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Metamath Proof Explorer |
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| Ref | Description |
| idi 1 | (_Note_: This inference r... |
| a1ii 2 | (_Note_: This inference r... |
| mp2 9 | A double modus ponens infe... |
| mp2b 10 | A double modus ponens infe... |
| a1i 11 | Inference introducing an a... |
| 2a1i 12 | Inference introducing two ... |
| mp1i 13 | Inference detaching an ant... |
| a2i 14 | Inference distributing an ... |
| mpd 15 | A modus ponens deduction. ... |
| imim2i 16 | Inference adding common an... |
| syl 17 | An inference version of th... |
| 3syl 18 | Inference chaining two syl... |
| 4syl 19 | Inference chaining three s... |
| mpi 20 | A nested modus ponens infe... |
| mpisyl 21 | A syllogism combined with ... |
| id 22 | Principle of identity. Th... |
| idALT 23 | Alternate proof of ~ id . ... |
| idd 24 | Principle of identity ~ id... |
| a1d 25 | Deduction introducing an e... |
| 2a1d 26 | Deduction introducing two ... |
| a1i13 27 | Add two antecedents to a w... |
| 2a1 28 | A double form of ~ ax-1 . ... |
| a2d 29 | Deduction distributing an ... |
| sylcom 30 | Syllogism inference with c... |
| syl5com 31 | Syllogism inference with c... |
| com12 32 | Inference that swaps (comm... |
| syl11 33 | A syllogism inference. Co... |
| syl5 34 | A syllogism rule of infere... |
| syl6 35 | A syllogism rule of infere... |
| syl56 36 | Combine ~ syl5 and ~ syl6 ... |
| syl6com 37 | Syllogism inference with c... |
| mpcom 38 | Modus ponens inference wit... |
| syli 39 | Syllogism inference with c... |
| syl2im 40 | Replace two antecedents. ... |
| syl2imc 41 | A commuted version of ~ sy... |
| pm2.27 42 | This theorem, sometimes ca... |
| mpdd 43 | A nested modus ponens dedu... |
| mpid 44 | A nested modus ponens dedu... |
| mpdi 45 | A nested modus ponens dedu... |
| mpii 46 | A doubly nested modus pone... |
| syld 47 | Syllogism deduction. Dedu... |
| syldc 48 | Syllogism deduction. Comm... |
| mp2d 49 | A double modus ponens dedu... |
| a1dd 50 | Double deduction introduci... |
| 2a1dd 51 | Double deduction introduci... |
| pm2.43i 52 | Inference absorbing redund... |
| pm2.43d 53 | Deduction absorbing redund... |
| pm2.43a 54 | Inference absorbing redund... |
| pm2.43b 55 | Inference absorbing redund... |
| pm2.43 56 | Absorption of redundant an... |
| imim2d 57 | Deduction adding nested an... |
| imim2 58 | A closed form of syllogism... |
| embantd 59 | Deduction embedding an ant... |
| 3syld 60 | Triple syllogism deduction... |
| sylsyld 61 | A double syllogism inferen... |
| imim12i 62 | Inference joining two impl... |
| imim1i 63 | Inference adding common co... |
| imim3i 64 | Inference adding three nes... |
| sylc 65 | A syllogism inference comb... |
| syl3c 66 | A syllogism inference comb... |
| syl6mpi 67 | A syllogism inference. (C... |
| mpsyl 68 | Modus ponens combined with... |
| mpsylsyld 69 | Modus ponens combined with... |
| syl6c 70 | Inference combining ~ syl6... |
| syl6ci 71 | A syllogism inference comb... |
| syldd 72 | Nested syllogism deduction... |
| syl5d 73 | A nested syllogism deducti... |
| syl7 74 | A syllogism rule of infere... |
| syl6d 75 | A nested syllogism deducti... |
| syl8 76 | A syllogism rule of infere... |
| syl9 77 | A nested syllogism inferen... |
| syl9r 78 | A nested syllogism inferen... |
| syl10 79 | A nested syllogism inferen... |
| a1ddd 80 | Triple deduction introduci... |
| imim12d 81 | Deduction combining antece... |
| imim1d 82 | Deduction adding nested co... |
| imim1 83 | A closed form of syllogism... |
| pm2.83 84 | Theorem *2.83 of [Whitehea... |
| peirceroll 85 | Over minimal implicational... |
| com23 86 | Commutation of antecedents... |
| com3r 87 | Commutation of antecedents... |
| com13 88 | Commutation of antecedents... |
| com3l 89 | Commutation of antecedents... |
| pm2.04 90 | Swap antecedents. Theorem... |
| com34 91 | Commutation of antecedents... |
| com4l 92 | Commutation of antecedents... |
| com4t 93 | Commutation of antecedents... |
| com4r 94 | Commutation of antecedents... |
| com24 95 | Commutation of antecedents... |
| com14 96 | Commutation of antecedents... |
| com45 97 | Commutation of antecedents... |
| com35 98 | Commutation of antecedents... |
| com25 99 | Commutation of antecedents... |
| com5l 100 | Commutation of antecedents... |
| com15 101 | Commutation of antecedents... |
| com52l 102 | Commutation of antecedents... |
| com52r 103 | Commutation of antecedents... |
| com5r 104 | Commutation of antecedents... |
| imim12 105 | Closed form of ~ imim12i a... |
| jarr 106 | Elimination of a nested an... |
| jarri 107 | Inference associated with ... |
| pm2.86d 108 | Deduction associated with ... |
| pm2.86 109 | Converse of Axiom ~ ax-2 .... |
| pm2.86i 110 | Inference associated with ... |
| loolin 111 | The Linearity Axiom of the... |
| loowoz 112 | An alternate for the Linea... |
| con4 113 | Alias for ~ ax-3 to be use... |
| con4i 114 | Inference associated with ... |
| con4d 115 | Deduction associated with ... |
| mt4 116 | The rule of modus tollens.... |
| mt4d 117 | Modus tollens deduction. ... |
| mt4i 118 | Modus tollens inference. ... |
| pm2.21i 119 | A contradiction implies an... |
| pm2.24ii 120 | A contradiction implies an... |
| pm2.21d 121 | A contradiction implies an... |
| pm2.21ddALT 122 | Alternate proof of ~ pm2.2... |
| pm2.21 123 | From a wff and its negatio... |
| pm2.24 124 | Theorem *2.24 of [Whitehea... |
| jarl 125 | Elimination of a nested an... |
| jarli 126 | Inference associated with ... |
| pm2.18d 127 | Deduction form of the Clav... |
| pm2.18 128 | Clavius law, or "consequen... |
| pm2.18i 129 | Inference associated with ... |
| notnotr 130 | Double negation eliminatio... |
| notnotri 131 | Inference associated with ... |
| notnotriALT 132 | Alternate proof of ~ notno... |
| notnotrd 133 | Deduction associated with ... |
| con2d 134 | A contraposition deduction... |
| con2 135 | Contraposition. Theorem *... |
| mt2d 136 | Modus tollens deduction. ... |
| mt2i 137 | Modus tollens inference. ... |
| nsyl3 138 | A negated syllogism infere... |
| con2i 139 | A contraposition inference... |
| nsyl 140 | A negated syllogism infere... |
| nsyl2 141 | A negated syllogism infere... |
| notnot 142 | Double negation introducti... |
| notnoti 143 | Inference associated with ... |
| notnotd 144 | Deduction associated with ... |
| con1d 145 | A contraposition deduction... |
| con1 146 | Contraposition. Theorem *... |
| con1i 147 | A contraposition inference... |
| mt3d 148 | Modus tollens deduction. ... |
| mt3i 149 | Modus tollens inference. ... |
| pm2.24i 150 | Inference associated with ... |
| pm2.24d 151 | Deduction form of ~ pm2.24... |
| con3d 152 | A contraposition deduction... |
| con3 153 | Contraposition. Theorem *... |
| con3i 154 | A contraposition inference... |
| con3rr3 155 | Rotate through consequent ... |
| nsyld 156 | A negated syllogism deduct... |
| nsyli 157 | A negated syllogism infere... |
| nsyl4 158 | A negated syllogism infere... |
| nsyl5 159 | A negated syllogism infere... |
| pm3.2im 160 | Theorem *3.2 of [Whitehead... |
| jc 161 | Deduction joining the cons... |
| jcn 162 | Theorem joining the conseq... |
| jcnd 163 | Deduction joining the cons... |
| impi 164 | An importation inference. ... |
| expi 165 | An exportation inference. ... |
| simprim 166 | Simplification. Similar t... |
| simplim 167 | Simplification. Similar t... |
| pm2.5g 168 | General instance of Theore... |
| pm2.5 169 | Theorem *2.5 of [Whitehead... |
| conax1 170 | Contrapositive of ~ ax-1 .... |
| conax1k 171 | Weakening of ~ conax1 . G... |
| pm2.51 172 | Theorem *2.51 of [Whitehea... |
| pm2.52 173 | Theorem *2.52 of [Whitehea... |
| pm2.521g 174 | A general instance of Theo... |
| pm2.521g2 175 | A general instance of Theo... |
| pm2.521 176 | Theorem *2.521 of [Whitehe... |
| expt 177 | Exportation theorem ~ pm3.... |
| impt 178 | Importation theorem ~ pm3.... |
| pm2.61d 179 | Deduction eliminating an a... |
| pm2.61d1 180 | Inference eliminating an a... |
| pm2.61d2 181 | Inference eliminating an a... |
| pm2.61i 182 | Inference eliminating an a... |
| pm2.61ii 183 | Inference eliminating two ... |
| pm2.61nii 184 | Inference eliminating two ... |
| pm2.61iii 185 | Inference eliminating thre... |
| ja 186 | Inference joining the ante... |
| jad 187 | Deduction form of ~ ja . ... |
| pm2.01 188 | Weak Clavius law. If a fo... |
| pm2.01d 189 | Deduction based on reducti... |
| pm2.6 190 | Theorem *2.6 of [Whitehead... |
| pm2.61 191 | Theorem *2.61 of [Whitehea... |
| pm2.65 192 | Theorem *2.65 of [Whitehea... |
| pm2.65i 193 | Inference for proof by con... |
| pm2.21dd 194 | A contradiction implies an... |
| pm2.65d 195 | Deduction for proof by con... |
| mto 196 | The rule of modus tollens.... |
| mtod 197 | Modus tollens deduction. ... |
| mtoi 198 | Modus tollens inference. ... |
| mt2 199 | A rule similar to modus to... |
| mt3 200 | A rule similar to modus to... |
| peirce 201 | Peirce's axiom. A non-int... |
| looinv 202 | The Inversion Axiom of the... |
| bijust0 203 | A self-implication (see ~ ... |
| bijust 204 | Theorem used to justify th... |
| impbi 207 | Property of the biconditio... |
| impbii 208 | Infer an equivalence from ... |
| impbidd 209 | Deduce an equivalence from... |
| impbid21d 210 | Deduce an equivalence from... |
| impbid 211 | Deduce an equivalence from... |
| dfbi1 212 | Relate the biconditional c... |
| dfbi1ALT 213 | Alternate proof of ~ dfbi1... |
| biimp 214 | Property of the biconditio... |
| biimpi 215 | Infer an implication from ... |
| sylbi 216 | A mixed syllogism inferenc... |
| sylib 217 | A mixed syllogism inferenc... |
| sylbb 218 | A mixed syllogism inferenc... |
| biimpr 219 | Property of the biconditio... |
| bicom1 220 | Commutative law for the bi... |
| bicom 221 | Commutative law for the bi... |
| bicomd 222 | Commute two sides of a bic... |
| bicomi 223 | Inference from commutative... |
| impbid1 224 | Infer an equivalence from ... |
| impbid2 225 | Infer an equivalence from ... |
| impcon4bid 226 | A variation on ~ impbid wi... |
| biimpri 227 | Infer a converse implicati... |
| biimpd 228 | Deduce an implication from... |
| mpbi 229 | An inference from a bicond... |
| mpbir 230 | An inference from a bicond... |
| mpbid 231 | A deduction from a bicondi... |
| mpbii 232 | An inference from a nested... |
| sylibr 233 | A mixed syllogism inferenc... |
| sylbir 234 | A mixed syllogism inferenc... |
| sylbbr 235 | A mixed syllogism inferenc... |
| sylbb1 236 | A mixed syllogism inferenc... |
| sylbb2 237 | A mixed syllogism inferenc... |
| sylibd 238 | A syllogism deduction. (C... |
| sylbid 239 | A syllogism deduction. (C... |
| mpbidi 240 | A deduction from a bicondi... |
| biimtrid 241 | A mixed syllogism inferenc... |
| biimtrrid 242 | A mixed syllogism inferenc... |
| imbitrid 243 | A mixed syllogism inferenc... |
| syl5ibcom 244 | A mixed syllogism inferenc... |
| imbitrrid 245 | A mixed syllogism inferenc... |
| syl5ibrcom 246 | A mixed syllogism inferenc... |
| biimprd 247 | Deduce a converse implicat... |
| biimpcd 248 | Deduce a commuted implicat... |
| biimprcd 249 | Deduce a converse commuted... |
| imbitrdi 250 | A mixed syllogism inferenc... |
| imbitrrdi 251 | A mixed syllogism inferenc... |
| biimtrdi 252 | A mixed syllogism inferenc... |
| syl6bi 253 | A mixed syllogism inferenc... |
| syl6bir 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... |
| simprbi 496 | Deduction eliminating a co... |
| simplbi 497 | Deduction eliminating a co... |
| simprbda 498 | Deduction eliminating a co... |
| simplbda 499 | Deduction eliminating a co... |
| simplbi2 500 | Deduction eliminating a co... |
| simplbi2comt 501 | Closed form of ~ simplbi2c... |
| simplbi2com 502 | A deduction eliminating a ... |
| simpl2im 503 | Implication from an elimin... |
| simplbiim 504 | Implication from an elimin... |
| impel 505 | An inference for implicati... |
| mpan9 506 | Modus ponens conjoining di... |
| sylan9 507 | Nested syllogism inference... |
| sylan9r 508 | Nested syllogism inference... |
| sylan9bb 509 | Nested syllogism inference... |
| sylan9bbr 510 | Nested syllogism inference... |
| jca 511 | Deduce conjunction of the ... |
| jcad 512 | Deduction conjoining the c... |
| jca2 513 | Inference conjoining the c... |
| jca31 514 | Join three consequents. (... |
| jca32 515 | Join three consequents. (... |
| jcai 516 | Deduction replacing implic... |
| jcab 517 | Distributive law for impli... |
| pm4.76 518 | Theorem *4.76 of [Whitehea... |
| jctil 519 | Inference conjoining a the... |
| jctir 520 | Inference conjoining a the... |
| jccir 521 | Inference conjoining a con... |
| jccil 522 | Inference conjoining a con... |
| jctl 523 | Inference conjoining a the... |
| jctr 524 | Inference conjoining a the... |
| jctild 525 | Deduction conjoining a the... |
| jctird 526 | Deduction conjoining a the... |
| iba 527 | Introduction of antecedent... |
| ibar 528 | Introduction of antecedent... |
| biantru 529 | A wff is equivalent to its... |
| biantrur 530 | A wff is equivalent to its... |
| biantrud 531 | A wff is equivalent to its... |
| biantrurd 532 | A wff is equivalent to its... |
| bianfi 533 | A wff conjoined with false... |
| bianfd 534 | A wff conjoined with false... |
| baib 535 | Move conjunction outside o... |
| baibr 536 | Move conjunction outside o... |
| rbaibr 537 | Move conjunction outside o... |
| rbaib 538 | Move conjunction outside o... |
| baibd 539 | Move conjunction outside o... |
| rbaibd 540 | Move conjunction outside o... |
| bianabs 541 | Absorb a hypothesis into t... |
| pm5.44 542 | Theorem *5.44 of [Whitehea... |
| pm5.42 543 | Theorem *5.42 of [Whitehea... |
| ancl 544 | Conjoin antecedent to left... |
| anclb 545 | Conjoin antecedent to left... |
| ancr 546 | Conjoin antecedent to righ... |
| ancrb 547 | Conjoin antecedent to righ... |
| ancli 548 | Deduction conjoining antec... |
| ancri 549 | Deduction conjoining antec... |
| ancld 550 | Deduction conjoining antec... |
| ancrd 551 | Deduction conjoining antec... |
| impac 552 | Importation with conjuncti... |
| anc2l 553 | Conjoin antecedent to left... |
| anc2r 554 | Conjoin antecedent to righ... |
| anc2li 555 | Deduction conjoining antec... |
| anc2ri 556 | Deduction conjoining antec... |
| pm4.71 557 | Implication in terms of bi... |
| pm4.71r 558 | Implication in terms of bi... |
| pm4.71i 559 | Inference converting an im... |
| pm4.71ri 560 | Inference converting an im... |
| pm4.71d 561 | Deduction converting an im... |
| pm4.71rd 562 | Deduction converting an im... |
| pm4.24 563 | Theorem *4.24 of [Whitehea... |
| anidm 564 | Idempotent law for conjunc... |
| anidmdbi 565 | Conjunction idempotence wi... |
| anidms 566 | Inference from idempotent ... |
| imdistan 567 | Distribution of implicatio... |
| imdistani 568 | Distribution of implicatio... |
| imdistanri 569 | Distribution of implicatio... |
| imdistand 570 | Distribution of implicatio... |
| imdistanda 571 | Distribution of implicatio... |
| pm5.3 572 | Theorem *5.3 of [Whitehead... |
| pm5.32 573 | Distribution of implicatio... |
| pm5.32i 574 | Distribution of implicatio... |
| pm5.32ri 575 | Distribution of implicatio... |
| pm5.32d 576 | Distribution of implicatio... |
| pm5.32rd 577 | Distribution of implicatio... |
| pm5.32da 578 | Distribution of implicatio... |
| sylan 579 | A syllogism inference. (C... |
| sylanb 580 | A syllogism inference. (C... |
| sylanbr 581 | A syllogism inference. (C... |
| sylanbrc 582 | Syllogism inference. (Con... |
| syl2anc 583 | Syllogism inference combin... |
| syl2anc2 584 | Double syllogism inference... |
| sylancl 585 | Syllogism inference combin... |
| sylancr 586 | Syllogism inference combin... |
| sylancom 587 | Syllogism inference with c... |
| sylanblc 588 | Syllogism inference combin... |
| sylanblrc 589 | Syllogism inference combin... |
| syldan 590 | A syllogism deduction with... |
| sylbida 591 | A syllogism deduction. (C... |
| sylan2 592 | A syllogism inference. (C... |
| sylan2b 593 | A syllogism inference. (C... |
| sylan2br 594 | A syllogism inference. (C... |
| syl2an 595 | A double syllogism inferen... |
| syl2anr 596 | A double syllogism inferen... |
| syl2anb 597 | A double syllogism inferen... |
| syl2anbr 598 | A double syllogism inferen... |
| sylancb 599 | A syllogism inference comb... |
| sylancbr 600 | A syllogism inference comb... |
| syldanl 601 | A syllogism deduction with... |
| syland 602 | A syllogism deduction. (C... |
| sylani 603 | A syllogism inference. (C... |
| sylan2d 604 | A syllogism deduction. (C... |
| sylan2i 605 | A syllogism inference. (C... |
| syl2ani 606 | A syllogism inference. (C... |
| syl2and 607 | A syllogism deduction. (C... |
| anim12d 608 | Conjoin antecedents and co... |
| anim12d1 609 | Variant of ~ anim12d where... |
| anim1d 610 | Add a conjunct to right of... |
| anim2d 611 | Add a conjunct to left of ... |
| anim12i 612 | Conjoin antecedents and co... |
| anim12ci 613 | Variant of ~ anim12i with ... |
| anim1i 614 | Introduce conjunct to both... |
| anim1ci 615 | Introduce conjunct to both... |
| anim2i 616 | Introduce conjunct to both... |
| anim12ii 617 | Conjoin antecedents and co... |
| anim12dan 618 | Conjoin antecedents and co... |
| im2anan9 619 | Deduction joining nested i... |
| im2anan9r 620 | Deduction joining nested i... |
| pm3.45 621 | Theorem *3.45 (Fact) of [W... |
| anbi2i 622 | Introduce a left conjunct ... |
| anbi1i 623 | Introduce a right conjunct... |
| anbi2ci 624 | Variant of ~ anbi2i with c... |
| anbi1ci 625 | Variant of ~ anbi1i with c... |
| anbi12i 626 | Conjoin both sides of two ... |
| anbi12ci 627 | Variant of ~ anbi12i with ... |
| anbi2d 628 | Deduction adding a left co... |
| anbi1d 629 | Deduction adding a right c... |
| anbi12d 630 | Deduction joining two equi... |
| anbi1 631 | Introduce a right conjunct... |
| anbi2 632 | Introduce a left conjunct ... |
| anbi1cd 633 | Introduce a proposition as... |
| an2anr 634 | Double commutation in conj... |
| pm4.38 635 | Theorem *4.38 of [Whitehea... |
| bi2anan9 636 | Deduction joining two equi... |
| bi2anan9r 637 | Deduction joining two equi... |
| bi2bian9 638 | Deduction joining two bico... |
| bianass 639 | An inference to merge two ... |
| bianassc 640 | An inference to merge two ... |
| an21 641 | Swap two conjuncts. (Cont... |
| an12 642 | Swap two conjuncts. Note ... |
| an32 643 | A rearrangement of conjunc... |
| an13 644 | A rearrangement of conjunc... |
| an31 645 | A rearrangement of conjunc... |
| an12s 646 | Swap two conjuncts in ante... |
| ancom2s 647 | Inference commuting a nest... |
| an13s 648 | Swap two conjuncts in ante... |
| an32s 649 | Swap two conjuncts in ante... |
| ancom1s 650 | Inference commuting a nest... |
| an31s 651 | Swap two conjuncts in ante... |
| anass1rs 652 | Commutative-associative la... |
| an4 653 | Rearrangement of 4 conjunc... |
| an42 654 | Rearrangement of 4 conjunc... |
| an43 655 | Rearrangement of 4 conjunc... |
| an3 656 | A rearrangement of conjunc... |
| an4s 657 | Inference rearranging 4 co... |
| an42s 658 | Inference rearranging 4 co... |
| anabs1 659 | Absorption into embedded c... |
| anabs5 660 | Absorption into embedded c... |
| anabs7 661 | Absorption into embedded c... |
| anabsan 662 | Absorption of antecedent w... |
| anabss1 663 | Absorption of antecedent i... |
| anabss4 664 | Absorption of antecedent i... |
| anabss5 665 | Absorption of antecedent i... |
| anabsi5 666 | Absorption of antecedent i... |
| anabsi6 667 | Absorption of antecedent i... |
| anabsi7 668 | Absorption of antecedent i... |
| anabsi8 669 | Absorption of antecedent i... |
| anabss7 670 | Absorption of antecedent i... |
| anabsan2 671 | Absorption of antecedent w... |
| anabss3 672 | Absorption of antecedent i... |
| anandi 673 | Distribution of conjunctio... |
| anandir 674 | Distribution of conjunctio... |
| anandis 675 | Inference that undistribut... |
| anandirs 676 | Inference that undistribut... |
| sylanl1 677 | A syllogism inference. (C... |
| sylanl2 678 | A syllogism inference. (C... |
| sylanr1 679 | A syllogism inference. (C... |
| sylanr2 680 | A syllogism inference. (C... |
| syl6an 681 | A syllogism deduction comb... |
| syl2an2r 682 | ~ syl2anr with antecedents... |
| syl2an2 683 | ~ syl2an with antecedents ... |
| mpdan 684 | An inference based on modu... |
| mpancom 685 | An inference based on modu... |
| mpidan 686 | A deduction which "stacks"... |
| mpan 687 | An inference based on modu... |
| mpan2 688 | An inference based on modu... |
| mp2an 689 | An inference based on modu... |
| mp4an 690 | An inference based on modu... |
| mpan2d 691 | A deduction based on modus... |
| mpand 692 | A deduction based on modus... |
| mpani 693 | An inference based on modu... |
| mpan2i 694 | An inference based on modu... |
| mp2ani 695 | An inference based on modu... |
| mp2and 696 | A deduction based on modus... |
| mpanl1 697 | An inference based on modu... |
| mpanl2 698 | An inference based on modu... |
| mpanl12 699 | An inference based on modu... |
| mpanr1 700 | An inference based on modu... |
| mpanr2 701 | An inference based on modu... |
| mpanr12 702 | An inference based on modu... |
| mpanlr1 703 | An inference based on modu... |
| mpbirand 704 | Detach truth from conjunct... |
| mpbiran2d 705 | Detach truth from conjunct... |
| mpbiran 706 | Detach truth from conjunct... |
| mpbiran2 707 | Detach truth from conjunct... |
| mpbir2an 708 | Detach a conjunction of tr... |
| mpbi2and 709 | Detach a conjunction of tr... |
| mpbir2and 710 | Detach a conjunction of tr... |
| adantll 711 | Deduction adding a conjunc... |
| adantlr 712 | Deduction adding a conjunc... |
| adantrl 713 | Deduction adding a conjunc... |
| adantrr 714 | Deduction adding a conjunc... |
| adantlll 715 | Deduction adding a conjunc... |
| adantllr 716 | Deduction adding a conjunc... |
| adantlrl 717 | Deduction adding a conjunc... |
| adantlrr 718 | Deduction adding a conjunc... |
| adantrll 719 | Deduction adding a conjunc... |
| adantrlr 720 | Deduction adding a conjunc... |
| adantrrl 721 | Deduction adding a conjunc... |
| adantrrr 722 | Deduction adding a conjunc... |
| ad2antrr 723 | Deduction adding two conju... |
| ad2antlr 724 | Deduction adding two conju... |
| ad2antrl 725 | Deduction adding two conju... |
| ad2antll 726 | Deduction adding conjuncts... |
| ad3antrrr 727 | Deduction adding three con... |
| ad3antlr 728 | Deduction adding three con... |
| ad4antr 729 | Deduction adding 4 conjunc... |
| ad4antlr 730 | Deduction adding 4 conjunc... |
| ad5antr 731 | Deduction adding 5 conjunc... |
| ad5antlr 732 | Deduction adding 5 conjunc... |
| ad6antr 733 | Deduction adding 6 conjunc... |
| ad6antlr 734 | Deduction adding 6 conjunc... |
| ad7antr 735 | Deduction adding 7 conjunc... |
| ad7antlr 736 | Deduction adding 7 conjunc... |
| ad8antr 737 | Deduction adding 8 conjunc... |
| ad8antlr 738 | Deduction adding 8 conjunc... |
| ad9antr 739 | Deduction adding 9 conjunc... |
| ad9antlr 740 | Deduction adding 9 conjunc... |
| ad10antr 741 | Deduction adding 10 conjun... |
| ad10antlr 742 | Deduction adding 10 conjun... |
| ad2ant2l 743 | Deduction adding two conju... |
| ad2ant2r 744 | Deduction adding two conju... |
| ad2ant2lr 745 | Deduction adding two conju... |
| ad2ant2rl 746 | Deduction adding two conju... |
| adantl3r 747 | Deduction adding 1 conjunc... |
| ad4ant13 748 | Deduction adding conjuncts... |
| ad4ant14 749 | Deduction adding conjuncts... |
| ad4ant23 750 | Deduction adding conjuncts... |
| ad4ant24 751 | Deduction adding conjuncts... |
| adantl4r 752 | Deduction adding 1 conjunc... |
| ad5ant12 753 | Deduction adding conjuncts... |
| ad5ant13 754 | Deduction adding conjuncts... |
| ad5ant14 755 | Deduction adding conjuncts... |
| ad5ant15 756 | Deduction adding conjuncts... |
| ad5ant23 757 | Deduction adding conjuncts... |
| ad5ant24 758 | Deduction adding conjuncts... |
| ad5ant25 759 | Deduction adding conjuncts... |
| adantl5r 760 | Deduction adding 1 conjunc... |
| adantl6r 761 | Deduction adding 1 conjunc... |
| pm3.33 762 | Theorem *3.33 (Syll) of [W... |
| pm3.34 763 | Theorem *3.34 (Syll) of [W... |
| simpll 764 | Simplification of a conjun... |
| simplld 765 | Deduction form of ~ simpll... |
| simplr 766 | Simplification of a conjun... |
| simplrd 767 | Deduction eliminating a do... |
| simprl 768 | Simplification of a conjun... |
| simprld 769 | Deduction eliminating a do... |
| simprr 770 | Simplification of a conjun... |
| simprrd 771 | Deduction form of ~ simprr... |
| simplll 772 | Simplification of a conjun... |
| simpllr 773 | Simplification of a conjun... |
| simplrl 774 | Simplification of a conjun... |
| simplrr 775 | Simplification of a conjun... |
| simprll 776 | Simplification of a conjun... |
| simprlr 777 | Simplification of a conjun... |
| simprrl 778 | Simplification of a conjun... |
| simprrr 779 | Simplification of a conjun... |
| simp-4l 780 | Simplification of a conjun... |
| simp-4r 781 | Simplification of a conjun... |
| simp-5l 782 | Simplification of a conjun... |
| simp-5r 783 | Simplification of a conjun... |
| simp-6l 784 | Simplification of a conjun... |
| simp-6r 785 | Simplification of a conjun... |
| simp-7l 786 | Simplification of a conjun... |
| simp-7r 787 | Simplification of a conjun... |
| simp-8l 788 | Simplification of a conjun... |
| simp-8r 789 | Simplification of a conjun... |
| simp-9l 790 | Simplification of a conjun... |
| simp-9r 791 | Simplification of a conjun... |
| simp-10l 792 | Simplification of a conjun... |
| simp-10r 793 | Simplification of a conjun... |
| simp-11l 794 | Simplification of a conjun... |
| simp-11r 795 | Simplification of a conjun... |
| pm2.01da 796 | Deduction based on reducti... |
| pm2.18da 797 | Deduction based on reducti... |
| impbida 798 | Deduce an equivalence from... |
| pm5.21nd 799 | Eliminate an antecedent im... |
| pm3.35 800 | Conjunctive detachment. T... |
| pm5.74da 801 | Distribution of implicatio... |
| bitr 802 | Theorem *4.22 of [Whitehea... |
| biantr 803 | A transitive law of equiva... |
| pm4.14 804 | Theorem *4.14 of [Whitehea... |
| pm3.37 805 | Theorem *3.37 (Transp) of ... |
| anim12 806 | Conjoin antecedents and co... |
| pm3.4 807 | Conjunction implies implic... |
| exbiri 808 | Inference form of ~ exbir ... |
| pm2.61ian 809 | Elimination of an antecede... |
| pm2.61dan 810 | Elimination of an antecede... |
| pm2.61ddan 811 | Elimination of two anteced... |
| pm2.61dda 812 | Elimination of two anteced... |
| mtand 813 | A modus tollens deduction.... |
| pm2.65da 814 | Deduction for proof by con... |
| condan 815 | Proof by contradiction. (... |
| biadan 816 | An implication is equivale... |
| biadani 817 | Inference associated with ... |
| biadaniALT 818 | Alternate proof of ~ biada... |
| biadanii 819 | Inference associated with ... |
| biadanid 820 | Deduction associated with ... |
| pm5.1 821 | Two propositions are equiv... |
| pm5.21 822 | Two propositions are equiv... |
| pm5.35 823 | Theorem *5.35 of [Whitehea... |
| abai 824 | Introduce one conjunct as ... |
| pm4.45im 825 | Conjunction with implicati... |
| impimprbi 826 | An implication and its rev... |
| nan 827 | Theorem to move a conjunct... |
| pm5.31 828 | Theorem *5.31 of [Whitehea... |
| pm5.31r 829 | Variant of ~ pm5.31 . (Co... |
| pm4.15 830 | Theorem *4.15 of [Whitehea... |
| pm5.36 831 | Theorem *5.36 of [Whitehea... |
| annotanannot 832 | A conjunction with a negat... |
| pm5.33 833 | Theorem *5.33 of [Whitehea... |
| syl12anc 834 | Syllogism combined with co... |
| syl21anc 835 | Syllogism combined with co... |
| syl22anc 836 | Syllogism combined with co... |
| syl1111anc 837 | Four-hypothesis eliminatio... |
| syldbl2 838 | Stacked hypotheseis implie... |
| mpsyl4anc 839 | An elimination deduction. ... |
| pm4.87 840 | Theorem *4.87 of [Whitehea... |
| bimsc1 841 | Removal of conjunct from o... |
| a2and 842 | Deduction distributing a c... |
| animpimp2impd 843 | Deduction deriving nested ... |
| pm4.64 846 | Theorem *4.64 of [Whitehea... |
| pm4.66 847 | Theorem *4.66 of [Whitehea... |
| pm2.53 848 | Theorem *2.53 of [Whitehea... |
| pm2.54 849 | Theorem *2.54 of [Whitehea... |
| imor 850 | Implication in terms of di... |
| imori 851 | Infer disjunction from imp... |
| imorri 852 | Infer implication from dis... |
| pm4.62 853 | Theorem *4.62 of [Whitehea... |
| jaoi 854 | Inference disjoining the a... |
| jao1i 855 | Add a disjunct in the ante... |
| jaod 856 | Deduction disjoining the a... |
| mpjaod 857 | Eliminate a disjunction in... |
| ori 858 | Infer implication from dis... |
| orri 859 | Infer disjunction from imp... |
| orrd 860 | Deduce disjunction from im... |
| ord 861 | Deduce implication from di... |
| orci 862 | Deduction introducing a di... |
| olci 863 | Deduction introducing a di... |
| orc 864 | Introduction of a disjunct... |
| olc 865 | Introduction of a disjunct... |
| pm1.4 866 | Axiom *1.4 of [WhiteheadRu... |
| orcom 867 | Commutative law for disjun... |
| orcomd 868 | Commutation of disjuncts i... |
| orcoms 869 | Commutation of disjuncts i... |
| orcd 870 | Deduction introducing a di... |
| olcd 871 | Deduction introducing a di... |
| orcs 872 | Deduction eliminating disj... |
| olcs 873 | Deduction eliminating disj... |
| olcnd 874 | A lemma for Conjunctive No... |
| unitreslOLD 875 | Obsolete version of ~ olcn... |
| orcnd 876 | A lemma for Conjunctive No... |
| mtord 877 | A modus tollens deduction ... |
| pm3.2ni 878 | Infer negated disjunction ... |
| pm2.45 879 | Theorem *2.45 of [Whitehea... |
| pm2.46 880 | Theorem *2.46 of [Whitehea... |
| pm2.47 881 | Theorem *2.47 of [Whitehea... |
| pm2.48 882 | Theorem *2.48 of [Whitehea... |
| pm2.49 883 | Theorem *2.49 of [Whitehea... |
| norbi 884 | If neither of two proposit... |
| nbior 885 | If two propositions are no... |
| orel1 886 | Elimination of disjunction... |
| pm2.25 887 | Theorem *2.25 of [Whitehea... |
| orel2 888 | Elimination of disjunction... |
| pm2.67-2 889 | Slight generalization of T... |
| pm2.67 890 | Theorem *2.67 of [Whitehea... |
| curryax 891 | A non-intuitionistic posit... |
| exmid 892 | Law of excluded middle, al... |
| exmidd 893 | Law of excluded middle in ... |
| pm2.1 894 | Theorem *2.1 of [Whitehead... |
| pm2.13 895 | Theorem *2.13 of [Whitehea... |
| pm2.621 896 | Theorem *2.621 of [Whitehe... |
| pm2.62 897 | Theorem *2.62 of [Whitehea... |
| pm2.68 898 | Theorem *2.68 of [Whitehea... |
| dfor2 899 | Logical 'or' expressed in ... |
| pm2.07 900 | Theorem *2.07 of [Whitehea... |
| pm1.2 901 | Axiom *1.2 of [WhiteheadRu... |
| oridm 902 | Idempotent law for disjunc... |
| pm4.25 903 | Theorem *4.25 of [Whitehea... |
| pm2.4 904 | Theorem *2.4 of [Whitehead... |
| pm2.41 905 | Theorem *2.41 of [Whitehea... |
| orim12i 906 | Disjoin antecedents and co... |
| orim1i 907 | Introduce disjunct to both... |
| orim2i 908 | Introduce disjunct to both... |
| orim12dALT 909 | Alternate proof of ~ orim1... |
| orbi2i 910 | Inference adding a left di... |
| orbi1i 911 | Inference adding a right d... |
| orbi12i 912 | Infer the disjunction of t... |
| orbi2d 913 | Deduction adding a left di... |
| orbi1d 914 | Deduction adding a right d... |
| orbi1 915 | Theorem *4.37 of [Whitehea... |
| orbi12d 916 | Deduction joining two equi... |
| pm1.5 917 | Axiom *1.5 (Assoc) of [Whi... |
| or12 918 | Swap two disjuncts. (Cont... |
| orass 919 | Associative law for disjun... |
| pm2.31 920 | Theorem *2.31 of [Whitehea... |
| pm2.32 921 | Theorem *2.32 of [Whitehea... |
| pm2.3 922 | Theorem *2.3 of [Whitehead... |
| or32 923 | A rearrangement of disjunc... |
| or4 924 | Rearrangement of 4 disjunc... |
| or42 925 | Rearrangement of 4 disjunc... |
| orordi 926 | Distribution of disjunctio... |
| orordir 927 | Distribution of disjunctio... |
| orimdi 928 | Disjunction distributes ov... |
| pm2.76 929 | Theorem *2.76 of [Whitehea... |
| pm2.85 930 | Theorem *2.85 of [Whitehea... |
| pm2.75 931 | Theorem *2.75 of [Whitehea... |
| pm4.78 932 | Implication distributes ov... |
| biort 933 | A disjunction with a true ... |
| biorf 934 | A wff is equivalent to its... |
| biortn 935 | A wff is equivalent to its... |
| biorfi 936 | A wff is equivalent to its... |
| pm2.26 937 | Theorem *2.26 of [Whitehea... |
| pm2.63 938 | Theorem *2.63 of [Whitehea... |
| pm2.64 939 | Theorem *2.64 of [Whitehea... |
| pm2.42 940 | Theorem *2.42 of [Whitehea... |
| pm5.11g 941 | A general instance of Theo... |
| pm5.11 942 | Theorem *5.11 of [Whitehea... |
| pm5.12 943 | Theorem *5.12 of [Whitehea... |
| pm5.14 944 | Theorem *5.14 of [Whitehea... |
| pm5.13 945 | Theorem *5.13 of [Whitehea... |
| pm5.55 946 | Theorem *5.55 of [Whitehea... |
| pm4.72 947 | Implication in terms of bi... |
| imimorb 948 | Simplify an implication be... |
| oibabs 949 | Absorption of disjunction ... |
| orbidi 950 | Disjunction distributes ov... |
| pm5.7 951 | Disjunction distributes ov... |
| jaao 952 | Inference conjoining and d... |
| jaoa 953 | Inference disjoining and c... |
| jaoian 954 | Inference disjoining the a... |
| jaodan 955 | Deduction disjoining the a... |
| mpjaodan 956 | Eliminate a disjunction in... |
| pm3.44 957 | Theorem *3.44 of [Whitehea... |
| jao 958 | Disjunction of antecedents... |
| jaob 959 | Disjunction of antecedents... |
| pm4.77 960 | Theorem *4.77 of [Whitehea... |
| pm3.48 961 | Theorem *3.48 of [Whitehea... |
| orim12d 962 | Disjoin antecedents and co... |
| orim1d 963 | Disjoin antecedents and co... |
| orim2d 964 | Disjoin antecedents and co... |
| orim2 965 | Axiom *1.6 (Sum) of [White... |
| pm2.38 966 | Theorem *2.38 of [Whitehea... |
| pm2.36 967 | Theorem *2.36 of [Whitehea... |
| pm2.37 968 | Theorem *2.37 of [Whitehea... |
| pm2.81 969 | Theorem *2.81 of [Whitehea... |
| pm2.8 970 | Theorem *2.8 of [Whitehead... |
| pm2.73 971 | Theorem *2.73 of [Whitehea... |
| pm2.74 972 | Theorem *2.74 of [Whitehea... |
| pm2.82 973 | Theorem *2.82 of [Whitehea... |
| pm4.39 974 | Theorem *4.39 of [Whitehea... |
| animorl 975 | Conjunction implies disjun... |
| animorr 976 | Conjunction implies disjun... |
| animorlr 977 | Conjunction implies disjun... |
| animorrl 978 | Conjunction implies disjun... |
| ianor 979 | Negated conjunction in ter... |
| anor 980 | Conjunction in terms of di... |
| ioran 981 | Negated disjunction in ter... |
| pm4.52 982 | Theorem *4.52 of [Whitehea... |
| pm4.53 983 | Theorem *4.53 of [Whitehea... |
| pm4.54 984 | Theorem *4.54 of [Whitehea... |
| pm4.55 985 | Theorem *4.55 of [Whitehea... |
| pm4.56 986 | Theorem *4.56 of [Whitehea... |
| oran 987 | Disjunction in terms of co... |
| pm4.57 988 | Theorem *4.57 of [Whitehea... |
| pm3.1 989 | Theorem *3.1 of [Whitehead... |
| pm3.11 990 | Theorem *3.11 of [Whitehea... |
| pm3.12 991 | Theorem *3.12 of [Whitehea... |
| pm3.13 992 | Theorem *3.13 of [Whitehea... |
| pm3.14 993 | Theorem *3.14 of [Whitehea... |
| pm4.44 994 | Theorem *4.44 of [Whitehea... |
| pm4.45 995 | Theorem *4.45 of [Whitehea... |
| orabs 996 | Absorption of redundant in... |
| oranabs 997 | Absorb a disjunct into a c... |
| pm5.61 998 | Theorem *5.61 of [Whitehea... |
| pm5.6 999 | Conjunction in antecedent ... |
| orcanai 1000 | Change disjunction in cons... |
| pm4.79 1001 | Theorem *4.79 of [Whitehea... |
| pm5.53 1002 | Theorem *5.53 of [Whitehea... |
| ordi 1003 | Distributive law for disju... |
| ordir 1004 | Distributive law for disju... |
| andi 1005 | Distributive law for conju... |
| andir 1006 | Distributive law for conju... |
| orddi 1007 | Double distributive law fo... |
| anddi 1008 | Double distributive law fo... |
| pm5.17 1009 | Theorem *5.17 of [Whitehea... |
| pm5.15 1010 | Theorem *5.15 of [Whitehea... |
| pm5.16 1011 | Theorem *5.16 of [Whitehea... |
| xor 1012 | Two ways to express exclus... |
| nbi2 1013 | Two ways to express "exclu... |
| xordi 1014 | Conjunction distributes ov... |
| pm5.54 1015 | Theorem *5.54 of [Whitehea... |
| pm5.62 1016 | Theorem *5.62 of [Whitehea... |
| pm5.63 1017 | Theorem *5.63 of [Whitehea... |
| niabn 1018 | Miscellaneous inference re... |
| ninba 1019 | Miscellaneous inference re... |
| pm4.43 1020 | Theorem *4.43 of [Whitehea... |
| pm4.82 1021 | Theorem *4.82 of [Whitehea... |
| pm4.83 1022 | Theorem *4.83 of [Whitehea... |
| pclem6 1023 | Negation inferred from emb... |
| bigolden 1024 | Dijkstra-Scholten's Golden... |
| pm5.71 1025 | Theorem *5.71 of [Whitehea... |
| pm5.75 1026 | Theorem *5.75 of [Whitehea... |
| ecase2d 1027 | Deduction for elimination ... |
| ecase2dOLD 1028 | Obsolete version of ~ ecas... |
| ecase3 1029 | Inference for elimination ... |
| ecase 1030 | Inference for elimination ... |
| ecase3d 1031 | Deduction for elimination ... |
| ecased 1032 | Deduction for elimination ... |
| ecase3ad 1033 | Deduction for elimination ... |
| ecase3adOLD 1034 | Obsolete version of ~ ecas... |
| ccase 1035 | Inference for combining ca... |
| ccased 1036 | Deduction for combining ca... |
| ccase2 1037 | Inference for combining ca... |
| 4cases 1038 | Inference eliminating two ... |
| 4casesdan 1039 | Deduction eliminating two ... |
| cases 1040 | Case disjunction according... |
| dedlem0a 1041 | Lemma for an alternate ver... |
| dedlem0b 1042 | Lemma for an alternate ver... |
| dedlema 1043 | Lemma for weak deduction t... |
| dedlemb 1044 | Lemma for weak deduction t... |
| cases2 1045 | Case disjunction according... |
| cases2ALT 1046 | Alternate proof of ~ cases... |
| dfbi3 1047 | An alternate definition of... |
| pm5.24 1048 | Theorem *5.24 of [Whitehea... |
| 4exmid 1049 | The disjunction of the fou... |
| consensus 1050 | The consensus theorem. Th... |
| pm4.42 1051 | Theorem *4.42 of [Whitehea... |
| prlem1 1052 | A specialized lemma for se... |
| prlem2 1053 | A specialized lemma for se... |
| oplem1 1054 | A specialized lemma for se... |
| dn1 1055 | A single axiom for Boolean... |
| bianir 1056 | A closed form of ~ mpbir ,... |
| jaoi2 1057 | Inference removing a negat... |
| jaoi3 1058 | Inference separating a dis... |
| ornld 1059 | Selecting one statement fr... |
| dfifp2 1062 | Alternate definition of th... |
| dfifp3 1063 | Alternate definition of th... |
| dfifp4 1064 | Alternate definition of th... |
| dfifp5 1065 | Alternate definition of th... |
| dfifp6 1066 | Alternate definition of th... |
| dfifp7 1067 | Alternate definition of th... |
| ifpdfbi 1068 | Define the biconditional a... |
| anifp 1069 | The conditional operator i... |
| ifpor 1070 | The conditional operator i... |
| ifpn 1071 | Conditional operator for t... |
| ifpnOLD 1072 | Obsolete version of ~ ifpn... |
| ifptru 1073 | Value of the conditional o... |
| ifpfal 1074 | Value of the conditional o... |
| ifpid 1075 | Value of the conditional o... |
| casesifp 1076 | Version of ~ cases express... |
| ifpbi123d 1077 | Equivalence deduction for ... |
| ifpbi123dOLD 1078 | Obsolete version of ~ ifpb... |
| ifpbi23d 1079 | Equivalence deduction for ... |
| ifpimpda 1080 | Separation of the values o... |
| 1fpid3 1081 | The value of the condition... |
| elimh 1082 | Hypothesis builder for the... |
| dedt 1083 | The weak deduction theorem... |
| con3ALT 1084 | Proof of ~ con3 from its a... |
| 3orass 1089 | Associative law for triple... |
| 3orel1 1090 | Partial elimination of a t... |
| 3orrot 1091 | Rotation law for triple di... |
| 3orcoma 1092 | Commutation law for triple... |
| 3orcomb 1093 | Commutation law for triple... |
| 3anass 1094 | Associative law for triple... |
| 3anan12 1095 | Convert triple conjunction... |
| 3anan32 1096 | Convert triple conjunction... |
| 3ancoma 1097 | Commutation law for triple... |
| 3ancomb 1098 | Commutation law for triple... |
| 3anrot 1099 | Rotation law for triple co... |
| 3anrev 1100 | Reversal law for triple co... |
| anandi3 1101 | Distribution of triple con... |
| anandi3r 1102 | Distribution of triple con... |
| 3anidm 1103 | Idempotent law for conjunc... |
| 3an4anass 1104 | Associative law for four c... |
| 3ioran 1105 | Negated triple disjunction... |
| 3ianor 1106 | Negated triple conjunction... |
| 3anor 1107 | Triple conjunction express... |
| 3oran 1108 | Triple disjunction in term... |
| 3impa 1109 | Importation from double to... |
| 3imp 1110 | Importation inference. (C... |
| 3imp31 1111 | The importation inference ... |
| 3imp231 1112 | Importation inference. (C... |
| 3imp21 1113 | The importation inference ... |
| 3impb 1114 | Importation from double to... |
| 3impib 1115 | Importation to triple conj... |
| 3impia 1116 | Importation to triple conj... |
| 3expa 1117 | Exportation from triple to... |
| 3exp 1118 | Exportation inference. (C... |
| 3expb 1119 | Exportation from triple to... |
| 3expia 1120 | Exportation from triple co... |
| 3expib 1121 | Exportation from triple co... |
| 3com12 1122 | Commutation in antecedent.... |
| 3com13 1123 | Commutation in antecedent.... |
| 3comr 1124 | Commutation in antecedent.... |
| 3com23 1125 | Commutation in antecedent.... |
| 3coml 1126 | Commutation in antecedent.... |
| 3jca 1127 | Join consequents with conj... |
| 3jcad 1128 | Deduction conjoining the c... |
| 3adant1 1129 | Deduction adding a conjunc... |
| 3adant2 1130 | Deduction adding a conjunc... |
| 3adant3 1131 | Deduction adding a conjunc... |
| 3ad2ant1 1132 | Deduction adding conjuncts... |
| 3ad2ant2 1133 | Deduction adding conjuncts... |
| 3ad2ant3 1134 | Deduction adding conjuncts... |
| simp1 1135 | Simplification of triple c... |
| simp2 1136 | Simplification of triple c... |
| simp3 1137 | Simplification of triple c... |
| simp1i 1138 | Infer a conjunct from a tr... |
| simp2i 1139 | Infer a conjunct from a tr... |
| simp3i 1140 | Infer a conjunct from a tr... |
| simp1d 1141 | Deduce a conjunct from a t... |
| simp2d 1142 | Deduce a conjunct from a t... |
| simp3d 1143 | Deduce a conjunct from a t... |
| simp1bi 1144 | Deduce a conjunct from a t... |
| simp2bi 1145 | Deduce a conjunct from a t... |
| simp3bi 1146 | Deduce a conjunct from a t... |
| 3simpa 1147 | Simplification of triple c... |
| 3simpb 1148 | Simplification of triple c... |
| 3simpc 1149 | Simplification of triple c... |
| 3anim123i 1150 | Join antecedents and conse... |
| 3anim1i 1151 | Add two conjuncts to antec... |
| 3anim2i 1152 | Add two conjuncts to antec... |
| 3anim3i 1153 | Add two conjuncts to antec... |
| 3anbi123i 1154 | Join 3 biconditionals with... |
| 3orbi123i 1155 | Join 3 biconditionals with... |
| 3anbi1i 1156 | Inference adding two conju... |
| 3anbi2i 1157 | Inference adding two conju... |
| 3anbi3i 1158 | Inference adding two conju... |
| syl3an 1159 | A triple syllogism inferen... |
| syl3anb 1160 | A triple syllogism inferen... |
| syl3anbr 1161 | A triple syllogism inferen... |
| syl3an1 1162 | A syllogism inference. (C... |
| syl3an2 1163 | A syllogism inference. (C... |
| syl3an3 1164 | A syllogism inference. (C... |
| 3adantl1 1165 | Deduction adding a conjunc... |
| 3adantl2 1166 | Deduction adding a conjunc... |
| 3adantl3 1167 | Deduction adding a conjunc... |
| 3adantr1 1168 | Deduction adding a conjunc... |
| 3adantr2 1169 | Deduction adding a conjunc... |
| 3adantr3 1170 | Deduction adding a conjunc... |
| ad4ant123 1171 | Deduction adding conjuncts... |
| ad4ant124 1172 | Deduction adding conjuncts... |
| ad4ant134 1173 | Deduction adding conjuncts... |
| ad4ant234 1174 | Deduction adding conjuncts... |
| 3adant1l 1175 | Deduction adding a conjunc... |
| 3adant1r 1176 | Deduction adding a conjunc... |
| 3adant2l 1177 | Deduction adding a conjunc... |
| 3adant2r 1178 | Deduction adding a conjunc... |
| 3adant3l 1179 | Deduction adding a conjunc... |
| 3adant3r 1180 | Deduction adding a conjunc... |
| 3adant3r1 1181 | Deduction adding a conjunc... |
| 3adant3r2 1182 | Deduction adding a conjunc... |
| 3adant3r3 1183 | Deduction adding a conjunc... |
| 3ad2antl1 1184 | Deduction adding conjuncts... |
| 3ad2antl2 1185 | Deduction adding conjuncts... |
| 3ad2antl3 1186 | Deduction adding conjuncts... |
| 3ad2antr1 1187 | Deduction adding conjuncts... |
| 3ad2antr2 1188 | Deduction adding conjuncts... |
| 3ad2antr3 1189 | Deduction adding conjuncts... |
| simpl1 1190 | Simplification of conjunct... |
| simpl2 1191 | Simplification of conjunct... |
| simpl3 1192 | Simplification of conjunct... |
| simpr1 1193 | Simplification of conjunct... |
| simpr2 1194 | Simplification of conjunct... |
| simpr3 1195 | Simplification of conjunct... |
| simp1l 1196 | Simplification of triple c... |
| simp1r 1197 | Simplification of triple c... |
| simp2l 1198 | Simplification of triple c... |
| simp2r 1199 | Simplification of triple c... |
| simp3l 1200 | Simplification of triple c... |
| simp3r 1201 | Simplification of triple c... |
| simp11 1202 | Simplification of doubly t... |
| simp12 1203 | Simplification of doubly t... |
| simp13 1204 | Simplification of doubly t... |
| simp21 1205 | Simplification of doubly t... |
| simp22 1206 | Simplification of doubly t... |
| simp23 1207 | Simplification of doubly t... |
| simp31 1208 | Simplification of doubly t... |
| simp32 1209 | Simplification of doubly t... |
| simp33 1210 | Simplification of doubly t... |
| simpll1 1211 | Simplification of conjunct... |
| simpll2 1212 | Simplification of conjunct... |
| simpll3 1213 | Simplification of conjunct... |
| simplr1 1214 | Simplification of conjunct... |
| simplr2 1215 | Simplification of conjunct... |
| simplr3 1216 | Simplification of conjunct... |
| simprl1 1217 | Simplification of conjunct... |
| simprl2 1218 | Simplification of conjunct... |
| simprl3 1219 | Simplification of conjunct... |
| simprr1 1220 | Simplification of conjunct... |
| simprr2 1221 | Simplification of conjunct... |
| simprr3 1222 | Simplification of conjunct... |
| simpl1l 1223 | Simplification of conjunct... |
| simpl1r 1224 | Simplification of conjunct... |
| simpl2l 1225 | Simplification of conjunct... |
| simpl2r 1226 | Simplification of conjunct... |
| simpl3l 1227 | Simplification of conjunct... |
| simpl3r 1228 | Simplification of conjunct... |
| simpr1l 1229 | Simplification of conjunct... |
| simpr1r 1230 | Simplification of conjunct... |
| simpr2l 1231 | Simplification of conjunct... |
| simpr2r 1232 | Simplification of conjunct... |
| simpr3l 1233 | Simplification of conjunct... |
| simpr3r 1234 | Simplification of conjunct... |
| simp1ll 1235 | Simplification of conjunct... |
| simp1lr 1236 | Simplification of conjunct... |
| simp1rl 1237 | Simplification of conjunct... |
| simp1rr 1238 | Simplification of conjunct... |
| simp2ll 1239 | Simplification of conjunct... |
| simp2lr 1240 | Simplification of conjunct... |
| simp2rl 1241 | Simplification of conjunct... |
| simp2rr 1242 | Simplification of conjunct... |
| simp3ll 1243 | Simplification of conjunct... |
| simp3lr 1244 | Simplification of conjunct... |
| simp3rl 1245 | Simplification of conjunct... |
| simp3rr 1246 | Simplification of conjunct... |
| simpl11 1247 | Simplification of conjunct... |
| simpl12 1248 | Simplification of conjunct... |
| simpl13 1249 | Simplification of conjunct... |
| simpl21 1250 | Simplification of conjunct... |
| simpl22 1251 | Simplification of conjunct... |
| simpl23 1252 | Simplification of conjunct... |
| simpl31 1253 | Simplification of conjunct... |
| simpl32 1254 | Simplification of conjunct... |
| simpl33 1255 | Simplification of conjunct... |
| simpr11 1256 | Simplification of conjunct... |
| simpr12 1257 | Simplification of conjunct... |
| simpr13 1258 | Simplification of conjunct... |
| simpr21 1259 | Simplification of conjunct... |
| simpr22 1260 | Simplification of conjunct... |
| simpr23 1261 | Simplification of conjunct... |
| simpr31 1262 | Simplification of conjunct... |
| simpr32 1263 | Simplification of conjunct... |
| simpr33 1264 | Simplification of conjunct... |
| simp1l1 1265 | Simplification of conjunct... |
| simp1l2 1266 | Simplification of conjunct... |
| simp1l3 1267 | Simplification of conjunct... |
| simp1r1 1268 | Simplification of conjunct... |
| simp1r2 1269 | Simplification of conjunct... |
| simp1r3 1270 | Simplification of conjunct... |
| simp2l1 1271 | Simplification of conjunct... |
| simp2l2 1272 | Simplification of conjunct... |
| simp2l3 1273 | Simplification of conjunct... |
| simp2r1 1274 | Simplification of conjunct... |
| simp2r2 1275 | Simplification of conjunct... |
| simp2r3 1276 | Simplification of conjunct... |
| simp3l1 1277 | Simplification of conjunct... |
| simp3l2 1278 | Simplification of conjunct... |
| simp3l3 1279 | Simplification of conjunct... |
| simp3r1 1280 | Simplification of conjunct... |
| simp3r2 1281 | Simplification of conjunct... |
| simp3r3 1282 | Simplification of conjunct... |
| simp11l 1283 | Simplification of conjunct... |
| simp11r 1284 | Simplification of conjunct... |
| simp12l 1285 | Simplification of conjunct... |
| simp12r 1286 | Simplification of conjunct... |
| simp13l 1287 | Simplification of conjunct... |
| simp13r 1288 | Simplification of conjunct... |
| simp21l 1289 | Simplification of conjunct... |
| simp21r 1290 | Simplification of conjunct... |
| simp22l 1291 | Simplification of conjunct... |
| simp22r 1292 | Simplification of conjunct... |
| simp23l 1293 | Simplification of conjunct... |
| simp23r 1294 | Simplification of conjunct... |
| simp31l 1295 | Simplification of conjunct... |
| simp31r 1296 | Simplification of conjunct... |
| simp32l 1297 | Simplification of conjunct... |
| simp32r 1298 | Simplification of conjunct... |
| simp33l 1299 | Simplification of conjunct... |
| simp33r 1300 | Simplification of conjunct... |
| simp111 1301 | Simplification of conjunct... |
| simp112 1302 | Simplification of conjunct... |
| simp113 1303 | Simplification of conjunct... |
| simp121 1304 | Simplification of conjunct... |
| simp122 1305 | Simplification of conjunct... |
| simp123 1306 | Simplification of conjunct... |
| simp131 1307 | Simplification of conjunct... |
| simp132 1308 | Simplification of conjunct... |
| simp133 1309 | Simplification of conjunct... |
| simp211 1310 | Simplification of conjunct... |
| simp212 1311 | Simplification of conjunct... |
| simp213 1312 | Simplification of conjunct... |
| simp221 1313 | Simplification of conjunct... |
| simp222 1314 | Simplification of conjunct... |
| simp223 1315 | Simplification of conjunct... |
| simp231 1316 | Simplification of conjunct... |
| simp232 1317 | Simplification of conjunct... |
| simp233 1318 | Simplification of conjunct... |
| simp311 1319 | Simplification of conjunct... |
| simp312 1320 | Simplification of conjunct... |
| simp313 1321 | Simplification of conjunct... |
| simp321 1322 | Simplification of conjunct... |
| simp322 1323 | Simplification of conjunct... |
| simp323 1324 | Simplification of conjunct... |
| simp331 1325 | Simplification of conjunct... |
| simp332 1326 | Simplification of conjunct... |
| simp333 1327 | Simplification of conjunct... |
| 3anibar 1328 | Remove a hypothesis from t... |
| 3mix1 1329 | Introduction in triple dis... |
| 3mix2 1330 | Introduction in triple dis... |
| 3mix3 1331 | Introduction in triple dis... |
| 3mix1i 1332 | Introduction in triple dis... |
| 3mix2i 1333 | Introduction in triple dis... |
| 3mix3i 1334 | Introduction in triple dis... |
| 3mix1d 1335 | Deduction introducing trip... |
| 3mix2d 1336 | Deduction introducing trip... |
| 3mix3d 1337 | Deduction introducing trip... |
| 3pm3.2i 1338 | Infer conjunction of premi... |
| pm3.2an3 1339 | Version of ~ pm3.2 for a t... |
| mpbir3an 1340 | Detach a conjunction of tr... |
| mpbir3and 1341 | Detach a conjunction of tr... |
| syl3anbrc 1342 | Syllogism inference. (Con... |
| syl21anbrc 1343 | Syllogism inference. (Con... |
| 3imp3i2an 1344 | An elimination deduction. ... |
| ex3 1345 | Apply ~ ex to a hypothesis... |
| 3imp1 1346 | Importation to left triple... |
| 3impd 1347 | Importation deduction for ... |
| 3imp2 1348 | Importation to right tripl... |
| 3impdi 1349 | Importation inference (und... |
| 3impdir 1350 | Importation inference (und... |
| 3exp1 1351 | Exportation from left trip... |
| 3expd 1352 | Exportation deduction for ... |
| 3exp2 1353 | Exportation from right tri... |
| exp5o 1354 | A triple exportation infer... |
| exp516 1355 | A triple exportation infer... |
| exp520 1356 | A triple exportation infer... |
| 3impexp 1357 | Version of ~ impexp for a ... |
| 3an1rs 1358 | Swap conjuncts. (Contribu... |
| 3anassrs 1359 | Associative law for conjun... |
| ad5ant245 1360 | Deduction adding conjuncts... |
| ad5ant234 1361 | Deduction adding conjuncts... |
| ad5ant235 1362 | Deduction adding conjuncts... |
| ad5ant123 1363 | Deduction adding conjuncts... |
| ad5ant124 1364 | Deduction adding conjuncts... |
| ad5ant125 1365 | Deduction adding conjuncts... |
| ad5ant134 1366 | Deduction adding conjuncts... |
| ad5ant135 1367 | Deduction adding conjuncts... |
| ad5ant145 1368 | Deduction adding conjuncts... |
| ad5ant2345 1369 | Deduction adding conjuncts... |
| syl3anc 1370 | Syllogism combined with co... |
| syl13anc 1371 | Syllogism combined with co... |
| syl31anc 1372 | Syllogism combined with co... |
| syl112anc 1373 | Syllogism combined with co... |
| syl121anc 1374 | Syllogism combined with co... |
| syl211anc 1375 | Syllogism combined with co... |
| syl23anc 1376 | Syllogism combined with co... |
| syl32anc 1377 | Syllogism combined with co... |
| syl122anc 1378 | Syllogism combined with co... |
| syl212anc 1379 | Syllogism combined with co... |
| syl221anc 1380 | Syllogism combined with co... |
| syl113anc 1381 | Syllogism combined with co... |
| syl131anc 1382 | Syllogism combined with co... |
| syl311anc 1383 | Syllogism combined with co... |
| syl33anc 1384 | Syllogism combined with co... |
| syl222anc 1385 | Syllogism combined with co... |
| syl123anc 1386 | Syllogism combined with co... |
| syl132anc 1387 | Syllogism combined with co... |
| syl213anc 1388 | Syllogism combined with co... |
| syl231anc 1389 | Syllogism combined with co... |
| syl312anc 1390 | Syllogism combined with co... |
| syl321anc 1391 | Syllogism combined with co... |
| syl133anc 1392 | Syllogism combined with co... |
| syl313anc 1393 | Syllogism combined with co... |
| syl331anc 1394 | Syllogism combined with co... |
| syl223anc 1395 | Syllogism combined with co... |
| syl232anc 1396 | Syllogism combined with co... |
| syl322anc 1397 | Syllogism combined with co... |
| syl233anc 1398 | Syllogism combined with co... |
| syl323anc 1399 | Syllogism combined with co... |
| syl332anc 1400 | Syllogism combined with co... |
| syl333anc 1401 | A syllogism inference comb... |
| syl3an1b 1402 | A syllogism inference. (C... |
| syl3an2b 1403 | A syllogism inference. (C... |
| syl3an3b 1404 | A syllogism inference. (C... |
| syl3an1br 1405 | A syllogism inference. (C... |
| syl3an2br 1406 | A syllogism inference. (C... |
| syl3an3br 1407 | A syllogism inference. (C... |
| syld3an3 1408 | A syllogism inference. (C... |
| syld3an1 1409 | A syllogism inference. (C... |
| syld3an2 1410 | A syllogism inference. (C... |
| syl3anl1 1411 | A syllogism inference. (C... |
| syl3anl2 1412 | A syllogism inference. (C... |
| syl3anl3 1413 | A syllogism inference. (C... |
| syl3anl 1414 | A triple syllogism inferen... |
| syl3anr1 1415 | A syllogism inference. (C... |
| syl3anr2 1416 | A syllogism inference. (C... |
| syl3anr3 1417 | A syllogism inference. (C... |
| 3anidm12 1418 | Inference from idempotent ... |
| 3anidm13 1419 | Inference from idempotent ... |
| 3anidm23 1420 | Inference from idempotent ... |
| syl2an3an 1421 | ~ syl3an with antecedents ... |
| syl2an23an 1422 | Deduction related to ~ syl... |
| 3ori 1423 | Infer implication from tri... |
| 3jao 1424 | Disjunction of three antec... |
| 3jaob 1425 | Disjunction of three antec... |
| 3jaoi 1426 | Disjunction of three antec... |
| 3jaod 1427 | Disjunction of three antec... |
| 3jaoian 1428 | Disjunction of three antec... |
| 3jaodan 1429 | Disjunction of three antec... |
| mpjao3dan 1430 | Eliminate a three-way disj... |
| mpjao3danOLD 1431 | Obsolete version of ~ mpja... |
| 3jaao 1432 | Inference conjoining and d... |
| syl3an9b 1433 | Nested syllogism inference... |
| 3orbi123d 1434 | Deduction joining 3 equiva... |
| 3anbi123d 1435 | Deduction joining 3 equiva... |
| 3anbi12d 1436 | Deduction conjoining and a... |
| 3anbi13d 1437 | Deduction conjoining and a... |
| 3anbi23d 1438 | Deduction conjoining and a... |
| 3anbi1d 1439 | Deduction adding conjuncts... |
| 3anbi2d 1440 | Deduction adding conjuncts... |
| 3anbi3d 1441 | Deduction adding conjuncts... |
| 3anim123d 1442 | Deduction joining 3 implic... |
| 3orim123d 1443 | Deduction joining 3 implic... |
| an6 1444 | Rearrangement of 6 conjunc... |
| 3an6 1445 | Analogue of ~ an4 for trip... |
| 3or6 1446 | Analogue of ~ or4 for trip... |
| mp3an1 1447 | An inference based on modu... |
| mp3an2 1448 | An inference based on modu... |
| mp3an3 1449 | An inference based on modu... |
| mp3an12 1450 | An inference based on modu... |
| mp3an13 1451 | An inference based on modu... |
| mp3an23 1452 | An inference based on modu... |
| mp3an1i 1453 | An inference based on modu... |
| mp3anl1 1454 | An inference based on modu... |
| mp3anl2 1455 | An inference based on modu... |
| mp3anl3 1456 | An inference based on modu... |
| mp3anr1 1457 | An inference based on modu... |
| mp3anr2 1458 | An inference based on modu... |
| mp3anr3 1459 | An inference based on modu... |
| mp3an 1460 | An inference based on modu... |
| mpd3an3 1461 | An inference based on modu... |
| mpd3an23 1462 | An inference based on modu... |
| mp3and 1463 | A deduction based on modus... |
| mp3an12i 1464 | ~ mp3an with antecedents i... |
| mp3an2i 1465 | ~ mp3an with antecedents i... |
| mp3an3an 1466 | ~ mp3an with antecedents i... |
| mp3an2ani 1467 | An elimination deduction. ... |
| biimp3a 1468 | Infer implication from a l... |
| biimp3ar 1469 | Infer implication from a l... |
| 3anandis 1470 | Inference that undistribut... |
| 3anandirs 1471 | Inference that undistribut... |
| ecase23d 1472 | Deduction for elimination ... |
| 3ecase 1473 | Inference for elimination ... |
| 3bior1fd 1474 | A disjunction is equivalen... |
| 3bior1fand 1475 | A disjunction is equivalen... |
| 3bior2fd 1476 | A wff is equivalent to its... |
| 3biant1d 1477 | A conjunction is equivalen... |
| intn3an1d 1478 | Introduction of a triple c... |
| intn3an2d 1479 | Introduction of a triple c... |
| intn3an3d 1480 | Introduction of a triple c... |
| an3andi 1481 | Distribution of conjunctio... |
| an33rean 1482 | Rearrange a 9-fold conjunc... |
| an33reanOLD 1483 | Obsolete version of ~ an33... |
| 3orel2 1484 | Partial elimination of a t... |
| 3orel3 1485 | Partial elimination of a t... |
| 3orel13 1486 | Elimination of two disjunc... |
| 3pm3.2ni 1487 | Triple negated disjunction... |
| nanan 1490 | Conjunction in terms of al... |
| dfnan2 1491 | Alternative denial in term... |
| nanor 1492 | Alternative denial in term... |
| nancom 1493 | Alternative denial is comm... |
| nannan 1494 | Nested alternative denials... |
| nanim 1495 | Implication in terms of al... |
| nannot 1496 | Negation in terms of alter... |
| nanbi 1497 | Biconditional in terms of ... |
| nanbi1 1498 | Introduce a right anti-con... |
| nanbi2 1499 | Introduce a left anti-conj... |
| nanbi12 1500 | Join two logical equivalen... |
| nanbi1i 1501 | Introduce a right anti-con... |
| nanbi2i 1502 | Introduce a left anti-conj... |
| nanbi12i 1503 | Join two logical equivalen... |
| nanbi1d 1504 | Introduce a right anti-con... |
| nanbi2d 1505 | Introduce a left anti-conj... |
| nanbi12d 1506 | Join two logical equivalen... |
| nanass 1507 | A characterization of when... |
| xnor 1510 | Two ways to write XNOR (ex... |
| xorcom 1511 | The connector ` \/_ ` is c... |
| xorcomOLD 1512 | Obsolete version of ~ xorc... |
| xorass 1513 | The connector ` \/_ ` is a... |
| excxor 1514 | This tautology shows that ... |
| xor2 1515 | Two ways to express "exclu... |
| xoror 1516 | Exclusive disjunction impl... |
| xornan 1517 | Exclusive disjunction impl... |
| xornan2 1518 | XOR implies NAND (written ... |
| xorneg2 1519 | The connector ` \/_ ` is n... |
| xorneg1 1520 | The connector ` \/_ ` is n... |
| xorneg 1521 | The connector ` \/_ ` is u... |
| xorbi12i 1522 | Equality property for excl... |
| xorbi12iOLD 1523 | Obsolete version of ~ xorb... |
| xorbi12d 1524 | Equality property for excl... |
| anxordi 1525 | Conjunction distributes ov... |
| xorexmid 1526 | Exclusive-or variant of th... |
| norcom 1529 | The connector ` -\/ ` is c... |
| norcomOLD 1530 | Obsolete version of ~ norc... |
| nornot 1531 | ` -. ` is expressible via ... |
| noran 1532 | ` /\ ` is expressible via ... |
| noror 1533 | ` \/ ` is expressible via ... |
| norasslem1 1534 | This lemma shows the equiv... |
| norasslem2 1535 | This lemma specializes ~ b... |
| norasslem3 1536 | This lemma specializes ~ b... |
| norass 1537 | A characterization of when... |
| trujust 1542 | Soundness justification th... |
| tru 1544 | The truth value ` T. ` is ... |
| dftru2 1545 | An alternate definition of... |
| trut 1546 | A proposition is equivalen... |
| mptru 1547 | Eliminate ` T. ` as an ant... |
| tbtru 1548 | A proposition is equivalen... |
| bitru 1549 | A theorem is equivalent to... |
| trud 1550 | Anything implies ` T. ` . ... |
| truan 1551 | True can be removed from a... |
| fal 1554 | The truth value ` F. ` is ... |
| nbfal 1555 | The negation of a proposit... |
| bifal 1556 | A contradiction is equival... |
| falim 1557 | The truth value ` F. ` imp... |
| falimd 1558 | The truth value ` F. ` imp... |
| dfnot 1559 | Given falsum ` F. ` , we c... |
| inegd 1560 | Negation introduction rule... |
| efald 1561 | Deduction based on reducti... |
| pm2.21fal 1562 | If a wff and its negation ... |
| truimtru 1563 | A ` -> ` identity. (Contr... |
| truimfal 1564 | A ` -> ` identity. (Contr... |
| falimtru 1565 | A ` -> ` identity. (Contr... |
| falimfal 1566 | A ` -> ` identity. (Contr... |
| nottru 1567 | A ` -. ` identity. (Contr... |
| notfal 1568 | A ` -. ` identity. (Contr... |
| trubitru 1569 | A ` <-> ` identity. (Cont... |
| falbitru 1570 | A ` <-> ` identity. (Cont... |
| trubifal 1571 | A ` <-> ` identity. (Cont... |
| falbifal 1572 | A ` <-> ` identity. (Cont... |
| truantru 1573 | A ` /\ ` identity. (Contr... |
| truanfal 1574 | A ` /\ ` identity. (Contr... |
| falantru 1575 | A ` /\ ` identity. (Contr... |
| falanfal 1576 | A ` /\ ` identity. (Contr... |
| truortru 1577 | A ` \/ ` identity. (Contr... |
| truorfal 1578 | A ` \/ ` identity. (Contr... |
| falortru 1579 | A ` \/ ` identity. (Contr... |
| falorfal 1580 | A ` \/ ` identity. (Contr... |
| trunantru 1581 | A ` -/\ ` identity. (Cont... |
| trunanfal 1582 | A ` -/\ ` identity. (Cont... |
| falnantru 1583 | A ` -/\ ` identity. (Cont... |
| falnanfal 1584 | A ` -/\ ` identity. (Cont... |
| truxortru 1585 | A ` \/_ ` identity. (Cont... |
| truxorfal 1586 | A ` \/_ ` identity. (Cont... |
| falxortru 1587 | A ` \/_ ` identity. (Cont... |
| falxorfal 1588 | A ` \/_ ` identity. (Cont... |
| trunortru 1589 | A ` -\/ ` identity. (Cont... |
| trunorfal 1590 | A ` -\/ ` identity. (Cont... |
| falnortru 1591 | A ` -\/ ` identity. (Cont... |
| falnorfal 1592 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1595 | Equality theorem for the a... |
| hadbi123i 1596 | Equality theorem for the a... |
| hadass 1597 | Associative law for the ad... |
| hadbi 1598 | The adder sum is the same ... |
| hadcoma 1599 | Commutative law for the ad... |
| hadcomb 1600 | Commutative law for the ad... |
| hadrot 1601 | Rotation law for the adder... |
| hadnot 1602 | The adder sum distributes ... |
| had1 1603 | If the first input is true... |
| had0 1604 | If the first input is fals... |
| hadifp 1605 | The value of the adder sum... |
| cador 1608 | The adder carry in disjunc... |
| cadan 1609 | The adder carry in conjunc... |
| cadbi123d 1610 | Equality theorem for the a... |
| cadbi123i 1611 | Equality theorem for the a... |
| cadcoma 1612 | Commutative law for the ad... |
| cadcomb 1613 | Commutative law for the ad... |
| cadrot 1614 | Rotation law for the adder... |
| cadnot 1615 | The adder carry distribute... |
| cad11 1616 | If (at least) two inputs a... |
| cad1 1617 | If one input is true, then... |
| cad0 1618 | If one input is false, the... |
| cad0OLD 1619 | Obsolete version of ~ cad0... |
| cadifp 1620 | The value of the carry is,... |
| cadtru 1621 | The adder carry is true as... |
| minimp 1622 | A single axiom for minimal... |
| minimp-syllsimp 1623 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1624 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1625 | Derivation of a commuted f... |
| minimp-ax2 1626 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1627 | Derivation of ~ pm2.43 (al... |
| impsingle 1628 | The shortest single axiom ... |
| impsingle-step4 1629 | Derivation of impsingle-st... |
| impsingle-step8 1630 | Derivation of impsingle-st... |
| impsingle-ax1 1631 | Derivation of impsingle-ax... |
| impsingle-step15 1632 | Derivation of impsingle-st... |
| impsingle-step18 1633 | Derivation of impsingle-st... |
| impsingle-step19 1634 | Derivation of impsingle-st... |
| impsingle-step20 1635 | Derivation of impsingle-st... |
| impsingle-step21 1636 | Derivation of impsingle-st... |
| impsingle-step22 1637 | Derivation of impsingle-st... |
| impsingle-step25 1638 | Derivation of impsingle-st... |
| impsingle-imim1 1639 | Derivation of impsingle-im... |
| impsingle-peirce 1640 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1641 | Derivation of ~ ax-2 from ... |
| meredith 1642 | Carew Meredith's sole axio... |
| merlem1 1643 | Step 3 of Meredith's proof... |
| merlem2 1644 | Step 4 of Meredith's proof... |
| merlem3 1645 | Step 7 of Meredith's proof... |
| merlem4 1646 | Step 8 of Meredith's proof... |
| merlem5 1647 | Step 11 of Meredith's proo... |
| merlem6 1648 | Step 12 of Meredith's proo... |
| merlem7 1649 | Between steps 14 and 15 of... |
| merlem8 1650 | Step 15 of Meredith's proo... |
| merlem9 1651 | Step 18 of Meredith's proo... |
| merlem10 1652 | Step 19 of Meredith's proo... |
| merlem11 1653 | Step 20 of Meredith's proo... |
| merlem12 1654 | Step 28 of Meredith's proo... |
| merlem13 1655 | Step 35 of Meredith's proo... |
| luk-1 1656 | 1 of 3 axioms for proposit... |
| luk-2 1657 | 2 of 3 axioms for proposit... |
| luk-3 1658 | 3 of 3 axioms for proposit... |
| luklem1 1659 | Used to rederive standard ... |
| luklem2 1660 | Used to rederive standard ... |
| luklem3 1661 | Used to rederive standard ... |
| luklem4 1662 | Used to rederive standard ... |
| luklem5 1663 | Used to rederive standard ... |
| luklem6 1664 | Used to rederive standard ... |
| luklem7 1665 | Used to rederive standard ... |
| luklem8 1666 | Used to rederive standard ... |
| ax1 1667 | Standard propositional axi... |
| ax2 1668 | Standard propositional axi... |
| ax3 1669 | Standard propositional axi... |
| nic-dfim 1670 | This theorem "defines" imp... |
| nic-dfneg 1671 | This theorem "defines" neg... |
| nic-mp 1672 | Derive Nicod's rule of mod... |
| nic-mpALT 1673 | A direct proof of ~ nic-mp... |
| nic-ax 1674 | Nicod's axiom derived from... |
| nic-axALT 1675 | A direct proof of ~ nic-ax... |
| nic-imp 1676 | Inference for ~ nic-mp usi... |
| nic-idlem1 1677 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1678 | Lemma for ~ nic-id . Infe... |
| nic-id 1679 | Theorem ~ id expressed wit... |
| nic-swap 1680 | The connector ` -/\ ` is s... |
| nic-isw1 1681 | Inference version of ~ nic... |
| nic-isw2 1682 | Inference for swapping nes... |
| nic-iimp1 1683 | Inference version of ~ nic... |
| nic-iimp2 1684 | Inference version of ~ nic... |
| nic-idel 1685 | Inference to remove the tr... |
| nic-ich 1686 | Chained inference. (Contr... |
| nic-idbl 1687 | Double the terms. Since d... |
| nic-bijust 1688 | Biconditional justificatio... |
| nic-bi1 1689 | Inference to extract one s... |
| nic-bi2 1690 | Inference to extract the o... |
| nic-stdmp 1691 | Derive the standard modus ... |
| nic-luk1 1692 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1693 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1694 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1695 | This alternative axiom for... |
| lukshefth1 1696 | Lemma for ~ renicax . (Co... |
| lukshefth2 1697 | Lemma for ~ renicax . (Co... |
| renicax 1698 | A rederivation of ~ nic-ax... |
| tbw-bijust 1699 | Justification for ~ tbw-ne... |
| tbw-negdf 1700 | The definition of negation... |
| tbw-ax1 1701 | The first of four axioms i... |
| tbw-ax2 1702 | The second of four axioms ... |
| tbw-ax3 1703 | The third of four axioms i... |
| tbw-ax4 1704 | The fourth of four axioms ... |
| tbwsyl 1705 | Used to rederive the Lukas... |
| tbwlem1 1706 | Used to rederive the Lukas... |
| tbwlem2 1707 | Used to rederive the Lukas... |
| tbwlem3 1708 | Used to rederive the Lukas... |
| tbwlem4 1709 | Used to rederive the Lukas... |
| tbwlem5 1710 | Used to rederive the Lukas... |
| re1luk1 1711 | ~ luk-1 derived from the T... |
| re1luk2 1712 | ~ luk-2 derived from the T... |
| re1luk3 1713 | ~ luk-3 derived from the T... |
| merco1 1714 | A single axiom for proposi... |
| merco1lem1 1715 | Used to rederive the Tarsk... |
| retbwax4 1716 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1717 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1718 | Used to rederive the Tarsk... |
| merco1lem3 1719 | Used to rederive the Tarsk... |
| merco1lem4 1720 | Used to rederive the Tarsk... |
| merco1lem5 1721 | Used to rederive the Tarsk... |
| merco1lem6 1722 | Used to rederive the Tarsk... |
| merco1lem7 1723 | Used to rederive the Tarsk... |
| retbwax3 1724 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1725 | Used to rederive the Tarsk... |
| merco1lem9 1726 | Used to rederive the Tarsk... |
| merco1lem10 1727 | Used to rederive the Tarsk... |
| merco1lem11 1728 | Used to rederive the Tarsk... |
| merco1lem12 1729 | Used to rederive the Tarsk... |
| merco1lem13 1730 | Used to rederive the Tarsk... |
| merco1lem14 1731 | Used to rederive the Tarsk... |
| merco1lem15 1732 | Used to rederive the Tarsk... |
| merco1lem16 1733 | Used to rederive the Tarsk... |
| merco1lem17 1734 | Used to rederive the Tarsk... |
| merco1lem18 1735 | Used to rederive the Tarsk... |
| retbwax1 1736 | ~ tbw-ax1 rederived from ~... |
| merco2 1737 | A single axiom for proposi... |
| mercolem1 1738 | Used to rederive the Tarsk... |
| mercolem2 1739 | Used to rederive the Tarsk... |
| mercolem3 1740 | Used to rederive the Tarsk... |
| mercolem4 1741 | Used to rederive the Tarsk... |
| mercolem5 1742 | Used to rederive the Tarsk... |
| mercolem6 1743 | Used to rederive the Tarsk... |
| mercolem7 1744 | Used to rederive the Tarsk... |
| mercolem8 1745 | Used to rederive the Tarsk... |
| re1tbw1 1746 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1747 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1748 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1749 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1750 | Justification for ~ rb-imd... |
| rb-imdf 1751 | The definition of implicat... |
| anmp 1752 | Modus ponens for ` { \/ , ... |
| rb-ax1 1753 | The first of four axioms i... |
| rb-ax2 1754 | The second of four axioms ... |
| rb-ax3 1755 | The third of four axioms i... |
| rb-ax4 1756 | The fourth of four axioms ... |
| rbsyl 1757 | Used to rederive the Lukas... |
| rblem1 1758 | Used to rederive the Lukas... |
| rblem2 1759 | Used to rederive the Lukas... |
| rblem3 1760 | Used to rederive the Lukas... |
| rblem4 1761 | Used to rederive the Lukas... |
| rblem5 1762 | Used to rederive the Lukas... |
| rblem6 1763 | Used to rederive the Lukas... |
| rblem7 1764 | Used to rederive the Lukas... |
| re1axmp 1765 | ~ ax-mp derived from Russe... |
| re2luk1 1766 | ~ luk-1 derived from Russe... |
| re2luk2 1767 | ~ luk-2 derived from Russe... |
| re2luk3 1768 | ~ luk-3 derived from Russe... |
| mptnan 1769 | Modus ponendo tollens 1, o... |
| mptxor 1770 | Modus ponendo tollens 2, o... |
| mtpor 1771 | Modus tollendo ponens (inc... |
| mtpxor 1772 | Modus tollendo ponens (ori... |
| stoic1a 1773 | Stoic logic Thema 1 (part ... |
| stoic1b 1774 | Stoic logic Thema 1 (part ... |
| stoic2a 1775 | Stoic logic Thema 2 versio... |
| stoic2b 1776 | Stoic logic Thema 2 versio... |
| stoic3 1777 | Stoic logic Thema 3. Stat... |
| stoic4a 1778 | Stoic logic Thema 4 versio... |
| stoic4b 1779 | Stoic logic Thema 4 versio... |
| alnex 1782 | Universal quantification o... |
| eximal 1783 | An equivalence between an ... |
| nf2 1786 | Alternate definition of no... |
| nf3 1787 | Alternate definition of no... |
| nf4 1788 | Alternate definition of no... |
| nfi 1789 | Deduce that ` x ` is not f... |
| nfri 1790 | Consequence of the definit... |
| nfd 1791 | Deduce that ` x ` is not f... |
| nfrd 1792 | Consequence of the definit... |
| nftht 1793 | Closed form of ~ nfth . (... |
| nfntht 1794 | Closed form of ~ nfnth . ... |
| nfntht2 1795 | Closed form of ~ nfnth . ... |
| gen2 1797 | Generalization applied twi... |
| mpg 1798 | Modus ponens combined with... |
| mpgbi 1799 | Modus ponens on biconditio... |
| mpgbir 1800 | Modus ponens on biconditio... |
| nex 1801 | Generalization rule for ne... |
| nfth 1802 | No variable is (effectivel... |
| nfnth 1803 | No variable is (effectivel... |
| hbth 1804 | No variable is (effectivel... |
| nftru 1805 | The true constant has no f... |
| nffal 1806 | The false constant has no ... |
| sptruw 1807 | Version of ~ sp when ` ph ... |
| altru 1808 | For all sets, ` T. ` is tr... |
| alfal 1809 | For all sets, ` -. F. ` is... |
| alim 1811 | Restatement of Axiom ~ ax-... |
| alimi 1812 | Inference quantifying both... |
| 2alimi 1813 | Inference doubly quantifyi... |
| ala1 1814 | Add an antecedent in a uni... |
| al2im 1815 | Closed form of ~ al2imi . ... |
| al2imi 1816 | Inference quantifying ante... |
| alanimi 1817 | Variant of ~ al2imi with c... |
| alimdh 1818 | Deduction form of Theorem ... |
| albi 1819 | Theorem 19.15 of [Margaris... |
| albii 1820 | Inference adding universal... |
| 2albii 1821 | Inference adding two unive... |
| 3albii 1822 | Inference adding three uni... |
| sylgt 1823 | Closed form of ~ sylg . (... |
| sylg 1824 | A syllogism combined with ... |
| alrimih 1825 | Inference form of Theorem ... |
| hbxfrbi 1826 | A utility lemma to transfe... |
| alex 1827 | Universal quantifier in te... |
| exnal 1828 | Existential quantification... |
| 2nalexn 1829 | Part of theorem *11.5 in [... |
| 2exnaln 1830 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1831 | Theorem *11.25 in [Whitehe... |
| alimex 1832 | An equivalence between an ... |
| aleximi 1833 | A variant of ~ al2imi : in... |
| alexbii 1834 | Biconditional form of ~ al... |
| exim 1835 | Theorem 19.22 of [Margaris... |
| eximi 1836 | Inference adding existenti... |
| 2eximi 1837 | Inference adding two exist... |
| eximii 1838 | Inference associated with ... |
| exa1 1839 | Add an antecedent in an ex... |
| 19.38 1840 | Theorem 19.38 of [Margaris... |
| 19.38a 1841 | Under a nonfreeness hypoth... |
| 19.38b 1842 | Under a nonfreeness hypoth... |
| imnang 1843 | Quantified implication in ... |
| alinexa 1844 | A transformation of quanti... |
| exnalimn 1845 | Existential quantification... |
| alexn 1846 | A relationship between two... |
| 2exnexn 1847 | Theorem *11.51 in [Whitehe... |
| exbi 1848 | Theorem 19.18 of [Margaris... |
| exbii 1849 | Inference adding existenti... |
| 2exbii 1850 | Inference adding two exist... |
| 3exbii 1851 | Inference adding three exi... |
| nfbiit 1852 | Equivalence theorem for th... |
| nfbii 1853 | Equality theorem for the n... |
| nfxfr 1854 | A utility lemma to transfe... |
| nfxfrd 1855 | A utility lemma to transfe... |
| nfnbi 1856 | A variable is nonfree in a... |
| nfnbiOLD 1857 | Obsolete version of ~ nfnb... |
| 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 ... |
| 19.36imvOLD 1948 | Obsolete version of ~ 19.3... |
| 19.36iv 1949 | Inference associated with ... |
| 19.37imv 1950 | One direction of ~ 19.37v ... |
| 19.37iv 1951 | Inference associated with ... |
| 19.41v 1952 | Version of ~ 19.41 with a ... |
| 19.41vv 1953 | Version of ~ 19.41 with tw... |
| 19.41vvv 1954 | Version of ~ 19.41 with th... |
| 19.41vvvv 1955 | Version of ~ 19.41 with fo... |
| 19.42v 1956 | Version of ~ 19.42 with a ... |
| exdistr 1957 | Distribution of existentia... |
| exdistrv 1958 | Distribute a pair of exist... |
| 4exdistrv 1959 | Distribute two pairs of ex... |
| 19.42vv 1960 | Version of ~ 19.42 with tw... |
| exdistr2 1961 | Distribution of existentia... |
| 19.42vvv 1962 | Version of ~ 19.42 with th... |
| 3exdistr 1963 | Distribution of existentia... |
| 4exdistr 1964 | Distribution of existentia... |
| weq 1965 | Extend wff definition to i... |
| speimfw 1966 | Specialization, with addit... |
| speimfwALT 1967 | Alternate proof of ~ speim... |
| spimfw 1968 | Specialization, with addit... |
| ax12i 1969 | Inference that has ~ ax-12... |
| ax6v 1971 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1972 | At least one individual ex... |
| spimw 1973 | Specialization. Lemma 8 o... |
| spimew 1974 | Existential introduction, ... |
| speiv 1975 | Inference from existential... |
| speivw 1976 | Version of ~ spei with a d... |
| exgen 1977 | Rule of existential genera... |
| extru 1978 | There exists a variable su... |
| 19.2 1979 | Theorem 19.2 of [Margaris]... |
| 19.2d 1980 | Deduction associated with ... |
| 19.8w 1981 | Weak version of ~ 19.8a an... |
| spnfw 1982 | Weak version of ~ sp . Us... |
| 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... |
| 19.39 1987 | Theorem 19.39 of [Margaris... |
| 19.24 1988 | Theorem 19.24 of [Margaris... |
| 19.34 1989 | Theorem 19.34 of [Margaris... |
| 19.36v 1990 | Version of ~ 19.36 with a ... |
| 19.12vvv 1991 | Version of ~ 19.12vv with ... |
| 19.27v 1992 | Version of ~ 19.27 with a ... |
| 19.28v 1993 | Version of ~ 19.28 with a ... |
| 19.37v 1994 | Version of ~ 19.37 with a ... |
| 19.44v 1995 | Version of ~ 19.44 with a ... |
| 19.45v 1996 | Version of ~ 19.45 with a ... |
| spimevw 1997 | Existential introduction, ... |
| spimvw 1998 | A weak form of specializat... |
| spvv 1999 | Specialization, using impl... |
| spfalw 2000 | Version of ~ sp when ` ph ... |
| chvarvv 2001 | Implicit substitution of `... |
| equs4v 2002 | Version of ~ equs4 with a ... |
| alequexv 2003 | Version of ~ equs4v with i... |
| exsbim 2004 | One direction of the equiv... |
| equsv 2005 | If a formula does not cont... |
| equsalvw 2006 | Version of ~ equsalv with ... |
| equsexvw 2007 | Version of ~ equsexv with ... |
| 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 ... |
| alcomiw 2045 | Weak version of ~ ax-11 . ... |
| alcomw 2046 | Weak version of ~ alcom an... |
| hbn1fw 2047 | Weak version of ~ ax-10 fr... |
| hbn1w 2048 | Weak version of ~ hbn1 . ... |
| hba1w 2049 | Weak version of ~ hba1 . ... |
| hbe1w 2050 | Weak version of ~ hbe1 . ... |
| hbalw 2051 | Weak version of ~ hbal . ... |
| 19.8aw 2052 | If a formula is true, then... |
| exexw 2053 | Existential quantification... |
| spaev 2054 | A special instance of ~ sp... |
| cbvaev 2055 | Change bound variable in a... |
| aevlem0 2056 | Lemma for ~ aevlem . Inst... |
| aevlem 2057 | Lemma for ~ aev and ~ axc1... |
| aeveq 2058 | The antecedent ` A. x x = ... |
| aev 2059 | A "distinctor elimination"... |
| aev2 2060 | A version of ~ aev with tw... |
| hbaev 2061 | All variables are effectiv... |
| naev 2062 | If some set variables can ... |
| naev2 2063 | Generalization of ~ hbnaev... |
| hbnaev 2064 | Any variable is free in ` ... |
| sbjust 2065 | Justification theorem for ... |
| sbt 2068 | A substitution into a theo... |
| sbtru 2069 | The result of substituting... |
| stdpc4 2070 | The specialization axiom o... |
| sbtALT 2071 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2072 | A double specialization us... |
| sbi1 2073 | Distribute substitution ov... |
| spsbim 2074 | Distribute substitution ov... |
| spsbbi 2075 | Biconditional property for... |
| sbimi 2076 | Distribute substitution ov... |
| sb2imi 2077 | Distribute substitution ov... |
| sbbii 2078 | Infer substitution into bo... |
| 2sbbii 2079 | Infer double substitution ... |
| sbimdv 2080 | Deduction substituting bot... |
| sbbidv 2081 | Deduction substituting bot... |
| sban 2082 | Conjunction inside and out... |
| sb3an 2083 | Threefold conjunction insi... |
| spsbe 2084 | Existential generalization... |
| sbequ 2085 | Equality property for subs... |
| sbequi 2086 | An equality theorem for su... |
| sb6 2087 | Alternate definition of su... |
| 2sb6 2088 | Equivalence for double sub... |
| sb1v 2089 | One direction of ~ sb5 , p... |
| sbv 2090 | Substitution for a variabl... |
| sbcom4 2091 | Commutativity law for subs... |
| pm11.07 2092 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2093 | Substitution in an implica... |
| sbievw 2094 | Conversion of implicit sub... |
| sbiedvw 2095 | Conversion of implicit sub... |
| 2sbievw 2096 | Conversion of double impli... |
| sbcom3vv 2097 | Substituting ` y ` for ` x... |
| sbievw2 2098 | ~ sbievw applied twice, av... |
| sbco2vv 2099 | A composition law for subs... |
| equsb3 2100 | Substitution in an equalit... |
| equsb3r 2101 | Substitution applied to th... |
| equsb1v 2102 | Substitution applied to an... |
| nsb 2103 | Any substitution in an alw... |
| sbn1 2104 | One direction of ~ sbn , u... |
| wel 2106 | Extend wff definition to i... |
| ax8v 2108 | Weakened version of ~ ax-8... |
| ax8v1 2109 | First of two weakened vers... |
| ax8v2 2110 | Second of two weakened ver... |
| ax8 2111 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2112 | An identity law for the no... |
| elsb1 2113 | Substitution for the first... |
| cleljust 2114 | When the class variables i... |
| ax9v 2116 | Weakened version of ~ ax-9... |
| ax9v1 2117 | First of two weakened vers... |
| ax9v2 2118 | Second of two weakened ver... |
| ax9 2119 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2120 | An identity law for the no... |
| elequ2g 2121 | A form of ~ elequ2 with a ... |
| elsb2 2122 | Substitution for the secon... |
| ax6dgen 2123 | Tarski's system uses the w... |
| ax10w 2124 | Weak version of ~ ax-10 fr... |
| ax11w 2125 | Weak version of ~ ax-11 fr... |
| ax11dgen 2126 | Degenerate instance of ~ a... |
| ax12wlem 2127 | Lemma for weak version of ... |
| ax12w 2128 | Weak version of ~ ax-12 fr... |
| ax12dgen 2129 | Degenerate instance of ~ a... |
| ax12wdemo 2130 | Example of an application ... |
| ax13w 2131 | Weak version (principal in... |
| ax13dgen1 2132 | Degenerate instance of ~ a... |
| ax13dgen2 2133 | Degenerate instance of ~ a... |
| ax13dgen3 2134 | Degenerate instance of ~ a... |
| ax13dgen4 2135 | Degenerate instance of ~ a... |
| hbn1 2137 | Alias for ~ ax-10 to be us... |
| hbe1 2138 | The setvar ` x ` is not fr... |
| hbe1a 2139 | Dual statement of ~ hbe1 .... |
| nf5-1 2140 | One direction of ~ nf5 can... |
| nf5i 2141 | Deduce that ` x ` is not f... |
| nf5dh 2142 | Deduce that ` x ` is not f... |
| nf5dv 2143 | Apply the definition of no... |
| nfnaew 2144 | All variables are effectiv... |
| nfnaewOLD 2145 | Obsolete version of ~ nfna... |
| nfe1 2146 | The setvar ` x ` is not fr... |
| nfa1 2147 | The setvar ` x ` is not fr... |
| nfna1 2148 | A convenience theorem part... |
| nfia1 2149 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2150 | The setvar ` x ` is not fr... |
| modal5 2151 | The analogue in our predic... |
| nfs1v 2152 | The setvar ` x ` is not fr... |
| alcoms 2154 | Swap quantifiers in an ant... |
| alcom 2155 | Theorem 19.5 of [Margaris]... |
| alrot3 2156 | Theorem *11.21 in [Whitehe... |
| alrot4 2157 | Rotate four universal quan... |
| sbal 2158 | Move universal quantifier ... |
| sbalv 2159 | Quantify with new variable... |
| sbcom2 2160 | Commutativity law for subs... |
| excom 2161 | Theorem 19.11 of [Margaris... |
| excomim 2162 | One direction of Theorem 1... |
| excom13 2163 | Swap 1st and 3rd existenti... |
| exrot3 2164 | Rotate existential quantif... |
| exrot4 2165 | Rotate existential quantif... |
| hbal 2166 | If ` x ` is not free in ` ... |
| hbald 2167 | Deduction form of bound-va... |
| hbsbw 2168 | If ` z ` is not free in ` ... |
| nfa2 2169 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2171 | This is essentially Axiom ... |
| ax12v2 2172 | It is possible to remove a... |
| 19.8a 2173 | If a wff is true, it is tr... |
| 19.8ad 2174 | If a wff is true, it is tr... |
| sp 2175 | Specialization. A univers... |
| spi 2176 | Inference rule of universa... |
| sps 2177 | Generalization of antecede... |
| 2sp 2178 | A double specialization (s... |
| spsd 2179 | Deduction generalizing ant... |
| 19.2g 2180 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2181 | Inference form of ~ 19.21 ... |
| 19.21bbi 2182 | Inference removing two uni... |
| 19.23bi 2183 | Inference form of Theorem ... |
| nexr 2184 | Inference associated with ... |
| qexmid 2185 | Quantified excluded middle... |
| nf5r 2186 | Consequence of the definit... |
| nf5ri 2187 | Consequence of the definit... |
| nf5rd 2188 | Consequence of the definit... |
| spimedv 2189 | Deduction version of ~ spi... |
| spimefv 2190 | Version of ~ spime with a ... |
| nfim1 2191 | A closed form of ~ nfim . ... |
| nfan1 2192 | A closed form of ~ nfan . ... |
| 19.3t 2193 | Closed form of ~ 19.3 and ... |
| 19.3 2194 | A wff may be quantified wi... |
| 19.9d 2195 | A deduction version of one... |
| 19.9t 2196 | Closed form of ~ 19.9 and ... |
| 19.9 2197 | A wff may be existentially... |
| 19.21t 2198 | Closed form of Theorem 19.... |
| 19.21 2199 | Theorem 19.21 of [Margaris... |
| stdpc5 2200 | An axiom scheme of standar... |
| 19.21-2 2201 | Version of ~ 19.21 with tw... |
| 19.23t 2202 | Closed form of Theorem 19.... |
| 19.23 2203 | Theorem 19.23 of [Margaris... |
| alimd 2204 | Deduction form of Theorem ... |
| alrimi 2205 | Inference form of Theorem ... |
| alrimdd 2206 | Deduction form of Theorem ... |
| alrimd 2207 | Deduction form of Theorem ... |
| eximd 2208 | Deduction form of Theorem ... |
| exlimi 2209 | Inference associated with ... |
| exlimd 2210 | Deduction form of Theorem ... |
| exlimimdd 2211 | Existential elimination ru... |
| exlimdd 2212 | Existential elimination ru... |
| nexd 2213 | Deduction for generalizati... |
| albid 2214 | Formula-building rule for ... |
| exbid 2215 | Formula-building rule for ... |
| nfbidf 2216 | An equality theorem for ef... |
| 19.16 2217 | Theorem 19.16 of [Margaris... |
| 19.17 2218 | Theorem 19.17 of [Margaris... |
| 19.27 2219 | Theorem 19.27 of [Margaris... |
| 19.28 2220 | Theorem 19.28 of [Margaris... |
| 19.19 2221 | Theorem 19.19 of [Margaris... |
| 19.36 2222 | Theorem 19.36 of [Margaris... |
| 19.36i 2223 | Inference associated with ... |
| 19.37 2224 | Theorem 19.37 of [Margaris... |
| 19.32 2225 | Theorem 19.32 of [Margaris... |
| 19.31 2226 | Theorem 19.31 of [Margaris... |
| 19.41 2227 | Theorem 19.41 of [Margaris... |
| 19.42 2228 | Theorem 19.42 of [Margaris... |
| 19.44 2229 | Theorem 19.44 of [Margaris... |
| 19.45 2230 | Theorem 19.45 of [Margaris... |
| spimfv 2231 | Specialization, using impl... |
| chvarfv 2232 | Implicit substitution of `... |
| cbv3v2 2233 | Version of ~ cbv3 with two... |
| sbalex 2234 | Equivalence of two ways to... |
| sb4av 2235 | Version of ~ sb4a with a d... |
| sbimd 2236 | Deduction substituting bot... |
| sbbid 2237 | Deduction substituting bot... |
| 2sbbid 2238 | Deduction doubly substitut... |
| sbequ1 2239 | An equality theorem for su... |
| sbequ2 2240 | An equality theorem for su... |
| stdpc7 2241 | One of the two equality ax... |
| sbequ12 2242 | An equality theorem for su... |
| sbequ12r 2243 | An equality theorem for su... |
| sbelx 2244 | Elimination of substitutio... |
| sbequ12a 2245 | An equality theorem for su... |
| sbid 2246 | An identity theorem for su... |
| sbcov 2247 | A composition law for subs... |
| sb6a 2248 | Equivalence for substituti... |
| sbid2vw 2249 | Reverting substitution yie... |
| axc16g 2250 | Generalization of ~ axc16 ... |
| axc16 2251 | Proof of older axiom ~ ax-... |
| axc16gb 2252 | Biconditional strengthenin... |
| axc16nf 2253 | If ~ dtru is false, then t... |
| axc11v 2254 | Version of ~ axc11 with a ... |
| axc11rv 2255 | Version of ~ axc11r with a... |
| drsb2 2256 | Formula-building lemma for... |
| equsalv 2257 | An equivalence related to ... |
| equsexv 2258 | An equivalence related to ... |
| equsexvOLD 2259 | Obsolete version of ~ equs... |
| sbft 2260 | Substitution has no effect... |
| sbf 2261 | Substitution for a variabl... |
| sbf2 2262 | Substitution has no effect... |
| sbh 2263 | Substitution for a variabl... |
| hbs1 2264 | The setvar ` x ` is not fr... |
| nfs1f 2265 | If ` x ` is not free in ` ... |
| sb5 2266 | Alternate definition of su... |
| sb5OLD 2267 | Obsolete version of ~ sb5 ... |
| sb56OLD 2268 | Obsolete version of ~ sbal... |
| equs5av 2269 | A property related to subs... |
| 2sb5 2270 | Equivalence for double sub... |
| sbco4lem 2271 | Lemma for ~ sbco4 . It re... |
| sbco4lemOLD 2272 | Obsolete version of ~ sbco... |
| sbco4 2273 | Two ways of exchanging two... |
| dfsb7 2274 | An alternate definition of... |
| sbn 2275 | Negation inside and outsid... |
| sbex 2276 | Move existential quantifie... |
| nf5 2277 | Alternate definition of ~ ... |
| nf6 2278 | An alternate definition of... |
| nf5d 2279 | Deduce that ` x ` is not f... |
| nf5di 2280 | Since the converse holds b... |
| 19.9h 2281 | A wff may be existentially... |
| 19.21h 2282 | Theorem 19.21 of [Margaris... |
| 19.23h 2283 | Theorem 19.23 of [Margaris... |
| exlimih 2284 | Inference associated with ... |
| exlimdh 2285 | Deduction form of Theorem ... |
| equsalhw 2286 | Version of ~ equsalh with ... |
| equsexhv 2287 | An equivalence related to ... |
| hba1 2288 | The setvar ` x ` is not fr... |
| hbnt 2289 | Closed theorem version of ... |
| hbn 2290 | If ` x ` is not free in ` ... |
| hbnd 2291 | Deduction form of bound-va... |
| hbim1 2292 | A closed form of ~ hbim . ... |
| hbimd 2293 | Deduction form of bound-va... |
| hbim 2294 | If ` x ` is not free in ` ... |
| hban 2295 | If ` x ` is not free in ` ... |
| hb3an 2296 | If ` x ` is not free in ` ... |
| sbi2 2297 | Introduction of implicatio... |
| sbim 2298 | Implication inside and out... |
| sbrim 2299 | Substitution in an implica... |
| sbrimOLD 2300 | Obsolete version of ~ sbri... |
| sblim 2301 | Substitution in an implica... |
| sbor 2302 | Disjunction inside and out... |
| sbbi 2303 | Equivalence inside and out... |
| sblbis 2304 | Introduce left bicondition... |
| sbrbis 2305 | Introduce right biconditio... |
| sbrbif 2306 | Introduce right biconditio... |
| sbiev 2307 | Conversion of implicit sub... |
| sbiedw 2308 | Conversion of implicit sub... |
| axc7 2309 | Show that the original axi... |
| axc7e 2310 | Abbreviated version of ~ a... |
| modal-b 2311 | The analogue in our predic... |
| 19.9ht 2312 | A closed version of ~ 19.9... |
| axc4 2313 | Show that the original axi... |
| axc4i 2314 | Inference version of ~ axc... |
| nfal 2315 | If ` x ` is not free in ` ... |
| nfex 2316 | If ` x ` is not free in ` ... |
| hbex 2317 | If ` x ` is not free in ` ... |
| nfnf 2318 | If ` x ` is not free in ` ... |
| 19.12 2319 | Theorem 19.12 of [Margaris... |
| nfald 2320 | Deduction form of ~ nfal .... |
| nfexd 2321 | If ` x ` is not free in ` ... |
| nfsbv 2322 | If ` z ` is not free in ` ... |
| nfsbvOLD 2323 | Obsolete version of ~ nfsb... |
| hbsbwOLD 2324 | Obsolete version of ~ hbsb... |
| sbco2v 2325 | A composition law for subs... |
| aaan 2326 | Distribute universal quant... |
| aaanOLD 2327 | Obsolete version of ~ aaan... |
| eeor 2328 | Distribute existential qua... |
| eeorOLD 2329 | Obsolete version of ~ eeor... |
| cbv3v 2330 | Rule used to change bound ... |
| cbv1v 2331 | Rule used to change bound ... |
| cbv2w 2332 | Rule used to change bound ... |
| cbvaldw 2333 | Deduction used to change b... |
| cbvexdw 2334 | Deduction used to change b... |
| cbv3hv 2335 | Rule used to change bound ... |
| cbvalv1 2336 | Rule used to change bound ... |
| cbvexv1 2337 | Rule used to change bound ... |
| cbval2v 2338 | Rule used to change bound ... |
| cbvex2v 2339 | Rule used to change bound ... |
| dvelimhw 2340 | Proof of ~ dvelimh without... |
| pm11.53 2341 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2342 | Special case of ~ 19.12 wh... |
| eean 2343 | Distribute existential qua... |
| eeanv 2344 | Distribute a pair of exist... |
| eeeanv 2345 | Distribute three existenti... |
| ee4anv 2346 | Distribute two pairs of ex... |
| sb8v 2347 | Substitution of variable i... |
| sb8f 2348 | Substitution of variable i... |
| sb8fOLD 2349 | Obsolete version of ~ sb8f... |
| sb8ef 2350 | Substitution of variable i... |
| 2sb8ef 2351 | An equivalent expression f... |
| sb6rfv 2352 | Reversed substitution. Ve... |
| sbnf2 2353 | Two ways of expressing " `... |
| exsb 2354 | An equivalent expression f... |
| 2exsb 2355 | An equivalent expression f... |
| sbbib 2356 | Reversal of substitution. ... |
| sbbibvv 2357 | Reversal of substitution. ... |
| cbvsbv 2358 | Change the bound variable ... |
| cbvsbvf 2359 | Change the bound variable ... |
| cleljustALT 2360 | Alternate proof of ~ clelj... |
| cleljustALT2 2361 | Alternate proof of ~ clelj... |
| equs5aALT 2362 | Alternate proof of ~ equs5... |
| equs5eALT 2363 | Alternate proof of ~ equs5... |
| axc11r 2364 | Same as ~ axc11 but with r... |
| dral1v 2365 | Formula-building lemma for... |
| dral1vOLD 2366 | Obsolete version of ~ dral... |
| drex1v 2367 | Formula-building lemma for... |
| drnf1v 2368 | Formula-building lemma for... |
| drnf1vOLD 2369 | Obsolete version of ~ drnf... |
| ax13v 2371 | A weaker version of ~ ax-1... |
| ax13lem1 2372 | A version of ~ ax13v with ... |
| ax13 2373 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2374 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2375 | An equation between setvar... |
| dveeq2 2376 | Quantifier introduction wh... |
| nfeqf1 2377 | An equation between setvar... |
| dveeq1 2378 | Quantifier introduction wh... |
| nfeqf 2379 | A variable is effectively ... |
| axc9 2380 | Derive set.mm's original ~... |
| ax6e 2381 | At least one individual ex... |
| ax6 2382 | Theorem showing that ~ ax-... |
| axc10 2383 | Show that the original axi... |
| spimt 2384 | Closed theorem form of ~ s... |
| spim 2385 | Specialization, using impl... |
| spimed 2386 | Deduction version of ~ spi... |
| spime 2387 | Existential introduction, ... |
| spimv 2388 | A version of ~ spim with a... |
| spimvALT 2389 | Alternate proof of ~ spimv... |
| spimev 2390 | Distinct-variable version ... |
| spv 2391 | Specialization, using impl... |
| spei 2392 | Inference from existential... |
| chvar 2393 | Implicit substitution of `... |
| chvarv 2394 | Implicit substitution of `... |
| cbv3 2395 | Rule used to change bound ... |
| cbval 2396 | Rule used to change bound ... |
| cbvex 2397 | Rule used to change bound ... |
| cbvalv 2398 | Rule used to change bound ... |
| cbvexv 2399 | Rule used to change bound ... |
| cbv1 2400 | Rule used to change bound ... |
| cbv2 2401 | Rule used to change bound ... |
| cbv3h 2402 | Rule used to change bound ... |
| cbv1h 2403 | Rule used to change bound ... |
| cbv2h 2404 | Rule used to change bound ... |
| cbvald 2405 | Deduction used to change b... |
| cbvexd 2406 | Deduction used to change b... |
| cbvaldva 2407 | Rule used to change the bo... |
| cbvexdva 2408 | Rule used to change the bo... |
| cbval2 2409 | Rule used to change bound ... |
| cbvex2 2410 | Rule used to change bound ... |
| cbval2vv 2411 | Rule used to change bound ... |
| cbvex2vv 2412 | Rule used to change bound ... |
| cbvex4v 2413 | Rule used to change bound ... |
| equs4 2414 | Lemma used in proofs of im... |
| equsal 2415 | An equivalence related to ... |
| equsex 2416 | An equivalence related to ... |
| equsexALT 2417 | Alternate proof of ~ equse... |
| equsalh 2418 | An equivalence related to ... |
| equsexh 2419 | An equivalence related to ... |
| axc15 2420 | Derivation of set.mm's ori... |
| ax12 2421 | Rederivation of Axiom ~ ax... |
| ax12b 2422 | A bidirectional version of... |
| ax13ALT 2423 | Alternate proof of ~ ax13 ... |
| axc11n 2424 | Derive set.mm's original ~... |
| aecom 2425 | Commutation law for identi... |
| aecoms 2426 | A commutation rule for ide... |
| naecoms 2427 | A commutation rule for dis... |
| axc11 2428 | Show that ~ ax-c11 can be ... |
| hbae 2429 | All variables are effectiv... |
| hbnae 2430 | All variables are effectiv... |
| nfae 2431 | All variables are effectiv... |
| nfnae 2432 | All variables are effectiv... |
| hbnaes 2433 | Rule that applies ~ hbnae ... |
| axc16i 2434 | Inference with ~ axc16 as ... |
| axc16nfALT 2435 | Alternate proof of ~ axc16... |
| dral2 2436 | Formula-building lemma for... |
| dral1 2437 | Formula-building lemma for... |
| dral1ALT 2438 | Alternate proof of ~ dral1... |
| drex1 2439 | Formula-building lemma for... |
| drex2 2440 | Formula-building lemma for... |
| drnf1 2441 | Formula-building lemma for... |
| drnf2 2442 | Formula-building lemma for... |
| nfald2 2443 | Variation on ~ nfald which... |
| nfexd2 2444 | Variation on ~ nfexd which... |
| exdistrf 2445 | Distribution of existentia... |
| dvelimf 2446 | Version of ~ dvelimv witho... |
| dvelimdf 2447 | Deduction form of ~ dvelim... |
| dvelimh 2448 | Version of ~ dvelim withou... |
| dvelim 2449 | This theorem can be used t... |
| dvelimv 2450 | Similar to ~ dvelim with f... |
| dvelimnf 2451 | Version of ~ dvelim using ... |
| dveeq2ALT 2452 | Alternate proof of ~ dveeq... |
| equvini 2453 | A variable introduction la... |
| equvel 2454 | A variable elimination law... |
| equs5a 2455 | A property related to subs... |
| equs5e 2456 | A property related to subs... |
| equs45f 2457 | Two ways of expressing sub... |
| equs5 2458 | Lemma used in proofs of su... |
| dveel1 2459 | Quantifier introduction wh... |
| dveel2 2460 | Quantifier introduction wh... |
| axc14 2461 | Axiom ~ ax-c14 is redundan... |
| sb6x 2462 | Equivalence involving subs... |
| sbequ5 2463 | Substitution does not chan... |
| sbequ6 2464 | Substitution does not chan... |
| sb5rf 2465 | Reversed substitution. Us... |
| sb6rf 2466 | Reversed substitution. Fo... |
| ax12vALT 2467 | Alternate proof of ~ ax12v... |
| 2ax6elem 2468 | We can always find values ... |
| 2ax6e 2469 | We can always find values ... |
| 2sb5rf 2470 | Reversed double substituti... |
| 2sb6rf 2471 | Reversed double substituti... |
| sbel2x 2472 | Elimination of double subs... |
| sb4b 2473 | Simplified definition of s... |
| sb3b 2474 | Simplified definition of s... |
| sb3 2475 | One direction of a simplif... |
| sb1 2476 | One direction of a simplif... |
| sb2 2477 | One direction of a simplif... |
| sb4a 2478 | A version of one implicati... |
| dfsb1 2479 | Alternate definition of su... |
| hbsb2 2480 | Bound-variable hypothesis ... |
| nfsb2 2481 | Bound-variable hypothesis ... |
| hbsb2a 2482 | Special case of a bound-va... |
| sb4e 2483 | One direction of a simplif... |
| hbsb2e 2484 | Special case of a bound-va... |
| hbsb3 2485 | If ` y ` is not free in ` ... |
| nfs1 2486 | If ` y ` is not free in ` ... |
| axc16ALT 2487 | Alternate proof of ~ axc16... |
| axc16gALT 2488 | Alternate proof of ~ axc16... |
| equsb1 2489 | Substitution applied to an... |
| equsb2 2490 | Substitution applied to an... |
| dfsb2 2491 | An alternate definition of... |
| dfsb3 2492 | An alternate definition of... |
| drsb1 2493 | Formula-building lemma for... |
| sb2ae 2494 | In the case of two success... |
| sb6f 2495 | Equivalence for substituti... |
| sb5f 2496 | Equivalence for substituti... |
| nfsb4t 2497 | A variable not free in a p... |
| nfsb4 2498 | A variable not free in a p... |
| sbequ8 2499 | Elimination of equality fr... |
| sbie 2500 | Conversion of implicit sub... |
| sbied 2501 | Conversion of implicit sub... |
| sbiedv 2502 | Conversion of implicit sub... |
| 2sbiev 2503 | Conversion of double impli... |
| sbcom3 2504 | Substituting ` y ` for ` x... |
| sbco 2505 | A composition law for subs... |
| sbid2 2506 | An identity law for substi... |
| sbid2v 2507 | An identity law for substi... |
| sbidm 2508 | An idempotent law for subs... |
| sbco2 2509 | A composition law for subs... |
| sbco2d 2510 | A composition law for subs... |
| sbco3 2511 | A composition law for subs... |
| sbcom 2512 | A commutativity law for su... |
| sbtrt 2513 | Partially closed form of ~... |
| sbtr 2514 | A partial converse to ~ sb... |
| sb8 2515 | Substitution of variable i... |
| sb8e 2516 | Substitution of variable i... |
| sb9 2517 | Commutation of quantificat... |
| sb9i 2518 | Commutation of quantificat... |
| sbhb 2519 | Two ways of expressing " `... |
| nfsbd 2520 | Deduction version of ~ nfs... |
| nfsb 2521 | If ` z ` is not free in ` ... |
| hbsb 2522 | If ` z ` is not free in ` ... |
| sb7f 2523 | This version of ~ dfsb7 do... |
| sb7h 2524 | This version of ~ dfsb7 do... |
| sb10f 2525 | Hao Wang's identity axiom ... |
| sbal1 2526 | Check out ~ sbal for a ver... |
| sbal2 2527 | Move quantifier in and out... |
| 2sb8e 2528 | An equivalent expression f... |
| dfmoeu 2529 | An elementary proof of ~ m... |
| dfeumo 2530 | An elementary proof showin... |
| mojust 2532 | Soundness justification th... |
| nexmo 2534 | Nonexistence implies uniqu... |
| exmo 2535 | Any proposition holds for ... |
| moabs 2536 | Absorption of existence co... |
| moim 2537 | The at-most-one quantifier... |
| moimi 2538 | The at-most-one quantifier... |
| moimdv 2539 | The at-most-one quantifier... |
| mobi 2540 | Equivalence theorem for th... |
| mobii 2541 | Formula-building rule for ... |
| mobidv 2542 | Formula-building rule for ... |
| mobid 2543 | Formula-building rule for ... |
| moa1 2544 | If an implication holds fo... |
| moan 2545 | "At most one" is still the... |
| moani 2546 | "At most one" is still tru... |
| moor 2547 | "At most one" is still the... |
| mooran1 2548 | "At most one" imports disj... |
| mooran2 2549 | "At most one" exports disj... |
| nfmo1 2550 | Bound-variable hypothesis ... |
| nfmod2 2551 | Bound-variable hypothesis ... |
| nfmodv 2552 | Bound-variable hypothesis ... |
| nfmov 2553 | Bound-variable hypothesis ... |
| nfmod 2554 | Bound-variable hypothesis ... |
| nfmo 2555 | Bound-variable hypothesis ... |
| mof 2556 | Version of ~ df-mo with di... |
| mo3 2557 | Alternate definition of th... |
| mo 2558 | Equivalent definitions of ... |
| mo4 2559 | At-most-one quantifier exp... |
| mo4f 2560 | At-most-one quantifier exp... |
| eu3v 2563 | An alternate way to expres... |
| eujust 2564 | Soundness justification th... |
| eujustALT 2565 | Alternate proof of ~ eujus... |
| eu6lem 2566 | Lemma of ~ eu6im . A diss... |
| eu6 2567 | Alternate definition of th... |
| eu6im 2568 | One direction of ~ eu6 nee... |
| euf 2569 | Version of ~ eu6 with disj... |
| euex 2570 | Existential uniqueness imp... |
| eumo 2571 | Existential uniqueness imp... |
| eumoi 2572 | Uniqueness inferred from e... |
| exmoeub 2573 | Existence implies that uni... |
| exmoeu 2574 | Existence is equivalent to... |
| moeuex 2575 | Uniqueness implies that ex... |
| moeu 2576 | Uniqueness is equivalent t... |
| eubi 2577 | Equivalence theorem for th... |
| eubii 2578 | Introduce unique existenti... |
| eubidv 2579 | Formula-building rule for ... |
| eubid 2580 | Formula-building rule for ... |
| nfeu1 2581 | Bound-variable hypothesis ... |
| nfeu1ALT 2582 | Alternate proof of ~ nfeu1... |
| nfeud2 2583 | Bound-variable hypothesis ... |
| nfeudw 2584 | Bound-variable hypothesis ... |
| nfeud 2585 | Bound-variable hypothesis ... |
| nfeuw 2586 | Bound-variable hypothesis ... |
| nfeu 2587 | Bound-variable hypothesis ... |
| dfeu 2588 | Rederive ~ df-eu from the ... |
| dfmo 2589 | Rederive ~ df-mo from the ... |
| euequ 2590 | There exists a unique set ... |
| sb8eulem 2591 | Lemma. Factor out the com... |
| sb8euv 2592 | Variable substitution in u... |
| sb8eu 2593 | Variable substitution in u... |
| sb8mo 2594 | Variable substitution for ... |
| cbvmovw 2595 | Change bound variable. Us... |
| cbvmow 2596 | Rule used to change bound ... |
| cbvmowOLD 2597 | Obsolete version of ~ cbvm... |
| cbvmo 2598 | Rule used to change bound ... |
| cbveuvw 2599 | Change bound variable. Us... |
| cbveuw 2600 | Version of ~ cbveu with a ... |
| cbveuwOLD 2601 | Obsolete version of ~ cbve... |
| cbveu 2602 | Rule used to change bound ... |
| cbveuALT 2603 | Alternative proof of ~ cbv... |
| eu2 2604 | An alternate way of defini... |
| eu1 2605 | An alternate way to expres... |
| euor 2606 | Introduce a disjunct into ... |
| euorv 2607 | Introduce a disjunct into ... |
| euor2 2608 | Introduce or eliminate a d... |
| sbmo 2609 | Substitution into an at-mo... |
| eu4 2610 | Uniqueness using implicit ... |
| euimmo 2611 | Existential uniqueness imp... |
| euim 2612 | Add unique existential qua... |
| moanimlem 2613 | Factor out the common proo... |
| moanimv 2614 | Introduction of a conjunct... |
| moanim 2615 | Introduction of a conjunct... |
| euan 2616 | Introduction of a conjunct... |
| moanmo 2617 | Nested at-most-one quantif... |
| moaneu 2618 | Nested at-most-one and uni... |
| euanv 2619 | Introduction of a conjunct... |
| mopick 2620 | "At most one" picks a vari... |
| moexexlem 2621 | Factor out the proof skele... |
| 2moexv 2622 | Double quantification with... |
| moexexvw 2623 | "At most one" double quant... |
| 2moswapv 2624 | A condition allowing to sw... |
| 2euswapv 2625 | A condition allowing to sw... |
| 2euexv 2626 | Double quantification with... |
| 2exeuv 2627 | Double existential uniquen... |
| eupick 2628 | Existential uniqueness "pi... |
| eupicka 2629 | Version of ~ eupick with c... |
| eupickb 2630 | Existential uniqueness "pi... |
| eupickbi 2631 | Theorem *14.26 in [Whitehe... |
| mopick2 2632 | "At most one" can show the... |
| moexex 2633 | "At most one" double quant... |
| moexexv 2634 | "At most one" double quant... |
| 2moex 2635 | Double quantification with... |
| 2euex 2636 | Double quantification with... |
| 2eumo 2637 | Nested unique existential ... |
| 2eu2ex 2638 | Double existential uniquen... |
| 2moswap 2639 | A condition allowing to sw... |
| 2euswap 2640 | A condition allowing to sw... |
| 2exeu 2641 | Double existential uniquen... |
| 2mo2 2642 | Two ways of expressing "th... |
| 2mo 2643 | Two ways of expressing "th... |
| 2mos 2644 | Double "there exists at mo... |
| 2eu1 2645 | Double existential uniquen... |
| 2eu1v 2646 | Double existential uniquen... |
| 2eu2 2647 | Double existential uniquen... |
| 2eu3 2648 | Double existential uniquen... |
| 2eu4 2649 | This theorem provides us w... |
| 2eu5 2650 | An alternate definition of... |
| 2eu6 2651 | Two equivalent expressions... |
| 2eu7 2652 | Two equivalent expressions... |
| 2eu8 2653 | Two equivalent expressions... |
| euae 2654 | Two ways to express "exact... |
| exists1 2655 | Two ways to express "exact... |
| exists2 2656 | A condition implying that ... |
| barbara 2657 | "Barbara", one of the fund... |
| celarent 2658 | "Celarent", one of the syl... |
| darii 2659 | "Darii", one of the syllog... |
| dariiALT 2660 | Alternate proof of ~ darii... |
| ferio 2661 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2662 | Lemma for ~ barbari and th... |
| barbari 2663 | "Barbari", one of the syll... |
| barbariALT 2664 | Alternate proof of ~ barba... |
| celaront 2665 | "Celaront", one of the syl... |
| cesare 2666 | "Cesare", one of the syllo... |
| camestres 2667 | "Camestres", one of the sy... |
| festino 2668 | "Festino", one of the syll... |
| festinoALT 2669 | Alternate proof of ~ festi... |
| baroco 2670 | "Baroco", one of the syllo... |
| barocoALT 2671 | Alternate proof of ~ festi... |
| cesaro 2672 | "Cesaro", one of the syllo... |
| camestros 2673 | "Camestros", one of the sy... |
| datisi 2674 | "Datisi", one of the syllo... |
| disamis 2675 | "Disamis", one of the syll... |
| ferison 2676 | "Ferison", one of the syll... |
| bocardo 2677 | "Bocardo", one of the syll... |
| darapti 2678 | "Darapti", one of the syll... |
| daraptiALT 2679 | Alternate proof of ~ darap... |
| felapton 2680 | "Felapton", one of the syl... |
| calemes 2681 | "Calemes", one of the syll... |
| dimatis 2682 | "Dimatis", one of the syll... |
| fresison 2683 | "Fresison", one of the syl... |
| calemos 2684 | "Calemos", one of the syll... |
| fesapo 2685 | "Fesapo", one of the syllo... |
| bamalip 2686 | "Bamalip", one of the syll... |
| axia1 2687 | Left 'and' elimination (in... |
| axia2 2688 | Right 'and' elimination (i... |
| axia3 2689 | 'And' introduction (intuit... |
| axin1 2690 | 'Not' introduction (intuit... |
| axin2 2691 | 'Not' elimination (intuiti... |
| axio 2692 | Definition of 'or' (intuit... |
| axi4 2693 | Specialization (intuitioni... |
| axi5r 2694 | Converse of ~ axc4 (intuit... |
| axial 2695 | The setvar ` x ` is not fr... |
| axie1 2696 | The setvar ` x ` is not fr... |
| axie2 2697 | A key property of existent... |
| axi9 2698 | Axiom of existence (intuit... |
| axi10 2699 | Axiom of Quantifier Substi... |
| axi12 2700 | Axiom of Quantifier Introd... |
| axbnd 2701 | Axiom of Bundling (intuiti... |
| axexte 2703 | The axiom of extensionalit... |
| axextg 2704 | A generalization of the ax... |
| axextb 2705 | A bidirectional version of... |
| axextmo 2706 | There exists at most one s... |
| nulmo 2707 | There exists at most one e... |
| eleq1ab 2710 | Extension (in the sense of... |
| cleljustab 2711 | Extension of ~ cleljust fr... |
| abid 2712 | Simplification of class ab... |
| vexwt 2713 | A standard theorem of pred... |
| vexw 2714 | If ` ph ` is a theorem, th... |
| vextru 2715 | Every setvar is a member o... |
| nfsab1 2716 | Bound-variable hypothesis ... |
| hbab1 2717 | Bound-variable hypothesis ... |
| hbab1OLD 2718 | Obsolete version of ~ hbab... |
| hbab 2719 | Bound-variable hypothesis ... |
| hbabg 2720 | Bound-variable hypothesis ... |
| nfsab 2721 | Bound-variable hypothesis ... |
| nfsabg 2722 | Bound-variable hypothesis ... |
| dfcleq 2724 | The defining characterizat... |
| cvjust 2725 | Every set is a class. Pro... |
| ax9ALT 2726 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2727 | A weaker version of ~ eleq... |
| eqriv 2728 | Infer equality of classes ... |
| eqrdv 2729 | Deduce equality of classes... |
| eqrdav 2730 | Deduce equality of classes... |
| eqid 2731 | Law of identity (reflexivi... |
| eqidd 2732 | Class identity law with an... |
| eqeq1d 2733 | Deduction from equality to... |
| eqeq1dALT 2734 | Alternate proof of ~ eqeq1... |
| eqeq1 2735 | Equality implies equivalen... |
| eqeq1i 2736 | Inference from equality to... |
| eqcomd 2737 | Deduction from commutative... |
| eqcom 2738 | Commutative law for class ... |
| eqcoms 2739 | Inference applying commuta... |
| eqcomi 2740 | Inference from commutative... |
| neqcomd 2741 | Commute an inequality. (C... |
| eqeq2d 2742 | Deduction from equality to... |
| eqeq2 2743 | Equality implies equivalen... |
| eqeq2i 2744 | Inference from equality to... |
| eqeqan12d 2745 | A useful inference for sub... |
| eqeqan12rd 2746 | A useful inference for sub... |
| eqeq12d 2747 | A useful inference for sub... |
| eqeq12 2748 | Equality relationship amon... |
| eqeq12i 2749 | A useful inference for sub... |
| eqeq12OLD 2750 | Obsolete version of ~ eqeq... |
| eqeq12dOLD 2751 | Obsolete version of ~ eqeq... |
| eqeqan12dOLD 2752 | Obsolete version of ~ eqeq... |
| eqeqan12dALT 2753 | Alternate proof of ~ eqeqa... |
| eqtr 2754 | Transitive law for class e... |
| eqtr2 2755 | A transitive law for class... |
| eqtr2OLD 2756 | Obsolete version of eqtr2 ... |
| eqtr3 2757 | A transitive law for class... |
| eqtr3OLD 2758 | Obsolete version of ~ eqtr... |
| eqtri 2759 | An equality transitivity i... |
| eqtr2i 2760 | An equality transitivity i... |
| eqtr3i 2761 | An equality transitivity i... |
| eqtr4i 2762 | An equality transitivity i... |
| 3eqtri 2763 | An inference from three ch... |
| 3eqtrri 2764 | An inference from three ch... |
| 3eqtr2i 2765 | An inference from three ch... |
| 3eqtr2ri 2766 | An inference from three ch... |
| 3eqtr3i 2767 | An inference from three ch... |
| 3eqtr3ri 2768 | An inference from three ch... |
| 3eqtr4i 2769 | An inference from three ch... |
| 3eqtr4ri 2770 | An inference from three ch... |
| eqtrd 2771 | An equality transitivity d... |
| eqtr2d 2772 | An equality transitivity d... |
| eqtr3d 2773 | An equality transitivity e... |
| eqtr4d 2774 | An equality transitivity e... |
| 3eqtrd 2775 | A deduction from three cha... |
| 3eqtrrd 2776 | A deduction from three cha... |
| 3eqtr2d 2777 | A deduction from three cha... |
| 3eqtr2rd 2778 | A deduction from three cha... |
| 3eqtr3d 2779 | A deduction from three cha... |
| 3eqtr3rd 2780 | A deduction from three cha... |
| 3eqtr4d 2781 | A deduction from three cha... |
| 3eqtr4rd 2782 | A deduction from three cha... |
| eqtrid 2783 | An equality transitivity d... |
| eqtr2id 2784 | An equality transitivity d... |
| eqtr3id 2785 | An equality transitivity d... |
| eqtr3di 2786 | An equality transitivity d... |
| eqtrdi 2787 | An equality transitivity d... |
| eqtr2di 2788 | An equality transitivity d... |
| eqtr4di 2789 | An equality transitivity d... |
| eqtr4id 2790 | An equality transitivity d... |
| sylan9eq 2791 | An equality transitivity d... |
| sylan9req 2792 | An equality transitivity d... |
| sylan9eqr 2793 | An equality transitivity d... |
| 3eqtr3g 2794 | A chained equality inferen... |
| 3eqtr3a 2795 | A chained equality inferen... |
| 3eqtr4g 2796 | A chained equality inferen... |
| 3eqtr4a 2797 | A chained equality inferen... |
| eq2tri 2798 | A compound transitive infe... |
| abbi 2799 | Equivalent formulas yield ... |
| abbidv 2800 | Equivalent wff's yield equ... |
| abbii 2801 | Equivalent wff's yield equ... |
| abbid 2802 | Equivalent wff's yield equ... |
| abbib 2803 | Equal class abstractions r... |
| cbvabv 2804 | Rule used to change bound ... |
| cbvabw 2805 | Rule used to change bound ... |
| cbvabwOLD 2806 | Obsolete version of ~ cbva... |
| cbvab 2807 | Rule used to change bound ... |
| eqabbw 2808 | Version of ~ eqabb using i... |
| dfclel 2810 | Characterization of the el... |
| elex2 2811 | If a class contains anothe... |
| issetlem 2812 | Lemma for ~ elisset and ~ ... |
| elissetv 2813 | An element of a class exis... |
| elisset 2814 | An element of a class exis... |
| eleq1w 2815 | Weaker version of ~ eleq1 ... |
| eleq2w 2816 | Weaker version of ~ eleq2 ... |
| eleq1d 2817 | Deduction from equality to... |
| eleq2d 2818 | Deduction from equality to... |
| eleq2dALT 2819 | Alternate proof of ~ eleq2... |
| eleq1 2820 | Equality implies equivalen... |
| eleq2 2821 | Equality implies equivalen... |
| eleq12 2822 | Equality implies equivalen... |
| eleq1i 2823 | Inference from equality to... |
| eleq2i 2824 | Inference from equality to... |
| eleq12i 2825 | Inference from equality to... |
| eleq12d 2826 | Deduction from equality to... |
| eleq1a 2827 | A transitive-type law rela... |
| eqeltri 2828 | Substitution of equal clas... |
| eqeltrri 2829 | Substitution of equal clas... |
| eleqtri 2830 | Substitution of equal clas... |
| eleqtrri 2831 | Substitution of equal clas... |
| eqeltrd 2832 | Substitution of equal clas... |
| eqeltrrd 2833 | Deduction that substitutes... |
| eleqtrd 2834 | Deduction that substitutes... |
| eleqtrrd 2835 | Deduction that substitutes... |
| eqeltrid 2836 | A membership and equality ... |
| eqeltrrid 2837 | A membership and equality ... |
| eleqtrid 2838 | A membership and equality ... |
| eleqtrrid 2839 | A membership and equality ... |
| eqeltrdi 2840 | A membership and equality ... |
| eqeltrrdi 2841 | A membership and equality ... |
| eleqtrdi 2842 | A membership and equality ... |
| eleqtrrdi 2843 | A membership and equality ... |
| 3eltr3i 2844 | Substitution of equal clas... |
| 3eltr4i 2845 | Substitution of equal clas... |
| 3eltr3d 2846 | Substitution of equal clas... |
| 3eltr4d 2847 | Substitution of equal clas... |
| 3eltr3g 2848 | Substitution of equal clas... |
| 3eltr4g 2849 | Substitution of equal clas... |
| eleq2s 2850 | Substitution of equal clas... |
| eqneltri 2851 | If a class is not an eleme... |
| eqneltrd 2852 | If a class is not an eleme... |
| eqneltrrd 2853 | If a class is not an eleme... |
| neleqtrd 2854 | If a class is not an eleme... |
| neleqtrrd 2855 | If a class is not an eleme... |
| nelneq 2856 | A way of showing two class... |
| nelneq2 2857 | A way of showing two class... |
| eqsb1 2858 | Substitution for the left-... |
| clelsb1 2859 | Substitution for the first... |
| clelsb2 2860 | Substitution for the secon... |
| clelsb2OLD 2861 | Obsolete version of ~ clel... |
| cleqh 2862 | Establish equality between... |
| hbxfreq 2863 | A utility lemma to transfe... |
| hblem 2864 | Change the free variable o... |
| hblemg 2865 | Change the free variable o... |
| eqabdv 2866 | Deduction from a wff to a ... |
| eqabcdv 2867 | Deduction from a wff to a ... |
| eqabi 2868 | Equality of a class variab... |
| abid1 2869 | Every class is equal to a ... |
| abid2 2870 | A simplification of class ... |
| eqab 2871 | One direction of ~ eqabb i... |
| eqabb 2872 | Equality of a class variab... |
| eqabbOLD 2873 | Obsolete version of ~ eqab... |
| eqabcb 2874 | Equality of a class variab... |
| eqabrd 2875 | Equality of a class variab... |
| eqabri 2876 | Equality of a class variab... |
| eqabcri 2877 | Equality of a class variab... |
| clelab 2878 | Membership of a class vari... |
| clelabOLD 2879 | Obsolete version of ~ clel... |
| clabel 2880 | Membership of a class abst... |
| sbab 2881 | The right-hand side of the... |
| nfcjust 2883 | Justification theorem for ... |
| nfci 2885 | Deduce that a class ` A ` ... |
| nfcii 2886 | Deduce that a class ` A ` ... |
| nfcr 2887 | Consequence of the not-fre... |
| nfcrALT 2888 | Alternate version of ~ nfc... |
| nfcri 2889 | Consequence of the not-fre... |
| nfcd 2890 | Deduce that a class ` A ` ... |
| nfcrd 2891 | Consequence of the not-fre... |
| nfcriOLD 2892 | Obsolete version of ~ nfcr... |
| nfcriOLDOLD 2893 | Obsolete version of ~ nfcr... |
| nfcrii 2894 | Consequence of the not-fre... |
| nfcriiOLD 2895 | Obsolete version of ~ nfcr... |
| nfcriOLDOLDOLD 2896 | Obsolete version of ~ nfcr... |
| nfceqdf 2897 | An equality theorem for ef... |
| nfceqdfOLD 2898 | Obsolete version of ~ nfce... |
| nfceqi 2899 | Equality theorem for class... |
| nfcxfr 2900 | A utility lemma to transfe... |
| nfcxfrd 2901 | A utility lemma to transfe... |
| nfcv 2902 | If ` x ` is disjoint from ... |
| nfcvd 2903 | If ` x ` is disjoint from ... |
| nfab1 2904 | Bound-variable hypothesis ... |
| nfnfc1 2905 | The setvar ` x ` is bound ... |
| clelsb1fw 2906 | Substitution for the first... |
| clelsb1f 2907 | Substitution for the first... |
| nfab 2908 | Bound-variable hypothesis ... |
| nfabg 2909 | Bound-variable hypothesis ... |
| nfaba1 2910 | Bound-variable hypothesis ... |
| nfaba1g 2911 | Bound-variable hypothesis ... |
| nfeqd 2912 | Hypothesis builder for equ... |
| nfeld 2913 | Hypothesis builder for ele... |
| nfnfc 2914 | Hypothesis builder for ` F... |
| nfeq 2915 | Hypothesis builder for equ... |
| nfel 2916 | Hypothesis builder for ele... |
| nfeq1 2917 | Hypothesis builder for equ... |
| nfel1 2918 | Hypothesis builder for ele... |
| nfeq2 2919 | Hypothesis builder for equ... |
| nfel2 2920 | Hypothesis builder for ele... |
| drnfc1 2921 | Formula-building lemma for... |
| drnfc1OLD 2922 | Obsolete version of ~ drnf... |
| drnfc2 2923 | Formula-building lemma for... |
| drnfc2OLD 2924 | Obsolete version of ~ drnf... |
| nfabdw 2925 | Bound-variable hypothesis ... |
| nfabdwOLD 2926 | Obsolete version of ~ nfab... |
| nfabd 2927 | Bound-variable hypothesis ... |
| nfabd2 2928 | Bound-variable hypothesis ... |
| dvelimdc 2929 | Deduction form of ~ dvelim... |
| dvelimc 2930 | Version of ~ dvelim for cl... |
| nfcvf 2931 | If ` x ` and ` y ` are dis... |
| nfcvf2 2932 | If ` x ` and ` y ` are dis... |
| cleqf 2933 | Establish equality between... |
| eqabf 2934 | Equality of a class variab... |
| abid2f 2935 | A simplification of class ... |
| abid2fOLD 2936 | Obsolete version of ~ abid... |
| sbabel 2937 | Theorem to move a substitu... |
| sbabelOLD 2938 | Obsolete version of ~ sbab... |
| neii 2941 | Inference associated with ... |
| neir 2942 | Inference associated with ... |
| nne 2943 | Negation of inequality. (... |
| neneqd 2944 | Deduction eliminating ineq... |
| neneq 2945 | From inequality to non-equ... |
| neqned 2946 | If it is not the case that... |
| neqne 2947 | From non-equality to inequ... |
| neirr 2948 | No class is unequal to its... |
| exmidne 2949 | Excluded middle with equal... |
| eqneqall 2950 | A contradiction concerning... |
| nonconne 2951 | Law of noncontradiction wi... |
| necon3ad 2952 | Contrapositive law deducti... |
| necon3bd 2953 | Contrapositive law deducti... |
| necon2ad 2954 | Contrapositive inference f... |
| necon2bd 2955 | Contrapositive inference f... |
| necon1ad 2956 | Contrapositive deduction f... |
| necon1bd 2957 | Contrapositive deduction f... |
| necon4ad 2958 | Contrapositive inference f... |
| necon4bd 2959 | Contrapositive inference f... |
| necon3d 2960 | Contrapositive law deducti... |
| necon1d 2961 | Contrapositive law deducti... |
| necon2d 2962 | Contrapositive inference f... |
| necon4d 2963 | Contrapositive inference f... |
| necon3ai 2964 | Contrapositive inference f... |
| necon3aiOLD 2965 | Obsolete version of ~ neco... |
| necon3bi 2966 | Contrapositive inference f... |
| necon1ai 2967 | Contrapositive inference f... |
| necon1bi 2968 | Contrapositive inference f... |
| necon2ai 2969 | Contrapositive inference f... |
| necon2bi 2970 | Contrapositive inference f... |
| necon4ai 2971 | Contrapositive inference f... |
| necon3i 2972 | Contrapositive inference f... |
| necon1i 2973 | Contrapositive inference f... |
| necon2i 2974 | Contrapositive inference f... |
| necon4i 2975 | Contrapositive inference f... |
| necon3abid 2976 | Deduction from equality to... |
| necon3bbid 2977 | Deduction from equality to... |
| necon1abid 2978 | Contrapositive deduction f... |
| necon1bbid 2979 | Contrapositive inference f... |
| necon4abid 2980 | Contrapositive law deducti... |
| necon4bbid 2981 | Contrapositive law deducti... |
| necon2abid 2982 | Contrapositive deduction f... |
| necon2bbid 2983 | Contrapositive deduction f... |
| necon3bid 2984 | Deduction from equality to... |
| necon4bid 2985 | Contrapositive law deducti... |
| necon3abii 2986 | Deduction from equality to... |
| necon3bbii 2987 | Deduction from equality to... |
| necon1abii 2988 | Contrapositive inference f... |
| necon1bbii 2989 | Contrapositive inference f... |
| necon2abii 2990 | Contrapositive inference f... |
| necon2bbii 2991 | Contrapositive inference f... |
| necon3bii 2992 | Inference from equality to... |
| necom 2993 | Commutation of inequality.... |
| necomi 2994 | Inference from commutative... |
| necomd 2995 | Deduction from commutative... |
| nesym 2996 | Characterization of inequa... |
| nesymi 2997 | Inference associated with ... |
| nesymir 2998 | Inference associated with ... |
| neeq1d 2999 | Deduction for inequality. ... |
| neeq2d 3000 | Deduction for inequality. ... |
| neeq12d 3001 | Deduction for inequality. ... |
| neeq1 3002 | Equality theorem for inequ... |
| neeq2 3003 | Equality theorem for inequ... |
| neeq1i 3004 | Inference for inequality. ... |
| neeq2i 3005 | Inference for inequality. ... |
| neeq12i 3006 | Inference for inequality. ... |
| eqnetrd 3007 | Substitution of equal clas... |
| eqnetrrd 3008 | Substitution of equal clas... |
| neeqtrd 3009 | Substitution of equal clas... |
| eqnetri 3010 | Substitution of equal clas... |
| eqnetrri 3011 | Substitution of equal clas... |
| neeqtri 3012 | Substitution of equal clas... |
| neeqtrri 3013 | Substitution of equal clas... |
| neeqtrrd 3014 | Substitution of equal clas... |
| eqnetrrid 3015 | A chained equality inferen... |
| 3netr3d 3016 | Substitution of equality i... |
| 3netr4d 3017 | Substitution of equality i... |
| 3netr3g 3018 | Substitution of equality i... |
| 3netr4g 3019 | Substitution of equality i... |
| nebi 3020 | Contraposition law for ine... |
| pm13.18 3021 | Theorem *13.18 in [Whitehe... |
| pm13.181 3022 | Theorem *13.181 in [Whiteh... |
| pm13.181OLD 3023 | Obsolete version of ~ pm13... |
| pm2.61ine 3024 | Inference eliminating an i... |
| pm2.21ddne 3025 | A contradiction implies an... |
| pm2.61ne 3026 | Deduction eliminating an i... |
| pm2.61dne 3027 | Deduction eliminating an i... |
| pm2.61dane 3028 | Deduction eliminating an i... |
| pm2.61da2ne 3029 | Deduction eliminating two ... |
| pm2.61da3ne 3030 | Deduction eliminating thre... |
| pm2.61iine 3031 | Equality version of ~ pm2.... |
| mteqand 3032 | A modus tollens deduction ... |
| neor 3033 | Logical OR with an equalit... |
| neanior 3034 | A De Morgan's law for ineq... |
| ne3anior 3035 | A De Morgan's law for ineq... |
| neorian 3036 | A De Morgan's law for ineq... |
| nemtbir 3037 | An inference from an inequ... |
| nelne1 3038 | Two classes are different ... |
| nelne2 3039 | Two classes are different ... |
| nelelne 3040 | Two classes are different ... |
| neneor 3041 | If two classes are differe... |
| nfne 3042 | Bound-variable hypothesis ... |
| nfned 3043 | Bound-variable hypothesis ... |
| nabbib 3044 | Not equivalent wff's corre... |
| neli 3047 | Inference associated with ... |
| nelir 3048 | Inference associated with ... |
| nelcon3d 3049 | Contrapositive law deducti... |
| neleq12d 3050 | Equality theorem for negat... |
| neleq1 3051 | Equality theorem for negat... |
| neleq2 3052 | Equality theorem for negat... |
| nfnel 3053 | Bound-variable hypothesis ... |
| nfneld 3054 | Bound-variable hypothesis ... |
| nnel 3055 | Negation of negated member... |
| elnelne1 3056 | Two classes are different ... |
| elnelne2 3057 | Two classes are different ... |
| pm2.24nel 3058 | A contradiction concerning... |
| pm2.61danel 3059 | Deduction eliminating an e... |
| rgen 3062 | Generalization rule for re... |
| ralel 3063 | All elements of a class ar... |
| rgenw 3064 | Generalization rule for re... |
| rgen2w 3065 | Generalization rule for re... |
| mprg 3066 | Modus ponens combined with... |
| mprgbir 3067 | Modus ponens on biconditio... |
| raln 3068 | Restricted universally qua... |
| ralnex 3071 | Relationship between restr... |
| dfrex2 3072 | Relationship between restr... |
| nrex 3073 | Inference adding restricte... |
| alral 3074 | Universal quantification i... |
| rexex 3075 | Restricted existence impli... |
| rextru 3076 | Two ways of expressing tha... |
| ralimi2 3077 | Inference quantifying both... |
| reximi2 3078 | Inference quantifying both... |
| ralimia 3079 | Inference quantifying both... |
| reximia 3080 | Inference quantifying both... |
| ralimiaa 3081 | Inference quantifying both... |
| ralimi 3082 | Inference quantifying both... |
| reximi 3083 | Inference quantifying both... |
| ral2imi 3084 | Inference quantifying ante... |
| ralim 3085 | Distribution of restricted... |
| rexim 3086 | Theorem 19.22 of [Margaris... |
| reximiaOLD 3087 | Obsolete version of ~ rexi... |
| ralbii2 3088 | Inference adding different... |
| rexbii2 3089 | Inference adding different... |
| ralbiia 3090 | Inference adding restricte... |
| rexbiia 3091 | Inference adding restricte... |
| ralbii 3092 | Inference adding restricte... |
| rexbii 3093 | Inference adding restricte... |
| ralanid 3094 | Cancellation law for restr... |
| rexanid 3095 | Cancellation law for restr... |
| ralcom3 3096 | A commutation law for rest... |
| ralcom3OLD 3097 | Obsolete version of ~ ralc... |
| dfral2 3098 | Relationship between restr... |
| rexnal 3099 | Relationship between restr... |
| ralinexa 3100 | A transformation of restri... |
| rexanali 3101 | A transformation of restri... |
| ralbi 3102 | Distribute a restricted un... |
| rexbi 3103 | Distribute restricted quan... |
| rexbiOLD 3104 | Obsolete version of ~ rexb... |
| ralrexbid 3105 | Formula-building rule for ... |
| ralrexbidOLD 3106 | Obsolete version of ~ ralr... |
| r19.35 3107 | Restricted quantifier vers... |
| r19.35OLD 3108 | Obsolete version of ~ 19.3... |
| r19.26m 3109 | Version of ~ 19.26 and ~ r... |
| r19.26 3110 | Restricted quantifier vers... |
| r19.26-3 3111 | Version of ~ r19.26 with t... |
| ralbiim 3112 | Split a biconditional and ... |
| r19.29 3113 | Restricted quantifier vers... |
| r19.29OLD 3114 | Obsolete version of ~ r19.... |
| r19.29r 3115 | Restricted quantifier vers... |
| r19.29rOLD 3116 | Obsolete version of ~ r19.... |
| r19.29imd 3117 | Theorem 19.29 of [Margaris... |
| r19.40 3118 | Restricted quantifier vers... |
| r19.30 3119 | Restricted quantifier vers... |
| r19.30OLD 3120 | Obsolete version of ~ 19.3... |
| r19.43 3121 | Restricted quantifier vers... |
| 2ralimi 3122 | Inference quantifying both... |
| 3ralimi 3123 | Inference quantifying both... |
| 4ralimi 3124 | Inference quantifying both... |
| 5ralimi 3125 | Inference quantifying both... |
| 6ralimi 3126 | Inference quantifying both... |
| 2ralbii 3127 | Inference adding two restr... |
| 2rexbii 3128 | Inference adding two restr... |
| 3ralbii 3129 | Inference adding three res... |
| 4ralbii 3130 | Inference adding four rest... |
| 2ralbiim 3131 | Split a biconditional and ... |
| ralnex2 3132 | Relationship between two r... |
| ralnex3 3133 | Relationship between three... |
| rexnal2 3134 | Relationship between two r... |
| rexnal3 3135 | Relationship between three... |
| nrexralim 3136 | Negation of a complex pred... |
| r19.26-2 3137 | Restricted quantifier vers... |
| 2r19.29 3138 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3139 | Theorem 19.29 of [Margaris... |
| r19.29d2rOLD 3140 | Obsolete version of ~ r19.... |
| r2allem 3141 | Lemma factoring out common... |
| r2exlem 3142 | Lemma factoring out common... |
| hbralrimi 3143 | Inference from Theorem 19.... |
| ralrimiv 3144 | Inference from Theorem 19.... |
| ralrimiva 3145 | Inference from Theorem 19.... |
| rexlimiva 3146 | Inference from Theorem 19.... |
| rexlimiv 3147 | Inference from Theorem 19.... |
| nrexdv 3148 | Deduction adding restricte... |
| ralrimivw 3149 | Inference from Theorem 19.... |
| rexlimivw 3150 | Weaker version of ~ rexlim... |
| ralrimdv 3151 | Inference from Theorem 19.... |
| rexlimdv 3152 | Inference from Theorem 19.... |
| ralrimdva 3153 | Inference from Theorem 19.... |
| rexlimdva 3154 | Inference from Theorem 19.... |
| rexlimdvaa 3155 | Inference from Theorem 19.... |
| rexlimdva2 3156 | Inference from Theorem 19.... |
| r19.29an 3157 | A commonly used pattern in... |
| rexlimdv3a 3158 | Inference from Theorem 19.... |
| rexlimdvw 3159 | Inference from Theorem 19.... |
| rexlimddv 3160 | Restricted existential eli... |
| r19.29a 3161 | A commonly used pattern in... |
| ralimdv2 3162 | Inference quantifying both... |
| reximdv2 3163 | Deduction quantifying both... |
| reximdvai 3164 | Deduction quantifying both... |
| reximdvaiOLD 3165 | Obsolete version of ~ rexi... |
| ralimdva 3166 | Deduction quantifying both... |
| reximdva 3167 | Deduction quantifying both... |
| ralimdv 3168 | Deduction quantifying both... |
| reximdv 3169 | Deduction from Theorem 19.... |
| reximddv 3170 | Deduction from Theorem 19.... |
| reximssdv 3171 | Derivation of a restricted... |
| ralbidv2 3172 | Formula-building rule for ... |
| rexbidv2 3173 | Formula-building rule for ... |
| ralbidva 3174 | Formula-building rule for ... |
| rexbidva 3175 | Formula-building rule for ... |
| ralbidv 3176 | Formula-building rule for ... |
| rexbidv 3177 | Formula-building rule for ... |
| r19.21v 3178 | Restricted quantifier vers... |
| r19.21vOLD 3179 | Obsolete version of ~ r19.... |
| r19.37v 3180 | Restricted quantifier vers... |
| r19.23v 3181 | Restricted quantifier vers... |
| r19.36v 3182 | Restricted quantifier vers... |
| rexlimivOLD 3183 | Obsolete version of ~ rexl... |
| rexlimivaOLD 3184 | Obsolete version of ~ rexl... |
| rexlimivwOLD 3185 | Obsolete version of ~ rexl... |
| r19.27v 3186 | Restricted quantitifer ver... |
| r19.41v 3187 | Restricted quantifier vers... |
| r19.28v 3188 | Restricted quantifier vers... |
| r19.42v 3189 | Restricted quantifier vers... |
| r19.32v 3190 | Restricted quantifier vers... |
| r19.45v 3191 | Restricted quantifier vers... |
| r19.44v 3192 | One direction of a restric... |
| r2al 3193 | Double restricted universa... |
| r2ex 3194 | Double restricted existent... |
| r3al 3195 | Triple restricted universa... |
| rgen2 3196 | Generalization rule for re... |
| ralrimivv 3197 | Inference from Theorem 19.... |
| rexlimivv 3198 | Inference from Theorem 19.... |
| ralrimivva 3199 | Inference from Theorem 19.... |
| ralrimdvv 3200 | Inference from Theorem 19.... |
| rgen3 3201 | Generalization rule for re... |
| ralrimivvva 3202 | Inference from Theorem 19.... |
| ralimdvva 3203 | Deduction doubly quantifyi... |
| reximdvva 3204 | Deduction doubly quantifyi... |
| ralimdvv 3205 | Deduction doubly quantifyi... |
| ralimd4v 3206 | Deduction quadrupally quan... |
| ralimd6v 3207 | Deduction sextupally quant... |
| ralrimdvva 3208 | Inference from Theorem 19.... |
| rexlimdvv 3209 | Inference from Theorem 19.... |
| rexlimdvva 3210 | Inference from Theorem 19.... |
| reximddv2 3211 | Double deduction from Theo... |
| r19.29vva 3212 | A commonly used pattern ba... |
| r19.29vvaOLD 3213 | Obsolete version of ~ r19.... |
| 2rexbiia 3214 | Inference adding two restr... |
| 2ralbidva 3215 | Formula-building rule for ... |
| 2rexbidva 3216 | Formula-building rule for ... |
| 2ralbidv 3217 | Formula-building rule for ... |
| 2rexbidv 3218 | Formula-building rule for ... |
| rexralbidv 3219 | Formula-building rule for ... |
| 3ralbidv 3220 | Formula-building rule for ... |
| 4ralbidv 3221 | Formula-building rule for ... |
| 6ralbidv 3222 | Formula-building rule for ... |
| r19.41vv 3223 | Version of ~ r19.41v with ... |
| reeanlem 3224 | Lemma factoring out common... |
| reeanv 3225 | Rearrange restricted exist... |
| 3reeanv 3226 | Rearrange three restricted... |
| 2ralor 3227 | Distribute restricted univ... |
| 2ralorOLD 3228 | Obsolete version of ~ 2ral... |
| risset 3229 | Two ways to say " ` A ` be... |
| nelb 3230 | A definition of ` -. A e. ... |
| nelbOLD 3231 | Obsolete version of ~ nelb... |
| rspw 3232 | Restricted specialization.... |
| cbvralvw 3233 | Change the bound variable ... |
| cbvrexvw 3234 | Change the bound variable ... |
| cbvraldva 3235 | Rule used to change the bo... |
| cbvrexdva 3236 | Rule used to change the bo... |
| cbvral2vw 3237 | Change bound variables of ... |
| cbvrex2vw 3238 | Change bound variables of ... |
| cbvral3vw 3239 | Change bound variables of ... |
| cbvral4vw 3240 | Change bound variables of ... |
| cbvral6vw 3241 | Change bound variables of ... |
| cbvral8vw 3242 | Change bound variables of ... |
| rsp 3243 | Restricted specialization.... |
| rspa 3244 | Restricted specialization.... |
| rspe 3245 | Restricted specialization.... |
| rspec 3246 | Specialization rule for re... |
| r19.21bi 3247 | Inference from Theorem 19.... |
| r19.21be 3248 | Inference from Theorem 19.... |
| r19.21t 3249 | Restricted quantifier vers... |
| r19.21 3250 | Restricted quantifier vers... |
| r19.23t 3251 | Closed theorem form of ~ r... |
| r19.23 3252 | Restricted quantifier vers... |
| ralrimi 3253 | Inference from Theorem 19.... |
| ralrimia 3254 | Inference from Theorem 19.... |
| rexlimi 3255 | Restricted quantifier vers... |
| ralimdaa 3256 | Deduction quantifying both... |
| reximdai 3257 | Deduction from Theorem 19.... |
| r19.37 3258 | Restricted quantifier vers... |
| r19.41 3259 | Restricted quantifier vers... |
| ralrimd 3260 | Inference from Theorem 19.... |
| rexlimd2 3261 | Version of ~ rexlimd with ... |
| rexlimd 3262 | Deduction form of ~ rexlim... |
| r19.29af2 3263 | A commonly used pattern ba... |
| r19.29af 3264 | A commonly used pattern ba... |
| reximd2a 3265 | Deduction quantifying both... |
| ralbida 3266 | Formula-building rule for ... |
| ralbidaOLD 3267 | Obsolete version of ~ ralb... |
| rexbida 3268 | Formula-building rule for ... |
| ralbid 3269 | Formula-building rule for ... |
| rexbid 3270 | Formula-building rule for ... |
| rexbidvALT 3271 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3272 | Alternate proof of ~ rexbi... |
| rsp2 3273 | Restricted specialization,... |
| rsp2e 3274 | Restricted specialization.... |
| rspec2 3275 | Specialization rule for re... |
| rspec3 3276 | Specialization rule for re... |
| r2alf 3277 | Double restricted universa... |
| r2exf 3278 | Double restricted existent... |
| 2ralbida 3279 | Formula-building rule for ... |
| nfra1 3280 | The setvar ` x ` is not fr... |
| nfre1 3281 | The setvar ` x ` is not fr... |
| ralcom4 3282 | Commutation of restricted ... |
| ralcom4OLD 3283 | Obsolete version of ~ ralc... |
| rexcom4 3284 | Commutation of restricted ... |
| ralcom 3285 | Commutation of restricted ... |
| rexcom 3286 | Commutation of restricted ... |
| rexcomOLD 3287 | Obsolete version of ~ rexc... |
| rexcom4a 3288 | Specialized existential co... |
| ralrot3 3289 | Rotate three restricted un... |
| ralcom13 3290 | Swap first and third restr... |
| ralcom13OLD 3291 | Obsolete version of ~ ralc... |
| rexcom13 3292 | Swap first and third restr... |
| rexrot4 3293 | Rotate four restricted exi... |
| 2ex2rexrot 3294 | Rotate two existential qua... |
| nfra2w 3295 | Similar to Lemma 24 of [Mo... |
| nfra2wOLD 3296 | Obsolete version of ~ nfra... |
| hbra1 3297 | The setvar ` x ` is not fr... |
| ralcomf 3298 | Commutation of restricted ... |
| rexcomf 3299 | Commutation of restricted ... |
| cbvralfw 3300 | Rule used to change bound ... |
| cbvrexfw 3301 | Rule used to change bound ... |
| cbvralw 3302 | Rule used to change bound ... |
| cbvrexw 3303 | Rule used to change bound ... |
| hbral 3304 | Bound-variable hypothesis ... |
| nfraldw 3305 | Deduction version of ~ nfr... |
| nfrexdw 3306 | Deduction version of ~ nfr... |
| nfralw 3307 | Bound-variable hypothesis ... |
| nfralwOLD 3308 | Obsolete version of ~ nfra... |
| nfrexw 3309 | Bound-variable hypothesis ... |
| r19.12 3310 | Restricted quantifier vers... |
| r19.12OLD 3311 | Obsolete version of ~ 19.1... |
| reean 3312 | Rearrange restricted exist... |
| cbvralsvw 3313 | Change bound variable by u... |
| cbvrexsvw 3314 | Change bound variable by u... |
| cbvralsvwOLD 3315 | Obsolete version of ~ cbvr... |
| cbvrexsvwOLD 3316 | Obsolete version of ~ cbvr... |
| nfraldwOLD 3317 | Obsolete version of ~ nfra... |
| nfra2wOLDOLD 3318 | Obsolete version of ~ nfra... |
| cbvralfwOLD 3319 | Obsolete version of ~ cbvr... |
| rexeq 3320 | Equality theorem for restr... |
| raleq 3321 | Equality theorem for restr... |
| raleqi 3322 | Equality inference for res... |
| rexeqi 3323 | Equality inference for res... |
| raleqdv 3324 | Equality deduction for res... |
| rexeqdv 3325 | Equality deduction for res... |
| raleqbidva 3326 | Equality deduction for res... |
| rexeqbidva 3327 | Equality deduction for res... |
| raleqbidvv 3328 | Version of ~ raleqbidv wit... |
| raleqbidvvOLD 3329 | Obsolete version of ~ rale... |
| rexeqbidvv 3330 | Version of ~ rexeqbidv wit... |
| rexeqbidvvOLD 3331 | Obsolete version of ~ rexe... |
| raleqbi1dv 3332 | Equality deduction for res... |
| rexeqbi1dv 3333 | Equality deduction for res... |
| raleqOLD 3334 | Obsolete version of ~ rale... |
| rexeqOLD 3335 | Obsolete version of ~ rale... |
| raleleq 3336 | All elements of a class ar... |
| raleqbii 3337 | Equality deduction for res... |
| rexeqbii 3338 | Equality deduction for res... |
| raleleqOLD 3339 | Obsolete version of ~ rale... |
| raleleqALT 3340 | Alternate proof of ~ ralel... |
| raleqbidv 3341 | Equality deduction for res... |
| rexeqbidv 3342 | Equality deduction for res... |
| cbvraldva2 3343 | Rule used to change the bo... |
| cbvrexdva2 3344 | Rule used to change the bo... |
| cbvrexdva2OLD 3345 | Obsolete version of ~ cbvr... |
| cbvraldvaOLD 3346 | Obsolete version of ~ cbvr... |
| cbvrexdvaOLD 3347 | Obsolete version of ~ cbvr... |
| raleqf 3348 | Equality theorem for restr... |
| rexeqf 3349 | Equality theorem for restr... |
| rexeqfOLD 3350 | Obsolete version of ~ rexe... |
| raleqbid 3351 | Equality deduction for res... |
| rexeqbid 3352 | Equality deduction for res... |
| sbralie 3353 | Implicit to explicit subst... |
| sbralieALT 3354 | Alternative shorter proof ... |
| cbvralf 3355 | Rule used to change bound ... |
| cbvrexf 3356 | Rule used to change bound ... |
| cbvral 3357 | Rule used to change bound ... |
| cbvrex 3358 | Rule used to change bound ... |
| cbvralv 3359 | Change the bound variable ... |
| cbvrexv 3360 | Change the bound variable ... |
| cbvralsv 3361 | Change bound variable by u... |
| cbvrexsv 3362 | Change bound variable by u... |
| cbvral2v 3363 | Change bound variables of ... |
| cbvrex2v 3364 | Change bound variables of ... |
| cbvral3v 3365 | Change bound variables of ... |
| rgen2a 3366 | Generalization rule for re... |
| nfrald 3367 | Deduction version of ~ nfr... |
| nfrexd 3368 | Deduction version of ~ nfr... |
| nfral 3369 | Bound-variable hypothesis ... |
| nfrex 3370 | Bound-variable hypothesis ... |
| nfra2 3371 | Similar to Lemma 24 of [Mo... |
| ralcom2 3372 | Commutation of restricted ... |
| reu5 3377 | Restricted uniqueness in t... |
| reurmo 3378 | Restricted existential uni... |
| reurex 3379 | Restricted unique existenc... |
| mormo 3380 | Unrestricted "at most one"... |
| rmobiia 3381 | Formula-building rule for ... |
| reubiia 3382 | Formula-building rule for ... |
| rmobii 3383 | Formula-building rule for ... |
| reubii 3384 | Formula-building rule for ... |
| rmoanid 3385 | Cancellation law for restr... |
| reuanid 3386 | Cancellation law for restr... |
| rmoanidOLD 3387 | Obsolete version of ~ rmoa... |
| reuanidOLD 3388 | Obsolete version of ~ reua... |
| 2reu2rex 3389 | Double restricted existent... |
| rmobidva 3390 | Formula-building rule for ... |
| reubidva 3391 | Formula-building rule for ... |
| rmobidv 3392 | Formula-building rule for ... |
| reubidv 3393 | Formula-building rule for ... |
| reueubd 3394 | Restricted existential uni... |
| rmo5 3395 | Restricted "at most one" i... |
| nrexrmo 3396 | Nonexistence implies restr... |
| moel 3397 | "At most one" element in a... |
| cbvrmovw 3398 | Change the bound variable ... |
| cbvreuvw 3399 | Change the bound variable ... |
| moelOLD 3400 | Obsolete version of ~ moel... |
| rmobida 3401 | Formula-building rule for ... |
| reubida 3402 | Formula-building rule for ... |
| rmobidvaOLD 3403 | Obsolete version of ~ rmob... |
| cbvrmow 3404 | Change the bound variable ... |
| cbvreuw 3405 | Change the bound variable ... |
| nfrmo1 3406 | The setvar ` x ` is not fr... |
| nfreu1 3407 | The setvar ` x ` is not fr... |
| nfrmow 3408 | Bound-variable hypothesis ... |
| nfreuw 3409 | Bound-variable hypothesis ... |
| cbvrmowOLD 3410 | Obsolete version of ~ cbvr... |
| cbvreuwOLD 3411 | Obsolete version of ~ cbvr... |
| cbvreuvwOLD 3412 | Obsolete version of ~ cbvr... |
| rmoeq1 3413 | Equality theorem for restr... |
| reueq1 3414 | Equality theorem for restr... |
| rmoeq1OLD 3415 | Obsolete version of ~ rmoe... |
| reueq1OLD 3416 | Obsolete version of ~ reue... |
| rmoeqd 3417 | Equality deduction for res... |
| reueqd 3418 | Equality deduction for res... |
| rmoeq1f 3419 | Equality theorem for restr... |
| reueq1f 3420 | Equality theorem for restr... |
| nfreuwOLD 3421 | Obsolete version of ~ nfre... |
| nfrmowOLD 3422 | Obsolete version of ~ nfrm... |
| cbvreu 3423 | Change the bound variable ... |
| cbvrmo 3424 | Change the bound variable ... |
| cbvrmov 3425 | Change the bound variable ... |
| cbvreuv 3426 | Change the bound variable ... |
| nfrmod 3427 | Deduction version of ~ nfr... |
| nfreud 3428 | Deduction version of ~ nfr... |
| nfrmo 3429 | Bound-variable hypothesis ... |
| nfreu 3430 | Bound-variable hypothesis ... |
| rabbidva2 3433 | Equivalent wff's yield equ... |
| rabbia2 3434 | Equivalent wff's yield equ... |
| rabbiia 3435 | Equivalent formulas yield ... |
| rabbiiaOLD 3436 | Obsolete version of ~ rabb... |
| rabbii 3437 | Equivalent wff's correspon... |
| rabbidva 3438 | Equivalent wff's yield equ... |
| rabbidv 3439 | Equivalent wff's yield equ... |
| rabswap 3440 | Swap with a membership rel... |
| cbvrabv 3441 | Rule to change the bound v... |
| rabeqcda 3442 | When ` ps ` is always true... |
| rabeqc 3443 | A restricted class abstrac... |
| rabeqi 3444 | Equality theorem for restr... |
| rabeq 3445 | Equality theorem for restr... |
| rabeqdv 3446 | Equality of restricted cla... |
| rabeqbidva 3447 | Equality of restricted cla... |
| rabeqbidv 3448 | Equality of restricted cla... |
| rabrabi 3449 | Abstract builder restricte... |
| nfrab1 3450 | The abstraction variable i... |
| rabid 3451 | An "identity" law of concr... |
| rabidim1 3452 | Membership in a restricted... |
| reqabi 3453 | Inference from equality of... |
| rabrab 3454 | Abstract builder restricte... |
| rabrabiOLD 3455 | Obsolete version of ~ rabr... |
| rabbida4 3456 | Version of ~ rabbidva2 wit... |
| rabbida 3457 | Equivalent wff's yield equ... |
| rabbid 3458 | Version of ~ rabbidv with ... |
| rabeqd 3459 | Deduction form of ~ rabeq ... |
| rabeqbida 3460 | Version of ~ rabeqbidva wi... |
| rabbi 3461 | Equivalent wff's correspon... |
| rabid2f 3462 | An "identity" law for rest... |
| rabid2 3463 | An "identity" law for rest... |
| rabid2OLD 3464 | Obsolete version of ~ rabi... |
| rabeqf 3465 | Equality theorem for restr... |
| cbvrabw 3466 | Rule to change the bound v... |
| nfrabw 3467 | A variable not free in a w... |
| nfrabwOLD 3468 | Obsolete version of ~ nfra... |
| rabbidaOLD 3469 | Obsolete version of ~ rabb... |
| rabeqiOLD 3470 | Obsolete version of ~ rabe... |
| nfrab 3471 | A variable not free in a w... |
| cbvrab 3472 | Rule to change the bound v... |
| vjust 3474 | Justification theorem for ... |
| dfv2 3476 | Alternate definition of th... |
| vex 3477 | All setvar variables are s... |
| vexOLD 3478 | Obsolete version of ~ vex ... |
| elv 3479 | If a proposition is implie... |
| elvd 3480 | If a proposition is implie... |
| el2v 3481 | If a proposition is implie... |
| eqv 3482 | The universe contains ever... |
| eqvf 3483 | The universe contains ever... |
| abv 3484 | The class of sets verifyin... |
| abvALT 3485 | Alternate proof of ~ abv ,... |
| isset 3486 | Two ways to express that "... |
| issetft 3487 | Closed theorem form of ~ i... |
| issetf 3488 | A version of ~ isset that ... |
| isseti 3489 | A way to say " ` A ` is a ... |
| issetri 3490 | A way to say " ` A ` is a ... |
| eqvisset 3491 | A class equal to a variabl... |
| elex 3492 | If a class is a member of ... |
| elexi 3493 | If a class is a member of ... |
| elexd 3494 | If a class is a member of ... |
| elex2OLD 3495 | Obsolete version of ~ elex... |
| elex22 3496 | If two classes each contai... |
| prcnel 3497 | A proper class doesn't bel... |
| ralv 3498 | A universal quantifier res... |
| rexv 3499 | An existential quantifier ... |
| reuv 3500 | A unique existential quant... |
| rmov 3501 | An at-most-one quantifier ... |
| rabab 3502 | A class abstraction restri... |
| rexcom4b 3503 | Specialized existential co... |
| ceqsal1t 3504 | One direction of ~ ceqsalt... |
| ceqsalt 3505 | Closed theorem version of ... |
| ceqsralt 3506 | Restricted quantifier vers... |
| ceqsalg 3507 | A representation of explic... |
| ceqsalgALT 3508 | Alternate proof of ~ ceqsa... |
| ceqsal 3509 | A representation of explic... |
| ceqsalALT 3510 | A representation of explic... |
| ceqsalv 3511 | A representation of explic... |
| ceqsalvOLD 3512 | Obsolete version of ~ ceqs... |
| ceqsralv 3513 | Restricted quantifier vers... |
| ceqsralvOLD 3514 | Obsolete version of ~ ceqs... |
| gencl 3515 | Implicit substitution for ... |
| 2gencl 3516 | Implicit substitution for ... |
| 3gencl 3517 | Implicit substitution for ... |
| cgsexg 3518 | Implicit substitution infe... |
| cgsex2g 3519 | Implicit substitution infe... |
| cgsex4g 3520 | An implicit substitution i... |
| cgsex4gOLD 3521 | Obsolete version of ~ cgse... |
| cgsex4gOLDOLD 3522 | Obsolete version of ~ cgse... |
| ceqsex 3523 | Elimination of an existent... |
| ceqsexOLD 3524 | Obsolete version of ~ ceqs... |
| ceqsexv 3525 | Elimination of an existent... |
| ceqsexvOLD 3526 | Obsolete version of ~ ceqs... |
| ceqsexvOLDOLD 3527 | Obsolete version of ~ ceqs... |
| ceqsexv2d 3528 | Elimination of an existent... |
| ceqsex2 3529 | Elimination of two existen... |
| ceqsex2v 3530 | Elimination of two existen... |
| ceqsex3v 3531 | Elimination of three exist... |
| ceqsex4v 3532 | Elimination of four existe... |
| ceqsex6v 3533 | Elimination of six existen... |
| ceqsex8v 3534 | Elimination of eight exist... |
| gencbvex 3535 | Change of bound variable u... |
| gencbvex2 3536 | Restatement of ~ gencbvex ... |
| gencbval 3537 | Change of bound variable u... |
| sbhypf 3538 | Introduce an explicit subs... |
| sbhypfOLD 3539 | Obsolete version of ~ sbhy... |
| vtoclgft 3540 | Closed theorem form of ~ v... |
| vtocleg 3541 | Implicit substitution of a... |
| vtoclg 3542 | Implicit substitution of a... |
| vtocle 3543 | Implicit substitution of a... |
| vtoclbg 3544 | Implicit substitution of a... |
| vtocl 3545 | Implicit substitution of a... |
| vtocldf 3546 | Implicit substitution of a... |
| vtocld 3547 | Implicit substitution of a... |
| vtocldOLD 3548 | Obsolete version of ~ vtoc... |
| vtocl2d 3549 | Implicit substitution of t... |
| vtoclef 3550 | Implicit substitution of a... |
| vtoclf 3551 | Implicit substitution of a... |
| vtoclfOLD 3552 | Obsolete version of ~ vtoc... |
| vtoclALT 3553 | Alternate proof of ~ vtocl... |
| vtocl2 3554 | Implicit substitution of c... |
| vtocl3 3555 | Implicit substitution of c... |
| vtoclb 3556 | Implicit substitution of a... |
| vtoclgf 3557 | Implicit substitution of a... |
| vtoclg1f 3558 | Version of ~ vtoclgf with ... |
| vtoclgOLD 3559 | Obsolete version of ~ vtoc... |
| vtoclgOLDOLD 3560 | Obsolete version of ~ vtoc... |
| vtocl2gf 3561 | Implicit substitution of a... |
| vtocl3gf 3562 | Implicit substitution of a... |
| vtocl2g 3563 | Implicit substitution of 2... |
| vtocl3g 3564 | Implicit substitution of a... |
| vtoclgaf 3565 | Implicit substitution of a... |
| vtoclga 3566 | Implicit substitution of a... |
| vtocl2ga 3567 | Implicit substitution of 2... |
| vtocl2gaf 3568 | Implicit substitution of 2... |
| vtocl3gaf 3569 | Implicit substitution of 3... |
| vtocl3ga 3570 | Implicit substitution of 3... |
| vtocl3gaOLD 3571 | Obsolete version of ~ vtoc... |
| vtocl4g 3572 | Implicit substitution of 4... |
| vtocl4ga 3573 | Implicit substitution of 4... |
| vtoclegft 3574 | Implicit substitution of a... |
| vtoclegftOLD 3575 | Obsolete version of ~ vtoc... |
| vtoclri 3576 | Implicit substitution of a... |
| spcimgft 3577 | A closed version of ~ spci... |
| spcgft 3578 | A closed version of ~ spcg... |
| spcimgf 3579 | Rule of specialization, us... |
| spcimegf 3580 | Existential specialization... |
| spcgf 3581 | Rule of specialization, us... |
| spcegf 3582 | Existential specialization... |
| spcimdv 3583 | Restricted specialization,... |
| spcdv 3584 | Rule of specialization, us... |
| spcimedv 3585 | Restricted existential spe... |
| spcgv 3586 | Rule of specialization, us... |
| spcegv 3587 | Existential specialization... |
| spcedv 3588 | Existential specialization... |
| spc2egv 3589 | Existential specialization... |
| spc2gv 3590 | Specialization with two qu... |
| spc2ed 3591 | Existential specialization... |
| spc2d 3592 | Specialization with 2 quan... |
| spc3egv 3593 | Existential specialization... |
| spc3gv 3594 | Specialization with three ... |
| spcv 3595 | Rule of specialization, us... |
| spcev 3596 | Existential specialization... |
| spc2ev 3597 | Existential specialization... |
| rspct 3598 | A closed version of ~ rspc... |
| rspcdf 3599 | Restricted specialization,... |
| rspc 3600 | Restricted specialization,... |
| rspce 3601 | Restricted existential spe... |
| rspcimdv 3602 | Restricted specialization,... |
| rspcimedv 3603 | Restricted existential spe... |
| rspcdv 3604 | Restricted specialization,... |
| rspcedv 3605 | Restricted existential spe... |
| rspcebdv 3606 | Restricted existential spe... |
| rspcdv2 3607 | Restricted specialization,... |
| rspcv 3608 | Restricted specialization,... |
| rspccv 3609 | Restricted specialization,... |
| rspcva 3610 | Restricted specialization,... |
| rspccva 3611 | Restricted specialization,... |
| rspcev 3612 | Restricted existential spe... |
| rspcdva 3613 | Restricted specialization,... |
| rspcedvd 3614 | Restricted existential spe... |
| rspcedvdw 3615 | Version of ~ rspcedvd wher... |
| rspcime 3616 | Prove a restricted existen... |
| rspceaimv 3617 | Restricted existential spe... |
| rspcedeq1vd 3618 | Restricted existential spe... |
| rspcedeq2vd 3619 | Restricted existential spe... |
| rspc2 3620 | Restricted specialization ... |
| rspc2gv 3621 | Restricted specialization ... |
| rspc2v 3622 | 2-variable restricted spec... |
| rspc2va 3623 | 2-variable restricted spec... |
| rspc2ev 3624 | 2-variable restricted exis... |
| 2rspcedvdw 3625 | Double application of ~ rs... |
| rspc2dv 3626 | 2-variable restricted spec... |
| rspc3v 3627 | 3-variable restricted spec... |
| rspc3ev 3628 | 3-variable restricted exis... |
| rspc3dv 3629 | 3-variable restricted spec... |
| rspc4v 3630 | 4-variable restricted spec... |
| rspc6v 3631 | 6-variable restricted spec... |
| rspc8v 3632 | 8-variable restricted spec... |
| rspceeqv 3633 | Restricted existential spe... |
| ralxpxfr2d 3634 | Transfer a universal quant... |
| rexraleqim 3635 | Statement following from e... |
| eqvincg 3636 | A variable introduction la... |
| eqvinc 3637 | A variable introduction la... |
| eqvincf 3638 | A variable introduction la... |
| alexeqg 3639 | Two ways to express substi... |
| ceqex 3640 | Equality implies equivalen... |
| ceqsexg 3641 | A representation of explic... |
| ceqsexgv 3642 | Elimination of an existent... |
| ceqsrexv 3643 | Elimination of a restricte... |
| ceqsrexbv 3644 | Elimination of a restricte... |
| ceqsralbv 3645 | Elimination of a restricte... |
| ceqsrex2v 3646 | Elimination of a restricte... |
| clel2g 3647 | Alternate definition of me... |
| clel2gOLD 3648 | Obsolete version of ~ clel... |
| clel2 3649 | Alternate definition of me... |
| clel3g 3650 | Alternate definition of me... |
| clel3 3651 | Alternate definition of me... |
| clel4g 3652 | Alternate definition of me... |
| clel4 3653 | Alternate definition of me... |
| clel4OLD 3654 | Obsolete version of ~ clel... |
| clel5 3655 | Alternate definition of cl... |
| pm13.183 3656 | Compare theorem *13.183 in... |
| rr19.3v 3657 | Restricted quantifier vers... |
| rr19.28v 3658 | Restricted quantifier vers... |
| elab6g 3659 | Membership in a class abst... |
| elabd2 3660 | Membership in a class abst... |
| elabd3 3661 | Membership in a class abst... |
| elabgt 3662 | Membership in a class abst... |
| elabgtOLD 3663 | Obsolete version of ~ elab... |
| elabgf 3664 | Membership in a class abst... |
| elabf 3665 | Membership in a class abst... |
| elabg 3666 | Membership in a class abst... |
| elabgOLD 3667 | Obsolete version of ~ elab... |
| elab 3668 | Membership in a class abst... |
| elabOLD 3669 | Obsolete version of ~ elab... |
| elab2g 3670 | Membership in a class abst... |
| elabd 3671 | Explicit demonstration the... |
| elab2 3672 | Membership in a class abst... |
| elab4g 3673 | Membership in a class abst... |
| elab3gf 3674 | Membership in a class abst... |
| elab3g 3675 | Membership in a class abst... |
| elab3 3676 | Membership in a class abst... |
| elrabi 3677 | Implication for the member... |
| elrabiOLD 3678 | Obsolete version of ~ elra... |
| elrabf 3679 | Membership in a restricted... |
| rabtru 3680 | Abstract builder using the... |
| rabeqcOLD 3681 | Obsolete version of ~ rabe... |
| elrab3t 3682 | Membership in a restricted... |
| elrab 3683 | Membership in a restricted... |
| elrab3 3684 | Membership in a restricted... |
| elrabd 3685 | Membership in a restricted... |
| elrab2 3686 | Membership in a restricted... |
| ralab 3687 | Universal quantification o... |
| ralabOLD 3688 | Obsolete version of ~ rala... |
| ralrab 3689 | Universal quantification o... |
| rexab 3690 | Existential quantification... |
| rexabOLD 3691 | Obsolete version of ~ rexa... |
| rexrab 3692 | Existential quantification... |
| ralab2 3693 | Universal quantification o... |
| ralrab2 3694 | Universal quantification o... |
| rexab2 3695 | Existential quantification... |
| rexrab2 3696 | Existential quantification... |
| reurab 3697 | Restricted existential uni... |
| abidnf 3698 | Identity used to create cl... |
| dedhb 3699 | A deduction theorem for co... |
| class2seteq 3700 | Writing a set as a class a... |
| nelrdva 3701 | Deduce negative membership... |
| eqeu 3702 | A condition which implies ... |
| moeq 3703 | There exists at most one s... |
| eueq 3704 | A class is a set if and on... |
| eueqi 3705 | There exists a unique set ... |
| eueq2 3706 | Equality has existential u... |
| eueq3 3707 | Equality has existential u... |
| moeq3 3708 | "At most one" property of ... |
| mosub 3709 | "At most one" remains true... |
| mo2icl 3710 | Theorem for inferring "at ... |
| mob2 3711 | Consequence of "at most on... |
| moi2 3712 | Consequence of "at most on... |
| mob 3713 | Equality implied by "at mo... |
| moi 3714 | Equality implied by "at mo... |
| morex 3715 | Derive membership from uni... |
| euxfr2w 3716 | Transfer existential uniqu... |
| euxfrw 3717 | Transfer existential uniqu... |
| euxfr2 3718 | Transfer existential uniqu... |
| euxfr 3719 | Transfer existential uniqu... |
| euind 3720 | Existential uniqueness via... |
| reu2 3721 | A way to express restricte... |
| reu6 3722 | A way to express restricte... |
| reu3 3723 | A way to express restricte... |
| reu6i 3724 | A condition which implies ... |
| eqreu 3725 | A condition which implies ... |
| rmo4 3726 | Restricted "at most one" u... |
| reu4 3727 | Restricted uniqueness usin... |
| reu7 3728 | Restricted uniqueness usin... |
| reu8 3729 | Restricted uniqueness usin... |
| rmo3f 3730 | Restricted "at most one" u... |
| rmo4f 3731 | Restricted "at most one" u... |
| reu2eqd 3732 | Deduce equality from restr... |
| reueq 3733 | Equality has existential u... |
| rmoeq 3734 | Equality's restricted exis... |
| rmoan 3735 | Restricted "at most one" s... |
| rmoim 3736 | Restricted "at most one" i... |
| rmoimia 3737 | Restricted "at most one" i... |
| rmoimi 3738 | Restricted "at most one" i... |
| rmoimi2 3739 | Restricted "at most one" i... |
| 2reu5a 3740 | Double restricted existent... |
| reuimrmo 3741 | Restricted uniqueness impl... |
| 2reuswap 3742 | A condition allowing swap ... |
| 2reuswap2 3743 | A condition allowing swap ... |
| reuxfrd 3744 | Transfer existential uniqu... |
| reuxfr 3745 | Transfer existential uniqu... |
| reuxfr1d 3746 | Transfer existential uniqu... |
| reuxfr1ds 3747 | Transfer existential uniqu... |
| reuxfr1 3748 | Transfer existential uniqu... |
| reuind 3749 | Existential uniqueness via... |
| 2rmorex 3750 | Double restricted quantifi... |
| 2reu5lem1 3751 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3752 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3753 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3754 | Double restricted existent... |
| 2reurmo 3755 | Double restricted quantifi... |
| 2reurex 3756 | Double restricted quantifi... |
| 2rmoswap 3757 | A condition allowing to sw... |
| 2rexreu 3758 | Double restricted existent... |
| cdeqi 3761 | Deduce conditional equalit... |
| cdeqri 3762 | Property of conditional eq... |
| cdeqth 3763 | Deduce conditional equalit... |
| cdeqnot 3764 | Distribute conditional equ... |
| cdeqal 3765 | Distribute conditional equ... |
| cdeqab 3766 | Distribute conditional equ... |
| cdeqal1 3767 | Distribute conditional equ... |
| cdeqab1 3768 | Distribute conditional equ... |
| cdeqim 3769 | Distribute conditional equ... |
| cdeqcv 3770 | Conditional equality for s... |
| cdeqeq 3771 | Distribute conditional equ... |
| cdeqel 3772 | Distribute conditional equ... |
| nfcdeq 3773 | If we have a conditional e... |
| nfccdeq 3774 | Variation of ~ nfcdeq for ... |
| rru 3775 | Relative version of Russel... |
| ru 3776 | Russell's Paradox. Propos... |
| dfsbcq 3779 | Proper substitution of a c... |
| dfsbcq2 3780 | This theorem, which is sim... |
| sbsbc 3781 | Show that ~ df-sb and ~ df... |
| sbceq1d 3782 | Equality theorem for class... |
| sbceq1dd 3783 | Equality theorem for class... |
| sbceqbid 3784 | Equality theorem for class... |
| sbc8g 3785 | This is the closest we can... |
| sbc2or 3786 | The disjunction of two equ... |
| sbcex 3787 | By our definition of prope... |
| sbceq1a 3788 | Equality theorem for class... |
| sbceq2a 3789 | Equality theorem for class... |
| spsbc 3790 | Specialization: if a formu... |
| spsbcd 3791 | Specialization: if a formu... |
| sbcth 3792 | A substitution into a theo... |
| sbcthdv 3793 | Deduction version of ~ sbc... |
| sbcid 3794 | An identity theorem for su... |
| nfsbc1d 3795 | Deduction version of ~ nfs... |
| nfsbc1 3796 | Bound-variable hypothesis ... |
| nfsbc1v 3797 | Bound-variable hypothesis ... |
| nfsbcdw 3798 | Deduction version of ~ nfs... |
| nfsbcw 3799 | Bound-variable hypothesis ... |
| sbccow 3800 | A composition law for clas... |
| nfsbcd 3801 | Deduction version of ~ nfs... |
| nfsbc 3802 | Bound-variable hypothesis ... |
| sbcco 3803 | A composition law for clas... |
| sbcco2 3804 | A composition law for clas... |
| sbc5 3805 | An equivalence for class s... |
| sbc5ALT 3806 | Alternate proof of ~ sbc5 ... |
| sbc6g 3807 | An equivalence for class s... |
| sbc6gOLD 3808 | Obsolete version of ~ sbc6... |
| sbc6 3809 | An equivalence for class s... |
| sbc7 3810 | An equivalence for class s... |
| cbvsbcw 3811 | Change bound variables in ... |
| cbvsbcvw 3812 | Change the bound variable ... |
| cbvsbc 3813 | Change bound variables in ... |
| cbvsbcv 3814 | Change the bound variable ... |
| sbciegft 3815 | Conversion of implicit sub... |
| sbciegf 3816 | Conversion of implicit sub... |
| sbcieg 3817 | Conversion of implicit sub... |
| sbciegOLD 3818 | Obsolete version of ~ sbci... |
| sbcie2g 3819 | Conversion of implicit sub... |
| sbcie 3820 | Conversion of implicit sub... |
| sbciedf 3821 | Conversion of implicit sub... |
| sbcied 3822 | Conversion of implicit sub... |
| sbciedOLD 3823 | Obsolete version of ~ sbci... |
| sbcied2 3824 | Conversion of implicit sub... |
| elrabsf 3825 | Membership in a restricted... |
| eqsbc1 3826 | Substitution for the left-... |
| sbcng 3827 | Move negation in and out o... |
| sbcimg 3828 | Distribution of class subs... |
| sbcan 3829 | Distribution of class subs... |
| sbcor 3830 | Distribution of class subs... |
| sbcbig 3831 | Distribution of class subs... |
| sbcn1 3832 | Move negation in and out o... |
| sbcim1 3833 | Distribution of class subs... |
| sbcim1OLD 3834 | Obsolete version of ~ sbci... |
| sbcbid 3835 | Formula-building deduction... |
| sbcbidv 3836 | Formula-building deduction... |
| sbcbii 3837 | Formula-building inference... |
| sbcbi1 3838 | Distribution of class subs... |
| sbcbi2 3839 | Substituting into equivale... |
| sbcbi2OLD 3840 | Obsolete proof of ~ sbcbi2... |
| sbcal 3841 | Move universal quantifier ... |
| sbcex2 3842 | Move existential quantifie... |
| sbceqal 3843 | Class version of one impli... |
| sbceqalOLD 3844 | Obsolete version of ~ sbce... |
| sbeqalb 3845 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3846 | Substitution for the right... |
| sbc3an 3847 | Distribution of class subs... |
| sbcel1v 3848 | Class substitution into a ... |
| sbcel2gv 3849 | Class substitution into a ... |
| sbcel21v 3850 | Class substitution into a ... |
| sbcimdv 3851 | Substitution analogue of T... |
| sbcimdvOLD 3852 | Obsolete version of ~ sbci... |
| sbctt 3853 | Substitution for a variabl... |
| sbcgf 3854 | Substitution for a variabl... |
| sbc19.21g 3855 | Substitution for a variabl... |
| sbcg 3856 | Substitution for a variabl... |
| sbcgOLD 3857 | Obsolete version of ~ sbcg... |
| sbcgfi 3858 | Substitution for a variabl... |
| sbc2iegf 3859 | Conversion of implicit sub... |
| sbc2ie 3860 | Conversion of implicit sub... |
| sbc2ieOLD 3861 | Obsolete version of ~ sbc2... |
| sbc2iedv 3862 | Conversion of implicit sub... |
| sbc3ie 3863 | Conversion of implicit sub... |
| sbccomlem 3864 | Lemma for ~ sbccom . (Con... |
| sbccom 3865 | Commutative law for double... |
| sbcralt 3866 | Interchange class substitu... |
| sbcrext 3867 | Interchange class substitu... |
| sbcralg 3868 | Interchange class substitu... |
| sbcrex 3869 | Interchange class substitu... |
| sbcreu 3870 | Interchange class substitu... |
| reu8nf 3871 | Restricted uniqueness usin... |
| sbcabel 3872 | Interchange class substitu... |
| rspsbc 3873 | Restricted quantifier vers... |
| rspsbca 3874 | Restricted quantifier vers... |
| rspesbca 3875 | Existence form of ~ rspsbc... |
| spesbc 3876 | Existence form of ~ spsbc ... |
| spesbcd 3877 | form of ~ spsbc . (Contri... |
| sbcth2 3878 | A substitution into a theo... |
| ra4v 3879 | Version of ~ ra4 with a di... |
| ra4 3880 | Restricted quantifier vers... |
| rmo2 3881 | Alternate definition of re... |
| rmo2i 3882 | Condition implying restric... |
| rmo3 3883 | Restricted "at most one" u... |
| rmob 3884 | Consequence of "at most on... |
| rmoi 3885 | Consequence of "at most on... |
| rmob2 3886 | Consequence of "restricted... |
| rmoi2 3887 | Consequence of "restricted... |
| rmoanim 3888 | Introduction of a conjunct... |
| rmoanimALT 3889 | Alternate proof of ~ rmoan... |
| reuan 3890 | Introduction of a conjunct... |
| 2reu1 3891 | Double restricted existent... |
| 2reu2 3892 | Double restricted existent... |
| csb2 3895 | Alternate expression for t... |
| csbeq1 3896 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3897 | Equality deduction for pro... |
| csbeq2 3898 | Substituting into equivale... |
| csbeq2d 3899 | Formula-building deduction... |
| csbeq2dv 3900 | Formula-building deduction... |
| csbeq2i 3901 | Formula-building inference... |
| csbeq12dv 3902 | Formula-building inference... |
| cbvcsbw 3903 | Change bound variables in ... |
| cbvcsb 3904 | Change bound variables in ... |
| cbvcsbv 3905 | Change the bound variable ... |
| csbid 3906 | Analogue of ~ sbid for pro... |
| csbeq1a 3907 | Equality theorem for prope... |
| csbcow 3908 | Composition law for chaine... |
| csbco 3909 | Composition law for chaine... |
| csbtt 3910 | Substitution doesn't affec... |
| csbconstgf 3911 | Substitution doesn't affec... |
| csbconstg 3912 | Substitution doesn't affec... |
| csbconstgOLD 3913 | Obsolete version of ~ csbc... |
| csbgfi 3914 | Substitution for a variabl... |
| csbconstgi 3915 | The proper substitution of... |
| nfcsb1d 3916 | Bound-variable hypothesis ... |
| nfcsb1 3917 | Bound-variable hypothesis ... |
| nfcsb1v 3918 | Bound-variable hypothesis ... |
| nfcsbd 3919 | Deduction version of ~ nfc... |
| nfcsbw 3920 | Bound-variable hypothesis ... |
| nfcsb 3921 | Bound-variable hypothesis ... |
| csbhypf 3922 | Introduce an explicit subs... |
| csbiebt 3923 | Conversion of implicit sub... |
| csbiedf 3924 | Conversion of implicit sub... |
| csbieb 3925 | Bidirectional conversion b... |
| csbiebg 3926 | Bidirectional conversion b... |
| csbiegf 3927 | Conversion of implicit sub... |
| csbief 3928 | Conversion of implicit sub... |
| csbie 3929 | Conversion of implicit sub... |
| csbieOLD 3930 | Obsolete version of ~ csbi... |
| csbied 3931 | Conversion of implicit sub... |
| csbiedOLD 3932 | Obsolete version of ~ csbi... |
| csbied2 3933 | Conversion of implicit sub... |
| csbie2t 3934 | Conversion of implicit sub... |
| csbie2 3935 | Conversion of implicit sub... |
| csbie2g 3936 | Conversion of implicit sub... |
| cbvrabcsfw 3937 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3938 | A more general version of ... |
| cbvrexcsf 3939 | A more general version of ... |
| cbvreucsf 3940 | A more general version of ... |
| cbvrabcsf 3941 | A more general version of ... |
| cbvralv2 3942 | Rule used to change the bo... |
| cbvrexv2 3943 | Rule used to change the bo... |
| rspc2vd 3944 | Deduction version of 2-var... |
| difjust 3950 | Soundness justification th... |
| unjust 3952 | Soundness justification th... |
| injust 3954 | Soundness justification th... |
| dfin5 3956 | Alternate definition for t... |
| dfdif2 3957 | Alternate definition of cl... |
| eldif 3958 | Expansion of membership in... |
| eldifd 3959 | If a class is in one class... |
| eldifad 3960 | If a class is in the diffe... |
| eldifbd 3961 | If a class is in the diffe... |
| elneeldif 3962 | The elements of a set diff... |
| velcomp 3963 | Characterization of setvar... |
| elin 3964 | Expansion of membership in... |
| dfss 3966 | Variant of subclass defini... |
| dfss2 3968 | Alternate definition of th... |
| dfss2OLD 3969 | Obsolete version of ~ dfss... |
| dfss3 3970 | Alternate definition of su... |
| dfss6 3971 | Alternate definition of su... |
| dfss2f 3972 | Equivalence for subclass r... |
| dfss3f 3973 | Equivalence for subclass r... |
| nfss 3974 | If ` x ` is not free in ` ... |
| ssel 3975 | Membership relationships f... |
| sselOLD 3976 | Obsolete version of ~ ssel... |
| ssel2 3977 | Membership relationships f... |
| sseli 3978 | Membership implication fro... |
| sselii 3979 | Membership inference from ... |
| sselid 3980 | Membership inference from ... |
| sseld 3981 | Membership deduction from ... |
| sselda 3982 | Membership deduction from ... |
| sseldd 3983 | Membership inference from ... |
| ssneld 3984 | If a class is not in anoth... |
| ssneldd 3985 | If an element is not in a ... |
| ssriv 3986 | Inference based on subclas... |
| ssrd 3987 | Deduction based on subclas... |
| ssrdv 3988 | Deduction based on subclas... |
| sstr2 3989 | Transitivity of subclass r... |
| sstr 3990 | Transitivity of subclass r... |
| sstri 3991 | Subclass transitivity infe... |
| sstrd 3992 | Subclass transitivity dedu... |
| sstrid 3993 | Subclass transitivity dedu... |
| sstrdi 3994 | Subclass transitivity dedu... |
| sylan9ss 3995 | A subclass transitivity de... |
| sylan9ssr 3996 | A subclass transitivity de... |
| eqss 3997 | The subclass relationship ... |
| eqssi 3998 | Infer equality from two su... |
| eqssd 3999 | Equality deduction from tw... |
| sssseq 4000 | If a class is a subclass o... |
| eqrd 4001 | Deduce equality of classes... |
| eqri 4002 | Infer equality of classes ... |
| eqelssd 4003 | Equality deduction from su... |
| ssid 4004 | Any class is a subclass of... |
| ssidd 4005 | Weakening of ~ ssid . (Co... |
| ssv 4006 | Any class is a subclass of... |
| sseq1 4007 | Equality theorem for subcl... |
| sseq2 4008 | Equality theorem for the s... |
| sseq12 4009 | Equality theorem for the s... |
| sseq1i 4010 | An equality inference for ... |
| sseq2i 4011 | An equality inference for ... |
| sseq12i 4012 | An equality inference for ... |
| sseq1d 4013 | An equality deduction for ... |
| sseq2d 4014 | An equality deduction for ... |
| sseq12d 4015 | An equality deduction for ... |
| eqsstri 4016 | Substitution of equality i... |
| eqsstrri 4017 | Substitution of equality i... |
| sseqtri 4018 | Substitution of equality i... |
| sseqtrri 4019 | Substitution of equality i... |
| eqsstrd 4020 | Substitution of equality i... |
| eqsstrrd 4021 | Substitution of equality i... |
| sseqtrd 4022 | Substitution of equality i... |
| sseqtrrd 4023 | Substitution of equality i... |
| 3sstr3i 4024 | Substitution of equality i... |
| 3sstr4i 4025 | Substitution of equality i... |
| 3sstr3g 4026 | Substitution of equality i... |
| 3sstr4g 4027 | Substitution of equality i... |
| 3sstr3d 4028 | Substitution of equality i... |
| 3sstr4d 4029 | Substitution of equality i... |
| eqsstrid 4030 | A chained subclass and equ... |
| eqsstrrid 4031 | A chained subclass and equ... |
| sseqtrdi 4032 | A chained subclass and equ... |
| sseqtrrdi 4033 | A chained subclass and equ... |
| sseqtrid 4034 | Subclass transitivity dedu... |
| sseqtrrid 4035 | Subclass transitivity dedu... |
| eqsstrdi 4036 | A chained subclass and equ... |
| eqsstrrdi 4037 | A chained subclass and equ... |
| eqimssd 4038 | Equality implies inclusion... |
| eqimsscd 4039 | Equality implies inclusion... |
| eqimss 4040 | Equality implies inclusion... |
| eqimss2 4041 | Equality implies inclusion... |
| eqimssi 4042 | Infer subclass relationshi... |
| eqimss2i 4043 | Infer subclass relationshi... |
| nssne1 4044 | Two classes are different ... |
| nssne2 4045 | Two classes are different ... |
| nss 4046 | Negation of subclass relat... |
| nelss 4047 | Demonstrate by witnesses t... |
| ssrexf 4048 | Restricted existential qua... |
| ssrmof 4049 | "At most one" existential ... |
| ssralv 4050 | Quantification restricted ... |
| ssrexv 4051 | Existential quantification... |
| ss2ralv 4052 | Two quantifications restri... |
| ss2rexv 4053 | Two existential quantifica... |
| ralss 4054 | Restricted universal quant... |
| rexss 4055 | Restricted existential qua... |
| ss2ab 4056 | Class abstractions in a su... |
| abss 4057 | Class abstraction in a sub... |
| ssab 4058 | Subclass of a class abstra... |
| ssabral 4059 | The relation for a subclas... |
| ss2abdv 4060 | Deduction of abstraction s... |
| ss2abdvALT 4061 | Alternate proof of ~ ss2ab... |
| ss2abdvOLD 4062 | Obsolete version of ~ ss2a... |
| ss2abi 4063 | Inference of abstraction s... |
| ss2abiOLD 4064 | Obsolete version of ~ ss2a... |
| abssdv 4065 | Deduction of abstraction s... |
| abssdvOLD 4066 | Obsolete version of ~ abss... |
| abssi 4067 | Inference of abstraction s... |
| ss2rab 4068 | Restricted abstraction cla... |
| rabss 4069 | Restricted class abstracti... |
| ssrab 4070 | Subclass of a restricted c... |
| ssrabdv 4071 | Subclass of a restricted c... |
| rabssdv 4072 | Subclass of a restricted c... |
| ss2rabdv 4073 | Deduction of restricted ab... |
| ss2rabi 4074 | Inference of restricted ab... |
| rabss2 4075 | Subclass law for restricte... |
| ssab2 4076 | Subclass relation for the ... |
| ssrab2 4077 | Subclass relation for a re... |
| ssrab2OLD 4078 | Obsolete version of ~ ssra... |
| rabss3d 4079 | Subclass law for restricte... |
| ssrab3 4080 | Subclass relation for a re... |
| rabssrabd 4081 | Subclass of a restricted c... |
| ssrabeq 4082 | If the restricting class o... |
| rabssab 4083 | A restricted class is a su... |
| uniiunlem 4084 | A subset relationship usef... |
| dfpss2 4085 | Alternate definition of pr... |
| dfpss3 4086 | Alternate definition of pr... |
| psseq1 4087 | Equality theorem for prope... |
| psseq2 4088 | Equality theorem for prope... |
| psseq1i 4089 | An equality inference for ... |
| psseq2i 4090 | An equality inference for ... |
| psseq12i 4091 | An equality inference for ... |
| psseq1d 4092 | An equality deduction for ... |
| psseq2d 4093 | An equality deduction for ... |
| psseq12d 4094 | An equality deduction for ... |
| pssss 4095 | A proper subclass is a sub... |
| pssne 4096 | Two classes in a proper su... |
| pssssd 4097 | Deduce subclass from prope... |
| pssned 4098 | Proper subclasses are uneq... |
| sspss 4099 | Subclass in terms of prope... |
| pssirr 4100 | Proper subclass is irrefle... |
| pssn2lp 4101 | Proper subclass has no 2-c... |
| sspsstri 4102 | Two ways of stating tricho... |
| ssnpss 4103 | Partial trichotomy law for... |
| psstr 4104 | Transitive law for proper ... |
| sspsstr 4105 | Transitive law for subclas... |
| psssstr 4106 | Transitive law for subclas... |
| psstrd 4107 | Proper subclass inclusion ... |
| sspsstrd 4108 | Transitivity involving sub... |
| psssstrd 4109 | Transitivity involving sub... |
| npss 4110 | A class is not a proper su... |
| ssnelpss 4111 | A subclass missing a membe... |
| ssnelpssd 4112 | Subclass inclusion with on... |
| ssexnelpss 4113 | If there is an element of ... |
| dfdif3 4114 | Alternate definition of cl... |
| difeq1 4115 | Equality theorem for class... |
| difeq2 4116 | Equality theorem for class... |
| difeq12 4117 | Equality theorem for class... |
| difeq1i 4118 | Inference adding differenc... |
| difeq2i 4119 | Inference adding differenc... |
| difeq12i 4120 | Equality inference for cla... |
| difeq1d 4121 | Deduction adding differenc... |
| difeq2d 4122 | Deduction adding differenc... |
| difeq12d 4123 | Equality deduction for cla... |
| difeqri 4124 | Inference from membership ... |
| nfdif 4125 | Bound-variable hypothesis ... |
| eldifi 4126 | Implication of membership ... |
| eldifn 4127 | Implication of membership ... |
| elndif 4128 | A set does not belong to a... |
| neldif 4129 | Implication of membership ... |
| difdif 4130 | Double class difference. ... |
| difss 4131 | Subclass relationship for ... |
| difssd 4132 | A difference of two classe... |
| difss2 4133 | If a class is contained in... |
| difss2d 4134 | If a class is contained in... |
| ssdifss 4135 | Preservation of a subclass... |
| ddif 4136 | Double complement under un... |
| ssconb 4137 | Contraposition law for sub... |
| sscon 4138 | Contraposition law for sub... |
| ssdif 4139 | Difference law for subsets... |
| ssdifd 4140 | If ` A ` is contained in `... |
| sscond 4141 | If ` A ` is contained in `... |
| ssdifssd 4142 | If ` A ` is contained in `... |
| ssdif2d 4143 | If ` A ` is contained in `... |
| raldifb 4144 | Restricted universal quant... |
| rexdifi 4145 | Restricted existential qua... |
| complss 4146 | Complementation reverses i... |
| compleq 4147 | Two classes are equal if a... |
| elun 4148 | Expansion of membership in... |
| elunnel1 4149 | A member of a union that i... |
| elunnel2 4150 | A member of a union that i... |
| uneqri 4151 | Inference from membership ... |
| unidm 4152 | Idempotent law for union o... |
| uncom 4153 | Commutative law for union ... |
| equncom 4154 | If a class equals the unio... |
| equncomi 4155 | Inference form of ~ equnco... |
| uneq1 4156 | Equality theorem for the u... |
| uneq2 4157 | Equality theorem for the u... |
| uneq12 4158 | Equality theorem for the u... |
| uneq1i 4159 | Inference adding union to ... |
| uneq2i 4160 | Inference adding union to ... |
| uneq12i 4161 | Equality inference for the... |
| uneq1d 4162 | Deduction adding union to ... |
| uneq2d 4163 | Deduction adding union to ... |
| uneq12d 4164 | Equality deduction for the... |
| nfun 4165 | Bound-variable hypothesis ... |
| unass 4166 | Associative law for union ... |
| un12 4167 | A rearrangement of union. ... |
| un23 4168 | A rearrangement of union. ... |
| un4 4169 | A rearrangement of the uni... |
| unundi 4170 | Union distributes over its... |
| unundir 4171 | Union distributes over its... |
| ssun1 4172 | Subclass relationship for ... |
| ssun2 4173 | Subclass relationship for ... |
| ssun3 4174 | Subclass law for union of ... |
| ssun4 4175 | Subclass law for union of ... |
| elun1 4176 | Membership law for union o... |
| elun2 4177 | Membership law for union o... |
| elunant 4178 | A statement is true for ev... |
| unss1 4179 | Subclass law for union of ... |
| ssequn1 4180 | A relationship between sub... |
| unss2 4181 | Subclass law for union of ... |
| unss12 4182 | Subclass law for union of ... |
| ssequn2 4183 | A relationship between sub... |
| unss 4184 | The union of two subclasse... |
| unssi 4185 | An inference showing the u... |
| unssd 4186 | A deduction showing the un... |
| unssad 4187 | If ` ( A u. B ) ` is conta... |
| unssbd 4188 | If ` ( A u. B ) ` is conta... |
| ssun 4189 | A condition that implies i... |
| rexun 4190 | Restricted existential qua... |
| ralunb 4191 | Restricted quantification ... |
| ralun 4192 | Restricted quantification ... |
| elini 4193 | Membership in an intersect... |
| elind 4194 | Deduce membership in an in... |
| elinel1 4195 | Membership in an intersect... |
| elinel2 4196 | Membership in an intersect... |
| elin2 4197 | Membership in a class defi... |
| elin1d 4198 | Elementhood in the first s... |
| elin2d 4199 | Elementhood in the first s... |
| elin3 4200 | Membership in a class defi... |
| incom 4201 | Commutative law for inters... |
| ineqcom 4202 | Two ways of expressing tha... |
| ineqcomi 4203 | Two ways of expressing tha... |
| ineqri 4204 | Inference from membership ... |
| ineq1 4205 | Equality theorem for inter... |
| ineq2 4206 | Equality theorem for inter... |
| ineq12 4207 | Equality theorem for inter... |
| ineq1i 4208 | Equality inference for int... |
| ineq2i 4209 | Equality inference for int... |
| ineq12i 4210 | Equality inference for int... |
| ineq1d 4211 | Equality deduction for int... |
| ineq2d 4212 | Equality deduction for int... |
| ineq12d 4213 | Equality deduction for int... |
| ineqan12d 4214 | Equality deduction for int... |
| sseqin2 4215 | A relationship between sub... |
| nfin 4216 | Bound-variable hypothesis ... |
| rabbi2dva 4217 | Deduction from a wff to a ... |
| inidm 4218 | Idempotent law for interse... |
| inass 4219 | Associative law for inters... |
| in12 4220 | A rearrangement of interse... |
| in32 4221 | A rearrangement of interse... |
| in13 4222 | A rearrangement of interse... |
| in31 4223 | A rearrangement of interse... |
| inrot 4224 | Rotate the intersection of... |
| in4 4225 | Rearrangement of intersect... |
| inindi 4226 | Intersection distributes o... |
| inindir 4227 | Intersection distributes o... |
| inss1 4228 | The intersection of two cl... |
| inss2 4229 | The intersection of two cl... |
| ssin 4230 | Subclass of intersection. ... |
| ssini 4231 | An inference showing that ... |
| ssind 4232 | A deduction showing that a... |
| ssrin 4233 | Add right intersection to ... |
| sslin 4234 | Add left intersection to s... |
| ssrind 4235 | Add right intersection to ... |
| ss2in 4236 | Intersection of subclasses... |
| ssinss1 4237 | Intersection preserves sub... |
| inss 4238 | Inclusion of an intersecti... |
| rexin 4239 | Restricted existential qua... |
| dfss7 4240 | Alternate definition of su... |
| symdifcom 4243 | Symmetric difference commu... |
| symdifeq1 4244 | Equality theorem for symme... |
| symdifeq2 4245 | Equality theorem for symme... |
| nfsymdif 4246 | Hypothesis builder for sym... |
| elsymdif 4247 | Membership in a symmetric ... |
| dfsymdif4 4248 | Alternate definition of th... |
| elsymdifxor 4249 | Membership in a symmetric ... |
| dfsymdif2 4250 | Alternate definition of th... |
| symdifass 4251 | Symmetric difference is as... |
| difsssymdif 4252 | The symmetric difference c... |
| difsymssdifssd 4253 | If the symmetric differenc... |
| unabs 4254 | Absorption law for union. ... |
| inabs 4255 | Absorption law for interse... |
| nssinpss 4256 | Negation of subclass expre... |
| nsspssun 4257 | Negation of subclass expre... |
| dfss4 4258 | Subclass defined in terms ... |
| dfun2 4259 | An alternate definition of... |
| dfin2 4260 | An alternate definition of... |
| difin 4261 | Difference with intersecti... |
| ssdifim 4262 | Implication of a class dif... |
| ssdifsym 4263 | Symmetric class difference... |
| dfss5 4264 | Alternate definition of su... |
| dfun3 4265 | Union defined in terms of ... |
| dfin3 4266 | Intersection defined in te... |
| dfin4 4267 | Alternate definition of th... |
| invdif 4268 | Intersection with universa... |
| indif 4269 | Intersection with class di... |
| indif2 4270 | Bring an intersection in a... |
| indif1 4271 | Bring an intersection in a... |
| indifcom 4272 | Commutation law for inters... |
| indi 4273 | Distributive law for inter... |
| undi 4274 | Distributive law for union... |
| indir 4275 | Distributive law for inter... |
| undir 4276 | Distributive law for union... |
| unineq 4277 | Infer equality from equali... |
| uneqin 4278 | Equality of union and inte... |
| difundi 4279 | Distributive law for class... |
| difundir 4280 | Distributive law for class... |
| difindi 4281 | Distributive law for class... |
| difindir 4282 | Distributive law for class... |
| indifdi 4283 | Distribute intersection ov... |
| indifdir 4284 | Distribute intersection ov... |
| indifdirOLD 4285 | Obsolete version of ~ indi... |
| difdif2 4286 | Class difference by a clas... |
| undm 4287 | De Morgan's law for union.... |
| indm 4288 | De Morgan's law for inters... |
| difun1 4289 | A relationship involving d... |
| undif3 4290 | An equality involving clas... |
| difin2 4291 | Represent a class differen... |
| dif32 4292 | Swap second and third argu... |
| difabs 4293 | Absorption-like law for cl... |
| sscon34b 4294 | Relative complementation r... |
| rcompleq 4295 | Two subclasses are equal i... |
| dfsymdif3 4296 | Alternate definition of th... |
| unabw 4297 | Union of two class abstrac... |
| unab 4298 | Union of two class abstrac... |
| inab 4299 | Intersection of two class ... |
| difab 4300 | Difference of two class ab... |
| abanssl 4301 | A class abstraction with a... |
| abanssr 4302 | A class abstraction with a... |
| notabw 4303 | A class abstraction define... |
| notab 4304 | A class abstraction define... |
| unrab 4305 | Union of two restricted cl... |
| inrab 4306 | Intersection of two restri... |
| inrab2 4307 | Intersection with a restri... |
| difrab 4308 | Difference of two restrict... |
| dfrab3 4309 | Alternate definition of re... |
| dfrab2 4310 | Alternate definition of re... |
| notrab 4311 | Complementation of restric... |
| dfrab3ss 4312 | Restricted class abstracti... |
| rabun2 4313 | Abstraction restricted to ... |
| reuun2 4314 | Transfer uniqueness to a s... |
| reuss2 4315 | Transfer uniqueness to a s... |
| reuss 4316 | Transfer uniqueness to a s... |
| reuun1 4317 | Transfer uniqueness to a s... |
| reupick 4318 | Restricted uniqueness "pic... |
| reupick3 4319 | Restricted uniqueness "pic... |
| reupick2 4320 | Restricted uniqueness "pic... |
| euelss 4321 | Transfer uniqueness of an ... |
| dfnul4 4324 | Alternate definition of th... |
| dfnul2 4325 | Alternate definition of th... |
| dfnul3 4326 | Alternate definition of th... |
| dfnul2OLD 4327 | Obsolete version of ~ dfnu... |
| dfnul3OLD 4328 | Obsolete version of ~ dfnu... |
| dfnul4OLD 4329 | Obsolete version of ~ dfnu... |
| noel 4330 | The empty set has no eleme... |
| noelOLD 4331 | Obsolete version of ~ noel... |
| nel02 4332 | The empty set has no eleme... |
| n0i 4333 | If a class has elements, t... |
| ne0i 4334 | If a class has elements, t... |
| ne0d 4335 | Deduction form of ~ ne0i .... |
| n0ii 4336 | If a class has elements, t... |
| ne0ii 4337 | If a class has elements, t... |
| vn0 4338 | The universal class is not... |
| vn0ALT 4339 | Alternate proof of ~ vn0 .... |
| eq0f 4340 | A class is equal to the em... |
| neq0f 4341 | A class is not empty if an... |
| n0f 4342 | A class is nonempty if and... |
| eq0 4343 | A class is equal to the em... |
| eq0ALT 4344 | Alternate proof of ~ eq0 .... |
| neq0 4345 | A class is not empty if an... |
| n0 4346 | A class is nonempty if and... |
| eq0OLDOLD 4347 | Obsolete version of ~ eq0 ... |
| neq0OLD 4348 | Obsolete version of ~ neq0... |
| n0OLD 4349 | Obsolete version of ~ n0 a... |
| nel0 4350 | From the general negation ... |
| reximdva0 4351 | Restricted existence deduc... |
| rspn0 4352 | Specialization for restric... |
| rspn0OLD 4353 | Obsolete version of ~ rspn... |
| n0rex 4354 | There is an element in a n... |
| ssn0rex 4355 | There is an element in a c... |
| n0moeu 4356 | A case of equivalence of "... |
| rex0 4357 | Vacuous restricted existen... |
| reu0 4358 | Vacuous restricted uniquen... |
| rmo0 4359 | Vacuous restricted at-most... |
| 0el 4360 | Membership of the empty se... |
| n0el 4361 | Negated membership of the ... |
| eqeuel 4362 | A condition which implies ... |
| ssdif0 4363 | Subclass expressed in term... |
| difn0 4364 | If the difference of two s... |
| pssdifn0 4365 | A proper subclass has a no... |
| pssdif 4366 | A proper subclass has a no... |
| ndisj 4367 | Express that an intersecti... |
| difin0ss 4368 | Difference, intersection, ... |
| inssdif0 4369 | Intersection, subclass, an... |
| difid 4370 | The difference between a c... |
| difidALT 4371 | Alternate proof of ~ difid... |
| dif0 4372 | The difference between a c... |
| ab0w 4373 | The class of sets verifyin... |
| ab0 4374 | The class of sets verifyin... |
| ab0OLD 4375 | Obsolete version of ~ ab0 ... |
| ab0ALT 4376 | Alternate proof of ~ ab0 ,... |
| dfnf5 4377 | Characterization of nonfre... |
| ab0orv 4378 | The class abstraction defi... |
| ab0orvALT 4379 | Alternate proof of ~ ab0or... |
| abn0 4380 | Nonempty class abstraction... |
| abn0OLD 4381 | Obsolete version of ~ abn0... |
| rab0 4382 | Any restricted class abstr... |
| rabeq0w 4383 | Condition for a restricted... |
| rabeq0 4384 | Condition for a restricted... |
| rabn0 4385 | Nonempty restricted class ... |
| rabxm 4386 | Law of excluded middle, in... |
| rabnc 4387 | Law of noncontradiction, i... |
| elneldisj 4388 | The set of elements ` s ` ... |
| elnelun 4389 | The union of the set of el... |
| un0 4390 | The union of a class with ... |
| in0 4391 | The intersection of a clas... |
| 0un 4392 | The union of the empty set... |
| 0in 4393 | The intersection of the em... |
| inv1 4394 | The intersection of a clas... |
| unv 4395 | The union of a class with ... |
| 0ss 4396 | The null set is a subset o... |
| ss0b 4397 | Any subset of the empty se... |
| ss0 4398 | Any subset of the empty se... |
| sseq0 4399 | A subclass of an empty cla... |
| ssn0 4400 | A class with a nonempty su... |
| 0dif 4401 | The difference between the... |
| abf 4402 | A class abstraction determ... |
| abfOLD 4403 | Obsolete version of ~ abf ... |
| eq0rdv 4404 | Deduction for equality to ... |
| eq0rdvALT 4405 | Alternate proof of ~ eq0rd... |
| csbprc 4406 | The proper substitution of... |
| csb0 4407 | The proper substitution of... |
| sbcel12 4408 | Distribute proper substitu... |
| sbceqg 4409 | Distribute proper substitu... |
| sbceqi 4410 | Distribution of class subs... |
| sbcnel12g 4411 | Distribute proper substitu... |
| sbcne12 4412 | Distribute proper substitu... |
| sbcel1g 4413 | Move proper substitution i... |
| sbceq1g 4414 | Move proper substitution t... |
| sbcel2 4415 | Move proper substitution i... |
| sbceq2g 4416 | Move proper substitution t... |
| csbcom 4417 | Commutative law for double... |
| sbcnestgfw 4418 | Nest the composition of tw... |
| csbnestgfw 4419 | Nest the composition of tw... |
| sbcnestgw 4420 | Nest the composition of tw... |
| csbnestgw 4421 | Nest the composition of tw... |
| sbcco3gw 4422 | Composition of two substit... |
| sbcnestgf 4423 | Nest the composition of tw... |
| csbnestgf 4424 | Nest the composition of tw... |
| sbcnestg 4425 | Nest the composition of tw... |
| csbnestg 4426 | Nest the composition of tw... |
| sbcco3g 4427 | Composition of two substit... |
| csbco3g 4428 | Composition of two class s... |
| csbnest1g 4429 | Nest the composition of tw... |
| csbidm 4430 | Idempotent law for class s... |
| csbvarg 4431 | The proper substitution of... |
| csbvargi 4432 | The proper substitution of... |
| sbccsb 4433 | Substitution into a wff ex... |
| sbccsb2 4434 | Substitution into a wff ex... |
| rspcsbela 4435 | Special case related to ~ ... |
| sbnfc2 4436 | Two ways of expressing " `... |
| csbab 4437 | Move substitution into a c... |
| csbun 4438 | Distribution of class subs... |
| csbin 4439 | Distribute proper substitu... |
| csbie2df 4440 | Conversion of implicit sub... |
| 2nreu 4441 | If there are two different... |
| un00 4442 | Two classes are empty iff ... |
| vss 4443 | Only the universal class h... |
| 0pss 4444 | The null set is a proper s... |
| npss0 4445 | No set is a proper subset ... |
| pssv 4446 | Any non-universal class is... |
| disj 4447 | Two ways of saying that tw... |
| disjOLD 4448 | Obsolete version of ~ disj... |
| disjr 4449 | Two ways of saying that tw... |
| disj1 4450 | Two ways of saying that tw... |
| reldisj 4451 | Two ways of saying that tw... |
| reldisjOLD 4452 | Obsolete version of ~ reld... |
| disj3 4453 | Two ways of saying that tw... |
| disjne 4454 | Members of disjoint sets a... |
| disjeq0 4455 | Two disjoint sets are equa... |
| disjel 4456 | A set can't belong to both... |
| disj2 4457 | Two ways of saying that tw... |
| disj4 4458 | Two ways of saying that tw... |
| ssdisj 4459 | Intersection with a subcla... |
| disjpss 4460 | A class is a proper subset... |
| undisj1 4461 | The union of disjoint clas... |
| undisj2 4462 | The union of disjoint clas... |
| ssindif0 4463 | Subclass expressed in term... |
| inelcm 4464 | The intersection of classe... |
| minel 4465 | A minimum element of a cla... |
| undif4 4466 | Distribute union over diff... |
| disjssun 4467 | Subset relation for disjoi... |
| vdif0 4468 | Universal class equality i... |
| difrab0eq 4469 | If the difference between ... |
| pssnel 4470 | A proper subclass has a me... |
| disjdif 4471 | A class and its relative c... |
| disjdifr 4472 | A class and its relative c... |
| difin0 4473 | The difference of a class ... |
| unvdif 4474 | The union of a class and i... |
| undif1 4475 | Absorption of difference b... |
| undif2 4476 | Absorption of difference b... |
| undifabs 4477 | Absorption of difference b... |
| inundif 4478 | The intersection and class... |
| disjdif2 4479 | The difference of a class ... |
| difun2 4480 | Absorption of union by dif... |
| undif 4481 | Union of complementary par... |
| undifr 4482 | Union of complementary par... |
| undifrOLD 4483 | Obsolete version of ~ undi... |
| undif5 4484 | An equality involving clas... |
| ssdifin0 4485 | A subset of a difference d... |
| ssdifeq0 4486 | A class is a subclass of i... |
| ssundif 4487 | A condition equivalent to ... |
| difcom 4488 | Swap the arguments of a cl... |
| pssdifcom1 4489 | Two ways to express overla... |
| pssdifcom2 4490 | Two ways to express non-co... |
| difdifdir 4491 | Distributive law for class... |
| uneqdifeq 4492 | Two ways to say that ` A `... |
| raldifeq 4493 | Equality theorem for restr... |
| r19.2z 4494 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4495 | A response to the notion t... |
| r19.3rz 4496 | Restricted quantification ... |
| r19.28z 4497 | Restricted quantifier vers... |
| r19.3rzv 4498 | Restricted quantification ... |
| r19.9rzv 4499 | Restricted quantification ... |
| r19.28zv 4500 | Restricted quantifier vers... |
| r19.37zv 4501 | Restricted quantifier vers... |
| r19.45zv 4502 | Restricted version of Theo... |
| r19.44zv 4503 | Restricted version of Theo... |
| r19.27z 4504 | Restricted quantifier vers... |
| r19.27zv 4505 | Restricted quantifier vers... |
| r19.36zv 4506 | Restricted quantifier vers... |
| ralidmw 4507 | Idempotent law for restric... |
| rzal 4508 | Vacuous quantification is ... |
| rzalALT 4509 | Alternate proof of ~ rzal ... |
| rexn0 4510 | Restricted existential qua... |
| ralidm 4511 | Idempotent law for restric... |
| ral0 4512 | Vacuous universal quantifi... |
| ralf0 4513 | The quantification of a fa... |
| rexn0OLD 4514 | Obsolete version of ~ rexn... |
| ralidmOLD 4515 | Obsolete version of ~ rali... |
| ral0OLD 4516 | Obsolete version of ~ ral0... |
| ralf0OLD 4517 | Obsolete version of ~ ralf... |
| ralnralall 4518 | A contradiction concerning... |
| falseral0 4519 | A false statement can only... |
| raaan 4520 | Rearrange restricted quant... |
| raaanv 4521 | Rearrange restricted quant... |
| sbss 4522 | Set substitution into the ... |
| sbcssg 4523 | Distribute proper substitu... |
| raaan2 4524 | Rearrange restricted quant... |
| 2reu4lem 4525 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4526 | Definition of double restr... |
| csbdif 4527 | Distribution of class subs... |
| dfif2 4530 | An alternate definition of... |
| dfif6 4531 | An alternate definition of... |
| ifeq1 4532 | Equality theorem for condi... |
| ifeq2 4533 | Equality theorem for condi... |
| iftrue 4534 | Value of the conditional o... |
| iftruei 4535 | Inference associated with ... |
| iftrued 4536 | Value of the conditional o... |
| iffalse 4537 | Value of the conditional o... |
| iffalsei 4538 | Inference associated with ... |
| iffalsed 4539 | Value of the conditional o... |
| ifnefalse 4540 | When values are unequal, b... |
| ifsb 4541 | Distribute a function over... |
| dfif3 4542 | Alternate definition of th... |
| dfif4 4543 | Alternate definition of th... |
| dfif5 4544 | Alternate definition of th... |
| ifssun 4545 | A conditional class is inc... |
| ifeq12 4546 | Equality theorem for condi... |
| ifeq1d 4547 | Equality deduction for con... |
| ifeq2d 4548 | Equality deduction for con... |
| ifeq12d 4549 | Equality deduction for con... |
| ifbi 4550 | Equivalence theorem for co... |
| ifbid 4551 | Equivalence deduction for ... |
| ifbieq1d 4552 | Equivalence/equality deduc... |
| ifbieq2i 4553 | Equivalence/equality infer... |
| ifbieq2d 4554 | Equivalence/equality deduc... |
| ifbieq12i 4555 | Equivalence deduction for ... |
| ifbieq12d 4556 | Equivalence deduction for ... |
| nfifd 4557 | Deduction form of ~ nfif .... |
| nfif 4558 | Bound-variable hypothesis ... |
| ifeq1da 4559 | Conditional equality. (Co... |
| ifeq2da 4560 | Conditional equality. (Co... |
| ifeq12da 4561 | Equivalence deduction for ... |
| ifbieq12d2 4562 | Equivalence deduction for ... |
| ifclda 4563 | Conditional closure. (Con... |
| ifeqda 4564 | Separation of the values o... |
| elimif 4565 | Elimination of a condition... |
| ifbothda 4566 | A wff ` th ` containing a ... |
| ifboth 4567 | A wff ` th ` containing a ... |
| ifid 4568 | Identical true and false a... |
| eqif 4569 | Expansion of an equality w... |
| ifval 4570 | Another expression of the ... |
| elif 4571 | Membership in a conditiona... |
| ifel 4572 | Membership of a conditiona... |
| ifcl 4573 | Membership (closure) of a ... |
| ifcld 4574 | Membership (closure) of a ... |
| ifcli 4575 | Inference associated with ... |
| ifexd 4576 | Existence of the condition... |
| ifexg 4577 | Existence of the condition... |
| ifex 4578 | Existence of the condition... |
| ifeqor 4579 | The possible values of a c... |
| ifnot 4580 | Negating the first argumen... |
| ifan 4581 | Rewrite a conjunction in a... |
| ifor 4582 | Rewrite a disjunction in a... |
| 2if2 4583 | Resolve two nested conditi... |
| ifcomnan 4584 | Commute the conditions in ... |
| csbif 4585 | Distribute proper substitu... |
| dedth 4586 | Weak deduction theorem tha... |
| dedth2h 4587 | Weak deduction theorem eli... |
| dedth3h 4588 | Weak deduction theorem eli... |
| dedth4h 4589 | Weak deduction theorem eli... |
| dedth2v 4590 | Weak deduction theorem for... |
| dedth3v 4591 | Weak deduction theorem for... |
| dedth4v 4592 | Weak deduction theorem for... |
| elimhyp 4593 | Eliminate a hypothesis con... |
| elimhyp2v 4594 | Eliminate a hypothesis con... |
| elimhyp3v 4595 | Eliminate a hypothesis con... |
| elimhyp4v 4596 | Eliminate a hypothesis con... |
| elimel 4597 | Eliminate a membership hyp... |
| elimdhyp 4598 | Version of ~ elimhyp where... |
| keephyp 4599 | Transform a hypothesis ` p... |
| keephyp2v 4600 | Keep a hypothesis containi... |
| keephyp3v 4601 | Keep a hypothesis containi... |
| pwjust 4603 | Soundness justification th... |
| elpwg 4605 | Membership in a power clas... |
| elpw 4606 | Membership in a power clas... |
| velpw 4607 | Setvar variable membership... |
| elpwd 4608 | Membership in a power clas... |
| elpwi 4609 | Subset relation implied by... |
| elpwb 4610 | Characterization of the el... |
| elpwid 4611 | An element of a power clas... |
| elelpwi 4612 | If ` A ` belongs to a part... |
| sspw 4613 | The powerclass preserves i... |
| sspwi 4614 | The powerclass preserves i... |
| sspwd 4615 | The powerclass preserves i... |
| pweq 4616 | Equality theorem for power... |
| pweqALT 4617 | Alternate proof of ~ pweq ... |
| pweqi 4618 | Equality inference for pow... |
| pweqd 4619 | Equality deduction for pow... |
| pwunss 4620 | The power class of the uni... |
| nfpw 4621 | Bound-variable hypothesis ... |
| pwidg 4622 | A set is an element of its... |
| pwidb 4623 | A class is an element of i... |
| pwid 4624 | A set is a member of its p... |
| pwss 4625 | Subclass relationship for ... |
| pwundif 4626 | Break up the power class o... |
| snjust 4627 | Soundness justification th... |
| sneq 4638 | Equality theorem for singl... |
| sneqi 4639 | Equality inference for sin... |
| sneqd 4640 | Equality deduction for sin... |
| dfsn2 4641 | Alternate definition of si... |
| elsng 4642 | There is exactly one eleme... |
| elsn 4643 | There is exactly one eleme... |
| velsn 4644 | There is only one element ... |
| elsni 4645 | There is at most one eleme... |
| absn 4646 | Condition for a class abst... |
| dfpr2 4647 | Alternate definition of a ... |
| dfsn2ALT 4648 | Alternate definition of si... |
| elprg 4649 | A member of a pair of clas... |
| elpri 4650 | If a class is an element o... |
| elpr 4651 | A member of a pair of clas... |
| elpr2g 4652 | A member of a pair of sets... |
| elpr2 4653 | A member of a pair of sets... |
| elpr2OLD 4654 | Obsolete version of ~ elpr... |
| nelpr2 4655 | If a class is not an eleme... |
| nelpr1 4656 | If a class is not an eleme... |
| nelpri 4657 | If an element doesn't matc... |
| prneli 4658 | If an element doesn't matc... |
| nelprd 4659 | If an element doesn't matc... |
| eldifpr 4660 | Membership in a set with t... |
| rexdifpr 4661 | Restricted existential qua... |
| snidg 4662 | A set is a member of its s... |
| snidb 4663 | A class is a set iff it is... |
| snid 4664 | A set is a member of its s... |
| vsnid 4665 | A setvar variable is a mem... |
| elsn2g 4666 | There is exactly one eleme... |
| elsn2 4667 | There is exactly one eleme... |
| nelsn 4668 | If a class is not equal to... |
| rabeqsn 4669 | Conditions for a restricte... |
| rabsssn 4670 | Conditions for a restricte... |
| rabeqsnd 4671 | Conditions for a restricte... |
| ralsnsg 4672 | Substitution expressed in ... |
| rexsns 4673 | Restricted existential qua... |
| rexsngf 4674 | Restricted existential qua... |
| ralsngf 4675 | Restricted universal quant... |
| reusngf 4676 | Restricted existential uni... |
| ralsng 4677 | Substitution expressed in ... |
| rexsng 4678 | Restricted existential qua... |
| reusng 4679 | Restricted existential uni... |
| 2ralsng 4680 | Substitution expressed in ... |
| ralsngOLD 4681 | Obsolete version of ~ rals... |
| rexsngOLD 4682 | Obsolete version of ~ rexs... |
| rexreusng 4683 | Restricted existential uni... |
| exsnrex 4684 | There is a set being the e... |
| ralsn 4685 | Convert a universal quanti... |
| rexsn 4686 | Convert an existential qua... |
| elpwunsn 4687 | Membership in an extension... |
| eqoreldif 4688 | An element of a set is eit... |
| eltpg 4689 | Members of an unordered tr... |
| eldiftp 4690 | Membership in a set with t... |
| eltpi 4691 | A member of an unordered t... |
| eltp 4692 | A member of an unordered t... |
| dftp2 4693 | Alternate definition of un... |
| nfpr 4694 | Bound-variable hypothesis ... |
| ifpr 4695 | Membership of a conditiona... |
| ralprgf 4696 | Convert a restricted unive... |
| rexprgf 4697 | Convert a restricted exist... |
| ralprg 4698 | Convert a restricted unive... |
| ralprgOLD 4699 | Obsolete version of ~ ralp... |
| rexprg 4700 | Convert a restricted exist... |
| rexprgOLD 4701 | Obsolete version of ~ rexp... |
| raltpg 4702 | Convert a restricted unive... |
| rextpg 4703 | Convert a restricted exist... |
| ralpr 4704 | Convert a restricted unive... |
| rexpr 4705 | Convert a restricted exist... |
| reuprg0 4706 | Convert a restricted exist... |
| reuprg 4707 | Convert a restricted exist... |
| reurexprg 4708 | Convert a restricted exist... |
| raltp 4709 | Convert a universal quanti... |
| rextp 4710 | Convert an existential qua... |
| nfsn 4711 | Bound-variable hypothesis ... |
| csbsng 4712 | Distribute proper substitu... |
| csbprg 4713 | Distribute proper substitu... |
| elinsn 4714 | If the intersection of two... |
| disjsn 4715 | Intersection with the sing... |
| disjsn2 4716 | Two distinct singletons ar... |
| disjpr2 4717 | Two completely distinct un... |
| disjprsn 4718 | The disjoint intersection ... |
| disjtpsn 4719 | The disjoint intersection ... |
| disjtp2 4720 | Two completely distinct un... |
| snprc 4721 | The singleton of a proper ... |
| snnzb 4722 | A singleton is nonempty if... |
| rmosn 4723 | A restricted at-most-one q... |
| r19.12sn 4724 | Special case of ~ r19.12 w... |
| rabsn 4725 | Condition where a restrict... |
| rabsnifsb 4726 | A restricted class abstrac... |
| rabsnif 4727 | A restricted class abstrac... |
| rabrsn 4728 | A restricted class abstrac... |
| euabsn2 4729 | Another way to express exi... |
| euabsn 4730 | Another way to express exi... |
| reusn 4731 | A way to express restricte... |
| absneu 4732 | Restricted existential uni... |
| rabsneu 4733 | Restricted existential uni... |
| eusn 4734 | Two ways to express " ` A ... |
| rabsnt 4735 | Truth implied by equality ... |
| prcom 4736 | Commutative law for unorde... |
| preq1 4737 | Equality theorem for unord... |
| preq2 4738 | Equality theorem for unord... |
| preq12 4739 | Equality theorem for unord... |
| preq1i 4740 | Equality inference for uno... |
| preq2i 4741 | Equality inference for uno... |
| preq12i 4742 | Equality inference for uno... |
| preq1d 4743 | Equality deduction for uno... |
| preq2d 4744 | Equality deduction for uno... |
| preq12d 4745 | Equality deduction for uno... |
| tpeq1 4746 | Equality theorem for unord... |
| tpeq2 4747 | Equality theorem for unord... |
| tpeq3 4748 | Equality theorem for unord... |
| tpeq1d 4749 | Equality theorem for unord... |
| tpeq2d 4750 | Equality theorem for unord... |
| tpeq3d 4751 | Equality theorem for unord... |
| tpeq123d 4752 | Equality theorem for unord... |
| tprot 4753 | Rotation of the elements o... |
| tpcoma 4754 | Swap 1st and 2nd members o... |
| tpcomb 4755 | Swap 2nd and 3rd members o... |
| tpass 4756 | Split off the first elemen... |
| qdass 4757 | Two ways to write an unord... |
| qdassr 4758 | Two ways to write an unord... |
| tpidm12 4759 | Unordered triple ` { A , A... |
| tpidm13 4760 | Unordered triple ` { A , B... |
| tpidm23 4761 | Unordered triple ` { A , B... |
| tpidm 4762 | Unordered triple ` { A , A... |
| tppreq3 4763 | An unordered triple is an ... |
| prid1g 4764 | An unordered pair contains... |
| prid2g 4765 | An unordered pair contains... |
| prid1 4766 | An unordered pair contains... |
| prid2 4767 | An unordered pair contains... |
| ifpprsnss 4768 | An unordered pair is a sin... |
| prprc1 4769 | A proper class vanishes in... |
| prprc2 4770 | A proper class vanishes in... |
| prprc 4771 | An unordered pair containi... |
| tpid1 4772 | One of the three elements ... |
| tpid1g 4773 | Closed theorem form of ~ t... |
| tpid2 4774 | One of the three elements ... |
| tpid2g 4775 | Closed theorem form of ~ t... |
| tpid3g 4776 | Closed theorem form of ~ t... |
| tpid3 4777 | One of the three elements ... |
| snnzg 4778 | The singleton of a set is ... |
| snn0d 4779 | The singleton of a set is ... |
| snnz 4780 | The singleton of a set is ... |
| prnz 4781 | A pair containing a set is... |
| prnzg 4782 | A pair containing a set is... |
| tpnz 4783 | An unordered triple contai... |
| tpnzd 4784 | An unordered triple contai... |
| raltpd 4785 | Convert a universal quanti... |
| snssb 4786 | Characterization of the in... |
| snssg 4787 | The singleton formed on a ... |
| snssgOLD 4788 | Obsolete version of ~ snss... |
| snss 4789 | The singleton of an elemen... |
| eldifsn 4790 | Membership in a set with a... |
| ssdifsn 4791 | Subset of a set with an el... |
| elpwdifsn 4792 | A subset of a set is an el... |
| eldifsni 4793 | Membership in a set with a... |
| eldifsnneq 4794 | An element of a difference... |
| neldifsn 4795 | The class ` A ` is not in ... |
| neldifsnd 4796 | The class ` A ` is not in ... |
| rexdifsn 4797 | Restricted existential qua... |
| raldifsni 4798 | Rearrangement of a propert... |
| raldifsnb 4799 | Restricted universal quant... |
| eldifvsn 4800 | A set is an element of the... |
| difsn 4801 | An element not in a set ca... |
| difprsnss 4802 | Removal of a singleton fro... |
| difprsn1 4803 | Removal of a singleton fro... |
| difprsn2 4804 | Removal of a singleton fro... |
| diftpsn3 4805 | Removal of a singleton fro... |
| difpr 4806 | Removing two elements as p... |
| tpprceq3 4807 | An unordered triple is an ... |
| tppreqb 4808 | An unordered triple is an ... |
| difsnb 4809 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4810 | ` ( B \ { A } ) ` is a pro... |
| snssi 4811 | The singleton of an elemen... |
| snssd 4812 | The singleton of an elemen... |
| difsnid 4813 | If we remove a single elem... |
| eldifeldifsn 4814 | An element of a difference... |
| pw0 4815 | Compute the power set of t... |
| pwpw0 4816 | Compute the power set of t... |
| snsspr1 4817 | A singleton is a subset of... |
| snsspr2 4818 | A singleton is a subset of... |
| snsstp1 4819 | A singleton is a subset of... |
| snsstp2 4820 | A singleton is a subset of... |
| snsstp3 4821 | A singleton is a subset of... |
| prssg 4822 | A pair of elements of a cl... |
| prss 4823 | A pair of elements of a cl... |
| prssi 4824 | A pair of elements of a cl... |
| prssd 4825 | Deduction version of ~ prs... |
| prsspwg 4826 | An unordered pair belongs ... |
| ssprss 4827 | A pair as subset of a pair... |
| ssprsseq 4828 | A proper pair is a subset ... |
| sssn 4829 | The subsets of a singleton... |
| ssunsn2 4830 | The property of being sand... |
| ssunsn 4831 | Possible values for a set ... |
| eqsn 4832 | Two ways to express that a... |
| issn 4833 | A sufficient condition for... |
| n0snor2el 4834 | A nonempty set is either a... |
| ssunpr 4835 | Possible values for a set ... |
| sspr 4836 | The subsets of a pair. (C... |
| sstp 4837 | The subsets of an unordere... |
| tpss 4838 | An unordered triple of ele... |
| tpssi 4839 | An unordered triple of ele... |
| sneqrg 4840 | Closed form of ~ sneqr . ... |
| sneqr 4841 | If the singletons of two s... |
| snsssn 4842 | If a singleton is a subset... |
| mosneq 4843 | There exists at most one s... |
| sneqbg 4844 | Two singletons of sets are... |
| snsspw 4845 | The singleton of a class i... |
| prsspw 4846 | An unordered pair belongs ... |
| preq1b 4847 | Biconditional equality lem... |
| preq2b 4848 | Biconditional equality lem... |
| preqr1 4849 | Reverse equality lemma for... |
| preqr2 4850 | Reverse equality lemma for... |
| preq12b 4851 | Equality relationship for ... |
| opthpr 4852 | An unordered pair has the ... |
| preqr1g 4853 | Reverse equality lemma for... |
| preq12bg 4854 | Closed form of ~ preq12b .... |
| prneimg 4855 | Two pairs are not equal if... |
| prnebg 4856 | A (proper) pair is not equ... |
| pr1eqbg 4857 | A (proper) pair is equal t... |
| pr1nebg 4858 | A (proper) pair is not equ... |
| preqsnd 4859 | Equivalence for a pair equ... |
| prnesn 4860 | A proper unordered pair is... |
| prneprprc 4861 | A proper unordered pair is... |
| preqsn 4862 | Equivalence for a pair equ... |
| preq12nebg 4863 | Equality relationship for ... |
| prel12g 4864 | Equality of two unordered ... |
| opthprneg 4865 | An unordered pair has the ... |
| elpreqprlem 4866 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4867 | Equality and membership ru... |
| elpreqprb 4868 | A set is an element of an ... |
| elpr2elpr 4869 | For an element ` A ` of an... |
| dfopif 4870 | Rewrite ~ df-op using ` if... |
| dfopg 4871 | Value of the ordered pair ... |
| dfop 4872 | Value of an ordered pair w... |
| opeq1 4873 | Equality theorem for order... |
| opeq2 4874 | Equality theorem for order... |
| opeq12 4875 | Equality theorem for order... |
| opeq1i 4876 | Equality inference for ord... |
| opeq2i 4877 | Equality inference for ord... |
| opeq12i 4878 | Equality inference for ord... |
| opeq1d 4879 | Equality deduction for ord... |
| opeq2d 4880 | Equality deduction for ord... |
| opeq12d 4881 | Equality deduction for ord... |
| oteq1 4882 | Equality theorem for order... |
| oteq2 4883 | Equality theorem for order... |
| oteq3 4884 | Equality theorem for order... |
| oteq1d 4885 | Equality deduction for ord... |
| oteq2d 4886 | Equality deduction for ord... |
| oteq3d 4887 | Equality deduction for ord... |
| oteq123d 4888 | Equality deduction for ord... |
| nfop 4889 | Bound-variable hypothesis ... |
| nfopd 4890 | Deduction version of bound... |
| csbopg 4891 | Distribution of class subs... |
| opidg 4892 | The ordered pair ` <. A , ... |
| opid 4893 | The ordered pair ` <. A , ... |
| ralunsn 4894 | Restricted quantification ... |
| 2ralunsn 4895 | Double restricted quantifi... |
| opprc 4896 | Expansion of an ordered pa... |
| opprc1 4897 | Expansion of an ordered pa... |
| opprc2 4898 | Expansion of an ordered pa... |
| oprcl 4899 | If an ordered pair has an ... |
| pwsn 4900 | The power set of a singlet... |
| pwsnOLD 4901 | Obsolete version of ~ pwsn... |
| pwpr 4902 | The power set of an unorde... |
| pwtp 4903 | The power set of an unorde... |
| pwpwpw0 4904 | Compute the power set of t... |
| pwv 4905 | The power class of the uni... |
| prproe 4906 | For an element of a proper... |
| 3elpr2eq 4907 | If there are three element... |
| dfuni2 4910 | Alternate definition of cl... |
| eluni 4911 | Membership in class union.... |
| eluni2 4912 | Membership in class union.... |
| elunii 4913 | Membership in class union.... |
| nfunid 4914 | Deduction version of ~ nfu... |
| nfuni 4915 | Bound-variable hypothesis ... |
| uniss 4916 | Subclass relationship for ... |
| unissi 4917 | Subclass relationship for ... |
| unissd 4918 | Subclass relationship for ... |
| unieq 4919 | Equality theorem for class... |
| unieqOLD 4920 | Obsolete version of ~ unie... |
| unieqi 4921 | Inference of equality of t... |
| unieqd 4922 | Deduction of equality of t... |
| eluniab 4923 | Membership in union of a c... |
| elunirab 4924 | Membership in union of a c... |
| uniprg 4925 | The union of a pair is the... |
| unipr 4926 | The union of a pair is the... |
| uniprOLD 4927 | Obsolete version of ~ unip... |
| uniprgOLD 4928 | Obsolete version of ~ unip... |
| unisng 4929 | A set equals the union of ... |
| unisn 4930 | A set equals the union of ... |
| unisnv 4931 | A set equals the union of ... |
| unisn3 4932 | Union of a singleton in th... |
| dfnfc2 4933 | An alternative statement o... |
| uniun 4934 | The class union of the uni... |
| uniin 4935 | The class union of the int... |
| ssuni 4936 | Subclass relationship for ... |
| uni0b 4937 | The union of a set is empt... |
| uni0c 4938 | The union of a set is empt... |
| uni0 4939 | The union of the empty set... |
| csbuni 4940 | Distribute proper substitu... |
| elssuni 4941 | An element of a class is a... |
| unissel 4942 | Condition turning a subcla... |
| unissb 4943 | Relationship involving mem... |
| unissbOLD 4944 | Obsolete version of ~ unis... |
| uniss2 4945 | A subclass condition on th... |
| unidif 4946 | If the difference ` A \ B ... |
| ssunieq 4947 | Relationship implying unio... |
| unimax 4948 | Any member of a class is t... |
| pwuni 4949 | A class is a subclass of t... |
| dfint2 4952 | Alternate definition of cl... |
| inteq 4953 | Equality law for intersect... |
| inteqi 4954 | Equality inference for cla... |
| inteqd 4955 | Equality deduction for cla... |
| elint 4956 | Membership in class inters... |
| elint2 4957 | Membership in class inters... |
| elintg 4958 | Membership in class inters... |
| elinti 4959 | Membership in class inters... |
| nfint 4960 | Bound-variable hypothesis ... |
| elintabg 4961 | Two ways of saying a set i... |
| elintab 4962 | Membership in the intersec... |
| elintabOLD 4963 | Obsolete version of ~ elin... |
| elintrab 4964 | Membership in the intersec... |
| elintrabg 4965 | Membership in the intersec... |
| int0 4966 | The intersection of the em... |
| intss1 4967 | An element of a class incl... |
| ssint 4968 | Subclass of a class inters... |
| ssintab 4969 | Subclass of the intersecti... |
| ssintub 4970 | Subclass of the least uppe... |
| ssmin 4971 | Subclass of the minimum va... |
| intmin 4972 | Any member of a class is t... |
| intss 4973 | Intersection of subclasses... |
| intssuni 4974 | The intersection of a none... |
| ssintrab 4975 | Subclass of the intersecti... |
| unissint 4976 | If the union of a class is... |
| intssuni2 4977 | Subclass relationship for ... |
| intminss 4978 | Under subset ordering, the... |
| intmin2 4979 | Any set is the smallest of... |
| intmin3 4980 | Under subset ordering, the... |
| intmin4 4981 | Elimination of a conjunct ... |
| intab 4982 | The intersection of a spec... |
| int0el 4983 | The intersection of a clas... |
| intun 4984 | The class intersection of ... |
| intprg 4985 | The intersection of a pair... |
| intpr 4986 | The intersection of a pair... |
| intprOLD 4987 | Obsolete version of ~ intp... |
| intprgOLD 4988 | Obsolete version of ~ intp... |
| intsng 4989 | Intersection of a singleto... |
| intsn 4990 | The intersection of a sing... |
| uniintsn 4991 | Two ways to express " ` A ... |
| uniintab 4992 | The union and the intersec... |
| intunsn 4993 | Theorem joining a singleto... |
| rint0 4994 | Relative intersection of a... |
| elrint 4995 | Membership in a restricted... |
| elrint2 4996 | Membership in a restricted... |
| eliun 5001 | Membership in indexed unio... |
| eliin 5002 | Membership in indexed inte... |
| eliuni 5003 | Membership in an indexed u... |
| iuncom 5004 | Commutation of indexed uni... |
| iuncom4 5005 | Commutation of union with ... |
| iunconst 5006 | Indexed union of a constan... |
| iinconst 5007 | Indexed intersection of a ... |
| iuneqconst 5008 | Indexed union of identical... |
| iuniin 5009 | Law combining indexed unio... |
| iinssiun 5010 | An indexed intersection is... |
| iunss1 5011 | Subclass theorem for index... |
| iinss1 5012 | Subclass theorem for index... |
| iuneq1 5013 | Equality theorem for index... |
| iineq1 5014 | Equality theorem for index... |
| ss2iun 5015 | Subclass theorem for index... |
| iuneq2 5016 | Equality theorem for index... |
| iineq2 5017 | Equality theorem for index... |
| iuneq2i 5018 | Equality inference for ind... |
| iineq2i 5019 | Equality inference for ind... |
| iineq2d 5020 | Equality deduction for ind... |
| iuneq2dv 5021 | Equality deduction for ind... |
| iineq2dv 5022 | Equality deduction for ind... |
| iuneq12df 5023 | Equality deduction for ind... |
| iuneq1d 5024 | Equality theorem for index... |
| iuneq12d 5025 | Equality deduction for ind... |
| iuneq2d 5026 | Equality deduction for ind... |
| nfiun 5027 | Bound-variable hypothesis ... |
| nfiin 5028 | Bound-variable hypothesis ... |
| nfiung 5029 | Bound-variable hypothesis ... |
| nfiing 5030 | Bound-variable hypothesis ... |
| nfiu1 5031 | Bound-variable hypothesis ... |
| nfii1 5032 | Bound-variable hypothesis ... |
| dfiun2g 5033 | Alternate definition of in... |
| dfiun2gOLD 5034 | Obsolete version of ~ dfiu... |
| dfiin2g 5035 | Alternate definition of in... |
| dfiun2 5036 | Alternate definition of in... |
| dfiin2 5037 | Alternate definition of in... |
| dfiunv2 5038 | Define double indexed unio... |
| cbviun 5039 | Rule used to change the bo... |
| cbviin 5040 | Change bound variables in ... |
| cbviung 5041 | Rule used to change the bo... |
| cbviing 5042 | Change bound variables in ... |
| cbviunv 5043 | Rule used to change the bo... |
| cbviinv 5044 | Change bound variables in ... |
| cbviunvg 5045 | Rule used to change the bo... |
| cbviinvg 5046 | Change bound variables in ... |
| iunssf 5047 | Subset theorem for an inde... |
| iunss 5048 | Subset theorem for an inde... |
| ssiun 5049 | Subset implication for an ... |
| ssiun2 5050 | Identity law for subset of... |
| ssiun2s 5051 | Subset relationship for an... |
| iunss2 5052 | A subclass condition on th... |
| iunssd 5053 | Subset theorem for an inde... |
| iunab 5054 | The indexed union of a cla... |
| iunrab 5055 | The indexed union of a res... |
| iunxdif2 5056 | Indexed union with a class... |
| ssiinf 5057 | Subset theorem for an inde... |
| ssiin 5058 | Subset theorem for an inde... |
| iinss 5059 | Subset implication for an ... |
| iinss2 5060 | An indexed intersection is... |
| uniiun 5061 | Class union in terms of in... |
| intiin 5062 | Class intersection in term... |
| iunid 5063 | An indexed union of single... |
| iunidOLD 5064 | Obsolete version of ~ iuni... |
| iun0 5065 | An indexed union of the em... |
| 0iun 5066 | An empty indexed union is ... |
| 0iin 5067 | An empty indexed intersect... |
| viin 5068 | Indexed intersection with ... |
| iunsn 5069 | Indexed union of a singlet... |
| iunn0 5070 | There is a nonempty class ... |
| iinab 5071 | Indexed intersection of a ... |
| iinrab 5072 | Indexed intersection of a ... |
| iinrab2 5073 | Indexed intersection of a ... |
| iunin2 5074 | Indexed union of intersect... |
| iunin1 5075 | Indexed union of intersect... |
| iinun2 5076 | Indexed intersection of un... |
| iundif2 5077 | Indexed union of class dif... |
| iindif1 5078 | Indexed intersection of cl... |
| 2iunin 5079 | Rearrange indexed unions o... |
| iindif2 5080 | Indexed intersection of cl... |
| iinin2 5081 | Indexed intersection of in... |
| iinin1 5082 | Indexed intersection of in... |
| iinvdif 5083 | The indexed intersection o... |
| elriin 5084 | Elementhood in a relative ... |
| riin0 5085 | Relative intersection of a... |
| riinn0 5086 | Relative intersection of a... |
| riinrab 5087 | Relative intersection of a... |
| symdif0 5088 | Symmetric difference with ... |
| symdifv 5089 | The symmetric difference w... |
| symdifid 5090 | The symmetric difference o... |
| iinxsng 5091 | A singleton index picks ou... |
| iinxprg 5092 | Indexed intersection with ... |
| iunxsng 5093 | A singleton index picks ou... |
| iunxsn 5094 | A singleton index picks ou... |
| iunxsngf 5095 | A singleton index picks ou... |
| iunun 5096 | Separate a union in an ind... |
| iunxun 5097 | Separate a union in the in... |
| iunxdif3 5098 | An indexed union where som... |
| iunxprg 5099 | A pair index picks out two... |
| iunxiun 5100 | Separate an indexed union ... |
| iinuni 5101 | A relationship involving u... |
| iununi 5102 | A relationship involving u... |
| sspwuni 5103 | Subclass relationship for ... |
| pwssb 5104 | Two ways to express a coll... |
| elpwpw 5105 | Characterization of the el... |
| pwpwab 5106 | The double power class wri... |
| pwpwssunieq 5107 | The class of sets whose un... |
| elpwuni 5108 | Relationship for power cla... |
| iinpw 5109 | The power class of an inte... |
| iunpwss 5110 | Inclusion of an indexed un... |
| intss2 5111 | A nonempty intersection of... |
| rintn0 5112 | Relative intersection of a... |
| dfdisj2 5115 | Alternate definition for d... |
| disjss2 5116 | If each element of a colle... |
| disjeq2 5117 | Equality theorem for disjo... |
| disjeq2dv 5118 | Equality deduction for dis... |
| disjss1 5119 | A subset of a disjoint col... |
| disjeq1 5120 | Equality theorem for disjo... |
| disjeq1d 5121 | Equality theorem for disjo... |
| disjeq12d 5122 | Equality theorem for disjo... |
| cbvdisj 5123 | Change bound variables in ... |
| cbvdisjv 5124 | Change bound variables in ... |
| nfdisjw 5125 | Bound-variable hypothesis ... |
| nfdisj 5126 | Bound-variable hypothesis ... |
| nfdisj1 5127 | Bound-variable hypothesis ... |
| disjor 5128 | Two ways to say that a col... |
| disjors 5129 | Two ways to say that a col... |
| disji2 5130 | Property of a disjoint col... |
| disji 5131 | Property of a disjoint col... |
| invdisj 5132 | If there is a function ` C... |
| invdisjrabw 5133 | Version of ~ invdisjrab wi... |
| invdisjrab 5134 | The restricted class abstr... |
| disjiun 5135 | A disjoint collection yiel... |
| disjord 5136 | Conditions for a collectio... |
| disjiunb 5137 | Two ways to say that a col... |
| disjiund 5138 | Conditions for a collectio... |
| sndisj 5139 | Any collection of singleto... |
| 0disj 5140 | Any collection of empty se... |
| disjxsn 5141 | A singleton collection is ... |
| disjx0 5142 | An empty collection is dis... |
| disjprgw 5143 | Version of ~ disjprg with ... |
| disjprg 5144 | A pair collection is disjo... |
| disjxiun 5145 | An indexed union of a disj... |
| disjxun 5146 | The union of two disjoint ... |
| disjss3 5147 | Expand a disjoint collecti... |
| breq 5150 | Equality theorem for binar... |
| breq1 5151 | Equality theorem for a bin... |
| breq2 5152 | Equality theorem for a bin... |
| breq12 5153 | Equality theorem for a bin... |
| breqi 5154 | Equality inference for bin... |
| breq1i 5155 | Equality inference for a b... |
| breq2i 5156 | Equality inference for a b... |
| breq12i 5157 | Equality inference for a b... |
| breq1d 5158 | Equality deduction for a b... |
| breqd 5159 | Equality deduction for a b... |
| breq2d 5160 | Equality deduction for a b... |
| breq12d 5161 | Equality deduction for a b... |
| breq123d 5162 | Equality deduction for a b... |
| breqdi 5163 | Equality deduction for a b... |
| breqan12d 5164 | Equality deduction for a b... |
| breqan12rd 5165 | Equality deduction for a b... |
| eqnbrtrd 5166 | Substitution of equal clas... |
| nbrne1 5167 | Two classes are different ... |
| nbrne2 5168 | Two classes are different ... |
| eqbrtri 5169 | Substitution of equal clas... |
| eqbrtrd 5170 | Substitution of equal clas... |
| eqbrtrri 5171 | Substitution of equal clas... |
| eqbrtrrd 5172 | Substitution of equal clas... |
| breqtri 5173 | Substitution of equal clas... |
| breqtrd 5174 | Substitution of equal clas... |
| breqtrri 5175 | Substitution of equal clas... |
| breqtrrd 5176 | Substitution of equal clas... |
| 3brtr3i 5177 | Substitution of equality i... |
| 3brtr4i 5178 | Substitution of equality i... |
| 3brtr3d 5179 | Substitution of equality i... |
| 3brtr4d 5180 | Substitution of equality i... |
| 3brtr3g 5181 | Substitution of equality i... |
| 3brtr4g 5182 | Substitution of equality i... |
| eqbrtrid 5183 | A chained equality inferen... |
| eqbrtrrid 5184 | A chained equality inferen... |
| breqtrid 5185 | A chained equality inferen... |
| breqtrrid 5186 | A chained equality inferen... |
| eqbrtrdi 5187 | A chained equality inferen... |
| eqbrtrrdi 5188 | A chained equality inferen... |
| breqtrdi 5189 | A chained equality inferen... |
| breqtrrdi 5190 | A chained equality inferen... |
| ssbrd 5191 | Deduction from a subclass ... |
| ssbr 5192 | Implication from a subclas... |
| ssbri 5193 | Inference from a subclass ... |
| nfbrd 5194 | Deduction version of bound... |
| nfbr 5195 | Bound-variable hypothesis ... |
| brab1 5196 | Relationship between a bin... |
| br0 5197 | The empty binary relation ... |
| brne0 5198 | If two sets are in a binar... |
| brun 5199 | The union of two binary re... |
| brin 5200 | The intersection of two re... |
| brdif 5201 | The difference of two bina... |
| sbcbr123 5202 | Move substitution in and o... |
| sbcbr 5203 | Move substitution in and o... |
| sbcbr12g 5204 | Move substitution in and o... |
| sbcbr1g 5205 | Move substitution in and o... |
| sbcbr2g 5206 | Move substitution in and o... |
| brsymdif 5207 | Characterization of the sy... |
| brralrspcev 5208 | Restricted existential spe... |
| brimralrspcev 5209 | Restricted existential spe... |
| opabss 5212 | The collection of ordered ... |
| opabbid 5213 | Equivalent wff's yield equ... |
| opabbidv 5214 | Equivalent wff's yield equ... |
| opabbii 5215 | Equivalent wff's yield equ... |
| nfopabd 5216 | Bound-variable hypothesis ... |
| nfopab 5217 | Bound-variable hypothesis ... |
| nfopab1 5218 | The first abstraction vari... |
| nfopab2 5219 | The second abstraction var... |
| cbvopab 5220 | Rule used to change bound ... |
| cbvopabv 5221 | Rule used to change bound ... |
| cbvopabvOLD 5222 | Obsolete version of ~ cbvo... |
| cbvopab1 5223 | Change first bound variabl... |
| cbvopab1g 5224 | Change first bound variabl... |
| cbvopab2 5225 | Change second bound variab... |
| cbvopab1s 5226 | Change first bound variabl... |
| cbvopab1v 5227 | Rule used to change the fi... |
| cbvopab1vOLD 5228 | Obsolete version of ~ cbvo... |
| cbvopab2v 5229 | Rule used to change the se... |
| unopab 5230 | Union of two ordered pair ... |
| mpteq12da 5233 | An equality inference for ... |
| mpteq12df 5234 | An equality inference for ... |
| mpteq12dfOLD 5235 | Obsolete version of ~ mpte... |
| mpteq12f 5236 | An equality theorem for th... |
| mpteq12dva 5237 | An equality inference for ... |
| mpteq12dvaOLD 5238 | Obsolete version of ~ mpte... |
| mpteq12dv 5239 | An equality inference for ... |
| mpteq12 5240 | An equality theorem for th... |
| mpteq1 5241 | An equality theorem for th... |
| mpteq1OLD 5242 | Obsolete version of ~ mpte... |
| mpteq1d 5243 | An equality theorem for th... |
| mpteq1i 5244 | An equality theorem for th... |
| mpteq1iOLD 5245 | Obsolete version of ~ mpte... |
| mpteq2da 5246 | Slightly more general equa... |
| mpteq2daOLD 5247 | Obsolete version of ~ mpte... |
| mpteq2dva 5248 | Slightly more general equa... |
| mpteq2dvaOLD 5249 | Obsolete version of ~ mpte... |
| mpteq2dv 5250 | An equality inference for ... |
| mpteq2ia 5251 | An equality inference for ... |
| mpteq2iaOLD 5252 | Obsolete version of ~ mpte... |
| mpteq2i 5253 | An equality inference for ... |
| mpteq12i 5254 | An equality inference for ... |
| nfmpt 5255 | Bound-variable hypothesis ... |
| nfmpt1 5256 | Bound-variable hypothesis ... |
| cbvmptf 5257 | Rule to change the bound v... |
| cbvmptfg 5258 | Rule to change the bound v... |
| cbvmpt 5259 | Rule to change the bound v... |
| cbvmptg 5260 | Rule to change the bound v... |
| cbvmptv 5261 | Rule to change the bound v... |
| cbvmptvOLD 5262 | Obsolete version of ~ cbvm... |
| cbvmptvg 5263 | Rule to change the bound v... |
| mptv 5264 | Function with universal do... |
| dftr2 5267 | An alternate way of defini... |
| dftr2c 5268 | Variant of ~ dftr2 with co... |
| dftr5 5269 | An alternate way of defini... |
| dftr5OLD 5270 | Obsolete version of ~ dftr... |
| dftr3 5271 | An alternate way of defini... |
| dftr4 5272 | An alternate way of defini... |
| treq 5273 | Equality theorem for the t... |
| trel 5274 | In a transitive class, the... |
| trel3 5275 | In a transitive class, the... |
| trss 5276 | An element of a transitive... |
| trin 5277 | The intersection of transi... |
| tr0 5278 | The empty set is transitiv... |
| trv 5279 | The universe is transitive... |
| triun 5280 | An indexed union of a clas... |
| truni 5281 | The union of a class of tr... |
| triin 5282 | An indexed intersection of... |
| trint 5283 | The intersection of a clas... |
| trintss 5284 | Any nonempty transitive cl... |
| axrep1 5286 | The version of the Axiom o... |
| axreplem 5287 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5288 | Axiom of Replacement expre... |
| axrep3 5289 | Axiom of Replacement sligh... |
| axrep4 5290 | A more traditional version... |
| axrep5 5291 | Axiom of Replacement (simi... |
| axrep6 5292 | A condensed form of ~ ax-r... |
| axrep6g 5293 | ~ axrep6 in class notation... |
| zfrepclf 5294 | An inference based on the ... |
| zfrep3cl 5295 | An inference based on the ... |
| zfrep4 5296 | A version of Replacement u... |
| axsepgfromrep 5297 | A more general version ~ a... |
| axsep 5298 | Axiom scheme of separation... |
| axsepg 5300 | A more general version of ... |
| zfauscl 5301 | Separation Scheme (Aussond... |
| bm1.3ii 5302 | Convert implication to equ... |
| ax6vsep 5303 | Derive ~ ax6v (a weakened ... |
| axnulALT 5304 | Alternate proof of ~ axnul... |
| axnul 5305 | The Null Set Axiom of ZF s... |
| 0ex 5307 | The Null Set Axiom of ZF s... |
| al0ssb 5308 | The empty set is the uniqu... |
| sseliALT 5309 | Alternate proof of ~ sseli... |
| csbexg 5310 | The existence of proper su... |
| csbex 5311 | The existence of proper su... |
| unisn2 5312 | A version of ~ unisn witho... |
| nalset 5313 | No set contains all sets. ... |
| vnex 5314 | The universal class does n... |
| vprc 5315 | The universal class is not... |
| nvel 5316 | The universal class does n... |
| inex1 5317 | Separation Scheme (Aussond... |
| inex2 5318 | Separation Scheme (Aussond... |
| inex1g 5319 | Closed-form, generalized S... |
| inex2g 5320 | Sufficient condition for a... |
| ssex 5321 | The subset of a set is als... |
| ssexi 5322 | The subset of a set is als... |
| ssexg 5323 | The subset of a set is als... |
| ssexd 5324 | A subclass of a set is a s... |
| prcssprc 5325 | The superclass of a proper... |
| sselpwd 5326 | Elementhood to a power set... |
| difexg 5327 | Existence of a difference.... |
| difexi 5328 | Existence of a difference,... |
| difexd 5329 | Existence of a difference.... |
| zfausab 5330 | Separation Scheme (Aussond... |
| rabexg 5331 | Separation Scheme in terms... |
| rabex 5332 | Separation Scheme in terms... |
| rabexd 5333 | Separation Scheme in terms... |
| rabex2 5334 | Separation Scheme in terms... |
| rab2ex 5335 | A class abstraction based ... |
| elssabg 5336 | Membership in a class abst... |
| intex 5337 | The intersection of a none... |
| intnex 5338 | If a class intersection is... |
| intexab 5339 | The intersection of a none... |
| intexrab 5340 | The intersection of a none... |
| iinexg 5341 | The existence of a class i... |
| intabs 5342 | Absorption of a redundant ... |
| inuni 5343 | The intersection of a unio... |
| elpw2g 5344 | Membership in a power clas... |
| elpw2 5345 | Membership in a power clas... |
| elpwi2 5346 | Membership in a power clas... |
| elpwi2OLD 5347 | Obsolete version of ~ elpw... |
| axpweq 5348 | Two equivalent ways to exp... |
| pwnss 5349 | The power set of a set is ... |
| pwne 5350 | No set equals its power se... |
| difelpw 5351 | A difference is an element... |
| rabelpw 5352 | A restricted class abstrac... |
| class2set 5353 | The class of elements of `... |
| 0elpw 5354 | Every power class contains... |
| pwne0 5355 | A power class is never emp... |
| 0nep0 5356 | The empty set and its powe... |
| 0inp0 5357 | Something cannot be equal ... |
| unidif0 5358 | The removal of the empty s... |
| eqsnuniex 5359 | If a class is equal to the... |
| iin0 5360 | An indexed intersection of... |
| notzfaus 5361 | In the Separation Scheme ~... |
| intv 5362 | The intersection of the un... |
| zfpow 5364 | Axiom of Power Sets expres... |
| axpow2 5365 | A variant of the Axiom of ... |
| axpow3 5366 | A variant of the Axiom of ... |
| elALT2 5367 | Alternate proof of ~ el us... |
| dtruALT2 5368 | Alternate proof of ~ dtru ... |
| dtrucor 5369 | Corollary of ~ dtru . Thi... |
| dtrucor2 5370 | The theorem form of the de... |
| dvdemo1 5371 | Demonstration of a theorem... |
| dvdemo2 5372 | Demonstration of a theorem... |
| nfnid 5373 | A setvar variable is not f... |
| nfcvb 5374 | The "distinctor" expressio... |
| vpwex 5375 | Power set axiom: the power... |
| pwexg 5376 | Power set axiom expressed ... |
| pwexd 5377 | Deduction version of the p... |
| pwex 5378 | Power set axiom expressed ... |
| pwel 5379 | Quantitative version of ~ ... |
| abssexg 5380 | Existence of a class of su... |
| snexALT 5381 | Alternate proof of ~ snex ... |
| p0ex 5382 | The power set of the empty... |
| p0exALT 5383 | Alternate proof of ~ p0ex ... |
| pp0ex 5384 | The power set of the power... |
| ord3ex 5385 | The ordinal number 3 is a ... |
| dtruALT 5386 | Alternate proof of ~ dtru ... |
| axc16b 5387 | This theorem shows that Ax... |
| eunex 5388 | Existential uniqueness imp... |
| eusv1 5389 | Two ways to express single... |
| eusvnf 5390 | Even if ` x ` is free in `... |
| eusvnfb 5391 | Two ways to say that ` A (... |
| eusv2i 5392 | Two ways to express single... |
| eusv2nf 5393 | Two ways to express single... |
| eusv2 5394 | Two ways to express single... |
| reusv1 5395 | Two ways to express single... |
| reusv2lem1 5396 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5397 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5398 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5399 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5400 | Lemma for ~ reusv2 . (Con... |
| reusv2 5401 | Two ways to express single... |
| reusv3i 5402 | Two ways of expressing exi... |
| reusv3 5403 | Two ways to express single... |
| eusv4 5404 | Two ways to express single... |
| alxfr 5405 | Transfer universal quantif... |
| ralxfrd 5406 | Transfer universal quantif... |
| rexxfrd 5407 | Transfer universal quantif... |
| ralxfr2d 5408 | Transfer universal quantif... |
| rexxfr2d 5409 | Transfer universal quantif... |
| ralxfrd2 5410 | Transfer universal quantif... |
| rexxfrd2 5411 | Transfer existence from a ... |
| ralxfr 5412 | Transfer universal quantif... |
| ralxfrALT 5413 | Alternate proof of ~ ralxf... |
| rexxfr 5414 | Transfer existence from a ... |
| rabxfrd 5415 | Membership in a restricted... |
| rabxfr 5416 | Membership in a restricted... |
| reuhypd 5417 | A theorem useful for elimi... |
| reuhyp 5418 | A theorem useful for elimi... |
| zfpair 5419 | The Axiom of Pairing of Ze... |
| axprALT 5420 | Alternate proof of ~ axpr ... |
| axprlem1 5421 | Lemma for ~ axpr . There ... |
| axprlem2 5422 | Lemma for ~ axpr . There ... |
| axprlem3 5423 | Lemma for ~ axpr . Elimin... |
| axprlem4 5424 | Lemma for ~ axpr . The fi... |
| axprlem5 5425 | Lemma for ~ axpr . The se... |
| axpr 5426 | Unabbreviated version of t... |
| zfpair2 5428 | Derive the abbreviated ver... |
| vsnex 5429 | A singleton built on a set... |
| snexg 5430 | A singleton built on a set... |
| snex 5431 | A singleton is a set. The... |
| prex 5432 | The Axiom of Pairing using... |
| exel 5433 | There exist two sets, one ... |
| exexneq 5434 | There exist two different ... |
| exneq 5435 | Given any set (the " ` y `... |
| dtru 5436 | Given any set (the " ` y `... |
| el 5437 | Any set is an element of s... |
| sels 5438 | If a class is a set, then ... |
| selsALT 5439 | Alternate proof of ~ sels ... |
| elALT 5440 | Alternate proof of ~ el , ... |
| dtruOLD 5441 | Obsolete proof of ~ dtru a... |
| snelpwg 5442 | A singleton of a set is a ... |
| snelpwi 5443 | If a set is a member of a ... |
| snelpwiOLD 5444 | Obsolete version of ~ snel... |
| snelpw 5445 | A singleton of a set is a ... |
| prelpw 5446 | An unordered pair of two s... |
| prelpwi 5447 | If two sets are members of... |
| rext 5448 | A theorem similar to exten... |
| sspwb 5449 | The powerclass constructio... |
| unipw 5450 | A class equals the union o... |
| univ 5451 | The union of the universe ... |
| pwtr 5452 | A class is transitive iff ... |
| ssextss 5453 | An extensionality-like pri... |
| ssext 5454 | An extensionality-like pri... |
| nssss 5455 | Negation of subclass relat... |
| pweqb 5456 | Classes are equal if and o... |
| intidg 5457 | The intersection of all se... |
| intidOLD 5458 | Obsolete version of ~ inti... |
| moabex 5459 | "At most one" existence im... |
| rmorabex 5460 | Restricted "at most one" e... |
| euabex 5461 | The abstraction of a wff w... |
| nnullss 5462 | A nonempty class (even if ... |
| exss 5463 | Restricted existence in a ... |
| opex 5464 | An ordered pair of classes... |
| otex 5465 | An ordered triple of class... |
| elopg 5466 | Characterization of the el... |
| elop 5467 | Characterization of the el... |
| opi1 5468 | One of the two elements in... |
| opi2 5469 | One of the two elements of... |
| opeluu 5470 | Each member of an ordered ... |
| op1stb 5471 | Extract the first member o... |
| brv 5472 | Two classes are always in ... |
| opnz 5473 | An ordered pair is nonempt... |
| opnzi 5474 | An ordered pair is nonempt... |
| opth1 5475 | Equality of the first memb... |
| opth 5476 | The ordered pair theorem. ... |
| opthg 5477 | Ordered pair theorem. ` C ... |
| opth1g 5478 | Equality of the first memb... |
| opthg2 5479 | Ordered pair theorem. (Co... |
| opth2 5480 | Ordered pair theorem. (Co... |
| opthneg 5481 | Two ordered pairs are not ... |
| opthne 5482 | Two ordered pairs are not ... |
| otth2 5483 | Ordered triple theorem, wi... |
| otth 5484 | Ordered triple theorem. (... |
| otthg 5485 | Ordered triple theorem, cl... |
| otthne 5486 | Contrapositive of the orde... |
| eqvinop 5487 | A variable introduction la... |
| sbcop1 5488 | The proper substitution of... |
| sbcop 5489 | The proper substitution of... |
| copsexgw 5490 | Version of ~ copsexg with ... |
| copsexg 5491 | Substitution of class ` A ... |
| copsex2t 5492 | Closed theorem form of ~ c... |
| copsex2g 5493 | Implicit substitution infe... |
| copsex2gOLD 5494 | Obsolete version of ~ cops... |
| copsex4g 5495 | An implicit substitution i... |
| 0nelop 5496 | A property of ordered pair... |
| opwo0id 5497 | An ordered pair is equal t... |
| opeqex 5498 | Equivalence of existence i... |
| oteqex2 5499 | Equivalence of existence i... |
| oteqex 5500 | Equivalence of existence i... |
| opcom 5501 | An ordered pair commutes i... |
| moop2 5502 | "At most one" property of ... |
| opeqsng 5503 | Equivalence for an ordered... |
| opeqsn 5504 | Equivalence for an ordered... |
| opeqpr 5505 | Equivalence for an ordered... |
| snopeqop 5506 | Equivalence for an ordered... |
| propeqop 5507 | Equivalence for an ordered... |
| propssopi 5508 | If a pair of ordered pairs... |
| snopeqopsnid 5509 | Equivalence for an ordered... |
| mosubopt 5510 | "At most one" remains true... |
| mosubop 5511 | "At most one" remains true... |
| euop2 5512 | Transfer existential uniqu... |
| euotd 5513 | Prove existential uniquene... |
| opthwiener 5514 | Justification theorem for ... |
| uniop 5515 | The union of an ordered pa... |
| uniopel 5516 | Ordered pair membership is... |
| opthhausdorff 5517 | Justification theorem for ... |
| opthhausdorff0 5518 | Justification theorem for ... |
| otsndisj 5519 | The singletons consisting ... |
| otiunsndisj 5520 | The union of singletons co... |
| iunopeqop 5521 | Implication of an ordered ... |
| brsnop 5522 | Binary relation for an ord... |
| brtp 5523 | A necessary and sufficient... |
| opabidw 5524 | The law of concretion. Sp... |
| opabid 5525 | The law of concretion. Sp... |
| elopabw 5526 | Membership in a class abst... |
| elopab 5527 | Membership in a class abst... |
| rexopabb 5528 | Restricted existential qua... |
| vopelopabsb 5529 | The law of concretion in t... |
| opelopabsb 5530 | The law of concretion in t... |
| brabsb 5531 | The law of concretion in t... |
| opelopabt 5532 | Closed theorem form of ~ o... |
| opelopabga 5533 | The law of concretion. Th... |
| brabga 5534 | The law of concretion for ... |
| opelopab2a 5535 | Ordered pair membership in... |
| opelopaba 5536 | The law of concretion. Th... |
| braba 5537 | The law of concretion for ... |
| opelopabg 5538 | The law of concretion. Th... |
| brabg 5539 | The law of concretion for ... |
| opelopabgf 5540 | The law of concretion. Th... |
| opelopab2 5541 | Ordered pair membership in... |
| opelopab 5542 | The law of concretion. Th... |
| brab 5543 | The law of concretion for ... |
| opelopabaf 5544 | The law of concretion. Th... |
| opelopabf 5545 | The law of concretion. Th... |
| ssopab2 5546 | Equivalence of ordered pai... |
| ssopab2bw 5547 | Equivalence of ordered pai... |
| eqopab2bw 5548 | Equivalence of ordered pai... |
| ssopab2b 5549 | Equivalence of ordered pai... |
| ssopab2i 5550 | Inference of ordered pair ... |
| ssopab2dv 5551 | Inference of ordered pair ... |
| eqopab2b 5552 | Equivalence of ordered pai... |
| opabn0 5553 | Nonempty ordered pair clas... |
| opab0 5554 | Empty ordered pair class a... |
| csbopab 5555 | Move substitution into a c... |
| csbopabgALT 5556 | Move substitution into a c... |
| csbmpt12 5557 | Move substitution into a m... |
| csbmpt2 5558 | Move substitution into the... |
| iunopab 5559 | Move indexed union inside ... |
| iunopabOLD 5560 | Obsolete version of ~ iuno... |
| elopabr 5561 | Membership in an ordered-p... |
| elopabran 5562 | Membership in an ordered-p... |
| elopabrOLD 5563 | Obsolete version of ~ elop... |
| rbropapd 5564 | Properties of a pair in an... |
| rbropap 5565 | Properties of a pair in a ... |
| 2rbropap 5566 | Properties of a pair in a ... |
| 0nelopab 5567 | The empty set is never an ... |
| 0nelopabOLD 5568 | Obsolete version of ~ 0nel... |
| brabv 5569 | If two classes are in a re... |
| pwin 5570 | The power class of the int... |
| pwssun 5571 | The power class of the uni... |
| pwun 5572 | The power class of the uni... |
| dfid4 5575 | The identity function expr... |
| dfid2 5576 | Alternate definition of th... |
| dfid3 5577 | A stronger version of ~ df... |
| dfid2OLD 5578 | Obsolete version of ~ dfid... |
| epelg 5581 | The membership relation an... |
| epeli 5582 | The membership relation an... |
| epel 5583 | The membership relation an... |
| 0sn0ep 5584 | An example for the members... |
| epn0 5585 | The membership relation is... |
| poss 5590 | Subset theorem for the par... |
| poeq1 5591 | Equality theorem for parti... |
| poeq2 5592 | Equality theorem for parti... |
| nfpo 5593 | Bound-variable hypothesis ... |
| nfso 5594 | Bound-variable hypothesis ... |
| pocl 5595 | Characteristic properties ... |
| poclOLD 5596 | Obsolete version of ~ pocl... |
| ispod 5597 | Sufficient conditions for ... |
| swopolem 5598 | Perform the substitutions ... |
| swopo 5599 | A strict weak order is a p... |
| poirr 5600 | A partial order is irrefle... |
| potr 5601 | A partial order is a trans... |
| po2nr 5602 | A partial order has no 2-c... |
| po3nr 5603 | A partial order has no 3-c... |
| po2ne 5604 | Two sets related by a part... |
| po0 5605 | Any relation is a partial ... |
| pofun 5606 | The inverse image of a par... |
| sopo 5607 | A strict linear order is a... |
| soss 5608 | Subset theorem for the str... |
| soeq1 5609 | Equality theorem for the s... |
| soeq2 5610 | Equality theorem for the s... |
| sonr 5611 | A strict order relation is... |
| sotr 5612 | A strict order relation is... |
| solin 5613 | A strict order relation is... |
| so2nr 5614 | A strict order relation ha... |
| so3nr 5615 | A strict order relation ha... |
| sotric 5616 | A strict order relation sa... |
| sotrieq 5617 | Trichotomy law for strict ... |
| sotrieq2 5618 | Trichotomy law for strict ... |
| soasym 5619 | Asymmetry law for strict o... |
| sotr2 5620 | A transitivity relation. ... |
| issod 5621 | An irreflexive, transitive... |
| issoi 5622 | An irreflexive, transitive... |
| isso2i 5623 | Deduce strict ordering fro... |
| so0 5624 | Any relation is a strict o... |
| somo 5625 | A totally ordered set has ... |
| sotrine 5626 | Trichotomy law for strict ... |
| sotr3 5627 | Transitivity law for stric... |
| dffr6 5634 | Alternate definition of ~ ... |
| frd 5635 | A nonempty subset of an ` ... |
| fri 5636 | A nonempty subset of an ` ... |
| friOLD 5637 | Obsolete version of ~ fri ... |
| seex 5638 | The ` R ` -preimage of an ... |
| exse 5639 | Any relation on a set is s... |
| dffr2 5640 | Alternate definition of we... |
| dffr2ALT 5641 | Alternate proof of ~ dffr2... |
| frc 5642 | Property of well-founded r... |
| frss 5643 | Subset theorem for the wel... |
| sess1 5644 | Subset theorem for the set... |
| sess2 5645 | Subset theorem for the set... |
| freq1 5646 | Equality theorem for the w... |
| freq2 5647 | Equality theorem for the w... |
| seeq1 5648 | Equality theorem for the s... |
| seeq2 5649 | Equality theorem for the s... |
| nffr 5650 | Bound-variable hypothesis ... |
| nfse 5651 | Bound-variable hypothesis ... |
| nfwe 5652 | Bound-variable hypothesis ... |
| frirr 5653 | A well-founded relation is... |
| fr2nr 5654 | A well-founded relation ha... |
| fr0 5655 | Any relation is well-found... |
| frminex 5656 | If an element of a well-fo... |
| efrirr 5657 | A well-founded class does ... |
| efrn2lp 5658 | A well-founded class conta... |
| epse 5659 | The membership relation is... |
| tz7.2 5660 | Similar to Theorem 7.2 of ... |
| dfepfr 5661 | An alternate way of saying... |
| epfrc 5662 | A subset of a well-founded... |
| wess 5663 | Subset theorem for the wel... |
| weeq1 5664 | Equality theorem for the w... |
| weeq2 5665 | Equality theorem for the w... |
| wefr 5666 | A well-ordering is well-fo... |
| weso 5667 | A well-ordering is a stric... |
| wecmpep 5668 | The elements of a class we... |
| wetrep 5669 | On a class well-ordered by... |
| wefrc 5670 | A nonempty subclass of a c... |
| we0 5671 | Any relation is a well-ord... |
| wereu 5672 | A nonempty subset of an ` ... |
| wereu2 5673 | A nonempty subclass of an ... |
| xpeq1 5690 | Equality theorem for Carte... |
| xpss12 5691 | Subset theorem for Cartesi... |
| xpss 5692 | A Cartesian product is inc... |
| inxpssres 5693 | Intersection with a Cartes... |
| relxp 5694 | A Cartesian product is a r... |
| xpss1 5695 | Subset relation for Cartes... |
| xpss2 5696 | Subset relation for Cartes... |
| xpeq2 5697 | Equality theorem for Carte... |
| elxpi 5698 | Membership in a Cartesian ... |
| elxp 5699 | Membership in a Cartesian ... |
| elxp2 5700 | Membership in a Cartesian ... |
| xpeq12 5701 | Equality theorem for Carte... |
| xpeq1i 5702 | Equality inference for Car... |
| xpeq2i 5703 | Equality inference for Car... |
| xpeq12i 5704 | Equality inference for Car... |
| xpeq1d 5705 | Equality deduction for Car... |
| xpeq2d 5706 | Equality deduction for Car... |
| xpeq12d 5707 | Equality deduction for Car... |
| sqxpeqd 5708 | Equality deduction for a C... |
| nfxp 5709 | Bound-variable hypothesis ... |
| 0nelxp 5710 | The empty set is not a mem... |
| 0nelelxp 5711 | A member of a Cartesian pr... |
| opelxp 5712 | Ordered pair membership in... |
| opelxpi 5713 | Ordered pair membership in... |
| opelxpii 5714 | Ordered pair membership in... |
| opelxpd 5715 | Ordered pair membership in... |
| opelvv 5716 | Ordered pair membership in... |
| opelvvg 5717 | Ordered pair membership in... |
| opelxp1 5718 | The first member of an ord... |
| opelxp2 5719 | The second member of an or... |
| otelxp 5720 | Ordered triple membership ... |
| otelxp1 5721 | The first member of an ord... |
| otel3xp 5722 | An ordered triple is an el... |
| opabssxpd 5723 | An ordered-pair class abst... |
| rabxp 5724 | Class abstraction restrict... |
| brxp 5725 | Binary relation on a Carte... |
| pwvrel 5726 | A set is a binary relation... |
| pwvabrel 5727 | The powerclass of the cart... |
| brrelex12 5728 | Two classes related by a b... |
| brrelex1 5729 | If two classes are related... |
| brrelex2 5730 | If two classes are related... |
| brrelex12i 5731 | Two classes that are relat... |
| brrelex1i 5732 | The first argument of a bi... |
| brrelex2i 5733 | The second argument of a b... |
| nprrel12 5734 | Proper classes are not rel... |
| nprrel 5735 | No proper class is related... |
| 0nelrel0 5736 | A binary relation does not... |
| 0nelrel 5737 | A binary relation does not... |
| fconstmpt 5738 | Representation of a consta... |
| vtoclr 5739 | Variable to class conversi... |
| opthprc 5740 | Justification theorem for ... |
| brel 5741 | Two things in a binary rel... |
| elxp3 5742 | Membership in a Cartesian ... |
| opeliunxp 5743 | Membership in a union of C... |
| xpundi 5744 | Distributive law for Carte... |
| xpundir 5745 | Distributive law for Carte... |
| xpiundi 5746 | Distributive law for Carte... |
| xpiundir 5747 | Distributive law for Carte... |
| iunxpconst 5748 | Membership in a union of C... |
| xpun 5749 | The Cartesian product of t... |
| elvv 5750 | Membership in universal cl... |
| elvvv 5751 | Membership in universal cl... |
| elvvuni 5752 | An ordered pair contains i... |
| brinxp2 5753 | Intersection of binary rel... |
| brinxp 5754 | Intersection of binary rel... |
| opelinxp 5755 | Ordered pair element in an... |
| poinxp 5756 | Intersection of partial or... |
| soinxp 5757 | Intersection of total orde... |
| frinxp 5758 | Intersection of well-found... |
| seinxp 5759 | Intersection of set-like r... |
| weinxp 5760 | Intersection of well-order... |
| posn 5761 | Partial ordering of a sing... |
| sosn 5762 | Strict ordering on a singl... |
| frsn 5763 | Founded relation on a sing... |
| wesn 5764 | Well-ordering of a singlet... |
| elopaelxp 5765 | Membership in an ordered-p... |
| elopaelxpOLD 5766 | Obsolete version of ~ elop... |
| bropaex12 5767 | Two classes related by an ... |
| opabssxp 5768 | An abstraction relation is... |
| brab2a 5769 | The law of concretion for ... |
| optocl 5770 | Implicit substitution of c... |
| 2optocl 5771 | Implicit substitution of c... |
| 3optocl 5772 | Implicit substitution of c... |
| opbrop 5773 | Ordered pair membership in... |
| 0xp 5774 | The Cartesian product with... |
| csbxp 5775 | Distribute proper substitu... |
| releq 5776 | Equality theorem for the r... |
| releqi 5777 | Equality inference for the... |
| releqd 5778 | Equality deduction for the... |
| nfrel 5779 | Bound-variable hypothesis ... |
| sbcrel 5780 | Distribute proper substitu... |
| relss 5781 | Subclass theorem for relat... |
| ssrel 5782 | A subclass relationship de... |
| ssrelOLD 5783 | Obsolete version of ~ ssre... |
| eqrel 5784 | Extensionality principle f... |
| ssrel2 5785 | A subclass relationship de... |
| ssrel3 5786 | Subclass relation in anoth... |
| relssi 5787 | Inference from subclass pr... |
| relssdv 5788 | Deduction from subclass pr... |
| eqrelriv 5789 | Inference from extensional... |
| eqrelriiv 5790 | Inference from extensional... |
| eqbrriv 5791 | Inference from extensional... |
| eqrelrdv 5792 | Deduce equality of relatio... |
| eqbrrdv 5793 | Deduction from extensional... |
| eqbrrdiv 5794 | Deduction from extensional... |
| eqrelrdv2 5795 | A version of ~ eqrelrdv . ... |
| ssrelrel 5796 | A subclass relationship de... |
| eqrelrel 5797 | Extensionality principle f... |
| elrel 5798 | A member of a relation is ... |
| rel0 5799 | The empty set is a relatio... |
| nrelv 5800 | The universal class is not... |
| relsng 5801 | A singleton is a relation ... |
| relsnb 5802 | An at-most-singleton is a ... |
| relsnopg 5803 | A singleton of an ordered ... |
| relsn 5804 | A singleton is a relation ... |
| relsnop 5805 | A singleton of an ordered ... |
| copsex2gb 5806 | Implicit substitution infe... |
| copsex2ga 5807 | Implicit substitution infe... |
| elopaba 5808 | Membership in an ordered-p... |
| xpsspw 5809 | A Cartesian product is inc... |
| unixpss 5810 | The double class union of ... |
| relun 5811 | The union of two relations... |
| relin1 5812 | The intersection with a re... |
| relin2 5813 | The intersection with a re... |
| relinxp 5814 | Intersection with a Cartes... |
| reldif 5815 | A difference cutting down ... |
| reliun 5816 | An indexed union is a rela... |
| reliin 5817 | An indexed intersection is... |
| reluni 5818 | The union of a class is a ... |
| relint 5819 | The intersection of a clas... |
| relopabiv 5820 | A class of ordered pairs i... |
| relopabv 5821 | A class of ordered pairs i... |
| relopabi 5822 | A class of ordered pairs i... |
| relopabiALT 5823 | Alternate proof of ~ relop... |
| relopab 5824 | A class of ordered pairs i... |
| mptrel 5825 | The maps-to notation alway... |
| reli 5826 | The identity relation is a... |
| rele 5827 | The membership relation is... |
| opabid2 5828 | A relation expressed as an... |
| inopab 5829 | Intersection of two ordere... |
| difopab 5830 | Difference of two ordered-... |
| difopabOLD 5831 | Obsolete version of ~ difo... |
| inxp 5832 | Intersection of two Cartes... |
| xpindi 5833 | Distributive law for Carte... |
| xpindir 5834 | Distributive law for Carte... |
| xpiindi 5835 | Distributive law for Carte... |
| xpriindi 5836 | Distributive law for Carte... |
| eliunxp 5837 | Membership in a union of C... |
| opeliunxp2 5838 | Membership in a union of C... |
| raliunxp 5839 | Write a double restricted ... |
| rexiunxp 5840 | Write a double restricted ... |
| ralxp 5841 | Universal quantification r... |
| rexxp 5842 | Existential quantification... |
| exopxfr 5843 | Transfer ordered-pair exis... |
| exopxfr2 5844 | Transfer ordered-pair exis... |
| djussxp 5845 | Disjoint union is a subset... |
| ralxpf 5846 | Version of ~ ralxp with bo... |
| rexxpf 5847 | Version of ~ rexxp with bo... |
| iunxpf 5848 | Indexed union on a Cartesi... |
| opabbi2dv 5849 | Deduce equality of a relat... |
| relop 5850 | A necessary and sufficient... |
| ideqg 5851 | For sets, the identity rel... |
| ideq 5852 | For sets, the identity rel... |
| ididg 5853 | A set is identical to itse... |
| issetid 5854 | Two ways of expressing set... |
| coss1 5855 | Subclass theorem for compo... |
| coss2 5856 | Subclass theorem for compo... |
| coeq1 5857 | Equality theorem for compo... |
| coeq2 5858 | Equality theorem for compo... |
| coeq1i 5859 | Equality inference for com... |
| coeq2i 5860 | Equality inference for com... |
| coeq1d 5861 | Equality deduction for com... |
| coeq2d 5862 | Equality deduction for com... |
| coeq12i 5863 | Equality inference for com... |
| coeq12d 5864 | Equality deduction for com... |
| nfco 5865 | Bound-variable hypothesis ... |
| brcog 5866 | Ordered pair membership in... |
| opelco2g 5867 | Ordered pair membership in... |
| brcogw 5868 | Ordered pair membership in... |
| eqbrrdva 5869 | Deduction from extensional... |
| brco 5870 | Binary relation on a compo... |
| opelco 5871 | Ordered pair membership in... |
| cnvss 5872 | Subset theorem for convers... |
| cnveq 5873 | Equality theorem for conve... |
| cnveqi 5874 | Equality inference for con... |
| cnveqd 5875 | Equality deduction for con... |
| elcnv 5876 | Membership in a converse r... |
| elcnv2 5877 | Membership in a converse r... |
| nfcnv 5878 | Bound-variable hypothesis ... |
| brcnvg 5879 | The converse of a binary r... |
| opelcnvg 5880 | Ordered-pair membership in... |
| opelcnv 5881 | Ordered-pair membership in... |
| brcnv 5882 | The converse of a binary r... |
| csbcnv 5883 | Move class substitution in... |
| csbcnvgALT 5884 | Move class substitution in... |
| cnvco 5885 | Distributive law of conver... |
| cnvuni 5886 | The converse of a class un... |
| dfdm3 5887 | Alternate definition of do... |
| dfrn2 5888 | Alternate definition of ra... |
| dfrn3 5889 | Alternate definition of ra... |
| elrn2g 5890 | Membership in a range. (C... |
| elrng 5891 | Membership in a range. (C... |
| elrn2 5892 | Membership in a range. (C... |
| elrn 5893 | Membership in a range. (C... |
| ssrelrn 5894 | If a relation is a subset ... |
| dfdm4 5895 | Alternate definition of do... |
| dfdmf 5896 | Definition of domain, usin... |
| csbdm 5897 | Distribute proper substitu... |
| eldmg 5898 | Domain membership. Theore... |
| eldm2g 5899 | Domain membership. Theore... |
| eldm 5900 | Membership in a domain. T... |
| eldm2 5901 | Membership in a domain. T... |
| dmss 5902 | Subset theorem for domain.... |
| dmeq 5903 | Equality theorem for domai... |
| dmeqi 5904 | Equality inference for dom... |
| dmeqd 5905 | Equality deduction for dom... |
| opeldmd 5906 | Membership of first of an ... |
| opeldm 5907 | Membership of first of an ... |
| breldm 5908 | Membership of first of a b... |
| breldmg 5909 | Membership of first of a b... |
| dmun 5910 | The domain of a union is t... |
| dmin 5911 | The domain of an intersect... |
| breldmd 5912 | Membership of first of a b... |
| dmiun 5913 | The domain of an indexed u... |
| dmuni 5914 | The domain of a union. Pa... |
| dmopab 5915 | The domain of a class of o... |
| dmopabelb 5916 | A set is an element of the... |
| dmopab2rex 5917 | The domain of an ordered p... |
| dmopabss 5918 | Upper bound for the domain... |
| dmopab3 5919 | The domain of a restricted... |
| dm0 5920 | The domain of the empty se... |
| dmi 5921 | The domain of the identity... |
| dmv 5922 | The domain of the universe... |
| dmep 5923 | The domain of the membersh... |
| dm0rn0 5924 | An empty domain is equival... |
| rn0 5925 | The range of the empty set... |
| rnep 5926 | The range of the membershi... |
| reldm0 5927 | A relation is empty iff it... |
| dmxp 5928 | The domain of a Cartesian ... |
| dmxpid 5929 | The domain of a Cartesian ... |
| dmxpin 5930 | The domain of the intersec... |
| xpid11 5931 | The Cartesian square is a ... |
| dmcnvcnv 5932 | The domain of the double c... |
| rncnvcnv 5933 | The range of the double co... |
| elreldm 5934 | The first member of an ord... |
| rneq 5935 | Equality theorem for range... |
| rneqi 5936 | Equality inference for ran... |
| rneqd 5937 | Equality deduction for ran... |
| rnss 5938 | Subset theorem for range. ... |
| rnssi 5939 | Subclass inference for ran... |
| brelrng 5940 | The second argument of a b... |
| brelrn 5941 | The second argument of a b... |
| opelrn 5942 | Membership of second membe... |
| releldm 5943 | The first argument of a bi... |
| relelrn 5944 | The second argument of a b... |
| releldmb 5945 | Membership in a domain. (... |
| relelrnb 5946 | Membership in a range. (C... |
| releldmi 5947 | The first argument of a bi... |
| relelrni 5948 | The second argument of a b... |
| dfrnf 5949 | Definition of range, using... |
| nfdm 5950 | Bound-variable hypothesis ... |
| nfrn 5951 | Bound-variable hypothesis ... |
| dmiin 5952 | Domain of an intersection.... |
| rnopab 5953 | The range of a class of or... |
| rnmpt 5954 | The range of a function in... |
| elrnmpt 5955 | The range of a function in... |
| elrnmpt1s 5956 | Elementhood in an image se... |
| elrnmpt1 5957 | Elementhood in an image se... |
| elrnmptg 5958 | Membership in the range of... |
| elrnmpti 5959 | Membership in the range of... |
| elrnmptd 5960 | The range of a function in... |
| elrnmptdv 5961 | Elementhood in the range o... |
| elrnmpt2d 5962 | Elementhood in the range o... |
| dfiun3g 5963 | Alternate definition of in... |
| dfiin3g 5964 | Alternate definition of in... |
| dfiun3 5965 | Alternate definition of in... |
| dfiin3 5966 | Alternate definition of in... |
| riinint 5967 | Express a relative indexed... |
| relrn0 5968 | A relation is empty iff it... |
| dmrnssfld 5969 | The domain and range of a ... |
| dmcoss 5970 | Domain of a composition. ... |
| rncoss 5971 | Range of a composition. (... |
| dmcosseq 5972 | Domain of a composition. ... |
| dmcoeq 5973 | Domain of a composition. ... |
| rncoeq 5974 | Range of a composition. (... |
| reseq1 5975 | Equality theorem for restr... |
| reseq2 5976 | Equality theorem for restr... |
| reseq1i 5977 | Equality inference for res... |
| reseq2i 5978 | Equality inference for res... |
| reseq12i 5979 | Equality inference for res... |
| reseq1d 5980 | Equality deduction for res... |
| reseq2d 5981 | Equality deduction for res... |
| reseq12d 5982 | Equality deduction for res... |
| nfres 5983 | Bound-variable hypothesis ... |
| csbres 5984 | Distribute proper substitu... |
| res0 5985 | A restriction to the empty... |
| dfres3 5986 | Alternate definition of re... |
| opelres 5987 | Ordered pair elementhood i... |
| brres 5988 | Binary relation on a restr... |
| opelresi 5989 | Ordered pair membership in... |
| brresi 5990 | Binary relation on a restr... |
| opres 5991 | Ordered pair membership in... |
| resieq 5992 | A restricted identity rela... |
| opelidres 5993 | ` <. A , A >. ` belongs to... |
| resres 5994 | The restriction of a restr... |
| resundi 5995 | Distributive law for restr... |
| resundir 5996 | Distributive law for restr... |
| resindi 5997 | Class restriction distribu... |
| resindir 5998 | Class restriction distribu... |
| inres 5999 | Move intersection into cla... |
| resdifcom 6000 | Commutative law for restri... |
| resiun1 6001 | Distribution of restrictio... |
| resiun2 6002 | Distribution of restrictio... |
| dmres 6003 | The domain of a restrictio... |
| ssdmres 6004 | A domain restricted to a s... |
| dmresexg 6005 | The domain of a restrictio... |
| resss 6006 | A class includes its restr... |
| rescom 6007 | Commutative law for restri... |
| ssres 6008 | Subclass theorem for restr... |
| ssres2 6009 | Subclass theorem for restr... |
| relres 6010 | A restriction is a relatio... |
| resabs1 6011 | Absorption law for restric... |
| resabs1d 6012 | Absorption law for restric... |
| resabs2 6013 | Absorption law for restric... |
| residm 6014 | Idempotent law for restric... |
| resima 6015 | A restriction to an image.... |
| resima2 6016 | Image under a restricted c... |
| rnresss 6017 | The range of a restriction... |
| xpssres 6018 | Restriction of a constant ... |
| elinxp 6019 | Membership in an intersect... |
| elres 6020 | Membership in a restrictio... |
| elsnres 6021 | Membership in restriction ... |
| relssres 6022 | Simplification law for res... |
| dmressnsn 6023 | The domain of a restrictio... |
| eldmressnsn 6024 | The element of the domain ... |
| eldmeldmressn 6025 | An element of the domain (... |
| resdm 6026 | A relation restricted to i... |
| resexg 6027 | The restriction of a set i... |
| resexd 6028 | The restriction of a set i... |
| resex 6029 | The restriction of a set i... |
| resindm 6030 | When restricting a relatio... |
| resdmdfsn 6031 | Restricting a relation to ... |
| reldisjun 6032 | Split a relation into two ... |
| relresdm1 6033 | Restriction of a disjoint ... |
| resopab 6034 | Restriction of a class abs... |
| iss 6035 | A subclass of the identity... |
| resopab2 6036 | Restriction of a class abs... |
| resmpt 6037 | Restriction of the mapping... |
| resmpt3 6038 | Unconditional restriction ... |
| resmptf 6039 | Restriction of the mapping... |
| resmptd 6040 | Restriction of the mapping... |
| dfres2 6041 | Alternate definition of th... |
| mptss 6042 | Sufficient condition for i... |
| elidinxp 6043 | Characterization of the el... |
| elidinxpid 6044 | Characterization of the el... |
| elrid 6045 | Characterization of the el... |
| idinxpres 6046 | The intersection of the id... |
| idinxpresid 6047 | The intersection of the id... |
| idssxp 6048 | A diagonal set as a subset... |
| opabresid 6049 | The restricted identity re... |
| mptresid 6050 | The restricted identity re... |
| dmresi 6051 | The domain of a restricted... |
| restidsing 6052 | Restriction of the identit... |
| iresn0n0 6053 | The identity function rest... |
| imaeq1 6054 | Equality theorem for image... |
| imaeq2 6055 | Equality theorem for image... |
| imaeq1i 6056 | Equality theorem for image... |
| imaeq2i 6057 | Equality theorem for image... |
| imaeq1d 6058 | Equality theorem for image... |
| imaeq2d 6059 | Equality theorem for image... |
| imaeq12d 6060 | Equality theorem for image... |
| dfima2 6061 | Alternate definition of im... |
| dfima3 6062 | Alternate definition of im... |
| elimag 6063 | Membership in an image. T... |
| elima 6064 | Membership in an image. T... |
| elima2 6065 | Membership in an image. T... |
| elima3 6066 | Membership in an image. T... |
| nfima 6067 | Bound-variable hypothesis ... |
| nfimad 6068 | Deduction version of bound... |
| imadmrn 6069 | The image of the domain of... |
| imassrn 6070 | The image of a class is a ... |
| mptima 6071 | Image of a function in map... |
| mptimass 6072 | Image of a function in map... |
| imai 6073 | Image under the identity r... |
| rnresi 6074 | The range of the restricte... |
| resiima 6075 | The image of a restriction... |
| ima0 6076 | Image of the empty set. T... |
| 0ima 6077 | Image under the empty rela... |
| csbima12 6078 | Move class substitution in... |
| imadisj 6079 | A class whose image under ... |
| cnvimass 6080 | A preimage under any class... |
| cnvimarndm 6081 | The preimage of the range ... |
| imasng 6082 | The image of a singleton. ... |
| relimasn 6083 | The image of a singleton. ... |
| elrelimasn 6084 | Elementhood in the image o... |
| elimasng1 6085 | Membership in an image of ... |
| elimasn1 6086 | Membership in an image of ... |
| elimasng 6087 | Membership in an image of ... |
| elimasn 6088 | Membership in an image of ... |
| elimasngOLD 6089 | Obsolete version of ~ elim... |
| elimasni 6090 | Membership in an image of ... |
| args 6091 | Two ways to express the cl... |
| elinisegg 6092 | Membership in the inverse ... |
| eliniseg 6093 | Membership in the inverse ... |
| epin 6094 | Any set is equal to its pr... |
| epini 6095 | Any set is equal to its pr... |
| iniseg 6096 | An idiom that signifies an... |
| inisegn0 6097 | Nonemptiness of an initial... |
| dffr3 6098 | Alternate definition of we... |
| dfse2 6099 | Alternate definition of se... |
| imass1 6100 | Subset theorem for image. ... |
| imass2 6101 | Subset theorem for image. ... |
| ndmima 6102 | The image of a singleton o... |
| relcnv 6103 | A converse is a relation. ... |
| relbrcnvg 6104 | When ` R ` is a relation, ... |
| eliniseg2 6105 | Eliminate the class existe... |
| relbrcnv 6106 | When ` R ` is a relation, ... |
| relco 6107 | A composition is a relatio... |
| cotrg 6108 | Two ways of saying that th... |
| cotrgOLD 6109 | Obsolete version of ~ cotr... |
| cotrgOLDOLD 6110 | Obsolete version of ~ cotr... |
| cotr 6111 | Two ways of saying a relat... |
| idrefALT 6112 | Alternate proof of ~ idref... |
| cnvsym 6113 | Two ways of saying a relat... |
| cnvsymOLD 6114 | Obsolete proof of ~ cnvsym... |
| cnvsymOLDOLD 6115 | Obsolete proof of ~ cnvsym... |
| intasym 6116 | Two ways of saying a relat... |
| asymref 6117 | Two ways of saying a relat... |
| asymref2 6118 | Two ways of saying a relat... |
| intirr 6119 | Two ways of saying a relat... |
| brcodir 6120 | Two ways of saying that tw... |
| codir 6121 | Two ways of saying a relat... |
| qfto 6122 | A quantifier-free way of e... |
| xpidtr 6123 | A Cartesian square is a tr... |
| trin2 6124 | The intersection of two tr... |
| poirr2 6125 | A partial order is irrefle... |
| trinxp 6126 | The relation induced by a ... |
| soirri 6127 | A strict order relation is... |
| sotri 6128 | A strict order relation is... |
| son2lpi 6129 | A strict order relation ha... |
| sotri2 6130 | A transitivity relation. ... |
| sotri3 6131 | A transitivity relation. ... |
| poleloe 6132 | Express "less than or equa... |
| poltletr 6133 | Transitive law for general... |
| somin1 6134 | Property of a minimum in a... |
| somincom 6135 | Commutativity of minimum i... |
| somin2 6136 | Property of a minimum in a... |
| soltmin 6137 | Being less than a minimum,... |
| cnvopab 6138 | The converse of a class ab... |
| mptcnv 6139 | The converse of a mapping ... |
| cnv0 6140 | The converse of the empty ... |
| cnvi 6141 | The converse of the identi... |
| cnvun 6142 | The converse of a union is... |
| cnvdif 6143 | Distributive law for conve... |
| cnvin 6144 | Distributive law for conve... |
| rnun 6145 | Distributive law for range... |
| rnin 6146 | The range of an intersecti... |
| rniun 6147 | The range of an indexed un... |
| rnuni 6148 | The range of a union. Par... |
| imaundi 6149 | Distributive law for image... |
| imaundir 6150 | The image of a union. (Co... |
| cnvimassrndm 6151 | The preimage of a superset... |
| dminss 6152 | An upper bound for interse... |
| imainss 6153 | An upper bound for interse... |
| inimass 6154 | The image of an intersecti... |
| inimasn 6155 | The intersection of the im... |
| cnvxp 6156 | The converse of a Cartesia... |
| xp0 6157 | The Cartesian product with... |
| xpnz 6158 | The Cartesian product of n... |
| xpeq0 6159 | At least one member of an ... |
| xpdisj1 6160 | Cartesian products with di... |
| xpdisj2 6161 | Cartesian products with di... |
| xpsndisj 6162 | Cartesian products with tw... |
| difxp 6163 | Difference of Cartesian pr... |
| difxp1 6164 | Difference law for Cartesi... |
| difxp2 6165 | Difference law for Cartesi... |
| djudisj 6166 | Disjoint unions with disjo... |
| xpdifid 6167 | The set of distinct couple... |
| resdisj 6168 | A double restriction to di... |
| rnxp 6169 | The range of a Cartesian p... |
| dmxpss 6170 | The domain of a Cartesian ... |
| rnxpss 6171 | The range of a Cartesian p... |
| rnxpid 6172 | The range of a Cartesian s... |
| ssxpb 6173 | A Cartesian product subcla... |
| xp11 6174 | The Cartesian product of n... |
| xpcan 6175 | Cancellation law for Carte... |
| xpcan2 6176 | Cancellation law for Carte... |
| ssrnres 6177 | Two ways to express surjec... |
| rninxp 6178 | Two ways to express surjec... |
| dminxp 6179 | Two ways to express totali... |
| imainrect 6180 | Image by a restricted and ... |
| xpima 6181 | Direct image by a Cartesia... |
| xpima1 6182 | Direct image by a Cartesia... |
| xpima2 6183 | Direct image by a Cartesia... |
| xpimasn 6184 | Direct image of a singleto... |
| sossfld 6185 | The base set of a strict o... |
| sofld 6186 | The base set of a nonempty... |
| cnvcnv3 6187 | The set of all ordered pai... |
| dfrel2 6188 | Alternate definition of re... |
| dfrel4v 6189 | A relation can be expresse... |
| dfrel4 6190 | A relation can be expresse... |
| cnvcnv 6191 | The double converse of a c... |
| cnvcnv2 6192 | The double converse of a c... |
| cnvcnvss 6193 | The double converse of a c... |
| cnvrescnv 6194 | Two ways to express the co... |
| cnveqb 6195 | Equality theorem for conve... |
| cnveq0 6196 | A relation empty iff its c... |
| dfrel3 6197 | Alternate definition of re... |
| elid 6198 | Characterization of the el... |
| dmresv 6199 | The domain of a universal ... |
| rnresv 6200 | The range of a universal r... |
| dfrn4 6201 | Range defined in terms of ... |
| csbrn 6202 | Distribute proper substitu... |
| rescnvcnv 6203 | The restriction of the dou... |
| cnvcnvres 6204 | The double converse of the... |
| imacnvcnv 6205 | The image of the double co... |
| dmsnn0 6206 | The domain of a singleton ... |
| rnsnn0 6207 | The range of a singleton i... |
| dmsn0 6208 | The domain of the singleto... |
| cnvsn0 6209 | The converse of the single... |
| dmsn0el 6210 | The domain of a singleton ... |
| relsn2 6211 | A singleton is a relation ... |
| dmsnopg 6212 | The domain of a singleton ... |
| dmsnopss 6213 | The domain of a singleton ... |
| dmpropg 6214 | The domain of an unordered... |
| dmsnop 6215 | The domain of a singleton ... |
| dmprop 6216 | The domain of an unordered... |
| dmtpop 6217 | The domain of an unordered... |
| cnvcnvsn 6218 | Double converse of a singl... |
| dmsnsnsn 6219 | The domain of the singleto... |
| rnsnopg 6220 | The range of a singleton o... |
| rnpropg 6221 | The range of a pair of ord... |
| cnvsng 6222 | Converse of a singleton of... |
| rnsnop 6223 | The range of a singleton o... |
| op1sta 6224 | Extract the first member o... |
| cnvsn 6225 | Converse of a singleton of... |
| op2ndb 6226 | Extract the second member ... |
| op2nda 6227 | Extract the second member ... |
| opswap 6228 | Swap the members of an ord... |
| cnvresima 6229 | An image under the convers... |
| resdm2 6230 | A class restricted to its ... |
| resdmres 6231 | Restriction to the domain ... |
| resresdm 6232 | A restriction by an arbitr... |
| imadmres 6233 | The image of the domain of... |
| resdmss 6234 | Subset relationship for th... |
| resdifdi 6235 | Distributive law for restr... |
| resdifdir 6236 | Distributive law for restr... |
| mptpreima 6237 | The preimage of a function... |
| mptiniseg 6238 | Converse singleton image o... |
| dmmpt 6239 | The domain of the mapping ... |
| dmmptss 6240 | The domain of a mapping is... |
| dmmptg 6241 | The domain of the mapping ... |
| rnmpt0f 6242 | The range of a function in... |
| rnmptn0 6243 | The range of a function in... |
| dfco2 6244 | Alternate definition of a ... |
| dfco2a 6245 | Generalization of ~ dfco2 ... |
| coundi 6246 | Class composition distribu... |
| coundir 6247 | Class composition distribu... |
| cores 6248 | Restricted first member of... |
| resco 6249 | Associative law for the re... |
| imaco 6250 | Image of the composition o... |
| rnco 6251 | The range of the compositi... |
| rnco2 6252 | The range of the compositi... |
| dmco 6253 | The domain of a compositio... |
| coeq0 6254 | A composition of two relat... |
| coiun 6255 | Composition with an indexe... |
| cocnvcnv1 6256 | A composition is not affec... |
| cocnvcnv2 6257 | A composition is not affec... |
| cores2 6258 | Absorption of a reverse (p... |
| co02 6259 | Composition with the empty... |
| co01 6260 | Composition with the empty... |
| coi1 6261 | Composition with the ident... |
| coi2 6262 | Composition with the ident... |
| coires1 6263 | Composition with a restric... |
| coass 6264 | Associative law for class ... |
| relcnvtrg 6265 | General form of ~ relcnvtr... |
| relcnvtr 6266 | A relation is transitive i... |
| relssdmrn 6267 | A relation is included in ... |
| relssdmrnOLD 6268 | Obsolete version of ~ rels... |
| resssxp 6269 | If the ` R ` -image of a c... |
| cnvssrndm 6270 | The converse is a subset o... |
| cossxp 6271 | Composition as a subset of... |
| relrelss 6272 | Two ways to describe the s... |
| unielrel 6273 | The membership relation fo... |
| relfld 6274 | The double union of a rela... |
| relresfld 6275 | Restriction of a relation ... |
| relcoi2 6276 | Composition with the ident... |
| relcoi1 6277 | Composition with the ident... |
| unidmrn 6278 | The double union of the co... |
| relcnvfld 6279 | if ` R ` is a relation, it... |
| dfdm2 6280 | Alternate definition of do... |
| unixp 6281 | The double class union of ... |
| unixp0 6282 | A Cartesian product is emp... |
| unixpid 6283 | Field of a Cartesian squar... |
| ressn 6284 | Restriction of a class to ... |
| cnviin 6285 | The converse of an interse... |
| cnvpo 6286 | The converse of a partial ... |
| cnvso 6287 | The converse of a strict o... |
| xpco 6288 | Composition of two Cartesi... |
| xpcoid 6289 | Composition of two Cartesi... |
| elsnxp 6290 | Membership in a Cartesian ... |
| reu3op 6291 | There is a unique ordered ... |
| reuop 6292 | There is a unique ordered ... |
| opreu2reurex 6293 | There is a unique ordered ... |
| opreu2reu 6294 | If there is a unique order... |
| dfpo2 6295 | Quantifier-free definition... |
| csbcog 6296 | Distribute proper substitu... |
| snres0 6297 | Condition for restriction ... |
| imaindm 6298 | The image is unaffected by... |
| predeq123 6301 | Equality theorem for the p... |
| predeq1 6302 | Equality theorem for the p... |
| predeq2 6303 | Equality theorem for the p... |
| predeq3 6304 | Equality theorem for the p... |
| nfpred 6305 | Bound-variable hypothesis ... |
| csbpredg 6306 | Move class substitution in... |
| predpredss 6307 | If ` A ` is a subset of ` ... |
| predss 6308 | The predecessor class of `... |
| sspred 6309 | Another subset/predecessor... |
| dfpred2 6310 | An alternate definition of... |
| dfpred3 6311 | An alternate definition of... |
| dfpred3g 6312 | An alternate definition of... |
| elpredgg 6313 | Membership in a predecesso... |
| elpredg 6314 | Membership in a predecesso... |
| elpredimg 6315 | Membership in a predecesso... |
| elpredim 6316 | Membership in a predecesso... |
| elpred 6317 | Membership in a predecesso... |
| predexg 6318 | The predecessor class exis... |
| predasetexOLD 6319 | Obsolete form of ~ predexg... |
| dffr4 6320 | Alternate definition of we... |
| predel 6321 | Membership in the predeces... |
| predbrg 6322 | Closed form of ~ elpredim ... |
| predtrss 6323 | If ` R ` is transitive ove... |
| predpo 6324 | Property of the predecesso... |
| predso 6325 | Property of the predecesso... |
| setlikespec 6326 | If ` R ` is set-like in ` ... |
| predidm 6327 | Idempotent law for the pre... |
| predin 6328 | Intersection law for prede... |
| predun 6329 | Union law for predecessor ... |
| preddif 6330 | Difference law for predece... |
| predep 6331 | The predecessor under the ... |
| trpred 6332 | The class of predecessors ... |
| preddowncl 6333 | A property of classes that... |
| predpoirr 6334 | Given a partial ordering, ... |
| predfrirr 6335 | Given a well-founded relat... |
| pred0 6336 | The predecessor class over... |
| dfse3 6337 | Alternate definition of se... |
| predrelss 6338 | Subset carries from relati... |
| predprc 6339 | The predecessor of a prope... |
| predres 6340 | Predecessor class is unaff... |
| frpomin 6341 | Every nonempty (possibly p... |
| frpomin2 6342 | Every nonempty (possibly p... |
| frpoind 6343 | The principle of well-foun... |
| frpoinsg 6344 | Well-Founded Induction Sch... |
| frpoins2fg 6345 | Well-Founded Induction sch... |
| frpoins2g 6346 | Well-Founded Induction sch... |
| frpoins3g 6347 | Well-Founded Induction sch... |
| tz6.26 6348 | All nonempty subclasses of... |
| tz6.26OLD 6349 | Obsolete proof of ~ tz6.26... |
| tz6.26i 6350 | All nonempty subclasses of... |
| wfi 6351 | The Principle of Well-Orde... |
| wfiOLD 6352 | Obsolete proof of ~ wfi as... |
| wfii 6353 | The Principle of Well-Orde... |
| wfisg 6354 | Well-Ordered Induction Sch... |
| wfisgOLD 6355 | Obsolete version of ~ wfis... |
| wfis 6356 | Well-Ordered Induction Sch... |
| wfis2fg 6357 | Well-Ordered Induction Sch... |
| wfis2fgOLD 6358 | Obsolete version of ~ wfis... |
| wfis2f 6359 | Well-Ordered Induction sch... |
| wfis2g 6360 | Well-Ordered Induction Sch... |
| wfis2 6361 | Well-Ordered Induction sch... |
| wfis3 6362 | Well-Ordered Induction sch... |
| ordeq 6371 | Equality theorem for the o... |
| elong 6372 | An ordinal number is an or... |
| elon 6373 | An ordinal number is an or... |
| eloni 6374 | An ordinal number has the ... |
| elon2 6375 | An ordinal number is an or... |
| limeq 6376 | Equality theorem for the l... |
| ordwe 6377 | Membership well-orders eve... |
| ordtr 6378 | An ordinal class is transi... |
| ordfr 6379 | Membership is well-founded... |
| ordelss 6380 | An element of an ordinal c... |
| trssord 6381 | A transitive subclass of a... |
| ordirr 6382 | No ordinal class is a memb... |
| nordeq 6383 | A member of an ordinal cla... |
| ordn2lp 6384 | An ordinal class cannot be... |
| tz7.5 6385 | A nonempty subclass of an ... |
| ordelord 6386 | An element of an ordinal c... |
| tron 6387 | The class of all ordinal n... |
| ordelon 6388 | An element of an ordinal c... |
| onelon 6389 | An element of an ordinal n... |
| tz7.7 6390 | A transitive class belongs... |
| ordelssne 6391 | For ordinal classes, membe... |
| ordelpss 6392 | For ordinal classes, membe... |
| ordsseleq 6393 | For ordinal classes, inclu... |
| ordin 6394 | The intersection of two or... |
| onin 6395 | The intersection of two or... |
| ordtri3or 6396 | A trichotomy law for ordin... |
| ordtri1 6397 | A trichotomy law for ordin... |
| ontri1 6398 | A trichotomy law for ordin... |
| ordtri2 6399 | A trichotomy law for ordin... |
| ordtri3 6400 | A trichotomy law for ordin... |
