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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... |
| syl5ibr 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... |
| syl6ib 250 | A mixed syllogism inferenc... |
| syl6ibr 251 | A mixed syllogism inferenc... |
| syl6bi 252 | A mixed syllogism inferenc... |
| syl6bir 253 | A mixed syllogism inferenc... |
| syl7bi 254 | A mixed syllogism inferenc... |
| syl8ib 255 | A syllogism rule of infere... |
| mpbird 256 | A deduction from a bicondi... |
| mpbiri 257 | An inference from a nested... |
| sylibrd 258 | A syllogism deduction. (C... |
| sylbird 259 | A syllogism deduction. (C... |
| biid 260 | Principle of identity for ... |
| biidd 261 | Principle of identity with... |
| pm5.1im 262 | Two propositions are equiv... |
| 2th 263 | Two truths are equivalent.... |
| 2thd 264 | Two truths are equivalent.... |
| monothetic 265 | Two self-implications (see... |
| ibi 266 | Inference that converts a ... |
| ibir 267 | Inference that converts a ... |
| ibd 268 | Deduction that converts a ... |
| pm5.74 269 | Distribution of implicatio... |
| pm5.74i 270 | Distribution of implicatio... |
| pm5.74ri 271 | Distribution of implicatio... |
| pm5.74d 272 | Distribution of implicatio... |
| pm5.74rd 273 | Distribution of implicatio... |
| bitri 274 | An inference from transiti... |
| bitr2i 275 | An inference from transiti... |
| bitr3i 276 | An inference from transiti... |
| bitr4i 277 | An inference from transiti... |
| bitrd 278 | Deduction form of ~ bitri ... |
| bitr2d 279 | Deduction form of ~ bitr2i... |
| bitr3d 280 | Deduction form of ~ bitr3i... |
| bitr4d 281 | Deduction form of ~ bitr4i... |
| bitrid 282 | A syllogism inference from... |
| bitr2id 283 | A syllogism inference from... |
| bitr3id 284 | A syllogism inference from... |
| bitr3di 285 | A syllogism inference from... |
| bitrdi 286 | A syllogism inference from... |
| bitr2di 287 | A syllogism inference from... |
| bitr4di 288 | A syllogism inference from... |
| bitr4id 289 | A syllogism inference from... |
| 3imtr3i 290 | A mixed syllogism inferenc... |
| 3imtr4i 291 | A mixed syllogism inferenc... |
| 3imtr3d 292 | More general version of ~ ... |
| 3imtr4d 293 | More general version of ~ ... |
| 3imtr3g 294 | More general version of ~ ... |
| 3imtr4g 295 | More general version of ~ ... |
| 3bitri 296 | A chained inference from t... |
| 3bitrri 297 | A chained inference from t... |
| 3bitr2i 298 | A chained inference from t... |
| 3bitr2ri 299 | A chained inference from t... |
| 3bitr3i 300 | A chained inference from t... |
| 3bitr3ri 301 | A chained inference from t... |
| 3bitr4i 302 | A chained inference from t... |
| 3bitr4ri 303 | A chained inference from t... |
| 3bitrd 304 | Deduction from transitivit... |
| 3bitrrd 305 | Deduction from transitivit... |
| 3bitr2d 306 | Deduction from transitivit... |
| 3bitr2rd 307 | Deduction from transitivit... |
| 3bitr3d 308 | Deduction from transitivit... |
| 3bitr3rd 309 | Deduction from transitivit... |
| 3bitr4d 310 | Deduction from transitivit... |
| 3bitr4rd 311 | Deduction from transitivit... |
| 3bitr3g 312 | More general version of ~ ... |
| 3bitr4g 313 | More general version of ~ ... |
| notnotb 314 | Double negation. Theorem ... |
| con34b 315 | A biconditional form of co... |
| con4bid 316 | A contraposition deduction... |
| notbid 317 | Deduction negating both si... |
| notbi 318 | Contraposition. Theorem *... |
| notbii 319 | Negate both sides of a log... |
| con4bii 320 | A contraposition inference... |
| mtbi 321 | An inference from a bicond... |
| mtbir 322 | An inference from a bicond... |
| mtbid 323 | A deduction from a bicondi... |
| mtbird 324 | A deduction from a bicondi... |
| mtbii 325 | An inference from a bicond... |
| mtbiri 326 | An inference from a bicond... |
| sylnib 327 | A mixed syllogism inferenc... |
| sylnibr 328 | A mixed syllogism inferenc... |
| sylnbi 329 | A mixed syllogism inferenc... |
| sylnbir 330 | A mixed syllogism inferenc... |
| xchnxbi 331 | Replacement of a subexpres... |
| xchnxbir 332 | Replacement of a subexpres... |
| xchbinx 333 | Replacement of a subexpres... |
| xchbinxr 334 | Replacement of a subexpres... |
| imbi2i 335 | Introduce an antecedent to... |
| jcndOLD 336 | Obsolete version of ~ jcnd... |
| 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... |
| 2falsedOLD 377 | Obsolete version of ~ 2fal... |
| pm5.21ni 378 | Two propositions implying ... |
| pm5.21nii 379 | Eliminate an antecedent im... |
| pm5.21ndd 380 | Eliminate an antecedent im... |
| bija 381 | Combine antecedents into a... |
| pm5.18 382 | Theorem *5.18 of [Whitehea... |
| xor3 383 | Two ways to express "exclu... |
| nbbn 384 | Move negation outside of b... |
| biass 385 | Associative law for the bi... |
| biluk 386 | Lukasiewicz's shortest axi... |
| pm5.19 387 | Theorem *5.19 of [Whitehea... |
| bi2.04 388 | Logical equivalence of com... |
| pm5.4 389 | Antecedent absorption impl... |
| imdi 390 | Distributive law for impli... |
| pm5.41 391 | Theorem *5.41 of [Whitehea... |
| imbibi 392 | The antecedent of one side... |
| pm4.8 393 | Theorem *4.8 of [Whitehead... |
| pm4.81 394 | A formula is equivalent to... |
| imim21b 395 | Simplify an implication be... |
| pm4.63 398 | Theorem *4.63 of [Whitehea... |
| pm4.67 399 | Theorem *4.67 of [Whitehea... |
| imnan 400 | Express an implication in ... |
| imnani 401 | Infer an implication from ... |
| iman 402 | Implication in terms of co... |
| pm3.24 403 | Law of noncontradiction. ... |
| annim 404 | Express a conjunction in t... |
| pm4.61 405 | Theorem *4.61 of [Whitehea... |
| pm4.65 406 | Theorem *4.65 of [Whitehea... |
| imp 407 | Importation inference. (C... |
| impcom 408 | Importation inference with... |
| con3dimp 409 | Variant of ~ con3d with im... |
| mpnanrd 410 | Eliminate the right side o... |
| impd 411 | Importation deduction. (C... |
| impcomd 412 | Importation deduction with... |
| ex 413 | Exportation inference. (T... |
| expcom 414 | Exportation inference with... |
| expdcom 415 | Commuted form of ~ expd . ... |
| expd 416 | Exportation deduction. (C... |
| expcomd 417 | Deduction form of ~ expcom... |
| imp31 418 | An importation inference. ... |
| imp32 419 | An importation inference. ... |
| exp31 420 | An exportation inference. ... |
| exp32 421 | An exportation inference. ... |
| imp4b 422 | An importation inference. ... |
| imp4a 423 | An importation inference. ... |
| imp4c 424 | An importation inference. ... |
| imp4d 425 | An importation inference. ... |
| imp41 426 | An importation inference. ... |
| imp42 427 | An importation inference. ... |
| imp43 428 | An importation inference. ... |
| imp44 429 | An importation inference. ... |
| imp45 430 | An importation inference. ... |
| exp4b 431 | An exportation inference. ... |
| exp4a 432 | An exportation inference. ... |
| exp4c 433 | An exportation inference. ... |
| exp4d 434 | An exportation inference. ... |
| exp41 435 | An exportation inference. ... |
| exp42 436 | An exportation inference. ... |
| exp43 437 | An exportation inference. ... |
| exp44 438 | An exportation inference. ... |
| exp45 439 | An exportation inference. ... |
| imp5d 440 | An importation inference. ... |
| imp5a 441 | An importation inference. ... |
| imp5g 442 | An importation inference. ... |
| imp55 443 | An importation inference. ... |
| imp511 444 | An importation inference. ... |
| exp5c 445 | An exportation inference. ... |
| exp5j 446 | An exportation inference. ... |
| exp5l 447 | An exportation inference. ... |
| exp53 448 | An exportation inference. ... |
| pm3.3 449 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 450 | Theorem *3.31 (Imp) of [Wh... |
| impexp 451 | Import-export theorem. Pa... |
| impancom 452 | Mixed importation/commutat... |
| expdimp 453 | A deduction version of exp... |
| expimpd 454 | Exportation followed by a ... |
| impr 455 | Import a wff into a right ... |
| impl 456 | Export a wff from a left c... |
| expr 457 | Export a wff from a right ... |
| expl 458 | Export a wff from a left c... |
| ancoms 459 | Inference commuting conjun... |
| pm3.22 460 | Theorem *3.22 of [Whitehea... |
| ancom 461 | Commutative law for conjun... |
| ancomd 462 | Commutation of conjuncts i... |
| biancomi 463 | Commuting conjunction in a... |
| biancomd 464 | Commuting conjunction in a... |
| ancomst 465 | Closed form of ~ ancoms . ... |
| ancomsd 466 | Deduction commuting conjun... |
| anasss 467 | Associative law for conjun... |
| anassrs 468 | Associative law for conjun... |
| anass 469 | Associative law for conjun... |
| pm3.2 470 | Join antecedents with conj... |
| pm3.2i 471 | Infer conjunction of premi... |
| pm3.21 472 | Join antecedents with conj... |
| pm3.43i 473 | Nested conjunction of ante... |
| pm3.43 474 | Theorem *3.43 (Comp) of [W... |
| dfbi2 475 | A theorem similar to the s... |
| dfbi 476 | Definition ~ df-bi rewritt... |
| biimpa 477 | Importation inference from... |
| biimpar 478 | Importation inference from... |
| biimpac 479 | Importation inference from... |
| biimparc 480 | Importation inference from... |
| adantr 481 | Inference adding a conjunc... |
| adantl 482 | Inference adding a conjunc... |
| simpl 483 | Elimination of a conjunct.... |
| simpli 484 | Inference eliminating a co... |
| simpr 485 | Elimination of a conjunct.... |
| simpri 486 | Inference eliminating a co... |
| intnan 487 | Introduction of conjunct i... |
| intnanr 488 | Introduction of conjunct i... |
| intnand 489 | Introduction of conjunct i... |
| intnanrd 490 | Introduction of conjunct i... |
| adantld 491 | Deduction adding a conjunc... |
| adantrd 492 | Deduction adding a conjunc... |
| pm3.41 493 | Theorem *3.41 of [Whitehea... |
| pm3.42 494 | Theorem *3.42 of [Whitehea... |
| simpld 495 | Deduction eliminating a co... |
| simprd 496 | Deduction eliminating a co... |
| simprbi 497 | Deduction eliminating a co... |
| simplbi 498 | Deduction eliminating a co... |
| simprbda 499 | Deduction eliminating a co... |
| simplbda 500 | Deduction eliminating a co... |
| simplbi2 501 | Deduction eliminating a co... |
| simplbi2comt 502 | Closed form of ~ simplbi2c... |
| simplbi2com 503 | A deduction eliminating a ... |
| simpl2im 504 | Implication from an elimin... |
| simplbiim 505 | Implication from an elimin... |
| impel 506 | An inference for implicati... |
| mpan9 507 | Modus ponens conjoining di... |
| sylan9 508 | Nested syllogism inference... |
| sylan9r 509 | Nested syllogism inference... |
| sylan9bb 510 | Nested syllogism inference... |
| sylan9bbr 511 | Nested syllogism inference... |
| jca 512 | Deduce conjunction of the ... |
| jcad 513 | Deduction conjoining the c... |
| jca2 514 | Inference conjoining the c... |
| jca31 515 | Join three consequents. (... |
| jca32 516 | Join three consequents. (... |
| jcai 517 | Deduction replacing implic... |
| jcab 518 | Distributive law for impli... |
| pm4.76 519 | Theorem *4.76 of [Whitehea... |
| jctil 520 | Inference conjoining a the... |
| jctir 521 | Inference conjoining a the... |
| jccir 522 | Inference conjoining a con... |
| jccil 523 | Inference conjoining a con... |
| jctl 524 | Inference conjoining a the... |
| jctr 525 | Inference conjoining a the... |
| jctild 526 | Deduction conjoining a the... |
| jctird 527 | Deduction conjoining a the... |
| iba 528 | Introduction of antecedent... |
| ibar 529 | Introduction of antecedent... |
| biantru 530 | A wff is equivalent to its... |
| biantrur 531 | A wff is equivalent to its... |
| biantrud 532 | A wff is equivalent to its... |
| biantrurd 533 | A wff is equivalent to its... |
| bianfi 534 | A wff conjoined with false... |
| bianfd 535 | A wff conjoined with false... |
| baib 536 | Move conjunction outside o... |
| baibr 537 | Move conjunction outside o... |
| rbaibr 538 | Move conjunction outside o... |
| rbaib 539 | Move conjunction outside o... |
| baibd 540 | Move conjunction outside o... |
| rbaibd 541 | Move conjunction outside o... |
| bianabs 542 | Absorb a hypothesis into t... |
| pm5.44 543 | Theorem *5.44 of [Whitehea... |
| pm5.42 544 | Theorem *5.42 of [Whitehea... |
| ancl 545 | Conjoin antecedent to left... |
| anclb 546 | Conjoin antecedent to left... |
| ancr 547 | Conjoin antecedent to righ... |
| ancrb 548 | Conjoin antecedent to righ... |
| ancli 549 | Deduction conjoining antec... |
| ancri 550 | Deduction conjoining antec... |
| ancld 551 | Deduction conjoining antec... |
| ancrd 552 | Deduction conjoining antec... |
| impac 553 | Importation with conjuncti... |
| anc2l 554 | Conjoin antecedent to left... |
| anc2r 555 | Conjoin antecedent to righ... |
| anc2li 556 | Deduction conjoining antec... |
| anc2ri 557 | Deduction conjoining antec... |
| pm4.71 558 | Implication in terms of bi... |
| pm4.71r 559 | Implication in terms of bi... |
| pm4.71i 560 | Inference converting an im... |
| pm4.71ri 561 | Inference converting an im... |
| pm4.71d 562 | Deduction converting an im... |
| pm4.71rd 563 | Deduction converting an im... |
| pm4.24 564 | Theorem *4.24 of [Whitehea... |
| anidm 565 | Idempotent law for conjunc... |
| anidmdbi 566 | Conjunction idempotence wi... |
| anidms 567 | Inference from idempotent ... |
| imdistan 568 | Distribution of implicatio... |
| imdistani 569 | Distribution of implicatio... |
| imdistanri 570 | Distribution of implicatio... |
| imdistand 571 | Distribution of implicatio... |
| imdistanda 572 | Distribution of implicatio... |
| pm5.3 573 | Theorem *5.3 of [Whitehead... |
| pm5.32 574 | Distribution of implicatio... |
| pm5.32i 575 | Distribution of implicatio... |
| pm5.32ri 576 | Distribution of implicatio... |
| pm5.32d 577 | Distribution of implicatio... |
| pm5.32rd 578 | Distribution of implicatio... |
| pm5.32da 579 | Distribution of implicatio... |
| sylan 580 | A syllogism inference. (C... |
| sylanb 581 | A syllogism inference. (C... |
| sylanbr 582 | A syllogism inference. (C... |
| sylanbrc 583 | Syllogism inference. (Con... |
| syl2anc 584 | Syllogism inference combin... |
| syl2anc2 585 | Double syllogism inference... |
| sylancl 586 | Syllogism inference combin... |
| sylancr 587 | Syllogism inference combin... |
| sylancom 588 | Syllogism inference with c... |
| sylanblc 589 | Syllogism inference combin... |
| sylanblrc 590 | Syllogism inference combin... |
| syldan 591 | A syllogism deduction with... |
| sylbida 592 | A syllogism deduction. (C... |
| sylan2 593 | A syllogism inference. (C... |
| sylan2b 594 | A syllogism inference. (C... |
| sylan2br 595 | A syllogism inference. (C... |
| syl2an 596 | A double syllogism inferen... |
| syl2anr 597 | A double syllogism inferen... |
| syl2anb 598 | A double syllogism inferen... |
| syl2anbr 599 | A double syllogism inferen... |
| sylancb 600 | A syllogism inference comb... |
| sylancbr 601 | A syllogism inference comb... |
| syldanl 602 | A syllogism deduction with... |
| syland 603 | A syllogism deduction. (C... |
| sylani 604 | A syllogism inference. (C... |
| sylan2d 605 | A syllogism deduction. (C... |
| sylan2i 606 | A syllogism inference. (C... |
| syl2ani 607 | A syllogism inference. (C... |
| syl2and 608 | A syllogism deduction. (C... |
| anim12d 609 | Conjoin antecedents and co... |
| anim12d1 610 | Variant of ~ anim12d where... |
| anim1d 611 | Add a conjunct to right of... |
| anim2d 612 | Add a conjunct to left of ... |
| anim12i 613 | Conjoin antecedents and co... |
| anim12ci 614 | Variant of ~ anim12i with ... |
| anim1i 615 | Introduce conjunct to both... |
| anim1ci 616 | Introduce conjunct to both... |
| anim2i 617 | Introduce conjunct to both... |
| anim12ii 618 | Conjoin antecedents and co... |
| anim12dan 619 | Conjoin antecedents and co... |
| im2anan9 620 | Deduction joining nested i... |
| im2anan9r 621 | Deduction joining nested i... |
| pm3.45 622 | Theorem *3.45 (Fact) of [W... |
| anbi2i 623 | Introduce a left conjunct ... |
| anbi1i 624 | Introduce a right conjunct... |
| anbi2ci 625 | Variant of ~ anbi2i with c... |
| anbi1ci 626 | Variant of ~ anbi1i with c... |
| anbi12i 627 | Conjoin both sides of two ... |
| anbi12ci 628 | Variant of ~ anbi12i with ... |
| anbi2d 629 | Deduction adding a left co... |
| anbi1d 630 | Deduction adding a right c... |
| anbi12d 631 | Deduction joining two equi... |
| anbi1 632 | Introduce a right conjunct... |
| anbi2 633 | Introduce a left conjunct ... |
| anbi1cd 634 | Introduce a proposition as... |
| an2anr 635 | Double commutation in conj... |
| pm4.38 636 | Theorem *4.38 of [Whitehea... |
| bi2anan9 637 | Deduction joining two equi... |
| bi2anan9r 638 | Deduction joining two equi... |
| bi2bian9 639 | Deduction joining two bico... |
| bianass 640 | An inference to merge two ... |
| bianassc 641 | An inference to merge two ... |
| an21 642 | Swap two conjuncts. (Cont... |
| an12 643 | Swap two conjuncts. Note ... |
| an32 644 | A rearrangement of conjunc... |
| an13 645 | A rearrangement of conjunc... |
| an31 646 | A rearrangement of conjunc... |
| an12s 647 | Swap two conjuncts in ante... |
| ancom2s 648 | Inference commuting a nest... |
| an13s 649 | Swap two conjuncts in ante... |
| an32s 650 | Swap two conjuncts in ante... |
| ancom1s 651 | Inference commuting a nest... |
| an31s 652 | Swap two conjuncts in ante... |
| anass1rs 653 | Commutative-associative la... |
| an4 654 | Rearrangement of 4 conjunc... |
| an42 655 | Rearrangement of 4 conjunc... |
| an43 656 | Rearrangement of 4 conjunc... |
| an3 657 | A rearrangement of conjunc... |
| an4s 658 | Inference rearranging 4 co... |
| an42s 659 | Inference rearranging 4 co... |
| anabs1 660 | Absorption into embedded c... |
| anabs5 661 | Absorption into embedded c... |
| anabs7 662 | Absorption into embedded c... |
| anabsan 663 | Absorption of antecedent w... |
| anabss1 664 | Absorption of antecedent i... |
| anabss4 665 | Absorption of antecedent i... |
| anabss5 666 | Absorption of antecedent i... |
| anabsi5 667 | Absorption of antecedent i... |
| anabsi6 668 | Absorption of antecedent i... |
| anabsi7 669 | Absorption of antecedent i... |
| anabsi8 670 | Absorption of antecedent i... |
| anabss7 671 | Absorption of antecedent i... |
| anabsan2 672 | Absorption of antecedent w... |
| anabss3 673 | Absorption of antecedent i... |
| anandi 674 | Distribution of conjunctio... |
| anandir 675 | Distribution of conjunctio... |
| anandis 676 | Inference that undistribut... |
| anandirs 677 | Inference that undistribut... |
| sylanl1 678 | A syllogism inference. (C... |
| sylanl2 679 | A syllogism inference. (C... |
| sylanr1 680 | A syllogism inference. (C... |
| sylanr2 681 | A syllogism inference. (C... |
| syl6an 682 | A syllogism deduction comb... |
| syl2an2r 683 | ~ syl2anr with antecedents... |
| syl2an2 684 | ~ syl2an with antecedents ... |
| mpdan 685 | An inference based on modu... |
| mpancom 686 | An inference based on modu... |
| mpidan 687 | A deduction which "stacks"... |
| mpan 688 | An inference based on modu... |
| mpan2 689 | An inference based on modu... |
| mp2an 690 | An inference based on modu... |
| mp4an 691 | An inference based on modu... |
| mpan2d 692 | A deduction based on modus... |
| mpand 693 | A deduction based on modus... |
| mpani 694 | An inference based on modu... |
| mpan2i 695 | An inference based on modu... |
| mp2ani 696 | An inference based on modu... |
| mp2and 697 | A deduction based on modus... |
| mpanl1 698 | An inference based on modu... |
| mpanl2 699 | An inference based on modu... |
| mpanl12 700 | An inference based on modu... |
| mpanr1 701 | An inference based on modu... |
| mpanr2 702 | An inference based on modu... |
| mpanr12 703 | An inference based on modu... |
| mpanlr1 704 | An inference based on modu... |
| mpbirand 705 | Detach truth from conjunct... |
| mpbiran2d 706 | Detach truth from conjunct... |
| mpbiran 707 | Detach truth from conjunct... |
| mpbiran2 708 | Detach truth from conjunct... |
| mpbir2an 709 | Detach a conjunction of tr... |
| mpbi2and 710 | Detach a conjunction of tr... |
| mpbir2and 711 | Detach a conjunction of tr... |
| adantll 712 | Deduction adding a conjunc... |
| adantlr 713 | Deduction adding a conjunc... |
| adantrl 714 | Deduction adding a conjunc... |
| adantrr 715 | Deduction adding a conjunc... |
| adantlll 716 | Deduction adding a conjunc... |
| adantllr 717 | Deduction adding a conjunc... |
| adantlrl 718 | Deduction adding a conjunc... |
| adantlrr 719 | Deduction adding a conjunc... |
| adantrll 720 | Deduction adding a conjunc... |
| adantrlr 721 | Deduction adding a conjunc... |
| adantrrl 722 | Deduction adding a conjunc... |
| adantrrr 723 | Deduction adding a conjunc... |
| ad2antrr 724 | Deduction adding two conju... |
| ad2antlr 725 | Deduction adding two conju... |
| ad2antrl 726 | Deduction adding two conju... |
| ad2antll 727 | Deduction adding conjuncts... |
| ad3antrrr 728 | Deduction adding three con... |
| ad3antlr 729 | Deduction adding three con... |
| ad4antr 730 | Deduction adding 4 conjunc... |
| ad4antlr 731 | Deduction adding 4 conjunc... |
| ad5antr 732 | Deduction adding 5 conjunc... |
| ad5antlr 733 | Deduction adding 5 conjunc... |
| ad6antr 734 | Deduction adding 6 conjunc... |
| ad6antlr 735 | Deduction adding 6 conjunc... |
| ad7antr 736 | Deduction adding 7 conjunc... |
| ad7antlr 737 | Deduction adding 7 conjunc... |
| ad8antr 738 | Deduction adding 8 conjunc... |
| ad8antlr 739 | Deduction adding 8 conjunc... |
| ad9antr 740 | Deduction adding 9 conjunc... |
| ad9antlr 741 | Deduction adding 9 conjunc... |
| ad10antr 742 | Deduction adding 10 conjun... |
| ad10antlr 743 | Deduction adding 10 conjun... |
| ad2ant2l 744 | Deduction adding two conju... |
| ad2ant2r 745 | Deduction adding two conju... |
| ad2ant2lr 746 | Deduction adding two conju... |
| ad2ant2rl 747 | Deduction adding two conju... |
| adantl3r 748 | Deduction adding 1 conjunc... |
| ad4ant13 749 | Deduction adding conjuncts... |
| ad4ant14 750 | Deduction adding conjuncts... |
| ad4ant23 751 | Deduction adding conjuncts... |
| ad4ant24 752 | Deduction adding conjuncts... |
| adantl4r 753 | Deduction adding 1 conjunc... |
| ad5ant12 754 | Deduction adding conjuncts... |
| ad5ant13 755 | Deduction adding conjuncts... |
| ad5ant14 756 | Deduction adding conjuncts... |
| ad5ant15 757 | Deduction adding conjuncts... |
| ad5ant23 758 | Deduction adding conjuncts... |
| ad5ant24 759 | Deduction adding conjuncts... |
| ad5ant25 760 | Deduction adding conjuncts... |
| adantl5r 761 | Deduction adding 1 conjunc... |
| adantl6r 762 | Deduction adding 1 conjunc... |
| pm3.33 763 | Theorem *3.33 (Syll) of [W... |
| pm3.34 764 | Theorem *3.34 (Syll) of [W... |
| simpll 765 | Simplification of a conjun... |
| simplld 766 | Deduction form of ~ simpll... |
| simplr 767 | Simplification of a conjun... |
| simplrd 768 | Deduction eliminating a do... |
| simprl 769 | Simplification of a conjun... |
| simprld 770 | Deduction eliminating a do... |
| simprr 771 | Simplification of a conjun... |
| simprrd 772 | Deduction form of ~ simprr... |
| simplll 773 | Simplification of a conjun... |
| simpllr 774 | Simplification of a conjun... |
| simplrl 775 | Simplification of a conjun... |
| simplrr 776 | Simplification of a conjun... |
| simprll 777 | Simplification of a conjun... |
| simprlr 778 | Simplification of a conjun... |
| simprrl 779 | Simplification of a conjun... |
| simprrr 780 | Simplification of a conjun... |
| simp-4l 781 | Simplification of a conjun... |
| simp-4r 782 | Simplification of a conjun... |
| simp-5l 783 | Simplification of a conjun... |
| simp-5r 784 | Simplification of a conjun... |
| simp-6l 785 | Simplification of a conjun... |
| simp-6r 786 | Simplification of a conjun... |
| simp-7l 787 | Simplification of a conjun... |
| simp-7r 788 | Simplification of a conjun... |
| simp-8l 789 | Simplification of a conjun... |
| simp-8r 790 | Simplification of a conjun... |
| simp-9l 791 | Simplification of a conjun... |
| simp-9r 792 | Simplification of a conjun... |
| simp-10l 793 | Simplification of a conjun... |
| simp-10r 794 | Simplification of a conjun... |
| simp-11l 795 | Simplification of a conjun... |
| simp-11r 796 | Simplification of a conjun... |
| pm2.01da 797 | Deduction based on reducti... |
| pm2.18da 798 | Deduction based on reducti... |
| impbida 799 | Deduce an equivalence from... |
| pm5.21nd 800 | Eliminate an antecedent im... |
| pm3.35 801 | Conjunctive detachment. T... |
| pm5.74da 802 | Distribution of implicatio... |
| bitr 803 | Theorem *4.22 of [Whitehea... |
| biantr 804 | A transitive law of equiva... |
| pm4.14 805 | Theorem *4.14 of [Whitehea... |
| pm3.37 806 | Theorem *3.37 (Transp) of ... |
| anim12 807 | Conjoin antecedents and co... |
| pm3.4 808 | Conjunction implies implic... |
| exbiri 809 | Inference form of ~ exbir ... |
| pm2.61ian 810 | Elimination of an antecede... |
| pm2.61dan 811 | Elimination of an antecede... |
| pm2.61ddan 812 | Elimination of two anteced... |
| pm2.61dda 813 | Elimination of two anteced... |
| mtand 814 | A modus tollens deduction.... |
| pm2.65da 815 | Deduction for proof by con... |
| condan 816 | Proof by contradiction. (... |
| biadan 817 | An implication is equivale... |
| biadani 818 | Inference associated with ... |
| biadaniALT 819 | Alternate proof of ~ biada... |
| biadanii 820 | Inference associated with ... |
| biadanid 821 | Deduction associated with ... |
| pm5.1 822 | Two propositions are equiv... |
| pm5.21 823 | Two propositions are equiv... |
| pm5.35 824 | Theorem *5.35 of [Whitehea... |
| abai 825 | Introduce one conjunct as ... |
| pm4.45im 826 | Conjunction with implicati... |
| impimprbi 827 | An implication and its rev... |
| nan 828 | Theorem to move a conjunct... |
| pm5.31 829 | Theorem *5.31 of [Whitehea... |
| pm5.31r 830 | Variant of ~ pm5.31 . (Co... |
| pm4.15 831 | Theorem *4.15 of [Whitehea... |
| pm5.36 832 | Theorem *5.36 of [Whitehea... |
| annotanannot 833 | A conjunction with a negat... |
| pm5.33 834 | Theorem *5.33 of [Whitehea... |
| syl12anc 835 | Syllogism combined with co... |
| syl21anc 836 | Syllogism combined with co... |
| syl22anc 837 | Syllogism combined with co... |
| syl1111anc 838 | Four-hypothesis eliminatio... |
| syldbl2 839 | Stacked hypotheseis implie... |
| mpsyl4anc 840 | An elimination deduction. ... |
| pm4.87 841 | Theorem *4.87 of [Whitehea... |
| bimsc1 842 | Removal of conjunct from o... |
| a2and 843 | Deduction distributing a c... |
| animpimp2impd 844 | Deduction deriving nested ... |
| pm4.64 847 | Theorem *4.64 of [Whitehea... |
| pm4.66 848 | Theorem *4.66 of [Whitehea... |
| pm2.53 849 | Theorem *2.53 of [Whitehea... |
| pm2.54 850 | Theorem *2.54 of [Whitehea... |
| imor 851 | Implication in terms of di... |
| imori 852 | Infer disjunction from imp... |
| imorri 853 | Infer implication from dis... |
| pm4.62 854 | Theorem *4.62 of [Whitehea... |
| jaoi 855 | Inference disjoining the a... |
| jao1i 856 | Add a disjunct in the ante... |
| jaod 857 | Deduction disjoining the a... |
| mpjaod 858 | Eliminate a disjunction in... |
| ori 859 | Infer implication from dis... |
| orri 860 | Infer disjunction from imp... |
| orrd 861 | Deduce disjunction from im... |
| ord 862 | Deduce implication from di... |
| orci 863 | Deduction introducing a di... |
| olci 864 | Deduction introducing a di... |
| orc 865 | Introduction of a disjunct... |
| olc 866 | Introduction of a disjunct... |
| pm1.4 867 | Axiom *1.4 of [WhiteheadRu... |
| orcom 868 | Commutative law for disjun... |
| orcomd 869 | Commutation of disjuncts i... |
| orcoms 870 | Commutation of disjuncts i... |
| orcd 871 | Deduction introducing a di... |
| olcd 872 | Deduction introducing a di... |
| orcs 873 | Deduction eliminating disj... |
| olcs 874 | Deduction eliminating disj... |
| olcnd 875 | A lemma for Conjunctive No... |
| unitreslOLD 876 | Obsolete version of ~ olcn... |
| orcnd 877 | A lemma for Conjunctive No... |
| mtord 878 | A modus tollens deduction ... |
| pm3.2ni 879 | Infer negated disjunction ... |
| pm2.45 880 | Theorem *2.45 of [Whitehea... |
| pm2.46 881 | Theorem *2.46 of [Whitehea... |
| pm2.47 882 | Theorem *2.47 of [Whitehea... |
| pm2.48 883 | Theorem *2.48 of [Whitehea... |
| pm2.49 884 | Theorem *2.49 of [Whitehea... |
| norbi 885 | If neither of two proposit... |
| nbior 886 | If two propositions are no... |
| orel1 887 | Elimination of disjunction... |
| pm2.25 888 | Theorem *2.25 of [Whitehea... |
| orel2 889 | Elimination of disjunction... |
| pm2.67-2 890 | Slight generalization of T... |
| pm2.67 891 | Theorem *2.67 of [Whitehea... |
| curryax 892 | A non-intuitionistic posit... |
| exmid 893 | Law of excluded middle, al... |
| exmidd 894 | Law of excluded middle in ... |
| pm2.1 895 | Theorem *2.1 of [Whitehead... |
| pm2.13 896 | Theorem *2.13 of [Whitehea... |
| pm2.621 897 | Theorem *2.621 of [Whitehe... |
| pm2.62 898 | Theorem *2.62 of [Whitehea... |
| pm2.68 899 | Theorem *2.68 of [Whitehea... |
| dfor2 900 | Logical 'or' expressed in ... |
| pm2.07 901 | Theorem *2.07 of [Whitehea... |
| pm1.2 902 | Axiom *1.2 of [WhiteheadRu... |
| oridm 903 | Idempotent law for disjunc... |
| pm4.25 904 | Theorem *4.25 of [Whitehea... |
| pm2.4 905 | Theorem *2.4 of [Whitehead... |
| pm2.41 906 | Theorem *2.41 of [Whitehea... |
| orim12i 907 | Disjoin antecedents and co... |
| orim1i 908 | Introduce disjunct to both... |
| orim2i 909 | Introduce disjunct to both... |
| orim12dALT 910 | Alternate proof of ~ orim1... |
| orbi2i 911 | Inference adding a left di... |
| orbi1i 912 | Inference adding a right d... |
| orbi12i 913 | Infer the disjunction of t... |
| orbi2d 914 | Deduction adding a left di... |
| orbi1d 915 | Deduction adding a right d... |
| orbi1 916 | Theorem *4.37 of [Whitehea... |
| orbi12d 917 | Deduction joining two equi... |
| pm1.5 918 | Axiom *1.5 (Assoc) of [Whi... |
| or12 919 | Swap two disjuncts. (Cont... |
| orass 920 | Associative law for disjun... |
| pm2.31 921 | Theorem *2.31 of [Whitehea... |
| pm2.32 922 | Theorem *2.32 of [Whitehea... |
| pm2.3 923 | Theorem *2.3 of [Whitehead... |
| or32 924 | A rearrangement of disjunc... |
| or4 925 | Rearrangement of 4 disjunc... |
| or42 926 | Rearrangement of 4 disjunc... |
| orordi 927 | Distribution of disjunctio... |
| orordir 928 | Distribution of disjunctio... |
| orimdi 929 | Disjunction distributes ov... |
| pm2.76 930 | Theorem *2.76 of [Whitehea... |
| pm2.85 931 | Theorem *2.85 of [Whitehea... |
| pm2.75 932 | Theorem *2.75 of [Whitehea... |
| pm4.78 933 | Implication distributes ov... |
| biort 934 | A disjunction with a true ... |
| biorf 935 | A wff is equivalent to its... |
| biortn 936 | A wff is equivalent to its... |
| biorfi 937 | A wff is equivalent to its... |
| pm2.26 938 | Theorem *2.26 of [Whitehea... |
| pm2.63 939 | Theorem *2.63 of [Whitehea... |
| pm2.64 940 | Theorem *2.64 of [Whitehea... |
| pm2.42 941 | Theorem *2.42 of [Whitehea... |
| pm5.11g 942 | A general instance of Theo... |
| pm5.11 943 | Theorem *5.11 of [Whitehea... |
| pm5.12 944 | Theorem *5.12 of [Whitehea... |
| pm5.14 945 | Theorem *5.14 of [Whitehea... |
| pm5.13 946 | Theorem *5.13 of [Whitehea... |
| pm5.55 947 | Theorem *5.55 of [Whitehea... |
| pm4.72 948 | Implication in terms of bi... |
| imimorb 949 | Simplify an implication be... |
| oibabs 950 | Absorption of disjunction ... |
| orbidi 951 | Disjunction distributes ov... |
| pm5.7 952 | Disjunction distributes ov... |
| jaao 953 | Inference conjoining and d... |
| jaoa 954 | Inference disjoining and c... |
| jaoian 955 | Inference disjoining the a... |
| jaodan 956 | Deduction disjoining the a... |
| mpjaodan 957 | Eliminate a disjunction in... |
| pm3.44 958 | Theorem *3.44 of [Whitehea... |
| jao 959 | Disjunction of antecedents... |
| jaob 960 | Disjunction of antecedents... |
| pm4.77 961 | Theorem *4.77 of [Whitehea... |
| pm3.48 962 | Theorem *3.48 of [Whitehea... |
| orim12d 963 | Disjoin antecedents and co... |
| orim1d 964 | Disjoin antecedents and co... |
| orim2d 965 | Disjoin antecedents and co... |
| orim2 966 | Axiom *1.6 (Sum) of [White... |
| pm2.38 967 | Theorem *2.38 of [Whitehea... |
| pm2.36 968 | Theorem *2.36 of [Whitehea... |
| pm2.37 969 | Theorem *2.37 of [Whitehea... |
| pm2.81 970 | Theorem *2.81 of [Whitehea... |
| pm2.8 971 | Theorem *2.8 of [Whitehead... |
| pm2.73 972 | Theorem *2.73 of [Whitehea... |
| pm2.74 973 | Theorem *2.74 of [Whitehea... |
| pm2.82 974 | Theorem *2.82 of [Whitehea... |
| pm4.39 975 | Theorem *4.39 of [Whitehea... |
| animorl 976 | Conjunction implies disjun... |
| animorr 977 | Conjunction implies disjun... |
| animorlr 978 | Conjunction implies disjun... |
| animorrl 979 | Conjunction implies disjun... |
| ianor 980 | Negated conjunction in ter... |
| anor 981 | Conjunction in terms of di... |
| ioran 982 | Negated disjunction in ter... |
| pm4.52 983 | Theorem *4.52 of [Whitehea... |
| pm4.53 984 | Theorem *4.53 of [Whitehea... |
| pm4.54 985 | Theorem *4.54 of [Whitehea... |
| pm4.55 986 | Theorem *4.55 of [Whitehea... |
| pm4.56 987 | Theorem *4.56 of [Whitehea... |
| oran 988 | Disjunction in terms of co... |
| pm4.57 989 | Theorem *4.57 of [Whitehea... |
| pm3.1 990 | Theorem *3.1 of [Whitehead... |
| pm3.11 991 | Theorem *3.11 of [Whitehea... |
| pm3.12 992 | Theorem *3.12 of [Whitehea... |
| pm3.13 993 | Theorem *3.13 of [Whitehea... |
| pm3.14 994 | Theorem *3.14 of [Whitehea... |
| pm4.44 995 | Theorem *4.44 of [Whitehea... |
| pm4.45 996 | Theorem *4.45 of [Whitehea... |
| orabs 997 | Absorption of redundant in... |
| oranabs 998 | Absorb a disjunct into a c... |
| pm5.61 999 | Theorem *5.61 of [Whitehea... |
| pm5.6 1000 | Conjunction in antecedent ... |
| orcanai 1001 | Change disjunction in cons... |
| pm4.79 1002 | Theorem *4.79 of [Whitehea... |
| pm5.53 1003 | Theorem *5.53 of [Whitehea... |
| ordi 1004 | Distributive law for disju... |
| ordir 1005 | Distributive law for disju... |
| andi 1006 | Distributive law for conju... |
| andir 1007 | Distributive law for conju... |
| orddi 1008 | Double distributive law fo... |
| anddi 1009 | Double distributive law fo... |
| pm5.17 1010 | Theorem *5.17 of [Whitehea... |
| pm5.15 1011 | Theorem *5.15 of [Whitehea... |
| pm5.16 1012 | Theorem *5.16 of [Whitehea... |
| xor 1013 | Two ways to express exclus... |
| nbi2 1014 | Two ways to express "exclu... |
| xordi 1015 | Conjunction distributes ov... |
| pm5.54 1016 | Theorem *5.54 of [Whitehea... |
| pm5.62 1017 | Theorem *5.62 of [Whitehea... |
| pm5.63 1018 | Theorem *5.63 of [Whitehea... |
| niabn 1019 | Miscellaneous inference re... |
| ninba 1020 | Miscellaneous inference re... |
| pm4.43 1021 | Theorem *4.43 of [Whitehea... |
| pm4.82 1022 | Theorem *4.82 of [Whitehea... |
| pm4.83 1023 | Theorem *4.83 of [Whitehea... |
| pclem6 1024 | Negation inferred from emb... |
| bigolden 1025 | Dijkstra-Scholten's Golden... |
| pm5.71 1026 | Theorem *5.71 of [Whitehea... |
| pm5.75 1027 | Theorem *5.75 of [Whitehea... |
| ecase2d 1028 | Deduction for elimination ... |
| ecase2dOLD 1029 | Obsolete version of ~ ecas... |
| ecase3 1030 | Inference for elimination ... |
| ecase 1031 | Inference for elimination ... |
| ecase3d 1032 | Deduction for elimination ... |
| ecased 1033 | Deduction for elimination ... |
| ecase3ad 1034 | Deduction for elimination ... |
| ecase3adOLD 1035 | Obsolete version of ~ ecas... |
| ccase 1036 | Inference for combining ca... |
| ccased 1037 | Deduction for combining ca... |
| ccase2 1038 | Inference for combining ca... |
| 4cases 1039 | Inference eliminating two ... |
| 4casesdan 1040 | Deduction eliminating two ... |
| cases 1041 | Case disjunction according... |
| dedlem0a 1042 | Lemma for an alternate ver... |
| dedlem0b 1043 | Lemma for an alternate ver... |
| dedlema 1044 | Lemma for weak deduction t... |
| dedlemb 1045 | Lemma for weak deduction t... |
| cases2 1046 | Case disjunction according... |
| cases2ALT 1047 | Alternate proof of ~ cases... |
| dfbi3 1048 | An alternate definition of... |
| pm5.24 1049 | Theorem *5.24 of [Whitehea... |
| 4exmid 1050 | The disjunction of the fou... |
| consensus 1051 | The consensus theorem. Th... |
| pm4.42 1052 | Theorem *4.42 of [Whitehea... |
| prlem1 1053 | A specialized lemma for se... |
| prlem2 1054 | A specialized lemma for se... |
| oplem1 1055 | A specialized lemma for se... |
| dn1 1056 | A single axiom for Boolean... |
| bianir 1057 | A closed form of ~ mpbir ,... |
| jaoi2 1058 | Inference removing a negat... |
| jaoi3 1059 | Inference separating a dis... |
| ornld 1060 | Selecting one statement fr... |
| dfifp2 1063 | Alternate definition of th... |
| dfifp3 1064 | Alternate definition of th... |
| dfifp4 1065 | Alternate definition of th... |
| dfifp5 1066 | Alternate definition of th... |
| dfifp6 1067 | Alternate definition of th... |
| dfifp7 1068 | Alternate definition of th... |
| ifpdfbi 1069 | Define the biconditional a... |
| anifp 1070 | The conditional operator i... |
| ifpor 1071 | The conditional operator i... |
| ifpn 1072 | Conditional operator for t... |
| ifpnOLD 1073 | Obsolete version of ~ ifpn... |
| ifptru 1074 | Value of the conditional o... |
| ifpfal 1075 | Value of the conditional o... |
| ifpid 1076 | Value of the conditional o... |
| casesifp 1077 | Version of ~ cases express... |
| ifpbi123d 1078 | Equivalence deduction for ... |
| ifpbi123dOLD 1079 | Obsolete version of ~ ifpb... |
| ifpbi23d 1080 | Equivalence deduction for ... |
| ifpimpda 1081 | Separation of the values o... |
| 1fpid3 1082 | The value of the condition... |
| elimh 1083 | Hypothesis builder for the... |
| dedt 1084 | The weak deduction theorem... |
| con3ALT 1085 | Proof of ~ con3 from its a... |
| 3orass 1090 | Associative law for triple... |
| 3orel1 1091 | Partial elimination of a t... |
| 3orrot 1092 | Rotation law for triple di... |
| 3orcoma 1093 | Commutation law for triple... |
| 3orcomb 1094 | Commutation law for triple... |
| 3anass 1095 | Associative law for triple... |
| 3anan12 1096 | Convert triple conjunction... |
| 3anan32 1097 | Convert triple conjunction... |
| 3ancoma 1098 | Commutation law for triple... |
| 3ancomb 1099 | Commutation law for triple... |
| 3anrot 1100 | Rotation law for triple co... |
| 3anrev 1101 | Reversal law for triple co... |
| anandi3 1102 | Distribution of triple con... |
| anandi3r 1103 | Distribution of triple con... |
| 3anidm 1104 | Idempotent law for conjunc... |
| 3an4anass 1105 | Associative law for four c... |
| 3ioran 1106 | Negated triple disjunction... |
| 3ianor 1107 | Negated triple conjunction... |
| 3anor 1108 | Triple conjunction express... |
| 3oran 1109 | Triple disjunction in term... |
| 3impa 1110 | Importation from double to... |
| 3imp 1111 | Importation inference. (C... |
| 3imp31 1112 | The importation inference ... |
| 3imp231 1113 | Importation inference. (C... |
| 3imp21 1114 | The importation inference ... |
| 3impb 1115 | Importation from double to... |
| 3impib 1116 | Importation to triple conj... |
| 3impia 1117 | Importation to triple conj... |
| 3expa 1118 | Exportation from triple to... |
| 3exp 1119 | Exportation inference. (C... |
| 3expb 1120 | Exportation from triple to... |
| 3expia 1121 | Exportation from triple co... |
| 3expib 1122 | Exportation from triple co... |
| 3com12 1123 | Commutation in antecedent.... |
| 3com13 1124 | Commutation in antecedent.... |
| 3comr 1125 | Commutation in antecedent.... |
| 3com23 1126 | Commutation in antecedent.... |
| 3coml 1127 | Commutation in antecedent.... |
| 3jca 1128 | Join consequents with conj... |
| 3jcad 1129 | Deduction conjoining the c... |
| 3adant1 1130 | Deduction adding a conjunc... |
| 3adant2 1131 | Deduction adding a conjunc... |
| 3adant3 1132 | Deduction adding a conjunc... |
| 3ad2ant1 1133 | Deduction adding conjuncts... |
| 3ad2ant2 1134 | Deduction adding conjuncts... |
| 3ad2ant3 1135 | Deduction adding conjuncts... |
| simp1 1136 | Simplification of triple c... |
| simp2 1137 | Simplification of triple c... |
| simp3 1138 | Simplification of triple c... |
| simp1i 1139 | Infer a conjunct from a tr... |
| simp2i 1140 | Infer a conjunct from a tr... |
| simp3i 1141 | Infer a conjunct from a tr... |
| simp1d 1142 | Deduce a conjunct from a t... |
| simp2d 1143 | Deduce a conjunct from a t... |
| simp3d 1144 | Deduce a conjunct from a t... |
| simp1bi 1145 | Deduce a conjunct from a t... |
| simp2bi 1146 | Deduce a conjunct from a t... |
| simp3bi 1147 | Deduce a conjunct from a t... |
| 3simpa 1148 | Simplification of triple c... |
| 3simpb 1149 | Simplification of triple c... |
| 3simpc 1150 | Simplification of triple c... |
| 3anim123i 1151 | Join antecedents and conse... |
| 3anim1i 1152 | Add two conjuncts to antec... |
| 3anim2i 1153 | Add two conjuncts to antec... |
| 3anim3i 1154 | Add two conjuncts to antec... |
| 3anbi123i 1155 | Join 3 biconditionals with... |
| 3orbi123i 1156 | Join 3 biconditionals with... |
| 3anbi1i 1157 | Inference adding two conju... |
| 3anbi2i 1158 | Inference adding two conju... |
| 3anbi3i 1159 | Inference adding two conju... |
| syl3an 1160 | A triple syllogism inferen... |
| syl3anb 1161 | A triple syllogism inferen... |
| syl3anbr 1162 | A triple syllogism inferen... |
| syl3an1 1163 | A syllogism inference. (C... |
| syl3an2 1164 | A syllogism inference. (C... |
| syl3an3 1165 | A syllogism inference. (C... |
| 3adantl1 1166 | Deduction adding a conjunc... |
| 3adantl2 1167 | Deduction adding a conjunc... |
| 3adantl3 1168 | Deduction adding a conjunc... |
| 3adantr1 1169 | Deduction adding a conjunc... |
| 3adantr2 1170 | Deduction adding a conjunc... |
| 3adantr3 1171 | Deduction adding a conjunc... |
| ad4ant123 1172 | Deduction adding conjuncts... |
| ad4ant124 1173 | Deduction adding conjuncts... |
| ad4ant134 1174 | Deduction adding conjuncts... |
| ad4ant234 1175 | Deduction adding conjuncts... |
| 3adant1l 1176 | Deduction adding a conjunc... |
| 3adant1r 1177 | Deduction adding a conjunc... |
| 3adant2l 1178 | Deduction adding a conjunc... |
| 3adant2r 1179 | Deduction adding a conjunc... |
| 3adant3l 1180 | Deduction adding a conjunc... |
| 3adant3r 1181 | Deduction adding a conjunc... |
| 3adant3r1 1182 | Deduction adding a conjunc... |
| 3adant3r2 1183 | Deduction adding a conjunc... |
| 3adant3r3 1184 | Deduction adding a conjunc... |
| 3ad2antl1 1185 | Deduction adding conjuncts... |
| 3ad2antl2 1186 | Deduction adding conjuncts... |
| 3ad2antl3 1187 | Deduction adding conjuncts... |
| 3ad2antr1 1188 | Deduction adding conjuncts... |
| 3ad2antr2 1189 | Deduction adding conjuncts... |
| 3ad2antr3 1190 | Deduction adding conjuncts... |
| simpl1 1191 | Simplification of conjunct... |
| simpl2 1192 | Simplification of conjunct... |
| simpl3 1193 | Simplification of conjunct... |
| simpr1 1194 | Simplification of conjunct... |
| simpr2 1195 | Simplification of conjunct... |
| simpr3 1196 | Simplification of conjunct... |
| simp1l 1197 | Simplification of triple c... |
| simp1r 1198 | Simplification of triple c... |
| simp2l 1199 | Simplification of triple c... |
| simp2r 1200 | Simplification of triple c... |
| simp3l 1201 | Simplification of triple c... |
| simp3r 1202 | Simplification of triple c... |
| simp11 1203 | Simplification of doubly t... |
| simp12 1204 | Simplification of doubly t... |
| simp13 1205 | Simplification of doubly t... |
| simp21 1206 | Simplification of doubly t... |
| simp22 1207 | Simplification of doubly t... |
| simp23 1208 | Simplification of doubly t... |
| simp31 1209 | Simplification of doubly t... |
| simp32 1210 | Simplification of doubly t... |
| simp33 1211 | Simplification of doubly t... |
| simpll1 1212 | Simplification of conjunct... |
| simpll2 1213 | Simplification of conjunct... |
| simpll3 1214 | Simplification of conjunct... |
| simplr1 1215 | Simplification of conjunct... |
| simplr2 1216 | Simplification of conjunct... |
| simplr3 1217 | Simplification of conjunct... |
| simprl1 1218 | Simplification of conjunct... |
| simprl2 1219 | Simplification of conjunct... |
| simprl3 1220 | Simplification of conjunct... |
| simprr1 1221 | Simplification of conjunct... |
| simprr2 1222 | Simplification of conjunct... |
| simprr3 1223 | Simplification of conjunct... |
| simpl1l 1224 | Simplification of conjunct... |
| simpl1r 1225 | Simplification of conjunct... |
| simpl2l 1226 | Simplification of conjunct... |
| simpl2r 1227 | Simplification of conjunct... |
| simpl3l 1228 | Simplification of conjunct... |
| simpl3r 1229 | Simplification of conjunct... |
| simpr1l 1230 | Simplification of conjunct... |
| simpr1r 1231 | Simplification of conjunct... |
| simpr2l 1232 | Simplification of conjunct... |
| simpr2r 1233 | Simplification of conjunct... |
| simpr3l 1234 | Simplification of conjunct... |
| simpr3r 1235 | Simplification of conjunct... |
| simp1ll 1236 | Simplification of conjunct... |
| simp1lr 1237 | Simplification of conjunct... |
| simp1rl 1238 | Simplification of conjunct... |
| simp1rr 1239 | Simplification of conjunct... |
| simp2ll 1240 | Simplification of conjunct... |
| simp2lr 1241 | Simplification of conjunct... |
| simp2rl 1242 | Simplification of conjunct... |
| simp2rr 1243 | Simplification of conjunct... |
| simp3ll 1244 | Simplification of conjunct... |
| simp3lr 1245 | Simplification of conjunct... |
| simp3rl 1246 | Simplification of conjunct... |
| simp3rr 1247 | Simplification of conjunct... |
| simpl11 1248 | Simplification of conjunct... |
| simpl12 1249 | Simplification of conjunct... |
| simpl13 1250 | Simplification of conjunct... |
| simpl21 1251 | Simplification of conjunct... |
| simpl22 1252 | Simplification of conjunct... |
| simpl23 1253 | Simplification of conjunct... |
| simpl31 1254 | Simplification of conjunct... |
| simpl32 1255 | Simplification of conjunct... |
| simpl33 1256 | Simplification of conjunct... |
| simpr11 1257 | Simplification of conjunct... |
| simpr12 1258 | Simplification of conjunct... |
| simpr13 1259 | Simplification of conjunct... |
| simpr21 1260 | Simplification of conjunct... |
| simpr22 1261 | Simplification of conjunct... |
| simpr23 1262 | Simplification of conjunct... |
| simpr31 1263 | Simplification of conjunct... |
| simpr32 1264 | Simplification of conjunct... |
| simpr33 1265 | Simplification of conjunct... |
| simp1l1 1266 | Simplification of conjunct... |
| simp1l2 1267 | Simplification of conjunct... |
| simp1l3 1268 | Simplification of conjunct... |
| simp1r1 1269 | Simplification of conjunct... |
| simp1r2 1270 | Simplification of conjunct... |
| simp1r3 1271 | Simplification of conjunct... |
| simp2l1 1272 | Simplification of conjunct... |
| simp2l2 1273 | Simplification of conjunct... |
| simp2l3 1274 | Simplification of conjunct... |
| simp2r1 1275 | Simplification of conjunct... |
| simp2r2 1276 | Simplification of conjunct... |
| simp2r3 1277 | Simplification of conjunct... |
| simp3l1 1278 | Simplification of conjunct... |
| simp3l2 1279 | Simplification of conjunct... |
| simp3l3 1280 | Simplification of conjunct... |
| simp3r1 1281 | Simplification of conjunct... |
| simp3r2 1282 | Simplification of conjunct... |
| simp3r3 1283 | Simplification of conjunct... |
| simp11l 1284 | Simplification of conjunct... |
| simp11r 1285 | Simplification of conjunct... |
| simp12l 1286 | Simplification of conjunct... |
| simp12r 1287 | Simplification of conjunct... |
| simp13l 1288 | Simplification of conjunct... |
| simp13r 1289 | Simplification of conjunct... |
| simp21l 1290 | Simplification of conjunct... |
| simp21r 1291 | Simplification of conjunct... |
| simp22l 1292 | Simplification of conjunct... |
| simp22r 1293 | Simplification of conjunct... |
| simp23l 1294 | Simplification of conjunct... |
| simp23r 1295 | Simplification of conjunct... |
| simp31l 1296 | Simplification of conjunct... |
| simp31r 1297 | Simplification of conjunct... |
| simp32l 1298 | Simplification of conjunct... |
| simp32r 1299 | Simplification of conjunct... |
| simp33l 1300 | Simplification of conjunct... |
| simp33r 1301 | Simplification of conjunct... |
| simp111 1302 | Simplification of conjunct... |
| simp112 1303 | Simplification of conjunct... |
| simp113 1304 | Simplification of conjunct... |
| simp121 1305 | Simplification of conjunct... |
| simp122 1306 | Simplification of conjunct... |
| simp123 1307 | Simplification of conjunct... |
| simp131 1308 | Simplification of conjunct... |
| simp132 1309 | Simplification of conjunct... |
| simp133 1310 | Simplification of conjunct... |
| simp211 1311 | Simplification of conjunct... |
| simp212 1312 | Simplification of conjunct... |
| simp213 1313 | Simplification of conjunct... |
| simp221 1314 | Simplification of conjunct... |
| simp222 1315 | Simplification of conjunct... |
| simp223 1316 | Simplification of conjunct... |
| simp231 1317 | Simplification of conjunct... |
| simp232 1318 | Simplification of conjunct... |
| simp233 1319 | Simplification of conjunct... |
| simp311 1320 | Simplification of conjunct... |
| simp312 1321 | Simplification of conjunct... |
| simp313 1322 | Simplification of conjunct... |
| simp321 1323 | Simplification of conjunct... |
| simp322 1324 | Simplification of conjunct... |
| simp323 1325 | Simplification of conjunct... |
| simp331 1326 | Simplification of conjunct... |
| simp332 1327 | Simplification of conjunct... |
| simp333 1328 | Simplification of conjunct... |
| 3anibar 1329 | Remove a hypothesis from t... |
| 3mix1 1330 | Introduction in triple dis... |
| 3mix2 1331 | Introduction in triple dis... |
| 3mix3 1332 | Introduction in triple dis... |
| 3mix1i 1333 | Introduction in triple dis... |
| 3mix2i 1334 | Introduction in triple dis... |
| 3mix3i 1335 | Introduction in triple dis... |
| 3mix1d 1336 | Deduction introducing trip... |
| 3mix2d 1337 | Deduction introducing trip... |
| 3mix3d 1338 | Deduction introducing trip... |
| 3pm3.2i 1339 | Infer conjunction of premi... |
| pm3.2an3 1340 | Version of ~ pm3.2 for a t... |
| mpbir3an 1341 | Detach a conjunction of tr... |
| mpbir3and 1342 | Detach a conjunction of tr... |
| syl3anbrc 1343 | Syllogism inference. (Con... |
| syl21anbrc 1344 | Syllogism inference. (Con... |
| 3imp3i2an 1345 | An elimination deduction. ... |
| ex3 1346 | Apply ~ ex to a hypothesis... |
| 3imp1 1347 | Importation to left triple... |
| 3impd 1348 | Importation deduction for ... |
| 3imp2 1349 | Importation to right tripl... |
| 3impdi 1350 | Importation inference (und... |
| 3impdir 1351 | Importation inference (und... |
| 3exp1 1352 | Exportation from left trip... |
| 3expd 1353 | Exportation deduction for ... |
| 3exp2 1354 | Exportation from right tri... |
| exp5o 1355 | A triple exportation infer... |
| exp516 1356 | A triple exportation infer... |
| exp520 1357 | A triple exportation infer... |
| 3impexp 1358 | Version of ~ impexp for a ... |
| 3an1rs 1359 | Swap conjuncts. (Contribu... |
| 3anassrs 1360 | Associative law for conjun... |
| ad5ant245 1361 | Deduction adding conjuncts... |
| ad5ant234 1362 | Deduction adding conjuncts... |
| ad5ant235 1363 | Deduction adding conjuncts... |
| ad5ant123 1364 | Deduction adding conjuncts... |
| ad5ant124 1365 | Deduction adding conjuncts... |
| ad5ant125 1366 | Deduction adding conjuncts... |
| ad5ant134 1367 | Deduction adding conjuncts... |
| ad5ant135 1368 | Deduction adding conjuncts... |
| ad5ant145 1369 | Deduction adding conjuncts... |
| ad5ant2345 1370 | Deduction adding conjuncts... |
| syl3anc 1371 | Syllogism combined with co... |
| syl13anc 1372 | Syllogism combined with co... |
| syl31anc 1373 | Syllogism combined with co... |
| syl112anc 1374 | Syllogism combined with co... |
| syl121anc 1375 | Syllogism combined with co... |
| syl211anc 1376 | Syllogism combined with co... |
| syl23anc 1377 | Syllogism combined with co... |
| syl32anc 1378 | Syllogism combined with co... |
| syl122anc 1379 | Syllogism combined with co... |
| syl212anc 1380 | Syllogism combined with co... |
| syl221anc 1381 | Syllogism combined with co... |
| syl113anc 1382 | Syllogism combined with co... |
| syl131anc 1383 | Syllogism combined with co... |
| syl311anc 1384 | Syllogism combined with co... |
| syl33anc 1385 | Syllogism combined with co... |
| syl222anc 1386 | Syllogism combined with co... |
| syl123anc 1387 | Syllogism combined with co... |
| syl132anc 1388 | Syllogism combined with co... |
| syl213anc 1389 | Syllogism combined with co... |
| syl231anc 1390 | Syllogism combined with co... |
| syl312anc 1391 | Syllogism combined with co... |
| syl321anc 1392 | Syllogism combined with co... |
| syl133anc 1393 | Syllogism combined with co... |
| syl313anc 1394 | Syllogism combined with co... |
| syl331anc 1395 | Syllogism combined with co... |
| syl223anc 1396 | Syllogism combined with co... |
| syl232anc 1397 | Syllogism combined with co... |
| syl322anc 1398 | Syllogism combined with co... |
| syl233anc 1399 | Syllogism combined with co... |
| syl323anc 1400 | Syllogism combined with co... |
| syl332anc 1401 | Syllogism combined with co... |
| syl333anc 1402 | A syllogism inference comb... |
| syl3an1b 1403 | A syllogism inference. (C... |
| syl3an2b 1404 | A syllogism inference. (C... |
| syl3an3b 1405 | A syllogism inference. (C... |
| syl3an1br 1406 | A syllogism inference. (C... |
| syl3an2br 1407 | A syllogism inference. (C... |
| syl3an3br 1408 | A syllogism inference. (C... |
| syld3an3 1409 | A syllogism inference. (C... |
| syld3an1 1410 | A syllogism inference. (C... |
| syld3an2 1411 | A syllogism inference. (C... |
| syl3anl1 1412 | A syllogism inference. (C... |
| syl3anl2 1413 | A syllogism inference. (C... |
| syl3anl3 1414 | A syllogism inference. (C... |
| syl3anl 1415 | A triple syllogism inferen... |
| syl3anr1 1416 | A syllogism inference. (C... |
| syl3anr2 1417 | A syllogism inference. (C... |
| syl3anr3 1418 | A syllogism inference. (C... |
| 3anidm12 1419 | Inference from idempotent ... |
| 3anidm13 1420 | Inference from idempotent ... |
| 3anidm23 1421 | Inference from idempotent ... |
| syl2an3an 1422 | ~ syl3an with antecedents ... |
| syl2an23an 1423 | Deduction related to ~ syl... |
| 3ori 1424 | Infer implication from tri... |
| 3jao 1425 | Disjunction of three antec... |
| 3jaob 1426 | Disjunction of three antec... |
| 3jaoi 1427 | Disjunction of three antec... |
| 3jaod 1428 | Disjunction of three antec... |
| 3jaoian 1429 | Disjunction of three antec... |
| 3jaodan 1430 | Disjunction of three antec... |
| mpjao3dan 1431 | Eliminate a three-way disj... |
| mpjao3danOLD 1432 | Obsolete version of ~ mpja... |
| 3jaao 1433 | Inference conjoining and d... |
| syl3an9b 1434 | Nested syllogism inference... |
| 3orbi123d 1435 | Deduction joining 3 equiva... |
| 3anbi123d 1436 | Deduction joining 3 equiva... |
| 3anbi12d 1437 | Deduction conjoining and a... |
| 3anbi13d 1438 | Deduction conjoining and a... |
| 3anbi23d 1439 | Deduction conjoining and a... |
| 3anbi1d 1440 | Deduction adding conjuncts... |
| 3anbi2d 1441 | Deduction adding conjuncts... |
| 3anbi3d 1442 | Deduction adding conjuncts... |
| 3anim123d 1443 | Deduction joining 3 implic... |
| 3orim123d 1444 | Deduction joining 3 implic... |
| an6 1445 | Rearrangement of 6 conjunc... |
| 3an6 1446 | Analogue of ~ an4 for trip... |
| 3or6 1447 | Analogue of ~ or4 for trip... |
| mp3an1 1448 | An inference based on modu... |
| mp3an2 1449 | An inference based on modu... |
| mp3an3 1450 | An inference based on modu... |
| mp3an12 1451 | An inference based on modu... |
| mp3an13 1452 | An inference based on modu... |
| mp3an23 1453 | An inference based on modu... |
| mp3an1i 1454 | An inference based on modu... |
| mp3anl1 1455 | An inference based on modu... |
| mp3anl2 1456 | An inference based on modu... |
| mp3anl3 1457 | An inference based on modu... |
| mp3anr1 1458 | An inference based on modu... |
| mp3anr2 1459 | An inference based on modu... |
| mp3anr3 1460 | An inference based on modu... |
| mp3an 1461 | An inference based on modu... |
| mpd3an3 1462 | An inference based on modu... |
| mpd3an23 1463 | An inference based on modu... |
| mp3and 1464 | A deduction based on modus... |
| mp3an12i 1465 | ~ mp3an with antecedents i... |
| mp3an2i 1466 | ~ mp3an with antecedents i... |
| mp3an3an 1467 | ~ mp3an with antecedents i... |
| mp3an2ani 1468 | An elimination deduction. ... |
| biimp3a 1469 | Infer implication from a l... |
| biimp3ar 1470 | Infer implication from a l... |
| 3anandis 1471 | Inference that undistribut... |
| 3anandirs 1472 | Inference that undistribut... |
| ecase23d 1473 | Deduction for elimination ... |
| 3ecase 1474 | Inference for elimination ... |
| 3bior1fd 1475 | A disjunction is equivalen... |
| 3bior1fand 1476 | A disjunction is equivalen... |
| 3bior2fd 1477 | A wff is equivalent to its... |
| 3biant1d 1478 | A conjunction is equivalen... |
| intn3an1d 1479 | Introduction of a triple c... |
| intn3an2d 1480 | Introduction of a triple c... |
| intn3an3d 1481 | Introduction of a triple c... |
| an3andi 1482 | Distribution of conjunctio... |
| an33rean 1483 | Rearrange a 9-fold conjunc... |
| an33reanOLD 1484 | Obsolete version of ~ an33... |
| 3orel2 1485 | Partial elimination of a t... |
| 3orel3 1486 | Partial elimination of a t... |
| 3orel13 1487 | Elimination of two disjunc... |
| 3pm3.2ni 1488 | Triple negated disjunction... |
| nanan 1491 | Conjunction in terms of al... |
| dfnan2 1492 | Alternative denial in term... |
| nanor 1493 | Alternative denial in term... |
| nancom 1494 | Alternative denial is comm... |
| nannan 1495 | Nested alternative denials... |
| nanim 1496 | Implication in terms of al... |
| nannot 1497 | Negation in terms of alter... |
| nanbi 1498 | Biconditional in terms of ... |
| nanbi1 1499 | Introduce a right anti-con... |
| nanbi2 1500 | Introduce a left anti-conj... |
| nanbi12 1501 | Join two logical equivalen... |
| nanbi1i 1502 | Introduce a right anti-con... |
| nanbi2i 1503 | Introduce a left anti-conj... |
| nanbi12i 1504 | Join two logical equivalen... |
| nanbi1d 1505 | Introduce a right anti-con... |
| nanbi2d 1506 | Introduce a left anti-conj... |
| nanbi12d 1507 | Join two logical equivalen... |
| nanass 1508 | A characterization of when... |
| xnor 1511 | Two ways to write XNOR (ex... |
| xorcom 1512 | The connector ` \/_ ` is c... |
| xorcomOLD 1513 | Obsolete version of ~ xorc... |
| xorass 1514 | The connector ` \/_ ` is a... |
| excxor 1515 | This tautology shows that ... |
| xor2 1516 | Two ways to express "exclu... |
| xoror 1517 | Exclusive disjunction impl... |
| xornan 1518 | Exclusive disjunction impl... |
| xornan2 1519 | XOR implies NAND (written ... |
| xorneg2 1520 | The connector ` \/_ ` is n... |
| xorneg1 1521 | The connector ` \/_ ` is n... |
| xorneg 1522 | The connector ` \/_ ` is u... |
| xorbi12i 1523 | Equality property for excl... |
| xorbi12iOLD 1524 | Obsolete version of ~ xorb... |
| xorbi12d 1525 | Equality property for excl... |
| anxordi 1526 | Conjunction distributes ov... |
| xorexmid 1527 | Exclusive-or variant of th... |
| norcom 1530 | The connector ` -\/ ` is c... |
| norcomOLD 1531 | Obsolete version of ~ norc... |
| nornot 1532 | ` -. ` is expressible via ... |
| noran 1533 | ` /\ ` is expressible via ... |
| noror 1534 | ` \/ ` is expressible via ... |
| norasslem1 1535 | This lemma shows the equiv... |
| norasslem2 1536 | This lemma specializes ~ b... |
| norasslem3 1537 | This lemma specializes ~ b... |
| norass 1538 | A characterization of when... |
| trujust 1543 | Soundness justification th... |
| tru 1545 | The truth value ` T. ` is ... |
| dftru2 1546 | An alternate definition of... |
| trut 1547 | A proposition is equivalen... |
| mptru 1548 | Eliminate ` T. ` as an ant... |
| tbtru 1549 | A proposition is equivalen... |
| bitru 1550 | A theorem is equivalent to... |
| trud 1551 | Anything implies ` T. ` . ... |
| truan 1552 | True can be removed from a... |
| fal 1555 | The truth value ` F. ` is ... |
| nbfal 1556 | The negation of a proposit... |
| bifal 1557 | A contradiction is equival... |
| falim 1558 | The truth value ` F. ` imp... |
| falimd 1559 | The truth value ` F. ` imp... |
| dfnot 1560 | Given falsum ` F. ` , we c... |
| inegd 1561 | Negation introduction rule... |
| efald 1562 | Deduction based on reducti... |
| pm2.21fal 1563 | If a wff and its negation ... |
| truimtru 1564 | A ` -> ` identity. (Contr... |
| truimfal 1565 | A ` -> ` identity. (Contr... |
| falimtru 1566 | A ` -> ` identity. (Contr... |
| falimfal 1567 | A ` -> ` identity. (Contr... |
| nottru 1568 | A ` -. ` identity. (Contr... |
| notfal 1569 | A ` -. ` identity. (Contr... |
| trubitru 1570 | A ` <-> ` identity. (Cont... |
| falbitru 1571 | A ` <-> ` identity. (Cont... |
| trubifal 1572 | A ` <-> ` identity. (Cont... |
| falbifal 1573 | A ` <-> ` identity. (Cont... |
| truantru 1574 | A ` /\ ` identity. (Contr... |
| truanfal 1575 | A ` /\ ` identity. (Contr... |
| falantru 1576 | A ` /\ ` identity. (Contr... |
| falanfal 1577 | A ` /\ ` identity. (Contr... |
| truortru 1578 | A ` \/ ` identity. (Contr... |
| truorfal 1579 | A ` \/ ` identity. (Contr... |
| falortru 1580 | A ` \/ ` identity. (Contr... |
| falorfal 1581 | A ` \/ ` identity. (Contr... |
| trunantru 1582 | A ` -/\ ` identity. (Cont... |
| trunanfal 1583 | A ` -/\ ` identity. (Cont... |
| falnantru 1584 | A ` -/\ ` identity. (Cont... |
| falnanfal 1585 | A ` -/\ ` identity. (Cont... |
| truxortru 1586 | A ` \/_ ` identity. (Cont... |
| truxorfal 1587 | A ` \/_ ` identity. (Cont... |
| falxortru 1588 | A ` \/_ ` identity. (Cont... |
| falxorfal 1589 | A ` \/_ ` identity. (Cont... |
| trunortru 1590 | A ` -\/ ` identity. (Cont... |
| trunorfal 1591 | A ` -\/ ` identity. (Cont... |
| falnortru 1592 | A ` -\/ ` identity. (Cont... |
| falnorfal 1593 | A ` -\/ ` identity. (Cont... |
| hadbi123d 1596 | Equality theorem for the a... |
| hadbi123i 1597 | Equality theorem for the a... |
| hadass 1598 | Associative law for the ad... |
| hadbi 1599 | The adder sum is the same ... |
| hadcoma 1600 | Commutative law for the ad... |
| hadcomb 1601 | Commutative law for the ad... |
| hadrot 1602 | Rotation law for the adder... |
| hadnot 1603 | The adder sum distributes ... |
| had1 1604 | If the first input is true... |
| had0 1605 | If the first input is fals... |
| hadifp 1606 | The value of the adder sum... |
| cador 1609 | The adder carry in disjunc... |
| cadan 1610 | The adder carry in conjunc... |
| cadbi123d 1611 | Equality theorem for the a... |
| cadbi123i 1612 | Equality theorem for the a... |
| cadcoma 1613 | Commutative law for the ad... |
| cadcomb 1614 | Commutative law for the ad... |
| cadrot 1615 | Rotation law for the adder... |
| cadnot 1616 | The adder carry distribute... |
| cad11 1617 | If (at least) two inputs a... |
| cad1 1618 | If one input is true, then... |
| cad0 1619 | If one input is false, the... |
| cad0OLD 1620 | Obsolete version of ~ cad0... |
| cadifp 1621 | The value of the carry is,... |
| cadtru 1622 | The adder carry is true as... |
| minimp 1623 | A single axiom for minimal... |
| minimp-syllsimp 1624 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1625 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1626 | Derivation of a commuted f... |
| minimp-ax2 1627 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1628 | Derivation of ~ pm2.43 (al... |
| impsingle 1629 | The shortest single axiom ... |
| impsingle-step4 1630 | Derivation of impsingle-st... |
| impsingle-step8 1631 | Derivation of impsingle-st... |
| impsingle-ax1 1632 | Derivation of impsingle-ax... |
| impsingle-step15 1633 | Derivation of impsingle-st... |
| impsingle-step18 1634 | Derivation of impsingle-st... |
| impsingle-step19 1635 | Derivation of impsingle-st... |
| impsingle-step20 1636 | Derivation of impsingle-st... |
| impsingle-step21 1637 | Derivation of impsingle-st... |
| impsingle-step22 1638 | Derivation of impsingle-st... |
| impsingle-step25 1639 | Derivation of impsingle-st... |
| impsingle-imim1 1640 | Derivation of impsingle-im... |
| impsingle-peirce 1641 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1642 | Derivation of ~ ax-2 from ... |
| meredith 1643 | Carew Meredith's sole axio... |
| merlem1 1644 | Step 3 of Meredith's proof... |
| merlem2 1645 | Step 4 of Meredith's proof... |
| merlem3 1646 | Step 7 of Meredith's proof... |
| merlem4 1647 | Step 8 of Meredith's proof... |
| merlem5 1648 | Step 11 of Meredith's proo... |
| merlem6 1649 | Step 12 of Meredith's proo... |
| merlem7 1650 | Between steps 14 and 15 of... |
| merlem8 1651 | Step 15 of Meredith's proo... |
| merlem9 1652 | Step 18 of Meredith's proo... |
| merlem10 1653 | Step 19 of Meredith's proo... |
| merlem11 1654 | Step 20 of Meredith's proo... |
| merlem12 1655 | Step 28 of Meredith's proo... |
| merlem13 1656 | Step 35 of Meredith's proo... |
| luk-1 1657 | 1 of 3 axioms for proposit... |
| luk-2 1658 | 2 of 3 axioms for proposit... |
| luk-3 1659 | 3 of 3 axioms for proposit... |
| luklem1 1660 | Used to rederive standard ... |
| luklem2 1661 | Used to rederive standard ... |
| luklem3 1662 | Used to rederive standard ... |
| luklem4 1663 | Used to rederive standard ... |
| luklem5 1664 | Used to rederive standard ... |
| luklem6 1665 | Used to rederive standard ... |
| luklem7 1666 | Used to rederive standard ... |
| luklem8 1667 | Used to rederive standard ... |
| ax1 1668 | Standard propositional axi... |
| ax2 1669 | Standard propositional axi... |
| ax3 1670 | Standard propositional axi... |
| nic-dfim 1671 | This theorem "defines" imp... |
| nic-dfneg 1672 | This theorem "defines" neg... |
| nic-mp 1673 | Derive Nicod's rule of mod... |
| nic-mpALT 1674 | A direct proof of ~ nic-mp... |
| nic-ax 1675 | Nicod's axiom derived from... |
| nic-axALT 1676 | A direct proof of ~ nic-ax... |
| nic-imp 1677 | Inference for ~ nic-mp usi... |
| nic-idlem1 1678 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1679 | Lemma for ~ nic-id . Infe... |
| nic-id 1680 | Theorem ~ id expressed wit... |
| nic-swap 1681 | The connector ` -/\ ` is s... |
| nic-isw1 1682 | Inference version of ~ nic... |
| nic-isw2 1683 | Inference for swapping nes... |
| nic-iimp1 1684 | Inference version of ~ nic... |
| nic-iimp2 1685 | Inference version of ~ nic... |
| nic-idel 1686 | Inference to remove the tr... |
| nic-ich 1687 | Chained inference. (Contr... |
| nic-idbl 1688 | Double the terms. Since d... |
| nic-bijust 1689 | Biconditional justificatio... |
| nic-bi1 1690 | Inference to extract one s... |
| nic-bi2 1691 | Inference to extract the o... |
| nic-stdmp 1692 | Derive the standard modus ... |
| nic-luk1 1693 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1694 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1695 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1696 | This alternative axiom for... |
| lukshefth1 1697 | Lemma for ~ renicax . (Co... |
| lukshefth2 1698 | Lemma for ~ renicax . (Co... |
| renicax 1699 | A rederivation of ~ nic-ax... |
| tbw-bijust 1700 | Justification for ~ tbw-ne... |
| tbw-negdf 1701 | The definition of negation... |
| tbw-ax1 1702 | The first of four axioms i... |
| tbw-ax2 1703 | The second of four axioms ... |
| tbw-ax3 1704 | The third of four axioms i... |
| tbw-ax4 1705 | The fourth of four axioms ... |
| tbwsyl 1706 | Used to rederive the Lukas... |
| tbwlem1 1707 | Used to rederive the Lukas... |
| tbwlem2 1708 | Used to rederive the Lukas... |
| tbwlem3 1709 | Used to rederive the Lukas... |
| tbwlem4 1710 | Used to rederive the Lukas... |
| tbwlem5 1711 | Used to rederive the Lukas... |
| re1luk1 1712 | ~ luk-1 derived from the T... |
| re1luk2 1713 | ~ luk-2 derived from the T... |
| re1luk3 1714 | ~ luk-3 derived from the T... |
| merco1 1715 | A single axiom for proposi... |
| merco1lem1 1716 | Used to rederive the Tarsk... |
| retbwax4 1717 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1718 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1719 | Used to rederive the Tarsk... |
| merco1lem3 1720 | Used to rederive the Tarsk... |
| merco1lem4 1721 | Used to rederive the Tarsk... |
| merco1lem5 1722 | Used to rederive the Tarsk... |
| merco1lem6 1723 | Used to rederive the Tarsk... |
| merco1lem7 1724 | Used to rederive the Tarsk... |
| retbwax3 1725 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1726 | Used to rederive the Tarsk... |
| merco1lem9 1727 | Used to rederive the Tarsk... |
| merco1lem10 1728 | Used to rederive the Tarsk... |
| merco1lem11 1729 | Used to rederive the Tarsk... |
| merco1lem12 1730 | Used to rederive the Tarsk... |
| merco1lem13 1731 | Used to rederive the Tarsk... |
| merco1lem14 1732 | Used to rederive the Tarsk... |
| merco1lem15 1733 | Used to rederive the Tarsk... |
| merco1lem16 1734 | Used to rederive the Tarsk... |
| merco1lem17 1735 | Used to rederive the Tarsk... |
| merco1lem18 1736 | Used to rederive the Tarsk... |
| retbwax1 1737 | ~ tbw-ax1 rederived from ~... |
| merco2 1738 | A single axiom for proposi... |
| mercolem1 1739 | Used to rederive the Tarsk... |
| mercolem2 1740 | Used to rederive the Tarsk... |
| mercolem3 1741 | Used to rederive the Tarsk... |
| mercolem4 1742 | Used to rederive the Tarsk... |
| mercolem5 1743 | Used to rederive the Tarsk... |
| mercolem6 1744 | Used to rederive the Tarsk... |
| mercolem7 1745 | Used to rederive the Tarsk... |
| mercolem8 1746 | Used to rederive the Tarsk... |
| re1tbw1 1747 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1748 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1749 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1750 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1751 | Justification for ~ rb-imd... |
| rb-imdf 1752 | The definition of implicat... |
| anmp 1753 | Modus ponens for ` { \/ , ... |
| rb-ax1 1754 | The first of four axioms i... |
| rb-ax2 1755 | The second of four axioms ... |
| rb-ax3 1756 | The third of four axioms i... |
| rb-ax4 1757 | The fourth of four axioms ... |
| rbsyl 1758 | Used to rederive the Lukas... |
| rblem1 1759 | Used to rederive the Lukas... |
| rblem2 1760 | Used to rederive the Lukas... |
| rblem3 1761 | Used to rederive the Lukas... |
| rblem4 1762 | Used to rederive the Lukas... |
| rblem5 1763 | Used to rederive the Lukas... |
| rblem6 1764 | Used to rederive the Lukas... |
| rblem7 1765 | Used to rederive the Lukas... |
| re1axmp 1766 | ~ ax-mp derived from Russe... |
| re2luk1 1767 | ~ luk-1 derived from Russe... |
| re2luk2 1768 | ~ luk-2 derived from Russe... |
| re2luk3 1769 | ~ luk-3 derived from Russe... |
| mptnan 1770 | Modus ponendo tollens 1, o... |
| mptxor 1771 | Modus ponendo tollens 2, o... |
| mtpor 1772 | Modus tollendo ponens (inc... |
| mtpxor 1773 | Modus tollendo ponens (ori... |
| stoic1a 1774 | Stoic logic Thema 1 (part ... |
| stoic1b 1775 | Stoic logic Thema 1 (part ... |
| stoic2a 1776 | Stoic logic Thema 2 versio... |
| stoic2b 1777 | Stoic logic Thema 2 versio... |
| stoic3 1778 | Stoic logic Thema 3. Stat... |
| stoic4a 1779 | Stoic logic Thema 4 versio... |
| stoic4b 1780 | Stoic logic Thema 4 versio... |
| alnex 1783 | Universal quantification o... |
| eximal 1784 | An equivalence between an ... |
| nf2 1787 | Alternate definition of no... |
| nf3 1788 | Alternate definition of no... |
| nf4 1789 | Alternate definition of no... |
| nfi 1790 | Deduce that ` x ` is not f... |
| nfri 1791 | Consequence of the definit... |
| nfd 1792 | Deduce that ` x ` is not f... |
| nfrd 1793 | Consequence of the definit... |
| nftht 1794 | Closed form of ~ nfth . (... |
| nfntht 1795 | Closed form of ~ nfnth . ... |
| nfntht2 1796 | Closed form of ~ nfnth . ... |
| gen2 1798 | Generalization applied twi... |
| mpg 1799 | Modus ponens combined with... |
| mpgbi 1800 | Modus ponens on biconditio... |
| mpgbir 1801 | Modus ponens on biconditio... |
| nex 1802 | Generalization rule for ne... |
| nfth 1803 | No variable is (effectivel... |
| nfnth 1804 | No variable is (effectivel... |
| hbth 1805 | No variable is (effectivel... |
| nftru 1806 | The true constant has no f... |
| nffal 1807 | The false constant has no ... |
| sptruw 1808 | Version of ~ sp when ` ph ... |
| altru 1809 | For all sets, ` T. ` is tr... |
| alfal 1810 | For all sets, ` -. F. ` is... |
| alim 1812 | Restatement of Axiom ~ ax-... |
| alimi 1813 | Inference quantifying both... |
| 2alimi 1814 | Inference doubly quantifyi... |
| ala1 1815 | Add an antecedent in a uni... |
| al2im 1816 | Closed form of ~ al2imi . ... |
| al2imi 1817 | Inference quantifying ante... |
| alanimi 1818 | Variant of ~ al2imi with c... |
| alimdh 1819 | Deduction form of Theorem ... |
| albi 1820 | Theorem 19.15 of [Margaris... |
| albii 1821 | Inference adding universal... |
| 2albii 1822 | Inference adding two unive... |
| 3albii 1823 | Inference adding three uni... |
| sylgt 1824 | Closed form of ~ sylg . (... |
| sylg 1825 | A syllogism combined with ... |
| alrimih 1826 | Inference form of Theorem ... |
| hbxfrbi 1827 | A utility lemma to transfe... |
| alex 1828 | Universal quantifier in te... |
| exnal 1829 | Existential quantification... |
| 2nalexn 1830 | Part of theorem *11.5 in [... |
| 2exnaln 1831 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1832 | Theorem *11.25 in [Whitehe... |
| alimex 1833 | An equivalence between an ... |
| aleximi 1834 | A variant of ~ al2imi : in... |
| alexbii 1835 | Biconditional form of ~ al... |
| exim 1836 | Theorem 19.22 of [Margaris... |
| eximi 1837 | Inference adding existenti... |
| 2eximi 1838 | Inference adding two exist... |
| eximii 1839 | Inference associated with ... |
| exa1 1840 | Add an antecedent in an ex... |
| 19.38 1841 | Theorem 19.38 of [Margaris... |
| 19.38a 1842 | Under a nonfreeness hypoth... |
| 19.38b 1843 | Under a nonfreeness hypoth... |
| imnang 1844 | Quantified implication in ... |
| alinexa 1845 | A transformation of quanti... |
| exnalimn 1846 | Existential quantification... |
| alexn 1847 | A relationship between two... |
| 2exnexn 1848 | Theorem *11.51 in [Whitehe... |
| exbi 1849 | Theorem 19.18 of [Margaris... |
| exbii 1850 | Inference adding existenti... |
| 2exbii 1851 | Inference adding two exist... |
| 3exbii 1852 | Inference adding three exi... |
| nfbiit 1853 | Equivalence theorem for th... |
| nfbii 1854 | Equality theorem for the n... |
| nfxfr 1855 | A utility lemma to transfe... |
| nfxfrd 1856 | A utility lemma to transfe... |
| nfnbi 1857 | A variable is nonfree in a... |
| nfnbiOLD 1858 | Obsolete version of ~ nfnb... |
| nfnt 1859 | If a variable is nonfree i... |
| nfn 1860 | Inference associated with ... |
| nfnd 1861 | Deduction associated with ... |
| exanali 1862 | A transformation of quanti... |
| 2exanali 1863 | Theorem *11.521 in [Whiteh... |
| exancom 1864 | Commutation of conjunction... |
| exan 1865 | Place a conjunct in the sc... |
| alrimdh 1866 | Deduction form of Theorem ... |
| eximdh 1867 | Deduction from Theorem 19.... |
| nexdh 1868 | Deduction for generalizati... |
| albidh 1869 | Formula-building rule for ... |
| exbidh 1870 | Formula-building rule for ... |
| exsimpl 1871 | Simplification of an exist... |
| exsimpr 1872 | Simplification of an exist... |
| 19.26 1873 | Theorem 19.26 of [Margaris... |
| 19.26-2 1874 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1875 | Theorem ~ 19.26 with tripl... |
| 19.29 1876 | Theorem 19.29 of [Margaris... |
| 19.29r 1877 | Variation of ~ 19.29 . (C... |
| 19.29r2 1878 | Variation of ~ 19.29r with... |
| 19.29x 1879 | Variation of ~ 19.29 with ... |
| 19.35 1880 | Theorem 19.35 of [Margaris... |
| 19.35i 1881 | Inference associated with ... |
| 19.35ri 1882 | Inference associated with ... |
| 19.25 1883 | Theorem 19.25 of [Margaris... |
| 19.30 1884 | Theorem 19.30 of [Margaris... |
| 19.43 1885 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1886 | Obsolete proof of ~ 19.43 ... |
| 19.33 1887 | Theorem 19.33 of [Margaris... |
| 19.33b 1888 | The antecedent provides a ... |
| 19.40 1889 | Theorem 19.40 of [Margaris... |
| 19.40-2 1890 | Theorem *11.42 in [Whitehe... |
| 19.40b 1891 | The antecedent provides a ... |
| albiim 1892 | Split a biconditional and ... |
| 2albiim 1893 | Split a biconditional and ... |
| exintrbi 1894 | Add/remove a conjunct in t... |
| exintr 1895 | Introduce a conjunct in th... |
| alsyl 1896 | Universally quantified and... |
| nfimd 1897 | If in a context ` x ` is n... |
| nfimt 1898 | Closed form of ~ nfim and ... |
| nfim 1899 | If ` x ` is not free in ` ... |
| nfand 1900 | If in a context ` x ` is n... |
| nf3and 1901 | Deduction form of bound-va... |
| nfan 1902 | If ` x ` is not free in ` ... |
| nfnan 1903 | If ` x ` is not free in ` ... |
| nf3an 1904 | If ` x ` is not free in ` ... |
| nfbid 1905 | If in a context ` x ` is n... |
| nfbi 1906 | If ` x ` is not free in ` ... |
| nfor 1907 | If ` x ` is not free in ` ... |
| nf3or 1908 | If ` x ` is not free in ` ... |
| empty 1909 | Two characterizations of t... |
| emptyex 1910 | On the empty domain, any e... |
| emptyal 1911 | On the empty domain, any u... |
| emptynf 1912 | On the empty domain, any v... |
| ax5d 1914 | Version of ~ ax-5 with ant... |
| ax5e 1915 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1916 | If a formula holds for som... |
| nfv 1917 | If ` x ` is not present in... |
| nfvd 1918 | ~ nfv with antecedent. Us... |
| alimdv 1919 | Deduction form of Theorem ... |
| eximdv 1920 | Deduction form of Theorem ... |
| 2alimdv 1921 | Deduction form of Theorem ... |
| 2eximdv 1922 | Deduction form of Theorem ... |
| albidv 1923 | Formula-building rule for ... |
| exbidv 1924 | Formula-building rule for ... |
| nfbidv 1925 | An equality theorem for no... |
| 2albidv 1926 | Formula-building rule for ... |
| 2exbidv 1927 | Formula-building rule for ... |
| 3exbidv 1928 | Formula-building rule for ... |
| 4exbidv 1929 | Formula-building rule for ... |
| alrimiv 1930 | Inference form of Theorem ... |
| alrimivv 1931 | Inference form of Theorem ... |
| alrimdv 1932 | Deduction form of Theorem ... |
| exlimiv 1933 | Inference form of Theorem ... |
| exlimiiv 1934 | Inference (Rule C) associa... |
| exlimivv 1935 | Inference form of Theorem ... |
| exlimdv 1936 | Deduction form of Theorem ... |
| exlimdvv 1937 | Deduction form of Theorem ... |
| exlimddv 1938 | Existential elimination ru... |
| nexdv 1939 | Deduction for generalizati... |
| 2ax5 1940 | Quantification of two vari... |
| stdpc5v 1941 | Version of ~ stdpc5 with a... |
| 19.21v 1942 | Version of ~ 19.21 with a ... |
| 19.32v 1943 | Version of ~ 19.32 with a ... |
| 19.31v 1944 | Version of ~ 19.31 with a ... |
| 19.23v 1945 | Version of ~ 19.23 with a ... |
| 19.23vv 1946 | Theorem ~ 19.23v extended ... |
| pm11.53v 1947 | Version of ~ pm11.53 with ... |
| 19.36imv 1948 | One direction of ~ 19.36v ... |
| 19.36imvOLD 1949 | Obsolete version of ~ 19.3... |
| 19.36iv 1950 | Inference associated with ... |
| 19.37imv 1951 | One direction of ~ 19.37v ... |
| 19.37iv 1952 | Inference associated with ... |
| 19.41v 1953 | Version of ~ 19.41 with a ... |
| 19.41vv 1954 | Version of ~ 19.41 with tw... |
| 19.41vvv 1955 | Version of ~ 19.41 with th... |
| 19.41vvvv 1956 | Version of ~ 19.41 with fo... |
| 19.42v 1957 | Version of ~ 19.42 with a ... |
| exdistr 1958 | Distribution of existentia... |
| exdistrv 1959 | Distribute a pair of exist... |
| 4exdistrv 1960 | Distribute two pairs of ex... |
| 19.42vv 1961 | Version of ~ 19.42 with tw... |
| exdistr2 1962 | Distribution of existentia... |
| 19.42vvv 1963 | Version of ~ 19.42 with th... |
| 3exdistr 1964 | Distribution of existentia... |
| 4exdistr 1965 | Distribution of existentia... |
| weq 1966 | Extend wff definition to i... |
| speimfw 1967 | Specialization, with addit... |
| speimfwALT 1968 | Alternate proof of ~ speim... |
| spimfw 1969 | Specialization, with addit... |
| ax12i 1970 | Inference that has ~ ax-12... |
| ax6v 1972 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1973 | At least one individual ex... |
| spimw 1974 | Specialization. Lemma 8 o... |
| spimew 1975 | Existential introduction, ... |
| speiv 1976 | Inference from existential... |
| speivw 1977 | Version of ~ spei with a d... |
| exgen 1978 | Rule of existential genera... |
| extru 1979 | There exists a variable su... |
| 19.2 1980 | Theorem 19.2 of [Margaris]... |
| 19.2d 1981 | Deduction associated with ... |
| 19.8w 1982 | Weak version of ~ 19.8a an... |
| spnfw 1983 | Weak version of ~ sp . Us... |
| spvw 1984 | Version of ~ sp when ` x `... |
| 19.3v 1985 | Version of ~ 19.3 with a d... |
| 19.8v 1986 | Version of ~ 19.8a with a ... |
| 19.9v 1987 | Version of ~ 19.9 with a d... |
| 19.39 1988 | Theorem 19.39 of [Margaris... |
| 19.24 1989 | Theorem 19.24 of [Margaris... |
| 19.34 1990 | Theorem 19.34 of [Margaris... |
| 19.36v 1991 | Version of ~ 19.36 with a ... |
| 19.12vvv 1992 | Version of ~ 19.12vv with ... |
| 19.27v 1993 | Version of ~ 19.27 with a ... |
| 19.28v 1994 | Version of ~ 19.28 with a ... |
| 19.37v 1995 | Version of ~ 19.37 with a ... |
| 19.44v 1996 | Version of ~ 19.44 with a ... |
| 19.45v 1997 | Version of ~ 19.45 with a ... |
| spimevw 1998 | Existential introduction, ... |
| spimvw 1999 | A weak form of specializat... |
| spvv 2000 | Specialization, using impl... |
| spfalw 2001 | Version of ~ sp when ` ph ... |
| chvarvv 2002 | Implicit substitution of `... |
| equs4v 2003 | Version of ~ equs4 with a ... |
| alequexv 2004 | Version of ~ equs4v with i... |
| exsbim 2005 | One direction of the equiv... |
| equsv 2006 | If a formula does not cont... |
| equsalvw 2007 | Version of ~ equsalv with ... |
| equsexvw 2008 | Version of ~ equsexv with ... |
| cbvaliw 2009 | Change bound variable. Us... |
| cbvalivw 2010 | Change bound variable. Us... |
| ax7v 2012 | Weakened version of ~ ax-7... |
| ax7v1 2013 | First of two weakened vers... |
| ax7v2 2014 | Second of two weakened ver... |
| equid 2015 | Identity law for equality.... |
| nfequid 2016 | Bound-variable hypothesis ... |
| equcomiv 2017 | Weaker form of ~ equcomi w... |
| ax6evr 2018 | A commuted form of ~ ax6ev... |
| ax7 2019 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2020 | Commutative law for equali... |
| equcom 2021 | Commutative law for equali... |
| equcomd 2022 | Deduction form of ~ equcom... |
| equcoms 2023 | An inference commuting equ... |
| equtr 2024 | A transitive law for equal... |
| equtrr 2025 | A transitive law for equal... |
| equeuclr 2026 | Commuted version of ~ eque... |
| equeucl 2027 | Equality is a left-Euclide... |
| equequ1 2028 | An equivalence law for equ... |
| equequ2 2029 | An equivalence law for equ... |
| equtr2 2030 | Equality is a left-Euclide... |
| stdpc6 2031 | One of the two equality ax... |
| equvinv 2032 | A variable introduction la... |
| equvinva 2033 | A modified version of the ... |
| equvelv 2034 | A biconditional form of ~ ... |
| ax13b 2035 | An equivalence between two... |
| spfw 2036 | Weak version of ~ sp . Us... |
| spw 2037 | Weak version of the specia... |
| cbvalw 2038 | Change bound variable. Us... |
| cbvalvw 2039 | Change bound variable. Us... |
| cbvexvw 2040 | Change bound variable. Us... |
| cbvaldvaw 2041 | Rule used to change the bo... |
| cbvexdvaw 2042 | Rule used to change the bo... |
| cbval2vw 2043 | Rule used to change bound ... |
| cbvex2vw 2044 | Rule used to change bound ... |
| cbvex4vw 2045 | Rule used to change bound ... |
| alcomiw 2046 | Weak version of ~ ax-11 . ... |
| alcomw 2047 | Weak version of ~ alcom an... |
| hbn1fw 2048 | Weak version of ~ ax-10 fr... |
| hbn1w 2049 | Weak version of ~ hbn1 . ... |
| hba1w 2050 | Weak version of ~ hba1 . ... |
| hbe1w 2051 | Weak version of ~ hbe1 . ... |
| hbalw 2052 | Weak version of ~ hbal . ... |
| 19.8aw 2053 | If a formula is true, then... |
| exexw 2054 | Existential quantification... |
| spaev 2055 | A special instance of ~ sp... |
| cbvaev 2056 | Change bound variable in a... |
| aevlem0 2057 | Lemma for ~ aevlem . Inst... |
| aevlem 2058 | Lemma for ~ aev and ~ axc1... |
| aeveq 2059 | The antecedent ` A. x x = ... |
| aev 2060 | A "distinctor elimination"... |
| aev2 2061 | A version of ~ aev with tw... |
| hbaev 2062 | All variables are effectiv... |
| naev 2063 | If some set variables can ... |
| naev2 2064 | Generalization of ~ hbnaev... |
| hbnaev 2065 | Any variable is free in ` ... |
| sbjust 2066 | Justification theorem for ... |
| sbt 2069 | A substitution into a theo... |
| sbtru 2070 | The result of substituting... |
| stdpc4 2071 | The specialization axiom o... |
| sbtALT 2072 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2073 | A double specialization us... |
| sbi1 2074 | Distribute substitution ov... |
| spsbim 2075 | Distribute substitution ov... |
| spsbbi 2076 | Biconditional property for... |
| sbimi 2077 | Distribute substitution ov... |
| sb2imi 2078 | Distribute substitution ov... |
| sbbii 2079 | Infer substitution into bo... |
| 2sbbii 2080 | Infer double substitution ... |
| sbimdv 2081 | Deduction substituting bot... |
| sbbidv 2082 | Deduction substituting bot... |
| sban 2083 | Conjunction inside and out... |
| sb3an 2084 | Threefold conjunction insi... |
| spsbe 2085 | Existential generalization... |
| sbequ 2086 | Equality property for subs... |
| sbequi 2087 | An equality theorem for su... |
| sb6 2088 | Alternate definition of su... |
| 2sb6 2089 | Equivalence for double sub... |
| sb1v 2090 | One direction of ~ sb5 , p... |
| sbv 2091 | Substitution for a variabl... |
