilem 20131 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20132 | Closure of the multiplicat... |
| srgass 20133 | Associative law for the mu... |
| srgideu 20134 | The unity element of a sem... |
| srgfcl 20135 | Functionality of the multi... |
| srgdi 20136 | Distributive law for the m... |
| srgdir 20137 | Distributive law for the m... |
| srgidcl 20138 | The unity element of a sem... |
| srg0cl 20139 | The zero element of a semi... |
| srgidmlem 20140 | Lemma for ~ srglidm and ~ ... |
| srglidm 20141 | The unity element of a sem... |
| srgridm 20142 | The unity element of a sem... |
| issrgid 20143 | Properties showing that an... |
| srgacl 20144 | Closure of the addition op... |
| srgcom 20145 | Commutativity of the addit... |
| srgrz 20146 | The zero of a semiring is ... |
| srglz 20147 | The zero of a semiring is ... |
| srgisid 20148 | In a semiring, the only le... |
| o2timesd 20149 | An element of a ring-like ... |
| rglcom4d 20150 | Restricted commutativity o... |
| srgo2times 20151 | A semiring element plus it... |
| srgcom4lem 20152 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20153 | Restricted commutativity o... |
| srg1zr 20154 | The only semiring with a b... |
| srgen1zr 20155 | The only semiring with one... |
| srgmulgass 20156 | An associative property be... |
| srgpcomp 20157 | If two elements of a semir... |
| srgpcompp 20158 | If two elements of a semir... |
| srgpcomppsc 20159 | If two elements of a semir... |
| srglmhm 20160 | Left-multiplication in a s... |
| srgrmhm 20161 | Right-multiplication in a ... |
| srgsummulcr 20162 | A finite semiring sum mult... |
| sgsummulcl 20163 | A finite semiring sum mult... |
| srg1expzeq1 20164 | The exponentiation (by a n... |
| srgbinomlem1 20165 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20166 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20167 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20168 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20169 | Lemma for ~ srgbinom . In... |
| srgbinom 20170 | The binomial theorem for c... |
| csrgbinom 20171 | The binomial theorem for c... |
| isring 20176 | The predicate "is a (unita... |
| ringgrp 20177 | A ring is a group. (Contr... |
| ringmgp 20178 | A ring is a monoid under m... |
| iscrng 20179 | A commutative ring is a ri... |
| crngmgp 20180 | A commutative ring's multi... |
| ringgrpd 20181 | A ring is a group. (Contr... |
| ringmnd 20182 | A ring is a monoid under a... |
| ringmgm 20183 | A ring is a magma. (Contr... |
| crngring 20184 | A commutative ring is a ri... |
| crngringd 20185 | A commutative ring is a ri... |
| crnggrpd 20186 | A commutative ring is a gr... |
| mgpf 20187 | Restricted functionality o... |
| ringdilem 20188 | Properties of a unital rin... |
| ringcl 20189 | Closure of the multiplicat... |
| crngcom 20190 | A commutative ring's multi... |
| iscrng2 20191 | A commutative ring is a ri... |
| ringass 20192 | Associative law for multip... |
| ringideu 20193 | The unity element of a rin... |
| crngcomd 20194 | Multiplication is commutat... |
| crngbascntr 20195 | The base set of a commutat... |
| ringassd 20196 | Associative law for multip... |
| crng12d 20197 | Commutative/associative la... |
| crng32d 20198 | Commutative/associative la... |
| ringcld 20199 | Closure of the multiplicat... |
| ringdi 20200 | Distributive law for the m... |
| ringdir 20201 | Distributive law for the m... |
| ringdid 20202 | Distributive law for the m... |
| ringdird 20203 | Distributive law for the m... |
| ringidcl 20204 | The unity element of a rin... |
| ringidcld 20205 | The unity element of a rin... |
| ring0cl 20206 | The zero element of a ring... |
| ringidmlem 20207 | Lemma for ~ ringlidm and ~... |
| ringlidm 20208 | The unity element of a rin... |
| ringridm 20209 | The unity element of a rin... |
| isringid 20210 | Properties showing that an... |
| ringlidmd 20211 | The unity element of a rin... |
| ringridmd 20212 | The unity element of a rin... |
| ringid 20213 | The multiplication operati... |
| ringo2times 20214 | A ring element plus itself... |
| ringadd2 20215 | A ring element plus itself... |
| ringidss 20216 | A subset of the multiplica... |
| ringacl 20217 | Closure of the addition op... |
| ringcomlem 20218 | Lemma for ~ ringcom . Thi... |
| ringcom 20219 | Commutativity of the addit... |
| ringabl 20220 | A ring is an Abelian group... |
| ringcmn 20221 | A ring is a commutative mo... |
| ringabld 20222 | A ring is an Abelian group... |
| ringcmnd 20223 | A ring is a commutative mo... |
| ringrng 20224 | A unital ring is a non-uni... |
| ringssrng 20225 | The unital rings are non-u... |
| isringrng 20226 | The predicate "is a unital... |
| ringpropd 20227 | If two structures have the... |
| crngpropd 20228 | If two structures have the... |
| ringprop 20229 | If two structures have the... |
| isringd 20230 | Properties that determine ... |
| iscrngd 20231 | Properties that determine ... |
| ringlz 20232 | The zero of a unital ring ... |
| ringrz 20233 | The zero of a unital ring ... |
| ringlzd 20234 | The zero of a unital ring ... |
| ringrzd 20235 | The zero of a unital ring ... |
| ringsrg 20236 | Any ring is also a semirin... |
| ring1eq0 20237 | If one and zero are equal,... |
| ring1ne0 20238 | If a ring has at least two... |
| ringinvnz1ne0 20239 | In a unital ring, a left i... |
| ringinvnzdiv 20240 | In a unital ring, a left i... |
| ringnegl 20241 | Negation in a ring is the ... |
| ringnegr 20242 | Negation in a ring is the ... |
| ringmneg1 20243 | Negation of a product in a... |
| ringmneg2 20244 | Negation of a product in a... |
| ringm2neg 20245 | Double negation of a produ... |
| ringsubdi 20246 | Ring multiplication distri... |
| ringsubdir 20247 | Ring multiplication distri... |
| mulgass2 20248 | An associative property be... |
| ring1 20249 | The (smallest) structure r... |
| ringn0 20250 | Rings exist. (Contributed... |
| ringlghm 20251 | Left-multiplication in a r... |
| ringrghm 20252 | Right-multiplication in a ... |
| gsummulc1 20253 | A finite ring sum multipli... |
| gsummulc2 20254 | A finite ring sum multipli... |
| gsummgp0 20255 | If one factor in a finite ... |
| gsumdixp 20256 | Distribute a binary produc... |
| prdsmulrcl 20257 | A structure product of rin... |
| prdsringd 20258 | A product of rings is a ri... |
| prdscrngd 20259 | A product of commutative r... |
| prds1 20260 | Value of the ring unity in... |
| pwsring 20261 | A structure power of a rin... |
| pws1 20262 | Value of the ring unity in... |
| pwscrng 20263 | A structure power of a com... |
| pwsmgp 20264 | The multiplicative group o... |
| pwspjmhmmgpd 20265 | The projection given by ~ ... |
| pwsexpg 20266 | Value of a group exponenti... |
| pwsgprod 20267 | Finite products in a power... |
| imasring 20268 | The image structure of a r... |
| imasringf1 20269 | The image of a ring under ... |
| xpsringd 20270 | A product of two rings is ... |
| xpsring1d 20271 | The multiplicative identit... |
| qusring2 20272 | The quotient structure of ... |
| crngbinom 20273 | The binomial theorem for c... |
| opprval 20276 | Value of the opposite ring... |
| opprmulfval 20277 | Value of the multiplicatio... |
| opprmul 20278 | Value of the multiplicatio... |
| crngoppr 20279 | In a commutative ring, the... |
| opprlem 20280 | Lemma for ~ opprbas and ~ ... |
| opprbas 20281 | Base set of an opposite ri... |
| oppradd 20282 | Addition operation of an o... |
| opprrng 20283 | An opposite non-unital rin... |
| opprrngb 20284 | A class is a non-unital ri... |
| opprring 20285 | An opposite ring is a ring... |
| opprringb 20286 | Bidirectional form of ~ op... |
| oppr0 20287 | Additive identity of an op... |
| oppr1 20288 | Multiplicative identity of... |
| opprneg 20289 | The negative function in a... |
| opprsubg 20290 | Being a subgroup is a symm... |
| mulgass3 20291 | An associative property be... |
| reldvdsr 20298 | The divides relation is a ... |
| dvdsrval 20299 | Value of the divides relat... |
| dvdsr 20300 | Value of the divides relat... |
| dvdsr2 20301 | Value of the divides relat... |
| dvdsrmul 20302 | A left-multiple of ` X ` i... |
| dvdsrcl 20303 | Closure of a dividing elem... |
| dvdsrcl2 20304 | Closure of a dividing elem... |
| dvdsrid 20305 | An element in a (unital) r... |
| dvdsrtr 20306 | Divisibility is transitive... |
| dvdsrmul1 20307 | The divisibility relation ... |
| dvdsrneg 20308 | An element divides its neg... |
| dvdsr01 20309 | In a ring, zero is divisib... |
| dvdsr02 20310 | Only zero is divisible by ... |
| isunit 20311 | Property of being a unit o... |
| 1unit 20312 | The multiplicative identit... |
| unitcl 20313 | A unit is an element of th... |
| unitss 20314 | The set of units is contai... |
| opprunit 20315 | Being a unit is a symmetri... |
| crngunit 20316 | Property of being a unit i... |
| dvdsunit 20317 | A divisor of a unit is a u... |
| unitmulcl 20318 | The product of units is a ... |
| unitmulclb 20319 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20320 | The base set of the group ... |
| unitgrp 20321 | The group of units is a gr... |
| unitabl 20322 | The group of units of a co... |
| unitgrpid 20323 | The identity of the group ... |
| unitsubm 20324 | The group of units is a su... |
| invrfval 20327 | Multiplicative inverse fun... |
| unitinvcl 20328 | The inverse of a unit exis... |
| unitinvinv 20329 | The inverse of the inverse... |
| ringinvcl 20330 | The inverse of a unit is a... |
| unitlinv 20331 | A unit times its inverse i... |
| unitrinv 20332 | A unit times its inverse i... |
| 1rinv 20333 | The inverse of the ring un... |
| 0unit 20334 | The additive identity is a... |
| unitnegcl 20335 | The negative of a unit is ... |
| ringunitnzdiv 20336 | In a unitary ring, a unit ... |
| ring1nzdiv 20337 | In a unitary ring, the rin... |
| dvrfval 20340 | Division operation in a ri... |
| dvrval 20341 | Division operation in a ri... |
| dvrcl 20342 | Closure of division operat... |
| unitdvcl 20343 | The units are closed under... |
| dvrid 20344 | A ring element divided by ... |
| dvr1 20345 | A ring element divided by ... |
| dvrass 20346 | An associative law for div... |
| dvrcan1 20347 | A cancellation law for div... |
| dvrcan3 20348 | A cancellation law for div... |
| dvreq1 20349 | Equality in terms of ratio... |
| dvrdir 20350 | Distributive law for the d... |
| rdivmuldivd 20351 | Multiplication of two rati... |
| ringinvdv 20352 | Write the inverse function... |
| rngidpropd 20353 | The ring unity depends onl... |
| dvdsrpropd 20354 | The divisibility relation ... |
| unitpropd 20355 | The set of units depends o... |
| invrpropd 20356 | The ring inverse function ... |
| isirred 20357 | An irreducible element of ... |
| isnirred 20358 | The property of being a no... |
| isirred2 20359 | Expand out the class diffe... |
| opprirred 20360 | Irreducibility is symmetri... |
| irredn0 20361 | The additive identity is n... |
| irredcl 20362 | An irreducible element is ... |
| irrednu 20363 | An irreducible element is ... |
| irredn1 20364 | The multiplicative identit... |
| irredrmul 20365 | The product of an irreduci... |
| irredlmul 20366 | The product of a unit and ... |
| irredmul 20367 | If product of two elements... |
| irredneg 20368 | The negative of an irreduc... |
| irrednegb 20369 | An element is irreducible ... |
| rnghmrcl 20376 | Reverse closure of a non-u... |
| rnghmfn 20377 | The mapping of two non-uni... |
| rnghmval 20378 | The set of the non-unital ... |
| isrnghm 20379 | A function is a non-unital... |
| isrnghmmul 20380 | A function is a non-unital... |
| rnghmmgmhm 20381 | A non-unital ring homomorp... |
| rnghmval2 20382 | The non-unital ring homomo... |
| isrngim 20383 | An isomorphism of non-unit... |
| rngimrcl 20384 | Reverse closure for an iso... |
| rnghmghm 20385 | A non-unital ring homomorp... |
| rnghmf 20386 | A ring homomorphism is a f... |
| rnghmmul 20387 | A homomorphism of non-unit... |
| isrnghm2d 20388 | Demonstration of non-unita... |
| isrnghmd 20389 | Demonstration of non-unita... |
| rnghmf1o 20390 | A non-unital ring homomorp... |
| isrngim2 20391 | An isomorphism of non-unit... |
| rngimf1o 20392 | An isomorphism of non-unit... |
| rngimrnghm 20393 | An isomorphism of non-unit... |
| rngimcnv 20394 | The converse of an isomorp... |
| rnghmco 20395 | The composition of non-uni... |
| idrnghm 20396 | The identity homomorphism ... |
| c0mgm 20397 | The constant mapping to ze... |
| c0mhm 20398 | The constant mapping to ze... |
| c0ghm 20399 | The constant mapping to ze... |
| c0snmgmhm 20400 | The constant mapping to ze... |
| c0snmhm 20401 | The constant mapping to ze... |
| c0snghm 20402 | The constant mapping to ze... |
| rngisomfv1 20403 | If there is a non-unital r... |
| rngisom1 20404 | If there is a non-unital r... |
| rngisomring 20405 | If there is a non-unital r... |
| rngisomring1 20406 | If there is a non-unital r... |
| dfrhm2 20412 | The property of a ring hom... |
| rhmrcl1 20414 | Reverse closure of a ring ... |
| rhmrcl2 20415 | Reverse closure of a ring ... |
| isrhm 20416 | A function is a ring homom... |
| rhmmhm 20417 | A ring homomorphism is a h... |
| rhmisrnghm 20418 | Each unital ring homomorph... |
| rimrcl 20419 | Reverse closure for an iso... |
| isrim0 20420 | A ring isomorphism is a ho... |
| rhmghm 20421 | A ring homomorphism is an ... |
| rhmf 20422 | A ring homomorphism is a f... |
| rhmmul 20423 | A homomorphism of rings pr... |
| isrhm2d 20424 | Demonstration of ring homo... |
| isrhmd 20425 | Demonstration of ring homo... |
| rhm1 20426 | Ring homomorphisms are req... |
| idrhm 20427 | The identity homomorphism ... |
| rhmf1o 20428 | A ring homomorphism is bij... |
| isrim 20429 | An isomorphism of rings is... |
| rimf1o 20430 | An isomorphism of rings is... |
| rimrhm 20431 | A ring isomorphism is a ho... |
| rimgim 20432 | An isomorphism of rings is... |
| rimisrngim 20433 | Each unital ring isomorphi... |
| rhmfn 20434 | The mapping of two rings t... |
| rhmval 20435 | The ring homomorphisms bet... |
| rhmco 20436 | The composition of ring ho... |
| pwsco1rhm 20437 | Right composition with a f... |
| pwsco2rhm 20438 | Left composition with a ri... |
| brric 20439 | The relation "is isomorphi... |
| brrici 20440 | Prove isomorphic by an exp... |
| brric2 20441 | The relation "is isomorphi... |
| ricgic 20442 | If two rings are (ring) is... |
| rhmdvdsr 20443 | A ring homomorphism preser... |
| rhmopp 20444 | A ring homomorphism is als... |
| elrhmunit 20445 | Ring homomorphisms preserv... |
| rhmunitinv 20446 | Ring homomorphisms preserv... |
| isnzr 20449 | Property of a nonzero ring... |
| nzrnz 20450 | One and zero are different... |
| nzrring 20451 | A nonzero ring is a ring. ... |
| nzrringOLD 20452 | Obsolete version of ~ nzrr... |
| isnzr2 20453 | Equivalent characterizatio... |
| isnzr2hash 20454 | Equivalent characterizatio... |
| nzrpropd 20455 | If two structures have the... |
| opprnzrb 20456 | The opposite of a nonzero ... |
| opprnzr 20457 | The opposite of a nonzero ... |
| ringelnzr 20458 | A ring is nonzero if it ha... |
| nzrunit 20459 | A unit is nonzero in any n... |
| 0ringnnzr 20460 | A ring is a zero ring iff ... |
| 0ring 20461 | If a ring has only one ele... |
| 0ringdif 20462 | A zero ring is a ring whic... |
| 0ringbas 20463 | The base set of a zero rin... |
| 0ring01eq 20464 | In a ring with only one el... |
| 01eq0ring 20465 | If the zero and the identi... |
| 01eq0ringOLD 20466 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20467 | In a unital ring the zero ... |
| 0ring1eq0 20468 | In a zero ring, a ring whi... |
| c0rhm 20469 | The constant mapping to ze... |
| c0rnghm 20470 | The constant mapping to ze... |
| zrrnghm 20471 | The constant mapping to ze... |
| nrhmzr 20472 | There is no ring homomorph... |
| islring 20475 | The predicate "is a local ... |
| lringnzr 20476 | A local ring is a nonzero ... |
| lringring 20477 | A local ring is a ring. (... |
| lringnz 20478 | A local ring is a nonzero ... |
| lringuplu 20479 | If the sum of two elements... |
| issubrng 20482 | The subring of non-unital ... |
| subrngss 20483 | A subring is a subset. (C... |
| subrngid 20484 | Every non-unital ring is a... |
| subrngrng 20485 | A subring is a non-unital ... |
| subrngrcl 20486 | Reverse closure for a subr... |
| subrngsubg 20487 | A subring is a subgroup. ... |
| subrngringnsg 20488 | A subring is a normal subg... |
| subrngbas 20489 | Base set of a subring stru... |
| subrng0 20490 | A subring always has the s... |
| subrngacl 20491 | A subring is closed under ... |
| subrngmcl 20492 | A subring is closed under ... |
| issubrng2 20493 | Characterize the subrings ... |
| opprsubrng 20494 | Being a subring is a symme... |
| subrngint 20495 | The intersection of a none... |
| subrngin 20496 | The intersection of two su... |
| subrngmre 20497 | The subrings of a non-unit... |
| subsubrng 20498 | A subring of a subring is ... |
| subsubrng2 20499 | The set of subrings of a s... |
| rhmimasubrnglem 20500 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20501 | The homomorphic image of a... |
| cntzsubrng 20502 | Centralizers in a non-unit... |
| subrngpropd 20503 | If two structures have the... |
| issubrg 20506 | The subring predicate. (C... |
| subrgss 20507 | A subring is a subset. (C... |
| subrgid 20508 | Every ring is a subring of... |
| subrgring 20509 | A subring is a ring. (Con... |
| subrgcrng 20510 | A subring of a commutative... |
| subrgrcl 20511 | Reverse closure for a subr... |
| subrgsubg 20512 | A subring is a subgroup. ... |
| subrgsubrng 20513 | A subring of a unital ring... |
| subrg0 20514 | A subring always has the s... |
| subrg1cl 20515 | A subring contains the mul... |
| subrgbas 20516 | Base set of a subring stru... |
| subrg1 20517 | A subring always has the s... |
| subrgacl 20518 | A subring is closed under ... |
| subrgmcl 20519 | A subring is closed under ... |
| subrgsubm 20520 | A subring is a submonoid o... |
| subrgdvds 20521 | If an element divides anot... |
| subrguss 20522 | A unit of a subring is a u... |
| subrginv 20523 | A subring always has the s... |
| subrgdv 20524 | A subring always has the s... |
| subrgunit 20525 | An element of a ring is a ... |
| subrgugrp 20526 | The units of a subring for... |
| issubrg2 20527 | Characterize the subrings ... |
| opprsubrg 20528 | Being a subring is a symme... |
| subrgnzr 20529 | A subring of a nonzero rin... |
| subrgint 20530 | The intersection of a none... |
| subrgin 20531 | The intersection of two su... |
| subrgmre 20532 | The subrings of a ring are... |
| subsubrg 20533 | A subring of a subring is ... |
| subsubrg2 20534 | The set of subrings of a s... |
| issubrg3 20535 | A subring is an additive s... |
| resrhm 20536 | Restriction of a ring homo... |
| resrhm2b 20537 | Restriction of the codomai... |
| rhmeql 20538 | The equalizer of two ring ... |
| rhmima 20539 | The homomorphic image of a... |
| rnrhmsubrg 20540 | The range of a ring homomo... |
| cntzsubr 20541 | Centralizers in a ring are... |
| pwsdiagrhm 20542 | Diagonal homomorphism into... |
| subrgpropd 20543 | If two structures have the... |
| rhmpropd 20544 | Ring homomorphism depends ... |
| rgspnval 20547 | Value of the ring-span of ... |
| rgspncl 20548 | The ring-span of a set is ... |
| rgspnssid 20549 | The ring-span of a set con... |
| rgspnmin 20550 | The ring-span is contained... |
| rngcval 20553 | Value of the category of n... |
| rnghmresfn 20554 | The class of non-unital ri... |
| rnghmresel 20555 | An element of the non-unit... |
| rngcbas 20556 | Set of objects of the cate... |
| rngchomfval 20557 | Set of arrows of the categ... |
| rngchom 20558 | Set of arrows of the categ... |
| elrngchom 20559 | A morphism of non-unital r... |
| rngchomfeqhom 20560 | The functionalized Hom-set... |
| rngccofval 20561 | Composition in the categor... |
| rngcco 20562 | Composition in the categor... |
| dfrngc2 20563 | Alternate definition of th... |
| rnghmsscmap2 20564 | The non-unital ring homomo... |
| rnghmsscmap 20565 | The non-unital ring homomo... |
| rnghmsubcsetclem1 20566 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 20567 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 20568 | The non-unital ring homomo... |
| rngccat 20569 | The category of non-unital... |
| rngcid 20570 | The identity arrow in the ... |
| rngcsect 20571 | A section in the category ... |
| rngcinv 20572 | An inverse in the category... |
| rngciso 20573 | An isomorphism in the cate... |
| rngcifuestrc 20574 | The "inclusion functor" fr... |
| funcrngcsetc 20575 | The "natural forgetful fun... |
| funcrngcsetcALT 20576 | Alternate proof of ~ funcr... |
| zrinitorngc 20577 | The zero ring is an initia... |
| zrtermorngc 20578 | The zero ring is a termina... |
| zrzeroorngc 20579 | The zero ring is a zero ob... |
| ringcval 20582 | Value of the category of u... |
| rhmresfn 20583 | The class of unital ring h... |
| rhmresel 20584 | An element of the unital r... |
| ringcbas 20585 | Set of objects of the cate... |
| ringchomfval 20586 | Set of arrows of the categ... |
| ringchom 20587 | Set of arrows of the categ... |
| elringchom 20588 | A morphism of unital rings... |
| ringchomfeqhom 20589 | The functionalized Hom-set... |
| ringccofval 20590 | Composition in the categor... |
| ringcco 20591 | Composition in the categor... |
| dfringc2 20592 | Alternate definition of th... |
| rhmsscmap2 20593 | The unital ring homomorphi... |
| rhmsscmap 20594 | The unital ring homomorphi... |
| rhmsubcsetclem1 20595 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 20596 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 20597 | The unital ring homomorphi... |
| ringccat 20598 | The category of unital rin... |
| ringcid 20599 | The identity arrow in the ... |
| rhmsscrnghm 20600 | The unital ring homomorphi... |
| rhmsubcrngclem1 20601 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 20602 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 20603 | The unital ring homomorphi... |
| rngcresringcat 20604 | The restriction of the cat... |
| ringcsect 20605 | A section in the category ... |
| ringcinv 20606 | An inverse in the category... |
| ringciso 20607 | An isomorphism in the cate... |
| ringcbasbas 20608 | An element of the base set... |
| funcringcsetc 20609 | The "natural forgetful fun... |
| zrtermoringc 20610 | The zero ring is a termina... |
| zrninitoringc 20611 | The zero ring is not an in... |
| srhmsubclem1 20612 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 20613 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 20614 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 20615 | According to ~ df-subc , t... |
| sringcat 20616 | The restriction of the cat... |
| crhmsubc 20617 | According to ~ df-subc , t... |
| cringcat 20618 | The restriction of the cat... |
| rngcrescrhm 20619 | The category of non-unital... |
| rhmsubclem1 20620 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 20621 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 20622 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 20623 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 20624 | According to ~ df-subc , t... |
| rhmsubccat 20625 | The restriction of the cat... |
| rrgval 20632 | Value of the set or left-r... |
| isrrg 20633 | Membership in the set of l... |
| rrgeq0i 20634 | Property of a left-regular... |
| rrgeq0 20635 | Left-multiplication by a l... |
| rrgsupp 20636 | Left multiplication by a l... |
| rrgss 20637 | Left-regular elements are ... |
| unitrrg 20638 | Units are regular elements... |
| rrgnz 20639 | In a nonzero ring, the zer... |
| isdomn 20640 | Expand definition of a dom... |
| domnnzr 20641 | A domain is a nonzero ring... |
| domnring 20642 | A domain is a ring. (Cont... |
| domneq0 20643 | In a domain, a product is ... |
| domnmuln0 20644 | In a domain, a product of ... |
| isdomn5 20645 | The equivalence between th... |
| isdomn2 20646 | A ring is a domain iff all... |
| isdomn2OLD 20647 | Obsolete version of ~ isdo... |
| domnrrg 20648 | In a domain, a nonzero ele... |
| isdomn6 20649 | A ring is a domain iff the... |
| isdomn3 20650 | Nonzero elements form a mu... |
| isdomn4 20651 | A ring is a domain iff it ... |
| opprdomnb 20652 | A class is a domain if and... |
| opprdomn 20653 | The opposite of a domain i... |
| isdomn4r 20654 | A ring is a domain iff it ... |
| domnlcanb 20655 | Left-cancellation law for ... |
| domnlcan 20656 | Left-cancellation law for ... |
| domnrcanb 20657 | Right-cancellation law for... |
| domnrcan 20658 | Right-cancellation law for... |
| domneq0r 20659 | Right multiplication by a ... |
| isidom 20660 | An integral domain is a co... |
| idomdomd 20661 | An integral domain is a do... |
| idomcringd 20662 | An integral domain is a co... |
| idomringd 20663 | An integral domain is a ri... |
| isdrng 20668 | The predicate "is a divisi... |
| drngunit 20669 | Elementhood in the set of ... |
| drngui 20670 | The set of units of a divi... |
| drngring 20671 | A division ring is a ring.... |
| drngringd 20672 | A division ring is a ring.... |
| drnggrpd 20673 | A division ring is a group... |
| drnggrp 20674 | A division ring is a group... |
| isfld 20675 | A field is a commutative d... |
| flddrngd 20676 | A field is a division ring... |
| fldcrngd 20677 | A field is a commutative r... |
| isdrng2 20678 | A division ring can equiva... |
| drngprop 20679 | If two structures have the... |
| drngmgp 20680 | A division ring contains a... |
| drngid 20681 | A division ring's unity is... |
| drngunz 20682 | A division ring's unity is... |
| drngnzr 20683 | A division ring is a nonze... |
| drngdomn 20684 | A division ring is a domai... |
| drngmcl 20685 | The product of two nonzero... |
| drngmclOLD 20686 | Obsolete version of ~ drng... |
| drngid2 20687 | Properties showing that an... |
| drnginvrcl 20688 | Closure of the multiplicat... |
| drnginvrn0 20689 | The multiplicative inverse... |
| drnginvrcld 20690 | Closure of the multiplicat... |
| drnginvrl 20691 | Property of the multiplica... |
| drnginvrr 20692 | Property of the multiplica... |
| drnginvrld 20693 | Property of the multiplica... |
| drnginvrrd 20694 | Property of the multiplica... |
| drngmul0or 20695 | A product is zero iff one ... |
| drngmul0orOLD 20696 | Obsolete version of ~ drng... |
| drngmulne0 20697 | A product is nonzero iff b... |
| drngmuleq0 20698 | An element is zero iff its... |
| opprdrng 20699 | The opposite of a division... |
| isdrngd 20700 | Properties that characteri... |
| isdrngrd 20701 | Properties that characteri... |
| isdrngdOLD 20702 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20703 | Obsolete version of ~ isdr... |
| drngpropd 20704 | If two structures have the... |
| fldpropd 20705 | If two structures have the... |
| fldidom 20706 | A field is an integral dom... |
| fidomndrnglem 20707 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20708 | A finite domain is a divis... |
| fiidomfld 20709 | A finite integral domain i... |
| rng1nnzr 20710 | The (smallest) structure r... |
| ring1zr 20711 | The only (unital) ring wit... |
| rngen1zr 20712 | The only (unital) ring wit... |
| ringen1zr 20713 | The only unital ring with ... |
| rng1nfld 20714 | The zero ring is not a fie... |
| issubdrg 20715 | Characterize the subfields... |
| drhmsubc 20716 | According to ~ df-subc , t... |
| drngcat 20717 | The restriction of the cat... |
| fldcat 20718 | The restriction of the cat... |
| fldc 20719 | The restriction of the cat... |
| fldhmsubc 20720 | According to ~ df-subc , t... |
| issdrg 20723 | Property of a division sub... |
| sdrgrcl 20724 | Reverse closure for a sub-... |
| sdrgdrng 20725 | A sub-division-ring is a d... |
| sdrgsubrg 20726 | A sub-division-ring is a s... |
| sdrgid 20727 | Every division ring is a d... |
| sdrgss 20728 | A division subring is a su... |
| sdrgbas 20729 | Base set of a sub-division... |
| issdrg2 20730 | Property of a division sub... |
| sdrgunit 20731 | A unit of a sub-division-r... |
| imadrhmcl 20732 | The image of a (nontrivial... |
| fldsdrgfld 20733 | A sub-division-ring of a f... |
| acsfn1p 20734 | Construction of a closure ... |
| subrgacs 20735 | Closure property of subrin... |
| sdrgacs 20736 | Closure property of divisi... |
| cntzsdrg 20737 | Centralizers in division r... |
| subdrgint 20738 | The intersection of a none... |
| sdrgint 20739 | The intersection of a none... |
| primefld 20740 | The smallest sub division ... |
| primefld0cl 20741 | The prime field contains t... |
| primefld1cl 20742 | The prime field contains t... |
| abvfval 20745 | Value of the set of absolu... |
| isabv 20746 | Elementhood in the set of ... |
| isabvd 20747 | Properties that determine ... |
| abvrcl 20748 | Reverse closure for the ab... |
| abvfge0 20749 | An absolute value is a fun... |
| abvf 20750 | An absolute value is a fun... |
| abvcl 20751 | An absolute value is a fun... |
| abvge0 20752 | The absolute value of a nu... |
| abveq0 20753 | The value of an absolute v... |
| abvne0 20754 | The absolute value of a no... |
| abvgt0 20755 | The absolute value of a no... |
| abvmul 20756 | An absolute value distribu... |
| abvtri 20757 | An absolute value satisfie... |
| abv0 20758 | The absolute value of zero... |
| abv1z 20759 | The absolute value of one ... |
| abv1 20760 | The absolute value of one ... |
| abvneg 20761 | The absolute value of a ne... |
| abvsubtri 20762 | An absolute value satisfie... |
| abvrec 20763 | The absolute value distrib... |
| abvdiv 20764 | The absolute value distrib... |
| abvdom 20765 | Any ring with an absolute ... |
| abvres 20766 | The restriction of an abso... |
| abvtrivd 20767 | The trivial absolute value... |
| abvtrivg 20768 | The trivial absolute value... |
| abvtriv 20769 | The trivial absolute value... |
| abvpropd 20770 | If two structures have the... |
| abvn0b 20771 | Another characterization o... |
| staffval 20776 | The functionalization of t... |
| stafval 20777 | The functionalization of t... |
| staffn 20778 | The functionalization is e... |
| issrng 20779 | The predicate "is a star r... |
| srngrhm 20780 | The involution function in... |
| srngring 20781 | A star ring is a ring. (C... |
| srngcnv 20782 | The involution function in... |
| srngf1o 20783 | The involution function in... |
| srngcl 20784 | The involution function in... |
| srngnvl 20785 | The involution function in... |
| srngadd 20786 | The involution function in... |
| srngmul 20787 | The involution function in... |
| srng1 20788 | The conjugate of the ring ... |
| srng0 20789 | The conjugate of the ring ... |
| issrngd 20790 | Properties that determine ... |
| idsrngd 20791 | A commutative ring is a st... |
| isorng 20796 | An ordered ring is a ring ... |
| orngring 20797 | An ordered ring is a ring.... |
| orngogrp 20798 | An ordered ring is an orde... |
| isofld 20799 | An ordered field is a fiel... |
| orngmul 20800 | In an ordered ring, the or... |
| orngsqr 20801 | In an ordered ring, all sq... |
| ornglmulle 20802 | In an ordered ring, multip... |
| orngrmulle 20803 | In an ordered ring, multip... |
| ornglmullt 20804 | In an ordered ring, multip... |
| orngrmullt 20805 | In an ordered ring, multip... |
| orngmullt 20806 | In an ordered ring, the st... |
| ofldfld 20807 | An ordered field is a fiel... |
| ofldtos 20808 | An ordered field is a tota... |
| orng0le1 20809 | In an ordered ring, the ri... |
| ofldlt1 20810 | In an ordered field, the r... |
| suborng 20811 | Every subring of an ordere... |
| subofld 20812 | Every subfield of an order... |
| islmod 20817 | The predicate "is a left m... |
| lmodlema 20818 | Lemma for properties of a ... |
| islmodd 20819 | Properties that determine ... |
| lmodgrp 20820 | A left module is a group. ... |
| lmodring 20821 | The scalar component of a ... |
| lmodfgrp 20822 | The scalar component of a ... |
| lmodgrpd 20823 | A left module is a group. ... |
| lmodbn0 20824 | The base set of a left mod... |
| lmodacl 20825 | Closure of ring addition f... |
| lmodmcl 20826 | Closure of ring multiplica... |
| lmodsn0 20827 | The set of scalars in a le... |
| lmodvacl 20828 | Closure of vector addition... |
| lmodass 20829 | Left module vector sum is ... |
| lmodlcan 20830 | Left cancellation law for ... |
| lmodvscl 20831 | Closure of scalar product ... |
| lmodvscld 20832 | Closure of scalar product ... |
| scaffval 20833 | The scalar multiplication ... |
| scafval 20834 | The scalar multiplication ... |
| scafeq 20835 | If the scalar multiplicati... |
| scaffn 20836 | The scalar multiplication ... |
| lmodscaf 20837 | The scalar multiplication ... |
| lmodvsdi 20838 | Distributive law for scala... |
| lmodvsdir 20839 | Distributive law for scala... |
| lmodvsass 20840 | Associative law for scalar... |
| lmod0cl 20841 | The ring zero in a left mo... |
| lmod1cl 20842 | The ring unity in a left m... |
| lmodvs1 20843 | Scalar product with the ri... |
| lmod0vcl 20844 | The zero vector is a vecto... |
| lmod0vlid 20845 | Left identity law for the ... |
| lmod0vrid 20846 | Right identity law for the... |
| lmod0vid 20847 | Identity equivalent to the... |
| lmod0vs 20848 | Zero times a vector is the... |
| lmodvs0 20849 | Anything times the zero ve... |
| lmodvsmmulgdi 20850 | Distributive law for a gro... |
| lmodfopnelem1 20851 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20852 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20853 | The (functionalized) opera... |
| lcomf 20854 | A linear-combination sum i... |
| lcomfsupp 20855 | A linear-combination sum i... |
| lmodvnegcl 20856 | Closure of vector negative... |
| lmodvnegid 20857 | Addition of a vector with ... |
| lmodvneg1 20858 | Minus 1 times a vector is ... |
| lmodvsneg 20859 | Multiplication of a vector... |
| lmodvsubcl 20860 | Closure of vector subtract... |
| lmodcom 20861 | Left module vector sum is ... |
| lmodabl 20862 | A left module is an abelia... |
| lmodcmn 20863 | A left module is a commuta... |
| lmodnegadd 20864 | Distribute negation throug... |
| lmod4 20865 | Commutative/associative la... |
| lmodvsubadd 20866 | Relationship between vecto... |
| lmodvaddsub4 20867 | Vector addition/subtractio... |
| lmodvpncan 20868 | Addition/subtraction cance... |
| lmodvnpcan 20869 | Cancellation law for vecto... |
| lmodvsubval2 20870 | Value of vector subtractio... |
| lmodsubvs 20871 | Subtraction of a scalar pr... |
| lmodsubdi 20872 | Scalar multiplication dist... |
| lmodsubdir 20873 | Scalar multiplication dist... |
| lmodsubeq0 20874 | If the difference between ... |
| lmodsubid 20875 | Subtraction of a vector fr... |
| lmodvsghm 20876 | Scalar multiplication of t... |
| lmodprop2d 20877 | If two structures have the... |
| lmodpropd 20878 | If two structures have the... |
| gsumvsmul 20879 | Pull a scalar multiplicati... |
| mptscmfsupp0 20880 | A mapping to a scalar prod... |
| mptscmfsuppd 20881 | A function mapping to a sc... |
| rmodislmodlem 20882 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20883 | The right module ` R ` ind... |
| lssset 20886 | The set of all (not necess... |
| islss 20887 | The predicate "is a subspa... |
| islssd 20888 | Properties that determine ... |
| lssss 20889 | A subspace is a set of vec... |
| lssel 20890 | A subspace member is a vec... |
| lss1 20891 | The set of vectors in a le... |
| lssuni 20892 | The union of all subspaces... |
| lssn0 20893 | A subspace is not empty. ... |
| 00lss 20894 | The empty structure has no... |
| lsscl 20895 | Closure property of a subs... |
| lssvacl 20896 | Closure of vector addition... |
| lssvsubcl 20897 | Closure of vector subtract... |
| lssvancl1 20898 | Non-closure: if one vector... |
| lssvancl2 20899 | Non-closure: if one vector... |
| lss0cl 20900 | The zero vector belongs to... |
| lsssn0 20901 | The singleton of the zero ... |
| lss0ss 20902 | The zero subspace is inclu... |
| lssle0 20903 | No subspace is smaller tha... |
| lssne0 20904 | A nonzero subspace has a n... |
| lssvneln0 20905 | A vector ` X ` which doesn... |
| lssneln0 20906 | A vector ` X ` which doesn... |
| lssssr 20907 | Conclude subspace ordering... |
| lssvscl 20908 | Closure of scalar product ... |
| lssvnegcl 20909 | Closure of negative vector... |
| lsssubg 20910 | All subspaces are subgroup... |
| lsssssubg 20911 | All subspaces are subgroup... |
| islss3 20912 | A linear subspace of a mod... |
| lsslmod 20913 | A submodule is a module. ... |
| lsslss 20914 | The subspaces of a subspac... |
| islss4 20915 | A linear subspace is a sub... |
| lss1d 20916 | One-dimensional subspace (... |
| lssintcl 20917 | The intersection of a none... |
| lssincl 20918 | The intersection of two su... |
| lssmre 20919 | The subspaces of a module ... |
| lssacs 20920 | Submodules are an algebrai... |
| prdsvscacl 20921 | Pointwise scalar multiplic... |
| prdslmodd 20922 | The product of a family of... |
| pwslmod 20923 | A structure power of a lef... |
| lspfval 20926 | The span function for a le... |
| lspf 20927 | The span function on a lef... |
| lspval 20928 | The span of a set of vecto... |
| lspcl 20929 | The span of a set of vecto... |
| lspsncl 20930 | The span of a singleton is... |
| lspprcl 20931 | The span of a pair is a su... |
| lsptpcl 20932 | The span of an unordered t... |
| lspsnsubg 20933 | The span of a singleton is... |
| 00lsp 20934 | ~ fvco4i lemma for linear ... |
| lspid 20935 | The span of a subspace is ... |
| lspssv 20936 | A span is a set of vectors... |
| lspss 20937 | Span preserves subset orde... |
| lspssid 20938 | A set of vectors is a subs... |
| lspidm 20939 | The span of a set of vecto... |
| lspun 20940 | The span of union is the s... |
| lspssp 20941 | If a set of vectors is a s... |
| mrclsp 20942 | Moore closure generalizes ... |
| lspsnss 20943 | The span of the singleton ... |
| ellspsn3 20944 | A member of the span of th... |
| lspprss 20945 | The span of a pair of vect... |
| lspsnid 20946 | A vector belongs to the sp... |
| ellspsn6 20947 | Relationship between a vec... |
| ellspsn5b 20948 | Relationship between a vec... |
| ellspsn5 20949 | Relationship between a vec... |
| lspprid1 20950 | A member of a pair of vect... |
| lspprid2 20951 | A member of a pair of vect... |
| lspprvacl 20952 | The sum of two vectors bel... |
| lssats2 20953 | A way to express atomistic... |
| ellspsni 20954 | A scalar product with a ve... |
| lspsn 20955 | Span of the singleton of a... |
| ellspsn 20956 | Member of span of the sing... |
| lspsnvsi 20957 | Span of a scalar product o... |
| lspsnss2 20958 | Comparable spans of single... |
| lspsnneg 20959 | Negation does not change t... |
| lspsnsub 20960 | Swapping subtraction order... |
| lspsn0 20961 | Span of the singleton of t... |
| lsp0 20962 | Span of the empty set. (C... |
| lspuni0 20963 | Union of the span of the e... |
| lspun0 20964 | The span of a union with t... |
| lspsneq0 20965 | Span of the singleton is t... |
| lspsneq0b 20966 | Equal singleton spans impl... |
| lmodindp1 20967 | Two independent (non-colin... |
| lsslsp 20968 | Spans in submodules corres... |
| lsslspOLD 20969 | Obsolete version of ~ lssl... |
| lss0v 20970 | The zero vector in a submo... |
| lsspropd 20971 | If two structures have the... |
| lsppropd 20972 | If two structures have the... |
| reldmlmhm 20979 | Lemma for module homomorph... |
| lmimfn 20980 | Lemma for module isomorphi... |
| islmhm 20981 | Property of being a homomo... |
| islmhm3 20982 | Property of a module homom... |
| lmhmlem 20983 | Non-quantified consequence... |
| lmhmsca 20984 | A homomorphism of left mod... |
| lmghm 20985 | A homomorphism of left mod... |
| lmhmlmod2 20986 | A homomorphism of left mod... |
| lmhmlmod1 20987 | A homomorphism of left mod... |
| lmhmf 20988 | A homomorphism of left mod... |
| lmhmlin 20989 | A homomorphism of left mod... |
| lmodvsinv 20990 | Multiplication of a vector... |
| lmodvsinv2 20991 | Multiplying a negated vect... |
| islmhm2 20992 | A one-equation proof of li... |
| islmhmd 20993 | Deduction for a module hom... |
| 0lmhm 20994 | The constant zero linear f... |
| idlmhm 20995 | The identity function on a... |
| invlmhm 20996 | The negative function on a... |
| lmhmco 20997 | The composition of two mod... |
| lmhmplusg 20998 | The pointwise sum of two l... |
| lmhmvsca 20999 | The pointwise scalar produ... |
| lmhmf1o 21000 | A bijective module homomor... |
| lmhmima 21001 | The image of a subspace un... |
| lmhmpreima 21002 | The inverse image of a sub... |
| lmhmlsp 21003 | Homomorphisms preserve spa... |
| lmhmrnlss 21004 | The range of a homomorphis... |
| lmhmkerlss 21005 | The kernel of a homomorphi... |
| reslmhm 21006 | Restriction of a homomorph... |
| reslmhm2 21007 | Expansion of the codomain ... |
| reslmhm2b 21008 | Expansion of the codomain ... |
| lmhmeql 21009 | The equalizer of two modul... |
| lspextmo 21010 | A linear function is compl... |
| pwsdiaglmhm 21011 | Diagonal homomorphism into... |
| pwssplit0 21012 | Splitting for structure po... |
| pwssplit1 21013 | Splitting for structure po... |
| pwssplit2 21014 | Splitting for structure po... |
| pwssplit3 21015 | Splitting for structure po... |
| islmim 21016 | An isomorphism of left mod... |
| lmimf1o 21017 | An isomorphism of left mod... |
| lmimlmhm 21018 | An isomorphism of modules ... |
| lmimgim 21019 | An isomorphism of modules ... |
| islmim2 21020 | An isomorphism of left mod... |
| lmimcnv 21021 | The converse of a bijectiv... |
| brlmic 21022 | The relation "is isomorphi... |
| brlmici 21023 | Prove isomorphic by an exp... |
| lmiclcl 21024 | Isomorphism implies the le... |
| lmicrcl 21025 | Isomorphism implies the ri... |
| lmicsym 21026 | Module isomorphism is symm... |
| lmhmpropd 21027 | Module homomorphism depend... |
| islbs 21030 | The predicate " ` B ` is a... |
| lbsss 21031 | A basis is a set of vector... |
| lbsel 21032 | An element of a basis is a... |
| lbssp 21033 | The span of a basis is the... |
| lbsind 21034 | A basis is linearly indepe... |
| lbsind2 21035 | A basis is linearly indepe... |
| lbspss 21036 | No proper subset of a basi... |
| lsmcl 21037 | The sum of two subspaces i... |
| lsmspsn 21038 | Member of subspace sum of ... |
| lsmelval2 21039 | Subspace sum membership in... |
| lsmsp 21040 | Subspace sum in terms of s... |
| lsmsp2 21041 | Subspace sum of spans of s... |
| lsmssspx 21042 | Subspace sum (in its exten... |
| lsmpr 21043 | The span of a pair of vect... |
| lsppreli 21044 | A vector expressed as a su... |
| lsmelpr 21045 | Two ways to say that a vec... |
| lsppr0 21046 | The span of a vector paire... |
| lsppr 21047 | Span of a pair of vectors.... |
| lspprel 21048 | Member of the span of a pa... |
| lspprabs 21049 | Absorption of vector sum i... |
| lspvadd 21050 | The span of a vector sum i... |
| lspsntri 21051 | Triangle-type inequality f... |
| lspsntrim 21052 | Triangle-type inequality f... |
| lbspropd 21053 | If two structures have the... |
| pj1lmhm 21054 | The left projection functi... |
| pj1lmhm2 21055 | The left projection functi... |
| islvec 21058 | The predicate "is a left v... |
| lvecdrng 21059 | The set of scalars of a le... |
| lveclmod 21060 | A left vector space is a l... |
| lveclmodd 21061 | A vector space is a left m... |
| lvecgrpd 21062 | A vector space is a group.... |
| lsslvec 21063 | A vector subspace is a vec... |
| lmhmlvec 21064 | The property for modules t... |
| lvecvs0or 21065 | If a scalar product is zer... |
| lvecvsn0 21066 | A scalar product is nonzer... |
| lssvs0or 21067 | If a scalar product belong... |
| lvecvscan 21068 | Cancellation law for scala... |
| lvecvscan2 21069 | Cancellation law for scala... |
| lvecinv 21070 | Invert coefficient of scal... |
| lspsnvs 21071 | A nonzero scalar product d... |
| lspsneleq 21072 | Membership relation that i... |
| lspsncmp 21073 | Comparable spans of nonzer... |
| lspsnne1 21074 | Two ways to express that v... |
| lspsnne2 21075 | Two ways to express that v... |
| lspsnnecom 21076 | Swap two vectors with diff... |
| lspabs2 21077 | Absorption law for span of... |
| lspabs3 21078 | Absorption law for span of... |
| lspsneq 21079 | Equal spans of singletons ... |
| lspsneu 21080 | Nonzero vectors with equal... |
| ellspsn4 21081 | A member of the span of th... |
| lspdisj 21082 | The span of a vector not i... |
| lspdisjb 21083 | A nonzero vector is not in... |
| lspdisj2 21084 | Unequal spans are disjoint... |
| lspfixed 21085 | Show membership in the spa... |
| lspexch 21086 | Exchange property for span... |
| lspexchn1 21087 | Exchange property for span... |
| lspexchn2 21088 | Exchange property for span... |
| lspindpi 21089 | Partial independence prope... |
| lspindp1 21090 | Alternate way to say 3 vec... |
| lspindp2l 21091 | Alternate way to say 3 vec... |
| lspindp2 21092 | Alternate way to say 3 vec... |
| lspindp3 21093 | Independence of 2 vectors ... |
| lspindp4 21094 | (Partial) independence of ... |
| lvecindp 21095 | Compute the ` X ` coeffici... |
| lvecindp2 21096 | Sums of independent vector... |
| lspsnsubn0 21097 | Unequal singleton spans im... |
| lsmcv 21098 | Subspace sum has the cover... |
| lspsolvlem 21099 | Lemma for ~ lspsolv . (Co... |
| lspsolv 21100 | If ` X ` is in the span of... |
| lssacsex 21101 | In a vector space, subspac... |
| lspsnat 21102 | There is no subspace stric... |
| lspsncv0 21103 | The span of a singleton co... |
| lsppratlem1 21104 | Lemma for ~ lspprat . Let... |
| lsppratlem2 21105 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 21106 | Lemma for ~ lspprat . In ... |
| lsppratlem4 21107 | Lemma for ~ lspprat . In ... |
| lsppratlem5 21108 | Lemma for ~ lspprat . Com... |
| lsppratlem6 21109 | Lemma for ~ lspprat . Neg... |
| lspprat 21110 | A proper subspace of the s... |
| islbs2 21111 | An equivalent formulation ... |
| islbs3 21112 | An equivalent formulation ... |
| lbsacsbs 21113 | Being a basis in a vector ... |
| lvecdim 21114 | The dimension theorem for ... |
| lbsextlem1 21115 | Lemma for ~ lbsext . The ... |
| lbsextlem2 21116 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 21117 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 21118 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 21119 | For any linearly independe... |
| lbsext 21120 | For any linearly independe... |
| lbsexg 21121 | Every vector space has a b... |
| lbsex 21122 | Every vector space has a b... |
| lvecprop2d 21123 | If two structures have the... |
| lvecpropd 21124 | If two structures have the... |
| sraval 21129 | Lemma for ~ srabase throug... |
| sralem 21130 | Lemma for ~ srabase and si... |
| srabase 21131 | Base set of a subring alge... |
| sraaddg 21132 | Additive operation of a su... |
| sramulr 21133 | Multiplicative operation o... |
| srasca 21134 | The set of scalars of a su... |
| sravsca 21135 | The scalar product operati... |
| sraip 21136 | The inner product operatio... |
| sratset 21137 | Topology component of a su... |
| sratopn 21138 | Topology component of a su... |
| srads 21139 | Distance function of a sub... |
| sraring 21140 | Condition for a subring al... |
| sralmod 21141 | The subring algebra is a l... |
| sralmod0 21142 | The subring module inherit... |
| issubrgd 21143 | Prove a subring by closure... |
| rlmfn 21144 | ` ringLMod ` is a function... |
| rlmval 21145 | Value of the ring module. ... |
| rlmval2 21146 | Value of the ring module e... |
| rlmbas 21147 | Base set of the ring modul... |
| rlmplusg 21148 | Vector addition in the rin... |
| rlm0 21149 | Zero vector in the ring mo... |
| rlmsub 21150 | Subtraction in the ring mo... |
| rlmmulr 21151 | Ring multiplication in the... |
| rlmsca 21152 | Scalars in the ring module... |
| rlmsca2 21153 | Scalars in the ring module... |
| rlmvsca 21154 | Scalar multiplication in t... |
| rlmtopn 21155 | Topology component of the ... |
| rlmds 21156 | Metric component of the ri... |
| rlmlmod 21157 | The ring module is a modul... |
| rlmlvec 21158 | The ring module over a div... |
| rlmlsm 21159 | Subgroup sum of the ring m... |
| rlmvneg 21160 | Vector negation in the rin... |
| rlmscaf 21161 | Functionalized scalar mult... |
| ixpsnbasval 21162 | The value of an infinite C... |
| lidlval 21167 | Value of the set of ring i... |
| rspval 21168 | Value of the ring span fun... |
| lidlss 21169 | An ideal is a subset of th... |
| lidlssbas 21170 | The base set of the restri... |
| lidlbas 21171 | A (left) ideal of a ring i... |
| islidl 21172 | Predicate of being a (left... |
| rnglidlmcl 21173 | A (left) ideal containing ... |
| rngridlmcl 21174 | A right ideal (which is a ... |
| dflidl2rng 21175 | Alternate (the usual textb... |
| isridlrng 21176 | A right ideal is a left id... |
| lidl0cl 21177 | An ideal contains 0. (Con... |
| lidlacl 21178 | An ideal is closed under a... |
| lidlnegcl 21179 | An ideal contains negative... |
| lidlsubg 21180 | An ideal is a subgroup of ... |
| lidlsubcl 21181 | An ideal is closed under s... |
| lidlmcl 21182 | An ideal is closed under l... |
| lidl1el 21183 | An ideal contains 1 iff it... |
| dflidl2 21184 | Alternate (the usual textb... |
| lidl0ALT 21185 | Alternate proof for ~ lidl... |
| rnglidl0 21186 | Every non-unital ring cont... |
| lidl0 21187 | Every ring contains a zero... |
| lidl1ALT 21188 | Alternate proof for ~ lidl... |
| rnglidl1 21189 | The base set of every non-... |
| lidl1 21190 | Every ring contains a unit... |
| lidlacs 21191 | The ideal system is an alg... |
| rspcl 21192 | The span of a set of ring ... |
| rspssid 21193 | The span of a set of ring ... |
| rsp1 21194 | The span of the identity e... |
| rsp0 21195 | The span of the zero eleme... |
| rspssp 21196 | The ideal span of a set of... |
| elrspsn 21197 | Membership in a principal ... |
| mrcrsp 21198 | Moore closure generalizes ... |
| lidlnz 21199 | A nonzero ideal contains a... |
| drngnidl 21200 | A division ring has only t... |
| lidlrsppropd 21201 | The left ideals and ring s... |
| rnglidlmmgm 21202 | The multiplicative group o... |
| rnglidlmsgrp 21203 | The multiplicative group o... |
| rnglidlrng 21204 | A (left) ideal of a non-un... |
| lidlnsg 21205 | An ideal is a normal subgr... |
| 2idlval 21208 | Definition of a two-sided ... |
| isridl 21209 | A right ideal is a left id... |
| 2idlelb 21210 | Membership in a two-sided ... |
| 2idllidld 21211 | A two-sided ideal is a lef... |
| 2idlridld 21212 | A two-sided ideal is a rig... |
| df2idl2rng 21213 | Alternate (the usual textb... |
| df2idl2 21214 | Alternate (the usual textb... |
| ridl0 21215 | Every ring contains a zero... |
| ridl1 21216 | Every ring contains a unit... |
| 2idl0 21217 | Every ring contains a zero... |
| 2idl1 21218 | Every ring contains a unit... |
| 2idlss 21219 | A two-sided ideal is a sub... |
| 2idlbas 21220 | The base set of a two-side... |
| 2idlelbas 21221 | The base set of a two-side... |
| rng2idlsubrng 21222 | A two-sided ideal of a non... |
| rng2idlnsg 21223 | A two-sided ideal of a non... |
| rng2idl0 21224 | The zero (additive identit... |
| rng2idlsubgsubrng 21225 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21226 | A two-sided ideal of a non... |
| rng2idlsubg0 21227 | The zero (additive identit... |
| 2idlcpblrng 21228 | The coset equivalence rela... |
| 2idlcpbl 21229 | The coset equivalence rela... |
| qus2idrng 21230 | The quotient of a non-unit... |
| qus1 21231 | The multiplicative identit... |
| qusring 21232 | If ` S ` is a two-sided id... |
| qusrhm 21233 | If ` S ` is a two-sided id... |
| rhmpreimaidl 21234 | The preimage of an ideal b... |
| kerlidl 21235 | The kernel of a ring homom... |
| qusmul2idl 21236 | Value of the ring operatio... |
| crngridl 21237 | In a commutative ring, the... |
| crng2idl 21238 | In a commutative ring, a t... |
| qusmulrng 21239 | Value of the multiplicatio... |
| quscrng 21240 | The quotient of a commutat... |
| qusmulcrng 21241 | Value of the ring operatio... |
| rhmqusnsg 21242 | The mapping ` J ` induced ... |
| rngqiprng1elbas 21243 | The ring unity of a two-si... |
| rngqiprngghmlem1 21244 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21245 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21246 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21247 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21248 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21249 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21250 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21251 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21252 | The base set of the produc... |
| rngqiprng 21253 | The product of the quotien... |
| rngqiprngimf 21254 | ` F ` is a function from (... |
| rngqiprngimfv 21255 | The value of the function ... |
| rngqiprngghm 21256 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21257 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21258 | ` F ` is a function from (... |
| rngqiprnglin 21259 | ` F ` is linear with respe... |
| rngqiprngho 21260 | ` F ` is a homomorphism of... |
| rngqiprngim 21261 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21262 | The unity of a two-sided i... |
| rngringbdlem1 21263 | In a unital ring, the quot... |
| rngringbdlem2 21264 | A non-unital ring is unita... |
| rngringbd 21265 | A non-unital ring is unita... |
| ring2idlqus 21266 | For every unital ring ther... |
| ring2idlqusb 21267 | A non-unital ring is unita... |
| rngqiprngfulem1 21268 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21269 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21270 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21271 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21272 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21273 | The ring unity of the prod... |
| rngqiprngfu 21274 | The function value of ` F ... |
| rngqiprngu 21275 | If a non-unital ring has a... |
| ring2idlqus1 21276 | If a non-unital ring has a... |
| lpival 21281 | Value of the set of princi... |
| islpidl 21282 | Property of being a princi... |
| lpi0 21283 | The zero ideal is always p... |
| lpi1 21284 | The unit ideal is always p... |
| islpir 21285 | Principal ideal rings are ... |
| lpiss 21286 | Principal ideals are a sub... |
| islpir2 21287 | Principal ideal rings are ... |
| lpirring 21288 | Principal ideal rings are ... |
| drnglpir 21289 | Division rings are princip... |
| rspsn 21290 | Membership in principal id... |
| lidldvgen 21291 | An element generates an id... |
| lpigen 21292 | An ideal is principal iff ... |
| cnfldstr 21313 | The field of complex numbe... |
| cnfldex 21314 | The field of complex numbe... |
| cnfldbas 21315 | The base set of the field ... |
| mpocnfldadd 21316 | The addition operation of ... |
| cnfldadd 21317 | The addition operation of ... |
| mpocnfldmul 21318 | The multiplication operati... |
| cnfldmul 21319 | The multiplication operati... |
| cnfldcj 21320 | The conjugation operation ... |
| cnfldtset 21321 | The topology component of ... |
| cnfldle 21322 | The ordering of the field ... |
| cnfldds 21323 | The metric of the field of... |
| cnfldunif 21324 | The uniform structure comp... |
| cnfldfun 21325 | The field of complex numbe... |
| cnfldfunALT 21326 | The field of complex numbe... |
| dfcnfldOLD 21327 | Obsolete version of ~ df-c... |
| cnfldstrOLD 21328 | Obsolete version of ~ cnfl... |
| cnfldexOLD 21329 | Obsolete version of ~ cnfl... |
| cnfldbasOLD 21330 | Obsolete version of ~ cnfl... |
| cnfldaddOLD 21331 | Obsolete version of ~ cnfl... |
| cnfldmulOLD 21332 | Obsolete version of ~ cnfl... |
| cnfldcjOLD 21333 | Obsolete version of ~ cnfl... |
| cnfldtsetOLD 21334 | Obsolete version of ~ cnfl... |
| cnfldleOLD 21335 | Obsolete version of ~ cnfl... |
| cnflddsOLD 21336 | Obsolete version of ~ cnfl... |
| cnfldunifOLD 21337 | Obsolete version of ~ cnfl... |
| cnfldfunOLD 21338 | Obsolete version of ~ cnfl... |
| cnfldfunALTOLD 21339 | Obsolete version of ~ cnfl... |
| xrsstr 21340 | The extended real structur... |
| xrsex 21341 | The extended real structur... |
| xrsadd 21342 | The addition operation of ... |
| xrsmul 21343 | The multiplication operati... |
| xrstset 21344 | The topology component of ... |
| cncrng 21345 | The complex numbers form a... |
| cncrngOLD 21346 | Obsolete version of ~ cncr... |
| cnring 21347 | The complex numbers form a... |
| xrsmcmn 21348 | The "multiplicative group"... |
| cnfld0 21349 | Zero is the zero element o... |
| cnfld1 21350 | One is the unity element o... |
| cnfld1OLD 21351 | Obsolete version of ~ cnfl... |
| cnfldneg 21352 | The additive inverse in th... |
| cnfldplusf 21353 | The functionalized additio... |
| cnfldsub 21354 | The subtraction operator i... |
| cndrng 21355 | The complex numbers form a... |
| cndrngOLD 21356 | Obsolete version of ~ cndr... |
| cnflddiv 21357 | The division operation in ... |
| cnflddivOLD 21358 | Obsolete version of ~ cnfl... |
| cnfldinv 21359 | The multiplicative inverse... |
| cnfldmulg 21360 | The group multiple functio... |
| cnfldexp 21361 | The exponentiation operato... |
| cnsrng 21362 | The complex numbers form a... |
| xrsmgm 21363 | The "additive group" of th... |
| xrsnsgrp 21364 | The "additive group" of th... |
| xrsmgmdifsgrp 21365 | The "additive group" of th... |
| xrsds 21366 | The metric of the extended... |
| xrsdsval 21367 | The metric of the extended... |
| xrsdsreval 21368 | The metric of the extended... |
| xrsdsreclblem 21369 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21370 | The metric of the extended... |
| cnsubmlem 21371 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21372 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21373 | Lemma for ~ resubdrg and f... |
| cnsubrglemOLD 21374 | Obsolete version of ~ cnsu... |
| cnsubdrglem 21375 | Lemma for ~ resubdrg and f... |
| qsubdrg 21376 | The rational numbers form ... |
| zsubrg 21377 | The integers form a subrin... |
| gzsubrg 21378 | The gaussian integers form... |
| nn0subm 21379 | The nonnegative integers f... |
| rege0subm 21380 | The nonnegative reals form... |
| absabv 21381 | The regular absolute value... |
| zsssubrg 21382 | The integers are a subset ... |
| qsssubdrg 21383 | The rational numbers are a... |
| cnsubrg 21384 | There are no subrings of t... |
| cnmgpabl 21385 | The unit group of the comp... |
| cnmgpid 21386 | The group identity element... |
| cnmsubglem 21387 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21388 | The positive reals form a ... |
| gzrngunitlem 21389 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21390 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21391 | Relate a group sum on ` CC... |
| regsumfsum 21392 | Relate a group sum on ` ( ... |
| expmhm 21393 | Exponentiation is a monoid... |
| nn0srg 21394 | The nonnegative integers f... |
| rge0srg 21395 | The nonnegative real numbe... |
| xrge0plusg 21396 | The additive law of the ex... |
| xrs1mnd 21397 | The extended real numbers,... |
| xrs10 21398 | The zero of the extended r... |
| xrs1cmn 21399 | The extended real numbers ... |
| xrge0subm 21400 | The nonnegative extended r... |
| xrge0cmn 21401 | The nonnegative extended r... |
| xrge0omnd 21402 | The nonnegative extended r... |
| zringcrng 21405 | The ring of integers is a ... |
| zringring 21406 | The ring of integers is a ... |
| zringrng 21407 | The ring of integers is a ... |
| zringabl 21408 | The ring of integers is an... |
| zringgrp 21409 | The ring of integers is an... |
| zringbas 21410 | The integers are the base ... |
| zringplusg 21411 | The addition operation of ... |
| zringsub 21412 | The subtraction of element... |
| zringmulg 21413 | The multiplication (group ... |
| zringmulr 21414 | The multiplication operati... |
| zring0 21415 | The zero element of the ri... |
| zring1 21416 | The unity element of the r... |
| zringnzr 21417 | The ring of integers is a ... |
| dvdsrzring 21418 | Ring divisibility in the r... |
| zringlpirlem1 21419 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21420 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21421 | Lemma for ~ zringlpir . A... |
| zringinvg 21422 | The additive inverse of an... |
| zringunit 21423 | The units of ` ZZ ` are th... |
| zringlpir 21424 | The integers are a princip... |
| zringndrg 21425 | The integers are not a div... |
| zringcyg 21426 | The integers are a cyclic ... |
| zringsubgval 21427 | Subtraction in the ring of... |
| zringmpg 21428 | The multiplicative group o... |
| prmirredlem 21429 | A positive integer is irre... |
| dfprm2 21430 | The positive irreducible e... |
| prmirred 21431 | The irreducible elements o... |
| expghm 21432 | Exponentiation is a group ... |
| mulgghm2 21433 | The powers of a group elem... |
| mulgrhm 21434 | The powers of the element ... |
| mulgrhm2 21435 | The powers of the element ... |
| irinitoringc 21436 | The ring of integers is an... |
| nzerooringczr 21437 | There is no zero object in... |
| pzriprnglem1 21438 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21439 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21440 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21441 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21442 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21443 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21444 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21445 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21446 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21447 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21448 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21449 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21450 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21451 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21452 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21453 | The ring unity of the ring... |
| pzriprng 21454 | The non-unital ring ` ( ZZ... |
| pzriprng1 21455 | The ring unity of the ring... |
| zrhval 21464 | Define the unique homomorp... |
| zrhval2 21465 | Alternate value of the ` Z... |
| zrhmulg 21466 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21467 | The ` ZRHom ` homomorphism... |
| zrhrhm 21468 | The ` ZRHom ` homomorphism... |
| zrh1 21469 | Interpretation of 1 in a r... |
| zrh0 21470 | Interpretation of 0 in a r... |
| zrhpropd 21471 | The ` ZZ ` ring homomorphi... |
| zlmval 21472 | Augment an abelian group w... |
| zlmlem 21473 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 21474 | Base set of a ` ZZ ` -modu... |
| zlmplusg 21475 | Group operation of a ` ZZ ... |
| zlmmulr 21476 | Ring operation of a ` ZZ `... |
| zlmsca 21477 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21478 | Scalar multiplication oper... |
| zlmlmod 21479 | The ` ZZ ` -module operati... |
| chrval 21480 | Definition substitution of... |
| chrcl 21481 | Closure of the characteris... |
| chrid 21482 | The canonical ` ZZ ` ring ... |
| chrdvds 21483 | The ` ZZ ` ring homomorphi... |
| chrcong 21484 | If two integers are congru... |
| dvdschrmulg 21485 | In a ring, any multiple of... |
| fermltlchr 21486 | A generalization of Fermat... |
| chrnzr 21487 | Nonzero rings are precisel... |
| chrrhm 21488 | The characteristic restric... |
| domnchr 21489 | The characteristic of a do... |
| znlidl 21490 | The set ` n ZZ ` is an ide... |
| zncrng2 21491 | Making a commutative ring ... |
| znval 21492 | The value of the ` Z/nZ ` ... |
| znle 21493 | The value of the ` Z/nZ ` ... |
| znval2 21494 | Self-referential expressio... |
| znbaslem 21495 | Lemma for ~ znbas . (Cont... |
| znbas2 21496 | The base set of ` Z/nZ ` i... |
| znadd 21497 | The additive structure of ... |
| znmul 21498 | The multiplicative structu... |
| znzrh 21499 | The ` ZZ ` ring homomorphi... |
| znbas 21500 | The base set of ` Z/nZ ` s... |
| zncrng 21501 | ` Z/nZ ` is a commutative ... |
| znzrh2 21502 | The ` ZZ ` ring homomorphi... |
| znzrhval 21503 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21504 | The ` ZZ ` ring homomorphi... |
| zncyg 21505 | The group ` ZZ / n ZZ ` is... |
| zndvds 21506 | Express equality of equiva... |
| zndvds0 21507 | Special case of ~ zndvds w... |
| znf1o 21508 | The function ` F ` enumera... |
| zzngim 21509 | The ` ZZ ` ring homomorphi... |
| znle2 21510 | The ordering of the ` Z/nZ... |
| znleval 21511 | The ordering of the ` Z/nZ... |
| znleval2 21512 | The ordering of the ` Z/nZ... |
| zntoslem 21513 | Lemma for ~ zntos . (Cont... |
| zntos 21514 | The ` Z/nZ ` structure is ... |
| znhash 21515 | The ` Z/nZ ` structure has... |
| znfi 21516 | The ` Z/nZ ` structure is ... |
| znfld 21517 | The ` Z/nZ ` structure is ... |
| znidomb 21518 | The ` Z/nZ ` structure is ... |
| znchr 21519 | Cyclic rings are defined b... |
| znunit 21520 | The units of ` Z/nZ ` are ... |
| znunithash 21521 | The size of the unit group... |
| znrrg 21522 | The regular elements of ` ... |
| cygznlem1 21523 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21524 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21525 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21526 | A cyclic group with ` n ` ... |
| cygzn 21527 | A cyclic group with ` n ` ... |
| cygth 21528 | The "fundamental theorem o... |
| cyggic 21529 | Cyclic groups are isomorph... |
| frgpcyg 21530 | A free group is cyclic iff... |
| freshmansdream 21531 | For a prime number ` P ` ,... |
| frobrhm 21532 | In a commutative ring with... |
| ofldchr 21533 | The characteristic of an o... |
| cnmsgnsubg 21534 | The signs form a multiplic... |
| cnmsgnbas 21535 | The base set of the sign s... |
| cnmsgngrp 21536 | The group of signs under m... |
| psgnghm 21537 | The sign is a homomorphism... |
| psgnghm2 21538 | The sign is a homomorphism... |
| psgninv 21539 | The sign of a permutation ... |
| psgnco 21540 | Multiplicativity of the pe... |
| zrhpsgnmhm 21541 | Embedding of permutation s... |
| zrhpsgninv 21542 | The embedded sign of a per... |
| evpmss 21543 | Even permutations are perm... |
| psgnevpmb 21544 | A class is an even permuta... |
| psgnodpm 21545 | A permutation which is odd... |
| psgnevpm 21546 | A permutation which is eve... |
| psgnodpmr 21547 | If a permutation has sign ... |
| zrhpsgnevpm 21548 | The sign of an even permut... |
| zrhpsgnodpm 21549 | The sign of an odd permuta... |
| cofipsgn 21550 | Composition of any class `... |
| zrhpsgnelbas 21551 | Embedding of permutation s... |
| zrhcopsgnelbas 21552 | Embedding of permutation s... |
| evpmodpmf1o 21553 | The function for performin... |
| pmtrodpm 21554 | A transposition is an odd ... |
| psgnfix1 21555 | A permutation of a finite ... |
| psgnfix2 21556 | A permutation of a finite ... |
| psgndiflemB 21557 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21558 | Lemma 2 for ~ psgndif . (... |
| psgndif 21559 | Embedding of permutation s... |
| copsgndif 21560 | Embedding of permutation s... |
| rebase 21563 | The base of the field of r... |
| remulg 21564 | The multiplication (group ... |
| resubdrg 21565 | The real numbers form a di... |
| resubgval 21566 | Subtraction in the field o... |
| replusg 21567 | The addition operation of ... |
| remulr 21568 | The multiplication operati... |
| re0g 21569 | The zero element of the fi... |
| re1r 21570 | The unity element of the f... |
| rele2 21571 | The ordering relation of t... |
| relt 21572 | The ordering relation of t... |
| reds 21573 | The distance of the field ... |
| redvr 21574 | The division operation of ... |
| retos 21575 | The real numbers are a tot... |
| refld 21576 | The real numbers form a fi... |
| refldcj 21577 | The conjugation operation ... |
| resrng 21578 | The real numbers form a st... |
| regsumsupp 21579 | The group sum over the rea... |
| rzgrp 21580 | The quotient group ` RR / ... |