| ordtri4 6401 | A trichotomy law for ordin... |
| orddisj 6402 | An ordinal class and its s... |
| onfr 6403 | The ordinal class is well-... |
| onelpss 6404 | Relationship between membe... |
| onsseleq 6405 | Relationship between subse... |
| onelss 6406 | An element of an ordinal n... |
| ordtr1 6407 | Transitive law for ordinal... |
| ordtr2 6408 | Transitive law for ordinal... |
| ordtr3 6409 | Transitive law for ordinal... |
| ontr1 6410 | Transitive law for ordinal... |
| ontr2 6411 | Transitive law for ordinal... |
| onelssex 6412 | Ordinal less than is equiv... |
| ordunidif 6413 | The union of an ordinal st... |
| ordintdif 6414 | If ` B ` is smaller than `... |
| onintss 6415 | If a property is true for ... |
| oneqmini 6416 | A way to show that an ordi... |
| ord0 6417 | The empty set is an ordina... |
| 0elon 6418 | The empty set is an ordina... |
| ord0eln0 6419 | A nonempty ordinal contain... |
| on0eln0 6420 | An ordinal number contains... |
| dflim2 6421 | An alternate definition of... |
| inton 6422 | The intersection of the cl... |
| nlim0 6423 | The empty set is not a lim... |
| limord 6424 | A limit ordinal is ordinal... |
| limuni 6425 | A limit ordinal is its own... |
| limuni2 6426 | The union of a limit ordin... |
| 0ellim 6427 | A limit ordinal contains t... |
| limelon 6428 | A limit ordinal class that... |
| onn0 6429 | The class of all ordinal n... |
| suceq 6430 | Equality of successors. (... |
| elsuci 6431 | Membership in a successor.... |
| elsucg 6432 | Membership in a successor.... |
| elsuc2g 6433 | Variant of membership in a... |
| elsuc 6434 | Membership in a successor.... |
| elsuc2 6435 | Membership in a successor.... |
| nfsuc 6436 | Bound-variable hypothesis ... |
| elelsuc 6437 | Membership in a successor.... |
| sucel 6438 | Membership of a successor ... |
| suc0 6439 | The successor of the empty... |
| sucprc 6440 | A proper class is its own ... |
| unisucs 6441 | The union of the successor... |
| unisucg 6442 | A transitive class is equa... |
| unisuc 6443 | A transitive class is equa... |
| sssucid 6444 | A class is included in its... |
| sucidg 6445 | Part of Proposition 7.23 o... |
| sucid 6446 | A set belongs to its succe... |
| nsuceq0 6447 | No successor is empty. (C... |
| eqelsuc 6448 | A set belongs to the succe... |
| iunsuc 6449 | Inductive definition for t... |
| suctr 6450 | The successor of a transit... |
| trsuc 6451 | A set whose successor belo... |
| trsucss 6452 | A member of the successor ... |
| ordsssuc 6453 | An ordinal is a subset of ... |
| onsssuc 6454 | A subset of an ordinal num... |
| ordsssuc2 6455 | An ordinal subset of an or... |
| onmindif 6456 | When its successor is subt... |
| ordnbtwn 6457 | There is no set between an... |
| onnbtwn 6458 | There is no set between an... |
| sucssel 6459 | A set whose successor is a... |
| orddif 6460 | Ordinal derived from its s... |
| orduniss 6461 | An ordinal class includes ... |
| ordtri2or 6462 | A trichotomy law for ordin... |
| ordtri2or2 6463 | A trichotomy law for ordin... |
| ordtri2or3 6464 | A consequence of total ord... |
| ordelinel 6465 | The intersection of two or... |
| ordssun 6466 | Property of a subclass of ... |
| ordequn 6467 | The maximum (i.e. union) o... |
| ordun 6468 | The maximum (i.e., union) ... |
| onunel 6469 | The union of two ordinals ... |
| ordunisssuc 6470 | A subclass relationship fo... |
| suc11 6471 | The successor operation be... |
| onun2 6472 | The union of two ordinals ... |
| ontr 6473 | An ordinal number is a tra... |
| onunisuc 6474 | An ordinal number is equal... |
| onordi 6475 | An ordinal number is an or... |
| ontrciOLD 6476 | Obsolete version of ~ ontr... |
| onirri 6477 | An ordinal number is not a... |
| oneli 6478 | A member of an ordinal num... |
| onelssi 6479 | A member of an ordinal num... |
| onssneli 6480 | An ordering law for ordina... |
| onssnel2i 6481 | An ordering law for ordina... |
| onelini 6482 | An element of an ordinal n... |
| oneluni 6483 | An ordinal number equals i... |
| onunisuci 6484 | An ordinal number is equal... |
| onsseli 6485 | Subset is equivalent to me... |
| onun2i 6486 | The union of two ordinal n... |
| unizlim 6487 | An ordinal equal to its ow... |
| on0eqel 6488 | An ordinal number either e... |
| snsn0non 6489 | The singleton of the singl... |
| onxpdisj 6490 | Ordinal numbers and ordere... |
| onnev 6491 | The class of ordinal numbe... |
| onnevOLD 6492 | Obsolete version of ~ onne... |
| iotajust 6494 | Soundness justification th... |
| dfiota2 6496 | Alternate definition for d... |
| nfiota1 6497 | Bound-variable hypothesis ... |
| nfiotadw 6498 | Deduction version of ~ nfi... |
| nfiotaw 6499 | Bound-variable hypothesis ... |
| nfiotad 6500 | Deduction version of ~ nfi... |
| nfiota 6501 | Bound-variable hypothesis ... |
| cbviotaw 6502 | Change bound variables in ... |
| cbviotavw 6503 | Change bound variables in ... |
| cbviotavwOLD 6504 | Obsolete version of ~ cbvi... |
| cbviota 6505 | Change bound variables in ... |
| cbviotav 6506 | Change bound variables in ... |
| sb8iota 6507 | Variable substitution in d... |
| iotaeq 6508 | Equality theorem for descr... |
| iotabi 6509 | Equivalence theorem for de... |
| uniabio 6510 | Part of Theorem 8.17 in [Q... |
| iotaval2 6511 | Version of ~ iotaval using... |
| iotauni2 6512 | Version of ~ iotauni using... |
| iotanul2 6513 | Version of ~ iotanul using... |
| iotaval 6514 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6515 | The ` iota ` class is a su... |
| iotaex 6516 | Theorem 8.23 in [Quine] p.... |
| iotavalOLD 6517 | Obsolete version of ~ iota... |
| iotauni 6518 | Equivalence between two di... |
| iotaint 6519 | Equivalence between two di... |
| iota1 6520 | Property of iota. (Contri... |
| iotanul 6521 | Theorem 8.22 in [Quine] p.... |
| iotassuniOLD 6522 | Obsolete version of ~ iota... |
| iotaexOLD 6523 | Obsolete version of ~ iota... |
| iota4 6524 | Theorem *14.22 in [Whitehe... |
| iota4an 6525 | Theorem *14.23 in [Whitehe... |
| iota5 6526 | A method for computing iot... |
| iotabidv 6527 | Formula-building deduction... |
| iotabii 6528 | Formula-building deduction... |
| iotacl 6529 | Membership law for descrip... |
| iota2df 6530 | A condition that allows to... |
| iota2d 6531 | A condition that allows to... |
| iota2 6532 | The unique element such th... |
| iotan0 6533 | Representation of "the uni... |
| sniota 6534 | A class abstraction with a... |
| dfiota4 6535 | The ` iota ` operation usi... |
| csbiota 6536 | Class substitution within ... |
| dffun2 6553 | Alternate definition of a ... |
| dffun2OLD 6554 | Obsolete version of ~ dffu... |
| dffun2OLDOLD 6555 | Obsolete version of ~ dffu... |
| dffun6 6556 | Alternate definition of a ... |
| dffun3 6557 | Alternate definition of fu... |
| dffun3OLD 6558 | Obsolete version of ~ dffu... |
| dffun4 6559 | Alternate definition of a ... |
| dffun5 6560 | Alternate definition of fu... |
| dffun6f 6561 | Definition of function, us... |
| dffun6OLD 6562 | Obsolete version of ~ dffu... |
| funmo 6563 | A function has at most one... |
| funmoOLD 6564 | Obsolete version of ~ funm... |
| funrel 6565 | A function is a relation. ... |
| 0nelfun 6566 | A function does not contai... |
| funss 6567 | Subclass theorem for funct... |
| funeq 6568 | Equality theorem for funct... |
| funeqi 6569 | Equality inference for the... |
| funeqd 6570 | Equality deduction for the... |
| nffun 6571 | Bound-variable hypothesis ... |
| sbcfung 6572 | Distribute proper substitu... |
| funeu 6573 | There is exactly one value... |
| funeu2 6574 | There is exactly one value... |
| dffun7 6575 | Alternate definition of a ... |
| dffun8 6576 | Alternate definition of a ... |
| dffun9 6577 | Alternate definition of a ... |
| funfn 6578 | A class is a function if a... |
| funfnd 6579 | A function is a function o... |
| funi 6580 | The identity relation is a... |
| nfunv 6581 | The universal class is not... |
| funopg 6582 | A Kuratowski ordered pair ... |
| funopab 6583 | A class of ordered pairs i... |
| funopabeq 6584 | A class of ordered pairs o... |
| funopab4 6585 | A class of ordered pairs o... |
| funmpt 6586 | A function in maps-to nota... |
| funmpt2 6587 | Functionality of a class g... |
| funco 6588 | The composition of two fun... |
| funresfunco 6589 | Composition of two functio... |
| funres 6590 | A restriction of a functio... |
| funresd 6591 | A restriction of a functio... |
| funssres 6592 | The restriction of a funct... |
| fun2ssres 6593 | Equality of restrictions o... |
| funun 6594 | The union of functions wit... |
| fununmo 6595 | If the union of classes is... |
| fununfun 6596 | If the union of classes is... |
| fundif 6597 | A function with removed el... |
| funcnvsn 6598 | The converse singleton of ... |
| funsng 6599 | A singleton of an ordered ... |
| fnsng 6600 | Functionality and domain o... |
| funsn 6601 | A singleton of an ordered ... |
| funprg 6602 | A set of two pairs is a fu... |
| funtpg 6603 | A set of three pairs is a ... |
| funpr 6604 | A function with a domain o... |
| funtp 6605 | A function with a domain o... |
| fnsn 6606 | Functionality and domain o... |
| fnprg 6607 | Function with a domain of ... |
| fntpg 6608 | Function with a domain of ... |
| fntp 6609 | A function with a domain o... |
| funcnvpr 6610 | The converse pair of order... |
| funcnvtp 6611 | The converse triple of ord... |
| funcnvqp 6612 | The converse quadruple of ... |
| fun0 6613 | The empty set is a functio... |
| funcnv0 6614 | The converse of the empty ... |
| funcnvcnv 6615 | The double converse of a f... |
| funcnv2 6616 | A simpler equivalence for ... |
| funcnv 6617 | The converse of a class is... |
| funcnv3 6618 | A condition showing a clas... |
| fun2cnv 6619 | The double converse of a c... |
| svrelfun 6620 | A single-valued relation i... |
| fncnv 6621 | Single-rootedness (see ~ f... |
| fun11 6622 | Two ways of stating that `... |
| fununi 6623 | The union of a chain (with... |
| funin 6624 | The intersection with a fu... |
| funres11 6625 | The restriction of a one-t... |
| funcnvres 6626 | The converse of a restrict... |
| cnvresid 6627 | Converse of a restricted i... |
| funcnvres2 6628 | The converse of a restrict... |
| funimacnv 6629 | The image of the preimage ... |
| funimass1 6630 | A kind of contraposition l... |
| funimass2 6631 | A kind of contraposition l... |
| imadif 6632 | The image of a difference ... |
| imain 6633 | The image of an intersecti... |
| funimaexg 6634 | Axiom of Replacement using... |
| funimaexgOLD 6635 | Obsolete version of ~ funi... |
| funimaex 6636 | The image of a set under a... |
| isarep1 6637 | Part of a study of the Axi... |
| isarep1OLD 6638 | Obsolete version of ~ isar... |
| isarep2 6639 | Part of a study of the Axi... |
| fneq1 6640 | Equality theorem for funct... |
| fneq2 6641 | Equality theorem for funct... |
| fneq1d 6642 | Equality deduction for fun... |
| fneq2d 6643 | Equality deduction for fun... |
| fneq12d 6644 | Equality deduction for fun... |
| fneq12 6645 | Equality theorem for funct... |
| fneq1i 6646 | Equality inference for fun... |
| fneq2i 6647 | Equality inference for fun... |
| nffn 6648 | Bound-variable hypothesis ... |
| fnfun 6649 | A function with domain is ... |
| fnfund 6650 | A function with domain is ... |
| fnrel 6651 | A function with domain is ... |
| fndm 6652 | The domain of a function. ... |
| fndmi 6653 | The domain of a function. ... |
| fndmd 6654 | The domain of a function. ... |
| funfni 6655 | Inference to convert a fun... |
| fndmu 6656 | A function has a unique do... |
| fnbr 6657 | The first argument of bina... |
| fnop 6658 | The first argument of an o... |
| fneu 6659 | There is exactly one value... |
| fneu2 6660 | There is exactly one value... |
| fnunres1 6661 | Restriction of a disjoint ... |
| fnunres2 6662 | Restriction of a disjoint ... |
| fnun 6663 | The union of two functions... |
| fnund 6664 | The union of two functions... |
| fnunop 6665 | Extension of a function wi... |
| fncofn 6666 | Composition of a function ... |
| fnco 6667 | Composition of two functio... |
| fncoOLD 6668 | Obsolete version of ~ fnco... |
| fnresdm 6669 | A function does not change... |
| fnresdisj 6670 | A function restricted to a... |
| 2elresin 6671 | Membership in two function... |
| fnssresb 6672 | Restriction of a function ... |
| fnssres 6673 | Restriction of a function ... |
| fnssresd 6674 | Restriction of a function ... |
| fnresin1 6675 | Restriction of a function'... |
| fnresin2 6676 | Restriction of a function'... |
| fnres 6677 | An equivalence for functio... |
| idfn 6678 | The identity relation is a... |
| fnresi 6679 | The restricted identity re... |
| fnima 6680 | The image of a function's ... |
| fn0 6681 | A function with empty doma... |
| fnimadisj 6682 | A class that is disjoint w... |
| fnimaeq0 6683 | Images under a function ne... |
| dfmpt3 6684 | Alternate definition for t... |
| mptfnf 6685 | The maps-to notation defin... |
| fnmptf 6686 | The maps-to notation defin... |
| fnopabg 6687 | Functionality and domain o... |
| fnopab 6688 | Functionality and domain o... |
| mptfng 6689 | The maps-to notation defin... |
| fnmpt 6690 | The maps-to notation defin... |
| fnmptd 6691 | The maps-to notation defin... |
| mpt0 6692 | A mapping operation with e... |
| fnmpti 6693 | Functionality and domain o... |
| dmmpti 6694 | Domain of the mapping oper... |
| dmmptd 6695 | The domain of the mapping ... |
| mptun 6696 | Union of mappings which ar... |
| partfun 6697 | Rewrite a function defined... |
| feq1 6698 | Equality theorem for funct... |
| feq2 6699 | Equality theorem for funct... |
| feq3 6700 | Equality theorem for funct... |
| feq23 6701 | Equality theorem for funct... |
| feq1d 6702 | Equality deduction for fun... |
| feq2d 6703 | Equality deduction for fun... |
| feq3d 6704 | Equality deduction for fun... |
| feq12d 6705 | Equality deduction for fun... |
| feq123d 6706 | Equality deduction for fun... |
| feq123 6707 | Equality theorem for funct... |
| feq1i 6708 | Equality inference for fun... |
| feq2i 6709 | Equality inference for fun... |
| feq12i 6710 | Equality inference for fun... |
| feq23i 6711 | Equality inference for fun... |
| feq23d 6712 | Equality deduction for fun... |
| nff 6713 | Bound-variable hypothesis ... |
| sbcfng 6714 | Distribute proper substitu... |
| sbcfg 6715 | Distribute proper substitu... |
| elimf 6716 | Eliminate a mapping hypoth... |
| ffn 6717 | A mapping is a function wi... |
| ffnd 6718 | A mapping is a function wi... |
| dffn2 6719 | Any function is a mapping ... |
| ffun 6720 | A mapping is a function. ... |
| ffund 6721 | A mapping is a function, d... |
| frel 6722 | A mapping is a relation. ... |
| freld 6723 | A mapping is a relation. ... |
| frn 6724 | The range of a mapping. (... |
| frnd 6725 | Deduction form of ~ frn . ... |
| fdm 6726 | The domain of a mapping. ... |
| fdmOLD 6727 | Obsolete version of ~ fdm ... |
| fdmd 6728 | Deduction form of ~ fdm . ... |
| fdmi 6729 | Inference associated with ... |
| dffn3 6730 | A function maps to its ran... |
| ffrn 6731 | A function maps to its ran... |
| ffrnb 6732 | Characterization of a func... |
| ffrnbd 6733 | A function maps to its ran... |
| fss 6734 | Expanding the codomain of ... |
| fssd 6735 | Expanding the codomain of ... |
| fssdmd 6736 | Expressing that a class is... |
| fssdm 6737 | Expressing that a class is... |
| fimass 6738 | The image of a class under... |
| fimacnv 6739 | The preimage of the codoma... |
| fcof 6740 | Composition of a function ... |
| fco 6741 | Composition of two functio... |
| fcoOLD 6742 | Obsolete version of ~ fco ... |
| fcod 6743 | Composition of two mapping... |
| fco2 6744 | Functionality of a composi... |
| fssxp 6745 | A mapping is a class of or... |
| funssxp 6746 | Two ways of specifying a p... |
| ffdm 6747 | A mapping is a partial fun... |
| ffdmd 6748 | The domain of a function. ... |
| fdmrn 6749 | A different way to write `... |
| funcofd 6750 | Composition of two functio... |
| fco3OLD 6751 | Obsolete version of ~ func... |
| opelf 6752 | The members of an ordered ... |
| fun 6753 | The union of two functions... |
| fun2 6754 | The union of two functions... |
| fun2d 6755 | The union of functions wit... |
| fnfco 6756 | Composition of two functio... |
| fssres 6757 | Restriction of a function ... |
| fssresd 6758 | Restriction of a function ... |
| fssres2 6759 | Restriction of a restricte... |
| fresin 6760 | An identity for the mappin... |
| resasplit 6761 | If two functions agree on ... |
| fresaun 6762 | The union of two functions... |
| fresaunres2 6763 | From the union of two func... |
| fresaunres1 6764 | From the union of two func... |
| fcoi1 6765 | Composition of a mapping a... |
| fcoi2 6766 | Composition of restricted ... |
| feu 6767 | There is exactly one value... |
| fcnvres 6768 | The converse of a restrict... |
| fimacnvdisj 6769 | The preimage of a class di... |
| fint 6770 | Function into an intersect... |
| fin 6771 | Mapping into an intersecti... |
| f0 6772 | The empty function. (Cont... |
| f00 6773 | A class is a function with... |
| f0bi 6774 | A function with empty doma... |
| f0dom0 6775 | A function is empty iff it... |
| f0rn0 6776 | If there is no element in ... |
| fconst 6777 | A Cartesian product with a... |
| fconstg 6778 | A Cartesian product with a... |
| fnconstg 6779 | A Cartesian product with a... |
| fconst6g 6780 | Constant function with loo... |
| fconst6 6781 | A constant function as a m... |
| f1eq1 6782 | Equality theorem for one-t... |
| f1eq2 6783 | Equality theorem for one-t... |
| f1eq3 6784 | Equality theorem for one-t... |
| nff1 6785 | Bound-variable hypothesis ... |
| dff12 6786 | Alternate definition of a ... |
| f1f 6787 | A one-to-one mapping is a ... |
| f1fn 6788 | A one-to-one mapping is a ... |
| f1fun 6789 | A one-to-one mapping is a ... |
| f1rel 6790 | A one-to-one onto mapping ... |
| f1dm 6791 | The domain of a one-to-one... |
| f1dmOLD 6792 | Obsolete version of ~ f1dm... |
| f1ss 6793 | A function that is one-to-... |
| f1ssr 6794 | A function that is one-to-... |
| f1ssres 6795 | A function that is one-to-... |
| f1resf1 6796 | The restriction of an inje... |
| f1cnvcnv 6797 | Two ways to express that a... |
| f1cof1 6798 | Composition of two one-to-... |
| f1co 6799 | Composition of one-to-one ... |
| f1coOLD 6800 | Obsolete version of ~ f1co... |
| foeq1 6801 | Equality theorem for onto ... |
| foeq2 6802 | Equality theorem for onto ... |
| foeq3 6803 | Equality theorem for onto ... |
| nffo 6804 | Bound-variable hypothesis ... |
| fof 6805 | An onto mapping is a mappi... |
| fofun 6806 | An onto mapping is a funct... |
| fofn 6807 | An onto mapping is a funct... |
| forn 6808 | The codomain of an onto fu... |
| dffo2 6809 | Alternate definition of an... |
| foima 6810 | The image of the domain of... |
| dffn4 6811 | A function maps onto its r... |
| funforn 6812 | A function maps its domain... |
| fodmrnu 6813 | An onto function has uniqu... |
| fimadmfo 6814 | A function is a function o... |
| fores 6815 | Restriction of an onto fun... |
| fimadmfoALT 6816 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6817 | The preimage of the codoma... |
| focofo 6818 | Composition of onto functi... |
| foco 6819 | Composition of onto functi... |
| foconst 6820 | A nonzero constant functio... |
| f1oeq1 6821 | Equality theorem for one-t... |
| f1oeq2 6822 | Equality theorem for one-t... |
| f1oeq3 6823 | Equality theorem for one-t... |
| f1oeq23 6824 | Equality theorem for one-t... |
| f1eq123d 6825 | Equality deduction for one... |
| foeq123d 6826 | Equality deduction for ont... |
| f1oeq123d 6827 | Equality deduction for one... |
| f1oeq1d 6828 | Equality deduction for one... |
| f1oeq2d 6829 | Equality deduction for one... |
| f1oeq3d 6830 | Equality deduction for one... |
| nff1o 6831 | Bound-variable hypothesis ... |
| f1of1 6832 | A one-to-one onto mapping ... |
| f1of 6833 | A one-to-one onto mapping ... |
| f1ofn 6834 | A one-to-one onto mapping ... |
| f1ofun 6835 | A one-to-one onto mapping ... |
| f1orel 6836 | A one-to-one onto mapping ... |
| f1odm 6837 | The domain of a one-to-one... |
| dff1o2 6838 | Alternate definition of on... |
| dff1o3 6839 | Alternate definition of on... |
| f1ofo 6840 | A one-to-one onto function... |
| dff1o4 6841 | Alternate definition of on... |
| dff1o5 6842 | Alternate definition of on... |
| f1orn 6843 | A one-to-one function maps... |
| f1f1orn 6844 | A one-to-one function maps... |
| f1ocnv 6845 | The converse of a one-to-o... |
| f1ocnvb 6846 | A relation is a one-to-one... |
| f1ores 6847 | The restriction of a one-t... |
| f1orescnv 6848 | The converse of a one-to-o... |
| f1imacnv 6849 | Preimage of an image. (Co... |
| foimacnv 6850 | A reverse version of ~ f1i... |
| foun 6851 | The union of two onto func... |
| f1oun 6852 | The union of two one-to-on... |
| f1un 6853 | The union of two one-to-on... |
| resdif 6854 | The restriction of a one-t... |
| resin 6855 | The restriction of a one-t... |
| f1oco 6856 | Composition of one-to-one ... |
| f1cnv 6857 | The converse of an injecti... |
| funcocnv2 6858 | Composition with the conve... |
| fococnv2 6859 | The composition of an onto... |
| f1ococnv2 6860 | The composition of a one-t... |
| f1cocnv2 6861 | Composition of an injectiv... |
| f1ococnv1 6862 | The composition of a one-t... |
| f1cocnv1 6863 | Composition of an injectiv... |
| funcoeqres 6864 | Express a constraint on a ... |
| f1ssf1 6865 | A subset of an injective f... |
| f10 6866 | The empty set maps one-to-... |
| f10d 6867 | The empty set maps one-to-... |
| f1o00 6868 | One-to-one onto mapping of... |
| fo00 6869 | Onto mapping of the empty ... |
| f1o0 6870 | One-to-one onto mapping of... |
| f1oi 6871 | A restriction of the ident... |
| f1ovi 6872 | The identity relation is a... |
| f1osn 6873 | A singleton of an ordered ... |
| f1osng 6874 | A singleton of an ordered ... |
| f1sng 6875 | A singleton of an ordered ... |
| fsnd 6876 | A singleton of an ordered ... |
| f1oprswap 6877 | A two-element swap is a bi... |
| f1oprg 6878 | An unordered pair of order... |
| tz6.12-2 6879 | Function value when ` F ` ... |
| fveu 6880 | The value of a function at... |
| brprcneu 6881 | If ` A ` is a proper class... |
| brprcneuALT 6882 | Alternate proof of ~ brprc... |
| fvprc 6883 | A function's value at a pr... |
| fvprcALT 6884 | Alternate proof of ~ fvprc... |
| rnfvprc 6885 | The range of a function va... |
| fv2 6886 | Alternate definition of fu... |
| dffv3 6887 | A definition of function v... |
| dffv4 6888 | The previous definition of... |
| elfv 6889 | Membership in a function v... |
| fveq1 6890 | Equality theorem for funct... |
| fveq2 6891 | Equality theorem for funct... |
| fveq1i 6892 | Equality inference for fun... |
| fveq1d 6893 | Equality deduction for fun... |
| fveq2i 6894 | Equality inference for fun... |
| fveq2d 6895 | Equality deduction for fun... |
| 2fveq3 6896 | Equality theorem for neste... |
| fveq12i 6897 | Equality deduction for fun... |
| fveq12d 6898 | Equality deduction for fun... |
| fveqeq2d 6899 | Equality deduction for fun... |
| fveqeq2 6900 | Equality deduction for fun... |
| nffv 6901 | Bound-variable hypothesis ... |
| nffvmpt1 6902 | Bound-variable hypothesis ... |
| nffvd 6903 | Deduction version of bound... |
| fvex 6904 | The value of a class exist... |
| fvexi 6905 | The value of a class exist... |
| fvexd 6906 | The value of a class exist... |
| fvif 6907 | Move a conditional outside... |
| iffv 6908 | Move a conditional outside... |
| fv3 6909 | Alternate definition of th... |
| fvres 6910 | The value of a restricted ... |
| fvresd 6911 | The value of a restricted ... |
| funssfv 6912 | The value of a member of t... |
| tz6.12c 6913 | Corollary of Theorem 6.12(... |
| tz6.12-1 6914 | Function value. Theorem 6... |
| tz6.12-1OLD 6915 | Obsolete version of ~ tz6.... |
| tz6.12 6916 | Function value. Theorem 6... |
| tz6.12f 6917 | Function value, using boun... |
| tz6.12cOLD 6918 | Obsolete version of ~ tz6.... |
| tz6.12i 6919 | Corollary of Theorem 6.12(... |
| fvbr0 6920 | Two possibilities for the ... |
| fvrn0 6921 | A function value is a memb... |
| fvn0fvelrn 6922 | If the value of a function... |
| elfvunirn 6923 | A function value is a subs... |
| fvssunirn 6924 | The result of a function v... |
| fvssunirnOLD 6925 | Obsolete version of ~ fvss... |
| ndmfv 6926 | The value of a class outsi... |
| ndmfvrcl 6927 | Reverse closure law for fu... |
| elfvdm 6928 | If a function value has a ... |
| elfvex 6929 | If a function value has a ... |
| elfvexd 6930 | If a function value has a ... |
| eliman0 6931 | A nonempty function value ... |
| nfvres 6932 | The value of a non-member ... |
| nfunsn 6933 | If the restriction of a cl... |
| fvfundmfvn0 6934 | If the "value of a class" ... |
| 0fv 6935 | Function value of the empt... |
| fv2prc 6936 | A function value of a func... |
| elfv2ex 6937 | If a function value of a f... |
| fveqres 6938 | Equal values imply equal v... |
| csbfv12 6939 | Move class substitution in... |
| csbfv2g 6940 | Move class substitution in... |
| csbfv 6941 | Substitution for a functio... |
| funbrfv 6942 | The second argument of a b... |
| funopfv 6943 | The second element in an o... |
| fnbrfvb 6944 | Equivalence of function va... |
| fnopfvb 6945 | Equivalence of function va... |
| funbrfvb 6946 | Equivalence of function va... |
| funopfvb 6947 | Equivalence of function va... |
| fnbrfvb2 6948 | Version of ~ fnbrfvb for f... |
| funbrfv2b 6949 | Function value in terms of... |
| dffn5 6950 | Representation of a functi... |
| fnrnfv 6951 | The range of a function ex... |
| fvelrnb 6952 | A member of a function's r... |
| foelcdmi 6953 | A member of a surjective f... |
| dfimafn 6954 | Alternate definition of th... |
| dfimafn2 6955 | Alternate definition of th... |
| funimass4 6956 | Membership relation for th... |
| fvelima 6957 | Function value in an image... |
| funimassd 6958 | Sufficient condition for t... |
| fvelimad 6959 | Function value in an image... |
| feqmptd 6960 | Deduction form of ~ dffn5 ... |
| feqresmpt 6961 | Express a restricted funct... |
| feqmptdf 6962 | Deduction form of ~ dffn5f... |
| dffn5f 6963 | Representation of a functi... |
| fvelimab 6964 | Function value in an image... |
| fvelimabd 6965 | Deduction form of ~ fvelim... |
| unima 6966 | Image of a union. (Contri... |
| fvi 6967 | The value of the identity ... |
| fviss 6968 | The value of the identity ... |
| fniinfv 6969 | The indexed intersection o... |
| fnsnfv 6970 | Singleton of function valu... |
| fnsnfvOLD 6971 | Obsolete version of ~ fnsn... |
| opabiotafun 6972 | Define a function whose va... |
| opabiotadm 6973 | Define a function whose va... |
| opabiota 6974 | Define a function whose va... |
| fnimapr 6975 | The image of a pair under ... |
| ssimaex 6976 | The existence of a subimag... |
| ssimaexg 6977 | The existence of a subimag... |
| funfv 6978 | A simplified expression fo... |
| funfv2 6979 | The value of a function. ... |
| funfv2f 6980 | The value of a function. ... |
| fvun 6981 | Value of the union of two ... |
| fvun1 6982 | The value of a union when ... |
| fvun2 6983 | The value of a union when ... |
| fvun1d 6984 | The value of a union when ... |
| fvun2d 6985 | The value of a union when ... |
| dffv2 6986 | Alternate definition of fu... |
| dmfco 6987 | Domains of a function comp... |
| fvco2 6988 | Value of a function compos... |
| fvco 6989 | Value of a function compos... |
| fvco3 6990 | Value of a function compos... |
| fvco3d 6991 | Value of a function compos... |
| fvco4i 6992 | Conditions for a compositi... |
| fvopab3g 6993 | Value of a function given ... |
| fvopab3ig 6994 | Value of a function given ... |
| brfvopabrbr 6995 | The binary relation of a f... |
| fvmptg 6996 | Value of a function given ... |
| fvmpti 6997 | Value of a function given ... |
| fvmpt 6998 | Value of a function given ... |
| fvmpt2f 6999 | Value of a function given ... |
| fvtresfn 7000 | Functionality of a tuple-r... |
| fvmpts 7001 | Value of a function given ... |
| fvmpt3 7002 | Value of a function given ... |
| fvmpt3i 7003 | Value of a function given ... |
| fvmptdf 7004 | Deduction version of ~ fvm... |
| fvmptd 7005 | Deduction version of ~ fvm... |
| fvmptd2 7006 | Deduction version of ~ fvm... |
| mptrcl 7007 | Reverse closure for a mapp... |
| fvmpt2i 7008 | Value of a function given ... |
| fvmpt2 7009 | Value of a function given ... |
| fvmptss 7010 | If all the values of the m... |
| fvmpt2d 7011 | Deduction version of ~ fvm... |
| fvmptex 7012 | Express a function ` F ` w... |
| fvmptd3f 7013 | Alternate deduction versio... |
| fvmptd2f 7014 | Alternate deduction versio... |
| fvmptdv 7015 | Alternate deduction versio... |
| fvmptdv2 7016 | Alternate deduction versio... |
| mpteqb 7017 | Bidirectional equality the... |
| fvmptt 7018 | Closed theorem form of ~ f... |
| fvmptf 7019 | Value of a function given ... |
| fvmptnf 7020 | The value of a function gi... |
| fvmptd3 7021 | Deduction version of ~ fvm... |
| fvmptn 7022 | This somewhat non-intuitiv... |
| fvmptss2 7023 | A mapping always evaluates... |
| elfvmptrab1w 7024 | Implications for the value... |
| elfvmptrab1 7025 | Implications for the value... |
| elfvmptrab 7026 | Implications for the value... |
| fvopab4ndm 7027 | Value of a function given ... |
| fvmptndm 7028 | Value of a function given ... |
| fvmptrabfv 7029 | Value of a function mappin... |
| fvopab5 7030 | The value of a function th... |
| fvopab6 7031 | Value of a function given ... |
| eqfnfv 7032 | Equality of functions is d... |
| eqfnfv2 7033 | Equality of functions is d... |
| eqfnfv3 7034 | Derive equality of functio... |
| eqfnfvd 7035 | Deduction for equality of ... |
| eqfnfv2f 7036 | Equality of functions is d... |
| eqfunfv 7037 | Equality of functions is d... |
| eqfnun 7038 | Two functions on ` A u. B ... |
| fvreseq0 7039 | Equality of restricted fun... |
| fvreseq1 7040 | Equality of a function res... |
| fvreseq 7041 | Equality of restricted fun... |
| fnmptfvd 7042 | A function with a given do... |
| fndmdif 7043 | Two ways to express the lo... |
| fndmdifcom 7044 | The difference set between... |
| fndmdifeq0 7045 | The difference set of two ... |
| fndmin 7046 | Two ways to express the lo... |
| fneqeql 7047 | Two functions are equal if... |
| fneqeql2 7048 | Two functions are equal if... |
| fnreseql 7049 | Two functions are equal on... |
| chfnrn 7050 | The range of a choice func... |
| funfvop 7051 | Ordered pair with function... |
| funfvbrb 7052 | Two ways to say that ` A `... |
| fvimacnvi 7053 | A member of a preimage is ... |
| fvimacnv 7054 | The argument of a function... |
| funimass3 7055 | A kind of contraposition l... |
| funimass5 7056 | A subclass of a preimage i... |
| funconstss 7057 | Two ways of specifying tha... |
| fvimacnvALT 7058 | Alternate proof of ~ fvima... |
| elpreima 7059 | Membership in the preimage... |
| elpreimad 7060 | Membership in the preimage... |
| fniniseg 7061 | Membership in the preimage... |
| fncnvima2 7062 | Inverse images under funct... |
| fniniseg2 7063 | Inverse point images under... |
| unpreima 7064 | Preimage of a union. (Con... |
| inpreima 7065 | Preimage of an intersectio... |
| difpreima 7066 | Preimage of a difference. ... |
| respreima 7067 | The preimage of a restrict... |
| cnvimainrn 7068 | The preimage of the inters... |
| sspreima 7069 | The preimage of a subset i... |
| iinpreima 7070 | Preimage of an intersectio... |
| intpreima 7071 | Preimage of an intersectio... |
| fimacnvOLD 7072 | Obsolete version of ~ fima... |
| fimacnvinrn 7073 | Taking the converse image ... |
| fimacnvinrn2 7074 | Taking the converse image ... |
| rescnvimafod 7075 | The restriction of a funct... |
| fvn0ssdmfun 7076 | If a class' function value... |
| fnopfv 7077 | Ordered pair with function... |
| fvelrn 7078 | A function's value belongs... |
| nelrnfvne 7079 | A function value cannot be... |
| fveqdmss 7080 | If the empty set is not co... |
| fveqressseq 7081 | If the empty set is not co... |
| fnfvelrn 7082 | A function's value belongs... |
| ffvelcdm 7083 | A function's value belongs... |
| fnfvelrnd 7084 | A function's value belongs... |
| ffvelcdmi 7085 | A function's value belongs... |
| ffvelcdmda 7086 | A function's value belongs... |
| ffvelcdmd 7087 | A function's value belongs... |
| rexrn 7088 | Restricted existential qua... |
| ralrn 7089 | Restricted universal quant... |
| elrnrexdm 7090 | For any element in the ran... |
| elrnrexdmb 7091 | For any element in the ran... |
| eldmrexrn 7092 | For any element in the dom... |
| eldmrexrnb 7093 | For any element in the dom... |
| fvcofneq 7094 | The values of two function... |
| ralrnmptw 7095 | A restricted quantifier ov... |
| rexrnmptw 7096 | A restricted quantifier ov... |
| ralrnmpt 7097 | A restricted quantifier ov... |
| rexrnmpt 7098 | A restricted quantifier ov... |
| f0cli 7099 | Unconditional closure of a... |
| dff2 7100 | Alternate definition of a ... |
| dff3 7101 | Alternate definition of a ... |
| dff4 7102 | Alternate definition of a ... |
| dffo3 7103 | An onto mapping expressed ... |
| dffo4 7104 | Alternate definition of an... |
| dffo5 7105 | Alternate definition of an... |
| exfo 7106 | A relation equivalent to t... |
| dffo3f 7107 | An onto mapping expressed ... |
| foelrn 7108 | Property of a surjective f... |
| foelrnf 7109 | Property of a surjective f... |
| foco2 7110 | If a composition of two fu... |
| fmpt 7111 | Functionality of the mappi... |
| f1ompt 7112 | Express bijection for a ma... |
| fmpti 7113 | Functionality of the mappi... |
| fvmptelcdm 7114 | The value of a function at... |
| fmptd 7115 | Domain and codomain of the... |
| fmpttd 7116 | Version of ~ fmptd with in... |
| fmpt3d 7117 | Domain and codomain of the... |
| fmptdf 7118 | A version of ~ fmptd using... |
| fompt 7119 | Express being onto for a m... |
| ffnfv 7120 | A function maps to a class... |
| ffnfvf 7121 | A function maps to a class... |
| fnfvrnss 7122 | An upper bound for range d... |
| fcdmssb 7123 | A function is a function i... |
| rnmptss 7124 | The range of an operation ... |
| fmpt2d 7125 | Domain and codomain of the... |
| ffvresb 7126 | A necessary and sufficient... |
| f1oresrab 7127 | Build a bijection between ... |
| f1ossf1o 7128 | Restricting a bijection, w... |
| fmptco 7129 | Composition of two functio... |
| fmptcof 7130 | Version of ~ fmptco where ... |
| fmptcos 7131 | Composition of two functio... |
| cofmpt 7132 | Express composition of a m... |
| fcompt 7133 | Express composition of two... |
| fcoconst 7134 | Composition with a constan... |
| fsn 7135 | A function maps a singleto... |
| fsn2 7136 | A function that maps a sin... |
| fsng 7137 | A function maps a singleto... |
| fsn2g 7138 | A function that maps a sin... |
| xpsng 7139 | The Cartesian product of t... |
| xpprsng 7140 | The Cartesian product of a... |
| xpsn 7141 | The Cartesian product of t... |
| f1o2sn 7142 | A singleton consisting in ... |
| residpr 7143 | Restriction of the identit... |
| dfmpt 7144 | Alternate definition for t... |
| fnasrn 7145 | A function expressed as th... |
| idref 7146 | Two ways to state that a r... |
| funiun 7147 | A function is a union of s... |
| funopsn 7148 | If a function is an ordere... |
| funop 7149 | An ordered pair is a funct... |
| funopdmsn 7150 | The domain of a function w... |
| funsndifnop 7151 | A singleton of an ordered ... |
| funsneqopb 7152 | A singleton of an ordered ... |
| ressnop0 7153 | If ` A ` is not in ` C ` ,... |
| fpr 7154 | A function with a domain o... |
| fprg 7155 | A function with a domain o... |
| ftpg 7156 | A function with a domain o... |
| ftp 7157 | A function with a domain o... |
| fnressn 7158 | A function restricted to a... |
| funressn 7159 | A function restricted to a... |
| fressnfv 7160 | The value of a function re... |
| fvrnressn 7161 | If the value of a function... |
| fvressn 7162 | The value of a function re... |
| fvn0fvelrnOLD 7163 | Obsolete version of ~ fvn0... |
| fvconst 7164 | The value of a constant fu... |
| fnsnr 7165 | If a class belongs to a fu... |
| fnsnb 7166 | A function whose domain is... |
| fmptsn 7167 | Express a singleton functi... |
| fmptsng 7168 | Express a singleton functi... |
| fmptsnd 7169 | Express a singleton functi... |
| fmptap 7170 | Append an additional value... |
| fmptapd 7171 | Append an additional value... |
| fmptpr 7172 | Express a pair function in... |
| fvresi 7173 | The value of a restricted ... |
| fninfp 7174 | Express the class of fixed... |
| fnelfp 7175 | Property of a fixed point ... |
| fndifnfp 7176 | Express the class of non-f... |
| fnelnfp 7177 | Property of a non-fixed po... |
| fnnfpeq0 7178 | A function is the identity... |
| fvunsn 7179 | Remove an ordered pair not... |
| fvsng 7180 | The value of a singleton o... |
| fvsn 7181 | The value of a singleton o... |
| fvsnun1 7182 | The value of a function wi... |
| fvsnun2 7183 | The value of a function wi... |
| fnsnsplit 7184 | Split a function into a si... |
| fsnunf 7185 | Adjoining a point to a fun... |
| fsnunf2 7186 | Adjoining a point to a pun... |
| fsnunfv 7187 | Recover the added point fr... |
| fsnunres 7188 | Recover the original funct... |
| funresdfunsn 7189 | Restricting a function to ... |
| fvpr1g 7190 | The value of a function wi... |
| fvpr2g 7191 | The value of a function wi... |
| fvpr2gOLD 7192 | Obsolete version of ~ fvpr... |
| fvpr1 7193 | The value of a function wi... |
| fvpr1OLD 7194 | Obsolete version of ~ fvpr... |
| fvpr2 7195 | The value of a function wi... |
| fvpr2OLD 7196 | Obsolete version of ~ fvpr... |
| fprb 7197 | A condition for functionho... |
| fvtp1 7198 | The first value of a funct... |
| fvtp2 7199 | The second value of a func... |
| fvtp3 7200 | The third value of a funct... |
| fvtp1g 7201 | The value of a function wi... |
| fvtp2g 7202 | The value of a function wi... |
| fvtp3g 7203 | The value of a function wi... |
| tpres 7204 | An unordered triple of ord... |
| fvconst2g 7205 | The value of a constant fu... |
| fconst2g 7206 | A constant function expres... |
| fvconst2 7207 | The value of a constant fu... |
| fconst2 7208 | A constant function expres... |
| fconst5 7209 | Two ways to express that a... |
| rnmptc 7210 | Range of a constant functi... |
| rnmptcOLD 7211 | Obsolete version of ~ rnmp... |
| fnprb 7212 | A function whose domain ha... |
| fntpb 7213 | A function whose domain ha... |
| fnpr2g 7214 | A function whose domain ha... |
| fpr2g 7215 | A function that maps a pai... |
| fconstfv 7216 | A constant function expres... |
| fconst3 7217 | Two ways to express a cons... |
| fconst4 7218 | Two ways to express a cons... |
| resfunexg 7219 | The restriction of a funct... |
| resiexd 7220 | The restriction of the ide... |
| fnex 7221 | If the domain of a functio... |
| fnexd 7222 | If the domain of a functio... |
| funex 7223 | If the domain of a functio... |
| opabex 7224 | Existence of a function ex... |
| mptexg 7225 | If the domain of a functio... |
| mptexgf 7226 | If the domain of a functio... |
| mptex 7227 | If the domain of a functio... |
| mptexd 7228 | If the domain of a functio... |
| mptrabex 7229 | If the domain of a functio... |
| fex 7230 | If the domain of a mapping... |
| fexd 7231 | If the domain of a mapping... |
| mptfvmpt 7232 | A function in maps-to nota... |
| eufnfv 7233 | A function is uniquely det... |
| funfvima 7234 | A function's value in a pr... |
| funfvima2 7235 | A function's value in an i... |
| funfvima2d 7236 | A function's value in a pr... |
| fnfvima 7237 | The function value of an o... |
| fnfvimad 7238 | A function's value belongs... |
| resfvresima 7239 | The value of the function ... |
| funfvima3 7240 | A class including a functi... |
| rexima 7241 | Existential quantification... |
| ralima 7242 | Universal quantification u... |
| fvclss 7243 | Upper bound for the class ... |
| elabrex 7244 | Elementhood in an image se... |
| elabrexg 7245 | Elementhood in an image se... |
| abrexco 7246 | Composition of two image m... |
| imaiun 7247 | The image of an indexed un... |
| imauni 7248 | The image of a union is th... |
| fniunfv 7249 | The indexed union of a fun... |
| funiunfv 7250 | The indexed union of a fun... |
| funiunfvf 7251 | The indexed union of a fun... |
| eluniima 7252 | Membership in the union of... |
| elunirn 7253 | Membership in the union of... |
| elunirnALT 7254 | Alternate proof of ~ eluni... |
| elunirn2OLD 7255 | Obsolete version of ~ elfv... |
| fnunirn 7256 | Membership in a union of s... |
| dff13 7257 | A one-to-one function in t... |
| dff13f 7258 | A one-to-one function in t... |
| f1veqaeq 7259 | If the values of a one-to-... |
| f1cofveqaeq 7260 | If the values of a composi... |
| f1cofveqaeqALT 7261 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7262 | If two one-to-one function... |
| f1mpt 7263 | Express injection for a ma... |
| f1fveq 7264 | Equality of function value... |
| f1elima 7265 | Membership in the image of... |
| f1imass 7266 | Taking images under a one-... |
| f1imaeq 7267 | Taking images under a one-... |
| f1imapss 7268 | Taking images under a one-... |
| fpropnf1 7269 | A function, given by an un... |
| f1dom3fv3dif 7270 | The function values for a ... |
| f1dom3el3dif 7271 | The codomain of a 1-1 func... |
| dff14a 7272 | A one-to-one function in t... |
| dff14b 7273 | A one-to-one function in t... |
| f12dfv 7274 | A one-to-one function with... |
| f13dfv 7275 | A one-to-one function with... |
| dff1o6 7276 | A one-to-one onto function... |
| f1ocnvfv1 7277 | The converse value of the ... |
| f1ocnvfv2 7278 | The value of the converse ... |
| f1ocnvfv 7279 | Relationship between the v... |
| f1ocnvfvb 7280 | Relationship between the v... |
| nvof1o 7281 | An involution is a bijecti... |
| nvocnv 7282 | The converse of an involut... |
| f1cdmsn 7283 | If a one-to-one function w... |
| fsnex 7284 | Relate a function with a s... |
| f1prex 7285 | Relate a one-to-one functi... |
| f1ocnvdm 7286 | The value of the converse ... |
| f1ocnvfvrneq 7287 | If the values of a one-to-... |
| fcof1 7288 | An application is injectiv... |
| fcofo 7289 | An application is surjecti... |
| cbvfo 7290 | Change bound variable betw... |
| cbvexfo 7291 | Change bound variable betw... |
| cocan1 7292 | An injection is left-cance... |
| cocan2 7293 | A surjection is right-canc... |
| fcof1oinvd 7294 | Show that a function is th... |
| fcof1od 7295 | A function is bijective if... |
| 2fcoidinvd 7296 | Show that a function is th... |
| fcof1o 7297 | Show that two functions ar... |
| 2fvcoidd 7298 | Show that the composition ... |
| 2fvidf1od 7299 | A function is bijective if... |
| 2fvidinvd 7300 | Show that two functions ar... |
| foeqcnvco 7301 | Condition for function equ... |
| f1eqcocnv 7302 | Condition for function equ... |
| f1eqcocnvOLD 7303 | Obsolete version of ~ f1eq... |
| fveqf1o 7304 | Given a bijection ` F ` , ... |
| nf1const 7305 | A constant function from a... |
| nf1oconst 7306 | A constant function from a... |
| f1ofvswap 7307 | Swapping two values in a b... |
| fliftrel 7308 | ` F ` , a function lift, i... |
| fliftel 7309 | Elementhood in the relatio... |
| fliftel1 7310 | Elementhood in the relatio... |
| fliftcnv 7311 | Converse of the relation `... |
| fliftfun 7312 | The function ` F ` is the ... |
| fliftfund 7313 | The function ` F ` is the ... |
| fliftfuns 7314 | The function ` F ` is the ... |
| fliftf 7315 | The domain and range of th... |
| fliftval 7316 | The value of the function ... |
| isoeq1 7317 | Equality theorem for isomo... |
| isoeq2 7318 | Equality theorem for isomo... |
| isoeq3 7319 | Equality theorem for isomo... |
| isoeq4 7320 | Equality theorem for isomo... |
| isoeq5 7321 | Equality theorem for isomo... |
| nfiso 7322 | Bound-variable hypothesis ... |
| isof1o 7323 | An isomorphism is a one-to... |
| isof1oidb 7324 | A function is a bijection ... |
| isof1oopb 7325 | A function is a bijection ... |
| isorel 7326 | An isomorphism connects bi... |
| soisores 7327 | Express the condition of i... |
| soisoi 7328 | Infer isomorphism from one... |
| isoid 7329 | Identity law for isomorphi... |
| isocnv 7330 | Converse law for isomorphi... |
| isocnv2 7331 | Converse law for isomorphi... |
| isocnv3 7332 | Complementation law for is... |
| isores2 7333 | An isomorphism from one we... |
| isores1 7334 | An isomorphism from one we... |
| isores3 7335 | Induced isomorphism on a s... |
| isotr 7336 | Composition (transitive) l... |
| isomin 7337 | Isomorphisms preserve mini... |
| isoini 7338 | Isomorphisms preserve init... |
| isoini2 7339 | Isomorphisms are isomorphi... |
| isofrlem 7340 | Lemma for ~ isofr . (Cont... |
| isoselem 7341 | Lemma for ~ isose . (Cont... |
| isofr 7342 | An isomorphism preserves w... |
| isose 7343 | An isomorphism preserves s... |
| isofr2 7344 | A weak form of ~ isofr tha... |
| isopolem 7345 | Lemma for ~ isopo . (Cont... |
| isopo 7346 | An isomorphism preserves t... |
| isosolem 7347 | Lemma for ~ isoso . (Cont... |
| isoso 7348 | An isomorphism preserves t... |
| isowe 7349 | An isomorphism preserves t... |
| isowe2 7350 | A weak form of ~ isowe tha... |
| f1oiso 7351 | Any one-to-one onto functi... |
| f1oiso2 7352 | Any one-to-one onto functi... |
| f1owe 7353 | Well-ordering of isomorphi... |
| weniso 7354 | A set-like well-ordering h... |
| weisoeq 7355 | Thus, there is at most one... |
| weisoeq2 7356 | Thus, there is at most one... |
| knatar 7357 | The Knaster-Tarski theorem... |
| fvresval 7358 | The value of a restricted ... |
| funeldmb 7359 | If ` (/) ` is not part of ... |
| eqfunresadj 7360 | Law for adjoining an eleme... |
| eqfunressuc 7361 | Law for equality of restri... |
| fnssintima 7362 | Condition for subset of an... |
| imaeqsexv 7363 | Substitute a function valu... |
| imaeqsalv 7364 | Substitute a function valu... |
| canth 7365 | No set ` A ` is equinumero... |
| ncanth 7366 | Cantor's theorem fails for... |
| riotaeqdv 7369 | Formula-building deduction... |
| riotabidv 7370 | Formula-building deduction... |
| riotaeqbidv 7371 | Equality deduction for res... |
| riotaex 7372 | Restricted iota is a set. ... |
| riotav 7373 | An iota restricted to the ... |
| riotauni 7374 | Restricted iota in terms o... |
| nfriota1 7375 | The abstraction variable i... |
| nfriotadw 7376 | Deduction version of ~ nfr... |
| cbvriotaw 7377 | Change bound variable in a... |
| cbvriotavw 7378 | Change bound variable in a... |
| cbvriotavwOLD 7379 | Obsolete version of ~ cbvr... |
| nfriotad 7380 | Deduction version of ~ nfr... |
| nfriota 7381 | A variable not free in a w... |
| cbvriota 7382 | Change bound variable in a... |
| cbvriotav 7383 | Change bound variable in a... |
| csbriota 7384 | Interchange class substitu... |
| riotacl2 7385 | Membership law for "the un... |
| riotacl 7386 | Closure of restricted iota... |
| riotasbc 7387 | Substitution law for descr... |
| riotabidva 7388 | Equivalent wff's yield equ... |
| riotabiia 7389 | Equivalent wff's yield equ... |
| riota1 7390 | Property of restricted iot... |
| riota1a 7391 | Property of iota. (Contri... |
| riota2df 7392 | A deduction version of ~ r... |
| riota2f 7393 | This theorem shows a condi... |
| riota2 7394 | This theorem shows a condi... |
| riotaeqimp 7395 | If two restricted iota des... |
| riotaprop 7396 | Properties of a restricted... |
| riota5f 7397 | A method for computing res... |
| riota5 7398 | A method for computing res... |
| riotass2 7399 | Restriction of a unique el... |
| riotass 7400 | Restriction of a unique el... |
| moriotass 7401 | Restriction of a unique el... |
| snriota 7402 | A restricted class abstrac... |
| riotaxfrd 7403 | Change the variable ` x ` ... |
| eusvobj2 7404 | Specify the same property ... |
| eusvobj1 7405 | Specify the same object in... |
| f1ofveu 7406 | There is one domain elemen... |
| f1ocnvfv3 7407 | Value of the converse of a... |
| riotaund 7408 | Restricted iota equals the... |
| riotassuni 7409 | The restricted iota class ... |
| riotaclb 7410 | Bidirectional closure of r... |
| riotarab 7411 | Restricted iota of a restr... |
| oveq 7418 | Equality theorem for opera... |
| oveq1 7419 | Equality theorem for opera... |
| oveq2 7420 | Equality theorem for opera... |
| oveq12 7421 | Equality theorem for opera... |
| oveq1i 7422 | Equality inference for ope... |
| oveq2i 7423 | Equality inference for ope... |
| oveq12i 7424 | Equality inference for ope... |
| oveqi 7425 | Equality inference for ope... |
| oveq123i 7426 | Equality inference for ope... |
| oveq1d 7427 | Equality deduction for ope... |
| oveq2d 7428 | Equality deduction for ope... |
| oveqd 7429 | Equality deduction for ope... |
| oveq12d 7430 | Equality deduction for ope... |
| oveqan12d 7431 | Equality deduction for ope... |
| oveqan12rd 7432 | Equality deduction for ope... |
| oveq123d 7433 | Equality deduction for ope... |
| fvoveq1d 7434 | Equality deduction for nes... |
| fvoveq1 7435 | Equality theorem for neste... |
| ovanraleqv 7436 | Equality theorem for a con... |
| imbrov2fvoveq 7437 | Equality theorem for neste... |
| ovrspc2v 7438 | If an operation value is e... |
| oveqrspc2v 7439 | Restricted specialization ... |
| oveqdr 7440 | Equality of two operations... |
| nfovd 7441 | Deduction version of bound... |
| nfov 7442 | Bound-variable hypothesis ... |
| oprabidw 7443 | The law of concretion. Sp... |
| oprabid 7444 | The law of concretion. Sp... |
| ovex 7445 | The result of an operation... |
| ovexi 7446 | The result of an operation... |
| ovexd 7447 | The result of an operation... |
| ovssunirn 7448 | The result of an operation... |
| 0ov 7449 | Operation value of the emp... |
| ovprc 7450 | The value of an operation ... |
| ovprc1 7451 | The value of an operation ... |
| ovprc2 7452 | The value of an operation ... |
| ovrcl 7453 | Reverse closure for an ope... |
| csbov123 7454 | Move class substitution in... |
| csbov 7455 | Move class substitution in... |
| csbov12g 7456 | Move class substitution in... |
| csbov1g 7457 | Move class substitution in... |
| csbov2g 7458 | Move class substitution in... |
| rspceov 7459 | A frequently used special ... |
| elovimad 7460 | Elementhood of the image s... |
| fnbrovb 7461 | Value of a binary operatio... |
| fnotovb 7462 | Equivalence of operation v... |
| opabbrex 7463 | A collection of ordered pa... |
| opabresex2 7464 | Restrictions of a collecti... |
| opabresex2d 7465 | Obsolete version of ~ opab... |
| fvmptopab 7466 | The function value of a ma... |
| fvmptopabOLD 7467 | Obsolete version of ~ fvmp... |
| f1opr 7468 | Condition for an operation... |
| brfvopab 7469 | The classes involved in a ... |
| dfoprab2 7470 | Class abstraction for oper... |
| reloprab 7471 | An operation class abstrac... |
| oprabv 7472 | If a pair and a class are ... |
| nfoprab1 7473 | The abstraction variables ... |
| nfoprab2 7474 | The abstraction variables ... |
| nfoprab3 7475 | The abstraction variables ... |
| nfoprab 7476 | Bound-variable hypothesis ... |
| oprabbid 7477 | Equivalent wff's yield equ... |
| oprabbidv 7478 | Equivalent wff's yield equ... |
| oprabbii 7479 | Equivalent wff's yield equ... |
| ssoprab2 7480 | Equivalence of ordered pai... |
| ssoprab2b 7481 | Equivalence of ordered pai... |
| eqoprab2bw 7482 | Equivalence of ordered pai... |
| eqoprab2b 7483 | Equivalence of ordered pai... |
| mpoeq123 7484 | An equality theorem for th... |
| mpoeq12 7485 | An equality theorem for th... |
| mpoeq123dva 7486 | An equality deduction for ... |
| mpoeq123dv 7487 | An equality deduction for ... |
| mpoeq123i 7488 | An equality inference for ... |
| mpoeq3dva 7489 | Slightly more general equa... |
| mpoeq3ia 7490 | An equality inference for ... |
| mpoeq3dv 7491 | An equality deduction for ... |
| nfmpo1 7492 | Bound-variable hypothesis ... |
| nfmpo2 7493 | Bound-variable hypothesis ... |
| nfmpo 7494 | Bound-variable hypothesis ... |
| 0mpo0 7495 | A mapping operation with e... |
| mpo0v 7496 | A mapping operation with e... |
| mpo0 7497 | A mapping operation with e... |
| oprab4 7498 | Two ways to state the doma... |
| cbvoprab1 7499 | Rule used to change first ... |
| cbvoprab2 7500 | Change the second bound va... |
| cbvoprab12 7501 | Rule used to change first ... |
| cbvoprab12v 7502 | Rule used to change first ... |
| cbvoprab3 7503 | Rule used to change the th... |
| cbvoprab3v 7504 | Rule used to change the th... |
| cbvmpox 7505 | Rule to change the bound v... |
| cbvmpo 7506 | Rule to change the bound v... |
| cbvmpov 7507 | Rule to change the bound v... |
| elimdelov 7508 | Eliminate a hypothesis whi... |
| ovif 7509 | Move a conditional outside... |
| ovif2 7510 | Move a conditional outside... |
| ovif12 7511 | Move a conditional outside... |
| ifov 7512 | Move a conditional outside... |
| dmoprab 7513 | The domain of an operation... |
| dmoprabss 7514 | The domain of an operation... |
| rnoprab 7515 | The range of an operation ... |
| rnoprab2 7516 | The range of a restricted ... |
| reldmoprab 7517 | The domain of an operation... |
| oprabss 7518 | Structure of an operation ... |
| eloprabga 7519 | The law of concretion for ... |
| eloprabgaOLD 7520 | Obsolete version of ~ elop... |
| eloprabg 7521 | The law of concretion for ... |
| ssoprab2i 7522 | Inference of operation cla... |
| mpov 7523 | Operation with universal d... |
| mpomptx 7524 | Express a two-argument fun... |
| mpompt 7525 | Express a two-argument fun... |
| mpodifsnif 7526 | A mapping with two argumen... |
| mposnif 7527 | A mapping with two argumen... |
| fconstmpo 7528 | Representation of a consta... |
| resoprab 7529 | Restriction of an operatio... |
| resoprab2 7530 | Restriction of an operator... |
| resmpo 7531 | Restriction of the mapping... |
| funoprabg 7532 | "At most one" is a suffici... |
| funoprab 7533 | "At most one" is a suffici... |
| fnoprabg 7534 | Functionality and domain o... |
| mpofun 7535 | The maps-to notation for a... |
| mpofunOLD 7536 | Obsolete version of ~ mpof... |
| fnoprab 7537 | Functionality and domain o... |
| ffnov 7538 | An operation maps to a cla... |
| fovcld 7539 | Closure law for an operati... |
| fovcl 7540 | Closure law for an operati... |
| eqfnov 7541 | Equality of two operations... |
| eqfnov2 7542 | Two operators with the sam... |
| fnov 7543 | Representation of a functi... |
| mpo2eqb 7544 | Bidirectional equality the... |
| rnmpo 7545 | The range of an operation ... |
| reldmmpo 7546 | The domain of an operation... |
| elrnmpog 7547 | Membership in the range of... |
| elrnmpo 7548 | Membership in the range of... |
| elrnmpores 7549 | Membership in the range of... |
| ralrnmpo 7550 | A restricted quantifier ov... |
| rexrnmpo 7551 | A restricted quantifier ov... |
| ovid 7552 | The value of an operation ... |
| ovidig 7553 | The value of an operation ... |
| ovidi 7554 | The value of an operation ... |
| ov 7555 | The value of an operation ... |
| ovigg 7556 | The value of an operation ... |
| ovig 7557 | The value of an operation ... |
| ovmpt4g 7558 | Value of a function given ... |
| ovmpos 7559 | Value of a function given ... |
| ov2gf 7560 | The value of an operation ... |
| ovmpodxf 7561 | Value of an operation give... |
| ovmpodx 7562 | Value of an operation give... |
| ovmpod 7563 | Value of an operation give... |
| ovmpox 7564 | The value of an operation ... |
| ovmpoga 7565 | Value of an operation give... |
| ovmpoa 7566 | Value of an operation give... |
| ovmpodf 7567 | Alternate deduction versio... |
| ovmpodv 7568 | Alternate deduction versio... |
| ovmpodv2 7569 | Alternate deduction versio... |
| ovmpog 7570 | Value of an operation give... |
| ovmpo 7571 | Value of an operation give... |
| ovmpot 7572 | The value of an operation ... |
| fvmpopr2d 7573 | Value of an operation give... |
| ov3 7574 | The value of an operation ... |
| ov6g 7575 | The value of an operation ... |
| ovg 7576 | The value of an operation ... |
| ovres 7577 | The value of a restricted ... |
| ovresd 7578 | Lemma for converting metri... |
| oprres 7579 | The restriction of an oper... |
| oprssov 7580 | The value of a member of t... |
| fovcdm 7581 | An operation's value belon... |
| fovcdmda 7582 | An operation's value belon... |
| fovcdmd 7583 | An operation's value belon... |
| fnrnov 7584 | The range of an operation ... |
| foov 7585 | An onto mapping of an oper... |
| fnovrn 7586 | An operation's value belon... |
| ovelrn 7587 | A member of an operation's... |
| funimassov 7588 | Membership relation for th... |
| ovelimab 7589 | Operation value in an imag... |
| ovima0 7590 | An operation value is a me... |
| ovconst2 7591 | The value of a constant op... |
| oprssdm 7592 | Domain of closure of an op... |
| nssdmovg 7593 | The value of an operation ... |
| ndmovg 7594 | The value of an operation ... |
| ndmov 7595 | The value of an operation ... |
| ndmovcl 7596 | The closure of an operatio... |
| ndmovrcl 7597 | Reverse closure law, when ... |
| ndmovcom 7598 | Any operation is commutati... |
| ndmovass 7599 | Any operation is associati... |
| ndmovdistr 7600 | Any operation is distribut... |
| ndmovord 7601 | Elimination of redundant a... |
| ndmovordi 7602 | Elimination of redundant a... |
| caovclg 7603 | Convert an operation closu... |
| caovcld 7604 | Convert an operation closu... |
| caovcl 7605 | Convert an operation closu... |
| caovcomg 7606 | Convert an operation commu... |
| caovcomd 7607 | Convert an operation commu... |
| caovcom 7608 | Convert an operation commu... |
| caovassg 7609 | Convert an operation assoc... |
| caovassd 7610 | Convert an operation assoc... |
| caovass 7611 | Convert an operation assoc... |
| caovcang 7612 | Convert an operation cance... |
| caovcand 7613 | Convert an operation cance... |
| caovcanrd 7614 | Commute the arguments of a... |
| caovcan 7615 | Convert an operation cance... |
| caovordig 7616 | Convert an operation order... |
| caovordid 7617 | Convert an operation order... |
| caovordg 7618 | Convert an operation order... |
| caovordd 7619 | Convert an operation order... |
| caovord2d 7620 | Operation ordering law wit... |
| caovord3d 7621 | Ordering law. (Contribute... |
| caovord 7622 | Convert an operation order... |
| caovord2 7623 | Operation ordering law wit... |
| caovord3 7624 | Ordering law. (Contribute... |
| caovdig 7625 | Convert an operation distr... |
| caovdid 7626 | Convert an operation distr... |
| caovdir2d 7627 | Convert an operation distr... |
| caovdirg 7628 | Convert an operation rever... |
| caovdird 7629 | Convert an operation distr... |
| caovdi 7630 | Convert an operation distr... |
| caov32d 7631 | Rearrange arguments in a c... |
| caov12d 7632 | Rearrange arguments in a c... |
| caov31d 7633 | Rearrange arguments in a c... |
| caov13d 7634 | Rearrange arguments in a c... |
| caov4d 7635 | Rearrange arguments in a c... |
| caov411d 7636 | Rearrange arguments in a c... |
| caov42d 7637 | Rearrange arguments in a c... |
| caov32 7638 | Rearrange arguments in a c... |
| caov12 7639 | Rearrange arguments in a c... |
| caov31 7640 | Rearrange arguments in a c... |
| caov13 7641 | Rearrange arguments in a c... |
| caov4 7642 | Rearrange arguments in a c... |
| caov411 7643 | Rearrange arguments in a c... |
| caov42 7644 | Rearrange arguments in a c... |
| caovdir 7645 | Reverse distributive law. ... |
| caovdilem 7646 | Lemma used by real number ... |
| caovlem2 7647 | Lemma used in real number ... |
| caovmo 7648 | Uniqueness of inverse elem... |
| imaeqexov 7649 | Substitute an operation va... |
| imaeqalov 7650 | Substitute an operation va... |
| mpondm0 7651 | The value of an operation ... |
| elmpocl 7652 | If a two-parameter class i... |
| elmpocl1 7653 | If a two-parameter class i... |
| elmpocl2 7654 | If a two-parameter class i... |
| elovmpo 7655 | Utility lemma for two-para... |
| elovmporab 7656 | Implications for the value... |
| elovmporab1w 7657 | Implications for the value... |
| elovmporab1 7658 | Implications for the value... |
| 2mpo0 7659 | If the operation value of ... |
| relmptopab 7660 | Any function to sets of or... |
| f1ocnvd 7661 | Describe an implicit one-t... |
| f1od 7662 | Describe an implicit one-t... |
| f1ocnv2d 7663 | Describe an implicit one-t... |
| f1o2d 7664 | Describe an implicit one-t... |
| f1opw2 7665 | A one-to-one mapping induc... |
| f1opw 7666 | A one-to-one mapping induc... |
| elovmpt3imp 7667 | If the value of a function... |
| ovmpt3rab1 7668 | The value of an operation ... |
| ovmpt3rabdm 7669 | If the value of a function... |
| elovmpt3rab1 7670 | Implications for the value... |
| elovmpt3rab 7671 | Implications for the value... |
| ofeqd 7676 | Equality theorem for funct... |
| ofeq 7677 | Equality theorem for funct... |
| ofreq 7678 | Equality theorem for funct... |
| ofexg 7679 | A function operation restr... |
| nfof 7680 | Hypothesis builder for fun... |
| nfofr 7681 | Hypothesis builder for fun... |
| ofrfvalg 7682 | Value of a relation applie... |
| offval 7683 | Value of an operation appl... |
| ofrfval 7684 | Value of a relation applie... |
| ofval 7685 | Evaluate a function operat... |
| ofrval 7686 | Exhibit a function relatio... |
| offn 7687 | The function operation pro... |
| offun 7688 | The function operation pro... |
| offval2f 7689 | The function operation exp... |
| ofmresval 7690 | Value of a restriction of ... |
| fnfvof 7691 | Function value of a pointw... |
| off 7692 | The function operation pro... |
| ofres 7693 | Restrict the operands of a... |
| offval2 7694 | The function operation exp... |
| ofrfval2 7695 | The function relation acti... |
| ofmpteq 7696 | Value of a pointwise opera... |
| ofco 7697 | The composition of a funct... |
| offveq 7698 | Convert an identity of the... |
| offveqb 7699 | Equivalent expressions for... |
| ofc1 7700 | Left operation by a consta... |
| ofc2 7701 | Right operation by a const... |
| ofc12 7702 | Function operation on two ... |
| caofref 7703 | Transfer a reflexive law t... |
| caofinvl 7704 | Transfer a left inverse la... |
| caofid0l 7705 | Transfer a left identity l... |
| caofid0r 7706 | Transfer a right identity ... |
| caofid1 7707 | Transfer a right absorptio... |
| caofid2 7708 | Transfer a right absorptio... |
| caofcom 7709 | Transfer a commutative law... |
| caofrss 7710 | Transfer a relation subset... |
| caofass 7711 | Transfer an associative la... |
| caoftrn 7712 | Transfer a transitivity la... |
| caofdi 7713 | Transfer a distributive la... |
| caofdir 7714 | Transfer a reverse distrib... |
| caonncan 7715 | Transfer ~ nncan -shaped l... |
| relrpss 7718 | The proper subset relation... |
| brrpssg 7719 | The proper subset relation... |
| brrpss 7720 | The proper subset relation... |
| porpss 7721 | Every class is partially o... |
| sorpss 7722 | Express strict ordering un... |
| sorpssi 7723 | Property of a chain of set... |
| sorpssun 7724 | A chain of sets is closed ... |
| sorpssin 7725 | A chain of sets is closed ... |
| sorpssuni 7726 | In a chain of sets, a maxi... |
| sorpssint 7727 | In a chain of sets, a mini... |
| sorpsscmpl 7728 | The componentwise compleme... |
| zfun 7730 | Axiom of Union expressed w... |
| axun2 7731 | A variant of the Axiom of ... |
| uniex2 7732 | The Axiom of Union using t... |
| vuniex 7733 | The union of a setvar is a... |
| uniexg 7734 | The ZF Axiom of Union in c... |
| uniex 7735 | The Axiom of Union in clas... |
| uniexd 7736 | Deduction version of the Z... |
| unex 7737 | The union of two sets is a... |
| tpex 7738 | An unordered triple of cla... |
| unexb 7739 | Existence of union is equi... |
| unexg 7740 | A union of two sets is a s... |
| xpexg 7741 | The Cartesian product of t... |
| xpexd 7742 | The Cartesian product of t... |
| 3xpexg 7743 | The Cartesian product of t... |
| xpex 7744 | The Cartesian product of t... |
| unexd 7745 | The union of two sets is a... |
| sqxpexg 7746 | The Cartesian square of a ... |
| abnexg 7747 | Sufficient condition for a... |
| abnex 7748 | Sufficient condition for a... |
| snnex 7749 | The class of all singleton... |
| pwnex 7750 | The class of all power set... |
| difex2 7751 | If the subtrahend of a cla... |
| difsnexi 7752 | If the difference of a cla... |
| uniuni 7753 | Expression for double unio... |
| uniexr 7754 | Converse of the Axiom of U... |
| uniexb 7755 | The Axiom of Union and its... |
| pwexr 7756 | Converse of the Axiom of P... |
| pwexb 7757 | The Axiom of Power Sets an... |
| elpwpwel 7758 | A class belongs to a doubl... |
| eldifpw 7759 | Membership in a power clas... |
| elpwun 7760 | Membership in the power cl... |
| pwuncl 7761 | Power classes are closed u... |
| iunpw 7762 | An indexed union of a powe... |
| fr3nr 7763 | A well-founded relation ha... |
| epne3 7764 | A well-founded class conta... |
| dfwe2 7765 | Alternate definition of we... |
| epweon 7766 | The membership relation we... |
| epweonALT 7767 | Alternate proof of ~ epweo... |
| ordon 7768 | The class of all ordinal n... |
| onprc 7769 | No set contains all ordina... |
| ssorduni 7770 | The union of a class of or... |
| ssonuni 7771 | The union of a set of ordi... |
| ssonunii 7772 | The union of a set of ordi... |
| ordeleqon 7773 | A way to express the ordin... |
| ordsson 7774 | Any ordinal class is a sub... |
| dford5 7775 | A class is ordinal iff it ... |
| onss 7776 | An ordinal number is a sub... |
| predon 7777 | The predecessor of an ordi... |
| predonOLD 7778 | Obsolete version of ~ pred... |
| ssonprc 7779 | Two ways of saying a class... |
| onuni 7780 | The union of an ordinal nu... |
| orduni 7781 | The union of an ordinal cl... |
| onint 7782 | The intersection (infimum)... |
| onint0 7783 | The intersection of a clas... |
| onssmin 7784 | A nonempty class of ordina... |
| onminesb 7785 | If a property is true for ... |
| onminsb 7786 | If a property is true for ... |
| oninton 7787 | The intersection of a none... |
| onintrab 7788 | The intersection of a clas... |
| onintrab2 7789 | An existence condition equ... |
| onnmin 7790 | No member of a set of ordi... |
| onnminsb 7791 | An ordinal number smaller ... |
| oneqmin 7792 | A way to show that an ordi... |
| uniordint 7793 | The union of a set of ordi... |
| onminex 7794 | If a wff is true for an or... |
| sucon 7795 | The class of all ordinal n... |
| sucexb 7796 | A successor exists iff its... |
| sucexg 7797 | The successor of a set is ... |
| sucex 7798 | The successor of a set is ... |
| onmindif2 7799 | The minimum of a class of ... |
| ordsuci 7800 | The successor of an ordina... |
| sucexeloni 7801 | If the successor of an ord... |
| sucexeloniOLD 7802 | Obsolete version of ~ suce... |
| onsuc 7803 | The successor of an ordina... |
| suceloniOLD 7804 | Obsolete version of ~ onsu... |
| ordsuc 7805 | A class is ordinal if and ... |
| ordsucOLD 7806 | Obsolete version of ~ ords... |
| ordpwsuc 7807 | The collection of ordinals... |
| onpwsuc 7808 | The collection of ordinal ... |
| onsucb 7809 | A class is an ordinal numb... |
| ordsucss 7810 | The successor of an elemen... |
| onpsssuc 7811 | An ordinal number is a pro... |
| ordelsuc 7812 | A set belongs to an ordina... |
| onsucmin 7813 | The successor of an ordina... |
| ordsucelsuc 7814 | Membership is inherited by... |
| ordsucsssuc 7815 | The subclass relationship ... |
| ordsucuniel 7816 | Given an element ` A ` of ... |
| ordsucun 7817 | The successor of the maxim... |
| ordunpr 7818 | The maximum of two ordinal... |
| ordunel 7819 | The maximum of two ordinal... |
| onsucuni 7820 | A class of ordinal numbers... |
| ordsucuni 7821 | An ordinal class is a subc... |
| orduniorsuc 7822 | An ordinal class is either... |
| unon 7823 | The class of all ordinal n... |
| ordunisuc 7824 | An ordinal class is equal ... |
| orduniss2 7825 | The union of the ordinal s... |
| onsucuni2 7826 | A successor ordinal is the... |
| 0elsuc 7827 | The successor of an ordina... |
| limon 7828 | The class of ordinal numbe... |
| onuniorsuc 7829 | An ordinal number is eithe... |
| onssi 7830 | An ordinal number is a sub... |
| onsuci 7831 | The successor of an ordina... |
| onuniorsuciOLD 7832 | Obsolete version of ~ onun... |
| onuninsuci 7833 | An ordinal is equal to its... |
| onsucssi 7834 | A set belongs to an ordina... |
| nlimsucg 7835 | A successor is not a limit... |
| orduninsuc 7836 | An ordinal class is equal ... |
| ordunisuc2 7837 | An ordinal equal to its un... |
| ordzsl 7838 | An ordinal is zero, a succ... |
| onzsl 7839 | An ordinal number is zero,... |
| dflim3 7840 | An alternate definition of... |
| dflim4 7841 | An alternate definition of... |
| limsuc 7842 | The successor of a member ... |
| limsssuc 7843 | A class includes a limit o... |
| nlimon 7844 | Two ways to express the cl... |
| limuni3 7845 | The union of a nonempty cl... |
| tfi 7846 | The Principle of Transfini... |
| tfisg 7847 | A closed form of ~ tfis . ... |
| tfis 7848 | Transfinite Induction Sche... |
| tfis2f 7849 | Transfinite Induction Sche... |
| tfis2 7850 | Transfinite Induction Sche... |
| tfis3 7851 | Transfinite Induction Sche... |
| tfisi 7852 | A transfinite induction sc... |
| tfinds 7853 | Principle of Transfinite I... |
| tfindsg 7854 | Transfinite Induction (inf... |
| tfindsg2 7855 | Transfinite Induction (inf... |
| tfindes 7856 | Transfinite Induction with... |
| tfinds2 7857 | Transfinite Induction (inf... |
| tfinds3 7858 | Principle of Transfinite I... |
| dfom2 7861 | An alternate definition of... |
| elom 7862 | Membership in omega. The ... |
| omsson 7863 | Omega is a subset of ` On ... |
| limomss 7864 | The class of natural numbe... |
| nnon 7865 | A natural number is an ord... |
| nnoni 7866 | A natural number is an ord... |
| nnord 7867 | A natural number is ordina... |
| trom 7868 | The class of finite ordina... |
| ordom 7869 | The class of finite ordina... |
| elnn 7870 | A member of a natural numb... |
| omon 7871 | The class of natural numbe... |
| omelon2 7872 | Omega is an ordinal number... |
| nnlim 7873 | A natural number is not a ... |
| omssnlim 7874 | The class of natural numbe... |
| limom 7875 | Omega is a limit ordinal. ... |
| peano2b 7876 | A class belongs to omega i... |
| nnsuc 7877 | A nonzero natural number i... |
| omsucne 7878 | A natural number is not th... |
| ssnlim 7879 | An ordinal subclass of non... |
| omsinds 7880 | Strong (or "total") induct... |
| omsindsOLD 7881 | Obsolete version of ~ omsi... |
| omun 7882 | The union of two finite or... |
| peano1 7883 | Zero is a natural number. ... |
| peano1OLD 7884 | Obsolete version of ~ pean... |
| peano2 7885 | The successor of any natur... |
| peano3 7886 | The successor of any natur... |
| peano4 7887 | Two natural numbers are eq... |
| peano5 7888 | The induction postulate: a... |
| peano5OLD 7889 | Obsolete version of ~ pean... |
| nn0suc 7890 | A natural number is either... |
| find 7891 | The Principle of Finite In... |
| findOLD 7892 | Obsolete version of ~ find... |
| finds 7893 | Principle of Finite Induct... |
| findsg 7894 | Principle of Finite Induct... |
| finds2 7895 | Principle of Finite Induct... |
| finds1 7896 | Principle of Finite Induct... |
| findes 7897 | Finite induction with expl... |
| dmexg 7898 | The domain of a set is a s... |
| rnexg 7899 | The range of a set is a se... |
| dmexd 7900 | The domain of a set is a s... |
| fndmexd 7901 | If a function is a set, it... |
| dmfex 7902 | If a mapping is a set, its... |
| fndmexb 7903 | The domain of a function i... |
| fdmexb 7904 | The domain of a function i... |
| dmfexALT 7905 | Alternate proof of ~ dmfex... |
| dmex 7906 | The domain of a set is a s... |
| rnex 7907 | The range of a set is a se... |
| iprc 7908 | The identity function is a... |
| resiexg 7909 | The existence of a restric... |
| imaexg 7910 | The image of a set is a se... |
| imaex 7911 | The image of a set is a se... |
| exse2 7912 | Any set relation is set-li... |
| xpexr 7913 | If a Cartesian product is ... |
| xpexr2 7914 | If a nonempty Cartesian pr... |
| xpexcnv 7915 | A condition where the conv... |
| soex 7916 | If the relation in a stric... |
| elxp4 7917 | Membership in a Cartesian ... |
| elxp5 7918 | Membership in a Cartesian ... |
| cnvexg 7919 | The converse of a set is a... |
| cnvex 7920 | The converse of a set is a... |
| relcnvexb 7921 | A relation is a set iff it... |
| f1oexrnex 7922 | If the range of a 1-1 onto... |
| f1oexbi 7923 | There is a one-to-one onto... |
| coexg 7924 | The composition of two set... |
| coex 7925 | The composition of two set... |
| funcnvuni 7926 | The union of a chain (with... |
| fun11uni 7927 | The union of a chain (with... |
| fex2 7928 | A function with bounded do... |
| fabexg 7929 | Existence of a set of func... |
| fabex 7930 | Existence of a set of func... |
| f1oabexg 7931 | The class of all 1-1-onto ... |
| fiunlem 7932 | Lemma for ~ fiun and ~ f1i... |
| fiun 7933 | The union of a chain (with... |
| f1iun 7934 | The union of a chain (with... |
| fviunfun 7935 | The function value of an i... |
| ffoss 7936 | Relationship between a map... |
| f11o 7937 | Relationship between one-t... |
| resfunexgALT 7938 | Alternate proof of ~ resfu... |
| cofunexg 7939 | Existence of a composition... |
| cofunex2g 7940 | Existence of a composition... |
| fnexALT 7941 | Alternate proof of ~ fnex ... |
| funexw 7942 | Weak version of ~ funex th... |
| mptexw 7943 | Weak version of ~ mptex th... |
| funrnex 7944 | If the domain of a functio... |
| zfrep6 7945 | A version of the Axiom of ... |
| focdmex 7946 | If the domain of an onto f... |
| f1dmex 7947 | If the codomain of a one-t... |
| f1ovv 7948 | The codomain/range of a 1-... |
| fvclex 7949 | Existence of the class of ... |
| fvresex 7950 | Existence of the class of ... |
| abrexexg 7951 | Existence of a class abstr... |
| abrexexgOLD 7952 | Obsolete version of ~ abre... |
| abrexex 7953 | Existence of a class abstr... |
| iunexg 7954 | The existence of an indexe... |
| abrexex2g 7955 | Existence of an existentia... |
| opabex3d 7956 | Existence of an ordered pa... |
| opabex3rd 7957 | Existence of an ordered pa... |
| opabex3 7958 | Existence of an ordered pa... |
| iunex 7959 | The existence of an indexe... |
| abrexex2 7960 | Existence of an existentia... |
| abexssex 7961 | Existence of a class abstr... |
| abexex 7962 | A condition where a class ... |
| f1oweALT 7963 | Alternate proof of ~ f1owe... |
| wemoiso 7964 | Thus, there is at most one... |
| wemoiso2 7965 | Thus, there is at most one... |
| oprabexd 7966 | Existence of an operator a... |
| oprabex 7967 | Existence of an operation ... |
| oprabex3 7968 | Existence of an operation ... |
| oprabrexex2 7969 | Existence of an existentia... |
| ab2rexex 7970 | Existence of a class abstr... |
| ab2rexex2 7971 | Existence of an existentia... |
| xpexgALT 7972 | Alternate proof of ~ xpexg... |
| offval3 7973 | General value of ` ( F oF ... |
| offres 7974 | Pointwise combination comm... |
| ofmres 7975 | Equivalent expressions for... |
| ofmresex 7976 | Existence of a restriction... |
| 1stval 7981 | The value of the function ... |
| 2ndval 7982 | The value of the function ... |
| 1stnpr 7983 | Value of the first-member ... |
| 2ndnpr 7984 | Value of the second-member... |
| 1st0 7985 | The value of the first-mem... |
| 2nd0 7986 | The value of the second-me... |
| op1st 7987 | Extract the first member o... |
| op2nd 7988 | Extract the second member ... |
| op1std 7989 | Extract the first member o... |
| op2ndd 7990 | Extract the second member ... |
| op1stg 7991 | Extract the first member o... |
| op2ndg 7992 | Extract the second member ... |
| ot1stg 7993 | Extract the first member o... |
| ot2ndg 7994 | Extract the second member ... |
| ot3rdg 7995 | Extract the third member o... |
| 1stval2 7996 | Alternate value of the fun... |
| 2ndval2 7997 | Alternate value of the fun... |
| oteqimp 7998 | The components of an order... |
| fo1st 7999 | The ` 1st ` function maps ... |
| fo2nd 8000 | The ` 2nd ` function maps ... |
| br1steqg 8001 | Uniqueness condition for t... |
| br2ndeqg 8002 | Uniqueness condition for t... |
| f1stres 8003 | Mapping of a restriction o... |
| f2ndres 8004 | Mapping of a restriction o... |
| fo1stres 8005 | Onto mapping of a restrict... |
| fo2ndres 8006 | Onto mapping of a restrict... |
| 1st2val 8007 | Value of an alternate defi... |
| 2nd2val 8008 | Value of an alternate defi... |
| 1stcof 8009 | Composition of the first m... |
| 2ndcof 8010 | Composition of the second ... |
| xp1st 8011 | Location of the first elem... |
| xp2nd 8012 | Location of the second ele... |
| elxp6 8013 | Membership in a Cartesian ... |
| elxp7 8014 | Membership in a Cartesian ... |
| eqopi 8015 | Equality with an ordered p... |
| xp2 8016 | Representation of Cartesia... |
| unielxp 8017 | The membership relation fo... |
| 1st2nd2 8018 | Reconstruction of a member... |
| 1st2ndb 8019 | Reconstruction of an order... |
| xpopth 8020 | An ordered pair theorem fo... |
| eqop 8021 | Two ways to express equali... |
| eqop2 8022 | Two ways to express equali... |
| op1steq 8023 | Two ways of expressing tha... |
| opreuopreu 8024 | There is a unique ordered ... |
| el2xptp 8025 | A member of a nested Carte... |
| el2xptp0 8026 | A member of a nested Carte... |
| el2xpss 8027 | Version of ~ elrel for tri... |
| 2nd1st 8028 | Swap the members of an ord... |
| 1st2nd 8029 | Reconstruction of a member... |
| 1stdm 8030 | The first ordered pair com... |
| 2ndrn 8031 | The second ordered pair co... |
| 1st2ndbr 8032 | Express an element of a re... |
| releldm2 8033 | Two ways of expressing mem... |
| reldm 8034 | An expression for the doma... |
| releldmdifi 8035 | One way of expressing memb... |
| funfv1st2nd 8036 | The function value for the... |
| funelss 8037 | If the first component of ... |
| funeldmdif 8038 | Two ways of expressing mem... |
| sbcopeq1a 8039 | Equality theorem for subst... |
| csbopeq1a 8040 | Equality theorem for subst... |
| sbcoteq1a 8041 | Equality theorem for subst... |
| dfopab2 8042 | A way to define an ordered... |
| dfoprab3s 8043 | A way to define an operati... |
| dfoprab3 8044 | Operation class abstractio... |
| dfoprab4 8045 | Operation class abstractio... |
| dfoprab4f 8046 | Operation class abstractio... |
| opabex2 8047 | Condition for an operation... |
| opabn1stprc 8048 | An ordered-pair class abst... |
| opiota 8049 | The property of a uniquely... |
| cnvoprab 8050 | The converse of a class ab... |
| dfxp3 8051 | Define the Cartesian produ... |
| elopabi 8052 | A consequence of membershi... |
| eloprabi 8053 | A consequence of membershi... |
| mpomptsx 8054 | Express a two-argument fun... |
| mpompts 8055 | Express a two-argument fun... |
| dmmpossx 8056 | The domain of a mapping is... |
| fmpox 8057 | Functionality, domain and ... |
| fmpo 8058 | Functionality, domain and ... |
| fnmpo 8059 | Functionality and domain o... |
| fnmpoi 8060 | Functionality and domain o... |
| dmmpo 8061 | Domain of a class given by... |
| ovmpoelrn 8062 | An operation's value belon... |
| dmmpoga 8063 | Domain of an operation giv... |
| dmmpogaOLD 8064 | Obsolete version of ~ dmmp... |
| dmmpog 8065 | Domain of an operation giv... |
| mpoexxg 8066 | Existence of an operation ... |
| mpoexg 8067 | Existence of an operation ... |
| mpoexga 8068 | If the domain of an operat... |
| mpoexw 8069 | Weak version of ~ mpoex th... |
| mpoex 8070 | If the domain of an operat... |
| mptmpoopabbrd 8071 | The operation value of a f... |
| mptmpoopabovd 8072 | The operation value of a f... |
| mptmpoopabbrdOLD 8073 | Obsolete version of ~ mptm... |
| mptmpoopabovdOLD 8074 | Obsolete version of ~ mptm... |
| el2mpocsbcl 8075 | If the operation value of ... |
| el2mpocl 8076 | If the operation value of ... |
| fnmpoovd 8077 | A function with a Cartesia... |
| offval22 8078 | The function operation exp... |
| brovpreldm 8079 | If a binary relation holds... |
| bropopvvv 8080 | If a binary relation holds... |
| bropfvvvvlem 8081 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8082 | If a binary relation holds... |
| ovmptss 8083 | If all the values of the m... |
| relmpoopab 8084 | Any function to sets of or... |
| fmpoco 8085 | Composition of two functio... |
| oprabco 8086 | Composition of a function ... |
| oprab2co 8087 | Composition of operator ab... |
| df1st2 8088 | An alternate possible defi... |
| df2nd2 8089 | An alternate possible defi... |
| 1stconst 8090 | The mapping of a restricti... |
| 2ndconst 8091 | The mapping of a restricti... |
| dfmpo 8092 | Alternate definition for t... |
| mposn 8093 | An operation (in maps-to n... |
| curry1 8094 | Composition with ` ``' ( 2... |
| curry1val 8095 | The value of a curried fun... |
| curry1f 8096 | Functionality of a curried... |
| curry2 8097 | Composition with ` ``' ( 1... |
| curry2f 8098 | Functionality of a curried... |
| curry2val 8099 | The value of a curried fun... |
| cnvf1olem 8100 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8101 | Describe a function that m... |
| fparlem1 8102 | Lemma for ~ fpar . (Contr... |
| fparlem2 8103 | Lemma for ~ fpar . (Contr... |
| fparlem3 8104 | Lemma for ~ fpar . (Contr... |
| fparlem4 8105 | Lemma for ~ fpar . (Contr... |
| fpar 8106 | Merge two functions in par... |
| fsplit 8107 | A function that can be use... |
| fsplitfpar 8108 | Merge two functions with a... |
| offsplitfpar 8109 | Express the function opera... |
| f2ndf 8110 | The ` 2nd ` (second compon... |
| fo2ndf 8111 | The ` 2nd ` (second compon... |
| f1o2ndf1 8112 | The ` 2nd ` (second compon... |
| opco1 8113 | Value of an operation prec... |
| opco2 8114 | Value of an operation prec... |
| opco1i 8115 | Inference form of ~ opco1 ... |
| frxp 8116 | A lexicographical ordering... |
| xporderlem 8117 | Lemma for lexicographical ... |
| poxp 8118 | A lexicographical ordering... |
| soxp 8119 | A lexicographical ordering... |
| wexp 8120 | A lexicographical ordering... |
| fnwelem 8121 | Lemma for ~ fnwe . (Contr... |
| fnwe 8122 | A variant on lexicographic... |
| fnse 8123 | Condition for the well-ord... |
| fvproj 8124 | Value of a function on ord... |
| fimaproj 8125 | Image of a cartesian produ... |
| ralxpes 8126 | A version of ~ ralxp with ... |
| ralxp3f 8127 | Restricted for all over a ... |
| ralxp3 8128 | Restricted for all over a ... |
| ralxp3es 8129 | Restricted for-all over a ... |
| frpoins3xpg 8130 | Special case of founded pa... |
| frpoins3xp3g 8131 | Special case of founded pa... |
| xpord2lem 8132 | Lemma for Cartesian produc... |
| poxp2 8133 | Another way of partially o... |
| frxp2 8134 | Another way of giving a we... |
| xpord2pred 8135 | Calculate the predecessor ... |
| sexp2 8136 | Condition for the relation... |
| xpord2indlem 8137 | Induction over the Cartesi... |
| xpord2ind 8138 | Induction over the Cartesi... |
| xpord3lem 8139 | Lemma for triple ordering.... |
| poxp3 8140 | Triple Cartesian product p... |
| frxp3 8141 | Give well-foundedness over... |
| xpord3pred 8142 | Calculate the predecsessor... |
| sexp3 8143 | Show that the triple order... |
| xpord3inddlem 8144 | Induction over the triple ... |
| xpord3indd 8145 | Induction over the triple ... |
| xpord3ind 8146 | Induction over the triple ... |
| orderseqlem 8147 | Lemma for ~ poseq and ~ so... |
| poseq 8148 | A partial ordering of ordi... |
| soseq 8149 | A linear ordering of ordin... |
| suppval 8152 | The value of the operation... |
| supp0prc 8153 | The support of a class is ... |
| suppvalbr 8154 | The value of the operation... |
| supp0 8155 | The support of the empty s... |
| suppval1 8156 | The value of the operation... |
| suppvalfng 8157 | The value of the operation... |
| suppvalfn 8158 | The value of the operation... |
| elsuppfng 8159 | An element of the support ... |
| elsuppfn 8160 | An element of the support ... |
| cnvimadfsn 8161 | The support of functions "... |
| suppimacnvss 8162 | The support of functions "... |
| suppimacnv 8163 | Support sets of functions ... |
| fsuppeq 8164 | Two ways of writing the su... |
| fsuppeqg 8165 | Version of ~ fsuppeq avoid... |
| suppssdm 8166 | The support of a function ... |
| suppsnop 8167 | The support of a singleton... |
| snopsuppss 8168 | The support of a singleton... |
| fvn0elsupp 8169 | If the function value for ... |
| fvn0elsuppb 8170 | The function value for a g... |
| rexsupp 8171 | Existential quantification... |
| ressuppss 8172 | The support of the restric... |
| suppun 8173 | The support of a class/fun... |
| ressuppssdif 8174 | The support of the restric... |
| mptsuppdifd 8175 | The support of a function ... |
| mptsuppd 8176 | The support of a function ... |
| extmptsuppeq 8177 | The support of an extended... |
| suppfnss 8178 | The support of a function ... |
| funsssuppss 8179 | The support of a function ... |
| fnsuppres 8180 | Two ways to express restri... |
| fnsuppeq0 8181 | The support of a function ... |
| fczsupp0 8182 | The support of a constant ... |
| suppss 8183 | Show that the support of a... |
| suppssOLD 8184 | Obsolete version of ~ supp... |
| suppssr 8185 | A function is zero outside... |
| suppssrg 8186 | A function is zero outside... |
| suppssov1 8187 | Formula building theorem f... |
| suppssof1 8188 | Formula building theorem f... |
| suppss2 8189 | Show that the support of a... |
| suppsssn 8190 | Show that the support of a... |
| suppssfv 8191 | Formula building theorem f... |
| suppofssd 8192 | Condition for the support ... |
| suppofss1d 8193 | Condition for the support ... |
| suppofss2d 8194 | Condition for the support ... |
| suppco 8195 | The support of the composi... |
| suppcoss 8196 | The support of the composi... |
| supp0cosupp0 8197 | The support of the composi... |
| imacosupp 8198 | The image of the support o... |
| opeliunxp2f 8199 | Membership in a union of C... |
| mpoxeldm 8200 | If there is an element of ... |
| mpoxneldm 8201 | If the first argument of a... |
| mpoxopn0yelv 8202 | If there is an element of ... |
| mpoxopynvov0g 8203 | If the second argument of ... |
| mpoxopxnop0 8204 | If the first argument of a... |
| mpoxopx0ov0 8205 | If the first argument of a... |
| mpoxopxprcov0 8206 | If the components of the f... |
| mpoxopynvov0 8207 | If the second argument of ... |
| mpoxopoveq 8208 | Value of an operation give... |
| mpoxopovel 8209 | Element of the value of an... |
| mpoxopoveqd 8210 | Value of an operation give... |
| brovex 8211 | A binary relation of the v... |
| brovmpoex 8212 | A binary relation of the v... |
| sprmpod 8213 | The extension of a binary ... |
| tposss 8216 | Subset theorem for transpo... |
| tposeq 8217 | Equality theorem for trans... |
| tposeqd 8218 | Equality theorem for trans... |
| tposssxp 8219 | The transposition is a sub... |
| reltpos 8220 | The transposition is a rel... |
| brtpos2 8221 | Value of the transposition... |
| brtpos0 8222 | The behavior of ` tpos ` w... |
| reldmtpos 8223 | Necessary and sufficient c... |
| brtpos 8224 | The transposition swaps ar... |
| ottpos 8225 | The transposition swaps th... |
| relbrtpos 8226 | The transposition swaps ar... |
| dmtpos 8227 | The domain of ` tpos F ` w... |
| rntpos 8228 | The range of ` tpos F ` wh... |
| tposexg 8229 | The transposition of a set... |
| ovtpos 8230 | The transposition swaps th... |
| tposfun 8231 | The transposition of a fun... |
| dftpos2 8232 | Alternate definition of ` ... |
| dftpos3 8233 | Alternate definition of ` ... |
| dftpos4 8234 | Alternate definition of ` ... |
| tpostpos 8235 | Value of the double transp... |
| tpostpos2 8236 | Value of the double transp... |
| tposfn2 8237 | The domain of a transposit... |
| tposfo2 8238 | Condition for a surjective... |
| tposf2 8239 | The domain and codomain of... |
| tposf12 8240 | Condition for an injective... |
| tposf1o2 8241 | Condition of a bijective t... |
| tposfo 8242 | The domain and codomain/ra... |
| tposf 8243 | The domain and codomain of... |
| tposfn 8244 | Functionality of a transpo... |
| tpos0 8245 | Transposition of the empty... |
| tposco 8246 | Transposition of a composi... |
| tpossym 8247 | Two ways to say a function... |
| tposeqi 8248 | Equality theorem for trans... |
| tposex 8249 | A transposition is a set. ... |
| nftpos 8250 | Hypothesis builder for tra... |
| tposoprab 8251 | Transposition of a class o... |
| tposmpo 8252 | Transposition of a two-arg... |
| tposconst 8253 | The transposition of a con... |
| mpocurryd 8258 | The currying of an operati... |
| mpocurryvald 8259 | The value of a curried ope... |
| fvmpocurryd 8260 | The value of the value of ... |
| pwuninel2 8263 | Direct proof of ~ pwuninel... |
| pwuninel 8264 | The power set of the union... |
| undefval 8265 | Value of the undefined val... |
| undefnel2 8266 | The undefined value genera... |
| undefnel 8267 | The undefined value genera... |
| undefne0 8268 | The undefined value genera... |
| frecseq123 8271 | Equality theorem for the w... |
| nffrecs 8272 | Bound-variable hypothesis ... |
| csbfrecsg 8273 | Move class substitution in... |
| fpr3g 8274 | Functions defined by well-... |
| frrlem1 8275 | Lemma for well-founded rec... |
| frrlem2 8276 | Lemma for well-founded rec... |
| frrlem3 8277 | Lemma for well-founded rec... |
| frrlem4 8278 | Lemma for well-founded rec... |
| frrlem5 8279 | Lemma for well-founded rec... |
| frrlem6 8280 | Lemma for well-founded rec... |
| frrlem7 8281 | Lemma for well-founded rec... |
| frrlem8 8282 | Lemma for well-founded rec... |
| frrlem9 8283 | Lemma for well-founded rec... |
| frrlem10 8284 | Lemma for well-founded rec... |
| frrlem11 8285 | Lemma for well-founded rec... |
| frrlem12 8286 | Lemma for well-founded rec... |
| frrlem13 8287 | Lemma for well-founded rec... |
| frrlem14 8288 | Lemma for well-founded rec... |
| fprlem1 8289 | Lemma for well-founded rec... |
| fprlem2 8290 | Lemma for well-founded rec... |
| fpr2a 8291 | Weak version of ~ fpr2 whi... |
| fpr1 8292 | Law of well-founded recurs... |
| fpr2 8293 | Law of well-founded recurs... |
| fpr3 8294 | Law of well-founded recurs... |
| frrrel 8295 | Show without using the axi... |
| frrdmss 8296 | Show without using the axi... |
| frrdmcl 8297 | Show without using the axi... |
| fprfung 8298 | A "function" defined by we... |
| fprresex 8299 | The restriction of a funct... |
| dfwrecsOLD 8302 | Obsolete definition of the... |
| wrecseq123 8303 | General equality theorem f... |
| wrecseq123OLD 8304 | Obsolete version of ~ wrec... |
| nfwrecs 8305 | Bound-variable hypothesis ... |
| nfwrecsOLD 8306 | Obsolete proof of ~ nfwrec... |
| wrecseq1 8307 | Equality theorem for the w... |
| wrecseq2 8308 | Equality theorem for the w... |
| wrecseq3 8309 | Equality theorem for the w... |
| csbwrecsg 8310 | Move class substitution in... |
| wfr3g 8311 | Functions defined by well-... |
| wfrlem1OLD 8312 | Lemma for well-ordered rec... |
| wfrlem2OLD 8313 | Lemma for well-ordered rec... |
| wfrlem3OLD 8314 | Lemma for well-ordered rec... |
| wfrlem3OLDa 8315 | Lemma for well-ordered rec... |
| wfrlem4OLD 8316 | Lemma for well-ordered rec... |
| wfrlem5OLD 8317 | Lemma for well-ordered rec... |
| wfrrelOLD 8318 | Obsolete proof of ~ wfrrel... |
| wfrdmssOLD 8319 | Obsolete proof of ~ wfrdms... |
| wfrlem8OLD 8320 | Lemma for well-ordered rec... |
| wfrdmclOLD 8321 | Obsolete version of ~ wfrd... |
| wfrlem10OLD 8322 | Lemma for well-ordered rec... |
| wfrfunOLD 8323 | Obsolete proof of ~ wfrfun... |
| wfrlem12OLD 8324 | Lemma for well-ordered rec... |
| wfrlem13OLD 8325 | Lemma for well-ordered rec... |
| wfrlem14OLD 8326 | Lemma for well-ordered rec... |
| wfrlem15OLD 8327 | Lemma for well-ordered rec... |
| wfrlem16OLD 8328 | Lemma for well-ordered rec... |
| wfrlem17OLD 8329 | Without using ~ ax-rep , s... |
| wfr2aOLD 8330 | Obsolete version of ~ wfr2... |
| wfr1OLD 8331 | Obsolete version of ~ wfr1... |
| wfr2OLD 8332 | Obsolete version of ~ wfr2... |
| wfrrel 8333 | The well-ordered recursion... |
| wfrdmss 8334 | The domain of the well-ord... |
| wfrdmcl 8335 | The predecessor class of a... |
| wfrfun 8336 | The "function" generated b... |
| wfrresex 8337 | Show without using the axi... |
| wfr2a 8338 | A weak version of ~ wfr2 w... |
| wfr1 8339 | The Principle of Well-Orde... |
| wfr2 8340 | The Principle of Well-Orde... |
| wfr3 8341 | The principle of Well-Orde... |
| wfr3OLD 8342 | Obsolete form of ~ wfr3 as... |
| iunon 8343 | The indexed union of a set... |
| iinon 8344 | The nonempty indexed inter... |
| onfununi 8345 | A property of functions on... |
| onovuni 8346 | A variant of ~ onfununi fo... |
| onoviun 8347 | A variant of ~ onovuni wit... |
| onnseq 8348 | There are no length ` _om ... |
| dfsmo2 8351 | Alternate definition of a ... |
| issmo 8352 | Conditions for which ` A `... |
| issmo2 8353 | Alternate definition of a ... |
| smoeq 8354 | Equality theorem for stric... |
| smodm 8355 | The domain of a strictly m... |
| smores 8356 | A strictly monotone functi... |
| smores3 8357 | A strictly monotone functi... |
| smores2 8358 | A strictly monotone ordina... |
| smodm2 8359 | The domain of a strictly m... |
| smofvon2 8360 | The function values of a s... |
| iordsmo 8361 | The identity relation rest... |
| smo0 8362 | The null set is a strictly... |
| smofvon 8363 | If ` B ` is a strictly mon... |
| smoel 8364 | If ` x ` is less than ` y ... |
| smoiun 8365 | The value of a strictly mo... |
| smoiso 8366 | If ` F ` is an isomorphism... |
| smoel2 8367 | A strictly monotone ordina... |
| smo11 8368 | A strictly monotone ordina... |
| smoord 8369 | A strictly monotone ordina... |
| smoword 8370 | A strictly monotone ordina... |
| smogt 8371 | A strictly monotone ordina... |
| smocdmdom 8372 | The codomain of a strictly... |
| smoiso2 8373 | The strictly monotone ordi... |
| dfrecs3 8376 | The old definition of tran... |
| dfrecs3OLD 8377 | Obsolete version of ~ dfre... |
| recseq 8378 | Equality theorem for ` rec... |
| nfrecs 8379 | Bound-variable hypothesis ... |
| tfrlem1 8380 | A technical lemma for tran... |
| tfrlem3a 8381 | Lemma for transfinite recu... |
| tfrlem3 8382 | Lemma for transfinite recu... |
| tfrlem4 8383 | Lemma for transfinite recu... |
| tfrlem5 8384 | Lemma for transfinite recu... |
| recsfval 8385 | Lemma for transfinite recu... |
| tfrlem6 8386 | Lemma for transfinite recu... |
| tfrlem7 8387 | Lemma for transfinite recu... |
| tfrlem8 8388 | Lemma for transfinite recu... |
| tfrlem9 8389 | Lemma for transfinite recu... |
| tfrlem9a 8390 | Lemma for transfinite recu... |
| tfrlem10 8391 | Lemma for transfinite recu... |
| tfrlem11 8392 | Lemma for transfinite recu... |
| tfrlem12 8393 | Lemma for transfinite recu... |
| tfrlem13 8394 | Lemma for transfinite recu... |
| tfrlem14 8395 | Lemma for transfinite recu... |
| tfrlem15 8396 | Lemma for transfinite recu... |
| tfrlem16 8397 | Lemma for finite recursion... |
| tfr1a 8398 | A weak version of ~ tfr1 w... |
| tfr2a 8399 | A weak version of ~ tfr2 w... |
| tfr2b 8400 | Without assuming ~ ax-rep ... |
| tfr1 8401 | Principle of Transfinite R... |
| tfr2 8402 | Principle of Transfinite R... |
| tfr3 8403 | Principle of Transfinite R... |
| tfr1ALT 8404 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8405 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8406 | Alternate proof of ~ tfr3 ... |
| recsfnon 8407 | Strong transfinite recursi... |
| recsval 8408 | Strong transfinite recursi... |
| tz7.44lem1 8409 | The ordered pair abstracti... |
| tz7.44-1 8410 | The value of ` F ` at ` (/... |
| tz7.44-2 8411 | The value of ` F ` at a su... |
| tz7.44-3 8412 | The value of ` F ` at a li... |
| rdgeq1 8415 | Equality theorem for the r... |
| rdgeq2 8416 | Equality theorem for the r... |
| rdgeq12 8417 | Equality theorem for the r... |
| nfrdg 8418 | Bound-variable hypothesis ... |
| rdglem1 8419 | Lemma used with the recurs... |
| rdgfun 8420 | The recursive definition g... |
| rdgdmlim 8421 | The domain of the recursiv... |
| rdgfnon 8422 | The recursive definition g... |
| rdgvalg 8423 | Value of the recursive def... |
| rdgval 8424 | Value of the recursive def... |
| rdg0 8425 | The initial value of the r... |
| rdgseg 8426 | The initial segments of th... |
| rdgsucg 8427 | The value of the recursive... |
| rdgsuc 8428 | The value of the recursive... |
| rdglimg 8429 | The value of the recursive... |
| rdglim 8430 | The value of the recursive... |
| rdg0g 8431 | The initial value of the r... |
| rdgsucmptf 8432 | The value of the recursive... |
| rdgsucmptnf 8433 | The value of the recursive... |
| rdgsucmpt2 8434 | This version of ~ rdgsucmp... |
| rdgsucmpt 8435 | The value of the recursive... |
| rdglim2 8436 | The value of the recursive... |
| rdglim2a 8437 | The value of the recursive... |
| rdg0n 8438 | If ` A ` is a proper class... |
| frfnom 8439 | The function generated by ... |
| fr0g 8440 | The initial value resultin... |
| frsuc 8441 | The successor value result... |
| frsucmpt 8442 | The successor value result... |
| frsucmptn 8443 | The value of the finite re... |
| frsucmpt2 8444 | The successor value result... |
| tz7.48lem 8445 | A way of showing an ordina... |
| tz7.48-2 8446 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8447 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8448 | Proposition 7.48(3) of [Ta... |
| tz7.49 8449 | Proposition 7.49 of [Takeu... |
| tz7.49c 8450 | Corollary of Proposition 7... |
| seqomlem0 8453 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8454 | Lemma for ` seqom ` . The... |
| seqomlem2 8455 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8456 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8457 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8458 | Equality theorem for ` seq... |
| fnseqom 8459 | An index-aware recursive d... |
| seqom0g 8460 | Value of an index-aware re... |
| seqomsuc 8461 | Value of an index-aware re... |
| omsucelsucb 8462 | Membership is inherited by... |
| df1o2 8477 | Expanded value of the ordi... |
| df2o3 8478 | Expanded value of the ordi... |
| df2o2 8479 | Expanded value of the ordi... |
| 1oex 8480 | Ordinal 1 is a set. (Cont... |
| 2oex 8481 | ` 2o ` is a set. (Contrib... |
| 1on 8482 | Ordinal 1 is an ordinal nu... |
| 1onOLD 8483 | Obsolete version of ~ 1on ... |
| 2on 8484 | Ordinal 2 is an ordinal nu... |
| 2onOLD 8485 | Obsolete version of ~ 2on ... |
| 2on0 8486 | Ordinal two is not zero. ... |
| ord3 8487 | Ordinal 3 is an ordinal cl... |
| 3on 8488 | Ordinal 3 is an ordinal nu... |
| 4on 8489 | Ordinal 4 is an ordinal nu... |
| 1oexOLD 8490 | Obsolete version of ~ 1oex... |
| 2oexOLD 8491 | Obsolete version of ~ 2oex... |
| 1n0 8492 | Ordinal one is not equal t... |
| nlim1 8493 | 1 is not a limit ordinal. ... |
| nlim2 8494 | 2 is not a limit ordinal. ... |
| xp01disj 8495 | Cartesian products with th... |
| xp01disjl 8496 | Cartesian products with th... |
| ordgt0ge1 8497 | Two ways to express that a... |
| ordge1n0 8498 | An ordinal greater than or... |
| el1o 8499 | Membership in ordinal one.... |
| ord1eln01 8500 | An ordinal that is not 0 o... |
| ord2eln012 8501 | An ordinal that is not 0, ... |
| 1ellim 8502 | A limit ordinal contains 1... |
| 2ellim 8503 | A limit ordinal contains 2... |
| dif1o 8504 | Two ways to say that ` A `... |
| ondif1 8505 | Two ways to say that ` A `... |
| ondif2 8506 | Two ways to say that ` A `... |
| 2oconcl 8507 | Closure of the pair swappi... |
| 0lt1o 8508 | Ordinal zero is less than ... |
| dif20el 8509 | An ordinal greater than on... |
| 0we1 8510 | The empty set is a well-or... |
| brwitnlem 8511 | Lemma for relations which ... |
| fnoa 8512 | Functionality and domain o... |
| fnom 8513 | Functionality and domain o... |
| fnoe 8514 | Functionality and domain o... |
| oav 8515 | Value of ordinal addition.... |
| omv 8516 | Value of ordinal multiplic... |
| oe0lem 8517 | A helper lemma for ~ oe0 a... |
| oev 8518 | Value of ordinal exponenti... |
| oevn0 8519 | Value of ordinal exponenti... |
| oa0 8520 | Addition with zero. Propo... |
| om0 8521 | Ordinal multiplication wit... |
| oe0m 8522 | Value of zero raised to an... |
| om0x 8523 | Ordinal multiplication wit... |
| oe0m0 8524 | Ordinal exponentiation wit... |
| oe0m1 8525 | Ordinal exponentiation wit... |
| oe0 8526 | Ordinal exponentiation wit... |
| oev2 8527 | Alternate value of ordinal... |
| oasuc 8528 | Addition with successor. ... |
| oesuclem 8529 | Lemma for ~ oesuc . (Cont... |
| omsuc 8530 | Multiplication with succes... |
| oesuc 8531 | Ordinal exponentiation wit... |
| onasuc 8532 | Addition with successor. ... |
| onmsuc 8533 | Multiplication with succes... |
| onesuc 8534 | Exponentiation with a succ... |
| oa1suc 8535 | Addition with 1 is same as... |
| oalim 8536 | Ordinal addition with a li... |
| omlim 8537 | Ordinal multiplication wit... |
| oelim 8538 | Ordinal exponentiation wit... |
| oacl 8539 | Closure law for ordinal ad... |
| omcl 8540 | Closure law for ordinal mu... |
| oecl 8541 | Closure law for ordinal ex... |
| oa0r 8542 | Ordinal addition with zero... |
| om0r 8543 | Ordinal multiplication wit... |
| o1p1e2 8544 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8545 | 2 + 2 = 4 for ordinal numb... |
| om1 8546 | Ordinal multiplication wit... |
| om1r 8547 | Ordinal multiplication wit... |
| oe1 8548 | Ordinal exponentiation wit... |
| oe1m 8549 | Ordinal exponentiation wit... |
| oaordi 8550 | Ordering property of ordin... |
| oaord 8551 | Ordering property of ordin... |
| oacan 8552 | Left cancellation law for ... |
| oaword 8553 | Weak ordering property of ... |
| oawordri 8554 | Weak ordering property of ... |
| oaord1 8555 | An ordinal is less than it... |
| oaword1 8556 | An ordinal is less than or... |
| oaword2 8557 | An ordinal is less than or... |
| oawordeulem 8558 | Lemma for ~ oawordex . (C... |
| oawordeu 8559 | Existence theorem for weak... |
| oawordexr 8560 | Existence theorem for weak... |
| oawordex 8561 | Existence theorem for weak... |
| oaordex 8562 | Existence theorem for orde... |
| oa00 8563 | An ordinal sum is zero iff... |
| oalimcl 8564 | The ordinal sum with a lim... |
| oaass 8565 | Ordinal addition is associ... |
| oarec 8566 | Recursive definition of or... |
| oaf1o 8567 | Left addition by a constan... |
| oacomf1olem 8568 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8569 | Define a bijection from ` ... |
| omordi 8570 | Ordering property of ordin... |
| omord2 8571 | Ordering property of ordin... |
| omord 8572 | Ordering property of ordin... |
| omcan 8573 | Left cancellation law for ... |
| omword 8574 | Weak ordering property of ... |
| omwordi 8575 | Weak ordering property of ... |
| omwordri 8576 | Weak ordering property of ... |
| omword1 8577 | An ordinal is less than or... |
| omword2 8578 | An ordinal is less than or... |
| om00 8579 | The product of two ordinal... |
| om00el 8580 | The product of two nonzero... |
| omordlim 8581 | Ordering involving the pro... |
| omlimcl 8582 | The product of any nonzero... |
| odi 8583 | Distributive law for ordin... |
| omass 8584 | Multiplication of ordinal ... |
| oneo 8585 | If an ordinal number is ev... |
| omeulem1 8586 | Lemma for ~ omeu : existen... |
| omeulem2 8587 | Lemma for ~ omeu : uniquen... |
| omopth2 8588 | An ordered pair-like theor... |
| omeu 8589 | The division algorithm for... |
| oen0 8590 | Ordinal exponentiation wit... |
| oeordi 8591 | Ordering law for ordinal e... |
| oeord 8592 | Ordering property of ordin... |
| oecan 8593 | Left cancellation law for ... |
| oeword 8594 | Weak ordering property of ... |
| oewordi 8595 | Weak ordering property of ... |
| oewordri 8596 | Weak ordering property of ... |
| oeworde 8597 | Ordinal exponentiation com... |
| oeordsuc 8598 | Ordering property of ordin... |
| oelim2 8599 | Ordinal exponentiation wit... |
| oeoalem 8600 | Lemma for ~ oeoa . (Contr... |
| oeoa 8601 | Sum of exponents law for o... |
| oeoelem 8602 | Lemma for ~ oeoe . (Contr... |
| oeoe 8603 | Product of exponents law f... |
| oelimcl 8604 | The ordinal exponential wi... |
| oeeulem 8605 | Lemma for ~ oeeu . (Contr... |
| oeeui 8606 | The division algorithm for... |
| oeeu 8607 | The division algorithm for... |
| nna0 8608 | Addition with zero. Theor... |
| nnm0 8609 | Multiplication with zero. ... |
| nnasuc 8610 | Addition with successor. ... |
| nnmsuc 8611 | Multiplication with succes... |
| nnesuc 8612 | Exponentiation with a succ... |
| nna0r 8613 | Addition to zero. Remark ... |
| nnm0r 8614 | Multiplication with zero. ... |
| nnacl 8615 | Closure of addition of nat... |
| nnmcl 8616 | Closure of multiplication ... |
| nnecl 8617 | Closure of exponentiation ... |
| nnacli 8618 | ` _om ` is closed under ad... |
| nnmcli 8619 | ` _om ` is closed under mu... |
| nnarcl 8620 | Reverse closure law for ad... |
| nnacom 8621 | Addition of natural number... |
| nnaordi 8622 | Ordering property of addit... |
| nnaord 8623 | Ordering property of addit... |
| nnaordr 8624 | Ordering property of addit... |
| nnawordi 8625 | Adding to both sides of an... |
| nnaass 8626 | Addition of natural number... |
| nndi 8627 | Distributive law for natur... |
| nnmass 8628 | Multiplication of natural ... |
| nnmsucr 8629 | Multiplication with succes... |
| nnmcom 8630 | Multiplication of natural ... |
| nnaword 8631 | Weak ordering property of ... |
| nnacan 8632 | Cancellation law for addit... |
| nnaword1 8633 | Weak ordering property of ... |
| nnaword2 8634 | Weak ordering property of ... |
| nnmordi 8635 | Ordering property of multi... |
| nnmord 8636 | Ordering property of multi... |
| nnmword 8637 | Weak ordering property of ... |
| nnmcan 8638 | Cancellation law for multi... |
| nnmwordi 8639 | Weak ordering property of ... |
| nnmwordri 8640 | Weak ordering property of ... |
| nnawordex 8641 | Equivalence for weak order... |
| nnaordex 8642 | Equivalence for ordering. ... |
| 1onn 8643 | The ordinal 1 is a natural... |
| 1onnALT 8644 | Shorter proof of ~ 1onn us... |
| 2onn 8645 | The ordinal 2 is a natural... |
| 2onnALT 8646 | Shorter proof of ~ 2onn us... |
| 3onn 8647 | The ordinal 3 is a natural... |
| 4onn 8648 | The ordinal 4 is a natural... |
| 1one2o 8649 | Ordinal one is not ordinal... |
| oaabslem 8650 | Lemma for ~ oaabs . (Cont... |
| oaabs 8651 | Ordinal addition absorbs a... |
| oaabs2 8652 | The absorption law ~ oaabs... |
| omabslem 8653 | Lemma for ~ omabs . (Cont... |
| omabs 8654 | Ordinal multiplication is ... |
| nnm1 8655 | Multiply an element of ` _... |
| nnm2 8656 | Multiply an element of ` _... |
| nn2m 8657 | Multiply an element of ` _... |
| nnneo 8658 | If a natural number is eve... |
| nneob 8659 | A natural number is even i... |
| omsmolem 8660 | Lemma for ~ omsmo . (Cont... |
| omsmo 8661 | A strictly monotonic ordin... |
| omopthlem1 8662 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8663 | Lemma for ~ omopthi . (Co... |
| omopthi 8664 | An ordered pair theorem fo... |
| omopth 8665 | An ordered pair theorem fo... |
| nnasmo 8666 | There is at most one left ... |
| eldifsucnn 8667 | Condition for membership i... |
| on2recsfn 8670 | Show that double recursion... |
| on2recsov 8671 | Calculate the value of the... |
| on2ind 8672 | Double induction over ordi... |
| on3ind 8673 | Triple induction over ordi... |
| coflton 8674 | Cofinality theorem for ord... |
| cofon1 8675 | Cofinality theorem for ord... |
| cofon2 8676 | Cofinality theorem for ord... |
| cofonr 8677 | Inverse cofinality law for... |
| naddfn 8678 | Natural addition is a func... |
| naddcllem 8679 | Lemma for ordinal addition... |
| naddcl 8680 | Closure law for natural ad... |
| naddov 8681 | The value of natural addit... |
| naddov2 8682 | Alternate expression for n... |
| naddov3 8683 | Alternate expression for n... |
| naddf 8684 | Function statement for nat... |
| naddcom 8685 | Natural addition commutes.... |
| naddrid 8686 | Ordinal zero is the additi... |
| naddlid 8687 | Ordinal zero is the additi... |
| naddssim 8688 | Ordinal less-than-or-equal... |
| naddelim 8689 | Ordinal less-than is prese... |
| naddel1 8690 | Ordinal less-than is not a... |
| naddel2 8691 | Ordinal less-than is not a... |
| naddss1 8692 | Ordinal less-than-or-equal... |
| naddss2 8693 | Ordinal less-than-or-equal... |
| naddword1 8694 | Weak-ordering principle fo... |
| naddword2 8695 | Weak-ordering principle fo... |
| naddunif 8696 | Uniformity theorem for nat... |
| naddasslem1 8697 | Lemma for ~ naddass . Exp... |
| naddasslem2 8698 | Lemma for ~ naddass . Exp... |
| naddass 8699 | Natural ordinal addition i... |
| nadd32 8700 | Commutative/associative la... |
| nadd4 8701 | Rearragement of terms in a... |
| nadd42 8702 | Rearragement of terms in a... |
| naddel12 8703 | Natural addition to both s... |
| dfer2 8708 | Alternate definition of eq... |
| dfec2 8710 | Alternate definition of ` ... |
| ecexg 8711 | An equivalence class modul... |
| ecexr 8712 | A nonempty equivalence cla... |
| ereq1 8714 | Equality theorem for equiv... |
| ereq2 8715 | Equality theorem for equiv... |
| errel 8716 | An equivalence relation is... |
| erdm 8717 | The domain of an equivalen... |
| ercl 8718 | Elementhood in the field o... |
| ersym 8719 | An equivalence relation is... |
| ercl2 8720 | Elementhood in the field o... |
| ersymb 8721 | An equivalence relation is... |
| ertr 8722 | An equivalence relation is... |
| ertrd 8723 | A transitivity relation fo... |
| ertr2d 8724 | A transitivity relation fo... |
| ertr3d 8725 | A transitivity relation fo... |
| ertr4d 8726 | A transitivity relation fo... |
| erref 8727 | An equivalence relation is... |
| ercnv 8728 | The converse of an equival... |
| errn 8729 | The range and domain of an... |
| erssxp 8730 | An equivalence relation is... |
| erex 8731 | An equivalence relation is... |
| erexb 8732 | An equivalence relation is... |
| iserd 8733 | A reflexive, symmetric, tr... |
| iseri 8734 | A reflexive, symmetric, tr... |
| iseriALT 8735 | Alternate proof of ~ iseri... |
| brdifun 8736 | Evaluate the incomparabili... |
| swoer 8737 | Incomparability under a st... |
| swoord1 8738 | The incomparability equiva... |
| swoord2 8739 | The incomparability equiva... |
| swoso 8740 | If the incomparability rel... |
| eqerlem 8741 | Lemma for ~ eqer . (Contr... |
| eqer 8742 | Equivalence relation invol... |
| ider 8743 | The identity relation is a... |
| 0er 8744 | The empty set is an equiva... |
| eceq1 8745 | Equality theorem for equiv... |
| eceq1d 8746 | Equality theorem for equiv... |
| eceq2 8747 | Equality theorem for equiv... |
| eceq2i 8748 | Equality theorem for the `... |
| eceq2d 8749 | Equality theorem for the `... |
| elecg 8750 | Membership in an equivalen... |
| elec 8751 | Membership in an equivalen... |
| relelec 8752 | Membership in an equivalen... |
| ecss 8753 | An equivalence class is a ... |
| ecdmn0 8754 | A representative of a none... |
| ereldm 8755 | Equality of equivalence cl... |
| erth 8756 | Basic property of equivale... |
| erth2 8757 | Basic property of equivale... |
| erthi 8758 | Basic property of equivale... |
| erdisj 8759 | Equivalence classes do not... |
| ecidsn 8760 | An equivalence class modul... |
| qseq1 8761 | Equality theorem for quoti... |
| qseq2 8762 | Equality theorem for quoti... |
| qseq2i 8763 | Equality theorem for quoti... |
| qseq2d 8764 | Equality theorem for quoti... |
| qseq12 8765 | Equality theorem for quoti... |
| elqsg 8766 | Closed form of ~ elqs . (... |
| elqs 8767 | Membership in a quotient s... |
| elqsi 8768 | Membership in a quotient s... |
| elqsecl 8769 | Membership in a quotient s... |
| ecelqsg 8770 | Membership of an equivalen... |
| ecelqsi 8771 | Membership of an equivalen... |
| ecopqsi 8772 | "Closure" law for equivale... |
| qsexg 8773 | A quotient set exists. (C... |
| qsex 8774 | A quotient set exists. (C... |
| uniqs 8775 | The union of a quotient se... |
| qsss 8776 | A quotient set is a set of... |
| uniqs2 8777 | The union of a quotient se... |
| snec 8778 | The singleton of an equiva... |
| ecqs 8779 | Equivalence class in terms... |
| ecid 8780 | A set is equal to its cose... |
| qsid 8781 | A set is equal to its quot... |
| ectocld 8782 | Implicit substitution of c... |
| ectocl 8783 | Implicit substitution of c... |
| elqsn0 8784 | A quotient set does not co... |
| ecelqsdm 8785 | Membership of an equivalen... |
| xpider 8786 | A Cartesian square is an e... |
| iiner 8787 | The intersection of a none... |
| riiner 8788 | The relative intersection ... |
| erinxp 8789 | A restricted equivalence r... |
| ecinxp 8790 | Restrict the relation in a... |
| qsinxp 8791 | Restrict the equivalence r... |
| qsdisj 8792 | Members of a quotient set ... |
| qsdisj2 8793 | A quotient set is a disjoi... |
| qsel 8794 | If an element of a quotien... |
| uniinqs 8795 | Class union distributes ov... |
| qliftlem 8796 | Lemma for theorems about a... |
| qliftrel 8797 | ` F ` , a function lift, i... |
| qliftel 8798 | Elementhood in the relatio... |
| qliftel1 8799 | Elementhood in the relatio... |
| qliftfun 8800 | The function ` F ` is the ... |
| qliftfund 8801 | The function ` F ` is the ... |
| qliftfuns 8802 | The function ` F ` is the ... |
| qliftf 8803 | The domain and codomain of... |
| qliftval 8804 | The value of the function ... |
| ecoptocl 8805 | Implicit substitution of c... |
| 2ecoptocl 8806 | Implicit substitution of c... |
| 3ecoptocl 8807 | Implicit substitution of c... |
| brecop 8808 | Binary relation on a quoti... |
| brecop2 8809 | Binary relation on a quoti... |
| eroveu 8810 | Lemma for ~ erov and ~ ero... |
| erovlem 8811 | Lemma for ~ erov and ~ ero... |
| erov 8812 | The value of an operation ... |
| eroprf 8813 | Functionality of an operat... |
| erov2 8814 | The value of an operation ... |
| eroprf2 8815 | Functionality of an operat... |
| ecopoveq 8816 | This is the first of sever... |
| ecopovsym 8817 | Assuming the operation ` F... |
| ecopovtrn 8818 | Assuming that operation ` ... |
| ecopover 8819 | Assuming that operation ` ... |
| eceqoveq 8820 | Equality of equivalence re... |
| ecovcom 8821 | Lemma used to transfer a c... |
| ecovass 8822 | Lemma used to transfer an ... |
| ecovdi 8823 | Lemma used to transfer a d... |
| mapprc 8828 | When ` A ` is a proper cla... |
| pmex 8829 | The class of all partial f... |
| mapex 8830 | The class of all functions... |
| fnmap 8831 | Set exponentiation has a u... |
| fnpm 8832 | Partial function exponenti... |
| reldmmap 8833 | Set exponentiation is a we... |
| mapvalg 8834 | The value of set exponenti... |
| pmvalg 8835 | The value of the partial m... |
| mapval 8836 | The value of set exponenti... |
| elmapg 8837 | Membership relation for se... |
| elmapd 8838 | Deduction form of ~ elmapg... |
| elmapdd 8839 | Deduction associated with ... |
| mapdm0 8840 | The empty set is the only ... |
| elpmg 8841 | The predicate "is a partia... |
| elpm2g 8842 | The predicate "is a partia... |
| elpm2r 8843 | Sufficient condition for b... |
| elpmi 8844 | A partial function is a fu... |
| pmfun 8845 | A partial function is a fu... |
| elmapex 8846 | Eliminate antecedent for m... |
| elmapi 8847 | A mapping is a function, f... |
| mapfset 8848 | If ` B ` is a set, the val... |
| mapssfset 8849 | The value of the set expon... |
| mapfoss 8850 | The value of the set expon... |
| fsetsspwxp 8851 | The class of all functions... |
| fset0 8852 | The set of functions from ... |
| fsetdmprc0 8853 | The set of functions with ... |
| fsetex 8854 | The set of functions betwe... |
| f1setex 8855 | The set of injections betw... |
| fosetex 8856 | The set of surjections bet... |
| f1osetex 8857 | The set of bijections betw... |
| fsetfcdm 8858 | The class of functions wit... |
| fsetfocdm 8859 | The class of functions wit... |
| fsetprcnex 8860 | The class of all functions... |
| fsetcdmex 8861 | The class of all functions... |
| fsetexb 8862 | The class of all functions... |
| elmapfn 8863 | A mapping is a function wi... |
| elmapfun 8864 | A mapping is always a func... |
| elmapssres 8865 | A restricted mapping is a ... |
| fpmg 8866 | A total function is a part... |
| pmss12g 8867 | Subset relation for the se... |
| pmresg 8868 | Elementhood of a restricte... |
| elmap 8869 | Membership relation for se... |
| mapval2 8870 | Alternate expression for t... |
| elpm 8871 | The predicate "is a partia... |
| elpm2 8872 | The predicate "is a partia... |
| fpm 8873 | A total function is a part... |
| mapsspm 8874 | Set exponentiation is a su... |
| pmsspw 8875 | Partial maps are a subset ... |
| mapsspw 8876 | Set exponentiation is a su... |
| mapfvd 8877 | The value of a function th... |
| elmapresaun 8878 | ~ fresaun transposed to ma... |
| fvmptmap 8879 | Special case of ~ fvmpt fo... |
| map0e 8880 | Set exponentiation with an... |
| map0b 8881 | Set exponentiation with an... |
| map0g 8882 | Set exponentiation is empt... |
| 0map0sn0 8883 | The set of mappings of the... |
| mapsnd 8884 | The value of set exponenti... |
| map0 8885 | Set exponentiation is empt... |
| mapsn 8886 | The value of set exponenti... |
| mapss 8887 | Subset inheritance for set... |
| fdiagfn 8888 | Functionality of the diago... |
| fvdiagfn 8889 | Functionality of the diago... |
| mapsnconst 8890 | Every singleton map is a c... |
| mapsncnv 8891 | Expression for the inverse... |
| mapsnf1o2 8892 | Explicit bijection between... |
| mapsnf1o3 8893 | Explicit bijection in the ... |
| ralxpmap 8894 | Quantification over functi... |
| dfixp 8897 | Eliminate the expression `... |
| ixpsnval 8898 | The value of an infinite C... |
| elixp2 8899 | Membership in an infinite ... |
| fvixp 8900 | Projection of a factor of ... |
| ixpfn 8901 | A nuple is a function. (C... |
| elixp 8902 | Membership in an infinite ... |
| elixpconst 8903 | Membership in an infinite ... |
| ixpconstg 8904 | Infinite Cartesian product... |
| ixpconst 8905 | Infinite Cartesian product... |
| ixpeq1 8906 | Equality theorem for infin... |
| ixpeq1d 8907 | Equality theorem for infin... |
| ss2ixp 8908 | Subclass theorem for infin... |
| ixpeq2 8909 | Equality theorem for infin... |
| ixpeq2dva 8910 | Equality theorem for infin... |
| ixpeq2dv 8911 | Equality theorem for infin... |
| cbvixp 8912 | Change bound variable in a... |
| cbvixpv 8913 | Change bound variable in a... |
| nfixpw 8914 | Bound-variable hypothesis ... |
| nfixp 8915 | Bound-variable hypothesis ... |
| nfixp1 8916 | The index variable in an i... |
| ixpprc 8917 | A cartesian product of pro... |
| ixpf 8918 | A member of an infinite Ca... |
| uniixp 8919 | The union of an infinite C... |
| ixpexg 8920 | The existence of an infini... |
| ixpin 8921 | The intersection of two in... |
| ixpiin 8922 | The indexed intersection o... |
| ixpint 8923 | The intersection of a coll... |
| ixp0x 8924 | An infinite Cartesian prod... |
| ixpssmap2g 8925 | An infinite Cartesian prod... |
| ixpssmapg 8926 | An infinite Cartesian prod... |
| 0elixp 8927 | Membership of the empty se... |
| ixpn0 8928 | The infinite Cartesian pro... |
| ixp0 8929 | The infinite Cartesian pro... |
| ixpssmap 8930 | An infinite Cartesian prod... |
| resixp 8931 | Restriction of an element ... |
| undifixp 8932 | Union of two projections o... |
| mptelixpg 8933 | Condition for an explicit ... |
| resixpfo 8934 | Restriction of elements of... |
| elixpsn 8935 | Membership in a class of s... |
| ixpsnf1o 8936 | A bijection between a clas... |
| mapsnf1o 8937 | A bijection between a set ... |
| boxriin 8938 | A rectangular subset of a ... |
| boxcutc 8939 | The relative complement of... |
| relen 8948 | Equinumerosity is a relati... |
| reldom 8949 | Dominance is a relation. ... |
| relsdom 8950 | Strict dominance is a rela... |
| encv 8951 | If two classes are equinum... |
| breng 8952 | Equinumerosity relation. ... |
| bren 8953 | Equinumerosity relation. ... |
| brenOLD 8954 | Obsolete version of ~ bren... |
| brdom2g 8955 | Dominance relation. This ... |
| brdomg 8956 | Dominance relation. (Cont... |
| brdomgOLD 8957 | Obsolete version of ~ brdo... |
| brdomi 8958 | Dominance relation. (Cont... |
| brdomiOLD 8959 | Obsolete version of ~ brdo... |
| brdom 8960 | Dominance relation. (Cont... |
| domen 8961 | Dominance in terms of equi... |
| domeng 8962 | Dominance in terms of equi... |
| ctex 8963 | A countable set is a set. ... |
| f1oen4g 8964 | The domain and range of a ... |
| f1dom4g 8965 | The domain of a one-to-one... |
| f1oen3g 8966 | The domain and range of a ... |
| f1dom3g 8967 | The domain of a one-to-one... |
| f1oen2g 8968 | The domain and range of a ... |
| f1dom2g 8969 | The domain of a one-to-one... |
| f1dom2gOLD 8970 | Obsolete version of ~ f1do... |
| f1oeng 8971 | The domain and range of a ... |
| f1domg 8972 | The domain of a one-to-one... |
| f1oen 8973 | The domain and range of a ... |
| f1dom 8974 | The domain of a one-to-one... |
| brsdom 8975 | Strict dominance relation,... |
| isfi 8976 | Express " ` A ` is finite"... |
| enssdom 8977 | Equinumerosity implies dom... |
| dfdom2 8978 | Alternate definition of do... |
| endom 8979 | Equinumerosity implies dom... |
| sdomdom 8980 | Strict dominance implies d... |
| sdomnen 8981 | Strict dominance implies n... |
| brdom2 8982 | Dominance in terms of stri... |
| bren2 8983 | Equinumerosity expressed i... |
| enrefg 8984 | Equinumerosity is reflexiv... |
| enref 8985 | Equinumerosity is reflexiv... |
| eqeng 8986 | Equality implies equinumer... |
| domrefg 8987 | Dominance is reflexive. (... |
| en2d 8988 | Equinumerosity inference f... |
| en3d 8989 | Equinumerosity inference f... |
| en2i 8990 | Equinumerosity inference f... |
| en3i 8991 | Equinumerosity inference f... |
| dom2lem 8992 | A mapping (first hypothesi... |
| dom2d 8993 | A mapping (first hypothesi... |
| dom3d 8994 | A mapping (first hypothesi... |
| dom2 8995 | A mapping (first hypothesi... |
| dom3 8996 | A mapping (first hypothesi... |
| idssen 8997 | Equality implies equinumer... |
| domssl 8998 | If ` A ` is a subset of ` ... |
| domssr 8999 | If ` C ` is a superset of ... |
| ssdomg 9000 | A set dominates its subset... |
| ener 9001 | Equinumerosity is an equiv... |
| ensymb 9002 | Symmetry of equinumerosity... |
| ensym 9003 | Symmetry of equinumerosity... |
| ensymi 9004 | Symmetry of equinumerosity... |
| ensymd 9005 | Symmetry of equinumerosity... |
| entr 9006 | Transitivity of equinumero... |
| domtr 9007 | Transitivity of dominance ... |
| entri 9008 | A chained equinumerosity i... |
| entr2i 9009 | A chained equinumerosity i... |
| entr3i 9010 | A chained equinumerosity i... |
| entr4i 9011 | A chained equinumerosity i... |
| endomtr 9012 | Transitivity of equinumero... |
| domentr 9013 | Transitivity of dominance ... |
| f1imaeng 9014 | If a function is one-to-on... |
| f1imaen2g 9015 | If a function is one-to-on... |
| f1imaen 9016 | If a function is one-to-on... |
| en0 9017 | The empty set is equinumer... |
| en0OLD 9018 | Obsolete version of ~ en0 ... |
| en0ALT 9019 | Shorter proof of ~ en0 , d... |
| en0r 9020 | The empty set is equinumer... |
| ensn1 9021 | A singleton is equinumerou... |
| ensn1OLD 9022 | Obsolete version of ~ ensn... |
| ensn1g 9023 | A singleton is equinumerou... |
| enpr1g 9024 | ` { A , A } ` has only one... |
| en1 9025 | A set is equinumerous to o... |
| en1OLD 9026 | Obsolete version of ~ en1 ... |
| en1b 9027 | A set is equinumerous to o... |
| en1bOLD 9028 | Obsolete version of ~ en1b... |
| reuen1 9029 | Two ways to express "exact... |
| euen1 9030 | Two ways to express "exact... |
| euen1b 9031 | Two ways to express " ` A ... |
| en1uniel 9032 | A singleton contains its s... |
| en1unielOLD 9033 | Obsolete version of ~ en1u... |
| 2dom 9034 | A set that dominates ordin... |
| fundmen 9035 | A function is equinumerous... |
| fundmeng 9036 | A function is equinumerous... |
| cnven 9037 | A relational set is equinu... |
| cnvct 9038 | If a set is countable, so ... |
| fndmeng 9039 | A function is equinumerate... |
| mapsnend 9040 | Set exponentiation to a si... |
| mapsnen 9041 | Set exponentiation to a si... |
| snmapen 9042 | Set exponentiation: a sing... |
| snmapen1 9043 | Set exponentiation: a sing... |
| map1 9044 | Set exponentiation: ordina... |
| en2sn 9045 | Two singletons are equinum... |
| en2snOLD 9046 | Obsolete version of ~ en2s... |
| en2snOLDOLD 9047 | Obsolete version of ~ en2s... |
| snfi 9048 | A singleton is finite. (C... |
| fiprc 9049 | The class of finite sets i... |
| unen 9050 | Equinumerosity of union of... |
| enrefnn 9051 | Equinumerosity is reflexiv... |
| en2prd 9052 | Two unordered pairs are eq... |
| enpr2d 9053 | A pair with distinct eleme... |
| enpr2dOLD 9054 | Obsolete version of ~ enpr... |
| ssct 9055 | Any subset of a countable ... |
| ssctOLD 9056 | Obsolete version of ~ ssct... |
| difsnen 9057 | All decrements of a set ar... |
| domdifsn 9058 | Dominance over a set with ... |
| xpsnen 9059 | A set is equinumerous to i... |
| xpsneng 9060 | A set is equinumerous to i... |
| xp1en 9061 | One times a cardinal numbe... |
| endisj 9062 | Any two sets are equinumer... |
| undom 9063 | Dominance law for union. ... |
| undomOLD 9064 | Obsolete version of ~ undo... |
| xpcomf1o 9065 | The canonical bijection fr... |
| xpcomco 9066 | Composition with the bijec... |
| xpcomen 9067 | Commutative law for equinu... |
| xpcomeng 9068 | Commutative law for equinu... |
| xpsnen2g 9069 | A set is equinumerous to i... |
| xpassen 9070 | Associative law for equinu... |
| xpdom2 9071 | Dominance law for Cartesia... |
| xpdom2g 9072 | Dominance law for Cartesia... |
| xpdom1g 9073 | Dominance law for Cartesia... |
| xpdom3 9074 | A set is dominated by its ... |
| xpdom1 9075 | Dominance law for Cartesia... |
| domunsncan 9076 | A singleton cancellation l... |
| omxpenlem 9077 | Lemma for ~ omxpen . (Con... |
| omxpen 9078 | The cardinal and ordinal p... |
| omf1o 9079 | Construct an explicit bije... |
| pw2f1olem 9080 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9081 | The power set of a set is ... |
| pw2eng 9082 | The power set of a set is ... |
| pw2en 9083 | The power set of a set is ... |
| fopwdom 9084 | Covering implies injection... |
| enfixsn 9085 | Given two equipollent sets... |
| sucdom2OLD 9086 | Obsolete version of ~ sucd... |
| sbthlem1 9087 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9088 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9089 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9090 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9091 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9092 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9093 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9094 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9095 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9096 | Lemma for ~ sbth . (Contr... |
| sbth 9097 | Schroeder-Bernstein Theore... |
| sbthb 9098 | Schroeder-Bernstein Theore... |
| sbthcl 9099 | Schroeder-Bernstein Theore... |
| dfsdom2 9100 | Alternate definition of st... |
| brsdom2 9101 | Alternate definition of st... |
| sdomnsym 9102 | Strict dominance is asymme... |
| domnsym 9103 | Theorem 22(i) of [Suppes] ... |
| 0domg 9104 | Any set dominates the empt... |
| 0domgOLD 9105 | Obsolete version of ~ 0dom... |
| dom0 9106 | A set dominated by the emp... |
| dom0OLD 9107 | Obsolete version of ~ dom0... |
| 0sdomg 9108 | A set strictly dominates t... |
| 0sdomgOLD 9109 | Obsolete version of ~ 0sdo... |
| 0dom 9110 | Any set dominates the empt... |
| 0sdom 9111 | A set strictly dominates t... |
| sdom0 9112 | The empty set does not str... |
| sdom0OLD 9113 | Obsolete version of ~ sdom... |
| sdomdomtr 9114 | Transitivity of strict dom... |
| sdomentr 9115 | Transitivity of strict dom... |
| domsdomtr 9116 | Transitivity of dominance ... |
| ensdomtr 9117 | Transitivity of equinumero... |
| sdomirr 9118 | Strict dominance is irrefl... |
| sdomtr 9119 | Strict dominance is transi... |
| sdomn2lp 9120 | Strict dominance has no 2-... |
| enen1 9121 | Equality-like theorem for ... |
| enen2 9122 | Equality-like theorem for ... |
| domen1 9123 | Equality-like theorem for ... |
| domen2 9124 | Equality-like theorem for ... |
| sdomen1 9125 | Equality-like theorem for ... |
| sdomen2 9126 | Equality-like theorem for ... |
| domtriord 9127 | Dominance is trichotomous ... |
| sdomel 9128 | For ordinals, strict domin... |
| sdomdif 9129 | The difference of a set fr... |
| onsdominel 9130 | An ordinal with more eleme... |
| domunsn 9131 | Dominance over a set with ... |
| fodomr 9132 | There exists a mapping fro... |
| pwdom 9133 | Injection of sets implies ... |
| canth2 9134 | Cantor's Theorem. No set ... |
| canth2g 9135 | Cantor's theorem with the ... |
| 2pwuninel 9136 | The power set of the power... |
| 2pwne 9137 | No set equals the power se... |
| disjen 9138 | A stronger form of ~ pwuni... |
| disjenex 9139 | Existence version of ~ dis... |
| domss2 9140 | A corollary of ~ disjenex ... |
| domssex2 9141 | A corollary of ~ disjenex ... |
| domssex 9142 | Weakening of ~ domssex2 to... |
| xpf1o 9143 | Construct a bijection on a... |
| xpen 9144 | Equinumerosity law for Car... |
| mapen 9145 | Two set exponentiations ar... |
| mapdom1 9146 | Order-preserving property ... |
| mapxpen 9147 | Equinumerosity law for dou... |
| xpmapenlem 9148 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9149 | Equinumerosity law for set... |
| mapunen 9150 | Equinumerosity law for set... |
| map2xp 9151 | A cardinal power with expo... |
| mapdom2 9152 | Order-preserving property ... |
| mapdom3 9153 | Set exponentiation dominat... |
| pwen 9154 | If two sets are equinumero... |
| ssenen 9155 | Equinumerosity of equinume... |
| limenpsi 9156 | A limit ordinal is equinum... |
| limensuci 9157 | A limit ordinal is equinum... |
| limensuc 9158 | A limit ordinal is equinum... |
| infensuc 9159 | Any infinite ordinal is eq... |
| dif1enlem 9160 | Lemma for ~ rexdif1en and ... |
| dif1enlemOLD 9161 | Obsolete version of ~ dif1... |
| rexdif1en 9162 | If a set is equinumerous t... |
| rexdif1enOLD 9163 | Obsolete version of ~ rexd... |
| dif1en 9164 | If a set ` A ` is equinume... |
| dif1ennn 9165 | If a set ` A ` is equinume... |
| dif1enOLD 9166 | Obsolete version of ~ dif1... |
| findcard 9167 | Schema for induction on th... |
| findcard2 9168 | Schema for induction on th... |
| findcard2s 9169 | Variation of ~ findcard2 r... |
| findcard2d 9170 | Deduction version of ~ fin... |
| nnfi 9171 | Natural numbers are finite... |
| pssnn 9172 | A proper subset of a natur... |
| ssnnfi 9173 | A subset of a natural numb... |
| ssnnfiOLD 9174 | Obsolete version of ~ ssnn... |
| 0fin 9175 | The empty set is finite. ... |
| unfi 9176 | The union of two finite se... |
| ssfi 9177 | A subset of a finite set i... |
| ssfiALT 9178 | Shorter proof of ~ ssfi us... |
| imafi 9179 | Images of finite sets are ... |
| pwfir 9180 | If the power set of a set ... |
| pwfilem 9181 | Lemma for ~ pwfi . (Contr... |
| pwfi 9182 | The power set of a finite ... |
| diffi 9183 | If ` A ` is finite, ` ( A ... |
| cnvfi 9184 | If a set is finite, its co... |
| fnfi 9185 | A version of ~ fnex for fi... |
| f1oenfi 9186 | If the domain of a one-to-... |
| f1oenfirn 9187 | If the range of a one-to-o... |
| f1domfi 9188 | If the codomain of a one-t... |
| f1domfi2 9189 | If the domain of a one-to-... |
| enreffi 9190 | Equinumerosity is reflexiv... |
| ensymfib 9191 | Symmetry of equinumerosity... |
| entrfil 9192 | Transitivity of equinumero... |
| enfii 9193 | A set equinumerous to a fi... |
| enfi 9194 | Equinumerous sets have the... |
| enfiALT 9195 | Shorter proof of ~ enfi us... |
| domfi 9196 | A set dominated by a finit... |
| entrfi 9197 | Transitivity of equinumero... |
| entrfir 9198 | Transitivity of equinumero... |
| domtrfil 9199 | Transitivity of dominance ... |
| domtrfi 9200 | Transitivity of dominance ... |
| domtrfir 9201 | Transitivity of dominance ... |
| f1imaenfi 9202 | If a function is one-to-on... |
| ssdomfi 9203 | A finite set dominates its... |
| ssdomfi2 9204 | A set dominates its finite... |
| sbthfilem 9205 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9206 | Schroeder-Bernstein Theore... |
| domnsymfi 9207 | If a set dominates a finit... |
| sdomdomtrfi 9208 | Transitivity of strict dom... |
| domsdomtrfi 9209 | Transitivity of dominance ... |
| sucdom2 9210 | Strict dominance of a set ... |
| phplem1 9211 | Lemma for Pigeonhole Princ... |
| phplem2 9212 | Lemma for Pigeonhole Princ... |
| nneneq 9213 | Two equinumerous natural n... |
| php 9214 | Pigeonhole Principle. A n... |
| php2 9215 | Corollary of Pigeonhole Pr... |
| php3 9216 | Corollary of Pigeonhole Pr... |
| php4 9217 | Corollary of the Pigeonhol... |
| php5 9218 | Corollary of the Pigeonhol... |
| phpeqd 9219 | Corollary of the Pigeonhol... |
| nndomog 9220 | Cardinal ordering agrees w... |
| phplem1OLD 9221 | Obsolete lemma for ~ php a... |
| phplem2OLD 9222 | Obsolete lemma for ~ php a... |
| phplem3OLD 9223 | Obsolete version of ~ phpl... |
| phplem4OLD 9224 | Obsolete version of ~ phpl... |
| nneneqOLD 9225 | Obsolete version of ~ nnen... |
| phpOLD 9226 | Obsolete version of ~ php ... |
| php2OLD 9227 | Obsolete version of ~ php2... |
| php3OLD 9228 | Obsolete version of ~ php3... |
| phpeqdOLD 9229 | Obsolete version of ~ phpe... |
| nndomogOLD 9230 | Obsolete version of ~ nndo... |
| snnen2oOLD 9231 | Obsolete version of ~ snne... |
| onomeneq 9232 | An ordinal number equinume... |
| onomeneqOLD 9233 | Obsolete version of ~ onom... |
| onfin 9234 | An ordinal number is finit... |
| onfin2 9235 | A set is a natural number ... |
| nnfiOLD 9236 | Obsolete version of ~ nnfi... |
| nndomo 9237 | Cardinal ordering agrees w... |
| nnsdomo 9238 | Cardinal ordering agrees w... |
| sucdom 9239 | Strict dominance of a set ... |
| sucdomOLD 9240 | Obsolete version of ~ sucd... |
| snnen2o 9241 | A singleton ` { A } ` is n... |
| 0sdom1dom 9242 | Strict dominance over 0 is... |
| 0sdom1domALT 9243 | Alternate proof of ~ 0sdom... |
| 1sdom2 9244 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9245 | Alternate proof of ~ 1sdom... |
| sdom1 9246 | A set has less than one me... |
| sdom1OLD 9247 | Obsolete version of ~ sdom... |
| modom 9248 | Two ways to express "at mo... |
| modom2 9249 | Two ways to express "at mo... |
| rex2dom 9250 | A set that has at least 2 ... |
| 1sdom2dom 9251 | Strict dominance over 1 is... |
| 1sdom 9252 | A set that strictly domina... |
| 1sdomOLD 9253 | Obsolete version of ~ 1sdo... |
| unxpdomlem1 9254 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9255 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9256 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9257 | Cartesian product dominate... |
| unxpdom2 9258 | Corollary of ~ unxpdom . ... |
| sucxpdom 9259 | Cartesian product dominate... |
| pssinf 9260 | A set equinumerous to a pr... |
| fisseneq 9261 | A finite set is equal to i... |
| ominf 9262 | The set of natural numbers... |
| ominfOLD 9263 | Obsolete version of ~ omin... |
| isinf 9264 | Any set that is not finite... |
| isinfOLD 9265 | Obsolete version of ~ isin... |
| fineqvlem 9266 | Lemma for ~ fineqv . (Con... |
| fineqv 9267 | If the Axiom of Infinity i... |
| enfiiOLD 9268 | Obsolete version of ~ enfi... |
| pssnnOLD 9269 | Obsolete version of ~ pssn... |
| xpfir 9270 | The components of a nonemp... |
| ssfid 9271 | A subset of a finite set i... |
| infi 9272 | The intersection of two se... |
| rabfi 9273 | A restricted class built f... |
| finresfin 9274 | The restriction of a finit... |
| f1finf1o 9275 | Any injection from one fin... |
| f1finf1oOLD 9276 | Obsolete version of ~ f1fi... |
| nfielex 9277 | If a class is not finite, ... |
| en1eqsn 9278 | A set with one element is ... |
| en1eqsnOLD 9279 | Obsolete version of ~ en1e... |
| en1eqsnbi 9280 | A set containing an elemen... |
| dif1ennnALT 9281 | Alternate proof of ~ dif1e... |
| enp1ilem 9282 | Lemma for uses of ~ enp1i ... |
| enp1i 9283 | Proof induction for ~ en2 ... |
| enp1iOLD 9284 | Obsolete version of ~ enp1... |
| en2 9285 | A set equinumerous to ordi... |
| en3 9286 | A set equinumerous to ordi... |
| en4 9287 | A set equinumerous to ordi... |
| findcard2OLD 9288 | Obsolete version of ~ find... |
| findcard3 9289 | Schema for strong inductio... |
| findcard3OLD 9290 | Obsolete version of ~ find... |
| ac6sfi 9291 | A version of ~ ac6s for fi... |
| frfi 9292 | A partial order is well-fo... |
| fimax2g 9293 | A finite set has a maximum... |
| fimaxg 9294 | A finite set has a maximum... |
| fisupg 9295 | Lemma showing existence an... |
| wofi 9296 | A total order on a finite ... |
| ordunifi 9297 | The maximum of a finite co... |
| nnunifi 9298 | The union (supremum) of a ... |
| unblem1 9299 | Lemma for ~ unbnn . After... |
| unblem2 9300 | Lemma for ~ unbnn . The v... |
| unblem3 9301 | Lemma for ~ unbnn . The v... |
| unblem4 9302 | Lemma for ~ unbnn . The f... |
| unbnn 9303 | Any unbounded subset of na... |
| unbnn2 9304 | Version of ~ unbnn that do... |
| isfinite2 9305 | Any set strictly dominated... |
| nnsdomg 9306 | Omega strictly dominates a... |
| nnsdomgOLD 9307 | Obsolete version of ~ nnsd... |
| isfiniteg 9308 | A set is finite iff it is ... |
| infsdomnn 9309 | An infinite set strictly d... |
| infsdomnnOLD 9310 | Obsolete version of ~ infs... |
| infn0 9311 | An infinite set is not emp... |
| infn0ALT 9312 | Shorter proof of ~ infn0 u... |
| fin2inf 9313 | This (useless) theorem, wh... |
| unfilem1 9314 | Lemma for proving that the... |
| unfilem2 9315 | Lemma for proving that the... |
| unfilem3 9316 | Lemma for proving that the... |
| unfiOLD 9317 | Obsolete version of ~ unfi... |
| unfir 9318 | If a union is finite, the ... |
| unfi2 9319 | The union of two finite se... |
| difinf 9320 | An infinite set ` A ` minu... |
| xpfi 9321 | The Cartesian product of t... |
| xpfiOLD 9322 | Obsolete version of ~ xpfi... |
| 3xpfi 9323 | The Cartesian product of t... |
| domunfican 9324 | A finite set union cancell... |
| infcntss 9325 | Every infinite set has a d... |
| prfi 9326 | An unordered pair is finit... |
| tpfi 9327 | An unordered triple is fin... |
| fiint 9328 | Equivalent ways of stating... |
| fodomfi 9329 | An onto function implies d... |
| fodomfib 9330 | Equivalence of an onto map... |
| fofinf1o 9331 | Any surjection from one fi... |
| rneqdmfinf1o 9332 | Any function from a finite... |
| fidomdm 9333 | Any finite set dominates i... |
| dmfi 9334 | The domain of a finite set... |
| fundmfibi 9335 | A function is finite if an... |
| resfnfinfin 9336 | The restriction of a funct... |
| residfi 9337 | A restricted identity func... |
| cnvfiALT 9338 | Shorter proof of ~ cnvfi u... |
| rnfi 9339 | The range of a finite set ... |
| f1dmvrnfibi 9340 | A one-to-one function whos... |
| f1vrnfibi 9341 | A one-to-one function whic... |
| fofi 9342 | If an onto function has a ... |
| f1fi 9343 | If a 1-to-1 function has a... |
| iunfi 9344 | The finite union of finite... |
| unifi 9345 | The finite union of finite... |
| unifi2 9346 | The finite union of finite... |
| infssuni 9347 | If an infinite set ` A ` i... |
| unirnffid 9348 | The union of the range of ... |
| imafiALT 9349 | Shorter proof of ~ imafi u... |
| pwfilemOLD 9350 | Obsolete version of ~ pwfi... |
| pwfiOLD 9351 | Obsolete version of ~ pwfi... |
| mapfi 9352 | Set exponentiation of fini... |
| ixpfi 9353 | A Cartesian product of fin... |
| ixpfi2 9354 | A Cartesian product of fin... |
| mptfi 9355 | A finite mapping set is fi... |
| abrexfi 9356 | An image set from a finite... |
| cnvimamptfin 9357 | A preimage of a mapping wi... |
| elfpw 9358 | Membership in a class of f... |
| unifpw 9359 | A set is the union of its ... |
| f1opwfi 9360 | A one-to-one mapping induc... |
| fissuni 9361 | A finite subset of a union... |
| fipreima 9362 | Given a finite subset ` A ... |
| finsschain 9363 | A finite subset of the uni... |
| indexfi 9364 | If for every element of a ... |
| relfsupp 9367 | The property of a function... |
| relprcnfsupp 9368 | A proper class is never fi... |
| isfsupp 9369 | The property of a class to... |
| isfsuppd 9370 | Deduction form of ~ isfsup... |
| funisfsupp 9371 | The property of a function... |
| fsuppimp 9372 | Implications of a class be... |
| fsuppimpd 9373 | A finitely supported funct... |
| fisuppfi 9374 | A function on a finite set... |
| fidmfisupp 9375 | A function with a finite d... |
| fdmfisuppfi 9376 | The support of a function ... |
| fdmfifsupp 9377 | A function with a finite d... |
| fsuppmptdm 9378 | A mapping with a finite do... |
| fndmfisuppfi 9379 | The support of a function ... |
| fndmfifsupp 9380 | A function with a finite d... |
| suppeqfsuppbi 9381 | If two functions have the ... |
| suppssfifsupp 9382 | If the support of a functi... |
| fsuppsssupp 9383 | If the support of a functi... |
| fsuppxpfi 9384 | The cartesian product of t... |
| fczfsuppd 9385 | A constant function with v... |
| fsuppun 9386 | The union of two finitely ... |
| fsuppunfi 9387 | The union of the support o... |
| fsuppunbi 9388 | If the union of two classe... |
| 0fsupp 9389 | The empty set is a finitel... |
| snopfsupp 9390 | A singleton containing an ... |
| funsnfsupp 9391 | Finite support for a funct... |
| fsuppres 9392 | The restriction of a finit... |
| fmptssfisupp 9393 | The restriction of a mappi... |
| ressuppfi 9394 | If the support of the rest... |
| resfsupp 9395 | If the restriction of a fu... |
| resfifsupp 9396 | The restriction of a funct... |
| ffsuppbi 9397 | Two ways of saying that a ... |
| fsuppmptif 9398 | A function mapping an argu... |
| sniffsupp 9399 | A function mapping all but... |
| fsuppcolem 9400 | Lemma for ~ fsuppco . For... |
| fsuppco 9401 | The composition of a 1-1 f... |
| fsuppco2 9402 | The composition of a funct... |
| fsuppcor 9403 | The composition of a funct... |
| mapfienlem1 9404 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9405 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9406 | Lemma 3 for ~ mapfien . (... |
| mapfien 9407 | A bijection of the base se... |
| mapfien2 9408 | Equinumerousity relation f... |
| fival 9411 | The set of all the finite ... |
| elfi 9412 | Specific properties of an ... |
| elfi2 9413 | The empty intersection nee... |
| elfir 9414 | Sufficient condition for a... |
| intrnfi 9415 | Sufficient condition for t... |
| iinfi 9416 | An indexed intersection of... |
| inelfi 9417 | The intersection of two se... |
| ssfii 9418 | Any element of a set ` A `... |
| fi0 9419 | The set of finite intersec... |
| fieq0 9420 | A set is empty iff the cla... |
| fiin 9421 | The elements of ` ( fi `` ... |
| dffi2 9422 | The set of finite intersec... |
| fiss 9423 | Subset relationship for fu... |
| inficl 9424 | A set which is closed unde... |
| fipwuni 9425 | The set of finite intersec... |
| fisn 9426 | A singleton is closed unde... |
| fiuni 9427 | The union of the finite in... |
| fipwss 9428 | If a set is a family of su... |
| elfiun 9429 | A finite intersection of e... |
| dffi3 9430 | The set of finite intersec... |
| fifo 9431 | Describe a surjection from... |
| marypha1lem 9432 | Core induction for Philip ... |
| marypha1 9433 | (Philip) Hall's marriage t... |
| marypha2lem1 9434 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9435 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9436 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9437 | Lemma for ~ marypha2 . Pr... |
| marypha2 9438 | Version of ~ marypha1 usin... |
| dfsup2 9443 | Quantifier-free definition... |
| supeq1 9444 | Equality theorem for supre... |
| supeq1d 9445 | Equality deduction for sup... |
| supeq1i 9446 | Equality inference for sup... |
| supeq2 9447 | Equality theorem for supre... |
| supeq3 9448 | Equality theorem for supre... |
| supeq123d 9449 | Equality deduction for sup... |
| nfsup 9450 | Hypothesis builder for sup... |
| supmo 9451 | Any class ` B ` has at mos... |
| supexd 9452 | A supremum is a set. (Con... |
| supeu 9453 | A supremum is unique. Sim... |
| supval2 9454 | Alternate expression for t... |
| eqsup 9455 | Sufficient condition for a... |
| eqsupd 9456 | Sufficient condition for a... |
| supcl 9457 | A supremum belongs to its ... |
| supub 9458 | A supremum is an upper bou... |
| suplub 9459 | A supremum is the least up... |
| suplub2 9460 | Bidirectional form of ~ su... |
| supnub 9461 | An upper bound is not less... |
| supex 9462 | A supremum is a set. (Con... |
| sup00 9463 | The supremum under an empt... |
| sup0riota 9464 | The supremum of an empty s... |
| sup0 9465 | The supremum of an empty s... |
| supmax 9466 | The greatest element of a ... |
| fisup2g 9467 | A finite set satisfies the... |
| fisupcl 9468 | A nonempty finite set cont... |
| supgtoreq 9469 | The supremum of a finite s... |
| suppr 9470 | The supremum of a pair. (... |
| supsn 9471 | The supremum of a singleto... |
| supisolem 9472 | Lemma for ~ supiso . (Con... |
| supisoex 9473 | Lemma for ~ supiso . (Con... |
| supiso 9474 | Image of a supremum under ... |
| infeq1 9475 | Equality theorem for infim... |
| infeq1d 9476 | Equality deduction for inf... |
| infeq1i 9477 | Equality inference for inf... |
| infeq2 9478 | Equality theorem for infim... |
| infeq3 9479 | Equality theorem for infim... |
| infeq123d 9480 | Equality deduction for inf... |
| nfinf 9481 | Hypothesis builder for inf... |
| infexd 9482 | An infimum is a set. (Con... |
| eqinf 9483 | Sufficient condition for a... |
| eqinfd 9484 | Sufficient condition for a... |
| infval 9485 | Alternate expression for t... |
| infcllem 9486 | Lemma for ~ infcl , ~ infl... |
| infcl 9487 | An infimum belongs to its ... |
| inflb 9488 | An infimum is a lower boun... |
| infglb 9489 | An infimum is the greatest... |
| infglbb 9490 | Bidirectional form of ~ in... |
| infnlb 9491 | A lower bound is not great... |
| infex 9492 | An infimum is a set. (Con... |
| infmin 9493 | The smallest element of a ... |
| infmo 9494 | Any class ` B ` has at mos... |
| infeu 9495 | An infimum is unique. (Co... |
| fimin2g 9496 | A finite set has a minimum... |
| fiming 9497 | A finite set has a minimum... |
| fiinfg 9498 | Lemma showing existence an... |
| fiinf2g 9499 | A finite set satisfies the... |
| fiinfcl 9500 | A nonempty finite set cont... |
| infltoreq 9501 | The infimum of a finite se... |
| infpr 9502 | The infimum of a pair. (C... |
| infsupprpr 9503 | The infimum of a proper pa... |
| infsn 9504 | The infimum of a singleton... |
| inf00 9505 | The infimum regarding an e... |
| infempty 9506 | The infimum of an empty se... |
| infiso 9507 | Image of an infimum under ... |
| dfoi 9510 | Rewrite ~ df-oi with abbre... |
| oieq1 9511 | Equality theorem for ordin... |
| oieq2 9512 | Equality theorem for ordin... |
| nfoi 9513 | Hypothesis builder for ord... |
| ordiso2 9514 | Generalize ~ ordiso to pro... |
| ordiso 9515 | Order-isomorphic ordinal n... |
| ordtypecbv 9516 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9517 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9518 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9519 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9520 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9521 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9522 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9523 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9524 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9525 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9526 | Lemma for ~ ordtype . Usi... |
| oi0 9527 | Definition of the ordinal ... |
| oicl 9528 | The order type of the well... |
| oif 9529 | The order isomorphism of t... |
| oiiso2 9530 | The order isomorphism of t... |
| ordtype 9531 | For any set-like well-orde... |
| oiiniseg 9532 | ` ran F ` is an initial se... |
| ordtype2 9533 | For any set-like well-orde... |
| oiexg 9534 | The order isomorphism on a... |
| oion 9535 | The order type of the well... |
| oiiso 9536 | The order isomorphism of t... |
| oien 9537 | The order type of a well-o... |
| oieu 9538 | Uniqueness of the unique o... |
| oismo 9539 | When ` A ` is a subclass o... |
| oiid 9540 | The order type of an ordin... |
| hartogslem1 9541 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9542 | Lemma for ~ hartogs . (Co... |
| hartogs 9543 | The class of ordinals domi... |
| wofib 9544 | The only sets which are we... |
| wemaplem1 9545 | Value of the lexicographic... |
| wemaplem2 9546 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9547 | Lemma for ~ wemapso . Tra... |
| wemappo 9548 | Construct lexicographic or... |
| wemapsolem 9549 | Lemma for ~ wemapso . (Co... |
| wemapso 9550 | Construct lexicographic or... |
| wemapso2lem 9551 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9552 | An alternative to having a... |
| card2on 9553 | The alternate definition o... |
| card2inf 9554 | The alternate definition o... |
| harf 9557 | Functionality of the Harto... |
| harcl 9558 | Values of the Hartogs func... |
| harval 9559 | Function value of the Hart... |
| elharval 9560 | The Hartogs number of a se... |
| harndom 9561 | The Hartogs number of a se... |
| harword 9562 | Weak ordering property of ... |
| relwdom 9565 | Weak dominance is a relati... |
| brwdom 9566 | Property of weak dominance... |
| brwdomi 9567 | Property of weak dominance... |
| brwdomn0 9568 | Weak dominance over nonemp... |
| 0wdom 9569 | Any set weakly dominates t... |
| fowdom 9570 | An onto function implies w... |
| wdomref 9571 | Reflexivity of weak domina... |
| brwdom2 9572 | Alternate characterization... |
| domwdom 9573 | Weak dominance is implied ... |
| wdomtr 9574 | Transitivity of weak domin... |
| wdomen1 9575 | Equality-like theorem for ... |
| wdomen2 9576 | Equality-like theorem for ... |
| wdompwdom 9577 | Weak dominance strengthens... |
| canthwdom 9578 | Cantor's Theorem, stated u... |
| wdom2d 9579 | Deduce weak dominance from... |
| wdomd 9580 | Deduce weak dominance from... |
| brwdom3 9581 | Condition for weak dominan... |
| brwdom3i 9582 | Weak dominance implies exi... |
| unwdomg 9583 | Weak dominance of a (disjo... |
| xpwdomg 9584 | Weak dominance of a Cartes... |
| wdomima2g 9585 | A set is weakly dominant o... |
| wdomimag 9586 | A set is weakly dominant o... |
| unxpwdom2 9587 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9588 | If a Cartesian product is ... |
| ixpiunwdom 9589 | Describe an onto function ... |
| harwdom 9590 | The value of the Hartogs f... |
| axreg2 9592 | Axiom of Regularity expres... |
| zfregcl 9593 | The Axiom of Regularity wi... |
| zfreg 9594 | The Axiom of Regularity us... |
| elirrv 9595 | The membership relation is... |
| elirr 9596 | No class is a member of it... |
| elneq 9597 | A class is not equal to an... |
| nelaneq 9598 | A class is not an element ... |
| epinid0 9599 | The membership relation an... |
| sucprcreg 9600 | A class is equal to its su... |
| ruv 9601 | The Russell class is equal... |
| ruALT 9602 | Alternate proof of ~ ru , ... |
| disjcsn 9603 | A class is disjoint from i... |
| zfregfr 9604 | The membership relation is... |
| en2lp 9605 | No class has 2-cycle membe... |
| elnanel 9606 | Two classes are not elemen... |
| cnvepnep 9607 | The membership (epsilon) r... |
| epnsym 9608 | The membership (epsilon) r... |
| elnotel 9609 | A class cannot be an eleme... |
| elnel 9610 | A class cannot be an eleme... |
| en3lplem1 9611 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9612 | Lemma for ~ en3lp . (Cont... |
| en3lp 9613 | No class has 3-cycle membe... |
| preleqg 9614 | Equality of two unordered ... |
| preleq 9615 | Equality of two unordered ... |
| preleqALT 9616 | Alternate proof of ~ prele... |
| opthreg 9617 | Theorem for alternate repr... |
| suc11reg 9618 | The successor operation be... |
| dford2 9619 | Assuming ~ ax-reg , an ord... |
| inf0 9620 | Existence of ` _om ` impli... |
| inf1 9621 | Variation of Axiom of Infi... |
| inf2 9622 | Variation of Axiom of Infi... |
| inf3lema 9623 | Lemma for our Axiom of Inf... |
| inf3lemb 9624 | Lemma for our Axiom of Inf... |
| inf3lemc 9625 | Lemma for our Axiom of Inf... |
| inf3lemd 9626 | Lemma for our Axiom of Inf... |
| inf3lem1 9627 | Lemma for our Axiom of Inf... |
| inf3lem2 9628 | Lemma for our Axiom of Inf... |
| inf3lem3 9629 | Lemma for our Axiom of Inf... |
| inf3lem4 9630 | Lemma for our Axiom of Inf... |
| inf3lem5 9631 | Lemma for our Axiom of Inf... |
| inf3lem6 9632 | Lemma for our Axiom of Inf... |
| inf3lem7 9633 | Lemma for our Axiom of Inf... |
| inf3 9634 | Our Axiom of Infinity ~ ax... |
| infeq5i 9635 | Half of ~ infeq5 . (Contr... |
| infeq5 9636 | The statement "there exist... |
| zfinf 9638 | Axiom of Infinity expresse... |
| axinf2 9639 | A standard version of Axio... |
| zfinf2 9641 | A standard version of the ... |
| omex 9642 | The existence of omega (th... |
| axinf 9643 | The first version of the A... |
| inf5 9644 | The statement "there exist... |
| omelon 9645 | Omega is an ordinal number... |
| dfom3 9646 | The class of natural numbe... |
| elom3 9647 | A simplification of ~ elom... |
| dfom4 9648 | A simplification of ~ df-o... |
| dfom5 9649 | ` _om ` is the smallest li... |
| oancom 9650 | Ordinal addition is not co... |
| isfinite 9651 | A set is finite iff it is ... |
| fict 9652 | A finite set is countable ... |
| nnsdom 9653 | A natural number is strict... |
| omenps 9654 | Omega is equinumerous to a... |
| omensuc 9655 | The set of natural numbers... |
| infdifsn 9656 | Removing a singleton from ... |
| infdiffi 9657 | Removing a finite set from... |
| unbnn3 9658 | Any unbounded subset of na... |
| noinfep 9659 | Using the Axiom of Regular... |
| cantnffval 9662 | The value of the Cantor no... |
| cantnfdm 9663 | The domain of the Cantor n... |
| cantnfvalf 9664 | Lemma for ~ cantnf . The ... |
| cantnfs 9665 | Elementhood in the set of ... |
| cantnfcl 9666 | Basic properties of the or... |
| cantnfval 9667 | The value of the Cantor no... |
| cantnfval2 9668 | Alternate expression for t... |
| cantnfsuc 9669 | The value of the recursive... |
| cantnfle 9670 | A lower bound on the ` CNF... |
| cantnflt 9671 | An upper bound on the part... |
| cantnflt2 9672 | An upper bound on the ` CN... |
| cantnff 9673 | The ` CNF ` function is a ... |
| cantnf0 9674 | The value of the zero func... |
| cantnfrescl 9675 | A function is finitely sup... |
| cantnfres 9676 | The ` CNF ` function respe... |
| cantnfp1lem1 9677 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9678 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9679 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9680 | If ` F ` is created by add... |
| oemapso 9681 | The relation ` T ` is a st... |
| oemapval 9682 | Value of the relation ` T ... |
| oemapvali 9683 | If ` F < G ` , then there ... |
| cantnflem1a 9684 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9685 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9686 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9687 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9688 | Lemma for ~ cantnf . This... |
| cantnflem2 9689 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9690 | Lemma for ~ cantnf . Here... |
| cantnflem4 9691 | Lemma for ~ cantnf . Comp... |
| cantnf 9692 | The Cantor Normal Form the... |
| oemapwe 9693 | The lexicographic order on... |
| cantnffval2 9694 | An alternate definition of... |
| cantnff1o 9695 | Simplify the isomorphism o... |
| wemapwe 9696 | Construct lexicographic or... |
| oef1o 9697 | A bijection of the base se... |
| cnfcomlem 9698 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9699 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9700 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9701 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9702 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9703 | Any infinite ordinal ` B `... |
| cnfcom3clem 9704 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9705 | Wrap the construction of ~... |
| ttrcleq 9708 | Equality theorem for trans... |
| nfttrcld 9709 | Bound variable hypothesis ... |
| nfttrcl 9710 | Bound variable hypothesis ... |
| relttrcl 9711 | The transitive closure of ... |
| brttrcl 9712 | Characterization of elemen... |
| brttrcl2 9713 | Characterization of elemen... |
| ssttrcl 9714 | If ` R ` is a relation, th... |
| ttrcltr 9715 | The transitive closure of ... |
| ttrclresv 9716 | The transitive closure of ... |
| ttrclco 9717 | Composition law for the tr... |
| cottrcl 9718 | Composition law for the tr... |
| ttrclss 9719 | If ` R ` is a subclass of ... |
| dmttrcl 9720 | The domain of a transitive... |
| rnttrcl 9721 | The range of a transitive ... |
| ttrclexg 9722 | If ` R ` is a set, then so... |
| dfttrcl2 9723 | When ` R ` is a set and a ... |
| ttrclselem1 9724 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9725 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9726 | If ` R ` is set-like over ... |
| trcl 9727 | For any set ` A ` , show t... |
| tz9.1 9728 | Every set has a transitive... |
| tz9.1c 9729 | Alternate expression for t... |
| epfrs 9730 | The strong form of the Axi... |
| zfregs 9731 | The strong form of the Axi... |
| zfregs2 9732 | Alternate strong form of t... |
| setind 9733 | Set (epsilon) induction. ... |
| setind2 9734 | Set (epsilon) induction, s... |
| tcvalg 9737 | Value of the transitive cl... |
| tcid 9738 | Defining property of the t... |
| tctr 9739 | Defining property of the t... |
| tcmin 9740 | Defining property of the t... |
| tc2 9741 | A variant of the definitio... |
| tcsni 9742 | The transitive closure of ... |
| tcss 9743 | The transitive closure fun... |
| tcel 9744 | The transitive closure fun... |
| tcidm 9745 | The transitive closure fun... |
| tc0 9746 | The transitive closure of ... |
| tc00 9747 | The transitive closure is ... |
| frmin 9748 | Every (possibly proper) su... |
| frind 9749 | A subclass of a well-found... |
| frinsg 9750 | Well-Founded Induction Sch... |
| frins 9751 | Well-Founded Induction Sch... |
| frins2f 9752 | Well-Founded Induction sch... |
| frins2 9753 | Well-Founded Induction sch... |
| frins3 9754 | Well-Founded Induction sch... |
| frr3g 9755 | Functions defined by well-... |
| frrlem15 9756 | Lemma for general well-fou... |
| frrlem16 9757 | Lemma for general well-fou... |
| frr1 9758 | Law of general well-founde... |
| frr2 9759 | Law of general well-founde... |
| frr3 9760 | Law of general well-founde... |
| r1funlim 9765 | The cumulative hierarchy o... |
| r1fnon 9766 | The cumulative hierarchy o... |
| r10 9767 | Value of the cumulative hi... |
| r1sucg 9768 | Value of the cumulative hi... |
| r1suc 9769 | Value of the cumulative hi... |
| r1limg 9770 | Value of the cumulative hi... |
| r1lim 9771 | Value of the cumulative hi... |
| r1fin 9772 | The first ` _om ` levels o... |
| r1sdom 9773 | Each stage in the cumulati... |
| r111 9774 | The cumulative hierarchy i... |
| r1tr 9775 | The cumulative hierarchy o... |
| r1tr2 9776 | The union of a cumulative ... |
| r1ordg 9777 | Ordering relation for the ... |
| r1ord3g 9778 | Ordering relation for the ... |
| r1ord 9779 | Ordering relation for the ... |
| r1ord2 9780 | Ordering relation for the ... |
| r1ord3 9781 | Ordering relation for the ... |
| r1sssuc 9782 | The value of the cumulativ... |
| r1pwss 9783 | Each set of the cumulative... |
| r1sscl 9784 | Each set of the cumulative... |
| r1val1 9785 | The value of the cumulativ... |
| tz9.12lem1 9786 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9787 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9788 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9789 | A set is well-founded if a... |
| tz9.13 9790 | Every set is well-founded,... |
| tz9.13g 9791 | Every set is well-founded,... |
| rankwflemb 9792 | Two ways of saying a set i... |
| rankf 9793 | The domain and codomain of... |
| rankon 9794 | The rank of a set is an or... |
| r1elwf 9795 | Any member of the cumulati... |
| rankvalb 9796 | Value of the rank function... |
| rankr1ai 9797 | One direction of ~ rankr1a... |
| rankvaln 9798 | Value of the rank function... |
| rankidb 9799 | Identity law for the rank ... |
| rankdmr1 9800 | A rank is a member of the ... |
| rankr1ag 9801 | A version of ~ rankr1a tha... |
| rankr1bg 9802 | A relationship between ran... |
| r1rankidb 9803 | Any set is a subset of the... |
| r1elssi 9804 | The range of the ` R1 ` fu... |
| r1elss 9805 | The range of the ` R1 ` fu... |
| pwwf 9806 | A power set is well-founde... |
| sswf 9807 | A subset of a well-founded... |
| snwf 9808 | A singleton is well-founde... |
| unwf 9809 | A binary union is well-fou... |
| prwf 9810 | An unordered pair is well-... |
| opwf 9811 | An ordered pair is well-fo... |
| unir1 9812 | The cumulative hierarchy o... |
| jech9.3 9813 | Every set belongs to some ... |
| rankwflem 9814 | Every set is well-founded,... |
| rankval 9815 | Value of the rank function... |
| rankvalg 9816 | Value of the rank function... |
| rankval2 9817 | Value of an alternate defi... |
| uniwf 9818 | A union is well-founded if... |
| rankr1clem 9819 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9820 | A relationship between the... |
| rankidn 9821 | A relationship between the... |
| rankpwi 9822 | The rank of a power set. ... |
| rankelb 9823 | The membership relation is... |
| wfelirr 9824 | A well-founded set is not ... |
| rankval3b 9825 | The value of the rank func... |
| ranksnb 9826 | The rank of a singleton. ... |
| rankonidlem 9827 | Lemma for ~ rankonid . (C... |
| rankonid 9828 | The rank of an ordinal num... |
| onwf 9829 | The ordinals are all well-... |
| onssr1 9830 | Initial segments of the or... |
| rankr1g 9831 | A relationship between the... |
| rankid 9832 | Identity law for the rank ... |
| rankr1 9833 | A relationship between the... |
| ssrankr1 9834 | A relationship between an ... |
| rankr1a 9835 | A relationship between ran... |
| r1val2 9836 | The value of the cumulativ... |
| r1val3 9837 | The value of the cumulativ... |
| rankel 9838 | The membership relation is... |
| rankval3 9839 | The value of the rank func... |
| bndrank 9840 | Any class whose elements h... |
| unbndrank 9841 | The elements of a proper c... |
| rankpw 9842 | The rank of a power set. ... |
| ranklim 9843 | The rank of a set belongs ... |
| r1pw 9844 | A stronger property of ` R... |
| r1pwALT 9845 | Alternate shorter proof of... |
| r1pwcl 9846 | The cumulative hierarchy o... |
| rankssb 9847 | The subset relation is inh... |
| rankss 9848 | The subset relation is inh... |
| rankunb 9849 | The rank of the union of t... |
| rankprb 9850 | The rank of an unordered p... |
| rankopb 9851 | The rank of an ordered pai... |
| rankuni2b 9852 | The value of the rank func... |
| ranksn 9853 | The rank of a singleton. ... |
| rankuni2 9854 | The rank of a union. Part... |
| rankun 9855 | The rank of the union of t... |
| rankpr 9856 | The rank of an unordered p... |
| rankop 9857 | The rank of an ordered pai... |
| r1rankid 9858 | Any set is a subset of the... |
| rankeq0b 9859 | A set is empty iff its ran... |
| rankeq0 9860 | A set is empty iff its ran... |
| rankr1id 9861 | The rank of the hierarchy ... |
| rankuni 9862 | The rank of a union. Part... |
| rankr1b 9863 | A relationship between ran... |
| ranksuc 9864 | The rank of a successor. ... |
| rankuniss 9865 | Upper bound of the rank of... |
| rankval4 9866 | The rank of a set is the s... |
| rankbnd 9867 | The rank of a set is bound... |
| rankbnd2 9868 | The rank of a set is bound... |
| rankc1 9869 | A relationship that can be... |
| rankc2 9870 | A relationship that can be... |
| rankelun 9871 | Rank membership is inherit... |
| rankelpr 9872 | Rank membership is inherit... |
| rankelop 9873 | Rank membership is inherit... |
| rankxpl 9874 | A lower bound on the rank ... |
| rankxpu 9875 | An upper bound on the rank... |
| rankfu 9876 | An upper bound on the rank... |
| rankmapu 9877 | An upper bound on the rank... |
| rankxplim 9878 | The rank of a Cartesian pr... |
| rankxplim2 9879 | If the rank of a Cartesian... |
| rankxplim3 9880 | The rank of a Cartesian pr... |
| rankxpsuc 9881 | The rank of a Cartesian pr... |
| tcwf 9882 | The transitive closure fun... |
| tcrank 9883 | This theorem expresses two... |
| scottex 9884 | Scott's trick collects all... |
| scott0 9885 | Scott's trick collects all... |
| scottexs 9886 | Theorem scheme version of ... |
| scott0s 9887 | Theorem scheme version of ... |
| cplem1 9888 | Lemma for the Collection P... |
| cplem2 9889 | Lemma for the Collection P... |
| cp 9890 | Collection Principle. Thi... |
| bnd 9891 | A very strong generalizati... |
| bnd2 9892 | A variant of the Boundedne... |
| kardex 9893 | The collection of all sets... |
| karden 9894 | If we allow the Axiom of R... |
| htalem 9895 | Lemma for defining an emul... |
| hta 9896 | A ZFC emulation of Hilbert... |
| djueq12 9903 | Equality theorem for disjo... |
| djueq1 9904 | Equality theorem for disjo... |
| djueq2 9905 | Equality theorem for disjo... |
| nfdju 9906 | Bound-variable hypothesis ... |
| djuex 9907 | The disjoint union of sets... |
| djuexb 9908 | The disjoint union of two ... |
| djulcl 9909 | Left closure of disjoint u... |
| djurcl 9910 | Right closure of disjoint ... |
| djulf1o 9911 | The left injection functio... |
| djurf1o 9912 | The right injection functi... |
| inlresf 9913 | The left injection restric... |
| inlresf1 9914 | The left injection restric... |
| inrresf 9915 | The right injection restri... |
| inrresf1 9916 | The right injection restri... |
| djuin 9917 | The images of any classes ... |
| djur 9918 | A member of a disjoint uni... |
| djuss 9919 | A disjoint union is a subc... |
| djuunxp 9920 | The union of a disjoint un... |
| djuexALT 9921 | Alternate proof of ~ djuex... |
| eldju1st 9922 | The first component of an ... |
| eldju2ndl 9923 | The second component of an... |
| eldju2ndr 9924 | The second component of an... |
| djuun 9925 | The disjoint union of two ... |
| 1stinl 9926 | The first component of the... |
| 2ndinl 9927 | The second component of th... |
| 1stinr 9928 | The first component of the... |
| 2ndinr 9929 | The second component of th... |
| updjudhf 9930 | The mapping of an element ... |
| updjudhcoinlf 9931 | The composition of the map... |
| updjudhcoinrg 9932 | The composition of the map... |
| updjud 9933 | Universal property of the ... |
| cardf2 9942 | The cardinality function i... |
| cardon 9943 | The cardinal number of a s... |
| isnum2 9944 | A way to express well-orde... |
| isnumi 9945 | A set equinumerous to an o... |
| ennum 9946 | Equinumerous sets are equi... |
| finnum 9947 | Every finite set is numera... |
| onenon 9948 | Every ordinal number is nu... |
| tskwe 9949 | A Tarski set is well-order... |
| xpnum 9950 | The cartesian product of n... |
| cardval3 9951 | An alternate definition of... |
| cardid2 9952 | Any numerable set is equin... |
| isnum3 9953 | A set is numerable iff it ... |
| oncardval 9954 | The value of the cardinal ... |
| oncardid 9955 | Any ordinal number is equi... |
| cardonle 9956 | The cardinal of an ordinal... |
| card0 9957 | The cardinality of the emp... |
| cardidm 9958 | The cardinality function i... |
| oncard 9959 | A set is a cardinal number... |
| ficardom 9960 | The cardinal number of a f... |
| ficardid 9961 | A finite set is equinumero... |
| cardnn 9962 | The cardinality of a natur... |
| cardnueq0 9963 | The empty set is the only ... |
| cardne 9964 | No member of a cardinal nu... |
| carden2a 9965 | If two sets have equal non... |
| carden2b 9966 | If two sets are equinumero... |
| card1 9967 | A set has cardinality one ... |
| cardsn 9968 | A singleton has cardinalit... |
| carddomi2 9969 | Two sets have the dominanc... |
| sdomsdomcardi 9970 | A set strictly dominates i... |
| cardlim 9971 | An infinite cardinal is a ... |
| cardsdomelir 9972 | A cardinal strictly domina... |
| cardsdomel 9973 | A cardinal strictly domina... |
| iscard 9974 | Two ways to express the pr... |
| iscard2 9975 | Two ways to express the pr... |
| carddom2 9976 | Two numerable sets have th... |
| harcard 9977 | The class of ordinal numbe... |
| cardprclem 9978 | Lemma for ~ cardprc . (Co... |
| cardprc 9979 | The class of all cardinal ... |
| carduni 9980 | The union of a set of card... |
| cardiun 9981 | The indexed union of a set... |
| cardennn 9982 | If ` A ` is equinumerous t... |
| cardsucinf 9983 | The cardinality of the suc... |
| cardsucnn 9984 | The cardinality of the suc... |
| cardom 9985 | The set of natural numbers... |
| carden2 9986 | Two numerable sets are equ... |
| cardsdom2 9987 | A numerable set is strictl... |
| domtri2 9988 | Trichotomy of dominance fo... |
| nnsdomel 9989 | Strict dominance and eleme... |
| cardval2 9990 | An alternate version of th... |
| isinffi 9991 | An infinite set contains s... |
| fidomtri 9992 | Trichotomy of dominance wi... |
| fidomtri2 9993 | Trichotomy of dominance wi... |
| harsdom 9994 | The Hartogs number of a we... |
| onsdom 9995 | Any well-orderable set is ... |
| harval2 9996 | An alternate expression fo... |
| harsucnn 9997 | The next cardinal after a ... |
| cardmin2 9998 | The smallest ordinal that ... |
| pm54.43lem 9999 | In Theorem *54.43 of [Whit... |
| pm54.43 10000 | Theorem *54.43 of [Whitehe... |
| enpr2 10001 | An unordered pair with dis... |
| pr2nelemOLD 10002 | Obsolete version of ~ enpr... |
| pr2ne 10003 | If an unordered pair has t... |
| pr2neOLD 10004 | Obsolete version of ~ pr2n... |
| prdom2 10005 | An unordered pair has at m... |
| en2eqpr 10006 | Building a set with two el... |
| en2eleq 10007 | Express a set of pair card... |
| en2other2 10008 | Taking the other element t... |
| dif1card 10009 | The cardinality of a nonem... |
| leweon 10010 | Lexicographical order is a... |
| r0weon 10011 | A set-like well-ordering o... |
| infxpenlem 10012 | Lemma for ~ infxpen . (Co... |
| infxpen 10013 | Every infinite ordinal is ... |
| xpomen 10014 | The Cartesian product of o... |
| xpct 10015 | The cartesian product of t... |
| infxpidm2 10016 | Every infinite well-ordera... |
| infxpenc 10017 | A canonical version of ~ i... |
| infxpenc2lem1 10018 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 10019 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 10020 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 10021 | Existence form of ~ infxpe... |
| iunmapdisj 10022 | The union ` U_ n e. C ( A ... |
| fseqenlem1 10023 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 10024 | Lemma for ~ fseqen . (Con... |
| fseqdom 10025 | One half of ~ fseqen . (C... |
| fseqen 10026 | A set that is equinumerous... |
| infpwfidom 10027 | The collection of finite s... |
| dfac8alem 10028 | Lemma for ~ dfac8a . If t... |
| dfac8a 10029 | Numeration theorem: every ... |
| dfac8b 10030 | The well-ordering theorem:... |
| dfac8clem 10031 | Lemma for ~ dfac8c . (Con... |
| dfac8c 10032 | If the union of a set is w... |
| ac10ct 10033 | A proof of the well-orderi... |
| ween 10034 | A set is numerable iff it ... |
| ac5num 10035 | A version of ~ ac5b with t... |
| ondomen 10036 | If a set is dominated by a... |
| numdom 10037 | A set dominated by a numer... |
| ssnum 10038 | A subset of a numerable se... |
| onssnum 10039 | All subsets of the ordinal... |
| indcardi 10040 | Indirect strong induction ... |
| acnrcl 10041 | Reverse closure for the ch... |
| acneq 10042 | Equality theorem for the c... |
| isacn 10043 | The property of being a ch... |
| acni 10044 | The property of being a ch... |
| acni2 10045 | The property of being a ch... |
| acni3 10046 | The property of being a ch... |
| acnlem 10047 | Construct a mapping satisf... |
| numacn 10048 | A well-orderable set has c... |
| finacn 10049 | Every set has finite choic... |
| acndom 10050 | A set with long choice seq... |
| acnnum 10051 | A set ` X ` which has choi... |
| acnen 10052 | The class of choice sets o... |
| acndom2 10053 | A set smaller than one wit... |
| acnen2 10054 | The class of sets with cho... |
| fodomacn 10055 | A version of ~ fodom that ... |
| fodomnum 10056 | A version of ~ fodom that ... |
| fonum 10057 | A surjection maps numerabl... |
| numwdom 10058 | A surjection maps numerabl... |
| fodomfi2 10059 | Onto functions define domi... |
| wdomfil 10060 | Weak dominance agrees with... |
| infpwfien 10061 | Any infinite well-orderabl... |
| inffien 10062 | The set of finite intersec... |
| wdomnumr 10063 | Weak dominance agrees with... |
| alephfnon 10064 | The aleph function is a fu... |
| aleph0 10065 | The first infinite cardina... |
| alephlim 10066 | Value of the aleph functio... |
| alephsuc 10067 | Value of the aleph functio... |
| alephon 10068 | An aleph is an ordinal num... |
| alephcard 10069 | Every aleph is a cardinal ... |
| alephnbtwn 10070 | No cardinal can be sandwic... |
| alephnbtwn2 10071 | No set has equinumerosity ... |
| alephordilem1 10072 | Lemma for ~ alephordi . (... |
| alephordi 10073 | Strict ordering property o... |
| alephord 10074 | Ordering property of the a... |
| alephord2 10075 | Ordering property of the a... |
| alephord2i 10076 | Ordering property of the a... |
| alephord3 10077 | Ordering property of the a... |
| alephsucdom 10078 | A set dominated by an alep... |
| alephsuc2 10079 | An alternate representatio... |
| alephdom 10080 | Relationship between inclu... |
| alephgeom 10081 | Every aleph is greater tha... |
| alephislim 10082 | Every aleph is a limit ord... |
| aleph11 10083 | The aleph function is one-... |
| alephf1 10084 | The aleph function is a on... |
| alephsdom 10085 | If an ordinal is smaller t... |
| alephdom2 10086 | A dominated initial ordina... |
| alephle 10087 | The argument of the aleph ... |
| cardaleph 10088 | Given any transfinite card... |
| cardalephex 10089 | Every transfinite cardinal... |
| infenaleph 10090 | An infinite numerable set ... |
| isinfcard 10091 | Two ways to express the pr... |
| iscard3 10092 | Two ways to express the pr... |
| cardnum 10093 | Two ways to express the cl... |
| alephinit 10094 | An infinite initial ordina... |
| carduniima 10095 | The union of the image of ... |
| cardinfima 10096 | If a mapping to cardinals ... |
| alephiso 10097 | Aleph is an order isomorph... |
| alephprc 10098 | The class of all transfini... |
| alephsson 10099 | The class of transfinite c... |
| unialeph 10100 | The union of the class of ... |
| alephsmo 10101 | The aleph function is stri... |
| alephf1ALT 10102 | Alternate proof of ~ aleph... |
| alephfplem1 10103 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10104 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10105 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10106 | Lemma for ~ alephfp . (Co... |
| alephfp 10107 | The aleph function has a f... |
| alephfp2 10108 | The aleph function has at ... |
| alephval3 10109 | An alternate way to expres... |
| alephsucpw2 10110 | The power set of an aleph ... |
| mappwen 10111 | Power rule for cardinal ar... |
| finnisoeu 10112 | A finite totally ordered s... |
| iunfictbso 10113 | Countability of a countabl... |
| aceq1 10116 | Equivalence of two version... |
| aceq0 10117 | Equivalence of two version... |
| aceq2 10118 | Equivalence of two version... |
| aceq3lem 10119 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10120 | Equivalence of two version... |
| dfac4 10121 | Equivalence of two version... |
| dfac5lem1 10122 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10123 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10124 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10125 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10126 | Lemma for ~ dfac5 . (Cont... |
| dfac5 10127 | Equivalence of two version... |
| dfac2a 10128 | Our Axiom of Choice (in th... |
| dfac2b 10129 | Axiom of Choice (first for... |
| dfac2 10130 | Axiom of Choice (first for... |
| dfac7 10131 | Equivalence of the Axiom o... |
| dfac0 10132 | Equivalence of two version... |
| dfac1 10133 | Equivalence of two version... |
| dfac8 10134 | A proof of the equivalency... |
| dfac9 10135 | Equivalence of the axiom o... |
| dfac10 10136 | Axiom of Choice equivalent... |
| dfac10c 10137 | Axiom of Choice equivalent... |
| dfac10b 10138 | Axiom of Choice equivalent... |
| acacni 10139 | A choice equivalent: every... |
| dfacacn 10140 | A choice equivalent: every... |
| dfac13 10141 | The axiom of choice holds ... |
| dfac12lem1 10142 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10143 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10144 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10145 | The axiom of choice holds ... |
| dfac12k 10146 | Equivalence of ~ dfac12 an... |
| dfac12a 10147 | The axiom of choice holds ... |
| dfac12 10148 | The axiom of choice holds ... |
| kmlem1 10149 | Lemma for 5-quantifier AC ... |
| kmlem2 10150 | Lemma for 5-quantifier AC ... |
| kmlem3 10151 | Lemma for 5-quantifier AC ... |
| kmlem4 10152 | Lemma for 5-quantifier AC ... |
| kmlem5 10153 | Lemma for 5-quantifier AC ... |
| kmlem6 10154 | Lemma for 5-quantifier AC ... |
| kmlem7 10155 | Lemma for 5-quantifier AC ... |
| kmlem8 10156 | Lemma for 5-quantifier AC ... |
| kmlem9 10157 | Lemma for 5-quantifier AC ... |
| kmlem10 10158 | Lemma for 5-quantifier AC ... |
| kmlem11 10159 | Lemma for 5-quantifier AC ... |
| kmlem12 10160 | Lemma for 5-quantifier AC ... |
| kmlem13 10161 | Lemma for 5-quantifier AC ... |
| kmlem14 10162 | Lemma for 5-quantifier AC ... |
| kmlem15 10163 | Lemma for 5-quantifier AC ... |
| kmlem16 10164 | Lemma for 5-quantifier AC ... |
| dfackm 10165 | Equivalence of the Axiom o... |
| undjudom 10166 | Cardinal addition dominate... |
| endjudisj 10167 | Equinumerosity of a disjoi... |
| djuen 10168 | Disjoint unions of equinum... |
| djuenun 10169 | Disjoint union is equinume... |
| dju1en 10170 | Cardinal addition with car... |
| dju1dif 10171 | Adding and subtracting one... |
| dju1p1e2 10172 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10173 | Alternate proof of ~ dju1p... |
| dju0en 10174 | Cardinal addition with car... |
| xp2dju 10175 | Two times a cardinal numbe... |
| djucomen 10176 | Commutative law for cardin... |
| djuassen 10177 | Associative law for cardin... |
| xpdjuen 10178 | Cardinal multiplication di... |
| mapdjuen 10179 | Sum of exponents law for c... |
| pwdjuen 10180 | Sum of exponents law for c... |
| djudom1 10181 | Ordering law for cardinal ... |
| djudom2 10182 | Ordering law for cardinal ... |
| djudoml 10183 | A set is dominated by its ... |
| djuxpdom 10184 | Cartesian product dominate... |
| djufi 10185 | The disjoint union of two ... |
| cdainflem 10186 | Any partition of omega int... |
| djuinf 10187 | A set is infinite iff the ... |
| infdju1 10188 | An infinite set is equinum... |
| pwdju1 10189 | The sum of a powerset with... |
| pwdjuidm 10190 | If the natural numbers inj... |
| djulepw 10191 | If ` A ` is idempotent und... |
| onadju 10192 | The cardinal and ordinal s... |
| cardadju 10193 | The cardinal sum is equinu... |
| djunum 10194 | The disjoint union of two ... |
| unnum 10195 | The union of two numerable... |
| nnadju 10196 | The cardinal and ordinal s... |
| nnadjuALT 10197 | Shorter proof of ~ nnadju ... |
| ficardadju 10198 | The disjoint union of fini... |
| ficardun 10199 | The cardinality of the uni... |
| ficardunOLD 10200 | Obsolete version of ~ fica... |
| ficardun2 10201 | The cardinality of the uni... |
| ficardun2OLD 10202 | Obsolete version of ~ fica... |
| pwsdompw 10203 | Lemma for ~ domtriom . Th... |
| unctb 10204 | The union of two countable... |
| infdjuabs 10205 | Absorption law for additio... |
| infunabs 10206 | An infinite set is equinum... |
| infdju 10207 | The sum of two cardinal nu... |
| infdif 10208 | The cardinality of an infi... |
| infdif2 10209 | Cardinality ordering for a... |
| infxpdom 10210 | Dominance law for multipli... |
| infxpabs 10211 | Absorption law for multipl... |
| infunsdom1 10212 | The union of two sets that... |
| infunsdom 10213 | The union of two sets that... |
| infxp 10214 | Absorption law for multipl... |
| pwdjudom 10215 | A property of dominance ov... |
| infpss 10216 | Every infinite set has an ... |
| infmap2 10217 | An exponentiation law for ... |
| ackbij2lem1 10218 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10219 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10220 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10221 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10222 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10223 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10224 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10225 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10226 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10227 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10228 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10229 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10230 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10231 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10232 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10233 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10234 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10235 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10236 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10237 | The Ackermann bijection, p... |
| ackbij1b 10238 | The Ackermann bijection, p... |
| ackbij2lem2 10239 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10240 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10241 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10242 | The Ackermann bijection, p... |
| r1om 10243 | The set of hereditarily fi... |
| fictb 10244 | A set is countable iff its... |
| cflem 10245 | A lemma used to simplify c... |
| cfval 10246 | Value of the cofinality fu... |
| cff 10247 | Cofinality is a function o... |
| cfub 10248 | An upper bound on cofinali... |
| cflm 10249 | Value of the cofinality fu... |
| cf0 10250 | Value of the cofinality fu... |
| cardcf 10251 | Cofinality is a cardinal n... |
| cflecard 10252 | Cofinality is bounded by t... |
| cfle 10253 | Cofinality is bounded by i... |
| cfon 10254 | The cofinality of any set ... |
| cfeq0 10255 | Only the ordinal zero has ... |
| cfsuc 10256 | Value of the cofinality fu... |
| cff1 10257 | There is always a map from... |
| cfflb 10258 | If there is a cofinal map ... |
| cfval2 10259 | Another expression for the... |
| coflim 10260 | A simpler expression for t... |
| cflim3 10261 | Another expression for the... |
| cflim2 10262 | The cofinality function is... |
| cfom 10263 | Value of the cofinality fu... |
| cfss 10264 | There is a cofinal subset ... |
| cfslb 10265 | Any cofinal subset of ` A ... |
| cfslbn 10266 | Any subset of ` A ` smalle... |
| cfslb2n 10267 | Any small collection of sm... |
| cofsmo 10268 | Any cofinal map implies th... |
| cfsmolem 10269 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10270 | The map in ~ cff1 can be a... |
| cfcoflem 10271 | Lemma for ~ cfcof , showin... |
| coftr 10272 | If there is a cofinal map ... |
| cfcof 10273 | If there is a cofinal map ... |
| cfidm 10274 | The cofinality function is... |
| alephsing 10275 | The cofinality of a limit ... |
| sornom 10276 | The range of a single-step... |
| isfin1a 10291 | Definition of a Ia-finite ... |
| fin1ai 10292 | Property of a Ia-finite se... |
| isfin2 10293 | Definition of a II-finite ... |
| fin2i 10294 | Property of a II-finite se... |
| isfin3 10295 | Definition of a III-finite... |
| isfin4 10296 | Definition of a IV-finite ... |
| fin4i 10297 | Infer that a set is IV-inf... |
| isfin5 10298 | Definition of a V-finite s... |
| isfin6 10299 | Definition of a VI-finite ... |
| isfin7 10300 | Definition of a VII-finite... |
| sdom2en01 10301 | A set with less than two e... |
| infpssrlem1 10302 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10303 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10304 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10305 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10306 | Lemma for ~ infpssr . (Co... |
| infpssr 10307 | Dedekind infinity implies ... |
| fin4en1 10308 | Dedekind finite is a cardi... |
| ssfin4 10309 | Dedekind finite sets have ... |
| domfin4 10310 | A set dominated by a Dedek... |
| ominf4 10311 | ` _om ` is Dedekind infini... |
| infpssALT 10312 | Alternate proof of ~ infps... |
| isfin4-2 10313 | Alternate definition of IV... |
| isfin4p1 10314 | Alternate definition of IV... |
| fin23lem7 10315 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10316 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10317 | A II-finite set contains m... |
| isfin2-2 10318 | ` Fin2 ` expressed in term... |
| ssfin2 10319 | A subset of a II-finite se... |
| enfin2i 10320 | II-finiteness is a cardina... |
| fin23lem24 10321 | Lemma for ~ fin23 . In a ... |
| fincssdom 10322 | In a chain of finite sets,... |
| fin23lem25 10323 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10324 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10325 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10326 | Lemma for ~ fin23 but coul... |
| fin23lem27 10327 | The mapping constructed in... |
| isfin3ds 10328 | Property of a III-finite s... |
| ssfin3ds 10329 | A subset of a III-finite s... |
| fin23lem12 10330 | The beginning of the proof... |
| fin23lem13 10331 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10332 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10333 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10334 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10335 | Lemma for ~ fin23 . The f... |
| fin23lem20 10336 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10337 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10338 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10339 | Lemma for ~ fin23 . The r... |
| fin23lem29 10340 | Lemma for ~ fin23 . The r... |
| fin23lem30 10341 | Lemma for ~ fin23 . The r... |
| fin23lem31 10342 | Lemma for ~ fin23 . The r... |
| fin23lem32 10343 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10344 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10345 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10346 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10347 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10348 | Lemma for ~ fin23 . The c... |
| fin23lem39 10349 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10350 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10351 | Lemma for ~ fin23 . A set... |
| isf32lem1 10352 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10353 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10354 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10355 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10356 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10357 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10358 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10359 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10360 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10361 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10362 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10363 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10364 | One half of ~ isfin3-2 . ... |
| isf33lem 10365 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10366 | Weakly Dedekind-infinite s... |
| isfin3-3 10367 | Weakly Dedekind-infinite s... |
| fin33i 10368 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10369 | Lemma for ~ compsscnv . (... |
| compsscnv 10370 | Complementation on a power... |
| isf34lem1 10371 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10372 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10373 | Complementation is an anti... |
| isf34lem3 10374 | Lemma for ~ isfin3-4 . (C... |
| compss 10375 | Express image under of the... |
| isf34lem4 10376 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10377 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10378 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10379 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10380 | Inference from ~ isfin3-4 ... |
| isfin3-4 10381 | Weakly Dedekind-infinite s... |
| fin11a 10382 | Every I-finite set is Ia-f... |
| enfin1ai 10383 | Ia-finiteness is a cardina... |
| isfin1-2 10384 | A set is finite in the usu... |
| isfin1-3 10385 | A set is I-finite iff ever... |
| isfin1-4 10386 | A set is I-finite iff ever... |
| dffin1-5 10387 | Compact quantifier-free ve... |
| fin23 10388 | Every II-finite set (every... |
| fin34 10389 | Every III-finite set is IV... |
| isfin5-2 10390 | Alternate definition of V-... |
| fin45 10391 | Every IV-finite set is V-f... |
| fin56 10392 | Every V-finite set is VI-f... |
| fin17 10393 | Every I-finite set is VII-... |
| fin67 10394 | Every VI-finite set is VII... |
| isfin7-2 10395 | A set is VII-finite iff it... |
| fin71num 10396 | A well-orderable set is VI... |
| dffin7-2 10397 | Class form of ~ isfin7-2 .... |
| dfacfin7 10398 | Axiom of Choice equivalent... |
| fin1a2lem1 10399 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10400 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10401 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10402 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10403 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10404 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10405 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10406 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10407 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10408 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10409 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10410 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10411 | Lemma for ~ fin1a2 . (Con... |
| fin12 10412 | Weak theorem which skips I... |
| fin1a2s 10413 | An II-infinite set can hav... |
| fin1a2 10414 | Every Ia-finite set is II-... |
| itunifval 10415 | Function value of iterated... |
| itunifn 10416 | Functionality of the itera... |
| ituni0 10417 | A zero-fold iterated union... |
| itunisuc 10418 | Successor iterated union. ... |
| itunitc1 10419 | Each union iterate is a me... |
| itunitc 10420 | The union of all union ite... |
| ituniiun 10421 | Unwrap an iterated union f... |
| hsmexlem7 10422 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10423 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10424 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10425 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10426 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10427 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10428 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10429 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10430 | Lemma for ~ hsmex . (Cont... |
| hsmex 10431 | The collection of heredita... |
| hsmex2 10432 | The set of hereditary size... |
| hsmex3 10433 | The set of hereditary size... |
| axcc2lem 10435 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10436 | A possibly more useful ver... |
| axcc3 10437 | A possibly more useful ver... |
| axcc4 10438 | A version of ~ axcc3 that ... |
| acncc 10439 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10440 | Relax the constraint on ~ ... |
| domtriomlem 10441 | Lemma for ~ domtriom . (C... |
| domtriom 10442 | Trichotomy of equinumerosi... |
| fin41 10443 | Under countable choice, th... |
| dominf 10444 | A nonempty set that is a s... |
| dcomex 10446 | The Axiom of Dependent Cho... |
| axdc2lem 10447 | Lemma for ~ axdc2 . We co... |
| axdc2 10448 | An apparent strengthening ... |
| axdc3lem 10449 | The class ` S ` of finite ... |
| axdc3lem2 10450 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10451 | Simple substitution lemma ... |
| axdc3lem4 10452 | Lemma for ~ axdc3 . We ha... |
| axdc3 10453 | Dependent Choice. Axiom D... |
| axdc4lem 10454 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10455 | A more general version of ... |
| axcclem 10456 | Lemma for ~ axcc . (Contr... |
| axcc 10457 | Although CC can be proven ... |
| zfac 10459 | Axiom of Choice expressed ... |
| ac2 10460 | Axiom of Choice equivalent... |
| ac3 10461 | Axiom of Choice using abbr... |
| axac3 10463 | This theorem asserts that ... |
| ackm 10464 | A remarkable equivalent to... |
| axac2 10465 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10466 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10467 | Apply a choice equivalent.... |
| cardeqv 10468 | All sets are well-orderabl... |
| numth3 10469 | All sets are well-orderabl... |
| numth2 10470 | Numeration theorem: any se... |
| numth 10471 | Numeration theorem: every ... |
| ac7 10472 | An Axiom of Choice equival... |
| ac7g 10473 | An Axiom of Choice equival... |
| ac4 10474 | Equivalent of Axiom of Cho... |
| ac4c 10475 | Equivalent of Axiom of Cho... |
| ac5 10476 | An Axiom of Choice equival... |
| ac5b 10477 | Equivalent of Axiom of Cho... |
| ac6num 10478 | A version of ~ ac6 which t... |
| ac6 10479 | Equivalent of Axiom of Cho... |
| ac6c4 10480 | Equivalent of Axiom of Cho... |
| ac6c5 10481 | Equivalent of Axiom of Cho... |
| ac9 10482 | An Axiom of Choice equival... |
| ac6s 10483 | Equivalent of Axiom of Cho... |
| ac6n 10484 | Equivalent of Axiom of Cho... |
| ac6s2 10485 | Generalization of the Axio... |
| ac6s3 10486 | Generalization of the Axio... |
| ac6sg 10487 | ~ ac6s with sethood as ant... |
| ac6sf 10488 | Version of ~ ac6 with boun... |
| ac6s4 10489 | Generalization of the Axio... |
| ac6s5 10490 | Generalization of the Axio... |
| ac8 10491 | An Axiom of Choice equival... |
| ac9s 10492 | An Axiom of Choice equival... |
| numthcor 10493 | Any set is strictly domina... |
| weth 10494 | Well-ordering theorem: any... |
| zorn2lem1 10495 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10496 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10497 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10498 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10499 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10500 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10501 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10502 | Zorn's Lemma of [Monk1] p.... |
| zorng 10503 | Zorn's Lemma. If the unio... |
| zornn0g 10504 | Variant of Zorn's lemma ~ ... |
| zorn2 10505 | Zorn's Lemma of [Monk1] p.... |
| zorn 10506 | Zorn's Lemma. If the unio... |
| zornn0 10507 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10508 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10509 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10510 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10511 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10512 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10513 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10514 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10515 | The Teichmüller-Tukey... |
| ttukeyg 10516 | The Teichmüller-Tukey... |
| ttukey 10517 | The Teichmüller-Tukey... |
| axdclem 10518 | Lemma for ~ axdc . (Contr... |
| axdclem2 10519 | Lemma for ~ axdc . Using ... |
| axdc 10520 | This theorem derives ~ ax-... |
| fodomg 10521 | An onto function implies d... |
| fodom 10522 | An onto function implies d... |
| dmct 10523 | The domain of a countable ... |
| rnct 10524 | The range of a countable s... |
| fodomb 10525 | Equivalence of an onto map... |
| wdomac 10526 | When assuming AC, weak and... |
| brdom3 10527 | Equivalence to a dominance... |
| brdom5 10528 | An equivalence to a domina... |
| brdom4 10529 | An equivalence to a domina... |
| brdom7disj 10530 | An equivalence to a domina... |
| brdom6disj 10531 | An equivalence to a domina... |
| fin71ac 10532 | Once we allow AC, the "str... |
| imadomg 10533 | An image of a function und... |
| fimact 10534 | The image by a function of... |
| fnrndomg 10535 | The range of a function is... |
| fnct 10536 | If the domain of a functio... |
| mptct 10537 | A countable mapping set is... |
| iunfo 10538 | Existence of an onto funct... |
| iundom2g 10539 | An upper bound for the car... |
| iundomg 10540 | An upper bound for the car... |
| iundom 10541 | An upper bound for the car... |
| unidom 10542 | An upper bound for the car... |
| uniimadom 10543 | An upper bound for the car... |
| uniimadomf 10544 | An upper bound for the car... |
| cardval 10545 | The value of the cardinal ... |
| cardid 10546 | Any set is equinumerous to... |
| cardidg 10547 | Any set is equinumerous to... |
| cardidd 10548 | Any set is equinumerous to... |
| cardf 10549 | The cardinality function i... |
| carden 10550 | Two sets are equinumerous ... |
| cardeq0 10551 | Only the empty set has car... |
| unsnen 10552 | Equinumerosity of a set wi... |
| carddom 10553 | Two sets have the dominanc... |
| cardsdom 10554 | Two sets have the strict d... |
| domtri 10555 | Trichotomy law for dominan... |
| entric 10556 | Trichotomy of equinumerosi... |
| entri2 10557 | Trichotomy of dominance an... |
| entri3 10558 | Trichotomy of dominance. ... |
| sdomsdomcard 10559 | A set strictly dominates i... |
| canth3 10560 | Cantor's theorem in terms ... |
| infxpidm 10561 | Every infinite class is eq... |
| ondomon 10562 | The class of ordinals domi... |
| cardmin 10563 | The smallest ordinal that ... |
| ficard 10564 | A set is finite iff its ca... |
| infinf 10565 | Equivalence between two in... |
| unirnfdomd 10566 | The union of the range of ... |
| konigthlem 10567 | Lemma for ~ konigth . (Co... |
| konigth 10568 | Konig's Theorem. If ` m (... |
| alephsucpw 10569 | The power set of an aleph ... |
| aleph1 10570 | The set exponentiation of ... |
| alephval2 10571 | An alternate way to expres... |
| dominfac 10572 | A nonempty set that is a s... |
| iunctb 10573 | The countable union of cou... |
| unictb 10574 | The countable union of cou... |
| infmap 10575 | An exponentiation law for ... |
| alephadd 10576 | The sum of two alephs is t... |
| alephmul 10577 | The product of two alephs ... |
| alephexp1 10578 | An exponentiation law for ... |
| alephsuc3 10579 | An alternate representatio... |
| alephexp2 10580 | An expression equinumerous... |
| alephreg 10581 | A successor aleph is regul... |
| pwcfsdom 10582 | A corollary of Konig's The... |
| cfpwsdom 10583 | A corollary of Konig's The... |
| alephom 10584 | From ~ canth2 , we know th... |
| smobeth 10585 | The beth function is stric... |
| nd1 10586 | A lemma for proving condit... |
| nd2 10587 | A lemma for proving condit... |
| nd3 10588 | A lemma for proving condit... |
| nd4 10589 | A lemma for proving condit... |
| axextnd 10590 | A version of the Axiom of ... |
| axrepndlem1 10591 | Lemma for the Axiom of Rep... |
| axrepndlem2 10592 | Lemma for the Axiom of Rep... |
| axrepnd 10593 | A version of the Axiom of ... |
| axunndlem1 10594 | Lemma for the Axiom of Uni... |
| axunnd 10595 | A version of the Axiom of ... |
| axpowndlem1 10596 | Lemma for the Axiom of Pow... |
| axpowndlem2 10597 | Lemma for the Axiom of Pow... |
| axpowndlem3 10598 | Lemma for the Axiom of Pow... |
| axpowndlem4 10599 | Lemma for the Axiom of Pow... |
| axpownd 10600 | A version of the Axiom of ... |
| axregndlem1 10601 | Lemma for the Axiom of Reg... |
| axregndlem2 10602 | Lemma for the Axiom of Reg... |
| axregnd 10603 | A version of the Axiom of ... |
| axinfndlem1 10604 | Lemma for the Axiom of Inf... |
| axinfnd 10605 | A version of the Axiom of ... |
| axacndlem1 10606 | Lemma for the Axiom of Cho... |
| axacndlem2 10607 | Lemma for the Axiom of Cho... |
| axacndlem3 10608 | Lemma for the Axiom of Cho... |
| axacndlem4 10609 | Lemma for the Axiom of Cho... |
| axacndlem5 10610 | Lemma for the Axiom of Cho... |
| axacnd 10611 | A version of the Axiom of ... |
| zfcndext 10612 | Axiom of Extensionality ~ ... |
| zfcndrep 10613 | Axiom of Replacement ~ ax-... |
| zfcndun 10614 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10615 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10616 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10617 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10618 | Axiom of Choice ~ ax-ac , ... |
| elgch 10621 | Elementhood in the collect... |
| fingch 10622 | A finite set is a GCH-set.... |
| gchi 10623 | The only GCH-sets which ha... |
| gchen1 10624 | If ` A <_ B < ~P A ` , and... |
| gchen2 10625 | If ` A < B <_ ~P A ` , and... |
| gchor 10626 | If ` A <_ B <_ ~P A ` , an... |
| engch 10627 | The property of being a GC... |
| gchdomtri 10628 | Under certain conditions, ... |
| fpwwe2cbv 10629 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10630 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10631 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10632 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10633 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10634 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10635 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10636 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10637 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10638 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10639 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10640 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10641 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10642 | Given any function ` F ` f... |
| fpwwecbv 10643 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10644 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10645 | Given any function ` F ` f... |
| canth4 10646 | An "effective" form of Can... |
| canthnumlem 10647 | Lemma for ~ canthnum . (C... |
| canthnum 10648 | The set of well-orderable ... |
| canthwelem 10649 | Lemma for ~ canthwe . (Co... |
| canthwe 10650 | The set of well-orders of ... |
| canthp1lem1 10651 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10652 | Lemma for ~ canthp1 . (Co... |
| canthp1 10653 | A slightly stronger form o... |
| finngch 10654 | The exclusion of finite se... |
| gchdju1 10655 | An infinite GCH-set is ide... |
| gchinf 10656 | An infinite GCH-set is Ded... |
| pwfseqlem1 10657 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10658 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10659 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10660 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10661 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10662 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10663 | The powerset of a Dedekind... |
| pwxpndom2 10664 | The powerset of a Dedekind... |
| pwxpndom 10665 | The powerset of a Dedekind... |
| pwdjundom 10666 | The powerset of a Dedekind... |
| gchdjuidm 10667 | An infinite GCH-set is ide... |
| gchxpidm 10668 | An infinite GCH-set is ide... |
| gchpwdom 10669 | A relationship between dom... |
| gchaleph 10670 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10671 | If ` ( aleph `` A ) ` and ... |
| hargch 10672 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10673 | If ` ( aleph `` suc A ) ` ... |
| gch2 10674 | It is sufficient to requir... |
| gch3 10675 | An equivalent formulation ... |
| gch-kn 10676 | The equivalence of two ver... |
| gchaclem 10677 | Lemma for ~ gchac (obsolet... |
| gchhar 10678 | A "local" form of ~ gchac ... |
| gchacg 10679 | A "local" form of ~ gchac ... |
| gchac 10680 | The Generalized Continuum ... |
| elwina 10685 | Conditions of weak inacces... |
| elina 10686 | Conditions of strong inacc... |
| winaon 10687 | A weakly inaccessible card... |
| inawinalem 10688 | Lemma for ~ inawina . (Co... |
| inawina 10689 | Every strongly inaccessibl... |
| omina 10690 | ` _om ` is a strongly inac... |
| winacard 10691 | A weakly inaccessible card... |
| winainflem 10692 | A weakly inaccessible card... |
| winainf 10693 | A weakly inaccessible card... |
| winalim 10694 | A weakly inaccessible card... |
| winalim2 10695 | A nontrivial weakly inacce... |
| winafp 10696 | A nontrivial weakly inacce... |
| winafpi 10697 | This theorem, which states... |
| gchina 10698 | Assuming the GCH, weakly a... |
| iswun 10703 | Properties of a weak unive... |
| wuntr 10704 | A weak universe is transit... |
| wununi 10705 | A weak universe is closed ... |
| wunpw 10706 | A weak universe is closed ... |
| wunelss 10707 | The elements of a weak uni... |
| wunpr 10708 | A weak universe is closed ... |
| wunun 10709 | A weak universe is closed ... |
| wuntp 10710 | A weak universe is closed ... |
| wunss 10711 | A weak universe is closed ... |
| wunin 10712 | A weak universe is closed ... |
| wundif 10713 | A weak universe is closed ... |
| wunint 10714 | A weak universe is closed ... |
| wunsn 10715 | A weak universe is closed ... |
| wunsuc 10716 | A weak universe is closed ... |
| wun0 10717 | A weak universe contains t... |
| wunr1om 10718 | A weak universe is infinit... |
| wunom 10719 | A weak universe contains a... |
| wunfi 10720 | A weak universe contains a... |
| wunop 10721 | A weak universe is closed ... |
| wunot 10722 | A weak universe is closed ... |
| wunxp 10723 | A weak universe is closed ... |
| wunpm 10724 | A weak universe is closed ... |
| wunmap 10725 | A weak universe is closed ... |
| wunf 10726 | A weak universe is closed ... |
| wundm 10727 | A weak universe is closed ... |
| wunrn 10728 | A weak universe is closed ... |
| wuncnv 10729 | A weak universe is closed ... |
| wunres 10730 | A weak universe is closed ... |
| wunfv 10731 | A weak universe is closed ... |
| wunco 10732 | A weak universe is closed ... |
| wuntpos 10733 | A weak universe is closed ... |
| intwun 10734 | The intersection of a coll... |
| r1limwun 10735 | Each limit stage in the cu... |
| r1wunlim 10736 | The weak universes in the ... |
| wunex2 10737 | Construct a weak universe ... |
| wunex 10738 | Construct a weak universe ... |
| uniwun 10739 | Every set is contained in ... |
| wunex3 10740 | Construct a weak universe ... |
| wuncval 10741 | Value of the weak universe... |
| wuncid 10742 | The weak universe closure ... |
| wunccl 10743 | The weak universe closure ... |
| wuncss 10744 | The weak universe closure ... |
| wuncidm 10745 | The weak universe closure ... |
| wuncval2 10746 | Our earlier expression for... |
| eltskg 10749 | Properties of a Tarski cla... |
| eltsk2g 10750 | Properties of a Tarski cla... |
| tskpwss 10751 | First axiom of a Tarski cl... |
| tskpw 10752 | Second axiom of a Tarski c... |
| tsken 10753 | Third axiom of a Tarski cl... |
| 0tsk 10754 | The empty set is a (transi... |
| tsksdom 10755 | An element of a Tarski cla... |
| tskssel 10756 | A part of a Tarski class s... |
| tskss 10757 | The subsets of an element ... |
| tskin 10758 | The intersection of two el... |
| tsksn 10759 | A singleton of an element ... |
| tsktrss 10760 | A transitive element of a ... |
| tsksuc 10761 | If an element of a Tarski ... |
| tsk0 10762 | A nonempty Tarski class co... |
| tsk1 10763 | One is an element of a non... |
| tsk2 10764 | Two is an element of a non... |
| 2domtsk 10765 | If a Tarski class is not e... |
| tskr1om 10766 | A nonempty Tarski class is... |
| tskr1om2 10767 | A nonempty Tarski class co... |
| tskinf 10768 | A nonempty Tarski class is... |
| tskpr 10769 | If ` A ` and ` B ` are mem... |
| tskop 10770 | If ` A ` and ` B ` are mem... |
| tskxpss 10771 | A Cartesian product of two... |
| tskwe2 10772 | A Tarski class is well-ord... |
| inttsk 10773 | The intersection of a coll... |
| inar1 10774 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10775 | Alternate proof of ~ r1om ... |
| rankcf 10776 | Any set must be at least a... |
| inatsk 10777 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10778 | The set of hereditarily fi... |
| tskord 10779 | A Tarski class contains al... |
| tskcard 10780 | An even more direct relati... |
| r1tskina 10781 | There is a direct relation... |
| tskuni 10782 | The union of an element of... |
| tskwun 10783 | A nonempty transitive Tars... |
| tskint 10784 | The intersection of an ele... |
| tskun 10785 | The union of two elements ... |
| tskxp 10786 | The Cartesian product of t... |
| tskmap 10787 | Set exponentiation is an e... |
| tskurn 10788 | A transitive Tarski class ... |
| elgrug 10791 | Properties of a Grothendie... |
| grutr 10792 | A Grothendieck universe is... |
| gruelss 10793 | A Grothendieck universe is... |
| grupw 10794 | A Grothendieck universe co... |
| gruss 10795 | Any subset of an element o... |
| grupr 10796 | A Grothendieck universe co... |
| gruurn 10797 | A Grothendieck universe co... |
| gruiun 10798 | If ` B ( x ) ` is a family... |
| gruuni 10799 | A Grothendieck universe co... |
| grurn 10800 | A Grothendieck universe co... |
| gruima 10801 | A Grothendieck universe co... |
| gruel 10802 | Any element of an element ... |
| grusn 10803 | A Grothendieck universe co... |
| gruop 10804 | A Grothendieck universe co... |
| gruun 10805 | A Grothendieck universe co... |
| gruxp 10806 | A Grothendieck universe co... |
| grumap 10807 | A Grothendieck universe co... |
| gruixp 10808 | A Grothendieck universe co... |
| gruiin 10809 | A Grothendieck universe co... |
| gruf 10810 | A Grothendieck universe co... |
| gruen 10811 | A Grothendieck universe co... |
| gruwun 10812 | A nonempty Grothendieck un... |
| intgru 10813 | The intersection of a fami... |
| ingru 10814 | The intersection of a univ... |
| wfgru 10815 | The wellfounded part of a ... |
| grudomon 10816 | Each ordinal that is compa... |
| gruina 10817 | If a Grothendieck universe... |
| grur1a 10818 | A characterization of Grot... |
| grur1 10819 | A characterization of Grot... |
| grutsk1 10820 | Grothendieck universes are... |
| grutsk 10821 | Grothendieck universes are... |
| axgroth5 10823 | The Tarski-Grothendieck ax... |
| axgroth2 10824 | Alternate version of the T... |
| grothpw 10825 | Derive the Axiom of Power ... |
| grothpwex 10826 | Derive the Axiom of Power ... |
| axgroth6 10827 | The Tarski-Grothendieck ax... |
| grothomex 10828 | The Tarski-Grothendieck Ax... |
| grothac 10829 | The Tarski-Grothendieck Ax... |
| axgroth3 10830 | Alternate version of the T... |
| axgroth4 10831 | Alternate version of the T... |
| grothprimlem 10832 | Lemma for ~ grothprim . E... |
| grothprim 10833 | The Tarski-Grothendieck Ax... |
| grothtsk 10834 | The Tarski-Grothendieck Ax... |
| inaprc 10835 | An equivalent to the Tarsk... |
| tskmval 10838 | Value of our tarski map. ... |
| tskmid 10839 | The set ` A ` is an elemen... |
| tskmcl 10840 | A Tarski class that contai... |
| sstskm 10841 | Being a part of ` ( tarski... |
| eltskm 10842 | Belonging to ` ( tarskiMap... |
| elni 10875 | Membership in the class of... |
| elni2 10876 | Membership in the class of... |
| pinn 10877 | A positive integer is a na... |
| pion 10878 | A positive integer is an o... |
| piord 10879 | A positive integer is ordi... |
| niex 10880 | The class of positive inte... |
| 0npi 10881 | The empty set is not a pos... |
| 1pi 10882 | Ordinal 'one' is a positiv... |
| addpiord 10883 | Positive integer addition ... |
| mulpiord 10884 | Positive integer multiplic... |
| mulidpi 10885 | 1 is an identity element f... |
| ltpiord 10886 | Positive integer 'less tha... |
| ltsopi 10887 | Positive integer 'less tha... |
| ltrelpi 10888 | Positive integer 'less tha... |
| dmaddpi 10889 | Domain of addition on posi... |
| dmmulpi 10890 | Domain of multiplication o... |
| addclpi 10891 | Closure of addition of pos... |
| mulclpi 10892 | Closure of multiplication ... |
| addcompi 10893 | Addition of positive integ... |
| addasspi 10894 | Addition of positive integ... |
| mulcompi 10895 | Multiplication of positive... |
| mulasspi 10896 | Multiplication of positive... |
| distrpi 10897 | Multiplication of positive... |
| addcanpi 10898 | Addition cancellation law ... |
| mulcanpi 10899 | Multiplication cancellatio... |
| addnidpi 10900 | There is no identity eleme... |
| ltexpi 10901 | Ordering on positive integ... |
| ltapi 10902 | Ordering property of addit... |
| ltmpi 10903 | Ordering property of multi... |
| 1lt2pi 10904 | One is less than two (one ... |
| nlt1pi 10905 | No positive integer is les... |
| indpi 10906 | Principle of Finite Induct... |
| enqbreq 10918 | Equivalence relation for p... |
| enqbreq2 10919 | Equivalence relation for p... |
| enqer 10920 | The equivalence relation f... |
| enqex 10921 | The equivalence relation f... |
| nqex 10922 | The class of positive frac... |
| 0nnq 10923 | The empty set is not a pos... |
| elpqn 10924 | Each positive fraction is ... |
| ltrelnq 10925 | Positive fraction 'less th... |
| pinq 10926 | The representatives of pos... |
| 1nq 10927 | The positive fraction 'one... |
| nqereu 10928 | There is a unique element ... |
| nqerf 10929 | Corollary of ~ nqereu : th... |
| nqercl 10930 | Corollary of ~ nqereu : cl... |
| nqerrel 10931 | Any member of ` ( N. X. N.... |
| nqerid 10932 | Corollary of ~ nqereu : th... |
| enqeq 10933 | Corollary of ~ nqereu : if... |
| nqereq 10934 | The function ` /Q ` acts a... |
| addpipq2 10935 | Addition of positive fract... |
| addpipq 10936 | Addition of positive fract... |
| addpqnq 10937 | Addition of positive fract... |
| mulpipq2 10938 | Multiplication of positive... |
| mulpipq 10939 | Multiplication of positive... |
| mulpqnq 10940 | Multiplication of positive... |
| ordpipq 10941 | Ordering of positive fract... |
| ordpinq 10942 | Ordering of positive fract... |
| addpqf 10943 | Closure of addition on pos... |
| addclnq 10944 | Closure of addition on pos... |
| mulpqf 10945 | Closure of multiplication ... |
| mulclnq 10946 | Closure of multiplication ... |
| addnqf 10947 | Domain of addition on posi... |
| mulnqf 10948 | Domain of multiplication o... |
| addcompq 10949 | Addition of positive fract... |
| addcomnq 10950 | Addition of positive fract... |
| mulcompq 10951 | Multiplication of positive... |
| mulcomnq 10952 | Multiplication of positive... |
| adderpqlem 10953 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10954 | Lemma for ~ mulerpq . (Co... |
| adderpq 10955 | Addition is compatible wit... |
| mulerpq 10956 | Multiplication is compatib... |
| addassnq 10957 | Addition of positive fract... |
| mulassnq 10958 | Multiplication of positive... |
| mulcanenq 10959 | Lemma for distributive law... |
| distrnq 10960 | Multiplication of positive... |
| 1nqenq 10961 | The equivalence class of r... |
| mulidnq 10962 | Multiplication identity el... |
| recmulnq 10963 | Relationship between recip... |
| recidnq 10964 | A positive fraction times ... |
| recclnq 10965 | Closure law for positive f... |
| recrecnq 10966 | Reciprocal of reciprocal o... |
| dmrecnq 10967 | Domain of reciprocal on po... |
| ltsonq 10968 | 'Less than' is a strict or... |
| lterpq 10969 | Compatibility of ordering ... |
| ltanq 10970 | Ordering property of addit... |
| ltmnq 10971 | Ordering property of multi... |
| 1lt2nq 10972 | One is less than two (one ... |
| ltaddnq 10973 | The sum of two fractions i... |
| ltexnq 10974 | Ordering on positive fract... |
| halfnq 10975 | One-half of any positive f... |
| nsmallnq 10976 | The is no smallest positiv... |
| ltbtwnnq 10977 | There exists a number betw... |
| ltrnq 10978 | Ordering property of recip... |
| archnq 10979 | For any fraction, there is... |
| npex 10985 | The class of positive real... |
| elnp 10986 | Membership in positive rea... |
| elnpi 10987 | Membership in positive rea... |
| prn0 10988 | A positive real is not emp... |
| prpssnq 10989 | A positive real is a subse... |
| elprnq 10990 | A positive real is a set o... |
| 0npr 10991 | The empty set is not a pos... |
| prcdnq 10992 | A positive real is closed ... |
| prub 10993 | A positive fraction not in... |
| prnmax 10994 | A positive real has no lar... |
| npomex 10995 | A simplifying observation,... |
| prnmadd 10996 | A positive real has no lar... |
| ltrelpr 10997 | Positive real 'less than' ... |
| genpv 10998 | Value of general operation... |
| genpelv 10999 | Membership in value of gen... |
| genpprecl 11000 | Pre-closure law for genera... |
| genpdm 11001 | Domain of general operatio... |
| genpn0 11002 | The result of an operation... |
| genpss 11003 | The result of an operation... |
| genpnnp 11004 | The result of an operation... |
| genpcd 11005 | Downward closure of an ope... |
| genpnmax 11006 | An operation on positive r... |
| genpcl 11007 | Closure of an operation on... |
| genpass 11008 | Associativity of an operat... |
| plpv 11009 | Value of addition on posit... |
| mpv 11010 | Value of multiplication on... |
| dmplp 11011 | Domain of addition on posi... |
| dmmp 11012 | Domain of multiplication o... |
| nqpr 11013 | The canonical embedding of... |
| 1pr 11014 | The positive real number '... |
| addclprlem1 11015 | Lemma to prove downward cl... |
| addclprlem2 11016 | Lemma to prove downward cl... |
| addclpr 11017 | Closure of addition on pos... |
| mulclprlem 11018 | Lemma to prove downward cl... |
| mulclpr 11019 | Closure of multiplication ... |
| addcompr 11020 | Addition of positive reals... |
| addasspr 11021 | Addition of positive reals... |
| mulcompr 11022 | Multiplication of positive... |
| mulasspr 11023 | Multiplication of positive... |
| distrlem1pr 11024 | Lemma for distributive law... |
| distrlem4pr 11025 | Lemma for distributive law... |
| distrlem5pr 11026 | Lemma for distributive law... |
| distrpr 11027 | Multiplication of positive... |
| 1idpr 11028 | 1 is an identity element f... |
| ltprord 11029 | Positive real 'less than' ... |
| psslinpr 11030 | Proper subset is a linear ... |
| ltsopr 11031 | Positive real 'less than' ... |
| prlem934 11032 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 11033 | The sum of two positive re... |
| ltaddpr2 11034 | The sum of two positive re... |
| ltexprlem1 11035 | Lemma for Proposition 9-3.... |
| ltexprlem2 11036 | Lemma for Proposition 9-3.... |
| ltexprlem3 11037 | Lemma for Proposition 9-3.... |
| ltexprlem4 11038 | Lemma for Proposition 9-3.... |
| ltexprlem5 11039 | Lemma for Proposition 9-3.... |
| ltexprlem6 11040 | Lemma for Proposition 9-3.... |
| ltexprlem7 11041 | Lemma for Proposition 9-3.... |
| ltexpri 11042 | Proposition 9-3.5(iv) of [... |
| ltaprlem 11043 | Lemma for Proposition 9-3.... |
| ltapr 11044 | Ordering property of addit... |
| addcanpr 11045 | Addition cancellation law ... |
| prlem936 11046 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 11047 | Lemma for Proposition 9-3.... |
| reclem3pr 11048 | Lemma for Proposition 9-3.... |
| reclem4pr 11049 | Lemma for Proposition 9-3.... |
| recexpr 11050 | The reciprocal of a positi... |
| suplem1pr 11051 | The union of a nonempty, b... |
| suplem2pr 11052 | The union of a set of posi... |
| supexpr 11053 | The union of a nonempty, b... |
| enrer 11062 | The equivalence relation f... |
| nrex1 11063 | The class of signed reals ... |
| enrbreq 11064 | Equivalence relation for s... |
| enreceq 11065 | Equivalence class equality... |
| enrex 11066 | The equivalence relation f... |
| ltrelsr 11067 | Signed real 'less than' is... |
| addcmpblnr 11068 | Lemma showing compatibilit... |
| mulcmpblnrlem 11069 | Lemma used in lemma showin... |
| mulcmpblnr 11070 | Lemma showing compatibilit... |
| prsrlem1 11071 | Decomposing signed reals i... |
| addsrmo 11072 | There is at most one resul... |
| mulsrmo 11073 | There is at most one resul... |
| addsrpr 11074 | Addition of signed reals i... |
| mulsrpr 11075 | Multiplication of signed r... |
| ltsrpr 11076 | Ordering of signed reals i... |
| gt0srpr 11077 | Greater than zero in terms... |
| 0nsr 11078 | The empty set is not a sig... |
| 0r 11079 | The constant ` 0R ` is a s... |
| 1sr 11080 | The constant ` 1R ` is a s... |
| m1r 11081 | The constant ` -1R ` is a ... |
| addclsr 11082 | Closure of addition on sig... |
| mulclsr 11083 | Closure of multiplication ... |
| dmaddsr 11084 | Domain of addition on sign... |
| dmmulsr 11085 | Domain of multiplication o... |
| addcomsr 11086 | Addition of signed reals i... |
| addasssr 11087 | Addition of signed reals i... |
| mulcomsr 11088 | Multiplication of signed r... |
| mulasssr 11089 | Multiplication of signed r... |
| distrsr 11090 | Multiplication of signed r... |
| m1p1sr 11091 | Minus one plus one is zero... |
| m1m1sr 11092 | Minus one times minus one ... |
| ltsosr 11093 | Signed real 'less than' is... |
| 0lt1sr 11094 | 0 is less than 1 for signe... |
| 1ne0sr 11095 | 1 and 0 are distinct for s... |
| 0idsr 11096 | The signed real number 0 i... |
| 1idsr 11097 | 1 is an identity element f... |
| 00sr 11098 | A signed real times 0 is 0... |
| ltasr 11099 | Ordering property of addit... |
| pn0sr 11100 | A signed real plus its neg... |
| negexsr 11101 | Existence of negative sign... |
| recexsrlem 11102 | The reciprocal of a positi... |
| addgt0sr 11103 | The sum of two positive si... |
| mulgt0sr 11104 | The product of two positiv... |
| sqgt0sr 11105 | The square of a nonzero si... |
| recexsr 11106 | The reciprocal of a nonzer... |
| mappsrpr 11107 | Mapping from positive sign... |
| ltpsrpr 11108 | Mapping of order from posi... |
| map2psrpr 11109 | Equivalence for positive s... |
| supsrlem 11110 | Lemma for supremum theorem... |
| supsr 11111 | A nonempty, bounded set of... |
| opelcn 11128 | Ordered pair membership in... |
| opelreal 11129 | Ordered pair membership in... |
| elreal 11130 | Membership in class of rea... |
| elreal2 11131 | Ordered pair membership in... |
| 0ncn 11132 | The empty set is not a com... |
| ltrelre 11133 | 'Less than' is a relation ... |
| addcnsr 11134 | Addition of complex number... |
| mulcnsr 11135 | Multiplication of complex ... |
| eqresr 11136 | Equality of real numbers i... |
| addresr 11137 | Addition of real numbers i... |
| mulresr 11138 | Multiplication of real num... |
| ltresr 11139 | Ordering of real subset of... |
| ltresr2 11140 | Ordering of real subset of... |
| dfcnqs 11141 | Technical trick to permit ... |
| addcnsrec 11142 | Technical trick to permit ... |
| mulcnsrec 11143 | Technical trick to permit ... |
| axaddf 11144 | Addition is an operation o... |
| axmulf 11145 | Multiplication is an opera... |
| axcnex 11146 | The complex numbers form a... |
| axresscn 11147 | The real numbers are a sub... |
| ax1cn 11148 | 1 is a complex number. Ax... |
| axicn 11149 | ` _i ` is a complex number... |
| axaddcl 11150 | Closure law for addition o... |
| axaddrcl 11151 | Closure law for addition i... |
| axmulcl 11152 | Closure law for multiplica... |
| axmulrcl 11153 | Closure law for multiplica... |
| axmulcom 11154 | Multiplication of complex ... |
| axaddass 11155 | Addition of complex number... |
| axmulass 11156 | Multiplication of complex ... |
| axdistr 11157 | Distributive law for compl... |
| axi2m1 11158 | i-squared equals -1 (expre... |
| ax1ne0 11159 | 1 and 0 are distinct. Axi... |
| ax1rid 11160 | ` 1 ` is an identity eleme... |
| axrnegex 11161 | Existence of negative of r... |
| axrrecex 11162 | Existence of reciprocal of... |
| axcnre 11163 | A complex number can be ex... |
| axpre-lttri 11164 | Ordering on reals satisfie... |
| axpre-lttrn 11165 | Ordering on reals is trans... |
| axpre-ltadd 11166 | Ordering property of addit... |
| axpre-mulgt0 11167 | The product of two positiv... |
| axpre-sup 11168 | A nonempty, bounded-above ... |
| wuncn 11169 | A weak universe containing... |
| cnex 11195 | Alias for ~ ax-cnex . See... |
| addcl 11196 | Alias for ~ ax-addcl , for... |
| readdcl 11197 | Alias for ~ ax-addrcl , fo... |
| mulcl 11198 | Alias for ~ ax-mulcl , for... |
| remulcl 11199 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11200 | Alias for ~ ax-mulcom , fo... |
| addass 11201 | Alias for ~ ax-addass , fo... |
| mulass 11202 | Alias for ~ ax-mulass , fo... |
| adddi 11203 | Alias for ~ ax-distr , for... |
| recn 11204 | A real number is a complex... |
| reex 11205 | The real numbers form a se... |
| reelprrecn 11206 | Reals are a subset of the ... |
| cnelprrecn 11207 | Complex numbers are a subs... |
| mpomulf 11208 | Multiplication is an opera... |
| elimne0 11209 | Hypothesis for weak deduct... |
| adddir 11210 | Distributive law for compl... |
| 0cn 11211 | Zero is a complex number. ... |
| 0cnd 11212 | Zero is a complex number, ... |
| c0ex 11213 | Zero is a set. (Contribut... |
| 1cnd 11214 | One is a complex number, d... |
| 1ex 11215 | One is a set. (Contribute... |
| cnre 11216 | Alias for ~ ax-cnre , for ... |
| mulrid 11217 | The number 1 is an identit... |
| mullid 11218 | Identity law for multiplic... |
| 1re 11219 | The number 1 is real. Thi... |
| 1red 11220 | The number 1 is real, dedu... |
| 0re 11221 | The number 0 is real. Rem... |
| 0red 11222 | The number 0 is real, dedu... |
| mulridi 11223 | Identity law for multiplic... |
| mullidi 11224 | Identity law for multiplic... |
| addcli 11225 | Closure law for addition. ... |
| mulcli 11226 | Closure law for multiplica... |
| mulcomi 11227 | Commutative law for multip... |
| mulcomli 11228 | Commutative law for multip... |
| addassi 11229 | Associative law for additi... |
| mulassi 11230 | Associative law for multip... |
| adddii 11231 | Distributive law (left-dis... |
| adddiri 11232 | Distributive law (right-di... |
| recni 11233 | A real number is a complex... |
| readdcli 11234 | Closure law for addition o... |
| remulcli 11235 | Closure law for multiplica... |
| mulridd 11236 | Identity law for multiplic... |
| mullidd 11237 | Identity law for multiplic... |
| addcld 11238 | Closure law for addition. ... |
| mulcld 11239 | Closure law for multiplica... |
| mulcomd 11240 | Commutative law for multip... |
| addassd 11241 | Associative law for additi... |
| mulassd 11242 | Associative law for multip... |
| adddid 11243 | Distributive law (left-dis... |
| adddird 11244 | Distributive law (right-di... |
| adddirp1d 11245 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11246 | Join AB+CB into (A+C) on L... |
| recnd 11247 | Deduction from real number... |
| readdcld 11248 | Closure law for addition o... |
| remulcld 11249 | Closure law for multiplica... |
| pnfnre 11260 | Plus infinity is not a rea... |
| pnfnre2 11261 | Plus infinity is not a rea... |
| mnfnre 11262 | Minus infinity is not a re... |
| ressxr 11263 | The standard reals are a s... |
| rexpssxrxp 11264 | The Cartesian product of s... |
| rexr 11265 | A standard real is an exte... |
| 0xr 11266 | Zero is an extended real. ... |
| renepnf 11267 | No (finite) real equals pl... |
| renemnf 11268 | No real equals minus infin... |
| rexrd 11269 | A standard real is an exte... |
| renepnfd 11270 | No (finite) real equals pl... |
| renemnfd 11271 | No real equals minus infin... |
| pnfex 11272 | Plus infinity exists. (Co... |
| pnfxr 11273 | Plus infinity belongs to t... |
| pnfnemnf 11274 | Plus and minus infinity ar... |
| mnfnepnf 11275 | Minus and plus infinity ar... |
| mnfxr 11276 | Minus infinity belongs to ... |
| rexri 11277 | A standard real is an exte... |
| 1xr 11278 | ` 1 ` is an extended real ... |
| renfdisj 11279 | The reals and the infiniti... |
| ltrelxr 11280 | "Less than" is a relation ... |
| ltrel 11281 | "Less than" is a relation.... |
| lerelxr 11282 | "Less than or equal to" is... |
| lerel 11283 | "Less than or equal to" is... |
| xrlenlt 11284 | "Less than or equal to" ex... |
| xrlenltd 11285 | "Less than or equal to" ex... |
| xrltnle 11286 | "Less than" expressed in t... |
| xrnltled 11287 | "Not less than" implies "l... |
| ssxr 11288 | The three (non-exclusive) ... |
| ltxrlt 11289 | The standard less-than ` <... |
| axlttri 11290 | Ordering on reals satisfie... |
| axlttrn 11291 | Ordering on reals is trans... |
| axltadd 11292 | Ordering property of addit... |
| axmulgt0 11293 | The product of two positiv... |
| axsup 11294 | A nonempty, bounded-above ... |
| lttr 11295 | Alias for ~ axlttrn , for ... |
| mulgt0 11296 | The product of two positiv... |
| lenlt 11297 | 'Less than or equal to' ex... |
| ltnle 11298 | 'Less than' expressed in t... |
| ltso 11299 | 'Less than' is a strict or... |
| gtso 11300 | 'Greater than' is a strict... |
| lttri2 11301 | Consequence of trichotomy.... |
| lttri3 11302 | Trichotomy law for 'less t... |
| lttri4 11303 | Trichotomy law for 'less t... |
| letri3 11304 | Trichotomy law. (Contribu... |
| leloe 11305 | 'Less than or equal to' ex... |
| eqlelt 11306 | Equality in terms of 'less... |
| ltle 11307 | 'Less than' implies 'less ... |
| leltne 11308 | 'Less than or equal to' im... |
| lelttr 11309 | Transitive law. (Contribu... |
| leltletr 11310 | Transitive law, weaker for... |
| ltletr 11311 | Transitive law. (Contribu... |
| ltleletr 11312 | Transitive law, weaker for... |
| letr 11313 | Transitive law. (Contribu... |
| ltnr 11314 | 'Less than' is irreflexive... |
| leid 11315 | 'Less than or equal to' is... |
| ltne 11316 | 'Less than' implies not eq... |
| ltnsym 11317 | 'Less than' is not symmetr... |
| ltnsym2 11318 | 'Less than' is antisymmetr... |
| letric 11319 | Trichotomy law. (Contribu... |
| ltlen 11320 | 'Less than' expressed in t... |
| eqle 11321 | Equality implies 'less tha... |
| eqled 11322 | Equality implies 'less tha... |
| ltadd2 11323 | Addition to both sides of ... |
| ne0gt0 11324 | A nonzero nonnegative numb... |
| lecasei 11325 | Ordering elimination by ca... |
| lelttric 11326 | Trichotomy law. (Contribu... |
| ltlecasei 11327 | Ordering elimination by ca... |
| ltnri 11328 | 'Less than' is irreflexive... |
| eqlei 11329 | Equality implies 'less tha... |
| eqlei2 11330 | Equality implies 'less tha... |
| gtneii 11331 | 'Less than' implies not eq... |
| ltneii 11332 | 'Greater than' implies not... |
| lttri2i 11333 | Consequence of trichotomy.... |
| lttri3i 11334 | Consequence of trichotomy.... |
| letri3i 11335 | Consequence of trichotomy.... |
| leloei 11336 | 'Less than or equal to' in... |
| ltleni 11337 | 'Less than' expressed in t... |
| ltnsymi 11338 | 'Less than' is not symmetr... |
| lenlti 11339 | 'Less than or equal to' in... |
| ltnlei 11340 | 'Less than' in terms of 'l... |
| ltlei 11341 | 'Less than' implies 'less ... |
| ltleii 11342 | 'Less than' implies 'less ... |
| ltnei 11343 | 'Less than' implies not eq... |
| letrii 11344 | Trichotomy law for 'less t... |
| lttri 11345 | 'Less than' is transitive.... |
| lelttri 11346 | 'Less than or equal to', '... |
| ltletri 11347 | 'Less than', 'less than or... |
| letri 11348 | 'Less than or equal to' is... |
| le2tri3i 11349 | Extended trichotomy law fo... |
| ltadd2i 11350 | Addition to both sides of ... |
| mulgt0i 11351 | The product of two positiv... |
| mulgt0ii 11352 | The product of two positiv... |
| ltnrd 11353 | 'Less than' is irreflexive... |
| gtned 11354 | 'Less than' implies not eq... |
| ltned 11355 | 'Greater than' implies not... |
| ne0gt0d 11356 | A nonzero nonnegative numb... |
| lttrid 11357 | Ordering on reals satisfie... |
| lttri2d 11358 | Consequence of trichotomy.... |
| lttri3d 11359 | Consequence of trichotomy.... |
| lttri4d 11360 | Trichotomy law for 'less t... |
| letri3d 11361 | Consequence of trichotomy.... |
| leloed 11362 | 'Less than or equal to' in... |
| eqleltd 11363 | Equality in terms of 'less... |
| ltlend 11364 | 'Less than' expressed in t... |
| lenltd 11365 | 'Less than or equal to' in... |
| ltnled 11366 | 'Less than' in terms of 'l... |
| ltled 11367 | 'Less than' implies 'less ... |
| ltnsymd 11368 | 'Less than' implies 'less ... |
| nltled 11369 | 'Not less than ' implies '... |
| lensymd 11370 | 'Less than or equal to' im... |
| letrid 11371 | Trichotomy law for 'less t... |
| leltned 11372 | 'Less than or equal to' im... |
| leneltd 11373 | 'Less than or equal to' an... |
| mulgt0d 11374 | The product of two positiv... |
| ltadd2d 11375 | Addition to both sides of ... |
| letrd 11376 | Transitive law deduction f... |
| lelttrd 11377 | Transitive law deduction f... |
| ltadd2dd 11378 | Addition to both sides of ... |
| ltletrd 11379 | Transitive law deduction f... |
| lttrd 11380 | Transitive law deduction f... |
| lelttrdi 11381 | If a number is less than a... |
| dedekind 11382 | The Dedekind cut theorem. ... |
| dedekindle 11383 | The Dedekind cut theorem, ... |
| mul12 11384 | Commutative/associative la... |
| mul32 11385 | Commutative/associative la... |
| mul31 11386 | Commutative/associative la... |
| mul4 11387 | Rearrangement of 4 factors... |
| mul4r 11388 | Rearrangement of 4 factors... |
| muladd11 11389 | A simple product of sums e... |
| 1p1times 11390 | Two times a number. (Cont... |
| peano2cn 11391 | A theorem for complex numb... |
| peano2re 11392 | A theorem for reals analog... |
| readdcan 11393 | Cancellation law for addit... |
| 00id 11394 | ` 0 ` is its own additive ... |
| mul02lem1 11395 | Lemma for ~ mul02 . If an... |
| mul02lem2 11396 | Lemma for ~ mul02 . Zero ... |
| mul02 11397 | Multiplication by ` 0 ` . ... |
| mul01 11398 | Multiplication by ` 0 ` . ... |
| addrid 11399 | ` 0 ` is an additive ident... |
| cnegex 11400 | Existence of the negative ... |
| cnegex2 11401 | Existence of a left invers... |
| addlid 11402 | ` 0 ` is a left identity f... |
| addcan 11403 | Cancellation law for addit... |
| addcan2 11404 | Cancellation law for addit... |
| addcom 11405 | Addition commutes. This u... |
| addridi 11406 | ` 0 ` is an additive ident... |
| addlidi 11407 | ` 0 ` is a left identity f... |
| mul02i 11408 | Multiplication by 0. Theo... |
| mul01i 11409 | Multiplication by ` 0 ` . ... |
| addcomi 11410 | Addition commutes. Based ... |
| addcomli 11411 | Addition commutes. (Contr... |
| addcani 11412 | Cancellation law for addit... |
| addcan2i 11413 | Cancellation law for addit... |
| mul12i 11414 | Commutative/associative la... |
| mul32i 11415 | Commutative/associative la... |
| mul4i 11416 | Rearrangement of 4 factors... |
| mul02d 11417 | Multiplication by 0. Theo... |
| mul01d 11418 | Multiplication by ` 0 ` . ... |
| addridd 11419 | ` 0 ` is an additive ident... |
| addlidd 11420 | ` 0 ` is a left identity f... |
| addcomd 11421 | Addition commutes. Based ... |
| addcand 11422 | Cancellation law for addit... |
| addcan2d 11423 | Cancellation law for addit... |
| addcanad 11424 | Cancelling a term on the l... |
| addcan2ad 11425 | Cancelling a term on the r... |
| addneintrd 11426 | Introducing a term on the ... |
| addneintr2d 11427 | Introducing a term on the ... |
| mul12d 11428 | Commutative/associative la... |
| mul32d 11429 | Commutative/associative la... |
| mul31d 11430 | Commutative/associative la... |
| mul4d 11431 | Rearrangement of 4 factors... |
| muladd11r 11432 | A simple product of sums e... |
| comraddd 11433 | Commute RHS addition, in d... |
| ltaddneg 11434 | Adding a negative number t... |
| ltaddnegr 11435 | Adding a negative number t... |
| add12 11436 | Commutative/associative la... |
| add32 11437 | Commutative/associative la... |
| add32r 11438 | Commutative/associative la... |
| add4 11439 | Rearrangement of 4 terms i... |
| add42 11440 | Rearrangement of 4 terms i... |
| add12i 11441 | Commutative/associative la... |
| add32i 11442 | Commutative/associative la... |
| add4i 11443 | Rearrangement of 4 terms i... |
| add42i 11444 | Rearrangement of 4 terms i... |
| add12d 11445 | Commutative/associative la... |
| add32d 11446 | Commutative/associative la... |
| add4d 11447 | Rearrangement of 4 terms i... |
| add42d 11448 | Rearrangement of 4 terms i... |
| 0cnALT 11453 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11454 | Alternate proof of ~ 0cnAL... |
| negeu 11455 | Existential uniqueness of ... |
| subval 11456 | Value of subtraction, whic... |
| negeq 11457 | Equality theorem for negat... |
| negeqi 11458 | Equality inference for neg... |
| negeqd 11459 | Equality deduction for neg... |
| nfnegd 11460 | Deduction version of ~ nfn... |
| nfneg 11461 | Bound-variable hypothesis ... |
| csbnegg 11462 | Move class substitution in... |
| negex 11463 | A negative is a set. (Con... |
| subcl 11464 | Closure law for subtractio... |
| negcl 11465 | Closure law for negative. ... |
| negicn 11466 | ` -u _i ` is a complex num... |
| subf 11467 | Subtraction is an operatio... |
| subadd 11468 | Relationship between subtr... |
| subadd2 11469 | Relationship between subtr... |
| subsub23 11470 | Swap subtrahend and result... |
| pncan 11471 | Cancellation law for subtr... |
| pncan2 11472 | Cancellation law for subtr... |
| pncan3 11473 | Subtraction and addition o... |
| npcan 11474 | Cancellation law for subtr... |
| addsubass 11475 | Associative-type law for a... |
| addsub 11476 | Law for addition and subtr... |
| subadd23 11477 | Commutative/associative la... |
| addsub12 11478 | Commutative/associative la... |
| 2addsub 11479 | Law for subtraction and ad... |
| addsubeq4 11480 | Relation between sums and ... |
| pncan3oi 11481 | Subtraction and addition o... |
| mvrraddi 11482 | Move the right term in a s... |
| mvlladdi 11483 | Move the left term in a su... |
| subid 11484 | Subtraction of a number fr... |
| subid1 11485 | Identity law for subtracti... |
| npncan 11486 | Cancellation law for subtr... |
| nppcan 11487 | Cancellation law for subtr... |
| nnpcan 11488 | Cancellation law for subtr... |
| nppcan3 11489 | Cancellation law for subtr... |
| subcan2 11490 | Cancellation law for subtr... |
| subeq0 11491 | If the difference between ... |
| npncan2 11492 | Cancellation law for subtr... |
| subsub2 11493 | Law for double subtraction... |
| nncan 11494 | Cancellation law for subtr... |
| subsub 11495 | Law for double subtraction... |
| nppcan2 11496 | Cancellation law for subtr... |
| subsub3 11497 | Law for double subtraction... |
| subsub4 11498 | Law for double subtraction... |
| sub32 11499 | Swap the second and third ... |
| nnncan 11500 | Cancellation law for subtr... |
| nnncan1 11501 | Cancellation law for subtr... |
| nnncan2 11502 | Cancellation law for subtr... |
| npncan3 11503 | Cancellation law for subtr... |
| pnpcan 11504 | Cancellation law for mixed... |
| pnpcan2 11505 | Cancellation law for mixed... |
| pnncan 11506 | Cancellation law for mixed... |
| ppncan 11507 | Cancellation law for mixed... |
| addsub4 11508 | Rearrangement of 4 terms i... |
| subadd4 11509 | Rearrangement of 4 terms i... |
| sub4 11510 | Rearrangement of 4 terms i... |
| neg0 11511 | Minus 0 equals 0. (Contri... |
| negid 11512 | Addition of a number and i... |
| negsub 11513 | Relationship between subtr... |
| subneg 11514 | Relationship between subtr... |
| negneg 11515 | A number is equal to the n... |
| neg11 11516 | Negative is one-to-one. (... |
| negcon1 11517 | Negative contraposition la... |
| negcon2 11518 | Negative contraposition la... |
| negeq0 11519 | A number is zero iff its n... |
| subcan 11520 | Cancellation law for subtr... |
| negsubdi 11521 | Distribution of negative o... |
| negdi 11522 | Distribution of negative o... |
| negdi2 11523 | Distribution of negative o... |
| negsubdi2 11524 | Distribution of negative o... |
| neg2sub 11525 | Relationship between subtr... |
| renegcli 11526 | Closure law for negative o... |
| resubcli 11527 | Closure law for subtractio... |
| renegcl 11528 | Closure law for negative o... |
| resubcl 11529 | Closure law for subtractio... |
| negreb 11530 | The negative of a real is ... |
| peano2cnm 11531 | "Reverse" second Peano pos... |
| peano2rem 11532 | "Reverse" second Peano pos... |
| negcli 11533 | Closure law for negative. ... |
| negidi 11534 | Addition of a number and i... |
| negnegi 11535 | A number is equal to the n... |
| subidi 11536 | Subtraction of a number fr... |
| subid1i 11537 | Identity law for subtracti... |
| negne0bi 11538 | A number is nonzero iff it... |
| negrebi 11539 | The negative of a real is ... |
| negne0i 11540 | The negative of a nonzero ... |
| subcli 11541 | Closure law for subtractio... |
| pncan3i 11542 | Subtraction and addition o... |
| negsubi 11543 | Relationship between subtr... |
| subnegi 11544 | Relationship between subtr... |
| subeq0i 11545 | If the difference between ... |
| neg11i 11546 | Negative is one-to-one. (... |
| negcon1i 11547 | Negative contraposition la... |
| negcon2i 11548 | Negative contraposition la... |
| negdii 11549 | Distribution of negative o... |
| negsubdii 11550 | Distribution of negative o... |
| negsubdi2i 11551 | Distribution of negative o... |
| subaddi 11552 | Relationship between subtr... |
| subadd2i 11553 | Relationship between subtr... |
| subaddrii 11554 | Relationship between subtr... |
| subsub23i 11555 | Swap subtrahend and result... |
| addsubassi 11556 | Associative-type law for s... |
| addsubi 11557 | Law for subtraction and ad... |
| subcani 11558 | Cancellation law for subtr... |
| subcan2i 11559 | Cancellation law for subtr... |
| pnncani 11560 | Cancellation law for mixed... |
| addsub4i 11561 | Rearrangement of 4 terms i... |
| 0reALT 11562 | Alternate proof of ~ 0re .... |
| negcld 11563 | Closure law for negative. ... |
| subidd 11564 | Subtraction of a number fr... |
| subid1d 11565 | Identity law for subtracti... |
| negidd 11566 | Addition of a number and i... |
| negnegd 11567 | A number is equal to the n... |
| negeq0d 11568 | A number is zero iff its n... |
| negne0bd 11569 | A number is nonzero iff it... |
| negcon1d 11570 | Contraposition law for una... |
| negcon1ad 11571 | Contraposition law for una... |
| neg11ad 11572 | The negatives of two compl... |
| negned 11573 | If two complex numbers are... |
| negne0d 11574 | The negative of a nonzero ... |
| negrebd 11575 | The negative of a real is ... |
| subcld 11576 | Closure law for subtractio... |
| pncand 11577 | Cancellation law for subtr... |
| pncan2d 11578 | Cancellation law for subtr... |
| pncan3d 11579 | Subtraction and addition o... |
| npcand 11580 | Cancellation law for subtr... |
| nncand 11581 | Cancellation law for subtr... |
| negsubd 11582 | Relationship between subtr... |
| subnegd 11583 | Relationship between subtr... |
| subeq0d 11584 | If the difference between ... |
| subne0d 11585 | Two unequal numbers have n... |
| subeq0ad 11586 | The difference of two comp... |
| subne0ad 11587 | If the difference of two c... |
| neg11d 11588 | If the difference between ... |
| negdid 11589 | Distribution of negative o... |
| negdi2d 11590 | Distribution of negative o... |
| negsubdid 11591 | Distribution of negative o... |
| negsubdi2d 11592 | Distribution of negative o... |
| neg2subd 11593 | Relationship between subtr... |
| subaddd 11594 | Relationship between subtr... |
| subadd2d 11595 | Relationship between subtr... |
| addsubassd 11596 | Associative-type law for s... |
| addsubd 11597 | Law for subtraction and ad... |
| subadd23d 11598 | Commutative/associative la... |
| addsub12d 11599 | Commutative/associative la... |
| npncand 11600 | Cancellation law for subtr... |
| nppcand 11601 | Cancellation law for subtr... |
| nppcan2d 11602 | Cancellation law for subtr... |
| nppcan3d 11603 | Cancellation law for subtr... |
| subsubd 11604 | Law for double subtraction... |
| subsub2d 11605 | Law for double subtraction... |
| subsub3d 11606 | Law for double subtraction... |
| subsub4d 11607 | Law for double subtraction... |
| sub32d 11608 | Swap the second and third ... |
| nnncand 11609 | Cancellation law for subtr... |
| nnncan1d 11610 | Cancellation law for subtr... |
| nnncan2d 11611 | Cancellation law for subtr... |
| npncan3d 11612 | Cancellation law for subtr... |
| pnpcand 11613 | Cancellation law for mixed... |
| pnpcan2d 11614 | Cancellation law for mixed... |
| pnncand 11615 | Cancellation law for mixed... |
| ppncand 11616 | Cancellation law for mixed... |
| subcand 11617 | Cancellation law for subtr... |
| subcan2d 11618 | Cancellation law for subtr... |
| subcanad 11619 | Cancellation law for subtr... |
| subneintrd 11620 | Introducing subtraction on... |
| subcan2ad 11621 | Cancellation law for subtr... |
| subneintr2d 11622 | Introducing subtraction on... |
| addsub4d 11623 | Rearrangement of 4 terms i... |
| subadd4d 11624 | Rearrangement of 4 terms i... |
| sub4d 11625 | Rearrangement of 4 terms i... |
| 2addsubd 11626 | Law for subtraction and ad... |
| addsubeq4d 11627 | Relation between sums and ... |
| subeqxfrd 11628 | Transfer two terms of a su... |
| mvlraddd 11629 | Move the right term in a s... |
| mvlladdd 11630 | Move the left term in a su... |
| mvrraddd 11631 | Move the right term in a s... |
| mvrladdd 11632 | Move the left term in a su... |
| assraddsubd 11633 | Associate RHS addition-sub... |
| subaddeqd 11634 | Transfer two terms of a su... |
| addlsub 11635 | Left-subtraction: Subtrac... |
| addrsub 11636 | Right-subtraction: Subtra... |
| subexsub 11637 | A subtraction law: Exchan... |
| addid0 11638 | If adding a number to a an... |
| addn0nid 11639 | Adding a nonzero number to... |
| pnpncand 11640 | Addition/subtraction cance... |
| subeqrev 11641 | Reverse the order of subtr... |
| addeq0 11642 | Two complex numbers add up... |
| pncan1 11643 | Cancellation law for addit... |
| npcan1 11644 | Cancellation law for subtr... |
| subeq0bd 11645 | If two complex numbers are... |
| renegcld 11646 | Closure law for negative o... |
| resubcld 11647 | Closure law for subtractio... |
| negn0 11648 | The image under negation o... |
| negf1o 11649 | Negation is an isomorphism... |
| kcnktkm1cn 11650 | k times k minus 1 is a com... |
| muladd 11651 | Product of two sums. (Con... |
| subdi 11652 | Distribution of multiplica... |
| subdir 11653 | Distribution of multiplica... |
| ine0 11654 | The imaginary unit ` _i ` ... |
| mulneg1 11655 | Product with negative is n... |
| mulneg2 11656 | The product with a negativ... |
| mulneg12 11657 | Swap the negative sign in ... |
| mul2neg 11658 | Product of two negatives. ... |
| submul2 11659 | Convert a subtraction to a... |
| mulm1 11660 | Product with minus one is ... |
| addneg1mul 11661 | Addition with product with... |
| mulsub 11662 | Product of two differences... |
| mulsub2 11663 | Swap the order of subtract... |
| mulm1i 11664 | Product with minus one is ... |
| mulneg1i 11665 | Product with negative is n... |
| mulneg2i 11666 | Product with negative is n... |
| mul2negi 11667 | Product of two negatives. ... |
| subdii 11668 | Distribution of multiplica... |
| subdiri 11669 | Distribution of multiplica... |
| muladdi 11670 | Product of two sums. (Con... |
| mulm1d 11671 | Product with minus one is ... |
| mulneg1d 11672 | Product with negative is n... |
| mulneg2d 11673 | Product with negative is n... |
| mul2negd 11674 | Product of two negatives. ... |
| subdid 11675 | Distribution of multiplica... |
| subdird 11676 | Distribution of multiplica... |
| muladdd 11677 | Product of two sums. (Con... |
| mulsubd 11678 | Product of two differences... |
| muls1d 11679 | Multiplication by one minu... |
| mulsubfacd 11680 | Multiplication followed by... |
| addmulsub 11681 | The product of a sum and a... |
| subaddmulsub 11682 | The difference with a prod... |
| mulsubaddmulsub 11683 | A special difference of a ... |
| gt0ne0 11684 | Positive implies nonzero. ... |
| lt0ne0 11685 | A number which is less tha... |
| ltadd1 11686 | Addition to both sides of ... |
| leadd1 11687 | Addition to both sides of ... |
| leadd2 11688 | Addition to both sides of ... |
| ltsubadd 11689 | 'Less than' relationship b... |
| ltsubadd2 11690 | 'Less than' relationship b... |
| lesubadd 11691 | 'Less than or equal to' re... |
| lesubadd2 11692 | 'Less than or equal to' re... |
| ltaddsub 11693 | 'Less than' relationship b... |
| ltaddsub2 11694 | 'Less than' relationship b... |
| leaddsub 11695 | 'Less than or equal to' re... |
| leaddsub2 11696 | 'Less than or equal to' re... |
| suble 11697 | Swap subtrahends in an ine... |
| lesub 11698 | Swap subtrahends in an ine... |
| ltsub23 11699 | 'Less than' relationship b... |
| ltsub13 11700 | 'Less than' relationship b... |
| le2add 11701 | Adding both sides of two '... |
| ltleadd 11702 | Adding both sides of two o... |
| leltadd 11703 | Adding both sides of two o... |
| lt2add 11704 | Adding both sides of two '... |
| addgt0 11705 | The sum of 2 positive numb... |
| addgegt0 11706 | The sum of nonnegative and... |
| addgtge0 11707 | The sum of nonnegative and... |
| addge0 11708 | The sum of 2 nonnegative n... |
| ltaddpos 11709 | Adding a positive number t... |
| ltaddpos2 11710 | Adding a positive number t... |
| ltsubpos 11711 | Subtracting a positive num... |
| posdif 11712 | Comparison of two numbers ... |
| lesub1 11713 | Subtraction from both side... |
| lesub2 11714 | Subtraction of both sides ... |
| ltsub1 11715 | Subtraction from both side... |
| ltsub2 11716 | Subtraction of both sides ... |
| lt2sub 11717 | Subtracting both sides of ... |
| le2sub 11718 | Subtracting both sides of ... |
| ltneg 11719 | Negative of both sides of ... |
| ltnegcon1 11720 | Contraposition of negative... |
| ltnegcon2 11721 | Contraposition of negative... |
| leneg 11722 | Negative of both sides of ... |
| lenegcon1 11723 | Contraposition of negative... |
| lenegcon2 11724 | Contraposition of negative... |
| lt0neg1 11725 | Comparison of a number and... |
| lt0neg2 11726 | Comparison of a number and... |
| le0neg1 11727 | Comparison of a number and... |
| le0neg2 11728 | Comparison of a number and... |
| addge01 11729 | A number is less than or e... |
| addge02 11730 | A number is less than or e... |
| add20 11731 | Two nonnegative numbers ar... |
| subge0 11732 | Nonnegative subtraction. ... |
| suble0 11733 | Nonpositive subtraction. ... |
| leaddle0 11734 | The sum of a real number a... |
| subge02 11735 | Nonnegative subtraction. ... |
| lesub0 11736 | Lemma to show a nonnegativ... |
| mulge0 11737 | The product of two nonnega... |
| mullt0 11738 | The product of two negativ... |
| msqgt0 11739 | A nonzero square is positi... |
| msqge0 11740 | A square is nonnegative. ... |
| 0lt1 11741 | 0 is less than 1. Theorem... |
| 0le1 11742 | 0 is less than or equal to... |
| relin01 11743 | An interval law for less t... |
| ltordlem 11744 | Lemma for ~ ltord1 . (Con... |
| ltord1 11745 | Infer an ordering relation... |
| leord1 11746 | Infer an ordering relation... |
| eqord1 11747 | A strictly increasing real... |
| ltord2 11748 | Infer an ordering relation... |
| leord2 11749 | Infer an ordering relation... |
| eqord2 11750 | A strictly decreasing real... |
| wloglei 11751 | Form of ~ wlogle where bot... |
| wlogle 11752 | If the predicate ` ch ( x ... |
| leidi 11753 | 'Less than or equal to' is... |
| gt0ne0i 11754 | Positive means nonzero (us... |
| gt0ne0ii 11755 | Positive implies nonzero. ... |
| msqgt0i 11756 | A nonzero square is positi... |
| msqge0i 11757 | A square is nonnegative. ... |
| addgt0i 11758 | Addition of 2 positive num... |
| addge0i 11759 | Addition of 2 nonnegative ... |
| addgegt0i 11760 | Addition of nonnegative an... |
| addgt0ii 11761 | Addition of 2 positive num... |
| add20i 11762 | Two nonnegative numbers ar... |
| ltnegi 11763 | Negative of both sides of ... |
| lenegi 11764 | Negative of both sides of ... |
| ltnegcon2i 11765 | Contraposition of negative... |
| mulge0i 11766 | The product of two nonnega... |
| lesub0i 11767 | Lemma to show a nonnegativ... |
| ltaddposi 11768 | Adding a positive number t... |
| posdifi 11769 | Comparison of two numbers ... |
| ltnegcon1i 11770 | Contraposition of negative... |
| lenegcon1i 11771 | Contraposition of negative... |
| subge0i 11772 | Nonnegative subtraction. ... |
| ltadd1i 11773 | Addition to both sides of ... |
| leadd1i 11774 | Addition to both sides of ... |
| leadd2i 11775 | Addition to both sides of ... |
| ltsubaddi 11776 | 'Less than' relationship b... |
| lesubaddi 11777 | 'Less than or equal to' re... |
| ltsubadd2i 11778 | 'Less than' relationship b... |
| lesubadd2i 11779 | 'Less than or equal to' re... |
| ltaddsubi 11780 | 'Less than' relationship b... |
| lt2addi 11781 | Adding both side of two in... |
| le2addi 11782 | Adding both side of two in... |
| gt0ne0d 11783 | Positive implies nonzero. ... |
| lt0ne0d 11784 | Something less than zero i... |
| leidd 11785 | 'Less than or equal to' is... |
| msqgt0d 11786 | A nonzero square is positi... |
| msqge0d 11787 | A square is nonnegative. ... |
| lt0neg1d 11788 | Comparison of a number and... |
| lt0neg2d 11789 | Comparison of a number and... |
| le0neg1d 11790 | Comparison of a number and... |
| le0neg2d 11791 | Comparison of a number and... |
| addgegt0d 11792 | Addition of nonnegative an... |
| addgtge0d 11793 | Addition of positive and n... |
| addgt0d 11794 | Addition of 2 positive num... |
| addge0d 11795 | Addition of 2 nonnegative ... |
| mulge0d 11796 | The product of two nonnega... |
| ltnegd 11797 | Negative of both sides of ... |
| lenegd 11798 | Negative of both sides of ... |
| ltnegcon1d 11799 | Contraposition of negative... |
| ltnegcon2d 11800 | Contraposition of negative... |
| lenegcon1d 11801 | Contraposition of negative... |
| lenegcon2d 11802 | Contraposition of negative... |
| ltaddposd 11803 | Adding a positive number t... |
| ltaddpos2d 11804 | Adding a positive number t... |
| ltsubposd 11805 | Subtracting a positive num... |
| posdifd 11806 | Comparison of two numbers ... |
| addge01d 11807 | A number is less than or e... |
| addge02d 11808 | A number is less than or e... |
| subge0d 11809 | Nonnegative subtraction. ... |
| suble0d 11810 | Nonpositive subtraction. ... |
| subge02d 11811 | Nonnegative subtraction. ... |
| ltadd1d 11812 | Addition to both sides of ... |
| leadd1d 11813 | Addition to both sides of ... |
| leadd2d 11814 | Addition to both sides of ... |
| ltsubaddd 11815 | 'Less than' relationship b... |
| lesubaddd 11816 | 'Less than or equal to' re... |
| ltsubadd2d 11817 | 'Less than' relationship b... |
| lesubadd2d 11818 | 'Less than or equal to' re... |
| ltaddsubd 11819 | 'Less than' relationship b... |
| ltaddsub2d 11820 | 'Less than' relationship b... |
| leaddsub2d 11821 | 'Less than or equal to' re... |
| subled 11822 | Swap subtrahends in an ine... |
| lesubd 11823 | Swap subtrahends in an ine... |
| ltsub23d 11824 | 'Less than' relationship b... |
| ltsub13d 11825 | 'Less than' relationship b... |
| lesub1d 11826 | Subtraction from both side... |
| lesub2d 11827 | Subtraction of both sides ... |
| ltsub1d 11828 | Subtraction from both side... |
| ltsub2d 11829 | Subtraction of both sides ... |
| ltadd1dd 11830 | Addition to both sides of ... |
| ltsub1dd 11831 | Subtraction from both side... |
| ltsub2dd 11832 | Subtraction of both sides ... |
| leadd1dd 11833 | Addition to both sides of ... |
| leadd2dd 11834 | Addition to both sides of ... |
| lesub1dd 11835 | Subtraction from both side... |
| lesub2dd 11836 | Subtraction of both sides ... |
| lesub3d 11837 | The result of subtracting ... |
| le2addd 11838 | Adding both side of two in... |
| le2subd 11839 | Subtracting both sides of ... |
| ltleaddd 11840 | Adding both sides of two o... |
| leltaddd 11841 | Adding both sides of two o... |
| lt2addd 11842 | Adding both side of two in... |
| lt2subd 11843 | Subtracting both sides of ... |
| possumd 11844 | Condition for a positive s... |
| sublt0d 11845 | When a subtraction gives a... |
| ltaddsublt 11846 | Addition and subtraction o... |
| 1le1 11847 | One is less than or equal ... |
| ixi 11848 | ` _i ` times itself is min... |
| recextlem1 11849 | Lemma for ~ recex . (Cont... |
| recextlem2 11850 | Lemma for ~ recex . (Cont... |
| recex 11851 | Existence of reciprocal of... |
| mulcand 11852 | Cancellation law for multi... |
| mulcan2d 11853 | Cancellation law for multi... |
| mulcanad 11854 | Cancellation of a nonzero ... |
| mulcan2ad 11855 | Cancellation of a nonzero ... |
| mulcan 11856 | Cancellation law for multi... |
| mulcan2 11857 | Cancellation law for multi... |
| mulcani 11858 | Cancellation law for multi... |
| mul0or 11859 | If a product is zero, one ... |
| mulne0b 11860 | The product of two nonzero... |
| mulne0 11861 | The product of two nonzero... |
| mulne0i 11862 | The product of two nonzero... |
| muleqadd 11863 | Property of numbers whose ... |
| receu 11864 | Existential uniqueness of ... |
| mulnzcnopr 11865 | Multiplication maps nonzer... |
| msq0i 11866 | A number is zero iff its s... |
| mul0ori 11867 | If a product is zero, one ... |
| msq0d 11868 | A number is zero iff its s... |
| mul0ord 11869 | If a product is zero, one ... |
| mulne0bd 11870 | The product of two nonzero... |
| mulne0d 11871 | The product of two nonzero... |
| mulcan1g 11872 | A generalized form of the ... |
| mulcan2g 11873 | A generalized form of the ... |
| mulne0bad 11874 | A factor of a nonzero comp... |
| mulne0bbd 11875 | A factor of a nonzero comp... |
| 1div0 11878 | You can't divide by zero, ... |
| divval 11879 | Value of division: if ` A ... |
| divmul 11880 | Relationship between divis... |
| divmul2 11881 | Relationship between divis... |
| divmul3 11882 | Relationship between divis... |
| divcl 11883 | Closure law for division. ... |
| reccl 11884 | Closure law for reciprocal... |
| divcan2 11885 | A cancellation law for div... |
| divcan1 11886 | A cancellation law for div... |
| diveq0 11887 | A ratio is zero iff the nu... |
| divne0b 11888 | The ratio of nonzero numbe... |
| divne0 11889 | The ratio of nonzero numbe... |
| recne0 11890 | The reciprocal of a nonzer... |
| recid 11891 | Multiplication of a number... |
| recid2 11892 | Multiplication of a number... |
| divrec 11893 | Relationship between divis... |
| divrec2 11894 | Relationship between divis... |
| divass 11895 | An associative law for div... |
| div23 11896 | A commutative/associative ... |
| div32 11897 | A commutative/associative ... |
| div13 11898 | A commutative/associative ... |
| div12 11899 | A commutative/associative ... |
| divmulass 11900 | An associative law for div... |
| divmulasscom 11901 | An associative/commutative... |
| divdir 11902 | Distribution of division o... |
| divcan3 11903 | A cancellation law for div... |
| divcan4 11904 | A cancellation law for div... |
| div11 11905 | One-to-one relationship fo... |
| divid 11906 | A number divided by itself... |
| div0 11907 | Division into zero is zero... |
| div1 11908 | A number divided by 1 is i... |
| 1div1e1 11909 | 1 divided by 1 is 1. (Con... |
| diveq1 11910 | Equality in terms of unit ... |
| divneg 11911 | Move negative sign inside ... |
| muldivdir 11912 | Distribution of division o... |
| divsubdir 11913 | Distribution of division o... |
| subdivcomb1 11914 | Bring a term in a subtract... |
| subdivcomb2 11915 | Bring a term in a subtract... |
| recrec 11916 | A number is equal to the r... |
| rec11 11917 | Reciprocal is one-to-one. ... |
| rec11r 11918 | Mutual reciprocals. (Cont... |
| divmuldiv 11919 | Multiplication of two rati... |
| divdivdiv 11920 | Division of two ratios. T... |
| divcan5 11921 | Cancellation of common fac... |
| divmul13 11922 | Swap the denominators in t... |
| divmul24 11923 | Swap the numerators in the... |
| divmuleq 11924 | Cross-multiply in an equal... |
| recdiv 11925 | The reciprocal of a ratio.... |
| divcan6 11926 | Cancellation of inverted f... |
| divdiv32 11927 | Swap denominators in a div... |
| divcan7 11928 | Cancel equal divisors in a... |
| dmdcan 11929 | Cancellation law for divis... |
| divdiv1 11930 | Division into a fraction. ... |
| divdiv2 11931 | Division by a fraction. (... |
| recdiv2 11932 | Division into a reciprocal... |
| ddcan 11933 | Cancellation in a double d... |
| divadddiv 11934 | Addition of two ratios. T... |
| divsubdiv 11935 | Subtraction of two ratios.... |
| conjmul 11936 | Two numbers whose reciproc... |
| rereccl 11937 | Closure law for reciprocal... |
| redivcl 11938 | Closure law for division o... |
| eqneg 11939 | A number equal to its nega... |
| eqnegd 11940 | A complex number equals it... |
| eqnegad 11941 | If a complex number equals... |
| div2neg 11942 | Quotient of two negatives.... |
| divneg2 11943 | Move negative sign inside ... |
| recclzi 11944 | Closure law for reciprocal... |
| recne0zi 11945 | The reciprocal of a nonzer... |
| recidzi 11946 | Multiplication of a number... |
| div1i 11947 | A number divided by 1 is i... |
| eqnegi 11948 | A number equal to its nega... |
| reccli 11949 | Closure law for reciprocal... |
| recidi 11950 | Multiplication of a number... |
| recreci 11951 | A number is equal to the r... |
| dividi 11952 | A number divided by itself... |
| div0i 11953 | Division into zero is zero... |
| divclzi 11954 | Closure law for division. ... |
| divcan1zi 11955 | A cancellation law for div... |
| divcan2zi 11956 | A cancellation law for div... |
| divreczi 11957 | Relationship between divis... |
| divcan3zi 11958 | A cancellation law for div... |
| divcan4zi 11959 | A cancellation law for div... |
| rec11i 11960 | Reciprocal is one-to-one. ... |
| divcli 11961 | Closure law for division. ... |
| divcan2i 11962 | A cancellation law for div... |
| divcan1i 11963 | A cancellation law for div... |
| divreci 11964 | Relationship between divis... |
| divcan3i 11965 | A cancellation law for div... |
| divcan4i 11966 | A cancellation law for div... |
| divne0i 11967 | The ratio of nonzero numbe... |
| rec11ii 11968 | Reciprocal is one-to-one. ... |
| divasszi 11969 | An associative law for div... |
| divmulzi 11970 | Relationship between divis... |
| divdirzi 11971 | Distribution of division o... |
| divdiv23zi 11972 | Swap denominators in a div... |
| divmuli 11973 | Relationship between divis... |
| divdiv32i 11974 | Swap denominators in a div... |
| divassi 11975 | An associative law for div... |
| divdiri 11976 | Distribution of division o... |
| div23i 11977 | A commutative/associative ... |
| div11i 11978 | One-to-one relationship fo... |
| divmuldivi 11979 | Multiplication of two rati... |
| divmul13i 11980 | Swap denominators of two r... |
| divadddivi 11981 | Addition of two ratios. T... |
| divdivdivi 11982 | Division of two ratios. T... |
| rerecclzi 11983 | Closure law for reciprocal... |
| rereccli 11984 | Closure law for reciprocal... |
| redivclzi 11985 | Closure law for division o... |
| redivcli 11986 | Closure law for division o... |
| div1d 11987 | A number divided by 1 is i... |
| reccld 11988 | Closure law for reciprocal... |
| recne0d 11989 | The reciprocal of a nonzer... |
| recidd 11990 | Multiplication of a number... |
| recid2d 11991 | Multiplication of a number... |
| recrecd 11992 | A number is equal to the r... |
| dividd 11993 | A number divided by itself... |
| div0d 11994 | Division into zero is zero... |
| divcld 11995 | Closure law for division. ... |
| divcan1d 11996 | A cancellation law for div... |
| divcan2d 11997 | A cancellation law for div... |
| divrecd 11998 | Relationship between divis... |
| divrec2d 11999 | Relationship between divis... |
| divcan3d 12000 | A cancellation law for div... |
| divcan4d 12001 | A cancellation law for div... |
| diveq0d 12002 | A ratio is zero iff the nu... |
| diveq1d 12003 | Equality in terms of unit ... |
| diveq1ad 12004 | The quotient of two comple... |
| diveq0ad 12005 | A fraction of complex numb... |
| divne1d 12006 | If two complex numbers are... |
| divne0bd 12007 | A ratio is zero iff the nu... |
| divnegd 12008 | Move negative sign inside ... |
| divneg2d 12009 | Move negative sign inside ... |
| div2negd 12010 | Quotient of two negatives.... |
| divne0d 12011 | The ratio of nonzero numbe... |
| recdivd 12012 | The reciprocal of a ratio.... |
| recdiv2d 12013 | Division into a reciprocal... |
| divcan6d 12014 | Cancellation of inverted f... |
| ddcand 12015 | Cancellation in a double d... |
| rec11d 12016 | Reciprocal is one-to-one. ... |
| divmuld 12017 | Relationship between divis... |
| div32d 12018 | A commutative/associative ... |
| div13d 12019 | A commutative/associative ... |
| divdiv32d 12020 | Swap denominators in a div... |
| divcan5d 12021 | Cancellation of common fac... |
| divcan5rd 12022 | Cancellation of common fac... |
| divcan7d 12023 | Cancel equal divisors in a... |
| dmdcand 12024 | Cancellation law for divis... |
| dmdcan2d 12025 | Cancellation law for divis... |
| divdiv1d 12026 | Division into a fraction. ... |
| divdiv2d 12027 | Division by a fraction. (... |
| divmul2d 12028 | Relationship between divis... |
| divmul3d 12029 | Relationship between divis... |
| divassd 12030 | An associative law for div... |
| div12d 12031 | A commutative/associative ... |
| div23d 12032 | A commutative/associative ... |
| divdird 12033 | Distribution of division o... |
| divsubdird 12034 | Distribution of division o... |
| div11d 12035 | One-to-one relationship fo... |
| divmuldivd 12036 | Multiplication of two rati... |
| divmul13d 12037 | Swap denominators of two r... |
| divmul24d 12038 | Swap the numerators in the... |
| divadddivd 12039 | Addition of two ratios. T... |
| divsubdivd 12040 | Subtraction of two ratios.... |
| divmuleqd 12041 | Cross-multiply in an equal... |
| divdivdivd 12042 | Division of two ratios. T... |
| diveq1bd 12043 | If two complex numbers are... |
| div2sub 12044 | Swap the order of subtract... |
| div2subd 12045 | Swap subtrahend and minuen... |
| rereccld 12046 | Closure law for reciprocal... |
| redivcld 12047 | Closure law for division o... |
| subrec 12048 | Subtraction of reciprocals... |
| subreci 12049 | Subtraction of reciprocals... |
| subrecd 12050 | Subtraction of reciprocals... |
| mvllmuld 12051 | Move the left term in a pr... |
| mvllmuli 12052 | Move the left term in a pr... |
| ldiv 12053 | Left-division. (Contribut... |
| rdiv 12054 | Right-division. (Contribu... |
| mdiv 12055 | A division law. (Contribu... |
| lineq 12056 | Solution of a (scalar) lin... |
| elimgt0 12057 | Hypothesis for weak deduct... |
| elimge0 12058 | Hypothesis for weak deduct... |
| ltp1 12059 | A number is less than itse... |
| lep1 12060 | A number is less than or e... |
| ltm1 12061 | A number minus 1 is less t... |
| lem1 12062 | A number minus 1 is less t... |
| letrp1 12063 | A transitive property of '... |
| p1le 12064 | A transitive property of p... |
| recgt0 12065 | The reciprocal of a positi... |
| prodgt0 12066 | Infer that a multiplicand ... |
| prodgt02 12067 | Infer that a multiplier is... |
| ltmul1a 12068 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12069 | Multiplication of both sid... |
| ltmul2 12070 | Multiplication of both sid... |
| lemul1 12071 | Multiplication of both sid... |
| lemul2 12072 | Multiplication of both sid... |
| lemul1a 12073 | Multiplication of both sid... |
| lemul2a 12074 | Multiplication of both sid... |
| ltmul12a 12075 | Comparison of product of t... |
| lemul12b 12076 | Comparison of product of t... |
| lemul12a 12077 | Comparison of product of t... |
| mulgt1 12078 | The product of two numbers... |
| ltmulgt11 12079 | Multiplication by a number... |
| ltmulgt12 12080 | Multiplication by a number... |
| lemulge11 12081 | Multiplication by a number... |
| lemulge12 12082 | Multiplication by a number... |
| ltdiv1 12083 | Division of both sides of ... |
| lediv1 12084 | Division of both sides of ... |
| gt0div 12085 | Division of a positive num... |
| ge0div 12086 | Division of a nonnegative ... |
| divgt0 12087 | The ratio of two positive ... |
| divge0 12088 | The ratio of nonnegative a... |
| mulge0b 12089 | A condition for multiplica... |
| mulle0b 12090 | A condition for multiplica... |
| mulsuble0b 12091 | A condition for multiplica... |
| ltmuldiv 12092 | 'Less than' relationship b... |
| ltmuldiv2 12093 | 'Less than' relationship b... |
| ltdivmul 12094 | 'Less than' relationship b... |
| ledivmul 12095 | 'Less than or equal to' re... |
| ltdivmul2 12096 | 'Less than' relationship b... |
| lt2mul2div 12097 | 'Less than' relationship b... |
| ledivmul2 12098 | 'Less than or equal to' re... |
| lemuldiv 12099 | 'Less than or equal' relat... |
| lemuldiv2 12100 | 'Less than or equal' relat... |
| ltrec 12101 | The reciprocal of both sid... |
| lerec 12102 | The reciprocal of both sid... |
| lt2msq1 12103 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12104 | Two nonnegative numbers co... |
| ltdiv2 12105 | Division of a positive num... |
| ltrec1 12106 | Reciprocal swap in a 'less... |
| lerec2 12107 | Reciprocal swap in a 'less... |
| ledivdiv 12108 | Invert ratios of positive ... |
| lediv2 12109 | Division of a positive num... |
| ltdiv23 12110 | Swap denominator with othe... |
| lediv23 12111 | Swap denominator with othe... |
| lediv12a 12112 | Comparison of ratio of two... |
| lediv2a 12113 | Division of both sides of ... |
| reclt1 12114 | The reciprocal of a positi... |
| recgt1 12115 | The reciprocal of a positi... |
| recgt1i 12116 | The reciprocal of a number... |
| recp1lt1 12117 | Construct a number less th... |
| recreclt 12118 | Given a positive number ` ... |
| le2msq 12119 | The square function on non... |
| msq11 12120 | The square of a nonnegativ... |
| ledivp1 12121 | "Less than or equal to" an... |
| squeeze0 12122 | If a nonnegative number is... |
| ltp1i 12123 | A number is less than itse... |
| recgt0i 12124 | The reciprocal of a positi... |
| recgt0ii 12125 | The reciprocal of a positi... |
| prodgt0i 12126 | Infer that a multiplicand ... |
| divgt0i 12127 | The ratio of two positive ... |
| divge0i 12128 | The ratio of nonnegative a... |
| ltreci 12129 | The reciprocal of both sid... |
| lereci 12130 | The reciprocal of both sid... |
| lt2msqi 12131 | The square function on non... |
| le2msqi 12132 | The square function on non... |
| msq11i 12133 | The square of a nonnegativ... |
| divgt0i2i 12134 | The ratio of two positive ... |
| ltrecii 12135 | The reciprocal of both sid... |
| divgt0ii 12136 | The ratio of two positive ... |
| ltmul1i 12137 | Multiplication of both sid... |
| ltdiv1i 12138 | Division of both sides of ... |
| ltmuldivi 12139 | 'Less than' relationship b... |
| ltmul2i 12140 | Multiplication of both sid... |
| lemul1i 12141 | Multiplication of both sid... |
| lemul2i 12142 | Multiplication of both sid... |
| ltdiv23i 12143 | Swap denominator with othe... |
| ledivp1i 12144 | "Less than or equal to" an... |
| ltdivp1i 12145 | Less-than and division rel... |
| ltdiv23ii 12146 | Swap denominator with othe... |
| ltmul1ii 12147 | Multiplication of both sid... |
| ltdiv1ii 12148 | Division of both sides of ... |
| ltp1d 12149 | A number is less than itse... |
| lep1d 12150 | A number is less than or e... |
| ltm1d 12151 | A number minus 1 is less t... |
| lem1d 12152 | A number minus 1 is less t... |
| recgt0d 12153 | The reciprocal of a positi... |
| divgt0d 12154 | The ratio of two positive ... |
| mulgt1d 12155 | The product of two numbers... |
| lemulge11d 12156 | Multiplication by a number... |
| lemulge12d 12157 | Multiplication by a number... |
| lemul1ad 12158 | Multiplication of both sid... |
| lemul2ad 12159 | Multiplication of both sid... |
| ltmul12ad 12160 | Comparison of product of t... |
| lemul12ad 12161 | Comparison of product of t... |
| lemul12bd 12162 | Comparison of product of t... |
| fimaxre 12163 | A finite set of real numbe... |
| fimaxre2 12164 | A nonempty finite set of r... |
| fimaxre3 12165 | A nonempty finite set of r... |
| fiminre 12166 | A nonempty finite set of r... |
| fiminre2 12167 | A nonempty finite set of r... |
| negfi 12168 | The negation of a finite s... |
| lbreu 12169 | If a set of reals contains... |
| lbcl 12170 | If a set of reals contains... |
| lble 12171 | If a set of reals contains... |
| lbinf 12172 | If a set of reals contains... |
| lbinfcl 12173 | If a set of reals contains... |
| lbinfle 12174 | If a set of reals contains... |
| sup2 12175 | A nonempty, bounded-above ... |
| sup3 12176 | A version of the completen... |
| infm3lem 12177 | Lemma for ~ infm3 . (Cont... |
| infm3 12178 | The completeness axiom for... |
| suprcl 12179 | Closure of supremum of a n... |
| suprub 12180 | A member of a nonempty bou... |
| suprubd 12181 | Natural deduction form of ... |
| suprcld 12182 | Natural deduction form of ... |
| suprlub 12183 | The supremum of a nonempty... |
| suprnub 12184 | An upper bound is not less... |
| suprleub 12185 | The supremum of a nonempty... |
| supaddc 12186 | The supremum function dist... |
| supadd 12187 | The supremum function dist... |
| supmul1 12188 | The supremum function dist... |
| supmullem1 12189 | Lemma for ~ supmul . (Con... |
| supmullem2 12190 | Lemma for ~ supmul . (Con... |
| supmul 12191 | The supremum function dist... |
| sup3ii 12192 | A version of the completen... |
| suprclii 12193 | Closure of supremum of a n... |
| suprubii 12194 | A member of a nonempty bou... |
| suprlubii 12195 | The supremum of a nonempty... |
| suprnubii 12196 | An upper bound is not less... |
| suprleubii 12197 | The supremum of a nonempty... |
| riotaneg 12198 | The negative of the unique... |
| negiso 12199 | Negation is an order anti-... |
| dfinfre 12200 | The infimum of a set of re... |
| infrecl 12201 | Closure of infimum of a no... |
| infrenegsup 12202 | The infimum of a set of re... |
| infregelb 12203 | Any lower bound of a nonem... |
| infrelb 12204 | If a nonempty set of real ... |
| infrefilb 12205 | The infimum of a finite se... |
| supfirege 12206 | The supremum of a finite s... |
| inelr 12207 | The imaginary unit ` _i ` ... |
| rimul 12208 | A real number times the im... |
| cru 12209 | The representation of comp... |
| crne0 12210 | The real representation of... |
| creur 12211 | The real part of a complex... |
| creui 12212 | The imaginary part of a co... |
| cju 12213 | The complex conjugate of a... |
| ofsubeq0 12214 | Function analogue of ~ sub... |
| ofnegsub 12215 | Function analogue of ~ neg... |
| ofsubge0 12216 | Function analogue of ~ sub... |
| nnexALT 12219 | Alternate proof of ~ nnex ... |
| peano5nni 12220 | Peano's inductive postulat... |
| nnssre 12221 | The positive integers are ... |
| nnsscn 12222 | The positive integers are ... |
| nnex 12223 | The set of positive intege... |
| nnre 12224 | A positive integer is a re... |
| nncn 12225 | A positive integer is a co... |
| nnrei 12226 | A positive integer is a re... |
| nncni 12227 | A positive integer is a co... |
| 1nn 12228 | Peano postulate: 1 is a po... |
| peano2nn 12229 | Peano postulate: a success... |
| dfnn2 12230 | Alternate definition of th... |
| dfnn3 12231 | Alternate definition of th... |
| nnred 12232 | A positive integer is a re... |
| nncnd 12233 | A positive integer is a co... |
| peano2nnd 12234 | Peano postulate: a success... |
| nnind 12235 | Principle of Mathematical ... |
| nnindALT 12236 | Principle of Mathematical ... |
| nnindd 12237 | Principle of Mathematical ... |
| nn1m1nn 12238 | Every positive integer is ... |
| nn1suc 12239 | If a statement holds for 1... |
| nnaddcl 12240 | Closure of addition of pos... |
| nnmulcl 12241 | Closure of multiplication ... |
| nnmulcli 12242 | Closure of multiplication ... |
| nnmtmip 12243 | "Minus times minus is plus... |
| nn2ge 12244 | There exists a positive in... |
| nnge1 12245 | A positive integer is one ... |
| nngt1ne1 12246 | A positive integer is grea... |
| nnle1eq1 12247 | A positive integer is less... |
| nngt0 12248 | A positive integer is posi... |
| nnnlt1 12249 | A positive integer is not ... |
| nnnle0 12250 | A positive integer is not ... |
| nnne0 12251 | A positive integer is nonz... |
| nnneneg 12252 | No positive integer is equ... |
| 0nnn 12253 | Zero is not a positive int... |
| 0nnnALT 12254 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12255 | Alternate version of ~ nnn... |
| nngt0i 12256 | A positive integer is posi... |
| nnne0i 12257 | A positive integer is nonz... |
| nndivre 12258 | The quotient of a real and... |
| nnrecre 12259 | The reciprocal of a positi... |
| nnrecgt0 12260 | The reciprocal of a positi... |
| nnsub 12261 | Subtraction of positive in... |
| nnsubi 12262 | Subtraction of positive in... |
| nndiv 12263 | Two ways to express " ` A ... |
| nndivtr 12264 | Transitive property of div... |
| nnge1d 12265 | A positive integer is one ... |
| nngt0d 12266 | A positive integer is posi... |
| nnne0d 12267 | A positive integer is nonz... |
| nnrecred 12268 | The reciprocal of a positi... |
| nnaddcld 12269 | Closure of addition of pos... |
| nnmulcld 12270 | Closure of multiplication ... |
| nndivred 12271 | A positive integer is one ... |
| 0ne1 12288 | Zero is different from one... |
| 1m1e0 12289 | One minus one equals zero.... |
| 2nn 12290 | 2 is a positive integer. ... |
| 2re 12291 | The number 2 is real. (Co... |
| 2cn 12292 | The number 2 is a complex ... |
| 2cnALT 12293 | Alternate proof of ~ 2cn .... |
| 2ex 12294 | The number 2 is a set. (C... |
| 2cnd 12295 | The number 2 is a complex ... |
| 3nn 12296 | 3 is a positive integer. ... |
| 3re 12297 | The number 3 is real. (Co... |
| 3cn 12298 | The number 3 is a complex ... |
| 3ex 12299 | The number 3 is a set. (C... |
| 4nn 12300 | 4 is a positive integer. ... |
| 4re 12301 | The number 4 is real. (Co... |
| 4cn 12302 | The number 4 is a complex ... |
| 5nn 12303 | 5 is a positive integer. ... |
| 5re 12304 | The number 5 is real. (Co... |
| 5cn 12305 | The number 5 is a complex ... |
| 6nn 12306 | 6 is a positive integer. ... |
| 6re 12307 | The number 6 is real. (Co... |
| 6cn 12308 | The number 6 is a complex ... |
| 7nn 12309 | 7 is a positive integer. ... |
| 7re 12310 | The number 7 is real. (Co... |
| 7cn 12311 | The number 7 is a complex ... |
| 8nn 12312 | 8 is a positive integer. ... |
| 8re 12313 | The number 8 is real. (Co... |
| 8cn 12314 | The number 8 is a complex ... |
| 9nn 12315 | 9 is a positive integer. ... |
| 9re 12316 | The number 9 is real. (Co... |
| 9cn 12317 | The number 9 is a complex ... |
| 0le0 12318 | Zero is nonnegative. (Con... |
| 0le2 12319 | The number 0 is less than ... |
| 2pos 12320 | The number 2 is positive. ... |
| 2ne0 12321 | The number 2 is nonzero. ... |
| 3pos 12322 | The number 3 is positive. ... |
| 3ne0 12323 | The number 3 is nonzero. ... |
| 4pos 12324 | The number 4 is positive. ... |
| 4ne0 12325 | The number 4 is nonzero. ... |
| 5pos 12326 | The number 5 is positive. ... |
| 6pos 12327 | The number 6 is positive. ... |
| 7pos 12328 | The number 7 is positive. ... |
| 8pos 12329 | The number 8 is positive. ... |
| 9pos 12330 | The number 9 is positive. ... |
| neg1cn 12331 | -1 is a complex number. (... |
| neg1rr 12332 | -1 is a real number. (Con... |
| neg1ne0 12333 | -1 is nonzero. (Contribut... |
| neg1lt0 12334 | -1 is less than 0. (Contr... |
| negneg1e1 12335 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 12336 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12337 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12338 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12339 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12340 | Function value at ` N + 1 ... |
| 1p0e1 12341 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12342 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12343 | 2 - 1 = 1. The result is ... |
| 1e2m1 12344 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12345 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12346 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12347 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12348 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12349 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12350 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12351 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12352 | Two plus two equals four. ... |
| 2times 12353 | Two times a number. (Cont... |
| times2 12354 | A number times 2. (Contri... |
| 2timesi 12355 | Two times a number. (Cont... |
| times2i 12356 | A number times 2. (Contri... |
| 2txmxeqx 12357 | Two times a complex number... |
| 2div2e1 12358 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12359 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12360 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12361 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12362 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12363 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12364 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12365 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12366 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12367 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12368 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12369 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12370 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12371 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12372 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12373 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12374 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12375 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12376 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12377 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12378 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12379 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12380 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12381 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12382 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12383 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12384 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12385 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12386 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12387 | One half of four is two. ... |
| 1lt2 12388 | 1 is less than 2. (Contri... |
| 2lt3 12389 | 2 is less than 3. (Contri... |
| 1lt3 12390 | 1 is less than 3. (Contri... |
| 3lt4 12391 | 3 is less than 4. (Contri... |
| 2lt4 12392 | 2 is less than 4. (Contri... |
| 1lt4 12393 | 1 is less than 4. (Contri... |
| 4lt5 12394 | 4 is less than 5. (Contri... |
| 3lt5 12395 | 3 is less than 5. (Contri... |
| 2lt5 12396 | 2 is less than 5. (Contri... |
| 1lt5 12397 | 1 is less than 5. (Contri... |
| 5lt6 12398 | 5 is less than 6. (Contri... |
| 4lt6 12399 | 4 is less than 6. (Contri... |
| 3lt6 12400 | 3 is less than 6. (Contri... |
| 2lt6 12401 | 2 is less than 6. (Contri... |
| 1lt6 12402 | 1 is less than 6. (Contri... |
| 6lt7 12403 | 6 is less than 7. (Contri... |
| 5lt7 12404 | 5 is less than 7. (Contri... |
| 4lt7 12405 | 4 is less than 7. (Contri... |
| 3lt7 12406 | 3 is less than 7. (Contri... |
| 2lt7 12407 | 2 is less than 7. (Contri... |
| 1lt7 12408 | 1 is less than 7. (Contri... |
| 7lt8 12409 | 7 is less than 8. (Contri... |
| 6lt8 12410 | 6 is less than 8. (Contri... |
| 5lt8 12411 | 5 is less than 8. (Contri... |
| 4lt8 12412 | 4 is less than 8. (Contri... |
| 3lt8 12413 | 3 is less than 8. (Contri... |
| 2lt8 12414 | 2 is less than 8. (Contri... |
| 1lt8 12415 | 1 is less than 8. (Contri... |
| 8lt9 12416 | 8 is less than 9. (Contri... |
| 7lt9 12417 | 7 is less than 9. (Contri... |
| 6lt9 12418 | 6 is less than 9. (Contri... |
| 5lt9 12419 | 5 is less than 9. (Contri... |
| 4lt9 12420 | 4 is less than 9. (Contri... |
| 3lt9 12421 | 3 is less than 9. (Contri... |
| 2lt9 12422 | 2 is less than 9. (Contri... |
| 1lt9 12423 | 1 is less than 9. (Contri... |
| 0ne2 12424 | 0 is not equal to 2. (Con... |
| 1ne2 12425 | 1 is not equal to 2. (Con... |
| 1le2 12426 | 1 is less than or equal to... |