| sbcom4 2092 | Commutativity law for subs... |
| pm11.07 2093 | Axiom *11.07 in [Whitehead... |
| sbrimvw 2094 | Substitution in an implica... |
| sbievw 2095 | Conversion of implicit sub... |
| sbiedvw 2096 | Conversion of implicit sub... |
| 2sbievw 2097 | Conversion of double impli... |
| sbcom3vv 2098 | Substituting ` y ` for ` x... |
| sbievw2 2099 | ~ sbievw applied twice, av... |
| sbco2vv 2100 | A composition law for subs... |
| equsb3 2101 | Substitution in an equalit... |
| equsb3r 2102 | Substitution applied to th... |
| equsb1v 2103 | Substitution applied to an... |
| nsb 2104 | Any substitution in an alw... |
| sbn1 2105 | One direction of ~ sbn , u... |
| wel 2107 | Extend wff definition to i... |
| ax8v 2109 | Weakened version of ~ ax-8... |
| ax8v1 2110 | First of two weakened vers... |
| ax8v2 2111 | Second of two weakened ver... |
| ax8 2112 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2113 | An identity law for the no... |
| elsb1 2114 | Substitution for the first... |
| cleljust 2115 | When the class variables i... |
| ax9v 2117 | Weakened version of ~ ax-9... |
| ax9v1 2118 | First of two weakened vers... |
| ax9v2 2119 | Second of two weakened ver... |
| ax9 2120 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2121 | An identity law for the no... |
| elequ2g 2122 | A form of ~ elequ2 with a ... |
| elsb2 2123 | Substitution for the secon... |
| ax6dgen 2124 | Tarski's system uses the w... |
| ax10w 2125 | Weak version of ~ ax-10 fr... |
| ax11w 2126 | Weak version of ~ ax-11 fr... |
| ax11dgen 2127 | Degenerate instance of ~ a... |
| ax12wlem 2128 | Lemma for weak version of ... |
| ax12w 2129 | Weak version of ~ ax-12 fr... |
| ax12dgen 2130 | Degenerate instance of ~ a... |
| ax12wdemo 2131 | Example of an application ... |
| ax13w 2132 | Weak version (principal in... |
| ax13dgen1 2133 | Degenerate instance of ~ a... |
| ax13dgen2 2134 | Degenerate instance of ~ a... |
| ax13dgen3 2135 | Degenerate instance of ~ a... |
| ax13dgen4 2136 | Degenerate instance of ~ a... |
| hbn1 2138 | Alias for ~ ax-10 to be us... |
| hbe1 2139 | The setvar ` x ` is not fr... |
| hbe1a 2140 | Dual statement of ~ hbe1 .... |
| nf5-1 2141 | One direction of ~ nf5 can... |
| nf5i 2142 | Deduce that ` x ` is not f... |
| nf5dh 2143 | Deduce that ` x ` is not f... |
| nf5dv 2144 | Apply the definition of no... |
| nfnaew 2145 | All variables are effectiv... |
| nfnaewOLD 2146 | Obsolete version of ~ nfna... |
| nfe1 2147 | The setvar ` x ` is not fr... |
| nfa1 2148 | The setvar ` x ` is not fr... |
| nfna1 2149 | A convenience theorem part... |
| nfia1 2150 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2151 | The setvar ` x ` is not fr... |
| modal5 2152 | The analogue in our predic... |
| nfs1v 2153 | The setvar ` x ` is not fr... |
| alcoms 2155 | Swap quantifiers in an ant... |
| alcom 2156 | Theorem 19.5 of [Margaris]... |
| alrot3 2157 | Theorem *11.21 in [Whitehe... |
| alrot4 2158 | Rotate four universal quan... |
| sbal 2159 | Move universal quantifier ... |
| sbalv 2160 | Quantify with new variable... |
| sbcom2 2161 | Commutativity law for subs... |
| excom 2162 | Theorem 19.11 of [Margaris... |
| excomim 2163 | One direction of Theorem 1... |
| excom13 2164 | Swap 1st and 3rd existenti... |
| exrot3 2165 | Rotate existential quantif... |
| exrot4 2166 | Rotate existential quantif... |
| hbal 2167 | If ` x ` is not free in ` ... |
| hbald 2168 | Deduction form of bound-va... |
| hbsbw 2169 | If ` z ` is not free in ` ... |
| nfa2 2170 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2172 | This is essentially Axiom ... |
| ax12v2 2173 | It is possible to remove a... |
| 19.8a 2174 | If a wff is true, it is tr... |
| 19.8ad 2175 | If a wff is true, it is tr... |
| sp 2176 | Specialization. A univers... |
| spi 2177 | Inference rule of universa... |
| sps 2178 | Generalization of antecede... |
| 2sp 2179 | A double specialization (s... |
| spsd 2180 | Deduction generalizing ant... |
| 19.2g 2181 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2182 | Inference form of ~ 19.21 ... |
| 19.21bbi 2183 | Inference removing two uni... |
| 19.23bi 2184 | Inference form of Theorem ... |
| nexr 2185 | Inference associated with ... |
| qexmid 2186 | Quantified excluded middle... |
| nf5r 2187 | Consequence of the definit... |
| nf5ri 2188 | Consequence of the definit... |
| nf5rd 2189 | Consequence of the definit... |
| spimedv 2190 | Deduction version of ~ spi... |
| spimefv 2191 | Version of ~ spime with a ... |
| nfim1 2192 | A closed form of ~ nfim . ... |
| nfan1 2193 | A closed form of ~ nfan . ... |
| 19.3t 2194 | Closed form of ~ 19.3 and ... |
| 19.3 2195 | A wff may be quantified wi... |
| 19.9d 2196 | A deduction version of one... |
| 19.9t 2197 | Closed form of ~ 19.9 and ... |
| 19.9 2198 | A wff may be existentially... |
| 19.21t 2199 | Closed form of Theorem 19.... |
| 19.21 2200 | Theorem 19.21 of [Margaris... |
| stdpc5 2201 | An axiom scheme of standar... |
| 19.21-2 2202 | Version of ~ 19.21 with tw... |
| 19.23t 2203 | Closed form of Theorem 19.... |
| 19.23 2204 | Theorem 19.23 of [Margaris... |
| alimd 2205 | Deduction form of Theorem ... |
| alrimi 2206 | Inference form of Theorem ... |
| alrimdd 2207 | Deduction form of Theorem ... |
| alrimd 2208 | Deduction form of Theorem ... |
| eximd 2209 | Deduction form of Theorem ... |
| exlimi 2210 | Inference associated with ... |
| exlimd 2211 | Deduction form of Theorem ... |
| exlimimdd 2212 | Existential elimination ru... |
| exlimdd 2213 | Existential elimination ru... |
| nexd 2214 | Deduction for generalizati... |
| albid 2215 | Formula-building rule for ... |
| exbid 2216 | Formula-building rule for ... |
| nfbidf 2217 | An equality theorem for ef... |
| 19.16 2218 | Theorem 19.16 of [Margaris... |
| 19.17 2219 | Theorem 19.17 of [Margaris... |
| 19.27 2220 | Theorem 19.27 of [Margaris... |
| 19.28 2221 | Theorem 19.28 of [Margaris... |
| 19.19 2222 | Theorem 19.19 of [Margaris... |
| 19.36 2223 | Theorem 19.36 of [Margaris... |
| 19.36i 2224 | Inference associated with ... |
| 19.37 2225 | Theorem 19.37 of [Margaris... |
| 19.32 2226 | Theorem 19.32 of [Margaris... |
| 19.31 2227 | Theorem 19.31 of [Margaris... |
| 19.41 2228 | Theorem 19.41 of [Margaris... |
| 19.42 2229 | Theorem 19.42 of [Margaris... |
| 19.44 2230 | Theorem 19.44 of [Margaris... |
| 19.45 2231 | Theorem 19.45 of [Margaris... |
| spimfv 2232 | Specialization, using impl... |
| chvarfv 2233 | Implicit substitution of `... |
| cbv3v2 2234 | Version of ~ cbv3 with two... |
| sbalex 2235 | Equivalence of two ways to... |
| sb4av 2236 | Version of ~ sb4a with a d... |
| sbimd 2237 | Deduction substituting bot... |
| sbbid 2238 | Deduction substituting bot... |
| 2sbbid 2239 | Deduction doubly substitut... |
| sbequ1 2240 | An equality theorem for su... |
| sbequ2 2241 | An equality theorem for su... |
| stdpc7 2242 | One of the two equality ax... |
| sbequ12 2243 | An equality theorem for su... |
| sbequ12r 2244 | An equality theorem for su... |
| sbelx 2245 | Elimination of substitutio... |
| sbequ12a 2246 | An equality theorem for su... |
| sbid 2247 | An identity theorem for su... |
| sbcov 2248 | A composition law for subs... |
| sb6a 2249 | Equivalence for substituti... |
| sbid2vw 2250 | Reverting substitution yie... |
| axc16g 2251 | Generalization of ~ axc16 ... |
| axc16 2252 | Proof of older axiom ~ ax-... |
| axc16gb 2253 | Biconditional strengthenin... |
| axc16nf 2254 | If ~ dtru is false, then t... |
| axc11v 2255 | Version of ~ axc11 with a ... |
| axc11rv 2256 | Version of ~ axc11r with a... |
| drsb2 2257 | Formula-building lemma for... |
| equsalv 2258 | An equivalence related to ... |
| equsexv 2259 | An equivalence related to ... |
| equsexvOLD 2260 | Obsolete version of ~ equs... |
| sbft 2261 | Substitution has no effect... |
| sbf 2262 | Substitution for a variabl... |
| sbf2 2263 | Substitution has no effect... |
| sbh 2264 | Substitution for a variabl... |
| hbs1 2265 | The setvar ` x ` is not fr... |
| nfs1f 2266 | If ` x ` is not free in ` ... |
| sb5 2267 | Alternate definition of su... |
| sb5OLD 2268 | Obsolete version of ~ sb5 ... |
| sb56OLD 2269 | Obsolete version of ~ sbal... |
| equs5av 2270 | A property related to subs... |
| 2sb5 2271 | Equivalence for double sub... |
| sbco4lem 2272 | Lemma for ~ sbco4 . It re... |
| sbco4lemOLD 2273 | Obsolete version of ~ sbco... |
| sbco4 2274 | Two ways of exchanging two... |
| dfsb7 2275 | An alternate definition of... |
| sbn 2276 | Negation inside and outsid... |
| sbex 2277 | Move existential quantifie... |
| nf5 2278 | Alternate definition of ~ ... |
| nf6 2279 | An alternate definition of... |
| nf5d 2280 | Deduce that ` x ` is not f... |
| nf5di 2281 | Since the converse holds b... |
| 19.9h 2282 | A wff may be existentially... |
| 19.21h 2283 | Theorem 19.21 of [Margaris... |
| 19.23h 2284 | Theorem 19.23 of [Margaris... |
| exlimih 2285 | Inference associated with ... |
| exlimdh 2286 | Deduction form of Theorem ... |
| equsalhw 2287 | Version of ~ equsalh with ... |
| equsexhv 2288 | An equivalence related to ... |
| hba1 2289 | The setvar ` x ` is not fr... |
| hbnt 2290 | Closed theorem version of ... |
| hbn 2291 | If ` x ` is not free in ` ... |
| hbnd 2292 | Deduction form of bound-va... |
| hbim1 2293 | A closed form of ~ hbim . ... |
| hbimd 2294 | Deduction form of bound-va... |
| hbim 2295 | If ` x ` is not free in ` ... |
| hban 2296 | If ` x ` is not free in ` ... |
| hb3an 2297 | If ` x ` is not free in ` ... |
| sbi2 2298 | Introduction of implicatio... |
| sbim 2299 | Implication inside and out... |
| sbrim 2300 | Substitution in an implica... |
| sbrimOLD 2301 | Obsolete version of ~ sbri... |
| sblim 2302 | Substitution in an implica... |
| sbor 2303 | Disjunction inside and out... |
| sbbi 2304 | Equivalence inside and out... |
| sblbis 2305 | Introduce left bicondition... |
| sbrbis 2306 | Introduce right biconditio... |
| sbrbif 2307 | Introduce right biconditio... |
| sbiev 2308 | Conversion of implicit sub... |
| sbiedw 2309 | Conversion of implicit sub... |
| axc7 2310 | Show that the original axi... |
| axc7e 2311 | Abbreviated version of ~ a... |
| modal-b 2312 | The analogue in our predic... |
| 19.9ht 2313 | A closed version of ~ 19.9... |
| axc4 2314 | Show that the original axi... |
| axc4i 2315 | Inference version of ~ axc... |
| nfal 2316 | If ` x ` is not free in ` ... |
| nfex 2317 | If ` x ` is not free in ` ... |
| hbex 2318 | If ` x ` is not free in ` ... |
| nfnf 2319 | If ` x ` is not free in ` ... |
| 19.12 2320 | Theorem 19.12 of [Margaris... |
| nfald 2321 | Deduction form of ~ nfal .... |
| nfexd 2322 | If ` x ` is not free in ` ... |
| nfsbv 2323 | If ` z ` is not free in ` ... |
| nfsbvOLD 2324 | Obsolete version of ~ nfsb... |
| hbsbwOLD 2325 | Obsolete version of ~ hbsb... |
| sbco2v 2326 | A composition law for subs... |
| aaan 2327 | Distribute universal quant... |
| aaanOLD 2328 | Obsolete version of ~ aaan... |
| eeor 2329 | Distribute existential qua... |
| eeorOLD 2330 | Obsolete version of ~ eeor... |
| cbv3v 2331 | Rule used to change bound ... |
| cbv1v 2332 | Rule used to change bound ... |
| cbv2w 2333 | Rule used to change bound ... |
| cbvaldw 2334 | Deduction used to change b... |
| cbvexdw 2335 | Deduction used to change b... |
| cbv3hv 2336 | Rule used to change bound ... |
| cbvalv1 2337 | Rule used to change bound ... |
| cbvexv1 2338 | Rule used to change bound ... |
| cbval2v 2339 | Rule used to change bound ... |
| cbvex2v 2340 | Rule used to change bound ... |
| dvelimhw 2341 | Proof of ~ dvelimh without... |
| pm11.53 2342 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2343 | Special case of ~ 19.12 wh... |
| eean 2344 | Distribute existential qua... |
| eeanv 2345 | Distribute a pair of exist... |
| eeeanv 2346 | Distribute three existenti... |
| ee4anv 2347 | Distribute two pairs of ex... |
| sb8v 2348 | Substitution of variable i... |
| sb8f 2349 | Substitution of variable i... |
| sb8fOLD 2350 | Obsolete version of ~ sb8f... |
| sb8ef 2351 | Substitution of variable i... |
| 2sb8ef 2352 | An equivalent expression f... |
| sb6rfv 2353 | Reversed substitution. Ve... |
| sbnf2 2354 | Two ways of expressing " `... |
| exsb 2355 | An equivalent expression f... |
| 2exsb 2356 | An equivalent expression f... |
| sbbib 2357 | Reversal of substitution. ... |
| sbbibvv 2358 | Reversal of substitution. ... |
| sbievg 2359 | Substitution applied to ex... |
| 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... |
| sb4bOLD 2474 | Obsolete version of ~ sb4b... |
| sb3b 2475 | Simplified definition of s... |
| sb3 2476 | One direction of a simplif... |
| sb1 2477 | One direction of a simplif... |
| sb2 2478 | One direction of a simplif... |
| sb3OLD 2479 | Obsolete version of ~ sb3 ... |
| sb1OLD 2480 | Obsolete version of ~ sb1 ... |
| sb3bOLD 2481 | Obsolete version of ~ sb3b... |
| sb4a 2482 | A version of one implicati... |
| dfsb1 2483 | Alternate definition of su... |
| hbsb2 2484 | Bound-variable hypothesis ... |
| nfsb2 2485 | Bound-variable hypothesis ... |
| hbsb2a 2486 | Special case of a bound-va... |
| sb4e 2487 | One direction of a simplif... |
| hbsb2e 2488 | Special case of a bound-va... |
| hbsb3 2489 | If ` y ` is not free in ` ... |
| nfs1 2490 | If ` y ` is not free in ` ... |
| axc16ALT 2491 | Alternate proof of ~ axc16... |
| axc16gALT 2492 | Alternate proof of ~ axc16... |
| equsb1 2493 | Substitution applied to an... |
| equsb2 2494 | Substitution applied to an... |
| dfsb2 2495 | An alternate definition of... |
| dfsb3 2496 | An alternate definition of... |
| drsb1 2497 | Formula-building lemma for... |
| sb2ae 2498 | In the case of two success... |
| sb6f 2499 | Equivalence for substituti... |
| sb5f 2500 | Equivalence for substituti... |
| nfsb4t 2501 | A variable not free in a p... |
| nfsb4 2502 | A variable not free in a p... |
| sbequ8 2503 | Elimination of equality fr... |
| sbie 2504 | Conversion of implicit sub... |
| sbied 2505 | Conversion of implicit sub... |
| sbiedv 2506 | Conversion of implicit sub... |
| 2sbiev 2507 | Conversion of double impli... |
| sbcom3 2508 | Substituting ` y ` for ` x... |
| sbco 2509 | A composition law for subs... |
| sbid2 2510 | An identity law for substi... |
| sbid2v 2511 | An identity law for substi... |
| sbidm 2512 | An idempotent law for subs... |
| sbco2 2513 | A composition law for subs... |
| sbco2d 2514 | A composition law for subs... |
| sbco3 2515 | A composition law for subs... |
| sbcom 2516 | A commutativity law for su... |
| sbtrt 2517 | Partially closed form of ~... |
| sbtr 2518 | A partial converse to ~ sb... |
| sb8 2519 | Substitution of variable i... |
| sb8e 2520 | Substitution of variable i... |
| sb9 2521 | Commutation of quantificat... |
| sb9i 2522 | Commutation of quantificat... |
| sbhb 2523 | Two ways of expressing " `... |
| nfsbd 2524 | Deduction version of ~ nfs... |
| nfsb 2525 | If ` z ` is not free in ` ... |
| nfsbOLD 2526 | Obsolete version of ~ nfsb... |
| hbsb 2527 | If ` z ` is not free in ` ... |
| sb7f 2528 | This version of ~ dfsb7 do... |
| sb7h 2529 | This version of ~ dfsb7 do... |
| sb10f 2530 | Hao Wang's identity axiom ... |
| sbal1 2531 | Check out ~ sbal for a ver... |
| sbal2 2532 | Move quantifier in and out... |
| 2sb8e 2533 | An equivalent expression f... |
| dfmoeu 2534 | An elementary proof of ~ m... |
| dfeumo 2535 | An elementary proof showin... |
| mojust 2537 | Soundness justification th... |
| nexmo 2539 | Nonexistence implies uniqu... |
| exmo 2540 | Any proposition holds for ... |
| moabs 2541 | Absorption of existence co... |
| moim 2542 | The at-most-one quantifier... |
| moimi 2543 | The at-most-one quantifier... |
| moimdv 2544 | The at-most-one quantifier... |
| mobi 2545 | Equivalence theorem for th... |
| mobii 2546 | Formula-building rule for ... |
| mobidv 2547 | Formula-building rule for ... |
| mobid 2548 | Formula-building rule for ... |
| moa1 2549 | If an implication holds fo... |
| moan 2550 | "At most one" is still the... |
| moani 2551 | "At most one" is still tru... |
| moor 2552 | "At most one" is still the... |
| mooran1 2553 | "At most one" imports disj... |
| mooran2 2554 | "At most one" exports disj... |
| nfmo1 2555 | Bound-variable hypothesis ... |
| nfmod2 2556 | Bound-variable hypothesis ... |
| nfmodv 2557 | Bound-variable hypothesis ... |
| nfmov 2558 | Bound-variable hypothesis ... |
| nfmod 2559 | Bound-variable hypothesis ... |
| nfmo 2560 | Bound-variable hypothesis ... |
| mof 2561 | Version of ~ df-mo with di... |
| mo3 2562 | Alternate definition of th... |
| mo 2563 | Equivalent definitions of ... |
| mo4 2564 | At-most-one quantifier exp... |
| mo4f 2565 | At-most-one quantifier exp... |
| eu3v 2568 | An alternate way to expres... |
| eujust 2569 | Soundness justification th... |
| eujustALT 2570 | Alternate proof of ~ eujus... |
| eu6lem 2571 | Lemma of ~ eu6im . A diss... |
| eu6 2572 | Alternate definition of th... |
| eu6im 2573 | One direction of ~ eu6 nee... |
| euf 2574 | Version of ~ eu6 with disj... |
| euex 2575 | Existential uniqueness imp... |
| eumo 2576 | Existential uniqueness imp... |
| eumoi 2577 | Uniqueness inferred from e... |
| exmoeub 2578 | Existence implies that uni... |
| exmoeu 2579 | Existence is equivalent to... |
| moeuex 2580 | Uniqueness implies that ex... |
| moeu 2581 | Uniqueness is equivalent t... |
| eubi 2582 | Equivalence theorem for th... |
| eubii 2583 | Introduce unique existenti... |
| eubidv 2584 | Formula-building rule for ... |
| eubid 2585 | Formula-building rule for ... |
| nfeu1 2586 | Bound-variable hypothesis ... |
| nfeu1ALT 2587 | Alternate proof of ~ nfeu1... |
| nfeud2 2588 | Bound-variable hypothesis ... |
| nfeudw 2589 | Bound-variable hypothesis ... |
| nfeud 2590 | Bound-variable hypothesis ... |
| nfeuw 2591 | Bound-variable hypothesis ... |
| nfeu 2592 | Bound-variable hypothesis ... |
| dfeu 2593 | Rederive ~ df-eu from the ... |
| dfmo 2594 | Rederive ~ df-mo from the ... |
| euequ 2595 | There exists a unique set ... |
| sb8eulem 2596 | Lemma. Factor out the com... |
| sb8euv 2597 | Variable substitution in u... |
| sb8eu 2598 | Variable substitution in u... |
| sb8mo 2599 | Variable substitution for ... |
| cbvmovw 2600 | Change bound variable. Us... |
| cbvmow 2601 | Rule used to change bound ... |
| cbvmowOLD 2602 | Obsolete version of ~ cbvm... |
| cbvmo 2603 | Rule used to change bound ... |
| cbveuvw 2604 | Change bound variable. Us... |
| cbveuw 2605 | Version of ~ cbveu with a ... |
| cbveuwOLD 2606 | Obsolete version of ~ cbve... |
| cbveu 2607 | Rule used to change bound ... |
| cbveuALT 2608 | Alternative proof of ~ cbv... |
| eu2 2609 | An alternate way of defini... |
| eu1 2610 | An alternate way to expres... |
| euor 2611 | Introduce a disjunct into ... |
| euorv 2612 | Introduce a disjunct into ... |
| euor2 2613 | Introduce or eliminate a d... |
| sbmo 2614 | Substitution into an at-mo... |
| eu4 2615 | Uniqueness using implicit ... |
| euimmo 2616 | Existential uniqueness imp... |
| euim 2617 | Add unique existential qua... |
| moanimlem 2618 | Factor out the common proo... |
| moanimv 2619 | Introduction of a conjunct... |
| moanim 2620 | Introduction of a conjunct... |
| euan 2621 | Introduction of a conjunct... |
| moanmo 2622 | Nested at-most-one quantif... |
| moaneu 2623 | Nested at-most-one and uni... |
| euanv 2624 | Introduction of a conjunct... |
| mopick 2625 | "At most one" picks a vari... |
| moexexlem 2626 | Factor out the proof skele... |
| 2moexv 2627 | Double quantification with... |
| moexexvw 2628 | "At most one" double quant... |
| 2moswapv 2629 | A condition allowing to sw... |
| 2euswapv 2630 | A condition allowing to sw... |
| 2euexv 2631 | Double quantification with... |
| 2exeuv 2632 | Double existential uniquen... |
| eupick 2633 | Existential uniqueness "pi... |
| eupicka 2634 | Version of ~ eupick with c... |
| eupickb 2635 | Existential uniqueness "pi... |
| eupickbi 2636 | Theorem *14.26 in [Whitehe... |
| mopick2 2637 | "At most one" can show the... |
| moexex 2638 | "At most one" double quant... |
| moexexv 2639 | "At most one" double quant... |
| 2moex 2640 | Double quantification with... |
| 2euex 2641 | Double quantification with... |
| 2eumo 2642 | Nested unique existential ... |
| 2eu2ex 2643 | Double existential uniquen... |
| 2moswap 2644 | A condition allowing to sw... |
| 2euswap 2645 | A condition allowing to sw... |
| 2exeu 2646 | Double existential uniquen... |
| 2mo2 2647 | Two ways of expressing "th... |
| 2mo 2648 | Two ways of expressing "th... |
| 2mos 2649 | Double "there exists at mo... |
| 2eu1 2650 | Double existential uniquen... |
| 2eu1v 2651 | Double existential uniquen... |
| 2eu2 2652 | Double existential uniquen... |
| 2eu3 2653 | Double existential uniquen... |
| 2eu4 2654 | This theorem provides us w... |
| 2eu5 2655 | An alternate definition of... |
| 2eu6 2656 | Two equivalent expressions... |
| 2eu7 2657 | Two equivalent expressions... |
| 2eu8 2658 | Two equivalent expressions... |
| euae 2659 | Two ways to express "exact... |
| exists1 2660 | Two ways to express "exact... |
| exists2 2661 | A condition implying that ... |
| barbara 2662 | "Barbara", one of the fund... |
| celarent 2663 | "Celarent", one of the syl... |
| darii 2664 | "Darii", one of the syllog... |
| dariiALT 2665 | Alternate proof of ~ darii... |
| ferio 2666 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2667 | Lemma for ~ barbari and th... |
| barbari 2668 | "Barbari", one of the syll... |
| barbariALT 2669 | Alternate proof of ~ barba... |
| celaront 2670 | "Celaront", one of the syl... |
| cesare 2671 | "Cesare", one of the syllo... |
| camestres 2672 | "Camestres", one of the sy... |
| festino 2673 | "Festino", one of the syll... |
| festinoALT 2674 | Alternate proof of ~ festi... |
| baroco 2675 | "Baroco", one of the syllo... |
| barocoALT 2676 | Alternate proof of ~ festi... |
| cesaro 2677 | "Cesaro", one of the syllo... |
| camestros 2678 | "Camestros", one of the sy... |
| datisi 2679 | "Datisi", one of the syllo... |
| disamis 2680 | "Disamis", one of the syll... |
| ferison 2681 | "Ferison", one of the syll... |
| bocardo 2682 | "Bocardo", one of the syll... |
| darapti 2683 | "Darapti", one of the syll... |
| daraptiALT 2684 | Alternate proof of ~ darap... |
| felapton 2685 | "Felapton", one of the syl... |
| calemes 2686 | "Calemes", one of the syll... |
| dimatis 2687 | "Dimatis", one of the syll... |
| fresison 2688 | "Fresison", one of the syl... |
| calemos 2689 | "Calemos", one of the syll... |
| fesapo 2690 | "Fesapo", one of the syllo... |
| bamalip 2691 | "Bamalip", one of the syll... |
| axia1 2692 | Left 'and' elimination (in... |
| axia2 2693 | Right 'and' elimination (i... |
| axia3 2694 | 'And' introduction (intuit... |
| axin1 2695 | 'Not' introduction (intuit... |
| axin2 2696 | 'Not' elimination (intuiti... |
| axio 2697 | Definition of 'or' (intuit... |
| axi4 2698 | Specialization (intuitioni... |
| axi5r 2699 | Converse of ~ axc4 (intuit... |
| axial 2700 | The setvar ` x ` is not fr... |
| axie1 2701 | The setvar ` x ` is not fr... |
| axie2 2702 | A key property of existent... |
| axi9 2703 | Axiom of existence (intuit... |
| axi10 2704 | Axiom of Quantifier Substi... |
| axi12 2705 | Axiom of Quantifier Introd... |
| axbnd 2706 | Axiom of Bundling (intuiti... |
| axexte 2708 | The axiom of extensionalit... |
| axextg 2709 | A generalization of the ax... |
| axextb 2710 | A bidirectional version of... |
| axextmo 2711 | There exists at most one s... |
| nulmo 2712 | There exists at most one e... |
| eleq1ab 2715 | Extension (in the sense of... |
| cleljustab 2716 | Extension of ~ cleljust fr... |
| abid 2717 | Simplification of class ab... |
| vexwt 2718 | A standard theorem of pred... |
| vexw 2719 | If ` ph ` is a theorem, th... |
| vextru 2720 | Every setvar is a member o... |
| nfsab1 2721 | Bound-variable hypothesis ... |
| hbab1 2722 | Bound-variable hypothesis ... |
| hbab1OLD 2723 | Obsolete version of ~ hbab... |
| hbab 2724 | Bound-variable hypothesis ... |
| hbabg 2725 | Bound-variable hypothesis ... |
| nfsab 2726 | Bound-variable hypothesis ... |
| nfsabg 2727 | Bound-variable hypothesis ... |
| dfcleq 2729 | The defining characterizat... |
| cvjust 2730 | Every set is a class. Pro... |
| ax9ALT 2731 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2732 | A weaker version of ~ eleq... |
| eqriv 2733 | Infer equality of classes ... |
| eqrdv 2734 | Deduce equality of classes... |
| eqrdav 2735 | Deduce equality of classes... |
| eqid 2736 | Law of identity (reflexivi... |
| eqidd 2737 | Class identity law with an... |
| eqeq1d 2738 | Deduction from equality to... |
| eqeq1dALT 2739 | Alternate proof of ~ eqeq1... |
| eqeq1 2740 | Equality implies equivalen... |
| eqeq1i 2741 | Inference from equality to... |
| eqcomd 2742 | Deduction from commutative... |
| eqcom 2743 | Commutative law for class ... |
| eqcoms 2744 | Inference applying commuta... |
| eqcomi 2745 | Inference from commutative... |
| neqcomd 2746 | Commute an inequality. (C... |
| eqeq2d 2747 | Deduction from equality to... |
| eqeq2 2748 | Equality implies equivalen... |
| eqeq2i 2749 | Inference from equality to... |
| eqeqan12d 2750 | A useful inference for sub... |
| eqeqan12rd 2751 | A useful inference for sub... |
| eqeq12d 2752 | A useful inference for sub... |
| eqeq12 2753 | Equality relationship amon... |
| eqeq12i 2754 | A useful inference for sub... |
| eqeq12OLD 2755 | Obsolete version of ~ eqeq... |
| eqeq12dOLD 2756 | Obsolete version of ~ eqeq... |
| eqeqan12dOLD 2757 | Obsolete version of ~ eqeq... |
| eqeqan12dALT 2758 | Alternate proof of ~ eqeqa... |
| eqtr 2759 | Transitive law for class e... |
| eqtr2 2760 | A transitive law for class... |
| eqtr2OLD 2761 | Obsolete version of eqtr2 ... |
| eqtr3 2762 | A transitive law for class... |
| eqtr3OLD 2763 | Obsolete version of ~ eqtr... |
| eqtri 2764 | An equality transitivity i... |
| eqtr2i 2765 | An equality transitivity i... |
| eqtr3i 2766 | An equality transitivity i... |
| eqtr4i 2767 | An equality transitivity i... |
| 3eqtri 2768 | An inference from three ch... |
| 3eqtrri 2769 | An inference from three ch... |
| 3eqtr2i 2770 | An inference from three ch... |
| 3eqtr2ri 2771 | An inference from three ch... |
| 3eqtr3i 2772 | An inference from three ch... |
| 3eqtr3ri 2773 | An inference from three ch... |
| 3eqtr4i 2774 | An inference from three ch... |
| 3eqtr4ri 2775 | An inference from three ch... |
| eqtrd 2776 | An equality transitivity d... |
| eqtr2d 2777 | An equality transitivity d... |
| eqtr3d 2778 | An equality transitivity e... |
| eqtr4d 2779 | An equality transitivity e... |
| 3eqtrd 2780 | A deduction from three cha... |
| 3eqtrrd 2781 | A deduction from three cha... |
| 3eqtr2d 2782 | A deduction from three cha... |
| 3eqtr2rd 2783 | A deduction from three cha... |
| 3eqtr3d 2784 | A deduction from three cha... |
| 3eqtr3rd 2785 | A deduction from three cha... |
| 3eqtr4d 2786 | A deduction from three cha... |
| 3eqtr4rd 2787 | A deduction from three cha... |
| eqtrid 2788 | An equality transitivity d... |
| eqtr2id 2789 | An equality transitivity d... |
| eqtr3id 2790 | An equality transitivity d... |
| eqtr3di 2791 | An equality transitivity d... |
| eqtrdi 2792 | An equality transitivity d... |
| eqtr2di 2793 | An equality transitivity d... |
| eqtr4di 2794 | An equality transitivity d... |
| eqtr4id 2795 | An equality transitivity d... |
| sylan9eq 2796 | An equality transitivity d... |
| sylan9req 2797 | An equality transitivity d... |
| sylan9eqr 2798 | An equality transitivity d... |
| 3eqtr3g 2799 | A chained equality inferen... |
| 3eqtr3a 2800 | A chained equality inferen... |
| 3eqtr4g 2801 | A chained equality inferen... |
| 3eqtr4a 2802 | A chained equality inferen... |
| eq2tri 2803 | A compound transitive infe... |
| abbi1 2804 | Equivalent formulas yield ... |
| abbidv 2805 | Equivalent wff's yield equ... |
| abbii 2806 | Equivalent wff's yield equ... |
| abbid 2807 | Equivalent wff's yield equ... |
| abbi 2808 | Equivalent formulas define... |
| cbvabv 2809 | Rule used to change bound ... |
| cbvabw 2810 | Rule used to change bound ... |
| cbvabwOLD 2811 | Obsolete version of ~ cbva... |
| cbvab 2812 | Rule used to change bound ... |
| eqabw 2813 | Version of ~ eqab using im... |
| dfclel 2815 | Characterization of the el... |
| elex2 2816 | If a class contains anothe... |
| issetlem 2817 | Lemma for ~ elisset and ~ ... |
| elissetv 2818 | An element of a class exis... |
| elisset 2819 | An element of a class exis... |
| eleq1w 2820 | Weaker version of ~ eleq1 ... |
| eleq2w 2821 | Weaker version of ~ eleq2 ... |
| eleq1d 2822 | Deduction from equality to... |
| eleq2d 2823 | Deduction from equality to... |
| eleq2dALT 2824 | Alternate proof of ~ eleq2... |
| eleq1 2825 | Equality implies equivalen... |
| eleq2 2826 | Equality implies equivalen... |
| eleq12 2827 | Equality implies equivalen... |
| eleq1i 2828 | Inference from equality to... |
| eleq2i 2829 | Inference from equality to... |
| eleq12i 2830 | Inference from equality to... |
| eqneltri 2831 | If a class is not an eleme... |
| eleq12d 2832 | Deduction from equality to... |
| eleq1a 2833 | A transitive-type law rela... |
| eqeltri 2834 | Substitution of equal clas... |
| eqeltrri 2835 | Substitution of equal clas... |
| eleqtri 2836 | Substitution of equal clas... |
| eleqtrri 2837 | Substitution of equal clas... |
| eqeltrd 2838 | Substitution of equal clas... |
| eqeltrrd 2839 | Deduction that substitutes... |
| eleqtrd 2840 | Deduction that substitutes... |
| eleqtrrd 2841 | Deduction that substitutes... |
| eqeltrid 2842 | A membership and equality ... |
| eqeltrrid 2843 | A membership and equality ... |
| eleqtrid 2844 | A membership and equality ... |
| eleqtrrid 2845 | A membership and equality ... |
| eqeltrdi 2846 | A membership and equality ... |
| eqeltrrdi 2847 | A membership and equality ... |
| eleqtrdi 2848 | A membership and equality ... |
| eleqtrrdi 2849 | A membership and equality ... |
| 3eltr3i 2850 | Substitution of equal clas... |
| 3eltr4i 2851 | Substitution of equal clas... |
| 3eltr3d 2852 | Substitution of equal clas... |
| 3eltr4d 2853 | Substitution of equal clas... |
| 3eltr3g 2854 | Substitution of equal clas... |
| 3eltr4g 2855 | Substitution of equal clas... |
| eleq2s 2856 | Substitution of equal clas... |
| eqneltrd 2857 | If a class is not an eleme... |
| eqneltrrd 2858 | If a class is not an eleme... |
| neleqtrd 2859 | If a class is not an eleme... |
| neleqtrrd 2860 | If a class is not an eleme... |
| nelneq 2861 | A way of showing two class... |
| nelneq2 2862 | A way of showing two class... |
| eqsb1 2863 | Substitution for the left-... |
| clelsb1 2864 | Substitution for the first... |
| clelsb2 2865 | Substitution for the secon... |
| clelsb2OLD 2866 | Obsolete version of ~ clel... |
| cleqh 2867 | Establish equality between... |
| hbxfreq 2868 | A utility lemma to transfe... |
| hblem 2869 | Change the free variable o... |
| hblemg 2870 | Change the free variable o... |
| abbi2dv 2871 | Deduction from a wff to a ... |
| abbi1dv 2872 | Deduction from a wff to a ... |
| abbi2i 2873 | Equality of a class variab... |
| abid1 2874 | Every class is equal to a ... |
| abid2 2875 | A simplification of class ... |
| eqabr 2876 | One direction of ~ eqab is... |
| eqab 2877 | Equality of a class variab... |
| eqabOLD 2878 | Obsolete version of ~ eqab... |
| eqabc 2879 | Equality of a class variab... |
| eqabd 2880 | Equality of a class variab... |
| eqabi 2881 | Equality of a class variab... |
| eqabci 2882 | Equality of a class variab... |
| clelab 2883 | Membership of a class vari... |
| clelabOLD 2884 | Obsolete version of ~ clel... |
| clabel 2885 | Membership of a class abst... |
| sbab 2886 | The right-hand side of the... |
| nfcjust 2888 | Justification theorem for ... |
| nfci 2890 | Deduce that a class ` A ` ... |
| nfcii 2891 | Deduce that a class ` A ` ... |
| nfcr 2892 | Consequence of the not-fre... |
| nfcrALT 2893 | Alternate version of ~ nfc... |
| nfcri 2894 | Consequence of the not-fre... |
| nfcd 2895 | Deduce that a class ` A ` ... |
| nfcrd 2896 | Consequence of the not-fre... |
| nfcriOLD 2897 | Obsolete version of ~ nfcr... |
| nfcriOLDOLD 2898 | Obsolete version of ~ nfcr... |
| nfcrii 2899 | Consequence of the not-fre... |
| nfcriiOLD 2900 | Obsolete version of ~ nfcr... |
| nfcriOLDOLDOLD 2901 | Obsolete version of ~ nfcr... |
| nfceqdf 2902 | An equality theorem for ef... |
| nfceqdfOLD 2903 | Obsolete version of ~ nfce... |
| nfceqi 2904 | Equality theorem for class... |
| nfcxfr 2905 | A utility lemma to transfe... |
| nfcxfrd 2906 | A utility lemma to transfe... |
| nfcv 2907 | If ` x ` is disjoint from ... |
| nfcvd 2908 | If ` x ` is disjoint from ... |
| nfab1 2909 | Bound-variable hypothesis ... |
| nfnfc1 2910 | The setvar ` x ` is bound ... |
| clelsb1fw 2911 | Substitution for the first... |
| clelsb1f 2912 | Substitution for the first... |
| nfab 2913 | Bound-variable hypothesis ... |
| nfabg 2914 | Bound-variable hypothesis ... |
| nfaba1 2915 | Bound-variable hypothesis ... |
| nfaba1g 2916 | Bound-variable hypothesis ... |
| nfeqd 2917 | Hypothesis builder for equ... |
| nfeld 2918 | Hypothesis builder for ele... |
| nfnfc 2919 | Hypothesis builder for ` F... |
| nfeq 2920 | Hypothesis builder for equ... |
| nfel 2921 | Hypothesis builder for ele... |
| nfeq1 2922 | Hypothesis builder for equ... |
| nfel1 2923 | Hypothesis builder for ele... |
| nfeq2 2924 | Hypothesis builder for equ... |
| nfel2 2925 | Hypothesis builder for ele... |
| drnfc1 2926 | Formula-building lemma for... |
| drnfc1OLD 2927 | Obsolete version of ~ drnf... |
| drnfc2 2928 | Formula-building lemma for... |
| drnfc2OLD 2929 | Obsolete version of ~ drnf... |
| nfabdw 2930 | Bound-variable hypothesis ... |
| nfabdwOLD 2931 | Obsolete version of ~ nfab... |
| nfabd 2932 | Bound-variable hypothesis ... |
| nfabd2 2933 | Bound-variable hypothesis ... |
| dvelimdc 2934 | Deduction form of ~ dvelim... |
| dvelimc 2935 | Version of ~ dvelim for cl... |
| nfcvf 2936 | If ` x ` and ` y ` are dis... |
| nfcvf2 2937 | If ` x ` and ` y ` are dis... |
| cleqf 2938 | Establish equality between... |
| abid2f 2939 | A simplification of class ... |
| eqabf 2940 | Equality of a class variab... |
| sbabel 2941 | Theorem to move a substitu... |
| sbabelOLD 2942 | Obsolete version of ~ sbab... |
| neii 2945 | Inference associated with ... |
| neir 2946 | Inference associated with ... |
| nne 2947 | Negation of inequality. (... |
| neneqd 2948 | Deduction eliminating ineq... |
| neneq 2949 | From inequality to non-equ... |
| neqned 2950 | If it is not the case that... |
| neqne 2951 | From non-equality to inequ... |
| neirr 2952 | No class is unequal to its... |
| exmidne 2953 | Excluded middle with equal... |
| eqneqall 2954 | A contradiction concerning... |
| nonconne 2955 | Law of noncontradiction wi... |
| necon3ad 2956 | Contrapositive law deducti... |
| necon3bd 2957 | Contrapositive law deducti... |
| necon2ad 2958 | Contrapositive inference f... |
| necon2bd 2959 | Contrapositive inference f... |
| necon1ad 2960 | Contrapositive deduction f... |
| necon1bd 2961 | Contrapositive deduction f... |
| necon4ad 2962 | Contrapositive inference f... |
| necon4bd 2963 | Contrapositive inference f... |
| necon3d 2964 | Contrapositive law deducti... |
| necon1d 2965 | Contrapositive law deducti... |
| necon2d 2966 | Contrapositive inference f... |
| necon4d 2967 | Contrapositive inference f... |
| necon3ai 2968 | Contrapositive inference f... |
| necon3aiOLD 2969 | Obsolete version of ~ neco... |
| necon3bi 2970 | Contrapositive inference f... |
| necon1ai 2971 | Contrapositive inference f... |
| necon1bi 2972 | Contrapositive inference f... |
| necon2ai 2973 | Contrapositive inference f... |
| necon2bi 2974 | Contrapositive inference f... |
| necon4ai 2975 | Contrapositive inference f... |
| necon3i 2976 | Contrapositive inference f... |
| necon1i 2977 | Contrapositive inference f... |
| necon2i 2978 | Contrapositive inference f... |
| necon4i 2979 | Contrapositive inference f... |
| necon3abid 2980 | Deduction from equality to... |
| necon3bbid 2981 | Deduction from equality to... |
| necon1abid 2982 | Contrapositive deduction f... |
| necon1bbid 2983 | Contrapositive inference f... |
| necon4abid 2984 | Contrapositive law deducti... |
| necon4bbid 2985 | Contrapositive law deducti... |
| necon2abid 2986 | Contrapositive deduction f... |
| necon2bbid 2987 | Contrapositive deduction f... |
| necon3bid 2988 | Deduction from equality to... |
| necon4bid 2989 | Contrapositive law deducti... |
| necon3abii 2990 | Deduction from equality to... |
| necon3bbii 2991 | Deduction from equality to... |
| necon1abii 2992 | Contrapositive inference f... |
| necon1bbii 2993 | Contrapositive inference f... |
| necon2abii 2994 | Contrapositive inference f... |
| necon2bbii 2995 | Contrapositive inference f... |
| necon3bii 2996 | Inference from equality to... |
| necom 2997 | Commutation of inequality.... |
| necomi 2998 | Inference from commutative... |
| necomd 2999 | Deduction from commutative... |
| nesym 3000 | Characterization of inequa... |
| nesymi 3001 | Inference associated with ... |
| nesymir 3002 | Inference associated with ... |
| neeq1d 3003 | Deduction for inequality. ... |
| neeq2d 3004 | Deduction for inequality. ... |
| neeq12d 3005 | Deduction for inequality. ... |
| neeq1 3006 | Equality theorem for inequ... |
| neeq2 3007 | Equality theorem for inequ... |
| neeq1i 3008 | Inference for inequality. ... |
| neeq2i 3009 | Inference for inequality. ... |
| neeq12i 3010 | Inference for inequality. ... |
| eqnetrd 3011 | Substitution of equal clas... |
| eqnetrrd 3012 | Substitution of equal clas... |
| neeqtrd 3013 | Substitution of equal clas... |
| eqnetri 3014 | Substitution of equal clas... |
| eqnetrri 3015 | Substitution of equal clas... |
| neeqtri 3016 | Substitution of equal clas... |
| neeqtrri 3017 | Substitution of equal clas... |
| neeqtrrd 3018 | Substitution of equal clas... |
| eqnetrrid 3019 | A chained equality inferen... |
| 3netr3d 3020 | Substitution of equality i... |
| 3netr4d 3021 | Substitution of equality i... |
| 3netr3g 3022 | Substitution of equality i... |
| 3netr4g 3023 | Substitution of equality i... |
| nebi 3024 | Contraposition law for ine... |
| pm13.18 3025 | Theorem *13.18 in [Whitehe... |
| pm13.181 3026 | Theorem *13.181 in [Whiteh... |
| pm13.181OLD 3027 | Obsolete version of ~ pm13... |
| pm2.61ine 3028 | Inference eliminating an i... |
| pm2.21ddne 3029 | A contradiction implies an... |
| pm2.61ne 3030 | Deduction eliminating an i... |
| pm2.61dne 3031 | Deduction eliminating an i... |
| pm2.61dane 3032 | Deduction eliminating an i... |
| pm2.61da2ne 3033 | Deduction eliminating two ... |
| pm2.61da3ne 3034 | Deduction eliminating thre... |
| pm2.61iine 3035 | Equality version of ~ pm2.... |
| neor 3036 | Logical OR with an equalit... |
| neanior 3037 | A De Morgan's law for ineq... |
| ne3anior 3038 | A De Morgan's law for ineq... |
| neorian 3039 | A De Morgan's law for ineq... |
| nemtbir 3040 | An inference from an inequ... |
| nelne1 3041 | Two classes are different ... |
| nelne2 3042 | Two classes are different ... |
| nelelne 3043 | Two classes are different ... |
| neneor 3044 | If two classes are differe... |
| nfne 3045 | Bound-variable hypothesis ... |
| nfned 3046 | Bound-variable hypothesis ... |
| nabbi 3047 | Not equivalent wff's corre... |
| mteqand 3048 | A modus tollens deduction ... |
| neli 3051 | Inference associated with ... |
| nelir 3052 | Inference associated with ... |
| neleq12d 3053 | Equality theorem for negat... |
| neleq1 3054 | Equality theorem for negat... |
| neleq2 3055 | Equality theorem for negat... |
| nfnel 3056 | Bound-variable hypothesis ... |
| nfneld 3057 | Bound-variable hypothesis ... |
| nnel 3058 | Negation of negated member... |
| elnelne1 3059 | Two classes are different ... |
| elnelne2 3060 | Two classes are different ... |
| nelcon3d 3061 | Contrapositive law deducti... |
| elnelall 3062 | A contradiction concerning... |
| pm2.61danel 3063 | Deduction eliminating an e... |
| rgen 3066 | Generalization rule for re... |
| ralel 3067 | All elements of a class ar... |
| rgenw 3068 | Generalization rule for re... |
| rgen2w 3069 | Generalization rule for re... |
| mprg 3070 | Modus ponens combined with... |
| mprgbir 3071 | Modus ponens on biconditio... |
| raln 3072 | Restricted universally qua... |
| ralnex 3075 | Relationship between restr... |
| dfrex2 3076 | Relationship between restr... |
| nrex 3077 | Inference adding restricte... |
| alral 3078 | Universal quantification i... |
| rexex 3079 | Restricted existence impli... |
| rextru 3080 | Two ways of expressing tha... |
| ralimi2 3081 | Inference quantifying both... |
| reximi2 3082 | Inference quantifying both... |
| ralimia 3083 | Inference quantifying both... |
| reximia 3084 | Inference quantifying both... |
| ralimiaa 3085 | Inference quantifying both... |
| ralimi 3086 | Inference quantifying both... |
| reximi 3087 | Inference quantifying both... |
| ral2imi 3088 | Inference quantifying ante... |
| ralim 3089 | Distribution of restricted... |
| rexim 3090 | Theorem 19.22 of [Margaris... |
| reximiaOLD 3091 | Obsolete version of ~ rexi... |
| ralbii2 3092 | Inference adding different... |
| rexbii2 3093 | Inference adding different... |
| ralbiia 3094 | Inference adding restricte... |
| rexbiia 3095 | Inference adding restricte... |
| ralbii 3096 | Inference adding restricte... |
| rexbii 3097 | Inference adding restricte... |
| ralanid 3098 | Cancellation law for restr... |
| rexanid 3099 | Cancellation law for restr... |
| ralcom3 3100 | A commutation law for rest... |
| ralcom3OLD 3101 | Obsolete version of ~ ralc... |
| dfral2 3102 | Relationship between restr... |
| rexnal 3103 | Relationship between restr... |
| ralinexa 3104 | A transformation of restri... |
| rexanali 3105 | A transformation of restri... |
| ralbi 3106 | Distribute a restricted un... |
| rexbi 3107 | Distribute restricted quan... |
| rexbiOLD 3108 | Obsolete version of ~ rexb... |
| ralrexbid 3109 | Formula-building rule for ... |
| ralrexbidOLD 3110 | Obsolete version of ~ ralr... |
| r19.35 3111 | Restricted quantifier vers... |
| r19.35OLD 3112 | Obsolete version of ~ 19.3... |
| r19.26m 3113 | Version of ~ 19.26 and ~ r... |
| r19.26 3114 | Restricted quantifier vers... |
| r19.26-3 3115 | Version of ~ r19.26 with t... |
| ralbiim 3116 | Split a biconditional and ... |
| r19.29 3117 | Restricted quantifier vers... |
| r19.29OLD 3118 | Obsolete version of ~ r19.... |
| r19.29r 3119 | Restricted quantifier vers... |
| r19.29rOLD 3120 | Obsolete version of ~ r19.... |
| r19.29imd 3121 | Theorem 19.29 of [Margaris... |
| r19.40 3122 | Restricted quantifier vers... |
| r19.30 3123 | Restricted quantifier vers... |
| r19.30OLD 3124 | Obsolete version of ~ 19.3... |
| r19.43 3125 | Restricted quantifier vers... |
| 2ralimi 3126 | Inference quantifying both... |
| 2ralbii 3127 | Inference adding two restr... |
| 2rexbii 3128 | Inference adding two restr... |
| 2ralbiim 3129 | Split a biconditional and ... |
| ralnex2 3130 | Relationship between two r... |
| ralnex3 3131 | Relationship between three... |
| rexnal2 3132 | Relationship between two r... |
| rexnal3 3133 | Relationship between three... |
| nrexralim 3134 | Negation of a complex pred... |
| r19.26-2 3135 | Restricted quantifier vers... |
| 2r19.29 3136 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3137 | Theorem 19.29 of [Margaris... |
| r19.29d2rOLD 3138 | Obsolete version of ~ r19.... |
| r2allem 3139 | Lemma factoring out common... |
| r2exlem 3140 | Lemma factoring out common... |
| hbralrimi 3141 | Inference from Theorem 19.... |
| ralrimiv 3142 | Inference from Theorem 19.... |
| ralrimiva 3143 | Inference from Theorem 19.... |
| rexlimiva 3144 | Inference from Theorem 19.... |
| rexlimiv 3145 | Inference from Theorem 19.... |
| nrexdv 3146 | Deduction adding restricte... |
| ralrimivw 3147 | Inference from Theorem 19.... |
| rexlimivw 3148 | Weaker version of ~ rexlim... |
| ralrimdv 3149 | Inference from Theorem 19.... |
| rexlimdv 3150 | Inference from Theorem 19.... |
| ralrimdva 3151 | Inference from Theorem 19.... |
| rexlimdva 3152 | Inference from Theorem 19.... |
| rexlimdvaa 3153 | Inference from Theorem 19.... |
| rexlimdva2 3154 | Inference from Theorem 19.... |
| r19.29an 3155 | A commonly used pattern in... |
| rexlimdv3a 3156 | Inference from Theorem 19.... |
| rexlimdvw 3157 | Inference from Theorem 19.... |
| rexlimddv 3158 | Restricted existential eli... |
| r19.29a 3159 | A commonly used pattern in... |
| ralimdv2 3160 | Inference quantifying both... |
| reximdv2 3161 | Deduction quantifying both... |
| reximdvai 3162 | Deduction quantifying both... |
| reximdvaiOLD 3163 | Obsolete version of ~ rexi... |
| ralimdva 3164 | Deduction quantifying both... |
| reximdva 3165 | Deduction quantifying both... |
| ralimdv 3166 | Deduction quantifying both... |
| reximdv 3167 | Deduction from Theorem 19.... |
| reximddv 3168 | Deduction from Theorem 19.... |
| reximssdv 3169 | Derivation of a restricted... |
| ralbidv2 3170 | Formula-building rule for ... |
| rexbidv2 3171 | Formula-building rule for ... |
| ralbidva 3172 | Formula-building rule for ... |
| rexbidva 3173 | Formula-building rule for ... |
| ralbidv 3174 | Formula-building rule for ... |
| rexbidv 3175 | Formula-building rule for ... |
| r19.21v 3176 | Restricted quantifier vers... |
| r19.21vOLD 3177 | Obsolete version of ~ r19.... |
| r19.37v 3178 | Restricted quantifier vers... |
| r19.23v 3179 | Restricted quantifier vers... |
| r19.36v 3180 | Restricted quantifier vers... |
| rexlimivOLD 3181 | Obsolete version of ~ rexl... |
| rexlimivaOLD 3182 | Obsolete version of ~ rexl... |
| rexlimivwOLD 3183 | Obsolete version of ~ rexl... |
| r19.27v 3184 | Restricted quantitifer ver... |
| r19.41v 3185 | Restricted quantifier vers... |
| r19.28v 3186 | Restricted quantifier vers... |
| r19.42v 3187 | Restricted quantifier vers... |
| r19.32v 3188 | Restricted quantifier vers... |
| r19.45v 3189 | Restricted quantifier vers... |
| r19.44v 3190 | One direction of a restric... |
| r2al 3191 | Double restricted universa... |
| r2ex 3192 | Double restricted existent... |
| r3al 3193 | Triple restricted universa... |
| rgen2 3194 | Generalization rule for re... |
| ralrimivv 3195 | Inference from Theorem 19.... |
| rexlimivv 3196 | Inference from Theorem 19.... |
| ralrimivva 3197 | Inference from Theorem 19.... |
| ralrimdvv 3198 | Inference from Theorem 19.... |
| rgen3 3199 | Generalization rule for re... |
| ralrimivvva 3200 | Inference from Theorem 19.... |
| ralimdvva 3201 | Deduction doubly quantifyi... |
| reximdvva 3202 | Deduction doubly quantifyi... |
| ralrimdvva 3203 | Inference from Theorem 19.... |
| rexlimdvv 3204 | Inference from Theorem 19.... |
| rexlimdvva 3205 | Inference from Theorem 19.... |
| reximddv2 3206 | Double deduction from Theo... |
| r19.29vva 3207 | A commonly used pattern ba... |
| r19.29vvaOLD 3208 | Obsolete version of ~ r19.... |
| 2rexbiia 3209 | Inference adding two restr... |
| 2ralbidva 3210 | Formula-building rule for ... |
| 2rexbidva 3211 | Formula-building rule for ... |
| 2ralbidv 3212 | Formula-building rule for ... |
| 2rexbidv 3213 | Formula-building rule for ... |
| rexralbidv 3214 | Formula-building rule for ... |
| r19.41vv 3215 | Version of ~ r19.41v with ... |
| reeanlem 3216 | Lemma factoring out common... |
| reeanv 3217 | Rearrange restricted exist... |
| 3reeanv 3218 | Rearrange three restricted... |
| 2ralor 3219 | Distribute restricted univ... |
| 2ralorOLD 3220 | Obsolete version of ~ 2ral... |
| risset 3221 | Two ways to say " ` A ` be... |
| nelb 3222 | A definition of ` -. A e. ... |
| nelbOLD 3223 | Obsolete version of ~ nelb... |
| rspw 3224 | Restricted specialization.... |
| cbvralvw 3225 | Change the bound variable ... |
| cbvrexvw 3226 | Change the bound variable ... |
| cbvral2vw 3227 | Change bound variables of ... |
| cbvrex2vw 3228 | Change bound variables of ... |
| cbvral3vw 3229 | Change bound variables of ... |
| rsp 3230 | Restricted specialization.... |
| rspa 3231 | Restricted specialization.... |
| rspe 3232 | Restricted specialization.... |
| rspec 3233 | Specialization rule for re... |
| r19.21bi 3234 | Inference from Theorem 19.... |
| r19.21be 3235 | Inference from Theorem 19.... |
| r19.21t 3236 | Restricted quantifier vers... |
| r19.21 3237 | Restricted quantifier vers... |
| r19.23t 3238 | Closed theorem form of ~ r... |
| r19.23 3239 | Restricted quantifier vers... |
| ralrimi 3240 | Inference from Theorem 19.... |
| ralrimia 3241 | Inference from Theorem 19.... |
| rexlimi 3242 | Restricted quantifier vers... |
| ralimdaa 3243 | Deduction quantifying both... |
| reximdai 3244 | Deduction from Theorem 19.... |
| r19.37 3245 | Restricted quantifier vers... |
| r19.41 3246 | Restricted quantifier vers... |
| ralrimd 3247 | Inference from Theorem 19.... |
| rexlimd2 3248 | Version of ~ rexlimd with ... |
| rexlimd 3249 | Deduction form of ~ rexlim... |
| r19.29af2 3250 | A commonly used pattern ba... |
| r19.29af 3251 | A commonly used pattern ba... |
| reximd2a 3252 | Deduction quantifying both... |
| ralbida 3253 | Formula-building rule for ... |
| ralbidaOLD 3254 | Obsolete version of ~ ralb... |
| rexbida 3255 | Formula-building rule for ... |
| ralbid 3256 | Formula-building rule for ... |
| rexbid 3257 | Formula-building rule for ... |
| rexbidvALT 3258 | Alternate proof of ~ rexbi... |
| rexbidvaALT 3259 | Alternate proof of ~ rexbi... |
| rsp2 3260 | Restricted specialization,... |
| rsp2e 3261 | Restricted specialization.... |
| rspec2 3262 | Specialization rule for re... |
| rspec3 3263 | Specialization rule for re... |
| r2alf 3264 | Double restricted universa... |
| r2exf 3265 | Double restricted existent... |
| 2ralbida 3266 | Formula-building rule for ... |
| nfra1 3267 | The setvar ` x ` is not fr... |
| nfre1 3268 | The setvar ` x ` is not fr... |
| ralcom4 3269 | Commutation of restricted ... |
| ralcom4OLD 3270 | Obsolete version of ~ ralc... |
| rexcom4 3271 | Commutation of restricted ... |
| ralcom 3272 | Commutation of restricted ... |
| rexcom 3273 | Commutation of restricted ... |
| rexcomOLD 3274 | Obsolete version of ~ rexc... |
| rexcom4a 3275 | Specialized existential co... |
| ralrot3 3276 | Rotate three restricted un... |
| ralcom13 3277 | Swap first and third restr... |
| ralcom13OLD 3278 | Obsolete version of ~ ralc... |
| rexcom13 3279 | Swap first and third restr... |
| rexrot4 3280 | Rotate four restricted exi... |
| 2ex2rexrot 3281 | Rotate two existential qua... |
| nfra2w 3282 | Similar to Lemma 24 of [Mo... |
| nfra2wOLD 3283 | Obsolete version of ~ nfra... |
| hbra1 3284 | The setvar ` x ` is not fr... |
| ralcomf 3285 | Commutation of restricted ... |
| rexcomf 3286 | Commutation of restricted ... |
| cbvralfw 3287 | Rule used to change bound ... |
| cbvrexfw 3288 | Rule used to change bound ... |
| cbvralw 3289 | Rule used to change bound ... |
| cbvrexw 3290 | Rule used to change bound ... |
| hbral 3291 | Bound-variable hypothesis ... |
| nfraldw 3292 | Deduction version of ~ nfr... |
| nfrexdw 3293 | Deduction version of ~ nfr... |
| nfralw 3294 | Bound-variable hypothesis ... |
| nfralwOLD 3295 | Obsolete version of ~ nfra... |
| nfrexw 3296 | Bound-variable hypothesis ... |
| r19.12 3297 | Restricted quantifier vers... |
| r19.12OLD 3298 | Obsolete version of ~ 19.1... |
| reean 3299 | Rearrange restricted exist... |
| cbvralsvw 3300 | Change bound variable by u... |
| cbvrexsvw 3301 | Change bound variable by u... |
| nfraldwOLD 3302 | Obsolete version of ~ nfra... |
| nfra2wOLDOLD 3303 | Obsolete version of ~ nfra... |
| cbvralfwOLD 3304 | Obsolete version of ~ cbvr... |
| raleqbidvv 3305 | Version of ~ raleqbidv wit... |
| rexeqbidvv 3306 | Version of ~ rexeqbidv wit... |
| raleqbi1dv 3307 | Equality deduction for res... |
| rexeqbi1dv 3308 | Equality deduction for res... |
| raleq 3309 | Equality theorem for restr... |
| rexeq 3310 | Equality theorem for restr... |
| raleqi 3311 | Equality inference for res... |
| rexeqi 3312 | Equality inference for res... |
| raleqdv 3313 | Equality deduction for res... |
| rexeqdv 3314 | Equality deduction for res... |
| raleqbii 3315 | Equality deduction for res... |
| rexeqbii 3316 | Equality deduction for res... |
| raleleq 3317 | All elements of a class ar... |
| raleleqALT 3318 | Alternate proof of ~ ralel... |
| raleqbidv 3319 | Equality deduction for res... |
| rexeqbidv 3320 | Equality deduction for res... |
| raleqbidva 3321 | Equality deduction for res... |
| rexeqbidva 3322 | Equality deduction for res... |
| cbvraldva2 3323 | Rule used to change the bo... |
| cbvrexdva2 3324 | Rule used to change the bo... |
| cbvrexdva2OLD 3325 | Obsolete version of ~ cbvr... |
| cbvraldva 3326 | Rule used to change the bo... |
| cbvrexdva 3327 | Rule used to change the bo... |
| raleqf 3328 | Equality theorem for restr... |
| rexeqf 3329 | Equality theorem for restr... |
| raleqbid 3330 | Equality deduction for res... |
| rexeqbid 3331 | Equality deduction for res... |
| sbralie 3332 | Implicit to explicit subst... |
| cbvralf 3333 | Rule used to change bound ... |
| cbvrexf 3334 | Rule used to change bound ... |
| cbvral 3335 | Rule used to change bound ... |
| cbvrex 3336 | Rule used to change bound ... |
| cbvralv 3337 | Change the bound variable ... |
| cbvrexv 3338 | Change the bound variable ... |
| cbvralsv 3339 | Change bound variable by u... |
| cbvrexsv 3340 | Change bound variable by u... |
| cbvral2v 3341 | Change bound variables of ... |
| cbvrex2v 3342 | Change bound variables of ... |
| cbvral3v 3343 | Change bound variables of ... |
| rgen2a 3344 | Generalization rule for re... |
| nfrald 3345 | Deduction version of ~ nfr... |
| nfrexd 3346 | Deduction version of ~ nfr... |
| nfral 3347 | Bound-variable hypothesis ... |
| nfrex 3348 | Bound-variable hypothesis ... |
| nfra2 3349 | Similar to Lemma 24 of [Mo... |
| ralcom2 3350 | Commutation of restricted ... |
| reu5 3355 | Restricted uniqueness in t... |
| reurmo 3356 | Restricted existential uni... |
| reurex 3357 | Restricted unique existenc... |
| mormo 3358 | Unrestricted "at most one"... |
| rmobiia 3359 | Formula-building rule for ... |
| reubiia 3360 | Formula-building rule for ... |
| rmobii 3361 | Formula-building rule for ... |
| reubii 3362 | Formula-building rule for ... |
| rmoanid 3363 | Cancellation law for restr... |
| reuanid 3364 | Cancellation law for restr... |
| rmoanidOLD 3365 | Obsolete version of ~ rmoa... |
| reuanidOLD 3366 | Obsolete version of ~ reua... |
| 2reu2rex 3367 | Double restricted existent... |
| rmobidva 3368 | Formula-building rule for ... |
| reubidva 3369 | Formula-building rule for ... |
| rmobidv 3370 | Formula-building rule for ... |
| reubidv 3371 | Formula-building rule for ... |
| reueubd 3372 | Restricted existential uni... |
| rmo5 3373 | Restricted "at most one" i... |
| nrexrmo 3374 | Nonexistence implies restr... |
| moel 3375 | "At most one" element in a... |
| cbvrmovw 3376 | Change the bound variable ... |
| cbvreuvw 3377 | Change the bound variable ... |
| moelOLD 3378 | Obsolete version of ~ moel... |
| rmobida 3379 | Formula-building rule for ... |
| reubida 3380 | Formula-building rule for ... |
| rmobidvaOLD 3381 | Obsolete version of ~ rmob... |
| cbvrmow 3382 | Change the bound variable ... |
| cbvreuw 3383 | Change the bound variable ... |
| nfrmo1 3384 | The setvar ` x ` is not fr... |
| nfreu1 3385 | The setvar ` x ` is not fr... |
| nfrmow 3386 | Bound-variable hypothesis ... |
| nfreuw 3387 | Bound-variable hypothesis ... |
| cbvrmowOLD 3388 | Obsolete version of ~ cbvr... |
| cbvreuwOLD 3389 | Obsolete version of ~ cbvr... |
| cbvreuvwOLD 3390 | Obsolete version of ~ cbvr... |
| rmoeq1 3391 | Equality theorem for restr... |
| reueq1 3392 | Equality theorem for restr... |
| rmoeqd 3393 | Equality deduction for res... |
| reueqd 3394 | Equality deduction for res... |
| rmoeq1f 3395 | Equality theorem for restr... |
| reueq1f 3396 | Equality theorem for restr... |
| nfreuwOLD 3397 | Obsolete version of ~ nfre... |
| nfrmowOLD 3398 | Obsolete version of ~ nfrm... |
| cbvreu 3399 | Change the bound variable ... |
| cbvrmo 3400 | Change the bound variable ... |
| cbvrmov 3401 | Change the bound variable ... |
| cbvreuv 3402 | Change the bound variable ... |
| nfrmod 3403 | Deduction version of ~ nfr... |
| nfreud 3404 | Deduction version of ~ nfr... |
| nfrmo 3405 | Bound-variable hypothesis ... |
| nfreu 3406 | Bound-variable hypothesis ... |
| rabbidva2 3409 | Equivalent wff's yield equ... |
| rabbia2 3410 | Equivalent wff's yield equ... |
| rabbiia 3411 | Equivalent formulas yield ... |
| rabbiiaOLD 3412 | Obsolete version of ~ rabb... |
| rabbii 3413 | Equivalent wff's correspon... |
| rabbidva 3414 | Equivalent wff's yield equ... |
| rabbidv 3415 | Equivalent wff's yield equ... |
| rabswap 3416 | Swap with a membership rel... |
| cbvrabv 3417 | Rule to change the bound v... |
| rabeqcda 3418 | When ` ps ` is always true... |
| rabeqc 3419 | A restricted class abstrac... |
| rabeqi 3420 | Equality theorem for restr... |
| rabeq 3421 | Equality theorem for restr... |
| rabeqdv 3422 | Equality of restricted cla... |
| rabeqbidva 3423 | Equality of restricted cla... |
| rabeqbidv 3424 | Equality of restricted cla... |
| rabrabi 3425 | Abstract builder restricte... |
| nfrab1 3426 | The abstraction variable i... |
| rabid 3427 | An "identity" law of concr... |
| rabidim1 3428 | Membership in a restricted... |
| reqabi 3429 | Inference from equality of... |
| rabrab 3430 | Abstract builder restricte... |
| rabrabiOLD 3431 | Obsolete version of ~ rabr... |
| rabbi 3432 | Equivalent wff's correspon... |
| rabbida 3433 | Equivalent wff's yield equ... |
| rabbid 3434 | Version of ~ rabbidv with ... |
| rabid2f 3435 | An "identity" law for rest... |
| rabid2 3436 | An "identity" law for rest... |
| rabid2OLD 3437 | Obsolete version of ~ rabi... |
| rabeqf 3438 | Equality theorem for restr... |
| cbvrabw 3439 | Rule to change the bound v... |
| nfrabw 3440 | A variable not free in a w... |
| nfrabwOLD 3441 | Obsolete version of ~ nfra... |
| rabeqiOLD 3442 | Obsolete version of ~ rabe... |
| nfrab 3443 | A variable not free in a w... |
| cbvrab 3444 | Rule to change the bound v... |
| vjust 3446 | Justification theorem for ... |
| dfv2 3448 | Alternate definition of th... |
| vex 3449 | All setvar variables are s... |
| vexOLD 3450 | Obsolete version of ~ vex ... |
| elv 3451 | If a proposition is implie... |
| elvd 3452 | If a proposition is implie... |
| el2v 3453 | If a proposition is implie... |
| eqv 3454 | The universe contains ever... |
| eqvf 3455 | The universe contains ever... |
| abv 3456 | The class of sets verifyin... |
| abvALT 3457 | Alternate proof of ~ abv ,... |
| isset 3458 | Two ways to express that "... |
| issetf 3459 | A version of ~ isset that ... |
| isseti 3460 | A way to say " ` A ` is a ... |
| issetri 3461 | A way to say " ` A ` is a ... |
| eqvisset 3462 | A class equal to a variabl... |
| elex 3463 | If a class is a member of ... |
| elexi 3464 | If a class is a member of ... |
| elexd 3465 | If a class is a member of ... |
| elex2OLD 3466 | Obsolete version of ~ elex... |
| elex22 3467 | If two classes each contai... |
| prcnel 3468 | A proper class doesn't bel... |
| ralv 3469 | A universal quantifier res... |
| rexv 3470 | An existential quantifier ... |
| reuv 3471 | A unique existential quant... |
| rmov 3472 | An at-most-one quantifier ... |
| rabab 3473 | A class abstraction restri... |
| rexcom4b 3474 | Specialized existential co... |
| ceqsalt 3475 | Closed theorem version of ... |
| ceqsralt 3476 | Restricted quantifier vers... |
| ceqsalg 3477 | A representation of explic... |
| ceqsalgALT 3478 | Alternate proof of ~ ceqsa... |
| ceqsal 3479 | A representation of explic... |
| ceqsalALT 3480 | A representation of explic... |
| ceqsalv 3481 | A representation of explic... |
| ceqsalvOLD 3482 | Obsolete version of ~ ceqs... |
| ceqsralv 3483 | Restricted quantifier vers... |
| ceqsralvOLD 3484 | Obsolete version of ~ ceqs... |
| gencl 3485 | Implicit substitution for ... |
| 2gencl 3486 | Implicit substitution for ... |
| 3gencl 3487 | Implicit substitution for ... |
| cgsexg 3488 | Implicit substitution infe... |
| cgsex2g 3489 | Implicit substitution infe... |
| cgsex4g 3490 | An implicit substitution i... |
| cgsex4gOLD 3491 | Obsolete version of ~ cgse... |
| ceqsex 3492 | Elimination of an existent... |
| ceqsexOLD 3493 | Obsolete version of ~ ceqs... |
| ceqsexv 3494 | Elimination of an existent... |
| ceqsexvOLD 3495 | Obsolete version of ~ ceqs... |
| ceqsexvOLDOLD 3496 | Obsolete version of ~ ceqs... |
| ceqsexv2d 3497 | Elimination of an existent... |
| ceqsex2 3498 | Elimination of two existen... |
| ceqsex2v 3499 | Elimination of two existen... |
| ceqsex3v 3500 | Elimination of three exist... |
| ceqsex4v 3501 | Elimination of four existe... |
| ceqsex6v 3502 | Elimination of six existen... |
| ceqsex8v 3503 | Elimination of eight exist... |
| gencbvex 3504 | Change of bound variable u... |
| gencbvex2 3505 | Restatement of ~ gencbvex ... |
| gencbval 3506 | Change of bound variable u... |
| sbhypf 3507 | Introduce an explicit subs... |
| sbhypfOLD 3508 | Obsolete version of ~ sbhy... |
| vtoclgft 3509 | Closed theorem form of ~ v... |
| vtocldf 3510 | Implicit substitution of a... |
| vtocld 3511 | Implicit substitution of a... |
| vtocldOLD 3512 | Obsolete version of ~ vtoc... |
| vtocl2d 3513 | Implicit substitution of t... |
| vtocleg 3514 | Implicit substitution of a... |
| vtoclef 3515 | Implicit substitution of a... |
| vtoclf 3516 | Implicit substitution of a... |
| vtoclfOLD 3517 | Obsolete version of ~ vtoc... |
| vtocl 3518 | Implicit substitution of a... |
| vtoclALT 3519 | Alternate proof of ~ vtocl... |
| vtocl2 3520 | Implicit substitution of c... |
| vtocl3 3521 | Implicit substitution of c... |
| vtoclb 3522 | Implicit substitution of a... |
| vtoclgf 3523 | Implicit substitution of a... |
| vtoclg1f 3524 | Version of ~ vtoclgf with ... |
| vtoclg 3525 | Implicit substitution of a... |
| vtoclgOLD 3526 | Obsolete version of ~ vtoc... |
| vtoclgOLDOLD 3527 | Obsolete version of ~ vtoc... |
| vtoclbg 3528 | Implicit substitution of a... |
| vtocl2gf 3529 | Implicit substitution of a... |
| vtocl3gf 3530 | Implicit substitution of a... |
| vtocl2g 3531 | Implicit substitution of 2... |
| vtocl3g 3532 | Implicit substitution of a... |
| vtoclgaf 3533 | Implicit substitution of a... |
| vtoclga 3534 | Implicit substitution of a... |
| vtocl2ga 3535 | Implicit substitution of 2... |
| vtocl2gaf 3536 | Implicit substitution of 2... |
| vtocl3gaf 3537 | Implicit substitution of 3... |
| vtocl3ga 3538 | Implicit substitution of 3... |
| vtocl3gaOLD 3539 | Obsolete version of ~ vtoc... |
| vtocl4g 3540 | Implicit substitution of 4... |
| vtocl4ga 3541 | Implicit substitution of 4... |
| vtoclegft 3542 | Implicit substitution of a... |
| vtoclegftOLD 3543 | Obsolete version of ~ vtoc... |
| vtocle 3544 | Implicit substitution of a... |
| vtoclri 3545 | Implicit substitution of a... |
| spcimgft 3546 | A closed version of ~ spci... |
| spcgft 3547 | A closed version of ~ spcg... |
| spcimgf 3548 | Rule of specialization, us... |
| spcimegf 3549 | Existential specialization... |
| spcgf 3550 | Rule of specialization, us... |
| spcegf 3551 | Existential specialization... |
| spcimdv 3552 | Restricted specialization,... |
| spcdv 3553 | Rule of specialization, us... |
| spcimedv 3554 | Restricted existential spe... |
| spcgv 3555 | Rule of specialization, us... |
| spcegv 3556 | Existential specialization... |
| spcedv 3557 | Existential specialization... |
| spc2egv 3558 | Existential specialization... |
| spc2gv 3559 | Specialization with two qu... |
| spc2ed 3560 | Existential specialization... |
| spc2d 3561 | Specialization with 2 quan... |
| spc3egv 3562 | Existential specialization... |
| spc3gv 3563 | Specialization with three ... |
| spcv 3564 | Rule of specialization, us... |
| spcev 3565 | Existential specialization... |
| spc2ev 3566 | Existential specialization... |
| rspct 3567 | A closed version of ~ rspc... |
| rspcdf 3568 | Restricted specialization,... |
| rspc 3569 | Restricted specialization,... |
| rspce 3570 | Restricted existential spe... |
| rspcimdv 3571 | Restricted specialization,... |
| rspcimedv 3572 | Restricted existential spe... |
| rspcdv 3573 | Restricted specialization,... |
| rspcedv 3574 | Restricted existential spe... |
| rspcebdv 3575 | Restricted existential spe... |
| rspcdv2 3576 | Restricted specialization,... |
| rspcv 3577 | Restricted specialization,... |
| rspccv 3578 | Restricted specialization,... |
| rspcva 3579 | Restricted specialization,... |
| rspccva 3580 | Restricted specialization,... |
| rspcev 3581 | Restricted existential spe... |
| rspcdva 3582 | Restricted specialization,... |
| rspcedvd 3583 | Restricted existential spe... |
| rspcime 3584 | Prove a restricted existen... |
| rspceaimv 3585 | Restricted existential spe... |
| rspcedeq1vd 3586 | Restricted existential spe... |
| rspcedeq2vd 3587 | Restricted existential spe... |
| rspc2 3588 | Restricted specialization ... |
| rspc2gv 3589 | Restricted specialization ... |
| rspc2v 3590 | 2-variable restricted spec... |
| rspc2va 3591 | 2-variable restricted spec... |
| rspc2ev 3592 | 2-variable restricted exis... |
| rspc3v 3593 | 3-variable restricted spec... |
| rspc3ev 3594 | 3-variable restricted exis... |
| rspceeqv 3595 | Restricted existential spe... |
| ralxpxfr2d 3596 | Transfer a universal quant... |
| rexraleqim 3597 | Statement following from e... |
| eqvincg 3598 | A variable introduction la... |
| eqvinc 3599 | A variable introduction la... |
| eqvincf 3600 | A variable introduction la... |
| alexeqg 3601 | Two ways to express substi... |
| ceqex 3602 | Equality implies equivalen... |
| ceqsexg 3603 | A representation of explic... |
| ceqsexgv 3604 | Elimination of an existent... |
| ceqsrexv 3605 | Elimination of a restricte... |
| ceqsrexbv 3606 | Elimination of a restricte... |
| ceqsralbv 3607 | Elimination of a restricte... |
| ceqsrex2v 3608 | Elimination of a restricte... |
| clel2g 3609 | Alternate definition of me... |
| clel2gOLD 3610 | Obsolete version of ~ clel... |
| clel2 3611 | Alternate definition of me... |
| clel3g 3612 | Alternate definition of me... |
| clel3 3613 | Alternate definition of me... |
| clel4g 3614 | Alternate definition of me... |
| clel4 3615 | Alternate definition of me... |
| clel4OLD 3616 | Obsolete version of ~ clel... |
| clel5 3617 | Alternate definition of cl... |
| pm13.183 3618 | Compare theorem *13.183 in... |
| rr19.3v 3619 | Restricted quantifier vers... |
| rr19.28v 3620 | Restricted quantifier vers... |
| elab6g 3621 | Membership in a class abst... |
| elabd2 3622 | Membership in a class abst... |
| elabd3 3623 | Membership in a class abst... |
| elabgt 3624 | Membership in a class abst... |
| elabgtOLD 3625 | Obsolete version of ~ elab... |
| elabgf 3626 | Membership in a class abst... |
| elabf 3627 | Membership in a class abst... |
| elabg 3628 | Membership in a class abst... |
| elabgOLD 3629 | Obsolete version of ~ elab... |
| elab 3630 | Membership in a class abst... |
| elabOLD 3631 | Obsolete version of ~ elab... |
| elab2g 3632 | Membership in a class abst... |
| elabd 3633 | Explicit demonstration the... |
| elab2 3634 | Membership in a class abst... |
| elab4g 3635 | Membership in a class abst... |
| elab3gf 3636 | Membership in a class abst... |
| elab3g 3637 | Membership in a class abst... |
| elab3 3638 | Membership in a class abst... |
| elrabi 3639 | Implication for the member... |
| elrabiOLD 3640 | Obsolete version of ~ elra... |
| elrabf 3641 | Membership in a restricted... |
| rabtru 3642 | Abstract builder using the... |
| rabeqcOLD 3643 | Obsolete version of ~ rabe... |
| elrab3t 3644 | Membership in a restricted... |
| elrab 3645 | Membership in a restricted... |
| elrab3 3646 | Membership in a restricted... |
| elrabd 3647 | Membership in a restricted... |
| elrab2 3648 | Membership in a restricted... |
| ralab 3649 | Universal quantification o... |
| ralabOLD 3650 | Obsolete version of ~ rala... |
| ralrab 3651 | Universal quantification o... |
| rexab 3652 | Existential quantification... |
| rexabOLD 3653 | Obsolete version of ~ rexa... |
| rexrab 3654 | Existential quantification... |
| ralab2 3655 | Universal quantification o... |
| ralrab2 3656 | Universal quantification o... |
| rexab2 3657 | Existential quantification... |
| rexrab2 3658 | Existential quantification... |
| reurab 3659 | Restricted existential uni... |
| abidnf 3660 | Identity used to create cl... |
| dedhb 3661 | A deduction theorem for co... |
| class2seteq 3662 | Writing a set as a class a... |
| nelrdva 3663 | Deduce negative membership... |
| eqeu 3664 | A condition which implies ... |
| moeq 3665 | There exists at most one s... |
| eueq 3666 | A class is a set if and on... |
| eueqi 3667 | There exists a unique set ... |
| eueq2 3668 | Equality has existential u... |
| eueq3 3669 | Equality has existential u... |
| moeq3 3670 | "At most one" property of ... |
| mosub 3671 | "At most one" remains true... |
| mo2icl 3672 | Theorem for inferring "at ... |
| mob2 3673 | Consequence of "at most on... |
| moi2 3674 | Consequence of "at most on... |
| mob 3675 | Equality implied by "at mo... |
| moi 3676 | Equality implied by "at mo... |
| morex 3677 | Derive membership from uni... |
| euxfr2w 3678 | Transfer existential uniqu... |
| euxfrw 3679 | Transfer existential uniqu... |
| euxfr2 3680 | Transfer existential uniqu... |
| euxfr 3681 | Transfer existential uniqu... |
| euind 3682 | Existential uniqueness via... |
| reu2 3683 | A way to express restricte... |
| reu6 3684 | A way to express restricte... |
| reu3 3685 | A way to express restricte... |
| reu6i 3686 | A condition which implies ... |
| eqreu 3687 | A condition which implies ... |
| rmo4 3688 | Restricted "at most one" u... |
| reu4 3689 | Restricted uniqueness usin... |
| reu7 3690 | Restricted uniqueness usin... |
| reu8 3691 | Restricted uniqueness usin... |
| rmo3f 3692 | Restricted "at most one" u... |
| rmo4f 3693 | Restricted "at most one" u... |
| reu2eqd 3694 | Deduce equality from restr... |
| reueq 3695 | Equality has existential u... |
| rmoeq 3696 | Equality's restricted exis... |
| rmoan 3697 | Restricted "at most one" s... |
| rmoim 3698 | Restricted "at most one" i... |
| rmoimia 3699 | Restricted "at most one" i... |
| rmoimi 3700 | Restricted "at most one" i... |
| rmoimi2 3701 | Restricted "at most one" i... |
| 2reu5a 3702 | Double restricted existent... |
| reuimrmo 3703 | Restricted uniqueness impl... |
| 2reuswap 3704 | A condition allowing swap ... |
| 2reuswap2 3705 | A condition allowing swap ... |
| reuxfrd 3706 | Transfer existential uniqu... |
| reuxfr 3707 | Transfer existential uniqu... |
| reuxfr1d 3708 | Transfer existential uniqu... |
| reuxfr1ds 3709 | Transfer existential uniqu... |
| reuxfr1 3710 | Transfer existential uniqu... |
| reuind 3711 | Existential uniqueness via... |
| 2rmorex 3712 | Double restricted quantifi... |
| 2reu5lem1 3713 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3714 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3715 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3716 | Double restricted existent... |
| 2reurmo 3717 | Double restricted quantifi... |
| 2reurex 3718 | Double restricted quantifi... |
| 2rmoswap 3719 | A condition allowing to sw... |
| 2rexreu 3720 | Double restricted existent... |
| cdeqi 3723 | Deduce conditional equalit... |
| cdeqri 3724 | Property of conditional eq... |
| cdeqth 3725 | Deduce conditional equalit... |
| cdeqnot 3726 | Distribute conditional equ... |
| cdeqal 3727 | Distribute conditional equ... |
| cdeqab 3728 | Distribute conditional equ... |
| cdeqal1 3729 | Distribute conditional equ... |
| cdeqab1 3730 | Distribute conditional equ... |
| cdeqim 3731 | Distribute conditional equ... |
| cdeqcv 3732 | Conditional equality for s... |
| cdeqeq 3733 | Distribute conditional equ... |
| cdeqel 3734 | Distribute conditional equ... |
| nfcdeq 3735 | If we have a conditional e... |
| nfccdeq 3736 | Variation of ~ nfcdeq for ... |
| rru 3737 | Relative version of Russel... |
| ru 3738 | Russell's Paradox. Propos... |
| dfsbcq 3741 | Proper substitution of a c... |
| dfsbcq2 3742 | This theorem, which is sim... |
| sbsbc 3743 | Show that ~ df-sb and ~ df... |
| sbceq1d 3744 | Equality theorem for class... |
| sbceq1dd 3745 | Equality theorem for class... |
| sbceqbid 3746 | Equality theorem for class... |
| sbc8g 3747 | This is the closest we can... |
| sbc2or 3748 | The disjunction of two equ... |
| sbcex 3749 | By our definition of prope... |
| sbceq1a 3750 | Equality theorem for class... |
| sbceq2a 3751 | Equality theorem for class... |
| spsbc 3752 | Specialization: if a formu... |
| spsbcd 3753 | Specialization: if a formu... |
| sbcth 3754 | A substitution into a theo... |
| sbcthdv 3755 | Deduction version of ~ sbc... |
| sbcid 3756 | An identity theorem for su... |
| nfsbc1d 3757 | Deduction version of ~ nfs... |
| nfsbc1 3758 | Bound-variable hypothesis ... |
| nfsbc1v 3759 | Bound-variable hypothesis ... |
| nfsbcdw 3760 | Deduction version of ~ nfs... |
| nfsbcw 3761 | Bound-variable hypothesis ... |
| sbccow 3762 | A composition law for clas... |
| nfsbcd 3763 | Deduction version of ~ nfs... |
| nfsbc 3764 | Bound-variable hypothesis ... |
| sbcco 3765 | A composition law for clas... |
| sbcco2 3766 | A composition law for clas... |
| sbc5 3767 | An equivalence for class s... |
| sbc5ALT 3768 | Alternate proof of ~ sbc5 ... |
| sbc6g 3769 | An equivalence for class s... |
| sbc6gOLD 3770 | Obsolete version of ~ sbc6... |
| sbc6 3771 | An equivalence for class s... |
| sbc7 3772 | An equivalence for class s... |
| cbvsbcw 3773 | Change bound variables in ... |
| cbvsbcvw 3774 | Change the bound variable ... |
| cbvsbc 3775 | Change bound variables in ... |
| cbvsbcv 3776 | Change the bound variable ... |
| sbciegft 3777 | Conversion of implicit sub... |
| sbciegf 3778 | Conversion of implicit sub... |
| sbcieg 3779 | Conversion of implicit sub... |
| sbciegOLD 3780 | Obsolete version of ~ sbci... |
| sbcie2g 3781 | Conversion of implicit sub... |
| sbcie 3782 | Conversion of implicit sub... |
| sbciedf 3783 | Conversion of implicit sub... |
| sbcied 3784 | Conversion of implicit sub... |
| sbciedOLD 3785 | Obsolete version of ~ sbci... |
| sbcied2 3786 | Conversion of implicit sub... |
| elrabsf 3787 | Membership in a restricted... |
| eqsbc1 3788 | Substitution for the left-... |
| sbcng 3789 | Move negation in and out o... |
| sbcimg 3790 | Distribution of class subs... |
| sbcan 3791 | Distribution of class subs... |
| sbcor 3792 | Distribution of class subs... |
| sbcbig 3793 | Distribution of class subs... |
| sbcn1 3794 | Move negation in and out o... |
| sbcim1 3795 | Distribution of class subs... |
| sbcim1OLD 3796 | Obsolete version of ~ sbci... |
| sbcbid 3797 | Formula-building deduction... |
| sbcbidv 3798 | Formula-building deduction... |
| sbcbii 3799 | Formula-building inference... |
| sbcbi1 3800 | Distribution of class subs... |
| sbcbi2 3801 | Substituting into equivale... |
| sbcbi2OLD 3802 | Obsolete proof of ~ sbcbi2... |
| sbcal 3803 | Move universal quantifier ... |
| sbcex2 3804 | Move existential quantifie... |
| sbceqal 3805 | Class version of one impli... |
| sbceqalOLD 3806 | Obsolete version of ~ sbce... |
| sbeqalb 3807 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3808 | Substitution for the right... |
| sbc3an 3809 | Distribution of class subs... |
| sbcel1v 3810 | Class substitution into a ... |
| sbcel2gv 3811 | Class substitution into a ... |
| sbcel21v 3812 | Class substitution into a ... |
| sbcimdv 3813 | Substitution analogue of T... |
| sbcimdvOLD 3814 | Obsolete version of ~ sbci... |
| sbctt 3815 | Substitution for a variabl... |
| sbcgf 3816 | Substitution for a variabl... |
| sbc19.21g 3817 | Substitution for a variabl... |
| sbcg 3818 | Substitution for a variabl... |
| sbcgOLD 3819 | Obsolete version of ~ sbcg... |
| sbcgfi 3820 | Substitution for a variabl... |
| sbc2iegf 3821 | Conversion of implicit sub... |
| sbc2ie 3822 | Conversion of implicit sub... |
| sbc2ieOLD 3823 | Obsolete version of ~ sbc2... |
| sbc2iedv 3824 | Conversion of implicit sub... |
| sbc3ie 3825 | Conversion of implicit sub... |
| sbccomlem 3826 | Lemma for ~ sbccom . (Con... |
| sbccom 3827 | Commutative law for double... |
| sbcralt 3828 | Interchange class substitu... |
| sbcrext 3829 | Interchange class substitu... |
| sbcralg 3830 | Interchange class substitu... |
| sbcrex 3831 | Interchange class substitu... |
| sbcreu 3832 | Interchange class substitu... |
| reu8nf 3833 | Restricted uniqueness usin... |
| sbcabel 3834 | Interchange class substitu... |
| rspsbc 3835 | Restricted quantifier vers... |
| rspsbca 3836 | Restricted quantifier vers... |
| rspesbca 3837 | Existence form of ~ rspsbc... |
| spesbc 3838 | Existence form of ~ spsbc ... |
| spesbcd 3839 | form of ~ spsbc . (Contri... |
| sbcth2 3840 | A substitution into a theo... |
| ra4v 3841 | Version of ~ ra4 with a di... |
| ra4 3842 | Restricted quantifier vers... |
| rmo2 3843 | Alternate definition of re... |
| rmo2i 3844 | Condition implying restric... |
| rmo3 3845 | Restricted "at most one" u... |
| rmob 3846 | Consequence of "at most on... |
| rmoi 3847 | Consequence of "at most on... |
| rmob2 3848 | Consequence of "restricted... |
| rmoi2 3849 | Consequence of "restricted... |
| rmoanim 3850 | Introduction of a conjunct... |
| rmoanimALT 3851 | Alternate proof of ~ rmoan... |
| reuan 3852 | Introduction of a conjunct... |
| 2reu1 3853 | Double restricted existent... |
| 2reu2 3854 | Double restricted existent... |
| csb2 3857 | Alternate expression for t... |
| csbeq1 3858 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3859 | Equality deduction for pro... |
| csbeq2 3860 | Substituting into equivale... |
| csbeq2d 3861 | Formula-building deduction... |
| csbeq2dv 3862 | Formula-building deduction... |
| csbeq2i 3863 | Formula-building inference... |
| csbeq12dv 3864 | Formula-building inference... |
| cbvcsbw 3865 | Change bound variables in ... |
| cbvcsb 3866 | Change bound variables in ... |
| cbvcsbv 3867 | Change the bound variable ... |
| csbid 3868 | Analogue of ~ sbid for pro... |
| csbeq1a 3869 | Equality theorem for prope... |
| csbcow 3870 | Composition law for chaine... |
| csbco 3871 | Composition law for chaine... |
| csbtt 3872 | Substitution doesn't affec... |
| csbconstgf 3873 | Substitution doesn't affec... |
| csbconstg 3874 | Substitution doesn't affec... |
| csbconstgOLD 3875 | Obsolete version of ~ csbc... |
| csbgfi 3876 | Substitution for a variabl... |
| csbconstgi 3877 | The proper substitution of... |
| nfcsb1d 3878 | Bound-variable hypothesis ... |
| nfcsb1 3879 | Bound-variable hypothesis ... |
| nfcsb1v 3880 | Bound-variable hypothesis ... |
| nfcsbd 3881 | Deduction version of ~ nfc... |
| nfcsbw 3882 | Bound-variable hypothesis ... |
| nfcsb 3883 | Bound-variable hypothesis ... |
| csbhypf 3884 | Introduce an explicit subs... |
| csbiebt 3885 | Conversion of implicit sub... |
| csbiedf 3886 | Conversion of implicit sub... |
| csbieb 3887 | Bidirectional conversion b... |
| csbiebg 3888 | Bidirectional conversion b... |
| csbiegf 3889 | Conversion of implicit sub... |
| csbief 3890 | Conversion of implicit sub... |
| csbie 3891 | Conversion of implicit sub... |
| csbieOLD 3892 | Obsolete version of ~ csbi... |
| csbied 3893 | Conversion of implicit sub... |
| csbiedOLD 3894 | Obsolete version of ~ csbi... |
| csbied2 3895 | Conversion of implicit sub... |
| csbie2t 3896 | Conversion of implicit sub... |
| csbie2 3897 | Conversion of implicit sub... |
| csbie2g 3898 | Conversion of implicit sub... |
| cbvrabcsfw 3899 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3900 | A more general version of ... |
| cbvrexcsf 3901 | A more general version of ... |
| cbvreucsf 3902 | A more general version of ... |
| cbvrabcsf 3903 | A more general version of ... |
| cbvralv2 3904 | Rule used to change the bo... |
| cbvrexv2 3905 | Rule used to change the bo... |
| rspc2vd 3906 | Deduction version of 2-var... |
| difjust 3912 | Soundness justification th... |
| unjust 3914 | Soundness justification th... |
| injust 3916 | Soundness justification th... |
| dfin5 3918 | Alternate definition for t... |
| dfdif2 3919 | Alternate definition of cl... |
| eldif 3920 | Expansion of membership in... |
| eldifd 3921 | If a class is in one class... |
| eldifad 3922 | If a class is in the diffe... |
| eldifbd 3923 | If a class is in the diffe... |
| elneeldif 3924 | The elements of a set diff... |
| velcomp 3925 | Characterization of setvar... |
| elin 3926 | Expansion of membership in... |
| dfss 3928 | Variant of subclass defini... |
| dfss2 3930 | Alternate definition of th... |
| dfss2OLD 3931 | Obsolete version of ~ dfss... |
| dfss3 3932 | Alternate definition of su... |
| dfss6 3933 | Alternate definition of su... |
| dfss2f 3934 | Equivalence for subclass r... |
| dfss3f 3935 | Equivalence for subclass r... |
| nfss 3936 | If ` x ` is not free in ` ... |
| ssel 3937 | Membership relationships f... |
| sselOLD 3938 | Obsolete version of ~ ssel... |
| ssel2 3939 | Membership relationships f... |
| sseli 3940 | Membership implication fro... |
| sselii 3941 | Membership inference from ... |
| sselid 3942 | Membership inference from ... |
| sseld 3943 | Membership deduction from ... |
| sselda 3944 | Membership deduction from ... |
| sseldd 3945 | Membership inference from ... |
| ssneld 3946 | If a class is not in anoth... |
| ssneldd 3947 | If an element is not in a ... |
| ssriv 3948 | Inference based on subclas... |
| ssrd 3949 | Deduction based on subclas... |
| ssrdv 3950 | Deduction based on subclas... |
| sstr2 3951 | Transitivity of subclass r... |
| sstr 3952 | Transitivity of subclass r... |
| sstri 3953 | Subclass transitivity infe... |
| sstrd 3954 | Subclass transitivity dedu... |
| sstrid 3955 | Subclass transitivity dedu... |
| sstrdi 3956 | Subclass transitivity dedu... |
| sylan9ss 3957 | A subclass transitivity de... |
| sylan9ssr 3958 | A subclass transitivity de... |
| eqss 3959 | The subclass relationship ... |
| eqssi 3960 | Infer equality from two su... |
| eqssd 3961 | Equality deduction from tw... |
| sssseq 3962 | If a class is a subclass o... |
| eqrd 3963 | Deduce equality of classes... |
| eqri 3964 | Infer equality of classes ... |
| eqelssd 3965 | Equality deduction from su... |
| ssid 3966 | Any class is a subclass of... |
| ssidd 3967 | Weakening of ~ ssid . (Co... |
| ssv 3968 | Any class is a subclass of... |
| sseq1 3969 | Equality theorem for subcl... |
| sseq2 3970 | Equality theorem for the s... |
| sseq12 3971 | Equality theorem for the s... |
| sseq1i 3972 | An equality inference for ... |
| sseq2i 3973 | An equality inference for ... |
| sseq12i 3974 | An equality inference for ... |
| sseq1d 3975 | An equality deduction for ... |
| sseq2d 3976 | An equality deduction for ... |
| sseq12d 3977 | An equality deduction for ... |
| eqsstri 3978 | Substitution of equality i... |
| eqsstrri 3979 | Substitution of equality i... |
| sseqtri 3980 | Substitution of equality i... |
| sseqtrri 3981 | Substitution of equality i... |
| eqsstrd 3982 | Substitution of equality i... |
| eqsstrrd 3983 | Substitution of equality i... |
| sseqtrd 3984 | Substitution of equality i... |
| sseqtrrd 3985 | Substitution of equality i... |
| 3sstr3i 3986 | Substitution of equality i... |
| 3sstr4i 3987 | Substitution of equality i... |
| 3sstr3g 3988 | Substitution of equality i... |
| 3sstr4g 3989 | Substitution of equality i... |
| 3sstr3d 3990 | Substitution of equality i... |
| 3sstr4d 3991 | Substitution of equality i... |
| eqsstrid 3992 | A chained subclass and equ... |
| eqsstrrid 3993 | A chained subclass and equ... |
| sseqtrdi 3994 | A chained subclass and equ... |
| sseqtrrdi 3995 | A chained subclass and equ... |
| sseqtrid 3996 | Subclass transitivity dedu... |
| sseqtrrid 3997 | Subclass transitivity dedu... |
| eqsstrdi 3998 | A chained subclass and equ... |
| eqsstrrdi 3999 | A chained subclass and equ... |
| eqimss 4000 | Equality implies inclusion... |
| eqimss2 4001 | Equality implies inclusion... |
| eqimssi 4002 | Infer subclass relationshi... |
| eqimss2i 4003 | Infer subclass relationshi... |
| nssne1 4004 | Two classes are different ... |
| nssne2 4005 | Two classes are different ... |
| nss 4006 | Negation of subclass relat... |
| nelss 4007 | Demonstrate by witnesses t... |
| ssrexf 4008 | Restricted existential qua... |
| ssrmof 4009 | "At most one" existential ... |
| ssralv 4010 | Quantification restricted ... |
| ssrexv 4011 | Existential quantification... |
| ss2ralv 4012 | Two quantifications restri... |