| isphl 21585 | The predicate "is a genera... |
| phllvec 21586 | A pre-Hilbert space is a l... |
| phllmod 21587 | A pre-Hilbert space is a l... |
| phlsrng 21588 | The scalar ring of a pre-H... |
| phllmhm 21589 | The inner product of a pre... |
| ipcl 21590 | Closure of the inner produ... |
| ipcj 21591 | Conjugate of an inner prod... |
| iporthcom 21592 | Orthogonality (meaning inn... |
| ip0l 21593 | Inner product with a zero ... |
| ip0r 21594 | Inner product with a zero ... |
| ipeq0 21595 | The inner product of a vec... |
| ipdir 21596 | Distributive law for inner... |
| ipdi 21597 | Distributive law for inner... |
| ip2di 21598 | Distributive law for inner... |
| ipsubdir 21599 | Distributive law for inner... |
| ipsubdi 21600 | Distributive law for inner... |
| ip2subdi 21601 | Distributive law for inner... |
| ipass 21602 | Associative law for inner ... |
| ipassr 21603 | "Associative" law for seco... |
| ipassr2 21604 | "Associative" law for inne... |
| ipffval 21605 | The inner product operatio... |
| ipfval 21606 | The inner product operatio... |
| ipfeq 21607 | If the inner product opera... |
| ipffn 21608 | The inner product operatio... |
| phlipf 21609 | The inner product operatio... |
| ip2eq 21610 | Two vectors are equal iff ... |
| isphld 21611 | Properties that determine ... |
| phlpropd 21612 | If two structures have the... |
| ssipeq 21613 | The inner product on a sub... |
| phssipval 21614 | The inner product on a sub... |
| phssip 21615 | The inner product (as a fu... |
| phlssphl 21616 | A subspace of an inner pro... |
| ocvfval 21623 | The orthocomplement operat... |
| ocvval 21624 | Value of the orthocompleme... |
| elocv 21625 | Elementhood in the orthoco... |
| ocvi 21626 | Property of a member of th... |
| ocvss 21627 | The orthocomplement of a s... |
| ocvocv 21628 | A set is contained in its ... |
| ocvlss 21629 | The orthocomplement of a s... |
| ocv2ss 21630 | Orthocomplements reverse s... |
| ocvin 21631 | An orthocomplement has tri... |
| ocvsscon 21632 | Two ways to say that ` S `... |
| ocvlsp 21633 | The orthocomplement of a l... |
| ocv0 21634 | The orthocomplement of the... |
| ocvz 21635 | The orthocomplement of the... |
| ocv1 21636 | The orthocomplement of the... |
| unocv 21637 | The orthocomplement of a u... |
| iunocv 21638 | The orthocomplement of an ... |
| cssval 21639 | The set of closed subspace... |
| iscss 21640 | The predicate "is a closed... |
| cssi 21641 | Property of a closed subsp... |
| cssss 21642 | A closed subspace is a sub... |
| iscss2 21643 | It is sufficient to prove ... |
| ocvcss 21644 | The orthocomplement of any... |
| cssincl 21645 | The zero subspace is a clo... |
| css0 21646 | The zero subspace is a clo... |
| css1 21647 | The whole space is a close... |
| csslss 21648 | A closed subspace of a pre... |
| lsmcss 21649 | A subset of a pre-Hilbert ... |
| cssmre 21650 | The closed subspaces of a ... |
| mrccss 21651 | The Moore closure correspo... |
| thlval 21652 | Value of the Hilbert latti... |
| thlbas 21653 | Base set of the Hilbert la... |
| thlle 21654 | Ordering on the Hilbert la... |
| thlleval 21655 | Ordering on the Hilbert la... |
| thloc 21656 | Orthocomplement on the Hil... |
| pjfval 21663 | The value of the projectio... |
| pjdm 21664 | A subspace is in the domai... |
| pjpm 21665 | The projection map is a pa... |
| pjfval2 21666 | Value of the projection ma... |
| pjval 21667 | Value of the projection ma... |
| pjdm2 21668 | A subspace is in the domai... |
| pjff 21669 | A projection is a linear o... |
| pjf 21670 | A projection is a function... |
| pjf2 21671 | A projection is a function... |
| pjfo 21672 | A projection is a surjecti... |
| pjcss 21673 | A projection subspace is a... |
| ocvpj 21674 | The orthocomplement of a p... |
| ishil 21675 | The predicate "is a Hilber... |
| ishil2 21676 | The predicate "is a Hilber... |
| isobs 21677 | The predicate "is an ortho... |
| obsip 21678 | The inner product of two e... |
| obsipid 21679 | A basis element has length... |
| obsrcl 21680 | Reverse closure for an ort... |
| obsss 21681 | An orthonormal basis is a ... |
| obsne0 21682 | A basis element is nonzero... |
| obsocv 21683 | An orthonormal basis has t... |
| obs2ocv 21684 | The double orthocomplement... |
| obselocv 21685 | A basis element is in the ... |
| obs2ss 21686 | A basis has no proper subs... |
| obslbs 21687 | An orthogonal basis is a l... |
| reldmdsmm 21690 | The direct sum is a well-b... |
| dsmmval 21691 | Value of the module direct... |
| dsmmbase 21692 | Base set of the module dir... |
| dsmmval2 21693 | Self-referential definitio... |
| dsmmbas2 21694 | Base set of the direct sum... |
| dsmmfi 21695 | For finite products, the d... |
| dsmmelbas 21696 | Membership in the finitely... |
| dsmm0cl 21697 | The all-zero vector is con... |
| dsmmacl 21698 | The finite hull is closed ... |
| prdsinvgd2 21699 | Negation of a single coord... |
| dsmmsubg 21700 | The finite hull of a produ... |
| dsmmlss 21701 | The finite hull of a produ... |
| dsmmlmod 21702 | The direct sum of a family... |
| frlmval 21705 | Value of the "free module"... |
| frlmlmod 21706 | The free module is a modul... |
| frlmpws 21707 | The free module as a restr... |
| frlmlss 21708 | The base set of the free m... |
| frlmpwsfi 21709 | The finite free module is ... |
| frlmsca 21710 | The ring of scalars of a f... |
| frlm0 21711 | Zero in a free module (rin... |
| frlmbas 21712 | Base set of the free modul... |
| frlmelbas 21713 | Membership in the base set... |
| frlmrcl 21714 | If a free module is inhabi... |
| frlmbasfsupp 21715 | Elements of the free modul... |
| frlmbasmap 21716 | Elements of the free modul... |
| frlmbasf 21717 | Elements of the free modul... |
| frlmlvec 21718 | The free module over a div... |
| frlmfibas 21719 | The base set of the finite... |
| elfrlmbasn0 21720 | If the dimension of a free... |
| frlmplusgval 21721 | Addition in a free module.... |
| frlmsubgval 21722 | Subtraction in a free modu... |
| frlmvscafval 21723 | Scalar multiplication in a... |
| frlmvplusgvalc 21724 | Coordinates of a sum with ... |
| frlmvscaval 21725 | Coordinates of a scalar mu... |
| frlmplusgvalb 21726 | Addition in a free module ... |
| frlmvscavalb 21727 | Scalar multiplication in a... |
| frlmvplusgscavalb 21728 | Addition combined with sca... |
| frlmgsum 21729 | Finite commutative sums in... |
| frlmsplit2 21730 | Restriction is homomorphic... |
| frlmsslss 21731 | A subset of a free module ... |
| frlmsslss2 21732 | A subset of a free module ... |
| frlmbas3 21733 | An element of the base set... |
| mpofrlmd 21734 | Elements of the free modul... |
| frlmip 21735 | The inner product of a fre... |
| frlmipval 21736 | The inner product of a fre... |
| frlmphllem 21737 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21738 | Conditions for a free modu... |
| uvcfval 21741 | Value of the unit-vector g... |
| uvcval 21742 | Value of a single unit vec... |
| uvcvval 21743 | Value of a unit vector coo... |
| uvcvvcl 21744 | A coordinate of a unit vec... |
| uvcvvcl2 21745 | A unit vector coordinate i... |
| uvcvv1 21746 | The unit vector is one at ... |
| uvcvv0 21747 | The unit vector is zero at... |
| uvcff 21748 | Domain and codomain of the... |
| uvcf1 21749 | In a nonzero ring, each un... |
| uvcresum 21750 | Any element of a free modu... |
| frlmssuvc1 21751 | A scalar multiple of a uni... |
| frlmssuvc2 21752 | A nonzero scalar multiple ... |
| frlmsslsp 21753 | A subset of a free module ... |
| frlmlbs 21754 | The unit vectors comprise ... |
| frlmup1 21755 | Any assignment of unit vec... |
| frlmup2 21756 | The evaluation map has the... |
| frlmup3 21757 | The range of such an evalu... |
| frlmup4 21758 | Universal property of the ... |
| ellspd 21759 | The elements of the span o... |
| elfilspd 21760 | Simplified version of ~ el... |
| rellindf 21765 | The independent-family pre... |
| islinds 21766 | Property of an independent... |
| linds1 21767 | An independent set of vect... |
| linds2 21768 | An independent set of vect... |
| islindf 21769 | Property of an independent... |
| islinds2 21770 | Expanded property of an in... |
| islindf2 21771 | Property of an independent... |
| lindff 21772 | Functional property of a l... |
| lindfind 21773 | A linearly independent fam... |
| lindsind 21774 | A linearly independent set... |
| lindfind2 21775 | In a linearly independent ... |
| lindsind2 21776 | In a linearly independent ... |
| lindff1 21777 | A linearly independent fam... |
| lindfrn 21778 | The range of an independen... |
| f1lindf 21779 | Rearranging and deleting e... |
| lindfres 21780 | Any restriction of an inde... |
| lindsss 21781 | Any subset of an independe... |
| f1linds 21782 | A family constructed from ... |
| islindf3 21783 | In a nonzero ring, indepen... |
| lindfmm 21784 | Linear independence of a f... |
| lindsmm 21785 | Linear independence of a s... |
| lindsmm2 21786 | The monomorphic image of a... |
| lsslindf 21787 | Linear independence is unc... |
| lsslinds 21788 | Linear independence is unc... |
| islbs4 21789 | A basis is an independent ... |
| lbslinds 21790 | A basis is independent. (... |
| islinds3 21791 | A subset is linearly indep... |
| islinds4 21792 | A set is independent in a ... |
| lmimlbs 21793 | The isomorphic image of a ... |
| lmiclbs 21794 | Having a basis is an isomo... |
| islindf4 21795 | A family is independent if... |
| islindf5 21796 | A family is independent if... |
| indlcim 21797 | An independent, spanning f... |
| lbslcic 21798 | A module with a basis is i... |
| lmisfree 21799 | A module has a basis iff i... |
| lvecisfrlm 21800 | Every vector space is isom... |
| lmimco 21801 | The composition of two iso... |
| lmictra 21802 | Module isomorphism is tran... |
| uvcf1o 21803 | In a nonzero ring, the map... |
| uvcendim 21804 | In a nonzero ring, the num... |
| frlmisfrlm 21805 | A free module is isomorphi... |
| frlmiscvec 21806 | Every free module is isomo... |
| isassa 21813 | The properties of an assoc... |
| assalem 21814 | The properties of an assoc... |
| assaass 21815 | Left-associative property ... |
| assaassr 21816 | Right-associative property... |
| assalmod 21817 | An associative algebra is ... |
| assaring 21818 | An associative algebra is ... |
| assasca 21819 | The scalars of an associat... |
| assa2ass 21820 | Left- and right-associativ... |
| assa2ass2 21821 | Left- and right-associativ... |
| isassad 21822 | Sufficient condition for b... |
| issubassa3 21823 | A subring that is also a s... |
| issubassa 21824 | The subalgebras of an asso... |
| sraassab 21825 | A subring algebra is an as... |
| sraassa 21826 | The subring algebra over a... |
| rlmassa 21827 | The ring module over a com... |
| assapropd 21828 | If two structures have the... |
| aspval 21829 | Value of the algebraic clo... |
| asplss 21830 | The algebraic span of a se... |
| aspid 21831 | The algebraic span of a su... |
| aspsubrg 21832 | The algebraic span of a se... |
| aspss 21833 | Span preserves subset orde... |
| aspssid 21834 | A set of vectors is a subs... |
| asclfval 21835 | Function value of the alge... |
| asclval 21836 | Value of a mapped algebra ... |
| asclfn 21837 | Unconditional functionalit... |
| asclf 21838 | The algebra scalar lifting... |
| asclghm 21839 | The algebra scalar lifting... |
| asclelbas 21840 | Lifted scalars are in the ... |
| ascl0 21841 | The scalar 0 embedded into... |
| ascl1 21842 | The scalar 1 embedded into... |
| asclmul1 21843 | Left multiplication by a l... |
| asclmul2 21844 | Right multiplication by a ... |
| ascldimul 21845 | The algebra scalar lifting... |
| asclinvg 21846 | The group inverse (negatio... |
| asclrhm 21847 | The algebra scalar lifting... |
| rnascl 21848 | The set of lifted scalars ... |
| issubassa2 21849 | A subring of a unital alge... |
| rnasclsubrg 21850 | The scalar multiples of th... |
| rnasclmulcl 21851 | (Vector) multiplication is... |
| rnasclassa 21852 | The scalar multiples of th... |
| ressascl 21853 | The lifting of scalars is ... |
| asclpropd 21854 | If two structures have the... |
| aspval2 21855 | The algebraic closure is t... |
| assamulgscmlem1 21856 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21857 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21858 | Exponentiation of a scalar... |
| asclmulg 21859 | Apply group multiplication... |
| zlmassa 21860 | The ` ZZ ` -module operati... |
| reldmpsr 21871 | The multivariate power ser... |
| psrval 21872 | Value of the multivariate ... |
| psrvalstr 21873 | The multivariate power ser... |
| psrbag 21874 | Elementhood in the set of ... |
| psrbagf 21875 | A finite bag is a function... |
| psrbagfsupp 21876 | Finite bags have finite su... |
| snifpsrbag 21877 | A bag containing one eleme... |
| fczpsrbag 21878 | The constant function equa... |
| psrbaglesupp 21879 | The support of a dominated... |
| psrbaglecl 21880 | The set of finite bags is ... |
| psrbagaddcl 21881 | The sum of two finite bags... |
| psrbagcon 21882 | The analogue of the statem... |
| psrbaglefi 21883 | There are finitely many ba... |
| psrbagconcl 21884 | The complement of a bag is... |
| psrbagleadd1 21885 | The analogue of " ` X <_ F... |
| psrbagconf1o 21886 | Bag complementation is a b... |
| gsumbagdiaglem 21887 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21888 | Two-dimensional commutatio... |
| psrass1lem 21889 | A group sum commutation us... |
| psrbas 21890 | The base set of the multiv... |
| psrelbas 21891 | An element of the set of p... |
| psrelbasfun 21892 | An element of the set of p... |
| psrplusg 21893 | The addition operation of ... |
| psradd 21894 | The addition operation of ... |
| psraddcl 21895 | Closure of the power serie... |
| psraddclOLD 21896 | Obsolete version of ~ psra... |
| rhmpsrlem1 21897 | Lemma for ~ rhmpsr et al. ... |
| rhmpsrlem2 21898 | Lemma for ~ rhmpsr et al. ... |
| psrmulr 21899 | The multiplication operati... |
| psrmulfval 21900 | The multiplication operati... |
| psrmulval 21901 | The multiplication operati... |
| psrmulcllem 21902 | Closure of the power serie... |
| psrmulcl 21903 | Closure of the power serie... |
| psrsca 21904 | The scalar field of the mu... |
| psrvscafval 21905 | The scalar multiplication ... |
| psrvsca 21906 | The scalar multiplication ... |
| psrvscaval 21907 | The scalar multiplication ... |
| psrvscacl 21908 | Closure of the power serie... |
| psr0cl 21909 | The zero element of the ri... |
| psr0lid 21910 | The zero element of the ri... |
| psrnegcl 21911 | The negative function in t... |
| psrlinv 21912 | The negative function in t... |
| psrgrp 21913 | The ring of power series i... |
| psr0 21914 | The zero element of the ri... |
| psrneg 21915 | The negative function of t... |
| psrlmod 21916 | The ring of power series i... |
| psr1cl 21917 | The identity element of th... |
| psrlidm 21918 | The identity element of th... |
| psrridm 21919 | The identity element of th... |
| psrass1 21920 | Associative identity for t... |
| psrdi 21921 | Distributive law for the r... |
| psrdir 21922 | Distributive law for the r... |
| psrass23l 21923 | Associative identity for t... |
| psrcom 21924 | Commutative law for the ri... |
| psrass23 21925 | Associative identities for... |
| psrring 21926 | The ring of power series i... |
| psr1 21927 | The identity element of th... |
| psrcrng 21928 | The ring of power series i... |
| psrassa 21929 | The ring of power series i... |
| resspsrbas 21930 | A restricted power series ... |
| resspsradd 21931 | A restricted power series ... |
| resspsrmul 21932 | A restricted power series ... |
| resspsrvsca 21933 | A restricted power series ... |
| subrgpsr 21934 | A subring of the base ring... |
| psrascl 21935 | Value of the scalar inject... |
| psrasclcl 21936 | A scalar is lifted into a ... |
| mvrfval 21937 | Value of the generating el... |
| mvrval 21938 | Value of the generating el... |
| mvrval2 21939 | Value of the generating el... |
| mvrid 21940 | The ` X i ` -th coefficien... |
| mvrf 21941 | The power series variable ... |
| mvrf1 21942 | The power series variable ... |
| mvrcl2 21943 | A power series variable is... |
| reldmmpl 21944 | The multivariate polynomia... |
| mplval 21945 | Value of the set of multiv... |
| mplbas 21946 | Base set of the set of mul... |
| mplelbas 21947 | Property of being a polyno... |
| mvrcl 21948 | A power series variable is... |
| mvrf2 21949 | The power series/polynomia... |
| mplrcl 21950 | Reverse closure for the po... |
| mplelsfi 21951 | A polynomial treated as a ... |
| mplval2 21952 | Self-referential expressio... |
| mplbasss 21953 | The set of polynomials is ... |
| mplelf 21954 | A polynomial is defined as... |
| mplsubglem 21955 | If ` A ` is an ideal of se... |
| mpllsslem 21956 | If ` A ` is an ideal of su... |
| mplsubglem2 21957 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21958 | The set of polynomials is ... |
| mpllss 21959 | The set of polynomials is ... |
| mplsubrglem 21960 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21961 | The set of polynomials is ... |
| mpl0 21962 | The zero polynomial. (Con... |
| mplplusg 21963 | Value of addition in a pol... |
| mplmulr 21964 | Value of multiplication in... |
| mpladd 21965 | The addition operation on ... |
| mplneg 21966 | The negative function on m... |
| mplmul 21967 | The multiplication operati... |
| mpl1 21968 | The identity element of th... |
| mplsca 21969 | The scalar field of a mult... |
| mplvsca2 21970 | The scalar multiplication ... |
| mplvsca 21971 | The scalar multiplication ... |
| mplvscaval 21972 | The scalar multiplication ... |
| mplgrp 21973 | The polynomial ring is a g... |
| mpllmod 21974 | The polynomial ring is a l... |
| mplring 21975 | The polynomial ring is a r... |
| mpllvec 21976 | The polynomial ring is a v... |
| mplcrng 21977 | The polynomial ring is a c... |
| mplassa 21978 | The polynomial ring is an ... |
| mplringd 21979 | The polynomial ring is a r... |
| mpllmodd 21980 | The polynomial ring is a l... |
| mplascl0 21981 | The zero scalar as a polyn... |
| mplascl1 21982 | The one scalar as a polyno... |
| ressmplbas2 21983 | The base set of a restrict... |
| ressmplbas 21984 | A restricted polynomial al... |
| ressmpladd 21985 | A restricted polynomial al... |
| ressmplmul 21986 | A restricted polynomial al... |
| ressmplvsca 21987 | A restricted power series ... |
| subrgmpl 21988 | A subring of the base ring... |
| subrgmvr 21989 | The variables in a subring... |
| subrgmvrf 21990 | The variables in a polynom... |
| mplmon 21991 | A monomial is a polynomial... |
| mplmonmul 21992 | The product of two monomia... |
| mplcoe1 21993 | Decompose a polynomial int... |
| mplcoe3 21994 | Decompose a monomial in on... |
| mplcoe5lem 21995 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21996 | Decompose a monomial into ... |
| mplcoe2 21997 | Decompose a monomial into ... |
| mplbas2 21998 | An alternative expression ... |
| ltbval 21999 | Value of the well-order on... |
| ltbwe 22000 | The finite bag order is a ... |
| reldmopsr 22001 | Lemma for ordered power se... |
| opsrval 22002 | The value of the "ordered ... |
| opsrle 22003 | An alternative expression ... |
| opsrval2 22004 | Self-referential expressio... |
| opsrbaslem 22005 | Get a component of the ord... |
| opsrbas 22006 | The base set of the ordere... |
| opsrplusg 22007 | The addition operation of ... |
| opsrmulr 22008 | The multiplication operati... |
| opsrvsca 22009 | The scalar product operati... |
| opsrsca 22010 | The scalar ring of the ord... |
| opsrtoslem1 22011 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 22012 | Lemma for ~ opsrtos . (Co... |
| opsrtos 22013 | The ordered power series s... |
| opsrso 22014 | The ordered power series s... |
| opsrcrng 22015 | The ring of ordered power ... |
| opsrassa 22016 | The ring of ordered power ... |
| mplmon2 22017 | Express a scaled monomial.... |
| psrbag0 22018 | The empty bag is a bag. (... |
| psrbagsn 22019 | A singleton bag is a bag. ... |
| mplascl 22020 | Value of the scalar inject... |
| mplasclf 22021 | The scalar injection is a ... |
| subrgascl 22022 | The scalar injection funct... |
| subrgasclcl 22023 | The scalars in a polynomia... |
| mplmon2cl 22024 | A scaled monomial is a pol... |
| mplmon2mul 22025 | Product of scaled monomial... |
| mplind 22026 | Prove a property of polyno... |
| mplcoe4 22027 | Decompose a polynomial int... |
| evlslem4 22032 | The support of a tensor pr... |
| psrbagev1 22033 | A bag of multipliers provi... |
| psrbagev2 22034 | Closure of a sum using a b... |
| evlslem2 22035 | A linear function on the p... |
| evlslem3 22036 | Lemma for ~ evlseu . Poly... |
| evlslem6 22037 | Lemma for ~ evlseu . Fini... |
| evlslem1 22038 | Lemma for ~ evlseu , give ... |
| evlseu 22039 | For a given interpretation... |
| reldmevls 22040 | Well-behaved binary operat... |
| mpfrcl 22041 | Reverse closure for the se... |
| evlsval 22042 | Value of the polynomial ev... |
| evlsval2 22043 | Characterizing properties ... |
| evlsrhm 22044 | Polynomial evaluation is a... |
| evlsval3 22045 | Give a formula for the pol... |
| evlsvval 22046 | Give a formula for the eva... |
| evlsvvvallem 22047 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 22048 | Lemma for theorems using ~... |
| evlsvvval 22049 | Give a formula for the eva... |
| evlssca 22050 | Polynomial evaluation maps... |
| evlsvar 22051 | Polynomial evaluation maps... |
| evlsgsumadd 22052 | Polynomial evaluation maps... |
| evlsgsummul 22053 | Polynomial evaluation maps... |
| evlspw 22054 | Polynomial evaluation for ... |
| evlsvarpw 22055 | Polynomial evaluation for ... |
| evlval 22056 | Value of the simple/same r... |
| evlrhm 22057 | The simple evaluation map ... |
| evlcl 22058 | A polynomial over the ring... |
| evladdval 22059 | Polynomial evaluation buil... |
| evlmulval 22060 | Polynomial evaluation buil... |
| evlsscasrng 22061 | The evaluation of a scalar... |
| evlsca 22062 | Simple polynomial evaluati... |
| evlsvarsrng 22063 | The evaluation of the vari... |
| evlvar 22064 | Simple polynomial evaluati... |
| mpfconst 22065 | Constants are multivariate... |
| mpfproj 22066 | Projections are multivaria... |
| mpfsubrg 22067 | Polynomial functions are a... |
| mpff 22068 | Polynomial functions are f... |
| mpfaddcl 22069 | The sum of multivariate po... |
| mpfmulcl 22070 | The product of multivariat... |
| mpfind 22071 | Prove a property of polyno... |
| selvffval 22077 | Value of the "variable sel... |
| selvfval 22078 | Value of the "variable sel... |
| selvval 22079 | Value of the "variable sel... |
| reldmmhp 22081 | The domain of the homogene... |
| mhpfval 22082 | Value of the "homogeneous ... |
| mhpval 22083 | Value of the "homogeneous ... |
| ismhp 22084 | Property of being a homoge... |
| ismhp2 22085 | Deduce a homogeneous polyn... |
| ismhp3 22086 | A polynomial is homogeneou... |
| mhprcl 22087 | Reverse closure for homoge... |
| mhpmpl 22088 | A homogeneous polynomial i... |
| mhpdeg 22089 | All nonzero terms of a hom... |
| mhp0cl 22090 | The zero polynomial is hom... |
| mhpsclcl 22091 | A scalar (or constant) pol... |
| mhpvarcl 22092 | A power series variable is... |
| mhpmulcl 22093 | A product of homogeneous p... |
| mhppwdeg 22094 | Degree of a homogeneous po... |
| mhpaddcl 22095 | Homogeneous polynomials ar... |
| mhpinvcl 22096 | Homogeneous polynomials ar... |
| mhpsubg 22097 | Homogeneous polynomials fo... |
| mhpvscacl 22098 | Homogeneous polynomials ar... |
| mhplss 22099 | Homogeneous polynomials fo... |
| psdffval 22101 | Value of the power series ... |
| psdfval 22102 | Give a map between power s... |
| psdval 22103 | Evaluate the partial deriv... |
| psdcoef 22104 | Coefficient of a term of t... |
| psdcl 22105 | The derivative of a power ... |
| psdmplcl 22106 | The derivative of a polyno... |
| psdadd 22107 | The derivative of a sum is... |
| psdvsca 22108 | The derivative of a scaled... |
| psdmullem 22109 | Lemma for ~ psdmul . Tran... |
| psdmul 22110 | Product rule for power ser... |
| psd1 22111 | The derivative of one is z... |
| psdascl 22112 | The derivative of a consta... |
| psdmvr 22113 | The partial derivative of ... |
| psdpw 22114 | Power rule for partial der... |
| psr1baslem 22126 | The set of finite bags on ... |
| psr1val 22127 | Value of the ring of univa... |
| psr1crng 22128 | The ring of univariate pow... |
| psr1assa 22129 | The ring of univariate pow... |
| psr1tos 22130 | The ordered power series s... |
| psr1bas2 22131 | The base set of the ring o... |
| psr1bas 22132 | The base set of the ring o... |
| vr1val 22133 | The value of the generator... |
| vr1cl2 22134 | The variable ` X ` is a me... |
| ply1val 22135 | The value of the set of un... |
| ply1bas 22136 | The value of the base set ... |
| ply1basOLD 22137 | Obsolete version of ~ ply1... |
| ply1lss 22138 | Univariate polynomials for... |
| ply1subrg 22139 | Univariate polynomials for... |
| ply1crng 22140 | The ring of univariate pol... |
| ply1assa 22141 | The ring of univariate pol... |
| psr1bascl 22142 | A univariate power series ... |
| psr1basf 22143 | Univariate power series ba... |
| ply1basf 22144 | Univariate polynomial base... |
| ply1bascl 22145 | A univariate polynomial is... |
| ply1bascl2 22146 | A univariate polynomial is... |
| coe1fval 22147 | Value of the univariate po... |
| coe1fv 22148 | Value of an evaluated coef... |
| fvcoe1 22149 | Value of a multivariate co... |
| coe1fval3 22150 | Univariate power series co... |
| coe1f2 22151 | Functionality of univariat... |
| coe1fval2 22152 | Univariate polynomial coef... |
| coe1f 22153 | Functionality of univariat... |
| coe1fvalcl 22154 | A coefficient of a univari... |
| coe1sfi 22155 | Finite support of univaria... |
| coe1fsupp 22156 | The coefficient vector of ... |
| mptcoe1fsupp 22157 | A mapping involving coeffi... |
| coe1ae0 22158 | The coefficient vector of ... |
| vr1cl 22159 | The generator of a univari... |
| opsr0 22160 | Zero in the ordered power ... |
| opsr1 22161 | One in the ordered power s... |
| psr1plusg 22162 | Value of addition in a uni... |
| psr1vsca 22163 | Value of scalar multiplica... |
| psr1mulr 22164 | Value of multiplication in... |
| ply1plusg 22165 | Value of addition in a uni... |
| ply1vsca 22166 | Value of scalar multiplica... |
| ply1mulr 22167 | Value of multiplication in... |
| ply1ass23l 22168 | Associative identity with ... |
| ressply1bas2 22169 | The base set of a restrict... |
| ressply1bas 22170 | A restricted polynomial al... |
| ressply1add 22171 | A restricted polynomial al... |
| ressply1mul 22172 | A restricted polynomial al... |
| ressply1vsca 22173 | A restricted power series ... |
| subrgply1 22174 | A subring of the base ring... |
| gsumply1subr 22175 | Evaluate a group sum in a ... |
| psrbaspropd 22176 | Property deduction for pow... |
| psrplusgpropd 22177 | Property deduction for pow... |
| mplbaspropd 22178 | Property deduction for pol... |
| psropprmul 22179 | Reversing multiplication i... |
| ply1opprmul 22180 | Reversing multiplication i... |
| 00ply1bas 22181 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 22182 | Protection compatibility o... |
| ply1plusgfvi 22183 | Protection compatibility o... |
| ply1baspropd 22184 | Property deduction for uni... |
| ply1plusgpropd 22185 | Property deduction for uni... |
| opsrring 22186 | Ordered power series form ... |
| opsrlmod 22187 | Ordered power series form ... |
| psr1ring 22188 | Univariate power series fo... |
| ply1ring 22189 | Univariate polynomials for... |
| psr1lmod 22190 | Univariate power series fo... |
| psr1sca 22191 | Scalars of a univariate po... |
| psr1sca2 22192 | Scalars of a univariate po... |
| ply1lmod 22193 | Univariate polynomials for... |
| ply1sca 22194 | Scalars of a univariate po... |
| ply1sca2 22195 | Scalars of a univariate po... |
| ply1ascl0 22196 | The zero scalar as a polyn... |
| ply1ascl1 22197 | The multiplicative identit... |
| ply1mpl0 22198 | The univariate polynomial ... |
| ply10s0 22199 | Zero times a univariate po... |
| ply1mpl1 22200 | The univariate polynomial ... |
| ply1ascl 22201 | The univariate polynomial ... |
| subrg1ascl 22202 | The scalar injection funct... |
| subrg1asclcl 22203 | The scalars in a polynomia... |
| subrgvr1 22204 | The variables in a subring... |
| subrgvr1cl 22205 | The variables in a polynom... |
| coe1z 22206 | The coefficient vector of ... |
| coe1add 22207 | The coefficient vector of ... |
| coe1addfv 22208 | A particular coefficient o... |
| coe1subfv 22209 | A particular coefficient o... |
| coe1mul2lem1 22210 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22211 | An equivalence for ~ coe1m... |
| coe1mul2 22212 | The coefficient vector of ... |
| coe1mul 22213 | The coefficient vector of ... |
| ply1moncl 22214 | Closure of the expression ... |
| ply1tmcl 22215 | Closure of the expression ... |
| coe1tm 22216 | Coefficient vector of a po... |
| coe1tmfv1 22217 | Nonzero coefficient of a p... |
| coe1tmfv2 22218 | Zero coefficient of a poly... |
| coe1tmmul2 22219 | Coefficient vector of a po... |
| coe1tmmul 22220 | Coefficient vector of a po... |
| coe1tmmul2fv 22221 | Function value of a right-... |
| coe1pwmul 22222 | Coefficient vector of a po... |
| coe1pwmulfv 22223 | Function value of a right-... |
| ply1scltm 22224 | A scalar is a term with ze... |
| coe1sclmul 22225 | Coefficient vector of a po... |
| coe1sclmulfv 22226 | A single coefficient of a ... |
| coe1sclmul2 22227 | Coefficient vector of a po... |
| ply1sclf 22228 | A scalar polynomial is a p... |
| ply1sclcl 22229 | The value of the algebra s... |
| coe1scl 22230 | Coefficient vector of a sc... |
| ply1sclid 22231 | Recover the base scalar fr... |
| ply1sclf1 22232 | The polynomial scalar func... |
| ply1scl0 22233 | The zero scalar is zero. ... |
| ply1scln0 22234 | Nonzero scalars create non... |
| ply1scl1 22235 | The one scalar is the unit... |
| coe1id 22236 | Coefficient vector of the ... |
| ply1idvr1 22237 | The identity of a polynomi... |
| ply1idvr1OLD 22238 | Obsolete version of ~ ply1... |
| cply1mul 22239 | The product of two constan... |
| ply1coefsupp 22240 | The decomposition of a uni... |
| ply1coe 22241 | Decompose a univariate pol... |
| eqcoe1ply1eq 22242 | Two polynomials over the s... |
| ply1coe1eq 22243 | Two polynomials over the s... |
| cply1coe0 22244 | All but the first coeffici... |
| cply1coe0bi 22245 | A polynomial is constant (... |
| coe1fzgsumdlem 22246 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22247 | Value of an evaluated coef... |
| ply1scleq 22248 | Equality of a constant pol... |
| ply1chr 22249 | The characteristic of a po... |
| gsumsmonply1 22250 | A finite group sum of scal... |
| gsummoncoe1 22251 | A coefficient of the polyn... |
| gsumply1eq 22252 | Two univariate polynomials... |
| lply1binom 22253 | The binomial theorem for l... |
| lply1binomsc 22254 | The binomial theorem for l... |
| ply1fermltlchr 22255 | Fermat's little theorem fo... |
| reldmevls1 22260 | Well-behaved binary operat... |
| ply1frcl 22261 | Reverse closure for the se... |
| evls1fval 22262 | Value of the univariate po... |
| evls1val 22263 | Value of the univariate po... |
| evls1rhmlem 22264 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22265 | Polynomial evaluation is a... |
| evls1sca 22266 | Univariate polynomial eval... |
| evls1gsumadd 22267 | Univariate polynomial eval... |
| evls1gsummul 22268 | Univariate polynomial eval... |
| evls1pw 22269 | Univariate polynomial eval... |
| evls1varpw 22270 | Univariate polynomial eval... |
| evl1fval 22271 | Value of the simple/same r... |
| evl1val 22272 | Value of the simple/same r... |
| evl1fval1lem 22273 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22274 | Value of the simple/same r... |
| evl1rhm 22275 | Polynomial evaluation is a... |
| fveval1fvcl 22276 | The function value of the ... |
| evl1sca 22277 | Polynomial evaluation maps... |
| evl1scad 22278 | Polynomial evaluation buil... |
| evl1var 22279 | Polynomial evaluation maps... |
| evl1vard 22280 | Polynomial evaluation buil... |
| evls1var 22281 | Univariate polynomial eval... |
| evls1scasrng 22282 | The evaluation of a scalar... |
| evls1varsrng 22283 | The evaluation of the vari... |
| evl1addd 22284 | Polynomial evaluation buil... |
| evl1subd 22285 | Polynomial evaluation buil... |
| evl1muld 22286 | Polynomial evaluation buil... |
| evl1vsd 22287 | Polynomial evaluation buil... |
| evl1expd 22288 | Polynomial evaluation buil... |
| pf1const 22289 | Constants are polynomial f... |
| pf1id 22290 | The identity is a polynomi... |
| pf1subrg 22291 | Polynomial functions are a... |
| pf1rcl 22292 | Reverse closure for the se... |
| pf1f 22293 | Polynomial functions are f... |
| mpfpf1 22294 | Convert a multivariate pol... |
| pf1mpf 22295 | Convert a univariate polyn... |
| pf1addcl 22296 | The sum of multivariate po... |
| pf1mulcl 22297 | The product of multivariat... |
| pf1ind 22298 | Prove a property of polyno... |
| evl1gsumdlem 22299 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22300 | Polynomial evaluation buil... |
| evl1gsumadd 22301 | Univariate polynomial eval... |
| evl1gsumaddval 22302 | Value of a univariate poly... |
| evl1gsummul 22303 | Univariate polynomial eval... |
| evl1varpw 22304 | Univariate polynomial eval... |
| evl1varpwval 22305 | Value of a univariate poly... |
| evl1scvarpw 22306 | Univariate polynomial eval... |
| evl1scvarpwval 22307 | Value of a univariate poly... |
| evl1gsummon 22308 | Value of a univariate poly... |
| evls1scafv 22309 | Value of the univariate po... |
| evls1expd 22310 | Univariate polynomial eval... |
| evls1varpwval 22311 | Univariate polynomial eval... |
| evls1fpws 22312 | Evaluation of a univariate... |
| ressply1evl 22313 | Evaluation of a univariate... |
| evls1addd 22314 | Univariate polynomial eval... |
| evls1muld 22315 | Univariate polynomial eval... |
| evls1vsca 22316 | Univariate polynomial eval... |
| asclply1subcl 22317 | Closure of the algebra sca... |
| evls1fvcl 22318 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 22319 | The function ` F ` mapping... |
| evls1maplmhm 22320 | The function ` F ` mapping... |
| evls1maprnss 22321 | The function ` F ` mapping... |
| evl1maprhm 22322 | The function ` F ` mapping... |
| mhmcompl 22323 | The composition of a monoi... |
| mhmcoaddmpl 22324 | Show that the ring homomor... |
| rhmcomulmpl 22325 | Show that the ring homomor... |
| rhmmpl 22326 | Provide a ring homomorphis... |
| ply1vscl 22327 | Closure of scalar multipli... |
| mhmcoply1 22328 | The composition of a monoi... |
| rhmply1 22329 | Provide a ring homomorphis... |
| rhmply1vr1 22330 | A ring homomorphism betwee... |
| rhmply1vsca 22331 | Apply a ring homomorphism ... |
| rhmply1mon 22332 | Apply a ring homomorphism ... |
| mamufval 22335 | Functional value of the ma... |
| mamuval 22336 | Multiplication of two matr... |
| mamufv 22337 | A cell in the multiplicati... |
| mamudm 22338 | The domain of the matrix m... |
| mamufacex 22339 | Every solution of the equa... |
| mamures 22340 | Rows in a matrix product a... |
| grpvlinv 22341 | Tuple-wise left inverse in... |
| grpvrinv 22342 | Tuple-wise right inverse i... |
| ringvcl 22343 | Tuple-wise multiplication ... |
| mamucl 22344 | Operation closure of matri... |
| mamuass 22345 | Matrix multiplication is a... |
| mamudi 22346 | Matrix multiplication dist... |
| mamudir 22347 | Matrix multiplication dist... |
| mamuvs1 22348 | Matrix multiplication dist... |
| mamuvs2 22349 | Matrix multiplication dist... |
| matbas0pc 22352 | There is no matrix with a ... |
| matbas0 22353 | There is no matrix for a n... |
| matval 22354 | Value of the matrix algebr... |
| matrcl 22355 | Reverse closure for the ma... |
| matbas 22356 | The matrix ring has the sa... |
| matplusg 22357 | The matrix ring has the sa... |
| matsca 22358 | The matrix ring has the sa... |
| matvsca 22359 | The matrix ring has the sa... |
| mat0 22360 | The matrix ring has the sa... |
| matinvg 22361 | The matrix ring has the sa... |
| mat0op 22362 | Value of a zero matrix as ... |
| matsca2 22363 | The scalars of the matrix ... |
| matbas2 22364 | The base set of the matrix... |
| matbas2i 22365 | A matrix is a function. (... |
| matbas2d 22366 | The base set of the matrix... |
| eqmat 22367 | Two square matrices of the... |
| matecl 22368 | Each entry (according to W... |
| matecld 22369 | Each entry (according to W... |
| matplusg2 22370 | Addition in the matrix rin... |
| matvsca2 22371 | Scalar multiplication in t... |
| matlmod 22372 | The matrix ring is a linea... |
| matgrp 22373 | The matrix ring is a group... |
| matvscl 22374 | Closure of the scalar mult... |
| matsubg 22375 | The matrix ring has the sa... |
| matplusgcell 22376 | Addition in the matrix rin... |
| matsubgcell 22377 | Subtraction in the matrix ... |
| matinvgcell 22378 | Additive inversion in the ... |
| matvscacell 22379 | Scalar multiplication in t... |
| matgsum 22380 | Finite commutative sums in... |
| matmulr 22381 | Multiplication in the matr... |
| mamumat1cl 22382 | The identity matrix (as op... |
| mat1comp 22383 | The components of the iden... |
| mamulid 22384 | The identity matrix (as op... |
| mamurid 22385 | The identity matrix (as op... |
| matring 22386 | Existence of the matrix ri... |
| matassa 22387 | Existence of the matrix al... |
| matmulcell 22388 | Multiplication in the matr... |
| mpomatmul 22389 | Multiplication of two N x ... |
| mat1 22390 | Value of an identity matri... |
| mat1ov 22391 | Entries of an identity mat... |
| mat1bas 22392 | The identity matrix is a m... |
| matsc 22393 | The identity matrix multip... |
| ofco2 22394 | Distribution law for the f... |
| oftpos 22395 | The transposition of the v... |
| mattposcl 22396 | The transpose of a square ... |
| mattpostpos 22397 | The transpose of the trans... |
| mattposvs 22398 | The transposition of a mat... |
| mattpos1 22399 | The transposition of the i... |
| tposmap 22400 | The transposition of an I ... |
| mamutpos 22401 | Behavior of transposes in ... |
| mattposm 22402 | Multiplying two transposed... |
| matgsumcl 22403 | Closure of a group sum ove... |
| madetsumid 22404 | The identity summand in th... |
| matepmcl 22405 | Each entry of a matrix wit... |
| matepm2cl 22406 | Each entry of a matrix wit... |
| madetsmelbas 22407 | A summand of the determina... |
| madetsmelbas2 22408 | A summand of the determina... |
| mat0dimbas0 22409 | The empty set is the one a... |
| mat0dim0 22410 | The zero of the algebra of... |
| mat0dimid 22411 | The identity of the algebr... |
| mat0dimscm 22412 | The scalar multiplication ... |
| mat0dimcrng 22413 | The algebra of matrices wi... |
| mat1dimelbas 22414 | A matrix with dimension 1 ... |
| mat1dimbas 22415 | A matrix with dimension 1 ... |
| mat1dim0 22416 | The zero of the algebra of... |
| mat1dimid 22417 | The identity of the algebr... |
| mat1dimscm 22418 | The scalar multiplication ... |
| mat1dimmul 22419 | The ring multiplication in... |
| mat1dimcrng 22420 | The algebra of matrices wi... |
| mat1f1o 22421 | There is a 1-1 function fr... |
| mat1rhmval 22422 | The value of the ring homo... |
| mat1rhmelval 22423 | The value of the ring homo... |
| mat1rhmcl 22424 | The value of the ring homo... |
| mat1f 22425 | There is a function from a... |
| mat1ghm 22426 | There is a group homomorph... |
| mat1mhm 22427 | There is a monoid homomorp... |
| mat1rhm 22428 | There is a ring homomorphi... |
| mat1rngiso 22429 | There is a ring isomorphis... |
| mat1ric 22430 | A ring is isomorphic to th... |
| dmatval 22435 | The set of ` N ` x ` N ` d... |
| dmatel 22436 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22437 | An ` N ` x ` N ` diagonal ... |
| dmatid 22438 | The identity matrix is a d... |
| dmatelnd 22439 | An extradiagonal entry of ... |
| dmatmul 22440 | The product of two diagona... |
| dmatsubcl 22441 | The difference of two diag... |
| dmatsgrp 22442 | The set of diagonal matric... |
| dmatmulcl 22443 | The product of two diagona... |
| dmatsrng 22444 | The set of diagonal matric... |
| dmatcrng 22445 | The subring of diagonal ma... |
| dmatscmcl 22446 | The multiplication of a di... |
| scmatval 22447 | The set of ` N ` x ` N ` s... |
| scmatel 22448 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22449 | A scalar matrix can be exp... |
| scmatscmide 22450 | An entry of a scalar matri... |
| scmatscmiddistr 22451 | Distributive law for scala... |
| scmatmat 22452 | An ` N ` x ` N ` scalar ma... |
| scmate 22453 | An entry of an ` N ` x ` N... |
| scmatmats 22454 | The set of an ` N ` x ` N ... |
| scmateALT 22455 | Alternate proof of ~ scmat... |
| scmatscm 22456 | The multiplication of a ma... |
| scmatid 22457 | The identity matrix is a s... |
| scmatdmat 22458 | A scalar matrix is a diago... |
| scmataddcl 22459 | The sum of two scalar matr... |
| scmatsubcl 22460 | The difference of two scal... |
| scmatmulcl 22461 | The product of two scalar ... |
| scmatsgrp 22462 | The set of scalar matrices... |
| scmatsrng 22463 | The set of scalar matrices... |
| scmatcrng 22464 | The subring of scalar matr... |
| scmatsgrp1 22465 | The set of scalar matrices... |
| scmatsrng1 22466 | The set of scalar matrices... |
| smatvscl 22467 | Closure of the scalar mult... |
| scmatlss 22468 | The set of scalar matrices... |
| scmatstrbas 22469 | The set of scalar matrices... |
| scmatrhmval 22470 | The value of the ring homo... |
| scmatrhmcl 22471 | The value of the ring homo... |
| scmatf 22472 | There is a function from a... |
| scmatfo 22473 | There is a function from a... |
| scmatf1 22474 | There is a 1-1 function fr... |
| scmatf1o 22475 | There is a bijection betwe... |
| scmatghm 22476 | There is a group homomorph... |
| scmatmhm 22477 | There is a monoid homomorp... |
| scmatrhm 22478 | There is a ring homomorphi... |
| scmatrngiso 22479 | There is a ring isomorphis... |
| scmatric 22480 | A ring is isomorphic to ev... |
| mat0scmat 22481 | The empty matrix over a ri... |
| mat1scmat 22482 | A 1-dimensional matrix ove... |
| mvmulfval 22485 | Functional value of the ma... |
| mvmulval 22486 | Multiplication of a vector... |
| mvmulfv 22487 | A cell/element in the vect... |
| mavmulval 22488 | Multiplication of a vector... |
| mavmulfv 22489 | A cell/element in the vect... |
| mavmulcl 22490 | Multiplication of an NxN m... |
| 1mavmul 22491 | Multiplication of the iden... |
| mavmulass 22492 | Associativity of the multi... |
| mavmuldm 22493 | The domain of the matrix v... |
| mavmulsolcl 22494 | Every solution of the equa... |
| mavmul0 22495 | Multiplication of a 0-dime... |
| mavmul0g 22496 | The result of the 0-dimens... |
| mvmumamul1 22497 | The multiplication of an M... |
| mavmumamul1 22498 | The multiplication of an N... |
| marrepfval 22503 | First substitution for the... |
| marrepval0 22504 | Second substitution for th... |
| marrepval 22505 | Third substitution for the... |
| marrepeval 22506 | An entry of a matrix with ... |
| marrepcl 22507 | Closure of the row replace... |
| marepvfval 22508 | First substitution for the... |
| marepvval0 22509 | Second substitution for th... |
| marepvval 22510 | Third substitution for the... |
| marepveval 22511 | An entry of a matrix with ... |
| marepvcl 22512 | Closure of the column repl... |
| ma1repvcl 22513 | Closure of the column repl... |
| ma1repveval 22514 | An entry of an identity ma... |
| mulmarep1el 22515 | Element by element multipl... |
| mulmarep1gsum1 22516 | The sum of element by elem... |
| mulmarep1gsum2 22517 | The sum of element by elem... |
| 1marepvmarrepid 22518 | Replacing the ith row by 0... |
| submabas 22521 | Any subset of the index se... |
| submafval 22522 | First substitution for a s... |
| submaval0 22523 | Second substitution for a ... |
| submaval 22524 | Third substitution for a s... |
| submaeval 22525 | An entry of a submatrix of... |
| 1marepvsma1 22526 | The submatrix of the ident... |
| mdetfval 22529 | First substitution for the... |
| mdetleib 22530 | Full substitution of our d... |
| mdetleib2 22531 | Leibniz' formula can also ... |
| nfimdetndef 22532 | The determinant is not def... |
| mdetfval1 22533 | First substitution of an a... |
| mdetleib1 22534 | Full substitution of an al... |
| mdet0pr 22535 | The determinant function f... |
| mdet0f1o 22536 | The determinant function f... |
| mdet0fv0 22537 | The determinant of the emp... |
| mdetf 22538 | Functionality of the deter... |
| mdetcl 22539 | The determinant evaluates ... |
| m1detdiag 22540 | The determinant of a 1-dim... |
| mdetdiaglem 22541 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22542 | The determinant of a diago... |
| mdetdiagid 22543 | The determinant of a diago... |
| mdet1 22544 | The determinant of the ide... |
| mdetrlin 22545 | The determinant function i... |
| mdetrsca 22546 | The determinant function i... |
| mdetrsca2 22547 | The determinant function i... |
| mdetr0 22548 | The determinant of a matri... |
| mdet0 22549 | The determinant of the zer... |
| mdetrlin2 22550 | The determinant function i... |
| mdetralt 22551 | The determinant function i... |
| mdetralt2 22552 | The determinant function i... |
| mdetero 22553 | The determinant function i... |
| mdettpos 22554 | Determinant is invariant u... |
| mdetunilem1 22555 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22556 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22557 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22558 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22559 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22560 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22561 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22562 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22563 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22564 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22565 | According to the definitio... |
| mdetmul 22566 | Multiplicativity of the de... |
| m2detleiblem1 22567 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22568 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22569 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22570 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22571 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22572 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22573 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22574 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22579 | Lemma for ~ maducoeval2 . ... |
| madufval 22580 | First substitution for the... |
| maduval 22581 | Second substitution for th... |
| maducoeval 22582 | An entry of the adjunct (c... |
| maducoeval2 22583 | An entry of the adjunct (c... |
| maduf 22584 | Creating the adjunct of ma... |
| madutpos 22585 | The adjuct of a transposed... |
| madugsum 22586 | The determinant of a matri... |
| madurid 22587 | Multiplying a matrix with ... |
| madulid 22588 | Multiplying the adjunct of... |
| minmar1fval 22589 | First substitution for the... |
| minmar1val0 22590 | Second substitution for th... |
| minmar1val 22591 | Third substitution for the... |
| minmar1eval 22592 | An entry of a matrix for a... |
| minmar1marrep 22593 | The minor matrix is a spec... |
| minmar1cl 22594 | Closure of the row replace... |
| maducoevalmin1 22595 | The coefficients of an adj... |
| symgmatr01lem 22596 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22597 | Applying a permutation tha... |
| gsummatr01lem1 22598 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22599 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22600 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22601 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22602 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22603 | Replacing a row of a squar... |
| smadiadetlem0 22604 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22605 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22606 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22607 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22608 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22609 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22610 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22611 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22612 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22613 | The determinant of a subma... |
| smadiadetglem1 22614 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22615 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22616 | The determinant of a squar... |
| smadiadetg0 22617 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22618 | The determinant of a squar... |
| invrvald 22619 | If a matrix multiplied wit... |
| matinv 22620 | The inverse of a matrix is... |
| matunit 22621 | A matrix is a unit in the ... |
| slesolvec 22622 | Every solution of a system... |
| slesolinv 22623 | The solution of a system o... |
| slesolinvbi 22624 | The solution of a system o... |
| slesolex 22625 | Every system of linear equ... |
| cramerimplem1 22626 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22627 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22628 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22629 | One direction of Cramer's ... |
| cramerlem1 22630 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22631 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22632 | Lemma 3 for ~ cramer . (C... |
| cramer0 22633 | Special case of Cramer's r... |
| cramer 22634 | Cramer's rule. According ... |
| pmatring 22635 | The set of polynomial matr... |
| pmatlmod 22636 | The set of polynomial matr... |
| pmatassa 22637 | The set of polynomial matr... |
| pmat0op 22638 | The zero polynomial matrix... |
| pmat1op 22639 | The identity polynomial ma... |
| pmat1ovd 22640 | Entries of the identity po... |
| pmat0opsc 22641 | The zero polynomial matrix... |
| pmat1opsc 22642 | The identity polynomial ma... |
| pmat1ovscd 22643 | Entries of the identity po... |
| pmatcoe1fsupp 22644 | For a polynomial matrix th... |
| 1pmatscmul 22645 | The scalar product of the ... |
| cpmat 22652 | Value of the constructor o... |
| cpmatpmat 22653 | A constant polynomial matr... |
| cpmatel 22654 | Property of a constant pol... |
| cpmatelimp 22655 | Implication of a set being... |
| cpmatel2 22656 | Another property of a cons... |
| cpmatelimp2 22657 | Another implication of a s... |
| 1elcpmat 22658 | The identity of the ring o... |
| cpmatacl 22659 | The set of all constant po... |
| cpmatinvcl 22660 | The set of all constant po... |
| cpmatmcllem 22661 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22662 | The set of all constant po... |
| cpmatsubgpmat 22663 | The set of all constant po... |
| cpmatsrgpmat 22664 | The set of all constant po... |
| 0elcpmat 22665 | The zero of the ring of al... |
| mat2pmatfval 22666 | Value of the matrix transf... |
| mat2pmatval 22667 | The result of a matrix tra... |
| mat2pmatvalel 22668 | A (matrix) element of the ... |
| mat2pmatbas 22669 | The result of a matrix tra... |
| mat2pmatbas0 22670 | The result of a matrix tra... |
| mat2pmatf 22671 | The matrix transformation ... |
| mat2pmatf1 22672 | The matrix transformation ... |
| mat2pmatghm 22673 | The transformation of matr... |
| mat2pmatmul 22674 | The transformation of matr... |
| mat2pmat1 22675 | The transformation of the ... |
| mat2pmatmhm 22676 | The transformation of matr... |
| mat2pmatrhm 22677 | The transformation of matr... |
| mat2pmatlin 22678 | The transformation of matr... |
| 0mat2pmat 22679 | The transformed zero matri... |
| idmatidpmat 22680 | The transformed identity m... |
| d0mat2pmat 22681 | The transformed empty set ... |
| d1mat2pmat 22682 | The transformation of a ma... |
| mat2pmatscmxcl 22683 | A transformed matrix multi... |
| m2cpm 22684 | The result of a matrix tra... |
| m2cpmf 22685 | The matrix transformation ... |
| m2cpmf1 22686 | The matrix transformation ... |
| m2cpmghm 22687 | The transformation of matr... |
| m2cpmmhm 22688 | The transformation of matr... |
| m2cpmrhm 22689 | The transformation of matr... |
| m2pmfzmap 22690 | The transformed values of ... |
| m2pmfzgsumcl 22691 | Closure of the sum of scal... |
| cpm2mfval 22692 | Value of the inverse matri... |
| cpm2mval 22693 | The result of an inverse m... |
| cpm2mvalel 22694 | A (matrix) element of the ... |
| cpm2mf 22695 | The inverse matrix transfo... |
| m2cpminvid 22696 | The inverse transformation... |
| m2cpminvid2lem 22697 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22698 | The transformation applied... |
| m2cpmfo 22699 | The matrix transformation ... |
| m2cpmf1o 22700 | The matrix transformation ... |
| m2cpmrngiso 22701 | The transformation of matr... |
| matcpmric 22702 | The ring of matrices over ... |
| m2cpminv 22703 | The inverse matrix transfo... |
| m2cpminv0 22704 | The inverse matrix transfo... |
| decpmatval0 22707 | The matrix consisting of t... |
| decpmatval 22708 | The matrix consisting of t... |
| decpmate 22709 | An entry of the matrix con... |
| decpmatcl 22710 | Closure of the decompositi... |
| decpmataa0 22711 | The matrix consisting of t... |
| decpmatfsupp 22712 | The mapping to the matrice... |
| decpmatid 22713 | The matrix consisting of t... |
| decpmatmullem 22714 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22715 | The matrix consisting of t... |
| decpmatmulsumfsupp 22716 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22717 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22718 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22719 | Write a polynomial matrix ... |
| pmatcollpw2lem 22720 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22721 | Write a polynomial matrix ... |
| monmatcollpw 22722 | The matrix consisting of t... |
| pmatcollpwlem 22723 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22724 | Write a polynomial matrix ... |
| pmatcollpwfi 22725 | Write a polynomial matrix ... |
| pmatcollpw3lem 22726 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22727 | Write a polynomial matrix ... |
| pmatcollpw3fi 22728 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22729 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22730 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22731 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22732 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22733 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22734 | Write a scalar matrix over... |
| pm2mpf1lem 22737 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22738 | Value of the transformatio... |
| pm2mpfval 22739 | A polynomial matrix transf... |
| pm2mpcl 22740 | The transformation of poly... |
| pm2mpf 22741 | The transformation of poly... |
| pm2mpf1 22742 | The transformation of poly... |
| pm2mpcoe1 22743 | A coefficient of the polyn... |
| idpm2idmp 22744 | The transformation of the ... |
| mptcoe1matfsupp 22745 | The mapping extracting the... |
| mply1topmatcllem 22746 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22747 | A polynomial over matrices... |
| mply1topmatcl 22748 | A polynomial over matrices... |
| mp2pm2mplem1 22749 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22750 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22751 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22752 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22753 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22754 | A polynomial over matrices... |
| pm2mpghmlem2 22755 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22756 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22757 | The transformation of poly... |
| pm2mpf1o 22758 | The transformation of poly... |
| pm2mpghm 22759 | The transformation of poly... |
| pm2mpgrpiso 22760 | The transformation of poly... |
| pm2mpmhmlem1 22761 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22762 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22763 | The transformation of poly... |
| pm2mprhm 22764 | The transformation of poly... |
| pm2mprngiso 22765 | The transformation of poly... |
| pmmpric 22766 | The ring of polynomial mat... |
| monmat2matmon 22767 | The transformation of a po... |
| pm2mp 22768 | The transformation of a su... |
| chmatcl 22771 | Closure of the characteris... |
| chmatval 22772 | The entries of the charact... |
| chpmatfval 22773 | Value of the characteristi... |
| chpmatval 22774 | The characteristic polynom... |
| chpmatply1 22775 | The characteristic polynom... |
| chpmatval2 22776 | The characteristic polynom... |
| chpmat0d 22777 | The characteristic polynom... |
| chpmat1dlem 22778 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22779 | The characteristic polynom... |
| chpdmatlem0 22780 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22781 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22782 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22783 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22784 | The characteristic polynom... |
| chpscmat 22785 | The characteristic polynom... |
| chpscmat0 22786 | The characteristic polynom... |
| chpscmatgsumbin 22787 | The characteristic polynom... |
| chpscmatgsummon 22788 | The characteristic polynom... |
| chp0mat 22789 | The characteristic polynom... |
| chpidmat 22790 | The characteristic polynom... |
| chmaidscmat 22791 | The characteristic polynom... |
| fvmptnn04if 22792 | The function values of a m... |
| fvmptnn04ifa 22793 | The function value of a ma... |
| fvmptnn04ifb 22794 | The function value of a ma... |
| fvmptnn04ifc 22795 | The function value of a ma... |
| fvmptnn04ifd 22796 | The function value of a ma... |
| chfacfisf 22797 | The "characteristic factor... |
| chfacfisfcpmat 22798 | The "characteristic factor... |
| chfacffsupp 22799 | The "characteristic factor... |
| chfacfscmulcl 22800 | Closure of a scaled value ... |
| chfacfscmul0 22801 | A scaled value of the "cha... |
| chfacfscmulfsupp 22802 | A mapping of scaled values... |
| chfacfscmulgsum 22803 | Breaking up a sum of value... |
| chfacfpmmulcl 22804 | Closure of the value of th... |
| chfacfpmmul0 22805 | The value of the "characte... |
| chfacfpmmulfsupp 22806 | A mapping of values of the... |
| chfacfpmmulgsum 22807 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22808 | Breaking up a sum of value... |
| cayhamlem1 22809 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22810 | The right-hand fundamental... |
| cpmidgsum 22811 | Representation of the iden... |
| cpmidgsumm2pm 22812 | Representation of the iden... |
| cpmidpmatlem1 22813 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22814 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22815 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22816 | Representation of the iden... |
| cpmadugsumlemB 22817 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22818 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22819 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22820 | The product of the charact... |
| cpmadugsum 22821 | The product of the charact... |
| cpmidgsum2 22822 | Representation of the iden... |
| cpmidg2sum 22823 | Equality of two sums repre... |
| cpmadumatpolylem1 22824 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22825 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22826 | The product of the charact... |
| cayhamlem2 22827 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22828 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22829 | The coefficients of the ch... |
| cayhamlem3 22830 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22831 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22832 | The Cayley-Hamilton theore... |
| cayleyhamilton 22833 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22834 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22835 | The Cayley-Hamilton theore... |
| istopg 22838 | Express the predicate " ` ... |
| istop2g 22839 | Express the predicate " ` ... |
| uniopn 22840 | The union of a subset of a... |
| iunopn 22841 | The indexed union of a sub... |
| inopn 22842 | The intersection of two op... |
| fitop 22843 | A topology is closed under... |
| fiinopn 22844 | The intersection of a none... |
| iinopn 22845 | The intersection of a none... |
| unopn 22846 | The union of two open sets... |
| 0opn 22847 | The empty set is an open s... |
| 0ntop 22848 | The empty set is not a top... |
| topopn 22849 | The underlying set of a to... |
| eltopss 22850 | A member of a topology is ... |
| riinopn 22851 | A finite indexed relative ... |
| rintopn 22852 | A finite relative intersec... |
| istopon 22855 | Property of being a topolo... |
| topontop 22856 | A topology on a given base... |
| toponuni 22857 | The base set of a topology... |
| topontopi 22858 | A topology on a given base... |
| toponunii 22859 | The base set of a topology... |
| toptopon 22860 | Alternative definition of ... |
| toptopon2 22861 | A topology is the same thi... |
| topontopon 22862 | A topology on a set is a t... |
| funtopon 22863 | The class ` TopOn ` is a f... |
| toponrestid 22864 | Given a topology on a set,... |
| toponsspwpw 22865 | The set of topologies on a... |
| dmtopon 22866 | The domain of ` TopOn ` is... |
| fntopon 22867 | The class ` TopOn ` is a f... |
| toprntopon 22868 | A topology is the same thi... |
| toponmax 22869 | The base set of a topology... |
| toponss 22870 | A member of a topology is ... |
| toponcom 22871 | If ` K ` is a topology on ... |
| toponcomb 22872 | Biconditional form of ~ to... |
| topgele 22873 | The topologies over the sa... |
| topsn 22874 | The only topology on a sin... |
| istps 22877 | Express the predicate "is ... |
| istps2 22878 | Express the predicate "is ... |
| tpsuni 22879 | The base set of a topologi... |
| tpstop 22880 | The topology extractor on ... |
| tpspropd 22881 | A topological space depend... |
| tpsprop2d 22882 | A topological space depend... |
| topontopn 22883 | Express the predicate "is ... |
| tsettps 22884 | If the topology component ... |
| istpsi 22885 | Properties that determine ... |
| eltpsg 22886 | Properties that determine ... |
| eltpsi 22887 | Properties that determine ... |
| isbasisg 22890 | Express the predicate "the... |
| isbasis2g 22891 | Express the predicate "the... |
| isbasis3g 22892 | Express the predicate "the... |
| basis1 22893 | Property of a basis. (Con... |
| basis2 22894 | Property of a basis. (Con... |
| fiinbas 22895 | If a set is closed under f... |
| basdif0 22896 | A basis is not affected by... |
| baspartn 22897 | A disjoint system of sets ... |
| tgval 22898 | The topology generated by ... |
| tgval2 22899 | Definition of a topology g... |
| eltg 22900 | Membership in a topology g... |
| eltg2 22901 | Membership in a topology g... |
| eltg2b 22902 | Membership in a topology g... |
| eltg4i 22903 | An open set in a topology ... |
| eltg3i 22904 | The union of a set of basi... |
| eltg3 22905 | Membership in a topology g... |
| tgval3 22906 | Alternate expression for t... |
| tg1 22907 | Property of a member of a ... |
| tg2 22908 | Property of a member of a ... |
| bastg 22909 | A member of a basis is a s... |
| unitg 22910 | The topology generated by ... |
| tgss 22911 | Subset relation for genera... |
| tgcl 22912 | Show that a basis generate... |
| tgclb 22913 | The property ~ tgcl can be... |
| tgtopon 22914 | A basis generates a topolo... |
| topbas 22915 | A topology is its own basi... |
| tgtop 22916 | A topology is its own basi... |
| eltop 22917 | Membership in a topology, ... |
| eltop2 22918 | Membership in a topology. ... |
| eltop3 22919 | Membership in a topology. ... |
| fibas 22920 | A collection of finite int... |
| tgdom 22921 | A space has no more open s... |
| tgiun 22922 | The indexed union of a set... |
| tgidm 22923 | The topology generator fun... |
| bastop 22924 | Two ways to express that a... |
| tgtop11 22925 | The topology generation fu... |
| 0top 22926 | The singleton of the empty... |
| en1top 22927 | ` { (/) } ` is the only to... |
| en2top 22928 | If a topology has two elem... |
| tgss3 22929 | A criterion for determinin... |
| tgss2 22930 | A criterion for determinin... |
| basgen 22931 | Given a topology ` J ` , s... |
| basgen2 22932 | Given a topology ` J ` , s... |
| 2basgen 22933 | Conditions that determine ... |
| tgfiss 22934 | If a subbase is included i... |
| tgdif0 22935 | A generated topology is no... |
| bastop1 22936 | A subset of a topology is ... |
| bastop2 22937 | A version of ~ bastop1 tha... |
| distop 22938 | The discrete topology on a... |
| topnex 22939 | The class of all topologie... |
| distopon 22940 | The discrete topology on a... |
| sn0topon 22941 | The singleton of the empty... |
| sn0top 22942 | The singleton of the empty... |
| indislem 22943 | A lemma to eliminate some ... |
| indistopon 22944 | The indiscrete topology on... |
| indistop 22945 | The indiscrete topology on... |
| indisuni 22946 | The base set of the indisc... |
| fctop 22947 | The finite complement topo... |
| fctop2 22948 | The finite complement topo... |
| cctop 22949 | The countable complement t... |
| ppttop 22950 | The particular point topol... |
| pptbas 22951 | The particular point topol... |
| epttop 22952 | The excluded point topolog... |
| indistpsx 22953 | The indiscrete topology on... |
| indistps 22954 | The indiscrete topology on... |
| indistps2 22955 | The indiscrete topology on... |
| indistpsALT 22956 | The indiscrete topology on... |
| indistps2ALT 22957 | The indiscrete topology on... |
| distps 22958 | The discrete topology on a... |
| fncld 22965 | The closed-set generator i... |
| cldval 22966 | The set of closed sets of ... |
| ntrfval 22967 | The interior function on t... |
| clsfval 22968 | The closure function on th... |
| cldrcl 22969 | Reverse closure of the clo... |
| iscld 22970 | The predicate "the class `... |
| iscld2 22971 | A subset of the underlying... |
| cldss 22972 | A closed set is a subset o... |
| cldss2 22973 | The set of closed sets is ... |
| cldopn 22974 | The complement of a closed... |
| isopn2 22975 | A subset of the underlying... |
| opncld 22976 | The complement of an open ... |
| difopn 22977 | The difference of a closed... |
| topcld 22978 | The underlying set of a to... |
| ntrval 22979 | The interior of a subset o... |
| clsval 22980 | The closure of a subset of... |
| 0cld 22981 | The empty set is closed. ... |
| iincld 22982 | The indexed intersection o... |
| intcld 22983 | The intersection of a set ... |
| uncld 22984 | The union of two closed se... |
| cldcls 22985 | A closed subset equals its... |
| incld 22986 | The intersection of two cl... |
| riincld 22987 | An indexed relative inters... |
| iuncld 22988 | A finite indexed union of ... |
| unicld 22989 | A finite union of closed s... |
| clscld 22990 | The closure of a subset of... |
| clsf 22991 | The closure function is a ... |
| ntropn 22992 | The interior of a subset o... |
| clsval2 22993 | Express closure in terms o... |
| ntrval2 22994 | Interior expressed in term... |
| ntrdif 22995 | An interior of a complemen... |
| clsdif 22996 | A closure of a complement ... |
| clsss 22997 | Subset relationship for cl... |
| ntrss 22998 | Subset relationship for in... |
| sscls 22999 | A subset of a topology's u... |
| ntrss2 23000 | A subset includes its inte... |
| ssntr 23001 | An open subset of a set is... |
| clsss3 23002 | The closure of a subset of... |
| ntrss3 23003 | The interior of a subset o... |
| ntrin 23004 | A pairwise intersection of... |
| cmclsopn 23005 | The complement of a closur... |
| cmntrcld 23006 | The complement of an inter... |
| iscld3 23007 | A subset is closed iff it ... |