| 2cnne0 12427 | 2 is a nonzero complex num... |
| 2rene0 12428 | 2 is a nonzero real number... |
| 1le3 12429 | 1 is less than or equal to... |
| neg1mulneg1e1 12430 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12431 | One-half is real. (Contri... |
| halfcn 12432 | One-half is a complex numb... |
| halfgt0 12433 | One-half is greater than z... |
| halfge0 12434 | One-half is not negative. ... |
| halflt1 12435 | One-half is less than one.... |
| 1mhlfehlf 12436 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12437 | An eighth of four thirds i... |
| halfpm6th 12438 | One half plus or minus one... |
| it0e0 12439 | i times 0 equals 0. (Cont... |
| 2mulicn 12440 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12441 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12442 | Closure of half of a numbe... |
| rehalfcl 12443 | Real closure of half. (Co... |
| half0 12444 | Half of a number is zero i... |
| 2halves 12445 | Two halves make a whole. ... |
| halfpos2 12446 | A number is positive iff i... |
| halfpos 12447 | A positive number is great... |
| halfnneg2 12448 | A number is nonnegative if... |
| halfaddsubcl 12449 | Closure of half-sum and ha... |
| halfaddsub 12450 | Sum and difference of half... |
| subhalfhalf 12451 | Subtracting the half of a ... |
| lt2halves 12452 | A sum is less than the who... |
| addltmul 12453 | Sum is less than product f... |
| nominpos 12454 | There is no smallest posit... |
| avglt1 12455 | Ordering property for aver... |
| avglt2 12456 | Ordering property for aver... |
| avgle1 12457 | Ordering property for aver... |
| avgle2 12458 | Ordering property for aver... |
| avgle 12459 | The average of two numbers... |
| 2timesd 12460 | Two times a number. (Cont... |
| times2d 12461 | A number times 2. (Contri... |
| halfcld 12462 | Closure of half of a numbe... |
| 2halvesd 12463 | Two halves make a whole. ... |
| rehalfcld 12464 | Real closure of half. (Co... |
| lt2halvesd 12465 | A sum is less than the who... |
| rehalfcli 12466 | Half a real number is real... |
| lt2addmuld 12467 | If two real numbers are le... |
| add1p1 12468 | Adding two times 1 to a nu... |
| sub1m1 12469 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12470 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12471 | A complex number increased... |
| div4p1lem1div2 12472 | An integer greater than 5,... |
| nnunb 12473 | The set of positive intege... |
| arch 12474 | Archimedean property of re... |
| nnrecl 12475 | There exists a positive in... |
| bndndx 12476 | A bounded real sequence ` ... |
| elnn0 12479 | Nonnegative integers expre... |
| nnssnn0 12480 | Positive naturals are a su... |
| nn0ssre 12481 | Nonnegative integers are a... |
| nn0sscn 12482 | Nonnegative integers are a... |
| nn0ex 12483 | The set of nonnegative int... |
| nnnn0 12484 | A positive integer is a no... |
| nnnn0i 12485 | A positive integer is a no... |
| nn0re 12486 | A nonnegative integer is a... |
| nn0cn 12487 | A nonnegative integer is a... |
| nn0rei 12488 | A nonnegative integer is a... |
| nn0cni 12489 | A nonnegative integer is a... |
| dfn2 12490 | The set of positive intege... |
| elnnne0 12491 | The positive integer prope... |
| 0nn0 12492 | 0 is a nonnegative integer... |
| 1nn0 12493 | 1 is a nonnegative integer... |
| 2nn0 12494 | 2 is a nonnegative integer... |
| 3nn0 12495 | 3 is a nonnegative integer... |
| 4nn0 12496 | 4 is a nonnegative integer... |
| 5nn0 12497 | 5 is a nonnegative integer... |
| 6nn0 12498 | 6 is a nonnegative integer... |
| 7nn0 12499 | 7 is a nonnegative integer... |
| 8nn0 12500 | 8 is a nonnegative integer... |
| 9nn0 12501 | 9 is a nonnegative integer... |
| nn0ge0 12502 | A nonnegative integer is g... |
| nn0nlt0 12503 | A nonnegative integer is n... |
| nn0ge0i 12504 | Nonnegative integers are n... |
| nn0le0eq0 12505 | A nonnegative integer is l... |
| nn0p1gt0 12506 | A nonnegative integer incr... |
| nnnn0addcl 12507 | A positive integer plus a ... |
| nn0nnaddcl 12508 | A nonnegative integer plus... |
| 0mnnnnn0 12509 | The result of subtracting ... |
| un0addcl 12510 | If ` S ` is closed under a... |
| un0mulcl 12511 | If ` S ` is closed under m... |
| nn0addcl 12512 | Closure of addition of non... |
| nn0mulcl 12513 | Closure of multiplication ... |
| nn0addcli 12514 | Closure of addition of non... |
| nn0mulcli 12515 | Closure of multiplication ... |
| nn0p1nn 12516 | A nonnegative integer plus... |
| peano2nn0 12517 | Second Peano postulate for... |
| nnm1nn0 12518 | A positive integer minus 1... |
| elnn0nn 12519 | The nonnegative integer pr... |
| elnnnn0 12520 | The positive integer prope... |
| elnnnn0b 12521 | The positive integer prope... |
| elnnnn0c 12522 | The positive integer prope... |
| nn0addge1 12523 | A number is less than or e... |
| nn0addge2 12524 | A number is less than or e... |
| nn0addge1i 12525 | A number is less than or e... |
| nn0addge2i 12526 | A number is less than or e... |
| nn0sub 12527 | Subtraction of nonnegative... |
| ltsubnn0 12528 | Subtracting a nonnegative ... |
| nn0negleid 12529 | A nonnegative integer is g... |
| difgtsumgt 12530 | If the difference of a rea... |
| nn0le2xi 12531 | A nonnegative integer is l... |
| nn0lele2xi 12532 | 'Less than or equal to' im... |
| fcdmnn0supp 12533 | Two ways to write the supp... |
| fcdmnn0fsupp 12534 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12535 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12536 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12537 | A positive integer is a no... |
| nn0red 12538 | A nonnegative integer is a... |
| nn0cnd 12539 | A nonnegative integer is a... |
| nn0ge0d 12540 | A nonnegative integer is g... |
| nn0addcld 12541 | Closure of addition of non... |
| nn0mulcld 12542 | Closure of multiplication ... |
| nn0readdcl 12543 | Closure law for addition o... |
| nn0n0n1ge2 12544 | A nonnegative integer whic... |
| nn0n0n1ge2b 12545 | A nonnegative integer is n... |
| nn0ge2m1nn 12546 | If a nonnegative integer i... |
| nn0ge2m1nn0 12547 | If a nonnegative integer i... |
| nn0nndivcl 12548 | Closure law for dividing o... |
| elxnn0 12551 | An extended nonnegative in... |
| nn0ssxnn0 12552 | The standard nonnegative i... |
| nn0xnn0 12553 | A standard nonnegative int... |
| xnn0xr 12554 | An extended nonnegative in... |
| 0xnn0 12555 | Zero is an extended nonneg... |
| pnf0xnn0 12556 | Positive infinity is an ex... |
| nn0nepnf 12557 | No standard nonnegative in... |
| nn0xnn0d 12558 | A standard nonnegative int... |
| nn0nepnfd 12559 | No standard nonnegative in... |
| xnn0nemnf 12560 | No extended nonnegative in... |
| xnn0xrnemnf 12561 | The extended nonnegative i... |
| xnn0nnn0pnf 12562 | An extended nonnegative in... |
| elz 12565 | Membership in the set of i... |
| nnnegz 12566 | The negative of a positive... |
| zre 12567 | An integer is a real. (Co... |
| zcn 12568 | An integer is a complex nu... |
| zrei 12569 | An integer is a real numbe... |
| zssre 12570 | The integers are a subset ... |
| zsscn 12571 | The integers are a subset ... |
| zex 12572 | The set of integers exists... |
| elnnz 12573 | Positive integer property ... |
| 0z 12574 | Zero is an integer. (Cont... |
| 0zd 12575 | Zero is an integer, deduct... |
| elnn0z 12576 | Nonnegative integer proper... |
| elznn0nn 12577 | Integer property expressed... |
| elznn0 12578 | Integer property expressed... |
| elznn 12579 | Integer property expressed... |
| zle0orge1 12580 | There is no integer in the... |
| elz2 12581 | Membership in the set of i... |
| dfz2 12582 | Alternative definition of ... |
| zexALT 12583 | Alternate proof of ~ zex .... |
| nnz 12584 | A positive integer is an i... |
| nnssz 12585 | Positive integers are a su... |
| nn0ssz 12586 | Nonnegative integers are a... |
| nnzOLD 12587 | Obsolete version of ~ nnz ... |
| nn0z 12588 | A nonnegative integer is a... |
| nn0zd 12589 | A nonnegative integer is a... |
| nnzd 12590 | A positive integer is an i... |
| nnzi 12591 | A positive integer is an i... |
| nn0zi 12592 | A nonnegative integer is a... |
| elnnz1 12593 | Positive integer property ... |
| znnnlt1 12594 | An integer is not a positi... |
| nnzrab 12595 | Positive integers expresse... |
| nn0zrab 12596 | Nonnegative integers expre... |
| 1z 12597 | One is an integer. (Contr... |
| 1zzd 12598 | One is an integer, deducti... |
| 2z 12599 | 2 is an integer. (Contrib... |
| 3z 12600 | 3 is an integer. (Contrib... |
| 4z 12601 | 4 is an integer. (Contrib... |
| znegcl 12602 | Closure law for negative i... |
| neg1z 12603 | -1 is an integer. (Contri... |
| znegclb 12604 | A complex number is an int... |
| nn0negz 12605 | The negative of a nonnegat... |
| nn0negzi 12606 | The negative of a nonnegat... |
| zaddcl 12607 | Closure of addition of int... |
| peano2z 12608 | Second Peano postulate gen... |
| zsubcl 12609 | Closure of subtraction of ... |
| peano2zm 12610 | "Reverse" second Peano pos... |
| zletr 12611 | Transitive law of ordering... |
| zrevaddcl 12612 | Reverse closure law for ad... |
| znnsub 12613 | The positive difference of... |
| znn0sub 12614 | The nonnegative difference... |
| nzadd 12615 | The sum of a real number n... |
| zmulcl 12616 | Closure of multiplication ... |
| zltp1le 12617 | Integer ordering relation.... |
| zleltp1 12618 | Integer ordering relation.... |
| zlem1lt 12619 | Integer ordering relation.... |
| zltlem1 12620 | Integer ordering relation.... |
| zgt0ge1 12621 | An integer greater than ` ... |
| nnleltp1 12622 | Positive integer ordering ... |
| nnltp1le 12623 | Positive integer ordering ... |
| nnaddm1cl 12624 | Closure of addition of pos... |
| nn0ltp1le 12625 | Nonnegative integer orderi... |
| nn0leltp1 12626 | Nonnegative integer orderi... |
| nn0ltlem1 12627 | Nonnegative integer orderi... |
| nn0sub2 12628 | Subtraction of nonnegative... |
| nn0lt10b 12629 | A nonnegative integer less... |
| nn0lt2 12630 | A nonnegative integer less... |
| nn0le2is012 12631 | A nonnegative integer whic... |
| nn0lem1lt 12632 | Nonnegative integer orderi... |
| nnlem1lt 12633 | Positive integer ordering ... |
| nnltlem1 12634 | Positive integer ordering ... |
| nnm1ge0 12635 | A positive integer decreas... |
| nn0ge0div 12636 | Division of a nonnegative ... |
| zdiv 12637 | Two ways to express " ` M ... |
| zdivadd 12638 | Property of divisibility: ... |
| zdivmul 12639 | Property of divisibility: ... |
| zextle 12640 | An extensionality-like pro... |
| zextlt 12641 | An extensionality-like pro... |
| recnz 12642 | The reciprocal of a number... |
| btwnnz 12643 | A number between an intege... |
| gtndiv 12644 | A larger number does not d... |
| halfnz 12645 | One-half is not an integer... |
| 3halfnz 12646 | Three halves is not an int... |
| suprzcl 12647 | The supremum of a bounded-... |
| prime 12648 | Two ways to express " ` A ... |
| msqznn 12649 | The square of a nonzero in... |
| zneo 12650 | No even integer equals an ... |
| nneo 12651 | A positive integer is even... |
| nneoi 12652 | A positive integer is even... |
| zeo 12653 | An integer is even or odd.... |
| zeo2 12654 | An integer is even or odd ... |
| peano2uz2 12655 | Second Peano postulate for... |
| peano5uzi 12656 | Peano's inductive postulat... |
| peano5uzti 12657 | Peano's inductive postulat... |
| dfuzi 12658 | An expression for the uppe... |
| uzind 12659 | Induction on the upper int... |
| uzind2 12660 | Induction on the upper int... |
| uzind3 12661 | Induction on the upper int... |
| nn0ind 12662 | Principle of Mathematical ... |
| nn0indALT 12663 | Principle of Mathematical ... |
| nn0indd 12664 | Principle of Mathematical ... |
| fzind 12665 | Induction on the integers ... |
| fnn0ind 12666 | Induction on the integers ... |
| nn0ind-raph 12667 | Principle of Mathematical ... |
| zindd 12668 | Principle of Mathematical ... |
| fzindd 12669 | Induction on the integers ... |
| btwnz 12670 | Any real number can be san... |
| zred 12671 | An integer is a real numbe... |
| zcnd 12672 | An integer is a complex nu... |
| znegcld 12673 | Closure law for negative i... |
| peano2zd 12674 | Deduction from second Pean... |
| zaddcld 12675 | Closure of addition of int... |
| zsubcld 12676 | Closure of subtraction of ... |
| zmulcld 12677 | Closure of multiplication ... |
| znnn0nn 12678 | The negative of a negative... |
| zadd2cl 12679 | Increasing an integer by 2... |
| zriotaneg 12680 | The negative of the unique... |
| suprfinzcl 12681 | The supremum of a nonempty... |
| 9p1e10 12684 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12685 | Version of the definition ... |
| decex 12686 | A decimal number is a set.... |
| deceq1 12687 | Equality theorem for the d... |
| deceq2 12688 | Equality theorem for the d... |
| deceq1i 12689 | Equality theorem for the d... |
| deceq2i 12690 | Equality theorem for the d... |
| deceq12i 12691 | Equality theorem for the d... |
| numnncl 12692 | Closure for a numeral (wit... |
| num0u 12693 | Add a zero in the units pl... |
| num0h 12694 | Add a zero in the higher p... |
| numcl 12695 | Closure for a decimal inte... |
| numsuc 12696 | The successor of a decimal... |
| deccl 12697 | Closure for a numeral. (C... |
| 10nn 12698 | 10 is a positive integer. ... |
| 10pos 12699 | The number 10 is positive.... |
| 10nn0 12700 | 10 is a nonnegative intege... |
| 10re 12701 | The number 10 is real. (C... |
| decnncl 12702 | Closure for a numeral. (C... |
| dec0u 12703 | Add a zero in the units pl... |
| dec0h 12704 | Add a zero in the higher p... |
| numnncl2 12705 | Closure for a decimal inte... |
| decnncl2 12706 | Closure for a decimal inte... |
| numlt 12707 | Comparing two decimal inte... |
| numltc 12708 | Comparing two decimal inte... |
| le9lt10 12709 | A "decimal digit" (i.e. a ... |
| declt 12710 | Comparing two decimal inte... |
| decltc 12711 | Comparing two decimal inte... |
| declth 12712 | Comparing two decimal inte... |
| decsuc 12713 | The successor of a decimal... |
| 3declth 12714 | Comparing two decimal inte... |
| 3decltc 12715 | Comparing two decimal inte... |
| decle 12716 | Comparing two decimal inte... |
| decleh 12717 | Comparing two decimal inte... |
| declei 12718 | Comparing a digit to a dec... |
| numlti 12719 | Comparing a digit to a dec... |
| declti 12720 | Comparing a digit to a dec... |
| decltdi 12721 | Comparing a digit to a dec... |
| numsucc 12722 | The successor of a decimal... |
| decsucc 12723 | The successor of a decimal... |
| 1e0p1 12724 | The successor of zero. (C... |
| dec10p 12725 | Ten plus an integer. (Con... |
| numma 12726 | Perform a multiply-add of ... |
| nummac 12727 | Perform a multiply-add of ... |
| numma2c 12728 | Perform a multiply-add of ... |
| numadd 12729 | Add two decimal integers `... |
| numaddc 12730 | Add two decimal integers `... |
| nummul1c 12731 | The product of a decimal i... |
| nummul2c 12732 | The product of a decimal i... |
| decma 12733 | Perform a multiply-add of ... |
| decmac 12734 | Perform a multiply-add of ... |
| decma2c 12735 | Perform a multiply-add of ... |
| decadd 12736 | Add two numerals ` M ` and... |
| decaddc 12737 | Add two numerals ` M ` and... |
| decaddc2 12738 | Add two numerals ` M ` and... |
| decrmanc 12739 | Perform a multiply-add of ... |
| decrmac 12740 | Perform a multiply-add of ... |
| decaddm10 12741 | The sum of two multiples o... |
| decaddi 12742 | Add two numerals ` M ` and... |
| decaddci 12743 | Add two numerals ` M ` and... |
| decaddci2 12744 | Add two numerals ` M ` and... |
| decsubi 12745 | Difference between a numer... |
| decmul1 12746 | The product of a numeral w... |
| decmul1c 12747 | The product of a numeral w... |
| decmul2c 12748 | The product of a numeral w... |
| decmulnc 12749 | The product of a numeral w... |
| 11multnc 12750 | The product of 11 (as nume... |
| decmul10add 12751 | A multiplication of a numb... |
| 6p5lem 12752 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12753 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12754 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12755 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12756 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12757 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12758 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12759 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12760 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12761 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12762 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12763 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12764 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12765 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12766 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12767 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12768 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12769 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12770 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12771 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12772 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12773 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12774 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12775 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12776 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12777 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12778 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12779 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12780 | 4 times 3 equals 12. (Con... |
| 4t4e16 12781 | 4 times 4 equals 16. (Con... |
| 5t2e10 12782 | 5 times 2 equals 10. (Con... |
| 5t3e15 12783 | 5 times 3 equals 15. (Con... |
| 5t4e20 12784 | 5 times 4 equals 20. (Con... |
| 5t5e25 12785 | 5 times 5 equals 25. (Con... |
| 6t2e12 12786 | 6 times 2 equals 12. (Con... |
| 6t3e18 12787 | 6 times 3 equals 18. (Con... |
| 6t4e24 12788 | 6 times 4 equals 24. (Con... |
| 6t5e30 12789 | 6 times 5 equals 30. (Con... |
| 6t6e36 12790 | 6 times 6 equals 36. (Con... |
| 7t2e14 12791 | 7 times 2 equals 14. (Con... |
| 7t3e21 12792 | 7 times 3 equals 21. (Con... |
| 7t4e28 12793 | 7 times 4 equals 28. (Con... |
| 7t5e35 12794 | 7 times 5 equals 35. (Con... |
| 7t6e42 12795 | 7 times 6 equals 42. (Con... |
| 7t7e49 12796 | 7 times 7 equals 49. (Con... |
| 8t2e16 12797 | 8 times 2 equals 16. (Con... |
| 8t3e24 12798 | 8 times 3 equals 24. (Con... |
| 8t4e32 12799 | 8 times 4 equals 32. (Con... |
| 8t5e40 12800 | 8 times 5 equals 40. (Con... |
| 8t6e48 12801 | 8 times 6 equals 48. (Con... |
| 8t7e56 12802 | 8 times 7 equals 56. (Con... |
| 8t8e64 12803 | 8 times 8 equals 64. (Con... |
| 9t2e18 12804 | 9 times 2 equals 18. (Con... |
| 9t3e27 12805 | 9 times 3 equals 27. (Con... |
| 9t4e36 12806 | 9 times 4 equals 36. (Con... |
| 9t5e45 12807 | 9 times 5 equals 45. (Con... |
| 9t6e54 12808 | 9 times 6 equals 54. (Con... |
| 9t7e63 12809 | 9 times 7 equals 63. (Con... |
| 9t8e72 12810 | 9 times 8 equals 72. (Con... |
| 9t9e81 12811 | 9 times 9 equals 81. (Con... |
| 9t11e99 12812 | 9 times 11 equals 99. (Co... |
| 9lt10 12813 | 9 is less than 10. (Contr... |
| 8lt10 12814 | 8 is less than 10. (Contr... |
| 7lt10 12815 | 7 is less than 10. (Contr... |
| 6lt10 12816 | 6 is less than 10. (Contr... |
| 5lt10 12817 | 5 is less than 10. (Contr... |
| 4lt10 12818 | 4 is less than 10. (Contr... |
| 3lt10 12819 | 3 is less than 10. (Contr... |
| 2lt10 12820 | 2 is less than 10. (Contr... |
| 1lt10 12821 | 1 is less than 10. (Contr... |
| decbin0 12822 | Decompose base 4 into base... |
| decbin2 12823 | Decompose base 4 into base... |
| decbin3 12824 | Decompose base 4 into base... |
| halfthird 12825 | Half minus a third. (Cont... |
| 5recm6rec 12826 | One fifth minus one sixth.... |
| uzval 12829 | The value of the upper int... |
| uzf 12830 | The domain and codomain of... |
| eluz1 12831 | Membership in the upper se... |
| eluzel2 12832 | Implication of membership ... |
| eluz2 12833 | Membership in an upper set... |
| eluzmn 12834 | Membership in an earlier u... |
| eluz1i 12835 | Membership in an upper set... |
| eluzuzle 12836 | An integer in an upper set... |
| eluzelz 12837 | A member of an upper set o... |
| eluzelre 12838 | A member of an upper set o... |
| eluzelcn 12839 | A member of an upper set o... |
| eluzle 12840 | Implication of membership ... |
| eluz 12841 | Membership in an upper set... |
| uzid 12842 | Membership of the least me... |
| uzidd 12843 | Membership of the least me... |
| uzn0 12844 | The upper integers are all... |
| uztrn 12845 | Transitive law for sets of... |
| uztrn2 12846 | Transitive law for sets of... |
| uzneg 12847 | Contraposition law for upp... |
| uzssz 12848 | An upper set of integers i... |
| uzssre 12849 | An upper set of integers i... |
| uzss 12850 | Subset relationship for tw... |
| uztric 12851 | Totality of the ordering r... |
| uz11 12852 | The upper integers functio... |
| eluzp1m1 12853 | Membership in the next upp... |
| eluzp1l 12854 | Strict ordering implied by... |
| eluzp1p1 12855 | Membership in the next upp... |
| eluzadd 12856 | Membership in a later uppe... |
| eluzsub 12857 | Membership in an earlier u... |
| eluzaddi 12858 | Membership in a later uppe... |
| eluzaddiOLD 12859 | Obsolete version of ~ eluz... |
| eluzsubi 12860 | Membership in an earlier u... |
| eluzsubiOLD 12861 | Obsolete version of ~ eluz... |
| eluzaddOLD 12862 | Obsolete version of ~ eluz... |
| eluzsubOLD 12863 | Obsolete version of ~ eluz... |
| subeluzsub 12864 | Membership of a difference... |
| uzm1 12865 | Choices for an element of ... |
| uznn0sub 12866 | The nonnegative difference... |
| uzin 12867 | Intersection of two upper ... |
| uzp1 12868 | Choices for an element of ... |
| nn0uz 12869 | Nonnegative integers expre... |
| nnuz 12870 | Positive integers expresse... |
| elnnuz 12871 | A positive integer express... |
| elnn0uz 12872 | A nonnegative integer expr... |
| eluz2nn 12873 | An integer greater than or... |
| eluz4eluz2 12874 | An integer greater than or... |
| eluz4nn 12875 | An integer greater than or... |
| eluzge2nn0 12876 | If an integer is greater t... |
| eluz2n0 12877 | An integer greater than or... |
| uzuzle23 12878 | An integer in the upper se... |
| eluzge3nn 12879 | If an integer is greater t... |
| uz3m2nn 12880 | An integer greater than or... |
| 1eluzge0 12881 | 1 is an integer greater th... |
| 2eluzge0 12882 | 2 is an integer greater th... |
| 2eluzge1 12883 | 2 is an integer greater th... |
| uznnssnn 12884 | The upper integers startin... |
| raluz 12885 | Restricted universal quant... |
| raluz2 12886 | Restricted universal quant... |
| rexuz 12887 | Restricted existential qua... |
| rexuz2 12888 | Restricted existential qua... |
| 2rexuz 12889 | Double existential quantif... |
| peano2uz 12890 | Second Peano postulate for... |
| peano2uzs 12891 | Second Peano postulate for... |
| peano2uzr 12892 | Reversed second Peano axio... |
| uzaddcl 12893 | Addition closure law for a... |
| nn0pzuz 12894 | The sum of a nonnegative i... |
| uzind4 12895 | Induction on the upper set... |
| uzind4ALT 12896 | Induction on the upper set... |
| uzind4s 12897 | Induction on the upper set... |
| uzind4s2 12898 | Induction on the upper set... |
| uzind4i 12899 | Induction on the upper int... |
| uzwo 12900 | Well-ordering principle: a... |
| uzwo2 12901 | Well-ordering principle: a... |
| nnwo 12902 | Well-ordering principle: a... |
| nnwof 12903 | Well-ordering principle: a... |
| nnwos 12904 | Well-ordering principle: a... |
| indstr 12905 | Strong Mathematical Induct... |
| eluznn0 12906 | Membership in a nonnegativ... |
| eluznn 12907 | Membership in a positive u... |
| eluz2b1 12908 | Two ways to say "an intege... |
| eluz2gt1 12909 | An integer greater than or... |
| eluz2b2 12910 | Two ways to say "an intege... |
| eluz2b3 12911 | Two ways to say "an intege... |
| uz2m1nn 12912 | One less than an integer g... |
| 1nuz2 12913 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12914 | A positive integer is eith... |
| uz2mulcl 12915 | Closure of multiplication ... |
| indstr2 12916 | Strong Mathematical Induct... |
| uzinfi 12917 | Extract the lower bound of... |
| nninf 12918 | The infimum of the set of ... |
| nn0inf 12919 | The infimum of the set of ... |
| infssuzle 12920 | The infimum of a subset of... |
| infssuzcl 12921 | The infimum of a subset of... |
| ublbneg 12922 | The image under negation o... |
| eqreznegel 12923 | Two ways to express the im... |
| supminf 12924 | The supremum of a bounded-... |
| lbzbi 12925 | If a set of reals is bound... |
| zsupss 12926 | Any nonempty bounded subse... |
| suprzcl2 12927 | The supremum of a bounded-... |
| suprzub 12928 | The supremum of a bounded-... |
| uzsupss 12929 | Any bounded subset of an u... |
| nn01to3 12930 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12931 | Alternate proof of ~ nn0ge... |
| uzwo3 12932 | Well-ordering principle: a... |
| zmin 12933 | There is a unique smallest... |
| zmax 12934 | There is a unique largest ... |
| zbtwnre 12935 | There is a unique integer ... |
| rebtwnz 12936 | There is a unique greatest... |
| elq 12939 | Membership in the set of r... |
| qmulz 12940 | If ` A ` is rational, then... |
| znq 12941 | The ratio of an integer an... |
| qre 12942 | A rational number is a rea... |
| zq 12943 | An integer is a rational n... |
| qred 12944 | A rational number is a rea... |
| zssq 12945 | The integers are a subset ... |
| nn0ssq 12946 | The nonnegative integers a... |
| nnssq 12947 | The positive integers are ... |
| qssre 12948 | The rationals are a subset... |
| qsscn 12949 | The rationals are a subset... |
| qex 12950 | The set of rational number... |
| nnq 12951 | A positive integer is rati... |
| qcn 12952 | A rational number is a com... |
| qexALT 12953 | Alternate proof of ~ qex .... |
| qaddcl 12954 | Closure of addition of rat... |
| qnegcl 12955 | Closure law for the negati... |
| qmulcl 12956 | Closure of multiplication ... |
| qsubcl 12957 | Closure of subtraction of ... |
| qreccl 12958 | Closure of reciprocal of r... |
| qdivcl 12959 | Closure of division of rat... |
| qrevaddcl 12960 | Reverse closure law for ad... |
| nnrecq 12961 | The reciprocal of a positi... |
| irradd 12962 | The sum of an irrational n... |
| irrmul 12963 | The product of an irration... |
| elpq 12964 | A positive rational is the... |
| elpqb 12965 | A class is a positive rati... |
| rpnnen1lem2 12966 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12967 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12968 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12969 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12970 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12971 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12972 | One half of ~ rpnnen , whe... |
| reexALT 12973 | Alternate proof of ~ reex ... |
| cnref1o 12974 | There is a natural one-to-... |
| cnexALT 12975 | The set of complex numbers... |
| xrex 12976 | The set of extended reals ... |
| addex 12977 | The addition operation is ... |
| mulex 12978 | The multiplication operati... |
| elrp 12981 | Membership in the set of p... |
| elrpii 12982 | Membership in the set of p... |
| 1rp 12983 | 1 is a positive real. (Co... |
| 2rp 12984 | 2 is a positive real. (Co... |
| 3rp 12985 | 3 is a positive real. (Co... |
| rpssre 12986 | The positive reals are a s... |
| rpre 12987 | A positive real is a real.... |
| rpxr 12988 | A positive real is an exte... |
| rpcn 12989 | A positive real is a compl... |
| nnrp 12990 | A positive integer is a po... |
| rpgt0 12991 | A positive real is greater... |
| rpge0 12992 | A positive real is greater... |
| rpregt0 12993 | A positive real is a posit... |
| rprege0 12994 | A positive real is a nonne... |
| rpne0 12995 | A positive real is nonzero... |
| rprene0 12996 | A positive real is a nonze... |
| rpcnne0 12997 | A positive real is a nonze... |
| rpcndif0 12998 | A positive real number is ... |
| ralrp 12999 | Quantification over positi... |
| rexrp 13000 | Quantification over positi... |
| rpaddcl 13001 | Closure law for addition o... |
| rpmulcl 13002 | Closure law for multiplica... |
| rpmtmip 13003 | "Minus times minus is plus... |
| rpdivcl 13004 | Closure law for division o... |
| rpreccl 13005 | Closure law for reciprocat... |
| rphalfcl 13006 | Closure law for half of a ... |
| rpgecl 13007 | A number greater than or e... |
| rphalflt 13008 | Half of a positive real is... |
| rerpdivcl 13009 | Closure law for division o... |
| ge0p1rp 13010 | A nonnegative number plus ... |
| rpneg 13011 | Either a nonzero real or i... |
| negelrp 13012 | Elementhood of a negation ... |
| negelrpd 13013 | The negation of a negative... |
| 0nrp 13014 | Zero is not a positive rea... |
| ltsubrp 13015 | Subtracting a positive rea... |
| ltaddrp 13016 | Adding a positive number t... |
| difrp 13017 | Two ways to say one number... |
| elrpd 13018 | Membership in the set of p... |
| nnrpd 13019 | A positive integer is a po... |
| zgt1rpn0n1 13020 | An integer greater than 1 ... |
| rpred 13021 | A positive real is a real.... |
| rpxrd 13022 | A positive real is an exte... |
| rpcnd 13023 | A positive real is a compl... |
| rpgt0d 13024 | A positive real is greater... |
| rpge0d 13025 | A positive real is greater... |
| rpne0d 13026 | A positive real is nonzero... |
| rpregt0d 13027 | A positive real is real an... |
| rprege0d 13028 | A positive real is real an... |
| rprene0d 13029 | A positive real is a nonze... |
| rpcnne0d 13030 | A positive real is a nonze... |
| rpreccld 13031 | Closure law for reciprocat... |
| rprecred 13032 | Closure law for reciprocat... |
| rphalfcld 13033 | Closure law for half of a ... |
| reclt1d 13034 | The reciprocal of a positi... |
| recgt1d 13035 | The reciprocal of a positi... |
| rpaddcld 13036 | Closure law for addition o... |
| rpmulcld 13037 | Closure law for multiplica... |
| rpdivcld 13038 | Closure law for division o... |
| ltrecd 13039 | The reciprocal of both sid... |
| lerecd 13040 | The reciprocal of both sid... |
| ltrec1d 13041 | Reciprocal swap in a 'less... |
| lerec2d 13042 | Reciprocal swap in a 'less... |
| lediv2ad 13043 | Division of both sides of ... |
| ltdiv2d 13044 | Division of a positive num... |
| lediv2d 13045 | Division of a positive num... |
| ledivdivd 13046 | Invert ratios of positive ... |
| divge1 13047 | The ratio of a number over... |
| divlt1lt 13048 | A real number divided by a... |
| divle1le 13049 | A real number divided by a... |
| ledivge1le 13050 | If a number is less than o... |
| ge0p1rpd 13051 | A nonnegative number plus ... |
| rerpdivcld 13052 | Closure law for division o... |
| ltsubrpd 13053 | Subtracting a positive rea... |
| ltaddrpd 13054 | Adding a positive number t... |
| ltaddrp2d 13055 | Adding a positive number t... |
| ltmulgt11d 13056 | Multiplication by a number... |
| ltmulgt12d 13057 | Multiplication by a number... |
| gt0divd 13058 | Division of a positive num... |
| ge0divd 13059 | Division of a nonnegative ... |
| rpgecld 13060 | A number greater than or e... |
| divge0d 13061 | The ratio of nonnegative a... |
| ltmul1d 13062 | The ratio of nonnegative a... |
| ltmul2d 13063 | Multiplication of both sid... |
| lemul1d 13064 | Multiplication of both sid... |
| lemul2d 13065 | Multiplication of both sid... |
| ltdiv1d 13066 | Division of both sides of ... |
| lediv1d 13067 | Division of both sides of ... |
| ltmuldivd 13068 | 'Less than' relationship b... |
| ltmuldiv2d 13069 | 'Less than' relationship b... |
| lemuldivd 13070 | 'Less than or equal to' re... |
| lemuldiv2d 13071 | 'Less than or equal to' re... |
| ltdivmuld 13072 | 'Less than' relationship b... |
| ltdivmul2d 13073 | 'Less than' relationship b... |
| ledivmuld 13074 | 'Less than or equal to' re... |
| ledivmul2d 13075 | 'Less than or equal to' re... |
| ltmul1dd 13076 | The ratio of nonnegative a... |
| ltmul2dd 13077 | Multiplication of both sid... |
| ltdiv1dd 13078 | Division of both sides of ... |
| lediv1dd 13079 | Division of both sides of ... |
| lediv12ad 13080 | Comparison of ratio of two... |
| mul2lt0rlt0 13081 | If the result of a multipl... |
| mul2lt0rgt0 13082 | If the result of a multipl... |
| mul2lt0llt0 13083 | If the result of a multipl... |
| mul2lt0lgt0 13084 | If the result of a multipl... |
| mul2lt0bi 13085 | If the result of a multipl... |
| prodge0rd 13086 | Infer that a multiplicand ... |
| prodge0ld 13087 | Infer that a multiplier is... |
| ltdiv23d 13088 | Swap denominator with othe... |
| lediv23d 13089 | Swap denominator with othe... |
| lt2mul2divd 13090 | The ratio of nonnegative a... |
| nnledivrp 13091 | Division of a positive int... |
| nn0ledivnn 13092 | Division of a nonnegative ... |
| addlelt 13093 | If the sum of a real numbe... |
| ltxr 13100 | The 'less than' binary rel... |
| elxr 13101 | Membership in the set of e... |
| xrnemnf 13102 | An extended real other tha... |
| xrnepnf 13103 | An extended real other tha... |
| xrltnr 13104 | The extended real 'less th... |
| ltpnf 13105 | Any (finite) real is less ... |
| ltpnfd 13106 | Any (finite) real is less ... |
| 0ltpnf 13107 | Zero is less than plus inf... |
| mnflt 13108 | Minus infinity is less tha... |
| mnfltd 13109 | Minus infinity is less tha... |
| mnflt0 13110 | Minus infinity is less tha... |
| mnfltpnf 13111 | Minus infinity is less tha... |
| mnfltxr 13112 | Minus infinity is less tha... |
| pnfnlt 13113 | No extended real is greate... |
| nltmnf 13114 | No extended real is less t... |
| pnfge 13115 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13116 | An extended nonnegative in... |
| 0lepnf 13117 | 0 less than or equal to po... |
| xnn0ge0 13118 | An extended nonnegative in... |
| mnfle 13119 | Minus infinity is less tha... |
| mnfled 13120 | Minus infinity is less tha... |
| xrltnsym 13121 | Ordering on the extended r... |
| xrltnsym2 13122 | 'Less than' is antisymmetr... |
| xrlttri 13123 | Ordering on the extended r... |
| xrlttr 13124 | Ordering on the extended r... |
| xrltso 13125 | 'Less than' is a strict or... |
| xrlttri2 13126 | Trichotomy law for 'less t... |
| xrlttri3 13127 | Trichotomy law for 'less t... |
| xrleloe 13128 | 'Less than or equal' expre... |
| xrleltne 13129 | 'Less than or equal to' im... |
| xrltlen 13130 | 'Less than' expressed in t... |
| dfle2 13131 | Alternative definition of ... |
| dflt2 13132 | Alternative definition of ... |
| xrltle 13133 | 'Less than' implies 'less ... |
| xrltled 13134 | 'Less than' implies 'less ... |
| xrleid 13135 | 'Less than or equal to' is... |
| xrleidd 13136 | 'Less than or equal to' is... |
| xrletri 13137 | Trichotomy law for extende... |
| xrletri3 13138 | Trichotomy law for extende... |
| xrletrid 13139 | Trichotomy law for extende... |
| xrlelttr 13140 | Transitive law for orderin... |
| xrltletr 13141 | Transitive law for orderin... |
| xrletr 13142 | Transitive law for orderin... |
| xrlttrd 13143 | Transitive law for orderin... |
| xrlelttrd 13144 | Transitive law for orderin... |
| xrltletrd 13145 | Transitive law for orderin... |
| xrletrd 13146 | Transitive law for orderin... |
| xrltne 13147 | 'Less than' implies not eq... |
| nltpnft 13148 | An extended real is not le... |
| xgepnf 13149 | An extended real which is ... |
| ngtmnft 13150 | An extended real is not gr... |
| xlemnf 13151 | An extended real which is ... |
| xrrebnd 13152 | An extended real is real i... |
| xrre 13153 | A way of proving that an e... |
| xrre2 13154 | An extended real between t... |
| xrre3 13155 | A way of proving that an e... |
| ge0gtmnf 13156 | A nonnegative extended rea... |
| ge0nemnf 13157 | A nonnegative extended rea... |
| xrrege0 13158 | A nonnegative extended rea... |
| xrmax1 13159 | An extended real is less t... |
| xrmax2 13160 | An extended real is less t... |
| xrmin1 13161 | The minimum of two extende... |
| xrmin2 13162 | The minimum of two extende... |
| xrmaxeq 13163 | The maximum of two extende... |
| xrmineq 13164 | The minimum of two extende... |
| xrmaxlt 13165 | Two ways of saying the max... |
| xrltmin 13166 | Two ways of saying an exte... |
| xrmaxle 13167 | Two ways of saying the max... |
| xrlemin 13168 | Two ways of saying a numbe... |
| max1 13169 | A number is less than or e... |
| max1ALT 13170 | A number is less than or e... |
| max2 13171 | A number is less than or e... |
| 2resupmax 13172 | The supremum of two real n... |
| min1 13173 | The minimum of two numbers... |
| min2 13174 | The minimum of two numbers... |
| maxle 13175 | Two ways of saying the max... |
| lemin 13176 | Two ways of saying a numbe... |
| maxlt 13177 | Two ways of saying the max... |
| ltmin 13178 | Two ways of saying a numbe... |
| lemaxle 13179 | A real number which is les... |
| max0sub 13180 | Decompose a real number in... |
| ifle 13181 | An if statement transforms... |
| z2ge 13182 | There exists an integer gr... |
| qbtwnre 13183 | The rational numbers are d... |
| qbtwnxr 13184 | The rational numbers are d... |
| qsqueeze 13185 | If a nonnegative real is l... |
| qextltlem 13186 | Lemma for ~ qextlt and qex... |
| qextlt 13187 | An extensionality-like pro... |
| qextle 13188 | An extensionality-like pro... |
| xralrple 13189 | Show that ` A ` is less th... |
| alrple 13190 | Show that ` A ` is less th... |
| xnegeq 13191 | Equality of two extended n... |
| xnegex 13192 | A negative extended real e... |
| xnegpnf 13193 | Minus ` +oo ` . Remark of... |
| xnegmnf 13194 | Minus ` -oo ` . Remark of... |
| rexneg 13195 | Minus a real number. Rema... |
| xneg0 13196 | The negative of zero. (Co... |
| xnegcl 13197 | Closure of extended real n... |
| xnegneg 13198 | Extended real version of ~... |
| xneg11 13199 | Extended real version of ~... |
| xltnegi 13200 | Forward direction of ~ xlt... |
| xltneg 13201 | Extended real version of ~... |
| xleneg 13202 | Extended real version of ~... |
| xlt0neg1 13203 | Extended real version of ~... |
| xlt0neg2 13204 | Extended real version of ~... |
| xle0neg1 13205 | Extended real version of ~... |
| xle0neg2 13206 | Extended real version of ~... |
| xaddval 13207 | Value of the extended real... |
| xaddf 13208 | The extended real addition... |
| xmulval 13209 | Value of the extended real... |
| xaddpnf1 13210 | Addition of positive infin... |
| xaddpnf2 13211 | Addition of positive infin... |
| xaddmnf1 13212 | Addition of negative infin... |
| xaddmnf2 13213 | Addition of negative infin... |
| pnfaddmnf 13214 | Addition of positive and n... |
| mnfaddpnf 13215 | Addition of negative and p... |
| rexadd 13216 | The extended real addition... |
| rexsub 13217 | Extended real subtraction ... |
| rexaddd 13218 | The extended real addition... |
| xnn0xaddcl 13219 | The extended nonnegative i... |
| xaddnemnf 13220 | Closure of extended real a... |
| xaddnepnf 13221 | Closure of extended real a... |
| xnegid 13222 | Extended real version of ~... |
| xaddcl 13223 | The extended real addition... |
| xaddcom 13224 | The extended real addition... |
| xaddrid 13225 | Extended real version of ~... |
| xaddlid 13226 | Extended real version of ~... |
| xaddridd 13227 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13228 | Extended nonnegative integ... |
| xnn0lenn0nn0 13229 | An extended nonnegative in... |
| xnn0le2is012 13230 | An extended nonnegative in... |
| xnn0xadd0 13231 | The sum of two extended no... |
| xnegdi 13232 | Extended real version of ~... |
| xaddass 13233 | Associativity of extended ... |
| xaddass2 13234 | Associativity of extended ... |
| xpncan 13235 | Extended real version of ~... |
| xnpcan 13236 | Extended real version of ~... |
| xleadd1a 13237 | Extended real version of ~... |
| xleadd2a 13238 | Commuted form of ~ xleadd1... |
| xleadd1 13239 | Weakened version of ~ xlea... |
| xltadd1 13240 | Extended real version of ~... |
| xltadd2 13241 | Extended real version of ~... |
| xaddge0 13242 | The sum of nonnegative ext... |
| xle2add 13243 | Extended real version of ~... |
| xlt2add 13244 | Extended real version of ~... |
| xsubge0 13245 | Extended real version of ~... |
| xposdif 13246 | Extended real version of ~... |
| xlesubadd 13247 | Under certain conditions, ... |
| xmullem 13248 | Lemma for ~ rexmul . (Con... |
| xmullem2 13249 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13250 | Extended real multiplicati... |
| xmul01 13251 | Extended real version of ~... |
| xmul02 13252 | Extended real version of ~... |
| xmulneg1 13253 | Extended real version of ~... |
| xmulneg2 13254 | Extended real version of ~... |
| rexmul 13255 | The extended real multipli... |
| xmulf 13256 | The extended real multipli... |
| xmulcl 13257 | Closure of extended real m... |
| xmulpnf1 13258 | Multiplication by plus inf... |
| xmulpnf2 13259 | Multiplication by plus inf... |
| xmulmnf1 13260 | Multiplication by minus in... |
| xmulmnf2 13261 | Multiplication by minus in... |
| xmulpnf1n 13262 | Multiplication by plus inf... |
| xmulrid 13263 | Extended real version of ~... |
| xmullid 13264 | Extended real version of ~... |
| xmulm1 13265 | Extended real version of ~... |
| xmulasslem2 13266 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13267 | Extended real version of ~... |
| xmulge0 13268 | Extended real version of ~... |
| xmulasslem 13269 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13270 | Lemma for ~ xmulass . (Co... |
| xmulass 13271 | Associativity of the exten... |
| xlemul1a 13272 | Extended real version of ~... |
| xlemul2a 13273 | Extended real version of ~... |
| xlemul1 13274 | Extended real version of ~... |
| xlemul2 13275 | Extended real version of ~... |
| xltmul1 13276 | Extended real version of ~... |
| xltmul2 13277 | Extended real version of ~... |
| xadddilem 13278 | Lemma for ~ xadddi . (Con... |
| xadddi 13279 | Distributive property for ... |
| xadddir 13280 | Commuted version of ~ xadd... |
| xadddi2 13281 | The assumption that the mu... |
| xadddi2r 13282 | Commuted version of ~ xadd... |
| x2times 13283 | Extended real version of ~... |
| xnegcld 13284 | Closure of extended real n... |
| xaddcld 13285 | The extended real addition... |
| xmulcld 13286 | Closure of extended real m... |
| xadd4d 13287 | Rearrangement of 4 terms i... |
| xnn0add4d 13288 | Rearrangement of 4 terms i... |
| xrsupexmnf 13289 | Adding minus infinity to a... |
| xrinfmexpnf 13290 | Adding plus infinity to a ... |
| xrsupsslem 13291 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13292 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13293 | Any subset of extended rea... |
| xrinfmss 13294 | Any subset of extended rea... |
| xrinfmss2 13295 | Any subset of extended rea... |
| xrub 13296 | By quantifying only over r... |
| supxr 13297 | The supremum of a set of e... |
| supxr2 13298 | The supremum of a set of e... |
| supxrcl 13299 | The supremum of an arbitra... |
| supxrun 13300 | The supremum of the union ... |
| supxrmnf 13301 | Adding minus infinity to a... |
| supxrpnf 13302 | The supremum of a set of e... |
| supxrunb1 13303 | The supremum of an unbound... |
| supxrunb2 13304 | The supremum of an unbound... |
| supxrbnd1 13305 | The supremum of a bounded-... |
| supxrbnd2 13306 | The supremum of a bounded-... |
| xrsup0 13307 | The supremum of an empty s... |
| supxrub 13308 | A member of a set of exten... |
| supxrlub 13309 | The supremum of a set of e... |
| supxrleub 13310 | The supremum of a set of e... |
| supxrre 13311 | The real and extended real... |
| supxrbnd 13312 | The supremum of a bounded-... |
| supxrgtmnf 13313 | The supremum of a nonempty... |
| supxrre1 13314 | The supremum of a nonempty... |
| supxrre2 13315 | The supremum of a nonempty... |
| supxrss 13316 | Smaller sets of extended r... |
| infxrcl 13317 | The infimum of an arbitrar... |
| infxrlb 13318 | A member of a set of exten... |
| infxrgelb 13319 | The infimum of a set of ex... |
| infxrre 13320 | The real and extended real... |
| infxrmnf 13321 | The infinimum of a set of ... |
| xrinf0 13322 | The infimum of the empty s... |
| infxrss 13323 | Larger sets of extended re... |
| reltre 13324 | For all real numbers there... |
| rpltrp 13325 | For all positive real numb... |
| reltxrnmnf 13326 | For all extended real numb... |
| infmremnf 13327 | The infimum of the reals i... |
| infmrp1 13328 | The infimum of the positiv... |
| ixxval 13337 | Value of the interval func... |
| elixx1 13338 | Membership in an interval ... |
| ixxf 13339 | The set of intervals of ex... |
| ixxex 13340 | The set of intervals of ex... |
| ixxssxr 13341 | The set of intervals of ex... |
| elixx3g 13342 | Membership in a set of ope... |
| ixxssixx 13343 | An interval is a subset of... |
| ixxdisj 13344 | Split an interval into dis... |
| ixxun 13345 | Split an interval into two... |
| ixxin 13346 | Intersection of two interv... |
| ixxss1 13347 | Subset relationship for in... |
| ixxss2 13348 | Subset relationship for in... |
| ixxss12 13349 | Subset relationship for in... |
| ixxub 13350 | Extract the upper bound of... |
| ixxlb 13351 | Extract the lower bound of... |
| iooex 13352 | The set of open intervals ... |
| iooval 13353 | Value of the open interval... |
| ioo0 13354 | An empty open interval of ... |
| ioon0 13355 | An open interval of extend... |
| ndmioo 13356 | The open interval function... |
| iooid 13357 | An open interval with iden... |
| elioo3g 13358 | Membership in a set of ope... |
| elioore 13359 | A member of an open interv... |
| lbioo 13360 | An open interval does not ... |
| ubioo 13361 | An open interval does not ... |
| iooval2 13362 | Value of the open interval... |
| iooin 13363 | Intersection of two open i... |
| iooss1 13364 | Subset relationship for op... |
| iooss2 13365 | Subset relationship for op... |
| iocval 13366 | Value of the open-below, c... |
| icoval 13367 | Value of the closed-below,... |
| iccval 13368 | Value of the closed interv... |
| elioo1 13369 | Membership in an open inte... |
| elioo2 13370 | Membership in an open inte... |
| elioc1 13371 | Membership in an open-belo... |
| elico1 13372 | Membership in a closed-bel... |
| elicc1 13373 | Membership in a closed int... |
| iccid 13374 | A closed interval with ide... |
| ico0 13375 | An empty open interval of ... |
| ioc0 13376 | An empty open interval of ... |
| icc0 13377 | An empty closed interval o... |
| dfrp2 13378 | Alternate definition of th... |
| elicod 13379 | Membership in a left-close... |
| icogelb 13380 | An element of a left-close... |
| elicore 13381 | A member of a left-closed ... |
| ubioc1 13382 | The upper bound belongs to... |
| lbico1 13383 | The lower bound belongs to... |
| iccleub 13384 | An element of a closed int... |
| iccgelb 13385 | An element of a closed int... |
| elioo5 13386 | Membership in an open inte... |
| eliooxr 13387 | A nonempty open interval s... |
| eliooord 13388 | Ordering implied by a memb... |
| elioo4g 13389 | Membership in an open inte... |
| ioossre 13390 | An open interval is a set ... |
| ioosscn 13391 | An open interval is a set ... |
| elioc2 13392 | Membership in an open-belo... |
| elico2 13393 | Membership in a closed-bel... |
| elicc2 13394 | Membership in a closed rea... |
| elicc2i 13395 | Inference for membership i... |
| elicc4 13396 | Membership in a closed rea... |
| iccss 13397 | Condition for a closed int... |
| iccssioo 13398 | Condition for a closed int... |
| icossico 13399 | Condition for a closed-bel... |
| iccss2 13400 | Condition for a closed int... |
| iccssico 13401 | Condition for a closed int... |
| iccssioo2 13402 | Condition for a closed int... |
| iccssico2 13403 | Condition for a closed int... |
| ioomax 13404 | The open interval from min... |
| iccmax 13405 | The closed interval from m... |
| ioopos 13406 | The set of positive reals ... |
| ioorp 13407 | The set of positive reals ... |
| iooshf 13408 | Shift the arguments of the... |
| iocssre 13409 | A closed-above interval wi... |
| icossre 13410 | A closed-below interval wi... |
| iccssre 13411 | A closed real interval is ... |
| iccssxr 13412 | A closed interval is a set... |
| iocssxr 13413 | An open-below, closed-abov... |
| icossxr 13414 | A closed-below, open-above... |
| ioossicc 13415 | An open interval is a subs... |
| iccssred 13416 | A closed real interval is ... |
| eliccxr 13417 | A member of a closed inter... |
| icossicc 13418 | A closed-below, open-above... |
| iocssicc 13419 | A closed-above, open-below... |
| ioossico 13420 | An open interval is a subs... |
| iocssioo 13421 | Condition for a closed int... |
| icossioo 13422 | Condition for a closed int... |
| ioossioo 13423 | Condition for an open inte... |
| iccsupr 13424 | A nonempty subset of a clo... |
| elioopnf 13425 | Membership in an unbounded... |
| elioomnf 13426 | Membership in an unbounded... |
| elicopnf 13427 | Membership in a closed unb... |
| repos 13428 | Two ways of saying that a ... |
| ioof 13429 | The set of open intervals ... |
| iccf 13430 | The set of closed interval... |
| unirnioo 13431 | The union of the range of ... |
| dfioo2 13432 | Alternate definition of th... |
| ioorebas 13433 | Open intervals are element... |
| xrge0neqmnf 13434 | A nonnegative extended rea... |
| xrge0nre 13435 | An extended real which is ... |
| elrege0 13436 | The predicate "is a nonneg... |
| nn0rp0 13437 | A nonnegative integer is a... |
| rge0ssre 13438 | Nonnegative real numbers a... |
| elxrge0 13439 | Elementhood in the set of ... |
| 0e0icopnf 13440 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13441 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13442 | The nonnegative reals are ... |
| ge0mulcl 13443 | The nonnegative reals are ... |
| ge0xaddcl 13444 | The nonnegative reals are ... |
| ge0xmulcl 13445 | The nonnegative extended r... |
| lbicc2 13446 | The lower bound of a close... |
| ubicc2 13447 | The upper bound of a close... |
| elicc01 13448 | Membership in the closed r... |
| elunitrn 13449 | The closed unit interval i... |
| elunitcn 13450 | The closed unit interval i... |
| 0elunit 13451 | Zero is an element of the ... |
| 1elunit 13452 | One is an element of the c... |
| iooneg 13453 | Membership in a negated op... |
| iccneg 13454 | Membership in a negated cl... |
| icoshft 13455 | A shifted real is a member... |
| icoshftf1o 13456 | Shifting a closed-below, o... |
| icoun 13457 | The union of two adjacent ... |
| icodisj 13458 | Adjacent left-closed right... |
| ioounsn 13459 | The union of an open inter... |
| snunioo 13460 | The closure of one end of ... |
| snunico 13461 | The closure of the open en... |
| snunioc 13462 | The closure of the open en... |
| prunioo 13463 | The closure of an open rea... |
| ioodisj 13464 | If the upper bound of one ... |
| ioojoin 13465 | Join two open intervals to... |
| difreicc 13466 | The class difference of ` ... |
| iccsplit 13467 | Split a closed interval in... |
| iccshftr 13468 | Membership in a shifted in... |
| iccshftri 13469 | Membership in a shifted in... |
| iccshftl 13470 | Membership in a shifted in... |
| iccshftli 13471 | Membership in a shifted in... |
| iccdil 13472 | Membership in a dilated in... |
| iccdili 13473 | Membership in a dilated in... |
| icccntr 13474 | Membership in a contracted... |
| icccntri 13475 | Membership in a contracted... |
| divelunit 13476 | A condition for a ratio to... |
| lincmb01cmp 13477 | A linear combination of tw... |
| iccf1o 13478 | Describe a bijection from ... |
| iccen 13479 | Any nontrivial closed inte... |
| xov1plusxeqvd 13480 | A complex number ` X ` is ... |
| unitssre 13481 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13482 | The closed unit interval i... |
| supicc 13483 | Supremum of a bounded set ... |
| supiccub 13484 | The supremum of a bounded ... |
| supicclub 13485 | The supremum of a bounded ... |
| supicclub2 13486 | The supremum of a bounded ... |
| zltaddlt1le 13487 | The sum of an integer and ... |
| xnn0xrge0 13488 | An extended nonnegative in... |
| fzval 13491 | The value of a finite set ... |
| fzval2 13492 | An alternative way of expr... |
| fzf 13493 | Establish the domain and c... |
| elfz1 13494 | Membership in a finite set... |
| elfz 13495 | Membership in a finite set... |
| elfz2 13496 | Membership in a finite set... |
| elfzd 13497 | Membership in a finite set... |
| elfz5 13498 | Membership in a finite set... |
| elfz4 13499 | Membership in a finite set... |
| elfzuzb 13500 | Membership in a finite set... |
| eluzfz 13501 | Membership in a finite set... |
| elfzuz 13502 | A member of a finite set o... |
| elfzuz3 13503 | Membership in a finite set... |
| elfzel2 13504 | Membership in a finite set... |
| elfzel1 13505 | Membership in a finite set... |
| elfzelz 13506 | A member of a finite set o... |
| elfzelzd 13507 | A member of a finite set o... |
| fzssz 13508 | A finite sequence of integ... |
| elfzle1 13509 | A member of a finite set o... |
| elfzle2 13510 | A member of a finite set o... |
| elfzuz2 13511 | Implication of membership ... |
| elfzle3 13512 | Membership in a finite set... |
| eluzfz1 13513 | Membership in a finite set... |
| eluzfz2 13514 | Membership in a finite set... |
| eluzfz2b 13515 | Membership in a finite set... |
| elfz3 13516 | Membership in a finite set... |
| elfz1eq 13517 | Membership in a finite set... |
| elfzubelfz 13518 | If there is a member in a ... |
| peano2fzr 13519 | A Peano-postulate-like the... |
| fzn0 13520 | Properties of a finite int... |
| fz0 13521 | A finite set of sequential... |
| fzn 13522 | A finite set of sequential... |
| fzen 13523 | A shifted finite set of se... |
| fz1n 13524 | A 1-based finite set of se... |
| 0nelfz1 13525 | 0 is not an element of a f... |
| 0fz1 13526 | Two ways to say a finite 1... |
| fz10 13527 | There are no integers betw... |
| uzsubsubfz 13528 | Membership of an integer g... |
| uzsubsubfz1 13529 | Membership of an integer g... |
| ige3m2fz 13530 | Membership of an integer g... |
| fzsplit2 13531 | Split a finite interval of... |
| fzsplit 13532 | Split a finite interval of... |
| fzdisj 13533 | Condition for two finite i... |
| fz01en 13534 | 0-based and 1-based finite... |
| elfznn 13535 | A member of a finite set o... |
| elfz1end 13536 | A nonempty finite range of... |
| fz1ssnn 13537 | A finite set of positive i... |
| fznn0sub 13538 | Subtraction closure for a ... |
| fzmmmeqm 13539 | Subtracting the difference... |
| fzaddel 13540 | Membership of a sum in a f... |
| fzadd2 13541 | Membership of a sum in a f... |
| fzsubel 13542 | Membership of a difference... |
| fzopth 13543 | A finite set of sequential... |
| fzass4 13544 | Two ways to express a nond... |
| fzss1 13545 | Subset relationship for fi... |
| fzss2 13546 | Subset relationship for fi... |
| fzssuz 13547 | A finite set of sequential... |
| fzsn 13548 | A finite interval of integ... |
| fzssp1 13549 | Subset relationship for fi... |
| fzssnn 13550 | Finite sets of sequential ... |
| ssfzunsnext 13551 | A subset of a finite seque... |
| ssfzunsn 13552 | A subset of a finite seque... |
| fzsuc 13553 | Join a successor to the en... |
| fzpred 13554 | Join a predecessor to the ... |
| fzpreddisj 13555 | A finite set of sequential... |
| elfzp1 13556 | Append an element to a fin... |
| fzp1ss 13557 | Subset relationship for fi... |
| fzelp1 13558 | Membership in a set of seq... |
| fzp1elp1 13559 | Add one to an element of a... |
| fznatpl1 13560 | Shift membership in a fini... |
| fzpr 13561 | A finite interval of integ... |
| fztp 13562 | A finite interval of integ... |
| fz12pr 13563 | An integer range between 1... |
| fzsuc2 13564 | Join a successor to the en... |
| fzp1disj 13565 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13566 | Remove a successor from th... |
| fzprval 13567 | Two ways of defining the f... |
| fztpval 13568 | Two ways of defining the f... |
| fzrev 13569 | Reversal of start and end ... |
| fzrev2 13570 | Reversal of start and end ... |
| fzrev2i 13571 | Reversal of start and end ... |
| fzrev3 13572 | The "complement" of a memb... |
| fzrev3i 13573 | The "complement" of a memb... |
| fznn 13574 | Finite set of sequential i... |
| elfz1b 13575 | Membership in a 1-based fi... |
| elfz1uz 13576 | Membership in a 1-based fi... |
| elfzm11 13577 | Membership in a finite set... |
| uzsplit 13578 | Express an upper integer s... |
| uzdisj 13579 | The first ` N ` elements o... |
| fseq1p1m1 13580 | Add/remove an item to/from... |
| fseq1m1p1 13581 | Add/remove an item to/from... |
| fz1sbc 13582 | Quantification over a one-... |
| elfzp1b 13583 | An integer is a member of ... |
| elfzm1b 13584 | An integer is a member of ... |
| elfzp12 13585 | Options for membership in ... |
| fzm1 13586 | Choices for an element of ... |
| fzneuz 13587 | No finite set of sequentia... |
| fznuz 13588 | Disjointness of the upper ... |
| uznfz 13589 | Disjointness of the upper ... |
| fzp1nel 13590 | One plus the upper bound o... |
| fzrevral 13591 | Reversal of scanning order... |
| fzrevral2 13592 | Reversal of scanning order... |
| fzrevral3 13593 | Reversal of scanning order... |
| fzshftral 13594 | Shift the scanning order i... |
| ige2m1fz1 13595 | Membership of an integer g... |
| ige2m1fz 13596 | Membership in a 0-based fi... |
| elfz2nn0 13597 | Membership in a finite set... |
| fznn0 13598 | Characterization of a fini... |
| elfznn0 13599 | A member of a finite set o... |
| elfz3nn0 13600 | The upper bound of a nonem... |
| fz0ssnn0 13601 | Finite sets of sequential ... |
| fz1ssfz0 13602 | Subset relationship for fi... |
| 0elfz 13603 | 0 is an element of a finit... |
| nn0fz0 13604 | A nonnegative integer is a... |
| elfz0add 13605 | An element of a finite set... |
| fz0sn 13606 | An integer range from 0 to... |
| fz0tp 13607 | An integer range from 0 to... |
| fz0to3un2pr 13608 | An integer range from 0 to... |
| fz0to4untppr 13609 | An integer range from 0 to... |
| elfz0ubfz0 13610 | An element of a finite set... |
| elfz0fzfz0 13611 | A member of a finite set o... |
| fz0fzelfz0 13612 | If a member of a finite se... |
| fznn0sub2 13613 | Subtraction closure for a ... |
| uzsubfz0 13614 | Membership of an integer g... |
| fz0fzdiffz0 13615 | The difference of an integ... |
| elfzmlbm 13616 | Subtracting the lower boun... |
| elfzmlbp 13617 | Subtracting the lower boun... |
| fzctr 13618 | Lemma for theorems about t... |
| difelfzle 13619 | The difference of two inte... |
| difelfznle 13620 | The difference of two inte... |
| nn0split 13621 | Express the set of nonnega... |
| nn0disj 13622 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13623 | A finite set of sequential... |
| fvffz0 13624 | The function value of a fu... |
| 1fv 13625 | A function on a singleton.... |
| 4fvwrd4 13626 | The first four function va... |
| 2ffzeq 13627 | Two functions over 0-based... |
| preduz 13628 | The value of the predecess... |
| prednn 13629 | The value of the predecess... |
| prednn0 13630 | The value of the predecess... |
| predfz 13631 | Calculate the predecessor ... |
| fzof 13634 | Functionality of the half-... |
| elfzoel1 13635 | Reverse closure for half-o... |
| elfzoel2 13636 | Reverse closure for half-o... |
| elfzoelz 13637 | Reverse closure for half-o... |
| fzoval 13638 | Value of the half-open int... |
| elfzo 13639 | Membership in a half-open ... |
| elfzo2 13640 | Membership in a half-open ... |
| elfzouz 13641 | Membership in a half-open ... |
| nelfzo 13642 | An integer not being a mem... |
| fzolb 13643 | The left endpoint of a hal... |
| fzolb2 13644 | The left endpoint of a hal... |
| elfzole1 13645 | A member in a half-open in... |
| elfzolt2 13646 | A member in a half-open in... |
| elfzolt3 13647 | Membership in a half-open ... |
| elfzolt2b 13648 | A member in a half-open in... |
| elfzolt3b 13649 | Membership in a half-open ... |
| elfzop1le2 13650 | A member in a half-open in... |
| fzonel 13651 | A half-open range does not... |
| elfzouz2 13652 | The upper bound of a half-... |
| elfzofz 13653 | A half-open range is conta... |
| elfzo3 13654 | Express membership in a ha... |
| fzon0 13655 | A half-open integer interv... |
| fzossfz 13656 | A half-open range is conta... |
| fzossz 13657 | A half-open integer interv... |
| fzon 13658 | A half-open set of sequent... |
| fzo0n 13659 | A half-open range of nonne... |
| fzonlt0 13660 | A half-open integer range ... |
| fzo0 13661 | Half-open sets with equal ... |
| fzonnsub 13662 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13663 | If ` M < N ` then ` N - M ... |
| fzoss1 13664 | Subset relationship for ha... |
| fzoss2 13665 | Subset relationship for ha... |
| fzossrbm1 13666 | Subset of a half-open rang... |
| fzo0ss1 13667 | Subset relationship for ha... |
| fzossnn0 13668 | A half-open integer range ... |
| fzospliti 13669 | One direction of splitting... |
| fzosplit 13670 | Split a half-open integer ... |
| fzodisj 13671 | Abutting half-open integer... |
| fzouzsplit 13672 | Split an upper integer set... |
| fzouzdisj 13673 | A half-open integer range ... |
| fzoun 13674 | A half-open integer range ... |
| fzodisjsn 13675 | A half-open integer range ... |
| prinfzo0 13676 | The intersection of a half... |
| lbfzo0 13677 | An integer is strictly gre... |
| elfzo0 13678 | Membership in a half-open ... |
| elfzo0z 13679 | Membership in a half-open ... |
| nn0p1elfzo 13680 | A nonnegative integer incr... |
| elfzo0le 13681 | A member in a half-open ra... |
| elfzonn0 13682 | A member of a half-open ra... |
| fzonmapblen 13683 | The result of subtracting ... |
| fzofzim 13684 | If a nonnegative integer i... |
| fz1fzo0m1 13685 | Translation of one between... |
| fzossnn 13686 | Half-open integer ranges s... |
| elfzo1 13687 | Membership in a half-open ... |
| fzo1fzo0n0 13688 | An integer between 1 and a... |
| fzo0n0 13689 | A half-open integer range ... |
| fzoaddel 13690 | Translate membership in a ... |
| fzo0addel 13691 | Translate membership in a ... |
| fzo0addelr 13692 | Translate membership in a ... |
| fzoaddel2 13693 | Translate membership in a ... |
| elfzoext 13694 | Membership of an integer i... |
| elincfzoext 13695 | Membership of an increased... |
| fzosubel 13696 | Translate membership in a ... |
| fzosubel2 13697 | Membership in a translated... |
| fzosubel3 13698 | Membership in a translated... |
| eluzgtdifelfzo 13699 | Membership of the differen... |
| ige2m2fzo 13700 | Membership of an integer g... |
| fzocatel 13701 | Translate membership in a ... |
| ubmelfzo 13702 | If an integer in a 1-based... |
| elfzodifsumelfzo 13703 | If an integer is in a half... |
| elfzom1elp1fzo 13704 | Membership of an integer i... |
| elfzom1elfzo 13705 | Membership in a half-open ... |
| fzval3 13706 | Expressing a closed intege... |
| fz0add1fz1 13707 | Translate membership in a ... |
| fzosn 13708 | Expressing a singleton as ... |
| elfzomin 13709 | Membership of an integer i... |
| zpnn0elfzo 13710 | Membership of an integer i... |
| zpnn0elfzo1 13711 | Membership of an integer i... |
| fzosplitsnm1 13712 | Removing a singleton from ... |
| elfzonlteqm1 13713 | If an element of a half-op... |
| fzonn0p1 13714 | A nonnegative integer is e... |
| fzossfzop1 13715 | A half-open range of nonne... |
| fzonn0p1p1 13716 | If a nonnegative integer i... |
| elfzom1p1elfzo 13717 | Increasing an element of a... |
| fzo0ssnn0 13718 | Half-open integer ranges s... |
| fzo01 13719 | Expressing the singleton o... |
| fzo12sn 13720 | A 1-based half-open intege... |
| fzo13pr 13721 | A 1-based half-open intege... |
| fzo0to2pr 13722 | A half-open integer range ... |
| fzo0to3tp 13723 | A half-open integer range ... |
| fzo0to42pr 13724 | A half-open integer range ... |
| fzo1to4tp 13725 | A half-open integer range ... |
| fzo0sn0fzo1 13726 | A half-open range of nonne... |
| elfzo0l 13727 | A member of a half-open ra... |
| fzoend 13728 | The endpoint of a half-ope... |
| fzo0end 13729 | The endpoint of a zero-bas... |
| ssfzo12 13730 | Subset relationship for ha... |
| ssfzoulel 13731 | If a half-open integer ran... |
| ssfzo12bi 13732 | Subset relationship for ha... |
| ubmelm1fzo 13733 | The result of subtracting ... |
| fzofzp1 13734 | If a point is in a half-op... |
| fzofzp1b 13735 | If a point is in a half-op... |
| elfzom1b 13736 | An integer is a member of ... |
| elfzom1elp1fzo1 13737 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13738 | Membership of a positive i... |
| elfzonelfzo 13739 | If an element of a half-op... |
| fzonfzoufzol 13740 | If an element of a half-op... |
| elfzomelpfzo 13741 | An integer increased by an... |
| elfznelfzo 13742 | A value in a finite set of... |
| elfznelfzob 13743 | A value in a finite set of... |
| peano2fzor 13744 | A Peano-postulate-like the... |
| fzosplitsn 13745 | Extending a half-open rang... |
| fzosplitpr 13746 | Extending a half-open inte... |
| fzosplitprm1 13747 | Extending a half-open inte... |
| fzosplitsni 13748 | Membership in a half-open ... |
| fzisfzounsn 13749 | A finite interval of integ... |
| elfzr 13750 | A member of a finite inter... |
| elfzlmr 13751 | A member of a finite inter... |
| elfz0lmr 13752 | A member of a finite inter... |
| fzostep1 13753 | Two possibilities for a nu... |
| fzoshftral 13754 | Shift the scanning order i... |
| fzind2 13755 | Induction on the integers ... |
| fvinim0ffz 13756 | The function values for th... |
| injresinjlem 13757 | Lemma for ~ injresinj . (... |
| injresinj 13758 | A function whose restricti... |
| subfzo0 13759 | The difference between two... |
| flval 13764 | Value of the floor (greate... |
| flcl 13765 | The floor (greatest intege... |
| reflcl 13766 | The floor (greatest intege... |
| fllelt 13767 | A basic property of the fl... |
| flcld 13768 | The floor (greatest intege... |
| flle 13769 | A basic property of the fl... |
| flltp1 13770 | A basic property of the fl... |
| fllep1 13771 | A basic property of the fl... |
| fraclt1 13772 | The fractional part of a r... |
| fracle1 13773 | The fractional part of a r... |
| fracge0 13774 | The fractional part of a r... |
| flge 13775 | The floor function value i... |
| fllt 13776 | The floor function value i... |
| flflp1 13777 | Move floor function betwee... |
| flid 13778 | An integer is its own floo... |
| flidm 13779 | The floor function is idem... |
| flidz 13780 | A real number equals its f... |
| flltnz 13781 | The floor of a non-integer... |
| flwordi 13782 | Ordering relation for the ... |
| flword2 13783 | Ordering relation for the ... |
| flval2 13784 | An alternate way to define... |
| flval3 13785 | An alternate way to define... |
| flbi 13786 | A condition equivalent to ... |
| flbi2 13787 | A condition equivalent to ... |
| adddivflid 13788 | The floor of a sum of an i... |
| ico01fl0 13789 | The floor of a real number... |
| flge0nn0 13790 | The floor of a number grea... |
| flge1nn 13791 | The floor of a number grea... |
| fldivnn0 13792 | The floor function of a di... |
| refldivcl 13793 | The floor function of a di... |
| divfl0 13794 | The floor of a fraction is... |
| fladdz 13795 | An integer can be moved in... |
| flzadd 13796 | An integer can be moved in... |
| flmulnn0 13797 | Move a nonnegative integer... |
| btwnzge0 13798 | A real bounded between an ... |
| 2tnp1ge0ge0 13799 | Two times an integer plus ... |
| flhalf 13800 | Ordering relation for the ... |
| fldivle 13801 | The floor function of a di... |
| fldivnn0le 13802 | The floor function of a di... |
| flltdivnn0lt 13803 | The floor function of a di... |
| ltdifltdiv 13804 | If the dividend of a divis... |
| fldiv4p1lem1div2 13805 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13806 | The floor of an integer gr... |
| fldiv4lem1div2 13807 | The floor of a positive in... |
| ceilval 13808 | The value of the ceiling f... |
| dfceil2 13809 | Alternative definition of ... |
| ceilval2 13810 | The value of the ceiling f... |
| ceicl 13811 | The ceiling function retur... |
| ceilcl 13812 | Closure of the ceiling fun... |
| ceilcld 13813 | Closure of the ceiling fun... |
| ceige 13814 | The ceiling of a real numb... |
| ceilge 13815 | The ceiling of a real numb... |
| ceilged 13816 | The ceiling of a real numb... |
| ceim1l 13817 | One less than the ceiling ... |
| ceilm1lt 13818 | One less than the ceiling ... |
| ceile 13819 | The ceiling of a real numb... |
| ceille 13820 | The ceiling of a real numb... |
| ceilid 13821 | An integer is its own ceil... |
| ceilidz 13822 | A real number equals its c... |
| flleceil 13823 | The floor of a real number... |
| fleqceilz 13824 | A real number is an intege... |
| quoremz 13825 | Quotient and remainder of ... |
| quoremnn0 13826 | Quotient and remainder of ... |
| quoremnn0ALT 13827 | Alternate proof of ~ quore... |
| intfrac2 13828 | Decompose a real into inte... |
| intfracq 13829 | Decompose a rational numbe... |
| fldiv 13830 | Cancellation of the embedd... |
| fldiv2 13831 | Cancellation of an embedde... |
| fznnfl 13832 | Finite set of sequential i... |
| uzsup 13833 | An upper set of integers i... |
| ioopnfsup 13834 | An upper set of reals is u... |
| icopnfsup 13835 | An upper set of reals is u... |
| rpsup 13836 | The positive reals are unb... |
| resup 13837 | The real numbers are unbou... |
| xrsup 13838 | The extended real numbers ... |
| modval 13841 | The value of the modulo op... |
| modvalr 13842 | The value of the modulo op... |
| modcl 13843 | Closure law for the modulo... |
| flpmodeq 13844 | Partition of a division in... |
| modcld 13845 | Closure law for the modulo... |
| mod0 13846 | ` A mod B ` is zero iff ` ... |
| mulmod0 13847 | The product of an integer ... |
| negmod0 13848 | ` A ` is divisible by ` B ... |
| modge0 13849 | The modulo operation is no... |
| modlt 13850 | The modulo operation is le... |
| modelico 13851 | Modular reduction produces... |
| moddiffl 13852 | Value of the modulo operat... |
| moddifz 13853 | The modulo operation diffe... |
| modfrac 13854 | The fractional part of a n... |
| flmod 13855 | The floor function express... |
| intfrac 13856 | Break a number into its in... |
| zmod10 13857 | An integer modulo 1 is 0. ... |
| zmod1congr 13858 | Two arbitrary integers are... |
| modmulnn 13859 | Move a positive integer in... |
| modvalp1 13860 | The value of the modulo op... |
| zmodcl 13861 | Closure law for the modulo... |
| zmodcld 13862 | Closure law for the modulo... |
| zmodfz 13863 | An integer mod ` B ` lies ... |
| zmodfzo 13864 | An integer mod ` B ` lies ... |
| zmodfzp1 13865 | An integer mod ` B ` lies ... |
| modid 13866 | Identity law for modulo. ... |
| modid0 13867 | A positive real number mod... |
| modid2 13868 | Identity law for modulo. ... |
| zmodid2 13869 | Identity law for modulo re... |
| zmodidfzo 13870 | Identity law for modulo re... |
| zmodidfzoimp 13871 | Identity law for modulo re... |
| 0mod 13872 | Special case: 0 modulo a p... |
| 1mod 13873 | Special case: 1 modulo a r... |
| modabs 13874 | Absorption law for modulo.... |
| modabs2 13875 | Absorption law for modulo.... |
| modcyc 13876 | The modulo operation is pe... |
| modcyc2 13877 | The modulo operation is pe... |
| modadd1 13878 | Addition property of the m... |
| modaddabs 13879 | Absorption law for modulo.... |
| modaddmod 13880 | The sum of a real number m... |
| muladdmodid 13881 | The sum of a positive real... |
| mulp1mod1 13882 | The product of an integer ... |
| modmuladd 13883 | Decomposition of an intege... |
| modmuladdim 13884 | Implication of a decomposi... |
| modmuladdnn0 13885 | Implication of a decomposi... |
| negmod 13886 | The negation of a number m... |
| m1modnnsub1 13887 | Minus one modulo a positiv... |
| m1modge3gt1 13888 | Minus one modulo an intege... |
| addmodid 13889 | The sum of a positive inte... |
| addmodidr 13890 | The sum of a positive inte... |
| modadd2mod 13891 | The sum of a real number m... |
| modm1p1mod0 13892 | If a real number modulo a ... |
| modltm1p1mod 13893 | If a real number modulo a ... |
| modmul1 13894 | Multiplication property of... |
| modmul12d 13895 | Multiplication property of... |
| modnegd 13896 | Negation property of the m... |
| modadd12d 13897 | Additive property of the m... |
| modsub12d 13898 | Subtraction property of th... |
| modsubmod 13899 | The difference of a real n... |
| modsubmodmod 13900 | The difference of a real n... |
| 2txmodxeq0 13901 | Two times a positive real ... |
| 2submod 13902 | If a real number is betwee... |
| modifeq2int 13903 | If a nonnegative integer i... |
| modaddmodup 13904 | The sum of an integer modu... |
| modaddmodlo 13905 | The sum of an integer modu... |
| modmulmod 13906 | The product of a real numb... |
| modmulmodr 13907 | The product of an integer ... |
| modaddmulmod 13908 | The sum of a real number a... |
| moddi 13909 | Distribute multiplication ... |
| modsubdir 13910 | Distribute the modulo oper... |
| modeqmodmin 13911 | A real number equals the d... |
| modirr 13912 | A number modulo an irratio... |
| modfzo0difsn 13913 | For a number within a half... |
| modsumfzodifsn 13914 | The sum of a number within... |
| modlteq 13915 | Two nonnegative integers l... |
| addmodlteq 13916 | Two nonnegative integers l... |
| om2uz0i 13917 | The mapping ` G ` is a one... |
| om2uzsuci 13918 | The value of ` G ` (see ~ ... |
| om2uzuzi 13919 | The value ` G ` (see ~ om2... |
| om2uzlti 13920 | Less-than relation for ` G... |
| om2uzlt2i 13921 | The mapping ` G ` (see ~ o... |
| om2uzrani 13922 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13923 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13924 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13925 | An alternative definition ... |
| om2uzrdg 13926 | A helper lemma for the val... |
| uzrdglem 13927 | A helper lemma for the val... |
| uzrdgfni 13928 | The recursive definition g... |
| uzrdg0i 13929 | Initial value of a recursi... |
| uzrdgsuci 13930 | Successor value of a recur... |
| ltweuz 13931 | ` < ` is a well-founded re... |
| ltwenn 13932 | Less than well-orders the ... |
| ltwefz 13933 | Less than well-orders a se... |
| uzenom 13934 | An upper integer set is de... |
| uzinf 13935 | An upper integer set is in... |
| nnnfi 13936 | The set of positive intege... |
| uzrdgxfr 13937 | Transfer the value of the ... |
| fzennn 13938 | The cardinality of a finit... |
| fzen2 13939 | The cardinality of a finit... |
| cardfz 13940 | The cardinality of a finit... |
| hashgf1o 13941 | ` G ` maps ` _om ` one-to-... |
| fzfi 13942 | A finite interval of integ... |
| fzfid 13943 | Commonly used special case... |
| fzofi 13944 | Half-open integer sets are... |
| fsequb 13945 | The values of a finite rea... |
| fsequb2 13946 | The values of a finite rea... |
| fseqsupcl 13947 | The values of a finite rea... |
| fseqsupubi 13948 | The values of a finite rea... |
| nn0ennn 13949 | The nonnegative integers a... |
| nnenom 13950 | The set of positive intege... |
| nnct 13951 | ` NN ` is countable. (Con... |
| uzindi 13952 | Indirect strong induction ... |
| axdc4uzlem 13953 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13954 | A version of ~ axdc4 that ... |
| ssnn0fi 13955 | A subset of the nonnegativ... |
| rabssnn0fi 13956 | A subset of the nonnegativ... |
| uzsinds 13957 | Strong (or "total") induct... |
| nnsinds 13958 | Strong (or "total") induct... |
| nn0sinds 13959 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13960 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13961 | If all functions of a fini... |
| fsuppmapnn0fiubex 13962 | If all functions of a fini... |
| fsuppmapnn0fiub0 13963 | If all functions of a fini... |
| suppssfz 13964 | Condition for a function o... |
| fsuppmapnn0ub 13965 | If a function over the non... |
| fsuppmapnn0fz 13966 | If a function over the non... |
| mptnn0fsupp 13967 | A mapping from the nonnega... |
| mptnn0fsuppd 13968 | A mapping from the nonnega... |
| mptnn0fsuppr 13969 | A finitely supported mappi... |
| f13idfv 13970 | A one-to-one function with... |
| seqex 13973 | Existence of the sequence ... |
| seqeq1 13974 | Equality theorem for the s... |
| seqeq2 13975 | Equality theorem for the s... |
| seqeq3 13976 | Equality theorem for the s... |
| seqeq1d 13977 | Equality deduction for the... |
| seqeq2d 13978 | Equality deduction for the... |
| seqeq3d 13979 | Equality deduction for the... |
| seqeq123d 13980 | Equality deduction for the... |
| nfseq 13981 | Hypothesis builder for the... |
| seqval 13982 | Value of the sequence buil... |
| seqfn 13983 | The sequence builder funct... |
| seq1 13984 | Value of the sequence buil... |
| seq1i 13985 | Value of the sequence buil... |
| seqp1 13986 | Value of the sequence buil... |
| seqexw 13987 | Weak version of ~ seqex th... |
| seqp1d 13988 | Value of the sequence buil... |
| seqp1iOLD 13989 | Obsolete version of ~ seqp... |
| seqm1 13990 | Value of the sequence buil... |
| seqcl2 13991 | Closure properties of the ... |
| seqf2 13992 | Range of the recursive seq... |
| seqcl 13993 | Closure properties of the ... |
| seqf 13994 | Range of the recursive seq... |
| seqfveq2 13995 | Equality of sequences. (C... |
| seqfeq2 13996 | Equality of sequences. (C... |
| seqfveq 13997 | Equality of sequences. (C... |
| seqfeq 13998 | Equality of sequences. (C... |
| seqshft2 13999 | Shifting the index set of ... |
| seqres 14000 | Restricting its characteri... |
| serf 14001 | An infinite series of comp... |
| serfre 14002 | An infinite series of real... |
| monoord 14003 | Ordering relation for a mo... |
| monoord2 14004 | Ordering relation for a mo... |
| sermono 14005 | The partial sums in an inf... |
| seqsplit 14006 | Split a sequence into two ... |
| seq1p 14007 | Removing the first term fr... |
| seqcaopr3 14008 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 14009 | The sum of two infinite se... |
| seqcaopr 14010 | The sum of two infinite se... |
| seqf1olem2a 14011 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 14012 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 14013 | Lemma for ~ seqf1o . (Con... |
| seqf1o 14014 | Rearrange a sum via an arb... |
| seradd 14015 | The sum of two infinite se... |
| sersub 14016 | The difference of two infi... |
| seqid3 14017 | A sequence that consists e... |
| seqid 14018 | Discarding the first few t... |
| seqid2 14019 | The last few partial sums ... |
| seqhomo 14020 | Apply a homomorphism to a ... |
| seqz 14021 | If the operation ` .+ ` ha... |
| seqfeq4 14022 | Equality of series under d... |
| seqfeq3 14023 | Equality of series under d... |
| seqdistr 14024 | The distributive property ... |
| ser0 14025 | The value of the partial s... |
| ser0f 14026 | A zero-valued infinite ser... |
| serge0 14027 | A finite sum of nonnegativ... |
| serle 14028 | Comparison of partial sums... |
| ser1const 14029 | Value of the partial serie... |
| seqof 14030 | Distribute function operat... |
| seqof2 14031 | Distribute function operat... |
| expval 14034 | Value of exponentiation to... |
| expnnval 14035 | Value of exponentiation to... |
| exp0 14036 | Value of a complex number ... |
| 0exp0e1 14037 | The zeroth power of zero e... |
| exp1 14038 | Value of a complex number ... |
| expp1 14039 | Value of a complex number ... |
| expneg 14040 | Value of a complex number ... |
| expneg2 14041 | Value of a complex number ... |
| expn1 14042 | A complex number raised to... |
| expcllem 14043 | Lemma for proving nonnegat... |
| expcl2lem 14044 | Lemma for proving integer ... |
| nnexpcl 14045 | Closure of exponentiation ... |
| nn0expcl 14046 | Closure of exponentiation ... |
| zexpcl 14047 | Closure of exponentiation ... |
| qexpcl 14048 | Closure of exponentiation ... |
| reexpcl 14049 | Closure of exponentiation ... |
| expcl 14050 | Closure law for nonnegativ... |
| rpexpcl 14051 | Closure law for integer ex... |
| qexpclz 14052 | Closure of integer exponen... |
| reexpclz 14053 | Closure of integer exponen... |
| expclzlem 14054 | Lemma for ~ expclz . (Con... |
| expclz 14055 | Closure law for integer ex... |
| m1expcl2 14056 | Closure of integer exponen... |
| m1expcl 14057 | Closure of exponentiation ... |
| zexpcld 14058 | Closure of exponentiation ... |
| nn0expcli 14059 | Closure of exponentiation ... |
| nn0sqcl 14060 | The square of a nonnegativ... |
| expm1t 14061 | Exponentiation in terms of... |
| 1exp 14062 | Value of 1 raised to an in... |
| expeq0 14063 | A positive integer power i... |
| expne0 14064 | A positive integer power i... |
| expne0i 14065 | An integer power is nonzer... |
| expgt0 14066 | A positive real raised to ... |
| expnegz 14067 | Value of a nonzero complex... |
| 0exp 14068 | Value of zero raised to a ... |
| expge0 14069 | A nonnegative real raised ... |
| expge1 14070 | A real greater than or equ... |
| expgt1 14071 | A real greater than 1 rais... |
| mulexp 14072 | Nonnegative integer expone... |
| mulexpz 14073 | Integer exponentiation of ... |
| exprec 14074 | Integer exponentiation of ... |
| expadd 14075 | Sum of exponents law for n... |
| expaddzlem 14076 | Lemma for ~ expaddz . (Co... |
| expaddz 14077 | Sum of exponents law for i... |
| expmul 14078 | Product of exponents law f... |
| expmulz 14079 | Product of exponents law f... |
| m1expeven 14080 | Exponentiation of negative... |
| expsub 14081 | Exponent subtraction law f... |
| expp1z 14082 | Value of a nonzero complex... |
| expm1 14083 | Value of a nonzero complex... |
| expdiv 14084 | Nonnegative integer expone... |
| sqval 14085 | Value of the square of a c... |
| sqneg 14086 | The square of the negative... |
| sqsubswap 14087 | Swap the order of subtract... |
| sqcl 14088 | Closure of square. (Contr... |
| sqmul 14089 | Distribution of squaring o... |
| sqeq0 14090 | A complex number is zero i... |
| sqdiv 14091 | Distribution of squaring o... |
| sqdivid 14092 | The square of a nonzero co... |
| sqne0 14093 | A complex number is nonzer... |
| resqcl 14094 | Closure of squaring in rea... |
| resqcld 14095 | Closure of squaring in rea... |
| sqgt0 14096 | The square of a nonzero re... |
| sqn0rp 14097 | The square of a nonzero re... |
| nnsqcl 14098 | The positive naturals are ... |
| zsqcl 14099 | Integers are closed under ... |
| qsqcl 14100 | The square of a rational i... |
| sq11 14101 | The square function is one... |
| nn0sq11 14102 | The square function is one... |
| lt2sq 14103 | The square function is inc... |
| le2sq 14104 | The square function is non... |
| le2sq2 14105 | The square function is non... |
| sqge0 14106 | The square of a real is no... |
| sqge0d 14107 | The square of a real is no... |
| zsqcl2 14108 | The square of an integer i... |
| 0expd 14109 | Value of zero raised to a ... |
| exp0d 14110 | Value of a complex number ... |
| exp1d 14111 | Value of a complex number ... |
| expeq0d 14112 | If a positive integer powe... |
| sqvald 14113 | Value of square. Inferenc... |
| sqcld 14114 | Closure of square. (Contr... |
| sqeq0d 14115 | A number is zero iff its s... |
| expcld 14116 | Closure law for nonnegativ... |
| expp1d 14117 | Value of a complex number ... |
| expaddd 14118 | Sum of exponents law for n... |
| expmuld 14119 | Product of exponents law f... |
| sqrecd 14120 | Square of reciprocal is re... |
| expclzd 14121 | Closure law for integer ex... |
| expne0d 14122 | A nonnegative integer powe... |
| expnegd 14123 | Value of a nonzero complex... |
| exprecd 14124 | An integer power of a reci... |
| expp1zd 14125 | Value of a nonzero complex... |
| expm1d 14126 | Value of a nonzero complex... |
| expsubd 14127 | Exponent subtraction law f... |
| sqmuld 14128 | Distribution of squaring o... |
| sqdivd 14129 | Distribution of squaring o... |
| expdivd 14130 | Nonnegative integer expone... |
| mulexpd 14131 | Nonnegative integer expone... |
| znsqcld 14132 | The square of a nonzero in... |
| reexpcld 14133 | Closure of exponentiation ... |
| expge0d 14134 | A nonnegative real raised ... |
| expge1d 14135 | A real greater than or equ... |
| ltexp2a 14136 | Exponent ordering relation... |
| expmordi 14137 | Base ordering relationship... |
| rpexpmord 14138 | Base ordering relationship... |
| expcan 14139 | Cancellation law for integ... |
| ltexp2 14140 | Strict ordering law for ex... |
| leexp2 14141 | Ordering law for exponenti... |
| leexp2a 14142 | Weak ordering relationship... |
| ltexp2r 14143 | The integer powers of a fi... |
| leexp2r 14144 | Weak ordering relationship... |
| leexp1a 14145 | Weak base ordering relatio... |
| exple1 14146 | A real between 0 and 1 inc... |
| expubnd 14147 | An upper bound on ` A ^ N ... |
| sumsqeq0 14148 | The sum of two squres of r... |
| sqvali 14149 | Value of square. Inferenc... |
| sqcli 14150 | Closure of square. (Contr... |
| sqeq0i 14151 | A complex number is zero i... |
| sqrecii 14152 | The square of a reciprocal... |
| sqmuli 14153 | Distribution of squaring o... |
| sqdivi 14154 | Distribution of squaring o... |
| resqcli 14155 | Closure of square in reals... |
| sqgt0i 14156 | The square of a nonzero re... |
| sqge0i 14157 | The square of a real is no... |
| lt2sqi 14158 | The square function on non... |
| le2sqi 14159 | The square function on non... |
| sq11i 14160 | The square function is one... |
| sq0 14161 | The square of 0 is 0. (Co... |
| sq0i 14162 | If a number is zero, then ... |
| sq0id 14163 | If a number is zero, then ... |
| sq1 14164 | The square of 1 is 1. (Co... |
| neg1sqe1 14165 | The square of ` -u 1 ` is ... |
| sq2 14166 | The square of 2 is 4. (Co... |
| sq3 14167 | The square of 3 is 9. (Co... |
| sq4e2t8 14168 | The square of 4 is 2 times... |
| cu2 14169 | The cube of 2 is 8. (Cont... |
| irec 14170 | The reciprocal of ` _i ` .... |
| i2 14171 | ` _i ` squared. (Contribu... |
| i3 14172 | ` _i ` cubed. (Contribute... |
| i4 14173 | ` _i ` to the fourth power... |
| nnlesq 14174 | A positive integer is less... |
| zzlesq 14175 | An integer is less than or... |
| iexpcyc 14176 | Taking ` _i ` to the ` K `... |
| expnass 14177 | A counterexample showing t... |
| sqlecan 14178 | Cancel one factor of a squ... |
| subsq 14179 | Factor the difference of t... |
| subsq2 14180 | Express the difference of ... |
| binom2i 14181 | The square of a binomial. ... |
| subsqi 14182 | Factor the difference of t... |
| sqeqori 14183 | The squares of two complex... |
| subsq0i 14184 | The two solutions to the d... |
| sqeqor 14185 | The squares of two complex... |
| binom2 14186 | The square of a binomial. ... |
| binom21 14187 | Special case of ~ binom2 w... |
| binom2sub 14188 | Expand the square of a sub... |
| binom2sub1 14189 | Special case of ~ binom2su... |
| binom2subi 14190 | Expand the square of a sub... |
| mulbinom2 14191 | The square of a binomial w... |
| binom3 14192 | The cube of a binomial. (... |
| sq01 14193 | If a complex number equals... |
| zesq 14194 | An integer is even iff its... |
| nnesq 14195 | A positive integer is even... |
| crreczi 14196 | Reciprocal of a complex nu... |
| bernneq 14197 | Bernoulli's inequality, du... |
| bernneq2 14198 | Variation of Bernoulli's i... |
| bernneq3 14199 | A corollary of ~ bernneq .... |
| expnbnd 14200 | Exponentiation with a base... |
| expnlbnd 14201 | The reciprocal of exponent... |
| expnlbnd2 14202 | The reciprocal of exponent... |
| expmulnbnd 14203 | Exponentiation with a base... |
| digit2 14204 | Two ways to express the ` ... |
| digit1 14205 | Two ways to express the ` ... |
| modexp 14206 | Exponentiation property of... |
| discr1 14207 | A nonnegative quadratic fo... |
| discr 14208 | If a quadratic polynomial ... |
| expnngt1 14209 | If an integer power with a... |
| expnngt1b 14210 | An integer power with an i... |
| sqoddm1div8 14211 | A squared odd number minus... |
| nnsqcld 14212 | The naturals are closed un... |
| nnexpcld 14213 | Closure of exponentiation ... |
| nn0expcld 14214 | Closure of exponentiation ... |
| rpexpcld 14215 | Closure law for exponentia... |
| ltexp2rd 14216 | The power of a positive nu... |
| reexpclzd 14217 | Closure of exponentiation ... |
| sqgt0d 14218 | The square of a nonzero re... |
| ltexp2d 14219 | Ordering relationship for ... |
| leexp2d 14220 | Ordering law for exponenti... |
| expcand 14221 | Ordering relationship for ... |
| leexp2ad 14222 | Ordering relationship for ... |
| leexp2rd 14223 | Ordering relationship for ... |
| lt2sqd 14224 | The square function on non... |
| le2sqd 14225 | The square function on non... |
| sq11d 14226 | The square function is one... |
| mulsubdivbinom2 14227 | The square of a binomial w... |
| muldivbinom2 14228 | The square of a binomial w... |
| sq10 14229 | The square of 10 is 100. ... |
| sq10e99m1 14230 | The square of 10 is 99 plu... |
| 3dec 14231 | A "decimal constructor" wh... |
| nn0le2msqi 14232 | The square function on non... |
| nn0opthlem1 14233 | A rather pretty lemma for ... |
| nn0opthlem2 14234 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14235 | An ordered pair theorem fo... |
| nn0opth2i 14236 | An ordered pair theorem fo... |
| nn0opth2 14237 | An ordered pair theorem fo... |
| facnn 14240 | Value of the factorial fun... |
| fac0 14241 | The factorial of 0. (Cont... |
| fac1 14242 | The factorial of 1. (Cont... |
| facp1 14243 | The factorial of a success... |
| fac2 14244 | The factorial of 2. (Cont... |
| fac3 14245 | The factorial of 3. (Cont... |
| fac4 14246 | The factorial of 4. (Cont... |
| facnn2 14247 | Value of the factorial fun... |
| faccl 14248 | Closure of the factorial f... |
| faccld 14249 | Closure of the factorial f... |
| facmapnn 14250 | The factorial function res... |
| facne0 14251 | The factorial function is ... |
| facdiv 14252 | A positive integer divides... |
| facndiv 14253 | No positive integer (great... |
| facwordi 14254 | Ordering property of facto... |
| faclbnd 14255 | A lower bound for the fact... |
| faclbnd2 14256 | A lower bound for the fact... |
| faclbnd3 14257 | A lower bound for the fact... |
| faclbnd4lem1 14258 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14259 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14260 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14261 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14262 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14263 | The factorial function gro... |
| faclbnd6 14264 | Geometric lower bound for ... |
| facubnd 14265 | An upper bound for the fac... |
| facavg 14266 | The product of two factori... |
| bcval 14269 | Value of the binomial coef... |
| bcval2 14270 | Value of the binomial coef... |
| bcval3 14271 | Value of the binomial coef... |
| bcval4 14272 | Value of the binomial coef... |
| bcrpcl 14273 | Closure of the binomial co... |
| bccmpl 14274 | "Complementing" its second... |
| bcn0 14275 | ` N ` choose 0 is 1. Rema... |
| bc0k 14276 | The binomial coefficient "... |
| bcnn 14277 | ` N ` choose ` N ` is 1. ... |
| bcn1 14278 | Binomial coefficient: ` N ... |
| bcnp1n 14279 | Binomial coefficient: ` N ... |
| bcm1k 14280 | The proportion of one bino... |
| bcp1n 14281 | The proportion of one bino... |
| bcp1nk 14282 | The proportion of one bino... |
| bcval5 14283 | Write out the top and bott... |
| bcn2 14284 | Binomial coefficient: ` N ... |
| bcp1m1 14285 | Compute the binomial coeff... |
| bcpasc 14286 | Pascal's rule for the bino... |
| bccl 14287 | A binomial coefficient, in... |
| bccl2 14288 | A binomial coefficient, in... |
| bcn2m1 14289 | Compute the binomial coeff... |
| bcn2p1 14290 | Compute the binomial coeff... |
| permnn 14291 | The number of permutations... |
| bcnm1 14292 | The binomial coefficent of... |
| 4bc3eq4 14293 | The value of four choose t... |
| 4bc2eq6 14294 | The value of four choose t... |
| hashkf 14297 | The finite part of the siz... |
| hashgval 14298 | The value of the ` # ` fun... |
| hashginv 14299 | The converse of ` G ` maps... |
| hashinf 14300 | The value of the ` # ` fun... |
| hashbnd 14301 | If ` A ` has size bounded ... |
| hashfxnn0 14302 | The size function is a fun... |
| hashf 14303 | The size function maps all... |
| hashxnn0 14304 | The value of the hash func... |
| hashresfn 14305 | Restriction of the domain ... |
| dmhashres 14306 | Restriction of the domain ... |
| hashnn0pnf 14307 | The value of the hash func... |
| hashnnn0genn0 14308 | If the size of a set is no... |
| hashnemnf 14309 | The size of a set is never... |
| hashv01gt1 14310 | The size of a set is eithe... |
| hashfz1 14311 | The set ` ( 1 ... N ) ` ha... |
| hashen 14312 | Two finite sets have the s... |
| hasheni 14313 | Equinumerous sets have the... |
| hasheqf1o 14314 | The size of two finite set... |
| fiinfnf1o 14315 | There is no bijection betw... |
| hasheqf1oi 14316 | The size of two sets is eq... |
| hashf1rn 14317 | The size of a finite set w... |
| hasheqf1od 14318 | The size of two sets is eq... |
| fz1eqb 14319 | Two possibly-empty 1-based... |
| hashcard 14320 | The size function of the c... |
| hashcl 14321 | Closure of the ` # ` funct... |
| hashxrcl 14322 | Extended real closure of t... |
| hashclb 14323 | Reverse closure of the ` #... |
| nfile 14324 | The size of any infinite s... |
| hashvnfin 14325 | A set of finite size is a ... |
| hashnfinnn0 14326 | The size of an infinite se... |
| isfinite4 14327 | A finite set is equinumero... |
| hasheq0 14328 | Two ways of saying a set i... |
| hashneq0 14329 | Two ways of saying a set i... |
| hashgt0n0 14330 | If the size of a set is gr... |
| hashnncl 14331 | Positive natural closure o... |
| hash0 14332 | The empty set has size zer... |
| hashelne0d 14333 | A set with an element has ... |
| hashsng 14334 | The size of a singleton. ... |
| hashen1 14335 | A set has size 1 if and on... |
| hash1elsn 14336 | A set of size 1 with a kno... |
| hashrabrsn 14337 | The size of a restricted c... |
| hashrabsn01 14338 | The size of a restricted c... |
| hashrabsn1 14339 | If the size of a restricte... |
| hashfn 14340 | A function is equinumerous... |
| fseq1hash 14341 | The value of the size func... |
| hashgadd 14342 | ` G ` maps ordinal additio... |
| hashgval2 14343 | A short expression for the... |
| hashdom 14344 | Dominance relation for the... |
| hashdomi 14345 | Non-strict order relation ... |
| hashsdom 14346 | Strict dominance relation ... |
| hashun 14347 | The size of the union of d... |
| hashun2 14348 | The size of the union of f... |
| hashun3 14349 | The size of the union of f... |
| hashinfxadd 14350 | The extended real addition... |
| hashunx 14351 | The size of the union of d... |
| hashge0 14352 | The cardinality of a set i... |
| hashgt0 14353 | The cardinality of a nonem... |
| hashge1 14354 | The cardinality of a nonem... |
| 1elfz0hash 14355 | 1 is an element of the fin... |
| hashnn0n0nn 14356 | If a nonnegative integer i... |
| hashunsng 14357 | The size of the union of a... |
| hashunsngx 14358 | The size of the union of a... |
| hashunsnggt 14359 | The size of a set is great... |
| hashprg 14360 | The size of an unordered p... |
| elprchashprn2 14361 | If one element of an unord... |
| hashprb 14362 | The size of an unordered p... |
| hashprdifel 14363 | The elements of an unorder... |
| prhash2ex 14364 | There is (at least) one se... |
| hashle00 14365 | If the size of a set is le... |
| hashgt0elex 14366 | If the size of a set is gr... |
| hashgt0elexb 14367 | The size of a set is great... |
| hashp1i 14368 | Size of a finite ordinal. ... |
| hash1 14369 | Size of a finite ordinal. ... |
| hash2 14370 | Size of a finite ordinal. ... |
| hash3 14371 | Size of a finite ordinal. ... |
| hash4 14372 | Size of a finite ordinal. ... |
| pr0hash2ex 14373 | There is (at least) one se... |
| hashss 14374 | The size of a subset is le... |
| prsshashgt1 14375 | The size of a superset of ... |
| hashin 14376 | The size of the intersecti... |
| hashssdif 14377 | The size of the difference... |
| hashdif 14378 | The size of the difference... |
| hashdifsn 14379 | The size of the difference... |
| hashdifpr 14380 | The size of the difference... |
| hashsn01 14381 | The size of a singleton is... |
| hashsnle1 14382 | The size of a singleton is... |
| hashsnlei 14383 | Get an upper bound on a co... |
| hash1snb 14384 | The size of a set is 1 if ... |
| euhash1 14385 | The size of a set is 1 in ... |
| hash1n0 14386 | If the size of a set is 1 ... |
| hashgt12el 14387 | In a set with more than on... |
| hashgt12el2 14388 | In a set with more than on... |
| hashgt23el 14389 | A set with more than two e... |
| hashunlei 14390 | Get an upper bound on a co... |
| hashsslei 14391 | Get an upper bound on a co... |
| hashfz 14392 | Value of the numeric cardi... |
| fzsdom2 14393 | Condition for finite range... |
| hashfzo 14394 | Cardinality of a half-open... |
| hashfzo0 14395 | Cardinality of a half-open... |
| hashfzp1 14396 | Value of the numeric cardi... |
| hashfz0 14397 | Value of the numeric cardi... |
| hashxplem 14398 | Lemma for ~ hashxp . (Con... |
| hashxp 14399 | The size of the Cartesian ... |
| hashmap 14400 | The size of the set expone... |
| hashpw 14401 | The size of the power set ... |
| hashfun 14402 | A finite set is a function... |
| hashres 14403 | The number of elements of ... |
| hashreshashfun 14404 | The number of elements of ... |
| hashimarn 14405 | The size of the image of a... |
| hashimarni 14406 | If the size of the image o... |
| hashfundm 14407 | The size of a set function... |
| hashf1dmrn 14408 | The size of the domain of ... |
| resunimafz0 14409 | TODO-AV: Revise using ` F... |
| fnfz0hash 14410 | The size of a function on ... |
| ffz0hash 14411 | The size of a function on ... |
| fnfz0hashnn0 14412 | The size of a function on ... |
| ffzo0hash 14413 | The size of a function on ... |
| fnfzo0hash 14414 | The size of a function on ... |
| fnfzo0hashnn0 14415 | The value of the size func... |
| hashbclem 14416 | Lemma for ~ hashbc : induc... |
| hashbc 14417 | The binomial coefficient c... |
| hashfacen 14418 | The number of bijections b... |
| hashfacenOLD 14419 | Obsolete version of ~ hash... |
| hashf1lem1 14420 | Lemma for ~ hashf1 . (Con... |
| hashf1lem1OLD 14421 | Obsolete version of ~ hash... |
| hashf1lem2 14422 | Lemma for ~ hashf1 . (Con... |
| hashf1 14423 | The permutation number ` |... |
| hashfac 14424 | A factorial counts the num... |
| leiso 14425 | Two ways to write a strict... |
| leisorel 14426 | Version of ~ isorel for st... |
| fz1isolem 14427 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14428 | Any finite ordered set has... |
| ishashinf 14429 | Any set that is not finite... |
| seqcoll 14430 | The function ` F ` contain... |
| seqcoll2 14431 | The function ` F ` contain... |
| phphashd 14432 | Corollary of the Pigeonhol... |
| phphashrd 14433 | Corollary of the Pigeonhol... |
| hashprlei 14434 | An unordered pair has at m... |
| hash2pr 14435 | A set of size two is an un... |
| hash2prde 14436 | A set of size two is an un... |
| hash2exprb 14437 | A set of size two is an un... |
| hash2prb 14438 | A set of size two is a pro... |
| prprrab 14439 | The set of proper pairs of... |
| nehash2 14440 | The cardinality of a set w... |
| hash2prd 14441 | A set of size two is an un... |
| hash2pwpr 14442 | If the size of a subset of... |
| hashle2pr 14443 | A nonempty set of size les... |
| hashle2prv 14444 | A nonempty subset of a pow... |
| pr2pwpr 14445 | The set of subsets of a pa... |
| hashge2el2dif 14446 | A set with size at least 2... |
| hashge2el2difr 14447 | A set with at least 2 diff... |
| hashge2el2difb 14448 | A set has size at least 2 ... |
| hashdmpropge2 14449 | The size of the domain of ... |
| hashtplei 14450 | An unordered triple has at... |
| hashtpg 14451 | The size of an unordered t... |
| hashge3el3dif 14452 | A set with size at least 3... |
| elss2prb 14453 | An element of the set of s... |
| hash2sspr 14454 | A subset of size two is an... |
| exprelprel 14455 | If there is an element of ... |
| hash3tr 14456 | A set of size three is an ... |
| hash1to3 14457 | If the size of a set is be... |
| fundmge2nop0 14458 | A function with a domain c... |
| fundmge2nop 14459 | A function with a domain c... |
| fun2dmnop0 14460 | A function with a domain c... |
| fun2dmnop 14461 | A function with a domain c... |
| hashdifsnp1 14462 | If the size of a set is a ... |
| fi1uzind 14463 | Properties of an ordered p... |
| brfi1uzind 14464 | Properties of a binary rel... |
| brfi1ind 14465 | Properties of a binary rel... |
| brfi1indALT 14466 | Alternate proof of ~ brfi1... |
| opfi1uzind 14467 | Properties of an ordered p... |
| opfi1ind 14468 | Properties of an ordered p... |
| iswrd 14471 | Property of being a word o... |
| wrdval 14472 | Value of the set of words ... |
| iswrdi 14473 | A zero-based sequence is a... |
| wrdf 14474 | A word is a zero-based seq... |
| iswrdb 14475 | A word over an alphabet is... |
| wrddm 14476 | The indices of a word (i.e... |
| sswrd 14477 | The set of words respects ... |
| snopiswrd 14478 | A singleton of an ordered ... |
| wrdexg 14479 | The set of words over a se... |
| wrdexb 14480 | The set of words over a se... |
| wrdexi 14481 | The set of words over a se... |
| wrdsymbcl 14482 | A symbol within a word ove... |
| wrdfn 14483 | A word is a function with ... |
| wrdv 14484 | A word over an alphabet is... |
| wrdlndm 14485 | The length of a word is no... |
| iswrdsymb 14486 | An arbitrary word is a wor... |
| wrdfin 14487 | A word is a finite set. (... |
| lencl 14488 | The length of a word is a ... |
| lennncl 14489 | The length of a nonempty w... |
| wrdffz 14490 | A word is a function from ... |
| wrdeq 14491 | Equality theorem for the s... |
| wrdeqi 14492 | Equality theorem for the s... |
| iswrddm0 14493 | A function with empty doma... |
| wrd0 14494 | The empty set is a word (t... |
| 0wrd0 14495 | The empty word is the only... |
| ffz0iswrd 14496 | A sequence with zero-based... |
| wrdsymb 14497 | A word is a word over the ... |
| nfwrd 14498 | Hypothesis builder for ` W... |
| csbwrdg 14499 | Class substitution for the... |
| wrdnval 14500 | Words of a fixed length ar... |
| wrdmap 14501 | Words as a mapping. (Cont... |
| hashwrdn 14502 | If there is only a finite ... |
| wrdnfi 14503 | If there is only a finite ... |
| wrdsymb0 14504 | A symbol at a position "ou... |
| wrdlenge1n0 14505 | A word with length at leas... |
| len0nnbi 14506 | The length of a word is a ... |
| wrdlenge2n0 14507 | A word with length at leas... |
| wrdsymb1 14508 | The first symbol of a none... |
| wrdlen1 14509 | A word of length 1 starts ... |
| fstwrdne 14510 | The first symbol of a none... |
| fstwrdne0 14511 | The first symbol of a none... |
| eqwrd 14512 | Two words are equal iff th... |
| elovmpowrd 14513 | Implications for the value... |
| elovmptnn0wrd 14514 | Implications for the value... |
| wrdred1 14515 | A word truncated by a symb... |
| wrdred1hash 14516 | The length of a word trunc... |
| lsw 14519 | Extract the last symbol of... |
| lsw0 14520 | The last symbol of an empt... |
| lsw0g 14521 | The last symbol of an empt... |
| lsw1 14522 | The last symbol of a word ... |
| lswcl 14523 | Closure of the last symbol... |
| lswlgt0cl 14524 | The last symbol of a nonem... |
| ccatfn 14527 | The concatenation operator... |
| ccatfval 14528 | Value of the concatenation... |
| ccatcl 14529 | The concatenation of two w... |
| ccatlen 14530 | The length of a concatenat... |
| ccat0 14531 | The concatenation of two w... |
| ccatval1 14532 | Value of a symbol in the l... |
| ccatval2 14533 | Value of a symbol in the r... |
| ccatval3 14534 | Value of a symbol in the r... |
| elfzelfzccat 14535 | An element of a finite set... |
| ccatvalfn 14536 | The concatenation of two w... |
| ccatsymb 14537 | The symbol at a given posi... |
| ccatfv0 14538 | The first symbol of a conc... |
| ccatval1lsw 14539 | The last symbol of the lef... |
| ccatval21sw 14540 | The first symbol of the ri... |
| ccatlid 14541 | Concatenation of a word by... |
| ccatrid 14542 | Concatenation of a word by... |
| ccatass 14543 | Associative law for concat... |
| ccatrn 14544 | The range of a concatenate... |
| ccatidid 14545 | Concatenation of the empty... |
| lswccatn0lsw 14546 | The last symbol of a word ... |
| lswccat0lsw 14547 | The last symbol of a word ... |
| ccatalpha 14548 | A concatenation of two arb... |
| ccatrcl1 14549 | Reverse closure of a conca... |
| ids1 14552 | Identity function protecti... |
| s1val 14553 | Value of a singleton word.... |
| s1rn 14554 | The range of a singleton w... |
| s1eq 14555 | Equality theorem for a sin... |
| s1eqd 14556 | Equality theorem for a sin... |
| s1cl 14557 | A singleton word is a word... |
| s1cld 14558 | A singleton word is a word... |
| s1prc 14559 | Value of a singleton word ... |
| s1cli 14560 | A singleton word is a word... |
| s1len 14561 | Length of a singleton word... |
| s1nz 14562 | A singleton word is not th... |
| s1dm 14563 | The domain of a singleton ... |
| s1dmALT 14564 | Alternate version of ~ s1d... |
| s1fv 14565 | Sole symbol of a singleton... |
| lsws1 14566 | The last symbol of a singl... |
| eqs1 14567 | A word of length 1 is a si... |
| wrdl1exs1 14568 | A word of length 1 is a si... |
| wrdl1s1 14569 | A word of length 1 is a si... |
| s111 14570 | The singleton word functio... |
| ccatws1cl 14571 | The concatenation of a wor... |
| ccatws1clv 14572 | The concatenation of a wor... |
| ccat2s1cl 14573 | The concatenation of two s... |
| ccats1alpha 14574 | A concatenation of a word ... |
| ccatws1len 14575 | The length of the concaten... |
| ccatws1lenp1b 14576 | The length of a word is ` ... |
| wrdlenccats1lenm1 14577 | The length of a word is th... |
| ccat2s1len 14578 | The length of the concaten... |
| ccatw2s1cl 14579 | The concatenation of a wor... |
| ccatw2s1len 14580 | The length of the concaten... |
| ccats1val1 14581 | Value of a symbol in the l... |
| ccats1val2 14582 | Value of the symbol concat... |
| ccat1st1st 14583 | The first symbol of a word... |
| ccat2s1p1 14584 | Extract the first of two c... |
| ccat2s1p2 14585 | Extract the second of two ... |
| ccatw2s1ass 14586 | Associative law for a conc... |
| ccatws1n0 14587 | The concatenation of a wor... |
| ccatws1ls 14588 | The last symbol of the con... |
| lswccats1 14589 | The last symbol of a word ... |
| lswccats1fst 14590 | The last symbol of a nonem... |
| ccatw2s1p1 14591 | Extract the symbol of the ... |
| ccatw2s1p2 14592 | Extract the second of two ... |
| ccat2s1fvw 14593 | Extract a symbol of a word... |
| ccat2s1fst 14594 | The first symbol of the co... |
| swrdnznd 14597 | The value of a subword ope... |
| swrdval 14598 | Value of a subword. (Cont... |
| swrd00 14599 | A zero length substring. ... |
| swrdcl 14600 | Closure of the subword ext... |
| swrdval2 14601 | Value of the subword extra... |
| swrdlen 14602 | Length of an extracted sub... |
| swrdfv 14603 | A symbol in an extracted s... |
| swrdfv0 14604 | The first symbol in an ext... |
| swrdf 14605 | A subword of a word is a f... |
| swrdvalfn 14606 | Value of the subword extra... |
| swrdrn 14607 | The range of a subword of ... |
| swrdlend 14608 | The value of the subword e... |
| swrdnd 14609 | The value of the subword e... |
| swrdnd2 14610 | Value of the subword extra... |
| swrdnnn0nd 14611 | The value of a subword ope... |
| swrdnd0 14612 | The value of a subword ope... |
| swrd0 14613 | A subword of an empty set ... |
| swrdrlen 14614 | Length of a right-anchored... |
| swrdlen2 14615 | Length of an extracted sub... |
| swrdfv2 14616 | A symbol in an extracted s... |
| swrdwrdsymb 14617 | A subword is a word over t... |
| swrdsb0eq 14618 | Two subwords with the same... |
| swrdsbslen 14619 | Two subwords with the same... |
| swrdspsleq 14620 | Two words have a common su... |
| swrds1 14621 | Extract a single symbol fr... |
| swrdlsw 14622 | Extract the last single sy... |
| ccatswrd 14623 | Joining two adjacent subwo... |
| swrdccat2 14624 | Recover the right half of ... |
| pfxnndmnd 14627 | The value of a prefix oper... |
| pfxval 14628 | Value of a prefix operatio... |
| pfx00 14629 | The zero length prefix is ... |
| pfx0 14630 | A prefix of an empty set i... |
| pfxval0 14631 | Value of a prefix operatio... |
| pfxcl 14632 | Closure of the prefix extr... |
| pfxmpt 14633 | Value of the prefix extrac... |
| pfxres 14634 | Value of the subword extra... |
| pfxf 14635 | A prefix of a word is a fu... |
| pfxfn 14636 | Value of the prefix extrac... |
| pfxfv 14637 | A symbol in a prefix of a ... |
| pfxlen 14638 | Length of a prefix. (Cont... |
| pfxid 14639 | A word is a prefix of itse... |
| pfxrn 14640 | The range of a prefix of a... |
| pfxn0 14641 | A prefix consisting of at ... |
| pfxnd 14642 | The value of a prefix oper... |
| pfxnd0 14643 | The value of a prefix oper... |
| pfxwrdsymb 14644 | A prefix of a word is a wo... |
| addlenrevpfx 14645 | The sum of the lengths of ... |
| addlenpfx 14646 | The sum of the lengths of ... |
| pfxfv0 14647 | The first symbol of a pref... |
| pfxtrcfv 14648 | A symbol in a word truncat... |
| pfxtrcfv0 14649 | The first symbol in a word... |
| pfxfvlsw 14650 | The last symbol in a nonem... |
| pfxeq 14651 | The prefixes of two words ... |
| pfxtrcfvl 14652 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14653 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14654 | Two (nonempty) words are e... |
| disjwrdpfx 14655 | Sets of words are disjoint... |
| ccatpfx 14656 | Concatenating a prefix wit... |
| pfxccat1 14657 | Recover the left half of a... |
| pfx1 14658 | The prefix of length one o... |
| swrdswrdlem 14659 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14660 | A subword of a subword is ... |
| pfxswrd 14661 | A prefix of a subword is a... |
| swrdpfx 14662 | A subword of a prefix is a... |
| pfxpfx 14663 | A prefix of a prefix is a ... |
| pfxpfxid 14664 | A prefix of a prefix with ... |
| pfxcctswrd 14665 | The concatenation of the p... |
| lenpfxcctswrd 14666 | The length of the concaten... |
| lenrevpfxcctswrd 14667 | The length of the concaten... |
| pfxlswccat 14668 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14669 | The last symbol of a word ... |
| ccats1pfxeqrex 14670 | There exists a symbol such... |
| ccatopth 14671 | An ~ opth -like theorem fo... |
| ccatopth2 14672 | An ~ opth -like theorem fo... |
| ccatlcan 14673 | Concatenation of words is ... |
| ccatrcan 14674 | Concatenation of words is ... |
| wrdeqs1cat 14675 | Decompose a nonempty word ... |
| cats1un 14676 | Express a word with an ext... |
| wrdind 14677 | Perform induction over the... |
| wrd2ind 14678 | Perform induction over the... |
| swrdccatfn 14679 | The subword of a concatena... |
| swrdccatin1 14680 | The subword of a concatena... |
| pfxccatin12lem4 14681 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14682 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14683 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14684 | The subword of a concatena... |
| pfxccatin12lem2c 14685 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14686 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14687 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14688 | The subword of a concatena... |
| pfxccat3 14689 | The subword of a concatena... |
| swrdccat 14690 | The subword of a concatena... |
| pfxccatpfx1 14691 | A prefix of a concatenatio... |
| pfxccatpfx2 14692 | A prefix of a concatenatio... |
| pfxccat3a 14693 | A prefix of a concatenatio... |
| swrdccat3blem 14694 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14695 | A suffix of a concatenatio... |
| pfxccatid 14696 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14697 | A word is a prefix of a wo... |
| swrdccatin1d 14698 | The subword of a concatena... |
| swrdccatin2d 14699 | The subword of a concatena... |
| pfxccatin12d 14700 | The subword of a concatena... |
| reuccatpfxs1lem 14701 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14702 | There is a unique word hav... |
| reuccatpfxs1v 14703 | There is a unique word hav... |
| splval 14706 | Value of the substring rep... |
| splcl 14707 | Closure of the substring r... |
| splid 14708 | Splicing a subword for the... |
| spllen 14709 | The length of a splice. (... |
| splfv1 14710 | Symbols to the left of a s... |
| splfv2a 14711 | Symbols within the replace... |
| splval2 14712 | Value of a splice, assumin... |
| revval 14715 | Value of the word reversin... |
| revcl 14716 | The reverse of a word is a... |
| revlen 14717 | The reverse of a word has ... |
| revfv 14718 | Reverse of a word at a poi... |
| rev0 14719 | The empty word is its own ... |
| revs1 14720 | Singleton words are their ... |
| revccat 14721 | Antiautomorphic property o... |
| revrev 14722 | Reversal is an involution ... |
| reps 14725 | Construct a function mappi... |
| repsundef 14726 | A function mapping a half-... |
| repsconst 14727 | Construct a function mappi... |
| repsf 14728 | The constructed function m... |
| repswsymb 14729 | The symbols of a "repeated... |
| repsw 14730 | A function mapping a half-... |
| repswlen 14731 | The length of a "repeated ... |
| repsw0 14732 | The "repeated symbol word"... |
| repsdf2 14733 | Alternative definition of ... |
| repswsymball 14734 | All the symbols of a "repe... |
| repswsymballbi 14735 | A word is a "repeated symb... |
| repswfsts 14736 | The first symbol of a none... |
| repswlsw 14737 | The last symbol of a nonem... |
| repsw1 14738 | The "repeated symbol word"... |
| repswswrd 14739 | A subword of a "repeated s... |
| repswpfx 14740 | A prefix of a repeated sym... |
| repswccat 14741 | The concatenation of two "... |
| repswrevw 14742 | The reverse of a "repeated... |
| cshfn 14745 | Perform a cyclical shift f... |
| cshword 14746 | Perform a cyclical shift f... |
| cshnz 14747 | A cyclical shift is the em... |
| 0csh0 14748 | Cyclically shifting an emp... |
| cshw0 14749 | A word cyclically shifted ... |
| cshwmodn 14750 | Cyclically shifting a word... |
| cshwsublen 14751 | Cyclically shifting a word... |
| cshwn 14752 | A word cyclically shifted ... |
| cshwcl 14753 | A cyclically shifted word ... |
| cshwlen 14754 | The length of a cyclically... |
| cshwf 14755 | A cyclically shifted word ... |
| cshwfn 14756 | A cyclically shifted word ... |
| cshwrn 14757 | The range of a cyclically ... |
| cshwidxmod 14758 | The symbol at a given inde... |
| cshwidxmodr 14759 | The symbol at a given inde... |
| cshwidx0mod 14760 | The symbol at index 0 of a... |
| cshwidx0 14761 | The symbol at index 0 of a... |
| cshwidxm1 14762 | The symbol at index ((n-N)... |
| cshwidxm 14763 | The symbol at index (n-N) ... |
| cshwidxn 14764 | The symbol at index (n-1) ... |
| cshf1 14765 | Cyclically shifting a word... |
| cshinj 14766 | If a word is injectiv (reg... |
| repswcshw 14767 | A cyclically shifted "repe... |
| 2cshw 14768 | Cyclically shifting a word... |
| 2cshwid 14769 | Cyclically shifting a word... |
| lswcshw 14770 | The last symbol of a word ... |
| 2cshwcom 14771 | Cyclically shifting a word... |
| cshwleneq 14772 | If the results of cyclical... |
| 3cshw 14773 | Cyclically shifting a word... |
| cshweqdif2 14774 | If cyclically shifting two... |
| cshweqdifid 14775 | If cyclically shifting a w... |
| cshweqrep 14776 | If cyclically shifting a w... |
| cshw1 14777 | If cyclically shifting a w... |
| cshw1repsw 14778 | If cyclically shifting a w... |
| cshwsexa 14779 | The class of (different!) ... |
| cshwsexaOLD 14780 | Obsolete version of ~ cshw... |
| 2cshwcshw 14781 | If a word is a cyclically ... |
| scshwfzeqfzo 14782 | For a nonempty word the se... |
| cshwcshid 14783 | A cyclically shifted word ... |
| cshwcsh2id 14784 | A cyclically shifted word ... |
| cshimadifsn 14785 | The image of a cyclically ... |
| cshimadifsn0 14786 | The image of a cyclically ... |
| wrdco 14787 | Mapping a word by a functi... |
| lenco 14788 | Length of a mapped word is... |
| s1co 14789 | Mapping of a singleton wor... |
| revco 14790 | Mapping of words (i.e., a ... |
| ccatco 14791 | Mapping of words commutes ... |
| cshco 14792 | Mapping of words commutes ... |
| swrdco 14793 | Mapping of words commutes ... |
| pfxco 14794 | Mapping of words commutes ... |
| lswco 14795 | Mapping of (nonempty) word... |
| repsco 14796 | Mapping of words commutes ... |
| cats1cld 14811 | Closure of concatenation w... |
| cats1co 14812 | Closure of concatenation w... |
| cats1cli 14813 | Closure of concatenation w... |
| cats1fvn 14814 | The last symbol of a conca... |
| cats1fv 14815 | A symbol other than the la... |
| cats1len 14816 | The length of concatenatio... |
| cats1cat 14817 | Closure of concatenation w... |
| cats2cat 14818 | Closure of concatenation o... |
| s2eqd 14819 | Equality theorem for a dou... |
| s3eqd 14820 | Equality theorem for a len... |
| s4eqd 14821 | Equality theorem for a len... |
| s5eqd 14822 | Equality theorem for a len... |
| s6eqd 14823 | Equality theorem for a len... |
| s7eqd 14824 | Equality theorem for a len... |
| s8eqd 14825 | Equality theorem for a len... |
| s3eq2 14826 | Equality theorem for a len... |
| s2cld 14827 | A doubleton word is a word... |
| s3cld 14828 | A length 3 string is a wor... |
| s4cld 14829 | A length 4 string is a wor... |
| s5cld 14830 | A length 5 string is a wor... |
| s6cld 14831 | A length 6 string is a wor... |
| s7cld 14832 | A length 7 string is a wor... |
| s8cld 14833 | A length 7 string is a wor... |
| s2cl 14834 | A doubleton word is a word... |
| s3cl 14835 | A length 3 string is a wor... |
| s2cli 14836 | A doubleton word is a word... |
| s3cli 14837 | A length 3 string is a wor... |
| s4cli 14838 | A length 4 string is a wor... |
| s5cli 14839 | A length 5 string is a wor... |
| s6cli 14840 | A length 6 string is a wor... |
| s7cli 14841 | A length 7 string is a wor... |
| s8cli 14842 | A length 8 string is a wor... |
| s2fv0 14843 | Extract the first symbol f... |
| s2fv1 14844 | Extract the second symbol ... |
| s2len 14845 | The length of a doubleton ... |
| s2dm 14846 | The domain of a doubleton ... |
| s3fv0 14847 | Extract the first symbol f... |
| s3fv1 14848 | Extract the second symbol ... |
| s3fv2 14849 | Extract the third symbol f... |
| s3len 14850 | The length of a length 3 s... |
| s4fv0 14851 | Extract the first symbol f... |
| s4fv1 14852 | Extract the second symbol ... |
| s4fv2 14853 | Extract the third symbol f... |
| s4fv3 14854 | Extract the fourth symbol ... |
| s4len 14855 | The length of a length 4 s... |
| s5len 14856 | The length of a length 5 s... |
| s6len 14857 | The length of a length 6 s... |
| s7len 14858 | The length of a length 7 s... |
| s8len 14859 | The length of a length 8 s... |
| lsws2 14860 | The last symbol of a doubl... |
| lsws3 14861 | The last symbol of a 3 let... |
| lsws4 14862 | The last symbol of a 4 let... |
| s2prop 14863 | A length 2 word is an unor... |
| s2dmALT 14864 | Alternate version of ~ s2d... |
| s3tpop 14865 | A length 3 word is an unor... |
| s4prop 14866 | A length 4 word is a union... |
| s3fn 14867 | A length 3 word is a funct... |
| funcnvs1 14868 | The converse of a singleto... |
| funcnvs2 14869 | The converse of a length 2... |
| funcnvs3 14870 | The converse of a length 3... |
| funcnvs4 14871 | The converse of a length 4... |
| s2f1o 14872 | A length 2 word with mutua... |
| f1oun2prg 14873 | A union of unordered pairs... |
| s4f1o 14874 | A length 4 word with mutua... |
| s4dom 14875 | The domain of a length 4 w... |
| s2co 14876 | Mapping a doubleton word b... |
| s3co 14877 | Mapping a length 3 string ... |
| s0s1 14878 | Concatenation of fixed len... |
| s1s2 14879 | Concatenation of fixed len... |
| s1s3 14880 | Concatenation of fixed len... |
| s1s4 14881 | Concatenation of fixed len... |
| s1s5 14882 | Concatenation of fixed len... |
| s1s6 14883 | Concatenation of fixed len... |
| s1s7 14884 | Concatenation of fixed len... |
| s2s2 14885 | Concatenation of fixed len... |
| s4s2 14886 | Concatenation of fixed len... |
| s4s3 14887 | Concatenation of fixed len... |
| s4s4 14888 | Concatenation of fixed len... |
| s3s4 14889 | Concatenation of fixed len... |
| s2s5 14890 | Concatenation of fixed len... |
| s5s2 14891 | Concatenation of fixed len... |
| s2eq2s1eq 14892 | Two length 2 words are equ... |
| s2eq2seq 14893 | Two length 2 words are equ... |
| s3eqs2s1eq 14894 | Two length 3 words are equ... |
| s3eq3seq 14895 | Two length 3 words are equ... |
| swrds2 14896 | Extract two adjacent symbo... |
| swrds2m 14897 | Extract two adjacent symbo... |
| wrdlen2i 14898 | Implications of a word of ... |
| wrd2pr2op 14899 | A word of length two repre... |
| wrdlen2 14900 | A word of length two. (Co... |
| wrdlen2s2 14901 | A word of length two as do... |
| wrdl2exs2 14902 | A word of length two is a ... |
| pfx2 14903 | A prefix of length two. (... |
| wrd3tpop 14904 | A word of length three rep... |
| wrdlen3s3 14905 | A word of length three as ... |
| repsw2 14906 | The "repeated symbol word"... |
| repsw3 14907 | The "repeated symbol word"... |
| swrd2lsw 14908 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14909 | Two words of length at lea... |
| ccatw2s1ccatws2 14910 | The concatenation of a wor... |
| ccat2s1fvwALT 14911 | Alternate proof of ~ ccat2... |
| wwlktovf 14912 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14913 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14914 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14915 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14916 | There is a bijection betwe... |
| eqwrds3 14917 | A word is equal with a len... |
| wrdl3s3 14918 | A word of length 3 is a le... |
| s3sndisj 14919 | The singletons consisting ... |
| s3iunsndisj 14920 | The union of singletons co... |
| ofccat 14921 | Letterwise operations on w... |
| ofs1 14922 | Letterwise operations on a... |
| ofs2 14923 | Letterwise operations on a... |
| coss12d 14924 | Subset deduction for compo... |
| trrelssd 14925 | The composition of subclas... |
| xpcogend 14926 | The most interesting case ... |
| xpcoidgend 14927 | If two classes are not dis... |
| cotr2g 14928 | Two ways of saying that th... |
| cotr2 14929 | Two ways of saying a relat... |
| cotr3 14930 | Two ways of saying a relat... |
| coemptyd 14931 | Deduction about compositio... |
| xptrrel 14932 | The cross product is alway... |
| 0trrel 14933 | The empty class is a trans... |
| cleq1lem 14934 | Equality implies bijection... |
| cleq1 14935 | Equality of relations impl... |
| clsslem 14936 | The closure of a subclass ... |
| trcleq1 14941 | Equality of relations impl... |
| trclsslem 14942 | The transitive closure (as... |
| trcleq2lem 14943 | Equality implies bijection... |
| cvbtrcl 14944 | Change of bound variable i... |
| trcleq12lem 14945 | Equality implies bijection... |
| trclexlem 14946 | Existence of relation impl... |
| trclublem 14947 | If a relation exists then ... |
| trclubi 14948 | The Cartesian product of t... |
| trclubgi 14949 | The union with the Cartesi... |
| trclub 14950 | The Cartesian product of t... |
| trclubg 14951 | The union with the Cartesi... |
| trclfv 14952 | The transitive closure of ... |
| brintclab 14953 | Two ways to express a bina... |
| brtrclfv 14954 | Two ways of expressing the... |
| brcnvtrclfv 14955 | Two ways of expressing the... |
| brtrclfvcnv 14956 | Two ways of expressing the... |
| brcnvtrclfvcnv 14957 | Two ways of expressing the... |
| trclfvss 14958 | The transitive closure (as... |
| trclfvub 14959 | The transitive closure of ... |
| trclfvlb 14960 | The transitive closure of ... |
| trclfvcotr 14961 | The transitive closure of ... |
| trclfvlb2 14962 | The transitive closure of ... |
| trclfvlb3 14963 | The transitive closure of ... |
| cotrtrclfv 14964 | The transitive closure of ... |
| trclidm 14965 | The transitive closure of ... |
| trclun 14966 | Transitive closure of a un... |
| trclfvg 14967 | The value of the transitiv... |
| trclfvcotrg 14968 | The value of the transitiv... |
| reltrclfv 14969 | The transitive closure of ... |
| dmtrclfv 14970 | The domain of the transiti... |
| reldmrelexp 14973 | The domain of the repeated... |
| relexp0g 14974 | A relation composed zero t... |
| relexp0 14975 | A relation composed zero t... |
| relexp0d 14976 | A relation composed zero t... |
| relexpsucnnr 14977 | A reduction for relation e... |
| relexp1g 14978 | A relation composed once i... |
| dfid5 14979 | Identity relation is equal... |
| dfid6 14980 | Identity relation expresse... |
| relexp1d 14981 | A relation composed once i... |
| relexpsucnnl 14982 | A reduction for relation e... |
| relexpsucl 14983 | A reduction for relation e... |
| relexpsucr 14984 | A reduction for relation e... |
| relexpsucrd 14985 | A reduction for relation e... |
| relexpsucld 14986 | A reduction for relation e... |
| relexpcnv 14987 | Commutation of converse an... |
| relexpcnvd 14988 | Commutation of converse an... |
| relexp0rel 14989 | The exponentiation of a cl... |
| relexprelg 14990 | The exponentiation of a cl... |
| relexprel 14991 | The exponentiation of a re... |
| relexpreld 14992 | The exponentiation of a re... |
| relexpnndm 14993 | The domain of an exponenti... |
| relexpdmg 14994 | The domain of an exponenti... |
| relexpdm 14995 | The domain of an exponenti... |
| relexpdmd 14996 | The domain of an exponenti... |
| relexpnnrn 14997 | The range of an exponentia... |
| relexprng 14998 | The range of an exponentia... |
| relexprn 14999 | The range of an exponentia... |
| relexprnd 15000 | The range of an exponentia... |
| relexpfld 15001 | The field of an exponentia... |
| relexpfldd 15002 | The field of an exponentia... |
| relexpaddnn 15003 | Relation composition becom... |
| relexpuzrel 15004 | The exponentiation of a cl... |
| relexpaddg 15005 | Relation composition becom... |
| relexpaddd 15006 | Relation composition becom... |
| rtrclreclem1 15009 | The reflexive, transitive ... |
| dfrtrclrec2 15010 | If two elements are connec... |
| rtrclreclem2 15011 | The reflexive, transitive ... |
| rtrclreclem3 15012 | The reflexive, transitive ... |
| rtrclreclem4 15013 | The reflexive, transitive ... |
| dfrtrcl2 15014 | The two definitions ` t* `... |
| relexpindlem 15015 | Principle of transitive in... |
| relexpind 15016 | Principle of transitive in... |
| rtrclind 15017 | Principle of transitive in... |
| shftlem 15020 | Two ways to write a shifte... |
| shftuz 15021 | A shift of the upper integ... |
| shftfval 15022 | The value of the sequence ... |
| shftdm 15023 | Domain of a relation shift... |
| shftfib 15024 | Value of a fiber of the re... |
| shftfn 15025 | Functionality and domain o... |
| shftval 15026 | Value of a sequence shifte... |
| shftval2 15027 | Value of a sequence shifte... |
| shftval3 15028 | Value of a sequence shifte... |
| shftval4 15029 | Value of a sequence shifte... |
| shftval5 15030 | Value of a shifted sequenc... |
| shftf 15031 | Functionality of a shifted... |
| 2shfti 15032 | Composite shift operations... |
| shftidt2 15033 | Identity law for the shift... |
| shftidt 15034 | Identity law for the shift... |
| shftcan1 15035 | Cancellation law for the s... |
| shftcan2 15036 | Cancellation law for the s... |
| seqshft 15037 | Shifting the index set of ... |
| sgnval 15040 | Value of the signum functi... |
| sgn0 15041 | The signum of 0 is 0. (Co... |
| sgnp 15042 | The signum of a positive e... |
| sgnrrp 15043 | The signum of a positive r... |
| sgn1 15044 | The signum of 1 is 1. (Co... |
| sgnpnf 15045 | The signum of ` +oo ` is 1... |
| sgnn 15046 | The signum of a negative e... |
| sgnmnf 15047 | The signum of ` -oo ` is -... |
| cjval 15054 | The value of the conjugate... |
| cjth 15055 | The defining property of t... |
| cjf 15056 | Domain and codomain of the... |
| cjcl 15057 | The conjugate of a complex... |
| reval 15058 | The value of the real part... |
| imval 15059 | The value of the imaginary... |
| imre 15060 | The imaginary part of a co... |
| reim 15061 | The real part of a complex... |
| recl 15062 | The real part of a complex... |
| imcl 15063 | The imaginary part of a co... |
| ref 15064 | Domain and codomain of the... |
| imf 15065 | Domain and codomain of the... |
| crre 15066 | The real part of a complex... |
| crim 15067 | The real part of a complex... |
| replim 15068 | Reconstruct a complex numb... |
| remim 15069 | Value of the conjugate of ... |
| reim0 15070 | The imaginary part of a re... |
| reim0b 15071 | A number is real iff its i... |
| rereb 15072 | A number is real iff it eq... |
| mulre 15073 | A product with a nonzero r... |
| rere 15074 | A real number equals its r... |
| cjreb 15075 | A number is real iff it eq... |
| recj 15076 | Real part of a complex con... |
| reneg 15077 | Real part of negative. (C... |
| readd 15078 | Real part distributes over... |
| resub 15079 | Real part distributes over... |
| remullem 15080 | Lemma for ~ remul , ~ immu... |
| remul 15081 | Real part of a product. (... |
| remul2 15082 | Real part of a product. (... |
| rediv 15083 | Real part of a division. ... |
| imcj 15084 | Imaginary part of a comple... |
| imneg 15085 | The imaginary part of a ne... |
| imadd 15086 | Imaginary part distributes... |
| imsub 15087 | Imaginary part distributes... |
| immul 15088 | Imaginary part of a produc... |
| immul2 15089 | Imaginary part of a produc... |
| imdiv 15090 | Imaginary part of a divisi... |
| cjre 15091 | A real number equals its c... |
| cjcj 15092 | The conjugate of the conju... |
| cjadd 15093 | Complex conjugate distribu... |
| cjmul 15094 | Complex conjugate distribu... |
| ipcnval 15095 | Standard inner product on ... |
| cjmulrcl 15096 | A complex number times its... |
| cjmulval 15097 | A complex number times its... |
| cjmulge0 15098 | A complex number times its... |
| cjneg 15099 | Complex conjugate of negat... |
| addcj 15100 | A number plus its conjugat... |
| cjsub 15101 | Complex conjugate distribu... |
| cjexp 15102 | Complex conjugate of posit... |
| imval2 15103 | The imaginary part of a nu... |
| re0 15104 | The real part of zero. (C... |
| im0 15105 | The imaginary part of zero... |
| re1 15106 | The real part of one. (Co... |
| im1 15107 | The imaginary part of one.... |
| rei 15108 | The real part of ` _i ` . ... |
| imi 15109 | The imaginary part of ` _i... |
| cj0 15110 | The conjugate of zero. (C... |
| cji 15111 | The complex conjugate of t... |
| cjreim 15112 | The conjugate of a represe... |
| cjreim2 15113 | The conjugate of the repre... |
| cj11 15114 | Complex conjugate is a one... |
| cjne0 15115 | A number is nonzero iff it... |
| cjdiv 15116 | Complex conjugate distribu... |
| cnrecnv 15117 | The inverse to the canonic... |
| sqeqd 15118 | A deduction for showing tw... |
| recli 15119 | The real part of a complex... |
| imcli 15120 | The imaginary part of a co... |
| cjcli 15121 | Closure law for complex co... |
| replimi 15122 | Construct a complex number... |
| cjcji 15123 | The conjugate of the conju... |
| reim0bi 15124 | A number is real iff its i... |
| rerebi 15125 | A real number equals its r... |
| cjrebi 15126 | A number is real iff it eq... |
| recji 15127 | Real part of a complex con... |
| imcji 15128 | Imaginary part of a comple... |
| cjmulrcli 15129 | A complex number times its... |
| cjmulvali 15130 | A complex number times its... |
| cjmulge0i 15131 | A complex number times its... |
| renegi 15132 | Real part of negative. (C... |
| imnegi 15133 | Imaginary part of negative... |
| cjnegi 15134 | Complex conjugate of negat... |
| addcji 15135 | A number plus its conjugat... |
| readdi 15136 | Real part distributes over... |
| imaddi 15137 | Imaginary part distributes... |
| remuli 15138 | Real part of a product. (... |
| immuli 15139 | Imaginary part of a produc... |
| cjaddi 15140 | Complex conjugate distribu... |
| cjmuli 15141 | Complex conjugate distribu... |
| ipcni 15142 | Standard inner product on ... |
| cjdivi 15143 | Complex conjugate distribu... |
| crrei 15144 | The real part of a complex... |
| crimi 15145 | The imaginary part of a co... |
| recld 15146 | The real part of a complex... |
| imcld 15147 | The imaginary part of a co... |
| cjcld 15148 | Closure law for complex co... |
| replimd 15149 | Construct a complex number... |
| remimd 15150 | Value of the conjugate of ... |
| cjcjd 15151 | The conjugate of the conju... |
| reim0bd 15152 | A number is real iff its i... |
| rerebd 15153 | A real number equals its r... |
| cjrebd 15154 | A number is real iff it eq... |
| cjne0d 15155 | A number is nonzero iff it... |
| recjd 15156 | Real part of a complex con... |
| imcjd 15157 | Imaginary part of a comple... |
| cjmulrcld 15158 | A complex number times its... |
| cjmulvald 15159 | A complex number times its... |
| cjmulge0d 15160 | A complex number times its... |
| renegd 15161 | Real part of negative. (C... |
| imnegd 15162 | Imaginary part of negative... |
| cjnegd 15163 | Complex conjugate of negat... |
| addcjd 15164 | A number plus its conjugat... |
| cjexpd 15165 | Complex conjugate of posit... |
| readdd 15166 | Real part distributes over... |
| imaddd 15167 | Imaginary part distributes... |
| resubd 15168 | Real part distributes over... |
| imsubd 15169 | Imaginary part distributes... |
| remuld 15170 | Real part of a product. (... |
| immuld 15171 | Imaginary part of a produc... |
| cjaddd 15172 | Complex conjugate distribu... |
| cjmuld 15173 | Complex conjugate distribu... |
| ipcnd 15174 | Standard inner product on ... |
| cjdivd 15175 | Complex conjugate distribu... |
| rered 15176 | A real number equals its r... |
| reim0d 15177 | The imaginary part of a re... |
| cjred 15178 | A real number equals its c... |
| remul2d 15179 | Real part of a product. (... |
| immul2d 15180 | Imaginary part of a produc... |
| redivd 15181 | Real part of a division. ... |
| imdivd 15182 | Imaginary part of a divisi... |
| crred 15183 | The real part of a complex... |
| crimd 15184 | The imaginary part of a co... |
| sqrtval 15189 | Value of square root funct... |
| absval 15190 | The absolute value (modulu... |
| rennim 15191 | A real number does not lie... |
| cnpart 15192 | The specification of restr... |
| sqrt0 15193 | The square root of zero is... |
| 01sqrexlem1 15194 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15195 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15196 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15197 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15198 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15199 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15200 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15201 | Existence of a square root... |
| resqrex 15202 | Existence of a square root... |
| sqrmo 15203 | Uniqueness for the square ... |
| resqreu 15204 | Existence and uniqueness f... |
| resqrtcl 15205 | Closure of the square root... |
| resqrtthlem 15206 | Lemma for ~ resqrtth . (C... |
| resqrtth 15207 | Square root theorem over t... |
| remsqsqrt 15208 | Square of square root. (C... |
| sqrtge0 15209 | The square root function i... |
| sqrtgt0 15210 | The square root function i... |
| sqrtmul 15211 | Square root distributes ov... |
| sqrtle 15212 | Square root is monotonic. ... |
| sqrtlt 15213 | Square root is strictly mo... |
| sqrt11 15214 | The square root function i... |
| sqrt00 15215 | A square root is zero iff ... |
| rpsqrtcl 15216 | The square root of a posit... |
| sqrtdiv 15217 | Square root distributes ov... |
| sqrtneglem 15218 | The square root of a negat... |
| sqrtneg 15219 | The square root of a negat... |
| sqrtsq2 15220 | Relationship between squar... |
| sqrtsq 15221 | Square root of square. (C... |
| sqrtmsq 15222 | Square root of square. (C... |
| sqrt1 15223 | The square root of 1 is 1.... |
| sqrt4 15224 | The square root of 4 is 2.... |
| sqrt9 15225 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15226 | The square root of 2 is bo... |
| sqrtm1 15227 | The imaginary unit is the ... |
| nn0sqeq1 15228 | A natural number with squa... |
| absneg 15229 | Absolute value of the oppo... |
| abscl 15230 | Real closure of absolute v... |
| abscj 15231 | The absolute value of a nu... |
| absvalsq 15232 | Square of value of absolut... |
| absvalsq2 15233 | Square of value of absolut... |
| sqabsadd 15234 | Square of absolute value o... |
| sqabssub 15235 | Square of absolute value o... |
| absval2 15236 | Value of absolute value fu... |
| abs0 15237 | The absolute value of 0. ... |
| absi 15238 | The absolute value of the ... |
| absge0 15239 | Absolute value is nonnegat... |
| absrpcl 15240 | The absolute value of a no... |
| abs00 15241 | The absolute value of a nu... |
| abs00ad 15242 | A complex number is zero i... |
| abs00bd 15243 | If a complex number is zer... |
| absreimsq 15244 | Square of the absolute val... |
| absreim 15245 | Absolute value of a number... |
| absmul 15246 | Absolute value distributes... |
| absdiv 15247 | Absolute value distributes... |
| absid 15248 | A nonnegative number is it... |
| abs1 15249 | The absolute value of one ... |
| absnid 15250 | A negative number is the n... |
| leabs 15251 | A real number is less than... |
| absor 15252 | The absolute value of a re... |
| absre 15253 | Absolute value of a real n... |
| absresq 15254 | Square of the absolute val... |
| absmod0 15255 | ` A ` is divisible by ` B ... |
| absexp 15256 | Absolute value of positive... |
| absexpz 15257 | Absolute value of integer ... |
| abssq 15258 | Square can be moved in and... |
| sqabs 15259 | The squares of two reals a... |
| absrele 15260 | The absolute value of a co... |
| absimle 15261 | The absolute value of a co... |
| max0add 15262 | The sum of the positive an... |
| absz 15263 | A real number is an intege... |
| nn0abscl 15264 | The absolute value of an i... |
| zabscl 15265 | The absolute value of an i... |
| abslt 15266 | Absolute value and 'less t... |
| absle 15267 | Absolute value and 'less t... |
| abssubne0 15268 | If the absolute value of a... |
| absdiflt 15269 | The absolute value of a di... |
| absdifle 15270 | The absolute value of a di... |
| elicc4abs 15271 | Membership in a symmetric ... |
| lenegsq 15272 | Comparison to a nonnegativ... |
| releabs 15273 | The real part of a number ... |
| recval 15274 | Reciprocal expressed with ... |
| absidm 15275 | The absolute value functio... |
| absgt0 15276 | The absolute value of a no... |
| nnabscl 15277 | The absolute value of a no... |
| abssub 15278 | Swapping order of subtract... |
| abssubge0 15279 | Absolute value of a nonneg... |
| abssuble0 15280 | Absolute value of a nonpos... |
| absmax 15281 | The maximum of two numbers... |
| abstri 15282 | Triangle inequality for ab... |
| abs3dif 15283 | Absolute value of differen... |
| abs2dif 15284 | Difference of absolute val... |
| abs2dif2 15285 | Difference of absolute val... |
| abs2difabs 15286 | Absolute value of differen... |
| abs1m 15287 | For any complex number, th... |
| recan 15288 | Cancellation law involving... |
| absf 15289 | Mapping domain and codomai... |
| abs3lem 15290 | Lemma involving absolute v... |
| abslem2 15291 | Lemma involving absolute v... |
| rddif 15292 | The difference between a r... |
| absrdbnd 15293 | Bound on the absolute valu... |
| fzomaxdiflem 15294 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15295 | A bound on the separation ... |
| uzin2 15296 | The upper integers are clo... |
| rexanuz 15297 | Combine two different uppe... |
| rexanre 15298 | Combine two different uppe... |
| rexfiuz 15299 | Combine finitely many diff... |
| rexuz3 15300 | Restrict the base of the u... |
| rexanuz2 15301 | Combine two different uppe... |
| r19.29uz 15302 | A version of ~ 19.29 for u... |
| r19.2uz 15303 | A version of ~ r19.2z for ... |
| rexuzre 15304 | Convert an upper real quan... |
| rexico 15305 | Restrict the base of an up... |
| cau3lem 15306 | Lemma for ~ cau3 . (Contr... |
| cau3 15307 | Convert between three-quan... |
| cau4 15308 | Change the base of a Cauch... |
| caubnd2 15309 | A Cauchy sequence of compl... |
| caubnd 15310 | A Cauchy sequence of compl... |
| sqreulem 15311 | Lemma for ~ sqreu : write ... |
| sqreu 15312 | Existence and uniqueness f... |
| sqrtcl 15313 | Closure of the square root... |
| sqrtthlem 15314 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15315 | Mapping domain and codomai... |
| sqrtth 15316 | Square root theorem over t... |
| sqrtrege0 15317 | The square root function m... |
| eqsqrtor 15318 | Solve an equation containi... |
| eqsqrtd 15319 | A deduction for showing th... |
| eqsqrt2d 15320 | A deduction for showing th... |
| amgm2 15321 | Arithmetic-geometric mean ... |
| sqrtthi 15322 | Square root theorem. Theo... |
| sqrtcli 15323 | The square root of a nonne... |
| sqrtgt0i 15324 | The square root of a posit... |
| sqrtmsqi 15325 | Square root of square. (C... |
| sqrtsqi 15326 | Square root of square. (C... |
| sqsqrti 15327 | Square of square root. (C... |
| sqrtge0i 15328 | The square root of a nonne... |
| absidi 15329 | A nonnegative number is it... |
| absnidi 15330 | A negative number is the n... |
| leabsi 15331 | A real number is less than... |
| absori 15332 | The absolute value of a re... |
| absrei 15333 | Absolute value of a real n... |
| sqrtpclii 15334 | The square root of a posit... |
| sqrtgt0ii 15335 | The square root of a posit... |
| sqrt11i 15336 | The square root function i... |
| sqrtmuli 15337 | Square root distributes ov... |
| sqrtmulii 15338 | Square root distributes ov... |
| sqrtmsq2i 15339 | Relationship between squar... |
| sqrtlei 15340 | Square root is monotonic. ... |
| sqrtlti 15341 | Square root is strictly mo... |
| abslti 15342 | Absolute value and 'less t... |
| abslei 15343 | Absolute value and 'less t... |
| cnsqrt00 15344 | A square root of a complex... |
| absvalsqi 15345 | Square of value of absolut... |
| absvalsq2i 15346 | Square of value of absolut... |
| abscli 15347 | Real closure of absolute v... |
| absge0i 15348 | Absolute value is nonnegat... |
| absval2i 15349 | Value of absolute value fu... |
| abs00i 15350 | The absolute value of a nu... |
| absgt0i 15351 | The absolute value of a no... |
| absnegi 15352 | Absolute value of negative... |
| abscji 15353 | The absolute value of a nu... |
| releabsi 15354 | The real part of a number ... |
| abssubi 15355 | Swapping order of subtract... |
| absmuli 15356 | Absolute value distributes... |
| sqabsaddi 15357 | Square of absolute value o... |
| sqabssubi 15358 | Square of absolute value o... |
| absdivzi 15359 | Absolute value distributes... |
| abstrii 15360 | Triangle inequality for ab... |
| abs3difi 15361 | Absolute value of differen... |
| abs3lemi 15362 | Lemma involving absolute v... |
| rpsqrtcld 15363 | The square root of a posit... |
| sqrtgt0d 15364 | The square root of a posit... |
| absnidd 15365 | A negative number is the n... |
| leabsd 15366 | A real number is less than... |
| absord 15367 | The absolute value of a re... |
| absred 15368 | Absolute value of a real n... |
| resqrtcld 15369 | The square root of a nonne... |
| sqrtmsqd 15370 | Square root of square. (C... |
| sqrtsqd 15371 | Square root of square. (C... |
| sqrtge0d 15372 | The square root of a nonne... |
| sqrtnegd 15373 | The square root of a negat... |
| absidd 15374 | A nonnegative number is it... |
| sqrtdivd 15375 | Square root distributes ov... |
| sqrtmuld 15376 | Square root distributes ov... |
| sqrtsq2d 15377 | Relationship between squar... |
| sqrtled 15378 | Square root is monotonic. ... |
| sqrtltd 15379 | Square root is strictly mo... |
| sqr11d 15380 | The square root function i... |
| absltd 15381 | Absolute value and 'less t... |
| absled 15382 | Absolute value and 'less t... |
| abssubge0d 15383 | Absolute value of a nonneg... |
| abssuble0d 15384 | Absolute value of a nonpos... |
| absdifltd 15385 | The absolute value of a di... |
| absdifled 15386 | The absolute value of a di... |
| icodiamlt 15387 | Two elements in a half-ope... |
| abscld 15388 | Real closure of absolute v... |
| sqrtcld 15389 | Closure of the square root... |
| sqrtrege0d 15390 | The real part of the squar... |
| sqsqrtd 15391 | Square root theorem. Theo... |
| msqsqrtd 15392 | Square root theorem. Theo... |
| sqr00d 15393 | A square root is zero iff ... |
| absvalsqd 15394 | Square of value of absolut... |
| absvalsq2d 15395 | Square of value of absolut... |
| absge0d 15396 | Absolute value is nonnegat... |
| absval2d 15397 | Value of absolute value fu... |
| abs00d 15398 | The absolute value of a nu... |
| absne0d 15399 | The absolute value of a nu... |
| absrpcld 15400 | The absolute value of a no... |
| absnegd 15401 | Absolute value of negative... |
| abscjd 15402 | The absolute value of a nu... |
| releabsd 15403 | The real part of a number ... |
| absexpd 15404 | Absolute value of positive... |
| abssubd 15405 | Swapping order of subtract... |
| absmuld 15406 | Absolute value distributes... |
| absdivd 15407 | Absolute value distributes... |
| abstrid 15408 | Triangle inequality for ab... |
| abs2difd 15409 | Difference of absolute val... |
| abs2dif2d 15410 | Difference of absolute val... |
| abs2difabsd 15411 | Absolute value of differen... |
| abs3difd 15412 | Absolute value of differen... |
| abs3lemd 15413 | Lemma involving absolute v... |
| reusq0 15414 | A complex number is the sq... |
| bhmafibid1cn 15415 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15416 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15417 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15418 | The Brahmagupta-Fibonacci ... |
| limsupgord 15421 | Ordering property of the s... |
| limsupcl 15422 | Closure of the superior li... |
| limsupval 15423 | The superior limit of an i... |
| limsupgf 15424 | Closure of the superior li... |
| limsupgval 15425 | Value of the superior limi... |
| limsupgle 15426 | The defining property of t... |
| limsuple 15427 | The defining property of t... |
| limsuplt 15428 | The defining property of t... |
| limsupval2 15429 | The superior limit, relati... |
| limsupgre 15430 | If a sequence of real numb... |
| limsupbnd1 15431 | If a sequence is eventuall... |
| limsupbnd2 15432 | If a sequence is eventuall... |
| climrel 15441 | The limit relation is a re... |
| rlimrel 15442 | The limit relation is a re... |
| clim 15443 | Express the predicate: Th... |
| rlim 15444 | Express the predicate: Th... |
| rlim2 15445 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15446 | Use strictly less-than in ... |
| rlim3 15447 | Restrict the range of the ... |
| climcl 15448 | Closure of the limit of a ... |
| rlimpm 15449 | Closure of a function with... |
| rlimf 15450 | Closure of a function with... |
| rlimss 15451 | Domain closure of a functi... |
| rlimcl 15452 | Closure of the limit of a ... |
| clim2 15453 | Express the predicate: Th... |
| clim2c 15454 | Express the predicate ` F ... |
| clim0 15455 | Express the predicate ` F ... |
| clim0c 15456 | Express the predicate ` F ... |
| rlim0 15457 | Express the predicate ` B ... |
| rlim0lt 15458 | Use strictly less-than in ... |
| climi 15459 | Convergence of a sequence ... |
| climi2 15460 | Convergence of a sequence ... |
| climi0 15461 | Convergence of a sequence ... |
| rlimi 15462 | Convergence at infinity of... |
| rlimi2 15463 | Convergence at infinity of... |
| ello1 15464 | Elementhood in the set of ... |
| ello12 15465 | Elementhood in the set of ... |
| ello12r 15466 | Sufficient condition for e... |
| lo1f 15467 | An eventually upper bounde... |
| lo1dm 15468 | An eventually upper bounde... |
| lo1bdd 15469 | The defining property of a... |
| ello1mpt 15470 | Elementhood in the set of ... |
| ello1mpt2 15471 | Elementhood in the set of ... |
| ello1d 15472 | Sufficient condition for e... |
| lo1bdd2 15473 | If an eventually bounded f... |
| lo1bddrp 15474 | Refine ~ o1bdd2 to give a ... |
| elo1 15475 | Elementhood in the set of ... |
| elo12 15476 | Elementhood in the set of ... |
| elo12r 15477 | Sufficient condition for e... |
| o1f 15478 | An eventually bounded func... |
| o1dm 15479 | An eventually bounded func... |
| o1bdd 15480 | The defining property of a... |
| lo1o1 15481 | A function is eventually b... |
| lo1o12 15482 | A function is eventually b... |
| elo1mpt 15483 | Elementhood in the set of ... |
| elo1mpt2 15484 | Elementhood in the set of ... |
| elo1d 15485 | Sufficient condition for e... |
| o1lo1 15486 | A real function is eventua... |
| o1lo12 15487 | A lower bounded real funct... |
| o1lo1d 15488 | A real eventually bounded ... |
| icco1 15489 | Derive eventual boundednes... |
| o1bdd2 15490 | If an eventually bounded f... |
| o1bddrp 15491 | Refine ~ o1bdd2 to give a ... |
| climconst 15492 | An (eventually) constant s... |
| rlimconst 15493 | A constant sequence conver... |
| rlimclim1 15494 | Forward direction of ~ rli... |
| rlimclim 15495 | A sequence on an upper int... |
| climrlim2 15496 | Produce a real limit from ... |
| climconst2 15497 | A constant sequence conver... |
| climz 15498 | The zero sequence converge... |
| rlimuni 15499 | A real function whose doma... |
| rlimdm 15500 | Two ways to express that a... |
| climuni 15501 | An infinite sequence of co... |
| fclim 15502 | The limit relation is func... |
| climdm 15503 | Two ways to express that a... |
| climeu 15504 | An infinite sequence of co... |
| climreu 15505 | An infinite sequence of co... |
| climmo 15506 | An infinite sequence of co... |
| rlimres 15507 | The restriction of a funct... |
| lo1res 15508 | The restriction of an even... |
| o1res 15509 | The restriction of an even... |
| rlimres2 15510 | The restriction of a funct... |
| lo1res2 15511 | The restriction of a funct... |
| o1res2 15512 | The restriction of a funct... |
| lo1resb 15513 | The restriction of a funct... |
| rlimresb 15514 | The restriction of a funct... |
| o1resb 15515 | The restriction of a funct... |
| climeq 15516 | Two functions that are eve... |
| lo1eq 15517 | Two functions that are eve... |
| rlimeq 15518 | Two functions that are eve... |
| o1eq 15519 | Two functions that are eve... |
| climmpt 15520 | Exhibit a function ` G ` w... |
| 2clim 15521 | If two sequences converge ... |
| climmpt2 15522 | Relate an integer limit on... |
| climshftlem 15523 | A shifted function converg... |
| climres 15524 | A function restricted to u... |
| climshft 15525 | A shifted function converg... |
| serclim0 15526 | The zero series converges ... |
| rlimcld2 15527 | If ` D ` is a closed set i... |
| rlimrege0 15528 | The limit of a sequence of... |
| rlimrecl 15529 | The limit of a real sequen... |
| rlimge0 15530 | The limit of a sequence of... |
| climshft2 15531 | A shifted function converg... |
| climrecl 15532 | The limit of a convergent ... |
| climge0 15533 | A nonnegative sequence con... |
| climabs0 15534 | Convergence to zero of the... |
| o1co 15535 | Sufficient condition for t... |
| o1compt 15536 | Sufficient condition for t... |
| rlimcn1 15537 | Image of a limit under a c... |
| rlimcn1b 15538 | Image of a limit under a c... |
| rlimcn3 15539 | Image of a limit under a c... |
| rlimcn2 15540 | Image of a limit under a c... |
| climcn1 15541 | Image of a limit under a c... |
| climcn2 15542 | Image of a limit under a c... |
| addcn2 15543 | Complex number addition is... |
| subcn2 15544 | Complex number subtraction... |
| mulcn2 15545 | Complex number multiplicat... |
| reccn2 15546 | The reciprocal function is... |
| cn1lem 15547 | A sufficient condition for... |
| abscn2 15548 | The absolute value functio... |
| cjcn2 15549 | The complex conjugate func... |
| recn2 15550 | The real part function is ... |
| imcn2 15551 | The imaginary part functio... |
| climcn1lem 15552 | The limit of a continuous ... |
| climabs 15553 | Limit of the absolute valu... |
| climcj 15554 | Limit of the complex conju... |
| climre 15555 | Limit of the real part of ... |
| climim 15556 | Limit of the imaginary par... |
| rlimmptrcl 15557 | Reverse closure for a real... |
| rlimabs 15558 | Limit of the absolute valu... |
| rlimcj 15559 | Limit of the complex conju... |
| rlimre 15560 | Limit of the real part of ... |
| rlimim 15561 | Limit of the imaginary par... |
| o1of2 15562 | Show that a binary operati... |
| o1add 15563 | The sum of two eventually ... |
| o1mul 15564 | The product of two eventua... |
| o1sub 15565 | The difference of two even... |
| rlimo1 15566 | Any function with a finite... |
| rlimdmo1 15567 | A convergent function is e... |
| o1rlimmul 15568 | The product of an eventual... |
| o1const 15569 | A constant function is eve... |
| lo1const 15570 | A constant function is eve... |
| lo1mptrcl 15571 | Reverse closure for an eve... |
| o1mptrcl 15572 | Reverse closure for an eve... |
| o1add2 15573 | The sum of two eventually ... |
| o1mul2 15574 | The product of two eventua... |
| o1sub2 15575 | The product of two eventua... |
| lo1add 15576 | The sum of two eventually ... |
| lo1mul 15577 | The product of an eventual... |
| lo1mul2 15578 | The product of an eventual... |
| o1dif 15579 | If the difference of two f... |
| lo1sub 15580 | The difference of an event... |
| climadd 15581 | Limit of the sum of two co... |
| climmul 15582 | Limit of the product of tw... |
| climsub 15583 | Limit of the difference of... |
| climaddc1 15584 | Limit of a constant ` C ` ... |
| climaddc2 15585 | Limit of a constant ` C ` ... |
| climmulc2 15586 | Limit of a sequence multip... |
| climsubc1 15587 | Limit of a constant ` C ` ... |
| climsubc2 15588 | Limit of a constant ` C ` ... |
| climle 15589 | Comparison of the limits o... |
| climsqz 15590 | Convergence of a sequence ... |
| climsqz2 15591 | Convergence of a sequence ... |
| rlimadd 15592 | Limit of the sum of two co... |
| rlimaddOLD 15593 | Obsolete version of ~ rlim... |
| rlimsub 15594 | Limit of the difference of... |
| rlimmul 15595 | Limit of the product of tw... |
| rlimmulOLD 15596 | Obsolete version of ~ rlim... |
| rlimdiv 15597 | Limit of the quotient of t... |
| rlimneg 15598 | Limit of the negative of a... |
| rlimle 15599 | Comparison of the limits o... |
| rlimsqzlem 15600 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15601 | Convergence of a sequence ... |
| rlimsqz2 15602 | Convergence of a sequence ... |
| lo1le 15603 | Transfer eventual upper bo... |
| o1le 15604 | Transfer eventual boundedn... |
| rlimno1 15605 | A function whose inverse c... |
| clim2ser 15606 | The limit of an infinite s... |
| clim2ser2 15607 | The limit of an infinite s... |
| iserex 15608 | An infinite series converg... |
| isermulc2 15609 | Multiplication of an infin... |
| climlec2 15610 | Comparison of a constant t... |
| iserle 15611 | Comparison of the limits o... |
| iserge0 15612 | The limit of an infinite s... |
| climub 15613 | The limit of a monotonic s... |
| climserle 15614 | The partial sums of a conv... |
| isershft 15615 | Index shift of the limit o... |
| isercolllem1 15616 | Lemma for ~ isercoll . (C... |
| isercolllem2 15617 | Lemma for ~ isercoll . (C... |
| isercolllem3 15618 | Lemma for ~ isercoll . (C... |
| isercoll 15619 | Rearrange an infinite seri... |
| isercoll2 15620 | Generalize ~ isercoll so t... |
| climsup 15621 | A bounded monotonic sequen... |
| climcau 15622 | A converging sequence of c... |
| climbdd 15623 | A converging sequence of c... |
| caucvgrlem 15624 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15625 | A Cauchy sequence of real ... |
| caucvgrlem2 15626 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15627 | A Cauchy sequence of compl... |
| caurcvg 15628 | A Cauchy sequence of real ... |
| caurcvg2 15629 | A Cauchy sequence of real ... |
| caucvg 15630 | A Cauchy sequence of compl... |
| caucvgb 15631 | A function is convergent i... |
| serf0 15632 | If an infinite series conv... |
| iseraltlem1 15633 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15634 | Lemma for ~ iseralt . The... |
| iseraltlem3 15635 | Lemma for ~ iseralt . Fro... |
| iseralt 15636 | The alternating series tes... |
| sumex 15639 | A sum is a set. (Contribu... |
| sumeq1 15640 | Equality theorem for a sum... |
| nfsum1 15641 | Bound-variable hypothesis ... |
| nfsum 15642 | Bound-variable hypothesis ... |
| sumeq2w 15643 | Equality theorem for sum, ... |
| sumeq2ii 15644 | Equality theorem for sum, ... |
| sumeq2 15645 | Equality theorem for sum. ... |
| cbvsum 15646 | Change bound variable in a... |
| cbvsumv 15647 | Change bound variable in a... |
| cbvsumi 15648 | Change bound variable in a... |
| sumeq1i 15649 | Equality inference for sum... |
| sumeq2i 15650 | Equality inference for sum... |
| sumeq12i 15651 | Equality inference for sum... |
| sumeq1d 15652 | Equality deduction for sum... |
| sumeq2d 15653 | Equality deduction for sum... |
| sumeq2dv 15654 | Equality deduction for sum... |
| sumeq2sdv 15655 | Equality deduction for sum... |
| 2sumeq2dv 15656 | Equality deduction for dou... |
| sumeq12dv 15657 | Equality deduction for sum... |
| sumeq12rdv 15658 | Equality deduction for sum... |
| sum2id 15659 | The second class argument ... |
| sumfc 15660 | A lemma to facilitate conv... |
| fz1f1o 15661 | A lemma for working with f... |
| sumrblem 15662 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15663 | The sequence of partial su... |
| sumrb 15664 | Rebase the starting point ... |
| summolem3 15665 | Lemma for ~ summo . (Cont... |
| summolem2a 15666 | Lemma for ~ summo . (Cont... |
| summolem2 15667 | Lemma for ~ summo . (Cont... |
| summo 15668 | A sum has at most one limi... |
| zsum 15669 | Series sum with index set ... |
| isum 15670 | Series sum with an upper i... |
| fsum 15671 | The value of a sum over a ... |
| sum0 15672 | Any sum over the empty set... |
| sumz 15673 | Any sum of zero over a sum... |
| fsumf1o 15674 | Re-index a finite sum usin... |
| sumss 15675 | Change the index set to a ... |
| fsumss 15676 | Change the index set to a ... |
| sumss2 15677 | Change the index set of a ... |
| fsumcvg2 15678 | The sequence of partial su... |
| fsumsers 15679 | Special case of series sum... |
| fsumcvg3 15680 | A finite sum is convergent... |
| fsumser 15681 | A finite sum expressed in ... |
| fsumcl2lem 15682 | - Lemma for finite sum clo... |
| fsumcllem 15683 | - Lemma for finite sum clo... |
| fsumcl 15684 | Closure of a finite sum of... |
| fsumrecl 15685 | Closure of a finite sum of... |
| fsumzcl 15686 | Closure of a finite sum of... |
| fsumnn0cl 15687 | Closure of a finite sum of... |
| fsumrpcl 15688 | Closure of a finite sum of... |
| fsumclf 15689 | Closure of a finite sum of... |
| fsumzcl2 15690 | A finite sum with integer ... |
| fsumadd 15691 | The sum of two finite sums... |
| fsumsplit 15692 | Split a sum into two parts... |
| fsumsplitf 15693 | Split a sum into two parts... |
| sumsnf 15694 | A sum of a singleton is th... |
| fsumsplitsn 15695 | Separate out a term in a f... |
| fsumsplit1 15696 | Separate out a term in a f... |
| sumsn 15697 | A sum of a singleton is th... |
| fsum1 15698 | The finite sum of ` A ( k ... |
| sumpr 15699 | A sum over a pair is the s... |
| sumtp 15700 | A sum over a triple is the... |
| sumsns 15701 | A sum of a singleton is th... |
| fsumm1 15702 | Separate out the last term... |
| fzosump1 15703 | Separate out the last term... |
| fsum1p 15704 | Separate out the first ter... |
| fsummsnunz 15705 | A finite sum all of whose ... |
| fsumsplitsnun 15706 | Separate out a term in a f... |
| fsump1 15707 | The addition of the next t... |
| isumclim 15708 | An infinite sum equals the... |
| isumclim2 15709 | A converging series conver... |
| isumclim3 15710 | The sequence of partial fi... |
| sumnul 15711 | The sum of a non-convergen... |
| isumcl 15712 | The sum of a converging in... |
| isummulc2 15713 | An infinite sum multiplied... |
| isummulc1 15714 | An infinite sum multiplied... |
| isumdivc 15715 | An infinite sum divided by... |
| isumrecl 15716 | The sum of a converging in... |
| isumge0 15717 | An infinite sum of nonnega... |
| isumadd 15718 | Addition of infinite sums.... |
| sumsplit 15719 | Split a sum into two parts... |
| fsump1i 15720 | Optimized version of ~ fsu... |
| fsum2dlem 15721 | Lemma for ~ fsum2d - induc... |
| fsum2d 15722 | Write a double sum as a su... |
| fsumxp 15723 | Combine two sums into a si... |
| fsumcnv 15724 | Transform a region of summ... |
| fsumcom2 15725 | Interchange order of summa... |
| fsumcom 15726 | Interchange order of summa... |
| fsum0diaglem 15727 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15728 | Two ways to express "the s... |
| mptfzshft 15729 | 1-1 onto function in maps-... |
| fsumrev 15730 | Reversal of a finite sum. ... |
| fsumshft 15731 | Index shift of a finite su... |
| fsumshftm 15732 | Negative index shift of a ... |
| fsumrev2 15733 | Reversal of a finite sum. ... |
| fsum0diag2 15734 | Two ways to express "the s... |
| fsummulc2 15735 | A finite sum multiplied by... |
| fsummulc1 15736 | A finite sum multiplied by... |
| fsumdivc 15737 | A finite sum divided by a ... |
| fsumneg 15738 | Negation of a finite sum. ... |
| fsumsub 15739 | Split a finite sum over a ... |
| fsum2mul 15740 | Separate the nested sum of... |
| fsumconst 15741 | The sum of constant terms ... |
| fsumdifsnconst 15742 | The sum of constant terms ... |
| modfsummodslem1 15743 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15744 | Induction step for ~ modfs... |
| modfsummod 15745 | A finite sum modulo a posi... |
| fsumge0 15746 | If all of the terms of a f... |
| fsumless 15747 | A shorter sum of nonnegati... |
| fsumge1 15748 | A sum of nonnegative numbe... |
| fsum00 15749 | A sum of nonnegative numbe... |
| fsumle 15750 | If all of the terms of fin... |
| fsumlt 15751 | If every term in one finit... |
| fsumabs 15752 | Generalized triangle inequ... |
| telfsumo 15753 | Sum of a telescoping serie... |
| telfsumo2 15754 | Sum of a telescoping serie... |
| telfsum 15755 | Sum of a telescoping serie... |
| telfsum2 15756 | Sum of a telescoping serie... |
| fsumparts 15757 | Summation by parts. (Cont... |
| fsumrelem 15758 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15759 | The real part of a sum. (... |
| fsumim 15760 | The imaginary part of a su... |
| fsumcj 15761 | The complex conjugate of a... |
| fsumrlim 15762 | Limit of a finite sum of c... |
| fsumo1 15763 | The finite sum of eventual... |
| o1fsum 15764 | If ` A ( k ) ` is O(1), th... |
| seqabs 15765 | Generalized triangle inequ... |
| iserabs 15766 | Generalized triangle inequ... |
| cvgcmp 15767 | A comparison test for conv... |
| cvgcmpub 15768 | An upper bound for the lim... |
| cvgcmpce 15769 | A comparison test for conv... |
| abscvgcvg 15770 | An absolutely convergent s... |
| climfsum 15771 | Limit of a finite sum of c... |
| fsumiun 15772 | Sum over a disjoint indexe... |
| hashiun 15773 | The cardinality of a disjo... |
| hash2iun 15774 | The cardinality of a neste... |
| hash2iun1dif1 15775 | The cardinality of a neste... |
| hashrabrex 15776 | The number of elements in ... |
| hashuni 15777 | The cardinality of a disjo... |
| qshash 15778 | The cardinality of a set w... |
| ackbijnn 15779 | Translate the Ackermann bi... |
| binomlem 15780 | Lemma for ~ binom (binomia... |
| binom 15781 | The binomial theorem: ` ( ... |
| binom1p 15782 | Special case of the binomi... |
| binom11 15783 | Special case of the binomi... |
| binom1dif 15784 | A summation for the differ... |
| bcxmaslem1 15785 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15786 | Parallel summation (Christ... |
| incexclem 15787 | Lemma for ~ incexc . (Con... |
| incexc 15788 | The inclusion/exclusion pr... |
| incexc2 15789 | The inclusion/exclusion pr... |
| isumshft 15790 | Index shift of an infinite... |
| isumsplit 15791 | Split off the first ` N ` ... |
| isum1p 15792 | The infinite sum of a conv... |
| isumnn0nn 15793 | Sum from 0 to infinity in ... |
| isumrpcl 15794 | The infinite sum of positi... |
| isumle 15795 | Comparison of two infinite... |
| isumless 15796 | A finite sum of nonnegativ... |
| isumsup2 15797 | An infinite sum of nonnega... |
| isumsup 15798 | An infinite sum of nonnega... |
| isumltss 15799 | A partial sum of a series ... |
| climcndslem1 15800 | Lemma for ~ climcnds : bou... |
| climcndslem2 15801 | Lemma for ~ climcnds : bou... |
| climcnds 15802 | The Cauchy condensation te... |
| divrcnv 15803 | The sequence of reciprocal... |
| divcnv 15804 | The sequence of reciprocal... |
| flo1 15805 | The floor function satisfi... |
| divcnvshft 15806 | Limit of a ratio function.... |
| supcvg 15807 | Extract a sequence ` f ` i... |
| infcvgaux1i 15808 | Auxiliary theorem for appl... |
| infcvgaux2i 15809 | Auxiliary theorem for appl... |
| harmonic 15810 | The harmonic series ` H ` ... |
| arisum 15811 | Arithmetic series sum of t... |
| arisum2 15812 | Arithmetic series sum of t... |
| trireciplem 15813 | Lemma for ~ trirecip . Sh... |
| trirecip 15814 | The sum of the reciprocals... |
| expcnv 15815 | A sequence of powers of a ... |
| explecnv 15816 | A sequence of terms conver... |
| geoserg 15817 | The value of the finite ge... |
| geoser 15818 | The value of the finite ge... |
| pwdif 15819 | The difference of two numb... |
| pwm1geoser 15820 | The n-th power of a number... |
| geolim 15821 | The partial sums in the in... |
| geolim2 15822 | The partial sums in the ge... |
| georeclim 15823 | The limit of a geometric s... |
| geo2sum 15824 | The value of the finite ge... |
| geo2sum2 15825 | The value of the finite ge... |
| geo2lim 15826 | The value of the infinite ... |
| geomulcvg 15827 | The geometric series conve... |
| geoisum 15828 | The infinite sum of ` 1 + ... |
| geoisumr 15829 | The infinite sum of recipr... |
| geoisum1 15830 | The infinite sum of ` A ^ ... |
| geoisum1c 15831 | The infinite sum of ` A x.... |
| 0.999... 15832 | The recurring decimal 0.99... |
| geoihalfsum 15833 | Prove that the infinite ge... |
| cvgrat 15834 | Ratio test for convergence... |
| mertenslem1 15835 | Lemma for ~ mertens . (Co... |
| mertenslem2 15836 | Lemma for ~ mertens . (Co... |
| mertens 15837 | Mertens' theorem. If ` A ... |
| prodf 15838 | An infinite product of com... |
| clim2prod 15839 | The limit of an infinite p... |
| clim2div 15840 | The limit of an infinite p... |
| prodfmul 15841 | The product of two infinit... |
| prodf1 15842 | The value of the partial p... |
| prodf1f 15843 | A one-valued infinite prod... |
| prodfclim1 15844 | The constant one product c... |
| prodfn0 15845 | No term of a nonzero infin... |
| prodfrec 15846 | The reciprocal of an infin... |
| prodfdiv 15847 | The quotient of two infini... |
| ntrivcvg 15848 | A non-trivially converging... |
| ntrivcvgn0 15849 | A product that converges t... |
| ntrivcvgfvn0 15850 | Any value of a product seq... |
| ntrivcvgtail 15851 | A tail of a non-trivially ... |
| ntrivcvgmullem 15852 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15853 | The product of two non-tri... |
| prodex 15856 | A product is a set. (Cont... |
| prodeq1f 15857 | Equality theorem for a pro... |
| prodeq1 15858 | Equality theorem for a pro... |
| nfcprod1 15859 | Bound-variable hypothesis ... |
| nfcprod 15860 | Bound-variable hypothesis ... |
| prodeq2w 15861 | Equality theorem for produ... |
| prodeq2ii 15862 | Equality theorem for produ... |
| prodeq2 15863 | Equality theorem for produ... |
| cbvprod 15864 | Change bound variable in a... |
| cbvprodv 15865 | Change bound variable in a... |
| cbvprodi 15866 | Change bound variable in a... |
| prodeq1i 15867 | Equality inference for pro... |
| prodeq2i 15868 | Equality inference for pro... |
| prodeq12i 15869 | Equality inference for pro... |
| prodeq1d 15870 | Equality deduction for pro... |
| prodeq2d 15871 | Equality deduction for pro... |
| prodeq2dv 15872 | Equality deduction for pro... |
| prodeq2sdv 15873 | Equality deduction for pro... |
| 2cprodeq2dv 15874 | Equality deduction for dou... |
| prodeq12dv 15875 | Equality deduction for pro... |
| prodeq12rdv 15876 | Equality deduction for pro... |
| prod2id 15877 | The second class argument ... |
| prodrblem 15878 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15879 | The sequence of partial pr... |
| prodrblem2 15880 | Lemma for ~ prodrb . (Con... |
| prodrb 15881 | Rebase the starting point ... |
| prodmolem3 15882 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15883 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15884 | Lemma for ~ prodmo . (Con... |
| prodmo 15885 | A product has at most one ... |
| zprod 15886 | Series product with index ... |
| iprod 15887 | Series product with an upp... |
| zprodn0 15888 | Nonzero series product wit... |
| iprodn0 15889 | Nonzero series product wit... |
| fprod 15890 | The value of a product ove... |
| fprodntriv 15891 | A non-triviality lemma for... |
| prod0 15892 | A product over the empty s... |
| prod1 15893 | Any product of one over a ... |
| prodfc 15894 | A lemma to facilitate conv... |
| fprodf1o 15895 | Re-index a finite product ... |
| prodss 15896 | Change the index set to a ... |
| fprodss 15897 | Change the index set to a ... |
| fprodser 15898 | A finite product expressed... |
| fprodcl2lem 15899 | Finite product closure lem... |
| fprodcllem 15900 | Finite product closure lem... |
| fprodcl 15901 | Closure of a finite produc... |
| fprodrecl 15902 | Closure of a finite produc... |
| fprodzcl 15903 | Closure of a finite produc... |
| fprodnncl 15904 | Closure of a finite produc... |
| fprodrpcl 15905 | Closure of a finite produc... |
| fprodnn0cl 15906 | Closure of a finite produc... |
| fprodcllemf 15907 | Finite product closure lem... |
| fprodreclf 15908 | Closure of a finite produc... |
| fprodmul 15909 | The product of two finite ... |
| fproddiv 15910 | The quotient of two finite... |
| prodsn 15911 | A product of a singleton i... |
| fprod1 15912 | A finite product of only o... |
| prodsnf 15913 | A product of a singleton i... |
| climprod1 15914 | The limit of a product ove... |
| fprodsplit 15915 | Split a finite product int... |
| fprodm1 15916 | Separate out the last term... |
| fprod1p 15917 | Separate out the first ter... |
| fprodp1 15918 | Multiply in the last term ... |
| fprodm1s 15919 | Separate out the last term... |
| fprodp1s 15920 | Multiply in the last term ... |
| prodsns 15921 | A product of the singleton... |
| fprodfac 15922 | Factorial using product no... |
| fprodabs 15923 | The absolute value of a fi... |
| fprodeq0 15924 | Any finite product contain... |
| fprodshft 15925 | Shift the index of a finit... |
| fprodrev 15926 | Reversal of a finite produ... |
| fprodconst 15927 | The product of constant te... |
| fprodn0 15928 | A finite product of nonzer... |
| fprod2dlem 15929 | Lemma for ~ fprod2d - indu... |
| fprod2d 15930 | Write a double product as ... |
| fprodxp 15931 | Combine two products into ... |
| fprodcnv 15932 | Transform a product region... |
| fprodcom2 15933 | Interchange order of multi... |
| fprodcom 15934 | Interchange product order.... |
| fprod0diag 15935 | Two ways to express "the p... |
| fproddivf 15936 | The quotient of two finite... |
| fprodsplitf 15937 | Split a finite product int... |
| fprodsplitsn 15938 | Separate out a term in a f... |
| fprodsplit1f 15939 | Separate out a term in a f... |
| fprodn0f 15940 | A finite product of nonzer... |
| fprodclf 15941 | Closure of a finite produc... |
| fprodge0 15942 | If all the terms of a fini... |
| fprodeq0g 15943 | Any finite product contain... |
| fprodge1 15944 | If all of the terms of a f... |
| fprodle 15945 | If all the terms of two fi... |
| fprodmodd 15946 | If all factors of two fini... |
| iprodclim 15947 | An infinite product equals... |
| iprodclim2 15948 | A converging product conve... |
| iprodclim3 15949 | The sequence of partial fi... |
| iprodcl 15950 | The product of a non-trivi... |
| iprodrecl 15951 | The product of a non-trivi... |
| iprodmul 15952 | Multiplication of infinite... |
| risefacval 15957 | The value of the rising fa... |
| fallfacval 15958 | The value of the falling f... |
| risefacval2 15959 | One-based value of rising ... |
| fallfacval2 15960 | One-based value of falling... |
| fallfacval3 15961 | A product representation o... |
| risefaccllem 15962 | Lemma for rising factorial... |
| fallfaccllem 15963 | Lemma for falling factoria... |
| risefaccl 15964 | Closure law for rising fac... |
| fallfaccl 15965 | Closure law for falling fa... |
| rerisefaccl 15966 | Closure law for rising fac... |
| refallfaccl 15967 | Closure law for falling fa... |
| nnrisefaccl 15968 | Closure law for rising fac... |
| zrisefaccl 15969 | Closure law for rising fac... |
| zfallfaccl 15970 | Closure law for falling fa... |
| nn0risefaccl 15971 | Closure law for rising fac... |
| rprisefaccl 15972 | Closure law for rising fac... |
| risefallfac 15973 | A relationship between ris... |
| fallrisefac 15974 | A relationship between fal... |
| risefall0lem 15975 | Lemma for ~ risefac0 and ~... |
| risefac0 15976 | The value of the rising fa... |
| fallfac0 15977 | The value of the falling f... |
| risefacp1 15978 | The value of the rising fa... |
| fallfacp1 15979 | The value of the falling f... |
| risefacp1d 15980 | The value of the rising fa... |
| fallfacp1d 15981 | The value of the falling f... |
| risefac1 15982 | The value of rising factor... |
| fallfac1 15983 | The value of falling facto... |
| risefacfac 15984 | Relate rising factorial to... |
| fallfacfwd 15985 | The forward difference of ... |
| 0fallfac 15986 | The value of the zero fall... |
| 0risefac 15987 | The value of the zero risi... |
| binomfallfaclem1 15988 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 15989 | Lemma for ~ binomfallfac .... |
| binomfallfac 15990 | A version of the binomial ... |
| binomrisefac 15991 | A version of the binomial ... |
| fallfacval4 15992 | Represent the falling fact... |
| bcfallfac 15993 | Binomial coefficient in te... |
| fallfacfac 15994 | Relate falling factorial t... |
| bpolylem 15997 | Lemma for ~ bpolyval . (C... |
| bpolyval 15998 | The value of the Bernoulli... |
| bpoly0 15999 | The value of the Bernoulli... |
| bpoly1 16000 | The value of the Bernoulli... |
| bpolycl 16001 | Closure law for Bernoulli ... |
| bpolysum 16002 | A sum for Bernoulli polyno... |
| bpolydiflem 16003 | Lemma for ~ bpolydif . (C... |
| bpolydif 16004 | Calculate the difference b... |
| fsumkthpow 16005 | A closed-form expression f... |
| bpoly2 16006 | The Bernoulli polynomials ... |
| bpoly3 16007 | The Bernoulli polynomials ... |
| bpoly4 16008 | The Bernoulli polynomials ... |
| fsumcube 16009 | Express the sum of cubes i... |
| eftcl 16022 | Closure of a term in the s... |
| reeftcl 16023 | The terms of the series ex... |
| eftabs 16024 | The absolute value of a te... |
| eftval 16025 | The value of a term in the... |
| efcllem 16026 | Lemma for ~ efcl . The se... |
| ef0lem 16027 | The series defining the ex... |
| efval 16028 | Value of the exponential f... |
| esum 16029 | Value of Euler's constant ... |
| eff 16030 | Domain and codomain of the... |
| efcl 16031 | Closure law for the expone... |
| efval2 16032 | Value of the exponential f... |
| efcvg 16033 | The series that defines th... |
| efcvgfsum 16034 | Exponential function conve... |
| reefcl 16035 | The exponential function i... |
| reefcld 16036 | The exponential function i... |
| ere 16037 | Euler's constant ` _e ` = ... |
| ege2le3 16038 | Lemma for ~ egt2lt3 . (Co... |
| ef0 16039 | Value of the exponential f... |
| efcj 16040 | The exponential of a compl... |
| efaddlem 16041 | Lemma for ~ efadd (exponen... |
| efadd 16042 | Sum of exponents law for e... |
| fprodefsum 16043 | Move the exponential funct... |
| efcan 16044 | Cancellation law for expon... |
| efne0 16045 | The exponential of a compl... |
| efneg 16046 | The exponential of the opp... |
| eff2 16047 | The exponential function m... |
| efsub 16048 | Difference of exponents la... |
| efexp 16049 | The exponential of an inte... |
| efzval 16050 | Value of the exponential f... |
| efgt0 16051 | The exponential of a real ... |
| rpefcl 16052 | The exponential of a real ... |
| rpefcld 16053 | The exponential of a real ... |
| eftlcvg 16054 | The tail series of the exp... |
| eftlcl 16055 | Closure of the sum of an i... |
| reeftlcl 16056 | Closure of the sum of an i... |
| eftlub 16057 | An upper bound on the abso... |
| efsep 16058 | Separate out the next term... |
| effsumlt 16059 | The partial sums of the se... |
| eft0val 16060 | The value of the first ter... |
| ef4p 16061 | Separate out the first fou... |
| efgt1p2 16062 | The exponential of a posit... |
| efgt1p 16063 | The exponential of a posit... |
| efgt1 16064 | The exponential of a posit... |
| eflt 16065 | The exponential function o... |
| efle 16066 | The exponential function o... |
| reef11 16067 | The exponential function o... |
| reeff1 16068 | The exponential function m... |
| eflegeo 16069 | The exponential function o... |
| sinval 16070 | Value of the sine function... |
| cosval 16071 | Value of the cosine functi... |
| sinf 16072 | Domain and codomain of the... |
| cosf 16073 | Domain and codomain of the... |
| sincl 16074 | Closure of the sine functi... |
| coscl 16075 | Closure of the cosine func... |
| tanval 16076 | Value of the tangent funct... |
| tancl 16077 | The closure of the tangent... |
| sincld 16078 | Closure of the sine functi... |
| coscld 16079 | Closure of the cosine func... |
| tancld 16080 | Closure of the tangent fun... |
| tanval2 16081 | Express the tangent functi... |
| tanval3 16082 | Express the tangent functi... |
| resinval 16083 | The sine of a real number ... |
| recosval 16084 | The cosine of a real numbe... |
| efi4p 16085 | Separate out the first fou... |
| resin4p 16086 | Separate out the first fou... |
| recos4p 16087 | Separate out the first fou... |
| resincl 16088 | The sine of a real number ... |
| recoscl 16089 | The cosine of a real numbe... |
| retancl 16090 | The closure of the tangent... |
| resincld 16091 | Closure of the sine functi... |
| recoscld 16092 | Closure of the cosine func... |
| retancld 16093 | Closure of the tangent fun... |
| sinneg 16094 | The sine of a negative is ... |
| cosneg 16095 | The cosines of a number an... |
| tanneg 16096 | The tangent of a negative ... |
| sin0 16097 | Value of the sine function... |
| cos0 16098 | Value of the cosine functi... |
| tan0 16099 | The value of the tangent f... |
| efival 16100 | The exponential function i... |
| efmival 16101 | The exponential function i... |
| sinhval 16102 | Value of the hyperbolic si... |
| coshval 16103 | Value of the hyperbolic co... |
| resinhcl 16104 | The hyperbolic sine of a r... |
| rpcoshcl 16105 | The hyperbolic cosine of a... |
| recoshcl 16106 | The hyperbolic cosine of a... |
| retanhcl 16107 | The hyperbolic tangent of ... |
| tanhlt1 16108 | The hyperbolic tangent of ... |
| tanhbnd 16109 | The hyperbolic tangent of ... |
| efeul 16110 | Eulerian representation of... |
| efieq 16111 | The exponentials of two im... |
| sinadd 16112 | Addition formula for sine.... |
| cosadd 16113 | Addition formula for cosin... |
| tanaddlem 16114 | A useful intermediate step... |
| tanadd 16115 | Addition formula for tange... |
| sinsub 16116 | Sine of difference. (Cont... |
| cossub 16117 | Cosine of difference. (Co... |
| addsin 16118 | Sum of sines. (Contribute... |
| subsin 16119 | Difference of sines. (Con... |
| sinmul 16120 | Product of sines can be re... |
| cosmul 16121 | Product of cosines can be ... |
| addcos 16122 | Sum of cosines. (Contribu... |
| subcos 16123 | Difference of cosines. (C... |
| sincossq 16124 | Sine squared plus cosine s... |
| sin2t 16125 | Double-angle formula for s... |
| cos2t 16126 | Double-angle formula for c... |
| cos2tsin 16127 | Double-angle formula for c... |
| sinbnd 16128 | The sine of a real number ... |
| cosbnd 16129 | The cosine of a real numbe... |
| sinbnd2 16130 | The sine of a real number ... |
| cosbnd2 16131 | The cosine of a real numbe... |
| ef01bndlem 16132 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16133 | Bounds on the sine of a po... |
| cos01bnd 16134 | Bounds on the cosine of a ... |
| cos1bnd 16135 | Bounds on the cosine of 1.... |
| cos2bnd 16136 | Bounds on the cosine of 2.... |
| sinltx 16137 | The sine of a positive rea... |
| sin01gt0 16138 | The sine of a positive rea... |
| cos01gt0 16139 | The cosine of a positive r... |
| sin02gt0 16140 | The sine of a positive rea... |
| sincos1sgn 16141 | The signs of the sine and ... |
| sincos2sgn 16142 | The signs of the sine and ... |
| sin4lt0 16143 | The sine of 4 is negative.... |
| absefi 16144 | The absolute value of the ... |
| absef 16145 | The absolute value of the ... |
| absefib 16146 | A complex number is real i... |
| efieq1re 16147 | A number whose imaginary e... |
| demoivre 16148 | De Moivre's Formula. Proo... |
| demoivreALT 16149 | Alternate proof of ~ demoi... |
| eirrlem 16152 | Lemma for ~ eirr . (Contr... |
| eirr 16153 | ` _e ` is irrational. (Co... |
| egt2lt3 16154 | Euler's constant ` _e ` = ... |
| epos 16155 | Euler's constant ` _e ` is... |
| epr 16156 | Euler's constant ` _e ` is... |
| ene0 16157 | ` _e ` is not 0. (Contrib... |
| ene1 16158 | ` _e ` is not 1. (Contrib... |
| xpnnen 16159 | The Cartesian product of t... |
| znnen 16160 | The set of integers and th... |
| qnnen 16161 | The rational numbers are c... |
| rpnnen2lem1 16162 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16163 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16164 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16165 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16166 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16167 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16168 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16169 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16170 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16171 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16172 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16173 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16174 | The other half of ~ rpnnen... |
| rpnnen 16175 | The cardinality of the con... |
| rexpen 16176 | The real numbers are equin... |
| cpnnen 16177 | The complex numbers are eq... |
| rucALT 16178 | Alternate proof of ~ ruc .... |
| ruclem1 16179 | Lemma for ~ ruc (the reals... |
| ruclem2 16180 | Lemma for ~ ruc . Orderin... |
| ruclem3 16181 | Lemma for ~ ruc . The con... |
| ruclem4 16182 | Lemma for ~ ruc . Initial... |
| ruclem6 16183 | Lemma for ~ ruc . Domain ... |
| ruclem7 16184 | Lemma for ~ ruc . Success... |
| ruclem8 16185 | Lemma for ~ ruc . The int... |
| ruclem9 16186 | Lemma for ~ ruc . The fir... |
| ruclem10 16187 | Lemma for ~ ruc . Every f... |
| ruclem11 16188 | Lemma for ~ ruc . Closure... |
| ruclem12 16189 | Lemma for ~ ruc . The sup... |
| ruclem13 16190 | Lemma for ~ ruc . There i... |
| ruc 16191 | The set of positive intege... |
| resdomq 16192 | The set of rationals is st... |
| aleph1re 16193 | There are at least aleph-o... |
| aleph1irr 16194 | There are at least aleph-o... |
| cnso 16195 | The complex numbers can be... |
| sqrt2irrlem 16196 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16197 | The square root of 2 is ir... |
| sqrt2re 16198 | The square root of 2 exist... |
| sqrt2irr0 16199 | The square root of 2 is an... |
| nthruc 16200 | The sequence ` NN ` , ` ZZ... |
| nthruz 16201 | The sequence ` NN ` , ` NN... |
| divides 16204 | Define the divides relatio... |
| dvdsval2 16205 | One nonzero integer divide... |
| dvdsval3 16206 | One nonzero integer divide... |
| dvdszrcl 16207 | Reverse closure for the di... |
| dvdsmod0 16208 | If a positive integer divi... |
| p1modz1 16209 | If a number greater than 1... |
| dvdsmodexp 16210 | If a positive integer divi... |
| nndivdvds 16211 | Strong form of ~ dvdsval2 ... |
| nndivides 16212 | Definition of the divides ... |
| moddvds 16213 | Two ways to say ` A == B `... |
| modm1div 16214 | An integer greater than on... |
| dvds0lem 16215 | A lemma to assist theorems... |
| dvds1lem 16216 | A lemma to assist theorems... |
| dvds2lem 16217 | A lemma to assist theorems... |
| iddvds 16218 | An integer divides itself.... |
| 1dvds 16219 | 1 divides any integer. Th... |
| dvds0 16220 | Any integer divides 0. Th... |
| negdvdsb 16221 | An integer divides another... |
| dvdsnegb 16222 | An integer divides another... |
| absdvdsb 16223 | An integer divides another... |
| dvdsabsb 16224 | An integer divides another... |
| 0dvds 16225 | Only 0 is divisible by 0. ... |
| dvdsmul1 16226 | An integer divides a multi... |
| dvdsmul2 16227 | An integer divides a multi... |
| iddvdsexp 16228 | An integer divides a posit... |
| muldvds1 16229 | If a product divides an in... |
| muldvds2 16230 | If a product divides an in... |
| dvdscmul 16231 | Multiplication by a consta... |
| dvdsmulc 16232 | Multiplication by a consta... |
| dvdscmulr 16233 | Cancellation law for the d... |
| dvdsmulcr 16234 | Cancellation law for the d... |
| summodnegmod 16235 | The sum of two integers mo... |
| modmulconst 16236 | Constant multiplication in... |
| dvds2ln 16237 | If an integer divides each... |
| dvds2add 16238 | If an integer divides each... |
| dvds2sub 16239 | If an integer divides each... |
| dvds2addd 16240 | Deduction form of ~ dvds2a... |
| dvds2subd 16241 | Deduction form of ~ dvds2s... |
| dvdstr 16242 | The divides relation is tr... |
| dvdstrd 16243 | The divides relation is tr... |
| dvdsmultr1 16244 | If an integer divides anot... |
| dvdsmultr1d 16245 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16246 | If an integer divides anot... |
| dvdsmultr2d 16247 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16248 | If an integer divides eith... |
| dvdssub2 16249 | If an integer divides a di... |
| dvdsadd 16250 | An integer divides another... |
| dvdsaddr 16251 | An integer divides another... |
| dvdssub 16252 | An integer divides another... |
| dvdssubr 16253 | An integer divides another... |
| dvdsadd2b 16254 | Adding a multiple of the b... |
| dvdsaddre2b 16255 | Adding a multiple of the b... |
| fsumdvds 16256 | If every term in a sum is ... |
| dvdslelem 16257 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16258 | The divisors of a positive... |
| dvdsleabs 16259 | The divisors of a nonzero ... |
| dvdsleabs2 16260 | Transfer divisibility to a... |
| dvdsabseq 16261 | If two integers divide eac... |
| dvdseq 16262 | If two nonnegative integer... |
| divconjdvds 16263 | If a nonzero integer ` M `... |
| dvdsdivcl 16264 | The complement of a diviso... |
| dvdsflip 16265 | An involution of the divis... |
| dvdsssfz1 16266 | The set of divisors of a n... |
| dvds1 16267 | The only nonnegative integ... |
| alzdvds 16268 | Only 0 is divisible by all... |
| dvdsext 16269 | Poset extensionality for d... |
| fzm1ndvds 16270 | No number between ` 1 ` an... |
| fzo0dvdseq 16271 | Zero is the only one of th... |
| fzocongeq 16272 | Two different elements of ... |
| addmodlteqALT 16273 | Two nonnegative integers l... |
| dvdsfac 16274 | A positive integer divides... |
| dvdsexp2im 16275 | If an integer divides anot... |
| dvdsexp 16276 | A power divides a power wi... |
| dvdsmod 16277 | Any number ` K ` whose mod... |
| mulmoddvds 16278 | If an integer is divisible... |
| 3dvds 16279 | A rule for divisibility by... |
| 3dvdsdec 16280 | A decimal number is divisi... |
| 3dvds2dec 16281 | A decimal number is divisi... |
| fprodfvdvdsd 16282 | A finite product of intege... |
| fproddvdsd 16283 | A finite product of intege... |
| evenelz 16284 | An even number is an integ... |
| zeo3 16285 | An integer is even or odd.... |
| zeo4 16286 | An integer is even or odd ... |
| zeneo 16287 | No even integer equals an ... |
| odd2np1lem 16288 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16289 | An integer is odd iff it i... |
| even2n 16290 | An integer is even iff it ... |
| oddm1even 16291 | An integer is odd iff its ... |
| oddp1even 16292 | An integer is odd iff its ... |
| oexpneg 16293 | The exponential of the neg... |
| mod2eq0even 16294 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16295 | An integer is 1 modulo 2 i... |
| oddnn02np1 16296 | A nonnegative integer is o... |
| oddge22np1 16297 | An integer greater than on... |
| evennn02n 16298 | A nonnegative integer is e... |
| evennn2n 16299 | A positive integer is even... |
| 2tp1odd 16300 | A number which is twice an... |
| mulsucdiv2z 16301 | An integer multiplied with... |
| sqoddm1div8z 16302 | A squared odd number minus... |
| 2teven 16303 | A number which is twice an... |
| zeo5 16304 | An integer is either even ... |
| evend2 16305 | An integer is even iff its... |
| oddp1d2 16306 | An integer is odd iff its ... |
| zob 16307 | Alternate characterization... |
| oddm1d2 16308 | An integer is odd iff its ... |
| ltoddhalfle 16309 | An integer is less than ha... |
| halfleoddlt 16310 | An integer is greater than... |
| opoe 16311 | The sum of two odds is eve... |
| omoe 16312 | The difference of two odds... |
| opeo 16313 | The sum of an odd and an e... |
| omeo 16314 | The difference of an odd a... |
| z0even 16315 | 2 divides 0. That means 0... |
| n2dvds1 16316 | 2 does not divide 1. That... |
| n2dvdsm1 16317 | 2 does not divide -1. Tha... |
| z2even 16318 | 2 divides 2. That means 2... |
| n2dvds3 16319 | 2 does not divide 3. That... |
| z4even 16320 | 2 divides 4. That means 4... |
| 4dvdseven 16321 | An integer which is divisi... |
| m1expe 16322 | Exponentiation of -1 by an... |
| m1expo 16323 | Exponentiation of -1 by an... |
| m1exp1 16324 | Exponentiation of negative... |
| nn0enne 16325 | A positive integer is an e... |
| nn0ehalf 16326 | The half of an even nonneg... |
| nnehalf 16327 | The half of an even positi... |
| nn0onn 16328 | An odd nonnegative integer... |
| nn0o1gt2 16329 | An odd nonnegative integer... |
| nno 16330 | An alternate characterizat... |
| nn0o 16331 | An alternate characterizat... |
| nn0ob 16332 | Alternate characterization... |
| nn0oddm1d2 16333 | A positive integer is odd ... |
| nnoddm1d2 16334 | A positive integer is odd ... |
| sumeven 16335 | If every term in a sum is ... |
| sumodd 16336 | If every term in a sum is ... |
| evensumodd 16337 | If every term in a sum wit... |
| oddsumodd 16338 | If every term in a sum wit... |
| pwp1fsum 16339 | The n-th power of a number... |
| oddpwp1fsum 16340 | An odd power of a number i... |
| divalglem0 16341 | Lemma for ~ divalg . (Con... |
| divalglem1 16342 | Lemma for ~ divalg . (Con... |
| divalglem2 16343 | Lemma for ~ divalg . (Con... |
| divalglem4 16344 | Lemma for ~ divalg . (Con... |
| divalglem5 16345 | Lemma for ~ divalg . (Con... |
| divalglem6 16346 | Lemma for ~ divalg . (Con... |
| divalglem7 16347 | Lemma for ~ divalg . (Con... |
| divalglem8 16348 | Lemma for ~ divalg . (Con... |
| divalglem9 16349 | Lemma for ~ divalg . (Con... |
| divalglem10 16350 | Lemma for ~ divalg . (Con... |
| divalg 16351 | The division algorithm (th... |
| divalgb 16352 | Express the division algor... |
| divalg2 16353 | The division algorithm (th... |
| divalgmod 16354 | The result of the ` mod ` ... |
| divalgmodcl 16355 | The result of the ` mod ` ... |
| modremain 16356 | The result of the modulo o... |
| ndvdssub 16357 | Corollary of the division ... |
| ndvdsadd 16358 | Corollary of the division ... |
| ndvdsp1 16359 | Special case of ~ ndvdsadd... |
| ndvdsi 16360 | A quick test for non-divis... |
| flodddiv4 16361 | The floor of an odd intege... |
| fldivndvdslt 16362 | The floor of an integer di... |
| flodddiv4lt 16363 | The floor of an odd number... |
| flodddiv4t2lthalf 16364 | The floor of an odd number... |
| bitsfval 16369 | Expand the definition of t... |
| bitsval 16370 | Expand the definition of t... |
| bitsval2 16371 | Expand the definition of t... |
| bitsss 16372 | The set of bits of an inte... |
| bitsf 16373 | The ` bits ` function is a... |
| bits0 16374 | Value of the zeroth bit. ... |
| bits0e 16375 | The zeroth bit of an even ... |
| bits0o 16376 | The zeroth bit of an odd n... |
| bitsp1 16377 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16378 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16379 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16380 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16381 | The bits of a number are a... |
| bitsmod 16382 | Truncating the bit sequenc... |
| bitsfi 16383 | Every number is associated... |
| bitscmp 16384 | The bit complement of ` N ... |
| 0bits 16385 | The bits of zero. (Contri... |
| m1bits 16386 | The bits of negative one. ... |
| bitsinv1lem 16387 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16388 | There is an explicit inver... |
| bitsinv2 16389 | There is an explicit inver... |
| bitsf1ocnv 16390 | The ` bits ` function rest... |
| bitsf1o 16391 | The ` bits ` function rest... |
| bitsf1 16392 | The ` bits ` function is a... |
| 2ebits 16393 | The bits of a power of two... |
| bitsinv 16394 | The inverse of the ` bits ... |
| bitsinvp1 16395 | Recursive definition of th... |
| sadadd2lem2 16396 | The core of the proof of ~... |
| sadfval 16398 | Define the addition of two... |
| sadcf 16399 | The carry sequence is a se... |
| sadc0 16400 | The initial element of the... |
| sadcp1 16401 | The carry sequence (which ... |
| sadval 16402 | The full adder sequence is... |
| sadcaddlem 16403 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16404 | Non-recursive definition o... |
| sadadd2lem 16405 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16406 | Sum of initial segments of... |
| sadadd3 16407 | Sum of initial segments of... |
| sadcl 16408 | The sum of two sequences i... |
| sadcom 16409 | The adder sequence functio... |
| saddisjlem 16410 | Lemma for ~ sadadd . (Con... |
| saddisj 16411 | The sum of disjoint sequen... |
| sadaddlem 16412 | Lemma for ~ sadadd . (Con... |
| sadadd 16413 | For sequences that corresp... |
| sadid1 16414 | The adder sequence functio... |
| sadid2 16415 | The adder sequence functio... |
| sadasslem 16416 | Lemma for ~ sadass . (Con... |
| sadass 16417 | Sequence addition is assoc... |
| sadeq 16418 | Any element of a sequence ... |
| bitsres 16419 | Restrict the bits of a num... |
| bitsuz 16420 | The bits of a number are a... |
| bitsshft 16421 | Shifting a bit sequence to... |
| smufval 16423 | The multiplication of two ... |
| smupf 16424 | The sequence of partial su... |
| smup0 16425 | The initial element of the... |
| smupp1 16426 | The initial element of the... |
| smuval 16427 | Define the addition of two... |
| smuval2 16428 | The partial sum sequence s... |
| smupvallem 16429 | If ` A ` only has elements... |
| smucl 16430 | The product of two sequenc... |
| smu01lem 16431 | Lemma for ~ smu01 and ~ sm... |
| smu01 16432 | Multiplication of a sequen... |
| smu02 16433 | Multiplication of a sequen... |
| smupval 16434 | Rewrite the elements of th... |
| smup1 16435 | Rewrite ~ smupp1 using onl... |
| smueqlem 16436 | Any element of a sequence ... |
| smueq 16437 | Any element of a sequence ... |
| smumullem 16438 | Lemma for ~ smumul . (Con... |
| smumul 16439 | For sequences that corresp... |
| gcdval 16442 | The value of the ` gcd ` o... |
| gcd0val 16443 | The value, by convention, ... |
| gcdn0val 16444 | The value of the ` gcd ` o... |
| gcdcllem1 16445 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16446 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16447 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16448 | Closure of the ` gcd ` ope... |
| gcddvds 16449 | The gcd of two integers di... |
| dvdslegcd 16450 | An integer which divides b... |
| nndvdslegcd 16451 | A positive integer which d... |
| gcdcl 16452 | Closure of the ` gcd ` ope... |
| gcdnncl 16453 | Closure of the ` gcd ` ope... |
| gcdcld 16454 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16455 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16456 | An integer is the product ... |
| divgcdz 16457 | An integer divided by the ... |
| gcdf 16458 | Domain and codomain of the... |
| gcdcom 16459 | The ` gcd ` operator is co... |
| gcdcomd 16460 | The ` gcd ` operator is co... |
| divgcdnn 16461 | A positive integer divided... |
| divgcdnnr 16462 | A positive integer divided... |
| gcdeq0 16463 | The gcd of two integers is... |
| gcdn0gt0 16464 | The gcd of two integers is... |
| gcd0id 16465 | The gcd of 0 and an intege... |
| gcdid0 16466 | The gcd of an integer and ... |
| nn0gcdid0 16467 | The gcd of a nonnegative i... |
| gcdneg 16468 | Negating one operand of th... |
| neggcd 16469 | Negating one operand of th... |
| gcdaddmlem 16470 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16471 | Adding a multiple of one o... |
| gcdadd 16472 | The GCD of two numbers is ... |
| gcdid 16473 | The gcd of a number and it... |
| gcd1 16474 | The gcd of a number with 1... |
| gcdabs1 16475 | ` gcd ` of the absolute va... |
| gcdabs2 16476 | ` gcd ` of the absolute va... |
| gcdabs 16477 | The gcd of two integers is... |
| gcdabsOLD 16478 | Obsolete version of ~ gcda... |
| modgcd 16479 | The gcd remains unchanged ... |
| 1gcd 16480 | The GCD of one and an inte... |
| gcdmultipled 16481 | The greatest common diviso... |
| gcdmultiplez 16482 | The GCD of a multiple of a... |
| gcdmultiple 16483 | The GCD of a multiple of a... |
| dvdsgcdidd 16484 | The greatest common diviso... |
| 6gcd4e2 16485 | The greatest common diviso... |
| bezoutlem1 16486 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16487 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16488 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16489 | Lemma for ~ bezout . (Con... |
| bezout 16490 | Bézout's identity: ... |
| dvdsgcd 16491 | An integer which divides e... |
| dvdsgcdb 16492 | Biconditional form of ~ dv... |
| dfgcd2 16493 | Alternate definition of th... |
| gcdass 16494 | Associative law for ` gcd ... |
| mulgcd 16495 | Distribute multiplication ... |
| absmulgcd 16496 | Distribute absolute value ... |
| mulgcdr 16497 | Reverse distribution law f... |
| gcddiv 16498 | Division law for GCD. (Con... |
| gcdzeq 16499 | A positive integer ` A ` i... |
| gcdeq 16500 | ` A ` is equal to its gcd ... |
| dvdssqim 16501 | Unidirectional form of ~ d... |
| dvdsmulgcd 16502 | A divisibility equivalent ... |
| rpmulgcd 16503 | If ` K ` and ` M ` are rel... |
| rplpwr 16504 | If ` A ` and ` B ` are rel... |
| rprpwr 16505 | If ` A ` and ` B ` are rel... |
| rppwr 16506 | If ` A ` and ` B ` are rel... |
| sqgcd 16507 | Square distributes over gc... |
| dvdssqlem 16508 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16509 | Two numbers are divisible ... |
| bezoutr 16510 | Partial converse to ~ bezo... |
| bezoutr1 16511 | Converse of ~ bezout for w... |
| nn0seqcvgd 16512 | A strictly-decreasing nonn... |
| seq1st 16513 | A sequence whose iteration... |
| algr0 16514 | The value of the algorithm... |
| algrf 16515 | An algorithm is a step fun... |
| algrp1 16516 | The value of the algorithm... |
| alginv 16517 | If ` I ` is an invariant o... |
| algcvg 16518 | One way to prove that an a... |
| algcvgblem 16519 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16520 | Two ways of expressing tha... |
| algcvga 16521 | The countdown function ` C... |
| algfx 16522 | If ` F ` reaches a fixed p... |
| eucalgval2 16523 | The value of the step func... |
| eucalgval 16524 | Euclid's Algorithm ~ eucal... |
| eucalgf 16525 | Domain and codomain of the... |
| eucalginv 16526 | The invariant of the step ... |
| eucalglt 16527 | The second member of the s... |
| eucalgcvga 16528 | Once Euclid's Algorithm ha... |
| eucalg 16529 | Euclid's Algorithm compute... |
| lcmval 16534 | Value of the ` lcm ` opera... |
| lcmcom 16535 | The ` lcm ` operator is co... |
| lcm0val 16536 | The value, by convention, ... |
| lcmn0val 16537 | The value of the ` lcm ` o... |
| lcmcllem 16538 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16539 | Closure of the ` lcm ` ope... |
| dvdslcm 16540 | The lcm of two integers is... |
| lcmledvds 16541 | A positive integer which b... |
| lcmeq0 16542 | The lcm of two integers is... |
| lcmcl 16543 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16544 | The greatest common diviso... |
| lcmneg 16545 | Negating one operand of th... |
| neglcm 16546 | Negating one operand of th... |
| lcmabs 16547 | The lcm of two integers is... |
| lcmgcdlem 16548 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16549 | The product of two numbers... |
| lcmdvds 16550 | The lcm of two integers di... |
| lcmid 16551 | The lcm of an integer and ... |
| lcm1 16552 | The lcm of an integer and ... |
| lcmgcdnn 16553 | The product of two positiv... |
| lcmgcdeq 16554 | Two integers' absolute val... |
| lcmdvdsb 16555 | Biconditional form of ~ lc... |
| lcmass 16556 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16557 | The least common multiple ... |
| 6lcm4e12 16558 | The least common multiple ... |
| absproddvds 16559 | The absolute value of the ... |
| absprodnn 16560 | The absolute value of the ... |
| fissn0dvds 16561 | For each finite subset of ... |
| fissn0dvdsn0 16562 | For each finite subset of ... |
| lcmfval 16563 | Value of the ` _lcm ` func... |
| lcmf0val 16564 | The value, by convention, ... |
| lcmfn0val 16565 | The value of the ` _lcm ` ... |
| lcmfnnval 16566 | The value of the ` _lcm ` ... |
| lcmfcllem 16567 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16568 | Closure of the ` _lcm ` fu... |
| lcmfpr 16569 | The value of the ` _lcm ` ... |
| lcmfcl 16570 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16571 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16572 | The least common multiple ... |
| dvdslcmf 16573 | The least common multiple ... |
| lcmfledvds 16574 | A positive integer which i... |
| lcmf 16575 | Characterization of the le... |
| lcmf0 16576 | The least common multiple ... |
| lcmfsn 16577 | The least common multiple ... |
| lcmftp 16578 | The least common multiple ... |
| lcmfunsnlem1 16579 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16580 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16581 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16582 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16583 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16584 | The least common multiple ... |
| lcmfdvdsb 16585 | Biconditional form of ~ lc... |
| lcmfunsn 16586 | The ` _lcm ` function for ... |
| lcmfun 16587 | The ` _lcm ` function for ... |
| lcmfass 16588 | Associative law for the ` ... |
| lcmf2a3a4e12 16589 | The least common multiple ... |
| lcmflefac 16590 | The least common multiple ... |
| coprmgcdb 16591 | Two positive integers are ... |
| ncoprmgcdne1b 16592 | Two positive integers are ... |
| ncoprmgcdgt1b 16593 | Two positive integers are ... |
| coprmdvds1 16594 | If two positive integers a... |
| coprmdvds 16595 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16596 | If an integer is divisible... |
| mulgcddvds 16597 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16598 | If ` M ` is relatively pri... |
| qredeq 16599 | Two equal reduced fraction... |
| qredeu 16600 | Every rational number has ... |
| rpmul 16601 | If ` K ` is relatively pri... |
| rpdvds 16602 | If ` K ` is relatively pri... |
| coprmprod 16603 | The product of the element... |
| coprmproddvdslem 16604 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16605 | If a positive integer is d... |
| congr 16606 | Definition of congruence b... |
| divgcdcoprm0 16607 | Integers divided by gcd ar... |
| divgcdcoprmex 16608 | Integers divided by gcd ar... |
| cncongr1 16609 | One direction of the bicon... |
| cncongr2 16610 | The other direction of the... |
| cncongr 16611 | Cancellability of Congruen... |
| cncongrcoprm 16612 | Corollary 1 of Cancellabil... |
| isprm 16615 | The predicate "is a prime ... |
| prmnn 16616 | A prime number is a positi... |
| prmz 16617 | A prime number is an integ... |
| prmssnn 16618 | The prime numbers are a su... |
| prmex 16619 | The set of prime numbers e... |
| 0nprm 16620 | 0 is not a prime number. ... |
| 1nprm 16621 | 1 is not a prime number. ... |
| 1idssfct 16622 | The positive divisors of a... |
| isprm2lem 16623 | Lemma for ~ isprm2 . (Con... |
| isprm2 16624 | The predicate "is a prime ... |
| isprm3 16625 | The predicate "is a prime ... |
| isprm4 16626 | The predicate "is a prime ... |
| prmind2 16627 | A variation on ~ prmind as... |
| prmind 16628 | Perform induction over the... |
| dvdsprime 16629 | If ` M ` divides a prime, ... |
| nprm 16630 | A product of two integers ... |
| nprmi 16631 | An inference for composite... |
| dvdsnprmd 16632 | If a number is divisible b... |
| prm2orodd 16633 | A prime number is either 2... |
| 2prm 16634 | 2 is a prime number. (Con... |
| 2mulprm 16635 | A multiple of two is prime... |
| 3prm 16636 | 3 is a prime number. (Con... |
| 4nprm 16637 | 4 is not a prime number. ... |
| prmuz2 16638 | A prime number is an integ... |
| prmgt1 16639 | A prime number is an integ... |
| prmm2nn0 16640 | Subtracting 2 from a prime... |
| oddprmgt2 16641 | An odd prime is greater th... |
| oddprmge3 16642 | An odd prime is greater th... |
| ge2nprmge4 16643 | A composite integer greate... |
| sqnprm 16644 | A square is never prime. ... |
| dvdsprm 16645 | An integer greater than or... |
| exprmfct 16646 | Every integer greater than... |
| prmdvdsfz 16647 | Each integer greater than ... |
| nprmdvds1 16648 | No prime number divides 1.... |
| isprm5 16649 | One need only check prime ... |
| isprm7 16650 | One need only check prime ... |
| maxprmfct 16651 | The set of prime factors o... |
| divgcdodd 16652 | Either ` A / ( A gcd B ) `... |
| coprm 16653 | A prime number either divi... |
| prmrp 16654 | Unequal prime numbers are ... |
| euclemma 16655 | Euclid's lemma. A prime n... |
| isprm6 16656 | A number is prime iff it s... |
| prmdvdsexp 16657 | A prime divides a positive... |
| prmdvdsexpb 16658 | A prime divides a positive... |
| prmdvdsexpr 16659 | If a prime divides a nonne... |
| prmdvdssq 16660 | Condition for a prime divi... |
| prmdvdssqOLD 16661 | Obsolete version of ~ prmd... |
| prmexpb 16662 | Two positive prime powers ... |
| prmfac1 16663 | The factorial of a number ... |
| rpexp 16664 | If two numbers ` A ` and `... |
| rpexp1i 16665 | Relative primality passes ... |
| rpexp12i 16666 | Relative primality passes ... |
| prmndvdsfaclt 16667 | A prime number does not di... |
| prmdvdsncoprmbd 16668 | Two positive integers are ... |
| ncoprmlnprm 16669 | If two positive integers a... |
| cncongrprm 16670 | Corollary 2 of Cancellabil... |
| isevengcd2 16671 | The predicate "is an even ... |
| isoddgcd1 16672 | The predicate "is an odd n... |
| 3lcm2e6 16673 | The least common multiple ... |
| qnumval 16678 | Value of the canonical num... |
| qdenval 16679 | Value of the canonical den... |
| qnumdencl 16680 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16681 | The canonical numerator of... |
| qdencl 16682 | The canonical denominator ... |
| fnum 16683 | Canonical numerator define... |
| fden 16684 | Canonical denominator defi... |
| qnumdenbi 16685 | Two numbers are the canoni... |
| qnumdencoprm 16686 | The canonical representati... |
| qeqnumdivden 16687 | Recover a rational number ... |
| qmuldeneqnum 16688 | Multiplying a rational by ... |
| divnumden 16689 | Calculate the reduced form... |
| divdenle 16690 | Reducing a quotient never ... |
| qnumgt0 16691 | A rational is positive iff... |
| qgt0numnn 16692 | A rational is positive iff... |
| nn0gcdsq 16693 | Squaring commutes with GCD... |
| zgcdsq 16694 | ~ nn0gcdsq extended to int... |
| numdensq 16695 | Squaring a rational square... |
| numsq 16696 | Square commutes with canon... |
| densq 16697 | Square commutes with canon... |
| qden1elz 16698 | A rational is an integer i... |
| zsqrtelqelz 16699 | If an integer has a ration... |
| nonsq 16700 | Any integer strictly betwe... |
| phival 16705 | Value of the Euler ` phi `... |
| phicl2 16706 | Bounds and closure for the... |
| phicl 16707 | Closure for the value of t... |
| phibndlem 16708 | Lemma for ~ phibnd . (Con... |
| phibnd 16709 | A slightly tighter bound o... |
| phicld 16710 | Closure for the value of t... |
| phi1 16711 | Value of the Euler ` phi `... |
| dfphi2 16712 | Alternate definition of th... |
| hashdvds 16713 | The number of numbers in a... |
| phiprmpw 16714 | Value of the Euler ` phi `... |
| phiprm 16715 | Value of the Euler ` phi `... |
| crth 16716 | The Chinese Remainder Theo... |
| phimullem 16717 | Lemma for ~ phimul . (Con... |
| phimul 16718 | The Euler ` phi ` function... |
| eulerthlem1 16719 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16720 | Lemma for ~ eulerth . (Co... |
| eulerth 16721 | Euler's theorem, a general... |
| fermltl 16722 | Fermat's little theorem. ... |
| prmdiv 16723 | Show an explicit expressio... |
| prmdiveq 16724 | The modular inverse of ` A... |
| prmdivdiv 16725 | The (modular) inverse of t... |
| hashgcdlem 16726 | A correspondence between e... |
| hashgcdeq 16727 | Number of initial positive... |
| phisum 16728 | The divisor sum identity o... |
| odzval 16729 | Value of the order functio... |
| odzcllem 16730 | - Lemma for ~ odzcl , show... |
| odzcl 16731 | The order of a group eleme... |
| odzid 16732 | Any element raised to the ... |
| odzdvds 16733 | The only powers of ` A ` t... |
| odzphi 16734 | The order of any group ele... |
| modprm1div 16735 | A prime number divides an ... |
| m1dvdsndvds 16736 | If an integer minus 1 is d... |
| modprminv 16737 | Show an explicit expressio... |
| modprminveq 16738 | The modular inverse of ` A... |
| vfermltl 16739 | Variant of Fermat's little... |
| vfermltlALT 16740 | Alternate proof of ~ vferm... |
| powm2modprm 16741 | If an integer minus 1 is d... |
| reumodprminv 16742 | For any prime number and f... |
| modprm0 16743 | For two positive integers ... |
| nnnn0modprm0 16744 | For a positive integer and... |
| modprmn0modprm0 16745 | For an integer not being 0... |
| coprimeprodsq 16746 | If three numbers are copri... |
| coprimeprodsq2 16747 | If three numbers are copri... |
| oddprm 16748 | A prime not equal to ` 2 `... |
| nnoddn2prm 16749 | A prime not equal to ` 2 `... |
| oddn2prm 16750 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16751 | A number is a prime number... |
| prm23lt5 16752 | A prime less than 5 is eit... |
| prm23ge5 16753 | A prime is either 2 or 3 o... |
| pythagtriplem1 16754 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16755 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16756 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16757 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16758 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16759 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16760 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16761 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16762 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16763 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16764 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16765 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16766 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16767 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16768 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16769 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16770 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16771 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16772 | Parameterize the Pythagore... |
| iserodd 16773 | Collect the odd terms in a... |
| pclem 16776 | - Lemma for the prime powe... |
| pcprecl 16777 | Closure of the prime power... |
| pcprendvds 16778 | Non-divisibility property ... |
| pcprendvds2 16779 | Non-divisibility property ... |
| pcpre1 16780 | Value of the prime power p... |
| pcpremul 16781 | Multiplicative property of... |
| pcval 16782 | The value of the prime pow... |
| pceulem 16783 | Lemma for ~ pceu . (Contr... |
| pceu 16784 | Uniqueness for the prime p... |
| pczpre 16785 | Connect the prime count pr... |
| pczcl 16786 | Closure of the prime power... |
| pccl 16787 | Closure of the prime power... |
| pccld 16788 | Closure of the prime power... |
| pcmul 16789 | Multiplication property of... |
| pcdiv 16790 | Division property of the p... |
| pcqmul 16791 | Multiplication property of... |
| pc0 16792 | The value of the prime pow... |
| pc1 16793 | Value of the prime count f... |
| pcqcl 16794 | Closure of the general pri... |
| pcqdiv 16795 | Division property of the p... |
| pcrec 16796 | Prime power of a reciproca... |
| pcexp 16797 | Prime power of an exponent... |
| pcxnn0cl 16798 | Extended nonnegative integ... |
| pcxcl 16799 | Extended real closure of t... |
| pcge0 16800 | The prime count of an inte... |
| pczdvds 16801 | Defining property of the p... |
| pcdvds 16802 | Defining property of the p... |
| pczndvds 16803 | Defining property of the p... |
| pcndvds 16804 | Defining property of the p... |
| pczndvds2 16805 | The remainder after dividi... |
| pcndvds2 16806 | The remainder after dividi... |
| pcdvdsb 16807 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16808 | There are a positive numbe... |
| pceq0 16809 | There are zero powers of a... |
| pcidlem 16810 | The prime count of a prime... |
| pcid 16811 | The prime count of a prime... |
| pcneg 16812 | The prime count of a negat... |
| pcabs 16813 | The prime count of an abso... |
| pcdvdstr 16814 | The prime count increases ... |
| pcgcd1 16815 | The prime count of a GCD i... |
| pcgcd 16816 | The prime count of a GCD i... |
| pc2dvds 16817 | A characterization of divi... |
| pc11 16818 | The prime count function, ... |
| pcz 16819 | The prime count function c... |
| pcprmpw2 16820 | Self-referential expressio... |
| pcprmpw 16821 | Self-referential expressio... |
| dvdsprmpweq 16822 | If a positive integer divi... |
| dvdsprmpweqnn 16823 | If an integer greater than... |
| dvdsprmpweqle 16824 | If a positive integer divi... |
| difsqpwdvds 16825 | If the difference of two s... |
| pcaddlem 16826 | Lemma for ~ pcadd . The o... |
| pcadd 16827 | An inequality for the prim... |
| pcadd2 16828 | The inequality of ~ pcadd ... |
| pcmptcl 16829 | Closure for the prime powe... |
| pcmpt 16830 | Construct a function with ... |
| pcmpt2 16831 | Dividing two prime count m... |
| pcmptdvds 16832 | The partial products of th... |
| pcprod 16833 | The product of the primes ... |
| sumhash 16834 | The sum of 1 over a set is... |
| fldivp1 16835 | The difference between the... |
| pcfaclem 16836 | Lemma for ~ pcfac . (Cont... |
| pcfac 16837 | Calculate the prime count ... |
| pcbc 16838 | Calculate the prime count ... |
| qexpz 16839 | If a power of a rational n... |
| expnprm 16840 | A second or higher power o... |
| oddprmdvds 16841 | Every positive integer whi... |
| prmpwdvds 16842 | A relation involving divis... |
| pockthlem 16843 | Lemma for ~ pockthg . (Co... |
| pockthg 16844 | The generalized Pocklingto... |
| pockthi 16845 | Pocklington's theorem, whi... |
| unbenlem 16846 | Lemma for ~ unben . (Cont... |
| unben 16847 | An unbounded set of positi... |
| infpnlem1 16848 | Lemma for ~ infpn . The s... |
| infpnlem2 16849 | Lemma for ~ infpn . For a... |
| infpn 16850 | There exist infinitely man... |
| infpn2 16851 | There exist infinitely man... |
| prmunb 16852 | The primes are unbounded. ... |
| prminf 16853 | There are an infinite numb... |
| prmreclem1 16854 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16855 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16856 | Lemma for ~ prmrec . The ... |
| prmreclem4 16857 | Lemma for ~ prmrec . Show... |
| prmreclem5 16858 | Lemma for ~ prmrec . Here... |
| prmreclem6 16859 | Lemma for ~ prmrec . If t... |
| prmrec 16860 | The sum of the reciprocals... |
| 1arithlem1 16861 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16862 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16863 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16864 | Lemma for ~ 1arith . (Con... |
| 1arith 16865 | Fundamental theorem of ari... |
| 1arith2 16866 | Fundamental theorem of ari... |
| elgz 16869 | Elementhood in the gaussia... |
| gzcn 16870 | A gaussian integer is a co... |
| zgz 16871 | An integer is a gaussian i... |
| igz 16872 | ` _i ` is a gaussian integ... |
| gznegcl 16873 | The gaussian integers are ... |
| gzcjcl 16874 | The gaussian integers are ... |
| gzaddcl 16875 | The gaussian integers are ... |
| gzmulcl 16876 | The gaussian integers are ... |
| gzreim 16877 | Construct a gaussian integ... |
| gzsubcl 16878 | The gaussian integers are ... |
| gzabssqcl 16879 | The squared norm of a gaus... |
| 4sqlem5 16880 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16881 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16882 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16883 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16884 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16885 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16886 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16887 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16888 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16889 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16890 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16891 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16892 | Euler's four-square identi... |
| 4sqlem11 16893 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16894 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16895 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16896 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16897 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16898 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16899 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16900 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16901 | Lemma for ~ 4sq . The pro... |
| 4sq 16902 | Lagrange's four-square the... |
| vdwapfval 16909 | Define the arithmetic prog... |
| vdwapf 16910 | The arithmetic progression... |
| vdwapval 16911 | Value of the arithmetic pr... |
| vdwapun 16912 | Remove the first element o... |
| vdwapid1 16913 | The first element of an ar... |
| vdwap0 16914 | Value of a length-1 arithm... |
| vdwap1 16915 | Value of a length-1 arithm... |
| vdwmc 16916 | The predicate " The ` <. R... |
| vdwmc2 16917 | Expand out the definition ... |
| vdwpc 16918 | The predicate " The colori... |
| vdwlem1 16919 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16920 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16921 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16922 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16923 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16924 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16925 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16926 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16927 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16928 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16929 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16930 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16931 | Lemma for ~ vdw . Main in... |
| vdw 16932 | Van der Waerden's theorem.... |
| vdwnnlem1 16933 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16934 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16935 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16936 | Van der Waerden's theorem,... |
| ramtlecl 16938 | The set ` T ` of numbers w... |
| hashbcval 16940 | Value of the "binomial set... |
| hashbccl 16941 | The binomial set is a fini... |
| hashbcss 16942 | Subset relation for the bi... |
| hashbc0 16943 | The set of subsets of size... |
| hashbc2 16944 | The size of the binomial s... |
| 0hashbc 16945 | There are no subsets of th... |
| ramval 16946 | The value of the Ramsey nu... |
| ramcl2lem 16947 | Lemma for extended real cl... |
| ramtcl 16948 | The Ramsey number has the ... |
| ramtcl2 16949 | The Ramsey number is an in... |
| ramtub 16950 | The Ramsey number is a low... |
| ramub 16951 | The Ramsey number is a low... |
| ramub2 16952 | It is sufficient to check ... |
| rami 16953 | The defining property of a... |
| ramcl2 16954 | The Ramsey number is eithe... |
| ramxrcl 16955 | The Ramsey number is an ex... |
| ramubcl 16956 | If the Ramsey number is up... |
| ramlb 16957 | Establish a lower bound on... |
| 0ram 16958 | The Ramsey number when ` M... |
| 0ram2 16959 | The Ramsey number when ` M... |
| ram0 16960 | The Ramsey number when ` R... |
| 0ramcl 16961 | Lemma for ~ ramcl : Exist... |
| ramz2 16962 | The Ramsey number when ` F... |
| ramz 16963 | The Ramsey number when ` F... |
| ramub1lem1 16964 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16965 | Lemma for ~ ramub1 . (Con... |
| ramub1 16966 | Inductive step for Ramsey'... |
| ramcl 16967 | Ramsey's theorem: the Rams... |
| ramsey 16968 | Ramsey's theorem with the ... |
| prmoval 16971 | Value of the primorial fun... |
| prmocl 16972 | Closure of the primorial f... |
| prmone0 16973 | The primorial function is ... |
| prmo0 16974 | The primorial of 0. (Cont... |
| prmo1 16975 | The primorial of 1. (Cont... |
| prmop1 16976 | The primorial of a success... |
| prmonn2 16977 | Value of the primorial fun... |
| prmo2 16978 | The primorial of 2. (Cont... |
| prmo3 16979 | The primorial of 3. (Cont... |
| prmdvdsprmo 16980 | The primorial of a number ... |
| prmdvdsprmop 16981 | The primorial of a number ... |
| fvprmselelfz 16982 | The value of the prime sel... |
| fvprmselgcd1 16983 | The greatest common diviso... |
| prmolefac 16984 | The primorial of a positiv... |
| prmodvdslcmf 16985 | The primorial of a nonnega... |
| prmolelcmf 16986 | The primorial of a positiv... |
| prmgaplem1 16987 | Lemma for ~ prmgap : The ... |
| prmgaplem2 16988 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 16989 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 16990 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 16991 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 16992 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 16993 | Lemma for ~ prmgap : for e... |
| prmgaplem6 16994 | Lemma for ~ prmgap : for e... |
| prmgaplem7 16995 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 16996 | Lemma for ~ prmgap . (Con... |
| prmgap 16997 | The prime gap theorem: for... |
| prmgaplcm 16998 | Alternate proof of ~ prmga... |
| prmgapprmolem 16999 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 17000 | Alternate proof of ~ prmga... |
| dec2dvds 17001 | Divisibility by two is obv... |
| dec5dvds 17002 | Divisibility by five is ob... |
| dec5dvds2 17003 | Divisibility by five is ob... |
| dec5nprm 17004 | Divisibility by five is ob... |
| dec2nprm 17005 | Divisibility by two is obv... |
| modxai 17006 | Add exponents in a power m... |
| mod2xi 17007 | Double exponents in a powe... |
| modxp1i 17008 | Add one to an exponent in ... |
| mod2xnegi 17009 | Version of ~ mod2xi with a... |
| modsubi 17010 | Subtract from within a mod... |
| gcdi 17011 | Calculate a GCD via Euclid... |
| gcdmodi 17012 | Calculate a GCD via Euclid... |
| decexp2 17013 | Calculate a power of two. ... |
| numexp0 17014 | Calculate an integer power... |
| numexp1 17015 | Calculate an integer power... |
| numexpp1 17016 | Calculate an integer power... |
| numexp2x 17017 | Double an integer power. ... |
| decsplit0b 17018 | Split a decimal number int... |
| decsplit0 17019 | Split a decimal number int... |
| decsplit1 17020 | Split a decimal number int... |
| decsplit 17021 | Split a decimal number int... |
| karatsuba 17022 | The Karatsuba multiplicati... |
| 2exp4 17023 | Two to the fourth power is... |
| 2exp5 17024 | Two to the fifth power is ... |
| 2exp6 17025 | Two to the sixth power is ... |
| 2exp7 17026 | Two to the seventh power i... |
| 2exp8 17027 | Two to the eighth power is... |
| 2exp11 17028 | Two to the eleventh power ... |
| 2exp16 17029 | Two to the sixteenth power... |
| 3exp3 17030 | Three to the third power i... |
| 2expltfac 17031 | The factorial grows faster... |
| cshwsidrepsw 17032 | If cyclically shifting a w... |
| cshwsidrepswmod0 17033 | If cyclically shifting a w... |
| cshwshashlem1 17034 | If cyclically shifting a w... |
| cshwshashlem2 17035 | If cyclically shifting a w... |
| cshwshashlem3 17036 | If cyclically shifting a w... |
| cshwsdisj 17037 | The singletons resulting b... |
| cshwsiun 17038 | The set of (different!) wo... |
| cshwsex 17039 | The class of (different!) ... |
| cshws0 17040 | The size of the set of (di... |
| cshwrepswhash1 17041 | The size of the set of (di... |
| cshwshashnsame 17042 | If a word (not consisting ... |
| cshwshash 17043 | If a word has a length bei... |
| prmlem0 17044 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 17045 | A quick proof skeleton to ... |
| prmlem1 17046 | A quick proof skeleton to ... |
| 5prm 17047 | 5 is a prime number. (Con... |
| 6nprm 17048 | 6 is not a prime number. ... |
| 7prm 17049 | 7 is a prime number. (Con... |
| 8nprm 17050 | 8 is not a prime number. ... |
| 9nprm 17051 | 9 is not a prime number. ... |
| 10nprm 17052 | 10 is not a prime number. ... |
| 11prm 17053 | 11 is a prime number. (Co... |
| 13prm 17054 | 13 is a prime number. (Co... |
| 17prm 17055 | 17 is a prime number. (Co... |
| 19prm 17056 | 19 is a prime number. (Co... |
| 23prm 17057 | 23 is a prime number. (Co... |
| prmlem2 17058 | Our last proving session g... |
| 37prm 17059 | 37 is a prime number. (Co... |
| 43prm 17060 | 43 is a prime number. (Co... |
| 83prm 17061 | 83 is a prime number. (Co... |
| 139prm 17062 | 139 is a prime number. (C... |
| 163prm 17063 | 163 is a prime number. (C... |
| 317prm 17064 | 317 is a prime number. (C... |
| 631prm 17065 | 631 is a prime number. (C... |
| prmo4 17066 | The primorial of 4. (Cont... |
| prmo5 17067 | The primorial of 5. (Cont... |
| prmo6 17068 | The primorial of 6. (Cont... |
| 1259lem1 17069 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17070 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17071 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17072 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17073 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17074 | 1259 is a prime number. (... |
| 2503lem1 17075 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17076 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17077 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17078 | 2503 is a prime number. (... |
| 4001lem1 17079 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17080 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17081 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17082 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17083 | 4001 is a prime number. (... |
| brstruct 17086 | The structure relation is ... |
| isstruct2 17087 | The property of being a st... |
| structex 17088 | A structure is a set. (Co... |
| structn0fun 17089 | A structure without the em... |
| isstruct 17090 | The property of being a st... |
| structcnvcnv 17091 | Two ways to express the re... |
| structfung 17092 | The converse of the conver... |
| structfun 17093 | Convert between two kinds ... |
| structfn 17094 | Convert between two kinds ... |
| strleun 17095 | Combine two structures int... |
| strle1 17096 | Make a structure from a si... |
| strle2 17097 | Make a structure from a pa... |
| strle3 17098 | Make a structure from a tr... |
| sbcie2s 17099 | A special version of class... |
| sbcie3s 17100 | A special version of class... |
| reldmsets 17103 | The structure override ope... |
| setsvalg 17104 | Value of the structure rep... |
| setsval 17105 | Value of the structure rep... |
| fvsetsid 17106 | The value of the structure... |
| fsets 17107 | The structure replacement ... |
| setsdm 17108 | The domain of a structure ... |
| setsfun 17109 | A structure with replaceme... |
| setsfun0 17110 | A structure with replaceme... |
| setsn0fun 17111 | The value of the structure... |
| setsstruct2 17112 | An extensible structure wi... |
| setsexstruct2 17113 | An extensible structure wi... |
| setsstruct 17114 | An extensible structure wi... |
| wunsets 17115 | Closure of structure repla... |
| setsres 17116 | The structure replacement ... |
| setsabs 17117 | Replacing the same compone... |
| setscom 17118 | Different components can b... |
| sloteq 17121 | Equality theorem for the `... |
| slotfn 17122 | A slot is a function on se... |
| strfvnd 17123 | Deduction version of ~ str... |
| strfvn 17124 | Value of a structure compo... |
| strfvss 17125 | A structure component extr... |
| wunstr 17126 | Closure of a structure ind... |
| str0 17127 | All components of the empt... |
| strfvi 17128 | Structure slot extractors ... |
| fveqprc 17129 | Lemma for showing the equa... |
| oveqprc 17130 | Lemma for showing the equa... |
| wunndx 17133 | Closure of the index extra... |
| ndxarg 17134 | Get the numeric argument f... |
| ndxid 17135 | A structure component extr... |
| strndxid 17136 | The value of a structure c... |
| setsidvald 17137 | Value of the structure rep... |
| setsidvaldOLD 17138 | Obsolete version of ~ sets... |
| strfvd 17139 | Deduction version of ~ str... |
| strfv2d 17140 | Deduction version of ~ str... |
| strfv2 17141 | A variation on ~ strfv to ... |
| strfv 17142 | Extract a structure compon... |
| strfv3 17143 | Variant on ~ strfv for lar... |
| strssd 17144 | Deduction version of ~ str... |
| strss 17145 | Propagate component extrac... |
| setsid 17146 | Value of the structure rep... |
| setsnid 17147 | Value of the structure rep... |
| setsnidOLD 17148 | Obsolete proof of ~ setsni... |
| baseval 17151 | Value of the base set extr... |
| baseid 17152 | Utility theorem: index-ind... |
| basfn 17153 | The base set extractor is ... |
| base0 17154 | The base set of the empty ... |
| elbasfv 17155 | Utility theorem: reverse c... |
| elbasov 17156 | Utility theorem: reverse c... |
| strov2rcl 17157 | Partial reverse closure fo... |
| basendx 17158 | Index value of the base se... |
| basendxnn 17159 | The index value of the bas... |
| basendxnnOLD 17160 | Obsolete proof of ~ basend... |
| basndxelwund 17161 | The index of the base set ... |
| basprssdmsets 17162 | The pair of the base index... |
| opelstrbas 17163 | The base set of a structur... |
| 1strstr 17164 | A constructed one-slot str... |
| 1strstr1 17165 | A constructed one-slot str... |
| 1strbas 17166 | The base set of a construc... |
| 1strbasOLD 17167 | Obsolete proof of ~ 1strba... |
| 1strwunbndx 17168 | A constructed one-slot str... |
| 1strwun 17169 | A constructed one-slot str... |
| 1strwunOLD 17170 | Obsolete version of ~ 1str... |
| 2strstr 17171 | A constructed two-slot str... |
| 2strbas 17172 | The base set of a construc... |
| 2strop 17173 | The other slot of a constr... |
| 2strstr1 17174 | A constructed two-slot str... |
| 2strstr1OLD 17175 | Obsolete version of ~ 2str... |
| 2strbas1 17176 | The base set of a construc... |
| 2strop1 17177 | The other slot of a constr... |
| reldmress 17180 | The structure restriction ... |
| ressval 17181 | Value of structure restric... |
| ressid2 17182 | General behavior of trivia... |
| ressval2 17183 | Value of nontrivial struct... |
| ressbas 17184 | Base set of a structure re... |
| ressbasOLD 17185 | Obsolete proof of ~ ressba... |
| ressbasssg 17186 | The base set of a restrict... |
| ressbas2 17187 | Base set of a structure re... |
| ressbasss 17188 | The base set of a restrict... |
| ressbasssOLD 17189 | Obsolete proof of ~ ressba... |
| ressbasss2 17190 | The base set of a restrict... |
| resseqnbas 17191 | The components of an exten... |
| resslemOLD 17192 | Obsolete version of ~ ress... |
| ress0 17193 | All restrictions of the nu... |
| ressid 17194 | Behavior of trivial restri... |
| ressinbas 17195 | Restriction only cares abo... |
| ressval3d 17196 | Value of structure restric... |
| ressval3dOLD 17197 | Obsolete version of ~ ress... |
| ressress 17198 | Restriction composition la... |
| ressabs 17199 | Restriction absorption law... |
| wunress 17200 | Closure of structure restr... |
| wunressOLD 17201 | Obsolete proof of ~ wunres... |
| plusgndx 17228 | Index value of the ~ df-pl... |
| plusgid 17229 | Utility theorem: index-ind... |
| plusgndxnn 17230 | The index of the slot for ... |
| basendxltplusgndx 17231 | The index of the slot for ... |
| basendxnplusgndx 17232 | The slot for the base set ... |
| basendxnplusgndxOLD 17233 | Obsolete version of ~ base... |
| grpstr 17234 | A constructed group is a s... |
| grpstrndx 17235 | A constructed group is a s... |
| grpbase 17236 | The base set of a construc... |
| grpbaseOLD 17237 | Obsolete version of ~ grpb... |
| grpplusg 17238 | The operation of a constru... |
| grpplusgOLD 17239 | Obsolete version of ~ grpp... |
| ressplusg 17240 | ` +g ` is unaffected by re... |
| grpbasex 17241 | The base of an explicitly ... |
| grpplusgx 17242 | The operation of an explic... |
| mulrndx 17243 | Index value of the ~ df-mu... |
| mulridx 17244 | Utility theorem: index-ind... |
| basendxnmulrndx 17245 | The slot for the base set ... |
| basendxnmulrndxOLD 17246 | Obsolete proof of ~ basend... |
| plusgndxnmulrndx 17247 | The slot for the group (ad... |
| rngstr 17248 | A constructed ring is a st... |
| rngbase 17249 | The base set of a construc... |
| rngplusg 17250 | The additive operation of ... |
| rngmulr 17251 | The multiplicative operati... |
| starvndx 17252 | Index value of the ~ df-st... |
| starvid 17253 | Utility theorem: index-ind... |
| starvndxnbasendx 17254 | The slot for the involutio... |
| starvndxnplusgndx 17255 | The slot for the involutio... |
| starvndxnmulrndx 17256 | The slot for the involutio... |
| ressmulr 17257 | ` .r ` is unaffected by re... |
| ressstarv 17258 | ` *r ` is unaffected by re... |
| srngstr 17259 | A constructed star ring is... |
| srngbase 17260 | The base set of a construc... |
| srngplusg 17261 | The addition operation of ... |
| srngmulr 17262 | The multiplication operati... |
| srnginvl 17263 | The involution function of... |
| scandx 17264 | Index value of the ~ df-sc... |
| scaid 17265 | Utility theorem: index-ind... |
| scandxnbasendx 17266 | The slot for the scalar is... |
| scandxnplusgndx 17267 | The slot for the scalar fi... |
| scandxnmulrndx 17268 | The slot for the scalar fi... |
| vscandx 17269 | Index value of the ~ df-vs... |
| vscaid 17270 | Utility theorem: index-ind... |
| vscandxnbasendx 17271 | The slot for the scalar pr... |
| vscandxnplusgndx 17272 | The slot for the scalar pr... |
| vscandxnmulrndx 17273 | The slot for the scalar pr... |
| vscandxnscandx 17274 | The slot for the scalar pr... |
| lmodstr 17275 | A constructed left module ... |
| lmodbase 17276 | The base set of a construc... |
| lmodplusg 17277 | The additive operation of ... |
| lmodsca 17278 | The set of scalars of a co... |
| lmodvsca 17279 | The scalar product operati... |
| ipndx 17280 | Index value of the ~ df-ip... |
| ipid 17281 | Utility theorem: index-ind... |
| ipndxnbasendx 17282 | The slot for the inner pro... |
| ipndxnplusgndx 17283 | The slot for the inner pro... |
| ipndxnmulrndx 17284 | The slot for the inner pro... |
| slotsdifipndx 17285 | The slot for the scalar is... |
| ipsstr 17286 | Lemma to shorten proofs of... |
| ipsbase 17287 | The base set of a construc... |
| ipsaddg 17288 | The additive operation of ... |
| ipsmulr 17289 | The multiplicative operati... |
| ipssca 17290 | The set of scalars of a co... |
| ipsvsca 17291 | The scalar product operati... |
| ipsip 17292 | The multiplicative operati... |
| resssca 17293 | ` Scalar ` is unaffected b... |
| ressvsca 17294 | ` .s ` is unaffected by re... |
| ressip 17295 | The inner product is unaff... |
| phlstr 17296 | A constructed pre-Hilbert ... |
| phlbase 17297 | The base set of a construc... |
| phlplusg 17298 | The additive operation of ... |
| phlsca 17299 | The ring of scalars of a c... |
| phlvsca 17300 | The scalar product operati... |
| phlip 17301 | The inner product (Hermiti... |
| tsetndx 17302 | Index value of the ~ df-ts... |
| tsetid 17303 | Utility theorem: index-ind... |
| tsetndxnn 17304 | The index of the slot for ... |
| basendxlttsetndx 17305 | The index of the slot for ... |
| tsetndxnbasendx 17306 | The slot for the topology ... |
| tsetndxnplusgndx 17307 | The slot for the topology ... |
| tsetndxnmulrndx 17308 | The slot for the topology ... |
| tsetndxnstarvndx 17309 | The slot for the topology ... |
| slotstnscsi 17310 | The slots ` Scalar ` , ` .... |
| topgrpstr 17311 | A constructed topological ... |
| topgrpbas 17312 | The base set of a construc... |
| topgrpplusg 17313 | The additive operation of ... |
| topgrptset 17314 | The topology of a construc... |
| resstset 17315 | ` TopSet ` is unaffected b... |
| plendx 17316 | Index value of the ~ df-pl... |
| pleid 17317 | Utility theorem: self-refe... |
| plendxnn 17318 | The index value of the ord... |
| basendxltplendx 17319 | The index value of the ` B... |
| plendxnbasendx 17320 | The slot for the order is ... |
| plendxnplusgndx 17321 | The slot for the "less tha... |
| plendxnmulrndx 17322 | The slot for the "less tha... |
| plendxnscandx 17323 | The slot for the "less tha... |
| plendxnvscandx 17324 | The slot for the "less tha... |
| slotsdifplendx 17325 | The index of the slot for ... |
| otpsstr 17326 | Functionality of a topolog... |
| otpsbas 17327 | The base set of a topologi... |
| otpstset 17328 | The open sets of a topolog... |
| otpsle 17329 | The order of a topological... |
| ressle 17330 | ` le ` is unaffected by re... |
| ocndx 17331 | Index value of the ~ df-oc... |
| ocid 17332 | Utility theorem: index-ind... |
| basendxnocndx 17333 | The slot for the orthocomp... |
| plendxnocndx 17334 | The slot for the orthocomp... |
| dsndx 17335 | Index value of the ~ df-ds... |
| dsid 17336 | Utility theorem: index-ind... |
| dsndxnn 17337 | The index of the slot for ... |
| basendxltdsndx 17338 | The index of the slot for ... |
| dsndxnbasendx 17339 | The slot for the distance ... |
| dsndxnplusgndx 17340 | The slot for the distance ... |
| dsndxnmulrndx 17341 | The slot for the distance ... |
| slotsdnscsi 17342 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17343 | The slot for the distance ... |
| slotsdifdsndx 17344 | The index of the slot for ... |
| unifndx 17345 | Index value of the ~ df-un... |
| unifid 17346 | Utility theorem: index-ind... |
| unifndxnn 17347 | The index of the slot for ... |
| basendxltunifndx 17348 | The index of the slot for ... |
| unifndxnbasendx 17349 | The slot for the uniform s... |
| unifndxntsetndx 17350 | The slot for the uniform s... |
| slotsdifunifndx 17351 | The index of the slot for ... |
| ressunif 17352 | ` UnifSet ` is unaffected ... |
| odrngstr 17353 | Functionality of an ordere... |
| odrngbas 17354 | The base set of an ordered... |
| odrngplusg 17355 | The addition operation of ... |
| odrngmulr 17356 | The multiplication operati... |
| odrngtset 17357 | The open sets of an ordere... |
| odrngle 17358 | The order of an ordered me... |
| odrngds 17359 | The metric of an ordered m... |
| ressds 17360 | ` dist ` is unaffected by ... |
| homndx 17361 | Index value of the ~ df-ho... |
| homid 17362 | Utility theorem: index-ind... |
| ccondx 17363 | Index value of the ~ df-cc... |
| ccoid 17364 | Utility theorem: index-ind... |
| slotsbhcdif 17365 | The slots ` Base ` , ` Hom... |
| slotsbhcdifOLD 17366 | Obsolete proof of ~ slotsb... |
| slotsdifplendx2 17367 | The index of the slot for ... |
| slotsdifocndx 17368 | The index of the slot for ... |
| resshom 17369 | ` Hom ` is unaffected by r... |
| ressco 17370 | ` comp ` is unaffected by ... |
| restfn 17375 | The subspace topology oper... |
| topnfn 17376 | The topology extractor fun... |
| restval 17377 | The subspace topology indu... |
| elrest 17378 | The predicate "is an open ... |
| elrestr 17379 | Sufficient condition for b... |
| 0rest 17380 | Value of the structure res... |
| restid2 17381 | The subspace topology over... |
| restsspw 17382 | The subspace topology is a... |
| firest 17383 | The finite intersections o... |
| restid 17384 | The subspace topology of t... |
| topnval 17385 | Value of the topology extr... |
| topnid 17386 | Value of the topology extr... |
| topnpropd 17387 | The topology extractor fun... |
| reldmprds 17399 | The structure product is a... |
| prdsbasex 17401 | Lemma for structure produc... |
| imasvalstr 17402 | An image structure value i... |
| prdsvalstr 17403 | Structure product value is... |
| prdsbaslem 17404 | Lemma for ~ prdsbas and si... |
| prdsvallem 17405 | Lemma for ~ prdsval . (Co... |
| prdsval 17406 | Value of the structure pro... |
| prdssca 17407 | Scalar ring of a structure... |
| prdsbas 17408 | Base set of a structure pr... |
| prdsplusg 17409 | Addition in a structure pr... |
| prdsmulr 17410 | Multiplication in a struct... |
| prdsvsca 17411 | Scalar multiplication in a... |
| prdsip 17412 | Inner product in a structu... |
| prdsle 17413 | Structure product weak ord... |
| prdsless 17414 | Closure of the order relat... |
| prdsds 17415 | Structure product distance... |
| prdsdsfn 17416 | Structure product distance... |
| prdstset 17417 | Structure product topology... |
| prdshom 17418 | Structure product hom-sets... |
| prdsco 17419 | Structure product composit... |
| prdsbas2 17420 | The base set of a structur... |
| prdsbasmpt 17421 | A constructed tuple is a p... |
| prdsbasfn 17422 | Points in the structure pr... |
| prdsbasprj 17423 | Each point in a structure ... |
| prdsplusgval 17424 | Value of a componentwise s... |
| prdsplusgfval 17425 | Value of a structure produ... |
| prdsmulrval 17426 | Value of a componentwise r... |
| prdsmulrfval 17427 | Value of a structure produ... |
| prdsleval 17428 | Value of the product order... |
| prdsdsval 17429 | Value of the metric in a s... |
| prdsvscaval 17430 | Scalar multiplication in a... |
| prdsvscafval 17431 | Scalar multiplication of a... |
| prdsbas3 17432 | The base set of an indexed... |
| prdsbasmpt2 17433 | A constructed tuple is a p... |
| prdsbascl 17434 | An element of the base has... |
| prdsdsval2 17435 | Value of the metric in a s... |
| prdsdsval3 17436 | Value of the metric in a s... |
| pwsval 17437 | Value of a structure power... |
| pwsbas 17438 | Base set of a structure po... |
| pwselbasb 17439 | Membership in the base set... |
| pwselbas 17440 | An element of a structure ... |
| pwsplusgval 17441 | Value of addition in a str... |
| pwsmulrval 17442 | Value of multiplication in... |
| pwsle 17443 | Ordering in a structure po... |
| pwsleval 17444 | Ordering in a structure po... |
| pwsvscafval 17445 | Scalar multiplication in a... |
| pwsvscaval 17446 | Scalar multiplication of a... |
| pwssca 17447 | The ring of scalars of a s... |
| pwsdiagel 17448 | Membership of diagonal ele... |
| pwssnf1o 17449 | Triviality of singleton po... |
| imasval 17462 | Value of an image structur... |
| imasbas 17463 | The base set of an image s... |
| imasds 17464 | The distance function of a... |
| imasdsfn 17465 | The distance function is a... |
| imasdsval 17466 | The distance function of a... |
| imasdsval2 17467 | The distance function of a... |
| imasplusg 17468 | The group operation in an ... |
| imasmulr 17469 | The ring multiplication in... |
| imassca 17470 | The scalar field of an ima... |
| imasvsca 17471 | The scalar multiplication ... |
| imasip 17472 | The inner product of an im... |
| imastset 17473 | The topology of an image s... |
| imasle 17474 | The ordering of an image s... |
| f1ocpbllem 17475 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17476 | An injection is compatible... |
| f1ovscpbl 17477 | An injection is compatible... |
| f1olecpbl 17478 | An injection is compatible... |
| imasaddfnlem 17479 | The image structure operat... |
| imasaddvallem 17480 | The operation of an image ... |
| imasaddflem 17481 | The image set operations a... |
| imasaddfn 17482 | The image structure's grou... |
| imasaddval 17483 | The value of an image stru... |
| imasaddf 17484 | The image structure's grou... |
| imasmulfn 17485 | The image structure's ring... |
| imasmulval 17486 | The value of an image stru... |
| imasmulf 17487 | The image structure's ring... |
| imasvscafn 17488 | The image structure's scal... |
| imasvscaval 17489 | The value of an image stru... |
| imasvscaf 17490 | The image structure's scal... |
| imasless 17491 | The order relation defined... |
| imasleval 17492 | The value of the image str... |
| qusval 17493 | Value of a quotient struct... |
| quslem 17494 | The function in ~ qusval i... |
| qusin 17495 | Restrict the equivalence r... |
| qusbas 17496 | Base set of a quotient str... |
| quss 17497 | The scalar field of a quot... |
| divsfval 17498 | Value of the function in ~... |
| ercpbllem 17499 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17500 | Translate the function com... |
| erlecpbl 17501 | Translate the relation com... |
| qusaddvallem 17502 | Value of an operation defi... |
| qusaddflem 17503 | The operation of a quotien... |
| qusaddval 17504 | The addition in a quotient... |
| qusaddf 17505 | The addition in a quotient... |
| qusmulval 17506 | The multiplication in a qu... |
| qusmulf 17507 | The multiplication in a qu... |
| fnpr2o 17508 | Function with a domain of ... |
| fnpr2ob 17509 | Biconditional version of ~... |
| fvpr0o 17510 | The value of a function wi... |
| fvpr1o 17511 | The value of a function wi... |
| fvprif 17512 | The value of the pair func... |
| xpsfrnel 17513 | Elementhood in the target ... |
| xpsfeq 17514 | A function on ` 2o ` is de... |
| xpsfrnel2 17515 | Elementhood in the target ... |
| xpscf 17516 | Equivalent condition for t... |
| xpsfval 17517 | The value of the function ... |
| xpsff1o 17518 | The function appearing in ... |
| xpsfrn 17519 | A short expression for the... |
| xpsff1o2 17520 | The function appearing in ... |
| xpsval 17521 | Value of the binary struct... |
| xpsrnbas 17522 | The indexed structure prod... |
| xpsbas 17523 | The base set of the binary... |
| xpsaddlem 17524 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17525 | Value of the addition oper... |
| xpsmul 17526 | Value of the multiplicatio... |
| xpssca 17527 | Value of the scalar field ... |
| xpsvsca 17528 | Value of the scalar multip... |
| xpsless 17529 | Closure of the ordering in... |
| xpsle 17530 | Value of the ordering in a... |
| ismre 17539 | Property of being a Moore ... |
| fnmre 17540 | The Moore collection gener... |
| mresspw 17541 | A Moore collection is a su... |
| mress 17542 | A Moore-closed subset is a... |
| mre1cl 17543 | In any Moore collection th... |
| mreintcl 17544 | A nonempty collection of c... |
| mreiincl 17545 | A nonempty indexed interse... |
| mrerintcl 17546 | The relative intersection ... |
| mreriincl 17547 | The relative intersection ... |
| mreincl 17548 | Two closed sets have a clo... |
| mreuni 17549 | Since the entire base set ... |
| mreunirn 17550 | Two ways to express the no... |
| ismred 17551 | Properties that determine ... |
| ismred2 17552 | Properties that determine ... |
| mremre 17553 | The Moore collections of s... |
| submre 17554 | The subcollection of a clo... |
| mrcflem 17555 | The domain and codomain of... |
| fnmrc 17556 | Moore-closure is a well-be... |
| mrcfval 17557 | Value of the function expr... |
| mrcf 17558 | The Moore closure is a fun... |
| mrcval 17559 | Evaluation of the Moore cl... |
| mrccl 17560 | The Moore closure of a set... |
| mrcsncl 17561 | The Moore closure of a sin... |
| mrcid 17562 | The closure of a closed se... |
| mrcssv 17563 | The closure of a set is a ... |
| mrcidb 17564 | A set is closed iff it is ... |
| mrcss 17565 | Closure preserves subset o... |
| mrcssid 17566 | The closure of a set is a ... |
| mrcidb2 17567 | A set is closed iff it con... |
| mrcidm 17568 | The closure operation is i... |
| mrcsscl 17569 | The closure is the minimal... |
| mrcuni 17570 | Idempotence of closure und... |
| mrcun 17571 | Idempotence of closure und... |
| mrcssvd 17572 | The Moore closure of a set... |
| mrcssd 17573 | Moore closure preserves su... |
| mrcssidd 17574 | A set is contained in its ... |
| mrcidmd 17575 | Moore closure is idempoten... |
| mressmrcd 17576 | In a Moore system, if a se... |
| submrc 17577 | In a closure system which ... |
| mrieqvlemd 17578 | In a Moore system, if ` Y ... |
| mrisval 17579 | Value of the set of indepe... |
| ismri 17580 | Criterion for a set to be ... |
| ismri2 17581 | Criterion for a subset of ... |
| ismri2d 17582 | Criterion for a subset of ... |
| ismri2dd 17583 | Definition of independence... |
| mriss 17584 | An independent set of a Mo... |
| mrissd 17585 | An independent set of a Mo... |
| ismri2dad 17586 | Consequence of a set in a ... |
| mrieqvd 17587 | In a Moore system, a set i... |
| mrieqv2d 17588 | In a Moore system, a set i... |
| mrissmrcd 17589 | In a Moore system, if an i... |
| mrissmrid 17590 | In a Moore system, subsets... |
| mreexd 17591 | In a Moore system, the clo... |
| mreexmrid 17592 | In a Moore system whose cl... |
| mreexexlemd 17593 | This lemma is used to gene... |
| mreexexlem2d 17594 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17595 | Base case of the induction... |
| mreexexlem4d 17596 | Induction step of the indu... |
| mreexexd 17597 | Exchange-type theorem. In... |
| mreexdomd 17598 | In a Moore system whose cl... |
| mreexfidimd 17599 | In a Moore system whose cl... |
| isacs 17600 | A set is an algebraic clos... |
| acsmre 17601 | Algebraic closure systems ... |
| isacs2 17602 | In the definition of an al... |
| acsfiel 17603 | A set is closed in an alge... |
| acsfiel2 17604 | A set is closed in an alge... |
| acsmred 17605 | An algebraic closure syste... |
| isacs1i 17606 | A closure system determine... |
| mreacs 17607 | Algebraicity is a composab... |
| acsfn 17608 | Algebraicity of a conditio... |
| acsfn0 17609 | Algebraicity of a point cl... |
| acsfn1 17610 | Algebraicity of a one-argu... |
| acsfn1c 17611 | Algebraicity of a one-argu... |
| acsfn2 17612 | Algebraicity of a two-argu... |
| iscat 17621 | The predicate "is a catego... |
| iscatd 17622 | Properties that determine ... |
| catidex 17623 | Each object in a category ... |
| catideu 17624 | Each object in a category ... |
| cidfval 17625 | Each object in a category ... |
| cidval 17626 | Each object in a category ... |
| cidffn 17627 | The identity arrow constru... |
| cidfn 17628 | The identity arrow operato... |
| catidd 17629 | Deduce the identity arrow ... |
| iscatd2 17630 | Version of ~ iscatd with a... |
| catidcl 17631 | Each object in a category ... |
| catlid 17632 | Left identity property of ... |
| catrid 17633 | Right identity property of... |
| catcocl 17634 | Closure of a composition a... |
| catass 17635 | Associativity of compositi... |
| catcone0 17636 | Composition of non-empty h... |
| 0catg 17637 | Any structure with an empt... |
| 0cat 17638 | The empty set is a categor... |
| homffval 17639 | Value of the functionalize... |
| fnhomeqhomf 17640 | If the Hom-set operation i... |
| homfval 17641 | Value of the functionalize... |
| homffn 17642 | The functionalized Hom-set... |
| homfeq 17643 | Condition for two categori... |
| homfeqd 17644 | If two structures have the... |
| homfeqbas 17645 | Deduce equality of base se... |
| homfeqval 17646 | Value of the functionalize... |
| comfffval 17647 | Value of the functionalize... |
| comffval 17648 | Value of the functionalize... |
| comfval 17649 | Value of the functionalize... |
| comfffval2 17650 | Value of the functionalize... |
| comffval2 17651 | Value of the functionalize... |
| comfval2 17652 | Value of the functionalize... |
| comfffn 17653 | The functionalized composi... |
| comffn 17654 | The functionalized composi... |
| comfeq 17655 | Condition for two categori... |
| comfeqd 17656 | Condition for two categori... |
| comfeqval 17657 | Equality of two compositio... |
| catpropd 17658 | Two structures with the sa... |
| cidpropd 17659 | Two structures with the sa... |
| oppcval 17662 | Value of the opposite cate... |
| oppchomfval 17663 | Hom-sets of the opposite c... |
| oppchomfvalOLD 17664 | Obsolete proof of ~ oppcho... |
| oppchom 17665 | Hom-sets of the opposite c... |
| oppccofval 17666 | Composition in the opposit... |
| oppcco 17667 | Composition in the opposit... |
| oppcbas 17668 | Base set of an opposite ca... |
| oppcbasOLD 17669 | Obsolete version of ~ oppc... |
| oppccatid 17670 | Lemma for ~ oppccat . (Co... |
| oppchomf 17671 | Hom-sets of the opposite c... |
| oppcid 17672 | Identity function of an op... |
| oppccat 17673 | An opposite category is a ... |
| 2oppcbas 17674 | The double opposite catego... |
| 2oppchomf 17675 | The double opposite catego... |
| 2oppccomf 17676 | The double opposite catego... |
| oppchomfpropd 17677 | If two categories have the... |
| oppccomfpropd 17678 | If two categories have the... |
| oppccatf 17679 | ` oppCat ` restricted to `... |
| monfval 17684 | Definition of a monomorphi... |
| ismon 17685 | Definition of a monomorphi... |
| ismon2 17686 | Write out the monomorphism... |
| monhom 17687 | A monomorphism is a morphi... |
| moni 17688 | Property of a monomorphism... |
| monpropd 17689 | If two categories have the... |
| oppcmon 17690 | A monomorphism in the oppo... |
| oppcepi 17691 | An epimorphism in the oppo... |
| isepi 17692 | Definition of an epimorphi... |
| isepi2 17693 | Write out the epimorphism ... |
| epihom 17694 | An epimorphism is a morphi... |
| epii 17695 | Property of an epimorphism... |
| sectffval 17702 | Value of the section opera... |
| sectfval 17703 | Value of the section relat... |
| sectss 17704 | The section relation is a ... |
| issect 17705 | The property " ` F ` is a ... |
| issect2 17706 | Property of being a sectio... |
| sectcan 17707 | If ` G ` is a section of `... |
| sectco 17708 | Composition of two section... |
| isofval 17709 | Function value of the func... |
| invffval 17710 | Value of the inverse relat... |
| invfval 17711 | Value of the inverse relat... |
| isinv 17712 | Value of the inverse relat... |
| invss 17713 | The inverse relation is a ... |
| invsym 17714 | The inverse relation is sy... |
| invsym2 17715 | The inverse relation is sy... |
| invfun 17716 | The inverse relation is a ... |
| isoval 17717 | The isomorphisms are the d... |
| inviso1 17718 | If ` G ` is an inverse to ... |
| inviso2 17719 | If ` G ` is an inverse to ... |
| invf 17720 | The inverse relation is a ... |
| invf1o 17721 | The inverse relation is a ... |
| invinv 17722 | The inverse of the inverse... |
| invco 17723 | The composition of two iso... |
| dfiso2 17724 | Alternate definition of an... |
| dfiso3 17725 | Alternate definition of an... |
| inveq 17726 | If there are two inverses ... |
| isofn 17727 | The function value of the ... |
| isohom 17728 | An isomorphism is a homomo... |
| isoco 17729 | The composition of two iso... |
| oppcsect 17730 | A section in the opposite ... |
| oppcsect2 17731 | A section in the opposite ... |
| oppcinv 17732 | An inverse in the opposite... |
| oppciso 17733 | An isomorphism in the oppo... |
| sectmon 17734 | If ` F ` is a section of `... |
| monsect 17735 | If ` F ` is a monomorphism... |
| sectepi 17736 | If ` F ` is a section of `... |
| episect 17737 | If ` F ` is an epimorphism... |
| sectid 17738 | The identity is a section ... |
| invid 17739 | The inverse of the identit... |
| idiso 17740 | The identity is an isomorp... |
| idinv 17741 | The inverse of the identit... |
| invisoinvl 17742 | The inverse of an isomorph... |
| invisoinvr 17743 | The inverse of an isomorph... |
| invcoisoid 17744 | The inverse of an isomorph... |
| isocoinvid 17745 | The inverse of an isomorph... |
| rcaninv 17746 | Right cancellation of an i... |
| cicfval 17749 | The set of isomorphic obje... |
| brcic 17750 | The relation "is isomorphi... |
| cic 17751 | Objects ` X ` and ` Y ` in... |
| brcici 17752 | Prove that two objects are... |
| cicref 17753 | Isomorphism is reflexive. ... |
| ciclcl 17754 | Isomorphism implies the le... |
| cicrcl 17755 | Isomorphism implies the ri... |
| cicsym 17756 | Isomorphism is symmetric. ... |
| cictr 17757 | Isomorphism is transitive.... |
| cicer 17758 | Isomorphism is an equivale... |
| sscrel 17765 | The subcategory subset rel... |
| brssc 17766 | The subcategory subset rel... |
| sscpwex 17767 | An analogue of ~ pwex for ... |
| subcrcl 17768 | Reverse closure for the su... |
| sscfn1 17769 | The subcategory subset rel... |
| sscfn2 17770 | The subcategory subset rel... |
| ssclem 17771 | Lemma for ~ ssc1 and simil... |
| isssc 17772 | Value of the subcategory s... |
| ssc1 17773 | Infer subset relation on o... |
| ssc2 17774 | Infer subset relation on m... |
| sscres 17775 | Any function restricted to... |
| sscid 17776 | The subcategory subset rel... |
| ssctr 17777 | The subcategory subset rel... |
| ssceq 17778 | The subcategory subset rel... |
| rescval 17779 | Value of the category rest... |
| rescval2 17780 | Value of the category rest... |
| rescbas 17781 | Base set of the category r... |
| rescbasOLD 17782 | Obsolete version of ~ resc... |
| reschom 17783 | Hom-sets of the category r... |
| reschomf 17784 | Hom-sets of the category r... |
| rescco 17785 | Composition in the categor... |
| resccoOLD 17786 | Obsolete proof of ~ rescco... |
| rescabs 17787 | Restriction absorption law... |
| rescabsOLD 17788 | Obsolete proof of ~ seqp1d... |
| rescabs2 17789 | Restriction absorption law... |
| issubc 17790 | Elementhood in the set of ... |
| issubc2 17791 | Elementhood in the set of ... |
| 0ssc 17792 | For any category ` C ` , t... |
| 0subcat 17793 | For any category ` C ` , t... |
| catsubcat 17794 | For any category ` C ` , `... |
| subcssc 17795 | An element in the set of s... |
| subcfn 17796 | An element in the set of s... |
| subcss1 17797 | The objects of a subcatego... |
| subcss2 17798 | The morphisms of a subcate... |
| subcidcl 17799 | The identity of the origin... |
| subccocl 17800 | A subcategory is closed un... |
| subccatid 17801 | A subcategory is a categor... |
| subcid 17802 | The identity in a subcateg... |
| subccat 17803 | A subcategory is a categor... |
| issubc3 17804 | Alternate definition of a ... |
| fullsubc 17805 | The full subcategory gener... |
| fullresc 17806 | The category formed by str... |
| resscat 17807 | A category restricted to a... |
| subsubc 17808 | A subcategory of a subcate... |
| relfunc 17817 | The set of functors is a r... |
| funcrcl 17818 | Reverse closure for a func... |
| isfunc 17819 | Value of the set of functo... |
| isfuncd 17820 | Deduce that an operation i... |
| funcf1 17821 | The object part of a funct... |
| funcixp 17822 | The morphism part of a fun... |
| funcf2 17823 | The morphism part of a fun... |
| funcfn2 17824 | The morphism part of a fun... |
| funcid 17825 | A functor maps each identi... |
| funcco 17826 | A functor maps composition... |
| funcsect 17827 | The image of a section und... |
| funcinv 17828 | The image of an inverse un... |
| funciso 17829 | The image of an isomorphis... |
| funcoppc 17830 | A functor on categories yi... |
| idfuval 17831 | Value of the identity func... |
| idfu2nd 17832 | Value of the morphism part... |
| idfu2 17833 | Value of the morphism part... |
| idfu1st 17834 | Value of the object part o... |
| idfu1 17835 | Value of the object part o... |
| idfucl 17836 | The identity functor is a ... |
| cofuval 17837 | Value of the composition o... |
| cofu1st 17838 | Value of the object part o... |
| cofu1 17839 | Value of the object part o... |
| cofu2nd 17840 | Value of the morphism part... |
| cofu2 17841 | Value of the morphism part... |
| cofuval2 17842 | Value of the composition o... |
| cofucl 17843 | The composition of two fun... |
| cofuass 17844 | Functor composition is ass... |
| cofulid 17845 | The identity functor is a ... |
| cofurid 17846 | The identity functor is a ... |
| resfval 17847 | Value of the functor restr... |
| resfval2 17848 | Value of the functor restr... |
| resf1st 17849 | Value of the functor restr... |
| resf2nd 17850 | Value of the functor restr... |
| funcres 17851 | A functor restricted to a ... |
| funcres2b 17852 | Condition for a functor to... |
| funcres2 17853 | A functor into a restricte... |
| wunfunc 17854 | A weak universe is closed ... |
| wunfuncOLD 17855 | Obsolete proof of ~ wunfun... |
| funcpropd 17856 | If two categories have the... |
| funcres2c 17857 | Condition for a functor to... |
| fullfunc 17862 | A full functor is a functo... |
| fthfunc 17863 | A faithful functor is a fu... |
| relfull 17864 | The set of full functors i... |
| relfth 17865 | The set of faithful functo... |
| isfull 17866 | Value of the set of full f... |
| isfull2 17867 | Equivalent condition for a... |
| fullfo 17868 | The morphism map of a full... |
| fulli 17869 | The morphism map of a full... |
| isfth 17870 | Value of the set of faithf... |
| isfth2 17871 | Equivalent condition for a... |
| isffth2 17872 | A fully faithful functor i... |
| fthf1 17873 | The morphism map of a fait... |
| fthi 17874 | The morphism map of a fait... |
| ffthf1o 17875 | The morphism map of a full... |
| fullpropd 17876 | If two categories have the... |
| fthpropd 17877 | If two categories have the... |
| fulloppc 17878 | The opposite functor of a ... |
| fthoppc 17879 | The opposite functor of a ... |
| ffthoppc 17880 | The opposite functor of a ... |
| fthsect 17881 | A faithful functor reflect... |
| fthinv 17882 | A faithful functor reflect... |
| fthmon 17883 | A faithful functor reflect... |
| fthepi 17884 | A faithful functor reflect... |
| ffthiso 17885 | A fully faithful functor r... |
| fthres2b 17886 | Condition for a faithful f... |
| fthres2c 17887 | Condition for a faithful f... |
| fthres2 17888 | A faithful functor into a ... |
| idffth 17889 | The identity functor is a ... |
| cofull 17890 | The composition of two ful... |
| cofth 17891 | The composition of two fai... |
| coffth 17892 | The composition of two ful... |
| rescfth 17893 | The inclusion functor from... |
| ressffth 17894 | The inclusion functor from... |
| fullres2c 17895 | Condition for a full funct... |
| ffthres2c 17896 | Condition for a fully fait... |
| fnfuc 17901 | The ` FuncCat ` operation ... |
| natfval 17902 | Value of the function givi... |
| isnat 17903 | Property of being a natura... |
| isnat2 17904 | Property of being a natura... |
| natffn 17905 | The natural transformation... |
| natrcl 17906 | Reverse closure for a natu... |
| nat1st2nd 17907 | Rewrite the natural transf... |
| natixp 17908 | A natural transformation i... |
| natcl 17909 | A component of a natural t... |
| natfn 17910 | A natural transformation i... |
| nati 17911 | Naturality property of a n... |
| wunnat 17912 | A weak universe is closed ... |
| wunnatOLD 17913 | Obsolete proof of ~ wunnat... |
| catstr 17914 | A category structure is a ... |
| fucval 17915 | Value of the functor categ... |
| fuccofval 17916 | Value of the functor categ... |
| fucbas 17917 | The objects of the functor... |
| fuchom 17918 | The morphisms in the funct... |
| fuchomOLD 17919 | Obsolete proof of ~ fuchom... |
| fucco 17920 | Value of the composition o... |
| fuccoval 17921 | Value of the functor categ... |
| fuccocl 17922 | The composition of two nat... |
| fucidcl 17923 | The identity natural trans... |
| fuclid 17924 | Left identity of natural t... |
| fucrid 17925 | Right identity of natural ... |
| fucass 17926 | Associativity of natural t... |
| fuccatid 17927 | The functor category is a ... |
| fuccat 17928 | The functor category is a ... |
| fucid 17929 | The identity morphism in t... |
| fucsect 17930 | Two natural transformation... |
| fucinv 17931 | Two natural transformation... |
| invfuc 17932 | If ` V ( x ) ` is an inver... |
| fuciso 17933 | A natural transformation i... |
| natpropd 17934 | If two categories have the... |
| fucpropd 17935 | If two categories have the... |
| initofn 17942 | ` InitO ` is a function on... |
| termofn 17943 | ` TermO ` is a function on... |
| zeroofn 17944 | ` ZeroO ` is a function on... |
| initorcl 17945 | Reverse closure for an ini... |
| termorcl 17946 | Reverse closure for a term... |
| zeroorcl 17947 | Reverse closure for a zero... |
| initoval 17948 | The value of the initial o... |
| termoval 17949 | The value of the terminal ... |
| zerooval 17950 | The value of the zero obje... |
| isinito 17951 | The predicate "is an initi... |
| istermo 17952 | The predicate "is a termin... |
| iszeroo 17953 | The predicate "is a zero o... |
| isinitoi 17954 | Implication of a class bei... |
| istermoi 17955 | Implication of a class bei... |
| initoid 17956 | For an initial object, the... |
| termoid 17957 | For a terminal object, the... |
| dfinito2 17958 | An initial object is a ter... |
| dftermo2 17959 | A terminal object is an in... |
| dfinito3 17960 | An alternate definition of... |
| dftermo3 17961 | An alternate definition of... |
| initoo 17962 | An initial object is an ob... |
| termoo 17963 | A terminal object is an ob... |
| iszeroi 17964 | Implication of a class bei... |
| 2initoinv 17965 | Morphisms between two init... |
| initoeu1 17966 | Initial objects are essent... |
| initoeu1w 17967 | Initial objects are essent... |
| initoeu2lem0 17968 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17969 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17970 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17971 | Initial objects are essent... |
| 2termoinv 17972 | Morphisms between two term... |
| termoeu1 17973 | Terminal objects are essen... |
| termoeu1w 17974 | Terminal objects are essen... |
| homarcl 17983 | Reverse closure for an arr... |
| homafval 17984 | Value of the disjointified... |
| homaf 17985 | Functionality of the disjo... |
| homaval 17986 | Value of the disjointified... |
| elhoma 17987 | Value of the disjointified... |
| elhomai 17988 | Produce an arrow from a mo... |
| elhomai2 17989 | Produce an arrow from a mo... |
| homarcl2 17990 | Reverse closure for the do... |
| homarel 17991 | An arrow is an ordered pai... |
| homa1 17992 | The first component of an ... |
| homahom2 17993 | The second component of an... |
| homahom 17994 | The second component of an... |
| homadm 17995 | The domain of an arrow wit... |
| homacd 17996 | The codomain of an arrow w... |
| homadmcd 17997 | Decompose an arrow into do... |
| arwval 17998 | The set of arrows is the u... |
| arwrcl 17999 | The first component of an ... |
| arwhoma 18000 | An arrow is contained in t... |
| homarw 18001 | A hom-set is a subset of t... |
| arwdm 18002 | The domain of an arrow is ... |
| arwcd 18003 | The codomain of an arrow i... |
| dmaf 18004 | The domain function is a f... |
| cdaf 18005 | The codomain function is a... |
| arwhom 18006 | The second component of an... |
| arwdmcd 18007 | Decompose an arrow into do... |
| idafval 18012 | Value of the identity arro... |
| idaval 18013 | Value of the identity arro... |
| ida2 18014 | Morphism part of the ident... |
| idahom 18015 | Domain and codomain of the... |
| idadm 18016 | Domain of the identity arr... |
| idacd 18017 | Codomain of the identity a... |
| idaf 18018 | The identity arrow functio... |
| coafval 18019 | The value of the compositi... |
| eldmcoa 18020 | A pair ` <. G , F >. ` is ... |
| dmcoass 18021 | The domain of composition ... |
| homdmcoa 18022 | If ` F : X --> Y ` and ` G... |
| coaval 18023 | Value of composition for c... |
| coa2 18024 | The morphism part of arrow... |
| coahom 18025 | The composition of two com... |
| coapm 18026 | Composition of arrows is a... |
| arwlid 18027 | Left identity of a categor... |
| arwrid 18028 | Right identity of a catego... |
| arwass 18029 | Associativity of compositi... |
| setcval 18032 | Value of the category of s... |
| setcbas 18033 | Set of objects of the cate... |
| setchomfval 18034 | Set of arrows of the categ... |
| setchom 18035 | Set of arrows of the categ... |
| elsetchom 18036 | A morphism of sets is a fu... |
| setccofval 18037 | Composition in the categor... |
| setcco 18038 | Composition in the categor... |
| setccatid 18039 | Lemma for ~ setccat . (Co... |
| setccat 18040 | The category of sets is a ... |
| setcid 18041 | The identity arrow in the ... |
| setcmon 18042 | A monomorphism of sets is ... |
| setcepi 18043 | An epimorphism of sets is ... |
| setcsect 18044 | A section in the category ... |
| setcinv 18045 | An inverse in the category... |
| setciso 18046 | An isomorphism in the cate... |
| resssetc 18047 | The restriction of the cat... |
| funcsetcres2 18048 | A functor into a smaller c... |
| setc2obas 18049 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 18050 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 18051 | The category of sets in a ... |
| cat1 18052 | The definition of category... |
| catcval 18055 | Value of the category of c... |
| catcbas 18056 | Set of objects of the cate... |
| catchomfval 18057 | Set of arrows of the categ... |
| catchom 18058 | Set of arrows of the categ... |
| catccofval 18059 | Composition in the categor... |
| catcco 18060 | Composition in the categor... |
| catccatid 18061 | Lemma for ~ catccat . (Co... |
| catcid 18062 | The identity arrow in the ... |
| catccat 18063 | The category of categories... |
| resscatc 18064 | The restriction of the cat... |
| catcisolem 18065 | Lemma for ~ catciso . (Co... |
| catciso 18066 | A functor is an isomorphis... |
| catcbascl 18067 | An element of the base set... |
| catcslotelcl 18068 | A slot entry of an element... |
| catcbaselcl 18069 | The base set of an element... |
| catchomcl 18070 | The Hom-set of an element ... |
| catcccocl 18071 | The composition operation ... |
| catcoppccl 18072 | The category of categories... |
| catcoppcclOLD 18073 | Obsolete proof of ~ catcop... |
| catcfuccl 18074 | The category of categories... |
| catcfucclOLD 18075 | Obsolete proof of ~ catcfu... |
| fncnvimaeqv 18076 | The inverse images of the ... |
| bascnvimaeqv 18077 | The inverse image of the u... |
| estrcval 18080 | Value of the category of e... |
| estrcbas 18081 | Set of objects of the cate... |
| estrchomfval 18082 | Set of morphisms ("arrows"... |
| estrchom 18083 | The morphisms between exte... |
| elestrchom 18084 | A morphism between extensi... |
| estrccofval 18085 | Composition in the categor... |
| estrcco 18086 | Composition in the categor... |
| estrcbasbas 18087 | An element of the base set... |
| estrccatid 18088 | Lemma for ~ estrccat . (C... |
| estrccat 18089 | The category of extensible... |
| estrcid 18090 | The identity arrow in the ... |
| estrchomfn 18091 | The Hom-set operation in t... |
| estrchomfeqhom 18092 | The functionalized Hom-set... |
| estrreslem1 18093 | Lemma 1 for ~ estrres . (... |
| estrreslem1OLD 18094 | Obsolete version of ~ estr... |
| estrreslem2 18095 | Lemma 2 for ~ estrres . (... |
| estrres 18096 | Any restriction of a categ... |
| funcestrcsetclem1 18097 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18098 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18099 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18100 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18101 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18102 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18103 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18104 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18105 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18106 | The "natural forgetful fun... |
| fthestrcsetc 18107 | The "natural forgetful fun... |
| fullestrcsetc 18108 | The "natural forgetful fun... |
| equivestrcsetc 18109 | The "natural forgetful fun... |
| setc1strwun 18110 | A constructed one-slot str... |
| funcsetcestrclem1 18111 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18112 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18113 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18114 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18115 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18116 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18117 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18118 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18119 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18120 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18121 | The "embedding functor" fr... |
| fthsetcestrc 18122 | The "embedding functor" fr... |
| fullsetcestrc 18123 | The "embedding functor" fr... |
| embedsetcestrc 18124 | The "embedding functor" fr... |
| fnxpc 18133 | The binary product of cate... |
| xpcval 18134 | Value of the binary produc... |
| xpcbas 18135 | Set of objects of the bina... |
| xpchomfval 18136 | Set of morphisms of the bi... |
| xpchom 18137 | Set of morphisms of the bi... |
| relxpchom 18138 | A hom-set in the binary pr... |
| xpccofval 18139 | Value of composition in th... |
| xpcco 18140 | Value of composition in th... |
| xpcco1st 18141 | Value of composition in th... |
| xpcco2nd 18142 | Value of composition in th... |
| xpchom2 18143 | Value of the set of morphi... |
| xpcco2 18144 | Value of composition in th... |
| xpccatid 18145 | The product of two categor... |
| xpcid 18146 | The identity morphism in t... |
| xpccat 18147 | The product of two categor... |
| 1stfval 18148 | Value of the first project... |
| 1stf1 18149 | Value of the first project... |
| 1stf2 18150 | Value of the first project... |
| 2ndfval 18151 | Value of the first project... |
| 2ndf1 18152 | Value of the first project... |
| 2ndf2 18153 | Value of the first project... |
| 1stfcl 18154 | The first projection funct... |
| 2ndfcl 18155 | The second projection func... |
| prfval 18156 | Value of the pairing funct... |
| prf1 18157 | Value of the pairing funct... |
| prf2fval 18158 | Value of the pairing funct... |
| prf2 18159 | Value of the pairing funct... |
| prfcl 18160 | The pairing of functors ` ... |
| prf1st 18161 | Cancellation of pairing wi... |
| prf2nd 18162 | Cancellation of pairing wi... |
| 1st2ndprf 18163 | Break a functor into a pro... |
| catcxpccl 18164 | The category of categories... |
| catcxpcclOLD 18165 | Obsolete proof of ~ catcxp... |
| xpcpropd 18166 | If two categories have the... |
| evlfval 18175 | Value of the evaluation fu... |
| evlf2 18176 | Value of the evaluation fu... |
| evlf2val 18177 | Value of the evaluation na... |
| evlf1 18178 | Value of the evaluation fu... |
| evlfcllem 18179 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18180 | The evaluation functor is ... |
| curfval 18181 | Value of the curry functor... |
| curf1fval 18182 | Value of the object part o... |
| curf1 18183 | Value of the object part o... |
| curf11 18184 | Value of the double evalua... |
| curf12 18185 | The partially evaluated cu... |
| curf1cl 18186 | The partially evaluated cu... |
| curf2 18187 | Value of the curry functor... |
| curf2val 18188 | Value of a component of th... |
| curf2cl 18189 | The curry functor at a mor... |
| curfcl 18190 | The curry functor of a fun... |
| curfpropd 18191 | If two categories have the... |
| uncfval 18192 | Value of the uncurry funct... |
| uncfcl 18193 | The uncurry operation take... |
| uncf1 18194 | Value of the uncurry funct... |
| uncf2 18195 | Value of the uncurry funct... |
| curfuncf 18196 | Cancellation of curry with... |
| uncfcurf 18197 | Cancellation of uncurry wi... |
| diagval 18198 | Define the diagonal functo... |
| diagcl 18199 | The diagonal functor is a ... |
| diag1cl 18200 | The constant functor of ` ... |
| diag11 18201 | Value of the constant func... |
| diag12 18202 | Value of the constant func... |
| diag2 18203 | Value of the diagonal func... |
| diag2cl 18204 | The diagonal functor at a ... |
| curf2ndf 18205 | As shown in ~ diagval , th... |
| hofval 18210 | Value of the Hom functor, ... |
| hof1fval 18211 | The object part of the Hom... |
| hof1 18212 | The object part of the Hom... |
| hof2fval 18213 | The morphism part of the H... |
| hof2val 18214 | The morphism part of the H... |
| hof2 18215 | The morphism part of the H... |
| hofcllem 18216 | Lemma for ~ hofcl . (Cont... |
| hofcl 18217 | Closure of the Hom functor... |
| oppchofcl 18218 | Closure of the opposite Ho... |
| yonval 18219 | Value of the Yoneda embedd... |
| yoncl 18220 | The Yoneda embedding is a ... |
| yon1cl 18221 | The Yoneda embedding at an... |
| yon11 18222 | Value of the Yoneda embedd... |
| yon12 18223 | Value of the Yoneda embedd... |
| yon2 18224 | Value of the Yoneda embedd... |
| hofpropd 18225 | If two categories have the... |
| yonpropd 18226 | If two categories have the... |
| oppcyon 18227 | Value of the opposite Yone... |
| oyoncl 18228 | The opposite Yoneda embedd... |
| oyon1cl 18229 | The opposite Yoneda embedd... |
| yonedalem1 18230 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18231 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18232 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18233 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18234 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18235 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18236 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18237 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18238 | Lemma for ~ yoneda . (Con... |
| yonedainv 18239 | The Yoneda Lemma with expl... |
| yonffthlem 18240 | Lemma for ~ yonffth . (Co... |
| yoneda 18241 | The Yoneda Lemma. There i... |
| yonffth 18242 | The Yoneda Lemma. The Yon... |
| yoniso 18243 | If the codomain is recover... |
| oduval 18246 | Value of an order dual str... |
| oduleval 18247 | Value of the less-equal re... |
| oduleg 18248 | Truth of the less-equal re... |
| odubas 18249 | Base set of an order dual ... |
| odubasOLD 18250 | Obsolete proof of ~ odubas... |
| isprs 18255 | Property of being a preord... |
| prslem 18256 | Lemma for ~ prsref and ~ p... |
| prsref 18257 | "Less than or equal to" is... |
| prstr 18258 | "Less than or equal to" is... |
| isdrs 18259 | Property of being a direct... |
| drsdir 18260 | Direction of a directed se... |
| drsprs 18261 | A directed set is a proset... |
| drsbn0 18262 | The base of a directed set... |
| drsdirfi 18263 | Any _finite_ number of ele... |
| isdrs2 18264 | Directed sets may be defin... |
| ispos 18272 | The predicate "is a poset"... |
| ispos2 18273 | A poset is an antisymmetri... |
| posprs 18274 | A poset is a proset. (Con... |
| posi 18275 | Lemma for poset properties... |
| posref 18276 | A poset ordering is reflex... |
| posasymb 18277 | A poset ordering is asymme... |
| postr 18278 | A poset ordering is transi... |
| 0pos 18279 | Technical lemma to simplif... |
| 0posOLD 18280 | Obsolete proof of ~ 0pos a... |
| isposd 18281 | Properties that determine ... |
| isposi 18282 | Properties that determine ... |
| isposix 18283 | Properties that determine ... |
| isposixOLD 18284 | Obsolete proof of ~ isposi... |
| pospropd 18285 | Posethood is determined on... |
| odupos 18286 | Being a poset is a self-du... |
| oduposb 18287 | Being a poset is a self-du... |
| pltfval 18289 | Value of the less-than rel... |
| pltval 18290 | Less-than relation. ( ~ d... |
| pltle 18291 | "Less than" implies "less ... |
| pltne 18292 | The "less than" relation i... |
| pltirr 18293 | The "less than" relation i... |
| pleval2i 18294 | One direction of ~ pleval2... |
| pleval2 18295 | "Less than or equal to" in... |
| pltnle 18296 | "Less than" implies not co... |
| pltval3 18297 | Alternate expression for t... |
| pltnlt 18298 | The less-than relation imp... |
| pltn2lp 18299 | The less-than relation has... |
| plttr 18300 | The less-than relation is ... |
| pltletr 18301 | Transitive law for chained... |
| plelttr 18302 | Transitive law for chained... |
| pospo 18303 | Write a poset structure in... |
| lubfval 18308 | Value of the least upper b... |
| lubdm 18309 | Domain of the least upper ... |
| lubfun 18310 | The LUB is a function. (C... |
| lubeldm 18311 | Member of the domain of th... |
| lubelss 18312 | A member of the domain of ... |
| lubeu 18313 | Unique existence proper of... |
| lubval 18314 | Value of the least upper b... |
| lubcl 18315 | The least upper bound func... |
| lubprop 18316 | Properties of greatest low... |
| luble 18317 | The greatest lower bound i... |
| lublecllem 18318 | Lemma for ~ lublecl and ~ ... |
| lublecl 18319 | The set of all elements le... |
| lubid 18320 | The LUB of elements less t... |
| glbfval 18321 | Value of the greatest lowe... |
| glbdm 18322 | Domain of the greatest low... |
| glbfun 18323 | The GLB is a function. (C... |
| glbeldm 18324 | Member of the domain of th... |
| glbelss 18325 | A member of the domain of ... |
| glbeu 18326 | Unique existence proper of... |
| glbval 18327 | Value of the greatest lowe... |
| glbcl 18328 | The least upper bound func... |
| glbprop 18329 | Properties of greatest low... |
| glble 18330 | The greatest lower bound i... |
| joinfval 18331 | Value of join function for... |
| joinfval2 18332 | Value of join function for... |
| joindm 18333 | Domain of join function fo... |
| joindef 18334 | Two ways to say that a joi... |
| joinval 18335 | Join value. Since both si... |
| joincl 18336 | Closure of join of element... |
| joindmss 18337 | Subset property of domain ... |
| joinval2lem 18338 | Lemma for ~ joinval2 and ~... |
| joinval2 18339 | Value of join for a poset ... |
| joineu 18340 | Uniqueness of join of elem... |
| joinlem 18341 | Lemma for join properties.... |
| lejoin1 18342 | A join's first argument is... |
| lejoin2 18343 | A join's second argument i... |
| joinle 18344 | A join is less than or equ... |
| meetfval 18345 | Value of meet function for... |
| meetfval2 18346 | Value of meet function for... |
| meetdm 18347 | Domain of meet function fo... |
| meetdef 18348 | Two ways to say that a mee... |
| meetval 18349 | Meet value. Since both si... |
| meetcl 18350 | Closure of meet of element... |
| meetdmss 18351 | Subset property of domain ... |
| meetval2lem 18352 | Lemma for ~ meetval2 and ~... |
| meetval2 18353 | Value of meet for a poset ... |
| meeteu 18354 | Uniqueness of meet of elem... |
| meetlem 18355 | Lemma for meet properties.... |
| lemeet1 18356 | A meet's first argument is... |
| lemeet2 18357 | A meet's second argument i... |
| meetle 18358 | A meet is less than or equ... |
| joincomALT 18359 | The join of a poset is com... |
| joincom 18360 | The join of a poset is com... |
| meetcomALT 18361 | The meet of a poset is com... |
| meetcom 18362 | The meet of a poset is com... |
| join0 18363 | Lemma for ~ odumeet . (Co... |
| meet0 18364 | Lemma for ~ odujoin . (Co... |
| odulub 18365 | Least upper bounds in a du... |
| odujoin 18366 | Joins in a dual order are ... |
| oduglb 18367 | Greatest lower bounds in a... |
| odumeet 18368 | Meets in a dual order are ... |
| poslubmo 18369 | Least upper bounds in a po... |
| posglbmo 18370 | Greatest lower bounds in a... |
| poslubd 18371 | Properties which determine... |
| poslubdg 18372 | Properties which determine... |
| posglbdg 18373 | Properties which determine... |
| istos 18376 | The predicate "is a toset"... |
| tosso 18377 | Write the totally ordered ... |
| tospos 18378 | A Toset is a Poset. (Cont... |