| ss2rexv 4013 | Two existential quantifica... |
| ralss 4014 | Restricted universal quant... |
| rexss 4015 | Restricted existential qua... |
| ss2ab 4016 | Class abstractions in a su... |
| abss 4017 | Class abstraction in a sub... |
| ssab 4018 | Subclass of a class abstra... |
| ssabral 4019 | The relation for a subclas... |
| ss2abdv 4020 | Deduction of abstraction s... |
| ss2abdvALT 4021 | Alternate proof of ~ ss2ab... |
| ss2abdvOLD 4022 | Obsolete version of ~ ss2a... |
| ss2abi 4023 | Inference of abstraction s... |
| ss2abiOLD 4024 | Obsolete version of ~ ss2a... |
| abssdv 4025 | Deduction of abstraction s... |
| abssdvOLD 4026 | Obsolete version of ~ abss... |
| abssi 4027 | Inference of abstraction s... |
| ss2rab 4028 | Restricted abstraction cla... |
| rabss 4029 | Restricted class abstracti... |
| ssrab 4030 | Subclass of a restricted c... |
| ssrabdv 4031 | Subclass of a restricted c... |
| rabssdv 4032 | Subclass of a restricted c... |
| ss2rabdv 4033 | Deduction of restricted ab... |
| ss2rabi 4034 | Inference of restricted ab... |
| rabss2 4035 | Subclass law for restricte... |
| ssab2 4036 | Subclass relation for the ... |
| ssrab2 4037 | Subclass relation for a re... |
| ssrab2OLD 4038 | Obsolete version of ~ ssra... |
| rabss3d 4039 | Subclass law for restricte... |
| ssrab3 4040 | Subclass relation for a re... |
| rabssrabd 4041 | Subclass of a restricted c... |
| ssrabeq 4042 | If the restricting class o... |
| rabssab 4043 | A restricted class is a su... |
| uniiunlem 4044 | A subset relationship usef... |
| dfpss2 4045 | Alternate definition of pr... |
| dfpss3 4046 | Alternate definition of pr... |
| psseq1 4047 | Equality theorem for prope... |
| psseq2 4048 | Equality theorem for prope... |
| psseq1i 4049 | An equality inference for ... |
| psseq2i 4050 | An equality inference for ... |
| psseq12i 4051 | An equality inference for ... |
| psseq1d 4052 | An equality deduction for ... |
| psseq2d 4053 | An equality deduction for ... |
| psseq12d 4054 | An equality deduction for ... |
| pssss 4055 | A proper subclass is a sub... |
| pssne 4056 | Two classes in a proper su... |
| pssssd 4057 | Deduce subclass from prope... |
| pssned 4058 | Proper subclasses are uneq... |
| sspss 4059 | Subclass in terms of prope... |
| pssirr 4060 | Proper subclass is irrefle... |
| pssn2lp 4061 | Proper subclass has no 2-c... |
| sspsstri 4062 | Two ways of stating tricho... |
| ssnpss 4063 | Partial trichotomy law for... |
| psstr 4064 | Transitive law for proper ... |
| sspsstr 4065 | Transitive law for subclas... |
| psssstr 4066 | Transitive law for subclas... |
| psstrd 4067 | Proper subclass inclusion ... |
| sspsstrd 4068 | Transitivity involving sub... |
| psssstrd 4069 | Transitivity involving sub... |
| npss 4070 | A class is not a proper su... |
| ssnelpss 4071 | A subclass missing a membe... |
| ssnelpssd 4072 | Subclass inclusion with on... |
| ssexnelpss 4073 | If there is an element of ... |
| dfdif3 4074 | Alternate definition of cl... |
| difeq1 4075 | Equality theorem for class... |
| difeq2 4076 | Equality theorem for class... |
| difeq12 4077 | Equality theorem for class... |
| difeq1i 4078 | Inference adding differenc... |
| difeq2i 4079 | Inference adding differenc... |
| difeq12i 4080 | Equality inference for cla... |
| difeq1d 4081 | Deduction adding differenc... |
| difeq2d 4082 | Deduction adding differenc... |
| difeq12d 4083 | Equality deduction for cla... |
| difeqri 4084 | Inference from membership ... |
| nfdif 4085 | Bound-variable hypothesis ... |
| eldifi 4086 | Implication of membership ... |
| eldifn 4087 | Implication of membership ... |
| elndif 4088 | A set does not belong to a... |
| neldif 4089 | Implication of membership ... |
| difdif 4090 | Double class difference. ... |
| difss 4091 | Subclass relationship for ... |
| difssd 4092 | A difference of two classe... |
| difss2 4093 | If a class is contained in... |
| difss2d 4094 | If a class is contained in... |
| ssdifss 4095 | Preservation of a subclass... |
| ddif 4096 | Double complement under un... |
| ssconb 4097 | Contraposition law for sub... |
| sscon 4098 | Contraposition law for sub... |
| ssdif 4099 | Difference law for subsets... |
| ssdifd 4100 | If ` A ` is contained in `... |
| sscond 4101 | If ` A ` is contained in `... |
| ssdifssd 4102 | If ` A ` is contained in `... |
| ssdif2d 4103 | If ` A ` is contained in `... |
| raldifb 4104 | Restricted universal quant... |
| rexdifi 4105 | Restricted existential qua... |
| complss 4106 | Complementation reverses i... |
| compleq 4107 | Two classes are equal if a... |
| elun 4108 | Expansion of membership in... |
| elunnel1 4109 | A member of a union that i... |
| elunnel2 4110 | A member of a union that i... |
| uneqri 4111 | Inference from membership ... |
| unidm 4112 | Idempotent law for union o... |
| uncom 4113 | Commutative law for union ... |
| equncom 4114 | If a class equals the unio... |
| equncomi 4115 | Inference form of ~ equnco... |
| uneq1 4116 | Equality theorem for the u... |
| uneq2 4117 | Equality theorem for the u... |
| uneq12 4118 | Equality theorem for the u... |
| uneq1i 4119 | Inference adding union to ... |
| uneq2i 4120 | Inference adding union to ... |
| uneq12i 4121 | Equality inference for the... |
| uneq1d 4122 | Deduction adding union to ... |
| uneq2d 4123 | Deduction adding union to ... |
| uneq12d 4124 | Equality deduction for the... |
| nfun 4125 | Bound-variable hypothesis ... |
| unass 4126 | Associative law for union ... |
| un12 4127 | A rearrangement of union. ... |
| un23 4128 | A rearrangement of union. ... |
| un4 4129 | A rearrangement of the uni... |
| unundi 4130 | Union distributes over its... |
| unundir 4131 | Union distributes over its... |
| ssun1 4132 | Subclass relationship for ... |
| ssun2 4133 | Subclass relationship for ... |
| ssun3 4134 | Subclass law for union of ... |
| ssun4 4135 | Subclass law for union of ... |
| elun1 4136 | Membership law for union o... |
| elun2 4137 | Membership law for union o... |
| elunant 4138 | A statement is true for ev... |
| unss1 4139 | Subclass law for union of ... |
| ssequn1 4140 | A relationship between sub... |
| unss2 4141 | Subclass law for union of ... |
| unss12 4142 | Subclass law for union of ... |
| ssequn2 4143 | A relationship between sub... |
| unss 4144 | The union of two subclasse... |
| unssi 4145 | An inference showing the u... |
| unssd 4146 | A deduction showing the un... |
| unssad 4147 | If ` ( A u. B ) ` is conta... |
| unssbd 4148 | If ` ( A u. B ) ` is conta... |
| ssun 4149 | A condition that implies i... |
| rexun 4150 | Restricted existential qua... |
| ralunb 4151 | Restricted quantification ... |
| ralun 4152 | Restricted quantification ... |
| elini 4153 | Membership in an intersect... |
| elind 4154 | Deduce membership in an in... |
| elinel1 4155 | Membership in an intersect... |
| elinel2 4156 | Membership in an intersect... |
| elin2 4157 | Membership in a class defi... |
| elin1d 4158 | Elementhood in the first s... |
| elin2d 4159 | Elementhood in the first s... |
| elin3 4160 | Membership in a class defi... |
| incom 4161 | Commutative law for inters... |
| ineqcom 4162 | Two ways of expressing tha... |
| ineqcomi 4163 | Two ways of expressing tha... |
| ineqri 4164 | Inference from membership ... |
| ineq1 4165 | Equality theorem for inter... |
| ineq2 4166 | Equality theorem for inter... |
| ineq12 4167 | Equality theorem for inter... |
| ineq1i 4168 | Equality inference for int... |
| ineq2i 4169 | Equality inference for int... |
| ineq12i 4170 | Equality inference for int... |
| ineq1d 4171 | Equality deduction for int... |
| ineq2d 4172 | Equality deduction for int... |
| ineq12d 4173 | Equality deduction for int... |
| ineqan12d 4174 | Equality deduction for int... |
| sseqin2 4175 | A relationship between sub... |
| nfin 4176 | Bound-variable hypothesis ... |
| rabbi2dva 4177 | Deduction from a wff to a ... |
| inidm 4178 | Idempotent law for interse... |
| inass 4179 | Associative law for inters... |
| in12 4180 | A rearrangement of interse... |
| in32 4181 | A rearrangement of interse... |
| in13 4182 | A rearrangement of interse... |
| in31 4183 | A rearrangement of interse... |
| inrot 4184 | Rotate the intersection of... |
| in4 4185 | Rearrangement of intersect... |
| inindi 4186 | Intersection distributes o... |
| inindir 4187 | Intersection distributes o... |
| inss1 4188 | The intersection of two cl... |
| inss2 4189 | The intersection of two cl... |
| ssin 4190 | Subclass of intersection. ... |
| ssini 4191 | An inference showing that ... |
| ssind 4192 | A deduction showing that a... |
| ssrin 4193 | Add right intersection to ... |
| sslin 4194 | Add left intersection to s... |
| ssrind 4195 | Add right intersection to ... |
| ss2in 4196 | Intersection of subclasses... |
| ssinss1 4197 | Intersection preserves sub... |
| inss 4198 | Inclusion of an intersecti... |
| rexin 4199 | Restricted existential qua... |
| dfss7 4200 | Alternate definition of su... |
| symdifcom 4203 | Symmetric difference commu... |
| symdifeq1 4204 | Equality theorem for symme... |
| symdifeq2 4205 | Equality theorem for symme... |
| nfsymdif 4206 | Hypothesis builder for sym... |
| elsymdif 4207 | Membership in a symmetric ... |
| dfsymdif4 4208 | Alternate definition of th... |
| elsymdifxor 4209 | Membership in a symmetric ... |
| dfsymdif2 4210 | Alternate definition of th... |
| symdifass 4211 | Symmetric difference is as... |
| difsssymdif 4212 | The symmetric difference c... |
| difsymssdifssd 4213 | If the symmetric differenc... |
| unabs 4214 | Absorption law for union. ... |
| inabs 4215 | Absorption law for interse... |
| nssinpss 4216 | Negation of subclass expre... |
| nsspssun 4217 | Negation of subclass expre... |
| dfss4 4218 | Subclass defined in terms ... |
| dfun2 4219 | An alternate definition of... |
| dfin2 4220 | An alternate definition of... |
| difin 4221 | Difference with intersecti... |
| ssdifim 4222 | Implication of a class dif... |
| ssdifsym 4223 | Symmetric class difference... |
| dfss5 4224 | Alternate definition of su... |
| dfun3 4225 | Union defined in terms of ... |
| dfin3 4226 | Intersection defined in te... |
| dfin4 4227 | Alternate definition of th... |
| invdif 4228 | Intersection with universa... |
| indif 4229 | Intersection with class di... |
| indif2 4230 | Bring an intersection in a... |
| indif1 4231 | Bring an intersection in a... |
| indifcom 4232 | Commutation law for inters... |
| indi 4233 | Distributive law for inter... |
| undi 4234 | Distributive law for union... |
| indir 4235 | Distributive law for inter... |
| undir 4236 | Distributive law for union... |
| unineq 4237 | Infer equality from equali... |
| uneqin 4238 | Equality of union and inte... |
| difundi 4239 | Distributive law for class... |
| difundir 4240 | Distributive law for class... |
| difindi 4241 | Distributive law for class... |
| difindir 4242 | Distributive law for class... |
| indifdi 4243 | Distribute intersection ov... |
| indifdir 4244 | Distribute intersection ov... |
| indifdirOLD 4245 | Obsolete version of ~ indi... |
| difdif2 4246 | Class difference by a clas... |
| undm 4247 | De Morgan's law for union.... |
| indm 4248 | De Morgan's law for inters... |
| difun1 4249 | A relationship involving d... |
| undif3 4250 | An equality involving clas... |
| difin2 4251 | Represent a class differen... |
| dif32 4252 | Swap second and third argu... |
| difabs 4253 | Absorption-like law for cl... |
| sscon34b 4254 | Relative complementation r... |
| rcompleq 4255 | Two subclasses are equal i... |
| dfsymdif3 4256 | Alternate definition of th... |
| unabw 4257 | Union of two class abstrac... |
| unab 4258 | Union of two class abstrac... |
| inab 4259 | Intersection of two class ... |
| difab 4260 | Difference of two class ab... |
| abanssl 4261 | A class abstraction with a... |
| abanssr 4262 | A class abstraction with a... |
| notabw 4263 | A class abstraction define... |
| notab 4264 | A class abstraction define... |
| unrab 4265 | Union of two restricted cl... |
| inrab 4266 | Intersection of two restri... |
| inrab2 4267 | Intersection with a restri... |
| difrab 4268 | Difference of two restrict... |
| dfrab3 4269 | Alternate definition of re... |
| dfrab2 4270 | Alternate definition of re... |
| notrab 4271 | Complementation of restric... |
| dfrab3ss 4272 | Restricted class abstracti... |
| rabun2 4273 | Abstraction restricted to ... |
| reuun2 4274 | Transfer uniqueness to a s... |
| reuss2 4275 | Transfer uniqueness to a s... |
| reuss 4276 | Transfer uniqueness to a s... |
| reuun1 4277 | Transfer uniqueness to a s... |
| reupick 4278 | Restricted uniqueness "pic... |
| reupick3 4279 | Restricted uniqueness "pic... |
| reupick2 4280 | Restricted uniqueness "pic... |
| euelss 4281 | Transfer uniqueness of an ... |
| dfnul4 4284 | Alternate definition of th... |
| dfnul2 4285 | Alternate definition of th... |
| dfnul3 4286 | Alternate definition of th... |
| dfnul2OLD 4287 | Obsolete version of ~ dfnu... |
| dfnul3OLD 4288 | Obsolete version of ~ dfnu... |
| dfnul4OLD 4289 | Obsolete version of ~ dfnu... |
| noel 4290 | The empty set has no eleme... |
| noelOLD 4291 | Obsolete version of ~ noel... |
| nel02 4292 | The empty set has no eleme... |
| n0i 4293 | If a class has elements, t... |
| ne0i 4294 | If a class has elements, t... |
| ne0d 4295 | Deduction form of ~ ne0i .... |
| n0ii 4296 | If a class has elements, t... |
| ne0ii 4297 | If a class has elements, t... |
| vn0 4298 | The universal class is not... |
| vn0ALT 4299 | Alternate proof of ~ vn0 .... |
| eq0f 4300 | A class is equal to the em... |
| neq0f 4301 | A class is not empty if an... |
| n0f 4302 | A class is nonempty if and... |
| eq0 4303 | A class is equal to the em... |
| eq0ALT 4304 | Alternate proof of ~ eq0 .... |
| neq0 4305 | A class is not empty if an... |
| n0 4306 | A class is nonempty if and... |
| eq0OLDOLD 4307 | Obsolete version of ~ eq0 ... |
| neq0OLD 4308 | Obsolete version of ~ neq0... |
| n0OLD 4309 | Obsolete version of ~ n0 a... |
| nel0 4310 | From the general negation ... |
| reximdva0 4311 | Restricted existence deduc... |
| rspn0 4312 | Specialization for restric... |
| rspn0OLD 4313 | Obsolete version of ~ rspn... |
| n0rex 4314 | There is an element in a n... |
| ssn0rex 4315 | There is an element in a c... |
| n0moeu 4316 | A case of equivalence of "... |
| rex0 4317 | Vacuous restricted existen... |
| reu0 4318 | Vacuous restricted uniquen... |
| rmo0 4319 | Vacuous restricted at-most... |
| 0el 4320 | Membership of the empty se... |
| n0el 4321 | Negated membership of the ... |
| eqeuel 4322 | A condition which implies ... |
| ssdif0 4323 | Subclass expressed in term... |
| difn0 4324 | If the difference of two s... |
| pssdifn0 4325 | A proper subclass has a no... |
| pssdif 4326 | A proper subclass has a no... |
| ndisj 4327 | Express that an intersecti... |
| difin0ss 4328 | Difference, intersection, ... |
| inssdif0 4329 | Intersection, subclass, an... |
| difid 4330 | The difference between a c... |
| difidALT 4331 | Alternate proof of ~ difid... |
| dif0 4332 | The difference between a c... |
| ab0w 4333 | The class of sets verifyin... |
| ab0 4334 | The class of sets verifyin... |
| ab0OLD 4335 | Obsolete version of ~ ab0 ... |
| ab0ALT 4336 | Alternate proof of ~ ab0 ,... |
| dfnf5 4337 | Characterization of nonfre... |
| ab0orv 4338 | The class abstraction defi... |
| ab0orvALT 4339 | Alternate proof of ~ ab0or... |
| abn0 4340 | Nonempty class abstraction... |
| abn0OLD 4341 | Obsolete version of ~ abn0... |
| rab0 4342 | Any restricted class abstr... |
| rabeq0w 4343 | Condition for a restricted... |
| rabeq0 4344 | Condition for a restricted... |
| rabn0 4345 | Nonempty restricted class ... |
| rabxm 4346 | Law of excluded middle, in... |
| rabnc 4347 | Law of noncontradiction, i... |
| elneldisj 4348 | The set of elements ` s ` ... |
| elnelun 4349 | The union of the set of el... |
| un0 4350 | The union of a class with ... |
| in0 4351 | The intersection of a clas... |
| 0un 4352 | The union of the empty set... |
| 0in 4353 | The intersection of the em... |
| inv1 4354 | The intersection of a clas... |
| unv 4355 | The union of a class with ... |
| 0ss 4356 | The null set is a subset o... |
| ss0b 4357 | Any subset of the empty se... |
| ss0 4358 | Any subset of the empty se... |
| sseq0 4359 | A subclass of an empty cla... |
| ssn0 4360 | A class with a nonempty su... |
| 0dif 4361 | The difference between the... |
| abf 4362 | A class abstraction determ... |
| abfOLD 4363 | Obsolete version of ~ abf ... |
| eq0rdv 4364 | Deduction for equality to ... |
| eq0rdvALT 4365 | Alternate proof of ~ eq0rd... |
| csbprc 4366 | The proper substitution of... |
| csb0 4367 | The proper substitution of... |
| sbcel12 4368 | Distribute proper substitu... |
| sbceqg 4369 | Distribute proper substitu... |
| sbceqi 4370 | Distribution of class subs... |
| sbcnel12g 4371 | Distribute proper substitu... |
| sbcne12 4372 | Distribute proper substitu... |
| sbcel1g 4373 | Move proper substitution i... |
| sbceq1g 4374 | Move proper substitution t... |
| sbcel2 4375 | Move proper substitution i... |
| sbceq2g 4376 | Move proper substitution t... |
| csbcom 4377 | Commutative law for double... |
| sbcnestgfw 4378 | Nest the composition of tw... |
| csbnestgfw 4379 | Nest the composition of tw... |
| sbcnestgw 4380 | Nest the composition of tw... |
| csbnestgw 4381 | Nest the composition of tw... |
| sbcco3gw 4382 | Composition of two substit... |
| sbcnestgf 4383 | Nest the composition of tw... |
| csbnestgf 4384 | Nest the composition of tw... |
| sbcnestg 4385 | Nest the composition of tw... |
| csbnestg 4386 | Nest the composition of tw... |
| sbcco3g 4387 | Composition of two substit... |
| csbco3g 4388 | Composition of two class s... |
| csbnest1g 4389 | Nest the composition of tw... |
| csbidm 4390 | Idempotent law for class s... |
| csbvarg 4391 | The proper substitution of... |
| csbvargi 4392 | The proper substitution of... |
| sbccsb 4393 | Substitution into a wff ex... |
| sbccsb2 4394 | Substitution into a wff ex... |
| rspcsbela 4395 | Special case related to ~ ... |
| sbnfc2 4396 | Two ways of expressing " `... |
| csbab 4397 | Move substitution into a c... |
| csbun 4398 | Distribution of class subs... |
| csbin 4399 | Distribute proper substitu... |
| csbie2df 4400 | Conversion of implicit sub... |
| 2nreu 4401 | If there are two different... |
| un00 4402 | Two classes are empty iff ... |
| vss 4403 | Only the universal class h... |
| 0pss 4404 | The null set is a proper s... |
| npss0 4405 | No set is a proper subset ... |
| pssv 4406 | Any non-universal class is... |
| disj 4407 | Two ways of saying that tw... |
| disjOLD 4408 | Obsolete version of ~ disj... |
| disjr 4409 | Two ways of saying that tw... |
| disj1 4410 | Two ways of saying that tw... |
| reldisj 4411 | Two ways of saying that tw... |
| reldisjOLD 4412 | Obsolete version of ~ reld... |
| disj3 4413 | Two ways of saying that tw... |
| disjne 4414 | Members of disjoint sets a... |
| disjeq0 4415 | Two disjoint sets are equa... |
| disjel 4416 | A set can't belong to both... |
| disj2 4417 | Two ways of saying that tw... |
| disj4 4418 | Two ways of saying that tw... |
| ssdisj 4419 | Intersection with a subcla... |
| disjpss 4420 | A class is a proper subset... |
| undisj1 4421 | The union of disjoint clas... |
| undisj2 4422 | The union of disjoint clas... |
| ssindif0 4423 | Subclass expressed in term... |
| inelcm 4424 | The intersection of classe... |
| minel 4425 | A minimum element of a cla... |
| undif4 4426 | Distribute union over diff... |
| disjssun 4427 | Subset relation for disjoi... |
| vdif0 4428 | Universal class equality i... |
| difrab0eq 4429 | If the difference between ... |
| pssnel 4430 | A proper subclass has a me... |
| disjdif 4431 | A class and its relative c... |
| disjdifr 4432 | A class and its relative c... |
| difin0 4433 | The difference of a class ... |
| unvdif 4434 | The union of a class and i... |
| undif1 4435 | Absorption of difference b... |
| undif2 4436 | Absorption of difference b... |
| undifabs 4437 | Absorption of difference b... |
| inundif 4438 | The intersection and class... |
| disjdif2 4439 | The difference of a class ... |
| difun2 4440 | Absorption of union by dif... |
| undif 4441 | Union of complementary par... |
| undif5 4442 | An equality involving clas... |
| ssdifin0 4443 | A subset of a difference d... |
| ssdifeq0 4444 | A class is a subclass of i... |
| ssundif 4445 | A condition equivalent to ... |
| difcom 4446 | Swap the arguments of a cl... |
| pssdifcom1 4447 | Two ways to express overla... |
| pssdifcom2 4448 | Two ways to express non-co... |
| difdifdir 4449 | Distributive law for class... |
| uneqdifeq 4450 | Two ways to say that ` A `... |
| raldifeq 4451 | Equality theorem for restr... |
| r19.2z 4452 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4453 | A response to the notion t... |
| r19.3rz 4454 | Restricted quantification ... |
| r19.28z 4455 | Restricted quantifier vers... |
| r19.3rzv 4456 | Restricted quantification ... |
| r19.9rzv 4457 | Restricted quantification ... |
| r19.28zv 4458 | Restricted quantifier vers... |
| r19.37zv 4459 | Restricted quantifier vers... |
| r19.45zv 4460 | Restricted version of Theo... |
| r19.44zv 4461 | Restricted version of Theo... |
| r19.27z 4462 | Restricted quantifier vers... |
| r19.27zv 4463 | Restricted quantifier vers... |
| r19.36zv 4464 | Restricted quantifier vers... |
| ralidmw 4465 | Idempotent law for restric... |
| rzal 4466 | Vacuous quantification is ... |
| rzalALT 4467 | Alternate proof of ~ rzal ... |
| rexn0 4468 | Restricted existential qua... |
| ralidm 4469 | Idempotent law for restric... |
| ral0 4470 | Vacuous universal quantifi... |
| ralf0 4471 | The quantification of a fa... |
| rexn0OLD 4472 | Obsolete version of ~ rexn... |
| ralidmOLD 4473 | Obsolete version of ~ rali... |
| ral0OLD 4474 | Obsolete version of ~ ral0... |
| ralf0OLD 4475 | Obsolete version of ~ ralf... |
| ralnralall 4476 | A contradiction concerning... |
| falseral0 4477 | A false statement can only... |
| raaan 4478 | Rearrange restricted quant... |
| raaanv 4479 | Rearrange restricted quant... |
| sbss 4480 | Set substitution into the ... |
| sbcssg 4481 | Distribute proper substitu... |
| raaan2 4482 | Rearrange restricted quant... |
| 2reu4lem 4483 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4484 | Definition of double restr... |
| csbdif 4485 | Distribution of class subs... |
| dfif2 4488 | An alternate definition of... |
| dfif6 4489 | An alternate definition of... |
| ifeq1 4490 | Equality theorem for condi... |
| ifeq2 4491 | Equality theorem for condi... |
| iftrue 4492 | Value of the conditional o... |
| iftruei 4493 | Inference associated with ... |
| iftrued 4494 | Value of the conditional o... |
| iffalse 4495 | Value of the conditional o... |
| iffalsei 4496 | Inference associated with ... |
| iffalsed 4497 | Value of the conditional o... |
| ifnefalse 4498 | When values are unequal, b... |
| ifsb 4499 | Distribute a function over... |
| dfif3 4500 | Alternate definition of th... |
| dfif4 4501 | Alternate definition of th... |
| dfif5 4502 | Alternate definition of th... |
| ifssun 4503 | A conditional class is inc... |
| ifeq12 4504 | Equality theorem for condi... |
| ifeq1d 4505 | Equality deduction for con... |
| ifeq2d 4506 | Equality deduction for con... |
| ifeq12d 4507 | Equality deduction for con... |
| ifbi 4508 | Equivalence theorem for co... |
| ifbid 4509 | Equivalence deduction for ... |
| ifbieq1d 4510 | Equivalence/equality deduc... |
| ifbieq2i 4511 | Equivalence/equality infer... |
| ifbieq2d 4512 | Equivalence/equality deduc... |
| ifbieq12i 4513 | Equivalence deduction for ... |
| ifbieq12d 4514 | Equivalence deduction for ... |
| nfifd 4515 | Deduction form of ~ nfif .... |
| nfif 4516 | Bound-variable hypothesis ... |
| ifeq1da 4517 | Conditional equality. (Co... |
| ifeq2da 4518 | Conditional equality. (Co... |
| ifeq12da 4519 | Equivalence deduction for ... |
| ifbieq12d2 4520 | Equivalence deduction for ... |
| ifclda 4521 | Conditional closure. (Con... |
| ifeqda 4522 | Separation of the values o... |
| elimif 4523 | Elimination of a condition... |
| ifbothda 4524 | A wff ` th ` containing a ... |
| ifboth 4525 | A wff ` th ` containing a ... |
| ifid 4526 | Identical true and false a... |
| eqif 4527 | Expansion of an equality w... |
| ifval 4528 | Another expression of the ... |
| elif 4529 | Membership in a conditiona... |
| ifel 4530 | Membership of a conditiona... |
| ifcl 4531 | Membership (closure) of a ... |
| ifcld 4532 | Membership (closure) of a ... |
| ifcli 4533 | Inference associated with ... |
| ifexd 4534 | Existence of the condition... |
| ifexg 4535 | Existence of the condition... |
| ifex 4536 | Existence of the condition... |
| ifeqor 4537 | The possible values of a c... |
| ifnot 4538 | Negating the first argumen... |
| ifan 4539 | Rewrite a conjunction in a... |
| ifor 4540 | Rewrite a disjunction in a... |
| 2if2 4541 | Resolve two nested conditi... |
| ifcomnan 4542 | Commute the conditions in ... |
| csbif 4543 | Distribute proper substitu... |
| dedth 4544 | Weak deduction theorem tha... |
| dedth2h 4545 | Weak deduction theorem eli... |
| dedth3h 4546 | Weak deduction theorem eli... |
| dedth4h 4547 | Weak deduction theorem eli... |
| dedth2v 4548 | Weak deduction theorem for... |
| dedth3v 4549 | Weak deduction theorem for... |
| dedth4v 4550 | Weak deduction theorem for... |
| elimhyp 4551 | Eliminate a hypothesis con... |
| elimhyp2v 4552 | Eliminate a hypothesis con... |
| elimhyp3v 4553 | Eliminate a hypothesis con... |
| elimhyp4v 4554 | Eliminate a hypothesis con... |
| elimel 4555 | Eliminate a membership hyp... |
| elimdhyp 4556 | Version of ~ elimhyp where... |
| keephyp 4557 | Transform a hypothesis ` p... |
| keephyp2v 4558 | Keep a hypothesis containi... |
| keephyp3v 4559 | Keep a hypothesis containi... |
| pwjust 4561 | Soundness justification th... |
| elpwg 4563 | Membership in a power clas... |
| elpw 4564 | Membership in a power clas... |
| velpw 4565 | Setvar variable membership... |
| elpwd 4566 | Membership in a power clas... |
| elpwi 4567 | Subset relation implied by... |
| elpwb 4568 | Characterization of the el... |
| elpwid 4569 | An element of a power clas... |
| elelpwi 4570 | If ` A ` belongs to a part... |
| sspw 4571 | The powerclass preserves i... |
| sspwi 4572 | The powerclass preserves i... |
| sspwd 4573 | The powerclass preserves i... |
| pweq 4574 | Equality theorem for power... |
| pweqALT 4575 | Alternate proof of ~ pweq ... |
| pweqi 4576 | Equality inference for pow... |
| pweqd 4577 | Equality deduction for pow... |
| pwunss 4578 | The power class of the uni... |
| nfpw 4579 | Bound-variable hypothesis ... |
| pwidg 4580 | A set is an element of its... |
| pwidb 4581 | A class is an element of i... |
| pwid 4582 | A set is a member of its p... |
| pwss 4583 | Subclass relationship for ... |
| pwundif 4584 | Break up the power class o... |
| snjust 4585 | Soundness justification th... |
| sneq 4596 | Equality theorem for singl... |
| sneqi 4597 | Equality inference for sin... |
| sneqd 4598 | Equality deduction for sin... |
| dfsn2 4599 | Alternate definition of si... |
| elsng 4600 | There is exactly one eleme... |
| elsn 4601 | There is exactly one eleme... |
| velsn 4602 | There is only one element ... |
| elsni 4603 | There is at most one eleme... |
| absn 4604 | Condition for a class abst... |
| dfpr2 4605 | Alternate definition of a ... |
| dfsn2ALT 4606 | Alternate definition of si... |
| elprg 4607 | A member of a pair of clas... |
| elpri 4608 | If a class is an element o... |
| elpr 4609 | A member of a pair of clas... |
| elpr2g 4610 | A member of a pair of sets... |
| elpr2 4611 | A member of a pair of sets... |
| elpr2OLD 4612 | Obsolete version of ~ elpr... |
| nelpr2 4613 | If a class is not an eleme... |
| nelpr1 4614 | If a class is not an eleme... |
| nelpri 4615 | If an element doesn't matc... |
| prneli 4616 | If an element doesn't matc... |
| nelprd 4617 | If an element doesn't matc... |
| eldifpr 4618 | Membership in a set with t... |
| rexdifpr 4619 | Restricted existential qua... |
| snidg 4620 | A set is a member of its s... |
| snidb 4621 | A class is a set iff it is... |
| snid 4622 | A set is a member of its s... |
| vsnid 4623 | A setvar variable is a mem... |
| elsn2g 4624 | There is exactly one eleme... |
| elsn2 4625 | There is exactly one eleme... |
| nelsn 4626 | If a class is not equal to... |
| rabeqsn 4627 | Conditions for a restricte... |
| rabsssn 4628 | Conditions for a restricte... |
| ralsnsg 4629 | Substitution expressed in ... |
| rexsns 4630 | Restricted existential qua... |
| rexsngf 4631 | Restricted existential qua... |
| ralsngf 4632 | Restricted universal quant... |
| reusngf 4633 | Restricted existential uni... |
| ralsng 4634 | Substitution expressed in ... |
| rexsng 4635 | Restricted existential qua... |
| reusng 4636 | Restricted existential uni... |
| 2ralsng 4637 | Substitution expressed in ... |
| ralsngOLD 4638 | Obsolete version of ~ rals... |
| rexsngOLD 4639 | Obsolete version of ~ rexs... |
| rexreusng 4640 | Restricted existential uni... |
| exsnrex 4641 | There is a set being the e... |
| ralsn 4642 | Convert a universal quanti... |
| rexsn 4643 | Convert an existential qua... |
| elpwunsn 4644 | Membership in an extension... |
| eqoreldif 4645 | An element of a set is eit... |
| eltpg 4646 | Members of an unordered tr... |
| eldiftp 4647 | Membership in a set with t... |
| eltpi 4648 | A member of an unordered t... |
| eltp 4649 | A member of an unordered t... |
| dftp2 4650 | Alternate definition of un... |
| nfpr 4651 | Bound-variable hypothesis ... |
| ifpr 4652 | Membership of a conditiona... |
| ralprgf 4653 | Convert a restricted unive... |
| rexprgf 4654 | Convert a restricted exist... |
| ralprg 4655 | Convert a restricted unive... |
| ralprgOLD 4656 | Obsolete version of ~ ralp... |
| rexprg 4657 | Convert a restricted exist... |
| rexprgOLD 4658 | Obsolete version of ~ rexp... |
| raltpg 4659 | Convert a restricted unive... |
| rextpg 4660 | Convert a restricted exist... |
| ralpr 4661 | Convert a restricted unive... |
| rexpr 4662 | Convert a restricted exist... |
| reuprg0 4663 | Convert a restricted exist... |
| reuprg 4664 | Convert a restricted exist... |
| reurexprg 4665 | Convert a restricted exist... |
| raltp 4666 | Convert a universal quanti... |
| rextp 4667 | Convert an existential qua... |
| nfsn 4668 | Bound-variable hypothesis ... |
| csbsng 4669 | Distribute proper substitu... |
| csbprg 4670 | Distribute proper substitu... |
| elinsn 4671 | If the intersection of two... |
| disjsn 4672 | Intersection with the sing... |
| disjsn2 4673 | Two distinct singletons ar... |
| disjpr2 4674 | Two completely distinct un... |
| disjprsn 4675 | The disjoint intersection ... |
| disjtpsn 4676 | The disjoint intersection ... |
| disjtp2 4677 | Two completely distinct un... |
| snprc 4678 | The singleton of a proper ... |
| snnzb 4679 | A singleton is nonempty if... |
| rmosn 4680 | A restricted at-most-one q... |
| r19.12sn 4681 | Special case of ~ r19.12 w... |
| rabsn 4682 | Condition where a restrict... |
| rabsnifsb 4683 | A restricted class abstrac... |
| rabsnif 4684 | A restricted class abstrac... |
| rabrsn 4685 | A restricted class abstrac... |
| euabsn2 4686 | Another way to express exi... |
| euabsn 4687 | Another way to express exi... |
| reusn 4688 | A way to express restricte... |
| absneu 4689 | Restricted existential uni... |
| rabsneu 4690 | Restricted existential uni... |
| eusn 4691 | Two ways to express " ` A ... |
| rabsnt 4692 | Truth implied by equality ... |
| prcom 4693 | Commutative law for unorde... |
| preq1 4694 | Equality theorem for unord... |
| preq2 4695 | Equality theorem for unord... |
| preq12 4696 | Equality theorem for unord... |
| preq1i 4697 | Equality inference for uno... |
| preq2i 4698 | Equality inference for uno... |
| preq12i 4699 | Equality inference for uno... |
| preq1d 4700 | Equality deduction for uno... |
| preq2d 4701 | Equality deduction for uno... |
| preq12d 4702 | Equality deduction for uno... |
| tpeq1 4703 | Equality theorem for unord... |
| tpeq2 4704 | Equality theorem for unord... |
| tpeq3 4705 | Equality theorem for unord... |
| tpeq1d 4706 | Equality theorem for unord... |
| tpeq2d 4707 | Equality theorem for unord... |
| tpeq3d 4708 | Equality theorem for unord... |
| tpeq123d 4709 | Equality theorem for unord... |
| tprot 4710 | Rotation of the elements o... |
| tpcoma 4711 | Swap 1st and 2nd members o... |
| tpcomb 4712 | Swap 2nd and 3rd members o... |
| tpass 4713 | Split off the first elemen... |
| qdass 4714 | Two ways to write an unord... |
| qdassr 4715 | Two ways to write an unord... |
| tpidm12 4716 | Unordered triple ` { A , A... |
| tpidm13 4717 | Unordered triple ` { A , B... |
| tpidm23 4718 | Unordered triple ` { A , B... |
| tpidm 4719 | Unordered triple ` { A , A... |
| tppreq3 4720 | An unordered triple is an ... |
| prid1g 4721 | An unordered pair contains... |
| prid2g 4722 | An unordered pair contains... |
| prid1 4723 | An unordered pair contains... |
| prid2 4724 | An unordered pair contains... |
| ifpprsnss 4725 | An unordered pair is a sin... |
| prprc1 4726 | A proper class vanishes in... |
| prprc2 4727 | A proper class vanishes in... |
| prprc 4728 | An unordered pair containi... |
| tpid1 4729 | One of the three elements ... |
| tpid1g 4730 | Closed theorem form of ~ t... |
| tpid2 4731 | One of the three elements ... |
| tpid2g 4732 | Closed theorem form of ~ t... |
| tpid3g 4733 | Closed theorem form of ~ t... |
| tpid3 4734 | One of the three elements ... |
| snnzg 4735 | The singleton of a set is ... |
| snn0d 4736 | The singleton of a set is ... |
| snnz 4737 | The singleton of a set is ... |
| prnz 4738 | A pair containing a set is... |
| prnzg 4739 | A pair containing a set is... |
| tpnz 4740 | An unordered triple contai... |
| tpnzd 4741 | An unordered triple contai... |
| raltpd 4742 | Convert a universal quanti... |
| snssb 4743 | Characterization of the in... |
| snssg 4744 | The singleton formed on a ... |
| snssgOLD 4745 | Obsolete version of ~ snss... |
| snss 4746 | The singleton of an elemen... |
| eldifsn 4747 | Membership in a set with a... |
| ssdifsn 4748 | Subset of a set with an el... |
| elpwdifsn 4749 | A subset of a set is an el... |
| eldifsni 4750 | Membership in a set with a... |
| eldifsnneq 4751 | An element of a difference... |
| neldifsn 4752 | The class ` A ` is not in ... |
| neldifsnd 4753 | The class ` A ` is not in ... |
| rexdifsn 4754 | Restricted existential qua... |
| raldifsni 4755 | Rearrangement of a propert... |
| raldifsnb 4756 | Restricted universal quant... |
| eldifvsn 4757 | A set is an element of the... |
| difsn 4758 | An element not in a set ca... |
| difprsnss 4759 | Removal of a singleton fro... |
| difprsn1 4760 | Removal of a singleton fro... |
| difprsn2 4761 | Removal of a singleton fro... |
| diftpsn3 4762 | Removal of a singleton fro... |
| difpr 4763 | Removing two elements as p... |
| tpprceq3 4764 | An unordered triple is an ... |
| tppreqb 4765 | An unordered triple is an ... |
| difsnb 4766 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4767 | ` ( B \ { A } ) ` is a pro... |
| snssi 4768 | The singleton of an elemen... |
| snssd 4769 | The singleton of an elemen... |
| difsnid 4770 | If we remove a single elem... |
| eldifeldifsn 4771 | An element of a difference... |
| pw0 4772 | Compute the power set of t... |
| pwpw0 4773 | Compute the power set of t... |
| snsspr1 4774 | A singleton is a subset of... |
| snsspr2 4775 | A singleton is a subset of... |
| snsstp1 4776 | A singleton is a subset of... |
| snsstp2 4777 | A singleton is a subset of... |
| snsstp3 4778 | A singleton is a subset of... |
| prssg 4779 | A pair of elements of a cl... |
| prss 4780 | A pair of elements of a cl... |
| prssi 4781 | A pair of elements of a cl... |
| prssd 4782 | Deduction version of ~ prs... |
| prsspwg 4783 | An unordered pair belongs ... |
| ssprss 4784 | A pair as subset of a pair... |
| ssprsseq 4785 | A proper pair is a subset ... |
| sssn 4786 | The subsets of a singleton... |
| ssunsn2 4787 | The property of being sand... |
| ssunsn 4788 | Possible values for a set ... |
| eqsn 4789 | Two ways to express that a... |
| issn 4790 | A sufficient condition for... |
| n0snor2el 4791 | A nonempty set is either a... |
| ssunpr 4792 | Possible values for a set ... |
| sspr 4793 | The subsets of a pair. (C... |
| sstp 4794 | The subsets of an unordere... |
| tpss 4795 | An unordered triple of ele... |
| tpssi 4796 | An unordered triple of ele... |
| sneqrg 4797 | Closed form of ~ sneqr . ... |
| sneqr 4798 | If the singletons of two s... |
| snsssn 4799 | If a singleton is a subset... |
| mosneq 4800 | There exists at most one s... |
| sneqbg 4801 | Two singletons of sets are... |
| snsspw 4802 | The singleton of a class i... |
| prsspw 4803 | An unordered pair belongs ... |
| preq1b 4804 | Biconditional equality lem... |
| preq2b 4805 | Biconditional equality lem... |
| preqr1 4806 | Reverse equality lemma for... |
| preqr2 4807 | Reverse equality lemma for... |
| preq12b 4808 | Equality relationship for ... |
| opthpr 4809 | An unordered pair has the ... |
| preqr1g 4810 | Reverse equality lemma for... |
| preq12bg 4811 | Closed form of ~ preq12b .... |
| prneimg 4812 | Two pairs are not equal if... |
| prnebg 4813 | A (proper) pair is not equ... |
| pr1eqbg 4814 | A (proper) pair is equal t... |
| pr1nebg 4815 | A (proper) pair is not equ... |
| preqsnd 4816 | Equivalence for a pair equ... |
| prnesn 4817 | A proper unordered pair is... |
| prneprprc 4818 | A proper unordered pair is... |
| preqsn 4819 | Equivalence for a pair equ... |
| preq12nebg 4820 | Equality relationship for ... |
| prel12g 4821 | Equality of two unordered ... |
| opthprneg 4822 | An unordered pair has the ... |
| elpreqprlem 4823 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4824 | Equality and membership ru... |
| elpreqprb 4825 | A set is an element of an ... |
| elpr2elpr 4826 | For an element ` A ` of an... |
| dfopif 4827 | Rewrite ~ df-op using ` if... |
| dfopg 4828 | Value of the ordered pair ... |
| dfop 4829 | Value of an ordered pair w... |
| opeq1 4830 | Equality theorem for order... |
| opeq2 4831 | Equality theorem for order... |
| opeq12 4832 | Equality theorem for order... |
| opeq1i 4833 | Equality inference for ord... |
| opeq2i 4834 | Equality inference for ord... |
| opeq12i 4835 | Equality inference for ord... |
| opeq1d 4836 | Equality deduction for ord... |
| opeq2d 4837 | Equality deduction for ord... |
| opeq12d 4838 | Equality deduction for ord... |
| oteq1 4839 | Equality theorem for order... |
| oteq2 4840 | Equality theorem for order... |
| oteq3 4841 | Equality theorem for order... |
| oteq1d 4842 | Equality deduction for ord... |
| oteq2d 4843 | Equality deduction for ord... |
| oteq3d 4844 | Equality deduction for ord... |
| oteq123d 4845 | Equality deduction for ord... |
| nfop 4846 | Bound-variable hypothesis ... |
| nfopd 4847 | Deduction version of bound... |
| csbopg 4848 | Distribution of class subs... |
| opidg 4849 | The ordered pair ` <. A , ... |
| opid 4850 | The ordered pair ` <. A , ... |
| ralunsn 4851 | Restricted quantification ... |
| 2ralunsn 4852 | Double restricted quantifi... |
| opprc 4853 | Expansion of an ordered pa... |
| opprc1 4854 | Expansion of an ordered pa... |
| opprc2 4855 | Expansion of an ordered pa... |
| oprcl 4856 | If an ordered pair has an ... |
| pwsn 4857 | The power set of a singlet... |
| pwsnOLD 4858 | Obsolete version of ~ pwsn... |
| pwpr 4859 | The power set of an unorde... |
| pwtp 4860 | The power set of an unorde... |
| pwpwpw0 4861 | Compute the power set of t... |
| pwv 4862 | The power class of the uni... |
| prproe 4863 | For an element of a proper... |
| 3elpr2eq 4864 | If there are three element... |
| dfuni2 4867 | Alternate definition of cl... |
| eluni 4868 | Membership in class union.... |
| eluni2 4869 | Membership in class union.... |
| elunii 4870 | Membership in class union.... |
| nfunid 4871 | Deduction version of ~ nfu... |
| nfuni 4872 | Bound-variable hypothesis ... |
| uniss 4873 | Subclass relationship for ... |
| unissi 4874 | Subclass relationship for ... |
| unissd 4875 | Subclass relationship for ... |
| unieq 4876 | Equality theorem for class... |
| unieqOLD 4877 | Obsolete version of ~ unie... |
| unieqi 4878 | Inference of equality of t... |
| unieqd 4879 | Deduction of equality of t... |
| eluniab 4880 | Membership in union of a c... |
| elunirab 4881 | Membership in union of a c... |
| uniprg 4882 | The union of a pair is the... |
| unipr 4883 | The union of a pair is the... |
| uniprOLD 4884 | Obsolete version of ~ unip... |
| uniprgOLD 4885 | Obsolete version of ~ unip... |
| unisng 4886 | A set equals the union of ... |
| unisn 4887 | A set equals the union of ... |
| unisnv 4888 | A set equals the union of ... |
| unisn3 4889 | Union of a singleton in th... |
| dfnfc2 4890 | An alternative statement o... |
| uniun 4891 | The class union of the uni... |
| uniin 4892 | The class union of the int... |
| ssuni 4893 | Subclass relationship for ... |
| uni0b 4894 | The union of a set is empt... |
| uni0c 4895 | The union of a set is empt... |
| uni0 4896 | The union of the empty set... |
| csbuni 4897 | Distribute proper substitu... |
| elssuni 4898 | An element of a class is a... |
| unissel 4899 | Condition turning a subcla... |
| unissb 4900 | Relationship involving mem... |
| unissbOLD 4901 | Obsolete version of ~ unis... |
| uniss2 4902 | A subclass condition on th... |
| unidif 4903 | If the difference ` A \ B ... |
| ssunieq 4904 | Relationship implying unio... |
| unimax 4905 | Any member of a class is t... |
| pwuni 4906 | A class is a subclass of t... |
| dfint2 4909 | Alternate definition of cl... |
| inteq 4910 | Equality law for intersect... |
| inteqi 4911 | Equality inference for cla... |
| inteqd 4912 | Equality deduction for cla... |
| elint 4913 | Membership in class inters... |
| elint2 4914 | Membership in class inters... |
| elintg 4915 | Membership in class inters... |
| elinti 4916 | Membership in class inters... |
| nfint 4917 | Bound-variable hypothesis ... |
| elintabg 4918 | Two ways of saying a set i... |
| elintab 4919 | Membership in the intersec... |
| elintabOLD 4920 | Obsolete version of ~ elin... |
| elintrab 4921 | Membership in the intersec... |
| elintrabg 4922 | Membership in the intersec... |
| int0 4923 | The intersection of the em... |
| intss1 4924 | An element of a class incl... |
| ssint 4925 | Subclass of a class inters... |
| ssintab 4926 | Subclass of the intersecti... |
| ssintub 4927 | Subclass of the least uppe... |
| ssmin 4928 | Subclass of the minimum va... |
| intmin 4929 | Any member of a class is t... |
| intss 4930 | Intersection of subclasses... |
| intssuni 4931 | The intersection of a none... |
| ssintrab 4932 | Subclass of the intersecti... |
| unissint 4933 | If the union of a class is... |
| intssuni2 4934 | Subclass relationship for ... |
| intminss 4935 | Under subset ordering, the... |
| intmin2 4936 | Any set is the smallest of... |
| intmin3 4937 | Under subset ordering, the... |
| intmin4 4938 | Elimination of a conjunct ... |
| intab 4939 | The intersection of a spec... |
| int0el 4940 | The intersection of a clas... |
| intun 4941 | The class intersection of ... |
| intprg 4942 | The intersection of a pair... |
| intpr 4943 | The intersection of a pair... |
| intprOLD 4944 | Obsolete version of ~ intp... |
| intprgOLD 4945 | Obsolete version of ~ intp... |
| intsng 4946 | Intersection of a singleto... |
| intsn 4947 | The intersection of a sing... |
| uniintsn 4948 | Two ways to express " ` A ... |
| uniintab 4949 | The union and the intersec... |
| intunsn 4950 | Theorem joining a singleto... |
| rint0 4951 | Relative intersection of a... |
| elrint 4952 | Membership in a restricted... |
| elrint2 4953 | Membership in a restricted... |
| eliun 4958 | Membership in indexed unio... |
| eliin 4959 | Membership in indexed inte... |
| eliuni 4960 | Membership in an indexed u... |
| iuncom 4961 | Commutation of indexed uni... |
| iuncom4 4962 | Commutation of union with ... |
| iunconst 4963 | Indexed union of a constan... |
| iinconst 4964 | Indexed intersection of a ... |
| iuneqconst 4965 | Indexed union of identical... |
| iuniin 4966 | Law combining indexed unio... |
| iinssiun 4967 | An indexed intersection is... |
| iunss1 4968 | Subclass theorem for index... |
| iinss1 4969 | Subclass theorem for index... |
| iuneq1 4970 | Equality theorem for index... |
| iineq1 4971 | Equality theorem for index... |
| ss2iun 4972 | Subclass theorem for index... |
| iuneq2 4973 | Equality theorem for index... |
| iineq2 4974 | Equality theorem for index... |
| iuneq2i 4975 | Equality inference for ind... |
| iineq2i 4976 | Equality inference for ind... |
| iineq2d 4977 | Equality deduction for ind... |
| iuneq2dv 4978 | Equality deduction for ind... |
| iineq2dv 4979 | Equality deduction for ind... |
| iuneq12df 4980 | Equality deduction for ind... |
| iuneq1d 4981 | Equality theorem for index... |
| iuneq12d 4982 | Equality deduction for ind... |
| iuneq2d 4983 | Equality deduction for ind... |
| nfiun 4984 | Bound-variable hypothesis ... |
| nfiin 4985 | Bound-variable hypothesis ... |
| nfiung 4986 | Bound-variable hypothesis ... |
| nfiing 4987 | Bound-variable hypothesis ... |
| nfiu1 4988 | Bound-variable hypothesis ... |
| nfii1 4989 | Bound-variable hypothesis ... |
| dfiun2g 4990 | Alternate definition of in... |
| dfiun2gOLD 4991 | Obsolete version of ~ dfiu... |
| dfiin2g 4992 | Alternate definition of in... |
| dfiun2 4993 | Alternate definition of in... |
| dfiin2 4994 | Alternate definition of in... |
| dfiunv2 4995 | Define double indexed unio... |
| cbviun 4996 | Rule used to change the bo... |
| cbviin 4997 | Change bound variables in ... |
| cbviung 4998 | Rule used to change the bo... |
| cbviing 4999 | Change bound variables in ... |
| cbviunv 5000 | Rule used to change the bo... |
| cbviinv 5001 | Change bound variables in ... |
| cbviunvg 5002 | Rule used to change the bo... |
| cbviinvg 5003 | Change bound variables in ... |
| iunssf 5004 | Subset theorem for an inde... |
| iunss 5005 | Subset theorem for an inde... |
| ssiun 5006 | Subset implication for an ... |
| ssiun2 5007 | Identity law for subset of... |
| ssiun2s 5008 | Subset relationship for an... |
| iunss2 5009 | A subclass condition on th... |
| iunssd 5010 | Subset theorem for an inde... |
| iunab 5011 | The indexed union of a cla... |
| iunrab 5012 | The indexed union of a res... |
| iunxdif2 5013 | Indexed union with a class... |
| ssiinf 5014 | Subset theorem for an inde... |
| ssiin 5015 | Subset theorem for an inde... |
| iinss 5016 | Subset implication for an ... |
| iinss2 5017 | An indexed intersection is... |
| uniiun 5018 | Class union in terms of in... |
| intiin 5019 | Class intersection in term... |
| iunid 5020 | An indexed union of single... |
| iunidOLD 5021 | Obsolete version of ~ iuni... |
| iun0 5022 | An indexed union of the em... |
| 0iun 5023 | An empty indexed union is ... |
| 0iin 5024 | An empty indexed intersect... |
| viin 5025 | Indexed intersection with ... |
| iunsn 5026 | Indexed union of a singlet... |
| iunn0 5027 | There is a nonempty class ... |
| iinab 5028 | Indexed intersection of a ... |
| iinrab 5029 | Indexed intersection of a ... |
| iinrab2 5030 | Indexed intersection of a ... |
| iunin2 5031 | Indexed union of intersect... |
| iunin1 5032 | Indexed union of intersect... |
| iinun2 5033 | Indexed intersection of un... |
| iundif2 5034 | Indexed union of class dif... |
| iindif1 5035 | Indexed intersection of cl... |
| 2iunin 5036 | Rearrange indexed unions o... |
| iindif2 5037 | Indexed intersection of cl... |
| iinin2 5038 | Indexed intersection of in... |
| iinin1 5039 | Indexed intersection of in... |
| iinvdif 5040 | The indexed intersection o... |
| elriin 5041 | Elementhood in a relative ... |
| riin0 5042 | Relative intersection of a... |
| riinn0 5043 | Relative intersection of a... |
| riinrab 5044 | Relative intersection of a... |
| symdif0 5045 | Symmetric difference with ... |
| symdifv 5046 | The symmetric difference w... |
| symdifid 5047 | The symmetric difference o... |
| iinxsng 5048 | A singleton index picks ou... |
| iinxprg 5049 | Indexed intersection with ... |
| iunxsng 5050 | A singleton index picks ou... |
| iunxsn 5051 | A singleton index picks ou... |
| iunxsngf 5052 | A singleton index picks ou... |
| iunun 5053 | Separate a union in an ind... |
| iunxun 5054 | Separate a union in the in... |
| iunxdif3 5055 | An indexed union where som... |
| iunxprg 5056 | A pair index picks out two... |
| iunxiun 5057 | Separate an indexed union ... |
| iinuni 5058 | A relationship involving u... |
| iununi 5059 | A relationship involving u... |
| sspwuni 5060 | Subclass relationship for ... |
| pwssb 5061 | Two ways to express a coll... |
| elpwpw 5062 | Characterization of the el... |
| pwpwab 5063 | The double power class wri... |
| pwpwssunieq 5064 | The class of sets whose un... |
| elpwuni 5065 | Relationship for power cla... |
| iinpw 5066 | The power class of an inte... |
| iunpwss 5067 | Inclusion of an indexed un... |
| intss2 5068 | A nonempty intersection of... |
| rintn0 5069 | Relative intersection of a... |
| dfdisj2 5072 | Alternate definition for d... |
| disjss2 5073 | If each element of a colle... |
| disjeq2 5074 | Equality theorem for disjo... |
| disjeq2dv 5075 | Equality deduction for dis... |
| disjss1 5076 | A subset of a disjoint col... |
| disjeq1 5077 | Equality theorem for disjo... |
| disjeq1d 5078 | Equality theorem for disjo... |
| disjeq12d 5079 | Equality theorem for disjo... |
| cbvdisj 5080 | Change bound variables in ... |
| cbvdisjv 5081 | Change bound variables in ... |
| nfdisjw 5082 | Bound-variable hypothesis ... |
| nfdisj 5083 | Bound-variable hypothesis ... |
| nfdisj1 5084 | Bound-variable hypothesis ... |
| disjor 5085 | Two ways to say that a col... |
| disjors 5086 | Two ways to say that a col... |
| disji2 5087 | Property of a disjoint col... |
| disji 5088 | Property of a disjoint col... |
| invdisj 5089 | If there is a function ` C... |
| invdisjrabw 5090 | Version of ~ invdisjrab wi... |
| invdisjrab 5091 | The restricted class abstr... |
| disjiun 5092 | A disjoint collection yiel... |
| disjord 5093 | Conditions for a collectio... |
| disjiunb 5094 | Two ways to say that a col... |
| disjiund 5095 | Conditions for a collectio... |
| sndisj 5096 | Any collection of singleto... |
| 0disj 5097 | Any collection of empty se... |
| disjxsn 5098 | A singleton collection is ... |
| disjx0 5099 | An empty collection is dis... |
| disjprgw 5100 | Version of ~ disjprg with ... |
| disjprg 5101 | A pair collection is disjo... |
| disjxiun 5102 | An indexed union of a disj... |
| disjxun 5103 | The union of two disjoint ... |
| disjss3 5104 | Expand a disjoint collecti... |
| breq 5107 | Equality theorem for binar... |
| breq1 5108 | Equality theorem for a bin... |
| breq2 5109 | Equality theorem for a bin... |
| breq12 5110 | Equality theorem for a bin... |
| breqi 5111 | Equality inference for bin... |
| breq1i 5112 | Equality inference for a b... |
| breq2i 5113 | Equality inference for a b... |
| breq12i 5114 | Equality inference for a b... |
| breq1d 5115 | Equality deduction for a b... |
| breqd 5116 | Equality deduction for a b... |
| breq2d 5117 | Equality deduction for a b... |
| breq12d 5118 | Equality deduction for a b... |
| breq123d 5119 | Equality deduction for a b... |
| breqdi 5120 | Equality deduction for a b... |
| breqan12d 5121 | Equality deduction for a b... |
| breqan12rd 5122 | Equality deduction for a b... |
| eqnbrtrd 5123 | Substitution of equal clas... |
| nbrne1 5124 | Two classes are different ... |
| nbrne2 5125 | Two classes are different ... |
| eqbrtri 5126 | Substitution of equal clas... |
| eqbrtrd 5127 | Substitution of equal clas... |
| eqbrtrri 5128 | Substitution of equal clas... |
| eqbrtrrd 5129 | Substitution of equal clas... |
| breqtri 5130 | Substitution of equal clas... |
| breqtrd 5131 | Substitution of equal clas... |
| breqtrri 5132 | Substitution of equal clas... |
| breqtrrd 5133 | Substitution of equal clas... |
| 3brtr3i 5134 | Substitution of equality i... |
| 3brtr4i 5135 | Substitution of equality i... |
| 3brtr3d 5136 | Substitution of equality i... |
| 3brtr4d 5137 | Substitution of equality i... |
| 3brtr3g 5138 | Substitution of equality i... |
| 3brtr4g 5139 | Substitution of equality i... |
| eqbrtrid 5140 | A chained equality inferen... |
| eqbrtrrid 5141 | A chained equality inferen... |
| breqtrid 5142 | A chained equality inferen... |
| breqtrrid 5143 | A chained equality inferen... |
| eqbrtrdi 5144 | A chained equality inferen... |
| eqbrtrrdi 5145 | A chained equality inferen... |
| breqtrdi 5146 | A chained equality inferen... |
| breqtrrdi 5147 | A chained equality inferen... |
| ssbrd 5148 | Deduction from a subclass ... |
| ssbr 5149 | Implication from a subclas... |
| ssbri 5150 | Inference from a subclass ... |
| nfbrd 5151 | Deduction version of bound... |
| nfbr 5152 | Bound-variable hypothesis ... |
| brab1 5153 | Relationship between a bin... |
| br0 5154 | The empty binary relation ... |
| brne0 5155 | If two sets are in a binar... |
| brun 5156 | The union of two binary re... |
| brin 5157 | The intersection of two re... |
| brdif 5158 | The difference of two bina... |
| sbcbr123 5159 | Move substitution in and o... |
| sbcbr 5160 | Move substitution in and o... |
| sbcbr12g 5161 | Move substitution in and o... |
| sbcbr1g 5162 | Move substitution in and o... |
| sbcbr2g 5163 | Move substitution in and o... |
| brsymdif 5164 | Characterization of the sy... |
| brralrspcev 5165 | Restricted existential spe... |
| brimralrspcev 5166 | Restricted existential spe... |
| opabss 5169 | The collection of ordered ... |
| opabbid 5170 | Equivalent wff's yield equ... |
| opabbidv 5171 | Equivalent wff's yield equ... |
| opabbii 5172 | Equivalent wff's yield equ... |
| nfopabd 5173 | Bound-variable hypothesis ... |
| nfopab 5174 | Bound-variable hypothesis ... |
| nfopab1 5175 | The first abstraction vari... |
| nfopab2 5176 | The second abstraction var... |
| cbvopab 5177 | Rule used to change bound ... |
| cbvopabv 5178 | Rule used to change bound ... |
| cbvopabvOLD 5179 | Obsolete version of ~ cbvo... |
| cbvopab1 5180 | Change first bound variabl... |
| cbvopab1g 5181 | Change first bound variabl... |
| cbvopab2 5182 | Change second bound variab... |
| cbvopab1s 5183 | Change first bound variabl... |
| cbvopab1v 5184 | Rule used to change the fi... |
| cbvopab1vOLD 5185 | Obsolete version of ~ cbvo... |
| cbvopab2v 5186 | Rule used to change the se... |
| unopab 5187 | Union of two ordered pair ... |
| mpteq12da 5190 | An equality inference for ... |
| mpteq12df 5191 | An equality inference for ... |
| mpteq12dfOLD 5192 | Obsolete version of ~ mpte... |
| mpteq12f 5193 | An equality theorem for th... |
| mpteq12dva 5194 | An equality inference for ... |
| mpteq12dvaOLD 5195 | Obsolete version of ~ mpte... |
| mpteq12dv 5196 | An equality inference for ... |
| mpteq12 5197 | An equality theorem for th... |
| mpteq1 5198 | An equality theorem for th... |
| mpteq1OLD 5199 | Obsolete version of ~ mpte... |
| mpteq1d 5200 | An equality theorem for th... |
| mpteq1i 5201 | An equality theorem for th... |
| mpteq1iOLD 5202 | An equality theorem for th... |
| mpteq2da 5203 | Slightly more general equa... |
| mpteq2daOLD 5204 | Obsolete version of ~ mpte... |
| mpteq2dva 5205 | Slightly more general equa... |
| mpteq2dvaOLD 5206 | Obsolete version of ~ mpte... |
| mpteq2dv 5207 | An equality inference for ... |
| mpteq2ia 5208 | An equality inference for ... |
| mpteq2iaOLD 5209 | Obsolete version of ~ mpte... |
| mpteq2i 5210 | An equality inference for ... |
| mpteq12i 5211 | An equality inference for ... |
| nfmpt 5212 | Bound-variable hypothesis ... |
| nfmpt1 5213 | Bound-variable hypothesis ... |
| cbvmptf 5214 | Rule to change the bound v... |
| cbvmptfg 5215 | Rule to change the bound v... |
| cbvmpt 5216 | Rule to change the bound v... |
| cbvmptg 5217 | Rule to change the bound v... |
| cbvmptv 5218 | Rule to change the bound v... |
| cbvmptvOLD 5219 | Obsolete version of ~ cbvm... |
| cbvmptvg 5220 | Rule to change the bound v... |
| mptv 5221 | Function with universal do... |
| dftr2 5224 | An alternate way of defini... |
| dftr2c 5225 | Variant of ~ dftr2 with co... |
| dftr5 5226 | An alternate way of defini... |
| dftr5OLD 5227 | Obsolete version of ~ dftr... |
| dftr3 5228 | An alternate way of defini... |
| dftr4 5229 | An alternate way of defini... |
| treq 5230 | Equality theorem for the t... |
| trel 5231 | In a transitive class, the... |
| trel3 5232 | In a transitive class, the... |
| trss 5233 | An element of a transitive... |
| trin 5234 | The intersection of transi... |
| tr0 5235 | The empty set is transitiv... |
| trv 5236 | The universe is transitive... |
| triun 5237 | An indexed union of a clas... |
| truni 5238 | The union of a class of tr... |
| triin 5239 | An indexed intersection of... |
| trint 5240 | The intersection of a clas... |
| trintss 5241 | Any nonempty transitive cl... |
| axrep1 5243 | The version of the Axiom o... |
| axreplem 5244 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5245 | Axiom of Replacement expre... |
| axrep3 5246 | Axiom of Replacement sligh... |
| axrep4 5247 | A more traditional version... |
| axrep5 5248 | Axiom of Replacement (simi... |
| axrep6 5249 | A condensed form of ~ ax-r... |
| axrep6g 5250 | ~ axrep6 in class notation... |
| zfrepclf 5251 | An inference based on the ... |
| zfrep3cl 5252 | An inference based on the ... |
| zfrep4 5253 | A version of Replacement u... |
| axsepgfromrep 5254 | A more general version ~ a... |
| axsep 5255 | Axiom scheme of separation... |
| axsepg 5257 | A more general version of ... |
| zfauscl 5258 | Separation Scheme (Aussond... |
| bm1.3ii 5259 | Convert implication to equ... |
| ax6vsep 5260 | Derive ~ ax6v (a weakened ... |
| axnulALT 5261 | Alternate proof of ~ axnul... |
| axnul 5262 | The Null Set Axiom of ZF s... |
| 0ex 5264 | The Null Set Axiom of ZF s... |
| al0ssb 5265 | The empty set is the uniqu... |
| sseliALT 5266 | Alternate proof of ~ sseli... |
| csbexg 5267 | The existence of proper su... |
| csbex 5268 | The existence of proper su... |
| unisn2 5269 | A version of ~ unisn witho... |
| nalset 5270 | No set contains all sets. ... |
| vnex 5271 | The universal class does n... |
| vprc 5272 | The universal class is not... |
| nvel 5273 | The universal class does n... |
| inex1 5274 | Separation Scheme (Aussond... |
| inex2 5275 | Separation Scheme (Aussond... |
| inex1g 5276 | Closed-form, generalized S... |
| inex2g 5277 | Sufficient condition for a... |
| ssex 5278 | The subset of a set is als... |
| ssexi 5279 | The subset of a set is als... |
| ssexg 5280 | The subset of a set is als... |
| ssexd 5281 | A subclass of a set is a s... |
| prcssprc 5282 | The superclass of a proper... |
| sselpwd 5283 | Elementhood to a power set... |
| difexg 5284 | Existence of a difference.... |
| difexi 5285 | Existence of a difference,... |
| difexd 5286 | Existence of a difference.... |
| zfausab 5287 | Separation Scheme (Aussond... |
| rabexg 5288 | Separation Scheme in terms... |
| rabex 5289 | Separation Scheme in terms... |
| rabexd 5290 | Separation Scheme in terms... |
| rabex2 5291 | Separation Scheme in terms... |
| rab2ex 5292 | A class abstraction based ... |
| elssabg 5293 | Membership in a class abst... |
| intex 5294 | The intersection of a none... |
| intnex 5295 | If a class intersection is... |
| intexab 5296 | The intersection of a none... |
| intexrab 5297 | The intersection of a none... |
| iinexg 5298 | The existence of a class i... |
| intabs 5299 | Absorption of a redundant ... |
| inuni 5300 | The intersection of a unio... |
| elpw2g 5301 | Membership in a power clas... |
| elpw2 5302 | Membership in a power clas... |
| elpwi2 5303 | Membership in a power clas... |
| elpwi2OLD 5304 | Obsolete version of ~ elpw... |
| axpweq 5305 | Two equivalent ways to exp... |
| pwnss 5306 | The power set of a set is ... |
| pwne 5307 | No set equals its power se... |
| difelpw 5308 | A difference is an element... |
| rabelpw 5309 | A restricted class abstrac... |
| class2set 5310 | The class of elements of `... |
| 0elpw 5311 | Every power class contains... |
| pwne0 5312 | A power class is never emp... |
| 0nep0 5313 | The empty set and its powe... |
| 0inp0 5314 | Something cannot be equal ... |
| unidif0 5315 | The removal of the empty s... |
| eqsnuniex 5316 | If a class is equal to the... |
| iin0 5317 | An indexed intersection of... |
| notzfaus 5318 | In the Separation Scheme ~... |
| intv 5319 | The intersection of the un... |
| zfpow 5321 | Axiom of Power Sets expres... |
| axpow2 5322 | A variant of the Axiom of ... |
| axpow3 5323 | A variant of the Axiom of ... |
| elALT2 5324 | Alternate proof of ~ el us... |
| dtruALT2 5325 | Alternate proof of ~ dtru ... |
| dtrucor 5326 | Corollary of ~ dtru . Thi... |
| dtrucor2 5327 | The theorem form of the de... |
| dvdemo1 5328 | Demonstration of a theorem... |
| dvdemo2 5329 | Demonstration of a theorem... |
| nfnid 5330 | A setvar variable is not f... |
| nfcvb 5331 | The "distinctor" expressio... |
| vpwex 5332 | Power set axiom: the power... |
| pwexg 5333 | Power set axiom expressed ... |
| pwexd 5334 | Deduction version of the p... |
| pwex 5335 | Power set axiom expressed ... |
| pwel 5336 | Quantitative version of ~ ... |
| abssexg 5337 | Existence of a class of su... |
| snexALT 5338 | Alternate proof of ~ snex ... |
| p0ex 5339 | The power set of the empty... |
| p0exALT 5340 | Alternate proof of ~ p0ex ... |
| pp0ex 5341 | The power set of the power... |
| ord3ex 5342 | The ordinal number 3 is a ... |
| dtruALT 5343 | Alternate proof of ~ dtru ... |
| axc16b 5344 | This theorem shows that Ax... |
| eunex 5345 | Existential uniqueness imp... |
| eusv1 5346 | Two ways to express single... |
| eusvnf 5347 | Even if ` x ` is free in `... |
| eusvnfb 5348 | Two ways to say that ` A (... |
| eusv2i 5349 | Two ways to express single... |
| eusv2nf 5350 | Two ways to express single... |
| eusv2 5351 | Two ways to express single... |
| reusv1 5352 | Two ways to express single... |
| reusv2lem1 5353 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5354 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5355 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5356 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5357 | Lemma for ~ reusv2 . (Con... |
| reusv2 5358 | Two ways to express single... |
| reusv3i 5359 | Two ways of expressing exi... |
| reusv3 5360 | Two ways to express single... |
| eusv4 5361 | Two ways to express single... |
| alxfr 5362 | Transfer universal quantif... |
| ralxfrd 5363 | Transfer universal quantif... |
| rexxfrd 5364 | Transfer universal quantif... |
| ralxfr2d 5365 | Transfer universal quantif... |
| rexxfr2d 5366 | Transfer universal quantif... |
| ralxfrd2 5367 | Transfer universal quantif... |
| rexxfrd2 5368 | Transfer existence from a ... |
| ralxfr 5369 | Transfer universal quantif... |
| ralxfrALT 5370 | Alternate proof of ~ ralxf... |
| rexxfr 5371 | Transfer existence from a ... |
| rabxfrd 5372 | Membership in a restricted... |
| rabxfr 5373 | Membership in a restricted... |
| reuhypd 5374 | A theorem useful for elimi... |
| reuhyp 5375 | A theorem useful for elimi... |
| zfpair 5376 | The Axiom of Pairing of Ze... |
| axprALT 5377 | Alternate proof of ~ axpr ... |
| axprlem1 5378 | Lemma for ~ axpr . There ... |
| axprlem2 5379 | Lemma for ~ axpr . There ... |
| axprlem3 5380 | Lemma for ~ axpr . Elimin... |
| axprlem4 5381 | Lemma for ~ axpr . The fi... |
| axprlem5 5382 | Lemma for ~ axpr . The se... |
| axpr 5383 | Unabbreviated version of t... |
| zfpair2 5385 | Derive the abbreviated ver... |
| vsnex 5386 | A singleton built on a set... |
| snexg 5387 | A singleton built on a set... |
| snex 5388 | A singleton is a set. The... |
| prex 5389 | The Axiom of Pairing using... |
| exel 5390 | There exist two sets, one ... |
| exexneq 5391 | There exist two different ... |
| exneq 5392 | Given any set (the " ` y `... |
| dtru 5393 | Given any set (the " ` y `... |
| el 5394 | Any set is an element of s... |
| sels 5395 | If a class is a set, then ... |
| selsALT 5396 | Alternate proof of ~ sels ... |
| elALT 5397 | Alternate proof of ~ el , ... |
| dtruOLD 5398 | Obsolete proof of ~ dtru a... |
| snelpwg 5399 | A singleton of a set is a ... |
| snelpwi 5400 | If a set is a member of a ... |
| snelpwiOLD 5401 | Obsolete version of ~ snel... |
| snelpw 5402 | A singleton of a set is a ... |
| prelpw 5403 | An unordered pair of two s... |
| prelpwi 5404 | If two sets are members of... |
| rext 5405 | A theorem similar to exten... |
| sspwb 5406 | The powerclass constructio... |
| unipw 5407 | A class equals the union o... |
| univ 5408 | The union of the universe ... |
| pwtr 5409 | A class is transitive iff ... |
| ssextss 5410 | An extensionality-like pri... |
| ssext 5411 | An extensionality-like pri... |
| nssss 5412 | Negation of subclass relat... |
| pweqb 5413 | Classes are equal if and o... |
| intidg 5414 | The intersection of all se... |
| intidOLD 5415 | Obsolete version of ~ inti... |
| moabex 5416 | "At most one" existence im... |
| rmorabex 5417 | Restricted "at most one" e... |
| euabex 5418 | The abstraction of a wff w... |
| nnullss 5419 | A nonempty class (even if ... |
| exss 5420 | Restricted existence in a ... |
| opex 5421 | An ordered pair of classes... |
| otex 5422 | An ordered triple of class... |
| elopg 5423 | Characterization of the el... |
| elop 5424 | Characterization of the el... |
| opi1 5425 | One of the two elements in... |
| opi2 5426 | One of the two elements of... |
| opeluu 5427 | Each member of an ordered ... |
| op1stb 5428 | Extract the first member o... |
| brv 5429 | Two classes are always in ... |
| opnz 5430 | An ordered pair is nonempt... |
| opnzi 5431 | An ordered pair is nonempt... |
| opth1 5432 | Equality of the first memb... |
| opth 5433 | The ordered pair theorem. ... |
| opthg 5434 | Ordered pair theorem. ` C ... |
| opth1g 5435 | Equality of the first memb... |
| opthg2 5436 | Ordered pair theorem. (Co... |
| opth2 5437 | Ordered pair theorem. (Co... |
| opthneg 5438 | Two ordered pairs are not ... |
| opthne 5439 | Two ordered pairs are not ... |
| otth2 5440 | Ordered triple theorem, wi... |
| otth 5441 | Ordered triple theorem. (... |
| otthg 5442 | Ordered triple theorem, cl... |
| otthne 5443 | Contrapositive of the orde... |
| eqvinop 5444 | A variable introduction la... |
| sbcop1 5445 | The proper substitution of... |
| sbcop 5446 | The proper substitution of... |
| copsexgw 5447 | Version of ~ copsexg with ... |
| copsexg 5448 | Substitution of class ` A ... |
| copsex2t 5449 | Closed theorem form of ~ c... |
| copsex2g 5450 | Implicit substitution infe... |
| copsex2gOLD 5451 | Obsolete version of ~ cops... |
| copsex4g 5452 | An implicit substitution i... |
| 0nelop 5453 | A property of ordered pair... |
| opwo0id 5454 | An ordered pair is equal t... |
| opeqex 5455 | Equivalence of existence i... |
| oteqex2 5456 | Equivalence of existence i... |
| oteqex 5457 | Equivalence of existence i... |
| opcom 5458 | An ordered pair commutes i... |
| moop2 5459 | "At most one" property of ... |
| opeqsng 5460 | Equivalence for an ordered... |
| opeqsn 5461 | Equivalence for an ordered... |
| opeqpr 5462 | Equivalence for an ordered... |
| snopeqop 5463 | Equivalence for an ordered... |
| propeqop 5464 | Equivalence for an ordered... |
| propssopi 5465 | If a pair of ordered pairs... |
| snopeqopsnid 5466 | Equivalence for an ordered... |
| mosubopt 5467 | "At most one" remains true... |
| mosubop 5468 | "At most one" remains true... |
| euop2 5469 | Transfer existential uniqu... |
| euotd 5470 | Prove existential uniquene... |
| opthwiener 5471 | Justification theorem for ... |
| uniop 5472 | The union of an ordered pa... |
| uniopel 5473 | Ordered pair membership is... |
| opthhausdorff 5474 | Justification theorem for ... |
| opthhausdorff0 5475 | Justification theorem for ... |
| otsndisj 5476 | The singletons consisting ... |
| otiunsndisj 5477 | The union of singletons co... |
| iunopeqop 5478 | Implication of an ordered ... |
| brsnop 5479 | Binary relation for an ord... |
| brtp 5480 | A necessary and sufficient... |
| opabidw 5481 | The law of concretion. Sp... |
| opabid 5482 | The law of concretion. Sp... |
| elopabw 5483 | Membership in a class abst... |
| elopab 5484 | Membership in a class abst... |
| rexopabb 5485 | Restricted existential qua... |
| vopelopabsb 5486 | The law of concretion in t... |
| opelopabsb 5487 | The law of concretion in t... |
| brabsb 5488 | The law of concretion in t... |
| opelopabt 5489 | Closed theorem form of ~ o... |
| opelopabga 5490 | The law of concretion. Th... |
| brabga 5491 | The law of concretion for ... |
| opelopab2a 5492 | Ordered pair membership in... |
| opelopaba 5493 | The law of concretion. Th... |
| braba 5494 | The law of concretion for ... |
| opelopabg 5495 | The law of concretion. Th... |
| brabg 5496 | The law of concretion for ... |
| opelopabgf 5497 | The law of concretion. Th... |
| opelopab2 5498 | Ordered pair membership in... |
| opelopab 5499 | The law of concretion. Th... |
| brab 5500 | The law of concretion for ... |
| opelopabaf 5501 | The law of concretion. Th... |
| opelopabf 5502 | The law of concretion. Th... |
| ssopab2 5503 | Equivalence of ordered pai... |
| ssopab2bw 5504 | Equivalence of ordered pai... |
| eqopab2bw 5505 | Equivalence of ordered pai... |
| ssopab2b 5506 | Equivalence of ordered pai... |
| ssopab2i 5507 | Inference of ordered pair ... |
| ssopab2dv 5508 | Inference of ordered pair ... |
| eqopab2b 5509 | Equivalence of ordered pai... |
| opabn0 5510 | Nonempty ordered pair clas... |
| opab0 5511 | Empty ordered pair class a... |
| csbopab 5512 | Move substitution into a c... |
| csbopabgALT 5513 | Move substitution into a c... |
| csbmpt12 5514 | Move substitution into a m... |
| csbmpt2 5515 | Move substitution into the... |
| iunopab 5516 | Move indexed union inside ... |
| iunopabOLD 5517 | Obsolete version of ~ iuno... |
| elopabr 5518 | Membership in an ordered-p... |
| elopabran 5519 | Membership in an ordered-p... |
| elopabrOLD 5520 | Obsolete version of ~ elop... |
| rbropapd 5521 | Properties of a pair in an... |
| rbropap 5522 | Properties of a pair in a ... |
| 2rbropap 5523 | Properties of a pair in a ... |
| 0nelopab 5524 | The empty set is never an ... |
| 0nelopabOLD 5525 | Obsolete version of ~ 0nel... |
| brabv 5526 | If two classes are in a re... |
| pwin 5527 | The power class of the int... |
| pwssun 5528 | The power class of the uni... |
| pwun 5529 | The power class of the uni... |
| dfid4 5532 | The identity function expr... |
| dfid2 5533 | Alternate definition of th... |
| dfid3 5534 | A stronger version of ~ df... |
| dfid2OLD 5535 | Obsolete version of ~ dfid... |
| epelg 5538 | The membership relation an... |
| epeli 5539 | The membership relation an... |
| epel 5540 | The membership relation an... |
| 0sn0ep 5541 | An example for the members... |
| epn0 5542 | The membership relation is... |
| poss 5547 | Subset theorem for the par... |
| poeq1 5548 | Equality theorem for parti... |
| poeq2 5549 | Equality theorem for parti... |
| nfpo 5550 | Bound-variable hypothesis ... |
| nfso 5551 | Bound-variable hypothesis ... |
| pocl 5552 | Characteristic properties ... |
| poclOLD 5553 | Obsolete version of ~ pocl... |
| ispod 5554 | Sufficient conditions for ... |
| swopolem 5555 | Perform the substitutions ... |
| swopo 5556 | A strict weak order is a p... |
| poirr 5557 | A partial order is irrefle... |
| potr 5558 | A partial order is a trans... |
| po2nr 5559 | A partial order has no 2-c... |
| po3nr 5560 | A partial order has no 3-c... |
| po2ne 5561 | Two sets related by a part... |
| po0 5562 | Any relation is a partial ... |
| pofun 5563 | The inverse image of a par... |
| sopo 5564 | A strict linear order is a... |
| soss 5565 | Subset theorem for the str... |
| soeq1 5566 | Equality theorem for the s... |
| soeq2 5567 | Equality theorem for the s... |
| sonr 5568 | A strict order relation is... |
| sotr 5569 | A strict order relation is... |
| solin 5570 | A strict order relation is... |
| so2nr 5571 | A strict order relation ha... |
| so3nr 5572 | A strict order relation ha... |
| sotric 5573 | A strict order relation sa... |
| sotrieq 5574 | Trichotomy law for strict ... |
| sotrieq2 5575 | Trichotomy law for strict ... |
| soasym 5576 | Asymmetry law for strict o... |
| sotr2 5577 | A transitivity relation. ... |
| issod 5578 | An irreflexive, transitive... |
| issoi 5579 | An irreflexive, transitive... |
| isso2i 5580 | Deduce strict ordering fro... |
| so0 5581 | Any relation is a strict o... |
| somo 5582 | A totally ordered set has ... |
| sotrine 5583 | Trichotomy law for strict ... |
| sotr3 5584 | Transitivity law for stric... |
| dffr6 5591 | Alternate definition of ~ ... |
| frd 5592 | A nonempty subset of an ` ... |
| fri 5593 | A nonempty subset of an ` ... |
| friOLD 5594 | Obsolete version of ~ fri ... |
| seex 5595 | The ` R ` -preimage of an ... |
| exse 5596 | Any relation on a set is s... |
| dffr2 5597 | Alternate definition of we... |
| dffr2ALT 5598 | Alternate proof of ~ dffr2... |
| frc 5599 | Property of well-founded r... |
| frss 5600 | Subset theorem for the wel... |
| sess1 5601 | Subset theorem for the set... |
| sess2 5602 | Subset theorem for the set... |
| freq1 5603 | Equality theorem for the w... |
| freq2 5604 | Equality theorem for the w... |
| seeq1 5605 | Equality theorem for the s... |
| seeq2 5606 | Equality theorem for the s... |
| nffr 5607 | Bound-variable hypothesis ... |
| nfse 5608 | Bound-variable hypothesis ... |
| nfwe 5609 | Bound-variable hypothesis ... |
| frirr 5610 | A well-founded relation is... |
| fr2nr 5611 | A well-founded relation ha... |
| fr0 5612 | Any relation is well-found... |
| frminex 5613 | If an element of a well-fo... |
| efrirr 5614 | A well-founded class does ... |
| efrn2lp 5615 | A well-founded class conta... |
| epse 5616 | The membership relation is... |
| tz7.2 5617 | Similar to Theorem 7.2 of ... |
| dfepfr 5618 | An alternate way of saying... |
| epfrc 5619 | A subset of a well-founded... |
| wess 5620 | Subset theorem for the wel... |
| weeq1 5621 | Equality theorem for the w... |
| weeq2 5622 | Equality theorem for the w... |
| wefr 5623 | A well-ordering is well-fo... |
| weso 5624 | A well-ordering is a stric... |
| wecmpep 5625 | The elements of a class we... |
| wetrep 5626 | On a class well-ordered by... |
| wefrc 5627 | A nonempty subclass of a c... |
| we0 5628 | Any relation is a well-ord... |
| wereu 5629 | A nonempty subset of an ` ... |
| wereu2 5630 | A nonempty subclass of an ... |
| xpeq1 5647 | Equality theorem for Carte... |
| xpss12 5648 | Subset theorem for Cartesi... |
| xpss 5649 | A Cartesian product is inc... |
| inxpssres 5650 | Intersection with a Cartes... |
| relxp 5651 | A Cartesian product is a r... |
| xpss1 5652 | Subset relation for Cartes... |
| xpss2 5653 | Subset relation for Cartes... |
| xpeq2 5654 | Equality theorem for Carte... |
| elxpi 5655 | Membership in a Cartesian ... |
| elxp 5656 | Membership in a Cartesian ... |
| elxp2 5657 | Membership in a Cartesian ... |
| xpeq12 5658 | Equality theorem for Carte... |
| xpeq1i 5659 | Equality inference for Car... |
| xpeq2i 5660 | Equality inference for Car... |
| xpeq12i 5661 | Equality inference for Car... |
| xpeq1d 5662 | Equality deduction for Car... |
| xpeq2d 5663 | Equality deduction for Car... |
| xpeq12d 5664 | Equality deduction for Car... |
| sqxpeqd 5665 | Equality deduction for a C... |
| nfxp 5666 | Bound-variable hypothesis ... |
| 0nelxp 5667 | The empty set is not a mem... |
| 0nelelxp 5668 | A member of a Cartesian pr... |
| opelxp 5669 | Ordered pair membership in... |
| opelxpi 5670 | Ordered pair membership in... |
| opelxpd 5671 | Ordered pair membership in... |
| opelvv 5672 | Ordered pair membership in... |
| opelvvg 5673 | Ordered pair membership in... |
| opelxp1 5674 | The first member of an ord... |
| opelxp2 5675 | The second member of an or... |
| otelxp 5676 | Ordered triple membership ... |
| otelxp1 5677 | The first member of an ord... |
| otel3xp 5678 | An ordered triple is an el... |
| opabssxpd 5679 | An ordered-pair class abst... |
| rabxp 5680 | Class abstraction restrict... |
| brxp 5681 | Binary relation on a Carte... |
| pwvrel 5682 | A set is a binary relation... |
| pwvabrel 5683 | The powerclass of the cart... |
| brrelex12 5684 | Two classes related by a b... |
| brrelex1 5685 | If two classes are related... |
| brrelex2 5686 | If two classes are related... |
| brrelex12i 5687 | Two classes that are relat... |
| brrelex1i 5688 | The first argument of a bi... |
| brrelex2i 5689 | The second argument of a b... |
| nprrel12 5690 | Proper classes are not rel... |
| nprrel 5691 | No proper class is related... |
| 0nelrel0 5692 | A binary relation does not... |
| 0nelrel 5693 | A binary relation does not... |
| fconstmpt 5694 | Representation of a consta... |
| vtoclr 5695 | Variable to class conversi... |
| opthprc 5696 | Justification theorem for ... |
| brel 5697 | Two things in a binary rel... |
| elxp3 5698 | Membership in a Cartesian ... |
| opeliunxp 5699 | Membership in a union of C... |
| xpundi 5700 | Distributive law for Carte... |
| xpundir 5701 | Distributive law for Carte... |
| xpiundi 5702 | Distributive law for Carte... |
| xpiundir 5703 | Distributive law for Carte... |
| iunxpconst 5704 | Membership in a union of C... |
| xpun 5705 | The Cartesian product of t... |
| elvv 5706 | Membership in universal cl... |
| elvvv 5707 | Membership in universal cl... |
| elvvuni 5708 | An ordered pair contains i... |
| brinxp2 5709 | Intersection of binary rel... |
| brinxp 5710 | Intersection of binary rel... |
| opelinxp 5711 | Ordered pair element in an... |
| poinxp 5712 | Intersection of partial or... |
| soinxp 5713 | Intersection of total orde... |
| frinxp 5714 | Intersection of well-found... |
| seinxp 5715 | Intersection of set-like r... |
| weinxp 5716 | Intersection of well-order... |
| posn 5717 | Partial ordering of a sing... |
| sosn 5718 | Strict ordering on a singl... |
| frsn 5719 | Founded relation on a sing... |
| wesn 5720 | Well-ordering of a singlet... |
| elopaelxp 5721 | Membership in an ordered-p... |
| elopaelxpOLD 5722 | Obsolete version of ~ elop... |
| bropaex12 5723 | Two classes related by an ... |
| opabssxp 5724 | An abstraction relation is... |
| brab2a 5725 | The law of concretion for ... |
| optocl 5726 | Implicit substitution of c... |
| 2optocl 5727 | Implicit substitution of c... |
| 3optocl 5728 | Implicit substitution of c... |
| opbrop 5729 | Ordered pair membership in... |
| 0xp 5730 | The Cartesian product with... |
| csbxp 5731 | Distribute proper substitu... |
| releq 5732 | Equality theorem for the r... |
| releqi 5733 | Equality inference for the... |
| releqd 5734 | Equality deduction for the... |
| nfrel 5735 | Bound-variable hypothesis ... |
| sbcrel 5736 | Distribute proper substitu... |
| relss 5737 | Subclass theorem for relat... |
| ssrel 5738 | A subclass relationship de... |
| ssrelOLD 5739 | Obsolete version of ~ ssre... |
| eqrel 5740 | Extensionality principle f... |
| ssrel2 5741 | A subclass relationship de... |
| ssrel3 5742 | Subclass relation in anoth... |
| relssi 5743 | Inference from subclass pr... |
| relssdv 5744 | Deduction from subclass pr... |
| eqrelriv 5745 | Inference from extensional... |
| eqrelriiv 5746 | Inference from extensional... |
| eqbrriv 5747 | Inference from extensional... |
| eqrelrdv 5748 | Deduce equality of relatio... |
| eqbrrdv 5749 | Deduction from extensional... |
| eqbrrdiv 5750 | Deduction from extensional... |
| eqrelrdv2 5751 | A version of ~ eqrelrdv . ... |
| ssrelrel 5752 | A subclass relationship de... |
| eqrelrel 5753 | Extensionality principle f... |
| elrel 5754 | A member of a relation is ... |
| rel0 5755 | The empty set is a relatio... |
| nrelv 5756 | The universal class is not... |
| relsng 5757 | A singleton is a relation ... |
| relsnb 5758 | An at-most-singleton is a ... |
| relsnopg 5759 | A singleton of an ordered ... |
| relsn 5760 | A singleton is a relation ... |
| relsnop 5761 | A singleton of an ordered ... |
| copsex2gb 5762 | Implicit substitution infe... |
| copsex2ga 5763 | Implicit substitution infe... |
| elopaba 5764 | Membership in an ordered-p... |
| xpsspw 5765 | A Cartesian product is inc... |
| unixpss 5766 | The double class union of ... |
| relun 5767 | The union of two relations... |
| relin1 5768 | The intersection with a re... |
| relin2 5769 | The intersection with a re... |
| relinxp 5770 | Intersection with a Cartes... |
| reldif 5771 | A difference cutting down ... |
| reliun 5772 | An indexed union is a rela... |
| reliin 5773 | An indexed intersection is... |
| reluni 5774 | The union of a class is a ... |
| relint 5775 | The intersection of a clas... |
| relopabiv 5776 | A class of ordered pairs i... |
| relopabv 5777 | A class of ordered pairs i... |
| relopabi 5778 | A class of ordered pairs i... |
| relopabiALT 5779 | Alternate proof of ~ relop... |
| relopab 5780 | A class of ordered pairs i... |
| mptrel 5781 | The maps-to notation alway... |
| reli 5782 | The identity relation is a... |
| rele 5783 | The membership relation is... |
| opabid2 5784 | A relation expressed as an... |
| inopab 5785 | Intersection of two ordere... |
| difopab 5786 | Difference of two ordered-... |
| difopabOLD 5787 | Obsolete version of ~ difo... |
| inxp 5788 | Intersection of two Cartes... |
| xpindi 5789 | Distributive law for Carte... |
| xpindir 5790 | Distributive law for Carte... |
| xpiindi 5791 | Distributive law for Carte... |
| xpriindi 5792 | Distributive law for Carte... |
| eliunxp 5793 | Membership in a union of C... |
| opeliunxp2 5794 | Membership in a union of C... |
| raliunxp 5795 | Write a double restricted ... |
| rexiunxp 5796 | Write a double restricted ... |
| ralxp 5797 | Universal quantification r... |
| rexxp 5798 | Existential quantification... |
| exopxfr 5799 | Transfer ordered-pair exis... |
| exopxfr2 5800 | Transfer ordered-pair exis... |
| djussxp 5801 | Disjoint union is a subset... |
| ralxpf 5802 | Version of ~ ralxp with bo... |
| rexxpf 5803 | Version of ~ rexxp with bo... |
| iunxpf 5804 | Indexed union on a Cartesi... |
| opabbi2dv 5805 | Deduce equality of a relat... |
| relop 5806 | A necessary and sufficient... |
| ideqg 5807 | For sets, the identity rel... |
| ideq 5808 | For sets, the identity rel... |
| ididg 5809 | A set is identical to itse... |
| issetid 5810 | Two ways of expressing set... |
| coss1 5811 | Subclass theorem for compo... |
| coss2 5812 | Subclass theorem for compo... |
| coeq1 5813 | Equality theorem for compo... |
| coeq2 5814 | Equality theorem for compo... |
| coeq1i 5815 | Equality inference for com... |
| coeq2i 5816 | Equality inference for com... |
| coeq1d 5817 | Equality deduction for com... |
| coeq2d 5818 | Equality deduction for com... |
| coeq12i 5819 | Equality inference for com... |
| coeq12d 5820 | Equality deduction for com... |
| nfco 5821 | Bound-variable hypothesis ... |
| brcog 5822 | Ordered pair membership in... |
| opelco2g 5823 | Ordered pair membership in... |
| brcogw 5824 | Ordered pair membership in... |
| eqbrrdva 5825 | Deduction from extensional... |
| brco 5826 | Binary relation on a compo... |
| opelco 5827 | Ordered pair membership in... |
| cnvss 5828 | Subset theorem for convers... |
| cnveq 5829 | Equality theorem for conve... |
| cnveqi 5830 | Equality inference for con... |
| cnveqd 5831 | Equality deduction for con... |
| elcnv 5832 | Membership in a converse r... |
| elcnv2 5833 | Membership in a converse r... |
| nfcnv 5834 | Bound-variable hypothesis ... |
| brcnvg 5835 | The converse of a binary r... |
| opelcnvg 5836 | Ordered-pair membership in... |
| opelcnv 5837 | Ordered-pair membership in... |
| brcnv 5838 | The converse of a binary r... |
| csbcnv 5839 | Move class substitution in... |
| csbcnvgALT 5840 | Move class substitution in... |
| cnvco 5841 | Distributive law of conver... |
| cnvuni 5842 | The converse of a class un... |
| dfdm3 5843 | Alternate definition of do... |
| dfrn2 5844 | Alternate definition of ra... |
| dfrn3 5845 | Alternate definition of ra... |
| elrn2g 5846 | Membership in a range. (C... |
| elrng 5847 | Membership in a range. (C... |
| elrn2 5848 | Membership in a range. (C... |
| elrn 5849 | Membership in a range. (C... |
| ssrelrn 5850 | If a relation is a subset ... |
| dfdm4 5851 | Alternate definition of do... |