| iscld4 23008 | A subset is closed iff it ... |
| isopn3 23009 | A subset is open iff it eq... |
| clsidm 23010 | The closure operation is i... |
| ntridm 23011 | The interior operation is ... |
| clstop 23012 | The closure of a topology'... |
| ntrtop 23013 | The interior of a topology... |
| 0ntr 23014 | A subset with an empty int... |
| clsss2 23015 | If a subset is included in... |
| elcls 23016 | Membership in a closure. ... |
| elcls2 23017 | Membership in a closure. ... |
| clsndisj 23018 | Any open set containing a ... |
| ntrcls0 23019 | A subset whose closure has... |
| ntreq0 23020 | Two ways to say that a sub... |
| cldmre 23021 | The closed sets of a topol... |
| mrccls 23022 | Moore closure generalizes ... |
| cls0 23023 | The closure of the empty s... |
| ntr0 23024 | The interior of the empty ... |
| isopn3i 23025 | An open subset equals its ... |
| elcls3 23026 | Membership in a closure in... |
| opncldf1 23027 | A bijection useful for con... |
| opncldf2 23028 | The values of the open-clo... |
| opncldf3 23029 | The values of the converse... |
| isclo 23030 | A set ` A ` is clopen iff ... |
| isclo2 23031 | A set ` A ` is clopen iff ... |
| discld 23032 | The open sets of a discret... |
| sn0cld 23033 | The closed sets of the top... |
| indiscld 23034 | The closed sets of an indi... |
| mretopd 23035 | A Moore collection which i... |
| toponmre 23036 | The topologies over a give... |
| cldmreon 23037 | The closed sets of a topol... |
| iscldtop 23038 | A family is the closed set... |
| mreclatdemoBAD 23039 | The closed subspaces of a ... |
| neifval 23042 | Value of the neighborhood ... |
| neif 23043 | The neighborhood function ... |
| neiss2 23044 | A set with a neighborhood ... |
| neival 23045 | Value of the set of neighb... |
| isnei 23046 | The predicate "the class `... |
| neiint 23047 | An intuitive definition of... |
| isneip 23048 | The predicate "the class `... |
| neii1 23049 | A neighborhood is included... |
| neisspw 23050 | The neighborhoods of any s... |
| neii2 23051 | Property of a neighborhood... |
| neiss 23052 | Any neighborhood of a set ... |
| ssnei 23053 | A set is included in any o... |
| elnei 23054 | A point belongs to any of ... |
| 0nnei 23055 | The empty set is not a nei... |
| neips 23056 | A neighborhood of a set is... |
| opnneissb 23057 | An open set is a neighborh... |
| opnssneib 23058 | Any superset of an open se... |
| ssnei2 23059 | Any subset ` M ` of ` X ` ... |
| neindisj 23060 | Any neighborhood of an ele... |
| opnneiss 23061 | An open set is a neighborh... |
| opnneip 23062 | An open set is a neighborh... |
| opnnei 23063 | A set is open iff it is a ... |
| tpnei 23064 | The underlying set of a to... |
| neiuni 23065 | The union of the neighborh... |
| neindisj2 23066 | A point ` P ` belongs to t... |
| topssnei 23067 | A finer topology has more ... |
| innei 23068 | The intersection of two ne... |
| opnneiid 23069 | Only an open set is a neig... |
| neissex 23070 | For any neighborhood ` N `... |
| 0nei 23071 | The empty set is a neighbo... |
| neipeltop 23072 | Lemma for ~ neiptopreu . ... |
| neiptopuni 23073 | Lemma for ~ neiptopreu . ... |
| neiptoptop 23074 | Lemma for ~ neiptopreu . ... |
| neiptopnei 23075 | Lemma for ~ neiptopreu . ... |
| neiptopreu 23076 | If, to each element ` P ` ... |
| lpfval 23081 | The limit point function o... |
| lpval 23082 | The set of limit points of... |
| islp 23083 | The predicate "the class `... |
| lpsscls 23084 | The limit points of a subs... |
| lpss 23085 | The limit points of a subs... |
| lpdifsn 23086 | ` P ` is a limit point of ... |
| lpss3 23087 | Subset relationship for li... |
| islp2 23088 | The predicate " ` P ` is a... |
| islp3 23089 | The predicate " ` P ` is a... |
| maxlp 23090 | A point is a limit point o... |
| clslp 23091 | The closure of a subset of... |
| islpi 23092 | A point belonging to a set... |
| cldlp 23093 | A subset of a topological ... |
| isperf 23094 | Definition of a perfect sp... |
| isperf2 23095 | Definition of a perfect sp... |
| isperf3 23096 | A perfect space is a topol... |
| perflp 23097 | The limit points of a perf... |
| perfi 23098 | Property of a perfect spac... |
| perftop 23099 | A perfect space is a topol... |
| restrcl 23100 | Reverse closure for the su... |
| restbas 23101 | A subspace topology basis ... |
| tgrest 23102 | A subspace can be generate... |
| resttop 23103 | A subspace topology is a t... |
| resttopon 23104 | A subspace topology is a t... |
| restuni 23105 | The underlying set of a su... |
| stoig 23106 | The topological space buil... |
| restco 23107 | Composition of subspaces. ... |
| restabs 23108 | Equivalence of being a sub... |
| restin 23109 | When the subspace region i... |
| restuni2 23110 | The underlying set of a su... |
| resttopon2 23111 | The underlying set of a su... |
| rest0 23112 | The subspace topology indu... |
| restsn 23113 | The only subspace topology... |
| restsn2 23114 | The subspace topology indu... |
| restcld 23115 | A closed set of a subspace... |
| restcldi 23116 | A closed set is closed in ... |
| restcldr 23117 | A set which is closed in t... |
| restopnb 23118 | If ` B ` is an open subset... |
| ssrest 23119 | If ` K ` is a finer topolo... |
| restopn2 23120 | If ` A ` is open, then ` B... |
| restdis 23121 | A subspace of a discrete t... |
| restfpw 23122 | The restriction of the set... |
| neitr 23123 | The neighborhood of a trac... |
| restcls 23124 | A closure in a subspace to... |
| restntr 23125 | An interior in a subspace ... |
| restlp 23126 | The limit points of a subs... |
| restperf 23127 | Perfection of a subspace. ... |
| perfopn 23128 | An open subset of a perfec... |
| resstopn 23129 | The topology of a restrict... |
| resstps 23130 | A restricted topological s... |
| ordtbaslem 23131 | Lemma for ~ ordtbas . In ... |
| ordtval 23132 | Value of the order topolog... |
| ordtuni 23133 | Value of the order topolog... |
| ordtbas2 23134 | Lemma for ~ ordtbas . (Co... |
| ordtbas 23135 | In a total order, the fini... |
| ordttopon 23136 | Value of the order topolog... |
| ordtopn1 23137 | An upward ray ` ( P , +oo ... |
| ordtopn2 23138 | A downward ray ` ( -oo , P... |
| ordtopn3 23139 | An open interval ` ( A , B... |
| ordtcld1 23140 | A downward ray ` ( -oo , P... |
| ordtcld2 23141 | An upward ray ` [ P , +oo ... |
| ordtcld3 23142 | A closed interval ` [ A , ... |
| ordttop 23143 | The order topology is a to... |
| ordtcnv 23144 | The order dual generates t... |
| ordtrest 23145 | The subspace topology of a... |
| ordtrest2lem 23146 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 23147 | An interval-closed set ` A... |
| letopon 23148 | The topology of the extend... |
| letop 23149 | The topology of the extend... |
| letopuni 23150 | The topology of the extend... |
| xrstopn 23151 | The topology component of ... |
| xrstps 23152 | The extended real number s... |
| leordtvallem1 23153 | Lemma for ~ leordtval . (... |
| leordtvallem2 23154 | Lemma for ~ leordtval . (... |
| leordtval2 23155 | The topology of the extend... |
| leordtval 23156 | The topology of the extend... |
| iccordt 23157 | A closed interval is close... |
| iocpnfordt 23158 | An unbounded above open in... |
| icomnfordt 23159 | An unbounded above open in... |
| iooordt 23160 | An open interval is open i... |
| reordt 23161 | The real numbers are an op... |
| lecldbas 23162 | The set of closed interval... |
| pnfnei 23163 | A neighborhood of ` +oo ` ... |
| mnfnei 23164 | A neighborhood of ` -oo ` ... |
| ordtrestixx 23165 | The restriction of the les... |
| ordtresticc 23166 | The restriction of the les... |
| lmrel 23173 | The topological space conv... |
| lmrcl 23174 | Reverse closure for the co... |
| lmfval 23175 | The relation "sequence ` f... |
| cnfval 23176 | The set of all continuous ... |
| cnpfval 23177 | The function mapping the p... |
| iscn 23178 | The predicate "the class `... |
| cnpval 23179 | The set of all functions f... |
| iscnp 23180 | The predicate "the class `... |
| iscn2 23181 | The predicate "the class `... |
| iscnp2 23182 | The predicate "the class `... |
| cntop1 23183 | Reverse closure for a cont... |
| cntop2 23184 | Reverse closure for a cont... |
| cnptop1 23185 | Reverse closure for a func... |
| cnptop2 23186 | Reverse closure for a func... |
| iscnp3 23187 | The predicate "the class `... |
| cnprcl 23188 | Reverse closure for a func... |
| cnf 23189 | A continuous function is a... |
| cnpf 23190 | A continuous function at p... |
| cnpcl 23191 | The value of a continuous ... |
| cnf2 23192 | A continuous function is a... |
| cnpf2 23193 | A continuous function at p... |
| cnprcl2 23194 | Reverse closure for a func... |
| tgcn 23195 | The continuity predicate w... |
| tgcnp 23196 | The "continuous at a point... |
| subbascn 23197 | The continuity predicate w... |
| ssidcn 23198 | The identity function is a... |
| cnpimaex 23199 | Property of a function con... |
| idcn 23200 | A restricted identity func... |
| lmbr 23201 | Express the binary relatio... |
| lmbr2 23202 | Express the binary relatio... |
| lmbrf 23203 | Express the binary relatio... |
| lmconst 23204 | A constant sequence conver... |
| lmcvg 23205 | Convergence property of a ... |
| iscnp4 23206 | The predicate "the class `... |
| cnpnei 23207 | A condition for continuity... |
| cnima 23208 | An open subset of the codo... |
| cnco 23209 | The composition of two con... |
| cnpco 23210 | The composition of a funct... |
| cnclima 23211 | A closed subset of the cod... |
| iscncl 23212 | A characterization of a co... |
| cncls2i 23213 | Property of the preimage o... |
| cnntri 23214 | Property of the preimage o... |
| cnclsi 23215 | Property of the image of a... |
| cncls2 23216 | Continuity in terms of clo... |
| cncls 23217 | Continuity in terms of clo... |
| cnntr 23218 | Continuity in terms of int... |
| cnss1 23219 | If the topology ` K ` is f... |
| cnss2 23220 | If the topology ` K ` is f... |
| cncnpi 23221 | A continuous function is c... |
| cnsscnp 23222 | The set of continuous func... |
| cncnp 23223 | A continuous function is c... |
| cncnp2 23224 | A continuous function is c... |
| cnnei 23225 | Continuity in terms of nei... |
| cnconst2 23226 | A constant function is con... |
| cnconst 23227 | A constant function is con... |
| cnrest 23228 | Continuity of a restrictio... |
| cnrest2 23229 | Equivalence of continuity ... |
| cnrest2r 23230 | Equivalence of continuity ... |
| cnpresti 23231 | One direction of ~ cnprest... |
| cnprest 23232 | Equivalence of continuity ... |
| cnprest2 23233 | Equivalence of point-conti... |
| cndis 23234 | Every function is continuo... |
| cnindis 23235 | Every function is continuo... |
| cnpdis 23236 | If ` A ` is an isolated po... |
| paste 23237 | Pasting lemma. If ` A ` a... |
| lmfpm 23238 | If ` F ` converges, then `... |
| lmfss 23239 | Inclusion of a function ha... |
| lmcl 23240 | Closure of a limit. (Cont... |
| lmss 23241 | Limit on a subspace. (Con... |
| sslm 23242 | A finer topology has fewer... |
| lmres 23243 | A function converges iff i... |
| lmff 23244 | If ` F ` converges, there ... |
| lmcls 23245 | Any convergent sequence of... |
| lmcld 23246 | Any convergent sequence of... |
| lmcnp 23247 | The image of a convergent ... |
| lmcn 23248 | The image of a convergent ... |
| ist0 23263 | The predicate "is a T_0 sp... |
| ist1 23264 | The predicate "is a T_1 sp... |
| ishaus 23265 | The predicate "is a Hausdo... |
| iscnrm 23266 | The property of being comp... |
| t0sep 23267 | Any two topologically indi... |
| t0dist 23268 | Any two distinct points in... |
| t1sncld 23269 | In a T_1 space, singletons... |
| t1ficld 23270 | In a T_1 space, finite set... |
| hausnei 23271 | Neighborhood property of a... |
| t0top 23272 | A T_0 space is a topologic... |
| t1top 23273 | A T_1 space is a topologic... |
| haustop 23274 | A Hausdorff space is a top... |
| isreg 23275 | The predicate "is a regula... |
| regtop 23276 | A regular space is a topol... |
| regsep 23277 | In a regular space, every ... |
| isnrm 23278 | The predicate "is a normal... |
| nrmtop 23279 | A normal space is a topolo... |
| cnrmtop 23280 | A completely normal space ... |
| iscnrm2 23281 | The property of being comp... |
| ispnrm 23282 | The property of being perf... |
| pnrmnrm 23283 | A perfectly normal space i... |
| pnrmtop 23284 | A perfectly normal space i... |
| pnrmcld 23285 | A closed set in a perfectl... |
| pnrmopn 23286 | An open set in a perfectly... |
| ist0-2 23287 | The predicate "is a T_0 sp... |
| ist0-3 23288 | The predicate "is a T_0 sp... |
| cnt0 23289 | The preimage of a T_0 topo... |
| ist1-2 23290 | An alternate characterizat... |
| t1t0 23291 | A T_1 space is a T_0 space... |
| ist1-3 23292 | A space is T_1 iff every p... |
| cnt1 23293 | The preimage of a T_1 topo... |
| ishaus2 23294 | Express the predicate " ` ... |
| haust1 23295 | A Hausdorff space is a T_1... |
| hausnei2 23296 | The Hausdorff condition st... |
| cnhaus 23297 | The preimage of a Hausdorf... |
| nrmsep3 23298 | In a normal space, given a... |
| nrmsep2 23299 | In a normal space, any two... |
| nrmsep 23300 | In a normal space, disjoin... |
| isnrm2 23301 | An alternate characterizat... |
| isnrm3 23302 | A topological space is nor... |
| cnrmi 23303 | A subspace of a completely... |
| cnrmnrm 23304 | A completely normal space ... |
| restcnrm 23305 | A subspace of a completely... |
| resthauslem 23306 | Lemma for ~ resthaus and s... |
| lpcls 23307 | The limit points of the cl... |
| perfcls 23308 | A subset of a perfect spac... |
| restt0 23309 | A subspace of a T_0 topolo... |
| restt1 23310 | A subspace of a T_1 topolo... |
| resthaus 23311 | A subspace of a Hausdorff ... |
| t1sep2 23312 | Any two points in a T_1 sp... |
| t1sep 23313 | Any two distinct points in... |
| sncld 23314 | A singleton is closed in a... |
| sshauslem 23315 | Lemma for ~ sshaus and sim... |
| sst0 23316 | A topology finer than a T_... |
| sst1 23317 | A topology finer than a T_... |
| sshaus 23318 | A topology finer than a Ha... |
| regsep2 23319 | In a regular space, a clos... |
| isreg2 23320 | A topological space is reg... |
| dnsconst 23321 | If a continuous mapping to... |
| ordtt1 23322 | The order topology is T_1 ... |
| lmmo 23323 | A sequence in a Hausdorff ... |
| lmfun 23324 | The convergence relation i... |
| dishaus 23325 | A discrete topology is Hau... |
| ordthauslem 23326 | Lemma for ~ ordthaus . (C... |
| ordthaus 23327 | The order topology of a to... |
| xrhaus 23328 | The topology of the extend... |
| iscmp 23331 | The predicate "is a compac... |
| cmpcov 23332 | An open cover of a compact... |
| cmpcov2 23333 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23334 | Combine ~ cmpcov with ~ ac... |
| cncmp 23335 | Compactness is respected b... |
| fincmp 23336 | A finite topology is compa... |
| 0cmp 23337 | The singleton of the empty... |
| cmptop 23338 | A compact topology is a to... |
| rncmp 23339 | The image of a compact set... |
| imacmp 23340 | The image of a compact set... |
| discmp 23341 | A discrete topology is com... |
| cmpsublem 23342 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23343 | Two equivalent ways of des... |
| tgcmp 23344 | A topology generated by a ... |
| cmpcld 23345 | A closed subset of a compa... |
| uncmp 23346 | The union of two compact s... |
| fiuncmp 23347 | A finite union of compact ... |
| sscmp 23348 | A subset of a compact topo... |
| hauscmplem 23349 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23350 | A compact subspace of a T2... |
| cmpfi 23351 | If a topology is compact a... |
| cmpfii 23352 | In a compact topology, a s... |
| bwth 23353 | The glorious Bolzano-Weier... |
| isconn 23356 | The predicate ` J ` is a c... |
| isconn2 23357 | The predicate ` J ` is a c... |
| connclo 23358 | The only nonempty clopen s... |
| conndisj 23359 | If a topology is connected... |
| conntop 23360 | A connected topology is a ... |
| indisconn 23361 | The indiscrete topology (o... |
| dfconn2 23362 | An alternate definition of... |
| connsuba 23363 | Connectedness for a subspa... |
| connsub 23364 | Two equivalent ways of say... |
| cnconn 23365 | Connectedness is respected... |
| nconnsubb 23366 | Disconnectedness for a sub... |
| connsubclo 23367 | If a clopen set meets a co... |
| connima 23368 | The image of a connected s... |
| conncn 23369 | A continuous function from... |
| iunconnlem 23370 | Lemma for ~ iunconn . (Co... |
| iunconn 23371 | The indexed union of conne... |
| unconn 23372 | The union of two connected... |
| clsconn 23373 | The closure of a connected... |
| conncompid 23374 | The connected component co... |
| conncompconn 23375 | The connected component co... |
| conncompss 23376 | The connected component co... |
| conncompcld 23377 | The connected component co... |
| conncompclo 23378 | The connected component co... |
| t1connperf 23379 | A connected T_1 space is p... |
| is1stc 23384 | The predicate "is a first-... |
| is1stc2 23385 | An equivalent way of sayin... |
| 1stctop 23386 | A first-countable topology... |
| 1stcclb 23387 | A property of points in a ... |
| 1stcfb 23388 | For any point ` A ` in a f... |
| is2ndc 23389 | The property of being seco... |
| 2ndctop 23390 | A second-countable topolog... |
| 2ndci 23391 | A countable basis generate... |
| 2ndcsb 23392 | Having a countable subbase... |
| 2ndcredom 23393 | A second-countable space h... |
| 2ndc1stc 23394 | A second-countable space i... |
| 1stcrestlem 23395 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23396 | A subspace of a first-coun... |
| 2ndcrest 23397 | A subspace of a second-cou... |
| 2ndcctbss 23398 | If a topology is second-co... |
| 2ndcdisj 23399 | Any disjoint family of ope... |
| 2ndcdisj2 23400 | Any disjoint collection of... |
| 2ndcomap 23401 | A surjective continuous op... |
| 2ndcsep 23402 | A second-countable topolog... |
| dis2ndc 23403 | A discrete space is second... |
| 1stcelcls 23404 | A point belongs to the clo... |
| 1stccnp 23405 | A mapping is continuous at... |
| 1stccn 23406 | A mapping ` X --> Y ` , wh... |
| islly 23411 | The property of being a lo... |
| isnlly 23412 | The property of being an n... |
| llyeq 23413 | Equality theorem for the `... |
| nllyeq 23414 | Equality theorem for the `... |
| llytop 23415 | A locally ` A ` space is a... |
| nllytop 23416 | A locally ` A ` space is a... |
| llyi 23417 | The property of a locally ... |
| nllyi 23418 | The property of an n-local... |
| nlly2i 23419 | Eliminate the neighborhood... |
| llynlly 23420 | A locally ` A ` space is n... |
| llyssnlly 23421 | A locally ` A ` space is n... |
| llyss 23422 | The "locally" predicate re... |
| nllyss 23423 | The "n-locally" predicate ... |
| subislly 23424 | The property of a subspace... |
| restnlly 23425 | If the property ` A ` pass... |
| restlly 23426 | If the property ` A ` pass... |
| islly2 23427 | An alternative expression ... |
| llyrest 23428 | An open subspace of a loca... |
| nllyrest 23429 | An open subspace of an n-l... |
| loclly 23430 | If ` A ` is a local proper... |
| llyidm 23431 | Idempotence of the "locall... |
| nllyidm 23432 | Idempotence of the "n-loca... |
| toplly 23433 | A topology is locally a to... |
| topnlly 23434 | A topology is n-locally a ... |
| hauslly 23435 | A Hausdorff space is local... |
| hausnlly 23436 | A Hausdorff space is n-loc... |
| hausllycmp 23437 | A compact Hausdorff space ... |
| cldllycmp 23438 | A closed subspace of a loc... |
| lly1stc 23439 | First-countability is a lo... |
| dislly 23440 | The discrete space ` ~P X ... |
| disllycmp 23441 | A discrete space is locall... |
| dis1stc 23442 | A discrete space is first-... |
| hausmapdom 23443 | If ` X ` is a first-counta... |
| hauspwdom 23444 | Simplify the cardinal ` A ... |
| refrel 23451 | Refinement is a relation. ... |
| isref 23452 | The property of being a re... |
| refbas 23453 | A refinement covers the sa... |
| refssex 23454 | Every set in a refinement ... |
| ssref 23455 | A subcover is a refinement... |
| refref 23456 | Reflexivity of refinement.... |
| reftr 23457 | Refinement is transitive. ... |
| refun0 23458 | Adding the empty set prese... |
| isptfin 23459 | The statement "is a point-... |
| islocfin 23460 | The statement "is a locall... |
| finptfin 23461 | A finite cover is a point-... |
| ptfinfin 23462 | A point covered by a point... |
| finlocfin 23463 | A finite cover of a topolo... |
| locfintop 23464 | A locally finite cover cov... |
| locfinbas 23465 | A locally finite cover mus... |
| locfinnei 23466 | A point covered by a local... |
| lfinpfin 23467 | A locally finite cover is ... |
| lfinun 23468 | Adding a finite set preser... |
| locfincmp 23469 | For a compact space, the l... |
| unisngl 23470 | Taking the union of the se... |
| dissnref 23471 | The set of singletons is a... |
| dissnlocfin 23472 | The set of singletons is l... |
| locfindis 23473 | The locally finite covers ... |
| locfincf 23474 | A locally finite cover in ... |
| comppfsc 23475 | A space where every open c... |
| kgenval 23478 | Value of the compact gener... |
| elkgen 23479 | Value of the compact gener... |
| kgeni 23480 | Property of the open sets ... |
| kgentopon 23481 | The compact generator gene... |
| kgenuni 23482 | The base set of the compac... |
| kgenftop 23483 | The compact generator gene... |
| kgenf 23484 | The compact generator is a... |
| kgentop 23485 | A compactly generated spac... |
| kgenss 23486 | The compact generator gene... |
| kgenhaus 23487 | The compact generator gene... |
| kgencmp 23488 | The compact generator topo... |
| kgencmp2 23489 | The compact generator topo... |
| kgenidm 23490 | The compact generator is i... |
| iskgen2 23491 | A space is compactly gener... |
| iskgen3 23492 | Derive the usual definitio... |
| llycmpkgen2 23493 | A locally compact space is... |
| cmpkgen 23494 | A compact space is compact... |
| llycmpkgen 23495 | A locally compact space is... |
| 1stckgenlem 23496 | The one-point compactifica... |
| 1stckgen 23497 | A first-countable space is... |
| kgen2ss 23498 | The compact generator pres... |
| kgencn 23499 | A function from a compactl... |
| kgencn2 23500 | A function ` F : J --> K `... |
| kgencn3 23501 | The set of continuous func... |
| kgen2cn 23502 | A continuous function is a... |
| txval 23507 | Value of the binary topolo... |
| txuni2 23508 | The underlying set of the ... |
| txbasex 23509 | The basis for the product ... |
| txbas 23510 | The set of Cartesian produ... |
| eltx 23511 | A set in a product is open... |
| txtop 23512 | The product of two topolog... |
| ptval 23513 | The value of the product t... |
| ptpjpre1 23514 | The preimage of a projecti... |
| elpt 23515 | Elementhood in the bases o... |
| elptr 23516 | A basic open set in the pr... |
| elptr2 23517 | A basic open set in the pr... |
| ptbasid 23518 | The base set of the produc... |
| ptuni2 23519 | The base set for the produ... |
| ptbasin 23520 | The basis for a product to... |
| ptbasin2 23521 | The basis for a product to... |
| ptbas 23522 | The basis for a product to... |
| ptpjpre2 23523 | The basis for a product to... |
| ptbasfi 23524 | The basis for the product ... |
| pttop 23525 | The product topology is a ... |
| ptopn 23526 | A basic open set in the pr... |
| ptopn2 23527 | A sub-basic open set in th... |
| xkotf 23528 | Functionality of function ... |
| xkobval 23529 | Alternative expression for... |
| xkoval 23530 | Value of the compact-open ... |
| xkotop 23531 | The compact-open topology ... |
| xkoopn 23532 | A basic open set of the co... |
| txtopi 23533 | The product of two topolog... |
| txtopon 23534 | The underlying set of the ... |
| txuni 23535 | The underlying set of the ... |
| txunii 23536 | The underlying set of the ... |
| ptuni 23537 | The base set for the produ... |
| ptunimpt 23538 | Base set of a product topo... |
| pttopon 23539 | The base set for the produ... |
| pttoponconst 23540 | The base set for a product... |
| ptuniconst 23541 | The base set for a product... |
| xkouni 23542 | The base set of the compac... |
| xkotopon 23543 | The base set of the compac... |
| ptval2 23544 | The value of the product t... |
| txopn 23545 | The product of two open se... |
| txcld 23546 | The product of two closed ... |
| txcls 23547 | Closure of a rectangle in ... |
| txss12 23548 | Subset property of the top... |
| txbasval 23549 | It is sufficient to consid... |
| neitx 23550 | The Cartesian product of t... |
| txcnpi 23551 | Continuity of a two-argume... |
| tx1cn 23552 | Continuity of the first pr... |
| tx2cn 23553 | Continuity of the second p... |
| ptpjcn 23554 | Continuity of a projection... |
| ptpjopn 23555 | The projection map is an o... |
| ptcld 23556 | A closed box in the produc... |
| ptcldmpt 23557 | A closed box in the produc... |
| ptclsg 23558 | The closure of a box in th... |
| ptcls 23559 | The closure of a box in th... |
| dfac14lem 23560 | Lemma for ~ dfac14 . By e... |
| dfac14 23561 | Theorem ~ ptcls is an equi... |
| xkoccn 23562 | The "constant function" fu... |
| txcnp 23563 | If two functions are conti... |
| ptcnplem 23564 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23565 | If every projection of a f... |
| upxp 23566 | Universal property of the ... |
| txcnmpt 23567 | A map into the product of ... |
| uptx 23568 | Universal property of the ... |
| txcn 23569 | A map into the product of ... |
| ptcn 23570 | If every projection of a f... |
| prdstopn 23571 | Topology of a structure pr... |
| prdstps 23572 | A structure product of top... |
| pwstps 23573 | A structure power of a top... |
| txrest 23574 | The subspace of a topologi... |
| txdis 23575 | The topological product of... |
| txindislem 23576 | Lemma for ~ txindis . (Co... |
| txindis 23577 | The topological product of... |
| txdis1cn 23578 | A function is jointly cont... |
| txlly 23579 | If the property ` A ` is p... |
| txnlly 23580 | If the property ` A ` is p... |
| pthaus 23581 | The product of a collectio... |
| ptrescn 23582 | Restriction is a continuou... |
| txtube 23583 | The "tube lemma". If ` X ... |
| txcmplem1 23584 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23585 | Lemma for ~ txcmp . (Cont... |
| txcmp 23586 | The topological product of... |
| txcmpb 23587 | The topological product of... |
| hausdiag 23588 | A topology is Hausdorff if... |
| hauseqlcld 23589 | In a Hausdorff topology, t... |
| txhaus 23590 | The topological product of... |
| txlm 23591 | Two sequences converge iff... |
| lmcn2 23592 | The image of a convergent ... |
| tx1stc 23593 | The topological product of... |
| tx2ndc 23594 | The topological product of... |
| txkgen 23595 | The topological product of... |
| xkohaus 23596 | If the codomain space is H... |
| xkoptsub 23597 | The compact-open topology ... |
| xkopt 23598 | The compact-open topology ... |
| xkopjcn 23599 | Continuity of a projection... |
| xkoco1cn 23600 | If ` F ` is a continuous f... |
| xkoco2cn 23601 | If ` F ` is a continuous f... |
| xkococnlem 23602 | Continuity of the composit... |
| xkococn 23603 | Continuity of the composit... |
| cnmptid 23604 | The identity function is c... |
| cnmptc 23605 | A constant function is con... |
| cnmpt11 23606 | The composition of continu... |
| cnmpt11f 23607 | The composition of continu... |
| cnmpt1t 23608 | The composition of continu... |
| cnmpt12f 23609 | The composition of continu... |
| cnmpt12 23610 | The composition of continu... |
| cnmpt1st 23611 | The projection onto the fi... |
| cnmpt2nd 23612 | The projection onto the se... |
| cnmpt2c 23613 | A constant function is con... |
| cnmpt21 23614 | The composition of continu... |
| cnmpt21f 23615 | The composition of continu... |
| cnmpt2t 23616 | The composition of continu... |
| cnmpt22 23617 | The composition of continu... |
| cnmpt22f 23618 | The composition of continu... |
| cnmpt1res 23619 | The restriction of a conti... |
| cnmpt2res 23620 | The restriction of a conti... |
| cnmptcom 23621 | The argument converse of a... |
| cnmptkc 23622 | The curried first projecti... |
| cnmptkp 23623 | The evaluation of the inne... |
| cnmptk1 23624 | The composition of a curri... |
| cnmpt1k 23625 | The composition of a one-a... |
| cnmptkk 23626 | The composition of two cur... |
| xkofvcn 23627 | Joint continuity of the fu... |
| cnmptk1p 23628 | The evaluation of a currie... |
| cnmptk2 23629 | The uncurrying of a currie... |
| xkoinjcn 23630 | Continuity of "injection",... |
| cnmpt2k 23631 | The currying of a two-argu... |
| txconn 23632 | The topological product of... |
| imasnopn 23633 | If a relation graph is ope... |
| imasncld 23634 | If a relation graph is clo... |
| imasncls 23635 | If a relation graph is clo... |
| qtopval 23638 | Value of the quotient topo... |
| qtopval2 23639 | Value of the quotient topo... |
| elqtop 23640 | Value of the quotient topo... |
| qtopres 23641 | The quotient topology is u... |
| qtoptop2 23642 | The quotient topology is a... |
| qtoptop 23643 | The quotient topology is a... |
| elqtop2 23644 | Value of the quotient topo... |
| qtopuni 23645 | The base set of the quotie... |
| elqtop3 23646 | Value of the quotient topo... |
| qtoptopon 23647 | The base set of the quotie... |
| qtopid 23648 | A quotient map is a contin... |
| idqtop 23649 | The quotient topology indu... |
| qtopcmplem 23650 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23651 | A quotient of a compact sp... |
| qtopconn 23652 | A quotient of a connected ... |
| qtopkgen 23653 | A quotient of a compactly ... |
| basqtop 23654 | An injection maps bases to... |
| tgqtop 23655 | An injection maps generate... |
| qtopcld 23656 | The property of being a cl... |
| qtopcn 23657 | Universal property of a qu... |
| qtopss 23658 | A surjective continuous fu... |
| qtopeu 23659 | Universal property of the ... |
| qtoprest 23660 | If ` A ` is a saturated op... |
| qtopomap 23661 | If ` F ` is a surjective c... |
| qtopcmap 23662 | If ` F ` is a surjective c... |
| imastopn 23663 | The topology of an image s... |
| imastps 23664 | The image of a topological... |
| qustps 23665 | A quotient structure is a ... |
| kqfval 23666 | Value of the function appe... |
| kqfeq 23667 | Two points in the Kolmogor... |
| kqffn 23668 | The topological indistingu... |
| kqval 23669 | Value of the quotient topo... |
| kqtopon 23670 | The Kolmogorov quotient is... |
| kqid 23671 | The topological indistingu... |
| ist0-4 23672 | The topological indistingu... |
| kqfvima 23673 | When the image set is open... |
| kqsat 23674 | Any open set is saturated ... |
| kqdisj 23675 | A version of ~ imain for t... |
| kqcldsat 23676 | Any closed set is saturate... |
| kqopn 23677 | The topological indistingu... |
| kqcld 23678 | The topological indistingu... |
| kqt0lem 23679 | Lemma for ~ kqt0 . (Contr... |
| isr0 23680 | The property " ` J ` is an... |
| r0cld 23681 | The analogue of the T_1 ax... |
| regr1lem 23682 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23683 | A Kolmogorov quotient of a... |
| kqreglem1 23684 | A Kolmogorov quotient of a... |
| kqreglem2 23685 | If the Kolmogorov quotient... |
| kqnrmlem1 23686 | A Kolmogorov quotient of a... |
| kqnrmlem2 23687 | If the Kolmogorov quotient... |
| kqtop 23688 | The Kolmogorov quotient is... |
| kqt0 23689 | The Kolmogorov quotient is... |
| kqf 23690 | The Kolmogorov quotient is... |
| r0sep 23691 | The separation property of... |
| nrmr0reg 23692 | A normal R_0 space is also... |
| regr1 23693 | A regular space is R_1, wh... |
| kqreg 23694 | The Kolmogorov quotient of... |
| kqnrm 23695 | The Kolmogorov quotient of... |
| hmeofn 23700 | The set of homeomorphisms ... |
| hmeofval 23701 | The set of all the homeomo... |
| ishmeo 23702 | The predicate F is a homeo... |
| hmeocn 23703 | A homeomorphism is continu... |
| hmeocnvcn 23704 | The converse of a homeomor... |
| hmeocnv 23705 | The converse of a homeomor... |
| hmeof1o2 23706 | A homeomorphism is a 1-1-o... |
| hmeof1o 23707 | A homeomorphism is a 1-1-o... |
| hmeoima 23708 | The image of an open set b... |
| hmeoopn 23709 | Homeomorphisms preserve op... |
| hmeocld 23710 | Homeomorphisms preserve cl... |
| hmeocls 23711 | Homeomorphisms preserve cl... |
| hmeontr 23712 | Homeomorphisms preserve in... |
| hmeoimaf1o 23713 | The function mapping open ... |
| hmeores 23714 | The restriction of a homeo... |
| hmeoco 23715 | The composite of two homeo... |
| idhmeo 23716 | The identity function is a... |
| hmeocnvb 23717 | The converse of a homeomor... |
| hmeoqtop 23718 | A homeomorphism is a quoti... |
| hmph 23719 | Express the predicate ` J ... |
| hmphi 23720 | If there is a homeomorphis... |
| hmphtop 23721 | Reverse closure for the ho... |
| hmphtop1 23722 | The relation "being homeom... |
| hmphtop2 23723 | The relation "being homeom... |
| hmphref 23724 | "Is homeomorphic to" is re... |
| hmphsym 23725 | "Is homeomorphic to" is sy... |
| hmphtr 23726 | "Is homeomorphic to" is tr... |
| hmpher 23727 | "Is homeomorphic to" is an... |
| hmphen 23728 | Homeomorphisms preserve th... |
| hmphsymb 23729 | "Is homeomorphic to" is sy... |
| haushmphlem 23730 | Lemma for ~ haushmph and s... |
| cmphmph 23731 | Compactness is a topologic... |
| connhmph 23732 | Connectedness is a topolog... |
| t0hmph 23733 | T_0 is a topological prope... |
| t1hmph 23734 | T_1 is a topological prope... |
| haushmph 23735 | Hausdorff-ness is a topolo... |
| reghmph 23736 | Regularity is a topologica... |
| nrmhmph 23737 | Normality is a topological... |
| hmph0 23738 | A topology homeomorphic to... |
| hmphdis 23739 | Homeomorphisms preserve to... |
| hmphindis 23740 | Homeomorphisms preserve to... |
| indishmph 23741 | Equinumerous sets equipped... |
| hmphen2 23742 | Homeomorphisms preserve th... |
| cmphaushmeo 23743 | A continuous bijection fro... |
| ordthmeolem 23744 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23745 | An order isomorphism is a ... |
| txhmeo 23746 | Lift a pair of homeomorphi... |
| txswaphmeolem 23747 | Show inverse for the "swap... |
| txswaphmeo 23748 | There is a homeomorphism f... |
| pt1hmeo 23749 | The canonical homeomorphis... |
| ptuncnv 23750 | Exhibit the converse funct... |
| ptunhmeo 23751 | Define a homeomorphism fro... |
| xpstopnlem1 23752 | The function ` F ` used in... |
| xpstps 23753 | A binary product of topolo... |
| xpstopnlem2 23754 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23755 | The topology on a binary p... |
| ptcmpfi 23756 | A topological product of f... |
| xkocnv 23757 | The inverse of the "curryi... |
| xkohmeo 23758 | The Exponential Law for to... |
| qtopf1 23759 | If a quotient map is injec... |
| qtophmeo 23760 | If two functions on a base... |
| t0kq 23761 | A topological space is T_0... |
| kqhmph 23762 | A topological space is T_0... |
| ist1-5lem 23763 | Lemma for ~ ist1-5 and sim... |
| t1r0 23764 | A T_1 space is R_0. That ... |
| ist1-5 23765 | A topological space is T_1... |
| ishaus3 23766 | A topological space is Hau... |
| nrmreg 23767 | A normal T_1 space is regu... |
| reghaus 23768 | A regular T_0 space is Hau... |
| nrmhaus 23769 | A T_1 normal space is Haus... |
| elmptrab 23770 | Membership in a one-parame... |
| elmptrab2 23771 | Membership in a one-parame... |
| isfbas 23772 | The predicate " ` F ` is a... |
| fbasne0 23773 | There are no empty filter ... |
| 0nelfb 23774 | No filter base contains th... |
| fbsspw 23775 | A filter base on a set is ... |
| fbelss 23776 | An element of the filter b... |
| fbdmn0 23777 | The domain of a filter bas... |
| isfbas2 23778 | The predicate " ` F ` is a... |
| fbasssin 23779 | A filter base contains sub... |
| fbssfi 23780 | A filter base contains sub... |
| fbssint 23781 | A filter base contains sub... |
| fbncp 23782 | A filter base does not con... |
| fbun 23783 | A necessary and sufficient... |
| fbfinnfr 23784 | No filter base containing ... |
| opnfbas 23785 | The collection of open sup... |
| trfbas2 23786 | Conditions for the trace o... |
| trfbas 23787 | Conditions for the trace o... |
| isfil 23790 | The predicate "is a filter... |
| filfbas 23791 | A filter is a filter base.... |
| 0nelfil 23792 | The empty set doesn't belo... |
| fileln0 23793 | An element of a filter is ... |
| filsspw 23794 | A filter is a subset of th... |
| filelss 23795 | An element of a filter is ... |
| filss 23796 | A filter is closed under t... |
| filin 23797 | A filter is closed under t... |
| filtop 23798 | The underlying set belongs... |
| isfil2 23799 | Derive the standard axioms... |
| isfildlem 23800 | Lemma for ~ isfild . (Con... |
| isfild 23801 | Sufficient condition for a... |
| filfi 23802 | A filter is closed under t... |
| filinn0 23803 | The intersection of two el... |
| filintn0 23804 | A filter has the finite in... |
| filn0 23805 | The empty set is not a fil... |
| infil 23806 | The intersection of two fi... |
| snfil 23807 | A singleton is a filter. ... |
| fbasweak 23808 | A filter base on any set i... |
| snfbas 23809 | Condition for a singleton ... |
| fsubbas 23810 | A condition for a set to g... |
| fbasfip 23811 | A filter base has the fini... |
| fbunfip 23812 | A helpful lemma for showin... |
| fgval 23813 | The filter generating clas... |
| elfg 23814 | A condition for elements o... |
| ssfg 23815 | A filter base is a subset ... |
| fgss 23816 | A bigger base generates a ... |
| fgss2 23817 | A condition for a filter t... |
| fgfil 23818 | A filter generates itself.... |
| elfilss 23819 | An element belongs to a fi... |
| filfinnfr 23820 | No filter containing a fin... |
| fgcl 23821 | A generated filter is a fi... |
| fgabs 23822 | Absorption law for filter ... |
| neifil 23823 | The neighborhoods of a non... |
| filunibas 23824 | Recover the base set from ... |
| filunirn 23825 | Two ways to express a filt... |
| filconn 23826 | A filter gives rise to a c... |
| fbasrn 23827 | Given a filter on a domain... |
| filuni 23828 | The union of a nonempty se... |
| trfil1 23829 | Conditions for the trace o... |
| trfil2 23830 | Conditions for the trace o... |
| trfil3 23831 | Conditions for the trace o... |
| trfilss 23832 | If ` A ` is a member of th... |
| fgtr 23833 | If ` A ` is a member of th... |
| trfg 23834 | The trace operation and th... |
| trnei 23835 | The trace, over a set ` A ... |
| cfinfil 23836 | Relative complements of th... |
| csdfil 23837 | The set of all elements wh... |
| supfil 23838 | The supersets of a nonempt... |
| zfbas 23839 | The set of upper sets of i... |
| uzrest 23840 | The restriction of the set... |
| uzfbas 23841 | The set of upper sets of i... |
| isufil 23846 | The property of being an u... |
| ufilfil 23847 | An ultrafilter is a filter... |
| ufilss 23848 | For any subset of the base... |
| ufilb 23849 | The complement is in an ul... |
| ufilmax 23850 | Any filter finer than an u... |
| isufil2 23851 | The maximal property of an... |
| ufprim 23852 | An ultrafilter is a prime ... |
| trufil 23853 | Conditions for the trace o... |
| filssufilg 23854 | A filter is contained in s... |
| filssufil 23855 | A filter is contained in s... |
| isufl 23856 | Define the (strong) ultraf... |
| ufli 23857 | Property of a set that sat... |
| numufl 23858 | Consequence of ~ filssufil... |
| fiufl 23859 | A finite set satisfies the... |
| acufl 23860 | The axiom of choice implie... |
| ssufl 23861 | If ` Y ` is a subset of ` ... |
| ufileu 23862 | If the ultrafilter contain... |
| filufint 23863 | A filter is equal to the i... |
| uffix 23864 | Lemma for ~ fixufil and ~ ... |
| fixufil 23865 | The condition describing a... |
| uffixfr 23866 | An ultrafilter is either f... |
| uffix2 23867 | A classification of fixed ... |
| uffixsn 23868 | The singleton of the gener... |
| ufildom1 23869 | An ultrafilter is generate... |
| uffinfix 23870 | An ultrafilter containing ... |
| cfinufil 23871 | An ultrafilter is free iff... |
| ufinffr 23872 | An infinite subset is cont... |
| ufilen 23873 | Any infinite set has an ul... |
| ufildr 23874 | An ultrafilter gives rise ... |
| fin1aufil 23875 | There are no definable fre... |
| fmval 23886 | Introduce a function that ... |
| fmfil 23887 | A mapping filter is a filt... |
| fmf 23888 | Pushing-forward via a func... |
| fmss 23889 | A finer filter produces a ... |
| elfm 23890 | An element of a mapping fi... |
| elfm2 23891 | An element of a mapping fi... |
| fmfg 23892 | The image filter of a filt... |
| elfm3 23893 | An alternate formulation o... |
| imaelfm 23894 | An image of a filter eleme... |
| rnelfmlem 23895 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23896 | A condition for a filter t... |
| fmfnfmlem1 23897 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23898 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23899 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23900 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23901 | A filter finer than an ima... |
| fmufil 23902 | An image filter of an ultr... |
| fmid 23903 | The filter map applied to ... |
| fmco 23904 | Composition of image filte... |
| ufldom 23905 | The ultrafilter lemma prop... |
| flimval 23906 | The set of limit points of... |
| elflim2 23907 | The predicate "is a limit ... |
| flimtop 23908 | Reverse closure for the li... |
| flimneiss 23909 | A filter contains the neig... |
| flimnei 23910 | A filter contains all of t... |
| flimelbas 23911 | A limit point of a filter ... |
| flimfil 23912 | Reverse closure for the li... |
| flimtopon 23913 | Reverse closure for the li... |
| elflim 23914 | The predicate "is a limit ... |
| flimss2 23915 | A limit point of a filter ... |
| flimss1 23916 | A limit point of a filter ... |
| neiflim 23917 | A point is a limit point o... |
| flimopn 23918 | The condition for being a ... |
| fbflim 23919 | A condition for a filter t... |
| fbflim2 23920 | A condition for a filter b... |
| flimclsi 23921 | The convergent points of a... |
| hausflimlem 23922 | If ` A ` and ` B ` are bot... |
| hausflimi 23923 | One direction of ~ hausfli... |
| hausflim 23924 | A condition for a topology... |
| flimcf 23925 | Fineness is properly chara... |
| flimrest 23926 | The set of limit points in... |
| flimclslem 23927 | Lemma for ~ flimcls . (Co... |
| flimcls 23928 | Closure in terms of filter... |
| flimsncls 23929 | If ` A ` is a limit point ... |
| hauspwpwf1 23930 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23931 | If ` X ` is a Hausdorff sp... |
| flffval 23932 | Given a topology and a fil... |
| flfval 23933 | Given a function from a fi... |
| flfnei 23934 | The property of being a li... |
| flfneii 23935 | A neighborhood of a limit ... |
| isflf 23936 | The property of being a li... |
| flfelbas 23937 | A limit point of a functio... |
| flffbas 23938 | Limit points of a function... |
| flftg 23939 | Limit points of a function... |
| hausflf 23940 | If a function has its valu... |
| hausflf2 23941 | If a convergent function h... |
| cnpflfi 23942 | Forward direction of ~ cnp... |
| cnpflf2 23943 | ` F ` is continuous at poi... |
| cnpflf 23944 | Continuity of a function a... |
| cnflf 23945 | A function is continuous i... |
| cnflf2 23946 | A function is continuous i... |
| flfcnp 23947 | A continuous function pres... |
| lmflf 23948 | The topological limit rela... |
| txflf 23949 | Two sequences converge in ... |
| flfcnp2 23950 | The image of a convergent ... |
| fclsval 23951 | The set of all cluster poi... |
| isfcls 23952 | A cluster point of a filte... |
| fclsfil 23953 | Reverse closure for the cl... |
| fclstop 23954 | Reverse closure for the cl... |
| fclstopon 23955 | Reverse closure for the cl... |
| isfcls2 23956 | A cluster point of a filte... |
| fclsopn 23957 | Write the cluster point co... |
| fclsopni 23958 | An open neighborhood of a ... |
| fclselbas 23959 | A cluster point is in the ... |
| fclsneii 23960 | A neighborhood of a cluste... |
| fclssscls 23961 | The set of cluster points ... |
| fclsnei 23962 | Cluster points in terms of... |
| supnfcls 23963 | The filter of supersets of... |
| fclsbas 23964 | Cluster points in terms of... |
| fclsss1 23965 | A finer topology has fewer... |
| fclsss2 23966 | A finer filter has fewer c... |
| fclsrest 23967 | The set of cluster points ... |
| fclscf 23968 | Characterization of finene... |
| flimfcls 23969 | A limit point is a cluster... |
| fclsfnflim 23970 | A filter clusters at a poi... |
| flimfnfcls 23971 | A filter converges to a po... |
| fclscmpi 23972 | Forward direction of ~ fcl... |
| fclscmp 23973 | A space is compact iff eve... |
| uffclsflim 23974 | The cluster points of an u... |
| ufilcmp 23975 | A space is compact iff eve... |
| fcfval 23976 | The set of cluster points ... |
| isfcf 23977 | The property of being a cl... |
| fcfnei 23978 | The property of being a cl... |
| fcfelbas 23979 | A cluster point of a funct... |
| fcfneii 23980 | A neighborhood of a cluste... |
| flfssfcf 23981 | A limit point of a functio... |
| uffcfflf 23982 | If the domain filter is an... |
| cnpfcfi 23983 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23984 | A function ` F ` is contin... |
| cnfcf 23985 | Continuity of a function i... |
| flfcntr 23986 | A continuous function's va... |
| alexsublem 23987 | Lemma for ~ alexsub . (Co... |
| alexsub 23988 | The Alexander Subbase Theo... |
| alexsubb 23989 | Biconditional form of the ... |
| alexsubALTlem1 23990 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23991 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23992 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23993 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23994 | The Alexander Subbase Theo... |
| ptcmplem1 23995 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23996 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23997 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23998 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23999 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 24000 | Tychonoff's theorem: The ... |
| ptcmp 24001 | Tychonoff's theorem: The ... |
| cnextval 24004 | The function applying cont... |
| cnextfval 24005 | The continuous extension o... |
| cnextrel 24006 | In the general case, a con... |
| cnextfun 24007 | If the target space is Hau... |
| cnextfvval 24008 | The value of the continuou... |
| cnextf 24009 | Extension by continuity. ... |
| cnextcn 24010 | Extension by continuity. ... |
| cnextfres1 24011 | ` F ` and its extension by... |
| cnextfres 24012 | ` F ` and its extension by... |
| istmd 24017 | The predicate "is a topolo... |
| tmdmnd 24018 | A topological monoid is a ... |
| tmdtps 24019 | A topological monoid is a ... |
| istgp 24020 | The predicate "is a topolo... |
| tgpgrp 24021 | A topological group is a g... |
| tgptmd 24022 | A topological group is a t... |
| tgptps 24023 | A topological group is a t... |
| tmdtopon 24024 | The topology of a topologi... |
| tgptopon 24025 | The topology of a topologi... |
| tmdcn 24026 | In a topological monoid, t... |
| tgpcn 24027 | In a topological group, th... |
| tgpinv 24028 | In a topological group, th... |
| grpinvhmeo 24029 | The inverse function in a ... |
| cnmpt1plusg 24030 | Continuity of the group su... |
| cnmpt2plusg 24031 | Continuity of the group su... |
| tmdcn2 24032 | Write out the definition o... |
| tgpsubcn 24033 | In a topological group, th... |
| istgp2 24034 | A group with a topology is... |
| tmdmulg 24035 | In a topological monoid, t... |
| tgpmulg 24036 | In a topological group, th... |
| tgpmulg2 24037 | In a topological monoid, t... |
| tmdgsum 24038 | In a topological monoid, t... |
| tmdgsum2 24039 | For any neighborhood ` U `... |
| oppgtmd 24040 | The opposite of a topologi... |
| oppgtgp 24041 | The opposite of a topologi... |
| distgp 24042 | Any group equipped with th... |
| indistgp 24043 | Any group equipped with th... |
| efmndtmd 24044 | The monoid of endofunction... |
| tmdlactcn 24045 | The left group action of e... |
| tgplacthmeo 24046 | The left group action of e... |
| submtmd 24047 | A submonoid of a topologic... |
| subgtgp 24048 | A subgroup of a topologica... |
| symgtgp 24049 | The symmetric group is a t... |
| subgntr 24050 | A subgroup of a topologica... |
| opnsubg 24051 | An open subgroup of a topo... |
| clssubg 24052 | The closure of a subgroup ... |
| clsnsg 24053 | The closure of a normal su... |
| cldsubg 24054 | A subgroup of finite index... |
| tgpconncompeqg 24055 | The connected component co... |
| tgpconncomp 24056 | The identity component, th... |
| tgpconncompss 24057 | The identity component is ... |
| ghmcnp 24058 | A group homomorphism on to... |
| snclseqg 24059 | The coset of the closure o... |
| tgphaus 24060 | A topological group is Hau... |
| tgpt1 24061 | Hausdorff and T1 are equiv... |
| tgpt0 24062 | Hausdorff and T0 are equiv... |
| qustgpopn 24063 | A quotient map in a topolo... |
| qustgplem 24064 | Lemma for ~ qustgp . (Con... |
| qustgp 24065 | The quotient of a topologi... |
| qustgphaus 24066 | The quotient of a topologi... |
| prdstmdd 24067 | The product of a family of... |
| prdstgpd 24068 | The product of a family of... |
| tsmsfbas 24071 | The collection of all sets... |
| tsmslem1 24072 | The finite partial sums of... |
| tsmsval2 24073 | Definition of the topologi... |
| tsmsval 24074 | Definition of the topologi... |
| tsmspropd 24075 | The group sum depends only... |
| eltsms 24076 | The property of being a su... |
| tsmsi 24077 | The property of being a su... |
| tsmscl 24078 | A sum in a topological gro... |
| haustsms 24079 | In a Hausdorff topological... |
| haustsms2 24080 | In a Hausdorff topological... |
| tsmscls 24081 | One half of ~ tgptsmscls ,... |
| tsmsgsum 24082 | The convergent points of a... |
| tsmsid 24083 | If a sum is finite, the us... |
| haustsmsid 24084 | In a Hausdorff topological... |
| tsms0 24085 | The sum of zero is zero. ... |
| tsmssubm 24086 | Evaluate an infinite group... |
| tsmsres 24087 | Extend an infinite group s... |
| tsmsf1o 24088 | Re-index an infinite group... |
| tsmsmhm 24089 | Apply a continuous group h... |
| tsmsadd 24090 | The sum of two infinite gr... |
| tsmsinv 24091 | Inverse of an infinite gro... |
| tsmssub 24092 | The difference of two infi... |
| tgptsmscls 24093 | A sum in a topological gro... |
| tgptsmscld 24094 | The set of limit points to... |
| tsmssplit 24095 | Split a topological group ... |
| tsmsxplem1 24096 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 24097 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 24098 | Write a sum over a two-dim... |
| istrg 24107 | Express the predicate " ` ... |
| trgtmd 24108 | The multiplicative monoid ... |
| istdrg 24109 | Express the predicate " ` ... |
| tdrgunit 24110 | The unit group of a topolo... |
| trgtgp 24111 | A topological ring is a to... |
| trgtmd2 24112 | A topological ring is a to... |
| trgtps 24113 | A topological ring is a to... |
| trgring 24114 | A topological ring is a ri... |
| trggrp 24115 | A topological ring is a gr... |
| tdrgtrg 24116 | A topological division rin... |
| tdrgdrng 24117 | A topological division rin... |
| tdrgring 24118 | A topological division rin... |
| tdrgtmd 24119 | A topological division rin... |
| tdrgtps 24120 | A topological division rin... |
| istdrg2 24121 | A topological-ring divisio... |
| mulrcn 24122 | The functionalization of t... |
| invrcn2 24123 | The multiplicative inverse... |
| invrcn 24124 | The multiplicative inverse... |
| cnmpt1mulr 24125 | Continuity of ring multipl... |
| cnmpt2mulr 24126 | Continuity of ring multipl... |
| dvrcn 24127 | The division function is c... |
| istlm 24128 | The predicate " ` W ` is a... |
| vscacn 24129 | The scalar multiplication ... |
| tlmtmd 24130 | A topological module is a ... |
| tlmtps 24131 | A topological module is a ... |
| tlmlmod 24132 | A topological module is a ... |
| tlmtrg 24133 | The scalar ring of a topol... |
| tlmscatps 24134 | The scalar ring of a topol... |
| istvc 24135 | A topological vector space... |
| tvctdrg 24136 | The scalar field of a topo... |
| cnmpt1vsca 24137 | Continuity of scalar multi... |
| cnmpt2vsca 24138 | Continuity of scalar multi... |
| tlmtgp 24139 | A topological vector space... |
| tvctlm 24140 | A topological vector space... |
| tvclmod 24141 | A topological vector space... |
| tvclvec 24142 | A topological vector space... |
| ustfn 24145 | The defined uniform struct... |
| ustval 24146 | The class of all uniform s... |
| isust 24147 | The predicate " ` U ` is a... |
| ustssxp 24148 | Entourages are subsets of ... |
| ustssel 24149 | A uniform structure is upw... |
| ustbasel 24150 | The full set is always an ... |
| ustincl 24151 | A uniform structure is clo... |
| ustdiag 24152 | The diagonal set is includ... |
| ustinvel 24153 | If ` V ` is an entourage, ... |
| ustexhalf 24154 | For each entourage ` V ` t... |
| ustrel 24155 | The elements of uniform st... |
| ustfilxp 24156 | A uniform structure on a n... |
| ustne0 24157 | A uniform structure cannot... |
| ustssco 24158 | In an uniform structure, a... |
| ustexsym 24159 | In an uniform structure, f... |
| ustex2sym 24160 | In an uniform structure, f... |
| ustex3sym 24161 | In an uniform structure, f... |
| ustref 24162 | Any element of the base se... |
| ust0 24163 | The unique uniform structu... |
| ustn0 24164 | The empty set is not an un... |
| ustund 24165 | If two intersecting sets `... |
| ustelimasn 24166 | Any point ` A ` is near en... |
| ustneism 24167 | For a point ` A ` in ` X `... |
| ustbas2 24168 | Second direction for ~ ust... |
| ustuni 24169 | The set union of a uniform... |
| ustbas 24170 | Recover the base of an uni... |
| ustimasn 24171 | Lemma for ~ ustuqtop . (C... |
| trust 24172 | The trace of a uniform str... |
| utopval 24175 | The topology induced by a ... |
| elutop 24176 | Open sets in the topology ... |
| utoptop 24177 | The topology induced by a ... |
| utopbas 24178 | The base of the topology i... |
| utoptopon 24179 | Topology induced by a unif... |
| restutop 24180 | Restriction of a topology ... |
| restutopopn 24181 | The restriction of the top... |
| ustuqtoplem 24182 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 24183 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 24184 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 24185 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 24186 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 24187 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 24188 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 24189 | For a given uniform struct... |
| utopsnneiplem 24190 | The neighborhoods of a poi... |
| utopsnneip 24191 | The neighborhoods of a poi... |
| utopsnnei 24192 | Images of singletons by en... |
| utop2nei 24193 | For any symmetrical entour... |
| utop3cls 24194 | Relation between a topolog... |
| utopreg 24195 | All Hausdorff uniform spac... |
| ussval 24202 | The uniform structure on u... |
| ussid 24203 | In case the base of the ` ... |
| isusp 24204 | The predicate ` W ` is a u... |
| ressuss 24205 | Value of the uniform struc... |
| ressust 24206 | The uniform structure of a... |
| ressusp 24207 | The restriction of a unifo... |
| tusval 24208 | The value of the uniform s... |
| tuslem 24209 | Lemma for ~ tusbas , ~ tus... |
| tusbas 24210 | The base set of a construc... |
| tusunif 24211 | The uniform structure of a... |
| tususs 24212 | The uniform structure of a... |
| tustopn 24213 | The topology induced by a ... |
| tususp 24214 | A constructed uniform spac... |
| tustps 24215 | A constructed uniform spac... |
| uspreg 24216 | If a uniform space is Haus... |
| ucnval 24219 | The set of all uniformly c... |
| isucn 24220 | The predicate " ` F ` is a... |
| isucn2 24221 | The predicate " ` F ` is a... |
| ucnimalem 24222 | Reformulate the ` G ` func... |
| ucnima 24223 | An equivalent statement of... |
| ucnprima 24224 | The preimage by a uniforml... |
| iducn 24225 | The identity is uniformly ... |
| cstucnd 24226 | A constant function is uni... |
| ucncn 24227 | Uniform continuity implies... |
| iscfilu 24230 | The predicate " ` F ` is a... |
| cfilufbas 24231 | A Cauchy filter base is a ... |
| cfiluexsm 24232 | For a Cauchy filter base a... |
| fmucndlem 24233 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24234 | The image of a Cauchy filt... |
| cfilufg 24235 | The filter generated by a ... |
| trcfilu 24236 | Condition for the trace of... |
| cfiluweak 24237 | A Cauchy filter base is al... |
| neipcfilu 24238 | In an uniform space, a nei... |
| iscusp 24241 | The predicate " ` W ` is a... |
| cuspusp 24242 | A complete uniform space i... |
| cuspcvg 24243 | In a complete uniform spac... |
| iscusp2 24244 | The predicate " ` W ` is a... |
| cnextucn 24245 | Extension by continuity. ... |
| ucnextcn 24246 | Extension by continuity. ... |
| ispsmet 24247 | Express the predicate " ` ... |
| psmetdmdm 24248 | Recover the base set from ... |
| psmetf 24249 | The distance function of a... |
| psmetcl 24250 | Closure of the distance fu... |
| psmet0 24251 | The distance function of a... |
| psmettri2 24252 | Triangle inequality for th... |
| psmetsym 24253 | The distance function of a... |
| psmettri 24254 | Triangle inequality for th... |
| psmetge0 24255 | The distance function of a... |
| psmetxrge0 24256 | The distance function of a... |
| psmetres2 24257 | Restriction of a pseudomet... |
| psmetlecl 24258 | Real closure of an extende... |
| distspace 24259 | A set ` X ` together with ... |
| ismet 24266 | Express the predicate " ` ... |
| isxmet 24267 | Express the predicate " ` ... |
| ismeti 24268 | Properties that determine ... |
| isxmetd 24269 | Properties that determine ... |
| isxmet2d 24270 | It is safe to only require... |
| metflem 24271 | Lemma for ~ metf and other... |
| xmetf 24272 | Mapping of the distance fu... |
| metf 24273 | Mapping of the distance fu... |
| xmetcl 24274 | Closure of the distance fu... |
| metcl 24275 | Closure of the distance fu... |
| ismet2 24276 | An extended metric is a me... |
| metxmet 24277 | A metric is an extended me... |
| xmetdmdm 24278 | Recover the base set from ... |
| metdmdm 24279 | Recover the base set from ... |
| xmetunirn 24280 | Two ways to express an ext... |
| xmeteq0 24281 | The value of an extended m... |
| meteq0 24282 | The value of a metric is z... |
| xmettri2 24283 | Triangle inequality for th... |
| mettri2 24284 | Triangle inequality for th... |
| xmet0 24285 | The distance function of a... |
| met0 24286 | The distance function of a... |
| xmetge0 24287 | The distance function of a... |
| metge0 24288 | The distance function of a... |
| xmetlecl 24289 | Real closure of an extende... |
| xmetsym 24290 | The distance function of a... |
| xmetpsmet 24291 | An extended metric is a ps... |
| xmettpos 24292 | The distance function of a... |
| metsym 24293 | The distance function of a... |
| xmettri 24294 | Triangle inequality for th... |
| mettri 24295 | Triangle inequality for th... |
| xmettri3 24296 | Triangle inequality for th... |
| mettri3 24297 | Triangle inequality for th... |
| xmetrtri 24298 | One half of the reverse tr... |
| xmetrtri2 24299 | The reverse triangle inequ... |
| metrtri 24300 | Reverse triangle inequalit... |
| xmetgt0 24301 | The distance function of a... |
| metgt0 24302 | The distance function of a... |
| metn0 24303 | A metric space is nonempty... |
| xmetres2 24304 | Restriction of an extended... |
| metreslem 24305 | Lemma for ~ metres . (Con... |
| metres2 24306 | Lemma for ~ metres . (Con... |
| xmetres 24307 | A restriction of an extend... |
| metres 24308 | A restriction of a metric ... |
| 0met 24309 | The empty metric. (Contri... |
| prdsdsf 24310 | The product metric is a fu... |
| prdsxmetlem 24311 | The product metric is an e... |
| prdsxmet 24312 | The product metric is an e... |
| prdsmet 24313 | The product metric is a me... |
| ressprdsds 24314 | Restriction of a product m... |
| resspwsds 24315 | Restriction of a power met... |
| imasdsf1olem 24316 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24317 | The distance function is t... |
| imasf1oxmet 24318 | The image of an extended m... |
| imasf1omet 24319 | The image of a metric is a... |
| xpsdsfn 24320 | Closure of the metric in a... |
| xpsdsfn2 24321 | Closure of the metric in a... |
| xpsxmetlem 24322 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24323 | A product metric of extend... |
| xpsdsval 24324 | Value of the metric in a b... |
| xpsmet 24325 | The direct product of two ... |
| blfvalps 24326 | The value of the ball func... |
| blfval 24327 | The value of the ball func... |
| blvalps 24328 | The ball around a point ` ... |
| blval 24329 | The ball around a point ` ... |
| elblps 24330 | Membership in a ball. (Co... |
| elbl 24331 | Membership in a ball. (Co... |
| elbl2ps 24332 | Membership in a ball. (Co... |
| elbl2 24333 | Membership in a ball. (Co... |
| elbl3ps 24334 | Membership in a ball, with... |
| elbl3 24335 | Membership in a ball, with... |
| blcomps 24336 | Commute the arguments to t... |
| blcom 24337 | Commute the arguments to t... |
| xblpnfps 24338 | The infinity ball in an ex... |
| xblpnf 24339 | The infinity ball in an ex... |
| blpnf 24340 | The infinity ball in a sta... |
| bldisj 24341 | Two balls are disjoint if ... |
| blgt0 24342 | A nonempty ball implies th... |
| bl2in 24343 | Two balls are disjoint if ... |
| xblss2ps 24344 | One ball is contained in a... |
| xblss2 24345 | One ball is contained in a... |
| blss2ps 24346 | One ball is contained in a... |
| blss2 24347 | One ball is contained in a... |
| blhalf 24348 | A ball of radius ` R / 2 `... |
| blfps 24349 | Mapping of a ball. (Contr... |
| blf 24350 | Mapping of a ball. (Contr... |
| blrnps 24351 | Membership in the range of... |
| blrn 24352 | Membership in the range of... |
| xblcntrps 24353 | A ball contains its center... |
| xblcntr 24354 | A ball contains its center... |
| blcntrps 24355 | A ball contains its center... |
| blcntr 24356 | A ball contains its center... |
| xbln0 24357 | A ball is nonempty iff the... |
| bln0 24358 | A ball is not empty. (Con... |
| blelrnps 24359 | A ball belongs to the set ... |
| blelrn 24360 | A ball belongs to the set ... |
| blssm 24361 | A ball is a subset of the ... |
| unirnblps 24362 | The union of the set of ba... |
| unirnbl 24363 | The union of the set of ba... |
| blin 24364 | The intersection of two ba... |
| ssblps 24365 | The size of a ball increas... |
| ssbl 24366 | The size of a ball increas... |
| blssps 24367 | Any point ` P ` in a ball ... |
| blss 24368 | Any point ` P ` in a ball ... |
| blssexps 24369 | Two ways to express the ex... |
| blssex 24370 | Two ways to express the ex... |
| ssblex 24371 | A nested ball exists whose... |
| blin2 24372 | Given any two balls and a ... |
| blbas 24373 | The balls of a metric spac... |
| blres 24374 | A ball in a restricted met... |
| xmeterval 24375 | Value of the "finitely sep... |
| xmeter 24376 | The "finitely separated" r... |
| xmetec 24377 | The equivalence classes un... |
| blssec 24378 | A ball centered at ` P ` i... |
| blpnfctr 24379 | The infinity ball in an ex... |
| xmetresbl 24380 | An extended metric restric... |
| mopnval 24381 | An open set is a subset of... |
| mopntopon 24382 | The set of open sets of a ... |
| mopntop 24383 | The set of open sets of a ... |
| mopnuni 24384 | The union of all open sets... |
| elmopn 24385 | The defining property of a... |
| mopnfss 24386 | The family of open sets of... |
| mopnm 24387 | The base set of a metric s... |
| elmopn2 24388 | A defining property of an ... |
| mopnss 24389 | An open set of a metric sp... |
| isxms 24390 | Express the predicate " ` ... |
| isxms2 24391 | Express the predicate " ` ... |
| isms 24392 | Express the predicate " ` ... |
| isms2 24393 | Express the predicate " ` ... |
| xmstopn 24394 | The topology component of ... |
| mstopn 24395 | The topology component of ... |
| xmstps 24396 | An extended metric space i... |
| msxms 24397 | A metric space is an exten... |
| mstps 24398 | A metric space is a topolo... |
| xmsxmet 24399 | The distance function, sui... |
| msmet 24400 | The distance function, sui... |
| msf 24401 | The distance function of a... |
| xmsxmet2 24402 | The distance function, sui... |
| msmet2 24403 | The distance function, sui... |
| mscl 24404 | Closure of the distance fu... |
| xmscl 24405 | Closure of the distance fu... |
| xmsge0 24406 | The distance function in a... |
| xmseq0 24407 | The distance between two p... |
| xmssym 24408 | The distance function in a... |
| xmstri2 24409 | Triangle inequality for th... |
| mstri2 24410 | Triangle inequality for th... |
| xmstri 24411 | Triangle inequality for th... |
| mstri 24412 | Triangle inequality for th... |
| xmstri3 24413 | Triangle inequality for th... |
| mstri3 24414 | Triangle inequality for th... |
| msrtri 24415 | Reverse triangle inequalit... |
| xmspropd 24416 | Property deduction for an ... |
| mspropd 24417 | Property deduction for a m... |
| setsmsbas 24418 | The base set of a construc... |
| setsmsds 24419 | The distance function of a... |
| setsmstset 24420 | The topology of a construc... |
| setsmstopn 24421 | The topology of a construc... |
| setsxms 24422 | The constructed metric spa... |
| setsms 24423 | The constructed metric spa... |
| tmsval 24424 | For any metric there is an... |
| tmslem 24425 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 24426 | The base set of a construc... |
| tmsds 24427 | The metric of a constructe... |
| tmstopn 24428 | The topology of a construc... |
| tmsxms 24429 | The constructed metric spa... |
| tmsms 24430 | The constructed metric spa... |
| imasf1obl 24431 | The image of a metric spac... |
| imasf1oxms 24432 | The image of a metric spac... |
| imasf1oms 24433 | The image of a metric spac... |
| prdsbl 24434 | A ball in the product metr... |
| mopni 24435 | An open set of a metric sp... |
| mopni2 24436 | An open set of a metric sp... |
| mopni3 24437 | An open set of a metric sp... |
| blssopn 24438 | The balls of a metric spac... |
| unimopn 24439 | The union of a collection ... |
| mopnin 24440 | The intersection of two op... |
| mopn0 24441 | The empty set is an open s... |
| rnblopn 24442 | A ball of a metric space i... |
| blopn 24443 | A ball of a metric space i... |
| neibl 24444 | The neighborhoods around a... |
| blnei 24445 | A ball around a point is a... |
| lpbl 24446 | Every ball around a limit ... |
| blsscls2 24447 | A smaller closed ball is c... |
| blcld 24448 | A "closed ball" in a metri... |
| blcls 24449 | The closure of an open bal... |
| blsscls 24450 | If two concentric balls ha... |
| metss 24451 | Two ways of saying that me... |
| metequiv 24452 | Two ways of saying that tw... |
| metequiv2 24453 | If there is a sequence of ... |
| metss2lem 24454 | Lemma for ~ metss2 . (Con... |
| metss2 24455 | If the metric ` D ` is "st... |
| comet 24456 | The composition of an exte... |
| stdbdmetval 24457 | Value of the standard boun... |
| stdbdxmet 24458 | The standard bounded metri... |
| stdbdmet 24459 | The standard bounded metri... |
| stdbdbl 24460 | The standard bounded metri... |
| stdbdmopn 24461 | The standard bounded metri... |
| mopnex 24462 | The topology generated by ... |
| methaus 24463 | The topology generated by ... |
| met1stc 24464 | The topology generated by ... |
| met2ndci 24465 | A separable metric space (... |
| met2ndc 24466 | A metric space is second-c... |
| metrest 24467 | Two alternate formulations... |
| ressxms 24468 | The restriction of a metri... |
| ressms 24469 | The restriction of a metri... |
| prdsmslem1 24470 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24471 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24472 | Lemma for ~ prdsxms . The... |