| tleile 18379 | In a Toset, any two elemen... |
| tltnle 18380 | In a Toset, "less than" is... |
| p0val 18385 | Value of poset zero. (Con... |
| p1val 18386 | Value of poset zero. (Con... |
| p0le 18387 | Any element is less than o... |
| ple1 18388 | Any element is less than o... |
| islat 18391 | The predicate "is a lattic... |
| odulatb 18392 | Being a lattice is self-du... |
| odulat 18393 | Being a lattice is self-du... |
| latcl2 18394 | The join and meet of any t... |
| latlem 18395 | Lemma for lattice properti... |
| latpos 18396 | A lattice is a poset. (Co... |
| latjcl 18397 | Closure of join operation ... |
| latmcl 18398 | Closure of meet operation ... |
| latref 18399 | A lattice ordering is refl... |
| latasymb 18400 | A lattice ordering is asym... |
| latasym 18401 | A lattice ordering is asym... |
| lattr 18402 | A lattice ordering is tran... |
| latasymd 18403 | Deduce equality from latti... |
| lattrd 18404 | A lattice ordering is tran... |
| latjcom 18405 | The join of a lattice comm... |
| latlej1 18406 | A join's first argument is... |
| latlej2 18407 | A join's second argument i... |
| latjle12 18408 | A join is less than or equ... |
| latleeqj1 18409 | "Less than or equal to" in... |
| latleeqj2 18410 | "Less than or equal to" in... |
| latjlej1 18411 | Add join to both sides of ... |
| latjlej2 18412 | Add join to both sides of ... |
| latjlej12 18413 | Add join to both sides of ... |
| latnlej 18414 | An idiom to express that a... |
| latnlej1l 18415 | An idiom to express that a... |
| latnlej1r 18416 | An idiom to express that a... |
| latnlej2 18417 | An idiom to express that a... |
| latnlej2l 18418 | An idiom to express that a... |
| latnlej2r 18419 | An idiom to express that a... |
| latjidm 18420 | Lattice join is idempotent... |
| latmcom 18421 | The join of a lattice comm... |
| latmle1 18422 | A meet is less than or equ... |
| latmle2 18423 | A meet is less than or equ... |
| latlem12 18424 | An element is less than or... |
| latleeqm1 18425 | "Less than or equal to" in... |
| latleeqm2 18426 | "Less than or equal to" in... |
| latmlem1 18427 | Add meet to both sides of ... |
| latmlem2 18428 | Add meet to both sides of ... |
| latmlem12 18429 | Add join to both sides of ... |
| latnlemlt 18430 | Negation of "less than or ... |
| latnle 18431 | Equivalent expressions for... |
| latmidm 18432 | Lattice meet is idempotent... |
| latabs1 18433 | Lattice absorption law. F... |
| latabs2 18434 | Lattice absorption law. F... |
| latledi 18435 | An ortholattice is distrib... |
| latmlej11 18436 | Ordering of a meet and joi... |
| latmlej12 18437 | Ordering of a meet and joi... |
| latmlej21 18438 | Ordering of a meet and joi... |
| latmlej22 18439 | Ordering of a meet and joi... |
| lubsn 18440 | The least upper bound of a... |
| latjass 18441 | Lattice join is associativ... |
| latj12 18442 | Swap 1st and 2nd members o... |
| latj32 18443 | Swap 2nd and 3rd members o... |
| latj13 18444 | Swap 1st and 3rd members o... |
| latj31 18445 | Swap 2nd and 3rd members o... |
| latjrot 18446 | Rotate lattice join of 3 c... |
| latj4 18447 | Rearrangement of lattice j... |
| latj4rot 18448 | Rotate lattice join of 4 c... |
| latjjdi 18449 | Lattice join distributes o... |
| latjjdir 18450 | Lattice join distributes o... |
| mod1ile 18451 | The weak direction of the ... |
| mod2ile 18452 | The weak direction of the ... |
| latmass 18453 | Lattice meet is associativ... |
| latdisdlem 18454 | Lemma for ~ latdisd . (Co... |
| latdisd 18455 | In a lattice, joins distri... |
| isclat 18458 | The predicate "is a comple... |
| clatpos 18459 | A complete lattice is a po... |
| clatlem 18460 | Lemma for properties of a ... |
| clatlubcl 18461 | Any subset of the base set... |
| clatlubcl2 18462 | Any subset of the base set... |
| clatglbcl 18463 | Any subset of the base set... |
| clatglbcl2 18464 | Any subset of the base set... |
| oduclatb 18465 | Being a complete lattice i... |
| clatl 18466 | A complete lattice is a la... |
| isglbd 18467 | Properties that determine ... |
| lublem 18468 | Lemma for the least upper ... |
| lubub 18469 | The LUB of a complete latt... |
| lubl 18470 | The LUB of a complete latt... |
| lubss 18471 | Subset law for least upper... |
| lubel 18472 | An element of a set is les... |
| lubun 18473 | The LUB of a union. (Cont... |
| clatglb 18474 | Properties of greatest low... |
| clatglble 18475 | The greatest lower bound i... |
| clatleglb 18476 | Two ways of expressing "le... |
| clatglbss 18477 | Subset law for greatest lo... |
| isdlat 18480 | Property of being a distri... |
| dlatmjdi 18481 | In a distributive lattice,... |
| dlatl 18482 | A distributive lattice is ... |
| odudlatb 18483 | The dual of a distributive... |
| dlatjmdi 18484 | In a distributive lattice,... |
| ipostr 18487 | The structure of ~ df-ipo ... |
| ipoval 18488 | Value of the inclusion pos... |
| ipobas 18489 | Base set of the inclusion ... |
| ipolerval 18490 | Relation of the inclusion ... |
| ipotset 18491 | Topology of the inclusion ... |
| ipole 18492 | Weak order condition of th... |
| ipolt 18493 | Strict order condition of ... |
| ipopos 18494 | The inclusion poset on a f... |
| isipodrs 18495 | Condition for a family of ... |
| ipodrscl 18496 | Direction by inclusion as ... |
| ipodrsfi 18497 | Finite upper bound propert... |
| fpwipodrs 18498 | The finite subsets of any ... |
| ipodrsima 18499 | The monotone image of a di... |
| isacs3lem 18500 | An algebraic closure syste... |
| acsdrsel 18501 | An algebraic closure syste... |
| isacs4lem 18502 | In a closure system in whi... |
| isacs5lem 18503 | If closure commutes with d... |
| acsdrscl 18504 | In an algebraic closure sy... |
| acsficl 18505 | A closure in an algebraic ... |
| isacs5 18506 | A closure system is algebr... |
| isacs4 18507 | A closure system is algebr... |
| isacs3 18508 | A closure system is algebr... |
| acsficld 18509 | In an algebraic closure sy... |
| acsficl2d 18510 | In an algebraic closure sy... |
| acsfiindd 18511 | In an algebraic closure sy... |
| acsmapd 18512 | In an algebraic closure sy... |
| acsmap2d 18513 | In an algebraic closure sy... |
| acsinfd 18514 | In an algebraic closure sy... |
| acsdomd 18515 | In an algebraic closure sy... |
| acsinfdimd 18516 | In an algebraic closure sy... |
| acsexdimd 18517 | In an algebraic closure sy... |
| mrelatglb 18518 | Greatest lower bounds in a... |
| mrelatglb0 18519 | The empty intersection in ... |
| mrelatlub 18520 | Least upper bounds in a Mo... |
| mreclatBAD 18521 | A Moore space is a complet... |
| isps 18526 | The predicate "is a poset"... |
| psrel 18527 | A poset is a relation. (C... |
| psref2 18528 | A poset is antisymmetric a... |
| pstr2 18529 | A poset is transitive. (C... |
| pslem 18530 | Lemma for ~ psref and othe... |
| psdmrn 18531 | The domain and range of a ... |
| psref 18532 | A poset is reflexive. (Co... |
| psrn 18533 | The range of a poset equal... |
| psasym 18534 | A poset is antisymmetric. ... |
| pstr 18535 | A poset is transitive. (C... |
| cnvps 18536 | The converse of a poset is... |
| cnvpsb 18537 | The converse of a poset is... |
| psss 18538 | Any subset of a partially ... |
| psssdm2 18539 | Field of a subposet. (Con... |
| psssdm 18540 | Field of a subposet. (Con... |
| istsr 18541 | The predicate is a toset. ... |
| istsr2 18542 | The predicate is a toset. ... |
| tsrlin 18543 | A toset is a linear order.... |
| tsrlemax 18544 | Two ways of saying a numbe... |
| tsrps 18545 | A toset is a poset. (Cont... |
| cnvtsr 18546 | The converse of a toset is... |
| tsrss 18547 | Any subset of a totally or... |
| ledm 18548 | The domain of ` <_ ` is ` ... |
| lern 18549 | The range of ` <_ ` is ` R... |
| lefld 18550 | The field of the 'less or ... |
| letsr 18551 | The "less than or equal to... |
| isdir 18556 | A condition for a relation... |
| reldir 18557 | A direction is a relation.... |
| dirdm 18558 | A direction's domain is eq... |
| dirref 18559 | A direction is reflexive. ... |
| dirtr 18560 | A direction is transitive.... |
| dirge 18561 | For any two elements of a ... |
| tsrdir 18562 | A totally ordered set is a... |
| 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... |
| grprinvlem 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... |
| prdsplusgcl 18691 | Structure product pointwis... |
| prdsidlem 18692 | Characterization of identi... |
| prdsmndd 18693 | The product of a family of... |
| prds0g 18694 | Zero in a product of monoi... |
| pwsmnd 18695 | The structure power of a m... |
| pws0g 18696 | Zero in a structure power ... |
| imasmnd2 18697 | The image structure of a m... |
| imasmnd 18698 | The image structure of a m... |
| imasmndf1 18699 | The image of a monoid unde... |
| xpsmnd 18700 | The binary product of mono... |
| xpsmnd0 18701 | The identity element of a ... |
| mnd1 18702 | The (smallest) structure r... |
| mnd1id 18703 | The singleton element of a... |
| ismhm 18708 | Property of a monoid homom... |
| ismhmd 18709 | Deduction version of ~ ism... |
| mhmrcl1 18710 | Reverse closure of a monoi... |
| mhmrcl2 18711 | Reverse closure of a monoi... |
| mhmf 18712 | A monoid homomorphism is a... |
| ismhm0 18713 | Property of a monoid homom... |
| mhmismgmhm 18714 | Each monoid homomorphism i... |
| mhmpropd 18715 | Monoid homomorphism depend... |
| mhmlin 18716 | A monoid homomorphism comm... |
| mhm0 18717 | A monoid homomorphism pres... |
| idmhm 18718 | The identity homomorphism ... |
| mhmf1o 18719 | A monoid homomorphism is b... |
| submrcl 18720 | Reverse closure for submon... |
| issubm 18721 | Expand definition of a sub... |
| issubm2 18722 | Submonoids are subsets tha... |
| issubmndb 18723 | The submonoid predicate. ... |
| issubmd 18724 | Deduction for proving a su... |
| mndissubm 18725 | If the base set of a monoi... |
| resmndismnd 18726 | If the base set of a monoi... |
| submss 18727 | Submonoids are subsets of ... |
| submid 18728 | Every monoid is trivially ... |
| subm0cl 18729 | Submonoids contain zero. ... |
| submcl 18730 | Submonoids are closed unde... |
| submmnd 18731 | Submonoids are themselves ... |
| submbas 18732 | The base set of a submonoi... |
| subm0 18733 | Submonoids have the same i... |
| subsubm 18734 | A submonoid of a submonoid... |
| 0subm 18735 | The zero submonoid of an a... |
| insubm 18736 | The intersection of two su... |
| 0mhm 18737 | The constant zero linear f... |
| resmhm 18738 | Restriction of a monoid ho... |
| resmhm2 18739 | One direction of ~ resmhm2... |
| resmhm2b 18740 | Restriction of the codomai... |
| mhmco 18741 | The composition of monoid ... |
| mhmimalem 18742 | Lemma for ~ mhmima and sim... |
| mhmima 18743 | The homomorphic image of a... |
| mhmeql 18744 | The equalizer of two monoi... |
| submacs 18745 | Submonoids are an algebrai... |
| mndind 18746 | Induction in a monoid. In... |
| prdspjmhm 18747 | A projection from a produc... |
| pwspjmhm 18748 | A projection from a struct... |
| pwsdiagmhm 18749 | Diagonal monoid homomorphi... |
| pwsco1mhm 18750 | Right composition with a f... |
| pwsco2mhm 18751 | Left composition with a mo... |
| gsumvallem2 18752 | Lemma for properties of th... |
| gsumsubm 18753 | Evaluate a group sum in a ... |
| gsumz 18754 | Value of a group sum over ... |
| gsumwsubmcl 18755 | Closure of the composite i... |
| gsumws1 18756 | A singleton composite reco... |
| gsumwcl 18757 | Closure of the composite o... |
| gsumsgrpccat 18758 | Homomorphic property of no... |
| gsumccat 18759 | Homomorphic property of co... |
| gsumws2 18760 | Valuation of a pair in a m... |
| gsumccatsn 18761 | Homomorphic property of co... |
| gsumspl 18762 | The primary purpose of the... |
| gsumwmhm 18763 | Behavior of homomorphisms ... |
| gsumwspan 18764 | The submonoid generated by... |
| frmdval 18769 | Value of the free monoid c... |
| frmdbas 18770 | The base set of a free mon... |
| frmdelbas 18771 | An element of the base set... |
| frmdplusg 18772 | The monoid operation of a ... |
| frmdadd 18773 | Value of the monoid operat... |
| vrmdfval 18774 | The canonical injection fr... |
| vrmdval 18775 | The value of the generatin... |
| vrmdf 18776 | The mapping from the index... |
| frmdmnd 18777 | A free monoid is a monoid.... |
| frmd0 18778 | The identity of the free m... |
| frmdsssubm 18779 | The set of words taking va... |
| frmdgsum 18780 | Any word in a free monoid ... |
| frmdss2 18781 | A subset of generators is ... |
| frmdup1 18782 | Any assignment of the gene... |
| frmdup2 18783 | The evaluation map has the... |
| frmdup3lem 18784 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18785 | Universal property of the ... |
| efmnd 18788 | The monoid of endofunction... |
| efmndbas 18789 | The base set of the monoid... |
| efmndbasabf 18790 | The base set of the monoid... |
| elefmndbas 18791 | Two ways of saying a funct... |
| elefmndbas2 18792 | Two ways of saying a funct... |
| efmndbasf 18793 | Elements in the monoid of ... |
| efmndhash 18794 | The monoid of endofunction... |
| efmndbasfi 18795 | The monoid of endofunction... |
| efmndfv 18796 | The function value of an e... |
| efmndtset 18797 | The topology of the monoid... |
| efmndplusg 18798 | The group operation of a m... |
| efmndov 18799 | The value of the group ope... |
| efmndcl 18800 | The group operation of the... |
| efmndtopn 18801 | The topology of the monoid... |
| symggrplem 18802 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18803 | The monoid of endofunction... |
| efmndsgrp 18804 | The monoid of endofunction... |
| ielefmnd 18805 | The identity function rest... |
| efmndid 18806 | The identity function rest... |
| efmndmnd 18807 | The monoid of endofunction... |
| efmnd0nmnd 18808 | Even the monoid of endofun... |
| efmndbas0 18809 | The base set of the monoid... |
| efmnd1hash 18810 | The monoid of endofunction... |
| efmnd1bas 18811 | The monoid of endofunction... |
| efmnd2hash 18812 | The monoid of endofunction... |
| submefmnd 18813 | If the base set of a monoi... |
| sursubmefmnd 18814 | The set of surjective endo... |
| injsubmefmnd 18815 | The set of injective endof... |
| idressubmefmnd 18816 | The singleton containing o... |
| idresefmnd 18817 | The structure with the sin... |
| smndex1ibas 18818 | The modulo function ` I ` ... |
| smndex1iidm 18819 | The modulo function ` I ` ... |
| smndex1gbas 18820 | The constant functions ` (... |
| smndex1gid 18821 | The composition of a const... |
| smndex1igid 18822 | The composition of the mod... |
| smndex1basss 18823 | The modulo function ` I ` ... |
| smndex1bas 18824 | The base set of the monoid... |
| smndex1mgm 18825 | The monoid of endofunction... |
| smndex1sgrp 18826 | The monoid of endofunction... |
| smndex1mndlem 18827 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18828 | The monoid of endofunction... |
| smndex1id 18829 | The modulo function ` I ` ... |
| smndex1n0mnd 18830 | The identity of the monoid... |
| nsmndex1 18831 | The base set ` B ` of the ... |
| smndex2dbas 18832 | The doubling function ` D ... |
| smndex2dnrinv 18833 | The doubling function ` D ... |
| smndex2hbas 18834 | The halving functions ` H ... |
| smndex2dlinvh 18835 | The halving functions ` H ... |
| mgm2nsgrplem1 18836 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18837 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18838 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18839 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18840 | A small magma (with two el... |
| sgrp2nmndlem1 18841 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18842 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18843 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18844 | A small semigroup (with tw... |
| sgrp2rid2ex 18845 | A small semigroup (with tw... |
| sgrp2nmndlem4 18846 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18847 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18848 | A small semigroup (with tw... |
| mgmnsgrpex 18849 | There is a magma which is ... |
| sgrpnmndex 18850 | There is a semigroup which... |
| sgrpssmgm 18851 | The class of all semigroup... |
| mndsssgrp 18852 | The class of all monoids i... |
| pwmndgplus 18853 | The operation of the monoi... |
| pwmndid 18854 | The identity of the monoid... |
| pwmnd 18855 | The power set of a class `... |
| isgrp 18862 | The predicate "is a group"... |
| grpmnd 18863 | A group is a monoid. (Con... |
| grpcl 18864 | Closure of the operation o... |
| grpass 18865 | A group operation is assoc... |
| grpinvex 18866 | Every member of a group ha... |
| grpideu 18867 | The two-sided identity ele... |
| grpassd 18868 | A group operation is assoc... |
| grpmndd 18869 | A group is a monoid. (Con... |
| grpcld 18870 | Closure of the operation o... |
| grpplusf 18871 | The group addition operati... |
| grpplusfo 18872 | The group addition operati... |
| resgrpplusfrn 18873 | The underlying set of a gr... |
| grppropd 18874 | If two structures have the... |
| grpprop 18875 | If two structures have the... |
| grppropstr 18876 | Generalize a specific 2-el... |
| grpss 18877 | Show that a structure exte... |
| isgrpd2e 18878 | Deduce a group from its pr... |
| isgrpd2 18879 | Deduce a group from its pr... |
| isgrpde 18880 | Deduce a group from its pr... |
| isgrpd 18881 | Deduce a group from its pr... |
| isgrpi 18882 | Properties that determine ... |
| grpsgrp 18883 | A group is a semigroup. (... |
| dfgrp2 18884 | Alternate definition of a ... |
| dfgrp2e 18885 | Alternate definition of a ... |
| isgrpix 18886 | Properties that determine ... |
| grpidcl 18887 | The identity element of a ... |
| grpbn0 18888 | The base set of a group is... |
| grplid 18889 | The identity element of a ... |
| grprid 18890 | The identity element of a ... |
| grplidd 18891 | The identity element of a ... |
| grpridd 18892 | The identity element of a ... |
| grpn0 18893 | A group is not empty. (Co... |
| hashfingrpnn 18894 | A finite group has positiv... |
| grprcan 18895 | Right cancellation law for... |
| grpinveu 18896 | The left inverse element o... |
| grpid 18897 | Two ways of saying that an... |
| isgrpid2 18898 | Properties showing that an... |
| grpidd2 18899 | Deduce the identity elemen... |
| grpinvfval 18900 | The inverse function of a ... |
| grpinvfvalALT 18901 | Shorter proof of ~ grpinvf... |
| grpinvval 18902 | The inverse of a group ele... |
| grpinvfn 18903 | Functionality of the group... |
| grpinvfvi 18904 | The group inverse function... |
| grpsubfval 18905 | Group subtraction (divisio... |
| grpsubfvalALT 18906 | Shorter proof of ~ grpsubf... |
| grpsubval 18907 | Group subtraction (divisio... |
| grpinvf 18908 | The group inversion operat... |
| grpinvcl 18909 | A group element's inverse ... |
| grpinvcld 18910 | A group element's inverse ... |
| grplinv 18911 | The left inverse of a grou... |
| grprinv 18912 | The right inverse of a gro... |
| grpinvid1 18913 | The inverse of a group ele... |
| grpinvid2 18914 | The inverse of a group ele... |
| isgrpinv 18915 | Properties showing that a ... |
| grplinvd 18916 | The left inverse of a grou... |
| grprinvd 18917 | The right inverse of a gro... |
| grplrinv 18918 | In a group, every member h... |
| grpidinv2 18919 | A group's properties using... |
| grpidinv 18920 | A group has a left and rig... |
| grpinvid 18921 | The inverse of the identit... |
| grplcan 18922 | Left cancellation law for ... |
| grpasscan1 18923 | An associative cancellatio... |
| grpasscan2 18924 | An associative cancellatio... |
| grpidrcan 18925 | If right adding an element... |
| grpidlcan 18926 | If left adding an element ... |
| grpinvinv 18927 | Double inverse law for gro... |
| grpinvcnv 18928 | The group inverse is its o... |
| grpinv11 18929 | The group inverse is one-t... |
| grpinvf1o 18930 | The group inverse is a one... |
| grpinvnz 18931 | The inverse of a nonzero g... |
| grpinvnzcl 18932 | The inverse of a nonzero g... |
| grpsubinv 18933 | Subtraction of an inverse.... |
| grplmulf1o 18934 | Left multiplication by a g... |
| grpinvpropd 18935 | If two structures have the... |
| grpidssd 18936 | If the base set of a group... |
| grpinvssd 18937 | If the base set of a group... |
| grpinvadd 18938 | The inverse of the group o... |
| grpsubf 18939 | Functionality of group sub... |
| grpsubcl 18940 | Closure of group subtracti... |
| grpsubrcan 18941 | Right cancellation law for... |
| grpinvsub 18942 | Inverse of a group subtrac... |
| grpinvval2 18943 | A ~ df-neg -like equation ... |
| grpsubid 18944 | Subtraction of a group ele... |
| grpsubid1 18945 | Subtraction of the identit... |
| grpsubeq0 18946 | If the difference between ... |
| grpsubadd0sub 18947 | Subtraction expressed as a... |
| grpsubadd 18948 | Relationship between group... |
| grpsubsub 18949 | Double group subtraction. ... |
| grpaddsubass 18950 | Associative-type law for g... |
| grppncan 18951 | Cancellation law for subtr... |
| grpnpcan 18952 | Cancellation law for subtr... |
| grpsubsub4 18953 | Double group subtraction (... |
| grppnpcan2 18954 | Cancellation law for mixed... |
| grpnpncan 18955 | Cancellation law for group... |
| grpnpncan0 18956 | Cancellation law for group... |
| grpnnncan2 18957 | Cancellation law for group... |
| dfgrp3lem 18958 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18959 | Alternate definition of a ... |
| dfgrp3e 18960 | Alternate definition of a ... |
| grplactfval 18961 | The left group action of e... |
| grplactval 18962 | The value of the left grou... |
| grplactcnv 18963 | The left group action of e... |
| grplactf1o 18964 | The left group action of e... |
| grpsubpropd 18965 | Weak property deduction fo... |
| grpsubpropd2 18966 | Strong property deduction ... |
| grp1 18967 | The (smallest) structure r... |
| grp1inv 18968 | The inverse function of th... |
| prdsinvlem 18969 | Characterization of invers... |
| prdsgrpd 18970 | The product of a family of... |
| prdsinvgd 18971 | Negation in a product of g... |
| pwsgrp 18972 | A structure power of a gro... |
| pwsinvg 18973 | Negation in a group power.... |
| pwssub 18974 | Subtraction in a group pow... |
| imasgrp2 18975 | The image structure of a g... |
| imasgrp 18976 | The image structure of a g... |
| imasgrpf1 18977 | The image of a group under... |
| qusgrp2 18978 | Prove that a quotient stru... |
| xpsgrp 18979 | The binary product of grou... |
| xpsinv 18980 | Value of the negation oper... |
| xpsgrpsub 18981 | Value of the subtraction o... |
| mhmlem 18982 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18983 | A surjective monoid morphi... |
| mhmmnd 18984 | The image of a monoid ` G ... |
| mhmfmhm 18985 | The function fulfilling th... |
| ghmgrp 18986 | The image of a group ` G `... |
| mulgfval 18989 | Group multiple (exponentia... |
| mulgfvalALT 18990 | Shorter proof of ~ mulgfva... |
| mulgval 18991 | Value of the group multipl... |
| mulgfn 18992 | Functionality of the group... |
| mulgfvi 18993 | The group multiple operati... |
| mulg0 18994 | Group multiple (exponentia... |
| mulgnn 18995 | Group multiple (exponentia... |
| mulgnngsum 18996 | Group multiple (exponentia... |
| mulgnn0gsum 18997 | Group multiple (exponentia... |
| mulg1 18998 | Group multiple (exponentia... |
| mulgnnp1 18999 | Group multiple (exponentia... |
| mulg2 19000 | Group multiple (exponentia... |
| mulgnegnn 19001 | Group multiple (exponentia... |
| mulgnn0p1 19002 | Group multiple (exponentia... |
| mulgnnsubcl 19003 | Closure of the group multi... |
| mulgnn0subcl 19004 | Closure of the group multi... |
| mulgsubcl 19005 | Closure of the group multi... |
| mulgnncl 19006 | Closure of the group multi... |
| mulgnn0cl 19007 | Closure of the group multi... |
| mulgcl 19008 | Closure of the group multi... |
| mulgneg 19009 | Group multiple (exponentia... |
| mulgnegneg 19010 | The inverse of a negative ... |
| mulgm1 19011 | Group multiple (exponentia... |
| mulgnn0cld 19012 | Closure of the group multi... |
| mulgcld 19013 | Deduction associated with ... |
| mulgaddcomlem 19014 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 19015 | The group multiple operato... |
| mulginvcom 19016 | The group multiple operato... |
| mulginvinv 19017 | The group multiple operato... |
| mulgnn0z 19018 | A group multiple of the id... |
| mulgz 19019 | A group multiple of the id... |
| mulgnndir 19020 | Sum of group multiples, fo... |
| mulgnn0dir 19021 | Sum of group multiples, ge... |
| mulgdirlem 19022 | Lemma for ~ mulgdir . (Co... |
| mulgdir 19023 | Sum of group multiples, ge... |
| mulgp1 19024 | Group multiple (exponentia... |
| mulgneg2 19025 | Group multiple (exponentia... |
| mulgnnass 19026 | Product of group multiples... |
| mulgnn0ass 19027 | Product of group multiples... |
| mulgass 19028 | Product of group multiples... |
| mulgassr 19029 | Reversed product of group ... |
| mulgmodid 19030 | Casting out multiples of t... |
| mulgsubdir 19031 | Distribution of group mult... |
| mhmmulg 19032 | A homomorphism of monoids ... |
| mulgpropd 19033 | Two structures with the sa... |
| submmulgcl 19034 | Closure of the group multi... |
| submmulg 19035 | A group multiple is the sa... |
| pwsmulg 19036 | Value of a group multiple ... |
| issubg 19043 | The subgroup predicate. (... |
| subgss 19044 | A subgroup is a subset. (... |
| subgid 19045 | A group is a subgroup of i... |
| subggrp 19046 | A subgroup is a group. (C... |
| subgbas 19047 | The base of the restricted... |
| subgrcl 19048 | Reverse closure for the su... |
| subg0 19049 | A subgroup of a group must... |
| subginv 19050 | The inverse of an element ... |
| subg0cl 19051 | The group identity is an e... |
| subginvcl 19052 | The inverse of an element ... |
| subgcl 19053 | A subgroup is closed under... |
| subgsubcl 19054 | A subgroup is closed under... |
| subgsub 19055 | The subtraction of element... |
| subgmulgcl 19056 | Closure of the group multi... |
| subgmulg 19057 | A group multiple is the sa... |
| issubg2 19058 | Characterize the subgroups... |
| issubgrpd2 19059 | Prove a subgroup by closur... |
| issubgrpd 19060 | Prove a subgroup by closur... |
| issubg3 19061 | A subgroup is a symmetric ... |
| issubg4 19062 | A subgroup is a nonempty s... |
| grpissubg 19063 | If the base set of a group... |
| resgrpisgrp 19064 | If the base set of a group... |
| subgsubm 19065 | A subgroup is a submonoid.... |
| subsubg 19066 | A subgroup of a subgroup i... |
| subgint 19067 | The intersection of a none... |
| 0subg 19068 | The zero subgroup of an ar... |
| 0subgOLD 19069 | Obsolete version of ~ 0sub... |
| trivsubgd 19070 | The only subgroup of a tri... |
| trivsubgsnd 19071 | The only subgroup of a tri... |
| isnsg 19072 | Property of being a normal... |
| isnsg2 19073 | Weaken the condition of ~ ... |
| nsgbi 19074 | Defining property of a nor... |
| nsgsubg 19075 | A normal subgroup is a sub... |
| nsgconj 19076 | The conjugation of an elem... |
| isnsg3 19077 | A subgroup is normal iff t... |
| subgacs 19078 | Subgroups are an algebraic... |
| nsgacs 19079 | Normal subgroups form an a... |
| elnmz 19080 | Elementhood in the normali... |
| nmzbi 19081 | Defining property of the n... |
| nmzsubg 19082 | The normalizer N_G(S) of a... |
| ssnmz 19083 | A subgroup is a subset of ... |
| isnsg4 19084 | A subgroup is normal iff i... |
| nmznsg 19085 | Any subgroup is a normal s... |
| 0nsg 19086 | The zero subgroup is norma... |
| nsgid 19087 | The whole group is a norma... |
| 0idnsgd 19088 | The whole group and the ze... |
| trivnsgd 19089 | The only normal subgroup o... |
| triv1nsgd 19090 | A trivial group has exactl... |
| 1nsgtrivd 19091 | A group with exactly one n... |
| releqg 19092 | The left coset equivalence... |
| eqgfval 19093 | Value of the subgroup left... |
| eqgval 19094 | Value of the subgroup left... |
| eqger 19095 | The subgroup coset equival... |
| eqglact 19096 | A left coset can be expres... |
| eqgid 19097 | The left coset containing ... |
| eqgen 19098 | Each coset is equipotent t... |
| eqgcpbl 19099 | The subgroup coset equival... |
| quselbas 19100 | Membership in the base set... |
| quseccl0 19101 | Closure of the quotient ma... |
| qusgrp 19102 | If ` Y ` is a normal subgr... |
| quseccl 19103 | Closure of the quotient ma... |
| qusadd 19104 | Value of the group operati... |
| qus0 19105 | Value of the group identit... |
| qusinv 19106 | Value of the group inverse... |
| qussub 19107 | Value of the group subtrac... |
| ecqusaddd 19108 | Addition of equivalence cl... |
| ecqusaddcl 19109 | Closure of the addition in... |
| lagsubg2 19110 | Lagrange's theorem for fin... |
| lagsubg 19111 | Lagrange's theorem for Gro... |
| eqg0subg 19112 | The coset equivalence rela... |
| eqg0subgecsn 19113 | The equivalence classes mo... |
| qus0subgbas 19114 | The base set of a quotient... |
| qus0subgadd 19115 | The addition in a quotient... |
| cycsubmel 19116 | Characterization of an ele... |
| cycsubmcl 19117 | The set of nonnegative int... |
| cycsubm 19118 | The set of nonnegative int... |
| cyccom 19119 | Condition for an operation... |
| cycsubmcom 19120 | The operation of a monoid ... |
| cycsubggend 19121 | The cyclic subgroup genera... |
| cycsubgcl 19122 | The set of integer powers ... |
| cycsubgss 19123 | The cyclic subgroup genera... |
| cycsubg 19124 | The cyclic group generated... |
| cycsubgcld 19125 | The cyclic subgroup genera... |
| cycsubg2 19126 | The subgroup generated by ... |
| cycsubg2cl 19127 | Any multiple of an element... |
| reldmghm 19130 | Lemma for group homomorphi... |
| isghm 19131 | Property of being a homomo... |
| isghm3 19132 | Property of a group homomo... |
| ghmgrp1 19133 | A group homomorphism is on... |
| ghmgrp2 19134 | A group homomorphism is on... |
| ghmf 19135 | A group homomorphism is a ... |
| ghmlin 19136 | A homomorphism of groups i... |
| ghmid 19137 | A homomorphism of groups p... |
| ghminv 19138 | A homomorphism of groups p... |
| ghmsub 19139 | Linearity of subtraction t... |
| isghmd 19140 | Deduction for a group homo... |
| ghmmhm 19141 | A group homomorphism is a ... |
| ghmmhmb 19142 | Group homomorphisms and mo... |
| ghmmulg 19143 | A homomorphism of monoids ... |
| ghmrn 19144 | The range of a homomorphis... |
| 0ghm 19145 | The constant zero linear f... |
| idghm 19146 | The identity homomorphism ... |
| resghm 19147 | Restriction of a homomorph... |
| resghm2 19148 | One direction of ~ resghm2... |
| resghm2b 19149 | Restriction of the codomai... |
| ghmghmrn 19150 | A group homomorphism from ... |
| ghmco 19151 | The composition of group h... |
| ghmima 19152 | The image of a subgroup un... |
| ghmpreima 19153 | The inverse image of a sub... |
| ghmeql 19154 | The equalizer of two group... |
| ghmnsgima 19155 | The image of a normal subg... |
| ghmnsgpreima 19156 | The inverse image of a nor... |
| ghmker 19157 | The kernel of a homomorphi... |
| ghmeqker 19158 | Two source points map to t... |
| pwsdiagghm 19159 | Diagonal homomorphism into... |
| f1ghm0to0 19160 | If a group homomorphism ` ... |
| ghmf1 19161 | Two ways of saying a group... |
| kerf1ghm 19162 | A group homomorphism ` F `... |
| ghmf1o 19163 | A bijective group homomorp... |
| conjghm 19164 | Conjugation is an automorp... |
| conjsubg 19165 | A conjugated subgroup is a... |
| conjsubgen 19166 | A conjugated subgroup is e... |
| conjnmz 19167 | A subgroup is unchanged un... |
| conjnmzb 19168 | Alternative condition for ... |
| conjnsg 19169 | A normal subgroup is uncha... |
| qusghm 19170 | If ` Y ` is a normal subgr... |
| ghmpropd 19171 | Group homomorphism depends... |
| gimfn 19176 | The group isomorphism func... |
| isgim 19177 | An isomorphism of groups i... |
| gimf1o 19178 | An isomorphism of groups i... |
| gimghm 19179 | An isomorphism of groups i... |
| isgim2 19180 | A group isomorphism is a h... |
| subggim 19181 | Behavior of subgroups unde... |
| gimcnv 19182 | The converse of a bijectiv... |
| gimco 19183 | The composition of group i... |
| gim0to0 19184 | A group isomorphism maps t... |
| brgic 19185 | The relation "is isomorphi... |
| brgici 19186 | Prove isomorphic by an exp... |
| gicref 19187 | Isomorphism is reflexive. ... |
| giclcl 19188 | Isomorphism implies the le... |
| gicrcl 19189 | Isomorphism implies the ri... |
| gicsym 19190 | Isomorphism is symmetric. ... |
| gictr 19191 | Isomorphism is transitive.... |
| gicer 19192 | Isomorphism is an equivale... |
| gicen 19193 | Isomorphic groups have equ... |
| gicsubgen 19194 | A less trivial example of ... |
| isga 19197 | The predicate "is a (left)... |
| gagrp 19198 | The left argument of a gro... |
| gaset 19199 | The right argument of a gr... |
| gagrpid 19200 | The identity of the group ... |
| gaf 19201 | The mapping of the group a... |
| gafo 19202 | A group action is onto its... |
| gaass 19203 | An "associative" property ... |
| ga0 19204 | The action of a group on t... |
| gaid 19205 | The trivial action of a gr... |
| subgga 19206 | A subgroup acts on its par... |
| gass 19207 | A subset of a group action... |
| gasubg 19208 | The restriction of a group... |
| gaid2 19209 | A group operation is a lef... |
| galcan 19210 | The action of a particular... |
| gacan 19211 | Group inverses cancel in a... |
| gapm 19212 | The action of a particular... |
| gaorb 19213 | The orbit equivalence rela... |
| gaorber 19214 | The orbit equivalence rela... |
| gastacl 19215 | The stabilizer subgroup in... |
| gastacos 19216 | Write the coset relation f... |
| orbstafun 19217 | Existence and uniqueness f... |
| orbstaval 19218 | Value of the function at a... |
| orbsta 19219 | The Orbit-Stabilizer theor... |
| orbsta2 19220 | Relation between the size ... |
| cntrval 19225 | Substitute definition of t... |
| cntzfval 19226 | First level substitution f... |
| cntzval 19227 | Definition substitution fo... |
| elcntz 19228 | Elementhood in the central... |
| cntzel 19229 | Membership in a centralize... |
| cntzsnval 19230 | Special substitution for t... |
| elcntzsn 19231 | Value of the centralizer o... |
| sscntz 19232 | A centralizer expression f... |
| cntzrcl 19233 | Reverse closure for elemen... |
| cntzssv 19234 | The centralizer is uncondi... |
| cntzi 19235 | Membership in a centralize... |
| elcntr 19236 | Elementhood in the center ... |
| cntrss 19237 | The center is a subset of ... |
| cntri 19238 | Defining property of the c... |
| resscntz 19239 | Centralizer in a substruct... |
| cntzsgrpcl 19240 | Centralizers are closed un... |
| cntz2ss 19241 | Centralizers reverse the s... |
| cntzrec 19242 | Reciprocity relationship f... |
| cntziinsn 19243 | Express any centralizer as... |
| cntzsubm 19244 | Centralizers in a monoid a... |
| cntzsubg 19245 | Centralizers in a group ar... |
| cntzidss 19246 | If the elements of ` S ` c... |
| cntzmhm 19247 | Centralizers in a monoid a... |
| cntzmhm2 19248 | Centralizers in a monoid a... |
| cntrsubgnsg 19249 | A central subgroup is norm... |
| cntrnsg 19250 | The center of a group is a... |
| oppgval 19253 | Value of the opposite grou... |
| oppgplusfval 19254 | Value of the addition oper... |
| oppgplus 19255 | Value of the addition oper... |
| setsplusg 19256 | The other components of an... |
| oppglemOLD 19257 | Obsolete version of ~ sets... |
| oppgbas 19258 | Base set of an opposite gr... |
| oppgbasOLD 19259 | Obsolete version of ~ oppg... |
| oppgtset 19260 | Topology of an opposite gr... |
| oppgtsetOLD 19261 | Obsolete version of ~ oppg... |
| oppgtopn 19262 | Topology of an opposite gr... |
| oppgmnd 19263 | The opposite of a monoid i... |
| oppgmndb 19264 | Bidirectional form of ~ op... |
| oppgid 19265 | Zero in a monoid is a symm... |
| oppggrp 19266 | The opposite of a group is... |
| oppggrpb 19267 | Bidirectional form of ~ op... |
| oppginv 19268 | Inverses in a group are a ... |
| invoppggim 19269 | The inverse is an antiauto... |
| oppggic 19270 | Every group is (naturally)... |
| oppgsubm 19271 | Being a submonoid is a sym... |
| oppgsubg 19272 | Being a subgroup is a symm... |
| oppgcntz 19273 | A centralizer in a group i... |
| oppgcntr 19274 | The center of a group is t... |
| gsumwrev 19275 | A sum in an opposite monoi... |
| symgval 19278 | The value of the symmetric... |
| permsetexOLD 19279 | Obsolete version of ~ f1os... |
| symgbas 19280 | The base set of the symmet... |
| symgbasexOLD 19281 | Obsolete as of 8-Aug-2024.... |
| elsymgbas2 19282 | Two ways of saying a funct... |
| elsymgbas 19283 | Two ways of saying a funct... |
| symgbasf1o 19284 | Elements in the symmetric ... |
| symgbasf 19285 | A permutation (element of ... |
| symgbasmap 19286 | A permutation (element of ... |
| symghash 19287 | The symmetric group on ` n... |
| symgbasfi 19288 | The symmetric group on a f... |
| symgfv 19289 | The function value of a pe... |
| symgfvne 19290 | The function values of a p... |
| symgressbas 19291 | The symmetric group on ` A... |
| symgplusg 19292 | The group operation of a s... |
| symgov 19293 | The value of the group ope... |
| symgcl 19294 | The group operation of the... |
| idresperm 19295 | The identity function rest... |
| symgmov1 19296 | For a permutation of a set... |
| symgmov2 19297 | For a permutation of a set... |
| symgbas0 19298 | The base set of the symmet... |
| symg1hash 19299 | The symmetric group on a s... |
| symg1bas 19300 | The symmetric group on a s... |
| symg2hash 19301 | The symmetric group on a (... |
| symg2bas 19302 | The symmetric group on a p... |
| 0symgefmndeq 19303 | The symmetric group on the... |
| snsymgefmndeq 19304 | The symmetric group on a s... |
| symgpssefmnd 19305 | For a set ` A ` with more ... |
| symgvalstruct 19306 | The value of the symmetric... |
| symgvalstructOLD 19307 | Obsolete proof of ~ symgva... |
| symgsubmefmnd 19308 | The symmetric group on a s... |
| symgtset 19309 | The topology of the symmet... |
| symggrp 19310 | The symmetric group on a s... |
| symgid 19311 | The group identity element... |
| symginv 19312 | The group inverse in the s... |
| symgsubmefmndALT 19313 | The symmetric group on a s... |
| galactghm 19314 | The currying of a group ac... |
| lactghmga 19315 | The converse of ~ galactgh... |
| symgtopn 19316 | The topology of the symmet... |
| symgga 19317 | The symmetric group induce... |
| pgrpsubgsymgbi 19318 | Every permutation group is... |
| pgrpsubgsymg 19319 | Every permutation group is... |
| idressubgsymg 19320 | The singleton containing o... |
| idrespermg 19321 | The structure with the sin... |
| cayleylem1 19322 | Lemma for ~ cayley . (Con... |
| cayleylem2 19323 | Lemma for ~ cayley . (Con... |
| cayley 19324 | Cayley's Theorem (construc... |
| cayleyth 19325 | Cayley's Theorem (existenc... |
| symgfix2 19326 | If a permutation does not ... |
| symgextf 19327 | The extension of a permuta... |
| symgextfv 19328 | The function value of the ... |
| symgextfve 19329 | The function value of the ... |
| symgextf1lem 19330 | Lemma for ~ symgextf1 . (... |
| symgextf1 19331 | The extension of a permuta... |
| symgextfo 19332 | The extension of a permuta... |
| symgextf1o 19333 | The extension of a permuta... |
| symgextsymg 19334 | The extension of a permuta... |
| symgextres 19335 | The restriction of the ext... |
| gsumccatsymgsn 19336 | Homomorphic property of co... |
| gsmsymgrfixlem1 19337 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19338 | The composition of permuta... |
| fvcosymgeq 19339 | The values of two composit... |
| gsmsymgreqlem1 19340 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19341 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19342 | Two combination of permuta... |
| symgfixelq 19343 | A permutation of a set fix... |
| symgfixels 19344 | The restriction of a permu... |
| symgfixelsi 19345 | The restriction of a permu... |
| symgfixf 19346 | The mapping of a permutati... |
| symgfixf1 19347 | The mapping of a permutati... |
| symgfixfolem1 19348 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19349 | The mapping of a permutati... |
| symgfixf1o 19350 | The mapping of a permutati... |
| f1omvdmvd 19353 | A permutation of any class... |
| f1omvdcnv 19354 | A permutation and its inve... |
| mvdco 19355 | Composing two permutations... |
| f1omvdconj 19356 | Conjugation of a permutati... |
| f1otrspeq 19357 | A transposition is charact... |
| f1omvdco2 19358 | If exactly one of two perm... |
| f1omvdco3 19359 | If a point is moved by exa... |
| pmtrfval 19360 | The function generating tr... |
| pmtrval 19361 | A generated transposition,... |
| pmtrfv 19362 | General value of mapping a... |
| pmtrprfv 19363 | In a transposition of two ... |
| pmtrprfv3 19364 | In a transposition of two ... |
| pmtrf 19365 | Functionality of a transpo... |
| pmtrmvd 19366 | A transposition moves prec... |
| pmtrrn 19367 | Transposing two points giv... |
| pmtrfrn 19368 | A transposition (as a kind... |
| pmtrffv 19369 | Mapping of a point under a... |
| pmtrrn2 19370 | For any transposition ther... |
| pmtrfinv 19371 | A transposition function i... |
| pmtrfmvdn0 19372 | A transposition moves at l... |
| pmtrff1o 19373 | A transposition function i... |
| pmtrfcnv 19374 | A transposition function i... |
| pmtrfb 19375 | An intrinsic characterizat... |
| pmtrfconj 19376 | Any conjugate of a transpo... |
| symgsssg 19377 | The symmetric group has su... |
| symgfisg 19378 | The symmetric group has a ... |
| symgtrf 19379 | Transpositions are element... |
| symggen 19380 | The span of the transposit... |
| symggen2 19381 | A finite permutation group... |
| symgtrinv 19382 | To invert a permutation re... |
| pmtr3ncomlem1 19383 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19384 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19385 | Transpositions over sets w... |
| pmtrdifellem1 19386 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19387 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19388 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19389 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19390 | A transposition of element... |
| pmtrdifwrdellem1 19391 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19392 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19393 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19394 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19395 | A sequence of transpositio... |
| pmtrdifwrdel2 19396 | A sequence of transpositio... |
| pmtrprfval 19397 | The transpositions on a pa... |
| pmtrprfvalrn 19398 | The range of the transposi... |
| psgnunilem1 19403 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19404 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19405 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19406 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19407 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19408 | Addition and subtraction o... |
| psgnuni 19409 | If the same permutation ca... |
| psgnfval 19410 | Function definition of the... |
| psgnfn 19411 | Functionality and domain o... |
| psgndmsubg 19412 | The finitary permutations ... |
| psgneldm 19413 | Property of being a finita... |
| psgneldm2 19414 | The finitary permutations ... |
| psgneldm2i 19415 | A sequence of transpositio... |
| psgneu 19416 | A finitary permutation has... |
| psgnval 19417 | Value of the permutation s... |
| psgnvali 19418 | A finitary permutation has... |
| psgnvalii 19419 | Any representation of a pe... |
| psgnpmtr 19420 | All transpositions are odd... |
| psgn0fv0 19421 | The permutation sign funct... |
| sygbasnfpfi 19422 | The class of non-fixed poi... |
| psgnfvalfi 19423 | Function definition of the... |
| psgnvalfi 19424 | Value of the permutation s... |
| psgnran 19425 | The range of the permutati... |
| gsmtrcl 19426 | The group sum of transposi... |
| psgnfitr 19427 | A permutation of a finite ... |
| psgnfieu 19428 | A permutation of a finite ... |
| pmtrsn 19429 | The value of the transposi... |
| psgnsn 19430 | The permutation sign funct... |
| psgnprfval 19431 | The permutation sign funct... |
| psgnprfval1 19432 | The permutation sign of th... |
| psgnprfval2 19433 | The permutation sign of th... |
| odfval 19442 | Value of the order functio... |
| odfvalALT 19443 | Shorter proof of ~ odfval ... |
| odval 19444 | Second substitution for th... |
| odlem1 19445 | The group element order is... |
| odcl 19446 | The order of a group eleme... |
| odf 19447 | Functionality of the group... |
| odid 19448 | Any element to the power o... |
| odlem2 19449 | Any positive annihilator o... |
| odmodnn0 19450 | Reduce the argument of a g... |
| mndodconglem 19451 | Lemma for ~ mndodcong . (... |
| mndodcong 19452 | If two multipliers are con... |
| mndodcongi 19453 | If two multipliers are con... |
| oddvdsnn0 19454 | The only multiples of ` A ... |
| odnncl 19455 | If a nonzero multiple of a... |
| odmod 19456 | Reduce the argument of a g... |
| oddvds 19457 | The only multiples of ` A ... |
| oddvdsi 19458 | Any group element is annih... |
| odcong 19459 | If two multipliers are con... |
| odeq 19460 | The ~ oddvds property uniq... |
| odval2 19461 | A non-conditional definiti... |
| odcld 19462 | The order of a group eleme... |
| odm1inv 19463 | The (order-1)th multiple o... |
| odmulgid 19464 | A relationship between the... |
| odmulg2 19465 | The order of a multiple di... |
| odmulg 19466 | Relationship between the o... |
| odmulgeq 19467 | A multiple of a point of f... |
| odbezout 19468 | If ` N ` is coprime to the... |
| od1 19469 | The order of the group ide... |
| odeq1 19470 | The group identity is the ... |
| odinv 19471 | The order of the inverse o... |
| odf1 19472 | The multiples of an elemen... |
| odinf 19473 | The multiples of an elemen... |
| dfod2 19474 | An alternative definition ... |
| odcl2 19475 | The order of an element of... |
| oddvds2 19476 | The order of an element of... |
| finodsubmsubg 19477 | A submonoid whose elements... |
| 0subgALT 19478 | A shorter proof of ~ 0subg... |
| submod 19479 | The order of an element is... |
| subgod 19480 | The order of an element is... |
| odsubdvds 19481 | The order of an element of... |
| odf1o1 19482 | An element with zero order... |
| odf1o2 19483 | An element with nonzero or... |
| odhash 19484 | An element of zero order g... |
| odhash2 19485 | If an element has nonzero ... |
| odhash3 19486 | An element which generates... |
| odngen 19487 | A cyclic subgroup of size ... |
| gexval 19488 | Value of the exponent of a... |
| gexlem1 19489 | The group element order is... |
| gexcl 19490 | The exponent of a group is... |
| gexid 19491 | Any element to the power o... |
| gexlem2 19492 | Any positive annihilator o... |
| gexdvdsi 19493 | Any group element is annih... |
| gexdvds 19494 | The only ` N ` that annihi... |
| gexdvds2 19495 | An integer divides the gro... |
| gexod 19496 | Any group element is annih... |
| gexcl3 19497 | If the order of every grou... |
| gexnnod 19498 | Every group element has fi... |
| gexcl2 19499 | The exponent of a finite g... |
| gexdvds3 19500 | The exponent of a finite g... |
| gex1 19501 | A group or monoid has expo... |
| ispgp 19502 | A group is a ` P ` -group ... |
| pgpprm 19503 | Reverse closure for the fi... |
| pgpgrp 19504 | Reverse closure for the se... |
| pgpfi1 19505 | A finite group with order ... |
| pgp0 19506 | The identity subgroup is a... |
| subgpgp 19507 | A subgroup of a p-group is... |
| sylow1lem1 19508 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19509 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19510 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19511 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19512 | Lemma for ~ sylow1 . Usin... |
| sylow1 19513 | Sylow's first theorem. If... |
| odcau 19514 | Cauchy's theorem for the o... |
| pgpfi 19515 | The converse to ~ pgpfi1 .... |
| pgpfi2 19516 | Alternate version of ~ pgp... |
| pgphash 19517 | The order of a p-group. (... |
| isslw 19518 | The property of being a Sy... |
| slwprm 19519 | Reverse closure for the fi... |
| slwsubg 19520 | A Sylow ` P ` -subgroup is... |
| slwispgp 19521 | Defining property of a Syl... |
| slwpss 19522 | A proper superset of a Syl... |
| slwpgp 19523 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19524 | Every ` P ` -subgroup is c... |
| slwn0 19525 | Every finite group contain... |
| subgslw 19526 | A Sylow subgroup that is c... |
| sylow2alem1 19527 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19528 | Lemma for ~ sylow2a . All... |
| sylow2a 19529 | A named lemma of Sylow's s... |
| sylow2blem1 19530 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19531 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19532 | Sylow's second theorem. P... |
| sylow2b 19533 | Sylow's second theorem. A... |
| slwhash 19534 | A sylow subgroup has cardi... |
| fislw 19535 | The sylow subgroups of a f... |
| sylow2 19536 | Sylow's second theorem. S... |
| sylow3lem1 19537 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19538 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19539 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19540 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19541 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19542 | Lemma for ~ sylow3 , secon... |
| sylow3 19543 | Sylow's third theorem. Th... |
| lsmfval 19548 | The subgroup sum function ... |
| lsmvalx 19549 | Subspace sum value (for a ... |
| lsmelvalx 19550 | Subspace sum membership (f... |
| lsmelvalix 19551 | Subspace sum membership (f... |
| oppglsm 19552 | The subspace sum operation... |
| lsmssv 19553 | Subgroup sum is a subset o... |
| lsmless1x 19554 | Subset implies subgroup su... |
| lsmless2x 19555 | Subset implies subgroup su... |
| lsmub1x 19556 | Subgroup sum is an upper b... |
| lsmub2x 19557 | Subgroup sum is an upper b... |
| lsmval 19558 | Subgroup sum value (for a ... |
| lsmelval 19559 | Subgroup sum membership (f... |
| lsmelvali 19560 | Subgroup sum membership (f... |
| lsmelvalm 19561 | Subgroup sum membership an... |
| lsmelvalmi 19562 | Membership of vector subtr... |
| lsmsubm 19563 | The sum of two commuting s... |
| lsmsubg 19564 | The sum of two commuting s... |
| lsmcom2 19565 | Subgroup sum commutes. (C... |
| smndlsmidm 19566 | The direct product is idem... |
| lsmub1 19567 | Subgroup sum is an upper b... |
| lsmub2 19568 | Subgroup sum is an upper b... |
| lsmunss 19569 | Union of subgroups is a su... |
| lsmless1 19570 | Subset implies subgroup su... |
| lsmless2 19571 | Subset implies subgroup su... |
| lsmless12 19572 | Subset implies subgroup su... |
| lsmidm 19573 | Subgroup sum is idempotent... |
| lsmlub 19574 | The least upper bound prop... |
| lsmss1 19575 | Subgroup sum with a subset... |
| lsmss1b 19576 | Subgroup sum with a subset... |
| lsmss2 19577 | Subgroup sum with a subset... |
| lsmss2b 19578 | Subgroup sum with a subset... |
| lsmass 19579 | Subgroup sum is associativ... |
| mndlsmidm 19580 | Subgroup sum is idempotent... |
| lsm01 19581 | Subgroup sum with the zero... |
| lsm02 19582 | Subgroup sum with the zero... |
| subglsm 19583 | The subgroup sum evaluated... |
| lssnle 19584 | Equivalent expressions for... |
| lsmmod 19585 | The modular law holds for ... |
| lsmmod2 19586 | Modular law dual for subgr... |
| lsmpropd 19587 | If two structures have the... |
| cntzrecd 19588 | Commute the "subgroups com... |
| lsmcntz 19589 | The "subgroups commute" pr... |
| lsmcntzr 19590 | The "subgroups commute" pr... |
| lsmdisj 19591 | Disjointness from a subgro... |
| lsmdisj2 19592 | Association of the disjoin... |
| lsmdisj3 19593 | Association of the disjoin... |
| lsmdisjr 19594 | Disjointness from a subgro... |
| lsmdisj2r 19595 | Association of the disjoin... |
| lsmdisj3r 19596 | Association of the disjoin... |
| lsmdisj2a 19597 | Association of the disjoin... |
| lsmdisj2b 19598 | Association of the disjoin... |
| lsmdisj3a 19599 | Association of the disjoin... |
| lsmdisj3b 19600 | Association of the disjoin... |
| subgdisj1 19601 | Vectors belonging to disjo... |
| subgdisj2 19602 | Vectors belonging to disjo... |
| subgdisjb 19603 | Vectors belonging to disjo... |
| pj1fval 19604 | The left projection functi... |
| pj1val 19605 | The left projection functi... |
| pj1eu 19606 | Uniqueness of a left proje... |
| pj1f 19607 | The left projection functi... |
| pj2f 19608 | The right projection funct... |
| pj1id 19609 | Any element of a direct su... |
| pj1eq 19610 | Any element of a direct su... |
| pj1lid 19611 | The left projection functi... |
| pj1rid 19612 | The left projection functi... |
| pj1ghm 19613 | The left projection functi... |
| pj1ghm2 19614 | The left projection functi... |
| lsmhash 19615 | The order of the direct pr... |
| efgmval 19622 | Value of the formal invers... |
| efgmf 19623 | The formal inverse operati... |
| efgmnvl 19624 | The inversion function on ... |
| efgrcl 19625 | Lemma for ~ efgval . (Con... |
| efglem 19626 | Lemma for ~ efgval . (Con... |
| efgval 19627 | Value of the free group co... |
| efger 19628 | Value of the free group co... |
| efgi 19629 | Value of the free group co... |
| efgi0 19630 | Value of the free group co... |
| efgi1 19631 | Value of the free group co... |
| efgtf 19632 | Value of the free group co... |
| efgtval 19633 | Value of the extension fun... |
| efgval2 19634 | Value of the free group co... |
| efgi2 19635 | Value of the free group co... |
| efgtlen 19636 | Value of the free group co... |
| efginvrel2 19637 | The inverse of the reverse... |
| efginvrel1 19638 | The inverse of the reverse... |
| efgsf 19639 | Value of the auxiliary fun... |
| efgsdm 19640 | Elementhood in the domain ... |
| efgsval 19641 | Value of the auxiliary fun... |
| efgsdmi 19642 | Property of the last link ... |
| efgsval2 19643 | Value of the auxiliary fun... |
| efgsrel 19644 | The start and end of any e... |
| efgs1 19645 | A singleton of an irreduci... |
| efgs1b 19646 | Every extension sequence e... |
| efgsp1 19647 | If ` F ` is an extension s... |
| efgsres 19648 | An initial segment of an e... |
| efgsfo 19649 | For any word, there is a s... |
| efgredlema 19650 | The reduced word that form... |
| efgredlemf 19651 | Lemma for ~ efgredleme . ... |
| efgredlemg 19652 | Lemma for ~ efgred . (Con... |
| efgredleme 19653 | Lemma for ~ efgred . (Con... |
| efgredlemd 19654 | The reduced word that form... |
| efgredlemc 19655 | The reduced word that form... |
| efgredlemb 19656 | The reduced word that form... |
| efgredlem 19657 | The reduced word that form... |
| efgred 19658 | The reduced word that form... |
| efgrelexlema 19659 | If two words ` A , B ` are... |
| efgrelexlemb 19660 | If two words ` A , B ` are... |
| efgrelex 19661 | If two words ` A , B ` are... |
| efgredeu 19662 | There is a unique reduced ... |
| efgred2 19663 | Two extension sequences ha... |
| efgcpbllema 19664 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19665 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19666 | Two extension sequences ha... |
| efgcpbl2 19667 | Two extension sequences ha... |
| frgpval 19668 | Value of the free group co... |
| frgpcpbl 19669 | Compatibility of the group... |
| frgp0 19670 | The free group is a group.... |
| frgpeccl 19671 | Closure of the quotient ma... |
| frgpgrp 19672 | The free group is a group.... |
| frgpadd 19673 | Addition in the free group... |
| frgpinv 19674 | The inverse of an element ... |
| frgpmhm 19675 | The "natural map" from wor... |
| vrgpfval 19676 | The canonical injection fr... |
| vrgpval 19677 | The value of the generatin... |
| vrgpf 19678 | The mapping from the index... |
| vrgpinv 19679 | The inverse of a generatin... |
| frgpuptf 19680 | Any assignment of the gene... |
| frgpuptinv 19681 | Any assignment of the gene... |
| frgpuplem 19682 | Any assignment of the gene... |
| frgpupf 19683 | Any assignment of the gene... |
| frgpupval 19684 | Any assignment of the gene... |
| frgpup1 19685 | Any assignment of the gene... |
| frgpup2 19686 | The evaluation map has the... |
| frgpup3lem 19687 | The evaluation map has the... |
| frgpup3 19688 | Universal property of the ... |
| 0frgp 19689 | The free group on zero gen... |
| isabl 19694 | The predicate "is an Abeli... |
| ablgrp 19695 | An Abelian group is a grou... |
| ablgrpd 19696 | An Abelian group is a grou... |
| ablcmn 19697 | An Abelian group is a comm... |
| ablcmnd 19698 | An Abelian group is a comm... |
| iscmn 19699 | The predicate "is a commut... |
| isabl2 19700 | The predicate "is an Abeli... |
| cmnpropd 19701 | If two structures have the... |
| ablpropd 19702 | If two structures have the... |
| ablprop 19703 | If two structures have the... |
| iscmnd 19704 | Properties that determine ... |
| isabld 19705 | Properties that determine ... |
| isabli 19706 | Properties that determine ... |
| cmnmnd 19707 | A commutative monoid is a ... |
| cmncom 19708 | A commutative monoid is co... |
| ablcom 19709 | An Abelian group operation... |
| cmn32 19710 | Commutative/associative la... |
| cmn4 19711 | Commutative/associative la... |
| cmn12 19712 | Commutative/associative la... |
| abl32 19713 | Commutative/associative la... |
| cmnmndd 19714 | A commutative monoid is a ... |
| cmnbascntr 19715 | The base set of a commutat... |
| rinvmod 19716 | Uniqueness of a right inve... |
| ablinvadd 19717 | The inverse of an Abelian ... |
| ablsub2inv 19718 | Abelian group subtraction ... |
| ablsubadd 19719 | Relationship between Abeli... |
| ablsub4 19720 | Commutative/associative su... |
| abladdsub4 19721 | Abelian group addition/sub... |
| abladdsub 19722 | Associative-type law for g... |
| ablsubadd23 19723 | Commutative/associative la... |
| ablsubaddsub 19724 | Double subtraction and add... |
| ablpncan2 19725 | Cancellation law for subtr... |
| ablpncan3 19726 | A cancellation law for Abe... |
| ablsubsub 19727 | Law for double subtraction... |
| ablsubsub4 19728 | Law for double subtraction... |
| ablpnpcan 19729 | Cancellation law for mixed... |
| ablnncan 19730 | Cancellation law for group... |
| ablsub32 19731 | Swap the second and third ... |
| ablnnncan 19732 | Cancellation law for group... |
| ablnnncan1 19733 | Cancellation law for group... |
| ablsubsub23 19734 | Swap subtrahend and result... |
| mulgnn0di 19735 | Group multiple of a sum, f... |
| mulgdi 19736 | Group multiple of a sum. ... |
| mulgmhm 19737 | The map from ` x ` to ` n ... |
| mulgghm 19738 | The map from ` x ` to ` n ... |
| mulgsubdi 19739 | Group multiple of a differ... |
| ghmfghm 19740 | The function fulfilling th... |
| ghmcmn 19741 | The image of a commutative... |
| ghmabl 19742 | The image of an abelian gr... |
| invghm 19743 | The inversion map is a gro... |
| eqgabl 19744 | Value of the subgroup cose... |
| qusecsub 19745 | Two subgroup cosets are eq... |
| subgabl 19746 | A subgroup of an abelian g... |
| subcmn 19747 | A submonoid of a commutati... |
| submcmn 19748 | A submonoid of a commutati... |
| submcmn2 19749 | A submonoid is commutative... |
| cntzcmn 19750 | The centralizer of any sub... |
| cntzcmnss 19751 | Any subset in a commutativ... |
| cntrcmnd 19752 | The center of a monoid is ... |
| cntrabl 19753 | The center of a group is a... |
| cntzspan 19754 | If the generators commute,... |
| cntzcmnf 19755 | Discharge the centralizer ... |
| ghmplusg 19756 | The pointwise sum of two l... |
| ablnsg 19757 | Every subgroup of an abeli... |
| odadd1 19758 | The order of a product in ... |
| odadd2 19759 | The order of a product in ... |
| odadd 19760 | The order of a product is ... |
| gex2abl 19761 | A group with exponent 2 (o... |
| gexexlem 19762 | Lemma for ~ gexex . (Cont... |
| gexex 19763 | In an abelian group with f... |
| torsubg 19764 | The set of all elements of... |
| oddvdssubg 19765 | The set of all elements wh... |
| lsmcomx 19766 | Subgroup sum commutes (ext... |
| ablcntzd 19767 | All subgroups in an abelia... |
| lsmcom 19768 | Subgroup sum commutes. (C... |
| lsmsubg2 19769 | The sum of two subgroups i... |
| lsm4 19770 | Commutative/associative la... |
| prdscmnd 19771 | The product of a family of... |
| prdsabld 19772 | The product of a family of... |
| pwscmn 19773 | The structure power on a c... |
| pwsabl 19774 | The structure power on an ... |
| qusabl 19775 | If ` Y ` is a subgroup of ... |
| abl1 19776 | The (smallest) structure r... |
| abln0 19777 | Abelian groups (and theref... |
| cnaddablx 19778 | The complex numbers are an... |
| cnaddabl 19779 | The complex numbers are an... |
| cnaddid 19780 | The group identity element... |
| cnaddinv 19781 | Value of the group inverse... |
| zaddablx 19782 | The integers are an Abelia... |
| frgpnabllem1 19783 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19784 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19785 | The free group on two or m... |
| imasabl 19786 | The image structure of an ... |
| iscyg 19789 | Definition of a cyclic gro... |
| iscyggen 19790 | The property of being a cy... |
| iscyggen2 19791 | The property of being a cy... |
| iscyg2 19792 | A cyclic group is a group ... |
| cyggeninv 19793 | The inverse of a cyclic ge... |
| cyggenod 19794 | An element is the generato... |
| cyggenod2 19795 | In an infinite cyclic grou... |
| iscyg3 19796 | Definition of a cyclic gro... |
| iscygd 19797 | Definition of a cyclic gro... |
| iscygodd 19798 | Show that a group with an ... |
| cycsubmcmn 19799 | The set of nonnegative int... |
| cyggrp 19800 | A cyclic group is a group.... |
| cygabl 19801 | A cyclic group is abelian.... |
| cygctb 19802 | A cyclic group is countabl... |
| 0cyg 19803 | The trivial group is cycli... |
| prmcyg 19804 | A group with prime order i... |
| lt6abl 19805 | A group with fewer than ` ... |
| ghmcyg 19806 | The image of a cyclic grou... |
| cyggex2 19807 | The exponent of a cyclic g... |
| cyggex 19808 | The exponent of a finite c... |
| cyggexb 19809 | A finite abelian group is ... |
| giccyg 19810 | Cyclicity is a group prope... |
| cycsubgcyg 19811 | The cyclic subgroup genera... |
| cycsubgcyg2 19812 | The cyclic subgroup genera... |
| gsumval3a 19813 | Value of the group sum ope... |
| gsumval3eu 19814 | The group sum as defined i... |
| gsumval3lem1 19815 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19816 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19817 | Value of the group sum ope... |
| gsumcllem 19818 | Lemma for ~ gsumcl and rel... |
| gsumzres 19819 | Extend a finite group sum ... |
| gsumzcl2 19820 | Closure of a finite group ... |
| gsumzcl 19821 | Closure of a finite group ... |
| gsumzf1o 19822 | Re-index a finite group su... |
| gsumres 19823 | Extend a finite group sum ... |
| gsumcl2 19824 | Closure of a finite group ... |
| gsumcl 19825 | Closure of a finite group ... |
| gsumf1o 19826 | Re-index a finite group su... |
| gsumreidx 19827 | Re-index a finite group su... |
| gsumzsubmcl 19828 | Closure of a group sum in ... |
| gsumsubmcl 19829 | Closure of a group sum in ... |
| gsumsubgcl 19830 | Closure of a group sum in ... |
| gsumzaddlem 19831 | The sum of two group sums.... |
| gsumzadd 19832 | The sum of two group sums.... |
| gsumadd 19833 | The sum of two group sums.... |
| gsummptfsadd 19834 | The sum of two group sums ... |
| gsummptfidmadd 19835 | The sum of two group sums ... |
| gsummptfidmadd2 19836 | The sum of two group sums ... |
| gsumzsplit 19837 | Split a group sum into two... |
| gsumsplit 19838 | Split a group sum into two... |
| gsumsplit2 19839 | Split a group sum into two... |
| gsummptfidmsplit 19840 | Split a group sum expresse... |
| gsummptfidmsplitres 19841 | Split a group sum expresse... |
| gsummptfzsplit 19842 | Split a group sum expresse... |
| gsummptfzsplitl 19843 | Split a group sum expresse... |
| gsumconst 19844 | Sum of a constant series. ... |
| gsumconstf 19845 | Sum of a constant series. ... |
| gsummptshft 19846 | Index shift of a finite gr... |
| gsumzmhm 19847 | Apply a group homomorphism... |
| gsummhm 19848 | Apply a group homomorphism... |
| gsummhm2 19849 | Apply a group homomorphism... |
| gsummptmhm 19850 | Apply a group homomorphism... |
| gsummulglem 19851 | Lemma for ~ gsummulg and ~... |
| gsummulg 19852 | Nonnegative multiple of a ... |
| gsummulgz 19853 | Integer multiple of a grou... |
| gsumzoppg 19854 | The opposite of a group su... |
| gsumzinv 19855 | Inverse of a group sum. (... |
| gsuminv 19856 | Inverse of a group sum. (... |
| gsummptfidminv 19857 | Inverse of a group sum exp... |
| gsumsub 19858 | The difference of two grou... |
| gsummptfssub 19859 | The difference of two grou... |
| gsummptfidmsub 19860 | The difference of two grou... |
| gsumsnfd 19861 | Group sum of a singleton, ... |
| gsumsnd 19862 | Group sum of a singleton, ... |
| gsumsnf 19863 | Group sum of a singleton, ... |
| gsumsn 19864 | Group sum of a singleton. ... |
| gsumpr 19865 | Group sum of a pair. (Con... |
| gsumzunsnd 19866 | Append an element to a fin... |
| gsumunsnfd 19867 | Append an element to a fin... |
| gsumunsnd 19868 | Append an element to a fin... |
| gsumunsnf 19869 | Append an element to a fin... |
| gsumunsn 19870 | Append an element to a fin... |
| gsumdifsnd 19871 | Extract a summand from a f... |
| gsumpt 19872 | Sum of a family that is no... |
| gsummptf1o 19873 | Re-index a finite group su... |
| gsummptun 19874 | Group sum of a disjoint un... |
| gsummpt1n0 19875 | If only one summand in a f... |
| gsummptif1n0 19876 | If only one summand in a f... |
| gsummptcl 19877 | Closure of a finite group ... |
| gsummptfif1o 19878 | Re-index a finite group su... |
| gsummptfzcl 19879 | Closure of a finite group ... |
| gsum2dlem1 19880 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19881 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19882 | Write a sum over a two-dim... |
| gsum2d2lem 19883 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19884 | Write a group sum over a t... |
| gsumcom2 19885 | Two-dimensional commutatio... |
| gsumxp 19886 | Write a group sum over a c... |
| gsumcom 19887 | Commute the arguments of a... |
| gsumcom3 19888 | A commutative law for fini... |
| gsumcom3fi 19889 | A commutative law for fini... |
| gsumxp2 19890 | Write a group sum over a c... |
| prdsgsum 19891 | Finite commutative sums in... |
| pwsgsum 19892 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19893 | Replacing a finitely suppo... |
| nn0gsumfz 19894 | Replacing a finitely suppo... |
| nn0gsumfz0 19895 | Replacing a finitely suppo... |
| gsummptnn0fz 19896 | A final group sum over a f... |
| gsummptnn0fzfv 19897 | A final group sum over a f... |
| telgsumfzslem 19898 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19899 | Telescoping group sum rang... |
| telgsumfz 19900 | Telescoping group sum rang... |
| telgsumfz0s 19901 | Telescoping finite group s... |
| telgsumfz0 19902 | Telescoping finite group s... |
| telgsums 19903 | Telescoping finitely suppo... |
| telgsum 19904 | Telescoping finitely suppo... |
| reldmdprd 19909 | The domain of the internal... |
| dmdprd 19910 | The domain of definition o... |
| dmdprdd 19911 | Show that a given family i... |
| dprddomprc 19912 | A family of subgroups inde... |
| dprddomcld 19913 | If a family of subgroups i... |
| dprdval0prc 19914 | The internal direct produc... |
| dprdval 19915 | The value of the internal ... |
| eldprd 19916 | A class ` A ` is an intern... |
| dprdgrp 19917 | Reverse closure for the in... |
| dprdf 19918 | The function ` S ` is a fa... |
| dprdf2 19919 | The function ` S ` is a fa... |
| dprdcntz 19920 | The function ` S ` is a fa... |
| dprddisj 19921 | The function ` S ` is a fa... |
| dprdw 19922 | The property of being a fi... |
| dprdwd 19923 | A mapping being a finitely... |
| dprdff 19924 | A finitely supported funct... |
| dprdfcl 19925 | A finitely supported funct... |
| dprdffsupp 19926 | A finitely supported funct... |
| dprdfcntz 19927 | A function on the elements... |
| dprdssv 19928 | The internal direct produc... |
| dprdfid 19929 | A function mapping all but... |
| eldprdi 19930 | The domain of definition o... |
| dprdfinv 19931 | Take the inverse of a grou... |
| dprdfadd 19932 | Take the sum of group sums... |
| dprdfsub 19933 | Take the difference of gro... |
| dprdfeq0 19934 | The zero function is the o... |
| dprdf11 19935 | Two group sums over a dire... |
| dprdsubg 19936 | The internal direct produc... |
| dprdub 19937 | Each factor is a subset of... |
| dprdlub 19938 | The direct product is smal... |
| dprdspan 19939 | The direct product is the ... |
| dprdres 19940 | Restriction of a direct pr... |
| dprdss 19941 | Create a direct product by... |
| dprdz 19942 | A family consisting entire... |
| dprd0 19943 | The empty family is an int... |
| dprdf1o 19944 | Rearrange the index set of... |
| dprdf1 19945 | Rearrange the index set of... |
| subgdmdprd 19946 | A direct product in a subg... |
| subgdprd 19947 | A direct product in a subg... |
| dprdsn 19948 | A singleton family is an i... |
| dmdprdsplitlem 19949 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19950 | The function ` S ` is a fa... |
| dprddisj2 19951 | The function ` S ` is a fa... |
| dprd2dlem2 19952 | The direct product of a co... |
| dprd2dlem1 19953 | The direct product of a co... |
| dprd2da 19954 | The direct product of a co... |
| dprd2db 19955 | The direct product of a co... |
| dprd2d2 19956 | The direct product of a co... |
| dmdprdsplit2lem 19957 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19958 | The direct product splits ... |
| dmdprdsplit 19959 | The direct product splits ... |
| dprdsplit 19960 | The direct product is the ... |
| dmdprdpr 19961 | A singleton family is an i... |
| dprdpr 19962 | A singleton family is an i... |
| dpjlem 19963 | Lemma for theorems about d... |
| dpjcntz 19964 | The two subgroups that app... |
| dpjdisj 19965 | The two subgroups that app... |
| dpjlsm 19966 | The two subgroups that app... |
| dpjfval 19967 | Value of the direct produc... |
| dpjval 19968 | Value of the direct produc... |
| dpjf 19969 | The ` X ` -th index projec... |
| dpjidcl 19970 | The key property of projec... |
| dpjeq 19971 | Decompose a group sum into... |
| dpjid 19972 | The key property of projec... |
| dpjlid 19973 | The ` X ` -th index projec... |
| dpjrid 19974 | The ` Y ` -th index projec... |
| dpjghm 19975 | The direct product is the ... |
| dpjghm2 19976 | The direct product is the ... |
| ablfacrplem 19977 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 19978 | A finite abelian group who... |
| ablfacrp2 19979 | The factors ` K , L ` of ~... |
| ablfac1lem 19980 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 19981 | The factors of ~ ablfac1b ... |
| ablfac1b 19982 | Any abelian group is the d... |
| ablfac1c 19983 | The factors of ~ ablfac1b ... |
| ablfac1eulem 19984 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 19985 | The factorization of ~ abl... |
| pgpfac1lem1 19986 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 19987 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 19988 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 19989 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 19990 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 19991 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 19992 | Factorization of a finite ... |
| pgpfaclem1 19993 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 19994 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 19995 | Lemma for ~ pgpfac . (Con... |
| pgpfac 19996 | Full factorization of a fi... |
| ablfaclem1 19997 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 19998 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 19999 | Lemma for ~ ablfac . (Con... |
| ablfac 20000 | The Fundamental Theorem of... |
| ablfac2 20001 | Choose generators for each... |
| issimpg 20004 | The predicate "is a simple... |
| issimpgd 20005 | Deduce a simple group from... |
| simpggrp 20006 | A simple group is a group.... |
| simpggrpd 20007 | A simple group is a group.... |
| simpg2nsg 20008 | A simple group has two nor... |
| trivnsimpgd 20009 | Trivial groups are not sim... |
| simpgntrivd 20010 | Simple groups are nontrivi... |
| simpgnideld 20011 | A simple group contains a ... |
| simpgnsgd 20012 | The only normal subgroups ... |
| simpgnsgeqd 20013 | A normal subgroup of a sim... |
| 2nsgsimpgd 20014 | If any normal subgroup of ... |
| simpgnsgbid 20015 | A nontrivial group is simp... |
| ablsimpnosubgd 20016 | A subgroup of an abelian s... |
| ablsimpg1gend 20017 | An abelian simple group is... |
| ablsimpgcygd 20018 | An abelian simple group is... |
| ablsimpgfindlem1 20019 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 20020 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 20021 | Relationship between the o... |
| ablsimpgfind 20022 | An abelian simple group is... |
| fincygsubgd 20023 | The subgroup referenced in... |
| fincygsubgodd 20024 | Calculate the order of a s... |
| fincygsubgodexd 20025 | A finite cyclic group has ... |
| prmgrpsimpgd 20026 | A group of prime order is ... |
| ablsimpgprmd 20027 | An abelian simple group ha... |
| ablsimpgd 20028 | An abelian group is simple... |
| fnmgp 20031 | The multiplicative group o... |
| mgpval 20032 | Value of the multiplicatio... |
| mgpplusg 20033 | Value of the group operati... |
| mgplemOLD 20034 | Obsolete version of ~ sets... |
| mgpbas 20035 | Base set of the multiplica... |
| mgpbasOLD 20036 | Obsolete version of ~ mgpb... |
| mgpsca 20037 | The multiplication monoid ... |
| mgpscaOLD 20038 | Obsolete version of ~ mgps... |
| mgptset 20039 | Topology component of the ... |
| mgptsetOLD 20040 | Obsolete version of ~ mgpt... |
| mgptopn 20041 | Topology of the multiplica... |
| mgpds 20042 | Distance function of the m... |
| mgpdsOLD 20043 | Obsolete version of ~ mgpd... |
| mgpress 20044 | Subgroup commutes with the... |
| mgpressOLD 20045 | Obsolete version of ~ mgpr... |
| prdsmgp 20046 | The multiplicative monoid ... |
| isrng 20049 | The predicate "is a non-un... |
| rngabl 20050 | A non-unital ring is an (a... |
| rngmgp 20051 | A non-unital ring is a sem... |
| rngmgpf 20052 | Restricted functionality o... |
| rnggrp 20053 | A non-unital ring is a (ad... |
| rngass 20054 | Associative law for the mu... |
| rngdi 20055 | Distributive law for the m... |
| rngdir 20056 | Distributive law for the m... |
| rngacl 20057 | Closure of the addition op... |
| rng0cl 20058 | The zero element of a non-... |
| rngcl 20059 | Closure of the multiplicat... |
| rnglz 20060 | The zero of a non-unital r... |
| rngrz 20061 | The zero of a non-unital r... |
| rngmneg1 20062 | Negation of a product in a... |
| rngmneg2 20063 | Negation of a product in a... |
| rngm2neg 20064 | Double negation of a produ... |
| rngansg 20065 | Every additive subgroup of... |
| rngsubdi 20066 | Ring multiplication distri... |
| rngsubdir 20067 | Ring multiplication distri... |
| isrngd 20068 | Properties that determine ... |
| rngpropd 20069 | If two structures have the... |
| prdsmulrngcl 20070 | Closure of the multiplicat... |
| prdsrngd 20071 | A product of non-unital ri... |
| imasrng 20072 | The image structure of a n... |
| imasrngf1 20073 | The image of a non-unital ... |
| xpsrngd 20074 | A product of two non-unita... |
| qusrng 20075 | The quotient structure of ... |
| ringidval 20078 | The value of the unity ele... |
| dfur2 20079 | The multiplicative identit... |
| ringurd 20080 | Deduce the unity element o... |
| issrg 20083 | The predicate "is a semiri... |
| srgcmn 20084 | A semiring is a commutativ... |
| srgmnd 20085 | A semiring is a monoid. (... |
| srgmgp 20086 | A semiring is a monoid und... |
| srgdilem 20087 | Lemma for ~ srgdi and ~ sr... |
| srgcl 20088 | Closure of the multiplicat... |
| srgass 20089 | Associative law for the mu... |
| srgideu 20090 | The unity element of a sem... |
| srgfcl 20091 | Functionality of the multi... |
| srgdi 20092 | Distributive law for the m... |
| srgdir 20093 | Distributive law for the m... |
| srgidcl 20094 | The unity element of a sem... |
| srg0cl 20095 | The zero element of a semi... |
| srgidmlem 20096 | Lemma for ~ srglidm and ~ ... |
| srglidm 20097 | The unity element of a sem... |
| srgridm 20098 | The unity element of a sem... |
| issrgid 20099 | Properties showing that an... |
| srgacl 20100 | Closure of the addition op... |
| srgcom 20101 | Commutativity of the addit... |
| srgrz 20102 | The zero of a semiring is ... |
| srglz 20103 | The zero of a semiring is ... |
| srgisid 20104 | In a semiring, the only le... |
| o2timesd 20105 | An element of a ring-like ... |
| rglcom4d 20106 | Restricted commutativity o... |
| srgo2times 20107 | A semiring element plus it... |
| srgcom4lem 20108 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 20109 | Restricted commutativity o... |
| srg1zr 20110 | The only semiring with a b... |
| srgen1zr 20111 | The only semiring with one... |
| srgmulgass 20112 | An associative property be... |
| srgpcomp 20113 | If two elements of a semir... |
| srgpcompp 20114 | If two elements of a semir... |
| srgpcomppsc 20115 | If two elements of a semir... |
| srglmhm 20116 | Left-multiplication in a s... |
| srgrmhm 20117 | Right-multiplication in a ... |
| srgsummulcr 20118 | A finite semiring sum mult... |
| sgsummulcl 20119 | A finite semiring sum mult... |
| srg1expzeq1 20120 | The exponentiation (by a n... |
| srgbinomlem1 20121 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 20122 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 20123 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 20124 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 20125 | Lemma for ~ srgbinom . In... |
| srgbinom 20126 | The binomial theorem for c... |
| csrgbinom 20127 | The binomial theorem for c... |
| isring 20132 | The predicate "is a (unita... |
| ringgrp 20133 | A ring is a group. (Contr... |
| ringmgp 20134 | A ring is a monoid under m... |
| iscrng 20135 | A commutative ring is a ri... |
| crngmgp 20136 | A commutative ring's multi... |
| ringgrpd 20137 | A ring is a group. (Contr... |
| ringmnd 20138 | A ring is a monoid under a... |
| ringmgm 20139 | A ring is a magma. (Contr... |
| crngring 20140 | A commutative ring is a ri... |
| crngringd 20141 | A commutative ring is a ri... |
| crnggrpd 20142 | A commutative ring is a gr... |
| mgpf 20143 | Restricted functionality o... |
| ringdilem 20144 | Properties of a unital rin... |
| ringcl 20145 | Closure of the multiplicat... |
| crngcom 20146 | A commutative ring's multi... |
| iscrng2 20147 | A commutative ring is a ri... |
| ringass 20148 | Associative law for multip... |
| ringideu 20149 | The unity element of a rin... |
| crngbascntr 20150 | The base set of a commutat... |
| ringassd 20151 | Associative law for multip... |
| ringcld 20152 | Closure of the multiplicat... |
| ringdi 20153 | Distributive law for the m... |
| ringdir 20154 | Distributive law for the m... |
| ringidcl 20155 | The unity element of a rin... |
| ring0cl 20156 | The zero element of a ring... |
| ringidmlem 20157 | Lemma for ~ ringlidm and ~... |
| ringlidm 20158 | The unity element of a rin... |
| ringridm 20159 | The unity element of a rin... |
| isringid 20160 | Properties showing that an... |
| ringlidmd 20161 | The unity element of a rin... |
| ringridmd 20162 | The unity element of a rin... |
| ringid 20163 | The multiplication operati... |
| ringo2times 20164 | A ring element plus itself... |
| ringadd2 20165 | A ring element plus itself... |
| ringidss 20166 | A subset of the multiplica... |
| ringacl 20167 | Closure of the addition op... |
| ringcomlem 20168 | Lemma for ~ ringcom . Thi... |
| ringcom 20169 | Commutativity of the addit... |
| ringabl 20170 | A ring is an Abelian group... |
| ringcmn 20171 | A ring is a commutative mo... |
| ringabld 20172 | A ring is an Abelian group... |
| ringcmnd 20173 | A ring is a commutative mo... |
| ringrng 20174 | A unital ring is a non-uni... |
| ringssrng 20175 | The unital rings are non-u... |
| isringrng 20176 | The predicate "is a unital... |
| ringpropd 20177 | If two structures have the... |
| crngpropd 20178 | If two structures have the... |
| ringprop 20179 | If two structures have the... |
| isringd 20180 | Properties that determine ... |
| iscrngd 20181 | Properties that determine ... |
| ringlz 20182 | The zero of a unital ring ... |
| ringrz 20183 | The zero of a unital ring ... |
| ringlzd 20184 | The zero of a unital ring ... |
| ringrzd 20185 | The zero of a unital ring ... |
| ringsrg 20186 | Any ring is also a semirin... |
| ring1eq0 20187 | If one and zero are equal,... |
| ring1ne0 20188 | If a ring has at least two... |
| ringinvnz1ne0 20189 | In a unital ring, a left i... |
| ringinvnzdiv 20190 | In a unital ring, a left i... |
| ringnegl 20191 | Negation in a ring is the ... |
| ringnegr 20192 | Negation in a ring is the ... |
| ringmneg1 20193 | Negation of a product in a... |
| ringmneg2 20194 | Negation of a product in a... |
| ringm2neg 20195 | Double negation of a produ... |
| ringsubdi 20196 | Ring multiplication distri... |
| ringsubdir 20197 | Ring multiplication distri... |
| mulgass2 20198 | An associative property be... |
| ring1 20199 | The (smallest) structure r... |
| ringn0 20200 | Rings exist. (Contributed... |
| ringlghm 20201 | Left-multiplication in a r... |
| ringrghm 20202 | Right-multiplication in a ... |
| gsummulc1OLD 20203 | Obsolete version of ~ gsum... |
| gsummulc2OLD 20204 | Obsolete version of ~ gsum... |
| gsummulc1 20205 | A finite ring sum multipli... |
| gsummulc2 20206 | A finite ring sum multipli... |
| gsummgp0 20207 | If one factor in a finite ... |
| gsumdixp 20208 | Distribute a binary produc... |
| prdsmulrcl 20209 | A structure product of rin... |
| prdsringd 20210 | A product of rings is a ri... |
| prdscrngd 20211 | A product of commutative r... |
| prds1 20212 | Value of the ring unity in... |
| pwsring 20213 | A structure power of a rin... |
| pws1 20214 | Value of the ring unity in... |
| pwscrng 20215 | A structure power of a com... |
| pwsmgp 20216 | The multiplicative group o... |
| pwspjmhmmgpd 20217 | The projection given by ~ ... |
| pwsexpg 20218 | Value of a group exponenti... |
| imasring 20219 | The image structure of a r... |
| imasringf1 20220 | The image of a ring under ... |
| xpsringd 20221 | A product of two rings is ... |
| xpsring1d 20222 | The multiplicative identit... |
| qusring2 20223 | The quotient structure of ... |
| crngbinom 20224 | The binomial theorem for c... |
| opprval 20227 | Value of the opposite ring... |
| opprmulfval 20228 | Value of the multiplicatio... |
| opprmul 20229 | Value of the multiplicatio... |
| crngoppr 20230 | In a commutative ring, the... |
| opprlem 20231 | Lemma for ~ opprbas and ~ ... |
| opprlemOLD 20232 | Obsolete version of ~ oppr... |
| opprbas 20233 | Base set of an opposite ri... |
| opprbasOLD 20234 | Obsolete proof of ~ opprba... |
| oppradd 20235 | Addition operation of an o... |
| oppraddOLD 20236 | Obsolete proof of ~ opprba... |
| opprrng 20237 | An opposite non-unital rin... |
| opprrngb 20238 | A class is a non-unital ri... |
| opprring 20239 | An opposite ring is a ring... |
| opprringb 20240 | Bidirectional form of ~ op... |
| oppr0 20241 | Additive identity of an op... |
| oppr1 20242 | Multiplicative identity of... |
| opprneg 20243 | The negative function in a... |
| opprsubg 20244 | Being a subgroup is a symm... |
| mulgass3 20245 | An associative property be... |
| reldvdsr 20252 | The divides relation is a ... |
| dvdsrval 20253 | Value of the divides relat... |
| dvdsr 20254 | Value of the divides relat... |
| dvdsr2 20255 | Value of the divides relat... |
| dvdsrmul 20256 | A left-multiple of ` X ` i... |
| dvdsrcl 20257 | Closure of a dividing elem... |
| dvdsrcl2 20258 | Closure of a dividing elem... |
| dvdsrid 20259 | An element in a (unital) r... |
| dvdsrtr 20260 | Divisibility is transitive... |
| dvdsrmul1 20261 | The divisibility relation ... |
| dvdsrneg 20262 | An element divides its neg... |
| dvdsr01 20263 | In a ring, zero is divisib... |
| dvdsr02 20264 | Only zero is divisible by ... |
| isunit 20265 | Property of being a unit o... |
| 1unit 20266 | The multiplicative identit... |
| unitcl 20267 | A unit is an element of th... |
| unitss 20268 | The set of units is contai... |
| opprunit 20269 | Being a unit is a symmetri... |
| crngunit 20270 | Property of being a unit i... |
| dvdsunit 20271 | A divisor of a unit is a u... |
| unitmulcl 20272 | The product of units is a ... |
| unitmulclb 20273 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20274 | The base set of the group ... |
| unitgrp 20275 | The group of units is a gr... |
| unitabl 20276 | The group of units of a co... |
| unitgrpid 20277 | The identity of the group ... |
| unitsubm 20278 | The group of units is a su... |
| invrfval 20281 | Multiplicative inverse fun... |
| unitinvcl 20282 | The inverse of a unit exis... |
| unitinvinv 20283 | The inverse of the inverse... |
| ringinvcl 20284 | The inverse of a unit is a... |
| unitlinv 20285 | A unit times its inverse i... |
| unitrinv 20286 | A unit times its inverse i... |
| 1rinv 20287 | The inverse of the ring un... |
| 0unit 20288 | The additive identity is a... |
| unitnegcl 20289 | The negative of a unit is ... |
| ringunitnzdiv 20290 | In a unitary ring, a unit ... |
| ring1nzdiv 20291 | In a unitary ring, the rin... |
| dvrfval 20294 | Division operation in a ri... |
| dvrval 20295 | Division operation in a ri... |
| dvrcl 20296 | Closure of division operat... |
| unitdvcl 20297 | The units are closed under... |
| dvrid 20298 | A ring element divided by ... |
| dvr1 20299 | A ring element divided by ... |
| dvrass 20300 | An associative law for div... |
| dvrcan1 20301 | A cancellation law for div... |
| dvrcan3 20302 | A cancellation law for div... |
| dvreq1 20303 | Equality in terms of ratio... |
| dvrdir 20304 | Distributive law for the d... |
| rdivmuldivd 20305 | Multiplication of two rati... |
| ringinvdv 20306 | Write the inverse function... |
| rngidpropd 20307 | The ring unity depends onl... |
| dvdsrpropd 20308 | The divisibility relation ... |
| unitpropd 20309 | The set of units depends o... |
| invrpropd 20310 | The ring inverse function ... |
| isirred 20311 | An irreducible element of ... |
| isnirred 20312 | The property of being a no... |
| isirred2 20313 | Expand out the class diffe... |
| opprirred 20314 | Irreducibility is symmetri... |
| irredn0 20315 | The additive identity is n... |
| irredcl 20316 | An irreducible element is ... |
| irrednu 20317 | An irreducible element is ... |
| irredn1 20318 | The multiplicative identit... |
| irredrmul 20319 | The product of an irreduci... |
| irredlmul 20320 | The product of a unit and ... |
| irredmul 20321 | If product of two elements... |
| irredneg 20322 | The negative of an irreduc... |
| irrednegb 20323 | An element is irreducible ... |
| rnghmrcl 20330 | Reverse closure of a non-u... |
| rnghmfn 20331 | The mapping of two non-uni... |
| rnghmval 20332 | The set of the non-unital ... |
| isrnghm 20333 | A function is a non-unital... |
| isrnghmmul 20334 | A function is a non-unital... |
| rnghmmgmhm 20335 | A non-unital ring homomorp... |
| rnghmval2 20336 | The non-unital ring homomo... |
| isrngim 20337 | An isomorphism of non-unit... |
| rngimrcl 20338 | Reverse closure for an iso... |
| rnghmghm 20339 | A non-unital ring homomorp... |
| rnghmf 20340 | A ring homomorphism is a f... |
| rnghmmul 20341 | A homomorphism of non-unit... |
| isrnghm2d 20342 | Demonstration of non-unita... |
| isrnghmd 20343 | Demonstration of non-unita... |
| rnghmf1o 20344 | A non-unital ring homomorp... |
| isrngim2 20345 | An isomorphism of non-unit... |
| rngimf1o 20346 | An isomorphism of non-unit... |
| rngimrnghm 20347 | An isomorphism of non-unit... |
| rngimcnv 20348 | The converse of an isomorp... |
| rnghmco 20349 | The composition of non-uni... |
| idrnghm 20350 | The identity homomorphism ... |
| c0mgm 20351 | The constant mapping to ze... |
| c0mhm 20352 | The constant mapping to ze... |
| c0ghm 20353 | The constant mapping to ze... |
| c0snmgmhm 20354 | The constant mapping to ze... |
| c0snmhm 20355 | The constant mapping to ze... |
| c0snghm 20356 | The constant mapping to ze... |
| rngisomfv1 20357 | If there is a non-unital r... |
| rngisom1 20358 | If there is a non-unital r... |
| rngisomring 20359 | If there is a non-unital r... |
| rngisomring1 20360 | If there is a non-unital r... |
| dfrhm2 20366 | The property of a ring hom... |
| rhmrcl1 20368 | Reverse closure of a ring ... |
| rhmrcl2 20369 | Reverse closure of a ring ... |
| isrhm 20370 | A function is a ring homom... |
| rhmmhm 20371 | A ring homomorphism is a h... |
| rhmisrnghm 20372 | Each unital ring homomorph... |
| isrim0OLD 20373 | Obsolete version of ~ isri... |
| rimrcl 20374 | Reverse closure for an iso... |
| isrim0 20375 | A ring isomorphism is a ho... |
| rhmghm 20376 | A ring homomorphism is an ... |
| rhmf 20377 | A ring homomorphism is a f... |
| rhmmul 20378 | A homomorphism of rings pr... |
| isrhm2d 20379 | Demonstration of ring homo... |
| isrhmd 20380 | Demonstration of ring homo... |
| rhm1 20381 | Ring homomorphisms are req... |
| idrhm 20382 | The identity homomorphism ... |
| rhmf1o 20383 | A ring homomorphism is bij... |
| isrim 20384 | An isomorphism of rings is... |
| isrimOLD 20385 | Obsolete version of ~ isri... |
| rimf1o 20386 | An isomorphism of rings is... |
| rimrhmOLD 20387 | Obsolete version of ~ rimr... |
| rimrhm 20388 | A ring isomorphism is a ho... |
| rimgim 20389 | An isomorphism of rings is... |
| rimisrngim 20390 | Each unital ring isomorphi... |
| rhmfn 20391 | The mapping of two rings t... |
| rhmval 20392 | The ring homomorphisms bet... |
| rhmco 20393 | The composition of ring ho... |
| pwsco1rhm 20394 | Right composition with a f... |
| pwsco2rhm 20395 | Left composition with a ri... |
| brric 20396 | The relation "is isomorphi... |
| brrici 20397 | Prove isomorphic by an exp... |
| brric2 20398 | The relation "is isomorphi... |
| ricgic 20399 | If two rings are (ring) is... |
| rhmdvdsr 20400 | A ring homomorphism preser... |
| rhmopp 20401 | A ring homomorphism is als... |
| elrhmunit 20402 | Ring homomorphisms preserv... |
| rhmunitinv 20403 | Ring homomorphisms preserv... |
| isnzr 20406 | Property of a nonzero ring... |
| nzrnz 20407 | One and zero are different... |
| nzrring 20408 | A nonzero ring is a ring. ... |
| nzrringOLD 20409 | Obsolete version of ~ nzrr... |
| isnzr2 20410 | Equivalent characterizatio... |
| isnzr2hash 20411 | Equivalent characterizatio... |
| opprnzr 20412 | The opposite of a nonzero ... |
| ringelnzr 20413 | A ring is nonzero if it ha... |
| nzrunit 20414 | A unit is nonzero in any n... |
| 0ringnnzr 20415 | A ring is a zero ring iff ... |
| 0ring 20416 | If a ring has only one ele... |
| 0ringdif 20417 | A zero ring is a ring whic... |
| 0ringbas 20418 | The base set of a zero rin... |
| 0ring01eq 20419 | In a ring with only one el... |
| 01eq0ring 20420 | If the zero and the identi... |
| 01eq0ringOLD 20421 | Obsolete version of ~ 01eq... |
| 0ring01eqbi 20422 | In a unital ring the zero ... |
| 0ring1eq0 20423 | In a zero ring, a ring whi... |
| c0rhm 20424 | The constant mapping to ze... |
| c0rnghm 20425 | The constant mapping to ze... |
| zrrnghm 20426 | The constant mapping to ze... |
| islring 20429 | The predicate "is a local ... |
| lringnzr 20430 | A local ring is a nonzero ... |
| lringring 20431 | A local ring is a ring. (... |
| lringnz 20432 | A local ring is a nonzero ... |
| lringuplu 20433 | If the sum of two elements... |
| issubrng 20436 | The subring of non-unital ... |
| subrngss 20437 | A subring is a subset. (C... |
| subrngid 20438 | Every non-unital ring is a... |
| subrngrng 20439 | A subring is a non-unital ... |
| subrngrcl 20440 | Reverse closure for a subr... |
| subrngsubg 20441 | A subring is a subgroup. ... |
| subrngringnsg 20442 | A subring is a normal subg... |
| subrngbas 20443 | Base set of a subring stru... |
| subrng0 20444 | A subring always has the s... |
| subrngacl 20445 | A subring is closed under ... |
| subrngmcl 20446 | A subgroup is closed under... |
| issubrng2 20447 | Characterize the subrings ... |
| opprsubrng 20448 | Being a subring is a symme... |
| subrngint 20449 | The intersection of a none... |
| subrngin 20450 | The intersection of two su... |
| subrngmre 20451 | The subrings of a non-unit... |
| subsubrng 20452 | A subring of a subring is ... |
| subsubrng2 20453 | The set of subrings of a s... |
| rhmimasubrnglem 20454 | Lemma for ~ rhmimasubrng :... |
| rhmimasubrng 20455 | The homomorphic image of a... |
| cntzsubrng 20456 | Centralizers in a non-unit... |
| subrngpropd 20457 | If two structures have the... |
| issubrg 20462 | The subring predicate. (C... |
| subrgss 20463 | A subring is a subset. (C... |
| subrgid 20464 | Every ring is a subring of... |
| subrgring 20465 | A subring is a ring. (Con... |
| subrgcrng 20466 | A subring of a commutative... |
| subrgrcl 20467 | Reverse closure for a subr... |
| subrgsubg 20468 | A subring is a subgroup. ... |
| subrgsubrng 20469 | A subring of a unital ring... |
| subrg0 20470 | A subring always has the s... |
| subrg1cl 20471 | A subring contains the mul... |
| subrgbas 20472 | Base set of a subring stru... |
| subrg1 20473 | A subring always has the s... |
| subrgacl 20474 | A subring is closed under ... |
| subrgmcl 20475 | A subgroup is closed under... |
| subrgsubm 20476 | A subring is a submonoid o... |
| subrgdvds 20477 | If an element divides anot... |
| subrguss 20478 | A unit of a subring is a u... |
| subrginv 20479 | A subring always has the s... |
| subrgdv 20480 | A subring always has the s... |
| subrgunit 20481 | An element of a ring is a ... |
| subrgugrp 20482 | The units of a subring for... |
| issubrg2 20483 | Characterize the subrings ... |
| opprsubrg 20484 | Being a subring is a symme... |
| subrgnzr 20485 | A subring of a nonzero rin... |
| subrgint 20486 | The intersection of a none... |
| subrgin 20487 | The intersection of two su... |
| subrgmre 20488 | The subrings of a ring are... |
| subsubrg 20489 | A subring of a subring is ... |
| subsubrg2 20490 | The set of subrings of a s... |
| issubrg3 20491 | A subring is an additive s... |
| resrhm 20492 | Restriction of a ring homo... |
| resrhm2b 20493 | Restriction of the codomai... |
| rhmeql 20494 | The equalizer of two ring ... |
| rhmima 20495 | The homomorphic image of a... |
| rnrhmsubrg 20496 | The range of a ring homomo... |
| cntzsubr 20497 | Centralizers in a ring are... |
| pwsdiagrhm 20498 | Diagonal homomorphism into... |
| subrgpropd 20499 | If two structures have the... |
| rhmpropd 20500 | Ring homomorphism depends ... |
| isdrng 20505 | The predicate "is a divisi... |
| drngunit 20506 | Elementhood in the set of ... |
| drngui 20507 | The set of units of a divi... |
| drngring 20508 | A division ring is a ring.... |
| drngringd 20509 | A division ring is a ring.... |
| drnggrpd 20510 | A division ring is a group... |
| drnggrp 20511 | A division ring is a group... |
| isfld 20512 | A field is a commutative d... |
| flddrngd 20513 | A field is a division ring... |
| fldcrngd 20514 | A field is a commutative r... |
| isdrng2 20515 | A division ring can equiva... |
| drngprop 20516 | If two structures have the... |
| drngmgp 20517 | A division ring contains a... |
| drngmcl 20518 | The product of two nonzero... |
| drngid 20519 | A division ring's unity is... |
| drngunz 20520 | A division ring's unity is... |
| drngnzr 20521 | All division rings are non... |
| drngid2 20522 | Properties showing that an... |
| drnginvrcl 20523 | Closure of the multiplicat... |
| drnginvrn0 20524 | The multiplicative inverse... |
| drnginvrcld 20525 | Closure of the multiplicat... |
| drnginvrl 20526 | Property of the multiplica... |
| drnginvrr 20527 | Property of the multiplica... |
| drnginvrld 20528 | Property of the multiplica... |
| drnginvrrd 20529 | Property of the multiplica... |
| drngmul0or 20530 | A product is zero iff one ... |
| drngmulne0 20531 | A product is nonzero iff b... |
| drngmuleq0 20532 | An element is zero iff its... |
| opprdrng 20533 | The opposite of a division... |
| isdrngd 20534 | Properties that characteri... |
| isdrngrd 20535 | Properties that characteri... |
| isdrngdOLD 20536 | Obsolete version of ~ isdr... |
| isdrngrdOLD 20537 | Obsolete version of ~ isdr... |
| drngpropd 20538 | If two structures have the... |
| fldpropd 20539 | If two structures have the... |
| rng1nnzr 20540 | The (smallest) structure r... |
| ring1zr 20541 | The only (unital) ring wit... |
| rngen1zr 20542 | The only (unital) ring wit... |
| ringen1zr 20543 | The only unital ring with ... |
| rng1nfld 20544 | The zero ring is not a fie... |
| issubdrg 20545 | Characterize the subfields... |
| issdrg 20548 | Property of a division sub... |
| sdrgrcl 20549 | Reverse closure for a sub-... |
| sdrgdrng 20550 | A sub-division-ring is a d... |
| sdrgsubrg 20551 | A sub-division-ring is a s... |
| sdrgid 20552 | Every division ring is a d... |
| sdrgss 20553 | A division subring is a su... |
| sdrgbas 20554 | Base set of a sub-division... |
| issdrg2 20555 | Property of a division sub... |
| sdrgunit 20556 | A unit of a sub-division-r... |
| imadrhmcl 20557 | The image of a (nontrivial... |
| fldsdrgfld 20558 | A sub-division-ring of a f... |
| acsfn1p 20559 | Construction of a closure ... |
| subrgacs 20560 | Closure property of subrin... |
| sdrgacs 20561 | Closure property of divisi... |
| cntzsdrg 20562 | Centralizers in division r... |
| subdrgint 20563 | The intersection of a none... |
| sdrgint 20564 | The intersection of a none... |
| primefld 20565 | The smallest sub division ... |
| primefld0cl 20566 | The prime field contains t... |
| primefld1cl 20567 | The prime field contains t... |
| abvfval 20570 | Value of the set of absolu... |
| isabv 20571 | Elementhood in the set of ... |
| isabvd 20572 | Properties that determine ... |
| abvrcl 20573 | Reverse closure for the ab... |
| abvfge0 20574 | An absolute value is a fun... |
| abvf 20575 | An absolute value is a fun... |
| abvcl 20576 | An absolute value is a fun... |
| abvge0 20577 | The absolute value of a nu... |
| abveq0 20578 | The value of an absolute v... |
| abvne0 20579 | The absolute value of a no... |
| abvgt0 20580 | The absolute value of a no... |
| abvmul 20581 | An absolute value distribu... |
| abvtri 20582 | An absolute value satisfie... |
| abv0 20583 | The absolute value of zero... |
| abv1z 20584 | The absolute value of one ... |
| abv1 20585 | The absolute value of one ... |
| abvneg 20586 | The absolute value of a ne... |
| abvsubtri 20587 | An absolute value satisfie... |
| abvrec 20588 | The absolute value distrib... |
| abvdiv 20589 | The absolute value distrib... |
| abvdom 20590 | Any ring with an absolute ... |
| abvres 20591 | The restriction of an abso... |
| abvtrivd 20592 | The trivial absolute value... |
| abvtriv 20593 | The trivial absolute value... |
| abvpropd 20594 | If two structures have the... |
| staffval 20599 | The functionalization of t... |
| stafval 20600 | The functionalization of t... |
| staffn 20601 | The functionalization is e... |
| issrng 20602 | The predicate "is a star r... |
| srngrhm 20603 | The involution function in... |
| srngring 20604 | A star ring is a ring. (C... |
| srngcnv 20605 | The involution function in... |
| srngf1o 20606 | The involution function in... |
| srngcl 20607 | The involution function in... |
| srngnvl 20608 | The involution function in... |
| srngadd 20609 | The involution function in... |
| srngmul 20610 | The involution function in... |
| srng1 20611 | The conjugate of the ring ... |
| srng0 20612 | The conjugate of the ring ... |
| issrngd 20613 | Properties that determine ... |
| idsrngd 20614 | A commutative ring is a st... |
| islmod 20619 | The predicate "is a left m... |
| lmodlema 20620 | Lemma for properties of a ... |
| islmodd 20621 | Properties that determine ... |
| lmodgrp 20622 | A left module is a group. ... |
| lmodring 20623 | The scalar component of a ... |
| lmodfgrp 20624 | The scalar component of a ... |
| lmodgrpd 20625 | A left module is a group. ... |
| lmodbn0 20626 | The base set of a left mod... |
| lmodacl 20627 | Closure of ring addition f... |
| lmodmcl 20628 | Closure of ring multiplica... |
| lmodsn0 20629 | The set of scalars in a le... |
| lmodvacl 20630 | Closure of vector addition... |
| lmodass 20631 | Left module vector sum is ... |
| lmodlcan 20632 | Left cancellation law for ... |
| lmodvscl 20633 | Closure of scalar product ... |
| lmodvscld 20634 | Closure of scalar product ... |
| scaffval 20635 | The scalar multiplication ... |
| scafval 20636 | The scalar multiplication ... |
| scafeq 20637 | If the scalar multiplicati... |
| scaffn 20638 | The scalar multiplication ... |
| lmodscaf 20639 | The scalar multiplication ... |
| lmodvsdi 20640 | Distributive law for scala... |
| lmodvsdir 20641 | Distributive law for scala... |
| lmodvsass 20642 | Associative law for scalar... |
| lmod0cl 20643 | The ring zero in a left mo... |
| lmod1cl 20644 | The ring unity in a left m... |
| lmodvs1 20645 | Scalar product with the ri... |
| lmod0vcl 20646 | The zero vector is a vecto... |
| lmod0vlid 20647 | Left identity law for the ... |
| lmod0vrid 20648 | Right identity law for the... |
| lmod0vid 20649 | Identity equivalent to the... |
| lmod0vs 20650 | Zero times a vector is the... |
| lmodvs0 20651 | Anything times the zero ve... |
| lmodvsmmulgdi 20652 | Distributive law for a gro... |
| lmodfopnelem1 20653 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20654 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20655 | The (functionalized) opera... |
| lcomf 20656 | A linear-combination sum i... |
| lcomfsupp 20657 | A linear-combination sum i... |
| lmodvnegcl 20658 | Closure of vector negative... |
| lmodvnegid 20659 | Addition of a vector with ... |
| lmodvneg1 20660 | Minus 1 times a vector is ... |
| lmodvsneg 20661 | Multiplication of a vector... |
| lmodvsubcl 20662 | Closure of vector subtract... |
| lmodcom 20663 | Left module vector sum is ... |
| lmodabl 20664 | A left module is an abelia... |
| lmodcmn 20665 | A left module is a commuta... |
| lmodnegadd 20666 | Distribute negation throug... |
| lmod4 20667 | Commutative/associative la... |
| lmodvsubadd 20668 | Relationship between vecto... |
| lmodvaddsub4 20669 | Vector addition/subtractio... |
| lmodvpncan 20670 | Addition/subtraction cance... |
| lmodvnpcan 20671 | Cancellation law for vecto... |
| lmodvsubval2 20672 | Value of vector subtractio... |
| lmodsubvs 20673 | Subtraction of a scalar pr... |
| lmodsubdi 20674 | Scalar multiplication dist... |
| lmodsubdir 20675 | Scalar multiplication dist... |
| lmodsubeq0 20676 | If the difference between ... |
| lmodsubid 20677 | Subtraction of a vector fr... |
| lmodvsghm 20678 | Scalar multiplication of t... |
| lmodprop2d 20679 | If two structures have the... |
| lmodpropd 20680 | If two structures have the... |
| gsumvsmul 20681 | Pull a scalar multiplicati... |
| mptscmfsupp0 20682 | A mapping to a scalar prod... |
| mptscmfsuppd 20683 | A function mapping to a sc... |
| rmodislmodlem 20684 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20685 | The right module ` R ` ind... |
| rmodislmodOLD 20686 | Obsolete version of ~ rmod... |
| lssset 20689 | The set of all (not necess... |
| islss 20690 | The predicate "is a subspa... |
| islssd 20691 | Properties that determine ... |
| lssss 20692 | A subspace is a set of vec... |
| lssel 20693 | A subspace member is a vec... |
| lss1 20694 | The set of vectors in a le... |
| lssuni 20695 | The union of all subspaces... |
| lssn0 20696 | A subspace is not empty. ... |
| 00lss 20697 | The empty structure has no... |
| lsscl 20698 | Closure property of a subs... |
| lssvsubcl 20699 | Closure of vector subtract... |
| lssvancl1 20700 | Non-closure: if one vector... |
| lssvancl2 20701 | Non-closure: if one vector... |
| lss0cl 20702 | The zero vector belongs to... |
| lsssn0 20703 | The singleton of the zero ... |
| lss0ss 20704 | The zero subspace is inclu... |
| lssle0 20705 | No subspace is smaller tha... |
| lssne0 20706 | A nonzero subspace has a n... |
| lssvneln0 20707 | A vector ` X ` which doesn... |
| lssneln0 20708 | A vector ` X ` which doesn... |
| lssssr 20709 | Conclude subspace ordering... |
| lssvacl 20710 | Closure of vector addition... |
| lssvscl 20711 | Closure of scalar product ... |
| lssvnegcl 20712 | Closure of negative vector... |
| lsssubg 20713 | All subspaces are subgroup... |
| lsssssubg 20714 | All subspaces are subgroup... |
| islss3 20715 | A linear subspace of a mod... |
| lsslmod 20716 | A submodule is a module. ... |
| lsslss 20717 | The subspaces of a subspac... |
| islss4 20718 | A linear subspace is a sub... |
| lss1d 20719 | One-dimensional subspace (... |
| lssintcl 20720 | The intersection of a none... |
| lssincl 20721 | The intersection of two su... |
| lssmre 20722 | The subspaces of a module ... |
| lssacs 20723 | Submodules are an algebrai... |
| prdsvscacl 20724 | Pointwise scalar multiplic... |
| prdslmodd 20725 | The product of a family of... |
| pwslmod 20726 | A structure power of a lef... |
| lspfval 20729 | The span function for a le... |
| lspf 20730 | The span function on a lef... |
| lspval 20731 | The span of a set of vecto... |
| lspcl 20732 | The span of a set of vecto... |
| lspsncl 20733 | The span of a singleton is... |
| lspprcl 20734 | The span of a pair is a su... |
| lsptpcl 20735 | The span of an unordered t... |
| lspsnsubg 20736 | The span of a singleton is... |
| 00lsp 20737 | ~ fvco4i lemma for linear ... |
| lspid 20738 | The span of a subspace is ... |
| lspssv 20739 | A span is a set of vectors... |
| lspss 20740 | Span preserves subset orde... |
| lspssid 20741 | A set of vectors is a subs... |
| lspidm 20742 | The span of a set of vecto... |
| lspun 20743 | The span of union is the s... |
| lspssp 20744 | If a set of vectors is a s... |
| mrclsp 20745 | Moore closure generalizes ... |
| lspsnss 20746 | The span of the singleton ... |
| lspsnel3 20747 | A member of the span of th... |
| lspprss 20748 | The span of a pair of vect... |
| lspsnid 20749 | A vector belongs to the sp... |
| lspsnel6 20750 | Relationship between a vec... |
| lspsnel5 20751 | Relationship between a vec... |
| lspsnel5a 20752 | Relationship between a vec... |
| lspprid1 20753 | A member of a pair of vect... |
| lspprid2 20754 | A member of a pair of vect... |
| lspprvacl 20755 | The sum of two vectors bel... |
| lssats2 20756 | A way to express atomistic... |
| lspsneli 20757 | A scalar product with a ve... |
| lspsn 20758 | Span of the singleton of a... |
| lspsnel 20759 | Member of span of the sing... |
| lspsnvsi 20760 | Span of a scalar product o... |
| lspsnss2 20761 | Comparable spans of single... |
| lspsnneg 20762 | Negation does not change t... |
| lspsnsub 20763 | Swapping subtraction order... |
| lspsn0 20764 | Span of the singleton of t... |
| lsp0 20765 | Span of the empty set. (C... |
| lspuni0 20766 | Union of the span of the e... |
| lspun0 20767 | The span of a union with t... |
| lspsneq0 20768 | Span of the singleton is t... |
| lspsneq0b 20769 | Equal singleton spans impl... |
| lmodindp1 20770 | Two independent (non-colin... |
| lsslsp 20771 | Spans in submodules corres... |
| lss0v 20772 | The zero vector in a submo... |
| lsspropd 20773 | If two structures have the... |
| lsppropd 20774 | If two structures have the... |
| reldmlmhm 20781 | Lemma for module homomorph... |
| lmimfn 20782 | Lemma for module isomorphi... |
| islmhm 20783 | Property of being a homomo... |
| islmhm3 20784 | Property of a module homom... |
| lmhmlem 20785 | Non-quantified consequence... |
| lmhmsca 20786 | A homomorphism of left mod... |
| lmghm 20787 | A homomorphism of left mod... |
| lmhmlmod2 20788 | A homomorphism of left mod... |
| lmhmlmod1 20789 | A homomorphism of left mod... |
| lmhmf 20790 | A homomorphism of left mod... |
| lmhmlin 20791 | A homomorphism of left mod... |
| lmodvsinv 20792 | Multiplication of a vector... |
| lmodvsinv2 20793 | Multiplying a negated vect... |
| islmhm2 20794 | A one-equation proof of li... |
| islmhmd 20795 | Deduction for a module hom... |
| 0lmhm 20796 | The constant zero linear f... |
| idlmhm 20797 | The identity function on a... |
| invlmhm 20798 | The negative function on a... |
| lmhmco 20799 | The composition of two mod... |
| lmhmplusg 20800 | The pointwise sum of two l... |
| lmhmvsca 20801 | The pointwise scalar produ... |
| lmhmf1o 20802 | A bijective module homomor... |
| lmhmima 20803 | The image of a subspace un... |
| lmhmpreima 20804 | The inverse image of a sub... |
| lmhmlsp 20805 | Homomorphisms preserve spa... |
| lmhmrnlss 20806 | The range of a homomorphis... |
| lmhmkerlss 20807 | The kernel of a homomorphi... |
| reslmhm 20808 | Restriction of a homomorph... |
| reslmhm2 20809 | Expansion of the codomain ... |
| reslmhm2b 20810 | Expansion of the codomain ... |
| lmhmeql 20811 | The equalizer of two modul... |
| lspextmo 20812 | A linear function is compl... |
| pwsdiaglmhm 20813 | Diagonal homomorphism into... |
| pwssplit0 20814 | Splitting for structure po... |
| pwssplit1 20815 | Splitting for structure po... |
| pwssplit2 20816 | Splitting for structure po... |
| pwssplit3 20817 | Splitting for structure po... |
| islmim 20818 | An isomorphism of left mod... |
| lmimf1o 20819 | An isomorphism of left mod... |
| lmimlmhm 20820 | An isomorphism of modules ... |
| lmimgim 20821 | An isomorphism of modules ... |
| islmim2 20822 | An isomorphism of left mod... |
| lmimcnv 20823 | The converse of a bijectiv... |
| brlmic 20824 | The relation "is isomorphi... |
| brlmici 20825 | Prove isomorphic by an exp... |
| lmiclcl 20826 | Isomorphism implies the le... |
| lmicrcl 20827 | Isomorphism implies the ri... |
| lmicsym 20828 | Module isomorphism is symm... |
| lmhmpropd 20829 | Module homomorphism depend... |
| islbs 20832 | The predicate " ` B ` is a... |
| lbsss 20833 | A basis is a set of vector... |
| lbsel 20834 | An element of a basis is a... |
| lbssp 20835 | The span of a basis is the... |
| lbsind 20836 | A basis is linearly indepe... |
| lbsind2 20837 | A basis is linearly indepe... |
| lbspss 20838 | No proper subset of a basi... |
| lsmcl 20839 | The sum of two subspaces i... |
| lsmspsn 20840 | Member of subspace sum of ... |
| lsmelval2 20841 | Subspace sum membership in... |
| lsmsp 20842 | Subspace sum in terms of s... |
| lsmsp2 20843 | Subspace sum of spans of s... |
| lsmssspx 20844 | Subspace sum (in its exten... |
| lsmpr 20845 | The span of a pair of vect... |
| lsppreli 20846 | A vector expressed as a su... |
| lsmelpr 20847 | Two ways to say that a vec... |
| lsppr0 20848 | The span of a vector paire... |
| lsppr 20849 | Span of a pair of vectors.... |
| lspprel 20850 | Member of the span of a pa... |
| lspprabs 20851 | Absorption of vector sum i... |
| lspvadd 20852 | The span of a vector sum i... |
| lspsntri 20853 | Triangle-type inequality f... |
| lspsntrim 20854 | Triangle-type inequality f... |
| lbspropd 20855 | If two structures have the... |
| pj1lmhm 20856 | The left projection functi... |
| pj1lmhm2 20857 | The left projection functi... |
| islvec 20860 | The predicate "is a left v... |
| lvecdrng 20861 | The set of scalars of a le... |
| lveclmod 20862 | A left vector space is a l... |
| lveclmodd 20863 | A vector space is a left m... |
| lvecgrpd 20864 | A vector space is a group.... |
| lsslvec 20865 | A vector subspace is a vec... |
| lmhmlvec 20866 | The property for modules t... |
| lvecvs0or 20867 | If a scalar product is zer... |
| lvecvsn0 20868 | A scalar product is nonzer... |
| lssvs0or 20869 | If a scalar product belong... |
| lvecvscan 20870 | Cancellation law for scala... |
| lvecvscan2 20871 | Cancellation law for scala... |
| lvecinv 20872 | Invert coefficient of scal... |
| lspsnvs 20873 | A nonzero scalar product d... |
| lspsneleq 20874 | Membership relation that i... |
| lspsncmp 20875 | Comparable spans of nonzer... |
| lspsnne1 20876 | Two ways to express that v... |
| lspsnne2 20877 | Two ways to express that v... |
| lspsnnecom 20878 | Swap two vectors with diff... |
| lspabs2 20879 | Absorption law for span of... |
| lspabs3 20880 | Absorption law for span of... |
| lspsneq 20881 | Equal spans of singletons ... |
| lspsneu 20882 | Nonzero vectors with equal... |
| lspsnel4 20883 | A member of the span of th... |
| lspdisj 20884 | The span of a vector not i... |
| lspdisjb 20885 | A nonzero vector is not in... |
| lspdisj2 20886 | Unequal spans are disjoint... |
| lspfixed 20887 | Show membership in the spa... |
| lspexch 20888 | Exchange property for span... |
| lspexchn1 20889 | Exchange property for span... |
| lspexchn2 20890 | Exchange property for span... |
| lspindpi 20891 | Partial independence prope... |
| lspindp1 20892 | Alternate way to say 3 vec... |
| lspindp2l 20893 | Alternate way to say 3 vec... |
| lspindp2 20894 | Alternate way to say 3 vec... |
| lspindp3 20895 | Independence of 2 vectors ... |
| lspindp4 20896 | (Partial) independence of ... |
| lvecindp 20897 | Compute the ` X ` coeffici... |
| lvecindp2 20898 | Sums of independent vector... |
| lspsnsubn0 20899 | Unequal singleton spans im... |
| lsmcv 20900 | Subspace sum has the cover... |
| lspsolvlem 20901 | Lemma for ~ lspsolv . (Co... |
| lspsolv 20902 | If ` X ` is in the span of... |
| lssacsex 20903 | In a vector space, subspac... |
| lspsnat 20904 | There is no subspace stric... |
| lspsncv0 20905 | The span of a singleton co... |
| lsppratlem1 20906 | Lemma for ~ lspprat . Let... |
| lsppratlem2 20907 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 20908 | Lemma for ~ lspprat . In ... |
| lsppratlem4 20909 | Lemma for ~ lspprat . In ... |
| lsppratlem5 20910 | Lemma for ~ lspprat . Com... |
| lsppratlem6 20911 | Lemma for ~ lspprat . Neg... |
| lspprat 20912 | A proper subspace of the s... |
| islbs2 20913 | An equivalent formulation ... |
| islbs3 20914 | An equivalent formulation ... |
| lbsacsbs 20915 | Being a basis in a vector ... |
| lvecdim 20916 | The dimension theorem for ... |
| lbsextlem1 20917 | Lemma for ~ lbsext . The ... |
| lbsextlem2 20918 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 20919 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 20920 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 20921 | For any linearly independe... |
| lbsext 20922 | For any linearly independe... |
| lbsexg 20923 | Every vector space has a b... |
| lbsex 20924 | Every vector space has a b... |
| lvecprop2d 20925 | If two structures have the... |
| lvecpropd 20926 | If two structures have the... |
| sraval 20935 | Lemma for ~ srabase throug... |
| sralem 20936 | Lemma for ~ srabase and si... |
| sralemOLD 20937 | Obsolete version of ~ sral... |
| srabase 20938 | Base set of a subring alge... |
| srabaseOLD 20939 | Obsolete proof of ~ srabas... |
| sraaddg 20940 | Additive operation of a su... |
| sraaddgOLD 20941 | Obsolete proof of ~ sraadd... |
| sramulr 20942 | Multiplicative operation o... |
| sramulrOLD 20943 | Obsolete proof of ~ sramul... |
| srasca 20944 | The set of scalars of a su... |
| srascaOLD 20945 | Obsolete proof of ~ srasca... |
| sravsca 20946 | The scalar product operati... |
| sravscaOLD 20947 | Obsolete proof of ~ sravsc... |
| sraip 20948 | The inner product operatio... |
| sratset 20949 | Topology component of a su... |
| sratsetOLD 20950 | Obsolete proof of ~ sratse... |
| sratopn 20951 | Topology component of a su... |
| srads 20952 | Distance function of a sub... |
| sradsOLD 20953 | Obsolete proof of ~ srads ... |
| sraring 20954 | Condition for a subring al... |
| sralmod 20955 | The subring algebra is a l... |
| sralmod0 20956 | The subring module inherit... |
| issubrgd 20957 | Prove a subring by closure... |
| rlmfn 20958 | ` ringLMod ` is a function... |
| rlmval 20959 | Value of the ring module. ... |
| lidlval 20960 | Value of the set of ring i... |
| rspval 20961 | Value of the ring span fun... |
| rlmval2 20962 | Value of the ring module e... |
| rlmbas 20963 | Base set of the ring modul... |
| rlmplusg 20964 | Vector addition in the rin... |
| rlm0 20965 | Zero vector in the ring mo... |
| rlmsub 20966 | Subtraction in the ring mo... |
| rlmmulr 20967 | Ring multiplication in the... |
| rlmsca 20968 | Scalars in the ring module... |
| rlmsca2 20969 | Scalars in the ring module... |
| rlmvsca 20970 | Scalar multiplication in t... |
| rlmtopn 20971 | Topology component of the ... |
| rlmds 20972 | Metric component of the ri... |
| rlmlmod 20973 | The ring module is a modul... |
| rlmlvec 20974 | The ring module over a div... |
| rlmlsm 20975 | Subgroup sum of the ring m... |
| rlmvneg 20976 | Vector negation in the rin... |
| rlmscaf 20977 | Functionalized scalar mult... |
| ixpsnbasval 20978 | The value of an infinite C... |
| lidlss 20979 | An ideal is a subset of th... |
| lidlssbas 20980 | The base set of the restri... |
| lidlbas 20981 | A (left) ideal of a ring i... |
| islidl 20982 | Predicate of being a (left... |
| rnglidlmcl 20983 | A (left) ideal containing ... |
| rngridlmcl 20984 | A right ideal (which is a ... |
| lidl0cl 20985 | An ideal contains 0. (Con... |
| lidlacl 20986 | An ideal is closed under a... |
| lidlnegcl 20987 | An ideal contains negative... |
| lidlsubg 20988 | An ideal is a subgroup of ... |
| lidlsubcl 20989 | An ideal is closed under s... |
| lidlmcl 20990 | An ideal is closed under l... |
| lidl1el 20991 | An ideal contains 1 iff it... |
| dflidl2lem 20992 | Lemma for ~ dflidl2 : a su... |
| dflidl2 20993 | Alternate (the usual textb... |
| lidl0 20994 | Every ring contains a zero... |
| lidl1 20995 | Every ring contains a unit... |
| lidlacs 20996 | The ideal system is an alg... |
| rspcl 20997 | The span of a set of ring ... |
| rspssid 20998 | The span of a set of ring ... |
| rsp1 20999 | The span of the identity e... |
| rsp0 21000 | The span of the zero eleme... |
| rspssp 21001 | The ideal span of a set of... |
| mrcrsp 21002 | Moore closure generalizes ... |
| lidlnz 21003 | A nonzero ideal contains a... |
| drngnidl 21004 | A division ring has only t... |
| lidlrsppropd 21005 | The left ideals and ring s... |
| 2idlval 21008 | Definition of a two-sided ... |
| isridl 21009 | A right ideal is a left id... |
| df2idl2 21010 | Alternate (the usual textb... |
| ridl0 21011 | Every ring contains a zero... |
| ridl1 21012 | Every ring contains a unit... |
| 2idl0 21013 | Every ring contains a zero... |
| 2idl1 21014 | Every ring contains a unit... |
| 2idlelb 21015 | Membership in a two-sided ... |
| 2idllidld 21016 | A two-sided ideal is a lef... |
| 2idlridld 21017 | A two-sided ideal is a rig... |
| 2idlss 21018 | A two-sided ideal is a sub... |
| 2idlbas 21019 | The base set of a two-side... |
| 2idlelbas 21020 | The base set of a two-side... |
| 2idlcpblrng 21021 | The coset equivalence rela... |
| 2idlcpbl 21022 | The coset equivalence rela... |
| qus1 21023 | The multiplicative identit... |
| qusring 21024 | If ` S ` is a two-sided id... |
| qusrhm 21025 | If ` S ` is a two-sided id... |
| qusmul2 21026 | Value of the ring operatio... |
| crngridl 21027 | In a commutative ring, the... |
| crng2idl 21028 | In a commutative ring, a t... |
| qusmulrng 21029 | Value of the multiplicatio... |
| quscrng 21030 | The quotient of a commutat... |
| dflidl2rng 21031 | Alternate (the usual textb... |
| isridlrng 21032 | A right ideal is a left id... |
| rnglidl0 21033 | Every non-unital ring cont... |
| rnglidl1 21034 | The base set of every non-... |
| rnglidlmmgm 21035 | The multiplicative group o... |
| rnglidlmsgrp 21036 | The multiplicative group o... |
| rnglidlrng 21037 | A (left) ideal of a non-un... |
| df2idl2rng 21038 | Alternate (the usual textb... |
| rng2idlsubrng 21039 | A two-sided ideal of a non... |
| rng2idlnsg 21040 | A two-sided ideal of a non... |
| rng2idl0 21041 | The zero (additive identit... |
| rng2idlsubgsubrng 21042 | A two-sided ideal of a non... |
| rng2idlsubgnsg 21043 | A two-sided ideal of a non... |
| rng2idlsubg0 21044 | The zero (additive identit... |
| qus2idrng 21045 | The quotient of a non-unit... |
| rngqiprng1elbas 21046 | The ring unity of a two-si... |
| rngqiprngghmlem1 21047 | Lemma 1 for ~ rngqiprngghm... |
| rngqiprngghmlem2 21048 | Lemma 2 for ~ rngqiprngghm... |
| rngqiprngghmlem3 21049 | Lemma 3 for ~ rngqiprngghm... |
| rngqiprngimfolem 21050 | Lemma for ~ rngqiprngimfo ... |
| rngqiprnglinlem1 21051 | Lemma 1 for ~ rngqiprnglin... |
| rngqiprnglinlem2 21052 | Lemma 2 for ~ rngqiprnglin... |
| rngqiprnglinlem3 21053 | Lemma 3 for ~ rngqiprnglin... |
| rngqiprngimf1lem 21054 | Lemma for ~ rngqiprngimf1 ... |
| rngqipbas 21055 | The base set of the produc... |
| rngqiprng 21056 | The product of the quotien... |
| rngqiprngimf 21057 | ` F ` is a function from (... |
| rngqiprngimfv 21058 | The value of the function ... |
| rngqiprngghm 21059 | ` F ` is a homomorphism of... |
| rngqiprngimf1 21060 | ` F ` is a one-to-one func... |
| rngqiprngimfo 21061 | ` F ` is a function from (... |
| rngqiprnglin 21062 | ` F ` is linear with respe... |
| rngqiprngho 21063 | ` F ` is a homomorphism of... |
| rngqiprngim 21064 | ` F ` is an isomorphism of... |
| rng2idl1cntr 21065 | The unity of a two-sided i... |
| rngringbdlem1 21066 | In a unital ring, the quot... |
| rngringbdlem2 21067 | A non-unital ring is unita... |
| rngringbd 21068 | A non-unital ring is unita... |
| ring2idlqus 21069 | For every unital ring ther... |
| ring2idlqusb 21070 | A non-unital ring is unita... |
| rngqiprngfulem1 21071 | Lemma 1 for ~ rngqiprngfu ... |
| rngqiprngfulem2 21072 | Lemma 2 for ~ rngqiprngfu ... |
| rngqiprngfulem3 21073 | Lemma 3 for ~ rngqiprngfu ... |
| rngqiprngfulem4 21074 | Lemma 4 for ~ rngqiprngfu ... |
| rngqiprngfulem5 21075 | Lemma 5 for ~ rngqiprngfu ... |
| rngqipring1 21076 | The ring unity of the prod... |
| rngqiprngfu 21077 | The function value of ` F ... |
| rngqiprngu 21078 | If a non-unital ring has a... |
| ring2idlqus1 21079 | If a non-unital ring has a... |
| lpival 21084 | Value of the set of princi... |
| islpidl 21085 | Property of being a princi... |
| lpi0 21086 | The zero ideal is always p... |
| lpi1 21087 | The unit ideal is always p... |
| islpir 21088 | Principal ideal rings are ... |
| lpiss 21089 | Principal ideals are a sub... |
| islpir2 21090 | Principal ideal rings are ... |
| lpirring 21091 | Principal ideal rings are ... |
| drnglpir 21092 | Division rings are princip... |
| rspsn 21093 | Membership in principal id... |
| lidldvgen 21094 | An element generates an id... |
| lpigen 21095 | An ideal is principal iff ... |
| rrgval 21104 | Value of the set or left-r... |
| isrrg 21105 | Membership in the set of l... |
| rrgeq0i 21106 | Property of a left-regular... |
| rrgeq0 21107 | Left-multiplication by a l... |
| rrgsupp 21108 | Left multiplication by a l... |
| rrgss 21109 | Left-regular elements are ... |
| unitrrg 21110 | Units are regular elements... |
| isdomn 21111 | Expand definition of a dom... |
| domnnzr 21112 | A domain is a nonzero ring... |
| domnring 21113 | A domain is a ring. (Cont... |
| domneq0 21114 | In a domain, a product is ... |
| domnmuln0 21115 | In a domain, a product of ... |
| isdomn2 21116 | A ring is a domain iff all... |
| domnrrg 21117 | In a domain, any nonzero e... |
| isdomn5 21118 | The right conjunct in the ... |
| isdomn4 21119 | A ring is a domain iff it ... |
| opprdomn 21120 | The opposite of a domain i... |
| abvn0b 21121 | Another characterization o... |
| drngdomn 21122 | A division ring is a domai... |
| isidom 21123 | An integral domain is a co... |
| fldidom 21124 | A field is an integral dom... |
| fldidomOLD 21125 | Obsolete version of ~ fldi... |
| fidomndrnglem 21126 | Lemma for ~ fidomndrng . ... |
| fidomndrng 21127 | A finite domain is a divis... |
| fiidomfld 21128 | A finite integral domain i... |
| cnfldstr 21147 | The field of complex numbe... |
| cnfldex 21148 | The field of complex numbe... |
| cnfldbas 21149 | The base set of the field ... |
| cnfldadd 21150 | The addition operation of ... |
| cnfldmul 21151 | The multiplication operati... |
| cnfldcj 21152 | The conjugation operation ... |
| cnfldtset 21153 | The topology component of ... |
| cnfldle 21154 | The ordering of the field ... |
| cnfldds 21155 | The metric of the field of... |
| cnfldunif 21156 | The uniform structure comp... |
| cnfldfun 21157 | The field of complex numbe... |
| cnfldfunALT 21158 | The field of complex numbe... |
| cnfldfunALTOLD 21159 | Obsolete proof of ~ cnfldf... |
| xrsstr 21160 | The extended real structur... |
| xrsex 21161 | The extended real structur... |
| xrsbas 21162 | The base set of the extend... |
| xrsadd 21163 | The addition operation of ... |
| xrsmul 21164 | The multiplication operati... |
| xrstset 21165 | The topology component of ... |
| xrsle 21166 | The ordering of the extend... |
| cncrng 21167 | The complex numbers form a... |
| cnring 21168 | The complex numbers form a... |
| xrsmcmn 21169 | The "multiplicative group"... |
| cnfld0 21170 | Zero is the zero element o... |
| cnfld1 21171 | One is the unity element o... |
| cnfldneg 21172 | The additive inverse in th... |
| cnfldplusf 21173 | The functionalized additio... |
| cnfldsub 21174 | The subtraction operator i... |
| cndrng 21175 | The complex numbers form a... |
| cnflddiv 21176 | The division operation in ... |
| cnfldinv 21177 | The multiplicative inverse... |
| cnfldmulg 21178 | The group multiple functio... |
| cnfldexp 21179 | The exponentiation operato... |
| cnsrng 21180 | The complex numbers form a... |
| xrsmgm 21181 | The "additive group" of th... |
| xrsnsgrp 21182 | The "additive group" of th... |
| xrsmgmdifsgrp 21183 | The "additive group" of th... |
| xrs1mnd 21184 | The extended real numbers,... |
| xrs10 21185 | The zero of the extended r... |
| xrs1cmn 21186 | The extended real numbers ... |
| xrge0subm 21187 | The nonnegative extended r... |
| xrge0cmn 21188 | The nonnegative extended r... |
| xrsds 21189 | The metric of the extended... |
| xrsdsval 21190 | The metric of the extended... |
| xrsdsreval 21191 | The metric of the extended... |
| xrsdsreclblem 21192 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 21193 | The metric of the extended... |
| cnsubmlem 21194 | Lemma for ~ nn0subm and fr... |
| cnsubglem 21195 | Lemma for ~ resubdrg and f... |
| cnsubrglem 21196 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 21197 | Lemma for ~ resubdrg and f... |
| qsubdrg 21198 | The rational numbers form ... |
| zsubrg 21199 | The integers form a subrin... |
| gzsubrg 21200 | The gaussian integers form... |
| nn0subm 21201 | The nonnegative integers f... |
| rege0subm 21202 | The nonnegative reals form... |
| absabv 21203 | The regular absolute value... |
| zsssubrg 21204 | The integers are a subset ... |
| qsssubdrg 21205 | The rational numbers are a... |
| cnsubrg 21206 | There are no subrings of t... |
| cnmgpabl 21207 | The unit group of the comp... |
| cnmgpid 21208 | The group identity element... |
| cnmsubglem 21209 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 21210 | The positive reals form a ... |
| gzrngunitlem 21211 | Lemma for ~ gzrngunit . (... |
| gzrngunit 21212 | The units on ` ZZ [ _i ] `... |
| gsumfsum 21213 | Relate a group sum on ` CC... |
| regsumfsum 21214 | Relate a group sum on ` ( ... |
| expmhm 21215 | Exponentiation is a monoid... |
| nn0srg 21216 | The nonnegative integers f... |
| rge0srg 21217 | The nonnegative real numbe... |
| zringcrng 21220 | The ring of integers is a ... |
| zringring 21221 | The ring of integers is a ... |
| zringrng 21222 | The ring of integers is a ... |
| zringabl 21223 | The ring of integers is an... |
| zringgrp 21224 | The ring of integers is an... |
| zringbas 21225 | The integers are the base ... |
| zringplusg 21226 | The addition operation of ... |
| zringsub 21227 | The subtraction of element... |
| zringmulg 21228 | The multiplication (group ... |
| zringmulr 21229 | The multiplication operati... |
| zring0 21230 | The zero element of the ri... |
| zring1 21231 | The unity element of the r... |
| zringnzr 21232 | The ring of integers is a ... |
| dvdsrzring 21233 | Ring divisibility in the r... |
| zringlpirlem1 21234 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 21235 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 21236 | Lemma for ~ zringlpir . A... |
| zringinvg 21237 | The additive inverse of an... |
| zringunit 21238 | The units of ` ZZ ` are th... |
| zringlpir 21239 | The integers are a princip... |
| zringndrg 21240 | The integers are not a div... |
| zringcyg 21241 | The integers are a cyclic ... |
| zringsubgval 21242 | Subtraction in the ring of... |
| zringmpg 21243 | The multiplicative group o... |
| prmirredlem 21244 | A positive integer is irre... |
| dfprm2 21245 | The positive irreducible e... |
| prmirred 21246 | The irreducible elements o... |
| expghm 21247 | Exponentiation is a group ... |
| mulgghm2 21248 | The powers of a group elem... |
| mulgrhm 21249 | The powers of the element ... |
| mulgrhm2 21250 | The powers of the element ... |
| pzriprnglem1 21251 | Lemma 1 for ~ pzriprng : `... |
| pzriprnglem2 21252 | Lemma 2 for ~ pzriprng : ... |
| pzriprnglem3 21253 | Lemma 3 for ~ pzriprng : ... |
| pzriprnglem4 21254 | Lemma 4 for ~ pzriprng : `... |
| pzriprnglem5 21255 | Lemma 5 for ~ pzriprng : `... |
| pzriprnglem6 21256 | Lemma 6 for ~ pzriprng : `... |
| pzriprnglem7 21257 | Lemma 7 for ~ pzriprng : `... |
| pzriprnglem8 21258 | Lemma 8 for ~ pzriprng : `... |
| pzriprnglem9 21259 | Lemma 9 for ~ pzriprng : ... |
| pzriprnglem10 21260 | Lemma 10 for ~ pzriprng : ... |
| pzriprnglem11 21261 | Lemma 11 for ~ pzriprng : ... |
| pzriprnglem12 21262 | Lemma 12 for ~ pzriprng : ... |
| pzriprnglem13 21263 | Lemma 13 for ~ pzriprng : ... |
| pzriprnglem14 21264 | Lemma 14 for ~ pzriprng : ... |
| pzriprngALT 21265 | The non-unital ring ` ( ZZ... |
| pzriprng1ALT 21266 | The ring unity of the ring... |
| pzriprng 21267 | The non-unital ring ` ( ZZ... |
| pzriprng1 21268 | The ring unity of the ring... |
| zrhval 21277 | Define the unique homomorp... |
| zrhval2 21278 | Alternate value of the ` Z... |
| zrhmulg 21279 | Value of the ` ZRHom ` hom... |
| zrhrhmb 21280 | The ` ZRHom ` homomorphism... |
| zrhrhm 21281 | The ` ZRHom ` homomorphism... |
| zrh1 21282 | Interpretation of 1 in a r... |
| zrh0 21283 | Interpretation of 0 in a r... |
| zrhpropd 21284 | The ` ZZ ` ring homomorphi... |
| zlmval 21285 | Augment an abelian group w... |
| zlmlem 21286 | Lemma for ~ zlmbas and ~ z... |
| zlmlemOLD 21287 | Obsolete version of ~ zlml... |
| zlmbas 21288 | Base set of a ` ZZ ` -modu... |
| zlmbasOLD 21289 | Obsolete version of ~ zlmb... |
| zlmplusg 21290 | Group operation of a ` ZZ ... |
| zlmplusgOLD 21291 | Obsolete version of ~ zlmb... |
| zlmmulr 21292 | Ring operation of a ` ZZ `... |
| zlmmulrOLD 21293 | Obsolete version of ~ zlmb... |
| zlmsca 21294 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 21295 | Scalar multiplication oper... |
| zlmlmod 21296 | The ` ZZ ` -module operati... |
| chrval 21297 | Definition substitution of... |
| chrcl 21298 | Closure of the characteris... |
| chrid 21299 | The canonical ` ZZ ` ring ... |
| chrdvds 21300 | The ` ZZ ` ring homomorphi... |
| chrcong 21301 | If two integers are congru... |
| chrnzr 21302 | Nonzero rings are precisel... |
| chrrhm 21303 | The characteristic restric... |
| domnchr 21304 | The characteristic of a do... |
| znlidl 21305 | The set ` n ZZ ` is an ide... |
| zncrng2 21306 | The value of the ` Z/nZ ` ... |
| znval 21307 | The value of the ` Z/nZ ` ... |
| znle 21308 | The value of the ` Z/nZ ` ... |
| znval2 21309 | Self-referential expressio... |
| znbaslem 21310 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 21311 | Obsolete version of ~ znba... |
| znbas2 21312 | The base set of ` Z/nZ ` i... |
| znbas2OLD 21313 | Obsolete version of ~ znba... |
| znadd 21314 | The additive structure of ... |
| znaddOLD 21315 | Obsolete version of ~ znad... |
| znmul 21316 | The multiplicative structu... |
| znmulOLD 21317 | Obsolete version of ~ znad... |
| znzrh 21318 | The ` ZZ ` ring homomorphi... |
| znbas 21319 | The base set of ` Z/nZ ` s... |
| zncrng 21320 | ` Z/nZ ` is a commutative ... |
| znzrh2 21321 | The ` ZZ ` ring homomorphi... |
| znzrhval 21322 | The ` ZZ ` ring homomorphi... |
| znzrhfo 21323 | The ` ZZ ` ring homomorphi... |
| zncyg 21324 | The group ` ZZ / n ZZ ` is... |
| zndvds 21325 | Express equality of equiva... |
| zndvds0 21326 | Special case of ~ zndvds w... |
| znf1o 21327 | The function ` F ` enumera... |
| zzngim 21328 | The ` ZZ ` ring homomorphi... |
| znle2 21329 | The ordering of the ` Z/nZ... |
| znleval 21330 | The ordering of the ` Z/nZ... |
| znleval2 21331 | The ordering of the ` Z/nZ... |
| zntoslem 21332 | Lemma for ~ zntos . (Cont... |
| zntos 21333 | The ` Z/nZ ` structure is ... |
| znhash 21334 | The ` Z/nZ ` structure has... |
| znfi 21335 | The ` Z/nZ ` structure is ... |
| znfld 21336 | The ` Z/nZ ` structure is ... |
| znidomb 21337 | The ` Z/nZ ` structure is ... |
| znchr 21338 | Cyclic rings are defined b... |
| znunit 21339 | The units of ` Z/nZ ` are ... |
| znunithash 21340 | The size of the unit group... |
| znrrg 21341 | The regular elements of ` ... |
| cygznlem1 21342 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 21343 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 21344 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 21345 | A cyclic group with ` n ` ... |
| cygzn 21346 | A cyclic group with ` n ` ... |
| cygth 21347 | The "fundamental theorem o... |
| cyggic 21348 | Cyclic groups are isomorph... |
| frgpcyg 21349 | A free group is cyclic iff... |
| cnmsgnsubg 21350 | The signs form a multiplic... |
| cnmsgnbas 21351 | The base set of the sign s... |
| cnmsgngrp 21352 | The group of signs under m... |
| psgnghm 21353 | The sign is a homomorphism... |
| psgnghm2 21354 | The sign is a homomorphism... |
| psgninv 21355 | The sign of a permutation ... |
| psgnco 21356 | Multiplicativity of the pe... |
| zrhpsgnmhm 21357 | Embedding of permutation s... |
| zrhpsgninv 21358 | The embedded sign of a per... |
| evpmss 21359 | Even permutations are perm... |
| psgnevpmb 21360 | A class is an even permuta... |
| psgnodpm 21361 | A permutation which is odd... |
| psgnevpm 21362 | A permutation which is eve... |
| psgnodpmr 21363 | If a permutation has sign ... |
| zrhpsgnevpm 21364 | The sign of an even permut... |
| zrhpsgnodpm 21365 | The sign of an odd permuta... |
| cofipsgn 21366 | Composition of any class `... |
| zrhpsgnelbas 21367 | Embedding of permutation s... |
| zrhcopsgnelbas 21368 | Embedding of permutation s... |
| evpmodpmf1o 21369 | The function for performin... |
| pmtrodpm 21370 | A transposition is an odd ... |
| psgnfix1 21371 | A permutation of a finite ... |
| psgnfix2 21372 | A permutation of a finite ... |
| psgndiflemB 21373 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21374 | Lemma 2 for ~ psgndif . (... |
| psgndif 21375 | Embedding of permutation s... |
| copsgndif 21376 | Embedding of permutation s... |
| rebase 21379 | The base of the field of r... |
| remulg 21380 | The multiplication (group ... |
| resubdrg 21381 | The real numbers form a di... |
| resubgval 21382 | Subtraction in the field o... |
| replusg 21383 | The addition operation of ... |
| remulr 21384 | The multiplication operati... |
| re0g 21385 | The zero element of the fi... |
| re1r 21386 | The unity element of the f... |
| rele2 21387 | The ordering relation of t... |
| relt 21388 | The ordering relation of t... |
| reds 21389 | The distance of the field ... |
| redvr 21390 | The division operation of ... |
| retos 21391 | The real numbers are a tot... |
| refld 21392 | The real numbers form a fi... |
| refldcj 21393 | The conjugation operation ... |
| resrng 21394 | The real numbers form a st... |
| regsumsupp 21395 | The group sum over the rea... |
| rzgrp 21396 | The quotient group ` RR / ... |
| isphl 21401 | The predicate "is a genera... |
| phllvec 21402 | A pre-Hilbert space is a l... |
| phllmod 21403 | A pre-Hilbert space is a l... |
| phlsrng 21404 | The scalar ring of a pre-H... |
| phllmhm 21405 | The inner product of a pre... |
| ipcl 21406 | Closure of the inner produ... |
| ipcj 21407 | Conjugate of an inner prod... |
| iporthcom 21408 | Orthogonality (meaning inn... |
| ip0l 21409 | Inner product with a zero ... |
| ip0r 21410 | Inner product with a zero ... |
| ipeq0 21411 | The inner product of a vec... |
| ipdir 21412 | Distributive law for inner... |
| ipdi 21413 | Distributive law for inner... |
| ip2di 21414 | Distributive law for inner... |
| ipsubdir 21415 | Distributive law for inner... |
| ipsubdi 21416 | Distributive law for inner... |
| ip2subdi 21417 | Distributive law for inner... |
| ipass 21418 | Associative law for inner ... |
| ipassr 21419 | "Associative" law for seco... |
| ipassr2 21420 | "Associative" law for inne... |
| ipffval 21421 | The inner product operatio... |
| ipfval 21422 | The inner product operatio... |
| ipfeq 21423 | If the inner product opera... |
| ipffn 21424 | The inner product operatio... |
| phlipf 21425 | The inner product operatio... |
| ip2eq 21426 | Two vectors are equal iff ... |
| isphld 21427 | Properties that determine ... |
| phlpropd 21428 | If two structures have the... |
| ssipeq 21429 | The inner product on a sub... |
| phssipval 21430 | The inner product on a sub... |
| phssip 21431 | The inner product (as a fu... |
| phlssphl 21432 | A subspace of an inner pro... |
| ocvfval 21439 | The orthocomplement operat... |
| ocvval 21440 | Value of the orthocompleme... |
| elocv 21441 | Elementhood in the orthoco... |
| ocvi 21442 | Property of a member of th... |
| ocvss 21443 | The orthocomplement of a s... |
| ocvocv 21444 | A set is contained in its ... |
| ocvlss 21445 | The orthocomplement of a s... |
| ocv2ss 21446 | Orthocomplements reverse s... |
| ocvin 21447 | An orthocomplement has tri... |
| ocvsscon 21448 | Two ways to say that ` S `... |
| ocvlsp 21449 | The orthocomplement of a l... |
| ocv0 21450 | The orthocomplement of the... |
| ocvz 21451 | The orthocomplement of the... |
| ocv1 21452 | The orthocomplement of the... |
| unocv 21453 | The orthocomplement of a u... |
| iunocv 21454 | The orthocomplement of an ... |
| cssval 21455 | The set of closed subspace... |
| iscss 21456 | The predicate "is a closed... |
| cssi 21457 | Property of a closed subsp... |
| cssss 21458 | A closed subspace is a sub... |
| iscss2 21459 | It is sufficient to prove ... |
| ocvcss 21460 | The orthocomplement of any... |
| cssincl 21461 | The zero subspace is a clo... |
| css0 21462 | The zero subspace is a clo... |
| css1 21463 | The whole space is a close... |
| csslss 21464 | A closed subspace of a pre... |
| lsmcss 21465 | A subset of a pre-Hilbert ... |
| cssmre 21466 | The closed subspaces of a ... |
| mrccss 21467 | The Moore closure correspo... |
| thlval 21468 | Value of the Hilbert latti... |
| thlbas 21469 | Base set of the Hilbert la... |
| thlbasOLD 21470 | Obsolete proof of ~ thlbas... |
| thlle 21471 | Ordering on the Hilbert la... |
| thlleOLD 21472 | Obsolete proof of ~ thlle ... |
| thlleval 21473 | Ordering on the Hilbert la... |
| thloc 21474 | Orthocomplement on the Hil... |
| pjfval 21481 | The value of the projectio... |
| pjdm 21482 | A subspace is in the domai... |
| pjpm 21483 | The projection map is a pa... |
| pjfval2 21484 | Value of the projection ma... |
| pjval 21485 | Value of the projection ma... |
| pjdm2 21486 | A subspace is in the domai... |
| pjff 21487 | A projection is a linear o... |
| pjf 21488 | A projection is a function... |
| pjf2 21489 | A projection is a function... |
| pjfo 21490 | A projection is a surjecti... |
| pjcss 21491 | A projection subspace is a... |
| ocvpj 21492 | The orthocomplement of a p... |
| ishil 21493 | The predicate "is a Hilber... |
| ishil2 21494 | The predicate "is a Hilber... |
| isobs 21495 | The predicate "is an ortho... |
| obsip 21496 | The inner product of two e... |
| obsipid 21497 | A basis element has length... |
| obsrcl 21498 | Reverse closure for an ort... |
| obsss 21499 | An orthonormal basis is a ... |
| obsne0 21500 | A basis element is nonzero... |
| obsocv 21501 | An orthonormal basis has t... |
| obs2ocv 21502 | The double orthocomplement... |
| obselocv 21503 | A basis element is in the ... |
| obs2ss 21504 | A basis has no proper subs... |
| obslbs 21505 | An orthogonal basis is a l... |
| reldmdsmm 21508 | The direct sum is a well-b... |
| dsmmval 21509 | Value of the module direct... |
| dsmmbase 21510 | Base set of the module dir... |
| dsmmval2 21511 | Self-referential definitio... |
| dsmmbas2 21512 | Base set of the direct sum... |
| dsmmfi 21513 | For finite products, the d... |
| dsmmelbas 21514 | Membership in the finitely... |
| dsmm0cl 21515 | The all-zero vector is con... |
| dsmmacl 21516 | The finite hull is closed ... |
| prdsinvgd2 21517 | Negation of a single coord... |
| dsmmsubg 21518 | The finite hull of a produ... |
| dsmmlss 21519 | The finite hull of a produ... |
| dsmmlmod 21520 | The direct sum of a family... |
| frlmval 21523 | Value of the "free module"... |
| frlmlmod 21524 | The free module is a modul... |
| frlmpws 21525 | The free module as a restr... |
| frlmlss 21526 | The base set of the free m... |
| frlmpwsfi 21527 | The finite free module is ... |
| frlmsca 21528 | The ring of scalars of a f... |
| frlm0 21529 | Zero in a free module (rin... |
| frlmbas 21530 | Base set of the free modul... |
| frlmelbas 21531 | Membership in the base set... |
| frlmrcl 21532 | If a free module is inhabi... |
| frlmbasfsupp 21533 | Elements of the free modul... |
| frlmbasmap 21534 | Elements of the free modul... |
| frlmbasf 21535 | Elements of the free modul... |
| frlmlvec 21536 | The free module over a div... |
| frlmfibas 21537 | The base set of the finite... |
| elfrlmbasn0 21538 | If the dimension of a free... |
| frlmplusgval 21539 | Addition in a free module.... |
| frlmsubgval 21540 | Subtraction in a free modu... |
| frlmvscafval 21541 | Scalar multiplication in a... |
| frlmvplusgvalc 21542 | Coordinates of a sum with ... |
| frlmvscaval 21543 | Coordinates of a scalar mu... |
| frlmplusgvalb 21544 | Addition in a free module ... |
| frlmvscavalb 21545 | Scalar multiplication in a... |
| frlmvplusgscavalb 21546 | Addition combined with sca... |
| frlmgsum 21547 | Finite commutative sums in... |
| frlmsplit2 21548 | Restriction is homomorphic... |
| frlmsslss 21549 | A subset of a free module ... |
| frlmsslss2 21550 | A subset of a free module ... |
| frlmbas3 21551 | An element of the base set... |
| mpofrlmd 21552 | Elements of the free modul... |
| frlmip 21553 | The inner product of a fre... |
| frlmipval 21554 | The inner product of a fre... |
| frlmphllem 21555 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21556 | Conditions for a free modu... |
| uvcfval 21559 | Value of the unit-vector g... |
| uvcval 21560 | Value of a single unit vec... |
| uvcvval 21561 | Value of a unit vector coo... |
| uvcvvcl 21562 | A coordinate of a unit vec... |
| uvcvvcl2 21563 | A unit vector coordinate i... |
| uvcvv1 21564 | The unit vector is one at ... |
| uvcvv0 21565 | The unit vector is zero at... |
| uvcff 21566 | Domain and codomain of the... |
| uvcf1 21567 | In a nonzero ring, each un... |
| uvcresum 21568 | Any element of a free modu... |
| frlmssuvc1 21569 | A scalar multiple of a uni... |
| frlmssuvc2 21570 | A nonzero scalar multiple ... |
| frlmsslsp 21571 | A subset of a free module ... |
| frlmlbs 21572 | The unit vectors comprise ... |
| frlmup1 21573 | Any assignment of unit vec... |
| frlmup2 21574 | The evaluation map has the... |
| frlmup3 21575 | The range of such an evalu... |
| frlmup4 21576 | Universal property of the ... |
| ellspd 21577 | The elements of the span o... |
| elfilspd 21578 | Simplified version of ~ el... |
| rellindf 21583 | The independent-family pre... |
| islinds 21584 | Property of an independent... |
| linds1 21585 | An independent set of vect... |
| linds2 21586 | An independent set of vect... |
| islindf 21587 | Property of an independent... |
| islinds2 21588 | Expanded property of an in... |
| islindf2 21589 | Property of an independent... |
| lindff 21590 | Functional property of a l... |
| lindfind 21591 | A linearly independent fam... |
| lindsind 21592 | A linearly independent set... |
| lindfind2 21593 | In a linearly independent ... |
| lindsind2 21594 | In a linearly independent ... |
| lindff1 21595 | A linearly independent fam... |
| lindfrn 21596 | The range of an independen... |
| f1lindf 21597 | Rearranging and deleting e... |
| lindfres 21598 | Any restriction of an inde... |
| lindsss 21599 | Any subset of an independe... |
| f1linds 21600 | A family constructed from ... |
| islindf3 21601 | In a nonzero ring, indepen... |
| lindfmm 21602 | Linear independence of a f... |
| lindsmm 21603 | Linear independence of a s... |
| lindsmm2 21604 | The monomorphic image of a... |
| lsslindf 21605 | Linear independence is unc... |
| lsslinds 21606 | Linear independence is unc... |
| islbs4 21607 | A basis is an independent ... |
| lbslinds 21608 | A basis is independent. (... |
| islinds3 21609 | A subset is linearly indep... |
| islinds4 21610 | A set is independent in a ... |
| lmimlbs 21611 | The isomorphic image of a ... |
| lmiclbs 21612 | Having a basis is an isomo... |
| islindf4 21613 | A family is independent if... |
| islindf5 21614 | A family is independent if... |
| indlcim 21615 | An independent, spanning f... |
| lbslcic 21616 | A module with a basis is i... |
| lmisfree 21617 | A module has a basis iff i... |
| lvecisfrlm 21618 | Every vector space is isom... |
| lmimco 21619 | The composition of two iso... |
| lmictra 21620 | Module isomorphism is tran... |
| uvcf1o 21621 | In a nonzero ring, the map... |
| uvcendim 21622 | In a nonzero ring, the num... |
| frlmisfrlm 21623 | A free module is isomorphi... |
| frlmiscvec 21624 | Every free module is isomo... |
| isassa 21631 | The properties of an assoc... |
| assalem 21632 | The properties of an assoc... |
| assaass 21633 | Left-associative property ... |
| assaassr 21634 | Right-associative property... |
| assalmod 21635 | An associative algebra is ... |
| assaring 21636 | An associative algebra is ... |
| assasca 21637 | The scalars of an associat... |
| assa2ass 21638 | Left- and right-associativ... |
| isassad 21639 | Sufficient condition for b... |
| issubassa3 21640 | A subring that is also a s... |
| issubassa 21641 | The subalgebras of an asso... |
| sraassab 21642 | A subring algebra is an as... |
| sraassa 21643 | The subring algebra over a... |
| sraassaOLD 21644 | Obsolete version of ~ sraa... |
| rlmassa 21645 | The ring module over a com... |
| assapropd 21646 | If two structures have the... |
| aspval 21647 | Value of the algebraic clo... |
| asplss 21648 | The algebraic span of a se... |
| aspid 21649 | The algebraic span of a su... |
| aspsubrg 21650 | The algebraic span of a se... |
| aspss 21651 | Span preserves subset orde... |
| aspssid 21652 | A set of vectors is a subs... |
| asclfval 21653 | Function value of the alge... |
| asclval 21654 | Value of a mapped algebra ... |
| asclfn 21655 | Unconditional functionalit... |
| asclf 21656 | The algebra scalars functi... |
| asclghm 21657 | The algebra scalars functi... |
| ascl0 21658 | The scalar 0 embedded into... |
| ascl1 21659 | The scalar 1 embedded into... |
| asclmul1 21660 | Left multiplication by a l... |
| asclmul2 21661 | Right multiplication by a ... |
| ascldimul 21662 | The algebra scalars functi... |
| asclinvg 21663 | The group inverse (negatio... |
| asclrhm 21664 | The scalar injection is a ... |
| rnascl 21665 | The set of injected scalar... |
| issubassa2 21666 | A subring of a unital alge... |
| rnasclsubrg 21667 | The scalar multiples of th... |
| rnasclmulcl 21668 | (Vector) multiplication is... |
| rnasclassa 21669 | The scalar multiples of th... |
| ressascl 21670 | The injection of scalars i... |
| asclpropd 21671 | If two structures have the... |
| aspval2 21672 | The algebraic closure is t... |
| assamulgscmlem1 21673 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21674 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21675 | Exponentiation of a scalar... |
| zlmassa 21676 | The ` ZZ ` -module operati... |
| reldmpsr 21687 | The multivariate power ser... |
| psrval 21688 | Value of the multivariate ... |
| psrvalstr 21689 | The multivariate power ser... |
| psrbag 21690 | Elementhood in the set of ... |
| psrbagf 21691 | A finite bag is a function... |
| psrbagfOLD 21692 | Obsolete version of ~ psrb... |
| psrbagfsupp 21693 | Finite bags have finite su... |
| psrbagfsuppOLD 21694 | Obsolete version of ~ psrb... |
| snifpsrbag 21695 | A bag containing one eleme... |
| fczpsrbag 21696 | The constant function equa... |
| psrbaglesupp 21697 | The support of a dominated... |
| psrbaglesuppOLD 21698 | Obsolete version of ~ psrb... |
| psrbaglecl 21699 | The set of finite bags is ... |
| psrbagleclOLD 21700 | Obsolete version of ~ psrb... |
| psrbagaddcl 21701 | The sum of two finite bags... |
| psrbagaddclOLD 21702 | Obsolete version of ~ psrb... |
| psrbagcon 21703 | The analogue of the statem... |
| psrbagconOLD 21704 | Obsolete version of ~ psrb... |
| psrbaglefi 21705 | There are finitely many ba... |
| psrbaglefiOLD 21706 | Obsolete version of ~ psrb... |
| psrbagconcl 21707 | The complement of a bag is... |
| psrbagconclOLD 21708 | Obsolete version of ~ psrb... |
| psrbagconf1o 21709 | Bag complementation is a b... |
| psrbagconf1oOLD 21710 | Obsolete version of ~ psrb... |
| gsumbagdiaglemOLD 21711 | Obsolete version of ~ gsum... |
| gsumbagdiagOLD 21712 | Obsolete version of ~ gsum... |
| psrass1lemOLD 21713 | Obsolete version of ~ psra... |
| gsumbagdiaglem 21714 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21715 | Two-dimensional commutatio... |
| psrass1lem 21716 | A group sum commutation us... |
| psrbas 21717 | The base set of the multiv... |
| psrelbas 21718 | An element of the set of p... |
| psrelbasfun 21719 | An element of the set of p... |
| psrplusg 21720 | The addition operation of ... |
| psradd 21721 | The addition operation of ... |
| psraddcl 21722 | Closure of the power serie... |
| psrmulr 21723 | The multiplication operati... |
| psrmulfval 21724 | The multiplication operati... |
| psrmulval 21725 | The multiplication operati... |
| psrmulcllem 21726 | Closure of the power serie... |
| psrmulcl 21727 | Closure of the power serie... |
| psrsca 21728 | The scalar field of the mu... |
| psrvscafval 21729 | The scalar multiplication ... |
| psrvsca 21730 | The scalar multiplication ... |
| psrvscaval 21731 | The scalar multiplication ... |
| psrvscacl 21732 | Closure of the power serie... |
| psr0cl 21733 | The zero element of the ri... |
| psr0lid 21734 | The zero element of the ri... |
| psrnegcl 21735 | The negative function in t... |
| psrlinv 21736 | The negative function in t... |
| psrgrp 21737 | The ring of power series i... |
| psrgrpOLD 21738 | Obsolete proof of ~ psrgrp... |
| psr0 21739 | The zero element of the ri... |
| psrneg 21740 | The negative function of t... |
| psrlmod 21741 | The ring of power series i... |
| psr1cl 21742 | The identity element of th... |
| psrlidm 21743 | The identity element of th... |
| psrridm 21744 | The identity element of th... |
| psrass1 21745 | Associative identity for t... |
| psrdi 21746 | Distributive law for the r... |
| psrdir 21747 | Distributive law for the r... |
| psrass23l 21748 | Associative identity for t... |
| psrcom 21749 | Commutative law for the ri... |
| psrass23 21750 | Associative identities for... |
| psrring 21751 | The ring of power series i... |
| psr1 21752 | The identity element of th... |
| psrcrng 21753 | The ring of power series i... |
| psrassa 21754 | The ring of power series i... |
| resspsrbas 21755 | A restricted power series ... |
| resspsradd 21756 | A restricted power series ... |
| resspsrmul 21757 | A restricted power series ... |
| resspsrvsca 21758 | A restricted power series ... |
| subrgpsr 21759 | A subring of the base ring... |
| mvrfval 21760 | Value of the generating el... |
| mvrval 21761 | Value of the generating el... |
| mvrval2 21762 | Value of the generating el... |
| mvrid 21763 | The ` X i ` -th coefficien... |
| mvrf 21764 | The power series variable ... |
| mvrf1 21765 | The power series variable ... |
| mvrcl2 21766 | A power series variable is... |
| reldmmpl 21767 | The multivariate polynomia... |
| mplval 21768 | Value of the set of multiv... |
| mplbas 21769 | Base set of the set of mul... |
| mplelbas 21770 | Property of being a polyno... |
| mvrcl 21771 | A power series variable is... |
| mvrf2 21772 | The power series/polynomia... |
| mplrcl 21773 | Reverse closure for the po... |
| mplelsfi 21774 | A polynomial treated as a ... |
| mplval2 21775 | Self-referential expressio... |
| mplbasss 21776 | The set of polynomials is ... |
| mplelf 21777 | A polynomial is defined as... |
| mplsubglem 21778 | If ` A ` is an ideal of se... |
| mpllsslem 21779 | If ` A ` is an ideal of su... |
| mplsubglem2 21780 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21781 | The set of polynomials is ... |
| mpllss 21782 | The set of polynomials is ... |
| mplsubrglem 21783 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21784 | The set of polynomials is ... |
| mpl0 21785 | The zero polynomial. (Con... |
| mplplusg 21786 | Value of addition in a pol... |
| mplmulr 21787 | Value of multiplication in... |
| mpladd 21788 | The addition operation on ... |
| mplneg 21789 | The negative function on m... |
| mplmul 21790 | The multiplication operati... |
| mpl1 21791 | The identity element of th... |
| mplsca 21792 | The scalar field of a mult... |
| mplvsca2 21793 | The scalar multiplication ... |
| mplvsca 21794 | The scalar multiplication ... |
| mplvscaval 21795 | The scalar multiplication ... |
| mplgrp 21796 | The polynomial ring is a g... |
| mpllmod 21797 | The polynomial ring is a l... |
| mplring 21798 | The polynomial ring is a r... |
| mpllvec 21799 | The polynomial ring is a v... |
| mplcrng 21800 | The polynomial ring is a c... |
| mplassa 21801 | The polynomial ring is an ... |
| ressmplbas2 21802 | The base set of a restrict... |
| ressmplbas 21803 | A restricted polynomial al... |
| ressmpladd 21804 | A restricted polynomial al... |
| ressmplmul 21805 | A restricted polynomial al... |
| ressmplvsca 21806 | A restricted power series ... |
| subrgmpl 21807 | A subring of the base ring... |
| subrgmvr 21808 | The variables in a subring... |
| subrgmvrf 21809 | The variables in a polynom... |
| mplmon 21810 | A monomial is a polynomial... |
| mplmonmul 21811 | The product of two monomia... |
| mplcoe1 21812 | Decompose a polynomial int... |
| mplcoe3 21813 | Decompose a monomial in on... |
| mplcoe5lem 21814 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21815 | Decompose a monomial into ... |
| mplcoe2 21816 | Decompose a monomial into ... |
| mplbas2 21817 | An alternative expression ... |
| ltbval 21818 | Value of the well-order on... |
| ltbwe 21819 | The finite bag order is a ... |
| reldmopsr 21820 | Lemma for ordered power se... |
| opsrval 21821 | The value of the "ordered ... |
| opsrle 21822 | An alternative expression ... |
| opsrval2 21823 | Self-referential expressio... |
| opsrbaslem 21824 | Get a component of the ord... |
| opsrbaslemOLD 21825 | Obsolete version of ~ opsr... |
| opsrbas 21826 | The base set of the ordere... |
| opsrbasOLD 21827 | Obsolete version of ~ opsr... |
| opsrplusg 21828 | The addition operation of ... |
| opsrplusgOLD 21829 | Obsolete version of ~ opsr... |
| opsrmulr 21830 | The multiplication operati... |
| opsrmulrOLD 21831 | Obsolete version of ~ opsr... |
| opsrvsca 21832 | The scalar product operati... |
| opsrvscaOLD 21833 | Obsolete version of ~ opsr... |
| opsrsca 21834 | The scalar ring of the ord... |
| opsrscaOLD 21835 | Obsolete version of ~ opsr... |
| opsrtoslem1 21836 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 21837 | Lemma for ~ opsrtos . (Co... |
| opsrtos 21838 | The ordered power series s... |
| opsrso 21839 | The ordered power series s... |
| opsrcrng 21840 | The ring of ordered power ... |
| opsrassa 21841 | The ring of ordered power ... |
| mplmon2 21842 | Express a scaled monomial.... |
| psrbag0 21843 | The empty bag is a bag. (... |
| psrbagsn 21844 | A singleton bag is a bag. ... |
| mplascl 21845 | Value of the scalar inject... |
| mplasclf 21846 | The scalar injection is a ... |
| subrgascl 21847 | The scalar injection funct... |
| subrgasclcl 21848 | The scalars in a polynomia... |
| mplmon2cl 21849 | A scaled monomial is a pol... |
| mplmon2mul 21850 | Product of scaled monomial... |
| mplind 21851 | Prove a property of polyno... |
| mplcoe4 21852 | Decompose a polynomial int... |
| evlslem4 21857 | The support of a tensor pr... |
| psrbagev1 21858 | A bag of multipliers provi... |
| psrbagev1OLD 21859 | Obsolete version of ~ psrb... |
| psrbagev2 21860 | Closure of a sum using a b... |
| psrbagev2OLD 21861 | Obsolete version of ~ psrb... |
| evlslem2 21862 | A linear function on the p... |
| evlslem3 21863 | Lemma for ~ evlseu . Poly... |
| evlslem6 21864 | Lemma for ~ evlseu . Fini... |
| evlslem1 21865 | Lemma for ~ evlseu , give ... |
| evlseu 21866 | For a given interpretation... |
| reldmevls 21867 | Well-behaved binary operat... |
| mpfrcl 21868 | Reverse closure for the se... |
| evlsval 21869 | Value of the polynomial ev... |
| evlsval2 21870 | Characterizing properties ... |
| evlsrhm 21871 | Polynomial evaluation is a... |
| evlssca 21872 | Polynomial evaluation maps... |
| evlsvar 21873 | Polynomial evaluation maps... |
| evlsgsumadd 21874 | Polynomial evaluation maps... |
| evlsgsummul 21875 | Polynomial evaluation maps... |
| evlspw 21876 | Polynomial evaluation for ... |
| evlsvarpw 21877 | Polynomial evaluation for ... |
| evlval 21878 | Value of the simple/same r... |
| evlrhm 21879 | The simple evaluation map ... |
| evlsscasrng 21880 | The evaluation of a scalar... |
| evlsca 21881 | Simple polynomial evaluati... |
| evlsvarsrng 21882 | The evaluation of the vari... |
| evlvar 21883 | Simple polynomial evaluati... |
| mpfconst 21884 | Constants are multivariate... |
| mpfproj 21885 | Projections are multivaria... |
| mpfsubrg 21886 | Polynomial functions are a... |
| mpff 21887 | Polynomial functions are f... |
| mpfaddcl 21888 | The sum of multivariate po... |
| mpfmulcl 21889 | The product of multivariat... |
| mpfind 21890 | Prove a property of polyno... |
| selvffval 21899 | Value of the "variable sel... |
| selvfval 21900 | Value of the "variable sel... |
| selvval 21901 | Value of the "variable sel... |
| mhpfval 21902 | Value of the "homogeneous ... |
| mhpval 21903 | Value of the "homogeneous ... |
| ismhp 21904 | Property of being a homoge... |
| ismhp2 21905 | Deduce a homogeneous polyn... |
| ismhp3 21906 | A polynomial is homogeneou... |
| mhpmpl 21907 | A homogeneous polynomial i... |
| mhpdeg 21908 | All nonzero terms of a hom... |
| mhp0cl 21909 | The zero polynomial is hom... |
| mhpsclcl 21910 | A scalar (or constant) pol... |
| mhpvarcl 21911 | A power series variable is... |
| mhpmulcl 21912 | A product of homogeneous p... |
| mhppwdeg 21913 | Degree of a homogeneous po... |
| mhpaddcl 21914 | Homogeneous polynomials ar... |
| mhpinvcl 21915 | Homogeneous polynomials ar... |
| mhpsubg 21916 | Homogeneous polynomials fo... |
| mhpvscacl 21917 | Homogeneous polynomials ar... |
| mhplss 21918 | Homogeneous polynomials fo... |
| psr1baslem 21929 | The set of finite bags on ... |
| psr1val 21930 | Value of the ring of univa... |
| psr1crng 21931 | The ring of univariate pow... |
| psr1assa 21932 | The ring of univariate pow... |
| psr1tos 21933 | The ordered power series s... |
| psr1bas2 21934 | The base set of the ring o... |
| psr1bas 21935 | The base set of the ring o... |
| vr1val 21936 | The value of the generator... |
| vr1cl2 21937 | The variable ` X ` is a me... |
| ply1val 21938 | The value of the set of un... |
| ply1bas 21939 | The value of the base set ... |
| ply1lss 21940 | Univariate polynomials for... |
| ply1subrg 21941 | Univariate polynomials for... |
| ply1crng 21942 | The ring of univariate pol... |
| ply1assa 21943 | The ring of univariate pol... |
| psr1bascl 21944 | A univariate power series ... |
| psr1basf 21945 | Univariate power series ba... |
| ply1basf 21946 | Univariate polynomial base... |
| ply1bascl 21947 | A univariate polynomial is... |
| ply1bascl2 21948 | A univariate polynomial is... |
| coe1fval 21949 | Value of the univariate po... |
| coe1fv 21950 | Value of an evaluated coef... |
| fvcoe1 21951 | Value of a multivariate co... |
| coe1fval3 21952 | Univariate power series co... |
| coe1f2 21953 | Functionality of univariat... |
| coe1fval2 21954 | Univariate polynomial coef... |
| coe1f 21955 | Functionality of univariat... |
| coe1fvalcl 21956 | A coefficient of a univari... |
| coe1sfi 21957 | Finite support of univaria... |
| coe1fsupp 21958 | The coefficient vector of ... |
| mptcoe1fsupp 21959 | A mapping involving coeffi... |
| coe1ae0 21960 | The coefficient vector of ... |
| vr1cl 21961 | The generator of a univari... |
| opsr0 21962 | Zero in the ordered power ... |
| opsr1 21963 | One in the ordered power s... |
| psr1plusg 21964 | Value of addition in a uni... |
| psr1vsca 21965 | Value of scalar multiplica... |
| psr1mulr 21966 | Value of multiplication in... |
| ply1plusg 21967 | Value of addition in a uni... |
| ply1vsca 21968 | Value of scalar multiplica... |
| ply1mulr 21969 | Value of multiplication in... |
| ply1ass23l 21970 | Associative identity with ... |
| ressply1bas2 21971 | The base set of a restrict... |
| ressply1bas 21972 | A restricted polynomial al... |
| ressply1add 21973 | A restricted polynomial al... |
| ressply1mul 21974 | A restricted polynomial al... |
| ressply1vsca 21975 | A restricted power series ... |
| subrgply1 21976 | A subring of the base ring... |
| gsumply1subr 21977 | Evaluate a group sum in a ... |
| psrbaspropd 21978 | Property deduction for pow... |
| psrplusgpropd 21979 | Property deduction for pow... |
| mplbaspropd 21980 | Property deduction for pol... |
| psropprmul 21981 | Reversing multiplication i... |
| ply1opprmul 21982 | Reversing multiplication i... |
| 00ply1bas 21983 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 21984 | Protection compatibility o... |
| ply1plusgfvi 21985 | Protection compatibility o... |
| ply1baspropd 21986 | Property deduction for uni... |
| ply1plusgpropd 21987 | Property deduction for uni... |
| opsrring 21988 | Ordered power series form ... |
| opsrlmod 21989 | Ordered power series form ... |
| psr1ring 21990 | Univariate power series fo... |
| ply1ring 21991 | Univariate polynomials for... |
| psr1lmod 21992 | Univariate power series fo... |
| psr1sca 21993 | Scalars of a univariate po... |
| psr1sca2 21994 | Scalars of a univariate po... |
| ply1lmod 21995 | Univariate polynomials for... |
| ply1sca 21996 | Scalars of a univariate po... |
| ply1sca2 21997 | Scalars of a univariate po... |
| ply1mpl0 21998 | The univariate polynomial ... |
| ply10s0 21999 | Zero times a univariate po... |
| ply1mpl1 22000 | The univariate polynomial ... |
| ply1ascl 22001 | The univariate polynomial ... |
| subrg1ascl 22002 | The scalar injection funct... |
| subrg1asclcl 22003 | The scalars in a polynomia... |
| subrgvr1 22004 | The variables in a subring... |
| subrgvr1cl 22005 | The variables in a polynom... |
| coe1z 22006 | The coefficient vector of ... |
| coe1add 22007 | The coefficient vector of ... |
| coe1addfv 22008 | A particular coefficient o... |
| coe1subfv 22009 | A particular coefficient o... |
| coe1mul2lem1 22010 | An equivalence for ~ coe1m... |
| coe1mul2lem2 22011 | An equivalence for ~ coe1m... |
| coe1mul2 22012 | The coefficient vector of ... |
| coe1mul 22013 | The coefficient vector of ... |
| ply1moncl 22014 | Closure of the expression ... |
| ply1tmcl 22015 | Closure of the expression ... |
| coe1tm 22016 | Coefficient vector of a po... |
| coe1tmfv1 22017 | Nonzero coefficient of a p... |
| coe1tmfv2 22018 | Zero coefficient of a poly... |
| coe1tmmul2 22019 | Coefficient vector of a po... |
| coe1tmmul 22020 | Coefficient vector of a po... |
| coe1tmmul2fv 22021 | Function value of a right-... |
| coe1pwmul 22022 | Coefficient vector of a po... |
| coe1pwmulfv 22023 | Function value of a right-... |
| ply1scltm 22024 | A scalar is a term with ze... |
| coe1sclmul 22025 | Coefficient vector of a po... |
| coe1sclmulfv 22026 | A single coefficient of a ... |
| coe1sclmul2 22027 | Coefficient vector of a po... |
| ply1sclf 22028 | A scalar polynomial is a p... |
| ply1sclcl 22029 | The value of the algebra s... |
| coe1scl 22030 | Coefficient vector of a sc... |
| ply1sclid 22031 | Recover the base scalar fr... |
| ply1sclf1 22032 | The polynomial scalar func... |
| ply1scl0 22033 | The zero scalar is zero. ... |
| ply1scl0OLD 22034 | Obsolete version of ~ ply1... |
| ply1scln0 22035 | Nonzero scalars create non... |
| ply1scl1 22036 | The one scalar is the unit... |
| ply1scl1OLD 22037 | Obsolete version of ~ ply1... |
| ply1idvr1 22038 | The identity of a polynomi... |
| cply1mul 22039 | The product of two constan... |
| ply1coefsupp 22040 | The decomposition of a uni... |
| ply1coe 22041 | Decompose a univariate pol... |
| eqcoe1ply1eq 22042 | Two polynomials over the s... |
| ply1coe1eq 22043 | Two polynomials over the s... |
| cply1coe0 22044 | All but the first coeffici... |
| cply1coe0bi 22045 | A polynomial is constant (... |
| coe1fzgsumdlem 22046 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 22047 | Value of an evaluated coef... |
| gsumsmonply1 22048 | A finite group sum of scal... |
| gsummoncoe1 22049 | A coefficient of the polyn... |
| gsumply1eq 22050 | Two univariate polynomials... |
| lply1binom 22051 | The binomial theorem for l... |
| lply1binomsc 22052 | The binomial theorem for l... |
| reldmevls1 22057 | Well-behaved binary operat... |
| ply1frcl 22058 | Reverse closure for the se... |
| evls1fval 22059 | Value of the univariate po... |
| evls1val 22060 | Value of the univariate po... |
| evls1rhmlem 22061 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 22062 | Polynomial evaluation is a... |
| evls1sca 22063 | Univariate polynomial eval... |
| evls1gsumadd 22064 | Univariate polynomial eval... |
| evls1gsummul 22065 | Univariate polynomial eval... |
| evls1pw 22066 | Univariate polynomial eval... |
| evls1varpw 22067 | Univariate polynomial eval... |
| evl1fval 22068 | Value of the simple/same r... |
| evl1val 22069 | Value of the simple/same r... |
| evl1fval1lem 22070 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 22071 | Value of the simple/same r... |
| evl1rhm 22072 | Polynomial evaluation is a... |
| fveval1fvcl 22073 | The function value of the ... |
| evl1sca 22074 | Polynomial evaluation maps... |
| evl1scad 22075 | Polynomial evaluation buil... |
| evl1var 22076 | Polynomial evaluation maps... |
| evl1vard 22077 | Polynomial evaluation buil... |
| evls1var 22078 | Univariate polynomial eval... |
| evls1scasrng 22079 | The evaluation of a scalar... |
| evls1varsrng 22080 | The evaluation of the vari... |
| evl1addd 22081 | Polynomial evaluation buil... |
| evl1subd 22082 | Polynomial evaluation buil... |
| evl1muld 22083 | Polynomial evaluation buil... |
| evl1vsd 22084 | Polynomial evaluation buil... |
| evl1expd 22085 | Polynomial evaluation buil... |
| pf1const 22086 | Constants are polynomial f... |
| pf1id 22087 | The identity is a polynomi... |
| pf1subrg 22088 | Polynomial functions are a... |
| pf1rcl 22089 | Reverse closure for the se... |
| pf1f 22090 | Polynomial functions are f... |
| mpfpf1 22091 | Convert a multivariate pol... |
| pf1mpf 22092 | Convert a univariate polyn... |
| pf1addcl 22093 | The sum of multivariate po... |
| pf1mulcl 22094 | The product of multivariat... |
| pf1ind 22095 | Prove a property of polyno... |
| evl1gsumdlem 22096 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 22097 | Polynomial evaluation buil... |
| evl1gsumadd 22098 | Univariate polynomial eval... |
| evl1gsumaddval 22099 | Value of a univariate poly... |
| evl1gsummul 22100 | Univariate polynomial eval... |
| evl1varpw 22101 | Univariate polynomial eval... |
| evl1varpwval 22102 | Value of a univariate poly... |
| evl1scvarpw 22103 | Univariate polynomial eval... |
| evl1scvarpwval 22104 | Value of a univariate poly... |
| evl1gsummon 22105 | Value of a univariate poly... |
| mamufval 22108 | Functional value of the ma... |
| mamuval 22109 | Multiplication of two matr... |
| mamufv 22110 | A cell in the multiplicati... |
| mamudm 22111 | The domain of the matrix m... |
| mamufacex 22112 | Every solution of the equa... |
| mamures 22113 | Rows in a matrix product a... |
| mndvcl 22114 | Tuple-wise additive closur... |
| mndvass 22115 | Tuple-wise associativity i... |
| mndvlid 22116 | Tuple-wise left identity i... |
| mndvrid 22117 | Tuple-wise right identity ... |
| grpvlinv 22118 | Tuple-wise left inverse in... |
| grpvrinv 22119 | Tuple-wise right inverse i... |
| mhmvlin 22120 | Tuple extension of monoid ... |
| ringvcl 22121 | Tuple-wise multiplication ... |
| mamucl 22122 | Operation closure of matri... |
| mamuass 22123 | Matrix multiplication is a... |
| mamudi 22124 | Matrix multiplication dist... |
| mamudir 22125 | Matrix multiplication dist... |
| mamuvs1 22126 | Matrix multiplication dist... |
| mamuvs2 22127 | Matrix multiplication dist... |
| matbas0pc 22130 | There is no matrix with a ... |
| matbas0 22131 | There is no matrix for a n... |
| matval 22132 | Value of the matrix algebr... |
| matrcl 22133 | Reverse closure for the ma... |
| matbas 22134 | The matrix ring has the sa... |
| matplusg 22135 | The matrix ring has the sa... |
| matsca 22136 | The matrix ring has the sa... |
| matscaOLD 22137 | Obsolete proof of ~ matsca... |
| matvsca 22138 | The matrix ring has the sa... |
| matvscaOLD 22139 | Obsolete proof of ~ matvsc... |
| mat0 22140 | The matrix ring has the sa... |
| matinvg 22141 | The matrix ring has the sa... |
| mat0op 22142 | Value of a zero matrix as ... |
| matsca2 22143 | The scalars of the matrix ... |
| matbas2 22144 | The base set of the matrix... |
| matbas2i 22145 | A matrix is a function. (... |
| matbas2d 22146 | The base set of the matrix... |
| eqmat 22147 | Two square matrices of the... |
| matecl 22148 | Each entry (according to W... |
| matecld 22149 | Each entry (according to W... |
| matplusg2 22150 | Addition in the matrix rin... |
| matvsca2 22151 | Scalar multiplication in t... |
| matlmod 22152 | The matrix ring is a linea... |
| matgrp 22153 | The matrix ring is a group... |
| matvscl 22154 | Closure of the scalar mult... |
| matsubg 22155 | The matrix ring has the sa... |
| matplusgcell 22156 | Addition in the matrix rin... |
| matsubgcell 22157 | Subtraction in the matrix ... |
| matinvgcell 22158 | Additive inversion in the ... |
| matvscacell 22159 | Scalar multiplication in t... |
| matgsum 22160 | Finite commutative sums in... |
| matmulr 22161 | Multiplication in the matr... |
| mamumat1cl 22162 | The identity matrix (as op... |
| mat1comp 22163 | The components of the iden... |
| mamulid 22164 | The identity matrix (as op... |
| mamurid 22165 | The identity matrix (as op... |
| matring 22166 | Existence of the matrix ri... |
| matassa 22167 | Existence of the matrix al... |
| matmulcell 22168 | Multiplication in the matr... |
| mpomatmul 22169 | Multiplication of two N x ... |
| mat1 22170 | Value of an identity matri... |
| mat1ov 22171 | Entries of an identity mat... |
| mat1bas 22172 | The identity matrix is a m... |
| matsc 22173 | The identity matrix multip... |
| ofco2 22174 | Distribution law for the f... |
| oftpos 22175 | The transposition of the v... |
| mattposcl 22176 | The transpose of a square ... |
| mattpostpos 22177 | The transpose of the trans... |
| mattposvs 22178 | The transposition of a mat... |
| mattpos1 22179 | The transposition of the i... |
| tposmap 22180 | The transposition of an I ... |
| mamutpos 22181 | Behavior of transposes in ... |
| mattposm 22182 | Multiplying two transposed... |
| matgsumcl 22183 | Closure of a group sum ove... |
| madetsumid 22184 | The identity summand in th... |
| matepmcl 22185 | Each entry of a matrix wit... |
| matepm2cl 22186 | Each entry of a matrix wit... |
| madetsmelbas 22187 | A summand of the determina... |
| madetsmelbas2 22188 | A summand of the determina... |
| mat0dimbas0 22189 | The empty set is the one a... |
| mat0dim0 22190 | The zero of the algebra of... |
| mat0dimid 22191 | The identity of the algebr... |
| mat0dimscm 22192 | The scalar multiplication ... |
| mat0dimcrng 22193 | The algebra of matrices wi... |
| mat1dimelbas 22194 | A matrix with dimension 1 ... |
| mat1dimbas 22195 | A matrix with dimension 1 ... |
| mat1dim0 22196 | The zero of the algebra of... |
| mat1dimid 22197 | The identity of the algebr... |
| mat1dimscm 22198 | The scalar multiplication ... |
| mat1dimmul 22199 | The ring multiplication in... |
| mat1dimcrng 22200 | The algebra of matrices wi... |
| mat1f1o 22201 | There is a 1-1 function fr... |
| mat1rhmval 22202 | The value of the ring homo... |
| mat1rhmelval 22203 | The value of the ring homo... |
| mat1rhmcl 22204 | The value of the ring homo... |
| mat1f 22205 | There is a function from a... |
| mat1ghm 22206 | There is a group homomorph... |
| mat1mhm 22207 | There is a monoid homomorp... |
| mat1rhm 22208 | There is a ring homomorphi... |
| mat1rngiso 22209 | There is a ring isomorphis... |
| mat1ric 22210 | A ring is isomorphic to th... |
| dmatval 22215 | The set of ` N ` x ` N ` d... |
| dmatel 22216 | A ` N ` x ` N ` diagonal m... |
| dmatmat 22217 | An ` N ` x ` N ` diagonal ... |
| dmatid 22218 | The identity matrix is a d... |
| dmatelnd 22219 | An extradiagonal entry of ... |
| dmatmul 22220 | The product of two diagona... |
| dmatsubcl 22221 | The difference of two diag... |
| dmatsgrp 22222 | The set of diagonal matric... |
| dmatmulcl 22223 | The product of two diagona... |
| dmatsrng 22224 | The set of diagonal matric... |
| dmatcrng 22225 | The subring of diagonal ma... |
| dmatscmcl 22226 | The multiplication of a di... |
| scmatval 22227 | The set of ` N ` x ` N ` s... |
| scmatel 22228 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 22229 | A scalar matrix can be exp... |
| scmatscmide 22230 | An entry of a scalar matri... |
| scmatscmiddistr 22231 | Distributive law for scala... |
| scmatmat 22232 | An ` N ` x ` N ` scalar ma... |
| scmate 22233 | An entry of an ` N ` x ` N... |
| scmatmats 22234 | The set of an ` N ` x ` N ... |
| scmateALT 22235 | Alternate proof of ~ scmat... |
| scmatscm 22236 | The multiplication of a ma... |
| scmatid 22237 | The identity matrix is a s... |
| scmatdmat 22238 | A scalar matrix is a diago... |
| scmataddcl 22239 | The sum of two scalar matr... |
| scmatsubcl 22240 | The difference of two scal... |
| scmatmulcl 22241 | The product of two scalar ... |
| scmatsgrp 22242 | The set of scalar matrices... |
| scmatsrng 22243 | The set of scalar matrices... |
| scmatcrng 22244 | The subring of scalar matr... |
| scmatsgrp1 22245 | The set of scalar matrices... |
| scmatsrng1 22246 | The set of scalar matrices... |
| smatvscl 22247 | Closure of the scalar mult... |
| scmatlss 22248 | The set of scalar matrices... |
| scmatstrbas 22249 | The set of scalar matrices... |
| scmatrhmval 22250 | The value of the ring homo... |
| scmatrhmcl 22251 | The value of the ring homo... |
| scmatf 22252 | There is a function from a... |
| scmatfo 22253 | There is a function from a... |
| scmatf1 22254 | There is a 1-1 function fr... |
| scmatf1o 22255 | There is a bijection betwe... |
| scmatghm 22256 | There is a group homomorph... |
| scmatmhm 22257 | There is a monoid homomorp... |
| scmatrhm 22258 | There is a ring homomorphi... |
| scmatrngiso 22259 | There is a ring isomorphis... |
| scmatric 22260 | A ring is isomorphic to ev... |
| mat0scmat 22261 | The empty matrix over a ri... |
| mat1scmat 22262 | A 1-dimensional matrix ove... |
| mvmulfval 22265 | Functional value of the ma... |
| mvmulval 22266 | Multiplication of a vector... |
| mvmulfv 22267 | A cell/element in the vect... |
| mavmulval 22268 | Multiplication of a vector... |
| mavmulfv 22269 | A cell/element in the vect... |
| mavmulcl 22270 | Multiplication of an NxN m... |
| 1mavmul 22271 | Multiplication of the iden... |
| mavmulass 22272 | Associativity of the multi... |
| mavmuldm 22273 | The domain of the matrix v... |
| mavmulsolcl 22274 | Every solution of the equa... |
| mavmul0 22275 | Multiplication of a 0-dime... |
| mavmul0g 22276 | The result of the 0-dimens... |
| mvmumamul1 22277 | The multiplication of an M... |
| mavmumamul1 22278 | The multiplication of an N... |
| marrepfval 22283 | First substitution for the... |
| marrepval0 22284 | Second substitution for th... |
| marrepval 22285 | Third substitution for the... |
| marrepeval 22286 | An entry of a matrix with ... |
| marrepcl 22287 | Closure of the row replace... |
| marepvfval 22288 | First substitution for the... |
| marepvval0 22289 | Second substitution for th... |
| marepvval 22290 | Third substitution for the... |
| marepveval 22291 | An entry of a matrix with ... |
| marepvcl 22292 | Closure of the column repl... |
| ma1repvcl 22293 | Closure of the column repl... |
| ma1repveval 22294 | An entry of an identity ma... |
| mulmarep1el 22295 | Element by element multipl... |
| mulmarep1gsum1 22296 | The sum of element by elem... |
| mulmarep1gsum2 22297 | The sum of element by elem... |
| 1marepvmarrepid 22298 | Replacing the ith row by 0... |
| submabas 22301 | Any subset of the index se... |
| submafval 22302 | First substitution for a s... |
| submaval0 22303 | Second substitution for a ... |
| submaval 22304 | Third substitution for a s... |
| submaeval 22305 | An entry of a submatrix of... |
| 1marepvsma1 22306 | The submatrix of the ident... |
| mdetfval 22309 | First substitution for the... |
| mdetleib 22310 | Full substitution of our d... |
| mdetleib2 22311 | Leibniz' formula can also ... |
| nfimdetndef 22312 | The determinant is not def... |
| mdetfval1 22313 | First substitution of an a... |
| mdetleib1 22314 | Full substitution of an al... |
| mdet0pr 22315 | The determinant function f... |
| mdet0f1o 22316 | The determinant function f... |
| mdet0fv0 22317 | The determinant of the emp... |
| mdetf 22318 | Functionality of the deter... |
| mdetcl 22319 | The determinant evaluates ... |
| m1detdiag 22320 | The determinant of a 1-dim... |
| mdetdiaglem 22321 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 22322 | The determinant of a diago... |
| mdetdiagid 22323 | The determinant of a diago... |
| mdet1 22324 | The determinant of the ide... |
| mdetrlin 22325 | The determinant function i... |
| mdetrsca 22326 | The determinant function i... |
| mdetrsca2 22327 | The determinant function i... |
| mdetr0 22328 | The determinant of a matri... |
| mdet0 22329 | The determinant of the zer... |
| mdetrlin2 22330 | The determinant function i... |
| mdetralt 22331 | The determinant function i... |
| mdetralt2 22332 | The determinant function i... |
| mdetero 22333 | The determinant function i... |
| mdettpos 22334 | Determinant is invariant u... |
| mdetunilem1 22335 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 22336 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 22337 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 22338 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 22339 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 22340 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 22341 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 22342 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 22343 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 22344 | Lemma for ~ mdetuni . (Co... |
| mdetuni 22345 | According to the definitio... |
| mdetmul 22346 | Multiplicativity of the de... |
| m2detleiblem1 22347 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 22348 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 22349 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 22350 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 22351 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 22352 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 22353 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 22354 | Leibniz' Formula for 2x2-m... |
| mndifsplit 22359 | Lemma for ~ maducoeval2 . ... |
| madufval 22360 | First substitution for the... |
| maduval 22361 | Second substitution for th... |
| maducoeval 22362 | An entry of the adjunct (c... |
| maducoeval2 22363 | An entry of the adjunct (c... |
| maduf 22364 | Creating the adjunct of ma... |
| madutpos 22365 | The adjuct of a transposed... |
| madugsum 22366 | The determinant of a matri... |
| madurid 22367 | Multiplying a matrix with ... |
| madulid 22368 | Multiplying the adjunct of... |
| minmar1fval 22369 | First substitution for the... |
| minmar1val0 22370 | Second substitution for th... |
| minmar1val 22371 | Third substitution for the... |
| minmar1eval 22372 | An entry of a matrix for a... |
| minmar1marrep 22373 | The minor matrix is a spec... |
| minmar1cl 22374 | Closure of the row replace... |
| maducoevalmin1 22375 | The coefficients of an adj... |
| symgmatr01lem 22376 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22377 | Applying a permutation tha... |
| gsummatr01lem1 22378 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22379 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22380 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22381 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22382 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22383 | Replacing a row of a squar... |
| smadiadetlem0 22384 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22385 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22386 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22387 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22388 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22389 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22390 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22391 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22392 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22393 | The determinant of a subma... |
| smadiadetglem1 22394 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22395 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22396 | The determinant of a squar... |
| smadiadetg0 22397 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22398 | The determinant of a squar... |
| invrvald 22399 | If a matrix multiplied wit... |
| matinv 22400 | The inverse of a matrix is... |
| matunit 22401 | A matrix is a unit in the ... |
| slesolvec 22402 | Every solution of a system... |
| slesolinv 22403 | The solution of a system o... |
| slesolinvbi 22404 | The solution of a system o... |
| slesolex 22405 | Every system of linear equ... |
| cramerimplem1 22406 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22407 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22408 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22409 | One direction of Cramer's ... |
| cramerlem1 22410 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22411 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22412 | Lemma 3 for ~ cramer . (C... |
| cramer0 22413 | Special case of Cramer's r... |
| cramer 22414 | Cramer's rule. According ... |
| pmatring 22415 | The set of polynomial matr... |
| pmatlmod 22416 | The set of polynomial matr... |
| pmatassa 22417 | The set of polynomial matr... |
| pmat0op 22418 | The zero polynomial matrix... |
| pmat1op 22419 | The identity polynomial ma... |
| pmat1ovd 22420 | Entries of the identity po... |
| pmat0opsc 22421 | The zero polynomial matrix... |
| pmat1opsc 22422 | The identity polynomial ma... |
| pmat1ovscd 22423 | Entries of the identity po... |
| pmatcoe1fsupp 22424 | For a polynomial matrix th... |
| 1pmatscmul 22425 | The scalar product of the ... |
| cpmat 22432 | Value of the constructor o... |
| cpmatpmat 22433 | A constant polynomial matr... |
| cpmatel 22434 | Property of a constant pol... |
| cpmatelimp 22435 | Implication of a set being... |
| cpmatel2 22436 | Another property of a cons... |
| cpmatelimp2 22437 | Another implication of a s... |
| 1elcpmat 22438 | The identity of the ring o... |
| cpmatacl 22439 | The set of all constant po... |
| cpmatinvcl 22440 | The set of all constant po... |
| cpmatmcllem 22441 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22442 | The set of all constant po... |
| cpmatsubgpmat 22443 | The set of all constant po... |
| cpmatsrgpmat 22444 | The set of all constant po... |
| 0elcpmat 22445 | The zero of the ring of al... |
| mat2pmatfval 22446 | Value of the matrix transf... |
| mat2pmatval 22447 | The result of a matrix tra... |
| mat2pmatvalel 22448 | A (matrix) element of the ... |
| mat2pmatbas 22449 | The result of a matrix tra... |
| mat2pmatbas0 22450 | The result of a matrix tra... |
| mat2pmatf 22451 | The matrix transformation ... |
| mat2pmatf1 22452 | The matrix transformation ... |
| mat2pmatghm 22453 | The transformation of matr... |
| mat2pmatmul 22454 | The transformation of matr... |
| mat2pmat1 22455 | The transformation of the ... |
| mat2pmatmhm 22456 | The transformation of matr... |
| mat2pmatrhm 22457 | The transformation of matr... |
| mat2pmatlin 22458 | The transformation of matr... |
| 0mat2pmat 22459 | The transformed zero matri... |
| idmatidpmat 22460 | The transformed identity m... |
| d0mat2pmat 22461 | The transformed empty set ... |
| d1mat2pmat 22462 | The transformation of a ma... |
| mat2pmatscmxcl 22463 | A transformed matrix multi... |
| m2cpm 22464 | The result of a matrix tra... |
| m2cpmf 22465 | The matrix transformation ... |
| m2cpmf1 22466 | The matrix transformation ... |
| m2cpmghm 22467 | The transformation of matr... |
| m2cpmmhm 22468 | The transformation of matr... |
| m2cpmrhm 22469 | The transformation of matr... |
| m2pmfzmap 22470 | The transformed values of ... |
| m2pmfzgsumcl 22471 | Closure of the sum of scal... |
| cpm2mfval 22472 | Value of the inverse matri... |
| cpm2mval 22473 | The result of an inverse m... |
| cpm2mvalel 22474 | A (matrix) element of the ... |
| cpm2mf 22475 | The inverse matrix transfo... |
| m2cpminvid 22476 | The inverse transformation... |
| m2cpminvid2lem 22477 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22478 | The transformation applied... |
| m2cpmfo 22479 | The matrix transformation ... |
| m2cpmf1o 22480 | The matrix transformation ... |
| m2cpmrngiso 22481 | The transformation of matr... |
| matcpmric 22482 | The ring of matrices over ... |
| m2cpminv 22483 | The inverse matrix transfo... |
| m2cpminv0 22484 | The inverse matrix transfo... |
| decpmatval0 22487 | The matrix consisting of t... |
| decpmatval 22488 | The matrix consisting of t... |
| decpmate 22489 | An entry of the matrix con... |
| decpmatcl 22490 | Closure of the decompositi... |
| decpmataa0 22491 | The matrix consisting of t... |
| decpmatfsupp 22492 | The mapping to the matrice... |
| decpmatid 22493 | The matrix consisting of t... |
| decpmatmullem 22494 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22495 | The matrix consisting of t... |
| decpmatmulsumfsupp 22496 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22497 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22498 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22499 | Write a polynomial matrix ... |
| pmatcollpw2lem 22500 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22501 | Write a polynomial matrix ... |
| monmatcollpw 22502 | The matrix consisting of t... |
| pmatcollpwlem 22503 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22504 | Write a polynomial matrix ... |
| pmatcollpwfi 22505 | Write a polynomial matrix ... |
| pmatcollpw3lem 22506 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22507 | Write a polynomial matrix ... |
| pmatcollpw3fi 22508 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22509 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22510 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22511 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22512 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22513 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22514 | Write a scalar matrix over... |
| pm2mpf1lem 22517 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22518 | Value of the transformatio... |
| pm2mpfval 22519 | A polynomial matrix transf... |
| pm2mpcl 22520 | The transformation of poly... |
| pm2mpf 22521 | The transformation of poly... |
| pm2mpf1 22522 | The transformation of poly... |
| pm2mpcoe1 22523 | A coefficient of the polyn... |
| idpm2idmp 22524 | The transformation of the ... |
| mptcoe1matfsupp 22525 | The mapping extracting the... |
| mply1topmatcllem 22526 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22527 | A polynomial over matrices... |
| mply1topmatcl 22528 | A polynomial over matrices... |
| mp2pm2mplem1 22529 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22530 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22531 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22532 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22533 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22534 | A polynomial over matrices... |
| pm2mpghmlem2 22535 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22536 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22537 | The transformation of poly... |
| pm2mpf1o 22538 | The transformation of poly... |
| pm2mpghm 22539 | The transformation of poly... |
| pm2mpgrpiso 22540 | The transformation of poly... |
| pm2mpmhmlem1 22541 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22542 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22543 | The transformation of poly... |
| pm2mprhm 22544 | The transformation of poly... |
| pm2mprngiso 22545 | The transformation of poly... |
| pmmpric 22546 | The ring of polynomial mat... |
| monmat2matmon 22547 | The transformation of a po... |
| pm2mp 22548 | The transformation of a su... |
| chmatcl 22551 | Closure of the characteris... |
| chmatval 22552 | The entries of the charact... |
| chpmatfval 22553 | Value of the characteristi... |
| chpmatval 22554 | The characteristic polynom... |
| chpmatply1 22555 | The characteristic polynom... |
| chpmatval2 22556 | The characteristic polynom... |
| chpmat0d 22557 | The characteristic polynom... |
| chpmat1dlem 22558 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22559 | The characteristic polynom... |
| chpdmatlem0 22560 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22561 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22562 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22563 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22564 | The characteristic polynom... |
| chpscmat 22565 | The characteristic polynom... |
| chpscmat0 22566 | The characteristic polynom... |
| chpscmatgsumbin 22567 | The characteristic polynom... |
| chpscmatgsummon 22568 | The characteristic polynom... |
| chp0mat 22569 | The characteristic polynom... |
| chpidmat 22570 | The characteristic polynom... |
| chmaidscmat 22571 | The characteristic polynom... |
| fvmptnn04if 22572 | The function values of a m... |
| fvmptnn04ifa 22573 | The function value of a ma... |
| fvmptnn04ifb 22574 | The function value of a ma... |
| fvmptnn04ifc 22575 | The function value of a ma... |
| fvmptnn04ifd 22576 | The function value of a ma... |
| chfacfisf 22577 | The "characteristic factor... |
| chfacfisfcpmat 22578 | The "characteristic factor... |
| chfacffsupp 22579 | The "characteristic factor... |
| chfacfscmulcl 22580 | Closure of a scaled value ... |
| chfacfscmul0 22581 | A scaled value of the "cha... |
| chfacfscmulfsupp 22582 | A mapping of scaled values... |
| chfacfscmulgsum 22583 | Breaking up a sum of value... |
| chfacfpmmulcl 22584 | Closure of the value of th... |
| chfacfpmmul0 22585 | The value of the "characte... |
| chfacfpmmulfsupp 22586 | A mapping of values of the... |
| chfacfpmmulgsum 22587 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22588 | Breaking up a sum of value... |
| cayhamlem1 22589 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22590 | The right-hand fundamental... |
| cpmidgsum 22591 | Representation of the iden... |
| cpmidgsumm2pm 22592 | Representation of the iden... |
| cpmidpmatlem1 22593 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22594 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22595 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22596 | Representation of the iden... |
| cpmadugsumlemB 22597 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22598 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22599 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22600 | The product of the charact... |
| cpmadugsum 22601 | The product of the charact... |
| cpmidgsum2 22602 | Representation of the iden... |
| cpmidg2sum 22603 | Equality of two sums repre... |
| cpmadumatpolylem1 22604 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22605 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22606 | The product of the charact... |
| cayhamlem2 22607 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22608 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22609 | The coefficients of the ch... |
| cayhamlem3 22610 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22611 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22612 | The Cayley-Hamilton theore... |
| cayleyhamilton 22613 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22614 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22615 | The Cayley-Hamilton theore... |
| istopg 22618 | Express the predicate " ` ... |
| istop2g 22619 | Express the predicate " ` ... |
| uniopn 22620 | The union of a subset of a... |
| iunopn 22621 | The indexed union of a sub... |
| inopn 22622 | The intersection of two op... |
| fitop 22623 | A topology is closed under... |
| fiinopn 22624 | The intersection of a none... |
| iinopn 22625 | The intersection of a none... |
| unopn 22626 | The union of two open sets... |
| 0opn 22627 | The empty set is an open s... |
| 0ntop 22628 | The empty set is not a top... |
| topopn 22629 | The underlying set of a to... |
| eltopss 22630 | A member of a topology is ... |
| riinopn 22631 | A finite indexed relative ... |
| rintopn 22632 | A finite relative intersec... |
| istopon 22635 | Property of being a topolo... |
| topontop 22636 | A topology on a given base... |
| toponuni 22637 | The base set of a topology... |
| topontopi 22638 | A topology on a given base... |
| toponunii 22639 | The base set of a topology... |
| toptopon 22640 | Alternative definition of ... |
| toptopon2 22641 | A topology is the same thi... |
| topontopon 22642 | A topology on a set is a t... |
| funtopon 22643 | The class ` TopOn ` is a f... |
| toponrestid 22644 | Given a topology on a set,... |
| toponsspwpw 22645 | The set of topologies on a... |
| dmtopon 22646 | The domain of ` TopOn ` is... |
| fntopon 22647 | The class ` TopOn ` is a f... |
| toprntopon 22648 | A topology is the same thi... |
| toponmax 22649 | The base set of a topology... |
| toponss 22650 | A member of a topology is ... |
| toponcom 22651 | If ` K ` is a topology on ... |
| toponcomb 22652 | Biconditional form of ~ to... |
| topgele 22653 | The topologies over the sa... |
| topsn 22654 | The only topology on a sin... |
| istps 22657 | Express the predicate "is ... |
| istps2 22658 | Express the predicate "is ... |
| tpsuni 22659 | The base set of a topologi... |
| tpstop 22660 | The topology extractor on ... |
| tpspropd 22661 | A topological space depend... |
| tpsprop2d 22662 | A topological space depend... |
| topontopn 22663 | Express the predicate "is ... |
| tsettps 22664 | If the topology component ... |
| istpsi 22665 | Properties that determine ... |
| eltpsg 22666 | Properties that determine ... |
| eltpsgOLD 22667 | Obsolete version of ~ eltp... |
| eltpsi 22668 | Properties that determine ... |
| isbasisg 22671 | Express the predicate "the... |
| isbasis2g 22672 | Express the predicate "the... |
| isbasis3g 22673 | Express the predicate "the... |
| basis1 22674 | Property of a basis. (Con... |
| basis2 22675 | Property of a basis. (Con... |
| fiinbas 22676 | If a set is closed under f... |
| basdif0 22677 | A basis is not affected by... |
| baspartn 22678 | A disjoint system of sets ... |
| tgval 22679 | The topology generated by ... |
| tgval2 22680 | Definition of a topology g... |
| eltg 22681 | Membership in a topology g... |
| eltg2 22682 | Membership in a topology g... |
| eltg2b 22683 | Membership in a topology g... |
| eltg4i 22684 | An open set in a topology ... |
| eltg3i 22685 | The union of a set of basi... |
| eltg3 22686 | Membership in a topology g... |
| tgval3 22687 | Alternate expression for t... |
| tg1 22688 | Property of a member of a ... |
| tg2 22689 | Property of a member of a ... |
| bastg 22690 | A member of a basis is a s... |
| unitg 22691 | The topology generated by ... |
| tgss 22692 | Subset relation for genera... |
| tgcl 22693 | Show that a basis generate... |
| tgclb 22694 | The property ~ tgcl can be... |
| tgtopon 22695 | A basis generates a topolo... |
| topbas 22696 | A topology is its own basi... |
| tgtop 22697 | A topology is its own basi... |
| eltop 22698 | Membership in a topology, ... |
| eltop2 22699 | Membership in a topology. ... |
| eltop3 22700 | Membership in a topology. ... |
| fibas 22701 | A collection of finite int... |
| tgdom 22702 | A space has no more open s... |
| tgiun 22703 | The indexed union of a set... |
| tgidm 22704 | The topology generator fun... |
| bastop 22705 | Two ways to express that a... |
| tgtop11 22706 | The topology generation fu... |
| 0top 22707 | The singleton of the empty... |
| en1top 22708 | ` { (/) } ` is the only to... |
| en2top 22709 | If a topology has two elem... |
| tgss3 22710 | A criterion for determinin... |
| tgss2 22711 | A criterion for determinin... |
| basgen 22712 | Given a topology ` J ` , s... |
| basgen2 22713 | Given a topology ` J ` , s... |
| 2basgen 22714 | Conditions that determine ... |
| tgfiss 22715 | If a subbase is included i... |
| tgdif0 22716 | A generated topology is no... |
| bastop1 22717 | A subset of a topology is ... |
| bastop2 22718 | A version of ~ bastop1 tha... |
| distop 22719 | The discrete topology on a... |
| topnex 22720 | The class of all topologie... |
| distopon 22721 | The discrete topology on a... |
| sn0topon 22722 | The singleton of the empty... |
| sn0top 22723 | The singleton of the empty... |
| indislem 22724 | A lemma to eliminate some ... |
| indistopon 22725 | The indiscrete topology on... |
| indistop 22726 | The indiscrete topology on... |
| indisuni 22727 | The base set of the indisc... |
| fctop 22728 | The finite complement topo... |
| fctop2 22729 | The finite complement topo... |
| cctop 22730 | The countable complement t... |
| ppttop 22731 | The particular point topol... |
| pptbas 22732 | The particular point topol... |
| epttop 22733 | The excluded point topolog... |
| indistpsx 22734 | The indiscrete topology on... |
| indistps 22735 | The indiscrete topology on... |
| indistps2 22736 | The indiscrete topology on... |
| indistpsALT 22737 | The indiscrete topology on... |
| indistpsALTOLD 22738 | Obsolete version of ~ indi... |
| indistps2ALT 22739 | The indiscrete topology on... |
| distps 22740 | The discrete topology on a... |
| fncld 22747 | The closed-set generator i... |
| cldval 22748 | The set of closed sets of ... |
| ntrfval 22749 | The interior function on t... |
| clsfval 22750 | The closure function on th... |
| cldrcl 22751 | Reverse closure of the clo... |
| iscld 22752 | The predicate "the class `... |
| iscld2 22753 | A subset of the underlying... |
| cldss 22754 | A closed set is a subset o... |
| cldss2 22755 | The set of closed sets is ... |
| cldopn 22756 | The complement of a closed... |
| isopn2 22757 | A subset of the underlying... |
| opncld 22758 | The complement of an open ... |
| difopn 22759 | The difference of a closed... |
| topcld 22760 | The underlying set of a to... |
| ntrval 22761 | The interior of a subset o... |
| clsval 22762 | The closure of a subset of... |
| 0cld 22763 | The empty set is closed. ... |
| iincld 22764 | The indexed intersection o... |
| intcld 22765 | The intersection of a set ... |
| uncld 22766 | The union of two closed se... |
| cldcls 22767 | A closed subset equals its... |
| incld 22768 | The intersection of two cl... |
| riincld 22769 | An indexed relative inters... |
| iuncld 22770 | A finite indexed union of ... |
| unicld 22771 | A finite union of closed s... |
| clscld 22772 | The closure of a subset of... |
| clsf 22773 | The closure function is a ... |
| ntropn 22774 | The interior of a subset o... |
| clsval2 22775 | Express closure in terms o... |
| ntrval2 22776 | Interior expressed in term... |
| ntrdif 22777 | An interior of a complemen... |
| clsdif 22778 | A closure of a complement ... |
| clsss 22779 | Subset relationship for cl... |
| ntrss 22780 | Subset relationship for in... |
| sscls 22781 | A subset of a topology's u... |
| ntrss2 22782 | A subset includes its inte... |
| ssntr 22783 | An open subset of a set is... |
| clsss3 22784 | The closure of a subset of... |
| ntrss3 22785 | The interior of a subset o... |
| ntrin 22786 | A pairwise intersection of... |
| cmclsopn 22787 | The complement of a closur... |
| cmntrcld 22788 | The complement of an inter... |
| iscld3 22789 | A subset is closed iff it ... |
| iscld4 22790 | A subset is closed iff it ... |
| isopn3 22791 | A subset is open iff it eq... |
| clsidm 22792 | The closure operation is i... |
| ntridm 22793 | The interior operation is ... |
| clstop 22794 | The closure of a topology'... |
| ntrtop 22795 | The interior of a topology... |
| 0ntr 22796 | A subset with an empty int... |
| clsss2 22797 | If a subset is included in... |
| elcls 22798 | Membership in a closure. ... |
| elcls2 22799 | Membership in a closure. ... |
| clsndisj 22800 | Any open set containing a ... |
| ntrcls0 22801 | A subset whose closure has... |
| ntreq0 22802 | Two ways to say that a sub... |
| cldmre 22803 | The closed sets of a topol... |
| mrccls 22804 | Moore closure generalizes ... |
| cls0 22805 | The closure of the empty s... |
| ntr0 22806 | The interior of the empty ... |
| isopn3i 22807 | An open subset equals its ... |
| elcls3 22808 | Membership in a closure in... |
| opncldf1 22809 | A bijection useful for con... |
| opncldf2 22810 | The values of the open-clo... |
| opncldf3 22811 | The values of the converse... |
| isclo 22812 | A set ` A ` is clopen iff ... |
| isclo2 22813 | A set ` A ` is clopen iff ... |
| discld 22814 | The open sets of a discret... |
| sn0cld 22815 | The closed sets of the top... |
| indiscld 22816 | The closed sets of an indi... |
| mretopd 22817 | A Moore collection which i... |
| toponmre 22818 | The topologies over a give... |
| cldmreon 22819 | The closed sets of a topol... |
| iscldtop 22820 | A family is the closed set... |
| mreclatdemoBAD 22821 | The closed subspaces of a ... |
| neifval 22824 | Value of the neighborhood ... |
| neif 22825 | The neighborhood function ... |
| neiss2 22826 | A set with a neighborhood ... |
| neival 22827 | Value of the set of neighb... |
| isnei 22828 | The predicate "the class `... |
| neiint 22829 | An intuitive definition of... |
| isneip 22830 | The predicate "the class `... |
| neii1 22831 | A neighborhood is included... |
| neisspw 22832 | The neighborhoods of any s... |
| neii2 22833 | Property of a neighborhood... |
| neiss 22834 | Any neighborhood of a set ... |
| ssnei 22835 | A set is included in any o... |
| elnei 22836 | A point belongs to any of ... |
| 0nnei 22837 | The empty set is not a nei... |
| neips 22838 | A neighborhood of a set is... |
| opnneissb 22839 | An open set is a neighborh... |
| opnssneib 22840 | Any superset of an open se... |
| ssnei2 22841 | Any subset ` M ` of ` X ` ... |
| neindisj 22842 | Any neighborhood of an ele... |
| opnneiss 22843 | An open set is a neighborh... |
| opnneip 22844 | An open set is a neighborh... |
| opnnei 22845 | A set is open iff it is a ... |
| tpnei 22846 | The underlying set of a to... |
| neiuni 22847 | The union of the neighborh... |
| neindisj2 22848 | A point ` P ` belongs to t... |
| topssnei 22849 | A finer topology has more ... |
| innei 22850 | The intersection of two ne... |
| opnneiid 22851 | Only an open set is a neig... |
| neissex 22852 | For any neighborhood ` N `... |
| 0nei 22853 | The empty set is a neighbo... |
| neipeltop 22854 | Lemma for ~ neiptopreu . ... |
| neiptopuni 22855 | Lemma for ~ neiptopreu . ... |
| neiptoptop 22856 | Lemma for ~ neiptopreu . ... |
| neiptopnei 22857 | Lemma for ~ neiptopreu . ... |
| neiptopreu 22858 | If, to each element ` P ` ... |
| lpfval 22863 | The limit point function o... |
| lpval 22864 | The set of limit points of... |
| islp 22865 | The predicate "the class `... |
| lpsscls 22866 | The limit points of a subs... |
| lpss 22867 | The limit points of a subs... |
| lpdifsn 22868 | ` P ` is a limit point of ... |
| lpss3 22869 | Subset relationship for li... |
| islp2 22870 | The predicate " ` P ` is a... |
| islp3 22871 | The predicate " ` P ` is a... |
| maxlp 22872 | A point is a limit point o... |
| clslp 22873 | The closure of a subset of... |
| islpi 22874 | A point belonging to a set... |
| cldlp 22875 | A subset of a topological ... |
| isperf 22876 | Definition of a perfect sp... |
| isperf2 22877 | Definition of a perfect sp... |
| isperf3 22878 | A perfect space is a topol... |
| perflp 22879 | The limit points of a perf... |
| perfi 22880 | Property of a perfect spac... |
| perftop 22881 | A perfect space is a topol... |
| restrcl 22882 | Reverse closure for the su... |
| restbas 22883 | A subspace topology basis ... |
| tgrest 22884 | A subspace can be generate... |
| resttop 22885 | A subspace topology is a t... |
| resttopon 22886 | A subspace topology is a t... |
| restuni 22887 | The underlying set of a su... |
| stoig 22888 | The topological space buil... |
| restco 22889 | Composition of subspaces. ... |
| restabs 22890 | Equivalence of being a sub... |
| restin 22891 | When the subspace region i... |
| restuni2 22892 | The underlying set of a su... |
| resttopon2 22893 | The underlying set of a su... |
| rest0 22894 | The subspace topology indu... |
| restsn 22895 | The only subspace topology... |
| restsn2 22896 | The subspace topology indu... |
| restcld 22897 | A closed set of a subspace... |
| restcldi 22898 | A closed set is closed in ... |
| restcldr 22899 | A set which is closed in t... |
| restopnb 22900 | If ` B ` is an open subset... |
| ssrest 22901 | If ` K ` is a finer topolo... |
| restopn2 22902 | If ` A ` is open, then ` B... |
| restdis 22903 | A subspace of a discrete t... |
| restfpw 22904 | The restriction of the set... |
| neitr 22905 | The neighborhood of a trac... |
| restcls 22906 | A closure in a subspace to... |
| restntr 22907 | An interior in a subspace ... |
| restlp 22908 | The limit points of a subs... |
| restperf 22909 | Perfection of a subspace. ... |
| perfopn 22910 | An open subset of a perfec... |
| resstopn 22911 | The topology of a restrict... |
| resstps 22912 | A restricted topological s... |
| ordtbaslem 22913 | Lemma for ~ ordtbas . In ... |
| ordtval 22914 | Value of the order topolog... |
| ordtuni 22915 | Value of the order topolog... |
| ordtbas2 22916 | Lemma for ~ ordtbas . (Co... |
| ordtbas 22917 | In a total order, the fini... |
| ordttopon 22918 | Value of the order topolog... |
| ordtopn1 22919 | An upward ray ` ( P , +oo ... |
| ordtopn2 22920 | A downward ray ` ( -oo , P... |
| ordtopn3 22921 | An open interval ` ( A , B... |
| ordtcld1 22922 | A downward ray ` ( -oo , P... |
| ordtcld2 22923 | An upward ray ` [ P , +oo ... |
| ordtcld3 22924 | A closed interval ` [ A , ... |
| ordttop 22925 | The order topology is a to... |
| ordtcnv 22926 | The order dual generates t... |
| ordtrest 22927 | The subspace topology of a... |
| ordtrest2lem 22928 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 22929 | An interval-closed set ` A... |
| letopon 22930 | The topology of the extend... |
| letop 22931 | The topology of the extend... |
| letopuni 22932 | The topology of the extend... |
| xrstopn 22933 | The topology component of ... |
| xrstps 22934 | The extended real number s... |
| leordtvallem1 22935 | Lemma for ~ leordtval . (... |
| leordtvallem2 22936 | Lemma for ~ leordtval . (... |
| leordtval2 22937 | The topology of the extend... |
| leordtval 22938 | The topology of the extend... |
| iccordt 22939 | A closed interval is close... |
| iocpnfordt 22940 | An unbounded above open in... |
| icomnfordt 22941 | An unbounded above open in... |
| iooordt 22942 | An open interval is open i... |
| reordt 22943 | The real numbers are an op... |
| lecldbas 22944 | The set of closed interval... |
| pnfnei 22945 | A neighborhood of ` +oo ` ... |
| mnfnei 22946 | A neighborhood of ` -oo ` ... |
| ordtrestixx 22947 | The restriction of the les... |
| ordtresticc 22948 | The restriction of the les... |
| lmrel 22955 | The topological space conv... |
| lmrcl 22956 | Reverse closure for the co... |
| lmfval 22957 | The relation "sequence ` f... |
| cnfval 22958 | The set of all continuous ... |
| cnpfval 22959 | The function mapping the p... |
| iscn 22960 | The predicate "the class `... |
| cnpval 22961 | The set of all functions f... |
| iscnp 22962 | The predicate "the class `... |
| iscn2 22963 | The predicate "the class `... |
| iscnp2 22964 | The predicate "the class `... |
| cntop1 22965 | Reverse closure for a cont... |
| cntop2 22966 | Reverse closure for a cont... |
| cnptop1 22967 | Reverse closure for a func... |
| cnptop2 22968 | Reverse closure for a func... |
| iscnp3 22969 | The predicate "the class `... |
| cnprcl 22970 | Reverse closure for a func... |
| cnf 22971 | A continuous function is a... |
| cnpf 22972 | A continuous function at p... |
| cnpcl 22973 | The value of a continuous ... |
| cnf2 22974 | A continuous function is a... |
| cnpf2 22975 | A continuous function at p... |
| cnprcl2 22976 | Reverse closure for a func... |
| tgcn 22977 | The continuity predicate w... |
| tgcnp 22978 | The "continuous at a point... |
| subbascn 22979 | The continuity predicate w... |
| ssidcn 22980 | The identity function is a... |
| cnpimaex 22981 | Property of a function con... |
| idcn 22982 | A restricted identity func... |
| lmbr 22983 | Express the binary relatio... |
| lmbr2 22984 | Express the binary relatio... |
| lmbrf 22985 | Express the binary relatio... |
| lmconst 22986 | A constant sequence conver... |
| lmcvg 22987 | Convergence property of a ... |
| iscnp4 22988 | The predicate "the class `... |
| cnpnei 22989 | A condition for continuity... |
| cnima 22990 | An open subset of the codo... |
| cnco 22991 | The composition of two con... |
| cnpco 22992 | The composition of a funct... |
| cnclima 22993 | A closed subset of the cod... |
| iscncl 22994 | A characterization of a co... |
| cncls2i 22995 | Property of the preimage o... |
| cnntri 22996 | Property of the preimage o... |
| cnclsi 22997 | Property of the image of a... |
| cncls2 22998 | Continuity in terms of clo... |
| cncls 22999 | Continuity in terms of clo... |
| cnntr 23000 | Continuity in terms of int... |
| cnss1 23001 | If the topology ` K ` is f... |
| cnss2 23002 | If the topology ` K ` is f... |
| cncnpi 23003 | A continuous function is c... |
| cnsscnp 23004 | The set of continuous func... |
| cncnp 23005 | A continuous function is c... |
| cncnp2 23006 | A continuous function is c... |
| cnnei 23007 | Continuity in terms of nei... |
| cnconst2 23008 | A constant function is con... |
| cnconst 23009 | A constant function is con... |
| cnrest 23010 | Continuity of a restrictio... |
| cnrest2 23011 | Equivalence of continuity ... |
| cnrest2r 23012 | Equivalence of continuity ... |
| cnpresti 23013 | One direction of ~ cnprest... |
| cnprest 23014 | Equivalence of continuity ... |
| cnprest2 23015 | Equivalence of point-conti... |
| cndis 23016 | Every function is continuo... |
| cnindis 23017 | Every function is continuo... |
| cnpdis 23018 | If ` A ` is an isolated po... |
| paste 23019 | Pasting lemma. If ` A ` a... |
| lmfpm 23020 | If ` F ` converges, then `... |
| lmfss 23021 | Inclusion of a function ha... |
| lmcl 23022 | Closure of a limit. (Cont... |
| lmss 23023 | Limit on a subspace. (Con... |
| sslm 23024 | A finer topology has fewer... |
| lmres 23025 | A function converges iff i... |
| lmff 23026 | If ` F ` converges, there ... |
| lmcls 23027 | Any convergent sequence of... |
| lmcld 23028 | Any convergent sequence of... |
| lmcnp 23029 | The image of a convergent ... |
| lmcn 23030 | The image of a convergent ... |
| ist0 23045 | The predicate "is a T_0 sp... |
| ist1 23046 | The predicate "is a T_1 sp... |
| ishaus 23047 | The predicate "is a Hausdo... |
| iscnrm 23048 | The property of being comp... |
| t0sep 23049 | Any two topologically indi... |
| t0dist 23050 | Any two distinct points in... |
| t1sncld 23051 | In a T_1 space, singletons... |
| t1ficld 23052 | In a T_1 space, finite set... |
| hausnei 23053 | Neighborhood property of a... |
| t0top 23054 | A T_0 space is a topologic... |
| t1top 23055 | A T_1 space is a topologic... |
| haustop 23056 | A Hausdorff space is a top... |
| isreg 23057 | The predicate "is a regula... |
| regtop 23058 | A regular space is a topol... |
| regsep 23059 | In a regular space, every ... |
| isnrm 23060 | The predicate "is a normal... |
| nrmtop 23061 | A normal space is a topolo... |
| cnrmtop 23062 | A completely normal space ... |
| iscnrm2 23063 | The property of being comp... |
| ispnrm 23064 | The property of being perf... |
| pnrmnrm 23065 | A perfectly normal space i... |
| pnrmtop 23066 | A perfectly normal space i... |
| pnrmcld 23067 | A closed set in a perfectl... |
| pnrmopn 23068 | An open set in a perfectly... |
| ist0-2 23069 | The predicate "is a T_0 sp... |
| ist0-3 23070 | The predicate "is a T_0 sp... |
| cnt0 23071 | The preimage of a T_0 topo... |
| ist1-2 23072 | An alternate characterizat... |
| t1t0 23073 | A T_1 space is a T_0 space... |
| ist1-3 23074 | A space is T_1 iff every p... |
| cnt1 23075 | The preimage of a T_1 topo... |
| ishaus2 23076 | Express the predicate " ` ... |
| haust1 23077 | A Hausdorff space is a T_1... |
| hausnei2 23078 | The Hausdorff condition st... |
| cnhaus 23079 | The preimage of a Hausdorf... |
| nrmsep3 23080 | In a normal space, given a... |
| nrmsep2 23081 | In a normal space, any two... |
| nrmsep 23082 | In a normal space, disjoin... |
| isnrm2 23083 | An alternate characterizat... |
| isnrm3 23084 | A topological space is nor... |
| cnrmi 23085 | A subspace of a completely... |
| cnrmnrm 23086 | A completely normal space ... |
| restcnrm 23087 | A subspace of a completely... |
| resthauslem 23088 | Lemma for ~ resthaus and s... |
| lpcls 23089 | The limit points of the cl... |
| perfcls 23090 | A subset of a perfect spac... |
| restt0 23091 | A subspace of a T_0 topolo... |
| restt1 23092 | A subspace of a T_1 topolo... |
| resthaus 23093 | A subspace of a Hausdorff ... |
| t1sep2 23094 | Any two points in a T_1 sp... |
| t1sep 23095 | Any two distinct points in... |
| sncld 23096 | A singleton is closed in a... |
| sshauslem 23097 | Lemma for ~ sshaus and sim... |
| sst0 23098 | A topology finer than a T_... |
| sst1 23099 | A topology finer than a T_... |
| sshaus 23100 | A topology finer than a Ha... |
| regsep2 23101 | In a regular space, a clos... |
| isreg2 23102 | A topological space is reg... |
| dnsconst 23103 | If a continuous mapping to... |
| ordtt1 23104 | The order topology is T_1 ... |
| lmmo 23105 | A sequence in a Hausdorff ... |
| lmfun 23106 | The convergence relation i... |
| dishaus 23107 | A discrete topology is Hau... |
| ordthauslem 23108 | Lemma for ~ ordthaus . (C... |
| ordthaus 23109 | The order topology of a to... |
| xrhaus 23110 | The topology of the extend... |
| iscmp 23113 | The predicate "is a compac... |
| cmpcov 23114 | An open cover of a compact... |
| cmpcov2 23115 | Rewrite ~ cmpcov for the c... |
| cmpcovf 23116 | Combine ~ cmpcov with ~ ac... |
| cncmp 23117 | Compactness is respected b... |
| fincmp 23118 | A finite topology is compa... |
| 0cmp 23119 | The singleton of the empty... |
| cmptop 23120 | A compact topology is a to... |
| rncmp 23121 | The image of a compact set... |
| imacmp 23122 | The image of a compact set... |
| discmp 23123 | A discrete topology is com... |
| cmpsublem 23124 | Lemma for ~ cmpsub . (Con... |
| cmpsub 23125 | Two equivalent ways of des... |
| tgcmp 23126 | A topology generated by a ... |
| cmpcld 23127 | A closed subset of a compa... |
| uncmp 23128 | The union of two compact s... |
| fiuncmp 23129 | A finite union of compact ... |
| sscmp 23130 | A subset of a compact topo... |
| hauscmplem 23131 | Lemma for ~ hauscmp . (Co... |
| hauscmp 23132 | A compact subspace of a T2... |
| cmpfi 23133 | If a topology is compact a... |
| cmpfii 23134 | In a compact topology, a s... |
| bwth 23135 | The glorious Bolzano-Weier... |
| isconn 23138 | The predicate ` J ` is a c... |
| isconn2 23139 | The predicate ` J ` is a c... |
| connclo 23140 | The only nonempty clopen s... |
| conndisj 23141 | If a topology is connected... |
| conntop 23142 | A connected topology is a ... |
| indisconn 23143 | The indiscrete topology (o... |
| dfconn2 23144 | An alternate definition of... |
| connsuba 23145 | Connectedness for a subspa... |
| connsub 23146 | Two equivalent ways of say... |
| cnconn 23147 | Connectedness is respected... |
| nconnsubb 23148 | Disconnectedness for a sub... |
| connsubclo 23149 | If a clopen set meets a co... |
| connima 23150 | The image of a connected s... |
| conncn 23151 | A continuous function from... |
| iunconnlem 23152 | Lemma for ~ iunconn . (Co... |
| iunconn 23153 | The indexed union of conne... |
| unconn 23154 | The union of two connected... |
| clsconn 23155 | The closure of a connected... |
| conncompid 23156 | The connected component co... |
| conncompconn 23157 | The connected component co... |
| conncompss 23158 | The connected component co... |
| conncompcld 23159 | The connected component co... |
| conncompclo 23160 | The connected component co... |
| t1connperf 23161 | A connected T_1 space is p... |
| is1stc 23166 | The predicate "is a first-... |
| is1stc2 23167 | An equivalent way of sayin... |
| 1stctop 23168 | A first-countable topology... |
| 1stcclb 23169 | A property of points in a ... |
| 1stcfb 23170 | For any point ` A ` in a f... |
| is2ndc 23171 | The property of being seco... |
| 2ndctop 23172 | A second-countable topolog... |
| 2ndci 23173 | A countable basis generate... |
| 2ndcsb 23174 | Having a countable subbase... |
| 2ndcredom 23175 | A second-countable space h... |
| 2ndc1stc 23176 | A second-countable space i... |
| 1stcrestlem 23177 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 23178 | A subspace of a first-coun... |
| 2ndcrest 23179 | A subspace of a second-cou... |
| 2ndcctbss 23180 | If a topology is second-co... |
| 2ndcdisj 23181 | Any disjoint family of ope... |
| 2ndcdisj2 23182 | Any disjoint collection of... |
| 2ndcomap 23183 | A surjective continuous op... |
| 2ndcsep 23184 | A second-countable topolog... |
| dis2ndc 23185 | A discrete space is second... |
| 1stcelcls 23186 | A point belongs to the clo... |
| 1stccnp 23187 | A mapping is continuous at... |
| 1stccn 23188 | A mapping ` X --> Y ` , wh... |
| islly 23193 | The property of being a lo... |
| isnlly 23194 | The property of being an n... |
| llyeq 23195 | Equality theorem for the `... |
| nllyeq 23196 | Equality theorem for the `... |
| llytop 23197 | A locally ` A ` space is a... |
| nllytop 23198 | A locally ` A ` space is a... |
| llyi 23199 | The property of a locally ... |
| nllyi 23200 | The property of an n-local... |
| nlly2i 23201 | Eliminate the neighborhood... |
| llynlly 23202 | A locally ` A ` space is n... |
| llyssnlly 23203 | A locally ` A ` space is n... |
| llyss 23204 | The "locally" predicate re... |
| nllyss 23205 | The "n-locally" predicate ... |
| subislly 23206 | The property of a subspace... |
| restnlly 23207 | If the property ` A ` pass... |
| restlly 23208 | If the property ` A ` pass... |
| islly2 23209 | An alternative expression ... |
| llyrest 23210 | An open subspace of a loca... |
| nllyrest 23211 | An open subspace of an n-l... |
| loclly 23212 | If ` A ` is a local proper... |
| llyidm 23213 | Idempotence of the "locall... |
| nllyidm 23214 | Idempotence of the "n-loca... |
| toplly 23215 | A topology is locally a to... |
| topnlly 23216 | A topology is n-locally a ... |
| hauslly 23217 | A Hausdorff space is local... |
| hausnlly 23218 | A Hausdorff space is n-loc... |
| hausllycmp 23219 | A compact Hausdorff space ... |
| cldllycmp 23220 | A closed subspace of a loc... |
| lly1stc 23221 | First-countability is a lo... |
| dislly 23222 | The discrete space ` ~P X ... |
| disllycmp 23223 | A discrete space is locall... |
| dis1stc 23224 | A discrete space is first-... |
| hausmapdom 23225 | If ` X ` is a first-counta... |
| hauspwdom 23226 | Simplify the cardinal ` A ... |
| refrel 23233 | Refinement is a relation. ... |
| isref 23234 | The property of being a re... |
| refbas 23235 | A refinement covers the sa... |
| refssex 23236 | Every set in a refinement ... |
| ssref 23237 | A subcover is a refinement... |
| refref 23238 | Reflexivity of refinement.... |
| reftr 23239 | Refinement is transitive. ... |
| refun0 23240 | Adding the empty set prese... |
| isptfin 23241 | The statement "is a point-... |
| islocfin 23242 | The statement "is a locall... |
| finptfin 23243 | A finite cover is a point-... |
| ptfinfin 23244 | A point covered by a point... |
| finlocfin 23245 | A finite cover of a topolo... |
| locfintop 23246 | A locally finite cover cov... |
| locfinbas 23247 | A locally finite cover mus... |
| locfinnei 23248 | A point covered by a local... |
| lfinpfin 23249 | A locally finite cover is ... |
| lfinun 23250 | Adding a finite set preser... |
| locfincmp 23251 | For a compact space, the l... |
| unisngl 23252 | Taking the union of the se... |
| dissnref 23253 | The set of singletons is a... |
| dissnlocfin 23254 | The set of singletons is l... |
| locfindis 23255 | The locally finite covers ... |
| locfincf 23256 | A locally finite cover in ... |
| comppfsc 23257 | A space where every open c... |
| kgenval 23260 | Value of the compact gener... |
| elkgen 23261 | Value of the compact gener... |
| kgeni 23262 | Property of the open sets ... |
| kgentopon 23263 | The compact generator gene... |
| kgenuni 23264 | The base set of the compac... |
| kgenftop 23265 | The compact generator gene... |
| kgenf 23266 | The compact generator is a... |
| kgentop 23267 | A compactly generated spac... |
| kgenss 23268 | The compact generator gene... |
| kgenhaus 23269 | The compact generator gene... |
| kgencmp 23270 | The compact generator topo... |
| kgencmp2 23271 | The compact generator topo... |
| kgenidm 23272 | The compact generator is i... |
| iskgen2 23273 | A space is compactly gener... |
| iskgen3 23274 | Derive the usual definitio... |
| llycmpkgen2 23275 | A locally compact space is... |
| cmpkgen 23276 | A compact space is compact... |
| llycmpkgen 23277 | A locally compact space is... |
| 1stckgenlem 23278 | The one-point compactifica... |
| 1stckgen 23279 | A first-countable space is... |
| kgen2ss 23280 | The compact generator pres... |
| kgencn 23281 | A function from a compactl... |
| kgencn2 23282 | A function ` F : J --> K `... |
| kgencn3 23283 | The set of continuous func... |
| kgen2cn 23284 | A continuous function is a... |
| txval 23289 | Value of the binary topolo... |
| txuni2 23290 | The underlying set of the ... |
| txbasex 23291 | The basis for the product ... |
| txbas 23292 | The set of Cartesian produ... |
| eltx 23293 | A set in a product is open... |
| txtop 23294 | The product of two topolog... |
| ptval 23295 | The value of the product t... |
| ptpjpre1 23296 | The preimage of a projecti... |
| elpt 23297 | Elementhood in the bases o... |
| elptr 23298 | A basic open set in the pr... |
| elptr2 23299 | A basic open set in the pr... |
| ptbasid 23300 | The base set of the produc... |
| ptuni2 23301 | The base set for the produ... |
| ptbasin 23302 | The basis for a product to... |
| ptbasin2 23303 | The basis for a product to... |
| ptbas 23304 | The basis for a product to... |
| ptpjpre2 23305 | The basis for a product to... |
| ptbasfi 23306 | The basis for the product ... |
| pttop 23307 | The product topology is a ... |
| ptopn 23308 | A basic open set in the pr... |
| ptopn2 23309 | A sub-basic open set in th... |
| xkotf 23310 | Functionality of function ... |
| xkobval 23311 | Alternative expression for... |
| xkoval 23312 | Value of the compact-open ... |
| xkotop 23313 | The compact-open topology ... |
| xkoopn 23314 | A basic open set of the co... |
| txtopi 23315 | The product of two topolog... |
| txtopon 23316 | The underlying set of the ... |
| txuni 23317 | The underlying set of the ... |
| txunii 23318 | The underlying set of the ... |
| ptuni 23319 | The base set for the produ... |
| ptunimpt 23320 | Base set of a product topo... |
| pttopon 23321 | The base set for the produ... |
| pttoponconst 23322 | The base set for a product... |
| ptuniconst 23323 | The base set for a product... |
| xkouni 23324 | The base set of the compac... |
| xkotopon 23325 | The base set of the compac... |
| ptval2 23326 | The value of the product t... |
| txopn 23327 | The product of two open se... |
| txcld 23328 | The product of two closed ... |
| txcls 23329 | Closure of a rectangle in ... |
| txss12 23330 | Subset property of the top... |
| txbasval 23331 | It is sufficient to consid... |
| neitx 23332 | The Cartesian product of t... |
| txcnpi 23333 | Continuity of a two-argume... |
| tx1cn 23334 | Continuity of the first pr... |
| tx2cn 23335 | Continuity of the second p... |
| ptpjcn 23336 | Continuity of a projection... |
| ptpjopn 23337 | The projection map is an o... |
| ptcld 23338 | A closed box in the produc... |
| ptcldmpt 23339 | A closed box in the produc... |
| ptclsg 23340 | The closure of a box in th... |
| ptcls 23341 | The closure of a box in th... |
| dfac14lem 23342 | Lemma for ~ dfac14 . By e... |
| dfac14 23343 | Theorem ~ ptcls is an equi... |
| xkoccn 23344 | The "constant function" fu... |
| txcnp 23345 | If two functions are conti... |
| ptcnplem 23346 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 23347 | If every projection of a f... |
| upxp 23348 | Universal property of the ... |
| txcnmpt 23349 | A map into the product of ... |
| uptx 23350 | Universal property of the ... |
| txcn 23351 | A map into the product of ... |
| ptcn 23352 | If every projection of a f... |
| prdstopn 23353 | Topology of a structure pr... |
| prdstps 23354 | A structure product of top... |
| pwstps 23355 | A structure power of a top... |
| txrest 23356 | The subspace of a topologi... |
| txdis 23357 | The topological product of... |
| txindislem 23358 | Lemma for ~ txindis . (Co... |
| txindis 23359 | The topological product of... |
| txdis1cn 23360 | A function is jointly cont... |
| txlly 23361 | If the property ` A ` is p... |
| txnlly 23362 | If the property ` A ` is p... |
| pthaus 23363 | The product of a collectio... |
| ptrescn 23364 | Restriction is a continuou... |
| txtube 23365 | The "tube lemma". If ` X ... |
| txcmplem1 23366 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 23367 | Lemma for ~ txcmp . (Cont... |
| txcmp 23368 | The topological product of... |
| txcmpb 23369 | The topological product of... |
| hausdiag 23370 | A topology is Hausdorff if... |
| hauseqlcld 23371 | In a Hausdorff topology, t... |
| txhaus 23372 | The topological product of... |
| txlm 23373 | Two sequences converge iff... |
| lmcn2 23374 | The image of a convergent ... |
| tx1stc 23375 | The topological product of... |
| tx2ndc 23376 | The topological product of... |
| txkgen 23377 | The topological product of... |
| xkohaus 23378 | If the codomain space is H... |
| xkoptsub 23379 | The compact-open topology ... |
| xkopt 23380 | The compact-open topology ... |
| xkopjcn 23381 | Continuity of a projection... |
| xkoco1cn 23382 | If ` F ` is a continuous f... |
| xkoco2cn 23383 | If ` F ` is a continuous f... |
| xkococnlem 23384 | Continuity of the composit... |
| xkococn 23385 | Continuity of the composit... |
| cnmptid 23386 | The identity function is c... |
| cnmptc 23387 | A constant function is con... |
| cnmpt11 23388 | The composition of continu... |
| cnmpt11f 23389 | The composition of continu... |
| cnmpt1t 23390 | The composition of continu... |
| cnmpt12f 23391 | The composition of continu... |
| cnmpt12 23392 | The composition of continu... |
| cnmpt1st 23393 | The projection onto the fi... |
| cnmpt2nd 23394 | The projection onto the se... |
| cnmpt2c 23395 | A constant function is con... |
| cnmpt21 23396 | The composition of continu... |
| cnmpt21f 23397 | The composition of continu... |
| cnmpt2t 23398 | The composition of continu... |
| cnmpt22 23399 | The composition of continu... |
| cnmpt22f 23400 | The composition of continu... |
| cnmpt1res 23401 | The restriction of a conti... |
| cnmpt2res 23402 | The restriction of a conti... |
| cnmptcom 23403 | The argument converse of a... |
| cnmptkc 23404 | The curried first projecti... |
| cnmptkp 23405 | The evaluation of the inne... |
| cnmptk1 23406 | The composition of a curri... |
| cnmpt1k 23407 | The composition of a one-a... |
| cnmptkk 23408 | The composition of two cur... |
| xkofvcn 23409 | Joint continuity of the fu... |
| cnmptk1p 23410 | The evaluation of a currie... |
| cnmptk2 23411 | The uncurrying of a currie... |
| xkoinjcn 23412 | Continuity of "injection",... |
| cnmpt2k 23413 | The currying of a two-argu... |
| txconn 23414 | The topological product of... |
| imasnopn 23415 | If a relation graph is ope... |
| imasncld 23416 | If a relation graph is clo... |
| imasncls 23417 | If a relation graph is clo... |
| qtopval 23420 | Value of the quotient topo... |
| qtopval2 23421 | Value of the quotient topo... |
| elqtop 23422 | Value of the quotient topo... |
| qtopres 23423 | The quotient topology is u... |
| qtoptop2 23424 | The quotient topology is a... |
| qtoptop 23425 | The quotient topology is a... |
| elqtop2 23426 | Value of the quotient topo... |
| qtopuni 23427 | The base set of the quotie... |
| elqtop3 23428 | Value of the quotient topo... |
| qtoptopon 23429 | The base set of the quotie... |
| qtopid 23430 | A quotient map is a contin... |
| idqtop 23431 | The quotient topology indu... |
| qtopcmplem 23432 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23433 | A quotient of a compact sp... |
| qtopconn 23434 | A quotient of a connected ... |
| qtopkgen 23435 | A quotient of a compactly ... |
| basqtop 23436 | An injection maps bases to... |
| tgqtop 23437 | An injection maps generate... |
| qtopcld 23438 | The property of being a cl... |
| qtopcn 23439 | Universal property of a qu... |
| qtopss 23440 | A surjective continuous fu... |
| qtopeu 23441 | Universal property of the ... |
| qtoprest 23442 | If ` A ` is a saturated op... |
| qtopomap 23443 | If ` F ` is a surjective c... |
| qtopcmap 23444 | If ` F ` is a surjective c... |
| imastopn 23445 | The topology of an image s... |
| imastps 23446 | The image of a topological... |
| qustps 23447 | A quotient structure is a ... |
| kqfval 23448 | Value of the function appe... |
| kqfeq 23449 | Two points in the Kolmogor... |
| kqffn 23450 | The topological indistingu... |
| kqval 23451 | Value of the quotient topo... |
| kqtopon 23452 | The Kolmogorov quotient is... |
| kqid 23453 | The topological indistingu... |
| ist0-4 23454 | The topological indistingu... |
| kqfvima 23455 | When the image set is open... |
| kqsat 23456 | Any open set is saturated ... |
| kqdisj 23457 | A version of ~ imain for t... |
| kqcldsat 23458 | Any closed set is saturate... |
| kqopn 23459 | The topological indistingu... |
| kqcld 23460 | The topological indistingu... |
| kqt0lem 23461 | Lemma for ~ kqt0 . (Contr... |
| isr0 23462 | The property " ` J ` is an... |
| r0cld 23463 | The analogue of the T_1 ax... |
| regr1lem 23464 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23465 | A Kolmogorov quotient of a... |
| kqreglem1 23466 | A Kolmogorov quotient of a... |
| kqreglem2 23467 | If the Kolmogorov quotient... |
| kqnrmlem1 23468 | A Kolmogorov quotient of a... |
| kqnrmlem2 23469 | If the Kolmogorov quotient... |
| kqtop 23470 | The Kolmogorov quotient is... |
| kqt0 23471 | The Kolmogorov quotient is... |
| kqf 23472 | The Kolmogorov quotient is... |
| r0sep 23473 | The separation property of... |
| nrmr0reg 23474 | A normal R_0 space is also... |
| regr1 23475 | A regular space is R_1, wh... |
| kqreg 23476 | The Kolmogorov quotient of... |
| kqnrm 23477 | The Kolmogorov quotient of... |
| hmeofn 23482 | The set of homeomorphisms ... |
| hmeofval 23483 | The set of all the homeomo... |
| ishmeo 23484 | The predicate F is a homeo... |
| hmeocn 23485 | A homeomorphism is continu... |
| hmeocnvcn 23486 | The converse of a homeomor... |
| hmeocnv 23487 | The converse of a homeomor... |
| hmeof1o2 23488 | A homeomorphism is a 1-1-o... |
| hmeof1o 23489 | A homeomorphism is a 1-1-o... |
| hmeoima 23490 | The image of an open set b... |
| hmeoopn 23491 | Homeomorphisms preserve op... |
| hmeocld 23492 | Homeomorphisms preserve cl... |
| hmeocls 23493 | Homeomorphisms preserve cl... |
| hmeontr 23494 | Homeomorphisms preserve in... |
| hmeoimaf1o 23495 | The function mapping open ... |
| hmeores 23496 | The restriction of a homeo... |
| hmeoco 23497 | The composite of two homeo... |
| idhmeo 23498 | The identity function is a... |
| hmeocnvb 23499 | The converse of a homeomor... |
| hmeoqtop 23500 | A homeomorphism is a quoti... |
| hmph 23501 | Express the predicate ` J ... |
| hmphi 23502 | If there is a homeomorphis... |
| hmphtop 23503 | Reverse closure for the ho... |
| hmphtop1 23504 | The relation "being homeom... |
| hmphtop2 23505 | The relation "being homeom... |
| hmphref 23506 | "Is homeomorphic to" is re... |
| hmphsym 23507 | "Is homeomorphic to" is sy... |
| hmphtr 23508 | "Is homeomorphic to" is tr... |
| hmpher 23509 | "Is homeomorphic to" is an... |
| hmphen 23510 | Homeomorphisms preserve th... |
| hmphsymb 23511 | "Is homeomorphic to" is sy... |
| haushmphlem 23512 | Lemma for ~ haushmph and s... |
| cmphmph 23513 | Compactness is a topologic... |
| connhmph 23514 | Connectedness is a topolog... |
| t0hmph 23515 | T_0 is a topological prope... |
| t1hmph 23516 | T_1 is a topological prope... |
| haushmph 23517 | Hausdorff-ness is a topolo... |
| reghmph 23518 | Regularity is a topologica... |
| nrmhmph 23519 | Normality is a topological... |
| hmph0 23520 | A topology homeomorphic to... |
| hmphdis 23521 | Homeomorphisms preserve to... |
| hmphindis 23522 | Homeomorphisms preserve to... |
| indishmph 23523 | Equinumerous sets equipped... |
| hmphen2 23524 | Homeomorphisms preserve th... |
| cmphaushmeo 23525 | A continuous bijection fro... |
| ordthmeolem 23526 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23527 | An order isomorphism is a ... |
| txhmeo 23528 | Lift a pair of homeomorphi... |
| txswaphmeolem 23529 | Show inverse for the "swap... |
| txswaphmeo 23530 | There is a homeomorphism f... |
| pt1hmeo 23531 | The canonical homeomorphis... |
| ptuncnv 23532 | Exhibit the converse funct... |
| ptunhmeo 23533 | Define a homeomorphism fro... |
| xpstopnlem1 23534 | The function ` F ` used in... |
| xpstps 23535 | A binary product of topolo... |
| xpstopnlem2 23536 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23537 | The topology on a binary p... |
| ptcmpfi 23538 | A topological product of f... |
| xkocnv 23539 | The inverse of the "curryi... |
| xkohmeo 23540 | The Exponential Law for to... |
| qtopf1 23541 | If a quotient map is injec... |
| qtophmeo 23542 | If two functions on a base... |
| t0kq 23543 | A topological space is T_0... |
| kqhmph 23544 | A topological space is T_0... |
| ist1-5lem 23545 | Lemma for ~ ist1-5 and sim... |
| t1r0 23546 | A T_1 space is R_0. That ... |
| ist1-5 23547 | A topological space is T_1... |
| ishaus3 23548 | A topological space is Hau... |
| nrmreg 23549 | A normal T_1 space is regu... |
| reghaus 23550 | A regular T_0 space is Hau... |
| nrmhaus 23551 | A T_1 normal space is Haus... |
| elmptrab 23552 | Membership in a one-parame... |
| elmptrab2 23553 | Membership in a one-parame... |
| isfbas 23554 | The predicate " ` F ` is a... |
| fbasne0 23555 | There are no empty filter ... |
| 0nelfb 23556 | No filter base contains th... |
| fbsspw 23557 | A filter base on a set is ... |
| fbelss 23558 | An element of the filter b... |
| fbdmn0 23559 | The domain of a filter bas... |
| isfbas2 23560 | The predicate " ` F ` is a... |
| fbasssin 23561 | A filter base contains sub... |
| fbssfi 23562 | A filter base contains sub... |
| fbssint 23563 | A filter base contains sub... |
| fbncp 23564 | A filter base does not con... |
| fbun 23565 | A necessary and sufficient... |
| fbfinnfr 23566 | No filter base containing ... |
| opnfbas 23567 | The collection of open sup... |
| trfbas2 23568 | Conditions for the trace o... |
| trfbas 23569 | Conditions for the trace o... |
| isfil 23572 | The predicate "is a filter... |
| filfbas 23573 | A filter is a filter base.... |
| 0nelfil 23574 | The empty set doesn't belo... |
| fileln0 23575 | An element of a filter is ... |
| filsspw 23576 | A filter is a subset of th... |
| filelss 23577 | An element of a filter is ... |
| filss 23578 | A filter is closed under t... |
| filin 23579 | A filter is closed under t... |
| filtop 23580 | The underlying set belongs... |
| isfil2 23581 | Derive the standard axioms... |
| isfildlem 23582 | Lemma for ~ isfild . (Con... |
| isfild 23583 | Sufficient condition for a... |
| filfi 23584 | A filter is closed under t... |
| filinn0 23585 | The intersection of two el... |
| filintn0 23586 | A filter has the finite in... |
| filn0 23587 | The empty set is not a fil... |
| infil 23588 | The intersection of two fi... |
| snfil 23589 | A singleton is a filter. ... |
| fbasweak 23590 | A filter base on any set i... |
| snfbas 23591 | Condition for a singleton ... |
| fsubbas 23592 | A condition for a set to g... |
| fbasfip 23593 | A filter base has the fini... |
| fbunfip 23594 | A helpful lemma for showin... |
| fgval 23595 | The filter generating clas... |
| elfg 23596 | A condition for elements o... |
| ssfg 23597 | A filter base is a subset ... |
| fgss 23598 | A bigger base generates a ... |
| fgss2 23599 | A condition for a filter t... |
| fgfil 23600 | A filter generates itself.... |
| elfilss 23601 | An element belongs to a fi... |
| filfinnfr 23602 | No filter containing a fin... |
| fgcl 23603 | A generated filter is a fi... |
| fgabs 23604 | Absorption law for filter ... |
| neifil 23605 | The neighborhoods of a non... |
| filunibas 23606 | Recover the base set from ... |
| filunirn 23607 | Two ways to express a filt... |
| filconn 23608 | A filter gives rise to a c... |
| fbasrn 23609 | Given a filter on a domain... |
| filuni 23610 | The union of a nonempty se... |
| trfil1 23611 | Conditions for the trace o... |
| trfil2 23612 | Conditions for the trace o... |
| trfil3 23613 | Conditions for the trace o... |
| trfilss 23614 | If ` A ` is a member of th... |
| fgtr 23615 | If ` A ` is a member of th... |
| trfg 23616 | The trace operation and th... |
| trnei 23617 | The trace, over a set ` A ... |
| cfinfil 23618 | Relative complements of th... |
| csdfil 23619 | The set of all elements wh... |
| supfil 23620 | The supersets of a nonempt... |
| zfbas 23621 | The set of upper sets of i... |
| uzrest 23622 | The restriction of the set... |
| uzfbas 23623 | The set of upper sets of i... |
| isufil 23628 | The property of being an u... |
| ufilfil 23629 | An ultrafilter is a filter... |
| ufilss 23630 | For any subset of the base... |
| ufilb 23631 | The complement is in an ul... |
| ufilmax 23632 | Any filter finer than an u... |
| isufil2 23633 | The maximal property of an... |
| ufprim 23634 | An ultrafilter is a prime ... |
| trufil 23635 | Conditions for the trace o... |
| filssufilg 23636 | A filter is contained in s... |
| filssufil 23637 | A filter is contained in s... |
| isufl 23638 | Define the (strong) ultraf... |
| ufli 23639 | Property of a set that sat... |
| numufl 23640 | Consequence of ~ filssufil... |
| fiufl 23641 | A finite set satisfies the... |
| acufl 23642 | The axiom of choice implie... |
| ssufl 23643 | If ` Y ` is a subset of ` ... |
| ufileu 23644 | If the ultrafilter contain... |
| filufint 23645 | A filter is equal to the i... |
| uffix 23646 | Lemma for ~ fixufil and ~ ... |
| fixufil 23647 | The condition describing a... |
| uffixfr 23648 | An ultrafilter is either f... |
| uffix2 23649 | A classification of fixed ... |
| uffixsn 23650 | The singleton of the gener... |
| ufildom1 23651 | An ultrafilter is generate... |
| uffinfix 23652 | An ultrafilter containing ... |
| cfinufil 23653 | An ultrafilter is free iff... |
| ufinffr 23654 | An infinite subset is cont... |
| ufilen 23655 | Any infinite set has an ul... |
| ufildr 23656 | An ultrafilter gives rise ... |
| fin1aufil 23657 | There are no definable fre... |
| fmval 23668 | Introduce a function that ... |
| fmfil 23669 | A mapping filter is a filt... |
| fmf 23670 | Pushing-forward via a func... |
| fmss 23671 | A finer filter produces a ... |
| elfm 23672 | An element of a mapping fi... |
| elfm2 23673 | An element of a mapping fi... |
| fmfg 23674 | The image filter of a filt... |
| elfm3 23675 | An alternate formulation o... |
| imaelfm 23676 | An image of a filter eleme... |
| rnelfmlem 23677 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23678 | A condition for a filter t... |
| fmfnfmlem1 23679 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23680 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23681 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23682 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23683 | A filter finer than an ima... |
| fmufil 23684 | An image filter of an ultr... |
| fmid 23685 | The filter map applied to ... |
| fmco 23686 | Composition of image filte... |
| ufldom 23687 | The ultrafilter lemma prop... |
| flimval 23688 | The set of limit points of... |
| elflim2 23689 | The predicate "is a limit ... |
| flimtop 23690 | Reverse closure for the li... |
| flimneiss 23691 | A filter contains the neig... |
| flimnei 23692 | A filter contains all of t... |
| flimelbas 23693 | A limit point of a filter ... |
| flimfil 23694 | Reverse closure for the li... |
| flimtopon 23695 | Reverse closure for the li... |
| elflim 23696 | The predicate "is a limit ... |
| flimss2 23697 | A limit point of a filter ... |
| flimss1 23698 | A limit point of a filter ... |
| neiflim 23699 | A point is a limit point o... |
| flimopn 23700 | The condition for being a ... |
| fbflim 23701 | A condition for a filter t... |
| fbflim2 23702 | A condition for a filter b... |
| flimclsi 23703 | The convergent points of a... |
| hausflimlem 23704 | If ` A ` and ` B ` are bot... |
| hausflimi 23705 | One direction of ~ hausfli... |
| hausflim 23706 | A condition for a topology... |
| flimcf 23707 | Fineness is properly chara... |
| flimrest 23708 | The set of limit points in... |
| flimclslem 23709 | Lemma for ~ flimcls . (Co... |
| flimcls 23710 | Closure in terms of filter... |
| flimsncls 23711 | If ` A ` is a limit point ... |
| hauspwpwf1 23712 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23713 | If ` X ` is a Hausdorff sp... |
| flffval 23714 | Given a topology and a fil... |
| flfval 23715 | Given a function from a fi... |
| flfnei 23716 | The property of being a li... |
| flfneii 23717 | A neighborhood of a limit ... |
| isflf 23718 | The property of being a li... |
| flfelbas 23719 | A limit point of a functio... |
| flffbas 23720 | Limit points of a function... |
| flftg 23721 | Limit points of a function... |
| hausflf 23722 | If a function has its valu... |
| hausflf2 23723 | If a convergent function h... |
| cnpflfi 23724 | Forward direction of ~ cnp... |
| cnpflf2 23725 | ` F ` is continuous at poi... |
| cnpflf 23726 | Continuity of a function a... |
| cnflf 23727 | A function is continuous i... |
| cnflf2 23728 | A function is continuous i... |
| flfcnp 23729 | A continuous function pres... |
| lmflf 23730 | The topological limit rela... |
| txflf 23731 | Two sequences converge in ... |
| flfcnp2 23732 | The image of a convergent ... |
| fclsval 23733 | The set of all cluster poi... |
| isfcls 23734 | A cluster point of a filte... |
| fclsfil 23735 | Reverse closure for the cl... |
| fclstop 23736 | Reverse closure for the cl... |
| fclstopon 23737 | Reverse closure for the cl... |
| isfcls2 23738 | A cluster point of a filte... |
| fclsopn 23739 | Write the cluster point co... |
| fclsopni 23740 | An open neighborhood of a ... |
| fclselbas 23741 | A cluster point is in the ... |
| fclsneii 23742 | A neighborhood of a cluste... |
| fclssscls 23743 | The set of cluster points ... |
| fclsnei 23744 | Cluster points in terms of... |
| supnfcls 23745 | The filter of supersets of... |
| fclsbas 23746 | Cluster points in terms of... |
| fclsss1 23747 | A finer topology has fewer... |
| fclsss2 23748 | A finer filter has fewer c... |
| fclsrest 23749 | The set of cluster points ... |
| fclscf 23750 | Characterization of finene... |
| flimfcls 23751 | A limit point is a cluster... |
| fclsfnflim 23752 | A filter clusters at a poi... |
| flimfnfcls 23753 | A filter converges to a po... |
| fclscmpi 23754 | Forward direction of ~ fcl... |
| fclscmp 23755 | A space is compact iff eve... |
| uffclsflim 23756 | The cluster points of an u... |
| ufilcmp 23757 | A space is compact iff eve... |
| fcfval 23758 | The set of cluster points ... |
| isfcf 23759 | The property of being a cl... |
| fcfnei 23760 | The property of being a cl... |
| fcfelbas 23761 | A cluster point of a funct... |
| fcfneii 23762 | A neighborhood of a cluste... |
| flfssfcf 23763 | A limit point of a functio... |
| uffcfflf 23764 | If the domain filter is an... |
| cnpfcfi 23765 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23766 | A function ` F ` is contin... |
| cnfcf 23767 | Continuity of a function i... |
| flfcntr 23768 | A continuous function's va... |
| alexsublem 23769 | Lemma for ~ alexsub . (Co... |
| alexsub 23770 | The Alexander Subbase Theo... |
| alexsubb 23771 | Biconditional form of the ... |
| alexsubALTlem1 23772 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23773 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23774 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23775 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23776 | The Alexander Subbase Theo... |
| ptcmplem1 23777 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23778 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23779 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23780 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23781 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 23782 | Tychonoff's theorem: The ... |
| ptcmp 23783 | Tychonoff's theorem: The ... |
| cnextval 23786 | The function applying cont... |
| cnextfval 23787 | The continuous extension o... |
| cnextrel 23788 | In the general case, a con... |
| cnextfun 23789 | If the target space is Hau... |
| cnextfvval 23790 | The value of the continuou... |
| cnextf 23791 | Extension by continuity. ... |
| cnextcn 23792 | Extension by continuity. ... |
| cnextfres1 23793 | ` F ` and its extension by... |
| cnextfres 23794 | ` F ` and its extension by... |
| istmd 23799 | The predicate "is a topolo... |
| tmdmnd 23800 | A topological monoid is a ... |
| tmdtps 23801 | A topological monoid is a ... |
| istgp 23802 | The predicate "is a topolo... |
| tgpgrp 23803 | A topological group is a g... |
| tgptmd 23804 | A topological group is a t... |
| tgptps 23805 | A topological group is a t... |
| tmdtopon 23806 | The topology of a topologi... |
| tgptopon 23807 | The topology of a topologi... |
| tmdcn 23808 | In a topological monoid, t... |
| tgpcn 23809 | In a topological group, th... |
| tgpinv 23810 | In a topological group, th... |
| grpinvhmeo 23811 | The inverse function in a ... |
| cnmpt1plusg 23812 | Continuity of the group su... |
| cnmpt2plusg 23813 | Continuity of the group su... |
| tmdcn2 23814 | Write out the definition o... |
| tgpsubcn 23815 | In a topological group, th... |
| istgp2 23816 | A group with a topology is... |
| tmdmulg 23817 | In a topological monoid, t... |
| tgpmulg 23818 | In a topological group, th... |
| tgpmulg2 23819 | In a topological monoid, t... |
| tmdgsum 23820 | In a topological monoid, t... |
| tmdgsum2 23821 | For any neighborhood ` U `... |
| oppgtmd 23822 | The opposite of a topologi... |
| oppgtgp 23823 | The opposite of a topologi... |
| distgp 23824 | Any group equipped with th... |
| indistgp 23825 | Any group equipped with th... |
| efmndtmd 23826 | The monoid of endofunction... |
| tmdlactcn 23827 | The left group action of e... |
| tgplacthmeo 23828 | The left group action of e... |
| submtmd 23829 | A submonoid of a topologic... |
| subgtgp 23830 | A subgroup of a topologica... |
| symgtgp 23831 | The symmetric group is a t... |
| subgntr 23832 | A subgroup of a topologica... |
| opnsubg 23833 | An open subgroup of a topo... |
| clssubg 23834 | The closure of a subgroup ... |
| clsnsg 23835 | The closure of a normal su... |
| cldsubg 23836 | A subgroup of finite index... |
| tgpconncompeqg 23837 | The connected component co... |
| tgpconncomp 23838 | The identity component, th... |
| tgpconncompss 23839 | The identity component is ... |
| ghmcnp 23840 | A group homomorphism on to... |
| snclseqg 23841 | The coset of the closure o... |
| tgphaus 23842 | A topological group is Hau... |
| tgpt1 23843 | Hausdorff and T1 are equiv... |
| tgpt0 23844 | Hausdorff and T0 are equiv... |
| qustgpopn 23845 | A quotient map in a topolo... |
| qustgplem 23846 | Lemma for ~ qustgp . (Con... |
| qustgp 23847 | The quotient of a topologi... |
| qustgphaus 23848 | The quotient of a topologi... |
| prdstmdd 23849 | The product of a family of... |
| prdstgpd 23850 | The product of a family of... |
| tsmsfbas 23853 | The collection of all sets... |
| tsmslem1 23854 | The finite partial sums of... |
| tsmsval2 23855 | Definition of the topologi... |
| tsmsval 23856 | Definition of the topologi... |
| tsmspropd 23857 | The group sum depends only... |
| eltsms 23858 | The property of being a su... |
| tsmsi 23859 | The property of being a su... |
| tsmscl 23860 | A sum in a topological gro... |
| haustsms 23861 | In a Hausdorff topological... |
| haustsms2 23862 | In a Hausdorff topological... |
| tsmscls 23863 | One half of ~ tgptsmscls ,... |
| tsmsgsum 23864 | The convergent points of a... |
| tsmsid 23865 | If a sum is finite, the us... |
| haustsmsid 23866 | In a Hausdorff topological... |
| tsms0 23867 | The sum of zero is zero. ... |
| tsmssubm 23868 | Evaluate an infinite group... |
| tsmsres 23869 | Extend an infinite group s... |
| tsmsf1o 23870 | Re-index an infinite group... |
| tsmsmhm 23871 | Apply a continuous group h... |
| tsmsadd 23872 | The sum of two infinite gr... |
| tsmsinv 23873 | Inverse of an infinite gro... |
| tsmssub 23874 | The difference of two infi... |
| tgptsmscls 23875 | A sum in a topological gro... |
| tgptsmscld 23876 | The set of limit points to... |
| tsmssplit 23877 | Split a topological group ... |
| tsmsxplem1 23878 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 23879 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 23880 | Write a sum over a two-dim... |
| istrg 23889 | Express the predicate " ` ... |
| trgtmd 23890 | The multiplicative monoid ... |
| istdrg 23891 | Express the predicate " ` ... |
| tdrgunit 23892 | The unit group of a topolo... |
| trgtgp 23893 | A topological ring is a to... |
| trgtmd2 23894 | A topological ring is a to... |
| trgtps 23895 | A topological ring is a to... |
| trgring 23896 | A topological ring is a ri... |
| trggrp 23897 | A topological ring is a gr... |
| tdrgtrg 23898 | A topological division rin... |
| tdrgdrng 23899 | A topological division rin... |
| tdrgring 23900 | A topological division rin... |
| tdrgtmd 23901 | A topological division rin... |
| tdrgtps 23902 | A topological division rin... |
| istdrg2 23903 | A topological-ring divisio... |
| mulrcn 23904 | The functionalization of t... |
| invrcn2 23905 | The multiplicative inverse... |
| invrcn 23906 | The multiplicative inverse... |
| cnmpt1mulr 23907 | Continuity of ring multipl... |
| cnmpt2mulr 23908 | Continuity of ring multipl... |
| dvrcn 23909 | The division function is c... |
| istlm 23910 | The predicate " ` W ` is a... |
| vscacn 23911 | The scalar multiplication ... |
| tlmtmd 23912 | A topological module is a ... |
| tlmtps 23913 | A topological module is a ... |
| tlmlmod 23914 | A topological module is a ... |
| tlmtrg 23915 | The scalar ring of a topol... |
| tlmscatps 23916 | The scalar ring of a topol... |
| istvc 23917 | A topological vector space... |
| tvctdrg 23918 | The scalar field of a topo... |
| cnmpt1vsca 23919 | Continuity of scalar multi... |
| cnmpt2vsca 23920 | Continuity of scalar multi... |
| tlmtgp 23921 | A topological vector space... |
| tvctlm 23922 | A topological vector space... |
| tvclmod 23923 | A topological vector space... |
| tvclvec 23924 | A topological vector space... |
| ustfn 23927 | The defined uniform struct... |
| ustval 23928 | The class of all uniform s... |
| isust 23929 | The predicate " ` U ` is a... |
| ustssxp 23930 | Entourages are subsets of ... |
| ustssel 23931 | A uniform structure is upw... |
| ustbasel 23932 | The full set is always an ... |
| ustincl 23933 | A uniform structure is clo... |
| ustdiag 23934 | The diagonal set is includ... |
| ustinvel 23935 | If ` V ` is an entourage, ... |
| ustexhalf 23936 | For each entourage ` V ` t... |
| ustrel 23937 | The elements of uniform st... |
| ustfilxp 23938 | A uniform structure on a n... |
| ustne0 23939 | A uniform structure cannot... |
| ustssco 23940 | In an uniform structure, a... |
| ustexsym 23941 | In an uniform structure, f... |
| ustex2sym 23942 | In an uniform structure, f... |
| ustex3sym 23943 | In an uniform structure, f... |
| ustref 23944 | Any element of the base se... |
| ust0 23945 | The unique uniform structu... |
| ustn0 23946 | The empty set is not an un... |
| ustund 23947 | If two intersecting sets `... |
| ustelimasn 23948 | Any point ` A ` is near en... |
| ustneism 23949 | For a point ` A ` in ` X `... |
| elrnustOLD 23950 | Obsolete version of ~ elfv... |
| ustbas2 23951 | Second direction for ~ ust... |
| ustuni 23952 | The set union of a uniform... |
| ustbas 23953 | Recover the base of an uni... |
| ustimasn 23954 | Lemma for ~ ustuqtop . (C... |
| trust 23955 | The trace of a uniform str... |
| utopval 23958 | The topology induced by a ... |
| elutop 23959 | Open sets in the topology ... |
| utoptop 23960 | The topology induced by a ... |
| utopbas 23961 | The base of the topology i... |
| utoptopon 23962 | Topology induced by a unif... |
| restutop 23963 | Restriction of a topology ... |
| restutopopn 23964 | The restriction of the top... |
| ustuqtoplem 23965 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 23966 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 23967 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 23968 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 23969 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 23970 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 23971 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 23972 | For a given uniform struct... |
| utopsnneiplem 23973 | The neighborhoods of a poi... |
| utopsnneip 23974 | The neighborhoods of a poi... |
| utopsnnei 23975 | Images of singletons by en... |
| utop2nei 23976 | For any symmetrical entour... |
| utop3cls 23977 | Relation between a topolog... |
| utopreg 23978 | All Hausdorff uniform spac... |
| ussval 23985 | The uniform structure on u... |
| ussid 23986 | In case the base of the ` ... |
| isusp 23987 | The predicate ` W ` is a u... |
| ressuss 23988 | Value of the uniform struc... |
| ressust 23989 | The uniform structure of a... |
| ressusp 23990 | The restriction of a unifo... |
| tusval 23991 | The value of the uniform s... |
| tuslem 23992 | Lemma for ~ tusbas , ~ tus... |
| tuslemOLD 23993 | Obsolete proof of ~ tuslem... |
| tusbas 23994 | The base set of a construc... |
| tusunif 23995 | The uniform structure of a... |
| tususs 23996 | The uniform structure of a... |
| tustopn 23997 | The topology induced by a ... |
| tususp 23998 | A constructed uniform spac... |
| tustps 23999 | A constructed uniform spac... |
| uspreg 24000 | If a uniform space is Haus... |
| ucnval 24003 | The set of all uniformly c... |
| isucn 24004 | The predicate " ` F ` is a... |
| isucn2 24005 | The predicate " ` F ` is a... |
| ucnimalem 24006 | Reformulate the ` G ` func... |
| ucnima 24007 | An equivalent statement of... |
| ucnprima 24008 | The preimage by a uniforml... |
| iducn 24009 | The identity is uniformly ... |
| cstucnd 24010 | A constant function is uni... |
| ucncn 24011 | Uniform continuity implies... |
| iscfilu 24014 | The predicate " ` F ` is a... |
| cfilufbas 24015 | A Cauchy filter base is a ... |
| cfiluexsm 24016 | For a Cauchy filter base a... |
| fmucndlem 24017 | Lemma for ~ fmucnd . (Con... |
| fmucnd 24018 | The image of a Cauchy filt... |
| cfilufg 24019 | The filter generated by a ... |
| trcfilu 24020 | Condition for the trace of... |
| cfiluweak 24021 | A Cauchy filter base is al... |
| neipcfilu 24022 | In an uniform space, a nei... |
| iscusp 24025 | The predicate " ` W ` is a... |
| cuspusp 24026 | A complete uniform space i... |
| cuspcvg 24027 | In a complete uniform spac... |
| iscusp2 24028 | The predicate " ` W ` is a... |
| cnextucn 24029 | Extension by continuity. ... |
| ucnextcn 24030 | Extension by continuity. ... |
| ispsmet 24031 | Express the predicate " ` ... |
| psmetdmdm 24032 | Recover the base set from ... |
| psmetf 24033 | The distance function of a... |
| psmetcl 24034 | Closure of the distance fu... |
| psmet0 24035 | The distance function of a... |
| psmettri2 24036 | Triangle inequality for th... |
| psmetsym 24037 | The distance function of a... |
| psmettri 24038 | Triangle inequality for th... |
| psmetge0 24039 | The distance function of a... |
| psmetxrge0 24040 | The distance function of a... |
| psmetres2 24041 | Restriction of a pseudomet... |
| psmetlecl 24042 | Real closure of an extende... |
| distspace 24043 | A set ` X ` together with ... |
| ismet 24050 | Express the predicate " ` ... |
| isxmet 24051 | Express the predicate " ` ... |
| ismeti 24052 | Properties that determine ... |
| isxmetd 24053 | Properties that determine ... |
| isxmet2d 24054 | It is safe to only require... |
| metflem 24055 | Lemma for ~ metf and other... |
| xmetf 24056 | Mapping of the distance fu... |
| metf 24057 | Mapping of the distance fu... |
| xmetcl 24058 | Closure of the distance fu... |
| metcl 24059 | Closure of the distance fu... |
| ismet2 24060 | An extended metric is a me... |
| metxmet 24061 | A metric is an extended me... |
| xmetdmdm 24062 | Recover the base set from ... |
| metdmdm 24063 | Recover the base set from ... |
| xmetunirn 24064 | Two ways to express an ext... |
| xmeteq0 24065 | The value of an extended m... |
| meteq0 24066 | The value of a metric is z... |
| xmettri2 24067 | Triangle inequality for th... |
| mettri2 24068 | Triangle inequality for th... |
| xmet0 24069 | The distance function of a... |
| met0 24070 | The distance function of a... |
| xmetge0 24071 | The distance function of a... |
| metge0 24072 | The distance function of a... |
| xmetlecl 24073 | Real closure of an extende... |
| xmetsym 24074 | The distance function of a... |
| xmetpsmet 24075 | An extended metric is a ps... |
| xmettpos 24076 | The distance function of a... |
| metsym 24077 | The distance function of a... |
| xmettri 24078 | Triangle inequality for th... |
| mettri 24079 | Triangle inequality for th... |
| xmettri3 24080 | Triangle inequality for th... |
| mettri3 24081 | Triangle inequality for th... |
| xmetrtri 24082 | One half of the reverse tr... |
| xmetrtri2 24083 | The reverse triangle inequ... |
| metrtri 24084 | Reverse triangle inequalit... |
| xmetgt0 24085 | The distance function of a... |
| metgt0 24086 | The distance function of a... |
| metn0 24087 | A metric space is nonempty... |
| xmetres2 24088 | Restriction of an extended... |
| metreslem 24089 | Lemma for ~ metres . (Con... |
| metres2 24090 | Lemma for ~ metres . (Con... |
| xmetres 24091 | A restriction of an extend... |
| metres 24092 | A restriction of a metric ... |
| 0met 24093 | The empty metric. (Contri... |
| prdsdsf 24094 | The product metric is a fu... |
| prdsxmetlem 24095 | The product metric is an e... |
| prdsxmet 24096 | The product metric is an e... |
| prdsmet 24097 | The product metric is a me... |
| ressprdsds 24098 | Restriction of a product m... |
| resspwsds 24099 | Restriction of a power met... |
| imasdsf1olem 24100 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 24101 | The distance function is t... |
| imasf1oxmet 24102 | The image of an extended m... |
| imasf1omet 24103 | The image of a metric is a... |
| xpsdsfn 24104 | Closure of the metric in a... |
| xpsdsfn2 24105 | Closure of the metric in a... |
| xpsxmetlem 24106 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 24107 | A product metric of extend... |
| xpsdsval 24108 | Value of the metric in a b... |
| xpsmet 24109 | The direct product of two ... |
| blfvalps 24110 | The value of the ball func... |
| blfval 24111 | The value of the ball func... |
| blvalps 24112 | The ball around a point ` ... |
| blval 24113 | The ball around a point ` ... |
| elblps 24114 | Membership in a ball. (Co... |
| elbl 24115 | Membership in a ball. (Co... |
| elbl2ps 24116 | Membership in a ball. (Co... |
| elbl2 24117 | Membership in a ball. (Co... |
| elbl3ps 24118 | Membership in a ball, with... |
| elbl3 24119 | Membership in a ball, with... |
| blcomps 24120 | Commute the arguments to t... |
| blcom 24121 | Commute the arguments to t... |
| xblpnfps 24122 | The infinity ball in an ex... |
| xblpnf 24123 | The infinity ball in an ex... |
| blpnf 24124 | The infinity ball in a sta... |
| bldisj 24125 | Two balls are disjoint if ... |
| blgt0 24126 | A nonempty ball implies th... |
| bl2in 24127 | Two balls are disjoint if ... |
| xblss2ps 24128 | One ball is contained in a... |
| xblss2 24129 | One ball is contained in a... |
| blss2ps 24130 | One ball is contained in a... |
| blss2 24131 | One ball is contained in a... |
| blhalf 24132 | A ball of radius ` R / 2 `... |
| blfps 24133 | Mapping of a ball. (Contr... |
| blf 24134 | Mapping of a ball. (Contr... |
| blrnps 24135 | Membership in the range of... |
| blrn 24136 | Membership in the range of... |
| xblcntrps 24137 | A ball contains its center... |
| xblcntr 24138 | A ball contains its center... |
| blcntrps 24139 | A ball contains its center... |
| blcntr 24140 | A ball contains its center... |
| xbln0 24141 | A ball is nonempty iff the... |
| bln0 24142 | A ball is not empty. (Con... |
| blelrnps 24143 | A ball belongs to the set ... |
| blelrn 24144 | A ball belongs to the set ... |
| blssm 24145 | A ball is a subset of the ... |
| unirnblps 24146 | The union of the set of ba... |
| unirnbl 24147 | The union of the set of ba... |
| blin 24148 | The intersection of two ba... |
| ssblps 24149 | The size of a ball increas... |
| ssbl 24150 | The size of a ball increas... |
| blssps 24151 | Any point ` P ` in a ball ... |
| blss 24152 | Any point ` P ` in a ball ... |
| blssexps 24153 | Two ways to express the ex... |
| blssex 24154 | Two ways to express the ex... |
| ssblex 24155 | A nested ball exists whose... |
| blin2 24156 | Given any two balls and a ... |
| blbas 24157 | The balls of a metric spac... |
| blres 24158 | A ball in a restricted met... |
| xmeterval 24159 | Value of the "finitely sep... |
| xmeter 24160 | The "finitely separated" r... |
| xmetec 24161 | The equivalence classes un... |
| blssec 24162 | A ball centered at ` P ` i... |
| blpnfctr 24163 | The infinity ball in an ex... |
| xmetresbl 24164 | An extended metric restric... |
| mopnval 24165 | An open set is a subset of... |
| mopntopon 24166 | The set of open sets of a ... |
| mopntop 24167 | The set of open sets of a ... |
| mopnuni 24168 | The union of all open sets... |
| elmopn 24169 | The defining property of a... |
| mopnfss 24170 | The family of open sets of... |
| mopnm 24171 | The base set of a metric s... |
| elmopn2 24172 | A defining property of an ... |
| mopnss 24173 | An open set of a metric sp... |
| isxms 24174 | Express the predicate " ` ... |
| isxms2 24175 | Express the predicate " ` ... |
| isms 24176 | Express the predicate " ` ... |
| isms2 24177 | Express the predicate " ` ... |
| xmstopn 24178 | The topology component of ... |
| mstopn 24179 | The topology component of ... |
| xmstps 24180 | An extended metric space i... |
| msxms 24181 | A metric space is an exten... |
| mstps 24182 | A metric space is a topolo... |
| xmsxmet 24183 | The distance function, sui... |
| msmet 24184 | The distance function, sui... |
| msf 24185 | The distance function of a... |
| xmsxmet2 24186 | The distance function, sui... |
| msmet2 24187 | The distance function, sui... |
| mscl 24188 | Closure of the distance fu... |
| xmscl 24189 | Closure of the distance fu... |
| xmsge0 24190 | The distance function in a... |
| xmseq0 24191 | The distance between two p... |
| xmssym 24192 | The distance function in a... |
| xmstri2 24193 | Triangle inequality for th... |
| mstri2 24194 | Triangle inequality for th... |
| xmstri 24195 | Triangle inequality for th... |
| mstri 24196 | Triangle inequality for th... |
| xmstri3 24197 | Triangle inequality for th... |
| mstri3 24198 | Triangle inequality for th... |
| msrtri 24199 | Reverse triangle inequalit... |
| xmspropd 24200 | Property deduction for an ... |
| mspropd 24201 | Property deduction for a m... |
| setsmsbas 24202 | The base set of a construc... |
| setsmsbasOLD 24203 | Obsolete proof of ~ setsms... |
| setsmsds 24204 | The distance function of a... |
| setsmsdsOLD 24205 | Obsolete proof of ~ setsms... |
| setsmstset 24206 | The topology of a construc... |
| setsmstopn 24207 | The topology of a construc... |
| setsxms 24208 | The constructed metric spa... |
| setsms 24209 | The constructed metric spa... |
| tmsval 24210 | For any metric there is an... |
| tmslem 24211 | Lemma for ~ tmsbas , ~ tms... |
| tmslemOLD 24212 | Obsolete version of ~ tmsl... |
| tmsbas 24213 | The base set of a construc... |
| tmsds 24214 | The metric of a constructe... |
| tmstopn 24215 | The topology of a construc... |
| tmsxms 24216 | The constructed metric spa... |
| tmsms 24217 | The constructed metric spa... |
| imasf1obl 24218 | The image of a metric spac... |
| imasf1oxms 24219 | The image of a metric spac... |
| imasf1oms 24220 | The image of a metric spac... |
| prdsbl 24221 | A ball in the product metr... |
| mopni 24222 | An open set of a metric sp... |
| mopni2 24223 | An open set of a metric sp... |
| mopni3 24224 | An open set of a metric sp... |
| blssopn 24225 | The balls of a metric spac... |
| unimopn 24226 | The union of a collection ... |
| mopnin 24227 | The intersection of two op... |
| mopn0 24228 | The empty set is an open s... |
| rnblopn 24229 | A ball of a metric space i... |
| blopn 24230 | A ball of a metric space i... |
| neibl 24231 | The neighborhoods around a... |
| blnei 24232 | A ball around a point is a... |
| lpbl 24233 | Every ball around a limit ... |
| blsscls2 24234 | A smaller closed ball is c... |
| blcld 24235 | A "closed ball" in a metri... |
| blcls 24236 | The closure of an open bal... |
| blsscls 24237 | If two concentric balls ha... |
| metss 24238 | Two ways of saying that me... |
| metequiv 24239 | Two ways of saying that tw... |
| metequiv2 24240 | If there is a sequence of ... |
| metss2lem 24241 | Lemma for ~ metss2 . (Con... |
| metss2 24242 | If the metric ` D ` is "st... |
| comet 24243 | The composition of an exte... |
| stdbdmetval 24244 | Value of the standard boun... |
| stdbdxmet 24245 | The standard bounded metri... |
| stdbdmet 24246 | The standard bounded metri... |
| stdbdbl 24247 | The standard bounded metri... |
| stdbdmopn 24248 | The standard bounded metri... |
| mopnex 24249 | The topology generated by ... |
| methaus 24250 | The topology generated by ... |
| met1stc 24251 | The topology generated by ... |
| met2ndci 24252 | A separable metric space (... |
| met2ndc 24253 | A metric space is second-c... |
| metrest 24254 | Two alternate formulations... |
| ressxms 24255 | The restriction of a metri... |
| ressms 24256 | The restriction of a metri... |
| prdsmslem1 24257 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 24258 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 24259 | Lemma for ~ prdsxms . The... |
| prdsxms 24260 | The indexed product struct... |
| prdsms 24261 | The indexed product struct... |
| pwsxms 24262 | A power of an extended met... |
| pwsms 24263 | A power of a metric space ... |
| xpsxms 24264 | A binary product of metric... |
| xpsms 24265 | A binary product of metric... |
| tmsxps 24266 | Express the product of two... |
| tmsxpsmopn 24267 | Express the product of two... |
| tmsxpsval 24268 | Value of the product of tw... |
| tmsxpsval2 24269 | Value of the product of tw... |
| metcnp3 24270 | Two ways to express that `... |
| metcnp 24271 | Two ways to say a mapping ... |
| metcnp2 24272 | Two ways to say a mapping ... |
| metcn 24273 | Two ways to say a mapping ... |
| metcnpi 24274 | Epsilon-delta property of ... |
| metcnpi2 24275 | Epsilon-delta property of ... |
| metcnpi3 24276 | Epsilon-delta property of ... |
| txmetcnp 24277 | Continuity of a binary ope... |
| txmetcn 24278 | Continuity of a binary ope... |
| metuval 24279 | Value of the uniform struc... |
| metustel 24280 | Define a filter base ` F `... |
| metustss 24281 | Range of the elements of t... |
| metustrel 24282 | Elements of the filter bas... |
| metustto 24283 | Any two elements of the fi... |
| metustid 24284 | The identity diagonal is i... |
| metustsym 24285 | Elements of the filter bas... |
| metustexhalf 24286 | For any element ` A ` of t... |
| metustfbas 24287 | The filter base generated ... |
| metust 24288 | The uniform structure gene... |
| cfilucfil 24289 | Given a metric ` D ` and a... |
| metuust 24290 | The uniform structure gene... |
| cfilucfil2 24291 | Given a metric ` D ` and a... |
| blval2 24292 | The ball around a point ` ... |
| elbl4 24293 | Membership in a ball, alte... |
| metuel 24294 | Elementhood in the uniform... |
| metuel2 24295 | Elementhood in the uniform... |
| metustbl 24296 | The "section" image of an ... |
| psmetutop 24297 | The topology induced by a ... |
| xmetutop 24298 | The topology induced by a ... |
| xmsusp 24299 | If the uniform set of a me... |
| restmetu 24300 | The uniform structure gene... |
| metucn 24301 | Uniform continuity in metr... |
| dscmet 24302 | The discrete metric on any... |
| dscopn 24303 | The discrete metric genera... |
| nrmmetd 24304 | Show that a group norm gen... |
| abvmet 24305 | An absolute value ` F ` ge... |
| nmfval 24318 | The value of the norm func... |
| nmval 24319 | The value of the norm as t... |
| nmfval0 24320 | The value of the norm func... |
| nmfval2 24321 | The value of the norm func... |
| nmval2 24322 | The value of the norm on a... |
| nmf2 24323 | The norm on a metric group... |
| nmpropd 24324 | Weak property deduction fo... |
| nmpropd2 24325 | Strong property deduction ... |
| isngp 24326 | The property of being a no... |
| isngp2 24327 | The property of being a no... |
| isngp3 24328 | The property of being a no... |
| ngpgrp 24329 | A normed group is a group.... |
| ngpms 24330 | A normed group is a metric... |
| ngpxms 24331 | A normed group is an exten... |
| ngptps 24332 | A normed group is a topolo... |
| ngpmet 24333 | The (induced) metric of a ... |
| ngpds 24334 | Value of the distance func... |
| ngpdsr 24335 | Value of the distance func... |
| ngpds2 24336 | Write the distance between... |
| ngpds2r 24337 | Write the distance between... |
| ngpds3 24338 | Write the distance between... |
| ngpds3r 24339 | Write the distance between... |
| ngprcan 24340 | Cancel right addition insi... |
| ngplcan 24341 | Cancel left addition insid... |
| isngp4 24342 | Express the property of be... |
| ngpinvds 24343 | Two elements are the same ... |
| ngpsubcan 24344 | Cancel right subtraction i... |
| nmf 24345 | The norm on a normed group... |
| nmcl 24346 | The norm of a normed group... |
| nmge0 24347 | The norm of a normed group... |
| nmeq0 24348 | The identity is the only e... |
| nmne0 24349 | The norm of a nonzero elem... |
| nmrpcl 24350 | The norm of a nonzero elem... |
| nminv 24351 | The norm of a negated elem... |
| nmmtri 24352 | The triangle inequality fo... |
| nmsub 24353 | The norm of the difference... |
| nmrtri 24354 | Reverse triangle inequalit... |
| nm2dif 24355 | Inequality for the differe... |
| nmtri 24356 | The triangle inequality fo... |
| nmtri2 24357 | Triangle inequality for th... |
| ngpi 24358 | The properties of a normed... |
| nm0 24359 | Norm of the identity eleme... |
| nmgt0 24360 | The norm of a nonzero elem... |
| sgrim 24361 | The induced metric on a su... |
| sgrimval 24362 | The induced metric on a su... |
| subgnm 24363 | The norm in a subgroup. (... |
| subgnm2 24364 | A substructure assigns the... |
| subgngp 24365 | A normed group restricted ... |
| ngptgp 24366 | A normed abelian group is ... |
| ngppropd 24367 | Property deduction for a n... |
| reldmtng 24368 | The function ` toNrmGrp ` ... |
| tngval 24369 | Value of the function whic... |
| tnglem 24370 | Lemma for ~ tngbas and sim... |
| tnglemOLD 24371 | Obsolete version of ~ tngl... |
| tngbas 24372 | The base set of a structur... |
| tngbasOLD 24373 | Obsolete proof of ~ tngbas... |
| tngplusg 24374 | The group addition of a st... |
| tngplusgOLD 24375 | Obsolete proof of ~ tngplu... |
| tng0 24376 | The group identity of a st... |
| tngmulr 24377 | The ring multiplication of... |
| tngmulrOLD 24378 | Obsolete proof of ~ tngmul... |
| tngsca 24379 | The scalar ring of a struc... |
| tngscaOLD 24380 | Obsolete proof of ~ tngsca... |
| tngvsca 24381 | The scalar multiplication ... |
| tngvscaOLD 24382 | Obsolete proof of ~ tngvsc... |
| tngip 24383 | The inner product operatio... |
| tngipOLD 24384 | Obsolete proof of ~ tngip ... |
| tngds 24385 | The metric function of a s... |
| tngdsOLD 24386 | Obsolete proof of ~ tngds ... |
| tngtset 24387 | The topology generated by ... |
| tngtopn 24388 | The topology generated by ... |
| tngnm 24389 | The topology generated by ... |
| tngngp2 24390 | A norm turns a group into ... |
| tngngpd 24391 | Derive the axioms for a no... |
| tngngp 24392 | Derive the axioms for a no... |
| tnggrpr 24393 | If a structure equipped wi... |
| tngngp3 24394 | Alternate definition of a ... |
| nrmtngdist 24395 | The augmentation of a norm... |
| nrmtngnrm 24396 | The augmentation of a norm... |
| tngngpim 24397 | The induced metric of a no... |
| isnrg 24398 | A normed ring is a ring wi... |
| nrgabv 24399 | The norm of a normed ring ... |
| nrgngp 24400 | A normed ring is a normed ... |
| nrgring 24401 | A normed ring is a ring. ... |
| nmmul 24402 | The norm of a product in a... |
| nrgdsdi 24403 | Distribute a distance calc... |
| nrgdsdir 24404 | Distribute a distance calc... |
| nm1 24405 | The norm of one in a nonze... |
| unitnmn0 24406 | The norm of a unit is nonz... |
| nminvr 24407 | The norm of an inverse in ... |
| nmdvr 24408 | The norm of a division in ... |
| nrgdomn 24409 | A nonzero normed ring is a... |
| nrgtgp 24410 | A normed ring is a topolog... |
| subrgnrg 24411 | A normed ring restricted t... |
| tngnrg 24412 | Given any absolute value o... |
| isnlm 24413 | A normed (left) module is ... |
| nmvs 24414 | Defining property of a nor... |
| nlmngp 24415 | A normed module is a norme... |
| nlmlmod 24416 | A normed module is a left ... |
| nlmnrg 24417 | The scalar component of a ... |
| nlmngp2 24418 | The scalar component of a ... |
| nlmdsdi 24419 | Distribute a distance calc... |
| nlmdsdir 24420 | Distribute a distance calc... |
| nlmmul0or 24421 | If a scalar product is zer... |
| sranlm 24422 | The subring algebra over a... |
| nlmvscnlem2 24423 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24424 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24425 | The scalar multiplication ... |
| rlmnlm 24426 | The ring module over a nor... |
| rlmnm 24427 | The norm function in the r... |
| nrgtrg 24428 | A normed ring is a topolog... |
| nrginvrcnlem 24429 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24430 | The ring inverse function ... |
| nrgtdrg 24431 | A normed division ring is ... |
| nlmtlm 24432 | A normed module is a topol... |
| isnvc 24433 | A normed vector space is j... |
| nvcnlm 24434 | A normed vector space is a... |
| nvclvec 24435 | A normed vector space is a... |
| nvclmod 24436 | A normed vector space is a... |
| isnvc2 24437 | A normed vector space is j... |
| nvctvc 24438 | A normed vector space is a... |
| lssnlm 24439 | A subspace of a normed mod... |
| lssnvc 24440 | A subspace of a normed vec... |
| rlmnvc 24441 | The ring module over a nor... |
| ngpocelbl 24442 | Membership of an off-cente... |
| nmoffn 24449 | The function producing ope... |
| reldmnghm 24450 | Lemma for normed group hom... |
| reldmnmhm 24451 | Lemma for module homomorph... |
| nmofval 24452 | Value of the operator norm... |
| nmoval 24453 | Value of the operator norm... |
| nmogelb 24454 | Property of the operator n... |
| nmolb 24455 | Any upper bound on the val... |
| nmolb2d 24456 | Any upper bound on the val... |
| nmof 24457 | The operator norm is a fun... |
| nmocl 24458 | The operator norm of an op... |
| nmoge0 24459 | The operator norm of an op... |
| nghmfval 24460 | A normed group homomorphis... |
| isnghm 24461 | A normed group homomorphis... |
| isnghm2 24462 | A normed group homomorphis... |
| isnghm3 24463 | A normed group homomorphis... |
| bddnghm 24464 | A bounded group homomorphi... |
| nghmcl 24465 | A normed group homomorphis... |
| nmoi 24466 | The operator norm achieves... |
| nmoix 24467 | The operator norm is a bou... |
| nmoi2 24468 | The operator norm is a bou... |
| nmoleub 24469 | The operator norm, defined... |
| nghmrcl1 24470 | Reverse closure for a norm... |
| nghmrcl2 24471 | Reverse closure for a norm... |
| nghmghm 24472 | A normed group homomorphis... |
| nmo0 24473 | The operator norm of the z... |
| nmoeq0 24474 | The operator norm is zero ... |
| nmoco 24475 | An upper bound on the oper... |
| nghmco 24476 | The composition of normed ... |
| nmotri 24477 | Triangle inequality for th... |
| nghmplusg 24478 | The sum of two bounded lin... |
| 0nghm 24479 | The zero operator is a nor... |
| nmoid 24480 | The operator norm of the i... |
| idnghm 24481 | The identity operator is a... |
| nmods 24482 | Upper bound for the distan... |
| nghmcn 24483 | A normed group homomorphis... |
| isnmhm 24484 | A normed module homomorphi... |
| nmhmrcl1 24485 | Reverse closure for a norm... |
| nmhmrcl2 24486 | Reverse closure for a norm... |
| nmhmlmhm 24487 | A normed module homomorphi... |
| nmhmnghm 24488 | A normed module homomorphi... |
| nmhmghm 24489 | A normed module homomorphi... |
| isnmhm2 24490 | A normed module homomorphi... |
| nmhmcl 24491 | A normed module homomorphi... |
| idnmhm 24492 | The identity operator is a... |
| 0nmhm 24493 | The zero operator is a bou... |
| nmhmco 24494 | The composition of bounded... |
| nmhmplusg 24495 | The sum of two bounded lin... |
| qtopbaslem 24496 | The set of open intervals ... |
| qtopbas 24497 | The set of open intervals ... |
| retopbas 24498 | A basis for the standard t... |
| retop 24499 | The standard topology on t... |
| uniretop 24500 | The underlying set of the ... |
| retopon 24501 | The standard topology on t... |
| retps 24502 | The standard topological s... |
| iooretop 24503 | Open intervals are open se... |
| icccld 24504 | Closed intervals are close... |
| icopnfcld 24505 | Right-unbounded closed int... |
| iocmnfcld 24506 | Left-unbounded closed inte... |
| qdensere 24507 | ` QQ ` is dense in the sta... |
| cnmetdval 24508 | Value of the distance func... |
| cnmet 24509 | The absolute value metric ... |
| cnxmet 24510 | The absolute value metric ... |
| cnbl0 24511 | Two ways to write the open... |
| cnblcld 24512 | Two ways to write the clos... |
| cnfldms 24513 | The complex number field i... |
| cnfldxms 24514 | The complex number field i... |
| cnfldtps 24515 | The complex number field i... |
| cnfldnm 24516 | The norm of the field of c... |
| cnngp 24517 | The complex numbers form a... |
| cnnrg 24518 | The complex numbers form a... |
| cnfldtopn 24519 | The topology of the comple... |
| cnfldtopon 24520 | The topology of the comple... |
| cnfldtop 24521 | The topology of the comple... |
| cnfldhaus 24522 | The topology of the comple... |
| unicntop 24523 | The underlying set of the ... |
| cnopn 24524 | The set of complex numbers... |
| zringnrg 24525 | The ring of integers is a ... |
| remetdval 24526 | Value of the distance func... |
| remet 24527 | The absolute value metric ... |
| rexmet 24528 | The absolute value metric ... |
| bl2ioo 24529 | A ball in terms of an open... |
| ioo2bl 24530 | An open interval of reals ... |
| ioo2blex 24531 | An open interval of reals ... |
| blssioo 24532 | The balls of the standard ... |
| tgioo 24533 | The topology generated by ... |
| qdensere2 24534 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24535 | An open ball in the comple... |
| rehaus 24536 | The standard topology on t... |
| tgqioo 24537 | The topology generated by ... |