| dfdmf 5852 | Definition of domain, usin... |
| csbdm 5853 | Distribute proper substitu... |
| eldmg 5854 | Domain membership. Theore... |
| eldm2g 5855 | Domain membership. Theore... |
| eldm 5856 | Membership in a domain. T... |
| eldm2 5857 | Membership in a domain. T... |
| dmss 5858 | Subset theorem for domain.... |
| dmeq 5859 | Equality theorem for domai... |
| dmeqi 5860 | Equality inference for dom... |
| dmeqd 5861 | Equality deduction for dom... |
| opeldmd 5862 | Membership of first of an ... |
| opeldm 5863 | Membership of first of an ... |
| breldm 5864 | Membership of first of a b... |
| breldmg 5865 | Membership of first of a b... |
| dmun 5866 | The domain of a union is t... |
| dmin 5867 | The domain of an intersect... |
| breldmd 5868 | Membership of first of a b... |
| dmiun 5869 | The domain of an indexed u... |
| dmuni 5870 | The domain of a union. Pa... |
| dmopab 5871 | The domain of a class of o... |
| dmopabelb 5872 | A set is an element of the... |
| dmopab2rex 5873 | The domain of an ordered p... |
| dmopabss 5874 | Upper bound for the domain... |
| dmopab3 5875 | The domain of a restricted... |
| dm0 5876 | The domain of the empty se... |
| dmi 5877 | The domain of the identity... |
| dmv 5878 | The domain of the universe... |
| dmep 5879 | The domain of the membersh... |
| dm0rn0 5880 | An empty domain is equival... |
| rn0 5881 | The range of the empty set... |
| rnep 5882 | The range of the membershi... |
| reldm0 5883 | A relation is empty iff it... |
| dmxp 5884 | The domain of a Cartesian ... |
| dmxpid 5885 | The domain of a Cartesian ... |
| dmxpin 5886 | The domain of the intersec... |
| xpid11 5887 | The Cartesian square is a ... |
| dmcnvcnv 5888 | The domain of the double c... |
| rncnvcnv 5889 | The range of the double co... |
| elreldm 5890 | The first member of an ord... |
| rneq 5891 | Equality theorem for range... |
| rneqi 5892 | Equality inference for ran... |
| rneqd 5893 | Equality deduction for ran... |
| rnss 5894 | Subset theorem for range. ... |
| rnssi 5895 | Subclass inference for ran... |
| brelrng 5896 | The second argument of a b... |
| brelrn 5897 | The second argument of a b... |
| opelrn 5898 | Membership of second membe... |
| releldm 5899 | The first argument of a bi... |
| relelrn 5900 | The second argument of a b... |
| releldmb 5901 | Membership in a domain. (... |
| relelrnb 5902 | Membership in a range. (C... |
| releldmi 5903 | The first argument of a bi... |
| relelrni 5904 | The second argument of a b... |
| dfrnf 5905 | Definition of range, using... |
| nfdm 5906 | Bound-variable hypothesis ... |
| nfrn 5907 | Bound-variable hypothesis ... |
| dmiin 5908 | Domain of an intersection.... |
| rnopab 5909 | The range of a class of or... |
| rnmpt 5910 | The range of a function in... |
| elrnmpt 5911 | The range of a function in... |
| elrnmpt1s 5912 | Elementhood in an image se... |
| elrnmpt1 5913 | Elementhood in an image se... |
| elrnmptg 5914 | Membership in the range of... |
| elrnmpti 5915 | Membership in the range of... |
| elrnmptd 5916 | The range of a function in... |
| elrnmptdv 5917 | Elementhood in the range o... |
| elrnmpt2d 5918 | Elementhood in the range o... |
| dfiun3g 5919 | Alternate definition of in... |
| dfiin3g 5920 | Alternate definition of in... |
| dfiun3 5921 | Alternate definition of in... |
| dfiin3 5922 | Alternate definition of in... |
| riinint 5923 | Express a relative indexed... |
| relrn0 5924 | A relation is empty iff it... |
| dmrnssfld 5925 | The domain and range of a ... |
| dmcoss 5926 | Domain of a composition. ... |
| rncoss 5927 | Range of a composition. (... |
| dmcosseq 5928 | Domain of a composition. ... |
| dmcoeq 5929 | Domain of a composition. ... |
| rncoeq 5930 | Range of a composition. (... |
| reseq1 5931 | Equality theorem for restr... |
| reseq2 5932 | Equality theorem for restr... |
| reseq1i 5933 | Equality inference for res... |
| reseq2i 5934 | Equality inference for res... |
| reseq12i 5935 | Equality inference for res... |
| reseq1d 5936 | Equality deduction for res... |
| reseq2d 5937 | Equality deduction for res... |
| reseq12d 5938 | Equality deduction for res... |
| nfres 5939 | Bound-variable hypothesis ... |
| csbres 5940 | Distribute proper substitu... |
| res0 5941 | A restriction to the empty... |
| dfres3 5942 | Alternate definition of re... |
| opelres 5943 | Ordered pair elementhood i... |
| brres 5944 | Binary relation on a restr... |
| opelresi 5945 | Ordered pair membership in... |
| brresi 5946 | Binary relation on a restr... |
| opres 5947 | Ordered pair membership in... |
| resieq 5948 | A restricted identity rela... |
| opelidres 5949 | ` <. A , A >. ` belongs to... |
| resres 5950 | The restriction of a restr... |
| resundi 5951 | Distributive law for restr... |
| resundir 5952 | Distributive law for restr... |
| resindi 5953 | Class restriction distribu... |
| resindir 5954 | Class restriction distribu... |
| inres 5955 | Move intersection into cla... |
| resdifcom 5956 | Commutative law for restri... |
| resiun1 5957 | Distribution of restrictio... |
| resiun2 5958 | Distribution of restrictio... |
| dmres 5959 | The domain of a restrictio... |
| ssdmres 5960 | A domain restricted to a s... |
| dmresexg 5961 | The domain of a restrictio... |
| resss 5962 | A class includes its restr... |
| rescom 5963 | Commutative law for restri... |
| ssres 5964 | Subclass theorem for restr... |
| ssres2 5965 | Subclass theorem for restr... |
| relres 5966 | A restriction is a relatio... |
| resabs1 5967 | Absorption law for restric... |
| resabs1d 5968 | Absorption law for restric... |
| resabs2 5969 | Absorption law for restric... |
| residm 5970 | Idempotent law for restric... |
| resima 5971 | A restriction to an image.... |
| resima2 5972 | Image under a restricted c... |
| rnresss 5973 | The range of a restriction... |
| xpssres 5974 | Restriction of a constant ... |
| elinxp 5975 | Membership in an intersect... |
| elres 5976 | Membership in a restrictio... |
| elsnres 5977 | Membership in restriction ... |
| relssres 5978 | Simplification law for res... |
| dmressnsn 5979 | The domain of a restrictio... |
| eldmressnsn 5980 | The element of the domain ... |
| eldmeldmressn 5981 | An element of the domain (... |
| resdm 5982 | A relation restricted to i... |
| resexg 5983 | The restriction of a set i... |
| resexd 5984 | The restriction of a set i... |
| resex 5985 | The restriction of a set i... |
| resindm 5986 | When restricting a relatio... |
| resdmdfsn 5987 | Restricting a relation to ... |
| resopab 5988 | Restriction of a class abs... |
| iss 5989 | A subclass of the identity... |
| resopab2 5990 | Restriction of a class abs... |
| resmpt 5991 | Restriction of the mapping... |
| resmpt3 5992 | Unconditional restriction ... |
| resmptf 5993 | Restriction of the mapping... |
| resmptd 5994 | Restriction of the mapping... |
| dfres2 5995 | Alternate definition of th... |
| mptss 5996 | Sufficient condition for i... |
| elidinxp 5997 | Characterization of the el... |
| elidinxpid 5998 | Characterization of the el... |
| elrid 5999 | Characterization of the el... |
| idinxpres 6000 | The intersection of the id... |
| idinxpresid 6001 | The intersection of the id... |
| idssxp 6002 | A diagonal set as a subset... |
| opabresid 6003 | The restricted identity re... |
| mptresid 6004 | The restricted identity re... |
| dmresi 6005 | The domain of a restricted... |
| restidsing 6006 | Restriction of the identit... |
| iresn0n0 6007 | The identity function rest... |
| imaeq1 6008 | Equality theorem for image... |
| imaeq2 6009 | Equality theorem for image... |
| imaeq1i 6010 | Equality theorem for image... |
| imaeq2i 6011 | Equality theorem for image... |
| imaeq1d 6012 | Equality theorem for image... |
| imaeq2d 6013 | Equality theorem for image... |
| imaeq12d 6014 | Equality theorem for image... |
| dfima2 6015 | Alternate definition of im... |
| dfima3 6016 | Alternate definition of im... |
| elimag 6017 | Membership in an image. T... |
| elima 6018 | Membership in an image. T... |
| elima2 6019 | Membership in an image. T... |
| elima3 6020 | Membership in an image. T... |
| nfima 6021 | Bound-variable hypothesis ... |
| nfimad 6022 | Deduction version of bound... |
| imadmrn 6023 | The image of the domain of... |
| imassrn 6024 | The image of a class is a ... |
| mptima 6025 | Image of a function in map... |
| imai 6026 | Image under the identity r... |
| rnresi 6027 | The range of the restricte... |
| resiima 6028 | The image of a restriction... |
| ima0 6029 | Image of the empty set. T... |
| 0ima 6030 | Image under the empty rela... |
| csbima12 6031 | Move class substitution in... |
| imadisj 6032 | A class whose image under ... |
| cnvimass 6033 | A preimage under any class... |
| cnvimarndm 6034 | The preimage of the range ... |
| imasng 6035 | The image of a singleton. ... |
| relimasn 6036 | The image of a singleton. ... |
| elrelimasn 6037 | Elementhood in the image o... |
| elimasng1 6038 | Membership in an image of ... |
| elimasn1 6039 | Membership in an image of ... |
| elimasng 6040 | Membership in an image of ... |
| elimasn 6041 | Membership in an image of ... |
| elimasngOLD 6042 | Obsolete version of ~ elim... |
| elimasni 6043 | Membership in an image of ... |
| args 6044 | Two ways to express the cl... |
| elinisegg 6045 | Membership in the inverse ... |
| eliniseg 6046 | Membership in the inverse ... |
| epin 6047 | Any set is equal to its pr... |
| epini 6048 | Any set is equal to its pr... |
| iniseg 6049 | An idiom that signifies an... |
| inisegn0 6050 | Nonemptiness of an initial... |
| dffr3 6051 | Alternate definition of we... |
| dfse2 6052 | Alternate definition of se... |
| imass1 6053 | Subset theorem for image. ... |
| imass2 6054 | Subset theorem for image. ... |
| ndmima 6055 | The image of a singleton o... |
| relcnv 6056 | A converse is a relation. ... |
| relbrcnvg 6057 | When ` R ` is a relation, ... |
| eliniseg2 6058 | Eliminate the class existe... |
| relbrcnv 6059 | When ` R ` is a relation, ... |
| relco 6060 | A composition is a relatio... |
| cotrg 6061 | Two ways of saying that th... |
| cotrgOLD 6062 | Obsolete version of ~ cotr... |
| cotrgOLDOLD 6063 | Obsolete version of ~ cotr... |
| cotr 6064 | Two ways of saying a relat... |
| idrefALT 6065 | Alternate proof of ~ idref... |
| cnvsym 6066 | Two ways of saying a relat... |
| cnvsymOLD 6067 | Obsolete proof of ~ cnvsym... |
| cnvsymOLDOLD 6068 | Obsolete proof of ~ cnvsym... |
| intasym 6069 | Two ways of saying a relat... |
| asymref 6070 | Two ways of saying a relat... |
| asymref2 6071 | Two ways of saying a relat... |
| intirr 6072 | Two ways of saying a relat... |
| brcodir 6073 | Two ways of saying that tw... |
| codir 6074 | Two ways of saying a relat... |
| qfto 6075 | A quantifier-free way of e... |
| xpidtr 6076 | A Cartesian square is a tr... |
| trin2 6077 | The intersection of two tr... |
| poirr2 6078 | A partial order is irrefle... |
| trinxp 6079 | The relation induced by a ... |
| soirri 6080 | A strict order relation is... |
| sotri 6081 | A strict order relation is... |
| son2lpi 6082 | A strict order relation ha... |
| sotri2 6083 | A transitivity relation. ... |
| sotri3 6084 | A transitivity relation. ... |
| poleloe 6085 | Express "less than or equa... |
| poltletr 6086 | Transitive law for general... |
| somin1 6087 | Property of a minimum in a... |
| somincom 6088 | Commutativity of minimum i... |
| somin2 6089 | Property of a minimum in a... |
| soltmin 6090 | Being less than a minimum,... |
| cnvopab 6091 | The converse of a class ab... |
| mptcnv 6092 | The converse of a mapping ... |
| cnv0 6093 | The converse of the empty ... |
| cnvi 6094 | The converse of the identi... |
| cnvun 6095 | The converse of a union is... |
| cnvdif 6096 | Distributive law for conve... |
| cnvin 6097 | Distributive law for conve... |
| rnun 6098 | Distributive law for range... |
| rnin 6099 | The range of an intersecti... |
| rniun 6100 | The range of an indexed un... |
| rnuni 6101 | The range of a union. Par... |
| imaundi 6102 | Distributive law for image... |
| imaundir 6103 | The image of a union. (Co... |
| cnvimassrndm 6104 | The preimage of a superset... |
| dminss 6105 | An upper bound for interse... |
| imainss 6106 | An upper bound for interse... |
| inimass 6107 | The image of an intersecti... |
| inimasn 6108 | The intersection of the im... |
| cnvxp 6109 | The converse of a Cartesia... |
| xp0 6110 | The Cartesian product with... |
| xpnz 6111 | The Cartesian product of n... |
| xpeq0 6112 | At least one member of an ... |
| xpdisj1 6113 | Cartesian products with di... |
| xpdisj2 6114 | Cartesian products with di... |
| xpsndisj 6115 | Cartesian products with tw... |
| difxp 6116 | Difference of Cartesian pr... |
| difxp1 6117 | Difference law for Cartesi... |
| difxp2 6118 | Difference law for Cartesi... |
| djudisj 6119 | Disjoint unions with disjo... |
| xpdifid 6120 | The set of distinct couple... |
| resdisj 6121 | A double restriction to di... |
| rnxp 6122 | The range of a Cartesian p... |
| dmxpss 6123 | The domain of a Cartesian ... |
| rnxpss 6124 | The range of a Cartesian p... |
| rnxpid 6125 | The range of a Cartesian s... |
| ssxpb 6126 | A Cartesian product subcla... |
| xp11 6127 | The Cartesian product of n... |
| xpcan 6128 | Cancellation law for Carte... |
| xpcan2 6129 | Cancellation law for Carte... |
| ssrnres 6130 | Two ways to express surjec... |
| rninxp 6131 | Two ways to express surjec... |
| dminxp 6132 | Two ways to express totali... |
| imainrect 6133 | Image by a restricted and ... |
| xpima 6134 | Direct image by a Cartesia... |
| xpima1 6135 | Direct image by a Cartesia... |
| xpima2 6136 | Direct image by a Cartesia... |
| xpimasn 6137 | Direct image of a singleto... |
| sossfld 6138 | The base set of a strict o... |
| sofld 6139 | The base set of a nonempty... |
| cnvcnv3 6140 | The set of all ordered pai... |
| dfrel2 6141 | Alternate definition of re... |
| dfrel4v 6142 | A relation can be expresse... |
| dfrel4 6143 | A relation can be expresse... |
| cnvcnv 6144 | The double converse of a c... |
| cnvcnv2 6145 | The double converse of a c... |
| cnvcnvss 6146 | The double converse of a c... |
| cnvrescnv 6147 | Two ways to express the co... |
| cnveqb 6148 | Equality theorem for conve... |
| cnveq0 6149 | A relation empty iff its c... |
| dfrel3 6150 | Alternate definition of re... |
| elid 6151 | Characterization of the el... |
| dmresv 6152 | The domain of a universal ... |
| rnresv 6153 | The range of a universal r... |
| dfrn4 6154 | Range defined in terms of ... |
| csbrn 6155 | Distribute proper substitu... |
| rescnvcnv 6156 | The restriction of the dou... |
| cnvcnvres 6157 | The double converse of the... |
| imacnvcnv 6158 | The image of the double co... |
| dmsnn0 6159 | The domain of a singleton ... |
| rnsnn0 6160 | The range of a singleton i... |
| dmsn0 6161 | The domain of the singleto... |
| cnvsn0 6162 | The converse of the single... |
| dmsn0el 6163 | The domain of a singleton ... |
| relsn2 6164 | A singleton is a relation ... |
| dmsnopg 6165 | The domain of a singleton ... |
| dmsnopss 6166 | The domain of a singleton ... |
| dmpropg 6167 | The domain of an unordered... |
| dmsnop 6168 | The domain of a singleton ... |
| dmprop 6169 | The domain of an unordered... |
| dmtpop 6170 | The domain of an unordered... |
| cnvcnvsn 6171 | Double converse of a singl... |
| dmsnsnsn 6172 | The domain of the singleto... |
| rnsnopg 6173 | The range of a singleton o... |
| rnpropg 6174 | The range of a pair of ord... |
| cnvsng 6175 | Converse of a singleton of... |
| rnsnop 6176 | The range of a singleton o... |
| op1sta 6177 | Extract the first member o... |
| cnvsn 6178 | Converse of a singleton of... |
| op2ndb 6179 | Extract the second member ... |
| op2nda 6180 | Extract the second member ... |
| opswap 6181 | Swap the members of an ord... |
| cnvresima 6182 | An image under the convers... |
| resdm2 6183 | A class restricted to its ... |
| resdmres 6184 | Restriction to the domain ... |
| resresdm 6185 | A restriction by an arbitr... |
| imadmres 6186 | The image of the domain of... |
| resdmss 6187 | Subset relationship for th... |
| resdifdi 6188 | Distributive law for restr... |
| resdifdir 6189 | Distributive law for restr... |
| mptpreima 6190 | The preimage of a function... |
| mptiniseg 6191 | Converse singleton image o... |
| dmmpt 6192 | The domain of the mapping ... |
| dmmptss 6193 | The domain of a mapping is... |
| dmmptg 6194 | The domain of the mapping ... |
| rnmpt0f 6195 | The range of a function in... |
| rnmptn0 6196 | The range of a function in... |
| dfco2 6197 | Alternate definition of a ... |
| dfco2a 6198 | Generalization of ~ dfco2 ... |
| coundi 6199 | Class composition distribu... |
| coundir 6200 | Class composition distribu... |
| cores 6201 | Restricted first member of... |
| resco 6202 | Associative law for the re... |
| imaco 6203 | Image of the composition o... |
| rnco 6204 | The range of the compositi... |
| rnco2 6205 | The range of the compositi... |
| dmco 6206 | The domain of a compositio... |
| coeq0 6207 | A composition of two relat... |
| coiun 6208 | Composition with an indexe... |
| cocnvcnv1 6209 | A composition is not affec... |
| cocnvcnv2 6210 | A composition is not affec... |
| cores2 6211 | Absorption of a reverse (p... |
| co02 6212 | Composition with the empty... |
| co01 6213 | Composition with the empty... |
| coi1 6214 | Composition with the ident... |
| coi2 6215 | Composition with the ident... |
| coires1 6216 | Composition with a restric... |
| coass 6217 | Associative law for class ... |
| relcnvtrg 6218 | General form of ~ relcnvtr... |
| relcnvtr 6219 | A relation is transitive i... |
| relssdmrn 6220 | A relation is included in ... |
| relssdmrnOLD 6221 | Obsolete version of ~ rels... |
| resssxp 6222 | If the ` R ` -image of a c... |
| cnvssrndm 6223 | The converse is a subset o... |
| cossxp 6224 | Composition as a subset of... |
| relrelss 6225 | Two ways to describe the s... |
| unielrel 6226 | The membership relation fo... |
| relfld 6227 | The double union of a rela... |
| relresfld 6228 | Restriction of a relation ... |
| relcoi2 6229 | Composition with the ident... |
| relcoi1 6230 | Composition with the ident... |
| unidmrn 6231 | The double union of the co... |
| relcnvfld 6232 | if ` R ` is a relation, it... |
| dfdm2 6233 | Alternate definition of do... |
| unixp 6234 | The double class union of ... |
| unixp0 6235 | A Cartesian product is emp... |
| unixpid 6236 | Field of a Cartesian squar... |
| ressn 6237 | Restriction of a class to ... |
| cnviin 6238 | The converse of an interse... |
| cnvpo 6239 | The converse of a partial ... |
| cnvso 6240 | The converse of a strict o... |
| xpco 6241 | Composition of two Cartesi... |
| xpcoid 6242 | Composition of two Cartesi... |
| elsnxp 6243 | Membership in a Cartesian ... |
| reu3op 6244 | There is a unique ordered ... |
| reuop 6245 | There is a unique ordered ... |
| opreu2reurex 6246 | There is a unique ordered ... |
| opreu2reu 6247 | If there is a unique order... |
| dfpo2 6248 | Quantifier-free definition... |
| csbcog 6249 | Distribute proper substitu... |
| snres0 6250 | Condition for restriction ... |
| imaindm 6251 | The image is unaffected by... |
| predeq123 6254 | Equality theorem for the p... |
| predeq1 6255 | Equality theorem for the p... |
| predeq2 6256 | Equality theorem for the p... |
| predeq3 6257 | Equality theorem for the p... |
| nfpred 6258 | Bound-variable hypothesis ... |
| csbpredg 6259 | Move class substitution in... |
| predpredss 6260 | If ` A ` is a subset of ` ... |
| predss 6261 | The predecessor class of `... |
| sspred 6262 | Another subset/predecessor... |
| dfpred2 6263 | An alternate definition of... |
| dfpred3 6264 | An alternate definition of... |
| dfpred3g 6265 | An alternate definition of... |
| elpredgg 6266 | Membership in a predecesso... |
| elpredg 6267 | Membership in a predecesso... |
| elpredimg 6268 | Membership in a predecesso... |
| elpredim 6269 | Membership in a predecesso... |
| elpred 6270 | Membership in a predecesso... |
| predexg 6271 | The predecessor class exis... |
| predasetexOLD 6272 | Obsolete form of ~ predexg... |
| dffr4 6273 | Alternate definition of we... |
| predel 6274 | Membership in the predeces... |
| predbrg 6275 | Closed form of ~ elpredim ... |
| predtrss 6276 | If ` R ` is transitive ove... |
| predpo 6277 | Property of the predecesso... |
| predso 6278 | Property of the predecesso... |
| setlikespec 6279 | If ` R ` is set-like in ` ... |
| predidm 6280 | Idempotent law for the pre... |
| predin 6281 | Intersection law for prede... |
| predun 6282 | Union law for predecessor ... |
| preddif 6283 | Difference law for predece... |
| predep 6284 | The predecessor under the ... |
| trpred 6285 | The class of predecessors ... |
| preddowncl 6286 | A property of classes that... |
| predpoirr 6287 | Given a partial ordering, ... |
| predfrirr 6288 | Given a well-founded relat... |
| pred0 6289 | The predecessor class over... |
| dfse3 6290 | Alternate definition of se... |
| predrelss 6291 | Subset carries from relati... |
| predprc 6292 | The predecessor of a prope... |
| predres 6293 | Predecessor class is unaff... |
| frpomin 6294 | Every nonempty (possibly p... |
| frpomin2 6295 | Every nonempty (possibly p... |
| frpoind 6296 | The principle of well-foun... |
| frpoinsg 6297 | Well-Founded Induction Sch... |
| frpoins2fg 6298 | Well-Founded Induction sch... |
| frpoins2g 6299 | Well-Founded Induction sch... |
| frpoins3g 6300 | Well-Founded Induction sch... |
| tz6.26 6301 | All nonempty subclasses of... |
| tz6.26OLD 6302 | Obsolete proof of ~ tz6.26... |
| tz6.26i 6303 | All nonempty subclasses of... |
| wfi 6304 | The Principle of Well-Orde... |
| wfiOLD 6305 | Obsolete proof of ~ wfi as... |
| wfii 6306 | The Principle of Well-Orde... |
| wfisg 6307 | Well-Ordered Induction Sch... |
| wfisgOLD 6308 | Obsolete proof of ~ wfisg ... |
| wfis 6309 | Well-Ordered Induction Sch... |
| wfis2fg 6310 | Well-Ordered Induction Sch... |
| wfis2fgOLD 6311 | Obsolete proof of ~ wfis2f... |
| wfis2f 6312 | Well-Ordered Induction sch... |
| wfis2g 6313 | Well-Ordered Induction Sch... |
| wfis2 6314 | Well-Ordered Induction sch... |
| wfis3 6315 | Well-Ordered Induction sch... |
| ordeq 6324 | Equality theorem for the o... |
| elong 6325 | An ordinal number is an or... |
| elon 6326 | An ordinal number is an or... |
| eloni 6327 | An ordinal number has the ... |
| elon2 6328 | An ordinal number is an or... |
| limeq 6329 | Equality theorem for the l... |
| ordwe 6330 | Membership well-orders eve... |
| ordtr 6331 | An ordinal class is transi... |
| ordfr 6332 | Membership is well-founded... |
| ordelss 6333 | An element of an ordinal c... |
| trssord 6334 | A transitive subclass of a... |
| ordirr 6335 | No ordinal class is a memb... |
| nordeq 6336 | A member of an ordinal cla... |
| ordn2lp 6337 | An ordinal class cannot be... |
| tz7.5 6338 | A nonempty subclass of an ... |
| ordelord 6339 | An element of an ordinal c... |
| tron 6340 | The class of all ordinal n... |
| ordelon 6341 | An element of an ordinal c... |
| onelon 6342 | An element of an ordinal n... |
| tz7.7 6343 | A transitive class belongs... |
| ordelssne 6344 | For ordinal classes, membe... |
| ordelpss 6345 | For ordinal classes, membe... |
| ordsseleq 6346 | For ordinal classes, inclu... |
| ordin 6347 | The intersection of two or... |
| onin 6348 | The intersection of two or... |
| ordtri3or 6349 | A trichotomy law for ordin... |
| ordtri1 6350 | A trichotomy law for ordin... |
| ontri1 6351 | A trichotomy law for ordin... |
| ordtri2 6352 | A trichotomy law for ordin... |
| ordtri3 6353 | A trichotomy law for ordin... |
| ordtri4 6354 | A trichotomy law for ordin... |
| orddisj 6355 | An ordinal class and its s... |
| onfr 6356 | The ordinal class is well-... |
| onelpss 6357 | Relationship between membe... |
| onsseleq 6358 | Relationship between subse... |
| onelss 6359 | An element of an ordinal n... |
| ordtr1 6360 | Transitive law for ordinal... |
| ordtr2 6361 | Transitive law for ordinal... |
| ordtr3 6362 | Transitive law for ordinal... |
| ontr1 6363 | Transitive law for ordinal... |
| ontr2 6364 | Transitive law for ordinal... |
| onelssex 6365 | Ordinal less than is equiv... |
| ordunidif 6366 | The union of an ordinal st... |
| ordintdif 6367 | If ` B ` is smaller than `... |
| onintss 6368 | If a property is true for ... |
| oneqmini 6369 | A way to show that an ordi... |
| ord0 6370 | The empty set is an ordina... |
| 0elon 6371 | The empty set is an ordina... |
| ord0eln0 6372 | A nonempty ordinal contain... |
| on0eln0 6373 | An ordinal number contains... |
| dflim2 6374 | An alternate definition of... |
| inton 6375 | The intersection of the cl... |
| nlim0 6376 | The empty set is not a lim... |
| limord 6377 | A limit ordinal is ordinal... |
| limuni 6378 | A limit ordinal is its own... |
| limuni2 6379 | The union of a limit ordin... |
| 0ellim 6380 | A limit ordinal contains t... |
| limelon 6381 | A limit ordinal class that... |
| onn0 6382 | The class of all ordinal n... |
| suceq 6383 | Equality of successors. (... |
| elsuci 6384 | Membership in a successor.... |
| elsucg 6385 | Membership in a successor.... |
| elsuc2g 6386 | Variant of membership in a... |
| elsuc 6387 | Membership in a successor.... |
| elsuc2 6388 | Membership in a successor.... |
| nfsuc 6389 | Bound-variable hypothesis ... |
| elelsuc 6390 | Membership in a successor.... |
| sucel 6391 | Membership of a successor ... |
| suc0 6392 | The successor of the empty... |
| sucprc 6393 | A proper class is its own ... |
| unisucs 6394 | The union of the successor... |
| unisucg 6395 | A transitive class is equa... |
| unisuc 6396 | A transitive class is equa... |
| sssucid 6397 | A class is included in its... |
| sucidg 6398 | Part of Proposition 7.23 o... |
| sucid 6399 | A set belongs to its succe... |
| nsuceq0 6400 | No successor is empty. (C... |
| eqelsuc 6401 | A set belongs to the succe... |
| iunsuc 6402 | Inductive definition for t... |
| suctr 6403 | The successor of a transit... |
| trsuc 6404 | A set whose successor belo... |
| trsucss 6405 | A member of the successor ... |
| ordsssuc 6406 | An ordinal is a subset of ... |
| onsssuc 6407 | A subset of an ordinal num... |
| ordsssuc2 6408 | An ordinal subset of an or... |
| onmindif 6409 | When its successor is subt... |
| ordnbtwn 6410 | There is no set between an... |
| onnbtwn 6411 | There is no set between an... |
| sucssel 6412 | A set whose successor is a... |
| orddif 6413 | Ordinal derived from its s... |
| orduniss 6414 | An ordinal class includes ... |
| ordtri2or 6415 | A trichotomy law for ordin... |
| ordtri2or2 6416 | A trichotomy law for ordin... |
| ordtri2or3 6417 | A consequence of total ord... |
| ordelinel 6418 | The intersection of two or... |
| ordssun 6419 | Property of a subclass of ... |
| ordequn 6420 | The maximum (i.e. union) o... |
| ordun 6421 | The maximum (i.e., union) ... |
| onunel 6422 | The union of two ordinals ... |
| ordunisssuc 6423 | A subclass relationship fo... |
| suc11 6424 | The successor operation be... |
| onun2 6425 | The union of two ordinals ... |
| ontr 6426 | An ordinal number is a tra... |
| onunisuc 6427 | An ordinal number is equal... |
| onordi 6428 | An ordinal number is an or... |
| ontrciOLD 6429 | Obsolete version of ~ ontr... |
| onirri 6430 | An ordinal number is not a... |
| oneli 6431 | A member of an ordinal num... |
| onelssi 6432 | A member of an ordinal num... |
| onssneli 6433 | An ordering law for ordina... |
| onssnel2i 6434 | An ordering law for ordina... |
| onelini 6435 | An element of an ordinal n... |
| oneluni 6436 | An ordinal number equals i... |
| onunisuci 6437 | An ordinal number is equal... |
| onsseli 6438 | Subset is equivalent to me... |
| onun2i 6439 | The union of two ordinal n... |
| unizlim 6440 | An ordinal equal to its ow... |
| on0eqel 6441 | An ordinal number either e... |
| snsn0non 6442 | The singleton of the singl... |
| onxpdisj 6443 | Ordinal numbers and ordere... |
| onnev 6444 | The class of ordinal numbe... |
| onnevOLD 6445 | Obsolete version of ~ onne... |
| iotajust 6447 | Soundness justification th... |
| dfiota2 6449 | Alternate definition for d... |
| nfiota1 6450 | Bound-variable hypothesis ... |
| nfiotadw 6451 | Deduction version of ~ nfi... |
| nfiotaw 6452 | Bound-variable hypothesis ... |
| nfiotad 6453 | Deduction version of ~ nfi... |
| nfiota 6454 | Bound-variable hypothesis ... |
| cbviotaw 6455 | Change bound variables in ... |
| cbviotavw 6456 | Change bound variables in ... |
| cbviotavwOLD 6457 | Obsolete version of ~ cbvi... |
| cbviota 6458 | Change bound variables in ... |
| cbviotav 6459 | Change bound variables in ... |
| sb8iota 6460 | Variable substitution in d... |
| iotaeq 6461 | Equality theorem for descr... |
| iotabi 6462 | Equivalence theorem for de... |
| uniabio 6463 | Part of Theorem 8.17 in [Q... |
| iotaval2 6464 | Version of ~ iotaval using... |
| iotauni2 6465 | Version of ~ iotauni using... |
| iotanul2 6466 | Version of ~ iotanul using... |
| iotaval 6467 | Theorem 8.19 in [Quine] p.... |
| iotassuni 6468 | The ` iota ` class is a su... |
| iotaex 6469 | Theorem 8.23 in [Quine] p.... |
| iotavalOLD 6470 | Obsolete version of ~ iota... |
| iotauni 6471 | Equivalence between two di... |
| iotaint 6472 | Equivalence between two di... |
| iota1 6473 | Property of iota. (Contri... |
| iotanul 6474 | Theorem 8.22 in [Quine] p.... |
| iotassuniOLD 6475 | Obsolete version of ~ iota... |
| iotaexOLD 6476 | Obsolete version of ~ iota... |
| iota4 6477 | Theorem *14.22 in [Whitehe... |
| iota4an 6478 | Theorem *14.23 in [Whitehe... |
| iota5 6479 | A method for computing iot... |
| iotabidv 6480 | Formula-building deduction... |
| iotabii 6481 | Formula-building deduction... |
| iotacl 6482 | Membership law for descrip... |
| iota2df 6483 | A condition that allows to... |
| iota2d 6484 | A condition that allows to... |
| iota2 6485 | The unique element such th... |
| iotan0 6486 | Representation of "the uni... |
| sniota 6487 | A class abstraction with a... |
| dfiota4 6488 | The ` iota ` operation usi... |
| csbiota 6489 | Class substitution within ... |
| dffun2 6506 | Alternate definition of a ... |
| dffun2OLD 6507 | Obsolete version of ~ dffu... |
| dffun2OLDOLD 6508 | Obsolete version of ~ dffu... |
| dffun6 6509 | Alternate definition of a ... |
| dffun3 6510 | Alternate definition of fu... |
| dffun3OLD 6511 | Obsolete version of ~ dffu... |
| dffun4 6512 | Alternate definition of a ... |
| dffun5 6513 | Alternate definition of fu... |
| dffun6f 6514 | Definition of function, us... |
| dffun6OLD 6515 | Obsolete version of ~ dffu... |
| funmo 6516 | A function has at most one... |
| funmoOLD 6517 | Obsolete version of ~ funm... |
| funrel 6518 | A function is a relation. ... |
| 0nelfun 6519 | A function does not contai... |
| funss 6520 | Subclass theorem for funct... |
| funeq 6521 | Equality theorem for funct... |
| funeqi 6522 | Equality inference for the... |
| funeqd 6523 | Equality deduction for the... |
| nffun 6524 | Bound-variable hypothesis ... |
| sbcfung 6525 | Distribute proper substitu... |
| funeu 6526 | There is exactly one value... |
| funeu2 6527 | There is exactly one value... |
| dffun7 6528 | Alternate definition of a ... |
| dffun8 6529 | Alternate definition of a ... |
| dffun9 6530 | Alternate definition of a ... |
| funfn 6531 | A class is a function if a... |
| funfnd 6532 | A function is a function o... |
| funi 6533 | The identity relation is a... |
| nfunv 6534 | The universal class is not... |
| funopg 6535 | A Kuratowski ordered pair ... |
| funopab 6536 | A class of ordered pairs i... |
| funopabeq 6537 | A class of ordered pairs o... |
| funopab4 6538 | A class of ordered pairs o... |
| funmpt 6539 | A function in maps-to nota... |
| funmpt2 6540 | Functionality of a class g... |
| funco 6541 | The composition of two fun... |
| funresfunco 6542 | Composition of two functio... |
| funres 6543 | A restriction of a functio... |
| funresd 6544 | A restriction of a functio... |
| funssres 6545 | The restriction of a funct... |
| fun2ssres 6546 | Equality of restrictions o... |
| funun 6547 | The union of functions wit... |
| fununmo 6548 | If the union of classes is... |
| fununfun 6549 | If the union of classes is... |
| fundif 6550 | A function with removed el... |
| funcnvsn 6551 | The converse singleton of ... |
| funsng 6552 | A singleton of an ordered ... |
| fnsng 6553 | Functionality and domain o... |
| funsn 6554 | A singleton of an ordered ... |
| funprg 6555 | A set of two pairs is a fu... |
| funtpg 6556 | A set of three pairs is a ... |
| funpr 6557 | A function with a domain o... |
| funtp 6558 | A function with a domain o... |
| fnsn 6559 | Functionality and domain o... |
| fnprg 6560 | Function with a domain of ... |
| fntpg 6561 | Function with a domain of ... |
| fntp 6562 | A function with a domain o... |
| funcnvpr 6563 | The converse pair of order... |
| funcnvtp 6564 | The converse triple of ord... |
| funcnvqp 6565 | The converse quadruple of ... |
| fun0 6566 | The empty set is a functio... |
| funcnv0 6567 | The converse of the empty ... |
| funcnvcnv 6568 | The double converse of a f... |
| funcnv2 6569 | A simpler equivalence for ... |
| funcnv 6570 | The converse of a class is... |
| funcnv3 6571 | A condition showing a clas... |
| fun2cnv 6572 | The double converse of a c... |
| svrelfun 6573 | A single-valued relation i... |
| fncnv 6574 | Single-rootedness (see ~ f... |
| fun11 6575 | Two ways of stating that `... |
| fununi 6576 | The union of a chain (with... |
| funin 6577 | The intersection with a fu... |
| funres11 6578 | The restriction of a one-t... |
| funcnvres 6579 | The converse of a restrict... |
| cnvresid 6580 | Converse of a restricted i... |
| funcnvres2 6581 | The converse of a restrict... |
| funimacnv 6582 | The image of the preimage ... |
| funimass1 6583 | A kind of contraposition l... |
| funimass2 6584 | A kind of contraposition l... |
| imadif 6585 | The image of a difference ... |
| imain 6586 | The image of an intersecti... |
| funimaexg 6587 | Axiom of Replacement using... |
| funimaexgOLD 6588 | Obsolete version of ~ funi... |
| funimaex 6589 | The image of a set under a... |
| isarep1 6590 | Part of a study of the Axi... |
| isarep1OLD 6591 | Obsolete version of ~ isar... |
| isarep2 6592 | Part of a study of the Axi... |
| fneq1 6593 | Equality theorem for funct... |
| fneq2 6594 | Equality theorem for funct... |
| fneq1d 6595 | Equality deduction for fun... |
| fneq2d 6596 | Equality deduction for fun... |
| fneq12d 6597 | Equality deduction for fun... |
| fneq12 6598 | Equality theorem for funct... |
| fneq1i 6599 | Equality inference for fun... |
| fneq2i 6600 | Equality inference for fun... |
| nffn 6601 | Bound-variable hypothesis ... |
| fnfun 6602 | A function with domain is ... |
| fnfund 6603 | A function with domain is ... |
| fnrel 6604 | A function with domain is ... |
| fndm 6605 | The domain of a function. ... |
| fndmi 6606 | The domain of a function. ... |
| fndmd 6607 | The domain of a function. ... |
| funfni 6608 | Inference to convert a fun... |
| fndmu 6609 | A function has a unique do... |
| fnbr 6610 | The first argument of bina... |
| fnop 6611 | The first argument of an o... |
| fneu 6612 | There is exactly one value... |
| fneu2 6613 | There is exactly one value... |
| fnun 6614 | The union of two functions... |
| fnund 6615 | The union of two functions... |
| fnunop 6616 | Extension of a function wi... |
| fncofn 6617 | Composition of a function ... |
| fnco 6618 | Composition of two functio... |
| fncoOLD 6619 | Obsolete version of ~ fnco... |
| fnresdm 6620 | A function does not change... |
| fnresdisj 6621 | A function restricted to a... |
| 2elresin 6622 | Membership in two function... |
| fnssresb 6623 | Restriction of a function ... |
| fnssres 6624 | Restriction of a function ... |
| fnssresd 6625 | Restriction of a function ... |
| fnresin1 6626 | Restriction of a function'... |
| fnresin2 6627 | Restriction of a function'... |
| fnres 6628 | An equivalence for functio... |
| idfn 6629 | The identity relation is a... |
| fnresi 6630 | The restricted identity re... |
| fnima 6631 | The image of a function's ... |
| fn0 6632 | A function with empty doma... |
| fnimadisj 6633 | A class that is disjoint w... |
| fnimaeq0 6634 | Images under a function ne... |
| dfmpt3 6635 | Alternate definition for t... |
| mptfnf 6636 | The maps-to notation defin... |
| fnmptf 6637 | The maps-to notation defin... |
| fnopabg 6638 | Functionality and domain o... |
| fnopab 6639 | Functionality and domain o... |
| mptfng 6640 | The maps-to notation defin... |
| fnmpt 6641 | The maps-to notation defin... |
| fnmptd 6642 | The maps-to notation defin... |
| mpt0 6643 | A mapping operation with e... |
| fnmpti 6644 | Functionality and domain o... |
| dmmpti 6645 | Domain of the mapping oper... |
| dmmptd 6646 | The domain of the mapping ... |
| mptun 6647 | Union of mappings which ar... |
| partfun 6648 | Rewrite a function defined... |
| feq1 6649 | Equality theorem for funct... |
| feq2 6650 | Equality theorem for funct... |
| feq3 6651 | Equality theorem for funct... |
| feq23 6652 | Equality theorem for funct... |
| feq1d 6653 | Equality deduction for fun... |
| feq2d 6654 | Equality deduction for fun... |
| feq3d 6655 | Equality deduction for fun... |
| feq12d 6656 | Equality deduction for fun... |
| feq123d 6657 | Equality deduction for fun... |
| feq123 6658 | Equality theorem for funct... |
| feq1i 6659 | Equality inference for fun... |
| feq2i 6660 | Equality inference for fun... |
| feq12i 6661 | Equality inference for fun... |
| feq23i 6662 | Equality inference for fun... |
| feq23d 6663 | Equality deduction for fun... |
| nff 6664 | Bound-variable hypothesis ... |
| sbcfng 6665 | Distribute proper substitu... |
| sbcfg 6666 | Distribute proper substitu... |
| elimf 6667 | Eliminate a mapping hypoth... |
| ffn 6668 | A mapping is a function wi... |
| ffnd 6669 | A mapping is a function wi... |
| dffn2 6670 | Any function is a mapping ... |
| ffun 6671 | A mapping is a function. ... |
| ffund 6672 | A mapping is a function, d... |
| frel 6673 | A mapping is a relation. ... |
| freld 6674 | A mapping is a relation. ... |
| frn 6675 | The range of a mapping. (... |
| frnd 6676 | Deduction form of ~ frn . ... |
| fdm 6677 | The domain of a mapping. ... |
| fdmOLD 6678 | Obsolete version of ~ fdm ... |
| fdmd 6679 | Deduction form of ~ fdm . ... |
| fdmi 6680 | Inference associated with ... |
| dffn3 6681 | A function maps to its ran... |
| ffrn 6682 | A function maps to its ran... |
| ffrnb 6683 | Characterization of a func... |
| ffrnbd 6684 | A function maps to its ran... |
| fss 6685 | Expanding the codomain of ... |
| fssd 6686 | Expanding the codomain of ... |
| fssdmd 6687 | Expressing that a class is... |
| fssdm 6688 | Expressing that a class is... |
| fimass 6689 | The image of a class under... |
| fimacnv 6690 | The preimage of the codoma... |
| fcof 6691 | Composition of a function ... |
| fco 6692 | Composition of two functio... |
| fcoOLD 6693 | Obsolete version of ~ fco ... |
| fcod 6694 | Composition of two mapping... |
| fco2 6695 | Functionality of a composi... |
| fssxp 6696 | A mapping is a class of or... |
| funssxp 6697 | Two ways of specifying a p... |
| ffdm 6698 | A mapping is a partial fun... |
| ffdmd 6699 | The domain of a function. ... |
| fdmrn 6700 | A different way to write `... |
| funcofd 6701 | Composition of two functio... |
| fco3OLD 6702 | Obsolete version of ~ func... |
| opelf 6703 | The members of an ordered ... |
| fun 6704 | The union of two functions... |
| fun2 6705 | The union of two functions... |
| fun2d 6706 | The union of functions wit... |
| fnfco 6707 | Composition of two functio... |
| fssres 6708 | Restriction of a function ... |
| fssresd 6709 | Restriction of a function ... |
| fssres2 6710 | Restriction of a restricte... |
| fresin 6711 | An identity for the mappin... |
| resasplit 6712 | If two functions agree on ... |
| fresaun 6713 | The union of two functions... |
| fresaunres2 6714 | From the union of two func... |
| fresaunres1 6715 | From the union of two func... |
| fcoi1 6716 | Composition of a mapping a... |
| fcoi2 6717 | Composition of restricted ... |
| feu 6718 | There is exactly one value... |
| fcnvres 6719 | The converse of a restrict... |
| fimacnvdisj 6720 | The preimage of a class di... |
| fint 6721 | Function into an intersect... |
| fin 6722 | Mapping into an intersecti... |
| f0 6723 | The empty function. (Cont... |
| f00 6724 | A class is a function with... |
| f0bi 6725 | A function with empty doma... |
| f0dom0 6726 | A function is empty iff it... |
| f0rn0 6727 | If there is no element in ... |
| fconst 6728 | A Cartesian product with a... |
| fconstg 6729 | A Cartesian product with a... |
| fnconstg 6730 | A Cartesian product with a... |
| fconst6g 6731 | Constant function with loo... |
| fconst6 6732 | A constant function as a m... |
| f1eq1 6733 | Equality theorem for one-t... |
| f1eq2 6734 | Equality theorem for one-t... |
| f1eq3 6735 | Equality theorem for one-t... |
| nff1 6736 | Bound-variable hypothesis ... |
| dff12 6737 | Alternate definition of a ... |
| f1f 6738 | A one-to-one mapping is a ... |
| f1fn 6739 | A one-to-one mapping is a ... |
| f1fun 6740 | A one-to-one mapping is a ... |
| f1rel 6741 | A one-to-one onto mapping ... |
| f1dm 6742 | The domain of a one-to-one... |
| f1dmOLD 6743 | Obsolete version of ~ f1dm... |
| f1ss 6744 | A function that is one-to-... |
| f1ssr 6745 | A function that is one-to-... |
| f1ssres 6746 | A function that is one-to-... |
| f1resf1 6747 | The restriction of an inje... |
| f1cnvcnv 6748 | Two ways to express that a... |
| f1cof1 6749 | Composition of two one-to-... |
| f1co 6750 | Composition of one-to-one ... |
| f1coOLD 6751 | Obsolete version of ~ f1co... |
| foeq1 6752 | Equality theorem for onto ... |
| foeq2 6753 | Equality theorem for onto ... |
| foeq3 6754 | Equality theorem for onto ... |
| nffo 6755 | Bound-variable hypothesis ... |
| fof 6756 | An onto mapping is a mappi... |
| fofun 6757 | An onto mapping is a funct... |
| fofn 6758 | An onto mapping is a funct... |
| forn 6759 | The codomain of an onto fu... |
| dffo2 6760 | Alternate definition of an... |
| foima 6761 | The image of the domain of... |
| dffn4 6762 | A function maps onto its r... |
| funforn 6763 | A function maps its domain... |
| fodmrnu 6764 | An onto function has uniqu... |
| fimadmfo 6765 | A function is a function o... |
| fores 6766 | Restriction of an onto fun... |
| fimadmfoALT 6767 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6768 | The preimage of the codoma... |
| focofo 6769 | Composition of onto functi... |
| foco 6770 | Composition of onto functi... |
| foconst 6771 | A nonzero constant functio... |
| f1oeq1 6772 | Equality theorem for one-t... |
| f1oeq2 6773 | Equality theorem for one-t... |
| f1oeq3 6774 | Equality theorem for one-t... |
| f1oeq23 6775 | Equality theorem for one-t... |
| f1eq123d 6776 | Equality deduction for one... |
| foeq123d 6777 | Equality deduction for ont... |
| f1oeq123d 6778 | Equality deduction for one... |
| f1oeq1d 6779 | Equality deduction for one... |
| f1oeq2d 6780 | Equality deduction for one... |
| f1oeq3d 6781 | Equality deduction for one... |
| nff1o 6782 | Bound-variable hypothesis ... |
| f1of1 6783 | A one-to-one onto mapping ... |
| f1of 6784 | A one-to-one onto mapping ... |
| f1ofn 6785 | A one-to-one onto mapping ... |
| f1ofun 6786 | A one-to-one onto mapping ... |
| f1orel 6787 | A one-to-one onto mapping ... |
| f1odm 6788 | The domain of a one-to-one... |
| dff1o2 6789 | Alternate definition of on... |
| dff1o3 6790 | Alternate definition of on... |
| f1ofo 6791 | A one-to-one onto function... |
| dff1o4 6792 | Alternate definition of on... |
| dff1o5 6793 | Alternate definition of on... |
| f1orn 6794 | A one-to-one function maps... |
| f1f1orn 6795 | A one-to-one function maps... |
| f1ocnv 6796 | The converse of a one-to-o... |
| f1ocnvb 6797 | A relation is a one-to-one... |
| f1ores 6798 | The restriction of a one-t... |
| f1orescnv 6799 | The converse of a one-to-o... |
| f1imacnv 6800 | Preimage of an image. (Co... |
| foimacnv 6801 | A reverse version of ~ f1i... |
| foun 6802 | The union of two onto func... |
| f1oun 6803 | The union of two one-to-on... |
| f1un 6804 | The union of two one-to-on... |
| resdif 6805 | The restriction of a one-t... |
| resin 6806 | The restriction of a one-t... |
| f1oco 6807 | Composition of one-to-one ... |
| f1cnv 6808 | The converse of an injecti... |
| funcocnv2 6809 | Composition with the conve... |
| fococnv2 6810 | The composition of an onto... |
| f1ococnv2 6811 | The composition of a one-t... |
| f1cocnv2 6812 | Composition of an injectiv... |
| f1ococnv1 6813 | The composition of a one-t... |
| f1cocnv1 6814 | Composition of an injectiv... |
| funcoeqres 6815 | Express a constraint on a ... |
| f1ssf1 6816 | A subset of an injective f... |
| f10 6817 | The empty set maps one-to-... |
| f10d 6818 | The empty set maps one-to-... |
| f1o00 6819 | One-to-one onto mapping of... |
| fo00 6820 | Onto mapping of the empty ... |
| f1o0 6821 | One-to-one onto mapping of... |
| f1oi 6822 | A restriction of the ident... |
| f1ovi 6823 | The identity relation is a... |
| f1osn 6824 | A singleton of an ordered ... |
| f1osng 6825 | A singleton of an ordered ... |
| f1sng 6826 | A singleton of an ordered ... |
| fsnd 6827 | A singleton of an ordered ... |
| f1oprswap 6828 | A two-element swap is a bi... |
| f1oprg 6829 | An unordered pair of order... |
| tz6.12-2 6830 | Function value when ` F ` ... |
| fveu 6831 | The value of a function at... |
| brprcneu 6832 | If ` A ` is a proper class... |
| brprcneuALT 6833 | Alternate proof of ~ brprc... |
| fvprc 6834 | A function's value at a pr... |
| fvprcALT 6835 | Alternate proof of ~ fvprc... |
| rnfvprc 6836 | The range of a function va... |
| fv2 6837 | Alternate definition of fu... |
| dffv3 6838 | A definition of function v... |
| dffv4 6839 | The previous definition of... |
| elfv 6840 | Membership in a function v... |
| fveq1 6841 | Equality theorem for funct... |
| fveq2 6842 | Equality theorem for funct... |
| fveq1i 6843 | Equality inference for fun... |
| fveq1d 6844 | Equality deduction for fun... |
| fveq2i 6845 | Equality inference for fun... |
| fveq2d 6846 | Equality deduction for fun... |
| 2fveq3 6847 | Equality theorem for neste... |
| fveq12i 6848 | Equality deduction for fun... |
| fveq12d 6849 | Equality deduction for fun... |
| fveqeq2d 6850 | Equality deduction for fun... |
| fveqeq2 6851 | Equality deduction for fun... |
| nffv 6852 | Bound-variable hypothesis ... |
| nffvmpt1 6853 | Bound-variable hypothesis ... |
| nffvd 6854 | Deduction version of bound... |
| fvex 6855 | The value of a class exist... |
| fvexi 6856 | The value of a class exist... |
| fvexd 6857 | The value of a class exist... |
| fvif 6858 | Move a conditional outside... |
| iffv 6859 | Move a conditional outside... |
| fv3 6860 | Alternate definition of th... |
| fvres 6861 | The value of a restricted ... |
| fvresd 6862 | The value of a restricted ... |
| funssfv 6863 | The value of a member of t... |
| tz6.12c 6864 | Corollary of Theorem 6.12(... |
| tz6.12-1 6865 | Function value. Theorem 6... |
| tz6.12-1OLD 6866 | Obsolete version of ~ tz6.... |
| tz6.12 6867 | Function value. Theorem 6... |
| tz6.12f 6868 | Function value, using boun... |
| tz6.12cOLD 6869 | Obsolete version of ~ tz6.... |
| tz6.12i 6870 | Corollary of Theorem 6.12(... |
| fvbr0 6871 | Two possibilities for the ... |
| fvrn0 6872 | A function value is a memb... |
| fvn0fvelrn 6873 | If the value of a function... |
| elfvunirn 6874 | A function value is a subs... |
| fvssunirn 6875 | The result of a function v... |
| fvssunirnOLD 6876 | Obsolete version of ~ fvss... |
| ndmfv 6877 | The value of a class outsi... |
| ndmfvrcl 6878 | Reverse closure law for fu... |
| elfvdm 6879 | If a function value has a ... |
| elfvex 6880 | If a function value has a ... |
| elfvexd 6881 | If a function value has a ... |
| eliman0 6882 | A nonempty function value ... |
| nfvres 6883 | The value of a non-member ... |
| nfunsn 6884 | If the restriction of a cl... |
| fvfundmfvn0 6885 | If the "value of a class" ... |
| 0fv 6886 | Function value of the empt... |
| fv2prc 6887 | A function value of a func... |
| elfv2ex 6888 | If a function value of a f... |
| fveqres 6889 | Equal values imply equal v... |
| csbfv12 6890 | Move class substitution in... |
| csbfv2g 6891 | Move class substitution in... |
| csbfv 6892 | Substitution for a functio... |
| funbrfv 6893 | The second argument of a b... |
| funopfv 6894 | The second element in an o... |
| fnbrfvb 6895 | Equivalence of function va... |
| fnopfvb 6896 | Equivalence of function va... |
| funbrfvb 6897 | Equivalence of function va... |
| funopfvb 6898 | Equivalence of function va... |
| fnbrfvb2 6899 | Version of ~ fnbrfvb for f... |
| funbrfv2b 6900 | Function value in terms of... |
| dffn5 6901 | Representation of a functi... |
| fnrnfv 6902 | The range of a function ex... |
| fvelrnb 6903 | A member of a function's r... |
| foelcdmi 6904 | A member of a surjective f... |
| dfimafn 6905 | Alternate definition of th... |
| dfimafn2 6906 | Alternate definition of th... |
| funimass4 6907 | Membership relation for th... |
| fvelima 6908 | Function value in an image... |
| fvelimad 6909 | Function value in an image... |
| feqmptd 6910 | Deduction form of ~ dffn5 ... |
| feqresmpt 6911 | Express a restricted funct... |
| feqmptdf 6912 | Deduction form of ~ dffn5f... |
| dffn5f 6913 | Representation of a functi... |
| fvelimab 6914 | Function value in an image... |
| fvelimabd 6915 | Deduction form of ~ fvelim... |
| unima 6916 | Image of a union. (Contri... |
| fvi 6917 | The value of the identity ... |
| fviss 6918 | The value of the identity ... |
| fniinfv 6919 | The indexed intersection o... |
| fnsnfv 6920 | Singleton of function valu... |
| fnsnfvOLD 6921 | Obsolete version of ~ fnsn... |
| opabiotafun 6922 | Define a function whose va... |
| opabiotadm 6923 | Define a function whose va... |
| opabiota 6924 | Define a function whose va... |
| fnimapr 6925 | The image of a pair under ... |
| ssimaex 6926 | The existence of a subimag... |
| ssimaexg 6927 | The existence of a subimag... |
| funfv 6928 | A simplified expression fo... |
| funfv2 6929 | The value of a function. ... |
| funfv2f 6930 | The value of a function. ... |
| fvun 6931 | Value of the union of two ... |
| fvun1 6932 | The value of a union when ... |
| fvun2 6933 | The value of a union when ... |
| fvun1d 6934 | The value of a union when ... |
| fvun2d 6935 | The value of a union when ... |
| dffv2 6936 | Alternate definition of fu... |
| dmfco 6937 | Domains of a function comp... |
| fvco2 6938 | Value of a function compos... |
| fvco 6939 | Value of a function compos... |
| fvco3 6940 | Value of a function compos... |
| fvco3d 6941 | Value of a function compos... |
| fvco4i 6942 | Conditions for a compositi... |
| fvopab3g 6943 | Value of a function given ... |
| fvopab3ig 6944 | Value of a function given ... |
| brfvopabrbr 6945 | The binary relation of a f... |
| fvmptg 6946 | Value of a function given ... |
| fvmpti 6947 | Value of a function given ... |
| fvmpt 6948 | Value of a function given ... |
| fvmpt2f 6949 | Value of a function given ... |
| fvtresfn 6950 | Functionality of a tuple-r... |
| fvmpts 6951 | Value of a function given ... |
| fvmpt3 6952 | Value of a function given ... |
| fvmpt3i 6953 | Value of a function given ... |
| fvmptdf 6954 | Deduction version of ~ fvm... |
| fvmptd 6955 | Deduction version of ~ fvm... |
| fvmptd2 6956 | Deduction version of ~ fvm... |
| mptrcl 6957 | Reverse closure for a mapp... |
| fvmpt2i 6958 | Value of a function given ... |
| fvmpt2 6959 | Value of a function given ... |
| fvmptss 6960 | If all the values of the m... |
| fvmpt2d 6961 | Deduction version of ~ fvm... |
| fvmptex 6962 | Express a function ` F ` w... |
| fvmptd3f 6963 | Alternate deduction versio... |
| fvmptd2f 6964 | Alternate deduction versio... |
| fvmptdv 6965 | Alternate deduction versio... |
| fvmptdv2 6966 | Alternate deduction versio... |
| mpteqb 6967 | Bidirectional equality the... |
| fvmptt 6968 | Closed theorem form of ~ f... |
| fvmptf 6969 | Value of a function given ... |
| fvmptnf 6970 | The value of a function gi... |
| fvmptd3 6971 | Deduction version of ~ fvm... |
| fvmptn 6972 | This somewhat non-intuitiv... |
| fvmptss2 6973 | A mapping always evaluates... |
| elfvmptrab1w 6974 | Implications for the value... |
| elfvmptrab1 6975 | Implications for the value... |
| elfvmptrab 6976 | Implications for the value... |
| fvopab4ndm 6977 | Value of a function given ... |
| fvmptndm 6978 | Value of a function given ... |
| fvmptrabfv 6979 | Value of a function mappin... |
| fvopab5 6980 | The value of a function th... |
| fvopab6 6981 | Value of a function given ... |
| eqfnfv 6982 | Equality of functions is d... |
| eqfnfv2 6983 | Equality of functions is d... |
| eqfnfv3 6984 | Derive equality of functio... |
| eqfnfvd 6985 | Deduction for equality of ... |
| eqfnfv2f 6986 | Equality of functions is d... |
| eqfunfv 6987 | Equality of functions is d... |
| fvreseq0 6988 | Equality of restricted fun... |
| fvreseq1 6989 | Equality of a function res... |
| fvreseq 6990 | Equality of restricted fun... |
| fnmptfvd 6991 | A function with a given do... |
| fndmdif 6992 | Two ways to express the lo... |
| fndmdifcom 6993 | The difference set between... |
| fndmdifeq0 6994 | The difference set of two ... |
| fndmin 6995 | Two ways to express the lo... |
| fneqeql 6996 | Two functions are equal if... |
| fneqeql2 6997 | Two functions are equal if... |
| fnreseql 6998 | Two functions are equal on... |
| chfnrn 6999 | The range of a choice func... |
| funfvop 7000 | Ordered pair with function... |
| funfvbrb 7001 | Two ways to say that ` A `... |
| fvimacnvi 7002 | A member of a preimage is ... |
| fvimacnv 7003 | The argument of a function... |
| funimass3 7004 | A kind of contraposition l... |
| funimass5 7005 | A subclass of a preimage i... |
| funconstss 7006 | Two ways of specifying tha... |
| fvimacnvALT 7007 | Alternate proof of ~ fvima... |
| elpreima 7008 | Membership in the preimage... |
| elpreimad 7009 | Membership in the preimage... |
| fniniseg 7010 | Membership in the preimage... |
| fncnvima2 7011 | Inverse images under funct... |
| fniniseg2 7012 | Inverse point images under... |
| unpreima 7013 | Preimage of a union. (Con... |
| inpreima 7014 | Preimage of an intersectio... |
| difpreima 7015 | Preimage of a difference. ... |
| respreima 7016 | The preimage of a restrict... |
| cnvimainrn 7017 | The preimage of the inters... |
| sspreima 7018 | The preimage of a subset i... |
| iinpreima 7019 | Preimage of an intersectio... |
| intpreima 7020 | Preimage of an intersectio... |
| fimacnvOLD 7021 | Obsolete version of ~ fima... |
| fimacnvinrn 7022 | Taking the converse image ... |
| fimacnvinrn2 7023 | Taking the converse image ... |
| rescnvimafod 7024 | The restriction of a funct... |
| fvn0ssdmfun 7025 | If a class' function value... |
| fnopfv 7026 | Ordered pair with function... |
| fvelrn 7027 | A function's value belongs... |
| nelrnfvne 7028 | A function value cannot be... |
| fveqdmss 7029 | If the empty set is not co... |
| fveqressseq 7030 | If the empty set is not co... |
| fnfvelrn 7031 | A function's value belongs... |
| ffvelcdm 7032 | A function's value belongs... |
| fnfvelrnd 7033 | A function's value belongs... |
| ffvelcdmi 7034 | A function's value belongs... |
| ffvelcdmda 7035 | A function's value belongs... |
| ffvelcdmd 7036 | A function's value belongs... |
| rexrn 7037 | Restricted existential qua... |
| ralrn 7038 | Restricted universal quant... |
| elrnrexdm 7039 | For any element in the ran... |
| elrnrexdmb 7040 | For any element in the ran... |
| eldmrexrn 7041 | For any element in the dom... |
| eldmrexrnb 7042 | For any element in the dom... |
| fvcofneq 7043 | The values of two function... |
| ralrnmptw 7044 | A restricted quantifier ov... |
| rexrnmptw 7045 | A restricted quantifier ov... |
| ralrnmpt 7046 | A restricted quantifier ov... |
| rexrnmpt 7047 | A restricted quantifier ov... |
| f0cli 7048 | Unconditional closure of a... |
| dff2 7049 | Alternate definition of a ... |
| dff3 7050 | Alternate definition of a ... |
| dff4 7051 | Alternate definition of a ... |
| dffo3 7052 | An onto mapping expressed ... |
| dffo4 7053 | Alternate definition of an... |
| dffo5 7054 | Alternate definition of an... |
| exfo 7055 | A relation equivalent to t... |
| foelrn 7056 | Property of a surjective f... |
| foco2 7057 | If a composition of two fu... |
| fmpt 7058 | Functionality of the mappi... |
| f1ompt 7059 | Express bijection for a ma... |
| fmpti 7060 | Functionality of the mappi... |
| fvmptelcdm 7061 | The value of a function at... |
| fmptd 7062 | Domain and codomain of the... |
| fmpttd 7063 | Version of ~ fmptd with in... |
| fmpt3d 7064 | Domain and codomain of the... |
| fmptdf 7065 | A version of ~ fmptd using... |
| ffnfv 7066 | A function maps to a class... |
| ffnfvf 7067 | A function maps to a class... |
| fnfvrnss 7068 | An upper bound for range d... |
| fcdmssb 7069 | A function is a function i... |
| rnmptss 7070 | The range of an operation ... |
| fmpt2d 7071 | Domain and codomain of the... |
| ffvresb 7072 | A necessary and sufficient... |
| f1oresrab 7073 | Build a bijection between ... |
| f1ossf1o 7074 | Restricting a bijection, w... |
| fmptco 7075 | Composition of two functio... |
| fmptcof 7076 | Version of ~ fmptco where ... |
| fmptcos 7077 | Composition of two functio... |
| cofmpt 7078 | Express composition of a m... |
| fcompt 7079 | Express composition of two... |
| fcoconst 7080 | Composition with a constan... |
| fsn 7081 | A function maps a singleto... |
| fsn2 7082 | A function that maps a sin... |
| fsng 7083 | A function maps a singleto... |
| fsn2g 7084 | A function that maps a sin... |
| xpsng 7085 | The Cartesian product of t... |
| xpprsng 7086 | The Cartesian product of a... |
| xpsn 7087 | The Cartesian product of t... |
| f1o2sn 7088 | A singleton consisting in ... |
| residpr 7089 | Restriction of the identit... |
| dfmpt 7090 | Alternate definition for t... |
| fnasrn 7091 | A function expressed as th... |
| idref 7092 | Two ways to state that a r... |
| funiun 7093 | A function is a union of s... |
| funopsn 7094 | If a function is an ordere... |
| funop 7095 | An ordered pair is a funct... |
| funopdmsn 7096 | The domain of a function w... |
| funsndifnop 7097 | A singleton of an ordered ... |
| funsneqopb 7098 | A singleton of an ordered ... |
| ressnop0 7099 | If ` A ` is not in ` C ` ,... |
| fpr 7100 | A function with a domain o... |
| fprg 7101 | A function with a domain o... |
| ftpg 7102 | A function with a domain o... |
| ftp 7103 | A function with a domain o... |
| fnressn 7104 | A function restricted to a... |
| funressn 7105 | A function restricted to a... |
| fressnfv 7106 | The value of a function re... |
| fvrnressn 7107 | If the value of a function... |
| fvressn 7108 | The value of a function re... |
| fvn0fvelrnOLD 7109 | Obsolete version of ~ fvn0... |
| fvconst 7110 | The value of a constant fu... |
| fnsnr 7111 | If a class belongs to a fu... |
| fnsnb 7112 | A function whose domain is... |
| fmptsn 7113 | Express a singleton functi... |
| fmptsng 7114 | Express a singleton functi... |
| fmptsnd 7115 | Express a singleton functi... |
| fmptap 7116 | Append an additional value... |
| fmptapd 7117 | Append an additional value... |
| fmptpr 7118 | Express a pair function in... |
| fvresi 7119 | The value of a restricted ... |
| fninfp 7120 | Express the class of fixed... |
| fnelfp 7121 | Property of a fixed point ... |
| fndifnfp 7122 | Express the class of non-f... |
| fnelnfp 7123 | Property of a non-fixed po... |
| fnnfpeq0 7124 | A function is the identity... |
| fvunsn 7125 | Remove an ordered pair not... |
| fvsng 7126 | The value of a singleton o... |
| fvsn 7127 | The value of a singleton o... |
| fvsnun1 7128 | The value of a function wi... |
| fvsnun2 7129 | The value of a function wi... |
| fnsnsplit 7130 | Split a function into a si... |
| fsnunf 7131 | Adjoining a point to a fun... |
| fsnunf2 7132 | Adjoining a point to a pun... |
| fsnunfv 7133 | Recover the added point fr... |
| fsnunres 7134 | Recover the original funct... |
| funresdfunsn 7135 | Restricting a function to ... |
| fvpr1g 7136 | The value of a function wi... |
| fvpr2g 7137 | The value of a function wi... |
| fvpr2gOLD 7138 | Obsolete version of ~ fvpr... |
| fvpr1 7139 | The value of a function wi... |
| fvpr1OLD 7140 | Obsolete version of ~ fvpr... |
| fvpr2 7141 | The value of a function wi... |
| fvpr2OLD 7142 | Obsolete version of ~ fvpr... |
| fprb 7143 | A condition for functionho... |
| fvtp1 7144 | The first value of a funct... |
| fvtp2 7145 | The second value of a func... |
| fvtp3 7146 | The third value of a funct... |
| fvtp1g 7147 | The value of a function wi... |
| fvtp2g 7148 | The value of a function wi... |
| fvtp3g 7149 | The value of a function wi... |
| tpres 7150 | An unordered triple of ord... |
| fvconst2g 7151 | The value of a constant fu... |
| fconst2g 7152 | A constant function expres... |
| fvconst2 7153 | The value of a constant fu... |
| fconst2 7154 | A constant function expres... |
| fconst5 7155 | Two ways to express that a... |
| rnmptc 7156 | Range of a constant functi... |
| rnmptcOLD 7157 | Obsolete version of ~ rnmp... |
| fnprb 7158 | A function whose domain ha... |
| fntpb 7159 | A function whose domain ha... |
| fnpr2g 7160 | A function whose domain ha... |
| fpr2g 7161 | A function that maps a pai... |
| fconstfv 7162 | A constant function expres... |
| fconst3 7163 | Two ways to express a cons... |
| fconst4 7164 | Two ways to express a cons... |
| resfunexg 7165 | The restriction of a funct... |
| resiexd 7166 | The restriction of the ide... |
| fnex 7167 | If the domain of a functio... |
| fnexd 7168 | If the domain of a functio... |
| funex 7169 | If the domain of a functio... |
| opabex 7170 | Existence of a function ex... |
| mptexg 7171 | If the domain of a functio... |
| mptexgf 7172 | If the domain of a functio... |
| mptex 7173 | If the domain of a functio... |
| mptexd 7174 | If the domain of a functio... |
| mptrabex 7175 | If the domain of a functio... |
| fex 7176 | If the domain of a mapping... |
| fexd 7177 | If the domain of a mapping... |
| mptfvmpt 7178 | A function in maps-to nota... |
| eufnfv 7179 | A function is uniquely det... |
| funfvima 7180 | A function's value in a pr... |
| funfvima2 7181 | A function's value in an i... |
| funfvima2d 7182 | A function's value in a pr... |
| fnfvima 7183 | The function value of an o... |
| fnfvimad 7184 | A function's value belongs... |
| resfvresima 7185 | The value of the function ... |
| funfvima3 7186 | A class including a functi... |
| rexima 7187 | Existential quantification... |
| ralima 7188 | Universal quantification u... |
| fvclss 7189 | Upper bound for the class ... |
| elabrex 7190 | Elementhood in an image se... |
| abrexco 7191 | Composition of two image m... |
| imaiun 7192 | The image of an indexed un... |
| imauni 7193 | The image of a union is th... |
| fniunfv 7194 | The indexed union of a fun... |
| funiunfv 7195 | The indexed union of a fun... |
| funiunfvf 7196 | The indexed union of a fun... |
| eluniima 7197 | Membership in the union of... |
| elunirn 7198 | Membership in the union of... |
| elunirnALT 7199 | Alternate proof of ~ eluni... |
| elunirn2OLD 7200 | Obsolete version of ~ elfv... |
| fnunirn 7201 | Membership in a union of s... |
| dff13 7202 | A one-to-one function in t... |
| dff13f 7203 | A one-to-one function in t... |
| f1veqaeq 7204 | If the values of a one-to-... |
| f1cofveqaeq 7205 | If the values of a composi... |
| f1cofveqaeqALT 7206 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7207 | If two one-to-one function... |
| f1mpt 7208 | Express injection for a ma... |
| f1fveq 7209 | Equality of function value... |
| f1elima 7210 | Membership in the image of... |
| f1imass 7211 | Taking images under a one-... |
| f1imaeq 7212 | Taking images under a one-... |
| f1imapss 7213 | Taking images under a one-... |
| fpropnf1 7214 | A function, given by an un... |
| f1dom3fv3dif 7215 | The function values for a ... |
| f1dom3el3dif 7216 | The codomain of a 1-1 func... |
| dff14a 7217 | A one-to-one function in t... |
| dff14b 7218 | A one-to-one function in t... |
| f12dfv 7219 | A one-to-one function with... |
| f13dfv 7220 | A one-to-one function with... |
| dff1o6 7221 | A one-to-one onto function... |
| f1ocnvfv1 7222 | The converse value of the ... |
| f1ocnvfv2 7223 | The value of the converse ... |
| f1ocnvfv 7224 | Relationship between the v... |
| f1ocnvfvb 7225 | Relationship between the v... |
| nvof1o 7226 | An involution is a bijecti... |
| nvocnv 7227 | The converse of an involut... |
| f1cdmsn 7228 | If a one-to-one function w... |
| fsnex 7229 | Relate a function with a s... |
| f1prex 7230 | Relate a one-to-one functi... |
| f1ocnvdm 7231 | The value of the converse ... |
| f1ocnvfvrneq 7232 | If the values of a one-to-... |
| fcof1 7233 | An application is injectiv... |
| fcofo 7234 | An application is surjecti... |
| cbvfo 7235 | Change bound variable betw... |
| cbvexfo 7236 | Change bound variable betw... |
| cocan1 7237 | An injection is left-cance... |
| cocan2 7238 | A surjection is right-canc... |
| fcof1oinvd 7239 | Show that a function is th... |
| fcof1od 7240 | A function is bijective if... |
| 2fcoidinvd 7241 | Show that a function is th... |
| fcof1o 7242 | Show that two functions ar... |
| 2fvcoidd 7243 | Show that the composition ... |
| 2fvidf1od 7244 | A function is bijective if... |
| 2fvidinvd 7245 | Show that two functions ar... |
| foeqcnvco 7246 | Condition for function equ... |
| f1eqcocnv 7247 | Condition for function equ... |
| f1eqcocnvOLD 7248 | Obsolete version of ~ f1eq... |
| fveqf1o 7249 | Given a bijection ` F ` , ... |
| nf1const 7250 | A constant function from a... |
| nf1oconst 7251 | A constant function from a... |
| f1ofvswap 7252 | Swapping two values in a b... |
| fliftrel 7253 | ` F ` , a function lift, i... |
| fliftel 7254 | Elementhood in the relatio... |
| fliftel1 7255 | Elementhood in the relatio... |
| fliftcnv 7256 | Converse of the relation `... |
| fliftfun 7257 | The function ` F ` is the ... |
| fliftfund 7258 | The function ` F ` is the ... |
| fliftfuns 7259 | The function ` F ` is the ... |
| fliftf 7260 | The domain and range of th... |
| fliftval 7261 | The value of the function ... |
| isoeq1 7262 | Equality theorem for isomo... |
| isoeq2 7263 | Equality theorem for isomo... |
| isoeq3 7264 | Equality theorem for isomo... |
| isoeq4 7265 | Equality theorem for isomo... |
| isoeq5 7266 | Equality theorem for isomo... |
| nfiso 7267 | Bound-variable hypothesis ... |
| isof1o 7268 | An isomorphism is a one-to... |
| isof1oidb 7269 | A function is a bijection ... |
| isof1oopb 7270 | A function is a bijection ... |
| isorel 7271 | An isomorphism connects bi... |
| soisores 7272 | Express the condition of i... |
| soisoi 7273 | Infer isomorphism from one... |
| isoid 7274 | Identity law for isomorphi... |
| isocnv 7275 | Converse law for isomorphi... |
| isocnv2 7276 | Converse law for isomorphi... |
| isocnv3 7277 | Complementation law for is... |
| isores2 7278 | An isomorphism from one we... |
| isores1 7279 | An isomorphism from one we... |
| isores3 7280 | Induced isomorphism on a s... |
| isotr 7281 | Composition (transitive) l... |
| isomin 7282 | Isomorphisms preserve mini... |
| isoini 7283 | Isomorphisms preserve init... |
| isoini2 7284 | Isomorphisms are isomorphi... |
| isofrlem 7285 | Lemma for ~ isofr . (Cont... |
| isoselem 7286 | Lemma for ~ isose . (Cont... |
| isofr 7287 | An isomorphism preserves w... |
| isose 7288 | An isomorphism preserves s... |
| isofr2 7289 | A weak form of ~ isofr tha... |
| isopolem 7290 | Lemma for ~ isopo . (Cont... |
| isopo 7291 | An isomorphism preserves t... |
| isosolem 7292 | Lemma for ~ isoso . (Cont... |
| isoso 7293 | An isomorphism preserves t... |
| isowe 7294 | An isomorphism preserves t... |
| isowe2 7295 | A weak form of ~ isowe tha... |
| f1oiso 7296 | Any one-to-one onto functi... |
| f1oiso2 7297 | Any one-to-one onto functi... |
| f1owe 7298 | Well-ordering of isomorphi... |
| weniso 7299 | A set-like well-ordering h... |
| weisoeq 7300 | Thus, there is at most one... |
| weisoeq2 7301 | Thus, there is at most one... |
| knatar 7302 | The Knaster-Tarski theorem... |
| fvresval 7303 | The value of a restricted ... |
| funeldmb 7304 | If ` (/) ` is not part of ... |
| eqfunresadj 7305 | Law for adjoining an eleme... |
| eqfunressuc 7306 | Law for equality of restri... |
| fnssintima 7307 | Condition for subset of an... |
| imaeqsexv 7308 | Substitute a function valu... |
| imaeqsalv 7309 | Substitute a function valu... |
| canth 7310 | No set ` A ` is equinumero... |
| ncanth 7311 | Cantor's theorem fails for... |
| riotaeqdv 7314 | Formula-building deduction... |
| riotabidv 7315 | Formula-building deduction... |
| riotaeqbidv 7316 | Equality deduction for res... |
| riotaex 7317 | Restricted iota is a set. ... |
| riotav 7318 | An iota restricted to the ... |
| riotauni 7319 | Restricted iota in terms o... |
| nfriota1 7320 | The abstraction variable i... |
| nfriotadw 7321 | Deduction version of ~ nfr... |
| cbvriotaw 7322 | Change bound variable in a... |
| cbvriotavw 7323 | Change bound variable in a... |
| cbvriotavwOLD 7324 | Obsolete version of ~ cbvr... |
| nfriotad 7325 | Deduction version of ~ nfr... |
| nfriota 7326 | A variable not free in a w... |
| cbvriota 7327 | Change bound variable in a... |
| cbvriotav 7328 | Change bound variable in a... |
| csbriota 7329 | Interchange class substitu... |
| riotacl2 7330 | Membership law for "the un... |
| riotacl 7331 | Closure of restricted iota... |
| riotasbc 7332 | Substitution law for descr... |
| riotabidva 7333 | Equivalent wff's yield equ... |
| riotabiia 7334 | Equivalent wff's yield equ... |
| riota1 7335 | Property of restricted iot... |
| riota1a 7336 | Property of iota. (Contri... |
| riota2df 7337 | A deduction version of ~ r... |
| riota2f 7338 | This theorem shows a condi... |
| riota2 7339 | This theorem shows a condi... |
| riotaeqimp 7340 | If two restricted iota des... |
| riotaprop 7341 | Properties of a restricted... |
| riota5f 7342 | A method for computing res... |
| riota5 7343 | A method for computing res... |
| riotass2 7344 | Restriction of a unique el... |
| riotass 7345 | Restriction of a unique el... |
| moriotass 7346 | Restriction of a unique el... |
| snriota 7347 | A restricted class abstrac... |
| riotaxfrd 7348 | Change the variable ` x ` ... |
| eusvobj2 7349 | Specify the same property ... |
| eusvobj1 7350 | Specify the same object in... |
| f1ofveu 7351 | There is one domain elemen... |
| f1ocnvfv3 7352 | Value of the converse of a... |
| riotaund 7353 | Restricted iota equals the... |
| riotassuni 7354 | The restricted iota class ... |
| riotaclb 7355 | Bidirectional closure of r... |
| riotarab 7356 | Restricted iota of a restr... |
| oveq 7363 | Equality theorem for opera... |
| oveq1 7364 | Equality theorem for opera... |
| oveq2 7365 | Equality theorem for opera... |
| oveq12 7366 | Equality theorem for opera... |
| oveq1i 7367 | Equality inference for ope... |
| oveq2i 7368 | Equality inference for ope... |
| oveq12i 7369 | Equality inference for ope... |
| oveqi 7370 | Equality inference for ope... |
| oveq123i 7371 | Equality inference for ope... |
| oveq1d 7372 | Equality deduction for ope... |
| oveq2d 7373 | Equality deduction for ope... |
| oveqd 7374 | Equality deduction for ope... |
| oveq12d 7375 | Equality deduction for ope... |
| oveqan12d 7376 | Equality deduction for ope... |
| oveqan12rd 7377 | Equality deduction for ope... |
| oveq123d 7378 | Equality deduction for ope... |
| fvoveq1d 7379 | Equality deduction for nes... |
| fvoveq1 7380 | Equality theorem for neste... |
| ovanraleqv 7381 | Equality theorem for a con... |
| imbrov2fvoveq 7382 | Equality theorem for neste... |
| ovrspc2v 7383 | If an operation value is e... |
| oveqrspc2v 7384 | Restricted specialization ... |
| oveqdr 7385 | Equality of two operations... |
| nfovd 7386 | Deduction version of bound... |
| nfov 7387 | Bound-variable hypothesis ... |
| oprabidw 7388 | The law of concretion. Sp... |
| oprabid 7389 | The law of concretion. Sp... |
| ovex 7390 | The result of an operation... |
| ovexi 7391 | The result of an operation... |
| ovexd 7392 | The result of an operation... |
| ovssunirn 7393 | The result of an operation... |
| 0ov 7394 | Operation value of the emp... |
| ovprc 7395 | The value of an operation ... |
| ovprc1 7396 | The value of an operation ... |
| ovprc2 7397 | The value of an operation ... |
| ovrcl 7398 | Reverse closure for an ope... |
| csbov123 7399 | Move class substitution in... |
| csbov 7400 | Move class substitution in... |
| csbov12g 7401 | Move class substitution in... |
| csbov1g 7402 | Move class substitution in... |
| csbov2g 7403 | Move class substitution in... |
| rspceov 7404 | A frequently used special ... |
| elovimad 7405 | Elementhood of the image s... |
| fnbrovb 7406 | Value of a binary operatio... |
| fnotovb 7407 | Equivalence of operation v... |
| opabbrex 7408 | A collection of ordered pa... |
| opabresex2 7409 | Restrictions of a collecti... |
| opabresex2d 7410 | Obsolete version of ~ opab... |
| fvmptopab 7411 | The function value of a ma... |
| fvmptopabOLD 7412 | Obsolete version of ~ fvmp... |
| f1opr 7413 | Condition for an operation... |
| brfvopab 7414 | The classes involved in a ... |
| dfoprab2 7415 | Class abstraction for oper... |
| reloprab 7416 | An operation class abstrac... |
| oprabv 7417 | If a pair and a class are ... |
| nfoprab1 7418 | The abstraction variables ... |
| nfoprab2 7419 | The abstraction variables ... |
| nfoprab3 7420 | The abstraction variables ... |
| nfoprab 7421 | Bound-variable hypothesis ... |
| oprabbid 7422 | Equivalent wff's yield equ... |
| oprabbidv 7423 | Equivalent wff's yield equ... |
| oprabbii 7424 | Equivalent wff's yield equ... |
| ssoprab2 7425 | Equivalence of ordered pai... |
| ssoprab2b 7426 | Equivalence of ordered pai... |
| eqoprab2bw 7427 | Equivalence of ordered pai... |
| eqoprab2b 7428 | Equivalence of ordered pai... |
| mpoeq123 7429 | An equality theorem for th... |
| mpoeq12 7430 | An equality theorem for th... |
| mpoeq123dva 7431 | An equality deduction for ... |
| mpoeq123dv 7432 | An equality deduction for ... |
| mpoeq123i 7433 | An equality inference for ... |
| mpoeq3dva 7434 | Slightly more general equa... |
| mpoeq3ia 7435 | An equality inference for ... |
| mpoeq3dv 7436 | An equality deduction for ... |
| nfmpo1 7437 | Bound-variable hypothesis ... |
| nfmpo2 7438 | Bound-variable hypothesis ... |
| nfmpo 7439 | Bound-variable hypothesis ... |
| 0mpo0 7440 | A mapping operation with e... |
| mpo0v 7441 | A mapping operation with e... |
| mpo0 7442 | A mapping operation with e... |
| oprab4 7443 | Two ways to state the doma... |
| cbvoprab1 7444 | Rule used to change first ... |
| cbvoprab2 7445 | Change the second bound va... |
| cbvoprab12 7446 | Rule used to change first ... |
| cbvoprab12v 7447 | Rule used to change first ... |
| cbvoprab3 7448 | Rule used to change the th... |
| cbvoprab3v 7449 | Rule used to change the th... |
| cbvmpox 7450 | Rule to change the bound v... |
| cbvmpo 7451 | Rule to change the bound v... |
| cbvmpov 7452 | Rule to change the bound v... |
| elimdelov 7453 | Eliminate a hypothesis whi... |
| ovif 7454 | Move a conditional outside... |
| ovif2 7455 | Move a conditional outside... |
| ovif12 7456 | Move a conditional outside... |
| ifov 7457 | Move a conditional outside... |
| dmoprab 7458 | The domain of an operation... |
| dmoprabss 7459 | The domain of an operation... |
| rnoprab 7460 | The range of an operation ... |
| rnoprab2 7461 | The range of a restricted ... |
| reldmoprab 7462 | The domain of an operation... |
| oprabss 7463 | Structure of an operation ... |
| eloprabga 7464 | The law of concretion for ... |
| eloprabgaOLD 7465 | Obsolete version of ~ elop... |
| eloprabg 7466 | The law of concretion for ... |
| ssoprab2i 7467 | Inference of operation cla... |
| mpov 7468 | Operation with universal d... |
| mpomptx 7469 | Express a two-argument fun... |
| mpompt 7470 | Express a two-argument fun... |
| mpodifsnif 7471 | A mapping with two argumen... |
| mposnif 7472 | A mapping with two argumen... |
| fconstmpo 7473 | Representation of a consta... |
| resoprab 7474 | Restriction of an operatio... |
| resoprab2 7475 | Restriction of an operator... |
| resmpo 7476 | Restriction of the mapping... |
| funoprabg 7477 | "At most one" is a suffici... |
| funoprab 7478 | "At most one" is a suffici... |
| fnoprabg 7479 | Functionality and domain o... |
| mpofun 7480 | The maps-to notation for a... |
| mpofunOLD 7481 | Obsolete version of ~ mpof... |
| fnoprab 7482 | Functionality and domain o... |
| ffnov 7483 | An operation maps to a cla... |
| fovcl 7484 | Closure law for an operati... |
| eqfnov 7485 | Equality of two operations... |
| eqfnov2 7486 | Two operators with the sam... |
| fnov 7487 | Representation of a functi... |
| mpo2eqb 7488 | Bidirectional equality the... |
| rnmpo 7489 | The range of an operation ... |
| reldmmpo 7490 | The domain of an operation... |
| elrnmpog 7491 | Membership in the range of... |
| elrnmpo 7492 | Membership in the range of... |
| elrnmpores 7493 | Membership in the range of... |
| ralrnmpo 7494 | A restricted quantifier ov... |
| rexrnmpo 7495 | A restricted quantifier ov... |
| ovid 7496 | The value of an operation ... |
| ovidig 7497 | The value of an operation ... |
| ovidi 7498 | The value of an operation ... |
| ov 7499 | The value of an operation ... |
| ovigg 7500 | The value of an operation ... |
| ovig 7501 | The value of an operation ... |
| ovmpt4g 7502 | Value of a function given ... |
| ovmpos 7503 | Value of a function given ... |
| ov2gf 7504 | The value of an operation ... |
| ovmpodxf 7505 | Value of an operation give... |
| ovmpodx 7506 | Value of an operation give... |
| ovmpod 7507 | Value of an operation give... |
| ovmpox 7508 | The value of an operation ... |
| ovmpoga 7509 | Value of an operation give... |
| ovmpoa 7510 | Value of an operation give... |
| ovmpodf 7511 | Alternate deduction versio... |
| ovmpodv 7512 | Alternate deduction versio... |
| ovmpodv2 7513 | Alternate deduction versio... |
| ovmpog 7514 | Value of an operation give... |
| ovmpo 7515 | Value of an operation give... |
| fvmpopr2d 7516 | Value of an operation give... |
| ov3 7517 | The value of an operation ... |
| ov6g 7518 | The value of an operation ... |
| ovg 7519 | The value of an operation ... |
| ovres 7520 | The value of a restricted ... |
| ovresd 7521 | Lemma for converting metri... |
| oprres 7522 | The restriction of an oper... |
| oprssov 7523 | The value of a member of t... |
| fovcdm 7524 | An operation's value belon... |
| fovcdmda 7525 | An operation's value belon... |
| fovcdmd 7526 | An operation's value belon... |
| fnrnov 7527 | The range of an operation ... |
| foov 7528 | An onto mapping of an oper... |
| fnovrn 7529 | An operation's value belon... |
| ovelrn 7530 | A member of an operation's... |
| funimassov 7531 | Membership relation for th... |
| ovelimab 7532 | Operation value in an imag... |
| ovima0 7533 | An operation value is a me... |
| ovconst2 7534 | The value of a constant op... |
| oprssdm 7535 | Domain of closure of an op... |
| nssdmovg 7536 | The value of an operation ... |
| ndmovg 7537 | The value of an operation ... |
| ndmov 7538 | The value of an operation ... |
| ndmovcl 7539 | The closure of an operatio... |
| ndmovrcl 7540 | Reverse closure law, when ... |
| ndmovcom 7541 | Any operation is commutati... |
| ndmovass 7542 | Any operation is associati... |
| ndmovdistr 7543 | Any operation is distribut... |
| ndmovord 7544 | Elimination of redundant a... |
| ndmovordi 7545 | Elimination of redundant a... |
| caovclg 7546 | Convert an operation closu... |
| caovcld 7547 | Convert an operation closu... |
| caovcl 7548 | Convert an operation closu... |
| caovcomg 7549 | Convert an operation commu... |
| caovcomd 7550 | Convert an operation commu... |
| caovcom 7551 | Convert an operation commu... |
| caovassg 7552 | Convert an operation assoc... |
| caovassd 7553 | Convert an operation assoc... |
| caovass 7554 | Convert an operation assoc... |
| caovcang 7555 | Convert an operation cance... |
| caovcand 7556 | Convert an operation cance... |
| caovcanrd 7557 | Commute the arguments of a... |
| caovcan 7558 | Convert an operation cance... |
| caovordig 7559 | Convert an operation order... |
| caovordid 7560 | Convert an operation order... |
| caovordg 7561 | Convert an operation order... |
| caovordd 7562 | Convert an operation order... |
| caovord2d 7563 | Operation ordering law wit... |
| caovord3d 7564 | Ordering law. (Contribute... |
| caovord 7565 | Convert an operation order... |
| caovord2 7566 | Operation ordering law wit... |
| caovord3 7567 | Ordering law. (Contribute... |
| caovdig 7568 | Convert an operation distr... |
| caovdid 7569 | Convert an operation distr... |
| caovdir2d 7570 | Convert an operation distr... |
| caovdirg 7571 | Convert an operation rever... |
| caovdird 7572 | Convert an operation distr... |
| caovdi 7573 | Convert an operation distr... |
| caov32d 7574 | Rearrange arguments in a c... |
| caov12d 7575 | Rearrange arguments in a c... |
| caov31d 7576 | Rearrange arguments in a c... |
| caov13d 7577 | Rearrange arguments in a c... |
| caov4d 7578 | Rearrange arguments in a c... |
| caov411d 7579 | Rearrange arguments in a c... |
| caov42d 7580 | Rearrange arguments in a c... |
| caov32 7581 | Rearrange arguments in a c... |
| caov12 7582 | Rearrange arguments in a c... |
| caov31 7583 | Rearrange arguments in a c... |
| caov13 7584 | Rearrange arguments in a c... |
| caov4 7585 | Rearrange arguments in a c... |
| caov411 7586 | Rearrange arguments in a c... |
| caov42 7587 | Rearrange arguments in a c... |
| caovdir 7588 | Reverse distributive law. ... |
| caovdilem 7589 | Lemma used by real number ... |
| caovlem2 7590 | Lemma used in real number ... |
| caovmo 7591 | Uniqueness of inverse elem... |
| imaeqexov 7592 | Substitute an operation va... |
| imaeqalov 7593 | Substitute an operation va... |
| mpondm0 7594 | The value of an operation ... |
| elmpocl 7595 | If a two-parameter class i... |
| elmpocl1 7596 | If a two-parameter class i... |
| elmpocl2 7597 | If a two-parameter class i... |
| elovmpo 7598 | Utility lemma for two-para... |
| elovmporab 7599 | Implications for the value... |
| elovmporab1w 7600 | Implications for the value... |
| elovmporab1 7601 | Implications for the value... |
| 2mpo0 7602 | If the operation value of ... |
| relmptopab 7603 | Any function to sets of or... |
| f1ocnvd 7604 | Describe an implicit one-t... |
| f1od 7605 | Describe an implicit one-t... |
| f1ocnv2d 7606 | Describe an implicit one-t... |
| f1o2d 7607 | Describe an implicit one-t... |
| f1opw2 7608 | A one-to-one mapping induc... |
| f1opw 7609 | A one-to-one mapping induc... |
| elovmpt3imp 7610 | If the value of a function... |
| ovmpt3rab1 7611 | The value of an operation ... |
| ovmpt3rabdm 7612 | If the value of a function... |
| elovmpt3rab1 7613 | Implications for the value... |
| elovmpt3rab 7614 | Implications for the value... |
| ofeqd 7619 | Equality theorem for funct... |
| ofeq 7620 | Equality theorem for funct... |
| ofreq 7621 | Equality theorem for funct... |
| ofexg 7622 | A function operation restr... |
| nfof 7623 | Hypothesis builder for fun... |
| nfofr 7624 | Hypothesis builder for fun... |
| ofrfvalg 7625 | Value of a relation applie... |
| offval 7626 | Value of an operation appl... |
| ofrfval 7627 | Value of a relation applie... |
| ofval 7628 | Evaluate a function operat... |
| ofrval 7629 | Exhibit a function relatio... |
| offn 7630 | The function operation pro... |
| offun 7631 | The function operation pro... |
| offval2f 7632 | The function operation exp... |
| ofmresval 7633 | Value of a restriction of ... |
| fnfvof 7634 | Function value of a pointw... |
| off 7635 | The function operation pro... |
| ofres 7636 | Restrict the operands of a... |
| offval2 7637 | The function operation exp... |
| ofrfval2 7638 | The function relation acti... |
| ofmpteq 7639 | Value of a pointwise opera... |
| ofco 7640 | The composition of a funct... |
| offveq 7641 | Convert an identity of the... |
| offveqb 7642 | Equivalent expressions for... |
| ofc1 7643 | Left operation by a consta... |
| ofc2 7644 | Right operation by a const... |