| prdsxms 24473 | The indexed product struct... |
| prdsms 24474 | The indexed product struct... |
| pwsxms 24475 | A power of an extended met... |
| pwsms 24476 | A power of a metric space ... |
| xpsxms 24477 | A binary product of metric... |
| xpsms 24478 | A binary product of metric... |
| tmsxps 24479 | Express the product of two... |
| tmsxpsmopn 24480 | Express the product of two... |
| tmsxpsval 24481 | Value of the product of tw... |
| tmsxpsval2 24482 | Value of the product of tw... |
| metcnp3 24483 | Two ways to express that `... |
| metcnp 24484 | Two ways to say a mapping ... |
| metcnp2 24485 | Two ways to say a mapping ... |
| metcn 24486 | Two ways to say a mapping ... |
| metcnpi 24487 | Epsilon-delta property of ... |
| metcnpi2 24488 | Epsilon-delta property of ... |
| metcnpi3 24489 | Epsilon-delta property of ... |
| txmetcnp 24490 | Continuity of a binary ope... |
| txmetcn 24491 | Continuity of a binary ope... |
| metuval 24492 | Value of the uniform struc... |
| metustel 24493 | Define a filter base ` F `... |
| metustss 24494 | Range of the elements of t... |
| metustrel 24495 | Elements of the filter bas... |
| metustto 24496 | Any two elements of the fi... |
| metustid 24497 | The identity diagonal is i... |
| metustsym 24498 | Elements of the filter bas... |
| metustexhalf 24499 | For any element ` A ` of t... |
| metustfbas 24500 | The filter base generated ... |
| metust 24501 | The uniform structure gene... |
| cfilucfil 24502 | Given a metric ` D ` and a... |
| metuust 24503 | The uniform structure gene... |
| cfilucfil2 24504 | Given a metric ` D ` and a... |
| blval2 24505 | The ball around a point ` ... |
| elbl4 24506 | Membership in a ball, alte... |
| metuel 24507 | Elementhood in the uniform... |
| metuel2 24508 | Elementhood in the uniform... |
| metustbl 24509 | The "section" image of an ... |
| psmetutop 24510 | The topology induced by a ... |
| xmetutop 24511 | The topology induced by a ... |
| xmsusp 24512 | If the uniform set of a me... |
| restmetu 24513 | The uniform structure gene... |
| metucn 24514 | Uniform continuity in metr... |
| dscmet 24515 | The discrete metric on any... |
| dscopn 24516 | The discrete metric genera... |
| nrmmetd 24517 | Show that a group norm gen... |
| abvmet 24518 | An absolute value ` F ` ge... |
| nmfval 24531 | The value of the norm func... |
| nmval 24532 | The value of the norm as t... |
| nmfval0 24533 | The value of the norm func... |
| nmfval2 24534 | The value of the norm func... |
| nmval2 24535 | The value of the norm on a... |
| nmf2 24536 | The norm on a metric group... |
| nmpropd 24537 | Weak property deduction fo... |
| nmpropd2 24538 | Strong property deduction ... |
| isngp 24539 | The property of being a no... |
| isngp2 24540 | The property of being a no... |
| isngp3 24541 | The property of being a no... |
| ngpgrp 24542 | A normed group is a group.... |
| ngpms 24543 | A normed group is a metric... |
| ngpxms 24544 | A normed group is an exten... |
| ngptps 24545 | A normed group is a topolo... |
| ngpmet 24546 | The (induced) metric of a ... |
| ngpds 24547 | Value of the distance func... |
| ngpdsr 24548 | Value of the distance func... |
| ngpds2 24549 | Write the distance between... |
| ngpds2r 24550 | Write the distance between... |
| ngpds3 24551 | Write the distance between... |
| ngpds3r 24552 | Write the distance between... |
| ngprcan 24553 | Cancel right addition insi... |
| ngplcan 24554 | Cancel left addition insid... |
| isngp4 24555 | Express the property of be... |
| ngpinvds 24556 | Two elements are the same ... |
| ngpsubcan 24557 | Cancel right subtraction i... |
| nmf 24558 | The norm on a normed group... |
| nmcl 24559 | The norm of a normed group... |
| nmge0 24560 | The norm of a normed group... |
| nmeq0 24561 | The identity is the only e... |
| nmne0 24562 | The norm of a nonzero elem... |
| nmrpcl 24563 | The norm of a nonzero elem... |
| nminv 24564 | The norm of a negated elem... |
| nmmtri 24565 | The triangle inequality fo... |
| nmsub 24566 | The norm of the difference... |
| nmrtri 24567 | Reverse triangle inequalit... |
| nm2dif 24568 | Inequality for the differe... |
| nmtri 24569 | The triangle inequality fo... |
| nmtri2 24570 | Triangle inequality for th... |
| ngpi 24571 | The properties of a normed... |
| nm0 24572 | Norm of the identity eleme... |
| nmgt0 24573 | The norm of a nonzero elem... |
| sgrim 24574 | The induced metric on a su... |
| sgrimval 24575 | The induced metric on a su... |
| subgnm 24576 | The norm in a subgroup. (... |
| subgnm2 24577 | A substructure assigns the... |
| subgngp 24578 | A normed group restricted ... |
| ngptgp 24579 | A normed abelian group is ... |
| ngppropd 24580 | Property deduction for a n... |
| reldmtng 24581 | The function ` toNrmGrp ` ... |
| tngval 24582 | Value of the function whic... |
| tnglem 24583 | Lemma for ~ tngbas and sim... |
| tngbas 24584 | The base set of a structur... |
| tngplusg 24585 | The group addition of a st... |
| tng0 24586 | The group identity of a st... |
| tngmulr 24587 | The ring multiplication of... |
| tngsca 24588 | The scalar ring of a struc... |
| tngvsca 24589 | The scalar multiplication ... |
| tngip 24590 | The inner product operatio... |
| tngds 24591 | The metric function of a s... |
| tngtset 24592 | The topology generated by ... |
| tngtopn 24593 | The topology generated by ... |
| tngnm 24594 | The topology generated by ... |
| tngngp2 24595 | A norm turns a group into ... |
| tngngpd 24596 | Derive the axioms for a no... |
| tngngp 24597 | Derive the axioms for a no... |
| tnggrpr 24598 | If a structure equipped wi... |
| tngngp3 24599 | Alternate definition of a ... |
| nrmtngdist 24600 | The augmentation of a norm... |
| nrmtngnrm 24601 | The augmentation of a norm... |
| tngngpim 24602 | The induced metric of a no... |
| isnrg 24603 | A normed ring is a ring wi... |
| nrgabv 24604 | The norm of a normed ring ... |
| nrgngp 24605 | A normed ring is a normed ... |
| nrgring 24606 | A normed ring is a ring. ... |
| nmmul 24607 | The norm of a product in a... |
| nrgdsdi 24608 | Distribute a distance calc... |
| nrgdsdir 24609 | Distribute a distance calc... |
| nm1 24610 | The norm of one in a nonze... |
| unitnmn0 24611 | The norm of a unit is nonz... |
| nminvr 24612 | The norm of an inverse in ... |
| nmdvr 24613 | The norm of a division in ... |
| nrgdomn 24614 | A nonzero normed ring is a... |
| nrgtgp 24615 | A normed ring is a topolog... |
| subrgnrg 24616 | A normed ring restricted t... |
| tngnrg 24617 | Given any absolute value o... |
| isnlm 24618 | A normed (left) module is ... |
| nmvs 24619 | Defining property of a nor... |
| nlmngp 24620 | A normed module is a norme... |
| nlmlmod 24621 | A normed module is a left ... |
| nlmnrg 24622 | The scalar component of a ... |
| nlmngp2 24623 | The scalar component of a ... |
| nlmdsdi 24624 | Distribute a distance calc... |
| nlmdsdir 24625 | Distribute a distance calc... |
| nlmmul0or 24626 | If a scalar product is zer... |
| sranlm 24627 | The subring algebra over a... |
| nlmvscnlem2 24628 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24629 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24630 | The scalar multiplication ... |
| rlmnlm 24631 | The ring module over a nor... |
| rlmnm 24632 | The norm function in the r... |
| nrgtrg 24633 | A normed ring is a topolog... |
| nrginvrcnlem 24634 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24635 | The ring inverse function ... |
| nrgtdrg 24636 | A normed division ring is ... |
| nlmtlm 24637 | A normed module is a topol... |
| isnvc 24638 | A normed vector space is j... |
| nvcnlm 24639 | A normed vector space is a... |
| nvclvec 24640 | A normed vector space is a... |
| nvclmod 24641 | A normed vector space is a... |
| isnvc2 24642 | A normed vector space is j... |
| nvctvc 24643 | A normed vector space is a... |
| lssnlm 24644 | A subspace of a normed mod... |
| lssnvc 24645 | A subspace of a normed vec... |
| rlmnvc 24646 | The ring module over a nor... |
| ngpocelbl 24647 | Membership of an off-cente... |
| nmoffn 24654 | The function producing ope... |
| reldmnghm 24655 | Lemma for normed group hom... |
| reldmnmhm 24656 | Lemma for module homomorph... |
| nmofval 24657 | Value of the operator norm... |
| nmoval 24658 | Value of the operator norm... |
| nmogelb 24659 | Property of the operator n... |
| nmolb 24660 | Any upper bound on the val... |
| nmolb2d 24661 | Any upper bound on the val... |
| nmof 24662 | The operator norm is a fun... |
| nmocl 24663 | The operator norm of an op... |
| nmoge0 24664 | The operator norm of an op... |
| nghmfval 24665 | A normed group homomorphis... |
| isnghm 24666 | A normed group homomorphis... |
| isnghm2 24667 | A normed group homomorphis... |
| isnghm3 24668 | A normed group homomorphis... |
| bddnghm 24669 | A bounded group homomorphi... |
| nghmcl 24670 | A normed group homomorphis... |
| nmoi 24671 | The operator norm achieves... |
| nmoix 24672 | The operator norm is a bou... |
| nmoi2 24673 | The operator norm is a bou... |
| nmoleub 24674 | The operator norm, defined... |
| nghmrcl1 24675 | Reverse closure for a norm... |
| nghmrcl2 24676 | Reverse closure for a norm... |
| nghmghm 24677 | A normed group homomorphis... |
| nmo0 24678 | The operator norm of the z... |
| nmoeq0 24679 | The operator norm is zero ... |
| nmoco 24680 | An upper bound on the oper... |
| nghmco 24681 | The composition of normed ... |
| nmotri 24682 | Triangle inequality for th... |
| nghmplusg 24683 | The sum of two bounded lin... |
| 0nghm 24684 | The zero operator is a nor... |
| nmoid 24685 | The operator norm of the i... |
| idnghm 24686 | The identity operator is a... |
| nmods 24687 | Upper bound for the distan... |
| nghmcn 24688 | A normed group homomorphis... |
| isnmhm 24689 | A normed module homomorphi... |
| nmhmrcl1 24690 | Reverse closure for a norm... |
| nmhmrcl2 24691 | Reverse closure for a norm... |
| nmhmlmhm 24692 | A normed module homomorphi... |
| nmhmnghm 24693 | A normed module homomorphi... |
| nmhmghm 24694 | A normed module homomorphi... |
| isnmhm2 24695 | A normed module homomorphi... |
| nmhmcl 24696 | A normed module homomorphi... |
| idnmhm 24697 | The identity operator is a... |
| 0nmhm 24698 | The zero operator is a bou... |
| nmhmco 24699 | The composition of bounded... |
| nmhmplusg 24700 | The sum of two bounded lin... |
| qtopbaslem 24701 | The set of open intervals ... |
| qtopbas 24702 | The set of open intervals ... |
| retopbas 24703 | A basis for the standard t... |
| retop 24704 | The standard topology on t... |
| uniretop 24705 | The underlying set of the ... |
| retopon 24706 | The standard topology on t... |
| retps 24707 | The standard topological s... |
| iooretop 24708 | Open intervals are open se... |
| icccld 24709 | Closed intervals are close... |
| icopnfcld 24710 | Right-unbounded closed int... |
| iocmnfcld 24711 | Left-unbounded closed inte... |
| qdensere 24712 | ` QQ ` is dense in the sta... |
| cnmetdval 24713 | Value of the distance func... |
| cnmet 24714 | The absolute value metric ... |
| cnxmet 24715 | The absolute value metric ... |
| cnbl0 24716 | Two ways to write the open... |
| cnblcld 24717 | Two ways to write the clos... |
| cnfldms 24718 | The complex number field i... |
| cnfldxms 24719 | The complex number field i... |
| cnfldtps 24720 | The complex number field i... |
| cnfldnm 24721 | The norm of the field of c... |
| cnngp 24722 | The complex numbers form a... |
| cnnrg 24723 | The complex numbers form a... |
| cnfldtopn 24724 | The topology of the comple... |
| cnfldtopon 24725 | The topology of the comple... |
| cnfldtop 24726 | The topology of the comple... |
| cnfldhaus 24727 | The topology of the comple... |
| unicntop 24728 | The underlying set of the ... |
| cnopn 24729 | The set of complex numbers... |
| cnn0opn 24730 | The set of nonzero complex... |
| zringnrg 24731 | The ring of integers is a ... |
| remetdval 24732 | Value of the distance func... |
| remet 24733 | The absolute value metric ... |
| rexmet 24734 | The absolute value metric ... |
| bl2ioo 24735 | A ball in terms of an open... |
| ioo2bl 24736 | An open interval of reals ... |
| ioo2blex 24737 | An open interval of reals ... |
| blssioo 24738 | The balls of the standard ... |
| tgioo 24739 | The topology generated by ... |
| qdensere2 24740 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24741 | An open ball in the comple... |
| rehaus 24742 | The standard topology on t... |
| tgqioo 24743 | The topology generated by ... |
| re2ndc 24744 | The standard topology on t... |
| resubmet 24745 | The subspace topology indu... |
| tgioo2 24746 | The standard topology on t... |
| rerest 24747 | The subspace topology indu... |
| tgioo4 24748 | The standard topology on t... |
| tgioo3 24749 | The standard topology on t... |
| xrtgioo 24750 | The topology on the extend... |
| xrrest 24751 | The subspace topology indu... |
| xrrest2 24752 | The subspace topology indu... |
| xrsxmet 24753 | The metric on the extended... |
| xrsdsre 24754 | The metric on the extended... |
| xrsblre 24755 | Any ball of the metric of ... |
| xrsmopn 24756 | The metric on the extended... |
| zcld 24757 | The integers are a closed ... |
| recld2 24758 | The real numbers are a clo... |
| zcld2 24759 | The integers are a closed ... |
| zdis 24760 | The integers are a discret... |
| sszcld 24761 | Every subset of the intege... |
| reperflem 24762 | A subset of the real numbe... |
| reperf 24763 | The real numbers are a per... |
| cnperf 24764 | The complex numbers are a ... |
| iccntr 24765 | The interior of a closed i... |
| icccmplem1 24766 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24767 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24768 | Lemma for ~ icccmp . (Con... |
| icccmp 24769 | A closed interval in ` RR ... |
| reconnlem1 24770 | Lemma for ~ reconn . Conn... |
| reconnlem2 24771 | Lemma for ~ reconn . (Con... |
| reconn 24772 | A subset of the reals is c... |
| retopconn 24773 | Corollary of ~ reconn . T... |
| iccconn 24774 | A closed interval is conne... |
| opnreen 24775 | Every nonempty open set is... |
| rectbntr0 24776 | A countable subset of the ... |
| xrge0gsumle 24777 | A finite sum in the nonneg... |
| xrge0tsms 24778 | Any finite or infinite sum... |
| xrge0tsms2 24779 | Any finite or infinite sum... |
| metdcnlem 24780 | The metric function of a m... |
| xmetdcn2 24781 | The metric function of an ... |
| xmetdcn 24782 | The metric function of an ... |
| metdcn2 24783 | The metric function of a m... |
| metdcn 24784 | The metric function of a m... |
| msdcn 24785 | The metric function of a m... |
| cnmpt1ds 24786 | Continuity of the metric f... |
| cnmpt2ds 24787 | Continuity of the metric f... |
| nmcn 24788 | The norm of a normed group... |
| ngnmcncn 24789 | The norm of a normed group... |
| abscn 24790 | The absolute value functio... |
| metdsval 24791 | Value of the "distance to ... |
| metdsf 24792 | The distance from a point ... |
| metdsge 24793 | The distance from the poin... |
| metds0 24794 | If a point is in a set, it... |
| metdstri 24795 | A generalization of the tr... |
| metdsle 24796 | The distance from a point ... |
| metdsre 24797 | The distance from a point ... |
| metdseq0 24798 | The distance from a point ... |
| metdscnlem 24799 | Lemma for ~ metdscn . (Co... |
| metdscn 24800 | The function ` F ` which g... |
| metdscn2 24801 | The function ` F ` which g... |
| metnrmlem1a 24802 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24803 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24804 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24805 | Lemma for ~ metnrm . (Con... |
| metnrm 24806 | A metric space is normal. ... |
| metreg 24807 | A metric space is regular.... |
| addcnlem 24808 | Lemma for ~ addcn , ~ subc... |
| addcn 24809 | Complex number addition is... |
| subcn 24810 | Complex number subtraction... |
| mulcn 24811 | Complex number multiplicat... |
| mpomulcn 24812 | Complex number multiplicat... |
| divcn 24813 | Complex number division is... |
| cnfldtgp 24814 | The complex numbers form a... |
| fsumcn 24815 | A finite sum of functions ... |
| fsum2cn 24816 | Version of ~ fsumcn for tw... |
| expcn 24817 | The power function on comp... |
| divccn 24818 | Division by a nonzero cons... |
| sqcn 24819 | The square function on com... |
| iitopon 24824 | The unit interval is a top... |
| iitop 24825 | The unit interval is a top... |
| iiuni 24826 | The base set of the unit i... |
| dfii2 24827 | Alternate definition of th... |
| dfii3 24828 | Alternate definition of th... |
| dfii4 24829 | Alternate definition of th... |
| dfii5 24830 | The unit interval expresse... |
| iicmp 24831 | The unit interval is compa... |
| iiconn 24832 | The unit interval is conne... |
| cncfval 24833 | The value of the continuou... |
| elcncf 24834 | Membership in the set of c... |
| elcncf2 24835 | Version of ~ elcncf with a... |
| cncfrss 24836 | Reverse closure of the con... |
| cncfrss2 24837 | Reverse closure of the con... |
| cncff 24838 | A continuous complex funct... |
| cncfi 24839 | Defining property of a con... |
| elcncf1di 24840 | Membership in the set of c... |
| elcncf1ii 24841 | Membership in the set of c... |
| rescncf 24842 | A continuous complex funct... |
| cncfcdm 24843 | Change the codomain of a c... |
| cncfss 24844 | The set of continuous func... |
| climcncf 24845 | Image of a limit under a c... |
| abscncf 24846 | Absolute value is continuo... |
| recncf 24847 | Real part is continuous. ... |
| imcncf 24848 | Imaginary part is continuo... |
| cjcncf 24849 | Complex conjugate is conti... |
| mulc1cncf 24850 | Multiplication by a consta... |
| divccncf 24851 | Division by a constant is ... |
| cncfco 24852 | The composition of two con... |
| cncfcompt2 24853 | Composition of continuous ... |
| cncfmet 24854 | Relate complex function co... |
| cncfcn 24855 | Relate complex function co... |
| cncfcn1 24856 | Relate complex function co... |
| cncfmptc 24857 | A constant function is a c... |
| cncfmptid 24858 | The identity function is a... |
| cncfmpt1f 24859 | Composition of continuous ... |
| cncfmpt2f 24860 | Composition of continuous ... |
| cncfmpt2ss 24861 | Composition of continuous ... |
| addccncf 24862 | Adding a constant is a con... |
| idcncf 24863 | The identity function is a... |
| sub1cncf 24864 | Subtracting a constant is ... |
| sub2cncf 24865 | Subtraction from a constan... |
| cdivcncf 24866 | Division with a constant n... |
| negcncf 24867 | The negative function is c... |
| negfcncf 24868 | The negative of a continuo... |
| abscncfALT 24869 | Absolute value is continuo... |
| cncfcnvcn 24870 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24871 | The power function on comp... |
| cnmptre 24872 | Lemma for ~ iirevcn and re... |
| cnmpopc 24873 | Piecewise definition of a ... |
| iirev 24874 | Reverse the unit interval.... |
| iirevcn 24875 | The reversion function is ... |
| iihalf1 24876 | Map the first half of ` II... |
| iihalf1cn 24877 | The first half function is... |
| iihalf2 24878 | Map the second half of ` I... |
| iihalf2cn 24879 | The second half function i... |
| elii1 24880 | Divide the unit interval i... |
| elii2 24881 | Divide the unit interval i... |
| iimulcl 24882 | The unit interval is close... |
| iimulcn 24883 | Multiplication is a contin... |
| icoopnst 24884 | A half-open interval start... |
| iocopnst 24885 | A half-open interval endin... |
| icchmeo 24886 | The natural bijection from... |
| icopnfcnv 24887 | Define a bijection from ` ... |
| icopnfhmeo 24888 | The defined bijection from... |
| iccpnfcnv 24889 | Define a bijection from ` ... |
| iccpnfhmeo 24890 | The defined bijection from... |
| xrhmeo 24891 | The bijection from ` [ -u ... |
| xrhmph 24892 | The extended reals are hom... |
| xrcmp 24893 | The topology of the extend... |
| xrconn 24894 | The topology of the extend... |
| icccvx 24895 | A linear combination of tw... |
| oprpiece1res1 24896 | Restriction to the first p... |
| oprpiece1res2 24897 | Restriction to the second ... |
| cnrehmeo 24898 | The canonical bijection fr... |
| cnheiborlem 24899 | Lemma for ~ cnheibor . (C... |
| cnheibor 24900 | Heine-Borel theorem for co... |
| cnllycmp 24901 | The topology on the comple... |
| rellycmp 24902 | The topology on the reals ... |
| bndth 24903 | The Boundedness Theorem. ... |
| evth 24904 | The Extreme Value Theorem.... |
| evth2 24905 | The Extreme Value Theorem,... |
| lebnumlem1 24906 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24907 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24908 | Lemma for ~ lebnum . By t... |
| lebnum 24909 | The Lebesgue number lemma,... |
| xlebnum 24910 | Generalize ~ lebnum to ext... |
| lebnumii 24911 | Specialize the Lebesgue nu... |
| ishtpy 24917 | Membership in the class of... |
| htpycn 24918 | A homotopy is a continuous... |
| htpyi 24919 | A homotopy evaluated at it... |
| ishtpyd 24920 | Deduction for membership i... |
| htpycom 24921 | Given a homotopy from ` F ... |
| htpyid 24922 | A homotopy from a function... |
| htpyco1 24923 | Compose a homotopy with a ... |
| htpyco2 24924 | Compose a homotopy with a ... |
| htpycc 24925 | Concatenate two homotopies... |
| isphtpy 24926 | Membership in the class of... |
| phtpyhtpy 24927 | A path homotopy is a homot... |
| phtpycn 24928 | A path homotopy is a conti... |
| phtpyi 24929 | Membership in the class of... |
| phtpy01 24930 | Two path-homotopic paths h... |
| isphtpyd 24931 | Deduction for membership i... |
| isphtpy2d 24932 | Deduction for membership i... |
| phtpycom 24933 | Given a homotopy from ` F ... |
| phtpyid 24934 | A homotopy from a path to ... |
| phtpyco2 24935 | Compose a path homotopy wi... |
| phtpycc 24936 | Concatenate two path homot... |
| phtpcrel 24938 | The path homotopy relation... |
| isphtpc 24939 | The relation "is path homo... |
| phtpcer 24940 | Path homotopy is an equiva... |
| phtpc01 24941 | Path homotopic paths have ... |
| reparphti 24942 | Lemma for ~ reparpht . (C... |
| reparpht 24943 | Reparametrization lemma. ... |
| phtpcco2 24944 | Compose a path homotopy wi... |
| pcofval 24955 | The value of the path conc... |
| pcoval 24956 | The concatenation of two p... |
| pcovalg 24957 | Evaluate the concatenation... |
| pcoval1 24958 | Evaluate the concatenation... |
| pco0 24959 | The starting point of a pa... |
| pco1 24960 | The ending point of a path... |
| pcoval2 24961 | Evaluate the concatenation... |
| pcocn 24962 | The concatenation of two p... |
| copco 24963 | The composition of a conca... |
| pcohtpylem 24964 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24965 | Homotopy invariance of pat... |
| pcoptcl 24966 | A constant function is a p... |
| pcopt 24967 | Concatenation with a point... |
| pcopt2 24968 | Concatenation with a point... |
| pcoass 24969 | Order of concatenation doe... |
| pcorevcl 24970 | Closure for a reversed pat... |
| pcorevlem 24971 | Lemma for ~ pcorev . Prov... |
| pcorev 24972 | Concatenation with the rev... |
| pcorev2 24973 | Concatenation with the rev... |
| pcophtb 24974 | The path homotopy equivale... |
| om1val 24975 | The definition of the loop... |
| om1bas 24976 | The base set of the loop s... |
| om1elbas 24977 | Elementhood in the base se... |
| om1addcl 24978 | Closure of the group opera... |
| om1plusg 24979 | The group operation (which... |
| om1tset 24980 | The topology of the loop s... |
| om1opn 24981 | The topology of the loop s... |
| pi1val 24982 | The definition of the fund... |
| pi1bas 24983 | The base set of the fundam... |
| pi1blem 24984 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24985 | Another way to write the l... |
| pi1bas2 24986 | The base set of the fundam... |
| pi1eluni 24987 | Elementhood in the base se... |
| pi1bas3 24988 | The base set of the fundam... |
| pi1cpbl 24989 | The group operation, loop ... |
| elpi1 24990 | The elements of the fundam... |
| elpi1i 24991 | The elements of the fundam... |
| pi1addf 24992 | The group operation of ` p... |
| pi1addval 24993 | The concatenation of two p... |
| pi1grplem 24994 | Lemma for ~ pi1grp . (Con... |
| pi1grp 24995 | The fundamental group is a... |
| pi1id 24996 | The identity element of th... |
| pi1inv 24997 | An inverse in the fundamen... |
| pi1xfrf 24998 | Functionality of the loop ... |
| pi1xfrval 24999 | The value of the loop tran... |
| pi1xfr 25000 | Given a path ` F ` and its... |
| pi1xfrcnvlem 25001 | Given a path ` F ` between... |
| pi1xfrcnv 25002 | Given a path ` F ` between... |
| pi1xfrgim 25003 | The mapping ` G ` between ... |
| pi1cof 25004 | Functionality of the loop ... |
| pi1coval 25005 | The value of the loop tran... |
| pi1coghm 25006 | The mapping ` G ` between ... |
| isclm 25009 | A subcomplex module is a l... |
| clmsca 25010 | The ring of scalars ` F ` ... |
| clmsubrg 25011 | The base set of the ring o... |
| clmlmod 25012 | A subcomplex module is a l... |
| clmgrp 25013 | A subcomplex module is an ... |
| clmabl 25014 | A subcomplex module is an ... |
| clmring 25015 | The scalar ring of a subco... |
| clmfgrp 25016 | The scalar ring of a subco... |
| clm0 25017 | The zero of the scalar rin... |
| clm1 25018 | The identity of the scalar... |
| clmadd 25019 | The addition of the scalar... |
| clmmul 25020 | The multiplication of the ... |
| clmcj 25021 | The conjugation of the sca... |
| isclmi 25022 | Reverse direction of ~ isc... |
| clmzss 25023 | The scalar ring of a subco... |
| clmsscn 25024 | The scalar ring of a subco... |
| clmsub 25025 | Subtraction in the scalar ... |
| clmneg 25026 | Negation in the scalar rin... |
| clmneg1 25027 | Minus one is in the scalar... |
| clmabs 25028 | Norm in the scalar ring of... |
| clmacl 25029 | Closure of ring addition f... |
| clmmcl 25030 | Closure of ring multiplica... |
| clmsubcl 25031 | Closure of ring subtractio... |
| lmhmclm 25032 | The domain of a linear ope... |
| clmvscl 25033 | Closure of scalar product ... |
| clmvsass 25034 | Associative law for scalar... |
| clmvscom 25035 | Commutative law for the sc... |
| clmvsdir 25036 | Distributive law for scala... |
| clmvsdi 25037 | Distributive law for scala... |
| clmvs1 25038 | Scalar product with ring u... |
| clmvs2 25039 | A vector plus itself is tw... |
| clm0vs 25040 | Zero times a vector is the... |
| clmopfne 25041 | The (functionalized) opera... |
| isclmp 25042 | The predicate "is a subcom... |
| isclmi0 25043 | Properties that determine ... |
| clmvneg1 25044 | Minus 1 times a vector is ... |
| clmvsneg 25045 | Multiplication of a vector... |
| clmmulg 25046 | The group multiple functio... |
| clmsubdir 25047 | Scalar multiplication dist... |
| clmpm1dir 25048 | Subtractive distributive l... |
| clmnegneg 25049 | Double negative of a vecto... |
| clmnegsubdi2 25050 | Distribution of negative o... |
| clmsub4 25051 | Rearrangement of 4 terms i... |
| clmvsrinv 25052 | A vector minus itself. (C... |
| clmvslinv 25053 | Minus a vector plus itself... |
| clmvsubval 25054 | Value of vector subtractio... |
| clmvsubval2 25055 | Value of vector subtractio... |
| clmvz 25056 | Two ways to express the ne... |
| zlmclm 25057 | The ` ZZ ` -module operati... |
| clmzlmvsca 25058 | The scalar product of a su... |
| nmoleub2lem 25059 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 25060 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 25061 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 25062 | The operator norm is the s... |
| nmoleub2b 25063 | The operator norm is the s... |
| nmoleub3 25064 | The operator norm is the s... |
| nmhmcn 25065 | A linear operator over a n... |
| cmodscexp 25066 | The powers of ` _i ` belon... |
| cmodscmulexp 25067 | The scalar product of a ve... |
| cvslvec 25070 | A subcomplex vector space ... |
| cvsclm 25071 | A subcomplex vector space ... |
| iscvs 25072 | A subcomplex vector space ... |
| iscvsp 25073 | The predicate "is a subcom... |
| iscvsi 25074 | Properties that determine ... |
| cvsi 25075 | The properties of a subcom... |
| cvsunit 25076 | Unit group of the scalar r... |
| cvsdiv 25077 | Division of the scalar rin... |
| cvsdivcl 25078 | The scalar field of a subc... |
| cvsmuleqdivd 25079 | An equality involving rati... |
| cvsdiveqd 25080 | An equality involving rati... |
| cnlmodlem1 25081 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 25082 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 25083 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 25084 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 25085 | The set of complex numbers... |
| cnstrcvs 25086 | The set of complex numbers... |
| cnrbas 25087 | The set of complex numbers... |
| cnrlmod 25088 | The complex left module of... |
| cnrlvec 25089 | The complex left module of... |
| cncvs 25090 | The complex left module of... |
| recvs 25091 | The field of the real numb... |
| qcvs 25092 | The field of rational numb... |
| zclmncvs 25093 | The ring of integers as le... |
| isncvsngp 25094 | A normed subcomplex vector... |
| isncvsngpd 25095 | Properties that determine ... |
| ncvsi 25096 | The properties of a normed... |
| ncvsprp 25097 | Proportionality property o... |
| ncvsge0 25098 | The norm of a scalar produ... |
| ncvsm1 25099 | The norm of the opposite o... |
| ncvsdif 25100 | The norm of the difference... |
| ncvspi 25101 | The norm of a vector plus ... |
| ncvs1 25102 | From any nonzero vector of... |
| cnrnvc 25103 | The module of complex numb... |
| cnncvs 25104 | The module of complex numb... |
| cnnm 25105 | The norm of the normed sub... |
| ncvspds 25106 | Value of the distance func... |
| cnindmet 25107 | The metric induced on the ... |
| cnncvsaddassdemo 25108 | Derive the associative law... |
| cnncvsmulassdemo 25109 | Derive the associative law... |
| cnncvsabsnegdemo 25110 | Derive the absolute value ... |
| iscph 25115 | A subcomplex pre-Hilbert s... |
| cphphl 25116 | A subcomplex pre-Hilbert s... |
| cphnlm 25117 | A subcomplex pre-Hilbert s... |
| cphngp 25118 | A subcomplex pre-Hilbert s... |
| cphlmod 25119 | A subcomplex pre-Hilbert s... |
| cphlvec 25120 | A subcomplex pre-Hilbert s... |
| cphnvc 25121 | A subcomplex pre-Hilbert s... |
| cphsubrglem 25122 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 25123 | Lemma for ~ cphreccl . (C... |
| cphsca 25124 | A subcomplex pre-Hilbert s... |
| cphsubrg 25125 | The scalar field of a subc... |
| cphreccl 25126 | The scalar field of a subc... |
| cphdivcl 25127 | The scalar field of a subc... |
| cphcjcl 25128 | The scalar field of a subc... |
| cphsqrtcl 25129 | The scalar field of a subc... |
| cphabscl 25130 | The scalar field of a subc... |
| cphsqrtcl2 25131 | The scalar field of a subc... |
| cphsqrtcl3 25132 | If the scalar field of a s... |
| cphqss 25133 | The scalar field of a subc... |
| cphclm 25134 | A subcomplex pre-Hilbert s... |
| cphnmvs 25135 | Norm of a scalar product. ... |
| cphipcl 25136 | An inner product is a memb... |
| cphnmfval 25137 | The value of the norm in a... |
| cphnm 25138 | The square of the norm is ... |
| nmsq 25139 | The square of the norm is ... |
| cphnmf 25140 | The norm of a vector is a ... |
| cphnmcl 25141 | The norm of a vector is a ... |
| reipcl 25142 | An inner product of an ele... |
| ipge0 25143 | The inner product in a sub... |
| cphipcj 25144 | Conjugate of an inner prod... |
| cphipipcj 25145 | An inner product times its... |
| cphorthcom 25146 | Orthogonality (meaning inn... |
| cphip0l 25147 | Inner product with a zero ... |
| cphip0r 25148 | Inner product with a zero ... |
| cphipeq0 25149 | The inner product of a vec... |
| cphdir 25150 | Distributive law for inner... |
| cphdi 25151 | Distributive law for inner... |
| cph2di 25152 | Distributive law for inner... |
| cphsubdir 25153 | Distributive law for inner... |
| cphsubdi 25154 | Distributive law for inner... |
| cph2subdi 25155 | Distributive law for inner... |
| cphass 25156 | Associative law for inner ... |
| cphassr 25157 | "Associative" law for seco... |
| cph2ass 25158 | Move scalar multiplication... |
| cphassi 25159 | Associative law for the fi... |
| cphassir 25160 | "Associative" law for the ... |
| cphpyth 25161 | The pythagorean theorem fo... |
| tcphex 25162 | Lemma for ~ tcphbas and si... |
| tcphval 25163 | Define a function to augme... |
| tcphbas 25164 | The base set of a subcompl... |
| tchplusg 25165 | The addition operation of ... |
| tcphsub 25166 | The subtraction operation ... |
| tcphmulr 25167 | The ring operation of a su... |
| tcphsca 25168 | The scalar field of a subc... |
| tcphvsca 25169 | The scalar multiplication ... |
| tcphip 25170 | The inner product of a sub... |
| tcphtopn 25171 | The topology of a subcompl... |
| tcphphl 25172 | Augmentation of a subcompl... |
| tchnmfval 25173 | The norm of a subcomplex p... |
| tcphnmval 25174 | The norm of a subcomplex p... |
| cphtcphnm 25175 | The norm of a norm-augment... |
| tcphds 25176 | The distance of a pre-Hilb... |
| phclm 25177 | A pre-Hilbert space whose ... |
| tcphcphlem3 25178 | Lemma for ~ tcphcph : real... |
| ipcau2 25179 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 25180 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 25181 | Lemma for ~ tcphcph : homo... |
| tcphcph 25182 | The standard definition of... |
| ipcau 25183 | The Cauchy-Schwarz inequal... |
| nmparlem 25184 | Lemma for ~ nmpar . (Cont... |
| nmpar 25185 | A subcomplex pre-Hilbert s... |
| cphipval2 25186 | Value of the inner product... |
| 4cphipval2 25187 | Four times the inner produ... |
| cphipval 25188 | Value of the inner product... |
| ipcnlem2 25189 | The inner product operatio... |
| ipcnlem1 25190 | The inner product operatio... |
| ipcn 25191 | The inner product operatio... |
| cnmpt1ip 25192 | Continuity of inner produc... |
| cnmpt2ip 25193 | Continuity of inner produc... |
| csscld 25194 | A "closed subspace" in a s... |
| clsocv 25195 | The orthogonal complement ... |
| cphsscph 25196 | A subspace of a subcomplex... |
| lmmbr 25203 | Express the binary relatio... |
| lmmbr2 25204 | Express the binary relatio... |
| lmmbr3 25205 | Express the binary relatio... |
| lmmcvg 25206 | Convergence property of a ... |
| lmmbrf 25207 | Express the binary relatio... |
| lmnn 25208 | A condition that implies c... |
| cfilfval 25209 | The set of Cauchy filters ... |
| iscfil 25210 | The property of being a Ca... |
| iscfil2 25211 | The property of being a Ca... |
| cfilfil 25212 | A Cauchy filter is a filte... |
| cfili 25213 | Property of a Cauchy filte... |
| cfil3i 25214 | A Cauchy filter contains b... |
| cfilss 25215 | A filter finer than a Cauc... |
| fgcfil 25216 | The Cauchy filter conditio... |
| fmcfil 25217 | The Cauchy filter conditio... |
| iscfil3 25218 | A filter is Cauchy iff it ... |
| cfilfcls 25219 | Similar to ultrafilters ( ... |
| caufval 25220 | The set of Cauchy sequence... |
| iscau 25221 | Express the property " ` F... |
| iscau2 25222 | Express the property " ` F... |
| iscau3 25223 | Express the Cauchy sequenc... |
| iscau4 25224 | Express the property " ` F... |
| iscauf 25225 | Express the property " ` F... |
| caun0 25226 | A metric with a Cauchy seq... |
| caufpm 25227 | Inclusion of a Cauchy sequ... |
| caucfil 25228 | A Cauchy sequence predicat... |
| iscmet 25229 | The property " ` D ` is a ... |
| cmetcvg 25230 | The convergence of a Cauch... |
| cmetmet 25231 | A complete metric space is... |
| cmetmeti 25232 | A complete metric space is... |
| cmetcaulem 25233 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25234 | The convergence of a Cauch... |
| iscmet3lem3 25235 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25236 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25237 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25238 | The property " ` D ` is a ... |
| iscmet2 25239 | A metric ` D ` is complete... |
| cfilresi 25240 | A Cauchy filter on a metri... |
| cfilres 25241 | Cauchy filter on a metric ... |
| caussi 25242 | Cauchy sequence on a metri... |
| causs 25243 | Cauchy sequence on a metri... |
| equivcfil 25244 | If the metric ` D ` is "st... |
| equivcau 25245 | If the metric ` D ` is "st... |
| lmle 25246 | If the distance from each ... |
| nglmle 25247 | If the norm of each member... |
| lmclim 25248 | Relate a limit on the metr... |
| lmclimf 25249 | Relate a limit on the metr... |
| metelcls 25250 | A point belongs to the clo... |
| metcld 25251 | A subset of a metric space... |
| metcld2 25252 | A subset of a metric space... |
| caubl 25253 | Sufficient condition to en... |
| caublcls 25254 | The convergent point of a ... |
| metcnp4 25255 | Two ways to say a mapping ... |
| metcn4 25256 | Two ways to say a mapping ... |
| iscmet3i 25257 | Properties that determine ... |
| lmcau 25258 | Every convergent sequence ... |
| flimcfil 25259 | Every convergent filter in... |
| metsscmetcld 25260 | A complete subspace of a m... |
| cmetss 25261 | A subspace of a complete m... |
| equivcmet 25262 | If two metrics are strongl... |
| relcmpcmet 25263 | If ` D ` is a metric space... |
| cmpcmet 25264 | A compact metric space is ... |
| cfilucfil3 25265 | Given a metric ` D ` and a... |
| cfilucfil4 25266 | Given a metric ` D ` and a... |
| cncmet 25267 | The set of complex numbers... |
| recmet 25268 | The real numbers are a com... |
| bcthlem1 25269 | Lemma for ~ bcth . Substi... |
| bcthlem2 25270 | Lemma for ~ bcth . The ba... |
| bcthlem3 25271 | Lemma for ~ bcth . The li... |
| bcthlem4 25272 | Lemma for ~ bcth . Given ... |
| bcthlem5 25273 | Lemma for ~ bcth . The pr... |
| bcth 25274 | Baire's Category Theorem. ... |
| bcth2 25275 | Baire's Category Theorem, ... |
| bcth3 25276 | Baire's Category Theorem, ... |
| isbn 25283 | A Banach space is a normed... |
| bnsca 25284 | The scalar field of a Bana... |
| bnnvc 25285 | A Banach space is a normed... |
| bnnlm 25286 | A Banach space is a normed... |
| bnngp 25287 | A Banach space is a normed... |
| bnlmod 25288 | A Banach space is a left m... |
| bncms 25289 | A Banach space is a comple... |
| iscms 25290 | A complete metric space is... |
| cmscmet 25291 | The induced metric on a co... |
| bncmet 25292 | The induced metric on Bana... |
| cmsms 25293 | A complete metric space is... |
| cmspropd 25294 | Property deduction for a c... |
| cmssmscld 25295 | The restriction of a metri... |
| cmsss 25296 | The restriction of a compl... |
| lssbn 25297 | A subspace of a Banach spa... |
| cmetcusp1 25298 | If the uniform set of a co... |
| cmetcusp 25299 | The uniform space generate... |
| cncms 25300 | The field of complex numbe... |
| cnflduss 25301 | The uniform structure of t... |
| cnfldcusp 25302 | The field of complex numbe... |
| resscdrg 25303 | The real numbers are a sub... |
| cncdrg 25304 | The only complete subfield... |
| srabn 25305 | The subring algebra over a... |
| rlmbn 25306 | The ring module over a com... |
| ishl 25307 | The predicate "is a subcom... |
| hlbn 25308 | Every subcomplex Hilbert s... |
| hlcph 25309 | Every subcomplex Hilbert s... |
| hlphl 25310 | Every subcomplex Hilbert s... |
| hlcms 25311 | Every subcomplex Hilbert s... |
| hlprlem 25312 | Lemma for ~ hlpr . (Contr... |
| hlress 25313 | The scalar field of a subc... |
| hlpr 25314 | The scalar field of a subc... |
| ishl2 25315 | A Hilbert space is a compl... |
| cphssphl 25316 | A Banach subspace of a sub... |
| cmslssbn 25317 | A complete linear subspace... |
| cmscsscms 25318 | A closed subspace of a com... |
| bncssbn 25319 | A closed subspace of a Ban... |
| cssbn 25320 | A complete subspace of a n... |
| csschl 25321 | A complete subspace of a c... |
| cmslsschl 25322 | A complete linear subspace... |
| chlcsschl 25323 | A closed subspace of a sub... |
| retopn 25324 | The topology of the real n... |
| recms 25325 | The real numbers form a co... |
| reust 25326 | The Uniform structure of t... |
| recusp 25327 | The real numbers form a co... |
| rrxval 25332 | Value of the generalized E... |
| rrxbase 25333 | The base of the generalize... |
| rrxprds 25334 | Expand the definition of t... |
| rrxip 25335 | The inner product of the g... |
| rrxnm 25336 | The norm of the generalize... |
| rrxcph 25337 | Generalized Euclidean real... |
| rrxds 25338 | The distance over generali... |
| rrxvsca 25339 | The scalar product over ge... |
| rrxplusgvscavalb 25340 | The result of the addition... |
| rrxsca 25341 | The field of real numbers ... |
| rrx0 25342 | The zero ("origin") in a g... |
| rrx0el 25343 | The zero ("origin") in a g... |
| csbren 25344 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25345 | Triangle inequality in R^n... |
| rrxf 25346 | Euclidean vectors as funct... |
| rrxfsupp 25347 | Euclidean vectors are of f... |
| rrxsuppss 25348 | Support of Euclidean vecto... |
| rrxmvallem 25349 | Support of the function us... |
| rrxmval 25350 | The value of the Euclidean... |
| rrxmfval 25351 | The value of the Euclidean... |
| rrxmetlem 25352 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25353 | Euclidean space is a metri... |
| rrxdstprj1 25354 | The distance between two p... |
| rrxbasefi 25355 | The base of the generalize... |
| rrxdsfi 25356 | The distance over generali... |
| rrxmetfi 25357 | Euclidean space is a metri... |
| rrxdsfival 25358 | The value of the Euclidean... |
| ehlval 25359 | Value of the Euclidean spa... |
| ehlbase 25360 | The base of the Euclidean ... |
| ehl0base 25361 | The base of the Euclidean ... |
| ehl0 25362 | The Euclidean space of dim... |
| ehleudis 25363 | The Euclidean distance fun... |
| ehleudisval 25364 | The value of the Euclidean... |
| ehl1eudis 25365 | The Euclidean distance fun... |
| ehl1eudisval 25366 | The value of the Euclidean... |
| ehl2eudis 25367 | The Euclidean distance fun... |
| ehl2eudisval 25368 | The value of the Euclidean... |
| minveclem1 25369 | Lemma for ~ minvec . The ... |
| minveclem4c 25370 | Lemma for ~ minvec . The ... |
| minveclem2 25371 | Lemma for ~ minvec . Any ... |
| minveclem3a 25372 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25373 | Lemma for ~ minvec . The ... |
| minveclem3 25374 | Lemma for ~ minvec . The ... |
| minveclem4a 25375 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25376 | Lemma for ~ minvec . The ... |
| minveclem4 25377 | Lemma for ~ minvec . The ... |
| minveclem5 25378 | Lemma for ~ minvec . Disc... |
| minveclem6 25379 | Lemma for ~ minvec . Any ... |
| minveclem7 25380 | Lemma for ~ minvec . Sinc... |
| minvec 25381 | Minimizing vector theorem,... |
| pjthlem1 25382 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25383 | Lemma for ~ pjth . (Contr... |
| pjth 25384 | Projection Theorem: Any H... |
| pjth2 25385 | Projection Theorem with ab... |
| cldcss 25386 | Corollary of the Projectio... |
| cldcss2 25387 | Corollary of the Projectio... |
| hlhil 25388 | Corollary of the Projectio... |
| addcncf 25389 | The addition of two contin... |
| subcncf 25390 | The subtraction of two con... |
| mulcncf 25391 | The multiplication of two ... |
| divcncf 25392 | The quotient of two contin... |
| pmltpclem1 25393 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25394 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25395 | Any function on the reals ... |
| ivthlem1 25396 | Lemma for ~ ivth . The se... |
| ivthlem2 25397 | Lemma for ~ ivth . Show t... |
| ivthlem3 25398 | Lemma for ~ ivth , the int... |
| ivth 25399 | The intermediate value the... |
| ivth2 25400 | The intermediate value the... |
| ivthle 25401 | The intermediate value the... |
| ivthle2 25402 | The intermediate value the... |
| ivthicc 25403 | The interval between any t... |
| evthicc 25404 | Specialization of the Extr... |
| evthicc2 25405 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25406 | A continuous function on a... |
| ovolfcl 25411 | Closure for the interval e... |
| ovolfioo 25412 | Unpack the interval coveri... |
| ovolficc 25413 | Unpack the interval coveri... |
| ovolficcss 25414 | Any (closed) interval cove... |
| ovolfsval 25415 | The value of the interval ... |
| ovolfsf 25416 | Closure for the interval l... |
| ovolsf 25417 | Closure for the partial su... |
| ovolval 25418 | The value of the outer mea... |
| elovolmlem 25419 | Lemma for ~ elovolm and re... |
| elovolm 25420 | Elementhood in the set ` M... |
| elovolmr 25421 | Sufficient condition for e... |
| ovolmge0 25422 | The set ` M ` is composed ... |
| ovolcl 25423 | The volume of a set is an ... |
| ovollb 25424 | The outer volume is a lowe... |
| ovolgelb 25425 | The outer volume is the gr... |
| ovolge0 25426 | The volume of a set is alw... |
| ovolf 25427 | The domain and codomain of... |
| ovollecl 25428 | If an outer volume is boun... |
| ovolsslem 25429 | Lemma for ~ ovolss . (Con... |
| ovolss 25430 | The volume of a set is mon... |
| ovolsscl 25431 | If a set is contained in a... |
| ovolssnul 25432 | A subset of a nullset is n... |
| ovollb2lem 25433 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25434 | It is often more convenien... |
| ovolctb 25435 | The volume of a denumerabl... |
| ovolq 25436 | The rational numbers have ... |
| ovolctb2 25437 | The volume of a countable ... |
| ovol0 25438 | The empty set has 0 outer ... |
| ovolfi 25439 | A finite set has 0 outer L... |
| ovolsn 25440 | A singleton has 0 outer Le... |
| ovolunlem1a 25441 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25442 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25443 | Lemma for ~ ovolun . (Con... |
| ovolun 25444 | The Lebesgue outer measure... |
| ovolunnul 25445 | Adding a nullset does not ... |
| ovolfiniun 25446 | The Lebesgue outer measure... |
| ovoliunlem1 25447 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25448 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25449 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25450 | The Lebesgue outer measure... |
| ovoliun2 25451 | The Lebesgue outer measure... |
| ovoliunnul 25452 | A countable union of nulls... |
| shft2rab 25453 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25454 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25455 | Lemma for ~ ovolshft . (C... |
| ovolshft 25456 | The Lebesgue outer measure... |
| sca2rab 25457 | If ` B ` is a scale of ` A... |
| ovolscalem1 25458 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25459 | Lemma for ~ ovolshft . (C... |
| ovolsca 25460 | The Lebesgue outer measure... |
| ovolicc1 25461 | The measure of a closed in... |
| ovolicc2lem1 25462 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25463 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25464 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25465 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25466 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25467 | The measure of a closed in... |
| ovolicc 25468 | The measure of a closed in... |
| ovolicopnf 25469 | The measure of a right-unb... |
| ovolre 25470 | The measure of the real nu... |
| ismbl 25471 | The predicate " ` A ` is L... |
| ismbl2 25472 | From ~ ovolun , it suffice... |
| volres 25473 | A self-referencing abbrevi... |
| volf 25474 | The domain and codomain of... |
| mblvol 25475 | The volume of a measurable... |
| mblss 25476 | A measurable set is a subs... |
| mblsplit 25477 | The defining property of m... |
| volss 25478 | The Lebesgue measure is mo... |
| cmmbl 25479 | The complement of a measur... |
| nulmbl 25480 | A nullset is measurable. ... |
| nulmbl2 25481 | A set of outer measure zer... |
| unmbl 25482 | A union of measurable sets... |
| shftmbl 25483 | A shift of a measurable se... |
| 0mbl 25484 | The empty set is measurabl... |
| rembl 25485 | The set of all real number... |
| unidmvol 25486 | The union of the Lebesgue ... |
| inmbl 25487 | An intersection of measura... |
| difmbl 25488 | A difference of measurable... |
| finiunmbl 25489 | A finite union of measurab... |
| volun 25490 | The Lebesgue measure funct... |
| volinun 25491 | Addition of non-disjoint s... |
| volfiniun 25492 | The volume of a disjoint f... |
| iundisj 25493 | Rewrite a countable union ... |
| iundisj2 25494 | A disjoint union is disjoi... |
| voliunlem1 25495 | Lemma for ~ voliun . (Con... |
| voliunlem2 25496 | Lemma for ~ voliun . (Con... |
| voliunlem3 25497 | Lemma for ~ voliun . (Con... |
| iunmbl 25498 | The measurable sets are cl... |
| voliun 25499 | The Lebesgue measure funct... |
| volsuplem 25500 | Lemma for ~ volsup . (Con... |
| volsup 25501 | The volume of the limit of... |
| iunmbl2 25502 | The measurable sets are cl... |
| ioombl1lem1 25503 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25504 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25505 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25506 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25507 | An open right-unbounded in... |
| icombl1 25508 | A closed unbounded-above i... |
| icombl 25509 | A closed-below, open-above... |
| ioombl 25510 | An open real interval is m... |
| iccmbl 25511 | A closed real interval is ... |
| iccvolcl 25512 | A closed real interval has... |
| ovolioo 25513 | The measure of an open int... |
| volioo 25514 | The measure of an open int... |
| ioovolcl 25515 | An open real interval has ... |
| ovolfs2 25516 | Alternative expression for... |
| ioorcl2 25517 | An open interval with fini... |
| ioorf 25518 | Define a function from ope... |
| ioorval 25519 | Define a function from ope... |
| ioorinv2 25520 | The function ` F ` is an "... |
| ioorinv 25521 | The function ` F ` is an "... |
| ioorcl 25522 | The function ` F ` does no... |
| uniiccdif 25523 | A union of closed interval... |
| uniioovol 25524 | A disjoint union of open i... |
| uniiccvol 25525 | An almost-disjoint union o... |
| uniioombllem1 25526 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25527 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25528 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25529 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25530 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25531 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25532 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25533 | Lemma for ~ uniioombl . (... |
| uniioombl 25534 | A disjoint union of open i... |
| uniiccmbl 25535 | An almost-disjoint union o... |
| dyadf 25536 | The function ` F ` returns... |
| dyadval 25537 | Value of the dyadic ration... |
| dyadovol 25538 | Volume of a dyadic rationa... |
| dyadss 25539 | Two closed dyadic rational... |
| dyaddisjlem 25540 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25541 | Two closed dyadic rational... |
| dyadmaxlem 25542 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25543 | Any nonempty set of dyadic... |
| dyadmbllem 25544 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25545 | Any union of dyadic ration... |
| opnmbllem 25546 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25547 | All open sets are measurab... |
| opnmblALT 25548 | All open sets are measurab... |
| subopnmbl 25549 | Sets which are open in a m... |
| volsup2 25550 | The volume of ` A ` is the... |
| volcn 25551 | The function formed by res... |
| volivth 25552 | The Intermediate Value The... |
| vitalilem1 25553 | Lemma for ~ vitali . (Con... |
| vitalilem2 25554 | Lemma for ~ vitali . (Con... |
| vitalilem3 25555 | Lemma for ~ vitali . (Con... |
| vitalilem4 25556 | Lemma for ~ vitali . (Con... |
| vitalilem5 25557 | Lemma for ~ vitali . (Con... |
| vitali 25558 | If the reals can be well-o... |
| ismbf1 25569 | The predicate " ` F ` is a... |
| mbff 25570 | A measurable function is a... |
| mbfdm 25571 | The domain of a measurable... |
| mbfconstlem 25572 | Lemma for ~ mbfconst and r... |
| ismbf 25573 | The predicate " ` F ` is a... |
| ismbfcn 25574 | A complex function is meas... |
| mbfima 25575 | Definitional property of a... |
| mbfimaicc 25576 | The preimage of any closed... |
| mbfimasn 25577 | The preimage of a point un... |
| mbfconst 25578 | A constant function is mea... |
| mbf0 25579 | The empty function is meas... |
| mbfid 25580 | The identity function is m... |
| mbfmptcl 25581 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25582 | The domain of a measurable... |
| ismbfcn2 25583 | A complex function is meas... |
| ismbfd 25584 | Deduction to prove measura... |
| ismbf2d 25585 | Deduction to prove measura... |
| mbfeqalem1 25586 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25587 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25588 | If two functions are equal... |
| mbfres 25589 | The restriction of a measu... |
| mbfres2 25590 | Measurability of a piecewi... |
| mbfss 25591 | Change the domain of a mea... |
| mbfmulc2lem 25592 | Multiplication by a consta... |
| mbfmulc2re 25593 | Multiplication by a consta... |
| mbfmax 25594 | The maximum of two functio... |
| mbfneg 25595 | The negative of a measurab... |
| mbfpos 25596 | The positive part of a mea... |
| mbfposr 25597 | Converse to ~ mbfpos . (C... |
| mbfposb 25598 | A function is measurable i... |
| ismbf3d 25599 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25600 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25601 | The preimage of any open s... |
| mbfimaopn2 25602 | The preimage of any set op... |
| cncombf 25603 | The composition of a conti... |
| cnmbf 25604 | A continuous function is m... |
| mbfaddlem 25605 | The sum of two measurable ... |
| mbfadd 25606 | The sum of two measurable ... |
| mbfsub 25607 | The difference of two meas... |
| mbfmulc2 25608 | A complex constant times a... |
| mbfsup 25609 | The supremum of a sequence... |
| mbfinf 25610 | The infimum of a sequence ... |
| mbflimsup 25611 | The limit supremum of a se... |
| mbflimlem 25612 | The pointwise limit of a s... |
| mbflim 25613 | The pointwise limit of a s... |
| 0pval 25616 | The zero function evaluate... |
| 0plef 25617 | Two ways to say that the f... |
| 0pledm 25618 | Adjust the domain of the l... |
| isi1f 25619 | The predicate " ` F ` is a... |
| i1fmbf 25620 | Simple functions are measu... |
| i1ff 25621 | A simple function is a fun... |
| i1frn 25622 | A simple function has fini... |
| i1fima 25623 | Any preimage of a simple f... |
| i1fima2 25624 | Any preimage of a simple f... |
| i1fima2sn 25625 | Preimage of a singleton. ... |
| i1fd 25626 | A simplified set of assump... |
| i1f0rn 25627 | Any simple function takes ... |
| itg1val 25628 | The value of the integral ... |
| itg1val2 25629 | The value of the integral ... |
| itg1cl 25630 | Closure of the integral on... |
| itg1ge0 25631 | Closure of the integral on... |
| i1f0 25632 | The zero function is simpl... |
| itg10 25633 | The zero function has zero... |
| i1f1lem 25634 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25635 | Base case simple functions... |
| itg11 25636 | The integral of an indicat... |
| itg1addlem1 25637 | Decompose a preimage, whic... |
| i1faddlem 25638 | Decompose the preimage of ... |
| i1fmullem 25639 | Decompose the preimage of ... |
| i1fadd 25640 | The sum of two simple func... |
| i1fmul 25641 | The pointwise product of t... |
| itg1addlem2 25642 | Lemma for ~ itg1add . The... |
| itg1addlem3 25643 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25644 | Lemma for ~ itg1add . (Co... |
| itg1addlem5 25645 | Lemma for ~ itg1add . (Co... |
| itg1add 25646 | The integral of a sum of s... |
| i1fmulclem 25647 | Decompose the preimage of ... |
| i1fmulc 25648 | A nonnegative constant tim... |
| itg1mulc 25649 | The integral of a constant... |
| i1fres 25650 | The "restriction" of a sim... |
| i1fpos 25651 | The positive part of a sim... |
| i1fposd 25652 | Deduction form of ~ i1fpos... |
| i1fsub 25653 | The difference of two simp... |
| itg1sub 25654 | The integral of a differen... |
| itg10a 25655 | The integral of a simple f... |
| itg1ge0a 25656 | The integral of an almost ... |
| itg1lea 25657 | Approximate version of ~ i... |
| itg1le 25658 | If one simple function dom... |
| itg1climres 25659 | Restricting the simple fun... |
| mbfi1fseqlem1 25660 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25661 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25662 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25663 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25664 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25665 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25666 | A characterization of meas... |
| mbfi1flimlem 25667 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25668 | Any real measurable functi... |
| mbfmullem2 25669 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25670 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25671 | The product of two measura... |
| itg2lcl 25672 | The set of lower sums is a... |
| itg2val 25673 | Value of the integral on n... |
| itg2l 25674 | Elementhood in the set ` L... |
| itg2lr 25675 | Sufficient condition for e... |
| xrge0f 25676 | A real function is a nonne... |
| itg2cl 25677 | The integral of a nonnegat... |
| itg2ub 25678 | The integral of a nonnegat... |
| itg2leub 25679 | Any upper bound on the int... |
| itg2ge0 25680 | The integral of a nonnegat... |
| itg2itg1 25681 | The integral of a nonnegat... |
| itg20 25682 | The integral of the zero f... |
| itg2lecl 25683 | If an ` S.2 ` integral is ... |
| itg2le 25684 | If one function dominates ... |
| itg2const 25685 | Integral of a constant fun... |
| itg2const2 25686 | When the base set of a con... |
| itg2seq 25687 | Definitional property of t... |
| itg2uba 25688 | Approximate version of ~ i... |
| itg2lea 25689 | Approximate version of ~ i... |
| itg2eqa 25690 | Approximate equality of in... |
| itg2mulclem 25691 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25692 | The integral of a nonnegat... |
| itg2splitlem 25693 | Lemma for ~ itg2split . (... |
| itg2split 25694 | The ` S.2 ` integral split... |
| itg2monolem1 25695 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25696 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25697 | Lemma for ~ itg2mono . (C... |
| itg2mono 25698 | The Monotone Convergence T... |
| itg2i1fseqle 25699 | Subject to the conditions ... |
| itg2i1fseq 25700 | Subject to the conditions ... |
| itg2i1fseq2 25701 | In an extension to the res... |
| itg2i1fseq3 25702 | Special case of ~ itg2i1fs... |
| itg2addlem 25703 | Lemma for ~ itg2add . (Co... |
| itg2add 25704 | The ` S.2 ` integral is li... |
| itg2gt0 25705 | If the function ` F ` is s... |
| itg2cnlem1 25706 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25707 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25708 | A sort of absolute continu... |
| ibllem 25709 | Conditioned equality theor... |
| isibl 25710 | The predicate " ` F ` is i... |
| isibl2 25711 | The predicate " ` F ` is i... |
| iblmbf 25712 | An integrable function is ... |
| iblitg 25713 | If a function is integrabl... |
| dfitg 25714 | Evaluate the class substit... |
| itgex 25715 | An integral is a set. (Co... |
| itgeq1f 25716 | Equality theorem for an in... |
| itgeq1fOLD 25717 | Obsolete version of ~ itge... |
| itgeq1 25718 | Equality theorem for an in... |
| nfitg1 25719 | Bound-variable hypothesis ... |
| nfitg 25720 | Bound-variable hypothesis ... |
| cbvitg 25721 | Change bound variable in a... |
| cbvitgv 25722 | Change bound variable in a... |
| itgeq2 25723 | Equality theorem for an in... |
| itgresr 25724 | The domain of an integral ... |
| itg0 25725 | The integral of anything o... |
| itgz 25726 | The integral of zero on an... |
| itgeq2dv 25727 | Equality theorem for an in... |
| itgmpt 25728 | Change bound variable in a... |
| itgcl 25729 | The integral of an integra... |
| itgvallem 25730 | Substitution lemma. (Cont... |
| itgvallem3 25731 | Lemma for ~ itgposval and ... |
| ibl0 25732 | The zero function is integ... |
| iblcnlem1 25733 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25734 | Expand out the universal q... |
| itgcnlem 25735 | Expand out the sum in ~ df... |
| iblrelem 25736 | Integrability of a real fu... |
| iblposlem 25737 | Lemma for ~ iblpos . (Con... |
| iblpos 25738 | Integrability of a nonnega... |
| iblre 25739 | Integrability of a real fu... |
| itgrevallem1 25740 | Lemma for ~ itgposval and ... |
| itgposval 25741 | The integral of a nonnegat... |
| itgreval 25742 | Decompose the integral of ... |
| itgrecl 25743 | Real closure of an integra... |
| iblcn 25744 | Integrability of a complex... |
| itgcnval 25745 | Decompose the integral of ... |
| itgre 25746 | Real part of an integral. ... |
| itgim 25747 | Imaginary part of an integ... |
| iblneg 25748 | The negative of an integra... |
| itgneg 25749 | Negation of an integral. ... |
| iblss 25750 | A subset of an integrable ... |
| iblss2 25751 | Change the domain of an in... |
| itgitg2 25752 | Transfer an integral using... |
| i1fibl 25753 | A simple function is integ... |
| itgitg1 25754 | Transfer an integral using... |
| itgle 25755 | Monotonicity of an integra... |
| itgge0 25756 | The integral of a positive... |
| itgss 25757 | Expand the set of an integ... |
| itgss2 25758 | Expand the set of an integ... |
| itgeqa 25759 | Approximate equality of in... |
| itgss3 25760 | Expand the set of an integ... |
| itgioo 25761 | Equality of integrals on o... |
| itgless 25762 | Expand the integral of a n... |
| iblconst 25763 | A constant function is int... |
| itgconst 25764 | Integral of a constant fun... |
| ibladdlem 25765 | Lemma for ~ ibladd . (Con... |
| ibladd 25766 | Add two integrals over the... |
| iblsub 25767 | Subtract two integrals ove... |
| itgaddlem1 25768 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25769 | Lemma for ~ itgadd . (Con... |
| itgadd 25770 | Add two integrals over the... |
| itgsub 25771 | Subtract two integrals ove... |
| itgfsum 25772 | Take a finite sum of integ... |
| iblabslem 25773 | Lemma for ~ iblabs . (Con... |
| iblabs 25774 | The absolute value of an i... |
| iblabsr 25775 | A measurable function is i... |
| iblmulc2 25776 | Multiply an integral by a ... |
| itgmulc2lem1 25777 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25778 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25779 | Multiply an integral by a ... |
| itgabs 25780 | The triangle inequality fo... |
| itgsplit 25781 | The ` S. ` integral splits... |
| itgspliticc 25782 | The ` S. ` integral splits... |
| itgsplitioo 25783 | The ` S. ` integral splits... |
| bddmulibl 25784 | A bounded function times a... |
| bddibl 25785 | A bounded function is inte... |
| cniccibl 25786 | A continuous function on a... |
| bddiblnc 25787 | Choice-free proof of ~ bdd... |
| cnicciblnc 25788 | Choice-free proof of ~ cni... |
| itggt0 25789 | The integral of a strictly... |
| itgcn 25790 | Transfer ~ itg2cn to the f... |
| ditgeq1 25793 | Equality theorem for the d... |
| ditgeq2 25794 | Equality theorem for the d... |
| ditgeq3 25795 | Equality theorem for the d... |
| ditgeq3dv 25796 | Equality theorem for the d... |
| ditgex 25797 | A directed integral is a s... |
| ditg0 25798 | Value of the directed inte... |
| cbvditg 25799 | Change bound variable in a... |
| cbvditgv 25800 | Change bound variable in a... |
| ditgpos 25801 | Value of the directed inte... |
| ditgneg 25802 | Value of the directed inte... |
| ditgcl 25803 | Closure of a directed inte... |
| ditgswap 25804 | Reverse a directed integra... |
| ditgsplitlem 25805 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25806 | This theorem is the raison... |
| reldv 25815 | The derivative function is... |
| limcvallem 25816 | Lemma for ~ ellimc . (Con... |
| limcfval 25817 | Value and set bounds on th... |
| ellimc 25818 | Value of the limit predica... |
| limcrcl 25819 | Reverse closure for the li... |
| limccl 25820 | Closure of the limit opera... |
| limcdif 25821 | It suffices to consider fu... |
| ellimc2 25822 | Write the definition of a ... |
| limcnlp 25823 | If ` B ` is not a limit po... |
| ellimc3 25824 | Write the epsilon-delta de... |
| limcflflem 25825 | Lemma for ~ limcflf . (Co... |
| limcflf 25826 | The limit operator can be ... |
| limcmo 25827 | If ` B ` is a limit point ... |
| limcmpt 25828 | Express the limit operator... |
| limcmpt2 25829 | Express the limit operator... |
| limcresi 25830 | Any limit of ` F ` is also... |
| limcres 25831 | If ` B ` is an interior po... |
| cnplimc 25832 | A function is continuous a... |
| cnlimc 25833 | ` F ` is a continuous func... |
| cnlimci 25834 | If ` F ` is a continuous f... |
| cnmptlimc 25835 | If ` F ` is a continuous f... |
| limccnp 25836 | If the limit of ` F ` at `... |
| limccnp2 25837 | The image of a convergent ... |
| limcco 25838 | Composition of two limits.... |
| limciun 25839 | A point is a limit of ` F ... |
| limcun 25840 | A point is a limit of ` F ... |
| dvlem 25841 | Closure for a difference q... |
| dvfval 25842 | Value and set bounds on th... |
| eldv 25843 | The differentiable predica... |
| dvcl 25844 | The derivative function ta... |
| dvbssntr 25845 | The set of differentiable ... |
| dvbss 25846 | The set of differentiable ... |
| dvbsss 25847 | The set of differentiable ... |
| perfdvf 25848 | The derivative is a functi... |
| recnprss 25849 | Both ` RR ` and ` CC ` are... |
| recnperf 25850 | Both ` RR ` and ` CC ` are... |
| dvfg 25851 | Explicitly write out the f... |
| dvf 25852 | The derivative is a functi... |
| dvfcn 25853 | The derivative is a functi... |
| dvreslem 25854 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25855 | Lemma for ~ dvres2 . (Con... |
| dvres 25856 | Restriction of a derivativ... |
| dvres2 25857 | Restriction of the base se... |
| dvres3 25858 | Restriction of a complex d... |
| dvres3a 25859 | Restriction of a complex d... |
| dvidlem 25860 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25861 | Derivative of a function r... |
| dvconst 25862 | Derivative of a constant f... |
| dvid 25863 | Derivative of the identity... |
| dvcnp 25864 | The difference quotient is... |
| dvcnp2 25865 | A function is continuous a... |
| dvcn 25866 | A differentiable function ... |
| dvnfval 25867 | Value of the iterated deri... |
| dvnff 25868 | The iterated derivative is... |
| dvn0 25869 | Zero times iterated deriva... |
| dvnp1 25870 | Successor iterated derivat... |
| dvn1 25871 | One times iterated derivat... |
| dvnf 25872 | The N-times derivative is ... |
| dvnbss 25873 | The set of N-times differe... |
| dvnadd 25874 | The ` N ` -th derivative o... |
| dvn2bss 25875 | An N-times differentiable ... |
| dvnres 25876 | Multiple derivative versio... |
| cpnfval 25877 | Condition for n-times cont... |
| fncpn 25878 | The ` C^n ` object is a fu... |
| elcpn 25879 | Condition for n-times cont... |
| cpnord 25880 | ` C^n ` conditions are ord... |
| cpncn 25881 | A ` C^n ` function is cont... |
| cpnres 25882 | The restriction of a ` C^n... |
| dvaddbr 25883 | The sum rule for derivativ... |
| dvmulbr 25884 | The product rule for deriv... |
| dvadd 25885 | The sum rule for derivativ... |
| dvmul 25886 | The product rule for deriv... |
| dvaddf 25887 | The sum rule for everywher... |
| dvmulf 25888 | The product rule for every... |
| dvcmul 25889 | The product rule when one ... |
| dvcmulf 25890 | The product rule when one ... |
| dvcobr 25891 | The chain rule for derivat... |
| dvco 25892 | The chain rule for derivat... |
| dvcof 25893 | The chain rule for everywh... |
| dvcjbr 25894 | The derivative of the conj... |
| dvcj 25895 | The derivative of the conj... |
| dvfre 25896 | The derivative of a real f... |
| dvnfre 25897 | The ` N ` -th derivative o... |
| dvexp 25898 | Derivative of a power func... |
| dvexp2 25899 | Derivative of an exponenti... |
| dvrec 25900 | Derivative of the reciproc... |
| dvmptres3 25901 | Function-builder for deriv... |