| re2ndc 24538 | The standard topology on t... |
| resubmet 24539 | The subspace topology indu... |
| tgioo2 24540 | The standard topology on t... |
| rerest 24541 | The subspace topology indu... |
| tgioo3 24542 | The standard topology on t... |
| xrtgioo 24543 | The topology on the extend... |
| xrrest 24544 | The subspace topology indu... |
| xrrest2 24545 | The subspace topology indu... |
| xrsxmet 24546 | The metric on the extended... |
| xrsdsre 24547 | The metric on the extended... |
| xrsblre 24548 | Any ball of the metric of ... |
| xrsmopn 24549 | The metric on the extended... |
| zcld 24550 | The integers are a closed ... |
| recld2 24551 | The real numbers are a clo... |
| zcld2 24552 | The integers are a closed ... |
| zdis 24553 | The integers are a discret... |
| sszcld 24554 | Every subset of the intege... |
| reperflem 24555 | A subset of the real numbe... |
| reperf 24556 | The real numbers are a per... |
| cnperf 24557 | The complex numbers are a ... |
| iccntr 24558 | The interior of a closed i... |
| icccmplem1 24559 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24560 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24561 | Lemma for ~ icccmp . (Con... |
| icccmp 24562 | A closed interval in ` RR ... |
| reconnlem1 24563 | Lemma for ~ reconn . Conn... |
| reconnlem2 24564 | Lemma for ~ reconn . (Con... |
| reconn 24565 | A subset of the reals is c... |
| retopconn 24566 | Corollary of ~ reconn . T... |
| iccconn 24567 | A closed interval is conne... |
| opnreen 24568 | Every nonempty open set is... |
| rectbntr0 24569 | A countable subset of the ... |
| xrge0gsumle 24570 | A finite sum in the nonneg... |
| xrge0tsms 24571 | Any finite or infinite sum... |
| xrge0tsms2 24572 | Any finite or infinite sum... |
| metdcnlem 24573 | The metric function of a m... |
| xmetdcn2 24574 | The metric function of an ... |
| xmetdcn 24575 | The metric function of an ... |
| metdcn2 24576 | The metric function of a m... |
| metdcn 24577 | The metric function of a m... |
| msdcn 24578 | The metric function of a m... |
| cnmpt1ds 24579 | Continuity of the metric f... |
| cnmpt2ds 24580 | Continuity of the metric f... |
| nmcn 24581 | The norm of a normed group... |
| ngnmcncn 24582 | The norm of a normed group... |
| abscn 24583 | The absolute value functio... |
| metdsval 24584 | Value of the "distance to ... |
| metdsf 24585 | The distance from a point ... |
| metdsge 24586 | The distance from the poin... |
| metds0 24587 | If a point is in a set, it... |
| metdstri 24588 | A generalization of the tr... |
| metdsle 24589 | The distance from a point ... |
| metdsre 24590 | The distance from a point ... |
| metdseq0 24591 | The distance from a point ... |
| metdscnlem 24592 | Lemma for ~ metdscn . (Co... |
| metdscn 24593 | The function ` F ` which g... |
| metdscn2 24594 | The function ` F ` which g... |
| metnrmlem1a 24595 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24596 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24597 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24598 | Lemma for ~ metnrm . (Con... |
| metnrm 24599 | A metric space is normal. ... |
| metreg 24600 | A metric space is regular.... |
| addcnlem 24601 | Lemma for ~ addcn , ~ subc... |
| addcn 24602 | Complex number addition is... |
| subcn 24603 | Complex number subtraction... |
| mulcn 24604 | Complex number multiplicat... |
| divcnOLD 24605 | Obsolete version of ~ divc... |
| mpomulcn 24606 | Complex number multiplicat... |
| divcn 24607 | Complex number division is... |
| cnfldtgp 24608 | The complex numbers form a... |
| fsumcn 24609 | A finite sum of functions ... |
| fsum2cn 24610 | Version of ~ fsumcn for tw... |
| expcn 24611 | The power function on comp... |
| divccn 24612 | Division by a nonzero cons... |
| expcnOLD 24613 | Obsolete version of ~ expc... |
| divccnOLD 24614 | Obsolete version of ~ divc... |
| sqcn 24615 | The square function on com... |
| iitopon 24620 | The unit interval is a top... |
| iitop 24621 | The unit interval is a top... |
| iiuni 24622 | The base set of the unit i... |
| dfii2 24623 | Alternate definition of th... |
| dfii3 24624 | Alternate definition of th... |
| dfii4 24625 | Alternate definition of th... |
| dfii5 24626 | The unit interval expresse... |
| iicmp 24627 | The unit interval is compa... |
| iiconn 24628 | The unit interval is conne... |
| cncfval 24629 | The value of the continuou... |
| elcncf 24630 | Membership in the set of c... |
| elcncf2 24631 | Version of ~ elcncf with a... |
| cncfrss 24632 | Reverse closure of the con... |
| cncfrss2 24633 | Reverse closure of the con... |
| cncff 24634 | A continuous complex funct... |
| cncfi 24635 | Defining property of a con... |
| elcncf1di 24636 | Membership in the set of c... |
| elcncf1ii 24637 | Membership in the set of c... |
| rescncf 24638 | A continuous complex funct... |
| cncfcdm 24639 | Change the codomain of a c... |
| cncfss 24640 | The set of continuous func... |
| climcncf 24641 | Image of a limit under a c... |
| abscncf 24642 | Absolute value is continuo... |
| recncf 24643 | Real part is continuous. ... |
| imcncf 24644 | Imaginary part is continuo... |
| cjcncf 24645 | Complex conjugate is conti... |
| mulc1cncf 24646 | Multiplication by a consta... |
| divccncf 24647 | Division by a constant is ... |
| cncfco 24648 | The composition of two con... |
| cncfcompt2 24649 | Composition of continuous ... |
| cncfmet 24650 | Relate complex function co... |
| cncfcn 24651 | Relate complex function co... |
| cncfcn1 24652 | Relate complex function co... |
| cncfmptc 24653 | A constant function is a c... |
| cncfmptid 24654 | The identity function is a... |
| cncfmpt1f 24655 | Composition of continuous ... |
| cncfmpt2f 24656 | Composition of continuous ... |
| cncfmpt2ss 24657 | Composition of continuous ... |
| addccncf 24658 | Adding a constant is a con... |
| idcncf 24659 | The identity function is a... |
| sub1cncf 24660 | Subtracting a constant is ... |
| sub2cncf 24661 | Subtraction from a constan... |
| cdivcncf 24662 | Division with a constant n... |
| negcncf 24663 | The negative function is c... |
| negcncfOLD 24664 | Obsolete version of ~ negc... |
| negfcncf 24665 | The negative of a continuo... |
| abscncfALT 24666 | Absolute value is continuo... |
| cncfcnvcn 24667 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24668 | The power function on comp... |
| cnmptre 24669 | Lemma for ~ iirevcn and re... |
| cnmpopc 24670 | Piecewise definition of a ... |
| iirev 24671 | Reverse the unit interval.... |
| iirevcn 24672 | The reversion function is ... |
| iihalf1 24673 | Map the first half of ` II... |
| iihalf1cn 24674 | The first half function is... |
| iihalf1cnOLD 24675 | Obsolete version of ~ iiha... |
| iihalf2 24676 | Map the second half of ` I... |
| iihalf2cn 24677 | The second half function i... |
| iihalf2cnOLD 24678 | Obsolete version of ~ iiha... |
| elii1 24679 | Divide the unit interval i... |
| elii2 24680 | Divide the unit interval i... |
| iimulcl 24681 | The unit interval is close... |
| iimulcn 24682 | Multiplication is a contin... |
| iimulcnOLD 24683 | Obsolete version of ~ iimu... |
| icoopnst 24684 | A half-open interval start... |
| iocopnst 24685 | A half-open interval endin... |
| icchmeo 24686 | The natural bijection from... |
| icchmeoOLD 24687 | Obsolete version of ~ icch... |
| icopnfcnv 24688 | Define a bijection from ` ... |
| icopnfhmeo 24689 | The defined bijection from... |
| iccpnfcnv 24690 | Define a bijection from ` ... |
| iccpnfhmeo 24691 | The defined bijection from... |
| xrhmeo 24692 | The bijection from ` [ -u ... |
| xrhmph 24693 | The extended reals are hom... |
| xrcmp 24694 | The topology of the extend... |
| xrconn 24695 | The topology of the extend... |
| icccvx 24696 | A linear combination of tw... |
| oprpiece1res1 24697 | Restriction to the first p... |
| oprpiece1res2 24698 | Restriction to the second ... |
| cnrehmeo 24699 | The canonical bijection fr... |
| cnrehmeoOLD 24700 | Obsolete version of ~ cnre... |
| cnheiborlem 24701 | Lemma for ~ cnheibor . (C... |
| cnheibor 24702 | Heine-Borel theorem for co... |
| cnllycmp 24703 | The topology on the comple... |
| rellycmp 24704 | The topology on the reals ... |
| bndth 24705 | The Boundedness Theorem. ... |
| evth 24706 | The Extreme Value Theorem.... |
| evth2 24707 | The Extreme Value Theorem,... |
| lebnumlem1 24708 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24709 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24710 | Lemma for ~ lebnum . By t... |
| lebnum 24711 | The Lebesgue number lemma,... |
| xlebnum 24712 | Generalize ~ lebnum to ext... |
| lebnumii 24713 | Specialize the Lebesgue nu... |
| ishtpy 24719 | Membership in the class of... |
| htpycn 24720 | A homotopy is a continuous... |
| htpyi 24721 | A homotopy evaluated at it... |
| ishtpyd 24722 | Deduction for membership i... |
| htpycom 24723 | Given a homotopy from ` F ... |
| htpyid 24724 | A homotopy from a function... |
| htpyco1 24725 | Compose a homotopy with a ... |
| htpyco2 24726 | Compose a homotopy with a ... |
| htpycc 24727 | Concatenate two homotopies... |
| isphtpy 24728 | Membership in the class of... |
| phtpyhtpy 24729 | A path homotopy is a homot... |
| phtpycn 24730 | A path homotopy is a conti... |
| phtpyi 24731 | Membership in the class of... |
| phtpy01 24732 | Two path-homotopic paths h... |
| isphtpyd 24733 | Deduction for membership i... |
| isphtpy2d 24734 | Deduction for membership i... |
| phtpycom 24735 | Given a homotopy from ` F ... |
| phtpyid 24736 | A homotopy from a path to ... |
| phtpyco2 24737 | Compose a path homotopy wi... |
| phtpycc 24738 | Concatenate two path homot... |
| phtpcrel 24740 | The path homotopy relation... |
| isphtpc 24741 | The relation "is path homo... |
| phtpcer 24742 | Path homotopy is an equiva... |
| phtpc01 24743 | Path homotopic paths have ... |
| reparphti 24744 | Lemma for ~ reparpht . (C... |
| reparphtiOLD 24745 | Obsolete version of ~ repa... |
| reparpht 24746 | Reparametrization lemma. ... |
| phtpcco2 24747 | Compose a path homotopy wi... |
| pcofval 24758 | The value of the path conc... |
| pcoval 24759 | The concatenation of two p... |
| pcovalg 24760 | Evaluate the concatenation... |
| pcoval1 24761 | Evaluate the concatenation... |
| pco0 24762 | The starting point of a pa... |
| pco1 24763 | The ending point of a path... |
| pcoval2 24764 | Evaluate the concatenation... |
| pcocn 24765 | The concatenation of two p... |
| copco 24766 | The composition of a conca... |
| pcohtpylem 24767 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24768 | Homotopy invariance of pat... |
| pcoptcl 24769 | A constant function is a p... |
| pcopt 24770 | Concatenation with a point... |
| pcopt2 24771 | Concatenation with a point... |
| pcoass 24772 | Order of concatenation doe... |
| pcorevcl 24773 | Closure for a reversed pat... |
| pcorevlem 24774 | Lemma for ~ pcorev . Prov... |
| pcorev 24775 | Concatenation with the rev... |
| pcorev2 24776 | Concatenation with the rev... |
| pcophtb 24777 | The path homotopy equivale... |
| om1val 24778 | The definition of the loop... |
| om1bas 24779 | The base set of the loop s... |
| om1elbas 24780 | Elementhood in the base se... |
| om1addcl 24781 | Closure of the group opera... |
| om1plusg 24782 | The group operation (which... |
| om1tset 24783 | The topology of the loop s... |
| om1opn 24784 | The topology of the loop s... |
| pi1val 24785 | The definition of the fund... |
| pi1bas 24786 | The base set of the fundam... |
| pi1blem 24787 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24788 | Another way to write the l... |
| pi1bas2 24789 | The base set of the fundam... |
| pi1eluni 24790 | Elementhood in the base se... |
| pi1bas3 24791 | The base set of the fundam... |
| pi1cpbl 24792 | The group operation, loop ... |
| elpi1 24793 | The elements of the fundam... |
| elpi1i 24794 | The elements of the fundam... |
| pi1addf 24795 | The group operation of ` p... |
| pi1addval 24796 | The concatenation of two p... |
| pi1grplem 24797 | Lemma for ~ pi1grp . (Con... |
| pi1grp 24798 | The fundamental group is a... |
| pi1id 24799 | The identity element of th... |
| pi1inv 24800 | An inverse in the fundamen... |
| pi1xfrf 24801 | Functionality of the loop ... |
| pi1xfrval 24802 | The value of the loop tran... |
| pi1xfr 24803 | Given a path ` F ` and its... |
| pi1xfrcnvlem 24804 | Given a path ` F ` between... |
| pi1xfrcnv 24805 | Given a path ` F ` between... |
| pi1xfrgim 24806 | The mapping ` G ` between ... |
| pi1cof 24807 | Functionality of the loop ... |
| pi1coval 24808 | The value of the loop tran... |
| pi1coghm 24809 | The mapping ` G ` between ... |
| isclm 24812 | A subcomplex module is a l... |
| clmsca 24813 | The ring of scalars ` F ` ... |
| clmsubrg 24814 | The base set of the ring o... |
| clmlmod 24815 | A subcomplex module is a l... |
| clmgrp 24816 | A subcomplex module is an ... |
| clmabl 24817 | A subcomplex module is an ... |
| clmring 24818 | The scalar ring of a subco... |
| clmfgrp 24819 | The scalar ring of a subco... |
| clm0 24820 | The zero of the scalar rin... |
| clm1 24821 | The identity of the scalar... |
| clmadd 24822 | The addition of the scalar... |
| clmmul 24823 | The multiplication of the ... |
| clmcj 24824 | The conjugation of the sca... |
| isclmi 24825 | Reverse direction of ~ isc... |
| clmzss 24826 | The scalar ring of a subco... |
| clmsscn 24827 | The scalar ring of a subco... |
| clmsub 24828 | Subtraction in the scalar ... |
| clmneg 24829 | Negation in the scalar rin... |
| clmneg1 24830 | Minus one is in the scalar... |
| clmabs 24831 | Norm in the scalar ring of... |
| clmacl 24832 | Closure of ring addition f... |
| clmmcl 24833 | Closure of ring multiplica... |
| clmsubcl 24834 | Closure of ring subtractio... |
| lmhmclm 24835 | The domain of a linear ope... |
| clmvscl 24836 | Closure of scalar product ... |
| clmvsass 24837 | Associative law for scalar... |
| clmvscom 24838 | Commutative law for the sc... |
| clmvsdir 24839 | Distributive law for scala... |
| clmvsdi 24840 | Distributive law for scala... |
| clmvs1 24841 | Scalar product with ring u... |
| clmvs2 24842 | A vector plus itself is tw... |
| clm0vs 24843 | Zero times a vector is the... |
| clmopfne 24844 | The (functionalized) opera... |
| isclmp 24845 | The predicate "is a subcom... |
| isclmi0 24846 | Properties that determine ... |
| clmvneg1 24847 | Minus 1 times a vector is ... |
| clmvsneg 24848 | Multiplication of a vector... |
| clmmulg 24849 | The group multiple functio... |
| clmsubdir 24850 | Scalar multiplication dist... |
| clmpm1dir 24851 | Subtractive distributive l... |
| clmnegneg 24852 | Double negative of a vecto... |
| clmnegsubdi2 24853 | Distribution of negative o... |
| clmsub4 24854 | Rearrangement of 4 terms i... |
| clmvsrinv 24855 | A vector minus itself. (C... |
| clmvslinv 24856 | Minus a vector plus itself... |
| clmvsubval 24857 | Value of vector subtractio... |
| clmvsubval2 24858 | Value of vector subtractio... |
| clmvz 24859 | Two ways to express the ne... |
| zlmclm 24860 | The ` ZZ ` -module operati... |
| clmzlmvsca 24861 | The scalar product of a su... |
| nmoleub2lem 24862 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 24863 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 24864 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 24865 | The operator norm is the s... |
| nmoleub2b 24866 | The operator norm is the s... |
| nmoleub3 24867 | The operator norm is the s... |
| nmhmcn 24868 | A linear operator over a n... |
| cmodscexp 24869 | The powers of ` _i ` belon... |
| cmodscmulexp 24870 | The scalar product of a ve... |
| cvslvec 24873 | A subcomplex vector space ... |
| cvsclm 24874 | A subcomplex vector space ... |
| iscvs 24875 | A subcomplex vector space ... |
| iscvsp 24876 | The predicate "is a subcom... |
| iscvsi 24877 | Properties that determine ... |
| cvsi 24878 | The properties of a subcom... |
| cvsunit 24879 | Unit group of the scalar r... |
| cvsdiv 24880 | Division of the scalar rin... |
| cvsdivcl 24881 | The scalar field of a subc... |
| cvsmuleqdivd 24882 | An equality involving rati... |
| cvsdiveqd 24883 | An equality involving rati... |
| cnlmodlem1 24884 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 24885 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 24886 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 24887 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 24888 | The set of complex numbers... |
| cnstrcvs 24889 | The set of complex numbers... |
| cnrbas 24890 | The set of complex numbers... |
| cnrlmod 24891 | The complex left module of... |
| cnrlvec 24892 | The complex left module of... |
| cncvs 24893 | The complex left module of... |
| recvs 24894 | The field of the real numb... |
| recvsOLD 24895 | Obsolete version of ~ recv... |
| qcvs 24896 | The field of rational numb... |
| zclmncvs 24897 | The ring of integers as le... |
| isncvsngp 24898 | A normed subcomplex vector... |
| isncvsngpd 24899 | Properties that determine ... |
| ncvsi 24900 | The properties of a normed... |
| ncvsprp 24901 | Proportionality property o... |
| ncvsge0 24902 | The norm of a scalar produ... |
| ncvsm1 24903 | The norm of the opposite o... |
| ncvsdif 24904 | The norm of the difference... |
| ncvspi 24905 | The norm of a vector plus ... |
| ncvs1 24906 | From any nonzero vector of... |
| cnrnvc 24907 | The module of complex numb... |
| cnncvs 24908 | The module of complex numb... |
| cnnm 24909 | The norm of the normed sub... |
| ncvspds 24910 | Value of the distance func... |
| cnindmet 24911 | The metric induced on the ... |
| cnncvsaddassdemo 24912 | Derive the associative law... |
| cnncvsmulassdemo 24913 | Derive the associative law... |
| cnncvsabsnegdemo 24914 | Derive the absolute value ... |
| iscph 24919 | A subcomplex pre-Hilbert s... |
| cphphl 24920 | A subcomplex pre-Hilbert s... |
| cphnlm 24921 | A subcomplex pre-Hilbert s... |
| cphngp 24922 | A subcomplex pre-Hilbert s... |
| cphlmod 24923 | A subcomplex pre-Hilbert s... |
| cphlvec 24924 | A subcomplex pre-Hilbert s... |
| cphnvc 24925 | A subcomplex pre-Hilbert s... |
| cphsubrglem 24926 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 24927 | Lemma for ~ cphreccl . (C... |
| cphsca 24928 | A subcomplex pre-Hilbert s... |
| cphsubrg 24929 | The scalar field of a subc... |
| cphreccl 24930 | The scalar field of a subc... |
| cphdivcl 24931 | The scalar field of a subc... |
| cphcjcl 24932 | The scalar field of a subc... |
| cphsqrtcl 24933 | The scalar field of a subc... |
| cphabscl 24934 | The scalar field of a subc... |
| cphsqrtcl2 24935 | The scalar field of a subc... |
| cphsqrtcl3 24936 | If the scalar field of a s... |
| cphqss 24937 | The scalar field of a subc... |
| cphclm 24938 | A subcomplex pre-Hilbert s... |
| cphnmvs 24939 | Norm of a scalar product. ... |
| cphipcl 24940 | An inner product is a memb... |
| cphnmfval 24941 | The value of the norm in a... |
| cphnm 24942 | The square of the norm is ... |
| nmsq 24943 | The square of the norm is ... |
| cphnmf 24944 | The norm of a vector is a ... |
| cphnmcl 24945 | The norm of a vector is a ... |
| reipcl 24946 | An inner product of an ele... |
| ipge0 24947 | The inner product in a sub... |
| cphipcj 24948 | Conjugate of an inner prod... |
| cphipipcj 24949 | An inner product times its... |
| cphorthcom 24950 | Orthogonality (meaning inn... |
| cphip0l 24951 | Inner product with a zero ... |
| cphip0r 24952 | Inner product with a zero ... |
| cphipeq0 24953 | The inner product of a vec... |
| cphdir 24954 | Distributive law for inner... |
| cphdi 24955 | Distributive law for inner... |
| cph2di 24956 | Distributive law for inner... |
| cphsubdir 24957 | Distributive law for inner... |
| cphsubdi 24958 | Distributive law for inner... |
| cph2subdi 24959 | Distributive law for inner... |
| cphass 24960 | Associative law for inner ... |
| cphassr 24961 | "Associative" law for seco... |
| cph2ass 24962 | Move scalar multiplication... |
| cphassi 24963 | Associative law for the fi... |
| cphassir 24964 | "Associative" law for the ... |
| cphpyth 24965 | The pythagorean theorem fo... |
| tcphex 24966 | Lemma for ~ tcphbas and si... |
| tcphval 24967 | Define a function to augme... |
| tcphbas 24968 | The base set of a subcompl... |
| tchplusg 24969 | The addition operation of ... |
| tcphsub 24970 | The subtraction operation ... |
| tcphmulr 24971 | The ring operation of a su... |
| tcphsca 24972 | The scalar field of a subc... |
| tcphvsca 24973 | The scalar multiplication ... |
| tcphip 24974 | The inner product of a sub... |
| tcphtopn 24975 | The topology of a subcompl... |
| tcphphl 24976 | Augmentation of a subcompl... |
| tchnmfval 24977 | The norm of a subcomplex p... |
| tcphnmval 24978 | The norm of a subcomplex p... |
| cphtcphnm 24979 | The norm of a norm-augment... |
| tcphds 24980 | The distance of a pre-Hilb... |
| phclm 24981 | A pre-Hilbert space whose ... |
| tcphcphlem3 24982 | Lemma for ~ tcphcph : real... |
| ipcau2 24983 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 24984 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 24985 | Lemma for ~ tcphcph : homo... |
| tcphcph 24986 | The standard definition of... |
| ipcau 24987 | The Cauchy-Schwarz inequal... |
| nmparlem 24988 | Lemma for ~ nmpar . (Cont... |
| nmpar 24989 | A subcomplex pre-Hilbert s... |
| cphipval2 24990 | Value of the inner product... |
| 4cphipval2 24991 | Four times the inner produ... |
| cphipval 24992 | Value of the inner product... |
| ipcnlem2 24993 | The inner product operatio... |
| ipcnlem1 24994 | The inner product operatio... |
| ipcn 24995 | The inner product operatio... |
| cnmpt1ip 24996 | Continuity of inner produc... |
| cnmpt2ip 24997 | Continuity of inner produc... |
| csscld 24998 | A "closed subspace" in a s... |
| clsocv 24999 | The orthogonal complement ... |
| cphsscph 25000 | A subspace of a subcomplex... |
| lmmbr 25007 | Express the binary relatio... |
| lmmbr2 25008 | Express the binary relatio... |
| lmmbr3 25009 | Express the binary relatio... |
| lmmcvg 25010 | Convergence property of a ... |
| lmmbrf 25011 | Express the binary relatio... |
| lmnn 25012 | A condition that implies c... |
| cfilfval 25013 | The set of Cauchy filters ... |
| iscfil 25014 | The property of being a Ca... |
| iscfil2 25015 | The property of being a Ca... |
| cfilfil 25016 | A Cauchy filter is a filte... |
| cfili 25017 | Property of a Cauchy filte... |
| cfil3i 25018 | A Cauchy filter contains b... |
| cfilss 25019 | A filter finer than a Cauc... |
| fgcfil 25020 | The Cauchy filter conditio... |
| fmcfil 25021 | The Cauchy filter conditio... |
| iscfil3 25022 | A filter is Cauchy iff it ... |
| cfilfcls 25023 | Similar to ultrafilters ( ... |
| caufval 25024 | The set of Cauchy sequence... |
| iscau 25025 | Express the property " ` F... |
| iscau2 25026 | Express the property " ` F... |
| iscau3 25027 | Express the Cauchy sequenc... |
| iscau4 25028 | Express the property " ` F... |
| iscauf 25029 | Express the property " ` F... |
| caun0 25030 | A metric with a Cauchy seq... |
| caufpm 25031 | Inclusion of a Cauchy sequ... |
| caucfil 25032 | A Cauchy sequence predicat... |
| iscmet 25033 | The property " ` D ` is a ... |
| cmetcvg 25034 | The convergence of a Cauch... |
| cmetmet 25035 | A complete metric space is... |
| cmetmeti 25036 | A complete metric space is... |
| cmetcaulem 25037 | Lemma for ~ cmetcau . (Co... |
| cmetcau 25038 | The convergence of a Cauch... |
| iscmet3lem3 25039 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 25040 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 25041 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 25042 | The property " ` D ` is a ... |
| iscmet2 25043 | A metric ` D ` is complete... |
| cfilresi 25044 | A Cauchy filter on a metri... |
| cfilres 25045 | Cauchy filter on a metric ... |
| caussi 25046 | Cauchy sequence on a metri... |
| causs 25047 | Cauchy sequence on a metri... |
| equivcfil 25048 | If the metric ` D ` is "st... |
| equivcau 25049 | If the metric ` D ` is "st... |
| lmle 25050 | If the distance from each ... |
| nglmle 25051 | If the norm of each member... |
| lmclim 25052 | Relate a limit on the metr... |
| lmclimf 25053 | Relate a limit on the metr... |
| metelcls 25054 | A point belongs to the clo... |
| metcld 25055 | A subset of a metric space... |
| metcld2 25056 | A subset of a metric space... |
| caubl 25057 | Sufficient condition to en... |
| caublcls 25058 | The convergent point of a ... |
| metcnp4 25059 | Two ways to say a mapping ... |
| metcn4 25060 | Two ways to say a mapping ... |
| iscmet3i 25061 | Properties that determine ... |
| lmcau 25062 | Every convergent sequence ... |
| flimcfil 25063 | Every convergent filter in... |
| metsscmetcld 25064 | A complete subspace of a m... |
| cmetss 25065 | A subspace of a complete m... |
| equivcmet 25066 | If two metrics are strongl... |
| relcmpcmet 25067 | If ` D ` is a metric space... |
| cmpcmet 25068 | A compact metric space is ... |
| cfilucfil3 25069 | Given a metric ` D ` and a... |
| cfilucfil4 25070 | Given a metric ` D ` and a... |
| cncmet 25071 | The set of complex numbers... |
| recmet 25072 | The real numbers are a com... |
| bcthlem1 25073 | Lemma for ~ bcth . Substi... |
| bcthlem2 25074 | Lemma for ~ bcth . The ba... |
| bcthlem3 25075 | Lemma for ~ bcth . The li... |
| bcthlem4 25076 | Lemma for ~ bcth . Given ... |
| bcthlem5 25077 | Lemma for ~ bcth . The pr... |
| bcth 25078 | Baire's Category Theorem. ... |
| bcth2 25079 | Baire's Category Theorem, ... |
| bcth3 25080 | Baire's Category Theorem, ... |
| isbn 25087 | A Banach space is a normed... |
| bnsca 25088 | The scalar field of a Bana... |
| bnnvc 25089 | A Banach space is a normed... |
| bnnlm 25090 | A Banach space is a normed... |
| bnngp 25091 | A Banach space is a normed... |
| bnlmod 25092 | A Banach space is a left m... |
| bncms 25093 | A Banach space is a comple... |
| iscms 25094 | A complete metric space is... |
| cmscmet 25095 | The induced metric on a co... |
| bncmet 25096 | The induced metric on Bana... |
| cmsms 25097 | A complete metric space is... |
| cmspropd 25098 | Property deduction for a c... |
| cmssmscld 25099 | The restriction of a metri... |
| cmsss 25100 | The restriction of a compl... |
| lssbn 25101 | A subspace of a Banach spa... |
| cmetcusp1 25102 | If the uniform set of a co... |
| cmetcusp 25103 | The uniform space generate... |
| cncms 25104 | The field of complex numbe... |
| cnflduss 25105 | The uniform structure of t... |
| cnfldcusp 25106 | The field of complex numbe... |
| resscdrg 25107 | The real numbers are a sub... |
| cncdrg 25108 | The only complete subfield... |
| srabn 25109 | The subring algebra over a... |
| rlmbn 25110 | The ring module over a com... |
| ishl 25111 | The predicate "is a subcom... |
| hlbn 25112 | Every subcomplex Hilbert s... |
| hlcph 25113 | Every subcomplex Hilbert s... |
| hlphl 25114 | Every subcomplex Hilbert s... |
| hlcms 25115 | Every subcomplex Hilbert s... |
| hlprlem 25116 | Lemma for ~ hlpr . (Contr... |
| hlress 25117 | The scalar field of a subc... |
| hlpr 25118 | The scalar field of a subc... |
| ishl2 25119 | A Hilbert space is a compl... |
| cphssphl 25120 | A Banach subspace of a sub... |
| cmslssbn 25121 | A complete linear subspace... |
| cmscsscms 25122 | A closed subspace of a com... |
| bncssbn 25123 | A closed subspace of a Ban... |
| cssbn 25124 | A complete subspace of a n... |
| csschl 25125 | A complete subspace of a c... |
| cmslsschl 25126 | A complete linear subspace... |
| chlcsschl 25127 | A closed subspace of a sub... |
| retopn 25128 | The topology of the real n... |
| recms 25129 | The real numbers form a co... |
| reust 25130 | The Uniform structure of t... |
| recusp 25131 | The real numbers form a co... |
| rrxval 25136 | Value of the generalized E... |
| rrxbase 25137 | The base of the generalize... |
| rrxprds 25138 | Expand the definition of t... |
| rrxip 25139 | The inner product of the g... |
| rrxnm 25140 | The norm of the generalize... |
| rrxcph 25141 | Generalized Euclidean real... |
| rrxds 25142 | The distance over generali... |
| rrxvsca 25143 | The scalar product over ge... |
| rrxplusgvscavalb 25144 | The result of the addition... |
| rrxsca 25145 | The field of real numbers ... |
| rrx0 25146 | The zero ("origin") in a g... |
| rrx0el 25147 | The zero ("origin") in a g... |
| csbren 25148 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 25149 | Triangle inequality in R^n... |
| rrxf 25150 | Euclidean vectors as funct... |
| rrxfsupp 25151 | Euclidean vectors are of f... |
| rrxsuppss 25152 | Support of Euclidean vecto... |
| rrxmvallem 25153 | Support of the function us... |
| rrxmval 25154 | The value of the Euclidean... |
| rrxmfval 25155 | The value of the Euclidean... |
| rrxmetlem 25156 | Lemma for ~ rrxmet . (Con... |
| rrxmet 25157 | Euclidean space is a metri... |
| rrxdstprj1 25158 | The distance between two p... |
| rrxbasefi 25159 | The base of the generalize... |
| rrxdsfi 25160 | The distance over generali... |
| rrxmetfi 25161 | Euclidean space is a metri... |
| rrxdsfival 25162 | The value of the Euclidean... |
| ehlval 25163 | Value of the Euclidean spa... |
| ehlbase 25164 | The base of the Euclidean ... |
| ehl0base 25165 | The base of the Euclidean ... |
| ehl0 25166 | The Euclidean space of dim... |
| ehleudis 25167 | The Euclidean distance fun... |
| ehleudisval 25168 | The value of the Euclidean... |
| ehl1eudis 25169 | The Euclidean distance fun... |
| ehl1eudisval 25170 | The value of the Euclidean... |
| ehl2eudis 25171 | The Euclidean distance fun... |
| ehl2eudisval 25172 | The value of the Euclidean... |
| minveclem1 25173 | Lemma for ~ minvec . The ... |
| minveclem4c 25174 | Lemma for ~ minvec . The ... |
| minveclem2 25175 | Lemma for ~ minvec . Any ... |
| minveclem3a 25176 | Lemma for ~ minvec . ` D `... |
| minveclem3b 25177 | Lemma for ~ minvec . The ... |
| minveclem3 25178 | Lemma for ~ minvec . The ... |
| minveclem4a 25179 | Lemma for ~ minvec . ` F `... |
| minveclem4b 25180 | Lemma for ~ minvec . The ... |
| minveclem4 25181 | Lemma for ~ minvec . The ... |
| minveclem5 25182 | Lemma for ~ minvec . Disc... |
| minveclem6 25183 | Lemma for ~ minvec . Any ... |
| minveclem7 25184 | Lemma for ~ minvec . Sinc... |
| minvec 25185 | Minimizing vector theorem,... |
| pjthlem1 25186 | Lemma for ~ pjth . (Contr... |
| pjthlem2 25187 | Lemma for ~ pjth . (Contr... |
| pjth 25188 | Projection Theorem: Any H... |
| pjth2 25189 | Projection Theorem with ab... |
| cldcss 25190 | Corollary of the Projectio... |
| cldcss2 25191 | Corollary of the Projectio... |
| hlhil 25192 | Corollary of the Projectio... |
| addcncf 25193 | The addition of two contin... |
| subcncf 25194 | The addition of two contin... |
| mulcncf 25195 | The multiplication of two ... |
| mulcncfOLD 25196 | Obsolete version of ~ mulc... |
| divcncf 25197 | The quotient of two contin... |
| pmltpclem1 25198 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 25199 | Lemma for ~ pmltpc . (Con... |
| pmltpc 25200 | Any function on the reals ... |
| ivthlem1 25201 | Lemma for ~ ivth . The se... |
| ivthlem2 25202 | Lemma for ~ ivth . Show t... |
| ivthlem3 25203 | Lemma for ~ ivth , the int... |
| ivth 25204 | The intermediate value the... |
| ivth2 25205 | The intermediate value the... |
| ivthle 25206 | The intermediate value the... |
| ivthle2 25207 | The intermediate value the... |
| ivthicc 25208 | The interval between any t... |
| evthicc 25209 | Specialization of the Extr... |
| evthicc2 25210 | Combine ~ ivthicc with ~ e... |
| cniccbdd 25211 | A continuous function on a... |
| ovolfcl 25216 | Closure for the interval e... |
| ovolfioo 25217 | Unpack the interval coveri... |
| ovolficc 25218 | Unpack the interval coveri... |
| ovolficcss 25219 | Any (closed) interval cove... |
| ovolfsval 25220 | The value of the interval ... |
| ovolfsf 25221 | Closure for the interval l... |
| ovolsf 25222 | Closure for the partial su... |
| ovolval 25223 | The value of the outer mea... |
| elovolmlem 25224 | Lemma for ~ elovolm and re... |
| elovolm 25225 | Elementhood in the set ` M... |
| elovolmr 25226 | Sufficient condition for e... |
| ovolmge0 25227 | The set ` M ` is composed ... |
| ovolcl 25228 | The volume of a set is an ... |
| ovollb 25229 | The outer volume is a lowe... |
| ovolgelb 25230 | The outer volume is the gr... |
| ovolge0 25231 | The volume of a set is alw... |
| ovolf 25232 | The domain and codomain of... |
| ovollecl 25233 | If an outer volume is boun... |
| ovolsslem 25234 | Lemma for ~ ovolss . (Con... |
| ovolss 25235 | The volume of a set is mon... |
| ovolsscl 25236 | If a set is contained in a... |
| ovolssnul 25237 | A subset of a nullset is n... |
| ovollb2lem 25238 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 25239 | It is often more convenien... |
| ovolctb 25240 | The volume of a denumerabl... |
| ovolq 25241 | The rational numbers have ... |
| ovolctb2 25242 | The volume of a countable ... |
| ovol0 25243 | The empty set has 0 outer ... |
| ovolfi 25244 | A finite set has 0 outer L... |
| ovolsn 25245 | A singleton has 0 outer Le... |
| ovolunlem1a 25246 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 25247 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 25248 | Lemma for ~ ovolun . (Con... |
| ovolun 25249 | The Lebesgue outer measure... |
| ovolunnul 25250 | Adding a nullset does not ... |
| ovolfiniun 25251 | The Lebesgue outer measure... |
| ovoliunlem1 25252 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 25253 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 25254 | Lemma for ~ ovoliun . (Co... |
| ovoliun 25255 | The Lebesgue outer measure... |
| ovoliun2 25256 | The Lebesgue outer measure... |
| ovoliunnul 25257 | A countable union of nulls... |
| shft2rab 25258 | If ` B ` is a shift of ` A... |
| ovolshftlem1 25259 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 25260 | Lemma for ~ ovolshft . (C... |
| ovolshft 25261 | The Lebesgue outer measure... |
| sca2rab 25262 | If ` B ` is a scale of ` A... |
| ovolscalem1 25263 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 25264 | Lemma for ~ ovolshft . (C... |
| ovolsca 25265 | The Lebesgue outer measure... |
| ovolicc1 25266 | The measure of a closed in... |
| ovolicc2lem1 25267 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 25268 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 25269 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 25270 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 25271 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 25272 | The measure of a closed in... |
| ovolicc 25273 | The measure of a closed in... |
| ovolicopnf 25274 | The measure of a right-unb... |
| ovolre 25275 | The measure of the real nu... |
| ismbl 25276 | The predicate " ` A ` is L... |
| ismbl2 25277 | From ~ ovolun , it suffice... |
| volres 25278 | A self-referencing abbrevi... |
| volf 25279 | The domain and codomain of... |
| mblvol 25280 | The volume of a measurable... |
| mblss 25281 | A measurable set is a subs... |
| mblsplit 25282 | The defining property of m... |
| volss 25283 | The Lebesgue measure is mo... |
| cmmbl 25284 | The complement of a measur... |
| nulmbl 25285 | A nullset is measurable. ... |
| nulmbl2 25286 | A set of outer measure zer... |
| unmbl 25287 | A union of measurable sets... |
| shftmbl 25288 | A shift of a measurable se... |
| 0mbl 25289 | The empty set is measurabl... |
| rembl 25290 | The set of all real number... |
| unidmvol 25291 | The union of the Lebesgue ... |
| inmbl 25292 | An intersection of measura... |
| difmbl 25293 | A difference of measurable... |
| finiunmbl 25294 | A finite union of measurab... |
| volun 25295 | The Lebesgue measure funct... |
| volinun 25296 | Addition of non-disjoint s... |
| volfiniun 25297 | The volume of a disjoint f... |
| iundisj 25298 | Rewrite a countable union ... |
| iundisj2 25299 | A disjoint union is disjoi... |
| voliunlem1 25300 | Lemma for ~ voliun . (Con... |
| voliunlem2 25301 | Lemma for ~ voliun . (Con... |
| voliunlem3 25302 | Lemma for ~ voliun . (Con... |
| iunmbl 25303 | The measurable sets are cl... |
| voliun 25304 | The Lebesgue measure funct... |
| volsuplem 25305 | Lemma for ~ volsup . (Con... |
| volsup 25306 | The volume of the limit of... |
| iunmbl2 25307 | The measurable sets are cl... |
| ioombl1lem1 25308 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 25309 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 25310 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 25311 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 25312 | An open right-unbounded in... |
| icombl1 25313 | A closed unbounded-above i... |
| icombl 25314 | A closed-below, open-above... |
| ioombl 25315 | An open real interval is m... |
| iccmbl 25316 | A closed real interval is ... |
| iccvolcl 25317 | A closed real interval has... |
| ovolioo 25318 | The measure of an open int... |
| volioo 25319 | The measure of an open int... |
| ioovolcl 25320 | An open real interval has ... |
| ovolfs2 25321 | Alternative expression for... |
| ioorcl2 25322 | An open interval with fini... |
| ioorf 25323 | Define a function from ope... |
| ioorval 25324 | Define a function from ope... |
| ioorinv2 25325 | The function ` F ` is an "... |
| ioorinv 25326 | The function ` F ` is an "... |
| ioorcl 25327 | The function ` F ` does no... |
| uniiccdif 25328 | A union of closed interval... |
| uniioovol 25329 | A disjoint union of open i... |
| uniiccvol 25330 | An almost-disjoint union o... |
| uniioombllem1 25331 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 25332 | Lemma for ~ uniioombl . (... |
| uniioombllem2 25333 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 25334 | Lemma for ~ uniioombl . (... |
| uniioombllem3 25335 | Lemma for ~ uniioombl . (... |
| uniioombllem4 25336 | Lemma for ~ uniioombl . (... |
| uniioombllem5 25337 | Lemma for ~ uniioombl . (... |
| uniioombllem6 25338 | Lemma for ~ uniioombl . (... |
| uniioombl 25339 | A disjoint union of open i... |
| uniiccmbl 25340 | An almost-disjoint union o... |
| dyadf 25341 | The function ` F ` returns... |
| dyadval 25342 | Value of the dyadic ration... |
| dyadovol 25343 | Volume of a dyadic rationa... |
| dyadss 25344 | Two closed dyadic rational... |
| dyaddisjlem 25345 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 25346 | Two closed dyadic rational... |
| dyadmaxlem 25347 | Lemma for ~ dyadmax . (Co... |
| dyadmax 25348 | Any nonempty set of dyadic... |
| dyadmbllem 25349 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 25350 | Any union of dyadic ration... |
| opnmbllem 25351 | Lemma for ~ opnmbl . (Con... |
| opnmbl 25352 | All open sets are measurab... |
| opnmblALT 25353 | All open sets are measurab... |
| subopnmbl 25354 | Sets which are open in a m... |
| volsup2 25355 | The volume of ` A ` is the... |
| volcn 25356 | The function formed by res... |
| volivth 25357 | The Intermediate Value The... |
| vitalilem1 25358 | Lemma for ~ vitali . (Con... |
| vitalilem2 25359 | Lemma for ~ vitali . (Con... |
| vitalilem3 25360 | Lemma for ~ vitali . (Con... |
| vitalilem4 25361 | Lemma for ~ vitali . (Con... |
| vitalilem5 25362 | Lemma for ~ vitali . (Con... |
| vitali 25363 | If the reals can be well-o... |
| ismbf1 25374 | The predicate " ` F ` is a... |
| mbff 25375 | A measurable function is a... |
| mbfdm 25376 | The domain of a measurable... |
| mbfconstlem 25377 | Lemma for ~ mbfconst and r... |
| ismbf 25378 | The predicate " ` F ` is a... |
| ismbfcn 25379 | A complex function is meas... |
| mbfima 25380 | Definitional property of a... |
| mbfimaicc 25381 | The preimage of any closed... |
| mbfimasn 25382 | The preimage of a point un... |
| mbfconst 25383 | A constant function is mea... |
| mbf0 25384 | The empty function is meas... |
| mbfid 25385 | The identity function is m... |
| mbfmptcl 25386 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25387 | The domain of a measurable... |
| ismbfcn2 25388 | A complex function is meas... |
| ismbfd 25389 | Deduction to prove measura... |
| ismbf2d 25390 | Deduction to prove measura... |
| mbfeqalem1 25391 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25392 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25393 | If two functions are equal... |
| mbfres 25394 | The restriction of a measu... |
| mbfres2 25395 | Measurability of a piecewi... |
| mbfss 25396 | Change the domain of a mea... |
| mbfmulc2lem 25397 | Multiplication by a consta... |
| mbfmulc2re 25398 | Multiplication by a consta... |
| mbfmax 25399 | The maximum of two functio... |
| mbfneg 25400 | The negative of a measurab... |
| mbfpos 25401 | The positive part of a mea... |
| mbfposr 25402 | Converse to ~ mbfpos . (C... |
| mbfposb 25403 | A function is measurable i... |
| ismbf3d 25404 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25405 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25406 | The preimage of any open s... |
| mbfimaopn2 25407 | The preimage of any set op... |
| cncombf 25408 | The composition of a conti... |
| cnmbf 25409 | A continuous function is m... |
| mbfaddlem 25410 | The sum of two measurable ... |
| mbfadd 25411 | The sum of two measurable ... |
| mbfsub 25412 | The difference of two meas... |
| mbfmulc2 25413 | A complex constant times a... |
| mbfsup 25414 | The supremum of a sequence... |
| mbfinf 25415 | The infimum of a sequence ... |
| mbflimsup 25416 | The limit supremum of a se... |
| mbflimlem 25417 | The pointwise limit of a s... |
| mbflim 25418 | The pointwise limit of a s... |
| 0pval 25421 | The zero function evaluate... |
| 0plef 25422 | Two ways to say that the f... |
| 0pledm 25423 | Adjust the domain of the l... |
| isi1f 25424 | The predicate " ` F ` is a... |
| i1fmbf 25425 | Simple functions are measu... |
| i1ff 25426 | A simple function is a fun... |
| i1frn 25427 | A simple function has fini... |
| i1fima 25428 | Any preimage of a simple f... |
| i1fima2 25429 | Any preimage of a simple f... |
| i1fima2sn 25430 | Preimage of a singleton. ... |
| i1fd 25431 | A simplified set of assump... |
| i1f0rn 25432 | Any simple function takes ... |
| itg1val 25433 | The value of the integral ... |
| itg1val2 25434 | The value of the integral ... |
| itg1cl 25435 | Closure of the integral on... |
| itg1ge0 25436 | Closure of the integral on... |
| i1f0 25437 | The zero function is simpl... |
| itg10 25438 | The zero function has zero... |
| i1f1lem 25439 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25440 | Base case simple functions... |
| itg11 25441 | The integral of an indicat... |
| itg1addlem1 25442 | Decompose a preimage, whic... |
| i1faddlem 25443 | Decompose the preimage of ... |
| i1fmullem 25444 | Decompose the preimage of ... |
| i1fadd 25445 | The sum of two simple func... |
| i1fmul 25446 | The pointwise product of t... |
| itg1addlem2 25447 | Lemma for ~ itg1add . The... |
| itg1addlem3 25448 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25449 | Lemma for ~ itg1add . (Co... |
| itg1addlem4OLD 25450 | Obsolete version of ~ itg1... |
| itg1addlem5 25451 | Lemma for ~ itg1add . (Co... |
| itg1add 25452 | The integral of a sum of s... |
| i1fmulclem 25453 | Decompose the preimage of ... |
| i1fmulc 25454 | A nonnegative constant tim... |
| itg1mulc 25455 | The integral of a constant... |
| i1fres 25456 | The "restriction" of a sim... |
| i1fpos 25457 | The positive part of a sim... |
| i1fposd 25458 | Deduction form of ~ i1fpos... |
| i1fsub 25459 | The difference of two simp... |
| itg1sub 25460 | The integral of a differen... |
| itg10a 25461 | The integral of a simple f... |
| itg1ge0a 25462 | The integral of an almost ... |
| itg1lea 25463 | Approximate version of ~ i... |
| itg1le 25464 | If one simple function dom... |
| itg1climres 25465 | Restricting the simple fun... |
| mbfi1fseqlem1 25466 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25467 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25468 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25469 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25470 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25471 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25472 | A characterization of meas... |
| mbfi1flimlem 25473 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25474 | Any real measurable functi... |
| mbfmullem2 25475 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25476 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25477 | The product of two measura... |
| itg2lcl 25478 | The set of lower sums is a... |
| itg2val 25479 | Value of the integral on n... |
| itg2l 25480 | Elementhood in the set ` L... |
| itg2lr 25481 | Sufficient condition for e... |
| xrge0f 25482 | A real function is a nonne... |
| itg2cl 25483 | The integral of a nonnegat... |
| itg2ub 25484 | The integral of a nonnegat... |
| itg2leub 25485 | Any upper bound on the int... |
| itg2ge0 25486 | The integral of a nonnegat... |
| itg2itg1 25487 | The integral of a nonnegat... |
| itg20 25488 | The integral of the zero f... |
| itg2lecl 25489 | If an ` S.2 ` integral is ... |
| itg2le 25490 | If one function dominates ... |
| itg2const 25491 | Integral of a constant fun... |
| itg2const2 25492 | When the base set of a con... |
| itg2seq 25493 | Definitional property of t... |
| itg2uba 25494 | Approximate version of ~ i... |
| itg2lea 25495 | Approximate version of ~ i... |
| itg2eqa 25496 | Approximate equality of in... |
| itg2mulclem 25497 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25498 | The integral of a nonnegat... |
| itg2splitlem 25499 | Lemma for ~ itg2split . (... |
| itg2split 25500 | The ` S.2 ` integral split... |
| itg2monolem1 25501 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25502 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25503 | Lemma for ~ itg2mono . (C... |
| itg2mono 25504 | The Monotone Convergence T... |
| itg2i1fseqle 25505 | Subject to the conditions ... |
| itg2i1fseq 25506 | Subject to the conditions ... |
| itg2i1fseq2 25507 | In an extension to the res... |
| itg2i1fseq3 25508 | Special case of ~ itg2i1fs... |
| itg2addlem 25509 | Lemma for ~ itg2add . (Co... |
| itg2add 25510 | The ` S.2 ` integral is li... |
| itg2gt0 25511 | If the function ` F ` is s... |
| itg2cnlem1 25512 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25513 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25514 | A sort of absolute continu... |
| ibllem 25515 | Conditioned equality theor... |
| isibl 25516 | The predicate " ` F ` is i... |
| isibl2 25517 | The predicate " ` F ` is i... |
| iblmbf 25518 | An integrable function is ... |
| iblitg 25519 | If a function is integrabl... |
| dfitg 25520 | Evaluate the class substit... |
| itgex 25521 | An integral is a set. (Co... |
| itgeq1f 25522 | Equality theorem for an in... |
| itgeq1 25523 | Equality theorem for an in... |
| nfitg1 25524 | Bound-variable hypothesis ... |
| nfitg 25525 | Bound-variable hypothesis ... |
| cbvitg 25526 | Change bound variable in a... |
| cbvitgv 25527 | Change bound variable in a... |
| itgeq2 25528 | Equality theorem for an in... |
| itgresr 25529 | The domain of an integral ... |
| itg0 25530 | The integral of anything o... |
| itgz 25531 | The integral of zero on an... |
| itgeq2dv 25532 | Equality theorem for an in... |
| itgmpt 25533 | Change bound variable in a... |
| itgcl 25534 | The integral of an integra... |
| itgvallem 25535 | Substitution lemma. (Cont... |
| itgvallem3 25536 | Lemma for ~ itgposval and ... |
| ibl0 25537 | The zero function is integ... |
| iblcnlem1 25538 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25539 | Expand out the universal q... |
| itgcnlem 25540 | Expand out the sum in ~ df... |
| iblrelem 25541 | Integrability of a real fu... |
| iblposlem 25542 | Lemma for ~ iblpos . (Con... |
| iblpos 25543 | Integrability of a nonnega... |
| iblre 25544 | Integrability of a real fu... |
| itgrevallem1 25545 | Lemma for ~ itgposval and ... |
| itgposval 25546 | The integral of a nonnegat... |
| itgreval 25547 | Decompose the integral of ... |
| itgrecl 25548 | Real closure of an integra... |
| iblcn 25549 | Integrability of a complex... |
| itgcnval 25550 | Decompose the integral of ... |
| itgre 25551 | Real part of an integral. ... |
| itgim 25552 | Imaginary part of an integ... |
| iblneg 25553 | The negative of an integra... |
| itgneg 25554 | Negation of an integral. ... |
| iblss 25555 | A subset of an integrable ... |
| iblss2 25556 | Change the domain of an in... |
| itgitg2 25557 | Transfer an integral using... |
| i1fibl 25558 | A simple function is integ... |
| itgitg1 25559 | Transfer an integral using... |
| itgle 25560 | Monotonicity of an integra... |
| itgge0 25561 | The integral of a positive... |
| itgss 25562 | Expand the set of an integ... |
| itgss2 25563 | Expand the set of an integ... |
| itgeqa 25564 | Approximate equality of in... |
| itgss3 25565 | Expand the set of an integ... |
| itgioo 25566 | Equality of integrals on o... |
| itgless 25567 | Expand the integral of a n... |
| iblconst 25568 | A constant function is int... |
| itgconst 25569 | Integral of a constant fun... |
| ibladdlem 25570 | Lemma for ~ ibladd . (Con... |
| ibladd 25571 | Add two integrals over the... |
| iblsub 25572 | Subtract two integrals ove... |
| itgaddlem1 25573 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25574 | Lemma for ~ itgadd . (Con... |
| itgadd 25575 | Add two integrals over the... |
| itgsub 25576 | Subtract two integrals ove... |
| itgfsum 25577 | Take a finite sum of integ... |
| iblabslem 25578 | Lemma for ~ iblabs . (Con... |
| iblabs 25579 | The absolute value of an i... |
| iblabsr 25580 | A measurable function is i... |
| iblmulc2 25581 | Multiply an integral by a ... |
| itgmulc2lem1 25582 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25583 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25584 | Multiply an integral by a ... |
| itgabs 25585 | The triangle inequality fo... |
| itgsplit 25586 | The ` S. ` integral splits... |
| itgspliticc 25587 | The ` S. ` integral splits... |
| itgsplitioo 25588 | The ` S. ` integral splits... |
| bddmulibl 25589 | A bounded function times a... |
| bddibl 25590 | A bounded function is inte... |
| cniccibl 25591 | A continuous function on a... |
| bddiblnc 25592 | Choice-free proof of ~ bdd... |
| cnicciblnc 25593 | Choice-free proof of ~ cni... |
| itggt0 25594 | The integral of a strictly... |
| itgcn 25595 | Transfer ~ itg2cn to the f... |
| ditgeq1 25598 | Equality theorem for the d... |
| ditgeq2 25599 | Equality theorem for the d... |
| ditgeq3 25600 | Equality theorem for the d... |
| ditgeq3dv 25601 | Equality theorem for the d... |
| ditgex 25602 | A directed integral is a s... |
| ditg0 25603 | Value of the directed inte... |
| cbvditg 25604 | Change bound variable in a... |
| cbvditgv 25605 | Change bound variable in a... |
| ditgpos 25606 | Value of the directed inte... |
| ditgneg 25607 | Value of the directed inte... |
| ditgcl 25608 | Closure of a directed inte... |
| ditgswap 25609 | Reverse a directed integra... |
| ditgsplitlem 25610 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25611 | This theorem is the raison... |
| reldv 25620 | The derivative function is... |
| limcvallem 25621 | Lemma for ~ ellimc . (Con... |
| limcfval 25622 | Value and set bounds on th... |
| ellimc 25623 | Value of the limit predica... |
| limcrcl 25624 | Reverse closure for the li... |
| limccl 25625 | Closure of the limit opera... |
| limcdif 25626 | It suffices to consider fu... |
| ellimc2 25627 | Write the definition of a ... |
| limcnlp 25628 | If ` B ` is not a limit po... |
| ellimc3 25629 | Write the epsilon-delta de... |
| limcflflem 25630 | Lemma for ~ limcflf . (Co... |
| limcflf 25631 | The limit operator can be ... |
| limcmo 25632 | If ` B ` is a limit point ... |
| limcmpt 25633 | Express the limit operator... |
| limcmpt2 25634 | Express the limit operator... |
| limcresi 25635 | Any limit of ` F ` is also... |
| limcres 25636 | If ` B ` is an interior po... |
| cnplimc 25637 | A function is continuous a... |
| cnlimc 25638 | ` F ` is a continuous func... |
| cnlimci 25639 | If ` F ` is a continuous f... |
| cnmptlimc 25640 | If ` F ` is a continuous f... |
| limccnp 25641 | If the limit of ` F ` at `... |
| limccnp2 25642 | The image of a convergent ... |
| limcco 25643 | Composition of two limits.... |
| limciun 25644 | A point is a limit of ` F ... |
| limcun 25645 | A point is a limit of ` F ... |
| dvlem 25646 | Closure for a difference q... |
| dvfval 25647 | Value and set bounds on th... |
| eldv 25648 | The differentiable predica... |
| dvcl 25649 | The derivative function ta... |
| dvbssntr 25650 | The set of differentiable ... |
| dvbss 25651 | The set of differentiable ... |
| dvbsss 25652 | The set of differentiable ... |
| perfdvf 25653 | The derivative is a functi... |
| recnprss 25654 | Both ` RR ` and ` CC ` are... |
| recnperf 25655 | Both ` RR ` and ` CC ` are... |
| dvfg 25656 | Explicitly write out the f... |
| dvf 25657 | The derivative is a functi... |
| dvfcn 25658 | The derivative is a functi... |
| dvreslem 25659 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25660 | Lemma for ~ dvres2 . (Con... |
| dvres 25661 | Restriction of a derivativ... |
| dvres2 25662 | Restriction of the base se... |
| dvres3 25663 | Restriction of a complex d... |
| dvres3a 25664 | Restriction of a complex d... |
| dvidlem 25665 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25666 | Derivative of a function r... |
| dvconst 25667 | Derivative of a constant f... |
| dvid 25668 | Derivative of the identity... |
| dvcnp 25669 | The difference quotient is... |
| dvcnp2 25670 | A function is continuous a... |
| dvcnp2OLD 25671 | Obsolete version of ~ dvcn... |
| dvcn 25672 | A differentiable function ... |
| dvnfval 25673 | Value of the iterated deri... |
| dvnff 25674 | The iterated derivative is... |
| dvn0 25675 | Zero times iterated deriva... |
| dvnp1 25676 | Successor iterated derivat... |
| dvn1 25677 | One times iterated derivat... |
| dvnf 25678 | The N-times derivative is ... |
| dvnbss 25679 | The set of N-times differe... |
| dvnadd 25680 | The ` N ` -th derivative o... |
| dvn2bss 25681 | An N-times differentiable ... |
| dvnres 25682 | Multiple derivative versio... |
| cpnfval 25683 | Condition for n-times cont... |
| fncpn 25684 | The ` C^n ` object is a fu... |
| elcpn 25685 | Condition for n-times cont... |
| cpnord 25686 | ` C^n ` conditions are ord... |
| cpncn 25687 | A ` C^n ` function is cont... |
| cpnres 25688 | The restriction of a ` C^n... |
| dvaddbr 25689 | The sum rule for derivativ... |
| dvmulbr 25690 | The product rule for deriv... |
| dvmulbrOLD 25691 | Obsolete version of ~ dvmu... |
| dvadd 25692 | The sum rule for derivativ... |
| dvmul 25693 | The product rule for deriv... |
| dvaddf 25694 | The sum rule for everywher... |
| dvmulf 25695 | The product rule for every... |
| dvcmul 25696 | The product rule when one ... |
| dvcmulf 25697 | The product rule when one ... |
| dvcobr 25698 | The chain rule for derivat... |
| dvcobrOLD 25699 | Obsolete version of ~ dvco... |
| dvco 25700 | The chain rule for derivat... |
| dvcof 25701 | The chain rule for everywh... |
| dvcjbr 25702 | The derivative of the conj... |
| dvcj 25703 | The derivative of the conj... |
| dvfre 25704 | The derivative of a real f... |
| dvnfre 25705 | The ` N ` -th derivative o... |
| dvexp 25706 | Derivative of a power func... |
| dvexp2 25707 | Derivative of an exponenti... |
| dvrec 25708 | Derivative of the reciproc... |
| dvmptres3 25709 | Function-builder for deriv... |
| dvmptid 25710 | Function-builder for deriv... |
| dvmptc 25711 | Function-builder for deriv... |
| dvmptcl 25712 | Closure lemma for ~ dvmptc... |
| dvmptadd 25713 | Function-builder for deriv... |
| dvmptmul 25714 | Function-builder for deriv... |
| dvmptres2 25715 | Function-builder for deriv... |
| dvmptres 25716 | Function-builder for deriv... |
| dvmptcmul 25717 | Function-builder for deriv... |
| dvmptdivc 25718 | Function-builder for deriv... |
| dvmptneg 25719 | Function-builder for deriv... |
| dvmptsub 25720 | Function-builder for deriv... |
| dvmptcj 25721 | Function-builder for deriv... |
| dvmptre 25722 | Function-builder for deriv... |
| dvmptim 25723 | Function-builder for deriv... |
| dvmptntr 25724 | Function-builder for deriv... |
| dvmptco 25725 | Function-builder for deriv... |
| dvrecg 25726 | Derivative of the reciproc... |
| dvmptdiv 25727 | Function-builder for deriv... |
| dvmptfsum 25728 | Function-builder for deriv... |
| dvcnvlem 25729 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25730 | A weak version of ~ dvcnvr... |
| dvexp3 25731 | Derivative of an exponenti... |
| dveflem 25732 | Derivative of the exponent... |
| dvef 25733 | Derivative of the exponent... |
| dvsincos 25734 | Derivative of the sine and... |
| dvsin 25735 | Derivative of the sine fun... |
| dvcos 25736 | Derivative of the cosine f... |
| dvferm1lem 25737 | Lemma for ~ dvferm . (Con... |
| dvferm1 25738 | One-sided version of ~ dvf... |
| dvferm2lem 25739 | Lemma for ~ dvferm . (Con... |
| dvferm2 25740 | One-sided version of ~ dvf... |
| dvferm 25741 | Fermat's theorem on statio... |
| rollelem 25742 | Lemma for ~ rolle . (Cont... |
| rolle 25743 | Rolle's theorem. If ` F `... |
| cmvth 25744 | Cauchy's Mean Value Theore... |
| mvth 25745 | The Mean Value Theorem. I... |
| dvlip 25746 | A function with derivative... |
| dvlipcn 25747 | A complex function with de... |
| dvlip2 25748 | Combine the results of ~ d... |
| c1liplem1 25749 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25750 | C^1 functions are Lipschit... |
| c1lip2 25751 | C^1 functions are Lipschit... |
| c1lip3 25752 | C^1 functions are Lipschit... |
| dveq0 25753 | If a continuous function h... |
| dv11cn 25754 | Two functions defined on a... |
| dvgt0lem1 25755 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25756 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25757 | A function on a closed int... |
| dvlt0 25758 | A function on a closed int... |
| dvge0 25759 | A function on a closed int... |
| dvle 25760 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25761 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25762 | Lemma for ~ dvivth . (Con... |
| dvivth 25763 | Darboux' theorem, or the i... |
| dvne0 25764 | A function on a closed int... |
| dvne0f1 25765 | A function on a closed int... |
| lhop1lem 25766 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25767 | L'Hôpital's Rule for... |
| lhop2 25768 | L'Hôpital's Rule for... |
| lhop 25769 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25770 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25771 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25772 | The derivative rule for in... |
| dvcvx 25773 | A real function with stric... |
| dvfsumle 25774 | Compare a finite sum to an... |
| dvfsumge 25775 | Compare a finite sum to an... |
| dvfsumabs 25776 | Compare a finite sum to an... |
| dvmptrecl 25777 | Real closure of a derivati... |
| dvfsumrlimf 25778 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25779 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25780 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 25781 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25782 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25783 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25784 | Compare a finite sum to an... |
| dvfsumrlim2 25785 | Compare a finite sum to an... |
| dvfsumrlim3 25786 | Conjoin the statements of ... |
| dvfsum2 25787 | The reverse of ~ dvfsumrli... |
| ftc1lem1 25788 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 25789 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 25790 | The Fundamental Theorem of... |
| ftc1lem3 25791 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 25792 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 25793 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 25794 | Lemma for ~ ftc1 . (Contr... |
| ftc1 25795 | The Fundamental Theorem of... |
| ftc1cn 25796 | Strengthen the assumptions... |
| ftc2 25797 | The Fundamental Theorem of... |
| ftc2ditglem 25798 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 25799 | Directed integral analogue... |
| itgparts 25800 | Integration by parts. If ... |
| itgsubstlem 25801 | Lemma for ~ itgsubst . (C... |
| itgsubst 25802 | Integration by ` u ` -subs... |
| itgpowd 25803 | The integral of a monomial... |
| reldmmdeg 25808 | Multivariate degree is a b... |
| tdeglem1 25809 | Functionality of the total... |
| tdeglem1OLD 25810 | Obsolete version of ~ tdeg... |
| tdeglem3 25811 | Additivity of the total de... |
| tdeglem3OLD 25812 | Obsolete version of ~ tdeg... |
| tdeglem4 25813 | There is only one multi-in... |
| tdeglem4OLD 25814 | Obsolete version of ~ tdeg... |
| tdeglem2 25815 | Simplification of total de... |
| mdegfval 25816 | Value of the multivariate ... |
| mdegval 25817 | Value of the multivariate ... |
| mdegleb 25818 | Property of being of limit... |
| mdeglt 25819 | If there is an upper limit... |
| mdegldg 25820 | A nonzero polynomial has s... |
| mdegxrcl 25821 | Closure of polynomial degr... |
| mdegxrf 25822 | Functionality of polynomia... |
| mdegcl 25823 | Sharp closure for multivar... |
| mdeg0 25824 | Degree of the zero polynom... |
| mdegnn0cl 25825 | Degree of a nonzero polyno... |
| degltlem1 25826 | Theorem on arithmetic of e... |
| degltp1le 25827 | Theorem on arithmetic of e... |
| mdegaddle 25828 | The degree of a sum is at ... |
| mdegvscale 25829 | The degree of a scalar mul... |
| mdegvsca 25830 | The degree of a scalar mul... |
| mdegle0 25831 | A polynomial has nonpositi... |
| mdegmullem 25832 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 25833 | The multivariate degree of... |
| deg1fval 25834 | Relate univariate polynomi... |
| deg1xrf 25835 | Functionality of univariat... |
| deg1xrcl 25836 | Closure of univariate poly... |
| deg1cl 25837 | Sharp closure of univariat... |
| mdegpropd 25838 | Property deduction for pol... |
| deg1fvi 25839 | Univariate polynomial degr... |
| deg1propd 25840 | Property deduction for pol... |
| deg1z 25841 | Degree of the zero univari... |
| deg1nn0cl 25842 | Degree of a nonzero univar... |
| deg1n0ima 25843 | Degree image of a set of p... |
| deg1nn0clb 25844 | A polynomial is nonzero if... |
| deg1lt0 25845 | A polynomial is zero iff i... |
| deg1ldg 25846 | A nonzero univariate polyn... |
| deg1ldgn 25847 | An index at which a polyno... |
| deg1ldgdomn 25848 | A nonzero univariate polyn... |
| deg1leb 25849 | Property of being of limit... |
| deg1val 25850 | Value of the univariate de... |
| deg1lt 25851 | If the degree of a univari... |
| deg1ge 25852 | Conversely, a nonzero coef... |
| coe1mul3 25853 | The coefficient vector of ... |
| coe1mul4 25854 | Value of the "leading" coe... |
| deg1addle 25855 | The degree of a sum is at ... |
| deg1addle2 25856 | If both factors have degre... |
| deg1add 25857 | Exact degree of a sum of t... |
| deg1vscale 25858 | The degree of a scalar tim... |
| deg1vsca 25859 | The degree of a scalar tim... |
| deg1invg 25860 | The degree of the negated ... |
| deg1suble 25861 | The degree of a difference... |
| deg1sub 25862 | Exact degree of a differen... |
| deg1mulle2 25863 | Produce a bound on the pro... |
| deg1sublt 25864 | Subtraction of two polynom... |
| deg1le0 25865 | A polynomial has nonpositi... |
| deg1sclle 25866 | A scalar polynomial has no... |
| deg1scl 25867 | A nonzero scalar polynomia... |
| deg1mul2 25868 | Degree of multiplication o... |
| deg1mul3 25869 | Degree of multiplication o... |
| deg1mul3le 25870 | Degree of multiplication o... |
| deg1tmle 25871 | Limiting degree of a polyn... |
| deg1tm 25872 | Exact degree of a polynomi... |
| deg1pwle 25873 | Limiting degree of a varia... |
| deg1pw 25874 | Exact degree of a variable... |
| ply1nz 25875 | Univariate polynomials ove... |
| ply1nzb 25876 | Univariate polynomials are... |
| ply1domn 25877 | Corollary of ~ deg1mul2 : ... |
| ply1idom 25878 | The ring of univariate pol... |
| ply1divmo 25889 | Uniqueness of a quotient i... |
| ply1divex 25890 | Lemma for ~ ply1divalg : e... |
| ply1divalg 25891 | The division algorithm for... |
| ply1divalg2 25892 | Reverse the order of multi... |
| uc1pval 25893 | Value of the set of unitic... |
| isuc1p 25894 | Being a unitic polynomial.... |
| mon1pval 25895 | Value of the set of monic ... |
| ismon1p 25896 | Being a monic polynomial. ... |
| uc1pcl 25897 | Unitic polynomials are pol... |
| mon1pcl 25898 | Monic polynomials are poly... |
| uc1pn0 25899 | Unitic polynomials are not... |
| mon1pn0 25900 | Monic polynomials are not ... |
| uc1pdeg 25901 | Unitic polynomials have no... |
| uc1pldg 25902 | Unitic polynomials have un... |
| mon1pldg 25903 | Unitic polynomials have on... |
| mon1puc1p 25904 | Monic polynomials are unit... |
| uc1pmon1p 25905 | Make a unitic polynomial m... |
| deg1submon1p 25906 | The difference of two moni... |
| q1pval 25907 | Value of the univariate po... |
| q1peqb 25908 | Characterizing property of... |
| q1pcl 25909 | Closure of the quotient by... |
| r1pval 25910 | Value of the polynomial re... |
| r1pcl 25911 | Closure of remainder follo... |
| r1pdeglt 25912 | The remainder has a degree... |
| r1pid 25913 | Express the original polyn... |
| dvdsq1p 25914 | Divisibility in a polynomi... |
| dvdsr1p 25915 | Divisibility in a polynomi... |
| ply1remlem 25916 | A term of the form ` x - N... |
| ply1rem 25917 | The polynomial remainder t... |
| facth1 25918 | The factor theorem and its... |
| fta1glem1 25919 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 25920 | Lemma for ~ fta1g . (Cont... |
| fta1g 25921 | The one-sided fundamental ... |
| fta1blem 25922 | Lemma for ~ fta1b . (Cont... |
| fta1b 25923 | The assumption that ` R ` ... |
| drnguc1p 25924 | Over a division ring, all ... |
| ig1peu 25925 | There is a unique monic po... |
| ig1pval 25926 | Substitutions for the poly... |
| ig1pval2 25927 | Generator of the zero idea... |
| ig1pval3 25928 | Characterizing properties ... |
| ig1pcl 25929 | The monic generator of an ... |
| ig1pdvds 25930 | The monic generator of an ... |
| ig1prsp 25931 | Any ideal of polynomials o... |
| ply1lpir 25932 | The ring of polynomials ov... |
| ply1pid 25933 | The polynomials over a fie... |
| plyco0 25942 | Two ways to say that a fun... |
| plyval 25943 | Value of the polynomial se... |
| plybss 25944 | Reverse closure of the par... |
| elply 25945 | Definition of a polynomial... |
| elply2 25946 | The coefficient function c... |
| plyun0 25947 | The set of polynomials is ... |
| plyf 25948 | The polynomial is a functi... |
| plyss 25949 | The polynomial set functio... |
| plyssc 25950 | Every polynomial ring is c... |
| elplyr 25951 | Sufficient condition for e... |
| elplyd 25952 | Sufficient condition for e... |
| ply1termlem 25953 | Lemma for ~ ply1term . (C... |
| ply1term 25954 | A one-term polynomial. (C... |
| plypow 25955 | A power is a polynomial. ... |
| plyconst 25956 | A constant function is a p... |
| ne0p 25957 | A test to show that a poly... |
| ply0 25958 | The zero function is a pol... |
| plyid 25959 | The identity function is a... |
| plyeq0lem 25960 | Lemma for ~ plyeq0 . If `... |
| plyeq0 25961 | If a polynomial is zero at... |
| plypf1 25962 | Write the set of complex p... |
| plyaddlem1 25963 | Derive the coefficient fun... |
| plymullem1 25964 | Derive the coefficient fun... |
| plyaddlem 25965 | Lemma for ~ plyadd . (Con... |
| plymullem 25966 | Lemma for ~ plymul . (Con... |
| plyadd 25967 | The sum of two polynomials... |
| plymul 25968 | The product of two polynom... |
| plysub 25969 | The difference of two poly... |
| plyaddcl 25970 | The sum of two polynomials... |
| plymulcl 25971 | The product of two polynom... |
| plysubcl 25972 | The difference of two poly... |
| coeval 25973 | Value of the coefficient f... |
| coeeulem 25974 | Lemma for ~ coeeu . (Cont... |
| coeeu 25975 | Uniqueness of the coeffici... |
| coelem 25976 | Lemma for properties of th... |
| coeeq 25977 | If ` A ` satisfies the pro... |
| dgrval 25978 | Value of the degree functi... |
| dgrlem 25979 | Lemma for ~ dgrcl and simi... |
| coef 25980 | The domain and codomain of... |
| coef2 25981 | The domain and codomain of... |
| coef3 25982 | The domain and codomain of... |
| dgrcl 25983 | The degree of any polynomi... |
| dgrub 25984 | If the ` M ` -th coefficie... |
| dgrub2 25985 | All the coefficients above... |
| dgrlb 25986 | If all the coefficients ab... |
| coeidlem 25987 | Lemma for ~ coeid . (Cont... |
| coeid 25988 | Reconstruct a polynomial a... |
| coeid2 25989 | Reconstruct a polynomial a... |
| coeid3 25990 | Reconstruct a polynomial a... |
| plyco 25991 | The composition of two pol... |
| coeeq2 25992 | Compute the coefficient fu... |
| dgrle 25993 | Given an explicit expressi... |
| dgreq 25994 | If the highest term in a p... |
| 0dgr 25995 | A constant function has de... |
| 0dgrb 25996 | A function has degree zero... |
| dgrnznn 25997 | A nonzero polynomial with ... |
| coefv0 25998 | The result of evaluating a... |
| coeaddlem 25999 | Lemma for ~ coeadd and ~ d... |
| coemullem 26000 | Lemma for ~ coemul and ~ d... |
| coeadd 26001 | The coefficient function o... |
| coemul 26002 | A coefficient of a product... |
| coe11 26003 | The coefficient function i... |
| coemulhi 26004 | The leading coefficient of... |
| coemulc 26005 | The coefficient function i... |
| coe0 26006 | The coefficients of the ze... |
| coesub 26007 | The coefficient function o... |
| coe1termlem 26008 | The coefficient function o... |
| coe1term 26009 | The coefficient function o... |
| dgr1term 26010 | The degree of a monomial. ... |
| plycn 26011 | A polynomial is a continuo... |
| plycnOLD 26012 | Obsolete version of ~ plyc... |
| dgr0 26013 | The degree of the zero pol... |
| coeidp 26014 | The coefficients of the id... |
| dgrid 26015 | The degree of the identity... |
| dgreq0 26016 | The leading coefficient of... |
| dgrlt 26017 | Two ways to say that the d... |
| dgradd 26018 | The degree of a sum of pol... |
| dgradd2 26019 | The degree of a sum of pol... |
| dgrmul2 26020 | The degree of a product of... |
| dgrmul 26021 | The degree of a product of... |
| dgrmulc 26022 | Scalar multiplication by a... |
| dgrsub 26023 | The degree of a difference... |
| dgrcolem1 26024 | The degree of a compositio... |
| dgrcolem2 26025 | Lemma for ~ dgrco . (Cont... |
| dgrco 26026 | The degree of a compositio... |
| plycjlem 26027 | Lemma for ~ plycj and ~ co... |
| plycj 26028 | The double conjugation of ... |
| coecj 26029 | Double conjugation of a po... |
| plyrecj 26030 | A polynomial with real coe... |
| plymul0or 26031 | Polynomial multiplication ... |
| ofmulrt 26032 | The set of roots of a prod... |
| plyreres 26033 | Real-coefficient polynomia... |
| dvply1 26034 | Derivative of a polynomial... |
| dvply2g 26035 | The derivative of a polyno... |
| dvply2 26036 | The derivative of a polyno... |
| dvnply2 26037 | Polynomials have polynomia... |
| dvnply 26038 | Polynomials have polynomia... |
| plycpn 26039 | Polynomials are smooth. (... |
| quotval 26042 | Value of the quotient func... |
| plydivlem1 26043 | Lemma for ~ plydivalg . (... |
| plydivlem2 26044 | Lemma for ~ plydivalg . (... |
| plydivlem3 26045 | Lemma for ~ plydivex . Ba... |
| plydivlem4 26046 | Lemma for ~ plydivex . In... |
| plydivex 26047 | Lemma for ~ plydivalg . (... |
| plydiveu 26048 | Lemma for ~ plydivalg . (... |
| plydivalg 26049 | The division algorithm on ... |
| quotlem 26050 | Lemma for properties of th... |
| quotcl 26051 | The quotient of two polyno... |
| quotcl2 26052 | Closure of the quotient fu... |
| quotdgr 26053 | Remainder property of the ... |
| plyremlem 26054 | Closure of a linear factor... |
| plyrem 26055 | The polynomial remainder t... |
| facth 26056 | The factor theorem. If a ... |
| fta1lem 26057 | Lemma for ~ fta1 . (Contr... |
| fta1 26058 | The easy direction of the ... |
| quotcan 26059 | Exact division with a mult... |
| vieta1lem1 26060 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 26061 | Lemma for ~ vieta1 : induc... |
| vieta1 26062 | The first-order Vieta's fo... |
| plyexmo 26063 | An infinite set of values ... |
| elaa 26066 | Elementhood in the set of ... |
| aacn 26067 | An algebraic number is a c... |
| aasscn 26068 | The algebraic numbers are ... |
| elqaalem1 26069 | Lemma for ~ elqaa . The f... |
| elqaalem2 26070 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 26071 | Lemma for ~ elqaa . (Cont... |
| elqaa 26072 | The set of numbers generat... |
| qaa 26073 | Every rational number is a... |
| qssaa 26074 | The rational numbers are c... |
| iaa 26075 | The imaginary unit is alge... |
| aareccl 26076 | The reciprocal of an algeb... |
| aacjcl 26077 | The conjugate of an algebr... |
| aannenlem1 26078 | Lemma for ~ aannen . (Con... |
| aannenlem2 26079 | Lemma for ~ aannen . (Con... |
| aannenlem3 26080 | The algebraic numbers are ... |
| aannen 26081 | The algebraic numbers are ... |
| aalioulem1 26082 | Lemma for ~ aaliou . An i... |
| aalioulem2 26083 | Lemma for ~ aaliou . (Con... |
| aalioulem3 26084 | Lemma for ~ aaliou . (Con... |
| aalioulem4 26085 | Lemma for ~ aaliou . (Con... |
| aalioulem5 26086 | Lemma for ~ aaliou . (Con... |
| aalioulem6 26087 | Lemma for ~ aaliou . (Con... |
| aaliou 26088 | Liouville's theorem on dio... |
| geolim3 26089 | Geometric series convergen... |
| aaliou2 26090 | Liouville's approximation ... |
| aaliou2b 26091 | Liouville's approximation ... |
| aaliou3lem1 26092 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 26093 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 26094 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 26095 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 26096 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 26097 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 26098 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 26099 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 26100 | Example of a "Liouville nu... |
| aaliou3 26101 | Example of a "Liouville nu... |
| taylfvallem1 26106 | Lemma for ~ taylfval . (C... |
| taylfvallem 26107 | Lemma for ~ taylfval . (C... |
| taylfval 26108 | Define the Taylor polynomi... |
| eltayl 26109 | Value of the Taylor series... |
| taylf 26110 | The Taylor series defines ... |
| tayl0 26111 | The Taylor series is alway... |
| taylplem1 26112 | Lemma for ~ taylpfval and ... |
| taylplem2 26113 | Lemma for ~ taylpfval and ... |
| taylpfval 26114 | Define the Taylor polynomi... |
| taylpf 26115 | The Taylor polynomial is a... |
| taylpval 26116 | Value of the Taylor polyno... |
| taylply2 26117 | The Taylor polynomial is a... |
| taylply 26118 | The Taylor polynomial is a... |
| dvtaylp 26119 | The derivative of the Tayl... |
| dvntaylp 26120 | The ` M ` -th derivative o... |
| dvntaylp0 26121 | The first ` N ` derivative... |
| taylthlem1 26122 | Lemma for ~ taylth . This... |
| taylthlem2 26123 | Lemma for ~ taylth . (Con... |
| taylth 26124 | Taylor's theorem. The Tay... |
| ulmrel 26127 | The uniform limit relation... |
| ulmscl 26128 | Closure of the base set in... |
| ulmval 26129 | Express the predicate: Th... |
| ulmcl 26130 | Closure of a uniform limit... |
| ulmf 26131 | Closure of a uniform limit... |
| ulmpm 26132 | Closure of a uniform limit... |
| ulmf2 26133 | Closure of a uniform limit... |
| ulm2 26134 | Simplify ~ ulmval when ` F... |
| ulmi 26135 | The uniform limit property... |
| ulmclm 26136 | A uniform limit of functio... |
| ulmres 26137 | A sequence of functions co... |
| ulmshftlem 26138 | Lemma for ~ ulmshft . (Co... |
| ulmshft 26139 | A sequence of functions co... |
| ulm0 26140 | Every function converges u... |
| ulmuni 26141 | A sequence of functions un... |
| ulmdm 26142 | Two ways to express that a... |
| ulmcaulem 26143 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 26144 | A sequence of functions co... |
| ulmcau2 26145 | A sequence of functions co... |
| ulmss 26146 | A uniform limit of functio... |
| ulmbdd 26147 | A uniform limit of bounded... |
| ulmcn 26148 | A uniform limit of continu... |
| ulmdvlem1 26149 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 26150 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 26151 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 26152 | If ` F ` is a sequence of ... |
| mtest 26153 | The Weierstrass M-test. I... |
| mtestbdd 26154 | Given the hypotheses of th... |
| mbfulm 26155 | A uniform limit of measura... |
| iblulm 26156 | A uniform limit of integra... |
| itgulm 26157 | A uniform limit of integra... |
| itgulm2 26158 | A uniform limit of integra... |
| pserval 26159 | Value of the function ` G ... |
| pserval2 26160 | Value of the function ` G ... |
| psergf 26161 | The sequence of terms in t... |
| radcnvlem1 26162 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 26163 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 26164 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 26165 | Zero is always a convergen... |
| radcnvcl 26166 | The radius of convergence ... |
| radcnvlt1 26167 | If ` X ` is within the ope... |
| radcnvlt2 26168 | If ` X ` is within the ope... |
| radcnvle 26169 | If ` X ` is a convergent p... |
| dvradcnv 26170 | The radius of convergence ... |
| pserulm 26171 | If ` S ` is a region conta... |
| psercn2 26172 | Since by ~ pserulm the ser... |
| psercnlem2 26173 | Lemma for ~ psercn . (Con... |
| psercnlem1 26174 | Lemma for ~ psercn . (Con... |
| psercn 26175 | An infinite series converg... |
| pserdvlem1 26176 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 26177 | Lemma for ~ pserdv . (Con... |
| pserdv 26178 | The derivative of a power ... |
| pserdv2 26179 | The derivative of a power ... |
| abelthlem1 26180 | Lemma for ~ abelth . (Con... |
| abelthlem2 26181 | Lemma for ~ abelth . The ... |
| abelthlem3 26182 | Lemma for ~ abelth . (Con... |
| abelthlem4 26183 | Lemma for ~ abelth . (Con... |
| abelthlem5 26184 | Lemma for ~ abelth . (Con... |
| abelthlem6 26185 | Lemma for ~ abelth . (Con... |
| abelthlem7a 26186 | Lemma for ~ abelth . (Con... |
| abelthlem7 26187 | Lemma for ~ abelth . (Con... |
| abelthlem8 26188 | Lemma for ~ abelth . (Con... |
| abelthlem9 26189 | Lemma for ~ abelth . By a... |
| abelth 26190 | Abel's theorem. If the po... |
| abelth2 26191 | Abel's theorem, restricted... |
| efcn 26192 | The exponential function i... |
| sincn 26193 | Sine is continuous. (Cont... |
| coscn 26194 | Cosine is continuous. (Co... |
| reeff1olem 26195 | Lemma for ~ reeff1o . (Co... |
| reeff1o 26196 | The real exponential funct... |
| reefiso 26197 | The exponential function o... |
| efcvx 26198 | The exponential function o... |
| reefgim 26199 | The exponential function i... |
| pilem1 26200 | Lemma for ~ pire , ~ pigt2... |
| pilem2 26201 | Lemma for ~ pire , ~ pigt2... |
| pilem3 26202 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 26203 | ` _pi ` is between 2 and 4... |
| sinpi 26204 | The sine of ` _pi ` is 0. ... |
| pire 26205 | ` _pi ` is a real number. ... |
| picn 26206 | ` _pi ` is a complex numbe... |
| pipos 26207 | ` _pi ` is positive. (Con... |
| pirp 26208 | ` _pi ` is a positive real... |
| negpicn 26209 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 26210 | Lemma for ~ sinhalfpi and ... |
| halfpire 26211 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 26212 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 26213 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 26214 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 26215 | The sine of ` _pi / 2 ` is... |
| coshalfpi 26216 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 26217 | The cosine of ` -u _pi / 2... |
| efhalfpi 26218 | The exponential of ` _i _p... |
| cospi 26219 | The cosine of ` _pi ` is `... |
| efipi 26220 | The exponential of ` _i x.... |
| eulerid 26221 | Euler's identity. (Contri... |
| sin2pi 26222 | The sine of ` 2 _pi ` is 0... |
| cos2pi 26223 | The cosine of ` 2 _pi ` is... |
| ef2pi 26224 | The exponential of ` 2 _pi... |
| ef2kpi 26225 | If ` K ` is an integer, th... |
| efper 26226 | The exponential function i... |
| sinperlem 26227 | Lemma for ~ sinper and ~ c... |
| sinper 26228 | The sine function is perio... |
| cosper 26229 | The cosine function is per... |
| sin2kpi 26230 | If ` K ` is an integer, th... |
| cos2kpi 26231 | If ` K ` is an integer, th... |
| sin2pim 26232 | Sine of a number subtracte... |
| cos2pim 26233 | Cosine of a number subtrac... |
| sinmpi 26234 | Sine of a number less ` _p... |
| cosmpi 26235 | Cosine of a number less ` ... |
| sinppi 26236 | Sine of a number plus ` _p... |
| cosppi 26237 | Cosine of a number plus ` ... |
| efimpi 26238 | The exponential function a... |
| sinhalfpip 26239 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 26240 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 26241 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 26242 | The cosine of ` _pi / 2 ` ... |
| ptolemy 26243 | Ptolemy's Theorem. This t... |
| sincosq1lem 26244 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 26245 | The signs of the sine and ... |
| sincosq2sgn 26246 | The signs of the sine and ... |
| sincosq3sgn 26247 | The signs of the sine and ... |
| sincosq4sgn 26248 | The signs of the sine and ... |
| coseq00topi 26249 | Location of the zeroes of ... |
| coseq0negpitopi 26250 | Location of the zeroes of ... |
| tanrpcl 26251 | Positive real closure of t... |
| tangtx 26252 | The tangent function is gr... |
| tanabsge 26253 | The tangent function is gr... |
| sinq12gt0 26254 | The sine of a number stric... |
| sinq12ge0 26255 | The sine of a number betwe... |
| sinq34lt0t 26256 | The sine of a number stric... |
| cosq14gt0 26257 | The cosine of a number str... |
| cosq14ge0 26258 | The cosine of a number bet... |
| sincosq1eq 26259 | Complementarity of the sin... |
| sincos4thpi 26260 | The sine and cosine of ` _... |
| tan4thpi 26261 | The tangent of ` _pi / 4 `... |
| sincos6thpi 26262 | The sine and cosine of ` _... |
| sincos3rdpi 26263 | The sine and cosine of ` _... |
| pigt3 26264 | ` _pi ` is greater than 3.... |
| pige3 26265 | ` _pi ` is greater than or... |
| pige3ALT 26266 | Alternate proof of ~ pige3... |
| abssinper 26267 | The absolute value of sine... |
| sinkpi 26268 | The sine of an integer mul... |
| coskpi 26269 | The absolute value of the ... |
| sineq0 26270 | A complex number whose sin... |
| coseq1 26271 | A complex number whose cos... |
| cos02pilt1 26272 | Cosine is less than one be... |
| cosq34lt1 26273 | Cosine is less than one in... |
| efeq1 26274 | A complex number whose exp... |
| cosne0 26275 | The cosine function has no... |
| cosordlem 26276 | Lemma for ~ cosord . (Con... |
| cosord 26277 | Cosine is decreasing over ... |
| cos0pilt1 26278 | Cosine is between minus on... |
| cos11 26279 | Cosine is one-to-one over ... |
| sinord 26280 | Sine is increasing over th... |
| recosf1o 26281 | The cosine function is a b... |
| resinf1o 26282 | The sine function is a bij... |
| tanord1 26283 | The tangent function is st... |
| tanord 26284 | The tangent function is st... |
| tanregt0 26285 | The real part of the tange... |
| negpitopissre 26286 | The interval ` ( -u _pi (,... |
| efgh 26287 | The exponential function o... |
| efif1olem1 26288 | Lemma for ~ efif1o . (Con... |
| efif1olem2 26289 | Lemma for ~ efif1o . (Con... |
| efif1olem3 26290 | Lemma for ~ efif1o . (Con... |
| efif1olem4 26291 | The exponential function o... |
| efif1o 26292 | The exponential function o... |
| efifo 26293 | The exponential function o... |
| eff1olem 26294 | The exponential function m... |
| eff1o 26295 | The exponential function m... |
| efabl 26296 | The image of a subgroup of... |
| efsubm 26297 | The image of a subgroup of... |
| circgrp 26298 | The circle group ` T ` is ... |
| circsubm 26299 | The circle group ` T ` is ... |
| logrn 26304 | The range of the natural l... |
| ellogrn 26305 | Write out the property ` A... |
| dflog2 26306 | The natural logarithm func... |
| relogrn 26307 | The range of the natural l... |
| logrncn 26308 | The range of the natural l... |
| eff1o2 26309 | The exponential function r... |
| logf1o 26310 | The natural logarithm func... |
| dfrelog 26311 | The natural logarithm func... |
| relogf1o 26312 | The natural logarithm func... |
| logrncl 26313 | Closure of the natural log... |
| logcl 26314 | Closure of the natural log... |
| logimcl 26315 | Closure of the imaginary p... |
| logcld 26316 | The logarithm of a nonzero... |
| logimcld 26317 | The imaginary part of the ... |
| logimclad 26318 | The imaginary part of the ... |
| abslogimle 26319 | The imaginary part of the ... |
| logrnaddcl 26320 | The range of the natural l... |
| relogcl 26321 | Closure of the natural log... |
| eflog 26322 | Relationship between the n... |
| logeq0im1 26323 | If the logarithm of a numb... |
| logccne0 26324 | The logarithm isn't 0 if i... |
| logne0 26325 | Logarithm of a non-1 posit... |
| reeflog 26326 | Relationship between the n... |
| logef 26327 | Relationship between the n... |
| relogef 26328 | Relationship between the n... |
| logeftb 26329 | Relationship between the n... |
| relogeftb 26330 | Relationship between the n... |
| log1 26331 | The natural logarithm of `... |
| loge 26332 | The natural logarithm of `... |
| logneg 26333 | The natural logarithm of a... |
| logm1 26334 | The natural logarithm of n... |
| lognegb 26335 | If a number has imaginary ... |
| relogoprlem 26336 | Lemma for ~ relogmul and ~... |
| relogmul 26337 | The natural logarithm of t... |
| relogdiv 26338 | The natural logarithm of t... |
| explog 26339 | Exponentiation of a nonzer... |
| reexplog 26340 | Exponentiation of a positi... |
| relogexp 26341 | The natural logarithm of p... |
| relog 26342 | Real part of a logarithm. ... |
| relogiso 26343 | The natural logarithm func... |
| reloggim 26344 | The natural logarithm is a... |
| logltb 26345 | The natural logarithm func... |
| logfac 26346 | The logarithm of a factori... |
| eflogeq 26347 | Solve an equation involvin... |
| logleb 26348 | Natural logarithm preserve... |
| rplogcl 26349 | Closure of the logarithm f... |
| logge0 26350 | The logarithm of a number ... |
| logcj 26351 | The natural logarithm dist... |
| efiarg 26352 | The exponential of the "ar... |
| cosargd 26353 | The cosine of the argument... |
| cosarg0d 26354 | The cosine of the argument... |
| argregt0 26355 | Closure of the argument of... |
| argrege0 26356 | Closure of the argument of... |
| argimgt0 26357 | Closure of the argument of... |
| argimlt0 26358 | Closure of the argument of... |
| logimul 26359 | Multiplying a number by ` ... |
| logneg2 26360 | The logarithm of the negat... |
| logmul2 26361 | Generalization of ~ relogm... |
| logdiv2 26362 | Generalization of ~ relogd... |
| abslogle 26363 | Bound on the magnitude of ... |
| tanarg 26364 | The basic relation between... |
| logdivlti 26365 | The ` log x / x ` function... |
| logdivlt 26366 | The ` log x / x ` function... |
| logdivle 26367 | The ` log x / x ` function... |
| relogcld 26368 | Closure of the natural log... |
| reeflogd 26369 | Relationship between the n... |
| relogmuld 26370 | The natural logarithm of t... |
| relogdivd 26371 | The natural logarithm of t... |
| logled 26372 | Natural logarithm preserve... |
| relogefd 26373 | Relationship between the n... |
| rplogcld 26374 | Closure of the logarithm f... |
| logge0d 26375 | The logarithm of a number ... |
| logge0b 26376 | The logarithm of a number ... |
| loggt0b 26377 | The logarithm of a number ... |
| logle1b 26378 | The logarithm of a number ... |
| loglt1b 26379 | The logarithm of a number ... |
| divlogrlim 26380 | The inverse logarithm func... |
| logno1 26381 | The logarithm function is ... |
| dvrelog 26382 | The derivative of the real... |
| relogcn 26383 | The real logarithm functio... |
| ellogdm 26384 | Elementhood in the "contin... |
| logdmn0 26385 | A number in the continuous... |
| logdmnrp 26386 | A number in the continuous... |
| logdmss 26387 | The continuity domain of `... |
| logcnlem2 26388 | Lemma for ~ logcn . (Cont... |
| logcnlem3 26389 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26390 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26391 | Lemma for ~ logcn . (Cont... |
| logcn 26392 | The logarithm function is ... |
| dvloglem 26393 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26394 | The "continuous domain" of... |
| logf1o2 26395 | The logarithm maps its con... |
| dvlog 26396 | The derivative of the comp... |
| dvlog2lem 26397 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26398 | The derivative of the comp... |
| advlog 26399 | The antiderivative of the ... |
| advlogexp 26400 | The antiderivative of a po... |
| efopnlem1 26401 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26402 | Lemma for ~ efopn . (Cont... |
| efopn 26403 | The exponential map is an ... |
| logtayllem 26404 | Lemma for ~ logtayl . (Co... |
| logtayl 26405 | The Taylor series for ` -u... |
| logtaylsum 26406 | The Taylor series for ` -u... |
| logtayl2 26407 | Power series expression fo... |
| logccv 26408 | The natural logarithm func... |
| cxpval 26409 | Value of the complex power... |
| cxpef 26410 | Value of the complex power... |
| 0cxp 26411 | Value of the complex power... |
| cxpexpz 26412 | Relate the complex power f... |
| cxpexp 26413 | Relate the complex power f... |
| logcxp 26414 | Logarithm of a complex pow... |
| cxp0 26415 | Value of the complex power... |
| cxp1 26416 | Value of the complex power... |
| 1cxp 26417 | Value of the complex power... |
| ecxp 26418 | Write the exponential func... |
| cxpcl 26419 | Closure of the complex pow... |
| recxpcl 26420 | Real closure of the comple... |
| rpcxpcl 26421 | Positive real closure of t... |
| cxpne0 26422 | Complex exponentiation is ... |
| cxpeq0 26423 | Complex exponentiation is ... |
| cxpadd 26424 | Sum of exponents law for c... |
| cxpp1 26425 | Value of a nonzero complex... |
| cxpneg 26426 | Value of a complex number ... |
| cxpsub 26427 | Exponent subtraction law f... |
| cxpge0 26428 | Nonnegative exponentiation... |
| mulcxplem 26429 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26430 | Complex exponentiation of ... |
| cxprec 26431 | Complex exponentiation of ... |
| divcxp 26432 | Complex exponentiation of ... |
| cxpmul 26433 | Product of exponents law f... |
| cxpmul2 26434 | Product of exponents law f... |
| cxproot 26435 | The complex power function... |
| cxpmul2z 26436 | Generalize ~ cxpmul2 to ne... |
| abscxp 26437 | Absolute value of a power,... |
| abscxp2 26438 | Absolute value of a power,... |
| cxplt 26439 | Ordering property for comp... |
| cxple 26440 | Ordering property for comp... |
| cxplea 26441 | Ordering property for comp... |
| cxple2 26442 | Ordering property for comp... |
| cxplt2 26443 | Ordering property for comp... |
| cxple2a 26444 | Ordering property for comp... |
| cxplt3 26445 | Ordering property for comp... |
| cxple3 26446 | Ordering property for comp... |
| cxpsqrtlem 26447 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26448 | The complex exponential fu... |
| logsqrt 26449 | Logarithm of a square root... |
| cxp0d 26450 | Value of the complex power... |
| cxp1d 26451 | Value of the complex power... |
| 1cxpd 26452 | Value of the complex power... |
| cxpcld 26453 | Closure of the complex pow... |
| cxpmul2d 26454 | Product of exponents law f... |
| 0cxpd 26455 | Value of the complex power... |
| cxpexpzd 26456 | Relate the complex power f... |
| cxpefd 26457 | Value of the complex power... |
| cxpne0d 26458 | Complex exponentiation is ... |
| cxpp1d 26459 | Value of a nonzero complex... |
| cxpnegd 26460 | Value of a complex number ... |
| cxpmul2zd 26461 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26462 | Sum of exponents law for c... |
| cxpsubd 26463 | Exponent subtraction law f... |
| cxpltd 26464 | Ordering property for comp... |
| cxpled 26465 | Ordering property for comp... |
| cxplead 26466 | Ordering property for comp... |
| divcxpd 26467 | Complex exponentiation of ... |
| recxpcld 26468 | Positive real closure of t... |
| cxpge0d 26469 | Nonnegative exponentiation... |
| cxple2ad 26470 | Ordering property for comp... |
| cxplt2d 26471 | Ordering property for comp... |
| cxple2d 26472 | Ordering property for comp... |
| mulcxpd 26473 | Complex exponentiation of ... |
| recxpf1lem 26474 | Complex exponentiation on ... |
| cxpsqrtth 26475 | Square root theorem over t... |
| 2irrexpq 26476 | There exist irrational num... |
| cxprecd 26477 | Complex exponentiation of ... |
| rpcxpcld 26478 | Positive real closure of t... |
| logcxpd 26479 | Logarithm of a complex pow... |
| cxplt3d 26480 | Ordering property for comp... |
| cxple3d 26481 | Ordering property for comp... |
| cxpmuld 26482 | Product of exponents law f... |
| cxpgt0d 26483 | A positive real raised to ... |
| cxpcom 26484 | Commutative law for real e... |
| dvcxp1 26485 | The derivative of a comple... |
| dvcxp2 26486 | The derivative of a comple... |
| dvsqrt 26487 | The derivative of the real... |
| dvcncxp1 26488 | Derivative of complex powe... |
| dvcnsqrt 26489 | Derivative of square root ... |
| cxpcn 26490 | Domain of continuity of th... |
| cxpcn2 26491 | Continuity of the complex ... |
| cxpcn3lem 26492 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26493 | Extend continuity of the c... |
| resqrtcn 26494 | Continuity of the real squ... |
| sqrtcn 26495 | Continuity of the square r... |
| cxpaddlelem 26496 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26497 | Ordering property for comp... |
| abscxpbnd 26498 | Bound on the absolute valu... |
| root1id 26499 | Property of an ` N ` -th r... |
| root1eq1 26500 | The only powers of an ` N ... |
| root1cj 26501 | Within the ` N ` -th roots... |
| cxpeq 26502 | Solve an equation involvin... |
| loglesqrt 26503 | An upper bound on the loga... |
| logreclem 26504 | Symmetry of the natural lo... |
| logrec 26505 | Logarithm of a reciprocal ... |
| logbval 26508 | Define the value of the ` ... |
| logbcl 26509 | General logarithm closure.... |
| logbid1 26510 | General logarithm is 1 whe... |
| logb1 26511 | The logarithm of ` 1 ` to ... |
| elogb 26512 | The general logarithm of a... |
| logbchbase 26513 | Change of base for logarit... |
| relogbval 26514 | Value of the general logar... |
| relogbcl 26515 | Closure of the general log... |
| relogbzcl 26516 | Closure of the general log... |
| relogbreexp 26517 | Power law for the general ... |
| relogbzexp 26518 | Power law for the general ... |
| relogbmul 26519 | The logarithm of the produ... |
| relogbmulexp 26520 | The logarithm of the produ... |
| relogbdiv 26521 | The logarithm of the quoti... |
| relogbexp 26522 | Identity law for general l... |
| nnlogbexp 26523 | Identity law for general l... |
| logbrec 26524 | Logarithm of a reciprocal ... |
| logbleb 26525 | The general logarithm func... |
| logblt 26526 | The general logarithm func... |
| relogbcxp 26527 | Identity law for the gener... |
| cxplogb 26528 | Identity law for the gener... |
| relogbcxpb 26529 | The logarithm is the inver... |
| logbmpt 26530 | The general logarithm to a... |
| logbf 26531 | The general logarithm to a... |
| logbfval 26532 | The general logarithm of a... |
| relogbf 26533 | The general logarithm to a... |
| logblog 26534 | The general logarithm to t... |
| logbgt0b 26535 | The logarithm of a positiv... |
| logbgcd1irr 26536 | The logarithm of an intege... |
| 2logb9irr 26537 | Example for ~ logbgcd1irr ... |
| logbprmirr 26538 | The logarithm of a prime t... |
| 2logb3irr 26539 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26540 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26541 | The square root of two to ... |
| 2irrexpqALT 26542 | Alternate proof of ~ 2irre... |
| angval 26543 | Define the angle function,... |
| angcan 26544 | Cancel a constant multipli... |
| angneg 26545 | Cancel a negative sign in ... |
| angvald 26546 | The (signed) angle between... |
| angcld 26547 | The (signed) angle between... |
| angrteqvd 26548 | Two vectors are at a right... |
| cosangneg2d 26549 | The cosine of the angle be... |
| angrtmuld 26550 | Perpendicularity of two ve... |
| ang180lem1 26551 | Lemma for ~ ang180 . Show... |
| ang180lem2 26552 | Lemma for ~ ang180 . Show... |
| ang180lem3 26553 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26554 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26555 | Lemma for ~ ang180 : Redu... |
| ang180 26556 | The sum of angles ` m A B ... |
| lawcoslem1 26557 | Lemma for ~ lawcos . Here... |
| lawcos 26558 | Law of cosines (also known... |
| pythag 26559 | Pythagorean theorem. Give... |
| isosctrlem1 26560 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26561 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26562 | Lemma for ~ isosctr . Cor... |
| isosctr 26563 | Isosceles triangle theorem... |
| ssscongptld 26564 | If two triangles have equa... |
| affineequiv 26565 | Equivalence between two wa... |
| affineequiv2 26566 | Equivalence between two wa... |
| affineequiv3 26567 | Equivalence between two wa... |
| affineequiv4 26568 | Equivalence between two wa... |
| affineequivne 26569 | Equivalence between two wa... |
| angpieqvdlem 26570 | Equivalence used in the pr... |
| angpieqvdlem2 26571 | Equivalence used in ~ angp... |
| angpined 26572 | If the angle at ABC is ` _... |
| angpieqvd 26573 | The angle ABC is ` _pi ` i... |
| chordthmlem 26574 | If ` M ` is the midpoint o... |
| chordthmlem2 26575 | If M is the midpoint of AB... |
| chordthmlem3 26576 | If M is the midpoint of AB... |
| chordthmlem4 26577 | If P is on the segment AB ... |
| chordthmlem5 26578 | If P is on the segment AB ... |
| chordthm 26579 | The intersecting chords th... |
| heron 26580 | Heron's formula gives the ... |
| quad2 26581 | The quadratic equation, wi... |
| quad 26582 | The quadratic equation. (... |
| 1cubrlem 26583 | The cube roots of unity. ... |
| 1cubr 26584 | The cube roots of unity. ... |
| dcubic1lem 26585 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26586 | Reverse direction of ~ dcu... |
| dcubic1 26587 | Forward direction of ~ dcu... |
| dcubic 26588 | Solutions to the depressed... |
| mcubic 26589 | Solutions to a monic cubic... |
| cubic2 26590 | The solution to the genera... |
| cubic 26591 | The cubic equation, which ... |
| binom4 26592 | Work out a quartic binomia... |
| dquartlem1 26593 | Lemma for ~ dquart . (Con... |
| dquartlem2 26594 | Lemma for ~ dquart . (Con... |
| dquart 26595 | Solve a depressed quartic ... |
| quart1cl 26596 | Closure lemmas for ~ quart... |
| quart1lem 26597 | Lemma for ~ quart1 . (Con... |
| quart1 26598 | Depress a quartic equation... |
| quartlem1 26599 | Lemma for ~ quart . (Cont... |
| quartlem2 26600 | Closure lemmas for ~ quart... |
| quartlem3 26601 | Closure lemmas for ~ quart... |
| quartlem4 26602 | Closure lemmas for ~ quart... |
| quart 26603 | The quartic equation, writ... |
| asinlem 26610 | The argument to the logari... |
| asinlem2 26611 | The argument to the logari... |
| asinlem3a 26612 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26613 | The argument to the logari... |
| asinf 26614 | Domain and codomain of the... |
| asincl 26615 | Closure for the arcsin fun... |
| acosf 26616 | Domain and codoamin of the... |
| acoscl 26617 | Closure for the arccos fun... |
| atandm 26618 | Since the property is a li... |
| atandm2 26619 | This form of ~ atandm is a... |
| atandm3 26620 | A compact form of ~ atandm... |
| atandm4 26621 | A compact form of ~ atandm... |
| atanf 26622 | Domain and codoamin of the... |
| atancl 26623 | Closure for the arctan fun... |
| asinval 26624 | Value of the arcsin functi... |
| acosval 26625 | Value of the arccos functi... |
| atanval 26626 | Value of the arctan functi... |
| atanre 26627 | A real number is in the do... |
| asinneg 26628 | The arcsine function is od... |
| acosneg 26629 | The negative symmetry rela... |
| efiasin 26630 | The exponential of the arc... |
| sinasin 26631 | The arcsine function is an... |
| cosacos 26632 | The arccosine function is ... |
| asinsinlem 26633 | Lemma for ~ asinsin . (Co... |
| asinsin 26634 | The arcsine function compo... |
| acoscos 26635 | The arccosine function is ... |
| asin1 26636 | The arcsine of ` 1 ` is ` ... |
| acos1 26637 | The arccosine of ` 1 ` is ... |
| reasinsin 26638 | The arcsine function compo... |
| asinsinb 26639 | Relationship between sine ... |
| acoscosb 26640 | Relationship between cosin... |
| asinbnd 26641 | The arcsine function has r... |
| acosbnd 26642 | The arccosine function has... |
| asinrebnd 26643 | Bounds on the arcsine func... |
| asinrecl 26644 | The arcsine function is re... |
| acosrecl 26645 | The arccosine function is ... |
| cosasin 26646 | The cosine of the arcsine ... |
| sinacos 26647 | The sine of the arccosine ... |
| atandmneg 26648 | The domain of the arctange... |
| atanneg 26649 | The arctangent function is... |
| atan0 26650 | The arctangent of zero is ... |
| atandmcj 26651 | The arctangent function di... |
| atancj 26652 | The arctangent function di... |
| atanrecl 26653 | The arctangent function is... |
| efiatan 26654 | Value of the exponential o... |
| atanlogaddlem 26655 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26656 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26657 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26658 | A variation on ~ atanlogad... |
| efiatan2 26659 | Value of the exponential o... |
| 2efiatan 26660 | Value of the exponential o... |
| tanatan 26661 | The arctangent function is... |
| atandmtan 26662 | The tangent function has r... |
| cosatan 26663 | The cosine of an arctangen... |
| cosatanne0 26664 | The arctangent function ha... |
| atantan 26665 | The arctangent function is... |
| atantanb 26666 | Relationship between tange... |
| atanbndlem 26667 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26668 | The arctangent function is... |
| atanord 26669 | The arctangent function is... |
| atan1 26670 | The arctangent of ` 1 ` is... |
| bndatandm 26671 | A point in the open unit d... |
| atans 26672 | The "domain of continuity"... |
| atans2 26673 | It suffices to show that `... |
| atansopn 26674 | The domain of continuity o... |
| atansssdm 26675 | The domain of continuity o... |
| ressatans 26676 | The real number line is a ... |
| dvatan 26677 | The derivative of the arct... |
| atancn 26678 | The arctangent is a contin... |
| atantayl 26679 | The Taylor series for ` ar... |
| atantayl2 26680 | The Taylor series for ` ar... |
| atantayl3 26681 | The Taylor series for ` ar... |
| leibpilem1 26682 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26683 | The Leibniz formula for ` ... |
| leibpi 26684 | The Leibniz formula for ` ... |
| leibpisum 26685 | The Leibniz formula for ` ... |
| log2cnv 26686 | Using the Taylor series fo... |
| log2tlbnd 26687 | Bound the error term in th... |
| log2ublem1 26688 | Lemma for ~ log2ub . The ... |
| log2ublem2 26689 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26690 | Lemma for ~ log2ub . In d... |
| log2ub 26691 | ` log 2 ` is less than ` 2... |
| log2le1 26692 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26693 | Lemma for ~ birthday . (C... |
| birthdaylem2 26694 | For general ` N ` and ` K ... |
| birthdaylem3 26695 | For general ` N ` and ` K ... |
| birthday 26696 | The Birthday Problem. The... |
| dmarea 26699 | The domain of the area fun... |
| areambl 26700 | The fibers of a measurable... |
| areass 26701 | A measurable region is a s... |
| dfarea 26702 | Rewrite ~ df-area self-ref... |
| areaf 26703 | Area measurement is a func... |
| areacl 26704 | The area of a measurable r... |
| areage0 26705 | The area of a measurable r... |
| areaval 26706 | The area of a measurable r... |
| rlimcnp 26707 | Relate a limit of a real-v... |
| rlimcnp2 26708 | Relate a limit of a real-v... |
| rlimcnp3 26709 | Relate a limit of a real-v... |
| xrlimcnp 26710 | Relate a limit of a real-v... |
| efrlim 26711 | The limit of the sequence ... |
| dfef2 26712 | The limit of the sequence ... |
| cxplim 26713 | A power to a negative expo... |
| sqrtlim 26714 | The inverse square root fu... |
| rlimcxp 26715 | Any power to a positive ex... |
| o1cxp 26716 | An eventually bounded func... |
| cxp2limlem 26717 | A linear factor grows slow... |
| cxp2lim 26718 | Any power grows slower tha... |
| cxploglim 26719 | The logarithm grows slower... |
| cxploglim2 26720 | Every power of the logarit... |
| divsqrtsumlem 26721 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26722 | The function ` F ` used in... |
| divsqrsum 26723 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26724 | A bound on the distance of... |
| divsqrtsumo1 26725 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26726 | Closure of a 0-1 linear co... |
| scvxcvx 26727 | A strictly convex function... |
| jensenlem1 26728 | Lemma for ~ jensen . (Con... |
| jensenlem2 26729 | Lemma for ~ jensen . (Con... |
| jensen 26730 | Jensen's inequality, a fin... |
| amgmlem 26731 | Lemma for ~ amgm . (Contr... |
| amgm 26732 | Inequality of arithmetic a... |
| logdifbnd 26735 | Bound on the difference of... |
| logdiflbnd 26736 | Lower bound on the differe... |
| emcllem1 26737 | Lemma for ~ emcl . The se... |
| emcllem2 26738 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26739 | Lemma for ~ emcl . The fu... |
| emcllem4 26740 | Lemma for ~ emcl . The di... |
| emcllem5 26741 | Lemma for ~ emcl . The pa... |
| emcllem6 26742 | Lemma for ~ emcl . By the... |
| emcllem7 26743 | Lemma for ~ emcl and ~ har... |
| emcl 26744 | Closure and bounds for the... |
| harmonicbnd 26745 | A bound on the harmonic se... |
| harmonicbnd2 26746 | A bound on the harmonic se... |
| emre 26747 | The Euler-Mascheroni const... |
| emgt0 26748 | The Euler-Mascheroni const... |
| harmonicbnd3 26749 | A bound on the harmonic se... |
| harmoniclbnd 26750 | A bound on the harmonic se... |
| harmonicubnd 26751 | A bound on the harmonic se... |
| harmonicbnd4 26752 | The asymptotic behavior of... |
| fsumharmonic 26753 | Bound a finite sum based o... |
| zetacvg 26756 | The zeta series is converg... |
| eldmgm 26763 | Elementhood in the set of ... |
| dmgmaddn0 26764 | If ` A ` is not a nonposit... |
| dmlogdmgm 26765 | If ` A ` is in the continu... |
| rpdmgm 26766 | A positive real number is ... |
| dmgmn0 26767 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26768 | If ` A ` is not a nonposit... |
| dmgmdivn0 26769 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26770 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26771 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26772 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26773 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26774 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26775 | The series ` G ` is unifor... |
| lgamgulm 26776 | The series ` G ` is unifor... |
| lgamgulm2 26777 | Rewrite the limit of the s... |
| lgambdd 26778 | The log-Gamma function is ... |
| lgamucov 26779 | The ` U ` regions used in ... |
| lgamucov2 26780 | The ` U ` regions used in ... |
| lgamcvglem 26781 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 26782 | The log-Gamma function is ... |
| lgamf 26783 | The log-Gamma function is ... |
| gamf 26784 | The Gamma function is a co... |
| gamcl 26785 | The exponential of the log... |
| eflgam 26786 | The exponential of the log... |
| gamne0 26787 | The Gamma function is neve... |
| igamval 26788 | Value of the inverse Gamma... |
| igamz 26789 | Value of the inverse Gamma... |
| igamgam 26790 | Value of the inverse Gamma... |
| igamlgam 26791 | Value of the inverse Gamma... |
| igamf 26792 | Closure of the inverse Gam... |
| igamcl 26793 | Closure of the inverse Gam... |
| gamigam 26794 | The Gamma function is the ... |
| lgamcvg 26795 | The series ` G ` converges... |
| lgamcvg2 26796 | The series ` G ` converges... |
| gamcvg 26797 | The pointwise exponential ... |
| lgamp1 26798 | The functional equation of... |
| gamp1 26799 | The functional equation of... |
| gamcvg2lem 26800 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 26801 | An infinite product expres... |
| regamcl 26802 | The Gamma function is real... |
| relgamcl 26803 | The log-Gamma function is ... |
| rpgamcl 26804 | The log-Gamma function is ... |
| lgam1 26805 | The log-Gamma function at ... |
| gam1 26806 | The log-Gamma function at ... |
| facgam 26807 | The Gamma function general... |
| gamfac 26808 | The Gamma function general... |
| wilthlem1 26809 | The only elements that are... |
| wilthlem2 26810 | Lemma for ~ wilth : induct... |
| wilthlem3 26811 | Lemma for ~ wilth . Here ... |
| wilth 26812 | Wilson's theorem. A numbe... |
| wilthimp 26813 | The forward implication of... |
| ftalem1 26814 | Lemma for ~ fta : "growth... |
| ftalem2 26815 | Lemma for ~ fta . There e... |
| ftalem3 26816 | Lemma for ~ fta . There e... |
| ftalem4 26817 | Lemma for ~ fta : Closure... |
| ftalem5 26818 | Lemma for ~ fta : Main pr... |
| ftalem6 26819 | Lemma for ~ fta : Dischar... |
| ftalem7 26820 | Lemma for ~ fta . Shift t... |
| fta 26821 | The Fundamental Theorem of... |
| basellem1 26822 | Lemma for ~ basel . Closu... |
| basellem2 26823 | Lemma for ~ basel . Show ... |
| basellem3 26824 | Lemma for ~ basel . Using... |
| basellem4 26825 | Lemma for ~ basel . By ~ ... |
| basellem5 26826 | Lemma for ~ basel . Using... |
| basellem6 26827 | Lemma for ~ basel . The f... |
| basellem7 26828 | Lemma for ~ basel . The f... |
| basellem8 26829 | Lemma for ~ basel . The f... |
| basellem9 26830 | Lemma for ~ basel . Since... |
| basel 26831 | The sum of the inverse squ... |
| efnnfsumcl 26844 | Finite sum closure in the ... |
| ppisval 26845 | The set of primes less tha... |
| ppisval2 26846 | The set of primes less tha... |
| ppifi 26847 | The set of primes less tha... |
| prmdvdsfi 26848 | The set of prime divisors ... |
| chtf 26849 | Domain and codoamin of the... |
| chtcl 26850 | Real closure of the Chebys... |
| chtval 26851 | Value of the Chebyshev fun... |
| efchtcl 26852 | The Chebyshev function is ... |
| chtge0 26853 | The Chebyshev function is ... |
| vmaval 26854 | Value of the von Mangoldt ... |
| isppw 26855 | Two ways to say that ` A `... |
| isppw2 26856 | Two ways to say that ` A `... |
| vmappw 26857 | Value of the von Mangoldt ... |
| vmaprm 26858 | Value of the von Mangoldt ... |
| vmacl 26859 | Closure for the von Mangol... |
| vmaf 26860 | Functionality of the von M... |
| efvmacl 26861 | The von Mangoldt is closed... |
| vmage0 26862 | The von Mangoldt function ... |
| chpval 26863 | Value of the second Chebys... |
| chpf 26864 | Functionality of the secon... |
| chpcl 26865 | Closure for the second Che... |
| efchpcl 26866 | The second Chebyshev funct... |
| chpge0 26867 | The second Chebyshev funct... |
| ppival 26868 | Value of the prime-countin... |
| ppival2 26869 | Value of the prime-countin... |
| ppival2g 26870 | Value of the prime-countin... |
| ppif 26871 | Domain and codomain of the... |
| ppicl 26872 | Real closure of the prime-... |
| muval 26873 | The value of the Möbi... |
| muval1 26874 | The value of the Möbi... |
| muval2 26875 | The value of the Möbi... |
| isnsqf 26876 | Two ways to say that a num... |
| issqf 26877 | Two ways to say that a num... |
| sqfpc 26878 | The prime count of a squar... |
| dvdssqf 26879 | A divisor of a squarefree ... |
| sqf11 26880 | A squarefree number is com... |
| muf 26881 | The Möbius function i... |
| mucl 26882 | Closure of the Möbius... |
| sgmval 26883 | The value of the divisor f... |
| sgmval2 26884 | The value of the divisor f... |
| 0sgm 26885 | The value of the sum-of-di... |
| sgmf 26886 | The divisor function is a ... |
| sgmcl 26887 | Closure of the divisor fun... |
| sgmnncl 26888 | Closure of the divisor fun... |
| mule1 26889 | The Möbius function t... |
| chtfl 26890 | The Chebyshev function doe... |
| chpfl 26891 | The second Chebyshev funct... |
| ppiprm 26892 | The prime-counting functio... |
| ppinprm 26893 | The prime-counting functio... |
| chtprm 26894 | The Chebyshev function at ... |
| chtnprm 26895 | The Chebyshev function at ... |
| chpp1 26896 | The second Chebyshev funct... |
| chtwordi 26897 | The Chebyshev function is ... |
| chpwordi 26898 | The second Chebyshev funct... |
| chtdif 26899 | The difference of the Cheb... |
| efchtdvds 26900 | The exponentiated Chebyshe... |
| ppifl 26901 | The prime-counting functio... |
| ppip1le 26902 | The prime-counting functio... |
| ppiwordi 26903 | The prime-counting functio... |
| ppidif 26904 | The difference of the prim... |
| ppi1 26905 | The prime-counting functio... |
| cht1 26906 | The Chebyshev function at ... |
| vma1 26907 | The von Mangoldt function ... |
| chp1 26908 | The second Chebyshev funct... |
| ppi1i 26909 | Inference form of ~ ppiprm... |
| ppi2i 26910 | Inference form of ~ ppinpr... |
| ppi2 26911 | The prime-counting functio... |
| ppi3 26912 | The prime-counting functio... |
| cht2 26913 | The Chebyshev function at ... |
| cht3 26914 | The Chebyshev function at ... |
| ppinncl 26915 | Closure of the prime-count... |
| chtrpcl 26916 | Closure of the Chebyshev f... |
| ppieq0 26917 | The prime-counting functio... |
| ppiltx 26918 | The prime-counting functio... |
| prmorcht 26919 | Relate the primorial (prod... |
| mumullem1 26920 | Lemma for ~ mumul . A mul... |
| mumullem2 26921 | Lemma for ~ mumul . The p... |
| mumul 26922 | The Möbius function i... |
| sqff1o 26923 | There is a bijection from ... |
| fsumdvdsdiaglem 26924 | A "diagonal commutation" o... |
| fsumdvdsdiag 26925 | A "diagonal commutation" o... |
| fsumdvdscom 26926 | A double commutation of di... |
| dvdsppwf1o 26927 | A bijection from the divis... |
| dvdsflf1o 26928 | A bijection from the numbe... |
| dvdsflsumcom 26929 | A sum commutation from ` s... |
| fsumfldivdiaglem 26930 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 26931 | The right-hand side of ~ d... |
| musum 26932 | The sum of the Möbius... |
| musumsum 26933 | Evaluate a collapsing sum ... |
| muinv 26934 | The Möbius inversion ... |
| dvdsmulf1o 26935 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 26936 | Product of two divisor sum... |
| sgmppw 26937 | The value of the divisor f... |
| 0sgmppw 26938 | A prime power ` P ^ K ` ha... |
| 1sgmprm 26939 | The sum of divisors for a ... |
| 1sgm2ppw 26940 | The sum of the divisors of... |
| sgmmul 26941 | The divisor function for f... |
| ppiublem1 26942 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 26943 | A prime greater than ` 3 `... |
| ppiub 26944 | An upper bound on the prim... |
| vmalelog 26945 | The von Mangoldt function ... |
| chtlepsi 26946 | The first Chebyshev functi... |
| chprpcl 26947 | Closure of the second Cheb... |
| chpeq0 26948 | The second Chebyshev funct... |
| chteq0 26949 | The first Chebyshev functi... |
| chtleppi 26950 | Upper bound on the ` theta... |
| chtublem 26951 | Lemma for ~ chtub . (Cont... |
| chtub 26952 | An upper bound on the Cheb... |
| fsumvma 26953 | Rewrite a sum over the von... |
| fsumvma2 26954 | Apply ~ fsumvma for the co... |
| pclogsum 26955 | The logarithmic analogue o... |
| vmasum 26956 | The sum of the von Mangold... |
| logfac2 26957 | Another expression for the... |
| chpval2 26958 | Express the second Chebysh... |
| chpchtsum 26959 | The second Chebyshev funct... |
| chpub 26960 | An upper bound on the seco... |
| logfacubnd 26961 | A simple upper bound on th... |
| logfaclbnd 26962 | A lower bound on the logar... |
| logfacbnd3 26963 | Show the stronger statemen... |
| logfacrlim 26964 | Combine the estimates ~ lo... |
| logexprlim 26965 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 26966 | Write out ~ logfacrlim as ... |
| mersenne 26967 | A Mersenne prime is a prim... |
| perfect1 26968 | Euclid's contribution to t... |
| perfectlem1 26969 | Lemma for ~ perfect . (Co... |
| perfectlem2 26970 | Lemma for ~ perfect . (Co... |
| perfect 26971 | The Euclid-Euler theorem, ... |
| dchrval 26974 | Value of the group of Diri... |
| dchrbas 26975 | Base set of the group of D... |
| dchrelbas 26976 | A Dirichlet character is a... |
| dchrelbas2 26977 | A Dirichlet character is a... |
| dchrelbas3 26978 | A Dirichlet character is a... |
| dchrelbasd 26979 | A Dirichlet character is a... |
| dchrrcl 26980 | Reverse closure for a Diri... |
| dchrmhm 26981 | A Dirichlet character is a... |
| dchrf 26982 | A Dirichlet character is a... |
| dchrelbas4 26983 | A Dirichlet character is a... |
| dchrzrh1 26984 | Value of a Dirichlet chara... |
| dchrzrhcl 26985 | A Dirichlet character take... |
| dchrzrhmul 26986 | A Dirichlet character is c... |
| dchrplusg 26987 | Group operation on the gro... |
| dchrmul 26988 | Group operation on the gro... |
| dchrmulcl 26989 | Closure of the group opera... |
| dchrn0 26990 | A Dirichlet character is n... |
| dchr1cl 26991 | Closure of the principal D... |
| dchrmullid 26992 | Left identity for the prin... |
| dchrinvcl 26993 | Closure of the group inver... |
| dchrabl 26994 | The set of Dirichlet chara... |
| dchrfi 26995 | The group of Dirichlet cha... |
| dchrghm 26996 | A Dirichlet character rest... |
| dchr1 26997 | Value of the principal Dir... |
| dchreq 26998 | A Dirichlet character is d... |
| dchrresb 26999 | A Dirichlet character is d... |
| dchrabs 27000 | A Dirichlet character take... |
| dchrinv 27001 | The inverse of a Dirichlet... |
| dchrabs2 27002 | A Dirichlet character take... |
| dchr1re 27003 | The principal Dirichlet ch... |
| dchrptlem1 27004 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 27005 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 27006 | Lemma for ~ dchrpt . (Con... |
| dchrpt 27007 | For any element other than... |
| dchrsum2 27008 | An orthogonality relation ... |
| dchrsum 27009 | An orthogonality relation ... |
| sumdchr2 27010 | Lemma for ~ sumdchr . (Co... |
| dchrhash 27011 | There are exactly ` phi ( ... |
| sumdchr 27012 | An orthogonality relation ... |
| dchr2sum 27013 | An orthogonality relation ... |
| sum2dchr 27014 | An orthogonality relation ... |
| bcctr 27015 | Value of the central binom... |
| pcbcctr 27016 | Prime count of a central b... |
| bcmono 27017 | The binomial coefficient i... |
| bcmax 27018 | The binomial coefficient t... |
| bcp1ctr 27019 | Ratio of two central binom... |
| bclbnd 27020 | A bound on the binomial co... |
| efexple 27021 | Convert a bound on a power... |
| bpos1lem 27022 | Lemma for ~ bpos1 . (Cont... |
| bpos1 27023 | Bertrand's postulate, chec... |
| bposlem1 27024 | An upper bound on the prim... |
| bposlem2 27025 | There are no odd primes in... |
| bposlem3 27026 | Lemma for ~ bpos . Since ... |
| bposlem4 27027 | Lemma for ~ bpos . (Contr... |
| bposlem5 27028 | Lemma for ~ bpos . Bound ... |
| bposlem6 27029 | Lemma for ~ bpos . By usi... |
| bposlem7 27030 | Lemma for ~ bpos . The fu... |
| bposlem8 27031 | Lemma for ~ bpos . Evalua... |
| bposlem9 27032 | Lemma for ~ bpos . Derive... |
| bpos 27033 | Bertrand's postulate: ther... |
| zabsle1 27036 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 27037 | When ` a ` is coprime to t... |
| lgslem2 27038 | The set ` Z ` of all integ... |
| lgslem3 27039 | The set ` Z ` of all integ... |
| lgslem4 27040 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 27041 | Value of the Legendre symb... |
| lgsfval 27042 | Value of the function ` F ... |
| lgsfcl2 27043 | The function ` F ` is clos... |
| lgscllem 27044 | The Legendre symbol is an ... |
| lgsfcl 27045 | Closure of the function ` ... |
| lgsfle1 27046 | The function ` F ` has mag... |
| lgsval2lem 27047 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 27048 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 27049 | The Legendre symbol is an ... |
| lgs0 27050 | The Legendre symbol when t... |
| lgscl 27051 | The Legendre symbol is an ... |
| lgsle1 27052 | The Legendre symbol has ab... |
| lgsval2 27053 | The Legendre symbol at a p... |
| lgs2 27054 | The Legendre symbol at ` 2... |
| lgsval3 27055 | The Legendre symbol at an ... |
| lgsvalmod 27056 | The Legendre symbol is equ... |
| lgsval4 27057 | Restate ~ lgsval for nonze... |
| lgsfcl3 27058 | Closure of the function ` ... |
| lgsval4a 27059 | Same as ~ lgsval4 for posi... |
| lgscl1 27060 | The value of the Legendre ... |
| lgsneg 27061 | The Legendre symbol is eit... |
| lgsneg1 27062 | The Legendre symbol for no... |
| lgsmod 27063 | The Legendre (Jacobi) symb... |
| lgsdilem 27064 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 27065 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 27066 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 27067 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 27068 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 27069 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 27070 | The Legendre symbol is com... |
| lgsdirprm 27071 | The Legendre symbol is com... |
| lgsdir 27072 | The Legendre symbol is com... |
| lgsdilem2 27073 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 27074 | The Legendre symbol is com... |
| lgsne0 27075 | The Legendre symbol is non... |
| lgsabs1 27076 | The Legendre symbol is non... |
| lgssq 27077 | The Legendre symbol at a s... |
| lgssq2 27078 | The Legendre symbol at a s... |
| lgsprme0 27079 | The Legendre symbol at any... |
| 1lgs 27080 | The Legendre symbol at ` 1... |
| lgs1 27081 | The Legendre symbol at ` 1... |
| lgsmodeq 27082 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 27083 | The Legendre (Jacobi) symb... |
| lgsdirnn0 27084 | Variation on ~ lgsdir vali... |
| lgsdinn0 27085 | Variation on ~ lgsdi valid... |
| lgsqrlem1 27086 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 27087 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 27088 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 27089 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 27090 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 27091 | The Legendre symbol for od... |
| lgsqrmod 27092 | If the Legendre symbol of ... |
| lgsqrmodndvds 27093 | If the Legendre symbol of ... |
| lgsdchrval 27094 | The Legendre symbol functi... |
| lgsdchr 27095 | The Legendre symbol functi... |
| gausslemma2dlem0a 27096 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 27097 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 27098 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 27099 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 27100 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 27101 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 27102 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 27103 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 27104 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 27105 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 27106 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 27107 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 27108 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 27109 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 27110 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 27111 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 27112 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 27113 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 27114 | Gauss' Lemma (see also the... |
| lgseisenlem1 27115 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 27116 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 27117 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 27118 | Lemma for ~ lgseisen . Th... |
| lgseisen 27119 | Eisenstein's lemma, an exp... |
| lgsquadlem1 27120 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 27121 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 27122 | Lemma for ~ lgsquad . (Co... |
| lgsquad 27123 | The Law of Quadratic Recip... |
| lgsquad2lem1 27124 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 27125 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 27126 | Extend ~ lgsquad to coprim... |
| lgsquad3 27127 | Extend ~ lgsquad2 to integ... |
| m1lgs 27128 | The first supplement to th... |
| 2lgslem1a1 27129 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 27130 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 27131 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 27132 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 27133 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 27134 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 27135 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 27136 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 27137 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 27138 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 27139 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 27140 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 27141 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 27142 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 27143 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 27144 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 27145 | The Legendre symbol for ` ... |
| 2lgslem4 27146 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 27147 | The second supplement to t... |
| 2lgsoddprmlem1 27148 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 27149 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 27150 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 27151 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 27152 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 27153 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 27154 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 27155 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 27156 | The second supplement to t... |
| 2sqlem1 27157 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 27158 | Lemma for ~ 2sq . (Contri... |
| mul2sq 27159 | Fibonacci's identity (actu... |
| 2sqlem3 27160 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 27161 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 27162 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 27163 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 27164 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 27165 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 27166 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 27167 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 27168 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 27169 | Lemma for ~ 2sq . (Contri... |
| 2sq 27170 | All primes of the form ` 4... |
| 2sqblem 27171 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 27172 | The converse to ~ 2sq . (... |
| 2sq2 27173 | ` 2 ` is the sum of square... |
| 2sqn0 27174 | If the sum of two squares ... |
| 2sqcoprm 27175 | If the sum of two squares ... |
| 2sqmod 27176 | Given two decompositions o... |
| 2sqmo 27177 | There exists at most one d... |
| 2sqnn0 27178 | All primes of the form ` 4... |
| 2sqnn 27179 | All primes of the form ` 4... |
| addsq2reu 27180 | For each complex number ` ... |
| addsqn2reu 27181 | For each complex number ` ... |
| addsqrexnreu 27182 | For each complex number, t... |
| addsqnreup 27183 | There is no unique decompo... |
| addsq2nreurex 27184 | For each complex number ` ... |
| addsqn2reurex2 27185 | For each complex number ` ... |
| 2sqreulem1 27186 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 27187 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 27188 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 27189 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 27190 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 27191 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 27192 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 27193 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 27194 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 27195 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 27196 | There exists a unique deco... |
| 2sqreunn 27197 | There exists a unique deco... |
| 2sqreult 27198 | There exists a unique deco... |
| 2sqreultb 27199 | There exists a unique deco... |
| 2sqreunnlt 27200 | There exists a unique deco... |
| 2sqreunnltb 27201 | There exists a unique deco... |
| 2sqreuop 27202 | There exists a unique deco... |
| 2sqreuopnn 27203 | There exists a unique deco... |
| 2sqreuoplt 27204 | There exists a unique deco... |
| 2sqreuopltb 27205 | There exists a unique deco... |
| 2sqreuopnnlt 27206 | There exists a unique deco... |
| 2sqreuopnnltb 27207 | There exists a unique deco... |
| 2sqreuopb 27208 | There exists a unique deco... |
| chebbnd1lem1 27209 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 27210 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 27211 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 27212 | The Chebyshev bound: The ... |
| chtppilimlem1 27213 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 27214 | Lemma for ~ chtppilim . (... |
| chtppilim 27215 | The ` theta ` function is ... |
| chto1ub 27216 | The ` theta ` function is ... |
| chebbnd2 27217 | The Chebyshev bound, part ... |
| chto1lb 27218 | The ` theta ` function is ... |
| chpchtlim 27219 | The ` psi ` and ` theta ` ... |
| chpo1ub 27220 | The ` psi ` function is up... |
| chpo1ubb 27221 | The ` psi ` function is up... |
| vmadivsum 27222 | The sum of the von Mangold... |
| vmadivsumb 27223 | Give a total bound on the ... |
| rplogsumlem1 27224 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 27225 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 27226 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 27227 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 27228 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 27229 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 27230 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 27231 | Lemma for ~ dchrisum . Le... |
| dchrisum 27232 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 27233 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 27234 | The sum of the Möbius... |
| dchrvmasumlem1 27235 | An alternative expression ... |
| dchrvmasum2lem 27236 | Give an expression for ` l... |
| dchrvmasum2if 27237 | Combine the results of ~ d... |
| dchrvmasumlem2 27238 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 27239 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 27240 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 27241 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 27242 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 27243 | An asymptotic approximatio... |
| dchrvmaeq0 27244 | The set ` W ` is the colle... |
| dchrisum0fval 27245 | Value of the function ` F ... |
| dchrisum0fmul 27246 | The function ` F ` , the d... |
| dchrisum0ff 27247 | The function ` F ` is a re... |
| dchrisum0flblem1 27248 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 27249 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 27250 | The divisor sum of a real ... |
| dchrisum0fno1 27251 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 27252 | A partial result along the... |
| dchrisum0re 27253 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 27254 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 27255 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 27256 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 27257 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 27258 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 27259 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 27260 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 27261 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 27262 | The sum of the Möbius... |
| dchrvmasumlem 27263 | The sum of the Möbius... |
| dchrmusum 27264 | The sum of the Möbius... |
| dchrvmasum 27265 | The sum of the von Mangold... |
| rpvmasum 27266 | The sum of the von Mangold... |
| rplogsum 27267 | The sum of ` log p / p ` o... |
| dirith2 27268 | Dirichlet's theorem: there... |
| dirith 27269 | Dirichlet's theorem: there... |
| mudivsum 27270 | Asymptotic formula for ` s... |
| mulogsumlem 27271 | Lemma for ~ mulogsum . (C... |
| mulogsum 27272 | Asymptotic formula for ... |
| logdivsum 27273 | Asymptotic analysis of ... |
| mulog2sumlem1 27274 | Asymptotic formula for ... |
| mulog2sumlem2 27275 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 27276 | Lemma for ~ mulog2sum . (... |
| mulog2sum 27277 | Asymptotic formula for ... |
| vmalogdivsum2 27278 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 27279 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 27280 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 27281 | The sum ` sum_ m n <_ x , ... |
| logsqvma 27282 | A formula for ` log ^ 2 ( ... |
| logsqvma2 27283 | The Möbius inverse of... |
| log2sumbnd 27284 | Bound on the difference be... |
| selberglem1 27285 | Lemma for ~ selberg . Est... |
| selberglem2 27286 | Lemma for ~ selberg . (Co... |
| selberglem3 27287 | Lemma for ~ selberg . Est... |
| selberg 27288 | Selberg's symmetry formula... |
| selbergb 27289 | Convert eventual boundedne... |
| selberg2lem 27290 | Lemma for ~ selberg2 . Eq... |
| selberg2 27291 | Selberg's symmetry formula... |
| selberg2b 27292 | Convert eventual boundedne... |
| chpdifbndlem1 27293 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 27294 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 27295 | A bound on the difference ... |
| logdivbnd 27296 | A bound on a sum of logs, ... |
| selberg3lem1 27297 | Introduce a log weighting ... |
| selberg3lem2 27298 | Lemma for ~ selberg3 . Eq... |
| selberg3 27299 | Introduce a log weighting ... |
| selberg4lem1 27300 | Lemma for ~ selberg4 . Eq... |
| selberg4 27301 | The Selberg symmetry formu... |
| pntrval 27302 | Define the residual of the... |
| pntrf 27303 | Functionality of the resid... |
| pntrmax 27304 | There is a bound on the re... |
| pntrsumo1 27305 | A bound on a sum over ` R ... |
| pntrsumbnd 27306 | A bound on a sum over ` R ... |
| pntrsumbnd2 27307 | A bound on a sum over ` R ... |
| selbergr 27308 | Selberg's symmetry formula... |
| selberg3r 27309 | Selberg's symmetry formula... |
| selberg4r 27310 | Selberg's symmetry formula... |
| selberg34r 27311 | The sum of ~ selberg3r and... |
| pntsval 27312 | Define the "Selberg functi... |
| pntsf 27313 | Functionality of the Selbe... |
| selbergs 27314 | Selberg's symmetry formula... |
| selbergsb 27315 | Selberg's symmetry formula... |
| pntsval2 27316 | The Selberg function can b... |
| pntrlog2bndlem1 27317 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 27318 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 27319 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 27320 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 27321 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 27322 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 27323 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 27324 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 27325 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 27326 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 27327 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 27328 | Lemma for ~ pnt . Establi... |
| pntibndlem1 27329 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 27330 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 27331 | Lemma for ~ pntibnd . The... |
| pntibndlem3 27332 | Lemma for ~ pntibnd . Pac... |
| pntibnd 27333 | Lemma for ~ pnt . Establi... |
| pntlemd 27334 | Lemma for ~ pnt . Closure... |
| pntlemc 27335 | Lemma for ~ pnt . Closure... |
| pntlema 27336 | Lemma for ~ pnt . Closure... |
| pntlemb 27337 | Lemma for ~ pnt . Unpack ... |
| pntlemg 27338 | Lemma for ~ pnt . Closure... |
| pntlemh 27339 | Lemma for ~ pnt . Bounds ... |
| pntlemn 27340 | Lemma for ~ pnt . The "na... |
| pntlemq 27341 | Lemma for ~ pntlemj . (Co... |
| pntlemr 27342 | Lemma for ~ pntlemj . (Co... |
| pntlemj 27343 | Lemma for ~ pnt . The ind... |
| pntlemi 27344 | Lemma for ~ pnt . Elimina... |
| pntlemf 27345 | Lemma for ~ pnt . Add up ... |
| pntlemk 27346 | Lemma for ~ pnt . Evaluat... |
| pntlemo 27347 | Lemma for ~ pnt . Combine... |
| pntleme 27348 | Lemma for ~ pnt . Package... |
| pntlem3 27349 | Lemma for ~ pnt . Equatio... |
| pntlemp 27350 | Lemma for ~ pnt . Wrappin... |
| pntleml 27351 | Lemma for ~ pnt . Equatio... |
| pnt3 27352 | The Prime Number Theorem, ... |
| pnt2 27353 | The Prime Number Theorem, ... |
| pnt 27354 | The Prime Number Theorem: ... |
| abvcxp 27355 | Raising an absolute value ... |
| padicfval 27356 | Value of the p-adic absolu... |
| padicval 27357 | Value of the p-adic absolu... |
| ostth2lem1 27358 | Lemma for ~ ostth2 , altho... |
| qrngbas 27359 | The base set of the field ... |
| qdrng 27360 | The rationals form a divis... |
| qrng0 27361 | The zero element of the fi... |
| qrng1 27362 | The unity element of the f... |
| qrngneg 27363 | The additive inverse in th... |
| qrngdiv 27364 | The division operation in ... |
| qabvle 27365 | By using induction on ` N ... |
| qabvexp 27366 | Induct the product rule ~ ... |
| ostthlem1 27367 | Lemma for ~ ostth . If tw... |
| ostthlem2 27368 | Lemma for ~ ostth . Refin... |
| qabsabv 27369 | The regular absolute value... |
| padicabv 27370 | The p-adic absolute value ... |
| padicabvf 27371 | The p-adic absolute value ... |
| padicabvcxp 27372 | All positive powers of the... |
| ostth1 27373 | - Lemma for ~ ostth : triv... |
| ostth2lem2 27374 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 27375 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 27376 | Lemma for ~ ostth2 . (Con... |
| ostth2 27377 | - Lemma for ~ ostth : regu... |
| ostth3 27378 | - Lemma for ~ ostth : p-ad... |
| ostth 27379 | Ostrowski's theorem, which... |
| elno 27386 | Membership in the surreals... |
| sltval 27387 | The value of the surreal l... |
| bdayval 27388 | The value of the birthday ... |
| nofun 27389 | A surreal is a function. ... |
| nodmon 27390 | The domain of a surreal is... |
| norn 27391 | The range of a surreal is ... |
| nofnbday 27392 | A surreal is a function ov... |
| nodmord 27393 | The domain of a surreal ha... |
| elno2 27394 | An alternative condition f... |
| elno3 27395 | Another condition for memb... |
| sltval2 27396 | Alternate expression for s... |
| nofv 27397 | The function value of a su... |
| nosgnn0 27398 | ` (/) ` is not a surreal s... |
| nosgnn0i 27399 | If ` X ` is a surreal sign... |
| noreson 27400 | The restriction of a surre... |
| sltintdifex 27401 |
If ` A |
| sltres 27402 | If the restrictions of two... |
| noxp1o 27403 | The Cartesian product of a... |
| noseponlem 27404 | Lemma for ~ nosepon . Con... |
| nosepon 27405 | Given two unequal surreals... |
| noextend 27406 | Extending a surreal by one... |
| noextendseq 27407 | Extend a surreal by a sequ... |
| noextenddif 27408 | Calculate the place where ... |
| noextendlt 27409 | Extending a surreal with a... |
| noextendgt 27410 | Extending a surreal with a... |
| nolesgn2o 27411 | Given ` A ` less-than or e... |
| nolesgn2ores 27412 | Given ` A ` less-than or e... |
| nogesgn1o 27413 | Given ` A ` greater than o... |
| nogesgn1ores 27414 | Given ` A ` greater than o... |
| sltsolem1 27415 | Lemma for ~ sltso . The "... |
| sltso 27416 | Less-than totally orders t... |
| bdayfo 27417 | The birthday function maps... |
| fvnobday 27418 | The value of a surreal at ... |
| nosepnelem 27419 | Lemma for ~ nosepne . (Co... |
| nosepne 27420 | The value of two non-equal... |
| nosep1o 27421 | If the value of a surreal ... |
| nosep2o 27422 | If the value of a surreal ... |
| nosepdmlem 27423 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27424 | The first place two surrea... |
| nosepeq 27425 | The values of two surreals... |
| nosepssdm 27426 | Given two non-equal surrea... |
| nodenselem4 27427 | Lemma for ~ nodense . Sho... |
| nodenselem5 27428 | Lemma for ~ nodense . If ... |
| nodenselem6 27429 | The restriction of a surre... |
| nodenselem7 27430 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27431 | Lemma for ~ nodense . Giv... |
| nodense 27432 | Given two distinct surreal... |
| bdayimaon 27433 | Lemma for full-eta propert... |
| nolt02olem 27434 | Lemma for ~ nolt02o . If ... |
| nolt02o 27435 | Given ` A ` less-than ` B ... |
| nogt01o 27436 | Given ` A ` greater than `... |
| noresle 27437 | Restriction law for surrea... |
| nomaxmo 27438 | A class of surreals has at... |
| nominmo 27439 | A class of surreals has at... |
| nosupprefixmo 27440 | In any class of surreals, ... |
| noinfprefixmo 27441 | In any class of surreals, ... |
| nosupcbv 27442 | Lemma to change bound vari... |
| nosupno 27443 | The next several theorems ... |
| nosupdm 27444 | The domain of the surreal ... |
| nosupbday 27445 | Birthday bounding law for ... |
| nosupfv 27446 | The value of surreal supre... |
| nosupres 27447 | A restriction law for surr... |
| nosupbnd1lem1 27448 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27449 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27450 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27451 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27452 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27453 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27454 | Bounding law from below fo... |
| nosupbnd2lem1 27455 | Bounding law from above wh... |
| nosupbnd2 27456 | Bounding law from above fo... |
| noinfcbv 27457 | Change bound variables for... |
| noinfno 27458 | The next several theorems ... |
| noinfdm 27459 | Next, we calculate the dom... |
| noinfbday 27460 | Birthday bounding law for ... |
| noinffv 27461 | The value of surreal infim... |
| noinfres 27462 | The restriction of surreal... |
| noinfbnd1lem1 27463 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27464 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27465 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27466 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27467 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27468 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27469 | Bounding law from above fo... |
| noinfbnd2lem1 27470 | Bounding law from below wh... |
| noinfbnd2 27471 | Bounding law from below fo... |
| nosupinfsep 27472 | Given two sets of surreals... |
| noetasuplem1 27473 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27474 | Lemma for ~ noeta . The r... |
| noetasuplem3 27475 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27476 | Lemma for ~ noeta . When ... |
| noetainflem1 27477 | Lemma for ~ noeta . Estab... |
| noetainflem2 27478 | Lemma for ~ noeta . The r... |
| noetainflem3 27479 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27480 | Lemma for ~ noeta . If ` ... |
| noetalem1 27481 | Lemma for ~ noeta . Eithe... |
| noetalem2 27482 | Lemma for ~ noeta . The f... |
| noeta 27483 | The full-eta axiom for the... |
| sltirr 27486 | Surreal less-than is irref... |
| slttr 27487 | Surreal less-than is trans... |
| sltasym 27488 | Surreal less-than is asymm... |
| sltlin 27489 | Surreal less-than obeys tr... |
| slttrieq2 27490 | Trichotomy law for surreal... |
| slttrine 27491 | Trichotomy law for surreal... |
| slenlt 27492 | Surreal less-than or equal... |
| sltnle 27493 | Surreal less-than in terms... |
| sleloe 27494 | Surreal less-than or equal... |
| sletri3 27495 | Trichotomy law for surreal... |
| sltletr 27496 | Surreal transitive law. (... |
| slelttr 27497 | Surreal transitive law. (... |
| sletr 27498 | Surreal transitive law. (... |
| slttrd 27499 | Surreal less-than is trans... |
| sltletrd 27500 | Surreal less-than is trans... |
| slelttrd 27501 | Surreal less-than is trans... |
| sletrd 27502 | Surreal less-than or equal... |
| slerflex 27503 | Surreal less-than or equal... |
| sletric 27504 | Surreal trichotomy law. (... |
| maxs1 27505 | A surreal is less than or ... |
| maxs2 27506 | A surreal is less than or ... |
| mins1 27507 | The minimum of two surreal... |
| mins2 27508 | The minimum of two surreal... |
| sltled 27509 | Surreal less-than implies ... |
| sltne 27510 | Surreal less-than implies ... |
| bdayfun 27511 | The birthday function is a... |
| bdayfn 27512 | The birthday function is a... |
| bdaydm 27513 | The birthday function's do... |
| bdayrn 27514 | The birthday function's ra... |
| bdayelon 27515 | The value of the birthday ... |
| nocvxminlem 27516 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27517 | Given a nonempty convex cl... |
| noprc 27518 | The surreal numbers are a ... |
| noeta2 27523 | A version of ~ noeta with ... |
| brsslt 27524 | Binary relation form of th... |
| ssltex1 27525 | The first argument of surr... |
| ssltex2 27526 | The second argument of sur... |
| ssltss1 27527 | The first argument of surr... |
| ssltss2 27528 | The second argument of sur... |
| ssltsep 27529 | The separation property of... |
| ssltd 27530 | Deduce surreal set less-th... |
| ssltsn 27531 | Surreal set less-than of t... |
| ssltsepc 27532 | Two elements of separated ... |
| ssltsepcd 27533 | Two elements of separated ... |
| sssslt1 27534 | Relation between surreal s... |
| sssslt2 27535 | Relation between surreal s... |
| nulsslt 27536 | The empty set is less-than... |
| nulssgt 27537 | The empty set is greater t... |
| conway 27538 | Conway's Simplicity Theore... |
| scutval 27539 | The value of the surreal c... |
| scutcut 27540 | Cut properties of the surr... |
| scutcl 27541 | Closure law for surreal cu... |
| scutcld 27542 | Closure law for surreal cu... |
| scutbday 27543 | The birthday of the surrea... |
| eqscut 27544 | Condition for equality to ... |
| eqscut2 27545 | Condition for equality to ... |
| sslttr 27546 | Transitive law for surreal... |
| ssltun1 27547 | Union law for surreal set ... |
| ssltun2 27548 | Union law for surreal set ... |
| scutun12 27549 | Union law for surreal cuts... |
| dmscut 27550 | The domain of the surreal ... |
| scutf 27551 | Functionality statement fo... |
| etasslt 27552 | A restatement of ~ noeta u... |
| etasslt2 27553 | A version of ~ etasslt wit... |
| scutbdaybnd 27554 | An upper bound on the birt... |
| scutbdaybnd2 27555 | An upper bound on the birt... |
| scutbdaybnd2lim 27556 | An upper bound on the birt... |
| scutbdaylt 27557 | If a surreal lies in a gap... |
| slerec 27558 | A comparison law for surre... |
| sltrec 27559 | A comparison law for surre... |
| ssltdisj 27560 | If ` A ` preceeds ` B ` , ... |
| 0sno 27565 | Surreal zero is a surreal.... |
| 1sno 27566 | Surreal one is a surreal. ... |
| bday0s 27567 | Calculate the birthday of ... |
| 0slt1s 27568 | Surreal zero is less than ... |
| bday0b 27569 | The only surreal with birt... |
| bday1s 27570 | The birthday of surreal on... |
| cuteq0 27571 | Condition for a surreal cu... |
| cuteq1 27572 | Condition for a surreal cu... |
| sgt0ne0 27573 | A positive surreal is not ... |
| sgt0ne0d 27574 | A positive surreal is not ... |
| madeval 27585 | The value of the made by f... |
| madeval2 27586 | Alternative characterizati... |
| oldval 27587 | The value of the old optio... |
| newval 27588 | The value of the new optio... |
| madef 27589 | The made function is a fun... |
| oldf 27590 | The older function is a fu... |
| newf 27591 | The new function is a func... |
| old0 27592 | No surreal is older than `... |
| madessno 27593 | Made sets are surreals. (... |
| oldssno 27594 | Old sets are surreals. (C... |
| newssno 27595 | New sets are surreals. (C... |
| leftval 27596 | The value of the left opti... |
| rightval 27597 | The value of the right opt... |
| leftf 27598 | The functionality of the l... |
| rightf 27599 | The functionality of the r... |
| elmade 27600 | Membership in the made fun... |
| elmade2 27601 | Membership in the made fun... |
| elold 27602 | Membership in an old set. ... |
| ssltleft 27603 | A surreal is greater than ... |
| ssltright 27604 | A surreal is less than its... |
| lltropt 27605 | The left options of a surr... |
| made0 27606 | The only surreal made on d... |
| new0 27607 | The only surreal new on da... |
| old1 27608 | The only surreal older tha... |
| madess 27609 | If ` A ` is less than or e... |
| oldssmade 27610 | The older-than set is a su... |
| leftssold 27611 | The left options are a sub... |
| rightssold 27612 | The right options are a su... |
| leftssno 27613 | The left set of a surreal ... |
| rightssno 27614 | The right set of a surreal... |
| madecut 27615 | Given a section that is a ... |
| madeun 27616 | The made set is the union ... |
| madeoldsuc 27617 | The made set is the old se... |
| oldsuc 27618 | The value of the old set a... |
| oldlim 27619 | The value of the old set a... |
| madebdayim 27620 | If a surreal is a member o... |
| oldbdayim 27621 | If ` X ` is in the old set... |
| oldirr 27622 | No surreal is a member of ... |
| leftirr 27623 | No surreal is a member of ... |
| rightirr 27624 | No surreal is a member of ... |
| left0s 27625 | The left set of ` 0s ` is ... |
| right0s 27626 | The right set of ` 0s ` is... |
| left1s 27627 | The left set of ` 1s ` is ... |
| right1s 27628 | The right set of ` 1s ` is... |
| lrold 27629 | The union of the left and ... |
| madebdaylemold 27630 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27631 | Lemma for ~ madebday . If... |
| madebday 27632 | A surreal is part of the s... |
| oldbday 27633 | A surreal is part of the s... |
| newbday 27634 | A surreal is an element of... |
| lrcut 27635 | A surreal is equal to the ... |
| scutfo 27636 | The surreal cut function i... |
| sltn0 27637 | If ` X ` is less than ` Y ... |
| lruneq 27638 | If two surreals share a bi... |
| sltlpss 27639 | If two surreals share a bi... |
| slelss 27640 | If two surreals ` A ` and ... |
| 0elold 27641 | Zero is in the old set of ... |
| 0elleft 27642 | Zero is in the left set of... |
| 0elright 27643 | Zero is in the right set o... |
| cofsslt 27644 | If every element of ` A ` ... |
| coinitsslt 27645 | If ` B ` is coinitial with... |
| cofcut1 27646 | If ` C ` is cofinal with `... |
| cofcut1d 27647 | If ` C ` is cofinal with `... |
| cofcut2 27648 | If ` A ` and ` C ` are mut... |
| cofcut2d 27649 | If ` A ` and ` C ` are mut... |
| cofcutr 27650 | If ` X ` is the cut of ` A... |
| cofcutr1d 27651 | If ` X ` is the cut of ` A... |
| cofcutr2d 27652 | If ` X ` is the cut of ` A... |
| cofcutrtime 27653 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27654 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27655 | If ` X ` is a timely cut o... |
| cofss 27656 | Cofinality for a subset. ... |
| coiniss 27657 | Coinitiality for a subset.... |
| cutlt 27658 | Eliminating all elements b... |
| cutpos 27659 | Reduce the elements of a c... |
| lrrecval 27662 | The next step in the devel... |
| lrrecval2 27663 | Next, we establish an alte... |
| lrrecpo 27664 | Now, we establish that ` R... |
| lrrecse 27665 | Next, we show that ` R ` i... |
| lrrecfr 27666 | Now we show that ` R ` is ... |
| lrrecpred 27667 | Finally, we calculate the ... |
| noinds 27668 | Induction principle for a ... |
| norecfn 27669 | Surreal recursion over one... |
| norecov 27670 | Calculate the value of the... |
| noxpordpo 27673 | To get through most of the... |
| noxpordfr 27674 | Next we establish the foun... |
| noxpordse 27675 | Next we establish the set-... |
| noxpordpred 27676 | Next we calculate the pred... |
| no2indslem 27677 | Double induction on surrea... |
| no2inds 27678 | Double induction on surrea... |
| norec2fn 27679 | The double-recursion opera... |
| norec2ov 27680 | The value of the double-re... |
| no3inds 27681 | Triple induction over surr... |
| addsfn 27684 | Surreal addition is a func... |
| addsval 27685 | The value of surreal addit... |
| addsval2 27686 | The value of surreal addit... |
| addsrid 27687 | Surreal addition to zero i... |
| addsridd 27688 | Surreal addition to zero i... |
| addscom 27689 | Surreal addition commutes.... |
| addscomd 27690 | Surreal addition commutes.... |
| addslid 27691 | Surreal addition to zero i... |
| addsproplem1 27692 | Lemma for surreal addition... |
| addsproplem2 27693 | Lemma for surreal addition... |
| addsproplem3 27694 | Lemma for surreal addition... |
| addsproplem4 27695 | Lemma for surreal addition... |
| addsproplem5 27696 | Lemma for surreal addition... |
| addsproplem6 27697 | Lemma for surreal addition... |
| addsproplem7 27698 | Lemma for surreal addition... |
| addsprop 27699 | Inductively show that surr... |
| addscutlem 27700 | Lemma for ~ addscut . Sho... |
| addscut 27701 | Demonstrate the cut proper... |
| addscut2 27702 | Show that the cut involved... |
| addscld 27703 | Surreal numbers are closed... |
| addscl 27704 | Surreal numbers are closed... |
| addsf 27705 | Function statement for sur... |
| addsfo 27706 | Surreal addition is onto. ... |
| peano2no 27707 | A theorem for surreals tha... |
| sltadd1im 27708 | Surreal less-than is prese... |
| sltadd2im 27709 | Surreal less-than is prese... |
| sleadd1im 27710 | Surreal less-than or equal... |
| sleadd2im 27711 | Surreal less-than or equal... |
| sleadd1 27712 | Addition to both sides of ... |
| sleadd2 27713 | Addition to both sides of ... |
| sltadd2 27714 | Addition to both sides of ... |
| sltadd1 27715 | Addition to both sides of ... |
| addscan2 27716 | Cancellation law for surre... |
| addscan1 27717 | Cancellation law for surre... |
| sleadd1d 27718 | Addition to both sides of ... |
| sleadd2d 27719 | Addition to both sides of ... |
| sltadd2d 27720 | Addition to both sides of ... |
| sltadd1d 27721 | Addition to both sides of ... |
| addscan2d 27722 | Cancellation law for surre... |
| addscan1d 27723 | Cancellation law for surre... |
| addsuniflem 27724 | Lemma for ~ addsunif . St... |
| addsunif 27725 | Uniformity theorem for sur... |
| addsasslem1 27726 | Lemma for addition associa... |
| addsasslem2 27727 | Lemma for addition associa... |
| addsass 27728 | Surreal addition is associ... |
| addsassd 27729 | Surreal addition is associ... |
| adds32d 27730 | Commutative/associative la... |
| adds12d 27731 | Commutative/associative la... |
| adds4d 27732 | Rearrangement of four term... |
| adds42d 27733 | Rearrangement of four term... |
| negsfn 27738 | Surreal negation is a func... |
| subsfn 27739 | Surreal subtraction is a f... |
| negsval 27740 | The value of the surreal n... |
| negs0s 27741 | Negative surreal zero is s... |
| negsproplem1 27742 | Lemma for surreal negation... |
| negsproplem2 27743 | Lemma for surreal negation... |
| negsproplem3 27744 | Lemma for surreal negation... |
| negsproplem4 27745 | Lemma for surreal negation... |
| negsproplem5 27746 | Lemma for surreal negation... |
| negsproplem6 27747 | Lemma for surreal negation... |
| negsproplem7 27748 | Lemma for surreal negation... |
| negsprop 27749 | Show closure and ordering ... |
| negscl 27750 | The surreals are closed un... |
| negscld 27751 | The surreals are closed un... |
| sltnegim 27752 | The forward direction of t... |
| negscut 27753 | The cut properties of surr... |
| negscut2 27754 | The cut that defines surre... |
| negsid 27755 | Surreal addition of a numb... |
| negsidd 27756 | Surreal addition of a numb... |
| negsex 27757 | Every surreal has a negati... |
| negnegs 27758 | A surreal is equal to the ... |
| sltneg 27759 | Negative of both sides of ... |
| sleneg 27760 | Negative of both sides of ... |
| sltnegd 27761 | Negative of both sides of ... |
| slenegd 27762 | Negative of both sides of ... |
| negs11 27763 | Surreal negation is one-to... |
| negsdi 27764 | Distribution of surreal ne... |
| slt0neg2d 27765 | Comparison of a surreal an... |
| negsf 27766 | Function statement for sur... |
| negsfo 27767 | Function statement for sur... |
| negsf1o 27768 | Surreal negation is a bije... |
| negsunif 27769 | Uniformity property for su... |
| negsbdaylem 27770 | Lemma for ~ negsbday . Bo... |
| negsbday 27771 | Negation of a surreal numb... |
| subsval 27772 | The value of surreal subtr... |
| subsvald 27773 | The value of surreal subtr... |
| subscl 27774 | Closure law for surreal su... |
| subscld 27775 | Closure law for surreal su... |
| subsid1 27776 | Identity law for subtracti... |
| subsid 27777 | Subtraction of a surreal f... |
| subadds 27778 | Relationship between addit... |
| subaddsd 27779 | Relationship between addit... |
| pncans 27780 | Cancellation law for surre... |
| pncan3s 27781 | Subtraction and addition o... |
| npcans 27782 | Cancellation law for surre... |
| sltsub1 27783 | Subtraction from both side... |
| sltsub2 27784 | Subtraction from both side... |
| sltsub1d 27785 | Subtraction from both side... |
| sltsub2d 27786 | Subtraction from both side... |
| negsubsdi2d 27787 | Distribution of negative o... |
| addsubsassd 27788 | Associative-type law for s... |
| addsubsd 27789 | Law for surreal addition a... |
| sltsubsubbd 27790 | Equivalence for the surrea... |
| sltsubsub2bd 27791 | Equivalence for the surrea... |
| sltsubsub3bd 27792 | Equivalence for the surrea... |
| slesubsubbd 27793 | Equivalence for the surrea... |
| slesubsub2bd 27794 | Equivalence for the surrea... |
| slesubsub3bd 27795 | Equivalence for the surrea... |
| sltsubaddd 27796 | Surreal less-than relation... |
| sltsubadd2d 27797 | Surreal less-than relation... |
| sltaddsubd 27798 | Surreal less-than relation... |
| sltaddsub2d 27799 | Surreal less-than relation... |
| subsubs4d 27800 | Law for double surreal sub... |
| posdifsd 27801 | Comparison of two surreals... |
| mulsfn 27804 | Surreal multiplication is ... |
| mulsval 27805 | The value of surreal multi... |
| mulsval2lem 27806 | Lemma for ~ mulsval2 . Ch... |
| mulsval2 27807 | The value of surreal multi... |
| muls01 27808 | Surreal multiplication by ... |
| mulsrid 27809 | Surreal one is a right ide... |
| mulsridd 27810 | Surreal one is a right ide... |
| mulsproplemcbv 27811 | Lemma for surreal multipli... |
| mulsproplem1 27812 | Lemma for surreal multipli... |
| mulsproplem2 27813 | Lemma for surreal multipli... |
| mulsproplem3 27814 | Lemma for surreal multipli... |
| mulsproplem4 27815 | Lemma for surreal multipli... |
| mulsproplem5 27816 | Lemma for surreal multipli... |
| mulsproplem6 27817 | Lemma for surreal multipli... |
| mulsproplem7 27818 | Lemma for surreal multipli... |
| mulsproplem8 27819 | Lemma for surreal multipli... |
| mulsproplem9 27820 | Lemma for surreal multipli... |
| mulsproplem10 27821 | Lemma for surreal multipli... |
| mulsproplem11 27822 | Lemma for surreal multipli... |
| mulsproplem12 27823 | Lemma for surreal multipli... |
| mulsproplem13 27824 | Lemma for surreal multipli... |
| mulsproplem14 27825 | Lemma for surreal multipli... |
| mulsprop 27826 | Surreals are closed under ... |
| mulscutlem 27827 | Lemma for ~ mulscut . Sta... |
| mulscut 27828 | Show the cut properties of... |
| mulscut2 27829 | Show that the cut involved... |
| mulscl 27830 | The surreals are closed un... |
| mulscld 27831 | The surreals are closed un... |
| sltmul 27832 | An ordering relationship f... |
| sltmuld 27833 | An ordering relationship f... |
| slemuld 27834 | An ordering relationship f... |
| mulscom 27835 | Surreal multiplication com... |
| mulscomd 27836 | Surreal multiplication com... |
| muls02 27837 | Surreal multiplication by ... |
| mulslid 27838 | Surreal one is a left iden... |
| mulslidd 27839 | Surreal one is a left iden... |
| mulsgt0 27840 | The product of two positiv... |
| mulsgt0d 27841 | The product of two positiv... |
| ssltmul1 27842 | One surreal set less-than ... |
| ssltmul2 27843 | One surreal set less-than ... |
| mulsuniflem 27844 | Lemma for ~ mulsunif . St... |
| mulsunif 27845 | Surreal multiplication has... |
| addsdilem1 27846 | Lemma for surreal distribu... |
| addsdilem2 27847 | Lemma for surreal distribu... |
| addsdilem3 27848 | Lemma for ~ addsdi . Show... |
| addsdilem4 27849 | Lemma for ~ addsdi . Show... |
| addsdi 27850 | Distributive law for surre... |
| addsdid 27851 | Distributive law for surre... |
| addsdird 27852 | Distributive law for surre... |
| subsdid 27853 | Distribution of surreal mu... |
| subsdird 27854 | Distribution of surreal mu... |
| mulnegs1d 27855 | Product with negative is n... |
| mulnegs2d 27856 | Product with negative is n... |
| mul2negsd 27857 | Surreal product of two neg... |
| mulsasslem1 27858 | Lemma for ~ mulsass . Exp... |
| mulsasslem2 27859 | Lemma for ~ mulsass . Exp... |
| mulsasslem3 27860 | Lemma for ~ mulsass . Dem... |
| mulsass 27861 | Associative law for surrea... |
| mulsassd 27862 | Associative law for surrea... |
| sltmul2 27863 | Multiplication of both sid... |
| sltmul2d 27864 | Multiplication of both sid... |
| sltmul1d 27865 | Multiplication of both sid... |
| slemul2d 27866 | Multiplication of both sid... |
| slemul1d 27867 | Multiplication of both sid... |
| sltmulneg1d 27868 | Multiplication of both sid... |
| sltmulneg2d 27869 | Multiplication of both sid... |
| mulscan2dlem 27870 | Lemma for ~ mulscan2d . C... |
| mulscan2d 27871 | Cancellation of surreal mu... |
| mulscan1d 27872 | Cancellation of surreal mu... |
| muls12d 27873 | Commutative/associative la... |
| divsmo 27874 | Uniqueness of surreal inve... |
| divsval 27877 | The value of surreal divis... |
| norecdiv 27878 | If a surreal has a recipro... |
| noreceuw 27879 | If a surreal has a recipro... |
| divsmulw 27880 | Relationship between surre... |
| divsmulwd 27881 | Relationship between surre... |
| divsclw 27882 | Weak division closure law.... |
| divsclwd 27883 | Weak division closure law.... |
| divscan2wd 27884 | A weak cancellation law fo... |
| divscan1wd 27885 | A weak cancellation law fo... |
| sltdivmulwd 27886 | Surreal less-than relation... |
| sltdivmul2wd 27887 | Surreal less-than relation... |
| sltmuldivwd 27888 | Surreal less-than relation... |
| sltmuldiv2wd 27889 | Surreal less-than relation... |
| divsasswd 27890 | An associative law for sur... |
| divs1 27891 | A surreal divided by one i... |
| precsexlemcbv 27892 | Lemma for surreal reciproc... |
| precsexlem1 27893 | Lemma for surreal reciproc... |
| precsexlem2 27894 | Lemma for surreal reciproc... |
| precsexlem3 27895 | Lemma for surreal reciproc... |
| precsexlem4 27896 | Lemma for surreal reciproc... |
| precsexlem5 27897 | Lemma for surreal reciproc... |
| precsexlem6 27898 | Lemma for surreal reciproc... |
| precsexlem7 27899 | Lemma for surreal reciproc... |
| precsexlem8 27900 | Lemma for surreal reciproc... |
| precsexlem9 27901 | Lemma for surreal reciproc... |
| precsexlem10 27902 | Lemma for surreal reciproc... |
| precsexlem11 27903 | Lemma for surreal reciproc... |
| precsex 27904 | Every positive surreal has... |
| recsex 27905 | A non-zero surreal has a r... |
| recsexd 27906 | A non-zero surreal has a r... |
| divsmul 27907 | Relationship between surre... |
| divsmuld 27908 | Relationship between surre... |
| divscl 27909 | Surreal division closure l... |
| divscld 27910 | Surreal division closure l... |
| divscan2d 27911 | A cancellation law for sur... |
| divscan1d 27912 | A cancellation law for sur... |
| sltdivmuld 27913 | Surreal less-than relation... |
| sltdivmul2d 27914 | Surreal less-than relation... |
| sltmuldivd 27915 | Surreal less-than relation... |
| sltmuldiv2d 27916 | Surreal less-than relation... |
| divsassd 27917 | An associative law for sur... |
| elons 27920 | Membership in the class of... |
| onssno 27921 | The surreal ordinals are a... |
| onsno 27922 | A surreal ordinal is a sur... |
| 0ons 27923 | Surreal zero is a surreal ... |
| 1ons 27924 | Surreal one is a surreal o... |
| elons2 27925 | A surreal is ordinal iff i... |
| elons2d 27926 | The cut of any set of surr... |
| sltonold 27927 | The class of ordinals less... |
| sltonex 27928 | The class of ordinals less... |
| onscutleft 27929 | A surreal ordinal is equal... |
| n0sex 27934 | The set of all non-negativ... |
| nnsex 27935 | The set of all positive su... |
| peano5n0s 27936 | Peano's inductive postulat... |
| n0ssno 27937 | The non-negative surreal i... |
| nnssn0s 27938 | The positive surreal integ... |
| nnssno 27939 | The positive surreal integ... |
| 0n0s 27940 | Peano postulate: ` 0s ` is... |
| peano2n0s 27941 | Peano postulate: the succe... |
| dfn0s2 27942 | Alternate definition of th... |
| n0sind 27943 | Principle of Mathematical ... |
| n0scut 27944 | A cut form for surreal nat... |
| n0ons 27945 | A surreal natural is a sur... |
| itvndx 27956 | Index value of the Interva... |
| lngndx 27957 | Index value of the "line" ... |
| itvid 27958 | Utility theorem: index-ind... |
| lngid 27959 | Utility theorem: index-ind... |
| slotsinbpsd 27960 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 27961 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 27962 | The slot for the line is n... |
| trkgstr 27963 | Functionality of a Tarski ... |
| trkgbas 27964 | The base set of a Tarski g... |
| trkgdist 27965 | The measure of a distance ... |
| trkgitv 27966 | The congruence relation in... |
| istrkgc 27973 | Property of being a Tarski... |
| istrkgb 27974 | Property of being a Tarski... |
| istrkgcb 27975 | Property of being a Tarski... |
| istrkge 27976 | Property of fulfilling Euc... |
| istrkgl 27977 | Building lines from the se... |
| istrkgld 27978 | Property of fulfilling the... |
| istrkg2ld 27979 | Property of fulfilling the... |
| istrkg3ld 27980 | Property of fulfilling the... |
| axtgcgrrflx 27981 | Axiom of reflexivity of co... |
| axtgcgrid 27982 | Axiom of identity of congr... |
| axtgsegcon 27983 | Axiom of segment construct... |
| axtg5seg 27984 | Five segments axiom, Axiom... |
| axtgbtwnid 27985 | Identity of Betweenness. ... |
| axtgpasch 27986 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 27987 | Axiom of Continuity. Axio... |
| axtgcont 27988 | Axiom of Continuity. Axio... |
| axtglowdim2 27989 | Lower dimension axiom for ... |
| axtgupdim2 27990 | Upper dimension axiom for ... |
| axtgeucl 27991 | Euclid's Axiom. Axiom A10... |
| tgjustf 27992 | Given any function ` F ` ,... |
| tgjustr 27993 | Given any equivalence rela... |
| tgjustc1 27994 | A justification for using ... |
| tgjustc2 27995 | A justification for using ... |
| tgcgrcomimp 27996 | Congruence commutes on the... |
| tgcgrcomr 27997 | Congruence commutes on the... |
| tgcgrcoml 27998 | Congruence commutes on the... |
| tgcgrcomlr 27999 | Congruence commutes on bot... |
| tgcgreqb 28000 | Congruence and equality. ... |
| tgcgreq 28001 | Congruence and equality. ... |
| tgcgrneq 28002 | Congruence and equality. ... |
| tgcgrtriv 28003 | Degenerate segments are co... |
| tgcgrextend 28004 | Link congruence over a pai... |
| tgsegconeq 28005 | Two points that satisfy th... |
| tgbtwntriv2 28006 | Betweenness always holds f... |
| tgbtwncom 28007 | Betweenness commutes. The... |
| tgbtwncomb 28008 | Betweenness commutes, bico... |
| tgbtwnne 28009 | Betweenness and inequality... |
| tgbtwntriv1 28010 | Betweenness always holds f... |
| tgbtwnswapid 28011 | If you can swap the first ... |
| tgbtwnintr 28012 | Inner transitivity law for... |
| tgbtwnexch3 28013 | Exchange the first endpoin... |
| tgbtwnouttr2 28014 | Outer transitivity law for... |
| tgbtwnexch2 28015 | Exchange the outer point o... |
| tgbtwnouttr 28016 | Outer transitivity law for... |
| tgbtwnexch 28017 | Outer transitivity law for... |
| tgtrisegint 28018 | A line segment between two... |
| tglowdim1 28019 | Lower dimension axiom for ... |
| tglowdim1i 28020 | Lower dimension axiom for ... |
| tgldimor 28021 | Excluded-middle like state... |
| tgldim0eq 28022 | In dimension zero, any two... |
| tgldim0itv 28023 | In dimension zero, any two... |
| tgldim0cgr 28024 | In dimension zero, any two... |
| tgbtwndiff 28025 | There is always a ` c ` di... |
| tgdim01 28026 | In geometries of dimension... |
| tgifscgr 28027 | Inner five segment congrue... |
| tgcgrsub 28028 | Removing identical parts f... |
| iscgrg 28031 | The congruence property fo... |
| iscgrgd 28032 | The property for two seque... |
| iscgrglt 28033 | The property for two seque... |
| trgcgrg 28034 | The property for two trian... |
| trgcgr 28035 | Triangle congruence. (Con... |
| ercgrg 28036 | The shape congruence relat... |
| tgcgrxfr 28037 | A line segment can be divi... |
| cgr3id 28038 | Reflexivity law for three-... |
| cgr3simp1 28039 | Deduce segment congruence ... |
| cgr3simp2 28040 | Deduce segment congruence ... |
| cgr3simp3 28041 | Deduce segment congruence ... |
| cgr3swap12 28042 | Permutation law for three-... |
| cgr3swap23 28043 | Permutation law for three-... |
| cgr3swap13 28044 | Permutation law for three-... |
| cgr3rotr 28045 | Permutation law for three-... |
| cgr3rotl 28046 | Permutation law for three-... |
| trgcgrcom 28047 | Commutative law for three-... |
| cgr3tr 28048 | Transitivity law for three... |
| tgbtwnxfr 28049 | A condition for extending ... |
| tgcgr4 28050 | Two quadrilaterals to be c... |
| isismt 28053 | Property of being an isome... |
| ismot 28054 | Property of being an isome... |
| motcgr 28055 | Property of a motion: dist... |
| idmot 28056 | The identity is a motion. ... |
| motf1o 28057 | Motions are bijections. (... |
| motcl 28058 | Closure of motions. (Cont... |
| motco 28059 | The composition of two mot... |
| cnvmot 28060 | The converse of a motion i... |
| motplusg 28061 | The operation for motions ... |
| motgrp 28062 | The motions of a geometry ... |
| motcgrg 28063 | Property of a motion: dist... |
| motcgr3 28064 | Property of a motion: dist... |
| tglng 28065 | Lines of a Tarski Geometry... |
| tglnfn 28066 | Lines as functions. (Cont... |
| tglnunirn 28067 | Lines are sets of points. ... |
| tglnpt 28068 | Lines are sets of points. ... |
| tglngne 28069 | It takes two different poi... |
| tglngval 28070 | The line going through poi... |
| tglnssp 28071 | Lines are subset of the ge... |
| tgellng 28072 | Property of lying on the l... |
| tgcolg 28073 | We choose the notation ` (... |
| btwncolg1 28074 | Betweenness implies coline... |
| btwncolg2 28075 | Betweenness implies coline... |
| btwncolg3 28076 | Betweenness implies coline... |
| colcom 28077 | Swapping the points defini... |
| colrot1 28078 | Rotating the points defini... |
| colrot2 28079 | Rotating the points defini... |
| ncolcom 28080 | Swapping non-colinear poin... |
| ncolrot1 28081 | Rotating non-colinear poin... |
| ncolrot2 28082 | Rotating non-colinear poin... |
| tgdim01ln 28083 | In geometries of dimension... |
| ncoltgdim2 28084 | If there are three non-col... |
| lnxfr 28085 | Transfer law for colineari... |
| lnext 28086 | Extend a line with a missi... |
| tgfscgr 28087 | Congruence law for the gen... |
| lncgr 28088 | Congruence rule for lines.... |
| lnid 28089 | Identity law for points on... |
| tgidinside 28090 | Law for finding a point in... |
| tgbtwnconn1lem1 28091 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 28092 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 28093 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 28094 | Connectivity law for betwe... |
| tgbtwnconn2 28095 | Another connectivity law f... |
| tgbtwnconn3 28096 | Inner connectivity law for... |
| tgbtwnconnln3 28097 | Derive colinearity from be... |
| tgbtwnconn22 28098 | Double connectivity law fo... |
| tgbtwnconnln1 28099 | Derive colinearity from be... |
| tgbtwnconnln2 28100 | Derive colinearity from be... |
| legval 28103 | Value of the less-than rel... |
| legov 28104 | Value of the less-than rel... |
| legov2 28105 | An equivalent definition o... |
| legid 28106 | Reflexivity of the less-th... |
| btwnleg 28107 | Betweenness implies less-t... |
| legtrd 28108 | Transitivity of the less-t... |
| legtri3 28109 | Equality from the less-tha... |
| legtrid 28110 | Trichotomy law for the les... |
| leg0 28111 | Degenerated (zero-length) ... |
| legeq 28112 | Deduce equality from "less... |
| legbtwn 28113 | Deduce betweenness from "l... |
| tgcgrsub2 28114 | Removing identical parts f... |
| ltgseg 28115 | The set ` E ` denotes the ... |
| ltgov 28116 | Strict "shorter than" geom... |
| legov3 28117 | An equivalent definition o... |
| legso 28118 | The "shorter than" relatio... |
| ishlg 28121 | Rays : Definition 6.1 of ... |
| hlcomb 28122 | The half-line relation com... |
| hlcomd 28123 | The half-line relation com... |
| hlne1 28124 | The half-line relation imp... |
| hlne2 28125 | The half-line relation imp... |
| hlln 28126 | The half-line relation imp... |
| hleqnid 28127 | The endpoint does not belo... |
| hlid 28128 | The half-line relation is ... |
| hltr 28129 | The half-line relation is ... |
| hlbtwn 28130 | Betweenness is a sufficien... |
| btwnhl1 28131 | Deduce half-line from betw... |
| btwnhl2 28132 | Deduce half-line from betw... |
| btwnhl 28133 | Swap betweenness for a hal... |
| lnhl 28134 | Either a point ` C ` on th... |
| hlcgrex 28135 | Construct a point on a hal... |
| hlcgreulem 28136 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 28137 | The point constructed in ~... |
| btwnlng1 28138 | Betweenness implies coline... |
| btwnlng2 28139 | Betweenness implies coline... |
| btwnlng3 28140 | Betweenness implies coline... |
| lncom 28141 | Swapping the points defini... |
| lnrot1 28142 | Rotating the points defini... |
| lnrot2 28143 | Rotating the points defini... |
| ncolne1 28144 | Non-colinear points are di... |
| ncolne2 28145 | Non-colinear points are di... |
| tgisline 28146 | The property of being a pr... |
| tglnne 28147 | It takes two different poi... |
| tglndim0 28148 | There are no lines in dime... |
| tgelrnln 28149 | The property of being a pr... |
| tglineeltr 28150 | Transitivity law for lines... |
| tglineelsb2 28151 | If ` S ` lies on PQ , then... |
| tglinerflx1 28152 | Reflexivity law for line m... |
| tglinerflx2 28153 | Reflexivity law for line m... |
| tglinecom 28154 | Commutativity law for line... |
| tglinethru 28155 | If ` A ` is a line contain... |
| tghilberti1 28156 | There is a line through an... |
| tghilberti2 28157 | There is at most one line ... |
| tglinethrueu 28158 | There is a unique line goi... |
| tglnne0 28159 | A line ` A ` has at least ... |
| tglnpt2 28160 | Find a second point on a l... |
| tglineintmo 28161 | Two distinct lines interse... |
| tglineineq 28162 | Two distinct lines interse... |
| tglineneq 28163 | Given three non-colinear p... |
| tglineinteq 28164 | Two distinct lines interse... |
| ncolncol 28165 | Deduce non-colinearity fro... |
| coltr 28166 | A transitivity law for col... |
| coltr3 28167 | A transitivity law for col... |
| colline 28168 | Three points are colinear ... |
| tglowdim2l 28169 | Reformulation of the lower... |
| tglowdim2ln 28170 | There is always one point ... |
| mirreu3 28173 | Existential uniqueness of ... |
| mirval 28174 | Value of the point inversi... |
| mirfv 28175 | Value of the point inversi... |
| mircgr 28176 | Property of the image by t... |
| mirbtwn 28177 | Property of the image by t... |
| ismir 28178 | Property of the image by t... |
| mirf 28179 | Point inversion as functio... |
| mircl 28180 | Closure of the point inver... |
| mirmir 28181 | The point inversion functi... |
| mircom 28182 | Variation on ~ mirmir . (... |
| mirreu 28183 | Any point has a unique ant... |
| mireq 28184 | Equality deduction for poi... |
| mirinv 28185 | The only invariant point o... |
| mirne 28186 | Mirror of non-center point... |
| mircinv 28187 | The center point is invari... |
| mirf1o 28188 | The point inversion functi... |
| miriso 28189 | The point inversion functi... |
| mirbtwni 28190 | Point inversion preserves ... |
| mirbtwnb 28191 | Point inversion preserves ... |
| mircgrs 28192 | Point inversion preserves ... |
| mirmir2 28193 | Point inversion of a point... |
| mirmot 28194 | Point investion is a motio... |
| mirln 28195 | If two points are on the s... |
| mirln2 28196 | If a point and its mirror ... |
| mirconn 28197 | Point inversion of connect... |
| mirhl 28198 | If two points ` X ` and ` ... |
| mirbtwnhl 28199 | If the center of the point... |
| mirhl2 28200 | Deduce half-line relation ... |
| mircgrextend 28201 | Link congruence over a pai... |
| mirtrcgr 28202 | Point inversion of one poi... |
| mirauto 28203 | Point inversion preserves ... |
| miduniq 28204 | Uniqueness of the middle p... |
| miduniq1 28205 | Uniqueness of the middle p... |
| miduniq2 28206 | If two point inversions co... |
| colmid 28207 | Colinearity and equidistan... |
| symquadlem 28208 | Lemma of the symetrial qua... |
| krippenlem 28209 | Lemma for ~ krippen . We ... |
| krippen 28210 | Krippenlemma (German for c... |
| midexlem 28211 | Lemma for the existence of... |
| israg 28216 | Property for 3 points A, B... |
| ragcom 28217 | Commutative rule for right... |
| ragcol 28218 | The right angle property i... |
| ragmir 28219 | Right angle property is pr... |
| mirrag 28220 | Right angle is conserved b... |
| ragtrivb 28221 | Trivial right angle. Theo... |
| ragflat2 28222 | Deduce equality from two r... |
| ragflat 28223 | Deduce equality from two r... |
| ragtriva 28224 | Trivial right angle. Theo... |
| ragflat3 28225 | Right angle and colinearit... |
| ragcgr 28226 | Right angle and colinearit... |
| motrag 28227 | Right angles are preserved... |
| ragncol 28228 | Right angle implies non-co... |
| perpln1 28229 | Derive a line from perpend... |
| perpln2 28230 | Derive a line from perpend... |
| isperp 28231 | Property for 2 lines A, B ... |
| perpcom 28232 | The "perpendicular" relati... |
| perpneq 28233 | Two perpendicular lines ar... |
| isperp2 28234 | Property for 2 lines A, B,... |
| isperp2d 28235 | One direction of ~ isperp2... |
| ragperp 28236 | Deduce that two lines are ... |
| footexALT 28237 | Alternative version of ~ f... |
| footexlem1 28238 | Lemma for ~ footex . (Con... |
| footexlem2 28239 | Lemma for ~ footex . (Con... |
| footex 28240 | From a point ` C ` outside... |
| foot 28241 | From a point ` C ` outside... |
| footne 28242 | Uniqueness of the foot poi... |
| footeq 28243 | Uniqueness of the foot poi... |
| hlperpnel 28244 | A point on a half-line whi... |
| perprag 28245 | Deduce a right angle from ... |
| perpdragALT 28246 | Deduce a right angle from ... |
| perpdrag 28247 | Deduce a right angle from ... |
| colperp 28248 | Deduce a perpendicularity ... |
| colperpexlem1 28249 | Lemma for ~ colperp . Fir... |
| colperpexlem2 28250 | Lemma for ~ colperpex . S... |
| colperpexlem3 28251 | Lemma for ~ colperpex . C... |
| colperpex 28252 | In dimension 2 and above, ... |
| mideulem2 28253 | Lemma for ~ opphllem , whi... |
| opphllem 28254 | Lemma 8.24 of [Schwabhause... |
| mideulem 28255 | Lemma for ~ mideu . We ca... |
| midex 28256 | Existence of the midpoint,... |
| mideu 28257 | Existence and uniqueness o... |
| islnopp 28258 | The property for two point... |
| islnoppd 28259 | Deduce that ` A ` and ` B ... |
| oppne1 28260 | Points lying on opposite s... |
| oppne2 28261 | Points lying on opposite s... |
| oppne3 28262 | Points lying on opposite s... |
| oppcom 28263 | Commutativity rule for "op... |
| opptgdim2 28264 | If two points opposite to ... |
| oppnid 28265 | The "opposite to a line" r... |
| opphllem1 28266 | Lemma for ~ opphl . (Cont... |
| opphllem2 28267 | Lemma for ~ opphl . Lemma... |
| opphllem3 28268 | Lemma for ~ opphl : We as... |
| opphllem4 28269 | Lemma for ~ opphl . (Cont... |
| opphllem5 28270 | Second part of Lemma 9.4 o... |
| opphllem6 28271 | First part of Lemma 9.4 of... |
| oppperpex 28272 | Restating ~ colperpex usin... |
| opphl 28273 | If two points ` A ` and ` ... |
| outpasch 28274 | Axiom of Pasch, outer form... |
| hlpasch 28275 | An application of the axio... |
| ishpg 28278 | Value of the half-plane re... |
| hpgbr 28279 | Half-planes : property for... |
| hpgne1 28280 | Points on the open half pl... |
| hpgne2 28281 | Points on the open half pl... |
| lnopp2hpgb 28282 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 28283 | If two points lie on the o... |
| hpgerlem 28284 | Lemma for the proof that t... |
| hpgid 28285 | The half-plane relation is... |
| hpgcom 28286 | The half-plane relation co... |
| hpgtr 28287 | The half-plane relation is... |
| colopp 28288 | Opposite sides of a line f... |
| colhp 28289 | Half-plane relation for co... |
| hphl 28290 | If two points are on the s... |
| midf 28295 | Midpoint as a function. (... |
| midcl 28296 | Closure of the midpoint. ... |
| ismidb 28297 | Property of the midpoint. ... |
| midbtwn 28298 | Betweenness of midpoint. ... |
| midcgr 28299 | Congruence of midpoint. (... |
| midid 28300 | Midpoint of a null segment... |
| midcom 28301 | Commutativity rule for the... |
| mirmid 28302 | Point inversion preserves ... |
| lmieu 28303 | Uniqueness of the line mir... |
| lmif 28304 | Line mirror as a function.... |
| lmicl 28305 | Closure of the line mirror... |
| islmib 28306 | Property of the line mirro... |
| lmicom 28307 | The line mirroring functio... |
| lmilmi 28308 | Line mirroring is an invol... |
| lmireu 28309 | Any point has a unique ant... |
| lmieq 28310 | Equality deduction for lin... |
| lmiinv 28311 | The invariants of the line... |
| lmicinv 28312 | The mirroring line is an i... |
| lmimid 28313 | If we have a right angle, ... |
| lmif1o 28314 | The line mirroring functio... |
| lmiisolem 28315 | Lemma for ~ lmiiso . (Con... |
| lmiiso 28316 | The line mirroring functio... |
| lmimot 28317 | Line mirroring is a motion... |
| hypcgrlem1 28318 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 28319 | Lemma for ~ hypcgr , case ... |
| hypcgr 28320 | If the catheti of two righ... |
| lmiopp 28321 | Line mirroring produces po... |
| lnperpex 28322 | Existence of a perpendicul... |
| trgcopy 28323 | Triangle construction: a c... |
| trgcopyeulem 28324 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 28325 | Triangle construction: a c... |
| iscgra 28328 | Property for two angles AB... |
| iscgra1 28329 | A special version of ~ isc... |
| iscgrad 28330 | Sufficient conditions for ... |
| cgrane1 28331 | Angles imply inequality. ... |
| cgrane2 28332 | Angles imply inequality. ... |
| cgrane3 28333 | Angles imply inequality. ... |
| cgrane4 28334 | Angles imply inequality. ... |
| cgrahl1 28335 | Angle congruence is indepe... |
| cgrahl2 28336 | Angle congruence is indepe... |
| cgracgr 28337 | First direction of proposi... |
| cgraid 28338 | Angle congruence is reflex... |
| cgraswap 28339 | Swap rays in a congruence ... |
| cgrcgra 28340 | Triangle congruence implie... |
| cgracom 28341 | Angle congruence commutes.... |
| cgratr 28342 | Angle congruence is transi... |
| flatcgra 28343 | Flat angles are congruent.... |
| cgraswaplr 28344 | Swap both side of angle co... |
| cgrabtwn 28345 | Angle congruence preserves... |
| cgrahl 28346 | Angle congruence preserves... |
| cgracol 28347 | Angle congruence preserves... |
| cgrancol 28348 | Angle congruence preserves... |
| dfcgra2 28349 | This is the full statement... |
| sacgr 28350 | Supplementary angles of co... |
| oacgr 28351 | Vertical angle theorem. V... |
| acopy 28352 | Angle construction. Theor... |
| acopyeu 28353 | Angle construction. Theor... |
| isinag 28357 | Property for point ` X ` t... |
| isinagd 28358 | Sufficient conditions for ... |
| inagflat 28359 | Any point lies in a flat a... |
| inagswap 28360 | Swap the order of the half... |
| inagne1 28361 | Deduce inequality from the... |
| inagne2 28362 | Deduce inequality from the... |
| inagne3 28363 | Deduce inequality from the... |
| inaghl 28364 | The "point lie in angle" r... |
| isleag 28366 | Geometrical "less than" pr... |
| isleagd 28367 | Sufficient condition for "... |
| leagne1 28368 | Deduce inequality from the... |
| leagne2 28369 | Deduce inequality from the... |
| leagne3 28370 | Deduce inequality from the... |
| leagne4 28371 | Deduce inequality from the... |
| cgrg3col4 28372 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 28373 | First congruence theorem: ... |
| tgsas 28374 | First congruence theorem: ... |
| tgsas2 28375 | First congruence theorem: ... |
| tgsas3 28376 | First congruence theorem: ... |
| tgasa1 28377 | Second congruence theorem:... |
| tgasa 28378 | Second congruence theorem:... |
| tgsss1 28379 | Third congruence theorem: ... |
| tgsss2 28380 | Third congruence theorem: ... |
| tgsss3 28381 | Third congruence theorem: ... |
| dfcgrg2 28382 | Congruence for two triangl... |
| isoas 28383 | Congruence theorem for iso... |
| iseqlg 28386 | Property of a triangle bei... |
| iseqlgd 28387 | Condition for a triangle t... |
| f1otrgds 28388 | Convenient lemma for ~ f1o... |
| f1otrgitv 28389 | Convenient lemma for ~ f1o... |
| f1otrg 28390 | A bijection between bases ... |
| f1otrge 28391 | A bijection between bases ... |
| ttgval 28394 | Define a function to augme... |
| ttgvalOLD 28395 | Obsolete proof of ~ ttgval... |
| ttglem 28396 | Lemma for ~ ttgbas , ~ ttg... |
| ttglemOLD 28397 | Obsolete version of ~ ttgl... |
| ttgbas 28398 | The base set of a subcompl... |
| ttgbasOLD 28399 | Obsolete proof of ~ ttgbas... |
| ttgplusg 28400 | The addition operation of ... |
| ttgplusgOLD 28401 | Obsolete proof of ~ ttgplu... |
| ttgsub 28402 | The subtraction operation ... |
| ttgvsca 28403 | The scalar product of a su... |
| ttgvscaOLD 28404 | Obsolete proof of ~ ttgvsc... |
| ttgds 28405 | The metric of a subcomplex... |
| ttgdsOLD 28406 | Obsolete proof of ~ ttgds ... |
| ttgitvval 28407 | Betweenness for a subcompl... |
| ttgelitv 28408 | Betweenness for a subcompl... |
| ttgbtwnid 28409 | Any subcomplex module equi... |
| ttgcontlem1 28410 | Lemma for % ttgcont . (Co... |
| xmstrkgc 28411 | Any metric space fulfills ... |
| cchhllem 28412 | Lemma for chlbas and chlvs... |
| cchhllemOLD 28413 | Obsolete version of ~ cchh... |
| elee 28420 | Membership in a Euclidean ... |
| mptelee 28421 | A condition for a mapping ... |
| eleenn 28422 | If ` A ` is in ` ( EE `` N... |
| eleei 28423 | The forward direction of ~... |
| eedimeq 28424 | A point belongs to at most... |
| brbtwn 28425 | The binary relation form o... |
| brcgr 28426 | The binary relation form o... |
| fveere 28427 | The function value of a po... |
| fveecn 28428 | The function value of a po... |
| eqeefv 28429 | Two points are equal iff t... |
| eqeelen 28430 | Two points are equal iff t... |
| brbtwn2 28431 | Alternate characterization... |
| colinearalglem1 28432 | Lemma for ~ colinearalg . ... |
| colinearalglem2 28433 | Lemma for ~ colinearalg . ... |
| colinearalglem3 28434 | Lemma for ~ colinearalg . ... |
| colinearalglem4 28435 | Lemma for ~ colinearalg . ... |
| colinearalg 28436 | An algebraic characterizat... |
| eleesub 28437 | Membership of a subtractio... |
| eleesubd 28438 | Membership of a subtractio... |
| axdimuniq 28439 | The unique dimension axiom... |
| axcgrrflx 28440 | ` A ` is as far from ` B `... |
| axcgrtr 28441 | Congruence is transitive. ... |
| axcgrid 28442 | If there is no distance be... |
| axsegconlem1 28443 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 28444 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 28445 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 28446 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 28447 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 28448 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 28449 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 28450 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 28451 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 28452 | Lemma for ~ axsegcon . Sh... |
| axsegcon 28453 | Any segment ` A B ` can be... |
| ax5seglem1 28454 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 28455 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 28456 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 28457 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 28458 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 28459 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 28460 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 28461 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 28462 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 28463 | Lemma for ~ ax5seg . Take... |
| ax5seg 28464 | The five segment axiom. T... |
| axbtwnid 28465 | Points are indivisible. T... |
| axpaschlem 28466 | Lemma for ~ axpasch . Set... |
| axpasch 28467 | The inner Pasch axiom. Ta... |
| axlowdimlem1 28468 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 28469 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 28470 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 28471 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 28472 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 28473 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 28474 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 28475 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 28476 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 28477 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 28478 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 28479 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 28480 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 28481 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 28482 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 28483 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 28484 | Lemma for ~ axlowdim . Es... |
| axlowdim1 28485 | The lower dimension axiom ... |
| axlowdim2 28486 | The lower two-dimensional ... |
| axlowdim 28487 | The general lower dimensio... |
| axeuclidlem 28488 | Lemma for ~ axeuclid . Ha... |
| axeuclid 28489 | Euclid's axiom. Take an a... |
| axcontlem1 28490 | Lemma for ~ axcont . Chan... |
| axcontlem2 28491 | Lemma for ~ axcont . The ... |
| axcontlem3 28492 | Lemma for ~ axcont . Give... |
| axcontlem4 28493 | Lemma for ~ axcont . Give... |
| axcontlem5 28494 | Lemma for ~ axcont . Comp... |
| axcontlem6 28495 | Lemma for ~ axcont . Stat... |
| axcontlem7 28496 | Lemma for ~ axcont . Give... |
| axcontlem8 28497 | Lemma for ~ axcont . A po... |
| axcontlem9 28498 | Lemma for ~ axcont . Give... |
| axcontlem10 28499 | Lemma for ~ axcont . Give... |
| axcontlem11 28500 | Lemma for ~ axcont . Elim... |
| axcontlem12 28501 | Lemma for ~ axcont . Elim... |
| axcont 28502 | The axiom of continuity. ... |
| eengv 28505 | The value of the Euclidean... |
| eengstr 28506 | The Euclidean geometry as ... |
| eengbas 28507 | The Base of the Euclidean ... |
| ebtwntg 28508 | The betweenness relation u... |
| ecgrtg 28509 | The congruence relation us... |
| elntg 28510 | The line definition in the... |
| elntg2 28511 | The line definition in the... |
| eengtrkg 28512 | The geometry structure for... |
| eengtrkge 28513 | The geometry structure for... |
| edgfid 28516 | Utility theorem: index-ind... |
| edgfndx 28517 | Index value of the ~ df-ed... |
| edgfndxnn 28518 | The index value of the edg... |
| edgfndxid 28519 | The value of the edge func... |
| edgfndxidOLD 28520 | Obsolete version of ~ edgf... |
| basendxltedgfndx 28521 | The index value of the ` B... |
| baseltedgfOLD 28522 | Obsolete proof of ~ basend... |
| basendxnedgfndx 28523 | The slots ` Base ` and ` .... |
| vtxval 28528 | The set of vertices of a g... |
| iedgval 28529 | The set of indexed edges o... |
| 1vgrex 28530 | A graph with at least one ... |
| opvtxval 28531 | The set of vertices of a g... |
| opvtxfv 28532 | The set of vertices of a g... |
| opvtxov 28533 | The set of vertices of a g... |
| opiedgval 28534 | The set of indexed edges o... |
| opiedgfv 28535 | The set of indexed edges o... |
| opiedgov 28536 | The set of indexed edges o... |
| opvtxfvi 28537 | The set of vertices of a g... |
| opiedgfvi 28538 | The set of indexed edges o... |
| funvtxdmge2val 28539 | The set of vertices of an ... |
| funiedgdmge2val 28540 | The set of indexed edges o... |
| funvtxdm2val 28541 | The set of vertices of an ... |
| funiedgdm2val 28542 | The set of indexed edges o... |
| funvtxval0 28543 | The set of vertices of an ... |
| basvtxval 28544 | The set of vertices of a g... |
| edgfiedgval 28545 | The set of indexed edges o... |
| funvtxval 28546 | The set of vertices of a g... |
| funiedgval 28547 | The set of indexed edges o... |
| structvtxvallem 28548 | Lemma for ~ structvtxval a... |
| structvtxval 28549 | The set of vertices of an ... |
| structiedg0val 28550 | The set of indexed edges o... |
| structgrssvtxlem 28551 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 28552 | The set of vertices of a g... |
| structgrssiedg 28553 | The set of indexed edges o... |
| struct2grstr 28554 | A graph represented as an ... |
| struct2grvtx 28555 | The set of vertices of a g... |
| struct2griedg 28556 | The set of indexed edges o... |
| graop 28557 | Any representation of a gr... |
| grastruct 28558 | Any representation of a gr... |
| gropd 28559 | If any representation of a... |
| grstructd 28560 | If any representation of a... |
| gropeld 28561 | If any representation of a... |
| grstructeld 28562 | If any representation of a... |
| setsvtx 28563 | The vertices of a structur... |
| setsiedg 28564 | The (indexed) edges of a s... |
| snstrvtxval 28565 | The set of vertices of a g... |
| snstriedgval 28566 | The set of indexed edges o... |
| vtxval0 28567 | Degenerated case 1 for ver... |
| iedgval0 28568 | Degenerated case 1 for edg... |
| vtxvalsnop 28569 | Degenerated case 2 for ver... |
| iedgvalsnop 28570 | Degenerated case 2 for edg... |
| vtxval3sn 28571 | Degenerated case 3 for ver... |
| iedgval3sn 28572 | Degenerated case 3 for edg... |
| vtxvalprc 28573 | Degenerated case 4 for ver... |
| iedgvalprc 28574 | Degenerated case 4 for edg... |
| edgval 28577 | The edges of a graph. (Co... |
| iedgedg 28578 | An indexed edge is an edge... |
| edgopval 28579 | The edges of a graph repre... |
| edgov 28580 | The edges of a graph repre... |
| edgstruct 28581 | The edges of a graph repre... |
| edgiedgb 28582 | A set is an edge iff it is... |
| edg0iedg0 28583 | There is no edge in a grap... |
| isuhgr 28588 | The predicate "is an undir... |
| isushgr 28589 | The predicate "is an undir... |
| uhgrf 28590 | The edge function of an un... |
| ushgrf 28591 | The edge function of an un... |
| uhgrss 28592 | An edge is a subset of ver... |
| uhgreq12g 28593 | If two sets have the same ... |
| uhgrfun 28594 | The edge function of an un... |
| uhgrn0 28595 | An edge is a nonempty subs... |
| lpvtx 28596 | The endpoints of a loop (w... |
| ushgruhgr 28597 | An undirected simple hyper... |
| isuhgrop 28598 | The property of being an u... |
| uhgr0e 28599 | The empty graph, with vert... |
| uhgr0vb 28600 | The null graph, with no ve... |
| uhgr0 28601 | The null graph represented... |
| uhgrun 28602 | The union ` U ` of two (un... |
| uhgrunop 28603 | The union of two (undirect... |
| ushgrun 28604 | The union ` U ` of two (un... |
| ushgrunop 28605 | The union of two (undirect... |
| uhgrstrrepe 28606 | Replacing (or adding) the ... |
| incistruhgr 28607 | An _incidence structure_ `... |
| isupgr 28612 | The property of being an u... |
| wrdupgr 28613 | The property of being an u... |
| upgrf 28614 | The edge function of an un... |
| upgrfn 28615 | The edge function of an un... |
| upgrss 28616 | An edge is a subset of ver... |
| upgrn0 28617 | An edge is a nonempty subs... |
| upgrle 28618 | An edge of an undirected p... |
| upgrfi 28619 | An edge is a finite subset... |
| upgrex 28620 | An edge is an unordered pa... |
| upgrbi 28621 | Show that an unordered pai... |
| upgrop 28622 | A pseudograph represented ... |
| isumgr 28623 | The property of being an u... |
| isumgrs 28624 | The simplified property of... |
| wrdumgr 28625 | The property of being an u... |
| umgrf 28626 | The edge function of an un... |
| umgrfn 28627 | The edge function of an un... |
| umgredg2 28628 | An edge of a multigraph ha... |
| umgrbi 28629 | Show that an unordered pai... |
| upgruhgr 28630 | An undirected pseudograph ... |
| umgrupgr 28631 | An undirected multigraph i... |
| umgruhgr 28632 | An undirected multigraph i... |
| upgrle2 28633 | An edge of an undirected p... |
| umgrnloopv 28634 | In a multigraph, there is ... |
| umgredgprv 28635 | In a multigraph, an edge i... |
| umgrnloop 28636 | In a multigraph, there is ... |
| umgrnloop0 28637 | A multigraph has no loops.... |
| umgr0e 28638 | The empty graph, with vert... |
| upgr0e 28639 | The empty graph, with vert... |
| upgr1elem 28640 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 28641 | A pseudograph with one edg... |
| upgr0eop 28642 | The empty graph, with vert... |
| upgr1eop 28643 | A pseudograph with one edg... |
| upgr0eopALT 28644 | Alternate proof of ~ upgr0... |
| upgr1eopALT 28645 | Alternate proof of ~ upgr1... |
| upgrun 28646 | The union ` U ` of two pse... |
| upgrunop 28647 | The union of two pseudogra... |
| umgrun 28648 | The union ` U ` of two mul... |
| umgrunop 28649 | The union of two multigrap... |
| umgrislfupgrlem 28650 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 28651 | A multigraph is a loop-fre... |
| lfgredgge2 28652 | An edge of a loop-free gra... |
| lfgrnloop 28653 | A loop-free graph has no l... |
| uhgredgiedgb 28654 | In a hypergraph, a set is ... |
| uhgriedg0edg0 28655 | A hypergraph has no edges ... |
| uhgredgn0 28656 | An edge of a hypergraph is... |
| edguhgr 28657 | An edge of a hypergraph is... |
| uhgredgrnv 28658 | An edge of a hypergraph co... |
| uhgredgss 28659 | The set of edges of a hype... |
| upgredgss 28660 | The set of edges of a pseu... |
| umgredgss 28661 | The set of edges of a mult... |
| edgupgr 28662 | Properties of an edge of a... |
| edgumgr 28663 | Properties of an edge of a... |
| uhgrvtxedgiedgb 28664 | In a hypergraph, a vertex ... |
| upgredg 28665 | For each edge in a pseudog... |
| umgredg 28666 | For each edge in a multigr... |
| upgrpredgv 28667 | An edge of a pseudograph a... |
| umgrpredgv 28668 | An edge of a multigraph al... |
| upgredg2vtx 28669 | For a vertex incident to a... |
| upgredgpr 28670 | If a proper pair (of verti... |
| edglnl 28671 | The edges incident with a ... |
| numedglnl 28672 | The number of edges incide... |
| umgredgne 28673 | An edge of a multigraph al... |
| umgrnloop2 28674 | A multigraph has no loops.... |
| umgredgnlp 28675 | An edge of a multigraph is... |
| isuspgr 28680 | The property of being a si... |
| isusgr 28681 | The property of being a si... |
| uspgrf 28682 | The edge function of a sim... |
| usgrf 28683 | The edge function of a sim... |
| isusgrs 28684 | The property of being a si... |
| usgrfs 28685 | The edge function of a sim... |
| usgrfun 28686 | The edge function of a sim... |
| usgredgss 28687 | The set of edges of a simp... |
| edgusgr 28688 | An edge of a simple graph ... |
| isuspgrop 28689 | The property of being an u... |
| isusgrop 28690 | The property of being an u... |
| usgrop 28691 | A simple graph represented... |
| isausgr 28692 | The property of an unorder... |
| ausgrusgrb 28693 | The equivalence of the def... |
| usgrausgri 28694 | A simple graph represented... |
| ausgrumgri 28695 | If an alternatively define... |
| ausgrusgri 28696 | The equivalence of the def... |
| usgrausgrb 28697 | The equivalence of the def... |
| usgredgop 28698 | An edge of a simple graph ... |
| usgrf1o 28699 | The edge function of a sim... |
| usgrf1 28700 | The edge function of a sim... |
| uspgrf1oedg 28701 | The edge function of a sim... |
| usgrss 28702 | An edge is a subset of ver... |
| uspgrushgr 28703 | A simple pseudograph is an... |
| uspgrupgr 28704 | A simple pseudograph is an... |
| uspgrupgrushgr 28705 | A graph is a simple pseudo... |
| usgruspgr 28706 | A simple graph is a simple... |
| usgrumgr 28707 | A simple graph is an undir... |
| usgrumgruspgr 28708 | A graph is a simple graph ... |
| usgruspgrb 28709 | A class is a simple graph ... |
| usgrupgr 28710 | A simple graph is an undir... |
| usgruhgr 28711 | A simple graph is an undir... |
| usgrislfuspgr 28712 | A simple graph is a loop-f... |
| uspgrun 28713 | The union ` U ` of two sim... |
| uspgrunop 28714 | The union of two simple ps... |
| usgrun 28715 | The union ` U ` of two sim... |
| usgrunop 28716 | The union of two simple gr... |
| usgredg2 28717 | The value of the "edge fun... |
| usgredg2ALT 28718 | Alternate proof of ~ usgre... |
| usgredgprv 28719 | In a simple graph, an edge... |
| usgredgprvALT 28720 | Alternate proof of ~ usgre... |
| usgredgppr 28721 | An edge of a simple graph ... |
| usgrpredgv 28722 | An edge of a simple graph ... |
| edgssv2 28723 | An edge of a simple graph ... |
| usgredg 28724 | For each edge in a simple ... |
| usgrnloopv 28725 | In a simple graph, there i... |
| usgrnloopvALT 28726 | Alternate proof of ~ usgrn... |
| usgrnloop 28727 | In a simple graph, there i... |
| usgrnloopALT 28728 | Alternate proof of ~ usgrn... |
| usgrnloop0 28729 | A simple graph has no loop... |
| usgrnloop0ALT 28730 | Alternate proof of ~ usgrn... |
| usgredgne 28731 | An edge of a simple graph ... |
| usgrf1oedg 28732 | The edge function of a sim... |
| uhgr2edg 28733 | If a vertex is adjacent to... |
| umgr2edg 28734 | If a vertex is adjacent to... |
| usgr2edg 28735 | If a vertex is adjacent to... |
| umgr2edg1 28736 | If a vertex is adjacent to... |
| usgr2edg1 28737 | If a vertex is adjacent to... |
| umgrvad2edg 28738 | If a vertex is adjacent to... |
| umgr2edgneu 28739 | If a vertex is adjacent to... |
| usgrsizedg 28740 | In a simple graph, the siz... |
| usgredg3 28741 | The value of the "edge fun... |
| usgredg4 28742 | For a vertex incident to a... |
| usgredgreu 28743 | For a vertex incident to a... |
| usgredg2vtx 28744 | For a vertex incident to a... |
| uspgredg2vtxeu 28745 | For a vertex incident to a... |
| usgredg2vtxeu 28746 | For a vertex incident to a... |
| usgredg2vtxeuALT 28747 | Alternate proof of ~ usgre... |
| uspgredg2vlem 28748 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 28749 | In a simple pseudograph, t... |
| usgredg2vlem1 28750 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 28751 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 28752 | In a simple graph, the map... |
| usgriedgleord 28753 | Alternate version of ~ usg... |
| ushgredgedg 28754 | In a simple hypergraph the... |
| usgredgedg 28755 | In a simple graph there is... |
| ushgredgedgloop 28756 | In a simple hypergraph the... |
| uspgredgleord 28757 | In a simple pseudograph th... |
| usgredgleord 28758 | In a simple graph the numb... |
| usgredgleordALT 28759 | Alternate proof for ~ usgr... |
| usgrstrrepe 28760 | Replacing (or adding) the ... |
| usgr0e 28761 | The empty graph, with vert... |
| usgr0vb 28762 | The null graph, with no ve... |
| uhgr0v0e 28763 | The null graph, with no ve... |
| uhgr0vsize0 28764 | The size of a hypergraph w... |
| uhgr0edgfi 28765 | A graph of order 0 (i.e. w... |
| usgr0v 28766 | The null graph, with no ve... |
| uhgr0vusgr 28767 | The null graph, with no ve... |
| usgr0 28768 | The null graph represented... |
| uspgr1e 28769 | A simple pseudograph with ... |
| usgr1e 28770 | A simple graph with one ed... |
| usgr0eop 28771 | The empty graph, with vert... |
| uspgr1eop 28772 | A simple pseudograph with ... |
| uspgr1ewop 28773 | A simple pseudograph with ... |
| uspgr1v1eop 28774 | A simple pseudograph with ... |
| usgr1eop 28775 | A simple graph with (at le... |
| uspgr2v1e2w 28776 | A simple pseudograph with ... |
| usgr2v1e2w 28777 | A simple graph with two ve... |
| edg0usgr 28778 | A class without edges is a... |
| lfuhgr1v0e 28779 | A loop-free hypergraph wit... |
| usgr1vr 28780 | A simple graph with one ve... |
| usgr1v 28781 | A class with one (or no) v... |
| usgr1v0edg 28782 | A class with one (or no) v... |
| usgrexmpldifpr 28783 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 28784 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 28785 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 28786 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 28787 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 28788 | ` G ` is a simple graph of... |
| griedg0prc 28789 | The class of empty graphs ... |
| griedg0ssusgr 28790 | The class of all simple gr... |
| usgrprc 28791 | The class of simple graphs... |
| relsubgr 28794 | The class of the subgraph ... |
| subgrv 28795 | If a class is a subgraph o... |
| issubgr 28796 | The property of a set to b... |
| issubgr2 28797 | The property of a set to b... |
| subgrprop 28798 | The properties of a subgra... |
| subgrprop2 28799 | The properties of a subgra... |
| uhgrissubgr 28800 | The property of a hypergra... |
| subgrprop3 28801 | The properties of a subgra... |
| egrsubgr 28802 | An empty graph consisting ... |
| 0grsubgr 28803 | The null graph (represente... |
| 0uhgrsubgr 28804 | The null graph (as hypergr... |
| uhgrsubgrself 28805 | A hypergraph is a subgraph... |
| subgrfun 28806 | The edge function of a sub... |
| subgruhgrfun 28807 | The edge function of a sub... |
| subgreldmiedg 28808 | An element of the domain o... |
| subgruhgredgd 28809 | An edge of a subgraph of a... |
| subumgredg2 28810 | An edge of a subgraph of a... |
| subuhgr 28811 | A subgraph of a hypergraph... |
| subupgr 28812 | A subgraph of a pseudograp... |
| subumgr 28813 | A subgraph of a multigraph... |
| subusgr 28814 | A subgraph of a simple gra... |
| uhgrspansubgrlem 28815 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 28816 | A spanning subgraph ` S ` ... |
| uhgrspan 28817 | A spanning subgraph ` S ` ... |
| upgrspan 28818 | A spanning subgraph ` S ` ... |
| umgrspan 28819 | A spanning subgraph ` S ` ... |
| usgrspan 28820 | A spanning subgraph ` S ` ... |
| uhgrspanop 28821 | A spanning subgraph of a h... |
| upgrspanop 28822 | A spanning subgraph of a p... |
| umgrspanop 28823 | A spanning subgraph of a m... |
| usgrspanop 28824 | A spanning subgraph of a s... |
| uhgrspan1lem1 28825 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 28826 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 28827 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 28828 | The induced subgraph ` S `... |
| upgrreslem 28829 | Lemma for ~ upgrres . (Co... |
| umgrreslem 28830 | Lemma for ~ umgrres and ~ ... |
| upgrres 28831 | A subgraph obtained by rem... |
| umgrres 28832 | A subgraph obtained by rem... |
| usgrres 28833 | A subgraph obtained by rem... |
| upgrres1lem1 28834 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 28835 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 28836 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 28837 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 28838 | A pseudograph obtained by ... |
| umgrres1 28839 | A multigraph obtained by r... |
| usgrres1 28840 | Restricting a simple graph... |
| isfusgr 28843 | The property of being a fi... |
| fusgrvtxfi 28844 | A finite simple graph has ... |
| isfusgrf1 28845 | The property of being a fi... |
| isfusgrcl 28846 | The property of being a fi... |
| fusgrusgr 28847 | A finite simple graph is a... |
| opfusgr 28848 | A finite simple graph repr... |
| usgredgffibi 28849 | The number of edges in a s... |
| fusgredgfi 28850 | In a finite simple graph t... |
| usgr1v0e 28851 | The size of a (finite) sim... |
| usgrfilem 28852 | In a finite simple graph, ... |
| fusgrfisbase 28853 | Induction base for ~ fusgr... |
| fusgrfisstep 28854 | Induction step in ~ fusgrf... |
| fusgrfis 28855 | A finite simple graph is o... |
| fusgrfupgrfs 28856 | A finite simple graph is a... |
| nbgrprc0 28859 | The set of neighbors is em... |
| nbgrcl 28860 | If a class ` X ` has at le... |
| nbgrval 28861 | The set of neighbors of a ... |
| dfnbgr2 28862 | Alternate definition of th... |
| dfnbgr3 28863 | Alternate definition of th... |
| nbgrnvtx0 28864 | If a class ` X ` is not a ... |
| nbgrel 28865 | Characterization of a neig... |
| nbgrisvtx 28866 | Every neighbor ` N ` of a ... |
| nbgrssvtx 28867 | The neighbors of a vertex ... |
| nbuhgr 28868 | The set of neighbors of a ... |
| nbupgr 28869 | The set of neighbors of a ... |
| nbupgrel 28870 | A neighbor of a vertex in ... |
| nbumgrvtx 28871 | The set of neighbors of a ... |
| nbumgr 28872 | The set of neighbors of an... |
| nbusgrvtx 28873 | The set of neighbors of a ... |
| nbusgr 28874 | The set of neighbors of an... |
| nbgr2vtx1edg 28875 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 28876 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 28877 | If a hypergraph has two ve... |
| nbusgreledg 28878 | A class/vertex is a neighb... |
| uhgrnbgr0nb 28879 | A vertex which is not endp... |
| nbgr0vtxlem 28880 | Lemma for ~ nbgr0vtx and ~... |
| nbgr0vtx 28881 | In a null graph (with no v... |
| nbgr0edg 28882 | In an empty graph (with no... |
| nbgr1vtx 28883 | In a graph with one vertex... |
| nbgrnself 28884 | A vertex in a graph is not... |
| nbgrnself2 28885 | A class ` X ` is not a nei... |
| nbgrssovtx 28886 | The neighbors of a vertex ... |
| nbgrssvwo2 28887 | The neighbors of a vertex ... |
| nbgrsym 28888 | In a graph, the neighborho... |
| nbupgrres 28889 | The neighborhood of a vert... |
| usgrnbcnvfv 28890 | Applying the edge function... |
| nbusgredgeu 28891 | For each neighbor of a ver... |
| edgnbusgreu 28892 | For each edge incident to ... |
| nbusgredgeu0 28893 | For each neighbor of a ver... |
| nbusgrf1o0 28894 | The mapping of neighbors o... |
| nbusgrf1o1 28895 | The set of neighbors of a ... |
| nbusgrf1o 28896 | The set of neighbors of a ... |
| nbedgusgr 28897 | The number of neighbors of... |
| edgusgrnbfin 28898 | The number of neighbors of... |
| nbusgrfi 28899 | The class of neighbors of ... |
| nbfiusgrfi 28900 | The class of neighbors of ... |
| hashnbusgrnn0 28901 | The number of neighbors of... |
| nbfusgrlevtxm1 28902 | The number of neighbors of... |
| nbfusgrlevtxm2 28903 | If there is a vertex which... |
| nbusgrvtxm1 28904 | If the number of neighbors... |
| nb3grprlem1 28905 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 28906 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 28907 | The neighbors of a vertex ... |
| nb3grpr2 28908 | The neighbors of a vertex ... |
| nb3gr2nb 28909 | If the neighbors of two ve... |
| uvtxval 28912 | The set of all universal v... |
| uvtxel 28913 | A universal vertex, i.e. a... |
| uvtxisvtx 28914 | A universal vertex is a ve... |
| uvtxssvtx 28915 | The set of the universal v... |
| vtxnbuvtx 28916 | A universal vertex has all... |
| uvtxnbgrss 28917 | A universal vertex has all... |
| uvtxnbgrvtx 28918 | A universal vertex is neig... |
| uvtx0 28919 | There is no universal vert... |
| isuvtx 28920 | The set of all universal v... |
| uvtxel1 28921 | Characterization of a univ... |
| uvtx01vtx 28922 | If a graph/class has no ed... |
| uvtx2vtx1edg 28923 | If a graph has two vertice... |
| uvtx2vtx1edgb 28924 | If a hypergraph has two ve... |
| uvtxnbgr 28925 | A universal vertex has all... |
| uvtxnbgrb 28926 | A vertex is universal iff ... |
| uvtxusgr 28927 | The set of all universal v... |
| uvtxusgrel 28928 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 28929 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 28930 | If the number of neighbors... |
| uvtxnbvtxm1 28931 | A universal vertex has ` n... |
| nbupgruvtxres 28932 | The neighborhood of a univ... |
| uvtxupgrres 28933 | A universal vertex is univ... |
| cplgruvtxb 28938 | A graph ` G ` is complete ... |
| prcliscplgr 28939 | A proper class (representi... |
| iscplgr 28940 | The property of being a co... |
| iscplgrnb 28941 | A graph is complete iff al... |
| iscplgredg 28942 | A graph ` G ` is complete ... |
| iscusgr 28943 | The property of being a co... |
| cusgrusgr 28944 | A complete simple graph is... |
| cusgrcplgr 28945 | A complete simple graph is... |
| iscusgrvtx 28946 | A simple graph is complete... |
| cusgruvtxb 28947 | A simple graph is complete... |
| iscusgredg 28948 | A simple graph is complete... |
| cusgredg 28949 | In a complete simple graph... |
| cplgr0 28950 | The null graph (with no ve... |
| cusgr0 28951 | The null graph (with no ve... |
| cplgr0v 28952 | A null graph (with no vert... |
| cusgr0v 28953 | A graph with no vertices a... |
| cplgr1vlem 28954 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 28955 | A graph with one vertex is... |
| cusgr1v 28956 | A graph with one vertex an... |
| cplgr2v 28957 | An undirected hypergraph w... |
| cplgr2vpr 28958 | An undirected hypergraph w... |
| nbcplgr 28959 | In a complete graph, each ... |
| cplgr3v 28960 | A pseudograph with three (... |
| cusgr3vnbpr 28961 | The neighbors of a vertex ... |
| cplgrop 28962 | A complete graph represent... |
| cusgrop 28963 | A complete simple graph re... |
| cusgrexilem1 28964 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 28965 | Lemma for ~ usgrexi . (Co... |
| usgrexi 28966 | An arbitrary set regarded ... |
| cusgrexilem2 28967 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 28968 | An arbitrary set ` V ` reg... |
| cusgrexg 28969 | For each set there is a se... |
| structtousgr 28970 | Any (extensible) structure... |
| structtocusgr 28971 | Any (extensible) structure... |
| cffldtocusgr 28972 | The field of complex numbe... |
| cusgrres 28973 | Restricting a complete sim... |
| cusgrsizeindb0 28974 | Base case of the induction... |
| cusgrsizeindb1 28975 | Base case of the induction... |
| cusgrsizeindslem 28976 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 28977 | Part 1 of induction step i... |
| cusgrsize2inds 28978 | Induction step in ~ cusgrs... |
| cusgrsize 28979 | The size of a finite compl... |
| cusgrfilem1 28980 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 28981 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 28982 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 28983 | If the size of a complete ... |
| usgredgsscusgredg 28984 | A simple graph is a subgra... |
| usgrsscusgr 28985 | A simple graph is a subgra... |
| sizusglecusglem1 28986 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 28987 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 28988 | The size of a simple graph... |
| fusgrmaxsize 28989 | The maximum size of a fini... |
| vtxdgfval 28992 | The value of the vertex de... |
| vtxdgval 28993 | The degree of a vertex. (... |
| vtxdgfival 28994 | The degree of a vertex for... |
| vtxdgop 28995 | The vertex degree expresse... |
| vtxdgf 28996 | The vertex degree function... |
| vtxdgelxnn0 28997 | The degree of a vertex is ... |
| vtxdg0v 28998 | The degree of a vertex in ... |
| vtxdg0e 28999 | The degree of a vertex in ... |
| vtxdgfisnn0 29000 | The degree of a vertex in ... |
| vtxdgfisf 29001 | The vertex degree function... |
| vtxdeqd 29002 | Equality theorem for the v... |
| vtxduhgr0e 29003 | The degree of a vertex in ... |
| vtxdlfuhgr1v 29004 | The degree of the vertex i... |
| vdumgr0 29005 | A vertex in a multigraph h... |
| vtxdun 29006 | The degree of a vertex in ... |
| vtxdfiun 29007 | The degree of a vertex in ... |
| vtxduhgrun 29008 | The degree of a vertex in ... |
| vtxduhgrfiun 29009 | The degree of a vertex in ... |
| vtxdlfgrval 29010 | The value of the vertex de... |
| vtxdumgrval 29011 | The value of the vertex de... |
| vtxdusgrval 29012 | The value of the vertex de... |
| vtxd0nedgb 29013 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 29014 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 29015 | The value of the vertex de... |
| vtxdusgrfvedg 29016 | The value of the vertex de... |
| vtxduhgr0nedg 29017 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 29018 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 29019 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 29020 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 29021 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 29022 | The vertex degree function... |
| vtxdgfusgr 29023 | In a finite simple graph, ... |
| fusgrn0degnn0 29024 | In a nonempty, finite grap... |
| 1loopgruspgr 29025 | A graph with one edge whic... |
| 1loopgredg 29026 | The set of edges in a grap... |
| 1loopgrnb0 29027 | In a graph (simple pseudog... |
| 1loopgrvd2 29028 | The vertex degree of a one... |
| 1loopgrvd0 29029 | The vertex degree of a one... |
| 1hevtxdg0 29030 | The vertex degree of verte... |
| 1hevtxdg1 29031 | The vertex degree of verte... |
| 1hegrvtxdg1 29032 | The vertex degree of a gra... |
| 1hegrvtxdg1r 29033 | The vertex degree of a gra... |
| 1egrvtxdg1 29034 | The vertex degree of a one... |
| 1egrvtxdg1r 29035 | The vertex degree of a one... |
| 1egrvtxdg0 29036 | The vertex degree of a one... |
| p1evtxdeqlem 29037 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 29038 | If an edge ` E ` which doe... |
| p1evtxdp1 29039 | If an edge ` E ` (not bein... |
| uspgrloopvtx 29040 | The set of vertices in a g... |
| uspgrloopvtxel 29041 | A vertex in a graph (simpl... |
| uspgrloopiedg 29042 | The set of edges in a grap... |
| uspgrloopedg 29043 | The set of edges in a grap... |
| uspgrloopnb0 29044 | In a graph (simple pseudog... |
| uspgrloopvd2 29045 | The vertex degree of a one... |
| umgr2v2evtx 29046 | The set of vertices in a m... |
| umgr2v2evtxel 29047 | A vertex in a multigraph w... |
| umgr2v2eiedg 29048 | The edge function in a mul... |
| umgr2v2eedg 29049 | The set of edges in a mult... |
| umgr2v2e 29050 | A multigraph with two edge... |
| umgr2v2enb1 29051 | In a multigraph with two e... |
| umgr2v2evd2 29052 | In a multigraph with two e... |
| hashnbusgrvd 29053 | In a simple graph, the num... |
| usgruvtxvdb 29054 | In a finite simple graph w... |
| vdiscusgrb 29055 | A finite simple graph with... |
| vdiscusgr 29056 | In a finite complete simpl... |
| vtxdusgradjvtx 29057 | The degree of a vertex in ... |
| usgrvd0nedg 29058 | If a vertex in a simple gr... |
| uhgrvd00 29059 | If every vertex in a hyper... |
| usgrvd00 29060 | If every vertex in a simpl... |
| vdegp1ai 29061 | The induction step for a v... |
| vdegp1bi 29062 | The induction step for a v... |
| vdegp1ci 29063 | The induction step for a v... |
| vtxdginducedm1lem1 29064 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 29065 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 29066 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 29067 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 29068 | The degree of a vertex ` v... |
| vtxdginducedm1fi 29069 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 29070 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 29071 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 29072 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 29073 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 29074 | Induction step of ~ finsum... |
| finsumvtxdg2size 29075 | The sum of the degrees of ... |
| fusgr1th 29076 | The sum of the degrees of ... |
| finsumvtxdgeven 29077 | The sum of the degrees of ... |
| vtxdgoddnumeven 29078 | The number of vertices of ... |
| fusgrvtxdgonume 29079 | The number of vertices of ... |
| isrgr 29084 | The property of a class be... |
| rgrprop 29085 | The properties of a k-regu... |
| isrusgr 29086 | The property of being a k-... |
| rusgrprop 29087 | The properties of a k-regu... |
| rusgrrgr 29088 | A k-regular simple graph i... |
| rusgrusgr 29089 | A k-regular simple graph i... |
| finrusgrfusgr 29090 | A finite regular simple gr... |
| isrusgr0 29091 | The property of being a k-... |
| rusgrprop0 29092 | The properties of a k-regu... |
| usgreqdrusgr 29093 | If all vertices in a simpl... |
| fusgrregdegfi 29094 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 29095 | If all vertices in a nonem... |
| frusgrnn0 29096 | In a nonempty finite k-reg... |
| 0edg0rgr 29097 | A graph is 0-regular if it... |
| uhgr0edg0rgr 29098 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 29099 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 29100 | A simple graph is 0-regula... |
| 0vtxrgr 29101 | A null graph (with no vert... |
| 0vtxrusgr 29102 | A graph with no vertices a... |
| 0uhgrrusgr 29103 | The null graph as hypergra... |
| 0grrusgr 29104 | The null graph represented... |
| 0grrgr 29105 | The null graph represented... |
| cusgrrusgr 29106 | A complete simple graph wi... |
| cusgrm1rusgr 29107 | A finite simple graph with... |
| rusgrpropnb 29108 | The properties of a k-regu... |
| rusgrpropedg 29109 | The properties of a k-regu... |
| rusgrpropadjvtx 29110 | The properties of a k-regu... |
| rusgrnumwrdl2 29111 | In a k-regular simple grap... |
| rusgr1vtxlem 29112 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 29113 | If a k-regular simple grap... |
| rgrusgrprc 29114 | The class of 0-regular sim... |
| rusgrprc 29115 | The class of 0-regular sim... |
| rgrprc 29116 | The class of 0-regular gra... |
| rgrprcx 29117 | The class of 0-regular gra... |
| rgrx0ndm 29118 | 0 is not in the domain of ... |
| rgrx0nd 29119 | The potentially alternativ... |
| ewlksfval 29126 | The set of s-walks of edge... |
| isewlk 29127 | Conditions for a function ... |
| ewlkprop 29128 | Properties of an s-walk of... |
| ewlkinedg 29129 | The intersection (common v... |
| ewlkle 29130 | An s-walk of edges is also... |
| upgrewlkle2 29131 | In a pseudograph, there is... |
| wkslem1 29132 | Lemma 1 for walks to subst... |
| wkslem2 29133 | Lemma 2 for walks to subst... |
| wksfval 29134 | The set of walks (in an un... |
| iswlk 29135 | Properties of a pair of fu... |
| wlkprop 29136 | Properties of a walk. (Co... |
| wlkv 29137 | The classes involved in a ... |
| iswlkg 29138 | Generalization of ~ iswlk ... |
| wlkf 29139 | The mapping enumerating th... |
| wlkcl 29140 | A walk has length ` # ( F ... |
| wlkp 29141 | The mapping enumerating th... |
| wlkpwrd 29142 | The sequence of vertices o... |
| wlklenvp1 29143 | The number of vertices of ... |
| wksv 29144 | The class of walks is a se... |
| wksvOLD 29145 | Obsolete version of ~ wksv... |
| wlkn0 29146 | The sequence of vertices o... |
| wlklenvm1 29147 | The number of edges of a w... |
| ifpsnprss 29148 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 29149 | Each pair of adjacent vert... |
| wlkvtxiedg 29150 | The vertices of a walk are... |
| relwlk 29151 | The set ` ( Walks `` G ) `... |
| wlkvv 29152 | If there is at least one w... |
| wlkop 29153 | A walk is an ordered pair.... |
| wlkcpr 29154 | A walk as class with two c... |
| wlk2f 29155 | If there is a walk ` W ` t... |
| wlkcomp 29156 | A walk expressed by proper... |
| wlkcompim 29157 | Implications for the prope... |
| wlkelwrd 29158 | The components of a walk a... |
| wlkeq 29159 | Conditions for two walks (... |
| edginwlk 29160 | The value of the edge func... |
| upgredginwlk 29161 | The value of the edge func... |
| iedginwlk 29162 | The value of the edge func... |
| wlkl1loop 29163 | A walk of length 1 from a ... |
| wlk1walk 29164 | A walk is a 1-walk "on the... |
| wlk1ewlk 29165 | A walk is an s-walk "on th... |
| upgriswlk 29166 | Properties of a pair of fu... |
| upgrwlkedg 29167 | The edges of a walk in a p... |
| upgrwlkcompim 29168 | Implications for the prope... |
| wlkvtxedg 29169 | The vertices of a walk are... |
| upgrwlkvtxedg 29170 | The pairs of connected ver... |
| uspgr2wlkeq 29171 | Conditions for two walks w... |
| uspgr2wlkeq2 29172 | Conditions for two walks w... |
| uspgr2wlkeqi 29173 | Conditions for two walks w... |
| umgrwlknloop 29174 | In a multigraph, each walk... |
| wlkResOLD 29175 | Obsolete version of ~ opab... |
| wlkv0 29176 | If there is a walk in the ... |
| g0wlk0 29177 | There is no walk in a null... |
| 0wlk0 29178 | There is no walk for the e... |
| wlk0prc 29179 | There is no walk in a null... |
| wlklenvclwlk 29180 | The number of vertices in ... |
| wlkson 29181 | The set of walks between t... |
| iswlkon 29182 | Properties of a pair of fu... |
| wlkonprop 29183 | Properties of a walk betwe... |
| wlkpvtx 29184 | A walk connects vertices. ... |
| wlkepvtx 29185 | The endpoints of a walk ar... |
| wlkoniswlk 29186 | A walk between two vertice... |
| wlkonwlk 29187 | A walk is a walk between i... |
| wlkonwlk1l 29188 | A walk is a walk from its ... |
| wlksoneq1eq2 29189 | Two walks with identical s... |
| wlkonl1iedg 29190 | If there is a walk between... |
| wlkon2n0 29191 | The length of a walk betwe... |
| 2wlklem 29192 | Lemma for theorems for wal... |
| upgr2wlk 29193 | Properties of a pair of fu... |
| wlkreslem 29194 | Lemma for ~ wlkres . (Con... |
| wlkres 29195 | The restriction ` <. H , Q... |
| redwlklem 29196 | Lemma for ~ redwlk . (Con... |
| redwlk 29197 | A walk ending at the last ... |
| wlkp1lem1 29198 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 29199 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 29200 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 29201 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 29202 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 29203 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 29204 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 29205 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 29206 | Append one path segment (e... |
| wlkdlem1 29207 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 29208 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 29209 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 29210 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 29211 | Two words representing a w... |
| lfgrwlkprop 29212 | Two adjacent vertices in a... |
| lfgriswlk 29213 | Conditions for a pair of f... |
| lfgrwlknloop 29214 | In a loop-free graph, each... |
| reltrls 29219 | The set ` ( Trails `` G ) ... |
| trlsfval 29220 | The set of trails (in an u... |
| istrl 29221 | Conditions for a pair of c... |
| trliswlk 29222 | A trail is a walk. (Contr... |
| trlf1 29223 | The enumeration ` F ` of a... |
| trlreslem 29224 | Lemma for ~ trlres . Form... |
| trlres 29225 | The restriction ` <. H , Q... |
| upgrtrls 29226 | The set of trails in a pse... |
| upgristrl 29227 | Properties of a pair of fu... |
| upgrf1istrl 29228 | Properties of a pair of a ... |
| wksonproplem 29229 | Lemma for theorems for pro... |
| wksonproplemOLD 29230 | Obsolete version of ~ wkso... |
| trlsonfval 29231 | The set of trails between ... |
| istrlson 29232 | Properties of a pair of fu... |
| trlsonprop 29233 | Properties of a trail betw... |
| trlsonistrl 29234 | A trail between two vertic... |
| trlsonwlkon 29235 | A trail between two vertic... |
| trlontrl 29236 | A trail is a trail between... |
| relpths 29245 | The set ` ( Paths `` G ) `... |
| pthsfval 29246 | The set of paths (in an un... |
| spthsfval 29247 | The set of simple paths (i... |
| ispth 29248 | Conditions for a pair of c... |
| isspth 29249 | Conditions for a pair of c... |
| pthistrl 29250 | A path is a trail (in an u... |
| spthispth 29251 | A simple path is a path (i... |
| pthiswlk 29252 | A path is a walk (in an un... |
| spthiswlk 29253 | A simple path is a walk (i... |
| pthdivtx 29254 | The inner vertices of a pa... |
| pthdadjvtx 29255 | The adjacent vertices of a... |
| 2pthnloop 29256 | A path of length at least ... |
| upgr2pthnlp 29257 | A path of length at least ... |
| spthdifv 29258 | The vertices of a simple p... |
| spthdep 29259 | A simple path (at least of... |
| pthdepisspth 29260 | A path with different star... |
| upgrwlkdvdelem 29261 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 29262 | In a pseudograph, all edge... |
| upgrspthswlk 29263 | The set of simple paths in... |
| upgrwlkdvspth 29264 | A walk consisting of diffe... |
| pthsonfval 29265 | The set of paths between t... |
| spthson 29266 | The set of simple paths be... |
| ispthson 29267 | Properties of a pair of fu... |
| isspthson 29268 | Properties of a pair of fu... |
| pthsonprop 29269 | Properties of a path betwe... |
| spthonprop 29270 | Properties of a simple pat... |
| pthonispth 29271 | A path between two vertice... |
| pthontrlon 29272 | A path between two vertice... |
| pthonpth 29273 | A path is a path between i... |
| isspthonpth 29274 | A pair of functions is a s... |
| spthonisspth 29275 | A simple path between to v... |
| spthonpthon 29276 | A simple path between two ... |
| spthonepeq 29277 | The endpoints of a simple ... |
| uhgrwkspthlem1 29278 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 29279 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 29280 | Any walk of length 1 betwe... |
| usgr2wlkneq 29281 | The vertices and edges are... |
| usgr2wlkspthlem1 29282 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 29283 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 29284 | In a simple graph, any wal... |
| usgr2trlncl 29285 | In a simple graph, any tra... |
| usgr2trlspth 29286 | In a simple graph, any tra... |
| usgr2pthspth 29287 | In a simple graph, any pat... |
| usgr2pthlem 29288 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 29289 | In a simple graph, there i... |
| usgr2pth0 29290 | In a simply graph, there i... |
| pthdlem1 29291 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 29292 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 29293 | Lemma 2 for ~ pthd . (Con... |
| pthd 29294 | Two words representing a t... |
| clwlks 29297 | The set of closed walks (i... |
| isclwlk 29298 | A pair of functions repres... |
| clwlkiswlk 29299 | A closed walk is a walk (i... |
| clwlkwlk 29300 | Closed walks are walks (in... |
| clwlkswks 29301 | Closed walks are walks (in... |
| isclwlke 29302 | Properties of a pair of fu... |
| isclwlkupgr 29303 | Properties of a pair of fu... |
| clwlkcomp 29304 | A closed walk expressed by... |
| clwlkcompim 29305 | Implications for the prope... |
| upgrclwlkcompim 29306 | Implications for the prope... |
| clwlkcompbp 29307 | Basic properties of the co... |
| clwlkl1loop 29308 | A closed walk of length 1 ... |
| crcts 29313 | The set of circuits (in an... |
| cycls 29314 | The set of cycles (in an u... |
| iscrct 29315 | Sufficient and necessary c... |
| iscycl 29316 | Sufficient and necessary c... |
| crctprop 29317 | The properties of a circui... |
| cyclprop 29318 | The properties of a cycle:... |
| crctisclwlk 29319 | A circuit is a closed walk... |
| crctistrl 29320 | A circuit is a trail. (Co... |
| crctiswlk 29321 | A circuit is a walk. (Con... |
| cyclispth 29322 | A cycle is a path. (Contr... |
| cycliswlk 29323 | A cycle is a walk. (Contr... |
| cycliscrct 29324 | A cycle is a circuit. (Co... |
| cyclnspth 29325 | A (non-trivial) cycle is n... |
| cyclispthon 29326 | A cycle is a path starting... |
| lfgrn1cycl 29327 | In a loop-free graph there... |
| usgr2trlncrct 29328 | In a simple graph, any tra... |
| umgrn1cycl 29329 | In a multigraph graph (wit... |
| uspgrn2crct 29330 | In a simple pseudograph th... |
| usgrn2cycl 29331 | In a simple graph there ar... |
| crctcshwlkn0lem1 29332 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 29333 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 29334 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 29335 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 29336 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 29337 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 29338 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 29339 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 29340 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 29341 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 29342 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 29343 | Cyclically shifting the in... |
| crctcshwlk 29344 | Cyclically shifting the in... |
| crctcshtrl 29345 | Cyclically shifting the in... |
| crctcsh 29346 | Cyclically shifting the in... |
| wwlks 29357 | The set of walks (in an un... |
| iswwlks 29358 | A word over the set of ver... |
| wwlksn 29359 | The set of walks (in an un... |
| iswwlksn 29360 | A word over the set of ver... |
| wwlksnprcl 29361 | Derivation of the length o... |
| iswwlksnx 29362 | Properties of a word to re... |
| wwlkbp 29363 | Basic properties of a walk... |
| wwlknbp 29364 | Basic properties of a walk... |
| wwlknp 29365 | Properties of a set being ... |
| wwlknbp1 29366 | Other basic properties of ... |
| wwlknvtx 29367 | The symbols of a word ` W ... |
| wwlknllvtx 29368 | If a word ` W ` represents... |
| wwlknlsw 29369 | If a word represents a wal... |
| wspthsn 29370 | The set of simple paths of... |
| iswspthn 29371 | An element of the set of s... |
| wspthnp 29372 | Properties of a set being ... |
| wwlksnon 29373 | The set of walks of a fixe... |
| wspthsnon 29374 | The set of simple paths of... |
| iswwlksnon 29375 | The set of walks of a fixe... |
| wwlksnon0 29376 | Sufficient conditions for ... |
| wwlksonvtx 29377 | If a word ` W ` represents... |
| iswspthsnon 29378 | The set of simple paths of... |
| wwlknon 29379 | An element of the set of w... |
| wspthnon 29380 | An element of the set of s... |
| wspthnonp 29381 | Properties of a set being ... |
| wspthneq1eq2 29382 | Two simple paths with iden... |
| wwlksn0s 29383 | The set of all walks as wo... |
| wwlkssswrd 29384 | Walks (represented by word... |
| wwlksn0 29385 | A walk of length 0 is repr... |
| 0enwwlksnge1 29386 | In graphs without edges, t... |
| wwlkswwlksn 29387 | A walk of a fixed length a... |
| wwlkssswwlksn 29388 | The walks of a fixed lengt... |
| wlkiswwlks1 29389 | The sequence of vertices i... |
| wlklnwwlkln1 29390 | The sequence of vertices i... |
| wlkiswwlks2lem1 29391 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 29392 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 29393 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 29394 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 29395 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 29396 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 29397 | A walk as word corresponds... |
| wlkiswwlks 29398 | A walk as word corresponds... |
| wlkiswwlksupgr2 29399 | A walk as word corresponds... |
| wlkiswwlkupgr 29400 | A walk as word corresponds... |
| wlkswwlksf1o 29401 | The mapping of (ordinary) ... |
| wlkswwlksen 29402 | The set of walks as words ... |
| wwlksm1edg 29403 | Removing the trailing edge... |
| wlklnwwlkln2lem 29404 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 29405 | A walk of length ` N ` as ... |
| wlklnwwlkn 29406 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 29407 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 29408 | A walk of length ` N ` as ... |
| wlknewwlksn 29409 | If a walk in a pseudograph... |
| wlknwwlksnbij 29410 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 29411 | In a simple pseudograph, t... |
| wlknwwlksneqs 29412 | The set of walks of a fixe... |
| wwlkseq 29413 | Equality of two walks (as ... |
| wwlksnred 29414 | Reduction of a walk (as wo... |
| wwlksnext 29415 | Extension of a walk (as wo... |
| wwlksnextbi 29416 | Extension of a walk (as wo... |
| wwlksnredwwlkn 29417 | For each walk (as word) of... |
| wwlksnredwwlkn0 29418 | For each walk (as word) of... |
| wwlksnextwrd 29419 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 29420 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 29421 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 29422 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 29423 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 29424 | There is a bijection betwe... |
| wwlksnexthasheq 29425 | The number of the extensio... |
| disjxwwlksn 29426 | Sets of walks (as words) e... |
| wwlksnndef 29427 | Conditions for ` WWalksN `... |
| wwlksnfi 29428 | The number of walks repres... |
| wlksnfi 29429 | The number of walks of fix... |
| wlksnwwlknvbij 29430 | There is a bijection betwe... |
| wwlksnextproplem1 29431 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 29432 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 29433 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 29434 | Adding additional properti... |
| disjxwwlkn 29435 | Sets of walks (as words) e... |
| hashwwlksnext 29436 | Number of walks (as words)... |
| wwlksnwwlksnon 29437 | A walk of fixed length is ... |
| wspthsnwspthsnon 29438 | A simple path of fixed len... |
| wspthsnonn0vne 29439 | If the set of simple paths... |
| wspthsswwlkn 29440 | The set of simple paths of... |
| wspthnfi 29441 | In a finite graph, the set... |
| wwlksnonfi 29442 | In a finite graph, the set... |
| wspthsswwlknon 29443 | The set of simple paths of... |
| wspthnonfi 29444 | In a finite graph, the set... |
| wspniunwspnon 29445 | The set of nonempty simple... |
| wspn0 29446 | If there are no vertices, ... |
| 2wlkdlem1 29447 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 29448 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 29449 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 29450 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 29451 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 29452 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 29453 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 29454 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 29455 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 29456 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 29457 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 29458 | Construction of a walk fro... |
| 2wlkond 29459 | A walk of length 2 from on... |
| 2trld 29460 | Construction of a trail fr... |
| 2trlond 29461 | A trail of length 2 from o... |
| 2pthd 29462 | A path of length 2 from on... |
| 2spthd 29463 | A simple path of length 2 ... |
| 2pthond 29464 | A simple path of length 2 ... |
| 2pthon3v 29465 | For a vertex adjacent to t... |
| umgr2adedgwlklem 29466 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 29467 | In a multigraph, two adjac... |
| umgr2adedgwlkon 29468 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 29469 | Alternate proof for ~ umgr... |
| umgr2adedgspth 29470 | In a multigraph, two adjac... |
| umgr2wlk 29471 | In a multigraph, there is ... |
| umgr2wlkon 29472 | For each pair of adjacent ... |
| elwwlks2s3 29473 | A walk of length 2 as word... |
| midwwlks2s3 29474 | There is a vertex between ... |
| wwlks2onv 29475 | If a length 3 string repre... |
| elwwlks2ons3im 29476 | A walk as word of length 2... |
| elwwlks2ons3 29477 | For each walk of length 2 ... |
| s3wwlks2on 29478 | A length 3 string which re... |
| umgrwwlks2on 29479 | A walk of length 2 between... |
| wwlks2onsym 29480 | There is a walk of length ... |
| elwwlks2on 29481 | A walk of length 2 between... |
| elwspths2on 29482 | A simple path of length 2 ... |
| wpthswwlks2on 29483 | For two different vertices... |
| 2wspdisj 29484 | All simple paths of length... |
| 2wspiundisj 29485 | All simple paths of length... |
| usgr2wspthons3 29486 | A simple path of length 2 ... |
| usgr2wspthon 29487 | A simple path of length 2 ... |
| elwwlks2 29488 | A walk of length 2 between... |
| elwspths2spth 29489 | A simple path of length 2 ... |
| rusgrnumwwlkl1 29490 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 29491 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 29492 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 29493 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 29494 | Induction base 1 for ~ rus... |
| rusgr0edg 29495 | Special case for graphs wi... |
| rusgrnumwwlks 29496 | Induction step for ~ rusgr... |
| rusgrnumwwlk 29497 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 29498 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 29499 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 29500 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 29501 | In a ` K `-regular graph, ... |
| clwwlk 29504 | The set of closed walks (i... |
| isclwwlk 29505 | Properties of a word to re... |
| clwwlkbp 29506 | Basic properties of a clos... |
| clwwlkgt0 29507 | There is no empty closed w... |
| clwwlksswrd 29508 | Closed walks (represented ... |
| clwwlk1loop 29509 | A closed walk of length 1 ... |
| clwwlkccatlem 29510 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 29511 | The concatenation of two w... |
| umgrclwwlkge2 29512 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 29513 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 29514 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 29515 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 29516 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 29517 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 29518 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 29519 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 29520 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 29521 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 29522 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 29523 | A closed walk as word of l... |
| clwlkclwwlk2 29524 | A closed walk corresponds ... |
| clwlkclwwlkflem 29525 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 29526 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 29527 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 29528 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 29529 | ` F ` is a function from t... |
| clwlkclwwlkfo 29530 | ` F ` is a function from t... |
| clwlkclwwlkf1 29531 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 29532 | ` F ` is a bijection betwe... |
| clwlkclwwlken 29533 | The set of the nonempty cl... |
| clwwisshclwwslemlem 29534 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 29535 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 29536 | Cyclically shifting a clos... |
| clwwisshclwwsn 29537 | Cyclically shifting a clos... |
| erclwwlkrel 29538 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 29539 | Two classes are equivalent... |
| erclwwlkeqlen 29540 | If two classes are equival... |
| erclwwlkref 29541 | ` .~ ` is a reflexive rela... |
| erclwwlksym 29542 | ` .~ ` is a symmetric rela... |
| erclwwlktr 29543 | ` .~ ` is a transitive rel... |
| erclwwlk 29544 | ` .~ ` is an equivalence r... |
| clwwlkn 29547 | The set of closed walks of... |
| isclwwlkn 29548 | A word over the set of ver... |
| clwwlkn0 29549 | There is no closed walk of... |
| clwwlkneq0 29550 | Sufficient conditions for ... |
| clwwlkclwwlkn 29551 | A closed walk of a fixed l... |
| clwwlksclwwlkn 29552 | The closed walks of a fixe... |
| clwwlknlen 29553 | The length of a word repre... |
| clwwlknnn 29554 | The length of a closed wal... |
| clwwlknwrd 29555 | A closed walk of a fixed l... |
| clwwlknbp 29556 | Basic properties of a clos... |
| isclwwlknx 29557 | Characterization of a word... |
| clwwlknp 29558 | Properties of a set being ... |
| clwwlknwwlksn 29559 | A word representing a clos... |
| clwwlknlbonbgr1 29560 | The last but one vertex in... |
| clwwlkinwwlk 29561 | If the initial vertex of a... |
| clwwlkn1 29562 | A closed walk of length 1 ... |
| loopclwwlkn1b 29563 | The singleton word consist... |
| clwwlkn1loopb 29564 | A word represents a closed... |
| clwwlkn2 29565 | A closed walk of length 2 ... |
| clwwlknfi 29566 | If there is only a finite ... |
| clwwlkel 29567 | Obtaining a closed walk (a... |
| clwwlkf 29568 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 29569 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 29570 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 29571 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 29572 | F is a 1-1 onto function, ... |
| clwwlken 29573 | The set of closed walks of... |
| clwwlknwwlkncl 29574 | Obtaining a closed walk (a... |
| clwwlkwwlksb 29575 | A nonempty word over verti... |
| clwwlknwwlksnb 29576 | A word over vertices repre... |
| clwwlkext2edg 29577 | If a word concatenated wit... |
| wwlksext2clwwlk 29578 | If a word represents a wal... |
| wwlksubclwwlk 29579 | Any prefix of a word repre... |
| clwwnisshclwwsn 29580 | Cyclically shifting a clos... |
| eleclclwwlknlem1 29581 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 29582 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 29583 | The set of cyclical shifts... |
| clwwlknccat 29584 | The concatenation of two w... |
| umgr2cwwk2dif 29585 | If a word represents a clo... |
| umgr2cwwkdifex 29586 | If a word represents a clo... |
| erclwwlknrel 29587 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 29588 | Two classes are equivalent... |
| erclwwlkneqlen 29589 | If two classes are equival... |
| erclwwlknref 29590 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 29591 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 29592 | ` .~ ` is a transitive rel... |
| erclwwlkn 29593 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 29594 | The quotient set of the se... |
| hashclwwlkn0 29595 | The number of closed walks... |
| eclclwwlkn1 29596 | An equivalence class accor... |
| eleclclwwlkn 29597 | A member of an equivalence... |
| hashecclwwlkn1 29598 | The size of every equivale... |
| umgrhashecclwwlk 29599 | The size of every equivale... |
| fusgrhashclwwlkn 29600 | The size of the set of clo... |
| clwwlkndivn 29601 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 29602 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 29603 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 29604 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 29605 | There is a one-to-one onto... |
| clwlkssizeeq 29606 | The size of the set of clo... |
| clwlksndivn 29607 | The size of the set of clo... |
| clwwlknonmpo 29610 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 29611 | The set of closed walks on... |
| isclwwlknon 29612 | A word over the set of ver... |
| clwwlk0on0 29613 | There is no word over the ... |
| clwwlknon0 29614 | Sufficient conditions for ... |
| clwwlknonfin 29615 | In a finite graph ` G ` , ... |
| clwwlknonel 29616 | Characterization of a word... |
| clwwlknonccat 29617 | The concatenation of two w... |
| clwwlknon1 29618 | The set of closed walks on... |
| clwwlknon1loop 29619 | If there is a loop at vert... |
| clwwlknon1nloop 29620 | If there is no loop at ver... |
| clwwlknon1sn 29621 | The set of (closed) walks ... |
| clwwlknon1le1 29622 | There is at most one (clos... |
| clwwlknon2 29623 | The set of closed walks on... |
| clwwlknon2x 29624 | The set of closed walks on... |
| s2elclwwlknon2 29625 | Sufficient conditions of a... |
| clwwlknon2num 29626 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 29627 | A word over vertices repre... |
| clwwlknonex2lem1 29628 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 29629 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 29630 | Extending a closed walk ` ... |
| clwwlknonex2e 29631 | Extending a closed walk ` ... |
| clwwlknondisj 29632 | The sets of closed walks o... |
| clwwlknun 29633 | The set of closed walks of... |
| clwwlkvbij 29634 | There is a bijection betwe... |
| 0ewlk 29635 | The empty set (empty seque... |
| 1ewlk 29636 | A sequence of 1 edge is an... |
| 0wlk 29637 | A pair of an empty set (of... |
| is0wlk 29638 | A pair of an empty set (of... |
| 0wlkonlem1 29639 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 29640 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 29641 | A walk of length 0 from a ... |
| 0wlkons1 29642 | A walk of length 0 from a ... |
| 0trl 29643 | A pair of an empty set (of... |
| is0trl 29644 | A pair of an empty set (of... |
| 0trlon 29645 | A trail of length 0 from a... |
| 0pth 29646 | A pair of an empty set (of... |
| 0spth 29647 | A pair of an empty set (of... |
| 0pthon 29648 | A path of length 0 from a ... |
| 0pthon1 29649 | A path of length 0 from a ... |
| 0pthonv 29650 | For each vertex there is a... |
| 0clwlk 29651 | A pair of an empty set (of... |
| 0clwlkv 29652 | Any vertex (more precisely... |
| 0clwlk0 29653 | There is no closed walk in... |
| 0crct 29654 | A pair of an empty set (of... |
| 0cycl 29655 | A pair of an empty set (of... |
| 1pthdlem1 29656 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 29657 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 29658 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 29659 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 29660 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 29661 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 29662 | In a graph with two vertic... |
| 1trld 29663 | In a graph with two vertic... |
| 1pthd 29664 | In a graph with two vertic... |
| 1pthond 29665 | In a graph with two vertic... |
| upgr1wlkdlem1 29666 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 29667 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 29668 | In a pseudograph with two ... |
| upgr1trld 29669 | In a pseudograph with two ... |
| upgr1pthd 29670 | In a pseudograph with two ... |
| upgr1pthond 29671 | In a pseudograph with two ... |
| lppthon 29672 | A loop (which is an edge a... |
| lp1cycl 29673 | A loop (which is an edge a... |
| 1pthon2v 29674 | For each pair of adjacent ... |
| 1pthon2ve 29675 | For each pair of adjacent ... |
| wlk2v2elem1 29676 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 29677 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 29678 | In a graph with two vertic... |
| ntrl2v2e 29679 | A walk which is not a trai... |
| 3wlkdlem1 29680 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 29681 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 29682 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 29683 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 29684 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 29685 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 29686 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 29687 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 29688 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 29689 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 29690 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 29691 | Construction of a walk fro... |
| 3wlkond 29692 | A walk of length 3 from on... |
| 3trld 29693 | Construction of a trail fr... |
| 3trlond 29694 | A trail of length 3 from o... |
| 3pthd 29695 | A path of length 3 from on... |
| 3pthond 29696 | A path of length 3 from on... |
| 3spthd 29697 | A simple path of length 3 ... |
| 3spthond 29698 | A simple path of length 3 ... |
| 3cycld 29699 | Construction of a 3-cycle ... |
| 3cyclpd 29700 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 29701 | If there is a cycle of len... |
| uhgr3cyclexlem 29702 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 29703 | If there are three differe... |
| umgr3cyclex 29704 | If there are three (differ... |
| umgr3v3e3cycl 29705 | If and only if there is a ... |
| upgr4cycl4dv4e 29706 | If there is a cycle of len... |
| dfconngr1 29709 | Alternative definition of ... |
| isconngr 29710 | The property of being a co... |
| isconngr1 29711 | The property of being a co... |
| cusconngr 29712 | A complete hypergraph is c... |
| 0conngr 29713 | A graph without vertices i... |
| 0vconngr 29714 | A graph without vertices i... |
| 1conngr 29715 | A graph with (at most) one... |
| conngrv2edg 29716 | A vertex in a connected gr... |
| vdn0conngrumgrv2 29717 | A vertex in a connected mu... |
| releupth 29720 | The set ` ( EulerPaths `` ... |
| eupths 29721 | The Eulerian paths on the ... |
| iseupth 29722 | The property " ` <. F , P ... |
| iseupthf1o 29723 | The property " ` <. F , P ... |
| eupthi 29724 | Properties of an Eulerian ... |
| eupthf1o 29725 | The ` F ` function in an E... |
| eupthfi 29726 | Any graph with an Eulerian... |
| eupthseg 29727 | The ` N ` -th edge in an e... |
| upgriseupth 29728 | The property " ` <. F , P ... |
| upgreupthi 29729 | Properties of an Eulerian ... |
| upgreupthseg 29730 | The ` N ` -th edge in an e... |
| eupthcl 29731 | An Eulerian path has lengt... |
| eupthistrl 29732 | An Eulerian path is a trai... |
| eupthiswlk 29733 | An Eulerian path is a walk... |
| eupthpf 29734 | The ` P ` function in an E... |
| eupth0 29735 | There is an Eulerian path ... |
| eupthres 29736 | The restriction ` <. H , Q... |
| eupthp1 29737 | Append one path segment to... |
| eupth2eucrct 29738 | Append one path segment to... |
| eupth2lem1 29739 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 29740 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 29741 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 29742 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 29743 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 29744 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 29745 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 29746 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 29747 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 29748 | Formerly part of proof of ... |
| eupth2lem3lem1 29749 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 29750 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 29751 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 29752 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 29753 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 29754 | Formerly part of proof of ... |
| eupth2lem3lem7 29755 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 29756 | Formerly part of proof of ... |
| eupth2lem3 29757 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 29758 | Lemma for ~ eupth2 (induct... |
| eupth2lems 29759 | Lemma for ~ eupth2 (induct... |
| eupth2 29760 | The only vertices of odd d... |
| eulerpathpr 29761 | A graph with an Eulerian p... |
| eulerpath 29762 | A pseudograph with an Eule... |
| eulercrct 29763 | A pseudograph with an Eule... |
| eucrctshift 29764 | Cyclically shifting the in... |
| eucrct2eupth1 29765 | Removing one edge ` ( I ``... |
| eucrct2eupth 29766 | Removing one edge ` ( I ``... |
| konigsbergvtx 29767 | The set of vertices of the... |
| konigsbergiedg 29768 | The indexed edges of the K... |
| konigsbergiedgw 29769 | The indexed edges of the K... |
| konigsbergssiedgwpr 29770 | Each subset of the indexed... |
| konigsbergssiedgw 29771 | Each subset of the indexed... |
| konigsbergumgr 29772 | The Königsberg graph ... |
| konigsberglem1 29773 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 29774 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 29775 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 29776 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 29777 | Lemma 5 for ~ konigsberg :... |
| konigsberg 29778 | The Königsberg Bridge... |
| isfrgr 29781 | The property of being a fr... |
| frgrusgr 29782 | A friendship graph is a si... |
| frgr0v 29783 | Any null graph (set with n... |
| frgr0vb 29784 | Any null graph (without ve... |
| frgruhgr0v 29785 | Any null graph (without ve... |
| frgr0 29786 | The null graph (graph with... |
| frcond1 29787 | The friendship condition: ... |
| frcond2 29788 | The friendship condition: ... |
| frgreu 29789 | Variant of ~ frcond2 : An... |
| frcond3 29790 | The friendship condition, ... |
| frcond4 29791 | The friendship condition, ... |
| frgr1v 29792 | Any graph with (at most) o... |
| nfrgr2v 29793 | Any graph with two (differ... |
| frgr3vlem1 29794 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 29795 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 29796 | Any graph with three verti... |
| 1vwmgr 29797 | Every graph with one verte... |
| 3vfriswmgrlem 29798 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 29799 | Every friendship graph wit... |
| 1to2vfriswmgr 29800 | Every friendship graph wit... |
| 1to3vfriswmgr 29801 | Every friendship graph wit... |
| 1to3vfriendship 29802 | The friendship theorem for... |
| 2pthfrgrrn 29803 | Between any two (different... |
| 2pthfrgrrn2 29804 | Between any two (different... |
| 2pthfrgr 29805 | Between any two (different... |
| 3cyclfrgrrn1 29806 | Every vertex in a friendsh... |
| 3cyclfrgrrn 29807 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 29808 | Every vertex in a friendsh... |
| 3cyclfrgr 29809 | Every vertex in a friendsh... |
| 4cycl2v2nb 29810 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 29811 | In a 4-cycle, two distinct... |
| n4cyclfrgr 29812 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 29813 | A graph with a 4-cycle is ... |
| frgrnbnb 29814 | If two neighbors ` U ` and... |
| frgrconngr 29815 | A friendship graph is conn... |
| vdgn0frgrv2 29816 | A vertex in a friendship g... |
| vdgn1frgrv2 29817 | Any vertex in a friendship... |
| vdgn1frgrv3 29818 | Any vertex in a friendship... |
| vdgfrgrgt2 29819 | Any vertex in a friendship... |
| frgrncvvdeqlem1 29820 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 29821 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 29822 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 29823 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 29824 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 29825 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 29826 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 29827 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 29828 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 29829 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 29830 | In a friendship graph, two... |
| frgrwopreglem4a 29831 | In a friendship graph any ... |
| frgrwopreglem5a 29832 | If a friendship graph has ... |
| frgrwopreglem1 29833 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 29834 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 29835 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 29836 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 29837 | According to statement 5 i... |
| frgrwopregbsn 29838 | According to statement 5 i... |
| frgrwopreg1 29839 | According to statement 5 i... |
| frgrwopreg2 29840 | According to statement 5 i... |
| frgrwopreglem5lem 29841 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 29842 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 29843 | Alternate direct proof of ... |
| frgrwopreg 29844 | In a friendship graph ther... |
| frgrregorufr0 29845 | In a friendship graph ther... |
| frgrregorufr 29846 | If there is a vertex havin... |
| frgrregorufrg 29847 | If there is a vertex havin... |
| frgr2wwlkeu 29848 | For two different vertices... |
| frgr2wwlkn0 29849 | In a friendship graph, the... |
| frgr2wwlk1 29850 | In a friendship graph, the... |
| frgr2wsp1 29851 | In a friendship graph, the... |
| frgr2wwlkeqm 29852 | If there is a (simple) pat... |
| frgrhash2wsp 29853 | The number of simple paths... |
| fusgreg2wsplem 29854 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 29855 | The set of paths of length... |
| fusgreghash2wspv 29856 | According to statement 7 i... |
| fusgreg2wsp 29857 | In a finite simple graph, ... |
| 2wspmdisj 29858 | The sets of paths of lengt... |
| fusgreghash2wsp 29859 | In a finite k-regular grap... |
| frrusgrord0lem 29860 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 29861 | If a nonempty finite frien... |
| frrusgrord 29862 | If a nonempty finite frien... |
| numclwwlk2lem1lem 29863 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 29864 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 29865 | If the initial vertex of a... |
| clwwnonrepclwwnon 29866 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 29867 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 29868 | Value of operation ` C ` ,... |
| 2clwwlk2 29869 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 29870 | Characterization of an ele... |
| 2clwwlk2clwwlk 29871 | An element of the value of... |
| numclwwlk1lem2foalem 29872 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 29873 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 29874 | Characterization of an ele... |
| numclwwlk1lem2foa 29875 | Going forth and back from ... |
| numclwwlk1lem2f 29876 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 29877 | Value of the function ` T ... |
| numclwwlk1lem2f1 29878 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 29879 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 29880 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 29881 | The set of double loops of... |
| numclwwlk1 29882 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 29883 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 29884 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 29885 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 29886 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 29887 | The sets of the two repres... |
| wlkl0 29888 | There is exactly one walk ... |
| clwlknon2num 29889 | There are k walks of lengt... |
| numclwlk1lem1 29890 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 29891 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 29892 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 29893 | Value of operation ` H ` ,... |
| numclwwlkovh 29894 | Value of operation ` H ` ,... |
| numclwwlkovq 29895 | Value of operation ` Q ` ,... |
| numclwwlkqhash 29896 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 29897 | In a friendship graph, for... |
| numclwlk2lem2f 29898 | ` R ` is a function mappin... |
| numclwlk2lem2fv 29899 | Value of the function ` R ... |
| numclwlk2lem2f1o 29900 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 29901 | In a friendship graph, the... |
| numclwwlk2 29902 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 29903 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 29904 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 29905 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 29906 | Statement 12 in [Huneke] p... |
| numclwwlk4 29907 | The total number of closed... |
| numclwwlk5lem 29908 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 29909 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 29910 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 29911 | For a prime divisor ` P ` ... |
| numclwwlk7 29912 | Statement 14 in [Huneke] p... |
| numclwwlk8 29913 | The size of the set of clo... |
| frgrreggt1 29914 | If a finite nonempty frien... |
| frgrreg 29915 | If a finite nonempty frien... |
| frgrregord013 29916 | If a finite friendship gra... |
| frgrregord13 29917 | If a nonempty finite frien... |
| frgrogt3nreg 29918 | If a finite friendship gra... |
| friendshipgt3 29919 | The friendship theorem for... |
| friendship 29920 | The friendship theorem: I... |
| conventions 29921 |
H... |
| conventions-labels 29922 |
... |
| conventions-comments 29923 |
... |
| natded 29924 | Here are typical n... |
| ex-natded5.2 29925 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 29926 | A more efficient proof of ... |
| ex-natded5.2i 29927 | The same as ~ ex-natded5.2... |
| ex-natded5.3 29928 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 29929 | A more efficient proof of ... |
| ex-natded5.3i 29930 | The same as ~ ex-natded5.3... |
| ex-natded5.5 29931 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 29932 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 29933 | A more efficient proof of ... |
| ex-natded5.8 29934 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 29935 | A more efficient proof of ... |
| ex-natded5.13 29936 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 29937 | A more efficient proof of ... |
| ex-natded9.20 29938 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 29939 | A more efficient proof of ... |
| ex-natded9.26 29940 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 29941 | A more efficient proof of ... |
| ex-or 29942 | Example for ~ df-or . Exa... |
| ex-an 29943 | Example for ~ df-an . Exa... |
| ex-dif 29944 | Example for ~ df-dif . Ex... |
| ex-un 29945 | Example for ~ df-un . Exa... |
| ex-in 29946 | Example for ~ df-in . Exa... |
| ex-uni 29947 | Example for ~ df-uni . Ex... |
| ex-ss 29948 | Example for ~ df-ss . Exa... |
| ex-pss 29949 | Example for ~ df-pss . Ex... |
| ex-pw 29950 | Example for ~ df-pw . Exa... |
| ex-pr 29951 | Example for ~ df-pr . (Co... |
| ex-br 29952 | Example for ~ df-br . Exa... |
| ex-opab 29953 | Example for ~ df-opab . E... |
| ex-eprel 29954 | Example for ~ df-eprel . ... |
| ex-id 29955 | Example for ~ df-id . Exa... |
| ex-po 29956 | Example for ~ df-po . Exa... |
| ex-xp 29957 | Example for ~ df-xp . Exa... |
| ex-cnv 29958 | Example for ~ df-cnv . Ex... |
| ex-co 29959 | Example for ~ df-co . Exa... |
| ex-dm 29960 | Example for ~ df-dm . Exa... |
| ex-rn 29961 | Example for ~ df-rn . Exa... |
| ex-res 29962 | Example for ~ df-res . Ex... |
| ex-ima 29963 | Example for ~ df-ima . Ex... |
| ex-fv 29964 | Example for ~ df-fv . Exa... |
| ex-1st 29965 | Example for ~ df-1st . Ex... |
| ex-2nd 29966 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 29967 | Example for ~ df-dec , 100... |
| ex-fl 29968 | Example for ~ df-fl . Exa... |
| ex-ceil 29969 | Example for ~ df-ceil . (... |
| ex-mod 29970 | Example for ~ df-mod . (C... |
| ex-exp 29971 | Example for ~ df-exp . (C... |
| ex-fac 29972 | Example for ~ df-fac . (C... |
| ex-bc 29973 | Example for ~ df-bc . (Co... |
| ex-hash 29974 | Example for ~ df-hash . (... |
| ex-sqrt 29975 | Example for ~ df-sqrt . (... |
| ex-abs 29976 | Example for ~ df-abs . (C... |
| ex-dvds 29977 | Example for ~ df-dvds : 3 ... |
| ex-gcd 29978 | Example for ~ df-gcd . (C... |
| ex-lcm 29979 | Example for ~ df-lcm . (C... |
| ex-prmo 29980 | Example for ~ df-prmo : ` ... |
| aevdemo 29981 | Proof illustrating the com... |
| ex-ind-dvds 29982 | Example of a proof by indu... |
| ex-fpar 29983 | Formalized example provide... |
| avril1 29984 | Poisson d'Avril's Theorem.... |
| 2bornot2b 29985 | The law of excluded middle... |
| helloworld 29986 | The classic "Hello world" ... |
| 1p1e2apr1 29987 | One plus one equals two. ... |
| eqid1 29988 | Law of identity (reflexivi... |
| 1div0apr 29989 | Division by zero is forbid... |
| topnfbey 29990 | Nothing seems to be imposs... |
| 9p10ne21 29991 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 29992 | 9 + 10 equals 21. This as... |
| nrt2irr 29994 | The ` N ` -th root of 2 is... |
| isplig 29997 | The predicate "is a planar... |
| ispligb 29998 | The predicate "is a planar... |
| tncp 29999 | In any planar incidence ge... |
| l2p 30000 | For any line in a planar i... |
| lpni 30001 | For any line in a planar i... |
| nsnlplig 30002 | There is no "one-point lin... |
| nsnlpligALT 30003 | Alternate version of ~ nsn... |
| n0lplig 30004 | There is no "empty line" i... |
| n0lpligALT 30005 | Alternate version of ~ n0l... |
| eulplig 30006 | Through two distinct point... |
| pliguhgr 30007 | Any planar incidence geome... |
| dummylink 30008 | Alias for ~ a1ii that may ... |
| id1 30009 | Alias for ~ idALT that may... |
| isgrpo 30018 | The predicate "is a group ... |
| isgrpoi 30019 | Properties that determine ... |
| grpofo 30020 | A group operation maps ont... |
| grpocl 30021 | Closure law for a group op... |
| grpolidinv 30022 | A group has a left identit... |
| grpon0 30023 | The base set of a group is... |
| grpoass 30024 | A group operation is assoc... |
| grpoidinvlem1 30025 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 30026 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 30027 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 30028 | Lemma for ~ grpoidinv . (... |
| grpoidinv 30029 | A group has a left and rig... |
| grpoideu 30030 | The left identity element ... |
| grporndm 30031 | A group's range in terms o... |
| 0ngrp 30032 | The empty set is not a gro... |
| gidval 30033 | The value of the identity ... |
| grpoidval 30034 | Lemma for ~ grpoidcl and o... |
| grpoidcl 30035 | The identity element of a ... |
| grpoidinv2 30036 | A group's properties using... |
| grpolid 30037 | The identity element of a ... |
| grporid 30038 | The identity element of a ... |
| grporcan 30039 | Right cancellation law for... |
| grpoinveu 30040 | The left inverse element o... |
| grpoid 30041 | Two ways of saying that an... |
| grporn 30042 | The range of a group opera... |
| grpoinvfval 30043 | The inverse function of a ... |
| grpoinvval 30044 | The inverse of a group ele... |
| grpoinvcl 30045 | A group element's inverse ... |
| grpoinv 30046 | The properties of a group ... |
| grpolinv 30047 | The left inverse of a grou... |
| grporinv 30048 | The right inverse of a gro... |
| grpoinvid1 30049 | The inverse of a group ele... |
| grpoinvid2 30050 | The inverse of a group ele... |
| grpolcan 30051 | Left cancellation law for ... |
| grpo2inv 30052 | Double inverse law for gro... |
| grpoinvf 30053 | Mapping of the inverse fun... |
| grpoinvop 30054 | The inverse of the group o... |
| grpodivfval 30055 | Group division (or subtrac... |
| grpodivval 30056 | Group division (or subtrac... |
| grpodivinv 30057 | Group division by an inver... |
| grpoinvdiv 30058 | Inverse of a group divisio... |
| grpodivf 30059 | Mapping for group division... |
| grpodivcl 30060 | Closure of group division ... |
| grpodivdiv 30061 | Double group division. (C... |
| grpomuldivass 30062 | Associative-type law for m... |
| grpodivid 30063 | Division of a group member... |
| grponpcan 30064 | Cancellation law for group... |
| isablo 30067 | The predicate "is an Abeli... |
| ablogrpo 30068 | An Abelian group operation... |
| ablocom 30069 | An Abelian group operation... |
| ablo32 30070 | Commutative/associative la... |
| ablo4 30071 | Commutative/associative la... |
| isabloi 30072 | Properties that determine ... |
| ablomuldiv 30073 | Law for group multiplicati... |
| ablodivdiv 30074 | Law for double group divis... |
| ablodivdiv4 30075 | Law for double group divis... |
| ablodiv32 30076 | Swap the second and third ... |
| ablonncan 30077 | Cancellation law for group... |
| ablonnncan1 30078 | Cancellation law for group... |
| vcrel 30081 | The class of all complex v... |
| vciOLD 30082 | Obsolete version of ~ cvsi... |
| vcsm 30083 | Functionality of th scalar... |
| vccl 30084 | Closure of the scalar prod... |
| vcidOLD 30085 | Identity element for the s... |
| vcdi 30086 | Distributive law for the s... |
| vcdir 30087 | Distributive law for the s... |
| vcass 30088 | Associative law for the sc... |
| vc2OLD 30089 | A vector plus itself is tw... |
| vcablo 30090 | Vector addition is an Abel... |
| vcgrp 30091 | Vector addition is a group... |
| vclcan 30092 | Left cancellation law for ... |
| vczcl 30093 | The zero vector is a vecto... |
| vc0rid 30094 | The zero vector is a right... |
| vc0 30095 | Zero times a vector is the... |
| vcz 30096 | Anything times the zero ve... |
| vcm 30097 | Minus 1 times a vector is ... |
| isvclem 30098 | Lemma for ~ isvcOLD . (Co... |
| vcex 30099 | The components of a comple... |
| isvcOLD 30100 | The predicate "is a comple... |
| isvciOLD 30101 | Properties that determine ... |
| cnaddabloOLD 30102 | Obsolete version of ~ cnad... |
| cnidOLD 30103 | Obsolete version of ~ cnad... |
| cncvcOLD 30104 | Obsolete version of ~ cncv... |
| nvss 30114 | Structure of the class of ... |
| nvvcop 30115 | A normed complex vector sp... |
| nvrel 30123 | The class of all normed co... |
| vafval 30124 | Value of the function for ... |
| bafval 30125 | Value of the function for ... |
| smfval 30126 | Value of the function for ... |
| 0vfval 30127 | Value of the function for ... |
| nmcvfval 30128 | Value of the norm function... |
| nvop2 30129 | A normed complex vector sp... |
| nvvop 30130 | The vector space component... |
| isnvlem 30131 | Lemma for ~ isnv . (Contr... |
| nvex 30132 | The components of a normed... |
| isnv 30133 | The predicate "is a normed... |
| isnvi 30134 | Properties that determine ... |
| nvi 30135 | The properties of a normed... |
| nvvc 30136 | The vector space component... |
| nvablo 30137 | The vector addition operat... |
| nvgrp 30138 | The vector addition operat... |
| nvgf 30139 | Mapping for the vector add... |
| nvsf 30140 | Mapping for the scalar mul... |
| nvgcl 30141 | Closure law for the vector... |
| nvcom 30142 | The vector addition (group... |
| nvass 30143 | The vector addition (group... |
| nvadd32 30144 | Commutative/associative la... |
| nvrcan 30145 | Right cancellation law for... |
| nvadd4 30146 | Rearrangement of 4 terms i... |
| nvscl 30147 | Closure law for the scalar... |
| nvsid 30148 | Identity element for the s... |
| nvsass 30149 | Associative law for the sc... |
| nvscom 30150 | Commutative law for the sc... |
| nvdi 30151 | Distributive law for the s... |
| nvdir 30152 | Distributive law for the s... |
| nv2 30153 | A vector plus itself is tw... |
| vsfval 30154 | Value of the function for ... |
| nvzcl 30155 | Closure law for the zero v... |
| nv0rid 30156 | The zero vector is a right... |
| nv0lid 30157 | The zero vector is a left ... |
| nv0 30158 | Zero times a vector is the... |
| nvsz 30159 | Anything times the zero ve... |
| nvinv 30160 | Minus 1 times a vector is ... |
| nvinvfval 30161 | Function for the negative ... |
| nvm 30162 | Vector subtraction in term... |
| nvmval 30163 | Value of vector subtractio... |
| nvmval2 30164 | Value of vector subtractio... |
| nvmfval 30165 | Value of the function for ... |
| nvmf 30166 | Mapping for the vector sub... |
| nvmcl 30167 | Closure law for the vector... |
| nvnnncan1 30168 | Cancellation law for vecto... |
| nvmdi 30169 | Distributive law for scala... |
| nvnegneg 30170 | Double negative of a vecto... |
| nvmul0or 30171 | If a scalar product is zer... |
| nvrinv 30172 | A vector minus itself. (C... |
| nvlinv 30173 | Minus a vector plus itself... |
| nvpncan2 30174 | Cancellation law for vecto... |
| nvpncan 30175 | Cancellation law for vecto... |
| nvaddsub 30176 | Commutative/associative la... |
| nvnpcan 30177 | Cancellation law for a nor... |
| nvaddsub4 30178 | Rearrangement of 4 terms i... |
| nvmeq0 30179 | The difference between two... |
| nvmid 30180 | A vector minus itself is t... |
| nvf 30181 | Mapping for the norm funct... |
| nvcl 30182 | The norm of a normed compl... |
| nvcli 30183 | The norm of a normed compl... |
| nvs 30184 | Proportionality property o... |
| nvsge0 30185 | The norm of a scalar produ... |
| nvm1 30186 | The norm of the negative o... |
| nvdif 30187 | The norm of the difference... |
| nvpi 30188 | The norm of a vector plus ... |
| nvz0 30189 | The norm of a zero vector ... |
| nvz 30190 | The norm of a vector is ze... |
| nvtri 30191 | Triangle inequality for th... |
| nvmtri 30192 | Triangle inequality for th... |
| nvabs 30193 | Norm difference property o... |
| nvge0 30194 | The norm of a normed compl... |
| nvgt0 30195 | A nonzero norm is positive... |
| nv1 30196 | From any nonzero vector, c... |
| nvop 30197 | A complex inner product sp... |
| cnnv 30198 | The set of complex numbers... |
| cnnvg 30199 | The vector addition (group... |
| cnnvba 30200 | The base set of the normed... |
| cnnvs 30201 | The scalar product operati... |
| cnnvnm 30202 | The norm operation of the ... |
| cnnvm 30203 | The vector subtraction ope... |
| elimnv 30204 | Hypothesis elimination lem... |
| elimnvu 30205 | Hypothesis elimination lem... |
| imsval 30206 | Value of the induced metri... |
| imsdval 30207 | Value of the induced metri... |
| imsdval2 30208 | Value of the distance func... |
| nvnd 30209 | The norm of a normed compl... |
| imsdf 30210 | Mapping for the induced me... |
| imsmetlem 30211 | Lemma for ~ imsmet . (Con... |
| imsmet 30212 | The induced metric of a no... |
| imsxmet 30213 | The induced metric of a no... |
| cnims 30214 | The metric induced on the ... |
| vacn 30215 | Vector addition is jointly... |
| nmcvcn 30216 | The norm of a normed compl... |
| nmcnc 30217 | The norm of a normed compl... |
| smcnlem 30218 | Lemma for ~ smcn . (Contr... |
| smcn 30219 | Scalar multiplication is j... |
| vmcn 30220 | Vector subtraction is join... |
| dipfval 30223 | The inner product function... |
| ipval 30224 | Value of the inner product... |
| ipval2lem2 30225 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 30226 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 30227 | Lemma for ~ ipval3 . (Con... |
| ipval2 30228 | Expansion of the inner pro... |
| 4ipval2 30229 | Four times the inner produ... |
| ipval3 30230 | Expansion of the inner pro... |
| ipidsq 30231 | The inner product of a vec... |
| ipnm 30232 | Norm expressed in terms of... |
| dipcl 30233 | An inner product is a comp... |
| ipf 30234 | Mapping for the inner prod... |
| dipcj 30235 | The complex conjugate of a... |
| ipipcj 30236 | An inner product times its... |
| diporthcom 30237 | Orthogonality (meaning inn... |
| dip0r 30238 | Inner product with a zero ... |
| dip0l 30239 | Inner product with a zero ... |
| ipz 30240 | The inner product of a vec... |
| dipcn 30241 | Inner product is jointly c... |
| sspval 30244 | The set of all subspaces o... |
| isssp 30245 | The predicate "is a subspa... |
| sspid 30246 | A normed complex vector sp... |
| sspnv 30247 | A subspace is a normed com... |
| sspba 30248 | The base set of a subspace... |
| sspg 30249 | Vector addition on a subsp... |
| sspgval 30250 | Vector addition on a subsp... |
| ssps 30251 | Scalar multiplication on a... |
| sspsval 30252 | Scalar multiplication on a... |
| sspmlem 30253 | Lemma for ~ sspm and other... |
| sspmval 30254 | Vector addition on a subsp... |
| sspm 30255 | Vector subtraction on a su... |
| sspz 30256 | The zero vector of a subsp... |
| sspn 30257 | The norm on a subspace is ... |
| sspnval 30258 | The norm on a subspace in ... |
| sspimsval 30259 | The induced metric on a su... |
| sspims 30260 | The induced metric on a su... |
| lnoval 30273 | The set of linear operator... |
| islno 30274 | The predicate "is a linear... |
| lnolin 30275 | Basic linearity property o... |
| lnof 30276 | A linear operator is a map... |
| lno0 30277 | The value of a linear oper... |
| lnocoi 30278 | The composition of two lin... |
| lnoadd 30279 | Addition property of a lin... |
| lnosub 30280 | Subtraction property of a ... |
| lnomul 30281 | Scalar multiplication prop... |
| nvo00 30282 | Two ways to express a zero... |
| nmoofval 30283 | The operator norm function... |
| nmooval 30284 | The operator norm function... |
| nmosetre 30285 | The set in the supremum of... |
| nmosetn0 30286 | The set in the supremum of... |
| nmoxr 30287 | The norm of an operator is... |
| nmooge0 30288 | The norm of an operator is... |
| nmorepnf 30289 | The norm of an operator is... |
| nmoreltpnf 30290 | The norm of any operator i... |
| nmogtmnf 30291 | The norm of an operator is... |
| nmoolb 30292 | A lower bound for an opera... |
| nmoubi 30293 | An upper bound for an oper... |
| nmoub3i 30294 | An upper bound for an oper... |
| nmoub2i 30295 | An upper bound for an oper... |
| nmobndi 30296 | Two ways to express that a... |
| nmounbi 30297 | Two ways two express that ... |
| nmounbseqi 30298 | An unbounded operator dete... |
| nmounbseqiALT 30299 | Alternate shorter proof of... |
| nmobndseqi 30300 | A bounded sequence determi... |
| nmobndseqiALT 30301 | Alternate shorter proof of... |
| bloval 30302 | The class of bounded linea... |
| isblo 30303 | The predicate "is a bounde... |
| isblo2 30304 | The predicate "is a bounde... |
| bloln 30305 | A bounded operator is a li... |
| blof 30306 | A bounded operator is an o... |
| nmblore 30307 | The norm of a bounded oper... |
| 0ofval 30308 | The zero operator between ... |
| 0oval 30309 | Value of the zero operator... |
| 0oo 30310 | The zero operator is an op... |
| 0lno 30311 | The zero operator is linea... |
| nmoo0 30312 | The operator norm of the z... |
| 0blo 30313 | The zero operator is a bou... |
| nmlno0lem 30314 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 30315 | The norm of a linear opera... |
| nmlno0 30316 | The norm of a linear opera... |
| nmlnoubi 30317 | An upper bound for the ope... |
| nmlnogt0 30318 | The norm of a nonzero line... |
| lnon0 30319 | The domain of a nonzero li... |
| nmblolbii 30320 | A lower bound for the norm... |
| nmblolbi 30321 | A lower bound for the norm... |
| isblo3i 30322 | The predicate "is a bounde... |
| blo3i 30323 | Properties that determine ... |
| blometi 30324 | Upper bound for the distan... |
| blocnilem 30325 | Lemma for ~ blocni and ~ l... |
| blocni 30326 | A linear operator is conti... |
| lnocni 30327 | If a linear operator is co... |
| blocn 30328 | A linear operator is conti... |
| blocn2 30329 | A bounded linear operator ... |
| ajfval 30330 | The adjoint function. (Co... |
| hmoval 30331 | The set of Hermitian (self... |
| ishmo 30332 | The predicate "is a hermit... |
| phnv 30335 | Every complex inner produc... |
| phrel 30336 | The class of all complex i... |
| phnvi 30337 | Every complex inner produc... |
| isphg 30338 | The predicate "is a comple... |
| phop 30339 | A complex inner product sp... |
| cncph 30340 | The set of complex numbers... |
| elimph 30341 | Hypothesis elimination lem... |
| elimphu 30342 | Hypothesis elimination lem... |
| isph 30343 | The predicate "is an inner... |
| phpar2 30344 | The parallelogram law for ... |
| phpar 30345 | The parallelogram law for ... |
| ip0i 30346 | A slight variant of Equati... |
| ip1ilem 30347 | Lemma for ~ ip1i . (Contr... |
| ip1i 30348 | Equation 6.47 of [Ponnusam... |
| ip2i 30349 | Equation 6.48 of [Ponnusam... |
| ipdirilem 30350 | Lemma for ~ ipdiri . (Con... |
| ipdiri 30351 | Distributive law for inner... |
| ipasslem1 30352 | Lemma for ~ ipassi . Show... |
| ipasslem2 30353 | Lemma for ~ ipassi . Show... |
| ipasslem3 30354 | Lemma for ~ ipassi . Show... |
| ipasslem4 30355 | Lemma for ~ ipassi . Show... |
| ipasslem5 30356 | Lemma for ~ ipassi . Show... |
| ipasslem7 30357 | Lemma for ~ ipassi . Show... |
| ipasslem8 30358 | Lemma for ~ ipassi . By ~... |
| ipasslem9 30359 | Lemma for ~ ipassi . Conc... |
| ipasslem10 30360 | Lemma for ~ ipassi . Show... |
| ipasslem11 30361 | Lemma for ~ ipassi . Show... |
| ipassi 30362 | Associative law for inner ... |
| dipdir 30363 | Distributive law for inner... |
| dipdi 30364 | Distributive law for inner... |
| ip2dii 30365 | Inner product of two sums.... |
| dipass 30366 | Associative law for inner ... |
| dipassr 30367 | "Associative" law for seco... |
| dipassr2 30368 | "Associative" law for inne... |
| dipsubdir 30369 | Distributive law for inner... |
| dipsubdi 30370 | Distributive law for inner... |
| pythi 30371 | The Pythagorean theorem fo... |
| siilem1 30372 | Lemma for ~ sii . (Contri... |
| siilem2 30373 | Lemma for ~ sii . (Contri... |
| siii 30374 | Inference from ~ sii . (C... |
| sii 30375 | Obsolete version of ~ ipca... |
| ipblnfi 30376 | A function ` F ` generated... |
| ip2eqi 30377 | Two vectors are equal iff ... |
| phoeqi 30378 | A condition implying that ... |
| ajmoi 30379 | Every operator has at most... |
| ajfuni 30380 | The adjoint function is a ... |
| ajfun 30381 | The adjoint function is a ... |