| ofc12 7645 | Function operation on two ... |
| caofref 7646 | Transfer a reflexive law t... |
| caofinvl 7647 | Transfer a left inverse la... |
| caofid0l 7648 | Transfer a left identity l... |
| caofid0r 7649 | Transfer a right identity ... |
| caofid1 7650 | Transfer a right absorptio... |
| caofid2 7651 | Transfer a right absorptio... |
| caofcom 7652 | Transfer a commutative law... |
| caofrss 7653 | Transfer a relation subset... |
| caofass 7654 | Transfer an associative la... |
| caoftrn 7655 | Transfer a transitivity la... |
| caofdi 7656 | Transfer a distributive la... |
| caofdir 7657 | Transfer a reverse distrib... |
| caonncan 7658 | Transfer ~ nncan -shaped l... |
| relrpss 7661 | The proper subset relation... |
| brrpssg 7662 | The proper subset relation... |
| brrpss 7663 | The proper subset relation... |
| porpss 7664 | Every class is partially o... |
| sorpss 7665 | Express strict ordering un... |
| sorpssi 7666 | Property of a chain of set... |
| sorpssun 7667 | A chain of sets is closed ... |
| sorpssin 7668 | A chain of sets is closed ... |
| sorpssuni 7669 | In a chain of sets, a maxi... |
| sorpssint 7670 | In a chain of sets, a mini... |
| sorpsscmpl 7671 | The componentwise compleme... |
| zfun 7673 | Axiom of Union expressed w... |
| axun2 7674 | A variant of the Axiom of ... |
| uniex2 7675 | The Axiom of Union using t... |
| vuniex 7676 | The union of a setvar is a... |
| uniexg 7677 | The ZF Axiom of Union in c... |
| uniex 7678 | The Axiom of Union in clas... |
| uniexd 7679 | Deduction version of the Z... |
| unex 7680 | The union of two sets is a... |
| tpex 7681 | An unordered triple of cla... |
| unexb 7682 | Existence of union is equi... |
| unexg 7683 | A union of two sets is a s... |
| xpexg 7684 | The Cartesian product of t... |
| xpexd 7685 | The Cartesian product of t... |
| 3xpexg 7686 | The Cartesian product of t... |
| xpex 7687 | The Cartesian product of t... |
| unexd 7688 | The union of two sets is a... |
| sqxpexg 7689 | The Cartesian square of a ... |
| abnexg 7690 | Sufficient condition for a... |
| abnex 7691 | Sufficient condition for a... |
| snnex 7692 | The class of all singleton... |
| pwnex 7693 | The class of all power set... |
| difex2 7694 | If the subtrahend of a cla... |
| difsnexi 7695 | If the difference of a cla... |
| uniuni 7696 | Expression for double unio... |
| uniexr 7697 | Converse of the Axiom of U... |
| uniexb 7698 | The Axiom of Union and its... |
| pwexr 7699 | Converse of the Axiom of P... |
| pwexb 7700 | The Axiom of Power Sets an... |
| elpwpwel 7701 | A class belongs to a doubl... |
| eldifpw 7702 | Membership in a power clas... |
| elpwun 7703 | Membership in the power cl... |
| pwuncl 7704 | Power classes are closed u... |
| iunpw 7705 | An indexed union of a powe... |
| fr3nr 7706 | A well-founded relation ha... |
| epne3 7707 | A well-founded class conta... |
| dfwe2 7708 | Alternate definition of we... |
| epweon 7709 | The membership relation we... |
| epweonALT 7710 | Alternate proof of ~ epweo... |
| ordon 7711 | The class of all ordinal n... |
| onprc 7712 | No set contains all ordina... |
| ssorduni 7713 | The union of a class of or... |
| ssonuni 7714 | The union of a set of ordi... |
| ssonunii 7715 | The union of a set of ordi... |
| ordeleqon 7716 | A way to express the ordin... |
| ordsson 7717 | Any ordinal class is a sub... |
| dford5 7718 | A class is ordinal iff it ... |
| onss 7719 | An ordinal number is a sub... |
| predon 7720 | The predecessor of an ordi... |
| predonOLD 7721 | Obsolete version of ~ pred... |
| ssonprc 7722 | Two ways of saying a class... |
| onuni 7723 | The union of an ordinal nu... |
| orduni 7724 | The union of an ordinal cl... |
| onint 7725 | The intersection (infimum)... |
| onint0 7726 | The intersection of a clas... |
| onssmin 7727 | A nonempty class of ordina... |
| onminesb 7728 | If a property is true for ... |
| onminsb 7729 | If a property is true for ... |
| oninton 7730 | The intersection of a none... |
| onintrab 7731 | The intersection of a clas... |
| onintrab2 7732 | An existence condition equ... |
| onnmin 7733 | No member of a set of ordi... |
| onnminsb 7734 | An ordinal number smaller ... |
| oneqmin 7735 | A way to show that an ordi... |
| uniordint 7736 | The union of a set of ordi... |
| onminex 7737 | If a wff is true for an or... |
| sucon 7738 | The class of all ordinal n... |
| sucexb 7739 | A successor exists iff its... |
| sucexg 7740 | The successor of a set is ... |
| sucex 7741 | The successor of a set is ... |
| onmindif2 7742 | The minimum of a class of ... |
| ordsuci 7743 | The successor of an ordina... |
| sucexeloni 7744 | If the successor of an ord... |
| sucexeloniOLD 7745 | Obsolete version of ~ suce... |
| onsuc 7746 | The successor of an ordina... |
| suceloniOLD 7747 | Obsolete version of ~ onsu... |
| ordsuc 7748 | A class is ordinal if and ... |
| ordsucOLD 7749 | Obsolete version of ~ ords... |
| ordpwsuc 7750 | The collection of ordinals... |
| onpwsuc 7751 | The collection of ordinal ... |
| onsucb 7752 | A class is an ordinal numb... |
| ordsucss 7753 | The successor of an elemen... |
| onpsssuc 7754 | An ordinal number is a pro... |
| ordelsuc 7755 | A set belongs to an ordina... |
| onsucmin 7756 | The successor of an ordina... |
| ordsucelsuc 7757 | Membership is inherited by... |
| ordsucsssuc 7758 | The subclass relationship ... |
| ordsucuniel 7759 | Given an element ` A ` of ... |
| ordsucun 7760 | The successor of the maxim... |
| ordunpr 7761 | The maximum of two ordinal... |
| ordunel 7762 | The maximum of two ordinal... |
| onsucuni 7763 | A class of ordinal numbers... |
| ordsucuni 7764 | An ordinal class is a subc... |
| orduniorsuc 7765 | An ordinal class is either... |
| unon 7766 | The class of all ordinal n... |
| ordunisuc 7767 | An ordinal class is equal ... |
| orduniss2 7768 | The union of the ordinal s... |
| onsucuni2 7769 | A successor ordinal is the... |
| 0elsuc 7770 | The successor of an ordina... |
| limon 7771 | The class of ordinal numbe... |
| onuniorsuc 7772 | An ordinal number is eithe... |
| onssi 7773 | An ordinal number is a sub... |
| onsuci 7774 | The successor of an ordina... |
| onuniorsuciOLD 7775 | Obsolete version of ~ onun... |
| onuninsuci 7776 | An ordinal is equal to its... |
| onsucssi 7777 | A set belongs to an ordina... |
| nlimsucg 7778 | A successor is not a limit... |
| orduninsuc 7779 | An ordinal class is equal ... |
| ordunisuc2 7780 | An ordinal equal to its un... |
| ordzsl 7781 | An ordinal is zero, a succ... |
| onzsl 7782 | An ordinal number is zero,... |
| dflim3 7783 | An alternate definition of... |
| dflim4 7784 | An alternate definition of... |
| limsuc 7785 | The successor of a member ... |
| limsssuc 7786 | A class includes a limit o... |
| nlimon 7787 | Two ways to express the cl... |
| limuni3 7788 | The union of a nonempty cl... |
| tfi 7789 | The Principle of Transfini... |
| tfisg 7790 | A closed form of ~ tfis . ... |
| tfis 7791 | Transfinite Induction Sche... |
| tfis2f 7792 | Transfinite Induction Sche... |
| tfis2 7793 | Transfinite Induction Sche... |
| tfis3 7794 | Transfinite Induction Sche... |
| tfisi 7795 | A transfinite induction sc... |
| tfinds 7796 | Principle of Transfinite I... |
| tfindsg 7797 | Transfinite Induction (inf... |
| tfindsg2 7798 | Transfinite Induction (inf... |
| tfindes 7799 | Transfinite Induction with... |
| tfinds2 7800 | Transfinite Induction (inf... |
| tfinds3 7801 | Principle of Transfinite I... |
| dfom2 7804 | An alternate definition of... |
| elom 7805 | Membership in omega. The ... |
| omsson 7806 | Omega is a subset of ` On ... |
| limomss 7807 | The class of natural numbe... |
| nnon 7808 | A natural number is an ord... |
| nnoni 7809 | A natural number is an ord... |
| nnord 7810 | A natural number is ordina... |
| trom 7811 | The class of finite ordina... |
| ordom 7812 | The class of finite ordina... |
| elnn 7813 | A member of a natural numb... |
| omon 7814 | The class of natural numbe... |
| omelon2 7815 | Omega is an ordinal number... |
| nnlim 7816 | A natural number is not a ... |
| omssnlim 7817 | The class of natural numbe... |
| limom 7818 | Omega is a limit ordinal. ... |
| peano2b 7819 | A class belongs to omega i... |
| nnsuc 7820 | A nonzero natural number i... |
| omsucne 7821 | A natural number is not th... |
| ssnlim 7822 | An ordinal subclass of non... |
| omsinds 7823 | Strong (or "total") induct... |
| omsindsOLD 7824 | Obsolete version of ~ omsi... |
| peano1 7825 | Zero is a natural number. ... |
| peano1OLD 7826 | Obsolete version of ~ pean... |
| peano2 7827 | The successor of any natur... |
| peano3 7828 | The successor of any natur... |
| peano4 7829 | Two natural numbers are eq... |
| peano5 7830 | The induction postulate: a... |
| peano5OLD 7831 | Obsolete version of ~ pean... |
| nn0suc 7832 | A natural number is either... |
| find 7833 | The Principle of Finite In... |
| findOLD 7834 | Obsolete version of ~ find... |
| finds 7835 | Principle of Finite Induct... |
| findsg 7836 | Principle of Finite Induct... |
| finds2 7837 | Principle of Finite Induct... |
| finds1 7838 | Principle of Finite Induct... |
| findes 7839 | Finite induction with expl... |
| dmexg 7840 | The domain of a set is a s... |
| rnexg 7841 | The range of a set is a se... |
| dmexd 7842 | The domain of a set is a s... |
| fndmexd 7843 | If a function is a set, it... |
| dmfex 7844 | If a mapping is a set, its... |
| fndmexb 7845 | The domain of a function i... |
| fdmexb 7846 | The domain of a function i... |
| dmfexALT 7847 | Alternate proof of ~ dmfex... |
| dmex 7848 | The domain of a set is a s... |
| rnex 7849 | The range of a set is a se... |
| iprc 7850 | The identity function is a... |
| resiexg 7851 | The existence of a restric... |
| imaexg 7852 | The image of a set is a se... |
| imaex 7853 | The image of a set is a se... |
| exse2 7854 | Any set relation is set-li... |
| xpexr 7855 | If a Cartesian product is ... |
| xpexr2 7856 | If a nonempty Cartesian pr... |
| xpexcnv 7857 | A condition where the conv... |
| soex 7858 | If the relation in a stric... |
| elxp4 7859 | Membership in a Cartesian ... |
| elxp5 7860 | Membership in a Cartesian ... |
| cnvexg 7861 | The converse of a set is a... |
| cnvex 7862 | The converse of a set is a... |
| relcnvexb 7863 | A relation is a set iff it... |
| f1oexrnex 7864 | If the range of a 1-1 onto... |
| f1oexbi 7865 | There is a one-to-one onto... |
| coexg 7866 | The composition of two set... |
| coex 7867 | The composition of two set... |
| funcnvuni 7868 | The union of a chain (with... |
| fun11uni 7869 | The union of a chain (with... |
| fex2 7870 | A function with bounded do... |
| fabexg 7871 | Existence of a set of func... |
| fabex 7872 | Existence of a set of func... |
| f1oabexg 7873 | The class of all 1-1-onto ... |
| fiunlem 7874 | Lemma for ~ fiun and ~ f1i... |
| fiun 7875 | The union of a chain (with... |
| f1iun 7876 | The union of a chain (with... |
| fviunfun 7877 | The function value of an i... |
| ffoss 7878 | Relationship between a map... |
| f11o 7879 | Relationship between one-t... |
| resfunexgALT 7880 | Alternate proof of ~ resfu... |
| cofunexg 7881 | Existence of a composition... |
| cofunex2g 7882 | Existence of a composition... |
| fnexALT 7883 | Alternate proof of ~ fnex ... |
| funexw 7884 | Weak version of ~ funex th... |
| mptexw 7885 | Weak version of ~ mptex th... |
| funrnex 7886 | If the domain of a functio... |
| zfrep6 7887 | A version of the Axiom of ... |
| focdmex 7888 | If the domain of an onto f... |
| f1dmex 7889 | If the codomain of a one-t... |
| f1ovv 7890 | The codomain/range of a 1-... |
| fvclex 7891 | Existence of the class of ... |
| fvresex 7892 | Existence of the class of ... |
| abrexexg 7893 | Existence of a class abstr... |
| abrexexgOLD 7894 | Obsolete version of ~ abre... |
| abrexex 7895 | Existence of a class abstr... |
| iunexg 7896 | The existence of an indexe... |
| abrexex2g 7897 | Existence of an existentia... |
| opabex3d 7898 | Existence of an ordered pa... |
| opabex3rd 7899 | Existence of an ordered pa... |
| opabex3 7900 | Existence of an ordered pa... |
| iunex 7901 | The existence of an indexe... |
| abrexex2 7902 | Existence of an existentia... |
| abexssex 7903 | Existence of a class abstr... |
| abexex 7904 | A condition where a class ... |
| f1oweALT 7905 | Alternate proof of ~ f1owe... |
| wemoiso 7906 | Thus, there is at most one... |
| wemoiso2 7907 | Thus, there is at most one... |
| oprabexd 7908 | Existence of an operator a... |
| oprabex 7909 | Existence of an operation ... |
| oprabex3 7910 | Existence of an operation ... |
| oprabrexex2 7911 | Existence of an existentia... |
| ab2rexex 7912 | Existence of a class abstr... |
| ab2rexex2 7913 | Existence of an existentia... |
| xpexgALT 7914 | Alternate proof of ~ xpexg... |
| offval3 7915 | General value of ` ( F oF ... |
| offres 7916 | Pointwise combination comm... |
| ofmres 7917 | Equivalent expressions for... |
| ofmresex 7918 | Existence of a restriction... |
| 1stval 7923 | The value of the function ... |
| 2ndval 7924 | The value of the function ... |
| 1stnpr 7925 | Value of the first-member ... |
| 2ndnpr 7926 | Value of the second-member... |
| 1st0 7927 | The value of the first-mem... |
| 2nd0 7928 | The value of the second-me... |
| op1st 7929 | Extract the first member o... |
| op2nd 7930 | Extract the second member ... |
| op1std 7931 | Extract the first member o... |
| op2ndd 7932 | Extract the second member ... |
| op1stg 7933 | Extract the first member o... |
| op2ndg 7934 | Extract the second member ... |
| ot1stg 7935 | Extract the first member o... |
| ot2ndg 7936 | Extract the second member ... |
| ot3rdg 7937 | Extract the third member o... |
| 1stval2 7938 | Alternate value of the fun... |
| 2ndval2 7939 | Alternate value of the fun... |
| oteqimp 7940 | The components of an order... |
| fo1st 7941 | The ` 1st ` function maps ... |
| fo2nd 7942 | The ` 2nd ` function maps ... |
| br1steqg 7943 | Uniqueness condition for t... |
| br2ndeqg 7944 | Uniqueness condition for t... |
| f1stres 7945 | Mapping of a restriction o... |
| f2ndres 7946 | Mapping of a restriction o... |
| fo1stres 7947 | Onto mapping of a restrict... |
| fo2ndres 7948 | Onto mapping of a restrict... |
| 1st2val 7949 | Value of an alternate defi... |
| 2nd2val 7950 | Value of an alternate defi... |
| 1stcof 7951 | Composition of the first m... |
| 2ndcof 7952 | Composition of the second ... |
| xp1st 7953 | Location of the first elem... |
| xp2nd 7954 | Location of the second ele... |
| elxp6 7955 | Membership in a Cartesian ... |
| elxp7 7956 | Membership in a Cartesian ... |
| eqopi 7957 | Equality with an ordered p... |
| xp2 7958 | Representation of Cartesia... |
| unielxp 7959 | The membership relation fo... |
| 1st2nd2 7960 | Reconstruction of a member... |
| 1st2ndb 7961 | Reconstruction of an order... |
| xpopth 7962 | An ordered pair theorem fo... |
| eqop 7963 | Two ways to express equali... |
| eqop2 7964 | Two ways to express equali... |
| op1steq 7965 | Two ways of expressing tha... |
| opreuopreu 7966 | There is a unique ordered ... |
| el2xptp 7967 | A member of a nested Carte... |
| el2xptp0 7968 | A member of a nested Carte... |
| el2xpss 7969 | Version of ~ elrel for tri... |
| 2nd1st 7970 | Swap the members of an ord... |
| 1st2nd 7971 | Reconstruction of a member... |
| 1stdm 7972 | The first ordered pair com... |
| 2ndrn 7973 | The second ordered pair co... |
| 1st2ndbr 7974 | Express an element of a re... |
| releldm2 7975 | Two ways of expressing mem... |
| reldm 7976 | An expression for the doma... |
| releldmdifi 7977 | One way of expressing memb... |
| funfv1st2nd 7978 | The function value for the... |
| funelss 7979 | If the first component of ... |
| funeldmdif 7980 | Two ways of expressing mem... |
| sbcopeq1a 7981 | Equality theorem for subst... |
| csbopeq1a 7982 | Equality theorem for subst... |
| sbcoteq1a 7983 | Equality theorem for subst... |
| dfopab2 7984 | A way to define an ordered... |
| dfoprab3s 7985 | A way to define an operati... |
| dfoprab3 7986 | Operation class abstractio... |
| dfoprab4 7987 | Operation class abstractio... |
| dfoprab4f 7988 | Operation class abstractio... |
| opabex2 7989 | Condition for an operation... |
| opabn1stprc 7990 | An ordered-pair class abst... |
| opiota 7991 | The property of a uniquely... |
| cnvoprab 7992 | The converse of a class ab... |
| dfxp3 7993 | Define the Cartesian produ... |
| elopabi 7994 | A consequence of membershi... |
| eloprabi 7995 | A consequence of membershi... |
| mpomptsx 7996 | Express a two-argument fun... |
| mpompts 7997 | Express a two-argument fun... |
| dmmpossx 7998 | The domain of a mapping is... |
| fmpox 7999 | Functionality, domain and ... |
| fmpo 8000 | Functionality, domain and ... |
| fnmpo 8001 | Functionality and domain o... |
| fnmpoi 8002 | Functionality and domain o... |
| dmmpo 8003 | Domain of a class given by... |
| ovmpoelrn 8004 | An operation's value belon... |
| dmmpoga 8005 | Domain of an operation giv... |
| dmmpogaOLD 8006 | Obsolete version of ~ dmmp... |
| dmmpog 8007 | Domain of an operation giv... |
| mpoexxg 8008 | Existence of an operation ... |
| mpoexg 8009 | Existence of an operation ... |
| mpoexga 8010 | If the domain of an operat... |
| mpoexw 8011 | Weak version of ~ mpoex th... |
| mpoex 8012 | If the domain of an operat... |
| mptmpoopabbrd 8013 | The operation value of a f... |
| mptmpoopabovd 8014 | The operation value of a f... |
| mptmpoopabbrdOLD 8015 | Obsolete version of ~ mptm... |
| mptmpoopabovdOLD 8016 | Obsolete version of ~ mptm... |
| el2mpocsbcl 8017 | If the operation value of ... |
| el2mpocl 8018 | If the operation value of ... |
| fnmpoovd 8019 | A function with a Cartesia... |
| offval22 8020 | The function operation exp... |
| brovpreldm 8021 | If a binary relation holds... |
| bropopvvv 8022 | If a binary relation holds... |
| bropfvvvvlem 8023 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 8024 | If a binary relation holds... |
| ovmptss 8025 | If all the values of the m... |
| relmpoopab 8026 | Any function to sets of or... |
| fmpoco 8027 | Composition of two functio... |
| oprabco 8028 | Composition of a function ... |
| oprab2co 8029 | Composition of operator ab... |
| df1st2 8030 | An alternate possible defi... |
| df2nd2 8031 | An alternate possible defi... |
| 1stconst 8032 | The mapping of a restricti... |
| 2ndconst 8033 | The mapping of a restricti... |
| dfmpo 8034 | Alternate definition for t... |
| mposn 8035 | An operation (in maps-to n... |
| curry1 8036 | Composition with ` ``' ( 2... |
| curry1val 8037 | The value of a curried fun... |
| curry1f 8038 | Functionality of a curried... |
| curry2 8039 | Composition with ` ``' ( 1... |
| curry2f 8040 | Functionality of a curried... |
| curry2val 8041 | The value of a curried fun... |
| cnvf1olem 8042 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 8043 | Describe a function that m... |
| fparlem1 8044 | Lemma for ~ fpar . (Contr... |
| fparlem2 8045 | Lemma for ~ fpar . (Contr... |
| fparlem3 8046 | Lemma for ~ fpar . (Contr... |
| fparlem4 8047 | Lemma for ~ fpar . (Contr... |
| fpar 8048 | Merge two functions in par... |
| fsplit 8049 | A function that can be use... |
| fsplitfpar 8050 | Merge two functions with a... |
| offsplitfpar 8051 | Express the function opera... |
| f2ndf 8052 | The ` 2nd ` (second compon... |
| fo2ndf 8053 | The ` 2nd ` (second compon... |
| f1o2ndf1 8054 | The ` 2nd ` (second compon... |
| opco1 8055 | Value of an operation prec... |
| opco2 8056 | Value of an operation prec... |
| opco1i 8057 | Inference form of ~ opco1 ... |
| frxp 8058 | A lexicographical ordering... |
| xporderlem 8059 | Lemma for lexicographical ... |
| poxp 8060 | A lexicographical ordering... |
| soxp 8061 | A lexicographical ordering... |
| wexp 8062 | A lexicographical ordering... |
| fnwelem 8063 | Lemma for ~ fnwe . (Contr... |
| fnwe 8064 | A variant on lexicographic... |
| fnse 8065 | Condition for the well-ord... |
| fvproj 8066 | Value of a function on ord... |
| fimaproj 8067 | Image of a cartesian produ... |
| ralxpes 8068 | A version of ~ ralxp with ... |
| ralxp3f 8069 | Restricted for all over a ... |
| ralxp3 8070 | Restricted for all over a ... |
| ralxp3es 8071 | Restricted for-all over a ... |
| frpoins3xpg 8072 | Special case of founded pa... |
| frpoins3xp3g 8073 | Special case of founded pa... |
| xpord2lem 8074 | Lemma for Cartesian produc... |
| poxp2 8075 | Another way of partially o... |
| frxp2 8076 | Another way of giving a we... |
| xpord2pred 8077 | Calculate the predecessor ... |
| sexp2 8078 | Condition for the relation... |
| xpord2indlem 8079 | Induction over the Cartesi... |
| xpord2ind 8080 | Induction over the Cartesi... |
| xpord3lem 8081 | Lemma for triple ordering.... |
| poxp3 8082 | Triple Cartesian product p... |
| frxp3 8083 | Give well-foundedness over... |
| xpord3pred 8084 | Calculate the predecsessor... |
| sexp3 8085 | Show that the triple order... |
| xpord3inddlem 8086 | Induction over the triple ... |
| xpord3indd 8087 | Induction over the triple ... |
| xpord3ind 8088 | Induction over the triple ... |
| orderseqlem 8089 | Lemma for ~ poseq and ~ so... |
| poseq 8090 | A partial ordering of ordi... |
| soseq 8091 | A linear ordering of ordin... |
| suppval 8094 | The value of the operation... |
| supp0prc 8095 | The support of a class is ... |
| suppvalbr 8096 | The value of the operation... |
| supp0 8097 | The support of the empty s... |
| suppval1 8098 | The value of the operation... |
| suppvalfng 8099 | The value of the operation... |
| suppvalfn 8100 | The value of the operation... |
| elsuppfng 8101 | An element of the support ... |
| elsuppfn 8102 | An element of the support ... |
| cnvimadfsn 8103 | The support of functions "... |
| suppimacnvss 8104 | The support of functions "... |
| suppimacnv 8105 | Support sets of functions ... |
| fsuppeq 8106 | Two ways of writing the su... |
| fsuppeqg 8107 | Version of ~ fsuppeq avoid... |
| suppssdm 8108 | The support of a function ... |
| suppsnop 8109 | The support of a singleton... |
| snopsuppss 8110 | The support of a singleton... |
| fvn0elsupp 8111 | If the function value for ... |
| fvn0elsuppb 8112 | The function value for a g... |
| rexsupp 8113 | Existential quantification... |
| ressuppss 8114 | The support of the restric... |
| suppun 8115 | The support of a class/fun... |
| ressuppssdif 8116 | The support of the restric... |
| mptsuppdifd 8117 | The support of a function ... |
| mptsuppd 8118 | The support of a function ... |
| extmptsuppeq 8119 | The support of an extended... |
| suppfnss 8120 | The support of a function ... |
| funsssuppss 8121 | The support of a function ... |
| fnsuppres 8122 | Two ways to express restri... |
| fnsuppeq0 8123 | The support of a function ... |
| fczsupp0 8124 | The support of a constant ... |
| suppss 8125 | Show that the support of a... |
| suppssOLD 8126 | Obsolete version of ~ supp... |
| suppssr 8127 | A function is zero outside... |
| suppssrg 8128 | A function is zero outside... |
| suppssov1 8129 | Formula building theorem f... |
| suppssof1 8130 | Formula building theorem f... |
| suppss2 8131 | Show that the support of a... |
| suppsssn 8132 | Show that the support of a... |
| suppssfv 8133 | Formula building theorem f... |
| suppofssd 8134 | Condition for the support ... |
| suppofss1d 8135 | Condition for the support ... |
| suppofss2d 8136 | Condition for the support ... |
| suppco 8137 | The support of the composi... |
| suppcoss 8138 | The support of the composi... |
| supp0cosupp0 8139 | The support of the composi... |
| imacosupp 8140 | The image of the support o... |
| opeliunxp2f 8141 | Membership in a union of C... |
| mpoxeldm 8142 | If there is an element of ... |
| mpoxneldm 8143 | If the first argument of a... |
| mpoxopn0yelv 8144 | If there is an element of ... |
| mpoxopynvov0g 8145 | If the second argument of ... |
| mpoxopxnop0 8146 | If the first argument of a... |
| mpoxopx0ov0 8147 | If the first argument of a... |
| mpoxopxprcov0 8148 | If the components of the f... |
| mpoxopynvov0 8149 | If the second argument of ... |
| mpoxopoveq 8150 | Value of an operation give... |
| mpoxopovel 8151 | Element of the value of an... |
| mpoxopoveqd 8152 | Value of an operation give... |
| brovex 8153 | A binary relation of the v... |
| brovmpoex 8154 | A binary relation of the v... |
| sprmpod 8155 | The extension of a binary ... |
| tposss 8158 | Subset theorem for transpo... |
| tposeq 8159 | Equality theorem for trans... |
| tposeqd 8160 | Equality theorem for trans... |
| tposssxp 8161 | The transposition is a sub... |
| reltpos 8162 | The transposition is a rel... |
| brtpos2 8163 | Value of the transposition... |
| brtpos0 8164 | The behavior of ` tpos ` w... |
| reldmtpos 8165 | Necessary and sufficient c... |
| brtpos 8166 | The transposition swaps ar... |
| ottpos 8167 | The transposition swaps th... |
| relbrtpos 8168 | The transposition swaps ar... |
| dmtpos 8169 | The domain of ` tpos F ` w... |
| rntpos 8170 | The range of ` tpos F ` wh... |
| tposexg 8171 | The transposition of a set... |
| ovtpos 8172 | The transposition swaps th... |
| tposfun 8173 | The transposition of a fun... |
| dftpos2 8174 | Alternate definition of ` ... |
| dftpos3 8175 | Alternate definition of ` ... |
| dftpos4 8176 | Alternate definition of ` ... |
| tpostpos 8177 | Value of the double transp... |
| tpostpos2 8178 | Value of the double transp... |
| tposfn2 8179 | The domain of a transposit... |
| tposfo2 8180 | Condition for a surjective... |
| tposf2 8181 | The domain and codomain of... |
| tposf12 8182 | Condition for an injective... |
| tposf1o2 8183 | Condition of a bijective t... |
| tposfo 8184 | The domain and codomain/ra... |
| tposf 8185 | The domain and codomain of... |
| tposfn 8186 | Functionality of a transpo... |
| tpos0 8187 | Transposition of the empty... |
| tposco 8188 | Transposition of a composi... |
| tpossym 8189 | Two ways to say a function... |
| tposeqi 8190 | Equality theorem for trans... |
| tposex 8191 | A transposition is a set. ... |
| nftpos 8192 | Hypothesis builder for tra... |
| tposoprab 8193 | Transposition of a class o... |
| tposmpo 8194 | Transposition of a two-arg... |
| tposconst 8195 | The transposition of a con... |
| mpocurryd 8200 | The currying of an operati... |
| mpocurryvald 8201 | The value of a curried ope... |
| fvmpocurryd 8202 | The value of the value of ... |
| pwuninel2 8205 | Direct proof of ~ pwuninel... |
| pwuninel 8206 | The power set of the union... |
| undefval 8207 | Value of the undefined val... |
| undefnel2 8208 | The undefined value genera... |
| undefnel 8209 | The undefined value genera... |
| undefne0 8210 | The undefined value genera... |
| frecseq123 8213 | Equality theorem for the w... |
| nffrecs 8214 | Bound-variable hypothesis ... |
| csbfrecsg 8215 | Move class substitution in... |
| fpr3g 8216 | Functions defined by well-... |
| frrlem1 8217 | Lemma for well-founded rec... |
| frrlem2 8218 | Lemma for well-founded rec... |
| frrlem3 8219 | Lemma for well-founded rec... |
| frrlem4 8220 | Lemma for well-founded rec... |
| frrlem5 8221 | Lemma for well-founded rec... |
| frrlem6 8222 | Lemma for well-founded rec... |
| frrlem7 8223 | Lemma for well-founded rec... |
| frrlem8 8224 | Lemma for well-founded rec... |
| frrlem9 8225 | Lemma for well-founded rec... |
| frrlem10 8226 | Lemma for well-founded rec... |
| frrlem11 8227 | Lemma for well-founded rec... |
| frrlem12 8228 | Lemma for well-founded rec... |
| frrlem13 8229 | Lemma for well-founded rec... |
| frrlem14 8230 | Lemma for well-founded rec... |
| fprlem1 8231 | Lemma for well-founded rec... |
| fprlem2 8232 | Lemma for well-founded rec... |
| fpr2a 8233 | Weak version of ~ fpr2 whi... |
| fpr1 8234 | Law of well-founded recurs... |
| fpr2 8235 | Law of well-founded recurs... |
| fpr3 8236 | Law of well-founded recurs... |
| frrrel 8237 | Show without using the axi... |
| frrdmss 8238 | Show without using the axi... |
| frrdmcl 8239 | Show without using the axi... |
| fprfung 8240 | A "function" defined by we... |
| fprresex 8241 | The restriction of a funct... |
| dfwrecsOLD 8244 | Obsolete definition of the... |
| wrecseq123 8245 | General equality theorem f... |
| wrecseq123OLD 8246 | Obsolete proof of ~ wrecse... |
| nfwrecs 8247 | Bound-variable hypothesis ... |
| nfwrecsOLD 8248 | Obsolete proof of ~ nfwrec... |
| wrecseq1 8249 | Equality theorem for the w... |
| wrecseq2 8250 | Equality theorem for the w... |
| wrecseq3 8251 | Equality theorem for the w... |
| csbwrecsg 8252 | Move class substitution in... |
| wfr3g 8253 | Functions defined by well-... |
| wfrlem1OLD 8254 | Lemma for well-ordered rec... |
| wfrlem2OLD 8255 | Lemma for well-ordered rec... |
| wfrlem3OLD 8256 | Lemma for well-ordered rec... |
| wfrlem3OLDa 8257 | Lemma for well-ordered rec... |
| wfrlem4OLD 8258 | Lemma for well-ordered rec... |
| wfrlem5OLD 8259 | Lemma for well-ordered rec... |
| wfrrelOLD 8260 | Obsolete proof of ~ wfrrel... |
| wfrdmssOLD 8261 | Obsolete proof of ~ wfrdms... |
| wfrlem8OLD 8262 | Lemma for well-ordered rec... |
| wfrdmclOLD 8263 | Obsolete proof of ~ wfrdmc... |
| wfrlem10OLD 8264 | Lemma for well-ordered rec... |
| wfrfunOLD 8265 | Obsolete proof of ~ wfrfun... |
| wfrlem12OLD 8266 | Lemma for well-ordered rec... |
| wfrlem13OLD 8267 | Lemma for well-ordered rec... |
| wfrlem14OLD 8268 | Lemma for well-ordered rec... |
| wfrlem15OLD 8269 | Lemma for well-ordered rec... |
| wfrlem16OLD 8270 | Lemma for well-ordered rec... |
| wfrlem17OLD 8271 | Without using ~ ax-rep , s... |
| wfr2aOLD 8272 | Obsolete proof of ~ wfr2a ... |
| wfr1OLD 8273 | Obsolete proof of ~ wfr1 a... |
| wfr2OLD 8274 | Obsolete proof of ~ wfr2 a... |
| wfrrel 8275 | The well-ordered recursion... |
| wfrdmss 8276 | The domain of the well-ord... |
| wfrdmcl 8277 | The predecessor class of a... |
| wfrfun 8278 | The "function" generated b... |
| wfrresex 8279 | Show without using the axi... |
| wfr2a 8280 | A weak version of ~ wfr2 w... |
| wfr1 8281 | The Principle of Well-Orde... |
| wfr2 8282 | The Principle of Well-Orde... |
| wfr3 8283 | The principle of Well-Orde... |
| wfr3OLD 8284 | Obsolete form of ~ wfr3 as... |
| iunon 8285 | The indexed union of a set... |
| iinon 8286 | The nonempty indexed inter... |
| onfununi 8287 | A property of functions on... |
| onovuni 8288 | A variant of ~ onfununi fo... |
| onoviun 8289 | A variant of ~ onovuni wit... |
| onnseq 8290 | There are no length ` _om ... |
| dfsmo2 8293 | Alternate definition of a ... |
| issmo 8294 | Conditions for which ` A `... |
| issmo2 8295 | Alternate definition of a ... |
| smoeq 8296 | Equality theorem for stric... |
| smodm 8297 | The domain of a strictly m... |
| smores 8298 | A strictly monotone functi... |
| smores3 8299 | A strictly monotone functi... |
| smores2 8300 | A strictly monotone ordina... |
| smodm2 8301 | The domain of a strictly m... |
| smofvon2 8302 | The function values of a s... |
| iordsmo 8303 | The identity relation rest... |
| smo0 8304 | The null set is a strictly... |
| smofvon 8305 | If ` B ` is a strictly mon... |
| smoel 8306 | If ` x ` is less than ` y ... |
| smoiun 8307 | The value of a strictly mo... |
| smoiso 8308 | If ` F ` is an isomorphism... |
| smoel2 8309 | A strictly monotone ordina... |
| smo11 8310 | A strictly monotone ordina... |
| smoord 8311 | A strictly monotone ordina... |
| smoword 8312 | A strictly monotone ordina... |
| smogt 8313 | A strictly monotone ordina... |
| smocdmdom 8314 | The codomain of a strictly... |
| smoiso2 8315 | The strictly monotone ordi... |
| dfrecs3 8318 | The old definition of tran... |
| dfrecs3OLD 8319 | Obsolete proof of ~ dfrecs... |
| recseq 8320 | Equality theorem for ` rec... |
| nfrecs 8321 | Bound-variable hypothesis ... |
| tfrlem1 8322 | A technical lemma for tran... |
| tfrlem3a 8323 | Lemma for transfinite recu... |
| tfrlem3 8324 | Lemma for transfinite recu... |
| tfrlem4 8325 | Lemma for transfinite recu... |
| tfrlem5 8326 | Lemma for transfinite recu... |
| recsfval 8327 | Lemma for transfinite recu... |
| tfrlem6 8328 | Lemma for transfinite recu... |
| tfrlem7 8329 | Lemma for transfinite recu... |
| tfrlem8 8330 | Lemma for transfinite recu... |
| tfrlem9 8331 | Lemma for transfinite recu... |
| tfrlem9a 8332 | Lemma for transfinite recu... |
| tfrlem10 8333 | Lemma for transfinite recu... |
| tfrlem11 8334 | Lemma for transfinite recu... |
| tfrlem12 8335 | Lemma for transfinite recu... |
| tfrlem13 8336 | Lemma for transfinite recu... |
| tfrlem14 8337 | Lemma for transfinite recu... |
| tfrlem15 8338 | Lemma for transfinite recu... |
| tfrlem16 8339 | Lemma for finite recursion... |
| tfr1a 8340 | A weak version of ~ tfr1 w... |
| tfr2a 8341 | A weak version of ~ tfr2 w... |
| tfr2b 8342 | Without assuming ~ ax-rep ... |
| tfr1 8343 | Principle of Transfinite R... |
| tfr2 8344 | Principle of Transfinite R... |
| tfr3 8345 | Principle of Transfinite R... |
| tfr1ALT 8346 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8347 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8348 | Alternate proof of ~ tfr3 ... |
| recsfnon 8349 | Strong transfinite recursi... |
| recsval 8350 | Strong transfinite recursi... |
| tz7.44lem1 8351 | The ordered pair abstracti... |
| tz7.44-1 8352 | The value of ` F ` at ` (/... |
| tz7.44-2 8353 | The value of ` F ` at a su... |
| tz7.44-3 8354 | The value of ` F ` at a li... |
| rdgeq1 8357 | Equality theorem for the r... |
| rdgeq2 8358 | Equality theorem for the r... |
| rdgeq12 8359 | Equality theorem for the r... |
| nfrdg 8360 | Bound-variable hypothesis ... |
| rdglem1 8361 | Lemma used with the recurs... |
| rdgfun 8362 | The recursive definition g... |
| rdgdmlim 8363 | The domain of the recursiv... |
| rdgfnon 8364 | The recursive definition g... |
| rdgvalg 8365 | Value of the recursive def... |
| rdgval 8366 | Value of the recursive def... |
| rdg0 8367 | The initial value of the r... |
| rdgseg 8368 | The initial segments of th... |
| rdgsucg 8369 | The value of the recursive... |
| rdgsuc 8370 | The value of the recursive... |
| rdglimg 8371 | The value of the recursive... |
| rdglim 8372 | The value of the recursive... |
| rdg0g 8373 | The initial value of the r... |
| rdgsucmptf 8374 | The value of the recursive... |
| rdgsucmptnf 8375 | The value of the recursive... |
| rdgsucmpt2 8376 | This version of ~ rdgsucmp... |
| rdgsucmpt 8377 | The value of the recursive... |
| rdglim2 8378 | The value of the recursive... |
| rdglim2a 8379 | The value of the recursive... |
| rdg0n 8380 | If ` A ` is a proper class... |
| frfnom 8381 | The function generated by ... |
| fr0g 8382 | The initial value resultin... |
| frsuc 8383 | The successor value result... |
| frsucmpt 8384 | The successor value result... |
| frsucmptn 8385 | The value of the finite re... |
| frsucmpt2 8386 | The successor value result... |
| tz7.48lem 8387 | A way of showing an ordina... |
| tz7.48-2 8388 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8389 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8390 | Proposition 7.48(3) of [Ta... |
| tz7.49 8391 | Proposition 7.49 of [Takeu... |
| tz7.49c 8392 | Corollary of Proposition 7... |
| seqomlem0 8395 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8396 | Lemma for ` seqom ` . The... |
| seqomlem2 8397 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8398 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8399 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8400 | Equality theorem for ` seq... |
| fnseqom 8401 | An index-aware recursive d... |
| seqom0g 8402 | Value of an index-aware re... |
| seqomsuc 8403 | Value of an index-aware re... |
| omsucelsucb 8404 | Membership is inherited by... |
| df1o2 8419 | Expanded value of the ordi... |
| df2o3 8420 | Expanded value of the ordi... |
| df2o2 8421 | Expanded value of the ordi... |
| 1oex 8422 | Ordinal 1 is a set. (Cont... |
| 2oex 8423 | ` 2o ` is a set. (Contrib... |
| 1on 8424 | Ordinal 1 is an ordinal nu... |
| 1onOLD 8425 | Obsolete version of ~ 1on ... |
| 2on 8426 | Ordinal 2 is an ordinal nu... |
| 2onOLD 8427 | Obsolete version of ~ 2on ... |
| 2on0 8428 | Ordinal two is not zero. ... |
| ord3 8429 | Ordinal 3 is an ordinal cl... |
| 3on 8430 | Ordinal 3 is an ordinal nu... |
| 4on 8431 | Ordinal 3 is an ordinal nu... |
| 1oexOLD 8432 | Obsolete version of ~ 1oex... |
| 2oexOLD 8433 | Obsolete version of ~ 2oex... |
| 1n0 8434 | Ordinal one is not equal t... |
| nlim1 8435 | 1 is not a limit ordinal. ... |
| nlim2 8436 | 2 is not a limit ordinal. ... |
| xp01disj 8437 | Cartesian products with th... |
| xp01disjl 8438 | Cartesian products with th... |
| ordgt0ge1 8439 | Two ways to express that a... |
| ordge1n0 8440 | An ordinal greater than or... |
| el1o 8441 | Membership in ordinal one.... |
| ord1eln01 8442 | An ordinal that is not 0 o... |
| ord2eln012 8443 | An ordinal that is not 0, ... |
| 1ellim 8444 | A limit ordinal contains 1... |
| 2ellim 8445 | A limit ordinal contains 2... |
| dif1o 8446 | Two ways to say that ` A `... |
| ondif1 8447 | Two ways to say that ` A `... |
| ondif2 8448 | Two ways to say that ` A `... |
| 2oconcl 8449 | Closure of the pair swappi... |
| 0lt1o 8450 | Ordinal zero is less than ... |
| dif20el 8451 | An ordinal greater than on... |
| 0we1 8452 | The empty set is a well-or... |
| brwitnlem 8453 | Lemma for relations which ... |
| fnoa 8454 | Functionality and domain o... |
| fnom 8455 | Functionality and domain o... |
| fnoe 8456 | Functionality and domain o... |
| oav 8457 | Value of ordinal addition.... |
| omv 8458 | Value of ordinal multiplic... |
| oe0lem 8459 | A helper lemma for ~ oe0 a... |
| oev 8460 | Value of ordinal exponenti... |
| oevn0 8461 | Value of ordinal exponenti... |
| oa0 8462 | Addition with zero. Propo... |
| om0 8463 | Ordinal multiplication wit... |
| oe0m 8464 | Value of zero raised to an... |
| om0x 8465 | Ordinal multiplication wit... |
| oe0m0 8466 | Ordinal exponentiation wit... |
| oe0m1 8467 | Ordinal exponentiation wit... |
| oe0 8468 | Ordinal exponentiation wit... |
| oev2 8469 | Alternate value of ordinal... |
| oasuc 8470 | Addition with successor. ... |
| oesuclem 8471 | Lemma for ~ oesuc . (Cont... |
| omsuc 8472 | Multiplication with succes... |
| oesuc 8473 | Ordinal exponentiation wit... |
| onasuc 8474 | Addition with successor. ... |
| onmsuc 8475 | Multiplication with succes... |
| onesuc 8476 | Exponentiation with a succ... |
| oa1suc 8477 | Addition with 1 is same as... |
| oalim 8478 | Ordinal addition with a li... |
| omlim 8479 | Ordinal multiplication wit... |
| oelim 8480 | Ordinal exponentiation wit... |
| oacl 8481 | Closure law for ordinal ad... |
| omcl 8482 | Closure law for ordinal mu... |
| oecl 8483 | Closure law for ordinal ex... |
| oa0r 8484 | Ordinal addition with zero... |
| om0r 8485 | Ordinal multiplication wit... |
| o1p1e2 8486 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8487 | 2 + 2 = 4 for ordinal numb... |
| o2p2e4OLD 8488 | Obsolete version of ~ o2p2... |
| om1 8489 | Ordinal multiplication wit... |
| om1r 8490 | Ordinal multiplication wit... |
| oe1 8491 | Ordinal exponentiation wit... |
| oe1m 8492 | Ordinal exponentiation wit... |
| oaordi 8493 | Ordering property of ordin... |
| oaord 8494 | Ordering property of ordin... |
| oacan 8495 | Left cancellation law for ... |
| oaword 8496 | Weak ordering property of ... |
| oawordri 8497 | Weak ordering property of ... |
| oaord1 8498 | An ordinal is less than it... |
| oaword1 8499 | An ordinal is less than or... |
| oaword2 8500 | An ordinal is less than or... |
| oawordeulem 8501 | Lemma for ~ oawordex . (C... |
| oawordeu 8502 | Existence theorem for weak... |
| oawordexr 8503 | Existence theorem for weak... |
| oawordex 8504 | Existence theorem for weak... |
| oaordex 8505 | Existence theorem for orde... |
| oa00 8506 | An ordinal sum is zero iff... |
| oalimcl 8507 | The ordinal sum with a lim... |
| oaass 8508 | Ordinal addition is associ... |
| oarec 8509 | Recursive definition of or... |
| oaf1o 8510 | Left addition by a constan... |
| oacomf1olem 8511 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8512 | Define a bijection from ` ... |
| omordi 8513 | Ordering property of ordin... |
| omord2 8514 | Ordering property of ordin... |
| omord 8515 | Ordering property of ordin... |
| omcan 8516 | Left cancellation law for ... |
| omword 8517 | Weak ordering property of ... |
| omwordi 8518 | Weak ordering property of ... |
| omwordri 8519 | Weak ordering property of ... |
| omword1 8520 | An ordinal is less than or... |
| omword2 8521 | An ordinal is less than or... |
| om00 8522 | The product of two ordinal... |
| om00el 8523 | The product of two nonzero... |
| omordlim 8524 | Ordering involving the pro... |
| omlimcl 8525 | The product of any nonzero... |
| odi 8526 | Distributive law for ordin... |
| omass 8527 | Multiplication of ordinal ... |
| oneo 8528 | If an ordinal number is ev... |
| omeulem1 8529 | Lemma for ~ omeu : existen... |
| omeulem2 8530 | Lemma for ~ omeu : uniquen... |
| omopth2 8531 | An ordered pair-like theor... |
| omeu 8532 | The division algorithm for... |
| oen0 8533 | Ordinal exponentiation wit... |
| oeordi 8534 | Ordering law for ordinal e... |
| oeord 8535 | Ordering property of ordin... |
| oecan 8536 | Left cancellation law for ... |
| oeword 8537 | Weak ordering property of ... |
| oewordi 8538 | Weak ordering property of ... |
| oewordri 8539 | Weak ordering property of ... |
| oeworde 8540 | Ordinal exponentiation com... |
| oeordsuc 8541 | Ordering property of ordin... |
| oelim2 8542 | Ordinal exponentiation wit... |
| oeoalem 8543 | Lemma for ~ oeoa . (Contr... |
| oeoa 8544 | Sum of exponents law for o... |
| oeoelem 8545 | Lemma for ~ oeoe . (Contr... |
| oeoe 8546 | Product of exponents law f... |
| oelimcl 8547 | The ordinal exponential wi... |
| oeeulem 8548 | Lemma for ~ oeeu . (Contr... |
| oeeui 8549 | The division algorithm for... |
| oeeu 8550 | The division algorithm for... |
| nna0 8551 | Addition with zero. Theor... |
| nnm0 8552 | Multiplication with zero. ... |
| nnasuc 8553 | Addition with successor. ... |
| nnmsuc 8554 | Multiplication with succes... |
| nnesuc 8555 | Exponentiation with a succ... |
| nna0r 8556 | Addition to zero. Remark ... |
| nnm0r 8557 | Multiplication with zero. ... |
| nnacl 8558 | Closure of addition of nat... |
| nnmcl 8559 | Closure of multiplication ... |
| nnecl 8560 | Closure of exponentiation ... |
| nnacli 8561 | ` _om ` is closed under ad... |
| nnmcli 8562 | ` _om ` is closed under mu... |
| nnarcl 8563 | Reverse closure law for ad... |
| nnacom 8564 | Addition of natural number... |
| nnaordi 8565 | Ordering property of addit... |
| nnaord 8566 | Ordering property of addit... |
| nnaordr 8567 | Ordering property of addit... |
| nnawordi 8568 | Adding to both sides of an... |
| nnaass 8569 | Addition of natural number... |
| nndi 8570 | Distributive law for natur... |
| nnmass 8571 | Multiplication of natural ... |
| nnmsucr 8572 | Multiplication with succes... |
| nnmcom 8573 | Multiplication of natural ... |
| nnaword 8574 | Weak ordering property of ... |
| nnacan 8575 | Cancellation law for addit... |
| nnaword1 8576 | Weak ordering property of ... |
| nnaword2 8577 | Weak ordering property of ... |
| nnmordi 8578 | Ordering property of multi... |
| nnmord 8579 | Ordering property of multi... |
| nnmword 8580 | Weak ordering property of ... |
| nnmcan 8581 | Cancellation law for multi... |
| nnmwordi 8582 | Weak ordering property of ... |
| nnmwordri 8583 | Weak ordering property of ... |
| nnawordex 8584 | Equivalence for weak order... |
| nnaordex 8585 | Equivalence for ordering. ... |
| 1onn 8586 | The ordinal 1 is a natural... |
| 1onnALT 8587 | Shorter proof of ~ 1onn us... |
| 2onn 8588 | The ordinal 2 is a natural... |
| 2onnALT 8589 | Shorter proof of ~ 2onn us... |
| 3onn 8590 | The ordinal 3 is a natural... |
| 4onn 8591 | The ordinal 4 is a natural... |
| 1one2o 8592 | Ordinal one is not ordinal... |
| oaabslem 8593 | Lemma for ~ oaabs . (Cont... |
| oaabs 8594 | Ordinal addition absorbs a... |
| oaabs2 8595 | The absorption law ~ oaabs... |
| omabslem 8596 | Lemma for ~ omabs . (Cont... |
| omabs 8597 | Ordinal multiplication is ... |
| nnm1 8598 | Multiply an element of ` _... |
| nnm2 8599 | Multiply an element of ` _... |
| nn2m 8600 | Multiply an element of ` _... |
| nnneo 8601 | If a natural number is eve... |
| nneob 8602 | A natural number is even i... |
| omsmolem 8603 | Lemma for ~ omsmo . (Cont... |
| omsmo 8604 | A strictly monotonic ordin... |
| omopthlem1 8605 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8606 | Lemma for ~ omopthi . (Co... |
| omopthi 8607 | An ordered pair theorem fo... |
| omopth 8608 | An ordered pair theorem fo... |
| nnasmo 8609 | There is at most one left ... |
| eldifsucnn 8610 | Condition for membership i... |
| on2recsfn 8613 | Show that double recursion... |
| on2recsov 8614 | Calculate the value of the... |
| on2ind 8615 | Double induction over ordi... |
| on3ind 8616 | Triple induction over ordi... |
| coflton 8617 | Cofinality theorem for ord... |
| cofon1 8618 | Cofinality theorem for ord... |
| cofon2 8619 | Cofinality theorem for ord... |
| cofonr 8620 | Inverse cofinality law for... |
| naddfn 8621 | Natural addition is a func... |
| naddcllem 8622 | Lemma for ordinal addition... |
| naddcl 8623 | Closure law for natural ad... |
| naddov 8624 | The value of natural addit... |
| naddov2 8625 | Alternate expression for n... |
| naddov3 8626 | Alternate expression for n... |
| naddf 8627 | Function statement for nat... |
| naddcom 8628 | Natural addition commutes.... |
| naddid1 8629 | Ordinal zero is the additi... |
| naddssim 8630 | Ordinal less-than-or-equal... |
| naddelim 8631 | Ordinal less-than is prese... |
| naddel1 8632 | Ordinal less-than is not a... |
| naddel2 8633 | Ordinal less-than is not a... |
| naddss1 8634 | Ordinal less-than-or-equal... |
| naddss2 8635 | Ordinal less-than-or-equal... |
| naddword1 8636 | Weak-ordering principle fo... |
| naddunif 8637 | Uniformity theorem for nat... |
| naddasslem1 8638 | Lemma for ~ naddass . Exp... |
| naddasslem2 8639 | Lemma for ~ naddass . Exp... |
| naddass 8640 | Natural ordinal addition i... |
| nadd32 8641 | Commutative/associative la... |
| nadd4 8642 | Rearragement of terms in a... |
| nadd42 8643 | Rearragement of terms in a... |
| naddel12 8644 | Natural addition to both s... |
| dfer2 8649 | Alternate definition of eq... |
| dfec2 8651 | Alternate definition of ` ... |
| ecexg 8652 | An equivalence class modul... |
| ecexr 8653 | A nonempty equivalence cla... |
| ereq1 8655 | Equality theorem for equiv... |
| ereq2 8656 | Equality theorem for equiv... |
| errel 8657 | An equivalence relation is... |
| erdm 8658 | The domain of an equivalen... |
| ercl 8659 | Elementhood in the field o... |
| ersym 8660 | An equivalence relation is... |
| ercl2 8661 | Elementhood in the field o... |
| ersymb 8662 | An equivalence relation is... |
| ertr 8663 | An equivalence relation is... |
| ertrd 8664 | A transitivity relation fo... |
| ertr2d 8665 | A transitivity relation fo... |
| ertr3d 8666 | A transitivity relation fo... |
| ertr4d 8667 | A transitivity relation fo... |
| erref 8668 | An equivalence relation is... |
| ercnv 8669 | The converse of an equival... |
| errn 8670 | The range and domain of an... |
| erssxp 8671 | An equivalence relation is... |
| erex 8672 | An equivalence relation is... |
| erexb 8673 | An equivalence relation is... |
| iserd 8674 | A reflexive, symmetric, tr... |
| iseri 8675 | A reflexive, symmetric, tr... |
| iseriALT 8676 | Alternate proof of ~ iseri... |
| brdifun 8677 | Evaluate the incomparabili... |
| swoer 8678 | Incomparability under a st... |
| swoord1 8679 | The incomparability equiva... |
| swoord2 8680 | The incomparability equiva... |
| swoso 8681 | If the incomparability rel... |
| eqerlem 8682 | Lemma for ~ eqer . (Contr... |
| eqer 8683 | Equivalence relation invol... |
| ider 8684 | The identity relation is a... |
| 0er 8685 | The empty set is an equiva... |
| eceq1 8686 | Equality theorem for equiv... |
| eceq1d 8687 | Equality theorem for equiv... |
| eceq2 8688 | Equality theorem for equiv... |
| eceq2i 8689 | Equality theorem for the `... |
| eceq2d 8690 | Equality theorem for the `... |
| elecg 8691 | Membership in an equivalen... |
| elec 8692 | Membership in an equivalen... |
| relelec 8693 | Membership in an equivalen... |
| ecss 8694 | An equivalence class is a ... |
| ecdmn0 8695 | A representative of a none... |
| ereldm 8696 | Equality of equivalence cl... |
| erth 8697 | Basic property of equivale... |
| erth2 8698 | Basic property of equivale... |
| erthi 8699 | Basic property of equivale... |
| erdisj 8700 | Equivalence classes do not... |
| ecidsn 8701 | An equivalence class modul... |
| qseq1 8702 | Equality theorem for quoti... |
| qseq2 8703 | Equality theorem for quoti... |
| qseq2i 8704 | Equality theorem for quoti... |
| qseq2d 8705 | Equality theorem for quoti... |
| qseq12 8706 | Equality theorem for quoti... |
| elqsg 8707 | Closed form of ~ elqs . (... |
| elqs 8708 | Membership in a quotient s... |
| elqsi 8709 | Membership in a quotient s... |
| elqsecl 8710 | Membership in a quotient s... |
| ecelqsg 8711 | Membership of an equivalen... |
| ecelqsi 8712 | Membership of an equivalen... |
| ecopqsi 8713 | "Closure" law for equivale... |
| qsexg 8714 | A quotient set exists. (C... |
| qsex 8715 | A quotient set exists. (C... |
| uniqs 8716 | The union of a quotient se... |
| qsss 8717 | A quotient set is a set of... |
| uniqs2 8718 | The union of a quotient se... |
| snec 8719 | The singleton of an equiva... |
| ecqs 8720 | Equivalence class in terms... |
| ecid 8721 | A set is equal to its cose... |
| qsid 8722 | A set is equal to its quot... |
| ectocld 8723 | Implicit substitution of c... |
| ectocl 8724 | Implicit substitution of c... |
| elqsn0 8725 | A quotient set does not co... |
| ecelqsdm 8726 | Membership of an equivalen... |
| xpider 8727 | A Cartesian square is an e... |
| iiner 8728 | The intersection of a none... |
| riiner 8729 | The relative intersection ... |
| erinxp 8730 | A restricted equivalence r... |
| ecinxp 8731 | Restrict the relation in a... |
| qsinxp 8732 | Restrict the equivalence r... |
| qsdisj 8733 | Members of a quotient set ... |
| qsdisj2 8734 | A quotient set is a disjoi... |
| qsel 8735 | If an element of a quotien... |
| uniinqs 8736 | Class union distributes ov... |
| qliftlem 8737 | Lemma for theorems about a... |
| qliftrel 8738 | ` F ` , a function lift, i... |
| qliftel 8739 | Elementhood in the relatio... |
| qliftel1 8740 | Elementhood in the relatio... |
| qliftfun 8741 | The function ` F ` is the ... |
| qliftfund 8742 | The function ` F ` is the ... |
| qliftfuns 8743 | The function ` F ` is the ... |
| qliftf 8744 | The domain and codomain of... |
| qliftval 8745 | The value of the function ... |
| ecoptocl 8746 | Implicit substitution of c... |
| 2ecoptocl 8747 | Implicit substitution of c... |
| 3ecoptocl 8748 | Implicit substitution of c... |
| brecop 8749 | Binary relation on a quoti... |
| brecop2 8750 | Binary relation on a quoti... |
| eroveu 8751 | Lemma for ~ erov and ~ ero... |
| erovlem 8752 | Lemma for ~ erov and ~ ero... |
| erov 8753 | The value of an operation ... |
| eroprf 8754 | Functionality of an operat... |
| erov2 8755 | The value of an operation ... |
| eroprf2 8756 | Functionality of an operat... |
| ecopoveq 8757 | This is the first of sever... |
| ecopovsym 8758 | Assuming the operation ` F... |
| ecopovtrn 8759 | Assuming that operation ` ... |
| ecopover 8760 | Assuming that operation ` ... |
| eceqoveq 8761 | Equality of equivalence re... |
| ecovcom 8762 | Lemma used to transfer a c... |
| ecovass 8763 | Lemma used to transfer an ... |
| ecovdi 8764 | Lemma used to transfer a d... |
| mapprc 8769 | When ` A ` is a proper cla... |
| pmex 8770 | The class of all partial f... |
| mapex 8771 | The class of all functions... |
| fnmap 8772 | Set exponentiation has a u... |
| fnpm 8773 | Partial function exponenti... |
| reldmmap 8774 | Set exponentiation is a we... |
| mapvalg 8775 | The value of set exponenti... |
| pmvalg 8776 | The value of the partial m... |
| mapval 8777 | The value of set exponenti... |
| elmapg 8778 | Membership relation for se... |
| elmapd 8779 | Deduction form of ~ elmapg... |
| mapdm0 8780 | The empty set is the only ... |
| elpmg 8781 | The predicate "is a partia... |
| elpm2g 8782 | The predicate "is a partia... |
| elpm2r 8783 | Sufficient condition for b... |
| elpmi 8784 | A partial function is a fu... |
| pmfun 8785 | A partial function is a fu... |
| elmapex 8786 | Eliminate antecedent for m... |
| elmapi 8787 | A mapping is a function, f... |
| mapfset 8788 | If ` B ` is a set, the val... |
| mapssfset 8789 | The value of the set expon... |
| mapfoss 8790 | The value of the set expon... |
| fsetsspwxp 8791 | The class of all functions... |
| fset0 8792 | The set of functions from ... |
| fsetdmprc0 8793 | The set of functions with ... |
| fsetex 8794 | The set of functions betwe... |
| f1setex 8795 | The set of injections betw... |
| fosetex 8796 | The set of surjections bet... |
| f1osetex 8797 | The set of bijections betw... |
| fsetfcdm 8798 | The class of functions wit... |
| fsetfocdm 8799 | The class of functions wit... |
| fsetprcnex 8800 | The class of all functions... |
| fsetcdmex 8801 | The class of all functions... |
| fsetexb 8802 | The class of all functions... |
| elmapfn 8803 | A mapping is a function wi... |
| elmapfun 8804 | A mapping is always a func... |
| elmapssres 8805 | A restricted mapping is a ... |
| fpmg 8806 | A total function is a part... |
| pmss12g 8807 | Subset relation for the se... |
| pmresg 8808 | Elementhood of a restricte... |
| elmap 8809 | Membership relation for se... |
| mapval2 8810 | Alternate expression for t... |
| elpm 8811 | The predicate "is a partia... |
| elpm2 8812 | The predicate "is a partia... |
| fpm 8813 | A total function is a part... |
| mapsspm 8814 | Set exponentiation is a su... |
| pmsspw 8815 | Partial maps are a subset ... |
| mapsspw 8816 | Set exponentiation is a su... |
| mapfvd 8817 | The value of a function th... |
| elmapresaun 8818 | ~ fresaun transposed to ma... |
| fvmptmap 8819 | Special case of ~ fvmpt fo... |
| map0e 8820 | Set exponentiation with an... |
| map0b 8821 | Set exponentiation with an... |
| map0g 8822 | Set exponentiation is empt... |
| 0map0sn0 8823 | The set of mappings of the... |
| mapsnd 8824 | The value of set exponenti... |
| map0 8825 | Set exponentiation is empt... |
| mapsn 8826 | The value of set exponenti... |
| mapss 8827 | Subset inheritance for set... |
| fdiagfn 8828 | Functionality of the diago... |
| fvdiagfn 8829 | Functionality of the diago... |
| mapsnconst 8830 | Every singleton map is a c... |
| mapsncnv 8831 | Expression for the inverse... |
| mapsnf1o2 8832 | Explicit bijection between... |
| mapsnf1o3 8833 | Explicit bijection in the ... |
| ralxpmap 8834 | Quantification over functi... |
| dfixp 8837 | Eliminate the expression `... |
| ixpsnval 8838 | The value of an infinite C... |
| elixp2 8839 | Membership in an infinite ... |
| fvixp 8840 | Projection of a factor of ... |
| ixpfn 8841 | A nuple is a function. (C... |
| elixp 8842 | Membership in an infinite ... |
| elixpconst 8843 | Membership in an infinite ... |
| ixpconstg 8844 | Infinite Cartesian product... |
| ixpconst 8845 | Infinite Cartesian product... |
| ixpeq1 8846 | Equality theorem for infin... |
| ixpeq1d 8847 | Equality theorem for infin... |
| ss2ixp 8848 | Subclass theorem for infin... |
| ixpeq2 8849 | Equality theorem for infin... |
| ixpeq2dva 8850 | Equality theorem for infin... |
| ixpeq2dv 8851 | Equality theorem for infin... |
| cbvixp 8852 | Change bound variable in a... |
| cbvixpv 8853 | Change bound variable in a... |
| nfixpw 8854 | Bound-variable hypothesis ... |
| nfixp 8855 | Bound-variable hypothesis ... |
| nfixp1 8856 | The index variable in an i... |
| ixpprc 8857 | A cartesian product of pro... |
| ixpf 8858 | A member of an infinite Ca... |
| uniixp 8859 | The union of an infinite C... |
| ixpexg 8860 | The existence of an infini... |
| ixpin 8861 | The intersection of two in... |
| ixpiin 8862 | The indexed intersection o... |
| ixpint 8863 | The intersection of a coll... |
| ixp0x 8864 | An infinite Cartesian prod... |
| ixpssmap2g 8865 | An infinite Cartesian prod... |
| ixpssmapg 8866 | An infinite Cartesian prod... |
| 0elixp 8867 | Membership of the empty se... |
| ixpn0 8868 | The infinite Cartesian pro... |
| ixp0 8869 | The infinite Cartesian pro... |
| ixpssmap 8870 | An infinite Cartesian prod... |
| resixp 8871 | Restriction of an element ... |
| undifixp 8872 | Union of two projections o... |
| mptelixpg 8873 | Condition for an explicit ... |
| resixpfo 8874 | Restriction of elements of... |
| elixpsn 8875 | Membership in a class of s... |
| ixpsnf1o 8876 | A bijection between a clas... |
| mapsnf1o 8877 | A bijection between a set ... |
| boxriin 8878 | A rectangular subset of a ... |
| boxcutc 8879 | The relative complement of... |
| relen 8888 | Equinumerosity is a relati... |
| reldom 8889 | Dominance is a relation. ... |
| relsdom 8890 | Strict dominance is a rela... |
| encv 8891 | If two classes are equinum... |
| breng 8892 | Equinumerosity relation. ... |
| bren 8893 | Equinumerosity relation. ... |
| brenOLD 8894 | Obsolete version of ~ bren... |
| brdom2g 8895 | Dominance relation. This ... |
| brdomg 8896 | Dominance relation. (Cont... |
| brdomgOLD 8897 | Obsolete version of ~ brdo... |
| brdomi 8898 | Dominance relation. (Cont... |
| brdomiOLD 8899 | Obsolete version of ~ brdo... |
| brdom 8900 | Dominance relation. (Cont... |
| domen 8901 | Dominance in terms of equi... |
| domeng 8902 | Dominance in terms of equi... |
| ctex 8903 | A countable set is a set. ... |
| f1oen4g 8904 | The domain and range of a ... |
| f1dom4g 8905 | The domain of a one-to-one... |
| f1oen3g 8906 | The domain and range of a ... |
| f1dom3g 8907 | The domain of a one-to-one... |
| f1oen2g 8908 | The domain and range of a ... |
| f1dom2g 8909 | The domain of a one-to-one... |
| f1dom2gOLD 8910 | Obsolete version of ~ f1do... |
| f1oeng 8911 | The domain and range of a ... |
| f1domg 8912 | The domain of a one-to-one... |
| f1oen 8913 | The domain and range of a ... |
| f1dom 8914 | The domain of a one-to-one... |
| brsdom 8915 | Strict dominance relation,... |
| isfi 8916 | Express " ` A ` is finite"... |
| enssdom 8917 | Equinumerosity implies dom... |
| dfdom2 8918 | Alternate definition of do... |
| endom 8919 | Equinumerosity implies dom... |
| sdomdom 8920 | Strict dominance implies d... |
| sdomnen 8921 | Strict dominance implies n... |
| brdom2 8922 | Dominance in terms of stri... |
| bren2 8923 | Equinumerosity expressed i... |
| enrefg 8924 | Equinumerosity is reflexiv... |
| enref 8925 | Equinumerosity is reflexiv... |
| eqeng 8926 | Equality implies equinumer... |
| domrefg 8927 | Dominance is reflexive. (... |
| en2d 8928 | Equinumerosity inference f... |
| en3d 8929 | Equinumerosity inference f... |
| en2i 8930 | Equinumerosity inference f... |
| en3i 8931 | Equinumerosity inference f... |
| dom2lem 8932 | A mapping (first hypothesi... |
| dom2d 8933 | A mapping (first hypothesi... |
| dom3d 8934 | A mapping (first hypothesi... |
| dom2 8935 | A mapping (first hypothesi... |
| dom3 8936 | A mapping (first hypothesi... |
| idssen 8937 | Equality implies equinumer... |
| domssl 8938 | If ` A ` is a subset of ` ... |
| domssr 8939 | If ` C ` is a superset of ... |
| ssdomg 8940 | A set dominates its subset... |
| ener 8941 | Equinumerosity is an equiv... |
| ensymb 8942 | Symmetry of equinumerosity... |
| ensym 8943 | Symmetry of equinumerosity... |
| ensymi 8944 | Symmetry of equinumerosity... |
| ensymd 8945 | Symmetry of equinumerosity... |
| entr 8946 | Transitivity of equinumero... |
| domtr 8947 | Transitivity of dominance ... |
| entri 8948 | A chained equinumerosity i... |
| entr2i 8949 | A chained equinumerosity i... |
| entr3i 8950 | A chained equinumerosity i... |
| entr4i 8951 | A chained equinumerosity i... |
| endomtr 8952 | Transitivity of equinumero... |
| domentr 8953 | Transitivity of dominance ... |
| f1imaeng 8954 | If a function is one-to-on... |
| f1imaen2g 8955 | If a function is one-to-on... |
| f1imaen 8956 | If a function is one-to-on... |
| en0 8957 | The empty set is equinumer... |
| en0OLD 8958 | Obsolete version of ~ en0 ... |
| en0ALT 8959 | Shorter proof of ~ en0 , d... |
| en0r 8960 | The empty set is equinumer... |
| ensn1 8961 | A singleton is equinumerou... |
| ensn1OLD 8962 | Obsolete version of ~ ensn... |
| ensn1g 8963 | A singleton is equinumerou... |
| enpr1g 8964 | ` { A , A } ` has only one... |
| en1 8965 | A set is equinumerous to o... |
| en1OLD 8966 | Obsolete version of ~ en1 ... |
| en1b 8967 | A set is equinumerous to o... |
| en1bOLD 8968 | Obsolete version of ~ en1b... |
| reuen1 8969 | Two ways to express "exact... |
| euen1 8970 | Two ways to express "exact... |
| euen1b 8971 | Two ways to express " ` A ... |
| en1uniel 8972 | A singleton contains its s... |
| en1unielOLD 8973 | Obsolete version of ~ en1u... |
| 2dom 8974 | A set that dominates ordin... |
| fundmen 8975 | A function is equinumerous... |
| fundmeng 8976 | A function is equinumerous... |
| cnven 8977 | A relational set is equinu... |
| cnvct 8978 | If a set is countable, so ... |
| fndmeng 8979 | A function is equinumerate... |
| mapsnend 8980 | Set exponentiation to a si... |
| mapsnen 8981 | Set exponentiation to a si... |
| snmapen 8982 | Set exponentiation: a sing... |
| snmapen1 8983 | Set exponentiation: a sing... |
| map1 8984 | Set exponentiation: ordina... |
| en2sn 8985 | Two singletons are equinum... |
| en2snOLD 8986 | Obsolete version of ~ en2s... |
| en2snOLDOLD 8987 | Obsolete version of ~ en2s... |
| snfi 8988 | A singleton is finite. (C... |
| fiprc 8989 | The class of finite sets i... |
| unen 8990 | Equinumerosity of union of... |
| enrefnn 8991 | Equinumerosity is reflexiv... |
| en2prd 8992 | Two unordered pairs are eq... |
| enpr2d 8993 | A pair with distinct eleme... |
| enpr2dOLD 8994 | Obsolete version of ~ enpr... |
| ssct 8995 | Any subset of a countable ... |
| ssctOLD 8996 | Obsolete version of ~ ssct... |
| difsnen 8997 | All decrements of a set ar... |
| domdifsn 8998 | Dominance over a set with ... |
| xpsnen 8999 | A set is equinumerous to i... |
| xpsneng 9000 | A set is equinumerous to i... |
| xp1en 9001 | One times a cardinal numbe... |
| endisj 9002 | Any two sets are equinumer... |
| undom 9003 | Dominance law for union. ... |
| undomOLD 9004 | Obsolete version of ~ undo... |
| xpcomf1o 9005 | The canonical bijection fr... |
| xpcomco 9006 | Composition with the bijec... |
| xpcomen 9007 | Commutative law for equinu... |
| xpcomeng 9008 | Commutative law for equinu... |
| xpsnen2g 9009 | A set is equinumerous to i... |
| xpassen 9010 | Associative law for equinu... |
| xpdom2 9011 | Dominance law for Cartesia... |
| xpdom2g 9012 | Dominance law for Cartesia... |
| xpdom1g 9013 | Dominance law for Cartesia... |
| xpdom3 9014 | A set is dominated by its ... |
| xpdom1 9015 | Dominance law for Cartesia... |
| domunsncan 9016 | A singleton cancellation l... |
| omxpenlem 9017 | Lemma for ~ omxpen . (Con... |
| omxpen 9018 | The cardinal and ordinal p... |
| omf1o 9019 | Construct an explicit bije... |
| pw2f1olem 9020 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 9021 | The power set of a set is ... |
| pw2eng 9022 | The power set of a set is ... |
| pw2en 9023 | The power set of a set is ... |
| fopwdom 9024 | Covering implies injection... |
| enfixsn 9025 | Given two equipollent sets... |
| sucdom2OLD 9026 | Obsolete version of ~ sucd... |
| sbthlem1 9027 | Lemma for ~ sbth . (Contr... |
| sbthlem2 9028 | Lemma for ~ sbth . (Contr... |
| sbthlem3 9029 | Lemma for ~ sbth . (Contr... |
| sbthlem4 9030 | Lemma for ~ sbth . (Contr... |
| sbthlem5 9031 | Lemma for ~ sbth . (Contr... |
| sbthlem6 9032 | Lemma for ~ sbth . (Contr... |
| sbthlem7 9033 | Lemma for ~ sbth . (Contr... |
| sbthlem8 9034 | Lemma for ~ sbth . (Contr... |
| sbthlem9 9035 | Lemma for ~ sbth . (Contr... |
| sbthlem10 9036 | Lemma for ~ sbth . (Contr... |
| sbth 9037 | Schroeder-Bernstein Theore... |
| sbthb 9038 | Schroeder-Bernstein Theore... |
| sbthcl 9039 | Schroeder-Bernstein Theore... |
| dfsdom2 9040 | Alternate definition of st... |
| brsdom2 9041 | Alternate definition of st... |
| sdomnsym 9042 | Strict dominance is asymme... |
| domnsym 9043 | Theorem 22(i) of [Suppes] ... |
| 0domg 9044 | Any set dominates the empt... |
| 0domgOLD 9045 | Obsolete version of ~ 0dom... |
| dom0 9046 | A set dominated by the emp... |
| dom0OLD 9047 | Obsolete version of ~ dom0... |
| 0sdomg 9048 | A set strictly dominates t... |
| 0sdomgOLD 9049 | Obsolete version of ~ 0sdo... |
| 0dom 9050 | Any set dominates the empt... |
| 0sdom 9051 | A set strictly dominates t... |
| sdom0 9052 | The empty set does not str... |
| sdom0OLD 9053 | Obsolete version of ~ sdom... |
| sdomdomtr 9054 | Transitivity of strict dom... |
| sdomentr 9055 | Transitivity of strict dom... |
| domsdomtr 9056 | Transitivity of dominance ... |
| ensdomtr 9057 | Transitivity of equinumero... |
| sdomirr 9058 | Strict dominance is irrefl... |
| sdomtr 9059 | Strict dominance is transi... |
| sdomn2lp 9060 | Strict dominance has no 2-... |
| enen1 9061 | Equality-like theorem for ... |
| enen2 9062 | Equality-like theorem for ... |
| domen1 9063 | Equality-like theorem for ... |
| domen2 9064 | Equality-like theorem for ... |
| sdomen1 9065 | Equality-like theorem for ... |
| sdomen2 9066 | Equality-like theorem for ... |
| domtriord 9067 | Dominance is trichotomous ... |
| sdomel 9068 | For ordinals, strict domin... |
| sdomdif 9069 | The difference of a set fr... |
| onsdominel 9070 | An ordinal with more eleme... |
| domunsn 9071 | Dominance over a set with ... |
| fodomr 9072 | There exists a mapping fro... |
| pwdom 9073 | Injection of sets implies ... |
| canth2 9074 | Cantor's Theorem. No set ... |
| canth2g 9075 | Cantor's theorem with the ... |
| 2pwuninel 9076 | The power set of the power... |
| 2pwne 9077 | No set equals the power se... |
| disjen 9078 | A stronger form of ~ pwuni... |
| disjenex 9079 | Existence version of ~ dis... |
| domss2 9080 | A corollary of ~ disjenex ... |
| domssex2 9081 | A corollary of ~ disjenex ... |
| domssex 9082 | Weakening of ~ domssex2 to... |
| xpf1o 9083 | Construct a bijection on a... |
| xpen 9084 | Equinumerosity law for Car... |
| mapen 9085 | Two set exponentiations ar... |
| mapdom1 9086 | Order-preserving property ... |
| mapxpen 9087 | Equinumerosity law for dou... |
| xpmapenlem 9088 | Lemma for ~ xpmapen . (Co... |
| xpmapen 9089 | Equinumerosity law for set... |
| mapunen 9090 | Equinumerosity law for set... |
| map2xp 9091 | A cardinal power with expo... |
| mapdom2 9092 | Order-preserving property ... |
| mapdom3 9093 | Set exponentiation dominat... |
| pwen 9094 | If two sets are equinumero... |
| ssenen 9095 | Equinumerosity of equinume... |
| limenpsi 9096 | A limit ordinal is equinum... |
| limensuci 9097 | A limit ordinal is equinum... |
| limensuc 9098 | A limit ordinal is equinum... |
| infensuc 9099 | Any infinite ordinal is eq... |
| dif1enlem 9100 | Lemma for ~ rexdif1en and ... |
| dif1enlemOLD 9101 | Obsolete version of ~ dif1... |
| rexdif1en 9102 | If a set is equinumerous t... |
| rexdif1enOLD 9103 | Obsolete version of ~ rexd... |
| dif1en 9104 | If a set ` A ` is equinume... |
| dif1ennn 9105 | If a set ` A ` is equinume... |
| dif1enOLD 9106 | Obsolete version of ~ dif1... |
| findcard 9107 | Schema for induction on th... |
| findcard2 9108 | Schema for induction on th... |
| findcard2s 9109 | Variation of ~ findcard2 r... |
| findcard2d 9110 | Deduction version of ~ fin... |
| nnfi 9111 | Natural numbers are finite... |
| pssnn 9112 | A proper subset of a natur... |
| ssnnfi 9113 | A subset of a natural numb... |
| ssnnfiOLD 9114 | Obsolete version of ~ ssnn... |
| 0fin 9115 | The empty set is finite. ... |
| unfi 9116 | The union of two finite se... |
| ssfi 9117 | A subset of a finite set i... |
| ssfiALT 9118 | Shorter proof of ~ ssfi us... |
| imafi 9119 | Images of finite sets are ... |
| pwfir 9120 | If the power set of a set ... |
| pwfilem 9121 | Lemma for ~ pwfi . (Contr... |
| pwfi 9122 | The power set of a finite ... |
| diffi 9123 | If ` A ` is finite, ` ( A ... |
| cnvfi 9124 | If a set is finite, its co... |
| fnfi 9125 | A version of ~ fnex for fi... |
| f1oenfi 9126 | If the domain of a one-to-... |
| f1oenfirn 9127 | If the range of a one-to-o... |
| f1domfi 9128 | If the codomain of a one-t... |
| f1domfi2 9129 | If the domain of a one-to-... |
| enreffi 9130 | Equinumerosity is reflexiv... |
| ensymfib 9131 | Symmetry of equinumerosity... |
| entrfil 9132 | Transitivity of equinumero... |
| enfii 9133 | A set equinumerous to a fi... |
| enfi 9134 | Equinumerous sets have the... |
| enfiALT 9135 | Shorter proof of ~ enfi us... |
| domfi 9136 | A set dominated by a finit... |
| entrfi 9137 | Transitivity of equinumero... |
| entrfir 9138 | Transitivity of equinumero... |
| domtrfil 9139 | Transitivity of dominance ... |
| domtrfi 9140 | Transitivity of dominance ... |
| domtrfir 9141 | Transitivity of dominance ... |
| f1imaenfi 9142 | If a function is one-to-on... |
| ssdomfi 9143 | A finite set dominates its... |
| ssdomfi2 9144 | A set dominates its finite... |
| sbthfilem 9145 | Lemma for ~ sbthfi . (Con... |
| sbthfi 9146 | Schroeder-Bernstein Theore... |
| domnsymfi 9147 | If a set dominates a finit... |
| sdomdomtrfi 9148 | Transitivity of strict dom... |
| domsdomtrfi 9149 | Transitivity of dominance ... |
| sucdom2 9150 | Strict dominance of a set ... |
| phplem1 9151 | Lemma for Pigeonhole Princ... |
| phplem2 9152 | Lemma for Pigeonhole Princ... |
| nneneq 9153 | Two equinumerous natural n... |
| php 9154 | Pigeonhole Principle. A n... |
| php2 9155 | Corollary of Pigeonhole Pr... |
| php3 9156 | Corollary of Pigeonhole Pr... |
| php4 9157 | Corollary of the Pigeonhol... |
| php5 9158 | Corollary of the Pigeonhol... |
| phpeqd 9159 | Corollary of the Pigeonhol... |
| nndomog 9160 | Cardinal ordering agrees w... |
| phplem1OLD 9161 | Obsolete lemma for ~ php .... |
| phplem2OLD 9162 | Obsolete lemma for ~ php .... |
| phplem3OLD 9163 | Obsolete version of ~ phpl... |
| phplem4OLD 9164 | Obsolete version of ~ phpl... |
| nneneqOLD 9165 | Obsolete version of ~ nnen... |
| phpOLD 9166 | Obsolete version of ~ php ... |
| php2OLD 9167 | Obsolete version of ~ php2... |
| php3OLD 9168 | Obsolete version of ~ php3... |
| phpeqdOLD 9169 | Obsolete version of ~ phpe... |
| nndomogOLD 9170 | Obsolete version of ~ nndo... |
| snnen2oOLD 9171 | Obsolete version of ~ snne... |
| onomeneq 9172 | An ordinal number equinume... |
| onomeneqOLD 9173 | Obsolete version of ~ onom... |
| onfin 9174 | An ordinal number is finit... |
| onfin2 9175 | A set is a natural number ... |
| nnfiOLD 9176 | Obsolete version of ~ nnfi... |
| nndomo 9177 | Cardinal ordering agrees w... |
| nnsdomo 9178 | Cardinal ordering agrees w... |
| sucdom 9179 | Strict dominance of a set ... |
| sucdomOLD 9180 | Obsolete version of ~ sucd... |
| snnen2o 9181 | A singleton ` { A } ` is n... |
| 0sdom1dom 9182 | Strict dominance over 0 is... |
| 0sdom1domALT 9183 | Alternate proof of ~ 0sdom... |
| 1sdom2 9184 | Ordinal 1 is strictly domi... |
| 1sdom2ALT 9185 | Alternate proof of ~ 1sdom... |
| sdom1 9186 | A set has less than one me... |
| sdom1OLD 9187 | Obsolete version of ~ sdom... |
| modom 9188 | Two ways to express "at mo... |
| modom2 9189 | Two ways to express "at mo... |
| rex2dom 9190 | A set that has at least 2 ... |
| 1sdom2dom 9191 | Strict dominance over 1 is... |
| 1sdom 9192 | A set that strictly domina... |
| 1sdomOLD 9193 | Obsolete version of ~ 1sdo... |
| unxpdomlem1 9194 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 9195 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 9196 | Lemma for ~ unxpdom . (Co... |
| unxpdom 9197 | Cartesian product dominate... |
| unxpdom2 9198 | Corollary of ~ unxpdom . ... |
| sucxpdom 9199 | Cartesian product dominate... |
| pssinf 9200 | A set equinumerous to a pr... |
| fisseneq 9201 | A finite set is equal to i... |
| ominf 9202 | The set of natural numbers... |
| ominfOLD 9203 | Obsolete version of ~ omin... |
| isinf 9204 | Any set that is not finite... |
| isinfOLD 9205 | Obsolete version of ~ isin... |
| fineqvlem 9206 | Lemma for ~ fineqv . (Con... |
| fineqv 9207 | If the Axiom of Infinity i... |
| enfiiOLD 9208 | Obsolete version of ~ enfi... |
| pssnnOLD 9209 | Obsolete version of ~ pssn... |
| xpfir 9210 | The components of a nonemp... |
| ssfid 9211 | A subset of a finite set i... |
| infi 9212 | The intersection of two se... |
| rabfi 9213 | A restricted class built f... |
| finresfin 9214 | The restriction of a finit... |
| f1finf1o 9215 | Any injection from one fin... |
| f1finf1oOLD 9216 | Obsolete version of ~ f1fi... |
| nfielex 9217 | If a class is not finite, ... |
| en1eqsn 9218 | A set with one element is ... |
| en1eqsnOLD 9219 | Obsolete version of ~ en1e... |
| en1eqsnbi 9220 | A set containing an elemen... |
| dif1ennnALT 9221 | Alternate proof of ~ dif1e... |
| enp1ilem 9222 | Lemma for uses of ~ enp1i ... |
| enp1i 9223 | Proof induction for ~ en2 ... |
| enp1iOLD 9224 | Obsolete version of ~ enp1... |
| en2 9225 | A set equinumerous to ordi... |
| en3 9226 | A set equinumerous to ordi... |
| en4 9227 | A set equinumerous to ordi... |
| findcard2OLD 9228 | Obsolete version of ~ find... |
| findcard3 9229 | Schema for strong inductio... |
| findcard3OLD 9230 | Obsolete version of ~ find... |
| ac6sfi 9231 | A version of ~ ac6s for fi... |
| frfi 9232 | A partial order is well-fo... |
| fimax2g 9233 | A finite set has a maximum... |
| fimaxg 9234 | A finite set has a maximum... |
| fisupg 9235 | Lemma showing existence an... |
| wofi 9236 | A total order on a finite ... |
| ordunifi 9237 | The maximum of a finite co... |
| nnunifi 9238 | The union (supremum) of a ... |
| unblem1 9239 | Lemma for ~ unbnn . After... |
| unblem2 9240 | Lemma for ~ unbnn . The v... |
| unblem3 9241 | Lemma for ~ unbnn . The v... |
| unblem4 9242 | Lemma for ~ unbnn . The f... |
| unbnn 9243 | Any unbounded subset of na... |
| unbnn2 9244 | Version of ~ unbnn that do... |
| isfinite2 9245 | Any set strictly dominated... |
| nnsdomg 9246 | Omega strictly dominates a... |
| nnsdomgOLD 9247 | Obsolete version of ~ nnsd... |
| isfiniteg 9248 | A set is finite iff it is ... |
| infsdomnn 9249 | An infinite set strictly d... |
| infsdomnnOLD 9250 | Obsolete version of ~ infs... |
| infn0 9251 | An infinite set is not emp... |
| infn0ALT 9252 | Shorter proof of ~ infn0 u... |
| fin2inf 9253 | This (useless) theorem, wh... |
| unfilem1 9254 | Lemma for proving that the... |
| unfilem2 9255 | Lemma for proving that the... |
| unfilem3 9256 | Lemma for proving that the... |
| unfiOLD 9257 | Obsolete version of ~ unfi... |
| unfir 9258 | If a union is finite, the ... |
| unfi2 9259 | The union of two finite se... |
| difinf 9260 | An infinite set ` A ` minu... |
| xpfi 9261 | The Cartesian product of t... |
| xpfiOLD 9262 | Obsolete version of ~ xpfi... |
| 3xpfi 9263 | The Cartesian product of t... |
| domunfican 9264 | A finite set union cancell... |
| infcntss 9265 | Every infinite set has a d... |
| prfi 9266 | An unordered pair is finit... |
| tpfi 9267 | An unordered triple is fin... |
| fiint 9268 | Equivalent ways of stating... |
| fodomfi 9269 | An onto function implies d... |
| fodomfib 9270 | Equivalence of an onto map... |
| fofinf1o 9271 | Any surjection from one fi... |
| rneqdmfinf1o 9272 | Any function from a finite... |
| fidomdm 9273 | Any finite set dominates i... |
| dmfi 9274 | The domain of a finite set... |
| fundmfibi 9275 | A function is finite if an... |
| resfnfinfin 9276 | The restriction of a funct... |
| residfi 9277 | A restricted identity func... |
| cnvfiALT 9278 | Shorter proof of ~ cnvfi u... |
| rnfi 9279 | The range of a finite set ... |
| f1dmvrnfibi 9280 | A one-to-one function whos... |
| f1vrnfibi 9281 | A one-to-one function whic... |
| fofi 9282 | If an onto function has a ... |
| f1fi 9283 | If a 1-to-1 function has a... |
| iunfi 9284 | The finite union of finite... |
| unifi 9285 | The finite union of finite... |
| unifi2 9286 | The finite union of finite... |
| infssuni 9287 | If an infinite set ` A ` i... |
| unirnffid 9288 | The union of the range of ... |
| imafiALT 9289 | Shorter proof of ~ imafi u... |
| pwfilemOLD 9290 | Obsolete version of ~ pwfi... |
| pwfiOLD 9291 | Obsolete version of ~ pwfi... |
| mapfi 9292 | Set exponentiation of fini... |
| ixpfi 9293 | A Cartesian product of fin... |
| ixpfi2 9294 | A Cartesian product of fin... |
| mptfi 9295 | A finite mapping set is fi... |
| abrexfi 9296 | An image set from a finite... |
| cnvimamptfin 9297 | A preimage of a mapping wi... |
| elfpw 9298 | Membership in a class of f... |
| unifpw 9299 | A set is the union of its ... |
| f1opwfi 9300 | A one-to-one mapping induc... |
| fissuni 9301 | A finite subset of a union... |
| fipreima 9302 | Given a finite subset ` A ... |
| finsschain 9303 | A finite subset of the uni... |
| indexfi 9304 | If for every element of a ... |
| relfsupp 9307 | The property of a function... |
| relprcnfsupp 9308 | A proper class is never fi... |
| isfsupp 9309 | The property of a class to... |
| funisfsupp 9310 | The property of a function... |
| fsuppimp 9311 | Implications of a class be... |
| fsuppimpd 9312 | A finitely supported funct... |
| fisuppfi 9313 | A function on a finite set... |
| fdmfisuppfi 9314 | The support of a function ... |
| fdmfifsupp 9315 | A function with a finite d... |
| fsuppmptdm 9316 | A mapping with a finite do... |
| fndmfisuppfi 9317 | The support of a function ... |
| fndmfifsupp 9318 | A function with a finite d... |
| suppeqfsuppbi 9319 | If two functions have the ... |
| suppssfifsupp 9320 | If the support of a functi... |
| fsuppsssupp 9321 | If the support of a functi... |
| fsuppxpfi 9322 | The cartesian product of t... |
| fczfsuppd 9323 | A constant function with v... |
| fsuppun 9324 | The union of two finitely ... |
| fsuppunfi 9325 | The union of the support o... |
| fsuppunbi 9326 | If the union of two classe... |
| 0fsupp 9327 | The empty set is a finitel... |
| snopfsupp 9328 | A singleton containing an ... |
| funsnfsupp 9329 | Finite support for a funct... |
| fsuppres 9330 | The restriction of a finit... |
| ressuppfi 9331 | If the support of the rest... |
| resfsupp 9332 | If the restriction of a fu... |
| resfifsupp 9333 | The restriction of a funct... |
| ffsuppbi 9334 | Two ways of saying that a ... |
| fsuppmptif 9335 | A function mapping an argu... |
| sniffsupp 9336 | A function mapping all but... |
| fsuppcolem 9337 | Lemma for ~ fsuppco . For... |
| fsuppco 9338 | The composition of a 1-1 f... |
| fsuppco2 9339 | The composition of a funct... |
| fsuppcor 9340 | The composition of a funct... |
| mapfienlem1 9341 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9342 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9343 | Lemma 3 for ~ mapfien . (... |
| mapfien 9344 | A bijection of the base se... |
| mapfien2 9345 | Equinumerousity relation f... |
| fival 9348 | The set of all the finite ... |
| elfi 9349 | Specific properties of an ... |
| elfi2 9350 | The empty intersection nee... |
| elfir 9351 | Sufficient condition for a... |
| intrnfi 9352 | Sufficient condition for t... |
| iinfi 9353 | An indexed intersection of... |
| inelfi 9354 | The intersection of two se... |
| ssfii 9355 | Any element of a set ` A `... |
| fi0 9356 | The set of finite intersec... |
| fieq0 9357 | A set is empty iff the cla... |
| fiin 9358 | The elements of ` ( fi `` ... |
| dffi2 9359 | The set of finite intersec... |
| fiss 9360 | Subset relationship for fu... |
| inficl 9361 | A set which is closed unde... |
| fipwuni 9362 | The set of finite intersec... |
| fisn 9363 | A singleton is closed unde... |
| fiuni 9364 | The union of the finite in... |
| fipwss 9365 | If a set is a family of su... |
| elfiun 9366 | A finite intersection of e... |
| dffi3 9367 | The set of finite intersec... |
| fifo 9368 | Describe a surjection from... |
| marypha1lem 9369 | Core induction for Philip ... |
| marypha1 9370 | (Philip) Hall's marriage t... |
| marypha2lem1 9371 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9372 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9373 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9374 | Lemma for ~ marypha2 . Pr... |
| marypha2 9375 | Version of ~ marypha1 usin... |
| dfsup2 9380 | Quantifier-free definition... |
| supeq1 9381 | Equality theorem for supre... |
| supeq1d 9382 | Equality deduction for sup... |
| supeq1i 9383 | Equality inference for sup... |
| supeq2 9384 | Equality theorem for supre... |
| supeq3 9385 | Equality theorem for supre... |
| supeq123d 9386 | Equality deduction for sup... |
| nfsup 9387 | Hypothesis builder for sup... |
| supmo 9388 | Any class ` B ` has at mos... |
| supexd 9389 | A supremum is a set. (Con... |
| supeu 9390 | A supremum is unique. Sim... |
| supval2 9391 | Alternate expression for t... |
| eqsup 9392 | Sufficient condition for a... |
| eqsupd 9393 | Sufficient condition for a... |
| supcl 9394 | A supremum belongs to its ... |
| supub 9395 | A supremum is an upper bou... |
| suplub 9396 | A supremum is the least up... |
| suplub2 9397 | Bidirectional form of ~ su... |
| supnub 9398 | An upper bound is not less... |
| supex 9399 | A supremum is a set. (Con... |
| sup00 9400 | The supremum under an empt... |
| sup0riota 9401 | The supremum of an empty s... |
| sup0 9402 | The supremum of an empty s... |
| supmax 9403 | The greatest element of a ... |
| fisup2g 9404 | A finite set satisfies the... |
| fisupcl 9405 | A nonempty finite set cont... |
| supgtoreq 9406 | The supremum of a finite s... |
| suppr 9407 | The supremum of a pair. (... |
| supsn 9408 | The supremum of a singleto... |
| supisolem 9409 | Lemma for ~ supiso . (Con... |
| supisoex 9410 | Lemma for ~ supiso . (Con... |
| supiso 9411 | Image of a supremum under ... |
| infeq1 9412 | Equality theorem for infim... |
| infeq1d 9413 | Equality deduction for inf... |
| infeq1i 9414 | Equality inference for inf... |
| infeq2 9415 | Equality theorem for infim... |
| infeq3 9416 | Equality theorem for infim... |
| infeq123d 9417 | Equality deduction for inf... |
| nfinf 9418 | Hypothesis builder for inf... |
| infexd 9419 | An infimum is a set. (Con... |
| eqinf 9420 | Sufficient condition for a... |
| eqinfd 9421 | Sufficient condition for a... |
| infval 9422 | Alternate expression for t... |
| infcllem 9423 | Lemma for ~ infcl , ~ infl... |
| infcl 9424 | An infimum belongs to its ... |
| inflb 9425 | An infimum is a lower boun... |
| infglb 9426 | An infimum is the greatest... |
| infglbb 9427 | Bidirectional form of ~ in... |
| infnlb 9428 | A lower bound is not great... |
| infex 9429 | An infimum is a set. (Con... |
| infmin 9430 | The smallest element of a ... |
| infmo 9431 | Any class ` B ` has at mos... |
| infeu 9432 | An infimum is unique. (Co... |
| fimin2g 9433 | A finite set has a minimum... |
| fiming 9434 | A finite set has a minimum... |
| fiinfg 9435 | Lemma showing existence an... |
| fiinf2g 9436 | A finite set satisfies the... |
| fiinfcl 9437 | A nonempty finite set cont... |
| infltoreq 9438 | The infimum of a finite se... |
| infpr 9439 | The infimum of a pair. (C... |
| infsupprpr 9440 | The infimum of a proper pa... |
| infsn 9441 | The infimum of a singleton... |
| inf00 9442 | The infimum regarding an e... |
| infempty 9443 | The infimum of an empty se... |
| infiso 9444 | Image of an infimum under ... |
| dfoi 9447 | Rewrite ~ df-oi with abbre... |
| oieq1 9448 | Equality theorem for ordin... |
| oieq2 9449 | Equality theorem for ordin... |
| nfoi 9450 | Hypothesis builder for ord... |
| ordiso2 9451 | Generalize ~ ordiso to pro... |
| ordiso 9452 | Order-isomorphic ordinal n... |
| ordtypecbv 9453 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9454 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9455 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9456 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9457 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9458 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9459 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9460 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9461 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9462 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9463 | Lemma for ~ ordtype . Usi... |
| oi0 9464 | Definition of the ordinal ... |
| oicl 9465 | The order type of the well... |
| oif 9466 | The order isomorphism of t... |