| dvmptid 25902 | Function-builder for deriv... |
| dvmptc 25903 | Function-builder for deriv... |
| dvmptcl 25904 | Closure lemma for ~ dvmptc... |
| dvmptadd 25905 | Function-builder for deriv... |
| dvmptmul 25906 | Function-builder for deriv... |
| dvmptres2 25907 | Function-builder for deriv... |
| dvmptres 25908 | Function-builder for deriv... |
| dvmptcmul 25909 | Function-builder for deriv... |
| dvmptdivc 25910 | Function-builder for deriv... |
| dvmptneg 25911 | Function-builder for deriv... |
| dvmptsub 25912 | Function-builder for deriv... |
| dvmptcj 25913 | Function-builder for deriv... |
| dvmptre 25914 | Function-builder for deriv... |
| dvmptim 25915 | Function-builder for deriv... |
| dvmptntr 25916 | Function-builder for deriv... |
| dvmptco 25917 | Function-builder for deriv... |
| dvrecg 25918 | Derivative of the reciproc... |
| dvmptdiv 25919 | Function-builder for deriv... |
| dvmptfsum 25920 | Function-builder for deriv... |
| dvcnvlem 25921 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25922 | A weak version of ~ dvcnvr... |
| dvexp3 25923 | Derivative of an exponenti... |
| dveflem 25924 | Derivative of the exponent... |
| dvef 25925 | Derivative of the exponent... |
| dvsincos 25926 | Derivative of the sine and... |
| dvsin 25927 | Derivative of the sine fun... |
| dvcos 25928 | Derivative of the cosine f... |
| dvferm1lem 25929 | Lemma for ~ dvferm . (Con... |
| dvferm1 25930 | One-sided version of ~ dvf... |
| dvferm2lem 25931 | Lemma for ~ dvferm . (Con... |
| dvferm2 25932 | One-sided version of ~ dvf... |
| dvferm 25933 | Fermat's theorem on statio... |
| rollelem 25934 | Lemma for ~ rolle . (Cont... |
| rolle 25935 | Rolle's theorem. If ` F `... |
| cmvth 25936 | Cauchy's Mean Value Theore... |
| cmvthOLD 25937 | Obsolete version of ~ cmvt... |
| mvth 25938 | The Mean Value Theorem. I... |
| dvlip 25939 | A function with derivative... |
| dvlipcn 25940 | A complex function with de... |
| dvlip2 25941 | Combine the results of ~ d... |
| c1liplem1 25942 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25943 | C^1 functions are Lipschit... |
| c1lip2 25944 | C^1 functions are Lipschit... |
| c1lip3 25945 | C^1 functions are Lipschit... |
| dveq0 25946 | If a continuous function h... |
| dv11cn 25947 | Two functions defined on a... |
| dvgt0lem1 25948 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25949 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25950 | A function on a closed int... |
| dvlt0 25951 | A function on a closed int... |
| dvge0 25952 | A function on a closed int... |
| dvle 25953 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25954 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25955 | Lemma for ~ dvivth . (Con... |
| dvivth 25956 | Darboux' theorem, or the i... |
| dvne0 25957 | A function on a closed int... |
| dvne0f1 25958 | A function on a closed int... |
| lhop1lem 25959 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25960 | L'Hôpital's Rule for... |
| lhop2 25961 | L'Hôpital's Rule for... |
| lhop 25962 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25963 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25964 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25965 | The derivative rule for in... |
| dvcvx 25966 | A real function with stric... |
| dvfsumle 25967 | Compare a finite sum to an... |
| dvfsumleOLD 25968 | Obsolete version of ~ dvfs... |
| dvfsumge 25969 | Compare a finite sum to an... |
| dvfsumabs 25970 | Compare a finite sum to an... |
| dvmptrecl 25971 | Real closure of a derivati... |
| dvfsumrlimf 25972 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25973 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25974 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2OLD 25975 | Obsolete version of ~ dvfs... |
| dvfsumlem3 25976 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25977 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25978 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25979 | Compare a finite sum to an... |
| dvfsumrlim2 25980 | Compare a finite sum to an... |
| dvfsumrlim3 25981 | Conjoin the statements of ... |
| dvfsum2 25982 | The reverse of ~ dvfsumrli... |
| ftc1lem1 25983 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 25984 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 25985 | The Fundamental Theorem of... |
| ftc1lem3 25986 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 25987 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 25988 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 25989 | Lemma for ~ ftc1 . (Contr... |
| ftc1 25990 | The Fundamental Theorem of... |
| ftc1cn 25991 | Strengthen the assumptions... |
| ftc2 25992 | The Fundamental Theorem of... |
| ftc2ditglem 25993 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 25994 | Directed integral analogue... |
| itgparts 25995 | Integration by parts. If ... |
| itgsubstlem 25996 | Lemma for ~ itgsubst . (C... |
| itgsubst 25997 | Integration by ` u ` -subs... |
| itgpowd 25998 | The integral of a monomial... |
| reldmmdeg 26003 | Multivariate degree is a b... |
| tdeglem1 26004 | Functionality of the total... |
| tdeglem3 26005 | Additivity of the total de... |
| tdeglem4 26006 | There is only one multi-in... |
| tdeglem2 26007 | Simplification of total de... |
| mdegfval 26008 | Value of the multivariate ... |
| mdegval 26009 | Value of the multivariate ... |
| mdegleb 26010 | Property of being of limit... |
| mdeglt 26011 | If there is an upper limit... |
| mdegldg 26012 | A nonzero polynomial has s... |
| mdegxrcl 26013 | Closure of polynomial degr... |
| mdegxrf 26014 | Functionality of polynomia... |
| mdegcl 26015 | Sharp closure for multivar... |
| mdeg0 26016 | Degree of the zero polynom... |
| mdegnn0cl 26017 | Degree of a nonzero polyno... |
| degltlem1 26018 | Theorem on arithmetic of e... |
| degltp1le 26019 | Theorem on arithmetic of e... |
| mdegaddle 26020 | The degree of a sum is at ... |
| mdegvscale 26021 | The degree of a scalar mul... |
| mdegvsca 26022 | The degree of a scalar mul... |
| mdegle0 26023 | A polynomial has nonpositi... |
| mdegmullem 26024 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 26025 | The multivariate degree of... |
| deg1fval 26026 | Relate univariate polynomi... |
| deg1xrf 26027 | Functionality of univariat... |
| deg1xrcl 26028 | Closure of univariate poly... |
| deg1cl 26029 | Sharp closure of univariat... |
| mdegpropd 26030 | Property deduction for pol... |
| deg1fvi 26031 | Univariate polynomial degr... |
| deg1propd 26032 | Property deduction for pol... |
| deg1z 26033 | Degree of the zero univari... |
| deg1nn0cl 26034 | Degree of a nonzero univar... |
| deg1n0ima 26035 | Degree image of a set of p... |
| deg1nn0clb 26036 | A polynomial is nonzero if... |
| deg1lt0 26037 | A polynomial is zero iff i... |
| deg1ldg 26038 | A nonzero univariate polyn... |
| deg1ldgn 26039 | An index at which a polyno... |
| deg1ldgdomn 26040 | A nonzero univariate polyn... |
| deg1leb 26041 | Property of being of limit... |
| deg1val 26042 | Value of the univariate de... |
| deg1lt 26043 | If the degree of a univari... |
| deg1ge 26044 | Conversely, a nonzero coef... |
| coe1mul3 26045 | The coefficient vector of ... |
| coe1mul4 26046 | Value of the "leading" coe... |
| deg1addle 26047 | The degree of a sum is at ... |
| deg1addle2 26048 | If both factors have degre... |
| deg1add 26049 | Exact degree of a sum of t... |
| deg1vscale 26050 | The degree of a scalar tim... |
| deg1vsca 26051 | The degree of a scalar tim... |
| deg1invg 26052 | The degree of the negated ... |
| deg1suble 26053 | The degree of a difference... |
| deg1sub 26054 | Exact degree of a differen... |
| deg1mulle2 26055 | Produce a bound on the pro... |
| deg1sublt 26056 | Subtraction of two polynom... |
| deg1le0 26057 | A polynomial has nonpositi... |
| deg1sclle 26058 | A scalar polynomial has no... |
| deg1scl 26059 | A nonzero scalar polynomia... |
| deg1mul2 26060 | Degree of multiplication o... |
| deg1mul 26061 | Degree of multiplication o... |
| deg1mul3 26062 | Degree of multiplication o... |
| deg1mul3le 26063 | Degree of multiplication o... |
| deg1tmle 26064 | Limiting degree of a polyn... |
| deg1tm 26065 | Exact degree of a polynomi... |
| deg1pwle 26066 | Limiting degree of a varia... |
| deg1pw 26067 | Exact degree of a variable... |
| ply1nz 26068 | Univariate polynomials ove... |
| ply1nzb 26069 | Univariate polynomials are... |
| ply1domn 26070 | Corollary of ~ deg1mul2 : ... |
| ply1idom 26071 | The ring of univariate pol... |
| ply1divmo 26082 | Uniqueness of a quotient i... |
| ply1divex 26083 | Lemma for ~ ply1divalg : e... |
| ply1divalg 26084 | The division algorithm for... |
| ply1divalg2 26085 | Reverse the order of multi... |
| uc1pval 26086 | Value of the set of unitic... |
| isuc1p 26087 | Being a unitic polynomial.... |
| mon1pval 26088 | Value of the set of monic ... |
| ismon1p 26089 | Being a monic polynomial. ... |
| uc1pcl 26090 | Unitic polynomials are pol... |
| mon1pcl 26091 | Monic polynomials are poly... |
| uc1pn0 26092 | Unitic polynomials are not... |
| mon1pn0 26093 | Monic polynomials are not ... |
| uc1pdeg 26094 | Unitic polynomials have no... |
| uc1pldg 26095 | Unitic polynomials have un... |
| mon1pldg 26096 | Unitic polynomials have on... |
| mon1puc1p 26097 | Monic polynomials are unit... |
| uc1pmon1p 26098 | Make a unitic polynomial m... |
| deg1submon1p 26099 | The difference of two moni... |
| mon1pid 26100 | Monicity and degree of the... |
| q1pval 26101 | Value of the univariate po... |
| q1peqb 26102 | Characterizing property of... |
| q1pcl 26103 | Closure of the quotient by... |
| r1pval 26104 | Value of the polynomial re... |
| r1pcl 26105 | Closure of remainder follo... |
| r1pdeglt 26106 | The remainder has a degree... |
| r1pid 26107 | Express the original polyn... |
| r1pid2 26108 | Identity law for polynomia... |
| dvdsq1p 26109 | Divisibility in a polynomi... |
| dvdsr1p 26110 | Divisibility in a polynomi... |
| ply1remlem 26111 | A term of the form ` x - N... |
| ply1rem 26112 | The polynomial remainder t... |
| facth1 26113 | The factor theorem and its... |
| fta1glem1 26114 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 26115 | Lemma for ~ fta1g . (Cont... |
| fta1g 26116 | The one-sided fundamental ... |
| fta1blem 26117 | Lemma for ~ fta1b . (Cont... |
| fta1b 26118 | The assumption that ` R ` ... |
| idomrootle 26119 | No element of an integral ... |
| drnguc1p 26120 | Over a division ring, all ... |
| ig1peu 26121 | There is a unique monic po... |
| ig1pval 26122 | Substitutions for the poly... |
| ig1pval2 26123 | Generator of the zero idea... |
| ig1pval3 26124 | Characterizing properties ... |
| ig1pcl 26125 | The monic generator of an ... |
| ig1pdvds 26126 | The monic generator of an ... |
| ig1prsp 26127 | Any ideal of polynomials o... |
| ply1lpir 26128 | The ring of polynomials ov... |
| ply1pid 26129 | The polynomials over a fie... |
| plyco0 26138 | Two ways to say that a fun... |
| plyval 26139 | Value of the polynomial se... |
| plybss 26140 | Reverse closure of the par... |
| elply 26141 | Definition of a polynomial... |
| elply2 26142 | The coefficient function c... |
| plyun0 26143 | The set of polynomials is ... |
| plyf 26144 | A polynomial is a function... |
| plyss 26145 | The polynomial set functio... |
| plyssc 26146 | Every polynomial ring is c... |
| elplyr 26147 | Sufficient condition for e... |
| elplyd 26148 | Sufficient condition for e... |
| ply1termlem 26149 | Lemma for ~ ply1term . (C... |
| ply1term 26150 | A one-term polynomial. (C... |
| plypow 26151 | A power is a polynomial. ... |
| plyconst 26152 | A constant function is a p... |
| ne0p 26153 | A test to show that a poly... |
| ply0 26154 | The zero function is a pol... |
| plyid 26155 | The identity function is a... |
| plyeq0lem 26156 | Lemma for ~ plyeq0 . If `... |
| plyeq0 26157 | If a polynomial is zero at... |
| plypf1 26158 | Write the set of complex p... |
| plyaddlem1 26159 | Derive the coefficient fun... |
| plymullem1 26160 | Derive the coefficient fun... |
| plyaddlem 26161 | Lemma for ~ plyadd . (Con... |
| plymullem 26162 | Lemma for ~ plymul . (Con... |
| plyadd 26163 | The sum of two polynomials... |
| plymul 26164 | The product of two polynom... |
| plysub 26165 | The difference of two poly... |
| plyaddcl 26166 | The sum of two polynomials... |
| plymulcl 26167 | The product of two polynom... |
| plysubcl 26168 | The difference of two poly... |
| coeval 26169 | Value of the coefficient f... |
| coeeulem 26170 | Lemma for ~ coeeu . (Cont... |
| coeeu 26171 | Uniqueness of the coeffici... |
| coelem 26172 | Lemma for properties of th... |
| coeeq 26173 | If ` A ` satisfies the pro... |
| dgrval 26174 | Value of the degree functi... |
| dgrlem 26175 | Lemma for ~ dgrcl and simi... |
| coef 26176 | The domain and codomain of... |
| coef2 26177 | The domain and codomain of... |
| coef3 26178 | The domain and codomain of... |
| dgrcl 26179 | The degree of any polynomi... |
| dgrub 26180 | If the ` M ` -th coefficie... |
| dgrub2 26181 | All the coefficients above... |
| dgrlb 26182 | If all the coefficients ab... |
| coeidlem 26183 | Lemma for ~ coeid . (Cont... |
| coeid 26184 | Reconstruct a polynomial a... |
| coeid2 26185 | Reconstruct a polynomial a... |
| coeid3 26186 | Reconstruct a polynomial a... |
| plyco 26187 | The composition of two pol... |
| coeeq2 26188 | Compute the coefficient fu... |
| dgrle 26189 | Given an explicit expressi... |
| dgreq 26190 | If the highest term in a p... |
| 0dgr 26191 | A constant function has de... |
| 0dgrb 26192 | A function has degree zero... |
| dgrnznn 26193 | A nonzero polynomial with ... |
| coefv0 26194 | The result of evaluating a... |
| coeaddlem 26195 | Lemma for ~ coeadd and ~ d... |
| coemullem 26196 | Lemma for ~ coemul and ~ d... |
| coeadd 26197 | The coefficient function o... |
| coemul 26198 | A coefficient of a product... |
| coe11 26199 | The coefficient function i... |
| coemulhi 26200 | The leading coefficient of... |
| coemulc 26201 | The coefficient function i... |
| coe0 26202 | The coefficients of the ze... |
| coesub 26203 | The coefficient function o... |
| coe1termlem 26204 | The coefficient function o... |
| coe1term 26205 | The coefficient function o... |
| dgr1term 26206 | The degree of a monomial. ... |
| plycn 26207 | A polynomial is a continuo... |
| dgr0 26208 | The degree of the zero pol... |
| coeidp 26209 | The coefficients of the id... |
| dgrid 26210 | The degree of the identity... |
| dgreq0 26211 | The leading coefficient of... |
| dgrlt 26212 | Two ways to say that the d... |
| dgradd 26213 | The degree of a sum of pol... |
| dgradd2 26214 | The degree of a sum of pol... |
| dgrmul2 26215 | The degree of a product of... |
| dgrmul 26216 | The degree of a product of... |
| dgrmulc 26217 | Scalar multiplication by a... |
| dgrsub 26218 | The degree of a difference... |
| dgrcolem1 26219 | The degree of a compositio... |
| dgrcolem2 26220 | Lemma for ~ dgrco . (Cont... |
| dgrco 26221 | The degree of a compositio... |
| plycjlem 26222 | Lemma for ~ plycj and ~ co... |
| plycj 26223 | The double conjugation of ... |
| coecj 26224 | Double conjugation of a po... |
| plycjOLD 26225 | Obsolete version of ~ plyc... |
| coecjOLD 26226 | Obsolete version of ~ coec... |
| plyrecj 26227 | A polynomial with real coe... |
| plymul0or 26228 | Polynomial multiplication ... |
| ofmulrt 26229 | The set of roots of a prod... |
| plyreres 26230 | Real-coefficient polynomia... |
| dvply1 26231 | Derivative of a polynomial... |
| dvply2g 26232 | The derivative of a polyno... |
| dvply2gOLD 26233 | Obsolete version of ~ dvpl... |
| dvply2 26234 | The derivative of a polyno... |
| dvnply2 26235 | Polynomials have polynomia... |
| dvnply 26236 | Polynomials have polynomia... |
| plycpn 26237 | Polynomials are smooth. (... |
| quotval 26240 | Value of the quotient func... |
| plydivlem1 26241 | Lemma for ~ plydivalg . (... |
| plydivlem2 26242 | Lemma for ~ plydivalg . (... |
| plydivlem3 26243 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26244 | Lemma for ~ plydivex . In... |
| plydivex 26245 | Lemma for ~ plydivalg . (... |
| plydiveu 26246 | Lemma for ~ plydivalg . (... |
| plydivalg 26247 | The division algorithm on ... |
| quotlem 26248 | Lemma for properties of th... |
| quotcl 26249 | The quotient of two polyno... |
| quotcl2 26250 | Closure of the quotient fu... |
| quotdgr 26251 | Remainder property of the ... |
| plyremlem 26252 | Closure of a linear factor... |
| plyrem 26253 | The polynomial remainder t... |
| facth 26254 | The factor theorem. If a ... |
| fta1lem 26255 | Lemma for ~ fta1 . (Contr... |
| fta1 26256 | The easy direction of the ... |
| quotcan 26257 | Exact division with a mult... |
| vieta1lem1 26258 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26259 | Lemma for ~ vieta1 : induc... |
| vieta1 26260 | The first-order Vieta's fo... |
| plyexmo 26261 | An infinite set of values ... |
| elaa 26264 | Elementhood in the set of ... |
| aacn 26265 | An algebraic number is a c... |
| aasscn 26266 | The algebraic numbers are ... |
| elqaalem1 26267 | Lemma for ~ elqaa . The f... |
| elqaalem2 26268 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26269 | Lemma for ~ elqaa . (Cont... |
| elqaa 26270 | The set of numbers generat... |
| qaa 26271 | Every rational number is a... |
| qssaa 26272 | The rational numbers are c... |
| iaa 26273 | The imaginary unit is alge... |
| aareccl 26274 | The reciprocal of an algeb... |
| aacjcl 26275 | The conjugate of an algebr... |
| aannenlem1 26276 | Lemma for ~ aannen . (Con... |
| aannenlem2 26277 | Lemma for ~ aannen . (Con... |
| aannenlem3 26278 | The algebraic numbers are ... |
| aannen 26279 | The algebraic numbers are ... |
| aalioulem1 26280 | Lemma for ~ aaliou . An i... |
| aalioulem2 26281 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26282 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26283 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26284 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26285 | Lemma for ~ aaliou . (Con... |
| aaliou 26286 | Liouville's theorem on dio... |
| geolim3 26287 | Geometric series convergen... |
| aaliou2 26288 | Liouville's approximation ... |
| aaliou2b 26289 | Liouville's approximation ... |
| aaliou3lem1 26290 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26291 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26292 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26293 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26294 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26295 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26296 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26297 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26298 | Example of a "Liouville nu... |
| aaliou3 26299 | Example of a "Liouville nu... |
| taylfvallem1 26304 | Lemma for ~ taylfval . (C... |
| taylfvallem 26305 | Lemma for ~ taylfval . (C... |
| taylfval 26306 | Define the Taylor polynomi... |
| eltayl 26307 | Value of the Taylor series... |
| taylf 26308 | The Taylor series defines ... |
| tayl0 26309 | The Taylor series is alway... |
| taylplem1 26310 | Lemma for ~ taylpfval and ... |
| taylplem2 26311 | Lemma for ~ taylpfval and ... |
| taylpfval 26312 | Define the Taylor polynomi... |
| taylpf 26313 | The Taylor polynomial is a... |
| taylpval 26314 | Value of the Taylor polyno... |
| taylply2 26315 | The Taylor polynomial is a... |
| taylply2OLD 26316 | Obsolete version of ~ tayl... |
| taylply 26317 | The Taylor polynomial is a... |
| dvtaylp 26318 | The derivative of the Tayl... |
| dvntaylp 26319 | The ` M ` -th derivative o... |
| dvntaylp0 26320 | The first ` N ` derivative... |
| taylthlem1 26321 | Lemma for ~ taylth . This... |
| taylthlem2 26322 | Lemma for ~ taylth . (Con... |
| taylthlem2OLD 26323 | Obsolete version of ~ tayl... |
| taylth 26324 | Taylor's theorem. The Tay... |
| ulmrel 26327 | The uniform limit relation... |
| ulmscl 26328 | Closure of the base set in... |
| ulmval 26329 | Express the predicate: Th... |
| ulmcl 26330 | Closure of a uniform limit... |
| ulmf 26331 | Closure of a uniform limit... |
| ulmpm 26332 | Closure of a uniform limit... |
| ulmf2 26333 | Closure of a uniform limit... |
| ulm2 26334 | Simplify ~ ulmval when ` F... |
| ulmi 26335 | The uniform limit property... |
| ulmclm 26336 | A uniform limit of functio... |
| ulmres 26337 | A sequence of functions co... |
| ulmshftlem 26338 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26339 | A sequence of functions co... |
| ulm0 26340 | Every function converges u... |
| ulmuni 26341 | A sequence of functions un... |
| ulmdm 26342 | Two ways to express that a... |
| ulmcaulem 26343 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26344 | A sequence of functions co... |
| ulmcau2 26345 | A sequence of functions co... |
| ulmss 26346 | A uniform limit of functio... |
| ulmbdd 26347 | A uniform limit of bounded... |
| ulmcn 26348 | A uniform limit of continu... |
| ulmdvlem1 26349 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26350 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26351 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26352 | If ` F ` is a sequence of ... |
| mtest 26353 | The Weierstrass M-test. I... |
| mtestbdd 26354 | Given the hypotheses of th... |
| mbfulm 26355 | A uniform limit of measura... |
| iblulm 26356 | A uniform limit of integra... |
| itgulm 26357 | A uniform limit of integra... |
| itgulm2 26358 | A uniform limit of integra... |
| pserval 26359 | Value of the function ` G ... |
| pserval2 26360 | Value of the function ` G ... |
| psergf 26361 | The sequence of terms in t... |
| radcnvlem1 26362 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26363 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26364 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26365 | Zero is always a convergen... |
| radcnvcl 26366 | The radius of convergence ... |
| radcnvlt1 26367 | If ` X ` is within the ope... |
| radcnvlt2 26368 | If ` X ` is within the ope... |
| radcnvle 26369 | If ` X ` is a convergent p... |
| dvradcnv 26370 | The radius of convergence ... |
| pserulm 26371 | If ` S ` is a region conta... |
| psercn2 26372 | Since by ~ pserulm the ser... |
| psercn2OLD 26373 | Obsolete version of ~ pser... |
| psercnlem2 26374 | Lemma for ~ psercn . (Con... |
| psercnlem1 26375 | Lemma for ~ psercn . (Con... |
| psercn 26376 | An infinite series converg... |
| pserdvlem1 26377 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26378 | Lemma for ~ pserdv . (Con... |
| pserdv 26379 | The derivative of a power ... |
| pserdv2 26380 | The derivative of a power ... |
| abelthlem1 26381 | Lemma for ~ abelth . (Con... |
| abelthlem2 26382 | Lemma for ~ abelth . The ... |
| abelthlem3 26383 | Lemma for ~ abelth . (Con... |
| abelthlem4 26384 | Lemma for ~ abelth . (Con... |
| abelthlem5 26385 | Lemma for ~ abelth . (Con... |
| abelthlem6 26386 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26387 | Lemma for ~ abelth . (Con... |
| abelthlem7 26388 | Lemma for ~ abelth . (Con... |
| abelthlem8 26389 | Lemma for ~ abelth . (Con... |
| abelthlem9 26390 | Lemma for ~ abelth . By a... |
| abelth 26391 | Abel's theorem. If the po... |
| abelth2 26392 | Abel's theorem, restricted... |
| efcn 26393 | The exponential function i... |
| sincn 26394 | Sine is continuous. (Cont... |
| coscn 26395 | Cosine is continuous. (Co... |
| reeff1olem 26396 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26397 | The real exponential funct... |
| reefiso 26398 | The exponential function o... |
| efcvx 26399 | The exponential function o... |
| reefgim 26400 | The exponential function i... |
| pilem1 26401 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26402 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26403 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26404 | ` _pi ` is between 2 and 4... |
| sinpi 26405 | The sine of ` _pi ` is 0. ... |
| pire 26406 | ` _pi ` is a real number. ... |
| picn 26407 | ` _pi ` is a complex numbe... |
| pipos 26408 | ` _pi ` is positive. (Con... |
| pine0 26409 | ` _pi ` is nonzero. (Cont... |
| pirp 26410 | ` _pi ` is a positive real... |
| negpicn 26411 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26412 | Lemma for ~ sinhalfpi and ... |
| halfpire 26413 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26414 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26415 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26416 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26417 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26418 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26419 | The cosine of ` -u _pi / 2... |
| efhalfpi 26420 | The exponential of ` _i _p... |
| cospi 26421 | The cosine of ` _pi ` is `... |
| efipi 26422 | The exponential of ` _i x.... |
| eulerid 26423 | Euler's identity. (Contri... |
| sin2pi 26424 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26425 | The cosine of ` 2 _pi ` is... |
| ef2pi 26426 | The exponential of ` 2 _pi... |
| ef2kpi 26427 | If ` K ` is an integer, th... |
| efper 26428 | The exponential function i... |
| sinperlem 26429 | Lemma for ~ sinper and ~ c... |
| sinper 26430 | The sine function is perio... |
| cosper 26431 | The cosine function is per... |
| sin2kpi 26432 | If ` K ` is an integer, th... |
| cos2kpi 26433 | If ` K ` is an integer, th... |
| sin2pim 26434 | Sine of a number subtracte... |
| cos2pim 26435 | Cosine of a number subtrac... |
| sinmpi 26436 | Sine of a number less ` _p... |
| cosmpi 26437 | Cosine of a number less ` ... |
| sinppi 26438 | Sine of a number plus ` _p... |
| cosppi 26439 | Cosine of a number plus ` ... |
| efimpi 26440 | The exponential function a... |
| sinhalfpip 26441 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26442 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26443 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26444 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26445 | Ptolemy's Theorem. This t... |
| sincosq1lem 26446 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26447 | The signs of the sine and ... |
| sincosq2sgn 26448 | The signs of the sine and ... |
| sincosq3sgn 26449 | The signs of the sine and ... |
| sincosq4sgn 26450 | The signs of the sine and ... |
| coseq00topi 26451 | Location of the zeroes of ... |
| coseq0negpitopi 26452 | Location of the zeroes of ... |
| tanrpcl 26453 | Positive real closure of t... |
| tangtx 26454 | The tangent function is gr... |
| tanabsge 26455 | The tangent function is gr... |
| sinq12gt0 26456 | The sine of a number stric... |
| sinq12ge0 26457 | The sine of a number betwe... |
| sinq34lt0t 26458 | The sine of a number stric... |
| cosq14gt0 26459 | The cosine of a number str... |
| cosq14ge0 26460 | The cosine of a number bet... |
| sincosq1eq 26461 | Complementarity of the sin... |
| sincos4thpi 26462 | The sine and cosine of ` _... |
| tan4thpi 26463 | The tangent of ` _pi / 4 `... |
| tan4thpiOLD 26464 | Obsolete version of ~ tan4... |
| sincos6thpi 26465 | The sine and cosine of ` _... |
| sincos3rdpi 26466 | The sine and cosine of ` _... |
| pigt3 26467 | ` _pi ` is greater than 3.... |
| pige3 26468 | ` _pi ` is greater than or... |
| pige3ALT 26469 | Alternate proof of ~ pige3... |
| abssinper 26470 | The absolute value of sine... |
| sinkpi 26471 | The sine of an integer mul... |
| coskpi 26472 | The absolute value of the ... |
| sineq0 26473 | A complex number whose sin... |
| coseq1 26474 | A complex number whose cos... |
| cos02pilt1 26475 | Cosine is less than one be... |
| cosq34lt1 26476 | Cosine is less than one in... |
| efeq1 26477 | A complex number whose exp... |
| cosne0 26478 | The cosine function has no... |
| cosordlem 26479 | Lemma for ~ cosord . (Con... |
| cosord 26480 | Cosine is decreasing over ... |
| cos0pilt1 26481 | Cosine is between minus on... |
| cos11 26482 | Cosine is one-to-one over ... |
| sinord 26483 | Sine is increasing over th... |
| recosf1o 26484 | The cosine function is a b... |
| resinf1o 26485 | The sine function is a bij... |
| tanord1 26486 | The tangent function is st... |
| tanord 26487 | The tangent function is st... |
| tanregt0 26488 | The real part of the tange... |
| negpitopissre 26489 | The interval ` ( -u _pi (,... |
| efgh 26490 | The exponential function o... |
| efif1olem1 26491 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26492 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26493 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26494 | The exponential function o... |
| efif1o 26495 | The exponential function o... |
| efifo 26496 | The exponential function o... |
| eff1olem 26497 | The exponential function m... |
| eff1o 26498 | The exponential function m... |
| efabl 26499 | The image of a subgroup of... |
| efsubm 26500 | The image of a subgroup of... |
| circgrp 26501 | The circle group ` T ` is ... |
| circsubm 26502 | The circle group ` T ` is ... |
| logrn 26507 | The range of the natural l... |
| ellogrn 26508 | Write out the property ` A... |
| dflog2 26509 | The natural logarithm func... |
| relogrn 26510 | The range of the natural l... |
| logrncn 26511 | The range of the natural l... |
| eff1o2 26512 | The exponential function r... |
| logf1o 26513 | The natural logarithm func... |
| dfrelog 26514 | The natural logarithm func... |
| relogf1o 26515 | The natural logarithm func... |
| logrncl 26516 | Closure of the natural log... |
| logcl 26517 | Closure of the natural log... |
| logimcl 26518 | Closure of the imaginary p... |
| logcld 26519 | The logarithm of a nonzero... |
| logimcld 26520 | The imaginary part of the ... |
| logimclad 26521 | The imaginary part of the ... |
| abslogimle 26522 | The imaginary part of the ... |
| logrnaddcl 26523 | The range of the natural l... |
| relogcl 26524 | Closure of the natural log... |
| eflog 26525 | Relationship between the n... |
| logeq0im1 26526 | If the logarithm of a numb... |
| logccne0 26527 | The logarithm isn't 0 if i... |
| logne0 26528 | Logarithm of a non-1 posit... |
| reeflog 26529 | Relationship between the n... |
| logef 26530 | Relationship between the n... |
| relogef 26531 | Relationship between the n... |
| logeftb 26532 | Relationship between the n... |
| relogeftb 26533 | Relationship between the n... |
| log1 26534 | The natural logarithm of `... |
| loge 26535 | The natural logarithm of `... |
| logi 26536 | The natural logarithm of `... |
| logneg 26537 | The natural logarithm of a... |
| logm1 26538 | The natural logarithm of n... |
| lognegb 26539 | If a number has imaginary ... |
| relogoprlem 26540 | Lemma for ~ relogmul and ~... |
| relogmul 26541 | The natural logarithm of t... |
| relogdiv 26542 | The natural logarithm of t... |
| explog 26543 | Exponentiation of a nonzer... |
| reexplog 26544 | Exponentiation of a positi... |
| relogexp 26545 | The natural logarithm of p... |
| relog 26546 | Real part of a logarithm. ... |
| relogiso 26547 | The natural logarithm func... |
| reloggim 26548 | The natural logarithm is a... |
| logltb 26549 | The natural logarithm func... |
| logfac 26550 | The logarithm of a factori... |
| eflogeq 26551 | Solve an equation involvin... |
| logleb 26552 | Natural logarithm preserve... |
| rplogcl 26553 | Closure of the logarithm f... |
| logge0 26554 | The logarithm of a number ... |
| logcj 26555 | The natural logarithm dist... |
| efiarg 26556 | The exponential of the "ar... |
| cosargd 26557 | The cosine of the argument... |
| cosarg0d 26558 | The cosine of the argument... |
| argregt0 26559 | Closure of the argument of... |
| argrege0 26560 | Closure of the argument of... |
| argimgt0 26561 | Closure of the argument of... |
| argimlt0 26562 | Closure of the argument of... |
| logimul 26563 | Multiplying a number by ` ... |
| logneg2 26564 | The logarithm of the negat... |
| logmul2 26565 | Generalization of ~ relogm... |
| logdiv2 26566 | Generalization of ~ relogd... |
| abslogle 26567 | Bound on the magnitude of ... |
| tanarg 26568 | The basic relation between... |
| logdivlti 26569 | The ` log x / x ` function... |
| logdivlt 26570 | The ` log x / x ` function... |
| logdivle 26571 | The ` log x / x ` function... |
| relogcld 26572 | Closure of the natural log... |
| reeflogd 26573 | Relationship between the n... |
| relogmuld 26574 | The natural logarithm of t... |
| relogdivd 26575 | The natural logarithm of t... |
| logled 26576 | Natural logarithm preserve... |
| relogefd 26577 | Relationship between the n... |
| rplogcld 26578 | Closure of the logarithm f... |
| logge0d 26579 | The logarithm of a number ... |
| logge0b 26580 | The logarithm of a number ... |
| loggt0b 26581 | The logarithm of a number ... |
| logle1b 26582 | The logarithm of a number ... |
| loglt1b 26583 | The logarithm of a number ... |
| divlogrlim 26584 | The inverse logarithm func... |
| logno1 26585 | The logarithm function is ... |
| dvrelog 26586 | The derivative of the real... |
| relogcn 26587 | The real logarithm functio... |
| ellogdm 26588 | Elementhood in the "contin... |
| logdmn0 26589 | A number in the continuous... |
| logdmnrp 26590 | A number in the continuous... |
| logdmss 26591 | The continuity domain of `... |
| logcnlem2 26592 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26593 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26594 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26595 | Lemma for ~ logcn . (Cont... |
| logcn 26596 | The logarithm function is ... |
| dvloglem 26597 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26598 | The "continuous domain" of... |
| logf1o2 26599 | The logarithm maps its con... |
| dvlog 26600 | The derivative of the comp... |
| dvlog2lem 26601 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26602 | The derivative of the comp... |
| advlog 26603 | The antiderivative of the ... |
| advlogexp 26604 | The antiderivative of a po... |
| efopnlem1 26605 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26606 | Lemma for ~ efopn . (Cont... |
| efopn 26607 | The exponential map is an ... |
| logtayllem 26608 | Lemma for ~ logtayl . (Co... |
| logtayl 26609 | The Taylor series for ` -u... |
| logtaylsum 26610 | The Taylor series for ` -u... |
| logtayl2 26611 | Power series expression fo... |
| logccv 26612 | The natural logarithm func... |
| cxpval 26613 | Value of the complex power... |
| cxpef 26614 | Value of the complex power... |
| 0cxp 26615 | Value of the complex power... |
| cxpexpz 26616 | Relate the complex power f... |
| cxpexp 26617 | Relate the complex power f... |
| logcxp 26618 | Logarithm of a complex pow... |
| cxp0 26619 | Value of the complex power... |
| cxp1 26620 | Value of the complex power... |
| 1cxp 26621 | Value of the complex power... |
| ecxp 26622 | Write the exponential func... |
| cxpcl 26623 | Closure of the complex pow... |
| recxpcl 26624 | Real closure of the comple... |
| rpcxpcl 26625 | Positive real closure of t... |
| cxpne0 26626 | Complex exponentiation is ... |
| cxpeq0 26627 | Complex exponentiation is ... |
| cxpadd 26628 | Sum of exponents law for c... |
| cxpp1 26629 | Value of a nonzero complex... |
| cxpneg 26630 | Value of a complex number ... |
| cxpsub 26631 | Exponent subtraction law f... |
| cxpge0 26632 | Nonnegative exponentiation... |
| mulcxplem 26633 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26634 | Complex exponentiation of ... |
| cxprec 26635 | Complex exponentiation of ... |
| divcxp 26636 | Complex exponentiation of ... |
| cxpmul 26637 | Product of exponents law f... |
| cxpmul2 26638 | Product of exponents law f... |
| cxproot 26639 | The complex power function... |
| cxpmul2z 26640 | Generalize ~ cxpmul2 to ne... |
| abscxp 26641 | Absolute value of a power,... |
| abscxp2 26642 | Absolute value of a power,... |
| cxplt 26643 | Ordering property for comp... |
| cxple 26644 | Ordering property for comp... |
| cxplea 26645 | Ordering property for comp... |
| cxple2 26646 | Ordering property for comp... |
| cxplt2 26647 | Ordering property for comp... |
| cxple2a 26648 | Ordering property for comp... |
| cxplt3 26649 | Ordering property for comp... |
| cxple3 26650 | Ordering property for comp... |
| cxpsqrtlem 26651 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26652 | The complex exponential fu... |
| logsqrt 26653 | Logarithm of a square root... |
| cxp0d 26654 | Value of the complex power... |
| cxp1d 26655 | Value of the complex power... |
| 1cxpd 26656 | Value of the complex power... |
| cxpcld 26657 | Closure of the complex pow... |
| cxpmul2d 26658 | Product of exponents law f... |
| 0cxpd 26659 | Value of the complex power... |
| cxpexpzd 26660 | Relate the complex power f... |
| cxpefd 26661 | Value of the complex power... |
| cxpne0d 26662 | Complex exponentiation is ... |
| cxpp1d 26663 | Value of a nonzero complex... |
| cxpnegd 26664 | Value of a complex number ... |
| cxpmul2zd 26665 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26666 | Sum of exponents law for c... |
| cxpsubd 26667 | Exponent subtraction law f... |
| cxpltd 26668 | Ordering property for comp... |
| cxpled 26669 | Ordering property for comp... |
| cxplead 26670 | Ordering property for comp... |
| divcxpd 26671 | Complex exponentiation of ... |
| recxpcld 26672 | Positive real closure of t... |
| cxpge0d 26673 | Nonnegative exponentiation... |
| cxple2ad 26674 | Ordering property for comp... |
| cxplt2d 26675 | Ordering property for comp... |
| cxple2d 26676 | Ordering property for comp... |
| mulcxpd 26677 | Complex exponentiation of ... |
| recxpf1lem 26678 | Complex exponentiation on ... |
| cxpsqrtth 26679 | Square root theorem over t... |
| 2irrexpq 26680 | There exist irrational num... |
| cxprecd 26681 | Complex exponentiation of ... |
| rpcxpcld 26682 | Positive real closure of t... |
| logcxpd 26683 | Logarithm of a complex pow... |
| cxplt3d 26684 | Ordering property for comp... |
| cxple3d 26685 | Ordering property for comp... |
| cxpmuld 26686 | Product of exponents law f... |
| cxpgt0d 26687 | A positive real raised to ... |
| cxpcom 26688 | Commutative law for real e... |
| dvcxp1 26689 | The derivative of a comple... |
| dvcxp2 26690 | The derivative of a comple... |
| dvsqrt 26691 | The derivative of the real... |
| dvcncxp1 26692 | Derivative of complex powe... |
| dvcnsqrt 26693 | Derivative of square root ... |
| cxpcn 26694 | Domain of continuity of th... |
| cxpcnOLD 26695 | Obsolete version of ~ cxpc... |
| cxpcn2 26696 | Continuity of the complex ... |
| cxpcn3lem 26697 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26698 | Extend continuity of the c... |
| resqrtcn 26699 | Continuity of the real squ... |
| sqrtcn 26700 | Continuity of the square r... |
| cxpaddlelem 26701 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26702 | Ordering property for comp... |
| abscxpbnd 26703 | Bound on the absolute valu... |
| root1id 26704 | Property of an ` N ` -th r... |
| root1eq1 26705 | The only powers of an ` N ... |
| root1cj 26706 | Within the ` N ` -th roots... |
| cxpeq 26707 | Solve an equation involvin... |
| zrtelqelz 26708 | If the ` N ` -th root of a... |
| zrtdvds 26709 | A positive integer root di... |
| rtprmirr 26710 | The root of a prime number... |
| loglesqrt 26711 | An upper bound on the loga... |
| logreclem 26712 | Symmetry of the natural lo... |
| logrec 26713 | Logarithm of a reciprocal ... |
| logbval 26716 | Define the value of the ` ... |
| logbcl 26717 | General logarithm closure.... |
| logbid1 26718 | General logarithm is 1 whe... |
| logb1 26719 | The logarithm of ` 1 ` to ... |
| elogb 26720 | The general logarithm of a... |
| logbchbase 26721 | Change of base for logarit... |
| relogbval 26722 | Value of the general logar... |
| relogbcl 26723 | Closure of the general log... |
| relogbzcl 26724 | Closure of the general log... |
| relogbreexp 26725 | Power law for the general ... |
| relogbzexp 26726 | Power law for the general ... |
| relogbmul 26727 | The logarithm of the produ... |
| relogbmulexp 26728 | The logarithm of the produ... |
| relogbdiv 26729 | The logarithm of the quoti... |
| relogbexp 26730 | Identity law for general l... |
| nnlogbexp 26731 | Identity law for general l... |
| logbrec 26732 | Logarithm of a reciprocal ... |
| logbleb 26733 | The general logarithm func... |
| logblt 26734 | The general logarithm func... |
| relogbcxp 26735 | Identity law for the gener... |
| cxplogb 26736 | Identity law for the gener... |
| relogbcxpb 26737 | The logarithm is the inver... |
| logbmpt 26738 | The general logarithm to a... |
| logbf 26739 | The general logarithm to a... |
| logbfval 26740 | The general logarithm of a... |
| relogbf 26741 | The general logarithm to a... |
| logblog 26742 | The general logarithm to t... |
| logbgt0b 26743 | The logarithm of a positiv... |
| logbgcd1irr 26744 | The logarithm of an intege... |
| 2logb9irr 26745 | Example for ~ logbgcd1irr ... |
| logbprmirr 26746 | The logarithm of a prime t... |
| 2logb3irr 26747 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26748 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26749 | The square root of two to ... |
| 2irrexpqALT 26750 | Alternate proof of ~ 2irre... |
| angval 26751 | Define the angle function,... |
| angcan 26752 | Cancel a constant multipli... |
| angneg 26753 | Cancel a negative sign in ... |
| angvald 26754 | The (signed) angle between... |
| angcld 26755 | The (signed) angle between... |
| angrteqvd 26756 | Two vectors are at a right... |
| cosangneg2d 26757 | The cosine of the angle be... |
| angrtmuld 26758 | Perpendicularity of two ve... |
| ang180lem1 26759 | Lemma for ~ ang180 . Show... |
| ang180lem2 26760 | Lemma for ~ ang180 . Show... |
| ang180lem3 26761 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26762 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26763 | Lemma for ~ ang180 : Redu... |
| ang180 26764 | The sum of angles ` m A B ... |
| lawcoslem1 26765 | Lemma for ~ lawcos . Here... |
| lawcos 26766 | Law of cosines (also known... |
| pythag 26767 | Pythagorean theorem. Give... |
| isosctrlem1 26768 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26769 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26770 | Lemma for ~ isosctr . Cor... |
| isosctr 26771 | Isosceles triangle theorem... |
| ssscongptld 26772 | If two triangles have equa... |
| affineequiv 26773 | Equivalence between two wa... |
| affineequiv2 26774 | Equivalence between two wa... |
| affineequiv3 26775 | Equivalence between two wa... |
| affineequiv4 26776 | Equivalence between two wa... |
| affineequivne 26777 | Equivalence between two wa... |
| angpieqvdlem 26778 | Equivalence used in the pr... |
| angpieqvdlem2 26779 | Equivalence used in ~ angp... |
| angpined 26780 | If the angle at ABC is ` _... |
| angpieqvd 26781 | The angle ABC is ` _pi ` i... |
| chordthmlem 26782 | If ` M ` is the midpoint o... |
| chordthmlem2 26783 | If M is the midpoint of AB... |
| chordthmlem3 26784 | If M is the midpoint of AB... |
| chordthmlem4 26785 | If P is on the segment AB ... |
| chordthmlem5 26786 | If P is on the segment AB ... |
| chordthm 26787 | The intersecting chords th... |
| heron 26788 | Heron's formula gives the ... |
| quad2 26789 | The quadratic equation, wi... |
| quad 26790 | The quadratic equation. (... |
| 1cubrlem 26791 | The cube roots of unity. ... |
| 1cubr 26792 | The cube roots of unity. ... |
| dcubic1lem 26793 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26794 | Reverse direction of ~ dcu... |
| dcubic1 26795 | Forward direction of ~ dcu... |
| dcubic 26796 | Solutions to the depressed... |
| mcubic 26797 | Solutions to a monic cubic... |
| cubic2 26798 | The solution to the genera... |
| cubic 26799 | The cubic equation, which ... |
| binom4 26800 | Work out a quartic binomia... |
| dquartlem1 26801 | Lemma for ~ dquart . (Con... |
| dquartlem2 26802 | Lemma for ~ dquart . (Con... |
| dquart 26803 | Solve a depressed quartic ... |
| quart1cl 26804 | Closure lemmas for ~ quart... |
| quart1lem 26805 | Lemma for ~ quart1 . (Con... |
| quart1 26806 | Depress a quartic equation... |
| quartlem1 26807 | Lemma for ~ quart . (Cont... |
| quartlem2 26808 | Closure lemmas for ~ quart... |
| quartlem3 26809 | Closure lemmas for ~ quart... |
| quartlem4 26810 | Closure lemmas for ~ quart... |
| quart 26811 | The quartic equation, writ... |
| asinlem 26818 | The argument to the logari... |
| asinlem2 26819 | The argument to the logari... |
| asinlem3a 26820 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26821 | The argument to the logari... |
| asinf 26822 | Domain and codomain of the... |
| asincl 26823 | Closure for the arcsin fun... |
| acosf 26824 | Domain and codoamin of the... |
| acoscl 26825 | Closure for the arccos fun... |
| atandm 26826 | Since the property is a li... |
| atandm2 26827 | This form of ~ atandm is a... |
| atandm3 26828 | A compact form of ~ atandm... |
| atandm4 26829 | A compact form of ~ atandm... |
| atanf 26830 | Domain and codoamin of the... |
| atancl 26831 | Closure for the arctan fun... |
| asinval 26832 | Value of the arcsin functi... |
| acosval 26833 | Value of the arccos functi... |
| atanval 26834 | Value of the arctan functi... |
| atanre 26835 | A real number is in the do... |
| asinneg 26836 | The arcsine function is od... |
| acosneg 26837 | The negative symmetry rela... |
| efiasin 26838 | The exponential of the arc... |
| sinasin 26839 | The arcsine function is an... |
| cosacos 26840 | The arccosine function is ... |
| asinsinlem 26841 | Lemma for ~ asinsin . (Co... |
| asinsin 26842 | The arcsine function compo... |
| acoscos 26843 | The arccosine function is ... |
| asin1 26844 | The arcsine of ` 1 ` is ` ... |
| acos1 26845 | The arccosine of ` 1 ` is ... |
| reasinsin 26846 | The arcsine function compo... |
| asinsinb 26847 | Relationship between sine ... |
| acoscosb 26848 | Relationship between cosin... |
| asinbnd 26849 | The arcsine function has r... |
| acosbnd 26850 | The arccosine function has... |
| asinrebnd 26851 | Bounds on the arcsine func... |
| asinrecl 26852 | The arcsine function is re... |
| acosrecl 26853 | The arccosine function is ... |
| cosasin 26854 | The cosine of the arcsine ... |
| sinacos 26855 | The sine of the arccosine ... |
| atandmneg 26856 | The domain of the arctange... |
| atanneg 26857 | The arctangent function is... |
| atan0 26858 | The arctangent of zero is ... |
| atandmcj 26859 | The arctangent function di... |
| atancj 26860 | The arctangent function di... |
| atanrecl 26861 | The arctangent function is... |
| efiatan 26862 | Value of the exponential o... |
| atanlogaddlem 26863 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26864 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26865 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26866 | A variation on ~ atanlogad... |
| efiatan2 26867 | Value of the exponential o... |
| 2efiatan 26868 | Value of the exponential o... |
| tanatan 26869 | The arctangent function is... |
| atandmtan 26870 | The tangent function has r... |
| cosatan 26871 | The cosine of an arctangen... |
| cosatanne0 26872 | The arctangent function ha... |
| atantan 26873 | The arctangent function is... |
| atantanb 26874 | Relationship between tange... |
| atanbndlem 26875 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26876 | The arctangent function is... |
| atanord 26877 | The arctangent function is... |
| atan1 26878 | The arctangent of ` 1 ` is... |
| bndatandm 26879 | A point in the open unit d... |
| atans 26880 | The "domain of continuity"... |
| atans2 26881 | It suffices to show that `... |
| atansopn 26882 | The domain of continuity o... |
| atansssdm 26883 | The domain of continuity o... |
| ressatans 26884 | The real number line is a ... |
| dvatan 26885 | The derivative of the arct... |
| atancn 26886 | The arctangent is a contin... |
| atantayl 26887 | The Taylor series for ` ar... |
| atantayl2 26888 | The Taylor series for ` ar... |
| atantayl3 26889 | The Taylor series for ` ar... |
| leibpilem1 26890 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26891 | The Leibniz formula for ` ... |
| leibpi 26892 | The Leibniz formula for ` ... |
| leibpisum 26893 | The Leibniz formula for ` ... |
| log2cnv 26894 | Using the Taylor series fo... |
| log2tlbnd 26895 | Bound the error term in th... |
| log2ublem1 26896 | Lemma for ~ log2ub . The ... |
| log2ublem2 26897 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26898 | Lemma for ~ log2ub . In d... |
| log2ub 26899 | ` log 2 ` is less than ` 2... |
| log2le1 26900 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26901 | Lemma for ~ birthday . (C... |
| birthdaylem2 26902 | For general ` N ` and ` K ... |
| birthdaylem3 26903 | For general ` N ` and ` K ... |
| birthday 26904 | The Birthday Problem. The... |
| dmarea 26907 | The domain of the area fun... |
| areambl 26908 | The fibers of a measurable... |
| areass 26909 | A measurable region is a s... |
| dfarea 26910 | Rewrite ~ df-area self-ref... |
| areaf 26911 | Area measurement is a func... |
| areacl 26912 | The area of a measurable r... |
| areage0 26913 | The area of a measurable r... |
| areaval 26914 | The area of a measurable r... |
| rlimcnp 26915 | Relate a limit of a real-v... |
| rlimcnp2 26916 | Relate a limit of a real-v... |
| rlimcnp3 26917 | Relate a limit of a real-v... |
| xrlimcnp 26918 | Relate a limit of a real-v... |
| efrlim 26919 | The limit of the sequence ... |
| efrlimOLD 26920 | Obsolete version of ~ efrl... |
| dfef2 26921 | The limit of the sequence ... |
| cxplim 26922 | A power to a negative expo... |
| sqrtlim 26923 | The inverse square root fu... |
| rlimcxp 26924 | Any power to a positive ex... |
| o1cxp 26925 | An eventually bounded func... |
| cxp2limlem 26926 | A linear factor grows slow... |
| cxp2lim 26927 | Any power grows slower tha... |
| cxploglim 26928 | The logarithm grows slower... |
| cxploglim2 26929 | Every power of the logarit... |
| divsqrtsumlem 26930 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26931 | The function ` F ` used in... |
| divsqrsum 26932 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26933 | A bound on the distance of... |
| divsqrtsumo1 26934 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26935 | Closure of a 0-1 linear co... |
| scvxcvx 26936 | A strictly convex function... |
| jensenlem1 26937 | Lemma for ~ jensen . (Con... |
| jensenlem2 26938 | Lemma for ~ jensen . (Con... |
| jensen 26939 | Jensen's inequality, a fin... |
| amgmlem 26940 | Lemma for ~ amgm . (Contr... |
| amgm 26941 | Inequality of arithmetic a... |
| logdifbnd 26944 | Bound on the difference of... |
| logdiflbnd 26945 | Lower bound on the differe... |
| emcllem1 26946 | Lemma for ~ emcl . The se... |
| emcllem2 26947 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26948 | Lemma for ~ emcl . The fu... |
| emcllem4 26949 | Lemma for ~ emcl . The di... |
| emcllem5 26950 | Lemma for ~ emcl . The pa... |
| emcllem6 26951 | Lemma for ~ emcl . By the... |
| emcllem7 26952 | Lemma for ~ emcl and ~ har... |
| emcl 26953 | Closure and bounds for the... |
| harmonicbnd 26954 | A bound on the harmonic se... |
| harmonicbnd2 26955 | A bound on the harmonic se... |
| emre 26956 | The Euler-Mascheroni const... |
| emgt0 26957 | The Euler-Mascheroni const... |
| harmonicbnd3 26958 | A bound on the harmonic se... |
| harmoniclbnd 26959 | A bound on the harmonic se... |
| harmonicubnd 26960 | A bound on the harmonic se... |
| harmonicbnd4 26961 | The asymptotic behavior of... |
| fsumharmonic 26962 | Bound a finite sum based o... |
| zetacvg 26965 | The zeta series is converg... |
| eldmgm 26972 | Elementhood in the set of ... |
| dmgmaddn0 26973 | If ` A ` is not a nonposit... |
| dmlogdmgm 26974 | If ` A ` is in the continu... |
| rpdmgm 26975 | A positive real number is ... |
| dmgmn0 26976 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26977 | If ` A ` is not a nonposit... |
| dmgmdivn0 26978 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26979 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26980 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26981 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26982 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26983 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26984 | The series ` G ` is unifor... |
| lgamgulm 26985 | The series ` G ` is unifor... |
| lgamgulm2 26986 | Rewrite the limit of the s... |
| lgambdd 26987 | The log-Gamma function is ... |
| lgamucov 26988 | The ` U ` regions used in ... |
| lgamucov2 26989 | The ` U ` regions used in ... |
| lgamcvglem 26990 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 26991 | The log-Gamma function is ... |
| lgamf 26992 | The log-Gamma function is ... |
| gamf 26993 | The Gamma function is a co... |
| gamcl 26994 | The exponential of the log... |
| eflgam 26995 | The exponential of the log... |
| gamne0 26996 | The Gamma function is neve... |
| igamval 26997 | Value of the inverse Gamma... |
| igamz 26998 | Value of the inverse Gamma... |
| igamgam 26999 | Value of the inverse Gamma... |
| igamlgam 27000 | Value of the inverse Gamma... |
| igamf 27001 | Closure of the inverse Gam... |
| igamcl 27002 | Closure of the inverse Gam... |
| gamigam 27003 | The Gamma function is the ... |
| lgamcvg 27004 | The series ` G ` converges... |
| lgamcvg2 27005 | The series ` G ` converges... |
| gamcvg 27006 | The pointwise exponential ... |
| lgamp1 27007 | The functional equation of... |
| gamp1 27008 | The functional equation of... |
| gamcvg2lem 27009 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 27010 | An infinite product expres... |
| regamcl 27011 | The Gamma function is real... |
| relgamcl 27012 | The log-Gamma function is ... |
| rpgamcl 27013 | The log-Gamma function is ... |
| lgam1 27014 | The log-Gamma function at ... |
| gam1 27015 | The log-Gamma function at ... |
| facgam 27016 | The Gamma function general... |
| gamfac 27017 | The Gamma function general... |
| wilthlem1 27018 | The only elements that are... |
| wilthlem2 27019 | Lemma for ~ wilth : induct... |
| wilthlem3 27020 | Lemma for ~ wilth . Here ... |
| wilth 27021 | Wilson's theorem. A numbe... |
| wilthimp 27022 | The forward implication of... |
| ftalem1 27023 | Lemma for ~ fta : "growth... |
| ftalem2 27024 | Lemma for ~ fta . There e... |
| ftalem3 27025 | Lemma for ~ fta . There e... |
| ftalem4 27026 | Lemma for ~ fta : Closure... |
| ftalem5 27027 | Lemma for ~ fta : Main pr... |
| ftalem6 27028 | Lemma for ~ fta : Dischar... |
| ftalem7 27029 | Lemma for ~ fta . Shift t... |
| fta 27030 | The Fundamental Theorem of... |
| basellem1 27031 | Lemma for ~ basel . Closu... |
| basellem2 27032 | Lemma for ~ basel . Show ... |
| basellem3 27033 | Lemma for ~ basel . Using... |
| basellem4 27034 | Lemma for ~ basel . By ~ ... |
| basellem5 27035 | Lemma for ~ basel . Using... |
| basellem6 27036 | Lemma for ~ basel . The f... |
| basellem7 27037 | Lemma for ~ basel . The f... |
| basellem8 27038 | Lemma for ~ basel . The f... |
| basellem9 27039 | Lemma for ~ basel . Since... |
| basel 27040 | The sum of the inverse squ... |
| efnnfsumcl 27053 | Finite sum closure in the ... |
| ppisval 27054 | The set of primes less tha... |
| ppisval2 27055 | The set of primes less tha... |
| ppifi 27056 | The set of primes less tha... |
| prmdvdsfi 27057 | The set of prime divisors ... |
| chtf 27058 | Domain and codoamin of the... |
| chtcl 27059 | Real closure of the Chebys... |
| chtval 27060 | Value of the Chebyshev fun... |
| efchtcl 27061 | The Chebyshev function is ... |
| chtge0 27062 | The Chebyshev function is ... |
| vmaval 27063 | Value of the von Mangoldt ... |
| isppw 27064 | Two ways to say that ` A `... |
| isppw2 27065 | Two ways to say that ` A `... |
| vmappw 27066 | Value of the von Mangoldt ... |
| vmaprm 27067 | Value of the von Mangoldt ... |
| vmacl 27068 | Closure for the von Mangol... |
| vmaf 27069 | Functionality of the von M... |
| efvmacl 27070 | The von Mangoldt is closed... |
| vmage0 27071 | The von Mangoldt function ... |
| chpval 27072 | Value of the second Chebys... |
| chpf 27073 | Functionality of the secon... |
| chpcl 27074 | Closure for the second Che... |
| efchpcl 27075 | The second Chebyshev funct... |
| chpge0 27076 | The second Chebyshev funct... |
| ppival 27077 | Value of the prime-countin... |
| ppival2 27078 | Value of the prime-countin... |
| ppival2g 27079 | Value of the prime-countin... |
| ppif 27080 | Domain and codomain of the... |
| ppicl 27081 | Real closure of the prime-... |
| muval 27082 | The value of the Möbi... |
| muval1 27083 | The value of the Möbi... |
| muval2 27084 | The value of the Möbi... |
| isnsqf 27085 | Two ways to say that a num... |
| issqf 27086 | Two ways to say that a num... |
| sqfpc 27087 | The prime count of a squar... |
| dvdssqf 27088 | A divisor of a squarefree ... |
| sqf11 27089 | A squarefree number is com... |
| muf 27090 | The Möbius function i... |
| mucl 27091 | Closure of the Möbius... |
| sgmval 27092 | The value of the divisor f... |
| sgmval2 27093 | The value of the divisor f... |
| 0sgm 27094 | The value of the sum-of-di... |
| sgmf 27095 | The divisor function is a ... |
| sgmcl 27096 | Closure of the divisor fun... |
| sgmnncl 27097 | Closure of the divisor fun... |
| mule1 27098 | The Möbius function t... |
| chtfl 27099 | The Chebyshev function doe... |
| chpfl 27100 | The second Chebyshev funct... |
| ppiprm 27101 | The prime-counting functio... |
| ppinprm 27102 | The prime-counting functio... |
| chtprm 27103 | The Chebyshev function at ... |
| chtnprm 27104 | The Chebyshev function at ... |
| chpp1 27105 | The second Chebyshev funct... |
| chtwordi 27106 | The Chebyshev function is ... |
| chpwordi 27107 | The second Chebyshev funct... |
| chtdif 27108 | The difference of the Cheb... |
| efchtdvds 27109 | The exponentiated Chebyshe... |
| ppifl 27110 | The prime-counting functio... |
| ppip1le 27111 | The prime-counting functio... |
| ppiwordi 27112 | The prime-counting functio... |
| ppidif 27113 | The difference of the prim... |
| ppi1 27114 | The prime-counting functio... |
| cht1 27115 | The Chebyshev function at ... |
| vma1 27116 | The von Mangoldt function ... |
| chp1 27117 | The second Chebyshev funct... |
| ppi1i 27118 | Inference form of ~ ppiprm... |
| ppi2i 27119 | Inference form of ~ ppinpr... |
| ppi2 27120 | The prime-counting functio... |
| ppi3 27121 | The prime-counting functio... |
| cht2 27122 | The Chebyshev function at ... |
| cht3 27123 | The Chebyshev function at ... |
| ppinncl 27124 | Closure of the prime-count... |
| chtrpcl 27125 | Closure of the Chebyshev f... |
| ppieq0 27126 | The prime-counting functio... |
| ppiltx 27127 | The prime-counting functio... |
| prmorcht 27128 | Relate the primorial (prod... |
| mumullem1 27129 | Lemma for ~ mumul . A mul... |
| mumullem2 27130 | Lemma for ~ mumul . The p... |
| mumul 27131 | The Möbius function i... |
| sqff1o 27132 | There is a bijection from ... |
| fsumdvdsdiaglem 27133 | A "diagonal commutation" o... |
| fsumdvdsdiag 27134 | A "diagonal commutation" o... |
| fsumdvdscom 27135 | A double commutation of di... |
| dvdsppwf1o 27136 | A bijection between the di... |
| dvdsflf1o 27137 | A bijection from the numbe... |
| dvdsflsumcom 27138 | A sum commutation from ` s... |
| fsumfldivdiaglem 27139 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 27140 | The right-hand side of ~ d... |
| musum 27141 | The sum of the Möbius... |
| musumsum 27142 | Evaluate a collapsing sum ... |
| muinv 27143 | The Möbius inversion ... |
| mpodvdsmulf1o 27144 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 27145 | Product of two divisor sum... |
| dvdsmulf1o 27146 | If ` M ` and ` N ` are two... |
| fsumdvdsmulOLD 27147 | Obsolete version of ~ fsum... |
| sgmppw 27148 | The value of the divisor f... |
| 0sgmppw 27149 | A prime power ` P ^ K ` ha... |
| 1sgmprm 27150 | The sum of divisors for a ... |
| 1sgm2ppw 27151 | The sum of the divisors of... |
| sgmmul 27152 | The divisor function for f... |
| ppiublem1 27153 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 27154 | A prime greater than ` 3 `... |
| ppiub 27155 | An upper bound on the prim... |
| vmalelog 27156 | The von Mangoldt function ... |
| chtlepsi 27157 | The first Chebyshev functi... |
| chprpcl 27158 | Closure of the second Cheb... |
| chpeq0 27159 | The second Chebyshev funct... |
| chteq0 27160 | The first Chebyshev functi... |
| chtleppi 27161 | Upper bound on the ` theta... |
| chtublem 27162 | Lemma for ~ chtub . (Cont... |
| chtub 27163 | An upper bound on the Cheb... |
| fsumvma 27164 | Rewrite a sum over the von... |
| fsumvma2 27165 | Apply ~ fsumvma for the co... |
| pclogsum 27166 | The logarithmic analogue o... |
| vmasum 27167 | The sum of the von Mangold... |
| logfac2 27168 | Another expression for the... |
| chpval2 27169 | Express the second Chebysh... |
| chpchtsum 27170 | The second Chebyshev funct... |
| chpub 27171 | An upper bound on the seco... |
| logfacubnd 27172 | A simple upper bound on th... |
| logfaclbnd 27173 | A lower bound on the logar... |
| logfacbnd3 27174 | Show the stronger statemen... |
| logfacrlim 27175 | Combine the estimates ~ lo... |
| logexprlim 27176 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 27177 | Write out ~ logfacrlim as ... |
| mersenne 27178 | A Mersenne prime is a prim... |
| perfect1 27179 | Euclid's contribution to t... |
| perfectlem1 27180 | Lemma for ~ perfect . (Co... |
| perfectlem2 27181 | Lemma for ~ perfect . (Co... |
| perfect 27182 | The Euclid-Euler theorem, ... |
| dchrval 27185 | Value of the group of Diri... |
| dchrbas 27186 | Base set of the group of D... |
| dchrelbas 27187 | A Dirichlet character is a... |
| dchrelbas2 27188 | A Dirichlet character is a... |
| dchrelbas3 27189 | A Dirichlet character is a... |
| dchrelbasd 27190 | A Dirichlet character is a... |
| dchrrcl 27191 | Reverse closure for a Diri... |
| dchrmhm 27192 | A Dirichlet character is a... |
| dchrf 27193 | A Dirichlet character is a... |
| dchrelbas4 27194 | A Dirichlet character is a... |
| dchrzrh1 27195 | Value of a Dirichlet chara... |
| dchrzrhcl 27196 | A Dirichlet character take... |
| dchrzrhmul 27197 | A Dirichlet character is c... |
| dchrplusg 27198 | Group operation on the gro... |
| dchrmul 27199 | Group operation on the gro... |
| dchrmulcl 27200 | Closure of the group opera... |
| dchrn0 27201 | A Dirichlet character is n... |
| dchr1cl 27202 | Closure of the principal D... |
| dchrmullid 27203 | Left identity for the prin... |
| dchrinvcl 27204 | Closure of the group inver... |
| dchrabl 27205 | The set of Dirichlet chara... |
| dchrfi 27206 | The group of Dirichlet cha... |
| dchrghm 27207 | A Dirichlet character rest... |
| dchr1 27208 | Value of the principal Dir... |
| dchreq 27209 | A Dirichlet character is d... |
| dchrresb 27210 | A Dirichlet character is d... |
| dchrabs 27211 | A Dirichlet character take... |
| dchrinv 27212 | The inverse of a Dirichlet... |
| dchrabs2 27213 | A Dirichlet character take... |
| dchr1re 27214 | The principal Dirichlet ch... |
| dchrptlem1 27215 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27216 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27217 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27218 | For any element other than... |
| dchrsum2 27219 | An orthogonality relation ... |
| dchrsum 27220 | An orthogonality relation ... |
| sumdchr2 27221 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27222 | There are exactly ` phi ( ... |
| sumdchr 27223 | An orthogonality relation ... |
| dchr2sum 27224 | An orthogonality relation ... |
| sum2dchr 27225 | An orthogonality relation ... |
| bcctr 27226 | Value of the central binom... |
| pcbcctr 27227 | Prime count of a central b... |
| bcmono 27228 | The binomial coefficient i... |
| bcmax 27229 | The binomial coefficient t... |
| bcp1ctr 27230 | Ratio of two central binom... |
| bclbnd 27231 | A bound on the binomial co... |
| efexple 27232 | Convert a bound on a power... |
| bpos1lem 27233 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27234 | Bertrand's postulate, chec... |
| bposlem1 27235 | An upper bound on the prim... |
| bposlem2 27236 | There are no odd primes in... |
| bposlem3 27237 | Lemma for ~ bpos . Since ... |
| bposlem4 27238 | Lemma for ~ bpos . (Contr... |
| bposlem5 27239 | Lemma for ~ bpos . Bound ... |
| bposlem6 27240 | Lemma for ~ bpos . By usi... |
| bposlem7 27241 | Lemma for ~ bpos . The fu... |
| bposlem8 27242 | Lemma for ~ bpos . Evalua... |
| bposlem9 27243 | Lemma for ~ bpos . Derive... |
| bpos 27244 | Bertrand's postulate: ther... |
| zabsle1 27247 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27248 | When ` a ` is coprime to t... |
| lgslem2 27249 | The set ` Z ` of all integ... |
| lgslem3 27250 | The set ` Z ` of all integ... |
| lgslem4 27251 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27252 | Value of the Legendre symb... |
| lgsfval 27253 | Value of the function ` F ... |
| lgsfcl2 27254 | The function ` F ` is clos... |
| lgscllem 27255 | The Legendre symbol is an ... |
| lgsfcl 27256 | Closure of the function ` ... |
| lgsfle1 27257 | The function ` F ` has mag... |
| lgsval2lem 27258 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27259 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27260 | The Legendre symbol is an ... |
| lgs0 27261 | The Legendre symbol when t... |
| lgscl 27262 | The Legendre symbol is an ... |
| lgsle1 27263 | The Legendre symbol has ab... |
| lgsval2 27264 | The Legendre symbol at a p... |
| lgs2 27265 | The Legendre symbol at ` 2... |
| lgsval3 27266 | The Legendre symbol at an ... |
| lgsvalmod 27267 | The Legendre symbol is equ... |
| lgsval4 27268 | Restate ~ lgsval for nonze... |
| lgsfcl3 27269 | Closure of the function ` ... |
| lgsval4a 27270 | Same as ~ lgsval4 for posi... |
| lgscl1 27271 | The value of the Legendre ... |
| lgsneg 27272 | The Legendre symbol is eit... |
| lgsneg1 27273 | The Legendre symbol for no... |
| lgsmod 27274 | The Legendre (Jacobi) symb... |
| lgsdilem 27275 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27276 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27277 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27278 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27279 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27280 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27281 | The Legendre symbol is com... |
| lgsdirprm 27282 | The Legendre symbol is com... |
| lgsdir 27283 | The Legendre symbol is com... |
| lgsdilem2 27284 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27285 | The Legendre symbol is com... |
| lgsne0 27286 | The Legendre symbol is non... |
| lgsabs1 27287 | The Legendre symbol is non... |
| lgssq 27288 | The Legendre symbol at a s... |
| lgssq2 27289 | The Legendre symbol at a s... |
| lgsprme0 27290 | The Legendre symbol at any... |
| 1lgs 27291 | The Legendre symbol at ` 1... |
| lgs1 27292 | The Legendre symbol at ` 1... |
| lgsmodeq 27293 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27294 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27295 | Variation on ~ lgsdir vali... |
| lgsdinn0 27296 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27297 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27298 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27299 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27300 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27301 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27302 | The Legendre symbol for od... |