| ajval 30382 | Value of the adjoint funct... |
| iscbn 30385 | A complex Banach space is ... |
| cbncms 30386 | The induced metric on comp... |
| bnnv 30387 | Every complex Banach space... |
| bnrel 30388 | The class of all complex B... |
| bnsscmcl 30389 | A subspace of a Banach spa... |
| cnbn 30390 | The set of complex numbers... |
| ubthlem1 30391 | Lemma for ~ ubth . The fu... |
| ubthlem2 30392 | Lemma for ~ ubth . Given ... |
| ubthlem3 30393 | Lemma for ~ ubth . Prove ... |
| ubth 30394 | Uniform Boundedness Theore... |
| minvecolem1 30395 | Lemma for ~ minveco . The... |
| minvecolem2 30396 | Lemma for ~ minveco . Any... |
| minvecolem3 30397 | Lemma for ~ minveco . The... |
| minvecolem4a 30398 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 30399 | Lemma for ~ minveco . The... |
| minvecolem4c 30400 | Lemma for ~ minveco . The... |
| minvecolem4 30401 | Lemma for ~ minveco . The... |
| minvecolem5 30402 | Lemma for ~ minveco . Dis... |
| minvecolem6 30403 | Lemma for ~ minveco . Any... |
| minvecolem7 30404 | Lemma for ~ minveco . Sin... |
| minveco 30405 | Minimizing vector theorem,... |
| ishlo 30408 | The predicate "is a comple... |
| hlobn 30409 | Every complex Hilbert spac... |
| hlph 30410 | Every complex Hilbert spac... |
| hlrel 30411 | The class of all complex H... |
| hlnv 30412 | Every complex Hilbert spac... |
| hlnvi 30413 | Every complex Hilbert spac... |
| hlvc 30414 | Every complex Hilbert spac... |
| hlcmet 30415 | The induced metric on a co... |
| hlmet 30416 | The induced metric on a co... |
| hlpar2 30417 | The parallelogram law sati... |
| hlpar 30418 | The parallelogram law sati... |
| hlex 30419 | The base set of a Hilbert ... |
| hladdf 30420 | Mapping for Hilbert space ... |
| hlcom 30421 | Hilbert space vector addit... |
| hlass 30422 | Hilbert space vector addit... |
| hl0cl 30423 | The Hilbert space zero vec... |
| hladdid 30424 | Hilbert space addition wit... |
| hlmulf 30425 | Mapping for Hilbert space ... |
| hlmulid 30426 | Hilbert space scalar multi... |
| hlmulass 30427 | Hilbert space scalar multi... |
| hldi 30428 | Hilbert space scalar multi... |
| hldir 30429 | Hilbert space scalar multi... |
| hlmul0 30430 | Hilbert space scalar multi... |
| hlipf 30431 | Mapping for Hilbert space ... |
| hlipcj 30432 | Conjugate law for Hilbert ... |
| hlipdir 30433 | Distributive law for Hilbe... |
| hlipass 30434 | Associative law for Hilber... |
| hlipgt0 30435 | The inner product of a Hil... |
| hlcompl 30436 | Completeness of a Hilbert ... |
| cnchl 30437 | The set of complex numbers... |
| htthlem 30438 | Lemma for ~ htth . The co... |
| htth 30439 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 30495 | The group (addition) opera... |
| h2hsm 30496 | The scalar product operati... |
| h2hnm 30497 | The norm function of Hilbe... |
| h2hvs 30498 | The vector subtraction ope... |
| h2hmetdval 30499 | Value of the distance func... |
| h2hcau 30500 | The Cauchy sequences of Hi... |
| h2hlm 30501 | The limit sequences of Hil... |
| axhilex-zf 30502 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 30503 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 30504 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 30505 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 30506 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 30507 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 30508 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 30509 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 30510 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 30511 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 30512 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 30513 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 30514 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 30515 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 30516 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 30517 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 30518 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 30519 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 30532 | The Hilbert space scalar p... |
| hvaddcl 30533 | Closure of vector addition... |
| hvmulcl 30534 | Closure of scalar multipli... |
| hvmulcli 30535 | Closure inference for scal... |
| hvsubf 30536 | Mapping domain and codomai... |
| hvsubval 30537 | Value of vector subtractio... |
| hvsubcl 30538 | Closure of vector subtract... |
| hvaddcli 30539 | Closure of vector addition... |
| hvcomi 30540 | Commutation of vector addi... |
| hvsubvali 30541 | Value of vector subtractio... |
| hvsubcli 30542 | Closure of vector subtract... |
| ifhvhv0 30543 | Prove ` if ( A e. ~H , A ,... |
| hvaddlid 30544 | Addition with the zero vec... |
| hvmul0 30545 | Scalar multiplication with... |
| hvmul0or 30546 | If a scalar product is zer... |
| hvsubid 30547 | Subtraction of a vector fr... |
| hvnegid 30548 | Addition of negative of a ... |
| hv2neg 30549 | Two ways to express the ne... |
| hvaddlidi 30550 | Addition with the zero vec... |
| hvnegidi 30551 | Addition of negative of a ... |
| hv2negi 30552 | Two ways to express the ne... |
| hvm1neg 30553 | Convert minus one times a ... |
| hvaddsubval 30554 | Value of vector addition i... |
| hvadd32 30555 | Commutative/associative la... |
| hvadd12 30556 | Commutative/associative la... |
| hvadd4 30557 | Hilbert vector space addit... |
| hvsub4 30558 | Hilbert vector space addit... |
| hvaddsub12 30559 | Commutative/associative la... |
| hvpncan 30560 | Addition/subtraction cance... |
| hvpncan2 30561 | Addition/subtraction cance... |
| hvaddsubass 30562 | Associativity of sum and d... |
| hvpncan3 30563 | Subtraction and addition o... |
| hvmulcom 30564 | Scalar multiplication comm... |
| hvsubass 30565 | Hilbert vector space assoc... |
| hvsub32 30566 | Hilbert vector space commu... |
| hvmulassi 30567 | Scalar multiplication asso... |
| hvmulcomi 30568 | Scalar multiplication comm... |
| hvmul2negi 30569 | Double negative in scalar ... |
| hvsubdistr1 30570 | Scalar multiplication dist... |
| hvsubdistr2 30571 | Scalar multiplication dist... |
| hvdistr1i 30572 | Scalar multiplication dist... |
| hvsubdistr1i 30573 | Scalar multiplication dist... |
| hvassi 30574 | Hilbert vector space assoc... |
| hvadd32i 30575 | Hilbert vector space commu... |
| hvsubassi 30576 | Hilbert vector space assoc... |
| hvsub32i 30577 | Hilbert vector space commu... |
| hvadd12i 30578 | Hilbert vector space commu... |
| hvadd4i 30579 | Hilbert vector space addit... |
| hvsubsub4i 30580 | Hilbert vector space addit... |
| hvsubsub4 30581 | Hilbert vector space addit... |
| hv2times 30582 | Two times a vector. (Cont... |
| hvnegdii 30583 | Distribution of negative o... |
| hvsubeq0i 30584 | If the difference between ... |
| hvsubcan2i 30585 | Vector cancellation law. ... |
| hvaddcani 30586 | Cancellation law for vecto... |
| hvsubaddi 30587 | Relationship between vecto... |
| hvnegdi 30588 | Distribution of negative o... |
| hvsubeq0 30589 | If the difference between ... |
| hvaddeq0 30590 | If the sum of two vectors ... |
| hvaddcan 30591 | Cancellation law for vecto... |
| hvaddcan2 30592 | Cancellation law for vecto... |
| hvmulcan 30593 | Cancellation law for scala... |
| hvmulcan2 30594 | Cancellation law for scala... |
| hvsubcan 30595 | Cancellation law for vecto... |
| hvsubcan2 30596 | Cancellation law for vecto... |
| hvsub0 30597 | Subtraction of a zero vect... |
| hvsubadd 30598 | Relationship between vecto... |
| hvaddsub4 30599 | Hilbert vector space addit... |
| hicl 30601 | Closure of inner product. ... |
| hicli 30602 | Closure inference for inne... |
| his5 30607 | Associative law for inner ... |
| his52 30608 | Associative law for inner ... |
| his35 30609 | Move scalar multiplication... |
| his35i 30610 | Move scalar multiplication... |
| his7 30611 | Distributive law for inner... |
| hiassdi 30612 | Distributive/associative l... |
| his2sub 30613 | Distributive law for inner... |
| his2sub2 30614 | Distributive law for inner... |
| hire 30615 | A necessary and sufficient... |
| hiidrcl 30616 | Real closure of inner prod... |
| hi01 30617 | Inner product with the 0 v... |
| hi02 30618 | Inner product with the 0 v... |
| hiidge0 30619 | Inner product with self is... |
| his6 30620 | Zero inner product with se... |
| his1i 30621 | Conjugate law for inner pr... |
| abshicom 30622 | Commuted inner products ha... |
| hial0 30623 | A vector whose inner produ... |
| hial02 30624 | A vector whose inner produ... |
| hisubcomi 30625 | Two vector subtractions si... |
| hi2eq 30626 | Lemma used to prove equali... |
| hial2eq 30627 | Two vectors whose inner pr... |
| hial2eq2 30628 | Two vectors whose inner pr... |
| orthcom 30629 | Orthogonality commutes. (... |
| normlem0 30630 | Lemma used to derive prope... |
| normlem1 30631 | Lemma used to derive prope... |
| normlem2 30632 | Lemma used to derive prope... |
| normlem3 30633 | Lemma used to derive prope... |
| normlem4 30634 | Lemma used to derive prope... |
| normlem5 30635 | Lemma used to derive prope... |
| normlem6 30636 | Lemma used to derive prope... |
| normlem7 30637 | Lemma used to derive prope... |
| normlem8 30638 | Lemma used to derive prope... |
| normlem9 30639 | Lemma used to derive prope... |
| normlem7tALT 30640 | Lemma used to derive prope... |
| bcseqi 30641 | Equality case of Bunjakova... |
| normlem9at 30642 | Lemma used to derive prope... |
| dfhnorm2 30643 | Alternate definition of th... |
| normf 30644 | The norm function maps fro... |
| normval 30645 | The value of the norm of a... |
| normcl 30646 | Real closure of the norm o... |
| normge0 30647 | The norm of a vector is no... |
| normgt0 30648 | The norm of nonzero vector... |
| norm0 30649 | The norm of a zero vector.... |
| norm-i 30650 | Theorem 3.3(i) of [Beran] ... |
| normne0 30651 | A norm is nonzero iff its ... |
| normcli 30652 | Real closure of the norm o... |
| normsqi 30653 | The square of a norm. (Co... |
| norm-i-i 30654 | Theorem 3.3(i) of [Beran] ... |
| normsq 30655 | The square of a norm. (Co... |
| normsub0i 30656 | Two vectors are equal iff ... |
| normsub0 30657 | Two vectors are equal iff ... |
| norm-ii-i 30658 | Triangle inequality for no... |
| norm-ii 30659 | Triangle inequality for no... |
| norm-iii-i 30660 | Theorem 3.3(iii) of [Beran... |
| norm-iii 30661 | Theorem 3.3(iii) of [Beran... |
| normsubi 30662 | Negative doesn't change th... |
| normpythi 30663 | Analogy to Pythagorean the... |
| normsub 30664 | Swapping order of subtract... |
| normneg 30665 | The norm of a vector equal... |
| normpyth 30666 | Analogy to Pythagorean the... |
| normpyc 30667 | Corollary to Pythagorean t... |
| norm3difi 30668 | Norm of differences around... |
| norm3adifii 30669 | Norm of differences around... |
| norm3lem 30670 | Lemma involving norm of di... |
| norm3dif 30671 | Norm of differences around... |
| norm3dif2 30672 | Norm of differences around... |
| norm3lemt 30673 | Lemma involving norm of di... |
| norm3adifi 30674 | Norm of differences around... |
| normpari 30675 | Parallelogram law for norm... |
| normpar 30676 | Parallelogram law for norm... |
| normpar2i 30677 | Corollary of parallelogram... |
| polid2i 30678 | Generalized polarization i... |
| polidi 30679 | Polarization identity. Re... |
| polid 30680 | Polarization identity. Re... |
| hilablo 30681 | Hilbert space vector addit... |
| hilid 30682 | The group identity element... |
| hilvc 30683 | Hilbert space is a complex... |
| hilnormi 30684 | Hilbert space norm in term... |
| hilhhi 30685 | Deduce the structure of Hi... |
| hhnv 30686 | Hilbert space is a normed ... |
| hhva 30687 | The group (addition) opera... |
| hhba 30688 | The base set of Hilbert sp... |
| hh0v 30689 | The zero vector of Hilbert... |
| hhsm 30690 | The scalar product operati... |
| hhvs 30691 | The vector subtraction ope... |
| hhnm 30692 | The norm function of Hilbe... |
| hhims 30693 | The induced metric of Hilb... |
| hhims2 30694 | Hilbert space distance met... |
| hhmet 30695 | The induced metric of Hilb... |
| hhxmet 30696 | The induced metric of Hilb... |
| hhmetdval 30697 | Value of the distance func... |
| hhip 30698 | The inner product operatio... |
| hhph 30699 | The Hilbert space of the H... |
| bcsiALT 30700 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 30701 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 30702 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 30703 | Corollary of the Bunjakova... |
| bcs3 30704 | Corollary of the Bunjakova... |
| hcau 30705 | Member of the set of Cauch... |
| hcauseq 30706 | A Cauchy sequences on a Hi... |
| hcaucvg 30707 | A Cauchy sequence on a Hil... |
| seq1hcau 30708 | A sequence on a Hilbert sp... |
| hlimi 30709 | Express the predicate: Th... |
| hlimseqi 30710 | A sequence with a limit on... |
| hlimveci 30711 | Closure of the limit of a ... |
| hlimconvi 30712 | Convergence of a sequence ... |
| hlim2 30713 | The limit of a sequence on... |
| hlimadd 30714 | Limit of the sum of two se... |
| hilmet 30715 | The Hilbert space norm det... |
| hilxmet 30716 | The Hilbert space norm det... |
| hilmetdval 30717 | Value of the distance func... |
| hilims 30718 | Hilbert space distance met... |
| hhcau 30719 | The Cauchy sequences of Hi... |
| hhlm 30720 | The limit sequences of Hil... |
| hhcmpl 30721 | Lemma used for derivation ... |
| hilcompl 30722 | Lemma used for derivation ... |
| hhcms 30724 | The Hilbert space induced ... |
| hhhl 30725 | The Hilbert space structur... |
| hilcms 30726 | The Hilbert space norm det... |
| hilhl 30727 | The Hilbert space of the H... |
| issh 30729 | Subspace ` H ` of a Hilber... |
| issh2 30730 | Subspace ` H ` of a Hilber... |
| shss 30731 | A subspace is a subset of ... |
| shel 30732 | A member of a subspace of ... |
| shex 30733 | The set of subspaces of a ... |
| shssii 30734 | A closed subspace of a Hil... |
| sheli 30735 | A member of a subspace of ... |
| shelii 30736 | A member of a subspace of ... |
| sh0 30737 | The zero vector belongs to... |
| shaddcl 30738 | Closure of vector addition... |
| shmulcl 30739 | Closure of vector scalar m... |
| issh3 30740 | Subspace ` H ` of a Hilber... |
| shsubcl 30741 | Closure of vector subtract... |
| isch 30743 | Closed subspace ` H ` of a... |
| isch2 30744 | Closed subspace ` H ` of a... |
| chsh 30745 | A closed subspace is a sub... |
| chsssh 30746 | Closed subspaces are subsp... |
| chex 30747 | The set of closed subspace... |
| chshii 30748 | A closed subspace is a sub... |
| ch0 30749 | The zero vector belongs to... |
| chss 30750 | A closed subspace of a Hil... |
| chel 30751 | A member of a closed subsp... |
| chssii 30752 | A closed subspace of a Hil... |
| cheli 30753 | A member of a closed subsp... |
| chelii 30754 | A member of a closed subsp... |
| chlimi 30755 | The limit property of a cl... |
| hlim0 30756 | The zero sequence in Hilbe... |
| hlimcaui 30757 | If a sequence in Hilbert s... |
| hlimf 30758 | Function-like behavior of ... |
| hlimuni 30759 | A Hilbert space sequence c... |
| hlimreui 30760 | The limit of a Hilbert spa... |
| hlimeui 30761 | The limit of a Hilbert spa... |
| isch3 30762 | A Hilbert subspace is clos... |
| chcompl 30763 | Completeness of a closed s... |
| helch 30764 | The Hilbert lattice one (w... |
| ifchhv 30765 | Prove ` if ( A e. CH , A ,... |
| helsh 30766 | Hilbert space is a subspac... |
| shsspwh 30767 | Subspaces are subsets of H... |
| chsspwh 30768 | Closed subspaces are subse... |
| hsn0elch 30769 | The zero subspace belongs ... |
| norm1 30770 | From any nonzero Hilbert s... |
| norm1exi 30771 | A normalized vector exists... |
| norm1hex 30772 | A normalized vector can ex... |
| elch0 30775 | Membership in zero for clo... |
| h0elch 30776 | The zero subspace is a clo... |
| h0elsh 30777 | The zero subspace is a sub... |
| hhssva 30778 | The vector addition operat... |
| hhsssm 30779 | The scalar multiplication ... |
| hhssnm 30780 | The norm operation on a su... |
| issubgoilem 30781 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 30782 | Lemma for ~ hhssabloi . F... |
| hhssabloi 30783 | Abelian group property of ... |
| hhssablo 30784 | Abelian group property of ... |
| hhssnv 30785 | Normed complex vector spac... |
| hhssnvt 30786 | Normed complex vector spac... |
| hhsst 30787 | A member of ` SH ` is a su... |
| hhshsslem1 30788 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 30789 | Lemma for ~ hhsssh . (Con... |
| hhsssh 30790 | The predicate " ` H ` is a... |
| hhsssh2 30791 | The predicate " ` H ` is a... |
| hhssba 30792 | The base set of a subspace... |
| hhssvs 30793 | The vector subtraction ope... |
| hhssvsf 30794 | Mapping of the vector subt... |
| hhssims 30795 | Induced metric of a subspa... |
| hhssims2 30796 | Induced metric of a subspa... |
| hhssmet 30797 | Induced metric of a subspa... |
| hhssmetdval 30798 | Value of the distance func... |
| hhsscms 30799 | The induced metric of a cl... |
| hhssbnOLD 30800 | Obsolete version of ~ cssb... |
| ocval 30801 | Value of orthogonal comple... |
| ocel 30802 | Membership in orthogonal c... |
| shocel 30803 | Membership in orthogonal c... |
| ocsh 30804 | The orthogonal complement ... |
| shocsh 30805 | The orthogonal complement ... |
| ocss 30806 | An orthogonal complement i... |
| shocss 30807 | An orthogonal complement i... |
| occon 30808 | Contraposition law for ort... |
| occon2 30809 | Double contraposition for ... |
| occon2i 30810 | Double contraposition for ... |
| oc0 30811 | The zero vector belongs to... |
| ocorth 30812 | Members of a subset and it... |
| shocorth 30813 | Members of a subspace and ... |
| ococss 30814 | Inclusion in complement of... |
| shococss 30815 | Inclusion in complement of... |
| shorth 30816 | Members of orthogonal subs... |
| ocin 30817 | Intersection of a Hilbert ... |
| occon3 30818 | Hilbert lattice contraposi... |
| ocnel 30819 | A nonzero vector in the co... |
| chocvali 30820 | Value of the orthogonal co... |
| shuni 30821 | Two subspaces with trivial... |
| chocunii 30822 | Lemma for uniqueness part ... |
| pjhthmo 30823 | Projection Theorem, unique... |
| occllem 30824 | Lemma for ~ occl . (Contr... |
| occl 30825 | Closure of complement of H... |
| shoccl 30826 | Closure of complement of H... |
| choccl 30827 | Closure of complement of H... |
| choccli 30828 | Closure of ` CH ` orthocom... |
| shsval 30833 | Value of subspace sum of t... |
| shsss 30834 | The subspace sum is a subs... |
| shsel 30835 | Membership in the subspace... |
| shsel3 30836 | Membership in the subspace... |
| shseli 30837 | Membership in subspace sum... |
| shscli 30838 | Closure of subspace sum. ... |
| shscl 30839 | Closure of subspace sum. ... |
| shscom 30840 | Commutative law for subspa... |
| shsva 30841 | Vector sum belongs to subs... |
| shsel1 30842 | A subspace sum contains a ... |
| shsel2 30843 | A subspace sum contains a ... |
| shsvs 30844 | Vector subtraction belongs... |
| shsub1 30845 | Subspace sum is an upper b... |
| shsub2 30846 | Subspace sum is an upper b... |
| choc0 30847 | The orthocomplement of the... |
| choc1 30848 | The orthocomplement of the... |
| chocnul 30849 | Orthogonal complement of t... |
| shintcli 30850 | Closure of intersection of... |
| shintcl 30851 | The intersection of a none... |
| chintcli 30852 | The intersection of a none... |
| chintcl 30853 | The intersection (infimum)... |
| spanval 30854 | Value of the linear span o... |
| hsupval 30855 | Value of supremum of set o... |
| chsupval 30856 | The value of the supremum ... |
| spancl 30857 | The span of a subset of Hi... |
| elspancl 30858 | A member of a span is a ve... |
| shsupcl 30859 | Closure of the subspace su... |
| hsupcl 30860 | Closure of supremum of set... |
| chsupcl 30861 | Closure of supremum of sub... |
| hsupss 30862 | Subset relation for suprem... |
| chsupss 30863 | Subset relation for suprem... |
| hsupunss 30864 | The union of a set of Hilb... |
| chsupunss 30865 | The union of a set of clos... |
| spanss2 30866 | A subset of Hilbert space ... |
| shsupunss 30867 | The union of a set of subs... |
| spanid 30868 | A subspace of Hilbert spac... |
| spanss 30869 | Ordering relationship for ... |
| spanssoc 30870 | The span of a subset of Hi... |
| sshjval 30871 | Value of join for subsets ... |
| shjval 30872 | Value of join in ` SH ` . ... |
| chjval 30873 | Value of join in ` CH ` . ... |
| chjvali 30874 | Value of join in ` CH ` . ... |
| sshjval3 30875 | Value of join for subsets ... |
| sshjcl 30876 | Closure of join for subset... |
| shjcl 30877 | Closure of join in ` SH ` ... |
| chjcl 30878 | Closure of join in ` CH ` ... |
| shjcom 30879 | Commutative law for Hilber... |
| shless 30880 | Subset implies subset of s... |
| shlej1 30881 | Add disjunct to both sides... |
| shlej2 30882 | Add disjunct to both sides... |
| shincli 30883 | Closure of intersection of... |
| shscomi 30884 | Commutative law for subspa... |
| shsvai 30885 | Vector sum belongs to subs... |
| shsel1i 30886 | A subspace sum contains a ... |
| shsel2i 30887 | A subspace sum contains a ... |
| shsvsi 30888 | Vector subtraction belongs... |
| shunssi 30889 | Union is smaller than subs... |
| shunssji 30890 | Union is smaller than Hilb... |
| shsleji 30891 | Subspace sum is smaller th... |
| shjcomi 30892 | Commutative law for join i... |
| shsub1i 30893 | Subspace sum is an upper b... |
| shsub2i 30894 | Subspace sum is an upper b... |
| shub1i 30895 | Hilbert lattice join is an... |
| shjcli 30896 | Closure of ` CH ` join. (... |
| shjshcli 30897 | ` SH ` closure of join. (... |
| shlessi 30898 | Subset implies subset of s... |
| shlej1i 30899 | Add disjunct to both sides... |
| shlej2i 30900 | Add disjunct to both sides... |
| shslej 30901 | Subspace sum is smaller th... |
| shincl 30902 | Closure of intersection of... |
| shub1 30903 | Hilbert lattice join is an... |
| shub2 30904 | A subspace is a subset of ... |
| shsidmi 30905 | Idempotent law for Hilbert... |
| shslubi 30906 | The least upper bound law ... |
| shlesb1i 30907 | Hilbert lattice ordering i... |
| shsval2i 30908 | An alternate way to expres... |
| shsval3i 30909 | An alternate way to expres... |
| shmodsi 30910 | The modular law holds for ... |
| shmodi 30911 | The modular law is implied... |
| pjhthlem1 30912 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 30913 | Lemma for ~ pjhth . (Cont... |
| pjhth 30914 | Projection Theorem: Any H... |
| pjhtheu 30915 | Projection Theorem: Any H... |
| pjhfval 30917 | The value of the projectio... |
| pjhval 30918 | Value of a projection. (C... |
| pjpreeq 30919 | Equality with a projection... |
| pjeq 30920 | Equality with a projection... |
| axpjcl 30921 | Closure of a projection in... |
| pjhcl 30922 | Closure of a projection in... |
| omlsilem 30923 | Lemma for orthomodular law... |
| omlsii 30924 | Subspace inference form of... |
| omlsi 30925 | Subspace form of orthomodu... |
| ococi 30926 | Complement of complement o... |
| ococ 30927 | Complement of complement o... |
| dfch2 30928 | Alternate definition of th... |
| ococin 30929 | The double complement is t... |
| hsupval2 30930 | Alternate definition of su... |
| chsupval2 30931 | The value of the supremum ... |
| sshjval2 30932 | Value of join in the set o... |
| chsupid 30933 | A subspace is the supremum... |
| chsupsn 30934 | Value of supremum of subse... |
| shlub 30935 | Hilbert lattice join is th... |
| shlubi 30936 | Hilbert lattice join is th... |
| pjhtheu2 30937 | Uniqueness of ` y ` for th... |
| pjcli 30938 | Closure of a projection in... |
| pjhcli 30939 | Closure of a projection in... |
| pjpjpre 30940 | Decomposition of a vector ... |
| axpjpj 30941 | Decomposition of a vector ... |
| pjclii 30942 | Closure of a projection in... |
| pjhclii 30943 | Closure of a projection in... |
| pjpj0i 30944 | Decomposition of a vector ... |
| pjpji 30945 | Decomposition of a vector ... |
| pjpjhth 30946 | Projection Theorem: Any H... |
| pjpjhthi 30947 | Projection Theorem: Any H... |
| pjop 30948 | Orthocomplement projection... |
| pjpo 30949 | Projection in terms of ort... |
| pjopi 30950 | Orthocomplement projection... |
| pjpoi 30951 | Projection in terms of ort... |
| pjoc1i 30952 | Projection of a vector in ... |
| pjchi 30953 | Projection of a vector in ... |
| pjoccl 30954 | The part of a vector that ... |
| pjoc1 30955 | Projection of a vector in ... |
| pjomli 30956 | Subspace form of orthomodu... |
| pjoml 30957 | Subspace form of orthomodu... |
| pjococi 30958 | Proof of orthocomplement t... |
| pjoc2i 30959 | Projection of a vector in ... |
| pjoc2 30960 | Projection of a vector in ... |
| sh0le 30961 | The zero subspace is the s... |
| ch0le 30962 | The zero subspace is the s... |
| shle0 30963 | No subspace is smaller tha... |
| chle0 30964 | No Hilbert lattice element... |
| chnlen0 30965 | A Hilbert lattice element ... |
| ch0pss 30966 | The zero subspace is a pro... |
| orthin 30967 | The intersection of orthog... |
| ssjo 30968 | The lattice join of a subs... |
| shne0i 30969 | A nonzero subspace has a n... |
| shs0i 30970 | Hilbert subspace sum with ... |
| shs00i 30971 | Two subspaces are zero iff... |
| ch0lei 30972 | The closed subspace zero i... |
| chle0i 30973 | No Hilbert closed subspace... |
| chne0i 30974 | A nonzero closed subspace ... |
| chocini 30975 | Intersection of a closed s... |
| chj0i 30976 | Join with lattice zero in ... |
| chm1i 30977 | Meet with lattice one in `... |
| chjcli 30978 | Closure of ` CH ` join. (... |
| chsleji 30979 | Subspace sum is smaller th... |
| chseli 30980 | Membership in subspace sum... |
| chincli 30981 | Closure of Hilbert lattice... |
| chsscon3i 30982 | Hilbert lattice contraposi... |
| chsscon1i 30983 | Hilbert lattice contraposi... |
| chsscon2i 30984 | Hilbert lattice contraposi... |
| chcon2i 30985 | Hilbert lattice contraposi... |
| chcon1i 30986 | Hilbert lattice contraposi... |
| chcon3i 30987 | Hilbert lattice contraposi... |
| chunssji 30988 | Union is smaller than ` CH... |
| chjcomi 30989 | Commutative law for join i... |
| chub1i 30990 | ` CH ` join is an upper bo... |
| chub2i 30991 | ` CH ` join is an upper bo... |
| chlubi 30992 | Hilbert lattice join is th... |
| chlubii 30993 | Hilbert lattice join is th... |
| chlej1i 30994 | Add join to both sides of ... |
| chlej2i 30995 | Add join to both sides of ... |
| chlej12i 30996 | Add join to both sides of ... |
| chlejb1i 30997 | Hilbert lattice ordering i... |
| chdmm1i 30998 | De Morgan's law for meet i... |
| chdmm2i 30999 | De Morgan's law for meet i... |
| chdmm3i 31000 | De Morgan's law for meet i... |
| chdmm4i 31001 | De Morgan's law for meet i... |
| chdmj1i 31002 | De Morgan's law for join i... |
| chdmj2i 31003 | De Morgan's law for join i... |
| chdmj3i 31004 | De Morgan's law for join i... |
| chdmj4i 31005 | De Morgan's law for join i... |
| chnlei 31006 | Equivalent expressions for... |
| chjassi 31007 | Associative law for Hilber... |
| chj00i 31008 | Two Hilbert lattice elemen... |
| chjoi 31009 | The join of a closed subsp... |
| chj1i 31010 | Join with Hilbert lattice ... |
| chm0i 31011 | Meet with Hilbert lattice ... |
| chm0 31012 | Meet with Hilbert lattice ... |
| shjshsi 31013 | Hilbert lattice join equal... |
| shjshseli 31014 | A closed subspace sum equa... |
| chne0 31015 | A nonzero closed subspace ... |
| chocin 31016 | Intersection of a closed s... |
| chssoc 31017 | A closed subspace less tha... |
| chj0 31018 | Join with Hilbert lattice ... |
| chslej 31019 | Subspace sum is smaller th... |
| chincl 31020 | Closure of Hilbert lattice... |
| chsscon3 31021 | Hilbert lattice contraposi... |
| chsscon1 31022 | Hilbert lattice contraposi... |
| chsscon2 31023 | Hilbert lattice contraposi... |
| chpsscon3 31024 | Hilbert lattice contraposi... |
| chpsscon1 31025 | Hilbert lattice contraposi... |
| chpsscon2 31026 | Hilbert lattice contraposi... |
| chjcom 31027 | Commutative law for Hilber... |
| chub1 31028 | Hilbert lattice join is gr... |
| chub2 31029 | Hilbert lattice join is gr... |
| chlub 31030 | Hilbert lattice join is th... |
| chlej1 31031 | Add join to both sides of ... |
| chlej2 31032 | Add join to both sides of ... |
| chlejb1 31033 | Hilbert lattice ordering i... |
| chlejb2 31034 | Hilbert lattice ordering i... |
| chnle 31035 | Equivalent expressions for... |
| chjo 31036 | The join of a closed subsp... |
| chabs1 31037 | Hilbert lattice absorption... |
| chabs2 31038 | Hilbert lattice absorption... |
| chabs1i 31039 | Hilbert lattice absorption... |
| chabs2i 31040 | Hilbert lattice absorption... |
| chjidm 31041 | Idempotent law for Hilbert... |
| chjidmi 31042 | Idempotent law for Hilbert... |
| chj12i 31043 | A rearrangement of Hilbert... |
| chj4i 31044 | Rearrangement of the join ... |
| chjjdiri 31045 | Hilbert lattice join distr... |
| chdmm1 31046 | De Morgan's law for meet i... |
| chdmm2 31047 | De Morgan's law for meet i... |
| chdmm3 31048 | De Morgan's law for meet i... |
| chdmm4 31049 | De Morgan's law for meet i... |
| chdmj1 31050 | De Morgan's law for join i... |
| chdmj2 31051 | De Morgan's law for join i... |
| chdmj3 31052 | De Morgan's law for join i... |
| chdmj4 31053 | De Morgan's law for join i... |
| chjass 31054 | Associative law for Hilber... |
| chj12 31055 | A rearrangement of Hilbert... |
| chj4 31056 | Rearrangement of the join ... |
| ledii 31057 | An ortholattice is distrib... |
| lediri 31058 | An ortholattice is distrib... |
| lejdii 31059 | An ortholattice is distrib... |
| lejdiri 31060 | An ortholattice is distrib... |
| ledi 31061 | An ortholattice is distrib... |
| spansn0 31062 | The span of the singleton ... |
| span0 31063 | The span of the empty set ... |
| elspani 31064 | Membership in the span of ... |
| spanuni 31065 | The span of a union is the... |
| spanun 31066 | The span of a union is the... |
| sshhococi 31067 | The join of two Hilbert sp... |
| hne0 31068 | Hilbert space has a nonzer... |
| chsup0 31069 | The supremum of the empty ... |
| h1deoi 31070 | Membership in orthocomplem... |
| h1dei 31071 | Membership in 1-dimensiona... |
| h1did 31072 | A generating vector belong... |
| h1dn0 31073 | A nonzero vector generates... |
| h1de2i 31074 | Membership in 1-dimensiona... |
| h1de2bi 31075 | Membership in 1-dimensiona... |
| h1de2ctlem 31076 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 31077 | Membership in 1-dimensiona... |
| spansni 31078 | The span of a singleton in... |
| elspansni 31079 | Membership in the span of ... |
| spansn 31080 | The span of a singleton in... |
| spansnch 31081 | The span of a Hilbert spac... |
| spansnsh 31082 | The span of a Hilbert spac... |
| spansnchi 31083 | The span of a singleton in... |
| spansnid 31084 | A vector belongs to the sp... |
| spansnmul 31085 | A scalar product with a ve... |
| elspansncl 31086 | A member of a span of a si... |
| elspansn 31087 | Membership in the span of ... |
| elspansn2 31088 | Membership in the span of ... |
| spansncol 31089 | The singletons of collinea... |
| spansneleqi 31090 | Membership relation implie... |
| spansneleq 31091 | Membership relation that i... |
| spansnss 31092 | The span of the singleton ... |
| elspansn3 31093 | A member of the span of th... |
| elspansn4 31094 | A span membership conditio... |
| elspansn5 31095 | A vector belonging to both... |
| spansnss2 31096 | The span of the singleton ... |
| normcan 31097 | Cancellation-type law that... |
| pjspansn 31098 | A projection on the span o... |
| spansnpji 31099 | A subset of Hilbert space ... |
| spanunsni 31100 | The span of the union of a... |
| spanpr 31101 | The span of a pair of vect... |
| h1datomi 31102 | A 1-dimensional subspace i... |
| h1datom 31103 | A 1-dimensional subspace i... |
| cmbr 31105 | Binary relation expressing... |
| pjoml2i 31106 | Variation of orthomodular ... |
| pjoml3i 31107 | Variation of orthomodular ... |
| pjoml4i 31108 | Variation of orthomodular ... |
| pjoml5i 31109 | The orthomodular law. Rem... |
| pjoml6i 31110 | An equivalent of the ortho... |
| cmbri 31111 | Binary relation expressing... |
| cmcmlem 31112 | Commutation is symmetric. ... |
| cmcmi 31113 | Commutation is symmetric. ... |
| cmcm2i 31114 | Commutation with orthocomp... |
| cmcm3i 31115 | Commutation with orthocomp... |
| cmcm4i 31116 | Commutation with orthocomp... |
| cmbr2i 31117 | Alternate definition of th... |
| cmcmii 31118 | Commutation is symmetric. ... |
| cmcm2ii 31119 | Commutation with orthocomp... |
| cmcm3ii 31120 | Commutation with orthocomp... |
| cmbr3i 31121 | Alternate definition for t... |
| cmbr4i 31122 | Alternate definition for t... |
| lecmi 31123 | Comparable Hilbert lattice... |
| lecmii 31124 | Comparable Hilbert lattice... |
| cmj1i 31125 | A Hilbert lattice element ... |
| cmj2i 31126 | A Hilbert lattice element ... |
| cmm1i 31127 | A Hilbert lattice element ... |
| cmm2i 31128 | A Hilbert lattice element ... |
| cmbr3 31129 | Alternate definition for t... |
| cm0 31130 | The zero Hilbert lattice e... |
| cmidi 31131 | The commutes relation is r... |
| pjoml2 31132 | Variation of orthomodular ... |
| pjoml3 31133 | Variation of orthomodular ... |
| pjoml5 31134 | The orthomodular law. Rem... |
| cmcm 31135 | Commutation is symmetric. ... |
| cmcm3 31136 | Commutation with orthocomp... |
| cmcm2 31137 | Commutation with orthocomp... |
| lecm 31138 | Comparable Hilbert lattice... |
| fh1 31139 | Foulis-Holland Theorem. I... |
| fh2 31140 | Foulis-Holland Theorem. I... |
| cm2j 31141 | A lattice element that com... |
| fh1i 31142 | Foulis-Holland Theorem. I... |
| fh2i 31143 | Foulis-Holland Theorem. I... |
| fh3i 31144 | Variation of the Foulis-Ho... |
| fh4i 31145 | Variation of the Foulis-Ho... |
| cm2ji 31146 | A lattice element that com... |
| cm2mi 31147 | A lattice element that com... |
| qlax1i 31148 | One of the equations showi... |
| qlax2i 31149 | One of the equations showi... |
| qlax3i 31150 | One of the equations showi... |
| qlax4i 31151 | One of the equations showi... |
| qlax5i 31152 | One of the equations showi... |
| qlaxr1i 31153 | One of the conditions show... |
| qlaxr2i 31154 | One of the conditions show... |
| qlaxr4i 31155 | One of the conditions show... |
| qlaxr5i 31156 | One of the conditions show... |
| qlaxr3i 31157 | A variation of the orthomo... |
| chscllem1 31158 | Lemma for ~ chscl . (Cont... |
| chscllem2 31159 | Lemma for ~ chscl . (Cont... |
| chscllem3 31160 | Lemma for ~ chscl . (Cont... |
| chscllem4 31161 | Lemma for ~ chscl . (Cont... |
| chscl 31162 | The subspace sum of two cl... |
| osumi 31163 | If two closed subspaces of... |
| osumcori 31164 | Corollary of ~ osumi . (C... |
| osumcor2i 31165 | Corollary of ~ osumi , sho... |
| osum 31166 | If two closed subspaces of... |
| spansnji 31167 | The subspace sum of a clos... |
| spansnj 31168 | The subspace sum of a clos... |
| spansnscl 31169 | The subspace sum of a clos... |
| sumspansn 31170 | The sum of two vectors bel... |
| spansnm0i 31171 | The meet of different one-... |
| nonbooli 31172 | A Hilbert lattice with two... |
| spansncvi 31173 | Hilbert space has the cove... |
| spansncv 31174 | Hilbert space has the cove... |
| 5oalem1 31175 | Lemma for orthoarguesian l... |
| 5oalem2 31176 | Lemma for orthoarguesian l... |
| 5oalem3 31177 | Lemma for orthoarguesian l... |
| 5oalem4 31178 | Lemma for orthoarguesian l... |
| 5oalem5 31179 | Lemma for orthoarguesian l... |
| 5oalem6 31180 | Lemma for orthoarguesian l... |
| 5oalem7 31181 | Lemma for orthoarguesian l... |
| 5oai 31182 | Orthoarguesian law 5OA. Th... |
| 3oalem1 31183 | Lemma for 3OA (weak) ortho... |
| 3oalem2 31184 | Lemma for 3OA (weak) ortho... |
| 3oalem3 31185 | Lemma for 3OA (weak) ortho... |
| 3oalem4 31186 | Lemma for 3OA (weak) ortho... |
| 3oalem5 31187 | Lemma for 3OA (weak) ortho... |
| 3oalem6 31188 | Lemma for 3OA (weak) ortho... |
| 3oai 31189 | 3OA (weak) orthoarguesian ... |
| pjorthi 31190 | Projection components on o... |
| pjch1 31191 | Property of identity proje... |
| pjo 31192 | The orthogonal projection.... |
| pjcompi 31193 | Component of a projection.... |
| pjidmi 31194 | A projection is idempotent... |
| pjadjii 31195 | A projection is self-adjoi... |
| pjaddii 31196 | Projection of vector sum i... |
| pjinormii 31197 | The inner product of a pro... |
| pjmulii 31198 | Projection of (scalar) pro... |
| pjsubii 31199 | Projection of vector diffe... |
| pjsslem 31200 | Lemma for subset relations... |
| pjss2i 31201 | Subset relationship for pr... |
| pjssmii 31202 | Projection meet property. ... |
| pjssge0ii 31203 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 31204 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 31205 | The projection on a subspa... |
| pjadji 31206 | A projection is self-adjoi... |
| pjaddi 31207 | Projection of vector sum i... |
| pjinormi 31208 | The inner product of a pro... |
| pjsubi 31209 | Projection of vector diffe... |
| pjmuli 31210 | Projection of scalar produ... |
| pjige0i 31211 | The inner product of a pro... |
| pjige0 31212 | The inner product of a pro... |
| pjcjt2 31213 | The projection on a subspa... |
| pj0i 31214 | The projection of the zero... |
| pjch 31215 | Projection of a vector in ... |
| pjid 31216 | The projection of a vector... |
| pjvec 31217 | The set of vectors belongi... |
| pjocvec 31218 | The set of vectors belongi... |
| pjocini 31219 | Membership of projection i... |
| pjini 31220 | Membership of projection i... |
| pjjsi 31221 | A sufficient condition for... |
| pjfni 31222 | Functionality of a project... |
| pjrni 31223 | The range of a projection.... |
| pjfoi 31224 | A projection maps onto its... |
| pjfi 31225 | The mapping of a projectio... |
| pjvi 31226 | The value of a projection ... |
| pjhfo 31227 | A projection maps onto its... |
| pjrn 31228 | The range of a projection.... |
| pjhf 31229 | The mapping of a projectio... |
| pjfn 31230 | Functionality of a project... |
| pjsumi 31231 | The projection on a subspa... |
| pj11i 31232 | One-to-one correspondence ... |
| pjdsi 31233 | Vector decomposition into ... |
| pjds3i 31234 | Vector decomposition into ... |
| pj11 31235 | One-to-one correspondence ... |
| pjmfn 31236 | Functionality of the proje... |
| pjmf1 31237 | The projector function map... |
| pjoi0 31238 | The inner product of proje... |
| pjoi0i 31239 | The inner product of proje... |
| pjopythi 31240 | Pythagorean theorem for pr... |
| pjopyth 31241 | Pythagorean theorem for pr... |
| pjnormi 31242 | The norm of the projection... |
| pjpythi 31243 | Pythagorean theorem for pr... |
| pjneli 31244 | If a vector does not belon... |
| pjnorm 31245 | The norm of the projection... |
| pjpyth 31246 | Pythagorean theorem for pr... |
| pjnel 31247 | If a vector does not belon... |
| pjnorm2 31248 | A vector belongs to the su... |
| mayete3i 31249 | Mayet's equation E_3. Par... |
| mayetes3i 31250 | Mayet's equation E^*_3, de... |
| hosmval 31256 | Value of the sum of two Hi... |
| hommval 31257 | Value of the scalar produc... |
| hodmval 31258 | Value of the difference of... |
| hfsmval 31259 | Value of the sum of two Hi... |
| hfmmval 31260 | Value of the scalar produc... |
| hosval 31261 | Value of the sum of two Hi... |
| homval 31262 | Value of the scalar produc... |
| hodval 31263 | Value of the difference of... |
| hfsval 31264 | Value of the sum of two Hi... |
| hfmval 31265 | Value of the scalar produc... |
| hoscl 31266 | Closure of the sum of two ... |
| homcl 31267 | Closure of the scalar prod... |
| hodcl 31268 | Closure of the difference ... |
| ho0val 31271 | Value of the zero Hilbert ... |
| ho0f 31272 | Functionality of the zero ... |
| df0op2 31273 | Alternate definition of Hi... |
| dfiop2 31274 | Alternate definition of Hi... |
| hoif 31275 | Functionality of the Hilbe... |
| hoival 31276 | The value of the Hilbert s... |
| hoico1 31277 | Composition with the Hilbe... |
| hoico2 31278 | Composition with the Hilbe... |
| hoaddcl 31279 | The sum of Hilbert space o... |
| homulcl 31280 | The scalar product of a Hi... |
| hoeq 31281 | Equality of Hilbert space ... |
| hoeqi 31282 | Equality of Hilbert space ... |
| hoscli 31283 | Closure of Hilbert space o... |
| hodcli 31284 | Closure of Hilbert space o... |
| hocoi 31285 | Composition of Hilbert spa... |
| hococli 31286 | Closure of composition of ... |
| hocofi 31287 | Mapping of composition of ... |
| hocofni 31288 | Functionality of compositi... |
| hoaddcli 31289 | Mapping of sum of Hilbert ... |
| hosubcli 31290 | Mapping of difference of H... |
| hoaddfni 31291 | Functionality of sum of Hi... |
| hosubfni 31292 | Functionality of differenc... |
| hoaddcomi 31293 | Commutativity of sum of Hi... |
| hosubcl 31294 | Mapping of difference of H... |
| hoaddcom 31295 | Commutativity of sum of Hi... |
| hodsi 31296 | Relationship between Hilbe... |
| hoaddassi 31297 | Associativity of sum of Hi... |
| hoadd12i 31298 | Commutative/associative la... |
| hoadd32i 31299 | Commutative/associative la... |
| hocadddiri 31300 | Distributive law for Hilbe... |
| hocsubdiri 31301 | Distributive law for Hilbe... |
| ho2coi 31302 | Double composition of Hilb... |
| hoaddass 31303 | Associativity of sum of Hi... |
| hoadd32 31304 | Commutative/associative la... |
| hoadd4 31305 | Rearrangement of 4 terms i... |
| hocsubdir 31306 | Distributive law for Hilbe... |
| hoaddridi 31307 | Sum of a Hilbert space ope... |
| hodidi 31308 | Difference of a Hilbert sp... |
| ho0coi 31309 | Composition of the zero op... |
| hoid1i 31310 | Composition of Hilbert spa... |
| hoid1ri 31311 | Composition of Hilbert spa... |
| hoaddrid 31312 | Sum of a Hilbert space ope... |
| hodid 31313 | Difference of a Hilbert sp... |
| hon0 31314 | A Hilbert space operator i... |
| hodseqi 31315 | Subtraction and addition o... |
| ho0subi 31316 | Subtraction of Hilbert spa... |
| honegsubi 31317 | Relationship between Hilbe... |
| ho0sub 31318 | Subtraction of Hilbert spa... |
| hosubid1 31319 | The zero operator subtract... |
| honegsub 31320 | Relationship between Hilbe... |
| homullid 31321 | An operator equals its sca... |
| homco1 31322 | Associative law for scalar... |
| homulass 31323 | Scalar product associative... |
| hoadddi 31324 | Scalar product distributiv... |
| hoadddir 31325 | Scalar product reverse dis... |
| homul12 31326 | Swap first and second fact... |
| honegneg 31327 | Double negative of a Hilbe... |
| hosubneg 31328 | Relationship between opera... |
| hosubdi 31329 | Scalar product distributiv... |
| honegdi 31330 | Distribution of negative o... |
| honegsubdi 31331 | Distribution of negative o... |
| honegsubdi2 31332 | Distribution of negative o... |
| hosubsub2 31333 | Law for double subtraction... |
| hosub4 31334 | Rearrangement of 4 terms i... |
| hosubadd4 31335 | Rearrangement of 4 terms i... |
| hoaddsubass 31336 | Associative-type law for a... |
| hoaddsub 31337 | Law for operator addition ... |
| hosubsub 31338 | Law for double subtraction... |
| hosubsub4 31339 | Law for double subtraction... |
| ho2times 31340 | Two times a Hilbert space ... |
| hoaddsubassi 31341 | Associativity of sum and d... |
| hoaddsubi 31342 | Law for sum and difference... |
| hosd1i 31343 | Hilbert space operator sum... |
| hosd2i 31344 | Hilbert space operator sum... |
| hopncani 31345 | Hilbert space operator can... |
| honpcani 31346 | Hilbert space operator can... |
| hosubeq0i 31347 | If the difference between ... |
| honpncani 31348 | Hilbert space operator can... |
| ho01i 31349 | A condition implying that ... |
| ho02i 31350 | A condition implying that ... |
| hoeq1 31351 | A condition implying that ... |
| hoeq2 31352 | A condition implying that ... |
| adjmo 31353 | Every Hilbert space operat... |
| adjsym 31354 | Symmetry property of an ad... |
| eigrei 31355 | A necessary and sufficient... |
| eigre 31356 | A necessary and sufficient... |
| eigposi 31357 | A sufficient condition (fi... |
| eigorthi 31358 | A necessary and sufficient... |
| eigorth 31359 | A necessary and sufficient... |
| nmopval 31377 | Value of the norm of a Hil... |
| elcnop 31378 | Property defining a contin... |
| ellnop 31379 | Property defining a linear... |
| lnopf 31380 | A linear Hilbert space ope... |
| elbdop 31381 | Property defining a bounde... |
| bdopln 31382 | A bounded linear Hilbert s... |
| bdopf 31383 | A bounded linear Hilbert s... |
| nmopsetretALT 31384 | The set in the supremum of... |
| nmopsetretHIL 31385 | The set in the supremum of... |
| nmopsetn0 31386 | The set in the supremum of... |
| nmopxr 31387 | The norm of a Hilbert spac... |
| nmoprepnf 31388 | The norm of a Hilbert spac... |
| nmopgtmnf 31389 | The norm of a Hilbert spac... |
| nmopreltpnf 31390 | The norm of a Hilbert spac... |
| nmopre 31391 | The norm of a bounded oper... |
| elbdop2 31392 | Property defining a bounde... |
| elunop 31393 | Property defining a unitar... |
| elhmop 31394 | Property defining a Hermit... |
| hmopf 31395 | A Hermitian operator is a ... |
| hmopex 31396 | The class of Hermitian ope... |
| nmfnval 31397 | Value of the norm of a Hil... |
| nmfnsetre 31398 | The set in the supremum of... |
| nmfnsetn0 31399 | The set in the supremum of... |
| nmfnxr 31400 | The norm of any Hilbert sp... |
| nmfnrepnf 31401 | The norm of a Hilbert spac... |
| nlfnval 31402 | Value of the null space of... |
| elcnfn 31403 | Property defining a contin... |
| ellnfn 31404 | Property defining a linear... |
| lnfnf 31405 | A linear Hilbert space fun... |
| dfadj2 31406 | Alternate definition of th... |
| funadj 31407 | Functionality of the adjoi... |
| dmadjss 31408 | The domain of the adjoint ... |
| dmadjop 31409 | A member of the domain of ... |
| adjeu 31410 | Elementhood in the domain ... |
| adjval 31411 | Value of the adjoint funct... |
| adjval2 31412 | Value of the adjoint funct... |
| cnvadj 31413 | The adjoint function equal... |
| funcnvadj 31414 | The converse of the adjoin... |
| adj1o 31415 | The adjoint function maps ... |
| dmadjrn 31416 | The adjoint of an operator... |
| eigvecval 31417 | The set of eigenvectors of... |
| eigvalfval 31418 | The eigenvalues of eigenve... |
| specval 31419 | The value of the spectrum ... |
| speccl 31420 | The spectrum of an operato... |
| hhlnoi 31421 | The linear operators of Hi... |
| hhnmoi 31422 | The norm of an operator in... |
| hhbloi 31423 | A bounded linear operator ... |
| hh0oi 31424 | The zero operator in Hilbe... |
| hhcno 31425 | The continuous operators o... |
| hhcnf 31426 | The continuous functionals... |
| dmadjrnb 31427 | The adjoint of an operator... |
| nmoplb 31428 | A lower bound for an opera... |
| nmopub 31429 | An upper bound for an oper... |
| nmopub2tALT 31430 | An upper bound for an oper... |
| nmopub2tHIL 31431 | An upper bound for an oper... |
| nmopge0 31432 | The norm of any Hilbert sp... |
| nmopgt0 31433 | A linear Hilbert space ope... |
| cnopc 31434 | Basic continuity property ... |
| lnopl 31435 | Basic linearity property o... |
| unop 31436 | Basic inner product proper... |
| unopf1o 31437 | A unitary operator in Hilb... |
| unopnorm 31438 | A unitary operator is idem... |
| cnvunop 31439 | The inverse (converse) of ... |
| unopadj 31440 | The inverse (converse) of ... |
| unoplin 31441 | A unitary operator is line... |
| counop 31442 | The composition of two uni... |
| hmop 31443 | Basic inner product proper... |
| hmopre 31444 | The inner product of the v... |
| nmfnlb 31445 | A lower bound for a functi... |
| nmfnleub 31446 | An upper bound for the nor... |
| nmfnleub2 31447 | An upper bound for the nor... |
| nmfnge0 31448 | The norm of any Hilbert sp... |
| elnlfn 31449 | Membership in the null spa... |
| elnlfn2 31450 | Membership in the null spa... |
| cnfnc 31451 | Basic continuity property ... |
| lnfnl 31452 | Basic linearity property o... |
| adjcl 31453 | Closure of the adjoint of ... |
| adj1 31454 | Property of an adjoint Hil... |
| adj2 31455 | Property of an adjoint Hil... |
| adjeq 31456 | A property that determines... |
| adjadj 31457 | Double adjoint. Theorem 3... |
| adjvalval 31458 | Value of the value of the ... |
| unopadj2 31459 | The adjoint of a unitary o... |
| hmopadj 31460 | A Hermitian operator is se... |
| hmdmadj 31461 | Every Hermitian operator h... |
| hmopadj2 31462 | An operator is Hermitian i... |
| hmoplin 31463 | A Hermitian operator is li... |
| brafval 31464 | The bra of a vector, expre... |
| braval 31465 | A bra-ket juxtaposition, e... |
| braadd 31466 | Linearity property of bra ... |
| bramul 31467 | Linearity property of bra ... |
| brafn 31468 | The bra function is a func... |
| bralnfn 31469 | The Dirac bra function is ... |
| bracl 31470 | Closure of the bra functio... |
| bra0 31471 | The Dirac bra of the zero ... |
| brafnmul 31472 | Anti-linearity property of... |
| kbfval 31473 | The outer product of two v... |
| kbop 31474 | The outer product of two v... |
| kbval 31475 | The value of the operator ... |
| kbmul 31476 | Multiplication property of... |
| kbpj 31477 | If a vector ` A ` has norm... |
| eleigvec 31478 | Membership in the set of e... |
| eleigvec2 31479 | Membership in the set of e... |
| eleigveccl 31480 | Closure of an eigenvector ... |
| eigvalval 31481 | The eigenvalue of an eigen... |
| eigvalcl 31482 | An eigenvalue is a complex... |
| eigvec1 31483 | Property of an eigenvector... |
| eighmre 31484 | The eigenvalues of a Hermi... |
| eighmorth 31485 | Eigenvectors of a Hermitia... |
| nmopnegi 31486 | Value of the norm of the n... |
| lnop0 31487 | The value of a linear Hilb... |
| lnopmul 31488 | Multiplicative property of... |
| lnopli 31489 | Basic scalar product prope... |
| lnopfi 31490 | A linear Hilbert space ope... |
| lnop0i 31491 | The value of a linear Hilb... |
| lnopaddi 31492 | Additive property of a lin... |
| lnopmuli 31493 | Multiplicative property of... |
| lnopaddmuli 31494 | Sum/product property of a ... |
| lnopsubi 31495 | Subtraction property for a... |
| lnopsubmuli 31496 | Subtraction/product proper... |
| lnopmulsubi 31497 | Product/subtraction proper... |
| homco2 31498 | Move a scalar product out ... |
| idunop 31499 | The identity function (res... |
| 0cnop 31500 | The identically zero funct... |
| 0cnfn 31501 | The identically zero funct... |
| idcnop 31502 | The identity function (res... |
| idhmop 31503 | The Hilbert space identity... |
| 0hmop 31504 | The identically zero funct... |
| 0lnop 31505 | The identically zero funct... |
| 0lnfn 31506 | The identically zero funct... |
| nmop0 31507 | The norm of the zero opera... |
| nmfn0 31508 | The norm of the identicall... |
| hmopbdoptHIL 31509 | A Hermitian operator is a ... |
| hoddii 31510 | Distributive law for Hilbe... |
| hoddi 31511 | Distributive law for Hilbe... |
| nmop0h 31512 | The norm of any operator o... |
| idlnop 31513 | The identity function (res... |
| 0bdop 31514 | The identically zero opera... |
| adj0 31515 | Adjoint of the zero operat... |
| nmlnop0iALT 31516 | A linear operator with a z... |
| nmlnop0iHIL 31517 | A linear operator with a z... |
| nmlnopgt0i 31518 | A linear Hilbert space ope... |
| nmlnop0 31519 | A linear operator with a z... |
| nmlnopne0 31520 | A linear operator with a n... |
| lnopmi 31521 | The scalar product of a li... |
| lnophsi 31522 | The sum of two linear oper... |
| lnophdi 31523 | The difference of two line... |
| lnopcoi 31524 | The composition of two lin... |
| lnopco0i 31525 | The composition of a linea... |
| lnopeq0lem1 31526 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 31527 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 31528 | A condition implying that ... |
| lnopeqi 31529 | Two linear Hilbert space o... |
| lnopeq 31530 | Two linear Hilbert space o... |
| lnopunilem1 31531 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 31532 | Lemma for ~ lnopunii . (C... |
| lnopunii 31533 | If a linear operator (whos... |
| elunop2 31534 | An operator is unitary iff... |
| nmopun 31535 | Norm of a unitary Hilbert ... |
| unopbd 31536 | A unitary operator is a bo... |
| lnophmlem1 31537 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 31538 | Lemma for ~ lnophmi . (Co... |
| lnophmi 31539 | A linear operator is Hermi... |
| lnophm 31540 | A linear operator is Hermi... |
| hmops 31541 | The sum of two Hermitian o... |
| hmopm 31542 | The scalar product of a He... |
| hmopd 31543 | The difference of two Herm... |
| hmopco 31544 | The composition of two com... |
| nmbdoplbi 31545 | A lower bound for the norm... |
| nmbdoplb 31546 | A lower bound for the norm... |
| nmcexi 31547 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 31548 | The norm of a continuous l... |
| nmcoplbi 31549 | A lower bound for the norm... |
| nmcopex 31550 | The norm of a continuous l... |
| nmcoplb 31551 | A lower bound for the norm... |
| nmophmi 31552 | The norm of the scalar pro... |
| bdophmi 31553 | The scalar product of a bo... |
| lnconi 31554 | Lemma for ~ lnopconi and ~... |
| lnopconi 31555 | A condition equivalent to ... |
| lnopcon 31556 | A condition equivalent to ... |
| lnopcnbd 31557 | A linear operator is conti... |
| lncnopbd 31558 | A continuous linear operat... |
| lncnbd 31559 | A continuous linear operat... |
| lnopcnre 31560 | A linear operator is conti... |
| lnfnli 31561 | Basic property of a linear... |
| lnfnfi 31562 | A linear Hilbert space fun... |
| lnfn0i 31563 | The value of a linear Hilb... |
| lnfnaddi 31564 | Additive property of a lin... |
| lnfnmuli 31565 | Multiplicative property of... |
| lnfnaddmuli 31566 | Sum/product property of a ... |
| lnfnsubi 31567 | Subtraction property for a... |
| lnfn0 31568 | The value of a linear Hilb... |
| lnfnmul 31569 | Multiplicative property of... |
| nmbdfnlbi 31570 | A lower bound for the norm... |
| nmbdfnlb 31571 | A lower bound for the norm... |
| nmcfnexi 31572 | The norm of a continuous l... |
| nmcfnlbi 31573 | A lower bound for the norm... |
| nmcfnex 31574 | The norm of a continuous l... |
| nmcfnlb 31575 | A lower bound of the norm ... |
| lnfnconi 31576 | A condition equivalent to ... |
| lnfncon 31577 | A condition equivalent to ... |
| lnfncnbd 31578 | A linear functional is con... |
| imaelshi 31579 | The image of a subspace un... |
| rnelshi 31580 | The range of a linear oper... |
| nlelshi 31581 | The null space of a linear... |
| nlelchi 31582 | The null space of a contin... |
| riesz3i 31583 | A continuous linear functi... |
| riesz4i 31584 | A continuous linear functi... |
| riesz4 31585 | A continuous linear functi... |
| riesz1 31586 | Part 1 of the Riesz repres... |
| riesz2 31587 | Part 2 of the Riesz repres... |
| cnlnadjlem1 31588 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 31589 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 31590 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 31591 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 31592 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 31593 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 31594 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 31595 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 31596 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 31597 | Every continuous linear op... |
| cnlnadjeui 31598 | Every continuous linear op... |
| cnlnadjeu 31599 | Every continuous linear op... |
| cnlnadj 31600 | Every continuous linear op... |
| cnlnssadj 31601 | Every continuous linear Hi... |
| bdopssadj 31602 | Every bounded linear Hilbe... |
| bdopadj 31603 | Every bounded linear Hilbe... |
| adjbdln 31604 | The adjoint of a bounded l... |
| adjbdlnb 31605 | An operator is bounded and... |
| adjbd1o 31606 | The mapping of adjoints of... |
| adjlnop 31607 | The adjoint of an operator... |
| adjsslnop 31608 | Every operator with an adj... |
| nmopadjlei 31609 | Property of the norm of an... |
| nmopadjlem 31610 | Lemma for ~ nmopadji . (C... |
| nmopadji 31611 | Property of the norm of an... |
| adjeq0 31612 | An operator is zero iff it... |
| adjmul 31613 | The adjoint of the scalar ... |
| adjadd 31614 | The adjoint of the sum of ... |
| nmoptrii 31615 | Triangle inequality for th... |
| nmopcoi 31616 | Upper bound for the norm o... |
| bdophsi 31617 | The sum of two bounded lin... |
| bdophdi 31618 | The difference between two... |
| bdopcoi 31619 | The composition of two bou... |
| nmoptri2i 31620 | Triangle-type inequality f... |
| adjcoi 31621 | The adjoint of a compositi... |
| nmopcoadji 31622 | The norm of an operator co... |
| nmopcoadj2i 31623 | The norm of an operator co... |
| nmopcoadj0i 31624 | An operator composed with ... |
| unierri 31625 | If we approximate a chain ... |
| branmfn 31626 | The norm of the bra functi... |
| brabn 31627 | The bra of a vector is a b... |
| rnbra 31628 | The set of bras equals the... |
| bra11 31629 | The bra function maps vect... |
| bracnln 31630 | A bra is a continuous line... |
| cnvbraval 31631 | Value of the converse of t... |
| cnvbracl 31632 | Closure of the converse of... |
| cnvbrabra 31633 | The converse bra of the br... |
| bracnvbra 31634 | The bra of the converse br... |
| bracnlnval 31635 | The vector that a continuo... |
| cnvbramul 31636 | Multiplication property of... |
| kbass1 31637 | Dirac bra-ket associative ... |
| kbass2 31638 | Dirac bra-ket associative ... |
| kbass3 31639 | Dirac bra-ket associative ... |
| kbass4 31640 | Dirac bra-ket associative ... |
| kbass5 31641 | Dirac bra-ket associative ... |
| kbass6 31642 | Dirac bra-ket associative ... |
| leopg 31643 | Ordering relation for posi... |
| leop 31644 | Ordering relation for oper... |
| leop2 31645 | Ordering relation for oper... |
| leop3 31646 | Operator ordering in terms... |
| leoppos 31647 | Binary relation defining a... |
| leoprf2 31648 | The ordering relation for ... |
| leoprf 31649 | The ordering relation for ... |
| leopsq 31650 | The square of a Hermitian ... |
| 0leop 31651 | The zero operator is a pos... |
| idleop 31652 | The identity operator is a... |
| leopadd 31653 | The sum of two positive op... |
| leopmuli 31654 | The scalar product of a no... |
| leopmul 31655 | The scalar product of a po... |
| leopmul2i 31656 | Scalar product applied to ... |
| leoptri 31657 | The positive operator orde... |
| leoptr 31658 | The positive operator orde... |
| leopnmid 31659 | A bounded Hermitian operat... |
| nmopleid 31660 | A nonzero, bounded Hermiti... |
| opsqrlem1 31661 | Lemma for opsqri . (Contr... |
| opsqrlem2 31662 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 31663 | Lemma for opsqri . (Contr... |
| opsqrlem4 31664 | Lemma for opsqri . (Contr... |
| opsqrlem5 31665 | Lemma for opsqri . (Contr... |
| opsqrlem6 31666 | Lemma for opsqri . (Contr... |
| pjhmopi 31667 | A projector is a Hermitian... |
| pjlnopi 31668 | A projector is a linear op... |
| pjnmopi 31669 | The operator norm of a pro... |
| pjbdlni 31670 | A projector is a bounded l... |
| pjhmop 31671 | A projection is a Hermitia... |
| hmopidmchi 31672 | An idempotent Hermitian op... |
| hmopidmpji 31673 | An idempotent Hermitian op... |
| hmopidmch 31674 | An idempotent Hermitian op... |
| hmopidmpj 31675 | An idempotent Hermitian op... |
| pjsdii 31676 | Distributive law for Hilbe... |
| pjddii 31677 | Distributive law for Hilbe... |
| pjsdi2i 31678 | Chained distributive law f... |
| pjcoi 31679 | Composition of projections... |
| pjcocli 31680 | Closure of composition of ... |
| pjcohcli 31681 | Closure of composition of ... |
| pjadjcoi 31682 | Adjoint of composition of ... |
| pjcofni 31683 | Functionality of compositi... |
| pjss1coi 31684 | Subset relationship for pr... |
| pjss2coi 31685 | Subset relationship for pr... |
| pjssmi 31686 | Projection meet property. ... |
| pjssge0i 31687 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 31688 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 31689 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 31690 | Composition of projections... |
| pjscji 31691 | The projection of orthogon... |
| pjssumi 31692 | The projection on a subspa... |
| pjssposi 31693 | Projector ordering can be ... |
| pjordi 31694 | The definition of projecto... |
| pjssdif2i 31695 | The projection subspace of... |
| pjssdif1i 31696 | A necessary and sufficient... |
| pjimai 31697 | The image of a projection.... |
| pjidmcoi 31698 | A projection is idempotent... |
| pjoccoi 31699 | Composition of projections... |
| pjtoi 31700 | Subspace sum of projection... |
| pjoci 31701 | Projection of orthocomplem... |
| pjidmco 31702 | A projection operator is i... |
| dfpjop 31703 | Definition of projection o... |
| pjhmopidm 31704 | Two ways to express the se... |
| elpjidm 31705 | A projection operator is i... |
| elpjhmop 31706 | A projection operator is H... |
| 0leopj 31707 | A projector is a positive ... |
| pjadj2 31708 | A projector is self-adjoin... |
| pjadj3 31709 | A projector is self-adjoin... |
| elpjch 31710 | Reconstruction of the subs... |
| elpjrn 31711 | Reconstruction of the subs... |
| pjinvari 31712 | A closed subspace ` H ` wi... |
| pjin1i 31713 | Lemma for Theorem 1.22 of ... |
| pjin2i 31714 | Lemma for Theorem 1.22 of ... |
| pjin3i 31715 | Lemma for Theorem 1.22 of ... |
| pjclem1 31716 | Lemma for projection commu... |
| pjclem2 31717 | Lemma for projection commu... |
| pjclem3 31718 | Lemma for projection commu... |
| pjclem4a 31719 | Lemma for projection commu... |
| pjclem4 31720 | Lemma for projection commu... |
| pjci 31721 | Two subspaces commute iff ... |
| pjcmul1i 31722 | A necessary and sufficient... |
| pjcmul2i 31723 | The projection subspace of... |
| pjcohocli 31724 | Closure of composition of ... |
| pjadj2coi 31725 | Adjoint of double composit... |
| pj2cocli 31726 | Closure of double composit... |
| pj3lem1 31727 | Lemma for projection tripl... |
| pj3si 31728 | Stronger projection triple... |
| pj3i 31729 | Projection triplet theorem... |
| pj3cor1i 31730 | Projection triplet corolla... |
| pjs14i 31731 | Theorem S-14 of Watanabe, ... |
| isst 31734 | Property of a state. (Con... |
| ishst 31735 | Property of a complex Hilb... |
| sticl 31736 | ` [ 0 , 1 ] ` closure of t... |
| stcl 31737 | Real closure of the value ... |
| hstcl 31738 | Closure of the value of a ... |
| hst1a 31739 | Unit value of a Hilbert-sp... |
| hstel2 31740 | Properties of a Hilbert-sp... |
| hstorth 31741 | Orthogonality property of ... |
| hstosum 31742 | Orthogonal sum property of... |
| hstoc 31743 | Sum of a Hilbert-space-val... |
| hstnmoc 31744 | Sum of norms of a Hilbert-... |
| stge0 31745 | The value of a state is no... |
| stle1 31746 | The value of a state is le... |
| hstle1 31747 | The norm of the value of a... |
| hst1h 31748 | The norm of a Hilbert-spac... |
| hst0h 31749 | The norm of a Hilbert-spac... |
| hstpyth 31750 | Pythagorean property of a ... |
| hstle 31751 | Ordering property of a Hil... |
| hstles 31752 | Ordering property of a Hil... |
| hstoh 31753 | A Hilbert-space-valued sta... |
| hst0 31754 | A Hilbert-space-valued sta... |
| sthil 31755 | The value of a state at th... |
| stj 31756 | The value of a state on a ... |
| sto1i 31757 | The state of a subspace pl... |
| sto2i 31758 | The state of the orthocomp... |
| stge1i 31759 | If a state is greater than... |
| stle0i 31760 | If a state is less than or... |
| stlei 31761 | Ordering law for states. ... |
| stlesi 31762 | Ordering law for states. ... |
| stji1i 31763 | Join of components of Sasa... |
| stm1i 31764 | State of component of unit... |
| stm1ri 31765 | State of component of unit... |
| stm1addi 31766 | Sum of states whose meet i... |
| staddi 31767 | If the sum of 2 states is ... |
| stm1add3i 31768 | Sum of states whose meet i... |
| stadd3i 31769 | If the sum of 3 states is ... |
| st0 31770 | The state of the zero subs... |
| strlem1 31771 | Lemma for strong state the... |
| strlem2 31772 | Lemma for strong state the... |
| strlem3a 31773 | Lemma for strong state the... |
| strlem3 31774 | Lemma for strong state the... |
| strlem4 31775 | Lemma for strong state the... |
| strlem5 31776 | Lemma for strong state the... |
| strlem6 31777 | Lemma for strong state the... |
| stri 31778 | Strong state theorem. The... |
| strb 31779 | Strong state theorem (bidi... |
| hstrlem2 31780 | Lemma for strong set of CH... |
| hstrlem3a 31781 | Lemma for strong set of CH... |
| hstrlem3 31782 | Lemma for strong set of CH... |
| hstrlem4 31783 | Lemma for strong set of CH... |
| hstrlem5 31784 | Lemma for strong set of CH... |
| hstrlem6 31785 | Lemma for strong set of CH... |
| hstri 31786 | Hilbert space admits a str... |
| hstrbi 31787 | Strong CH-state theorem (b... |
| largei 31788 | A Hilbert lattice admits a... |
| jplem1 31789 | Lemma for Jauch-Piron theo... |
| jplem2 31790 | Lemma for Jauch-Piron theo... |
| jpi 31791 | The function ` S ` , that ... |
| golem1 31792 | Lemma for Godowski's equat... |
| golem2 31793 | Lemma for Godowski's equat... |
| goeqi 31794 | Godowski's equation, shown... |
| stcltr1i 31795 | Property of a strong class... |
| stcltr2i 31796 | Property of a strong class... |
| stcltrlem1 31797 | Lemma for strong classical... |
| stcltrlem2 31798 | Lemma for strong classical... |
| stcltrthi 31799 | Theorem for classically st... |
| cvbr 31803 | Binary relation expressing... |
| cvbr2 31804 | Binary relation expressing... |
| cvcon3 31805 | Contraposition law for the... |
| cvpss 31806 | The covers relation implie... |
| cvnbtwn 31807 | The covers relation implie... |
| cvnbtwn2 31808 | The covers relation implie... |
| cvnbtwn3 31809 | The covers relation implie... |
| cvnbtwn4 31810 | The covers relation implie... |
| cvnsym 31811 | The covers relation is not... |
| cvnref 31812 | The covers relation is not... |
| cvntr 31813 | The covers relation is not... |
| spansncv2 31814 | Hilbert space has the cove... |
| mdbr 31815 | Binary relation expressing... |
| mdi 31816 | Consequence of the modular... |
| mdbr2 31817 | Binary relation expressing... |
| mdbr3 31818 | Binary relation expressing... |
| mdbr4 31819 | Binary relation expressing... |
| dmdbr 31820 | Binary relation expressing... |
| dmdmd 31821 | The dual modular pair prop... |
| mddmd 31822 | The modular pair property ... |
| dmdi 31823 | Consequence of the dual mo... |
| dmdbr2 31824 | Binary relation expressing... |
| dmdi2 31825 | Consequence of the dual mo... |
| dmdbr3 31826 | Binary relation expressing... |
| dmdbr4 31827 | Binary relation expressing... |
| dmdi4 31828 | Consequence of the dual mo... |
| dmdbr5 31829 | Binary relation expressing... |
| mddmd2 31830 | Relationship between modul... |
| mdsl0 31831 | A sublattice condition tha... |
| ssmd1 31832 | Ordering implies the modul... |
| ssmd2 31833 | Ordering implies the modul... |
| ssdmd1 31834 | Ordering implies the dual ... |
| ssdmd2 31835 | Ordering implies the dual ... |
| dmdsl3 31836 | Sublattice mapping for a d... |
| mdsl3 31837 | Sublattice mapping for a m... |
| mdslle1i 31838 | Order preservation of the ... |
| mdslle2i 31839 | Order preservation of the ... |
| mdslj1i 31840 | Join preservation of the o... |
| mdslj2i 31841 | Meet preservation of the r... |
| mdsl1i 31842 | If the modular pair proper... |
| mdsl2i 31843 | If the modular pair proper... |
| mdsl2bi 31844 | If the modular pair proper... |
| cvmdi 31845 | The covering property impl... |
| mdslmd1lem1 31846 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 31847 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 31848 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 31849 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 31850 | Preservation of the modula... |
| mdslmd2i 31851 | Preservation of the modula... |
| mdsldmd1i 31852 | Preservation of the dual m... |
| mdslmd3i 31853 | Modular pair conditions th... |
| mdslmd4i 31854 | Modular pair condition tha... |
| csmdsymi 31855 | Cross-symmetry implies M-s... |
| mdexchi 31856 | An exchange lemma for modu... |
| cvmd 31857 | The covering property impl... |
| cvdmd 31858 | The covering property impl... |
| ela 31860 | Atoms in a Hilbert lattice... |
| elat2 31861 | Expanded membership relati... |
| elatcv0 31862 | A Hilbert lattice element ... |
| atcv0 31863 | An atom covers the zero su... |
| atssch 31864 | Atoms are a subset of the ... |
| atelch 31865 | An atom is a Hilbert latti... |
| atne0 31866 | An atom is not the Hilbert... |
| atss 31867 | A lattice element smaller ... |
| atsseq 31868 | Two atoms in a subset rela... |
| atcveq0 31869 | A Hilbert lattice element ... |
| h1da 31870 | A 1-dimensional subspace i... |
| spansna 31871 | The span of the singleton ... |
| sh1dle 31872 | A 1-dimensional subspace i... |
| ch1dle 31873 | A 1-dimensional subspace i... |
| atom1d 31874 | The 1-dimensional subspace... |
| superpos 31875 | Superposition Principle. ... |
| chcv1 31876 | The Hilbert lattice has th... |
| chcv2 31877 | The Hilbert lattice has th... |
| chjatom 31878 | The join of a closed subsp... |
| shatomici 31879 | The lattice of Hilbert sub... |
| hatomici 31880 | The Hilbert lattice is ato... |
| hatomic 31881 | A Hilbert lattice is atomi... |
| shatomistici 31882 | The lattice of Hilbert sub... |
| hatomistici 31883 | ` CH ` is atomistic, i.e. ... |
| chpssati 31884 | Two Hilbert lattice elemen... |
| chrelati 31885 | The Hilbert lattice is rel... |
| chrelat2i 31886 | A consequence of relative ... |
| cvati 31887 | If a Hilbert lattice eleme... |
| cvbr4i 31888 | An alternate way to expres... |
| cvexchlem 31889 | Lemma for ~ cvexchi . (Co... |
| cvexchi 31890 | The Hilbert lattice satisf... |
| chrelat2 31891 | A consequence of relative ... |
| chrelat3 31892 | A consequence of relative ... |
| chrelat3i 31893 | A consequence of the relat... |
| chrelat4i 31894 | A consequence of relative ... |
| cvexch 31895 | The Hilbert lattice satisf... |
| cvp 31896 | The Hilbert lattice satisf... |
| atnssm0 31897 | The meet of a Hilbert latt... |
| atnemeq0 31898 | The meet of distinct atoms... |
| atssma 31899 | The meet with an atom's su... |
| atcv0eq 31900 | Two atoms covering the zer... |
| atcv1 31901 | Two atoms covering the zer... |
| atexch 31902 | The Hilbert lattice satisf... |
| atomli 31903 | An assertion holding in at... |
| atoml2i 31904 | An assertion holding in at... |
| atordi 31905 | An ordering law for a Hilb... |
| atcvatlem 31906 | Lemma for ~ atcvati . (Co... |
| atcvati 31907 | A nonzero Hilbert lattice ... |
| atcvat2i 31908 | A Hilbert lattice element ... |
| atord 31909 | An ordering law for a Hilb... |
| atcvat2 31910 | A Hilbert lattice element ... |
| chirredlem1 31911 | Lemma for ~ chirredi . (C... |
| chirredlem2 31912 | Lemma for ~ chirredi . (C... |
| chirredlem3 31913 | Lemma for ~ chirredi . (C... |
| chirredlem4 31914 | Lemma for ~ chirredi . (C... |
| chirredi 31915 | The Hilbert lattice is irr... |
| chirred 31916 | The Hilbert lattice is irr... |
| atcvat3i 31917 | A condition implying that ... |
| atcvat4i 31918 | A condition implying exist... |
| atdmd 31919 | Two Hilbert lattice elemen... |
| atmd 31920 | Two Hilbert lattice elemen... |
| atmd2 31921 | Two Hilbert lattice elemen... |
| atabsi 31922 | Absorption of an incompara... |
| atabs2i 31923 | Absorption of an incompara... |
| mdsymlem1 31924 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 31925 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 31926 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 31927 | Lemma for ~ mdsymi . This... |
| mdsymlem5 31928 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 31929 | Lemma for ~ mdsymi . This... |
| mdsymlem7 31930 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 31931 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 31932 | M-symmetry of the Hilbert ... |
| mdsym 31933 | M-symmetry of the Hilbert ... |
| dmdsym 31934 | Dual M-symmetry of the Hil... |
| atdmd2 31935 | Two Hilbert lattice elemen... |
| sumdmdii 31936 | If the subspace sum of two... |
| cmmdi 31937 | Commuting subspaces form a... |
| cmdmdi 31938 | Commuting subspaces form a... |
| sumdmdlem 31939 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 31940 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 31941 | The subspace sum of two Hi... |
| dmdbr4ati 31942 | Dual modular pair property... |
| dmdbr5ati 31943 | Dual modular pair property... |
| dmdbr6ati 31944 | Dual modular pair property... |
| dmdbr7ati 31945 | Dual modular pair property... |
| mdoc1i 31946 | Orthocomplements form a mo... |
| mdoc2i 31947 | Orthocomplements form a mo... |
| dmdoc1i 31948 | Orthocomplements form a du... |
| dmdoc2i 31949 | Orthocomplements form a du... |
| mdcompli 31950 | A condition equivalent to ... |
| dmdcompli 31951 | A condition equivalent to ... |
| mddmdin0i 31952 | If dual modular implies mo... |
| cdjreui 31953 | A member of the sum of dis... |
| cdj1i 31954 | Two ways to express " ` A ... |
| cdj3lem1 31955 | A property of " ` A ` and ... |
| cdj3lem2 31956 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 31957 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 31958 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 31959 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 31960 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 31961 | Lemma for ~ cdj3i . The s... |
| cdj3i 31962 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 31963 | (_This theorem is a dummy ... |
| sa-abvi 31964 | A theorem about the univer... |
| xfree 31965 | A partial converse to ~ 19... |
| xfree2 31966 | A partial converse to ~ 19... |
| addltmulALT 31967 | A proof readability experi... |
| bian1d 31968 | Adding a superfluous conju... |
| or3di 31969 | Distributive law for disju... |
| or3dir 31970 | Distributive law for disju... |
| 3o1cs 31971 | Deduction eliminating disj... |
| 3o2cs 31972 | Deduction eliminating disj... |
| 3o3cs 31973 | Deduction eliminating disj... |
| 13an22anass 31974 | Associative law for four c... |
| sbc2iedf 31975 | Conversion of implicit sub... |
| rspc2daf 31976 | Double restricted speciali... |
| ralcom4f 31977 | Commutation of restricted ... |
| rexcom4f 31978 | Commutation of restricted ... |
| 19.9d2rf 31979 | A deduction version of one... |
| 19.9d2r 31980 | A deduction version of one... |
| r19.29ffa 31981 | A commonly used pattern ba... |
| eqtrb 31982 | A transposition of equalit... |
| eqelbid 31983 | A variable elimination law... |
| opsbc2ie 31984 | Conversion of implicit sub... |
| opreu2reuALT 31985 | Correspondence between uni... |
| 2reucom 31988 | Double restricted existent... |
| 2reu2rex1 31989 | Double restricted existent... |
| 2reureurex 31990 | Double restricted existent... |
| 2reu2reu2 31991 | Double restricted existent... |
| opreu2reu1 31992 | Equivalent definition of t... |
| sq2reunnltb 31993 | There exists a unique deco... |
| addsqnot2reu 31994 | For each complex number ` ... |
| sbceqbidf 31995 | Equality theorem for class... |
| sbcies 31996 | A special version of class... |
| mo5f 31997 | Alternate definition of "a... |
| nmo 31998 | Negation of "at most one".... |
| reuxfrdf 31999 | Transfer existential uniqu... |
| rexunirn 32000 | Restricted existential qua... |
| rmoxfrd 32001 | Transfer "at most one" res... |
| rmoun 32002 | "At most one" restricted e... |
| rmounid 32003 | A case where an "at most o... |
| riotaeqbidva 32004 | Equivalent wff's yield equ... |
| dmrab 32005 | Domain of a restricted cla... |
| difrab2 32006 | Difference of two restrict... |
| rabexgfGS 32007 | Separation Scheme in terms... |
| rabsnel 32008 | Truth implied by equality ... |
| eqrrabd 32009 | Deduce equality with a res... |
| foresf1o 32010 | From a surjective function... |
| rabfodom 32011 | Domination relation for re... |
| abrexdomjm 32012 | An indexed set is dominate... |
| abrexdom2jm 32013 | An indexed set is dominate... |
| abrexexd 32014 | Existence of a class abstr... |
| elabreximd 32015 | Class substitution in an i... |
| elabreximdv 32016 | Class substitution in an i... |
| abrexss 32017 | A necessary condition for ... |
| elunsn 32018 | Elementhood to a union wit... |
| nelun 32019 | Negated membership for a u... |
| snsssng 32020 | If a singleton is a subset... |
| inin 32021 | Intersection with an inter... |
| inindif 32022 | See ~ inundif . (Contribu... |
| difininv 32023 | Condition for the intersec... |
| difeq 32024 | Rewriting an equation with... |
| eqdif 32025 | If both set differences of... |
| indifbi 32026 | Two ways to express equali... |
| diffib 32027 | Case where ~ diffi is a bi... |
| difxp1ss 32028 | Difference law for Cartesi... |
| difxp2ss 32029 | Difference law for Cartesi... |
| indifundif 32030 | A remarkable equation with... |
| elpwincl1 32031 | Closure of intersection wi... |
| elpwdifcl 32032 | Closure of class differenc... |
| elpwiuncl 32033 | Closure of indexed union w... |
| eqsnd 32034 | Deduce that a set is a sin... |
| elpreq 32035 | Equality wihin a pair. (C... |
| nelpr 32036 | A set ` A ` not in a pair ... |
| inpr0 32037 | Rewrite an empty intersect... |
| neldifpr1 32038 | The first element of a pai... |
| neldifpr2 32039 | The second element of a pa... |
| unidifsnel 32040 | The other element of a pai... |
| unidifsnne 32041 | The other element of a pai... |
| ifeqeqx 32042 | An equality theorem tailor... |
| elimifd 32043 | Elimination of a condition... |
| elim2if 32044 | Elimination of two conditi... |
| elim2ifim 32045 | Elimination of two conditi... |
| ifeq3da 32046 | Given an expression ` C ` ... |
| ifnetrue 32047 | Deduce truth from a condit... |
| ifnefals 32048 | Deduce falsehood from a co... |
| ifnebib 32049 | The converse of ~ ifbi hol... |
| uniinn0 32050 | Sufficient and necessary c... |
| uniin1 32051 | Union of intersection. Ge... |
| uniin2 32052 | Union of intersection. Ge... |
| difuncomp 32053 | Express a class difference... |
| elpwunicl 32054 | Closure of a set union wit... |
| cbviunf 32055 | Rule used to change the bo... |
| iuneq12daf 32056 | Equality deduction for ind... |
| iunin1f 32057 | Indexed union of intersect... |
| ssiun3 32058 | Subset equivalence for an ... |
| ssiun2sf 32059 | Subset relationship for an... |
| iuninc 32060 | The union of an increasing... |
| iundifdifd 32061 | The intersection of a set ... |
| iundifdif 32062 | The intersection of a set ... |
| iunrdx 32063 | Re-index an indexed union.... |
| iunpreima 32064 | Preimage of an indexed uni... |
| iunrnmptss 32065 | A subset relation for an i... |
| iunxunsn 32066 | Appending a set to an inde... |
| iunxunpr 32067 | Appending two sets to an i... |
| iinabrex 32068 | Rewriting an indexed inter... |
| disjnf 32069 | In case ` x ` is not free ... |
| cbvdisjf 32070 | Change bound variables in ... |
| disjss1f 32071 | A subset of a disjoint col... |
| disjeq1f 32072 | Equality theorem for disjo... |
| disjxun0 32073 | Simplify a disjoint union.... |
| disjdifprg 32074 | A trivial partition into a... |
| disjdifprg2 32075 | A trivial partition of a s... |
| disji2f 32076 | Property of a disjoint col... |
| disjif 32077 | Property of a disjoint col... |
| disjorf 32078 | Two ways to say that a col... |
| disjorsf 32079 | Two ways to say that a col... |
| disjif2 32080 | Property of a disjoint col... |
| disjabrex 32081 | Rewriting a disjoint colle... |
| disjabrexf 32082 | Rewriting a disjoint colle... |
| disjpreima 32083 | A preimage of a disjoint s... |
| disjrnmpt 32084 | Rewriting a disjoint colle... |
| disjin 32085 | If a collection is disjoin... |
| disjin2 32086 | If a collection is disjoin... |
| disjxpin 32087 | Derive a disjunction over ... |
| iundisjf 32088 | Rewrite a countable union ... |
| iundisj2f 32089 | A disjoint union is disjoi... |
| disjrdx 32090 | Re-index a disjunct collec... |
| disjex 32091 | Two ways to say that two c... |
| disjexc 32092 | A variant of ~ disjex , ap... |
| disjunsn 32093 | Append an element to a dis... |
| disjun0 32094 | Adding the empty element p... |
| disjiunel 32095 | A set of elements B of a d... |
| disjuniel 32096 | A set of elements B of a d... |
| xpdisjres 32097 | Restriction of a constant ... |
| opeldifid 32098 | Ordered pair elementhood o... |
| difres 32099 | Case when class difference... |
| imadifxp 32100 | Image of the difference wi... |
| relfi 32101 | A relation (set) is finite... |
| 0res 32102 | Restriction of the empty f... |
| fcoinver 32103 | Build an equivalence relat... |
| fcoinvbr 32104 | Binary relation for the eq... |
| brabgaf 32105 | The law of concretion for ... |
| brelg 32106 | Two things in a binary rel... |
| br8d 32107 | Substitution for an eight-... |
| opabdm 32108 | Domain of an ordered-pair ... |
| opabrn 32109 | Range of an ordered-pair c... |
| opabssi 32110 | Sufficient condition for a... |
| opabid2ss 32111 | One direction of ~ opabid2... |
| ssrelf 32112 | A subclass relationship de... |
| eqrelrd2 32113 | A version of ~ eqrelrdv2 w... |
| erbr3b 32114 | Biconditional for equivale... |
| iunsnima 32115 | Image of a singleton by an... |
| iunsnima2 32116 | Version of ~ iunsnima with... |
| ac6sf2 32117 | Alternate version of ~ ac6... |
| fnresin 32118 | Restriction of a function ... |
| f1o3d 32119 | Describe an implicit one-t... |
| eldmne0 32120 | A function of nonempty dom... |
| f1rnen 32121 | Equinumerosity of the rang... |
| rinvf1o 32122 | Sufficient conditions for ... |
| fresf1o 32123 | Conditions for a restricti... |
| nfpconfp 32124 | The set of fixed points of... |
| fmptco1f1o 32125 | The action of composing (t... |
| cofmpt2 32126 | Express composition of a m... |
| f1mptrn 32127 | Express injection for a ma... |
| dfimafnf 32128 | Alternate definition of th... |
| funimass4f 32129 | Membership relation for th... |
| elimampt 32130 | Membership in the image of... |
| suppss2f 32131 | Show that the support of a... |
| ofrn 32132 | The range of the function ... |
| ofrn2 32133 | The range of the function ... |
| off2 32134 | The function operation pro... |
| ofresid 32135 | Applying an operation rest... |
| fimarab 32136 | Expressing the image of a ... |
| unipreima 32137 | Preimage of a class union.... |
| opfv 32138 | Value of a function produc... |
| xppreima 32139 | The preimage of a Cartesia... |
| 2ndimaxp 32140 | Image of a cartesian produ... |
| djussxp2 32141 | Stronger version of ~ djus... |
| 2ndresdju 32142 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 32143 | The ` 2nd ` function restr... |
| xppreima2 32144 | The preimage of a Cartesia... |
| abfmpunirn 32145 | Membership in a union of a... |
| rabfmpunirn 32146 | Membership in a union of a... |
| abfmpeld 32147 | Membership in an element o... |
| abfmpel 32148 | Membership in an element o... |
| fmptdF 32149 | Domain and codomain of the... |
| fmptcof2 32150 | Composition of two functio... |
| fcomptf 32151 | Express composition of two... |
| acunirnmpt 32152 | Axiom of choice for the un... |
| acunirnmpt2 32153 | Axiom of choice for the un... |
| acunirnmpt2f 32154 | Axiom of choice for the un... |
| aciunf1lem 32155 | Choice in an index union. ... |
| aciunf1 32156 | Choice in an index union. ... |
| ofoprabco 32157 | Function operation as a co... |
| ofpreima 32158 | Express the preimage of a ... |
| ofpreima2 32159 | Express the preimage of a ... |
| funcnvmpt 32160 | Condition for a function i... |
| funcnv5mpt 32161 | Two ways to say that a fun... |
| funcnv4mpt 32162 | Two ways to say that a fun... |
| preimane 32163 | Different elements have di... |
| fnpreimac 32164 | Choose a set ` x ` contain... |
| fgreu 32165 | Exactly one point of a fun... |
| fcnvgreu 32166 | If the converse of a relat... |
| rnmposs 32167 | The range of an operation ... |
| mptssALT 32168 | Deduce subset relation of ... |
| dfcnv2 32169 | Alternative definition of ... |
| fnimatp 32170 | The image of an unordered ... |
| rnexd 32171 | The range of a set is a se... |
| imaexd 32172 | The image of a set is a se... |
| mpomptxf 32173 | Express a two-argument fun... |
| suppovss 32174 | A bound for the support of... |
| fvdifsupp 32175 | Function value is zero out... |
| suppiniseg 32176 | Relation between the suppo... |
| fsuppinisegfi 32177 | The initial segment ` ( ``... |
| fressupp 32178 | The restriction of a funct... |
| fdifsuppconst 32179 | A function is a zero const... |
| ressupprn 32180 | The range of a function re... |
| supppreima 32181 | Express the support of a f... |
| fsupprnfi 32182 | Finite support implies fin... |
| mptiffisupp 32183 | Conditions for a mapping f... |
| cosnopne 32184 | Composition of two ordered... |
| cosnop 32185 | Composition of two ordered... |
| cnvprop 32186 | Converse of a pair of orde... |
| brprop 32187 | Binary relation for a pair... |
| mptprop 32188 | Rewrite pairs of ordered p... |
| coprprop 32189 | Composition of two pairs o... |
| gtiso 32190 | Two ways to write a strict... |
| isoun 32191 | Infer an isomorphism from ... |
| disjdsct 32192 | A disjoint collection is d... |
| df1stres 32193 | Definition for a restricti... |
| df2ndres 32194 | Definition for a restricti... |
| 1stpreimas 32195 | The preimage of a singleto... |
| 1stpreima 32196 | The preimage by ` 1st ` is... |
| 2ndpreima 32197 | The preimage by ` 2nd ` is... |
| curry2ima 32198 | The image of a curried fun... |
| preiman0 32199 | The preimage of a nonempty... |
| intimafv 32200 | The intersection of an ima... |
| ecref 32201 | All elements are in their ... |
| supssd 32202 | Inequality deduction for s... |
| infssd 32203 | Inequality deduction for i... |
| imafi2 32204 | The image by a finite set ... |
| unifi3 32205 | If a union is finite, then... |
| snct 32206 | A singleton is countable. ... |
| prct 32207 | An unordered pair is count... |
| mpocti 32208 | An operation is countable ... |
| abrexct 32209 | An image set of a countabl... |
| mptctf 32210 | A countable mapping set is... |
| abrexctf 32211 | An image set of a countabl... |
| padct 32212 | Index a countable set with... |
| cnvoprabOLD 32213 | The converse of a class ab... |
| f1od2 32214 | Sufficient condition for a... |
| fcobij 32215 | Composing functions with a... |
| fcobijfs 32216 | Composing finitely support... |
| suppss3 32217 | Deduce a function's suppor... |
| fsuppcurry1 32218 | Finite support of a currie... |
| fsuppcurry2 32219 | Finite support of a currie... |
| offinsupp1 32220 | Finite support for a funct... |
| ffs2 32221 | Rewrite a function's suppo... |
| ffsrn 32222 | The range of a finitely su... |
| resf1o 32223 | Restriction of functions t... |
| maprnin 32224 | Restricting the range of t... |
| fpwrelmapffslem 32225 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 32226 | Define a canonical mapping... |
| fpwrelmapffs 32227 | Define a canonical mapping... |
| creq0 32228 | The real representation of... |
| 1nei 32229 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 32230 | An integer unit times itse... |
| nnmulge 32231 | Multiplying by a positive ... |
| lt2addrd 32232 | If the right-hand side of ... |
| xrlelttric 32233 | Trichotomy law for extende... |
| xaddeq0 32234 | Two extended reals which a... |
| xrinfm 32235 | The extended real numbers ... |
| le2halvesd 32236 | A sum is less than the who... |
| xraddge02 32237 | A number is less than or e... |
| xrge0addge 32238 | A number is less than or e... |
| xlt2addrd 32239 | If the right-hand side of ... |
| xrsupssd 32240 | Inequality deduction for s... |
| xrge0infss 32241 | Any subset of nonnegative ... |
| xrge0infssd 32242 | Inequality deduction for i... |
| xrge0addcld 32243 | Nonnegative extended reals... |
| xrge0subcld 32244 | Condition for closure of n... |
| infxrge0lb 32245 | A member of a set of nonne... |
| infxrge0glb 32246 | The infimum of a set of no... |
| infxrge0gelb 32247 | The infimum of a set of no... |
| xrofsup 32248 | The supremum is preserved ... |
| supxrnemnf 32249 | The supremum of a nonempty... |
| xnn0gt0 32250 | Nonzero extended nonnegati... |
| xnn01gt 32251 | An extended nonnegative in... |
| nn0xmulclb 32252 | Finite multiplication in t... |
| joiniooico 32253 | Disjoint joining an open i... |
| ubico 32254 | A right-open interval does... |
| xeqlelt 32255 | Equality in terms of 'less... |
| eliccelico 32256 | Relate elementhood to a cl... |
| elicoelioo 32257 | Relate elementhood to a cl... |
| iocinioc2 32258 | Intersection between two o... |
| xrdifh 32259 | Class difference of a half... |
| iocinif 32260 | Relate intersection of two... |
| difioo 32261 | The difference between two... |
| difico 32262 | The difference between two... |
| uzssico 32263 | Upper integer sets are a s... |
| fz2ssnn0 32264 | A finite set of sequential... |
| nndiffz1 32265 | Upper set of the positive ... |
| ssnnssfz 32266 | For any finite subset of `... |
| fzne1 32267 | Elementhood in a finite se... |
| fzm1ne1 32268 | Elementhood of an integer ... |
| fzspl 32269 | Split the last element of ... |
| fzdif2 32270 | Split the last element of ... |
| fzodif2 32271 | Split the last element of ... |
| fzodif1 32272 | Set difference of two half... |
| fzsplit3 32273 | Split a finite interval of... |
| bcm1n 32274 | The proportion of one bino... |
| iundisjfi 32275 | Rewrite a countable union ... |
| iundisj2fi 32276 | A disjoint union is disjoi... |
| iundisjcnt 32277 | Rewrite a countable union ... |
| iundisj2cnt 32278 | A countable disjoint union... |
| fzone1 32279 | Elementhood in a half-open... |
| fzom1ne1 32280 | Elementhood in a half-open... |
| f1ocnt 32281 | Given a countable set ` A ... |
| fz1nnct 32282 | NN and integer ranges star... |
| fz1nntr 32283 | NN and integer ranges star... |
| nn0difffzod 32284 | A nonnegative integer that... |
| suppssnn0 32285 | Show that the support of a... |
| hashunif 32286 | The cardinality of a disjo... |
| hashxpe 32287 | The size of the Cartesian ... |
| hashgt1 32288 | Restate "set contains at l... |
| dvdszzq 32289 | Divisibility for an intege... |
| prmdvdsbc 32290 | Condition for a prime numb... |
| numdenneg 32291 | Numerator and denominator ... |
| divnumden2 32292 | Calculate the reduced form... |
| nnindf 32293 | Principle of Mathematical ... |
| nn0min 32294 | Extracting the minimum pos... |
| subne0nn 32295 | A nonnegative difference i... |
| ltesubnnd 32296 | Subtracting an integer num... |
| fprodeq02 32297 | If one of the factors is z... |
| pr01ssre 32298 | The range of the indicator... |
| fprodex01 32299 | A product of factors equal... |
| prodpr 32300 | A product over a pair is t... |
| prodtp 32301 | A product over a triple is... |
| fsumub 32302 | An upper bound for a term ... |
| fsumiunle 32303 | Upper bound for a sum of n... |
| dfdec100 32304 | Split the hundreds from a ... |
| dp2eq1 32307 | Equality theorem for the d... |
| dp2eq2 32308 | Equality theorem for the d... |
| dp2eq1i 32309 | Equality theorem for the d... |
| dp2eq2i 32310 | Equality theorem for the d... |
| dp2eq12i 32311 | Equality theorem for the d... |
| dp20u 32312 | Add a zero in the tenths (... |
| dp20h 32313 | Add a zero in the unit pla... |
| dp2cl 32314 | Closure for the decimal fr... |
| dp2clq 32315 | Closure for a decimal frac... |
| rpdp2cl 32316 | Closure for a decimal frac... |
| rpdp2cl2 32317 | Closure for a decimal frac... |
| dp2lt10 32318 | Decimal fraction builds re... |
| dp2lt 32319 | Comparing two decimal frac... |
| dp2ltsuc 32320 | Comparing a decimal fracti... |
| dp2ltc 32321 | Comparing two decimal expa... |
| dpval 32324 | Define the value of the de... |
| dpcl 32325 | Prove that the closure of ... |
| dpfrac1 32326 | Prove a simple equivalence... |
| dpval2 32327 | Value of the decimal point... |
| dpval3 32328 | Value of the decimal point... |
| dpmul10 32329 | Multiply by 10 a decimal e... |
| decdiv10 32330 | Divide a decimal number by... |
| dpmul100 32331 | Multiply by 100 a decimal ... |
| dp3mul10 32332 | Multiply by 10 a decimal e... |
| dpmul1000 32333 | Multiply by 1000 a decimal... |
| dpval3rp 32334 | Value of the decimal point... |
| dp0u 32335 | Add a zero in the tenths p... |
| dp0h 32336 | Remove a zero in the units... |
| rpdpcl 32337 | Closure of the decimal poi... |
| dplt 32338 | Comparing two decimal expa... |
| dplti 32339 | Comparing a decimal expans... |
| dpgti 32340 | Comparing a decimal expans... |
| dpltc 32341 | Comparing two decimal inte... |
| dpexpp1 32342 | Add one zero to the mantis... |
| 0dp2dp 32343 | Multiply by 10 a decimal e... |
| dpadd2 32344 | Addition with one decimal,... |
| dpadd 32345 | Addition with one decimal.... |
| dpadd3 32346 | Addition with two decimals... |
| dpmul 32347 | Multiplication with one de... |
| dpmul4 32348 | An upper bound to multipli... |
| threehalves 32349 | Example theorem demonstrat... |
| 1mhdrd 32350 | Example theorem demonstrat... |
| xdivval 32353 | Value of division: the (un... |
| xrecex 32354 | Existence of reciprocal of... |
| xmulcand 32355 | Cancellation law for exten... |
| xreceu 32356 | Existential uniqueness of ... |
| xdivcld 32357 | Closure law for the extend... |
| xdivcl 32358 | Closure law for the extend... |
| xdivmul 32359 | Relationship between divis... |
| rexdiv 32360 | The extended real division... |
| xdivrec 32361 | Relationship between divis... |
| xdivid 32362 | A number divided by itself... |
| xdiv0 32363 | Division into zero is zero... |
| xdiv0rp 32364 | Division into zero is zero... |
| eliccioo 32365 | Membership in a closed int... |
| elxrge02 32366 | Elementhood in the set of ... |
| xdivpnfrp 32367 | Plus infinity divided by a... |
| rpxdivcld 32368 | Closure law for extended d... |
| xrpxdivcld 32369 | Closure law for extended d... |
| wrdfd 32370 | A word is a zero-based seq... |
| wrdres 32371 | Condition for the restrict... |
| wrdsplex 32372 | Existence of a split of a ... |
| pfx1s2 32373 | The prefix of length 1 of ... |
| pfxrn2 32374 | The range of a prefix of a... |
| pfxrn3 32375 | Express the range of a pre... |
| pfxf1 32376 | Condition for a prefix to ... |
| s1f1 32377 | Conditions for a length 1 ... |
| s2rn 32378 | Range of a length 2 string... |
| s2f1 32379 | Conditions for a length 2 ... |
| s3rn 32380 | Range of a length 3 string... |
| s3f1 32381 | Conditions for a length 3 ... |
| s3clhash 32382 | Closure of the words of le... |
| ccatf1 32383 | Conditions for a concatena... |
| pfxlsw2ccat 32384 | Reconstruct a word from it... |
| wrdt2ind 32385 | Perform an induction over ... |
| swrdrn2 32386 | The range of a subword is ... |
| swrdrn3 32387 | Express the range of a sub... |
| swrdf1 32388 | Condition for a subword to... |
| swrdrndisj 32389 | Condition for the range of... |
| splfv3 32390 | Symbols to the right of a ... |
| 1cshid 32391 | Cyclically shifting a sing... |
| cshw1s2 32392 | Cyclically shifting a leng... |
| cshwrnid 32393 | Cyclically shifting a word... |
| cshf1o 32394 | Condition for the cyclic s... |
| ressplusf 32395 | The group operation functi... |
| ressnm 32396 | The norm in a restricted s... |
| abvpropd2 32397 | Weaker version of ~ abvpro... |
| oppgle 32398 | less-than relation of an o... |
| oppgleOLD 32399 | Obsolete version of ~ oppg... |
| oppglt 32400 | less-than relation of an o... |
| ressprs 32401 | The restriction of a prose... |
| oduprs 32402 | Being a proset is a self-d... |
| posrasymb 32403 | A poset ordering is asymet... |
| resspos 32404 | The restriction of a Poset... |
| resstos 32405 | The restriction of a Toset... |
| odutos 32406 | Being a toset is a self-du... |
| tlt2 32407 | In a Toset, two elements m... |
| tlt3 32408 | In a Toset, two elements m... |
| trleile 32409 | In a Toset, two elements m... |
| toslublem 32410 | Lemma for ~ toslub and ~ x... |
| toslub 32411 | In a toset, the lowest upp... |
| tosglblem 32412 | Lemma for ~ tosglb and ~ x... |
| tosglb 32413 | Same theorem as ~ toslub ,... |
| clatp0cl 32414 | The poset zero of a comple... |
| clatp1cl 32415 | The poset one of a complet... |
| mntoval 32420 | Operation value of the mon... |
| ismnt 32421 | Express the statement " ` ... |
| ismntd 32422 | Property of being a monoto... |
| mntf 32423 | A monotone function is a f... |
| mgcoval 32424 | Operation value of the mon... |
| mgcval 32425 | Monotone Galois connection... |
| mgcf1 32426 | The lower adjoint ` F ` of... |
| mgcf2 32427 | The upper adjoint ` G ` of... |
| mgccole1 32428 | An inequality for the kern... |
| mgccole2 32429 | Inequality for the closure... |
| mgcmnt1 32430 | The lower adjoint ` F ` of... |
| mgcmnt2 32431 | The upper adjoint ` G ` of... |
| mgcmntco 32432 | A Galois connection like s... |
| dfmgc2lem 32433 | Lemma for dfmgc2, backward... |
| dfmgc2 32434 | Alternate definition of th... |
| mgcmnt1d 32435 | Galois connection implies ... |
| mgcmnt2d 32436 | Galois connection implies ... |
| mgccnv 32437 | The inverse Galois connect... |
| pwrssmgc 32438 | Given a function ` F ` , e... |
| mgcf1olem1 32439 | Property of a Galois conne... |
| mgcf1olem2 32440 | Property of a Galois conne... |
| mgcf1o 32441 | Given a Galois connection,... |
| xrs0 32444 | The zero of the extended r... |
| xrslt 32445 | The "strictly less than" r... |
| xrsinvgval 32446 | The inversion operation in... |
| xrsmulgzz 32447 | The "multiple" function in... |
| xrstos 32448 | The extended real numbers ... |
| xrsclat 32449 | The extended real numbers ... |
| xrsp0 32450 | The poset 0 of the extende... |
| xrsp1 32451 | The poset 1 of the extende... |
| ressmulgnn 32452 | Values for the group multi... |
| ressmulgnn0 32453 | Values for the group multi... |
| xrge0base 32454 | The base of the extended n... |
| xrge00 32455 | The zero of the extended n... |
| xrge0plusg 32456 | The additive law of the ex... |
| xrge0le 32457 | The "less than or equal to... |
| xrge0mulgnn0 32458 | The group multiple functio... |
| xrge0addass 32459 | Associativity of extended ... |
| xrge0addgt0 32460 | The sum of nonnegative and... |
| xrge0adddir 32461 | Right-distributivity of ex... |
| xrge0adddi 32462 | Left-distributivity of ext... |
| xrge0npcan 32463 | Extended nonnegative real ... |
| fsumrp0cl 32464 | Closure of a finite sum of... |
| abliso 32465 | The image of an Abelian gr... |
| lmhmghmd 32466 | A module homomorphism is a... |
| mhmimasplusg 32467 | Value of the operation of ... |
| lmhmimasvsca 32468 | Value of the scalar produc... |
| gsumsubg 32469 | The group sum in a subgrou... |
| gsumsra 32470 | The group sum in a subring... |
| gsummpt2co 32471 | Split a finite sum into a ... |
| gsummpt2d 32472 | Express a finite sum over ... |
| lmodvslmhm 32473 | Scalar multiplication in a... |
| gsumvsmul1 32474 | Pull a scalar multiplicati... |
| gsummptres 32475 | Extend a finite group sum ... |
| gsummptres2 32476 | Extend a finite group sum ... |
| gsumzresunsn 32477 | Append an element to a fin... |
| gsumpart 32478 | Express a group sum as a d... |
| gsumhashmul 32479 | Express a group sum by gro... |
| xrge0tsmsd 32480 | Any finite or infinite sum... |
| xrge0tsmsbi 32481 | Any limit of a finite or i... |
| xrge0tsmseq 32482 | Any limit of a finite or i... |
| cntzun 32483 | The centralizer of a union... |
| cntzsnid 32484 | The centralizer of the ide... |
| cntrcrng 32485 | The center of a ring is a ... |
| isomnd 32490 | A (left) ordered monoid is... |
| isogrp 32491 | A (left-)ordered group is ... |
| ogrpgrp 32492 | A left-ordered group is a ... |
| omndmnd 32493 | A left-ordered monoid is a... |
| omndtos 32494 | A left-ordered monoid is a... |
| omndadd 32495 | In an ordered monoid, the ... |
| omndaddr 32496 | In a right ordered monoid,... |
| omndadd2d 32497 | In a commutative left orde... |
| omndadd2rd 32498 | In a left- and right- orde... |
| submomnd 32499 | A submonoid of an ordered ... |
| xrge0omnd 32500 | The nonnegative extended r... |
| omndmul2 32501 | In an ordered monoid, the ... |
| omndmul3 32502 | In an ordered monoid, the ... |
| omndmul 32503 | In a commutative ordered m... |
| ogrpinv0le 32504 | In an ordered group, the o... |
| ogrpsub 32505 | In an ordered group, the o... |
| ogrpaddlt 32506 | In an ordered group, stric... |
| ogrpaddltbi 32507 | In a right ordered group, ... |
| ogrpaddltrd 32508 | In a right ordered group, ... |
| ogrpaddltrbid 32509 | In a right ordered group, ... |
| ogrpsublt 32510 | In an ordered group, stric... |
| ogrpinv0lt 32511 | In an ordered group, the o... |
| ogrpinvlt 32512 | In an ordered group, the o... |
| gsumle 32513 | A finite sum in an ordered... |
| symgfcoeu 32514 | Uniqueness property of per... |
| symgcom 32515 | Two permutations ` X ` and... |
| symgcom2 32516 | Two permutations ` X ` and... |
| symgcntz 32517 | All elements of a (finite)... |
| odpmco 32518 | The composition of two odd... |
| symgsubg 32519 | The value of the group sub... |
| pmtrprfv2 32520 | In a transposition of two ... |
| pmtrcnel 32521 | Composing a permutation ` ... |
| pmtrcnel2 32522 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 32523 | Composing a permutation ` ... |
| pmtridf1o 32524 | Transpositions of ` X ` an... |
| pmtridfv1 32525 | Value at X of the transpos... |
| pmtridfv2 32526 | Value at Y of the transpos... |
| psgnid 32527 | Permutation sign of the id... |
| psgndmfi 32528 | For a finite base set, the... |
| pmtrto1cl 32529 | Useful lemma for the follo... |
| psgnfzto1stlem 32530 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 32531 | Value of our permutation `... |
| fzto1st1 32532 | Special case where the per... |
| fzto1st 32533 | The function moving one el... |
| fzto1stinvn 32534 | Value of the inverse of ou... |
| psgnfzto1st 32535 | The permutation sign for m... |
| tocycval 32538 | Value of the cycle builder... |
| tocycfv 32539 | Function value of a permut... |
| tocycfvres1 32540 | A cyclic permutation is a ... |
| tocycfvres2 32541 | A cyclic permutation is th... |
| cycpmfvlem 32542 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 32543 | Value of a cycle function ... |
| cycpmfv2 32544 | Value of a cycle function ... |
| cycpmfv3 32545 | Values outside of the orbi... |
| cycpmcl 32546 | Cyclic permutations are pe... |
| tocycf 32547 | The permutation cycle buil... |
| tocyc01 32548 | Permutation cycles built f... |
| cycpm2tr 32549 | A cyclic permutation of 2 ... |
| cycpm2cl 32550 | Closure for the 2-cycles. ... |
| cyc2fv1 32551 | Function value of a 2-cycl... |
| cyc2fv2 32552 | Function value of a 2-cycl... |
| trsp2cyc 32553 | Exhibit the word a transpo... |
| cycpmco2f1 32554 | The word U used in ~ cycpm... |
| cycpmco2rn 32555 | The orbit of the compositi... |
| cycpmco2lem1 32556 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 32557 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 32558 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 32559 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 32560 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 32561 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 32562 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 32563 | The composition of a cycli... |
| cyc2fvx 32564 | Function value of a 2-cycl... |
| cycpm3cl 32565 | Closure of the 3-cycles in... |
| cycpm3cl2 32566 | Closure of the 3-cycles in... |
| cyc3fv1 32567 | Function value of a 3-cycl... |
| cyc3fv2 32568 | Function value of a 3-cycl... |
| cyc3fv3 32569 | Function value of a 3-cycl... |
| cyc3co2 32570 | Represent a 3-cycle as a c... |
| cycpmconjvlem 32571 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 32572 | A formula for computing co... |
| cycpmrn 32573 | The range of the word used... |
| tocyccntz 32574 | All elements of a (finite)... |
| evpmval 32575 | Value of the set of even p... |
| cnmsgn0g 32576 | The neutral element of the... |
| evpmsubg 32577 | The alternating group is a... |
| evpmid 32578 | The identity is an even pe... |
| altgnsg 32579 | The alternating group ` ( ... |
| cyc3evpm 32580 | 3-Cycles are even permutat... |
| cyc3genpmlem 32581 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 32582 | The alternating group ` A ... |
| cycpmgcl 32583 | Cyclic permutations are pe... |
| cycpmconjslem1 32584 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 32585 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 32586 | All cycles of the same len... |
| cyc3conja 32587 | All 3-cycles are conjugate... |
| sgnsv 32590 | The sign mapping. (Contri... |
| sgnsval 32591 | The sign value. (Contribu... |
| sgnsf 32592 | The sign function. (Contr... |
| inftmrel 32597 | The infinitesimal relation... |
| isinftm 32598 | Express ` x ` is infinites... |
| isarchi 32599 | Express the predicate " ` ... |
| pnfinf 32600 | Plus infinity is an infini... |
| xrnarchi 32601 | The completed real line is... |
| isarchi2 32602 | Alternative way to express... |
| submarchi 32603 | A submonoid is archimedean... |
| isarchi3 32604 | This is the usual definiti... |
| archirng 32605 | Property of Archimedean or... |
| archirngz 32606 | Property of Archimedean le... |
| archiexdiv 32607 | In an Archimedean group, g... |
| archiabllem1a 32608 | Lemma for ~ archiabl : In... |
| archiabllem1b 32609 | Lemma for ~ archiabl . (C... |
| archiabllem1 32610 | Archimedean ordered groups... |
| archiabllem2a 32611 | Lemma for ~ archiabl , whi... |
| archiabllem2c 32612 | Lemma for ~ archiabl . (C... |
| archiabllem2b 32613 | Lemma for ~ archiabl . (C... |
| archiabllem2 32614 | Archimedean ordered groups... |
| archiabl 32615 | Archimedean left- and righ... |
| isslmd 32618 | The predicate "is a semimo... |
| slmdlema 32619 | Lemma for properties of a ... |
| lmodslmd 32620 | Left semimodules generaliz... |
| slmdcmn 32621 | A semimodule is a commutat... |
| slmdmnd 32622 | A semimodule is a monoid. ... |
| slmdsrg 32623 | The scalar component of a ... |
| slmdbn0 32624 | The base set of a semimodu... |
| slmdacl 32625 | Closure of ring addition f... |
| slmdmcl 32626 | Closure of ring multiplica... |
| slmdsn0 32627 | The set of scalars in a se... |
| slmdvacl 32628 | Closure of vector addition... |
| slmdass 32629 | Semiring left module vecto... |
| slmdvscl 32630 | Closure of scalar product ... |
| slmdvsdi 32631 | Distributive law for scala... |
| slmdvsdir 32632 | Distributive law for scala... |
| slmdvsass 32633 | Associative law for scalar... |
| slmd0cl 32634 | The ring zero in a semimod... |
| slmd1cl 32635 | The ring unity in a semiri... |
| slmdvs1 32636 | Scalar product with ring u... |
| slmd0vcl 32637 | The zero vector is a vecto... |
| slmd0vlid 32638 | Left identity law for the ... |
| slmd0vrid 32639 | Right identity law for the... |
| slmd0vs 32640 | Zero times a vector is the... |
| slmdvs0 32641 | Anything times the zero ve... |
| gsumvsca1 32642 | Scalar product of a finite... |
| gsumvsca2 32643 | Scalar product of a finite... |
| prmsimpcyc 32644 | A group of prime order is ... |
| idomdomd 32645 | An integral domain is a do... |
| idomringd 32646 | An integral domain is a ri... |
| domnlcan 32647 | Left-cancellation law for ... |
| idomrcan 32648 | Right-cancellation law for... |
| urpropd 32649 | Sufficient condition for r... |
| 0ringsubrg 32650 | A subring of a zero ring i... |
| dvdschrmulg 32651 | In a ring, any multiple of... |
| freshmansdream 32652 | For a prime number ` P ` ,... |
| frobrhm 32653 | In a commutative ring with... |
| ress1r 32654 | ` 1r ` is unaffected by re... |
| ringinvval 32655 | The ring inverse expressed... |
| dvrcan5 32656 | Cancellation law for commo... |
| subrgchr 32657 | If ` A ` is a subring of `... |
| rmfsupp2 32658 | A mapping of a multiplicat... |
| eufndx 32661 | Index value of the Euclide... |
| eufid 32662 | Utility theorem: index-ind... |
| ringinveu 32665 | If a ring unit element ` X... |
| isdrng4 32666 | A division ring is a ring ... |
| rndrhmcl 32667 | The image of a division ri... |
| sdrgdvcl 32668 | A sub-division-ring is clo... |
| sdrginvcl 32669 | A sub-division-ring is clo... |
| primefldchr 32670 | The characteristic of a pr... |
| fldgenval 32673 | Value of the field generat... |
| fldgenssid 32674 | The field generated by a s... |
| fldgensdrg 32675 | A generated subfield is a ... |
| fldgenssv 32676 | A generated subfield is a ... |
| fldgenss 32677 | Generated subfields preser... |
| fldgenidfld 32678 | The subfield generated by ... |
| fldgenssp 32679 | The field generated by a s... |
| fldgenid 32680 | The subfield of a field ` ... |
| fldgenfld 32681 | A generated subfield is a ... |
| primefldgen1 32682 | The prime field of a divis... |
| 1fldgenq 32683 | The field of rational numb... |
| isorng 32688 | An ordered ring is a ring ... |
| orngring 32689 | An ordered ring is a ring.... |
| orngogrp 32690 | An ordered ring is an orde... |
| isofld 32691 | An ordered field is a fiel... |
| orngmul 32692 | In an ordered ring, the or... |
| orngsqr 32693 | In an ordered ring, all sq... |
| ornglmulle 32694 | In an ordered ring, multip... |
| orngrmulle 32695 | In an ordered ring, multip... |
| ornglmullt 32696 | In an ordered ring, multip... |
| orngrmullt 32697 | In an ordered ring, multip... |
| orngmullt 32698 | In an ordered ring, the st... |
| ofldfld 32699 | An ordered field is a fiel... |
| ofldtos 32700 | An ordered field is a tota... |
| orng0le1 32701 | In an ordered ring, the ri... |
| ofldlt1 32702 | In an ordered field, the r... |
| ofldchr 32703 | The characteristic of an o... |
| suborng 32704 | Every subring of an ordere... |
| subofld 32705 | Every subfield of an order... |
| isarchiofld 32706 | Axiom of Archimedes : a ch... |
| rhmdvd 32707 | A ring homomorphism preser... |
| kerunit 32708 | If a unit element lies in ... |
| reldmresv 32711 | The scalar restriction is ... |
| resvval 32712 | Value of structure restric... |
| resvid2 32713 | General behavior of trivia... |
| resvval2 32714 | Value of nontrivial struct... |
| resvsca 32715 | Base set of a structure re... |
| resvlem 32716 | Other elements of a scalar... |
| resvlemOLD 32717 | Obsolete version of ~ resv... |
| resvbas 32718 | ` Base ` is unaffected by ... |
| resvbasOLD 32719 | Obsolete proof of ~ resvba... |
| resvplusg 32720 | ` +g ` is unaffected by sc... |
| resvplusgOLD 32721 | Obsolete proof of ~ resvpl... |
| resvvsca 32722 | ` .s ` is unaffected by sc... |
| resvvscaOLD 32723 | Obsolete proof of ~ resvvs... |
| resvmulr 32724 | ` .r ` is unaffected by sc... |
| resvmulrOLD 32725 | Obsolete proof of ~ resvmu... |
| resv0g 32726 | ` 0g ` is unaffected by sc... |
| resv1r 32727 | ` 1r ` is unaffected by sc... |
| resvcmn 32728 | Scalar restriction preserv... |
| gzcrng 32729 | The gaussian integers form... |
| reofld 32730 | The real numbers form an o... |
| nn0omnd 32731 | The nonnegative integers f... |
| rearchi 32732 | The field of the real numb... |
| nn0archi 32733 | The monoid of the nonnegat... |
| xrge0slmod 32734 | The extended nonnegative r... |
| qusker 32735 | The kernel of a quotient m... |
| eqgvscpbl 32736 | The left coset equivalence... |
| qusvscpbl 32737 | The quotient map distribut... |
| qusvsval 32738 | Value of the scalar multip... |
| imaslmod 32739 | The image structure of a l... |
| imasmhm 32740 | Given a function ` F ` wit... |
| imasghm 32741 | Given a function ` F ` wit... |
| imasrhm 32742 | Given a function ` F ` wit... |
| imaslmhm 32743 | Given a function ` F ` wit... |
| quslmod 32744 | If ` G ` is a submodule in... |
| quslmhm 32745 | If ` G ` is a submodule of... |
| quslvec 32746 | If ` S ` is a vector subsp... |
| ecxpid 32747 | The equivalence class of a... |
| eqg0el 32748 | Equivalence class of a quo... |
| qsxpid 32749 | The quotient set of a cart... |
| qusxpid 32750 | The Group quotient equival... |
| qustriv 32751 | The quotient of a group ` ... |
| qustrivr 32752 | Converse of ~ qustriv . (... |
| fermltlchr 32753 | A generalization of Fermat... |
| znfermltl 32754 | Fermat's little theorem in... |
| islinds5 32755 | A set is linearly independ... |
| ellspds 32756 | Variation on ~ ellspd . (... |
| 0ellsp 32757 | Zero is in all spans. (Co... |
| 0nellinds 32758 | The group identity cannot ... |
| rspsnel 32759 | Membership in a principal ... |
| rspsnid 32760 | A principal ideal contains... |
| elrsp 32761 | Write the elements of a ri... |
| rspidlid 32762 | The ideal span of an ideal... |
| pidlnz 32763 | A principal ideal generate... |
| dvdsruassoi 32764 | If two elements ` X ` and ... |
| dvdsruasso 32765 | Two elements ` X ` and ` Y... |
| dvdsrspss 32766 | In a ring, an element ` X ... |
| rspsnasso 32767 | Two elements ` X ` and ` Y... |
| lbslsp 32768 | Any element of a left modu... |
| lindssn 32769 | Any singleton of a nonzero... |
| lindflbs 32770 | Conditions for an independ... |
| islbs5 32771 | An equivalent formulation ... |
| linds2eq 32772 | Deduce equality of element... |
| lindfpropd 32773 | Property deduction for lin... |
| lindspropd 32774 | Property deduction for lin... |
| elgrplsmsn 32775 | Membership in a sumset wit... |
| lsmsnorb 32776 | The sumset of a group with... |
| lsmsnorb2 32777 | The sumset of a single ele... |
| elringlsm 32778 | Membership in a product of... |
| elringlsmd 32779 | Membership in a product of... |
| ringlsmss 32780 | Closure of the product of ... |
| ringlsmss1 32781 | The product of an ideal ` ... |
| ringlsmss2 32782 | The product with an ideal ... |
| lsmsnpridl 32783 | The product of the ring wi... |
| lsmsnidl 32784 | The product of the ring wi... |
| lsmidllsp 32785 | The sum of two ideals is t... |
| lsmidl 32786 | The sum of two ideals is a... |
| lsmssass 32787 | Group sum is associative, ... |
| grplsm0l 32788 | Sumset with the identity s... |
| grplsmid 32789 | The direct sum of an eleme... |
| qusmul 32790 | Value of the ring operatio... |
| quslsm 32791 | Express the image by the q... |
| qusbas2 32792 | Alternate definition of th... |
| qus0g 32793 | The identity element of a ... |
| qusima 32794 | The image of a subgroup by... |
| qusrn 32795 | The natural map from eleme... |
| nsgqus0 32796 | A normal subgroup ` N ` is... |
| nsgmgclem 32797 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 32798 | There is a monotone Galois... |
| nsgqusf1olem1 32799 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 32800 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 32801 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 32802 | The canonical projection h... |
| ghmquskerlem1 32803 | Lemma for ~ ghmqusker (Con... |
| ghmquskerco 32804 | In the case of theorem ~ g... |
| ghmquskerlem2 32805 | Lemma for ~ ghmqusker . (... |
| ghmquskerlem3 32806 | The mapping ` H ` induced ... |
| ghmqusker 32807 | A surjective group homomor... |
| gicqusker 32808 | The image ` H ` of a group... |
| lmhmqusker 32809 | A surjective module homomo... |
| lmicqusker 32810 | The image ` H ` of a modul... |
| intlidl 32811 | The intersection of a none... |
| rhmpreimaidl 32812 | The preimage of an ideal b... |
| kerlidl 32813 | The kernel of a ring homom... |
| 0ringidl 32814 | The zero ideal is the only... |
| pidlnzb 32815 | A principal ideal is nonze... |
| lidlunitel 32816 | If an ideal ` I ` contains... |
| unitpidl1 32817 | The ideal ` I ` generated ... |
| rhmquskerlem 32818 | The mapping ` J ` induced ... |
| rhmqusker 32819 | A surjective ring homomorp... |
| ricqusker 32820 | The image ` H ` of a ring ... |
| elrspunidl 32821 | Elementhood in the span of... |
| elrspunsn 32822 | Membership to the span of ... |
| lidlincl 32823 | Ideals are closed under in... |
| idlinsubrg 32824 | The intersection between a... |
| rhmimaidl 32825 | The image of an ideal ` I ... |
| drngidl 32826 | A nonzero ring is a divisi... |
| drngidlhash 32827 | A ring is a division ring ... |
| prmidlval 32830 | The class of prime ideals ... |
| isprmidl 32831 | The predicate "is a prime ... |
| prmidlnr 32832 | A prime ideal is a proper ... |
| prmidl 32833 | The main property of a pri... |
| prmidl2 32834 | A condition that shows an ... |
| idlmulssprm 32835 | Let ` P ` be a prime ideal... |
| pridln1 32836 | A proper ideal cannot cont... |
| prmidlidl 32837 | A prime ideal is an ideal.... |
| prmidlssidl 32838 | Prime ideals as a subset o... |
| lidlnsg 32839 | An ideal is a normal subgr... |
| cringm4 32840 | Commutative/associative la... |
| isprmidlc 32841 | The predicate "is prime id... |
| prmidlc 32842 | Property of a prime ideal ... |
| 0ringprmidl 32843 | The trivial ring does not ... |
| prmidl0 32844 | The zero ideal of a commut... |
| rhmpreimaprmidl 32845 | The preimage of a prime id... |
| qsidomlem1 32846 | If the quotient ring of a ... |
| qsidomlem2 32847 | A quotient by a prime idea... |
| qsidom 32848 | An ideal ` I ` in the comm... |
| qsnzr 32849 | A quotient of a non-zero r... |
| mxidlval 32852 | The set of maximal ideals ... |
| ismxidl 32853 | The predicate "is a maxima... |
| mxidlidl 32854 | A maximal ideal is an idea... |
| mxidlnr 32855 | A maximal ideal is proper.... |
| mxidlmax 32856 | A maximal ideal is a maxim... |
| mxidln1 32857 | One is not contained in an... |
| mxidlnzr 32858 | A ring with a maximal idea... |
| mxidlmaxv 32859 | An ideal ` I ` strictly co... |
| crngmxidl 32860 | In a commutative ring, max... |
| mxidlprm 32861 | Every maximal ideal is pri... |
| mxidlirredi 32862 | In an integral domain, the... |
| mxidlirred 32863 | In a principal ideal domai... |
| ssmxidllem 32864 | The set ` P ` used in the ... |
| ssmxidl 32865 | Let ` R ` be a ring, and l... |
| drnglidl1ne0 32866 | In a nonzero ring, the zer... |
| drng0mxidl 32867 | In a division ring, the ze... |
| drngmxidl 32868 | The zero ideal is the only... |
| krull 32869 | Krull's theorem: Any nonz... |
| mxidlnzrb 32870 | A ring is nonzero if and o... |
| opprabs 32871 | The opposite ring of the o... |
| oppreqg 32872 | Group coset equivalence re... |
| opprnsg 32873 | Normal subgroups of the op... |
| opprlidlabs 32874 | The ideals of the opposite... |
| oppr2idl 32875 | Two sided ideal of the opp... |
| opprmxidlabs 32876 | The maximal ideal of the o... |
| opprqusbas 32877 | The base of the quotient o... |
| opprqusplusg 32878 | The group operation of the... |
| opprqus0g 32879 | The group identity element... |
| opprqusmulr 32880 | The multiplication operati... |
| opprqus1r 32881 | The ring unity of the quot... |
| opprqusdrng 32882 | The quotient of the opposi... |
| qsdrngilem 32883 | Lemma for ~ qsdrngi . (Co... |
| qsdrngi 32884 | A quotient by a maximal le... |
| qsdrnglem2 32885 | Lemma for ~ qsdrng . (Con... |
| qsdrng 32886 | An ideal ` M ` is both lef... |
| qsfld 32887 | An ideal ` M ` in the comm... |
| mxidlprmALT 32888 | Every maximal ideal is pri... |
| idlsrgstr 32891 | A constructed semiring of ... |
| idlsrgval 32892 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 32893 | Base of the ideals of a ri... |
| idlsrgplusg 32894 | Additive operation of the ... |
| idlsrg0g 32895 | The zero ideal is the addi... |
| idlsrgmulr 32896 | Multiplicative operation o... |
| idlsrgtset 32897 | Topology component of the ... |
| idlsrgmulrval 32898 | Value of the ring multipli... |
| idlsrgmulrcl 32899 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 32900 | In a commutative ring, the... |
| idlsrgmulrss2 32901 | The product of two ideals ... |
| idlsrgmulrssin 32902 | In a commutative ring, the... |
| idlsrgmnd 32903 | The ideals of a ring form ... |
| idlsrgcmnd 32904 | The ideals of a ring form ... |
| isufd 32907 | The property of being a Un... |
| rprmval 32908 | The prime elements of a ri... |
| isrprm 32909 | Property for ` P ` to be a... |
| asclmulg 32910 | Apply group multiplication... |
| 0ringmon1p 32911 | There are no monic polynom... |
| fply1 32912 | Conditions for a function ... |
| ply1lvec 32913 | In a division ring, the un... |
| ply1scleq 32914 | Equality of a constant pol... |
| evls1fn 32915 | Functionality of the subri... |
| evls1dm 32916 | The domain of the subring ... |
| evls1fvf 32917 | The subring evaluation fun... |
| evls1scafv 32918 | Value of the univariate po... |
| evls1expd 32919 | Univariate polynomial eval... |
| evls1varpwval 32920 | Univariate polynomial eval... |
| evls1fpws 32921 | Evaluation of a univariate... |
| ressply1evl 32922 | Evaluation of a univariate... |
| evls1addd 32923 | Univariate polynomial eval... |
| evls1muld 32924 | Univariate polynomial eval... |
| evls1vsca 32925 | Univariate polynomial eval... |
| ressdeg1 32926 | The degree of a univariate... |
| ply1ascl0 32927 | The zero scalar as a polyn... |
| ply1ascl1 32928 | The multiplicative unit sc... |
| ply1asclunit 32929 | A non-zero scalar polynomi... |
| deg1le0eq0 32930 | A polynomial with nonposit... |
| ressply10g 32931 | A restricted polynomial al... |
| ressply1mon1p 32932 | The monic polynomials of a... |
| ressply1invg 32933 | An element of a restricted... |
| ressply1sub 32934 | A restricted polynomial al... |
| asclply1subcl 32935 | Closure of the algebra sca... |
| ply1chr 32936 | The characteristic of a po... |
| ply1fermltlchr 32937 | Fermat's little theorem fo... |
| ply1fermltl 32938 | Fermat's little theorem fo... |
| coe1mon 32939 | Coefficient vector of a mo... |
| ply1moneq 32940 | Two monomials are equal if... |
| ply1degltel 32941 | Characterize elementhood i... |
| ply1degleel 32942 | Characterize elementhood i... |
| ply1degltlss 32943 | The space ` S ` of the uni... |
| gsummoncoe1fzo 32944 | A coefficient of the polyn... |
| ply1gsumz 32945 | If a polynomial given as a... |
| deg1addlt 32946 | If both factors have degre... |
| ig1pnunit 32947 | The polynomial ideal gener... |
| ig1pmindeg 32948 | The polynomial ideal gener... |
| q1pdir 32949 | Distribution of univariate... |
| q1pvsca 32950 | Scalar multiplication prop... |
| r1pvsca 32951 | Scalar multiplication prop... |
| r1p0 32952 | Polynomial remainder opera... |
| r1pcyc 32953 | The polynomial remainder o... |
| r1padd1 32954 | Addition property of the p... |
| r1pid2 32955 | Identity law for polynomia... |
| r1plmhm 32956 | The univariate polynomial ... |
| r1pquslmic 32957 | The univariate polynomial ... |
| sra1r 32958 | The unity element of a sub... |
| sradrng 32959 | Condition for a subring al... |
| srasubrg 32960 | A subring of the original ... |
| sralvec 32961 | Given a sub division ring ... |
| srafldlvec 32962 | Given a subfield ` F ` of ... |
| resssra 32963 | The subring algebra of a r... |
| lsssra 32964 | A subring is a subspace of... |
| drgext0g 32965 | The additive neutral eleme... |
| drgextvsca 32966 | The scalar multiplication ... |
| drgext0gsca 32967 | The additive neutral eleme... |
| drgextsubrg 32968 | The scalar field is a subr... |
| drgextlsp 32969 | The scalar field is a subs... |
| drgextgsum 32970 | Group sum in a division ri... |
| lvecdimfi 32971 | Finite version of ~ lvecdi... |
| dimval 32974 | The dimension of a vector ... |
| dimvalfi 32975 | The dimension of a vector ... |
| dimcl 32976 | Closure of the vector spac... |
| lmimdim 32977 | Module isomorphisms preser... |
| lmicdim 32978 | Module isomorphisms preser... |
| lvecdim0i 32979 | A vector space of dimensio... |
| lvecdim0 32980 | A vector space of dimensio... |
| lssdimle 32981 | The dimension of a linear ... |
| dimpropd 32982 | If two structures have the... |
| rlmdim 32983 | The left vector space indu... |
| rgmoddimOLD 32984 | Obsolete version of ~ rlmd... |
| frlmdim 32985 | Dimension of a free left m... |
| tnglvec 32986 | Augmenting a structure wit... |
| tngdim 32987 | Dimension of a left vector... |
| rrxdim 32988 | Dimension of the generaliz... |
| matdim 32989 | Dimension of the space of ... |
| lbslsat 32990 | A nonzero vector ` X ` is ... |
| lsatdim 32991 | A line, spanned by a nonze... |
| drngdimgt0 32992 | The dimension of a vector ... |
| lmhmlvec2 32993 | A homomorphism of left vec... |
| kerlmhm 32994 | The kernel of a vector spa... |
| imlmhm 32995 | The image of a vector spac... |
| ply1degltdimlem 32996 | Lemma for ~ ply1degltdim .... |
| ply1degltdim 32997 | The space ` S ` of the uni... |
| lindsunlem 32998 | Lemma for ~ lindsun . (Co... |
| lindsun 32999 | Condition for the union of... |
| lbsdiflsp0 33000 | The linear spans of two di... |
| dimkerim 33001 | Given a linear map ` F ` b... |
| qusdimsum 33002 | Let ` W ` be a vector spac... |
| fedgmullem1 33003 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 33004 | Lemma for ~ fedgmul . (Co... |
| fedgmul 33005 | The multiplicativity formu... |
| relfldext 33014 | The field extension is a r... |
| brfldext 33015 | The field extension relati... |
| ccfldextrr 33016 | The field of the complex n... |
| fldextfld1 33017 | A field extension is only ... |
| fldextfld2 33018 | A field extension is only ... |
| fldextsubrg 33019 | Field extension implies a ... |
| fldextress 33020 | Field extension implies a ... |
| brfinext 33021 | The finite field extension... |
| extdgval 33022 | Value of the field extensi... |
| fldextsralvec 33023 | The subring algebra associ... |
| extdgcl 33024 | Closure of the field exten... |
| extdggt0 33025 | Degrees of field extension... |
| fldexttr 33026 | Field extension is a trans... |
| fldextid 33027 | The field extension relati... |
| extdgid 33028 | A trivial field extension ... |
| extdgmul 33029 | The multiplicativity formu... |
| finexttrb 33030 | The extension ` E ` of ` K... |
| extdg1id 33031 | If the degree of the exten... |
| extdg1b 33032 | The degree of the extensio... |
| fldextchr 33033 | The characteristic of a su... |
| evls1fldgencl 33034 | Closure of the subring pol... |
| ccfldsrarelvec 33035 | The subring algebra of the... |
| ccfldextdgrr 33036 | The degree of the field ex... |
| irngval 33039 | The elements of a field ` ... |
| elirng 33040 | Property for an element ` ... |
| irngss 33041 | All elements of a subring ... |
| irngssv 33042 | An integral element is an ... |
| 0ringirng 33043 | A zero ring ` R ` has no i... |
| irngnzply1lem 33044 | In the case of a field ` E... |
| irngnzply1 33045 | In the case of a field ` E... |
| evls1fvcl 33048 | Variant of ~ fveval1fvcl f... |
| evls1maprhm 33049 | The function ` F ` mapping... |
| evls1maplmhm 33050 | The function ` F ` mapping... |
| evls1maprnss 33051 | The function ` F ` mapping... |
| ply1annidllem 33052 | Write the set ` Q ` of pol... |
| ply1annidl 33053 | The set ` Q ` of polynomia... |
| ply1annnr 33054 | The set ` Q ` of polynomia... |
| ply1annig1p 33055 | The ideal ` Q ` of polynom... |
| minplyval 33056 | Expand the value of the mi... |
| minplycl 33057 | The minimal polynomial is ... |
| ply1annprmidl 33058 | The set ` Q ` of polynomia... |
| minplyirredlem 33059 | Lemma for ~ minplyirred . ... |
| minplyirred 33060 | A nonzero minimal polynomi... |
| irngnminplynz 33061 | Integral elements have non... |
| minplym1p 33062 | A minimal polynomial is mo... |
| algextdeglem1 33063 | Lemma for ~ algextdeg . (... |
| algextdeglem2 33064 | Lemma for ~ algextdeg . (... |
| algextdeglem3 33065 | Lemma for ~ algextdeg . (... |
| algextdeglem4 33066 | Lemma for ~ algextdeg . (... |
| algextdeglem5 33067 | Lemma for ~ algextdeg . (... |
| algextdeglem6 33068 | Lemma for ~ algextdeg . (... |
| algextdeglem7 33069 | Lemma for ~ algextdeg . (... |
| algextdeglem8 33070 | Lemma for ~ algextdeg . (... |
| algextdeg 33071 | The degree of an algebraic... |
| smatfval 33074 | Value of the submatrix. (... |
| smatrcl 33075 | Closure of the rectangular... |
| smatlem 33076 | Lemma for the next theorem... |
| smattl 33077 | Entries of a submatrix, to... |
| smattr 33078 | Entries of a submatrix, to... |
| smatbl 33079 | Entries of a submatrix, bo... |
| smatbr 33080 | Entries of a submatrix, bo... |
| smatcl 33081 | Closure of the square subm... |
| matmpo 33082 | Write a square matrix as a... |
| 1smat1 33083 | The submatrix of the ident... |
| submat1n 33084 | One case where the submatr... |
| submatres 33085 | Special case where the sub... |
| submateqlem1 33086 | Lemma for ~ submateq . (C... |
| submateqlem2 33087 | Lemma for ~ submateq . (C... |
| submateq 33088 | Sufficient condition for t... |
| submatminr1 33089 | If we take a submatrix by ... |
| lmatval 33092 | Value of the literal matri... |
| lmatfval 33093 | Entries of a literal matri... |
| lmatfvlem 33094 | Useful lemma to extract li... |
| lmatcl 33095 | Closure of the literal mat... |
| lmat22lem 33096 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 33097 | Entry of a 2x2 literal mat... |
| lmat22e12 33098 | Entry of a 2x2 literal mat... |
| lmat22e21 33099 | Entry of a 2x2 literal mat... |
| lmat22e22 33100 | Entry of a 2x2 literal mat... |
| lmat22det 33101 | The determinant of a liter... |
| mdetpmtr1 33102 | The determinant of a matri... |
| mdetpmtr2 33103 | The determinant of a matri... |
| mdetpmtr12 33104 | The determinant of a matri... |
| mdetlap1 33105 | A Laplace expansion of the... |
| madjusmdetlem1 33106 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 33107 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 33108 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 33109 | Lemma for ~ madjusmdet . ... |
| madjusmdet 33110 | Express the cofactor of th... |
| mdetlap 33111 | Laplace expansion of the d... |
| ist0cld 33112 | The predicate "is a T_0 sp... |
| txomap 33113 | Given two open maps ` F ` ... |
| qtopt1 33114 | If every equivalence class... |
| qtophaus 33115 | If an open map's graph in ... |
| circtopn 33116 | The topology of the unit c... |
| circcn 33117 | The function gluing the re... |
| reff 33118 | For any cover refinement, ... |
| locfinreflem 33119 | A locally finite refinemen... |
| locfinref 33120 | A locally finite refinemen... |
| iscref 33123 | The property that every op... |
| crefeq 33124 | Equality theorem for the "... |
| creftop 33125 | A space where every open c... |
| crefi 33126 | The property that every op... |
| crefdf 33127 | A formulation of ~ crefi e... |
| crefss 33128 | The "every open cover has ... |
| cmpcref 33129 | Equivalent definition of c... |
| cmpfiref 33130 | Every open cover of a Comp... |
| ldlfcntref 33133 | Every open cover of a Lind... |
| ispcmp 33136 | The predicate "is a paraco... |
| cmppcmp 33137 | Every compact space is par... |
| dispcmp 33138 | Every discrete space is pa... |
| pcmplfin 33139 | Given a paracompact topolo... |
| pcmplfinf 33140 | Given a paracompact topolo... |
| rspecval 33143 | Value of the spectrum of t... |
| rspecbas 33144 | The prime ideals form the ... |
| rspectset 33145 | Topology component of the ... |
| rspectopn 33146 | The topology component of ... |
| zarcls0 33147 | The closure of the identit... |
| zarcls1 33148 | The unit ideal ` B ` is th... |
| zarclsun 33149 | The union of two closed se... |
| zarclsiin 33150 | In a Zariski topology, the... |
| zarclsint 33151 | The intersection of a fami... |
| zarclssn 33152 | The closed points of Zaris... |
| zarcls 33153 | The open sets of the Zaris... |
| zartopn 33154 | The Zariski topology is a ... |
| zartop 33155 | The Zariski topology is a ... |
| zartopon 33156 | The points of the Zariski ... |
| zar0ring 33157 | The Zariski Topology of th... |
| zart0 33158 | The Zariski topology is T_... |
| zarmxt1 33159 | The Zariski topology restr... |
| zarcmplem 33160 | Lemma for ~ zarcmp . (Con... |
| zarcmp 33161 | The Zariski topology is co... |
| rspectps 33162 | The spectrum of a ring ` R... |
| rhmpreimacnlem 33163 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 33164 | The function mapping a pri... |
| metidval 33169 | Value of the metric identi... |
| metidss 33170 | As a relation, the metric ... |
| metidv 33171 | ` A ` and ` B ` identify b... |
| metideq 33172 | Basic property of the metr... |
| metider 33173 | The metric identification ... |
| pstmval 33174 | Value of the metric induce... |
| pstmfval 33175 | Function value of the metr... |
| pstmxmet 33176 | The metric induced by a ps... |
| hauseqcn 33177 | In a Hausdorff topology, t... |
| elunitge0 33178 | An element of the closed u... |
| unitssxrge0 33179 | The closed unit interval i... |
| unitdivcld 33180 | Necessary conditions for a... |
| iistmd 33181 | The closed unit interval f... |
| unicls 33182 | The union of the closed se... |
| tpr2tp 33183 | The usual topology on ` ( ... |
| tpr2uni 33184 | The usual topology on ` ( ... |
| xpinpreima 33185 | Rewrite the cartesian prod... |
| xpinpreima2 33186 | Rewrite the cartesian prod... |
| sqsscirc1 33187 | The complex square of side... |
| sqsscirc2 33188 | The complex square of side... |
| cnre2csqlem 33189 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 33190 | Image of a centered square... |
| tpr2rico 33191 | For any point of an open s... |
| cnvordtrestixx 33192 | The restriction of the 'gr... |
| prsdm 33193 | Domain of the relation of ... |
| prsrn 33194 | Range of the relation of a... |
| prsss 33195 | Relation of a subproset. ... |
| prsssdm 33196 | Domain of a subproset rela... |
| ordtprsval 33197 | Value of the order topolog... |
| ordtprsuni 33198 | Value of the order topolog... |
| ordtcnvNEW 33199 | The order dual generates t... |
| ordtrestNEW 33200 | The subspace topology of a... |
| ordtrest2NEWlem 33201 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 33202 | An interval-closed set ` A... |
| ordtconnlem1 33203 | Connectedness in the order... |
| ordtconn 33204 | Connectedness in the order... |
| mndpluscn 33205 | A mapping that is both a h... |
| mhmhmeotmd 33206 | Deduce a Topological Monoi... |
| rmulccn 33207 | Multiplication by a real c... |
| raddcn 33208 | Addition in the real numbe... |
| xrmulc1cn 33209 | The operation multiplying ... |
| fmcncfil 33210 | The image of a Cauchy filt... |
| xrge0hmph 33211 | The extended nonnegative r... |
| xrge0iifcnv 33212 | Define a bijection from ` ... |
| xrge0iifcv 33213 | The defined function's val... |
| xrge0iifiso 33214 | The defined bijection from... |
| xrge0iifhmeo 33215 | Expose a homeomorphism fro... |
| xrge0iifhom 33216 | The defined function from ... |
| xrge0iif1 33217 | Condition for the defined ... |
| xrge0iifmhm 33218 | The defined function from ... |
| xrge0pluscn 33219 | The addition operation of ... |
| xrge0mulc1cn 33220 | The operation multiplying ... |
| xrge0tps 33221 | The extended nonnegative r... |
| xrge0topn 33222 | The topology of the extend... |
| xrge0haus 33223 | The topology of the extend... |
| xrge0tmd 33224 | The extended nonnegative r... |
| xrge0tmdALT 33225 | Alternate proof of ~ xrge0... |
| lmlim 33226 | Relate a limit in a given ... |
| lmlimxrge0 33227 | Relate a limit in the nonn... |
| rge0scvg 33228 | Implication of convergence... |
| fsumcvg4 33229 | A serie with finite suppor... |
| pnfneige0 33230 | A neighborhood of ` +oo ` ... |
| lmxrge0 33231 | Express "sequence ` F ` co... |
| lmdvg 33232 | If a monotonic sequence of... |
| lmdvglim 33233 | If a monotonic real number... |
| pl1cn 33234 | A univariate polynomial is... |
| zringnm 33237 | The norm (function) for a ... |
| zzsnm 33238 | The norm of the ring of th... |
| zlm0 33239 | Zero of a ` ZZ ` -module. ... |
| zlm1 33240 | Unity element of a ` ZZ ` ... |
| zlmds 33241 | Distance in a ` ZZ ` -modu... |
| zlmdsOLD 33242 | Obsolete proof of ~ zlmds ... |
| zlmtset 33243 | Topology in a ` ZZ ` -modu... |
| zlmtsetOLD 33244 | Obsolete proof of ~ zlmtse... |
| zlmnm 33245 | Norm of a ` ZZ ` -module (... |
| zhmnrg 33246 | The ` ZZ ` -module built f... |
| nmmulg 33247 | The norm of a group produc... |
| zrhnm 33248 | The norm of the image by `... |
| cnzh 33249 | The ` ZZ ` -module of ` CC... |
| rezh 33250 | The ` ZZ ` -module of ` RR... |
| qqhval 33253 | Value of the canonical hom... |
| zrhf1ker 33254 | The kernel of the homomorp... |
| zrhchr 33255 | The kernel of the homomorp... |
| zrhker 33256 | The kernel of the homomorp... |
| zrhunitpreima 33257 | The preimage by ` ZRHom ` ... |
| elzrhunit 33258 | Condition for the image by... |
| elzdif0 33259 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 33260 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 33261 | Value of the canonical hom... |
| qqhvval 33262 | Value of the canonical hom... |
| qqh0 33263 | The image of ` 0 ` by the ... |
| qqh1 33264 | The image of ` 1 ` by the ... |
| qqhf 33265 | ` QQHom ` as a function. ... |
| qqhvq 33266 | The image of a quotient by... |
| qqhghm 33267 | The ` QQHom ` homomorphism... |
| qqhrhm 33268 | The ` QQHom ` homomorphism... |
| qqhnm 33269 | The norm of the image by `... |
| qqhcn 33270 | The ` QQHom ` homomorphism... |
| qqhucn 33271 | The ` QQHom ` homomorphism... |
| rrhval 33275 | Value of the canonical hom... |
| rrhcn 33276 | If the topology of ` R ` i... |
| rrhf 33277 | If the topology of ` R ` i... |
| isrrext 33279 | Express the property " ` R... |
| rrextnrg 33280 | An extension of ` RR ` is ... |
| rrextdrg 33281 | An extension of ` RR ` is ... |
| rrextnlm 33282 | The norm of an extension o... |
| rrextchr 33283 | The ring characteristic of... |
| rrextcusp 33284 | An extension of ` RR ` is ... |
| rrexttps 33285 | An extension of ` RR ` is ... |
| rrexthaus 33286 | The topology of an extensi... |
| rrextust 33287 | The uniformity of an exten... |
| rerrext 33288 | The field of the real numb... |
| cnrrext 33289 | The field of the complex n... |
| qqtopn 33290 | The topology of the field ... |
| rrhfe 33291 | If ` R ` is an extension o... |
| rrhcne 33292 | If ` R ` is an extension o... |
| rrhqima 33293 | The ` RRHom ` homomorphism... |
| rrh0 33294 | The image of ` 0 ` by the ... |
| xrhval 33297 | The value of the embedding... |
| zrhre 33298 | The ` ZRHom ` homomorphism... |
| qqhre 33299 | The ` QQHom ` homomorphism... |
| rrhre 33300 | The ` RRHom ` homomorphism... |
| relmntop 33303 | Manifold is a relation. (... |
| ismntoplly 33304 | Property of being a manifo... |
| ismntop 33305 | Property of being a manifo... |
| nexple 33306 | A lower bound for an expon... |
| indv 33309 | Value of the indicator fun... |
| indval 33310 | Value of the indicator fun... |
| indval2 33311 | Alternate value of the ind... |
| indf 33312 | An indicator function as a... |
| indfval 33313 | Value of the indicator fun... |
| ind1 33314 | Value of the indicator fun... |
| ind0 33315 | Value of the indicator fun... |
| ind1a 33316 | Value of the indicator fun... |
| indpi1 33317 | Preimage of the singleton ... |
| indsum 33318 | Finite sum of a product wi... |
| indsumin 33319 | Finite sum of a product wi... |
| prodindf 33320 | The product of indicators ... |
| indf1o 33321 | The bijection between a po... |
| indpreima 33322 | A function with range ` { ... |
| indf1ofs 33323 | The bijection between fini... |
| esumex 33326 | An extended sum is a set b... |
| esumcl 33327 | Closure for extended sum i... |
| esumeq12dvaf 33328 | Equality deduction for ext... |
| esumeq12dva 33329 | Equality deduction for ext... |
| esumeq12d 33330 | Equality deduction for ext... |
| esumeq1 33331 | Equality theorem for an ex... |
| esumeq1d 33332 | Equality theorem for an ex... |
| esumeq2 33333 | Equality theorem for exten... |
| esumeq2d 33334 | Equality deduction for ext... |
| esumeq2dv 33335 | Equality deduction for ext... |
| esumeq2sdv 33336 | Equality deduction for ext... |
| nfesum1 33337 | Bound-variable hypothesis ... |
| nfesum2 33338 | Bound-variable hypothesis ... |
| cbvesum 33339 | Change bound variable in a... |
| cbvesumv 33340 | Change bound variable in a... |
| esumid 33341 | Identify the extended sum ... |
| esumgsum 33342 | A finite extended sum is t... |
| esumval 33343 | Develop the value of the e... |
| esumel 33344 | The extended sum is a limi... |
| esumnul 33345 | Extended sum over the empt... |
| esum0 33346 | Extended sum of zero. (Co... |
| esumf1o 33347 | Re-index an extended sum u... |
| esumc 33348 | Convert from the collectio... |
| esumrnmpt 33349 | Rewrite an extended sum in... |
| esumsplit 33350 | Split an extended sum into... |
| esummono 33351 | Extended sum is monotonic.... |
| esumpad 33352 | Extend an extended sum by ... |
| esumpad2 33353 | Remove zeroes from an exte... |
| esumadd 33354 | Addition of infinite sums.... |
| esumle 33355 | If all of the terms of an ... |
| gsumesum 33356 | Relate a group sum on ` ( ... |
| esumlub 33357 | The extended sum is the lo... |
| esumaddf 33358 | Addition of infinite sums.... |
| esumlef 33359 | If all of the terms of an ... |
| esumcst 33360 | The extended sum of a cons... |
| esumsnf 33361 | The extended sum of a sing... |
| esumsn 33362 | The extended sum of a sing... |
| esumpr 33363 | Extended sum over a pair. ... |
| esumpr2 33364 | Extended sum over a pair, ... |
| esumrnmpt2 33365 | Rewrite an extended sum in... |
| esumfzf 33366 | Formulating a partial exte... |
| esumfsup 33367 | Formulating an extended su... |
| esumfsupre 33368 | Formulating an extended su... |
| esumss 33369 | Change the index set to a ... |
| esumpinfval 33370 | The value of the extended ... |
| esumpfinvallem 33371 | Lemma for ~ esumpfinval . ... |
| esumpfinval 33372 | The value of the extended ... |
| esumpfinvalf 33373 | Same as ~ esumpfinval , mi... |
| esumpinfsum 33374 | The value of the extended ... |
| esumpcvgval 33375 | The value of the extended ... |
| esumpmono 33376 | The partial sums in an ext... |
| esumcocn 33377 | Lemma for ~ esummulc2 and ... |
| esummulc1 33378 | An extended sum multiplied... |
| esummulc2 33379 | An extended sum multiplied... |
| esumdivc 33380 | An extended sum divided by... |
| hashf2 33381 | Lemma for ~ hasheuni . (C... |
| hasheuni 33382 | The cardinality of a disjo... |
| esumcvg 33383 | The sequence of partial su... |
| esumcvg2 33384 | Simpler version of ~ esumc... |
| esumcvgsum 33385 | The value of the extended ... |
| esumsup 33386 | Express an extended sum as... |
| esumgect 33387 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 33388 | All terms of a converging ... |
| esum2dlem 33389 | Lemma for ~ esum2d (finite... |
| esum2d 33390 | Write a double extended su... |
| esumiun 33391 | Sum over a nonnecessarily ... |
| ofceq 33394 | Equality theorem for funct... |
| ofcfval 33395 | Value of an operation appl... |
| ofcval 33396 | Evaluate a function/consta... |
| ofcfn 33397 | The function operation pro... |
| ofcfeqd2 33398 | Equality theorem for funct... |
| ofcfval3 33399 | General value of ` ( F oFC... |
| ofcf 33400 | The function/constant oper... |
| ofcfval2 33401 | The function operation exp... |
| ofcfval4 33402 | The function/constant oper... |
| ofcc 33403 | Left operation by a consta... |
| ofcof 33404 | Relate function operation ... |
| sigaex 33407 | Lemma for ~ issiga and ~ i... |
| sigaval 33408 | The set of sigma-algebra w... |
| issiga 33409 | An alternative definition ... |
| isrnsiga 33410 | The property of being a si... |
| 0elsiga 33411 | A sigma-algebra contains t... |
| baselsiga 33412 | A sigma-algebra contains i... |
| sigasspw 33413 | A sigma-algebra is a set o... |
| sigaclcu 33414 | A sigma-algebra is closed ... |
| sigaclcuni 33415 | A sigma-algebra is closed ... |
| sigaclfu 33416 | A sigma-algebra is closed ... |
| sigaclcu2 33417 | A sigma-algebra is closed ... |
| sigaclfu2 33418 | A sigma-algebra is closed ... |
| sigaclcu3 33419 | A sigma-algebra is closed ... |
| issgon 33420 | Property of being a sigma-... |
| sgon 33421 | A sigma-algebra is a sigma... |
| elsigass 33422 | An element of a sigma-alge... |
| elrnsiga 33423 | Dropping the base informat... |
| isrnsigau 33424 | The property of being a si... |
| unielsiga 33425 | A sigma-algebra contains i... |
| dmvlsiga 33426 | Lebesgue-measurable subset... |
| pwsiga 33427 | Any power set forms a sigm... |
| prsiga 33428 | The smallest possible sigm... |
| sigaclci 33429 | A sigma-algebra is closed ... |
| difelsiga 33430 | A sigma-algebra is closed ... |
| unelsiga 33431 | A sigma-algebra is closed ... |
| inelsiga 33432 | A sigma-algebra is closed ... |
| sigainb 33433 | Building a sigma-algebra f... |
| insiga 33434 | The intersection of a coll... |
| sigagenval 33437 | Value of the generated sig... |
| sigagensiga 33438 | A generated sigma-algebra ... |
| sgsiga 33439 | A generated sigma-algebra ... |
| unisg 33440 | The sigma-algebra generate... |
| dmsigagen 33441 | A sigma-algebra can be gen... |
| sssigagen 33442 | A set is a subset of the s... |
| sssigagen2 33443 | A subset of the generating... |
| elsigagen 33444 | Any element of a set is al... |
| elsigagen2 33445 | Any countable union of ele... |
| sigagenss 33446 | The generated sigma-algebr... |
| sigagenss2 33447 | Sufficient condition for i... |
| sigagenid 33448 | The sigma-algebra generate... |
| ispisys 33449 | The property of being a pi... |
| ispisys2 33450 | The property of being a pi... |
| inelpisys 33451 | Pi-systems are closed unde... |
| sigapisys 33452 | All sigma-algebras are pi-... |
| isldsys 33453 | The property of being a la... |
| pwldsys 33454 | The power set of the unive... |
| unelldsys 33455 | Lambda-systems are closed ... |
| sigaldsys 33456 | All sigma-algebras are lam... |
| ldsysgenld 33457 | The intersection of all la... |
| sigapildsyslem 33458 | Lemma for ~ sigapildsys . ... |
| sigapildsys 33459 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 33460 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 33461 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 33462 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 33463 | The lambda system ` E ` ge... |
| dynkin 33464 | Dynkin's lambda-pi theorem... |
| isros 33465 | The property of being a ri... |
| rossspw 33466 | A ring of sets is a collec... |
| 0elros 33467 | A ring of sets contains th... |
| unelros 33468 | A ring of sets is closed u... |
| difelros 33469 | A ring of sets is closed u... |
| inelros 33470 | A ring of sets is closed u... |
| fiunelros 33471 | A ring of sets is closed u... |
| issros 33472 | The property of being a se... |
| srossspw 33473 | A semiring of sets is a co... |
| 0elsros 33474 | A semiring of sets contain... |
| inelsros 33475 | A semiring of sets is clos... |
| diffiunisros 33476 | In semiring of sets, compl... |
| rossros 33477 | Rings of sets are semiring... |
| brsiga 33480 | The Borel Algebra on real ... |
| brsigarn 33481 | The Borel Algebra is a sig... |
| brsigasspwrn 33482 | The Borel Algebra is a set... |
| unibrsiga 33483 | The union of the Borel Alg... |
| cldssbrsiga 33484 | A Borel Algebra contains a... |
| sxval 33487 | Value of the product sigma... |
| sxsiga 33488 | A product sigma-algebra is... |
| sxsigon 33489 | A product sigma-algebra is... |
| sxuni 33490 | The base set of a product ... |
| elsx 33491 | The cartesian product of t... |
| measbase 33494 | The base set of a measure ... |
| measval 33495 | The value of the ` measure... |
| ismeas 33496 | The property of being a me... |
| isrnmeas 33497 | The property of being a me... |
| dmmeas 33498 | The domain of a measure is... |
| measbasedom 33499 | The base set of a measure ... |
| measfrge0 33500 | A measure is a function ov... |
| measfn 33501 | A measure is a function on... |
| measvxrge0 33502 | The values of a measure ar... |
| measvnul 33503 | The measure of the empty s... |
| measge0 33504 | A measure is nonnegative. ... |
| measle0 33505 | If the measure of a given ... |
| measvun 33506 | The measure of a countable... |
| measxun2 33507 | The measure the union of t... |
| measun 33508 | The measure the union of t... |
| measvunilem 33509 | Lemma for ~ measvuni . (C... |
| measvunilem0 33510 | Lemma for ~ measvuni . (C... |
| measvuni 33511 | The measure of a countable... |
| measssd 33512 | A measure is monotone with... |
| measunl 33513 | A measure is sub-additive ... |
| measiuns 33514 | The measure of the union o... |
| measiun 33515 | A measure is sub-additive.... |
| meascnbl 33516 | A measure is continuous fr... |
| measinblem 33517 | Lemma for ~ measinb . (Co... |
| measinb 33518 | Building a measure restric... |
| measres 33519 | Building a measure restric... |
| measinb2 33520 | Building a measure restric... |
| measdivcst 33521 | Division of a measure by a... |
| measdivcstALTV 33522 | Alternate version of ~ mea... |
| cntmeas 33523 | The Counting measure is a ... |
| pwcntmeas 33524 | The counting measure is a ... |
| cntnevol 33525 | Counting and Lebesgue meas... |
| voliune 33526 | The Lebesgue measure funct... |
| volfiniune 33527 | The Lebesgue measure funct... |
| volmeas 33528 | The Lebesgue measure is a ... |
| ddeval1 33531 | Value of the delta measure... |
| ddeval0 33532 | Value of the delta measure... |
| ddemeas 33533 | The Dirac delta measure is... |
| relae 33537 | 'almost everywhere' is a r... |
| brae 33538 | 'almost everywhere' relati... |
| braew 33539 | 'almost everywhere' relati... |
| truae 33540 | A truth holds almost every... |
| aean 33541 | A conjunction holds almost... |
| faeval 33543 | Value of the 'almost every... |
| relfae 33544 | The 'almost everywhere' bu... |
| brfae 33545 | 'almost everywhere' relati... |
| ismbfm 33548 | The predicate " ` F ` is a... |
| elunirnmbfm 33549 | The property of being a me... |
| mbfmfun 33550 | A measurable function is a... |
| mbfmf 33551 | A measurable function as a... |
| isanmbfmOLD 33552 | Obsolete version of ~ isan... |
| mbfmcnvima 33553 | The preimage by a measurab... |
| isanmbfm 33554 | The predicate to be a meas... |
| mbfmbfmOLD 33555 | A measurable function to a... |
| mbfmbfm 33556 | A measurable function to a... |
| mbfmcst 33557 | A constant function is mea... |
| 1stmbfm 33558 | The first projection map i... |
| 2ndmbfm 33559 | The second projection map ... |
| imambfm 33560 | If the sigma-algebra in th... |
| cnmbfm 33561 | A continuous function is m... |
| mbfmco 33562 | The composition of two mea... |
| mbfmco2 33563 | The pair building of two m... |
| mbfmvolf 33564 | Measurable functions with ... |
| elmbfmvol2 33565 | Measurable functions with ... |
| mbfmcnt 33566 | All functions are measurab... |
| br2base 33567 | The base set for the gener... |
| dya2ub 33568 | An upper bound for a dyadi... |
| sxbrsigalem0 33569 | The closed half-spaces of ... |
| sxbrsigalem3 33570 | The sigma-algebra generate... |
| dya2iocival 33571 | The function ` I ` returns... |
| dya2iocress 33572 | Dyadic intervals are subse... |
| dya2iocbrsiga 33573 | Dyadic intervals are Borel... |
| dya2icobrsiga 33574 | Dyadic intervals are Borel... |
| dya2icoseg 33575 | For any point and any clos... |
| dya2icoseg2 33576 | For any point and any open... |
| dya2iocrfn 33577 | The function returning dya... |
| dya2iocct 33578 | The dyadic rectangle set i... |
| dya2iocnrect 33579 | For any point of an open r... |
| dya2iocnei 33580 | For any point of an open s... |
| dya2iocuni 33581 | Every open set of ` ( RR X... |
| dya2iocucvr 33582 | The dyadic rectangular set... |
| sxbrsigalem1 33583 | The Borel algebra on ` ( R... |
| sxbrsigalem2 33584 | The sigma-algebra generate... |
| sxbrsigalem4 33585 | The Borel algebra on ` ( R... |
| sxbrsigalem5 33586 | First direction for ~ sxbr... |
| sxbrsigalem6 33587 | First direction for ~ sxbr... |
| sxbrsiga 33588 | The product sigma-algebra ... |
| omsval 33591 | Value of the function mapp... |
| omsfval 33592 | Value of the outer measure... |
| omscl 33593 | A closure lemma for the co... |
| omsf 33594 | A constructed outer measur... |
| oms0 33595 | A constructed outer measur... |
| omsmon 33596 | A constructed outer measur... |
| omssubaddlem 33597 | For any small margin ` E `... |
| omssubadd 33598 | A constructed outer measur... |
| carsgval 33601 | Value of the Caratheodory ... |
| carsgcl 33602 | Closure of the Caratheodor... |
| elcarsg 33603 | Property of being a Carath... |
| baselcarsg 33604 | The universe set, ` O ` , ... |
| 0elcarsg 33605 | The empty set is Caratheod... |
| carsguni 33606 | The union of all Caratheod... |
| elcarsgss 33607 | Caratheodory measurable se... |
| difelcarsg 33608 | The Caratheodory measurabl... |
| inelcarsg 33609 | The Caratheodory measurabl... |
| unelcarsg 33610 | The Caratheodory-measurabl... |
| difelcarsg2 33611 | The Caratheodory-measurabl... |
| carsgmon 33612 | Utility lemma: Apply mono... |
| carsgsigalem 33613 | Lemma for the following th... |
| fiunelcarsg 33614 | The Caratheodory measurabl... |
| carsgclctunlem1 33615 | Lemma for ~ carsgclctun . ... |
| carsggect 33616 | The outer measure is count... |
| carsgclctunlem2 33617 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 33618 | Lemma for ~ carsgclctun . ... |
| carsgclctun 33619 | The Caratheodory measurabl... |
| carsgsiga 33620 | The Caratheodory measurabl... |
| omsmeas 33621 | The restriction of a const... |
| pmeasmono 33622 | This theorem's hypotheses ... |
| pmeasadd 33623 | A premeasure on a ring of ... |
| itgeq12dv 33624 | Equality theorem for an in... |
| sitgval 33630 | Value of the simple functi... |
| issibf 33631 | The predicate " ` F ` is a... |
| sibf0 33632 | The constant zero function... |
| sibfmbl 33633 | A simple function is measu... |
| sibff 33634 | A simple function is a fun... |
| sibfrn 33635 | A simple function has fini... |
| sibfima 33636 | Any preimage of a singleto... |
| sibfinima 33637 | The measure of the interse... |
| sibfof 33638 | Applying function operatio... |
| sitgfval 33639 | Value of the Bochner integ... |
| sitgclg 33640 | Closure of the Bochner int... |
| sitgclbn 33641 | Closure of the Bochner int... |
| sitgclcn 33642 | Closure of the Bochner int... |
| sitgclre 33643 | Closure of the Bochner int... |
| sitg0 33644 | The integral of the consta... |
| sitgf 33645 | The integral for simple fu... |
| sitgaddlemb 33646 | Lemma for * sitgadd . (Co... |
| sitmval 33647 | Value of the simple functi... |
| sitmfval 33648 | Value of the integral dist... |
| sitmcl 33649 | Closure of the integral di... |
| sitmf 33650 | The integral metric as a f... |
| oddpwdc 33652 | Lemma for ~ eulerpart . T... |
| oddpwdcv 33653 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 33654 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 33655 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 33656 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 33657 | Lemma for ~ eulerpart . (... |
| eulerpartlems 33658 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 33659 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 33660 | Lemma for ~ eulerpart . (... |
| eulerpartleme 33661 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 33662 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 33663 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 33664 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 33665 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 33666 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 33667 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 33668 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 33669 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 33670 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 33671 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 33672 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 33673 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 33674 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 33675 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 33676 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 33677 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 33678 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 33679 | Lemma for ~ eulerpart . (... |
| eulerpart 33680 | Euler's theorem on partiti... |
| subiwrd 33683 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 33684 | Length of a subword of an ... |
| iwrdsplit 33685 | Lemma for ~ sseqp1 . (Con... |
| sseqval 33686 | Value of the strong sequen... |
| sseqfv1 33687 | Value of the strong sequen... |
| sseqfn 33688 | A strong recursive sequenc... |
| sseqmw 33689 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 33690 | A strong recursive sequenc... |
| sseqfres 33691 | The first elements in the ... |
| sseqfv2 33692 | Value of the strong sequen... |
| sseqp1 33693 | Value of the strong sequen... |
| fiblem 33696 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 33697 | Value of the Fibonacci seq... |
| fib1 33698 | Value of the Fibonacci seq... |
| fibp1 33699 | Value of the Fibonacci seq... |
| fib2 33700 | Value of the Fibonacci seq... |
| fib3 33701 | Value of the Fibonacci seq... |
| fib4 33702 | Value of the Fibonacci seq... |
| fib5 33703 | Value of the Fibonacci seq... |
| fib6 33704 | Value of the Fibonacci seq... |
| elprob 33707 | The property of being a pr... |
| domprobmeas 33708 | A probability measure is a... |
| domprobsiga 33709 | The domain of a probabilit... |
| probtot 33710 | The probability of the uni... |
| prob01 33711 | A probability is an elemen... |
| probnul 33712 | The probability of the emp... |
| unveldomd 33713 | The universe is an element... |
| unveldom 33714 | The universe is an element... |
| nuleldmp 33715 | The empty set is an elemen... |
| probcun 33716 | The probability of the uni... |
| probun 33717 | The probability of the uni... |
| probdif 33718 | The probability of the dif... |
| probinc 33719 | A probability law is incre... |
| probdsb 33720 | The probability of the com... |
| probmeasd 33721 | A probability measure is a... |
| probvalrnd 33722 | The value of a probability... |
| probtotrnd 33723 | The probability of the uni... |
| totprobd 33724 | Law of total probability, ... |
| totprob 33725 | Law of total probability. ... |
| probfinmeasb 33726 | Build a probability measur... |
| probfinmeasbALTV 33727 | Alternate version of ~ pro... |
| probmeasb 33728 | Build a probability from a... |
| cndprobval 33731 | The value of the condition... |
| cndprobin 33732 | An identity linking condit... |
| cndprob01 33733 | The conditional probabilit... |
| cndprobtot 33734 | The conditional probabilit... |
| cndprobnul 33735 | The conditional probabilit... |
| cndprobprob 33736 | The conditional probabilit... |
| bayesth 33737 | Bayes Theorem. (Contribut... |
| rrvmbfm 33740 | A real-valued random varia... |
| isrrvv 33741 | Elementhood to the set of ... |
| rrvvf 33742 | A real-valued random varia... |
| rrvfn 33743 | A real-valued random varia... |
| rrvdm 33744 | The domain of a random var... |
| rrvrnss 33745 | The range of a random vari... |
| rrvf2 33746 | A real-valued random varia... |
| rrvdmss 33747 | The domain of a random var... |
| rrvfinvima 33748 | For a real-value random va... |
| 0rrv 33749 | The constant function equa... |
| rrvadd 33750 | The sum of two random vari... |
| rrvmulc 33751 | A random variable multipli... |
| rrvsum 33752 | An indexed sum of random v... |
| orvcval 33755 | Value of the preimage mapp... |
| orvcval2 33756 | Another way to express the... |
| elorvc 33757 | Elementhood of a preimage.... |
| orvcval4 33758 | The value of the preimage ... |
| orvcoel 33759 | If the relation produces o... |
| orvccel 33760 | If the relation produces c... |
| elorrvc 33761 | Elementhood of a preimage ... |
| orrvcval4 33762 | The value of the preimage ... |
| orrvcoel 33763 | If the relation produces o... |
| orrvccel 33764 | If the relation produces c... |
| orvcgteel 33765 | Preimage maps produced by ... |
| orvcelval 33766 | Preimage maps produced by ... |
| orvcelel 33767 | Preimage maps produced by ... |
| dstrvval 33768 | The value of the distribut... |
| dstrvprob 33769 | The distribution of a rand... |
| orvclteel 33770 | Preimage maps produced by ... |
| dstfrvel 33771 | Elementhood of preimage ma... |
| dstfrvunirn 33772 | The limit of all preimage ... |
| orvclteinc 33773 | Preimage maps produced by ... |
| dstfrvinc 33774 | A cumulative distribution ... |
| dstfrvclim1 33775 | The limit of the cumulativ... |
| coinfliplem 33776 | Division in the extended r... |
| coinflipprob 33777 | The ` P ` we defined for c... |
| coinflipspace 33778 | The space of our coin-flip... |
| coinflipuniv 33779 | The universe of our coin-f... |
| coinfliprv 33780 | The ` X ` we defined for c... |
| coinflippv 33781 | The probability of heads i... |
| coinflippvt 33782 | The probability of tails i... |
| ballotlemoex 33783 | ` O ` is a set. (Contribu... |
| ballotlem1 33784 | The size of the universe i... |
| ballotlemelo 33785 | Elementhood in ` O ` . (C... |
| ballotlem2 33786 | The probability that the f... |
| ballotlemfval 33787 | The value of ` F ` . (Con... |
| ballotlemfelz 33788 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 33789 | If the ` J ` th ballot is ... |
| ballotlemfc0 33790 | ` F ` takes value 0 betwee... |
| ballotlemfcc 33791 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 33792 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 33793 | ` ( F `` C ) ` always star... |
| ballotleme 33794 | Elements of ` E ` . (Cont... |
| ballotlemodife 33795 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 33796 | If the first pick is a vot... |
| ballotlem5 33797 | If A is not ahead througho... |
| ballotlemi 33798 | Value of ` I ` for a given... |
| ballotlemiex 33799 | Properties of ` ( I `` C )... |
| ballotlemi1 33800 | The first tie cannot be re... |
| ballotlemii 33801 | The first tie cannot be re... |
| ballotlemsup 33802 | The set of zeroes of ` F `... |
| ballotlemimin 33803 | ` ( I `` C ) ` is the firs... |
| ballotlemic 33804 | If the first vote is for B... |
| ballotlem1c 33805 | If the first vote is for A... |
| ballotlemsval 33806 | Value of ` S ` . (Contrib... |
| ballotlemsv 33807 | Value of ` S ` evaluated a... |
| ballotlemsgt1 33808 | ` S ` maps values less tha... |
| ballotlemsdom 33809 | Domain of ` S ` for a give... |
| ballotlemsel1i 33810 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 33811 | The defined ` S ` is a bij... |
| ballotlemsi 33812 | The image by ` S ` of the ... |
| ballotlemsima 33813 | The image by ` S ` of an i... |
| ballotlemieq 33814 | If two countings share the... |
| ballotlemrval 33815 | Value of ` R ` . (Contrib... |
| ballotlemscr 33816 | The image of ` ( R `` C ) ... |
| ballotlemrv 33817 | Value of ` R ` evaluated a... |
| ballotlemrv1 33818 | Value of ` R ` before the ... |
| ballotlemrv2 33819 | Value of ` R ` after the t... |
| ballotlemro 33820 | Range of ` R ` is included... |
| ballotlemgval 33821 | Expand the value of ` .^ `... |
| ballotlemgun 33822 | A property of the defined ... |
| ballotlemfg 33823 | Express the value of ` ( F... |
| ballotlemfrc 33824 | Express the value of ` ( F... |
| ballotlemfrci 33825 | Reverse counting preserves... |
| ballotlemfrceq 33826 | Value of ` F ` for a rever... |
| ballotlemfrcn0 33827 | Value of ` F ` for a rever... |
| ballotlemrc 33828 | Range of ` R ` . (Contrib... |
| ballotlemirc 33829 | Applying ` R ` does not ch... |
| ballotlemrinv0 33830 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 33831 | ` R ` is its own inverse :... |
| ballotlem1ri 33832 | When the vote on the first... |
| ballotlem7 33833 | ` R ` is a bijection betwe... |
| ballotlem8 33834 | There are as many counting... |
| ballotth 33835 | Bertrand's ballot problem ... |
| sgncl 33836 | Closure of the signum. (C... |
| sgnclre 33837 | Closure of the signum. (C... |
| sgnneg 33838 | Negation of the signum. (... |
| sgn3da 33839 | A conditional containing a... |
| sgnmul 33840 | Signum of a product. (Con... |
| sgnmulrp2 33841 | Multiplication by a positi... |
| sgnsub 33842 | Subtraction of a number of... |
| sgnnbi 33843 | Negative signum. (Contrib... |
| sgnpbi 33844 | Positive signum. (Contrib... |
| sgn0bi 33845 | Zero signum. (Contributed... |
| sgnsgn 33846 | Signum is idempotent. (Co... |
| sgnmulsgn 33847 | If two real numbers are of... |
| sgnmulsgp 33848 | If two real numbers are of... |
| fzssfzo 33849 | Condition for an integer i... |
| gsumncl 33850 | Closure of a group sum in ... |
| gsumnunsn 33851 | Closure of a group sum in ... |
| ccatmulgnn0dir 33852 | Concatenation of words fol... |
| ofcccat 33853 | Letterwise operations on w... |
| ofcs1 33854 | Letterwise operations on a... |
| ofcs2 33855 | Letterwise operations on a... |
| plymul02 33856 | Product of a polynomial wi... |
| plymulx0 33857 | Coefficients of a polynomi... |
| plymulx 33858 | Coefficients of a polynomi... |
| plyrecld 33859 | Closure of a polynomial wi... |
| signsplypnf 33860 | The quotient of a polynomi... |
| signsply0 33861 | Lemma for the rule of sign... |
| signspval 33862 | The value of the skipping ... |
| signsw0glem 33863 | Neutral element property o... |
| signswbase 33864 | The base of ` W ` is the u... |
| signswplusg 33865 | The operation of ` W ` . ... |
| signsw0g 33866 | The neutral element of ` W... |
| signswmnd 33867 | ` W ` is a monoid structur... |
| signswrid 33868 | The zero-skipping operatio... |
| signswlid 33869 | The zero-skipping operatio... |
| signswn0 33870 | The zero-skipping operatio... |
| signswch 33871 | The zero-skipping operatio... |
| signslema 33872 | Computational part of ~~? ... |
| signstfv 33873 | Value of the zero-skipping... |
| signstfval 33874 | Value of the zero-skipping... |
| signstcl 33875 | Closure of the zero skippi... |
| signstf 33876 | The zero skipping sign wor... |
| signstlen 33877 | Length of the zero skippin... |
| signstf0 33878 | Sign of a single letter wo... |
| signstfvn 33879 | Zero-skipping sign in a wo... |
| signsvtn0 33880 | If the last letter is nonz... |
| signstfvp 33881 | Zero-skipping sign in a wo... |
| signstfvneq0 33882 | In case the first letter i... |
| signstfvcl 33883 | Closure of the zero skippi... |
| signstfvc 33884 | Zero-skipping sign in a wo... |
| signstres 33885 | Restriction of a zero skip... |
| signstfveq0a 33886 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 33887 | In case the last letter is... |
| signsvvfval 33888 | The value of ` V ` , which... |
| signsvvf 33889 | ` V ` is a function. (Con... |
| signsvf0 33890 | There is no change of sign... |
| signsvf1 33891 | In a single-letter word, w... |
| signsvfn 33892 | Number of changes in a wor... |
| signsvtp 33893 | Adding a letter of the sam... |
| signsvtn 33894 | Adding a letter of a diffe... |
| signsvfpn 33895 | Adding a letter of the sam... |
| signsvfnn 33896 | Adding a letter of a diffe... |
| signlem0 33897 | Adding a zero as the highe... |
| signshf 33898 | ` H ` , corresponding to t... |
| signshwrd 33899 | ` H ` , corresponding to t... |
| signshlen 33900 | Length of ` H ` , correspo... |
| signshnz 33901 | ` H ` is not the empty wor... |
| efcld 33902 | Closure law for the expone... |
| iblidicc 33903 | The identity function is i... |
| rpsqrtcn 33904 | Continuity of the real pos... |
| divsqrtid 33905 | A real number divided by i... |
| cxpcncf1 33906 | The power function on comp... |
| efmul2picn 33907 | Multiplying by ` ( _i x. (... |
| fct2relem 33908 | Lemma for ~ ftc2re . (Con... |
| ftc2re 33909 | The Fundamental Theorem of... |
| fdvposlt 33910 | Functions with a positive ... |
| fdvneggt 33911 | Functions with a negative ... |
| fdvposle 33912 | Functions with a nonnegati... |
| fdvnegge 33913 | Functions with a nonpositi... |
| prodfzo03 33914 | A product of three factors... |
| actfunsnf1o 33915 | The action ` F ` of extend... |
| actfunsnrndisj 33916 | The action ` F ` of extend... |
| itgexpif 33917 | The basis for the circle m... |
| fsum2dsub 33918 | Lemma for ~ breprexp - Re-... |
| reprval 33921 | Value of the representatio... |
| repr0 33922 | There is exactly one repre... |
| reprf 33923 | Members of the representat... |
| reprsum 33924 | Sums of values of the memb... |
| reprle 33925 | Upper bound to the terms i... |
| reprsuc 33926 | Express the representation... |
| reprfi 33927 | Bounded representations ar... |
| reprss 33928 | Representations with terms... |
| reprinrn 33929 | Representations with term ... |
| reprlt 33930 | There are no representatio... |
| hashreprin 33931 | Express a sum of represent... |
| reprgt 33932 | There are no representatio... |
| reprinfz1 33933 | For the representation of ... |
| reprfi2 33934 | Corollary of ~ reprinfz1 .... |
| reprfz1 33935 | Corollary of ~ reprinfz1 .... |
| hashrepr 33936 | Develop the number of repr... |
| reprpmtf1o 33937 | Transposing ` 0 ` and ` X ... |
| reprdifc 33938 | Express the representation... |
| chpvalz 33939 | Value of the second Chebys... |
| chtvalz 33940 | Value of the Chebyshev fun... |
| breprexplema 33941 | Lemma for ~ breprexp (indu... |
| breprexplemb 33942 | Lemma for ~ breprexp (clos... |
| breprexplemc 33943 | Lemma for ~ breprexp (indu... |
| breprexp 33944 | Express the ` S ` th power... |
| breprexpnat 33945 | Express the ` S ` th power... |
| vtsval 33948 | Value of the Vinogradov tr... |
| vtscl 33949 | Closure of the Vinogradov ... |
| vtsprod 33950 | Express the Vinogradov tri... |
| circlemeth 33951 | The Hardy, Littlewood and ... |
| circlemethnat 33952 | The Hardy, Littlewood and ... |
| circlevma 33953 | The Circle Method, where t... |
| circlemethhgt 33954 | The circle method, where t... |
| hgt750lemc 33958 | An upper bound to the summ... |
| hgt750lemd 33959 | An upper bound to the summ... |
| hgt749d 33960 | A deduction version of ~ a... |
| logdivsqrle 33961 | Conditions for ` ( ( log `... |
| hgt750lem 33962 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 33963 | Decimal multiplication gal... |
| hgt750lemf 33964 | Lemma for the statement 7.... |
| hgt750lemg 33965 | Lemma for the statement 7.... |
| oddprm2 33966 | Two ways to write the set ... |
| hgt750lemb 33967 | An upper bound on the cont... |
| hgt750lema 33968 | An upper bound on the cont... |
| hgt750leme 33969 | An upper bound on the cont... |
| tgoldbachgnn 33970 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 33971 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 33972 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 33973 | Odd integers greater than ... |
| tgoldbachgt 33974 | Odd integers greater than ... |
| istrkg2d 33977 | Property of fulfilling dim... |
| axtglowdim2ALTV 33978 | Alternate version of ~ axt... |
| axtgupdim2ALTV 33979 | Alternate version of ~ axt... |
| afsval 33982 | Value of the AFS relation ... |
| brafs 33983 | Binary relation form of th... |
| tg5segofs 33984 | Rephrase ~ axtg5seg using ... |
| lpadval 33987 | Value of the ` leftpad ` f... |
| lpadlem1 33988 | Lemma for the ` leftpad ` ... |
| lpadlem3 33989 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 33990 | Length of a left-padded wo... |
| lpadlem2 33991 | Lemma for the ` leftpad ` ... |
| lpadlen2 33992 | Length of a left-padded wo... |
| lpadmax 33993 | Length of a left-padded wo... |
| lpadleft 33994 | The contents of prefix of ... |
| lpadright 33995 | The suffix of a left-padde... |
| bnj170 34008 | ` /\ ` -manipulation. (Co... |
| bnj240 34009 | ` /\ ` -manipulation. (Co... |
| bnj248 34010 | ` /\ ` -manipulation. (Co... |
| bnj250 34011 | ` /\ ` -manipulation. (Co... |
| bnj251 34012 | ` /\ ` -manipulation. (Co... |
| bnj252 34013 | ` /\ ` -manipulation. (Co... |
| bnj253 34014 | ` /\ ` -manipulation. (Co... |
| bnj255 34015 | ` /\ ` -manipulation. (Co... |
| bnj256 34016 | ` /\ ` -manipulation. (Co... |
| bnj257 34017 | ` /\ ` -manipulation. (Co... |
| bnj258 34018 | ` /\ ` -manipulation. (Co... |
| bnj268 34019 | ` /\ ` -manipulation. (Co... |
| bnj290 34020 | ` /\ ` -manipulation. (Co... |
| bnj291 34021 | ` /\ ` -manipulation. (Co... |
| bnj312 34022 | ` /\ ` -manipulation. (Co... |
| bnj334 34023 | ` /\ ` -manipulation. (Co... |
| bnj345 34024 | ` /\ ` -manipulation. (Co... |
| bnj422 34025 | ` /\ ` -manipulation. (Co... |
| bnj432 34026 | ` /\ ` -manipulation. (Co... |
| bnj446 34027 | ` /\ ` -manipulation. (Co... |
| bnj23 34028 | First-order logic and set ... |
| bnj31 34029 | First-order logic and set ... |
| bnj62 34030 | First-order logic and set ... |
| bnj89 34031 | First-order logic and set ... |
| bnj90 34032 | First-order logic and set ... |
| bnj101 34033 | First-order logic and set ... |
| bnj105 34034 | First-order logic and set ... |
| bnj115 34035 | First-order logic and set ... |
| bnj132 34036 | First-order logic and set ... |
| bnj133 34037 | First-order logic and set ... |
| bnj156 34038 | First-order logic and set ... |
| bnj158 34039 | First-order logic and set ... |
| bnj168 34040 | First-order logic and set ... |
| bnj206 34041 | First-order logic and set ... |
| bnj216 34042 | First-order logic and set ... |
| bnj219 34043 | First-order logic and set ... |
| bnj226 34044 | First-order logic and set ... |
| bnj228 34045 | First-order logic and set ... |
| bnj519 34046 | First-order logic and set ... |
| bnj524 34047 | First-order logic and set ... |
| bnj525 34048 | First-order logic and set ... |
| bnj534 34049 | First-order logic and set ... |
| bnj538 34050 | First-order logic and set ... |
| bnj529 34051 | First-order logic and set ... |
| bnj551 34052 | First-order logic and set ... |
| bnj563 34053 | First-order logic and set ... |
| bnj564 34054 | First-order logic and set ... |
| bnj593 34055 | First-order logic and set ... |
| bnj596 34056 | First-order logic and set ... |
| bnj610 34057 | Pass from equality ( ` x =... |
| bnj642 34058 | ` /\ ` -manipulation. (Co... |
| bnj643 34059 | ` /\ ` -manipulation. (Co... |
| bnj645 34060 | ` /\ ` -manipulation. (Co... |
| bnj658 34061 | ` /\ ` -manipulation. (Co... |
| bnj667 34062 | ` /\ ` -manipulation. (Co... |
| bnj705 34063 | ` /\ ` -manipulation. (Co... |
| bnj706 34064 | ` /\ ` -manipulation. (Co... |
| bnj707 34065 | ` /\ ` -manipulation. (Co... |
| bnj708 34066 | ` /\ ` -manipulation. (Co... |
| bnj721 34067 | ` /\ ` -manipulation. (Co... |
| bnj832 34068 | ` /\ ` -manipulation. (Co... |
| bnj835 34069 | ` /\ ` -manipulation. (Co... |
| bnj836 34070 | ` /\ ` -manipulation. (Co... |
| bnj837 34071 | ` /\ ` -manipulation. (Co... |
| bnj769 34072 | ` /\ ` -manipulation. (Co... |
| bnj770 34073 | ` /\ ` -manipulation. (Co... |
| bnj771 34074 | ` /\ ` -manipulation. (Co... |
| bnj887 34075 | ` /\ ` -manipulation. (Co... |
| bnj918 34076 | First-order logic and set ... |
| bnj919 34077 | First-order logic and set ... |
| bnj923 34078 | First-order logic and set ... |
| bnj927 34079 | First-order logic and set ... |
| bnj931 34080 | First-order logic and set ... |
| bnj937 34081 | First-order logic and set ... |
| bnj941 34082 | First-order logic and set ... |
| bnj945 34083 | Technical lemma for ~ bnj6... |
| bnj946 34084 | First-order logic and set ... |
| bnj951 34085 | ` /\ ` -manipulation. (Co... |
| bnj956 34086 | First-order logic and set ... |
| bnj976 34087 | First-order logic and set ... |
| bnj982 34088 | First-order logic and set ... |
| bnj1019 34089 | First-order logic and set ... |
| bnj1023 34090 | First-order logic and set ... |
| bnj1095 34091 | First-order logic and set ... |
| bnj1096 34092 | First-order logic and set ... |
| bnj1098 34093 | First-order logic and set ... |
| bnj1101 34094 | First-order logic and set ... |
| bnj1113 34095 | First-order logic and set ... |
| bnj1109 34096 | First-order logic and set ... |
| bnj1131 34097 | First-order logic and set ... |
| bnj1138 34098 | First-order logic and set ... |
| bnj1142 34099 | First-order logic and set ... |
| bnj1143 34100 | First-order logic and set ... |
| bnj1146 34101 | First-order logic and set ... |
| bnj1149 34102 | First-order logic and set ... |
| bnj1185 34103 | First-order logic and set ... |
| bnj1196 34104 | First-order logic and set ... |
| bnj1198 34105 | First-order logic and set ... |
| bnj1209 34106 | First-order logic and set ... |
| bnj1211 34107 | First-order logic and set ... |
| bnj1213 34108 | First-order logic and set ... |
| bnj1212 34109 | First-order logic and set ... |
| bnj1219 34110 | First-order logic and set ... |
| bnj1224 34111 | First-order logic and set ... |
| bnj1230 34112 | First-order logic and set ... |
| bnj1232 34113 | First-order logic and set ... |
| bnj1235 34114 | First-order logic and set ... |
| bnj1239 34115 | First-order logic and set ... |
| bnj1238 34116 | First-order logic and set ... |
| bnj1241 34117 | First-order logic and set ... |
| bnj1247 34118 | First-order logic and set ... |
| bnj1254 34119 | First-order logic and set ... |
| bnj1262 34120 | First-order logic and set ... |
| bnj1266 34121 | First-order logic and set ... |
| bnj1265 34122 | First-order logic and set ... |
| bnj1275 34123 | First-order logic and set ... |
| bnj1276 34124 | First-order logic and set ... |
| bnj1292 34125 | First-order logic and set ... |
| bnj1293 34126 | First-order logic and set ... |
| bnj1294 34127 | First-order logic and set ... |
| bnj1299 34128 | First-order logic and set ... |
| bnj1304 34129 | First-order logic and set ... |
| bnj1316 34130 | First-order logic and set ... |
| bnj1317 34131 | First-order logic and set ... |
| bnj1322 34132 | First-order logic and set ... |
| bnj1340 34133 | First-order logic and set ... |
| bnj1345 34134 | First-order logic and set ... |
| bnj1350 34135 | First-order logic and set ... |
| bnj1351 34136 | First-order logic and set ... |
| bnj1352 34137 | First-order logic and set ... |
| bnj1361 34138 | First-order logic and set ... |
| bnj1366 34139 | First-order logic and set ... |
| bnj1379 34140 | First-order logic and set ... |
| bnj1383 34141 | First-order logic and set ... |
| bnj1385 34142 | First-order logic and set ... |
| bnj1386 34143 | First-order logic and set ... |
| bnj1397 34144 | First-order logic and set ... |
| bnj1400 34145 | First-order logic and set ... |
| bnj1405 34146 | First-order logic and set ... |
| bnj1422 34147 | First-order logic and set ... |
| bnj1424 34148 | First-order logic and set ... |
| bnj1436 34149 | First-order logic and set ... |
| bnj1441 34150 | First-order logic and set ... |
| bnj1441g 34151 | First-order logic and set ... |
| bnj1454 34152 | First-order logic and set ... |
| bnj1459 34153 | First-order logic and set ... |
| bnj1464 34154 | Conversion of implicit sub... |
| bnj1465 34155 | First-order logic and set ... |
| bnj1468 34156 | Conversion of implicit sub... |
| bnj1476 34157 | First-order logic and set ... |
| bnj1502 34158 | First-order logic and set ... |
| bnj1503 34159 | First-order logic and set ... |
| bnj1517 34160 | First-order logic and set ... |
| bnj1521 34161 | First-order logic and set ... |
| bnj1533 34162 | First-order logic and set ... |
| bnj1534 34163 | First-order logic and set ... |
| bnj1536 34164 | First-order logic and set ... |
| bnj1538 34165 | First-order logic and set ... |
| bnj1541 34166 | First-order logic and set ... |
| bnj1542 34167 | First-order logic and set ... |
| bnj110 34168 | Well-founded induction res... |
| bnj157 34169 | Well-founded induction res... |
| bnj66 34170 | Technical lemma for ~ bnj6... |
| bnj91 34171 | First-order logic and set ... |
| bnj92 34172 | First-order logic and set ... |
| bnj93 34173 | Technical lemma for ~ bnj9... |
| bnj95 34174 | Technical lemma for ~ bnj1... |
| bnj96 34175 | Technical lemma for ~ bnj1... |
| bnj97 34176 | Technical lemma for ~ bnj1... |
| bnj98 34177 | Technical lemma for ~ bnj1... |
| bnj106 34178 | First-order logic and set ... |
| bnj118 34179 | First-order logic and set ... |
| bnj121 34180 | First-order logic and set ... |
| bnj124 34181 | Technical lemma for ~ bnj1... |
| bnj125 34182 | Technical lemma for ~ bnj1... |
| bnj126 34183 | Technical lemma for ~ bnj1... |
| bnj130 34184 | Technical lemma for ~ bnj1... |
| bnj149 34185 | Technical lemma for ~ bnj1... |
| bnj150 34186 | Technical lemma for ~ bnj1... |
| bnj151 34187 | Technical lemma for ~ bnj1... |
| bnj154 34188 | Technical lemma for ~ bnj1... |
| bnj155 34189 | Technical lemma for ~ bnj1... |
| bnj153 34190 | Technical lemma for ~ bnj8... |
| bnj207 34191 | Technical lemma for ~ bnj8... |
| bnj213 34192 | First-order logic and set ... |
| bnj222 34193 | Technical lemma for ~ bnj2... |
| bnj229 34194 | Technical lemma for ~ bnj5... |
| bnj517 34195 | Technical lemma for ~ bnj5... |
| bnj518 34196 | Technical lemma for ~ bnj8... |
| bnj523 34197 | Technical lemma for ~ bnj8... |
| bnj526 34198 | Technical lemma for ~ bnj8... |
| bnj528 34199 | Technical lemma for ~ bnj8... |
| bnj535 34200 | Technical lemma for ~ bnj8... |
| bnj539 34201 | Technical lemma for ~ bnj8... |
| bnj540 34202 | Technical lemma for ~ bnj8... |
| bnj543 34203 | Technical lemma for ~ bnj8... |
| bnj544 34204 | Technical lemma for ~ bnj8... |
| bnj545 34205 | Technical lemma for ~ bnj8... |
| bnj546 34206 | Technical lemma for ~ bnj8... |
| bnj548 34207 | Technical lemma for ~ bnj8... |
| bnj553 34208 | Technical lemma for ~ bnj8... |
| bnj554 34209 | Technical lemma for ~ bnj8... |
| bnj556 34210 | Technical lemma for ~ bnj8... |
| bnj557 34211 | Technical lemma for ~ bnj8... |
| bnj558 34212 | Technical lemma for ~ bnj8... |
| bnj561 34213 | Technical lemma for ~ bnj8... |
| bnj562 34214 | Technical lemma for ~ bnj8... |
| bnj570 34215 | Technical lemma for ~ bnj8... |
| bnj571 34216 | Technical lemma for ~ bnj8... |
| bnj605 34217 | Technical lemma. This lem... |
| bnj581 34218 | Technical lemma for ~ bnj5... |
| bnj589 34219 | Technical lemma for ~ bnj8... |
| bnj590 34220 | Technical lemma for ~ bnj8... |
| bnj591 34221 | Technical lemma for ~ bnj8... |
| bnj594 34222 | Technical lemma for ~ bnj8... |
| bnj580 34223 | Technical lemma for ~ bnj5... |
| bnj579 34224 | Technical lemma for ~ bnj8... |
| bnj602 34225 | Equality theorem for the `... |
| bnj607 34226 | Technical lemma for ~ bnj8... |
| bnj609 34227 | Technical lemma for ~ bnj8... |
| bnj611 34228 | Technical lemma for ~ bnj8... |
| bnj600 34229 | Technical lemma for ~ bnj8... |
| bnj601 34230 | Technical lemma for ~ bnj8... |
| bnj852 34231 | Technical lemma for ~ bnj6... |
| bnj864 34232 | Technical lemma for ~ bnj6... |
| bnj865 34233 | Technical lemma for ~ bnj6... |
| bnj873 34234 | Technical lemma for ~ bnj6... |
| bnj849 34235 | Technical lemma for ~ bnj6... |
| bnj882 34236 | Definition (using hypothes... |
| bnj18eq1 34237 | Equality theorem for trans... |
| bnj893 34238 | Property of ` _trCl ` . U... |
| bnj900 34239 | Technical lemma for ~ bnj6... |
| bnj906 34240 | Property of ` _trCl ` . (... |
| bnj908 34241 | Technical lemma for ~ bnj6... |
| bnj911 34242 | Technical lemma for ~ bnj6... |
| bnj916 34243 | Technical lemma for ~ bnj6... |
| bnj917 34244 | Technical lemma for ~ bnj6... |
| bnj934 34245 | Technical lemma for ~ bnj6... |
| bnj929 34246 | Technical lemma for ~ bnj6... |
| bnj938 34247 | Technical lemma for ~ bnj6... |
| bnj944 34248 | Technical lemma for ~ bnj6... |
| bnj953 34249 | Technical lemma for ~ bnj6... |
| bnj958 34250 | Technical lemma for ~ bnj6... |
| bnj1000 34251 | Technical lemma for ~ bnj8... |
| bnj965 34252 | Technical lemma for ~ bnj8... |
| bnj964 34253 | Technical lemma for ~ bnj6... |
| bnj966 34254 | Technical lemma for ~ bnj6... |
| bnj967 34255 | Technical lemma for ~ bnj6... |
| bnj969 34256 | Technical lemma for ~ bnj6... |
| bnj970 34257 | Technical lemma for ~ bnj6... |
| bnj910 34258 | Technical lemma for ~ bnj6... |
| bnj978 34259 | Technical lemma for ~ bnj6... |
| bnj981 34260 | Technical lemma for ~ bnj6... |
| bnj983 34261 | Technical lemma for ~ bnj6... |
| bnj984 34262 | Technical lemma for ~ bnj6... |
| bnj985v 34263 | Version of ~ bnj985 with a... |
| bnj985 34264 | Technical lemma for ~ bnj6... |
| bnj986 34265 | Technical lemma for ~ bnj6... |
| bnj996 34266 | Technical lemma for ~ bnj6... |
| bnj998 34267 | Technical lemma for ~ bnj6... |
| bnj999 34268 | Technical lemma for ~ bnj6... |
| bnj1001 34269 | Technical lemma for ~ bnj6... |
| bnj1006 34270 | Technical lemma for ~ bnj6... |
| bnj1014 34271 | Technical lemma for ~ bnj6... |
| bnj1015 34272 | Technical lemma for ~ bnj6... |
| bnj1018g 34273 | Version of ~ bnj1018 with ... |
| bnj1018 34274 | Technical lemma for ~ bnj6... |
| bnj1020 34275 | Technical lemma for ~ bnj6... |
| bnj1021 34276 | Technical lemma for ~ bnj6... |
| bnj907 34277 | Technical lemma for ~ bnj6... |
| bnj1029 34278 | Property of ` _trCl ` . (... |
| bnj1033 34279 | Technical lemma for ~ bnj6... |
| bnj1034 34280 | Technical lemma for ~ bnj6... |
| bnj1039 34281 | Technical lemma for ~ bnj6... |
| bnj1040 34282 | Technical lemma for ~ bnj6... |
| bnj1047 34283 | Technical lemma for ~ bnj6... |
| bnj1049 34284 | Technical lemma for ~ bnj6... |
| bnj1052 34285 | Technical lemma for ~ bnj6... |
| bnj1053 34286 | Technical lemma for ~ bnj6... |
| bnj1071 34287 | Technical lemma for ~ bnj6... |
| bnj1083 34288 | Technical lemma for ~ bnj6... |
| bnj1090 34289 | Technical lemma for ~ bnj6... |
| bnj1093 34290 | Technical lemma for ~ bnj6... |
| bnj1097 34291 | Technical lemma for ~ bnj6... |
| bnj1110 34292 | Technical lemma for ~ bnj6... |
| bnj1112 34293 | Technical lemma for ~ bnj6... |
| bnj1118 34294 | Technical lemma for ~ bnj6... |
| bnj1121 34295 | Technical lemma for ~ bnj6... |
| bnj1123 34296 | Technical lemma for ~ bnj6... |
| bnj1030 34297 | Technical lemma for ~ bnj6... |
| bnj1124 34298 | Property of ` _trCl ` . (... |
| bnj1133 34299 | Technical lemma for ~ bnj6... |
| bnj1128 34300 | Technical lemma for ~ bnj6... |
| bnj1127 34301 | Property of ` _trCl ` . (... |
| bnj1125 34302 | Property of ` _trCl ` . (... |
| bnj1145 34303 | Technical lemma for ~ bnj6... |
| bnj1147 34304 | Property of ` _trCl ` . (... |
| bnj1137 34305 | Property of ` _trCl ` . (... |
| bnj1148 34306 | Property of ` _pred ` . (... |
| bnj1136 34307 | Technical lemma for ~ bnj6... |
| bnj1152 34308 | Technical lemma for ~ bnj6... |
| bnj1154 34309 | Property of ` Fr ` . (Con... |
| bnj1171 34310 | Technical lemma for ~ bnj6... |
| bnj1172 34311 | Technical lemma for ~ bnj6... |
| bnj1173 34312 | Technical lemma for ~ bnj6... |
| bnj1174 34313 | Technical lemma for ~ bnj6... |
| bnj1175 34314 | Technical lemma for ~ bnj6... |
| bnj1176 34315 | Technical lemma for ~ bnj6... |
| bnj1177 34316 | Technical lemma for ~ bnj6... |
| bnj1186 34317 | Technical lemma for ~ bnj6... |
| bnj1190 34318 | Technical lemma for ~ bnj6... |
| bnj1189 34319 | Technical lemma for ~ bnj6... |
| bnj69 34320 | Existence of a minimal ele... |
| bnj1228 34321 | Existence of a minimal ele... |
| bnj1204 34322 | Well-founded induction. T... |
| bnj1234 34323 | Technical lemma for ~ bnj6... |
| bnj1245 34324 | Technical lemma for ~ bnj6... |
| bnj1256 34325 | Technical lemma for ~ bnj6... |
| bnj1259 34326 | Technical lemma for ~ bnj6... |
| bnj1253 34327 | Technical lemma for ~ bnj6... |
| bnj1279 34328 | Technical lemma for ~ bnj6... |
| bnj1286 34329 | Technical lemma for ~ bnj6... |
| bnj1280 34330 | Technical lemma for ~ bnj6... |
| bnj1296 34331 | Technical lemma for ~ bnj6... |
| bnj1309 34332 | Technical lemma for ~ bnj6... |
| bnj1307 34333 | Technical lemma for ~ bnj6... |
| bnj1311 34334 | Technical lemma for ~ bnj6... |
| bnj1318 34335 | Technical lemma for ~ bnj6... |
| bnj1326 34336 | Technical lemma for ~ bnj6... |
| bnj1321 34337 | Technical lemma for ~ bnj6... |
| bnj1364 34338 | Property of ` _FrSe ` . (... |
| bnj1371 34339 | Technical lemma for ~ bnj6... |
| bnj1373 34340 | Technical lemma for ~ bnj6... |
| bnj1374 34341 | Technical lemma for ~ bnj6... |
| bnj1384 34342 | Technical lemma for ~ bnj6... |
| bnj1388 34343 | Technical lemma for ~ bnj6... |
| bnj1398 34344 | Technical lemma for ~ bnj6... |
| bnj1413 34345 | Property of ` _trCl ` . (... |
| bnj1408 34346 | Technical lemma for ~ bnj1... |
| bnj1414 34347 | Property of ` _trCl ` . (... |
| bnj1415 34348 | Technical lemma for ~ bnj6... |
| bnj1416 34349 | Technical lemma for ~ bnj6... |
| bnj1418 34350 | Property of ` _pred ` . (... |
| bnj1417 34351 | Technical lemma for ~ bnj6... |
| bnj1421 34352 | Technical lemma for ~ bnj6... |
| bnj1444 34353 | Technical lemma for ~ bnj6... |
| bnj1445 34354 | Technical lemma for ~ bnj6... |
| bnj1446 34355 | Technical lemma for ~ bnj6... |
| bnj1447 34356 | Technical lemma for ~ bnj6... |
| bnj1448 34357 | Technical lemma for ~ bnj6... |
| bnj1449 34358 | Technical lemma for ~ bnj6... |
| bnj1442 34359 | Technical lemma for ~ bnj6... |
| bnj1450 34360 | Technical lemma for ~ bnj6... |
| bnj1423 34361 | Technical lemma for ~ bnj6... |
| bnj1452 34362 | Technical lemma for ~ bnj6... |
| bnj1466 34363 | Technical lemma for ~ bnj6... |
| bnj1467 34364 | Technical lemma for ~ bnj6... |
| bnj1463 34365 | Technical lemma for ~ bnj6... |
| bnj1489 34366 | Technical lemma for ~ bnj6... |
| bnj1491 34367 | Technical lemma for ~ bnj6... |
| bnj1312 34368 | Technical lemma for ~ bnj6... |
| bnj1493 34369 | Technical lemma for ~ bnj6... |
| bnj1497 34370 | Technical lemma for ~ bnj6... |
| bnj1498 34371 | Technical lemma for ~ bnj6... |
| bnj60 34372 | Well-founded recursion, pa... |
| bnj1514 34373 | Technical lemma for ~ bnj1... |
| bnj1518 34374 | Technical lemma for ~ bnj1... |
| bnj1519 34375 | Technical lemma for ~ bnj1... |
| bnj1520 34376 | Technical lemma for ~ bnj1... |
| bnj1501 34377 | Technical lemma for ~ bnj1... |
| bnj1500 34378 | Well-founded recursion, pa... |
| bnj1525 34379 | Technical lemma for ~ bnj1... |
| bnj1529 34380 | Technical lemma for ~ bnj1... |
| bnj1523 34381 | Technical lemma for ~ bnj1... |
| bnj1522 34382 | Well-founded recursion, pa... |
| exdifsn 34383 | There exists an element in... |
| srcmpltd 34384 | If a statement is true for... |
| prsrcmpltd 34385 | If a statement is true for... |
| dff15 34386 | A one-to-one function in t... |
| f1resveqaeq 34387 | If a function restricted t... |
| f1resrcmplf1dlem 34388 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 34389 | If a function's restrictio... |
| funen1cnv 34390 | If a function is equinumer... |
| fnrelpredd 34391 | A function that preserves ... |
| cardpred 34392 | The cardinality function p... |
| nummin 34393 | Every nonempty class of nu... |
| fineqvrep 34394 | If the Axiom of Infinity i... |
| fineqvpow 34395 | If the Axiom of Infinity i... |
| fineqvac 34396 | If the Axiom of Infinity i... |
| fineqvacALT 34397 | Shorter proof of ~ fineqva... |
| zltp1ne 34398 | Integer ordering relation.... |
| nnltp1ne 34399 | Positive integer ordering ... |
| nn0ltp1ne 34400 | Nonnegative integer orderi... |
| 0nn0m1nnn0 34401 | A number is zero if and on... |
| f1resfz0f1d 34402 | If a function with a seque... |
| fisshasheq 34403 | A finite set is equal to i... |
| hashf1dmcdm 34404 | The size of the domain of ... |
| revpfxsfxrev 34405 | The reverse of a prefix of... |
| swrdrevpfx 34406 | A subword expressed in ter... |
| lfuhgr 34407 | A hypergraph is loop-free ... |
| lfuhgr2 34408 | A hypergraph is loop-free ... |
| lfuhgr3 34409 | A hypergraph is loop-free ... |
| cplgredgex 34410 | Any two (distinct) vertice... |
| cusgredgex 34411 | Any two (distinct) vertice... |
| cusgredgex2 34412 | Any two distinct vertices ... |
| pfxwlk 34413 | A prefix of a walk is a wa... |
| revwlk 34414 | The reverse of a walk is a... |
| revwlkb 34415 | Two words represent a walk... |
| swrdwlk 34416 | Two matching subwords of a... |
| pthhashvtx 34417 | A graph containing a path ... |
| pthisspthorcycl 34418 | A path is either a simple ... |
| spthcycl 34419 | A walk is a trivial path i... |
| usgrgt2cycl 34420 | A non-trivial cycle in a s... |
| usgrcyclgt2v 34421 | A simple graph with a non-... |
| subgrwlk 34422 | If a walk exists in a subg... |
| subgrtrl 34423 | If a trail exists in a sub... |
| subgrpth 34424 | If a path exists in a subg... |
| subgrcycl 34425 | If a cycle exists in a sub... |
| cusgr3cyclex 34426 | Every complete simple grap... |
| loop1cycl 34427 | A hypergraph has a cycle o... |
| 2cycld 34428 | Construction of a 2-cycle ... |
| 2cycl2d 34429 | Construction of a 2-cycle ... |
| umgr2cycllem 34430 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 34431 | A multigraph with two dist... |
| dfacycgr1 34434 | An alternate definition of... |
| isacycgr 34435 | The property of being an a... |
| isacycgr1 34436 | The property of being an a... |
| acycgrcycl 34437 | Any cycle in an acyclic gr... |
| acycgr0v 34438 | A null graph (with no vert... |
| acycgr1v 34439 | A multigraph with one vert... |
| acycgr2v 34440 | A simple graph with two ve... |
| prclisacycgr 34441 | A proper class (representi... |
| acycgrislfgr 34442 | An acyclic hypergraph is a... |
| upgracycumgr 34443 | An acyclic pseudograph is ... |
| umgracycusgr 34444 | An acyclic multigraph is a... |
| upgracycusgr 34445 | An acyclic pseudograph is ... |
| cusgracyclt3v 34446 | A complete simple graph is... |
| pthacycspth 34447 | A path in an acyclic graph... |
| acycgrsubgr 34448 | The subgraph of an acyclic... |
| quartfull 34455 | The quartic equation, writ... |
| deranglem 34456 | Lemma for derangements. (... |
| derangval 34457 | Define the derangement fun... |
| derangf 34458 | The derangement number is ... |
| derang0 34459 | The derangement number of ... |
| derangsn 34460 | The derangement number of ... |
| derangenlem 34461 | One half of ~ derangen . ... |
| derangen 34462 | The derangement number is ... |
| subfacval 34463 | The subfactorial is define... |
| derangen2 34464 | Write the derangement numb... |
| subfacf 34465 | The subfactorial is a func... |
| subfaclefac 34466 | The subfactorial is less t... |
| subfac0 34467 | The subfactorial at zero. ... |
| subfac1 34468 | The subfactorial at one. ... |
| subfacp1lem1 34469 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 34470 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 34471 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 34472 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 34473 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 34474 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 34475 | Lemma for ~ subfacp1 . By... |
| subfacp1 34476 | A two-term recurrence for ... |
| subfacval2 34477 | A closed-form expression f... |
| subfaclim 34478 | The subfactorial converges... |
| subfacval3 34479 | Another closed form expres... |
| derangfmla 34480 | The derangements formula, ... |
| erdszelem1 34481 | Lemma for ~ erdsze . (Con... |
| erdszelem2 34482 | Lemma for ~ erdsze . (Con... |
| erdszelem3 34483 | Lemma for ~ erdsze . (Con... |
| erdszelem4 34484 | Lemma for ~ erdsze . (Con... |
| erdszelem5 34485 | Lemma for ~ erdsze . (Con... |
| erdszelem6 34486 | Lemma for ~ erdsze . (Con... |
| erdszelem7 34487 | Lemma for ~ erdsze . (Con... |
| erdszelem8 34488 | Lemma for ~ erdsze . (Con... |
| erdszelem9 34489 | Lemma for ~ erdsze . (Con... |
| erdszelem10 34490 | Lemma for ~ erdsze . (Con... |
| erdszelem11 34491 | Lemma for ~ erdsze . (Con... |
| erdsze 34492 | The Erdős-Szekeres th... |
| erdsze2lem1 34493 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 34494 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 34495 | Generalize the statement o... |
| kur14lem1 34496 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 34497 | Lemma for ~ kur14 . Write... |
| kur14lem3 34498 | Lemma for ~ kur14 . A clo... |
| kur14lem4 34499 | Lemma for ~ kur14 . Compl... |
| kur14lem5 34500 | Lemma for ~ kur14 . Closu... |
| kur14lem6 34501 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 34502 | Lemma for ~ kur14 : main p... |
| kur14lem8 34503 | Lemma for ~ kur14 . Show ... |
| kur14lem9 34504 | Lemma for ~ kur14 . Since... |
| kur14lem10 34505 | Lemma for ~ kur14 . Disch... |
| kur14 34506 | Kuratowski's closure-compl... |
| ispconn 34513 | The property of being a pa... |
| pconncn 34514 | The property of being a pa... |
| pconntop 34515 | A simply connected space i... |
| issconn 34516 | The property of being a si... |
| sconnpconn 34517 | A simply connected space i... |
| sconntop 34518 | A simply connected space i... |
| sconnpht 34519 | A closed path in a simply ... |
| cnpconn 34520 | An image of a path-connect... |
| pconnconn 34521 | A path-connected space is ... |
| txpconn 34522 | The topological product of... |
| ptpconn 34523 | The topological product of... |
| indispconn 34524 | The indiscrete topology (o... |
| connpconn 34525 | A connected and locally pa... |
| qtoppconn 34526 | A quotient of a path-conne... |
| pconnpi1 34527 | All fundamental groups in ... |
| sconnpht2 34528 | Any two paths in a simply ... |
| sconnpi1 34529 | A path-connected topologic... |
| txsconnlem 34530 | Lemma for ~ txsconn . (Co... |
| txsconn 34531 | The topological product of... |
| cvxpconn 34532 | A convex subset of the com... |
| cvxsconn 34533 | A convex subset of the com... |
| blsconn 34534 | An open ball in the comple... |
| cnllysconn 34535 | The topology of the comple... |
| resconn 34536 | A subset of ` RR ` is simp... |
| ioosconn 34537 | An open interval is simply... |
| iccsconn 34538 | A closed interval is simpl... |
| retopsconn 34539 | The real numbers are simpl... |
| iccllysconn 34540 | A closed interval is local... |
| rellysconn 34541 | The real numbers are local... |
| iisconn 34542 | The unit interval is simpl... |
| iillysconn 34543 | The unit interval is local... |
| iinllyconn 34544 | The unit interval is local... |
| fncvm 34547 | Lemma for covering maps. ... |
| cvmscbv 34548 | Change bound variables in ... |
| iscvm 34549 | The property of being a co... |
| cvmtop1 34550 | Reverse closure for a cove... |
| cvmtop2 34551 | Reverse closure for a cove... |
| cvmcn 34552 | A covering map is a contin... |
| cvmcov 34553 | Property of a covering map... |
| cvmsrcl 34554 | Reverse closure for an eve... |
| cvmsi 34555 | One direction of ~ cvmsval... |
| cvmsval 34556 | Elementhood in the set ` S... |
| cvmsss 34557 | An even covering is a subs... |
| cvmsn0 34558 | An even covering is nonemp... |
| cvmsuni 34559 | An even covering of ` U ` ... |
| cvmsdisj 34560 | An even covering of ` U ` ... |
| cvmshmeo 34561 | Every element of an even c... |
| cvmsf1o 34562 | ` F ` , localized to an el... |
| cvmscld 34563 | The sets of an even coveri... |
| cvmsss2 34564 | An open subset of an evenl... |
| cvmcov2 34565 | The covering map property ... |
| cvmseu 34566 | Every element in ` U. T ` ... |
| cvmsiota 34567 | Identify the unique elemen... |
| cvmopnlem 34568 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 34569 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 34570 | A covering map is an open ... |
| cvmliftmolem1 34571 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 34572 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 34573 | A lift of a continuous fun... |
| cvmliftmo 34574 | A lift of a continuous fun... |
| cvmliftlem1 34575 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 34576 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 34577 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 34578 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 34579 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 34580 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 34581 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 34582 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 34583 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 34584 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 34585 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 34586 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 34587 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 34588 | Lemma for ~ cvmlift . Dis... |
| cvmlift 34589 | One of the important prope... |
| cvmfo 34590 | A covering map is an onto ... |
| cvmliftiota 34591 | Write out a function ` H `... |
| cvmlift2lem1 34592 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 34593 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 34594 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 34595 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 34596 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 34597 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 34598 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 34599 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 34600 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 34601 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 34602 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 34603 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 34604 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 34605 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 34606 | A two-dimensional version ... |
| cvmliftphtlem 34607 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 34608 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 34609 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 34610 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 34611 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 34612 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 34613 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 34614 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 34615 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 34616 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 34617 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 34618 | A general version of ~ cvm... |
| snmlff 34619 | The function ` F ` from ~ ... |
| snmlfval 34620 | The function ` F ` from ~ ... |
| snmlval 34621 | The property " ` A ` is si... |
| snmlflim 34622 | If ` A ` is simply normal,... |
| goel 34637 | A "Godel-set of membership... |
| goelel3xp 34638 | A "Godel-set of membership... |
| goeleq12bg 34639 | Two "Godel-set of membersh... |
| gonafv 34640 | The "Godel-set for the She... |
| goaleq12d 34641 | Equality of the "Godel-set... |
| gonanegoal 34642 | The Godel-set for the Shef... |
| satf 34643 | The satisfaction predicate... |
| satfsucom 34644 | The satisfaction predicate... |
| satfn 34645 | The satisfaction predicate... |
| satom 34646 | The satisfaction predicate... |
| satfvsucom 34647 | The satisfaction predicate... |
| satfv0 34648 | The value of the satisfact... |
| satfvsuclem1 34649 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 34650 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 34651 | The value of the satisfact... |
| satfv1lem 34652 | Lemma for ~ satfv1 . (Con... |
| satfv1 34653 | The value of the satisfact... |
| satfsschain 34654 | The binary relation of a s... |
| satfvsucsuc 34655 | The satisfaction predicate... |
| satfbrsuc 34656 | The binary relation of a s... |
| satfrel 34657 | The value of the satisfact... |
| satfdmlem 34658 | Lemma for ~ satfdm . (Con... |
| satfdm 34659 | The domain of the satisfac... |
| satfrnmapom 34660 | The range of the satisfact... |
| satfv0fun 34661 | The value of the satisfact... |
| satf0 34662 | The satisfaction predicate... |
| satf0sucom 34663 | The satisfaction predicate... |
| satf00 34664 | The value of the satisfact... |
| satf0suclem 34665 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 34666 | The value of the satisfact... |
| satf0op 34667 | An element of a value of t... |
| satf0n0 34668 | The value of the satisfact... |
| sat1el2xp 34669 | The first component of an ... |
| fmlafv 34670 | The valid Godel formulas o... |
| fmla 34671 | The set of all valid Godel... |
| fmla0 34672 | The valid Godel formulas o... |
| fmla0xp 34673 | The valid Godel formulas o... |
| fmlasuc0 34674 | The valid Godel formulas o... |
| fmlafvel 34675 | A class is a valid Godel f... |
| fmlasuc 34676 | The valid Godel formulas o... |
| fmla1 34677 | The valid Godel formulas o... |
| isfmlasuc 34678 | The characterization of a ... |
| fmlasssuc 34679 | The Godel formulas of heig... |
| fmlaomn0 34680 | The empty set is not a God... |
| fmlan0 34681 | The empty set is not a God... |
| gonan0 34682 | The "Godel-set of NAND" is... |
| goaln0 34683 | The "Godel-set of universa... |
| gonarlem 34684 | Lemma for ~ gonar (inducti... |
| gonar 34685 | If the "Godel-set of NAND"... |
| goalrlem 34686 | Lemma for ~ goalr (inducti... |
| goalr 34687 | If the "Godel-set of unive... |
| fmla0disjsuc 34688 | The set of valid Godel for... |
| fmlasucdisj 34689 | The valid Godel formulas o... |
| satfdmfmla 34690 | The domain of the satisfac... |
| satffunlem 34691 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 34692 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 34693 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 34694 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 34695 | The domain of an ordered p... |
| satffunlem2lem2 34696 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 34697 | Lemma 1 for ~ satffun : in... |
| satffunlem2 34698 | Lemma 2 for ~ satffun : in... |
| satffun 34699 | The value of the satisfact... |
| satff 34700 | The satisfaction predicate... |
| satfun 34701 | The satisfaction predicate... |
| satfvel 34702 | An element of the value of... |
| satfv0fvfmla0 34703 | The value of the satisfact... |
| satefv 34704 | The simplified satisfactio... |
| sate0 34705 | The simplified satisfactio... |
| satef 34706 | The simplified satisfactio... |
| sate0fv0 34707 | A simplified satisfaction ... |
| satefvfmla0 34708 | The simplified satisfactio... |
| sategoelfvb 34709 | Characterization of a valu... |
| sategoelfv 34710 | Condition of a valuation `... |
| ex-sategoelel 34711 | Example of a valuation of ... |
| ex-sategoel 34712 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 34713 | The value of the satisfact... |
| 2goelgoanfmla1 34714 | Two Godel-sets of membersh... |
| satefvfmla1 34715 | The simplified satisfactio... |
| ex-sategoelelomsuc 34716 | Example of a valuation of ... |
| ex-sategoelel12 34717 | Example of a valuation of ... |
| prv 34718 | The "proves" relation on a... |
| elnanelprv 34719 | The wff ` ( A e. B -/\ B e... |
| prv0 34720 | Every wff encoded as ` U `... |
| prv1n 34721 | No wff encoded as a Godel-... |
| mvtval 34790 | The set of variable typeco... |
| mrexval 34791 | The set of "raw expression... |
| mexval 34792 | The set of expressions, wh... |
| mexval2 34793 | The set of expressions, wh... |
| mdvval 34794 | The set of disjoint variab... |
| mvrsval 34795 | The set of variables in an... |
| mvrsfpw 34796 | The set of variables in an... |
| mrsubffval 34797 | The substitution of some v... |
| mrsubfval 34798 | The substitution of some v... |
| mrsubval 34799 | The substitution of some v... |
| mrsubcv 34800 | The value of a substituted... |
| mrsubvr 34801 | The value of a substituted... |
| mrsubff 34802 | A substitution is a functi... |
| mrsubrn 34803 | Although it is defined for... |
| mrsubff1 34804 | When restricted to complet... |
| mrsubff1o 34805 | When restricted to complet... |
| mrsub0 34806 | The value of the substitut... |
| mrsubf 34807 | A substitution is a functi... |
| mrsubccat 34808 | Substitution distributes o... |
| mrsubcn 34809 | A substitution does not ch... |
| elmrsubrn 34810 | Characterization of the su... |
| mrsubco 34811 | The composition of two sub... |
| mrsubvrs 34812 | The set of variables in a ... |
| msubffval 34813 | A substitution applied to ... |
| msubfval 34814 | A substitution applied to ... |
| msubval 34815 | A substitution applied to ... |
| msubrsub 34816 | A substitution applied to ... |
| msubty 34817 | The type of a substituted ... |
| elmsubrn 34818 | Characterization of substi... |
| msubrn 34819 | Although it is defined for... |
| msubff 34820 | A substitution is a functi... |
| msubco 34821 | The composition of two sub... |
| msubf 34822 | A substitution is a functi... |
| mvhfval 34823 | Value of the function mapp... |
| mvhval 34824 | Value of the function mapp... |
| mpstval 34825 | A pre-statement is an orde... |
| elmpst 34826 | Property of being a pre-st... |
| msrfval 34827 | Value of the reduct of a p... |
| msrval 34828 | Value of the reduct of a p... |
| mpstssv 34829 | A pre-statement is an orde... |
| mpst123 34830 | Decompose a pre-statement ... |
| mpstrcl 34831 | The elements of a pre-stat... |
| msrf 34832 | The reduct of a pre-statem... |
| msrrcl 34833 | If ` X ` and ` Y ` have th... |
| mstaval 34834 | Value of the set of statem... |
| msrid 34835 | The reduct of a statement ... |
| msrfo 34836 | The reduct of a pre-statem... |
| mstapst 34837 | A statement is a pre-state... |
| elmsta 34838 | Property of being a statem... |
| ismfs 34839 | A formal system is a tuple... |
| mfsdisj 34840 | The constants and variable... |
| mtyf2 34841 | The type function maps var... |
| mtyf 34842 | The type function maps var... |
| mvtss 34843 | The set of variable typeco... |
| maxsta 34844 | An axiom is a statement. ... |
| mvtinf 34845 | Each variable typecode has... |
| msubff1 34846 | When restricted to complet... |
| msubff1o 34847 | When restricted to complet... |
| mvhf 34848 | The function mapping varia... |
| mvhf1 34849 | The function mapping varia... |
| msubvrs 34850 | The set of variables in a ... |
| mclsrcl 34851 | Reverse closure for the cl... |
| mclsssvlem 34852 | Lemma for ~ mclsssv . (Co... |
| mclsval 34853 | The function mapping varia... |
| mclsssv 34854 | The closure of a set of ex... |
| ssmclslem 34855 | Lemma for ~ ssmcls . (Con... |
| vhmcls 34856 | All variable hypotheses ar... |
| ssmcls 34857 | The original expressions a... |
| ss2mcls 34858 | The closure is monotonic u... |
| mclsax 34859 | The closure is closed unde... |
| mclsind 34860 | Induction theorem for clos... |
| mppspstlem 34861 | Lemma for ~ mppspst . (Co... |
| mppsval 34862 | Definition of a provable p... |
| elmpps 34863 | Definition of a provable p... |
| mppspst 34864 | A provable pre-statement i... |
| mthmval 34865 | A theorem is a pre-stateme... |
| elmthm 34866 | A theorem is a pre-stateme... |
| mthmi 34867 | A statement whose reduct i... |
| mthmsta 34868 | A theorem is a pre-stateme... |
| mppsthm 34869 | A provable pre-statement i... |
| mthmblem 34870 | Lemma for ~ mthmb . (Cont... |
| mthmb 34871 | If two statements have the... |
| mthmpps 34872 | Given a theorem, there is ... |
| mclsppslem 34873 | The closure is closed unde... |
| mclspps 34874 | The closure is closed unde... |
| problem1 34949 | Practice problem 1. Clues... |
| problem2 34950 | Practice problem 2. Clues... |
| problem3 34951 | Practice problem 3. Clues... |
| problem4 34952 | Practice problem 4. Clues... |
| problem5 34953 | Practice problem 5. Clues... |
| quad3 34954 | Variant of quadratic equat... |
| climuzcnv 34955 | Utility lemma to convert b... |
| sinccvglem 34956 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 34957 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 34958 | The circumference of a cir... |
| elfzm12 34959 | Membership in a curtailed ... |
| nn0seqcvg 34960 | A strictly-decreasing nonn... |
| lediv2aALT 34961 | Division of both sides of ... |
| abs2sqlei 34962 | The absolute values of two... |
| abs2sqlti 34963 | The absolute values of two... |
| abs2sqle 34964 | The absolute values of two... |
| abs2sqlt 34965 | The absolute values of two... |
| abs2difi 34966 | Difference of absolute val... |
| abs2difabsi 34967 | Absolute value of differen... |
| currybi 34968 | Biconditional version of C... |
| axextprim 34975 | ~ ax-ext without distinct ... |
| axrepprim 34976 | ~ ax-rep without distinct ... |
| axunprim 34977 | ~ ax-un without distinct v... |
| axpowprim 34978 | ~ ax-pow without distinct ... |
| axregprim 34979 | ~ ax-reg without distinct ... |
| axinfprim 34980 | ~ ax-inf without distinct ... |
| axacprim 34981 | ~ ax-ac without distinct v... |
| untelirr 34982 | We call a class "untanged"... |
| untuni 34983 | The union of a class is un... |
| untsucf 34984 | If a class is untangled, t... |
| unt0 34985 | The null set is untangled.... |
| untint 34986 | If there is an untangled e... |
| efrunt 34987 | If ` A ` is well-founded b... |
| untangtr 34988 | A transitive class is unta... |
| 3jaodd 34989 | Double deduction form of ~... |
| 3orit 34990 | Closed form of ~ 3ori . (... |
| biimpexp 34991 | A biconditional in the ant... |
| nepss 34992 | Two classes are unequal if... |
| 3ccased 34993 | Triple disjunction form of... |
| dfso3 34994 | Expansion of the definitio... |
| brtpid1 34995 | A binary relation involvin... |
| brtpid2 34996 | A binary relation involvin... |
| brtpid3 34997 | A binary relation involvin... |
| iota5f 34998 | A method for computing iot... |
| jath 34999 | Closed form of ~ ja . Pro... |
| xpab 35000 | Cartesian product of two c... |
| nnuni 35001 | The union of a finite ordi... |
| sqdivzi 35002 | Distribution of square ove... |
| supfz 35003 | The supremum of a finite s... |
| inffz 35004 | The infimum of a finite se... |
| fz0n 35005 | The sequence ` ( 0 ... ( N... |
| shftvalg 35006 | Value of a sequence shifte... |
| divcnvlin 35007 | Limit of the ratio of two ... |
| climlec3 35008 | Comparison of a constant t... |
| logi 35009 | Calculate the logarithm of... |
| iexpire 35010 | ` _i ` raised to itself is... |
| bcneg1 35011 | The binomial coefficent ov... |
| bcm1nt 35012 | The proportion of one bion... |
| bcprod 35013 | A product identity for bin... |
| bccolsum 35014 | A column-sum rule for bino... |
| iprodefisumlem 35015 | Lemma for ~ iprodefisum . ... |
| iprodefisum 35016 | Applying the exponential f... |
| iprodgam 35017 | An infinite product versio... |
| faclimlem1 35018 | Lemma for ~ faclim . Clos... |
| faclimlem2 35019 | Lemma for ~ faclim . Show... |
| faclimlem3 35020 | Lemma for ~ faclim . Alge... |
| faclim 35021 | An infinite product expres... |
| iprodfac 35022 | An infinite product expres... |
| faclim2 35023 | Another factorial limit du... |
| gcd32 35024 | Swap the second and third ... |
| gcdabsorb 35025 | Absorption law for gcd. (... |
| dftr6 35026 | A potential definition of ... |
| coep 35027 | Composition with the membe... |
| coepr 35028 | Composition with the conve... |
| dffr5 35029 | A quantifier-free definiti... |
| dfso2 35030 | Quantifier-free definition... |
| br8 35031 | Substitution for an eight-... |
| br6 35032 | Substitution for a six-pla... |
| br4 35033 | Substitution for a four-pl... |
| cnvco1 35034 | Another distributive law o... |
| cnvco2 35035 | Another distributive law o... |
| eldm3 35036 | Quantifier-free definition... |
| elrn3 35037 | Quantifier-free definition... |
| pocnv 35038 | The converse of a partial ... |
| socnv 35039 | The converse of a strict o... |
| sotrd 35040 | Transitivity law for stric... |
| elintfv 35041 | Membership in an intersect... |
| funpsstri 35042 | A condition for subset tri... |
| fundmpss 35043 | If a class ` F ` is a prop... |
| funsseq 35044 | Given two functions with e... |
| fununiq 35045 | The uniqueness condition o... |
| funbreq 35046 | An equality condition for ... |
| br1steq 35047 | Uniqueness condition for t... |
| br2ndeq 35048 | Uniqueness condition for t... |
| dfdm5 35049 | Definition of domain in te... |
| dfrn5 35050 | Definition of range in ter... |
| opelco3 35051 | Alternate way of saying th... |
| elima4 35052 | Quantifier-free expression... |
| fv1stcnv 35053 | The value of the converse ... |
| fv2ndcnv 35054 | The value of the converse ... |
| setinds 35055 | Principle of set induction... |
| setinds2f 35056 | ` _E ` induction schema, u... |
| setinds2 35057 | ` _E ` induction schema, u... |
| elpotr 35058 | A class of transitive sets... |
| dford5reg 35059 | Given ~ ax-reg , an ordina... |
| dfon2lem1 35060 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 35061 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 35062 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 35063 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 35064 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 35065 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 35066 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 35067 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 35068 | Lemma for ~ dfon2 . A cla... |
| dfon2 35069 | ` On ` consists of all set... |
| rdgprc0 35070 | The value of the recursive... |
| rdgprc 35071 | The value of the recursive... |
| dfrdg2 35072 | Alternate definition of th... |
| dfrdg3 35073 | Generalization of ~ dfrdg2... |
| axextdfeq 35074 | A version of ~ ax-ext for ... |
| ax8dfeq 35075 | A version of ~ ax-8 for us... |
| axextdist 35076 | ~ ax-ext with distinctors ... |
| axextbdist 35077 | ~ axextb with distinctors ... |
| 19.12b 35078 | Version of ~ 19.12vv with ... |
| exnel 35079 | There is always a set not ... |
| distel 35080 | Distinctors in terms of me... |
| axextndbi 35081 | ~ axextnd as a bicondition... |
| hbntg 35082 | A more general form of ~ h... |
| hbimtg 35083 | A more general and closed ... |
| hbaltg 35084 | A more general and closed ... |
| hbng 35085 | A more general form of ~ h... |
| hbimg 35086 | A more general form of ~ h... |
| wsuceq123 35091 | Equality theorem for well-... |
| wsuceq1 35092 | Equality theorem for well-... |
| wsuceq2 35093 | Equality theorem for well-... |
| wsuceq3 35094 | Equality theorem for well-... |
| nfwsuc 35095 | Bound-variable hypothesis ... |
| wlimeq12 35096 | Equality theorem for the l... |
| wlimeq1 35097 | Equality theorem for the l... |
| wlimeq2 35098 | Equality theorem for the l... |
| nfwlim 35099 | Bound-variable hypothesis ... |
| elwlim 35100 | Membership in the limit cl... |
| wzel 35101 | The zero of a well-founded... |
| wsuclem 35102 | Lemma for the supremum pro... |
| wsucex 35103 | Existence theorem for well... |
| wsuccl 35104 | If ` X ` is a set with an ... |
| wsuclb 35105 | A well-founded successor i... |
| wlimss 35106 | The class of limit points ... |
| txpss3v 35155 | A tail Cartesian product i... |
| txprel 35156 | A tail Cartesian product i... |
| brtxp 35157 | Characterize a ternary rel... |
| brtxp2 35158 | The binary relation over a... |
| dfpprod2 35159 | Expanded definition of par... |
| pprodcnveq 35160 | A converse law for paralle... |
| pprodss4v 35161 | The parallel product is a ... |
| brpprod 35162 | Characterize a quaternary ... |
| brpprod3a 35163 | Condition for parallel pro... |
| brpprod3b 35164 | Condition for parallel pro... |
| relsset 35165 | The subset class is a bina... |
| brsset 35166 | For sets, the ` SSet ` bin... |
| idsset 35167 | ` _I ` is equal to the int... |
| eltrans 35168 | Membership in the class of... |
| dfon3 35169 | A quantifier-free definiti... |
| dfon4 35170 | Another quantifier-free de... |
| brtxpsd 35171 | Expansion of a common form... |
| brtxpsd2 35172 | Another common abbreviatio... |
| brtxpsd3 35173 | A third common abbreviatio... |
| relbigcup 35174 | The ` Bigcup ` relationshi... |
| brbigcup 35175 | Binary relation over ` Big... |
| dfbigcup2 35176 | ` Bigcup ` using maps-to n... |
| fobigcup 35177 | ` Bigcup ` maps the univer... |
| fnbigcup 35178 | ` Bigcup ` is a function o... |
| fvbigcup 35179 | For sets, ` Bigcup ` yield... |
| elfix 35180 | Membership in the fixpoint... |
| elfix2 35181 | Alternative membership in ... |
| dffix2 35182 | The fixpoints of a class i... |
| fixssdm 35183 | The fixpoints of a class a... |
| fixssrn 35184 | The fixpoints of a class a... |
| fixcnv 35185 | The fixpoints of a class a... |
| fixun 35186 | The fixpoint operator dist... |
| ellimits 35187 | Membership in the class of... |
| limitssson 35188 | The class of all limit ord... |
| dfom5b 35189 | A quantifier-free definiti... |
| sscoid 35190 | A condition for subset and... |
| dffun10 35191 | Another potential definiti... |
| elfuns 35192 | Membership in the class of... |
| elfunsg 35193 | Closed form of ~ elfuns . ... |
| brsingle 35194 | The binary relation form o... |
| elsingles 35195 | Membership in the class of... |
| fnsingle 35196 | The singleton relationship... |
| fvsingle 35197 | The value of the singleton... |
| dfsingles2 35198 | Alternate definition of th... |
| snelsingles 35199 | A singleton is a member of... |
| dfiota3 35200 | A definition of iota using... |
| dffv5 35201 | Another quantifier-free de... |
| unisnif 35202 | Express union of singleton... |
| brimage 35203 | Binary relation form of th... |
| brimageg 35204 | Closed form of ~ brimage .... |
| funimage 35205 | ` Image A ` is a function.... |
| fnimage 35206 | ` Image R ` is a function ... |
| imageval 35207 | The image functor in maps-... |
| fvimage 35208 | Value of the image functor... |
| brcart 35209 | Binary relation form of th... |
| brdomain 35210 | Binary relation form of th... |
| brrange 35211 | Binary relation form of th... |
| brdomaing 35212 | Closed form of ~ brdomain ... |
| brrangeg 35213 | Closed form of ~ brrange .... |
| brimg 35214 | Binary relation form of th... |
| brapply 35215 | Binary relation form of th... |
| brcup 35216 | Binary relation form of th... |
| brcap 35217 | Binary relation form of th... |
| brsuccf 35218 | Binary relation form of th... |
| funpartlem 35219 | Lemma for ~ funpartfun . ... |
| funpartfun 35220 | The functional part of ` F... |
| funpartss 35221 | The functional part of ` F... |
| funpartfv 35222 | The function value of the ... |
| fullfunfnv 35223 | The full functional part o... |
| fullfunfv 35224 | The function value of the ... |
| brfullfun 35225 | A binary relation form con... |
| brrestrict 35226 | Binary relation form of th... |
| dfrecs2 35227 | A quantifier-free definiti... |
| dfrdg4 35228 | A quantifier-free definiti... |
| dfint3 35229 | Quantifier-free definition... |
| imagesset 35230 | The Image functor applied ... |
| brub 35231 | Binary relation form of th... |
| brlb 35232 | Binary relation form of th... |
| altopex 35237 | Alternative ordered pairs ... |
| altopthsn 35238 | Two alternate ordered pair... |
| altopeq12 35239 | Equality for alternate ord... |
| altopeq1 35240 | Equality for alternate ord... |
| altopeq2 35241 | Equality for alternate ord... |
| altopth1 35242 | Equality of the first memb... |
| altopth2 35243 | Equality of the second mem... |
| altopthg 35244 | Alternate ordered pair the... |
| altopthbg 35245 | Alternate ordered pair the... |
| altopth 35246 | The alternate ordered pair... |
| altopthb 35247 | Alternate ordered pair the... |
| altopthc 35248 | Alternate ordered pair the... |
| altopthd 35249 | Alternate ordered pair the... |
| altxpeq1 35250 | Equality for alternate Car... |
| altxpeq2 35251 | Equality for alternate Car... |
| elaltxp 35252 | Membership in alternate Ca... |
| altopelaltxp 35253 | Alternate ordered pair mem... |
| altxpsspw 35254 | An inclusion rule for alte... |
| altxpexg 35255 | The alternate Cartesian pr... |
| rankaltopb 35256 | Compute the rank of an alt... |
| nfaltop 35257 | Bound-variable hypothesis ... |
| sbcaltop 35258 | Distribution of class subs... |
| cgrrflx2d 35261 | Deduction form of ~ axcgrr... |
| cgrtr4d 35262 | Deduction form of ~ axcgrt... |
| cgrtr4and 35263 | Deduction form of ~ axcgrt... |
| cgrrflx 35264 | Reflexivity law for congru... |
| cgrrflxd 35265 | Deduction form of ~ cgrrfl... |
| cgrcomim 35266 | Congruence commutes on the... |
| cgrcom 35267 | Congruence commutes betwee... |
| cgrcomand 35268 | Deduction form of ~ cgrcom... |
| cgrtr 35269 | Transitivity law for congr... |
| cgrtrand 35270 | Deduction form of ~ cgrtr ... |
| cgrtr3 35271 | Transitivity law for congr... |
| cgrtr3and 35272 | Deduction form of ~ cgrtr3... |
| cgrcoml 35273 | Congruence commutes on the... |
| cgrcomr 35274 | Congruence commutes on the... |
| cgrcomlr 35275 | Congruence commutes on bot... |
| cgrcomland 35276 | Deduction form of ~ cgrcom... |
| cgrcomrand 35277 | Deduction form of ~ cgrcom... |
| cgrcomlrand 35278 | Deduction form of ~ cgrcom... |
| cgrtriv 35279 | Degenerate segments are co... |
| cgrid2 35280 | Identity law for congruenc... |
| cgrdegen 35281 | Two congruent segments are... |
| brofs 35282 | Binary relation form of th... |
| 5segofs 35283 | Rephrase ~ ax5seg using th... |
| ofscom 35284 | The outer five segment pre... |
| cgrextend 35285 | Link congruence over a pai... |
| cgrextendand 35286 | Deduction form of ~ cgrext... |
| segconeq 35287 | Two points that satisfy th... |
| segconeu 35288 | Existential uniqueness ver... |
| btwntriv2 35289 | Betweenness always holds f... |
| btwncomim 35290 | Betweenness commutes. Imp... |
| btwncom 35291 | Betweenness commutes. (Co... |
| btwncomand 35292 | Deduction form of ~ btwnco... |
| btwntriv1 35293 | Betweenness always holds f... |
| btwnswapid 35294 | If you can swap the first ... |
| btwnswapid2 35295 | If you can swap arguments ... |
| btwnintr 35296 | Inner transitivity law for... |
| btwnexch3 35297 | Exchange the first endpoin... |
| btwnexch3and 35298 | Deduction form of ~ btwnex... |
| btwnouttr2 35299 | Outer transitivity law for... |
| btwnexch2 35300 | Exchange the outer point o... |
| btwnouttr 35301 | Outer transitivity law for... |
| btwnexch 35302 | Outer transitivity law for... |
| btwnexchand 35303 | Deduction form of ~ btwnex... |
| btwndiff 35304 | There is always a ` c ` di... |
| trisegint 35305 | A line segment between two... |
| funtransport 35308 | The ` TransportTo ` relati... |
| fvtransport 35309 | Calculate the value of the... |
| transportcl 35310 | Closure law for segment tr... |
| transportprops 35311 | Calculate the defining pro... |
| brifs 35320 | Binary relation form of th... |
| ifscgr 35321 | Inner five segment congrue... |
| cgrsub 35322 | Removing identical parts f... |
| brcgr3 35323 | Binary relation form of th... |
| cgr3permute3 35324 | Permutation law for three-... |
| cgr3permute1 35325 | Permutation law for three-... |
| cgr3permute2 35326 | Permutation law for three-... |
| cgr3permute4 35327 | Permutation law for three-... |
| cgr3permute5 35328 | Permutation law for three-... |
| cgr3tr4 35329 | Transitivity law for three... |
| cgr3com 35330 | Commutativity law for thre... |
| cgr3rflx 35331 | Identity law for three-pla... |
| cgrxfr 35332 | A line segment can be divi... |
| btwnxfr 35333 | A condition for extending ... |
| colinrel 35334 | Colinearity is a relations... |
| brcolinear2 35335 | Alternate colinearity bina... |
| brcolinear 35336 | The binary relation form o... |
| colinearex 35337 | The colinear predicate exi... |
| colineardim1 35338 | If ` A ` is colinear with ... |
| colinearperm1 35339 | Permutation law for coline... |
| colinearperm3 35340 | Permutation law for coline... |
| colinearperm2 35341 | Permutation law for coline... |
| colinearperm4 35342 | Permutation law for coline... |
| colinearperm5 35343 | Permutation law for coline... |
| colineartriv1 35344 | Trivial case of colinearit... |
| colineartriv2 35345 | Trivial case of colinearit... |
| btwncolinear1 35346 | Betweenness implies coline... |
| btwncolinear2 35347 | Betweenness implies coline... |
| btwncolinear3 35348 | Betweenness implies coline... |
| btwncolinear4 35349 | Betweenness implies coline... |
| btwncolinear5 35350 | Betweenness implies coline... |
| btwncolinear6 35351 | Betweenness implies coline... |
| colinearxfr 35352 | Transfer law for colineari... |
| lineext 35353 | Extend a line with a missi... |
| brofs2 35354 | Change some conditions for... |
| brifs2 35355 | Change some conditions for... |
| brfs 35356 | Binary relation form of th... |
| fscgr 35357 | Congruence law for the gen... |
| linecgr 35358 | Congruence rule for lines.... |
| linecgrand 35359 | Deduction form of ~ linecg... |
| lineid 35360 | Identity law for points on... |
| idinside 35361 | Law for finding a point in... |
| endofsegid 35362 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 35363 | Deduction form of ~ endofs... |
| btwnconn1lem1 35364 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 35365 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 35366 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 35367 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 35368 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 35369 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 35370 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 35371 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 35372 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 35373 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 35374 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 35375 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 35376 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 35377 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 35378 | Connectitivy law for betwe... |
| btwnconn2 35379 | Another connectivity law f... |
| btwnconn3 35380 | Inner connectivity law for... |
| midofsegid 35381 | If two points fall in the ... |
| segcon2 35382 | Generalization of ~ axsegc... |
| brsegle 35385 | Binary relation form of th... |
| brsegle2 35386 | Alternate characterization... |
| seglecgr12im 35387 | Substitution law for segme... |
| seglecgr12 35388 | Substitution law for segme... |
| seglerflx 35389 | Segment comparison is refl... |
| seglemin 35390 | Any segment is at least as... |
| segletr 35391 | Segment less than is trans... |
| segleantisym 35392 | Antisymmetry law for segme... |
| seglelin 35393 | Linearity law for segment ... |
| btwnsegle 35394 | If ` B ` falls between ` A... |
| colinbtwnle 35395 | Given three colinear point... |
| broutsideof 35398 | Binary relation form of ` ... |
| broutsideof2 35399 | Alternate form of ` Outsid... |
| outsidene1 35400 | Outsideness implies inequa... |
| outsidene2 35401 | Outsideness implies inequa... |
| btwnoutside 35402 | A principle linking outsid... |
| broutsideof3 35403 | Characterization of outsid... |
| outsideofrflx 35404 | Reflexivity of outsideness... |
| outsideofcom 35405 | Commutativity law for outs... |
| outsideoftr 35406 | Transitivity law for outsi... |
| outsideofeq 35407 | Uniqueness law for ` Outsi... |
| outsideofeu 35408 | Given a nondegenerate ray,... |
| outsidele 35409 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 35410 | Outside of implies colinea... |
| funray 35417 | Show that the ` Ray ` rela... |
| fvray 35418 | Calculate the value of the... |
| funline 35419 | Show that the ` Line ` rel... |
| linedegen 35420 | When ` Line ` is applied w... |
| fvline 35421 | Calculate the value of the... |
| liness 35422 | A line is a subset of the ... |
| fvline2 35423 | Alternate definition of a ... |
| lineunray 35424 | A line is composed of a po... |
| lineelsb2 35425 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 35426 | Reflexivity law for line m... |
| linecom 35427 | Commutativity law for line... |
| linerflx2 35428 | Reflexivity law for line m... |
| ellines 35429 | Membership in the set of a... |
| linethru 35430 | If ` A ` is a line contain... |
| hilbert1.1 35431 | There is a line through an... |
| hilbert1.2 35432 | There is at most one line ... |
| linethrueu 35433 | There is a unique line goi... |
| lineintmo 35434 | Two distinct lines interse... |
| fwddifval 35439 | Calculate the value of the... |
| fwddifnval 35440 | The value of the forward d... |
| fwddifn0 35441 | The value of the n-iterate... |
| fwddifnp1 35442 | The value of the n-iterate... |
| rankung 35443 | The rank of the union of t... |
| ranksng 35444 | The rank of a singleton. ... |
| rankelg 35445 | The membership relation is... |
| rankpwg 35446 | The rank of a power set. ... |
| rank0 35447 | The rank of the empty set ... |
| rankeq1o 35448 | The only set with rank ` 1... |
| elhf 35451 | Membership in the heredita... |
| elhf2 35452 | Alternate form of membersh... |
| elhf2g 35453 | Hereditarily finiteness vi... |
| 0hf 35454 | The empty set is a heredit... |
| hfun 35455 | The union of two HF sets i... |
| hfsn 35456 | The singleton of an HF set... |
| hfadj 35457 | Adjoining one HF element t... |
| hfelhf 35458 | Any member of an HF set is... |
| hftr 35459 | The class of all hereditar... |
| hfext 35460 | Extensionality for HF sets... |
| hfuni 35461 | The union of an HF set is ... |
| hfpw 35462 | The power class of an HF s... |
| hfninf 35463 | ` _om ` is not hereditaril... |
| mpomulex 35464 | The multiplication operati... |
| gg-psercn2 35465 | Since by ~ pserulm the ser... |
| gg-rmulccn 35466 | Multiplication by a real c... |
| gg-cmvth 35467 | Cauchy's Mean Value Theore... |
| gg-cnfldex 35468 | The field of complex numbe... |
| gg-dvfsumle 35469 | Compare a finite sum to an... |
| gg-dvfsumlem2 35470 | Lemma for ~ dvfsumrlim . ... |
| gg-cxpcn 35471 | Domain of continuity of th... |
| mpoaddf 35472 | Addition is an operation o... |
| mpoaddex 35473 | The addition operation is ... |
| gg-dfcnfld 35474 | Alternative definition of ... |
| gg-cnfldstr 35475 | The field of complex numbe... |
| gg-cnfldbas 35476 | The base set of the field ... |
| mpocnfldadd 35477 | The addition operation of ... |
| mpocnfldmul 35478 | The multiplication operati... |
| gg-cnfldcj 35479 | The conjugation operation ... |
| gg-cnfldtset 35480 | The topology component of ... |
| gg-cnfldle 35481 | The ordering of the field ... |
| gg-cnfldds 35482 | The metric of the field of... |
| gg-cnfldunif 35483 | The uniform structure comp... |
| gg-cnfldfun 35484 | The field of complex numbe... |
| gg-cnfldfunALT 35485 | The field of complex numbe... |
| gg-cffldtocusgr 35486 | The field of complex numbe... |
| gg-cncrng 35487 | The complex numbers form a... |
| gg-cnfld1 35488 | One is the unity element o... |
| a1i14 35489 | Add two antecedents to a w... |
| a1i24 35490 | Add two antecedents to a w... |
| exp5d 35491 | An exportation inference. ... |
| exp5g 35492 | An exportation inference. ... |
| exp5k 35493 | An exportation inference. ... |
| exp56 35494 | An exportation inference. ... |
| exp58 35495 | An exportation inference. ... |
| exp510 35496 | An exportation inference. ... |
| exp511 35497 | An exportation inference. ... |
| exp512 35498 | An exportation inference. ... |
| 3com12d 35499 | Commutation in consequent.... |
| imp5p 35500 | A triple importation infer... |
| imp5q 35501 | A triple importation infer... |
| ecase13d 35502 | Deduction for elimination ... |
| subtr 35503 | Transitivity of implicit s... |
| subtr2 35504 | Transitivity of implicit s... |
| trer 35505 | A relation intersected wit... |
| elicc3 35506 | An equivalent membership c... |
| finminlem 35507 | A useful lemma about finit... |
| gtinf 35508 | Any number greater than an... |
| opnrebl 35509 | A set is open in the stand... |
| opnrebl2 35510 | A set is open in the stand... |
| nn0prpwlem 35511 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 35512 | Two nonnegative integers a... |
| topbnd 35513 | Two equivalent expressions... |
| opnbnd 35514 | A set is open iff it is di... |
| cldbnd 35515 | A set is closed iff it con... |
| ntruni 35516 | A union of interiors is a ... |
| clsun 35517 | A pairwise union of closur... |
| clsint2 35518 | The closure of an intersec... |
| opnregcld 35519 | A set is regularly closed ... |
| cldregopn 35520 | A set if regularly open if... |
| neiin 35521 | Two neighborhoods intersec... |
| hmeoclda 35522 | Homeomorphisms preserve cl... |
| hmeocldb 35523 | Homeomorphisms preserve cl... |
| ivthALT 35524 | An alternate proof of the ... |
| fnerel 35527 | Fineness is a relation. (... |
| isfne 35528 | The predicate " ` B ` is f... |
| isfne4 35529 | The predicate " ` B ` is f... |
| isfne4b 35530 | A condition for a topology... |
| isfne2 35531 | The predicate " ` B ` is f... |
| isfne3 35532 | The predicate " ` B ` is f... |
| fnebas 35533 | A finer cover covers the s... |
| fnetg 35534 | A finer cover generates a ... |
| fnessex 35535 | If ` B ` is finer than ` A... |
| fneuni 35536 | If ` B ` is finer than ` A... |
| fneint 35537 | If a cover is finer than a... |
| fness 35538 | A cover is finer than its ... |
| fneref 35539 | Reflexivity of the finenes... |
| fnetr 35540 | Transitivity of the finene... |
| fneval 35541 | Two covers are finer than ... |
| fneer 35542 | Fineness intersected with ... |
| topfne 35543 | Fineness for covers corres... |
| topfneec 35544 | A cover is equivalent to a... |
| topfneec2 35545 | A topology is precisely id... |
| fnessref 35546 | A cover is finer iff it ha... |
| refssfne 35547 | A cover is a refinement if... |
| neibastop1 35548 | A collection of neighborho... |
| neibastop2lem 35549 | Lemma for ~ neibastop2 . ... |
| neibastop2 35550 | In the topology generated ... |
| neibastop3 35551 | The topology generated by ... |
| topmtcl 35552 | The meet of a collection o... |
| topmeet 35553 | Two equivalent formulation... |
| topjoin 35554 | Two equivalent formulation... |
| fnemeet1 35555 | The meet of a collection o... |
| fnemeet2 35556 | The meet of equivalence cl... |
| fnejoin1 35557 | Join of equivalence classe... |
| fnejoin2 35558 | Join of equivalence classe... |
| fgmin 35559 | Minimality property of a g... |
| neifg 35560 | The neighborhood filter of... |
| tailfval 35561 | The tail function for a di... |
| tailval 35562 | The tail of an element in ... |
| eltail 35563 | An element of a tail. (Co... |
| tailf 35564 | The tail function of a dir... |
| tailini 35565 | A tail contains its initia... |
| tailfb 35566 | The collection of tails of... |
| filnetlem1 35567 | Lemma for ~ filnet . Chan... |
| filnetlem2 35568 | Lemma for ~ filnet . The ... |
| filnetlem3 35569 | Lemma for ~ filnet . (Con... |
| filnetlem4 35570 | Lemma for ~ filnet . (Con... |
| filnet 35571 | A filter has the same conv... |
| tb-ax1 35572 | The first of three axioms ... |
| tb-ax2 35573 | The second of three axioms... |
| tb-ax3 35574 | The third of three axioms ... |
| tbsyl 35575 | The weak syllogism from Ta... |
| re1ax2lem 35576 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 35577 | ~ ax-2 rederived from the ... |
| naim1 35578 | Constructor theorem for ` ... |
| naim2 35579 | Constructor theorem for ` ... |
| naim1i 35580 | Constructor rule for ` -/\... |
| naim2i 35581 | Constructor rule for ` -/\... |
| naim12i 35582 | Constructor rule for ` -/\... |
| nabi1i 35583 | Constructor rule for ` -/\... |
| nabi2i 35584 | Constructor rule for ` -/\... |
| nabi12i 35585 | Constructor rule for ` -/\... |
| df3nandALT1 35588 | The double nand expressed ... |
| df3nandALT2 35589 | The double nand expressed ... |
| andnand1 35590 | Double and in terms of dou... |
| imnand2 35591 | An ` -> ` nand relation. ... |
| nalfal 35592 | Not all sets hold ` F. ` a... |
| nexntru 35593 | There does not exist a set... |
| nexfal 35594 | There does not exist a set... |
| neufal 35595 | There does not exist exact... |
| neutru 35596 | There does not exist exact... |
| nmotru 35597 | There does not exist at mo... |
| mofal 35598 | There exist at most one se... |
| nrmo 35599 | "At most one" restricted e... |
| meran1 35600 | A single axiom for proposi... |
| meran2 35601 | A single axiom for proposi... |
| meran3 35602 | A single axiom for proposi... |
| waj-ax 35603 | A single axiom for proposi... |
| lukshef-ax2 35604 | A single axiom for proposi... |
| arg-ax 35605 | A single axiom for proposi... |
| negsym1 35606 | In the paper "On Variable ... |
| imsym1 35607 | A symmetry with ` -> ` . ... |
| bisym1 35608 | A symmetry with ` <-> ` . ... |
| consym1 35609 | A symmetry with ` /\ ` . ... |
| dissym1 35610 | A symmetry with ` \/ ` . ... |
| nandsym1 35611 | A symmetry with ` -/\ ` . ... |
| unisym1 35612 | A symmetry with ` A. ` . ... |
| exisym1 35613 | A symmetry with ` E. ` . ... |
| unqsym1 35614 | A symmetry with ` E! ` . ... |
| amosym1 35615 | A symmetry with ` E* ` . ... |
| subsym1 35616 | A symmetry with ` [ x / y ... |
| ontopbas 35617 | An ordinal number is a top... |
| onsstopbas 35618 | The class of ordinal numbe... |
| onpsstopbas 35619 | The class of ordinal numbe... |
| ontgval 35620 | The topology generated fro... |
| ontgsucval 35621 | The topology generated fro... |
| onsuctop 35622 | A successor ordinal number... |
| onsuctopon 35623 | One of the topologies on a... |
| ordtoplem 35624 | Membership of the class of... |
| ordtop 35625 | An ordinal is a topology i... |
| onsucconni 35626 | A successor ordinal number... |
| onsucconn 35627 | A successor ordinal number... |
| ordtopconn 35628 | An ordinal topology is con... |
| onintopssconn 35629 | An ordinal topology is con... |
| onsuct0 35630 | A successor ordinal number... |
| ordtopt0 35631 | An ordinal topology is T_0... |
| onsucsuccmpi 35632 | The successor of a success... |
| onsucsuccmp 35633 | The successor of a success... |
| limsucncmpi 35634 | The successor of a limit o... |
| limsucncmp 35635 | The successor of a limit o... |
| ordcmp 35636 | An ordinal topology is com... |
| ssoninhaus 35637 | The ordinal topologies ` 1... |
| onint1 35638 | The ordinal T_1 spaces are... |
| oninhaus 35639 | The ordinal Hausdorff spac... |
| fveleq 35640 | Please add description her... |
| findfvcl 35641 | Please add description her... |
| findreccl 35642 | Please add description her... |
| findabrcl 35643 | Please add description her... |
| nnssi2 35644 | Convert a theorem for real... |
| nnssi3 35645 | Convert a theorem for real... |
| nndivsub 35646 | Please add description her... |
| nndivlub 35647 | A factor of a positive int... |
| ee7.2aOLD 35650 | Lemma for Euclid's Element... |
| dnival 35651 | Value of the "distance to ... |
| dnicld1 35652 | Closure theorem for the "d... |
| dnicld2 35653 | Closure theorem for the "d... |
| dnif 35654 | The "distance to nearest i... |
| dnizeq0 35655 | The distance to nearest in... |
| dnizphlfeqhlf 35656 | The distance to nearest in... |
| rddif2 35657 | Variant of ~ rddif . (Con... |
| dnibndlem1 35658 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 35659 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 35660 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 35661 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 35662 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 35663 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 35664 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 35665 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 35666 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 35667 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 35668 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 35669 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 35670 | Lemma for ~ dnibnd . (Con... |
| dnibnd 35671 | The "distance to nearest i... |
| dnicn 35672 | The "distance to nearest i... |
| knoppcnlem1 35673 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 35674 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 35675 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 35676 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 35677 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 35678 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 35679 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 35680 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 35681 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 35682 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 35683 | Lemma for ~ knoppcn . (Co... |
| knoppcn 35684 | The continuous nowhere dif... |
| knoppcld 35685 | Closure theorem for Knopp'... |
| unblimceq0lem 35686 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 35687 | If ` F ` is unbounded near... |
| unbdqndv1 35688 | If the difference quotient... |
| unbdqndv2lem1 35689 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 35690 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 35691 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 35692 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 35693 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 35694 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 35695 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 35696 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 35697 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 35698 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 35699 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 35700 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 35701 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 35702 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 35703 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 35704 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 35705 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 35706 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 35707 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 35708 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 35709 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 35710 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 35711 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 35712 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 35713 | Lemma for ~ knoppndv . (C... |
| knoppndv 35714 | The continuous nowhere dif... |
| knoppf 35715 | Knopp's function is a func... |
| knoppcn2 35716 | Variant of ~ knoppcn with ... |
| cnndvlem1 35717 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 35718 | Lemma for ~ cnndv . (Cont... |
| cnndv 35719 | There exists a continuous ... |
| bj-mp2c 35720 | A double modus ponens infe... |
| bj-mp2d 35721 | A double modus ponens infe... |
| bj-0 35722 | A syntactic theorem. See ... |
| bj-1 35723 | In this proof, the use of ... |
| bj-a1k 35724 | Weakening of ~ ax-1 . As ... |
| bj-poni 35725 | Inference associated with ... |
| bj-nnclav 35726 | When ` F. ` is substituted... |
| bj-nnclavi 35727 | Inference associated with ... |
| bj-nnclavc 35728 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 35729 | Inference associated with ... |
| bj-jarrii 35730 | Inference associated with ... |
| bj-imim21 35731 | The propositional function... |
| bj-imim21i 35732 | Inference associated with ... |
| bj-peircestab 35733 | Over minimal implicational... |
| bj-stabpeirce 35734 | This minimal implicational... |
| bj-syl66ib 35735 | A mixed syllogism inferenc... |
| bj-orim2 35736 | Proof of ~ orim2 from the ... |
| bj-currypeirce 35737 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 35738 | Peirce's axiom ~ peirce im... |
| bj-animbi 35739 | Conjunction in terms of im... |
| bj-currypara 35740 | Curry's paradox. Note tha... |
| bj-con2com 35741 | A commuted form of the con... |
| bj-con2comi 35742 | Inference associated with ... |
| bj-pm2.01i 35743 | Inference associated with ... |
| bj-nimn 35744 | If a formula is true, then... |
| bj-nimni 35745 | Inference associated with ... |
| bj-peircei 35746 | Inference associated with ... |
| bj-looinvi 35747 | Inference associated with ... |
| bj-looinvii 35748 | Inference associated with ... |
| bj-mt2bi 35749 | Version of ~ mt2 where the... |
| bj-ntrufal 35750 | The negation of a theorem ... |
| bj-fal 35751 | Shortening of ~ fal using ... |
| bj-jaoi1 35752 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 35753 | Shortens ~ consensus (110>... |
| bj-dfbi4 35754 | Alternate definition of th... |
| bj-dfbi5 35755 | Alternate definition of th... |
| bj-dfbi6 35756 | Alternate definition of th... |
| bj-bijust0ALT 35757 | Alternate proof of ~ bijus... |
| bj-bijust00 35758 | A self-implication does no... |
| bj-consensus 35759 | Version of ~ consensus exp... |
| bj-consensusALT 35760 | Alternate proof of ~ bj-co... |
| bj-df-ifc 35761 | Candidate definition for t... |
| bj-dfif 35762 | Alternate definition of th... |
| bj-ififc 35763 | A biconditional connecting... |
| bj-imbi12 35764 | Uncurried (imported) form ... |
| bj-biorfi 35765 | This should be labeled "bi... |
| bj-falor 35766 | Dual of ~ truan (which has... |
| bj-falor2 35767 | Dual of ~ truan . (Contri... |
| bj-bibibi 35768 | A property of the bicondit... |
| bj-imn3ani 35769 | Duplication of ~ bnj1224 .... |
| bj-andnotim 35770 | Two ways of expressing a c... |
| bj-bi3ant 35771 | This used to be in the mai... |
| bj-bisym 35772 | This used to be in the mai... |
| bj-bixor 35773 | Equivalence of two ternary... |
| bj-axdd2 35774 | This implication, proved u... |
| bj-axd2d 35775 | This implication, proved u... |
| bj-axtd 35776 | This implication, proved f... |
| bj-gl4 35777 | In a normal modal logic, t... |
| bj-axc4 35778 | Over minimal calculus, the... |
| prvlem1 35783 | An elementary property of ... |
| prvlem2 35784 | An elementary property of ... |
| bj-babygodel 35785 | See the section header com... |
| bj-babylob 35786 | See the section header com... |
| bj-godellob 35787 | Proof of Gödel's theo... |
| bj-genr 35788 | Generalization rule on the... |
| bj-genl 35789 | Generalization rule on the... |
| bj-genan 35790 | Generalization rule on a c... |
| bj-mpgs 35791 | From a closed form theorem... |
| bj-2alim 35792 | Closed form of ~ 2alimi . ... |
| bj-2exim 35793 | Closed form of ~ 2eximi . ... |
| bj-alanim 35794 | Closed form of ~ alanimi .... |
| bj-2albi 35795 | Closed form of ~ 2albii . ... |
| bj-notalbii 35796 | Equivalence of universal q... |
| bj-2exbi 35797 | Closed form of ~ 2exbii . ... |
| bj-3exbi 35798 | Closed form of ~ 3exbii . ... |
| bj-sylgt2 35799 | Uncurried (imported) form ... |
| bj-alrimg 35800 | The general form of the *a... |
| bj-alrimd 35801 | A slightly more general ~ ... |
| bj-sylget 35802 | Dual statement of ~ sylgt ... |
| bj-sylget2 35803 | Uncurried (imported) form ... |
| bj-exlimg 35804 | The general form of the *e... |
| bj-sylge 35805 | Dual statement of ~ sylg (... |
| bj-exlimd 35806 | A slightly more general ~ ... |
| bj-nfimexal 35807 | A weak from of nonfreeness... |
| bj-alexim 35808 | Closed form of ~ aleximi .... |
| bj-nexdh 35809 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 35810 | Uncurried (imported) form ... |
| bj-hbxfrbi 35811 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 35812 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 35813 | Distribute quantifiers ove... |
| bj-exalimi 35814 | An inference for distribut... |
| bj-exalims 35815 | Distributing quantifiers o... |
| bj-exalimsi 35816 | An inference for distribut... |
| bj-ax12ig 35817 | A lemma used to prove a we... |
| bj-ax12i 35818 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 35819 | Closed form of ~ nfim and ... |
| bj-cbvalimt 35820 | A lemma in closed form use... |
| bj-cbveximt 35821 | A lemma in closed form use... |
| bj-eximALT 35822 | Alternate proof of ~ exim ... |
| bj-aleximiALT 35823 | Alternate proof of ~ alexi... |
| bj-eximcom 35824 | A commuted form of ~ exim ... |
| bj-ax12wlem 35825 | A lemma used to prove a we... |
| bj-cbvalim 35826 | A lemma used to prove ~ bj... |
| bj-cbvexim 35827 | A lemma used to prove ~ bj... |
| bj-cbvalimi 35828 | An equality-free general i... |
| bj-cbveximi 35829 | An equality-free general i... |
| bj-cbval 35830 | Changing a bound variable ... |
| bj-cbvex 35831 | Changing a bound variable ... |
| bj-ssbeq 35834 | Substitution in an equalit... |
| bj-ssblem1 35835 | A lemma for the definiens ... |
| bj-ssblem2 35836 | An instance of ~ ax-11 pro... |
| bj-ax12v 35837 | A weaker form of ~ ax-12 a... |
| bj-ax12 35838 | Remove a DV condition from... |
| bj-ax12ssb 35839 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 35840 | Special case of ~ 19.41 pr... |
| bj-equsexval 35841 | Special case of ~ equsexv ... |
| bj-subst 35842 | Proof of ~ sbalex from cor... |
| bj-ssbid2 35843 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 35844 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 35845 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 35846 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 35847 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 35848 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 35849 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 35850 | Closed form of ~ spimvw . ... |
| bj-spnfw 35851 | Theorem close to a closed ... |
| bj-cbvexiw 35852 | Change bound variable. Th... |
| bj-cbvexivw 35853 | Change bound variable. Th... |
| bj-modald 35854 | A short form of the axiom ... |
| bj-denot 35855 | A weakening of ~ ax-6 and ... |
| bj-eqs 35856 | A lemma for substitutions,... |
| bj-cbvexw 35857 | Change bound variable. Th... |
| bj-ax12w 35858 | The general statement that... |
| bj-ax89 35859 | A theorem which could be u... |
| bj-elequ12 35860 | An identity law for the no... |
| bj-cleljusti 35861 | One direction of ~ cleljus... |
| bj-alcomexcom 35862 | Commutation of two existen... |
| bj-hbalt 35863 | Closed form of ~ hbal . W... |
| axc11n11 35864 | Proof of ~ axc11n from { ~... |
| axc11n11r 35865 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 35866 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 35867 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 35868 | Alternate proof of ~ bj-ax... |
| bj-sb 35869 | A weak variant of ~ sbid2 ... |
| bj-modalbe 35870 | The predicate-calculus ver... |
| bj-spst 35871 | Closed form of ~ sps . On... |
| bj-19.21bit 35872 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 35873 | Closed form of ~ 19.23bi .... |
| bj-nexrt 35874 | Closed form of ~ nexr . C... |
| bj-alrim 35875 | Closed form of ~ alrimi . ... |
| bj-alrim2 35876 | Uncurried (imported) form ... |
| bj-nfdt0 35877 | A theorem close to a close... |
| bj-nfdt 35878 | Closed form of ~ nf5d and ... |
| bj-nexdt 35879 | Closed form of ~ nexd . (... |
| bj-nexdvt 35880 | Closed form of ~ nexdv . ... |
| bj-alexbiex 35881 | Adding a second quantifier... |
| bj-exexbiex 35882 | Adding a second quantifier... |
| bj-alalbial 35883 | Adding a second quantifier... |
| bj-exalbial 35884 | Adding a second quantifier... |
| bj-19.9htbi 35885 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 35886 | Strengthening ~ hbnt by re... |
| bj-biexal1 35887 | A general FOL biconditiona... |
| bj-biexal2 35888 | When ` ph ` is substituted... |
| bj-biexal3 35889 | When ` ph ` is substituted... |
| bj-bialal 35890 | When ` ph ` is substituted... |
| bj-biexex 35891 | When ` ph ` is substituted... |
| bj-hbext 35892 | Closed form of ~ hbex . (... |
| bj-nfalt 35893 | Closed form of ~ nfal . (... |
| bj-nfext 35894 | Closed form of ~ nfex . (... |
| bj-eeanvw 35895 | Version of ~ exdistrv with... |
| bj-modal4 35896 | First-order logic form of ... |
| bj-modal4e 35897 | First-order logic form of ... |
| bj-modalb 35898 | A short form of the axiom ... |
| bj-wnf1 35899 | When ` ph ` is substituted... |
| bj-wnf2 35900 | When ` ph ` is substituted... |
| bj-wnfanf 35901 | When ` ph ` is substituted... |
| bj-wnfenf 35902 | When ` ph ` is substituted... |
| bj-substax12 35903 | Equivalent form of the axi... |
| bj-substw 35904 | Weak form of the LHS of ~ ... |
| bj-nnfbi 35907 | If two formulas are equiva... |
| bj-nnfbd 35908 | If two formulas are equiva... |
| bj-nnfbii 35909 | If two formulas are equiva... |
| bj-nnfa 35910 | Nonfreeness implies the eq... |
| bj-nnfad 35911 | Nonfreeness implies the eq... |
| bj-nnfai 35912 | Nonfreeness implies the eq... |
| bj-nnfe 35913 | Nonfreeness implies the eq... |
| bj-nnfed 35914 | Nonfreeness implies the eq... |
| bj-nnfei 35915 | Nonfreeness implies the eq... |
| bj-nnfea 35916 | Nonfreeness implies the eq... |
| bj-nnfead 35917 | Nonfreeness implies the eq... |
| bj-nnfeai 35918 | Nonfreeness implies the eq... |
| bj-dfnnf2 35919 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 35920 | New nonfreeness implies ol... |
| bj-wnfnf 35921 | When ` ph ` is substituted... |
| bj-nnfnt 35922 | A variable is nonfree in a... |
| bj-nnftht 35923 | A variable is nonfree in a... |
| bj-nnfth 35924 | A variable is nonfree in a... |
| bj-nnfnth 35925 | A variable is nonfree in t... |
| bj-nnfim1 35926 | A consequence of nonfreene... |
| bj-nnfim2 35927 | A consequence of nonfreene... |
| bj-nnfim 35928 | Nonfreeness in the anteced... |
| bj-nnfimd 35929 | Nonfreeness in the anteced... |
| bj-nnfan 35930 | Nonfreeness in both conjun... |
| bj-nnfand 35931 | Nonfreeness in both conjun... |
| bj-nnfor 35932 | Nonfreeness in both disjun... |
| bj-nnford 35933 | Nonfreeness in both disjun... |
| bj-nnfbit 35934 | Nonfreeness in both sides ... |
| bj-nnfbid 35935 | Nonfreeness in both sides ... |
| bj-nnfv 35936 | A non-occurring variable i... |
| bj-nnf-alrim 35937 | Proof of the closed form o... |
| bj-nnf-exlim 35938 | Proof of the closed form o... |
| bj-dfnnf3 35939 | Alternate definition of no... |
| bj-nfnnfTEMP 35940 | New nonfreeness is equival... |
| bj-nnfa1 35941 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 35942 | See ~ nfe1 . (Contributed... |
| bj-19.12 35943 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 35944 | One of four lemmas for non... |
| bj-nnflemee 35945 | One of four lemmas for non... |
| bj-nnflemae 35946 | One of four lemmas for non... |
| bj-nnflemea 35947 | One of four lemmas for non... |
| bj-nnfalt 35948 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 35949 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 35950 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 35951 | Statement ~ 19.21t proved ... |
| bj-19.23t 35952 | Statement ~ 19.23t proved ... |
| bj-19.36im 35953 | One direction of ~ 19.36 f... |
| bj-19.37im 35954 | One direction of ~ 19.37 f... |
| bj-19.42t 35955 | Closed form of ~ 19.42 fro... |
| bj-19.41t 35956 | Closed form of ~ 19.41 fro... |
| bj-sbft 35957 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 35958 | Version of ~ pm11.53v with... |
| bj-pm11.53v 35959 | Version of ~ pm11.53v with... |
| bj-pm11.53a 35960 | A variant of ~ pm11.53v . ... |
| bj-equsvt 35961 | A variant of ~ equsv . (C... |
| bj-equsalvwd 35962 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 35963 | Variant of ~ equsexvw . (... |
| bj-sbievwd 35964 | Variant of ~ sbievw . (Co... |
| bj-axc10 35965 | Alternate proof of ~ axc10... |
| bj-alequex 35966 | A fol lemma. See ~ aleque... |
| bj-spimt2 35967 | A step in the proof of ~ s... |
| bj-cbv3ta 35968 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 35969 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 35970 | A theorem close to a close... |
| bj-hbsb3 35971 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 35972 | A theorem close to a close... |
| bj-nfs1t2 35973 | A theorem close to a close... |
| bj-nfs1 35974 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 35975 | Version of ~ axc10 with a ... |
| bj-spimtv 35976 | Version of ~ spimt with a ... |
| bj-cbv3hv2 35977 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 35978 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 35979 | Version of ~ cbv2h with a ... |
| bj-cbv2v 35980 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 35981 | Version of ~ cbvald with a... |
| bj-cbvexdv 35982 | Version of ~ cbvexd with a... |
| bj-cbval2vv 35983 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 35984 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 35985 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 35986 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 35987 | Version of ~ cbvex4v with ... |
| bj-equsalhv 35988 | Version of ~ equsalh with ... |
| bj-axc11nv 35989 | Version of ~ axc11n with a... |
| bj-aecomsv 35990 | Version of ~ aecoms with a... |
| bj-axc11v 35991 | Version of ~ axc11 with a ... |
| bj-drnf2v 35992 | Version of ~ drnf2 with a ... |
| bj-equs45fv 35993 | Version of ~ equs45f with ... |
| bj-hbs1 35994 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 35995 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 35996 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 35997 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 35998 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 35999 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 36000 | Biconditional version of a... |
| bj-hbaeb 36001 | Biconditional version of ~... |
| bj-hbnaeb 36002 | Biconditional version of ~... |
| bj-dvv 36003 | A special instance of ~ bj... |
| bj-equsal1t 36004 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 36005 | Inference associated with ... |
| bj-equsal1 36006 | One direction of ~ equsal ... |
| bj-equsal2 36007 | One direction of ~ equsal ... |
| bj-equsal 36008 | Shorter proof of ~ equsal ... |
| stdpc5t 36009 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 36010 | More direct proof of ~ std... |
| 2stdpc5 36011 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 36012 | Proof of ~ 19.21t from ~ s... |
| exlimii 36013 | Inference associated with ... |
| ax11-pm 36014 | Proof of ~ ax-11 similar t... |
| ax6er 36015 | Commuted form of ~ ax6e . ... |
| exlimiieq1 36016 | Inferring a theorem when i... |
| exlimiieq2 36017 | Inferring a theorem when i... |
| ax11-pm2 36018 | Proof of ~ ax-11 from the ... |
| bj-sbsb 36019 | Biconditional showing two ... |
| bj-dfsb2 36020 | Alternate (dual) definitio... |
| bj-sbf3 36021 | Substitution has no effect... |
| bj-sbf4 36022 | Substitution has no effect... |
| bj-sbnf 36023 | Move nonfree predicate in ... |
| bj-eu3f 36024 | Version of ~ eu3v where th... |
| bj-sblem1 36025 | Lemma for substitution. (... |
| bj-sblem2 36026 | Lemma for substitution. (... |
| bj-sblem 36027 | Lemma for substitution. (... |
| bj-sbievw1 36028 | Lemma for substitution. (... |
| bj-sbievw2 36029 | Lemma for substitution. (... |
| bj-sbievw 36030 | Lemma for substitution. C... |
| bj-sbievv 36031 | Version of ~ sbie with a s... |
| bj-moeub 36032 | Uniqueness is equivalent t... |
| bj-sbidmOLD 36033 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 36034 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 36035 | Curried (exported) form of... |
| bj-dvelimv 36036 | A version of ~ dvelim usin... |
| bj-nfeel2 36037 | Nonfreeness in a membershi... |
| bj-axc14nf 36038 | Proof of a version of ~ ax... |
| bj-axc14 36039 | Alternate proof of ~ axc14... |
| mobidvALT 36040 | Alternate proof of ~ mobid... |
| sbn1ALT 36041 | Alternate proof of ~ sbn1 ... |
| eliminable1 36042 | A theorem used to prove th... |
| eliminable2a 36043 | A theorem used to prove th... |
| eliminable2b 36044 | A theorem used to prove th... |
| eliminable2c 36045 | A theorem used to prove th... |
| eliminable3a 36046 | A theorem used to prove th... |
| eliminable3b 36047 | A theorem used to prove th... |
| eliminable-velab 36048 | A theorem used to prove th... |
| eliminable-veqab 36049 | A theorem used to prove th... |
| eliminable-abeqv 36050 | A theorem used to prove th... |
| eliminable-abeqab 36051 | A theorem used to prove th... |
| eliminable-abelv 36052 | A theorem used to prove th... |
| eliminable-abelab 36053 | A theorem used to prove th... |
| bj-denoteslem 36054 | Lemma for ~ bj-denotes . ... |
| bj-denotes 36055 | This would be the justific... |
| bj-issettru 36056 | Weak version of ~ isset wi... |
| bj-elabtru 36057 | This is as close as we can... |
| bj-issetwt 36058 | Closed form of ~ bj-issetw... |
| bj-issetw 36059 | The closest one can get to... |
| bj-elissetALT 36060 | Alternate proof of ~ eliss... |
| bj-issetiv 36061 | Version of ~ bj-isseti wit... |
| bj-isseti 36062 | Version of ~ isseti with a... |
| bj-ralvw 36063 | A weak version of ~ ralv n... |
| bj-rexvw 36064 | A weak version of ~ rexv n... |
| bj-rababw 36065 | A weak version of ~ rabab ... |
| bj-rexcom4bv 36066 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 36067 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 36068 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 36069 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 36070 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 36071 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 36072 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 36073 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 36074 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 36075 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 36076 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 36077 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 36078 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 36079 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 36080 | Remove from ~ spcimdv depe... |
| elelb 36081 | Equivalence between two co... |
| bj-pwvrelb 36082 | Characterization of the el... |
| bj-nfcsym 36083 | The nonfreeness quantifier... |
| bj-sbeqALT 36084 | Substitution in an equalit... |
| bj-sbeq 36085 | Distribute proper substitu... |
| bj-sbceqgALT 36086 | Distribute proper substitu... |
| bj-csbsnlem 36087 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 36088 | Substitution in a singleto... |
| bj-sbel1 36089 | Version of ~ sbcel1g when ... |
| bj-abv 36090 | The class of sets verifyin... |
| bj-abvALT 36091 | Alternate version of ~ bj-... |
| bj-ab0 36092 | The class of sets verifyin... |
| bj-abf 36093 | Shorter proof of ~ abf (wh... |
| bj-csbprc 36094 | More direct proof of ~ csb... |
| bj-exlimvmpi 36095 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 36096 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 36097 | Lemma for theorems of the ... |
| bj-exlimmpbir 36098 | Lemma for theorems of the ... |
| bj-vtoclf 36099 | Remove dependency on ~ ax-... |
| bj-vtocl 36100 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 36101 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 36102 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 36103 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 36104 | A version of ~ vtoclg with... |
| bj-rabeqbid 36105 | Version of ~ rabeqbidv wit... |
| bj-seex 36106 | Version of ~ seex with a d... |
| bj-nfcf 36107 | Version of ~ df-nfc with a... |
| bj-zfauscl 36108 | General version of ~ zfaus... |
| bj-elabd2ALT 36109 | Alternate proof of ~ elabd... |
| bj-unrab 36110 | Generalization of ~ unrab ... |
| bj-inrab 36111 | Generalization of ~ inrab ... |
| bj-inrab2 36112 | Shorter proof of ~ inrab .... |
| bj-inrab3 36113 | Generalization of ~ dfrab3... |
| bj-rabtr 36114 | Restricted class abstracti... |
| bj-rabtrALT 36115 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 36116 | Proof of ~ bj-rabtr found ... |
| bj-gabss 36119 | Inclusion of generalized c... |
| bj-gabssd 36120 | Inclusion of generalized c... |
| bj-gabeqd 36121 | Equality of generalized cl... |
| bj-gabeqis 36122 | Equality of generalized cl... |
| bj-elgab 36123 | Elements of a generalized ... |
| bj-gabima 36124 | Generalized class abstract... |
| bj-ru0 36127 | The FOL part of Russell's ... |
| bj-ru1 36128 | A version of Russell's par... |
| bj-ru 36129 | Remove dependency on ~ ax-... |
| currysetlem 36130 | Lemma for ~ currysetlem , ... |
| curryset 36131 | Curry's paradox in set the... |
| currysetlem1 36132 | Lemma for ~ currysetALT . ... |
| currysetlem2 36133 | Lemma for ~ currysetALT . ... |
| currysetlem3 36134 | Lemma for ~ currysetALT . ... |
| currysetALT 36135 | Alternate proof of ~ curry... |
| bj-n0i 36136 | Inference associated with ... |
| bj-disjsn01 36137 | Disjointness of the single... |
| bj-0nel1 36138 | The empty set does not bel... |
| bj-1nel0 36139 | ` 1o ` does not belong to ... |
| bj-xpimasn 36140 | The image of a singleton, ... |
| bj-xpima1sn 36141 | The image of a singleton b... |
| bj-xpima1snALT 36142 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 36143 | The image of a singleton b... |
| bj-xpnzex 36144 | If the first factor of a p... |
| bj-xpexg2 36145 | Curried (exported) form of... |
| bj-xpnzexb 36146 | If the first factor of a p... |
| bj-cleq 36147 | Substitution property for ... |
| bj-snsetex 36148 | The class of sets "whose s... |
| bj-clexab 36149 | Sethood of certain classes... |
| bj-sngleq 36152 | Substitution property for ... |
| bj-elsngl 36153 | Characterization of the el... |
| bj-snglc 36154 | Characterization of the el... |
| bj-snglss 36155 | The singletonization of a ... |
| bj-0nelsngl 36156 | The empty set is not a mem... |
| bj-snglinv 36157 | Inverse of singletonizatio... |
| bj-snglex 36158 | A class is a set if and on... |
| bj-tageq 36161 | Substitution property for ... |
| bj-eltag 36162 | Characterization of the el... |
| bj-0eltag 36163 | The empty set belongs to t... |
| bj-tagn0 36164 | The tagging of a class is ... |
| bj-tagss 36165 | The tagging of a class is ... |
| bj-snglsstag 36166 | The singletonization is in... |
| bj-sngltagi 36167 | The singletonization is in... |
| bj-sngltag 36168 | The singletonization and t... |
| bj-tagci 36169 | Characterization of the el... |
| bj-tagcg 36170 | Characterization of the el... |
| bj-taginv 36171 | Inverse of tagging. (Cont... |
| bj-tagex 36172 | A class is a set if and on... |
| bj-xtageq 36173 | The products of a given cl... |
| bj-xtagex 36174 | The product of a set and t... |
| bj-projeq 36177 | Substitution property for ... |
| bj-projeq2 36178 | Substitution property for ... |
| bj-projun 36179 | The class projection on a ... |
| bj-projex 36180 | Sethood of the class proje... |
| bj-projval 36181 | Value of the class project... |
| bj-1upleq 36184 | Substitution property for ... |
| bj-pr1eq 36187 | Substitution property for ... |
| bj-pr1un 36188 | The first projection prese... |
| bj-pr1val 36189 | Value of the first project... |
| bj-pr11val 36190 | Value of the first project... |
| bj-pr1ex 36191 | Sethood of the first proje... |
| bj-1uplth 36192 | The characteristic propert... |
| bj-1uplex 36193 | A monuple is a set if and ... |
| bj-1upln0 36194 | A monuple is nonempty. (C... |
| bj-2upleq 36197 | Substitution property for ... |
| bj-pr21val 36198 | Value of the first project... |
| bj-pr2eq 36201 | Substitution property for ... |
| bj-pr2un 36202 | The second projection pres... |
| bj-pr2val 36203 | Value of the second projec... |
| bj-pr22val 36204 | Value of the second projec... |
| bj-pr2ex 36205 | Sethood of the second proj... |
| bj-2uplth 36206 | The characteristic propert... |
| bj-2uplex 36207 | A couple is a set if and o... |
| bj-2upln0 36208 | A couple is nonempty. (Co... |
| bj-2upln1upl 36209 | A couple is never equal to... |
| bj-rcleqf 36210 | Relative version of ~ cleq... |
| bj-rcleq 36211 | Relative version of ~ dfcl... |
| bj-reabeq 36212 | Relative form of ~ eqabb .... |
| bj-disj2r 36213 | Relative version of ~ ssdi... |
| bj-sscon 36214 | Contraposition law for rel... |
| bj-abex 36215 | Two ways of stating that t... |
| bj-clex 36216 | Two ways of stating that a... |
| bj-axsn 36217 | Two ways of stating the ax... |
| bj-snexg 36219 | A singleton built on a set... |
| bj-snex 36220 | A singleton is a set. See... |
| bj-axbun 36221 | Two ways of stating the ax... |
| bj-unexg 36223 | Existence of binary unions... |
| bj-prexg 36224 | Existence of unordered pai... |
| bj-prex 36225 | Existence of unordered pai... |
| bj-axadj 36226 | Two ways of stating the ax... |
| bj-adjg1 36228 | Existence of the result of... |
| bj-snfromadj 36229 | Singleton from adjunction ... |
| bj-prfromadj 36230 | Unordered pair from adjunc... |
| bj-adjfrombun 36231 | Adjunction from singleton ... |
| eleq2w2ALT 36232 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 36233 | Alternate proof of ~ clel3... |
| bj-pw0ALT 36234 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 36235 | Quantitative version of ~ ... |
| bj-unirel 36236 | Quantitative version of ~ ... |
| bj-elpwg 36237 | If the intersection of two... |
| bj-velpwALT 36238 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 36239 | Alternate proof of ~ elpwg... |
| bj-vjust 36240 | Justification theorem for ... |
| bj-nul 36241 | Two formulations of the ax... |
| bj-nuliota 36242 | Definition of the empty se... |
| bj-nuliotaALT 36243 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 36244 | Alternate proof of ~ vtocl... |
| bj-elsn12g 36245 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 36246 | Biconditional version of ~... |
| bj-pwcfsdom 36247 | Remove hypothesis from ~ p... |
| bj-grur1 36248 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 36249 | The extension of a predica... |
| bj-dfid2ALT 36250 | Alternate version of ~ dfi... |
| bj-0nelopab 36251 | The empty set is never an ... |
| bj-brrelex12ALT 36252 | Two classes related by a b... |
| bj-epelg 36253 | The membership relation an... |
| bj-epelb 36254 | Two classes are related by... |
| bj-nsnid 36255 | A set does not contain the... |
| bj-rdg0gALT 36256 | Alternate proof of ~ rdg0g... |
| bj-evaleq 36257 | Equality theorem for the `... |
| bj-evalfun 36258 | The evaluation at a class ... |
| bj-evalfn 36259 | The evaluation at a class ... |
| bj-evalval 36260 | Value of the evaluation at... |
| bj-evalid 36261 | The evaluation at a set of... |
| bj-ndxarg 36262 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 36263 | Closed general form of ~ s... |
| bj-rest00 36266 | An elementwise intersectio... |
| bj-restsn 36267 | An elementwise intersectio... |
| bj-restsnss 36268 | Special case of ~ bj-rests... |
| bj-restsnss2 36269 | Special case of ~ bj-rests... |
| bj-restsn0 36270 | An elementwise intersectio... |
| bj-restsn10 36271 | Special case of ~ bj-rests... |
| bj-restsnid 36272 | The elementwise intersecti... |
| bj-rest10 36273 | An elementwise intersectio... |
| bj-rest10b 36274 | Alternate version of ~ bj-... |
| bj-restn0 36275 | An elementwise intersectio... |
| bj-restn0b 36276 | Alternate version of ~ bj-... |
| bj-restpw 36277 | The elementwise intersecti... |
| bj-rest0 36278 | An elementwise intersectio... |
| bj-restb 36279 | An elementwise intersectio... |
| bj-restv 36280 | An elementwise intersectio... |
| bj-resta 36281 | An elementwise intersectio... |
| bj-restuni 36282 | The union of an elementwis... |
| bj-restuni2 36283 | The union of an elementwis... |
| bj-restreg 36284 | A reformulation of the axi... |
| bj-raldifsn 36285 | All elements in a set sati... |
| bj-0int 36286 | If ` A ` is a collection o... |
| bj-mooreset 36287 | A Moore collection is a se... |
| bj-ismoore 36290 | Characterization of Moore ... |
| bj-ismoored0 36291 | Necessary condition to be ... |
| bj-ismoored 36292 | Necessary condition to be ... |
| bj-ismoored2 36293 | Necessary condition to be ... |
| bj-ismooredr 36294 | Sufficient condition to be... |
| bj-ismooredr2 36295 | Sufficient condition to be... |
| bj-discrmoore 36296 | The powerclass ` ~P A ` is... |
| bj-0nmoore 36297 | The empty set is not a Moo... |
| bj-snmoore 36298 | A singleton is a Moore col... |
| bj-snmooreb 36299 | A singleton is a Moore col... |
| bj-prmoore 36300 | A pair formed of two neste... |
| bj-0nelmpt 36301 | The empty set is not an el... |
| bj-mptval 36302 | Value of a function given ... |
| bj-dfmpoa 36303 | An equivalent definition o... |
| bj-mpomptALT 36304 | Alternate proof of ~ mpomp... |
| setsstrset 36321 | Relation between ~ df-sets... |
| bj-nfald 36322 | Variant of ~ nfald . (Con... |
| bj-nfexd 36323 | Variant of ~ nfexd . (Con... |
| copsex2d 36324 | Implicit substitution dedu... |
| copsex2b 36325 | Biconditional form of ~ co... |
| opelopabd 36326 | Membership of an ordere pa... |
| opelopabb 36327 | Membership of an ordered p... |
| opelopabbv 36328 | Membership of an ordered p... |
| bj-opelrelex 36329 | The coordinates of an orde... |
| bj-opelresdm 36330 | If an ordered pair is in a... |
| bj-brresdm 36331 | If two classes are related... |
| brabd0 36332 | Expressing that two sets a... |
| brabd 36333 | Expressing that two sets a... |
| bj-brab2a1 36334 | "Unbounded" version of ~ b... |
| bj-opabssvv 36335 | A variant of ~ relopabiv (... |
| bj-funidres 36336 | The restricted identity re... |
| bj-opelidb 36337 | Characterization of the or... |
| bj-opelidb1 36338 | Characterization of the or... |
| bj-inexeqex 36339 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 36340 | If the intersection of two... |
| bj-opelid 36341 | Characterization of the or... |
| bj-ideqg 36342 | Characterization of the cl... |
| bj-ideqgALT 36343 | Alternate proof of ~ bj-id... |
| bj-ideqb 36344 | Characterization of classe... |
| bj-idres 36345 | Alternate expression for t... |
| bj-opelidres 36346 | Characterization of the or... |
| bj-idreseq 36347 | Sufficient condition for t... |
| bj-idreseqb 36348 | Characterization for two c... |
| bj-ideqg1 36349 | For sets, the identity rel... |
| bj-ideqg1ALT 36350 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 36351 | Characterization of the co... |
| bj-elid3 36352 | Characterization of the co... |
| bj-elid4 36353 | Characterization of the el... |
| bj-elid5 36354 | Characterization of the el... |
| bj-elid6 36355 | Characterization of the el... |
| bj-elid7 36356 | Characterization of the el... |
| bj-diagval 36359 | Value of the functionalize... |
| bj-diagval2 36360 | Value of the functionalize... |
| bj-eldiag 36361 | Characterization of the el... |
| bj-eldiag2 36362 | Characterization of the el... |
| bj-imdirvallem 36365 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 36366 | Value of the functionalize... |
| bj-imdirval2lem 36367 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 36368 | Value of the functionalize... |
| bj-imdirval3 36369 | Value of the functionalize... |
| bj-imdiridlem 36370 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 36371 | Functorial property of the... |
| bj-opelopabid 36372 | Membership in an ordered-p... |
| bj-opabco 36373 | Composition of ordered-pai... |
| bj-xpcossxp 36374 | The composition of two Car... |
| bj-imdirco 36375 | Functorial property of the... |
| bj-iminvval 36378 | Value of the functionalize... |
| bj-iminvval2 36379 | Value of the functionalize... |
| bj-iminvid 36380 | Functorial property of the... |
| bj-inftyexpitaufo 36387 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 36390 | An element of the circle a... |
| bj-inftyexpiinv 36393 | Utility theorem for the in... |
| bj-inftyexpiinj 36394 | Injectivity of the paramet... |
| bj-inftyexpidisj 36395 | An element of the circle a... |
| bj-ccinftydisj 36398 | The circle at infinity is ... |
| bj-elccinfty 36399 | A lemma for infinite exten... |
| bj-ccssccbar 36402 | Complex numbers are extend... |
| bj-ccinftyssccbar 36403 | Infinite extended complex ... |
| bj-pinftyccb 36406 | The class ` pinfty ` is an... |
| bj-pinftynrr 36407 | The extended complex numbe... |
| bj-minftyccb 36410 | The class ` minfty ` is an... |
| bj-minftynrr 36411 | The extended complex numbe... |
| bj-pinftynminfty 36412 | The extended complex numbe... |
| bj-rrhatsscchat 36421 | The real projective line i... |
| bj-imafv 36436 | If the direct image of a s... |
| bj-funun 36437 | Value of a function expres... |
| bj-fununsn1 36438 | Value of a function expres... |
| bj-fununsn2 36439 | Value of a function expres... |
| bj-fvsnun1 36440 | The value of a function wi... |
| bj-fvsnun2 36441 | The value of a function wi... |
| bj-fvmptunsn1 36442 | Value of a function expres... |
| bj-fvmptunsn2 36443 | Value of a function expres... |
| bj-iomnnom 36444 | The canonical bijection fr... |
| bj-smgrpssmgm 36453 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 36454 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 36455 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 36456 | Monoids are semigroups (el... |
| bj-cmnssmnd 36457 | Commutative monoids are mo... |
| bj-cmnssmndel 36458 | Commutative monoids are mo... |
| bj-grpssmnd 36459 | Groups are monoids. (Cont... |
| bj-grpssmndel 36460 | Groups are monoids (elemen... |
| bj-ablssgrp 36461 | Abelian groups are groups.... |
| bj-ablssgrpel 36462 | Abelian groups are groups ... |
| bj-ablsscmn 36463 | Abelian groups are commuta... |
| bj-ablsscmnel 36464 | Abelian groups are commuta... |
| bj-modssabl 36465 | (The additive groups of) m... |
| bj-vecssmod 36466 | Vector spaces are modules.... |
| bj-vecssmodel 36467 | Vector spaces are modules ... |
| bj-finsumval0 36470 | Value of a finite sum. (C... |
| bj-fvimacnv0 36471 | Variant of ~ fvimacnv wher... |
| bj-isvec 36472 | The predicate "is a vector... |
| bj-fldssdrng 36473 | Fields are division rings.... |
| bj-flddrng 36474 | Fields are division rings ... |
| bj-rrdrg 36475 | The field of real numbers ... |
| bj-isclm 36476 | The predicate "is a subcom... |
| bj-isrvec 36479 | The predicate "is a real v... |
| bj-rvecmod 36480 | Real vector spaces are mod... |
| bj-rvecssmod 36481 | Real vector spaces are mod... |
| bj-rvecrr 36482 | The field of scalars of a ... |
| bj-isrvecd 36483 | The predicate "is a real v... |
| bj-rvecvec 36484 | Real vector spaces are vec... |
| bj-isrvec2 36485 | The predicate "is a real v... |
| bj-rvecssvec 36486 | Real vector spaces are vec... |
| bj-rveccmod 36487 | Real vector spaces are sub... |
| bj-rvecsscmod 36488 | Real vector spaces are sub... |
| bj-rvecsscvec 36489 | Real vector spaces are sub... |
| bj-rveccvec 36490 | Real vector spaces are sub... |
| bj-rvecssabl 36491 | (The additive groups of) r... |
| bj-rvecabl 36492 | (The additive groups of) r... |
| bj-subcom 36493 | A consequence of commutati... |
| bj-lineqi 36494 | Solution of a (scalar) lin... |
| bj-bary1lem 36495 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 36496 | Lemma for bj-bary1: comput... |
| bj-bary1 36497 | Barycentric coordinates in... |
| bj-endval 36500 | Value of the monoid of end... |
| bj-endbase 36501 | Base set of the monoid of ... |
| bj-endcomp 36502 | Composition law of the mon... |
| bj-endmnd 36503 | The monoid of endomorphism... |
| taupilem3 36504 | Lemma for tau-related theo... |
| taupilemrplb 36505 | A set of positive reals ha... |
| taupilem1 36506 | Lemma for ~ taupi . A pos... |
| taupilem2 36507 | Lemma for ~ taupi . The s... |
| taupi 36508 | Relationship between ` _ta... |
| dfgcd3 36509 | Alternate definition of th... |
| irrdifflemf 36510 | Lemma for ~ irrdiff . The... |
| irrdiff 36511 | The irrationals are exactl... |
| iccioo01 36512 | The closed unit interval i... |
| csbrecsg 36513 | Move class substitution in... |
| csbrdgg 36514 | Move class substitution in... |
| csboprabg 36515 | Move class substitution in... |
| csbmpo123 36516 | Move class substitution in... |
| con1bii2 36517 | A contraposition inference... |
| con2bii2 36518 | A contraposition inference... |
| vtoclefex 36519 | Implicit substitution of a... |
| rnmptsn 36520 | The range of a function ma... |
| f1omptsnlem 36521 | This is the core of the pr... |
| f1omptsn 36522 | A function mapping to sing... |
| mptsnunlem 36523 | This is the core of the pr... |
| mptsnun 36524 | A class ` B ` is equal to ... |
| dissneqlem 36525 | This is the core of the pr... |
| dissneq 36526 | Any topology that contains... |
| exlimim 36527 | Closed form of ~ exlimimd ... |
| exlimimd 36528 | Existential elimination ru... |
| exellim 36529 | Closed form of ~ exellimdd... |
| exellimddv 36530 | Eliminate an antecedent wh... |
| topdifinfindis 36531 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 36532 | This is the core of the pr... |
| topdifinffin 36533 | Part of Exercise 3 of [Mun... |
| topdifinf 36534 | Part of Exercise 3 of [Mun... |
| topdifinfeq 36535 | Two different ways of defi... |
| icorempo 36536 | Closed-below, open-above i... |
| icoreresf 36537 | Closed-below, open-above i... |
| icoreval 36538 | Value of the closed-below,... |
| icoreelrnab 36539 | Elementhood in the set of ... |
| isbasisrelowllem1 36540 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 36541 | Lemma for ~ isbasisrelowl ... |
| icoreclin 36542 | The set of closed-below, o... |
| isbasisrelowl 36543 | The set of all closed-belo... |
| icoreunrn 36544 | The union of all closed-be... |
| istoprelowl 36545 | The set of all closed-belo... |
| icoreelrn 36546 | A class abstraction which ... |
| iooelexlt 36547 | An element of an open inte... |
| relowlssretop 36548 | The lower limit topology o... |
| relowlpssretop 36549 | The lower limit topology o... |
| sucneqond 36550 | Inequality of an ordinal s... |
| sucneqoni 36551 | Inequality of an ordinal s... |
| onsucuni3 36552 | If an ordinal number has a... |
| 1oequni2o 36553 | The ordinal number ` 1o ` ... |
| rdgsucuni 36554 | If an ordinal number has a... |
| rdgeqoa 36555 | If a recursive function wi... |
| elxp8 36556 | Membership in a Cartesian ... |
| cbveud 36557 | Deduction used to change b... |
| cbvreud 36558 | Deduction used to change b... |
| difunieq 36559 | The difference of unions i... |