| oiiso2 9467 | The order isomorphism of t... |
| ordtype 9468 | For any set-like well-orde... |
| oiiniseg 9469 | ` ran F ` is an initial se... |
| ordtype2 9470 | For any set-like well-orde... |
| oiexg 9471 | The order isomorphism on a... |
| oion 9472 | The order type of the well... |
| oiiso 9473 | The order isomorphism of t... |
| oien 9474 | The order type of a well-o... |
| oieu 9475 | Uniqueness of the unique o... |
| oismo 9476 | When ` A ` is a subclass o... |
| oiid 9477 | The order type of an ordin... |
| hartogslem1 9478 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9479 | Lemma for ~ hartogs . (Co... |
| hartogs 9480 | The class of ordinals domi... |
| wofib 9481 | The only sets which are we... |
| wemaplem1 9482 | Value of the lexicographic... |
| wemaplem2 9483 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9484 | Lemma for ~ wemapso . Tra... |
| wemappo 9485 | Construct lexicographic or... |
| wemapsolem 9486 | Lemma for ~ wemapso . (Co... |
| wemapso 9487 | Construct lexicographic or... |
| wemapso2lem 9488 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9489 | An alternative to having a... |
| card2on 9490 | The alternate definition o... |
| card2inf 9491 | The alternate definition o... |
| harf 9494 | Functionality of the Harto... |
| harcl 9495 | Values of the Hartogs func... |
| harval 9496 | Function value of the Hart... |
| elharval 9497 | The Hartogs number of a se... |
| harndom 9498 | The Hartogs number of a se... |
| harword 9499 | Weak ordering property of ... |
| relwdom 9502 | Weak dominance is a relati... |
| brwdom 9503 | Property of weak dominance... |
| brwdomi 9504 | Property of weak dominance... |
| brwdomn0 9505 | Weak dominance over nonemp... |
| 0wdom 9506 | Any set weakly dominates t... |
| fowdom 9507 | An onto function implies w... |
| wdomref 9508 | Reflexivity of weak domina... |
| brwdom2 9509 | Alternate characterization... |
| domwdom 9510 | Weak dominance is implied ... |
| wdomtr 9511 | Transitivity of weak domin... |
| wdomen1 9512 | Equality-like theorem for ... |
| wdomen2 9513 | Equality-like theorem for ... |
| wdompwdom 9514 | Weak dominance strengthens... |
| canthwdom 9515 | Cantor's Theorem, stated u... |
| wdom2d 9516 | Deduce weak dominance from... |
| wdomd 9517 | Deduce weak dominance from... |
| brwdom3 9518 | Condition for weak dominan... |
| brwdom3i 9519 | Weak dominance implies exi... |
| unwdomg 9520 | Weak dominance of a (disjo... |
| xpwdomg 9521 | Weak dominance of a Cartes... |
| wdomima2g 9522 | A set is weakly dominant o... |
| wdomimag 9523 | A set is weakly dominant o... |
| unxpwdom2 9524 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9525 | If a Cartesian product is ... |
| ixpiunwdom 9526 | Describe an onto function ... |
| harwdom 9527 | The value of the Hartogs f... |
| axreg2 9529 | Axiom of Regularity expres... |
| zfregcl 9530 | The Axiom of Regularity wi... |
| zfreg 9531 | The Axiom of Regularity us... |
| elirrv 9532 | The membership relation is... |
| elirr 9533 | No class is a member of it... |
| elneq 9534 | A class is not equal to an... |
| nelaneq 9535 | A class is not an element ... |
| epinid0 9536 | The membership relation an... |
| sucprcreg 9537 | A class is equal to its su... |
| ruv 9538 | The Russell class is equal... |
| ruALT 9539 | Alternate proof of ~ ru , ... |
| disjcsn 9540 | A class is disjoint from i... |
| zfregfr 9541 | The membership relation is... |
| en2lp 9542 | No class has 2-cycle membe... |
| elnanel 9543 | Two classes are not elemen... |
| cnvepnep 9544 | The membership (epsilon) r... |
| epnsym 9545 | The membership (epsilon) r... |
| elnotel 9546 | A class cannot be an eleme... |
| elnel 9547 | A class cannot be an eleme... |
| en3lplem1 9548 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9549 | Lemma for ~ en3lp . (Cont... |
| en3lp 9550 | No class has 3-cycle membe... |
| preleqg 9551 | Equality of two unordered ... |
| preleq 9552 | Equality of two unordered ... |
| preleqALT 9553 | Alternate proof of ~ prele... |
| opthreg 9554 | Theorem for alternate repr... |
| suc11reg 9555 | The successor operation be... |
| dford2 9556 | Assuming ~ ax-reg , an ord... |
| inf0 9557 | Existence of ` _om ` impli... |
| inf1 9558 | Variation of Axiom of Infi... |
| inf2 9559 | Variation of Axiom of Infi... |
| inf3lema 9560 | Lemma for our Axiom of Inf... |
| inf3lemb 9561 | Lemma for our Axiom of Inf... |
| inf3lemc 9562 | Lemma for our Axiom of Inf... |
| inf3lemd 9563 | Lemma for our Axiom of Inf... |
| inf3lem1 9564 | Lemma for our Axiom of Inf... |
| inf3lem2 9565 | Lemma for our Axiom of Inf... |
| inf3lem3 9566 | Lemma for our Axiom of Inf... |
| inf3lem4 9567 | Lemma for our Axiom of Inf... |
| inf3lem5 9568 | Lemma for our Axiom of Inf... |
| inf3lem6 9569 | Lemma for our Axiom of Inf... |
| inf3lem7 9570 | Lemma for our Axiom of Inf... |
| inf3 9571 | Our Axiom of Infinity ~ ax... |
| infeq5i 9572 | Half of ~ infeq5 . (Contr... |
| infeq5 9573 | The statement "there exist... |
| zfinf 9575 | Axiom of Infinity expresse... |
| axinf2 9576 | A standard version of Axio... |
| zfinf2 9578 | A standard version of the ... |
| omex 9579 | The existence of omega (th... |
| axinf 9580 | The first version of the A... |
| inf5 9581 | The statement "there exist... |
| omelon 9582 | Omega is an ordinal number... |
| dfom3 9583 | The class of natural numbe... |
| elom3 9584 | A simplification of ~ elom... |
| dfom4 9585 | A simplification of ~ df-o... |
| dfom5 9586 | ` _om ` is the smallest li... |
| oancom 9587 | Ordinal addition is not co... |
| isfinite 9588 | A set is finite iff it is ... |
| fict 9589 | A finite set is countable ... |
| nnsdom 9590 | A natural number is strict... |
| omenps 9591 | Omega is equinumerous to a... |
| omensuc 9592 | The set of natural numbers... |
| infdifsn 9593 | Removing a singleton from ... |
| infdiffi 9594 | Removing a finite set from... |
| unbnn3 9595 | Any unbounded subset of na... |
| noinfep 9596 | Using the Axiom of Regular... |
| cantnffval 9599 | The value of the Cantor no... |
| cantnfdm 9600 | The domain of the Cantor n... |
| cantnfvalf 9601 | Lemma for ~ cantnf . The ... |
| cantnfs 9602 | Elementhood in the set of ... |
| cantnfcl 9603 | Basic properties of the or... |
| cantnfval 9604 | The value of the Cantor no... |
| cantnfval2 9605 | Alternate expression for t... |
| cantnfsuc 9606 | The value of the recursive... |
| cantnfle 9607 | A lower bound on the ` CNF... |
| cantnflt 9608 | An upper bound on the part... |
| cantnflt2 9609 | An upper bound on the ` CN... |
| cantnff 9610 | The ` CNF ` function is a ... |
| cantnf0 9611 | The value of the zero func... |
| cantnfrescl 9612 | A function is finitely sup... |
| cantnfres 9613 | The ` CNF ` function respe... |
| cantnfp1lem1 9614 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9615 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9616 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9617 | If ` F ` is created by add... |
| oemapso 9618 | The relation ` T ` is a st... |
| oemapval 9619 | Value of the relation ` T ... |
| oemapvali 9620 | If ` F < G ` , then there ... |
| cantnflem1a 9621 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9622 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9623 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9624 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9625 | Lemma for ~ cantnf . This... |
| cantnflem2 9626 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9627 | Lemma for ~ cantnf . Here... |
| cantnflem4 9628 | Lemma for ~ cantnf . Comp... |
| cantnf 9629 | The Cantor Normal Form the... |
| oemapwe 9630 | The lexicographic order on... |
| cantnffval2 9631 | An alternate definition of... |
| cantnff1o 9632 | Simplify the isomorphism o... |
| wemapwe 9633 | Construct lexicographic or... |
| oef1o 9634 | A bijection of the base se... |
| cnfcomlem 9635 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9636 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9637 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9638 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9639 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9640 | Any infinite ordinal ` B `... |
| cnfcom3clem 9641 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9642 | Wrap the construction of ~... |
| ttrcleq 9645 | Equality theorem for trans... |
| nfttrcld 9646 | Bound variable hypothesis ... |
| nfttrcl 9647 | Bound variable hypothesis ... |
| relttrcl 9648 | The transitive closure of ... |
| brttrcl 9649 | Characterization of elemen... |
| brttrcl2 9650 | Characterization of elemen... |
| ssttrcl 9651 | If ` R ` is a relation, th... |
| ttrcltr 9652 | The transitive closure of ... |
| ttrclresv 9653 | The transitive closure of ... |
| ttrclco 9654 | Composition law for the tr... |
| cottrcl 9655 | Composition law for the tr... |
| ttrclss 9656 | If ` R ` is a subclass of ... |
| dmttrcl 9657 | The domain of a transitive... |
| rnttrcl 9658 | The range of a transitive ... |
| ttrclexg 9659 | If ` R ` is a set, then so... |
| dfttrcl2 9660 | When ` R ` is a set and a ... |
| ttrclselem1 9661 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 9662 | Lemma for ~ ttrclse . Sho... |
| ttrclse 9663 | If ` R ` is set-like over ... |
| trcl 9664 | For any set ` A ` , show t... |
| tz9.1 9665 | Every set has a transitive... |
| tz9.1c 9666 | Alternate expression for t... |
| epfrs 9667 | The strong form of the Axi... |
| zfregs 9668 | The strong form of the Axi... |
| zfregs2 9669 | Alternate strong form of t... |
| setind 9670 | Set (epsilon) induction. ... |
| setind2 9671 | Set (epsilon) induction, s... |
| tcvalg 9674 | Value of the transitive cl... |
| tcid 9675 | Defining property of the t... |
| tctr 9676 | Defining property of the t... |
| tcmin 9677 | Defining property of the t... |
| tc2 9678 | A variant of the definitio... |
| tcsni 9679 | The transitive closure of ... |
| tcss 9680 | The transitive closure fun... |
| tcel 9681 | The transitive closure fun... |
| tcidm 9682 | The transitive closure fun... |
| tc0 9683 | The transitive closure of ... |
| tc00 9684 | The transitive closure is ... |
| frmin 9685 | Every (possibly proper) su... |
| frind 9686 | A subclass of a well-found... |
| frinsg 9687 | Well-Founded Induction Sch... |
| frins 9688 | Well-Founded Induction Sch... |
| frins2f 9689 | Well-Founded Induction sch... |
| frins2 9690 | Well-Founded Induction sch... |
| frins3 9691 | Well-Founded Induction sch... |
| frr3g 9692 | Functions defined by well-... |
| frrlem15 9693 | Lemma for general well-fou... |
| frrlem16 9694 | Lemma for general well-fou... |
| frr1 9695 | Law of general well-founde... |
| frr2 9696 | Law of general well-founde... |
| frr3 9697 | Law of general well-founde... |
| r1funlim 9702 | The cumulative hierarchy o... |
| r1fnon 9703 | The cumulative hierarchy o... |
| r10 9704 | Value of the cumulative hi... |
| r1sucg 9705 | Value of the cumulative hi... |
| r1suc 9706 | Value of the cumulative hi... |
| r1limg 9707 | Value of the cumulative hi... |
| r1lim 9708 | Value of the cumulative hi... |
| r1fin 9709 | The first ` _om ` levels o... |
| r1sdom 9710 | Each stage in the cumulati... |
| r111 9711 | The cumulative hierarchy i... |
| r1tr 9712 | The cumulative hierarchy o... |
| r1tr2 9713 | The union of a cumulative ... |
| r1ordg 9714 | Ordering relation for the ... |
| r1ord3g 9715 | Ordering relation for the ... |
| r1ord 9716 | Ordering relation for the ... |
| r1ord2 9717 | Ordering relation for the ... |
| r1ord3 9718 | Ordering relation for the ... |
| r1sssuc 9719 | The value of the cumulativ... |
| r1pwss 9720 | Each set of the cumulative... |
| r1sscl 9721 | Each set of the cumulative... |
| r1val1 9722 | The value of the cumulativ... |
| tz9.12lem1 9723 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9724 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9725 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9726 | A set is well-founded if a... |
| tz9.13 9727 | Every set is well-founded,... |
| tz9.13g 9728 | Every set is well-founded,... |
| rankwflemb 9729 | Two ways of saying a set i... |
| rankf 9730 | The domain and codomain of... |
| rankon 9731 | The rank of a set is an or... |
| r1elwf 9732 | Any member of the cumulati... |
| rankvalb 9733 | Value of the rank function... |
| rankr1ai 9734 | One direction of ~ rankr1a... |
| rankvaln 9735 | Value of the rank function... |
| rankidb 9736 | Identity law for the rank ... |
| rankdmr1 9737 | A rank is a member of the ... |
| rankr1ag 9738 | A version of ~ rankr1a tha... |
| rankr1bg 9739 | A relationship between ran... |
| r1rankidb 9740 | Any set is a subset of the... |
| r1elssi 9741 | The range of the ` R1 ` fu... |
| r1elss 9742 | The range of the ` R1 ` fu... |
| pwwf 9743 | A power set is well-founde... |
| sswf 9744 | A subset of a well-founded... |
| snwf 9745 | A singleton is well-founde... |
| unwf 9746 | A binary union is well-fou... |
| prwf 9747 | An unordered pair is well-... |
| opwf 9748 | An ordered pair is well-fo... |
| unir1 9749 | The cumulative hierarchy o... |
| jech9.3 9750 | Every set belongs to some ... |
| rankwflem 9751 | Every set is well-founded,... |
| rankval 9752 | Value of the rank function... |
| rankvalg 9753 | Value of the rank function... |
| rankval2 9754 | Value of an alternate defi... |
| uniwf 9755 | A union is well-founded if... |
| rankr1clem 9756 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9757 | A relationship between the... |
| rankidn 9758 | A relationship between the... |
| rankpwi 9759 | The rank of a power set. ... |
| rankelb 9760 | The membership relation is... |
| wfelirr 9761 | A well-founded set is not ... |
| rankval3b 9762 | The value of the rank func... |
| ranksnb 9763 | The rank of a singleton. ... |
| rankonidlem 9764 | Lemma for ~ rankonid . (C... |
| rankonid 9765 | The rank of an ordinal num... |
| onwf 9766 | The ordinals are all well-... |
| onssr1 9767 | Initial segments of the or... |
| rankr1g 9768 | A relationship between the... |
| rankid 9769 | Identity law for the rank ... |
| rankr1 9770 | A relationship between the... |
| ssrankr1 9771 | A relationship between an ... |
| rankr1a 9772 | A relationship between ran... |
| r1val2 9773 | The value of the cumulativ... |
| r1val3 9774 | The value of the cumulativ... |
| rankel 9775 | The membership relation is... |
| rankval3 9776 | The value of the rank func... |
| bndrank 9777 | Any class whose elements h... |
| unbndrank 9778 | The elements of a proper c... |
| rankpw 9779 | The rank of a power set. ... |
| ranklim 9780 | The rank of a set belongs ... |
| r1pw 9781 | A stronger property of ` R... |
| r1pwALT 9782 | Alternate shorter proof of... |
| r1pwcl 9783 | The cumulative hierarchy o... |
| rankssb 9784 | The subset relation is inh... |
| rankss 9785 | The subset relation is inh... |
| rankunb 9786 | The rank of the union of t... |
| rankprb 9787 | The rank of an unordered p... |
| rankopb 9788 | The rank of an ordered pai... |
| rankuni2b 9789 | The value of the rank func... |
| ranksn 9790 | The rank of a singleton. ... |
| rankuni2 9791 | The rank of a union. Part... |
| rankun 9792 | The rank of the union of t... |
| rankpr 9793 | The rank of an unordered p... |
| rankop 9794 | The rank of an ordered pai... |
| r1rankid 9795 | Any set is a subset of the... |
| rankeq0b 9796 | A set is empty iff its ran... |
| rankeq0 9797 | A set is empty iff its ran... |
| rankr1id 9798 | The rank of the hierarchy ... |
| rankuni 9799 | The rank of a union. Part... |
| rankr1b 9800 | A relationship between ran... |
| ranksuc 9801 | The rank of a successor. ... |
| rankuniss 9802 | Upper bound of the rank of... |
| rankval4 9803 | The rank of a set is the s... |
| rankbnd 9804 | The rank of a set is bound... |
| rankbnd2 9805 | The rank of a set is bound... |
| rankc1 9806 | A relationship that can be... |
| rankc2 9807 | A relationship that can be... |
| rankelun 9808 | Rank membership is inherit... |
| rankelpr 9809 | Rank membership is inherit... |
| rankelop 9810 | Rank membership is inherit... |
| rankxpl 9811 | A lower bound on the rank ... |
| rankxpu 9812 | An upper bound on the rank... |
| rankfu 9813 | An upper bound on the rank... |
| rankmapu 9814 | An upper bound on the rank... |
| rankxplim 9815 | The rank of a Cartesian pr... |
| rankxplim2 9816 | If the rank of a Cartesian... |
| rankxplim3 9817 | The rank of a Cartesian pr... |
| rankxpsuc 9818 | The rank of a Cartesian pr... |
| tcwf 9819 | The transitive closure fun... |
| tcrank 9820 | This theorem expresses two... |
| scottex 9821 | Scott's trick collects all... |
| scott0 9822 | Scott's trick collects all... |
| scottexs 9823 | Theorem scheme version of ... |
| scott0s 9824 | Theorem scheme version of ... |
| cplem1 9825 | Lemma for the Collection P... |
| cplem2 9826 | Lemma for the Collection P... |
| cp 9827 | Collection Principle. Thi... |
| bnd 9828 | A very strong generalizati... |
| bnd2 9829 | A variant of the Boundedne... |
| kardex 9830 | The collection of all sets... |
| karden 9831 | If we allow the Axiom of R... |
| htalem 9832 | Lemma for defining an emul... |
| hta 9833 | A ZFC emulation of Hilbert... |
| djueq12 9840 | Equality theorem for disjo... |
| djueq1 9841 | Equality theorem for disjo... |
| djueq2 9842 | Equality theorem for disjo... |
| nfdju 9843 | Bound-variable hypothesis ... |
| djuex 9844 | The disjoint union of sets... |
| djuexb 9845 | The disjoint union of two ... |
| djulcl 9846 | Left closure of disjoint u... |
| djurcl 9847 | Right closure of disjoint ... |
| djulf1o 9848 | The left injection functio... |
| djurf1o 9849 | The right injection functi... |
| inlresf 9850 | The left injection restric... |
| inlresf1 9851 | The left injection restric... |
| inrresf 9852 | The right injection restri... |
| inrresf1 9853 | The right injection restri... |
| djuin 9854 | The images of any classes ... |
| djur 9855 | A member of a disjoint uni... |
| djuss 9856 | A disjoint union is a subc... |
| djuunxp 9857 | The union of a disjoint un... |
| djuexALT 9858 | Alternate proof of ~ djuex... |
| eldju1st 9859 | The first component of an ... |
| eldju2ndl 9860 | The second component of an... |
| eldju2ndr 9861 | The second component of an... |
| djuun 9862 | The disjoint union of two ... |
| 1stinl 9863 | The first component of the... |
| 2ndinl 9864 | The second component of th... |
| 1stinr 9865 | The first component of the... |
| 2ndinr 9866 | The second component of th... |
| updjudhf 9867 | The mapping of an element ... |
| updjudhcoinlf 9868 | The composition of the map... |
| updjudhcoinrg 9869 | The composition of the map... |
| updjud 9870 | Universal property of the ... |
| cardf2 9879 | The cardinality function i... |
| cardon 9880 | The cardinal number of a s... |
| isnum2 9881 | A way to express well-orde... |
| isnumi 9882 | A set equinumerous to an o... |
| ennum 9883 | Equinumerous sets are equi... |
| finnum 9884 | Every finite set is numera... |
| onenon 9885 | Every ordinal number is nu... |
| tskwe 9886 | A Tarski set is well-order... |
| xpnum 9887 | The cartesian product of n... |
| cardval3 9888 | An alternate definition of... |
| cardid2 9889 | Any numerable set is equin... |
| isnum3 9890 | A set is numerable iff it ... |
| oncardval 9891 | The value of the cardinal ... |
| oncardid 9892 | Any ordinal number is equi... |
| cardonle 9893 | The cardinal of an ordinal... |
| card0 9894 | The cardinality of the emp... |
| cardidm 9895 | The cardinality function i... |
| oncard 9896 | A set is a cardinal number... |
| ficardom 9897 | The cardinal number of a f... |
| ficardid 9898 | A finite set is equinumero... |
| cardnn 9899 | The cardinality of a natur... |
| cardnueq0 9900 | The empty set is the only ... |
| cardne 9901 | No member of a cardinal nu... |
| carden2a 9902 | If two sets have equal non... |
| carden2b 9903 | If two sets are equinumero... |
| card1 9904 | A set has cardinality one ... |
| cardsn 9905 | A singleton has cardinalit... |
| carddomi2 9906 | Two sets have the dominanc... |
| sdomsdomcardi 9907 | A set strictly dominates i... |
| cardlim 9908 | An infinite cardinal is a ... |
| cardsdomelir 9909 | A cardinal strictly domina... |
| cardsdomel 9910 | A cardinal strictly domina... |
| iscard 9911 | Two ways to express the pr... |
| iscard2 9912 | Two ways to express the pr... |
| carddom2 9913 | Two numerable sets have th... |
| harcard 9914 | The class of ordinal numbe... |
| cardprclem 9915 | Lemma for ~ cardprc . (Co... |
| cardprc 9916 | The class of all cardinal ... |
| carduni 9917 | The union of a set of card... |
| cardiun 9918 | The indexed union of a set... |
| cardennn 9919 | If ` A ` is equinumerous t... |
| cardsucinf 9920 | The cardinality of the suc... |
| cardsucnn 9921 | The cardinality of the suc... |
| cardom 9922 | The set of natural numbers... |
| carden2 9923 | Two numerable sets are equ... |
| cardsdom2 9924 | A numerable set is strictl... |
| domtri2 9925 | Trichotomy of dominance fo... |
| nnsdomel 9926 | Strict dominance and eleme... |
| cardval2 9927 | An alternate version of th... |
| isinffi 9928 | An infinite set contains s... |
| fidomtri 9929 | Trichotomy of dominance wi... |
| fidomtri2 9930 | Trichotomy of dominance wi... |
| harsdom 9931 | The Hartogs number of a we... |
| onsdom 9932 | Any well-orderable set is ... |
| harval2 9933 | An alternate expression fo... |
| harsucnn 9934 | The next cardinal after a ... |
| cardmin2 9935 | The smallest ordinal that ... |
| pm54.43lem 9936 | In Theorem *54.43 of [Whit... |
| pm54.43 9937 | Theorem *54.43 of [Whitehe... |
| enpr2 9938 | An unordered pair with dis... |
| pr2nelemOLD 9939 | Obsolete version of ~ enpr... |
| pr2ne 9940 | If an unordered pair has t... |
| pr2neOLD 9941 | Obsolete version of ~ pr2n... |
| prdom2 9942 | An unordered pair has at m... |
| en2eqpr 9943 | Building a set with two el... |
| en2eleq 9944 | Express a set of pair card... |
| en2other2 9945 | Taking the other element t... |
| dif1card 9946 | The cardinality of a nonem... |
| leweon 9947 | Lexicographical order is a... |
| r0weon 9948 | A set-like well-ordering o... |
| infxpenlem 9949 | Lemma for ~ infxpen . (Co... |
| infxpen 9950 | Every infinite ordinal is ... |
| xpomen 9951 | The Cartesian product of o... |
| xpct 9952 | The cartesian product of t... |
| infxpidm2 9953 | Every infinite well-ordera... |
| infxpenc 9954 | A canonical version of ~ i... |
| infxpenc2lem1 9955 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9956 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9957 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9958 | Existence form of ~ infxpe... |
| iunmapdisj 9959 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9960 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9961 | Lemma for ~ fseqen . (Con... |
| fseqdom 9962 | One half of ~ fseqen . (C... |
| fseqen 9963 | A set that is equinumerous... |
| infpwfidom 9964 | The collection of finite s... |
| dfac8alem 9965 | Lemma for ~ dfac8a . If t... |
| dfac8a 9966 | Numeration theorem: every ... |
| dfac8b 9967 | The well-ordering theorem:... |
| dfac8clem 9968 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9969 | If the union of a set is w... |
| ac10ct 9970 | A proof of the well-orderi... |
| ween 9971 | A set is numerable iff it ... |
| ac5num 9972 | A version of ~ ac5b with t... |
| ondomen 9973 | If a set is dominated by a... |
| numdom 9974 | A set dominated by a numer... |
| ssnum 9975 | A subset of a numerable se... |
| onssnum 9976 | All subsets of the ordinal... |
| indcardi 9977 | Indirect strong induction ... |
| acnrcl 9978 | Reverse closure for the ch... |
| acneq 9979 | Equality theorem for the c... |
| isacn 9980 | The property of being a ch... |
| acni 9981 | The property of being a ch... |
| acni2 9982 | The property of being a ch... |
| acni3 9983 | The property of being a ch... |
| acnlem 9984 | Construct a mapping satisf... |
| numacn 9985 | A well-orderable set has c... |
| finacn 9986 | Every set has finite choic... |
| acndom 9987 | A set with long choice seq... |
| acnnum 9988 | A set ` X ` which has choi... |
| acnen 9989 | The class of choice sets o... |
| acndom2 9990 | A set smaller than one wit... |
| acnen2 9991 | The class of sets with cho... |
| fodomacn 9992 | A version of ~ fodom that ... |
| fodomnum 9993 | A version of ~ fodom that ... |
| fonum 9994 | A surjection maps numerabl... |
| numwdom 9995 | A surjection maps numerabl... |
| fodomfi2 9996 | Onto functions define domi... |
| wdomfil 9997 | Weak dominance agrees with... |
| infpwfien 9998 | Any infinite well-orderabl... |
| inffien 9999 | The set of finite intersec... |
| wdomnumr 10000 | Weak dominance agrees with... |
| alephfnon 10001 | The aleph function is a fu... |
| aleph0 10002 | The first infinite cardina... |
| alephlim 10003 | Value of the aleph functio... |
| alephsuc 10004 | Value of the aleph functio... |
| alephon 10005 | An aleph is an ordinal num... |
| alephcard 10006 | Every aleph is a cardinal ... |
| alephnbtwn 10007 | No cardinal can be sandwic... |
| alephnbtwn2 10008 | No set has equinumerosity ... |
| alephordilem1 10009 | Lemma for ~ alephordi . (... |
| alephordi 10010 | Strict ordering property o... |
| alephord 10011 | Ordering property of the a... |
| alephord2 10012 | Ordering property of the a... |
| alephord2i 10013 | Ordering property of the a... |
| alephord3 10014 | Ordering property of the a... |
| alephsucdom 10015 | A set dominated by an alep... |
| alephsuc2 10016 | An alternate representatio... |
| alephdom 10017 | Relationship between inclu... |
| alephgeom 10018 | Every aleph is greater tha... |
| alephislim 10019 | Every aleph is a limit ord... |
| aleph11 10020 | The aleph function is one-... |
| alephf1 10021 | The aleph function is a on... |
| alephsdom 10022 | If an ordinal is smaller t... |
| alephdom2 10023 | A dominated initial ordina... |
| alephle 10024 | The argument of the aleph ... |
| cardaleph 10025 | Given any transfinite card... |
| cardalephex 10026 | Every transfinite cardinal... |
| infenaleph 10027 | An infinite numerable set ... |
| isinfcard 10028 | Two ways to express the pr... |
| iscard3 10029 | Two ways to express the pr... |
| cardnum 10030 | Two ways to express the cl... |
| alephinit 10031 | An infinite initial ordina... |
| carduniima 10032 | The union of the image of ... |
| cardinfima 10033 | If a mapping to cardinals ... |
| alephiso 10034 | Aleph is an order isomorph... |
| alephprc 10035 | The class of all transfini... |
| alephsson 10036 | The class of transfinite c... |
| unialeph 10037 | The union of the class of ... |
| alephsmo 10038 | The aleph function is stri... |
| alephf1ALT 10039 | Alternate proof of ~ aleph... |
| alephfplem1 10040 | Lemma for ~ alephfp . (Co... |
| alephfplem2 10041 | Lemma for ~ alephfp . (Co... |
| alephfplem3 10042 | Lemma for ~ alephfp . (Co... |
| alephfplem4 10043 | Lemma for ~ alephfp . (Co... |
| alephfp 10044 | The aleph function has a f... |
| alephfp2 10045 | The aleph function has at ... |
| alephval3 10046 | An alternate way to expres... |
| alephsucpw2 10047 | The power set of an aleph ... |
| mappwen 10048 | Power rule for cardinal ar... |
| finnisoeu 10049 | A finite totally ordered s... |
| iunfictbso 10050 | Countability of a countabl... |
| aceq1 10053 | Equivalence of two version... |
| aceq0 10054 | Equivalence of two version... |
| aceq2 10055 | Equivalence of two version... |
| aceq3lem 10056 | Lemma for ~ dfac3 . (Cont... |
| dfac3 10057 | Equivalence of two version... |
| dfac4 10058 | Equivalence of two version... |
| dfac5lem1 10059 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 10060 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 10061 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 10062 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 10063 | Lemma for ~ dfac5 . (Cont... |
| dfac5 10064 | Equivalence of two version... |
| dfac2a 10065 | Our Axiom of Choice (in th... |
| dfac2b 10066 | Axiom of Choice (first for... |
| dfac2 10067 | Axiom of Choice (first for... |
| dfac7 10068 | Equivalence of the Axiom o... |
| dfac0 10069 | Equivalence of two version... |
| dfac1 10070 | Equivalence of two version... |
| dfac8 10071 | A proof of the equivalency... |
| dfac9 10072 | Equivalence of the axiom o... |
| dfac10 10073 | Axiom of Choice equivalent... |
| dfac10c 10074 | Axiom of Choice equivalent... |
| dfac10b 10075 | Axiom of Choice equivalent... |
| acacni 10076 | A choice equivalent: every... |
| dfacacn 10077 | A choice equivalent: every... |
| dfac13 10078 | The axiom of choice holds ... |
| dfac12lem1 10079 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 10080 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 10081 | Lemma for ~ dfac12 . (Con... |
| dfac12r 10082 | The axiom of choice holds ... |
| dfac12k 10083 | Equivalence of ~ dfac12 an... |
| dfac12a 10084 | The axiom of choice holds ... |
| dfac12 10085 | The axiom of choice holds ... |
| kmlem1 10086 | Lemma for 5-quantifier AC ... |
| kmlem2 10087 | Lemma for 5-quantifier AC ... |
| kmlem3 10088 | Lemma for 5-quantifier AC ... |
| kmlem4 10089 | Lemma for 5-quantifier AC ... |
| kmlem5 10090 | Lemma for 5-quantifier AC ... |
| kmlem6 10091 | Lemma for 5-quantifier AC ... |
| kmlem7 10092 | Lemma for 5-quantifier AC ... |
| kmlem8 10093 | Lemma for 5-quantifier AC ... |
| kmlem9 10094 | Lemma for 5-quantifier AC ... |
| kmlem10 10095 | Lemma for 5-quantifier AC ... |
| kmlem11 10096 | Lemma for 5-quantifier AC ... |
| kmlem12 10097 | Lemma for 5-quantifier AC ... |
| kmlem13 10098 | Lemma for 5-quantifier AC ... |
| kmlem14 10099 | Lemma for 5-quantifier AC ... |
| kmlem15 10100 | Lemma for 5-quantifier AC ... |
| kmlem16 10101 | Lemma for 5-quantifier AC ... |
| dfackm 10102 | Equivalence of the Axiom o... |
| undjudom 10103 | Cardinal addition dominate... |
| endjudisj 10104 | Equinumerosity of a disjoi... |
| djuen 10105 | Disjoint unions of equinum... |
| djuenun 10106 | Disjoint union is equinume... |
| dju1en 10107 | Cardinal addition with car... |
| dju1dif 10108 | Adding and subtracting one... |
| dju1p1e2 10109 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 10110 | Alternate proof of ~ dju1p... |
| dju0en 10111 | Cardinal addition with car... |
| xp2dju 10112 | Two times a cardinal numbe... |
| djucomen 10113 | Commutative law for cardin... |
| djuassen 10114 | Associative law for cardin... |
| xpdjuen 10115 | Cardinal multiplication di... |
| mapdjuen 10116 | Sum of exponents law for c... |
| pwdjuen 10117 | Sum of exponents law for c... |
| djudom1 10118 | Ordering law for cardinal ... |
| djudom2 10119 | Ordering law for cardinal ... |
| djudoml 10120 | A set is dominated by its ... |
| djuxpdom 10121 | Cartesian product dominate... |
| djufi 10122 | The disjoint union of two ... |
| cdainflem 10123 | Any partition of omega int... |
| djuinf 10124 | A set is infinite iff the ... |
| infdju1 10125 | An infinite set is equinum... |
| pwdju1 10126 | The sum of a powerset with... |
| pwdjuidm 10127 | If the natural numbers inj... |
| djulepw 10128 | If ` A ` is idempotent und... |
| onadju 10129 | The cardinal and ordinal s... |
| cardadju 10130 | The cardinal sum is equinu... |
| djunum 10131 | The disjoint union of two ... |
| unnum 10132 | The union of two numerable... |
| nnadju 10133 | The cardinal and ordinal s... |
| nnadjuALT 10134 | Shorter proof of ~ nnadju ... |
| ficardadju 10135 | The disjoint union of fini... |
| ficardun 10136 | The cardinality of the uni... |
| ficardunOLD 10137 | Obsolete version of ~ fica... |
| ficardun2 10138 | The cardinality of the uni... |
| ficardun2OLD 10139 | Obsolete version of ~ fica... |
| pwsdompw 10140 | Lemma for ~ domtriom . Th... |
| unctb 10141 | The union of two countable... |
| infdjuabs 10142 | Absorption law for additio... |
| infunabs 10143 | An infinite set is equinum... |
| infdju 10144 | The sum of two cardinal nu... |
| infdif 10145 | The cardinality of an infi... |
| infdif2 10146 | Cardinality ordering for a... |
| infxpdom 10147 | Dominance law for multipli... |
| infxpabs 10148 | Absorption law for multipl... |
| infunsdom1 10149 | The union of two sets that... |
| infunsdom 10150 | The union of two sets that... |
| infxp 10151 | Absorption law for multipl... |
| pwdjudom 10152 | A property of dominance ov... |
| infpss 10153 | Every infinite set has an ... |
| infmap2 10154 | An exponentiation law for ... |
| ackbij2lem1 10155 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 10156 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 10157 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 10158 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 10159 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 10160 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 10161 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 10162 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 10163 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 10164 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 10165 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 10166 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 10167 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 10168 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 10169 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 10170 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 10171 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 10172 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 10173 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 10174 | The Ackermann bijection, p... |
| ackbij1b 10175 | The Ackermann bijection, p... |
| ackbij2lem2 10176 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 10177 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 10178 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 10179 | The Ackermann bijection, p... |
| r1om 10180 | The set of hereditarily fi... |
| fictb 10181 | A set is countable iff its... |
| cflem 10182 | A lemma used to simplify c... |
| cfval 10183 | Value of the cofinality fu... |
| cff 10184 | Cofinality is a function o... |
| cfub 10185 | An upper bound on cofinali... |
| cflm 10186 | Value of the cofinality fu... |
| cf0 10187 | Value of the cofinality fu... |
| cardcf 10188 | Cofinality is a cardinal n... |
| cflecard 10189 | Cofinality is bounded by t... |
| cfle 10190 | Cofinality is bounded by i... |
| cfon 10191 | The cofinality of any set ... |
| cfeq0 10192 | Only the ordinal zero has ... |
| cfsuc 10193 | Value of the cofinality fu... |
| cff1 10194 | There is always a map from... |
| cfflb 10195 | If there is a cofinal map ... |
| cfval2 10196 | Another expression for the... |
| coflim 10197 | A simpler expression for t... |
| cflim3 10198 | Another expression for the... |
| cflim2 10199 | The cofinality function is... |
| cfom 10200 | Value of the cofinality fu... |
| cfss 10201 | There is a cofinal subset ... |
| cfslb 10202 | Any cofinal subset of ` A ... |
| cfslbn 10203 | Any subset of ` A ` smalle... |
| cfslb2n 10204 | Any small collection of sm... |
| cofsmo 10205 | Any cofinal map implies th... |
| cfsmolem 10206 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 10207 | The map in ~ cff1 can be a... |
| cfcoflem 10208 | Lemma for ~ cfcof , showin... |
| coftr 10209 | If there is a cofinal map ... |
| cfcof 10210 | If there is a cofinal map ... |
| cfidm 10211 | The cofinality function is... |
| alephsing 10212 | The cofinality of a limit ... |
| sornom 10213 | The range of a single-step... |
| isfin1a 10228 | Definition of a Ia-finite ... |
| fin1ai 10229 | Property of a Ia-finite se... |
| isfin2 10230 | Definition of a II-finite ... |
| fin2i 10231 | Property of a II-finite se... |
| isfin3 10232 | Definition of a III-finite... |
| isfin4 10233 | Definition of a IV-finite ... |
| fin4i 10234 | Infer that a set is IV-inf... |
| isfin5 10235 | Definition of a V-finite s... |
| isfin6 10236 | Definition of a VI-finite ... |
| isfin7 10237 | Definition of a VII-finite... |
| sdom2en01 10238 | A set with less than two e... |
| infpssrlem1 10239 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 10240 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 10241 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 10242 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 10243 | Lemma for ~ infpssr . (Co... |
| infpssr 10244 | Dedekind infinity implies ... |
| fin4en1 10245 | Dedekind finite is a cardi... |
| ssfin4 10246 | Dedekind finite sets have ... |
| domfin4 10247 | A set dominated by a Dedek... |
| ominf4 10248 | ` _om ` is Dedekind infini... |
| infpssALT 10249 | Alternate proof of ~ infps... |
| isfin4-2 10250 | Alternate definition of IV... |
| isfin4p1 10251 | Alternate definition of IV... |
| fin23lem7 10252 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10253 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10254 | A II-finite set contains m... |
| isfin2-2 10255 | ` Fin2 ` expressed in term... |
| ssfin2 10256 | A subset of a II-finite se... |
| enfin2i 10257 | II-finiteness is a cardina... |
| fin23lem24 10258 | Lemma for ~ fin23 . In a ... |
| fincssdom 10259 | In a chain of finite sets,... |
| fin23lem25 10260 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10261 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10262 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10263 | Lemma for ~ fin23 but coul... |
| fin23lem27 10264 | The mapping constructed in... |
| isfin3ds 10265 | Property of a III-finite s... |
| ssfin3ds 10266 | A subset of a III-finite s... |
| fin23lem12 10267 | The beginning of the proof... |
| fin23lem13 10268 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10269 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10270 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10271 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10272 | Lemma for ~ fin23 . The f... |
| fin23lem20 10273 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10274 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10275 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10276 | Lemma for ~ fin23 . The r... |
| fin23lem29 10277 | Lemma for ~ fin23 . The r... |
| fin23lem30 10278 | Lemma for ~ fin23 . The r... |
| fin23lem31 10279 | Lemma for ~ fin23 . The r... |
| fin23lem32 10280 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10281 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10282 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10283 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10284 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10285 | Lemma for ~ fin23 . The c... |
| fin23lem39 10286 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10287 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10288 | Lemma for ~ fin23 . A set... |
| isf32lem1 10289 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10290 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10291 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10292 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10293 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10294 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10295 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10296 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10297 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10298 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10299 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10300 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10301 | One half of ~ isfin3-2 . ... |
| isf33lem 10302 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10303 | Weakly Dedekind-infinite s... |
| isfin3-3 10304 | Weakly Dedekind-infinite s... |
| fin33i 10305 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10306 | Lemma for ~ compsscnv . (... |
| compsscnv 10307 | Complementation on a power... |
| isf34lem1 10308 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10309 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10310 | Complementation is an anti... |
| isf34lem3 10311 | Lemma for ~ isfin3-4 . (C... |
| compss 10312 | Express image under of the... |
| isf34lem4 10313 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10314 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10315 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10316 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10317 | Inference from ~ isfin3-4 ... |
| isfin3-4 10318 | Weakly Dedekind-infinite s... |
| fin11a 10319 | Every I-finite set is Ia-f... |
| enfin1ai 10320 | Ia-finiteness is a cardina... |
| isfin1-2 10321 | A set is finite in the usu... |
| isfin1-3 10322 | A set is I-finite iff ever... |
| isfin1-4 10323 | A set is I-finite iff ever... |
| dffin1-5 10324 | Compact quantifier-free ve... |
| fin23 10325 | Every II-finite set (every... |
| fin34 10326 | Every III-finite set is IV... |
| isfin5-2 10327 | Alternate definition of V-... |
| fin45 10328 | Every IV-finite set is V-f... |
| fin56 10329 | Every V-finite set is VI-f... |
| fin17 10330 | Every I-finite set is VII-... |
| fin67 10331 | Every VI-finite set is VII... |
| isfin7-2 10332 | A set is VII-finite iff it... |
| fin71num 10333 | A well-orderable set is VI... |
| dffin7-2 10334 | Class form of ~ isfin7-2 .... |
| dfacfin7 10335 | Axiom of Choice equivalent... |
| fin1a2lem1 10336 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10337 | Lemma for ~ fin1a2 . The ... |
| fin1a2lem3 10338 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10339 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10340 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10341 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10342 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10343 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10344 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10345 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10346 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10347 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10348 | Lemma for ~ fin1a2 . (Con... |
| fin12 10349 | Weak theorem which skips I... |
| fin1a2s 10350 | An II-infinite set can hav... |
| fin1a2 10351 | Every Ia-finite set is II-... |
| itunifval 10352 | Function value of iterated... |
| itunifn 10353 | Functionality of the itera... |
| ituni0 10354 | A zero-fold iterated union... |
| itunisuc 10355 | Successor iterated union. ... |
| itunitc1 10356 | Each union iterate is a me... |
| itunitc 10357 | The union of all union ite... |
| ituniiun 10358 | Unwrap an iterated union f... |
| hsmexlem7 10359 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10360 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10361 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10362 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10363 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10364 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10365 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10366 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10367 | Lemma for ~ hsmex . (Cont... |
| hsmex 10368 | The collection of heredita... |
| hsmex2 10369 | The set of hereditary size... |
| hsmex3 10370 | The set of hereditary size... |
| axcc2lem 10372 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10373 | A possibly more useful ver... |
| axcc3 10374 | A possibly more useful ver... |
| axcc4 10375 | A version of ~ axcc3 that ... |
| acncc 10376 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10377 | Relax the constraint on ~ ... |
| domtriomlem 10378 | Lemma for ~ domtriom . (C... |
| domtriom 10379 | Trichotomy of equinumerosi... |
| fin41 10380 | Under countable choice, th... |
| dominf 10381 | A nonempty set that is a s... |
| dcomex 10383 | The Axiom of Dependent Cho... |
| axdc2lem 10384 | Lemma for ~ axdc2 . We co... |
| axdc2 10385 | An apparent strengthening ... |
| axdc3lem 10386 | The class ` S ` of finite ... |
| axdc3lem2 10387 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10388 | Simple substitution lemma ... |
| axdc3lem4 10389 | Lemma for ~ axdc3 . We ha... |
| axdc3 10390 | Dependent Choice. Axiom D... |
| axdc4lem 10391 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10392 | A more general version of ... |
| axcclem 10393 | Lemma for ~ axcc . (Contr... |
| axcc 10394 | Although CC can be proven ... |
| zfac 10396 | Axiom of Choice expressed ... |
| ac2 10397 | Axiom of Choice equivalent... |
| ac3 10398 | Axiom of Choice using abbr... |
| axac3 10400 | This theorem asserts that ... |
| ackm 10401 | A remarkable equivalent to... |
| axac2 10402 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10403 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10404 | Apply a choice equivalent.... |
| cardeqv 10405 | All sets are well-orderabl... |
| numth3 10406 | All sets are well-orderabl... |
| numth2 10407 | Numeration theorem: any se... |
| numth 10408 | Numeration theorem: every ... |
| ac7 10409 | An Axiom of Choice equival... |
| ac7g 10410 | An Axiom of Choice equival... |
| ac4 10411 | Equivalent of Axiom of Cho... |
| ac4c 10412 | Equivalent of Axiom of Cho... |
| ac5 10413 | An Axiom of Choice equival... |
| ac5b 10414 | Equivalent of Axiom of Cho... |
| ac6num 10415 | A version of ~ ac6 which t... |
| ac6 10416 | Equivalent of Axiom of Cho... |
| ac6c4 10417 | Equivalent of Axiom of Cho... |
| ac6c5 10418 | Equivalent of Axiom of Cho... |
| ac9 10419 | An Axiom of Choice equival... |
| ac6s 10420 | Equivalent of Axiom of Cho... |
| ac6n 10421 | Equivalent of Axiom of Cho... |
| ac6s2 10422 | Generalization of the Axio... |
| ac6s3 10423 | Generalization of the Axio... |
| ac6sg 10424 | ~ ac6s with sethood as ant... |
| ac6sf 10425 | Version of ~ ac6 with boun... |
| ac6s4 10426 | Generalization of the Axio... |
| ac6s5 10427 | Generalization of the Axio... |
| ac8 10428 | An Axiom of Choice equival... |
| ac9s 10429 | An Axiom of Choice equival... |
| numthcor 10430 | Any set is strictly domina... |
| weth 10431 | Well-ordering theorem: any... |
| zorn2lem1 10432 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10433 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10434 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10435 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10436 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10437 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10438 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10439 | Zorn's Lemma of [Monk1] p.... |
| zorng 10440 | Zorn's Lemma. If the unio... |
| zornn0g 10441 | Variant of Zorn's lemma ~ ... |
| zorn2 10442 | Zorn's Lemma of [Monk1] p.... |
| zorn 10443 | Zorn's Lemma. If the unio... |
| zornn0 10444 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10445 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10446 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10447 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10448 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10449 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10450 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10451 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10452 | The Teichmüller-Tukey... |
| ttukeyg 10453 | The Teichmüller-Tukey... |
| ttukey 10454 | The Teichmüller-Tukey... |
| axdclem 10455 | Lemma for ~ axdc . (Contr... |
| axdclem2 10456 | Lemma for ~ axdc . Using ... |
| axdc 10457 | This theorem derives ~ ax-... |
| fodomg 10458 | An onto function implies d... |
| fodom 10459 | An onto function implies d... |
| dmct 10460 | The domain of a countable ... |
| rnct 10461 | The range of a countable s... |
| fodomb 10462 | Equivalence of an onto map... |
| wdomac 10463 | When assuming AC, weak and... |
| brdom3 10464 | Equivalence to a dominance... |
| brdom5 10465 | An equivalence to a domina... |
| brdom4 10466 | An equivalence to a domina... |
| brdom7disj 10467 | An equivalence to a domina... |
| brdom6disj 10468 | An equivalence to a domina... |
| fin71ac 10469 | Once we allow AC, the "str... |
| imadomg 10470 | An image of a function und... |
| fimact 10471 | The image by a function of... |
| fnrndomg 10472 | The range of a function is... |
| fnct 10473 | If the domain of a functio... |
| mptct 10474 | A countable mapping set is... |
| iunfo 10475 | Existence of an onto funct... |
| iundom2g 10476 | An upper bound for the car... |
| iundomg 10477 | An upper bound for the car... |
| iundom 10478 | An upper bound for the car... |
| unidom 10479 | An upper bound for the car... |
| uniimadom 10480 | An upper bound for the car... |
| uniimadomf 10481 | An upper bound for the car... |
| cardval 10482 | The value of the cardinal ... |
| cardid 10483 | Any set is equinumerous to... |
| cardidg 10484 | Any set is equinumerous to... |
| cardidd 10485 | Any set is equinumerous to... |
| cardf 10486 | The cardinality function i... |
| carden 10487 | Two sets are equinumerous ... |
| cardeq0 10488 | Only the empty set has car... |
| unsnen 10489 | Equinumerosity of a set wi... |
| carddom 10490 | Two sets have the dominanc... |
| cardsdom 10491 | Two sets have the strict d... |
| domtri 10492 | Trichotomy law for dominan... |
| entric 10493 | Trichotomy of equinumerosi... |
| entri2 10494 | Trichotomy of dominance an... |
| entri3 10495 | Trichotomy of dominance. ... |
| sdomsdomcard 10496 | A set strictly dominates i... |
| canth3 10497 | Cantor's theorem in terms ... |
| infxpidm 10498 | Every infinite class is eq... |
| ondomon 10499 | The class of ordinals domi... |
| cardmin 10500 | The smallest ordinal that ... |
| ficard 10501 | A set is finite iff its ca... |
| infinf 10502 | Equivalence between two in... |
| unirnfdomd 10503 | The union of the range of ... |
| konigthlem 10504 | Lemma for ~ konigth . (Co... |
| konigth 10505 | Konig's Theorem. If ` m (... |
| alephsucpw 10506 | The power set of an aleph ... |
| aleph1 10507 | The set exponentiation of ... |
| alephval2 10508 | An alternate way to expres... |
| dominfac 10509 | A nonempty set that is a s... |
| iunctb 10510 | The countable union of cou... |
| unictb 10511 | The countable union of cou... |
| infmap 10512 | An exponentiation law for ... |
| alephadd 10513 | The sum of two alephs is t... |
| alephmul 10514 | The product of two alephs ... |
| alephexp1 10515 | An exponentiation law for ... |
| alephsuc3 10516 | An alternate representatio... |
| alephexp2 10517 | An expression equinumerous... |
| alephreg 10518 | A successor aleph is regul... |
| pwcfsdom 10519 | A corollary of Konig's The... |
| cfpwsdom 10520 | A corollary of Konig's The... |
| alephom 10521 | From ~ canth2 , we know th... |
| smobeth 10522 | The beth function is stric... |
| nd1 10523 | A lemma for proving condit... |
| nd2 10524 | A lemma for proving condit... |
| nd3 10525 | A lemma for proving condit... |
| nd4 10526 | A lemma for proving condit... |
| axextnd 10527 | A version of the Axiom of ... |
| axrepndlem1 10528 | Lemma for the Axiom of Rep... |
| axrepndlem2 10529 | Lemma for the Axiom of Rep... |
| axrepnd 10530 | A version of the Axiom of ... |
| axunndlem1 10531 | Lemma for the Axiom of Uni... |
| axunnd 10532 | A version of the Axiom of ... |
| axpowndlem1 10533 | Lemma for the Axiom of Pow... |
| axpowndlem2 10534 | Lemma for the Axiom of Pow... |
| axpowndlem3 10535 | Lemma for the Axiom of Pow... |
| axpowndlem4 10536 | Lemma for the Axiom of Pow... |
| axpownd 10537 | A version of the Axiom of ... |
| axregndlem1 10538 | Lemma for the Axiom of Reg... |
| axregndlem2 10539 | Lemma for the Axiom of Reg... |
| axregnd 10540 | A version of the Axiom of ... |
| axinfndlem1 10541 | Lemma for the Axiom of Inf... |
| axinfnd 10542 | A version of the Axiom of ... |
| axacndlem1 10543 | Lemma for the Axiom of Cho... |
| axacndlem2 10544 | Lemma for the Axiom of Cho... |
| axacndlem3 10545 | Lemma for the Axiom of Cho... |
| axacndlem4 10546 | Lemma for the Axiom of Cho... |
| axacndlem5 10547 | Lemma for the Axiom of Cho... |
| axacnd 10548 | A version of the Axiom of ... |
| zfcndext 10549 | Axiom of Extensionality ~ ... |
| zfcndrep 10550 | Axiom of Replacement ~ ax-... |
| zfcndun 10551 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10552 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10553 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10554 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10555 | Axiom of Choice ~ ax-ac , ... |
| elgch 10558 | Elementhood in the collect... |
| fingch 10559 | A finite set is a GCH-set.... |
| gchi 10560 | The only GCH-sets which ha... |
| gchen1 10561 | If ` A <_ B < ~P A ` , and... |
| gchen2 10562 | If ` A < B <_ ~P A ` , and... |
| gchor 10563 | If ` A <_ B <_ ~P A ` , an... |
| engch 10564 | The property of being a GC... |
| gchdomtri 10565 | Under certain conditions, ... |
| fpwwe2cbv 10566 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10567 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10568 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10569 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10570 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10571 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10572 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10573 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10574 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10575 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10576 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10577 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10578 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10579 | Given any function ` F ` f... |
| fpwwecbv 10580 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10581 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10582 | Given any function ` F ` f... |
| canth4 10583 | An "effective" form of Can... |
| canthnumlem 10584 | Lemma for ~ canthnum . (C... |
| canthnum 10585 | The set of well-orderable ... |
| canthwelem 10586 | Lemma for ~ canthwe . (Co... |
| canthwe 10587 | The set of well-orders of ... |
| canthp1lem1 10588 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10589 | Lemma for ~ canthp1 . (Co... |
| canthp1 10590 | A slightly stronger form o... |
| finngch 10591 | The exclusion of finite se... |
| gchdju1 10592 | An infinite GCH-set is ide... |
| gchinf 10593 | An infinite GCH-set is Ded... |
| pwfseqlem1 10594 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10595 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10596 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10597 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10598 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10599 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10600 | The powerset of a Dedekind... |
| pwxpndom2 10601 | The powerset of a Dedekind... |
| pwxpndom 10602 | The powerset of a Dedekind... |
| pwdjundom 10603 | The powerset of a Dedekind... |
| gchdjuidm 10604 | An infinite GCH-set is ide... |
| gchxpidm 10605 | An infinite GCH-set is ide... |
| gchpwdom 10606 | A relationship between dom... |
| gchaleph 10607 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10608 | If ` ( aleph `` A ) ` and ... |
| hargch 10609 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10610 | If ` ( aleph `` suc A ) ` ... |
| gch2 10611 | It is sufficient to requir... |
| gch3 10612 | An equivalent formulation ... |
| gch-kn 10613 | The equivalence of two ver... |
| gchaclem 10614 | Lemma for ~ gchac (obsolet... |
| gchhar 10615 | A "local" form of ~ gchac ... |
| gchacg 10616 | A "local" form of ~ gchac ... |
| gchac 10617 | The Generalized Continuum ... |
| elwina 10622 | Conditions of weak inacces... |
| elina 10623 | Conditions of strong inacc... |
| winaon 10624 | A weakly inaccessible card... |
| inawinalem 10625 | Lemma for ~ inawina . (Co... |
| inawina 10626 | Every strongly inaccessibl... |
| omina 10627 | ` _om ` is a strongly inac... |
| winacard 10628 | A weakly inaccessible card... |
| winainflem 10629 | A weakly inaccessible card... |
| winainf 10630 | A weakly inaccessible card... |
| winalim 10631 | A weakly inaccessible card... |
| winalim2 10632 | A nontrivial weakly inacce... |
| winafp 10633 | A nontrivial weakly inacce... |
| winafpi 10634 | This theorem, which states... |
| gchina 10635 | Assuming the GCH, weakly a... |
| iswun 10640 | Properties of a weak unive... |
| wuntr 10641 | A weak universe is transit... |
| wununi 10642 | A weak universe is closed ... |
| wunpw 10643 | A weak universe is closed ... |
| wunelss 10644 | The elements of a weak uni... |
| wunpr 10645 | A weak universe is closed ... |
| wunun 10646 | A weak universe is closed ... |
| wuntp 10647 | A weak universe is closed ... |
| wunss 10648 | A weak universe is closed ... |
| wunin 10649 | A weak universe is closed ... |
| wundif 10650 | A weak universe is closed ... |
| wunint 10651 | A weak universe is closed ... |
| wunsn 10652 | A weak universe is closed ... |
| wunsuc 10653 | A weak universe is closed ... |
| wun0 10654 | A weak universe contains t... |
| wunr1om 10655 | A weak universe is infinit... |
| wunom 10656 | A weak universe contains a... |
| wunfi 10657 | A weak universe contains a... |
| wunop 10658 | A weak universe is closed ... |
| wunot 10659 | A weak universe is closed ... |
| wunxp 10660 | A weak universe is closed ... |
| wunpm 10661 | A weak universe is closed ... |
| wunmap 10662 | A weak universe is closed ... |
| wunf 10663 | A weak universe is closed ... |
| wundm 10664 | A weak universe is closed ... |
| wunrn 10665 | A weak universe is closed ... |
| wuncnv 10666 | A weak universe is closed ... |
| wunres 10667 | A weak universe is closed ... |
| wunfv 10668 | A weak universe is closed ... |
| wunco 10669 | A weak universe is closed ... |
| wuntpos 10670 | A weak universe is closed ... |
| intwun 10671 | The intersection of a coll... |
| r1limwun 10672 | Each limit stage in the cu... |
| r1wunlim 10673 | The weak universes in the ... |
| wunex2 10674 | Construct a weak universe ... |
| wunex 10675 | Construct a weak universe ... |
| uniwun 10676 | Every set is contained in ... |
| wunex3 10677 | Construct a weak universe ... |
| wuncval 10678 | Value of the weak universe... |
| wuncid 10679 | The weak universe closure ... |
| wunccl 10680 | The weak universe closure ... |
| wuncss 10681 | The weak universe closure ... |
| wuncidm 10682 | The weak universe closure ... |
| wuncval2 10683 | Our earlier expression for... |
| eltskg 10686 | Properties of a Tarski cla... |
| eltsk2g 10687 | Properties of a Tarski cla... |
| tskpwss 10688 | First axiom of a Tarski cl... |
| tskpw 10689 | Second axiom of a Tarski c... |
| tsken 10690 | Third axiom of a Tarski cl... |
| 0tsk 10691 | The empty set is a (transi... |
| tsksdom 10692 | An element of a Tarski cla... |
| tskssel 10693 | A part of a Tarski class s... |
| tskss 10694 | The subsets of an element ... |
| tskin 10695 | The intersection of two el... |
| tsksn 10696 | A singleton of an element ... |
| tsktrss 10697 | A transitive element of a ... |
| tsksuc 10698 | If an element of a Tarski ... |
| tsk0 10699 | A nonempty Tarski class co... |
| tsk1 10700 | One is an element of a non... |
| tsk2 10701 | Two is an element of a non... |
| 2domtsk 10702 | If a Tarski class is not e... |
| tskr1om 10703 | A nonempty Tarski class is... |
| tskr1om2 10704 | A nonempty Tarski class co... |
| tskinf 10705 | A nonempty Tarski class is... |
| tskpr 10706 | If ` A ` and ` B ` are mem... |
| tskop 10707 | If ` A ` and ` B ` are mem... |
| tskxpss 10708 | A Cartesian product of two... |
| tskwe2 10709 | A Tarski class is well-ord... |
| inttsk 10710 | The intersection of a coll... |
| inar1 10711 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10712 | Alternate proof of ~ r1om ... |
| rankcf 10713 | Any set must be at least a... |
| inatsk 10714 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10715 | The set of hereditarily fi... |
| tskord 10716 | A Tarski class contains al... |
| tskcard 10717 | An even more direct relati... |
| r1tskina 10718 | There is a direct relation... |
| tskuni 10719 | The union of an element of... |
| tskwun 10720 | A nonempty transitive Tars... |
| tskint 10721 | The intersection of an ele... |
| tskun 10722 | The union of two elements ... |
| tskxp 10723 | The Cartesian product of t... |
| tskmap 10724 | Set exponentiation is an e... |
| tskurn 10725 | A transitive Tarski class ... |
| elgrug 10728 | Properties of a Grothendie... |
| grutr 10729 | A Grothendieck universe is... |
| gruelss 10730 | A Grothendieck universe is... |
| grupw 10731 | A Grothendieck universe co... |
| gruss 10732 | Any subset of an element o... |
| grupr 10733 | A Grothendieck universe co... |
| gruurn 10734 | A Grothendieck universe co... |
| gruiun 10735 | If ` B ( x ) ` is a family... |
| gruuni 10736 | A Grothendieck universe co... |
| grurn 10737 | A Grothendieck universe co... |
| gruima 10738 | A Grothendieck universe co... |
| gruel 10739 | Any element of an element ... |
| grusn 10740 | A Grothendieck universe co... |
| gruop 10741 | A Grothendieck universe co... |
| gruun 10742 | A Grothendieck universe co... |
| gruxp 10743 | A Grothendieck universe co... |
| grumap 10744 | A Grothendieck universe co... |
| gruixp 10745 | A Grothendieck universe co... |
| gruiin 10746 | A Grothendieck universe co... |
| gruf 10747 | A Grothendieck universe co... |
| gruen 10748 | A Grothendieck universe co... |
| gruwun 10749 | A nonempty Grothendieck un... |
| intgru 10750 | The intersection of a fami... |
| ingru 10751 | The intersection of a univ... |
| wfgru 10752 | The wellfounded part of a ... |
| grudomon 10753 | Each ordinal that is compa... |
| gruina 10754 | If a Grothendieck universe... |
| grur1a 10755 | A characterization of Grot... |
| grur1 10756 | A characterization of Grot... |
| grutsk1 10757 | Grothendieck universes are... |
| grutsk 10758 | Grothendieck universes are... |
| axgroth5 10760 | The Tarski-Grothendieck ax... |
| axgroth2 10761 | Alternate version of the T... |
| grothpw 10762 | Derive the Axiom of Power ... |
| grothpwex 10763 | Derive the Axiom of Power ... |
| axgroth6 10764 | The Tarski-Grothendieck ax... |
| grothomex 10765 | The Tarski-Grothendieck Ax... |
| grothac 10766 | The Tarski-Grothendieck Ax... |
| axgroth3 10767 | Alternate version of the T... |
| axgroth4 10768 | Alternate version of the T... |
| grothprimlem 10769 | Lemma for ~ grothprim . E... |
| grothprim 10770 | The Tarski-Grothendieck Ax... |
| grothtsk 10771 | The Tarski-Grothendieck Ax... |
| inaprc 10772 | An equivalent to the Tarsk... |
| tskmval 10775 | Value of our tarski map. ... |
| tskmid 10776 | The set ` A ` is an elemen... |
| tskmcl 10777 | A Tarski class that contai... |
| sstskm 10778 | Being a part of ` ( tarski... |
| eltskm 10779 | Belonging to ` ( tarskiMap... |
| elni 10812 | Membership in the class of... |
| elni2 10813 | Membership in the class of... |
| pinn 10814 | A positive integer is a na... |
| pion 10815 | A positive integer is an o... |
| piord 10816 | A positive integer is ordi... |
| niex 10817 | The class of positive inte... |
| 0npi 10818 | The empty set is not a pos... |
| 1pi 10819 | Ordinal 'one' is a positiv... |
| addpiord 10820 | Positive integer addition ... |
| mulpiord 10821 | Positive integer multiplic... |
| mulidpi 10822 | 1 is an identity element f... |
| ltpiord 10823 | Positive integer 'less tha... |
| ltsopi 10824 | Positive integer 'less tha... |
| ltrelpi 10825 | Positive integer 'less tha... |
| dmaddpi 10826 | Domain of addition on posi... |
| dmmulpi 10827 | Domain of multiplication o... |
| addclpi 10828 | Closure of addition of pos... |
| mulclpi 10829 | Closure of multiplication ... |
| addcompi 10830 | Addition of positive integ... |
| addasspi 10831 | Addition of positive integ... |
| mulcompi 10832 | Multiplication of positive... |
| mulasspi 10833 | Multiplication of positive... |
| distrpi 10834 | Multiplication of positive... |
| addcanpi 10835 | Addition cancellation law ... |
| mulcanpi 10836 | Multiplication cancellatio... |
| addnidpi 10837 | There is no identity eleme... |
| ltexpi 10838 | Ordering on positive integ... |
| ltapi 10839 | Ordering property of addit... |
| ltmpi 10840 | Ordering property of multi... |
| 1lt2pi 10841 | One is less than two (one ... |
| nlt1pi 10842 | No positive integer is les... |
| indpi 10843 | Principle of Finite Induct... |
| enqbreq 10855 | Equivalence relation for p... |
| enqbreq2 10856 | Equivalence relation for p... |
| enqer 10857 | The equivalence relation f... |
| enqex 10858 | The equivalence relation f... |
| nqex 10859 | The class of positive frac... |
| 0nnq 10860 | The empty set is not a pos... |
| elpqn 10861 | Each positive fraction is ... |
| ltrelnq 10862 | Positive fraction 'less th... |
| pinq 10863 | The representatives of pos... |
| 1nq 10864 | The positive fraction 'one... |
| nqereu 10865 | There is a unique element ... |
| nqerf 10866 | Corollary of ~ nqereu : th... |
| nqercl 10867 | Corollary of ~ nqereu : cl... |
| nqerrel 10868 | Any member of ` ( N. X. N.... |
| nqerid 10869 | Corollary of ~ nqereu : th... |
| enqeq 10870 | Corollary of ~ nqereu : if... |
| nqereq 10871 | The function ` /Q ` acts a... |
| addpipq2 10872 | Addition of positive fract... |
| addpipq 10873 | Addition of positive fract... |
| addpqnq 10874 | Addition of positive fract... |
| mulpipq2 10875 | Multiplication of positive... |
| mulpipq 10876 | Multiplication of positive... |
| mulpqnq 10877 | Multiplication of positive... |
| ordpipq 10878 | Ordering of positive fract... |
| ordpinq 10879 | Ordering of positive fract... |
| addpqf 10880 | Closure of addition on pos... |
| addclnq 10881 | Closure of addition on pos... |
| mulpqf 10882 | Closure of multiplication ... |
| mulclnq 10883 | Closure of multiplication ... |
| addnqf 10884 | Domain of addition on posi... |
| mulnqf 10885 | Domain of multiplication o... |
| addcompq 10886 | Addition of positive fract... |
| addcomnq 10887 | Addition of positive fract... |
| mulcompq 10888 | Multiplication of positive... |
| mulcomnq 10889 | Multiplication of positive... |
| adderpqlem 10890 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10891 | Lemma for ~ mulerpq . (Co... |
| adderpq 10892 | Addition is compatible wit... |
| mulerpq 10893 | Multiplication is compatib... |
| addassnq 10894 | Addition of positive fract... |
| mulassnq 10895 | Multiplication of positive... |
| mulcanenq 10896 | Lemma for distributive law... |
| distrnq 10897 | Multiplication of positive... |
| 1nqenq 10898 | The equivalence class of r... |
| mulidnq 10899 | Multiplication identity el... |
| recmulnq 10900 | Relationship between recip... |
| recidnq 10901 | A positive fraction times ... |
| recclnq 10902 | Closure law for positive f... |
| recrecnq 10903 | Reciprocal of reciprocal o... |
| dmrecnq 10904 | Domain of reciprocal on po... |
| ltsonq 10905 | 'Less than' is a strict or... |
| lterpq 10906 | Compatibility of ordering ... |
| ltanq 10907 | Ordering property of addit... |
| ltmnq 10908 | Ordering property of multi... |
| 1lt2nq 10909 | One is less than two (one ... |
| ltaddnq 10910 | The sum of two fractions i... |
| ltexnq 10911 | Ordering on positive fract... |
| halfnq 10912 | One-half of any positive f... |
| nsmallnq 10913 | The is no smallest positiv... |
| ltbtwnnq 10914 | There exists a number betw... |
| ltrnq 10915 | Ordering property of recip... |
| archnq 10916 | For any fraction, there is... |
| npex 10922 | The class of positive real... |
| elnp 10923 | Membership in positive rea... |
| elnpi 10924 | Membership in positive rea... |
| prn0 10925 | A positive real is not emp... |
| prpssnq 10926 | A positive real is a subse... |
| elprnq 10927 | A positive real is a set o... |
| 0npr 10928 | The empty set is not a pos... |
| prcdnq 10929 | A positive real is closed ... |
| prub 10930 | A positive fraction not in... |
| prnmax 10931 | A positive real has no lar... |
| npomex 10932 | A simplifying observation,... |
| prnmadd 10933 | A positive real has no lar... |
| ltrelpr 10934 | Positive real 'less than' ... |
| genpv 10935 | Value of general operation... |
| genpelv 10936 | Membership in value of gen... |
| genpprecl 10937 | Pre-closure law for genera... |
| genpdm 10938 | Domain of general operatio... |
| genpn0 10939 | The result of an operation... |
| genpss 10940 | The result of an operation... |
| genpnnp 10941 | The result of an operation... |
| genpcd 10942 | Downward closure of an ope... |
| genpnmax 10943 | An operation on positive r... |
| genpcl 10944 | Closure of an operation on... |
| genpass 10945 | Associativity of an operat... |
| plpv 10946 | Value of addition on posit... |
| mpv 10947 | Value of multiplication on... |
| dmplp 10948 | Domain of addition on posi... |
| dmmp 10949 | Domain of multiplication o... |
| nqpr 10950 | The canonical embedding of... |
| 1pr 10951 | The positive real number '... |
| addclprlem1 10952 | Lemma to prove downward cl... |
| addclprlem2 10953 | Lemma to prove downward cl... |
| addclpr 10954 | Closure of addition on pos... |
| mulclprlem 10955 | Lemma to prove downward cl... |
| mulclpr 10956 | Closure of multiplication ... |
| addcompr 10957 | Addition of positive reals... |
| addasspr 10958 | Addition of positive reals... |
| mulcompr 10959 | Multiplication of positive... |
| mulasspr 10960 | Multiplication of positive... |
| distrlem1pr 10961 | Lemma for distributive law... |
| distrlem4pr 10962 | Lemma for distributive law... |
| distrlem5pr 10963 | Lemma for distributive law... |
| distrpr 10964 | Multiplication of positive... |
| 1idpr 10965 | 1 is an identity element f... |
| ltprord 10966 | Positive real 'less than' ... |
| psslinpr 10967 | Proper subset is a linear ... |
| ltsopr 10968 | Positive real 'less than' ... |
| prlem934 10969 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10970 | The sum of two positive re... |
| ltaddpr2 10971 | The sum of two positive re... |
| ltexprlem1 10972 | Lemma for Proposition 9-3.... |
| ltexprlem2 10973 | Lemma for Proposition 9-3.... |
| ltexprlem3 10974 | Lemma for Proposition 9-3.... |
| ltexprlem4 10975 | Lemma for Proposition 9-3.... |
| ltexprlem5 10976 | Lemma for Proposition 9-3.... |
| ltexprlem6 10977 | Lemma for Proposition 9-3.... |
| ltexprlem7 10978 | Lemma for Proposition 9-3.... |
| ltexpri 10979 | Proposition 9-3.5(iv) of [... |
| ltaprlem 10980 | Lemma for Proposition 9-3.... |
| ltapr 10981 | Ordering property of addit... |
| addcanpr 10982 | Addition cancellation law ... |
| prlem936 10983 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 10984 | Lemma for Proposition 9-3.... |
| reclem3pr 10985 | Lemma for Proposition 9-3.... |
| reclem4pr 10986 | Lemma for Proposition 9-3.... |
| recexpr 10987 | The reciprocal of a positi... |
| suplem1pr 10988 | The union of a nonempty, b... |
| suplem2pr 10989 | The union of a set of posi... |
| supexpr 10990 | The union of a nonempty, b... |
| enrer 10999 | The equivalence relation f... |
| nrex1 11000 | The class of signed reals ... |
| enrbreq 11001 | Equivalence relation for s... |
| enreceq 11002 | Equivalence class equality... |
| enrex 11003 | The equivalence relation f... |
| ltrelsr 11004 | Signed real 'less than' is... |
| addcmpblnr 11005 | Lemma showing compatibilit... |
| mulcmpblnrlem 11006 | Lemma used in lemma showin... |
| mulcmpblnr 11007 | Lemma showing compatibilit... |
| prsrlem1 11008 | Decomposing signed reals i... |
| addsrmo 11009 | There is at most one resul... |
| mulsrmo 11010 | There is at most one resul... |
| addsrpr 11011 | Addition of signed reals i... |
| mulsrpr 11012 | Multiplication of signed r... |
| ltsrpr 11013 | Ordering of signed reals i... |
| gt0srpr 11014 | Greater than zero in terms... |
| 0nsr 11015 | The empty set is not a sig... |
| 0r 11016 | The constant ` 0R ` is a s... |
| 1sr 11017 | The constant ` 1R ` is a s... |
| m1r 11018 | The constant ` -1R ` is a ... |
| addclsr 11019 | Closure of addition on sig... |
| mulclsr 11020 | Closure of multiplication ... |
| dmaddsr 11021 | Domain of addition on sign... |
| dmmulsr 11022 | Domain of multiplication o... |
| addcomsr 11023 | Addition of signed reals i... |
| addasssr 11024 | Addition of signed reals i... |
| mulcomsr 11025 | Multiplication of signed r... |
| mulasssr 11026 | Multiplication of signed r... |
| distrsr 11027 | Multiplication of signed r... |
| m1p1sr 11028 | Minus one plus one is zero... |
| m1m1sr 11029 | Minus one times minus one ... |
| ltsosr 11030 | Signed real 'less than' is... |
| 0lt1sr 11031 | 0 is less than 1 for signe... |
| 1ne0sr 11032 | 1 and 0 are distinct for s... |
| 0idsr 11033 | The signed real number 0 i... |
| 1idsr 11034 | 1 is an identity element f... |
| 00sr 11035 | A signed real times 0 is 0... |
| ltasr 11036 | Ordering property of addit... |
| pn0sr 11037 | A signed real plus its neg... |
| negexsr 11038 | Existence of negative sign... |
| recexsrlem 11039 | The reciprocal of a positi... |
| addgt0sr 11040 | The sum of two positive si... |
| mulgt0sr 11041 | The product of two positiv... |
| sqgt0sr 11042 | The square of a nonzero si... |
| recexsr 11043 | The reciprocal of a nonzer... |
| mappsrpr 11044 | Mapping from positive sign... |
| ltpsrpr 11045 | Mapping of order from posi... |
| map2psrpr 11046 | Equivalence for positive s... |
| supsrlem 11047 | Lemma for supremum theorem... |
| supsr 11048 | A nonempty, bounded set of... |
| opelcn 11065 | Ordered pair membership in... |
| opelreal 11066 | Ordered pair membership in... |
| elreal 11067 | Membership in class of rea... |
| elreal2 11068 | Ordered pair membership in... |
| 0ncn 11069 | The empty set is not a com... |
| ltrelre 11070 | 'Less than' is a relation ... |
| addcnsr 11071 | Addition of complex number... |
| mulcnsr 11072 | Multiplication of complex ... |
| eqresr 11073 | Equality of real numbers i... |
| addresr 11074 | Addition of real numbers i... |
| mulresr 11075 | Multiplication of real num... |
| ltresr 11076 | Ordering of real subset of... |
| ltresr2 11077 | Ordering of real subset of... |
| dfcnqs 11078 | Technical trick to permit ... |
| addcnsrec 11079 | Technical trick to permit ... |
| mulcnsrec 11080 | Technical trick to permit ... |
| axaddf 11081 | Addition is an operation o... |
| axmulf 11082 | Multiplication is an opera... |
| axcnex 11083 | The complex numbers form a... |
| axresscn 11084 | The real numbers are a sub... |
| ax1cn 11085 | 1 is a complex number. Ax... |
| axicn 11086 | ` _i ` is a complex number... |
| axaddcl 11087 | Closure law for addition o... |
| axaddrcl 11088 | Closure law for addition i... |
| axmulcl 11089 | Closure law for multiplica... |
| axmulrcl 11090 | Closure law for multiplica... |
| axmulcom 11091 | Multiplication of complex ... |
| axaddass 11092 | Addition of complex number... |
| axmulass 11093 | Multiplication of complex ... |
| axdistr 11094 | Distributive law for compl... |
| axi2m1 11095 | i-squared equals -1 (expre... |
| ax1ne0 11096 | 1 and 0 are distinct. Axi... |
| ax1rid 11097 | ` 1 ` is an identity eleme... |
| axrnegex 11098 | Existence of negative of r... |
| axrrecex 11099 | Existence of reciprocal of... |
| axcnre 11100 | A complex number can be ex... |
| axpre-lttri 11101 | Ordering on reals satisfie... |
| axpre-lttrn 11102 | Ordering on reals is trans... |
| axpre-ltadd 11103 | Ordering property of addit... |
| axpre-mulgt0 11104 | The product of two positiv... |
| axpre-sup 11105 | A nonempty, bounded-above ... |
| wuncn 11106 | A weak universe containing... |
| cnex 11132 | Alias for ~ ax-cnex . See... |
| addcl 11133 | Alias for ~ ax-addcl , for... |
| readdcl 11134 | Alias for ~ ax-addrcl , fo... |
| mulcl 11135 | Alias for ~ ax-mulcl , for... |
| remulcl 11136 | Alias for ~ ax-mulrcl , fo... |
| mulcom 11137 | Alias for ~ ax-mulcom , fo... |
| addass 11138 | Alias for ~ ax-addass , fo... |
| mulass 11139 | Alias for ~ ax-mulass , fo... |
| adddi 11140 | Alias for ~ ax-distr , for... |
| recn 11141 | A real number is a complex... |
| reex 11142 | The real numbers form a se... |
| reelprrecn 11143 | Reals are a subset of the ... |
| cnelprrecn 11144 | Complex numbers are a subs... |
| elimne0 11145 | Hypothesis for weak deduct... |
| adddir 11146 | Distributive law for compl... |
| 0cn 11147 | Zero is a complex number. ... |
| 0cnd 11148 | Zero is a complex number, ... |
| c0ex 11149 | Zero is a set. (Contribut... |
| 1cnd 11150 | One is a complex number, d... |
| 1ex 11151 | One is a set. (Contribute... |
| cnre 11152 | Alias for ~ ax-cnre , for ... |
| mulid1 11153 | The number 1 is an identit... |
| mulid2 11154 | Identity law for multiplic... |
| 1re 11155 | The number 1 is real. Thi... |
| 1red 11156 | The number 1 is real, dedu... |
| 0re 11157 | The number 0 is real. Rem... |
| 0red 11158 | The number 0 is real, dedu... |
| mulid1i 11159 | Identity law for multiplic... |
| mulid2i 11160 | Identity law for multiplic... |
| addcli 11161 | Closure law for addition. ... |
| mulcli 11162 | Closure law for multiplica... |
| mulcomi 11163 | Commutative law for multip... |
| mulcomli 11164 | Commutative law for multip... |
| addassi 11165 | Associative law for additi... |
| mulassi 11166 | Associative law for multip... |
| adddii 11167 | Distributive law (left-dis... |
| adddiri 11168 | Distributive law (right-di... |
| recni 11169 | A real number is a complex... |
| readdcli 11170 | Closure law for addition o... |
| remulcli 11171 | Closure law for multiplica... |
| mulid1d 11172 | Identity law for multiplic... |
| mulid2d 11173 | Identity law for multiplic... |
| addcld 11174 | Closure law for addition. ... |
| mulcld 11175 | Closure law for multiplica... |
| mulcomd 11176 | Commutative law for multip... |
| addassd 11177 | Associative law for additi... |
| mulassd 11178 | Associative law for multip... |
| adddid 11179 | Distributive law (left-dis... |
| adddird 11180 | Distributive law (right-di... |
| adddirp1d 11181 | Distributive law, plus 1 v... |
| joinlmuladdmuld 11182 | Join AB+CB into (A+C) on L... |
| recnd 11183 | Deduction from real number... |
| readdcld 11184 | Closure law for addition o... |
| remulcld 11185 | Closure law for multiplica... |
| pnfnre 11196 | Plus infinity is not a rea... |
| pnfnre2 11197 | Plus infinity is not a rea... |
| mnfnre 11198 | Minus infinity is not a re... |
| ressxr 11199 | The standard reals are a s... |
| rexpssxrxp 11200 | The Cartesian product of s... |
| rexr 11201 | A standard real is an exte... |
| 0xr 11202 | Zero is an extended real. ... |
| renepnf 11203 | No (finite) real equals pl... |
| renemnf 11204 | No real equals minus infin... |
| rexrd 11205 | A standard real is an exte... |
| renepnfd 11206 | No (finite) real equals pl... |
| renemnfd 11207 | No real equals minus infin... |
| pnfex 11208 | Plus infinity exists. (Co... |
| pnfxr 11209 | Plus infinity belongs to t... |
| pnfnemnf 11210 | Plus and minus infinity ar... |
| mnfnepnf 11211 | Minus and plus infinity ar... |
| mnfxr 11212 | Minus infinity belongs to ... |
| rexri 11213 | A standard real is an exte... |
| 1xr 11214 | ` 1 ` is an extended real ... |
| renfdisj 11215 | The reals and the infiniti... |
| ltrelxr 11216 | "Less than" is a relation ... |
| ltrel 11217 | "Less than" is a relation.... |
| lerelxr 11218 | "Less than or equal to" is... |
| lerel 11219 | "Less than or equal to" is... |
| xrlenlt 11220 | "Less than or equal to" ex... |
| xrlenltd 11221 | "Less than or equal to" ex... |
| xrltnle 11222 | "Less than" expressed in t... |
| xrnltled 11223 | "Not less than" implies "l... |
| ssxr 11224 | The three (non-exclusive) ... |
| ltxrlt 11225 | The standard less-than ` <... |
| axlttri 11226 | Ordering on reals satisfie... |
| axlttrn 11227 | Ordering on reals is trans... |
| axltadd 11228 | Ordering property of addit... |
| axmulgt0 11229 | The product of two positiv... |
| axsup 11230 | A nonempty, bounded-above ... |
| lttr 11231 | Alias for ~ axlttrn , for ... |
| mulgt0 11232 | The product of two positiv... |
| lenlt 11233 | 'Less than or equal to' ex... |
| ltnle 11234 | 'Less than' expressed in t... |
| ltso 11235 | 'Less than' is a strict or... |
| gtso 11236 | 'Greater than' is a strict... |
| lttri2 11237 | Consequence of trichotomy.... |
| lttri3 11238 | Trichotomy law for 'less t... |
| lttri4 11239 | Trichotomy law for 'less t... |
| letri3 11240 | Trichotomy law. (Contribu... |
| leloe 11241 | 'Less than or equal to' ex... |
| eqlelt 11242 | Equality in terms of 'less... |
| ltle 11243 | 'Less than' implies 'less ... |
| leltne 11244 | 'Less than or equal to' im... |
| lelttr 11245 | Transitive law. (Contribu... |
| leltletr 11246 | Transitive law, weaker for... |
| ltletr 11247 | Transitive law. (Contribu... |
| ltleletr 11248 | Transitive law, weaker for... |
| letr 11249 | Transitive law. (Contribu... |
| ltnr 11250 | 'Less than' is irreflexive... |
| leid 11251 | 'Less than or equal to' is... |
| ltne 11252 | 'Less than' implies not eq... |
| ltnsym 11253 | 'Less than' is not symmetr... |
| ltnsym2 11254 | 'Less than' is antisymmetr... |
| letric 11255 | Trichotomy law. (Contribu... |
| ltlen 11256 | 'Less than' expressed in t... |
| eqle 11257 | Equality implies 'less tha... |
| eqled 11258 | Equality implies 'less tha... |
| ltadd2 11259 | Addition to both sides of ... |
| ne0gt0 11260 | A nonzero nonnegative numb... |
| lecasei 11261 | Ordering elimination by ca... |
| lelttric 11262 | Trichotomy law. (Contribu... |
| ltlecasei 11263 | Ordering elimination by ca... |
| ltnri 11264 | 'Less than' is irreflexive... |
| eqlei 11265 | Equality implies 'less tha... |
| eqlei2 11266 | Equality implies 'less tha... |
| gtneii 11267 | 'Less than' implies not eq... |
| ltneii 11268 | 'Greater than' implies not... |
| lttri2i 11269 | Consequence of trichotomy.... |
| lttri3i 11270 | Consequence of trichotomy.... |
| letri3i 11271 | Consequence of trichotomy.... |
| leloei 11272 | 'Less than or equal to' in... |
| ltleni 11273 | 'Less than' expressed in t... |
| ltnsymi 11274 | 'Less than' is not symmetr... |
| lenlti 11275 | 'Less than or equal to' in... |
| ltnlei 11276 | 'Less than' in terms of 'l... |
| ltlei 11277 | 'Less than' implies 'less ... |
| ltleii 11278 | 'Less than' implies 'less ... |
| ltnei 11279 | 'Less than' implies not eq... |
| letrii 11280 | Trichotomy law for 'less t... |
| lttri 11281 | 'Less than' is transitive.... |
| lelttri 11282 | 'Less than or equal to', '... |
| ltletri 11283 | 'Less than', 'less than or... |
| letri 11284 | 'Less than or equal to' is... |
| le2tri3i 11285 | Extended trichotomy law fo... |
| ltadd2i 11286 | Addition to both sides of ... |
| mulgt0i 11287 | The product of two positiv... |
| mulgt0ii 11288 | The product of two positiv... |
| ltnrd 11289 | 'Less than' is irreflexive... |
| gtned 11290 | 'Less than' implies not eq... |
| ltned 11291 | 'Greater than' implies not... |
| ne0gt0d 11292 | A nonzero nonnegative numb... |
| lttrid 11293 | Ordering on reals satisfie... |
| lttri2d 11294 | Consequence of trichotomy.... |
| lttri3d 11295 | Consequence of trichotomy.... |
| lttri4d 11296 | Trichotomy law for 'less t... |
| letri3d 11297 | Consequence of trichotomy.... |
| leloed 11298 | 'Less than or equal to' in... |
| eqleltd 11299 | Equality in terms of 'less... |
| ltlend 11300 | 'Less than' expressed in t... |
| lenltd 11301 | 'Less than or equal to' in... |
| ltnled 11302 | 'Less than' in terms of 'l... |
| ltled 11303 | 'Less than' implies 'less ... |
| ltnsymd 11304 | 'Less than' implies 'less ... |
| nltled 11305 | 'Not less than ' implies '... |
| lensymd 11306 | 'Less than or equal to' im... |
| letrid 11307 | Trichotomy law for 'less t... |
| leltned 11308 | 'Less than or equal to' im... |
| leneltd 11309 | 'Less than or equal to' an... |
| mulgt0d 11310 | The product of two positiv... |
| ltadd2d 11311 | Addition to both sides of ... |
| letrd 11312 | Transitive law deduction f... |
| lelttrd 11313 | Transitive law deduction f... |
| ltadd2dd 11314 | Addition to both sides of ... |
| ltletrd 11315 | Transitive law deduction f... |
| lttrd 11316 | Transitive law deduction f... |
| lelttrdi 11317 | If a number is less than a... |
| dedekind 11318 | The Dedekind cut theorem. ... |
| dedekindle 11319 | The Dedekind cut theorem, ... |
| mul12 11320 | Commutative/associative la... |
| mul32 11321 | Commutative/associative la... |
| mul31 11322 | Commutative/associative la... |
| mul4 11323 | Rearrangement of 4 factors... |
| mul4r 11324 | Rearrangement of 4 factors... |
| muladd11 11325 | A simple product of sums e... |
| 1p1times 11326 | Two times a number. (Cont... |
| peano2cn 11327 | A theorem for complex numb... |
| peano2re 11328 | A theorem for reals analog... |
| readdcan 11329 | Cancellation law for addit... |
| 00id 11330 | ` 0 ` is its own additive ... |
| mul02lem1 11331 | Lemma for ~ mul02 . If an... |
| mul02lem2 11332 | Lemma for ~ mul02 . Zero ... |
| mul02 11333 | Multiplication by ` 0 ` . ... |
| mul01 11334 | Multiplication by ` 0 ` . ... |
| addid1 11335 | ` 0 ` is an additive ident... |
| cnegex 11336 | Existence of the negative ... |