| lgsqrmod 27303 | If the Legendre symbol of ... |
| lgsqrmodndvds 27304 | If the Legendre symbol of ... |
| lgsdchrval 27305 | The Legendre symbol functi... |
| lgsdchr 27306 | The Legendre symbol functi... |
| gausslemma2dlem0a 27307 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27308 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27309 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27310 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27311 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27312 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27313 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27314 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27315 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27316 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27317 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27318 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27319 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27320 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27321 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27322 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27323 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27324 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27325 | Gauss' Lemma (see also the... |
| lgseisenlem1 27326 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27327 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27328 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27329 | Lemma for ~ lgseisen . (C... |
| lgseisen 27330 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27331 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27332 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27333 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27334 | The Law of Quadratic Recip... |
| lgsquad2lem1 27335 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27336 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27337 | Extend ~ lgsquad to coprim... |
| lgsquad3 27338 | Extend ~ lgsquad2 to integ... |
| m1lgs 27339 | The first supplement to th... |
| 2lgslem1a1 27340 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27341 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27342 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27343 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27344 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27345 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27346 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27347 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27348 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27349 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27350 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27351 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27352 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27353 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27354 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27355 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27356 | The Legendre symbol for ` ... |
| 2lgslem4 27357 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27358 | The second supplement to t... |
| 2lgsoddprmlem1 27359 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27360 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27361 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27362 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27363 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27364 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27365 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27366 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27367 | The second supplement to t... |
| 2sqlem1 27368 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27369 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27370 | Fibonacci's identity (actu... |
| 2sqlem3 27371 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27372 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27373 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27374 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27375 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27376 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27377 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27378 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27379 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27380 | Lemma for ~ 2sq . (Contri... |
| 2sq 27381 | All primes of the form ` 4... |
| 2sqblem 27382 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27383 | The converse to ~ 2sq . (... |
| 2sq2 27384 | ` 2 ` is the sum of square... |
| 2sqn0 27385 | If the sum of two squares ... |
| 2sqcoprm 27386 | If the sum of two squares ... |
| 2sqmod 27387 | Given two decompositions o... |
| 2sqmo 27388 | There exists at most one d... |
| 2sqnn0 27389 | All primes of the form ` 4... |
| 2sqnn 27390 | All primes of the form ` 4... |
| addsq2reu 27391 | For each complex number ` ... |
| addsqn2reu 27392 | For each complex number ` ... |
| addsqrexnreu 27393 | For each complex number, t... |
| addsqnreup 27394 | There is no unique decompo... |
| addsq2nreurex 27395 | For each complex number ` ... |
| addsqn2reurex2 27396 | For each complex number ` ... |
| 2sqreulem1 27397 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27398 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27399 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27400 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27401 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27402 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27403 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27404 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27405 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27406 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27407 | There exists a unique deco... |
| 2sqreunn 27408 | There exists a unique deco... |
| 2sqreult 27409 | There exists a unique deco... |
| 2sqreultb 27410 | There exists a unique deco... |
| 2sqreunnlt 27411 | There exists a unique deco... |
| 2sqreunnltb 27412 | There exists a unique deco... |
| 2sqreuop 27413 | There exists a unique deco... |
| 2sqreuopnn 27414 | There exists a unique deco... |
| 2sqreuoplt 27415 | There exists a unique deco... |
| 2sqreuopltb 27416 | There exists a unique deco... |
| 2sqreuopnnlt 27417 | There exists a unique deco... |
| 2sqreuopnnltb 27418 | There exists a unique deco... |
| 2sqreuopb 27419 | There exists a unique deco... |
| chebbnd1lem1 27420 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27421 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27422 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27423 | The Chebyshev bound: The ... |
| chtppilimlem1 27424 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27425 | Lemma for ~ chtppilim . (... |
| chtppilim 27426 | The ` theta ` function is ... |
| chto1ub 27427 | The ` theta ` function is ... |
| chebbnd2 27428 | The Chebyshev bound, part ... |
| chto1lb 27429 | The ` theta ` function is ... |
| chpchtlim 27430 | The ` psi ` and ` theta ` ... |
| chpo1ub 27431 | The ` psi ` function is up... |
| chpo1ubb 27432 | The ` psi ` function is up... |
| vmadivsum 27433 | The sum of the von Mangold... |
| vmadivsumb 27434 | Give a total bound on the ... |
| rplogsumlem1 27435 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27436 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27437 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27438 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27439 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27440 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27441 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27442 | Lemma for ~ dchrisum . Le... |
| dchrisum 27443 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27444 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27445 | The sum of the Möbius... |
| dchrvmasumlem1 27446 | An alternative expression ... |
| dchrvmasum2lem 27447 | Give an expression for ` l... |
| dchrvmasum2if 27448 | Combine the results of ~ d... |
| dchrvmasumlem2 27449 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27450 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27451 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27452 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27453 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27454 | An asymptotic approximatio... |
| dchrvmaeq0 27455 | The set ` W ` is the colle... |
| dchrisum0fval 27456 | Value of the function ` F ... |
| dchrisum0fmul 27457 | The function ` F ` , the d... |
| dchrisum0ff 27458 | The function ` F ` is a re... |
| dchrisum0flblem1 27459 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27460 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27461 | The divisor sum of a real ... |
| dchrisum0fno1 27462 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27463 | A partial result along the... |
| dchrisum0re 27464 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27465 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27466 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27467 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27468 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27469 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27470 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27471 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27472 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27473 | The sum of the Möbius... |
| dchrvmasumlem 27474 | The sum of the Möbius... |
| dchrmusum 27475 | The sum of the Möbius... |
| dchrvmasum 27476 | The sum of the von Mangold... |
| rpvmasum 27477 | The sum of the von Mangold... |
| rplogsum 27478 | The sum of ` log p / p ` o... |
| dirith2 27479 | Dirichlet's theorem: there... |
| dirith 27480 | Dirichlet's theorem: there... |
| mudivsum 27481 | Asymptotic formula for ` s... |
| mulogsumlem 27482 | Lemma for ~ mulogsum . (C... |
| mulogsum 27483 | Asymptotic formula for ... |
| logdivsum 27484 | Asymptotic analysis of ... |
| mulog2sumlem1 27485 | Asymptotic formula for ... |
| mulog2sumlem2 27486 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27487 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27488 | Asymptotic formula for ... |
| vmalogdivsum2 27489 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27490 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27491 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27492 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27493 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27494 | The Möbius inverse of... |
| log2sumbnd 27495 | Bound on the difference be... |
| selberglem1 27496 | Lemma for ~ selberg . Est... |
| selberglem2 27497 | Lemma for ~ selberg . (Co... |
| selberglem3 27498 | Lemma for ~ selberg . Est... |
| selberg 27499 | Selberg's symmetry formula... |
| selbergb 27500 | Convert eventual boundedne... |
| selberg2lem 27501 | Lemma for ~ selberg2 . Eq... |
| selberg2 27502 | Selberg's symmetry formula... |
| selberg2b 27503 | Convert eventual boundedne... |
| chpdifbndlem1 27504 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27505 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27506 | A bound on the difference ... |
| logdivbnd 27507 | A bound on a sum of logs, ... |
| selberg3lem1 27508 | Introduce a log weighting ... |
| selberg3lem2 27509 | Lemma for ~ selberg3 . Eq... |
| selberg3 27510 | Introduce a log weighting ... |
| selberg4lem1 27511 | Lemma for ~ selberg4 . Eq... |
| selberg4 27512 | The Selberg symmetry formu... |
| pntrval 27513 | Define the residual of the... |
| pntrf 27514 | Functionality of the resid... |
| pntrmax 27515 | There is a bound on the re... |
| pntrsumo1 27516 | A bound on a sum over ` R ... |
| pntrsumbnd 27517 | A bound on a sum over ` R ... |
| pntrsumbnd2 27518 | A bound on a sum over ` R ... |
| selbergr 27519 | Selberg's symmetry formula... |
| selberg3r 27520 | Selberg's symmetry formula... |
| selberg4r 27521 | Selberg's symmetry formula... |
| selberg34r 27522 | The sum of ~ selberg3r and... |
| pntsval 27523 | Define the "Selberg functi... |
| pntsf 27524 | Functionality of the Selbe... |
| selbergs 27525 | Selberg's symmetry formula... |
| selbergsb 27526 | Selberg's symmetry formula... |
| pntsval2 27527 | The Selberg function can b... |
| pntrlog2bndlem1 27528 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27529 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27530 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27531 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27532 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27533 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27534 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27535 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27536 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27537 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27538 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27539 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27540 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27541 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27542 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27543 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27544 | Lemma for ~ pnt . Establi... |
| pntlemd 27545 | Lemma for ~ pnt . Closure... |
| pntlemc 27546 | Lemma for ~ pnt . Closure... |
| pntlema 27547 | Lemma for ~ pnt . Closure... |
| pntlemb 27548 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27549 | Lemma for ~ pnt . Closure... |
| pntlemh 27550 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27551 | Lemma for ~ pnt . The "na... |
| pntlemq 27552 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27553 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27554 | Lemma for ~ pnt . The ind... |
| pntlemi 27555 | Lemma for ~ pnt . Elimina... |
| pntlemf 27556 | Lemma for ~ pnt . Add up ... |
| pntlemk 27557 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27558 | Lemma for ~ pnt . Combine... |
| pntleme 27559 | Lemma for ~ pnt . Package... |
| pntlem3 27560 | Lemma for ~ pnt . Equatio... |
| pntlemp 27561 | Lemma for ~ pnt . Wrappin... |
| pntleml 27562 | Lemma for ~ pnt . Equatio... |
| pnt3 27563 | The Prime Number Theorem, ... |
| pnt2 27564 | The Prime Number Theorem, ... |
| pnt 27565 | The Prime Number Theorem: ... |
| abvcxp 27566 | Raising an absolute value ... |
| padicfval 27567 | Value of the p-adic absolu... |
| padicval 27568 | Value of the p-adic absolu... |
| ostth2lem1 27569 | Lemma for ~ ostth2 , altho... |
| qrngbas 27570 | The base set of the field ... |
| qdrng 27571 | The rationals form a divis... |
| qrng0 27572 | The zero element of the fi... |
| qrng1 27573 | The unity element of the f... |
| qrngneg 27574 | The additive inverse in th... |
| qrngdiv 27575 | The division operation in ... |
| qabvle 27576 | By using induction on ` N ... |
| qabvexp 27577 | Induct the product rule ~ ... |
| ostthlem1 27578 | Lemma for ~ ostth . If tw... |
| ostthlem2 27579 | Lemma for ~ ostth . Refin... |
| qabsabv 27580 | The regular absolute value... |
| padicabv 27581 | The p-adic absolute value ... |
| padicabvf 27582 | The p-adic absolute value ... |
| padicabvcxp 27583 | All positive powers of the... |
| ostth1 27584 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27585 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27586 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27587 | Lemma for ~ ostth2 . (Con... |
| ostth2 27588 | - Lemma for ~ ostth : regu... |
| ostth3 27589 | - Lemma for ~ ostth : p-ad... |
| ostth 27590 | Ostrowski's theorem, which... |
| elno 27597 | Membership in the surreals... |
| elnoOLD 27598 | Obsolete version of ~ elno... |
| ltsval 27599 | The value of the surreal l... |
| bdayval 27600 | The value of the birthday ... |
| nofun 27601 | A surreal is a function. ... |
| nodmon 27602 | The domain of a surreal is... |
| norn 27603 | The range of a surreal is ... |
| nofnbday 27604 | A surreal is a function ov... |
| nodmord 27605 | The domain of a surreal ha... |
| elno2 27606 | An alternative condition f... |
| elno3 27607 | Another condition for memb... |
| ltsval2 27608 | Alternate expression for s... |
| nofv 27609 | The function value of a su... |
| nosgnn0 27610 | ` (/) ` is not a surreal s... |
| nosgnn0i 27611 | If ` X ` is a surreal sign... |
| noreson 27612 | The restriction of a surre... |
| ltsintdifex 27613 |
If ` A |
| ltsres 27614 | If the restrictions of two... |
| noxp1o 27615 | The Cartesian product of a... |
| noseponlem 27616 | Lemma for ~ nosepon . Con... |
| nosepon 27617 | Given two unequal surreals... |
| noextend 27618 | Extending a surreal by one... |
| noextendseq 27619 | Extend a surreal by a sequ... |
| noextenddif 27620 | Calculate the place where ... |
| noextendlt 27621 | Extending a surreal with a... |
| noextendgt 27622 | Extending a surreal with a... |
| nolesgn2o 27623 | Given ` A ` less-than or e... |
| nolesgn2ores 27624 | Given ` A ` less-than or e... |
| nogesgn1o 27625 | Given ` A ` greater than o... |
| nogesgn1ores 27626 | Given ` A ` greater than o... |
| ltssolem1 27627 | Lemma for ~ ltsso . The "... |
| ltsso 27628 | Less-than totally orders t... |
| bdayfo 27629 | The birthday function maps... |
| fvnobday 27630 | The value of a surreal at ... |
| nosepnelem 27631 | Lemma for ~ nosepne . (Co... |
| nosepne 27632 | The value of two non-equal... |
| nosep1o 27633 | If the value of a surreal ... |
| nosep2o 27634 | If the value of a surreal ... |
| nosepdmlem 27635 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27636 | The first place two surrea... |
| nosepeq 27637 | The values of two surreals... |
| nosepssdm 27638 | Given two non-equal surrea... |
| nodenselem4 27639 | Lemma for ~ nodense . Sho... |
| nodenselem5 27640 | Lemma for ~ nodense . If ... |
| nodenselem6 27641 | The restriction of a surre... |
| nodenselem7 27642 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27643 | Lemma for ~ nodense . Giv... |
| nodense 27644 | Given two distinct surreal... |
| bdayimaon 27645 | Lemma for full-eta propert... |
| nolt02olem 27646 | Lemma for ~ nolt02o . If ... |
| nolt02o 27647 | Given ` A ` less-than ` B ... |
| nogt01o 27648 | Given ` A ` greater than `... |
| noresle 27649 | Restriction law for surrea... |
| nomaxmo 27650 | A class of surreals has at... |
| nominmo 27651 | A class of surreals has at... |
| nosupprefixmo 27652 | In any class of surreals, ... |
| noinfprefixmo 27653 | In any class of surreals, ... |
| nosupcbv 27654 | Lemma to change bound vari... |
| nosupno 27655 | The next several theorems ... |
| nosupdm 27656 | The domain of the surreal ... |
| nosupbday 27657 | Birthday bounding law for ... |
| nosupfv 27658 | The value of surreal supre... |
| nosupres 27659 | A restriction law for surr... |
| nosupbnd1lem1 27660 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27661 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27662 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27663 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27664 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27665 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27666 | Bounding law from below fo... |
| nosupbnd2lem1 27667 | Bounding law from above wh... |
| nosupbnd2 27668 | Bounding law from above fo... |
| noinfcbv 27669 | Change bound variables for... |
| noinfno 27670 | The next several theorems ... |
| noinfdm 27671 | Next, we calculate the dom... |
| noinfbday 27672 | Birthday bounding law for ... |
| noinffv 27673 | The value of surreal infim... |
| noinfres 27674 | The restriction of surreal... |
| noinfbnd1lem1 27675 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27676 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27677 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27678 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27679 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27680 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27681 | Bounding law from above fo... |
| noinfbnd2lem1 27682 | Bounding law from below wh... |
| noinfbnd2 27683 | Bounding law from below fo... |
| nosupinfsep 27684 | Given two sets of surreals... |
| noetasuplem1 27685 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27686 | Lemma for ~ noeta . The r... |
| noetasuplem3 27687 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27688 | Lemma for ~ noeta . When ... |
| noetainflem1 27689 | Lemma for ~ noeta . Estab... |
| noetainflem2 27690 | Lemma for ~ noeta . The r... |
| noetainflem3 27691 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27692 | Lemma for ~ noeta . If ` ... |
| noetalem1 27693 | Lemma for ~ noeta . Eithe... |
| noetalem2 27694 | Lemma for ~ noeta . The f... |
| noeta 27695 | The full-eta axiom for the... |
| ltsirr 27698 | Surreal less-than is irref... |
| ltstr 27699 | Surreal less-than is trans... |
| ltsasym 27700 | Surreal less-than is asymm... |
| ltslin 27701 | Surreal less-than obeys tr... |
| ltstrieq2 27702 | Trichotomy law for surreal... |
| ltstrine 27703 | Trichotomy law for surreal... |
| lenlts 27704 | Surreal less-than or equal... |
| ltnles 27705 | Surreal less-than in terms... |
| lesloe 27706 | Surreal less-than or equal... |
| lestri3 27707 | Trichotomy law for surreal... |
| lesnltd 27708 | Surreal less-than or equal... |
| ltsnled 27709 | Surreal less-than in terms... |
| lesloed 27710 | Surreal less-than or equal... |
| lestri3d 27711 | Trichotomy law for surreal... |
| ltlestr 27712 | Surreal transitive law. (... |
| leltstr 27713 | Surreal transitive law. (... |
| lestr 27714 | Surreal transitive law. (... |
| ltstrd 27715 | Surreal less-than is trans... |
| ltlestrd 27716 | Surreal less-than is trans... |
| leltstrd 27717 | Surreal less-than is trans... |
| lestrd 27718 | Surreal less-than or equal... |
| lesid 27719 | Surreal less-than or equal... |
| lestric 27720 | Surreal trichotomy law. (... |
| maxs1 27721 | A surreal is less than or ... |
| maxs2 27722 | A surreal is less than or ... |
| mins1 27723 | The minimum of two surreal... |
| mins2 27724 | The minimum of two surreal... |
| ltlesd 27725 | Surreal less-than implies ... |
| ltsne 27726 | Surreal less-than implies ... |
| ltlesnd 27727 | Surreal less-than in terms... |
| bdayfun 27728 | The birthday function is a... |
| bdayfn 27729 | The birthday function is a... |
| bdaydm 27730 | The birthday function's do... |
| bdayrn 27731 | The birthday function's ra... |
| bdayon 27732 | The value of the birthday ... |
| nobdaymin 27733 | Any non-empty class of sur... |
| nocvxminlem 27734 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27735 | Given a nonempty convex cl... |
| noprc 27736 | The surreal numbers are a ... |
| noeta2 27741 | A version of ~ noeta with ... |
| brslts 27742 | Binary relation form of th... |
| sltsex1 27743 | The first argument of surr... |
| sltsex2 27744 | The second argument of sur... |
| sltsss1 27745 | The first argument of surr... |
| sltsss2 27746 | The second argument of sur... |
| sltssep 27747 | The separation property of... |
| sltsd 27748 | Deduce surreal set less-th... |
| sltssnb 27749 | Surreal set less-than of t... |
| sltssn 27750 | Surreal set less-than of t... |
| sltssepc 27751 | Two elements of separated ... |
| sltssepcd 27752 | Two elements of separated ... |
| ssslts1 27753 | Relation between surreal s... |
| ssslts2 27754 | Relation between surreal s... |
| nulslts 27755 | The empty set is less-than... |
| nulsgts 27756 | The empty set is greater t... |
| nulsltsd 27757 | The empty set is less-than... |
| nulsgtsd 27758 | The empty set is greater t... |
| conway 27759 | Conway's Simplicity Theore... |
| cutsval 27760 | The value of the surreal c... |
| cutcuts 27761 | Cut properties of the surr... |
| cutscl 27762 | Closure law for surreal cu... |
| cutscld 27763 | Closure law for surreal cu... |
| cutbday 27764 | The birthday of the surrea... |
| eqcuts 27765 | Condition for equality to ... |
| eqcuts2 27766 | Condition for equality to ... |
| sltstr 27767 | Transitive law for surreal... |
| sltsun1 27768 | Union law for surreal set ... |
| sltsun2 27769 | Union law for surreal set ... |
| cutsun12 27770 | Union law for surreal cuts... |
| dmcuts 27771 | The domain of the surreal ... |
| cutsf 27772 | Functionality statement fo... |
| etaslts 27773 | A restatement of ~ noeta u... |
| etaslts2 27774 | A version of ~ etaslts wit... |
| cutbdaybnd 27775 | An upper bound on the birt... |
| cutbdaybnd2 27776 | An upper bound on the birt... |
| cutbdaybnd2lim 27777 | An upper bound on the birt... |
| cutbdaylt 27778 | If a surreal lies in a gap... |
| lesrec 27779 | A comparison law for surre... |
| lesrecd 27780 | A comparison law for surre... |
| ltsrec 27781 | A comparison law for surre... |
| ltsrecd 27782 | A comparison law for surre... |
| sltsdisj 27783 | If ` A ` preceeds ` B ` , ... |
| eqcuts3 27784 | A variant of the simplicit... |
| 0no 27789 | Surreal zero is a surreal.... |
| 1no 27790 | Surreal one is a surreal. ... |
| bday0 27791 | Calculate the birthday of ... |
| 0lt1s 27792 | Surreal zero is less than ... |
| bday0b 27793 | The only surreal with birt... |
| bday1 27794 | The birthday of surreal on... |
| cuteq0 27795 | Condition for a surreal cu... |
| cutneg 27796 | The simplest number greate... |
| cuteq1 27797 | Condition for a surreal cu... |
| gt0ne0s 27798 | A positive surreal is not ... |
| gt0ne0sd 27799 | A positive surreal is not ... |
| 1ne0s 27800 | Surreal zero does not equa... |
| rightge0 27801 | A surreal is non-negative ... |
| madeval 27812 | The value of the made by f... |
| madeval2 27813 | Alternative characterizati... |
| oldval 27814 | The value of the old optio... |
| newval 27815 | The value of the new optio... |
| madef 27816 | The made function is a fun... |
| oldf 27817 | The older function is a fu... |
| newf 27818 | The new function is a func... |
| old0 27819 | No surreal is older than `... |
| madessno 27820 | Made sets are surreals. (... |
| oldssno 27821 | Old sets are surreals. (C... |
| newssno 27822 | New sets are surreals. (C... |
| madeno 27823 | An element of a made set i... |
| oldno 27824 | An element of an old set i... |
| newno 27825 | An element of a new set is... |
| madenod 27826 | An element of a made set i... |
| oldnod 27827 | An element of an old set i... |
| newnod 27828 | An element of a new set is... |
| leftval 27829 | The value of the left opti... |
| rightval 27830 | The value of the right opt... |
| elleft 27831 | Membership in the left set... |
| elright 27832 | Membership in the right se... |
| leftlt 27833 | A member of a surreal's le... |
| rightgt 27834 | A member of a surreal's ri... |
| leftf 27835 | The functionality of the l... |
| rightf 27836 | The functionality of the r... |
| elmade 27837 | Membership in the made fun... |
| elmade2 27838 | Membership in the made fun... |
| elold 27839 | Membership in an old set. ... |
| sltsleft 27840 | A surreal is greater than ... |
| sltsright 27841 | A surreal is less than its... |
| lltr 27842 | The left options of a surr... |
| made0 27843 | The only surreal made on d... |
| new0 27844 | The only surreal new on da... |
| old1 27845 | The only surreal older tha... |
| madess 27846 | If ` A ` is less than or e... |
| oldssmade 27847 | The older-than set is a su... |
| oldmade 27848 | An element of an old set i... |
| oldmaded 27849 | An element of an old set i... |
| oldss 27850 | If ` A ` is less than or e... |
| leftssold 27851 | The left options are a sub... |
| rightssold 27852 | The right options are a su... |
| leftssno 27853 | The left set of a surreal ... |
| rightssno 27854 | The right set of a surreal... |
| leftold 27855 | An element of a left set i... |
| rightold 27856 | An element of a right set ... |
| leftno 27857 | An element of a left set i... |
| rightno 27858 | An element of a right set ... |
| leftoldd 27859 | An element of a left set i... |
| leftnod 27860 | An element of a left set i... |
| rightoldd 27861 | An element of a right set ... |
| rightnod 27862 | An element of a right set ... |
| madecut 27863 | Given a section that is a ... |
| madeun 27864 | The made set is the union ... |
| madeoldsuc 27865 | The made set is the old se... |
| oldsuc 27866 | The value of the old set a... |
| oldlim 27867 | The value of the old set a... |
| madebdayim 27868 | If a surreal is a member o... |
| oldbdayim 27869 | If ` X ` is in the old set... |
| oldirr 27870 | No surreal is a member of ... |
| leftirr 27871 | No surreal is a member of ... |
| rightirr 27872 | No surreal is a member of ... |
| left0s 27873 | The left set of ` 0s ` is ... |
| right0s 27874 | The right set of ` 0s ` is... |
| left1s 27875 | The left set of ` 1s ` is ... |
| right1s 27876 | The right set of ` 1s ` is... |
| lrold 27877 | The union of the left and ... |
| madebdaylemold 27878 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27879 | Lemma for ~ madebday . If... |
| madebday 27880 | A surreal is part of the s... |
| oldbday 27881 | A surreal is part of the s... |
| newbday 27882 | A surreal is an element of... |
| newbdayim 27883 | One direction of the bicon... |
| lrcut 27884 | A surreal is equal to the ... |
| cutsfo 27885 | The surreal cut function i... |
| ltsn0 27886 | If ` X ` is less than ` Y ... |
| lruneq 27887 | If two surreals share a bi... |
| ltslpss 27888 | If two surreals share a bi... |
| leslss 27889 | If two surreals ` A ` and ... |
| 0elold 27890 | Zero is in the old set of ... |
| 0elleft 27891 | Zero is in the left set of... |
| 0elright 27892 | Zero is in the right set o... |
| madefi 27893 | The made set of an ordinal... |
| oldfi 27894 | The old set of an ordinal ... |
| bdayiun 27895 | The birthday of a surreal ... |
| bdayle 27896 | A condition for bounding a... |
| sltsbday 27897 | Birthday comparison rule f... |
| cofslts 27898 | If every element of ` A ` ... |
| coinitslts 27899 | If ` B ` is coinitial with... |
| cofcut1 27900 | If ` C ` is cofinal with `... |
| cofcut1d 27901 | If ` C ` is cofinal with `... |
| cofcut2 27902 | If ` A ` and ` C ` are mut... |
| cofcut2d 27903 | If ` A ` and ` C ` are mut... |
| cofcutr 27904 | If ` X ` is the cut of ` A... |
| cofcutr1d 27905 | If ` X ` is the cut of ` A... |
| cofcutr2d 27906 | If ` X ` is the cut of ` A... |
| cofcutrtime 27907 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27908 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27909 | If ` X ` is a timely cut o... |
| cofss 27910 | Cofinality for a subset. ... |
| coiniss 27911 | Coinitiality for a subset.... |
| cutlt 27912 | Eliminating all elements b... |
| cutpos 27913 | Reduce the elements of a c... |
| cutmax 27914 | If ` A ` has a maximum, th... |
| cutmin 27915 | If ` B ` has a minimum, th... |
| cutminmax 27916 | If the left set of ` X ` h... |
| lrrecval 27919 | The next step in the devel... |
| lrrecval2 27920 | Next, we establish an alte... |
| lrrecpo 27921 | Now, we establish that ` R... |
| lrrecse 27922 | Next, we show that ` R ` i... |
| lrrecfr 27923 | Now we show that ` R ` is ... |
| lrrecpred 27924 | Finally, we calculate the ... |
| noinds 27925 | Induction principle for a ... |
| norecfn 27926 | Surreal recursion over one... |
| norecov 27927 | Calculate the value of the... |
| noxpordpo 27930 | To get through most of the... |
| noxpordfr 27931 | Next we establish the foun... |
| noxpordse 27932 | Next we establish the set-... |
| noxpordpred 27933 | Next we calculate the pred... |
| no2indlesm 27934 | Double induction on surrea... |
| no2inds 27935 | Double induction on surrea... |
| norec2fn 27936 | The double-recursion opera... |
| norec2ov 27937 | The value of the double-re... |
| no3inds 27938 | Triple induction over surr... |
| addsfn 27941 | Surreal addition is a func... |
| addsval 27942 | The value of surreal addit... |
| addsval2 27943 | The value of surreal addit... |
| addsrid 27944 | Surreal addition to zero i... |
| addsridd 27945 | Surreal addition to zero i... |
| addscom 27946 | Surreal addition commutes.... |
| addscomd 27947 | Surreal addition commutes.... |
| addslid 27948 | Surreal addition to zero i... |
| addsproplem1 27949 | Lemma for surreal addition... |
| addsproplem2 27950 | Lemma for surreal addition... |
| addsproplem3 27951 | Lemma for surreal addition... |
| addsproplem4 27952 | Lemma for surreal addition... |
| addsproplem5 27953 | Lemma for surreal addition... |
| addsproplem6 27954 | Lemma for surreal addition... |
| addsproplem7 27955 | Lemma for surreal addition... |
| addsprop 27956 | Inductively show that surr... |
| addcutslem 27957 | Lemma for ~ addcuts . Sho... |
| addcuts 27958 | Demonstrate the cut proper... |
| addcuts2 27959 | Show that the cut involved... |
| addscld 27960 | Surreal numbers are closed... |
| addscl 27961 | Surreal numbers are closed... |
| addsf 27962 | Function statement for sur... |
| addsfo 27963 | Surreal addition is onto. ... |
| peano2no 27964 | A theorem for surreals tha... |
| ltadds1im 27965 | Surreal less-than is prese... |
| ltadds2im 27966 | Surreal less-than is prese... |
| leadds1im 27967 | Surreal less-than or equal... |
| leadds2im 27968 | Surreal less-than or equal... |
| leadds1 27969 | Addition to both sides of ... |
| leadds2 27970 | Addition to both sides of ... |
| ltadds2 27971 | Addition to both sides of ... |
| ltadds1 27972 | Addition to both sides of ... |
| addscan2 27973 | Cancellation law for surre... |
| addscan1 27974 | Cancellation law for surre... |
| leadds1d 27975 | Addition to both sides of ... |
| leadds2d 27976 | Addition to both sides of ... |
| ltadds2d 27977 | Addition to both sides of ... |
| ltadds1d 27978 | Addition to both sides of ... |
| addscan2d 27979 | Cancellation law for surre... |
| addscan1d 27980 | Cancellation law for surre... |
| addsuniflem 27981 | Lemma for ~ addsunif . St... |
| addsunif 27982 | Uniformity theorem for sur... |
| addsasslem1 27983 | Lemma for addition associa... |
| addsasslem2 27984 | Lemma for addition associa... |
| addsass 27985 | Surreal addition is associ... |
| addsassd 27986 | Surreal addition is associ... |
| adds32d 27987 | Commutative/associative la... |
| adds12d 27988 | Commutative/associative la... |
| adds4d 27989 | Rearrangement of four term... |
| adds42d 27990 | Rearrangement of four term... |
| ltaddspos1d 27991 | Addition of a positive num... |
| ltaddspos2d 27992 | Addition of a positive num... |
| lt2addsd 27993 | Adding both sides of two s... |
| addsgt0d 27994 | The sum of two positive su... |
| ltsp1d 27995 | A surreal is less than its... |
| addsge01d 27996 | A surreal is less-than or ... |
| addbdaylem 27997 | Lemma for ~ addbday . (Co... |
| addbday 27998 | The birthday of the sum of... |
| negsfn 28003 | Surreal negation is a func... |
| subsfn 28004 | Surreal subtraction is a f... |
| negsval 28005 | The value of the surreal n... |
| neg0s 28006 | Negative surreal zero is s... |
| neg1s 28007 | An expression for negative... |
| negsproplem1 28008 | Lemma for surreal negation... |
| negsproplem2 28009 | Lemma for surreal negation... |
| negsproplem3 28010 | Lemma for surreal negation... |
| negsproplem4 28011 | Lemma for surreal negation... |
| negsproplem5 28012 | Lemma for surreal negation... |
| negsproplem6 28013 | Lemma for surreal negation... |
| negsproplem7 28014 | Lemma for surreal negation... |
| negsprop 28015 | Show closure and ordering ... |
| negscl 28016 | The surreals are closed un... |
| negscld 28017 | The surreals are closed un... |
| ltnegsim 28018 | The forward direction of t... |
| negcut 28019 | The cut properties of surr... |
| negcut2 28020 | The cut that defines surre... |
| negsid 28021 | Surreal addition of a numb... |
| negsidd 28022 | Surreal addition of a numb... |
| negsex 28023 | Every surreal has a negati... |
| negnegs 28024 | A surreal is equal to the ... |
| ltnegs 28025 | Negative of both sides of ... |
| lenegs 28026 | Negative of both sides of ... |
| ltnegsd 28027 | Negative of both sides of ... |
| lenegsd 28028 | Negative of both sides of ... |
| negs11 28029 | Surreal negation is one-to... |
| negsdi 28030 | Distribution of surreal ne... |
| lt0negs2d 28031 | Comparison of a surreal an... |
| negsf 28032 | Function statement for sur... |
| negsfo 28033 | Function statement for sur... |
| negsf1o 28034 | Surreal negation is a bije... |
| negsunif 28035 | Uniformity property for su... |
| negbdaylem 28036 | Lemma for ~ negbday . Bou... |
| negbday 28037 | Negation of a surreal numb... |
| negleft 28038 | The left set of the negati... |
| negright 28039 | The right set of the negat... |
| subsval 28040 | The value of surreal subtr... |
| subsvald 28041 | The value of surreal subtr... |
| subscl 28042 | Closure law for surreal su... |
| subscld 28043 | Closure law for surreal su... |
| subsf 28044 | Function statement for sur... |
| subsfo 28045 | Surreal subtraction is an ... |
| negsval2 28046 | Surreal negation in terms ... |
| negsval2d 28047 | Surreal negation in terms ... |
| subsid1 28048 | Identity law for subtracti... |
| subsid 28049 | Subtraction of a surreal f... |
| subadds 28050 | Relationship between addit... |
| subaddsd 28051 | Relationship between addit... |
| pncans 28052 | Cancellation law for surre... |
| pncan3s 28053 | Subtraction and addition o... |
| pncan2s 28054 | Cancellation law for surre... |
| npcans 28055 | Cancellation law for surre... |
| ltsubs1 28056 | Subtraction from both side... |
| ltsubs2 28057 | Subtraction from both side... |
| ltsubs1d 28058 | Subtraction from both side... |
| ltsubs2d 28059 | Subtraction from both side... |
| negsubsdi2d 28060 | Distribution of negative o... |
| addsubsassd 28061 | Associative-type law for s... |
| addsubsd 28062 | Law for surreal addition a... |
| ltsubsubsbd 28063 | Equivalence for the surrea... |
| ltsubsubs2bd 28064 | Equivalence for the surrea... |
| ltsubsubs3bd 28065 | Equivalence for the surrea... |
| lesubsubsbd 28066 | Equivalence for the surrea... |
| lesubsubs2bd 28067 | Equivalence for the surrea... |
| lesubsubs3bd 28068 | Equivalence for the surrea... |
| ltsubaddsd 28069 | Surreal less-than relation... |
| ltsubadds2d 28070 | Surreal less-than relation... |
| ltaddsubsd 28071 | Surreal less-than relation... |
| ltaddsubs2d 28072 | Surreal less-than relation... |
| lesubaddsd 28073 | Surreal less-than or equal... |
| subsubs4d 28074 | Law for double surreal sub... |
| subsubs2d 28075 | Law for double surreal sub... |
| lesubsd 28076 | Swap subtrahends in a surr... |
| nncansd 28077 | Cancellation law for surre... |
| posdifsd 28078 | Comparison of two surreals... |
| ltsubsposd 28079 | Subtraction of a positive ... |
| subsge0d 28080 | Non-negative subtraction. ... |
| addsubs4d 28081 | Rearrangement of four term... |
| ltsm1d 28082 | A surreal is greater than ... |
| subscan1d 28083 | Cancellation law for surre... |
| subscan2d 28084 | Cancellation law for surre... |
| subseq0d 28085 | The difference between two... |
| mulsfn 28088 | Surreal multiplication is ... |
| mulsval 28089 | The value of surreal multi... |
| mulsval2lem 28090 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 28091 | The value of surreal multi... |
| muls01 28092 | Surreal multiplication by ... |
| mulsrid 28093 | Surreal one is a right ide... |
| mulsridd 28094 | Surreal one is a right ide... |
| mulsproplemcbv 28095 | Lemma for surreal multipli... |
| mulsproplem1 28096 | Lemma for surreal multipli... |
| mulsproplem2 28097 | Lemma for surreal multipli... |
| mulsproplem3 28098 | Lemma for surreal multipli... |
| mulsproplem4 28099 | Lemma for surreal multipli... |
| mulsproplem5 28100 | Lemma for surreal multipli... |
| mulsproplem6 28101 | Lemma for surreal multipli... |
| mulsproplem7 28102 | Lemma for surreal multipli... |
| mulsproplem8 28103 | Lemma for surreal multipli... |
| mulsproplem9 28104 | Lemma for surreal multipli... |
| mulsproplem10 28105 | Lemma for surreal multipli... |
| mulsproplem11 28106 | Lemma for surreal multipli... |
| mulsproplem12 28107 | Lemma for surreal multipli... |
| mulsproplem13 28108 | Lemma for surreal multipli... |
| mulsproplem14 28109 | Lemma for surreal multipli... |
| mulsprop 28110 | Surreals are closed under ... |
| mulcutlem 28111 | Lemma for ~ mulcut . Stat... |
| mulcut 28112 | Show the cut properties of... |
| mulcut2 28113 | Show that the cut involved... |
| mulscl 28114 | The surreals are closed un... |
| mulscld 28115 | The surreals are closed un... |
| ltmuls 28116 | An ordering relationship f... |
| ltmulsd 28117 | An ordering relationship f... |
| lemulsd 28118 | An ordering relationship f... |
| mulscom 28119 | Surreal multiplication com... |
| mulscomd 28120 | Surreal multiplication com... |
| muls02 28121 | Surreal multiplication by ... |
| mulslid 28122 | Surreal one is a left iden... |
| mulslidd 28123 | Surreal one is a left iden... |
| mulsgt0 28124 | The product of two positiv... |
| mulsgt0d 28125 | The product of two positiv... |
| mulsge0d 28126 | The product of two non-neg... |
| sltmuls1 28127 | One surreal set less-than ... |
| sltmuls2 28128 | One surreal set less-than ... |
| mulsuniflem 28129 | Lemma for ~ mulsunif . St... |
| mulsunif 28130 | Surreal multiplication has... |
| addsdilem1 28131 | Lemma for surreal distribu... |
| addsdilem2 28132 | Lemma for surreal distribu... |
| addsdilem3 28133 | Lemma for ~ addsdi . Show... |
| addsdilem4 28134 | Lemma for ~ addsdi . Show... |
| addsdi 28135 | Distributive law for surre... |
| addsdid 28136 | Distributive law for surre... |
| addsdird 28137 | Distributive law for surre... |
| subsdid 28138 | Distribution of surreal mu... |
| subsdird 28139 | Distribution of surreal mu... |
| mulnegs1d 28140 | Product with negative is n... |
| mulnegs2d 28141 | Product with negative is n... |
| mul2negsd 28142 | Surreal product of two neg... |
| mulsasslem1 28143 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 28144 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 28145 | Lemma for ~ mulsass . Dem... |
| mulsass 28146 | Associative law for surrea... |
| mulsassd 28147 | Associative law for surrea... |
| muls4d 28148 | Rearrangement of four surr... |
| mulsunif2lem 28149 | Lemma for ~ mulsunif2 . S... |
| mulsunif2 28150 | Alternate expression for s... |
| ltmuls2 28151 | Multiplication of both sid... |
| ltmuls2d 28152 | Multiplication of both sid... |
| ltmuls1d 28153 | Multiplication of both sid... |
| lemuls2d 28154 | Multiplication of both sid... |
| lemuls1d 28155 | Multiplication of both sid... |
| ltmulnegs1d 28156 | Multiplication of both sid... |
| ltmulnegs2d 28157 | Multiplication of both sid... |
| mulscan2dlem 28158 | Lemma for ~ mulscan2d . C... |
| mulscan2d 28159 | Cancellation of surreal mu... |
| mulscan1d 28160 | Cancellation of surreal mu... |
| muls12d 28161 | Commutative/associative la... |
| lemuls1ad 28162 | Multiplication of both sid... |
| ltmuls12ad 28163 | Comparison of the product ... |
| divsmo 28164 | Uniqueness of surreal inve... |
| muls0ord 28165 | If a surreal product is ze... |
| mulsne0bd 28166 | The product of two nonzero... |
| divsval 28169 | The value of surreal divis... |
| norecdiv 28170 | If a surreal has a recipro... |
| noreceuw 28171 | If a surreal has a recipro... |
| recsne0 28172 | If a surreal has a recipro... |
| divmulsw 28173 | Relationship between surre... |
| divmulswd 28174 | Relationship between surre... |
| divsclw 28175 | Weak division closure law.... |
| divsclwd 28176 | Weak division closure law.... |
| divscan2wd 28177 | A weak cancellation law fo... |
| divscan1wd 28178 | A weak cancellation law fo... |
| ltdivmulswd 28179 | Surreal less-than relation... |
| ltdivmuls2wd 28180 | Surreal less-than relation... |
| ltmuldivswd 28181 | Surreal less-than relation... |
| ltmuldivs2wd 28182 | Surreal less-than relation... |
| divsasswd 28183 | An associative law for sur... |
| divs1 28184 | A surreal divided by one i... |
| divs1d 28185 | A surreal divided by one i... |
| precsexlemcbv 28186 | Lemma for surreal reciproc... |
| precsexlem1 28187 | Lemma for surreal reciproc... |
| precsexlem2 28188 | Lemma for surreal reciproc... |
| precsexlem3 28189 | Lemma for surreal reciproc... |
| precsexlem4 28190 | Lemma for surreal reciproc... |
| precsexlem5 28191 | Lemma for surreal reciproc... |
| precsexlem6 28192 | Lemma for surreal reciproc... |
| precsexlem7 28193 | Lemma for surreal reciproc... |
| precsexlem8 28194 | Lemma for surreal reciproc... |
| precsexlem9 28195 | Lemma for surreal reciproc... |
| precsexlem10 28196 | Lemma for surreal reciproc... |
| precsexlem11 28197 | Lemma for surreal reciproc... |
| precsex 28198 | Every positive surreal has... |
| recsex 28199 | A nonzero surreal has a re... |
| recsexd 28200 | A nonzero surreal has a re... |
| divmuls 28201 | Relationship between surre... |
| divmulsd 28202 | Relationship between surre... |
| divscl 28203 | Surreal division closure l... |
| divscld 28204 | Surreal division closure l... |
| divscan2d 28205 | A cancellation law for sur... |
| divscan1d 28206 | A cancellation law for sur... |
| ltdivmulsd 28207 | Surreal less-than relation... |
| ltdivmuls2d 28208 | Surreal less-than relation... |
| ltmuldivsd 28209 | Surreal less-than relation... |
| ltmuldivs2d 28210 | Surreal less-than relation... |
| divsassd 28211 | An associative law for sur... |
| divmuldivsd 28212 | Multiplication of two surr... |
| divdivs1d 28213 | Surreal division into a fr... |
| divsrecd 28214 | Relationship between surre... |
| divsdird 28215 | Distribution of surreal di... |
| divscan3d 28216 | A cancellation law for sur... |
| abssval 28219 | The value of surreal absol... |
| absscl 28220 | Closure law for surreal ab... |
| abssid 28221 | The absolute value of a no... |
| abs0s 28222 | The absolute value of surr... |
| abssnid 28223 | For a negative surreal, it... |
| absmuls 28224 | Surreal absolute value dis... |
| abssge0 28225 | The absolute value of a su... |
| abssor 28226 | The absolute value of a su... |
| absnegs 28227 | Surreal absolute value of ... |
| leabss 28228 | A surreal is less than or ... |
| abslts 28229 | Surreal absolute value and... |
| abssubs 28230 | Swapping order of surreal ... |
| elons 28233 | Membership in the class of... |
| onssno 28234 | The surreal ordinals are a... |
| onno 28235 | A surreal ordinal is a sur... |
| 0ons 28236 | Surreal zero is a surreal ... |
| 1ons 28237 | Surreal one is a surreal o... |
| elons2 28238 | A surreal is ordinal iff i... |
| elons2d 28239 | The cut of any set of surr... |
| onleft 28240 | The left set of a surreal ... |
| ltonold 28241 | The class of ordinals less... |
| ltonsex 28242 | The class of ordinals less... |
| oncutleft 28243 | A surreal ordinal is equal... |
| oncutlt 28244 | A surreal ordinal is the s... |
| bday11on 28245 | The birthday function is o... |
| onnolt 28246 | If a surreal ordinal is le... |
| onlts 28247 | Less-than is the same as b... |
| onles 28248 | Less-than or equal is the ... |
| onltsd 28249 | Less-than is the same as b... |
| onlesd 28250 | Less-than or equal is the ... |
| oniso 28251 | The birthday function rest... |
| onswe 28252 | Surreal less-than well-ord... |
| onsse 28253 | Surreal less-than is set-l... |
| onsis 28254 | Transfinite induction sche... |
| ons2ind 28255 | Double induction schema fo... |
| bdayons 28256 | The birthday of a surreal ... |
| onaddscl 28257 | The surreal ordinals are c... |
| onmulscl 28258 | The surreal ordinals are c... |
| addonbday 28259 | The birthday of the sum of... |
| peano2ons 28260 | The successor of a surreal... |
| onsbnd 28261 | The surreals of a given bi... |
| onsbnd2 28262 | The surreals of a given bi... |
| seqsex 28265 | Existence of the surreal s... |
| seqseq123d 28266 | Equality deduction for the... |
| nfseqs 28267 | Hypothesis builder for the... |
| seqsval 28268 | The value of the surreal s... |
| noseqex 28269 | The next several theorems ... |
| noseq0 28270 | The surreal ` A ` is a mem... |
| noseqp1 28271 | One plus an element of ` Z... |
| noseqind 28272 | Peano's inductive postulat... |
| noseqinds 28273 | Induction schema for surre... |
| noseqssno 28274 | A surreal sequence is a su... |
| noseqno 28275 | An element of a surreal se... |
| om2noseq0 28276 | The mapping ` G ` is a one... |
| om2noseqsuc 28277 | The value of ` G ` at a su... |
| om2noseqfo 28278 | Function statement for ` G... |
| om2noseqlt 28279 | Surreal less-than relation... |
| om2noseqlt2 28280 | The mapping ` G ` preserve... |
| om2noseqf1o 28281 | ` G ` is a bijection. (Co... |
| om2noseqiso 28282 | ` G ` is an isomorphism fr... |
| om2noseqoi 28283 | An alternative definition ... |
| om2noseqrdg 28284 | A helper lemma for the val... |
| noseqrdglem 28285 | A helper lemma for the val... |
| noseqrdgfn 28286 | The recursive definition g... |
| noseqrdg0 28287 | Initial value of a recursi... |
| noseqrdgsuc 28288 | Successor value of a recur... |
| seqsfn 28289 | The surreal sequence build... |
| seqs1 28290 | The value of the surreal s... |
| seqsp1 28291 | The value of the surreal s... |
| n0sexg 28296 | The set of all non-negativ... |
| n0sex 28297 | The set of all non-negativ... |
| nnsex 28298 | The set of all positive su... |
| peano5n0s 28299 | Peano's inductive postulat... |
| n0ssno 28300 | The non-negative surreal i... |
| nnssn0s 28301 | The positive surreal integ... |
| nnssno 28302 | The positive surreal integ... |
| n0no 28303 | A non-negative surreal int... |
| nnno 28304 | A positive surreal integer... |
| n0nod 28305 | A non-negative surreal int... |
| nnnod 28306 | A positive surreal integer... |
| nnn0s 28307 | A positive surreal integer... |
| nnn0sd 28308 | A positive surreal integer... |
| 0n0s 28309 | Peano postulate: ` 0s ` is... |
| peano2n0s 28310 | Peano postulate: the succe... |
| peano2n0sd 28311 | Peano postulate: the succe... |
| dfn0s2 28312 | Alternate definition of th... |
| n0sind 28313 | Principle of Mathematical ... |
| n0cut 28314 | A cut form for non-negativ... |
| n0cut2 28315 | A cut form for the success... |
| n0on 28316 | A surreal natural is a sur... |
| nnne0s 28317 | A surreal positive integer... |
| n0sge0 28318 | A non-negative integer is ... |
| nnsgt0 28319 | A positive integer is grea... |
| elnns 28320 | Membership in the positive... |
| elnns2 28321 | A positive surreal integer... |
| n0s0suc 28322 | A non-negative surreal int... |
| nnsge1 28323 | A positive surreal integer... |
| n0addscl 28324 | The non-negative surreal i... |
| n0mulscl 28325 | The non-negative surreal i... |
| nnaddscl 28326 | The positive surreal integ... |
| nnmulscl 28327 | The positive surreal integ... |
| 1n0s 28328 | Surreal one is a non-negat... |
| 1nns 28329 | Surreal one is a positive ... |
| peano2nns 28330 | Peano postulate for positi... |
| nnsrecgt0d 28331 | The reciprocal of a positi... |
| n0bday 28332 | A non-negative surreal int... |
| n0ssoldg 28333 | The non-negative surreal i... |
| n0ssold 28334 | The non-negative surreal i... |
| n0fincut 28335 | The simplest number greate... |
| onsfi 28336 | A surreal ordinal with a f... |
| eln0s2 28337 | A non-negative surreal int... |
| onltn0s 28338 | A surreal ordinal that is ... |
| n0cutlt 28339 | A non-negative surreal int... |
| seqn0sfn 28340 | The surreal sequence build... |
| eln0s 28341 | A non-negative surreal int... |
| n0s0m1 28342 | Every non-negative surreal... |
| n0subs 28343 | Subtraction of non-negativ... |
| n0subs2 28344 | Subtraction of non-negativ... |
| n0ltsp1le 28345 | Non-negative surreal order... |
| n0lesltp1 28346 | Non-negative surreal order... |
| n0lesm1lt 28347 | Non-negative surreal order... |
| n0lts1e0 28348 | A non-negative surreal int... |
| bdayn0p1 28349 | The birthday of ` A +s 1s ... |
| bdayn0sf1o 28350 | The birthday function rest... |
| n0p1nns 28351 | One plus a non-negative su... |
| dfnns2 28352 | Alternate definition of th... |
| nnsind 28353 | Principle of Mathematical ... |
| nn1m1nns 28354 | Every positive surreal int... |
| nnm1n0s 28355 | A positive surreal integer... |
| eucliddivs 28356 | Euclid's division lemma fo... |
| oldfib 28357 | The old set of an ordinal ... |
| zsex 28360 | The surreal integers form ... |
| zssno 28361 | The surreal integers are a... |
| zno 28362 | A surreal integer is a sur... |
| znod 28363 | A surreal integer is a sur... |
| elzs 28364 | Membership in the set of s... |
| nnzsubs 28365 | The difference of two surr... |
| nnzs 28366 | A positive surreal integer... |
| nnzsd 28367 | A positive surreal integer... |
| 0zs 28368 | Zero is a surreal integer.... |
| n0zs 28369 | A non-negative surreal int... |
| n0zsd 28370 | A non-negative surreal int... |
| 1zs 28371 | One is a surreal integer. ... |
| znegscl 28372 | The surreal integers are c... |
| znegscld 28373 | The surreal integers are c... |
| zaddscl 28374 | The surreal integers are c... |
| zaddscld 28375 | The surreal integers are c... |
| zsubscld 28376 | The surreal integers are c... |
| zmulscld 28377 | The surreal integers are c... |
| elzn0s 28378 | A surreal integer is a sur... |
| elzs2 28379 | A surreal integer is eithe... |
| eln0zs 28380 | Non-negative surreal integ... |
| elnnzs 28381 | Positive surreal integer p... |
| elznns 28382 | Surreal integer property e... |
| zn0subs 28383 | The non-negative differenc... |
| peano5uzs 28384 | Peano's inductive postulat... |
| uzsind 28385 | Induction on the upper sur... |
| zsbday 28386 | A surreal integer has a fi... |
| zcuts 28387 | A cut expression for surre... |
| zcuts0 28388 | Either the left or right s... |
| zsoring 28389 | The surreal integers form ... |
| 1p1e2s 28396 | One plus one is two. Surr... |
| no2times 28397 | Version of ~ 2times for su... |
| 2nns 28398 | Surreal two is a surreal n... |
| 2no 28399 | Surreal two is a surreal n... |
| 2ne0s 28400 | Surreal two is nonzero. (... |
| n0seo 28401 | A non-negative surreal int... |
| zseo 28402 | A surreal integer is eithe... |
| twocut 28403 | Two times the cut of zero ... |
| nohalf 28404 | An explicit expression for... |
| expsval 28405 | The value of surreal expon... |
| expnnsval 28406 | Value of surreal exponenti... |
| exps0 28407 | Surreal exponentiation to ... |
| exps1 28408 | Surreal exponentiation to ... |
| expsp1 28409 | Value of a surreal number ... |
| expscllem 28410 | Lemma for proving non-nega... |
| expscl 28411 | Closure law for surreal ex... |
| n0expscl 28412 | Closure law for non-negati... |
| nnexpscl 28413 | Closure law for positive s... |
| zexpscl 28414 | Closure law for surreal in... |
| expadds 28415 | Sum of exponents law for s... |
| expsne0 28416 | A non-negative surreal int... |
| expsgt0 28417 | A non-negative surreal int... |
| pw2recs 28418 | Any power of two has a mul... |
| pw2divscld 28419 | Division closure for power... |
| pw2divmulsd 28420 | Relationship between surre... |
| pw2divscan3d 28421 | Cancellation law for surre... |
| pw2divscan2d 28422 | A cancellation law for sur... |
| pw2divsassd 28423 | An associative law for div... |
| pw2divscan4d 28424 | Cancellation law for divis... |
| pw2gt0divsd 28425 | Division of a positive sur... |
| pw2ge0divsd 28426 | Divison of a non-negative ... |
| pw2divsrecd 28427 | Relationship between surre... |
| pw2divsdird 28428 | Distribution of surreal di... |
| pw2divsnegd 28429 | Move negative sign inside ... |
| pw2ltdivmulsd 28430 | Surreal less-than relation... |
| pw2ltmuldivs2d 28431 | Surreal less-than relation... |
| pw2ltsdiv1d 28432 | Surreal less-than relation... |
| avglts1d 28433 | Ordering property for aver... |
| avglts2d 28434 | Ordering property for aver... |
| pw2divs0d 28435 | Division into zero is zero... |
| pw2divsidd 28436 | Identity law for division ... |
| pw2ltdivmuls2d 28437 | Surreal less-than relation... |
| halfcut 28438 | Relate the cut of twice of... |
| addhalfcut 28439 | The cut of a surreal non-n... |
| pw2cut 28440 | Extend ~ halfcut to arbitr... |
| pw2cutp1 28441 | Simplify ~ pw2cut in the c... |
| pw2cut2 28442 | Cut expression for powers ... |
| bdaypw2n0bndlem 28443 | Lemma for ~ bdaypw2n0bnd .... |
| bdaypw2n0bnd 28444 | Upper bound for the birthd... |
| bdaypw2bnd 28445 | Birthday bounding rule for... |
| bdayfinbndcbv 28446 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem1 28447 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbndlem2 28448 | Lemma for ~ bdayfinbnd . ... |
| bdayfinbnd 28449 | Given a non-negative integ... |
| z12bdaylem1 28450 | Lemma for ~ z12bday . Pro... |
| z12bdaylem2 28451 | Lemma for ~ z12bday . Sho... |
| elz12s 28452 | Membership in the dyadic f... |
| elz12si 28453 | Inference form of membersh... |
| z12sex 28454 | The class of dyadic fracti... |
| zz12s 28455 | A surreal integer is a dya... |
| z12no 28456 | A dyadic is a surreal. (C... |
| z12addscl 28457 | The dyadics are closed und... |
| z12negscl 28458 | The dyadics are closed und... |
| z12subscl 28459 | The dyadics are closed und... |
| z12shalf 28460 | Half of a dyadic is a dyad... |
| z12negsclb 28461 | A surreal is a dyadic frac... |
| z12zsodd 28462 | A dyadic fraction is eithe... |
| z12sge0 28463 | An expression for non-nega... |
| z12bdaylem 28464 | Lemma for ~ z12bday . Han... |
| z12bday 28465 | A dyadic fraction has a fi... |
| bdayfinlem 28466 | Lemma for ~ bdayfin . Han... |
| bdayfin 28467 | A surreal has a finite bir... |
| dfz12s2 28468 | The set of dyadic fraction... |
| elreno 28471 | Membership in the set of s... |
| reno 28472 | A surreal real is a surrea... |
| renod 28473 | A surreal real is a surrea... |
| recut 28474 | The cut involved in defini... |
| elreno2 28475 | Alternate characterization... |
| 0reno 28476 | Surreal zero is a surreal ... |
| 1reno 28477 | Surreal one is a surreal r... |
| renegscl 28478 | The surreal reals are clos... |
| readdscl 28479 | The surreal reals are clos... |
| remulscllem1 28480 | Lemma for ~ remulscl . Sp... |
| remulscllem2 28481 | Lemma for ~ remulscl . Bo... |
| remulscl 28482 | The surreal reals are clos... |
| itvndx 28493 | Index value of the Interva... |
| lngndx 28494 | Index value of the "line" ... |
| itvid 28495 | Utility theorem: index-ind... |
| lngid 28496 | Utility theorem: index-ind... |
| slotsinbpsd 28497 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 28498 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 28499 | The slot for the line is n... |
| trkgstr 28500 | Functionality of a Tarski ... |
| trkgbas 28501 | The base set of a Tarski g... |
| trkgdist 28502 | The measure of a distance ... |
| trkgitv 28503 | The congruence relation in... |
| istrkgc 28510 | Property of being a Tarski... |
| istrkgb 28511 | Property of being a Tarski... |
| istrkgcb 28512 | Property of being a Tarski... |
| istrkge 28513 | Property of fulfilling Euc... |
| istrkgl 28514 | Building lines from the se... |
| istrkgld 28515 | Property of fulfilling the... |
| istrkg2ld 28516 | Property of fulfilling the... |
| istrkg3ld 28517 | Property of fulfilling the... |
| axtgcgrrflx 28518 | Axiom of reflexivity of co... |
| axtgcgrid 28519 | Axiom of identity of congr... |
| axtgsegcon 28520 | Axiom of segment construct... |
| axtg5seg 28521 | Five segments axiom, Axiom... |
| axtgbtwnid 28522 | Identity of Betweenness. ... |
| axtgpasch 28523 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 28524 | Axiom of Continuity. Axio... |
| axtgcont 28525 | Axiom of Continuity. Axio... |
| axtglowdim2 28526 | Lower dimension axiom for ... |
| axtgupdim2 28527 | Upper dimension axiom for ... |
| axtgeucl 28528 | Euclid's Axiom. Axiom A10... |
| tgjustf 28529 | Given any function ` F ` ,... |
| tgjustr 28530 | Given any equivalence rela... |
| tgjustc1 28531 | A justification for using ... |
| tgjustc2 28532 | A justification for using ... |
| tgcgrcomimp 28533 | Congruence commutes on the... |
| tgcgrcomr 28534 | Congruence commutes on the... |
| tgcgrcoml 28535 | Congruence commutes on the... |
| tgcgrcomlr 28536 | Congruence commutes on bot... |
| tgcgreqb 28537 | Congruence and equality. ... |
| tgcgreq 28538 | Congruence and equality. ... |
| tgcgrneq 28539 | Congruence and equality. ... |
| tgcgrtriv 28540 | Degenerate segments are co... |
| tgcgrextend 28541 | Link congruence over a pai... |
| tgsegconeq 28542 | Two points that satisfy th... |
| tgbtwntriv2 28543 | Betweenness always holds f... |
| tgbtwncom 28544 | Betweenness commutes. The... |
| tgbtwncomb 28545 | Betweenness commutes, bico... |
| tgbtwnne 28546 | Betweenness and inequality... |
| tgbtwntriv1 28547 | Betweenness always holds f... |
| tgbtwnswapid 28548 | If you can swap the first ... |
| tgbtwnintr 28549 | Inner transitivity law for... |
| tgbtwnexch3 28550 | Exchange the first endpoin... |
| tgbtwnouttr2 28551 | Outer transitivity law for... |
| tgbtwnexch2 28552 | Exchange the outer point o... |
| tgbtwnouttr 28553 | Outer transitivity law for... |
| tgbtwnexch 28554 | Outer transitivity law for... |
| tgtrisegint 28555 | A line segment between two... |
| tglowdim1 28556 | Lower dimension axiom for ... |
| tglowdim1i 28557 | Lower dimension axiom for ... |
| tgldimor 28558 | Excluded-middle like state... |
| tgldim0eq 28559 | In dimension zero, any two... |
| tgldim0itv 28560 | In dimension zero, any two... |
| tgldim0cgr 28561 | In dimension zero, any two... |
| tgbtwndiff 28562 | There is always a ` c ` di... |
| tgdim01 28563 | In geometries of dimension... |
| tgifscgr 28564 | Inner five segment congrue... |
| tgcgrsub 28565 | Removing identical parts f... |
| iscgrg 28568 | The congruence property fo... |
| iscgrgd 28569 | The property for two seque... |
| iscgrglt 28570 | The property for two seque... |
| trgcgrg 28571 | The property for two trian... |
| trgcgr 28572 | Triangle congruence. (Con... |
| ercgrg 28573 | The shape congruence relat... |
| tgcgrxfr 28574 | A line segment can be divi... |
| cgr3id 28575 | Reflexivity law for three-... |
| cgr3simp1 28576 | Deduce segment congruence ... |
| cgr3simp2 28577 | Deduce segment congruence ... |
| cgr3simp3 28578 | Deduce segment congruence ... |
| cgr3swap12 28579 | Permutation law for three-... |
| cgr3swap23 28580 | Permutation law for three-... |
| cgr3swap13 28581 | Permutation law for three-... |
| cgr3rotr 28582 | Permutation law for three-... |
| cgr3rotl 28583 | Permutation law for three-... |
| trgcgrcom 28584 | Commutative law for three-... |
| cgr3tr 28585 | Transitivity law for three... |
| tgbtwnxfr 28586 | A condition for extending ... |
| tgcgr4 28587 | Two quadrilaterals to be c... |
| isismt 28590 | Property of being an isome... |
| ismot 28591 | Property of being an isome... |
| motcgr 28592 | Property of a motion: dist... |
| idmot 28593 | The identity is a motion. ... |
| motf1o 28594 | Motions are bijections. (... |
| motcl 28595 | Closure of motions. (Cont... |
| motco 28596 | The composition of two mot... |
| cnvmot 28597 | The converse of a motion i... |
| motplusg 28598 | The operation for motions ... |
| motgrp 28599 | The motions of a geometry ... |
| motcgrg 28600 | Property of a motion: dist... |
| motcgr3 28601 | Property of a motion: dist... |
| tglng 28602 | Lines of a Tarski Geometry... |
| tglnfn 28603 | Lines as functions. (Cont... |
| tglnunirn 28604 | Lines are sets of points. ... |
| tglnpt 28605 | Lines are sets of points. ... |
| tglngne 28606 | It takes two different poi... |
| tglngval 28607 | The line going through poi... |
| tglnssp 28608 | Lines are subset of the ge... |
| tgellng 28609 | Property of lying on the l... |
| tgcolg 28610 | We choose the notation ` (... |
| btwncolg1 28611 | Betweenness implies coline... |
| btwncolg2 28612 | Betweenness implies coline... |
| btwncolg3 28613 | Betweenness implies coline... |
| colcom 28614 | Swapping the points defini... |
| colrot1 28615 | Rotating the points defini... |
| colrot2 28616 | Rotating the points defini... |
| ncolcom 28617 | Swapping non-colinear poin... |
| ncolrot1 28618 | Rotating non-colinear poin... |
| ncolrot2 28619 | Rotating non-colinear poin... |
| tgdim01ln 28620 | In geometries of dimension... |
| ncoltgdim2 28621 | If there are three non-col... |
| lnxfr 28622 | Transfer law for colineari... |
| lnext 28623 | Extend a line with a missi... |
| tgfscgr 28624 | Congruence law for the gen... |
| lncgr 28625 | Congruence rule for lines.... |
| lnid 28626 | Identity law for points on... |
| tgidinside 28627 | Law for finding a point in... |
| tgbtwnconn1lem1 28628 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28629 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28630 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28631 | Connectivity law for betwe... |
| tgbtwnconn2 28632 | Another connectivity law f... |
| tgbtwnconn3 28633 | Inner connectivity law for... |
| tgbtwnconnln3 28634 | Derive colinearity from be... |
| tgbtwnconn22 28635 | Double connectivity law fo... |
| tgbtwnconnln1 28636 | Derive colinearity from be... |
| tgbtwnconnln2 28637 | Derive colinearity from be... |
| legval 28640 | Value of the less-than rel... |
| legov 28641 | Value of the less-than rel... |
| legov2 28642 | An equivalent definition o... |
| legid 28643 | Reflexivity of the less-th... |
| btwnleg 28644 | Betweenness implies less-t... |
| legtrd 28645 | Transitivity of the less-t... |
| legtri3 28646 | Equality from the less-tha... |
| legtrid 28647 | Trichotomy law for the les... |
| leg0 28648 | Degenerated (zero-length) ... |
| legeq 28649 | Deduce equality from "less... |
| legbtwn 28650 | Deduce betweenness from "l... |
| tgcgrsub2 28651 | Removing identical parts f... |
| ltgseg 28652 | The set ` E ` denotes the ... |
| ltgov 28653 | Strict "shorter than" geom... |
| legov3 28654 | An equivalent definition o... |
| legso 28655 | The "shorter than" relatio... |
| ishlg 28658 | Rays : Definition 6.1 of ... |
| hlcomb 28659 | The half-line relation com... |
| hlcomd 28660 | The half-line relation com... |
| hlne1 28661 | The half-line relation imp... |
| hlne2 28662 | The half-line relation imp... |
| hlln 28663 | The half-line relation imp... |
| hleqnid 28664 | The endpoint does not belo... |
| hlid 28665 | The half-line relation is ... |
| hltr 28666 | The half-line relation is ... |
| hlbtwn 28667 | Betweenness is a sufficien... |
| btwnhl1 28668 | Deduce half-line from betw... |
| btwnhl2 28669 | Deduce half-line from betw... |
| btwnhl 28670 | Swap betweenness for a hal... |
| lnhl 28671 | Either a point ` C ` on th... |
| hlcgrex 28672 | Construct a point on a hal... |
| hlcgreulem 28673 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28674 | The point constructed in ~... |
| btwnlng1 28675 | Betweenness implies coline... |
| btwnlng2 28676 | Betweenness implies coline... |
| btwnlng3 28677 | Betweenness implies coline... |
| lncom 28678 | Swapping the points defini... |
| lnrot1 28679 | Rotating the points defini... |
| lnrot2 28680 | Rotating the points defini... |
| ncolne1 28681 | Non-colinear points are di... |
| ncolne2 28682 | Non-colinear points are di... |
| tgisline 28683 | The property of being a pr... |
| tglnne 28684 | It takes two different poi... |
| tglndim0 28685 | There are no lines in dime... |
| tgelrnln 28686 | The property of being a pr... |
| tglineeltr 28687 | Transitivity law for lines... |
| tglineelsb2 28688 | If ` S ` lies on PQ , then... |
| tglinerflx1 28689 | Reflexivity law for line m... |
| tglinerflx2 28690 | Reflexivity law for line m... |
| tglinecom 28691 | Commutativity law for line... |
| tglinethru 28692 | If ` A ` is a line contain... |
| tghilberti1 28693 | There is a line through an... |
| tghilberti2 28694 | There is at most one line ... |