| inunissunidif 36560 | Theorem about subsets of t... |
| rdgellim 36561 | Elementhood in a recursive... |
| rdglimss 36562 | A recursive definition at ... |
| rdgssun 36563 | In a recursive definition ... |
| exrecfnlem 36564 | Lemma for ~ exrecfn . (Co... |
| exrecfn 36565 | Theorem about the existenc... |
| exrecfnpw 36566 | For any base set, a set wh... |
| finorwe 36567 | If the Axiom of Infinity i... |
| dffinxpf 36570 | This theorem is the same a... |
| finxpeq1 36571 | Equality theorem for Carte... |
| finxpeq2 36572 | Equality theorem for Carte... |
| csbfinxpg 36573 | Distribute proper substitu... |
| finxpreclem1 36574 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 36575 | Lemma for ` ^^ ` recursion... |
| finxp0 36576 | The value of Cartesian exp... |
| finxp1o 36577 | The value of Cartesian exp... |
| finxpreclem3 36578 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 36579 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 36580 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 36581 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 36582 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 36583 | The value of Cartesian exp... |
| finxp2o 36584 | The value of Cartesian exp... |
| finxp3o 36585 | The value of Cartesian exp... |
| finxpnom 36586 | Cartesian exponentiation w... |
| finxp00 36587 | Cartesian exponentiation o... |
| iunctb2 36588 | Using the axiom of countab... |
| domalom 36589 | A class which dominates ev... |
| isinf2 36590 | The converse of ~ isinf . ... |
| ctbssinf 36591 | Using the axiom of choice,... |
| ralssiun 36592 | The index set of an indexe... |
| nlpineqsn 36593 | For every point ` p ` of a... |
| nlpfvineqsn 36594 | Given a subset ` A ` of ` ... |
| fvineqsnf1 36595 | A theorem about functions ... |
| fvineqsneu 36596 | A theorem about functions ... |
| fvineqsneq 36597 | A theorem about functions ... |
| pibp16 36598 | Property P000016 of pi-bas... |
| pibp19 36599 | Property P000019 of pi-bas... |
| pibp21 36600 | Property P000021 of pi-bas... |
| pibt1 36601 | Theorem T000001 of pi-base... |
| pibt2 36602 | Theorem T000002 of pi-base... |
| wl-section-prop 36603 | Intuitionistic logic is no... |
| wl-section-boot 36607 | In this section, I provide... |
| wl-luk-imim1i 36608 | Inference adding common co... |
| wl-luk-syl 36609 | An inference version of th... |
| wl-luk-imtrid 36610 | A syllogism rule of infere... |
| wl-luk-pm2.18d 36611 | Deduction based on reducti... |
| wl-luk-con4i 36612 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 36613 | Inference rule. Copy of ~... |
| wl-luk-a1i 36614 | Inference rule. Copy of ~... |
| wl-luk-mpi 36615 | A nested modus ponens infe... |
| wl-luk-imim2i 36616 | Inference adding common an... |
| wl-luk-imtrdi 36617 | A syllogism rule of infere... |
| wl-luk-ax3 36618 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 36619 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 36620 | This theorem, called "Asse... |
| wl-luk-com12 36621 | Inference that swaps (comm... |
| wl-luk-pm2.21 36622 | From a wff and its negatio... |
| wl-luk-con1i 36623 | A contraposition inference... |
| wl-luk-ja 36624 | Inference joining the ante... |
| wl-luk-imim2 36625 | A closed form of syllogism... |
| wl-luk-a1d 36626 | Deduction introducing an e... |
| wl-luk-ax2 36627 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 36628 | Principle of identity. Th... |
| wl-luk-notnotr 36629 | Converse of double negatio... |
| wl-luk-pm2.04 36630 | Swap antecedents. Theorem... |
| wl-section-impchain 36631 | An implication like ` ( ps... |
| wl-impchain-mp-x 36632 | This series of theorems pr... |
| wl-impchain-mp-0 36633 | This theorem is the start ... |
| wl-impchain-mp-1 36634 | This theorem is in fact a ... |
| wl-impchain-mp-2 36635 | This theorem is in fact a ... |
| wl-impchain-com-1.x 36636 | It is often convenient to ... |
| wl-impchain-com-1.1 36637 | A degenerate form of antec... |
| wl-impchain-com-1.2 36638 | This theorem is in fact a ... |
| wl-impchain-com-1.3 36639 | This theorem is in fact a ... |
| wl-impchain-com-1.4 36640 | This theorem is in fact a ... |
| wl-impchain-com-n.m 36641 | This series of theorems al... |
| wl-impchain-com-2.3 36642 | This theorem is in fact a ... |
| wl-impchain-com-2.4 36643 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 36644 | This theorem is in fact a ... |
| wl-impchain-a1-x 36645 | If an implication chain is... |
| wl-impchain-a1-1 36646 | Inference rule, a copy of ... |
| wl-impchain-a1-2 36647 | Inference rule, a copy of ... |
| wl-impchain-a1-3 36648 | Inference rule, a copy of ... |
| wl-ifp-ncond1 36649 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 36650 | If one case of an ` if- ` ... |
| wl-ifpimpr 36651 | If one case of an ` if- ` ... |
| wl-ifp4impr 36652 | If one case of an ` if- ` ... |
| wl-df-3xor 36653 | Alternative definition of ... |
| wl-df3xor2 36654 | Alternative definition of ... |
| wl-df3xor3 36655 | Alternative form of ~ wl-d... |
| wl-3xortru 36656 | If the first input is true... |
| wl-3xorfal 36657 | If the first input is fals... |
| wl-3xorbi 36658 | Triple xor can be replaced... |
| wl-3xorbi2 36659 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 36660 | Equivalence theorem for tr... |
| wl-3xorbi123i 36661 | Equivalence theorem for tr... |
| wl-3xorrot 36662 | Rotation law for triple xo... |
| wl-3xorcoma 36663 | Commutative law for triple... |
| wl-3xorcomb 36664 | Commutative law for triple... |
| wl-3xornot1 36665 | Flipping the first input f... |
| wl-3xornot 36666 | Triple xor distributes ove... |
| wl-1xor 36667 | In the recursive scheme ... |
| wl-2xor 36668 | In the recursive scheme ... |
| wl-df-3mintru2 36669 | Alternative definition of ... |
| wl-df2-3mintru2 36670 | The adder carry in disjunc... |
| wl-df3-3mintru2 36671 | The adder carry in conjunc... |
| wl-df4-3mintru2 36672 | An alternative definition ... |
| wl-1mintru1 36673 | Using the recursion formul... |
| wl-1mintru2 36674 | Using the recursion formul... |
| wl-2mintru1 36675 | Using the recursion formul... |
| wl-2mintru2 36676 | Using the recursion formul... |
| wl-df3maxtru1 36677 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 36679 | A version of ~ ax-wl-13v w... |
| wl-mps 36680 | Replacing a nested consequ... |
| wl-syls1 36681 | Replacing a nested consequ... |
| wl-syls2 36682 | Replacing a nested anteced... |
| wl-embant 36683 | A true wff can always be a... |
| wl-orel12 36684 | In a conjunctive normal fo... |
| wl-cases2-dnf 36685 | A particular instance of ~... |
| wl-cbvmotv 36686 | Change bound variable. Us... |
| wl-moteq 36687 | Change bound variable. Us... |
| wl-motae 36688 | Change bound variable. Us... |
| wl-moae 36689 | Two ways to express "at mo... |
| wl-euae 36690 | Two ways to express "exact... |
| wl-nax6im 36691 | The following series of th... |
| wl-hbae1 36692 | This specialization of ~ h... |
| wl-naevhba1v 36693 | An instance of ~ hbn1w app... |
| wl-spae 36694 | Prove an instance of ~ sp ... |
| wl-speqv 36695 | Under the assumption ` -. ... |
| wl-19.8eqv 36696 | Under the assumption ` -. ... |
| wl-19.2reqv 36697 | Under the assumption ` -. ... |
| wl-nfalv 36698 | If ` x ` is not present in... |
| wl-nfimf1 36699 | An antecedent is irrelevan... |
| wl-nfae1 36700 | Unlike ~ nfae , this speci... |
| wl-nfnae1 36701 | Unlike ~ nfnae , this spec... |
| wl-aetr 36702 | A transitive law for varia... |
| wl-axc11r 36703 | Same as ~ axc11r , but usi... |
| wl-dral1d 36704 | A version of ~ dral1 with ... |
| wl-cbvalnaed 36705 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 36706 | A more general version of ... |
| wl-exeq 36707 | The semantics of ` E. x y ... |
| wl-aleq 36708 | The semantics of ` A. x y ... |
| wl-nfeqfb 36709 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 36710 | If ` y ` is not free in ` ... |
| wl-equsalvw 36711 | Version of ~ equsalv with ... |
| wl-equsald 36712 | Deduction version of ~ equ... |
| wl-equsal 36713 | A useful equivalence relat... |
| wl-equsal1t 36714 | The expression ` x = y ` i... |
| wl-equsalcom 36715 | This simple equivalence ea... |
| wl-equsal1i 36716 | The antecedent ` x = y ` i... |
| wl-sb6rft 36717 | A specialization of ~ wl-e... |
| wl-cbvalsbi 36718 | Change bounded variables i... |
| wl-sbrimt 36719 | Substitution with a variab... |
| wl-sblimt 36720 | Substitution with a variab... |
| wl-sb8t 36721 | Substitution of variable i... |
| wl-sb8et 36722 | Substitution of variable i... |
| wl-sbhbt 36723 | Closed form of ~ sbhb . C... |
| wl-sbnf1 36724 | Two ways expressing that `... |
| wl-equsb3 36725 | ~ equsb3 with a distinctor... |
| wl-equsb4 36726 | Substitution applied to an... |
| wl-2sb6d 36727 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 36728 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 36729 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 36730 | Version of ~ sbcom2 with a... |
| wl-sbalnae 36731 | A theorem used in eliminat... |
| wl-sbal1 36732 | A theorem used in eliminat... |
| wl-sbal2 36733 | Move quantifier in and out... |
| wl-2spsbbi 36734 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 36735 | This theorem provides a ba... |
| wl-lem-nexmo 36736 | This theorem provides a ba... |
| wl-lem-moexsb 36737 | The antecedent ` A. x ( ph... |
| wl-alanbii 36738 | This theorem extends ~ ala... |
| wl-mo2df 36739 | Version of ~ mof with a co... |
| wl-mo2tf 36740 | Closed form of ~ mof with ... |
| wl-eudf 36741 | Version of ~ eu6 with a co... |
| wl-eutf 36742 | Closed form of ~ eu6 with ... |
| wl-euequf 36743 | ~ euequ proved with a dist... |
| wl-mo2t 36744 | Closed form of ~ mof . (C... |
| wl-mo3t 36745 | Closed form of ~ mo3 . (C... |
| wl-sb8eut 36746 | Substitution of variable i... |
| wl-sb8mot 36747 | Substitution of variable i... |
| wl-issetft 36748 | A closed form of ~ issetf ... |
| wl-axc11rc11 36749 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 36751 | A transitive law for varia... |
| wl-ax11-lem2 36752 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 36753 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 36754 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 36755 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 36756 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 36757 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 36758 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 36759 | The easy part when ` x ` c... |
| wl-ax11-lem10 36760 | We now have prepared every... |
| wl-clabv 36761 | Variant of ~ df-clab , whe... |
| wl-dfclab 36762 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 36763 | Using class abstraction in... |
| wl-clabt 36764 | Using class abstraction in... |
| rabiun 36765 | Abstraction restricted to ... |
| iundif1 36766 | Indexed union of class dif... |
| imadifss 36767 | The difference of images i... |
| cureq 36768 | Equality theorem for curry... |
| unceq 36769 | Equality theorem for uncur... |
| curf 36770 | Functional property of cur... |
| uncf 36771 | Functional property of unc... |
| curfv 36772 | Value of currying. (Contr... |
| uncov 36773 | Value of uncurrying. (Con... |
| curunc 36774 | Currying of uncurrying. (... |
| unccur 36775 | Uncurrying of currying. (... |
| phpreu 36776 | Theorem related to pigeonh... |
| finixpnum 36777 | A finite Cartesian product... |
| fin2solem 36778 | Lemma for ~ fin2so . (Con... |
| fin2so 36779 | Any totally ordered Tarski... |
| ltflcei 36780 | Theorem to move the floor ... |
| leceifl 36781 | Theorem to move the floor ... |
| sin2h 36782 | Half-angle rule for sine. ... |
| cos2h 36783 | Half-angle rule for cosine... |
| tan2h 36784 | Half-angle rule for tangen... |
| lindsadd 36785 | In a vector space, the uni... |
| lindsdom 36786 | A linearly independent set... |
| lindsenlbs 36787 | A maximal linearly indepen... |
| matunitlindflem1 36788 | One direction of ~ matunit... |
| matunitlindflem2 36789 | One direction of ~ matunit... |
| matunitlindf 36790 | A matrix over a field is i... |
| ptrest 36791 | Expressing a restriction o... |
| ptrecube 36792 | Any point in an open set o... |
| poimirlem1 36793 | Lemma for ~ poimir - the v... |
| poimirlem2 36794 | Lemma for ~ poimir - conse... |
| poimirlem3 36795 | Lemma for ~ poimir to add ... |
| poimirlem4 36796 | Lemma for ~ poimir connect... |
| poimirlem5 36797 | Lemma for ~ poimir to esta... |
| poimirlem6 36798 | Lemma for ~ poimir establi... |
| poimirlem7 36799 | Lemma for ~ poimir , simil... |
| poimirlem8 36800 | Lemma for ~ poimir , estab... |
| poimirlem9 36801 | Lemma for ~ poimir , estab... |
| poimirlem10 36802 | Lemma for ~ poimir establi... |
| poimirlem11 36803 | Lemma for ~ poimir connect... |
| poimirlem12 36804 | Lemma for ~ poimir connect... |
| poimirlem13 36805 | Lemma for ~ poimir - for a... |
| poimirlem14 36806 | Lemma for ~ poimir - for a... |
| poimirlem15 36807 | Lemma for ~ poimir , that ... |
| poimirlem16 36808 | Lemma for ~ poimir establi... |
| poimirlem17 36809 | Lemma for ~ poimir establi... |
| poimirlem18 36810 | Lemma for ~ poimir stating... |
| poimirlem19 36811 | Lemma for ~ poimir establi... |
| poimirlem20 36812 | Lemma for ~ poimir establi... |
| poimirlem21 36813 | Lemma for ~ poimir stating... |
| poimirlem22 36814 | Lemma for ~ poimir , that ... |
| poimirlem23 36815 | Lemma for ~ poimir , two w... |
| poimirlem24 36816 | Lemma for ~ poimir , two w... |
| poimirlem25 36817 | Lemma for ~ poimir stating... |
| poimirlem26 36818 | Lemma for ~ poimir showing... |
| poimirlem27 36819 | Lemma for ~ poimir showing... |
| poimirlem28 36820 | Lemma for ~ poimir , a var... |
| poimirlem29 36821 | Lemma for ~ poimir connect... |
| poimirlem30 36822 | Lemma for ~ poimir combini... |
| poimirlem31 36823 | Lemma for ~ poimir , assig... |
| poimirlem32 36824 | Lemma for ~ poimir , combi... |
| poimir 36825 | Poincare-Miranda theorem. ... |
| broucube 36826 | Brouwer - or as Kulpa call... |
| heicant 36827 | Heine-Cantor theorem: a co... |
| opnmbllem0 36828 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 36829 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 36830 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 36831 | The difference between two... |
| mblfinlem4 36832 | Backward direction of ~ is... |
| ismblfin 36833 | Measurability in terms of ... |
| ovoliunnfl 36834 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 36835 | Demonstration of ~ ovoliun... |
| voliunnfl 36836 | ~ voliun is incompatible w... |
| volsupnfl 36837 | ~ volsup is incompatible w... |
| mbfresfi 36838 | Measurability of a piecewi... |
| mbfposadd 36839 | If the sum of two measurab... |
| cnambfre 36840 | A real-valued, a.e. contin... |
| dvtanlem 36841 | Lemma for ~ dvtan - the do... |
| dvtan 36842 | Derivative of tangent. (C... |
| itg2addnclem 36843 | An alternate expression fo... |
| itg2addnclem2 36844 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 36845 | Lemma incomprehensible in ... |
| itg2addnc 36846 | Alternate proof of ~ itg2a... |
| itg2gt0cn 36847 | ~ itg2gt0 holds on functio... |
| ibladdnclem 36848 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 36849 | Choice-free analogue of ~ ... |
| itgaddnclem1 36850 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 36851 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 36852 | Choice-free analogue of ~ ... |
| iblsubnc 36853 | Choice-free analogue of ~ ... |
| itgsubnc 36854 | Choice-free analogue of ~ ... |
| iblabsnclem 36855 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 36856 | Choice-free analogue of ~ ... |
| iblmulc2nc 36857 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 36858 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 36859 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 36860 | Choice-free analogue of ~ ... |
| itgabsnc 36861 | Choice-free analogue of ~ ... |
| itggt0cn 36862 | ~ itggt0 holds for continu... |
| ftc1cnnclem 36863 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 36864 | Choice-free proof of ~ ftc... |
| ftc1anclem1 36865 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 36866 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 36867 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 36868 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 36869 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 36870 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 36871 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 36872 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 36873 | ~ ftc1a holds for function... |
| ftc2nc 36874 | Choice-free proof of ~ ftc... |
| asindmre 36875 | Real part of domain of dif... |
| dvasin 36876 | Derivative of arcsine. (C... |
| dvacos 36877 | Derivative of arccosine. ... |
| dvreasin 36878 | Real derivative of arcsine... |
| dvreacos 36879 | Real derivative of arccosi... |
| areacirclem1 36880 | Antiderivative of cross-se... |
| areacirclem2 36881 | Endpoint-inclusive continu... |
| areacirclem3 36882 | Integrability of cross-sec... |
| areacirclem4 36883 | Endpoint-inclusive continu... |
| areacirclem5 36884 | Finding the cross-section ... |
| areacirc 36885 | The area of a circle of ra... |
| unirep 36886 | Define a quantity whose de... |
| cover2 36887 | Two ways of expressing the... |
| cover2g 36888 | Two ways of expressing the... |
| brabg2 36889 | Relation by a binary relat... |
| opelopab3 36890 | Ordered pair membership in... |
| cocanfo 36891 | Cancellation of a surjecti... |
| brresi2 36892 | Restriction of a binary re... |
| fnopabeqd 36893 | Equality deduction for fun... |
| fvopabf4g 36894 | Function value of an opera... |
| fnopabco 36895 | Composition of a function ... |
| opropabco 36896 | Composition of an operator... |
| cocnv 36897 | Composition with a functio... |
| f1ocan1fv 36898 | Cancel a composition by a ... |
| f1ocan2fv 36899 | Cancel a composition by th... |
| inixp 36900 | Intersection of Cartesian ... |
| upixp 36901 | Universal property of the ... |
| abrexdom 36902 | An indexed set is dominate... |
| abrexdom2 36903 | An indexed set is dominate... |
| ac6gf 36904 | Axiom of Choice. (Contrib... |
| indexa 36905 | If for every element of an... |
| indexdom 36906 | If for every element of an... |
| frinfm 36907 | A subset of a well-founded... |
| welb 36908 | A nonempty subset of a wel... |
| supex2g 36909 | Existence of supremum. (C... |
| supclt 36910 | Closure of supremum. (Con... |
| supubt 36911 | Upper bound property of su... |
| filbcmb 36912 | Combine a finite set of lo... |
| fzmul 36913 | Membership of a product in... |
| sdclem2 36914 | Lemma for ~ sdc . (Contri... |
| sdclem1 36915 | Lemma for ~ sdc . (Contri... |
| sdc 36916 | Strong dependent choice. ... |
| fdc 36917 | Finite version of dependen... |
| fdc1 36918 | Variant of ~ fdc with no s... |
| seqpo 36919 | Two ways to say that a seq... |
| incsequz 36920 | An increasing sequence of ... |
| incsequz2 36921 | An increasing sequence of ... |
| nnubfi 36922 | A bounded above set of pos... |
| nninfnub 36923 | An infinite set of positiv... |
| subspopn 36924 | An open set is open in the... |
| neificl 36925 | Neighborhoods are closed u... |
| lpss2 36926 | Limit points of a subset a... |
| metf1o 36927 | Use a bijection with a met... |
| blssp 36928 | A ball in the subspace met... |
| mettrifi 36929 | Generalized triangle inequ... |
| lmclim2 36930 | A sequence in a metric spa... |
| geomcau 36931 | If the distance between co... |
| caures 36932 | The restriction of a Cauch... |
| caushft 36933 | A shifted Cauchy sequence ... |
| constcncf 36934 | A constant function is a c... |
| cnres2 36935 | The restriction of a conti... |
| cnresima 36936 | A continuous function is c... |
| cncfres 36937 | A continuous function on c... |
| istotbnd 36941 | The predicate "is a totall... |
| istotbnd2 36942 | The predicate "is a totall... |
| istotbnd3 36943 | A metric space is totally ... |
| totbndmet 36944 | The predicate "totally bou... |
| 0totbnd 36945 | The metric (there is only ... |
| sstotbnd2 36946 | Condition for a subset of ... |
| sstotbnd 36947 | Condition for a subset of ... |
| sstotbnd3 36948 | Use a net that is not nece... |
| totbndss 36949 | A subset of a totally boun... |
| equivtotbnd 36950 | If the metric ` M ` is "st... |
| isbnd 36952 | The predicate "is a bounde... |
| bndmet 36953 | A bounded metric space is ... |
| isbndx 36954 | A "bounded extended metric... |
| isbnd2 36955 | The predicate "is a bounde... |
| isbnd3 36956 | A metric space is bounded ... |
| isbnd3b 36957 | A metric space is bounded ... |
| bndss 36958 | A subset of a bounded metr... |
| blbnd 36959 | A ball is bounded. (Contr... |
| ssbnd 36960 | A subset of a metric space... |
| totbndbnd 36961 | A totally bounded metric s... |
| equivbnd 36962 | If the metric ` M ` is "st... |
| bnd2lem 36963 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 36964 | If balls are totally bound... |
| prdsbnd 36965 | The product metric over fi... |
| prdstotbnd 36966 | The product metric over fi... |
| prdsbnd2 36967 | If balls are totally bound... |
| cntotbnd 36968 | A subset of the complex nu... |
| cnpwstotbnd 36969 | A subset of ` A ^ I ` , wh... |
| ismtyval 36972 | The set of isometries betw... |
| isismty 36973 | The condition "is an isome... |
| ismtycnv 36974 | The inverse of an isometry... |
| ismtyima 36975 | The image of a ball under ... |
| ismtyhmeolem 36976 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 36977 | An isometry is a homeomorp... |
| ismtybndlem 36978 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 36979 | Isometries preserve bounde... |
| ismtyres 36980 | A restriction of an isomet... |
| heibor1lem 36981 | Lemma for ~ heibor1 . A c... |
| heibor1 36982 | One half of ~ heibor , tha... |
| heiborlem1 36983 | Lemma for ~ heibor . We w... |
| heiborlem2 36984 | Lemma for ~ heibor . Subs... |
| heiborlem3 36985 | Lemma for ~ heibor . Usin... |
| heiborlem4 36986 | Lemma for ~ heibor . Usin... |
| heiborlem5 36987 | Lemma for ~ heibor . The ... |
| heiborlem6 36988 | Lemma for ~ heibor . Sinc... |
| heiborlem7 36989 | Lemma for ~ heibor . Sinc... |
| heiborlem8 36990 | Lemma for ~ heibor . The ... |
| heiborlem9 36991 | Lemma for ~ heibor . Disc... |
| heiborlem10 36992 | Lemma for ~ heibor . The ... |
| heibor 36993 | Generalized Heine-Borel Th... |
| bfplem1 36994 | Lemma for ~ bfp . The seq... |
| bfplem2 36995 | Lemma for ~ bfp . Using t... |
| bfp 36996 | Banach fixed point theorem... |
| rrnval 36999 | The n-dimensional Euclidea... |
| rrnmval 37000 | The value of the Euclidean... |
| rrnmet 37001 | Euclidean space is a metri... |
| rrndstprj1 37002 | The distance between two p... |
| rrndstprj2 37003 | Bound on the distance betw... |
| rrncmslem 37004 | Lemma for ~ rrncms . (Con... |
| rrncms 37005 | Euclidean space is complet... |
| repwsmet 37006 | The supremum metric on ` R... |
| rrnequiv 37007 | The supremum metric on ` R... |
| rrntotbnd 37008 | A set in Euclidean space i... |
| rrnheibor 37009 | Heine-Borel theorem for Eu... |
| ismrer1 37010 | An isometry between ` RR `... |
| reheibor 37011 | Heine-Borel theorem for re... |
| iccbnd 37012 | A closed interval in ` RR ... |
| icccmpALT 37013 | A closed interval in ` RR ... |
| isass 37018 | The predicate "is an assoc... |
| isexid 37019 | The predicate ` G ` has a ... |
| ismgmOLD 37022 | Obsolete version of ~ ismg... |
| clmgmOLD 37023 | Obsolete version of ~ mgmc... |
| opidonOLD 37024 | Obsolete version of ~ mndp... |
| rngopidOLD 37025 | Obsolete version of ~ mndp... |
| opidon2OLD 37026 | Obsolete version of ~ mndp... |
| isexid2 37027 | If ` G e. ( Magma i^i ExId... |
| exidu1 37028 | Uniqueness of the left and... |
| idrval 37029 | The value of the identity ... |
| iorlid 37030 | A magma right and left ide... |
| cmpidelt 37031 | A magma right and left ide... |
| smgrpismgmOLD 37034 | Obsolete version of ~ sgrp... |
| issmgrpOLD 37035 | Obsolete version of ~ issg... |
| smgrpmgm 37036 | A semigroup is a magma. (... |
| smgrpassOLD 37037 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 37040 | Obsolete version of ~ mnds... |
| mndoisexid 37041 | A monoid has an identity e... |
| mndoismgmOLD 37042 | Obsolete version of ~ mndm... |
| mndomgmid 37043 | A monoid is a magma with a... |
| ismndo 37044 | The predicate "is a monoid... |
| ismndo1 37045 | The predicate "is a monoid... |
| ismndo2 37046 | The predicate "is a monoid... |
| grpomndo 37047 | A group is a monoid. (Con... |
| exidcl 37048 | Closure of the binary oper... |
| exidreslem 37049 | Lemma for ~ exidres and ~ ... |
| exidres 37050 | The restriction of a binar... |
| exidresid 37051 | The restriction of a binar... |
| ablo4pnp 37052 | A commutative/associative ... |
| grpoeqdivid 37053 | Two group elements are equ... |
| grposnOLD 37054 | The group operation for th... |
| elghomlem1OLD 37057 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 37058 | Obsolete as of 15-Mar-2020... |
| elghomOLD 37059 | Obsolete version of ~ isgh... |
| ghomlinOLD 37060 | Obsolete version of ~ ghml... |
| ghomidOLD 37061 | Obsolete version of ~ ghmi... |
| ghomf 37062 | Mapping property of a grou... |
| ghomco 37063 | The composition of two gro... |
| ghomdiv 37064 | Group homomorphisms preser... |
| grpokerinj 37065 | A group homomorphism is in... |
| relrngo 37068 | The class of all unital ri... |
| isrngo 37069 | The predicate "is a (unita... |
| isrngod 37070 | Conditions that determine ... |
| rngoi 37071 | The properties of a unital... |
| rngosm 37072 | Functionality of the multi... |
| rngocl 37073 | Closure of the multiplicat... |
| rngoid 37074 | The multiplication operati... |
| rngoideu 37075 | The unity element of a rin... |
| rngodi 37076 | Distributive law for the m... |
| rngodir 37077 | Distributive law for the m... |
| rngoass 37078 | Associative law for the mu... |
| rngo2 37079 | A ring element plus itself... |
| rngoablo 37080 | A ring's addition operatio... |
| rngoablo2 37081 | In a unital ring the addit... |
| rngogrpo 37082 | A ring's addition operatio... |
| rngone0 37083 | The base set of a ring is ... |
| rngogcl 37084 | Closure law for the additi... |
| rngocom 37085 | The addition operation of ... |
| rngoaass 37086 | The addition operation of ... |
| rngoa32 37087 | The addition operation of ... |
| rngoa4 37088 | Rearrangement of 4 terms i... |
| rngorcan 37089 | Right cancellation law for... |
| rngolcan 37090 | Left cancellation law for ... |
| rngo0cl 37091 | A ring has an additive ide... |
| rngo0rid 37092 | The additive identity of a... |
| rngo0lid 37093 | The additive identity of a... |
| rngolz 37094 | The zero of a unital ring ... |
| rngorz 37095 | The zero of a unital ring ... |
| rngosn3 37096 | Obsolete as of 25-Jan-2020... |
| rngosn4 37097 | Obsolete as of 25-Jan-2020... |
| rngosn6 37098 | Obsolete as of 25-Jan-2020... |
| rngonegcl 37099 | A ring is closed under neg... |
| rngoaddneg1 37100 | Adding the negative in a r... |
| rngoaddneg2 37101 | Adding the negative in a r... |
| rngosub 37102 | Subtraction in a ring, in ... |
| rngmgmbs4 37103 | The range of an internal o... |
| rngodm1dm2 37104 | In a unital ring the domai... |
| rngorn1 37105 | In a unital ring the range... |
| rngorn1eq 37106 | In a unital ring the range... |
| rngomndo 37107 | In a unital ring the multi... |
| rngoidmlem 37108 | The unity element of a rin... |
| rngolidm 37109 | The unity element of a rin... |
| rngoridm 37110 | The unity element of a rin... |
| rngo1cl 37111 | The unity element of a rin... |
| rngoueqz 37112 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 37113 | Negation in a ring is the ... |
| rngonegmn1r 37114 | Negation in a ring is the ... |
| rngoneglmul 37115 | Negation of a product in a... |
| rngonegrmul 37116 | Negation of a product in a... |
| rngosubdi 37117 | Ring multiplication distri... |
| rngosubdir 37118 | Ring multiplication distri... |
| zerdivemp1x 37119 | In a unital ring a left in... |
| isdivrngo 37122 | The predicate "is a divisi... |
| drngoi 37123 | The properties of a divisi... |
| gidsn 37124 | Obsolete as of 23-Jan-2020... |
| zrdivrng 37125 | The zero ring is not a div... |
| dvrunz 37126 | In a division ring the rin... |
| isgrpda 37127 | Properties that determine ... |
| isdrngo1 37128 | The predicate "is a divisi... |
| divrngcl 37129 | The product of two nonzero... |
| isdrngo2 37130 | A division ring is a ring ... |
| isdrngo3 37131 | A division ring is a ring ... |
| rngohomval 37136 | The set of ring homomorphi... |
| isrngohom 37137 | The predicate "is a ring h... |
| rngohomf 37138 | A ring homomorphism is a f... |
| rngohomcl 37139 | Closure law for a ring hom... |
| rngohom1 37140 | A ring homomorphism preser... |
| rngohomadd 37141 | Ring homomorphisms preserv... |
| rngohommul 37142 | Ring homomorphisms preserv... |
| rngogrphom 37143 | A ring homomorphism is a g... |
| rngohom0 37144 | A ring homomorphism preser... |
| rngohomsub 37145 | Ring homomorphisms preserv... |
| rngohomco 37146 | The composition of two rin... |
| rngokerinj 37147 | A ring homomorphism is inj... |
| rngoisoval 37149 | The set of ring isomorphis... |
| isrngoiso 37150 | The predicate "is a ring i... |
| rngoiso1o 37151 | A ring isomorphism is a bi... |
| rngoisohom 37152 | A ring isomorphism is a ri... |
| rngoisocnv 37153 | The inverse of a ring isom... |
| rngoisoco 37154 | The composition of two rin... |
| isriscg 37156 | The ring isomorphism relat... |
| isrisc 37157 | The ring isomorphism relat... |
| risc 37158 | The ring isomorphism relat... |
| risci 37159 | Determine that two rings a... |
| riscer 37160 | Ring isomorphism is an equ... |
| iscom2 37167 | A device to add commutativ... |
| iscrngo 37168 | The predicate "is a commut... |
| iscrngo2 37169 | The predicate "is a commut... |
| iscringd 37170 | Conditions that determine ... |
| flddivrng 37171 | A field is a division ring... |
| crngorngo 37172 | A commutative ring is a ri... |
| crngocom 37173 | The multiplication operati... |
| crngm23 37174 | Commutative/associative la... |
| crngm4 37175 | Commutative/associative la... |
| fldcrngo 37176 | A field is a commutative r... |
| isfld2 37177 | The predicate "is a field"... |
| crngohomfo 37178 | The image of a homomorphis... |
| idlval 37185 | The class of ideals of a r... |
| isidl 37186 | The predicate "is an ideal... |
| isidlc 37187 | The predicate "is an ideal... |
| idlss 37188 | An ideal of ` R ` is a sub... |
| idlcl 37189 | An element of an ideal is ... |
| idl0cl 37190 | An ideal contains ` 0 ` . ... |
| idladdcl 37191 | An ideal is closed under a... |
| idllmulcl 37192 | An ideal is closed under m... |
| idlrmulcl 37193 | An ideal is closed under m... |
| idlnegcl 37194 | An ideal is closed under n... |
| idlsubcl 37195 | An ideal is closed under s... |
| rngoidl 37196 | A ring ` R ` is an ` R ` i... |
| 0idl 37197 | The set containing only ` ... |
| 1idl 37198 | Two ways of expressing the... |
| 0rngo 37199 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 37200 | The only ideals in a divis... |
| intidl 37201 | The intersection of a none... |
| inidl 37202 | The intersection of two id... |
| unichnidl 37203 | The union of a nonempty ch... |
| keridl 37204 | The kernel of a ring homom... |
| pridlval 37205 | The class of prime ideals ... |
| ispridl 37206 | The predicate "is a prime ... |
| pridlidl 37207 | A prime ideal is an ideal.... |
| pridlnr 37208 | A prime ideal is a proper ... |
| pridl 37209 | The main property of a pri... |
| ispridl2 37210 | A condition that shows an ... |
| maxidlval 37211 | The set of maximal ideals ... |
| ismaxidl 37212 | The predicate "is a maxima... |
| maxidlidl 37213 | A maximal ideal is an idea... |
| maxidlnr 37214 | A maximal ideal is proper.... |
| maxidlmax 37215 | A maximal ideal is a maxim... |
| maxidln1 37216 | One is not contained in an... |
| maxidln0 37217 | A ring with a maximal idea... |
| isprrngo 37222 | The predicate "is a prime ... |
| prrngorngo 37223 | A prime ring is a ring. (... |
| smprngopr 37224 | A simple ring (one whose o... |
| divrngpr 37225 | A division ring is a prime... |
| isdmn 37226 | The predicate "is a domain... |
| isdmn2 37227 | The predicate "is a domain... |
| dmncrng 37228 | A domain is a commutative ... |
| dmnrngo 37229 | A domain is a ring. (Cont... |
| flddmn 37230 | A field is a domain. (Con... |
| igenval 37233 | The ideal generated by a s... |
| igenss 37234 | A set is a subset of the i... |
| igenidl 37235 | The ideal generated by a s... |
| igenmin 37236 | The ideal generated by a s... |
| igenidl2 37237 | The ideal generated by an ... |
| igenval2 37238 | The ideal generated by a s... |
| prnc 37239 | A principal ideal (an idea... |
| isfldidl 37240 | Determine if a ring is a f... |
| isfldidl2 37241 | Determine if a ring is a f... |
| ispridlc 37242 | The predicate "is a prime ... |
| pridlc 37243 | Property of a prime ideal ... |
| pridlc2 37244 | Property of a prime ideal ... |
| pridlc3 37245 | Property of a prime ideal ... |
| isdmn3 37246 | The predicate "is a domain... |
| dmnnzd 37247 | A domain has no zero-divis... |
| dmncan1 37248 | Cancellation law for domai... |
| dmncan2 37249 | Cancellation law for domai... |
| efald2 37250 | A proof by contradiction. ... |
| notbinot1 37251 | Simplification rule of neg... |
| bicontr 37252 | Biconditional of its own n... |
| impor 37253 | An equivalent formula for ... |
| orfa 37254 | The falsum ` F. ` can be r... |
| notbinot2 37255 | Commutation rule between n... |
| biimpor 37256 | A rewriting rule for bicon... |
| orfa1 37257 | Add a contradicting disjun... |
| orfa2 37258 | Remove a contradicting dis... |
| bifald 37259 | Infer the equivalence to a... |
| orsild 37260 | A lemma for not-or-not eli... |
| orsird 37261 | A lemma for not-or-not eli... |
| cnf1dd 37262 | A lemma for Conjunctive No... |
| cnf2dd 37263 | A lemma for Conjunctive No... |
| cnfn1dd 37264 | A lemma for Conjunctive No... |
| cnfn2dd 37265 | A lemma for Conjunctive No... |
| or32dd 37266 | A rearrangement of disjunc... |
| notornotel1 37267 | A lemma for not-or-not eli... |
| notornotel2 37268 | A lemma for not-or-not eli... |
| contrd 37269 | A proof by contradiction, ... |
| an12i 37270 | An inference from commutin... |
| exmid2 37271 | An excluded middle law. (... |
| selconj 37272 | An inference for selecting... |
| truconj 37273 | Add true as a conjunct. (... |
| orel 37274 | An inference for disjuncti... |
| negel 37275 | An inference for negation ... |
| botel 37276 | An inference for bottom el... |
| tradd 37277 | Add top ad a conjunct. (C... |
| gm-sbtru 37278 | Substitution does not chan... |
| sbfal 37279 | Substitution does not chan... |
| sbcani 37280 | Distribution of class subs... |
| sbcori 37281 | Distribution of class subs... |
| sbcimi 37282 | Distribution of class subs... |
| sbcni 37283 | Move class substitution in... |
| sbali 37284 | Discard class substitution... |
| sbexi 37285 | Discard class substitution... |
| sbcalf 37286 | Move universal quantifier ... |
| sbcexf 37287 | Move existential quantifie... |
| sbcalfi 37288 | Move universal quantifier ... |
| sbcexfi 37289 | Move existential quantifie... |
| spsbcdi 37290 | A lemma for eliminating a ... |
| alrimii 37291 | A lemma for introducing a ... |
| spesbcdi 37292 | A lemma for introducing an... |
| exlimddvf 37293 | A lemma for eliminating an... |
| exlimddvfi 37294 | A lemma for eliminating an... |
| sbceq1ddi 37295 | A lemma for eliminating in... |
| sbccom2lem 37296 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 37297 | Commutative law for double... |
| sbccom2f 37298 | Commutative law for double... |
| sbccom2fi 37299 | Commutative law for double... |
| csbcom2fi 37300 | Commutative law for double... |
| fald 37301 | Refutation of falsity, in ... |
| tsim1 37302 | A Tseitin axiom for logica... |
| tsim2 37303 | A Tseitin axiom for logica... |
| tsim3 37304 | A Tseitin axiom for logica... |
| tsbi1 37305 | A Tseitin axiom for logica... |
| tsbi2 37306 | A Tseitin axiom for logica... |
| tsbi3 37307 | A Tseitin axiom for logica... |
| tsbi4 37308 | A Tseitin axiom for logica... |
| tsxo1 37309 | A Tseitin axiom for logica... |
| tsxo2 37310 | A Tseitin axiom for logica... |
| tsxo3 37311 | A Tseitin axiom for logica... |
| tsxo4 37312 | A Tseitin axiom for logica... |
| tsan1 37313 | A Tseitin axiom for logica... |
| tsan2 37314 | A Tseitin axiom for logica... |
| tsan3 37315 | A Tseitin axiom for logica... |
| tsna1 37316 | A Tseitin axiom for logica... |
| tsna2 37317 | A Tseitin axiom for logica... |
| tsna3 37318 | A Tseitin axiom for logica... |
| tsor1 37319 | A Tseitin axiom for logica... |
| tsor2 37320 | A Tseitin axiom for logica... |
| tsor3 37321 | A Tseitin axiom for logica... |
| ts3an1 37322 | A Tseitin axiom for triple... |
| ts3an2 37323 | A Tseitin axiom for triple... |
| ts3an3 37324 | A Tseitin axiom for triple... |
| ts3or1 37325 | A Tseitin axiom for triple... |
| ts3or2 37326 | A Tseitin axiom for triple... |
| ts3or3 37327 | A Tseitin axiom for triple... |
| iuneq2f 37328 | Equality deduction for ind... |
| rabeq12f 37329 | Equality deduction for res... |
| csbeq12 37330 | Equality deduction for sub... |
| sbeqi 37331 | Equality deduction for sub... |
| ralbi12f 37332 | Equality deduction for res... |
| oprabbi 37333 | Equality deduction for cla... |
| mpobi123f 37334 | Equality deduction for map... |
| iuneq12f 37335 | Equality deduction for ind... |
| iineq12f 37336 | Equality deduction for ind... |
| opabbi 37337 | Equality deduction for cla... |
| mptbi12f 37338 | Equality deduction for map... |
| orcomdd 37339 | Commutativity of logic dis... |
| scottexf 37340 | A version of ~ scottex wit... |
| scott0f 37341 | A version of ~ scott0 with... |
| scottn0f 37342 | A version of ~ scott0f wit... |
| ac6s3f 37343 | Generalization of the Axio... |
| ac6s6 37344 | Generalization of the Axio... |
| ac6s6f 37345 | Generalization of the Axio... |
| el2v1 37389 | New way ( ~ elv , and the ... |
| el3v 37390 | New way ( ~ elv , and the ... |
| el3v1 37391 | New way ( ~ elv , and the ... |
| el3v2 37392 | New way ( ~ elv , and the ... |
| el3v3 37393 | New way ( ~ elv , and the ... |
| el3v12 37394 | New way ( ~ elv , and the ... |
| el3v13 37395 | New way ( ~ elv , and the ... |
| el3v23 37396 | New way ( ~ elv , and the ... |
| anan 37397 | Multiple commutations in c... |
| triantru3 37398 | A wff is equivalent to its... |
| bianbi 37399 | Exchanging conjunction in ... |
| bianim 37400 | Exchanging conjunction in ... |
| biorfd 37401 | A wff is equivalent to its... |
| eqbrtr 37402 | Substitution of equal clas... |
| eqbrb 37403 | Substitution of equal clas... |
| eqeltr 37404 | Substitution of equal clas... |
| eqelb 37405 | Substitution of equal clas... |
| eqeqan2d 37406 | Implication of introducing... |
| suceqsneq 37407 | One-to-one relationship be... |
| sucdifsn2 37408 | Absorption of union with a... |
| sucdifsn 37409 | The difference between the... |
| disjresin 37410 | The restriction to a disjo... |
| disjresdisj 37411 | The intersection of restri... |
| disjresdif 37412 | The difference between res... |
| disjresundif 37413 | Lemma for ~ ressucdifsn2 .... |
| ressucdifsn2 37414 | The difference between res... |
| ressucdifsn 37415 | The difference between res... |
| inres2 37416 | Two ways of expressing the... |
| coideq 37417 | Equality theorem for compo... |
| nexmo1 37418 | If there is no case where ... |
| ralin 37419 | Restricted universal quant... |
| r2alan 37420 | Double restricted universa... |
| ssrabi 37421 | Inference of restricted ab... |
| rabbieq 37422 | Equivalent wff's correspon... |
| rabimbieq 37423 | Restricted equivalent wff'... |
| abeqin 37424 | Intersection with class ab... |
| abeqinbi 37425 | Intersection with class ab... |
| rabeqel 37426 | Class element of a restric... |
| eqrelf 37427 | The equality connective be... |
| br1cnvinxp 37428 | Binary relation on the con... |
| releleccnv 37429 | Elementhood in a converse ... |
| releccnveq 37430 | Equality of converse ` R `... |
| opelvvdif 37431 | Negated elementhood of ord... |
| vvdifopab 37432 | Ordered-pair class abstrac... |
| brvdif 37433 | Binary relation with unive... |
| brvdif2 37434 | Binary relation with unive... |
| brvvdif 37435 | Binary relation with the c... |
| brvbrvvdif 37436 | Binary relation with the c... |
| brcnvep 37437 | The converse of the binary... |
| elecALTV 37438 | Elementhood in the ` R ` -... |
| brcnvepres 37439 | Restricted converse epsilo... |
| brres2 37440 | Binary relation on a restr... |
| br1cnvres 37441 | Binary relation on the con... |
| eldmres 37442 | Elementhood in the domain ... |
| elrnres 37443 | Element of the range of a ... |
| eldmressnALTV 37444 | Element of the domain of a... |
| elrnressn 37445 | Element of the range of a ... |
| eldm4 37446 | Elementhood in a domain. ... |
| eldmres2 37447 | Elementhood in the domain ... |
| eceq1i 37448 | Equality theorem for ` C `... |
| elecres 37449 | Elementhood in the restric... |
| ecres 37450 | Restricted coset of ` B ` ... |
| ecres2 37451 | The restricted coset of ` ... |
| eccnvepres 37452 | Restricted converse epsilo... |
| eleccnvep 37453 | Elementhood in the convers... |
| eccnvep 37454 | The converse epsilon coset... |
| extep 37455 | Property of epsilon relati... |
| disjeccnvep 37456 | Property of the epsilon re... |
| eccnvepres2 37457 | The restricted converse ep... |
| eccnvepres3 37458 | Condition for a restricted... |
| eldmqsres 37459 | Elementhood in a restricte... |
| eldmqsres2 37460 | Elementhood in a restricte... |
| qsss1 37461 | Subclass theorem for quoti... |
| qseq1i 37462 | Equality theorem for quoti... |
| qseq1d 37463 | Equality theorem for quoti... |
| brinxprnres 37464 | Binary relation on a restr... |
| inxprnres 37465 | Restriction of a class as ... |
| dfres4 37466 | Alternate definition of th... |
| exan3 37467 | Equivalent expressions wit... |
| exanres 37468 | Equivalent expressions wit... |
| exanres3 37469 | Equivalent expressions wit... |
| exanres2 37470 | Equivalent expressions wit... |
| cnvepres 37471 | Restricted converse epsilo... |
| eqrel2 37472 | Equality of relations. (C... |
| rncnv 37473 | Range of converse is the d... |
| dfdm6 37474 | Alternate definition of do... |
| dfrn6 37475 | Alternate definition of ra... |
| rncnvepres 37476 | The range of the restricte... |
| dmecd 37477 | Equality of the coset of `... |
| dmec2d 37478 | Equality of the coset of `... |
| brid 37479 | Property of the identity b... |
| ideq2 37480 | For sets, the identity bin... |
| idresssidinxp 37481 | Condition for the identity... |
| idreseqidinxp 37482 | Condition for the identity... |
| extid 37483 | Property of identity relat... |
| inxpss 37484 | Two ways to say that an in... |
| idinxpss 37485 | Two ways to say that an in... |
| ref5 37486 | Two ways to say that an in... |
| inxpss3 37487 | Two ways to say that an in... |
| inxpss2 37488 | Two ways to say that inter... |
| inxpssidinxp 37489 | Two ways to say that inter... |
| idinxpssinxp 37490 | Two ways to say that inter... |
| idinxpssinxp2 37491 | Identity intersection with... |
| idinxpssinxp3 37492 | Identity intersection with... |
| idinxpssinxp4 37493 | Identity intersection with... |
| relcnveq3 37494 | Two ways of saying a relat... |
| relcnveq 37495 | Two ways of saying a relat... |
| relcnveq2 37496 | Two ways of saying a relat... |
| relcnveq4 37497 | Two ways of saying a relat... |
| qsresid 37498 | Simplification of a specia... |
| n0elqs 37499 | Two ways of expressing tha... |
| n0elqs2 37500 | Two ways of expressing tha... |
| ecex2 37501 | Condition for a coset to b... |
| uniqsALTV 37502 | The union of a quotient se... |
| imaexALTV 37503 | Existence of an image of a... |
| ecexALTV 37504 | Existence of a coset, like... |
| rnresequniqs 37505 | The range of a restriction... |
| n0el2 37506 | Two ways of expressing tha... |
| cnvepresex 37507 | Sethood condition for the ... |
| eccnvepex 37508 | The converse epsilon coset... |
| cnvepimaex 37509 | The image of converse epsi... |
| cnvepima 37510 | The image of converse epsi... |
| inex3 37511 | Sufficient condition for t... |
| inxpex 37512 | Sufficient condition for a... |
| eqres 37513 | Converting a class constan... |
| brrabga 37514 | The law of concretion for ... |
| brcnvrabga 37515 | The law of concretion for ... |
| opideq 37516 | Equality conditions for or... |
| iss2 37517 | A subclass of the identity... |
| eldmcnv 37518 | Elementhood in a domain of... |
| dfrel5 37519 | Alternate definition of th... |
| dfrel6 37520 | Alternate definition of th... |
| cnvresrn 37521 | Converse restricted to ran... |
| relssinxpdmrn 37522 | Subset of restriction, spe... |
| cnvref4 37523 | Two ways to say that a rel... |
| cnvref5 37524 | Two ways to say that a rel... |
| ecin0 37525 | Two ways of saying that th... |
| ecinn0 37526 | Two ways of saying that th... |
| ineleq 37527 | Equivalence of restricted ... |
| inecmo 37528 | Equivalence of a double re... |
| inecmo2 37529 | Equivalence of a double re... |
| ineccnvmo 37530 | Equivalence of a double re... |
| alrmomorn 37531 | Equivalence of an "at most... |
| alrmomodm 37532 | Equivalence of an "at most... |
| ineccnvmo2 37533 | Equivalence of a double un... |
| inecmo3 37534 | Equivalence of a double un... |
| moeu2 37535 | Uniqueness is equivalent t... |
| mopickr 37536 | "At most one" picks a vari... |
| moantr 37537 | Sufficient condition for t... |
| brabidgaw 37538 | The law of concretion for ... |
| brabidga 37539 | The law of concretion for ... |
| inxp2 37540 | Intersection with a Cartes... |
| opabf 37541 | A class abstraction of a c... |
| ec0 37542 | The empty-coset of a class... |
| 0qs 37543 | Quotient set with the empt... |
| brcnvin 37544 | Intersection with a conver... |
| xrnss3v 37546 | A range Cartesian product ... |
| xrnrel 37547 | A range Cartesian product ... |
| brxrn 37548 | Characterize a ternary rel... |
| brxrn2 37549 | A characterization of the ... |
| dfxrn2 37550 | Alternate definition of th... |
| xrneq1 37551 | Equality theorem for the r... |
| xrneq1i 37552 | Equality theorem for the r... |
| xrneq1d 37553 | Equality theorem for the r... |
| xrneq2 37554 | Equality theorem for the r... |
| xrneq2i 37555 | Equality theorem for the r... |
| xrneq2d 37556 | Equality theorem for the r... |
| xrneq12 37557 | Equality theorem for the r... |
| xrneq12i 37558 | Equality theorem for the r... |
| xrneq12d 37559 | Equality theorem for the r... |
| elecxrn 37560 | Elementhood in the ` ( R |... |
| ecxrn 37561 | The ` ( R |X. S ) ` -coset... |
| disjressuc2 37562 | Double restricted quantifi... |
| disjecxrn 37563 | Two ways of saying that ` ... |
| disjecxrncnvep 37564 | Two ways of saying that co... |
| disjsuc2 37565 | Double restricted quantifi... |
| xrninxp 37566 | Intersection of a range Ca... |
| xrninxp2 37567 | Intersection of a range Ca... |
| xrninxpex 37568 | Sufficient condition for t... |
| inxpxrn 37569 | Two ways to express the in... |
| br1cnvxrn2 37570 | The converse of a binary r... |
| elec1cnvxrn2 37571 | Elementhood in the convers... |
| rnxrn 37572 | Range of the range Cartesi... |
| rnxrnres 37573 | Range of a range Cartesian... |
| rnxrncnvepres 37574 | Range of a range Cartesian... |
| rnxrnidres 37575 | Range of a range Cartesian... |
| xrnres 37576 | Two ways to express restri... |
| xrnres2 37577 | Two ways to express restri... |
| xrnres3 37578 | Two ways to express restri... |
| xrnres4 37579 | Two ways to express restri... |
| xrnresex 37580 | Sufficient condition for a... |
| xrnidresex 37581 | Sufficient condition for a... |
| xrncnvepresex 37582 | Sufficient condition for a... |
| brin2 37583 | Binary relation on an inte... |
| brin3 37584 | Binary relation on an inte... |
| dfcoss2 37587 | Alternate definition of th... |
| dfcoss3 37588 | Alternate definition of th... |
| dfcoss4 37589 | Alternate definition of th... |
| cosscnv 37590 | Class of cosets by the con... |
| coss1cnvres 37591 | Class of cosets by the con... |
| coss2cnvepres 37592 | Special case of ~ coss1cnv... |
| cossex 37593 | If ` A ` is a set then the... |
| cosscnvex 37594 | If ` A ` is a set then the... |
| 1cosscnvepresex 37595 | Sufficient condition for a... |
| 1cossxrncnvepresex 37596 | Sufficient condition for a... |
| relcoss 37597 | Cosets by ` R ` is a relat... |
| relcoels 37598 | Coelements on ` A ` is a r... |
| cossss 37599 | Subclass theorem for the c... |
| cosseq 37600 | Equality theorem for the c... |
| cosseqi 37601 | Equality theorem for the c... |
| cosseqd 37602 | Equality theorem for the c... |
| 1cossres 37603 | The class of cosets by a r... |
| dfcoels 37604 | Alternate definition of th... |
| brcoss 37605 | ` A ` and ` B ` are cosets... |
| brcoss2 37606 | Alternate form of the ` A ... |
| brcoss3 37607 | Alternate form of the ` A ... |
| brcosscnvcoss 37608 | For sets, the ` A ` and ` ... |
| brcoels 37609 | ` B ` and ` C ` are coelem... |
| cocossss 37610 | Two ways of saying that co... |
| cnvcosseq 37611 | The converse of cosets by ... |
| br2coss 37612 | Cosets by ` ,~ R ` binary ... |
| br1cossres 37613 | ` B ` and ` C ` are cosets... |
| br1cossres2 37614 | ` B ` and ` C ` are cosets... |
| brressn 37615 | Binary relation on a restr... |
| ressn2 37616 | A class ' R ' restricted t... |
| refressn 37617 | Any class ' R ' restricted... |
| antisymressn 37618 | Every class ' R ' restrict... |
| trressn 37619 | Any class ' R ' restricted... |
| relbrcoss 37620 | ` A ` and ` B ` are cosets... |
| br1cossinres 37621 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 37622 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 37623 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 37624 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 37625 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 37626 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 37627 | The domain of cosets is th... |
| dmcoss2 37628 | The domain of cosets is th... |
| rncossdmcoss 37629 | The range of cosets is the... |
| dm1cosscnvepres 37630 | The domain of cosets of th... |
| dmcoels 37631 | The domain of coelements i... |
| eldmcoss 37632 | Elementhood in the domain ... |
| eldmcoss2 37633 | Elementhood in the domain ... |
| eldm1cossres 37634 | Elementhood in the domain ... |
| eldm1cossres2 37635 | Elementhood in the domain ... |
| refrelcosslem 37636 | Lemma for the left side of... |
| refrelcoss3 37637 | The class of cosets by ` R... |
| refrelcoss2 37638 | The class of cosets by ` R... |
| symrelcoss3 37639 | The class of cosets by ` R... |
| symrelcoss2 37640 | The class of cosets by ` R... |
| cossssid 37641 | Equivalent expressions for... |
| cossssid2 37642 | Equivalent expressions for... |
| cossssid3 37643 | Equivalent expressions for... |
| cossssid4 37644 | Equivalent expressions for... |
| cossssid5 37645 | Equivalent expressions for... |
| brcosscnv 37646 | ` A ` and ` B ` are cosets... |
| brcosscnv2 37647 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 37648 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 37649 | Cosets by the converse ran... |
| cosscnvssid3 37650 | Equivalent expressions for... |
| cosscnvssid4 37651 | Equivalent expressions for... |
| cosscnvssid5 37652 | Equivalent expressions for... |
| coss0 37653 | Cosets by the empty set ar... |
| cossid 37654 | Cosets by the identity rel... |
| cosscnvid 37655 | Cosets by the converse ide... |
| trcoss 37656 | Sufficient condition for t... |
| eleccossin 37657 | Two ways of saying that th... |
| trcoss2 37658 | Equivalent expressions for... |
| elrels2 37660 | The element of the relatio... |
| elrelsrel 37661 | The element of the relatio... |
| elrelsrelim 37662 | The element of the relatio... |
| elrels5 37663 | Equivalent expressions for... |
| elrels6 37664 | Equivalent expressions for... |
| elrelscnveq3 37665 | Two ways of saying a relat... |
| elrelscnveq 37666 | Two ways of saying a relat... |
| elrelscnveq2 37667 | Two ways of saying a relat... |
| elrelscnveq4 37668 | Two ways of saying a relat... |
| cnvelrels 37669 | The converse of a set is a... |
| cosselrels 37670 | Cosets of sets are element... |
| cosscnvelrels 37671 | Cosets of converse sets ar... |
| dfssr2 37673 | Alternate definition of th... |
| relssr 37674 | The subset relation is a r... |
| brssr 37675 | The subset relation and su... |
| brssrid 37676 | Any set is a subset of its... |
| issetssr 37677 | Two ways of expressing set... |
| brssrres 37678 | Restricted subset binary r... |
| br1cnvssrres 37679 | Restricted converse subset... |
| brcnvssr 37680 | The converse of a subset r... |
| brcnvssrid 37681 | Any set is a converse subs... |
| br1cossxrncnvssrres 37682 | ` <. B , C >. ` and ` <. D... |
| extssr 37683 | Property of subset relatio... |
| dfrefrels2 37687 | Alternate definition of th... |
| dfrefrels3 37688 | Alternate definition of th... |
| dfrefrel2 37689 | Alternate definition of th... |
| dfrefrel3 37690 | Alternate definition of th... |
| dfrefrel5 37691 | Alternate definition of th... |
| elrefrels2 37692 | Element of the class of re... |
| elrefrels3 37693 | Element of the class of re... |
| elrefrelsrel 37694 | For sets, being an element... |
| refreleq 37695 | Equality theorem for refle... |
| refrelid 37696 | Identity relation is refle... |
| refrelcoss 37697 | The class of cosets by ` R... |
| refrelressn 37698 | Any class ' R ' restricted... |
| dfcnvrefrels2 37702 | Alternate definition of th... |
| dfcnvrefrels3 37703 | Alternate definition of th... |
| dfcnvrefrel2 37704 | Alternate definition of th... |
| dfcnvrefrel3 37705 | Alternate definition of th... |
| dfcnvrefrel4 37706 | Alternate definition of th... |
| dfcnvrefrel5 37707 | Alternate definition of th... |
| elcnvrefrels2 37708 | Element of the class of co... |
| elcnvrefrels3 37709 | Element of the class of co... |
| elcnvrefrelsrel 37710 | For sets, being an element... |
| cnvrefrelcoss2 37711 | Necessary and sufficient c... |
| cosselcnvrefrels2 37712 | Necessary and sufficient c... |
| cosselcnvrefrels3 37713 | Necessary and sufficient c... |
| cosselcnvrefrels4 37714 | Necessary and sufficient c... |
| cosselcnvrefrels5 37715 | Necessary and sufficient c... |
| dfsymrels2 37719 | Alternate definition of th... |
| dfsymrels3 37720 | Alternate definition of th... |
| dfsymrels4 37721 | Alternate definition of th... |
| dfsymrels5 37722 | Alternate definition of th... |
| dfsymrel2 37723 | Alternate definition of th... |
| dfsymrel3 37724 | Alternate definition of th... |
| dfsymrel4 37725 | Alternate definition of th... |
| dfsymrel5 37726 | Alternate definition of th... |
| elsymrels2 37727 | Element of the class of sy... |
| elsymrels3 37728 | Element of the class of sy... |
| elsymrels4 37729 | Element of the class of sy... |
| elsymrels5 37730 | Element of the class of sy... |
| elsymrelsrel 37731 | For sets, being an element... |
| symreleq 37732 | Equality theorem for symme... |
| symrelim 37733 | Symmetric relation implies... |
| symrelcoss 37734 | The class of cosets by ` R... |
| idsymrel 37735 | The identity relation is s... |
| epnsymrel 37736 | The membership (epsilon) r... |
| symrefref2 37737 | Symmetry is a sufficient c... |
| symrefref3 37738 | Symmetry is a sufficient c... |
| refsymrels2 37739 | Elements of the class of r... |
| refsymrels3 37740 | Elements of the class of r... |
| refsymrel2 37741 | A relation which is reflex... |
| refsymrel3 37742 | A relation which is reflex... |
| elrefsymrels2 37743 | Elements of the class of r... |
| elrefsymrels3 37744 | Elements of the class of r... |
| elrefsymrelsrel 37745 | For sets, being an element... |
| dftrrels2 37749 | Alternate definition of th... |
| dftrrels3 37750 | Alternate definition of th... |
| dftrrel2 37751 | Alternate definition of th... |
| dftrrel3 37752 | Alternate definition of th... |
| eltrrels2 37753 | Element of the class of tr... |
| eltrrels3 37754 | Element of the class of tr... |
| eltrrelsrel 37755 | For sets, being an element... |
| trreleq 37756 | Equality theorem for the t... |
| trrelressn 37757 | Any class ' R ' restricted... |
| dfeqvrels2 37762 | Alternate definition of th... |
| dfeqvrels3 37763 | Alternate definition of th... |
| dfeqvrel2 37764 | Alternate definition of th... |
| dfeqvrel3 37765 | Alternate definition of th... |
| eleqvrels2 37766 | Element of the class of eq... |
| eleqvrels3 37767 | Element of the class of eq... |
| eleqvrelsrel 37768 | For sets, being an element... |
| elcoeleqvrels 37769 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 37770 | For sets, being an element... |
| eqvrelrel 37771 | An equivalence relation is... |
| eqvrelrefrel 37772 | An equivalence relation is... |
| eqvrelsymrel 37773 | An equivalence relation is... |
| eqvreltrrel 37774 | An equivalence relation is... |
| eqvrelim 37775 | Equivalence relation impli... |
| eqvreleq 37776 | Equality theorem for equiv... |
| eqvreleqi 37777 | Equality theorem for equiv... |
| eqvreleqd 37778 | Equality theorem for equiv... |
| eqvrelsym 37779 | An equivalence relation is... |
| eqvrelsymb 37780 | An equivalence relation is... |
| eqvreltr 37781 | An equivalence relation is... |
| eqvreltrd 37782 | A transitivity relation fo... |
| eqvreltr4d 37783 | A transitivity relation fo... |
| eqvrelref 37784 | An equivalence relation is... |
| eqvrelth 37785 | Basic property of equivale... |
| eqvrelcl 37786 | Elementhood in the field o... |
| eqvrelthi 37787 | Basic property of equivale... |
| eqvreldisj 37788 | Equivalence classes do not... |
| qsdisjALTV 37789 | Elements of a quotient set... |
| eqvrelqsel 37790 | If an element of a quotien... |
| eqvrelcoss 37791 | Two ways to express equiva... |
| eqvrelcoss3 37792 | Two ways to express equiva... |
| eqvrelcoss2 37793 | Two ways to express equiva... |
| eqvrelcoss4 37794 | Two ways to express equiva... |
| dfcoeleqvrels 37795 | Alternate definition of th... |
| dfcoeleqvrel 37796 | Alternate definition of th... |
| brredunds 37800 | Binary relation on the cla... |
| brredundsredund 37801 | For sets, binary relation ... |
| redundss3 37802 | Implication of redundancy ... |
| redundeq1 37803 | Equivalence of redundancy ... |
| redundpim3 37804 | Implication of redundancy ... |
| redundpbi1 37805 | Equivalence of redundancy ... |
| refrelsredund4 37806 | The naive version of the c... |
| refrelsredund2 37807 | The naive version of the c... |
| refrelsredund3 37808 | The naive version of the c... |
| refrelredund4 37809 | The naive version of the d... |
| refrelredund2 37810 | The naive version of the d... |
| refrelredund3 37811 | The naive version of the d... |
| dmqseq 37814 | Equality theorem for domai... |
| dmqseqi 37815 | Equality theorem for domai... |
| dmqseqd 37816 | Equality theorem for domai... |
| dmqseqeq1 37817 | Equality theorem for domai... |
| dmqseqeq1i 37818 | Equality theorem for domai... |
| dmqseqeq1d 37819 | Equality theorem for domai... |
| brdmqss 37820 | The domain quotient binary... |
| brdmqssqs 37821 | If ` A ` and ` R ` are set... |
| n0eldmqs 37822 | The empty set is not an el... |
| n0eldmqseq 37823 | The empty set is not an el... |
| n0elim 37824 | Implication of that the em... |
| n0el3 37825 | Two ways of expressing tha... |
| cnvepresdmqss 37826 | The domain quotient binary... |
| cnvepresdmqs 37827 | The domain quotient predic... |
| unidmqs 37828 | The range of a relation is... |
| unidmqseq 37829 | The union of the domain qu... |
| dmqseqim 37830 | If the domain quotient of ... |
| dmqseqim2 37831 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 37832 | Elementhood in the domain ... |
| eldmqs1cossres 37833 | Elementhood in the domain ... |
| releldmqscoss 37834 | Elementhood in the domain ... |
| dmqscoelseq 37835 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 37836 | Two ways to express the eq... |
| brers 37841 | Binary equivalence relatio... |
| dferALTV2 37842 | Equivalence relation with ... |
| erALTVeq1 37843 | Equality theorem for equiv... |
| erALTVeq1i 37844 | Equality theorem for equiv... |
| erALTVeq1d 37845 | Equality theorem for equiv... |
| dfcomember 37846 | Alternate definition of th... |
| dfcomember2 37847 | Alternate definition of th... |
| dfcomember3 37848 | Alternate definition of th... |
| eqvreldmqs 37849 | Two ways to express comemb... |
| eqvreldmqs2 37850 | Two ways to express comemb... |
| brerser 37851 | Binary equivalence relatio... |
| erimeq2 37852 | Equivalence relation on it... |
| erimeq 37853 | Equivalence relation on it... |
| dffunsALTV 37857 | Alternate definition of th... |
| dffunsALTV2 37858 | Alternate definition of th... |
| dffunsALTV3 37859 | Alternate definition of th... |
| dffunsALTV4 37860 | Alternate definition of th... |
| dffunsALTV5 37861 | Alternate definition of th... |
| dffunALTV2 37862 | Alternate definition of th... |
| dffunALTV3 37863 | Alternate definition of th... |
| dffunALTV4 37864 | Alternate definition of th... |
| dffunALTV5 37865 | Alternate definition of th... |
| elfunsALTV 37866 | Elementhood in the class o... |
| elfunsALTV2 37867 | Elementhood in the class o... |
| elfunsALTV3 37868 | Elementhood in the class o... |
| elfunsALTV4 37869 | Elementhood in the class o... |
| elfunsALTV5 37870 | Elementhood in the class o... |
| elfunsALTVfunALTV 37871 | The element of the class o... |
| funALTVfun 37872 | Our definition of the func... |
| funALTVss 37873 | Subclass theorem for funct... |
| funALTVeq 37874 | Equality theorem for funct... |
| funALTVeqi 37875 | Equality inference for the... |
| funALTVeqd 37876 | Equality deduction for the... |
| dfdisjs 37882 | Alternate definition of th... |
| dfdisjs2 37883 | Alternate definition of th... |
| dfdisjs3 37884 | Alternate definition of th... |
| dfdisjs4 37885 | Alternate definition of th... |
| dfdisjs5 37886 | Alternate definition of th... |
| dfdisjALTV 37887 | Alternate definition of th... |
| dfdisjALTV2 37888 | Alternate definition of th... |
| dfdisjALTV3 37889 | Alternate definition of th... |
| dfdisjALTV4 37890 | Alternate definition of th... |
| dfdisjALTV5 37891 | Alternate definition of th... |
| dfeldisj2 37892 | Alternate definition of th... |
| dfeldisj3 37893 | Alternate definition of th... |
| dfeldisj4 37894 | Alternate definition of th... |
| dfeldisj5 37895 | Alternate definition of th... |
| eldisjs 37896 | Elementhood in the class o... |
| eldisjs2 37897 | Elementhood in the class o... |
| eldisjs3 37898 | Elementhood in the class o... |
| eldisjs4 37899 | Elementhood in the class o... |
| eldisjs5 37900 | Elementhood in the class o... |
| eldisjsdisj 37901 | The element of the class o... |
| eleldisjs 37902 | Elementhood in the disjoin... |
| eleldisjseldisj 37903 | The element of the disjoin... |
| disjrel 37904 | Disjoint relation is a rel... |
| disjss 37905 | Subclass theorem for disjo... |
| disjssi 37906 | Subclass theorem for disjo... |
| disjssd 37907 | Subclass theorem for disjo... |
| disjeq 37908 | Equality theorem for disjo... |
| disjeqi 37909 | Equality theorem for disjo... |
| disjeqd 37910 | Equality theorem for disjo... |
| disjdmqseqeq1 37911 | Lemma for the equality the... |
| eldisjss 37912 | Subclass theorem for disjo... |
| eldisjssi 37913 | Subclass theorem for disjo... |
| eldisjssd 37914 | Subclass theorem for disjo... |
| eldisjeq 37915 | Equality theorem for disjo... |
| eldisjeqi 37916 | Equality theorem for disjo... |
| eldisjeqd 37917 | Equality theorem for disjo... |
| disjres 37918 | Disjoint restriction. (Co... |
| eldisjn0elb 37919 | Two forms of disjoint elem... |
| disjxrn 37920 | Two ways of saying that a ... |
| disjxrnres5 37921 | Disjoint range Cartesian p... |
| disjorimxrn 37922 | Disjointness condition for... |
| disjimxrn 37923 | Disjointness condition for... |
| disjimres 37924 | Disjointness condition for... |
| disjimin 37925 | Disjointness condition for... |
| disjiminres 37926 | Disjointness condition for... |
| disjimxrnres 37927 | Disjointness condition for... |
| disjALTV0 37928 | The null class is disjoint... |
| disjALTVid 37929 | The class of identity rela... |
| disjALTVidres 37930 | The class of identity rela... |
| disjALTVinidres 37931 | The intersection with rest... |
| disjALTVxrnidres 37932 | The class of range Cartesi... |
| disjsuc 37933 | Disjoint range Cartesian p... |
| dfantisymrel4 37935 | Alternate definition of th... |
| dfantisymrel5 37936 | Alternate definition of th... |
| antisymrelres 37937 | (Contributed by Peter Mazs... |
| antisymrelressn 37938 | (Contributed by Peter Mazs... |
| dfpart2 37943 | Alternate definition of th... |
| dfmembpart2 37944 | Alternate definition of th... |
| brparts 37945 | Binary partitions relation... |
| brparts2 37946 | Binary partitions relation... |
| brpartspart 37947 | Binary partition and the p... |
| parteq1 37948 | Equality theorem for parti... |
| parteq2 37949 | Equality theorem for parti... |
| parteq12 37950 | Equality theorem for parti... |
| parteq1i 37951 | Equality theorem for parti... |
| parteq1d 37952 | Equality theorem for parti... |
| partsuc2 37953 | Property of the partition.... |
| partsuc 37954 | Property of the partition.... |
| disjim 37955 | The "Divide et Aequivalere... |
| disjimi 37956 | Every disjoint relation ge... |
| detlem 37957 | If a relation is disjoint,... |
| eldisjim 37958 | If the elements of ` A ` a... |
| eldisjim2 37959 | Alternate form of ~ eldisj... |
| eqvrel0 37960 | The null class is an equiv... |
| det0 37961 | The cosets by the null cla... |
| eqvrelcoss0 37962 | The cosets by the null cla... |
| eqvrelid 37963 | The identity relation is a... |
| eqvrel1cossidres 37964 | The cosets by a restricted... |
| eqvrel1cossinidres 37965 | The cosets by an intersect... |
| eqvrel1cossxrnidres 37966 | The cosets by a range Cart... |
| detid 37967 | The cosets by the identity... |
| eqvrelcossid 37968 | The cosets by the identity... |
| detidres 37969 | The cosets by the restrict... |
| detinidres 37970 | The cosets by the intersec... |
| detxrnidres 37971 | The cosets by the range Ca... |
| disjlem14 37972 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 37973 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 37974 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 37975 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 37976 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 37977 | Lemma for ~ disjdmqseq via... |
| disjdmqs 37978 | If a relation is disjoint,... |
| disjdmqseq 37979 | If a relation is disjoint,... |
| eldisjn0el 37980 | Special case of ~ disjdmqs... |
| partim2 37981 | Disjoint relation on its n... |
| partim 37982 | Partition implies equivale... |
| partimeq 37983 | Partition implies that the... |
| eldisjlem19 37984 | Special case of ~ disjlem1... |
| membpartlem19 37985 | Together with ~ disjlem19 ... |
| petlem 37986 | If you can prove that the ... |
| petlemi 37987 | If you can prove disjointn... |
| pet02 37988 | Class ` A ` is a partition... |
| pet0 37989 | Class ` A ` is a partition... |
| petid2 37990 | Class ` A ` is a partition... |
| petid 37991 | A class is a partition by ... |
| petidres2 37992 | Class ` A ` is a partition... |
| petidres 37993 | A class is a partition by ... |
| petinidres2 37994 | Class ` A ` is a partition... |
| petinidres 37995 | A class is a partition by ... |
| petxrnidres2 37996 | Class ` A ` is a partition... |
| petxrnidres 37997 | A class is a partition by ... |
| eqvreldisj1 37998 | The elements of the quotie... |
| eqvreldisj2 37999 | The elements of the quotie... |
| eqvreldisj3 38000 | The elements of the quotie... |
| eqvreldisj4 38001 | Intersection with the conv... |
| eqvreldisj5 38002 | Range Cartesian product wi... |
| eqvrelqseqdisj2 38003 | Implication of ~ eqvreldis... |
| fences3 38004 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 38005 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 38006 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 38007 | Lemma for the Partition-Eq... |
| mainer 38008 | The Main Theorem of Equiva... |
| partimcomember 38009 | Partition with general ` R... |
| mpet3 38010 | Member Partition-Equivalen... |
| cpet2 38011 | The conventional form of t... |
| cpet 38012 | The conventional form of M... |
| mpet 38013 | Member Partition-Equivalen... |
| mpet2 38014 | Member Partition-Equivalen... |
| mpets2 38015 | Member Partition-Equivalen... |
| mpets 38016 | Member Partition-Equivalen... |
| mainpart 38017 | Partition with general ` R... |
| fences 38018 | The Theorem of Fences by E... |
| fences2 38019 | The Theorem of Fences by E... |
| mainer2 38020 | The Main Theorem of Equiva... |
| mainerim 38021 | Every equivalence relation... |
| petincnvepres2 38022 | A partition-equivalence th... |
| petincnvepres 38023 | The shortest form of a par... |
| pet2 38024 | Partition-Equivalence Theo... |
| pet 38025 | Partition-Equivalence Theo... |
| pets 38026 | Partition-Equivalence Theo... |
| prtlem60 38027 | Lemma for ~ prter3 . (Con... |
| bicomdd 38028 | Commute two sides of a bic... |
| jca2r 38029 | Inference conjoining the c... |
| jca3 38030 | Inference conjoining the c... |
| prtlem70 38031 | Lemma for ~ prter3 : a rea... |
| ibdr 38032 | Reverse of ~ ibd . (Contr... |
| prtlem100 38033 | Lemma for ~ prter3 . (Con... |
| prtlem5 38034 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 38035 | Lemma for ~ prter2 . (Con... |
| brabsb2 38036 | A closed form of ~ brabsb ... |
| eqbrrdv2 38037 | Other version of ~ eqbrrdi... |
| prtlem9 38038 | Lemma for ~ prter3 . (Con... |
| prtlem10 38039 | Lemma for ~ prter3 . (Con... |
| prtlem11 38040 | Lemma for ~ prter2 . (Con... |
| prtlem12 38041 | Lemma for ~ prtex and ~ pr... |
| prtlem13 38042 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 38043 | Lemma for ~ prtex , ~ prte... |
| prtlem400 38044 | Lemma for ~ prter2 and als... |
| erprt 38047 | The quotient set of an equ... |
| prtlem14 38048 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 38049 | Lemma for ~ prter1 and ~ p... |
| prtlem17 38050 | Lemma for ~ prter2 . (Con... |
| prtlem18 38051 | Lemma for ~ prter2 . (Con... |
| prtlem19 38052 | Lemma for ~ prter2 . (Con... |
| prter1 38053 | Every partition generates ... |
| prtex 38054 | The equivalence relation g... |
| prter2 38055 | The quotient set of the eq... |
| prter3 38056 | For every partition there ... |
| axc5 38067 | This theorem repeats ~ sp ... |
| ax4fromc4 38068 | Rederivation of Axiom ~ ax... |
| ax10fromc7 38069 | Rederivation of Axiom ~ ax... |
| ax6fromc10 38070 | Rederivation of Axiom ~ ax... |
| hba1-o 38071 | The setvar ` x ` is not fr... |
| axc4i-o 38072 | Inference version of ~ ax-... |
| equid1 38073 | Proof of ~ equid from our ... |
| equcomi1 38074 | Proof of ~ equcomi from ~ ... |
| aecom-o 38075 | Commutation law for identi... |
| aecoms-o 38076 | A commutation rule for ide... |
| hbae-o 38077 | All variables are effectiv... |
| dral1-o 38078 | Formula-building lemma for... |
| ax12fromc15 38079 | Rederivation of Axiom ~ ax... |
| ax13fromc9 38080 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 38081 | Axiom to quantify a variab... |
| sps-o 38082 | Generalization of antecede... |
| hbequid 38083 | Bound-variable hypothesis ... |
| nfequid-o 38084 | Bound-variable hypothesis ... |
| axc5c7 38085 | Proof of a single axiom th... |
| axc5c7toc5 38086 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 38087 | Rederivation of ~ ax-c7 fr... |
| axc711 38088 | Proof of a single axiom th... |
| nfa1-o 38089 | ` x ` is not free in ` A. ... |
| axc711toc7 38090 | Rederivation of ~ ax-c7 fr... |
| axc711to11 38091 | Rederivation of ~ ax-11 fr... |
| axc5c711 38092 | Proof of a single axiom th... |
| axc5c711toc5 38093 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 38094 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 38095 | Rederivation of ~ ax-11 fr... |
| equidqe 38096 | ~ equid with existential q... |
| axc5sp1 38097 | A special case of ~ ax-c5 ... |
| equidq 38098 | ~ equid with universal qua... |
| equid1ALT 38099 | Alternate proof of ~ equid... |
| axc11nfromc11 38100 | Rederivation of ~ ax-c11n ... |
| naecoms-o 38101 | A commutation rule for dis... |
| hbnae-o 38102 | All variables are effectiv... |
| dvelimf-o 38103 | Proof of ~ dvelimh that us... |
| dral2-o 38104 | Formula-building lemma for... |
| aev-o 38105 | A "distinctor elimination"... |
| ax5eq 38106 | Theorem to add distinct qu... |
| dveeq2-o 38107 | Quantifier introduction wh... |
| axc16g-o 38108 | A generalization of Axiom ... |
| dveeq1-o 38109 | Quantifier introduction wh... |
| dveeq1-o16 38110 | Version of ~ dveeq1 using ... |
| ax5el 38111 | Theorem to add distinct qu... |
| axc11n-16 38112 | This theorem shows that, g... |
| dveel2ALT 38113 | Alternate proof of ~ dveel... |
| ax12f 38114 | Basis step for constructin... |
| ax12eq 38115 | Basis step for constructin... |
| ax12el 38116 | Basis step for constructin... |
| ax12indn 38117 | Induction step for constru... |
| ax12indi 38118 | Induction step for constru... |
| ax12indalem 38119 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 38120 | Alternate proof of ~ ax12i... |
| ax12inda2 38121 | Induction step for constru... |
| ax12inda 38122 | Induction step for constru... |
| ax12v2-o 38123 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 38124 | Derive ~ ax-c15 from a hyp... |
| axc11-o 38125 | Show that ~ ax-c11 can be ... |
| fsumshftd 38126 | Index shift of a finite su... |
| riotaclbgBAD 38128 | Closure of restricted iota... |
| riotaclbBAD 38129 | Closure of restricted iota... |
| riotasvd 38130 | Deduction version of ~ rio... |
| riotasv2d 38131 | Value of description binde... |
| riotasv2s 38132 | The value of description b... |
| riotasv 38133 | Value of description binde... |
| riotasv3d 38134 | A property ` ch ` holding ... |
| elimhyps 38135 | A version of ~ elimhyp usi... |
| dedths 38136 | A version of weak deductio... |
| renegclALT 38137 | Closure law for negative o... |
| elimhyps2 38138 | Generalization of ~ elimhy... |
| dedths2 38139 | Generalization of ~ dedths... |
| nfcxfrdf 38140 | A utility lemma to transfe... |
| nfded 38141 | A deduction theorem that c... |
| nfded2 38142 | A deduction theorem that c... |
| nfunidALT2 38143 | Deduction version of ~ nfu... |
| nfunidALT 38144 | Deduction version of ~ nfu... |
| nfopdALT 38145 | Deduction version of bound... |
| cnaddcom 38146 | Recover the commutative la... |
| toycom 38147 | Show the commutative law f... |
| lshpset 38152 | The set of all hyperplanes... |
| islshp 38153 | The predicate "is a hyperp... |
| islshpsm 38154 | Hyperplane properties expr... |
| lshplss 38155 | A hyperplane is a subspace... |
| lshpne 38156 | A hyperplane is not equal ... |
| lshpnel 38157 | A hyperplane's generating ... |
| lshpnelb 38158 | The subspace sum of a hype... |
| lshpnel2N 38159 | Condition that determines ... |
| lshpne0 38160 | The member of the span in ... |
| lshpdisj 38161 | A hyperplane and the span ... |
| lshpcmp 38162 | If two hyperplanes are com... |
| lshpinN 38163 | The intersection of two di... |
| lsatset 38164 | The set of all 1-dim subsp... |
| islsat 38165 | The predicate "is a 1-dim ... |
| lsatlspsn2 38166 | The span of a nonzero sing... |
| lsatlspsn 38167 | The span of a nonzero sing... |
| islsati 38168 | A 1-dim subspace (atom) (o... |
| lsateln0 38169 | A 1-dim subspace (atom) (o... |
| lsatlss 38170 | The set of 1-dim subspaces... |
| lsatlssel 38171 | An atom is a subspace. (C... |
| lsatssv 38172 | An atom is a set of vector... |
| lsatn0 38173 | A 1-dim subspace (atom) of... |
| lsatspn0 38174 | The span of a vector is an... |
| lsator0sp 38175 | The span of a vector is ei... |
| lsatssn0 38176 | A subspace (or any class) ... |
| lsatcmp 38177 | If two atoms are comparabl... |
| lsatcmp2 38178 | If an atom is included in ... |
| lsatel 38179 | A nonzero vector in an ato... |
| lsatelbN 38180 | A nonzero vector in an ato... |
| lsat2el 38181 | Two atoms sharing a nonzer... |
| lsmsat 38182 | Convert comparison of atom... |
| lsatfixedN 38183 | Show equality with the spa... |
| lsmsatcv 38184 | Subspace sum has the cover... |
| lssatomic 38185 | The lattice of subspaces i... |
| lssats 38186 | The lattice of subspaces i... |
| lpssat 38187 | Two subspaces in a proper ... |
| lrelat 38188 | Subspaces are relatively a... |
| lssatle 38189 | The ordering of two subspa... |
| lssat 38190 | Two subspaces in a proper ... |
| islshpat 38191 | Hyperplane properties expr... |
| lcvfbr 38194 | The covers relation for a ... |
| lcvbr 38195 | The covers relation for a ... |
| lcvbr2 38196 | The covers relation for a ... |
| lcvbr3 38197 | The covers relation for a ... |
| lcvpss 38198 | The covers relation implie... |
| lcvnbtwn 38199 | The covers relation implie... |
| lcvntr 38200 | The covers relation is not... |
| lcvnbtwn2 38201 | The covers relation implie... |
| lcvnbtwn3 38202 | The covers relation implie... |
| lsmcv2 38203 | Subspace sum has the cover... |
| lcvat 38204 | If a subspace covers anoth... |
| lsatcv0 38205 | An atom covers the zero su... |
| lsatcveq0 38206 | A subspace covered by an a... |
| lsat0cv 38207 | A subspace is an atom iff ... |
| lcvexchlem1 38208 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 38209 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 38210 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 38211 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 38212 | Lemma for ~ lcvexch . (Co... |
| lcvexch 38213 | Subspaces satisfy the exch... |
| lcvp 38214 | Covering property of Defin... |
| lcv1 38215 | Covering property of a sub... |
| lcv2 38216 | Covering property of a sub... |
| lsatexch 38217 | The atom exchange property... |
| lsatnle 38218 | The meet of a subspace and... |
| lsatnem0 38219 | The meet of distinct atoms... |
| lsatexch1 38220 | The atom exch1ange propert... |
| lsatcv0eq 38221 | If the sum of two atoms co... |
| lsatcv1 38222 | Two atoms covering the zer... |
| lsatcvatlem 38223 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 38224 | A nonzero subspace less th... |
| lsatcvat2 38225 | A subspace covered by the ... |
| lsatcvat3 38226 | A condition implying that ... |
| islshpcv 38227 | Hyperplane properties expr... |
| l1cvpat 38228 | A subspace covered by the ... |
| l1cvat 38229 | Create an atom under an el... |
| lshpat 38230 | Create an atom under a hyp... |
| lflset 38233 | The set of linear function... |
| islfl 38234 | The predicate "is a linear... |
| lfli 38235 | Property of a linear funct... |
| islfld 38236 | Properties that determine ... |
| lflf 38237 | A linear functional is a f... |
| lflcl 38238 | A linear functional value ... |
| lfl0 38239 | A linear functional is zer... |
| lfladd 38240 | Property of a linear funct... |
| lflsub 38241 | Property of a linear funct... |
| lflmul 38242 | Property of a linear funct... |
| lfl0f 38243 | The zero function is a fun... |
| lfl1 38244 | A nonzero functional has a... |
| lfladdcl 38245 | Closure of addition of two... |
| lfladdcom 38246 | Commutativity of functiona... |
| lfladdass 38247 | Associativity of functiona... |
| lfladd0l 38248 | Functional addition with t... |
| lflnegcl 38249 | Closure of the negative of... |
| lflnegl 38250 | A functional plus its nega... |
| lflvscl 38251 | Closure of a scalar produc... |
| lflvsdi1 38252 | Distributive law for (righ... |
| lflvsdi2 38253 | Reverse distributive law f... |
| lflvsdi2a 38254 | Reverse distributive law f... |
| lflvsass 38255 | Associative law for (right... |
| lfl0sc 38256 | The (right vector space) s... |
| lflsc0N 38257 | The scalar product with th... |
| lfl1sc 38258 | The (right vector space) s... |
| lkrfval 38261 | The kernel of a functional... |
| lkrval 38262 | Value of the kernel of a f... |
| ellkr 38263 | Membership in the kernel o... |
| lkrval2 38264 | Value of the kernel of a f... |
| ellkr2 38265 | Membership in the kernel o... |
| lkrcl 38266 | A member of the kernel of ... |
| lkrf0 38267 | The value of a functional ... |
| lkr0f 38268 | The kernel of the zero fun... |
| lkrlss 38269 | The kernel of a linear fun... |
| lkrssv 38270 | The kernel of a linear fun... |
| lkrsc 38271 | The kernel of a nonzero sc... |
| lkrscss 38272 | The kernel of a scalar pro... |
| eqlkr 38273 | Two functionals with the s... |
| eqlkr2 38274 | Two functionals with the s... |
| eqlkr3 38275 | Two functionals with the s... |
| lkrlsp 38276 | The subspace sum of a kern... |
| lkrlsp2 38277 | The subspace sum of a kern... |
| lkrlsp3 38278 | The subspace sum of a kern... |
| lkrshp 38279 | The kernel of a nonzero fu... |
| lkrshp3 38280 | The kernels of nonzero fun... |
| lkrshpor 38281 | The kernel of a functional... |
| lkrshp4 38282 | A kernel is a hyperplane i... |
| lshpsmreu 38283 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 38284 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 38285 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 38286 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 38287 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 38288 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 38289 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 38290 | The set ` G ` defined by h... |
| lshpkr 38291 | The kernel of functional `... |
| lshpkrex 38292 | There exists a functional ... |
| lshpset2N 38293 | The set of all hyperplanes... |
| islshpkrN 38294 | The predicate "is a hyperp... |
| lfl1dim 38295 | Equivalent expressions for... |
| lfl1dim2N 38296 | Equivalent expressions for... |
| ldualset 38299 | Define the (left) dual of ... |
| ldualvbase 38300 | The vectors of a dual spac... |
| ldualelvbase 38301 | Utility theorem for conver... |
| ldualfvadd 38302 | Vector addition in the dua... |
| ldualvadd 38303 | Vector addition in the dua... |
| ldualvaddcl 38304 | The value of vector additi... |
| ldualvaddval 38305 | The value of the value of ... |
| ldualsca 38306 | The ring of scalars of the... |
| ldualsbase 38307 | Base set of scalar ring fo... |
| ldualsaddN 38308 | Scalar addition for the du... |
| ldualsmul 38309 | Scalar multiplication for ... |
| ldualfvs 38310 | Scalar product operation f... |
| ldualvs 38311 | Scalar product operation v... |
| ldualvsval 38312 | Value of scalar product op... |
| ldualvscl 38313 | The scalar product operati... |
| ldualvaddcom 38314 | Commutative law for vector... |
| ldualvsass 38315 | Associative law for scalar... |
| ldualvsass2 38316 | Associative law for scalar... |
| ldualvsdi1 38317 | Distributive law for scala... |
| ldualvsdi2 38318 | Reverse distributive law f... |
| ldualgrplem 38319 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 38320 | The dual of a vector space... |
| ldual0 38321 | The zero scalar of the dua... |
| ldual1 38322 | The unit scalar of the dua... |
| ldualneg 38323 | The negative of a scalar o... |
| ldual0v 38324 | The zero vector of the dua... |
| ldual0vcl 38325 | The dual zero vector is a ... |
| lduallmodlem 38326 | Lemma for ~ lduallmod . (... |
| lduallmod 38327 | The dual of a left module ... |
| lduallvec 38328 | The dual of a left vector ... |
| ldualvsub 38329 | The value of vector subtra... |
| ldualvsubcl 38330 | Closure of vector subtract... |
| ldualvsubval 38331 | The value of the value of ... |
| ldualssvscl 38332 | Closure of scalar product ... |
| ldualssvsubcl 38333 | Closure of vector subtract... |
| ldual0vs 38334 | Scalar zero times a functi... |
| lkr0f2 38335 | The kernel of the zero fun... |
| lduallkr3 38336 | The kernels of nonzero fun... |
| lkrpssN 38337 | Proper subset relation bet... |
| lkrin 38338 | Intersection of the kernel... |
| eqlkr4 38339 | Two functionals with the s... |
| ldual1dim 38340 | Equivalent expressions for... |
| ldualkrsc 38341 | The kernel of a nonzero sc... |
| lkrss 38342 | The kernel of a scalar pro... |
| lkrss2N 38343 | Two functionals with kerne... |
| lkreqN 38344 | Proportional functionals h... |
| lkrlspeqN 38345 | Condition for colinear fun... |
| isopos 38354 | The predicate "is an ortho... |
| opposet 38355 | Every orthoposet is a pose... |
| oposlem 38356 | Lemma for orthoposet prope... |
| op01dm 38357 | Conditions necessary for z... |
| op0cl 38358 | An orthoposet has a zero e... |
| op1cl 38359 | An orthoposet has a unity ... |
| op0le 38360 | Orthoposet zero is less th... |
| ople0 38361 | An element less than or eq... |
| opnlen0 38362 | An element not less than a... |
| lub0N 38363 | The least upper bound of t... |
| opltn0 38364 | A lattice element greater ... |
| ople1 38365 | Any element is less than t... |
| op1le 38366 | If the orthoposet unity is... |
| glb0N 38367 | The greatest lower bound o... |
| opoccl 38368 | Closure of orthocomplement... |
| opococ 38369 | Double negative law for or... |
| opcon3b 38370 | Contraposition law for ort... |
| opcon2b 38371 | Orthocomplement contraposi... |
| opcon1b 38372 | Orthocomplement contraposi... |
| oplecon3 38373 | Contraposition law for ort... |
| oplecon3b 38374 | Contraposition law for ort... |
| oplecon1b 38375 | Contraposition law for str... |
| opoc1 38376 | Orthocomplement of orthopo... |
| opoc0 38377 | Orthocomplement of orthopo... |
| opltcon3b 38378 | Contraposition law for str... |
| opltcon1b 38379 | Contraposition law for str... |
| opltcon2b 38380 | Contraposition law for str... |
| opexmid 38381 | Law of excluded middle for... |
| opnoncon 38382 | Law of contradiction for o... |
| riotaocN 38383 | The orthocomplement of the... |
| cmtfvalN 38384 | Value of commutes relation... |
| cmtvalN 38385 | Equivalence for commutes r... |
| isolat 38386 | The predicate "is an ortho... |
| ollat 38387 | An ortholattice is a latti... |
| olop 38388 | An ortholattice is an orth... |
| olposN 38389 | An ortholattice is a poset... |
| isolatiN 38390 | Properties that determine ... |
| oldmm1 38391 | De Morgan's law for meet i... |
| oldmm2 38392 | De Morgan's law for meet i... |
| oldmm3N 38393 | De Morgan's law for meet i... |
| oldmm4 38394 | De Morgan's law for meet i... |
| oldmj1 38395 | De Morgan's law for join i... |
| oldmj2 38396 | De Morgan's law for join i... |
| oldmj3 38397 | De Morgan's law for join i... |
| oldmj4 38398 | De Morgan's law for join i... |
| olj01 38399 | An ortholattice element jo... |
| olj02 38400 | An ortholattice element jo... |
| olm11 38401 | The meet of an ortholattic... |
| olm12 38402 | The meet of an ortholattic... |
| latmassOLD 38403 | Ortholattice meet is assoc... |
| latm12 38404 | A rearrangement of lattice... |
| latm32 38405 | A rearrangement of lattice... |
| latmrot 38406 | Rotate lattice meet of 3 c... |
| latm4 38407 | Rearrangement of lattice m... |
| latmmdiN 38408 | Lattice meet distributes o... |
| latmmdir 38409 | Lattice meet distributes o... |
| olm01 38410 | Meet with lattice zero is ... |
| olm02 38411 | Meet with lattice zero is ... |
| isoml 38412 | The predicate "is an ortho... |
| isomliN 38413 | Properties that determine ... |
| omlol 38414 | An orthomodular lattice is... |
| omlop 38415 | An orthomodular lattice is... |
| omllat 38416 | An orthomodular lattice is... |
| omllaw 38417 | The orthomodular law. (Co... |
| omllaw2N 38418 | Variation of orthomodular ... |
| omllaw3 38419 | Orthomodular law equivalen... |
| omllaw4 38420 | Orthomodular law equivalen... |
| omllaw5N 38421 | The orthomodular law. Rem... |
| cmtcomlemN 38422 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 38423 | Commutation is symmetric. ... |
| cmt2N 38424 | Commutation with orthocomp... |
| cmt3N 38425 | Commutation with orthocomp... |
| cmt4N 38426 | Commutation with orthocomp... |
| cmtbr2N 38427 | Alternate definition of th... |
| cmtbr3N 38428 | Alternate definition for t... |
| cmtbr4N 38429 | Alternate definition for t... |
| lecmtN 38430 | Ordered elements commute. ... |
| cmtidN 38431 | Any element commutes with ... |
| omlfh1N 38432 | Foulis-Holland Theorem, pa... |
| omlfh3N 38433 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 38434 | Analogue of modular law ~ ... |
| omlspjN 38435 | Contraction of a Sasaki pr... |
| cvrfval 38442 | Value of covers relation "... |
| cvrval 38443 | Binary relation expressing... |
| cvrlt 38444 | The covers relation implie... |
| cvrnbtwn 38445 | There is no element betwee... |
| ncvr1 38446 | No element covers the latt... |
| cvrletrN 38447 | Property of an element abo... |
| cvrval2 38448 | Binary relation expressing... |
| cvrnbtwn2 38449 | The covers relation implie... |
| cvrnbtwn3 38450 | The covers relation implie... |
| cvrcon3b 38451 | Contraposition law for the... |
| cvrle 38452 | The covers relation implie... |
| cvrnbtwn4 38453 | The covers relation implie... |
| cvrnle 38454 | The covers relation implie... |
| cvrne 38455 | The covers relation implie... |
| cvrnrefN 38456 | The covers relation is not... |
| cvrcmp 38457 | If two lattice elements th... |
| cvrcmp2 38458 | If two lattice elements co... |
| pats 38459 | The set of atoms in a pose... |
| isat 38460 | The predicate "is an atom"... |
| isat2 38461 | The predicate "is an atom"... |
| atcvr0 38462 | An atom covers zero. ( ~ ... |
| atbase 38463 | An atom is a member of the... |
| atssbase 38464 | The set of atoms is a subs... |
| 0ltat 38465 | An atom is greater than ze... |
| leatb 38466 | A poset element less than ... |
| leat 38467 | A poset element less than ... |
| leat2 38468 | A nonzero poset element le... |
| leat3 38469 | A poset element less than ... |
| meetat 38470 | The meet of any element wi... |
| meetat2 38471 | The meet of any element wi... |
| isatl 38473 | The predicate "is an atomi... |
| atllat 38474 | An atomic lattice is a lat... |
| atlpos 38475 | An atomic lattice is a pos... |
| atl0dm 38476 | Condition necessary for ze... |
| atl0cl 38477 | An atomic lattice has a ze... |
| atl0le 38478 | Orthoposet zero is less th... |
| atlle0 38479 | An element less than or eq... |
| atlltn0 38480 | A lattice element greater ... |
| isat3 38481 | The predicate "is an atom"... |
| atn0 38482 | An atom is not zero. ( ~ ... |
| atnle0 38483 | An atom is not less than o... |
| atlen0 38484 | A lattice element is nonze... |
| atcmp 38485 | If two atoms are comparabl... |
| atncmp 38486 | Frequently-used variation ... |
| atnlt 38487 | Two atoms cannot satisfy t... |
| atcvreq0 38488 | An element covered by an a... |
| atncvrN 38489 | Two atoms cannot satisfy t... |
| atlex 38490 | Every nonzero element of a... |
| atnle 38491 | Two ways of expressing "an... |
| atnem0 38492 | The meet of distinct atoms... |
| atlatmstc 38493 | An atomic, complete, ortho... |
| atlatle 38494 | The ordering of two Hilber... |
| atlrelat1 38495 | An atomistic lattice with ... |
| iscvlat 38497 | The predicate "is an atomi... |
| iscvlat2N 38498 | The predicate "is an atomi... |
| cvlatl 38499 | An atomic lattice with the... |
| cvllat 38500 | An atomic lattice with the... |
| cvlposN 38501 | An atomic lattice with the... |
| cvlexch1 38502 | An atomic covering lattice... |
| cvlexch2 38503 | An atomic covering lattice... |
| cvlexchb1 38504 | An atomic covering lattice... |
| cvlexchb2 38505 | An atomic covering lattice... |
| cvlexch3 38506 | An atomic covering lattice... |
| cvlexch4N 38507 | An atomic covering lattice... |
| cvlatexchb1 38508 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 38509 | A version of ~ cvlexchb2 f... |
| cvlatexch1 38510 | Atom exchange property. (... |
| cvlatexch2 38511 | Atom exchange property. (... |
| cvlatexch3 38512 | Atom exchange property. (... |
| cvlcvr1 38513 | The covering property. Pr... |
| cvlcvrp 38514 | A Hilbert lattice satisfie... |
| cvlatcvr1 38515 | An atom is covered by its ... |
| cvlatcvr2 38516 | An atom is covered by its ... |
| cvlsupr2 38517 | Two equivalent ways of exp... |
| cvlsupr3 38518 | Two equivalent ways of exp... |
| cvlsupr4 38519 | Consequence of superpositi... |
| cvlsupr5 38520 | Consequence of superpositi... |
| cvlsupr6 38521 | Consequence of superpositi... |
| cvlsupr7 38522 | Consequence of superpositi... |
| cvlsupr8 38523 | Consequence of superpositi... |
| ishlat1 38526 | The predicate "is a Hilber... |
| ishlat2 38527 | The predicate "is a Hilber... |
| ishlat3N 38528 | The predicate "is a Hilber... |
| ishlatiN 38529 | Properties that determine ... |
| hlomcmcv 38530 | A Hilbert lattice is ortho... |
| hloml 38531 | A Hilbert lattice is ortho... |
| hlclat 38532 | A Hilbert lattice is compl... |
| hlcvl 38533 | A Hilbert lattice is an at... |
| hlatl 38534 | A Hilbert lattice is atomi... |
| hlol 38535 | A Hilbert lattice is an or... |
| hlop 38536 | A Hilbert lattice is an or... |
| hllat 38537 | A Hilbert lattice is a lat... |
| hllatd 38538 | Deduction form of ~ hllat ... |
| hlomcmat 38539 | A Hilbert lattice is ortho... |
| hlpos 38540 | A Hilbert lattice is a pos... |
| hlatjcl 38541 | Closure of join operation.... |
| hlatjcom 38542 | Commutatitivity of join op... |
| hlatjidm 38543 | Idempotence of join operat... |
| hlatjass 38544 | Lattice join is associativ... |
| hlatj12 38545 | Swap 1st and 2nd members o... |
| hlatj32 38546 | Swap 2nd and 3rd members o... |
| hlatjrot 38547 | Rotate lattice join of 3 c... |
| hlatj4 38548 | Rearrangement of lattice j... |
| hlatlej1 38549 | A join's first argument is... |
| hlatlej2 38550 | A join's second argument i... |
| glbconN 38551 | De Morgan's law for GLB an... |
| glbconNOLD 38552 | Obsolete version of ~ glbc... |
| glbconxN 38553 | De Morgan's law for GLB an... |
| atnlej1 38554 | If an atom is not less tha... |
| atnlej2 38555 | If an atom is not less tha... |
| hlsuprexch 38556 | A Hilbert lattice has the ... |
| hlexch1 38557 | A Hilbert lattice has the ... |
| hlexch2 38558 | A Hilbert lattice has the ... |
| hlexchb1 38559 | A Hilbert lattice has the ... |
| hlexchb2 38560 | A Hilbert lattice has the ... |
| hlsupr 38561 | A Hilbert lattice has the ... |
| hlsupr2 38562 | A Hilbert lattice has the ... |
| hlhgt4 38563 | A Hilbert lattice has a he... |
| hlhgt2 38564 | A Hilbert lattice has a he... |
| hl0lt1N 38565 | Lattice 0 is less than lat... |
| hlexch3 38566 | A Hilbert lattice has the ... |
| hlexch4N 38567 | A Hilbert lattice has the ... |
| hlatexchb1 38568 | A version of ~ hlexchb1 fo... |
| hlatexchb2 38569 | A version of ~ hlexchb2 fo... |
| hlatexch1 38570 | Atom exchange property. (... |
| hlatexch2 38571 | Atom exchange property. (... |
| hlatmstcOLDN 38572 | An atomic, complete, ortho... |
| hlatle 38573 | The ordering of two Hilber... |
| hlateq 38574 | The equality of two Hilber... |
| hlrelat1 38575 | An atomistic lattice with ... |
| hlrelat5N 38576 | An atomistic lattice with ... |
| hlrelat 38577 | A Hilbert lattice is relat... |
| hlrelat2 38578 | A consequence of relative ... |
| exatleN 38579 | A condition for an atom to... |
| hl2at 38580 | A Hilbert lattice has at l... |
| atex 38581 | At least one atom exists. ... |
| intnatN 38582 | If the intersection with a... |
| 2llnne2N 38583 | Condition implying that tw... |
| 2llnneN 38584 | Condition implying that tw... |
| cvr1 38585 | A Hilbert lattice has the ... |
| cvr2N 38586 | Less-than and covers equiv... |
| hlrelat3 38587 | The Hilbert lattice is rel... |
| cvrval3 38588 | Binary relation expressing... |
| cvrval4N 38589 | Binary relation expressing... |
| cvrval5 38590 | Binary relation expressing... |
| cvrp 38591 | A Hilbert lattice satisfie... |
| atcvr1 38592 | An atom is covered by its ... |
| atcvr2 38593 | An atom is covered by its ... |
| cvrexchlem 38594 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 38595 | A Hilbert lattice satisfie... |
| cvratlem 38596 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 38597 | A nonzero Hilbert lattice ... |
| ltltncvr 38598 | A chained strong ordering ... |
| ltcvrntr 38599 | Non-transitive condition f... |
| cvrntr 38600 | The covers relation is not... |
| atcvr0eq 38601 | The covers relation is not... |
| lnnat 38602 | A line (the join of two di... |
| atcvrj0 38603 | Two atoms covering the zer... |
| cvrat2 38604 | A Hilbert lattice element ... |
| atcvrneN 38605 | Inequality derived from at... |
| atcvrj1 38606 | Condition for an atom to b... |
| atcvrj2b 38607 | Condition for an atom to b... |
| atcvrj2 38608 | Condition for an atom to b... |
| atleneN 38609 | Inequality derived from at... |
| atltcvr 38610 | An equivalence of less-tha... |
| atle 38611 | Any nonzero element has an... |
| atlt 38612 | Two atoms are unequal iff ... |
| atlelt 38613 | Transfer less-than relatio... |
| 2atlt 38614 | Given an atom less than an... |
| atexchcvrN 38615 | Atom exchange property. V... |
| atexchltN 38616 | Atom exchange property. V... |
| cvrat3 38617 | A condition implying that ... |
| cvrat4 38618 | A condition implying exist... |
| cvrat42 38619 | Commuted version of ~ cvra... |
| 2atjm 38620 | The meet of a line (expres... |
| atbtwn 38621 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 38622 | There exists a 3rd atom ` ... |
| atbtwnex 38623 | Given atoms ` P ` in ` X `... |
| 3noncolr2 38624 | Two ways to express 3 non-... |
| 3noncolr1N 38625 | Two ways to express 3 non-... |
| hlatcon3 38626 | Atom exchange combined wit... |
| hlatcon2 38627 | Atom exchange combined wit... |
| 4noncolr3 38628 | A way to express 4 non-col... |
| 4noncolr2 38629 | A way to express 4 non-col... |
| 4noncolr1 38630 | A way to express 4 non-col... |
| athgt 38631 | A Hilbert lattice, whose h... |
| 3dim0 38632 | There exists a 3-dimension... |
| 3dimlem1 38633 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 38634 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 38635 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 38636 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 38637 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 38638 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 38639 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 38640 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 38641 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 38642 | Construct a 3-dimensional ... |
| 3dim2 38643 | Construct 2 new layers on ... |
| 3dim3 38644 | Construct a new layer on t... |
| 2dim 38645 | Generate a height-3 elemen... |
| 1dimN 38646 | An atom is covered by a he... |
| 1cvrco 38647 | The orthocomplement of an ... |
| 1cvratex 38648 | There exists an atom less ... |
| 1cvratlt 38649 | An atom less than or equal... |
| 1cvrjat 38650 | An element covered by the ... |
| 1cvrat 38651 | Create an atom under an el... |
| ps-1 38652 | The join of two atoms ` R ... |
| ps-2 38653 | Lattice analogue for the p... |
| 2atjlej 38654 | Two atoms are different if... |
| hlatexch3N 38655 | Rearrange join of atoms in... |
| hlatexch4 38656 | Exchange 2 atoms. (Contri... |
| ps-2b 38657 | Variation of projective ge... |
| 3atlem1 38658 | Lemma for ~ 3at . (Contri... |
| 3atlem2 38659 | Lemma for ~ 3at . (Contri... |
| 3atlem3 38660 | Lemma for ~ 3at . (Contri... |
| 3atlem4 38661 | Lemma for ~ 3at . (Contri... |
| 3atlem5 38662 | Lemma for ~ 3at . (Contri... |
| 3atlem6 38663 | Lemma for ~ 3at . (Contri... |
| 3atlem7 38664 | Lemma for ~ 3at . (Contri... |
| 3at 38665 | Any three non-colinear ato... |
| llnset 38680 | The set of lattice lines i... |
| islln 38681 | The predicate "is a lattic... |
| islln4 38682 | The predicate "is a lattic... |
| llni 38683 | Condition implying a latti... |
| llnbase 38684 | A lattice line is a lattic... |
| islln3 38685 | The predicate "is a lattic... |
| islln2 38686 | The predicate "is a lattic... |
| llni2 38687 | The join of two different ... |
| llnnleat 38688 | An atom cannot majorize a ... |
| llnneat 38689 | A lattice line is not an a... |
| 2atneat 38690 | The join of two distinct a... |
| llnn0 38691 | A lattice line is nonzero.... |
| islln2a 38692 | The predicate "is a lattic... |
| llnle 38693 | Any element greater than 0... |
| atcvrlln2 38694 | An atom under a line is co... |
| atcvrlln 38695 | An element covering an ato... |
| llnexatN 38696 | Given an atom on a line, t... |
| llncmp 38697 | If two lattice lines are c... |
| llnnlt 38698 | Two lattice lines cannot s... |
| 2llnmat 38699 | Two intersecting lines int... |
| 2at0mat0 38700 | Special case of ~ 2atmat0 ... |
| 2atmat0 38701 | The meet of two unequal li... |
| 2atm 38702 | An atom majorized by two d... |
| ps-2c 38703 | Variation of projective ge... |
| lplnset 38704 | The set of lattice planes ... |
| islpln 38705 | The predicate "is a lattic... |
| islpln4 38706 | The predicate "is a lattic... |
| lplni 38707 | Condition implying a latti... |
| islpln3 38708 | The predicate "is a lattic... |
| lplnbase 38709 | A lattice plane is a latti... |
| islpln5 38710 | The predicate "is a lattic... |
| islpln2 38711 | The predicate "is a lattic... |
| lplni2 38712 | The join of 3 different at... |
| lvolex3N 38713 | There is an atom outside o... |
| llnmlplnN 38714 | The intersection of a line... |
| lplnle 38715 | Any element greater than 0... |
| lplnnle2at 38716 | A lattice line (or atom) c... |
| lplnnleat 38717 | A lattice plane cannot maj... |
| lplnnlelln 38718 | A lattice plane is not les... |
| 2atnelpln 38719 | The join of two atoms is n... |
| lplnneat 38720 | No lattice plane is an ato... |
| lplnnelln 38721 | No lattice plane is a latt... |
| lplnn0N 38722 | A lattice plane is nonzero... |
| islpln2a 38723 | The predicate "is a lattic... |
| islpln2ah 38724 | The predicate "is a lattic... |
| lplnriaN 38725 | Property of a lattice plan... |
| lplnribN 38726 | Property of a lattice plan... |
| lplnric 38727 | Property of a lattice plan... |
| lplnri1 38728 | Property of a lattice plan... |
| lplnri2N 38729 | Property of a lattice plan... |
| lplnri3N 38730 | Property of a lattice plan... |
| lplnllnneN 38731 | Two lattice lines defined ... |
| llncvrlpln2 38732 | A lattice line under a lat... |
| llncvrlpln 38733 | An element covering a latt... |
| 2lplnmN 38734 | If the join of two lattice... |
| 2llnmj 38735 | The meet of two lattice li... |
| 2atmat 38736 | The meet of two intersecti... |
| lplncmp 38737 | If two lattice planes are ... |
| lplnexatN 38738 | Given a lattice line on a ... |
| lplnexllnN 38739 | Given an atom on a lattice... |
| lplnnlt 38740 | Two lattice planes cannot ... |
| 2llnjaN 38741 | The join of two different ... |
| 2llnjN 38742 | The join of two different ... |
| 2llnm2N 38743 | The meet of two different ... |
| 2llnm3N 38744 | Two lattice lines in a lat... |
| 2llnm4 38745 | Two lattice lines that maj... |
| 2llnmeqat 38746 | An atom equals the interse... |
| lvolset 38747 | The set of 3-dim lattice v... |
| islvol 38748 | The predicate "is a 3-dim ... |
| islvol4 38749 | The predicate "is a 3-dim ... |
| lvoli 38750 | Condition implying a 3-dim... |
| islvol3 38751 | The predicate "is a 3-dim ... |
| lvoli3 38752 | Condition implying a 3-dim... |
| lvolbase 38753 | A 3-dim lattice volume is ... |
| islvol5 38754 | The predicate "is a 3-dim ... |
| islvol2 38755 | The predicate "is a 3-dim ... |
| lvoli2 38756 | The join of 4 different at... |
| lvolnle3at 38757 | A lattice plane (or lattic... |
| lvolnleat 38758 | An atom cannot majorize a ... |
| lvolnlelln 38759 | A lattice line cannot majo... |
| lvolnlelpln 38760 | A lattice plane cannot maj... |
| 3atnelvolN 38761 | The join of 3 atoms is not... |
| 2atnelvolN 38762 | The join of two atoms is n... |
| lvolneatN 38763 | No lattice volume is an at... |
| lvolnelln 38764 | No lattice volume is a lat... |
| lvolnelpln 38765 | No lattice volume is a lat... |
| lvoln0N 38766 | A lattice volume is nonzer... |
| islvol2aN 38767 | The predicate "is a lattic... |
| 4atlem0a 38768 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 38769 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 38770 | Lemma for ~ 4at . (Contri... |
| 4atlem3 38771 | Lemma for ~ 4at . Break i... |
| 4atlem3a 38772 | Lemma for ~ 4at . Break i... |
| 4atlem3b 38773 | Lemma for ~ 4at . Break i... |
| 4atlem4a 38774 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 38775 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 38776 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 38777 | Lemma for ~ 4at . Frequen... |
| 4atlem9 38778 | Lemma for ~ 4at . Substit... |
| 4atlem10a 38779 | Lemma for ~ 4at . Substit... |
| 4atlem10b 38780 | Lemma for ~ 4at . Substit... |
| 4atlem10 38781 | Lemma for ~ 4at . Combine... |
| 4atlem11a 38782 | Lemma for ~ 4at . Substit... |
| 4atlem11b 38783 | Lemma for ~ 4at . Substit... |
| 4atlem11 38784 | Lemma for ~ 4at . Combine... |
| 4atlem12a 38785 | Lemma for ~ 4at . Substit... |
| 4atlem12b 38786 | Lemma for ~ 4at . Substit... |
| 4atlem12 38787 | Lemma for ~ 4at . Combine... |
| 4at 38788 | Four atoms determine a lat... |
| 4at2 38789 | Four atoms determine a lat... |
| lplncvrlvol2 38790 | A lattice line under a lat... |
| lplncvrlvol 38791 | An element covering a latt... |
| lvolcmp 38792 | If two lattice planes are ... |
| lvolnltN 38793 | Two lattice volumes cannot... |
| 2lplnja 38794 | The join of two different ... |
| 2lplnj 38795 | The join of two different ... |
| 2lplnm2N 38796 | The meet of two different ... |
| 2lplnmj 38797 | The meet of two lattice pl... |
| dalemkehl 38798 | Lemma for ~ dath . Freque... |
| dalemkelat 38799 | Lemma for ~ dath . Freque... |
| dalemkeop 38800 | Lemma for ~ dath . Freque... |
| dalempea 38801 | Lemma for ~ dath . Freque... |
| dalemqea 38802 | Lemma for ~ dath . Freque... |
| dalemrea 38803 | Lemma for ~ dath . Freque... |
| dalemsea 38804 | Lemma for ~ dath . Freque... |
| dalemtea 38805 | Lemma for ~ dath . Freque... |
| dalemuea 38806 | Lemma for ~ dath . Freque... |
| dalemyeo 38807 | Lemma for ~ dath . Freque... |
| dalemzeo 38808 | Lemma for ~ dath . Freque... |
| dalemclpjs 38809 | Lemma for ~ dath . Freque... |
| dalemclqjt 38810 | Lemma for ~ dath . Freque... |
| dalemclrju 38811 | Lemma for ~ dath . Freque... |
| dalem-clpjq 38812 | Lemma for ~ dath . Freque... |
| dalemceb 38813 | Lemma for ~ dath . Freque... |
| dalempeb 38814 | Lemma for ~ dath . Freque... |
| dalemqeb 38815 | Lemma for ~ dath . Freque... |
| dalemreb 38816 | Lemma for ~ dath . Freque... |
| dalemseb 38817 | Lemma for ~ dath . Freque... |
| dalemteb 38818 | Lemma for ~ dath . Freque... |
| dalemueb 38819 | Lemma for ~ dath . Freque... |
| dalempjqeb 38820 | Lemma for ~ dath . Freque... |
| dalemsjteb 38821 | Lemma for ~ dath . Freque... |
| dalemtjueb 38822 | Lemma for ~ dath . Freque... |
| dalemqrprot 38823 | Lemma for ~ dath . Freque... |
| dalemyeb 38824 | Lemma for ~ dath . Freque... |
| dalemcnes 38825 | Lemma for ~ dath . Freque... |
| dalempnes 38826 | Lemma for ~ dath . Freque... |
| dalemqnet 38827 | Lemma for ~ dath . Freque... |
| dalempjsen 38828 | Lemma for ~ dath . Freque... |
| dalemply 38829 | Lemma for ~ dath . Freque... |
| dalemsly 38830 | Lemma for ~ dath . Freque... |
| dalemswapyz 38831 | Lemma for ~ dath . Swap t... |
| dalemrot 38832 | Lemma for ~ dath . Rotate... |
| dalemrotyz 38833 | Lemma for ~ dath . Rotate... |
| dalem1 38834 | Lemma for ~ dath . Show t... |
| dalemcea 38835 | Lemma for ~ dath . Freque... |
| dalem2 38836 | Lemma for ~ dath . Show t... |
| dalemdea 38837 | Lemma for ~ dath . Freque... |
| dalemeea 38838 | Lemma for ~ dath . Freque... |
| dalem3 38839 | Lemma for ~ dalemdnee . (... |
| dalem4 38840 | Lemma for ~ dalemdnee . (... |
| dalemdnee 38841 | Lemma for ~ dath . Axis o... |
| dalem5 38842 | Lemma for ~ dath . Atom `... |
| dalem6 38843 | Lemma for ~ dath . Analog... |
| dalem7 38844 | Lemma for ~ dath . Analog... |
| dalem8 38845 | Lemma for ~ dath . Plane ... |
| dalem-cly 38846 | Lemma for ~ dalem9 . Cent... |
| dalem9 38847 | Lemma for ~ dath . Since ... |
| dalem10 38848 | Lemma for ~ dath . Atom `... |
| dalem11 38849 | Lemma for ~ dath . Analog... |
| dalem12 38850 | Lemma for ~ dath . Analog... |
| dalem13 38851 | Lemma for ~ dalem14 . (Co... |
| dalem14 38852 | Lemma for ~ dath . Planes... |
| dalem15 38853 | Lemma for ~ dath . The ax... |
| dalem16 38854 | Lemma for ~ dath . The at... |
| dalem17 38855 | Lemma for ~ dath . When p... |
| dalem18 38856 | Lemma for ~ dath . Show t... |
| dalem19 38857 | Lemma for ~ dath . Show t... |
| dalemccea 38858 | Lemma for ~ dath . Freque... |
| dalemddea 38859 | Lemma for ~ dath . Freque... |
| dalem-ccly 38860 | Lemma for ~ dath . Freque... |
| dalem-ddly 38861 | Lemma for ~ dath . Freque... |
| dalemccnedd 38862 | Lemma for ~ dath . Freque... |
| dalemclccjdd 38863 | Lemma for ~ dath . Freque... |
| dalemcceb 38864 | Lemma for ~ dath . Freque... |
| dalemswapyzps 38865 | Lemma for ~ dath . Swap t... |
| dalemrotps 38866 | Lemma for ~ dath . Rotate... |
| dalemcjden 38867 | Lemma for ~ dath . Show t... |
| dalem20 38868 | Lemma for ~ dath . Show t... |
| dalem21 38869 | Lemma for ~ dath . Show t... |
| dalem22 38870 | Lemma for ~ dath . Show t... |
| dalem23 38871 | Lemma for ~ dath . Show t... |
| dalem24 38872 | Lemma for ~ dath . Show t... |
| dalem25 38873 | Lemma for ~ dath . Show t... |
| dalem27 38874 | Lemma for ~ dath . Show t... |
| dalem28 38875 | Lemma for ~ dath . Lemma ... |
| dalem29 38876 | Lemma for ~ dath . Analog... |
| dalem30 38877 | Lemma for ~ dath . Analog... |
| dalem31N 38878 | Lemma for ~ dath . Analog... |
| dalem32 38879 | Lemma for ~ dath . Analog... |
| dalem33 38880 | Lemma for ~ dath . Analog... |
| dalem34 38881 | Lemma for ~ dath . Analog... |
| dalem35 38882 | Lemma for ~ dath . Analog... |
| dalem36 38883 | Lemma for ~ dath . Analog... |
| dalem37 38884 | Lemma for ~ dath . Analog... |
| dalem38 38885 | Lemma for ~ dath . Plane ... |
| dalem39 38886 | Lemma for ~ dath . Auxili... |
| dalem40 38887 | Lemma for ~ dath . Analog... |
| dalem41 38888 | Lemma for ~ dath . (Contr... |
| dalem42 38889 | Lemma for ~ dath . Auxili... |
| dalem43 38890 | Lemma for ~ dath . Planes... |
| dalem44 38891 | Lemma for ~ dath . Dummy ... |
| dalem45 38892 | Lemma for ~ dath . Dummy ... |
| dalem46 38893 | Lemma for ~ dath . Analog... |
| dalem47 38894 | Lemma for ~ dath . Analog... |
| dalem48 38895 | Lemma for ~ dath . Analog... |
| dalem49 38896 | Lemma for ~ dath . Analog... |
| dalem50 38897 | Lemma for ~ dath . Analog... |
| dalem51 38898 | Lemma for ~ dath . Constr... |
| dalem52 38899 | Lemma for ~ dath . Lines ... |
| dalem53 38900 | Lemma for ~ dath . The au... |
| dalem54 38901 | Lemma for ~ dath . Line `... |
| dalem55 38902 | Lemma for ~ dath . Lines ... |
| dalem56 38903 | Lemma for ~ dath . Analog... |
| dalem57 38904 | Lemma for ~ dath . Axis o... |
| dalem58 38905 | Lemma for ~ dath . Analog... |
| dalem59 38906 | Lemma for ~ dath . Analog... |
| dalem60 38907 | Lemma for ~ dath . ` B ` i... |
| dalem61 38908 | Lemma for ~ dath . Show t... |
| dalem62 38909 | Lemma for ~ dath . Elimin... |
| dalem63 38910 | Lemma for ~ dath . Combin... |
| dath 38911 | Desargues's theorem of pro... |
| dath2 38912 | Version of Desargues's the... |
| lineset 38913 | The set of lines in a Hilb... |
| isline 38914 | The predicate "is a line".... |
| islinei 38915 | Condition implying "is a l... |
| pointsetN 38916 | The set of points in a Hil... |
| ispointN 38917 | The predicate "is a point"... |
| atpointN 38918 | The singleton of an atom i... |
| psubspset 38919 | The set of projective subs... |
| ispsubsp 38920 | The predicate "is a projec... |
| ispsubsp2 38921 | The predicate "is a projec... |
| psubspi 38922 | Property of a projective s... |
| psubspi2N 38923 | Property of a projective s... |
| 0psubN 38924 | The empty set is a project... |
| snatpsubN 38925 | The singleton of an atom i... |
| pointpsubN 38926 | A point (singleton of an a... |
| linepsubN 38927 | A line is a projective sub... |
| atpsubN 38928 | The set of all atoms is a ... |
| psubssat 38929 | A projective subspace cons... |
| psubatN 38930 | A member of a projective s... |
| pmapfval 38931 | The projective map of a Hi... |
| pmapval 38932 | Value of the projective ma... |
| elpmap 38933 | Member of a projective map... |
| pmapssat 38934 | The projective map of a Hi... |
| pmapssbaN 38935 | A weakening of ~ pmapssat ... |
| pmaple 38936 | The projective map of a Hi... |
| pmap11 38937 | The projective map of a Hi... |
| pmapat 38938 | The projective map of an a... |
| elpmapat 38939 | Member of the projective m... |
| pmap0 38940 | Value of the projective ma... |
| pmapeq0 38941 | A projective map value is ... |
| pmap1N 38942 | Value of the projective ma... |
| pmapsub 38943 | The projective map of a Hi... |
| pmapglbx 38944 | The projective map of the ... |
| pmapglb 38945 | The projective map of the ... |
| pmapglb2N 38946 | The projective map of the ... |
| pmapglb2xN 38947 | The projective map of the ... |
| pmapmeet 38948 | The projective map of a me... |
| isline2 38949 | Definition of line in term... |
| linepmap 38950 | A line described with a pr... |
| isline3 38951 | Definition of line in term... |
| isline4N 38952 | Definition of line in term... |
| lneq2at 38953 | A line equals the join of ... |
| lnatexN 38954 | There is an atom in a line... |
| lnjatN 38955 | Given an atom in a line, t... |
| lncvrelatN 38956 | A lattice element covered ... |
| lncvrat 38957 | A line covers the atoms it... |
| lncmp 38958 | If two lines are comparabl... |
| 2lnat 38959 | Two intersecting lines int... |
| 2atm2atN 38960 | Two joins with a common at... |
| 2llnma1b 38961 | Generalization of ~ 2llnma... |
| 2llnma1 38962 | Two different intersecting... |
| 2llnma3r 38963 | Two different intersecting... |
| 2llnma2 38964 | Two different intersecting... |
| 2llnma2rN 38965 | Two different intersecting... |
| cdlema1N 38966 | A condition for required f... |
| cdlema2N 38967 | A condition for required f... |
| cdlemblem 38968 | Lemma for ~ cdlemb . (Con... |
| cdlemb 38969 | Given two atoms not less t... |
| paddfval 38972 | Projective subspace sum op... |
| paddval 38973 | Projective subspace sum op... |
| elpadd 38974 | Member of a projective sub... |
| elpaddn0 38975 | Member of projective subsp... |
| paddvaln0N 38976 | Projective subspace sum op... |
| elpaddri 38977 | Condition implying members... |
| elpaddatriN 38978 | Condition implying members... |
| elpaddat 38979 | Membership in a projective... |
| elpaddatiN 38980 | Consequence of membership ... |
| elpadd2at 38981 | Membership in a projective... |
| elpadd2at2 38982 | Membership in a projective... |
| paddunssN 38983 | Projective subspace sum in... |
| elpadd0 38984 | Member of projective subsp... |
| paddval0 38985 | Projective subspace sum wi... |
| padd01 38986 | Projective subspace sum wi... |
| padd02 38987 | Projective subspace sum wi... |
| paddcom 38988 | Projective subspace sum co... |
| paddssat 38989 | A projective subspace sum ... |
| sspadd1 38990 | A projective subspace sum ... |
| sspadd2 38991 | A projective subspace sum ... |
| paddss1 38992 | Subset law for projective ... |
| paddss2 38993 | Subset law for projective ... |
| paddss12 38994 | Subset law for projective ... |
| paddasslem1 38995 | Lemma for ~ paddass . (Co... |
| paddasslem2 38996 | Lemma for ~ paddass . (Co... |
| paddasslem3 38997 | Lemma for ~ paddass . Res... |
| paddasslem4 38998 | Lemma for ~ paddass . Com... |
| paddasslem5 38999 | Lemma for ~ paddass . Sho... |
| paddasslem6 39000 | Lemma for ~ paddass . (Co... |
| paddasslem7 39001 | Lemma for ~ paddass . Com... |
| paddasslem8 39002 | Lemma for ~ paddass . (Co... |
| paddasslem9 39003 | Lemma for ~ paddass . Com... |
| paddasslem10 39004 | Lemma for ~ paddass . Use... |
| paddasslem11 39005 | Lemma for ~ paddass . The... |
| paddasslem12 39006 | Lemma for ~ paddass . The... |
| paddasslem13 39007 | Lemma for ~ paddass . The... |
| paddasslem14 39008 | Lemma for ~ paddass . Rem... |
| paddasslem15 39009 | Lemma for ~ paddass . Use... |
| paddasslem16 39010 | Lemma for ~ paddass . Use... |
| paddasslem17 39011 | Lemma for ~ paddass . The... |
| paddasslem18 39012 | Lemma for ~ paddass . Com... |
| paddass 39013 | Projective subspace sum is... |
| padd12N 39014 | Commutative/associative la... |
| padd4N 39015 | Rearrangement of 4 terms i... |
| paddidm 39016 | Projective subspace sum is... |
| paddclN 39017 | The projective sum of two ... |
| paddssw1 39018 | Subset law for projective ... |
| paddssw2 39019 | Subset law for projective ... |
| paddss 39020 | Subset law for projective ... |
| pmodlem1 39021 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 39022 | Lemma for ~ pmod1i . (Con... |
| pmod1i 39023 | The modular law holds in a... |
| pmod2iN 39024 | Dual of the modular law. ... |
| pmodN 39025 | The modular law for projec... |
| pmodl42N 39026 | Lemma derived from modular... |
| pmapjoin 39027 | The projective map of the ... |
| pmapjat1 39028 | The projective map of the ... |
| pmapjat2 39029 | The projective map of the ... |
| pmapjlln1 39030 | The projective map of the ... |
| hlmod1i 39031 | A version of the modular l... |
| atmod1i1 39032 | Version of modular law ~ p... |
| atmod1i1m 39033 | Version of modular law ~ p... |
| atmod1i2 39034 | Version of modular law ~ p... |
| llnmod1i2 39035 | Version of modular law ~ p... |
| atmod2i1 39036 | Version of modular law ~ p... |
| atmod2i2 39037 | Version of modular law ~ p... |
| llnmod2i2 39038 | Version of modular law ~ p... |
| atmod3i1 39039 | Version of modular law tha... |
| atmod3i2 39040 | Version of modular law tha... |
| atmod4i1 39041 | Version of modular law tha... |
| atmod4i2 39042 | Version of modular law tha... |
| llnexchb2lem 39043 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 39044 | Line exchange property (co... |
| llnexch2N 39045 | Line exchange property (co... |
| dalawlem1 39046 | Lemma for ~ dalaw . Speci... |
| dalawlem2 39047 | Lemma for ~ dalaw . Utili... |
| dalawlem3 39048 | Lemma for ~ dalaw . First... |
| dalawlem4 39049 | Lemma for ~ dalaw . Secon... |
| dalawlem5 39050 | Lemma for ~ dalaw . Speci... |
| dalawlem6 39051 | Lemma for ~ dalaw . First... |
| dalawlem7 39052 | Lemma for ~ dalaw . Secon... |
| dalawlem8 39053 | Lemma for ~ dalaw . Speci... |
| dalawlem9 39054 | Lemma for ~ dalaw . Speci... |
| dalawlem10 39055 | Lemma for ~ dalaw . Combi... |
| dalawlem11 39056 | Lemma for ~ dalaw . First... |
| dalawlem12 39057 | Lemma for ~ dalaw . Secon... |
| dalawlem13 39058 | Lemma for ~ dalaw . Speci... |
| dalawlem14 39059 | Lemma for ~ dalaw . Combi... |
| dalawlem15 39060 | Lemma for ~ dalaw . Swap ... |
| dalaw 39061 | Desargues's law, derived f... |
| pclfvalN 39064 | The projective subspace cl... |
| pclvalN 39065 | Value of the projective su... |
| pclclN 39066 | Closure of the projective ... |
| elpclN 39067 | Membership in the projecti... |
| elpcliN 39068 | Implication of membership ... |
| pclssN 39069 | Ordering is preserved by s... |
| pclssidN 39070 | A set of atoms is included... |
| pclidN 39071 | The projective subspace cl... |
| pclbtwnN 39072 | A projective subspace sand... |
| pclunN 39073 | The projective subspace cl... |
| pclun2N 39074 | The projective subspace cl... |
| pclfinN 39075 | The projective subspace cl... |
| pclcmpatN 39076 | The set of projective subs... |
| polfvalN 39079 | The projective subspace po... |
| polvalN 39080 | Value of the projective su... |
| polval2N 39081 | Alternate expression for v... |
| polsubN 39082 | The polarity of a set of a... |
| polssatN 39083 | The polarity of a set of a... |
| pol0N 39084 | The polarity of the empty ... |
| pol1N 39085 | The polarity of the whole ... |
| 2pol0N 39086 | The closed subspace closur... |
| polpmapN 39087 | The polarity of a projecti... |
| 2polpmapN 39088 | Double polarity of a proje... |
| 2polvalN 39089 | Value of double polarity. ... |
| 2polssN 39090 | A set of atoms is a subset... |
| 3polN 39091 | Triple polarity cancels to... |
| polcon3N 39092 | Contraposition law for pol... |
| 2polcon4bN 39093 | Contraposition law for pol... |
| polcon2N 39094 | Contraposition law for pol... |
| polcon2bN 39095 | Contraposition law for pol... |
| pclss2polN 39096 | The projective subspace cl... |
| pcl0N 39097 | The projective subspace cl... |
| pcl0bN 39098 | The projective subspace cl... |
| pmaplubN 39099 | The LUB of a projective ma... |
| sspmaplubN 39100 | A set of atoms is a subset... |
| 2pmaplubN 39101 | Double projective map of a... |
| paddunN 39102 | The closure of the project... |
| poldmj1N 39103 | De Morgan's law for polari... |
| pmapj2N 39104 | The projective map of the ... |
| pmapocjN 39105 | The projective map of the ... |
| polatN 39106 | The polarity of the single... |
| 2polatN 39107 | Double polarity of the sin... |
| pnonsingN 39108 | The intersection of a set ... |
| psubclsetN 39111 | The set of closed projecti... |
| ispsubclN 39112 | The predicate "is a closed... |
| psubcliN 39113 | Property of a closed proje... |
| psubcli2N 39114 | Property of a closed proje... |
| psubclsubN 39115 | A closed projective subspa... |
| psubclssatN 39116 | A closed projective subspa... |
| pmapidclN 39117 | Projective map of the LUB ... |
| 0psubclN 39118 | The empty set is a closed ... |
| 1psubclN 39119 | The set of all atoms is a ... |
| atpsubclN 39120 | A point (singleton of an a... |
| pmapsubclN 39121 | A projective map value is ... |
| ispsubcl2N 39122 | Alternate predicate for "i... |
| psubclinN 39123 | The intersection of two cl... |
| paddatclN 39124 | The projective sum of a cl... |
| pclfinclN 39125 | The projective subspace cl... |
| linepsubclN 39126 | A line is a closed project... |
| polsubclN 39127 | A polarity is a closed pro... |
| poml4N 39128 | Orthomodular law for proje... |
| poml5N 39129 | Orthomodular law for proje... |
| poml6N 39130 | Orthomodular law for proje... |
| osumcllem1N 39131 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 39132 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 39133 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 39134 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 39135 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 39136 | Lemma for ~ osumclN . Use... |
| osumcllem7N 39137 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 39138 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 39139 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 39140 | Lemma for ~ osumclN . Con... |
| osumcllem11N 39141 | Lemma for ~ osumclN . (Co... |
| osumclN 39142 | Closure of orthogonal sum.... |
| pmapojoinN 39143 | For orthogonal elements, p... |
| pexmidN 39144 | Excluded middle law for cl... |
| pexmidlem1N 39145 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 39146 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 39147 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 39148 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 39149 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 39150 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 39151 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 39152 | Lemma for ~ pexmidN . The... |
| pexmidALTN 39153 | Excluded middle law for cl... |
| pl42lem1N 39154 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 39155 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 39156 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 39157 | Lemma for ~ pl42N . (Cont... |
| pl42N 39158 | Law holding in a Hilbert l... |
| watfvalN 39167 | The W atoms function. (Co... |
| watvalN 39168 | Value of the W atoms funct... |
| iswatN 39169 | The predicate "is a W atom... |
| lhpset 39170 | The set of co-atoms (latti... |
| islhp 39171 | The predicate "is a co-ato... |
| islhp2 39172 | The predicate "is a co-ato... |
| lhpbase 39173 | A co-atom is a member of t... |
| lhp1cvr 39174 | The lattice unity covers a... |
| lhplt 39175 | An atom under a co-atom is... |
| lhp2lt 39176 | The join of two atoms unde... |
| lhpexlt 39177 | There exists an atom less ... |
| lhp0lt 39178 | A co-atom is greater than ... |
| lhpn0 39179 | A co-atom is nonzero. TOD... |
| lhpexle 39180 | There exists an atom under... |
| lhpexnle 39181 | There exists an atom not u... |
| lhpexle1lem 39182 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 39183 | There exists an atom under... |
| lhpexle2lem 39184 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 39185 | There exists atom under a ... |
| lhpexle3lem 39186 | There exists atom under a ... |
| lhpexle3 39187 | There exists atom under a ... |
| lhpex2leN 39188 | There exist at least two d... |
| lhpoc 39189 | The orthocomplement of a c... |
| lhpoc2N 39190 | The orthocomplement of an ... |
| lhpocnle 39191 | The orthocomplement of a c... |
| lhpocat 39192 | The orthocomplement of a c... |
| lhpocnel 39193 | The orthocomplement of a c... |
| lhpocnel2 39194 | The orthocomplement of a c... |
| lhpjat1 39195 | The join of a co-atom (hyp... |
| lhpjat2 39196 | The join of a co-atom (hyp... |
| lhpj1 39197 | The join of a co-atom (hyp... |
| lhpmcvr 39198 | The meet of a lattice hype... |
| lhpmcvr2 39199 | Alternate way to express t... |
| lhpmcvr3 39200 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 39201 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 39202 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 39203 | Specialization of ~ lhpmcv... |
| lhpm0atN 39204 | If the meet of a lattice h... |
| lhpmat 39205 | An element covered by the ... |
| lhpmatb 39206 | An element covered by the ... |
| lhp2at0 39207 | Join and meet with differe... |
| lhp2atnle 39208 | Inequality for 2 different... |
| lhp2atne 39209 | Inequality for joins with ... |
| lhp2at0nle 39210 | Inequality for 2 different... |
| lhp2at0ne 39211 | Inequality for joins with ... |
| lhpelim 39212 | Eliminate an atom not unde... |
| lhpmod2i2 39213 | Modular law for hyperplane... |
| lhpmod6i1 39214 | Modular law for hyperplane... |
| lhprelat3N 39215 | The Hilbert lattice is rel... |
| cdlemb2 39216 | Given two atoms not under ... |
| lhple 39217 | Property of a lattice elem... |
| lhpat 39218 | Create an atom under a co-... |
| lhpat4N 39219 | Property of an atom under ... |
| lhpat2 39220 | Create an atom under a co-... |
| lhpat3 39221 | There is only one atom und... |
| 4atexlemk 39222 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 39223 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 39224 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 39225 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 39226 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 39227 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 39228 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 39229 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 39230 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 39231 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 39232 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 39233 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 39234 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 39235 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 39236 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 39237 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 39238 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 39239 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 39240 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 39241 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 39242 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 39243 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 39244 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 39245 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 39246 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 39247 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 39248 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 39249 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 39250 | Whenever there are at leas... |
| 4atex 39251 | Whenever there are at leas... |
| 4atex2 39252 | More general version of ~ ... |
| 4atex2-0aOLDN 39253 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 39254 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 39255 | Same as ~ 4atex2 except th... |
| 4atex3 39256 | More general version of ~ ... |
| lautset 39257 | The set of lattice automor... |
| islaut 39258 | The predicate "is a lattic... |
| lautle 39259 | Less-than or equal propert... |
| laut1o 39260 | A lattice automorphism is ... |
| laut11 39261 | One-to-one property of a l... |
| lautcl 39262 | A lattice automorphism val... |
| lautcnvclN 39263 | Reverse closure of a latti... |
| lautcnvle 39264 | Less-than or equal propert... |
| lautcnv 39265 | The converse of a lattice ... |
| lautlt 39266 | Less-than property of a la... |
| lautcvr 39267 | Covering property of a lat... |
| lautj 39268 | Meet property of a lattice... |
| lautm 39269 | Meet property of a lattice... |
| lauteq 39270 | A lattice automorphism arg... |
| idlaut 39271 | The identity function is a... |
| lautco 39272 | The composition of two lat... |
| pautsetN 39273 | The set of projective auto... |
| ispautN 39274 | The predicate "is a projec... |
| ldilfset 39283 | The mapping from fiducial ... |
| ldilset 39284 | The set of lattice dilatio... |
| isldil 39285 | The predicate "is a lattic... |
| ldillaut 39286 | A lattice dilation is an a... |
| ldil1o 39287 | A lattice dilation is a on... |
| ldilval 39288 | Value of a lattice dilatio... |
| idldil 39289 | The identity function is a... |
| ldilcnv 39290 | The converse of a lattice ... |
| ldilco 39291 | The composition of two lat... |
| ltrnfset 39292 | The set of all lattice tra... |
| ltrnset 39293 | The set of lattice transla... |
| isltrn 39294 | The predicate "is a lattic... |
| isltrn2N 39295 | The predicate "is a lattic... |
| ltrnu 39296 | Uniqueness property of a l... |
| ltrnldil 39297 | A lattice translation is a... |
| ltrnlaut 39298 | A lattice translation is a... |
| ltrn1o 39299 | A lattice translation is a... |
| ltrncl 39300 | Closure of a lattice trans... |
| ltrn11 39301 | One-to-one property of a l... |
| ltrncnvnid 39302 | If a translation is differ... |
| ltrncoidN 39303 | Two translations are equal... |
| ltrnle 39304 | Less-than or equal propert... |
| ltrncnvleN 39305 | Less-than or equal propert... |
| ltrnm 39306 | Lattice translation of a m... |
| ltrnj 39307 | Lattice translation of a m... |
| ltrncvr 39308 | Covering property of a lat... |
| ltrnval1 39309 | Value of a lattice transla... |
| ltrnid 39310 | A lattice translation is t... |
| ltrnnid 39311 | If a lattice translation i... |
| ltrnatb 39312 | The lattice translation of... |
| ltrncnvatb 39313 | The converse of the lattic... |
| ltrnel 39314 | The lattice translation of... |
| ltrnat 39315 | The lattice translation of... |
| ltrncnvat 39316 | The converse of the lattic... |
| ltrncnvel 39317 | The converse of the lattic... |
| ltrncoelN 39318 | Composition of lattice tra... |
| ltrncoat 39319 | Composition of lattice tra... |
| ltrncoval 39320 | Two ways to express value ... |
| ltrncnv 39321 | The converse of a lattice ... |
| ltrn11at 39322 | Frequently used one-to-one... |
| ltrneq2 39323 | The equality of two transl... |
| ltrneq 39324 | The equality of two transl... |
| idltrn 39325 | The identity function is a... |
| ltrnmw 39326 | Property of lattice transl... |
| dilfsetN 39327 | The mapping from fiducial ... |
| dilsetN 39328 | The set of dilations for a... |
| isdilN 39329 | The predicate "is a dilati... |
| trnfsetN 39330 | The mapping from fiducial ... |
| trnsetN 39331 | The set of translations fo... |
| istrnN 39332 | The predicate "is a transl... |
| trlfset 39335 | The set of all traces of l... |
| trlset 39336 | The set of traces of latti... |
| trlval 39337 | The value of the trace of ... |
| trlval2 39338 | The value of the trace of ... |
| trlcl 39339 | Closure of the trace of a ... |
| trlcnv 39340 | The trace of the converse ... |
| trljat1 39341 | The value of a translation... |
| trljat2 39342 | The value of a translation... |
| trljat3 39343 | The value of a translation... |
| trlat 39344 | If an atom differs from it... |
| trl0 39345 | If an atom not under the f... |
| trlator0 39346 | The trace of a lattice tra... |
| trlatn0 39347 | The trace of a lattice tra... |
| trlnidat 39348 | The trace of a lattice tra... |
| ltrnnidn 39349 | If a lattice translation i... |
| ltrnideq 39350 | Property of the identity l... |
| trlid0 39351 | The trace of the identity ... |
| trlnidatb 39352 | A lattice translation is n... |
| trlid0b 39353 | A lattice translation is t... |
| trlnid 39354 | Different translations wit... |
| ltrn2ateq 39355 | Property of the equality o... |
| ltrnateq 39356 | If any atom (under ` W ` )... |
| ltrnatneq 39357 | If any atom (under ` W ` )... |
| ltrnatlw 39358 | If the value of an atom eq... |
| trlle 39359 | The trace of a lattice tra... |
| trlne 39360 | The trace of a lattice tra... |
| trlnle 39361 | The atom not under the fid... |
| trlval3 39362 | The value of the trace of ... |
| trlval4 39363 | The value of the trace of ... |
| trlval5 39364 | The value of the trace of ... |
| arglem1N 39365 | Lemma for Desargues's law.... |
| cdlemc1 39366 | Part of proof of Lemma C i... |
| cdlemc2 39367 | Part of proof of Lemma C i... |
| cdlemc3 39368 | Part of proof of Lemma C i... |
| cdlemc4 39369 | Part of proof of Lemma C i... |
| cdlemc5 39370 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 39371 | Lemma for ~ cdlemc . (Con... |
| cdlemc 39372 | Lemma C in [Crawley] p. 11... |
| cdlemd1 39373 | Part of proof of Lemma D i... |
| cdlemd2 39374 | Part of proof of Lemma D i... |
| cdlemd3 39375 | Part of proof of Lemma D i... |
| cdlemd4 39376 | Part of proof of Lemma D i... |
| cdlemd5 39377 | Part of proof of Lemma D i... |
| cdlemd6 39378 | Part of proof of Lemma D i... |
| cdlemd7 39379 | Part of proof of Lemma D i... |
| cdlemd8 39380 | Part of proof of Lemma D i... |
| cdlemd9 39381 | Part of proof of Lemma D i... |
| cdlemd 39382 | If two translations agree ... |
| ltrneq3 39383 | Two translations agree at ... |
| cdleme00a 39384 | Part of proof of Lemma E i... |
| cdleme0aa 39385 | Part of proof of Lemma E i... |
| cdleme0a 39386 | Part of proof of Lemma E i... |
| cdleme0b 39387 | Part of proof of Lemma E i... |
| cdleme0c 39388 | Part of proof of Lemma E i... |
| cdleme0cp 39389 | Part of proof of Lemma E i... |
| cdleme0cq 39390 | Part of proof of Lemma E i... |
| cdleme0dN 39391 | Part of proof of Lemma E i... |
| cdleme0e 39392 | Part of proof of Lemma E i... |
| cdleme0fN 39393 | Part of proof of Lemma E i... |
| cdleme0gN 39394 | Part of proof of Lemma E i... |
| cdlemeulpq 39395 | Part of proof of Lemma E i... |
| cdleme01N 39396 | Part of proof of Lemma E i... |
| cdleme02N 39397 | Part of proof of Lemma E i... |
| cdleme0ex1N 39398 | Part of proof of Lemma E i... |
| cdleme0ex2N 39399 | Part of proof of Lemma E i... |
| cdleme0moN 39400 | Part of proof of Lemma E i... |
| cdleme1b 39401 | Part of proof of Lemma E i... |
| cdleme1 39402 | Part of proof of Lemma E i... |
| cdleme2 39403 | Part of proof of Lemma E i... |
| cdleme3b 39404 | Part of proof of Lemma E i... |
| cdleme3c 39405 | Part of proof of Lemma E i... |
| cdleme3d 39406 | Part of proof of Lemma E i... |
| cdleme3e 39407 | Part of proof of Lemma E i... |
| cdleme3fN 39408 | Part of proof of Lemma E i... |
| cdleme3g 39409 | Part of proof of Lemma E i... |
| cdleme3h 39410 | Part of proof of Lemma E i... |
| cdleme3fa 39411 | Part of proof of Lemma E i... |
| cdleme3 39412 | Part of proof of Lemma E i... |
| cdleme4 39413 | Part of proof of Lemma E i... |
| cdleme4a 39414 | Part of proof of Lemma E i... |
| cdleme5 39415 | Part of proof of Lemma E i... |
| cdleme6 39416 | Part of proof of Lemma E i... |
| cdleme7aa 39417 | Part of proof of Lemma E i... |
| cdleme7a 39418 | Part of proof of Lemma E i... |
| cdleme7b 39419 | Part of proof of Lemma E i... |
| cdleme7c 39420 | Part of proof of Lemma E i... |
| cdleme7d 39421 | Part of proof of Lemma E i... |
| cdleme7e 39422 | Part of proof of Lemma E i... |
| cdleme7ga 39423 | Part of proof of Lemma E i... |
| cdleme7 39424 | Part of proof of Lemma E i... |
| cdleme8 39425 | Part of proof of Lemma E i... |
| cdleme9a 39426 | Part of proof of Lemma E i... |
| cdleme9b 39427 | Utility lemma for Lemma E ... |
| cdleme9 39428 | Part of proof of Lemma E i... |
| cdleme10 39429 | Part of proof of Lemma E i... |
| cdleme8tN 39430 | Part of proof of Lemma E i... |
| cdleme9taN 39431 | Part of proof of Lemma E i... |
| cdleme9tN 39432 | Part of proof of Lemma E i... |
| cdleme10tN 39433 | Part of proof of Lemma E i... |
| cdleme16aN 39434 | Part of proof of Lemma E i... |
| cdleme11a 39435 | Part of proof of Lemma E i... |
| cdleme11c 39436 | Part of proof of Lemma E i... |
| cdleme11dN 39437 | Part of proof of Lemma E i... |
| cdleme11e 39438 | Part of proof of Lemma E i... |
| cdleme11fN 39439 | Part of proof of Lemma E i... |
| cdleme11g 39440 | Part of proof of Lemma E i... |
| cdleme11h 39441 | Part of proof of Lemma E i... |
| cdleme11j 39442 | Part of proof of Lemma E i... |
| cdleme11k 39443 | Part of proof of Lemma E i... |
| cdleme11l 39444 | Part of proof of Lemma E i... |
| cdleme11 39445 | Part of proof of Lemma E i... |
| cdleme12 39446 | Part of proof of Lemma E i... |
| cdleme13 39447 | Part of proof of Lemma E i... |
| cdleme14 39448 | Part of proof of Lemma E i... |
| cdleme15a 39449 | Part of proof of Lemma E i... |
| cdleme15b 39450 | Part of proof of Lemma E i... |
| cdleme15c 39451 | Part of proof of Lemma E i... |
| cdleme15d 39452 | Part of proof of Lemma E i... |
| cdleme15 39453 | Part of proof of Lemma E i... |
| cdleme16b 39454 | Part of proof of Lemma E i... |
| cdleme16c 39455 | Part of proof of Lemma E i... |
| cdleme16d 39456 | Part of proof of Lemma E i... |
| cdleme16e 39457 | Part of proof of Lemma E i... |
| cdleme16f 39458 | Part of proof of Lemma E i... |
| cdleme16g 39459 | Part of proof of Lemma E i... |
| cdleme16 39460 | Part of proof of Lemma E i... |
| cdleme17a 39461 | Part of proof of Lemma E i... |
| cdleme17b 39462 | Lemma leading to ~ cdleme1... |
| cdleme17c 39463 | Part of proof of Lemma E i... |
| cdleme17d1 39464 | Part of proof of Lemma E i... |
| cdleme0nex 39465 | Part of proof of Lemma E i... |
| cdleme18a 39466 | Part of proof of Lemma E i... |
| cdleme18b 39467 | Part of proof of Lemma E i... |
| cdleme18c 39468 | Part of proof of Lemma E i... |
| cdleme22gb 39469 | Utility lemma for Lemma E ... |
| cdleme18d 39470 | Part of proof of Lemma E i... |
| cdlemesner 39471 | Part of proof of Lemma E i... |
| cdlemedb 39472 | Part of proof of Lemma E i... |
| cdlemeda 39473 | Part of proof of Lemma E i... |
| cdlemednpq 39474 | Part of proof of Lemma E i... |
| cdlemednuN 39475 | Part of proof of Lemma E i... |
| cdleme20zN 39476 | Part of proof of Lemma E i... |
| cdleme20y 39477 | Part of proof of Lemma E i... |
| cdleme19a 39478 | Part of proof of Lemma E i... |
| cdleme19b 39479 | Part of proof of Lemma E i... |
| cdleme19c 39480 | Part of proof of Lemma E i... |
| cdleme19d 39481 | Part of proof of Lemma E i... |
| cdleme19e 39482 | Part of proof of Lemma E i... |
| cdleme19f 39483 | Part of proof of Lemma E i... |
| cdleme20aN 39484 | Part of proof of Lemma E i... |
| cdleme20bN 39485 | Part of proof of Lemma E i... |
| cdleme20c 39486 | Part of proof of Lemma E i... |
| cdleme20d 39487 | Part of proof of Lemma E i... |
| cdleme20e 39488 | Part of proof of Lemma E i... |
| cdleme20f 39489 | Part of proof of Lemma E i... |
| cdleme20g 39490 | Part of proof of Lemma E i... |
| cdleme20h 39491 | Part of proof of Lemma E i... |
| cdleme20i 39492 | Part of proof of Lemma E i... |
| cdleme20j 39493 | Part of proof of Lemma E i... |
| cdleme20k 39494 | Part of proof of Lemma E i... |
| cdleme20l1 39495 | Part of proof of Lemma E i... |
| cdleme20l2 39496 | Part of proof of Lemma E i... |
| cdleme20l 39497 | Part of proof of Lemma E i... |
| cdleme20m 39498 | Part of proof of Lemma E i... |
| cdleme20 39499 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 39500 | Part of proof of Lemma E i... |
| cdleme21b 39501 | Part of proof of Lemma E i... |
| cdleme21c 39502 | Part of proof of Lemma E i... |
| cdleme21at 39503 | Part of proof of Lemma E i... |
| cdleme21ct 39504 | Part of proof of Lemma E i... |
| cdleme21d 39505 | Part of proof of Lemma E i... |
| cdleme21e 39506 | Part of proof of Lemma E i... |
| cdleme21f 39507 | Part of proof of Lemma E i... |
| cdleme21g 39508 | Part of proof of Lemma E i... |
| cdleme21h 39509 | Part of proof of Lemma E i... |
| cdleme21i 39510 | Part of proof of Lemma E i... |
| cdleme21j 39511 | Combine ~ cdleme20 and ~ c... |
| cdleme21 39512 | Part of proof of Lemma E i... |
| cdleme21k 39513 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 39514 | Part of proof of Lemma E i... |
| cdleme22a 39515 | Part of proof of Lemma E i... |
| cdleme22b 39516 | Part of proof of Lemma E i... |
| cdleme22cN 39517 | Part of proof of Lemma E i... |
| cdleme22d 39518 | Part of proof of Lemma E i... |
| cdleme22e 39519 | Part of proof of Lemma E i... |
| cdleme22eALTN 39520 | Part of proof of Lemma E i... |
| cdleme22f 39521 | Part of proof of Lemma E i... |
| cdleme22f2 39522 | Part of proof of Lemma E i... |
| cdleme22g 39523 | Part of proof of Lemma E i... |
| cdleme23a 39524 | Part of proof of Lemma E i... |
| cdleme23b 39525 | Part of proof of Lemma E i... |
| cdleme23c 39526 | Part of proof of Lemma E i... |
| cdleme24 39527 | Quantified version of ~ cd... |
| cdleme25a 39528 | Lemma for ~ cdleme25b . (... |
| cdleme25b 39529 | Transform ~ cdleme24 . TO... |
| cdleme25c 39530 | Transform ~ cdleme25b . (... |
| cdleme25dN 39531 | Transform ~ cdleme25c . (... |
| cdleme25cl 39532 | Show closure of the unique... |
| cdleme25cv 39533 | Change bound variables in ... |
| cdleme26e 39534 | Part of proof of Lemma E i... |
| cdleme26ee 39535 | Part of proof of Lemma E i... |
| cdleme26eALTN 39536 | Part of proof of Lemma E i... |
| cdleme26fALTN 39537 | Part of proof of Lemma E i... |
| cdleme26f 39538 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 39539 | Part of proof of Lemma E i... |
| cdleme26f2 39540 | Part of proof of Lemma E i... |
| cdleme27cl 39541 | Part of proof of Lemma E i... |
| cdleme27a 39542 | Part of proof of Lemma E i... |
| cdleme27b 39543 | Lemma for ~ cdleme27N . (... |
| cdleme27N 39544 | Part of proof of Lemma E i... |
| cdleme28a 39545 | Lemma for ~ cdleme25b . T... |
| cdleme28b 39546 | Lemma for ~ cdleme25b . T... |
| cdleme28c 39547 | Part of proof of Lemma E i... |
| cdleme28 39548 | Quantified version of ~ cd... |
| cdleme29ex 39549 | Lemma for ~ cdleme29b . (... |
| cdleme29b 39550 | Transform ~ cdleme28 . (C... |
| cdleme29c 39551 | Transform ~ cdleme28b . (... |
| cdleme29cl 39552 | Show closure of the unique... |
| cdleme30a 39553 | Part of proof of Lemma E i... |
| cdleme31so 39554 | Part of proof of Lemma E i... |
| cdleme31sn 39555 | Part of proof of Lemma E i... |
| cdleme31sn1 39556 | Part of proof of Lemma E i... |
| cdleme31se 39557 | Part of proof of Lemma D i... |
| cdleme31se2 39558 | Part of proof of Lemma D i... |
| cdleme31sc 39559 | Part of proof of Lemma E i... |
| cdleme31sde 39560 | Part of proof of Lemma D i... |
| cdleme31snd 39561 | Part of proof of Lemma D i... |
| cdleme31sdnN 39562 | Part of proof of Lemma E i... |
| cdleme31sn1c 39563 | Part of proof of Lemma E i... |
| cdleme31sn2 39564 | Part of proof of Lemma E i... |
| cdleme31fv 39565 | Part of proof of Lemma E i... |
| cdleme31fv1 39566 | Part of proof of Lemma E i... |
| cdleme31fv1s 39567 | Part of proof of Lemma E i... |
| cdleme31fv2 39568 | Part of proof of Lemma E i... |
| cdleme31id 39569 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 39570 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 39571 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 39572 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 39573 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 39574 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 39575 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 39576 | Part of proof of Lemma E i... |
| cdlemefr29exN 39577 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 39578 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 39579 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 39580 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 39581 | TODO fix comment. (Contri... |
| cdlemefr29clN 39582 | Show closure of the unique... |
| cdleme43frv1snN 39583 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 39584 | Part of proof of Lemma E i... |
| cdlemefr32fva1 39585 | Part of proof of Lemma E i... |
| cdlemefr31fv1 39586 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 39587 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 39588 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 39589 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 39590 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 39591 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 39592 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 39593 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 39594 | Show closure of the unique... |
| cdleme43fsv1snlem 39595 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 39596 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 39597 | Part of proof of Lemma E i... |
| cdlemefs32fva1 39598 | Part of proof of Lemma E i... |
| cdlemefs31fv1 39599 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 39600 | Value of f(r) when r is an... |
| cdlemefs44 39601 | Value of f_s(r) when r is ... |
| cdlemefr45 39602 | Value of f(r) when r is an... |
| cdlemefr45e 39603 | Explicit expansion of ~ cd... |
| cdlemefs45 39604 | Value of f_s(r) when r is ... |
| cdlemefs45ee 39605 | Explicit expansion of ~ cd... |
| cdlemefs45eN 39606 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 39607 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 39608 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 39609 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 39610 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 39611 | Show closure of ` [_ R / s... |
| cdleme32fva 39612 | Part of proof of Lemma D i... |
| cdleme32fva1 39613 | Part of proof of Lemma D i... |
| cdleme32fvaw 39614 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 39615 | Part of proof of Lemma D i... |
| cdleme32a 39616 | Part of proof of Lemma D i... |
| cdleme32b 39617 | Part of proof of Lemma D i... |
| cdleme32c 39618 | Part of proof of Lemma D i... |
| cdleme32d 39619 | Part of proof of Lemma D i... |
| cdleme32e 39620 | Part of proof of Lemma D i... |
| cdleme32f 39621 | Part of proof of Lemma D i... |
| cdleme32le 39622 | Part of proof of Lemma D i... |
| cdleme35a 39623 | Part of proof of Lemma E i... |
| cdleme35fnpq 39624 | Part of proof of Lemma E i... |
| cdleme35b 39625 | Part of proof of Lemma E i... |
| cdleme35c 39626 | Part of proof of Lemma E i... |
| cdleme35d 39627 | Part of proof of Lemma E i... |
| cdleme35e 39628 | Part of proof of Lemma E i... |
| cdleme35f 39629 | Part of proof of Lemma E i... |
| cdleme35g 39630 | Part of proof of Lemma E i... |
| cdleme35h 39631 | Part of proof of Lemma E i... |
| cdleme35h2 39632 | Part of proof of Lemma E i... |
| cdleme35sn2aw 39633 | Part of proof of Lemma E i... |
| cdleme35sn3a 39634 | Part of proof of Lemma E i... |
| cdleme36a 39635 | Part of proof of Lemma E i... |
| cdleme36m 39636 | Part of proof of Lemma E i... |
| cdleme37m 39637 | Part of proof of Lemma E i... |
| cdleme38m 39638 | Part of proof of Lemma E i... |
| cdleme38n 39639 | Part of proof of Lemma E i... |
| cdleme39a 39640 | Part of proof of Lemma E i... |
| cdleme39n 39641 | Part of proof of Lemma E i... |
| cdleme40m 39642 | Part of proof of Lemma E i... |
| cdleme40n 39643 | Part of proof of Lemma E i... |
| cdleme40v 39644 | Part of proof of Lemma E i... |
| cdleme40w 39645 | Part of proof of Lemma E i... |
| cdleme42a 39646 | Part of proof of Lemma E i... |
| cdleme42c 39647 | Part of proof of Lemma E i... |
| cdleme42d 39648 | Part of proof of Lemma E i... |
| cdleme41sn3aw 39649 | Part of proof of Lemma E i... |
| cdleme41sn4aw 39650 | Part of proof of Lemma E i... |
| cdleme41snaw 39651 | Part of proof of Lemma E i... |
| cdleme41fva11 39652 | Part of proof of Lemma E i... |
| cdleme42b 39653 | Part of proof of Lemma E i... |
| cdleme42e 39654 | Part of proof of Lemma E i... |
| cdleme42f 39655 | Part of proof of Lemma E i... |
| cdleme42g 39656 | Part of proof of Lemma E i... |
| cdleme42h 39657 | Part of proof of Lemma E i... |
| cdleme42i 39658 | Part of proof of Lemma E i... |
| cdleme42k 39659 | Part of proof of Lemma E i... |
| cdleme42ke 39660 | Part of proof of Lemma E i... |
| cdleme42keg 39661 | Part of proof of Lemma E i... |
| cdleme42mN 39662 | Part of proof of Lemma E i... |
| cdleme42mgN 39663 | Part of proof of Lemma E i... |
| cdleme43aN 39664 | Part of proof of Lemma E i... |
| cdleme43bN 39665 | Lemma for Lemma E in [Craw... |
| cdleme43cN 39666 | Part of proof of Lemma E i... |
| cdleme43dN 39667 | Part of proof of Lemma E i... |
| cdleme46f2g2 39668 | Conversion for ` G ` to re... |
| cdleme46f2g1 39669 | Conversion for ` G ` to re... |
| cdleme17d2 39670 | Part of proof of Lemma E i... |
| cdleme17d3 39671 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 39672 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 39673 | Part of proof of Lemma E i... |
| cdleme48fv 39674 | Part of proof of Lemma D i... |
| cdleme48fvg 39675 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 39676 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 39677 | TODO: fix comment. TODO: ... |
| cdleme48b 39678 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 39679 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 39680 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 39681 | TODO: fix comment. (Contr... |
| cdleme4gfv 39682 | Part of proof of Lemma D i... |
| cdlemeg47b 39683 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 39684 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 39685 | Value of g_s(r) when r is ... |
| cdlemeg49le 39686 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 39687 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 39688 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 39689 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 39690 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 39691 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 39692 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 39693 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 39694 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 39695 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 39696 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 39697 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 39698 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 39699 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 39700 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 39701 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 39702 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 39703 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 39704 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 39705 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 39706 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 39707 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 39708 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 39709 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 39710 | TODO: fix comment. (Contr... |
| cdleme48gfv1 39711 | TODO: fix comment. (Contr... |
| cdleme48gfv 39712 | TODO: fix comment. (Contr... |
| cdleme48fgv 39713 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 39714 | Part of proof of Lemma D i... |
| cdleme50lebi 39715 | Part of proof of Lemma D i... |
| cdleme50eq 39716 | Part of proof of Lemma D i... |
| cdleme50f 39717 | Part of proof of Lemma D i... |
| cdleme50f1 39718 | Part of proof of Lemma D i... |
| cdleme50rnlem 39719 | Part of proof of Lemma D i... |
| cdleme50rn 39720 | Part of proof of Lemma D i... |
| cdleme50f1o 39721 | Part of proof of Lemma D i... |
| cdleme50laut 39722 | Part of proof of Lemma D i... |
| cdleme50ldil 39723 | Part of proof of Lemma D i... |
| cdleme50trn1 39724 | Part of proof that ` F ` i... |
| cdleme50trn2a 39725 | Part of proof that ` F ` i... |
| cdleme50trn2 39726 | Part of proof that ` F ` i... |
| cdleme50trn12 39727 | Part of proof that ` F ` i... |
| cdleme50trn3 39728 | Part of proof that ` F ` i... |
| cdleme50trn123 39729 | Part of proof that ` F ` i... |
| cdleme51finvfvN 39730 | Part of proof of Lemma E i... |
| cdleme51finvN 39731 | Part of proof of Lemma E i... |
| cdleme50ltrn 39732 | Part of proof of Lemma E i... |
| cdleme51finvtrN 39733 | Part of proof of Lemma E i... |
| cdleme50ex 39734 | Part of Lemma E in [Crawle... |
| cdleme 39735 | Lemma E in [Crawley] p. 11... |
| cdlemf1 39736 | Part of Lemma F in [Crawle... |
| cdlemf2 39737 | Part of Lemma F in [Crawle... |
| cdlemf 39738 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 39739 | ~ cdlemf with additional c... |
| cdlemftr3 39740 | Special case of ~ cdlemf s... |
| cdlemftr2 39741 | Special case of ~ cdlemf s... |
| cdlemftr1 39742 | Part of proof of Lemma G o... |
| cdlemftr0 39743 | Special case of ~ cdlemf s... |
| trlord 39744 | The ordering of two Hilber... |
| cdlemg1a 39745 | Shorter expression for ` G... |
| cdlemg1b2 39746 | This theorem can be used t... |
| cdlemg1idlemN 39747 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 39748 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 39749 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 39750 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 39751 | This theorem can be used t... |
| cdlemg1idN 39752 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 39753 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 39754 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 39755 | Version of ~ cdleme51finvt... |
| ltrniotaval 39756 | Value of the unique transl... |
| ltrniotacnvval 39757 | Converse value of the uniq... |
| ltrniotaidvalN 39758 | Value of the unique transl... |
| ltrniotavalbN 39759 | Value of the unique transl... |
| cdlemeiota 39760 | A translation is uniquely ... |
| cdlemg1ci2 39761 | Any function of the form o... |
| cdlemg1cN 39762 | Any translation belongs to... |
| cdlemg1cex 39763 | Any translation is one of ... |
| cdlemg2cN 39764 | Any translation belongs to... |
| cdlemg2dN 39765 | This theorem can be used t... |
| cdlemg2cex 39766 | Any translation is one of ... |
| cdlemg2ce 39767 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 39768 | Part of proof of Lemma E i... |
| cdlemg2fvlem 39769 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 39770 | ~ cdleme42keg with simpler... |
| cdlemg2idN 39771 | Version of ~ cdleme31id wi... |
| cdlemg3a 39772 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 39773 | TODO: Replace this with ~... |
| cdlemg2fv 39774 | Value of a translation in ... |
| cdlemg2fv2 39775 | Value of a translation in ... |
| cdlemg2k 39776 | ~ cdleme42keg with simpler... |
| cdlemg2kq 39777 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 39778 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 39779 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 39780 | TODO: Is there a simpler ... |
| cdlemb3 39781 | Given two atoms not under ... |
| cdlemg7fvbwN 39782 | Properties of a translatio... |
| cdlemg4a 39783 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 39784 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 39785 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 39786 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 39787 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 39788 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 39789 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 39790 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 39791 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 39792 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 39793 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 39794 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 39795 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 39796 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 39797 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 39798 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 39799 | Value of a translation com... |
| cdlemg7aN 39800 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 39801 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 39802 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 39803 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 39804 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 39805 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 39806 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 39807 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 39808 | The triples ` <. P , ( F `... |
| cdlemg9 39809 | The triples ` <. P , ( F `... |
| cdlemg10b 39810 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 39811 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 39812 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 39813 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 39814 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 39815 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 39816 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 39817 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 39818 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 39819 | The triples ` <. P , ( F `... |
| cdlemg12c 39820 | The triples ` <. P , ( F `... |
| cdlemg12d 39821 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 39822 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 39823 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 39824 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 39825 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 39826 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 39827 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 39828 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 39829 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 39830 | Eliminate the ` ( F `` P )... |
| cdlemg15 39831 | Eliminate the ` ( (... |
| cdlemg16 39832 | Part of proof of Lemma G o... |
| cdlemg16ALTN 39833 | This version of ~ cdlemg16... |
| cdlemg16z 39834 | Eliminate ` ( ( F `... |
| cdlemg16zz 39835 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 39836 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 39837 | Part of proof of Lemma G i... |
| cdlemg17dN 39838 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 39839 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 39840 | TODO: fix comment. (Contr... |
| cdlemg17f 39841 | TODO: fix comment. (Contr... |
| cdlemg17g 39842 | TODO: fix comment. (Contr... |
| cdlemg17h 39843 | TODO: fix comment. (Contr... |
| cdlemg17i 39844 | TODO: fix comment. (Contr... |
| cdlemg17ir 39845 | TODO: fix comment. (Contr... |
| cdlemg17j 39846 | TODO: fix comment. (Contr... |
| cdlemg17pq 39847 | Utility theorem for swappi... |
| cdlemg17bq 39848 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 39849 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 39850 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 39851 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 39852 | Part of Lemma G of [Crawle... |
| cdlemg18a 39853 | Show two lines are differe... |
| cdlemg18b 39854 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 39855 | Show two lines intersect a... |
| cdlemg18d 39856 | Show two lines intersect a... |
| cdlemg18 39857 | Show two lines intersect a... |
| cdlemg19a 39858 | Show two lines intersect a... |
| cdlemg19 39859 | Show two lines intersect a... |
| cdlemg20 39860 | Show two lines intersect a... |
| cdlemg21 39861 | Version of cdlemg19 with `... |
| cdlemg22 39862 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 39863 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 39864 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 39865 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 39866 | ~ cdlemg16zz restated for ... |
| cdlemg27a 39867 | For use with case when ` (... |
| cdlemg28a 39868 | Part of proof of Lemma G o... |
| cdlemg31b0N 39869 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 39870 | TODO: Fix comment. (Cont... |
| cdlemg27b 39871 | TODO: Fix comment. (Cont... |
| cdlemg31a 39872 | TODO: fix comment. (Contr... |
| cdlemg31b 39873 | TODO: fix comment. (Contr... |
| cdlemg31c 39874 | Show that when ` N ` is an... |
| cdlemg31d 39875 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 39876 | TODO: Fix comment. (Cont... |
| cdlemg33c0 39877 | TODO: Fix comment. (Cont... |
| cdlemg28b 39878 | Part of proof of Lemma G o... |
| cdlemg28 39879 | Part of proof of Lemma G o... |
| cdlemg29 39880 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 39881 | TODO: Fix comment. (Cont... |
| cdlemg33b 39882 | TODO: Fix comment. (Cont... |
| cdlemg33c 39883 | TODO: Fix comment. (Cont... |
| cdlemg33d 39884 | TODO: Fix comment. (Cont... |
| cdlemg33e 39885 | TODO: Fix comment. (Cont... |
| cdlemg33 39886 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 39887 | Use cdlemg33 to eliminate ... |
| cdlemg35 39888 | TODO: Fix comment. TODO:... |
| cdlemg36 39889 | Use cdlemg35 to eliminate ... |
| cdlemg38 39890 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 39891 | Eliminate ` =/= ` conditio... |
| cdlemg40 39892 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 39893 | Convert ~ cdlemg40 to func... |
| ltrnco 39894 | The composition of two tra... |
| trlcocnv 39895 | Swap the arguments of the ... |
| trlcoabs 39896 | Absorption into a composit... |
| trlcoabs2N 39897 | Absorption of the trace of... |
| trlcoat 39898 | The trace of a composition... |
| trlcocnvat 39899 | Commonly used special case... |
| trlconid 39900 | The composition of two dif... |
| trlcolem 39901 | Lemma for ~ trlco . (Cont... |
| trlco 39902 | The trace of a composition... |
| trlcone 39903 | If two translations have d... |
| cdlemg42 39904 | Part of proof of Lemma G o... |
| cdlemg43 39905 | Part of proof of Lemma G o... |
| cdlemg44a 39906 | Part of proof of Lemma G o... |
| cdlemg44b 39907 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 39908 | Part of proof of Lemma G o... |
| cdlemg47a 39909 | TODO: fix comment. TODO: ... |
| cdlemg46 39910 | Part of proof of Lemma G o... |
| cdlemg47 39911 | Part of proof of Lemma G o... |
| cdlemg48 39912 | Eliminate ` h ` from ~ cdl... |
| ltrncom 39913 | Composition is commutative... |
| ltrnco4 39914 | Rearrange a composition of... |
| trljco 39915 | Trace joined with trace of... |
| trljco2 39916 | Trace joined with trace of... |
| tgrpfset 39919 | The translation group maps... |
| tgrpset 39920 | The translation group for ... |
| tgrpbase 39921 | The base set of the transl... |
| tgrpopr 39922 | The group operation of the... |
| tgrpov 39923 | The group operation value ... |
| tgrpgrplem 39924 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 39925 | The translation group is a... |
| tgrpabl 39926 | The translation group is a... |
| tendofset 39933 | The set of all trace-prese... |
| tendoset 39934 | The set of trace-preservin... |
| istendo 39935 | The predicate "is a trace-... |
| tendotp 39936 | Trace-preserving property ... |
| istendod 39937 | Deduce the predicate "is a... |
| tendof 39938 | Functionality of a trace-p... |
| tendoeq1 39939 | Condition determining equa... |
| tendovalco 39940 | Value of composition of tr... |
| tendocoval 39941 | Value of composition of en... |
| tendocl 39942 | Closure of a trace-preserv... |
| tendoco2 39943 | Distribution of compositio... |
| tendoidcl 39944 | The identity is a trace-pr... |
| tendo1mul 39945 | Multiplicative identity mu... |
| tendo1mulr 39946 | Multiplicative identity mu... |
| tendococl 39947 | The composition of two tra... |
| tendoid 39948 | The identity value of a tr... |
| tendoeq2 39949 | Condition determining equa... |
| tendoplcbv 39950 | Define sum operation for t... |
| tendopl 39951 | Value of endomorphism sum ... |
| tendopl2 39952 | Value of result of endomor... |
| tendoplcl2 39953 | Value of result of endomor... |
| tendoplco2 39954 | Value of result of endomor... |
| tendopltp 39955 | Trace-preserving property ... |
| tendoplcl 39956 | Endomorphism sum is a trac... |
| tendoplcom 39957 | The endomorphism sum opera... |
| tendoplass 39958 | The endomorphism sum opera... |
| tendodi1 39959 | Endomorphism composition d... |
| tendodi2 39960 | Endomorphism composition d... |
| tendo0cbv 39961 | Define additive identity f... |
| tendo02 39962 | Value of additive identity... |
| tendo0co2 39963 | The additive identity trac... |
| tendo0tp 39964 | Trace-preserving property ... |
| tendo0cl 39965 | The additive identity is a... |
| tendo0pl 39966 | Property of the additive i... |
| tendo0plr 39967 | Property of the additive i... |
| tendoicbv 39968 | Define inverse function fo... |
| tendoi 39969 | Value of inverse endomorph... |
| tendoi2 39970 | Value of additive inverse ... |
| tendoicl 39971 | Closure of the additive in... |
| tendoipl 39972 | Property of the additive i... |
| tendoipl2 39973 | Property of the additive i... |
| erngfset 39974 | The division rings on trac... |
| erngset 39975 | The division ring on trace... |
| erngbase 39976 | The base set of the divisi... |
| erngfplus 39977 | Ring addition operation. ... |
| erngplus 39978 | Ring addition operation. ... |
| erngplus2 39979 | Ring addition operation. ... |
| erngfmul 39980 | Ring multiplication operat... |
| erngmul 39981 | Ring addition operation. ... |
| erngfset-rN 39982 | The division rings on trac... |
| erngset-rN 39983 | The division ring on trace... |
| erngbase-rN 39984 | The base set of the divisi... |
| erngfplus-rN 39985 | Ring addition operation. ... |
| erngplus-rN 39986 | Ring addition operation. ... |
| erngplus2-rN 39987 | Ring addition operation. ... |
| erngfmul-rN 39988 | Ring multiplication operat... |
| erngmul-rN 39989 | Ring addition operation. ... |
| cdlemh1 39990 | Part of proof of Lemma H o... |
| cdlemh2 39991 | Part of proof of Lemma H o... |
| cdlemh 39992 | Lemma H of [Crawley] p. 11... |
| cdlemi1 39993 | Part of proof of Lemma I o... |
| cdlemi2 39994 | Part of proof of Lemma I o... |
| cdlemi 39995 | Lemma I of [Crawley] p. 11... |
| cdlemj1 39996 | Part of proof of Lemma J o... |
| cdlemj2 39997 | Part of proof of Lemma J o... |
| cdlemj3 39998 | Part of proof of Lemma J o... |
| tendocan 39999 | Cancellation law: if the v... |
| tendoid0 40000 | A trace-preserving endomor... |
| tendo0mul 40001 | Additive identity multipli... |
| tendo0mulr 40002 | Additive identity multipli... |
| tendo1ne0 40003 | The identity (unity) is no... |
| tendoconid 40004 | The composition (product) ... |
| tendotr 40005 | The trace of the value of ... |
| cdlemk1 40006 | Part of proof of Lemma K o... |
| cdlemk2 40007 | Part of proof of Lemma K o... |
| cdlemk3 40008 | Part of proof of Lemma K o... |
| cdlemk4 40009 | Part of proof of Lemma K o... |
| cdlemk5a 40010 | Part of proof of Lemma K o... |
| cdlemk5 40011 | Part of proof of Lemma K o... |
| cdlemk6 40012 | Part of proof of Lemma K o... |
| cdlemk8 40013 | Part of proof of Lemma K o... |
| cdlemk9 40014 | Part of proof of Lemma K o... |
| cdlemk9bN 40015 | Part of proof of Lemma K o... |
| cdlemki 40016 | Part of proof of Lemma K o... |
| cdlemkvcl 40017 | Part of proof of Lemma K o... |
| cdlemk10 40018 | Part of proof of Lemma K o... |
| cdlemksv 40019 | Part of proof of Lemma K o... |
| cdlemksel 40020 | Part of proof of Lemma K o... |
| cdlemksat 40021 | Part of proof of Lemma K o... |
| cdlemksv2 40022 | Part of proof of Lemma K o... |
| cdlemk7 40023 | Part of proof of Lemma K o... |
| cdlemk11 40024 | Part of proof of Lemma K o... |
| cdlemk12 40025 | Part of proof of Lemma K o... |
| cdlemkoatnle 40026 | Utility lemma. (Contribut... |
| cdlemk13 40027 | Part of proof of Lemma K o... |
| cdlemkole 40028 | Utility lemma. (Contribut... |
| cdlemk14 40029 | Part of proof of Lemma K o... |
| cdlemk15 40030 | Part of proof of Lemma K o... |
| cdlemk16a 40031 | Part of proof of Lemma K o... |
| cdlemk16 40032 | Part of proof of Lemma K o... |
| cdlemk17 40033 | Part of proof of Lemma K o... |
| cdlemk1u 40034 | Part of proof of Lemma K o... |
| cdlemk5auN 40035 | Part of proof of Lemma K o... |
| cdlemk5u 40036 | Part of proof of Lemma K o... |
| cdlemk6u 40037 | Part of proof of Lemma K o... |
| cdlemkj 40038 | Part of proof of Lemma K o... |
| cdlemkuvN 40039 | Part of proof of Lemma K o... |
| cdlemkuel 40040 | Part of proof of Lemma K o... |
| cdlemkuat 40041 | Part of proof of Lemma K o... |
| cdlemkuv2 40042 | Part of proof of Lemma K o... |
| cdlemk18 40043 | Part of proof of Lemma K o... |
| cdlemk19 40044 | Part of proof of Lemma K o... |
| cdlemk7u 40045 | Part of proof of Lemma K o... |
| cdlemk11u 40046 | Part of proof of Lemma K o... |
| cdlemk12u 40047 | Part of proof of Lemma K o... |
| cdlemk21N 40048 | Part of proof of Lemma K o... |
| cdlemk20 40049 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 40050 | Utility lemma. (Contribut... |
| cdlemk13-2N 40051 | Part of proof of Lemma K o... |
| cdlemkole-2N 40052 | Utility lemma. (Contribut... |
| cdlemk14-2N 40053 | Part of proof of Lemma K o... |
| cdlemk15-2N 40054 | Part of proof of Lemma K o... |
| cdlemk16-2N 40055 | Part of proof of Lemma K o... |
| cdlemk17-2N 40056 | Part of proof of Lemma K o... |
| cdlemkj-2N 40057 | Part of proof of Lemma K o... |
| cdlemkuv-2N 40058 | Part of proof of Lemma K o... |
| cdlemkuel-2N 40059 | Part of proof of Lemma K o... |
| cdlemkuv2-2 40060 | Part of proof of Lemma K o... |
| cdlemk18-2N 40061 | Part of proof of Lemma K o... |
| cdlemk19-2N 40062 | Part of proof of Lemma K o... |
| cdlemk7u-2N 40063 | Part of proof of Lemma K o... |
| cdlemk11u-2N 40064 | Part of proof of Lemma K o... |
| cdlemk12u-2N 40065 | Part of proof of Lemma K o... |
| cdlemk21-2N 40066 | Part of proof of Lemma K o... |
| cdlemk20-2N 40067 | Part of proof of Lemma K o... |
| cdlemk22 40068 | Part of proof of Lemma K o... |
| cdlemk30 40069 | Part of proof of Lemma K o... |
| cdlemkuu 40070 | Convert between function a... |
| cdlemk31 40071 | Part of proof of Lemma K o... |
| cdlemk32 40072 | Part of proof of Lemma K o... |
| cdlemkuel-3 40073 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 40074 | Part of proof of Lemma K o... |
| cdlemk18-3N 40075 | Part of proof of Lemma K o... |
| cdlemk22-3 40076 | Part of proof of Lemma K o... |
| cdlemk23-3 40077 | Part of proof of Lemma K o... |
| cdlemk24-3 40078 | Part of proof of Lemma K o... |
| cdlemk25-3 40079 | Part of proof of Lemma K o... |
| cdlemk26b-3 40080 | Part of proof of Lemma K o... |
| cdlemk26-3 40081 | Part of proof of Lemma K o... |
| cdlemk27-3 40082 | Part of proof of Lemma K o... |
| cdlemk28-3 40083 | Part of proof of Lemma K o... |
| cdlemk33N 40084 | Part of proof of Lemma K o... |
| cdlemk34 40085 | Part of proof of Lemma K o... |
| cdlemk29-3 40086 | Part of proof of Lemma K o... |
| cdlemk35 40087 | Part of proof of Lemma K o... |
| cdlemk36 40088 | Part of proof of Lemma K o... |
| cdlemk37 40089 | Part of proof of Lemma K o... |
| cdlemk38 40090 | Part of proof of Lemma K o... |
| cdlemk39 40091 | Part of proof of Lemma K o... |
| cdlemk40 40092 | TODO: fix comment. (Contr... |
| cdlemk40t 40093 | TODO: fix comment. (Contr... |
| cdlemk40f 40094 | TODO: fix comment. (Contr... |
| cdlemk41 40095 | Part of proof of Lemma K o... |
| cdlemkfid1N 40096 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 40097 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 40098 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 40099 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 40100 | TODO: is this useful or sh... |
| cdlemky 40101 | Part of proof of Lemma K o... |
| cdlemkyu 40102 | Convert between function a... |
| cdlemkyuu 40103 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 40104 | Part of proof of Lemma K o... |
| cdlemk19ylem 40105 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 40106 | Part of proof of Lemma K o... |
| cdlemk19y 40107 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 40108 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 40109 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 40110 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 40111 | The value of the tau funct... |
| cdlemk35s 40112 | Substitution version of ~ ... |
| cdlemk35s-id 40113 | Substitution version of ~ ... |
| cdlemk39s 40114 | Substitution version of ~ ... |
| cdlemk39s-id 40115 | Substitution version of ~ ... |
| cdlemk42 40116 | Part of proof of Lemma K o... |
| cdlemk19xlem 40117 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 40118 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 40119 | Part of proof of Lemma K o... |
| cdlemk11tc 40120 | Part of proof of Lemma K o... |
| cdlemk11t 40121 | Part of proof of Lemma K o... |
| cdlemk45 40122 | Part of proof of Lemma K o... |
| cdlemk46 40123 | Part of proof of Lemma K o... |
| cdlemk47 40124 | Part of proof of Lemma K o... |
| cdlemk48 40125 | Part of proof of Lemma K o... |
| cdlemk49 40126 | Part of proof of Lemma K o... |
| cdlemk50 40127 | Part of proof of Lemma K o... |
| cdlemk51 40128 | Part of proof of Lemma K o... |
| cdlemk52 40129 | Part of proof of Lemma K o... |
| cdlemk53a 40130 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 40131 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 40132 | Part of proof of Lemma K o... |
| cdlemk54 40133 | Part of proof of Lemma K o... |
| cdlemk55a 40134 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 40135 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 40136 | Part of proof of Lemma K o... |
| cdlemkyyN 40137 | Part of proof of Lemma K o... |
| cdlemk43N 40138 | Part of proof of Lemma K o... |
| cdlemk35u 40139 | Substitution version of ~ ... |
| cdlemk55u1 40140 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 40141 | Part of proof of Lemma K o... |
| cdlemk39u1 40142 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 40143 | Part of proof of Lemma K o... |
| cdlemk19u1 40144 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 40145 | Part of Lemma K of [Crawle... |
| cdlemk56 40146 | Part of Lemma K of [Crawle... |
| cdlemk19w 40147 | Use a fixed element to eli... |
| cdlemk56w 40148 | Use a fixed element to eli... |
| cdlemk 40149 | Lemma K of [Crawley] p. 11... |
| tendoex 40150 | Generalization of Lemma K ... |
| cdleml1N 40151 | Part of proof of Lemma L o... |
| cdleml2N 40152 | Part of proof of Lemma L o... |
| cdleml3N 40153 | Part of proof of Lemma L o... |
| cdleml4N 40154 | Part of proof of Lemma L o... |
| cdleml5N 40155 | Part of proof of Lemma L o... |
| cdleml6 40156 | Part of proof of Lemma L o... |
| cdleml7 40157 | Part of proof of Lemma L o... |
| cdleml8 40158 | Part of proof of Lemma L o... |
| cdleml9 40159 | Part of proof of Lemma L o... |
| dva1dim 40160 | Two expressions for the 1-... |
| dvhb1dimN 40161 | Two expressions for the 1-... |
| erng1lem 40162 | Value of the endomorphism ... |
| erngdvlem1 40163 | Lemma for ~ eringring . (... |
| erngdvlem2N 40164 | Lemma for ~ eringring . (... |
| erngdvlem3 40165 | Lemma for ~ eringring . (... |
| erngdvlem4 40166 | Lemma for ~ erngdv . (Con... |
| eringring 40167 | An endomorphism ring is a ... |
| erngdv 40168 | An endomorphism ring is a ... |
| erng0g 40169 | The division ring zero of ... |
| erng1r 40170 | The division ring unity of... |
| erngdvlem1-rN 40171 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 40172 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 40173 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 40174 | Lemma for ~ erngdv . (Con... |
| erngring-rN 40175 | An endomorphism ring is a ... |
| erngdv-rN 40176 | An endomorphism ring is a ... |
| dvafset 40179 | The constructed partial ve... |
| dvaset 40180 | The constructed partial ve... |
| dvasca 40181 | The ring base set of the c... |
| dvabase 40182 | The ring base set of the c... |
| dvafplusg 40183 | Ring addition operation fo... |
| dvaplusg 40184 | Ring addition operation fo... |
| dvaplusgv 40185 | Ring addition operation fo... |
| dvafmulr 40186 | Ring multiplication operat... |
| dvamulr 40187 | Ring multiplication operat... |
| dvavbase 40188 | The vectors (vector base s... |
| dvafvadd 40189 | The vector sum operation f... |
| dvavadd 40190 | Ring addition operation fo... |
| dvafvsca 40191 | Ring addition operation fo... |
| dvavsca 40192 | Ring addition operation fo... |
| tendospcl 40193 | Closure of endomorphism sc... |
| tendospass 40194 | Associative law for endomo... |
| tendospdi1 40195 | Forward distributive law f... |
| tendocnv 40196 | Converse of a trace-preser... |
| tendospdi2 40197 | Reverse distributive law f... |
| tendospcanN 40198 | Cancellation law for trace... |
| dvaabl 40199 | The constructed partial ve... |
| dvalveclem 40200 | Lemma for ~ dvalvec . (Co... |
| dvalvec 40201 | The constructed partial ve... |
| dva0g 40202 | The zero vector of partial... |
| diaffval 40205 | The partial isomorphism A ... |
| diafval 40206 | The partial isomorphism A ... |
| diaval 40207 | The partial isomorphism A ... |
| diaelval 40208 | Member of the partial isom... |
| diafn 40209 | Functionality and domain o... |
| diadm 40210 | Domain of the partial isom... |
| diaeldm 40211 | Member of domain of the pa... |
| diadmclN 40212 | A member of domain of the ... |
| diadmleN 40213 | A member of domain of the ... |
| dian0 40214 | The value of the partial i... |
| dia0eldmN 40215 | The lattice zero belongs t... |
| dia1eldmN 40216 | The fiducial hyperplane (t... |
| diass 40217 | The value of the partial i... |
| diael 40218 | A member of the value of t... |
| diatrl 40219 | Trace of a member of the p... |
| diaelrnN 40220 | Any value of the partial i... |
| dialss 40221 | The value of partial isomo... |
| diaord 40222 | The partial isomorphism A ... |
| dia11N 40223 | The partial isomorphism A ... |
| diaf11N 40224 | The partial isomorphism A ... |
| diaclN 40225 | Closure of partial isomorp... |
| diacnvclN 40226 | Closure of partial isomorp... |
| dia0 40227 | The value of the partial i... |
| dia1N 40228 | The value of the partial i... |
| dia1elN 40229 | The largest subspace in th... |
| diaglbN 40230 | Partial isomorphism A of a... |
| diameetN 40231 | Partial isomorphism A of a... |
| diainN 40232 | Inverse partial isomorphis... |
| diaintclN 40233 | The intersection of partia... |
| diasslssN 40234 | The partial isomorphism A ... |
| diassdvaN 40235 | The partial isomorphism A ... |
| dia1dim 40236 | Two expressions for the 1-... |
| dia1dim2 40237 | Two expressions for a 1-di... |
| dia1dimid 40238 | A vector (translation) bel... |
| dia2dimlem1 40239 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 40240 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 40241 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 40242 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 40243 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 40244 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 40245 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 40246 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 40247 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 40248 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 40249 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 40250 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 40251 | Lemma for ~ dia2dim . Eli... |
| dia2dim 40252 | A two-dimensional subspace... |
| dvhfset 40255 | The constructed full vecto... |
| dvhset 40256 | The constructed full vecto... |
| dvhsca 40257 | The ring of scalars of the... |
| dvhbase 40258 | The ring base set of the c... |
| dvhfplusr 40259 | Ring addition operation fo... |
| dvhfmulr 40260 | Ring multiplication operat... |
| dvhmulr 40261 | Ring multiplication operat... |
| dvhvbase 40262 | The vectors (vector base s... |
| dvhelvbasei 40263 | Vector membership in the c... |
| dvhvaddcbv 40264 | Change bound variables to ... |
| dvhvaddval 40265 | The vector sum operation f... |
| dvhfvadd 40266 | The vector sum operation f... |
| dvhvadd 40267 | The vector sum operation f... |
| dvhopvadd 40268 | The vector sum operation f... |
| dvhopvadd2 40269 | The vector sum operation f... |
| dvhvaddcl 40270 | Closure of the vector sum ... |
| dvhvaddcomN 40271 | Commutativity of vector su... |
| dvhvaddass 40272 | Associativity of vector su... |
| dvhvscacbv 40273 | Change bound variables to ... |
| dvhvscaval 40274 | The scalar product operati... |
| dvhfvsca 40275 | Scalar product operation f... |
| dvhvsca 40276 | Scalar product operation f... |
| dvhopvsca 40277 | Scalar product operation f... |
| dvhvscacl 40278 | Closure of the scalar prod... |
| tendoinvcl 40279 | Closure of multiplicative ... |
| tendolinv 40280 | Left multiplicative invers... |
| tendorinv 40281 | Right multiplicative inver... |
| dvhgrp 40282 | The full vector space ` U ... |
| dvhlveclem 40283 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 40284 | The full vector space ` U ... |
| dvhlmod 40285 | The full vector space ` U ... |
| dvh0g 40286 | The zero vector of vector ... |
| dvheveccl 40287 | Properties of a unit vecto... |
| dvhopclN 40288 | Closure of a ` DVecH ` vec... |
| dvhopaddN 40289 | Sum of ` DVecH ` vectors e... |
| dvhopspN 40290 | Scalar product of ` DVecH ... |
| dvhopN 40291 | Decompose a ` DVecH ` vect... |
| dvhopellsm 40292 | Ordered pair membership in... |
| cdlemm10N 40293 | The image of the map ` G `... |
| docaffvalN 40296 | Subspace orthocomplement f... |
| docafvalN 40297 | Subspace orthocomplement f... |
| docavalN 40298 | Subspace orthocomplement f... |
| docaclN 40299 | Closure of subspace orthoc... |
| diaocN 40300 | Value of partial isomorphi... |
| doca2N 40301 | Double orthocomplement of ... |
| doca3N 40302 | Double orthocomplement of ... |
| dvadiaN 40303 | Any closed subspace is a m... |
| diarnN 40304 | Partial isomorphism A maps... |
| diaf1oN 40305 | The partial isomorphism A ... |
| djaffvalN 40308 | Subspace join for ` DVecA ... |
| djafvalN 40309 | Subspace join for ` DVecA ... |
| djavalN 40310 | Subspace join for ` DVecA ... |
| djaclN 40311 | Closure of subspace join f... |
| djajN 40312 | Transfer lattice join to `... |
| dibffval 40315 | The partial isomorphism B ... |
| dibfval 40316 | The partial isomorphism B ... |
| dibval 40317 | The partial isomorphism B ... |
| dibopelvalN 40318 | Member of the partial isom... |
| dibval2 40319 | Value of the partial isomo... |
| dibopelval2 40320 | Member of the partial isom... |
| dibval3N 40321 | Value of the partial isomo... |
| dibelval3 40322 | Member of the partial isom... |
| dibopelval3 40323 | Member of the partial isom... |
| dibelval1st 40324 | Membership in value of the... |
| dibelval1st1 40325 | Membership in value of the... |
| dibelval1st2N 40326 | Membership in value of the... |
| dibelval2nd 40327 | Membership in value of the... |
| dibn0 40328 | The value of the partial i... |
| dibfna 40329 | Functionality and domain o... |
| dibdiadm 40330 | Domain of the partial isom... |
| dibfnN 40331 | Functionality and domain o... |
| dibdmN 40332 | Domain of the partial isom... |
| dibeldmN 40333 | Member of domain of the pa... |
| dibord 40334 | The isomorphism B for a la... |
| dib11N 40335 | The isomorphism B for a la... |
| dibf11N 40336 | The partial isomorphism A ... |
| dibclN 40337 | Closure of partial isomorp... |
| dibvalrel 40338 | The value of partial isomo... |
| dib0 40339 | The value of partial isomo... |
| dib1dim 40340 | Two expressions for the 1-... |
| dibglbN 40341 | Partial isomorphism B of a... |
| dibintclN 40342 | The intersection of partia... |
| dib1dim2 40343 | Two expressions for a 1-di... |
| dibss 40344 | The partial isomorphism B ... |
| diblss 40345 | The value of partial isomo... |
| diblsmopel 40346 | Membership in subspace sum... |
| dicffval 40349 | The partial isomorphism C ... |
| dicfval 40350 | The partial isomorphism C ... |
| dicval 40351 | The partial isomorphism C ... |
| dicopelval 40352 | Membership in value of the... |
| dicelvalN 40353 | Membership in value of the... |
| dicval2 40354 | The partial isomorphism C ... |
| dicelval3 40355 | Member of the partial isom... |
| dicopelval2 40356 | Membership in value of the... |
| dicelval2N 40357 | Membership in value of the... |
| dicfnN 40358 | Functionality and domain o... |
| dicdmN 40359 | Domain of the partial isom... |
| dicvalrelN 40360 | The value of partial isomo... |
| dicssdvh 40361 | The partial isomorphism C ... |
| dicelval1sta 40362 | Membership in value of the... |
| dicelval1stN 40363 | Membership in value of the... |
| dicelval2nd 40364 | Membership in value of the... |
| dicvaddcl 40365 | Membership in value of the... |
| dicvscacl 40366 | Membership in value of the... |
| dicn0 40367 | The value of the partial i... |
| diclss 40368 | The value of partial isomo... |
| diclspsn 40369 | The value of isomorphism C... |
| cdlemn2 40370 | Part of proof of Lemma N o... |
| cdlemn2a 40371 | Part of proof of Lemma N o... |
| cdlemn3 40372 | Part of proof of Lemma N o... |
| cdlemn4 40373 | Part of proof of Lemma N o... |
| cdlemn4a 40374 | Part of proof of Lemma N o... |
| cdlemn5pre 40375 | Part of proof of Lemma N o... |
| cdlemn5 40376 | Part of proof of Lemma N o... |
| cdlemn6 40377 | Part of proof of Lemma N o... |
| cdlemn7 40378 | Part of proof of Lemma N o... |
| cdlemn8 40379 | Part of proof of Lemma N o... |
| cdlemn9 40380 | Part of proof of Lemma N o... |
| cdlemn10 40381 | Part of proof of Lemma N o... |
| cdlemn11a 40382 | Part of proof of Lemma N o... |
| cdlemn11b 40383 | Part of proof of Lemma N o... |
| cdlemn11c 40384 | Part of proof of Lemma N o... |
| cdlemn11pre 40385 | Part of proof of Lemma N o... |
| cdlemn11 40386 | Part of proof of Lemma N o... |
| cdlemn 40387 | Lemma N of [Crawley] p. 12... |
| dihordlem6 40388 | Part of proof of Lemma N o... |
| dihordlem7 40389 | Part of proof of Lemma N o... |
| dihordlem7b 40390 | Part of proof of Lemma N o... |
| dihjustlem 40391 | Part of proof after Lemma ... |
| dihjust 40392 | Part of proof after Lemma ... |
| dihord1 40393 | Part of proof after Lemma ... |
| dihord2a 40394 | Part of proof after Lemma ... |
| dihord2b 40395 | Part of proof after Lemma ... |
| dihord2cN 40396 | Part of proof after Lemma ... |
| dihord11b 40397 | Part of proof after Lemma ... |
| dihord10 40398 | Part of proof after Lemma ... |
| dihord11c 40399 | Part of proof after Lemma ... |
| dihord2pre 40400 | Part of proof after Lemma ... |
| dihord2pre2 40401 | Part of proof after Lemma ... |
| dihord2 40402 | Part of proof after Lemma ... |
| dihffval 40405 | The isomorphism H for a la... |
| dihfval 40406 | Isomorphism H for a lattic... |
| dihval 40407 | Value of isomorphism H for... |
| dihvalc 40408 | Value of isomorphism H for... |
| dihlsscpre 40409 | Closure of isomorphism H f... |
| dihvalcqpre 40410 | Value of isomorphism H for... |
| dihvalcq 40411 | Value of isomorphism H for... |
| dihvalb 40412 | Value of isomorphism H for... |
| dihopelvalbN 40413 | Ordered pair member of the... |
| dihvalcqat 40414 | Value of isomorphism H for... |
| dih1dimb 40415 | Two expressions for a 1-di... |
| dih1dimb2 40416 | Isomorphism H at an atom u... |
| dih1dimc 40417 | Isomorphism H at an atom n... |
| dib2dim 40418 | Extend ~ dia2dim to partia... |
| dih2dimb 40419 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 40420 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 40421 | Ordered pair member of the... |
| dihvalcq2 40422 | Value of isomorphism H for... |
| dihopelvalcpre 40423 | Member of value of isomorp... |
| dihopelvalc 40424 | Member of value of isomorp... |
| dihlss 40425 | The value of isomorphism H... |
| dihss 40426 | The value of isomorphism H... |
| dihssxp 40427 | An isomorphism H value is ... |
| dihopcl 40428 | Closure of an ordered pair... |
| xihopellsmN 40429 | Ordered pair membership in... |
| dihopellsm 40430 | Ordered pair membership in... |
| dihord6apre 40431 | Part of proof that isomorp... |
| dihord3 40432 | The isomorphism H for a la... |
| dihord4 40433 | The isomorphism H for a la... |
| dihord5b 40434 | Part of proof that isomorp... |
| dihord6b 40435 | Part of proof that isomorp... |
| dihord6a 40436 | Part of proof that isomorp... |
| dihord5apre 40437 | Part of proof that isomorp... |
| dihord5a 40438 | Part of proof that isomorp... |
| dihord 40439 | The isomorphism H is order... |
| dih11 40440 | The isomorphism H is one-t... |
| dihf11lem 40441 | Functionality of the isomo... |
| dihf11 40442 | The isomorphism H for a la... |
| dihfn 40443 | Functionality and domain o... |
| dihdm 40444 | Domain of isomorphism H. (... |
| dihcl 40445 | Closure of isomorphism H. ... |
| dihcnvcl 40446 | Closure of isomorphism H c... |
| dihcnvid1 40447 | The converse isomorphism o... |
| dihcnvid2 40448 | The isomorphism of a conve... |
| dihcnvord 40449 | Ordering property for conv... |
| dihcnv11 40450 | The converse of isomorphis... |
| dihsslss 40451 | The isomorphism H maps to ... |
| dihrnlss 40452 | The isomorphism H maps to ... |
| dihrnss 40453 | The isomorphism H maps to ... |
| dihvalrel 40454 | The value of isomorphism H... |
| dih0 40455 | The value of isomorphism H... |
| dih0bN 40456 | A lattice element is zero ... |
| dih0vbN 40457 | A vector is zero iff its s... |
| dih0cnv 40458 | The isomorphism H converse... |
| dih0rn 40459 | The zero subspace belongs ... |
| dih0sb 40460 | A subspace is zero iff the... |
| dih1 40461 | The value of isomorphism H... |
| dih1rn 40462 | The full vector space belo... |
| dih1cnv 40463 | The isomorphism H converse... |
| dihwN 40464 | Value of isomorphism H at ... |
| dihmeetlem1N 40465 | Isomorphism H of a conjunc... |
| dihglblem5apreN 40466 | A conjunction property of ... |
| dihglblem5aN 40467 | A conjunction property of ... |
| dihglblem2aN 40468 | Lemma for isomorphism H of... |
| dihglblem2N 40469 | The GLB of a set of lattic... |
| dihglblem3N 40470 | Isomorphism H of a lattice... |
| dihglblem3aN 40471 | Isomorphism H of a lattice... |
| dihglblem4 40472 | Isomorphism H of a lattice... |
| dihglblem5 40473 | Isomorphism H of a lattice... |
| dihmeetlem2N 40474 | Isomorphism H of a conjunc... |
| dihglbcpreN 40475 | Isomorphism H of a lattice... |
| dihglbcN 40476 | Isomorphism H of a lattice... |
| dihmeetcN 40477 | Isomorphism H of a lattice... |
| dihmeetbN 40478 | Isomorphism H of a lattice... |
| dihmeetbclemN 40479 | Lemma for isomorphism H of... |
| dihmeetlem3N 40480 | Lemma for isomorphism H of... |
| dihmeetlem4preN 40481 | Lemma for isomorphism H of... |
| dihmeetlem4N 40482 | Lemma for isomorphism H of... |
| dihmeetlem5 40483 | Part of proof that isomorp... |
| dihmeetlem6 40484 | Lemma for isomorphism H of... |
| dihmeetlem7N 40485 | Lemma for isomorphism H of... |
| dihjatc1 40486 | Lemma for isomorphism H of... |
| dihjatc2N 40487 | Isomorphism H of join with... |
| dihjatc3 40488 | Isomorphism H of join with... |
| dihmeetlem8N 40489 | Lemma for isomorphism H of... |
| dihmeetlem9N 40490 | Lemma for isomorphism H of... |
| dihmeetlem10N 40491 | Lemma for isomorphism H of... |
| dihmeetlem11N 40492 | Lemma for isomorphism H of... |
| dihmeetlem12N 40493 | Lemma for isomorphism H of... |
| dihmeetlem13N 40494 | Lemma for isomorphism H of... |
| dihmeetlem14N 40495 | Lemma for isomorphism H of... |
| dihmeetlem15N 40496 | Lemma for isomorphism H of... |
| dihmeetlem16N 40497 | Lemma for isomorphism H of... |
| dihmeetlem17N 40498 | Lemma for isomorphism H of... |
| dihmeetlem18N 40499 | Lemma for isomorphism H of... |
| dihmeetlem19N 40500 | Lemma for isomorphism H of... |
| dihmeetlem20N 40501 | Lemma for isomorphism H of... |
| dihmeetALTN 40502 | Isomorphism H of a lattice... |
| dih1dimatlem0 40503 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 40504 | Lemma for ~ dih1dimat . (... |
| dih1dimat 40505 | Any 1-dimensional subspace... |
| dihlsprn 40506 | The span of a vector belon... |
| dihlspsnssN 40507 | A subspace included in a 1... |
| dihlspsnat 40508 | The inverse isomorphism H ... |
| dihatlat 40509 | The isomorphism H of an at... |
| dihat 40510 | There exists at least one ... |
| dihpN 40511 | The value of isomorphism H... |
| dihlatat 40512 | The reverse isomorphism H ... |
| dihatexv 40513 | There is a nonzero vector ... |
| dihatexv2 40514 | There is a nonzero vector ... |
| dihglblem6 40515 | Isomorphism H of a lattice... |
| dihglb 40516 | Isomorphism H of a lattice... |
| dihglb2 40517 | Isomorphism H of a lattice... |
| dihmeet 40518 | Isomorphism H of a lattice... |
| dihintcl 40519 | The intersection of closed... |
| dihmeetcl 40520 | Closure of closed subspace... |
| dihmeet2 40521 | Reverse isomorphism H of a... |
| dochffval 40524 | Subspace orthocomplement f... |
| dochfval 40525 | Subspace orthocomplement f... |
| dochval 40526 | Subspace orthocomplement f... |
| dochval2 40527 | Subspace orthocomplement f... |
| dochcl 40528 | Closure of subspace orthoc... |
| dochlss 40529 | A subspace orthocomplement... |
| dochssv 40530 | A subspace orthocomplement... |
| dochfN 40531 | Domain and codomain of the... |
| dochvalr 40532 | Orthocomplement of a close... |
| doch0 40533 | Orthocomplement of the zer... |
| doch1 40534 | Orthocomplement of the uni... |
| dochoc0 40535 | The zero subspace is close... |
| dochoc1 40536 | The unit subspace (all vec... |
| dochvalr2 40537 | Orthocomplement of a close... |
| dochvalr3 40538 | Orthocomplement of a close... |
| doch2val2 40539 | Double orthocomplement for... |
| dochss 40540 | Subset law for orthocomple... |
| dochocss 40541 | Double negative law for or... |
| dochoc 40542 | Double negative law for or... |
| dochsscl 40543 | If a set of vectors is inc... |
| dochoccl 40544 | A set of vectors is closed... |
| dochord 40545 | Ordering law for orthocomp... |
| dochord2N 40546 | Ordering law for orthocomp... |
| dochord3 40547 | Ordering law for orthocomp... |
| doch11 40548 | Orthocomplement is one-to-... |
| dochsordN 40549 | Strict ordering law for or... |
| dochn0nv 40550 | An orthocomplement is nonz... |
| dihoml4c 40551 | Version of ~ dihoml4 with ... |
| dihoml4 40552 | Orthomodular law for const... |
| dochspss 40553 | The span of a set of vecto... |
| dochocsp 40554 | The span of an orthocomple... |
| dochspocN 40555 | The span of an orthocomple... |
| dochocsn 40556 | The double orthocomplement... |
| dochsncom 40557 | Swap vectors in an orthoco... |
| dochsat 40558 | The double orthocomplement... |
| dochshpncl 40559 | If a hyperplane is not clo... |
| dochlkr 40560 | Equivalent conditions for ... |
| dochkrshp 40561 | The closure of a kernel is... |
| dochkrshp2 40562 | Properties of the closure ... |
| dochkrshp3 40563 | Properties of the closure ... |
| dochkrshp4 40564 | Properties of the closure ... |
| dochdmj1 40565 | De Morgan-like law for sub... |
| dochnoncon 40566 | Law of noncontradiction. ... |
| dochnel2 40567 | A nonzero member of a subs... |
| dochnel 40568 | A nonzero vector doesn't b... |
| djhffval 40571 | Subspace join for ` DVecH ... |
| djhfval 40572 | Subspace join for ` DVecH ... |
| djhval 40573 | Subspace join for ` DVecH ... |
| djhval2 40574 | Value of subspace join for... |
| djhcl 40575 | Closure of subspace join f... |
| djhlj 40576 | Transfer lattice join to `... |
| djhljjN 40577 | Lattice join in terms of `... |
| djhjlj 40578 | ` DVecH ` vector space clo... |
| djhj 40579 | ` DVecH ` vector space clo... |
| djhcom 40580 | Subspace join commutes. (... |
| djhspss 40581 | Subspace span of union is ... |
| djhsumss 40582 | Subspace sum is a subset o... |
| dihsumssj 40583 | The subspace sum of two is... |
| djhunssN 40584 | Subspace union is a subset... |
| dochdmm1 40585 | De Morgan-like law for clo... |
| djhexmid 40586 | Excluded middle property o... |
| djh01 40587 | Closed subspace join with ... |
| djh02 40588 | Closed subspace join with ... |
| djhlsmcl 40589 | A closed subspace sum equa... |
| djhcvat42 40590 | A covering property. ( ~ ... |
| dihjatb 40591 | Isomorphism H of lattice j... |
| dihjatc 40592 | Isomorphism H of lattice j... |
| dihjatcclem1 40593 | Lemma for isomorphism H of... |
| dihjatcclem2 40594 | Lemma for isomorphism H of... |
| dihjatcclem3 40595 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 40596 | Lemma for isomorphism H of... |
| dihjatcc 40597 | Isomorphism H of lattice j... |
| dihjat 40598 | Isomorphism H of lattice j... |
| dihprrnlem1N 40599 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 40600 | Lemma for ~ dihprrn . (Co... |
| dihprrn 40601 | The span of a vector pair ... |
| djhlsmat 40602 | The sum of two subspace at... |
| dihjat1lem 40603 | Subspace sum of a closed s... |
| dihjat1 40604 | Subspace sum of a closed s... |
| dihsmsprn 40605 | Subspace sum of a closed s... |
| dihjat2 40606 | The subspace sum of a clos... |
| dihjat3 40607 | Isomorphism H of lattice j... |
| dihjat4 40608 | Transfer the subspace sum ... |
| dihjat6 40609 | Transfer the subspace sum ... |
| dihsmsnrn 40610 | The subspace sum of two si... |
| dihsmatrn 40611 | The subspace sum of a clos... |
| dihjat5N 40612 | Transfer lattice join with... |
| dvh4dimat 40613 | There is an atom that is o... |
| dvh3dimatN 40614 | There is an atom that is o... |
| dvh2dimatN 40615 | Given an atom, there exist... |
| dvh1dimat 40616 | There exists an atom. (Co... |
| dvh1dim 40617 | There exists a nonzero vec... |
| dvh4dimlem 40618 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 40619 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 40620 | There is a vector that is ... |
| dvh3dim 40621 | There is a vector that is ... |
| dvh4dimN 40622 | There is a vector that is ... |
| dvh3dim2 40623 | There is a vector that is ... |
| dvh3dim3N 40624 | There is a vector that is ... |
| dochsnnz 40625 | The orthocomplement of a s... |
| dochsatshp 40626 | The orthocomplement of a s... |
| dochsatshpb 40627 | The orthocomplement of a s... |
| dochsnshp 40628 | The orthocomplement of a n... |
| dochshpsat 40629 | A hyperplane is closed iff... |
| dochkrsat 40630 | The orthocomplement of a k... |
| dochkrsat2 40631 | The orthocomplement of a k... |
| dochsat0 40632 | The orthocomplement of a k... |
| dochkrsm 40633 | The subspace sum of a clos... |
| dochexmidat 40634 | Special case of excluded m... |
| dochexmidlem1 40635 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 40636 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 40637 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 40638 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 40639 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 40640 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 40641 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 40642 | Lemma for ~ dochexmid . T... |
| dochexmid 40643 | Excluded middle law for cl... |
| dochsnkrlem1 40644 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 40645 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 40646 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 40647 | A (closed) kernel expresse... |
| dochsnkr2 40648 | Kernel of the explicit fun... |
| dochsnkr2cl 40649 | The ` X ` determining func... |
| dochflcl 40650 | Closure of the explicit fu... |
| dochfl1 40651 | The value of the explicit ... |
| dochfln0 40652 | The value of a functional ... |
| dochkr1 40653 | A nonzero functional has a... |
| dochkr1OLDN 40654 | A nonzero functional has a... |
| lpolsetN 40657 | The set of polarities of a... |
| islpolN 40658 | The predicate "is a polari... |
| islpoldN 40659 | Properties that determine ... |
| lpolfN 40660 | Functionality of a polarit... |
| lpolvN 40661 | The polarity of the whole ... |
| lpolconN 40662 | Contraposition property of... |
| lpolsatN 40663 | The polarity of an atomic ... |
| lpolpolsatN 40664 | Property of a polarity. (... |
| dochpolN 40665 | The subspace orthocompleme... |
| lcfl1lem 40666 | Property of a functional w... |
| lcfl1 40667 | Property of a functional w... |
| lcfl2 40668 | Property of a functional w... |
| lcfl3 40669 | Property of a functional w... |
| lcfl4N 40670 | Property of a functional w... |
| lcfl5 40671 | Property of a functional w... |
| lcfl5a 40672 | Property of a functional w... |
| lcfl6lem 40673 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 40674 | Lemma for ~ lcfl7N . If t... |
| lcfl6 40675 | Property of a functional w... |
| lcfl7N 40676 | Property of a functional w... |
| lcfl8 40677 | Property of a functional w... |
| lcfl8a 40678 | Property of a functional w... |
| lcfl8b 40679 | Property of a nonzero func... |
| lcfl9a 40680 | Property implying that a f... |
| lclkrlem1 40681 | The set of functionals hav... |
| lclkrlem2a 40682 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 40683 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 40684 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 40685 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 40686 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 40687 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 40688 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 40689 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 40690 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 40691 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 40692 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 40693 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 40694 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 40695 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 40696 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 40697 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 40698 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 40699 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 40700 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 40701 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 40702 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 40703 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 40704 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 40705 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 40706 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 40707 | The set of functionals hav... |
| lclkr 40708 | The set of functionals wit... |
| lcfls1lem 40709 | Property of a functional w... |
| lcfls1N 40710 | Property of a functional w... |
| lcfls1c 40711 | Property of a functional w... |
| lclkrslem1 40712 | The set of functionals hav... |
| lclkrslem2 40713 | The set of functionals hav... |
| lclkrs 40714 | The set of functionals hav... |
| lclkrs2 40715 | The set of functionals wit... |
| lcfrvalsnN 40716 | Reconstruction from the du... |
| lcfrlem1 40717 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 40718 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 40719 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 40720 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 40721 | Lemma for ~ lcfr . The se... |
| lcfrlem6 40722 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 40723 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 40724 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 40725 | Lemma for ~ lcf1o . (This... |
| lcf1o 40726 | Define a function ` J ` th... |
| lcfrlem10 40727 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 40728 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 40729 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 40730 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 40731 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 40732 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 40733 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 40734 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 40735 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 40736 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 40737 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 40738 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 40739 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 40740 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 40741 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 40742 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 40743 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 40744 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 40745 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 40746 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 40747 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 40748 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 40749 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 40750 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 40751 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 40752 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 40753 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 40754 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 40755 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 40756 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 40757 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 40758 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 40759 | Lemma for ~ lcfr . Elimin... |
| lcfr 40760 | Reconstruction of a subspa... |
| lcdfval 40763 | Dual vector space of funct... |
| lcdval 40764 | Dual vector space of funct... |
| lcdval2 40765 | Dual vector space of funct... |
| lcdlvec 40766 | The dual vector space of f... |
| lcdlmod 40767 | The dual vector space of f... |
| lcdvbase 40768 | Vector base set of a dual ... |
| lcdvbasess 40769 | The vector base set of the... |
| lcdvbaselfl 40770 | A vector in the base set o... |
| lcdvbasecl 40771 | Closure of the value of a ... |
| lcdvadd 40772 | Vector addition for the cl... |
| lcdvaddval 40773 | The value of the value of ... |
| lcdsca 40774 | The ring of scalars of the... |
| lcdsbase 40775 | Base set of scalar ring fo... |
| lcdsadd 40776 | Scalar addition for the cl... |
| lcdsmul 40777 | Scalar multiplication for ... |
| lcdvs 40778 | Scalar product for the clo... |
| lcdvsval 40779 | Value of scalar product op... |
| lcdvscl 40780 | The scalar product operati... |
| lcdlssvscl 40781 | Closure of scalar product ... |
| lcdvsass 40782 | Associative law for scalar... |
| lcd0 40783 | The zero scalar of the clo... |
| lcd1 40784 | The unit scalar of the clo... |
| lcdneg 40785 | The unit scalar of the clo... |
| lcd0v 40786 | The zero functional in the... |
| lcd0v2 40787 | The zero functional in the... |
| lcd0vvalN 40788 | Value of the zero function... |
| lcd0vcl 40789 | Closure of the zero functi... |
| lcd0vs 40790 | A scalar zero times a func... |
| lcdvs0N 40791 | A scalar times the zero fu... |
| lcdvsub 40792 | The value of vector subtra... |
| lcdvsubval 40793 | The value of the value of ... |
| lcdlss 40794 | Subspaces of a dual vector... |
| lcdlss2N 40795 | Subspaces of a dual vector... |
| lcdlsp 40796 | Span in the set of functio... |
| lcdlkreqN 40797 | Colinear functionals have ... |
| lcdlkreq2N 40798 | Colinear functionals have ... |
| mapdffval 40801 | Projectivity from vector s... |
| mapdfval 40802 | Projectivity from vector s... |
| mapdval 40803 | Value of projectivity from... |
| mapdvalc 40804 | Value of projectivity from... |
| mapdval2N 40805 | Value of projectivity from... |
| mapdval3N 40806 | Value of projectivity from... |
| mapdval4N 40807 | Value of projectivity from... |
| mapdval5N 40808 | Value of projectivity from... |
| mapdordlem1a 40809 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 40810 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 40811 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 40812 | Lemma for ~ mapdord . Ord... |
| mapdord 40813 | Ordering property of the m... |
| mapd11 40814 | The map defined by ~ df-ma... |
| mapddlssN 40815 | The mapping of a subspace ... |
| mapdsn 40816 | Value of the map defined b... |
| mapdsn2 40817 | Value of the map defined b... |
| mapdsn3 40818 | Value of the map defined b... |
| mapd1dim2lem1N 40819 | Value of the map defined b... |
| mapdrvallem2 40820 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 40821 | Lemma for ~ mapdrval . (C... |
| mapdrval 40822 | Given a dual subspace ` R ... |
| mapd1o 40823 | The map defined by ~ df-ma... |
| mapdrn 40824 | Range of the map defined b... |
| mapdunirnN 40825 | Union of the range of the ... |
| mapdrn2 40826 | Range of the map defined b... |
| mapdcnvcl 40827 | Closure of the converse of... |
| mapdcl 40828 | Closure the value of the m... |
| mapdcnvid1N 40829 | Converse of the value of t... |
| mapdsord 40830 | Strong ordering property o... |
| mapdcl2 40831 | The mapping of a subspace ... |
| mapdcnvid2 40832 | Value of the converse of t... |
| mapdcnvordN 40833 | Ordering property of the c... |
| mapdcnv11N 40834 | The converse of the map de... |
| mapdcv 40835 | Covering property of the c... |
| mapdincl 40836 | Closure of dual subspace i... |
| mapdin 40837 | Subspace intersection is p... |
| mapdlsmcl 40838 | Closure of dual subspace s... |
| mapdlsm 40839 | Subspace sum is preserved ... |
| mapd0 40840 | Projectivity map of the ze... |
| mapdcnvatN 40841 | Atoms are preserved by the... |
| mapdat 40842 | Atoms are preserved by the... |
| mapdspex 40843 | The map of a span equals t... |
| mapdn0 40844 | Transfer nonzero property ... |
| mapdncol 40845 | Transfer non-colinearity f... |
| mapdindp 40846 | Transfer (part of) vector ... |
| mapdpglem1 40847 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 40848 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 40849 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 40850 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 40851 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 40852 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 40853 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 40854 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 40855 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 40856 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 40857 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 40858 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 40859 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 40860 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 40861 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 40862 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 40863 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 40864 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 40865 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 40866 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 40867 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 40868 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 40869 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 40870 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 40871 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 40872 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 40873 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 40874 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 40875 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 40876 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 40877 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 40878 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 40879 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 40880 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 40881 | Part 1 of proof of the fir... |
| baerlem3lem1 40882 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 40883 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 40884 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 40885 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 40886 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 40887 | Lemma for ~ baerlem5b . (... |
| baerlem3 40888 | An equality that holds whe... |
| baerlem5a 40889 | An equality that holds whe... |
| baerlem5b 40890 | An equality that holds whe... |
| baerlem5amN 40891 | An equality that holds whe... |
| baerlem5bmN 40892 | An equality that holds whe... |
| baerlem5abmN 40893 | An equality that holds whe... |
| mapdindp0 40894 | Vector independence lemma.... |
| mapdindp1 40895 | Vector independence lemma.... |
| mapdindp2 40896 | Vector independence lemma.... |
| mapdindp3 40897 | Vector independence lemma.... |
| mapdindp4 40898 | Vector independence lemma.... |
| mapdhval 40899 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 40900 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 40901 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 40902 | Lemmma for ~~? mapdh . (C... |
| mapdheq 40903 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 40904 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 40905 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 40906 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 40907 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 40908 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 40909 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 40910 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 40911 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 40912 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 40913 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 40914 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 40915 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 40916 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 40917 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 40918 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 40919 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 40920 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 40921 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 40922 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 40923 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 40924 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 40925 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 40926 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 40927 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 40928 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 40929 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 40930 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 40933 | Map from nonzero vectors t... |
| hvmapfval 40934 | Map from nonzero vectors t... |
| hvmapval 40935 | Value of map from nonzero ... |
| hvmapvalvalN 40936 | Value of value of map (i.e... |
| hvmapidN 40937 | The value of the vector to... |
| hvmap1o 40938 | The vector to functional m... |
| hvmapclN 40939 | Closure of the vector to f... |
| hvmap1o2 40940 | The vector to functional m... |
| hvmapcl2 40941 | Closure of the vector to f... |
| hvmaplfl 40942 | The vector to functional m... |
| hvmaplkr 40943 | Kernel of the vector to fu... |
| mapdhvmap 40944 | Relationship between ` map... |
| lspindp5 40945 | Obtain an independent vect... |
| hdmaplem1 40946 | Lemma to convert a frequen... |
| hdmaplem2N 40947 | Lemma to convert a frequen... |
| hdmaplem3 40948 | Lemma to convert a frequen... |
| hdmaplem4 40949 | Lemma to convert a frequen... |
| mapdh8a 40950 | Part of Part (8) in [Baer]... |
| mapdh8aa 40951 | Part of Part (8) in [Baer]... |
| mapdh8ab 40952 | Part of Part (8) in [Baer]... |
| mapdh8ac 40953 | Part of Part (8) in [Baer]... |
| mapdh8ad 40954 | Part of Part (8) in [Baer]... |
| mapdh8b 40955 | Part of Part (8) in [Baer]... |
| mapdh8c 40956 | Part of Part (8) in [Baer]... |
| mapdh8d0N 40957 | Part of Part (8) in [Baer]... |
| mapdh8d 40958 | Part of Part (8) in [Baer]... |
| mapdh8e 40959 | Part of Part (8) in [Baer]... |
| mapdh8g 40960 | Part of Part (8) in [Baer]... |
| mapdh8i 40961 | Part of Part (8) in [Baer]... |
| mapdh8j 40962 | Part of Part (8) in [Baer]... |
| mapdh8 40963 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 40964 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 40965 | Lemma for part (9) in [Bae... |
| hdmap1ffval 40970 | Preliminary map from vecto... |
| hdmap1fval 40971 | Preliminary map from vecto... |
| hdmap1vallem 40972 | Value of preliminary map f... |
| hdmap1val 40973 | Value of preliminary map f... |
| hdmap1val0 40974 | Value of preliminary map f... |
| hdmap1val2 40975 | Value of preliminary map f... |
| hdmap1eq 40976 | The defining equation for ... |
| hdmap1cbv 40977 | Frequently used lemma to c... |
| hdmap1valc 40978 | Connect the value of the p... |
| hdmap1cl 40979 | Convert closure theorem ~ ... |
| hdmap1eq2 40980 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 40981 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 40982 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 40983 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 40984 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 40985 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 40986 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 40987 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 40988 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 40989 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 40990 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 40991 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 40992 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 40993 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 40994 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 40995 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 40996 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 40997 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 40998 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 40999 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 41000 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 41001 | Map from vectors to functi... |
| hdmapfval 41002 | Map from vectors to functi... |
| hdmapval 41003 | Value of map from vectors ... |
| hdmapfnN 41004 | Functionality of map from ... |
| hdmapcl 41005 | Closure of map from vector... |
| hdmapval2lem 41006 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 41007 | Value of map from vectors ... |
| hdmapval0 41008 | Value of map from vectors ... |
| hdmapeveclem 41009 | Lemma for ~ hdmapevec . T... |
| hdmapevec 41010 | Value of map from vectors ... |
| hdmapevec2 41011 | The inner product of the r... |
| hdmapval3lemN 41012 | Value of map from vectors ... |
| hdmapval3N 41013 | Value of map from vectors ... |
| hdmap10lem 41014 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 41015 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 41016 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 41017 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 41018 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 41019 | Part of proof of part 12 i... |
| hdmapnzcl 41020 | Nonzero vector closure of ... |
| hdmapneg 41021 | Part of proof of part 12 i... |
| hdmapsub 41022 | Part of proof of part 12 i... |
| hdmap11 41023 | Part of proof of part 12 i... |
| hdmaprnlem1N 41024 | Part of proof of part 12 i... |
| hdmaprnlem3N 41025 | Part of proof of part 12 i... |
| hdmaprnlem3uN 41026 | Part of proof of part 12 i... |
| hdmaprnlem4tN 41027 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 41028 | Part of proof of part 12 i... |
| hdmaprnlem6N 41029 | Part of proof of part 12 i... |
| hdmaprnlem7N 41030 | Part of proof of part 12 i... |
| hdmaprnlem8N 41031 | Part of proof of part 12 i... |
| hdmaprnlem9N 41032 | Part of proof of part 12 i... |
| hdmaprnlem3eN 41033 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 41034 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 41035 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 41036 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 41037 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 41038 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 41039 | Part of proof of part 12 i... |
| hdmapf1oN 41040 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 41041 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 41042 | Prior to part 14 in [Baer]... |
| hdmap14lem1 41043 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 41044 | Prior to part 14 in [Baer]... |
| hdmap14lem3 41045 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 41046 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 41047 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 41048 | Case where ` F ` is zero. ... |
| hdmap14lem7 41049 | Combine cases of ` F ` . ... |
| hdmap14lem8 41050 | Part of proof of part 14 i... |
| hdmap14lem9 41051 | Part of proof of part 14 i... |
| hdmap14lem10 41052 | Part of proof of part 14 i... |
| hdmap14lem11 41053 | Part of proof of part 14 i... |
| hdmap14lem12 41054 | Lemma for proof of part 14... |
| hdmap14lem13 41055 | Lemma for proof of part 14... |
| hdmap14lem14 41056 | Part of proof of part 14 i... |
| hdmap14lem15 41057 | Part of proof of part 14 i... |
| hgmapffval 41060 | Map from the scalar divisi... |
| hgmapfval 41061 | Map from the scalar divisi... |
| hgmapval 41062 | Value of map from the scal... |
| hgmapfnN 41063 | Functionality of scalar si... |
| hgmapcl 41064 | Closure of scalar sigma ma... |
| hgmapdcl 41065 | Closure of the vector spac... |
| hgmapvs 41066 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 41067 | Value of the scalar sigma ... |
| hgmapval1 41068 | Value of the scalar sigma ... |
| hgmapadd 41069 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 41070 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 41071 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 41072 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 41073 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 41074 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 41075 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 41076 | Part of proof of part 16 i... |
| hgmap11 41077 | The scalar sigma map is on... |
| hgmapf1oN 41078 | The scalar sigma map is a ... |
| hgmapeq0 41079 | The scalar sigma map is ze... |
| hdmapipcl 41080 | The inner product (Hermiti... |
| hdmapln1 41081 | Linearity property that wi... |
| hdmaplna1 41082 | Additive property of first... |
| hdmaplns1 41083 | Subtraction property of fi... |
| hdmaplnm1 41084 | Multiplicative property of... |
| hdmaplna2 41085 | Additive property of secon... |
| hdmapglnm2 41086 | g-linear property of secon... |
| hdmapgln2 41087 | g-linear property that wil... |
| hdmaplkr 41088 | Kernel of the vector to du... |
| hdmapellkr 41089 | Membership in the kernel (... |
| hdmapip0 41090 | Zero property that will be... |
| hdmapip1 41091 | Construct a proportional v... |
| hdmapip0com 41092 | Commutation property of Ba... |
| hdmapinvlem1 41093 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 41094 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 41095 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 41096 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 41097 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 41098 | Involution property of sca... |
| hgmapvvlem2 41099 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 41100 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 41101 | Value of a double involuti... |
| hdmapglem7a 41102 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 41103 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 41104 | Lemma for ~ hdmapg . Line... |
| hdmapg 41105 | Apply the scalar sigma fun... |
| hdmapoc 41106 | Express our constructed or... |
| hlhilset 41109 | The final Hilbert space co... |
| hlhilsca 41110 | The scalar of the final co... |
| hlhilbase 41111 | The base set of the final ... |
| hlhilplus 41112 | The vector addition for th... |
| hlhilslem 41113 | Lemma for ~ hlhilsbase etc... |
| hlhilslemOLD 41114 | Obsolete version of ~ hlhi... |
| hlhilsbase 41115 | The scalar base set of the... |
| hlhilsbaseOLD 41116 | Obsolete version of ~ hlhi... |
| hlhilsplus 41117 | Scalar addition for the fi... |
| hlhilsplusOLD 41118 | Obsolete version of ~ hlhi... |
| hlhilsmul 41119 | Scalar multiplication for ... |
| hlhilsmulOLD 41120 | Obsolete version of ~ hlhi... |
| hlhilsbase2 41121 | The scalar base set of the... |
| hlhilsplus2 41122 | Scalar addition for the fi... |
| hlhilsmul2 41123 | Scalar multiplication for ... |
| hlhils0 41124 | The scalar ring zero for t... |
| hlhils1N 41125 | The scalar ring unity for ... |
| hlhilvsca 41126 | The scalar product for the... |
| hlhilip 41127 | Inner product operation fo... |
| hlhilipval 41128 | Value of inner product ope... |
| hlhilnvl 41129 | The involution operation o... |
| hlhillvec 41130 | The final constructed Hilb... |
| hlhildrng 41131 | The star division ring for... |
| hlhilsrnglem 41132 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 41133 | The star division ring for... |
| hlhil0 41134 | The zero vector for the fi... |
| hlhillsm 41135 | The vector sum operation f... |
| hlhilocv 41136 | The orthocomplement for th... |
| hlhillcs 41137 | The closed subspaces of th... |
| hlhilphllem 41138 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 41139 | Lemma for ~ hlhil . (Cont... |
| hlathil 41140 | Construction of a Hilbert ... |
| iscsrg 41143 | A commutative semiring is ... |
| leexp1ad 41144 | Weak base ordering relatio... |
| relogbcld 41145 | Closure of the general log... |
| relogbexpd 41146 | Identity law for general l... |
| relogbzexpd 41147 | Power law for the general ... |
| logblebd 41148 | The general logarithm is m... |
| uzindd 41149 | Induction on the upper int... |
| fzadd2d 41150 | Membership of a sum in a f... |
| zltlem1d 41151 | Integer ordering relation,... |
| zltp1led 41152 | Integer ordering relation,... |
| fzne2d 41153 | Elementhood in a finite se... |
| eqfnfv2d2 41154 | Equality of functions is d... |
| fzsplitnd 41155 | Split a finite interval of... |
| fzsplitnr 41156 | Split a finite interval of... |
| addassnni 41157 | Associative law for additi... |
| addcomnni 41158 | Commutative law for additi... |
| mulassnni 41159 | Associative law for multip... |
| mulcomnni 41160 | Commutative law for multip... |
| gcdcomnni 41161 | Commutative law for gcd. ... |
| gcdnegnni 41162 | Negation invariance for gc... |
| neggcdnni 41163 | Negation invariance for gc... |
| bccl2d 41164 | Closure of the binomial co... |
| recbothd 41165 | Take reciprocal on both si... |
| gcdmultiplei 41166 | The GCD of a multiple of a... |
| gcdaddmzz2nni 41167 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 41168 | Adding a multiple of one o... |
| gcdnncli 41169 | Closure of the gcd operato... |
| muldvds1d 41170 | If a product divides an in... |
| muldvds2d 41171 | If a product divides an in... |
| nndivdvdsd 41172 | A positive integer divides... |
| nnproddivdvdsd 41173 | A product of natural numbe... |
| coprmdvds2d 41174 | If an integer is divisible... |
| 12gcd5e1 41175 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 41176 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 41177 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 41178 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 41179 | Calculate the least common... |
| 12lcm5e60 41180 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 41181 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 41182 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 41183 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 41184 | Useful equation to calcula... |
| lcm1un 41185 | Least common multiple of n... |
| lcm2un 41186 | Least common multiple of n... |
| lcm3un 41187 | Least common multiple of n... |
| lcm4un 41188 | Least common multiple of n... |
| lcm5un 41189 | Least common multiple of n... |
| lcm6un 41190 | Least common multiple of n... |
| lcm7un 41191 | Least common multiple of n... |
| lcm8un 41192 | Least common multiple of n... |
| 3factsumint1 41193 | Move constants out of inte... |
| 3factsumint2 41194 | Move constants out of inte... |
| 3factsumint3 41195 | Move constants out of inte... |
| 3factsumint4 41196 | Move constants out of inte... |
| 3factsumint 41197 | Helpful equation for lcm i... |
| resopunitintvd 41198 | Restrict continuous functi... |
| resclunitintvd 41199 | Restrict continuous functi... |
| resdvopclptsd 41200 | Restrict derivative on uni... |
| lcmineqlem1 41201 | Part of lcm inequality lem... |
| lcmineqlem2 41202 | Part of lcm inequality lem... |
| lcmineqlem3 41203 | Part of lcm inequality lem... |
| lcmineqlem4 41204 | Part of lcm inequality lem... |
| lcmineqlem5 41205 | Technical lemma for recipr... |
| lcmineqlem6 41206 | Part of lcm inequality lem... |
| lcmineqlem7 41207 | Derivative of 1-x for chai... |
| lcmineqlem8 41208 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 41209 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 41210 | Induction step of ~ lcmine... |
| lcmineqlem11 41211 | Induction step, continuati... |
| lcmineqlem12 41212 | Base case for induction. ... |
| lcmineqlem13 41213 | Induction proof for lcm in... |
| lcmineqlem14 41214 | Technical lemma for inequa... |
| lcmineqlem15 41215 | F times the least common m... |
| lcmineqlem16 41216 | Technical divisibility lem... |
| lcmineqlem17 41217 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 41218 | Technical lemma to shift f... |
| lcmineqlem19 41219 | Dividing implies inequalit... |
| lcmineqlem20 41220 | Inequality for lcm lemma. ... |
| lcmineqlem21 41221 | The lcm inequality lemma w... |
| lcmineqlem22 41222 | The lcm inequality lemma w... |
| lcmineqlem23 41223 | Penultimate step to the lc... |
| lcmineqlem 41224 | The least common multiple ... |
| 3exp7 41225 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 41226 | First inequality in inequa... |
| 3lexlogpow5ineq2 41227 | Second inequality in inequ... |
| 3lexlogpow5ineq4 41228 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 41229 | Combined inequality chain ... |
| 3lexlogpow2ineq1 41230 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 41231 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 41232 | Result for bound in AKS in... |
| intlewftc 41233 | Inequality inference by in... |
| aks4d1lem1 41234 | Technical lemma to reduce ... |
| aks4d1p1p1 41235 | Exponential law for finite... |
| dvrelog2 41236 | The derivative of the loga... |
| dvrelog3 41237 | The derivative of the loga... |
| dvrelog2b 41238 | Derivative of the binary l... |
| 0nonelalab 41239 | Technical lemma for open i... |
| dvrelogpow2b 41240 | Derivative of the power of... |
| aks4d1p1p3 41241 | Bound of a ceiling of the ... |
| aks4d1p1p2 41242 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 41243 | Technical step for inequal... |
| dvle2 41244 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 41245 | Inequality lift to differe... |
| aks4d1p1p7 41246 | Bound of intermediary of i... |
| aks4d1p1p5 41247 | Show inequality for existe... |
| aks4d1p1 41248 | Show inequality for existe... |
| aks4d1p2 41249 | Technical lemma for existe... |
| aks4d1p3 41250 | There exists a small enoug... |
| aks4d1p4 41251 | There exists a small enoug... |
| aks4d1p5 41252 | Show that ` N ` and ` R ` ... |
| aks4d1p6 41253 | The maximal prime power ex... |
| aks4d1p7d1 41254 | Technical step in AKS lemm... |
| aks4d1p7 41255 | Technical step in AKS lemm... |
| aks4d1p8d1 41256 | If a prime divides one num... |
| aks4d1p8d2 41257 | Any prime power dividing a... |
| aks4d1p8d3 41258 | The remainder of a divisio... |
| aks4d1p8 41259 | Show that ` N ` and ` R ` ... |
| aks4d1p9 41260 | Show that the order is bou... |
| aks4d1 41261 | Lemma 4.1 from ~ https://w... |
| fldhmf1 41262 | A field homomorphism is in... |
| aks6d1c2p1 41263 | In the AKS-theorem the sub... |
| aks6d1c2p2 41264 | Injective condition for co... |
| 5bc2eq10 41265 | The value of 5 choose 2. ... |
| facp2 41266 | The factorial of a success... |
| 2np3bcnp1 41267 | Part of induction step for... |
| 2ap1caineq 41268 | Inequality for Theorem 6.6... |
| sticksstones1 41269 | Different strictly monoton... |
| sticksstones2 41270 | The range function on stri... |
| sticksstones3 41271 | The range function on stri... |
| sticksstones4 41272 | Equinumerosity lemma for s... |
| sticksstones5 41273 | Count the number of strict... |
| sticksstones6 41274 | Function induces an order ... |
| sticksstones7 41275 | Closure property of sticks... |
| sticksstones8 41276 | Establish mapping between ... |
| sticksstones9 41277 | Establish mapping between ... |
| sticksstones10 41278 | Establish mapping between ... |
| sticksstones11 41279 | Establish bijective mappin... |
| sticksstones12a 41280 | Establish bijective mappin... |
| sticksstones12 41281 | Establish bijective mappin... |
| sticksstones13 41282 | Establish bijective mappin... |
| sticksstones14 41283 | Sticks and stones with def... |
| sticksstones15 41284 | Sticks and stones with alm... |
| sticksstones16 41285 | Sticks and stones with col... |
| sticksstones17 41286 | Extend sticks and stones t... |
| sticksstones18 41287 | Extend sticks and stones t... |
| sticksstones19 41288 | Extend sticks and stones t... |
| sticksstones20 41289 | Lift sticks and stones to ... |
| sticksstones21 41290 | Lift sticks and stones to ... |
| sticksstones22 41291 | Non-exhaustive sticks and ... |
| metakunt1 41292 | A is an endomapping. (Con... |
| metakunt2 41293 | A is an endomapping. (Con... |
| metakunt3 41294 | Value of A. (Contributed b... |
| metakunt4 41295 | Value of A. (Contributed b... |
| metakunt5 41296 | C is the left inverse for ... |
| metakunt6 41297 | C is the left inverse for ... |
| metakunt7 41298 | C is the left inverse for ... |
| metakunt8 41299 | C is the left inverse for ... |
| metakunt9 41300 | C is the left inverse for ... |
| metakunt10 41301 | C is the right inverse for... |
| metakunt11 41302 | C is the right inverse for... |
| metakunt12 41303 | C is the right inverse for... |
| metakunt13 41304 | C is the right inverse for... |
| metakunt14 41305 | A is a primitive permutati... |
| metakunt15 41306 | Construction of another pe... |
| metakunt16 41307 | Construction of another pe... |
| metakunt17 41308 | The union of three disjoin... |
| metakunt18 41309 | Disjoint domains and codom... |
| metakunt19 41310 | Domains on restrictions of... |
| metakunt20 41311 | Show that B coincides on t... |
| metakunt21 41312 | Show that B coincides on t... |
| metakunt22 41313 | Show that B coincides on t... |
| metakunt23 41314 | B coincides on the union o... |
| metakunt24 41315 | Technical condition such t... |
| metakunt25 41316 | B is a permutation. (Cont... |
| metakunt26 41317 | Construction of one soluti... |
| metakunt27 41318 | Construction of one soluti... |
| metakunt28 41319 | Construction of one soluti... |
| metakunt29 41320 | Construction of one soluti... |
| metakunt30 41321 | Construction of one soluti... |
| metakunt31 41322 | Construction of one soluti... |
| metakunt32 41323 | Construction of one soluti... |
| metakunt33 41324 | Construction of one soluti... |
| metakunt34 41325 | ` D ` is a permutation. (... |
| andiff 41326 | Adding biconditional when ... |
| fac2xp3 41327 | Factorial of 2x+3, sublemm... |
| prodsplit 41328 | Product split into two fac... |
| 2xp3dxp2ge1d 41329 | 2x+3 is greater than or eq... |
| factwoffsmonot 41330 | A factorial with offset is... |
| ioin9i8 41331 | Miscellaneous inference cr... |
| jaodd 41332 | Double deduction form of ~... |
| syl3an12 41333 | A double syllogism inferen... |
| sbtd 41334 | A true statement is true u... |
| sbor2 41335 | One direction of ~ sbor , ... |
| 19.9dev 41336 | ~ 19.9d in the case of an ... |
| 3rspcedvdw 41337 | Triple application of ~ rs... |
| 3rspcedvd 41338 | Triple application of ~ rs... |
| rabdif 41339 | Move difference in and out... |
| sn-axrep5v 41340 | A condensed form of ~ axre... |
| sn-axprlem3 41341 | ~ axprlem3 using only Tars... |
| sn-exelALT 41342 | Alternate proof of ~ exel ... |
| ss2ab1 41343 | Class abstractions in a su... |
| ssabdv 41344 | Deduction of abstraction s... |
| sn-iotalem 41345 | An unused lemma showing th... |
| sn-iotalemcor 41346 | Corollary of ~ sn-iotalem ... |
| abbi1sn 41347 | Originally part of ~ uniab... |
| brif1 41348 | Move a relation inside and... |
| brif2 41349 | Move a relation inside and... |
| brif12 41350 | Move a relation inside and... |
| pssexg 41351 | The proper subset of a set... |
| pssn0 41352 | A proper superset is nonem... |
| psspwb 41353 | Classes are proper subclas... |
| xppss12 41354 | Proper subset theorem for ... |
| coexd 41355 | The composition of two set... |
| elpwbi 41356 | Membership in a power set,... |
| imaopab 41357 | The image of a class of or... |
| fnsnbt 41358 | A function's domain is a s... |
| fnimasnd 41359 | The image of a function by... |
| fvmptd4 41360 | Deduction version of ~ fvm... |
| eqresfnbd 41361 | Property of being the rest... |
| f1o2d2 41362 | Sufficient condition for a... |
| fmpocos 41363 | Composition of two functio... |
| ovmpogad 41364 | Value of an operation give... |
| ofun 41365 | A function operation of un... |
| dfqs2 41366 | Alternate definition of qu... |
| dfqs3 41367 | Alternate definition of qu... |
| qseq12d 41368 | Equality theorem for quoti... |
| qsalrel 41369 | The quotient set is equal ... |
| fsuppfund 41370 | A finitely supported funct... |
| fsuppsssuppgd 41371 | If the support of a functi... |
| fsuppss 41372 | A subset of a finitely sup... |
| elmapssresd 41373 | A restricted mapping is a ... |
| mapcod 41374 | Compose two mappings. (Co... |
| fzosumm1 41375 | Separate out the last term... |
| ccatcan2d 41376 | Cancellation law for conca... |
| nelsubginvcld 41377 | The inverse of a non-subgr... |
| nelsubgcld 41378 | A non-subgroup-member plus... |
| nelsubgsubcld 41379 | A non-subgroup-member minu... |
| rnasclg 41380 | The set of injected scalar... |
| frlmfielbas 41381 | The vectors of a finite fr... |
| frlmfzwrd 41382 | A vector of a module with ... |
| frlmfzowrd 41383 | A vector of a module with ... |
| frlmfzolen 41384 | The dimension of a vector ... |
| frlmfzowrdb 41385 | The vectors of a module wi... |
| frlmfzoccat 41386 | The concatenation of two v... |
| frlmvscadiccat 41387 | Scalar multiplication dist... |
| grpasscan2d 41388 | An associative cancellatio... |
| grpcominv1 41389 | If two elements commute, t... |
| grpcominv2 41390 | If two elements commute, t... |
| finsubmsubg 41391 | A submonoid of a finite gr... |
| crngcomd 41392 | Multiplication is commutat... |
| crng12d 41393 | Commutative/associative la... |
| imacrhmcl 41394 | The image of a commutative... |
| rimrcl1 41395 | Reverse closure of a ring ... |
| rimrcl2 41396 | Reverse closure of a ring ... |
| rimcnv 41397 | The converse of a ring iso... |
| rimco 41398 | The composition of ring is... |
| ricsym 41399 | Ring isomorphism is symmet... |
| rictr 41400 | Ring isomorphism is transi... |
| riccrng1 41401 | Ring isomorphism preserves... |
| riccrng 41402 | A ring is commutative if a... |
| drnginvrn0d 41403 | A multiplicative inverse i... |
| drngmulcanad 41404 | Cancellation of a nonzero ... |
| drngmulcan2ad 41405 | Cancellation of a nonzero ... |
| drnginvmuld 41406 | Inverse of a nonzero produ... |
| ricdrng1 41407 | A ring isomorphism maps a ... |
| ricdrng 41408 | A ring is a division ring ... |
| ricfld 41409 | A ring is a field if and o... |
| lvecgrp 41410 | A vector space is a group.... |
| lvecring 41411 | The scalar component of a ... |
| frlm0vald 41412 | All coordinates of the zer... |
| frlmsnic 41413 | Given a free module with a... |
| uvccl 41414 | A unit vector is a vector.... |
| uvcn0 41415 | A unit vector is nonzero. ... |
| pwselbasr 41416 | The reverse direction of ~... |
| pwsgprod 41417 | Finite products in a power... |
| psrbagres 41418 | Restrict a bag of variable... |
| mpllmodd 41419 | The polynomial ring is a l... |
| mplringd 41420 | The polynomial ring is a r... |
| mplcrngd 41421 | The polynomial ring is a c... |
| mplsubrgcl 41422 | An element of a polynomial... |
| mhmcompl 41423 | The composition of a monoi... |
| rhmmpllem1 41424 | Lemma for ~ rhmmpl . A su... |
| rhmmpllem2 41425 | Lemma for ~ rhmmpl . A su... |
| mhmcoaddmpl 41426 | Show that the ring homomor... |
| rhmcomulmpl 41427 | Show that the ring homomor... |
| rhmmpl 41428 | Provide a ring homomorphis... |
| mplascl0 41429 | The zero scalar as a polyn... |
| mplascl1 41430 | The one scalar as a polyno... |
| mplmapghm 41431 | The function ` H ` mapping... |
| evl0 41432 | The zero polynomial evalua... |
| evlscl 41433 | A polynomial over the ring... |
| evlsval3 41434 | Give a formula for the pol... |
| evlsvval 41435 | Give a formula for the eva... |
| evlsvvvallem 41436 | Lemma for ~ evlsvvval akin... |
| evlsvvvallem2 41437 | Lemma for theorems using ~... |
| evlsvvval 41438 | Give a formula for the eva... |
| evlsscaval 41439 | Polynomial evaluation buil... |
| evlsvarval 41440 | Polynomial evaluation buil... |
| evlsbagval 41441 | Polynomial evaluation buil... |
| evlsexpval 41442 | Polynomial evaluation buil... |
| evlsaddval 41443 | Polynomial evaluation buil... |
| evlsmulval 41444 | Polynomial evaluation buil... |
| evlsmaprhm 41445 | The function ` F ` mapping... |
| evlsevl 41446 | Evaluation in a subring is... |
| evlcl 41447 | A polynomial over the ring... |
| evlvvval 41448 | Give a formula for the eva... |
| evlvvvallem 41449 | Lemma for theorems using ~... |
| evladdval 41450 | Polynomial evaluation buil... |
| evlmulval 41451 | Polynomial evaluation buil... |
| selvcllem1 41452 | ` T ` is an associative al... |
| selvcllem2 41453 | ` D ` is a ring homomorphi... |
| selvcllem3 41454 | The third argument passed ... |
| selvcllemh 41455 | Apply the third argument (... |
| selvcllem4 41456 | The fourth argument passed... |
| selvcllem5 41457 | The fifth argument passed ... |
| selvcl 41458 | Closure of the "variable s... |
| selvval2 41459 | Value of the "variable sel... |
| selvvvval 41460 | Recover the original polyn... |
| evlselvlem 41461 | Lemma for ~ evlselv . Use... |
| evlselv 41462 | Evaluating a selection of ... |
| selvadd 41463 | The "variable selection" f... |
| selvmul 41464 | The "variable selection" f... |
| fsuppind 41465 | Induction on functions ` F... |
| fsuppssindlem1 41466 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 41467 | Lemma for ~ fsuppssind . ... |
| fsuppssind 41468 | Induction on functions ` F... |
| mhpind 41469 | The homogeneous polynomial... |
| evlsmhpvvval 41470 | Give a formula for the eva... |
| mhphflem 41471 | Lemma for ~ mhphf . Add s... |
| mhphf 41472 | A homogeneous polynomial d... |
| mhphf2 41473 | A homogeneous polynomial d... |
| mhphf3 41474 | A homogeneous polynomial d... |
| mhphf4 41475 | A homogeneous polynomial d... |
| c0exALT 41476 | Alternate proof of ~ c0ex ... |
| 0cnALT3 41477 | Alternate proof of ~ 0cn u... |
| elre0re 41478 | Specialized version of ~ 0... |
| 1t1e1ALT 41479 | Alternate proof of ~ 1t1e1... |
| remulcan2d 41480 | ~ mulcan2d for real number... |
| readdridaddlidd 41481 | Given some real number ` B... |
| sn-1ne2 41482 | A proof of ~ 1ne2 without ... |
| nnn1suc 41483 | A positive integer that is... |
| nnadd1com 41484 | Addition with 1 is commuta... |
| nnaddcom 41485 | Addition is commutative fo... |
| nnaddcomli 41486 | Version of ~ addcomli for ... |
| nnadddir 41487 | Right-distributivity for n... |
| nnmul1com 41488 | Multiplication with 1 is c... |
| nnmulcom 41489 | Multiplication is commutat... |
| mvrrsubd 41490 | Move a subtraction in the ... |
| laddrotrd 41491 | Rotate the variables right... |
| raddcom12d 41492 | Swap the first two variabl... |
| lsubrotld 41493 | Rotate the variables left ... |
| lsubcom23d 41494 | Swap the second and third ... |
| addsubeq4com 41495 | Relation between sums and ... |
| sqsumi 41496 | A sum squared. (Contribut... |
| negn0nposznnd 41497 | Lemma for ~ dffltz . (Con... |
| sqmid3api 41498 | Value of the square of the... |
| decaddcom 41499 | Commute ones place in addi... |
| sqn5i 41500 | The square of a number end... |
| sqn5ii 41501 | The square of a number end... |
| decpmulnc 41502 | Partial products algorithm... |
| decpmul 41503 | Partial products algorithm... |
| sqdeccom12 41504 | The square of a number in ... |
| sq3deccom12 41505 | Variant of ~ sqdeccom12 wi... |
| 4t5e20 41506 | 4 times 5 equals 20. (Con... |
| sq9 41507 | The square of 9 is 81. (C... |
| 235t711 41508 | Calculate a product by lon... |
| ex-decpmul 41509 | Example usage of ~ decpmul... |
| fz1sumconst 41510 | The sum of ` N ` constant ... |
| fz1sump1 41511 | Add one more term to a sum... |
| oddnumth 41512 | The Odd Number Theorem. T... |
| nicomachus 41513 | Nicomachus's Theorem. The... |
| sumcubes 41514 | The sum of the first ` N `... |
| oexpreposd 41515 | Lemma for ~ dffltz . TODO... |
| ltexp1d 41516 | ~ ltmul1d for exponentiati... |
| ltexp1dd 41517 | Raising both sides of 'les... |
| exp11nnd 41518 | ~ sq11d for positive real ... |
| exp11d 41519 | ~ exp11nnd for nonzero int... |
| 0dvds0 41520 | 0 divides 0. (Contributed... |
| absdvdsabsb 41521 | Divisibility is invariant ... |
| dvdsexpim 41522 | ~ dvdssqim generalized to ... |
| gcdnn0id 41523 | The ` gcd ` of a nonnegati... |
| gcdle1d 41524 | The greatest common diviso... |
| gcdle2d 41525 | The greatest common diviso... |
| dvdsexpad 41526 | Deduction associated with ... |
| nn0rppwr 41527 | If ` A ` and ` B ` are rel... |
| expgcd 41528 | Exponentiation distributes... |
| nn0expgcd 41529 | Exponentiation distributes... |
| zexpgcd 41530 | Exponentiation distributes... |
| numdenexp 41531 | ~ numdensq extended to non... |
| numexp 41532 | ~ numsq extended to nonneg... |
| denexp 41533 | ~ densq extended to nonneg... |
| dvdsexpnn 41534 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 41535 | ~ dvdsexpnn generalized to... |
| dvdsexpb 41536 | ~ dvdssq generalized to po... |
| posqsqznn 41537 | When a positive rational s... |
| zrtelqelz 41538 | ~ zsqrtelqelz generalized ... |
| zrtdvds 41539 | A positive integer root di... |
| rtprmirr 41540 | The root of a prime number... |
| resubval 41543 | Value of real subtraction,... |
| renegeulemv 41544 | Lemma for ~ renegeu and si... |
| renegeulem 41545 | Lemma for ~ renegeu and si... |
| renegeu 41546 | Existential uniqueness of ... |
| rernegcl 41547 | Closure law for negative r... |
| renegadd 41548 | Relationship between real ... |
| renegid 41549 | Addition of a real number ... |
| reneg0addlid 41550 | Negative zero is a left ad... |
| resubeulem1 41551 | Lemma for ~ resubeu . A v... |
| resubeulem2 41552 | Lemma for ~ resubeu . A v... |
| resubeu 41553 | Existential uniqueness of ... |
| rersubcl 41554 | Closure for real subtracti... |
| resubadd 41555 | Relation between real subt... |
| resubaddd 41556 | Relationship between subtr... |
| resubf 41557 | Real subtraction is an ope... |
| repncan2 41558 | Addition and subtraction o... |
| repncan3 41559 | Addition and subtraction o... |
| readdsub 41560 | Law for addition and subtr... |
| reladdrsub 41561 | Move LHS of a sum into RHS... |
| reltsub1 41562 | Subtraction from both side... |
| reltsubadd2 41563 | 'Less than' relationship b... |
| resubcan2 41564 | Cancellation law for real ... |
| resubsub4 41565 | Law for double subtraction... |
| rennncan2 41566 | Cancellation law for real ... |
| renpncan3 41567 | Cancellation law for real ... |
| repnpcan 41568 | Cancellation law for addit... |
| reppncan 41569 | Cancellation law for mixed... |
| resubidaddlidlem 41570 | Lemma for ~ resubidaddlid ... |
| resubidaddlid 41571 | Any real number subtracted... |
| resubdi 41572 | Distribution of multiplica... |
| re1m1e0m0 41573 | Equality of two left-addit... |
| sn-00idlem1 41574 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 41575 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 41576 | Lemma for ~ sn-00id . (Co... |
| sn-00id 41577 | ~ 00id proven without ~ ax... |
| re0m0e0 41578 | Real number version of ~ 0... |
| readdlid 41579 | Real number version of ~ a... |
| sn-addlid 41580 | ~ addlid without ~ ax-mulc... |
| remul02 41581 | Real number version of ~ m... |
| sn-0ne2 41582 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 41583 | Real number version of ~ m... |
| resubid 41584 | Subtraction of a real numb... |
| readdrid 41585 | Real number version of ~ a... |
| resubid1 41586 | Real number version of ~ s... |
| renegneg 41587 | A real number is equal to ... |
| readdcan2 41588 | Commuted version of ~ read... |
| renegid2 41589 | Commuted version of ~ rene... |
| remulneg2d 41590 | Product with negative is n... |
| sn-it0e0 41591 | Proof of ~ it0e0 without ~... |
| sn-negex12 41592 | A combination of ~ cnegex ... |
| sn-negex 41593 | Proof of ~ cnegex without ... |
| sn-negex2 41594 | Proof of ~ cnegex2 without... |
| sn-addcand 41595 | ~ addcand without ~ ax-mul... |
| sn-addrid 41596 | ~ addrid without ~ ax-mulc... |
| sn-addcan2d 41597 | ~ addcan2d without ~ ax-mu... |
| reixi 41598 | ~ ixi without ~ ax-mulcom ... |
| rei4 41599 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 41600 | A number that sums to itse... |
| sn-mul01 41601 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 41602 | ~ negeu without ~ ax-mulco... |
| sn-subcl 41603 | ~ subcl without ~ ax-mulco... |
| sn-subf 41604 | ~ subf without ~ ax-mulcom... |
| resubeqsub 41605 | Equivalence between real s... |
| subresre 41606 | Subtraction restricted to ... |
| addinvcom 41607 | A number commutes with its... |
| remulinvcom 41608 | A left multiplicative inve... |
| remullid 41609 | Commuted version of ~ ax-1... |
| sn-1ticom 41610 | Lemma for ~ sn-mullid and ... |
| sn-mullid 41611 | ~ mullid without ~ ax-mulc... |
| it1ei 41612 | ` 1 ` is a multiplicative ... |
| ipiiie0 41613 | The multiplicative inverse... |
| remulcand 41614 | Commuted version of ~ remu... |
| sn-0tie0 41615 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 41616 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 41617 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 41618 | ~ ltaddneg without ~ ax-mu... |
| reposdif 41619 | Comparison of two numbers ... |
| relt0neg1 41620 | Comparison of a real and i... |
| relt0neg2 41621 | Comparison of a real and i... |
| sn-addlt0d 41622 | The sum of negative number... |
| sn-addgt0d 41623 | The sum of positive number... |
| sn-nnne0 41624 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 41625 | ~ elznn0nn restated using ... |
| nn0addcom 41626 | Addition is commutative fo... |
| zaddcomlem 41627 | Lemma for ~ zaddcom . (Co... |
| zaddcom 41628 | Addition is commutative fo... |
| renegmulnnass 41629 | Move multiplication by a n... |
| nn0mulcom 41630 | Multiplication is commutat... |
| zmulcomlem 41631 | Lemma for ~ zmulcom . (Co... |
| zmulcom 41632 | Multiplication is commutat... |
| mulgt0con1dlem 41633 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 41634 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 41635 | Lemma for ~ mulgt0b2d and ... |
| mulgt0b2d 41636 | Biconditional, deductive f... |
| sn-ltmul2d 41637 | ~ ltmul2d without ~ ax-mul... |
| sn-0lt1 41638 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 41639 | ~ ltp1 without ~ ax-mulcom... |
| reneg1lt0 41640 | Lemma for ~ sn-inelr . (C... |
| sn-inelr 41641 | ~ inelr without ~ ax-mulco... |
| itrere 41642 | ` _i ` times a real is rea... |
| retire 41643 | Commuted version of ~ itre... |
| cnreeu 41644 | The reals in the expressio... |
| sn-sup2 41645 | ~ sup2 with exactly the sa... |
| prjspval 41648 | Value of the projective sp... |
| prjsprel 41649 | Utility theorem regarding ... |
| prjspertr 41650 | The relation in ` PrjSp ` ... |
| prjsperref 41651 | The relation in ` PrjSp ` ... |
| prjspersym 41652 | The relation in ` PrjSp ` ... |
| prjsper 41653 | The relation used to defin... |
| prjspreln0 41654 | Two nonzero vectors are eq... |
| prjspvs 41655 | A nonzero multiple of a ve... |
| prjsprellsp 41656 | Two vectors are equivalent... |
| prjspeclsp 41657 | The vectors equivalent to ... |
| prjspval2 41658 | Alternate definition of pr... |
| prjspnval 41661 | Value of the n-dimensional... |
| prjspnerlem 41662 | A lemma showing that the e... |
| prjspnval2 41663 | Value of the n-dimensional... |
| prjspner 41664 | The relation used to defin... |
| prjspnvs 41665 | A nonzero multiple of a ve... |
| prjspnssbas 41666 | A projective point spans a... |
| prjspnn0 41667 | A projective point is none... |
| 0prjspnlem 41668 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 41669 | Any vector is equivalent t... |
| prjspner01 41670 | Any vector is equivalent t... |
| prjspner1 41671 | Two vectors whose zeroth c... |
| 0prjspnrel 41672 | In the zero-dimensional pr... |
| 0prjspn 41673 | A zero-dimensional project... |
| prjcrvfval 41676 | Value of the projective cu... |
| prjcrvval 41677 | Value of the projective cu... |
| prjcrv0 41678 | The "curve" (zero set) cor... |
| dffltz 41679 | Fermat's Last Theorem (FLT... |
| fltmul 41680 | A counterexample to FLT st... |
| fltdiv 41681 | A counterexample to FLT st... |
| flt0 41682 | A counterexample for FLT d... |
| fltdvdsabdvdsc 41683 | Any factor of both ` A ` a... |
| fltabcoprmex 41684 | A counterexample to FLT im... |
| fltaccoprm 41685 | A counterexample to FLT wi... |
| fltbccoprm 41686 | A counterexample to FLT wi... |
| fltabcoprm 41687 | A counterexample to FLT wi... |
| infdesc 41688 | Infinite descent. The hyp... |
| fltne 41689 | If a counterexample to FLT... |
| flt4lem 41690 | Raising a number to the fo... |
| flt4lem1 41691 | Satisfy the antecedent use... |
| flt4lem2 41692 | If ` A ` is even, ` B ` is... |
| flt4lem3 41693 | Equivalent to ~ pythagtrip... |
| flt4lem4 41694 | If the product of two copr... |
| flt4lem5 41695 | In the context of the lemm... |
| flt4lem5elem 41696 | Version of ~ fltaccoprm an... |
| flt4lem5a 41697 | Part 1 of Equation 1 of ... |
| flt4lem5b 41698 | Part 2 of Equation 1 of ... |
| flt4lem5c 41699 | Part 2 of Equation 2 of ... |
| flt4lem5d 41700 | Part 3 of Equation 2 of ... |
| flt4lem5e 41701 | Satisfy the hypotheses of ... |
| flt4lem5f 41702 | Final equation of ~... |
| flt4lem6 41703 | Remove shared factors in a... |
| flt4lem7 41704 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 41705 | Strengthening of Fermat's ... |
| fltltc 41706 | ` ( C ^ N ) ` is the large... |
| fltnltalem 41707 | Lemma for ~ fltnlta . A l... |
| fltnlta 41708 | In a Fermat counterexample... |
| iddii 41709 | Version of ~ a1ii with the... |
| bicomdALT 41710 | Alternate proof of ~ bicom... |
| elabgw 41711 | Membership in a class abst... |
| elab2gw 41712 | Membership in a class abst... |
| elrab2w 41713 | Membership in a restricted... |
| ruvALT 41714 | Alternate proof of ~ ruv w... |
| sn-wcdeq 41715 | Alternative to ~ wcdeq and... |
| sq45 41716 | 45 squared is 2025. (Cont... |
| sum9cubes 41717 | The sum of the first nine ... |
| acos1half 41718 | The arccosine of ` 1 / 2 `... |
| aprilfools2025 41719 | An abuse of notation. (Co... |
| binom2d 41720 | Deduction form of binom2. ... |
| cu3addd 41721 | Cube of sum of three numbe... |
| sqnegd 41722 | The square of the negative... |
| negexpidd 41723 | The sum of a real number t... |
| rexlimdv3d 41724 | An extended version of ~ r... |
| 3cubeslem1 41725 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 41726 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 41727 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 41728 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 41729 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 41730 | Lemma for ~ 3cubes . This... |
| 3cubes 41731 | Every rational number is a... |
| rntrclfvOAI 41732 | The range of the transitiv... |
| moxfr 41733 | Transfer at-most-one betwe... |
| imaiinfv 41734 | Indexed intersection of an... |
| elrfi 41735 | Elementhood in a set of re... |
| elrfirn 41736 | Elementhood in a set of re... |
| elrfirn2 41737 | Elementhood in a set of re... |
| cmpfiiin 41738 | In a compact topology, a s... |
| ismrcd1 41739 | Any function from the subs... |
| ismrcd2 41740 | Second half of ~ ismrcd1 .... |
| istopclsd 41741 | A closure function which s... |
| ismrc 41742 | A function is a Moore clos... |
| isnacs 41745 | Expand definition of Noeth... |
| nacsfg 41746 | In a Noetherian-type closu... |
| isnacs2 41747 | Express Noetherian-type cl... |
| mrefg2 41748 | Slight variation on finite... |
| mrefg3 41749 | Slight variation on finite... |
| nacsacs 41750 | A closure system of Noethe... |
| isnacs3 41751 | A choice-free order equiva... |
| incssnn0 41752 | Transitivity induction of ... |
| nacsfix 41753 | An increasing sequence of ... |
| constmap 41754 | A constant (represented wi... |
| mapco2g 41755 | Renaming indices in a tupl... |
| mapco2 41756 | Post-composition (renaming... |
| mapfzcons 41757 | Extending a one-based mapp... |
| mapfzcons1 41758 | Recover prefix mapping fro... |
| mapfzcons1cl 41759 | A nonempty mapping has a p... |
| mapfzcons2 41760 | Recover added element from... |
| mptfcl 41761 | Interpret range of a maps-... |
| mzpclval 41766 | Substitution lemma for ` m... |
| elmzpcl 41767 | Double substitution lemma ... |
| mzpclall 41768 | The set of all functions w... |
| mzpcln0 41769 | Corollary of ~ mzpclall : ... |
| mzpcl1 41770 | Defining property 1 of a p... |
| mzpcl2 41771 | Defining property 2 of a p... |
| mzpcl34 41772 | Defining properties 3 and ... |
| mzpval 41773 | Value of the ` mzPoly ` fu... |
| dmmzp 41774 | ` mzPoly ` is defined for ... |
| mzpincl 41775 | Polynomial closedness is a... |
| mzpconst 41776 | Constant functions are pol... |
| mzpf 41777 | A polynomial function is a... |
| mzpproj 41778 | A projection function is p... |
| mzpadd 41779 | The pointwise sum of two p... |
| mzpmul 41780 | The pointwise product of t... |
| mzpconstmpt 41781 | A constant function expres... |
| mzpaddmpt 41782 | Sum of polynomial function... |
| mzpmulmpt 41783 | Product of polynomial func... |
| mzpsubmpt 41784 | The difference of two poly... |
| mzpnegmpt 41785 | Negation of a polynomial f... |
| mzpexpmpt 41786 | Raise a polynomial functio... |
| mzpindd 41787 | "Structural" induction to ... |
| mzpmfp 41788 | Relationship between multi... |
| mzpsubst 41789 | Substituting polynomials f... |
| mzprename 41790 | Simplified version of ~ mz... |
| mzpresrename 41791 | A polynomial is a polynomi... |
| mzpcompact2lem 41792 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 41793 | Polynomials are finitary o... |
| coeq0i 41794 | ~ coeq0 but without explic... |
| fzsplit1nn0 41795 | Split a finite 1-based set... |
| eldiophb 41798 | Initial expression of Diop... |
| eldioph 41799 | Condition for a set to be ... |
| diophrw 41800 | Renaming and adding unused... |
| eldioph2lem1 41801 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 41802 | Lemma for ~ eldioph2 . Co... |
| eldioph2 41803 | Construct a Diophantine se... |
| eldioph2b 41804 | While Diophantine sets wer... |
| eldiophelnn0 41805 | Remove antecedent on ` B `... |
| eldioph3b 41806 | Define Diophantine sets in... |
| eldioph3 41807 | Inference version of ~ eld... |
| ellz1 41808 | Membership in a lower set ... |
| lzunuz 41809 | The union of a lower set o... |
| fz1eqin 41810 | Express a one-based finite... |
| lzenom 41811 | Lower integers are countab... |
| elmapresaunres2 41812 | ~ fresaunres2 transposed t... |
| diophin 41813 | If two sets are Diophantin... |
| diophun 41814 | If two sets are Diophantin... |
| eldiophss 41815 | Diophantine sets are sets ... |
| diophrex 41816 | Projecting a Diophantine s... |
| eq0rabdioph 41817 | This is the first of a num... |
| eqrabdioph 41818 | Diophantine set builder fo... |
| 0dioph 41819 | The null set is Diophantin... |
| vdioph 41820 | The "universal" set (as la... |
| anrabdioph 41821 | Diophantine set builder fo... |
| orrabdioph 41822 | Diophantine set builder fo... |
| 3anrabdioph 41823 | Diophantine set builder fo... |
| 3orrabdioph 41824 | Diophantine set builder fo... |
| 2sbcrex 41825 | Exchange an existential qu... |
| sbcrexgOLD 41826 | Interchange class substitu... |
| 2sbcrexOLD 41827 | Exchange an existential qu... |
| sbc2rex 41828 | Exchange a substitution wi... |
| sbc2rexgOLD 41829 | Exchange a substitution wi... |
| sbc4rex 41830 | Exchange a substitution wi... |
| sbc4rexgOLD 41831 | Exchange a substitution wi... |
| sbcrot3 41832 | Rotate a sequence of three... |
| sbcrot5 41833 | Rotate a sequence of five ... |
| sbccomieg 41834 | Commute two explicit subst... |
| rexrabdioph 41835 | Diophantine set builder fo... |
| rexfrabdioph 41836 | Diophantine set builder fo... |
| 2rexfrabdioph 41837 | Diophantine set builder fo... |
| 3rexfrabdioph 41838 | Diophantine set builder fo... |
| 4rexfrabdioph 41839 | Diophantine set builder fo... |
| 6rexfrabdioph 41840 | Diophantine set builder fo... |
| 7rexfrabdioph 41841 | Diophantine set builder fo... |
| rabdiophlem1 41842 | Lemma for arithmetic dioph... |
| rabdiophlem2 41843 | Lemma for arithmetic dioph... |
| elnn0rabdioph 41844 | Diophantine set builder fo... |
| rexzrexnn0 41845 | Rewrite an existential qua... |
| lerabdioph 41846 | Diophantine set builder fo... |
| eluzrabdioph 41847 | Diophantine set builder fo... |
| elnnrabdioph 41848 | Diophantine set builder fo... |
| ltrabdioph 41849 | Diophantine set builder fo... |
| nerabdioph 41850 | Diophantine set builder fo... |
| dvdsrabdioph 41851 | Divisibility is a Diophant... |
| eldioph4b 41852 | Membership in ` Dioph ` ex... |
| eldioph4i 41853 | Forward-only version of ~ ... |
| diophren 41854 | Change variables in a Diop... |
| rabrenfdioph 41855 | Change variable numbers in... |
| rabren3dioph 41856 | Change variable numbers in... |
| fphpd 41857 | Pigeonhole principle expre... |
| fphpdo 41858 | Pigeonhole principle for s... |
| ctbnfien 41859 | An infinite subset of a co... |
| fiphp3d 41860 | Infinite pigeonhole princi... |
| rencldnfilem 41861 | Lemma for ~ rencldnfi . (... |
| rencldnfi 41862 | A set of real numbers whic... |
| irrapxlem1 41863 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 41864 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 41865 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 41866 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 41867 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 41868 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 41869 | Dirichlet's approximation ... |
| pellexlem1 41870 | Lemma for ~ pellex . Arit... |
| pellexlem2 41871 | Lemma for ~ pellex . Arit... |
| pellexlem3 41872 | Lemma for ~ pellex . To e... |
| pellexlem4 41873 | Lemma for ~ pellex . Invo... |
| pellexlem5 41874 | Lemma for ~ pellex . Invo... |
| pellexlem6 41875 | Lemma for ~ pellex . Doin... |
| pellex 41876 | Every Pell equation has a ... |
| pell1qrval 41887 | Value of the set of first-... |
| elpell1qr 41888 | Membership in a first-quad... |
| pell14qrval 41889 | Value of the set of positi... |
| elpell14qr 41890 | Membership in the set of p... |
| pell1234qrval 41891 | Value of the set of genera... |
| elpell1234qr 41892 | Membership in the set of g... |
| pell1234qrre 41893 | General Pell solutions are... |
| pell1234qrne0 41894 | No solution to a Pell equa... |
| pell1234qrreccl 41895 | General solutions of the P... |
| pell1234qrmulcl 41896 | General solutions of the P... |
| pell14qrss1234 41897 | A positive Pell solution i... |
| pell14qrre 41898 | A positive Pell solution i... |
| pell14qrne0 41899 | A positive Pell solution i... |
| pell14qrgt0 41900 | A positive Pell solution i... |
| pell14qrrp 41901 | A positive Pell solution i... |
| pell1234qrdich 41902 | A general Pell solution is... |
| elpell14qr2 41903 | A number is a positive Pel... |
| pell14qrmulcl 41904 | Positive Pell solutions ar... |
| pell14qrreccl 41905 | Positive Pell solutions ar... |
| pell14qrdivcl 41906 | Positive Pell solutions ar... |
| pell14qrexpclnn0 41907 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 41908 | Positive Pell solutions ar... |
| pell1qrss14 41909 | First-quadrant Pell soluti... |
| pell14qrdich 41910 | A positive Pell solution i... |
| pell1qrge1 41911 | A Pell solution in the fir... |
| pell1qr1 41912 | 1 is a Pell solution and i... |
| elpell1qr2 41913 | The first quadrant solutio... |
| pell1qrgaplem 41914 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 41915 | First-quadrant Pell soluti... |
| pell14qrgap 41916 | Positive Pell solutions ar... |
| pell14qrgapw 41917 | Positive Pell solutions ar... |
| pellqrexplicit 41918 | Condition for a calculated... |
| infmrgelbi 41919 | Any lower bound of a nonem... |
| pellqrex 41920 | There is a nontrivial solu... |
| pellfundval 41921 | Value of the fundamental s... |
| pellfundre 41922 | The fundamental solution o... |
| pellfundge 41923 | Lower bound on the fundame... |
| pellfundgt1 41924 | Weak lower bound on the Pe... |
| pellfundlb 41925 | A nontrivial first quadran... |
| pellfundglb 41926 | If a real is larger than t... |
| pellfundex 41927 | The fundamental solution a... |
| pellfund14gap 41928 | There are no solutions bet... |
| pellfundrp 41929 | The fundamental Pell solut... |
| pellfundne1 41930 | The fundamental Pell solut... |
| reglogcl 41931 | General logarithm is a rea... |
| reglogltb 41932 | General logarithm preserve... |
| reglogleb 41933 | General logarithm preserve... |
| reglogmul 41934 | Multiplication law for gen... |
| reglogexp 41935 | Power law for general log.... |
| reglogbas 41936 | General log of the base is... |
| reglog1 41937 | General log of 1 is 0. (C... |
| reglogexpbas 41938 | General log of a power of ... |
| pellfund14 41939 | Every positive Pell soluti... |
| pellfund14b 41940 | The positive Pell solution... |
| rmxfval 41945 | Value of the X sequence. ... |
| rmyfval 41946 | Value of the Y sequence. ... |
| rmspecsqrtnq 41947 | The discriminant used to d... |
| rmspecnonsq 41948 | The discriminant used to d... |
| qirropth 41949 | This lemma implements the ... |
| rmspecfund 41950 | The base of exponent used ... |
| rmxyelqirr 41951 | The solutions used to cons... |
| rmxyelqirrOLD 41952 | Obsolete version of ~ rmxy... |
| rmxypairf1o 41953 | The function used to extra... |
| rmxyelxp 41954 | Lemma for ~ frmx and ~ frm... |
| frmx 41955 | The X sequence is a nonneg... |
| frmy 41956 | The Y sequence is an integ... |
| rmxyval 41957 | Main definition of the X a... |
| rmspecpos 41958 | The discriminant used to d... |
| rmxycomplete 41959 | The X and Y sequences take... |
| rmxynorm 41960 | The X and Y sequences defi... |
| rmbaserp 41961 | The base of exponentiation... |
| rmxyneg 41962 | Negation law for X and Y s... |
| rmxyadd 41963 | Addition formula for X and... |
| rmxy1 41964 | Value of the X and Y seque... |
| rmxy0 41965 | Value of the X and Y seque... |
| rmxneg 41966 | Negation law (even functio... |
| rmx0 41967 | Value of X sequence at 0. ... |
| rmx1 41968 | Value of X sequence at 1. ... |
| rmxadd 41969 | Addition formula for X seq... |
| rmyneg 41970 | Negation formula for Y seq... |
| rmy0 41971 | Value of Y sequence at 0. ... |
| rmy1 41972 | Value of Y sequence at 1. ... |
| rmyadd 41973 | Addition formula for Y seq... |
| rmxp1 41974 | Special addition-of-1 form... |
| rmyp1 41975 | Special addition of 1 form... |
| rmxm1 41976 | Subtraction of 1 formula f... |
| rmym1 41977 | Subtraction of 1 formula f... |
| rmxluc 41978 | The X sequence is a Lucas ... |
| rmyluc 41979 | The Y sequence is a Lucas ... |
| rmyluc2 41980 | Lucas sequence property of... |
| rmxdbl 41981 | "Double-angle formula" for... |
| rmydbl 41982 | "Double-angle formula" for... |
| monotuz 41983 | A function defined on an u... |
| monotoddzzfi 41984 | A function which is odd an... |
| monotoddzz 41985 | A function (given implicit... |
| oddcomabszz 41986 | An odd function which take... |
| 2nn0ind 41987 | Induction on nonnegative i... |
| zindbi 41988 | Inductively transfer a pro... |
| rmxypos 41989 | For all nonnegative indice... |
| ltrmynn0 41990 | The Y-sequence is strictly... |
| ltrmxnn0 41991 | The X-sequence is strictly... |
| lermxnn0 41992 | The X-sequence is monotoni... |
| rmxnn 41993 | The X-sequence is defined ... |
| ltrmy 41994 | The Y-sequence is strictly... |
| rmyeq0 41995 | Y is zero only at zero. (... |
| rmyeq 41996 | Y is one-to-one. (Contrib... |
| lermy 41997 | Y is monotonic (non-strict... |
| rmynn 41998 | ` rmY ` is positive for po... |
| rmynn0 41999 | ` rmY ` is nonnegative for... |
| rmyabs 42000 | ` rmY ` commutes with ` ab... |
| jm2.24nn 42001 | X(n) is strictly greater t... |
| jm2.17a 42002 | First half of lemma 2.17 o... |
| jm2.17b 42003 | Weak form of the second ha... |
| jm2.17c 42004 | Second half of lemma 2.17 ... |
| jm2.24 42005 | Lemma 2.24 of [JonesMatija... |
| rmygeid 42006 | Y(n) increases faster than... |
| congtr 42007 | A wff of the form ` A || (... |
| congadd 42008 | If two pairs of numbers ar... |
| congmul 42009 | If two pairs of numbers ar... |
| congsym 42010 | Congruence mod ` A ` is a ... |
| congneg 42011 | If two integers are congru... |
| congsub 42012 | If two pairs of numbers ar... |
| congid 42013 | Every integer is congruent... |
| mzpcong 42014 | Polynomials commute with c... |
| congrep 42015 | Every integer is congruent... |
| congabseq 42016 | If two integers are congru... |
| acongid 42017 | A wff like that in this th... |
| acongsym 42018 | Symmetry of alternating co... |
| acongneg2 42019 | Negate right side of alter... |
| acongtr 42020 | Transitivity of alternatin... |
| acongeq12d 42021 | Substitution deduction for... |
| acongrep 42022 | Every integer is alternati... |
| fzmaxdif 42023 | Bound on the difference be... |
| fzneg 42024 | Reflection of a finite ran... |
| acongeq 42025 | Two numbers in the fundame... |
| dvdsacongtr 42026 | Alternating congruence pas... |
| coprmdvdsb 42027 | Multiplication by a coprim... |
| modabsdifz 42028 | Divisibility in terms of m... |
| dvdsabsmod0 42029 | Divisibility in terms of m... |
| jm2.18 42030 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 42031 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 42032 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 42033 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 42034 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 42035 | Lemma 2.19 of [JonesMatija... |
| jm2.21 42036 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 42037 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 42038 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 42039 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 42040 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 42041 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 42042 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 42043 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 42044 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 42045 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 42046 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 42047 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 42048 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 42049 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 42050 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 42051 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 42052 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 42053 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 42054 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 42055 | Lemma for ~ rmydioph . Us... |
| rmydioph 42056 | ~ jm2.27 restated in terms... |
| rmxdiophlem 42057 | X can be expressed in term... |
| rmxdioph 42058 | X is a Diophantine functio... |
| jm3.1lem1 42059 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 42060 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 42061 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 42062 | Diophantine expression for... |
| expdiophlem1 42063 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 42064 | Lemma for ~ expdioph . Ex... |
| expdioph 42065 | The exponential function i... |
| setindtr 42066 | Set induction for sets con... |
| setindtrs 42067 | Set induction scheme witho... |
| dford3lem1 42068 | Lemma for ~ dford3 . (Con... |
| dford3lem2 42069 | Lemma for ~ dford3 . (Con... |
| dford3 42070 | Ordinals are precisely the... |
| dford4 42071 | ~ dford3 expressed in prim... |
| wopprc 42072 | Unrelated: Wiener pairs t... |
| rpnnen3lem 42073 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 42074 | Dedekind cut injection of ... |
| axac10 42075 | Characterization of choice... |
| harinf 42076 | The Hartogs number of an i... |
| wdom2d2 42077 | Deduction for weak dominan... |
| ttac 42078 | Tarski's theorem about cho... |
| pw2f1ocnv 42079 | Define a bijection between... |
| pw2f1o2 42080 | Define a bijection between... |
| pw2f1o2val 42081 | Function value of the ~ pw... |
| pw2f1o2val2 42082 | Membership in a mapped set... |
| soeq12d 42083 | Equality deduction for tot... |
| freq12d 42084 | Equality deduction for fou... |
| weeq12d 42085 | Equality deduction for wel... |
| limsuc2 42086 | Limit ordinals in the sens... |
| wepwsolem 42087 | Transfer an ordering on ch... |
| wepwso 42088 | A well-ordering induces a ... |
| dnnumch1 42089 | Define an enumeration of a... |
| dnnumch2 42090 | Define an enumeration (wea... |
| dnnumch3lem 42091 | Value of the ordinal injec... |
| dnnumch3 42092 | Define an injection from a... |
| dnwech 42093 | Define a well-ordering fro... |
| fnwe2val 42094 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 42095 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 42096 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 42097 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 42098 | A well-ordering can be con... |
| aomclem1 42099 | Lemma for ~ dfac11 . This... |
| aomclem2 42100 | Lemma for ~ dfac11 . Succ... |
| aomclem3 42101 | Lemma for ~ dfac11 . Succ... |
| aomclem4 42102 | Lemma for ~ dfac11 . Limi... |
| aomclem5 42103 | Lemma for ~ dfac11 . Comb... |
| aomclem6 42104 | Lemma for ~ dfac11 . Tran... |
| aomclem7 42105 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 42106 | Lemma for ~ dfac11 . Perf... |
| dfac11 42107 | The right-hand side of thi... |
| kelac1 42108 | Kelley's choice, basic for... |
| kelac2lem 42109 | Lemma for ~ kelac2 and ~ d... |
| kelac2 42110 | Kelley's choice, most comm... |
| dfac21 42111 | Tychonoff's theorem is a c... |
| islmodfg 42114 | Property of a finitely gen... |
| islssfg 42115 | Property of a finitely gen... |
| islssfg2 42116 | Property of a finitely gen... |
| islssfgi 42117 | Finitely spanned subspaces... |
| fglmod 42118 | Finitely generated left mo... |
| lsmfgcl 42119 | The sum of two finitely ge... |
| islnm 42122 | Property of being a Noethe... |
| islnm2 42123 | Property of being a Noethe... |
| lnmlmod 42124 | A Noetherian left module i... |
| lnmlssfg 42125 | A submodule of Noetherian ... |
| lnmlsslnm 42126 | All submodules of a Noethe... |
| lnmfg 42127 | A Noetherian left module i... |
| kercvrlsm 42128 | The domain of a linear fun... |
| lmhmfgima 42129 | A homomorphism maps finite... |
| lnmepi 42130 | Epimorphic images of Noeth... |
| lmhmfgsplit 42131 | If the kernel and range of... |
| lmhmlnmsplit 42132 | If the kernel and range of... |
| lnmlmic 42133 | Noetherian is an invariant... |
| pwssplit4 42134 | Splitting for structure po... |
| filnm 42135 | Finite left modules are No... |
| pwslnmlem0 42136 | Zeroeth powers are Noether... |
| pwslnmlem1 42137 | First powers are Noetheria... |
| pwslnmlem2 42138 | A sum of powers is Noether... |
| pwslnm 42139 | Finite powers of Noetheria... |
| unxpwdom3 42140 | Weaker version of ~ unxpwd... |
| pwfi2f1o 42141 | The ~ pw2f1o bijection rel... |
| pwfi2en 42142 | Finitely supported indicat... |
| frlmpwfi 42143 | Formal linear combinations... |
| gicabl 42144 | Being Abelian is a group i... |
| imasgim 42145 | A relabeling of the elemen... |
| isnumbasgrplem1 42146 | A set which is equipollent... |
| harn0 42147 | The Hartogs number of a se... |
| numinfctb 42148 | A numerable infinite set c... |
| isnumbasgrplem2 42149 | If the (to be thought of a... |
| isnumbasgrplem3 42150 | Every nonempty numerable s... |
| isnumbasabl 42151 | A set is numerable iff it ... |
| isnumbasgrp 42152 | A set is numerable iff it ... |
| dfacbasgrp 42153 | A choice equivalent in abs... |
| islnr 42156 | Property of a left-Noether... |
| lnrring 42157 | Left-Noetherian rings are ... |
| lnrlnm 42158 | Left-Noetherian rings have... |
| islnr2 42159 | Property of being a left-N... |
| islnr3 42160 | Relate left-Noetherian rin... |
| lnr2i 42161 | Given an ideal in a left-N... |
| lpirlnr 42162 | Left principal ideal rings... |
| lnrfrlm 42163 | Finite-dimensional free mo... |
| lnrfg 42164 | Finitely-generated modules... |
| lnrfgtr 42165 | A submodule of a finitely ... |
| hbtlem1 42168 | Value of the leading coeff... |
| hbtlem2 42169 | Leading coefficient ideals... |
| hbtlem7 42170 | Functionality of leading c... |
| hbtlem4 42171 | The leading ideal function... |
| hbtlem3 42172 | The leading ideal function... |
| hbtlem5 42173 | The leading ideal function... |
| hbtlem6 42174 | There is a finite set of p... |
| hbt 42175 | The Hilbert Basis Theorem ... |
| dgrsub2 42180 | Subtracting two polynomial... |
| elmnc 42181 | Property of a monic polyno... |
| mncply 42182 | A monic polynomial is a po... |
| mnccoe 42183 | A monic polynomial has lea... |
| mncn0 42184 | A monic polynomial is not ... |
| dgraaval 42189 | Value of the degree functi... |
| dgraalem 42190 | Properties of the degree o... |
| dgraacl 42191 | Closure of the degree func... |
| dgraaf 42192 | Degree function on algebra... |
| dgraaub 42193 | Upper bound on degree of a... |
| dgraa0p 42194 | A rational polynomial of d... |
| mpaaeu 42195 | An algebraic number has ex... |
| mpaaval 42196 | Value of the minimal polyn... |
| mpaalem 42197 | Properties of the minimal ... |
| mpaacl 42198 | Minimal polynomial is a po... |
| mpaadgr 42199 | Minimal polynomial has deg... |
| mpaaroot 42200 | The minimal polynomial of ... |
| mpaamn 42201 | Minimal polynomial is moni... |
| itgoval 42206 | Value of the integral-over... |
| aaitgo 42207 | The standard algebraic num... |
| itgoss 42208 | An integral element is int... |
| itgocn 42209 | All integral elements are ... |
| cnsrexpcl 42210 | Exponentiation is closed i... |
| fsumcnsrcl 42211 | Finite sums are closed in ... |
| cnsrplycl 42212 | Polynomials are closed in ... |
| rgspnval 42213 | Value of the ring-span of ... |
| rgspncl 42214 | The ring-span of a set is ... |
| rgspnssid 42215 | The ring-span of a set con... |
| rgspnmin 42216 | The ring-span is contained... |
| rgspnid 42217 | The span of a subring is i... |
| rngunsnply 42218 | Adjoining one element to a... |
| flcidc 42219 | Finite linear combinations... |
| algstr 42222 | Lemma to shorten proofs of... |
| algbase 42223 | The base set of a construc... |
| algaddg 42224 | The additive operation of ... |
| algmulr 42225 | The multiplicative operati... |
| algsca 42226 | The set of scalars of a co... |
| algvsca 42227 | The scalar product operati... |
| mendval 42228 | Value of the module endomo... |
| mendbas 42229 | Base set of the module end... |
| mendplusgfval 42230 | Addition in the module end... |
| mendplusg 42231 | A specific addition in the... |
| mendmulrfval 42232 | Multiplication in the modu... |
| mendmulr 42233 | A specific multiplication ... |
| mendsca 42234 | The module endomorphism al... |
| mendvscafval 42235 | Scalar multiplication in t... |
| mendvsca 42236 | A specific scalar multipli... |
| mendring 42237 | The module endomorphism al... |
| mendlmod 42238 | The module endomorphism al... |
| mendassa 42239 | The module endomorphism al... |
| idomrootle 42240 | No element of an integral ... |
| idomodle 42241 | Limit on the number of ` N... |
| fiuneneq 42242 | Two finite sets of equal s... |
| idomsubgmo 42243 | The units of an integral d... |
| proot1mul 42244 | Any primitive ` N ` -th ro... |
| proot1hash 42245 | If an integral domain has ... |
| proot1ex 42246 | The complex field has prim... |
| isdomn3 42249 | Nonzero elements form a mu... |
| mon1pid 42250 | Monicity and degree of the... |
| mon1psubm 42251 | Monic polynomials are a mu... |
| deg1mhm 42252 | Homomorphic property of th... |
| cytpfn 42253 | Functionality of the cyclo... |
| cytpval 42254 | Substitutions for the Nth ... |
| fgraphopab 42255 | Express a function as a su... |
| fgraphxp 42256 | Express a function as a su... |
| hausgraph 42257 | The graph of a continuous ... |
| r1sssucd 42262 | Deductive form of ~ r1sssu... |
| iocunico 42263 | Split an open interval int... |
| iocinico 42264 | The intersection of two se... |
| iocmbl 42265 | An open-below, closed-abov... |
| cnioobibld 42266 | A bounded, continuous func... |
| arearect 42267 | The area of a rectangle wh... |
| areaquad 42268 | The area of a quadrilatera... |
| uniel 42269 | Two ways to say a union is... |
| unielss 42270 | Two ways to say the union ... |
| unielid 42271 | Two ways to say the union ... |
| ssunib 42272 | Two ways to say a class is... |
| rp-intrabeq 42273 | Equality theorem for supre... |
| rp-unirabeq 42274 | Equality theorem for infim... |
| onmaxnelsup 42275 | Two ways to say the maximu... |
| onsupneqmaxlim0 42276 | If the supremum of a class... |
| onsupcl2 42277 | The supremum of a set of o... |
| onuniintrab 42278 | The union of a set of ordi... |
| onintunirab 42279 | The intersection of a non-... |
| onsupnmax 42280 | If the union of a class of... |
| onsupuni 42281 | The supremum of a set of o... |
| onsupuni2 42282 | The supremum of a set of o... |
| onsupintrab 42283 | The supremum of a set of o... |
| onsupintrab2 42284 | The supremum of a set of o... |
| onsupcl3 42285 | The supremum of a set of o... |
| onsupex3 42286 | The supremum of a set of o... |
| onuniintrab2 42287 | The union of a set of ordi... |
| oninfint 42288 | The infimum of a non-empty... |
| oninfunirab 42289 | The infimum of a non-empty... |
| oninfcl2 42290 | The infimum of a non-empty... |
| onsupmaxb 42291 | The union of a class of or... |
| onexgt 42292 | For any ordinal, there is ... |
| onexomgt 42293 | For any ordinal, there is ... |
| omlimcl2 42294 | The product of a limit ord... |
| onexlimgt 42295 | For any ordinal, there is ... |
| onexoegt 42296 | For any ordinal, there is ... |
| oninfex2 42297 | The infimum of a non-empty... |
| onsupeqmax 42298 | Condition when the supremu... |
| onsupeqnmax 42299 | Condition when the supremu... |
| onsuplub 42300 | The supremum of a set of o... |
| onsupnub 42301 | An upper bound of a set of... |
| onfisupcl 42302 | Sufficient condition when ... |
| onelord 42303 | Every element of a ordinal... |
| onepsuc 42304 | Every ordinal is less than... |