| cnegex2 11337 | Existence of a left invers... |
| addid2 11338 | ` 0 ` is a left identity f... |
| addcan 11339 | Cancellation law for addit... |
| addcan2 11340 | Cancellation law for addit... |
| addcom 11341 | Addition commutes. This u... |
| addid1i 11342 | ` 0 ` is an additive ident... |
| addid2i 11343 | ` 0 ` is a left identity f... |
| mul02i 11344 | Multiplication by 0. Theo... |
| mul01i 11345 | Multiplication by ` 0 ` . ... |
| addcomi 11346 | Addition commutes. Based ... |
| addcomli 11347 | Addition commutes. (Contr... |
| addcani 11348 | Cancellation law for addit... |
| addcan2i 11349 | Cancellation law for addit... |
| mul12i 11350 | Commutative/associative la... |
| mul32i 11351 | Commutative/associative la... |
| mul4i 11352 | Rearrangement of 4 factors... |
| mul02d 11353 | Multiplication by 0. Theo... |
| mul01d 11354 | Multiplication by ` 0 ` . ... |
| addid1d 11355 | ` 0 ` is an additive ident... |
| addid2d 11356 | ` 0 ` is a left identity f... |
| addcomd 11357 | Addition commutes. Based ... |
| addcand 11358 | Cancellation law for addit... |
| addcan2d 11359 | Cancellation law for addit... |
| addcanad 11360 | Cancelling a term on the l... |
| addcan2ad 11361 | Cancelling a term on the r... |
| addneintrd 11362 | Introducing a term on the ... |
| addneintr2d 11363 | Introducing a term on the ... |
| mul12d 11364 | Commutative/associative la... |
| mul32d 11365 | Commutative/associative la... |
| mul31d 11366 | Commutative/associative la... |
| mul4d 11367 | Rearrangement of 4 factors... |
| muladd11r 11368 | A simple product of sums e... |
| comraddd 11369 | Commute RHS addition, in d... |
| ltaddneg 11370 | Adding a negative number t... |
| ltaddnegr 11371 | Adding a negative number t... |
| add12 11372 | Commutative/associative la... |
| add32 11373 | Commutative/associative la... |
| add32r 11374 | Commutative/associative la... |
| add4 11375 | Rearrangement of 4 terms i... |
| add42 11376 | Rearrangement of 4 terms i... |
| add12i 11377 | Commutative/associative la... |
| add32i 11378 | Commutative/associative la... |
| add4i 11379 | Rearrangement of 4 terms i... |
| add42i 11380 | Rearrangement of 4 terms i... |
| add12d 11381 | Commutative/associative la... |
| add32d 11382 | Commutative/associative la... |
| add4d 11383 | Rearrangement of 4 terms i... |
| add42d 11384 | Rearrangement of 4 terms i... |
| 0cnALT 11389 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11390 | Alternate proof of ~ 0cnAL... |
| negeu 11391 | Existential uniqueness of ... |
| subval 11392 | Value of subtraction, whic... |
| negeq 11393 | Equality theorem for negat... |
| negeqi 11394 | Equality inference for neg... |
| negeqd 11395 | Equality deduction for neg... |
| nfnegd 11396 | Deduction version of ~ nfn... |
| nfneg 11397 | Bound-variable hypothesis ... |
| csbnegg 11398 | Move class substitution in... |
| negex 11399 | A negative is a set. (Con... |
| subcl 11400 | Closure law for subtractio... |
| negcl 11401 | Closure law for negative. ... |
| negicn 11402 | ` -u _i ` is a complex num... |
| subf 11403 | Subtraction is an operatio... |
| subadd 11404 | Relationship between subtr... |
| subadd2 11405 | Relationship between subtr... |
| subsub23 11406 | Swap subtrahend and result... |
| pncan 11407 | Cancellation law for subtr... |
| pncan2 11408 | Cancellation law for subtr... |
| pncan3 11409 | Subtraction and addition o... |
| npcan 11410 | Cancellation law for subtr... |
| addsubass 11411 | Associative-type law for a... |
| addsub 11412 | Law for addition and subtr... |
| subadd23 11413 | Commutative/associative la... |
| addsub12 11414 | Commutative/associative la... |
| 2addsub 11415 | Law for subtraction and ad... |
| addsubeq4 11416 | Relation between sums and ... |
| pncan3oi 11417 | Subtraction and addition o... |
| mvrraddi 11418 | Move the right term in a s... |
| mvlladdi 11419 | Move the left term in a su... |
| subid 11420 | Subtraction of a number fr... |
| subid1 11421 | Identity law for subtracti... |
| npncan 11422 | Cancellation law for subtr... |
| nppcan 11423 | Cancellation law for subtr... |
| nnpcan 11424 | Cancellation law for subtr... |
| nppcan3 11425 | Cancellation law for subtr... |
| subcan2 11426 | Cancellation law for subtr... |
| subeq0 11427 | If the difference between ... |
| npncan2 11428 | Cancellation law for subtr... |
| subsub2 11429 | Law for double subtraction... |
| nncan 11430 | Cancellation law for subtr... |
| subsub 11431 | Law for double subtraction... |
| nppcan2 11432 | Cancellation law for subtr... |
| subsub3 11433 | Law for double subtraction... |
| subsub4 11434 | Law for double subtraction... |
| sub32 11435 | Swap the second and third ... |
| nnncan 11436 | Cancellation law for subtr... |
| nnncan1 11437 | Cancellation law for subtr... |
| nnncan2 11438 | Cancellation law for subtr... |
| npncan3 11439 | Cancellation law for subtr... |
| pnpcan 11440 | Cancellation law for mixed... |
| pnpcan2 11441 | Cancellation law for mixed... |
| pnncan 11442 | Cancellation law for mixed... |
| ppncan 11443 | Cancellation law for mixed... |
| addsub4 11444 | Rearrangement of 4 terms i... |
| subadd4 11445 | Rearrangement of 4 terms i... |
| sub4 11446 | Rearrangement of 4 terms i... |
| neg0 11447 | Minus 0 equals 0. (Contri... |
| negid 11448 | Addition of a number and i... |
| negsub 11449 | Relationship between subtr... |
| subneg 11450 | Relationship between subtr... |
| negneg 11451 | A number is equal to the n... |
| neg11 11452 | Negative is one-to-one. (... |
| negcon1 11453 | Negative contraposition la... |
| negcon2 11454 | Negative contraposition la... |
| negeq0 11455 | A number is zero iff its n... |
| subcan 11456 | Cancellation law for subtr... |
| negsubdi 11457 | Distribution of negative o... |
| negdi 11458 | Distribution of negative o... |
| negdi2 11459 | Distribution of negative o... |
| negsubdi2 11460 | Distribution of negative o... |
| neg2sub 11461 | Relationship between subtr... |
| renegcli 11462 | Closure law for negative o... |
| resubcli 11463 | Closure law for subtractio... |
| renegcl 11464 | Closure law for negative o... |
| resubcl 11465 | Closure law for subtractio... |
| negreb 11466 | The negative of a real is ... |
| peano2cnm 11467 | "Reverse" second Peano pos... |
| peano2rem 11468 | "Reverse" second Peano pos... |
| negcli 11469 | Closure law for negative. ... |
| negidi 11470 | Addition of a number and i... |
| negnegi 11471 | A number is equal to the n... |
| subidi 11472 | Subtraction of a number fr... |
| subid1i 11473 | Identity law for subtracti... |
| negne0bi 11474 | A number is nonzero iff it... |
| negrebi 11475 | The negative of a real is ... |
| negne0i 11476 | The negative of a nonzero ... |
| subcli 11477 | Closure law for subtractio... |
| pncan3i 11478 | Subtraction and addition o... |
| negsubi 11479 | Relationship between subtr... |
| subnegi 11480 | Relationship between subtr... |
| subeq0i 11481 | If the difference between ... |
| neg11i 11482 | Negative is one-to-one. (... |
| negcon1i 11483 | Negative contraposition la... |
| negcon2i 11484 | Negative contraposition la... |
| negdii 11485 | Distribution of negative o... |
| negsubdii 11486 | Distribution of negative o... |
| negsubdi2i 11487 | Distribution of negative o... |
| subaddi 11488 | Relationship between subtr... |
| subadd2i 11489 | Relationship between subtr... |
| subaddrii 11490 | Relationship between subtr... |
| subsub23i 11491 | Swap subtrahend and result... |
| addsubassi 11492 | Associative-type law for s... |
| addsubi 11493 | Law for subtraction and ad... |
| subcani 11494 | Cancellation law for subtr... |
| subcan2i 11495 | Cancellation law for subtr... |
| pnncani 11496 | Cancellation law for mixed... |
| addsub4i 11497 | Rearrangement of 4 terms i... |
| 0reALT 11498 | Alternate proof of ~ 0re .... |
| negcld 11499 | Closure law for negative. ... |
| subidd 11500 | Subtraction of a number fr... |
| subid1d 11501 | Identity law for subtracti... |
| negidd 11502 | Addition of a number and i... |
| negnegd 11503 | A number is equal to the n... |
| negeq0d 11504 | A number is zero iff its n... |
| negne0bd 11505 | A number is nonzero iff it... |
| negcon1d 11506 | Contraposition law for una... |
| negcon1ad 11507 | Contraposition law for una... |
| neg11ad 11508 | The negatives of two compl... |
| negned 11509 | If two complex numbers are... |
| negne0d 11510 | The negative of a nonzero ... |
| negrebd 11511 | The negative of a real is ... |
| subcld 11512 | Closure law for subtractio... |
| pncand 11513 | Cancellation law for subtr... |
| pncan2d 11514 | Cancellation law for subtr... |
| pncan3d 11515 | Subtraction and addition o... |
| npcand 11516 | Cancellation law for subtr... |
| nncand 11517 | Cancellation law for subtr... |
| negsubd 11518 | Relationship between subtr... |
| subnegd 11519 | Relationship between subtr... |
| subeq0d 11520 | If the difference between ... |
| subne0d 11521 | Two unequal numbers have n... |
| subeq0ad 11522 | The difference of two comp... |
| subne0ad 11523 | If the difference of two c... |
| neg11d 11524 | If the difference between ... |
| negdid 11525 | Distribution of negative o... |
| negdi2d 11526 | Distribution of negative o... |
| negsubdid 11527 | Distribution of negative o... |
| negsubdi2d 11528 | Distribution of negative o... |
| neg2subd 11529 | Relationship between subtr... |
| subaddd 11530 | Relationship between subtr... |
| subadd2d 11531 | Relationship between subtr... |
| addsubassd 11532 | Associative-type law for s... |
| addsubd 11533 | Law for subtraction and ad... |
| subadd23d 11534 | Commutative/associative la... |
| addsub12d 11535 | Commutative/associative la... |
| npncand 11536 | Cancellation law for subtr... |
| nppcand 11537 | Cancellation law for subtr... |
| nppcan2d 11538 | Cancellation law for subtr... |
| nppcan3d 11539 | Cancellation law for subtr... |
| subsubd 11540 | Law for double subtraction... |
| subsub2d 11541 | Law for double subtraction... |
| subsub3d 11542 | Law for double subtraction... |
| subsub4d 11543 | Law for double subtraction... |
| sub32d 11544 | Swap the second and third ... |
| nnncand 11545 | Cancellation law for subtr... |
| nnncan1d 11546 | Cancellation law for subtr... |
| nnncan2d 11547 | Cancellation law for subtr... |
| npncan3d 11548 | Cancellation law for subtr... |
| pnpcand 11549 | Cancellation law for mixed... |
| pnpcan2d 11550 | Cancellation law for mixed... |
| pnncand 11551 | Cancellation law for mixed... |
| ppncand 11552 | Cancellation law for mixed... |
| subcand 11553 | Cancellation law for subtr... |
| subcan2d 11554 | Cancellation law for subtr... |
| subcanad 11555 | Cancellation law for subtr... |
| subneintrd 11556 | Introducing subtraction on... |
| subcan2ad 11557 | Cancellation law for subtr... |
| subneintr2d 11558 | Introducing subtraction on... |
| addsub4d 11559 | Rearrangement of 4 terms i... |
| subadd4d 11560 | Rearrangement of 4 terms i... |
| sub4d 11561 | Rearrangement of 4 terms i... |
| 2addsubd 11562 | Law for subtraction and ad... |
| addsubeq4d 11563 | Relation between sums and ... |
| subeqxfrd 11564 | Transfer two terms of a su... |
| mvlraddd 11565 | Move the right term in a s... |
| mvlladdd 11566 | Move the left term in a su... |
| mvrraddd 11567 | Move the right term in a s... |
| mvrladdd 11568 | Move the left term in a su... |
| assraddsubd 11569 | Associate RHS addition-sub... |
| subaddeqd 11570 | Transfer two terms of a su... |
| addlsub 11571 | Left-subtraction: Subtrac... |
| addrsub 11572 | Right-subtraction: Subtra... |
| subexsub 11573 | A subtraction law: Exchan... |
| addid0 11574 | If adding a number to a an... |
| addn0nid 11575 | Adding a nonzero number to... |
| pnpncand 11576 | Addition/subtraction cance... |
| subeqrev 11577 | Reverse the order of subtr... |
| addeq0 11578 | Two complex numbers add up... |
| pncan1 11579 | Cancellation law for addit... |
| npcan1 11580 | Cancellation law for subtr... |
| subeq0bd 11581 | If two complex numbers are... |
| renegcld 11582 | Closure law for negative o... |
| resubcld 11583 | Closure law for subtractio... |
| negn0 11584 | The image under negation o... |
| negf1o 11585 | Negation is an isomorphism... |
| kcnktkm1cn 11586 | k times k minus 1 is a com... |
| muladd 11587 | Product of two sums. (Con... |
| subdi 11588 | Distribution of multiplica... |
| subdir 11589 | Distribution of multiplica... |
| ine0 11590 | The imaginary unit ` _i ` ... |
| mulneg1 11591 | Product with negative is n... |
| mulneg2 11592 | The product with a negativ... |
| mulneg12 11593 | Swap the negative sign in ... |
| mul2neg 11594 | Product of two negatives. ... |
| submul2 11595 | Convert a subtraction to a... |
| mulm1 11596 | Product with minus one is ... |
| addneg1mul 11597 | Addition with product with... |
| mulsub 11598 | Product of two differences... |
| mulsub2 11599 | Swap the order of subtract... |
| mulm1i 11600 | Product with minus one is ... |
| mulneg1i 11601 | Product with negative is n... |
| mulneg2i 11602 | Product with negative is n... |
| mul2negi 11603 | Product of two negatives. ... |
| subdii 11604 | Distribution of multiplica... |
| subdiri 11605 | Distribution of multiplica... |
| muladdi 11606 | Product of two sums. (Con... |
| mulm1d 11607 | Product with minus one is ... |
| mulneg1d 11608 | Product with negative is n... |
| mulneg2d 11609 | Product with negative is n... |
| mul2negd 11610 | Product of two negatives. ... |
| subdid 11611 | Distribution of multiplica... |
| subdird 11612 | Distribution of multiplica... |
| muladdd 11613 | Product of two sums. (Con... |
| mulsubd 11614 | Product of two differences... |
| muls1d 11615 | Multiplication by one minu... |
| mulsubfacd 11616 | Multiplication followed by... |
| addmulsub 11617 | The product of a sum and a... |
| subaddmulsub 11618 | The difference with a prod... |
| mulsubaddmulsub 11619 | A special difference of a ... |
| gt0ne0 11620 | Positive implies nonzero. ... |
| lt0ne0 11621 | A number which is less tha... |
| ltadd1 11622 | Addition to both sides of ... |
| leadd1 11623 | Addition to both sides of ... |
| leadd2 11624 | Addition to both sides of ... |
| ltsubadd 11625 | 'Less than' relationship b... |
| ltsubadd2 11626 | 'Less than' relationship b... |
| lesubadd 11627 | 'Less than or equal to' re... |
| lesubadd2 11628 | 'Less than or equal to' re... |
| ltaddsub 11629 | 'Less than' relationship b... |
| ltaddsub2 11630 | 'Less than' relationship b... |
| leaddsub 11631 | 'Less than or equal to' re... |
| leaddsub2 11632 | 'Less than or equal to' re... |
| suble 11633 | Swap subtrahends in an ine... |
| lesub 11634 | Swap subtrahends in an ine... |
| ltsub23 11635 | 'Less than' relationship b... |
| ltsub13 11636 | 'Less than' relationship b... |
| le2add 11637 | Adding both sides of two '... |
| ltleadd 11638 | Adding both sides of two o... |
| leltadd 11639 | Adding both sides of two o... |
| lt2add 11640 | Adding both sides of two '... |
| addgt0 11641 | The sum of 2 positive numb... |
| addgegt0 11642 | The sum of nonnegative and... |
| addgtge0 11643 | The sum of nonnegative and... |
| addge0 11644 | The sum of 2 nonnegative n... |
| ltaddpos 11645 | Adding a positive number t... |
| ltaddpos2 11646 | Adding a positive number t... |
| ltsubpos 11647 | Subtracting a positive num... |
| posdif 11648 | Comparison of two numbers ... |
| lesub1 11649 | Subtraction from both side... |
| lesub2 11650 | Subtraction of both sides ... |
| ltsub1 11651 | Subtraction from both side... |
| ltsub2 11652 | Subtraction of both sides ... |
| lt2sub 11653 | Subtracting both sides of ... |
| le2sub 11654 | Subtracting both sides of ... |
| ltneg 11655 | Negative of both sides of ... |
| ltnegcon1 11656 | Contraposition of negative... |
| ltnegcon2 11657 | Contraposition of negative... |
| leneg 11658 | Negative of both sides of ... |
| lenegcon1 11659 | Contraposition of negative... |
| lenegcon2 11660 | Contraposition of negative... |
| lt0neg1 11661 | Comparison of a number and... |
| lt0neg2 11662 | Comparison of a number and... |
| le0neg1 11663 | Comparison of a number and... |
| le0neg2 11664 | Comparison of a number and... |
| addge01 11665 | A number is less than or e... |
| addge02 11666 | A number is less than or e... |
| add20 11667 | Two nonnegative numbers ar... |
| subge0 11668 | Nonnegative subtraction. ... |
| suble0 11669 | Nonpositive subtraction. ... |
| leaddle0 11670 | The sum of a real number a... |
| subge02 11671 | Nonnegative subtraction. ... |
| lesub0 11672 | Lemma to show a nonnegativ... |
| mulge0 11673 | The product of two nonnega... |
| mullt0 11674 | The product of two negativ... |
| msqgt0 11675 | A nonzero square is positi... |
| msqge0 11676 | A square is nonnegative. ... |
| 0lt1 11677 | 0 is less than 1. Theorem... |
| 0le1 11678 | 0 is less than or equal to... |
| relin01 11679 | An interval law for less t... |
| ltordlem 11680 | Lemma for ~ ltord1 . (Con... |
| ltord1 11681 | Infer an ordering relation... |
| leord1 11682 | Infer an ordering relation... |
| eqord1 11683 | A strictly increasing real... |
| ltord2 11684 | Infer an ordering relation... |
| leord2 11685 | Infer an ordering relation... |
| eqord2 11686 | A strictly decreasing real... |
| wloglei 11687 | Form of ~ wlogle where bot... |
| wlogle 11688 | If the predicate ` ch ( x ... |
| leidi 11689 | 'Less than or equal to' is... |
| gt0ne0i 11690 | Positive means nonzero (us... |
| gt0ne0ii 11691 | Positive implies nonzero. ... |
| msqgt0i 11692 | A nonzero square is positi... |
| msqge0i 11693 | A square is nonnegative. ... |
| addgt0i 11694 | Addition of 2 positive num... |
| addge0i 11695 | Addition of 2 nonnegative ... |
| addgegt0i 11696 | Addition of nonnegative an... |
| addgt0ii 11697 | Addition of 2 positive num... |
| add20i 11698 | Two nonnegative numbers ar... |
| ltnegi 11699 | Negative of both sides of ... |
| lenegi 11700 | Negative of both sides of ... |
| ltnegcon2i 11701 | Contraposition of negative... |
| mulge0i 11702 | The product of two nonnega... |
| lesub0i 11703 | Lemma to show a nonnegativ... |
| ltaddposi 11704 | Adding a positive number t... |
| posdifi 11705 | Comparison of two numbers ... |
| ltnegcon1i 11706 | Contraposition of negative... |
| lenegcon1i 11707 | Contraposition of negative... |
| subge0i 11708 | Nonnegative subtraction. ... |
| ltadd1i 11709 | Addition to both sides of ... |
| leadd1i 11710 | Addition to both sides of ... |
| leadd2i 11711 | Addition to both sides of ... |
| ltsubaddi 11712 | 'Less than' relationship b... |
| lesubaddi 11713 | 'Less than or equal to' re... |
| ltsubadd2i 11714 | 'Less than' relationship b... |
| lesubadd2i 11715 | 'Less than or equal to' re... |
| ltaddsubi 11716 | 'Less than' relationship b... |
| lt2addi 11717 | Adding both side of two in... |
| le2addi 11718 | Adding both side of two in... |
| gt0ne0d 11719 | Positive implies nonzero. ... |
| lt0ne0d 11720 | Something less than zero i... |
| leidd 11721 | 'Less than or equal to' is... |
| msqgt0d 11722 | A nonzero square is positi... |
| msqge0d 11723 | A square is nonnegative. ... |
| lt0neg1d 11724 | Comparison of a number and... |
| lt0neg2d 11725 | Comparison of a number and... |
| le0neg1d 11726 | Comparison of a number and... |
| le0neg2d 11727 | Comparison of a number and... |
| addgegt0d 11728 | Addition of nonnegative an... |
| addgtge0d 11729 | Addition of positive and n... |
| addgt0d 11730 | Addition of 2 positive num... |
| addge0d 11731 | Addition of 2 nonnegative ... |
| mulge0d 11732 | The product of two nonnega... |
| ltnegd 11733 | Negative of both sides of ... |
| lenegd 11734 | Negative of both sides of ... |
| ltnegcon1d 11735 | Contraposition of negative... |
| ltnegcon2d 11736 | Contraposition of negative... |
| lenegcon1d 11737 | Contraposition of negative... |
| lenegcon2d 11738 | Contraposition of negative... |
| ltaddposd 11739 | Adding a positive number t... |
| ltaddpos2d 11740 | Adding a positive number t... |
| ltsubposd 11741 | Subtracting a positive num... |
| posdifd 11742 | Comparison of two numbers ... |
| addge01d 11743 | A number is less than or e... |
| addge02d 11744 | A number is less than or e... |
| subge0d 11745 | Nonnegative subtraction. ... |
| suble0d 11746 | Nonpositive subtraction. ... |
| subge02d 11747 | Nonnegative subtraction. ... |
| ltadd1d 11748 | Addition to both sides of ... |
| leadd1d 11749 | Addition to both sides of ... |
| leadd2d 11750 | Addition to both sides of ... |
| ltsubaddd 11751 | 'Less than' relationship b... |
| lesubaddd 11752 | 'Less than or equal to' re... |
| ltsubadd2d 11753 | 'Less than' relationship b... |
| lesubadd2d 11754 | 'Less than or equal to' re... |
| ltaddsubd 11755 | 'Less than' relationship b... |
| ltaddsub2d 11756 | 'Less than' relationship b... |
| leaddsub2d 11757 | 'Less than or equal to' re... |
| subled 11758 | Swap subtrahends in an ine... |
| lesubd 11759 | Swap subtrahends in an ine... |
| ltsub23d 11760 | 'Less than' relationship b... |
| ltsub13d 11761 | 'Less than' relationship b... |
| lesub1d 11762 | Subtraction from both side... |
| lesub2d 11763 | Subtraction of both sides ... |
| ltsub1d 11764 | Subtraction from both side... |
| ltsub2d 11765 | Subtraction of both sides ... |
| ltadd1dd 11766 | Addition to both sides of ... |
| ltsub1dd 11767 | Subtraction from both side... |
| ltsub2dd 11768 | Subtraction of both sides ... |
| leadd1dd 11769 | Addition to both sides of ... |
| leadd2dd 11770 | Addition to both sides of ... |
| lesub1dd 11771 | Subtraction from both side... |
| lesub2dd 11772 | Subtraction of both sides ... |
| lesub3d 11773 | The result of subtracting ... |
| le2addd 11774 | Adding both side of two in... |
| le2subd 11775 | Subtracting both sides of ... |
| ltleaddd 11776 | Adding both sides of two o... |
| leltaddd 11777 | Adding both sides of two o... |
| lt2addd 11778 | Adding both side of two in... |
| lt2subd 11779 | Subtracting both sides of ... |
| possumd 11780 | Condition for a positive s... |
| sublt0d 11781 | When a subtraction gives a... |
| ltaddsublt 11782 | Addition and subtraction o... |
| 1le1 11783 | One is less than or equal ... |
| ixi 11784 | ` _i ` times itself is min... |
| recextlem1 11785 | Lemma for ~ recex . (Cont... |
| recextlem2 11786 | Lemma for ~ recex . (Cont... |
| recex 11787 | Existence of reciprocal of... |
| mulcand 11788 | Cancellation law for multi... |
| mulcan2d 11789 | Cancellation law for multi... |
| mulcanad 11790 | Cancellation of a nonzero ... |
| mulcan2ad 11791 | Cancellation of a nonzero ... |
| mulcan 11792 | Cancellation law for multi... |
| mulcan2 11793 | Cancellation law for multi... |
| mulcani 11794 | Cancellation law for multi... |
| mul0or 11795 | If a product is zero, one ... |
| mulne0b 11796 | The product of two nonzero... |
| mulne0 11797 | The product of two nonzero... |
| mulne0i 11798 | The product of two nonzero... |
| muleqadd 11799 | Property of numbers whose ... |
| receu 11800 | Existential uniqueness of ... |
| mulnzcnopr 11801 | Multiplication maps nonzer... |
| msq0i 11802 | A number is zero iff its s... |
| mul0ori 11803 | If a product is zero, one ... |
| msq0d 11804 | A number is zero iff its s... |
| mul0ord 11805 | If a product is zero, one ... |
| mulne0bd 11806 | The product of two nonzero... |
| mulne0d 11807 | The product of two nonzero... |
| mulcan1g 11808 | A generalized form of the ... |
| mulcan2g 11809 | A generalized form of the ... |
| mulne0bad 11810 | A factor of a nonzero comp... |
| mulne0bbd 11811 | A factor of a nonzero comp... |
| 1div0 11814 | You can't divide by zero, ... |
| divval 11815 | Value of division: if ` A ... |
| divmul 11816 | Relationship between divis... |
| divmul2 11817 | Relationship between divis... |
| divmul3 11818 | Relationship between divis... |
| divcl 11819 | Closure law for division. ... |
| reccl 11820 | Closure law for reciprocal... |
| divcan2 11821 | A cancellation law for div... |
| divcan1 11822 | A cancellation law for div... |
| diveq0 11823 | A ratio is zero iff the nu... |
| divne0b 11824 | The ratio of nonzero numbe... |
| divne0 11825 | The ratio of nonzero numbe... |
| recne0 11826 | The reciprocal of a nonzer... |
| recid 11827 | Multiplication of a number... |
| recid2 11828 | Multiplication of a number... |
| divrec 11829 | Relationship between divis... |
| divrec2 11830 | Relationship between divis... |
| divass 11831 | An associative law for div... |
| div23 11832 | A commutative/associative ... |
| div32 11833 | A commutative/associative ... |
| div13 11834 | A commutative/associative ... |
| div12 11835 | A commutative/associative ... |
| divmulass 11836 | An associative law for div... |
| divmulasscom 11837 | An associative/commutative... |
| divdir 11838 | Distribution of division o... |
| divcan3 11839 | A cancellation law for div... |
| divcan4 11840 | A cancellation law for div... |
| div11 11841 | One-to-one relationship fo... |
| divid 11842 | A number divided by itself... |
| div0 11843 | Division into zero is zero... |
| div1 11844 | A number divided by 1 is i... |
| 1div1e1 11845 | 1 divided by 1 is 1. (Con... |
| diveq1 11846 | Equality in terms of unit ... |
| divneg 11847 | Move negative sign inside ... |
| muldivdir 11848 | Distribution of division o... |
| divsubdir 11849 | Distribution of division o... |
| subdivcomb1 11850 | Bring a term in a subtract... |
| subdivcomb2 11851 | Bring a term in a subtract... |
| recrec 11852 | A number is equal to the r... |
| rec11 11853 | Reciprocal is one-to-one. ... |
| rec11r 11854 | Mutual reciprocals. (Cont... |
| divmuldiv 11855 | Multiplication of two rati... |
| divdivdiv 11856 | Division of two ratios. T... |
| divcan5 11857 | Cancellation of common fac... |
| divmul13 11858 | Swap the denominators in t... |
| divmul24 11859 | Swap the numerators in the... |
| divmuleq 11860 | Cross-multiply in an equal... |
| recdiv 11861 | The reciprocal of a ratio.... |
| divcan6 11862 | Cancellation of inverted f... |
| divdiv32 11863 | Swap denominators in a div... |
| divcan7 11864 | Cancel equal divisors in a... |
| dmdcan 11865 | Cancellation law for divis... |
| divdiv1 11866 | Division into a fraction. ... |
| divdiv2 11867 | Division by a fraction. (... |
| recdiv2 11868 | Division into a reciprocal... |
| ddcan 11869 | Cancellation in a double d... |
| divadddiv 11870 | Addition of two ratios. T... |
| divsubdiv 11871 | Subtraction of two ratios.... |
| conjmul 11872 | Two numbers whose reciproc... |
| rereccl 11873 | Closure law for reciprocal... |
| redivcl 11874 | Closure law for division o... |
| eqneg 11875 | A number equal to its nega... |
| eqnegd 11876 | A complex number equals it... |
| eqnegad 11877 | If a complex number equals... |
| div2neg 11878 | Quotient of two negatives.... |
| divneg2 11879 | Move negative sign inside ... |
| recclzi 11880 | Closure law for reciprocal... |
| recne0zi 11881 | The reciprocal of a nonzer... |
| recidzi 11882 | Multiplication of a number... |
| div1i 11883 | A number divided by 1 is i... |
| eqnegi 11884 | A number equal to its nega... |
| reccli 11885 | Closure law for reciprocal... |
| recidi 11886 | Multiplication of a number... |
| recreci 11887 | A number is equal to the r... |
| dividi 11888 | A number divided by itself... |
| div0i 11889 | Division into zero is zero... |
| divclzi 11890 | Closure law for division. ... |
| divcan1zi 11891 | A cancellation law for div... |
| divcan2zi 11892 | A cancellation law for div... |
| divreczi 11893 | Relationship between divis... |
| divcan3zi 11894 | A cancellation law for div... |
| divcan4zi 11895 | A cancellation law for div... |
| rec11i 11896 | Reciprocal is one-to-one. ... |
| divcli 11897 | Closure law for division. ... |
| divcan2i 11898 | A cancellation law for div... |
| divcan1i 11899 | A cancellation law for div... |
| divreci 11900 | Relationship between divis... |
| divcan3i 11901 | A cancellation law for div... |
| divcan4i 11902 | A cancellation law for div... |
| divne0i 11903 | The ratio of nonzero numbe... |
| rec11ii 11904 | Reciprocal is one-to-one. ... |
| divasszi 11905 | An associative law for div... |
| divmulzi 11906 | Relationship between divis... |
| divdirzi 11907 | Distribution of division o... |
| divdiv23zi 11908 | Swap denominators in a div... |
| divmuli 11909 | Relationship between divis... |
| divdiv32i 11910 | Swap denominators in a div... |
| divassi 11911 | An associative law for div... |
| divdiri 11912 | Distribution of division o... |
| div23i 11913 | A commutative/associative ... |
| div11i 11914 | One-to-one relationship fo... |
| divmuldivi 11915 | Multiplication of two rati... |
| divmul13i 11916 | Swap denominators of two r... |
| divadddivi 11917 | Addition of two ratios. T... |
| divdivdivi 11918 | Division of two ratios. T... |
| rerecclzi 11919 | Closure law for reciprocal... |
| rereccli 11920 | Closure law for reciprocal... |
| redivclzi 11921 | Closure law for division o... |
| redivcli 11922 | Closure law for division o... |
| div1d 11923 | A number divided by 1 is i... |
| reccld 11924 | Closure law for reciprocal... |
| recne0d 11925 | The reciprocal of a nonzer... |
| recidd 11926 | Multiplication of a number... |
| recid2d 11927 | Multiplication of a number... |
| recrecd 11928 | A number is equal to the r... |
| dividd 11929 | A number divided by itself... |
| div0d 11930 | Division into zero is zero... |
| divcld 11931 | Closure law for division. ... |
| divcan1d 11932 | A cancellation law for div... |
| divcan2d 11933 | A cancellation law for div... |
| divrecd 11934 | Relationship between divis... |
| divrec2d 11935 | Relationship between divis... |
| divcan3d 11936 | A cancellation law for div... |
| divcan4d 11937 | A cancellation law for div... |
| diveq0d 11938 | A ratio is zero iff the nu... |
| diveq1d 11939 | Equality in terms of unit ... |
| diveq1ad 11940 | The quotient of two comple... |
| diveq0ad 11941 | A fraction of complex numb... |
| divne1d 11942 | If two complex numbers are... |
| divne0bd 11943 | A ratio is zero iff the nu... |
| divnegd 11944 | Move negative sign inside ... |
| divneg2d 11945 | Move negative sign inside ... |
| div2negd 11946 | Quotient of two negatives.... |
| divne0d 11947 | The ratio of nonzero numbe... |
| recdivd 11948 | The reciprocal of a ratio.... |
| recdiv2d 11949 | Division into a reciprocal... |
| divcan6d 11950 | Cancellation of inverted f... |
| ddcand 11951 | Cancellation in a double d... |
| rec11d 11952 | Reciprocal is one-to-one. ... |
| divmuld 11953 | Relationship between divis... |
| div32d 11954 | A commutative/associative ... |
| div13d 11955 | A commutative/associative ... |
| divdiv32d 11956 | Swap denominators in a div... |
| divcan5d 11957 | Cancellation of common fac... |
| divcan5rd 11958 | Cancellation of common fac... |
| divcan7d 11959 | Cancel equal divisors in a... |
| dmdcand 11960 | Cancellation law for divis... |
| dmdcan2d 11961 | Cancellation law for divis... |
| divdiv1d 11962 | Division into a fraction. ... |
| divdiv2d 11963 | Division by a fraction. (... |
| divmul2d 11964 | Relationship between divis... |
| divmul3d 11965 | Relationship between divis... |
| divassd 11966 | An associative law for div... |
| div12d 11967 | A commutative/associative ... |
| div23d 11968 | A commutative/associative ... |
| divdird 11969 | Distribution of division o... |
| divsubdird 11970 | Distribution of division o... |
| div11d 11971 | One-to-one relationship fo... |
| divmuldivd 11972 | Multiplication of two rati... |
| divmul13d 11973 | Swap denominators of two r... |
| divmul24d 11974 | Swap the numerators in the... |
| divadddivd 11975 | Addition of two ratios. T... |
| divsubdivd 11976 | Subtraction of two ratios.... |
| divmuleqd 11977 | Cross-multiply in an equal... |
| divdivdivd 11978 | Division of two ratios. T... |
| diveq1bd 11979 | If two complex numbers are... |
| div2sub 11980 | Swap the order of subtract... |
| div2subd 11981 | Swap subtrahend and minuen... |
| rereccld 11982 | Closure law for reciprocal... |
| redivcld 11983 | Closure law for division o... |
| subrec 11984 | Subtraction of reciprocals... |
| subreci 11985 | Subtraction of reciprocals... |
| subrecd 11986 | Subtraction of reciprocals... |
| mvllmuld 11987 | Move the left term in a pr... |
| mvllmuli 11988 | Move the left term in a pr... |
| ldiv 11989 | Left-division. (Contribut... |
| rdiv 11990 | Right-division. (Contribu... |
| mdiv 11991 | A division law. (Contribu... |
| lineq 11992 | Solution of a (scalar) lin... |
| elimgt0 11993 | Hypothesis for weak deduct... |
| elimge0 11994 | Hypothesis for weak deduct... |
| ltp1 11995 | A number is less than itse... |
| lep1 11996 | A number is less than or e... |
| ltm1 11997 | A number minus 1 is less t... |
| lem1 11998 | A number minus 1 is less t... |
| letrp1 11999 | A transitive property of '... |
| p1le 12000 | A transitive property of p... |
| recgt0 12001 | The reciprocal of a positi... |
| prodgt0 12002 | Infer that a multiplicand ... |
| prodgt02 12003 | Infer that a multiplier is... |
| ltmul1a 12004 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 12005 | Multiplication of both sid... |
| ltmul2 12006 | Multiplication of both sid... |
| lemul1 12007 | Multiplication of both sid... |
| lemul2 12008 | Multiplication of both sid... |
| lemul1a 12009 | Multiplication of both sid... |
| lemul2a 12010 | Multiplication of both sid... |
| ltmul12a 12011 | Comparison of product of t... |
| lemul12b 12012 | Comparison of product of t... |
| lemul12a 12013 | Comparison of product of t... |
| mulgt1 12014 | The product of two numbers... |
| ltmulgt11 12015 | Multiplication by a number... |
| ltmulgt12 12016 | Multiplication by a number... |
| lemulge11 12017 | Multiplication by a number... |
| lemulge12 12018 | Multiplication by a number... |
| ltdiv1 12019 | Division of both sides of ... |
| lediv1 12020 | Division of both sides of ... |
| gt0div 12021 | Division of a positive num... |
| ge0div 12022 | Division of a nonnegative ... |
| divgt0 12023 | The ratio of two positive ... |
| divge0 12024 | The ratio of nonnegative a... |
| mulge0b 12025 | A condition for multiplica... |
| mulle0b 12026 | A condition for multiplica... |
| mulsuble0b 12027 | A condition for multiplica... |
| ltmuldiv 12028 | 'Less than' relationship b... |
| ltmuldiv2 12029 | 'Less than' relationship b... |
| ltdivmul 12030 | 'Less than' relationship b... |
| ledivmul 12031 | 'Less than or equal to' re... |
| ltdivmul2 12032 | 'Less than' relationship b... |
| lt2mul2div 12033 | 'Less than' relationship b... |
| ledivmul2 12034 | 'Less than or equal to' re... |
| lemuldiv 12035 | 'Less than or equal' relat... |
| lemuldiv2 12036 | 'Less than or equal' relat... |
| ltrec 12037 | The reciprocal of both sid... |
| lerec 12038 | The reciprocal of both sid... |
| lt2msq1 12039 | Lemma for ~ lt2msq . (Con... |
| lt2msq 12040 | Two nonnegative numbers co... |
| ltdiv2 12041 | Division of a positive num... |
| ltrec1 12042 | Reciprocal swap in a 'less... |
| lerec2 12043 | Reciprocal swap in a 'less... |
| ledivdiv 12044 | Invert ratios of positive ... |
| lediv2 12045 | Division of a positive num... |
| ltdiv23 12046 | Swap denominator with othe... |
| lediv23 12047 | Swap denominator with othe... |
| lediv12a 12048 | Comparison of ratio of two... |
| lediv2a 12049 | Division of both sides of ... |
| reclt1 12050 | The reciprocal of a positi... |
| recgt1 12051 | The reciprocal of a positi... |
| recgt1i 12052 | The reciprocal of a number... |
| recp1lt1 12053 | Construct a number less th... |
| recreclt 12054 | Given a positive number ` ... |
| le2msq 12055 | The square function on non... |
| msq11 12056 | The square of a nonnegativ... |
| ledivp1 12057 | "Less than or equal to" an... |
| squeeze0 12058 | If a nonnegative number is... |
| ltp1i 12059 | A number is less than itse... |
| recgt0i 12060 | The reciprocal of a positi... |
| recgt0ii 12061 | The reciprocal of a positi... |
| prodgt0i 12062 | Infer that a multiplicand ... |
| divgt0i 12063 | The ratio of two positive ... |
| divge0i 12064 | The ratio of nonnegative a... |
| ltreci 12065 | The reciprocal of both sid... |
| lereci 12066 | The reciprocal of both sid... |
| lt2msqi 12067 | The square function on non... |
| le2msqi 12068 | The square function on non... |
| msq11i 12069 | The square of a nonnegativ... |
| divgt0i2i 12070 | The ratio of two positive ... |
| ltrecii 12071 | The reciprocal of both sid... |
| divgt0ii 12072 | The ratio of two positive ... |
| ltmul1i 12073 | Multiplication of both sid... |
| ltdiv1i 12074 | Division of both sides of ... |
| ltmuldivi 12075 | 'Less than' relationship b... |
| ltmul2i 12076 | Multiplication of both sid... |
| lemul1i 12077 | Multiplication of both sid... |
| lemul2i 12078 | Multiplication of both sid... |
| ltdiv23i 12079 | Swap denominator with othe... |
| ledivp1i 12080 | "Less than or equal to" an... |
| ltdivp1i 12081 | Less-than and division rel... |
| ltdiv23ii 12082 | Swap denominator with othe... |
| ltmul1ii 12083 | Multiplication of both sid... |
| ltdiv1ii 12084 | Division of both sides of ... |
| ltp1d 12085 | A number is less than itse... |
| lep1d 12086 | A number is less than or e... |
| ltm1d 12087 | A number minus 1 is less t... |
| lem1d 12088 | A number minus 1 is less t... |
| recgt0d 12089 | The reciprocal of a positi... |
| divgt0d 12090 | The ratio of two positive ... |
| mulgt1d 12091 | The product of two numbers... |
| lemulge11d 12092 | Multiplication by a number... |
| lemulge12d 12093 | Multiplication by a number... |
| lemul1ad 12094 | Multiplication of both sid... |
| lemul2ad 12095 | Multiplication of both sid... |
| ltmul12ad 12096 | Comparison of product of t... |
| lemul12ad 12097 | Comparison of product of t... |
| lemul12bd 12098 | Comparison of product of t... |
| fimaxre 12099 | A finite set of real numbe... |
| fimaxre2 12100 | A nonempty finite set of r... |
| fimaxre3 12101 | A nonempty finite set of r... |
| fiminre 12102 | A nonempty finite set of r... |
| fiminre2 12103 | A nonempty finite set of r... |
| negfi 12104 | The negation of a finite s... |
| lbreu 12105 | If a set of reals contains... |
| lbcl 12106 | If a set of reals contains... |
| lble 12107 | If a set of reals contains... |
| lbinf 12108 | If a set of reals contains... |
| lbinfcl 12109 | If a set of reals contains... |
| lbinfle 12110 | If a set of reals contains... |
| sup2 12111 | A nonempty, bounded-above ... |
| sup3 12112 | A version of the completen... |
| infm3lem 12113 | Lemma for ~ infm3 . (Cont... |
| infm3 12114 | The completeness axiom for... |
| suprcl 12115 | Closure of supremum of a n... |
| suprub 12116 | A member of a nonempty bou... |
| suprubd 12117 | Natural deduction form of ... |
| suprcld 12118 | Natural deduction form of ... |
| suprlub 12119 | The supremum of a nonempty... |
| suprnub 12120 | An upper bound is not less... |
| suprleub 12121 | The supremum of a nonempty... |
| supaddc 12122 | The supremum function dist... |
| supadd 12123 | The supremum function dist... |
| supmul1 12124 | The supremum function dist... |
| supmullem1 12125 | Lemma for ~ supmul . (Con... |
| supmullem2 12126 | Lemma for ~ supmul . (Con... |
| supmul 12127 | The supremum function dist... |
| sup3ii 12128 | A version of the completen... |
| suprclii 12129 | Closure of supremum of a n... |
| suprubii 12130 | A member of a nonempty bou... |
| suprlubii 12131 | The supremum of a nonempty... |
| suprnubii 12132 | An upper bound is not less... |
| suprleubii 12133 | The supremum of a nonempty... |
| riotaneg 12134 | The negative of the unique... |
| negiso 12135 | Negation is an order anti-... |
| dfinfre 12136 | The infimum of a set of re... |
| infrecl 12137 | Closure of infimum of a no... |
| infrenegsup 12138 | The infimum of a set of re... |
| infregelb 12139 | Any lower bound of a nonem... |
| infrelb 12140 | If a nonempty set of real ... |
| infrefilb 12141 | The infimum of a finite se... |
| supfirege 12142 | The supremum of a finite s... |
| inelr 12143 | The imaginary unit ` _i ` ... |
| rimul 12144 | A real number times the im... |
| cru 12145 | The representation of comp... |
| crne0 12146 | The real representation of... |
| creur 12147 | The real part of a complex... |
| creui 12148 | The imaginary part of a co... |
| cju 12149 | The complex conjugate of a... |
| ofsubeq0 12150 | Function analogue of ~ sub... |
| ofnegsub 12151 | Function analogue of ~ neg... |
| ofsubge0 12152 | Function analogue of ~ sub... |
| nnexALT 12155 | Alternate proof of ~ nnex ... |
| peano5nni 12156 | Peano's inductive postulat... |
| nnssre 12157 | The positive integers are ... |
| nnsscn 12158 | The positive integers are ... |
| nnex 12159 | The set of positive intege... |
| nnre 12160 | A positive integer is a re... |
| nncn 12161 | A positive integer is a co... |
| nnrei 12162 | A positive integer is a re... |
| nncni 12163 | A positive integer is a co... |
| 1nn 12164 | Peano postulate: 1 is a po... |
| peano2nn 12165 | Peano postulate: a success... |
| dfnn2 12166 | Alternate definition of th... |
| dfnn3 12167 | Alternate definition of th... |
| nnred 12168 | A positive integer is a re... |
| nncnd 12169 | A positive integer is a co... |
| peano2nnd 12170 | Peano postulate: a success... |
| nnind 12171 | Principle of Mathematical ... |
| nnindALT 12172 | Principle of Mathematical ... |
| nnindd 12173 | Principle of Mathematical ... |
| nn1m1nn 12174 | Every positive integer is ... |
| nn1suc 12175 | If a statement holds for 1... |
| nnaddcl 12176 | Closure of addition of pos... |
| nnmulcl 12177 | Closure of multiplication ... |
| nnmulcli 12178 | Closure of multiplication ... |
| nnmtmip 12179 | "Minus times minus is plus... |
| nn2ge 12180 | There exists a positive in... |
| nnge1 12181 | A positive integer is one ... |
| nngt1ne1 12182 | A positive integer is grea... |
| nnle1eq1 12183 | A positive integer is less... |
| nngt0 12184 | A positive integer is posi... |
| nnnlt1 12185 | A positive integer is not ... |
| nnnle0 12186 | A positive integer is not ... |
| nnne0 12187 | A positive integer is nonz... |
| nnneneg 12188 | No positive integer is equ... |
| 0nnn 12189 | Zero is not a positive int... |
| 0nnnALT 12190 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 12191 | Alternate version of ~ nnn... |
| nngt0i 12192 | A positive integer is posi... |
| nnne0i 12193 | A positive integer is nonz... |
| nndivre 12194 | The quotient of a real and... |
| nnrecre 12195 | The reciprocal of a positi... |
| nnrecgt0 12196 | The reciprocal of a positi... |
| nnsub 12197 | Subtraction of positive in... |
| nnsubi 12198 | Subtraction of positive in... |
| nndiv 12199 | Two ways to express " ` A ... |
| nndivtr 12200 | Transitive property of div... |
| nnge1d 12201 | A positive integer is one ... |
| nngt0d 12202 | A positive integer is posi... |
| nnne0d 12203 | A positive integer is nonz... |
| nnrecred 12204 | The reciprocal of a positi... |
| nnaddcld 12205 | Closure of addition of pos... |
| nnmulcld 12206 | Closure of multiplication ... |
| nndivred 12207 | A positive integer is one ... |
| 0ne1 12224 | Zero is different from one... |
| 1m1e0 12225 | One minus one equals zero.... |
| 2nn 12226 | 2 is a positive integer. ... |
| 2re 12227 | The number 2 is real. (Co... |
| 2cn 12228 | The number 2 is a complex ... |
| 2cnALT 12229 | Alternate proof of ~ 2cn .... |
| 2ex 12230 | The number 2 is a set. (C... |
| 2cnd 12231 | The number 2 is a complex ... |
| 3nn 12232 | 3 is a positive integer. ... |
| 3re 12233 | The number 3 is real. (Co... |
| 3cn 12234 | The number 3 is a complex ... |
| 3ex 12235 | The number 3 is a set. (C... |
| 4nn 12236 | 4 is a positive integer. ... |
| 4re 12237 | The number 4 is real. (Co... |
| 4cn 12238 | The number 4 is a complex ... |
| 5nn 12239 | 5 is a positive integer. ... |
| 5re 12240 | The number 5 is real. (Co... |
| 5cn 12241 | The number 5 is a complex ... |
| 6nn 12242 | 6 is a positive integer. ... |
| 6re 12243 | The number 6 is real. (Co... |
| 6cn 12244 | The number 6 is a complex ... |
| 7nn 12245 | 7 is a positive integer. ... |
| 7re 12246 | The number 7 is real. (Co... |
| 7cn 12247 | The number 7 is a complex ... |
| 8nn 12248 | 8 is a positive integer. ... |
| 8re 12249 | The number 8 is real. (Co... |
| 8cn 12250 | The number 8 is a complex ... |
| 9nn 12251 | 9 is a positive integer. ... |
| 9re 12252 | The number 9 is real. (Co... |
| 9cn 12253 | The number 9 is a complex ... |
| 0le0 12254 | Zero is nonnegative. (Con... |
| 0le2 12255 | The number 0 is less than ... |
| 2pos 12256 | The number 2 is positive. ... |
| 2ne0 12257 | The number 2 is nonzero. ... |
| 3pos 12258 | The number 3 is positive. ... |
| 3ne0 12259 | The number 3 is nonzero. ... |
| 4pos 12260 | The number 4 is positive. ... |
| 4ne0 12261 | The number 4 is nonzero. ... |
| 5pos 12262 | The number 5 is positive. ... |
| 6pos 12263 | The number 6 is positive. ... |
| 7pos 12264 | The number 7 is positive. ... |
| 8pos 12265 | The number 8 is positive. ... |
| 9pos 12266 | The number 9 is positive. ... |
| neg1cn 12267 | -1 is a complex number. (... |
| neg1rr 12268 | -1 is a real number. (Con... |
| neg1ne0 12269 | -1 is nonzero. (Contribut... |
| neg1lt0 12270 | -1 is less than 0. (Contr... |
| negneg1e1 12271 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 12272 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12273 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12274 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12275 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12276 | Function value at ` N + 1 ... |
| 1p0e1 12277 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12278 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12279 | 2 - 1 = 1. The result is ... |
| 1e2m1 12280 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12281 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12282 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12283 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12284 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12285 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12286 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12287 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12288 | Two plus two equals four. ... |
| 2times 12289 | Two times a number. (Cont... |
| times2 12290 | A number times 2. (Contri... |
| 2timesi 12291 | Two times a number. (Cont... |
| times2i 12292 | A number times 2. (Contri... |
| 2txmxeqx 12293 | Two times a complex number... |
| 2div2e1 12294 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12295 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12296 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12297 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12298 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12299 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12300 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12301 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12302 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12303 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12304 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12305 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12306 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12307 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12308 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12309 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12310 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12311 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12312 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12313 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12314 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12315 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12316 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12317 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12318 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12319 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12320 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12321 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12322 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12323 | One half of four is two. ... |
| 1lt2 12324 | 1 is less than 2. (Contri... |
| 2lt3 12325 | 2 is less than 3. (Contri... |
| 1lt3 12326 | 1 is less than 3. (Contri... |
| 3lt4 12327 | 3 is less than 4. (Contri... |
| 2lt4 12328 | 2 is less than 4. (Contri... |
| 1lt4 12329 | 1 is less than 4. (Contri... |
| 4lt5 12330 | 4 is less than 5. (Contri... |
| 3lt5 12331 | 3 is less than 5. (Contri... |
| 2lt5 12332 | 2 is less than 5. (Contri... |
| 1lt5 12333 | 1 is less than 5. (Contri... |
| 5lt6 12334 | 5 is less than 6. (Contri... |
| 4lt6 12335 | 4 is less than 6. (Contri... |
| 3lt6 12336 | 3 is less than 6. (Contri... |
| 2lt6 12337 | 2 is less than 6. (Contri... |
| 1lt6 12338 | 1 is less than 6. (Contri... |
| 6lt7 12339 | 6 is less than 7. (Contri... |
| 5lt7 12340 | 5 is less than 7. (Contri... |
| 4lt7 12341 | 4 is less than 7. (Contri... |
| 3lt7 12342 | 3 is less than 7. (Contri... |
| 2lt7 12343 | 2 is less than 7. (Contri... |
| 1lt7 12344 | 1 is less than 7. (Contri... |
| 7lt8 12345 | 7 is less than 8. (Contri... |
| 6lt8 12346 | 6 is less than 8. (Contri... |
| 5lt8 12347 | 5 is less than 8. (Contri... |
| 4lt8 12348 | 4 is less than 8. (Contri... |
| 3lt8 12349 | 3 is less than 8. (Contri... |
| 2lt8 12350 | 2 is less than 8. (Contri... |
| 1lt8 12351 | 1 is less than 8. (Contri... |
| 8lt9 12352 | 8 is less than 9. (Contri... |
| 7lt9 12353 | 7 is less than 9. (Contri... |
| 6lt9 12354 | 6 is less than 9. (Contri... |
| 5lt9 12355 | 5 is less than 9. (Contri... |
| 4lt9 12356 | 4 is less than 9. (Contri... |
| 3lt9 12357 | 3 is less than 9. (Contri... |
| 2lt9 12358 | 2 is less than 9. (Contri... |
| 1lt9 12359 | 1 is less than 9. (Contri... |
| 0ne2 12360 | 0 is not equal to 2. (Con... |
| 1ne2 12361 | 1 is not equal to 2. (Con... |
| 1le2 12362 | 1 is less than or equal to... |
| 2cnne0 12363 | 2 is a nonzero complex num... |
| 2rene0 12364 | 2 is a nonzero real number... |
| 1le3 12365 | 1 is less than or equal to... |
| neg1mulneg1e1 12366 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12367 | One-half is real. (Contri... |
| halfcn 12368 | One-half is a complex numb... |
| halfgt0 12369 | One-half is greater than z... |
| halfge0 12370 | One-half is not negative. ... |
| halflt1 12371 | One-half is less than one.... |
| 1mhlfehlf 12372 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12373 | An eighth of four thirds i... |
| halfpm6th 12374 | One half plus or minus one... |
| it0e0 12375 | i times 0 equals 0. (Cont... |
| 2mulicn 12376 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12377 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12378 | Closure of half of a numbe... |
| rehalfcl 12379 | Real closure of half. (Co... |
| half0 12380 | Half of a number is zero i... |
| 2halves 12381 | Two halves make a whole. ... |
| halfpos2 12382 | A number is positive iff i... |
| halfpos 12383 | A positive number is great... |
| halfnneg2 12384 | A number is nonnegative if... |
| halfaddsubcl 12385 | Closure of half-sum and ha... |
| halfaddsub 12386 | Sum and difference of half... |
| subhalfhalf 12387 | Subtracting the half of a ... |
| lt2halves 12388 | A sum is less than the who... |
| addltmul 12389 | Sum is less than product f... |
| nominpos 12390 | There is no smallest posit... |
| avglt1 12391 | Ordering property for aver... |
| avglt2 12392 | Ordering property for aver... |
| avgle1 12393 | Ordering property for aver... |
| avgle2 12394 | Ordering property for aver... |
| avgle 12395 | The average of two numbers... |
| 2timesd 12396 | Two times a number. (Cont... |
| times2d 12397 | A number times 2. (Contri... |
| halfcld 12398 | Closure of half of a numbe... |
| 2halvesd 12399 | Two halves make a whole. ... |
| rehalfcld 12400 | Real closure of half. (Co... |
| lt2halvesd 12401 | A sum is less than the who... |
| rehalfcli 12402 | Half a real number is real... |
| lt2addmuld 12403 | If two real numbers are le... |
| add1p1 12404 | Adding two times 1 to a nu... |
| sub1m1 12405 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12406 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12407 | A complex number increased... |
| div4p1lem1div2 12408 | An integer greater than 5,... |
| nnunb 12409 | The set of positive intege... |
| arch 12410 | Archimedean property of re... |
| nnrecl 12411 | There exists a positive in... |
| bndndx 12412 | A bounded real sequence ` ... |
| elnn0 12415 | Nonnegative integers expre... |
| nnssnn0 12416 | Positive naturals are a su... |
| nn0ssre 12417 | Nonnegative integers are a... |
| nn0sscn 12418 | Nonnegative integers are a... |
| nn0ex 12419 | The set of nonnegative int... |
| nnnn0 12420 | A positive integer is a no... |
| nnnn0i 12421 | A positive integer is a no... |
| nn0re 12422 | A nonnegative integer is a... |
| nn0cn 12423 | A nonnegative integer is a... |
| nn0rei 12424 | A nonnegative integer is a... |
| nn0cni 12425 | A nonnegative integer is a... |
| dfn2 12426 | The set of positive intege... |
| elnnne0 12427 | The positive integer prope... |
| 0nn0 12428 | 0 is a nonnegative integer... |
| 1nn0 12429 | 1 is a nonnegative integer... |
| 2nn0 12430 | 2 is a nonnegative integer... |
| 3nn0 12431 | 3 is a nonnegative integer... |
| 4nn0 12432 | 4 is a nonnegative integer... |
| 5nn0 12433 | 5 is a nonnegative integer... |
| 6nn0 12434 | 6 is a nonnegative integer... |
| 7nn0 12435 | 7 is a nonnegative integer... |
| 8nn0 12436 | 8 is a nonnegative integer... |
| 9nn0 12437 | 9 is a nonnegative integer... |
| nn0ge0 12438 | A nonnegative integer is g... |
| nn0nlt0 12439 | A nonnegative integer is n... |
| nn0ge0i 12440 | Nonnegative integers are n... |
| nn0le0eq0 12441 | A nonnegative integer is l... |
| nn0p1gt0 12442 | A nonnegative integer incr... |
| nnnn0addcl 12443 | A positive integer plus a ... |
| nn0nnaddcl 12444 | A nonnegative integer plus... |
| 0mnnnnn0 12445 | The result of subtracting ... |
| un0addcl 12446 | If ` S ` is closed under a... |
| un0mulcl 12447 | If ` S ` is closed under m... |
| nn0addcl 12448 | Closure of addition of non... |
| nn0mulcl 12449 | Closure of multiplication ... |
| nn0addcli 12450 | Closure of addition of non... |
| nn0mulcli 12451 | Closure of multiplication ... |
| nn0p1nn 12452 | A nonnegative integer plus... |
| peano2nn0 12453 | Second Peano postulate for... |
| nnm1nn0 12454 | A positive integer minus 1... |
| elnn0nn 12455 | The nonnegative integer pr... |
| elnnnn0 12456 | The positive integer prope... |
| elnnnn0b 12457 | The positive integer prope... |
| elnnnn0c 12458 | The positive integer prope... |
| nn0addge1 12459 | A number is less than or e... |
| nn0addge2 12460 | A number is less than or e... |
| nn0addge1i 12461 | A number is less than or e... |
| nn0addge2i 12462 | A number is less than or e... |
| nn0sub 12463 | Subtraction of nonnegative... |
| ltsubnn0 12464 | Subtracting a nonnegative ... |
| nn0negleid 12465 | A nonnegative integer is g... |
| difgtsumgt 12466 | If the difference of a rea... |
| nn0le2xi 12467 | A nonnegative integer is l... |
| nn0lele2xi 12468 | 'Less than or equal to' im... |
| fcdmnn0supp 12469 | Two ways to write the supp... |
| fcdmnn0fsupp 12470 | A function into ` NN0 ` is... |
| fcdmnn0suppg 12471 | Version of ~ fcdmnn0supp a... |
| fcdmnn0fsuppg 12472 | Version of ~ fcdmnn0fsupp ... |
| nnnn0d 12473 | A positive integer is a no... |
| nn0red 12474 | A nonnegative integer is a... |
| nn0cnd 12475 | A nonnegative integer is a... |
| nn0ge0d 12476 | A nonnegative integer is g... |
| nn0addcld 12477 | Closure of addition of non... |
| nn0mulcld 12478 | Closure of multiplication ... |
| nn0readdcl 12479 | Closure law for addition o... |
| nn0n0n1ge2 12480 | A nonnegative integer whic... |
| nn0n0n1ge2b 12481 | A nonnegative integer is n... |
| nn0ge2m1nn 12482 | If a nonnegative integer i... |
| nn0ge2m1nn0 12483 | If a nonnegative integer i... |
| nn0nndivcl 12484 | Closure law for dividing o... |
| elxnn0 12487 | An extended nonnegative in... |
| nn0ssxnn0 12488 | The standard nonnegative i... |
| nn0xnn0 12489 | A standard nonnegative int... |
| xnn0xr 12490 | An extended nonnegative in... |
| 0xnn0 12491 | Zero is an extended nonneg... |
| pnf0xnn0 12492 | Positive infinity is an ex... |
| nn0nepnf 12493 | No standard nonnegative in... |
| nn0xnn0d 12494 | A standard nonnegative int... |
| nn0nepnfd 12495 | No standard nonnegative in... |
| xnn0nemnf 12496 | No extended nonnegative in... |
| xnn0xrnemnf 12497 | The extended nonnegative i... |
| xnn0nnn0pnf 12498 | An extended nonnegative in... |
| elz 12501 | Membership in the set of i... |
| nnnegz 12502 | The negative of a positive... |
| zre 12503 | An integer is a real. (Co... |
| zcn 12504 | An integer is a complex nu... |
| zrei 12505 | An integer is a real numbe... |
| zssre 12506 | The integers are a subset ... |
| zsscn 12507 | The integers are a subset ... |
| zex 12508 | The set of integers exists... |
| elnnz 12509 | Positive integer property ... |
| 0z 12510 | Zero is an integer. (Cont... |
| 0zd 12511 | Zero is an integer, deduct... |
| elnn0z 12512 | Nonnegative integer proper... |
| elznn0nn 12513 | Integer property expressed... |
| elznn0 12514 | Integer property expressed... |
| elznn 12515 | Integer property expressed... |
| zle0orge1 12516 | There is no integer in the... |
| elz2 12517 | Membership in the set of i... |
| dfz2 12518 | Alternative definition of ... |
| zexALT 12519 | Alternate proof of ~ zex .... |
| nnz 12520 | A positive integer is an i... |
| nnssz 12521 | Positive integers are a su... |
| nn0ssz 12522 | Nonnegative integers are a... |
| nnzOLD 12523 | Obsolete version of ~ nnz ... |
| nn0z 12524 | A nonnegative integer is a... |
| nn0zd 12525 | A nonnegative integer is a... |
| nnzd 12526 | A positive integer is an i... |
| nnzi 12527 | A positive integer is an i... |
| nn0zi 12528 | A nonnegative integer is a... |
| elnnz1 12529 | Positive integer property ... |
| znnnlt1 12530 | An integer is not a positi... |
| nnzrab 12531 | Positive integers expresse... |
| nn0zrab 12532 | Nonnegative integers expre... |
| 1z 12533 | One is an integer. (Contr... |
| 1zzd 12534 | One is an integer, deducti... |
| 2z 12535 | 2 is an integer. (Contrib... |
| 3z 12536 | 3 is an integer. (Contrib... |
| 4z 12537 | 4 is an integer. (Contrib... |
| znegcl 12538 | Closure law for negative i... |
| neg1z 12539 | -1 is an integer. (Contri... |
| znegclb 12540 | A complex number is an int... |
| nn0negz 12541 | The negative of a nonnegat... |
| nn0negzi 12542 | The negative of a nonnegat... |
| zaddcl 12543 | Closure of addition of int... |
| peano2z 12544 | Second Peano postulate gen... |
| zsubcl 12545 | Closure of subtraction of ... |
| peano2zm 12546 | "Reverse" second Peano pos... |
| zletr 12547 | Transitive law of ordering... |
| zrevaddcl 12548 | Reverse closure law for ad... |
| znnsub 12549 | The positive difference of... |
| znn0sub 12550 | The nonnegative difference... |
| nzadd 12551 | The sum of a real number n... |
| zmulcl 12552 | Closure of multiplication ... |
| zltp1le 12553 | Integer ordering relation.... |
| zleltp1 12554 | Integer ordering relation.... |
| zlem1lt 12555 | Integer ordering relation.... |
| zltlem1 12556 | Integer ordering relation.... |
| zgt0ge1 12557 | An integer greater than ` ... |
| nnleltp1 12558 | Positive integer ordering ... |
| nnltp1le 12559 | Positive integer ordering ... |
| nnaddm1cl 12560 | Closure of addition of pos... |
| nn0ltp1le 12561 | Nonnegative integer orderi... |
| nn0leltp1 12562 | Nonnegative integer orderi... |
| nn0ltlem1 12563 | Nonnegative integer orderi... |
| nn0sub2 12564 | Subtraction of nonnegative... |
| nn0lt10b 12565 | A nonnegative integer less... |
| nn0lt2 12566 | A nonnegative integer less... |
| nn0le2is012 12567 | A nonnegative integer whic... |
| nn0lem1lt 12568 | Nonnegative integer orderi... |
| nnlem1lt 12569 | Positive integer ordering ... |
| nnltlem1 12570 | Positive integer ordering ... |
| nnm1ge0 12571 | A positive integer decreas... |
| nn0ge0div 12572 | Division of a nonnegative ... |
| zdiv 12573 | Two ways to express " ` M ... |
| zdivadd 12574 | Property of divisibility: ... |
| zdivmul 12575 | Property of divisibility: ... |
| zextle 12576 | An extensionality-like pro... |
| zextlt 12577 | An extensionality-like pro... |
| recnz 12578 | The reciprocal of a number... |
| btwnnz 12579 | A number between an intege... |
| gtndiv 12580 | A larger number does not d... |
| halfnz 12581 | One-half is not an integer... |
| 3halfnz 12582 | Three halves is not an int... |
| suprzcl 12583 | The supremum of a bounded-... |
| prime 12584 | Two ways to express " ` A ... |
| msqznn 12585 | The square of a nonzero in... |
| zneo 12586 | No even integer equals an ... |
| nneo 12587 | A positive integer is even... |
| nneoi 12588 | A positive integer is even... |
| zeo 12589 | An integer is even or odd.... |
| zeo2 12590 | An integer is even or odd ... |
| peano2uz2 12591 | Second Peano postulate for... |
| peano5uzi 12592 | Peano's inductive postulat... |
| peano5uzti 12593 | Peano's inductive postulat... |
| dfuzi 12594 | An expression for the uppe... |
| uzind 12595 | Induction on the upper int... |
| uzind2 12596 | Induction on the upper int... |
| uzind3 12597 | Induction on the upper int... |
| nn0ind 12598 | Principle of Mathematical ... |
| nn0indALT 12599 | Principle of Mathematical ... |
| nn0indd 12600 | Principle of Mathematical ... |
| fzind 12601 | Induction on the integers ... |
| fnn0ind 12602 | Induction on the integers ... |
| nn0ind-raph 12603 | Principle of Mathematical ... |
| zindd 12604 | Principle of Mathematical ... |
| fzindd 12605 | Induction on the integers ... |
| btwnz 12606 | Any real number can be san... |
| zred 12607 | An integer is a real numbe... |
| zcnd 12608 | An integer is a complex nu... |
| znegcld 12609 | Closure law for negative i... |
| peano2zd 12610 | Deduction from second Pean... |
| zaddcld 12611 | Closure of addition of int... |
| zsubcld 12612 | Closure of subtraction of ... |
| zmulcld 12613 | Closure of multiplication ... |
| znnn0nn 12614 | The negative of a negative... |
| zadd2cl 12615 | Increasing an integer by 2... |
| zriotaneg 12616 | The negative of the unique... |
| suprfinzcl 12617 | The supremum of a nonempty... |
| 9p1e10 12620 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12621 | Version of the definition ... |
| decex 12622 | A decimal number is a set.... |
| deceq1 12623 | Equality theorem for the d... |
| deceq2 12624 | Equality theorem for the d... |
| deceq1i 12625 | Equality theorem for the d... |
| deceq2i 12626 | Equality theorem for the d... |
| deceq12i 12627 | Equality theorem for the d... |
| numnncl 12628 | Closure for a numeral (wit... |
| num0u 12629 | Add a zero in the units pl... |
| num0h 12630 | Add a zero in the higher p... |
| numcl 12631 | Closure for a decimal inte... |
| numsuc 12632 | The successor of a decimal... |
| deccl 12633 | Closure for a numeral. (C... |
| 10nn 12634 | 10 is a positive integer. ... |
| 10pos 12635 | The number 10 is positive.... |
| 10nn0 12636 | 10 is a nonnegative intege... |
| 10re 12637 | The number 10 is real. (C... |
| decnncl 12638 | Closure for a numeral. (C... |
| dec0u 12639 | Add a zero in the units pl... |
| dec0h 12640 | Add a zero in the higher p... |
| numnncl2 12641 | Closure for a decimal inte... |
| decnncl2 12642 | Closure for a decimal inte... |
| numlt 12643 | Comparing two decimal inte... |
| numltc 12644 | Comparing two decimal inte... |
| le9lt10 12645 | A "decimal digit" (i.e. a ... |
| declt 12646 | Comparing two decimal inte... |
| decltc 12647 | Comparing two decimal inte... |
| declth 12648 | Comparing two decimal inte... |
| decsuc 12649 | The successor of a decimal... |
| 3declth 12650 | Comparing two decimal inte... |
| 3decltc 12651 | Comparing two decimal inte... |
| decle 12652 | Comparing two decimal inte... |
| decleh 12653 | Comparing two decimal inte... |
| declei 12654 | Comparing a digit to a dec... |
| numlti 12655 | Comparing a digit to a dec... |
| declti 12656 | Comparing a digit to a dec... |
| decltdi 12657 | Comparing a digit to a dec... |
| numsucc 12658 | The successor of a decimal... |
| decsucc 12659 | The successor of a decimal... |
| 1e0p1 12660 | The successor of zero. (C... |
| dec10p 12661 | Ten plus an integer. (Con... |
| numma 12662 | Perform a multiply-add of ... |
| nummac 12663 | Perform a multiply-add of ... |
| numma2c 12664 | Perform a multiply-add of ... |
| numadd 12665 | Add two decimal integers `... |
| numaddc 12666 | Add two decimal integers `... |
| nummul1c 12667 | The product of a decimal i... |
| nummul2c 12668 | The product of a decimal i... |
| decma 12669 | Perform a multiply-add of ... |
| decmac 12670 | Perform a multiply-add of ... |
| decma2c 12671 | Perform a multiply-add of ... |
| decadd 12672 | Add two numerals ` M ` and... |
| decaddc 12673 | Add two numerals ` M ` and... |
| decaddc2 12674 | Add two numerals ` M ` and... |
| decrmanc 12675 | Perform a multiply-add of ... |
| decrmac 12676 | Perform a multiply-add of ... |
| decaddm10 12677 | The sum of two multiples o... |
| decaddi 12678 | Add two numerals ` M ` and... |
| decaddci 12679 | Add two numerals ` M ` and... |
| decaddci2 12680 | Add two numerals ` M ` and... |
| decsubi 12681 | Difference between a numer... |
| decmul1 12682 | The product of a numeral w... |
| decmul1c 12683 | The product of a numeral w... |
| decmul2c 12684 | The product of a numeral w... |
| decmulnc 12685 | The product of a numeral w... |
| 11multnc 12686 | The product of 11 (as nume... |
| decmul10add 12687 | A multiplication of a numb... |
| 6p5lem 12688 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12689 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12690 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12691 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12692 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12693 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12694 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12695 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12696 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12697 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12698 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12699 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12700 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12701 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12702 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12703 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12704 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12705 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12706 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12707 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12708 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12709 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12710 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12711 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12712 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12713 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12714 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12715 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12716 | 4 times 3 equals 12. (Con... |
| 4t4e16 12717 | 4 times 4 equals 16. (Con... |
| 5t2e10 12718 | 5 times 2 equals 10. (Con... |
| 5t3e15 12719 | 5 times 3 equals 15. (Con... |
| 5t4e20 12720 | 5 times 4 equals 20. (Con... |
| 5t5e25 12721 | 5 times 5 equals 25. (Con... |
| 6t2e12 12722 | 6 times 2 equals 12. (Con... |
| 6t3e18 12723 | 6 times 3 equals 18. (Con... |
| 6t4e24 12724 | 6 times 4 equals 24. (Con... |
| 6t5e30 12725 | 6 times 5 equals 30. (Con... |
| 6t6e36 12726 | 6 times 6 equals 36. (Con... |
| 7t2e14 12727 | 7 times 2 equals 14. (Con... |
| 7t3e21 12728 | 7 times 3 equals 21. (Con... |
| 7t4e28 12729 | 7 times 4 equals 28. (Con... |
| 7t5e35 12730 | 7 times 5 equals 35. (Con... |
| 7t6e42 12731 | 7 times 6 equals 42. (Con... |
| 7t7e49 12732 | 7 times 7 equals 49. (Con... |
| 8t2e16 12733 | 8 times 2 equals 16. (Con... |
| 8t3e24 12734 | 8 times 3 equals 24. (Con... |
| 8t4e32 12735 | 8 times 4 equals 32. (Con... |
| 8t5e40 12736 | 8 times 5 equals 40. (Con... |
| 8t6e48 12737 | 8 times 6 equals 48. (Con... |
| 8t7e56 12738 | 8 times 7 equals 56. (Con... |
| 8t8e64 12739 | 8 times 8 equals 64. (Con... |
| 9t2e18 12740 | 9 times 2 equals 18. (Con... |
| 9t3e27 12741 | 9 times 3 equals 27. (Con... |
| 9t4e36 12742 | 9 times 4 equals 36. (Con... |
| 9t5e45 12743 | 9 times 5 equals 45. (Con... |
| 9t6e54 12744 | 9 times 6 equals 54. (Con... |
| 9t7e63 12745 | 9 times 7 equals 63. (Con... |
| 9t8e72 12746 | 9 times 8 equals 72. (Con... |
| 9t9e81 12747 | 9 times 9 equals 81. (Con... |
| 9t11e99 12748 | 9 times 11 equals 99. (Co... |
| 9lt10 12749 | 9 is less than 10. (Contr... |
| 8lt10 12750 | 8 is less than 10. (Contr... |
| 7lt10 12751 | 7 is less than 10. (Contr... |
| 6lt10 12752 | 6 is less than 10. (Contr... |
| 5lt10 12753 | 5 is less than 10. (Contr... |
| 4lt10 12754 | 4 is less than 10. (Contr... |
| 3lt10 12755 | 3 is less than 10. (Contr... |
| 2lt10 12756 | 2 is less than 10. (Contr... |
| 1lt10 12757 | 1 is less than 10. (Contr... |
| decbin0 12758 | Decompose base 4 into base... |
| decbin2 12759 | Decompose base 4 into base... |
| decbin3 12760 | Decompose base 4 into base... |
| halfthird 12761 | Half minus a third. (Cont... |
| 5recm6rec 12762 | One fifth minus one sixth.... |
| uzval 12765 | The value of the upper int... |
| uzf 12766 | The domain and codomain of... |
| eluz1 12767 | Membership in the upper se... |
| eluzel2 12768 | Implication of membership ... |
| eluz2 12769 | Membership in an upper set... |
| eluzmn 12770 | Membership in an earlier u... |
| eluz1i 12771 | Membership in an upper set... |
| eluzuzle 12772 | An integer in an upper set... |
| eluzelz 12773 | A member of an upper set o... |
| eluzelre 12774 | A member of an upper set o... |
| eluzelcn 12775 | A member of an upper set o... |
| eluzle 12776 | Implication of membership ... |
| eluz 12777 | Membership in an upper set... |
| uzid 12778 | Membership of the least me... |
| uzidd 12779 | Membership of the least me... |
| uzn0 12780 | The upper integers are all... |
| uztrn 12781 | Transitive law for sets of... |
| uztrn2 12782 | Transitive law for sets of... |
| uzneg 12783 | Contraposition law for upp... |
| uzssz 12784 | An upper set of integers i... |
| uzssre 12785 | An upper set of integers i... |
| uzss 12786 | Subset relationship for tw... |
| uztric 12787 | Totality of the ordering r... |
| uz11 12788 | The upper integers functio... |
| eluzp1m1 12789 | Membership in the next upp... |
| eluzp1l 12790 | Strict ordering implied by... |
| eluzp1p1 12791 | Membership in the next upp... |
| eluzadd 12792 | Membership in a later uppe... |
| eluzsub 12793 | Membership in an earlier u... |
| eluzaddi 12794 | Membership in a later uppe... |
| eluzaddiOLD 12795 | Obsolete version of ~ eluz... |
| eluzsubi 12796 | Membership in an earlier u... |
| eluzsubiOLD 12797 | Obsolete version of ~ eluz... |
| eluzaddOLD 12798 | Obsolete version of ~ eluz... |
| eluzsubOLD 12799 | Obsolete version of ~ eluz... |
| subeluzsub 12800 | Membership of a difference... |
| uzm1 12801 | Choices for an element of ... |
| uznn0sub 12802 | The nonnegative difference... |
| uzin 12803 | Intersection of two upper ... |
| uzp1 12804 | Choices for an element of ... |
| nn0uz 12805 | Nonnegative integers expre... |
| nnuz 12806 | Positive integers expresse... |
| elnnuz 12807 | A positive integer express... |
| elnn0uz 12808 | A nonnegative integer expr... |
| eluz2nn 12809 | An integer greater than or... |
| eluz4eluz2 12810 | An integer greater than or... |
| eluz4nn 12811 | An integer greater than or... |
| eluzge2nn0 12812 | If an integer is greater t... |
| eluz2n0 12813 | An integer greater than or... |
| uzuzle23 12814 | An integer in the upper se... |
| eluzge3nn 12815 | If an integer is greater t... |
| uz3m2nn 12816 | An integer greater than or... |
| 1eluzge0 12817 | 1 is an integer greater th... |
| 2eluzge0 12818 | 2 is an integer greater th... |
| 2eluzge1 12819 | 2 is an integer greater th... |
| uznnssnn 12820 | The upper integers startin... |
| raluz 12821 | Restricted universal quant... |
| raluz2 12822 | Restricted universal quant... |
| rexuz 12823 | Restricted existential qua... |
| rexuz2 12824 | Restricted existential qua... |
| 2rexuz 12825 | Double existential quantif... |
| peano2uz 12826 | Second Peano postulate for... |
| peano2uzs 12827 | Second Peano postulate for... |
| peano2uzr 12828 | Reversed second Peano axio... |
| uzaddcl 12829 | Addition closure law for a... |
| nn0pzuz 12830 | The sum of a nonnegative i... |
| uzind4 12831 | Induction on the upper set... |
| uzind4ALT 12832 | Induction on the upper set... |
| uzind4s 12833 | Induction on the upper set... |
| uzind4s2 12834 | Induction on the upper set... |
| uzind4i 12835 | Induction on the upper int... |
| uzwo 12836 | Well-ordering principle: a... |
| uzwo2 12837 | Well-ordering principle: a... |
| nnwo 12838 | Well-ordering principle: a... |
| nnwof 12839 | Well-ordering principle: a... |
| nnwos 12840 | Well-ordering principle: a... |
| indstr 12841 | Strong Mathematical Induct... |
| eluznn0 12842 | Membership in a nonnegativ... |
| eluznn 12843 | Membership in a positive u... |
| eluz2b1 12844 | Two ways to say "an intege... |
| eluz2gt1 12845 | An integer greater than or... |
| eluz2b2 12846 | Two ways to say "an intege... |
| eluz2b3 12847 | Two ways to say "an intege... |
| uz2m1nn 12848 | One less than an integer g... |
| 1nuz2 12849 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12850 | A positive integer is eith... |
| uz2mulcl 12851 | Closure of multiplication ... |
| indstr2 12852 | Strong Mathematical Induct... |
| uzinfi 12853 | Extract the lower bound of... |
| nninf 12854 | The infimum of the set of ... |
| nn0inf 12855 | The infimum of the set of ... |
| infssuzle 12856 | The infimum of a subset of... |
| infssuzcl 12857 | The infimum of a subset of... |
| ublbneg 12858 | The image under negation o... |
| eqreznegel 12859 | Two ways to express the im... |
| supminf 12860 | The supremum of a bounded-... |
| lbzbi 12861 | If a set of reals is bound... |
| zsupss 12862 | Any nonempty bounded subse... |
| suprzcl2 12863 | The supremum of a bounded-... |
| suprzub 12864 | The supremum of a bounded-... |
| uzsupss 12865 | Any bounded subset of an u... |
| nn01to3 12866 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12867 | Alternate proof of ~ nn0ge... |
| uzwo3 12868 | Well-ordering principle: a... |
| zmin 12869 | There is a unique smallest... |
| zmax 12870 | There is a unique largest ... |
| zbtwnre 12871 | There is a unique integer ... |
| rebtwnz 12872 | There is a unique greatest... |
| elq 12875 | Membership in the set of r... |
| qmulz 12876 | If ` A ` is rational, then... |
| znq 12877 | The ratio of an integer an... |
| qre 12878 | A rational number is a rea... |
| zq 12879 | An integer is a rational n... |
| qred 12880 | A rational number is a rea... |
| zssq 12881 | The integers are a subset ... |
| nn0ssq 12882 | The nonnegative integers a... |
| nnssq 12883 | The positive integers are ... |
| qssre 12884 | The rationals are a subset... |
| qsscn 12885 | The rationals are a subset... |
| qex 12886 | The set of rational number... |
| nnq 12887 | A positive integer is rati... |
| qcn 12888 | A rational number is a com... |
| qexALT 12889 | Alternate proof of ~ qex .... |
| qaddcl 12890 | Closure of addition of rat... |
| qnegcl 12891 | Closure law for the negati... |
| qmulcl 12892 | Closure of multiplication ... |
| qsubcl 12893 | Closure of subtraction of ... |
| qreccl 12894 | Closure of reciprocal of r... |
| qdivcl 12895 | Closure of division of rat... |
| qrevaddcl 12896 | Reverse closure law for ad... |
| nnrecq 12897 | The reciprocal of a positi... |
| irradd 12898 | The sum of an irrational n... |
| irrmul 12899 | The product of an irration... |
| elpq 12900 | A positive rational is the... |
| elpqb 12901 | A class is a positive rati... |
| rpnnen1lem2 12902 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12903 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12904 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12905 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12906 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12907 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12908 | One half of ~ rpnnen , whe... |
| reexALT 12909 | Alternate proof of ~ reex ... |
| cnref1o 12910 | There is a natural one-to-... |
| cnexALT 12911 | The set of complex numbers... |
| xrex 12912 | The set of extended reals ... |
| addex 12913 | The addition operation is ... |
| mulex 12914 | The multiplication operati... |
| elrp 12917 | Membership in the set of p... |
| elrpii 12918 | Membership in the set of p... |
| 1rp 12919 | 1 is a positive real. (Co... |
| 2rp 12920 | 2 is a positive real. (Co... |
| 3rp 12921 | 3 is a positive real. (Co... |
| rpssre 12922 | The positive reals are a s... |
| rpre 12923 | A positive real is a real.... |
| rpxr 12924 | A positive real is an exte... |
| rpcn 12925 | A positive real is a compl... |
| nnrp 12926 | A positive integer is a po... |
| rpgt0 12927 | A positive real is greater... |
| rpge0 12928 | A positive real is greater... |
| rpregt0 12929 | A positive real is a posit... |
| rprege0 12930 | A positive real is a nonne... |
| rpne0 12931 | A positive real is nonzero... |
| rprene0 12932 | A positive real is a nonze... |
| rpcnne0 12933 | A positive real is a nonze... |
| rpcndif0 12934 | A positive real number is ... |
| ralrp 12935 | Quantification over positi... |
| rexrp 12936 | Quantification over positi... |
| rpaddcl 12937 | Closure law for addition o... |
| rpmulcl 12938 | Closure law for multiplica... |
| rpmtmip 12939 | "Minus times minus is plus... |
| rpdivcl 12940 | Closure law for division o... |
| rpreccl 12941 | Closure law for reciprocat... |
| rphalfcl 12942 | Closure law for half of a ... |
| rpgecl 12943 | A number greater than or e... |
| rphalflt 12944 | Half of a positive real is... |
| rerpdivcl 12945 | Closure law for division o... |
| ge0p1rp 12946 | A nonnegative number plus ... |
| rpneg 12947 | Either a nonzero real or i... |
| negelrp 12948 | Elementhood of a negation ... |
| negelrpd 12949 | The negation of a negative... |
| 0nrp 12950 | Zero is not a positive rea... |
| ltsubrp 12951 | Subtracting a positive rea... |
| ltaddrp 12952 | Adding a positive number t... |
| difrp 12953 | Two ways to say one number... |
| elrpd 12954 | Membership in the set of p... |
| nnrpd 12955 | A positive integer is a po... |
| zgt1rpn0n1 12956 | An integer greater than 1 ... |
| rpred 12957 | A positive real is a real.... |
| rpxrd 12958 | A positive real is an exte... |
| rpcnd 12959 | A positive real is a compl... |
| rpgt0d 12960 | A positive real is greater... |
| rpge0d 12961 | A positive real is greater... |
| rpne0d 12962 | A positive real is nonzero... |
| rpregt0d 12963 | A positive real is real an... |
| rprege0d 12964 | A positive real is real an... |
| rprene0d 12965 | A positive real is a nonze... |
| rpcnne0d 12966 | A positive real is a nonze... |
| rpreccld 12967 | Closure law for reciprocat... |
| rprecred 12968 | Closure law for reciprocat... |
| rphalfcld 12969 | Closure law for half of a ... |
| reclt1d 12970 | The reciprocal of a positi... |
| recgt1d 12971 | The reciprocal of a positi... |
| rpaddcld 12972 | Closure law for addition o... |
| rpmulcld 12973 | Closure law for multiplica... |
| rpdivcld 12974 | Closure law for division o... |
| ltrecd 12975 | The reciprocal of both sid... |
| lerecd 12976 | The reciprocal of both sid... |
| ltrec1d 12977 | Reciprocal swap in a 'less... |
| lerec2d 12978 | Reciprocal swap in a 'less... |
| lediv2ad 12979 | Division of both sides of ... |
| ltdiv2d 12980 | Division of a positive num... |
| lediv2d 12981 | Division of a positive num... |
| ledivdivd 12982 | Invert ratios of positive ... |
| divge1 12983 | The ratio of a number over... |
| divlt1lt 12984 | A real number divided by a... |
| divle1le 12985 | A real number divided by a... |
| ledivge1le 12986 | If a number is less than o... |
| ge0p1rpd 12987 | A nonnegative number plus ... |
| rerpdivcld 12988 | Closure law for division o... |
| ltsubrpd 12989 | Subtracting a positive rea... |
| ltaddrpd 12990 | Adding a positive number t... |
| ltaddrp2d 12991 | Adding a positive number t... |
| ltmulgt11d 12992 | Multiplication by a number... |
| ltmulgt12d 12993 | Multiplication by a number... |
| gt0divd 12994 | Division of a positive num... |
| ge0divd 12995 | Division of a nonnegative ... |
| rpgecld 12996 | A number greater than or e... |
| divge0d 12997 | The ratio of nonnegative a... |
| ltmul1d 12998 | The ratio of nonnegative a... |
| ltmul2d 12999 | Multiplication of both sid... |
| lemul1d 13000 | Multiplication of both sid... |
| lemul2d 13001 | Multiplication of both sid... |
| ltdiv1d 13002 | Division of both sides of ... |
| lediv1d 13003 | Division of both sides of ... |
| ltmuldivd 13004 | 'Less than' relationship b... |
| ltmuldiv2d 13005 | 'Less than' relationship b... |
| lemuldivd 13006 | 'Less than or equal to' re... |
| lemuldiv2d 13007 | 'Less than or equal to' re... |
| ltdivmuld 13008 | 'Less than' relationship b... |
| ltdivmul2d 13009 | 'Less than' relationship b... |
| ledivmuld 13010 | 'Less than or equal to' re... |
| ledivmul2d 13011 | 'Less than or equal to' re... |
| ltmul1dd 13012 | The ratio of nonnegative a... |
| ltmul2dd 13013 | Multiplication of both sid... |
| ltdiv1dd 13014 | Division of both sides of ... |
| lediv1dd 13015 | Division of both sides of ... |
| lediv12ad 13016 | Comparison of ratio of two... |
| mul2lt0rlt0 13017 | If the result of a multipl... |
| mul2lt0rgt0 13018 | If the result of a multipl... |
| mul2lt0llt0 13019 | If the result of a multipl... |
| mul2lt0lgt0 13020 | If the result of a multipl... |
| mul2lt0bi 13021 | If the result of a multipl... |
| prodge0rd 13022 | Infer that a multiplicand ... |
| prodge0ld 13023 | Infer that a multiplier is... |
| ltdiv23d 13024 | Swap denominator with othe... |
| lediv23d 13025 | Swap denominator with othe... |
| lt2mul2divd 13026 | The ratio of nonnegative a... |
| nnledivrp 13027 | Division of a positive int... |
| nn0ledivnn 13028 | Division of a nonnegative ... |
| addlelt 13029 | If the sum of a real numbe... |
| ltxr 13036 | The 'less than' binary rel... |
| elxr 13037 | Membership in the set of e... |
| xrnemnf 13038 | An extended real other tha... |
| xrnepnf 13039 | An extended real other tha... |
| xrltnr 13040 | The extended real 'less th... |
| ltpnf 13041 | Any (finite) real is less ... |
| ltpnfd 13042 | Any (finite) real is less ... |
| 0ltpnf 13043 | Zero is less than plus inf... |
| mnflt 13044 | Minus infinity is less tha... |
| mnfltd 13045 | Minus infinity is less tha... |
| mnflt0 13046 | Minus infinity is less tha... |
| mnfltpnf 13047 | Minus infinity is less tha... |
| mnfltxr 13048 | Minus infinity is less tha... |
| pnfnlt 13049 | No extended real is greate... |
| nltmnf 13050 | No extended real is less t... |
| pnfge 13051 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 13052 | An extended nonnegative in... |
| 0lepnf 13053 | 0 less than or equal to po... |
| xnn0ge0 13054 | An extended nonnegative in... |
| mnfle 13055 | Minus infinity is less tha... |
| xrltnsym 13056 | Ordering on the extended r... |
| xrltnsym2 13057 | 'Less than' is antisymmetr... |
| xrlttri 13058 | Ordering on the extended r... |
| xrlttr 13059 | Ordering on the extended r... |
| xrltso 13060 | 'Less than' is a strict or... |
| xrlttri2 13061 | Trichotomy law for 'less t... |
| xrlttri3 13062 | Trichotomy law for 'less t... |
| xrleloe 13063 | 'Less than or equal' expre... |
| xrleltne 13064 | 'Less than or equal to' im... |
| xrltlen 13065 | 'Less than' expressed in t... |
| dfle2 13066 | Alternative definition of ... |
| dflt2 13067 | Alternative definition of ... |
| xrltle 13068 | 'Less than' implies 'less ... |
| xrltled 13069 | 'Less than' implies 'less ... |
| xrleid 13070 | 'Less than or equal to' is... |
| xrleidd 13071 | 'Less than or equal to' is... |
| xrletri 13072 | Trichotomy law for extende... |
| xrletri3 13073 | Trichotomy law for extende... |
| xrletrid 13074 | Trichotomy law for extende... |
| xrlelttr 13075 | Transitive law for orderin... |
| xrltletr 13076 | Transitive law for orderin... |
| xrletr 13077 | Transitive law for orderin... |
| xrlttrd 13078 | Transitive law for orderin... |
| xrlelttrd 13079 | Transitive law for orderin... |
| xrltletrd 13080 | Transitive law for orderin... |
| xrletrd 13081 | Transitive law for orderin... |
| xrltne 13082 | 'Less than' implies not eq... |
| nltpnft 13083 | An extended real is not le... |
| xgepnf 13084 | An extended real which is ... |