| tglinethrueu 28695 | There is a unique line goi... |
| tglnne0 28696 | A line ` A ` has at least ... |
| tglnpt2 28697 | Find a second point on a l... |
| tglineintmo 28698 | Two distinct lines interse... |
| tglineineq 28699 | Two distinct lines interse... |
| tglineneq 28700 | Given three non-colinear p... |
| tglineinteq 28701 | Two distinct lines interse... |
| ncolncol 28702 | Deduce non-colinearity fro... |
| coltr 28703 | A transitivity law for col... |
| coltr3 28704 | A transitivity law for col... |
| colline 28705 | Three points are colinear ... |
| tglowdim2l 28706 | Reformulation of the lower... |
| tglowdim2ln 28707 | There is always one point ... |
| mirreu3 28710 | Existential uniqueness of ... |
| mirval 28711 | Value of the point inversi... |
| mirfv 28712 | Value of the point inversi... |
| mircgr 28713 | Property of the image by t... |
| mirbtwn 28714 | Property of the image by t... |
| ismir 28715 | Property of the image by t... |
| mirf 28716 | Point inversion as functio... |
| mircl 28717 | Closure of the point inver... |
| mirmir 28718 | The point inversion functi... |
| mircom 28719 | Variation on ~ mirmir . (... |
| mirreu 28720 | Any point has a unique ant... |
| mireq 28721 | Equality deduction for poi... |
| mirinv 28722 | The only invariant point o... |
| mirne 28723 | Mirror of non-center point... |
| mircinv 28724 | The center point is invari... |
| mirf1o 28725 | The point inversion functi... |
| miriso 28726 | The point inversion functi... |
| mirbtwni 28727 | Point inversion preserves ... |
| mirbtwnb 28728 | Point inversion preserves ... |
| mircgrs 28729 | Point inversion preserves ... |
| mirmir2 28730 | Point inversion of a point... |
| mirmot 28731 | Point investion is a motio... |
| mirln 28732 | If two points are on the s... |
| mirln2 28733 | If a point and its mirror ... |
| mirconn 28734 | Point inversion of connect... |
| mirhl 28735 | If two points ` X ` and ` ... |
| mirbtwnhl 28736 | If the center of the point... |
| mirhl2 28737 | Deduce half-line relation ... |
| mircgrextend 28738 | Link congruence over a pai... |
| mirtrcgr 28739 | Point inversion of one poi... |
| mirauto 28740 | Point inversion preserves ... |
| miduniq 28741 | Uniqueness of the middle p... |
| miduniq1 28742 | Uniqueness of the middle p... |
| miduniq2 28743 | If two point inversions co... |
| colmid 28744 | Colinearity and equidistan... |
| symquadlem 28745 | Lemma of the symmetrical q... |
| krippenlem 28746 | Lemma for ~ krippen . We ... |
| krippen 28747 | Krippenlemma (German for c... |
| midexlem 28748 | Lemma for the existence of... |
| israg 28753 | Property for 3 points A, B... |
| ragcom 28754 | Commutative rule for right... |
| ragcol 28755 | The right angle property i... |
| ragmir 28756 | Right angle property is pr... |
| mirrag 28757 | Right angle is conserved b... |
| ragtrivb 28758 | Trivial right angle. Theo... |
| ragflat2 28759 | Deduce equality from two r... |
| ragflat 28760 | Deduce equality from two r... |
| ragtriva 28761 | Trivial right angle. Theo... |
| ragflat3 28762 | Right angle and colinearit... |
| ragcgr 28763 | Right angle and colinearit... |
| motrag 28764 | Right angles are preserved... |
| ragncol 28765 | Right angle implies non-co... |
| perpln1 28766 | Derive a line from perpend... |
| perpln2 28767 | Derive a line from perpend... |
| isperp 28768 | Property for 2 lines A, B ... |
| perpcom 28769 | The "perpendicular" relati... |
| perpneq 28770 | Two perpendicular lines ar... |
| isperp2 28771 | Property for 2 lines A, B,... |
| isperp2d 28772 | One direction of ~ isperp2... |
| ragperp 28773 | Deduce that two lines are ... |
| footexALT 28774 | Alternative version of ~ f... |
| footexlem1 28775 | Lemma for ~ footex . (Con... |
| footexlem2 28776 | Lemma for ~ footex . (Con... |
| footex 28777 | From a point ` C ` outside... |
| foot 28778 | From a point ` C ` outside... |
| footne 28779 | Uniqueness of the foot poi... |
| footeq 28780 | Uniqueness of the foot poi... |
| hlperpnel 28781 | A point on a half-line whi... |
| perprag 28782 | Deduce a right angle from ... |
| perpdragALT 28783 | Deduce a right angle from ... |
| perpdrag 28784 | Deduce a right angle from ... |
| colperp 28785 | Deduce a perpendicularity ... |
| colperpexlem1 28786 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28787 | Lemma for ~ colperpex . S... |
| colperpexlem3 28788 | Lemma for ~ colperpex . C... |
| colperpex 28789 | In dimension 2 and above, ... |
| mideulem2 28790 | Lemma for ~ opphllem , whi... |
| opphllem 28791 | Lemma 8.24 of [Schwabhause... |
| mideulem 28792 | Lemma for ~ mideu . We ca... |
| midex 28793 | Existence of the midpoint,... |
| mideu 28794 | Existence and uniqueness o... |
| islnopp 28795 | The property for two point... |
| islnoppd 28796 | Deduce that ` A ` and ` B ... |
| oppne1 28797 | Points lying on opposite s... |
| oppne2 28798 | Points lying on opposite s... |
| oppne3 28799 | Points lying on opposite s... |
| oppcom 28800 | Commutativity rule for "op... |
| opptgdim2 28801 | If two points opposite to ... |
| oppnid 28802 | The "opposite to a line" r... |
| opphllem1 28803 | Lemma for ~ opphl . (Cont... |
| opphllem2 28804 | Lemma for ~ opphl . Lemma... |
| opphllem3 28805 | Lemma for ~ opphl : We as... |
| opphllem4 28806 | Lemma for ~ opphl . (Cont... |
| opphllem5 28807 | Second part of Lemma 9.4 o... |
| opphllem6 28808 | First part of Lemma 9.4 of... |
| oppperpex 28809 | Restating ~ colperpex usin... |
| opphl 28810 | If two points ` A ` and ` ... |
| outpasch 28811 | Axiom of Pasch, outer form... |
| hlpasch 28812 | An application of the axio... |
| ishpg 28815 | Value of the half-plane re... |
| hpgbr 28816 | Half-planes : property for... |
| hpgne1 28817 | Points on the open half pl... |
| hpgne2 28818 | Points on the open half pl... |
| lnopp2hpgb 28819 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28820 | If two points lie on the o... |
| hpgerlem 28821 | Lemma for the proof that t... |
| hpgid 28822 | The half-plane relation is... |
| hpgcom 28823 | The half-plane relation co... |
| hpgtr 28824 | The half-plane relation is... |
| colopp 28825 | Opposite sides of a line f... |
| colhp 28826 | Half-plane relation for co... |
| hphl 28827 | If two points are on the s... |
| midf 28832 | Midpoint as a function. (... |
| midcl 28833 | Closure of the midpoint. ... |
| ismidb 28834 | Property of the midpoint. ... |
| midbtwn 28835 | Betweenness of midpoint. ... |
| midcgr 28836 | Congruence of midpoint. (... |
| midid 28837 | Midpoint of a null segment... |
| midcom 28838 | Commutativity rule for the... |
| mirmid 28839 | Point inversion preserves ... |
| lmieu 28840 | Uniqueness of the line mir... |
| lmif 28841 | Line mirror as a function.... |
| lmicl 28842 | Closure of the line mirror... |
| islmib 28843 | Property of the line mirro... |
| lmicom 28844 | The line mirroring functio... |
| lmilmi 28845 | Line mirroring is an invol... |
| lmireu 28846 | Any point has a unique ant... |
| lmieq 28847 | Equality deduction for lin... |
| lmiinv 28848 | The invariants of the line... |
| lmicinv 28849 | The mirroring line is an i... |
| lmimid 28850 | If we have a right angle, ... |
| lmif1o 28851 | The line mirroring functio... |
| lmiisolem 28852 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28853 | The line mirroring functio... |
| lmimot 28854 | Line mirroring is a motion... |
| hypcgrlem1 28855 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28856 | Lemma for ~ hypcgr , case ... |
| hypcgr 28857 | If the catheti of two righ... |
| lmiopp 28858 | Line mirroring produces po... |
| lnperpex 28859 | Existence of a perpendicul... |
| trgcopy 28860 | Triangle construction: a c... |
| trgcopyeulem 28861 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28862 | Triangle construction: a c... |
| iscgra 28865 | Property for two angles AB... |
| iscgra1 28866 | A special version of ~ isc... |
| iscgrad 28867 | Sufficient conditions for ... |
| cgrane1 28868 | Angles imply inequality. ... |
| cgrane2 28869 | Angles imply inequality. ... |
| cgrane3 28870 | Angles imply inequality. ... |
| cgrane4 28871 | Angles imply inequality. ... |
| cgrahl1 28872 | Angle congruence is indepe... |
| cgrahl2 28873 | Angle congruence is indepe... |
| cgracgr 28874 | First direction of proposi... |
| cgraid 28875 | Angle congruence is reflex... |
| cgraswap 28876 | Swap rays in a congruence ... |
| cgrcgra 28877 | Triangle congruence implie... |
| cgracom 28878 | Angle congruence commutes.... |
| cgratr 28879 | Angle congruence is transi... |
| flatcgra 28880 | Flat angles are congruent.... |
| cgraswaplr 28881 | Swap both side of angle co... |
| cgrabtwn 28882 | Angle congruence preserves... |
| cgrahl 28883 | Angle congruence preserves... |
| cgracol 28884 | Angle congruence preserves... |
| cgrancol 28885 | Angle congruence preserves... |
| dfcgra2 28886 | This is the full statement... |
| sacgr 28887 | Supplementary angles of co... |
| oacgr 28888 | Vertical angle theorem. V... |
| acopy 28889 | Angle construction. Theor... |
| acopyeu 28890 | Angle construction. Theor... |
| isinag 28894 | Property for point ` X ` t... |
| isinagd 28895 | Sufficient conditions for ... |
| inagflat 28896 | Any point lies in a flat a... |
| inagswap 28897 | Swap the order of the half... |
| inagne1 28898 | Deduce inequality from the... |
| inagne2 28899 | Deduce inequality from the... |
| inagne3 28900 | Deduce inequality from the... |
| inaghl 28901 | The "point lie in angle" r... |
| isleag 28903 | Geometrical "less than" pr... |
| isleagd 28904 | Sufficient condition for "... |
| leagne1 28905 | Deduce inequality from the... |
| leagne2 28906 | Deduce inequality from the... |
| leagne3 28907 | Deduce inequality from the... |
| leagne4 28908 | Deduce inequality from the... |
| cgrg3col4 28909 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28910 | First congruence theorem: ... |
| tgsas 28911 | First congruence theorem: ... |
| tgsas2 28912 | First congruence theorem: ... |
| tgsas3 28913 | First congruence theorem: ... |
| tgasa1 28914 | Second congruence theorem:... |
| tgasa 28915 | Second congruence theorem:... |
| tgsss1 28916 | Third congruence theorem: ... |
| tgsss2 28917 | Third congruence theorem: ... |
| tgsss3 28918 | Third congruence theorem: ... |
| dfcgrg2 28919 | Congruence for two triangl... |
| isoas 28920 | Congruence theorem for iso... |
| iseqlg 28923 | Property of a triangle bei... |
| iseqlgd 28924 | Condition for a triangle t... |
| f1otrgds 28925 | Convenient lemma for ~ f1o... |
| f1otrgitv 28926 | Convenient lemma for ~ f1o... |
| f1otrg 28927 | A bijection between bases ... |
| f1otrge 28928 | A bijection between bases ... |
| ttgval 28931 | Define a function to augme... |
| ttglem 28932 | Lemma for ~ ttgbas , ~ ttg... |
| ttgbas 28933 | The base set of a subcompl... |
| ttgplusg 28934 | The addition operation of ... |
| ttgsub 28935 | The subtraction operation ... |
| ttgvsca 28936 | The scalar product of a su... |
| ttgds 28937 | The metric of a subcomplex... |
| ttgitvval 28938 | Betweenness for a subcompl... |
| ttgelitv 28939 | Betweenness for a subcompl... |
| ttgbtwnid 28940 | Any subcomplex module equi... |
| ttgcontlem1 28941 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28942 | Any metric space fulfills ... |
| cchhllem 28943 | Lemma for chlbas and chlvs... |
| elee 28950 | Membership in a Euclidean ... |
| mptelee 28951 | A condition for a mapping ... |
| mpteleeOLD 28952 | Obsolete version of ~ mpte... |
| eleenn 28953 | If ` A ` is in ` ( EE `` N... |
| eleei 28954 | The forward direction of ~... |
| eedimeq 28955 | A point belongs to at most... |
| brbtwn 28956 | The binary relation form o... |
| brcgr 28957 | The binary relation form o... |
| fveere 28958 | The function value of a po... |
| fveecn 28959 | The function value of a po... |
| eqeefv 28960 | Two points are equal iff t... |
| eqeelen 28961 | Two points are equal iff t... |
| brbtwn2 28962 | Alternate characterization... |
| colinearalglem1 28963 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28964 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28965 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28966 | Lemma for ~ colinearalg . ... |
| colinearalg 28967 | An algebraic characterizat... |
| eleesub 28968 | Membership of a subtractio... |
| eleesubd 28969 | Membership of a subtractio... |
| axdimuniq 28970 | The unique dimension axiom... |
| axcgrrflx 28971 | ` A ` is as far from ` B `... |
| axcgrtr 28972 | Congruence is transitive. ... |
| axcgrid 28973 | If there is no distance be... |
| axsegconlem1 28974 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28975 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28976 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28977 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28978 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28979 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28980 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28981 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28982 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28983 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28984 | Any segment ` A B ` can be... |
| ax5seglem1 28985 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28986 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28987 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28988 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28989 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28990 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28991 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28992 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28993 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28994 | Lemma for ~ ax5seg . Take... |
| ax5seg 28995 | The five segment axiom. T... |
| axbtwnid 28996 | Points are indivisible. T... |
| axpaschlem 28997 | Lemma for ~ axpasch . Set... |
| axpasch 28998 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28999 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 29000 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 29001 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 29002 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 29003 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 29004 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 29005 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 29006 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 29007 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 29008 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 29009 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 29010 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 29011 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 29012 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 29013 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 29014 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 29015 | Lemma for ~ axlowdim . Es... |
| axlowdim1 29016 | The lower dimension axiom ... |
| axlowdim2 29017 | The lower two-dimensional ... |
| axlowdim 29018 | The general lower dimensio... |
| axeuclidlem 29019 | Lemma for ~ axeuclid . Ha... |
| axeuclid 29020 | Euclid's axiom. Take an a... |
| axcontlem1 29021 | Lemma for ~ axcont . Chan... |
| axcontlem2 29022 | Lemma for ~ axcont . The ... |
| axcontlem3 29023 | Lemma for ~ axcont . Give... |
| axcontlem4 29024 | Lemma for ~ axcont . Give... |
| axcontlem5 29025 | Lemma for ~ axcont . Comp... |
| axcontlem6 29026 | Lemma for ~ axcont . Stat... |
| axcontlem7 29027 | Lemma for ~ axcont . Give... |
| axcontlem8 29028 | Lemma for ~ axcont . A po... |
| axcontlem9 29029 | Lemma for ~ axcont . Give... |
| axcontlem10 29030 | Lemma for ~ axcont . Give... |
| axcontlem11 29031 | Lemma for ~ axcont . Elim... |
| axcontlem12 29032 | Lemma for ~ axcont . Elim... |
| axcont 29033 | The axiom of continuity. ... |
| eengv 29036 | The value of the Euclidean... |
| eengstr 29037 | The Euclidean geometry as ... |
| eengbas 29038 | The Base of the Euclidean ... |
| ebtwntg 29039 | The betweenness relation u... |
| ecgrtg 29040 | The congruence relation us... |
| elntg 29041 | The line definition in the... |
| elntg2 29042 | The line definition in the... |
| eengtrkg 29043 | The geometry structure for... |
| eengtrkge 29044 | The geometry structure for... |
| edgfid 29047 | Utility theorem: index-ind... |
| edgfndx 29048 | Index value of the ~ df-ed... |
| edgfndxnn 29049 | The index value of the edg... |
| edgfndxid 29050 | The value of the edge func... |
| basendxltedgfndx 29051 | The index value of the ` B... |
| basendxnedgfndx 29052 | The slots ` Base ` and ` .... |
| vtxval 29057 | The set of vertices of a g... |
| iedgval 29058 | The set of indexed edges o... |
| 1vgrex 29059 | A graph with at least one ... |
| opvtxval 29060 | The set of vertices of a g... |
| opvtxfv 29061 | The set of vertices of a g... |
| opvtxov 29062 | The set of vertices of a g... |
| opiedgval 29063 | The set of indexed edges o... |
| opiedgfv 29064 | The set of indexed edges o... |
| opiedgov 29065 | The set of indexed edges o... |
| opvtxfvi 29066 | The set of vertices of a g... |
| opiedgfvi 29067 | The set of indexed edges o... |
| funvtxdmge2val 29068 | The set of vertices of an ... |
| funiedgdmge2val 29069 | The set of indexed edges o... |
| funvtxdm2val 29070 | The set of vertices of an ... |
| funiedgdm2val 29071 | The set of indexed edges o... |
| funvtxval0 29072 | The set of vertices of an ... |
| basvtxval 29073 | The set of vertices of a g... |
| edgfiedgval 29074 | The set of indexed edges o... |
| funvtxval 29075 | The set of vertices of a g... |
| funiedgval 29076 | The set of indexed edges o... |
| structvtxvallem 29077 | Lemma for ~ structvtxval a... |
| structvtxval 29078 | The set of vertices of an ... |
| structiedg0val 29079 | The set of indexed edges o... |
| structgrssvtxlem 29080 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 29081 | The set of vertices of a g... |
| structgrssiedg 29082 | The set of indexed edges o... |
| struct2grstr 29083 | A graph represented as an ... |
| struct2grvtx 29084 | The set of vertices of a g... |
| struct2griedg 29085 | The set of indexed edges o... |
| graop 29086 | Any representation of a gr... |
| grastruct 29087 | Any representation of a gr... |
| gropd 29088 | If any representation of a... |
| grstructd 29089 | If any representation of a... |
| gropeld 29090 | If any representation of a... |
| grstructeld 29091 | If any representation of a... |
| setsvtx 29092 | The vertices of a structur... |
| setsiedg 29093 | The (indexed) edges of a s... |
| snstrvtxval 29094 | The set of vertices of a g... |
| snstriedgval 29095 | The set of indexed edges o... |
| vtxval0 29096 | Degenerated case 1 for ver... |
| iedgval0 29097 | Degenerated case 1 for edg... |
| vtxvalsnop 29098 | Degenerated case 2 for ver... |
| iedgvalsnop 29099 | Degenerated case 2 for edg... |
| vtxval3sn 29100 | Degenerated case 3 for ver... |
| iedgval3sn 29101 | Degenerated case 3 for edg... |
| vtxvalprc 29102 | Degenerated case 4 for ver... |
| iedgvalprc 29103 | Degenerated case 4 for edg... |
| edgval 29106 | The edges of a graph. (Co... |
| iedgedg 29107 | An indexed edge is an edge... |
| edgopval 29108 | The edges of a graph repre... |
| edgov 29109 | The edges of a graph repre... |
| edgstruct 29110 | The edges of a graph repre... |
| edgiedgb 29111 | A set is an edge iff it is... |
| edg0iedg0 29112 | There is no edge in a grap... |
| isuhgr 29117 | The predicate "is an undir... |
| isushgr 29118 | The predicate "is an undir... |
| uhgrf 29119 | The edge function of an un... |
| ushgrf 29120 | The edge function of an un... |
| uhgrss 29121 | An edge is a subset of ver... |
| uhgreq12g 29122 | If two sets have the same ... |
| uhgrfun 29123 | The edge function of an un... |
| uhgrn0 29124 | An edge is a nonempty subs... |
| lpvtx 29125 | The endpoints of a loop (w... |
| ushgruhgr 29126 | An undirected simple hyper... |
| isuhgrop 29127 | The property of being an u... |
| uhgr0e 29128 | The empty graph, with vert... |
| uhgr0vb 29129 | The null graph, with no ve... |
| uhgr0 29130 | The null graph represented... |
| uhgrun 29131 | The union ` U ` of two (un... |
| uhgrunop 29132 | The union of two (undirect... |
| ushgrun 29133 | The union ` U ` of two (un... |
| ushgrunop 29134 | The union of two (undirect... |
| uhgrstrrepe 29135 | Replacing (or adding) the ... |
| incistruhgr 29136 | An _incidence structure_ `... |
| isupgr 29141 | The property of being an u... |
| wrdupgr 29142 | The property of being an u... |
| upgrf 29143 | The edge function of an un... |
| upgrfn 29144 | The edge function of an un... |
| upgrss 29145 | An edge is a subset of ver... |
| upgrn0 29146 | An edge is a nonempty subs... |
| upgrle 29147 | An edge of an undirected p... |
| upgrfi 29148 | An edge is a finite subset... |
| upgrex 29149 | An edge is an unordered pa... |
| upgrbi 29150 | Show that an unordered pai... |
| upgrop 29151 | A pseudograph represented ... |
| isumgr 29152 | The property of being an u... |
| isumgrs 29153 | The simplified property of... |
| wrdumgr 29154 | The property of being an u... |
| umgrf 29155 | The edge function of an un... |
| umgrfn 29156 | The edge function of an un... |
| umgredg2 29157 | An edge of a multigraph ha... |
| umgrbi 29158 | Show that an unordered pai... |
| upgruhgr 29159 | An undirected pseudograph ... |
| umgrupgr 29160 | An undirected multigraph i... |
| umgruhgr 29161 | An undirected multigraph i... |
| upgrle2 29162 | An edge of an undirected p... |
| umgrnloopv 29163 | In a multigraph, there is ... |
| umgredgprv 29164 | In a multigraph, an edge i... |
| umgrnloop 29165 | In a multigraph, there is ... |
| umgrnloop0 29166 | A multigraph has no loops.... |
| umgr0e 29167 | The empty graph, with vert... |
| upgr0e 29168 | The empty graph, with vert... |
| upgr1elem 29169 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 29170 | A pseudograph with one edg... |
| upgr0eop 29171 | The empty graph, with vert... |
| upgr1eop 29172 | A pseudograph with one edg... |
| upgr0eopALT 29173 | Alternate proof of ~ upgr0... |
| upgr1eopALT 29174 | Alternate proof of ~ upgr1... |
| upgrun 29175 | The union ` U ` of two pse... |
| upgrunop 29176 | The union of two pseudogra... |
| umgrun 29177 | The union ` U ` of two mul... |
| umgrunop 29178 | The union of two multigrap... |
| umgrislfupgrlem 29179 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 29180 | A multigraph is a loop-fre... |
| lfgredgge2 29181 | An edge of a loop-free gra... |
| lfgrnloop 29182 | A loop-free graph has no l... |
| uhgredgiedgb 29183 | In a hypergraph, a set is ... |
| uhgriedg0edg0 29184 | A hypergraph has no edges ... |
| uhgredgn0 29185 | An edge of a hypergraph is... |
| edguhgr 29186 | An edge of a hypergraph is... |
| uhgredgrnv 29187 | An edge of a hypergraph co... |
| uhgredgss 29188 | The set of edges of a hype... |
| upgredgss 29189 | The set of edges of a pseu... |
| umgredgss 29190 | The set of edges of a mult... |
| edgupgr 29191 | Properties of an edge of a... |
| edgumgr 29192 | Properties of an edge of a... |
| uhgrvtxedgiedgb 29193 | In a hypergraph, a vertex ... |
| upgredg 29194 | For each edge in a pseudog... |
| umgredg 29195 | For each edge in a multigr... |
| upgrpredgv 29196 | An edge of a pseudograph a... |
| umgrpredgv 29197 | An edge of a multigraph al... |
| upgredg2vtx 29198 | For a vertex incident to a... |
| upgredgpr 29199 | If a proper pair (of verti... |
| edglnl 29200 | The edges incident with a ... |
| numedglnl 29201 | The number of edges incide... |
| umgredgne 29202 | An edge of a multigraph al... |
| umgrnloop2 29203 | A multigraph has no loops.... |
| umgredgnlp 29204 | An edge of a multigraph is... |
| isuspgr 29209 | The property of being a si... |
| isusgr 29210 | The property of being a si... |
| uspgrf 29211 | The edge function of a sim... |
| usgrf 29212 | The edge function of a sim... |
| isusgrs 29213 | The property of being a si... |
| usgrfs 29214 | The edge function of a sim... |
| usgrfun 29215 | The edge function of a sim... |
| usgredgss 29216 | The set of edges of a simp... |
| edgusgr 29217 | An edge of a simple graph ... |
| isuspgrop 29218 | The property of being an u... |
| isusgrop 29219 | The property of being an u... |
| usgrop 29220 | A simple graph represented... |
| isausgr 29221 | The property of an ordered... |
| ausgrusgrb 29222 | The equivalence of the def... |
| usgrausgri 29223 | A simple graph represented... |
| ausgrumgri 29224 | If an alternatively define... |
| ausgrusgri 29225 | The equivalence of the def... |
| usgrausgrb 29226 | The equivalence of the def... |
| usgredgop 29227 | An edge of a simple graph ... |
| usgrf1o 29228 | The edge function of a sim... |
| usgrf1 29229 | The edge function of a sim... |
| uspgrf1oedg 29230 | The edge function of a sim... |
| usgrss 29231 | An edge is a subset of ver... |
| uspgredgiedg 29232 | In a simple pseudograph, f... |
| uspgriedgedg 29233 | In a simple pseudograph, f... |
| uspgrushgr 29234 | A simple pseudograph is an... |
| uspgrupgr 29235 | A simple pseudograph is an... |
| uspgrupgrushgr 29236 | A graph is a simple pseudo... |
| usgruspgr 29237 | A simple graph is a simple... |
| usgrumgr 29238 | A simple graph is an undir... |
| usgrumgruspgr 29239 | A graph is a simple graph ... |
| usgruspgrb 29240 | A class is a simple graph ... |
| uspgruhgr 29241 | An undirected simple pseud... |
| usgrupgr 29242 | A simple graph is an undir... |
| usgruhgr 29243 | A simple graph is an undir... |
| usgrislfuspgr 29244 | A simple graph is a loop-f... |
| uspgrun 29245 | The union ` U ` of two sim... |
| uspgrunop 29246 | The union of two simple ps... |
| usgrun 29247 | The union ` U ` of two sim... |
| usgrunop 29248 | The union of two simple gr... |
| usgredg2 29249 | The value of the "edge fun... |
| usgredg2ALT 29250 | Alternate proof of ~ usgre... |
| usgredgprv 29251 | In a simple graph, an edge... |
| usgredgprvALT 29252 | Alternate proof of ~ usgre... |
| usgredgppr 29253 | An edge of a simple graph ... |
| usgrpredgv 29254 | An edge of a simple graph ... |
| edgssv2 29255 | An edge of a simple graph ... |
| usgredg 29256 | For each edge in a simple ... |
| usgrnloopv 29257 | In a simple graph, there i... |
| usgrnloopvALT 29258 | Alternate proof of ~ usgrn... |
| usgrnloop 29259 | In a simple graph, there i... |
| usgrnloopALT 29260 | Alternate proof of ~ usgrn... |
| usgrnloop0 29261 | A simple graph has no loop... |
| usgrnloop0ALT 29262 | Alternate proof of ~ usgrn... |
| usgredgne 29263 | An edge of a simple graph ... |
| usgrf1oedg 29264 | The edge function of a sim... |
| uhgr2edg 29265 | If a vertex is adjacent to... |
| umgr2edg 29266 | If a vertex is adjacent to... |
| usgr2edg 29267 | If a vertex is adjacent to... |
| umgr2edg1 29268 | If a vertex is adjacent to... |
| usgr2edg1 29269 | If a vertex is adjacent to... |
| umgrvad2edg 29270 | If a vertex is adjacent to... |
| umgr2edgneu 29271 | If a vertex is adjacent to... |
| usgrsizedg 29272 | In a simple graph, the siz... |
| usgredg3 29273 | The value of the "edge fun... |
| usgredg4 29274 | For a vertex incident to a... |
| usgredgreu 29275 | For a vertex incident to a... |
| usgredg2vtx 29276 | For a vertex incident to a... |
| uspgredg2vtxeu 29277 | For a vertex incident to a... |
| usgredg2vtxeu 29278 | For a vertex incident to a... |
| usgredg2vtxeuALT 29279 | Alternate proof of ~ usgre... |
| uspgredg2vlem 29280 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 29281 | In a simple pseudograph, t... |
| usgredg2vlem1 29282 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 29283 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 29284 | In a simple graph, the map... |
| usgriedgleord 29285 | Alternate version of ~ usg... |
| ushgredgedg 29286 | In a simple hypergraph the... |
| usgredgedg 29287 | In a simple graph there is... |
| ushgredgedgloop 29288 | In a simple hypergraph the... |
| uspgredgleord 29289 | In a simple pseudograph th... |
| usgredgleord 29290 | In a simple graph the numb... |
| usgredgleordALT 29291 | Alternate proof for ~ usgr... |
| usgrstrrepe 29292 | Replacing (or adding) the ... |
| usgr0e 29293 | The empty graph, with vert... |
| usgr0vb 29294 | The null graph, with no ve... |
| uhgr0v0e 29295 | The null graph, with no ve... |
| uhgr0vsize0 29296 | The size of a hypergraph w... |
| uhgr0edgfi 29297 | A graph of order 0 (i.e. w... |
| usgr0v 29298 | The null graph, with no ve... |
| uhgr0vusgr 29299 | The null graph, with no ve... |
| usgr0 29300 | The null graph represented... |
| uspgr1e 29301 | A simple pseudograph with ... |
| usgr1e 29302 | A simple graph with one ed... |
| usgr0eop 29303 | The empty graph, with vert... |
| uspgr1eop 29304 | A simple pseudograph with ... |
| uspgr1ewop 29305 | A simple pseudograph with ... |
| uspgr1v1eop 29306 | A simple pseudograph with ... |
| usgr1eop 29307 | A simple graph with (at le... |
| uspgr2v1e2w 29308 | A simple pseudograph with ... |
| usgr2v1e2w 29309 | A simple graph with two ve... |
| edg0usgr 29310 | A class without edges is a... |
| lfuhgr1v0e 29311 | A loop-free hypergraph wit... |
| usgr1vr 29312 | A simple graph with one ve... |
| usgr1v 29313 | A class with one (or no) v... |
| usgr1v0edg 29314 | A class with one (or no) v... |
| usgrexmpldifpr 29315 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 29316 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 29317 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 29318 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 29319 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 29320 | ` G ` is a simple graph of... |
| griedg0prc 29321 | The class of empty graphs ... |
| griedg0ssusgr 29322 | The class of all simple gr... |
| usgrprc 29323 | The class of simple graphs... |
| relsubgr 29326 | The class of the subgraph ... |
| subgrv 29327 | If a class is a subgraph o... |
| issubgr 29328 | The property of a set to b... |
| issubgr2 29329 | The property of a set to b... |
| subgrprop 29330 | The properties of a subgra... |
| subgrprop2 29331 | The properties of a subgra... |
| uhgrissubgr 29332 | The property of a hypergra... |
| subgrprop3 29333 | The properties of a subgra... |
| egrsubgr 29334 | An empty graph consisting ... |
| 0grsubgr 29335 | The null graph (represente... |
| 0uhgrsubgr 29336 | The null graph (as hypergr... |
| uhgrsubgrself 29337 | A hypergraph is a subgraph... |
| subgrfun 29338 | The edge function of a sub... |
| subgruhgrfun 29339 | The edge function of a sub... |
| subgreldmiedg 29340 | An element of the domain o... |
| subgruhgredgd 29341 | An edge of a subgraph of a... |
| subumgredg2 29342 | An edge of a subgraph of a... |
| subuhgr 29343 | A subgraph of a hypergraph... |
| subupgr 29344 | A subgraph of a pseudograp... |
| subumgr 29345 | A subgraph of a multigraph... |
| subusgr 29346 | A subgraph of a simple gra... |
| uhgrspansubgrlem 29347 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 29348 | A spanning subgraph ` S ` ... |
| uhgrspan 29349 | A spanning subgraph ` S ` ... |
| upgrspan 29350 | A spanning subgraph ` S ` ... |
| umgrspan 29351 | A spanning subgraph ` S ` ... |
| usgrspan 29352 | A spanning subgraph ` S ` ... |
| uhgrspanop 29353 | A spanning subgraph of a h... |
| upgrspanop 29354 | A spanning subgraph of a p... |
| umgrspanop 29355 | A spanning subgraph of a m... |
| usgrspanop 29356 | A spanning subgraph of a s... |
| uhgrspan1lem1 29357 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 29358 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 29359 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 29360 | The induced subgraph ` S `... |
| upgrreslem 29361 | Lemma for ~ upgrres . (Co... |
| umgrreslem 29362 | Lemma for ~ umgrres and ~ ... |
| upgrres 29363 | A subgraph obtained by rem... |
| umgrres 29364 | A subgraph obtained by rem... |
| usgrres 29365 | A subgraph obtained by rem... |
| upgrres1lem1 29366 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 29367 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 29368 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 29369 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 29370 | A pseudograph obtained by ... |
| umgrres1 29371 | A multigraph obtained by r... |
| usgrres1 29372 | Restricting a simple graph... |
| isfusgr 29375 | The property of being a fi... |
| fusgrvtxfi 29376 | A finite simple graph has ... |
| isfusgrf1 29377 | The property of being a fi... |
| isfusgrcl 29378 | The property of being a fi... |
| fusgrusgr 29379 | A finite simple graph is a... |
| opfusgr 29380 | A finite simple graph repr... |
| usgredgffibi 29381 | The number of edges in a s... |
| fusgredgfi 29382 | In a finite simple graph t... |
| usgr1v0e 29383 | The size of a (finite) sim... |
| usgrfilem 29384 | In a finite simple graph, ... |
| fusgrfisbase 29385 | Induction base for ~ fusgr... |
| fusgrfisstep 29386 | Induction step in ~ fusgrf... |
| fusgrfis 29387 | A finite simple graph is o... |
| fusgrfupgrfs 29388 | A finite simple graph is a... |
| nbgrprc0 29391 | The set of neighbors is em... |
| nbgrcl 29392 | If a class ` X ` has at le... |
| nbgrval 29393 | The set of neighbors of a ... |
| dfnbgr2 29394 | Alternate definition of th... |
| dfnbgr3 29395 | Alternate definition of th... |
| nbgrnvtx0 29396 | If a class ` X ` is not a ... |
| nbgrel 29397 | Characterization of a neig... |
| nbgrisvtx 29398 | Every neighbor ` N ` of a ... |
| nbgrssvtx 29399 | The neighbors of a vertex ... |
| nbuhgr 29400 | The set of neighbors of a ... |
| nbupgr 29401 | The set of neighbors of a ... |
| nbupgrel 29402 | A neighbor of a vertex in ... |
| nbumgrvtx 29403 | The set of neighbors of a ... |
| nbumgr 29404 | The set of neighbors of an... |
| nbusgrvtx 29405 | The set of neighbors of a ... |
| nbusgr 29406 | The set of neighbors of an... |
| nbgr2vtx1edg 29407 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 29408 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 29409 | If a hypergraph has two ve... |
| nbusgreledg 29410 | A class/vertex is a neighb... |
| uhgrnbgr0nb 29411 | A vertex which is not endp... |
| nbgr0vtx 29412 | In a null graph (with no v... |
| nbgr0edglem 29413 | Lemma for ~ nbgr0edg and ~... |
| nbgr0edg 29414 | In an empty graph (with no... |
| nbgr1vtx 29415 | In a graph with one vertex... |
| nbgrnself 29416 | A vertex in a graph is not... |
| nbgrnself2 29417 | A class ` X ` is not a nei... |
| nbgrssovtx 29418 | The neighbors of a vertex ... |
| nbgrssvwo2 29419 | The neighbors of a vertex ... |
| nbgrsym 29420 | In a graph, the neighborho... |
| nbupgrres 29421 | The neighborhood of a vert... |
| usgrnbcnvfv 29422 | Applying the edge function... |
| nbusgredgeu 29423 | For each neighbor of a ver... |
| edgnbusgreu 29424 | For each edge incident to ... |
| nbusgredgeu0 29425 | For each neighbor of a ver... |
| nbusgrf1o0 29426 | The mapping of neighbors o... |
| nbusgrf1o1 29427 | The set of neighbors of a ... |
| nbusgrf1o 29428 | The set of neighbors of a ... |
| nbedgusgr 29429 | The number of neighbors of... |
| edgusgrnbfin 29430 | The number of neighbors of... |
| nbusgrfi 29431 | The class of neighbors of ... |
| nbfiusgrfi 29432 | The class of neighbors of ... |
| hashnbusgrnn0 29433 | The number of neighbors of... |
| nbfusgrlevtxm1 29434 | The number of neighbors of... |
| nbfusgrlevtxm2 29435 | If there is a vertex which... |
| nbusgrvtxm1 29436 | If the number of neighbors... |
| nb3grprlem1 29437 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 29438 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 29439 | The neighbors of a vertex ... |
| nb3grpr2 29440 | The neighbors of a vertex ... |
| nb3gr2nb 29441 | If the neighbors of two ve... |
| uvtxval 29444 | The set of all universal v... |
| uvtxel 29445 | A universal vertex, i.e. a... |
| uvtxisvtx 29446 | A universal vertex is a ve... |
| uvtxssvtx 29447 | The set of the universal v... |
| vtxnbuvtx 29448 | A universal vertex has all... |
| uvtxnbgrss 29449 | A universal vertex has all... |
| uvtxnbgrvtx 29450 | A universal vertex is neig... |
| uvtx0 29451 | There is no universal vert... |
| isuvtx 29452 | The set of all universal v... |
| uvtxel1 29453 | Characterization of a univ... |
| uvtx01vtx 29454 | If a graph/class has no ed... |
| uvtx2vtx1edg 29455 | If a graph has two vertice... |
| uvtx2vtx1edgb 29456 | If a hypergraph has two ve... |
| uvtxnbgr 29457 | A universal vertex has all... |
| uvtxnbgrb 29458 | A vertex is universal iff ... |
| uvtxusgr 29459 | The set of all universal v... |
| uvtxusgrel 29460 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 29461 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 29462 | If the number of neighbors... |
| uvtxnbvtxm1 29463 | A universal vertex has ` n... |
| nbupgruvtxres 29464 | The neighborhood of a univ... |
| uvtxupgrres 29465 | A universal vertex is univ... |
| cplgruvtxb 29470 | A graph ` G ` is complete ... |
| prcliscplgr 29471 | A proper class (representi... |
| iscplgr 29472 | The property of being a co... |
| iscplgrnb 29473 | A graph is complete iff al... |
| iscplgredg 29474 | A graph ` G ` is complete ... |
| iscusgr 29475 | The property of being a co... |
| cusgrusgr 29476 | A complete simple graph is... |
| cusgrcplgr 29477 | A complete simple graph is... |
| iscusgrvtx 29478 | A simple graph is complete... |
| cusgruvtxb 29479 | A simple graph is complete... |
| iscusgredg 29480 | A simple graph is complete... |
| cusgredg 29481 | In a complete simple graph... |
| cplgr0 29482 | The null graph (with no ve... |
| cusgr0 29483 | The null graph (with no ve... |
| cplgr0v 29484 | A null graph (with no vert... |
| cusgr0v 29485 | A graph with no vertices a... |
| cplgr1vlem 29486 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 29487 | A graph with one vertex is... |
| cusgr1v 29488 | A graph with one vertex an... |
| cplgr2v 29489 | An undirected hypergraph w... |
| cplgr2vpr 29490 | An undirected hypergraph w... |
| nbcplgr 29491 | In a complete graph, each ... |
| cplgr3v 29492 | A pseudograph with three (... |
| cusgr3vnbpr 29493 | The neighbors of a vertex ... |
| cplgrop 29494 | A complete graph represent... |
| cusgrop 29495 | A complete simple graph re... |
| cusgrexilem1 29496 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 29497 | Lemma for ~ usgrexi . (Co... |
| usgrexi 29498 | An arbitrary set regarded ... |
| cusgrexilem2 29499 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 29500 | An arbitrary set ` V ` reg... |
| cusgrexg 29501 | For each set there is a se... |
| structtousgr 29502 | Any (extensible) structure... |
| structtocusgr 29503 | Any (extensible) structure... |
| cffldtocusgr 29504 | The field of complex numbe... |
| cffldtocusgrOLD 29505 | Obsolete version of ~ cffl... |
| cusgrres 29506 | Restricting a complete sim... |
| cusgrsizeindb0 29507 | Base case of the induction... |
| cusgrsizeindb1 29508 | Base case of the induction... |
| cusgrsizeindslem 29509 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 29510 | Part 1 of induction step i... |
| cusgrsize2inds 29511 | Induction step in ~ cusgrs... |
| cusgrsize 29512 | The size of a finite compl... |
| cusgrfilem1 29513 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 29514 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 29515 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 29516 | If the size of a complete ... |
| usgredgsscusgredg 29517 | A simple graph is a subgra... |
| usgrsscusgr 29518 | A simple graph is a subgra... |
| sizusglecusglem1 29519 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 29520 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 29521 | The size of a simple graph... |
| fusgrmaxsize 29522 | The maximum size of a fini... |
| vtxdgfval 29525 | The value of the vertex de... |
| vtxdgval 29526 | The degree of a vertex. (... |
| vtxdgfival 29527 | The degree of a vertex for... |
| vtxdgop 29528 | The vertex degree expresse... |
| vtxdgf 29529 | The vertex degree function... |
| vtxdgelxnn0 29530 | The degree of a vertex is ... |
| vtxdg0v 29531 | The degree of a vertex in ... |
| vtxdg0e 29532 | The degree of a vertex in ... |
| vtxdgfisnn0 29533 | The degree of a vertex in ... |
| vtxdgfisf 29534 | The vertex degree function... |
| vtxdeqd 29535 | Equality theorem for the v... |
| vtxduhgr0e 29536 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29537 | The degree of the vertex i... |
| vdumgr0 29538 | A vertex in a multigraph h... |
| vtxdun 29539 | The degree of a vertex in ... |
| vtxdfiun 29540 | The degree of a vertex in ... |
| vtxduhgrun 29541 | The degree of a vertex in ... |
| vtxduhgrfiun 29542 | The degree of a vertex in ... |
| vtxdlfgrval 29543 | The value of the vertex de... |
| vtxdumgrval 29544 | The value of the vertex de... |
| vtxdusgrval 29545 | The value of the vertex de... |
| vtxd0nedgb 29546 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29547 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29548 | The value of the vertex de... |
| vtxdusgrfvedg 29549 | The value of the vertex de... |
| vtxduhgr0nedg 29550 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29551 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29552 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29553 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29554 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29555 | The vertex degree function... |
| vtxdgfusgr 29556 | In a finite simple graph, ... |
| fusgrn0degnn0 29557 | In a nonempty, finite grap... |
| 1loopgruspgr 29558 | A graph with one edge whic... |
| 1loopgredg 29559 | The set of edges in a grap... |
| 1loopgrnb0 29560 | In a graph (simple pseudog... |
| 1loopgrvd2 29561 | The vertex degree of a one... |
| 1loopgrvd0 29562 | The vertex degree of a one... |
| 1hevtxdg0 29563 | The vertex degree of verte... |
| 1hevtxdg1 29564 | The vertex degree of verte... |
| 1hegrvtxdg1 29565 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29566 | The vertex degree of a gra... |
| 1egrvtxdg1 29567 | The vertex degree of a one... |
| 1egrvtxdg1r 29568 | The vertex degree of a one... |
| 1egrvtxdg0 29569 | The vertex degree of a one... |
| p1evtxdeqlem 29570 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29571 | If an edge ` E ` which doe... |
| p1evtxdp1 29572 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29573 | The set of vertices in a g... |
| uspgrloopvtxel 29574 | A vertex in a graph (simpl... |
| uspgrloopiedg 29575 | The set of edges in a grap... |
| uspgrloopedg 29576 | The set of edges in a grap... |
| uspgrloopnb0 29577 | In a graph (simple pseudog... |
| uspgrloopvd2 29578 | The vertex degree of a one... |
| umgr2v2evtx 29579 | The set of vertices in a m... |
| umgr2v2evtxel 29580 | A vertex in a multigraph w... |
| umgr2v2eiedg 29581 | The edge function in a mul... |
| umgr2v2eedg 29582 | The set of edges in a mult... |
| umgr2v2e 29583 | A multigraph with two edge... |
| umgr2v2enb1 29584 | In a multigraph with two e... |
| umgr2v2evd2 29585 | In a multigraph with two e... |
| hashnbusgrvd 29586 | In a simple graph, the num... |
| usgruvtxvdb 29587 | In a finite simple graph w... |
| vdiscusgrb 29588 | A finite simple graph with... |
| vdiscusgr 29589 | In a finite complete simpl... |
| vtxdusgradjvtx 29590 | The degree of a vertex in ... |
| usgrvd0nedg 29591 | If a vertex in a simple gr... |
| uhgrvd00 29592 | If every vertex in a hyper... |
| usgrvd00 29593 | If every vertex in a simpl... |
| vdegp1ai 29594 | The induction step for a v... |
| vdegp1bi 29595 | The induction step for a v... |
| vdegp1ci 29596 | The induction step for a v... |
| vtxdginducedm1lem1 29597 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29598 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29599 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29600 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29601 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29602 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29603 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29604 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29605 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29606 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29607 | Induction step of ~ finsum... |
| finsumvtxdg2size 29608 | The sum of the degrees of ... |
| fusgr1th 29609 | The sum of the degrees of ... |
| finsumvtxdgeven 29610 | The sum of the degrees of ... |
| vtxdgoddnumeven 29611 | The number of vertices of ... |
| fusgrvtxdgonume 29612 | The number of vertices of ... |
| isrgr 29617 | The property of a class be... |
| rgrprop 29618 | The properties of a k-regu... |
| isrusgr 29619 | The property of being a k-... |
| rusgrprop 29620 | The properties of a k-regu... |
| rusgrrgr 29621 | A k-regular simple graph i... |
| rusgrusgr 29622 | A k-regular simple graph i... |
| finrusgrfusgr 29623 | A finite regular simple gr... |
| isrusgr0 29624 | The property of being a k-... |
| rusgrprop0 29625 | The properties of a k-regu... |
| usgreqdrusgr 29626 | If all vertices in a simpl... |
| fusgrregdegfi 29627 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29628 | If all vertices in a nonem... |
| frusgrnn0 29629 | In a nonempty finite k-reg... |
| 0edg0rgr 29630 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29631 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29632 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29633 | A simple graph is 0-regula... |
| 0vtxrgr 29634 | A null graph (with no vert... |
| 0vtxrusgr 29635 | A graph with no vertices a... |
| 0uhgrrusgr 29636 | The null graph as hypergra... |
| 0grrusgr 29637 | The null graph represented... |
| 0grrgr 29638 | The null graph represented... |
| cusgrrusgr 29639 | A complete simple graph wi... |
| cusgrm1rusgr 29640 | A finite simple graph with... |
| rusgrpropnb 29641 | The properties of a k-regu... |
| rusgrpropedg 29642 | The properties of a k-regu... |
| rusgrpropadjvtx 29643 | The properties of a k-regu... |
| rusgrnumwrdl2 29644 | In a k-regular simple grap... |
| rusgr1vtxlem 29645 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29646 | If a k-regular simple grap... |
| rgrusgrprc 29647 | The class of 0-regular sim... |
| rusgrprc 29648 | The class of 0-regular sim... |
| rgrprc 29649 | The class of 0-regular gra... |
| rgrprcx 29650 | The class of 0-regular gra... |
| rgrx0ndm 29651 | 0 is not in the domain of ... |
| rgrx0nd 29652 | The potentially alternativ... |
| ewlksfval 29659 | The set of s-walks of edge... |
| isewlk 29660 | Conditions for a function ... |
| ewlkprop 29661 | Properties of an s-walk of... |
| ewlkinedg 29662 | The intersection (common v... |
| ewlkle 29663 | An s-walk of edges is also... |
| upgrewlkle2 29664 | In a pseudograph, there is... |
| wkslem1 29665 | Lemma 1 for walks to subst... |
| wkslem2 29666 | Lemma 2 for walks to subst... |
| wksfval 29667 | The set of walks (in an un... |
| iswlk 29668 | Properties of a pair of fu... |
| wlkprop 29669 | Properties of a walk. (Co... |
| wlkv 29670 | The classes involved in a ... |
| iswlkg 29671 | Generalization of ~ iswlk ... |
| wlkf 29672 | The mapping enumerating th... |
| wlkcl 29673 | A walk has length ` # ( F ... |
| wlkp 29674 | The mapping enumerating th... |
| wlkpwrd 29675 | The sequence of vertices o... |
| wlklenvp1 29676 | The number of vertices of ... |
| wksv 29677 | The class of walks is a se... |
| wlkn0 29678 | The sequence of vertices o... |
| wlklenvm1 29679 | The number of edges of a w... |
| ifpsnprss 29680 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29681 | Each pair of adjacent vert... |
| wlkvtxiedg 29682 | The vertices of a walk are... |
| relwlk 29683 | The set ` ( Walks `` G ) `... |
| wlkvv 29684 | If there is at least one w... |
| wlkop 29685 | A walk is an ordered pair.... |
| wlkcpr 29686 | A walk as class with two c... |
| wlk2f 29687 | If there is a walk ` W ` t... |
| wlkcomp 29688 | A walk expressed by proper... |
| wlkcompim 29689 | Implications for the prope... |
| wlkelwrd 29690 | The components of a walk a... |
| wlkeq 29691 | Conditions for two walks (... |
| edginwlk 29692 | The value of the edge func... |
| upgredginwlk 29693 | The value of the edge func... |
| iedginwlk 29694 | The value of the edge func... |
| wlkl1loop 29695 | A walk of length 1 from a ... |
| wlk1walk 29696 | A walk is a 1-walk "on the... |
| wlk1ewlk 29697 | A walk is an s-walk "on th... |
| upgriswlk 29698 | Properties of a pair of fu... |
| upgrwlkedg 29699 | The edges of a walk in a p... |
| upgrwlkcompim 29700 | Implications for the prope... |
| wlkvtxedg 29701 | The vertices of a walk are... |
| upgrwlkvtxedg 29702 | The pairs of connected ver... |
| uspgr2wlkeq 29703 | Conditions for two walks w... |
| uspgr2wlkeq2 29704 | Conditions for two walks w... |
| uspgr2wlkeqi 29705 | Conditions for two walks w... |
| umgrwlknloop 29706 | In a multigraph, each walk... |
| wlkv0 29707 | If there is a walk in the ... |
| g0wlk0 29708 | There is no walk in a null... |
| 0wlk0 29709 | There is no walk for the e... |
| wlk0prc 29710 | There is no walk in a null... |
| wlklenvclwlk 29711 | The number of vertices in ... |
| wlkson 29712 | The set of walks between t... |
| iswlkon 29713 | Properties of a pair of fu... |
| wlkonprop 29714 | Properties of a walk betwe... |
| wlkpvtx 29715 | A walk connects vertices. ... |
| wlkepvtx 29716 | The endpoints of a walk ar... |
| wlkoniswlk 29717 | A walk between two vertice... |
| wlkonwlk 29718 | A walk is a walk between i... |
| wlkonwlk1l 29719 | A walk is a walk from its ... |
| wlksoneq1eq2 29720 | Two walks with identical s... |
| wlkonl1iedg 29721 | If there is a walk between... |
| wlkon2n0 29722 | The length of a walk betwe... |
| 2wlklem 29723 | Lemma for theorems for wal... |
| upgr2wlk 29724 | Properties of a pair of fu... |
| wlkreslem 29725 | Lemma for ~ wlkres . (Con... |
| wlkres 29726 | The restriction ` <. H , Q... |
| redwlklem 29727 | Lemma for ~ redwlk . (Con... |
| redwlk 29728 | A walk ending at the last ... |
| wlkp1lem1 29729 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29730 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29731 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29732 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29733 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29734 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29735 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29736 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29737 | Append one path segment (e... |
| wlkdlem1 29738 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29739 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29740 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29741 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29742 | Two words representing a w... |
| lfgrwlkprop 29743 | Two adjacent vertices in a... |
| lfgriswlk 29744 | Conditions for a pair of f... |
| lfgrwlknloop 29745 | In a loop-free graph, each... |
| reltrls 29750 | The set ` ( Trails `` G ) ... |
| trlsfval 29751 | The set of trails (in an u... |
| istrl 29752 | Conditions for a pair of c... |
| trliswlk 29753 | A trail is a walk. (Contr... |
| trlf1 29754 | The enumeration ` F ` of a... |
| trlreslem 29755 | Lemma for ~ trlres . Form... |
| trlres 29756 | The restriction ` <. H , Q... |
| upgrtrls 29757 | The set of trails in a pse... |
| upgristrl 29758 | Properties of a pair of fu... |
| upgrf1istrl 29759 | Properties of a pair of a ... |
| wksonproplem 29760 | Lemma for theorems for pro... |
| trlsonfval 29761 | The set of trails between ... |
| istrlson 29762 | Properties of a pair of fu... |
| trlsonprop 29763 | Properties of a trail betw... |
| trlsonistrl 29764 | A trail between two vertic... |
| trlsonwlkon 29765 | A trail between two vertic... |
| trlontrl 29766 | A trail is a trail between... |
| relpths 29775 | The set ` ( Paths `` G ) `... |
| pthsfval 29776 | The set of paths (in an un... |
| spthsfval 29777 | The set of simple paths (i... |
| ispth 29778 | Conditions for a pair of c... |
| isspth 29779 | Conditions for a pair of c... |
| pthistrl 29780 | A path is a trail (in an u... |
| spthispth 29781 | A simple path is a path (i... |
| pthiswlk 29782 | A path is a walk (in an un... |
| spthiswlk 29783 | A simple path is a walk (i... |
| pthdivtx 29784 | The inner vertices of a pa... |
| pthdadjvtx 29785 | The adjacent vertices of a... |
| dfpth2 29786 | Alternate definition for a... |
| pthdifv 29787 | The vertices of a path are... |
| 2pthnloop 29788 | A path of length at least ... |
| upgr2pthnlp 29789 | A path of length at least ... |
| spthdifv 29790 | The vertices of a simple p... |
| spthdep 29791 | A simple path (at least of... |
| pthdepisspth 29792 | A path with different star... |
| upgrwlkdvdelem 29793 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29794 | In a pseudograph, all edge... |
| upgrspthswlk 29795 | The set of simple paths in... |
| upgrwlkdvspth 29796 | A walk consisting of diffe... |
| pthsonfval 29797 | The set of paths between t... |
| spthson 29798 | The set of simple paths be... |
| ispthson 29799 | Properties of a pair of fu... |
| isspthson 29800 | Properties of a pair of fu... |
| pthsonprop 29801 | Properties of a path betwe... |
| spthonprop 29802 | Properties of a simple pat... |
| pthonispth 29803 | A path between two vertice... |
| pthontrlon 29804 | A path between two vertice... |
| pthonpth 29805 | A path is a path between i... |
| isspthonpth 29806 | A pair of functions is a s... |
| spthonisspth 29807 | A simple path between to v... |
| spthonpthon 29808 | A simple path between two ... |
| spthonepeq 29809 | The endpoints of a simple ... |
| uhgrwkspthlem1 29810 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29811 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29812 | Any walk of length 1 betwe... |
| usgr2wlkneq 29813 | The vertices and edges are... |
| usgr2wlkspthlem1 29814 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29815 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29816 | In a simple graph, any wal... |
| usgr2trlncl 29817 | In a simple graph, any tra... |
| usgr2trlspth 29818 | In a simple graph, any tra... |
| usgr2pthspth 29819 | In a simple graph, any pat... |
| usgr2pthlem 29820 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29821 | In a simple graph, there i... |
| usgr2pth0 29822 | In a simply graph, there i... |
| pthdlem1 29823 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29824 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29825 | Lemma 2 for ~ pthd . (Con... |
| pthd 29826 | Two words representing a t... |
| clwlks 29829 | The set of closed walks (i... |
| isclwlk 29830 | A pair of functions repres... |
| clwlkiswlk 29831 | A closed walk is a walk (i... |
| clwlkwlk 29832 | Closed walks are walks (in... |
| clwlkswks 29833 | Closed walks are walks (in... |
| isclwlke 29834 | Properties of a pair of fu... |
| isclwlkupgr 29835 | Properties of a pair of fu... |
| clwlkcomp 29836 | A closed walk expressed by... |
| clwlkcompim 29837 | Implications for the prope... |
| upgrclwlkcompim 29838 | Implications for the prope... |
| clwlkcompbp 29839 | Basic properties of the co... |
| clwlkl1loop 29840 | A closed walk of length 1 ... |
| crcts 29845 | The set of circuits (in an... |
| cycls 29846 | The set of cycles (in an u... |
| iscrct 29847 | Sufficient and necessary c... |
| iscycl 29848 | Sufficient and necessary c... |
| crctprop 29849 | The properties of a circui... |
| cyclprop 29850 | The properties of a cycle:... |
| crctisclwlk 29851 | A circuit is a closed walk... |
| crctistrl 29852 | A circuit is a trail. (Co... |
| crctiswlk 29853 | A circuit is a walk. (Con... |
| cyclispth 29854 | A cycle is a path. (Contr... |
| cycliswlk 29855 | A cycle is a walk. (Contr... |
| cycliscrct 29856 | A cycle is a circuit. (Co... |
| cyclnumvtx 29857 | The number of vertices of ... |
| cyclnspth 29858 | A (non-trivial) cycle is n... |
| pthisspthorcycl 29859 | A path is either a simple ... |
| pthspthcyc 29860 | A pair ` <. F , P >. ` rep... |
| cyclispthon 29861 | A cycle is a path starting... |
| lfgrn1cycl 29862 | In a loop-free graph there... |
| usgr2trlncrct 29863 | In a simple graph, any tra... |
| umgrn1cycl 29864 | In a multigraph graph (wit... |
| uspgrn2crct 29865 | In a simple pseudograph th... |
| usgrn2cycl 29866 | In a simple graph there ar... |
| crctcshwlkn0lem1 29867 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29868 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29869 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29870 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29871 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29872 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29873 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29874 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29875 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29876 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29877 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29878 | Cyclically shifting the in... |
| crctcshwlk 29879 | Cyclically shifting the in... |
| crctcshtrl 29880 | Cyclically shifting the in... |
| crctcsh 29881 | Cyclically shifting the in... |
| wwlks 29892 | The set of walks (in an un... |
| iswwlks 29893 | A word over the set of ver... |
| wwlksn 29894 | The set of walks (in an un... |
| iswwlksn 29895 | A word over the set of ver... |
| wwlksnprcl 29896 | Derivation of the length o... |
| iswwlksnx 29897 | Properties of a word to re... |
| wwlkbp 29898 | Basic properties of a walk... |
| wwlknbp 29899 | Basic properties of a walk... |
| wwlknp 29900 | Properties of a set being ... |
| wwlknbp1 29901 | Other basic properties of ... |
| wwlknvtx 29902 | The symbols of a word ` W ... |
| wwlknllvtx 29903 | If a word ` W ` represents... |
| wwlknlsw 29904 | If a word represents a wal... |
| wspthsn 29905 | The set of simple paths of... |
| iswspthn 29906 | An element of the set of s... |
| wspthnp 29907 | Properties of a set being ... |
| wwlksnon 29908 | The set of walks of a fixe... |
| wspthsnon 29909 | The set of simple paths of... |
| iswwlksnon 29910 | The set of walks of a fixe... |
| wwlksnon0 29911 | Sufficient conditions for ... |
| wwlksonvtx 29912 | If a word ` W ` represents... |
| iswspthsnon 29913 | The set of simple paths of... |
| wwlknon 29914 | An element of the set of w... |
| wspthnon 29915 | An element of the set of s... |
| wspthnonp 29916 | Properties of a set being ... |
| wspthneq1eq2 29917 | Two simple paths with iden... |
| wwlksn0s 29918 | The set of all walks as wo... |
| wwlkssswrd 29919 | Walks (represented by word... |
| wwlksn0 29920 | A walk of length 0 is repr... |
| 0enwwlksnge1 29921 | In graphs without edges, t... |
| wwlkswwlksn 29922 | A walk of a fixed length a... |
| wwlkssswwlksn 29923 | The walks of a fixed lengt... |
| wlkiswwlks1 29924 | The sequence of vertices i... |
| wlklnwwlkln1 29925 | The sequence of vertices i... |
| wlkiswwlks2lem1 29926 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29927 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29928 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29929 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29930 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29931 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29932 | A walk as word corresponds... |
| wlkiswwlks 29933 | A walk as word corresponds... |
| wlkiswwlksupgr2 29934 | A walk as word corresponds... |
| wlkiswwlkupgr 29935 | A walk as word corresponds... |
| wlkswwlksf1o 29936 | The mapping of (ordinary) ... |
| wlkswwlksen 29937 | The set of walks as words ... |
| wwlksm1edg 29938 | Removing the trailing edge... |
| wlklnwwlkln2lem 29939 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29940 | A walk of length ` N ` as ... |
| wlklnwwlkn 29941 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29942 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29943 | A walk of length ` N ` as ... |
| wlknewwlksn 29944 | If a walk in a pseudograph... |
| wlknwwlksnbij 29945 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29946 | In a simple pseudograph, t... |
| wlknwwlksneqs 29947 | The set of walks of a fixe... |
| wwlkseq 29948 | Equality of two walks (as ... |
| wwlksnred 29949 | Reduction of a walk (as wo... |
| wwlksnext 29950 | Extension of a walk (as wo... |
| wwlksnextbi 29951 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29952 | For each walk (as word) of... |
| wwlksnredwwlkn0 29953 | For each walk (as word) of... |
| wwlksnextwrd 29954 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29955 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29956 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29957 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29958 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29959 | There is a bijection betwe... |
| wwlksnexthasheq 29960 | The number of the extensio... |
| disjxwwlksn 29961 | Sets of walks (as words) e... |
| wwlksnndef 29962 | Conditions for ` WWalksN `... |
| wwlksnfi 29963 | The number of walks repres... |
| wlksnfi 29964 | The number of walks of fix... |
| wlksnwwlknvbij 29965 | There is a bijection betwe... |
| wwlksnextproplem1 29966 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29967 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29968 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29969 | Adding additional properti... |
| disjxwwlkn 29970 | Sets of walks (as words) e... |
| hashwwlksnext 29971 | Number of walks (as words)... |
| wwlksnwwlksnon 29972 | A walk of fixed length is ... |
| wspthsnwspthsnon 29973 | A simple path of fixed len... |
| wspthsnonn0vne 29974 | If the set of simple paths... |
| wspthsswwlkn 29975 | The set of simple paths of... |
| wspthnfi 29976 | In a finite graph, the set... |
| wwlksnonfi 29977 | In a finite graph, the set... |
| wspthsswwlknon 29978 | The set of simple paths of... |
| wspthnonfi 29979 | In a finite graph, the set... |
| wspniunwspnon 29980 | The set of nonempty simple... |
| wspn0 29981 | If there are no vertices, ... |
| 2wlkdlem1 29982 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29983 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29984 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29985 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29986 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29987 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29988 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29989 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29990 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29991 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29992 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29993 | Construction of a walk fro... |
| 2wlkond 29994 | A walk of length 2 from on... |
| 2trld 29995 | Construction of a trail fr... |
| 2trlond 29996 | A trail of length 2 from o... |
| 2pthd 29997 | A path of length 2 from on... |
| 2spthd 29998 | A simple path of length 2 ... |
| 2pthond 29999 | A simple path of length 2 ... |
| 2pthon3v 30000 | For a vertex adjacent to t... |
| umgr2adedgwlklem 30001 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 30002 | In a multigraph, two adjac... |
| umgr2adedgwlkon 30003 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 30004 | Alternate proof for ~ umgr... |
| umgr2adedgspth 30005 | In a multigraph, two adjac... |
| umgr2wlk 30006 | In a multigraph, there is ... |
| umgr2wlkon 30007 | For each pair of adjacent ... |
| elwwlks2s3 30008 | A walk of length 2 as word... |
| midwwlks2s3 30009 | There is a vertex between ... |
| wwlks2onv 30010 | If a length 3 string repre... |
| elwwlks2ons3im 30011 | A walk as word of length 2... |
| elwwlks2ons3 30012 | For each walk of length 2 ... |
| s3wwlks2on 30013 | A length 3 string which re... |
| sps3wwlks2on 30014 | A length 3 string which re... |
| usgrwwlks2on 30015 | A walk of length 2 between... |
| umgrwwlks2on 30016 | A walk of length 2 between... |
| wwlks2onsym 30017 | There is a walk of length ... |
| elwwlks2on 30018 | A walk of length 2 between... |
| elwspths2on 30019 | A simple path of length 2 ... |
| elwspths2onw 30020 | A simple path of length 2 ... |
| wpthswwlks2on 30021 | For two different vertices... |
| 2wspdisj 30022 | All simple paths of length... |
| 2wspiundisj 30023 | All simple paths of length... |
| usgr2wspthons3 30024 | A simple path of length 2 ... |
| usgr2wspthon 30025 | A simple path of length 2 ... |
| elwwlks2 30026 | A walk of length 2 between... |
| elwspths2spth 30027 | A simple path of length 2 ... |
| rusgrnumwwlkl1 30028 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 30029 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 30030 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 30031 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 30032 | Induction base 1 for ~ rus... |
| rusgr0edg 30033 | Special case for graphs wi... |
| rusgrnumwwlks 30034 | Induction step for ~ rusgr... |
| rusgrnumwwlk 30035 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 30036 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 30037 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 30038 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 30039 | In a ` K `-regular graph, ... |
| clwwlk 30042 | The set of closed walks (i... |
| isclwwlk 30043 | Properties of a word to re... |
| clwwlkbp 30044 | Basic properties of a clos... |
| clwwlkgt0 30045 | There is no empty closed w... |
| clwwlksswrd 30046 | Closed walks (represented ... |
| clwwlk1loop 30047 | A closed walk of length 1 ... |
| clwwlkccatlem 30048 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 30049 | The concatenation of two w... |
| umgrclwwlkge2 30050 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 30051 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 30052 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 30053 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 30054 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 30055 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 30056 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 30057 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 30058 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 30059 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 30060 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 30061 | A closed walk as word of l... |
| clwlkclwwlk2 30062 | A closed walk corresponds ... |
| clwlkclwwlkflem 30063 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 30064 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 30065 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 30066 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 30067 | ` F ` is a function from t... |
| clwlkclwwlkfo 30068 | ` F ` is a function from t... |
| clwlkclwwlkf1 30069 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 30070 | ` F ` is a bijection betwe... |
| clwlkclwwlken 30071 | The set of the nonempty cl... |
| clwwisshclwwslemlem 30072 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 30073 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 30074 | Cyclically shifting a clos... |
| clwwisshclwwsn 30075 | Cyclically shifting a clos... |
| erclwwlkrel 30076 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 30077 | Two classes are equivalent... |
| erclwwlkeqlen 30078 | If two classes are equival... |