| epsoon 42305 | The ordinals are strictly ... |
| epirron 42306 | The strict order on the or... |
| oneptr 42307 | The strict order on the or... |
| oneltr 42308 | The elementhood relation o... |
| oneptri 42309 | The strict, complete (line... |
| oneltri 42310 | The elementhood relation o... |
| ordeldif 42311 | Membership in the differen... |
| ordeldifsucon 42312 | Membership in the differen... |
| ordeldif1o 42313 | Membership in the differen... |
| ordne0gt0 42314 | Ordinal zero is less than ... |
| ondif1i 42315 | Ordinal zero is less than ... |
| onsucelab 42316 | The successor of every ord... |
| dflim6 42317 | A limit ordinal is a non-z... |
| limnsuc 42318 | A limit ordinal is not an ... |
| onsucss 42319 | If one ordinal is less tha... |
| ordnexbtwnsuc 42320 | For any distinct pair of o... |
| orddif0suc 42321 | For any distinct pair of o... |
| onsucf1lem 42322 | For ordinals, the successo... |
| onsucf1olem 42323 | The successor operation is... |
| onsucrn 42324 | The successor operation is... |
| onsucf1o 42325 | The successor operation is... |
| dflim7 42326 | A limit ordinal is a non-z... |
| onov0suclim 42327 | Compactly express rules fo... |
| oa0suclim 42328 | Closed form expression of ... |
| om0suclim 42329 | Closed form expression of ... |
| oe0suclim 42330 | Closed form expression of ... |
| oaomoecl 42331 | The operations of addition... |
| onsupsucismax 42332 | If the union of a set of o... |
| onsssupeqcond 42333 | If for every element of a ... |
| limexissup 42334 | An ordinal which is a limi... |
| limiun 42335 | A limit ordinal is the uni... |
| limexissupab 42336 | An ordinal which is a limi... |
| om1om1r 42337 | Ordinal one is both a left... |
| oe0rif 42338 | Ordinal zero raised to any... |
| oasubex 42339 | While subtraction can't be... |
| nnamecl 42340 | Natural numbers are closed... |
| onsucwordi 42341 | The successor operation pr... |
| oalim2cl 42342 | The ordinal sum of any ord... |
| oaltublim 42343 | Given ` C ` is a limit ord... |
| oaordi3 42344 | Ordinal addition of the sa... |
| oaord3 42345 | When the same ordinal is a... |
| 1oaomeqom 42346 | Ordinal one plus omega is ... |
| oaabsb 42347 | The right addend absorbs t... |
| oaordnrex 42348 | When omega is added on the... |
| oaordnr 42349 | When the same ordinal is a... |
| omge1 42350 | Any non-zero ordinal produ... |
| omge2 42351 | Any non-zero ordinal produ... |
| omlim2 42352 | The non-zero product with ... |
| omord2lim 42353 | Given a limit ordinal, the... |
| omord2i 42354 | Ordinal multiplication of ... |
| omord2com 42355 | When the same non-zero ord... |
| 2omomeqom 42356 | Ordinal two times omega is... |
| omnord1ex 42357 | When omega is multiplied o... |
| omnord1 42358 | When the same non-zero ord... |
| oege1 42359 | Any non-zero ordinal power... |
| oege2 42360 | Any power of an ordinal at... |
| rp-oelim2 42361 | The power of an ordinal at... |
| oeord2lim 42362 | Given a limit ordinal, the... |
| oeord2i 42363 | Ordinal exponentiation of ... |
| oeord2com 42364 | When the same base at leas... |
| nnoeomeqom 42365 | Any natural number at leas... |
| df3o2 42366 | Ordinal 3 is the unordered... |
| df3o3 42367 | Ordinal 3, fully expanded.... |
| oenord1ex 42368 | When ordinals two and thre... |
| oenord1 42369 | When two ordinals (both at... |
| oaomoencom 42370 | Ordinal addition, multipli... |
| oenassex 42371 | Ordinal two raised to two ... |
| oenass 42372 | Ordinal exponentiation is ... |
| cantnftermord 42373 | For terms of the form of a... |
| cantnfub 42374 | Given a finite number of t... |
| cantnfub2 42375 | Given a finite number of t... |
| bropabg 42376 | Equivalence for two classe... |
| cantnfresb 42377 | A Cantor normal form which... |
| cantnf2 42378 | For every ordinal, ` A ` ,... |
| oawordex2 42379 | If ` C ` is between ` A ` ... |
| nnawordexg 42380 | If an ordinal, ` B ` , is ... |
| succlg 42381 | Closure law for ordinal su... |
| dflim5 42382 | A limit ordinal is either ... |
| oacl2g 42383 | Closure law for ordinal ad... |
| onmcl 42384 | If an ordinal is less than... |
| omabs2 42385 | Ordinal multiplication by ... |
| omcl2 42386 | Closure law for ordinal mu... |
| omcl3g 42387 | Closure law for ordinal mu... |
| ordsssucb 42388 | An ordinal number is less ... |
| tfsconcatlem 42389 | Lemma for ~ tfsconcatun . ... |
| tfsconcatun 42390 | The concatenation of two t... |
| tfsconcatfn 42391 | The concatenation of two t... |
| tfsconcatfv1 42392 | An early value of the conc... |
| tfsconcatfv2 42393 | A latter value of the conc... |
| tfsconcatfv 42394 | The value of the concatena... |
| tfsconcatrn 42395 | The range of the concatena... |
| tfsconcatfo 42396 | The concatenation of two t... |
| tfsconcatb0 42397 | The concatentation with th... |
| tfsconcat0i 42398 | The concatentation with th... |
| tfsconcat0b 42399 | The concatentation with th... |
| tfsconcat00 42400 | The concatentation of two ... |
| tfsconcatrev 42401 | If the domain of a transfi... |
| tfsconcatrnss12 42402 | The range of the concatena... |
| tfsconcatrnss 42403 | The concatenation of trans... |
| tfsconcatrnsson 42404 | The concatenation of trans... |
| tfsnfin 42405 | A transfinite sequence is ... |
| rp-tfslim 42406 | The limit of a sequence of... |
| ofoafg 42407 | Addition operator for func... |
| ofoaf 42408 | Addition operator for func... |
| ofoafo 42409 | Addition operator for func... |
| ofoacl 42410 | Closure law for component ... |
| ofoaid1 42411 | Identity law for component... |
| ofoaid2 42412 | Identity law for component... |
| ofoaass 42413 | Component-wise addition of... |
| ofoacom 42414 | Component-wise addition of... |
| naddcnff 42415 | Addition operator for Cant... |
| naddcnffn 42416 | Addition operator for Cant... |
| naddcnffo 42417 | Addition of Cantor normal ... |
| naddcnfcl 42418 | Closure law for component-... |
| naddcnfcom 42419 | Component-wise ordinal add... |
| naddcnfid1 42420 | Identity law for component... |
| naddcnfid2 42421 | Identity law for component... |
| naddcnfass 42422 | Component-wise addition of... |
| onsucunifi 42423 | The successor to the union... |
| sucunisn 42424 | The successor to the union... |
| onsucunipr 42425 | The successor to the union... |
| onsucunitp 42426 | The successor to the union... |
| oaun3lem1 42427 | The class of all ordinal s... |
| oaun3lem2 42428 | The class of all ordinal s... |
| oaun3lem3 42429 | The class of all ordinal s... |
| oaun3lem4 42430 | The class of all ordinal s... |
| rp-abid 42431 | Two ways to express a clas... |
| oadif1lem 42432 | Express the set difference... |
| oadif1 42433 | Express the set difference... |
| oaun2 42434 | Ordinal addition as a unio... |
| oaun3 42435 | Ordinal addition as a unio... |
| naddov4 42436 | Alternate expression for n... |
| nadd2rabtr 42437 | The set of ordinals which ... |
| nadd2rabord 42438 | The set of ordinals which ... |
| nadd2rabex 42439 | The class of ordinals whic... |
| nadd2rabon 42440 | The set of ordinals which ... |
| nadd1rabtr 42441 | The set of ordinals which ... |
| nadd1rabord 42442 | The set of ordinals which ... |
| nadd1rabex 42443 | The class of ordinals whic... |
| nadd1rabon 42444 | The set of ordinals which ... |
| nadd1suc 42445 | Natural addition with 1 is... |
| naddsuc2 42446 | Natural addition with succ... |
| naddass1 42447 | Natural addition of ordina... |
| naddgeoa 42448 | Natural addition results i... |
| naddonnn 42449 | Natural addition with a na... |
| naddwordnexlem0 42450 | When ` A ` is the sum of a... |
| naddwordnexlem1 42451 | When ` A ` is the sum of a... |
| naddwordnexlem2 42452 | When ` A ` is the sum of a... |
| naddwordnexlem3 42453 | When ` A ` is the sum of a... |
| oawordex3 42454 | When ` A ` is the sum of a... |
| naddwordnexlem4 42455 | When ` A ` is the sum of a... |
| ordsssucim 42456 | If an ordinal is less than... |
| insucid 42457 | The intersection of a clas... |
| om2 42458 | Two ways to double an ordi... |
| oaltom 42459 | Multiplication eventually ... |
| oe2 42460 | Two ways to square an ordi... |
| omltoe 42461 | Exponentiation eventually ... |
| abeqabi 42462 | Generalized condition for ... |
| abpr 42463 | Condition for a class abst... |
| abtp 42464 | Condition for a class abst... |
| ralopabb 42465 | Restricted universal quant... |
| fpwfvss 42466 | Functions into a powerset ... |
| sdomne0 42467 | A class that strictly domi... |
| sdomne0d 42468 | A class that strictly domi... |
| safesnsupfiss 42469 | If ` B ` is a finite subse... |
| safesnsupfiub 42470 | If ` B ` is a finite subse... |
| safesnsupfidom1o 42471 | If ` B ` is a finite subse... |
| safesnsupfilb 42472 | If ` B ` is a finite subse... |
| isoeq145d 42473 | Equality deduction for iso... |
| resisoeq45d 42474 | Equality deduction for equ... |
| negslem1 42475 | An equivalence between ide... |
| nvocnvb 42476 | Equivalence to saying the ... |
| rp-brsslt 42477 | Binary relation form of a ... |
| nla0002 42478 | Extending a linear order t... |
| nla0003 42479 | Extending a linear order t... |
| nla0001 42480 | Extending a linear order t... |
| faosnf0.11b 42481 | ` B ` is called a non-limi... |
| dfno2 42482 | A surreal number, in the f... |
| onnog 42483 | Every ordinal maps to a su... |
| onnobdayg 42484 | Every ordinal maps to a su... |
| bdaybndex 42485 | Bounds formed from the bir... |
| bdaybndbday 42486 | Bounds formed from the bir... |
| onno 42487 | Every ordinal maps to a su... |
| onnoi 42488 | Every ordinal maps to a su... |
| 0no 42489 | Ordinal zero maps to a sur... |
| 1no 42490 | Ordinal one maps to a surr... |
| 2no 42491 | Ordinal two maps to a surr... |
| 3no 42492 | Ordinal three maps to a su... |
| 4no 42493 | Ordinal four maps to a sur... |
| fnimafnex 42494 | The functional image of a ... |
| nlimsuc 42495 | A successor is not a limit... |
| nlim1NEW 42496 | 1 is not a limit ordinal. ... |
| nlim2NEW 42497 | 2 is not a limit ordinal. ... |
| nlim3 42498 | 3 is not a limit ordinal. ... |
| nlim4 42499 | 4 is not a limit ordinal. ... |
| oa1un 42500 | Given ` A e. On ` , let ` ... |
| oa1cl 42501 | ` A +o 1o ` is in ` On ` .... |
| 0finon 42502 | 0 is a finite ordinal. Se... |
| 1finon 42503 | 1 is a finite ordinal. Se... |
| 2finon 42504 | 2 is a finite ordinal. Se... |
| 3finon 42505 | 3 is a finite ordinal. Se... |
| 4finon 42506 | 4 is a finite ordinal. Se... |
| finona1cl 42507 | The finite ordinals are cl... |
| finonex 42508 | The finite ordinals are a ... |
| fzunt 42509 | Union of two adjacent fini... |
| fzuntd 42510 | Union of two adjacent fini... |
| fzunt1d 42511 | Union of two overlapping f... |
| fzuntgd 42512 | Union of two adjacent or o... |
| ifpan123g 42513 | Conjunction of conditional... |
| ifpan23 42514 | Conjunction of conditional... |
| ifpdfor2 42515 | Define or in terms of cond... |
| ifporcor 42516 | Corollary of commutation o... |
| ifpdfan2 42517 | Define and with conditiona... |
| ifpancor 42518 | Corollary of commutation o... |
| ifpdfor 42519 | Define or in terms of cond... |
| ifpdfan 42520 | Define and with conditiona... |
| ifpbi2 42521 | Equivalence theorem for co... |
| ifpbi3 42522 | Equivalence theorem for co... |
| ifpim1 42523 | Restate implication as con... |
| ifpnot 42524 | Restate negated wff as con... |
| ifpid2 42525 | Restate wff as conditional... |
| ifpim2 42526 | Restate implication as con... |
| ifpbi23 42527 | Equivalence theorem for co... |
| ifpbiidcor 42528 | Restatement of ~ biid . (... |
| ifpbicor 42529 | Corollary of commutation o... |
| ifpxorcor 42530 | Corollary of commutation o... |
| ifpbi1 42531 | Equivalence theorem for co... |
| ifpnot23 42532 | Negation of conditional lo... |
| ifpnotnotb 42533 | Factor conditional logic o... |
| ifpnorcor 42534 | Corollary of commutation o... |
| ifpnancor 42535 | Corollary of commutation o... |
| ifpnot23b 42536 | Negation of conditional lo... |
| ifpbiidcor2 42537 | Restatement of ~ biid . (... |
| ifpnot23c 42538 | Negation of conditional lo... |
| ifpnot23d 42539 | Negation of conditional lo... |
| ifpdfnan 42540 | Define nand as conditional... |
| ifpdfxor 42541 | Define xor as conditional ... |
| ifpbi12 42542 | Equivalence theorem for co... |
| ifpbi13 42543 | Equivalence theorem for co... |
| ifpbi123 42544 | Equivalence theorem for co... |
| ifpidg 42545 | Restate wff as conditional... |
| ifpid3g 42546 | Restate wff as conditional... |
| ifpid2g 42547 | Restate wff as conditional... |
| ifpid1g 42548 | Restate wff as conditional... |
| ifpim23g 42549 | Restate implication as con... |
| ifpim3 42550 | Restate implication as con... |
| ifpnim1 42551 | Restate negated implicatio... |
| ifpim4 42552 | Restate implication as con... |
| ifpnim2 42553 | Restate negated implicatio... |
| ifpim123g 42554 | Implication of conditional... |
| ifpim1g 42555 | Implication of conditional... |
| ifp1bi 42556 | Substitute the first eleme... |
| ifpbi1b 42557 | When the first variable is... |
| ifpimimb 42558 | Factor conditional logic o... |
| ifpororb 42559 | Factor conditional logic o... |
| ifpananb 42560 | Factor conditional logic o... |
| ifpnannanb 42561 | Factor conditional logic o... |
| ifpor123g 42562 | Disjunction of conditional... |
| ifpimim 42563 | Consequnce of implication.... |
| ifpbibib 42564 | Factor conditional logic o... |
| ifpxorxorb 42565 | Factor conditional logic o... |
| rp-fakeimass 42566 | A special case where impli... |
| rp-fakeanorass 42567 | A special case where a mix... |
| rp-fakeoranass 42568 | A special case where a mix... |
| rp-fakeinunass 42569 | A special case where a mix... |
| rp-fakeuninass 42570 | A special case where a mix... |
| rp-isfinite5 42571 | A set is said to be finite... |
| rp-isfinite6 42572 | A set is said to be finite... |
| intabssd 42573 | When for each element ` y ... |
| eu0 42574 | There is only one empty se... |
| epelon2 42575 | Over the ordinal numbers, ... |
| ontric3g 42576 | For all ` x , y e. On ` , ... |
| dfsucon 42577 | ` A ` is called a successo... |
| snen1g 42578 | A singleton is equinumerou... |
| snen1el 42579 | A singleton is equinumerou... |
| sn1dom 42580 | A singleton is dominated b... |
| pr2dom 42581 | An unordered pair is domin... |
| tr3dom 42582 | An unordered triple is dom... |
| ensucne0 42583 | A class equinumerous to a ... |
| ensucne0OLD 42584 | A class equinumerous to a ... |
| dfom6 42585 | Let ` _om ` be defined to ... |
| infordmin 42586 | ` _om ` is the smallest in... |
| iscard4 42587 | Two ways to express the pr... |
| minregex 42588 | Given any cardinal number ... |
| minregex2 42589 | Given any cardinal number ... |
| iscard5 42590 | Two ways to express the pr... |
| elrncard 42591 | Let us define a cardinal n... |
| harval3 42592 | ` ( har `` A ) ` is the le... |
| harval3on 42593 | For any ordinal number ` A... |
| omssrncard 42594 | All natural numbers are ca... |
| 0iscard 42595 | 0 is a cardinal number. (... |
| 1iscard 42596 | 1 is a cardinal number. (... |
| omiscard 42597 | ` _om ` is a cardinal numb... |
| sucomisnotcard 42598 | ` _om +o 1o ` is not a car... |
| nna1iscard 42599 | For any natural number, th... |
| har2o 42600 | The least cardinal greater... |
| en2pr 42601 | A class is equinumerous to... |
| pr2cv 42602 | If an unordered pair is eq... |
| pr2el1 42603 | If an unordered pair is eq... |
| pr2cv1 42604 | If an unordered pair is eq... |
| pr2el2 42605 | If an unordered pair is eq... |
| pr2cv2 42606 | If an unordered pair is eq... |
| pren2 42607 | An unordered pair is equin... |
| pr2eldif1 42608 | If an unordered pair is eq... |
| pr2eldif2 42609 | If an unordered pair is eq... |
| pren2d 42610 | A pair of two distinct set... |
| aleph1min 42611 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 42612 | ` aleph ` is a strictly or... |
| alephiso3 42613 | ` aleph ` is a strictly or... |
| pwelg 42614 | The powerclass is an eleme... |
| pwinfig 42615 | The powerclass of an infin... |
| pwinfi2 42616 | The powerclass of an infin... |
| pwinfi3 42617 | The powerclass of an infin... |
| pwinfi 42618 | The powerclass of an infin... |
| fipjust 42619 | A definition of the finite... |
| cllem0 42620 | The class of all sets with... |
| superficl 42621 | The class of all supersets... |
| superuncl 42622 | The class of all supersets... |
| ssficl 42623 | The class of all subsets o... |
| ssuncl 42624 | The class of all subsets o... |
| ssdifcl 42625 | The class of all subsets o... |
| sssymdifcl 42626 | The class of all subsets o... |
| fiinfi 42627 | If two classes have the fi... |
| rababg 42628 | Condition when restricted ... |
| elinintab 42629 | Two ways of saying a set i... |
| elmapintrab 42630 | Two ways to say a set is a... |
| elinintrab 42631 | Two ways of saying a set i... |
| inintabss 42632 | Upper bound on intersectio... |
| inintabd 42633 | Value of the intersection ... |
| xpinintabd 42634 | Value of the intersection ... |
| relintabex 42635 | If the intersection of a c... |
| elcnvcnvintab 42636 | Two ways of saying a set i... |
| relintab 42637 | Value of the intersection ... |
| nonrel 42638 | A non-relation is equal to... |
| elnonrel 42639 | Only an ordered pair where... |
| cnvssb 42640 | Subclass theorem for conve... |
| relnonrel 42641 | The non-relation part of a... |
| cnvnonrel 42642 | The converse of the non-re... |
| brnonrel 42643 | A non-relation cannot rela... |
| dmnonrel 42644 | The domain of the non-rela... |
| rnnonrel 42645 | The range of the non-relat... |
| resnonrel 42646 | A restriction of the non-r... |
| imanonrel 42647 | An image under the non-rel... |
| cononrel1 42648 | Composition with the non-r... |
| cononrel2 42649 | Composition with the non-r... |
| elmapintab 42650 | Two ways to say a set is a... |
| fvnonrel 42651 | The function value of any ... |
| elinlem 42652 | Two ways to say a set is a... |
| elcnvcnvlem 42653 | Two ways to say a set is a... |
| cnvcnvintabd 42654 | Value of the relationship ... |
| elcnvlem 42655 | Two ways to say a set is a... |
| elcnvintab 42656 | Two ways of saying a set i... |
| cnvintabd 42657 | Value of the converse of t... |
| undmrnresiss 42658 | Two ways of saying the ide... |
| reflexg 42659 | Two ways of saying a relat... |
| cnvssco 42660 | A condition weaker than re... |
| refimssco 42661 | Reflexive relations are su... |
| cleq2lem 42662 | Equality implies bijection... |
| cbvcllem 42663 | Change of bound variable i... |
| clublem 42664 | If a superset ` Y ` of ` X... |
| clss2lem 42665 | The closure of a property ... |
| dfid7 42666 | Definition of identity rel... |
| mptrcllem 42667 | Show two versions of a clo... |
| cotrintab 42668 | The intersection of a clas... |
| rclexi 42669 | The reflexive closure of a... |
| rtrclexlem 42670 | Existence of relation impl... |
| rtrclex 42671 | The reflexive-transitive c... |
| trclubgNEW 42672 | If a relation exists then ... |
| trclubNEW 42673 | If a relation exists then ... |
| trclexi 42674 | The transitive closure of ... |
| rtrclexi 42675 | The reflexive-transitive c... |
| clrellem 42676 | When the property ` ps ` h... |
| clcnvlem 42677 | When ` A ` , an upper boun... |
| cnvtrucl0 42678 | The converse of the trivia... |
| cnvrcl0 42679 | The converse of the reflex... |
| cnvtrcl0 42680 | The converse of the transi... |
| dmtrcl 42681 | The domain of the transiti... |
| rntrcl 42682 | The range of the transitiv... |
| dfrtrcl5 42683 | Definition of reflexive-tr... |
| trcleq2lemRP 42684 | Equality implies bijection... |
| sqrtcvallem1 42685 | Two ways of saying a compl... |
| reabsifneg 42686 | Alternate expression for t... |
| reabsifnpos 42687 | Alternate expression for t... |
| reabsifpos 42688 | Alternate expression for t... |
| reabsifnneg 42689 | Alternate expression for t... |
| reabssgn 42690 | Alternate expression for t... |
| sqrtcvallem2 42691 | Equivalent to saying that ... |
| sqrtcvallem3 42692 | Equivalent to saying that ... |
| sqrtcvallem4 42693 | Equivalent to saying that ... |
| sqrtcvallem5 42694 | Equivalent to saying that ... |
| sqrtcval 42695 | Explicit formula for the c... |
| sqrtcval2 42696 | Explicit formula for the c... |
| resqrtval 42697 | Real part of the complex s... |
| imsqrtval 42698 | Imaginary part of the comp... |
| resqrtvalex 42699 | Example for ~ resqrtval . ... |
| imsqrtvalex 42700 | Example for ~ imsqrtval . ... |
| al3im 42701 | Version of ~ ax-4 for a ne... |
| intima0 42702 | Two ways of expressing the... |
| elimaint 42703 | Element of image of inters... |
| cnviun 42704 | Converse of indexed union.... |
| imaiun1 42705 | The image of an indexed un... |
| coiun1 42706 | Composition with an indexe... |
| elintima 42707 | Element of intersection of... |
| intimass 42708 | The image under the inters... |
| intimass2 42709 | The image under the inters... |
| intimag 42710 | Requirement for the image ... |
| intimasn 42711 | Two ways to express the im... |
| intimasn2 42712 | Two ways to express the im... |
| ss2iundf 42713 | Subclass theorem for index... |
| ss2iundv 42714 | Subclass theorem for index... |
| cbviuneq12df 42715 | Rule used to change the bo... |
| cbviuneq12dv 42716 | Rule used to change the bo... |
| conrel1d 42717 | Deduction about compositio... |
| conrel2d 42718 | Deduction about compositio... |
| trrelind 42719 | The intersection of transi... |
| xpintrreld 42720 | The intersection of a tran... |
| restrreld 42721 | The restriction of a trans... |
| trrelsuperreldg 42722 | Concrete construction of a... |
| trficl 42723 | The class of all transitiv... |
| cnvtrrel 42724 | The converse of a transiti... |
| trrelsuperrel2dg 42725 | Concrete construction of a... |
| dfrcl2 42728 | Reflexive closure of a rel... |
| dfrcl3 42729 | Reflexive closure of a rel... |
| dfrcl4 42730 | Reflexive closure of a rel... |
| relexp2 42731 | A set operated on by the r... |
| relexpnul 42732 | If the domain and range of... |
| eliunov2 42733 | Membership in the indexed ... |
| eltrclrec 42734 | Membership in the indexed ... |
| elrtrclrec 42735 | Membership in the indexed ... |
| briunov2 42736 | Two classes related by the... |
| brmptiunrelexpd 42737 | If two elements are connec... |
| fvmptiunrelexplb0d 42738 | If the indexed union range... |
| fvmptiunrelexplb0da 42739 | If the indexed union range... |
| fvmptiunrelexplb1d 42740 | If the indexed union range... |
| brfvid 42741 | If two elements are connec... |
| brfvidRP 42742 | If two elements are connec... |
| fvilbd 42743 | A set is a subset of its i... |
| fvilbdRP 42744 | A set is a subset of its i... |
| brfvrcld 42745 | If two elements are connec... |
| brfvrcld2 42746 | If two elements are connec... |
| fvrcllb0d 42747 | A restriction of the ident... |
| fvrcllb0da 42748 | A restriction of the ident... |
| fvrcllb1d 42749 | A set is a subset of its i... |
| brtrclrec 42750 | Two classes related by the... |
| brrtrclrec 42751 | Two classes related by the... |
| briunov2uz 42752 | Two classes related by the... |
| eliunov2uz 42753 | Membership in the indexed ... |
| ov2ssiunov2 42754 | Any particular operator va... |
| relexp0eq 42755 | The zeroth power of relati... |
| iunrelexp0 42756 | Simplification of zeroth p... |
| relexpxpnnidm 42757 | Any positive power of a Ca... |
| relexpiidm 42758 | Any power of any restricti... |
| relexpss1d 42759 | The relational power of a ... |
| comptiunov2i 42760 | The composition two indexe... |
| corclrcl 42761 | The reflexive closure is i... |
| iunrelexpmin1 42762 | The indexed union of relat... |
| relexpmulnn 42763 | With exponents limited to ... |
| relexpmulg 42764 | With ordered exponents, th... |
| trclrelexplem 42765 | The union of relational po... |
| iunrelexpmin2 42766 | The indexed union of relat... |
| relexp01min 42767 | With exponents limited to ... |
| relexp1idm 42768 | Repeated raising a relatio... |
| relexp0idm 42769 | Repeated raising a relatio... |
| relexp0a 42770 | Absorption law for zeroth ... |
| relexpxpmin 42771 | The composition of powers ... |
| relexpaddss 42772 | The composition of two pow... |
| iunrelexpuztr 42773 | The indexed union of relat... |
| dftrcl3 42774 | Transitive closure of a re... |
| brfvtrcld 42775 | If two elements are connec... |
| fvtrcllb1d 42776 | A set is a subset of its i... |
| trclfvcom 42777 | The transitive closure of ... |
| cnvtrclfv 42778 | The converse of the transi... |
| cotrcltrcl 42779 | The transitive closure is ... |
| trclimalb2 42780 | Lower bound for image unde... |
| brtrclfv2 42781 | Two ways to indicate two e... |
| trclfvdecomr 42782 | The transitive closure of ... |
| trclfvdecoml 42783 | The transitive closure of ... |
| dmtrclfvRP 42784 | The domain of the transiti... |
| rntrclfvRP 42785 | The range of the transitiv... |
| rntrclfv 42786 | The range of the transitiv... |
| dfrtrcl3 42787 | Reflexive-transitive closu... |
| brfvrtrcld 42788 | If two elements are connec... |
| fvrtrcllb0d 42789 | A restriction of the ident... |
| fvrtrcllb0da 42790 | A restriction of the ident... |
| fvrtrcllb1d 42791 | A set is a subset of its i... |
| dfrtrcl4 42792 | Reflexive-transitive closu... |
| corcltrcl 42793 | The composition of the ref... |
| cortrcltrcl 42794 | Composition with the refle... |
| corclrtrcl 42795 | Composition with the refle... |
| cotrclrcl 42796 | The composition of the ref... |
| cortrclrcl 42797 | Composition with the refle... |
| cotrclrtrcl 42798 | Composition with the refle... |
| cortrclrtrcl 42799 | The reflexive-transitive c... |
| frege77d 42800 | If the images of both ` { ... |
| frege81d 42801 | If the image of ` U ` is a... |
| frege83d 42802 | If the image of the union ... |
| frege96d 42803 | If ` C ` follows ` A ` in ... |
| frege87d 42804 | If the images of both ` { ... |
| frege91d 42805 | If ` B ` follows ` A ` in ... |
| frege97d 42806 | If ` A ` contains all elem... |
| frege98d 42807 | If ` C ` follows ` A ` and... |
| frege102d 42808 | If either ` A ` and ` C ` ... |
| frege106d 42809 | If ` B ` follows ` A ` in ... |
| frege108d 42810 | If either ` A ` and ` C ` ... |
| frege109d 42811 | If ` A ` contains all elem... |
| frege114d 42812 | If either ` R ` relates ` ... |
| frege111d 42813 | If either ` A ` and ` C ` ... |
| frege122d 42814 | If ` F ` is a function, ` ... |
| frege124d 42815 | If ` F ` is a function, ` ... |
| frege126d 42816 | If ` F ` is a function, ` ... |
| frege129d 42817 | If ` F ` is a function and... |
| frege131d 42818 | If ` F ` is a function and... |
| frege133d 42819 | If ` F ` is a function and... |
| dfxor4 42820 | Express exclusive-or in te... |
| dfxor5 42821 | Express exclusive-or in te... |
| df3or2 42822 | Express triple-or in terms... |
| df3an2 42823 | Express triple-and in term... |
| nev 42824 | Express that not every set... |
| 0pssin 42825 | Express that an intersecti... |
| dfhe2 42828 | The property of relation `... |
| dfhe3 42829 | The property of relation `... |
| heeq12 42830 | Equality law for relations... |
| heeq1 42831 | Equality law for relations... |
| heeq2 42832 | Equality law for relations... |
| sbcheg 42833 | Distribute proper substitu... |
| hess 42834 | Subclass law for relations... |
| xphe 42835 | Any Cartesian product is h... |
| 0he 42836 | The empty relation is here... |
| 0heALT 42837 | The empty relation is here... |
| he0 42838 | Any relation is hereditary... |
| unhe1 42839 | The union of two relations... |
| snhesn 42840 | Any singleton is hereditar... |
| idhe 42841 | The identity relation is h... |
| psshepw 42842 | The relation between sets ... |
| sshepw 42843 | The relation between sets ... |
| rp-simp2-frege 42846 | Simplification of triple c... |
| rp-simp2 42847 | Simplification of triple c... |
| rp-frege3g 42848 | Add antecedent to ~ ax-fre... |
| frege3 42849 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 42850 | Double-use of ~ ax-frege2 ... |
| rp-frege24 42851 | Introducing an embedded an... |
| rp-frege4g 42852 | Deduction related to distr... |
| frege4 42853 | Special case of closed for... |
| frege5 42854 | A closed form of ~ syl . ... |
| rp-7frege 42855 | Distribute antecedent and ... |
| rp-4frege 42856 | Elimination of a nested an... |
| rp-6frege 42857 | Elimination of a nested an... |
| rp-8frege 42858 | Eliminate antecedent when ... |
| rp-frege25 42859 | Closed form for ~ a1dd . ... |
| frege6 42860 | A closed form of ~ imim2d ... |
| axfrege8 42861 | Swap antecedents. Identic... |
| frege7 42862 | A closed form of ~ syl6 . ... |
| frege26 42864 | Identical to ~ idd . Prop... |
| frege27 42865 | We cannot (at the same tim... |
| frege9 42866 | Closed form of ~ syl with ... |
| frege12 42867 | A closed form of ~ com23 .... |
| frege11 42868 | Elimination of a nested an... |
| frege24 42869 | Closed form for ~ a1d . D... |
| frege16 42870 | A closed form of ~ com34 .... |
| frege25 42871 | Closed form for ~ a1dd . ... |
| frege18 42872 | Closed form of a syllogism... |
| frege22 42873 | A closed form of ~ com45 .... |
| frege10 42874 | Result commuting anteceden... |
| frege17 42875 | A closed form of ~ com3l .... |
| frege13 42876 | A closed form of ~ com3r .... |
| frege14 42877 | Closed form of a deduction... |
| frege19 42878 | A closed form of ~ syl6 . ... |
| frege23 42879 | Syllogism followed by rota... |
| frege15 42880 | A closed form of ~ com4r .... |
| frege21 42881 | Replace antecedent in ante... |
| frege20 42882 | A closed form of ~ syl8 . ... |
| axfrege28 42883 | Contraposition. Identical... |
| frege29 42885 | Closed form of ~ con3d . ... |
| frege30 42886 | Commuted, closed form of ~... |
| axfrege31 42887 | Identical to ~ notnotr . ... |
| frege32 42889 | Deduce ~ con1 from ~ con3 ... |
| frege33 42890 | If ` ph ` or ` ps ` takes ... |
| frege34 42891 | If as a consequence of the... |
| frege35 42892 | Commuted, closed form of ~... |
| frege36 42893 | The case in which ` ps ` i... |
| frege37 42894 | If ` ch ` is a necessary c... |
| frege38 42895 | Identical to ~ pm2.21 . P... |
| frege39 42896 | Syllogism between ~ pm2.18... |
| frege40 42897 | Anything implies ~ pm2.18 ... |
| axfrege41 42898 | Identical to ~ notnot . A... |
| frege42 42900 | Not not ~ id . Propositio... |
| frege43 42901 | If there is a choice only ... |
| frege44 42902 | Similar to a commuted ~ pm... |
| frege45 42903 | Deduce ~ pm2.6 from ~ con1... |
| frege46 42904 | If ` ps ` holds when ` ph ... |
| frege47 42905 | Deduce consequence follows... |
| frege48 42906 | Closed form of syllogism w... |
| frege49 42907 | Closed form of deduction w... |
| frege50 42908 | Closed form of ~ jaoi . P... |
| frege51 42909 | Compare with ~ jaod . Pro... |
| axfrege52a 42910 | Justification for ~ ax-fre... |
| frege52aid 42912 | The case when the content ... |
| frege53aid 42913 | Specialization of ~ frege5... |
| frege53a 42914 | Lemma for ~ frege55a . Pr... |
| axfrege54a 42915 | Justification for ~ ax-fre... |
| frege54cor0a 42917 | Synonym for logical equiva... |
| frege54cor1a 42918 | Reflexive equality. (Cont... |
| frege55aid 42919 | Lemma for ~ frege57aid . ... |
| frege55lem1a 42920 | Necessary deduction regard... |
| frege55lem2a 42921 | Core proof of Proposition ... |
| frege55a 42922 | Proposition 55 of [Frege18... |
| frege55cor1a 42923 | Proposition 55 of [Frege18... |
| frege56aid 42924 | Lemma for ~ frege57aid . ... |
| frege56a 42925 | Proposition 56 of [Frege18... |
| frege57aid 42926 | This is the all imporant f... |
| frege57a 42927 | Analogue of ~ frege57aid .... |
| axfrege58a 42928 | Identical to ~ anifp . Ju... |
| frege58acor 42930 | Lemma for ~ frege59a . (C... |
| frege59a 42931 | A kind of Aristotelian inf... |
| frege60a 42932 | Swap antecedents of ~ ax-f... |
| frege61a 42933 | Lemma for ~ frege65a . Pr... |
| frege62a 42934 | A kind of Aristotelian inf... |
| frege63a 42935 | Proposition 63 of [Frege18... |
| frege64a 42936 | Lemma for ~ frege65a . Pr... |
| frege65a 42937 | A kind of Aristotelian inf... |
| frege66a 42938 | Swap antecedents of ~ freg... |
| frege67a 42939 | Lemma for ~ frege68a . Pr... |
| frege68a 42940 | Combination of applying a ... |
| axfrege52c 42941 | Justification for ~ ax-fre... |
| frege52b 42943 | The case when the content ... |
| frege53b 42944 | Lemma for frege102 (via ~ ... |
| axfrege54c 42945 | Reflexive equality of clas... |
| frege54b 42947 | Reflexive equality of sets... |
| frege54cor1b 42948 | Reflexive equality. (Cont... |
| frege55lem1b 42949 | Necessary deduction regard... |
| frege55lem2b 42950 | Lemma for ~ frege55b . Co... |
| frege55b 42951 | Lemma for ~ frege57b . Pr... |
| frege56b 42952 | Lemma for ~ frege57b . Pr... |
| frege57b 42953 | Analogue of ~ frege57aid .... |
| axfrege58b 42954 | If ` A. x ph ` is affirmed... |
| frege58bid 42956 | If ` A. x ph ` is affirmed... |
| frege58bcor 42957 | Lemma for ~ frege59b . (C... |
| frege59b 42958 | A kind of Aristotelian inf... |
| frege60b 42959 | Swap antecedents of ~ ax-f... |
| frege61b 42960 | Lemma for ~ frege65b . Pr... |
| frege62b 42961 | A kind of Aristotelian inf... |
| frege63b 42962 | Lemma for ~ frege91 . Pro... |
| frege64b 42963 | Lemma for ~ frege65b . Pr... |
| frege65b 42964 | A kind of Aristotelian inf... |
| frege66b 42965 | Swap antecedents of ~ freg... |
| frege67b 42966 | Lemma for ~ frege68b . Pr... |
| frege68b 42967 | Combination of applying a ... |
| frege53c 42968 | Proposition 53 of [Frege18... |
| frege54cor1c 42969 | Reflexive equality. (Cont... |
| frege55lem1c 42970 | Necessary deduction regard... |
| frege55lem2c 42971 | Core proof of Proposition ... |
| frege55c 42972 | Proposition 55 of [Frege18... |
| frege56c 42973 | Lemma for ~ frege57c . Pr... |
| frege57c 42974 | Swap order of implication ... |
| frege58c 42975 | Principle related to ~ sp ... |
| frege59c 42976 | A kind of Aristotelian inf... |
| frege60c 42977 | Swap antecedents of ~ freg... |
| frege61c 42978 | Lemma for ~ frege65c . Pr... |
| frege62c 42979 | A kind of Aristotelian inf... |
| frege63c 42980 | Analogue of ~ frege63b . ... |
| frege64c 42981 | Lemma for ~ frege65c . Pr... |
| frege65c 42982 | A kind of Aristotelian inf... |
| frege66c 42983 | Swap antecedents of ~ freg... |
| frege67c 42984 | Lemma for ~ frege68c . Pr... |
| frege68c 42985 | Combination of applying a ... |
| dffrege69 42986 | If from the proposition th... |
| frege70 42987 | Lemma for ~ frege72 . Pro... |
| frege71 42988 | Lemma for ~ frege72 . Pro... |
| frege72 42989 | If property ` A ` is hered... |
| frege73 42990 | Lemma for ~ frege87 . Pro... |
| frege74 42991 | If ` X ` has a property ` ... |
| frege75 42992 | If from the proposition th... |
| dffrege76 42993 | If from the two propositio... |
| frege77 42994 | If ` Y ` follows ` X ` in ... |
| frege78 42995 | Commuted form of ~ frege77... |
| frege79 42996 | Distributed form of ~ freg... |
| frege80 42997 | Add additional condition t... |
| frege81 42998 | If ` X ` has a property ` ... |
| frege82 42999 | Closed-form deduction base... |
| frege83 43000 | Apply commuted form of ~ f... |
| frege84 43001 | Commuted form of ~ frege81... |
| frege85 43002 | Commuted form of ~ frege77... |
| frege86 43003 | Conclusion about element o... |
| frege87 43004 | If ` Z ` is a result of an... |
| frege88 43005 | Commuted form of ~ frege87... |
| frege89 43006 | One direction of ~ dffrege... |
| frege90 43007 | Add antecedent to ~ frege8... |
| frege91 43008 | Every result of an applica... |
| frege92 43009 | Inference from ~ frege91 .... |
| frege93 43010 | Necessary condition for tw... |
| frege94 43011 | Looking one past a pair re... |
| frege95 43012 | Looking one past a pair re... |
| frege96 43013 | Every result of an applica... |
| frege97 43014 | The property of following ... |
| frege98 43015 | If ` Y ` follows ` X ` and... |
| dffrege99 43016 | If ` Z ` is identical with... |
| frege100 43017 | One direction of ~ dffrege... |
| frege101 43018 | Lemma for ~ frege102 . Pr... |
| frege102 43019 | If ` Z ` belongs to the ` ... |
| frege103 43020 | Proposition 103 of [Frege1... |
| frege104 43021 | Proposition 104 of [Frege1... |
| frege105 43022 | Proposition 105 of [Frege1... |
| frege106 43023 | Whatever follows ` X ` in ... |
| frege107 43024 | Proposition 107 of [Frege1... |
| frege108 43025 | If ` Y ` belongs to the ` ... |
| frege109 43026 | The property of belonging ... |
| frege110 43027 | Proposition 110 of [Frege1... |
| frege111 43028 | If ` Y ` belongs to the ` ... |
| frege112 43029 | Identity implies belonging... |
| frege113 43030 | Proposition 113 of [Frege1... |
| frege114 43031 | If ` X ` belongs to the ` ... |
| dffrege115 43032 | If from the circumstance t... |
| frege116 43033 | One direction of ~ dffrege... |
| frege117 43034 | Lemma for ~ frege118 . Pr... |
| frege118 43035 | Simplified application of ... |
| frege119 43036 | Lemma for ~ frege120 . Pr... |
| frege120 43037 | Simplified application of ... |
| frege121 43038 | Lemma for ~ frege122 . Pr... |
| frege122 43039 | If ` X ` is a result of an... |
| frege123 43040 | Lemma for ~ frege124 . Pr... |
| frege124 43041 | If ` X ` is a result of an... |
| frege125 43042 | Lemma for ~ frege126 . Pr... |
| frege126 43043 | If ` M ` follows ` Y ` in ... |
| frege127 43044 | Communte antecedents of ~ ... |
| frege128 43045 | Lemma for ~ frege129 . Pr... |
| frege129 43046 | If the procedure ` R ` is ... |
| frege130 43047 | Lemma for ~ frege131 . Pr... |
| frege131 43048 | If the procedure ` R ` is ... |
| frege132 43049 | Lemma for ~ frege133 . Pr... |
| frege133 43050 | If the procedure ` R ` is ... |
| enrelmap 43051 | The set of all possible re... |
| enrelmapr 43052 | The set of all possible re... |
| enmappw 43053 | The set of all mappings fr... |
| enmappwid 43054 | The set of all mappings fr... |
| rfovd 43055 | Value of the operator, ` (... |
| rfovfvd 43056 | Value of the operator, ` (... |
| rfovfvfvd 43057 | Value of the operator, ` (... |
| rfovcnvf1od 43058 | Properties of the operator... |
| rfovcnvd 43059 | Value of the converse of t... |
| rfovf1od 43060 | The value of the operator,... |
| rfovcnvfvd 43061 | Value of the converse of t... |
| fsovd 43062 | Value of the operator, ` (... |
| fsovrfovd 43063 | The operator which gives a... |
| fsovfvd 43064 | Value of the operator, ` (... |
| fsovfvfvd 43065 | Value of the operator, ` (... |
| fsovfd 43066 | The operator, ` ( A O B ) ... |
| fsovcnvlem 43067 | The ` O ` operator, which ... |
| fsovcnvd 43068 | The value of the converse ... |
| fsovcnvfvd 43069 | The value of the converse ... |
| fsovf1od 43070 | The value of ` ( A O B ) `... |
| dssmapfvd 43071 | Value of the duality opera... |
| dssmapfv2d 43072 | Value of the duality opera... |
| dssmapfv3d 43073 | Value of the duality opera... |
| dssmapnvod 43074 | For any base set ` B ` the... |
| dssmapf1od 43075 | For any base set ` B ` the... |
| dssmap2d 43076 | For any base set ` B ` the... |
| or3or 43077 | Decompose disjunction into... |
| andi3or 43078 | Distribute over triple dis... |
| uneqsn 43079 | If a union of classes is e... |
| brfvimex 43080 | If a binary relation holds... |
| brovmptimex 43081 | If a binary relation holds... |
| brovmptimex1 43082 | If a binary relation holds... |
| brovmptimex2 43083 | If a binary relation holds... |
| brcoffn 43084 | Conditions allowing the de... |
| brcofffn 43085 | Conditions allowing the de... |
| brco2f1o 43086 | Conditions allowing the de... |
| brco3f1o 43087 | Conditions allowing the de... |
| ntrclsbex 43088 | If (pseudo-)interior and (... |
| ntrclsrcomplex 43089 | The relative complement of... |
| neik0imk0p 43090 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 43091 | If an interior function is... |
| ntrkbimka 43092 | If the interiors of disjoi... |
| ntrk0kbimka 43093 | If the interiors of disjoi... |
| clsk3nimkb 43094 | If the base set is not emp... |
| clsk1indlem0 43095 | The ansatz closure functio... |
| clsk1indlem2 43096 | The ansatz closure functio... |
| clsk1indlem3 43097 | The ansatz closure functio... |
| clsk1indlem4 43098 | The ansatz closure functio... |
| clsk1indlem1 43099 | The ansatz closure functio... |
| clsk1independent 43100 | For generalized closure fu... |
| neik0pk1imk0 43101 | Kuratowski's K0' and K1 ax... |
| isotone1 43102 | Two different ways to say ... |
| isotone2 43103 | Two different ways to say ... |
| ntrk1k3eqk13 43104 | An interior function is bo... |
| ntrclsf1o 43105 | If (pseudo-)interior and (... |
| ntrclsnvobr 43106 | If (pseudo-)interior and (... |
| ntrclsiex 43107 | If (pseudo-)interior and (... |
| ntrclskex 43108 | If (pseudo-)interior and (... |
| ntrclsfv1 43109 | If (pseudo-)interior and (... |
| ntrclsfv2 43110 | If (pseudo-)interior and (... |
| ntrclselnel1 43111 | If (pseudo-)interior and (... |
| ntrclselnel2 43112 | If (pseudo-)interior and (... |
| ntrclsfv 43113 | The value of the interior ... |
| ntrclsfveq1 43114 | If interior and closure fu... |
| ntrclsfveq2 43115 | If interior and closure fu... |
| ntrclsfveq 43116 | If interior and closure fu... |
| ntrclsss 43117 | If interior and closure fu... |
| ntrclsneine0lem 43118 | If (pseudo-)interior and (... |
| ntrclsneine0 43119 | If (pseudo-)interior and (... |
| ntrclscls00 43120 | If (pseudo-)interior and (... |
| ntrclsiso 43121 | If (pseudo-)interior and (... |
| ntrclsk2 43122 | An interior function is co... |
| ntrclskb 43123 | The interiors of disjoint ... |
| ntrclsk3 43124 | The intersection of interi... |
| ntrclsk13 43125 | The interior of the inters... |
| ntrclsk4 43126 | Idempotence of the interio... |
| ntrneibex 43127 | If (pseudo-)interior and (... |
| ntrneircomplex 43128 | The relative complement of... |
| ntrneif1o 43129 | If (pseudo-)interior and (... |
| ntrneiiex 43130 | If (pseudo-)interior and (... |
| ntrneinex 43131 | If (pseudo-)interior and (... |
| ntrneicnv 43132 | If (pseudo-)interior and (... |
| ntrneifv1 43133 | If (pseudo-)interior and (... |
| ntrneifv2 43134 | If (pseudo-)interior and (... |
| ntrneiel 43135 | If (pseudo-)interior and (... |
| ntrneifv3 43136 | The value of the neighbors... |
| ntrneineine0lem 43137 | If (pseudo-)interior and (... |
| ntrneineine1lem 43138 | If (pseudo-)interior and (... |
| ntrneifv4 43139 | The value of the interior ... |
| ntrneiel2 43140 | Membership in iterated int... |
| ntrneineine0 43141 | If (pseudo-)interior and (... |
| ntrneineine1 43142 | If (pseudo-)interior and (... |
| ntrneicls00 43143 | If (pseudo-)interior and (... |
| ntrneicls11 43144 | If (pseudo-)interior and (... |
| ntrneiiso 43145 | If (pseudo-)interior and (... |
| ntrneik2 43146 | An interior function is co... |
| ntrneix2 43147 | An interior (closure) func... |
| ntrneikb 43148 | The interiors of disjoint ... |
| ntrneixb 43149 | The interiors (closures) o... |
| ntrneik3 43150 | The intersection of interi... |
| ntrneix3 43151 | The closure of the union o... |
| ntrneik13 43152 | The interior of the inters... |
| ntrneix13 43153 | The closure of the union o... |
| ntrneik4w 43154 | Idempotence of the interio... |
| ntrneik4 43155 | Idempotence of the interio... |
| clsneibex 43156 | If (pseudo-)closure and (p... |
| clsneircomplex 43157 | The relative complement of... |
| clsneif1o 43158 | If a (pseudo-)closure func... |
| clsneicnv 43159 | If a (pseudo-)closure func... |
| clsneikex 43160 | If closure and neighborhoo... |
| clsneinex 43161 | If closure and neighborhoo... |
| clsneiel1 43162 | If a (pseudo-)closure func... |
| clsneiel2 43163 | If a (pseudo-)closure func... |
| clsneifv3 43164 | Value of the neighborhoods... |
| clsneifv4 43165 | Value of the closure (inte... |
| neicvgbex 43166 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 43167 | The relative complement of... |
| neicvgf1o 43168 | If neighborhood and conver... |
| neicvgnvo 43169 | If neighborhood and conver... |
| neicvgnvor 43170 | If neighborhood and conver... |
| neicvgmex 43171 | If the neighborhoods and c... |
| neicvgnex 43172 | If the neighborhoods and c... |
| neicvgel1 43173 | A subset being an element ... |
| neicvgel2 43174 | The complement of a subset... |
| neicvgfv 43175 | The value of the neighborh... |
| ntrrn 43176 | The range of the interior ... |
| ntrf 43177 | The interior function of a... |
| ntrf2 43178 | The interior function is a... |
| ntrelmap 43179 | The interior function is a... |
| clsf2 43180 | The closure function is a ... |
| clselmap 43181 | The closure function is a ... |
| dssmapntrcls 43182 | The interior and closure o... |
| dssmapclsntr 43183 | The closure and interior o... |
| gneispa 43184 | Each point ` p ` of the ne... |
| gneispb 43185 | Given a neighborhood ` N `... |
| gneispace2 43186 | The predicate that ` F ` i... |
| gneispace3 43187 | The predicate that ` F ` i... |
| gneispace 43188 | The predicate that ` F ` i... |
| gneispacef 43189 | A generic neighborhood spa... |
| gneispacef2 43190 | A generic neighborhood spa... |
| gneispacefun 43191 | A generic neighborhood spa... |
| gneispacern 43192 | A generic neighborhood spa... |
| gneispacern2 43193 | A generic neighborhood spa... |
| gneispace0nelrn 43194 | A generic neighborhood spa... |
| gneispace0nelrn2 43195 | A generic neighborhood spa... |
| gneispace0nelrn3 43196 | A generic neighborhood spa... |
| gneispaceel 43197 | Every neighborhood of a po... |
| gneispaceel2 43198 | Every neighborhood of a po... |
| gneispacess 43199 | All supersets of a neighbo... |
| gneispacess2 43200 | All supersets of a neighbo... |
| k0004lem1 43201 | Application of ~ ssin to r... |
| k0004lem2 43202 | A mapping with a particula... |
| k0004lem3 43203 | When the value of a mappin... |
| k0004val 43204 | The topological simplex of... |
| k0004ss1 43205 | The topological simplex of... |
| k0004ss2 43206 | The topological simplex of... |
| k0004ss3 43207 | The topological simplex of... |
| k0004val0 43208 | The topological simplex of... |
| inductionexd 43209 | Simple induction example. ... |
| wwlemuld 43210 | Natural deduction form of ... |
| leeq1d 43211 | Specialization of ~ breq1d... |
| leeq2d 43212 | Specialization of ~ breq2d... |
| absmulrposd 43213 | Specialization of absmuld ... |
| imadisjld 43214 | Natural dduction form of o... |
| imadisjlnd 43215 | Natural deduction form of ... |
| wnefimgd 43216 | The image of a mapping fro... |
| fco2d 43217 | Natural deduction form of ... |
| wfximgfd 43218 | The value of a function on... |
| extoimad 43219 | If |f(x)| <= C for all x t... |
| imo72b2lem0 43220 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 43221 | Natural deduction form of ... |
| imo72b2lem2 43222 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 43223 | Natural deduction form of ... |
| imo72b2lem1 43224 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 43225 | 'Less than or equal to' re... |
| lemuldiv4d 43226 | 'Less than or equal to' re... |
| imo72b2 43227 | IMO 1972 B2. (14th Intern... |
| int-addcomd 43228 | AdditionCommutativity gene... |
| int-addassocd 43229 | AdditionAssociativity gene... |
| int-addsimpd 43230 | AdditionSimplification gen... |
| int-mulcomd 43231 | MultiplicationCommutativit... |
| int-mulassocd 43232 | MultiplicationAssociativit... |
| int-mulsimpd 43233 | MultiplicationSimplificati... |
| int-leftdistd 43234 | AdditionMultiplicationLeft... |
| int-rightdistd 43235 | AdditionMultiplicationRigh... |
| int-sqdefd 43236 | SquareDefinition generator... |
| int-mul11d 43237 | First MultiplicationOne ge... |
| int-mul12d 43238 | Second MultiplicationOne g... |
| int-add01d 43239 | First AdditionZero generat... |
| int-add02d 43240 | Second AdditionZero genera... |
| int-sqgeq0d 43241 | SquareGEQZero generator ru... |
| int-eqprincd 43242 | PrincipleOfEquality genera... |
| int-eqtransd 43243 | EqualityTransitivity gener... |
| int-eqmvtd 43244 | EquMoveTerm generator rule... |
| int-eqineqd 43245 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 43246 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 43247 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 43248 | SecondPrincipleOfInequalit... |
| int-ineqtransd 43249 | InequalityTransitivity gen... |
| unitadd 43250 | Theorem used in conjunctio... |
| gsumws3 43251 | Valuation of a length 3 wo... |
| gsumws4 43252 | Valuation of a length 4 wo... |
| amgm2d 43253 | Arithmetic-geometric mean ... |
| amgm3d 43254 | Arithmetic-geometric mean ... |
| amgm4d 43255 | Arithmetic-geometric mean ... |
| spALT 43256 | ~ sp can be proven from th... |
| elnelneqd 43257 | Two classes are not equal ... |
| elnelneq2d 43258 | Two classes are not equal ... |
| rr-spce 43259 | Prove an existential. (Co... |
| rexlimdvaacbv 43260 | Unpack a restricted existe... |
| rexlimddvcbvw 43261 | Unpack a restricted existe... |
| rexlimddvcbv 43262 | Unpack a restricted existe... |
| rr-elrnmpt3d 43263 | Elementhood in an image se... |
| finnzfsuppd 43264 | If a function is zero outs... |
| rr-phpd 43265 | Equivalent of ~ php withou... |
| suceqd 43266 | Deduction associated with ... |
| tfindsd 43267 | Deduction associated with ... |
| mnringvald 43270 | Value of the monoid ring f... |
| mnringnmulrd 43271 | Components of a monoid rin... |
| mnringnmulrdOLD 43272 | Obsolete version of ~ mnri... |
| mnringbased 43273 | The base set of a monoid r... |
| mnringbasedOLD 43274 | Obsolete version of ~ mnri... |
| mnringbaserd 43275 | The base set of a monoid r... |
| mnringelbased 43276 | Membership in the base set... |
| mnringbasefd 43277 | Elements of a monoid ring ... |
| mnringbasefsuppd 43278 | Elements of a monoid ring ... |
| mnringaddgd 43279 | The additive operation of ... |
| mnringaddgdOLD 43280 | Obsolete version of ~ mnri... |
| mnring0gd 43281 | The additive identity of a... |
| mnring0g2d 43282 | The additive identity of a... |
| mnringmulrd 43283 | The ring product of a mono... |
| mnringscad 43284 | The scalar ring of a monoi... |
| mnringscadOLD 43285 | Obsolete version of ~ mnri... |
| mnringvscad 43286 | The scalar product of a mo... |
| mnringvscadOLD 43287 | Obsolete version of ~ mnri... |
| mnringlmodd 43288 | Monoid rings are left modu... |
| mnringmulrvald 43289 | Value of multiplication in... |
| mnringmulrcld 43290 | Monoid rings are closed un... |
| gru0eld 43291 | A nonempty Grothendieck un... |
| grusucd 43292 | Grothendieck universes are... |
| r1rankcld 43293 | Any rank of the cumulative... |
| grur1cld 43294 | Grothendieck universes are... |
| grurankcld 43295 | Grothendieck universes are... |
| grurankrcld 43296 | If a Grothendieck universe... |
| scotteqd 43299 | Equality theorem for the S... |
| scotteq 43300 | Closed form of ~ scotteqd ... |
| nfscott 43301 | Bound-variable hypothesis ... |
| scottabf 43302 | Value of the Scott operati... |
| scottab 43303 | Value of the Scott operati... |
| scottabes 43304 | Value of the Scott operati... |
| scottss 43305 | Scott's trick produces a s... |
| elscottab 43306 | An element of the output o... |
| scottex2 43307 | ~ scottex expressed using ... |
| scotteld 43308 | The Scott operation sends ... |
| scottelrankd 43309 | Property of a Scott's tric... |
| scottrankd 43310 | Rank of a nonempty Scott's... |
| gruscottcld 43311 | If a Grothendieck universe... |
| dfcoll2 43314 | Alternate definition of th... |
| colleq12d 43315 | Equality theorem for the c... |
| colleq1 43316 | Equality theorem for the c... |
| colleq2 43317 | Equality theorem for the c... |
| nfcoll 43318 | Bound-variable hypothesis ... |
| collexd 43319 | The output of the collecti... |
| cpcolld 43320 | Property of the collection... |
| cpcoll2d 43321 | ~ cpcolld with an extra ex... |
| grucollcld 43322 | A Grothendieck universe co... |
| ismnu 43323 | The hypothesis of this the... |
| mnuop123d 43324 | Operations of a minimal un... |
| mnussd 43325 | Minimal universes are clos... |
| mnuss2d 43326 | ~ mnussd with arguments pr... |
| mnu0eld 43327 | A nonempty minimal univers... |
| mnuop23d 43328 | Second and third operation... |
| mnupwd 43329 | Minimal universes are clos... |
| mnusnd 43330 | Minimal universes are clos... |
| mnuprssd 43331 | A minimal universe contain... |
| mnuprss2d 43332 | Special case of ~ mnuprssd... |
| mnuop3d 43333 | Third operation of a minim... |
| mnuprdlem1 43334 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 43335 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 43336 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 43337 | Lemma for ~ mnuprd . Gene... |
| mnuprd 43338 | Minimal universes are clos... |
| mnuunid 43339 | Minimal universes are clos... |
| mnuund 43340 | Minimal universes are clos... |
| mnutrcld 43341 | Minimal universes contain ... |
| mnutrd 43342 | Minimal universes are tran... |
| mnurndlem1 43343 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 43344 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 43345 | Minimal universes contain ... |
| mnugrud 43346 | Minimal universes are Grot... |
| grumnudlem 43347 | Lemma for ~ grumnud . (Co... |
| grumnud 43348 | Grothendieck universes are... |
| grumnueq 43349 | The class of Grothendieck ... |
| expandan 43350 | Expand conjunction to prim... |
| expandexn 43351 | Expand an existential quan... |
| expandral 43352 | Expand a restricted univer... |
| expandrexn 43353 | Expand a restricted existe... |
| expandrex 43354 | Expand a restricted existe... |
| expanduniss 43355 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 43356 | Express the predicate on `... |
| rr-grothprimbi 43357 | Express "every set is cont... |
| inagrud 43358 | Inaccessible levels of the... |
| inaex 43359 | Assuming the Tarski-Grothe... |
| gruex 43360 | Assuming the Tarski-Grothe... |
| rr-groth 43361 | An equivalent of ~ ax-grot... |
| rr-grothprim 43362 | An equivalent of ~ ax-grot... |
| ismnushort 43363 | Express the predicate on `... |
| dfuniv2 43364 | Alternative definition of ... |
| rr-grothshortbi 43365 | Express "every set is cont... |
| rr-grothshort 43366 | A shorter equivalent of ~ ... |
| nanorxor 43367 | 'nand' is equivalent to th... |
| undisjrab 43368 | Union of two disjoint rest... |
| iso0 43369 | The empty set is an ` R , ... |
| ssrecnpr 43370 | ` RR ` is a subset of both... |
| seff 43371 | Let set ` S ` be the real ... |
| sblpnf 43372 | The infinity ball in the a... |
| prmunb2 43373 | The primes are unbounded. ... |
| dvgrat 43374 | Ratio test for divergence ... |
| cvgdvgrat 43375 | Ratio test for convergence... |
| radcnvrat 43376 | Let ` L ` be the limit, if... |
| reldvds 43377 | The divides relation is in... |
| nznngen 43378 | All positive integers in t... |
| nzss 43379 | The set of multiples of _m... |
| nzin 43380 | The intersection of the se... |
| nzprmdif 43381 | Subtract one prime's multi... |
| hashnzfz 43382 | Special case of ~ hashdvds... |
| hashnzfz2 43383 | Special case of ~ hashnzfz... |
| hashnzfzclim 43384 | As the upper bound ` K ` o... |
| caofcan 43385 | Transfer a cancellation la... |
| ofsubid 43386 | Function analogue of ~ sub... |
| ofmul12 43387 | Function analogue of ~ mul... |
| ofdivrec 43388 | Function analogue of ~ div... |
| ofdivcan4 43389 | Function analogue of ~ div... |
| ofdivdiv2 43390 | Function analogue of ~ div... |
| lhe4.4ex1a 43391 | Example of the Fundamental... |
| dvsconst 43392 | Derivative of a constant f... |
| dvsid 43393 | Derivative of the identity... |
| dvsef 43394 | Derivative of the exponent... |
| expgrowthi 43395 | Exponential growth and dec... |
| dvconstbi 43396 | The derivative of a functi... |
| expgrowth 43397 | Exponential growth and dec... |
| bccval 43400 | Value of the generalized b... |
| bcccl 43401 | Closure of the generalized... |
| bcc0 43402 | The generalized binomial c... |
| bccp1k 43403 | Generalized binomial coeff... |
| bccm1k 43404 | Generalized binomial coeff... |
| bccn0 43405 | Generalized binomial coeff... |
| bccn1 43406 | Generalized binomial coeff... |
| bccbc 43407 | The binomial coefficient a... |
| uzmptshftfval 43408 | When ` F ` is a maps-to fu... |
| dvradcnv2 43409 | The radius of convergence ... |
| binomcxplemwb 43410 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 43411 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 43412 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 43413 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 43414 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 43415 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 43416 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 43417 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 43418 | Lemma for ~ binomcxp . Wh... |
| binomcxp 43419 | Generalize the binomial th... |
| pm10.12 43420 | Theorem *10.12 in [Whitehe... |
| pm10.14 43421 | Theorem *10.14 in [Whitehe... |
| pm10.251 43422 | Theorem *10.251 in [Whiteh... |
| pm10.252 43423 | Theorem *10.252 in [Whiteh... |
| pm10.253 43424 | Theorem *10.253 in [Whiteh... |
| albitr 43425 | Theorem *10.301 in [Whiteh... |
| pm10.42 43426 | Theorem *10.42 in [Whitehe... |
| pm10.52 43427 | Theorem *10.52 in [Whitehe... |
| pm10.53 43428 | Theorem *10.53 in [Whitehe... |
| pm10.541 43429 | Theorem *10.541 in [Whiteh... |
| pm10.542 43430 | Theorem *10.542 in [Whiteh... |
| pm10.55 43431 | Theorem *10.55 in [Whitehe... |
| pm10.56 43432 | Theorem *10.56 in [Whitehe... |
| pm10.57 43433 | Theorem *10.57 in [Whitehe... |
| 2alanimi 43434 | Removes two universal quan... |
| 2al2imi 43435 | Removes two universal quan... |
| pm11.11 43436 | Theorem *11.11 in [Whitehe... |
| pm11.12 43437 | Theorem *11.12 in [Whitehe... |
| 19.21vv 43438 | Compare Theorem *11.3 in [... |
| 2alim 43439 | Theorem *11.32 in [Whitehe... |
| 2albi 43440 | Theorem *11.33 in [Whitehe... |
| 2exim 43441 | Theorem *11.34 in [Whitehe... |
| 2exbi 43442 | Theorem *11.341 in [Whiteh... |
| spsbce-2 43443 | Theorem *11.36 in [Whitehe... |
| 19.33-2 43444 | Theorem *11.421 in [Whiteh... |
| 19.36vv 43445 | Theorem *11.43 in [Whitehe... |
| 19.31vv 43446 | Theorem *11.44 in [Whitehe... |
| 19.37vv 43447 | Theorem *11.46 in [Whitehe... |
| 19.28vv 43448 | Theorem *11.47 in [Whitehe... |
| pm11.52 43449 | Theorem *11.52 in [Whitehe... |
| aaanv 43450 | Theorem *11.56 in [Whitehe... |
| pm11.57 43451 | Theorem *11.57 in [Whitehe... |
| pm11.58 43452 | Theorem *11.58 in [Whitehe... |
| pm11.59 43453 | Theorem *11.59 in [Whitehe... |
| pm11.6 43454 | Theorem *11.6 in [Whitehea... |
| pm11.61 43455 | Theorem *11.61 in [Whitehe... |
| pm11.62 43456 | Theorem *11.62 in [Whitehe... |
| pm11.63 43457 | Theorem *11.63 in [Whitehe... |
| pm11.7 43458 | Theorem *11.7 in [Whitehea... |
| pm11.71 43459 | Theorem *11.71 in [Whitehe... |
| sbeqal1 43460 | If ` x = y ` always implie... |
| sbeqal1i 43461 | Suppose you know ` x = y `... |
| sbeqal2i 43462 | If ` x = y ` implies ` x =... |
| axc5c4c711 43463 | Proof of a theorem that ca... |
| axc5c4c711toc5 43464 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 43465 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 43466 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 43467 | Rederivation of ~ ax-11 fr... |
| axc11next 43468 | This theorem shows that, g... |
| pm13.13a 43469 | One result of theorem *13.... |
| pm13.13b 43470 | Theorem *13.13 in [Whitehe... |
| pm13.14 43471 | Theorem *13.14 in [Whitehe... |
| pm13.192 43472 | Theorem *13.192 in [Whiteh... |
| pm13.193 43473 | Theorem *13.193 in [Whiteh... |
| pm13.194 43474 | Theorem *13.194 in [Whiteh... |
| pm13.195 43475 | Theorem *13.195 in [Whiteh... |
| pm13.196a 43476 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 43477 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 43478 | Theorem *13.22 in [Whitehe... |
| iotain 43479 | Equivalence between two di... |
| iotaexeu 43480 | The iota class exists. Th... |
| iotasbc 43481 | Definition *14.01 in [Whit... |
| iotasbc2 43482 | Theorem *14.111 in [Whiteh... |
| pm14.12 43483 | Theorem *14.12 in [Whitehe... |
| pm14.122a 43484 | Theorem *14.122 in [Whiteh... |
| pm14.122b 43485 | Theorem *14.122 in [Whiteh... |
| pm14.122c 43486 | Theorem *14.122 in [Whiteh... |
| pm14.123a 43487 | Theorem *14.123 in [Whiteh... |
| pm14.123b 43488 | Theorem *14.123 in [Whiteh... |
| pm14.123c 43489 | Theorem *14.123 in [Whiteh... |
| pm14.18 43490 | Theorem *14.18 in [Whitehe... |
| iotaequ 43491 | Theorem *14.2 in [Whitehea... |
| iotavalb 43492 | Theorem *14.202 in [Whiteh... |
| iotasbc5 43493 | Theorem *14.205 in [Whiteh... |
| pm14.24 43494 | Theorem *14.24 in [Whitehe... |
| iotavalsb 43495 | Theorem *14.242 in [Whiteh... |
| sbiota1 43496 | Theorem *14.25 in [Whitehe... |
| sbaniota 43497 | Theorem *14.26 in [Whitehe... |
| eubiOLD 43498 | Obsolete proof of ~ eubi a... |
| iotasbcq 43499 | Theorem *14.272 in [Whiteh... |
| elnev 43500 | Any set that contains one ... |
| rusbcALT 43501 | A version of Russell's par... |
| compeq 43502 | Equality between two ways ... |
| compne 43503 | The complement of ` A ` is... |
| compab 43504 | Two ways of saying "the co... |
| conss2 43505 | Contrapositive law for sub... |
| conss1 43506 | Contrapositive law for sub... |
| ralbidar 43507 | More general form of ~ ral... |
| rexbidar 43508 | More general form of ~ rex... |
| dropab1 43509 | Theorem to aid use of the ... |
| dropab2 43510 | Theorem to aid use of the ... |
| ipo0 43511 | If the identity relation p... |
| ifr0 43512 | A class that is founded by... |
| ordpss 43513 | ~ ordelpss with an anteced... |
| fvsb 43514 | Explicit substitution of a... |
| fveqsb 43515 | Implicit substitution of a... |
| xpexb 43516 | A Cartesian product exists... |
| trelpss 43517 | An element of a transitive... |
| addcomgi 43518 | Generalization of commutat... |
| addrval 43528 | Value of the operation of ... |
| subrval 43529 | Value of the operation of ... |
| mulvval 43530 | Value of the operation of ... |
| addrfv 43531 | Vector addition at a value... |
| subrfv 43532 | Vector subtraction at a va... |
| mulvfv 43533 | Scalar multiplication at a... |
| addrfn 43534 | Vector addition produces a... |
| subrfn 43535 | Vector subtraction produce... |
| mulvfn 43536 | Scalar multiplication prod... |
| addrcom 43537 | Vector addition is commuta... |
| idiALT 43541 | Placeholder for ~ idi . T... |
| exbir 43542 | Exportation implication al... |
| 3impexpbicom 43543 | Version of ~ 3impexp where... |
| 3impexpbicomi 43544 | Inference associated with ... |
| bi1imp 43545 | Importation inference simi... |
| bi2imp 43546 | Importation inference simi... |
| bi3impb 43547 | Similar to ~ 3impb with im... |
| bi3impa 43548 | Similar to ~ 3impa with im... |
| bi23impib 43549 | ~ 3impib with the inner im... |
| bi13impib 43550 | ~ 3impib with the outer im... |
| bi123impib 43551 | ~ 3impib with the implicat... |
| bi13impia 43552 | ~ 3impia with the outer im... |
| bi123impia 43553 | ~ 3impia with the implicat... |
| bi33imp12 43554 | ~ 3imp with innermost impl... |
| bi23imp13 43555 | ~ 3imp with middle implica... |
| bi13imp23 43556 | ~ 3imp with outermost impl... |
| bi13imp2 43557 | Similar to ~ 3imp except t... |
| bi12imp3 43558 | Similar to ~ 3imp except a... |
| bi23imp1 43559 | Similar to ~ 3imp except a... |
| bi123imp0 43560 | Similar to ~ 3imp except a... |
| 4animp1 43561 | A single hypothesis unific... |
| 4an31 43562 | A rearrangement of conjunc... |
| 4an4132 43563 | A rearrangement of conjunc... |
| expcomdg 43564 | Biconditional form of ~ ex... |
| iidn3 43565 | ~ idn3 without virtual ded... |
| ee222 43566 | ~ e222 without virtual ded... |
| ee3bir 43567 | Right-biconditional form o... |
| ee13 43568 | ~ e13 without virtual dedu... |
| ee121 43569 | ~ e121 without virtual ded... |
| ee122 43570 | ~ e122 without virtual ded... |
| ee333 43571 | ~ e333 without virtual ded... |
| ee323 43572 | ~ e323 without virtual ded... |
| 3ornot23 43573 | If the second and third di... |
| orbi1r 43574 | ~ orbi1 with order of disj... |
| 3orbi123 43575 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 43576 | Closed form of ~ syl5 . D... |
| impexpd 43577 | The following User's Proof... |
| com3rgbi 43578 | The following User's Proof... |
| impexpdcom 43579 | The following User's Proof... |
| ee1111 43580 | Non-virtual deduction form... |
| pm2.43bgbi 43581 | Logical equivalence of a 2... |
| pm2.43cbi 43582 | Logical equivalence of a 3... |
| ee233 43583 | Non-virtual deduction form... |
| imbi13 43584 | Join three logical equival... |
| ee33 43585 | Non-virtual deduction form... |
| con5 43586 | Biconditional contrapositi... |
| con5i 43587 | Inference form of ~ con5 .... |
| exlimexi 43588 | Inference similar to Theor... |
| sb5ALT 43589 | Equivalence for substituti... |
| eexinst01 43590 | ~ exinst01 without virtual... |
| eexinst11 43591 | ~ exinst11 without virtual... |
| vk15.4j 43592 | Excercise 4j of Unit 15 of... |
| notnotrALT 43593 | Converse of double negatio... |
| con3ALT2 43594 | Contraposition. Alternate... |
| ssralv2 43595 | Quantification restricted ... |
| sbc3or 43596 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 43597 | Closed form of ~ alrimi wi... |
| rspsbc2 43598 | ~ rspsbc with two quantify... |
| sbcoreleleq 43599 | Substitution of a setvar v... |
| tratrb 43600 | If a class is transitive a... |
| ordelordALT 43601 | An element of an ordinal c... |
| sbcim2g 43602 | Distribution of class subs... |
| sbcbi 43603 | Implication form of ~ sbcb... |
| trsbc 43604 | Formula-building inference... |
| truniALT 43605 | The union of a class of tr... |
| onfrALTlem5 43606 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 43607 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 43608 | Lemma for ~ onfrALT . (Co... |
| ggen31 43609 | ~ gen31 without virtual de... |
| onfrALTlem2 43610 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 43611 | A theorem pertaining to th... |
| onfrALTlem1 43612 | Lemma for ~ onfrALT . (Co... |
| onfrALT 43613 | The membership relation is... |
| 19.41rg 43614 | Closed form of right-to-le... |
| opelopab4 43615 | Ordered pair membership in... |
| 2pm13.193 43616 | ~ pm13.193 for two variabl... |
| hbntal 43617 | A closed form of ~ hbn . ~... |
| hbimpg 43618 | A closed form of ~ hbim . ... |
| hbalg 43619 | Closed form of ~ hbal . D... |
| hbexg 43620 | Closed form of ~ nfex . D... |
| ax6e2eq 43621 | Alternate form of ~ ax6e f... |
| ax6e2nd 43622 | If at least two sets exist... |
| ax6e2ndeq 43623 | "At least two sets exist" ... |
| 2sb5nd 43624 | Equivalence for double sub... |
| 2uasbanh 43625 | Distribute the unabbreviat... |
| 2uasban 43626 | Distribute the unabbreviat... |
| e2ebind 43627 | Absorption of an existenti... |
| elpwgded 43628 | ~ elpwgdedVD in convention... |
| trelded 43629 | Deduction form of ~ trel .... |
| jaoded 43630 | Deduction form of ~ jao . ... |
| sbtT 43631 | A substitution into a theo... |
| not12an2impnot1 43632 | If a double conjunction is... |
| in1 43635 | Inference form of ~ df-vd1... |
| iin1 43636 | ~ in1 without virtual dedu... |
| dfvd1ir 43637 | Inference form of ~ df-vd1... |
| idn1 43638 | Virtual deduction identity... |
| dfvd1imp 43639 | Left-to-right part of defi... |
| dfvd1impr 43640 | Right-to-left part of defi... |
| dfvd2 43643 | Definition of a 2-hypothes... |
| dfvd2an 43646 | Definition of a 2-hypothes... |
| dfvd2ani 43647 | Inference form of ~ dfvd2a... |
| dfvd2anir 43648 | Right-to-left inference fo... |
| dfvd2i 43649 | Inference form of ~ dfvd2 ... |
| dfvd2ir 43650 | Right-to-left inference fo... |
| dfvd3 43655 | Definition of a 3-hypothes... |
| dfvd3i 43656 | Inference form of ~ dfvd3 ... |
| dfvd3ir 43657 | Right-to-left inference fo... |
| dfvd3an 43658 | Definition of a 3-hypothes... |
| dfvd3ani 43659 | Inference form of ~ dfvd3a... |
| dfvd3anir 43660 | Right-to-left inference fo... |
| vd01 43661 | A virtual hypothesis virtu... |
| vd02 43662 | Two virtual hypotheses vir... |
| vd03 43663 | A theorem is virtually inf... |
| vd12 43664 | A virtual deduction with 1... |
| vd13 43665 | A virtual deduction with 1... |
| vd23 43666 | A virtual deduction with 2... |
| dfvd2imp 43667 | The virtual deduction form... |
| dfvd2impr 43668 | A 2-antecedent nested impl... |
| in2 43669 | The virtual deduction intr... |
| int2 43670 | The virtual deduction intr... |
| iin2 43671 | ~ in2 without virtual dedu... |
| in2an 43672 | The virtual deduction intr... |
| in3 43673 | The virtual deduction intr... |
| iin3 43674 | ~ in3 without virtual dedu... |
| in3an 43675 | The virtual deduction intr... |
| int3 43676 | The virtual deduction intr... |
| idn2 43677 | Virtual deduction identity... |
| iden2 43678 | Virtual deduction identity... |
| idn3 43679 | Virtual deduction identity... |
| gen11 43680 | Virtual deduction generali... |
| gen11nv 43681 | Virtual deduction generali... |
| gen12 43682 | Virtual deduction generali... |
| gen21 43683 | Virtual deduction generali... |
| gen21nv 43684 | Virtual deduction form of ... |
| gen31 43685 | Virtual deduction generali... |
| gen22 43686 | Virtual deduction generali... |
| ggen22 43687 | ~ gen22 without virtual de... |
| exinst 43688 | Existential Instantiation.... |
| exinst01 43689 | Existential Instantiation.... |
| exinst11 43690 | Existential Instantiation.... |
| e1a 43691 | A Virtual deduction elimin... |
| el1 43692 | A Virtual deduction elimin... |
| e1bi 43693 | Biconditional form of ~ e1... |
| e1bir 43694 | Right biconditional form o... |
| e2 43695 | A virtual deduction elimin... |
| e2bi 43696 | Biconditional form of ~ e2... |
| e2bir 43697 | Right biconditional form o... |
| ee223 43698 | ~ e223 without virtual ded... |
| e223 43699 | A virtual deduction elimin... |
| e222 43700 | A virtual deduction elimin... |
| e220 43701 | A virtual deduction elimin... |
| ee220 43702 | ~ e220 without virtual ded... |
| e202 43703 | A virtual deduction elimin... |
| ee202 43704 | ~ e202 without virtual ded... |
| e022 43705 | A virtual deduction elimin... |
| ee022 43706 | ~ e022 without virtual ded... |
| e002 43707 | A virtual deduction elimin... |
| ee002 43708 | ~ e002 without virtual ded... |
| e020 43709 | A virtual deduction elimin... |
| ee020 43710 | ~ e020 without virtual ded... |
| e200 43711 | A virtual deduction elimin... |
| ee200 43712 | ~ e200 without virtual ded... |
| e221 43713 | A virtual deduction elimin... |
| ee221 43714 | ~ e221 without virtual ded... |
| e212 43715 | A virtual deduction elimin... |
| ee212 43716 | ~ e212 without virtual ded... |
| e122 43717 | A virtual deduction elimin... |
| e112 43718 | A virtual deduction elimin... |
| ee112 43719 | ~ e112 without virtual ded... |
| e121 43720 | A virtual deduction elimin... |
| e211 43721 | A virtual deduction elimin... |
| ee211 43722 | ~ e211 without virtual ded... |
| e210 43723 | A virtual deduction elimin... |
| ee210 43724 | ~ e210 without virtual ded... |
| e201 43725 | A virtual deduction elimin... |
| ee201 43726 | ~ e201 without virtual ded... |
| e120 43727 | A virtual deduction elimin... |
| ee120 43728 | Virtual deduction rule ~ e... |
| e021 43729 | A virtual deduction elimin... |
| ee021 43730 | ~ e021 without virtual ded... |
| e012 43731 | A virtual deduction elimin... |
| ee012 43732 | ~ e012 without virtual ded... |
| e102 43733 | A virtual deduction elimin... |
| ee102 43734 | ~ e102 without virtual ded... |
| e22 43735 | A virtual deduction elimin... |
| e22an 43736 | Conjunction form of ~ e22 ... |
| ee22an 43737 | ~ e22an without virtual de... |
| e111 43738 | A virtual deduction elimin... |
| e1111 43739 | A virtual deduction elimin... |
| e110 43740 | A virtual deduction elimin... |
| ee110 43741 | ~ e110 without virtual ded... |
| e101 43742 | A virtual deduction elimin... |
| ee101 43743 | ~ e101 without virtual ded... |
| e011 43744 | A virtual deduction elimin... |
| ee011 43745 | ~ e011 without virtual ded... |
| e100 43746 | A virtual deduction elimin... |
| ee100 43747 | ~ e100 without virtual ded... |
| e010 43748 | A virtual deduction elimin... |
| ee010 43749 | ~ e010 without virtual ded... |
| e001 43750 | A virtual deduction elimin... |
| ee001 43751 | ~ e001 without virtual ded... |
| e11 43752 | A virtual deduction elimin... |
| e11an 43753 | Conjunction form of ~ e11 ... |
| ee11an 43754 | ~ e11an without virtual de... |
| e01 43755 | A virtual deduction elimin... |
| e01an 43756 | Conjunction form of ~ e01 ... |
| ee01an 43757 | ~ e01an without virtual de... |
| e10 43758 | A virtual deduction elimin... |
| e10an 43759 | Conjunction form of ~ e10 ... |
| ee10an 43760 | ~ e10an without virtual de... |
| e02 43761 | A virtual deduction elimin... |
| e02an 43762 | Conjunction form of ~ e02 ... |
| ee02an 43763 | ~ e02an without virtual de... |
| eel021old 43764 | ~ el021old without virtual... |
| el021old 43765 | A virtual deduction elimin... |
| eel132 43766 | ~ syl2an with antecedents ... |
| eel000cT 43767 | An elimination deduction. ... |
| eel0TT 43768 | An elimination deduction. ... |
| eelT00 43769 | An elimination deduction. ... |
| eelTTT 43770 | An elimination deduction. ... |
| eelT11 43771 | An elimination deduction. ... |
| eelT1 43772 | Syllogism inference combin... |
| eelT12 43773 | An elimination deduction. ... |
| eelTT1 43774 | An elimination deduction. ... |
| eelT01 43775 | An elimination deduction. ... |
| eel0T1 43776 | An elimination deduction. ... |
| eel12131 43777 | An elimination deduction. ... |
| eel2131 43778 | ~ syl2an with antecedents ... |
| eel3132 43779 | ~ syl2an with antecedents ... |
| eel0321old 43780 | ~ el0321old without virtua... |
| el0321old 43781 | A virtual deduction elimin... |
| eel2122old 43782 | ~ el2122old without virtua... |
| el2122old 43783 | A virtual deduction elimin... |
| eel0000 43784 | Elimination rule similar t... |
| eel00001 43785 | An elimination deduction. ... |
| eel00000 43786 | Elimination rule similar ~... |
| eel11111 43787 | Five-hypothesis eliminatio... |
| e12 43788 | A virtual deduction elimin... |
| e12an 43789 | Conjunction form of ~ e12 ... |
| el12 43790 | Virtual deduction form of ... |
| e20 43791 | A virtual deduction elimin... |
| e20an 43792 | Conjunction form of ~ e20 ... |
| ee20an 43793 | ~ e20an without virtual de... |
| e21 43794 | A virtual deduction elimin... |
| e21an 43795 | Conjunction form of ~ e21 ... |
| ee21an 43796 | ~ e21an without virtual de... |
| e333 43797 | A virtual deduction elimin... |
| e33 43798 | A virtual deduction elimin... |
| e33an 43799 | Conjunction form of ~ e33 ... |
| ee33an 43800 | ~ e33an without virtual de... |
| e3 43801 | Meta-connective form of ~ ... |
| e3bi 43802 | Biconditional form of ~ e3... |
| e3bir 43803 | Right biconditional form o... |
| e03 43804 | A virtual deduction elimin... |
| ee03 43805 | ~ e03 without virtual dedu... |
| e03an 43806 | Conjunction form of ~ e03 ... |
| ee03an 43807 | Conjunction form of ~ ee03... |
| e30 43808 | A virtual deduction elimin... |
| ee30 43809 | ~ e30 without virtual dedu... |
| e30an 43810 | A virtual deduction elimin... |
| ee30an 43811 | Conjunction form of ~ ee30... |
| e13 43812 | A virtual deduction elimin... |
| e13an 43813 | A virtual deduction elimin... |
| ee13an 43814 | ~ e13an without virtual de... |
| e31 43815 | A virtual deduction elimin... |
| ee31 43816 | ~ e31 without virtual dedu... |
| e31an 43817 | A virtual deduction elimin... |
| ee31an 43818 | ~ e31an without virtual de... |
| e23 43819 | A virtual deduction elimin... |
| e23an 43820 | A virtual deduction elimin... |
| ee23an 43821 | ~ e23an without virtual de... |
| e32 43822 | A virtual deduction elimin... |
| ee32 43823 | ~ e32 without virtual dedu... |
| e32an 43824 | A virtual deduction elimin... |
| ee32an 43825 | ~ e33an without virtual de... |
| e123 43826 | A virtual deduction elimin... |
| ee123 43827 | ~ e123 without virtual ded... |
| el123 43828 | A virtual deduction elimin... |
| e233 43829 | A virtual deduction elimin... |
| e323 43830 | A virtual deduction elimin... |
| e000 43831 | A virtual deduction elimin... |
| e00 43832 | Elimination rule identical... |
| e00an 43833 | Elimination rule identical... |
| eel00cT 43834 | An elimination deduction. ... |
| eelTT 43835 | An elimination deduction. ... |
| e0a 43836 | Elimination rule identical... |
| eelT 43837 | An elimination deduction. ... |
| eel0cT 43838 | An elimination deduction. ... |
| eelT0 43839 | An elimination deduction. ... |
| e0bi 43840 | Elimination rule identical... |
| e0bir 43841 | Elimination rule identical... |
| uun0.1 43842 | Convention notation form o... |
| un0.1 43843 | ` T. ` is the constant tru... |
| uunT1 43844 | A deduction unionizing a n... |
| uunT1p1 43845 | A deduction unionizing a n... |
| uunT21 43846 | A deduction unionizing a n... |
| uun121 43847 | A deduction unionizing a n... |
| uun121p1 43848 | A deduction unionizing a n... |
| uun132 43849 | A deduction unionizing a n... |
| uun132p1 43850 | A deduction unionizing a n... |
| anabss7p1 43851 | A deduction unionizing a n... |
| un10 43852 | A unionizing deduction. (... |
| un01 43853 | A unionizing deduction. (... |
| un2122 43854 | A deduction unionizing a n... |
| uun2131 43855 | A deduction unionizing a n... |
| uun2131p1 43856 | A deduction unionizing a n... |
| uunTT1 43857 | A deduction unionizing a n... |
| uunTT1p1 43858 | A deduction unionizing a n... |
| uunTT1p2 43859 | A deduction unionizing a n... |
| uunT11 43860 | A deduction unionizing a n... |
| uunT11p1 43861 | A deduction unionizing a n... |
| uunT11p2 43862 | A deduction unionizing a n... |
| uunT12 43863 | A deduction unionizing a n... |
| uunT12p1 43864 | A deduction unionizing a n... |
| uunT12p2 43865 | A deduction unionizing a n... |
| uunT12p3 43866 | A deduction unionizing a n... |
| uunT12p4 43867 | A deduction unionizing a n... |
| uunT12p5 43868 | A deduction unionizing a n... |
| uun111 43869 | A deduction unionizing a n... |
| 3anidm12p1 43870 | A deduction unionizing a n... |
| 3anidm12p2 43871 | A deduction unionizing a n... |
| uun123 43872 | A deduction unionizing a n... |
| uun123p1 43873 | A deduction unionizing a n... |
| uun123p2 43874 | A deduction unionizing a n... |
| uun123p3 43875 | A deduction unionizing a n... |
| uun123p4 43876 | A deduction unionizing a n... |
| uun2221 43877 | A deduction unionizing a n... |
| uun2221p1 43878 | A deduction unionizing a n... |
| uun2221p2 43879 | A deduction unionizing a n... |
| 3impdirp1 43880 | A deduction unionizing a n... |
| 3impcombi 43881 | A 1-hypothesis proposition... |
| trsspwALT 43882 | Virtual deduction proof of... |
| trsspwALT2 43883 | Virtual deduction proof of... |
| trsspwALT3 43884 | Short predicate calculus p... |
| sspwtr 43885 | Virtual deduction proof of... |
| sspwtrALT 43886 | Virtual deduction proof of... |
| sspwtrALT2 43887 | Short predicate calculus p... |
| pwtrVD 43888 | Virtual deduction proof of... |
| pwtrrVD 43889 | Virtual deduction proof of... |
| suctrALT 43890 | The successor of a transit... |
| snssiALTVD 43891 | Virtual deduction proof of... |
| snssiALT 43892 | If a class is an element o... |
| snsslVD 43893 | Virtual deduction proof of... |
| snssl 43894 | If a singleton is a subcla... |
| snelpwrVD 43895 | Virtual deduction proof of... |
| unipwrVD 43896 | Virtual deduction proof of... |
| unipwr 43897 | A class is a subclass of t... |
| sstrALT2VD 43898 | Virtual deduction proof of... |
| sstrALT2 43899 | Virtual deduction proof of... |
| suctrALT2VD 43900 | Virtual deduction proof of... |
| suctrALT2 43901 | Virtual deduction proof of... |
| elex2VD 43902 | Virtual deduction proof of... |
| elex22VD 43903 | Virtual deduction proof of... |
| eqsbc2VD 43904 | Virtual deduction proof of... |
| zfregs2VD 43905 | Virtual deduction proof of... |
| tpid3gVD 43906 | Virtual deduction proof of... |
| en3lplem1VD 43907 | Virtual deduction proof of... |
| en3lplem2VD 43908 | Virtual deduction proof of... |
| en3lpVD 43909 | Virtual deduction proof of... |
| simplbi2VD 43910 | Virtual deduction proof of... |
| 3ornot23VD 43911 | Virtual deduction proof of... |
| orbi1rVD 43912 | Virtual deduction proof of... |
| bitr3VD 43913 | Virtual deduction proof of... |
| 3orbi123VD 43914 | Virtual deduction proof of... |
| sbc3orgVD 43915 | Virtual deduction proof of... |
| 19.21a3con13vVD 43916 | Virtual deduction proof of... |
| exbirVD 43917 | Virtual deduction proof of... |
| exbiriVD 43918 | Virtual deduction proof of... |
| rspsbc2VD 43919 | Virtual deduction proof of... |
| 3impexpVD 43920 | Virtual deduction proof of... |
| 3impexpbicomVD 43921 | Virtual deduction proof of... |
| 3impexpbicomiVD 43922 | Virtual deduction proof of... |
| sbcoreleleqVD 43923 | Virtual deduction proof of... |
| hbra2VD 43924 | Virtual deduction proof of... |
| tratrbVD 43925 | Virtual deduction proof of... |
| al2imVD 43926 | Virtual deduction proof of... |
| syl5impVD 43927 | Virtual deduction proof of... |
| idiVD 43928 | Virtual deduction proof of... |
| ancomstVD 43929 | Closed form of ~ ancoms . ... |
| ssralv2VD 43930 | Quantification restricted ... |
| ordelordALTVD 43931 | An element of an ordinal c... |
| equncomVD 43932 | If a class equals the unio... |
| equncomiVD 43933 | Inference form of ~ equnco... |
| sucidALTVD 43934 | A set belongs to its succe... |
| sucidALT 43935 | A set belongs to its succe... |
| sucidVD 43936 | A set belongs to its succe... |
| imbi12VD 43937 | Implication form of ~ imbi... |
| imbi13VD 43938 | Join three logical equival... |
| sbcim2gVD 43939 | Distribution of class subs... |
| sbcbiVD 43940 | Implication form of ~ sbcb... |
| trsbcVD 43941 | Formula-building inference... |
| truniALTVD 43942 | The union of a class of tr... |
| ee33VD 43943 | Non-virtual deduction form... |
| trintALTVD 43944 | The intersection of a clas... |
| trintALT 43945 | The intersection of a clas... |
| undif3VD 43946 | The first equality of Exer... |
| sbcssgVD 43947 | Virtual deduction proof of... |
| csbingVD 43948 | Virtual deduction proof of... |
| onfrALTlem5VD 43949 | Virtual deduction proof of... |
| onfrALTlem4VD 43950 | Virtual deduction proof of... |
| onfrALTlem3VD 43951 | Virtual deduction proof of... |
| simplbi2comtVD 43952 | Virtual deduction proof of... |
| onfrALTlem2VD 43953 | Virtual deduction proof of... |
| onfrALTlem1VD 43954 | Virtual deduction proof of... |
| onfrALTVD 43955 | Virtual deduction proof of... |
| csbeq2gVD 43956 | Virtual deduction proof of... |
| csbsngVD 43957 | Virtual deduction proof of... |
| csbxpgVD 43958 | Virtual deduction proof of... |
| csbresgVD 43959 | Virtual deduction proof of... |
| csbrngVD 43960 | Virtual deduction proof of... |
| csbima12gALTVD 43961 | Virtual deduction proof of... |
| csbunigVD 43962 | Virtual deduction proof of... |
| csbfv12gALTVD 43963 | Virtual deduction proof of... |
| con5VD 43964 | Virtual deduction proof of... |
| relopabVD 43965 | Virtual deduction proof of... |
| 19.41rgVD 43966 | Virtual deduction proof of... |
| 2pm13.193VD 43967 | Virtual deduction proof of... |
| hbimpgVD 43968 | Virtual deduction proof of... |
| hbalgVD 43969 | Virtual deduction proof of... |
| hbexgVD 43970 | Virtual deduction proof of... |
| ax6e2eqVD 43971 | The following User's Proof... |
| ax6e2ndVD 43972 | The following User's Proof... |
| ax6e2ndeqVD 43973 | The following User's Proof... |
| 2sb5ndVD 43974 | The following User's Proof... |
| 2uasbanhVD 43975 | The following User's Proof... |
| e2ebindVD 43976 | The following User's Proof... |
| sb5ALTVD 43977 | The following User's Proof... |
| vk15.4jVD 43978 | The following User's Proof... |
| notnotrALTVD 43979 | The following User's Proof... |
| con3ALTVD 43980 | The following User's Proof... |
| elpwgdedVD 43981 | Membership in a power clas... |
| sspwimp 43982 | If a class is a subclass o... |
| sspwimpVD 43983 | The following User's Proof... |
| sspwimpcf 43984 | If a class is a subclass o... |
| sspwimpcfVD 43985 | The following User's Proof... |
| suctrALTcf 43986 | The sucessor of a transiti... |
| suctrALTcfVD 43987 | The following User's Proof... |
| suctrALT3 43988 | The successor of a transit... |
| sspwimpALT 43989 | If a class is a subclass o... |
| unisnALT 43990 | A set equals the union of ... |
| notnotrALT2 43991 | Converse of double negatio... |
| sspwimpALT2 43992 | If a class is a subclass o... |
| e2ebindALT 43993 | Absorption of an existenti... |
| ax6e2ndALT 43994 | If at least two sets exist... |
| ax6e2ndeqALT 43995 | "At least two sets exist" ... |
| 2sb5ndALT 43996 | Equivalence for double sub... |
| chordthmALT 43997 | The intersecting chords th... |
| isosctrlem1ALT 43998 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 43999 | The indexed union of conne... |
| iunconnALT 44000 | The indexed union of conne... |
| sineq0ALT 44001 | A complex number whose sin... |
| evth2f 44002 | A version of ~ evth2 using... |
| elunif 44003 | A version of ~ eluni using... |
| rzalf 44004 | A version of ~ rzal using ... |
| fvelrnbf 44005 | A version of ~ fvelrnb usi... |
| rfcnpre1 44006 | If F is a continuous funct... |
| ubelsupr 44007 | If U belongs to A and U is... |
| fsumcnf 44008 | A finite sum of functions ... |
| mulltgt0 44009 | The product of a negative ... |
| rspcegf 44010 | A version of ~ rspcev usin... |
| rabexgf 44011 | A version of ~ rabexg usin... |
| fcnre 44012 | A function continuous with... |
| sumsnd 44013 | A sum of a singleton is th... |
| evthf 44014 | A version of ~ evth using ... |
| cnfex 44015 | The class of continuous fu... |
| fnchoice 44016 | For a finite set, a choice... |
| refsumcn 44017 | A finite sum of continuous... |
| rfcnpre2 44018 | If ` F ` is a continuous f... |
| cncmpmax 44019 | When the hypothesis for th... |
| rfcnpre3 44020 | If F is a continuous funct... |
| rfcnpre4 44021 | If F is a continuous funct... |
| sumpair 44022 | Sum of two distinct comple... |
| rfcnnnub 44023 | Given a real continuous fu... |
| refsum2cnlem1 44024 | This is the core Lemma for... |
| refsum2cn 44025 | The sum of two continuus r... |
| adantlllr 44026 | Deduction adding a conjunc... |
| 3adantlr3 44027 | Deduction adding a conjunc... |
| 3adantll2 44028 | Deduction adding a conjunc... |
| 3adantll3 44029 | Deduction adding a conjunc... |
| ssnel 44030 | If not element of a set, t... |
| sncldre 44031 | A singleton is closed w.r.... |
| n0p 44032 | A polynomial with a nonzer... |
| pm2.65ni 44033 | Inference rule for proof b... |
| pwssfi 44034 | Every element of the power... |
| iuneq2df 44035 | Equality deduction for ind... |
| nnfoctb 44036 | There exists a mapping fro... |
| ssinss1d 44037 | Intersection preserves sub... |
| elpwinss 44038 | An element of the powerset... |
| unidmex 44039 | If ` F ` is a set, then ` ... |
| ndisj2 44040 | A non-disjointness conditi... |
| zenom 44041 | The set of integer numbers... |
| uzwo4 44042 | Well-ordering principle: a... |
| unisn0 44043 | The union of the singleton... |
| ssin0 44044 | If two classes are disjoin... |
| inabs3 44045 | Absorption law for interse... |
| pwpwuni 44046 | Relationship between power... |
| disjiun2 44047 | In a disjoint collection, ... |
| 0pwfi 44048 | The empty set is in any po... |
| ssinss2d 44049 | Intersection preserves sub... |
| zct 44050 | The set of integer numbers... |
| pwfin0 44051 | A finite set always belong... |
| uzct 44052 | An upper integer set is co... |
| iunxsnf 44053 | A singleton index picks ou... |
| fiiuncl 44054 | If a set is closed under t... |
| iunp1 44055 | The addition of the next s... |
| fiunicl 44056 | If a set is closed under t... |
| ixpeq2d 44057 | Equality theorem for infin... |
| disjxp1 44058 | The sets of a cartesian pr... |
| disjsnxp 44059 | The sets in the cartesian ... |
| eliind 44060 | Membership in indexed inte... |
| rspcef 44061 | Restricted existential spe... |
| inn0f 44062 | A nonempty intersection. ... |
| ixpssmapc 44063 | An infinite Cartesian prod... |
| inn0 44064 | A nonempty intersection. ... |
| elintd 44065 | Membership in class inters... |
| ssdf 44066 | A sufficient condition for... |
| brneqtrd 44067 | Substitution of equal clas... |
| ssnct 44068 | A set containing an uncoun... |
| ssuniint 44069 | Sufficient condition for b... |
| elintdv 44070 | Membership in class inters... |
| ssd 44071 | A sufficient condition for... |
| ralimralim 44072 | Introducing any antecedent... |
| snelmap 44073 | Membership of the element ... |
| xrnmnfpnf 44074 | An extended real that is n... |
| nelrnmpt 44075 | Non-membership in the rang... |
| iuneq1i 44076 | Equality theorem for index... |
| nssrex 44077 | Negation of subclass relat... |
| ssinc 44078 | Inclusion relation for a m... |
| ssdec 44079 | Inclusion relation for a m... |
| elixpconstg 44080 | Membership in an infinite ... |
| iineq1d 44081 | Equality theorem for index... |
| metpsmet 44082 | A metric is a pseudometric... |
| ixpssixp 44083 | Subclass theorem for infin... |
| ballss3 44084 | A sufficient condition for... |
| iunincfi 44085 | Given a sequence of increa... |
| nsstr 44086 | If it's not a subclass, it... |
| rexanuz3 44087 | Combine two different uppe... |
| cbvmpo2 44088 | Rule to change the second ... |
| cbvmpo1 44089 | Rule to change the first b... |
| eliuniin 44090 | Indexed union of indexed i... |
| ssabf 44091 | Subclass of a class abstra... |
| pssnssi 44092 | A proper subclass does not... |
| rabidim2 44093 | Membership in a restricted... |
| eluni2f 44094 | Membership in class union.... |
| eliin2f 44095 | Membership in indexed inte... |
| nssd 44096 | Negation of subclass relat... |
| iineq12dv 44097 | Equality deduction for ind... |
| supxrcld 44098 | The supremum of an arbitra... |
| elrestd 44099 | A sufficient condition for... |
| eliuniincex 44100 | Counterexample to show tha... |
| eliincex 44101 | Counterexample to show tha... |
| eliinid 44102 | Membership in an indexed i... |
| abssf 44103 | Class abstraction in a sub... |
| supxrubd 44104 | A member of a set of exten... |
| ssrabf 44105 | Subclass of a restricted c... |
| ssrabdf 44106 | Subclass of a restricted c... |
| eliin2 44107 | Membership in indexed inte... |
| ssrab2f 44108 | Subclass relation for a re... |
| restuni3 44109 | The underlying set of a su... |
| rabssf 44110 | Restricted class abstracti... |
| eliuniin2 44111 | Indexed union of indexed i... |
| restuni4 44112 | The underlying set of a su... |
| restuni6 44113 | The underlying set of a su... |
| restuni5 44114 | The underlying set of a su... |
| unirestss 44115 | The union of an elementwis... |
| iniin1 44116 | Indexed intersection of in... |
| iniin2 44117 | Indexed intersection of in... |
| cbvrabv2 44118 | A more general version of ... |
| cbvrabv2w 44119 | A more general version of ... |
| iinssiin 44120 | Subset implication for an ... |
| eliind2 44121 | Membership in indexed inte... |
| iinssd 44122 | Subset implication for an ... |
| rabbida2 44123 | Equivalent wff's yield equ... |
| iinexd 44124 | The existence of an indexe... |
| rabexf 44125 | Separation Scheme in terms... |
| rabbida3 44126 | Equivalent wff's yield equ... |
| r19.36vf 44127 | Restricted quantifier vers... |
| raleqd 44128 | Equality deduction for res... |
| iinssf 44129 | Subset implication for an ... |
| iinssdf 44130 | Subset implication for an ... |
| resabs2i 44131 | Absorption law for restric... |
| ssdf2 44132 | A sufficient condition for... |
| rabssd 44133 | Restricted class abstracti... |
| rexnegd 44134 | Minus a real number. (Con... |
| rexlimd3 44135 | * Inference from Theorem 1... |
| resabs1i 44136 | Absorption law for restric... |
| nel1nelin 44137 | Membership in an intersect... |
| nel2nelin 44138 | Membership in an intersect... |
| nel1nelini 44139 | Membership in an intersect... |
| nel2nelini 44140 | Membership in an intersect... |
| eliunid 44141 | Membership in indexed unio... |
| reximddv3 44142 | Deduction from Theorem 19.... |
| reximdd 44143 | Deduction from Theorem 19.... |
| unfid 44144 | The union of two finite se... |
| inopnd 44145 | The intersection of two op... |
| ss2rabdf 44146 | Deduction of restricted ab... |
| restopn3 44147 | If ` A ` is open, then ` A... |
| restopnssd 44148 | A topology restricted to a... |
| restsubel 44149 | A subset belongs in the sp... |
| toprestsubel 44150 | A subset is open in the to... |
| rabidd 44151 | An "identity" law of concr... |
| iunssdf 44152 | Subset theorem for an inde... |
| iinss2d 44153 | Subset implication for an ... |
| r19.3rzf 44154 | Restricted quantification ... |
| r19.28zf 44155 | Restricted quantifier vers... |
| iindif2f 44156 | Indexed intersection of cl... |
| ralfal 44157 | Two ways of expressing emp... |
| archd 44158 | Archimedean property of re... |
| eliund 44159 | Membership in indexed unio... |
| nimnbi 44160 | If an implication is false... |
| nimnbi2 44161 | If an implication is false... |
| notbicom 44162 | Commutative law for the ne... |
| rexeqif 44163 | Equality inference for res... |
| rspced 44164 | Restricted existential spe... |
| feq1dd 44165 | Equality deduction for fun... |
| fnresdmss 44166 | A function does not change... |
| fmptsnxp 44167 | Maps-to notation and Carte... |
| fvmpt2bd 44168 | Value of a function given ... |
| rnmptfi 44169 | The range of a function wi... |
| fresin2 44170 | Restriction of a function ... |
| ffi 44171 | A function with finite dom... |
| suprnmpt 44172 | An explicit bound for the ... |
| rnffi 44173 | The range of a function wi... |
| mptelpm 44174 | A function in maps-to nota... |
| rnmptpr 44175 | Range of a function define... |
| resmpti 44176 | Restriction of the mapping... |
| founiiun 44177 | Union expressed as an inde... |
| rnresun 44178 | Distribution law for range... |
| elrnmptf 44179 | The range of a function in... |
| rnmptssrn 44180 | Inclusion relation for two... |
| disjf1 44181 | A 1 to 1 mapping built fro... |
| rnsnf 44182 | The range of a function wh... |
| wessf1ornlem 44183 | Given a function ` F ` on ... |
| wessf1orn 44184 | Given a function ` F ` on ... |
| nelrnres 44185 | If ` A ` is not in the ran... |
| disjrnmpt2 44186 | Disjointness of the range ... |
| elrnmpt1sf 44187 | Elementhood in an image se... |
| founiiun0 44188 | Union expressed as an inde... |
| disjf1o 44189 | A bijection built from dis... |
| disjinfi 44190 | Only a finite number of di... |
| fvovco 44191 | Value of the composition o... |
| ssnnf1octb 44192 | There exists a bijection b... |
| nnf1oxpnn 44193 | There is a bijection betwe... |
| rnmptssd 44194 | The range of a function gi... |
| projf1o 44195 | A biijection from a set to... |
| fvmap 44196 | Function value for a membe... |
| fvixp2 44197 | Projection of a factor of ... |
| choicefi 44198 | For a finite set, a choice... |
| mpct 44199 | The exponentiation of a co... |
| cnmetcoval 44200 | Value of the distance func... |
| fcomptss 44201 | Express composition of two... |
| elmapsnd 44202 | Membership in a set expone... |
| mapss2 44203 | Subset inheritance for set... |
| fsneq 44204 | Equality condition for two... |
| difmap 44205 | Difference of two sets exp... |
| unirnmap 44206 | Given a subset of a set ex... |
| inmap 44207 | Intersection of two sets e... |
| fcoss 44208 | Composition of two mapping... |
| fsneqrn 44209 | Equality condition for two... |
| difmapsn 44210 | Difference of two sets exp... |
| mapssbi 44211 | Subset inheritance for set... |
| unirnmapsn 44212 | Equality theorem for a sub... |
| iunmapss 44213 | The indexed union of set e... |
| ssmapsn 44214 | A subset ` C ` of a set ex... |
| iunmapsn 44215 | The indexed union of set e... |
| absfico 44216 | Mapping domain and codomai... |
| icof 44217 | The set of left-closed rig... |
| elpmrn 44218 | The range of a partial fun... |
| imaexi 44219 | The image of a set is a se... |
| axccdom 44220 | Relax the constraint on ax... |
| dmmptdff 44221 | The domain of the mapping ... |
| dmmptdf 44222 | The domain of the mapping ... |
| elpmi2 44223 | The domain of a partial fu... |
| dmrelrnrel 44224 | A relation preserving func... |
| fvcod 44225 | Value of a function compos... |
| elrnmpoid 44226 | Membership in the range of... |
| axccd 44227 | An alternative version of ... |
| axccd2 44228 | An alternative version of ... |
| fimassd 44229 | The image of a class is a ... |
| feqresmptf 44230 | Express a restricted funct... |
| elrnmpt1d 44231 | Elementhood in an image se... |
| dmresss 44232 | The domain of a restrictio... |
| dmmptssf 44233 | The domain of a mapping is... |
| dmmptdf2 44234 | The domain of the mapping ... |
| dmuz 44235 | Domain of the upper intege... |
| fmptd2f 44236 | Domain and codomain of the... |
| mpteq1df 44237 | An equality theorem for th... |
| mpteq1dfOLD 44238 | Obsolete version of ~ mpte... |
| mptexf 44239 | If the domain of a functio... |
| fvmpt4 44240 | Value of a function given ... |
| fmptf 44241 | Functionality of the mappi... |
| resimass 44242 | The image of a restriction... |
| mptssid 44243 | The mapping operation expr... |
| mptfnd 44244 | The maps-to notation defin... |
| mpteq12daOLD 44245 | Obsolete version of ~ mpte... |
| rnmptlb 44246 | Boundness below of the ran... |
| rnmptbddlem 44247 | Boundness of the range of ... |
| rnmptbdd 44248 | Boundness of the range of ... |
| funimaeq 44249 | Membership relation for th... |
| rnmptssf 44250 | The range of a function gi... |
| rnmptbd2lem 44251 | Boundness below of the ran... |
| rnmptbd2 44252 | Boundness below of the ran... |
| infnsuprnmpt 44253 | The indexed infimum of rea... |
| suprclrnmpt 44254 | Closure of the indexed sup... |
| suprubrnmpt2 44255 | A member of a nonempty ind... |
| suprubrnmpt 44256 | A member of a nonempty ind... |
| rnmptssdf 44257 | The range of a function gi... |
| rnmptbdlem 44258 | Boundness above of the ran... |
| rnmptbd 44259 | Boundness above of the ran... |
| rnmptss2 44260 | The range of a function gi... |
| elmptima 44261 | The image of a function in... |
| ralrnmpt3 44262 | A restricted quantifier ov... |
| fvelima2 44263 | Function value in an image... |
| rnmptssbi 44264 | The range of a function gi... |
| imass2d 44265 | Subset theorem for image. ... |
| imassmpt 44266 | Membership relation for th... |
| fpmd 44267 | A total function is a part... |
| fconst7 44268 | An alternative way to expr... |
| fnmptif 44269 | Functionality and domain o... |
| dmmptif 44270 | Domain of the mapping oper... |
| mpteq2dfa 44271 | Slightly more general equa... |
| dmmpt1 44272 | The domain of the mapping ... |
| fmptff 44273 | Functionality of the mappi... |
| fvmptelcdmf 44274 | The value of a function at... |
| fmptdff 44275 | A version of ~ fmptd using... |
| fvmpt2df 44276 | Deduction version of ~ fvm... |
| rn1st 44277 | The range of a function wi... |
| rnmptssff 44278 | The range of a function gi... |
| rnmptssdff 44279 | The range of a function gi... |
| fvmpt4d 44280 | Value of a function given ... |
| sub2times 44281 | Subtracting from a number,... |
| nnxrd 44282 | A natural number is an ext... |
| nnxr 44283 | A natural number is an ext... |
| abssubrp 44284 | The distance of two distin... |
| elfzfzo 44285 | Relationship between membe... |
| oddfl 44286 | Odd number representation ... |
| abscosbd 44287 | Bound for the absolute val... |
| mul13d 44288 | Commutative/associative la... |
| negpilt0 44289 | Negative ` _pi ` is negati... |
| dstregt0 44290 | A complex number ` A ` tha... |
| subadd4b 44291 | Rearrangement of 4 terms i... |
| xrlttri5d 44292 | Not equal and not larger i... |
| neglt 44293 | The negative of a positive... |
| zltlesub 44294 | If an integer ` N ` is les... |
| divlt0gt0d 44295 | The ratio of a negative nu... |
| subsub23d 44296 | Swap subtrahend and result... |
| 2timesgt 44297 | Double of a positive real ... |
| reopn 44298 | The reals are open with re... |
| sub31 44299 | Swap the first and third t... |
| nnne1ge2 44300 | A positive integer which i... |
| lefldiveq 44301 | A closed enough, smaller r... |
| negsubdi3d 44302 | Distribution of negative o... |
| ltdiv2dd 44303 | Division of a positive num... |
| abssinbd 44304 | Bound for the absolute val... |
| halffl 44305 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 44306 | Ordering relation for a st... |
| hashssle 44307 | The size of a subset of a ... |
| lttri5d 44308 | Not equal and not larger i... |
| fzisoeu 44309 | A finite ordered set has a... |
| lt3addmuld 44310 | If three real numbers are ... |
| absnpncan2d 44311 | Triangular inequality, com... |
| fperiodmullem 44312 | A function with period ` T... |
| fperiodmul 44313 | A function with period T i... |
| upbdrech 44314 | Choice of an upper bound f... |
| lt4addmuld 44315 | If four real numbers are l... |
| absnpncan3d 44316 | Triangular inequality, com... |
| upbdrech2 44317 | Choice of an upper bound f... |
| ssfiunibd 44318 | A finite union of bounded ... |
| fzdifsuc2 44319 | Remove a successor from th... |
| fzsscn 44320 | A finite sequence of integ... |
| divcan8d 44321 | A cancellation law for div... |
| dmmcand 44322 | Cancellation law for divis... |
| fzssre 44323 | A finite sequence of integ... |
| bccld 44324 | A binomial coefficient, in... |
| leadd12dd 44325 | Addition to both sides of ... |
| fzssnn0 44326 | A finite set of sequential... |
| xreqle 44327 | Equality implies 'less tha... |
| xaddlidd 44328 | ` 0 ` is a left identity f... |
| xadd0ge 44329 | A number is less than or e... |
| elfzolem1 44330 | A member in a half-open in... |
| xrgtned 44331 | 'Greater than' implies not... |
| xrleneltd 44332 | 'Less than or equal to' an... |
| xaddcomd 44333 | The extended real addition... |
| supxrre3 44334 | The supremum of a nonempty... |
| uzfissfz 44335 | For any finite subset of t... |
| xleadd2d 44336 | Addition of extended reals... |
| suprltrp 44337 | The supremum of a nonempty... |
| xleadd1d 44338 | Addition of extended reals... |
| xreqled 44339 | Equality implies 'less tha... |
| xrgepnfd 44340 | An extended real greater t... |
| xrge0nemnfd 44341 | A nonnegative extended rea... |
| supxrgere 44342 | If a real number can be ap... |
| iuneqfzuzlem 44343 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 44344 | If two unions indexed by u... |
| xle2addd 44345 | Adding both side of two in... |
| supxrgelem 44346 | If an extended real number... |
| supxrge 44347 | If an extended real number... |
| suplesup 44348 | If any element of ` A ` ca... |
| infxrglb 44349 | The infimum of a set of ex... |
| xadd0ge2 44350 | A number is less than or e... |
| nepnfltpnf 44351 | An extended real that is n... |
| ltadd12dd 44352 | Addition to both sides of ... |
| nemnftgtmnft 44353 | An extended real that is n... |
| xrgtso 44354 | 'Greater than' is a strict... |
| rpex 44355 | The positive reals form a ... |
| xrge0ge0 44356 | A nonnegative extended rea... |
| xrssre 44357 | A subset of extended reals... |
| ssuzfz 44358 | A finite subset of the upp... |
| absfun 44359 | The absolute value is a fu... |
| infrpge 44360 | The infimum of a nonempty,... |
| xrlexaddrp 44361 | If an extended real number... |
| supsubc 44362 | The supremum function dist... |
| xralrple2 44363 | Show that ` A ` is less th... |
| nnuzdisj 44364 | The first ` N ` elements o... |
| ltdivgt1 44365 | Divsion by a number greate... |
| xrltned 44366 | 'Less than' implies not eq... |
| nnsplit 44367 | Express the set of positiv... |
| divdiv3d 44368 | Division into a fraction. ... |
| abslt2sqd 44369 | Comparison of the square o... |
| qenom 44370 | The set of rational number... |
| qct 44371 | The set of rational number... |
| xrltnled 44372 | 'Less than' in terms of 'l... |
| lenlteq 44373 | 'less than or equal to' bu... |
| xrred 44374 | An extended real that is n... |
| rr2sscn2 44375 | The cartesian square of ` ... |
| infxr 44376 | The infimum of a set of ex... |
| infxrunb2 44377 | The infimum of an unbounde... |
| infxrbnd2 44378 | The infimum of a bounded-b... |
| infleinflem1 44379 | Lemma for ~ infleinf , cas... |
| infleinflem2 44380 | Lemma for ~ infleinf , whe... |
| infleinf 44381 | If any element of ` B ` ca... |
| xralrple4 44382 | Show that ` A ` is less th... |
| xralrple3 44383 | Show that ` A ` is less th... |
| eluzelzd 44384 | A member of an upper set o... |
| suplesup2 44385 | If any element of ` A ` is... |
| recnnltrp 44386 | ` N ` is a natural number ... |
| nnn0 44387 | The set of positive intege... |
| fzct 44388 | A finite set of sequential... |
| rpgtrecnn 44389 | Any positive real number i... |
| fzossuz 44390 | A half-open integer interv... |
| infxrrefi 44391 | The real and extended real... |
| xrralrecnnle 44392 | Show that ` A ` is less th... |
| fzoct 44393 | A finite set of sequential... |
| frexr 44394 | A function taking real val... |
| nnrecrp 44395 | The reciprocal of a positi... |
| reclt0d 44396 | The reciprocal of a negati... |
| lt0neg1dd 44397 | If a number is negative, i... |
| infxrcld 44398 | The infimum of an arbitrar... |
| xrralrecnnge 44399 | Show that ` A ` is less th... |
| reclt0 44400 | The reciprocal of a negati... |
| ltmulneg 44401 | Multiplying by a negative ... |
| allbutfi 44402 | For all but finitely many.... |
| ltdiv23neg 44403 | Swap denominator with othe... |
| xreqnltd 44404 | A consequence of trichotom... |
| mnfnre2 44405 | Minus infinity is not a re... |
| zssxr 44406 | The integers are a subset ... |
| fisupclrnmpt 44407 | A nonempty finite indexed ... |
| supxrunb3 44408 | The supremum of an unbound... |
| elfzod 44409 | Membership in a half-open ... |
| fimaxre4 44410 | A nonempty finite set of r... |
| ren0 44411 | The set of reals is nonemp... |
| eluzelz2 44412 | A member of an upper set o... |
| resabs2d 44413 | Absorption law for restric... |
| uzid2 44414 | Membership of the least me... |
| supxrleubrnmpt 44415 | The supremum of a nonempty... |
| uzssre2 44416 | An upper set of integers i... |
| uzssd 44417 | Subset relationship for tw... |
| eluzd 44418 | Membership in an upper set... |
| infxrlbrnmpt2 44419 | A member of a nonempty ind... |
| xrre4 44420 | An extended real is real i... |
| uz0 44421 | The upper integers functio... |
| eluzelz2d 44422 | A member of an upper set o... |
| infleinf2 44423 | If any element in ` B ` is... |
| unb2ltle 44424 | "Unbounded below" expresse... |
| uzidd2 44425 | Membership of the least me... |
| uzssd2 44426 | Subset relationship for tw... |
| rexabslelem 44427 | An indexed set of absolute... |
| rexabsle 44428 | An indexed set of absolute... |
| allbutfiinf 44429 | Given a "for all but finit... |
| supxrrernmpt 44430 | The real and extended real... |
| suprleubrnmpt 44431 | The supremum of a nonempty... |
| infrnmptle 44432 | An indexed infimum of exte... |
| infxrunb3 44433 | The infimum of an unbounde... |
| uzn0d 44434 | The upper integers are all... |
| uzssd3 44435 | Subset relationship for tw... |
| rexabsle2 44436 | An indexed set of absolute... |
| infxrunb3rnmpt 44437 | The infimum of an unbounde... |
| supxrre3rnmpt 44438 | The indexed supremum of a ... |
| uzublem 44439 | A set of reals, indexed by... |
| uzub 44440 | A set of reals, indexed by... |
| ssrexr 44441 | A subset of the reals is a... |
| supxrmnf2 44442 | Removing minus infinity fr... |
| supxrcli 44443 | The supremum of an arbitra... |
| uzid3 44444 | Membership of the least me... |
| infxrlesupxr 44445 | The supremum of a nonempty... |
| xnegeqd 44446 | Equality of two extended n... |
| xnegrecl 44447 | The extended real negative... |
| xnegnegi 44448 | Extended real version of ~... |
| xnegeqi 44449 | Equality of two extended n... |
| nfxnegd 44450 | Deduction version of ~ nfx... |
| xnegnegd 44451 | Extended real version of ~... |
| uzred 44452 | An upper integer is a real... |
| xnegcli 44453 | Closure of extended real n... |
| supminfrnmpt 44454 | The indexed supremum of a ... |
| infxrpnf 44455 | Adding plus infinity to a ... |
| infxrrnmptcl 44456 | The infimum of an arbitrar... |
| leneg2d 44457 | Negative of one side of 'l... |
| supxrltinfxr 44458 | The supremum of the empty ... |
| max1d 44459 | A number is less than or e... |
| supxrleubrnmptf 44460 | The supremum of a nonempty... |
| nleltd 44461 | 'Not less than or equal to... |
| zxrd 44462 | An integer is an extended ... |
| infxrgelbrnmpt 44463 | The infimum of an indexed ... |
| rphalfltd 44464 | Half of a positive real is... |
| uzssz2 44465 | An upper set of integers i... |
| leneg3d 44466 | Negative of one side of 'l... |
| max2d 44467 | A number is less than or e... |
| uzn0bi 44468 | The upper integers functio... |
| xnegrecl2 44469 | If the extended real negat... |
| nfxneg 44470 | Bound-variable hypothesis ... |
| uzxrd 44471 | An upper integer is an ext... |
| infxrpnf2 44472 | Removing plus infinity fro... |
| supminfxr 44473 | The extended real suprema ... |
| infrpgernmpt 44474 | The infimum of a nonempty,... |
| xnegre 44475 | An extended real is real i... |
| xnegrecl2d 44476 | If the extended real negat... |
| uzxr 44477 | An upper integer is an ext... |
| supminfxr2 44478 | The extended real suprema ... |
| xnegred 44479 | An extended real is real i... |
| supminfxrrnmpt 44480 | The indexed supremum of a ... |
| min1d 44481 | The minimum of two numbers... |
| min2d 44482 | The minimum of two numbers... |
| pnfged 44483 | Plus infinity is an upper ... |
| xrnpnfmnf 44484 | An extended real that is n... |
| uzsscn 44485 | An upper set of integers i... |
| absimnre 44486 | The absolute value of the ... |
| uzsscn2 44487 | An upper set of integers i... |
| xrtgcntopre 44488 | The standard topologies on... |
| absimlere 44489 | The absolute value of the ... |
| rpssxr 44490 | The positive reals are a s... |
| monoordxrv 44491 | Ordering relation for a mo... |
| monoordxr 44492 | Ordering relation for a mo... |
| monoord2xrv 44493 | Ordering relation for a mo... |
| monoord2xr 44494 | Ordering relation for a mo... |
| xrpnf 44495 | An extended real is plus i... |
| xlenegcon1 44496 | Extended real version of ~... |
| xlenegcon2 44497 | Extended real version of ~... |
| pimxrneun 44498 | The preimage of a set of e... |
| caucvgbf 44499 | A function is convergent i... |
| cvgcau 44500 | A convergent function is C... |
| cvgcaule 44501 | A convergent function is C... |
| rexanuz2nf 44502 | A simple counterexample re... |
| gtnelioc 44503 | A real number larger than ... |
| ioossioc 44504 | An open interval is a subs... |
| ioondisj2 44505 | A condition for two open i... |
| ioondisj1 44506 | A condition for two open i... |
| ioogtlb 44507 | An element of a closed int... |
| evthiccabs 44508 | Extreme Value Theorem on y... |
| ltnelicc 44509 | A real number smaller than... |
| eliood 44510 | Membership in an open real... |
| iooabslt 44511 | An upper bound for the dis... |
| gtnelicc 44512 | A real number greater than... |
| iooinlbub 44513 | An open interval has empty... |
| iocgtlb 44514 | An element of a left-open ... |
| iocleub 44515 | An element of a left-open ... |
| eliccd 44516 | Membership in a closed rea... |
| eliccre 44517 | A member of a closed inter... |
| eliooshift 44518 | Element of an open interva... |
| eliocd 44519 | Membership in a left-open ... |
| icoltub 44520 | An element of a left-close... |
| eliocre 44521 | A member of a left-open ri... |
| iooltub 44522 | An element of an open inte... |
| ioontr 44523 | The interior of an interva... |
| snunioo1 44524 | The closure of one end of ... |
| lbioc 44525 | A left-open right-closed i... |
| ioomidp 44526 | The midpoint is an element... |
| iccdifioo 44527 | If the open inverval is re... |
| iccdifprioo 44528 | An open interval is the cl... |
| ioossioobi 44529 | Biconditional form of ~ io... |
| iccshift 44530 | A closed interval shifted ... |
| iccsuble 44531 | An upper bound to the dist... |
| iocopn 44532 | A left-open right-closed i... |
| eliccelioc 44533 | Membership in a closed int... |
| iooshift 44534 | An open interval shifted b... |
| iccintsng 44535 | Intersection of two adiace... |
| icoiccdif 44536 | Left-closed right-open int... |
| icoopn 44537 | A left-closed right-open i... |
| icoub 44538 | A left-closed, right-open ... |
| eliccxrd 44539 | Membership in a closed rea... |
| pnfel0pnf 44540 | ` +oo ` is a nonnegative e... |
| eliccnelico 44541 | An element of a closed int... |
| eliccelicod 44542 | A member of a closed inter... |
| ge0xrre 44543 | A nonnegative extended rea... |
| ge0lere 44544 | A nonnegative extended Rea... |
| elicores 44545 | Membership in a left-close... |
| inficc 44546 | The infimum of a nonempty ... |
| qinioo 44547 | The rational numbers are d... |
| lenelioc 44548 | A real number smaller than... |
| ioonct 44549 | A nonempty open interval i... |
| xrgtnelicc 44550 | A real number greater than... |
| iccdificc 44551 | The difference of two clos... |
| iocnct 44552 | A nonempty left-open, righ... |
| iccnct 44553 | A closed interval, with mo... |
| iooiinicc 44554 | A closed interval expresse... |
| iccgelbd 44555 | An element of a closed int... |
| iooltubd 44556 | An element of an open inte... |
| icoltubd 44557 | An element of a left-close... |
| qelioo 44558 | The rational numbers are d... |
| tgqioo2 44559 | Every open set of reals is... |
| iccleubd 44560 | An element of a closed int... |
| elioored 44561 | A member of an open interv... |
| ioogtlbd 44562 | An element of a closed int... |
| ioofun 44563 | ` (,) ` is a function. (C... |
| icomnfinre 44564 | A left-closed, right-open,... |
| sqrlearg 44565 | The square compared with i... |
| ressiocsup 44566 | If the supremum belongs to... |
| ressioosup 44567 | If the supremum does not b... |
| iooiinioc 44568 | A left-open, right-closed ... |
| ressiooinf 44569 | If the infimum does not be... |
| icogelbd 44570 | An element of a left-close... |
| iocleubd 44571 | An element of a left-open ... |
| uzinico 44572 | An upper interval of integ... |
| preimaiocmnf 44573 | Preimage of a right-closed... |
| uzinico2 44574 | An upper interval of integ... |
| uzinico3 44575 | An upper interval of integ... |
| icossico2 44576 | Condition for a closed-bel... |
| dmico 44577 | The domain of the closed-b... |
| ndmico 44578 | The closed-below, open-abo... |
| uzubioo 44579 | The upper integers are unb... |
| uzubico 44580 | The upper integers are unb... |
| uzubioo2 44581 | The upper integers are unb... |
| uzubico2 44582 | The upper integers are unb... |
| iocgtlbd 44583 | An element of a left-open ... |
| xrtgioo2 44584 | The topology on the extend... |
| tgioo4 44585 | The standard topology on t... |
| fsummulc1f 44586 | Closure of a finite sum of... |
| fsumnncl 44587 | Closure of a nonempty, fin... |
| fsumge0cl 44588 | The finite sum of nonnegat... |
| fsumf1of 44589 | Re-index a finite sum usin... |
| fsumiunss 44590 | Sum over a disjoint indexe... |
| fsumreclf 44591 | Closure of a finite sum of... |
| fsumlessf 44592 | A shorter sum of nonnegati... |
| fsumsupp0 44593 | Finite sum of function val... |
| fsumsermpt 44594 | A finite sum expressed in ... |
| fmul01 44595 | Multiplying a finite numbe... |
| fmulcl 44596 | If ' Y ' is closed under t... |
| fmuldfeqlem1 44597 | induction step for the pro... |
| fmuldfeq 44598 | X and Z are two equivalent... |
| fmul01lt1lem1 44599 | Given a finite multiplicat... |
| fmul01lt1lem2 44600 | Given a finite multiplicat... |
| fmul01lt1 44601 | Given a finite multiplicat... |
| cncfmptss 44602 | A continuous complex funct... |
| rrpsscn 44603 | The positive reals are a s... |
| mulc1cncfg 44604 | A version of ~ mulc1cncf u... |
| infrglb 44605 | The infimum of a nonempty ... |
| expcnfg 44606 | If ` F ` is a complex cont... |
| prodeq2ad 44607 | Equality deduction for pro... |
| fprodsplit1 44608 | Separate out a term in a f... |
| fprodexp 44609 | Positive integer exponenti... |
| fprodabs2 44610 | The absolute value of a fi... |
| fprod0 44611 | A finite product with a ze... |
| mccllem 44612 | * Induction step for ~ mcc... |
| mccl 44613 | A multinomial coefficient,... |
| fprodcnlem 44614 | A finite product of functi... |
| fprodcn 44615 | A finite product of functi... |
| clim1fr1 44616 | A class of sequences of fr... |
| isumneg 44617 | Negation of a converging s... |
| climrec 44618 | Limit of the reciprocal of... |
| climmulf 44619 | A version of ~ climmul usi... |
| climexp 44620 | The limit of natural power... |
| climinf 44621 | A bounded monotonic noninc... |
| climsuselem1 44622 | The subsequence index ` I ... |
| climsuse 44623 | A subsequence ` G ` of a c... |
| climrecf 44624 | A version of ~ climrec usi... |
| climneg 44625 | Complex limit of the negat... |
| climinff 44626 | A version of ~ climinf usi... |
| climdivf 44627 | Limit of the ratio of two ... |
| climreeq 44628 | If ` F ` is a real functio... |
| ellimciota 44629 | An explicit value for the ... |
| climaddf 44630 | A version of ~ climadd usi... |
| mullimc 44631 | Limit of the product of tw... |
| ellimcabssub0 44632 | An equivalent condition fo... |
| limcdm0 44633 | If a function has empty do... |
| islptre 44634 | An equivalence condition f... |
| limccog 44635 | Limit of the composition o... |
| limciccioolb 44636 | The limit of a function at... |
| climf 44637 | Express the predicate: Th... |
| mullimcf 44638 | Limit of the multiplicatio... |
| constlimc 44639 | Limit of constant function... |
| rexlim2d 44640 | Inference removing two res... |
| idlimc 44641 | Limit of the identity func... |
| divcnvg 44642 | The sequence of reciprocal... |
| limcperiod 44643 | If ` F ` is a periodic fun... |
| limcrecl 44644 | If ` F ` is a real-valued ... |
| sumnnodd 44645 | A series indexed by ` NN `... |
| lptioo2 44646 | The upper bound of an open... |
| lptioo1 44647 | The lower bound of an open... |
| elprn1 44648 | A member of an unordered p... |
| elprn2 44649 | A member of an unordered p... |
| limcmptdm 44650 | The domain of a maps-to fu... |
| clim2f 44651 | Express the predicate: Th... |
| limcicciooub 44652 | The limit of a function at... |
| ltmod 44653 | A sufficient condition for... |
| islpcn 44654 | A characterization for a l... |
| lptre2pt 44655 | If a set in the real line ... |
| limsupre 44656 | If a sequence is bounded, ... |
| limcresiooub 44657 | The left limit doesn't cha... |
| limcresioolb 44658 | The right limit doesn't ch... |
| limcleqr 44659 | If the left and the right ... |
| lptioo2cn 44660 | The upper bound of an open... |
| lptioo1cn 44661 | The lower bound of an open... |
| neglimc 44662 | Limit of the negative func... |
| addlimc 44663 | Sum of two limits. (Contr... |
| 0ellimcdiv 44664 | If the numerator converges... |
| clim2cf 44665 | Express the predicate ` F ... |
| limclner 44666 | For a limit point, both fr... |
| sublimc 44667 | Subtraction of two limits.... |
| reclimc 44668 | Limit of the reciprocal of... |
| clim0cf 44669 | Express the predicate ` F ... |
| limclr 44670 | For a limit point, both fr... |
| divlimc 44671 | Limit of the quotient of t... |
| expfac 44672 | Factorial grows faster tha... |
| climconstmpt 44673 | A constant sequence conver... |
| climresmpt 44674 | A function restricted to u... |
| climsubmpt 44675 | Limit of the difference of... |
| climsubc2mpt 44676 | Limit of the difference of... |
| climsubc1mpt 44677 | Limit of the difference of... |
| fnlimfv 44678 | The value of the limit fun... |
| climreclf 44679 | The limit of a convergent ... |
| climeldmeq 44680 | Two functions that are eve... |
| climf2 44681 | Express the predicate: Th... |
| fnlimcnv 44682 | The sequence of function v... |
| climeldmeqmpt 44683 | Two functions that are eve... |
| climfveq 44684 | Two functions that are eve... |
| clim2f2 44685 | Express the predicate: Th... |
| climfveqmpt 44686 | Two functions that are eve... |
| climd 44687 | Express the predicate: Th... |
| clim2d 44688 | The limit of complex numbe... |
| fnlimfvre 44689 | The limit function of real... |
| allbutfifvre 44690 | Given a sequence of real-v... |
| climleltrp 44691 | The limit of complex numbe... |
| fnlimfvre2 44692 | The limit function of real... |
| fnlimf 44693 | The limit function of real... |
| fnlimabslt 44694 | A sequence of function val... |
| climfveqf 44695 | Two functions that are eve... |
| climmptf 44696 | Exhibit a function ` G ` w... |
| climfveqmpt3 44697 | Two functions that are eve... |
| climeldmeqf 44698 | Two functions that are eve... |
| climreclmpt 44699 | The limit of B convergent ... |
| limsupref 44700 | If a sequence is bounded, ... |
| limsupbnd1f 44701 | If a sequence is eventuall... |
| climbddf 44702 | A converging sequence of c... |
| climeqf 44703 | Two functions that are eve... |
| climeldmeqmpt3 44704 | Two functions that are eve... |
| limsupcld 44705 | Closure of the superior li... |
| climfv 44706 | The limit of a convergent ... |
| limsupval3 44707 | The superior limit of an i... |
| climfveqmpt2 44708 | Two functions that are eve... |
| limsup0 44709 | The superior limit of the ... |
| climeldmeqmpt2 44710 | Two functions that are eve... |
| limsupresre 44711 | The supremum limit of a fu... |
| climeqmpt 44712 | Two functions that are eve... |
| climfvd 44713 | The limit of a convergent ... |
| limsuplesup 44714 | An upper bound for the sup... |
| limsupresico 44715 | The superior limit doesn't... |
| limsuppnfdlem 44716 | If the restriction of a fu... |
| limsuppnfd 44717 | If the restriction of a fu... |
| limsupresuz 44718 | If the real part of the do... |
| limsupub 44719 | If the limsup is not ` +oo... |
| limsupres 44720 | The superior limit of a re... |
| climinf2lem 44721 | A convergent, nonincreasin... |
| climinf2 44722 | A convergent, nonincreasin... |
| limsupvaluz 44723 | The superior limit, when t... |
| limsupresuz2 44724 | If the domain of a functio... |
| limsuppnflem 44725 | If the restriction of a fu... |
| limsuppnf 44726 | If the restriction of a fu... |
| limsupubuzlem 44727 | If the limsup is not ` +oo... |
| limsupubuz 44728 | For a real-valued function... |
| climinf2mpt 44729 | A bounded below, monotonic... |
| climinfmpt 44730 | A bounded below, monotonic... |
| climinf3 44731 | A convergent, nonincreasin... |
| limsupvaluzmpt 44732 | The superior limit, when t... |
| limsupequzmpt2 44733 | Two functions that are eve... |
| limsupubuzmpt 44734 | If the limsup is not ` +oo... |
| limsupmnflem 44735 | The superior limit of a fu... |
| limsupmnf 44736 | The superior limit of a fu... |
| limsupequzlem 44737 | Two functions that are eve... |
| limsupequz 44738 | Two functions that are eve... |
| limsupre2lem 44739 | Given a function on the ex... |
| limsupre2 44740 | Given a function on the ex... |
| limsupmnfuzlem 44741 | The superior limit of a fu... |
| limsupmnfuz 44742 | The superior limit of a fu... |
| limsupequzmptlem 44743 | Two functions that are eve... |
| limsupequzmpt 44744 | Two functions that are eve... |
| limsupre2mpt 44745 | Given a function on the ex... |
| limsupequzmptf 44746 | Two functions that are eve... |
| limsupre3lem 44747 | Given a function on the ex... |
| limsupre3 44748 | Given a function on the ex... |
| limsupre3mpt 44749 | Given a function on the ex... |
| limsupre3uzlem 44750 | Given a function on the ex... |
| limsupre3uz 44751 | Given a function on the ex... |
| limsupreuz 44752 | Given a function on the re... |
| limsupvaluz2 44753 | The superior limit, when t... |
| limsupreuzmpt 44754 | Given a function on the re... |
| supcnvlimsup 44755 | If a function on a set of ... |
| supcnvlimsupmpt 44756 | If a function on a set of ... |
| 0cnv 44757 | If ` (/) ` is a complex nu... |
| climuzlem 44758 | Express the predicate: Th... |
| climuz 44759 | Express the predicate: Th... |
| lmbr3v 44760 | Express the binary relatio... |
| climisp 44761 | If a sequence converges to... |
| lmbr3 44762 | Express the binary relatio... |
| climrescn 44763 | A sequence converging w.r.... |
| climxrrelem 44764 | If a sequence ranging over... |
| climxrre 44765 | If a sequence ranging over... |
| limsuplt2 44768 | The defining property of t... |
| liminfgord 44769 | Ordering property of the i... |
| limsupvald 44770 | The superior limit of a se... |
| limsupresicompt 44771 | The superior limit doesn't... |
| limsupcli 44772 | Closure of the superior li... |
| liminfgf 44773 | Closure of the inferior li... |
| liminfval 44774 | The inferior limit of a se... |
| climlimsup 44775 | A sequence of real numbers... |
| limsupge 44776 | The defining property of t... |
| liminfgval 44777 | Value of the inferior limi... |
| liminfcl 44778 | Closure of the inferior li... |
| liminfvald 44779 | The inferior limit of a se... |
| liminfval5 44780 | The inferior limit of an i... |
| limsupresxr 44781 | The superior limit of a fu... |
| liminfresxr 44782 | The inferior limit of a fu... |
| liminfval2 44783 | The superior limit, relati... |
| climlimsupcex 44784 | Counterexample for ~ climl... |
| liminfcld 44785 | Closure of the inferior li... |
| liminfresico 44786 | The inferior limit doesn't... |
| limsup10exlem 44787 | The range of the given fun... |
| limsup10ex 44788 | The superior limit of a fu... |
| liminf10ex 44789 | The inferior limit of a fu... |
| liminflelimsuplem 44790 | The superior limit is grea... |
| liminflelimsup 44791 | The superior limit is grea... |
| limsupgtlem 44792 | For any positive real, the... |
| limsupgt 44793 | Given a sequence of real n... |
| liminfresre 44794 | The inferior limit of a fu... |
| liminfresicompt 44795 | The inferior limit doesn't... |
| liminfltlimsupex 44796 | An example where the ` lim... |
| liminfgelimsup 44797 | The inferior limit is grea... |
| liminfvalxr 44798 | Alternate definition of ` ... |
| liminfresuz 44799 | If the real part of the do... |
| liminflelimsupuz 44800 | The superior limit is grea... |
| liminfvalxrmpt 44801 | Alternate definition of ` ... |
| liminfresuz2 44802 | If the domain of a functio... |
| liminfgelimsupuz 44803 | The inferior limit is grea... |
| liminfval4 44804 | Alternate definition of ` ... |
| liminfval3 44805 | Alternate definition of ` ... |
| liminfequzmpt2 44806 | Two functions that are eve... |
| liminfvaluz 44807 | Alternate definition of ` ... |
| liminf0 44808 | The inferior limit of the ... |
| limsupval4 44809 | Alternate definition of ` ... |
| liminfvaluz2 44810 | Alternate definition of ` ... |
| liminfvaluz3 44811 | Alternate definition of ` ... |
| liminflelimsupcex 44812 | A counterexample for ~ lim... |
| limsupvaluz3 44813 | Alternate definition of ` ... |
| liminfvaluz4 44814 | Alternate definition of ` ... |
| limsupvaluz4 44815 | Alternate definition of ` ... |
| climliminflimsupd 44816 | If a sequence of real numb... |
| liminfreuzlem 44817 | Given a function on the re... |
| liminfreuz 44818 | Given a function on the re... |
| liminfltlem 44819 | Given a sequence of real n... |
| liminflt 44820 | Given a sequence of real n... |
| climliminf 44821 | A sequence of real numbers... |
| liminflimsupclim 44822 | A sequence of real numbers... |
| climliminflimsup 44823 | A sequence of real numbers... |
| climliminflimsup2 44824 | A sequence of real numbers... |
| climliminflimsup3 44825 | A sequence of real numbers... |
| climliminflimsup4 44826 | A sequence of real numbers... |
| limsupub2 44827 | A extended real valued fun... |
| limsupubuz2 44828 | A sequence with values in ... |
| xlimpnfxnegmnf 44829 | A sequence converges to ` ... |
| liminflbuz2 44830 | A sequence with values in ... |
| liminfpnfuz 44831 | The inferior limit of a fu... |
| liminflimsupxrre 44832 | A sequence with values in ... |
| xlimrel 44835 | The limit on extended real... |
| xlimres 44836 | A function converges iff i... |
| xlimcl 44837 | The limit of a sequence of... |
| rexlimddv2 44838 | Restricted existential eli... |
| xlimclim 44839 | Given a sequence of reals,... |
| xlimconst 44840 | A constant sequence conver... |
| climxlim 44841 | A converging sequence in t... |
| xlimbr 44842 | Express the binary relatio... |
| fuzxrpmcn 44843 | A function mapping from an... |
| cnrefiisplem 44844 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 44845 | A non-real, complex number... |
| xlimxrre 44846 | If a sequence ranging over... |
| xlimmnfvlem1 44847 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 44848 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 44849 | A function converges to mi... |
| xlimconst2 44850 | A sequence that eventually... |
| xlimpnfvlem1 44851 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 44852 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 44853 | A function converges to pl... |
| xlimclim2lem 44854 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 44855 | Given a sequence of extend... |
| xlimmnf 44856 | A function converges to mi... |
| xlimpnf 44857 | A function converges to pl... |
| xlimmnfmpt 44858 | A function converges to pl... |
| xlimpnfmpt 44859 | A function converges to pl... |
| climxlim2lem 44860 | In this lemma for ~ climxl... |
| climxlim2 44861 | A sequence of extended rea... |
| dfxlim2v 44862 | An alternative definition ... |
| dfxlim2 44863 | An alternative definition ... |
| climresd 44864 | A function restricted to u... |
| climresdm 44865 | A real function converges ... |
| dmclimxlim 44866 | A real valued sequence tha... |
| xlimmnflimsup2 44867 | A sequence of extended rea... |
| xlimuni 44868 | An infinite sequence conve... |
| xlimclimdm 44869 | A sequence of extended rea... |
| xlimfun 44870 | The convergence relation o... |
| xlimmnflimsup 44871 | If a sequence of extended ... |
| xlimdm 44872 | Two ways to express that a... |
| xlimpnfxnegmnf2 44873 | A sequence converges to ` ... |
| xlimresdm 44874 | A function converges in th... |
| xlimpnfliminf 44875 | If a sequence of extended ... |
| xlimpnfliminf2 44876 | A sequence of extended rea... |
| xlimliminflimsup 44877 | A sequence of extended rea... |
| xlimlimsupleliminf 44878 | A sequence of extended rea... |
| coseq0 44879 | A complex number whose cos... |
| sinmulcos 44880 | Multiplication formula for... |
| coskpi2 44881 | The cosine of an integer m... |
| cosnegpi 44882 | The cosine of negative ` _... |
| sinaover2ne0 44883 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 44884 | The cosine of an integer m... |
| mulcncff 44885 | The multiplication of two ... |
| cncfmptssg 44886 | A continuous complex funct... |
| constcncfg 44887 | A constant function is a c... |
| idcncfg 44888 | The identity function is a... |
| cncfshift 44889 | A periodic continuous func... |
| resincncf 44890 | ` sin ` restricted to real... |
| addccncf2 44891 | Adding a constant is a con... |
| 0cnf 44892 | The empty set is a continu... |
| fsumcncf 44893 | The finite sum of continuo... |
| cncfperiod 44894 | A periodic continuous func... |
| subcncff 44895 | The subtraction of two con... |
| negcncfg 44896 | The opposite of a continuo... |
| cnfdmsn 44897 | A function with a singleto... |
| cncfcompt 44898 | Composition of continuous ... |
| addcncff 44899 | The sum of two continuous ... |
| ioccncflimc 44900 | Limit at the upper bound o... |
| cncfuni 44901 | A complex function on a su... |
| icccncfext 44902 | A continuous function on a... |
| cncficcgt0 44903 | A the absolute value of a ... |
| icocncflimc 44904 | Limit at the lower bound, ... |
| cncfdmsn 44905 | A complex function with a ... |
| divcncff 44906 | The quotient of two contin... |
| cncfshiftioo 44907 | A periodic continuous func... |
| cncfiooicclem1 44908 | A continuous function ` F ... |
| cncfiooicc 44909 | A continuous function ` F ... |
| cncfiooiccre 44910 | A continuous function ` F ... |
| cncfioobdlem 44911 | ` G ` actually extends ` F... |
| cncfioobd 44912 | A continuous function ` F ... |
| jumpncnp 44913 | Jump discontinuity or disc... |
| cxpcncf2 44914 | The complex power function... |
| fprodcncf 44915 | The finite product of cont... |
| add1cncf 44916 | Addition to a constant is ... |
| add2cncf 44917 | Addition to a constant is ... |
| sub1cncfd 44918 | Subtracting a constant is ... |
| sub2cncfd 44919 | Subtraction from a constan... |
| fprodsub2cncf 44920 | ` F ` is continuous. (Con... |
| fprodadd2cncf 44921 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 44922 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 44923 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 44924 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 44925 | The sequence ` S ` of fini... |
| dvsinexp 44926 | The derivative of sin^N . ... |
| dvcosre 44927 | The real derivative of the... |
| dvsinax 44928 | Derivative exercise: the d... |
| dvsubf 44929 | The subtraction rule for e... |
| dvmptconst 44930 | Function-builder for deriv... |
| dvcnre 44931 | From complex differentiati... |
| dvmptidg 44932 | Function-builder for deriv... |
| dvresntr 44933 | Function-builder for deriv... |
| fperdvper 44934 | The derivative of a period... |
| dvasinbx 44935 | Derivative exercise: the d... |
| dvresioo 44936 | Restriction of a derivativ... |
| dvdivf 44937 | The quotient rule for ever... |
| dvdivbd 44938 | A sufficient condition for... |
| dvsubcncf 44939 | A sufficient condition for... |
| dvmulcncf 44940 | A sufficient condition for... |
| dvcosax 44941 | Derivative exercise: the d... |
| dvdivcncf 44942 | A sufficient condition for... |
| dvbdfbdioolem1 44943 | Given a function with boun... |
| dvbdfbdioolem2 44944 | A function on an open inte... |
| dvbdfbdioo 44945 | A function on an open inte... |
| ioodvbdlimc1lem1 44946 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 44947 | Limit at the lower bound o... |
| ioodvbdlimc1 44948 | A real function with bound... |
| ioodvbdlimc2lem 44949 | Limit at the upper bound o... |
| ioodvbdlimc2 44950 | A real function with bound... |
| dvdmsscn 44951 | ` X ` is a subset of ` CC ... |
| dvmptmulf 44952 | Function-builder for deriv... |
| dvnmptdivc 44953 | Function-builder for itera... |
| dvdsn1add 44954 | If ` K ` divides ` N ` but... |
| dvxpaek 44955 | Derivative of the polynomi... |
| dvnmptconst 44956 | The ` N ` -th derivative o... |
| dvnxpaek 44957 | The ` n ` -th derivative o... |
| dvnmul 44958 | Function-builder for the `... |
| dvmptfprodlem 44959 | Induction step for ~ dvmpt... |
| dvmptfprod 44960 | Function-builder for deriv... |
| dvnprodlem1 44961 | ` D ` is bijective. (Cont... |
| dvnprodlem2 44962 | Induction step for ~ dvnpr... |
| dvnprodlem3 44963 | The multinomial formula fo... |
| dvnprod 44964 | The multinomial formula fo... |
| itgsin0pilem1 44965 | Calculation of the integra... |
| ibliccsinexp 44966 | sin^n on a closed interval... |
| itgsin0pi 44967 | Calculation of the integra... |
| iblioosinexp 44968 | sin^n on an open integral ... |
| itgsinexplem1 44969 | Integration by parts is ap... |
| itgsinexp 44970 | A recursive formula for th... |
| iblconstmpt 44971 | A constant function is int... |
| itgeq1d 44972 | Equality theorem for an in... |
| mbfres2cn 44973 | Measurability of a piecewi... |
| vol0 44974 | The measure of the empty s... |
| ditgeqiooicc 44975 | A function ` F ` on an ope... |
| volge0 44976 | The volume of a set is alw... |
| cnbdibl 44977 | A continuous bounded funct... |
| snmbl 44978 | A singleton is measurable.... |
| ditgeq3d 44979 | Equality theorem for the d... |
| iblempty 44980 | The empty function is inte... |
| iblsplit 44981 | The union of two integrabl... |
| volsn 44982 | A singleton has 0 Lebesgue... |
| itgvol0 44983 | If the domani is negligibl... |
| itgcoscmulx 44984 | Exercise: the integral of ... |
| iblsplitf 44985 | A version of ~ iblsplit us... |
| ibliooicc 44986 | If a function is integrabl... |
| volioc 44987 | The measure of a left-open... |
| iblspltprt 44988 | If a function is integrabl... |
| itgsincmulx 44989 | Exercise: the integral of ... |
| itgsubsticclem 44990 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 44991 | Integration by u-substitut... |
| itgioocnicc 44992 | The integral of a piecewis... |
| iblcncfioo 44993 | A continuous function ` F ... |
| itgspltprt 44994 | The ` S. ` integral splits... |
| itgiccshift 44995 | The integral of a function... |
| itgperiod 44996 | The integral of a periodic... |
| itgsbtaddcnst 44997 | Integral substitution, add... |
| volico 44998 | The measure of left-closed... |
| sublevolico 44999 | The Lebesgue measure of a ... |
| dmvolss 45000 | Lebesgue measurable sets a... |
| ismbl3 45001 | The predicate " ` A ` is L... |
| volioof 45002 | The function that assigns ... |
| ovolsplit 45003 | The Lebesgue outer measure... |
| fvvolioof 45004 | The function value of the ... |
| volioore 45005 | The measure of an open int... |
| fvvolicof 45006 | The function value of the ... |
| voliooico 45007 | An open interval and a lef... |
| ismbl4 45008 | The predicate " ` A ` is L... |
| volioofmpt 45009 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 45010 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 45011 | The Lebesgue measure of op... |
| volicofmpt 45012 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 45013 | The Lebesgue measure of a ... |
| voliccico 45014 | A closed interval and a le... |
| mbfdmssre 45015 | The domain of a measurable... |
| stoweidlem1 45016 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 45017 | lemma for ~ stoweid : here... |
| stoweidlem3 45018 | Lemma for ~ stoweid : if `... |
| stoweidlem4 45019 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 45020 | There exists a δ as ... |
| stoweidlem6 45021 | Lemma for ~ stoweid : two ... |
| stoweidlem7 45022 | This lemma is used to prov... |
| stoweidlem8 45023 | Lemma for ~ stoweid : two ... |
| stoweidlem9 45024 | Lemma for ~ stoweid : here... |
| stoweidlem10 45025 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 45026 | This lemma is used to prov... |
| stoweidlem12 45027 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 45028 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 45029 | There exists a ` k ` as in... |
| stoweidlem15 45030 | This lemma is used to prov... |
| stoweidlem16 45031 | Lemma for ~ stoweid . The... |
| stoweidlem17 45032 | This lemma proves that the... |
| stoweidlem18 45033 | This theorem proves Lemma ... |
| stoweidlem19 45034 | If a set of real functions... |
| stoweidlem20 45035 | If a set A of real functio... |
| stoweidlem21 45036 | Once the Stone Weierstrass... |
| stoweidlem22 45037 | If a set of real functions... |
| stoweidlem23 45038 | This lemma is used to prov... |
| stoweidlem24 45039 | This lemma proves that for... |
| stoweidlem25 45040 | This lemma proves that for... |
| stoweidlem26 45041 | This lemma is used to prov... |
| stoweidlem27 45042 | This lemma is used to prov... |
| stoweidlem28 45043 | There exists a δ as ... |
| stoweidlem29 45044 | When the hypothesis for th... |
| stoweidlem30 45045 | This lemma is used to prov... |
| stoweidlem31 45046 | This lemma is used to prov... |
| stoweidlem32 45047 | If a set A of real functio... |
| stoweidlem33 45048 | If a set of real functions... |
| stoweidlem34 45049 | This lemma proves that for... |
| stoweidlem35 45050 | This lemma is used to prov... |
| stoweidlem36 45051 | This lemma is used to prov... |
| stoweidlem37 45052 | This lemma is used to prov... |
| stoweidlem38 45053 | This lemma is used to prov... |
| stoweidlem39 45054 | This lemma is used to prov... |
| stoweidlem40 45055 | This lemma proves that q_n... |
| stoweidlem41 45056 | This lemma is used to prov... |
| stoweidlem42 45057 | This lemma is used to prov... |
| stoweidlem43 45058 | This lemma is used to prov... |
| stoweidlem44 45059 | This lemma is used to prov... |
| stoweidlem45 45060 | This lemma proves that, gi... |
| stoweidlem46 45061 | This lemma proves that set... |
| stoweidlem47 45062 | Subtracting a constant fro... |
| stoweidlem48 45063 | This lemma is used to prov... |
| stoweidlem49 45064 | There exists a function q_... |
| stoweidlem50 45065 | This lemma proves that set... |
| stoweidlem51 45066 | There exists a function x ... |
| stoweidlem52 45067 | There exists a neighborhoo... |
| stoweidlem53 45068 | This lemma is used to prov... |
| stoweidlem54 45069 | There exists a function ` ... |
| stoweidlem55 45070 | This lemma proves the exis... |
| stoweidlem56 45071 | This theorem proves Lemma ... |
| stoweidlem57 45072 | There exists a function x ... |
| stoweidlem58 45073 | This theorem proves Lemma ... |
| stoweidlem59 45074 | This lemma proves that the... |
| stoweidlem60 45075 | This lemma proves that the... |
| stoweidlem61 45076 | This lemma proves that the... |
| stoweidlem62 45077 | This theorem proves the St... |
| stoweid 45078 | This theorem proves the St... |
| stowei 45079 | This theorem proves the St... |
| wallispilem1 45080 | ` I ` is monotone: increas... |
| wallispilem2 45081 | A first set of properties ... |
| wallispilem3 45082 | I maps to real values. (C... |
| wallispilem4 45083 | ` F ` maps to explicit exp... |
| wallispilem5 45084 | The sequence ` H ` converg... |
| wallispi 45085 | Wallis' formula for π :... |
| wallispi2lem1 45086 | An intermediate step betwe... |
| wallispi2lem2 45087 | Two expressions are proven... |
| wallispi2 45088 | An alternative version of ... |
| stirlinglem1 45089 | A simple limit of fraction... |
| stirlinglem2 45090 | ` A ` maps to positive rea... |
| stirlinglem3 45091 | Long but simple algebraic ... |
| stirlinglem4 45092 | Algebraic manipulation of ... |
| stirlinglem5 45093 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 45094 | A series that converges to... |
| stirlinglem7 45095 | Algebraic manipulation of ... |
| stirlinglem8 45096 | If ` A ` converges to ` C ... |
| stirlinglem9 45097 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 45098 | A bound for any B(N)-B(N +... |
| stirlinglem11 45099 | ` B ` is decreasing. (Con... |
| stirlinglem12 45100 | The sequence ` B ` is boun... |
| stirlinglem13 45101 | ` B ` is decreasing and ha... |
| stirlinglem14 45102 | The sequence ` A ` converg... |
| stirlinglem15 45103 | The Stirling's formula is ... |
| stirling 45104 | Stirling's approximation f... |
| stirlingr 45105 | Stirling's approximation f... |
| dirkerval 45106 | The N_th Dirichlet Kernel.... |
| dirker2re 45107 | The Dirichlet Kernel value... |
| dirkerdenne0 45108 | The Dirichlet Kernel denom... |
| dirkerval2 45109 | The N_th Dirichlet Kernel ... |
| dirkerre 45110 | The Dirichlet Kernel at an... |
| dirkerper 45111 | the Dirichlet Kernel has p... |
| dirkerf 45112 | For any natural number ` N... |
| dirkertrigeqlem1 45113 | Sum of an even number of a... |
| dirkertrigeqlem2 45114 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 45115 | Trigonometric equality lem... |
| dirkertrigeq 45116 | Trigonometric equality for... |
| dirkeritg 45117 | The definite integral of t... |
| dirkercncflem1 45118 | If ` Y ` is a multiple of ... |
| dirkercncflem2 45119 | Lemma used to prove that t... |
| dirkercncflem3 45120 | The Dirichlet Kernel is co... |
| dirkercncflem4 45121 | The Dirichlet Kernel is co... |
| dirkercncf 45122 | For any natural number ` N... |
| fourierdlem1 45123 | A partition interval is a ... |
| fourierdlem2 45124 | Membership in a partition.... |
| fourierdlem3 45125 | Membership in a partition.... |
| fourierdlem4 45126 | ` E ` is a function that m... |
| fourierdlem5 45127 | ` S ` is a function. (Con... |
| fourierdlem6 45128 | ` X ` is in the periodic p... |
| fourierdlem7 45129 | The difference between the... |
| fourierdlem8 45130 | A partition interval is a ... |
| fourierdlem9 45131 | ` H ` is a complex functio... |
| fourierdlem10 45132 | Condition on the bounds of... |
| fourierdlem11 45133 | If there is a partition, t... |
| fourierdlem12 45134 | A point of a partition is ... |
| fourierdlem13 45135 | Value of ` V ` in terms of... |
| fourierdlem14 45136 | Given the partition ` V ` ... |
| fourierdlem15 45137 | The range of the partition... |
| fourierdlem16 45138 | The coefficients of the fo... |
| fourierdlem17 45139 | The defined ` L ` is actua... |
| fourierdlem18 45140 | The function ` S ` is cont... |
| fourierdlem19 45141 | If two elements of ` D ` h... |
| fourierdlem20 45142 | Every interval in the part... |
| fourierdlem21 45143 | The coefficients of the fo... |
| fourierdlem22 45144 | The coefficients of the fo... |
| fourierdlem23 45145 | If ` F ` is continuous and... |
| fourierdlem24 45146 | A sufficient condition for... |
| fourierdlem25 45147 | If ` C ` is not in the ran... |
| fourierdlem26 45148 | Periodic image of a point ... |
| fourierdlem27 45149 | A partition open interval ... |
| fourierdlem28 45150 | Derivative of ` ( F `` ( X... |
| fourierdlem29 45151 | Explicit function value fo... |
| fourierdlem30 45152 | Sum of three small pieces ... |
| fourierdlem31 45153 | If ` A ` is finite and for... |
| fourierdlem32 45154 | Limit of a continuous func... |
| fourierdlem33 45155 | Limit of a continuous func... |
| fourierdlem34 45156 | A partition is one to one.... |
| fourierdlem35 45157 | There is a single point in... |
| fourierdlem36 45158 | ` F ` is an isomorphism. ... |
| fourierdlem37 45159 | ` I ` is a function that m... |
| fourierdlem38 45160 | The function ` F ` is cont... |
| fourierdlem39 45161 | Integration by parts of ... |
| fourierdlem40 45162 | ` H ` is a continuous func... |
| fourierdlem41 45163 | Lemma used to prove that e... |
| fourierdlem42 45164 | The set of points in a mov... |
| fourierdlem43 45165 | ` K ` is a real function. ... |
| fourierdlem44 45166 | A condition for having ` (... |
| fourierdlem46 45167 | The function ` F ` has a l... |
| fourierdlem47 45168 | For ` r ` large enough, th... |
| fourierdlem48 45169 | The given periodic functio... |
| fourierdlem49 45170 | The given periodic functio... |
| fourierdlem50 45171 | Continuity of ` O ` and it... |
| fourierdlem51 45172 | ` X ` is in the periodic p... |
| fourierdlem52 45173 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 45174 | The limit of ` F ( s ) ` a... |
| fourierdlem54 45175 | Given a partition ` Q ` an... |
| fourierdlem55 45176 | ` U ` is a real function. ... |
| fourierdlem56 45177 | Derivative of the ` K ` fu... |
| fourierdlem57 45178 | The derivative of ` O ` . ... |
| fourierdlem58 45179 | The derivative of ` K ` is... |
| fourierdlem59 45180 | The derivative of ` H ` is... |
| fourierdlem60 45181 | Given a differentiable fun... |
| fourierdlem61 45182 | Given a differentiable fun... |
| fourierdlem62 45183 | The function ` K ` is cont... |
| fourierdlem63 45184 | The upper bound of interva... |
| fourierdlem64 45185 | The partition ` V ` is fin... |
| fourierdlem65 45186 | The distance of two adjace... |
| fourierdlem66 45187 | Value of the ` G ` functio... |
| fourierdlem67 45188 | ` G ` is a function. (Con... |
| fourierdlem68 45189 | The derivative of ` O ` is... |
| fourierdlem69 45190 | A piecewise continuous fun... |
| fourierdlem70 45191 | A piecewise continuous fun... |
| fourierdlem71 45192 | A periodic piecewise conti... |
| fourierdlem72 45193 | The derivative of ` O ` is... |
| fourierdlem73 45194 | A version of the Riemann L... |
| fourierdlem74 45195 | Given a piecewise smooth f... |
| fourierdlem75 45196 | Given a piecewise smooth f... |
| fourierdlem76 45197 | Continuity of ` O ` and it... |
| fourierdlem77 45198 | If ` H ` is bounded, then ... |
| fourierdlem78 45199 | ` G ` is continuous when r... |
| fourierdlem79 45200 | ` E ` projects every inter... |
| fourierdlem80 45201 | The derivative of ` O ` is... |
| fourierdlem81 45202 | The integral of a piecewis... |
| fourierdlem82 45203 | Integral by substitution, ... |
| fourierdlem83 45204 | The fourier partial sum fo... |
| fourierdlem84 45205 | If ` F ` is piecewise coni... |
| fourierdlem85 45206 | Limit of the function ` G ... |
| fourierdlem86 45207 | Continuity of ` O ` and it... |
| fourierdlem87 45208 | The integral of ` G ` goes... |
| fourierdlem88 45209 | Given a piecewise continuo... |
| fourierdlem89 45210 | Given a piecewise continuo... |
| fourierdlem90 45211 | Given a piecewise continuo... |
| fourierdlem91 45212 | Given a piecewise continuo... |
| fourierdlem92 45213 | The integral of a piecewis... |
| fourierdlem93 45214 | Integral by substitution (... |
| fourierdlem94 45215 | For a piecewise smooth fun... |
| fourierdlem95 45216 | Algebraic manipulation of ... |
| fourierdlem96 45217 | limit for ` F ` at the low... |
| fourierdlem97 45218 | ` F ` is continuous on the... |
| fourierdlem98 45219 | ` F ` is continuous on the... |
| fourierdlem99 45220 | limit for ` F ` at the upp... |
| fourierdlem100 45221 | A piecewise continuous fun... |
| fourierdlem101 45222 | Integral by substitution f... |
| fourierdlem102 45223 | For a piecewise smooth fun... |
| fourierdlem103 45224 | The half lower part of the... |
| fourierdlem104 45225 | The half upper part of the... |
| fourierdlem105 45226 | A piecewise continuous fun... |
| fourierdlem106 45227 | For a piecewise smooth fun... |
| fourierdlem107 45228 | The integral of a piecewis... |
| fourierdlem108 45229 | The integral of a piecewis... |
| fourierdlem109 45230 | The integral of a piecewis... |
| fourierdlem110 45231 | The integral of a piecewis... |
| fourierdlem111 45232 | The fourier partial sum fo... |
| fourierdlem112 45233 | Here abbreviations (local ... |
| fourierdlem113 45234 | Fourier series convergence... |
| fourierdlem114 45235 | Fourier series convergence... |
| fourierdlem115 45236 | Fourier serier convergence... |
| fourierd 45237 | Fourier series convergence... |
| fourierclimd 45238 | Fourier series convergence... |
| fourierclim 45239 | Fourier series convergence... |
| fourier 45240 | Fourier series convergence... |
| fouriercnp 45241 | If ` F ` is continuous at ... |
| fourier2 45242 | Fourier series convergence... |
| sqwvfoura 45243 | Fourier coefficients for t... |
| sqwvfourb 45244 | Fourier series ` B ` coeff... |
| fourierswlem 45245 | The Fourier series for the... |
| fouriersw 45246 | Fourier series convergence... |
| fouriercn 45247 | If the derivative of ` F `... |
| elaa2lem 45248 | Elementhood in the set of ... |
| elaa2 45249 | Elementhood in the set of ... |
| etransclem1 45250 | ` H ` is a function. (Con... |
| etransclem2 45251 | Derivative of ` G ` . (Co... |
| etransclem3 45252 | The given ` if ` term is a... |
| etransclem4 45253 | ` F ` expressed as a finit... |
| etransclem5 45254 | A change of bound variable... |
| etransclem6 45255 | A change of bound variable... |
| etransclem7 45256 | The given product is an in... |
| etransclem8 45257 | ` F ` is a function. (Con... |
| etransclem9 45258 | If ` K ` divides ` N ` but... |
| etransclem10 45259 | The given ` if ` term is a... |
| etransclem11 45260 | A change of bound variable... |
| etransclem12 45261 | ` C ` applied to ` N ` . ... |
| etransclem13 45262 | ` F ` applied to ` Y ` . ... |
| etransclem14 45263 | Value of the term ` T ` , ... |
| etransclem15 45264 | Value of the term ` T ` , ... |
| etransclem16 45265 | Every element in the range... |
| etransclem17 45266 | The ` N ` -th derivative o... |
| etransclem18 45267 | The given function is inte... |
| etransclem19 45268 | The ` N ` -th derivative o... |
| etransclem20 45269 | ` H ` is smooth. (Contrib... |
| etransclem21 45270 | The ` N ` -th derivative o... |
| etransclem22 45271 | The ` N ` -th derivative o... |
| etransclem23 45272 | This is the claim proof in... |
| etransclem24 45273 | ` P ` divides the I -th de... |
| etransclem25 45274 | ` P ` factorial divides th... |
| etransclem26 45275 | Every term in the sum of t... |
| etransclem27 45276 | The ` N ` -th derivative o... |
| etransclem28 45277 | ` ( P - 1 ) ` factorial di... |
| etransclem29 45278 | The ` N ` -th derivative o... |
| etransclem30 45279 | The ` N ` -th derivative o... |
| etransclem31 45280 | The ` N ` -th derivative o... |
| etransclem32 45281 | This is the proof for the ... |
| etransclem33 45282 | ` F ` is smooth. (Contrib... |
| etransclem34 45283 | The ` N ` -th derivative o... |
| etransclem35 45284 | ` P ` does not divide the ... |
| etransclem36 45285 | The ` N ` -th derivative o... |
| etransclem37 45286 | ` ( P - 1 ) ` factorial di... |
| etransclem38 45287 | ` P ` divides the I -th de... |
| etransclem39 45288 | ` G ` is a function. (Con... |
| etransclem40 45289 | The ` N ` -th derivative o... |
| etransclem41 45290 | ` P ` does not divide the ... |
| etransclem42 45291 | The ` N ` -th derivative o... |
| etransclem43 45292 | ` G ` is a continuous func... |
| etransclem44 45293 | The given finite sum is no... |
| etransclem45 45294 | ` K ` is an integer. (Con... |
| etransclem46 45295 | This is the proof for equa... |
| etransclem47 45296 | ` _e ` is transcendental. ... |
| etransclem48 45297 | ` _e ` is transcendental. ... |
| etransc 45298 | ` _e ` is transcendental. ... |
| rrxtopn 45299 | The topology of the genera... |
| rrxngp 45300 | Generalized Euclidean real... |
| rrxtps 45301 | Generalized Euclidean real... |
| rrxtopnfi 45302 | The topology of the n-dime... |
| rrxtopon 45303 | The topology on generalize... |
| rrxtop 45304 | The topology on generalize... |
| rrndistlt 45305 | Given two points in the sp... |
| rrxtoponfi 45306 | The topology on n-dimensio... |
| rrxunitopnfi 45307 | The base set of the standa... |
| rrxtopn0 45308 | The topology of the zero-d... |
| qndenserrnbllem 45309 | n-dimensional rational num... |
| qndenserrnbl 45310 | n-dimensional rational num... |
| rrxtopn0b 45311 | The topology of the zero-d... |
| qndenserrnopnlem 45312 | n-dimensional rational num... |
| qndenserrnopn 45313 | n-dimensional rational num... |
| qndenserrn 45314 | n-dimensional rational num... |
| rrxsnicc 45315 | A multidimensional singlet... |
| rrnprjdstle 45316 | The distance between two p... |
| rrndsmet 45317 | ` D ` is a metric for the ... |
| rrndsxmet 45318 | ` D ` is an extended metri... |
| ioorrnopnlem 45319 | The a point in an indexed ... |
| ioorrnopn 45320 | The indexed product of ope... |
| ioorrnopnxrlem 45321 | Given a point ` F ` that b... |
| ioorrnopnxr 45322 | The indexed product of ope... |
| issal 45329 | Express the predicate " ` ... |
| pwsal 45330 | The power set of a given s... |
| salunicl 45331 | SAlg sigma-algebra is clos... |
| saluncl 45332 | The union of two sets in a... |
| prsal 45333 | The pair of the empty set ... |
| saldifcl 45334 | The complement of an eleme... |
| 0sal 45335 | The empty set belongs to e... |
| salgenval 45336 | The sigma-algebra generate... |
| saliunclf 45337 | SAlg sigma-algebra is clos... |
| saliuncl 45338 | SAlg sigma-algebra is clos... |
| salincl 45339 | The intersection of two se... |
| saluni 45340 | A set is an element of any... |
| saliinclf 45341 | SAlg sigma-algebra is clos... |
| saliincl 45342 | SAlg sigma-algebra is clos... |
| saldifcl2 45343 | The difference of two elem... |
| intsaluni 45344 | The union of an arbitrary ... |
| intsal 45345 | The arbitrary intersection... |
| salgenn0 45346 | The set used in the defini... |
| salgencl 45347 | ` SalGen ` actually genera... |
| issald 45348 | Sufficient condition to pr... |
| salexct 45349 | An example of nontrivial s... |
| sssalgen 45350 | A set is a subset of the s... |
| salgenss 45351 | The sigma-algebra generate... |
| salgenuni 45352 | The base set of the sigma-... |
| issalgend 45353 | One side of ~ dfsalgen2 . ... |
| salexct2 45354 | An example of a subset tha... |
| unisalgen 45355 | The union of a set belongs... |
| dfsalgen2 45356 | Alternate characterization... |
| salexct3 45357 | An example of a sigma-alge... |
| salgencntex 45358 | This counterexample shows ... |
| salgensscntex 45359 | This counterexample shows ... |
| issalnnd 45360 | Sufficient condition to pr... |
| dmvolsal 45361 | Lebesgue measurable sets f... |
| saldifcld 45362 | The complement of an eleme... |
| saluncld 45363 | The union of two sets in a... |
| salgencld 45364 | ` SalGen ` actually genera... |
| 0sald 45365 | The empty set belongs to e... |
| iooborel 45366 | An open interval is a Bore... |
| salincld 45367 | The intersection of two se... |
| salunid 45368 | A set is an element of any... |
| unisalgen2 45369 | The union of a set belongs... |
| bor1sal 45370 | The Borel sigma-algebra on... |
| iocborel 45371 | A left-open, right-closed ... |
| subsaliuncllem 45372 | A subspace sigma-algebra i... |
| subsaliuncl 45373 | A subspace sigma-algebra i... |
| subsalsal 45374 | A subspace sigma-algebra i... |
| subsaluni 45375 | A set belongs to the subsp... |
| salrestss 45376 | A sigma-algebra restricted... |
| sge0rnre 45379 | When ` sum^ ` is applied t... |
| fge0icoicc 45380 | If ` F ` maps to nonnegati... |
| sge0val 45381 | The value of the sum of no... |
| fge0npnf 45382 | If ` F ` maps to nonnegati... |
| sge0rnn0 45383 | The range used in the defi... |
| sge0vald 45384 | The value of the sum of no... |
| fge0iccico 45385 | A range of nonnegative ext... |
| gsumge0cl 45386 | Closure of group sum, for ... |
| sge0reval 45387 | Value of the sum of nonneg... |
| sge0pnfval 45388 | If a term in the sum of no... |
| fge0iccre 45389 | A range of nonnegative ext... |
| sge0z 45390 | Any nonnegative extended s... |
| sge00 45391 | The sum of nonnegative ext... |
| fsumlesge0 45392 | Every finite subsum of non... |
| sge0revalmpt 45393 | Value of the sum of nonneg... |
| sge0sn 45394 | A sum of a nonnegative ext... |
| sge0tsms 45395 | ` sum^ ` applied to a nonn... |
| sge0cl 45396 | The arbitrary sum of nonne... |
| sge0f1o 45397 | Re-index a nonnegative ext... |
| sge0snmpt 45398 | A sum of a nonnegative ext... |
| sge0ge0 45399 | The sum of nonnegative ext... |
| sge0xrcl 45400 | The arbitrary sum of nonne... |
| sge0repnf 45401 | The of nonnegative extende... |
| sge0fsum 45402 | The arbitrary sum of a fin... |
| sge0rern 45403 | If the sum of nonnegative ... |
| sge0supre 45404 | If the arbitrary sum of no... |
| sge0fsummpt 45405 | The arbitrary sum of a fin... |
| sge0sup 45406 | The arbitrary sum of nonne... |
| sge0less 45407 | A shorter sum of nonnegati... |
| sge0rnbnd 45408 | The range used in the defi... |
| sge0pr 45409 | Sum of a pair of nonnegati... |
| sge0gerp 45410 | The arbitrary sum of nonne... |
| sge0pnffigt 45411 | If the sum of nonnegative ... |
| sge0ssre 45412 | If a sum of nonnegative ex... |
| sge0lefi 45413 | A sum of nonnegative exten... |
| sge0lessmpt 45414 | A shorter sum of nonnegati... |
| sge0ltfirp 45415 | If the sum of nonnegative ... |
| sge0prle 45416 | The sum of a pair of nonne... |
| sge0gerpmpt 45417 | The arbitrary sum of nonne... |
| sge0resrnlem 45418 | The sum of nonnegative ext... |
| sge0resrn 45419 | The sum of nonnegative ext... |
| sge0ssrempt 45420 | If a sum of nonnegative ex... |
| sge0resplit 45421 | ` sum^ ` splits into two p... |
| sge0le 45422 | If all of the terms of sum... |
| sge0ltfirpmpt 45423 | If the extended sum of non... |
| sge0split 45424 | Split a sum of nonnegative... |
| sge0lempt 45425 | If all of the terms of sum... |
| sge0splitmpt 45426 | Split a sum of nonnegative... |
| sge0ss 45427 | Change the index set to a ... |
| sge0iunmptlemfi 45428 | Sum of nonnegative extende... |
| sge0p1 45429 | The addition of the next t... |
| sge0iunmptlemre 45430 | Sum of nonnegative extende... |
| sge0fodjrnlem 45431 | Re-index a nonnegative ext... |
| sge0fodjrn 45432 | Re-index a nonnegative ext... |
| sge0iunmpt 45433 | Sum of nonnegative extende... |
| sge0iun 45434 | Sum of nonnegative extende... |
| sge0nemnf 45435 | The generalized sum of non... |
| sge0rpcpnf 45436 | The sum of an infinite num... |
| sge0rernmpt 45437 | If the sum of nonnegative ... |
| sge0lefimpt 45438 | A sum of nonnegative exten... |
| nn0ssge0 45439 | Nonnegative integers are n... |
| sge0clmpt 45440 | The generalized sum of non... |
| sge0ltfirpmpt2 45441 | If the extended sum of non... |
| sge0isum 45442 | If a series of nonnegative... |
| sge0xrclmpt 45443 | The generalized sum of non... |
| sge0xp 45444 | Combine two generalized su... |
| sge0isummpt 45445 | If a series of nonnegative... |
| sge0ad2en 45446 | The value of the infinite ... |
| sge0isummpt2 45447 | If a series of nonnegative... |
| sge0xaddlem1 45448 | The extended addition of t... |
| sge0xaddlem2 45449 | The extended addition of t... |
| sge0xadd 45450 | The extended addition of t... |
| sge0fsummptf 45451 | The generalized sum of a f... |
| sge0snmptf 45452 | A sum of a nonnegative ext... |
| sge0ge0mpt 45453 | The sum of nonnegative ext... |
| sge0repnfmpt 45454 | The of nonnegative extende... |
| sge0pnffigtmpt 45455 | If the generalized sum of ... |
| sge0splitsn 45456 | Separate out a term in a g... |
| sge0pnffsumgt 45457 | If the sum of nonnegative ... |
| sge0gtfsumgt 45458 | If the generalized sum of ... |
| sge0uzfsumgt 45459 | If a real number is smalle... |
| sge0pnfmpt 45460 | If a term in the sum of no... |
| sge0seq 45461 | A series of nonnegative re... |
| sge0reuz 45462 | Value of the generalized s... |
| sge0reuzb 45463 | Value of the generalized s... |
| ismea 45466 | Express the predicate " ` ... |
| dmmeasal 45467 | The domain of a measure is... |
| meaf 45468 | A measure is a function th... |
| mea0 45469 | The measure of the empty s... |
| nnfoctbdjlem 45470 | There exists a mapping fro... |
| nnfoctbdj 45471 | There exists a mapping fro... |
| meadjuni 45472 | The measure of the disjoin... |
| meacl 45473 | The measure of a set is a ... |
| iundjiunlem 45474 | The sets in the sequence `... |
| iundjiun 45475 | Given a sequence ` E ` of ... |
| meaxrcl 45476 | The measure of a set is an... |
| meadjun 45477 | The measure of the union o... |
| meassle 45478 | The measure of a set is gr... |
| meaunle 45479 | The measure of the union o... |
| meadjiunlem 45480 | The sum of nonnegative ext... |
| meadjiun 45481 | The measure of the disjoin... |
| ismeannd 45482 | Sufficient condition to pr... |
| meaiunlelem 45483 | The measure of the union o... |
| meaiunle 45484 | The measure of the union o... |
| psmeasurelem 45485 | ` M ` applied to a disjoin... |
| psmeasure 45486 | Point supported measure, R... |
| voliunsge0lem 45487 | The Lebesgue measure funct... |
| voliunsge0 45488 | The Lebesgue measure funct... |
| volmea 45489 | The Lebesgue measure on th... |
| meage0 45490 | If the measure of a measur... |
| meadjunre 45491 | The measure of the union o... |
| meassre 45492 | If the measure of a measur... |
| meale0eq0 45493 | A measure that is less tha... |
| meadif 45494 | The measure of the differe... |
| meaiuninclem 45495 | Measures are continuous fr... |
| meaiuninc 45496 | Measures are continuous fr... |
| meaiuninc2 45497 | Measures are continuous fr... |
| meaiunincf 45498 | Measures are continuous fr... |
| meaiuninc3v 45499 | Measures are continuous fr... |
| meaiuninc3 45500 | Measures are continuous fr... |
| meaiininclem 45501 | Measures are continuous fr... |
| meaiininc 45502 | Measures are continuous fr... |
| meaiininc2 45503 | Measures are continuous fr... |
| caragenval 45508 | The sigma-algebra generate... |
| isome 45509 | Express the predicate " ` ... |
| caragenel 45510 | Membership in the Caratheo... |
| omef 45511 | An outer measure is a func... |
| ome0 45512 | The outer measure of the e... |
| omessle 45513 | The outer measure of a set... |
| omedm 45514 | The domain of an outer mea... |
| caragensplit 45515 | If ` E ` is in the set gen... |
| caragenelss 45516 | An element of the Caratheo... |
| carageneld 45517 | Membership in the Caratheo... |
| omecl 45518 | The outer measure of a set... |
| caragenss 45519 | The sigma-algebra generate... |
| omeunile 45520 | The outer measure of the u... |
| caragen0 45521 | The empty set belongs to a... |
| omexrcl 45522 | The outer measure of a set... |
| caragenunidm 45523 | The base set of an outer m... |
| caragensspw 45524 | The sigma-algebra generate... |
| omessre 45525 | If the outer measure of a ... |
| caragenuni 45526 | The base set of the sigma-... |
| caragenuncllem 45527 | The Caratheodory's constru... |
| caragenuncl 45528 | The Caratheodory's constru... |
| caragendifcl 45529 | The Caratheodory's constru... |
| caragenfiiuncl 45530 | The Caratheodory's constru... |
| omeunle 45531 | The outer measure of the u... |
| omeiunle 45532 | The outer measure of the i... |
| omelesplit 45533 | The outer measure of a set... |
| omeiunltfirp 45534 | If the outer measure of a ... |
| omeiunlempt 45535 | The outer measure of the i... |
| carageniuncllem1 45536 | The outer measure of ` A i... |
| carageniuncllem2 45537 | The Caratheodory's constru... |
| carageniuncl 45538 | The Caratheodory's constru... |
| caragenunicl 45539 | The Caratheodory's constru... |
| caragensal 45540 | Caratheodory's method gene... |
| caratheodorylem1 45541 | Lemma used to prove that C... |
| caratheodorylem2 45542 | Caratheodory's constructio... |
| caratheodory 45543 | Caratheodory's constructio... |
| 0ome 45544 | The map that assigns 0 to ... |
| isomenndlem 45545 | ` O ` is sub-additive w.r.... |
| isomennd 45546 | Sufficient condition to pr... |
| caragenel2d 45547 | Membership in the Caratheo... |
| omege0 45548 | If the outer measure of a ... |
| omess0 45549 | If the outer measure of a ... |
| caragencmpl 45550 | A measure built with the C... |
| vonval 45555 | Value of the Lebesgue meas... |
| ovnval 45556 | Value of the Lebesgue oute... |
| elhoi 45557 | Membership in a multidimen... |
| icoresmbl 45558 | A closed-below, open-above... |
| hoissre 45559 | The projection of a half-o... |
| ovnval2 45560 | Value of the Lebesgue oute... |
| volicorecl 45561 | The Lebesgue measure of a ... |
| hoiprodcl 45562 | The pre-measure of half-op... |
| hoicvr 45563 | ` I ` is a countable set o... |
| hoissrrn 45564 | A half-open interval is a ... |
| ovn0val 45565 | The Lebesgue outer measure... |
| ovnn0val 45566 | The value of a (multidimen... |
| ovnval2b 45567 | Value of the Lebesgue oute... |
| volicorescl 45568 | The Lebesgue measure of a ... |
| ovnprodcl 45569 | The product used in the de... |
| hoiprodcl2 45570 | The pre-measure of half-op... |
| hoicvrrex 45571 | Any subset of the multidim... |
| ovnsupge0 45572 | The set used in the defini... |
| ovnlecvr 45573 | Given a subset of multidim... |
| ovnpnfelsup 45574 | ` +oo ` is an element of t... |
| ovnsslelem 45575 | The (multidimensional, non... |
| ovnssle 45576 | The (multidimensional) Leb... |
| ovnlerp 45577 | The Lebesgue outer measure... |
| ovnf 45578 | The Lebesgue outer measure... |
| ovncvrrp 45579 | The Lebesgue outer measure... |
| ovn0lem 45580 | For any finite dimension, ... |
| ovn0 45581 | For any finite dimension, ... |
| ovncl 45582 | The Lebesgue outer measure... |
| ovn02 45583 | For the zero-dimensional s... |
| ovnxrcl 45584 | The Lebesgue outer measure... |
| ovnsubaddlem1 45585 | The Lebesgue outer measure... |
| ovnsubaddlem2 45586 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 45587 | ` ( voln* `` X ) ` is suba... |
| ovnome 45588 | ` ( voln* `` X ) ` is an o... |
| vonmea 45589 | ` ( voln `` X ) ` is a mea... |
| volicon0 45590 | The measure of a nonempty ... |
| hsphoif 45591 | ` H ` is a function (that ... |
| hoidmvval 45592 | The dimensional volume of ... |
| hoissrrn2 45593 | A half-open interval is a ... |
| hsphoival 45594 | ` H ` is a function (that ... |
| hoiprodcl3 45595 | The pre-measure of half-op... |
| volicore 45596 | The Lebesgue measure of a ... |
| hoidmvcl 45597 | The dimensional volume of ... |
| hoidmv0val 45598 | The dimensional volume of ... |
| hoidmvn0val 45599 | The dimensional volume of ... |
| hsphoidmvle2 45600 | The dimensional volume of ... |
| hsphoidmvle 45601 | The dimensional volume of ... |
| hoidmvval0 45602 | The dimensional volume of ... |
| hoiprodp1 45603 | The dimensional volume of ... |
| sge0hsphoire 45604 | If the generalized sum of ... |
| hoidmvval0b 45605 | The dimensional volume of ... |
| hoidmv1lelem1 45606 | The supremum of ` U ` belo... |
| hoidmv1lelem2 45607 | This is the contradiction ... |
| hoidmv1lelem3 45608 | The dimensional volume of ... |
| hoidmv1le 45609 | The dimensional volume of ... |
| hoidmvlelem1 45610 | The supremum of ` U ` belo... |
| hoidmvlelem2 45611 | This is the contradiction ... |
| hoidmvlelem3 45612 | This is the contradiction ... |
| hoidmvlelem4 45613 | The dimensional volume of ... |
| hoidmvlelem5 45614 | The dimensional volume of ... |
| hoidmvle 45615 | The dimensional volume of ... |
| ovnhoilem1 45616 | The Lebesgue outer measure... |
| ovnhoilem2 45617 | The Lebesgue outer measure... |
| ovnhoi 45618 | The Lebesgue outer measure... |
| dmovn 45619 | The domain of the Lebesgue... |
| hoicoto2 45620 | The half-open interval exp... |
| dmvon 45621 | Lebesgue measurable n-dime... |
| hoi2toco 45622 | The half-open interval exp... |
| hoidifhspval 45623 | ` D ` is a function that r... |
| hspval 45624 | The value of the half-spac... |
| ovnlecvr2 45625 | Given a subset of multidim... |
| ovncvr2 45626 | ` B ` and ` T ` are the le... |
| dmovnsal 45627 | The domain of the Lebesgue... |
| unidmovn 45628 | Base set of the n-dimensio... |
| rrnmbl 45629 | The set of n-dimensional R... |
| hoidifhspval2 45630 | ` D ` is a function that r... |
| hspdifhsp 45631 | A n-dimensional half-open ... |
| unidmvon 45632 | Base set of the n-dimensio... |
| hoidifhspf 45633 | ` D ` is a function that r... |
| hoidifhspval3 45634 | ` D ` is a function that r... |
| hoidifhspdmvle 45635 | The dimensional volume of ... |
| voncmpl 45636 | The Lebesgue measure is co... |
| hoiqssbllem1 45637 | The center of the n-dimens... |
| hoiqssbllem2 45638 | The center of the n-dimens... |
| hoiqssbllem3 45639 | A n-dimensional ball conta... |
| hoiqssbl 45640 | A n-dimensional ball conta... |
| hspmbllem1 45641 | Any half-space of the n-di... |
| hspmbllem2 45642 | Any half-space of the n-di... |
| hspmbllem3 45643 | Any half-space of the n-di... |
| hspmbl 45644 | Any half-space of the n-di... |
| hoimbllem 45645 | Any n-dimensional half-ope... |
| hoimbl 45646 | Any n-dimensional half-ope... |
| opnvonmbllem1 45647 | The half-open interval exp... |
| opnvonmbllem2 45648 | An open subset of the n-di... |
| opnvonmbl 45649 | An open subset of the n-di... |
| opnssborel 45650 | Open sets of a generalized... |
| borelmbl 45651 | All Borel subsets of the n... |
| volicorege0 45652 | The Lebesgue measure of a ... |
| isvonmbl 45653 | The predicate " ` A ` is m... |
| mblvon 45654 | The n-dimensional Lebesgue... |
| vonmblss 45655 | n-dimensional Lebesgue mea... |
| volico2 45656 | The measure of left-closed... |
| vonmblss2 45657 | n-dimensional Lebesgue mea... |
| ovolval2lem 45658 | The value of the Lebesgue ... |
| ovolval2 45659 | The value of the Lebesgue ... |
| ovnsubadd2lem 45660 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 45661 | ` ( voln* `` X ) ` is suba... |
| ovolval3 45662 | The value of the Lebesgue ... |
| ovnsplit 45663 | The n-dimensional Lebesgue... |
| ovolval4lem1 45664 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 45665 | The value of the Lebesgue ... |
| ovolval4 45666 | The value of the Lebesgue ... |
| ovolval5lem1 45667 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 45668 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 45669 | The value of the Lebesgue ... |
| ovolval5 45670 | The value of the Lebesgue ... |
| ovnovollem1 45671 | if ` F ` is a cover of ` B... |
| ovnovollem2 45672 | if ` I ` is a cover of ` (... |
| ovnovollem3 45673 | The 1-dimensional Lebesgue... |
| ovnovol 45674 | The 1-dimensional Lebesgue... |
| vonvolmbllem 45675 | If a subset ` B ` of real ... |
| vonvolmbl 45676 | A subset of Real numbers i... |
| vonvol 45677 | The 1-dimensional Lebesgue... |
| vonvolmbl2 45678 | A subset ` X ` of the spac... |
| vonvol2 45679 | The 1-dimensional Lebesgue... |
| hoimbl2 45680 | Any n-dimensional half-ope... |
| voncl 45681 | The Lebesgue measure of a ... |
| vonhoi 45682 | The Lebesgue outer measure... |
| vonxrcl 45683 | The Lebesgue measure of a ... |
| ioosshoi 45684 | A n-dimensional open inter... |
| vonn0hoi 45685 | The Lebesgue outer measure... |
| von0val 45686 | The Lebesgue measure (for ... |
| vonhoire 45687 | The Lebesgue measure of a ... |
| iinhoiicclem 45688 | A n-dimensional closed int... |
| iinhoiicc 45689 | A n-dimensional closed int... |
| iunhoiioolem 45690 | A n-dimensional open inter... |
| iunhoiioo 45691 | A n-dimensional open inter... |
| ioovonmbl 45692 | Any n-dimensional open int... |
| iccvonmbllem 45693 | Any n-dimensional closed i... |
| iccvonmbl 45694 | Any n-dimensional closed i... |
| vonioolem1 45695 | The sequence of the measur... |
| vonioolem2 45696 | The n-dimensional Lebesgue... |
| vonioo 45697 | The n-dimensional Lebesgue... |
| vonicclem1 45698 | The sequence of the measur... |
| vonicclem2 45699 | The n-dimensional Lebesgue... |
| vonicc 45700 | The n-dimensional Lebesgue... |
| snvonmbl 45701 | A n-dimensional singleton ... |
| vonn0ioo 45702 | The n-dimensional Lebesgue... |
| vonn0icc 45703 | The n-dimensional Lebesgue... |
| ctvonmbl 45704 | Any n-dimensional countabl... |
| vonn0ioo2 45705 | The n-dimensional Lebesgue... |
| vonsn 45706 | The n-dimensional Lebesgue... |
| vonn0icc2 45707 | The n-dimensional Lebesgue... |
| vonct 45708 | The n-dimensional Lebesgue... |
| vitali2 45709 | There are non-measurable s... |
| pimltmnf2f 45712 | Given a real-valued functi... |
| pimltmnf2 45713 | Given a real-valued functi... |
| preimagelt 45714 | The preimage of a right-op... |
| preimalegt 45715 | The preimage of a left-ope... |
| pimconstlt0 45716 | Given a constant function,... |
| pimconstlt1 45717 | Given a constant function,... |
| pimltpnff 45718 | Given a real-valued functi... |
| pimltpnf 45719 | Given a real-valued functi... |
| pimgtpnf2f 45720 | Given a real-valued functi... |
| pimgtpnf2 45721 | Given a real-valued functi... |
| salpreimagelt 45722 | If all the preimages of le... |
| pimrecltpos 45723 | The preimage of an unbound... |
| salpreimalegt 45724 | If all the preimages of ri... |
| pimiooltgt 45725 | The preimage of an open in... |
| preimaicomnf 45726 | Preimage of an open interv... |
| pimltpnf2f 45727 | Given a real-valued functi... |
| pimltpnf2 45728 | Given a real-valued functi... |
| pimgtmnf2 45729 | Given a real-valued functi... |
| pimdecfgtioc 45730 | Given a nonincreasing func... |
| pimincfltioc 45731 | Given a nondecreasing func... |
| pimdecfgtioo 45732 | Given a nondecreasing func... |
| pimincfltioo 45733 | Given a nondecreasing func... |
| preimaioomnf 45734 | Preimage of an open interv... |
| preimageiingt 45735 | A preimage of a left-close... |
| preimaleiinlt 45736 | A preimage of a left-open,... |
| pimgtmnff 45737 | Given a real-valued functi... |
| pimgtmnf 45738 | Given a real-valued functi... |
| pimrecltneg 45739 | The preimage of an unbound... |
| salpreimagtge 45740 | If all the preimages of le... |
| salpreimaltle 45741 | If all the preimages of ri... |
| issmflem 45742 | The predicate " ` F ` is a... |
| issmf 45743 | The predicate " ` F ` is a... |
| salpreimalelt 45744 | If all the preimages of ri... |
| salpreimagtlt 45745 | If all the preimages of le... |
| smfpreimalt 45746 | Given a function measurabl... |
| smff 45747 | A function measurable w.r.... |
| smfdmss 45748 | The domain of a function m... |
| issmff 45749 | The predicate " ` F ` is a... |
| issmfd 45750 | A sufficient condition for... |
| smfpreimaltf 45751 | Given a function measurabl... |
| issmfdf 45752 | A sufficient condition for... |
| sssmf 45753 | The restriction of a sigma... |
| mbfresmf 45754 | A real-valued measurable f... |
| cnfsmf 45755 | A continuous function is m... |
| incsmflem 45756 | A nondecreasing function i... |
| incsmf 45757 | A real-valued, nondecreasi... |
| smfsssmf 45758 | If a function is measurabl... |
| issmflelem 45759 | The predicate " ` F ` is a... |
| issmfle 45760 | The predicate " ` F ` is a... |
| smfpimltmpt 45761 | Given a function measurabl... |
| smfpimltxr 45762 | Given a function measurabl... |
| issmfdmpt 45763 | A sufficient condition for... |
| smfconst 45764 | Given a sigma-algebra over... |
| sssmfmpt 45765 | The restriction of a sigma... |
| cnfrrnsmf 45766 | A function, continuous fro... |
| smfid 45767 | The identity function is B... |
| bormflebmf 45768 | A Borel measurable functio... |
| smfpreimale 45769 | Given a function measurabl... |
| issmfgtlem 45770 | The predicate " ` F ` is a... |
| issmfgt 45771 | The predicate " ` F ` is a... |
| issmfled 45772 | A sufficient condition for... |
| smfpimltxrmptf 45773 | Given a function measurabl... |
| smfpimltxrmpt 45774 | Given a function measurabl... |
| smfmbfcex 45775 | A constant function, with ... |
| issmfgtd 45776 | A sufficient condition for... |
| smfpreimagt 45777 | Given a function measurabl... |
| smfaddlem1 45778 | Given the sum of two funct... |
| smfaddlem2 45779 | The sum of two sigma-measu... |
| smfadd 45780 | The sum of two sigma-measu... |
| decsmflem 45781 | A nonincreasing function i... |
| decsmf 45782 | A real-valued, nonincreasi... |
| smfpreimagtf 45783 | Given a function measurabl... |
| issmfgelem 45784 | The predicate " ` F ` is a... |
| issmfge 45785 | The predicate " ` F ` is a... |
| smflimlem1 45786 | Lemma for the proof that t... |
| smflimlem2 45787 | Lemma for the proof that t... |
| smflimlem3 45788 | The limit of sigma-measura... |
| smflimlem4 45789 | Lemma for the proof that t... |
| smflimlem5 45790 | Lemma for the proof that t... |
| smflimlem6 45791 | Lemma for the proof that t... |
| smflim 45792 | The limit of sigma-measura... |
| nsssmfmbflem 45793 | The sigma-measurable funct... |
| nsssmfmbf 45794 | The sigma-measurable funct... |
| smfpimgtxr 45795 | Given a function measurabl... |
| smfpimgtmpt 45796 | Given a function measurabl... |
| smfpreimage 45797 | Given a function measurabl... |
| mbfpsssmf 45798 | Real-valued measurable fun... |
| smfpimgtxrmptf 45799 | Given a function measurabl... |
| smfpimgtxrmpt 45800 | Given a function measurabl... |
| smfpimioompt 45801 | Given a function measurabl... |
| smfpimioo 45802 | Given a function measurabl... |
| smfresal 45803 | Given a sigma-measurable f... |
| smfrec 45804 | The reciprocal of a sigma-... |
| smfres 45805 | The restriction of sigma-m... |
| smfmullem1 45806 | The multiplication of two ... |
| smfmullem2 45807 | The multiplication of two ... |
| smfmullem3 45808 | The multiplication of two ... |
| smfmullem4 45809 | The multiplication of two ... |
| smfmul 45810 | The multiplication of two ... |
| smfmulc1 45811 | A sigma-measurable functio... |
| smfdiv 45812 | The fraction of two sigma-... |
| smfpimbor1lem1 45813 | Every open set belongs to ... |
| smfpimbor1lem2 45814 | Given a sigma-measurable f... |
| smfpimbor1 45815 | Given a sigma-measurable f... |
| smf2id 45816 | Twice the identity functio... |
| smfco 45817 | The composition of a Borel... |
| smfneg 45818 | The negative of a sigma-me... |
| smffmptf 45819 | A function measurable w.r.... |
| smffmpt 45820 | A function measurable w.r.... |
| smflim2 45821 | The limit of a sequence of... |
| smfpimcclem 45822 | Lemma for ~ smfpimcc given... |
| smfpimcc 45823 | Given a countable set of s... |
| issmfle2d 45824 | A sufficient condition for... |
| smflimmpt 45825 | The limit of a sequence of... |
| smfsuplem1 45826 | The supremum of a countabl... |
| smfsuplem2 45827 | The supremum of a countabl... |
| smfsuplem3 45828 | The supremum of a countabl... |
| smfsup 45829 | The supremum of a countabl... |
| smfsupmpt 45830 | The supremum of a countabl... |
| smfsupxr 45831 | The supremum of a countabl... |
| smfinflem 45832 | The infimum of a countable... |
| smfinf 45833 | The infimum of a countable... |
| smfinfmpt 45834 | The infimum of a countable... |
| smflimsuplem1 45835 | If ` H ` converges, the ` ... |
| smflimsuplem2 45836 | The superior limit of a se... |
| smflimsuplem3 45837 | The limit of the ` ( H `` ... |
| smflimsuplem4 45838 | If ` H ` converges, the ` ... |
| smflimsuplem5 45839 | ` H ` converges to the sup... |
| smflimsuplem6 45840 | The superior limit of a se... |
| smflimsuplem7 45841 | The superior limit of a se... |
| smflimsuplem8 45842 | The superior limit of a se... |
| smflimsup 45843 | The superior limit of a se... |
| smflimsupmpt 45844 | The superior limit of a se... |
| smfliminflem 45845 | The inferior limit of a co... |
| smfliminf 45846 | The inferior limit of a co... |
| smfliminfmpt 45847 | The inferior limit of a co... |
| adddmmbl 45848 | If two functions have doma... |
| adddmmbl2 45849 | If two functions have doma... |
| muldmmbl 45850 | If two functions have doma... |
| muldmmbl2 45851 | If two functions have doma... |
| smfdmmblpimne 45852 | If a measurable function w... |
| smfdivdmmbl 45853 | If a functions and a sigma... |
| smfpimne 45854 | Given a function measurabl... |
| smfpimne2 45855 | Given a function measurabl... |
| smfdivdmmbl2 45856 | If a functions and a sigma... |
| fsupdm 45857 | The domain of the sup func... |
| fsupdm2 45858 | The domain of the sup func... |
| smfsupdmmbllem 45859 | If a countable set of sigm... |
| smfsupdmmbl 45860 | If a countable set of sigm... |
| finfdm 45861 | The domain of the inf func... |
| finfdm2 45862 | The domain of the inf func... |
| smfinfdmmbllem 45863 | If a countable set of sigm... |
| smfinfdmmbl 45864 | If a countable set of sigm... |
| sigarval 45865 | Define the signed area by ... |
| sigarim 45866 | Signed area takes value in... |
| sigarac 45867 | Signed area is anticommuta... |
| sigaraf 45868 | Signed area is additive by... |
| sigarmf 45869 | Signed area is additive (w... |
| sigaras 45870 | Signed area is additive by... |
| sigarms 45871 | Signed area is additive (w... |
| sigarls 45872 | Signed area is linear by t... |
| sigarid 45873 | Signed area of a flat para... |
| sigarexp 45874 | Expand the signed area for... |
| sigarperm 45875 | Signed area ` ( A - C ) G ... |
| sigardiv 45876 | If signed area between vec... |
| sigarimcd 45877 | Signed area takes value in... |
| sigariz 45878 | If signed area is zero, th... |
| sigarcol 45879 | Given three points ` A ` ,... |
| sharhght 45880 | Let ` A B C ` be a triangl... |
| sigaradd 45881 | Subtracting (double) area ... |
| cevathlem1 45882 | Ceva's theorem first lemma... |
| cevathlem2 45883 | Ceva's theorem second lemm... |
| cevath 45884 | Ceva's theorem. Let ` A B... |
| simpcntrab 45885 | The center of a simple gro... |
| et-ltneverrefl 45886 | Less-than class is never r... |
| et-equeucl 45887 | Alternative proof that equ... |
| et-sqrtnegnre 45888 | The square root of a negat... |
| natlocalincr 45889 | Global monotonicity on hal... |
| natglobalincr 45890 | Local monotonicity on half... |
| upwordnul 45893 | Empty set is an increasing... |
| upwordisword 45894 | Any increasing sequence is... |
| singoutnword 45895 | Singleton with character o... |
| singoutnupword 45896 | Singleton with character o... |
| upwordsing 45897 | Singleton is an increasing... |
| upwordsseti 45898 | Strictly increasing sequen... |
| tworepnotupword 45899 | Concatenation of identical... |
| upwrdfi 45900 | There is a finite number o... |
| hirstL-ax3 45901 | The third axiom of a syste... |
| ax3h 45902 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 45903 | A closed form showing (a i... |
| aibandbiaiaiffb 45904 | A closed form showing (a i... |
| notatnand 45905 | Do not use. Use intnanr i... |
| aistia 45906 | Given a is equivalent to `... |
| aisfina 45907 | Given a is equivalent to `... |
| bothtbothsame 45908 | Given both a, b are equiva... |
| bothfbothsame 45909 | Given both a, b are equiva... |
| aiffbbtat 45910 | Given a is equivalent to b... |
| aisbbisfaisf 45911 | Given a is equivalent to b... |
| axorbtnotaiffb 45912 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 45913 | Given a is equivalent to (... |
| axorbciffatcxorb 45914 | Given a is equivalent to (... |
| aibnbna 45915 | Given a implies b, (not b)... |
| aibnbaif 45916 | Given a implies b, not b, ... |
| aiffbtbat 45917 | Given a is equivalent to b... |
| astbstanbst 45918 | Given a is equivalent to T... |
| aistbistaandb 45919 | Given a is equivalent to T... |
| aisbnaxb 45920 | Given a is equivalent to b... |
| atbiffatnnb 45921 | If a implies b, then a imp... |
| bisaiaisb 45922 | Application of bicom1 with... |
| atbiffatnnbalt 45923 | If a implies b, then a imp... |
| abnotbtaxb 45924 | Assuming a, not b, there e... |
| abnotataxb 45925 | Assuming not a, b, there e... |
| conimpf 45926 | Assuming a, not b, and a i... |
| conimpfalt 45927 | Assuming a, not b, and a i... |
| aistbisfiaxb 45928 | Given a is equivalent to T... |
| aisfbistiaxb 45929 | Given a is equivalent to F... |
| aifftbifffaibif 45930 | Given a is equivalent to T... |
| aifftbifffaibifff 45931 | Given a is equivalent to T... |
| atnaiana 45932 | Given a, it is not the cas... |
| ainaiaandna 45933 | Given a, a implies it is n... |
| abcdta 45934 | Given (((a and b) and c) a... |
| abcdtb 45935 | Given (((a and b) and c) a... |
| abcdtc 45936 | Given (((a and b) and c) a... |
| abcdtd 45937 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 45938 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 45939 | Operands in a biconditiona... |
| nabctnabc 45940 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 45941 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 45942 | From one negated implicati... |
| twonotinotbothi 45943 | From these two negated imp... |
| clifte 45944 | show d is the same as an i... |
| cliftet 45945 | show d is the same as an i... |
| clifteta 45946 | show d is the same as an i... |
| cliftetb 45947 | show d is the same as an i... |
| confun 45948 | Given the hypotheses there... |
| confun2 45949 | Confun simplified to two p... |
| confun3 45950 | Confun's more complex form... |
| confun4 45951 | An attempt at derivative. ... |
| confun5 45952 | An attempt at derivative. ... |
| plcofph 45953 | Given, a,b and a "definiti... |
| pldofph 45954 | Given, a,b c, d, "definiti... |
| plvcofph 45955 | Given, a,b,d, and "definit... |
| plvcofphax 45956 | Given, a,b,d, and "definit... |
| plvofpos 45957 | rh is derivable because ON... |
| mdandyv0 45958 | Given the equivalences set... |
| mdandyv1 45959 | Given the equivalences set... |
| mdandyv2 45960 | Given the equivalences set... |
| mdandyv3 45961 | Given the equivalences set... |
| mdandyv4 45962 | Given the equivalences set... |
| mdandyv5 45963 | Given the equivalences set... |
| mdandyv6 45964 | Given the equivalences set... |
| mdandyv7 45965 | Given the equivalences set... |
| mdandyv8 45966 | Given the equivalences set... |
| mdandyv9 45967 | Given the equivalences set... |
| mdandyv10 45968 | Given the equivalences set... |
| mdandyv11 45969 | Given the equivalences set... |
| mdandyv12 45970 | Given the equivalences set... |
| mdandyv13 45971 | Given the equivalences set... |
| mdandyv14 45972 | Given the equivalences set... |
| mdandyv15 45973 | Given the equivalences set... |
| mdandyvr0 45974 | Given the equivalences set... |
| mdandyvr1 45975 | Given the equivalences set... |
| mdandyvr2 45976 | Given the equivalences set... |
| mdandyvr3 45977 | Given the equivalences set... |
| mdandyvr4 45978 | Given the equivalences set... |
| mdandyvr5 45979 | Given the equivalences set... |
| mdandyvr6 45980 | Given the equivalences set... |
| mdandyvr7 45981 | Given the equivalences set... |
| mdandyvr8 45982 | Given the equivalences set... |
| mdandyvr9 45983 | Given the equivalences set... |
| mdandyvr10 45984 | Given the equivalences set... |
| mdandyvr11 45985 | Given the equivalences set... |
| mdandyvr12 45986 | Given the equivalences set... |
| mdandyvr13 45987 | Given the equivalences set... |
| mdandyvr14 45988 | Given the equivalences set... |
| mdandyvr15 45989 | Given the equivalences set... |
| mdandyvrx0 45990 | Given the exclusivities se... |
| mdandyvrx1 45991 | Given the exclusivities se... |
| mdandyvrx2 45992 | Given the exclusivities se... |
| mdandyvrx3 45993 | Given the exclusivities se... |
| mdandyvrx4 45994 | Given the exclusivities se... |
| mdandyvrx5 45995 | Given the exclusivities se... |
| mdandyvrx6 45996 | Given the exclusivities se... |
| mdandyvrx7 45997 | Given the exclusivities se... |
| mdandyvrx8 45998 | Given the exclusivities se... |
| mdandyvrx9 45999 | Given the exclusivities se... |
| mdandyvrx10 46000 | Given the exclusivities se... |
| mdandyvrx11 46001 | Given the exclusivities se... |
| mdandyvrx12 46002 | Given the exclusivities se... |
| mdandyvrx13 46003 | Given the exclusivities se... |
| mdandyvrx14 46004 | Given the exclusivities se... |
| mdandyvrx15 46005 | Given the exclusivities se... |
| H15NH16TH15IH16 46006 | Given 15 hypotheses and a ... |
| dandysum2p2e4 46007 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 46008 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 46009 | Replacement of a nested an... |
| adh-minim 46010 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 46011 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 46012 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 46013 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 46014 | Fourth lemma for the deriv... |
| adh-minim-ax1 46015 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 46016 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 46017 | Sixth lemma for the deriva... |
| adh-minim-ax2c 46018 | Derivation of a commuted f... |
| adh-minim-ax2 46019 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 46020 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 46021 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 46022 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 46023 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 46024 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 46025 | Third lemma for the deriva... |
| adh-minimp-sylsimp 46026 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 46027 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 46028 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 46029 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 46030 | Fourth lemma for the deriv... |
| adh-minimp-ax2 46031 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 46032 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 46033 | Derivation of ~ pm2.43 Whi... |
| n0nsn2el 46034 | If a class with one elemen... |
| eusnsn 46035 | There is a unique element ... |
| absnsb 46036 | If the class abstraction `... |
| euabsneu 46037 | Another way to express exi... |
| elprneb 46038 | An element of a proper uno... |
| oppr 46039 | Equality for ordered pairs... |
| opprb 46040 | Equality for unordered pai... |
| or2expropbilem1 46041 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 46042 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 46043 | If two classes are strictl... |
| eubrv 46044 | If there is a unique set w... |
| eubrdm 46045 | If there is a unique set w... |
| eldmressn 46046 | Element of the domain of a... |
| iota0def 46047 | Example for a defined iota... |
| iota0ndef 46048 | Example for an undefined i... |
| fveqvfvv 46049 | If a function's value at a... |
| fnresfnco 46050 | Composition of two functio... |
| funcoressn 46051 | A composition restricted t... |
| funressnfv 46052 | A restriction to a singlet... |
| funressndmfvrn 46053 | The value of a function ` ... |
| funressnvmo 46054 | A function restricted to a... |
| funressnmo 46055 | A function restricted to a... |
| funressneu 46056 | There is exactly one value... |
| fresfo 46057 | Conditions for a restricti... |
| fsetsniunop 46058 | The class of all functions... |
| fsetabsnop 46059 | The class of all functions... |
| fsetsnf 46060 | The mapping of an element ... |
| fsetsnf1 46061 | The mapping of an element ... |
| fsetsnfo 46062 | The mapping of an element ... |
| fsetsnf1o 46063 | The mapping of an element ... |
| fsetsnprcnex 46064 | The class of all functions... |
| cfsetssfset 46065 | The class of constant func... |
| cfsetsnfsetfv 46066 | The function value of the ... |
| cfsetsnfsetf 46067 | The mapping of the class o... |
| cfsetsnfsetf1 46068 | The mapping of the class o... |
| cfsetsnfsetfo 46069 | The mapping of the class o... |
| cfsetsnfsetf1o 46070 | The mapping of the class o... |
| fsetprcnexALT 46071 | First version of proof for... |
| fcoreslem1 46072 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 46073 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 46074 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 46075 | Lemma 4 for ~ fcores . (C... |
| fcores 46076 | Every composite function `... |
| fcoresf1lem 46077 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 46078 | If a composition is inject... |
| fcoresf1b 46079 | A composition is injective... |
| fcoresfo 46080 | If a composition is surjec... |
| fcoresfob 46081 | A composition is surjectiv... |
| fcoresf1ob 46082 | A composition is bijective... |
| f1cof1blem 46083 | Lemma for ~ f1cof1b and ~ ... |
| f1cof1b 46084 | If the range of ` F ` equa... |
| funfocofob 46085 | If the domain of a functio... |
| fnfocofob 46086 | If the domain of a functio... |
| focofob 46087 | If the domain of a functio... |
| f1ocof1ob 46088 | If the range of ` F ` equa... |
| f1ocof1ob2 46089 | If the range of ` F ` equa... |
| aiotajust 46091 | Soundness justification th... |
| dfaiota2 46093 | Alternate definition of th... |
| reuabaiotaiota 46094 | The iota and the alternate... |
| reuaiotaiota 46095 | The iota and the alternate... |
| aiotaexb 46096 | The alternate iota over a ... |
| aiotavb 46097 | The alternate iota over a ... |
| aiotaint 46098 | This is to ~ df-aiota what... |
| dfaiota3 46099 | Alternate definition of ` ... |
| iotan0aiotaex 46100 | If the iota over a wff ` p... |
| aiotaexaiotaiota 46101 | The alternate iota over a ... |
| aiotaval 46102 | Theorem 8.19 in [Quine] p.... |
| aiota0def 46103 | Example for a defined alte... |
| aiota0ndef 46104 | Example for an undefined a... |
| r19.32 46105 | Theorem 19.32 of [Margaris... |
| rexsb 46106 | An equivalent expression f... |
| rexrsb 46107 | An equivalent expression f... |
| 2rexsb 46108 | An equivalent expression f... |
| 2rexrsb 46109 | An equivalent expression f... |
| cbvral2 46110 | Change bound variables of ... |
| cbvrex2 46111 | Change bound variables of ... |
| ralndv1 46112 | Example for a theorem abou... |
| ralndv2 46113 | Second example for a theor... |
| reuf1odnf 46114 | There is exactly one eleme... |
| reuf1od 46115 | There is exactly one eleme... |
| euoreqb 46116 | There is a set which is eq... |
| 2reu3 46117 | Double restricted existent... |
| 2reu7 46118 | Two equivalent expressions... |
| 2reu8 46119 | Two equivalent expressions... |
| 2reu8i 46120 | Implication of a double re... |
| 2reuimp0 46121 | Implication of a double re... |
| 2reuimp 46122 | Implication of a double re... |
| ralbinrald 46129 | Elemination of a restricte... |
| nvelim 46130 | If a class is the universa... |
| alneu 46131 | If a statement holds for a... |
| eu2ndop1stv 46132 | If there is a unique secon... |
| dfateq12d 46133 | Equality deduction for "de... |
| nfdfat 46134 | Bound-variable hypothesis ... |
| dfdfat2 46135 | Alternate definition of th... |
| fundmdfat 46136 | A function is defined at a... |
| dfatprc 46137 | A function is not defined ... |
| dfatelrn 46138 | The value of a function ` ... |
| dfafv2 46139 | Alternative definition of ... |
| afveq12d 46140 | Equality deduction for fun... |
| afveq1 46141 | Equality theorem for funct... |
| afveq2 46142 | Equality theorem for funct... |
| nfafv 46143 | Bound-variable hypothesis ... |
| csbafv12g 46144 | Move class substitution in... |
| afvfundmfveq 46145 | If a class is a function r... |
| afvnfundmuv 46146 | If a set is not in the dom... |
| ndmafv 46147 | The value of a class outsi... |
| afvvdm 46148 | If the function value of a... |
| nfunsnafv 46149 | If the restriction of a cl... |
| afvvfunressn 46150 | If the function value of a... |
| afvprc 46151 | A function's value at a pr... |
| afvvv 46152 | If a function's value at a... |
| afvpcfv0 46153 | If the value of the altern... |
| afvnufveq 46154 | The value of the alternati... |
| afvvfveq 46155 | The value of the alternati... |
| afv0fv0 46156 | If the value of the altern... |
| afvfvn0fveq 46157 | If the function's value at... |
| afv0nbfvbi 46158 | The function's value at an... |
| afvfv0bi 46159 | The function's value at an... |
| afveu 46160 | The value of a function at... |
| fnbrafvb 46161 | Equivalence of function va... |
| fnopafvb 46162 | Equivalence of function va... |
| funbrafvb 46163 | Equivalence of function va... |
| funopafvb 46164 | Equivalence of function va... |
| funbrafv 46165 | The second argument of a b... |
| funbrafv2b 46166 | Function value in terms of... |
| dfafn5a 46167 | Representation of a functi... |
| dfafn5b 46168 | Representation of a functi... |
| fnrnafv 46169 | The range of a function ex... |
| afvelrnb 46170 | A member of a function's r... |
| afvelrnb0 46171 | A member of a function's r... |
| dfaimafn 46172 | Alternate definition of th... |
| dfaimafn2 46173 | Alternate definition of th... |
| afvelima 46174 | Function value in an image... |
| afvelrn 46175 | A function's value belongs... |
| fnafvelrn 46176 | A function's value belongs... |
| fafvelcdm 46177 | A function's value belongs... |
| ffnafv 46178 | A function maps to a class... |
| afvres 46179 | The value of a restricted ... |
| tz6.12-afv 46180 | Function value. Theorem 6... |
| tz6.12-1-afv 46181 | Function value (Theorem 6.... |
| dmfcoafv 46182 | Domains of a function comp... |
| afvco2 46183 | Value of a function compos... |
| rlimdmafv 46184 | Two ways to express that a... |
| aoveq123d 46185 | Equality deduction for ope... |
| nfaov 46186 | Bound-variable hypothesis ... |
| csbaovg 46187 | Move class substitution in... |
| aovfundmoveq 46188 | If a class is a function r... |
| aovnfundmuv 46189 | If an ordered pair is not ... |
| ndmaov 46190 | The value of an operation ... |
| ndmaovg 46191 | The value of an operation ... |
| aovvdm 46192 | If the operation value of ... |
| nfunsnaov 46193 | If the restriction of a cl... |
| aovvfunressn 46194 | If the operation value of ... |
| aovprc 46195 | The value of an operation ... |
| aovrcl 46196 | Reverse closure for an ope... |
| aovpcov0 46197 | If the alternative value o... |
| aovnuoveq 46198 | The alternative value of t... |
| aovvoveq 46199 | The alternative value of t... |
| aov0ov0 46200 | If the alternative value o... |
| aovovn0oveq 46201 | If the operation's value a... |
| aov0nbovbi 46202 | The operation's value on a... |
| aovov0bi 46203 | The operation's value on a... |
| rspceaov 46204 | A frequently used special ... |
| fnotaovb 46205 | Equivalence of operation v... |
| ffnaov 46206 | An operation maps to a cla... |
| faovcl 46207 | Closure law for an operati... |
| aovmpt4g 46208 | Value of a function given ... |
| aoprssdm 46209 | Domain of closure of an op... |
| ndmaovcl 46210 | The "closure" of an operat... |
| ndmaovrcl 46211 | Reverse closure law, in co... |
| ndmaovcom 46212 | Any operation is commutati... |
| ndmaovass 46213 | Any operation is associati... |
| ndmaovdistr 46214 | Any operation is distribut... |
| dfatafv2iota 46217 | If a function is defined a... |
| ndfatafv2 46218 | The alternate function val... |
| ndfatafv2undef 46219 | The alternate function val... |
| dfatafv2ex 46220 | The alternate function val... |
| afv2ex 46221 | The alternate function val... |
| afv2eq12d 46222 | Equality deduction for fun... |
| afv2eq1 46223 | Equality theorem for funct... |
| afv2eq2 46224 | Equality theorem for funct... |
| nfafv2 46225 | Bound-variable hypothesis ... |
| csbafv212g 46226 | Move class substitution in... |
| fexafv2ex 46227 | The alternate function val... |
| ndfatafv2nrn 46228 | The alternate function val... |
| ndmafv2nrn 46229 | The value of a class outsi... |
| funressndmafv2rn 46230 | The alternate function val... |
| afv2ndefb 46231 | Two ways to say that an al... |
| nfunsnafv2 46232 | If the restriction of a cl... |
| afv2prc 46233 | A function's value at a pr... |
| dfatafv2rnb 46234 | The alternate function val... |
| afv2orxorb 46235 | If a set is in the range o... |
| dmafv2rnb 46236 | The alternate function val... |
| fundmafv2rnb 46237 | The alternate function val... |
| afv2elrn 46238 | An alternate function valu... |
| afv20defat 46239 | If the alternate function ... |
| fnafv2elrn 46240 | An alternate function valu... |
| fafv2elcdm 46241 | An alternate function valu... |
| fafv2elrnb 46242 | An alternate function valu... |
| fcdmvafv2v 46243 | If the codomain of a funct... |
| tz6.12-2-afv2 46244 | Function value when ` F ` ... |
| afv2eu 46245 | The value of a function at... |
| afv2res 46246 | The value of a restricted ... |
| tz6.12-afv2 46247 | Function value (Theorem 6.... |
| tz6.12-1-afv2 46248 | Function value (Theorem 6.... |
| tz6.12c-afv2 46249 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 46250 | Corollary of Theorem 6.12(... |
| funressnbrafv2 46251 | The second argument of a b... |
| dfatbrafv2b 46252 | Equivalence of function va... |
| dfatopafv2b 46253 | Equivalence of function va... |
| funbrafv2 46254 | The second argument of a b... |
| fnbrafv2b 46255 | Equivalence of function va... |
| fnopafv2b 46256 | Equivalence of function va... |
| funbrafv22b 46257 | Equivalence of function va... |
| funopafv2b 46258 | Equivalence of function va... |
| dfatsnafv2 46259 | Singleton of function valu... |
| dfafv23 46260 | A definition of function v... |
| dfatdmfcoafv2 46261 | Domain of a function compo... |
| dfatcolem 46262 | Lemma for ~ dfatco . (Con... |
| dfatco 46263 | The predicate "defined at"... |
| afv2co2 46264 | Value of a function compos... |
| rlimdmafv2 46265 | Two ways to express that a... |
| dfafv22 46266 | Alternate definition of ` ... |
| afv2ndeffv0 46267 | If the alternate function ... |
| dfatafv2eqfv 46268 | If a function is defined a... |
| afv2rnfveq 46269 | If the alternate function ... |
| afv20fv0 46270 | If the alternate function ... |
| afv2fvn0fveq 46271 | If the function's value at... |
| afv2fv0 46272 | If the function's value at... |
| afv2fv0b 46273 | The function's value at an... |
| afv2fv0xorb 46274 | If a set is in the range o... |
| an4com24 46275 | Rearrangement of 4 conjunc... |
| 3an4ancom24 46276 | Commutative law for a conj... |
| 4an21 46277 | Rearrangement of 4 conjunc... |
| dfnelbr2 46280 | Alternate definition of th... |
| nelbr 46281 | The binary relation of a s... |
| nelbrim 46282 | If a set is related to ano... |
| nelbrnel 46283 | A set is related to anothe... |
| nelbrnelim 46284 | If a set is related to ano... |
| ralralimp 46285 | Selecting one of two alter... |
| otiunsndisjX 46286 | The union of singletons co... |
| fvifeq 46287 | Equality of function value... |
| rnfdmpr 46288 | The range of a one-to-one ... |
| imarnf1pr 46289 | The image of the range of ... |
| funop1 46290 | A function is an ordered p... |
| fun2dmnopgexmpl 46291 | A function with a domain c... |
| opabresex0d 46292 | A collection of ordered pa... |
| opabbrfex0d 46293 | A collection of ordered pa... |
| opabresexd 46294 | A collection of ordered pa... |
| opabbrfexd 46295 | A collection of ordered pa... |
| f1oresf1orab 46296 | Build a bijection by restr... |
| f1oresf1o 46297 | Build a bijection by restr... |
| f1oresf1o2 46298 | Build a bijection by restr... |
| fvmptrab 46299 | Value of a function mappin... |
| fvmptrabdm 46300 | Value of a function mappin... |
| cnambpcma 46301 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 46302 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 46303 | The sum of two complex num... |
| leaddsuble 46304 | Addition and subtraction o... |
| 2leaddle2 46305 | If two real numbers are le... |
| ltnltne 46306 | Variant of trichotomy law ... |
| p1lep2 46307 | A real number increasd by ... |
| ltsubsubaddltsub 46308 | If the result of subtracti... |
| zm1nn 46309 | An integer minus 1 is posi... |
| readdcnnred 46310 | The sum of a real number a... |
| resubcnnred 46311 | The difference of a real n... |
| recnmulnred 46312 | The product of a real numb... |
| cndivrenred 46313 | The quotient of an imagina... |
| sqrtnegnre 46314 | The square root of a negat... |
| nn0resubcl 46315 | Closure law for subtractio... |
| zgeltp1eq 46316 | If an integer is between a... |
| 1t10e1p1e11 46317 | 11 is 1 times 10 to the po... |
| deccarry 46318 | Add 1 to a 2 digit number ... |
| eluzge0nn0 46319 | If an integer is greater t... |
| nltle2tri 46320 | Negated extended trichotom... |
| ssfz12 46321 | Subset relationship for fi... |
| elfz2z 46322 | Membership of an integer i... |
| 2elfz3nn0 46323 | If there are two elements ... |
| fz0addcom 46324 | The addition of two member... |
| 2elfz2melfz 46325 | If the sum of two integers... |
| fz0addge0 46326 | The sum of two integers in... |
| elfzlble 46327 | Membership of an integer i... |
| elfzelfzlble 46328 | Membership of an element o... |
| fzopred 46329 | Join a predecessor to the ... |
| fzopredsuc 46330 | Join a predecessor and a s... |
| 1fzopredsuc 46331 | Join 0 and a successor to ... |
| el1fzopredsuc 46332 | An element of an open inte... |
| subsubelfzo0 46333 | Subtracting a difference f... |
| fzoopth 46334 | A half-open integer range ... |
| 2ffzoeq 46335 | Two functions over a half-... |
| m1mod0mod1 46336 | An integer decreased by 1 ... |
| elmod2 46337 | An integer modulo 2 is eit... |
| smonoord 46338 | Ordering relation for a st... |
| fsummsndifre 46339 | A finite sum with one of i... |
| fsumsplitsndif 46340 | Separate out a term in a f... |
| fsummmodsndifre 46341 | A finite sum of summands m... |
| fsummmodsnunz 46342 | A finite sum of summands m... |
| setsidel 46343 | The injected slot is an el... |
| setsnidel 46344 | The injected slot is an el... |
| setsv 46345 | The value of the structure... |
| preimafvsnel 46346 | The preimage of a function... |
| preimafvn0 46347 | The preimage of a function... |
| uniimafveqt 46348 | The union of the image of ... |
| uniimaprimaeqfv 46349 | The union of the image of ... |
| setpreimafvex 46350 | The class ` P ` of all pre... |
| elsetpreimafvb 46351 | The characterization of an... |
| elsetpreimafv 46352 | An element of the class ` ... |
| elsetpreimafvssdm 46353 | An element of the class ` ... |
| fvelsetpreimafv 46354 | There is an element in a p... |
| preimafvelsetpreimafv 46355 | The preimage of a function... |
| preimafvsspwdm 46356 | The class ` P ` of all pre... |
| 0nelsetpreimafv 46357 | The empty set is not an el... |
| elsetpreimafvbi 46358 | An element of the preimage... |
| elsetpreimafveqfv 46359 | The elements of the preima... |
| eqfvelsetpreimafv 46360 | If an element of the domai... |
| elsetpreimafvrab 46361 | An element of the preimage... |
| imaelsetpreimafv 46362 | The image of an element of... |
| uniimaelsetpreimafv 46363 | The union of the image of ... |
| elsetpreimafveq 46364 | If two preimages of functi... |
| fundcmpsurinjlem1 46365 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 46366 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 46367 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 46368 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 46369 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 46370 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 46371 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 46372 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 46373 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 46374 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 46375 | Every function ` F : A -->... |
| fundcmpsurinj 46376 | Every function ` F : A -->... |
| fundcmpsurbijinj 46377 | Every function ` F : A -->... |
| fundcmpsurinjimaid 46378 | Every function ` F : A -->... |
| fundcmpsurinjALT 46379 | Alternate proof of ~ fundc... |
| iccpval 46382 | Partition consisting of a ... |
| iccpart 46383 | A special partition. Corr... |
| iccpartimp 46384 | Implications for a class b... |
| iccpartres 46385 | The restriction of a parti... |
| iccpartxr 46386 | If there is a partition, t... |
| iccpartgtprec 46387 | If there is a partition, t... |
| iccpartipre 46388 | If there is a partition, t... |
| iccpartiltu 46389 | If there is a partition, t... |
| iccpartigtl 46390 | If there is a partition, t... |
| iccpartlt 46391 | If there is a partition, t... |
| iccpartltu 46392 | If there is a partition, t... |
| iccpartgtl 46393 | If there is a partition, t... |
| iccpartgt 46394 | If there is a partition, t... |
| iccpartleu 46395 | If there is a partition, t... |
| iccpartgel 46396 | If there is a partition, t... |
| iccpartrn 46397 | If there is a partition, t... |
| iccpartf 46398 | The range of the partition... |
| iccpartel 46399 | If there is a partition, t... |
| iccelpart 46400 | An element of any partitio... |
| iccpartiun 46401 | A half-open interval of ex... |
| icceuelpartlem 46402 | Lemma for ~ icceuelpart . ... |
| icceuelpart 46403 | An element of a partitione... |
| iccpartdisj 46404 | The segments of a partitio... |
| iccpartnel 46405 | A point of a partition is ... |
| fargshiftfv 46406 | If a class is a function, ... |
| fargshiftf 46407 | If a class is a function, ... |
| fargshiftf1 46408 | If a function is 1-1, then... |
| fargshiftfo 46409 | If a function is onto, the... |
| fargshiftfva 46410 | The values of a shifted fu... |
| lswn0 46411 | The last symbol of a not e... |
| nfich1 46414 | The first interchangeable ... |
| nfich2 46415 | The second interchangeable... |
| ichv 46416 | Setvar variables are inter... |
| ichf 46417 | Setvar variables are inter... |
| ichid 46418 | A setvar variable is alway... |
| icht 46419 | A theorem is interchangeab... |
| ichbidv 46420 | Formula building rule for ... |
| ichcircshi 46421 | The setvar variables are i... |
| ichan 46422 | If two setvar variables ar... |
| ichn 46423 | Negation does not affect i... |
| ichim 46424 | Formula building rule for ... |
| dfich2 46425 | Alternate definition of th... |
| ichcom 46426 | The interchangeability of ... |
| ichbi12i 46427 | Equivalence for interchang... |
| icheqid 46428 | In an equality for the sam... |
| icheq 46429 | In an equality of setvar v... |
| ichnfimlem 46430 | Lemma for ~ ichnfim : A s... |
| ichnfim 46431 | If in an interchangeabilit... |
| ichnfb 46432 | If ` x ` and ` y ` are int... |
| ichal 46433 | Move a universal quantifie... |
| ich2al 46434 | Two setvar variables are a... |
| ich2ex 46435 | Two setvar variables are a... |
| ichexmpl1 46436 | Example for interchangeabl... |
| ichexmpl2 46437 | Example for interchangeabl... |
| ich2exprop 46438 | If the setvar variables ar... |
| ichnreuop 46439 | If the setvar variables ar... |
| ichreuopeq 46440 | If the setvar variables ar... |
| sprid 46441 | Two identical representati... |
| elsprel 46442 | An unordered pair is an el... |
| spr0nelg 46443 | The empty set is not an el... |
| sprval 46446 | The set of all unordered p... |
| sprvalpw 46447 | The set of all unordered p... |
| sprssspr 46448 | The set of all unordered p... |
| spr0el 46449 | The empty set is not an un... |
| sprvalpwn0 46450 | The set of all unordered p... |
| sprel 46451 | An element of the set of a... |
| prssspr 46452 | An element of a subset of ... |
| prelspr 46453 | An unordered pair of eleme... |
| prsprel 46454 | The elements of a pair fro... |
| prsssprel 46455 | The elements of a pair fro... |
| sprvalpwle2 46456 | The set of all unordered p... |
| sprsymrelfvlem 46457 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 46458 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 46459 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 46460 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 46461 | The value of the function ... |
| sprsymrelf 46462 | The mapping ` F ` is a fun... |
| sprsymrelf1 46463 | The mapping ` F ` is a one... |
| sprsymrelfo 46464 | The mapping ` F ` is a fun... |
| sprsymrelf1o 46465 | The mapping ` F ` is a bij... |
| sprbisymrel 46466 | There is a bijection betwe... |
| sprsymrelen 46467 | The class ` P ` of subsets... |
| prpair 46468 | Characterization of a prop... |
| prproropf1olem0 46469 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 46470 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 46471 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 46472 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 46473 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 46474 | There is a bijection betwe... |
| prproropen 46475 | The set of proper pairs an... |
| prproropreud 46476 | There is exactly one order... |
| pairreueq 46477 | Two equivalent representat... |
| paireqne 46478 | Two sets are not equal iff... |
| prprval 46481 | The set of all proper unor... |
| prprvalpw 46482 | The set of all proper unor... |
| prprelb 46483 | An element of the set of a... |
| prprelprb 46484 | A set is an element of the... |
| prprspr2 46485 | The set of all proper unor... |
| prprsprreu 46486 | There is a unique proper u... |
| prprreueq 46487 | There is a unique proper u... |
| sbcpr 46488 | The proper substitution of... |
| reupr 46489 | There is a unique unordere... |
| reuprpr 46490 | There is a unique proper u... |
| poprelb 46491 | Equality for unordered pai... |
| 2exopprim 46492 | The existence of an ordere... |
| reuopreuprim 46493 | There is a unique unordere... |
| fmtno 46496 | The ` N ` th Fermat number... |
| fmtnoge3 46497 | Each Fermat number is grea... |
| fmtnonn 46498 | Each Fermat number is a po... |
| fmtnom1nn 46499 | A Fermat number minus one ... |
| fmtnoodd 46500 | Each Fermat number is odd.... |
| fmtnorn 46501 | A Fermat number is a funct... |
| fmtnof1 46502 | The enumeration of the Fer... |
| fmtnoinf 46503 | The set of Fermat numbers ... |
| fmtnorec1 46504 | The first recurrence relat... |
| sqrtpwpw2p 46505 | The floor of the square ro... |
| fmtnosqrt 46506 | The floor of the square ro... |
| fmtno0 46507 | The ` 0 ` th Fermat number... |
| fmtno1 46508 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 46509 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 46510 | The second recurrence rela... |
| fmtnodvds 46511 | Any Fermat number divides ... |
| goldbachthlem1 46512 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 46513 | Lemma 2 for ~ goldbachth .... |
| goldbachth 46514 | Goldbach's theorem: Two d... |
| fmtnorec3 46515 | The third recurrence relat... |
| fmtnorec4 46516 | The fourth recurrence rela... |
| fmtno2 46517 | The ` 2 ` nd Fermat number... |
| fmtno3 46518 | The ` 3 ` rd Fermat number... |
| fmtno4 46519 | The ` 4 ` th Fermat number... |
| fmtno5lem1 46520 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 46521 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 46522 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 46523 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 46524 | The ` 5 ` th Fermat number... |
| fmtno0prm 46525 | The ` 0 ` th Fermat number... |
| fmtno1prm 46526 | The ` 1 ` st Fermat number... |
| fmtno2prm 46527 | The ` 2 ` nd Fermat number... |
| 257prm 46528 | 257 is a prime number (the... |
| fmtno3prm 46529 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 46530 | Any power of two is coprim... |
| fmtnoprmfac1lem 46531 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 46532 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 46533 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 46534 | Divisor of Fermat number (... |
| fmtnofac2lem 46535 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 46536 | Divisor of Fermat number (... |
| fmtnofac1 46537 | Divisor of Fermat number (... |
| fmtno4sqrt 46538 | The floor of the square ro... |
| fmtno4prmfac 46539 | If P was a (prime) factor ... |
| fmtno4prmfac193 46540 | If P was a (prime) factor ... |
| fmtno4nprmfac193 46541 | 193 is not a (prime) facto... |
| fmtno4prm 46542 | The ` 4 `-th Fermat number... |
| 65537prm 46543 | 65537 is a prime number (t... |
| fmtnofz04prm 46544 | The first five Fermat numb... |
| fmtnole4prm 46545 | The first five Fermat numb... |
| fmtno5faclem1 46546 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 46547 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 46548 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 46549 | The factorisation of the `... |
| fmtno5nprm 46550 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 46551 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 46552 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 46553 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 46554 | The mapping of a Fermat nu... |
| prminf2 46555 | The set of prime numbers i... |
| 2pwp1prm 46556 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 46557 | Every prime number of the ... |
| m2prm 46558 | The second Mersenne number... |
| m3prm 46559 | The third Mersenne number ... |
| flsqrt 46560 | A condition equivalent to ... |
| flsqrt5 46561 | The floor of the square ro... |
| 3ndvds4 46562 | 3 does not divide 4. (Con... |
| 139prmALT 46563 | 139 is a prime number. In... |
| 31prm 46564 | 31 is a prime number. In ... |
| m5prm 46565 | The fifth Mersenne number ... |
| 127prm 46566 | 127 is a prime number. (C... |
| m7prm 46567 | The seventh Mersenne numbe... |
| m11nprm 46568 | The eleventh Mersenne numb... |
| mod42tp1mod8 46569 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 46570 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 46571 | If ` P ` is a Sophie Germa... |
| lighneallem1 46572 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 46573 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 46574 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 46575 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 46576 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 46577 | Lemma 3 for ~ lighneal . ... |
| lighneal 46578 | If a power of a prime ` P ... |
| modexp2m1d 46579 | The square of an integer w... |
| proththdlem 46580 | Lemma for ~ proththd . (C... |
| proththd 46581 | Proth's theorem (1878). I... |
| 5tcu2e40 46582 | 5 times the cube of 2 is 4... |
| 3exp4mod41 46583 | 3 to the fourth power is -... |
| 41prothprmlem1 46584 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 46585 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 46586 | 41 is a _Proth prime_. (C... |
| quad1 46587 | A condition for a quadrati... |
| requad01 46588 | A condition for a quadrati... |
| requad1 46589 | A condition for a quadrati... |
| requad2 46590 | A condition for a quadrati... |
| iseven 46595 | The predicate "is an even ... |
| isodd 46596 | The predicate "is an odd n... |
| evenz 46597 | An even number is an integ... |
| oddz 46598 | An odd number is an intege... |
| evendiv2z 46599 | The result of dividing an ... |
| oddp1div2z 46600 | The result of dividing an ... |
| oddm1div2z 46601 | The result of dividing an ... |
| isodd2 46602 | The predicate "is an odd n... |
| dfodd2 46603 | Alternate definition for o... |
| dfodd6 46604 | Alternate definition for o... |
| dfeven4 46605 | Alternate definition for e... |
| evenm1odd 46606 | The predecessor of an even... |
| evenp1odd 46607 | The successor of an even n... |
| oddp1eveni 46608 | The successor of an odd nu... |
| oddm1eveni 46609 | The predecessor of an odd ... |
| evennodd 46610 | An even number is not an o... |
| oddneven 46611 | An odd number is not an ev... |
| enege 46612 | The negative of an even nu... |
| onego 46613 | The negative of an odd num... |
| m1expevenALTV 46614 | Exponentiation of -1 by an... |
| m1expoddALTV 46615 | Exponentiation of -1 by an... |
| dfeven2 46616 | Alternate definition for e... |
| dfodd3 46617 | Alternate definition for o... |
| iseven2 46618 | The predicate "is an even ... |
| isodd3 46619 | The predicate "is an odd n... |
| 2dvdseven 46620 | 2 divides an even number. ... |
| m2even 46621 | A multiple of 2 is an even... |
| 2ndvdsodd 46622 | 2 does not divide an odd n... |
| 2dvdsoddp1 46623 | 2 divides an odd number in... |
| 2dvdsoddm1 46624 | 2 divides an odd number de... |
| dfeven3 46625 | Alternate definition for e... |
| dfodd4 46626 | Alternate definition for o... |
| dfodd5 46627 | Alternate definition for o... |
| zefldiv2ALTV 46628 | The floor of an even numbe... |
| zofldiv2ALTV 46629 | The floor of an odd numer ... |
| oddflALTV 46630 | Odd number representation ... |
| iseven5 46631 | The predicate "is an even ... |
| isodd7 46632 | The predicate "is an odd n... |
| dfeven5 46633 | Alternate definition for e... |
| dfodd7 46634 | Alternate definition for o... |
| gcd2odd1 46635 | The greatest common diviso... |
| zneoALTV 46636 | No even integer equals an ... |
| zeoALTV 46637 | An integer is even or odd.... |
| zeo2ALTV 46638 | An integer is even or odd ... |
| nneoALTV 46639 | A positive integer is even... |
| nneoiALTV 46640 | A positive integer is even... |
| odd2np1ALTV 46641 | An integer is odd iff it i... |
| oddm1evenALTV 46642 | An integer is odd iff its ... |
| oddp1evenALTV 46643 | An integer is odd iff its ... |
| oexpnegALTV 46644 | The exponential of the neg... |
| oexpnegnz 46645 | The exponential of the neg... |
| bits0ALTV 46646 | Value of the zeroth bit. ... |
| bits0eALTV 46647 | The zeroth bit of an even ... |
| bits0oALTV 46648 | The zeroth bit of an odd n... |
| divgcdoddALTV 46649 | Either ` A / ( A gcd B ) `... |
| opoeALTV 46650 | The sum of two odds is eve... |
| opeoALTV 46651 | The sum of an odd and an e... |
| omoeALTV 46652 | The difference of two odds... |
| omeoALTV 46653 | The difference of an odd a... |
| oddprmALTV 46654 | A prime not equal to ` 2 `... |
| 0evenALTV 46655 | 0 is an even number. (Con... |
| 0noddALTV 46656 | 0 is not an odd number. (... |
| 1oddALTV 46657 | 1 is an odd number. (Cont... |
| 1nevenALTV 46658 | 1 is not an even number. ... |
| 2evenALTV 46659 | 2 is an even number. (Con... |
| 2noddALTV 46660 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 46661 | An odd nonnegative integer... |
| nnoALTV 46662 | An alternate characterizat... |
| nn0oALTV 46663 | An alternate characterizat... |
| nn0e 46664 | An alternate characterizat... |
| nneven 46665 | An alternate characterizat... |
| nn0onn0exALTV 46666 | For each odd nonnegative i... |
| nn0enn0exALTV 46667 | For each even nonnegative ... |
| nnennexALTV 46668 | For each even positive int... |
| nnpw2evenALTV 46669 | 2 to the power of a positi... |
| epoo 46670 | The sum of an even and an ... |
| emoo 46671 | The difference of an even ... |
| epee 46672 | The sum of two even number... |
| emee 46673 | The difference of two even... |
| evensumeven 46674 | If a summand is even, the ... |
| 3odd 46675 | 3 is an odd number. (Cont... |
| 4even 46676 | 4 is an even number. (Con... |
| 5odd 46677 | 5 is an odd number. (Cont... |
| 6even 46678 | 6 is an even number. (Con... |
| 7odd 46679 | 7 is an odd number. (Cont... |
| 8even 46680 | 8 is an even number. (Con... |
| evenprm2 46681 | A prime number is even iff... |
| oddprmne2 46682 | Every prime number not bei... |
| oddprmuzge3 46683 | A prime number which is od... |
| evenltle 46684 | If an even number is great... |
| odd2prm2 46685 | If an odd number is the su... |
| even3prm2 46686 | If an even number is the s... |
| mogoldbblem 46687 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 46688 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 46689 | Lemma for ~ perfectALTV . ... |
| perfectALTV 46690 | The Euclid-Euler theorem, ... |
| fppr 46693 | The set of Fermat pseudopr... |
| fpprmod 46694 | The set of Fermat pseudopr... |
| fpprel 46695 | A Fermat pseudoprime to th... |
| fpprbasnn 46696 | The base of a Fermat pseud... |
| fpprnn 46697 | A Fermat pseudoprime to th... |
| fppr2odd 46698 | A Fermat pseudoprime to th... |
| 11t31e341 46699 | 341 is the product of 11 a... |
| 2exp340mod341 46700 | Eight to the eighth power ... |
| 341fppr2 46701 | 341 is the (smallest) _Pou... |
| 4fppr1 46702 | 4 is the (smallest) Fermat... |
| 8exp8mod9 46703 | Eight to the eighth power ... |
| 9fppr8 46704 | 9 is the (smallest) Fermat... |
| dfwppr 46705 | Alternate definition of a ... |
| fpprwppr 46706 | A Fermat pseudoprime to th... |
| fpprwpprb 46707 | An integer ` X ` which is ... |
| fpprel2 46708 | An alternate definition fo... |
| nfermltl8rev 46709 | Fermat's little theorem wi... |
| nfermltl2rev 46710 | Fermat's little theorem wi... |
| nfermltlrev 46711 | Fermat's little theorem re... |
| isgbe 46718 | The predicate "is an even ... |
| isgbow 46719 | The predicate "is a weak o... |
| isgbo 46720 | The predicate "is an odd G... |
| gbeeven 46721 | An even Goldbach number is... |
| gbowodd 46722 | A weak odd Goldbach number... |
| gbogbow 46723 | A (strong) odd Goldbach nu... |
| gboodd 46724 | An odd Goldbach number is ... |
| gbepos 46725 | Any even Goldbach number i... |
| gbowpos 46726 | Any weak odd Goldbach numb... |
| gbopos 46727 | Any odd Goldbach number is... |
| gbegt5 46728 | Any even Goldbach number i... |
| gbowgt5 46729 | Any weak odd Goldbach numb... |
| gbowge7 46730 | Any weak odd Goldbach numb... |
| gboge9 46731 | Any odd Goldbach number is... |
| gbege6 46732 | Any even Goldbach number i... |
| gbpart6 46733 | The Goldbach partition of ... |
| gbpart7 46734 | The (weak) Goldbach partit... |
| gbpart8 46735 | The Goldbach partition of ... |
| gbpart9 46736 | The (strong) Goldbach part... |
| gbpart11 46737 | The (strong) Goldbach part... |
| 6gbe 46738 | 6 is an even Goldbach numb... |
| 7gbow 46739 | 7 is a weak odd Goldbach n... |
| 8gbe 46740 | 8 is an even Goldbach numb... |
| 9gbo 46741 | 9 is an odd Goldbach numbe... |
| 11gbo 46742 | 11 is an odd Goldbach numb... |
| stgoldbwt 46743 | If the strong ternary Gold... |
| sbgoldbwt 46744 | If the strong binary Goldb... |
| sbgoldbst 46745 | If the strong binary Goldb... |
| sbgoldbaltlem1 46746 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 46747 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 46748 | An alternate (related to t... |
| sbgoldbb 46749 | If the strong binary Goldb... |
| sgoldbeven3prm 46750 | If the binary Goldbach con... |
| sbgoldbm 46751 | If the strong binary Goldb... |
| mogoldbb 46752 | If the modern version of t... |
| sbgoldbmb 46753 | The strong binary Goldbach... |
| sbgoldbo 46754 | If the strong binary Goldb... |
| nnsum3primes4 46755 | 4 is the sum of at most 3 ... |
| nnsum4primes4 46756 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 46757 | Every prime is "the sum of... |
| nnsum4primesprm 46758 | Every prime is "the sum of... |
| nnsum3primesgbe 46759 | Any even Goldbach number i... |
| nnsum4primesgbe 46760 | Any even Goldbach number i... |
| nnsum3primesle9 46761 | Every integer greater than... |
| nnsum4primesle9 46762 | Every integer greater than... |
| nnsum4primesodd 46763 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 46764 | If the (strong) ternary Go... |
| evengpop3 46765 | If the (weak) ternary Gold... |
| evengpoap3 46766 | If the (strong) ternary Go... |
| nnsum4primeseven 46767 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 46768 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 46769 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 46770 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 46771 | If the binary Goldbach con... |
| bgoldbtbndlem1 46772 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 46773 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 46774 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 46775 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 46776 | If the binary Goldbach con... |
| tgoldbachgtALTV 46779 | Variant of Thierry Arnoux'... |
| bgoldbachlt 46780 | The binary Goldbach conjec... |
| tgblthelfgott 46782 | The ternary Goldbach conje... |
| tgoldbachlt 46783 | The ternary Goldbach conje... |
| tgoldbach 46784 | The ternary Goldbach conje... |
| isomgrrel 46789 | The isomorphy relation for... |
| isomgr 46790 | The isomorphy relation for... |
| isisomgr 46791 | Implications of two graphs... |
| isomgreqve 46792 | A set is isomorphic to a h... |
| isomushgr 46793 | The isomorphy relation for... |
| isomuspgrlem1 46794 | Lemma 1 for ~ isomuspgr . ... |
| isomuspgrlem2a 46795 | Lemma 1 for ~ isomuspgrlem... |
| isomuspgrlem2b 46796 | Lemma 2 for ~ isomuspgrlem... |
| isomuspgrlem2c 46797 | Lemma 3 for ~ isomuspgrlem... |
| isomuspgrlem2d 46798 | Lemma 4 for ~ isomuspgrlem... |
| isomuspgrlem2e 46799 | Lemma 5 for ~ isomuspgrlem... |
| isomuspgrlem2 46800 | Lemma 2 for ~ isomuspgr . ... |
| isomuspgr 46801 | The isomorphy relation for... |
| isomgrref 46802 | The isomorphy relation is ... |
| isomgrsym 46803 | The isomorphy relation is ... |
| isomgrsymb 46804 | The isomorphy relation is ... |
| isomgrtrlem 46805 | Lemma for ~ isomgrtr . (C... |
| isomgrtr 46806 | The isomorphy relation is ... |
| strisomgrop 46807 | A graph represented as an ... |
| ushrisomgr 46808 | A simple hypergraph (with ... |
| 1hegrlfgr 46809 | A graph ` G ` with one hyp... |
| upwlksfval 46812 | The set of simple walks (i... |
| isupwlk 46813 | Properties of a pair of fu... |
| isupwlkg 46814 | Generalization of ~ isupwl... |
| upwlkbprop 46815 | Basic properties of a simp... |
| upwlkwlk 46816 | A simple walk is a walk. ... |
| upgrwlkupwlk 46817 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 46818 | In a pseudograph, the defi... |
| upgrisupwlkALT 46819 | Alternate proof of ~ upgri... |
| upgredgssspr 46820 | The set of edges of a pseu... |
| uspgropssxp 46821 | The set ` G ` of "simple p... |
| uspgrsprfv 46822 | The value of the function ... |
| uspgrsprf 46823 | The mapping ` F ` is a fun... |
| uspgrsprf1 46824 | The mapping ` F ` is a one... |
| uspgrsprfo 46825 | The mapping ` F ` is a fun... |
| uspgrsprf1o 46826 | The mapping ` F ` is a bij... |
| uspgrex 46827 | The class ` G ` of all "si... |
| uspgrbispr 46828 | There is a bijection betwe... |
| uspgrspren 46829 | The set ` G ` of the "simp... |
| uspgrymrelen 46830 | The set ` G ` of the "simp... |
| uspgrbisymrel 46831 | There is a bijection betwe... |
| uspgrbisymrelALT 46832 | Alternate proof of ~ uspgr... |
| ovn0dmfun 46833 | If a class operation value... |
| xpsnopab 46834 | A Cartesian product with a... |
| xpiun 46835 | A Cartesian product expres... |
| ovn0ssdmfun 46836 | If a class' operation valu... |
| fnxpdmdm 46837 | The domain of the domain o... |
| cnfldsrngbas 46838 | The base set of a subring ... |
| cnfldsrngadd 46839 | The group addition operati... |
| cnfldsrngmul 46840 | The ring multiplication op... |
| plusfreseq 46841 | If the empty set is not co... |
| mgmplusfreseq 46842 | If the empty set is not co... |
| 0mgm 46843 | A set with an empty base s... |
| opmpoismgm 46844 | A structure with a group a... |
| copissgrp 46845 | A structure with a constan... |
| copisnmnd 46846 | A structure with a constan... |
| 0nodd 46847 | 0 is not an odd integer. ... |
| 1odd 46848 | 1 is an odd integer. (Con... |
| 2nodd 46849 | 2 is not an odd integer. ... |
| oddibas 46850 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 46851 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 46852 | The structure of all odd i... |
| nnsgrpmgm 46853 | The structure of positive ... |
| nnsgrp 46854 | The structure of positive ... |
| nnsgrpnmnd 46855 | The structure of positive ... |
| nn0mnd 46856 | The set of nonnegative int... |
| gsumsplit2f 46857 | Split a group sum into two... |
| gsumdifsndf 46858 | Extract a summand from a f... |
| gsumfsupp 46859 | A group sum of a family ca... |
| iscllaw 46866 | The predicate "is a closed... |
| iscomlaw 46867 | The predicate "is a commut... |
| clcllaw 46868 | Closure of a closed operat... |
| isasslaw 46869 | The predicate "is an assoc... |
| asslawass 46870 | Associativity of an associ... |
| mgmplusgiopALT 46871 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 46872 | Slot 2 (group operation) o... |
| intopval 46879 | The internal (binary) oper... |
| intop 46880 | An internal (binary) opera... |
| clintopval 46881 | The closed (internal binar... |
| assintopval 46882 | The associative (closed in... |
| assintopmap 46883 | The associative (closed in... |
| isclintop 46884 | The predicate "is a closed... |
| clintop 46885 | A closed (internal binary)... |
| assintop 46886 | An associative (closed int... |
| isassintop 46887 | The predicate "is an assoc... |
| clintopcllaw 46888 | The closure law holds for ... |
| assintopcllaw 46889 | The closure low holds for ... |
| assintopasslaw 46890 | The associative low holds ... |
| assintopass 46891 | An associative (closed int... |
| ismgmALT 46900 | The predicate "is a magma"... |
| iscmgmALT 46901 | The predicate "is a commut... |
| issgrpALT 46902 | The predicate "is a semigr... |
| iscsgrpALT 46903 | The predicate "is a commut... |
| mgm2mgm 46904 | Equivalence of the two def... |
| sgrp2sgrp 46905 | Equivalence of the two def... |
| idfusubc0 46906 | The identity functor for a... |
| idfusubc 46907 | The identity functor for a... |
| inclfusubc 46908 | The "inclusion functor" fr... |
| lmod0rng 46909 | If the scalar ring of a mo... |
| nzrneg1ne0 46910 | The additive inverse of th... |
| nrhmzr 46911 | There is no ring homomorph... |
| lidldomn1 46912 | If a (left) ideal (which i... |
| lidlabl 46913 | A (left) ideal of a ring i... |
| lidlrng 46914 | A (left) ideal of a ring i... |
| zlidlring 46915 | The zero (left) ideal of a... |
| uzlidlring 46916 | Only the zero (left) ideal... |
| lidldomnnring 46917 | A (left) ideal of a domain... |
| 0even 46918 | 0 is an even integer. (Co... |
| 1neven 46919 | 1 is not an even integer. ... |
| 2even 46920 | 2 is an even integer. (Co... |
| 2zlidl 46921 | The even integers are a (l... |
| 2zrng 46922 | The ring of integers restr... |
| 2zrngbas 46923 | The base set of R is the s... |
| 2zrngadd 46924 | The group addition operati... |
| 2zrng0 46925 | The additive identity of R... |
| 2zrngamgm 46926 | R is an (additive) magma. ... |
| 2zrngasgrp 46927 | R is an (additive) semigro... |
| 2zrngamnd 46928 | R is an (additive) monoid.... |
| 2zrngacmnd 46929 | R is a commutative (additi... |
| 2zrngagrp 46930 | R is an (additive) group. ... |
| 2zrngaabl 46931 | R is an (additive) abelian... |
| 2zrngmul 46932 | The ring multiplication op... |
| 2zrngmmgm 46933 | R is a (multiplicative) ma... |
| 2zrngmsgrp 46934 | R is a (multiplicative) se... |
| 2zrngALT 46935 | The ring of integers restr... |
| 2zrngnmlid 46936 | R has no multiplicative (l... |
| 2zrngnmrid 46937 | R has no multiplicative (r... |
| 2zrngnmlid2 46938 | R has no multiplicative (l... |
| 2zrngnring 46939 | R is not a unital ring. (... |
| cznrnglem 46940 | Lemma for ~ cznrng : The ... |
| cznabel 46941 | The ring constructed from ... |
| cznrng 46942 | The ring constructed from ... |
| cznnring 46943 | The ring constructed from ... |
| rngcvalALTV 46948 | Value of the category of n... |
| rngcval 46949 | Value of the category of n... |
| rnghmresfn 46950 | The class of non-unital ri... |
| rnghmresel 46951 | An element of the non-unit... |
| rngcbas 46952 | Set of objects of the cate... |
| rngchomfval 46953 | Set of arrows of the categ... |
| rngchom 46954 | Set of arrows of the categ... |
| elrngchom 46955 | A morphism of non-unital r... |
| rngchomfeqhom 46956 | The functionalized Hom-set... |
| rngccofval 46957 | Composition in the categor... |
| rngcco 46958 | Composition in the categor... |
| dfrngc2 46959 | Alternate definition of th... |
| rnghmsscmap2 46960 | The non-unital ring homomo... |
| rnghmsscmap 46961 | The non-unital ring homomo... |
| rnghmsubcsetclem1 46962 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 46963 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 46964 | The non-unital ring homomo... |
| rngccat 46965 | The category of non-unital... |
| rngcid 46966 | The identity arrow in the ... |
| rngcsect 46967 | A section in the category ... |
| rngcinv 46968 | An inverse in the category... |
| rngciso 46969 | An isomorphism in the cate... |
| rngcbasALTV 46970 | Set of objects of the cate... |
| rngchomfvalALTV 46971 | Set of arrows of the categ... |
| rngchomALTV 46972 | Set of arrows of the categ... |
| elrngchomALTV 46973 | A morphism of non-unital r... |
| rngccofvalALTV 46974 | Composition in the categor... |
| rngccoALTV 46975 | Composition in the categor... |
| rngccatidALTV 46976 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 46977 | The category of non-unital... |
| rngcidALTV 46978 | The identity arrow in the ... |
| rngcsectALTV 46979 | A section in the category ... |
| rngcinvALTV 46980 | An inverse in the category... |
| rngcisoALTV 46981 | An isomorphism in the cate... |
| rngchomffvalALTV 46982 | The value of the functiona... |
| rngchomrnghmresALTV 46983 | The value of the functiona... |
| rngcifuestrc 46984 | The "inclusion functor" fr... |
| funcrngcsetc 46985 | The "natural forgetful fun... |
| funcrngcsetcALT 46986 | Alternate proof of ~ funcr... |
| zrinitorngc 46987 | The zero ring is an initia... |
| zrtermorngc 46988 | The zero ring is a termina... |
| zrzeroorngc 46989 | The zero ring is a zero ob... |
| ringcvalALTV 46994 | Value of the category of r... |
| ringcval 46995 | Value of the category of u... |
| rhmresfn 46996 | The class of unital ring h... |
| rhmresel 46997 | An element of the unital r... |
| ringcbas 46998 | Set of objects of the cate... |
| ringchomfval 46999 | Set of arrows of the categ... |
| ringchom 47000 | Set of arrows of the categ... |
| elringchom 47001 | A morphism of unital rings... |
| ringchomfeqhom 47002 | The functionalized Hom-set... |
| ringccofval 47003 | Composition in the categor... |
| ringcco 47004 | Composition in the categor... |
| dfringc2 47005 | Alternate definition of th... |
| rhmsscmap2 47006 | The unital ring homomorphi... |
| rhmsscmap 47007 | The unital ring homomorphi... |
| rhmsubcsetclem1 47008 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 47009 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 47010 | The unital ring homomorphi... |
| ringccat 47011 | The category of unital rin... |
| ringcid 47012 | The identity arrow in the ... |
| rhmsscrnghm 47013 | The unital ring homomorphi... |
| rhmsubcrngclem1 47014 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 47015 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 47016 | The unital ring homomorphi... |
| rngcresringcat 47017 | The restriction of the cat... |
| ringcsect 47018 | A section in the category ... |
| ringcinv 47019 | An inverse in the category... |
| ringciso 47020 | An isomorphism in the cate... |
| ringcbasbas 47021 | An element of the base set... |
| funcringcsetc 47022 | The "natural forgetful fun... |
| funcringcsetcALTV2lem1 47023 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 47024 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 47025 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 47026 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 47027 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 47028 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 47029 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 47030 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 47031 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 47032 | The "natural forgetful fun... |
| ringcbasALTV 47033 | Set of objects of the cate... |
| ringchomfvalALTV 47034 | Set of arrows of the categ... |
| ringchomALTV 47035 | Set of arrows of the categ... |
| elringchomALTV 47036 | A morphism of rings is a f... |
| ringccofvalALTV 47037 | Composition in the categor... |
| ringccoALTV 47038 | Composition in the categor... |
| ringccatidALTV 47039 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 47040 | The category of rings is a... |
| ringcidALTV 47041 | The identity arrow in the ... |
| ringcsectALTV 47042 | A section in the category ... |
| ringcinvALTV 47043 | An inverse in the category... |
| ringcisoALTV 47044 | An isomorphism in the cate... |
| ringcbasbasALTV 47045 | An element of the base set... |
| funcringcsetclem1ALTV 47046 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 47047 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 47048 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 47049 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 47050 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 47051 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 47052 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 47053 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 47054 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 47055 | The "natural forgetful fun... |
| irinitoringc 47056 | The ring of integers is an... |
| zrtermoringc 47057 | The zero ring is a termina... |
| zrninitoringc 47058 | The zero ring is not an in... |
| nzerooringczr 47059 | There is no zero object in... |
| srhmsubclem1 47060 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 47061 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 47062 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 47063 | According to ~ df-subc , t... |
| sringcat 47064 | The restriction of the cat... |
| crhmsubc 47065 | According to ~ df-subc , t... |
| cringcat 47066 | The restriction of the cat... |
| drhmsubc 47067 | According to ~ df-subc , t... |
| drngcat 47068 | The restriction of the cat... |
| fldcat 47069 | The restriction of the cat... |
| fldc 47070 | The restriction of the cat... |
| fldhmsubc 47071 | According to ~ df-subc , t... |
| rngcrescrhm 47072 | The category of non-unital... |
| rhmsubclem1 47073 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 47074 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 47075 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 47076 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 47077 | According to ~ df-subc , t... |
| rhmsubccat 47078 | The restriction of the cat... |
| srhmsubcALTVlem1 47079 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 47080 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 47081 | According to ~ df-subc , t... |
| sringcatALTV 47082 | The restriction of the cat... |
| crhmsubcALTV 47083 | According to ~ df-subc , t... |
| cringcatALTV 47084 | The restriction of the cat... |
| drhmsubcALTV 47085 | According to ~ df-subc , t... |
| drngcatALTV 47086 | The restriction of the cat... |
| fldcatALTV 47087 | The restriction of the cat... |
| fldcALTV 47088 | The restriction of the cat... |
| fldhmsubcALTV 47089 | According to ~ df-subc , t... |
| rngcrescrhmALTV 47090 | The category of non-unital... |
| rhmsubcALTVlem1 47091 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 47092 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 47093 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 47094 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 47095 | According to ~ df-subc , t... |
| rhmsubcALTVcat 47096 | The restriction of the cat... |
| opeliun2xp 47097 | Membership of an ordered p... |
| eliunxp2 47098 | Membership in a union of C... |
| mpomptx2 47099 | Express a two-argument fun... |
| cbvmpox2 47100 | Rule to change the bound v... |
| dmmpossx2 47101 | The domain of a mapping is... |
| mpoexxg2 47102 | Existence of an operation ... |
| ovmpordxf 47103 | Value of an operation give... |
| ovmpordx 47104 | Value of an operation give... |
| ovmpox2 47105 | The value of an operation ... |
| fdmdifeqresdif 47106 | The restriction of a condi... |
| offvalfv 47107 | The function operation exp... |
| ofaddmndmap 47108 | The function operation app... |
| mapsnop 47109 | A singleton of an ordered ... |
| fprmappr 47110 | A function with a domain o... |
| mapprop 47111 | An unordered pair containi... |
| ztprmneprm 47112 | A prime is not an integer ... |
| 2t6m3t4e0 47113 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 47114 | For any finite subset of `... |
| nn0sumltlt 47115 | If the sum of two nonnegat... |
| bcpascm1 47116 | Pascal's rule for the bino... |
| altgsumbc 47117 | The sum of binomial coeffi... |
| altgsumbcALT 47118 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 47119 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 47120 | An element of the (base se... |
| zlmodzxz0 47121 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 47122 | The scalar multiplication ... |
| zlmodzxzadd 47123 | The addition of the ` ZZ `... |
| zlmodzxzsubm 47124 | The subtraction of the ` Z... |
| zlmodzxzsub 47125 | The subtraction of the ` Z... |
| mgpsumunsn 47126 | Extract a summand/factor f... |
| mgpsumz 47127 | If the group sum for the m... |
| mgpsumn 47128 | If the group sum for the m... |
| exple2lt6 47129 | A nonnegative integer to t... |
| pgrple2abl 47130 | Every symmetric group on a... |
| pgrpgt2nabl 47131 | Every symmetric group on a... |
| invginvrid 47132 | Identity for a multiplicat... |
| rmsupp0 47133 | The support of a mapping o... |
| domnmsuppn0 47134 | The support of a mapping o... |
| rmsuppss 47135 | The support of a mapping o... |
| mndpsuppss 47136 | The support of a mapping o... |
| scmsuppss 47137 | The support of a mapping o... |
| rmsuppfi 47138 | The support of a mapping o... |
| rmfsupp 47139 | A mapping of a multiplicat... |
| mndpsuppfi 47140 | The support of a mapping o... |
| mndpfsupp 47141 | A mapping of a scalar mult... |
| scmsuppfi 47142 | The support of a mapping o... |
| scmfsupp 47143 | A mapping of a scalar mult... |
| suppmptcfin 47144 | The support of a mapping w... |
| mptcfsupp 47145 | A mapping with value 0 exc... |
| fsuppmptdmf 47146 | A mapping with a finite do... |
| lmodvsmdi 47147 | Multiple distributive law ... |
| gsumlsscl 47148 | Closure of a group sum in ... |
| assaascl0 47149 | The scalar 0 embedded into... |
| assaascl1 47150 | The scalar 1 embedded into... |
| ply1vr1smo 47151 | The variable in a polynomi... |
| ply1sclrmsm 47152 | The ring multiplication of... |
| coe1id 47153 | Coefficient vector of the ... |
| coe1sclmulval 47154 | The value of the coefficie... |
| ply1mulgsumlem1 47155 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 47156 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 47157 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 47158 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 47159 | The product of two polynom... |
| evl1at0 47160 | Polynomial evaluation for ... |
| evl1at1 47161 | Polynomial evaluation for ... |
| linply1 47162 | A term of the form ` x - C... |
| lineval 47163 | A term of the form ` x - C... |
| linevalexample 47164 | The polynomial ` x - 3 ` o... |
| dmatALTval 47169 | The algebra of ` N ` x ` N... |
| dmatALTbas 47170 | The base set of the algebr... |
| dmatALTbasel 47171 | An element of the base set... |
| dmatbas 47172 | The set of all ` N ` x ` N... |
| lincop 47177 | A linear combination as op... |
| lincval 47178 | The value of a linear comb... |
| dflinc2 47179 | Alternative definition of ... |
| lcoop 47180 | A linear combination as op... |
| lcoval 47181 | The value of a linear comb... |
| lincfsuppcl 47182 | A linear combination of ve... |
| linccl 47183 | A linear combination of ve... |
| lincval0 47184 | The value of an empty line... |
| lincvalsng 47185 | The linear combination ove... |
| lincvalsn 47186 | The linear combination ove... |
| lincvalpr 47187 | The linear combination ove... |
| lincval1 47188 | The linear combination ove... |
| lcosn0 47189 | Properties of a linear com... |
| lincvalsc0 47190 | The linear combination whe... |
| lcoc0 47191 | Properties of a linear com... |
| linc0scn0 47192 | If a set contains the zero... |
| lincdifsn 47193 | A vector is a linear combi... |
| linc1 47194 | A vector is a linear combi... |
| lincellss 47195 | A linear combination of a ... |
| lco0 47196 | The set of empty linear co... |
| lcoel0 47197 | The zero vector is always ... |
| lincsum 47198 | The sum of two linear comb... |
| lincscm 47199 | A linear combinations mult... |
| lincsumcl 47200 | The sum of two linear comb... |
| lincscmcl 47201 | The multiplication of a li... |
| lincsumscmcl 47202 | The sum of a linear combin... |
| lincolss 47203 | According to the statement... |
| ellcoellss 47204 | Every linear combination o... |
| lcoss 47205 | A set of vectors of a modu... |
| lspsslco 47206 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 47207 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 47208 | Equivalence of a _span_ of... |
| rellininds 47212 | The class defining the rel... |
| linindsv 47214 | The classes of the module ... |
| islininds 47215 | The property of being a li... |
| linindsi 47216 | The implications of being ... |
| linindslinci 47217 | The implications of being ... |
| islinindfis 47218 | The property of being a li... |
| islinindfiss 47219 | The property of being a li... |
| linindscl 47220 | A linearly independent set... |
| lindepsnlininds 47221 | A linearly dependent subse... |
| islindeps 47222 | The property of being a li... |
| lincext1 47223 | Property 1 of an extension... |
| lincext2 47224 | Property 2 of an extension... |
| lincext3 47225 | Property 3 of an extension... |
| lindslinindsimp1 47226 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 47227 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 47228 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 47229 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 47230 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 47231 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 47232 | Implication 2 for ~ lindsl... |
| lindslininds 47233 | Equivalence of definitions... |
| linds0 47234 | The empty set is always a ... |
| el0ldep 47235 | A set containing the zero ... |
| el0ldepsnzr 47236 | A set containing the zero ... |
| lindsrng01 47237 | Any subset of a module is ... |
| lindszr 47238 | Any subset of a module ove... |
| snlindsntorlem 47239 | Lemma for ~ snlindsntor . ... |
| snlindsntor 47240 | A singleton is linearly in... |
| ldepsprlem 47241 | Lemma for ~ ldepspr . (Co... |
| ldepspr 47242 | If a vector is a scalar mu... |
| lincresunit3lem3 47243 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 47244 | Lemma 1 for properties of ... |
| lincresunitlem2 47245 | Lemma for properties of a ... |
| lincresunit1 47246 | Property 1 of a specially ... |
| lincresunit2 47247 | Property 2 of a specially ... |
| lincresunit3lem1 47248 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 47249 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 47250 | Property 3 of a specially ... |
| lincreslvec3 47251 | Property 3 of a specially ... |
| islindeps2 47252 | Conditions for being a lin... |
| islininds2 47253 | Implication of being a lin... |
| isldepslvec2 47254 | Alternative definition of ... |
| lindssnlvec 47255 | A singleton not containing... |
| lmod1lem1 47256 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 47257 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 47258 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 47259 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 47260 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 47261 | The (smallest) structure r... |
| lmod1zr 47262 | The (smallest) structure r... |
| lmod1zrnlvec 47263 | There is a (left) module (... |
| lmodn0 47264 | Left modules exist. (Cont... |
| zlmodzxzequa 47265 | Example of an equation wit... |
| zlmodzxznm 47266 | Example of a linearly depe... |
| zlmodzxzldeplem 47267 | A and B are not equal. (C... |
| zlmodzxzequap 47268 | Example of an equation wit... |
| zlmodzxzldeplem1 47269 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 47270 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 47271 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 47272 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 47273 | { A , B } is a linearly de... |
| ldepsnlinclem1 47274 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 47275 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 47276 | The class of all (left) ve... |
| ldepsnlinc 47277 | The reverse implication of... |
| ldepslinc 47278 | For (left) vector spaces, ... |
| suppdm 47279 | If the range of a function... |
| eluz2cnn0n1 47280 | An integer greater than 1 ... |
| divge1b 47281 | The ratio of a real number... |
| divgt1b 47282 | The ratio of a real number... |
| ltsubaddb 47283 | Equivalence for the "less ... |
| ltsubsubb 47284 | Equivalence for the "less ... |
| ltsubadd2b 47285 | Equivalence for the "less ... |
| divsub1dir 47286 | Distribution of division o... |
| expnegico01 47287 | An integer greater than 1 ... |
| elfzolborelfzop1 47288 | An element of a half-open ... |
| pw2m1lepw2m1 47289 | 2 to the power of a positi... |
| zgtp1leeq 47290 | If an integer is between a... |
| flsubz 47291 | An integer can be moved in... |
| fldivmod 47292 | Expressing the floor of a ... |
| mod0mul 47293 | If an integer is 0 modulo ... |
| modn0mul 47294 | If an integer is not 0 mod... |
| m1modmmod 47295 | An integer decreased by 1 ... |
| difmodm1lt 47296 | The difference between an ... |
| nn0onn0ex 47297 | For each odd nonnegative i... |
| nn0enn0ex 47298 | For each even nonnegative ... |
| nnennex 47299 | For each even positive int... |
| nneop 47300 | A positive integer is even... |
| nneom 47301 | A positive integer is even... |
| nn0eo 47302 | A nonnegative integer is e... |
| nnpw2even 47303 | 2 to the power of a positi... |
| zefldiv2 47304 | The floor of an even integ... |
| zofldiv2 47305 | The floor of an odd intege... |
| nn0ofldiv2 47306 | The floor of an odd nonneg... |
| flnn0div2ge 47307 | The floor of a positive in... |
| flnn0ohalf 47308 | The floor of the half of a... |
| logcxp0 47309 | Logarithm of a complex pow... |
| regt1loggt0 47310 | The natural logarithm for ... |
| fdivval 47313 | The quotient of two functi... |
| fdivmpt 47314 | The quotient of two functi... |
| fdivmptf 47315 | The quotient of two functi... |
| refdivmptf 47316 | The quotient of two functi... |
| fdivpm 47317 | The quotient of two functi... |
| refdivpm 47318 | The quotient of two functi... |
| fdivmptfv 47319 | The function value of a qu... |
| refdivmptfv 47320 | The function value of a qu... |
| bigoval 47323 | Set of functions of order ... |
| elbigofrcl 47324 | Reverse closure of the "bi... |
| elbigo 47325 | Properties of a function o... |
| elbigo2 47326 | Properties of a function o... |
| elbigo2r 47327 | Sufficient condition for a... |
| elbigof 47328 | A function of order G(x) i... |
| elbigodm 47329 | The domain of a function o... |
| elbigoimp 47330 | The defining property of a... |
| elbigolo1 47331 | A function (into the posit... |
| rege1logbrege0 47332 | The general logarithm, wit... |
| rege1logbzge0 47333 | The general logarithm, wit... |
| fllogbd 47334 | A real number is between t... |
| relogbmulbexp 47335 | The logarithm of the produ... |
| relogbdivb 47336 | The logarithm of the quoti... |
| logbge0b 47337 | The logarithm of a number ... |
| logblt1b 47338 | The logarithm of a number ... |
| fldivexpfllog2 47339 | The floor of a positive re... |
| nnlog2ge0lt1 47340 | A positive integer is 1 if... |
| logbpw2m1 47341 | The floor of the binary lo... |
| fllog2 47342 | The floor of the binary lo... |
| blenval 47345 | The binary length of an in... |
| blen0 47346 | The binary length of 0. (... |
| blenn0 47347 | The binary length of a "nu... |
| blenre 47348 | The binary length of a pos... |
| blennn 47349 | The binary length of a pos... |
| blennnelnn 47350 | The binary length of a pos... |
| blennn0elnn 47351 | The binary length of a non... |
| blenpw2 47352 | The binary length of a pow... |
| blenpw2m1 47353 | The binary length of a pow... |
| nnpw2blen 47354 | A positive integer is betw... |
| nnpw2blenfzo 47355 | A positive integer is betw... |
| nnpw2blenfzo2 47356 | A positive integer is eith... |
| nnpw2pmod 47357 | Every positive integer can... |
| blen1 47358 | The binary length of 1. (... |
| blen2 47359 | The binary length of 2. (... |
| nnpw2p 47360 | Every positive integer can... |
| nnpw2pb 47361 | A number is a positive int... |
| blen1b 47362 | The binary length of a non... |
| blennnt2 47363 | The binary length of a pos... |
| nnolog2flm1 47364 | The floor of the binary lo... |
| blennn0em1 47365 | The binary length of the h... |
| blennngt2o2 47366 | The binary length of an od... |
| blengt1fldiv2p1 47367 | The binary length of an in... |
| blennn0e2 47368 | The binary length of an ev... |
| digfval 47371 | Operation to obtain the ` ... |
| digval 47372 | The ` K ` th digit of a no... |
| digvalnn0 47373 | The ` K ` th digit of a no... |
| nn0digval 47374 | The ` K ` th digit of a no... |
| dignn0fr 47375 | The digits of the fraction... |
| dignn0ldlem 47376 | Lemma for ~ dignnld . (Co... |
| dignnld 47377 | The leading digits of a po... |
| dig2nn0ld 47378 | The leading digits of a po... |
| dig2nn1st 47379 | The first (relevant) digit... |
| dig0 47380 | All digits of 0 are 0. (C... |
| digexp 47381 | The ` K ` th digit of a po... |
| dig1 47382 | All but one digits of 1 ar... |
| 0dig1 47383 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 47384 | The integers 0 and 1 corre... |
| dig2nn0 47385 | A digit of a nonnegative i... |
| 0dig2nn0e 47386 | The last bit of an even in... |
| 0dig2nn0o 47387 | The last bit of an odd int... |
| dig2bits 47388 | The ` K ` th digit of a no... |
| dignn0flhalflem1 47389 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 47390 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 47391 | The digits of the half of ... |
| dignn0flhalf 47392 | The digits of the rounded ... |
| nn0sumshdiglemA 47393 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 47394 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 47395 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 47396 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 47397 | A nonnegative integer can ... |
| nn0mulfsum 47398 | Trivial algorithm to calcu... |
| nn0mullong 47399 | Standard algorithm (also k... |
| naryfval 47402 | The set of the n-ary (endo... |
| naryfvalixp 47403 | The set of the n-ary (endo... |
| naryfvalel 47404 | An n-ary (endo)function on... |
| naryrcl 47405 | Reverse closure for n-ary ... |
| naryfvalelfv 47406 | The value of an n-ary (end... |
| naryfvalelwrdf 47407 | An n-ary (endo)function on... |
| 0aryfvalel 47408 | A nullary (endo)function o... |
| 0aryfvalelfv 47409 | The value of a nullary (en... |
| 1aryfvalel 47410 | A unary (endo)function on ... |
| fv1arycl 47411 | Closure of a unary (endo)f... |
| 1arympt1 47412 | A unary (endo)function in ... |
| 1arympt1fv 47413 | The value of a unary (endo... |
| 1arymaptfv 47414 | The value of the mapping o... |
| 1arymaptf 47415 | The mapping of unary (endo... |
| 1arymaptf1 47416 | The mapping of unary (endo... |
| 1arymaptfo 47417 | The mapping of unary (endo... |
| 1arymaptf1o 47418 | The mapping of unary (endo... |
| 1aryenef 47419 | The set of unary (endo)fun... |
| 1aryenefmnd 47420 | The set of unary (endo)fun... |
| 2aryfvalel 47421 | A binary (endo)function on... |
| fv2arycl 47422 | Closure of a binary (endo)... |
| 2arympt 47423 | A binary (endo)function in... |
| 2arymptfv 47424 | The value of a binary (end... |
| 2arymaptfv 47425 | The value of the mapping o... |
| 2arymaptf 47426 | The mapping of binary (end... |
| 2arymaptf1 47427 | The mapping of binary (end... |
| 2arymaptfo 47428 | The mapping of binary (end... |
| 2arymaptf1o 47429 | The mapping of binary (end... |
| 2aryenef 47430 | The set of binary (endo)fu... |
| itcoval 47435 | The value of the function ... |
| itcoval0 47436 | A function iterated zero t... |
| itcoval1 47437 | A function iterated once. ... |
| itcoval2 47438 | A function iterated twice.... |
| itcoval3 47439 | A function iterated three ... |
| itcoval0mpt 47440 | A mapping iterated zero ti... |
| itcovalsuc 47441 | The value of the function ... |
| itcovalsucov 47442 | The value of the function ... |
| itcovalendof 47443 | The n-th iterate of an end... |
| itcovalpclem1 47444 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 47445 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 47446 | The value of the function ... |
| itcovalt2lem2lem1 47447 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 47448 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 47449 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 47450 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 47451 | The value of the function ... |
| ackvalsuc1mpt 47452 | The Ackermann function at ... |
| ackvalsuc1 47453 | The Ackermann function at ... |
| ackval0 47454 | The Ackermann function at ... |
| ackval1 47455 | The Ackermann function at ... |
| ackval2 47456 | The Ackermann function at ... |
| ackval3 47457 | The Ackermann function at ... |
| ackendofnn0 47458 | The Ackermann function at ... |
| ackfnnn0 47459 | The Ackermann function at ... |
| ackval0val 47460 | The Ackermann function at ... |
| ackvalsuc0val 47461 | The Ackermann function at ... |
| ackvalsucsucval 47462 | The Ackermann function at ... |
| ackval0012 47463 | The Ackermann function at ... |
| ackval1012 47464 | The Ackermann function at ... |
| ackval2012 47465 | The Ackermann function at ... |
| ackval3012 47466 | The Ackermann function at ... |
| ackval40 47467 | The Ackermann function at ... |
| ackval41a 47468 | The Ackermann function at ... |
| ackval41 47469 | The Ackermann function at ... |
| ackval42 47470 | The Ackermann function at ... |
| ackval42a 47471 | The Ackermann function at ... |
| ackval50 47472 | The Ackermann function at ... |
| fv1prop 47473 | The function value of unor... |
| fv2prop 47474 | The function value of unor... |
| submuladdmuld 47475 | Transformation of a sum of... |
| affinecomb1 47476 | Combination of two real af... |
| affinecomb2 47477 | Combination of two real af... |
| affineid 47478 | Identity of an affine comb... |
| 1subrec1sub 47479 | Subtract the reciprocal of... |
| resum2sqcl 47480 | The sum of two squares of ... |
| resum2sqgt0 47481 | The sum of the square of a... |
| resum2sqrp 47482 | The sum of the square of a... |
| resum2sqorgt0 47483 | The sum of the square of t... |
| reorelicc 47484 | Membership in and outside ... |
| rrx2pxel 47485 | The x-coordinate of a poin... |
| rrx2pyel 47486 | The y-coordinate of a poin... |
| prelrrx2 47487 | An unordered pair of order... |
| prelrrx2b 47488 | An unordered pair of order... |
| rrx2pnecoorneor 47489 | If two different points ` ... |
| rrx2pnedifcoorneor 47490 | If two different points ` ... |
| rrx2pnedifcoorneorr 47491 | If two different points ` ... |
| rrx2xpref1o 47492 | There is a bijection betwe... |
| rrx2xpreen 47493 | The set of points in the t... |
| rrx2plord 47494 | The lexicographical orderi... |
| rrx2plord1 47495 | The lexicographical orderi... |
| rrx2plord2 47496 | The lexicographical orderi... |
| rrx2plordisom 47497 | The set of points in the t... |
| rrx2plordso 47498 | The lexicographical orderi... |
| ehl2eudisval0 47499 | The Euclidean distance of ... |
| ehl2eudis0lt 47500 | An upper bound of the Eucl... |
| lines 47505 | The lines passing through ... |
| line 47506 | The line passing through t... |
| rrxlines 47507 | Definition of lines passin... |
| rrxline 47508 | The line passing through t... |
| rrxlinesc 47509 | Definition of lines passin... |
| rrxlinec 47510 | The line passing through t... |
| eenglngeehlnmlem1 47511 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 47512 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 47513 | The line definition in the... |
| rrx2line 47514 | The line passing through t... |
| rrx2vlinest 47515 | The vertical line passing ... |
| rrx2linest 47516 | The line passing through t... |
| rrx2linesl 47517 | The line passing through t... |
| rrx2linest2 47518 | The line passing through t... |
| elrrx2linest2 47519 | The line passing through t... |
| spheres 47520 | The spheres for given cent... |
| sphere 47521 | A sphere with center ` X `... |
| rrxsphere 47522 | The sphere with center ` M... |
| 2sphere 47523 | The sphere with center ` M... |
| 2sphere0 47524 | The sphere around the orig... |
| line2ylem 47525 | Lemma for ~ line2y . This... |
| line2 47526 | Example for a line ` G ` p... |
| line2xlem 47527 | Lemma for ~ line2x . This... |
| line2x 47528 | Example for a horizontal l... |
| line2y 47529 | Example for a vertical lin... |
| itsclc0lem1 47530 | Lemma for theorems about i... |
| itsclc0lem2 47531 | Lemma for theorems about i... |
| itsclc0lem3 47532 | Lemma for theorems about i... |
| itscnhlc0yqe 47533 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 47534 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 47535 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 47536 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 47537 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 47538 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 47539 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 47540 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 47541 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 47542 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 47543 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 47544 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 47545 | The intersection points of... |
| itsclc0b 47546 | The intersection points of... |
| itsclinecirc0 47547 | The intersection points of... |
| itsclinecirc0b 47548 | The intersection points of... |
| itsclinecirc0in 47549 | The intersection points of... |
| itsclquadb 47550 | Quadratic equation for the... |
| itsclquadeu 47551 | Quadratic equation for the... |
| 2itscplem1 47552 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 47553 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 47554 | Lemma D for ~ 2itscp . (C... |
| 2itscp 47555 | A condition for a quadrati... |
| itscnhlinecirc02plem1 47556 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 47557 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 47558 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 47559 | Intersection of a nonhoriz... |
| inlinecirc02plem 47560 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 47561 | Intersection of a line wit... |
| inlinecirc02preu 47562 | Intersection of a line wit... |
| pm4.71da 47563 | Deduction converting a bic... |
| logic1 47564 | Distribution of implicatio... |
| logic1a 47565 | Variant of ~ logic1 . (Co... |
| logic2 47566 | Variant of ~ logic1 . (Co... |
| pm5.32dav 47567 | Distribution of implicatio... |
| pm5.32dra 47568 | Reverse distribution of im... |
| exp12bd 47569 | The import-export theorem ... |
| mpbiran3d 47570 | Equivalence with a conjunc... |
| mpbiran4d 47571 | Equivalence with a conjunc... |
| dtrucor3 47572 | An example of how ~ ax-5 w... |
| ralbidb 47573 | Formula-building rule for ... |
| ralbidc 47574 | Formula-building rule for ... |
| r19.41dv 47575 | A complex deduction form o... |
| rspceb2dv 47576 | Restricted existential spe... |
| rmotru 47577 | Two ways of expressing "at... |
| reutru 47578 | Two ways of expressing "ex... |
| reutruALT 47579 | Alternate proof for ~ reut... |
| ssdisjd 47580 | Subset preserves disjointn... |
| ssdisjdr 47581 | Subset preserves disjointn... |
| disjdifb 47582 | Relative complement is ant... |
| predisj 47583 | Preimages of disjoint sets... |
| vsn 47584 | The singleton of the unive... |
| mosn 47585 | "At most one" element in a... |
| mo0 47586 | "At most one" element in a... |
| mosssn 47587 | "At most one" element in a... |
| mo0sn 47588 | Two ways of expressing "at... |
| mosssn2 47589 | Two ways of expressing "at... |
| unilbss 47590 | Superclass of the greatest... |
| inpw 47591 | Two ways of expressing a c... |
| mof0 47592 | There is at most one funct... |
| mof02 47593 | A variant of ~ mof0 . (Co... |
| mof0ALT 47594 | Alternate proof for ~ mof0... |
| eufsnlem 47595 | There is exactly one funct... |
| eufsn 47596 | There is exactly one funct... |
| eufsn2 47597 | There is exactly one funct... |
| mofsn 47598 | There is at most one funct... |
| mofsn2 47599 | There is at most one funct... |
| mofsssn 47600 | There is at most one funct... |
| mofmo 47601 | There is at most one funct... |
| mofeu 47602 | The uniqueness of a functi... |
| elfvne0 47603 | If a function value has a ... |
| fdomne0 47604 | A function with non-empty ... |
| f1sn2g 47605 | A function that maps a sin... |
| f102g 47606 | A function that maps the e... |
| f1mo 47607 | A function that maps a set... |
| f002 47608 | A function with an empty c... |
| map0cor 47609 | A function exists iff an e... |
| fvconstr 47610 | Two ways of expressing ` A... |
| fvconstrn0 47611 | Two ways of expressing ` A... |
| fvconstr2 47612 | Two ways of expressing ` A... |
| fvconst0ci 47613 | A constant function's valu... |
| fvconstdomi 47614 | A constant function's valu... |
| f1omo 47615 | There is at most one eleme... |
| f1omoALT 47616 | There is at most one eleme... |
| iccin 47617 | Intersection of two closed... |
| iccdisj2 47618 | If the upper bound of one ... |
| iccdisj 47619 | If the upper bound of one ... |
| mreuniss 47620 | The union of a collection ... |
| clduni 47621 | The union of closed sets i... |
| opncldeqv 47622 | Conditions on open sets ar... |
| opndisj 47623 | Two ways of saying that tw... |
| clddisj 47624 | Two ways of saying that tw... |
| neircl 47625 | Reverse closure of the nei... |
| opnneilem 47626 | Lemma factoring out common... |
| opnneir 47627 | If something is true for a... |
| opnneirv 47628 | A variant of ~ opnneir wit... |
| opnneilv 47629 | The converse of ~ opnneir ... |
| opnneil 47630 | A variant of ~ opnneilv . ... |
| opnneieqv 47631 | The equivalence between ne... |
| opnneieqvv 47632 | The equivalence between ne... |
| restcls2lem 47633 | A closed set in a subspace... |
| restcls2 47634 | A closed set in a subspace... |
| restclsseplem 47635 | Lemma for ~ restclssep . ... |
| restclssep 47636 | Two disjoint closed sets i... |
| cnneiima 47637 | Given a continuous functio... |
| iooii 47638 | Open intervals are open se... |
| icccldii 47639 | Closed intervals are close... |
| i0oii 47640 | ` ( 0 [,) A ) ` is open in... |
| io1ii 47641 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 47642 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 47643 | Open neighborhood and neig... |
| sepnsepo 47644 | Open neighborhood and neig... |
| sepdisj 47645 | Separated sets are disjoin... |
| seposep 47646 | If two sets are separated ... |
| sepcsepo 47647 | If two sets are separated ... |
| sepfsepc 47648 | If two sets are separated ... |
| seppsepf 47649 | If two sets are precisely ... |
| seppcld 47650 | If two sets are precisely ... |
| isnrm4 47651 | A topological space is nor... |
| dfnrm2 47652 | A topological space is nor... |
| dfnrm3 47653 | A topological space is nor... |
| iscnrm3lem1 47654 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 47655 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem3 47656 | Lemma for ~ iscnrm3lem4 . ... |
| iscnrm3lem4 47657 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 47658 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 47659 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 47660 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 47661 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 47662 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 47663 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 47664 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 47665 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 47666 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 47667 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 47668 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 47669 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 47670 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 47671 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 47672 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 47673 | A completely normal topolo... |
| iscnrm3v 47674 | A topology is completely n... |
| iscnrm4 47675 | A completely normal topolo... |
| isprsd 47676 | Property of being a preord... |
| lubeldm2 47677 | Member of the domain of th... |
| glbeldm2 47678 | Member of the domain of th... |
| lubeldm2d 47679 | Member of the domain of th... |
| glbeldm2d 47680 | Member of the domain of th... |
| lubsscl 47681 | If a subset of ` S ` conta... |
| glbsscl 47682 | If a subset of ` S ` conta... |
| lubprlem 47683 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 47684 | The set of two comparable ... |
| lubpr 47685 | The LUB of the set of two ... |
| glbprlem 47686 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 47687 | The set of two comparable ... |
| glbpr 47688 | The GLB of the set of two ... |
| joindm2 47689 | The join of any two elemen... |
| joindm3 47690 | The join of any two elemen... |
| meetdm2 47691 | The meet of any two elemen... |
| meetdm3 47692 | The meet of any two elemen... |
| posjidm 47693 | Poset join is idempotent. ... |
| posmidm 47694 | Poset meet is idempotent. ... |
| toslat 47695 | A toset is a lattice. (Co... |
| isclatd 47696 | The predicate "is a comple... |
| intubeu 47697 | Existential uniqueness of ... |
| unilbeu 47698 | Existential uniqueness of ... |
| ipolublem 47699 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 47700 | The domain of the LUB of t... |
| ipolub 47701 | The LUB of the inclusion p... |
| ipoglblem 47702 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 47703 | The domain of the GLB of t... |
| ipoglb 47704 | The GLB of the inclusion p... |
| ipolub0 47705 | The LUB of the empty set i... |
| ipolub00 47706 | The LUB of the empty set i... |
| ipoglb0 47707 | The GLB of the empty set i... |
| mrelatlubALT 47708 | Least upper bounds in a Mo... |
| mrelatglbALT 47709 | Greatest lower bounds in a... |
| mreclat 47710 | A Moore space is a complet... |
| topclat 47711 | A topology is a complete l... |
| toplatglb0 47712 | The empty intersection in ... |
| toplatlub 47713 | Least upper bounds in a to... |
| toplatglb 47714 | Greatest lower bounds in a... |
| toplatjoin 47715 | Joins in a topology are re... |
| toplatmeet 47716 | Meets in a topology are re... |
| topdlat 47717 | A topology is a distributi... |
| catprslem 47718 | Lemma for ~ catprs . (Con... |
| catprs 47719 | A preorder can be extracte... |
| catprs2 47720 | A category equipped with t... |
| catprsc 47721 | A construction of the preo... |
| catprsc2 47722 | An alternate construction ... |
| endmndlem 47723 | A diagonal hom-set in a ca... |
| idmon 47724 | An identity arrow, or an i... |
| idepi 47725 | An identity arrow, or an i... |
| funcf2lem 47726 | A utility theorem for prov... |
| isthinc 47729 | The predicate "is a thin c... |
| isthinc2 47730 | A thin category is a categ... |
| isthinc3 47731 | A thin category is a categ... |
| thincc 47732 | A thin category is a categ... |
| thinccd 47733 | A thin category is a categ... |
| thincssc 47734 | A thin category is a categ... |
| isthincd2lem1 47735 | Lemma for ~ isthincd2 and ... |
| thincmo2 47736 | Morphisms in the same hom-... |
| thincmo 47737 | There is at most one morph... |
| thincmoALT 47738 | Alternate proof for ~ thin... |
| thincmod 47739 | At most one morphism in ea... |
| thincn0eu 47740 | In a thin category, a hom-... |
| thincid 47741 | In a thin category, a morp... |
| thincmon 47742 | In a thin category, all mo... |
| thincepi 47743 | In a thin category, all mo... |
| isthincd2lem2 47744 | Lemma for ~ isthincd2 . (... |
| isthincd 47745 | The predicate "is a thin c... |
| isthincd2 47746 | The predicate " ` C ` is a... |
| oppcthin 47747 | The opposite category of a... |
| subthinc 47748 | A subcategory of a thin ca... |
| functhinclem1 47749 | Lemma for ~ functhinc . G... |
| functhinclem2 47750 | Lemma for ~ functhinc . (... |
| functhinclem3 47751 | Lemma for ~ functhinc . T... |
| functhinclem4 47752 | Lemma for ~ functhinc . O... |
| functhinc 47753 | A functor to a thin catego... |
| fullthinc 47754 | A functor to a thin catego... |
| fullthinc2 47755 | A full functor to a thin c... |
| thincfth 47756 | A functor from a thin cate... |
| thincciso 47757 | Two thin categories are is... |
| 0thincg 47758 | Any structure with an empt... |
| 0thinc 47759 | The empty category (see ~ ... |
| indthinc 47760 | An indiscrete category in ... |
| indthincALT 47761 | An alternate proof for ~ i... |
| prsthinc 47762 | Preordered sets as categor... |
| setcthin 47763 | A category of sets all of ... |
| setc2othin 47764 | The category ` ( SetCat ``... |
| thincsect 47765 | In a thin category, one mo... |
| thincsect2 47766 | In a thin category, ` F ` ... |
| thincinv 47767 | In a thin category, ` F ` ... |
| thinciso 47768 | In a thin category, ` F : ... |
| thinccic 47769 | In a thin category, two ob... |
| prstcval 47772 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 47773 | Lemma for ~ prstcnid and ~... |
| prstcnid 47774 | Components other than ` Ho... |
| prstcbas 47775 | The base set is unchanged.... |
| prstcleval 47776 | Value of the less-than-or-... |
| prstclevalOLD 47777 | Obsolete proof of ~ prstcl... |
| prstcle 47778 | Value of the less-than-or-... |
| prstcocval 47779 | Orthocomplementation is un... |
| prstcocvalOLD 47780 | Obsolete proof of ~ prstco... |
| prstcoc 47781 | Orthocomplementation is un... |
| prstchomval 47782 | Hom-sets of the constructe... |
| prstcprs 47783 | The category is a preorder... |
| prstcthin 47784 | The preordered set is equi... |
| prstchom 47785 | Hom-sets of the constructe... |
| prstchom2 47786 | Hom-sets of the constructe... |
| prstchom2ALT 47787 | Hom-sets of the constructe... |
| postcpos 47788 | The converted category is ... |
| postcposALT 47789 | Alternate proof for ~ post... |
| postc 47790 | The converted category is ... |
| mndtcval 47793 | Value of the category buil... |
| mndtcbasval 47794 | The base set of the catego... |
| mndtcbas 47795 | The category built from a ... |
| mndtcob 47796 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 47797 | Two objects in a category ... |
| mndtchom 47798 | The only hom-set of the ca... |
| mndtcco 47799 | The composition of the cat... |
| mndtcco2 47800 | The composition of the cat... |
| mndtccatid 47801 | Lemma for ~ mndtccat and ~... |
| mndtccat 47802 | The function value is a ca... |
| mndtcid 47803 | The identity morphism, or ... |
| grptcmon 47804 | All morphisms in a categor... |
| grptcepi 47805 | All morphisms in a categor... |
| nfintd 47806 | Bound-variable hypothesis ... |
| nfiund 47807 | Bound-variable hypothesis ... |
| nfiundg 47808 | Bound-variable hypothesis ... |
| iunord 47809 | The indexed union of a col... |
| iunordi 47810 | The indexed union of a col... |
| spd 47811 | Specialization deduction, ... |
| spcdvw 47812 | A version of ~ spcdv where... |
| tfis2d 47813 | Transfinite Induction Sche... |
| bnd2d 47814 | Deduction form of ~ bnd2 .... |
| dffun3f 47815 | Alternate definition of fu... |
| setrecseq 47818 | Equality theorem for set r... |
| nfsetrecs 47819 | Bound-variable hypothesis ... |
| setrec1lem1 47820 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 47821 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 47822 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 47823 | Lemma for ~ setrec1 . If ... |
| setrec1 47824 | This is the first of two f... |
| setrec2fun 47825 | This is the second of two ... |
| setrec2lem1 47826 | Lemma for ~ setrec2 . The... |
| setrec2lem2 47827 | Lemma for ~ setrec2 . The... |
| setrec2 47828 | This is the second of two ... |
| setrec2v 47829 | Version of ~ setrec2 with ... |
| setrec2mpt 47830 | Version of ~ setrec2 where... |
| setis 47831 | Version of ~ setrec2 expre... |
| elsetrecslem 47832 | Lemma for ~ elsetrecs . A... |
| elsetrecs 47833 | A set ` A ` is an element ... |
| setrecsss 47834 | The ` setrecs ` operator r... |
| setrecsres 47835 | A recursively generated cl... |
| vsetrec 47836 | Construct ` _V ` using set... |
| 0setrec 47837 | If a function sends the em... |
| onsetreclem1 47838 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 47839 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 47840 | Lemma for ~ onsetrec . (C... |
| onsetrec 47841 | Construct ` On ` using set... |
| elpglem1 47844 | Lemma for ~ elpg . (Contr... |
| elpglem2 47845 | Lemma for ~ elpg . (Contr... |
| elpglem3 47846 | Lemma for ~ elpg . (Contr... |
| elpg 47847 | Membership in the class of... |
| pgindlem 47848 | Lemma for ~ pgind . (Cont... |
| pgindnf 47849 | Version of ~ pgind with ex... |
| pgind 47850 | Induction on partizan game... |
| sbidd 47851 | An identity theorem for su... |
| sbidd-misc 47852 | An identity theorem for su... |
| gte-lte 47857 | Simple relationship betwee... |
| gt-lt 47858 | Simple relationship betwee... |
| gte-lteh 47859 | Relationship between ` <_ ... |
| gt-lth 47860 | Relationship between ` < `... |
| ex-gt 47861 | Simple example of ` > ` , ... |
| ex-gte 47862 | Simple example of ` >_ ` ,... |
| sinhval-named 47869 | Value of the named sinh fu... |
| coshval-named 47870 | Value of the named cosh fu... |
| tanhval-named 47871 | Value of the named tanh fu... |
| sinh-conventional 47872 | Conventional definition of... |
| sinhpcosh 47873 | Prove that ` ( sinh `` A )... |
| secval 47880 | Value of the secant functi... |
| cscval 47881 | Value of the cosecant func... |
| cotval 47882 | Value of the cotangent fun... |
| seccl 47883 | The closure of the secant ... |
| csccl 47884 | The closure of the cosecan... |
| cotcl 47885 | The closure of the cotange... |
| reseccl 47886 | The closure of the secant ... |
| recsccl 47887 | The closure of the cosecan... |
| recotcl 47888 | The closure of the cotange... |
| recsec 47889 | The reciprocal of secant i... |
| reccsc 47890 | The reciprocal of cosecant... |
| reccot 47891 | The reciprocal of cotangen... |
| rectan 47892 | The reciprocal of tangent ... |
| sec0 47893 | The value of the secant fu... |
| onetansqsecsq 47894 | Prove the tangent squared ... |
| cotsqcscsq 47895 | Prove the tangent squared ... |
| ifnmfalse 47896 | If A is not a member of B,... |
| logb2aval 47897 | Define the value of the ` ... |
| comraddi 47904 | Commute RHS addition. See... |
| mvlraddi 47905 | Move the right term in a s... |
| mvrladdi 47906 | Move the left term in a su... |
| assraddsubi 47907 | Associate RHS addition-sub... |
| joinlmuladdmuli 47908 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 47909 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 47910 | Join AB-CB into (A-C) on L... |
| mvlrmuld 47911 | Move the right term in a p... |
| mvlrmuli 47912 | Move the right term in a p... |
| i2linesi 47913 | Solve for the intersection... |
| i2linesd 47914 | Solve for the intersection... |
| alimp-surprise 47915 | Demonstrate that when usin... |
| alimp-no-surprise 47916 | There is no "surprise" in ... |
| empty-surprise 47917 | Demonstrate that when usin... |
| empty-surprise2 47918 | "Prove" that false is true... |
| eximp-surprise 47919 | Show what implication insi... |
| eximp-surprise2 47920 | Show that "there exists" w... |
| alsconv 47925 | There is an equivalence be... |
| alsi1d 47926 | Deduction rule: Given "al... |
| alsi2d 47927 | Deduction rule: Given "al... |
| alsc1d 47928 | Deduction rule: Given "al... |
| alsc2d 47929 | Deduction rule: Given "al... |
| alscn0d 47930 | Deduction rule: Given "al... |
| alsi-no-surprise 47931 | Demonstrate that there is ... |
| 5m4e1 47932 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 47933 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 47934 | Resolution rule. This is ... |
| testable 47935 | In classical logic all wff... |
| aacllem 47936 | Lemma for other theorems a... |
| amgmwlem 47937 | Weighted version of ~ amgm... |
| amgmlemALT 47938 | Alternate proof of ~ amgml... |
| amgmw2d 47939 | Weighted arithmetic-geomet... |
| young2d 47940 | Young's inequality for ` n... |
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