| erclwwlkref 30079 | ` .~ ` is a reflexive rela... |
| erclwwlksym 30080 | ` .~ ` is a symmetric rela... |
| erclwwlktr 30081 | ` .~ ` is a transitive rel... |
| erclwwlk 30082 | ` .~ ` is an equivalence r... |
| clwwlkn 30085 | The set of closed walks of... |
| isclwwlkn 30086 | A word over the set of ver... |
| clwwlkn0 30087 | There is no closed walk of... |
| clwwlkneq0 30088 | Sufficient conditions for ... |
| clwwlkclwwlkn 30089 | A closed walk of a fixed l... |
| clwwlksclwwlkn 30090 | The closed walks of a fixe... |
| clwwlknlen 30091 | The length of a word repre... |
| clwwlknnn 30092 | The length of a closed wal... |
| clwwlknwrd 30093 | A closed walk of a fixed l... |
| clwwlknbp 30094 | Basic properties of a clos... |
| isclwwlknx 30095 | Characterization of a word... |
| clwwlknp 30096 | Properties of a set being ... |
| clwwlknwwlksn 30097 | A word representing a clos... |
| clwwlknlbonbgr1 30098 | The last but one vertex in... |
| clwwlkinwwlk 30099 | If the initial vertex of a... |
| clwwlkn1 30100 | A closed walk of length 1 ... |
| loopclwwlkn1b 30101 | The singleton word consist... |
| clwwlkn1loopb 30102 | A word represents a closed... |
| clwwlkn2 30103 | A closed walk of length 2 ... |
| clwwlknfi 30104 | If there is only a finite ... |
| clwwlkel 30105 | Obtaining a closed walk (a... |
| clwwlkf 30106 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 30107 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 30108 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 30109 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 30110 | F is a 1-1 onto function, ... |
| clwwlken 30111 | The set of closed walks of... |
| clwwlknwwlkncl 30112 | Obtaining a closed walk (a... |
| clwwlkwwlksb 30113 | A nonempty word over verti... |
| clwwlknwwlksnb 30114 | A word over vertices repre... |
| clwwlkext2edg 30115 | If a word concatenated wit... |
| wwlksext2clwwlk 30116 | If a word represents a wal... |
| wwlksubclwwlk 30117 | Any prefix of a word repre... |
| clwwnisshclwwsn 30118 | Cyclically shifting a clos... |
| eleclclwwlknlem1 30119 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 30120 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 30121 | The set of cyclical shifts... |
| clwwlknccat 30122 | The concatenation of two w... |
| umgr2cwwk2dif 30123 | If a word represents a clo... |
| umgr2cwwkdifex 30124 | If a word represents a clo... |
| erclwwlknrel 30125 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 30126 | Two classes are equivalent... |
| erclwwlkneqlen 30127 | If two classes are equival... |
| erclwwlknref 30128 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 30129 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 30130 | ` .~ ` is a transitive rel... |
| erclwwlkn 30131 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 30132 | The quotient set of the se... |
| hashclwwlkn0 30133 | The number of closed walks... |
| eclclwwlkn1 30134 | An equivalence class accor... |
| eleclclwwlkn 30135 | A member of an equivalence... |
| hashecclwwlkn1 30136 | The size of every equivale... |
| umgrhashecclwwlk 30137 | The size of every equivale... |
| fusgrhashclwwlkn 30138 | The size of the set of clo... |
| clwwlkndivn 30139 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 30140 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 30141 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 30142 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 30143 | There is a one-to-one onto... |
| clwlkssizeeq 30144 | The size of the set of clo... |
| clwlksndivn 30145 | The size of the set of clo... |
| clwwlknonmpo 30148 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 30149 | The set of closed walks on... |
| isclwwlknon 30150 | A word over the set of ver... |
| clwwlk0on0 30151 | There is no word over the ... |
| clwwlknon0 30152 | Sufficient conditions for ... |
| clwwlknonfin 30153 | In a finite graph ` G ` , ... |
| clwwlknonel 30154 | Characterization of a word... |
| clwwlknonccat 30155 | The concatenation of two w... |
| clwwlknon1 30156 | The set of closed walks on... |
| clwwlknon1loop 30157 | If there is a loop at vert... |
| clwwlknon1nloop 30158 | If there is no loop at ver... |
| clwwlknon1sn 30159 | The set of (closed) walks ... |
| clwwlknon1le1 30160 | There is at most one (clos... |
| clwwlknon2 30161 | The set of closed walks on... |
| clwwlknon2x 30162 | The set of closed walks on... |
| s2elclwwlknon2 30163 | Sufficient conditions of a... |
| clwwlknon2num 30164 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 30165 | A word over vertices repre... |
| clwwlknonex2lem1 30166 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 30167 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 30168 | Extending a closed walk ` ... |
| clwwlknonex2e 30169 | Extending a closed walk ` ... |
| clwwlknondisj 30170 | The sets of closed walks o... |
| clwwlknun 30171 | The set of closed walks of... |
| clwwlkvbij 30172 | There is a bijection betwe... |
| 0ewlk 30173 | The empty set (empty seque... |
| 1ewlk 30174 | A sequence of 1 edge is an... |
| 0wlk 30175 | A pair of an empty set (of... |
| is0wlk 30176 | A pair of an empty set (of... |
| 0wlkonlem1 30177 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 30178 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 30179 | A walk of length 0 from a ... |
| 0wlkons1 30180 | A walk of length 0 from a ... |
| 0trl 30181 | A pair of an empty set (of... |
| is0trl 30182 | A pair of an empty set (of... |
| 0trlon 30183 | A trail of length 0 from a... |
| 0pth 30184 | A pair of an empty set (of... |
| 0spth 30185 | A pair of an empty set (of... |
| 0pthon 30186 | A path of length 0 from a ... |
| 0pthon1 30187 | A path of length 0 from a ... |
| 0pthonv 30188 | For each vertex there is a... |
| 0clwlk 30189 | A pair of an empty set (of... |
| 0clwlkv 30190 | Any vertex (more precisely... |
| 0clwlk0 30191 | There is no closed walk in... |
| 0crct 30192 | A pair of an empty set (of... |
| 0cycl 30193 | A pair of an empty set (of... |
| 1pthdlem1 30194 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 30195 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 30196 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 30197 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 30198 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 30199 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 30200 | In a graph with two vertic... |
| 1trld 30201 | In a graph with two vertic... |
| 1pthd 30202 | In a graph with two vertic... |
| 1pthond 30203 | In a graph with two vertic... |
| upgr1wlkdlem1 30204 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 30205 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 30206 | In a pseudograph with two ... |
| upgr1trld 30207 | In a pseudograph with two ... |
| upgr1pthd 30208 | In a pseudograph with two ... |
| upgr1pthond 30209 | In a pseudograph with two ... |
| lppthon 30210 | A loop (which is an edge a... |
| lp1cycl 30211 | A loop (which is an edge a... |
| 1pthon2v 30212 | For each pair of adjacent ... |
| 1pthon2ve 30213 | For each pair of adjacent ... |
| wlk2v2elem1 30214 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 30215 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 30216 | In a graph with two vertic... |
| ntrl2v2e 30217 | A walk which is not a trai... |
| 3wlkdlem1 30218 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 30219 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 30220 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 30221 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 30222 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 30223 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 30224 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 30225 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 30226 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 30227 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 30228 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 30229 | Construction of a walk fro... |
| 3wlkond 30230 | A walk of length 3 from on... |
| 3trld 30231 | Construction of a trail fr... |
| 3trlond 30232 | A trail of length 3 from o... |
| 3pthd 30233 | A path of length 3 from on... |
| 3pthond 30234 | A path of length 3 from on... |
| 3spthd 30235 | A simple path of length 3 ... |
| 3spthond 30236 | A simple path of length 3 ... |
| 3cycld 30237 | Construction of a 3-cycle ... |
| 3cyclpd 30238 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 30239 | If there is a cycle of len... |
| uhgr3cyclexlem 30240 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 30241 | If there are three differe... |
| umgr3cyclex 30242 | If there are three (differ... |
| umgr3v3e3cycl 30243 | If and only if there is a ... |
| upgr4cycl4dv4e 30244 | If there is a cycle of len... |
| dfconngr1 30247 | Alternative definition of ... |
| isconngr 30248 | The property of being a co... |
| isconngr1 30249 | The property of being a co... |
| cusconngr 30250 | A complete hypergraph is c... |
| 0conngr 30251 | A graph without vertices i... |
| 0vconngr 30252 | A graph without vertices i... |
| 1conngr 30253 | A graph with (at most) one... |
| conngrv2edg 30254 | A vertex in a connected gr... |
| vdn0conngrumgrv2 30255 | A vertex in a connected mu... |
| releupth 30258 | The set ` ( EulerPaths `` ... |
| eupths 30259 | The Eulerian paths on the ... |
| iseupth 30260 | The property " ` <. F , P ... |
| iseupthf1o 30261 | The property " ` <. F , P ... |
| eupthi 30262 | Properties of an Eulerian ... |
| eupthf1o 30263 | The ` F ` function in an E... |
| eupthfi 30264 | Any graph with an Eulerian... |
| eupthseg 30265 | The ` N ` -th edge in an e... |
| upgriseupth 30266 | The property " ` <. F , P ... |
| upgreupthi 30267 | Properties of an Eulerian ... |
| upgreupthseg 30268 | The ` N ` -th edge in an e... |
| eupthcl 30269 | An Eulerian path has lengt... |
| eupthistrl 30270 | An Eulerian path is a trai... |
| eupthiswlk 30271 | An Eulerian path is a walk... |
| eupthpf 30272 | The ` P ` function in an E... |
| eupth0 30273 | There is an Eulerian path ... |
| eupthres 30274 | The restriction ` <. H , Q... |
| eupthp1 30275 | Append one path segment to... |
| eupth2eucrct 30276 | Append one path segment to... |
| eupth2lem1 30277 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 30278 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 30279 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 30280 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 30281 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 30282 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 30283 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 30284 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 30285 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 30286 | Formerly part of proof of ... |
| eupth2lem3lem1 30287 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 30288 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 30289 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 30290 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 30291 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 30292 | Formerly part of proof of ... |
| eupth2lem3lem7 30293 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 30294 | Formerly part of proof of ... |
| eupth2lem3 30295 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 30296 | Lemma for ~ eupth2 (induct... |
| eupth2lems 30297 | Lemma for ~ eupth2 (induct... |
| eupth2 30298 | The only vertices of odd d... |
| eulerpathpr 30299 | A graph with an Eulerian p... |
| eulerpath 30300 | A pseudograph with an Eule... |
| eulercrct 30301 | A pseudograph with an Eule... |
| eucrctshift 30302 | Cyclically shifting the in... |
| eucrct2eupth1 30303 | Removing one edge ` ( I ``... |
| eucrct2eupth 30304 | Removing one edge ` ( I ``... |
| konigsbergvtx 30305 | The set of vertices of the... |
| konigsbergiedg 30306 | The indexed edges of the K... |
| konigsbergiedgw 30307 | The indexed edges of the K... |
| konigsbergssiedgwpr 30308 | Each subset of the indexed... |
| konigsbergssiedgw 30309 | Each subset of the indexed... |
| konigsbergumgr 30310 | The Königsberg graph ... |
| konigsberglem1 30311 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 30312 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 30313 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 30314 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 30315 | Lemma 5 for ~ konigsberg :... |
| konigsberg 30316 | The Königsberg Bridge... |
| isfrgr 30319 | The property of being a fr... |
| frgrusgr 30320 | A friendship graph is a si... |
| frgr0v 30321 | Any null graph (set with n... |
| frgr0vb 30322 | Any null graph (without ve... |
| frgruhgr0v 30323 | Any null graph (without ve... |
| frgr0 30324 | The null graph (graph with... |
| frcond1 30325 | The friendship condition: ... |
| frcond2 30326 | The friendship condition: ... |
| frgreu 30327 | Variant of ~ frcond2 : An... |
| frcond3 30328 | The friendship condition, ... |
| frcond4 30329 | The friendship condition, ... |
| frgr1v 30330 | Any graph with (at most) o... |
| nfrgr2v 30331 | Any graph with two (differ... |
| frgr3vlem1 30332 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 30333 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 30334 | Any graph with three verti... |
| 1vwmgr 30335 | Every graph with one verte... |
| 3vfriswmgrlem 30336 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 30337 | Every friendship graph wit... |
| 1to2vfriswmgr 30338 | Every friendship graph wit... |
| 1to3vfriswmgr 30339 | Every friendship graph wit... |
| 1to3vfriendship 30340 | The friendship theorem for... |
| 2pthfrgrrn 30341 | Between any two (different... |
| 2pthfrgrrn2 30342 | Between any two (different... |
| 2pthfrgr 30343 | Between any two (different... |
| 3cyclfrgrrn1 30344 | Every vertex in a friendsh... |
| 3cyclfrgrrn 30345 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 30346 | Every vertex in a friendsh... |
| 3cyclfrgr 30347 | Every vertex in a friendsh... |
| 4cycl2v2nb 30348 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 30349 | In a 4-cycle, two distinct... |
| n4cyclfrgr 30350 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 30351 | A graph with a 4-cycle is ... |
| frgrnbnb 30352 | If two neighbors ` U ` and... |
| frgrconngr 30353 | A friendship graph is conn... |
| vdgn0frgrv2 30354 | A vertex in a friendship g... |
| vdgn1frgrv2 30355 | Any vertex in a friendship... |
| vdgn1frgrv3 30356 | Any vertex in a friendship... |
| vdgfrgrgt2 30357 | Any vertex in a friendship... |
| frgrncvvdeqlem1 30358 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 30359 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 30360 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 30361 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 30362 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 30363 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 30364 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 30365 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 30366 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 30367 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 30368 | In a friendship graph, two... |
| frgrwopreglem4a 30369 | In a friendship graph any ... |
| frgrwopreglem5a 30370 | If a friendship graph has ... |
| frgrwopreglem1 30371 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 30372 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 30373 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 30374 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 30375 | According to statement 5 i... |
| frgrwopregbsn 30376 | According to statement 5 i... |
| frgrwopreg1 30377 | According to statement 5 i... |
| frgrwopreg2 30378 | According to statement 5 i... |
| frgrwopreglem5lem 30379 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 30380 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 30381 | Alternate direct proof of ... |
| frgrwopreg 30382 | In a friendship graph ther... |
| frgrregorufr0 30383 | In a friendship graph ther... |
| frgrregorufr 30384 | If there is a vertex havin... |
| frgrregorufrg 30385 | If there is a vertex havin... |
| frgr2wwlkeu 30386 | For two different vertices... |
| frgr2wwlkn0 30387 | In a friendship graph, the... |
| frgr2wwlk1 30388 | In a friendship graph, the... |
| frgr2wsp1 30389 | In a friendship graph, the... |
| frgr2wwlkeqm 30390 | If there is a (simple) pat... |
| frgrhash2wsp 30391 | The number of simple paths... |
| fusgreg2wsplem 30392 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 30393 | The set of paths of length... |
| fusgreghash2wspv 30394 | According to statement 7 i... |
| fusgreg2wsp 30395 | In a finite simple graph, ... |
| 2wspmdisj 30396 | The sets of paths of lengt... |
| fusgreghash2wsp 30397 | In a finite k-regular grap... |
| frrusgrord0lem 30398 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 30399 | If a nonempty finite frien... |
| frrusgrord 30400 | If a nonempty finite frien... |
| numclwwlk2lem1lem 30401 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 30402 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 30403 | If the initial vertex of a... |
| clwwnonrepclwwnon 30404 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 30405 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 30406 | Value of operation ` C ` ,... |
| 2clwwlk2 30407 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 30408 | Characterization of an ele... |
| 2clwwlk2clwwlk 30409 | An element of the value of... |
| numclwwlk1lem2foalem 30410 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 30411 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 30412 | Characterization of an ele... |
| numclwwlk1lem2foa 30413 | Going forth and back from ... |
| numclwwlk1lem2f 30414 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 30415 | Value of the function ` T ... |
| numclwwlk1lem2f1 30416 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 30417 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 30418 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 30419 | The set of double loops of... |
| numclwwlk1 30420 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 30421 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 30422 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 30423 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 30424 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 30425 | The sets of the two repres... |
| wlkl0 30426 | There is exactly one walk ... |
| clwlknon2num 30427 | There are k walks of lengt... |
| numclwlk1lem1 30428 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 30429 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 30430 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 30431 | Value of operation ` H ` ,... |
| numclwwlkovh 30432 | Value of operation ` H ` ,... |
| numclwwlkovq 30433 | Value of operation ` Q ` ,... |
| numclwwlkqhash 30434 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 30435 | In a friendship graph, for... |
| numclwlk2lem2f 30436 | ` R ` is a function mappin... |
| numclwlk2lem2fv 30437 | Value of the function ` R ... |
| numclwlk2lem2f1o 30438 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 30439 | In a friendship graph, the... |
| numclwwlk2 30440 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 30441 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 30442 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 30443 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 30444 | Statement 12 in [Huneke] p... |
| numclwwlk4 30445 | The total number of closed... |
| numclwwlk5lem 30446 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 30447 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 30448 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 30449 | For a prime divisor ` P ` ... |
| numclwwlk7 30450 | Statement 14 in [Huneke] p... |
| numclwwlk8 30451 | The size of the set of clo... |
| frgrreggt1 30452 | If a finite nonempty frien... |
| frgrreg 30453 | If a finite nonempty frien... |
| frgrregord013 30454 | If a finite friendship gra... |
| frgrregord13 30455 | If a nonempty finite frien... |
| frgrogt3nreg 30456 | If a finite friendship gra... |
| friendshipgt3 30457 | The friendship theorem for... |
| friendship 30458 | The friendship theorem: I... |
| conventions 30459 |
H... |
| conventions-labels 30460 |
... |
| conventions-comments 30461 |
... |
| natded 30462 | Here are typical n... |
| ex-natded5.2 30463 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 30464 | A more efficient proof of ... |
| ex-natded5.2i 30465 | The same as ~ ex-natded5.2... |
| ex-natded5.3 30466 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 30467 | A more efficient proof of ... |
| ex-natded5.3i 30468 | The same as ~ ex-natded5.3... |
| ex-natded5.5 30469 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 30470 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 30471 | A more efficient proof of ... |
| ex-natded5.8 30472 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 30473 | A more efficient proof of ... |
| ex-natded5.13 30474 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 30475 | A more efficient proof of ... |
| ex-natded9.20 30476 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 30477 | A more efficient proof of ... |
| ex-natded9.26 30478 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 30479 | A more efficient proof of ... |
| ex-or 30480 | Example for ~ df-or . Exa... |
| ex-an 30481 | Example for ~ df-an . Exa... |
| ex-dif 30482 | Example for ~ df-dif . Ex... |
| ex-un 30483 | Example for ~ df-un . Exa... |
| ex-in 30484 | Example for ~ df-in . Exa... |
| ex-uni 30485 | Example for ~ df-uni . Ex... |
| ex-ss 30486 | Example for ~ df-ss . Exa... |
| ex-pss 30487 | Example for ~ df-pss . Ex... |
| ex-pw 30488 | Example for ~ df-pw . Exa... |
| ex-pr 30489 | Example for ~ df-pr . (Co... |
| ex-br 30490 | Example for ~ df-br . Exa... |
| ex-opab 30491 | Example for ~ df-opab . E... |
| ex-eprel 30492 | Example for ~ df-eprel . ... |
| ex-id 30493 | Example for ~ df-id . Exa... |
| ex-po 30494 | Example for ~ df-po . Exa... |
| ex-xp 30495 | Example for ~ df-xp . Exa... |
| ex-cnv 30496 | Example for ~ df-cnv . Ex... |
| ex-co 30497 | Example for ~ df-co . Exa... |
| ex-dm 30498 | Example for ~ df-dm . Exa... |
| ex-rn 30499 | Example for ~ df-rn . Exa... |
| ex-res 30500 | Example for ~ df-res . Ex... |
| ex-ima 30501 | Example for ~ df-ima . Ex... |
| ex-fv 30502 | Example for ~ df-fv . Exa... |
| ex-1st 30503 | Example for ~ df-1st . Ex... |
| ex-2nd 30504 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 30505 | Example for ~ df-dec , 100... |
| ex-fl 30506 | Example for ~ df-fl . Exa... |
| ex-ceil 30507 | Example for ~ df-ceil . (... |
| ex-mod 30508 | Example for ~ df-mod . (C... |
| ex-exp 30509 | Example for ~ df-exp . (C... |
| ex-fac 30510 | Example for ~ df-fac . (C... |
| ex-bc 30511 | Example for ~ df-bc . (Co... |
| ex-hash 30512 | Example for ~ df-hash . (... |
| ex-sqrt 30513 | Example for ~ df-sqrt . (... |
| ex-abs 30514 | Example for ~ df-abs . (C... |
| ex-dvds 30515 | Example for ~ df-dvds : 3 ... |
| ex-gcd 30516 | Example for ~ df-gcd . (C... |
| ex-lcm 30517 | Example for ~ df-lcm . (C... |
| ex-prmo 30518 | Example for ~ df-prmo : ` ... |
| aevdemo 30519 | Proof illustrating the com... |
| ex-ind-dvds 30520 | Example of a proof by indu... |
| ex-fpar 30521 | Formalized example provide... |
| avril1 30522 | Poisson d'Avril's Theorem.... |
| 2bornot2b 30523 | The law of excluded middle... |
| helloworld 30524 | The classic "Hello world" ... |
| 1p1e2apr1 30525 | One plus one equals two. ... |
| eqid1 30526 | Law of identity (reflexivi... |
| 1div0apr 30527 | Division by zero is forbid... |
| topnfbey 30528 | Nothing seems to be imposs... |
| 9p10ne21 30529 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 30530 | 9 + 10 equals 21. This as... |
| nrt2irr 30532 | The ` N ` -th root of 2 is... |
| nowisdomv 30533 | One's wisdom on matters of... |
| isplig 30536 | The predicate "is a planar... |
| ispligb 30537 | The predicate "is a planar... |
| tncp 30538 | In any planar incidence ge... |
| l2p 30539 | For any line in a planar i... |
| lpni 30540 | For any line in a planar i... |
| nsnlplig 30541 | There is no "one-point lin... |
| nsnlpligALT 30542 | Alternate version of ~ nsn... |
| n0lplig 30543 | There is no "empty line" i... |
| n0lpligALT 30544 | Alternate version of ~ n0l... |
| eulplig 30545 | Through two distinct point... |
| pliguhgr 30546 | Any planar incidence geome... |
| dummylink 30547 | Alias for ~ a1ii that may ... |
| id1 30548 | Alias for ~ idALT that may... |
| isgrpo 30557 | The predicate "is a group ... |
| isgrpoi 30558 | Properties that determine ... |
| grpofo 30559 | A group operation maps ont... |
| grpocl 30560 | Closure law for a group op... |
| grpolidinv 30561 | A group has a left identit... |
| grpon0 30562 | The base set of a group is... |
| grpoass 30563 | A group operation is assoc... |
| grpoidinvlem1 30564 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30565 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30566 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30567 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30568 | A group has a left and rig... |
| grpoideu 30569 | The left identity element ... |
| grporndm 30570 | A group's range in terms o... |
| 0ngrp 30571 | The empty set is not a gro... |
| gidval 30572 | The value of the identity ... |
| grpoidval 30573 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30574 | The identity element of a ... |
| grpoidinv2 30575 | A group's properties using... |
| grpolid 30576 | The identity element of a ... |
| grporid 30577 | The identity element of a ... |
| grporcan 30578 | Right cancellation law for... |
| grpoinveu 30579 | The left inverse element o... |
| grpoid 30580 | Two ways of saying that an... |
| grporn 30581 | The range of a group opera... |
| grpoinvfval 30582 | The inverse function of a ... |
| grpoinvval 30583 | The inverse of a group ele... |
| grpoinvcl 30584 | A group element's inverse ... |
| grpoinv 30585 | The properties of a group ... |
| grpolinv 30586 | The left inverse of a grou... |
| grporinv 30587 | The right inverse of a gro... |
| grpoinvid1 30588 | The inverse of a group ele... |
| grpoinvid2 30589 | The inverse of a group ele... |
| grpolcan 30590 | Left cancellation law for ... |
| grpo2inv 30591 | Double inverse law for gro... |
| grpoinvf 30592 | Mapping of the inverse fun... |
| grpoinvop 30593 | The inverse of the group o... |
| grpodivfval 30594 | Group division (or subtrac... |
| grpodivval 30595 | Group division (or subtrac... |
| grpodivinv 30596 | Group division by an inver... |
| grpoinvdiv 30597 | Inverse of a group divisio... |
| grpodivf 30598 | Mapping for group division... |
| grpodivcl 30599 | Closure of group division ... |
| grpodivdiv 30600 | Double group division. (C... |
| grpomuldivass 30601 | Associative-type law for m... |
| grpodivid 30602 | Division of a group member... |
| grponpcan 30603 | Cancellation law for group... |
| isablo 30606 | The predicate "is an Abeli... |
| ablogrpo 30607 | An Abelian group operation... |
| ablocom 30608 | An Abelian group operation... |
| ablo32 30609 | Commutative/associative la... |
| ablo4 30610 | Commutative/associative la... |
| isabloi 30611 | Properties that determine ... |
| ablomuldiv 30612 | Law for group multiplicati... |
| ablodivdiv 30613 | Law for double group divis... |
| ablodivdiv4 30614 | Law for double group divis... |
| ablodiv32 30615 | Swap the second and third ... |
| ablonncan 30616 | Cancellation law for group... |
| ablonnncan1 30617 | Cancellation law for group... |
| vcrel 30620 | The class of all complex v... |
| vciOLD 30621 | Obsolete version of ~ cvsi... |
| vcsm 30622 | Functionality of th scalar... |
| vccl 30623 | Closure of the scalar prod... |
| vcidOLD 30624 | Identity element for the s... |
| vcdi 30625 | Distributive law for the s... |
| vcdir 30626 | Distributive law for the s... |
| vcass 30627 | Associative law for the sc... |
| vc2OLD 30628 | A vector plus itself is tw... |
| vcablo 30629 | Vector addition is an Abel... |
| vcgrp 30630 | Vector addition is a group... |
| vclcan 30631 | Left cancellation law for ... |
| vczcl 30632 | The zero vector is a vecto... |
| vc0rid 30633 | The zero vector is a right... |
| vc0 30634 | Zero times a vector is the... |
| vcz 30635 | Anything times the zero ve... |
| vcm 30636 | Minus 1 times a vector is ... |
| isvclem 30637 | Lemma for ~ isvcOLD . (Co... |
| vcex 30638 | The components of a comple... |
| isvcOLD 30639 | The predicate "is a comple... |
| isvciOLD 30640 | Properties that determine ... |
| cnaddabloOLD 30641 | Obsolete version of ~ cnad... |
| cnidOLD 30642 | Obsolete version of ~ cnad... |
| cncvcOLD 30643 | Obsolete version of ~ cncv... |
| nvss 30653 | Structure of the class of ... |
| nvvcop 30654 | A normed complex vector sp... |
| nvrel 30662 | The class of all normed co... |
| vafval 30663 | Value of the function for ... |
| bafval 30664 | Value of the function for ... |
| smfval 30665 | Value of the function for ... |
| 0vfval 30666 | Value of the function for ... |
| nmcvfval 30667 | Value of the norm function... |
| nvop2 30668 | A normed complex vector sp... |
| nvvop 30669 | The vector space component... |
| isnvlem 30670 | Lemma for ~ isnv . (Contr... |
| nvex 30671 | The components of a normed... |
| isnv 30672 | The predicate "is a normed... |
| isnvi 30673 | Properties that determine ... |
| nvi 30674 | The properties of a normed... |
| nvvc 30675 | The vector space component... |
| nvablo 30676 | The vector addition operat... |
| nvgrp 30677 | The vector addition operat... |
| nvgf 30678 | Mapping for the vector add... |
| nvsf 30679 | Mapping for the scalar mul... |
| nvgcl 30680 | Closure law for the vector... |
| nvcom 30681 | The vector addition (group... |
| nvass 30682 | The vector addition (group... |
| nvadd32 30683 | Commutative/associative la... |
| nvrcan 30684 | Right cancellation law for... |
| nvadd4 30685 | Rearrangement of 4 terms i... |
| nvscl 30686 | Closure law for the scalar... |
| nvsid 30687 | Identity element for the s... |
| nvsass 30688 | Associative law for the sc... |
| nvscom 30689 | Commutative law for the sc... |
| nvdi 30690 | Distributive law for the s... |
| nvdir 30691 | Distributive law for the s... |
| nv2 30692 | A vector plus itself is tw... |
| vsfval 30693 | Value of the function for ... |
| nvzcl 30694 | Closure law for the zero v... |
| nv0rid 30695 | The zero vector is a right... |
| nv0lid 30696 | The zero vector is a left ... |
| nv0 30697 | Zero times a vector is the... |
| nvsz 30698 | Anything times the zero ve... |
| nvinv 30699 | Minus 1 times a vector is ... |
| nvinvfval 30700 | Function for the negative ... |
| nvm 30701 | Vector subtraction in term... |
| nvmval 30702 | Value of vector subtractio... |
| nvmval2 30703 | Value of vector subtractio... |
| nvmfval 30704 | Value of the function for ... |
| nvmf 30705 | Mapping for the vector sub... |
| nvmcl 30706 | Closure law for the vector... |
| nvnnncan1 30707 | Cancellation law for vecto... |
| nvmdi 30708 | Distributive law for scala... |
| nvnegneg 30709 | Double negative of a vecto... |
| nvmul0or 30710 | If a scalar product is zer... |
| nvrinv 30711 | A vector minus itself. (C... |
| nvlinv 30712 | Minus a vector plus itself... |
| nvpncan2 30713 | Cancellation law for vecto... |
| nvpncan 30714 | Cancellation law for vecto... |
| nvaddsub 30715 | Commutative/associative la... |
| nvnpcan 30716 | Cancellation law for a nor... |
| nvaddsub4 30717 | Rearrangement of 4 terms i... |
| nvmeq0 30718 | The difference between two... |
| nvmid 30719 | A vector minus itself is t... |
| nvf 30720 | Mapping for the norm funct... |
| nvcl 30721 | The norm of a normed compl... |
| nvcli 30722 | The norm of a normed compl... |
| nvs 30723 | Proportionality property o... |
| nvsge0 30724 | The norm of a scalar produ... |
| nvm1 30725 | The norm of the negative o... |
| nvdif 30726 | The norm of the difference... |
| nvpi 30727 | The norm of a vector plus ... |
| nvz0 30728 | The norm of a zero vector ... |
| nvz 30729 | The norm of a vector is ze... |
| nvtri 30730 | Triangle inequality for th... |
| nvmtri 30731 | Triangle inequality for th... |
| nvabs 30732 | Norm difference property o... |
| nvge0 30733 | The norm of a normed compl... |
| nvgt0 30734 | A nonzero norm is positive... |
| nv1 30735 | From any nonzero vector, c... |
| nvop 30736 | A complex inner product sp... |
| cnnv 30737 | The set of complex numbers... |
| cnnvg 30738 | The vector addition (group... |
| cnnvba 30739 | The base set of the normed... |
| cnnvs 30740 | The scalar product operati... |
| cnnvnm 30741 | The norm operation of the ... |
| cnnvm 30742 | The vector subtraction ope... |
| elimnv 30743 | Hypothesis elimination lem... |
| elimnvu 30744 | Hypothesis elimination lem... |
| imsval 30745 | Value of the induced metri... |
| imsdval 30746 | Value of the induced metri... |
| imsdval2 30747 | Value of the distance func... |
| nvnd 30748 | The norm of a normed compl... |
| imsdf 30749 | Mapping for the induced me... |
| imsmetlem 30750 | Lemma for ~ imsmet . (Con... |
| imsmet 30751 | The induced metric of a no... |
| imsxmet 30752 | The induced metric of a no... |
| cnims 30753 | The metric induced on the ... |
| vacn 30754 | Vector addition is jointly... |
| nmcvcn 30755 | The norm of a normed compl... |
| nmcnc 30756 | The norm of a normed compl... |
| smcnlem 30757 | Lemma for ~ smcn . (Contr... |
| smcn 30758 | Scalar multiplication is j... |
| vmcn 30759 | Vector subtraction is join... |
| dipfval 30762 | The inner product function... |
| ipval 30763 | Value of the inner product... |
| ipval2lem2 30764 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30765 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30766 | Lemma for ~ ipval3 . (Con... |
| ipval2 30767 | Expansion of the inner pro... |
| 4ipval2 30768 | Four times the inner produ... |
| ipval3 30769 | Expansion of the inner pro... |
| ipidsq 30770 | The inner product of a vec... |
| ipnm 30771 | Norm expressed in terms of... |
| dipcl 30772 | An inner product is a comp... |
| ipf 30773 | Mapping for the inner prod... |
| dipcj 30774 | The complex conjugate of a... |
| ipipcj 30775 | An inner product times its... |
| diporthcom 30776 | Orthogonality (meaning inn... |
| dip0r 30777 | Inner product with a zero ... |
| dip0l 30778 | Inner product with a zero ... |
| ipz 30779 | The inner product of a vec... |
| dipcn 30780 | Inner product is jointly c... |
| sspval 30783 | The set of all subspaces o... |
| isssp 30784 | The predicate "is a subspa... |
| sspid 30785 | A normed complex vector sp... |
| sspnv 30786 | A subspace is a normed com... |
| sspba 30787 | The base set of a subspace... |
| sspg 30788 | Vector addition on a subsp... |
| sspgval 30789 | Vector addition on a subsp... |
| ssps 30790 | Scalar multiplication on a... |
| sspsval 30791 | Scalar multiplication on a... |
| sspmlem 30792 | Lemma for ~ sspm and other... |
| sspmval 30793 | Vector addition on a subsp... |
| sspm 30794 | Vector subtraction on a su... |
| sspz 30795 | The zero vector of a subsp... |
| sspn 30796 | The norm on a subspace is ... |
| sspnval 30797 | The norm on a subspace in ... |
| sspimsval 30798 | The induced metric on a su... |
| sspims 30799 | The induced metric on a su... |
| lnoval 30812 | The set of linear operator... |
| islno 30813 | The predicate "is a linear... |
| lnolin 30814 | Basic linearity property o... |
| lnof 30815 | A linear operator is a map... |
| lno0 30816 | The value of a linear oper... |
| lnocoi 30817 | The composition of two lin... |
| lnoadd 30818 | Addition property of a lin... |
| lnosub 30819 | Subtraction property of a ... |
| lnomul 30820 | Scalar multiplication prop... |
| nvo00 30821 | Two ways to express a zero... |
| nmoofval 30822 | The operator norm function... |
| nmooval 30823 | The operator norm function... |
| nmosetre 30824 | The set in the supremum of... |
| nmosetn0 30825 | The set in the supremum of... |
| nmoxr 30826 | The norm of an operator is... |
| nmooge0 30827 | The norm of an operator is... |
| nmorepnf 30828 | The norm of an operator is... |
| nmoreltpnf 30829 | The norm of any operator i... |
| nmogtmnf 30830 | The norm of an operator is... |
| nmoolb 30831 | A lower bound for an opera... |
| nmoubi 30832 | An upper bound for an oper... |
| nmoub3i 30833 | An upper bound for an oper... |
| nmoub2i 30834 | An upper bound for an oper... |
| nmobndi 30835 | Two ways to express that a... |
| nmounbi 30836 | Two ways two express that ... |
| nmounbseqi 30837 | An unbounded operator dete... |
| nmounbseqiALT 30838 | Alternate shorter proof of... |
| nmobndseqi 30839 | A bounded sequence determi... |
| nmobndseqiALT 30840 | Alternate shorter proof of... |
| bloval 30841 | The class of bounded linea... |
| isblo 30842 | The predicate "is a bounde... |
| isblo2 30843 | The predicate "is a bounde... |
| bloln 30844 | A bounded operator is a li... |
| blof 30845 | A bounded operator is an o... |
| nmblore 30846 | The norm of a bounded oper... |
| 0ofval 30847 | The zero operator between ... |
| 0oval 30848 | Value of the zero operator... |
| 0oo 30849 | The zero operator is an op... |
| 0lno 30850 | The zero operator is linea... |
| nmoo0 30851 | The operator norm of the z... |
| 0blo 30852 | The zero operator is a bou... |
| nmlno0lem 30853 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30854 | The norm of a linear opera... |
| nmlno0 30855 | The norm of a linear opera... |
| nmlnoubi 30856 | An upper bound for the ope... |
| nmlnogt0 30857 | The norm of a nonzero line... |
| lnon0 30858 | The domain of a nonzero li... |
| nmblolbii 30859 | A lower bound for the norm... |
| nmblolbi 30860 | A lower bound for the norm... |
| isblo3i 30861 | The predicate "is a bounde... |
| blo3i 30862 | Properties that determine ... |
| blometi 30863 | Upper bound for the distan... |
| blocnilem 30864 | Lemma for ~ blocni and ~ l... |
| blocni 30865 | A linear operator is conti... |
| lnocni 30866 | If a linear operator is co... |
| blocn 30867 | A linear operator is conti... |
| blocn2 30868 | A bounded linear operator ... |
| ajfval 30869 | The adjoint function. (Co... |
| hmoval 30870 | The set of Hermitian (self... |
| ishmo 30871 | The predicate "is a hermit... |
| phnv 30874 | Every complex inner produc... |
| phrel 30875 | The class of all complex i... |
| phnvi 30876 | Every complex inner produc... |
| isphg 30877 | The predicate "is a comple... |
| phop 30878 | A complex inner product sp... |
| cncph 30879 | The set of complex numbers... |
| elimph 30880 | Hypothesis elimination lem... |
| elimphu 30881 | Hypothesis elimination lem... |
| isph 30882 | The predicate "is an inner... |
| phpar2 30883 | The parallelogram law for ... |
| phpar 30884 | The parallelogram law for ... |
| ip0i 30885 | A slight variant of Equati... |
| ip1ilem 30886 | Lemma for ~ ip1i . (Contr... |
| ip1i 30887 | Equation 6.47 of [Ponnusam... |
| ip2i 30888 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30889 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30890 | Distributive law for inner... |
| ipasslem1 30891 | Lemma for ~ ipassi . Show... |
| ipasslem2 30892 | Lemma for ~ ipassi . Show... |
| ipasslem3 30893 | Lemma for ~ ipassi . Show... |
| ipasslem4 30894 | Lemma for ~ ipassi . Show... |
| ipasslem5 30895 | Lemma for ~ ipassi . Show... |
| ipasslem7 30896 | Lemma for ~ ipassi . Show... |
| ipasslem8 30897 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30898 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30899 | Lemma for ~ ipassi . Show... |
| ipasslem11 30900 | Lemma for ~ ipassi . Show... |
| ipassi 30901 | Associative law for inner ... |
| dipdir 30902 | Distributive law for inner... |
| dipdi 30903 | Distributive law for inner... |
| ip2dii 30904 | Inner product of two sums.... |
| dipass 30905 | Associative law for inner ... |
| dipassr 30906 | "Associative" law for seco... |
| dipassr2 30907 | "Associative" law for inne... |
| dipsubdir 30908 | Distributive law for inner... |
| dipsubdi 30909 | Distributive law for inner... |
| pythi 30910 | The Pythagorean theorem fo... |
| siilem1 30911 | Lemma for ~ sii . (Contri... |
| siilem2 30912 | Lemma for ~ sii . (Contri... |
| siii 30913 | Inference from ~ sii . (C... |
| sii 30914 | Obsolete version of ~ ipca... |
| ipblnfi 30915 | A function ` F ` generated... |
| ip2eqi 30916 | Two vectors are equal iff ... |
| phoeqi 30917 | A condition implying that ... |
| ajmoi 30918 | Every operator has at most... |
| ajfuni 30919 | The adjoint function is a ... |
| ajfun 30920 | The adjoint function is a ... |
| ajval 30921 | Value of the adjoint funct... |
| iscbn 30924 | A complex Banach space is ... |
| cbncms 30925 | The induced metric on comp... |
| bnnv 30926 | Every complex Banach space... |
| bnrel 30927 | The class of all complex B... |
| bnsscmcl 30928 | A subspace of a Banach spa... |
| cnbn 30929 | The set of complex numbers... |
| ubthlem1 30930 | Lemma for ~ ubth . The fu... |
| ubthlem2 30931 | Lemma for ~ ubth . Given ... |
| ubthlem3 30932 | Lemma for ~ ubth . Prove ... |
| ubth 30933 | Uniform Boundedness Theore... |
| minvecolem1 30934 | Lemma for ~ minveco . The... |
| minvecolem2 30935 | Lemma for ~ minveco . Any... |
| minvecolem3 30936 | Lemma for ~ minveco . The... |
| minvecolem4a 30937 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30938 | Lemma for ~ minveco . The... |
| minvecolem4c 30939 | Lemma for ~ minveco . The... |
| minvecolem4 30940 | Lemma for ~ minveco . The... |
| minvecolem5 30941 | Lemma for ~ minveco . Dis... |
| minvecolem6 30942 | Lemma for ~ minveco . Any... |
| minvecolem7 30943 | Lemma for ~ minveco . Sin... |
| minveco 30944 | Minimizing vector theorem,... |
| ishlo 30947 | The predicate "is a comple... |
| hlobn 30948 | Every complex Hilbert spac... |
| hlph 30949 | Every complex Hilbert spac... |
| hlrel 30950 | The class of all complex H... |
| hlnv 30951 | Every complex Hilbert spac... |
| hlnvi 30952 | Every complex Hilbert spac... |
| hlvc 30953 | Every complex Hilbert spac... |
| hlcmet 30954 | The induced metric on a co... |
| hlmet 30955 | The induced metric on a co... |
| hlpar2 30956 | The parallelogram law sati... |
| hlpar 30957 | The parallelogram law sati... |
| hlex 30958 | The base set of a Hilbert ... |
| hladdf 30959 | Mapping for Hilbert space ... |
| hlcom 30960 | Hilbert space vector addit... |
| hlass 30961 | Hilbert space vector addit... |
| hl0cl 30962 | The Hilbert space zero vec... |
| hladdid 30963 | Hilbert space addition wit... |
| hlmulf 30964 | Mapping for Hilbert space ... |
| hlmulid 30965 | Hilbert space scalar multi... |
| hlmulass 30966 | Hilbert space scalar multi... |
| hldi 30967 | Hilbert space scalar multi... |
| hldir 30968 | Hilbert space scalar multi... |
| hlmul0 30969 | Hilbert space scalar multi... |
| hlipf 30970 | Mapping for Hilbert space ... |
| hlipcj 30971 | Conjugate law for Hilbert ... |
| hlipdir 30972 | Distributive law for Hilbe... |
| hlipass 30973 | Associative law for Hilber... |
| hlipgt0 30974 | The inner product of a Hil... |
| hlcompl 30975 | Completeness of a Hilbert ... |
| cnchl 30976 | The set of complex numbers... |
| htthlem 30977 | Lemma for ~ htth . The co... |
| htth 30978 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 31034 | The group (addition) opera... |
| h2hsm 31035 | The scalar product operati... |
| h2hnm 31036 | The norm function of Hilbe... |
| h2hvs 31037 | The vector subtraction ope... |
| h2hmetdval 31038 | Value of the distance func... |
| h2hcau 31039 | The Cauchy sequences of Hi... |
| h2hlm 31040 | The limit sequences of Hil... |
| axhilex-zf 31041 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 31042 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 31043 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 31044 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 31045 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 31046 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 31047 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 31048 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 31049 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 31050 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 31051 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 31052 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 31053 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 31054 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 31055 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 31056 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 31057 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 31058 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 31071 | The Hilbert space scalar p... |
| hvaddcl 31072 | Closure of vector addition... |
| hvmulcl 31073 | Closure of scalar multipli... |
| hvmulcli 31074 | Closure inference for scal... |
| hvsubf 31075 | Mapping domain and codomai... |
| hvsubval 31076 | Value of vector subtractio... |
| hvsubcl 31077 | Closure of vector subtract... |
| hvaddcli 31078 | Closure of vector addition... |
| hvcomi 31079 | Commutation of vector addi... |
| hvsubvali 31080 | Value of vector subtractio... |
| hvsubcli 31081 | Closure of vector subtract... |
| ifhvhv0 31082 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 31083 | Addition with the zero vec... |
| hvmul0 31084 | Scalar multiplication with... |
| hvmul0or 31085 | If a scalar product is zer... |
| hvsubid 31086 | Subtraction of a vector fr... |
| hvnegid 31087 | Addition of negative of a ... |
| hv2neg 31088 | Two ways to express the ne... |
| hvaddlidi 31089 | Addition with the zero vec... |
| hvnegidi 31090 | Addition of negative of a ... |
| hv2negi 31091 | Two ways to express the ne... |
| hvm1neg 31092 | Convert minus one times a ... |
| hvaddsubval 31093 | Value of vector addition i... |
| hvadd32 31094 | Commutative/associative la... |
| hvadd12 31095 | Commutative/associative la... |
| hvadd4 31096 | Hilbert vector space addit... |
| hvsub4 31097 | Hilbert vector space addit... |
| hvaddsub12 31098 | Commutative/associative la... |
| hvpncan 31099 | Addition/subtraction cance... |
| hvpncan2 31100 | Addition/subtraction cance... |
| hvaddsubass 31101 | Associativity of sum and d... |
| hvpncan3 31102 | Subtraction and addition o... |
| hvmulcom 31103 | Scalar multiplication comm... |
| hvsubass 31104 | Hilbert vector space assoc... |
| hvsub32 31105 | Hilbert vector space commu... |
| hvmulassi 31106 | Scalar multiplication asso... |
| hvmulcomi 31107 | Scalar multiplication comm... |
| hvmul2negi 31108 | Double negative in scalar ... |
| hvsubdistr1 31109 | Scalar multiplication dist... |
| hvsubdistr2 31110 | Scalar multiplication dist... |
| hvdistr1i 31111 | Scalar multiplication dist... |
| hvsubdistr1i 31112 | Scalar multiplication dist... |
| hvassi 31113 | Hilbert vector space assoc... |
| hvadd32i 31114 | Hilbert vector space commu... |
| hvsubassi 31115 | Hilbert vector space assoc... |
| hvsub32i 31116 | Hilbert vector space commu... |
| hvadd12i 31117 | Hilbert vector space commu... |
| hvadd4i 31118 | Hilbert vector space addit... |
| hvsubsub4i 31119 | Hilbert vector space addit... |
| hvsubsub4 31120 | Hilbert vector space addit... |
| hv2times 31121 | Two times a vector. (Cont... |
| hvnegdii 31122 | Distribution of negative o... |
| hvsubeq0i 31123 | If the difference between ... |
| hvsubcan2i 31124 | Vector cancellation law. ... |
| hvaddcani 31125 | Cancellation law for vecto... |
| hvsubaddi 31126 | Relationship between vecto... |
| hvnegdi 31127 | Distribution of negative o... |
| hvsubeq0 31128 | If the difference between ... |
| hvaddeq0 31129 | If the sum of two vectors ... |
| hvaddcan 31130 | Cancellation law for vecto... |
| hvaddcan2 31131 | Cancellation law for vecto... |
| hvmulcan 31132 | Cancellation law for scala... |
| hvmulcan2 31133 | Cancellation law for scala... |
| hvsubcan 31134 | Cancellation law for vecto... |
| hvsubcan2 31135 | Cancellation law for vecto... |
| hvsub0 31136 | Subtraction of a zero vect... |
| hvsubadd 31137 | Relationship between vecto... |
| hvaddsub4 31138 | Hilbert vector space addit... |
| hicl 31140 | Closure of inner product. ... |
| hicli 31141 | Closure inference for inne... |
| his5 31146 | Associative law for inner ... |
| his52 31147 | Associative law for inner ... |
| his35 31148 | Move scalar multiplication... |
| his35i 31149 | Move scalar multiplication... |
| his7 31150 | Distributive law for inner... |
| hiassdi 31151 | Distributive/associative l... |
| his2sub 31152 | Distributive law for inner... |
| his2sub2 31153 | Distributive law for inner... |
| hire 31154 | A necessary and sufficient... |
| hiidrcl 31155 | Real closure of inner prod... |
| hi01 31156 | Inner product with the 0 v... |
| hi02 31157 | Inner product with the 0 v... |
| hiidge0 31158 | Inner product with self is... |
| his6 31159 | Zero inner product with se... |
| his1i 31160 | Conjugate law for inner pr... |
| abshicom 31161 | Commuted inner products ha... |
| hial0 31162 | A vector whose inner produ... |
| hial02 31163 | A vector whose inner produ... |
| hisubcomi 31164 | Two vector subtractions si... |
| hi2eq 31165 | Lemma used to prove equali... |
| hial2eq 31166 | Two vectors whose inner pr... |
| hial2eq2 31167 | Two vectors whose inner pr... |
| orthcom 31168 | Orthogonality commutes. (... |
| normlem0 31169 | Lemma used to derive prope... |
| normlem1 31170 | Lemma used to derive prope... |
| normlem2 31171 | Lemma used to derive prope... |
| normlem3 31172 | Lemma used to derive prope... |
| normlem4 31173 | Lemma used to derive prope... |
| normlem5 31174 | Lemma used to derive prope... |
| normlem6 31175 | Lemma used to derive prope... |
| normlem7 31176 | Lemma used to derive prope... |
| normlem8 31177 | Lemma used to derive prope... |
| normlem9 31178 | Lemma used to derive prope... |
| normlem7tALT 31179 | Lemma used to derive prope... |
| bcseqi 31180 | Equality case of Bunjakova... |
| normlem9at 31181 | Lemma used to derive prope... |
| dfhnorm2 31182 | Alternate definition of th... |
| normf 31183 | The norm function maps fro... |
| normval 31184 | The value of the norm of a... |
| normcl 31185 | Real closure of the norm o... |
| normge0 31186 | The norm of a vector is no... |
| normgt0 31187 | The norm of nonzero vector... |
| norm0 31188 | The norm of a zero vector.... |
| norm-i 31189 | Theorem 3.3(i) of [Beran] ... |
| normne0 31190 | A norm is nonzero iff its ... |
| normcli 31191 | Real closure of the norm o... |
| normsqi 31192 | The square of a norm. (Co... |
| norm-i-i 31193 | Theorem 3.3(i) of [Beran] ... |
| normsq 31194 | The square of a norm. (Co... |
| normsub0i 31195 | Two vectors are equal iff ... |
| normsub0 31196 | Two vectors are equal iff ... |
| norm-ii-i 31197 | Triangle inequality for no... |
| norm-ii 31198 | Triangle inequality for no... |
| norm-iii-i 31199 | Theorem 3.3(iii) of [Beran... |
| norm-iii 31200 | Theorem 3.3(iii) of [Beran... |
| normsubi 31201 | Negative doesn't change th... |
| normpythi 31202 | Analogy to Pythagorean the... |
| normsub 31203 | Swapping order of subtract... |
| normneg 31204 | The norm of a vector equal... |
| normpyth 31205 | Analogy to Pythagorean the... |
| normpyc 31206 | Corollary to Pythagorean t... |
| norm3difi 31207 | Norm of differences around... |
| norm3adifii 31208 | Norm of differences around... |
| norm3lem 31209 | Lemma involving norm of di... |
| norm3dif 31210 | Norm of differences around... |
| norm3dif2 31211 | Norm of differences around... |
| norm3lemt 31212 | Lemma involving norm of di... |
| norm3adifi 31213 | Norm of differences around... |
| normpari 31214 | Parallelogram law for norm... |
| normpar 31215 | Parallelogram law for norm... |
| normpar2i 31216 | Corollary of parallelogram... |
| polid2i 31217 | Generalized polarization i... |
| polidi 31218 | Polarization identity. Re... |
| polid 31219 | Polarization identity. Re... |
| hilablo 31220 | Hilbert space vector addit... |
| hilid 31221 | The group identity element... |
| hilvc 31222 | Hilbert space is a complex... |
| hilnormi 31223 | Hilbert space norm in term... |
| hilhhi 31224 | Deduce the structure of Hi... |
| hhnv 31225 | Hilbert space is a normed ... |
| hhva 31226 | The group (addition) opera... |
| hhba 31227 | The base set of Hilbert sp... |
| hh0v 31228 | The zero vector of Hilbert... |
| hhsm 31229 | The scalar product operati... |
| hhvs 31230 | The vector subtraction ope... |
| hhnm 31231 | The norm function of Hilbe... |
| hhims 31232 | The induced metric of Hilb... |
| hhims2 31233 | Hilbert space distance met... |
| hhmet 31234 | The induced metric of Hilb... |
| hhxmet 31235 | The induced metric of Hilb... |
| hhmetdval 31236 | Value of the distance func... |
| hhip 31237 | The inner product operatio... |
| hhph 31238 | The Hilbert space of the H... |
| bcsiALT 31239 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 31240 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 31241 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 31242 | Corollary of the Bunjakova... |
| bcs3 31243 | Corollary of the Bunjakova... |
| hcau 31244 | Member of the set of Cauch... |
| hcauseq 31245 | A Cauchy sequences on a Hi... |
| hcaucvg 31246 | A Cauchy sequence on a Hil... |
| seq1hcau 31247 | A sequence on a Hilbert sp... |
| hlimi 31248 | Express the predicate: Th... |
| hlimseqi 31249 | A sequence with a limit on... |
| hlimveci 31250 | Closure of the limit of a ... |
| hlimconvi 31251 | Convergence of a sequence ... |
| hlim2 31252 | The limit of a sequence on... |
| hlimadd 31253 | Limit of the sum of two se... |
| hilmet 31254 | The Hilbert space norm det... |
| hilxmet 31255 | The Hilbert space norm det... |
| hilmetdval 31256 | Value of the distance func... |
| hilims 31257 | Hilbert space distance met... |
| hhcau 31258 | The Cauchy sequences of Hi... |
| hhlm 31259 | The limit sequences of Hil... |
| hhcmpl 31260 | Lemma used for derivation ... |
| hilcompl 31261 | Lemma used for derivation ... |
| hhcms 31263 | The Hilbert space induced ... |
| hhhl 31264 | The Hilbert space structur... |
| hilcms 31265 | The Hilbert space norm det... |
| hilhl 31266 | The Hilbert space of the H... |
| issh 31268 | Subspace ` H ` of a Hilber... |
| issh2 31269 | Subspace ` H ` of a Hilber... |
| shss 31270 | A subspace is a subset of ... |
| shel 31271 | A member of a subspace of ... |
| shex 31272 | The set of subspaces of a ... |
| shssii 31273 | A closed subspace of a Hil... |
| sheli 31274 | A member of a subspace of ... |
| shelii 31275 | A member of a subspace of ... |
| sh0 31276 | The zero vector belongs to... |
| shaddcl 31277 | Closure of vector addition... |
| shmulcl 31278 | Closure of vector scalar m... |
| issh3 31279 | Subspace ` H ` of a Hilber... |
| shsubcl 31280 | Closure of vector subtract... |
| isch 31282 | Closed subspace ` H ` of a... |
| isch2 31283 | Closed subspace ` H ` of a... |
| chsh 31284 | A closed subspace is a sub... |
| chsssh 31285 | Closed subspaces are subsp... |
| chex 31286 | The set of closed subspace... |
| chshii 31287 | A closed subspace is a sub... |
| ch0 31288 | The zero vector belongs to... |
| chss 31289 | A closed subspace of a Hil... |
| chel 31290 | A member of a closed subsp... |
| chssii 31291 | A closed subspace of a Hil... |
| cheli 31292 | A member of a closed subsp... |
| chelii 31293 | A member of a closed subsp... |
| chlimi 31294 | The limit property of a cl... |
| hlim0 31295 | The zero sequence in Hilbe... |
| hlimcaui 31296 | If a sequence in Hilbert s... |
| hlimf 31297 | Function-like behavior of ... |
| hlimuni 31298 | A Hilbert space sequence c... |
| hlimreui 31299 | The limit of a Hilbert spa... |
| hlimeui 31300 | The limit of a Hilbert spa... |
| isch3 31301 | A Hilbert subspace is clos... |
| chcompl 31302 | Completeness of a closed s... |
| helch 31303 | The Hilbert lattice one (w... |
| ifchhv 31304 | Prove ` if ( A e. CH , A ,... |
| helsh 31305 | Hilbert space is a subspac... |
| shsspwh 31306 | Subspaces are subsets of H... |
| chsspwh 31307 | Closed subspaces are subse... |
| hsn0elch 31308 | The zero subspace belongs ... |
| norm1 31309 | From any nonzero Hilbert s... |
| norm1exi 31310 | A normalized vector exists... |
| norm1hex 31311 | A normalized vector can ex... |
| elch0 31314 | Membership in zero for clo... |
| h0elch 31315 | The zero subspace is a clo... |
| h0elsh 31316 | The zero subspace is a sub... |
| hhssva 31317 | The vector addition operat... |
| hhsssm 31318 | The scalar multiplication ... |
| hhssnm 31319 | The norm operation on a su... |
| issubgoilem 31320 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 31321 | Lemma for ~ hhssabloi . F... |
| hhssabloi 31322 | Abelian group property of ... |
| hhssablo 31323 | Abelian group property of ... |
| hhssnv 31324 | Normed complex vector spac... |
| hhssnvt 31325 | Normed complex vector spac... |
| hhsst 31326 | A member of ` SH ` is a su... |
| hhshsslem1 31327 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 31328 | Lemma for ~ hhsssh . (Con... |
| hhsssh 31329 | The predicate " ` H ` is a... |
| hhsssh2 31330 | The predicate " ` H ` is a... |
| hhssba 31331 | The base set of a subspace... |
| hhssvs 31332 | The vector subtraction ope... |
| hhssvsf 31333 | Mapping of the vector subt... |
| hhssims 31334 | Induced metric of a subspa... |
| hhssims2 31335 | Induced metric of a subspa... |
| hhssmet 31336 | Induced metric of a subspa... |
| hhssmetdval 31337 | Value of the distance func... |
| hhsscms 31338 | The induced metric of a cl... |
| hhssbnOLD 31339 | Obsolete version of ~ cssb... |
| ocval 31340 | Value of orthogonal comple... |
| ocel 31341 | Membership in orthogonal c... |
| shocel 31342 | Membership in orthogonal c... |
| ocsh 31343 | The orthogonal complement ... |
| shocsh 31344 | The orthogonal complement ... |
| ocss 31345 | An orthogonal complement i... |
| shocss 31346 | An orthogonal complement i... |
| occon 31347 | Contraposition law for ort... |
| occon2 31348 | Double contraposition for ... |
| occon2i 31349 | Double contraposition for ... |
| oc0 31350 | The zero vector belongs to... |
| ocorth 31351 | Members of a subset and it... |
| shocorth 31352 | Members of a subspace and ... |
| ococss 31353 | Inclusion in complement of... |
| shococss 31354 | Inclusion in complement of... |
| shorth 31355 | Members of orthogonal subs... |
| ocin 31356 | Intersection of a Hilbert ... |
| occon3 31357 | Hilbert lattice contraposi... |
| ocnel 31358 | A nonzero vector in the co... |
| chocvali 31359 | Value of the orthogonal co... |
| shuni 31360 | Two subspaces with trivial... |
| chocunii 31361 | Lemma for uniqueness part ... |
| pjhthmo 31362 | Projection Theorem, unique... |
| occllem 31363 | Lemma for ~ occl . (Contr... |
| occl 31364 | Closure of complement of H... |
| shoccl 31365 | Closure of complement of H... |
| choccl 31366 | Closure of complement of H... |
| choccli 31367 | Closure of ` CH ` orthocom... |
| shsval 31372 | Value of subspace sum of t... |
| shsss 31373 | The subspace sum is a subs... |
| shsel 31374 | Membership in the subspace... |
| shsel3 31375 | Membership in the subspace... |
| shseli 31376 | Membership in subspace sum... |
| shscli 31377 | Closure of subspace sum. ... |
| shscl 31378 | Closure of subspace sum. ... |
| shscom 31379 | Commutative law for subspa... |
| shsva 31380 | Vector sum belongs to subs... |
| shsel1 31381 | A subspace sum contains a ... |
| shsel2 31382 | A subspace sum contains a ... |
| shsvs 31383 | Vector subtraction belongs... |
| shsub1 31384 | Subspace sum is an upper b... |
| shsub2 31385 | Subspace sum is an upper b... |
| choc0 31386 | The orthocomplement of the... |
| choc1 31387 | The orthocomplement of the... |
| chocnul 31388 | Orthogonal complement of t... |
| shintcli 31389 | Closure of intersection of... |
| shintcl 31390 | The intersection of a none... |
| chintcli 31391 | The intersection of a none... |
| chintcl 31392 | The intersection (infimum)... |
| spanval 31393 | Value of the linear span o... |
| hsupval 31394 | Value of supremum of set o... |
| chsupval 31395 | The value of the supremum ... |
| spancl 31396 | The span of a subset of Hi... |
| elspancl 31397 | A member of a span is a ve... |
| shsupcl 31398 | Closure of the subspace su... |
| hsupcl 31399 | Closure of supremum of set... |
| chsupcl 31400 | Closure of supremum of sub... |
| hsupss 31401 | Subset relation for suprem... |
| chsupss 31402 | Subset relation for suprem... |
| hsupunss 31403 | The union of a set of Hilb... |
| chsupunss 31404 | The union of a set of clos... |
| spanss2 31405 | A subset of Hilbert space ... |
| shsupunss 31406 | The union of a set of subs... |
| spanid 31407 | A subspace of Hilbert spac... |
| spanss 31408 | Ordering relationship for ... |
| spanssoc 31409 | The span of a subset of Hi... |
| sshjval 31410 | Value of join for subsets ... |
| shjval 31411 | Value of join in ` SH ` . ... |
| chjval 31412 | Value of join in ` CH ` . ... |
| chjvali 31413 | Value of join in ` CH ` . ... |
| sshjval3 31414 | Value of join for subsets ... |
| sshjcl 31415 | Closure of join for subset... |
| shjcl 31416 | Closure of join in ` SH ` ... |
| chjcl 31417 | Closure of join in ` CH ` ... |
| shjcom 31418 | Commutative law for Hilber... |
| shless 31419 | Subset implies subset of s... |
| shlej1 31420 | Add disjunct to both sides... |
| shlej2 31421 | Add disjunct to both sides... |
| shincli 31422 | Closure of intersection of... |
| shscomi 31423 | Commutative law for subspa... |
| shsvai 31424 | Vector sum belongs to subs... |
| shsel1i 31425 | A subspace sum contains a ... |
| shsel2i 31426 | A subspace sum contains a ... |
| shsvsi 31427 | Vector subtraction belongs... |
| shunssi 31428 | Union is smaller than subs... |
| shunssji 31429 | Union is smaller than Hilb... |
| shsleji 31430 | Subspace sum is smaller th... |
| shjcomi 31431 | Commutative law for join i... |
| shsub1i 31432 | Subspace sum is an upper b... |
| shsub2i 31433 | Subspace sum is an upper b... |
| shub1i 31434 | Hilbert lattice join is an... |
| shjcli 31435 | Closure of ` CH ` join. (... |
| shjshcli 31436 | ` SH ` closure of join. (... |
| shlessi 31437 | Subset implies subset of s... |
| shlej1i 31438 | Add disjunct to both sides... |
| shlej2i 31439 | Add disjunct to both sides... |
| shslej 31440 | Subspace sum is smaller th... |
| shincl 31441 | Closure of intersection of... |
| shub1 31442 | Hilbert lattice join is an... |
| shub2 31443 | A subspace is a subset of ... |
| shsidmi 31444 | Idempotent law for Hilbert... |
| shslubi 31445 | The least upper bound law ... |
| shlesb1i 31446 | Hilbert lattice ordering i... |
| shsval2i 31447 | An alternate way to expres... |
| shsval3i 31448 | An alternate way to expres... |
| shmodsi 31449 | The modular law holds for ... |
| shmodi 31450 | The modular law is implied... |
| pjhthlem1 31451 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 31452 | Lemma for ~ pjhth . (Cont... |
| pjhth 31453 | Projection Theorem: Any H... |
| pjhtheu 31454 | Projection Theorem: Any H... |
| pjhfval 31456 | The value of the projectio... |
| pjhval 31457 | Value of a projection. (C... |
| pjpreeq 31458 | Equality with a projection... |
| pjeq 31459 | Equality with a projection... |
| axpjcl 31460 | Closure of a projection in... |
| pjhcl 31461 | Closure of a projection in... |
| omlsilem 31462 | Lemma for orthomodular law... |
| omlsii 31463 | Subspace inference form of... |
| omlsi 31464 | Subspace form of orthomodu... |
| ococi 31465 | Complement of complement o... |
| ococ 31466 | Complement of complement o... |
| dfch2 31467 | Alternate definition of th... |
| ococin 31468 | The double complement is t... |
| hsupval2 31469 | Alternate definition of su... |
| chsupval2 31470 | The value of the supremum ... |
| sshjval2 31471 | Value of join in the set o... |
| chsupid 31472 | A subspace is the supremum... |
| chsupsn 31473 | Value of supremum of subse... |
| shlub 31474 | Hilbert lattice join is th... |
| shlubi 31475 | Hilbert lattice join is th... |
| pjhtheu2 31476 | Uniqueness of ` y ` for th... |
| pjcli 31477 | Closure of a projection in... |
| pjhcli 31478 | Closure of a projection in... |
| pjpjpre 31479 | Decomposition of a vector ... |
| axpjpj 31480 | Decomposition of a vector ... |
| pjclii 31481 | Closure of a projection in... |
| pjhclii 31482 | Closure of a projection in... |
| pjpj0i 31483 | Decomposition of a vector ... |
| pjpji 31484 | Decomposition of a vector ... |
| pjpjhth 31485 | Projection Theorem: Any H... |
| pjpjhthi 31486 | Projection Theorem: Any H... |
| pjop 31487 | Orthocomplement projection... |
| pjpo 31488 | Projection in terms of ort... |
| pjopi 31489 | Orthocomplement projection... |
| pjpoi 31490 | Projection in terms of ort... |
| pjoc1i 31491 | Projection of a vector in ... |
| pjchi 31492 | Projection of a vector in ... |
| pjoccl 31493 | The part of a vector that ... |
| pjoc1 31494 | Projection of a vector in ... |
| pjomli 31495 | Subspace form of orthomodu... |
| pjoml 31496 | Subspace form of orthomodu... |
| pjococi 31497 | Proof of orthocomplement t... |
| pjoc2i 31498 | Projection of a vector in ... |
| pjoc2 31499 | Projection of a vector in ... |
| sh0le 31500 | The zero subspace is the s... |
| ch0le 31501 | The zero subspace is the s... |
| shle0 31502 | No subspace is smaller tha... |
| chle0 31503 | No Hilbert lattice element... |
| chnlen0 31504 | A Hilbert lattice element ... |
| ch0pss 31505 | The zero subspace is a pro... |
| orthin 31506 | The intersection of orthog... |
| ssjo 31507 | The lattice join of a subs... |
| shne0i 31508 | A nonzero subspace has a n... |
| shs0i 31509 | Hilbert subspace sum with ... |
| shs00i 31510 | Two subspaces are zero iff... |
| ch0lei 31511 | The closed subspace zero i... |
| chle0i 31512 | No Hilbert closed subspace... |
| chne0i 31513 | A nonzero closed subspace ... |
| chocini 31514 | Intersection of a closed s... |
| chj0i 31515 | Join with lattice zero in ... |
| chm1i 31516 | Meet with lattice one in `... |
| chjcli 31517 | Closure of ` CH ` join. (... |
| chsleji 31518 | Subspace sum is smaller th... |
| chseli 31519 | Membership in subspace sum... |
| chincli 31520 | Closure of Hilbert lattice... |
| chsscon3i 31521 | Hilbert lattice contraposi... |
| chsscon1i 31522 | Hilbert lattice contraposi... |
| chsscon2i 31523 | Hilbert lattice contraposi... |
| chcon2i 31524 | Hilbert lattice contraposi... |
| chcon1i 31525 | Hilbert lattice contraposi... |
| chcon3i 31526 | Hilbert lattice contraposi... |
| chunssji 31527 | Union is smaller than ` CH... |
| chjcomi 31528 | Commutative law for join i... |
| chub1i 31529 | ` CH ` join is an upper bo... |
| chub2i 31530 | ` CH ` join is an upper bo... |
| chlubi 31531 | Hilbert lattice join is th... |
| chlubii 31532 | Hilbert lattice join is th... |
| chlej1i 31533 | Add join to both sides of ... |
| chlej2i 31534 | Add join to both sides of ... |
| chlej12i 31535 | Add join to both sides of ... |
| chlejb1i 31536 | Hilbert lattice ordering i... |
| chdmm1i 31537 | De Morgan's law for meet i... |
| chdmm2i 31538 | De Morgan's law for meet i... |
| chdmm3i 31539 | De Morgan's law for meet i... |
| chdmm4i 31540 | De Morgan's law for meet i... |
| chdmj1i 31541 | De Morgan's law for join i... |
| chdmj2i 31542 | De |