| ngtmnft 13085 | An extended real is not gr... |
| xlemnf 13086 | An extended real which is ... |
| xrrebnd 13087 | An extended real is real i... |
| xrre 13088 | A way of proving that an e... |
| xrre2 13089 | An extended real between t... |
| xrre3 13090 | A way of proving that an e... |
| ge0gtmnf 13091 | A nonnegative extended rea... |
| ge0nemnf 13092 | A nonnegative extended rea... |
| xrrege0 13093 | A nonnegative extended rea... |
| xrmax1 13094 | An extended real is less t... |
| xrmax2 13095 | An extended real is less t... |
| xrmin1 13096 | The minimum of two extende... |
| xrmin2 13097 | The minimum of two extende... |
| xrmaxeq 13098 | The maximum of two extende... |
| xrmineq 13099 | The minimum of two extende... |
| xrmaxlt 13100 | Two ways of saying the max... |
| xrltmin 13101 | Two ways of saying an exte... |
| xrmaxle 13102 | Two ways of saying the max... |
| xrlemin 13103 | Two ways of saying a numbe... |
| max1 13104 | A number is less than or e... |
| max1ALT 13105 | A number is less than or e... |
| max2 13106 | A number is less than or e... |
| 2resupmax 13107 | The supremum of two real n... |
| min1 13108 | The minimum of two numbers... |
| min2 13109 | The minimum of two numbers... |
| maxle 13110 | Two ways of saying the max... |
| lemin 13111 | Two ways of saying a numbe... |
| maxlt 13112 | Two ways of saying the max... |
| ltmin 13113 | Two ways of saying a numbe... |
| lemaxle 13114 | A real number which is les... |
| max0sub 13115 | Decompose a real number in... |
| ifle 13116 | An if statement transforms... |
| z2ge 13117 | There exists an integer gr... |
| qbtwnre 13118 | The rational numbers are d... |
| qbtwnxr 13119 | The rational numbers are d... |
| qsqueeze 13120 | If a nonnegative real is l... |
| qextltlem 13121 | Lemma for ~ qextlt and qex... |
| qextlt 13122 | An extensionality-like pro... |
| qextle 13123 | An extensionality-like pro... |
| xralrple 13124 | Show that ` A ` is less th... |
| alrple 13125 | Show that ` A ` is less th... |
| xnegeq 13126 | Equality of two extended n... |
| xnegex 13127 | A negative extended real e... |
| xnegpnf 13128 | Minus ` +oo ` . Remark of... |
| xnegmnf 13129 | Minus ` -oo ` . Remark of... |
| rexneg 13130 | Minus a real number. Rema... |
| xneg0 13131 | The negative of zero. (Co... |
| xnegcl 13132 | Closure of extended real n... |
| xnegneg 13133 | Extended real version of ~... |
| xneg11 13134 | Extended real version of ~... |
| xltnegi 13135 | Forward direction of ~ xlt... |
| xltneg 13136 | Extended real version of ~... |
| xleneg 13137 | Extended real version of ~... |
| xlt0neg1 13138 | Extended real version of ~... |
| xlt0neg2 13139 | Extended real version of ~... |
| xle0neg1 13140 | Extended real version of ~... |
| xle0neg2 13141 | Extended real version of ~... |
| xaddval 13142 | Value of the extended real... |
| xaddf 13143 | The extended real addition... |
| xmulval 13144 | Value of the extended real... |
| xaddpnf1 13145 | Addition of positive infin... |
| xaddpnf2 13146 | Addition of positive infin... |
| xaddmnf1 13147 | Addition of negative infin... |
| xaddmnf2 13148 | Addition of negative infin... |
| pnfaddmnf 13149 | Addition of positive and n... |
| mnfaddpnf 13150 | Addition of negative and p... |
| rexadd 13151 | The extended real addition... |
| rexsub 13152 | Extended real subtraction ... |
| rexaddd 13153 | The extended real addition... |
| xnn0xaddcl 13154 | The extended nonnegative i... |
| xaddnemnf 13155 | Closure of extended real a... |
| xaddnepnf 13156 | Closure of extended real a... |
| xnegid 13157 | Extended real version of ~... |
| xaddcl 13158 | The extended real addition... |
| xaddcom 13159 | The extended real addition... |
| xaddid1 13160 | Extended real version of ~... |
| xaddid2 13161 | Extended real version of ~... |
| xaddid1d 13162 | ` 0 ` is a right identity ... |
| xnn0lem1lt 13163 | Extended nonnegative integ... |
| xnn0lenn0nn0 13164 | An extended nonnegative in... |
| xnn0le2is012 13165 | An extended nonnegative in... |
| xnn0xadd0 13166 | The sum of two extended no... |
| xnegdi 13167 | Extended real version of ~... |
| xaddass 13168 | Associativity of extended ... |
| xaddass2 13169 | Associativity of extended ... |
| xpncan 13170 | Extended real version of ~... |
| xnpcan 13171 | Extended real version of ~... |
| xleadd1a 13172 | Extended real version of ~... |
| xleadd2a 13173 | Commuted form of ~ xleadd1... |
| xleadd1 13174 | Weakened version of ~ xlea... |
| xltadd1 13175 | Extended real version of ~... |
| xltadd2 13176 | Extended real version of ~... |
| xaddge0 13177 | The sum of nonnegative ext... |
| xle2add 13178 | Extended real version of ~... |
| xlt2add 13179 | Extended real version of ~... |
| xsubge0 13180 | Extended real version of ~... |
| xposdif 13181 | Extended real version of ~... |
| xlesubadd 13182 | Under certain conditions, ... |
| xmullem 13183 | Lemma for ~ rexmul . (Con... |
| xmullem2 13184 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 13185 | Extended real multiplicati... |
| xmul01 13186 | Extended real version of ~... |
| xmul02 13187 | Extended real version of ~... |
| xmulneg1 13188 | Extended real version of ~... |
| xmulneg2 13189 | Extended real version of ~... |
| rexmul 13190 | The extended real multipli... |
| xmulf 13191 | The extended real multipli... |
| xmulcl 13192 | Closure of extended real m... |
| xmulpnf1 13193 | Multiplication by plus inf... |
| xmulpnf2 13194 | Multiplication by plus inf... |
| xmulmnf1 13195 | Multiplication by minus in... |
| xmulmnf2 13196 | Multiplication by minus in... |
| xmulpnf1n 13197 | Multiplication by plus inf... |
| xmulid1 13198 | Extended real version of ~... |
| xmulid2 13199 | Extended real version of ~... |
| xmulm1 13200 | Extended real version of ~... |
| xmulasslem2 13201 | Lemma for ~ xmulass . (Co... |
| xmulgt0 13202 | Extended real version of ~... |
| xmulge0 13203 | Extended real version of ~... |
| xmulasslem 13204 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 13205 | Lemma for ~ xmulass . (Co... |
| xmulass 13206 | Associativity of the exten... |
| xlemul1a 13207 | Extended real version of ~... |
| xlemul2a 13208 | Extended real version of ~... |
| xlemul1 13209 | Extended real version of ~... |
| xlemul2 13210 | Extended real version of ~... |
| xltmul1 13211 | Extended real version of ~... |
| xltmul2 13212 | Extended real version of ~... |
| xadddilem 13213 | Lemma for ~ xadddi . (Con... |
| xadddi 13214 | Distributive property for ... |
| xadddir 13215 | Commuted version of ~ xadd... |
| xadddi2 13216 | The assumption that the mu... |
| xadddi2r 13217 | Commuted version of ~ xadd... |
| x2times 13218 | Extended real version of ~... |
| xnegcld 13219 | Closure of extended real n... |
| xaddcld 13220 | The extended real addition... |
| xmulcld 13221 | Closure of extended real m... |
| xadd4d 13222 | Rearrangement of 4 terms i... |
| xnn0add4d 13223 | Rearrangement of 4 terms i... |
| xrsupexmnf 13224 | Adding minus infinity to a... |
| xrinfmexpnf 13225 | Adding plus infinity to a ... |
| xrsupsslem 13226 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 13227 | Lemma for ~ xrinfmss . (C... |
| xrsupss 13228 | Any subset of extended rea... |
| xrinfmss 13229 | Any subset of extended rea... |
| xrinfmss2 13230 | Any subset of extended rea... |
| xrub 13231 | By quantifying only over r... |
| supxr 13232 | The supremum of a set of e... |
| supxr2 13233 | The supremum of a set of e... |
| supxrcl 13234 | The supremum of an arbitra... |
| supxrun 13235 | The supremum of the union ... |
| supxrmnf 13236 | Adding minus infinity to a... |
| supxrpnf 13237 | The supremum of a set of e... |
| supxrunb1 13238 | The supremum of an unbound... |
| supxrunb2 13239 | The supremum of an unbound... |
| supxrbnd1 13240 | The supremum of a bounded-... |
| supxrbnd2 13241 | The supremum of a bounded-... |
| xrsup0 13242 | The supremum of an empty s... |
| supxrub 13243 | A member of a set of exten... |
| supxrlub 13244 | The supremum of a set of e... |
| supxrleub 13245 | The supremum of a set of e... |
| supxrre 13246 | The real and extended real... |
| supxrbnd 13247 | The supremum of a bounded-... |
| supxrgtmnf 13248 | The supremum of a nonempty... |
| supxrre1 13249 | The supremum of a nonempty... |
| supxrre2 13250 | The supremum of a nonempty... |
| supxrss 13251 | Smaller sets of extended r... |
| infxrcl 13252 | The infimum of an arbitrar... |
| infxrlb 13253 | A member of a set of exten... |
| infxrgelb 13254 | The infimum of a set of ex... |
| infxrre 13255 | The real and extended real... |
| infxrmnf 13256 | The infinimum of a set of ... |
| xrinf0 13257 | The infimum of the empty s... |
| infxrss 13258 | Larger sets of extended re... |
| reltre 13259 | For all real numbers there... |
| rpltrp 13260 | For all positive real numb... |
| reltxrnmnf 13261 | For all extended real numb... |
| infmremnf 13262 | The infimum of the reals i... |
| infmrp1 13263 | The infimum of the positiv... |
| ixxval 13272 | Value of the interval func... |
| elixx1 13273 | Membership in an interval ... |
| ixxf 13274 | The set of intervals of ex... |
| ixxex 13275 | The set of intervals of ex... |
| ixxssxr 13276 | The set of intervals of ex... |
| elixx3g 13277 | Membership in a set of ope... |
| ixxssixx 13278 | An interval is a subset of... |
| ixxdisj 13279 | Split an interval into dis... |
| ixxun 13280 | Split an interval into two... |
| ixxin 13281 | Intersection of two interv... |
| ixxss1 13282 | Subset relationship for in... |
| ixxss2 13283 | Subset relationship for in... |
| ixxss12 13284 | Subset relationship for in... |
| ixxub 13285 | Extract the upper bound of... |
| ixxlb 13286 | Extract the lower bound of... |
| iooex 13287 | The set of open intervals ... |
| iooval 13288 | Value of the open interval... |
| ioo0 13289 | An empty open interval of ... |
| ioon0 13290 | An open interval of extend... |
| ndmioo 13291 | The open interval function... |
| iooid 13292 | An open interval with iden... |
| elioo3g 13293 | Membership in a set of ope... |
| elioore 13294 | A member of an open interv... |
| lbioo 13295 | An open interval does not ... |
| ubioo 13296 | An open interval does not ... |
| iooval2 13297 | Value of the open interval... |
| iooin 13298 | Intersection of two open i... |
| iooss1 13299 | Subset relationship for op... |
| iooss2 13300 | Subset relationship for op... |
| iocval 13301 | Value of the open-below, c... |
| icoval 13302 | Value of the closed-below,... |
| iccval 13303 | Value of the closed interv... |
| elioo1 13304 | Membership in an open inte... |
| elioo2 13305 | Membership in an open inte... |
| elioc1 13306 | Membership in an open-belo... |
| elico1 13307 | Membership in a closed-bel... |
| elicc1 13308 | Membership in a closed int... |
| iccid 13309 | A closed interval with ide... |
| ico0 13310 | An empty open interval of ... |
| ioc0 13311 | An empty open interval of ... |
| icc0 13312 | An empty closed interval o... |
| dfrp2 13313 | Alternate definition of th... |
| elicod 13314 | Membership in a left-close... |
| icogelb 13315 | An element of a left-close... |
| elicore 13316 | A member of a left-closed ... |
| ubioc1 13317 | The upper bound belongs to... |
| lbico1 13318 | The lower bound belongs to... |
| iccleub 13319 | An element of a closed int... |
| iccgelb 13320 | An element of a closed int... |
| elioo5 13321 | Membership in an open inte... |
| eliooxr 13322 | A nonempty open interval s... |
| eliooord 13323 | Ordering implied by a memb... |
| elioo4g 13324 | Membership in an open inte... |
| ioossre 13325 | An open interval is a set ... |
| ioosscn 13326 | An open interval is a set ... |
| elioc2 13327 | Membership in an open-belo... |
| elico2 13328 | Membership in a closed-bel... |
| elicc2 13329 | Membership in a closed rea... |
| elicc2i 13330 | Inference for membership i... |
| elicc4 13331 | Membership in a closed rea... |
| iccss 13332 | Condition for a closed int... |
| iccssioo 13333 | Condition for a closed int... |
| icossico 13334 | Condition for a closed-bel... |
| iccss2 13335 | Condition for a closed int... |
| iccssico 13336 | Condition for a closed int... |
| iccssioo2 13337 | Condition for a closed int... |
| iccssico2 13338 | Condition for a closed int... |
| ioomax 13339 | The open interval from min... |
| iccmax 13340 | The closed interval from m... |
| ioopos 13341 | The set of positive reals ... |
| ioorp 13342 | The set of positive reals ... |
| iooshf 13343 | Shift the arguments of the... |
| iocssre 13344 | A closed-above interval wi... |
| icossre 13345 | A closed-below interval wi... |
| iccssre 13346 | A closed real interval is ... |
| iccssxr 13347 | A closed interval is a set... |
| iocssxr 13348 | An open-below, closed-abov... |
| icossxr 13349 | A closed-below, open-above... |
| ioossicc 13350 | An open interval is a subs... |
| iccssred 13351 | A closed real interval is ... |
| eliccxr 13352 | A member of a closed inter... |
| icossicc 13353 | A closed-below, open-above... |
| iocssicc 13354 | A closed-above, open-below... |
| ioossico 13355 | An open interval is a subs... |
| iocssioo 13356 | Condition for a closed int... |
| icossioo 13357 | Condition for a closed int... |
| ioossioo 13358 | Condition for an open inte... |
| iccsupr 13359 | A nonempty subset of a clo... |
| elioopnf 13360 | Membership in an unbounded... |
| elioomnf 13361 | Membership in an unbounded... |
| elicopnf 13362 | Membership in a closed unb... |
| repos 13363 | Two ways of saying that a ... |
| ioof 13364 | The set of open intervals ... |
| iccf 13365 | The set of closed interval... |
| unirnioo 13366 | The union of the range of ... |
| dfioo2 13367 | Alternate definition of th... |
| ioorebas 13368 | Open intervals are element... |
| xrge0neqmnf 13369 | A nonnegative extended rea... |
| xrge0nre 13370 | An extended real which is ... |
| elrege0 13371 | The predicate "is a nonneg... |
| nn0rp0 13372 | A nonnegative integer is a... |
| rge0ssre 13373 | Nonnegative real numbers a... |
| elxrge0 13374 | Elementhood in the set of ... |
| 0e0icopnf 13375 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13376 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13377 | The nonnegative reals are ... |
| ge0mulcl 13378 | The nonnegative reals are ... |
| ge0xaddcl 13379 | The nonnegative reals are ... |
| ge0xmulcl 13380 | The nonnegative extended r... |
| lbicc2 13381 | The lower bound of a close... |
| ubicc2 13382 | The upper bound of a close... |
| elicc01 13383 | Membership in the closed r... |
| elunitrn 13384 | The closed unit interval i... |
| elunitcn 13385 | The closed unit interval i... |
| 0elunit 13386 | Zero is an element of the ... |
| 1elunit 13387 | One is an element of the c... |
| iooneg 13388 | Membership in a negated op... |
| iccneg 13389 | Membership in a negated cl... |
| icoshft 13390 | A shifted real is a member... |
| icoshftf1o 13391 | Shifting a closed-below, o... |
| icoun 13392 | The union of two adjacent ... |
| icodisj 13393 | Adjacent left-closed right... |
| ioounsn 13394 | The union of an open inter... |
| snunioo 13395 | The closure of one end of ... |
| snunico 13396 | The closure of the open en... |
| snunioc 13397 | The closure of the open en... |
| prunioo 13398 | The closure of an open rea... |
| ioodisj 13399 | If the upper bound of one ... |
| ioojoin 13400 | Join two open intervals to... |
| difreicc 13401 | The class difference of ` ... |
| iccsplit 13402 | Split a closed interval in... |
| iccshftr 13403 | Membership in a shifted in... |
| iccshftri 13404 | Membership in a shifted in... |
| iccshftl 13405 | Membership in a shifted in... |
| iccshftli 13406 | Membership in a shifted in... |
| iccdil 13407 | Membership in a dilated in... |
| iccdili 13408 | Membership in a dilated in... |
| icccntr 13409 | Membership in a contracted... |
| icccntri 13410 | Membership in a contracted... |
| divelunit 13411 | A condition for a ratio to... |
| lincmb01cmp 13412 | A linear combination of tw... |
| iccf1o 13413 | Describe a bijection from ... |
| iccen 13414 | Any nontrivial closed inte... |
| xov1plusxeqvd 13415 | A complex number ` X ` is ... |
| unitssre 13416 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13417 | The closed unit interval i... |
| supicc 13418 | Supremum of a bounded set ... |
| supiccub 13419 | The supremum of a bounded ... |
| supicclub 13420 | The supremum of a bounded ... |
| supicclub2 13421 | The supremum of a bounded ... |
| zltaddlt1le 13422 | The sum of an integer and ... |
| xnn0xrge0 13423 | An extended nonnegative in... |
| fzval 13426 | The value of a finite set ... |
| fzval2 13427 | An alternative way of expr... |
| fzf 13428 | Establish the domain and c... |
| elfz1 13429 | Membership in a finite set... |
| elfz 13430 | Membership in a finite set... |
| elfz2 13431 | Membership in a finite set... |
| elfzd 13432 | Membership in a finite set... |
| elfz5 13433 | Membership in a finite set... |
| elfz4 13434 | Membership in a finite set... |
| elfzuzb 13435 | Membership in a finite set... |
| eluzfz 13436 | Membership in a finite set... |
| elfzuz 13437 | A member of a finite set o... |
| elfzuz3 13438 | Membership in a finite set... |
| elfzel2 13439 | Membership in a finite set... |
| elfzel1 13440 | Membership in a finite set... |
| elfzelz 13441 | A member of a finite set o... |
| elfzelzd 13442 | A member of a finite set o... |
| fzssz 13443 | A finite sequence of integ... |
| elfzle1 13444 | A member of a finite set o... |
| elfzle2 13445 | A member of a finite set o... |
| elfzuz2 13446 | Implication of membership ... |
| elfzle3 13447 | Membership in a finite set... |
| eluzfz1 13448 | Membership in a finite set... |
| eluzfz2 13449 | Membership in a finite set... |
| eluzfz2b 13450 | Membership in a finite set... |
| elfz3 13451 | Membership in a finite set... |
| elfz1eq 13452 | Membership in a finite set... |
| elfzubelfz 13453 | If there is a member in a ... |
| peano2fzr 13454 | A Peano-postulate-like the... |
| fzn0 13455 | Properties of a finite int... |
| fz0 13456 | A finite set of sequential... |
| fzn 13457 | A finite set of sequential... |
| fzen 13458 | A shifted finite set of se... |
| fz1n 13459 | A 1-based finite set of se... |
| 0nelfz1 13460 | 0 is not an element of a f... |
| 0fz1 13461 | Two ways to say a finite 1... |
| fz10 13462 | There are no integers betw... |
| uzsubsubfz 13463 | Membership of an integer g... |
| uzsubsubfz1 13464 | Membership of an integer g... |
| ige3m2fz 13465 | Membership of an integer g... |
| fzsplit2 13466 | Split a finite interval of... |
| fzsplit 13467 | Split a finite interval of... |
| fzdisj 13468 | Condition for two finite i... |
| fz01en 13469 | 0-based and 1-based finite... |
| elfznn 13470 | A member of a finite set o... |
| elfz1end 13471 | A nonempty finite range of... |
| fz1ssnn 13472 | A finite set of positive i... |
| fznn0sub 13473 | Subtraction closure for a ... |
| fzmmmeqm 13474 | Subtracting the difference... |
| fzaddel 13475 | Membership of a sum in a f... |
| fzadd2 13476 | Membership of a sum in a f... |
| fzsubel 13477 | Membership of a difference... |
| fzopth 13478 | A finite set of sequential... |
| fzass4 13479 | Two ways to express a nond... |
| fzss1 13480 | Subset relationship for fi... |
| fzss2 13481 | Subset relationship for fi... |
| fzssuz 13482 | A finite set of sequential... |
| fzsn 13483 | A finite interval of integ... |
| fzssp1 13484 | Subset relationship for fi... |
| fzssnn 13485 | Finite sets of sequential ... |
| ssfzunsnext 13486 | A subset of a finite seque... |
| ssfzunsn 13487 | A subset of a finite seque... |
| fzsuc 13488 | Join a successor to the en... |
| fzpred 13489 | Join a predecessor to the ... |
| fzpreddisj 13490 | A finite set of sequential... |
| elfzp1 13491 | Append an element to a fin... |
| fzp1ss 13492 | Subset relationship for fi... |
| fzelp1 13493 | Membership in a set of seq... |
| fzp1elp1 13494 | Add one to an element of a... |
| fznatpl1 13495 | Shift membership in a fini... |
| fzpr 13496 | A finite interval of integ... |
| fztp 13497 | A finite interval of integ... |
| fz12pr 13498 | An integer range between 1... |
| fzsuc2 13499 | Join a successor to the en... |
| fzp1disj 13500 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13501 | Remove a successor from th... |
| fzprval 13502 | Two ways of defining the f... |
| fztpval 13503 | Two ways of defining the f... |
| fzrev 13504 | Reversal of start and end ... |
| fzrev2 13505 | Reversal of start and end ... |
| fzrev2i 13506 | Reversal of start and end ... |
| fzrev3 13507 | The "complement" of a memb... |
| fzrev3i 13508 | The "complement" of a memb... |
| fznn 13509 | Finite set of sequential i... |
| elfz1b 13510 | Membership in a 1-based fi... |
| elfz1uz 13511 | Membership in a 1-based fi... |
| elfzm11 13512 | Membership in a finite set... |
| uzsplit 13513 | Express an upper integer s... |
| uzdisj 13514 | The first ` N ` elements o... |
| fseq1p1m1 13515 | Add/remove an item to/from... |
| fseq1m1p1 13516 | Add/remove an item to/from... |
| fz1sbc 13517 | Quantification over a one-... |
| elfzp1b 13518 | An integer is a member of ... |
| elfzm1b 13519 | An integer is a member of ... |
| elfzp12 13520 | Options for membership in ... |
| fzm1 13521 | Choices for an element of ... |
| fzneuz 13522 | No finite set of sequentia... |
| fznuz 13523 | Disjointness of the upper ... |
| uznfz 13524 | Disjointness of the upper ... |
| fzp1nel 13525 | One plus the upper bound o... |
| fzrevral 13526 | Reversal of scanning order... |
| fzrevral2 13527 | Reversal of scanning order... |
| fzrevral3 13528 | Reversal of scanning order... |
| fzshftral 13529 | Shift the scanning order i... |
| ige2m1fz1 13530 | Membership of an integer g... |
| ige2m1fz 13531 | Membership in a 0-based fi... |
| elfz2nn0 13532 | Membership in a finite set... |
| fznn0 13533 | Characterization of a fini... |
| elfznn0 13534 | A member of a finite set o... |
| elfz3nn0 13535 | The upper bound of a nonem... |
| fz0ssnn0 13536 | Finite sets of sequential ... |
| fz1ssfz0 13537 | Subset relationship for fi... |
| 0elfz 13538 | 0 is an element of a finit... |
| nn0fz0 13539 | A nonnegative integer is a... |
| elfz0add 13540 | An element of a finite set... |
| fz0sn 13541 | An integer range from 0 to... |
| fz0tp 13542 | An integer range from 0 to... |
| fz0to3un2pr 13543 | An integer range from 0 to... |
| fz0to4untppr 13544 | An integer range from 0 to... |
| elfz0ubfz0 13545 | An element of a finite set... |
| elfz0fzfz0 13546 | A member of a finite set o... |
| fz0fzelfz0 13547 | If a member of a finite se... |
| fznn0sub2 13548 | Subtraction closure for a ... |
| uzsubfz0 13549 | Membership of an integer g... |
| fz0fzdiffz0 13550 | The difference of an integ... |
| elfzmlbm 13551 | Subtracting the lower boun... |
| elfzmlbp 13552 | Subtracting the lower boun... |
| fzctr 13553 | Lemma for theorems about t... |
| difelfzle 13554 | The difference of two inte... |
| difelfznle 13555 | The difference of two inte... |
| nn0split 13556 | Express the set of nonnega... |
| nn0disj 13557 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13558 | A finite set of sequential... |
| fvffz0 13559 | The function value of a fu... |
| 1fv 13560 | A function on a singleton.... |
| 4fvwrd4 13561 | The first four function va... |
| 2ffzeq 13562 | Two functions over 0-based... |
| preduz 13563 | The value of the predecess... |
| prednn 13564 | The value of the predecess... |
| prednn0 13565 | The value of the predecess... |
| predfz 13566 | Calculate the predecessor ... |
| fzof 13569 | Functionality of the half-... |
| elfzoel1 13570 | Reverse closure for half-o... |
| elfzoel2 13571 | Reverse closure for half-o... |
| elfzoelz 13572 | Reverse closure for half-o... |
| fzoval 13573 | Value of the half-open int... |
| elfzo 13574 | Membership in a half-open ... |
| elfzo2 13575 | Membership in a half-open ... |
| elfzouz 13576 | Membership in a half-open ... |
| nelfzo 13577 | An integer not being a mem... |
| fzolb 13578 | The left endpoint of a hal... |
| fzolb2 13579 | The left endpoint of a hal... |
| elfzole1 13580 | A member in a half-open in... |
| elfzolt2 13581 | A member in a half-open in... |
| elfzolt3 13582 | Membership in a half-open ... |
| elfzolt2b 13583 | A member in a half-open in... |
| elfzolt3b 13584 | Membership in a half-open ... |
| elfzop1le2 13585 | A member in a half-open in... |
| fzonel 13586 | A half-open range does not... |
| elfzouz2 13587 | The upper bound of a half-... |
| elfzofz 13588 | A half-open range is conta... |
| elfzo3 13589 | Express membership in a ha... |
| fzon0 13590 | A half-open integer interv... |
| fzossfz 13591 | A half-open range is conta... |
| fzossz 13592 | A half-open integer interv... |
| fzon 13593 | A half-open set of sequent... |
| fzo0n 13594 | A half-open range of nonne... |
| fzonlt0 13595 | A half-open integer range ... |
| fzo0 13596 | Half-open sets with equal ... |
| fzonnsub 13597 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13598 | If ` M < N ` then ` N - M ... |
| fzoss1 13599 | Subset relationship for ha... |
| fzoss2 13600 | Subset relationship for ha... |
| fzossrbm1 13601 | Subset of a half-open rang... |
| fzo0ss1 13602 | Subset relationship for ha... |
| fzossnn0 13603 | A half-open integer range ... |
| fzospliti 13604 | One direction of splitting... |
| fzosplit 13605 | Split a half-open integer ... |
| fzodisj 13606 | Abutting half-open integer... |
| fzouzsplit 13607 | Split an upper integer set... |
| fzouzdisj 13608 | A half-open integer range ... |
| fzoun 13609 | A half-open integer range ... |
| fzodisjsn 13610 | A half-open integer range ... |
| prinfzo0 13611 | The intersection of a half... |
| lbfzo0 13612 | An integer is strictly gre... |
| elfzo0 13613 | Membership in a half-open ... |
| elfzo0z 13614 | Membership in a half-open ... |
| nn0p1elfzo 13615 | A nonnegative integer incr... |
| elfzo0le 13616 | A member in a half-open ra... |
| elfzonn0 13617 | A member of a half-open ra... |
| fzonmapblen 13618 | The result of subtracting ... |
| fzofzim 13619 | If a nonnegative integer i... |
| fz1fzo0m1 13620 | Translation of one between... |
| fzossnn 13621 | Half-open integer ranges s... |
| elfzo1 13622 | Membership in a half-open ... |
| fzo1fzo0n0 13623 | An integer between 1 and a... |
| fzo0n0 13624 | A half-open integer range ... |
| fzoaddel 13625 | Translate membership in a ... |
| fzo0addel 13626 | Translate membership in a ... |
| fzo0addelr 13627 | Translate membership in a ... |
| fzoaddel2 13628 | Translate membership in a ... |
| elfzoext 13629 | Membership of an integer i... |
| elincfzoext 13630 | Membership of an increased... |
| fzosubel 13631 | Translate membership in a ... |
| fzosubel2 13632 | Membership in a translated... |
| fzosubel3 13633 | Membership in a translated... |
| eluzgtdifelfzo 13634 | Membership of the differen... |
| ige2m2fzo 13635 | Membership of an integer g... |
| fzocatel 13636 | Translate membership in a ... |
| ubmelfzo 13637 | If an integer in a 1-based... |
| elfzodifsumelfzo 13638 | If an integer is in a half... |
| elfzom1elp1fzo 13639 | Membership of an integer i... |
| elfzom1elfzo 13640 | Membership in a half-open ... |
| fzval3 13641 | Expressing a closed intege... |
| fz0add1fz1 13642 | Translate membership in a ... |
| fzosn 13643 | Expressing a singleton as ... |
| elfzomin 13644 | Membership of an integer i... |
| zpnn0elfzo 13645 | Membership of an integer i... |
| zpnn0elfzo1 13646 | Membership of an integer i... |
| fzosplitsnm1 13647 | Removing a singleton from ... |
| elfzonlteqm1 13648 | If an element of a half-op... |
| fzonn0p1 13649 | A nonnegative integer is e... |
| fzossfzop1 13650 | A half-open range of nonne... |
| fzonn0p1p1 13651 | If a nonnegative integer i... |
| elfzom1p1elfzo 13652 | Increasing an element of a... |
| fzo0ssnn0 13653 | Half-open integer ranges s... |
| fzo01 13654 | Expressing the singleton o... |
| fzo12sn 13655 | A 1-based half-open intege... |
| fzo13pr 13656 | A 1-based half-open intege... |
| fzo0to2pr 13657 | A half-open integer range ... |
| fzo0to3tp 13658 | A half-open integer range ... |
| fzo0to42pr 13659 | A half-open integer range ... |
| fzo1to4tp 13660 | A half-open integer range ... |
| fzo0sn0fzo1 13661 | A half-open range of nonne... |
| elfzo0l 13662 | A member of a half-open ra... |
| fzoend 13663 | The endpoint of a half-ope... |
| fzo0end 13664 | The endpoint of a zero-bas... |
| ssfzo12 13665 | Subset relationship for ha... |
| ssfzoulel 13666 | If a half-open integer ran... |
| ssfzo12bi 13667 | Subset relationship for ha... |
| ubmelm1fzo 13668 | The result of subtracting ... |
| fzofzp1 13669 | If a point is in a half-op... |
| fzofzp1b 13670 | If a point is in a half-op... |
| elfzom1b 13671 | An integer is a member of ... |
| elfzom1elp1fzo1 13672 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13673 | Membership of a positive i... |
| elfzonelfzo 13674 | If an element of a half-op... |
| fzonfzoufzol 13675 | If an element of a half-op... |
| elfzomelpfzo 13676 | An integer increased by an... |
| elfznelfzo 13677 | A value in a finite set of... |
| elfznelfzob 13678 | A value in a finite set of... |
| peano2fzor 13679 | A Peano-postulate-like the... |
| fzosplitsn 13680 | Extending a half-open rang... |
| fzosplitpr 13681 | Extending a half-open inte... |
| fzosplitprm1 13682 | Extending a half-open inte... |
| fzosplitsni 13683 | Membership in a half-open ... |
| fzisfzounsn 13684 | A finite interval of integ... |
| elfzr 13685 | A member of a finite inter... |
| elfzlmr 13686 | A member of a finite inter... |
| elfz0lmr 13687 | A member of a finite inter... |
| fzostep1 13688 | Two possibilities for a nu... |
| fzoshftral 13689 | Shift the scanning order i... |
| fzind2 13690 | Induction on the integers ... |
| fvinim0ffz 13691 | The function values for th... |
| injresinjlem 13692 | Lemma for ~ injresinj . (... |
| injresinj 13693 | A function whose restricti... |
| subfzo0 13694 | The difference between two... |
| flval 13699 | Value of the floor (greate... |
| flcl 13700 | The floor (greatest intege... |
| reflcl 13701 | The floor (greatest intege... |
| fllelt 13702 | A basic property of the fl... |
| flcld 13703 | The floor (greatest intege... |
| flle 13704 | A basic property of the fl... |
| flltp1 13705 | A basic property of the fl... |
| fllep1 13706 | A basic property of the fl... |
| fraclt1 13707 | The fractional part of a r... |
| fracle1 13708 | The fractional part of a r... |
| fracge0 13709 | The fractional part of a r... |
| flge 13710 | The floor function value i... |
| fllt 13711 | The floor function value i... |
| flflp1 13712 | Move floor function betwee... |
| flid 13713 | An integer is its own floo... |
| flidm 13714 | The floor function is idem... |
| flidz 13715 | A real number equals its f... |
| flltnz 13716 | The floor of a non-integer... |
| flwordi 13717 | Ordering relation for the ... |
| flword2 13718 | Ordering relation for the ... |
| flval2 13719 | An alternate way to define... |
| flval3 13720 | An alternate way to define... |
| flbi 13721 | A condition equivalent to ... |
| flbi2 13722 | A condition equivalent to ... |
| adddivflid 13723 | The floor of a sum of an i... |
| ico01fl0 13724 | The floor of a real number... |
| flge0nn0 13725 | The floor of a number grea... |
| flge1nn 13726 | The floor of a number grea... |
| fldivnn0 13727 | The floor function of a di... |
| refldivcl 13728 | The floor function of a di... |
| divfl0 13729 | The floor of a fraction is... |
| fladdz 13730 | An integer can be moved in... |
| flzadd 13731 | An integer can be moved in... |
| flmulnn0 13732 | Move a nonnegative integer... |
| btwnzge0 13733 | A real bounded between an ... |
| 2tnp1ge0ge0 13734 | Two times an integer plus ... |
| flhalf 13735 | Ordering relation for the ... |
| fldivle 13736 | The floor function of a di... |
| fldivnn0le 13737 | The floor function of a di... |
| flltdivnn0lt 13738 | The floor function of a di... |
| ltdifltdiv 13739 | If the dividend of a divis... |
| fldiv4p1lem1div2 13740 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13741 | The floor of an integer gr... |
| fldiv4lem1div2 13742 | The floor of a positive in... |
| ceilval 13743 | The value of the ceiling f... |
| dfceil2 13744 | Alternative definition of ... |
| ceilval2 13745 | The value of the ceiling f... |
| ceicl 13746 | The ceiling function retur... |
| ceilcl 13747 | Closure of the ceiling fun... |
| ceilcld 13748 | Closure of the ceiling fun... |
| ceige 13749 | The ceiling of a real numb... |
| ceilge 13750 | The ceiling of a real numb... |
| ceilged 13751 | The ceiling of a real numb... |
| ceim1l 13752 | One less than the ceiling ... |
| ceilm1lt 13753 | One less than the ceiling ... |
| ceile 13754 | The ceiling of a real numb... |
| ceille 13755 | The ceiling of a real numb... |
| ceilid 13756 | An integer is its own ceil... |
| ceilidz 13757 | A real number equals its c... |
| flleceil 13758 | The floor of a real number... |
| fleqceilz 13759 | A real number is an intege... |
| quoremz 13760 | Quotient and remainder of ... |
| quoremnn0 13761 | Quotient and remainder of ... |
| quoremnn0ALT 13762 | Alternate proof of ~ quore... |
| intfrac2 13763 | Decompose a real into inte... |
| intfracq 13764 | Decompose a rational numbe... |
| fldiv 13765 | Cancellation of the embedd... |
| fldiv2 13766 | Cancellation of an embedde... |
| fznnfl 13767 | Finite set of sequential i... |
| uzsup 13768 | An upper set of integers i... |
| ioopnfsup 13769 | An upper set of reals is u... |
| icopnfsup 13770 | An upper set of reals is u... |
| rpsup 13771 | The positive reals are unb... |
| resup 13772 | The real numbers are unbou... |
| xrsup 13773 | The extended real numbers ... |
| modval 13776 | The value of the modulo op... |
| modvalr 13777 | The value of the modulo op... |
| modcl 13778 | Closure law for the modulo... |
| flpmodeq 13779 | Partition of a division in... |
| modcld 13780 | Closure law for the modulo... |
| mod0 13781 | ` A mod B ` is zero iff ` ... |
| mulmod0 13782 | The product of an integer ... |
| negmod0 13783 | ` A ` is divisible by ` B ... |
| modge0 13784 | The modulo operation is no... |
| modlt 13785 | The modulo operation is le... |
| modelico 13786 | Modular reduction produces... |
| moddiffl 13787 | Value of the modulo operat... |
| moddifz 13788 | The modulo operation diffe... |
| modfrac 13789 | The fractional part of a n... |
| flmod 13790 | The floor function express... |
| intfrac 13791 | Break a number into its in... |
| zmod10 13792 | An integer modulo 1 is 0. ... |
| zmod1congr 13793 | Two arbitrary integers are... |
| modmulnn 13794 | Move a positive integer in... |
| modvalp1 13795 | The value of the modulo op... |
| zmodcl 13796 | Closure law for the modulo... |
| zmodcld 13797 | Closure law for the modulo... |
| zmodfz 13798 | An integer mod ` B ` lies ... |
| zmodfzo 13799 | An integer mod ` B ` lies ... |
| zmodfzp1 13800 | An integer mod ` B ` lies ... |
| modid 13801 | Identity law for modulo. ... |
| modid0 13802 | A positive real number mod... |
| modid2 13803 | Identity law for modulo. ... |
| zmodid2 13804 | Identity law for modulo re... |
| zmodidfzo 13805 | Identity law for modulo re... |
| zmodidfzoimp 13806 | Identity law for modulo re... |
| 0mod 13807 | Special case: 0 modulo a p... |
| 1mod 13808 | Special case: 1 modulo a r... |
| modabs 13809 | Absorption law for modulo.... |
| modabs2 13810 | Absorption law for modulo.... |
| modcyc 13811 | The modulo operation is pe... |
| modcyc2 13812 | The modulo operation is pe... |
| modadd1 13813 | Addition property of the m... |
| modaddabs 13814 | Absorption law for modulo.... |
| modaddmod 13815 | The sum of a real number m... |
| muladdmodid 13816 | The sum of a positive real... |
| mulp1mod1 13817 | The product of an integer ... |
| modmuladd 13818 | Decomposition of an intege... |
| modmuladdim 13819 | Implication of a decomposi... |
| modmuladdnn0 13820 | Implication of a decomposi... |
| negmod 13821 | The negation of a number m... |
| m1modnnsub1 13822 | Minus one modulo a positiv... |
| m1modge3gt1 13823 | Minus one modulo an intege... |
| addmodid 13824 | The sum of a positive inte... |
| addmodidr 13825 | The sum of a positive inte... |
| modadd2mod 13826 | The sum of a real number m... |
| modm1p1mod0 13827 | If a real number modulo a ... |
| modltm1p1mod 13828 | If a real number modulo a ... |
| modmul1 13829 | Multiplication property of... |
| modmul12d 13830 | Multiplication property of... |
| modnegd 13831 | Negation property of the m... |
| modadd12d 13832 | Additive property of the m... |
| modsub12d 13833 | Subtraction property of th... |
| modsubmod 13834 | The difference of a real n... |
| modsubmodmod 13835 | The difference of a real n... |
| 2txmodxeq0 13836 | Two times a positive real ... |
| 2submod 13837 | If a real number is betwee... |
| modifeq2int 13838 | If a nonnegative integer i... |
| modaddmodup 13839 | The sum of an integer modu... |
| modaddmodlo 13840 | The sum of an integer modu... |
| modmulmod 13841 | The product of a real numb... |
| modmulmodr 13842 | The product of an integer ... |
| modaddmulmod 13843 | The sum of a real number a... |
| moddi 13844 | Distribute multiplication ... |
| modsubdir 13845 | Distribute the modulo oper... |
| modeqmodmin 13846 | A real number equals the d... |
| modirr 13847 | A number modulo an irratio... |
| modfzo0difsn 13848 | For a number within a half... |
| modsumfzodifsn 13849 | The sum of a number within... |
| modlteq 13850 | Two nonnegative integers l... |
| addmodlteq 13851 | Two nonnegative integers l... |
| om2uz0i 13852 | The mapping ` G ` is a one... |
| om2uzsuci 13853 | The value of ` G ` (see ~ ... |
| om2uzuzi 13854 | The value ` G ` (see ~ om2... |
| om2uzlti 13855 | Less-than relation for ` G... |
| om2uzlt2i 13856 | The mapping ` G ` (see ~ o... |
| om2uzrani 13857 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13858 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13859 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13860 | An alternative definition ... |
| om2uzrdg 13861 | A helper lemma for the val... |
| uzrdglem 13862 | A helper lemma for the val... |
| uzrdgfni 13863 | The recursive definition g... |
| uzrdg0i 13864 | Initial value of a recursi... |
| uzrdgsuci 13865 | Successor value of a recur... |
| ltweuz 13866 | ` < ` is a well-founded re... |
| ltwenn 13867 | Less than well-orders the ... |
| ltwefz 13868 | Less than well-orders a se... |
| uzenom 13869 | An upper integer set is de... |
| uzinf 13870 | An upper integer set is in... |
| nnnfi 13871 | The set of positive intege... |
| uzrdgxfr 13872 | Transfer the value of the ... |
| fzennn 13873 | The cardinality of a finit... |
| fzen2 13874 | The cardinality of a finit... |
| cardfz 13875 | The cardinality of a finit... |
| hashgf1o 13876 | ` G ` maps ` _om ` one-to-... |
| fzfi 13877 | A finite interval of integ... |
| fzfid 13878 | Commonly used special case... |
| fzofi 13879 | Half-open integer sets are... |
| fsequb 13880 | The values of a finite rea... |
| fsequb2 13881 | The values of a finite rea... |
| fseqsupcl 13882 | The values of a finite rea... |
| fseqsupubi 13883 | The values of a finite rea... |
| nn0ennn 13884 | The nonnegative integers a... |
| nnenom 13885 | The set of positive intege... |
| nnct 13886 | ` NN ` is countable. (Con... |
| uzindi 13887 | Indirect strong induction ... |
| axdc4uzlem 13888 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13889 | A version of ~ axdc4 that ... |
| ssnn0fi 13890 | A subset of the nonnegativ... |
| rabssnn0fi 13891 | A subset of the nonnegativ... |
| uzsinds 13892 | Strong (or "total") induct... |
| nnsinds 13893 | Strong (or "total") induct... |
| nn0sinds 13894 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13895 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13896 | If all functions of a fini... |
| fsuppmapnn0fiubex 13897 | If all functions of a fini... |
| fsuppmapnn0fiub0 13898 | If all functions of a fini... |
| suppssfz 13899 | Condition for a function o... |
| fsuppmapnn0ub 13900 | If a function over the non... |
| fsuppmapnn0fz 13901 | If a function over the non... |
| mptnn0fsupp 13902 | A mapping from the nonnega... |
| mptnn0fsuppd 13903 | A mapping from the nonnega... |
| mptnn0fsuppr 13904 | A finitely supported mappi... |
| f13idfv 13905 | A one-to-one function with... |
| seqex 13908 | Existence of the sequence ... |
| seqeq1 13909 | Equality theorem for the s... |
| seqeq2 13910 | Equality theorem for the s... |
| seqeq3 13911 | Equality theorem for the s... |
| seqeq1d 13912 | Equality deduction for the... |
| seqeq2d 13913 | Equality deduction for the... |
| seqeq3d 13914 | Equality deduction for the... |
| seqeq123d 13915 | Equality deduction for the... |
| nfseq 13916 | Hypothesis builder for the... |
| seqval 13917 | Value of the sequence buil... |
| seqfn 13918 | The sequence builder funct... |
| seq1 13919 | Value of the sequence buil... |
| seq1i 13920 | Value of the sequence buil... |
| seqp1 13921 | Value of the sequence buil... |
| seqexw 13922 | Weak version of ~ seqex th... |
| seqp1d 13923 | Value of the sequence buil... |
| seqp1iOLD 13924 | Obsolete version of ~ seqp... |
| seqm1 13925 | Value of the sequence buil... |
| seqcl2 13926 | Closure properties of the ... |
| seqf2 13927 | Range of the recursive seq... |
| seqcl 13928 | Closure properties of the ... |
| seqf 13929 | Range of the recursive seq... |
| seqfveq2 13930 | Equality of sequences. (C... |
| seqfeq2 13931 | Equality of sequences. (C... |
| seqfveq 13932 | Equality of sequences. (C... |
| seqfeq 13933 | Equality of sequences. (C... |
| seqshft2 13934 | Shifting the index set of ... |
| seqres 13935 | Restricting its characteri... |
| serf 13936 | An infinite series of comp... |
| serfre 13937 | An infinite series of real... |
| monoord 13938 | Ordering relation for a mo... |
| monoord2 13939 | Ordering relation for a mo... |
| sermono 13940 | The partial sums in an inf... |
| seqsplit 13941 | Split a sequence into two ... |
| seq1p 13942 | Removing the first term fr... |
| seqcaopr3 13943 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 13944 | The sum of two infinite se... |
| seqcaopr 13945 | The sum of two infinite se... |
| seqf1olem2a 13946 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 13947 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 13948 | Lemma for ~ seqf1o . (Con... |
| seqf1o 13949 | Rearrange a sum via an arb... |
| seradd 13950 | The sum of two infinite se... |
| sersub 13951 | The difference of two infi... |
| seqid3 13952 | A sequence that consists e... |
| seqid 13953 | Discarding the first few t... |
| seqid2 13954 | The last few partial sums ... |
| seqhomo 13955 | Apply a homomorphism to a ... |
| seqz 13956 | If the operation ` .+ ` ha... |
| seqfeq4 13957 | Equality of series under d... |
| seqfeq3 13958 | Equality of series under d... |
| seqdistr 13959 | The distributive property ... |
| ser0 13960 | The value of the partial s... |
| ser0f 13961 | A zero-valued infinite ser... |
| serge0 13962 | A finite sum of nonnegativ... |
| serle 13963 | Comparison of partial sums... |
| ser1const 13964 | Value of the partial serie... |
| seqof 13965 | Distribute function operat... |
| seqof2 13966 | Distribute function operat... |
| expval 13969 | Value of exponentiation to... |
| expnnval 13970 | Value of exponentiation to... |
| exp0 13971 | Value of a complex number ... |
| 0exp0e1 13972 | The zeroth power of zero e... |
| exp1 13973 | Value of a complex number ... |
| expp1 13974 | Value of a complex number ... |
| expneg 13975 | Value of a complex number ... |
| expneg2 13976 | Value of a complex number ... |
| expn1 13977 | A complex number raised to... |
| expcllem 13978 | Lemma for proving nonnegat... |
| expcl2lem 13979 | Lemma for proving integer ... |
| nnexpcl 13980 | Closure of exponentiation ... |
| nn0expcl 13981 | Closure of exponentiation ... |
| zexpcl 13982 | Closure of exponentiation ... |
| qexpcl 13983 | Closure of exponentiation ... |
| reexpcl 13984 | Closure of exponentiation ... |
| expcl 13985 | Closure law for nonnegativ... |
| rpexpcl 13986 | Closure law for integer ex... |
| qexpclz 13987 | Closure of integer exponen... |
| reexpclz 13988 | Closure of integer exponen... |
| expclzlem 13989 | Lemma for ~ expclz . (Con... |
| expclz 13990 | Closure law for integer ex... |
| m1expcl2 13991 | Closure of integer exponen... |
| m1expcl 13992 | Closure of exponentiation ... |
| zexpcld 13993 | Closure of exponentiation ... |
| nn0expcli 13994 | Closure of exponentiation ... |
| nn0sqcl 13995 | The square of a nonnegativ... |
| expm1t 13996 | Exponentiation in terms of... |
| 1exp 13997 | Value of 1 raised to an in... |
| expeq0 13998 | A positive integer power i... |
| expne0 13999 | A positive integer power i... |
| expne0i 14000 | An integer power is nonzer... |
| expgt0 14001 | A positive real raised to ... |
| expnegz 14002 | Value of a nonzero complex... |
| 0exp 14003 | Value of zero raised to a ... |
| expge0 14004 | A nonnegative real raised ... |
| expge1 14005 | A real greater than or equ... |
| expgt1 14006 | A real greater than 1 rais... |
| mulexp 14007 | Nonnegative integer expone... |
| mulexpz 14008 | Integer exponentiation of ... |
| exprec 14009 | Integer exponentiation of ... |
| expadd 14010 | Sum of exponents law for n... |
| expaddzlem 14011 | Lemma for ~ expaddz . (Co... |
| expaddz 14012 | Sum of exponents law for i... |
| expmul 14013 | Product of exponents law f... |
| expmulz 14014 | Product of exponents law f... |
| m1expeven 14015 | Exponentiation of negative... |
| expsub 14016 | Exponent subtraction law f... |
| expp1z 14017 | Value of a nonzero complex... |
| expm1 14018 | Value of a nonzero complex... |
| expdiv 14019 | Nonnegative integer expone... |
| sqval 14020 | Value of the square of a c... |
| sqneg 14021 | The square of the negative... |
| sqsubswap 14022 | Swap the order of subtract... |
| sqcl 14023 | Closure of square. (Contr... |
| sqmul 14024 | Distribution of squaring o... |
| sqeq0 14025 | A complex number is zero i... |
| sqdiv 14026 | Distribution of squaring o... |
| sqdivid 14027 | The square of a nonzero co... |
| sqne0 14028 | A complex number is nonzer... |
| resqcl 14029 | Closure of squaring in rea... |
| resqcld 14030 | Closure of squaring in rea... |
| sqgt0 14031 | The square of a nonzero re... |
| sqn0rp 14032 | The square of a nonzero re... |
| nnsqcl 14033 | The positive naturals are ... |
| zsqcl 14034 | Integers are closed under ... |
| qsqcl 14035 | The square of a rational i... |
| sq11 14036 | The square function is one... |
| nn0sq11 14037 | The square function is one... |
| lt2sq 14038 | The square function is inc... |
| le2sq 14039 | The square function is non... |
| le2sq2 14040 | The square function is non... |
| sqge0 14041 | The square of a real is no... |
| sqge0d 14042 | The square of a real is no... |
| zsqcl2 14043 | The square of an integer i... |
| 0expd 14044 | Value of zero raised to a ... |
| exp0d 14045 | Value of a complex number ... |
| exp1d 14046 | Value of a complex number ... |
| expeq0d 14047 | If a positive integer powe... |
| sqvald 14048 | Value of square. Inferenc... |
| sqcld 14049 | Closure of square. (Contr... |
| sqeq0d 14050 | A number is zero iff its s... |
| expcld 14051 | Closure law for nonnegativ... |
| expp1d 14052 | Value of a complex number ... |
| expaddd 14053 | Sum of exponents law for n... |
| expmuld 14054 | Product of exponents law f... |
| sqrecd 14055 | Square of reciprocal is re... |
| expclzd 14056 | Closure law for integer ex... |
| expne0d 14057 | A nonnegative integer powe... |
| expnegd 14058 | Value of a nonzero complex... |
| exprecd 14059 | An integer power of a reci... |
| expp1zd 14060 | Value of a nonzero complex... |
| expm1d 14061 | Value of a nonzero complex... |
| expsubd 14062 | Exponent subtraction law f... |
| sqmuld 14063 | Distribution of squaring o... |
| sqdivd 14064 | Distribution of squaring o... |
| expdivd 14065 | Nonnegative integer expone... |
| mulexpd 14066 | Nonnegative integer expone... |
| znsqcld 14067 | The square of a nonzero in... |
| reexpcld 14068 | Closure of exponentiation ... |
| expge0d 14069 | A nonnegative real raised ... |
| expge1d 14070 | A real greater than or equ... |
| ltexp2a 14071 | Exponent ordering relation... |
| expmordi 14072 | Base ordering relationship... |
| rpexpmord 14073 | Base ordering relationship... |
| expcan 14074 | Cancellation law for integ... |
| ltexp2 14075 | Strict ordering law for ex... |
| leexp2 14076 | Ordering law for exponenti... |
| leexp2a 14077 | Weak ordering relationship... |
| ltexp2r 14078 | The integer powers of a fi... |
| leexp2r 14079 | Weak ordering relationship... |
| leexp1a 14080 | Weak base ordering relatio... |
| exple1 14081 | A real between 0 and 1 inc... |
| expubnd 14082 | An upper bound on ` A ^ N ... |
| sumsqeq0 14083 | The sum of two squres of r... |
| sqvali 14084 | Value of square. Inferenc... |
| sqcli 14085 | Closure of square. (Contr... |
| sqeq0i 14086 | A complex number is zero i... |
| sqrecii 14087 | The square of a reciprocal... |
| sqmuli 14088 | Distribution of squaring o... |
| sqdivi 14089 | Distribution of squaring o... |
| resqcli 14090 | Closure of square in reals... |
| sqgt0i 14091 | The square of a nonzero re... |
| sqge0i 14092 | The square of a real is no... |
| lt2sqi 14093 | The square function on non... |
| le2sqi 14094 | The square function on non... |
| sq11i 14095 | The square function is one... |
| sq0 14096 | The square of 0 is 0. (Co... |
| sq0i 14097 | If a number is zero, then ... |
| sq0id 14098 | If a number is zero, then ... |
| sq1 14099 | The square of 1 is 1. (Co... |
| neg1sqe1 14100 | The square of ` -u 1 ` is ... |
| sq2 14101 | The square of 2 is 4. (Co... |
| sq3 14102 | The square of 3 is 9. (Co... |
| sq4e2t8 14103 | The square of 4 is 2 times... |
| cu2 14104 | The cube of 2 is 8. (Cont... |
| irec 14105 | The reciprocal of ` _i ` .... |
| i2 14106 | ` _i ` squared. (Contribu... |
| i3 14107 | ` _i ` cubed. (Contribute... |
| i4 14108 | ` _i ` to the fourth power... |
| nnlesq 14109 | A positive integer is less... |
| zzlesq 14110 | An integer is less than or... |
| iexpcyc 14111 | Taking ` _i ` to the ` K `... |
| expnass 14112 | A counterexample showing t... |
| sqlecan 14113 | Cancel one factor of a squ... |
| subsq 14114 | Factor the difference of t... |
| subsq2 14115 | Express the difference of ... |
| binom2i 14116 | The square of a binomial. ... |
| subsqi 14117 | Factor the difference of t... |
| sqeqori 14118 | The squares of two complex... |
| subsq0i 14119 | The two solutions to the d... |
| sqeqor 14120 | The squares of two complex... |
| binom2 14121 | The square of a binomial. ... |
| binom21 14122 | Special case of ~ binom2 w... |
| binom2sub 14123 | Expand the square of a sub... |
| binom2sub1 14124 | Special case of ~ binom2su... |
| binom2subi 14125 | Expand the square of a sub... |
| mulbinom2 14126 | The square of a binomial w... |
| binom3 14127 | The cube of a binomial. (... |
| sq01 14128 | If a complex number equals... |
| zesq 14129 | An integer is even iff its... |
| nnesq 14130 | A positive integer is even... |
| crreczi 14131 | Reciprocal of a complex nu... |
| bernneq 14132 | Bernoulli's inequality, du... |
| bernneq2 14133 | Variation of Bernoulli's i... |
| bernneq3 14134 | A corollary of ~ bernneq .... |
| expnbnd 14135 | Exponentiation with a base... |
| expnlbnd 14136 | The reciprocal of exponent... |
| expnlbnd2 14137 | The reciprocal of exponent... |
| expmulnbnd 14138 | Exponentiation with a base... |
| digit2 14139 | Two ways to express the ` ... |
| digit1 14140 | Two ways to express the ` ... |
| modexp 14141 | Exponentiation property of... |
| discr1 14142 | A nonnegative quadratic fo... |
| discr 14143 | If a quadratic polynomial ... |
| expnngt1 14144 | If an integer power with a... |
| expnngt1b 14145 | An integer power with an i... |
| sqoddm1div8 14146 | A squared odd number minus... |
| nnsqcld 14147 | The naturals are closed un... |
| nnexpcld 14148 | Closure of exponentiation ... |
| nn0expcld 14149 | Closure of exponentiation ... |
| rpexpcld 14150 | Closure law for exponentia... |
| ltexp2rd 14151 | The power of a positive nu... |
| reexpclzd 14152 | Closure of exponentiation ... |
| sqgt0d 14153 | The square of a nonzero re... |
| ltexp2d 14154 | Ordering relationship for ... |
| leexp2d 14155 | Ordering law for exponenti... |
| expcand 14156 | Ordering relationship for ... |
| leexp2ad 14157 | Ordering relationship for ... |
| leexp2rd 14158 | Ordering relationship for ... |
| lt2sqd 14159 | The square function on non... |
| le2sqd 14160 | The square function on non... |
| sq11d 14161 | The square function is one... |
| mulsubdivbinom2 14162 | The square of a binomial w... |
| muldivbinom2 14163 | The square of a binomial w... |
| sq10 14164 | The square of 10 is 100. ... |
| sq10e99m1 14165 | The square of 10 is 99 plu... |
| 3dec 14166 | A "decimal constructor" wh... |
| nn0le2msqi 14167 | The square function on non... |
| nn0opthlem1 14168 | A rather pretty lemma for ... |
| nn0opthlem2 14169 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 14170 | An ordered pair theorem fo... |
| nn0opth2i 14171 | An ordered pair theorem fo... |
| nn0opth2 14172 | An ordered pair theorem fo... |
| facnn 14175 | Value of the factorial fun... |
| fac0 14176 | The factorial of 0. (Cont... |
| fac1 14177 | The factorial of 1. (Cont... |
| facp1 14178 | The factorial of a success... |
| fac2 14179 | The factorial of 2. (Cont... |
| fac3 14180 | The factorial of 3. (Cont... |
| fac4 14181 | The factorial of 4. (Cont... |
| facnn2 14182 | Value of the factorial fun... |
| faccl 14183 | Closure of the factorial f... |
| faccld 14184 | Closure of the factorial f... |
| facmapnn 14185 | The factorial function res... |
| facne0 14186 | The factorial function is ... |
| facdiv 14187 | A positive integer divides... |
| facndiv 14188 | No positive integer (great... |
| facwordi 14189 | Ordering property of facto... |
| faclbnd 14190 | A lower bound for the fact... |
| faclbnd2 14191 | A lower bound for the fact... |
| faclbnd3 14192 | A lower bound for the fact... |
| faclbnd4lem1 14193 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 14194 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 14195 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 14196 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 14197 | Variant of ~ faclbnd5 prov... |
| faclbnd5 14198 | The factorial function gro... |
| faclbnd6 14199 | Geometric lower bound for ... |
| facubnd 14200 | An upper bound for the fac... |
| facavg 14201 | The product of two factori... |
| bcval 14204 | Value of the binomial coef... |
| bcval2 14205 | Value of the binomial coef... |
| bcval3 14206 | Value of the binomial coef... |
| bcval4 14207 | Value of the binomial coef... |
| bcrpcl 14208 | Closure of the binomial co... |
| bccmpl 14209 | "Complementing" its second... |
| bcn0 14210 | ` N ` choose 0 is 1. Rema... |
| bc0k 14211 | The binomial coefficient "... |
| bcnn 14212 | ` N ` choose ` N ` is 1. ... |
| bcn1 14213 | Binomial coefficient: ` N ... |
| bcnp1n 14214 | Binomial coefficient: ` N ... |
| bcm1k 14215 | The proportion of one bino... |
| bcp1n 14216 | The proportion of one bino... |
| bcp1nk 14217 | The proportion of one bino... |
| bcval5 14218 | Write out the top and bott... |
| bcn2 14219 | Binomial coefficient: ` N ... |
| bcp1m1 14220 | Compute the binomial coeff... |
| bcpasc 14221 | Pascal's rule for the bino... |
| bccl 14222 | A binomial coefficient, in... |
| bccl2 14223 | A binomial coefficient, in... |
| bcn2m1 14224 | Compute the binomial coeff... |
| bcn2p1 14225 | Compute the binomial coeff... |
| permnn 14226 | The number of permutations... |
| bcnm1 14227 | The binomial coefficent of... |
| 4bc3eq4 14228 | The value of four choose t... |
| 4bc2eq6 14229 | The value of four choose t... |
| hashkf 14232 | The finite part of the siz... |
| hashgval 14233 | The value of the ` # ` fun... |
| hashginv 14234 | The converse of ` G ` maps... |
| hashinf 14235 | The value of the ` # ` fun... |
| hashbnd 14236 | If ` A ` has size bounded ... |
| hashfxnn0 14237 | The size function is a fun... |
| hashf 14238 | The size function maps all... |
| hashxnn0 14239 | The value of the hash func... |
| hashresfn 14240 | Restriction of the domain ... |
| dmhashres 14241 | Restriction of the domain ... |
| hashnn0pnf 14242 | The value of the hash func... |
| hashnnn0genn0 14243 | If the size of a set is no... |
| hashnemnf 14244 | The size of a set is never... |
| hashv01gt1 14245 | The size of a set is eithe... |
| hashfz1 14246 | The set ` ( 1 ... N ) ` ha... |
| hashen 14247 | Two finite sets have the s... |
| hasheni 14248 | Equinumerous sets have the... |
| hasheqf1o 14249 | The size of two finite set... |
| fiinfnf1o 14250 | There is no bijection betw... |
| hasheqf1oi 14251 | The size of two sets is eq... |
| hashf1rn 14252 | The size of a finite set w... |
| hasheqf1od 14253 | The size of two sets is eq... |
| fz1eqb 14254 | Two possibly-empty 1-based... |
| hashcard 14255 | The size function of the c... |
| hashcl 14256 | Closure of the ` # ` funct... |
| hashxrcl 14257 | Extended real closure of t... |
| hashclb 14258 | Reverse closure of the ` #... |
| nfile 14259 | The size of any infinite s... |
| hashvnfin 14260 | A set of finite size is a ... |
| hashnfinnn0 14261 | The size of an infinite se... |
| isfinite4 14262 | A finite set is equinumero... |
| hasheq0 14263 | Two ways of saying a finit... |
| hashneq0 14264 | Two ways of saying a set i... |
| hashgt0n0 14265 | If the size of a set is gr... |
| hashnncl 14266 | Positive natural closure o... |
| hash0 14267 | The empty set has size zer... |
| hashelne0d 14268 | A set with an element has ... |
| hashsng 14269 | The size of a singleton. ... |
| hashen1 14270 | A set has size 1 if and on... |
| hash1elsn 14271 | A set of size 1 with a kno... |
| hashrabrsn 14272 | The size of a restricted c... |
| hashrabsn01 14273 | The size of a restricted c... |
| hashrabsn1 14274 | If the size of a restricte... |
| hashfn 14275 | A function is equinumerous... |
| fseq1hash 14276 | The value of the size func... |
| hashgadd 14277 | ` G ` maps ordinal additio... |
| hashgval2 14278 | A short expression for the... |
| hashdom 14279 | Dominance relation for the... |
| hashdomi 14280 | Non-strict order relation ... |
| hashsdom 14281 | Strict dominance relation ... |
| hashun 14282 | The size of the union of d... |
| hashun2 14283 | The size of the union of f... |
| hashun3 14284 | The size of the union of f... |
| hashinfxadd 14285 | The extended real addition... |
| hashunx 14286 | The size of the union of d... |
| hashge0 14287 | The cardinality of a set i... |
| hashgt0 14288 | The cardinality of a nonem... |
| hashge1 14289 | The cardinality of a nonem... |
| 1elfz0hash 14290 | 1 is an element of the fin... |
| hashnn0n0nn 14291 | If a nonnegative integer i... |
| hashunsng 14292 | The size of the union of a... |
| hashunsngx 14293 | The size of the union of a... |
| hashunsnggt 14294 | The size of a set is great... |
| hashprg 14295 | The size of an unordered p... |
| elprchashprn2 14296 | If one element of an unord... |
| hashprb 14297 | The size of an unordered p... |
| hashprdifel 14298 | The elements of an unorder... |
| prhash2ex 14299 | There is (at least) one se... |
| hashle00 14300 | If the size of a set is le... |
| hashgt0elex 14301 | If the size of a set is gr... |
| hashgt0elexb 14302 | The size of a set is great... |
| hashp1i 14303 | Size of a finite ordinal. ... |
| hash1 14304 | Size of a finite ordinal. ... |
| hash2 14305 | Size of a finite ordinal. ... |
| hash3 14306 | Size of a finite ordinal. ... |
| hash4 14307 | Size of a finite ordinal. ... |
| pr0hash2ex 14308 | There is (at least) one se... |
| hashss 14309 | The size of a subset is le... |
| prsshashgt1 14310 | The size of a superset of ... |
| hashin 14311 | The size of the intersecti... |
| hashssdif 14312 | The size of the difference... |
| hashdif 14313 | The size of the difference... |
| hashdifsn 14314 | The size of the difference... |
| hashdifpr 14315 | The size of the difference... |
| hashsn01 14316 | The size of a singleton is... |
| hashsnle1 14317 | The size of a singleton is... |
| hashsnlei 14318 | Get an upper bound on a co... |
| hash1snb 14319 | The size of a set is 1 if ... |
| euhash1 14320 | The size of a set is 1 in ... |
| hash1n0 14321 | If the size of a set is 1 ... |
| hashgt12el 14322 | In a set with more than on... |
| hashgt12el2 14323 | In a set with more than on... |
| hashgt23el 14324 | A set with more than two e... |
| hashunlei 14325 | Get an upper bound on a co... |
| hashsslei 14326 | Get an upper bound on a co... |
| hashfz 14327 | Value of the numeric cardi... |
| fzsdom2 14328 | Condition for finite range... |
| hashfzo 14329 | Cardinality of a half-open... |
| hashfzo0 14330 | Cardinality of a half-open... |
| hashfzp1 14331 | Value of the numeric cardi... |
| hashfz0 14332 | Value of the numeric cardi... |
| hashxplem 14333 | Lemma for ~ hashxp . (Con... |
| hashxp 14334 | The size of the Cartesian ... |
| hashmap 14335 | The size of the set expone... |
| hashpw 14336 | The size of the power set ... |
| hashfun 14337 | A finite set is a function... |
| hashres 14338 | The number of elements of ... |
| hashreshashfun 14339 | The number of elements of ... |
| hashimarn 14340 | The size of the image of a... |
| hashimarni 14341 | If the size of the image o... |
| resunimafz0 14342 | TODO-AV: Revise using ` F... |
| fnfz0hash 14343 | The size of a function on ... |
| ffz0hash 14344 | The size of a function on ... |
| fnfz0hashnn0 14345 | The size of a function on ... |
| ffzo0hash 14346 | The size of a function on ... |
| fnfzo0hash 14347 | The size of a function on ... |
| fnfzo0hashnn0 14348 | The value of the size func... |
| hashbclem 14349 | Lemma for ~ hashbc : induc... |
| hashbc 14350 | The binomial coefficient c... |
| hashfacen 14351 | The number of bijections b... |
| hashfacenOLD 14352 | Obsolete version of ~ hash... |
| hashf1lem1 14353 | Lemma for ~ hashf1 . (Con... |
| hashf1lem1OLD 14354 | Obsolete version of ~ hash... |
| hashf1lem2 14355 | Lemma for ~ hashf1 . (Con... |
| hashf1 14356 | The permutation number ` |... |
| hashfac 14357 | A factorial counts the num... |
| leiso 14358 | Two ways to write a strict... |
| leisorel 14359 | Version of ~ isorel for st... |
| fz1isolem 14360 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14361 | Any finite ordered set has... |
| ishashinf 14362 | Any set that is not finite... |
| seqcoll 14363 | The function ` F ` contain... |
| seqcoll2 14364 | The function ` F ` contain... |
| phphashd 14365 | Corollary of the Pigeonhol... |
| phphashrd 14366 | Corollary of the Pigeonhol... |
| hashprlei 14367 | An unordered pair has at m... |
| hash2pr 14368 | A set of size two is an un... |
| hash2prde 14369 | A set of size two is an un... |
| hash2exprb 14370 | A set of size two is an un... |
| hash2prb 14371 | A set of size two is a pro... |
| prprrab 14372 | The set of proper pairs of... |
| nehash2 14373 | The cardinality of a set w... |
| hash2prd 14374 | A set of size two is an un... |
| hash2pwpr 14375 | If the size of a subset of... |
| hashle2pr 14376 | A nonempty set of size les... |
| hashle2prv 14377 | A nonempty subset of a pow... |
| pr2pwpr 14378 | The set of subsets of a pa... |
| hashge2el2dif 14379 | A set with size at least 2... |
| hashge2el2difr 14380 | A set with at least 2 diff... |
| hashge2el2difb 14381 | A set has size at least 2 ... |
| hashdmpropge2 14382 | The size of the domain of ... |
| hashtplei 14383 | An unordered triple has at... |
| hashtpg 14384 | The size of an unordered t... |
| hashge3el3dif 14385 | A set with size at least 3... |
| elss2prb 14386 | An element of the set of s... |
| hash2sspr 14387 | A subset of size two is an... |
| exprelprel 14388 | If there is an element of ... |
| hash3tr 14389 | A set of size three is an ... |
| hash1to3 14390 | If the size of a set is be... |
| fundmge2nop0 14391 | A function with a domain c... |
| fundmge2nop 14392 | A function with a domain c... |
| fun2dmnop0 14393 | A function with a domain c... |
| fun2dmnop 14394 | A function with a domain c... |
| hashdifsnp1 14395 | If the size of a set is a ... |
| fi1uzind 14396 | Properties of an ordered p... |
| brfi1uzind 14397 | Properties of a binary rel... |
| brfi1ind 14398 | Properties of a binary rel... |
| brfi1indALT 14399 | Alternate proof of ~ brfi1... |
| opfi1uzind 14400 | Properties of an ordered p... |
| opfi1ind 14401 | Properties of an ordered p... |
| iswrd 14404 | Property of being a word o... |
| wrdval 14405 | Value of the set of words ... |
| iswrdi 14406 | A zero-based sequence is a... |
| wrdf 14407 | A word is a zero-based seq... |
| iswrdb 14408 | A word over an alphabet is... |
| wrddm 14409 | The indices of a word (i.e... |
| sswrd 14410 | The set of words respects ... |
| snopiswrd 14411 | A singleton of an ordered ... |
| wrdexg 14412 | The set of words over a se... |
| wrdexb 14413 | The set of words over a se... |
| wrdexi 14414 | The set of words over a se... |
| wrdsymbcl 14415 | A symbol within a word ove... |
| wrdfn 14416 | A word is a function with ... |
| wrdv 14417 | A word over an alphabet is... |
| wrdlndm 14418 | The length of a word is no... |
| iswrdsymb 14419 | An arbitrary word is a wor... |
| wrdfin 14420 | A word is a finite set. (... |
| lencl 14421 | The length of a word is a ... |
| lennncl 14422 | The length of a nonempty w... |
| wrdffz 14423 | A word is a function from ... |
| wrdeq 14424 | Equality theorem for the s... |
| wrdeqi 14425 | Equality theorem for the s... |
| iswrddm0 14426 | A function with empty doma... |
| wrd0 14427 | The empty set is a word (t... |
| 0wrd0 14428 | The empty word is the only... |
| ffz0iswrd 14429 | A sequence with zero-based... |
| wrdsymb 14430 | A word is a word over the ... |
| nfwrd 14431 | Hypothesis builder for ` W... |
| csbwrdg 14432 | Class substitution for the... |
| wrdnval 14433 | Words of a fixed length ar... |
| wrdmap 14434 | Words as a mapping. (Cont... |
| hashwrdn 14435 | If there is only a finite ... |
| wrdnfi 14436 | If there is only a finite ... |
| wrdsymb0 14437 | A symbol at a position "ou... |
| wrdlenge1n0 14438 | A word with length at leas... |
| len0nnbi 14439 | The length of a word is a ... |
| wrdlenge2n0 14440 | A word with length at leas... |
| wrdsymb1 14441 | The first symbol of a none... |
| wrdlen1 14442 | A word of length 1 starts ... |
| fstwrdne 14443 | The first symbol of a none... |
| fstwrdne0 14444 | The first symbol of a none... |
| eqwrd 14445 | Two words are equal iff th... |
| elovmpowrd 14446 | Implications for the value... |
| elovmptnn0wrd 14447 | Implications for the value... |
| wrdred1 14448 | A word truncated by a symb... |
| wrdred1hash 14449 | The length of a word trunc... |
| lsw 14452 | Extract the last symbol of... |
| lsw0 14453 | The last symbol of an empt... |
| lsw0g 14454 | The last symbol of an empt... |
| lsw1 14455 | The last symbol of a word ... |
| lswcl 14456 | Closure of the last symbol... |
| lswlgt0cl 14457 | The last symbol of a nonem... |
| ccatfn 14460 | The concatenation operator... |
| ccatfval 14461 | Value of the concatenation... |
| ccatcl 14462 | The concatenation of two w... |
| ccatlen 14463 | The length of a concatenat... |
| ccat0 14464 | The concatenation of two w... |
| ccatval1 14465 | Value of a symbol in the l... |
| ccatval2 14466 | Value of a symbol in the r... |
| ccatval3 14467 | Value of a symbol in the r... |
| elfzelfzccat 14468 | An element of a finite set... |
| ccatvalfn 14469 | The concatenation of two w... |
| ccatsymb 14470 | The symbol at a given posi... |
| ccatfv0 14471 | The first symbol of a conc... |
| ccatval1lsw 14472 | The last symbol of the lef... |
| ccatval21sw 14473 | The first symbol of the ri... |
| ccatlid 14474 | Concatenation of a word by... |
| ccatrid 14475 | Concatenation of a word by... |
| ccatass 14476 | Associative law for concat... |
| ccatrn 14477 | The range of a concatenate... |
| ccatidid 14478 | Concatenation of the empty... |
| lswccatn0lsw 14479 | The last symbol of a word ... |
| lswccat0lsw 14480 | The last symbol of a word ... |
| ccatalpha 14481 | A concatenation of two arb... |
| ccatrcl1 14482 | Reverse closure of a conca... |
| ids1 14485 | Identity function protecti... |
| s1val 14486 | Value of a singleton word.... |
| s1rn 14487 | The range of a singleton w... |
| s1eq 14488 | Equality theorem for a sin... |
| s1eqd 14489 | Equality theorem for a sin... |
| s1cl 14490 | A singleton word is a word... |
| s1cld 14491 | A singleton word is a word... |
| s1prc 14492 | Value of a singleton word ... |
| s1cli 14493 | A singleton word is a word... |
| s1len 14494 | Length of a singleton word... |
| s1nz 14495 | A singleton word is not th... |
| s1dm 14496 | The domain of a singleton ... |
| s1dmALT 14497 | Alternate version of ~ s1d... |
| s1fv 14498 | Sole symbol of a singleton... |
| lsws1 14499 | The last symbol of a singl... |
| eqs1 14500 | A word of length 1 is a si... |
| wrdl1exs1 14501 | A word of length 1 is a si... |
| wrdl1s1 14502 | A word of length 1 is a si... |
| s111 14503 | The singleton word functio... |
| ccatws1cl 14504 | The concatenation of a wor... |
| ccatws1clv 14505 | The concatenation of a wor... |
| ccat2s1cl 14506 | The concatenation of two s... |
| ccats1alpha 14507 | A concatenation of a word ... |
| ccatws1len 14508 | The length of the concaten... |
| ccatws1lenp1b 14509 | The length of a word is ` ... |
| wrdlenccats1lenm1 14510 | The length of a word is th... |
| ccat2s1len 14511 | The length of the concaten... |
| ccatw2s1cl 14512 | The concatenation of a wor... |
| ccatw2s1len 14513 | The length of the concaten... |
| ccats1val1 14514 | Value of a symbol in the l... |
| ccats1val2 14515 | Value of the symbol concat... |
| ccat1st1st 14516 | The first symbol of a word... |
| ccat2s1p1 14517 | Extract the first of two c... |
| ccat2s1p2 14518 | Extract the second of two ... |
| ccatw2s1ass 14519 | Associative law for a conc... |
| ccatws1n0 14520 | The concatenation of a wor... |
| ccatws1ls 14521 | The last symbol of the con... |
| lswccats1 14522 | The last symbol of a word ... |
| lswccats1fst 14523 | The last symbol of a nonem... |
| ccatw2s1p1 14524 | Extract the symbol of the ... |
| ccatw2s1p2 14525 | Extract the second of two ... |
| ccat2s1fvw 14526 | Extract a symbol of a word... |
| ccat2s1fst 14527 | The first symbol of the co... |
| swrdnznd 14530 | The value of a subword ope... |
| swrdval 14531 | Value of a subword. (Cont... |
| swrd00 14532 | A zero length substring. ... |
| swrdcl 14533 | Closure of the subword ext... |
| swrdval2 14534 | Value of the subword extra... |
| swrdlen 14535 | Length of an extracted sub... |
| swrdfv 14536 | A symbol in an extracted s... |
| swrdfv0 14537 | The first symbol in an ext... |
| swrdf 14538 | A subword of a word is a f... |
| swrdvalfn 14539 | Value of the subword extra... |
| swrdrn 14540 | The range of a subword of ... |
| swrdlend 14541 | The value of the subword e... |
| swrdnd 14542 | The value of the subword e... |
| swrdnd2 14543 | Value of the subword extra... |
| swrdnnn0nd 14544 | The value of a subword ope... |
| swrdnd0 14545 | The value of a subword ope... |
| swrd0 14546 | A subword of an empty set ... |
| swrdrlen 14547 | Length of a right-anchored... |
| swrdlen2 14548 | Length of an extracted sub... |
| swrdfv2 14549 | A symbol in an extracted s... |
| swrdwrdsymb 14550 | A subword is a word over t... |
| swrdsb0eq 14551 | Two subwords with the same... |
| swrdsbslen 14552 | Two subwords with the same... |
| swrdspsleq 14553 | Two words have a common su... |
| swrds1 14554 | Extract a single symbol fr... |
| swrdlsw 14555 | Extract the last single sy... |
| ccatswrd 14556 | Joining two adjacent subwo... |
| swrdccat2 14557 | Recover the right half of ... |
| pfxnndmnd 14560 | The value of a prefix oper... |
| pfxval 14561 | Value of a prefix operatio... |
| pfx00 14562 | The zero length prefix is ... |
| pfx0 14563 | A prefix of an empty set i... |
| pfxval0 14564 | Value of a prefix operatio... |
| pfxcl 14565 | Closure of the prefix extr... |
| pfxmpt 14566 | Value of the prefix extrac... |
| pfxres 14567 | Value of the subword extra... |
| pfxf 14568 | A prefix of a word is a fu... |
| pfxfn 14569 | Value of the prefix extrac... |
| pfxfv 14570 | A symbol in a prefix of a ... |
| pfxlen 14571 | Length of a prefix. (Cont... |
| pfxid 14572 | A word is a prefix of itse... |
| pfxrn 14573 | The range of a prefix of a... |
| pfxn0 14574 | A prefix consisting of at ... |
| pfxnd 14575 | The value of a prefix oper... |
| pfxnd0 14576 | The value of a prefix oper... |
| pfxwrdsymb 14577 | A prefix of a word is a wo... |
| addlenrevpfx 14578 | The sum of the lengths of ... |
| addlenpfx 14579 | The sum of the lengths of ... |
| pfxfv0 14580 | The first symbol of a pref... |
| pfxtrcfv 14581 | A symbol in a word truncat... |
| pfxtrcfv0 14582 | The first symbol in a word... |
| pfxfvlsw 14583 | The last symbol in a nonem... |
| pfxeq 14584 | The prefixes of two words ... |
| pfxtrcfvl 14585 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14586 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14587 | Two (nonempty) words are e... |
| disjwrdpfx 14588 | Sets of words are disjoint... |
| ccatpfx 14589 | Concatenating a prefix wit... |
| pfxccat1 14590 | Recover the left half of a... |
| pfx1 14591 | The prefix of length one o... |
| swrdswrdlem 14592 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14593 | A subword of a subword is ... |
| pfxswrd 14594 | A prefix of a subword is a... |
| swrdpfx 14595 | A subword of a prefix is a... |
| pfxpfx 14596 | A prefix of a prefix is a ... |
| pfxpfxid 14597 | A prefix of a prefix with ... |
| pfxcctswrd 14598 | The concatenation of the p... |
| lenpfxcctswrd 14599 | The length of the concaten... |
| lenrevpfxcctswrd 14600 | The length of the concaten... |
| pfxlswccat 14601 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14602 | The last symbol of a word ... |
| ccats1pfxeqrex 14603 | There exists a symbol such... |
| ccatopth 14604 | An ~ opth -like theorem fo... |
| ccatopth2 14605 | An ~ opth -like theorem fo... |
| ccatlcan 14606 | Concatenation of words is ... |
| ccatrcan 14607 | Concatenation of words is ... |
| wrdeqs1cat 14608 | Decompose a nonempty word ... |
| cats1un 14609 | Express a word with an ext... |
| wrdind 14610 | Perform induction over the... |
| wrd2ind 14611 | Perform induction over the... |
| swrdccatfn 14612 | The subword of a concatena... |
| swrdccatin1 14613 | The subword of a concatena... |
| pfxccatin12lem4 14614 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14615 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14616 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14617 | The subword of a concatena... |
| pfxccatin12lem2c 14618 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14619 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14620 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14621 | The subword of a concatena... |
| pfxccat3 14622 | The subword of a concatena... |
| swrdccat 14623 | The subword of a concatena... |
| pfxccatpfx1 14624 | A prefix of a concatenatio... |
| pfxccatpfx2 14625 | A prefix of a concatenatio... |
| pfxccat3a 14626 | A prefix of a concatenatio... |
| swrdccat3blem 14627 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14628 | A suffix of a concatenatio... |
| pfxccatid 14629 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14630 | A word is a prefix of a wo... |
| swrdccatin1d 14631 | The subword of a concatena... |
| swrdccatin2d 14632 | The subword of a concatena... |
| pfxccatin12d 14633 | The subword of a concatena... |
| reuccatpfxs1lem 14634 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14635 | There is a unique word hav... |
| reuccatpfxs1v 14636 | There is a unique word hav... |
| splval 14639 | Value of the substring rep... |
| splcl 14640 | Closure of the substring r... |
| splid 14641 | Splicing a subword for the... |
| spllen 14642 | The length of a splice. (... |
| splfv1 14643 | Symbols to the left of a s... |
| splfv2a 14644 | Symbols within the replace... |
| splval2 14645 | Value of a splice, assumin... |
| revval 14648 | Value of the word reversin... |
| revcl 14649 | The reverse of a word is a... |
| revlen 14650 | The reverse of a word has ... |
| revfv 14651 | Reverse of a word at a poi... |
| rev0 14652 | The empty word is its own ... |
| revs1 14653 | Singleton words are their ... |
| revccat 14654 | Antiautomorphic property o... |
| revrev 14655 | Reversal is an involution ... |
| reps 14658 | Construct a function mappi... |
| repsundef 14659 | A function mapping a half-... |
| repsconst 14660 | Construct a function mappi... |
| repsf 14661 | The constructed function m... |
| repswsymb 14662 | The symbols of a "repeated... |
| repsw 14663 | A function mapping a half-... |
| repswlen 14664 | The length of a "repeated ... |
| repsw0 14665 | The "repeated symbol word"... |
| repsdf2 14666 | Alternative definition of ... |
| repswsymball 14667 | All the symbols of a "repe... |
| repswsymballbi 14668 | A word is a "repeated symb... |
| repswfsts 14669 | The first symbol of a none... |
| repswlsw 14670 | The last symbol of a nonem... |
| repsw1 14671 | The "repeated symbol word"... |
| repswswrd 14672 | A subword of a "repeated s... |
| repswpfx 14673 | A prefix of a repeated sym... |
| repswccat 14674 | The concatenation of two "... |
| repswrevw 14675 | The reverse of a "repeated... |
| cshfn 14678 | Perform a cyclical shift f... |
| cshword 14679 | Perform a cyclical shift f... |
| cshnz 14680 | A cyclical shift is the em... |
| 0csh0 14681 | Cyclically shifting an emp... |
| cshw0 14682 | A word cyclically shifted ... |
| cshwmodn 14683 | Cyclically shifting a word... |
| cshwsublen 14684 | Cyclically shifting a word... |
| cshwn 14685 | A word cyclically shifted ... |
| cshwcl 14686 | A cyclically shifted word ... |
| cshwlen 14687 | The length of a cyclically... |
| cshwf 14688 | A cyclically shifted word ... |
| cshwfn 14689 | A cyclically shifted word ... |
| cshwrn 14690 | The range of a cyclically ... |
| cshwidxmod 14691 | The symbol at a given inde... |
| cshwidxmodr 14692 | The symbol at a given inde... |
| cshwidx0mod 14693 | The symbol at index 0 of a... |
| cshwidx0 14694 | The symbol at index 0 of a... |
| cshwidxm1 14695 | The symbol at index ((n-N)... |
| cshwidxm 14696 | The symbol at index (n-N) ... |
| cshwidxn 14697 | The symbol at index (n-1) ... |
| cshf1 14698 | Cyclically shifting a word... |
| cshinj 14699 | If a word is injectiv (reg... |
| repswcshw 14700 | A cyclically shifted "repe... |
| 2cshw 14701 | Cyclically shifting a word... |
| 2cshwid 14702 | Cyclically shifting a word... |
| lswcshw 14703 | The last symbol of a word ... |
| 2cshwcom 14704 | Cyclically shifting a word... |
| cshwleneq 14705 | If the results of cyclical... |
| 3cshw 14706 | Cyclically shifting a word... |
| cshweqdif2 14707 | If cyclically shifting two... |
| cshweqdifid 14708 | If cyclically shifting a w... |
| cshweqrep 14709 | If cyclically shifting a w... |
| cshw1 14710 | If cyclically shifting a w... |
| cshw1repsw 14711 | If cyclically shifting a w... |
| cshwsexa 14712 | The class of (different!) ... |
| cshwsexaOLD 14713 | Obsolete version of ~ cshw... |
| 2cshwcshw 14714 | If a word is a cyclically ... |
| scshwfzeqfzo 14715 | For a nonempty word the se... |
| cshwcshid 14716 | A cyclically shifted word ... |
| cshwcsh2id 14717 | A cyclically shifted word ... |
| cshimadifsn 14718 | The image of a cyclically ... |
| cshimadifsn0 14719 | The image of a cyclically ... |
| wrdco 14720 | Mapping a word by a functi... |
| lenco 14721 | Length of a mapped word is... |
| s1co 14722 | Mapping of a singleton wor... |
| revco 14723 | Mapping of words (i.e., a ... |
| ccatco 14724 | Mapping of words commutes ... |
| cshco 14725 | Mapping of words commutes ... |
| swrdco 14726 | Mapping of words commutes ... |
| pfxco 14727 | Mapping of words commutes ... |
| lswco 14728 | Mapping of (nonempty) word... |
| repsco 14729 | Mapping of words commutes ... |
| cats1cld 14744 | Closure of concatenation w... |
| cats1co 14745 | Closure of concatenation w... |
| cats1cli 14746 | Closure of concatenation w... |
| cats1fvn 14747 | The last symbol of a conca... |
| cats1fv 14748 | A symbol other than the la... |
| cats1len 14749 | The length of concatenatio... |
| cats1cat 14750 | Closure of concatenation w... |
| cats2cat 14751 | Closure of concatenation o... |
| s2eqd 14752 | Equality theorem for a dou... |
| s3eqd 14753 | Equality theorem for a len... |
| s4eqd 14754 | Equality theorem for a len... |
| s5eqd 14755 | Equality theorem for a len... |
| s6eqd 14756 | Equality theorem for a len... |
| s7eqd 14757 | Equality theorem for a len... |
| s8eqd 14758 | Equality theorem for a len... |
| s3eq2 14759 | Equality theorem for a len... |
| s2cld 14760 | A doubleton word is a word... |
| s3cld 14761 | A length 3 string is a wor... |
| s4cld 14762 | A length 4 string is a wor... |
| s5cld 14763 | A length 5 string is a wor... |
| s6cld 14764 | A length 6 string is a wor... |
| s7cld 14765 | A length 7 string is a wor... |
| s8cld 14766 | A length 7 string is a wor... |
| s2cl 14767 | A doubleton word is a word... |
| s3cl 14768 | A length 3 string is a wor... |
| s2cli 14769 | A doubleton word is a word... |
| s3cli 14770 | A length 3 string is a wor... |
| s4cli 14771 | A length 4 string is a wor... |
| s5cli 14772 | A length 5 string is a wor... |
| s6cli 14773 | A length 6 string is a wor... |
| s7cli 14774 | A length 7 string is a wor... |
| s8cli 14775 | A length 8 string is a wor... |
| s2fv0 14776 | Extract the first symbol f... |
| s2fv1 14777 | Extract the second symbol ... |
| s2len 14778 | The length of a doubleton ... |
| s2dm 14779 | The domain of a doubleton ... |
| s3fv0 14780 | Extract the first symbol f... |
| s3fv1 14781 | Extract the second symbol ... |
| s3fv2 14782 | Extract the third symbol f... |
| s3len 14783 | The length of a length 3 s... |
| s4fv0 14784 | Extract the first symbol f... |
| s4fv1 14785 | Extract the second symbol ... |
| s4fv2 14786 | Extract the third symbol f... |
| s4fv3 14787 | Extract the fourth symbol ... |
| s4len 14788 | The length of a length 4 s... |
| s5len 14789 | The length of a length 5 s... |
| s6len 14790 | The length of a length 6 s... |
| s7len 14791 | The length of a length 7 s... |
| s8len 14792 | The length of a length 8 s... |
| lsws2 14793 | The last symbol of a doubl... |
| lsws3 14794 | The last symbol of a 3 let... |
| lsws4 14795 | The last symbol of a 4 let... |
| s2prop 14796 | A length 2 word is an unor... |
| s2dmALT 14797 | Alternate version of ~ s2d... |
| s3tpop 14798 | A length 3 word is an unor... |
| s4prop 14799 | A length 4 word is a union... |
| s3fn 14800 | A length 3 word is a funct... |
| funcnvs1 14801 | The converse of a singleto... |
| funcnvs2 14802 | The converse of a length 2... |
| funcnvs3 14803 | The converse of a length 3... |
| funcnvs4 14804 | The converse of a length 4... |
| s2f1o 14805 | A length 2 word with mutua... |
| f1oun2prg 14806 | A union of unordered pairs... |
| s4f1o 14807 | A length 4 word with mutua... |
| s4dom 14808 | The domain of a length 4 w... |
| s2co 14809 | Mapping a doubleton word b... |
| s3co 14810 | Mapping a length 3 string ... |
| s0s1 14811 | Concatenation of fixed len... |
| s1s2 14812 | Concatenation of fixed len... |
| s1s3 14813 | Concatenation of fixed len... |
| s1s4 14814 | Concatenation of fixed len... |
| s1s5 14815 | Concatenation of fixed len... |
| s1s6 14816 | Concatenation of fixed len... |
| s1s7 14817 | Concatenation of fixed len... |
| s2s2 14818 | Concatenation of fixed len... |
| s4s2 14819 | Concatenation of fixed len... |
| s4s3 14820 | Concatenation of fixed len... |
| s4s4 14821 | Concatenation of fixed len... |
| s3s4 14822 | Concatenation of fixed len... |
| s2s5 14823 | Concatenation of fixed len... |
| s5s2 14824 | Concatenation of fixed len... |
| s2eq2s1eq 14825 | Two length 2 words are equ... |
| s2eq2seq 14826 | Two length 2 words are equ... |
| s3eqs2s1eq 14827 | Two length 3 words are equ... |
| s3eq3seq 14828 | Two length 3 words are equ... |
| swrds2 14829 | Extract two adjacent symbo... |
| swrds2m 14830 | Extract two adjacent symbo... |
| wrdlen2i 14831 | Implications of a word of ... |
| wrd2pr2op 14832 | A word of length two repre... |
| wrdlen2 14833 | A word of length two. (Co... |
| wrdlen2s2 14834 | A word of length two as do... |
| wrdl2exs2 14835 | A word of length two is a ... |
| pfx2 14836 | A prefix of length two. (... |
| wrd3tpop 14837 | A word of length three rep... |
| wrdlen3s3 14838 | A word of length three as ... |
| repsw2 14839 | The "repeated symbol word"... |
| repsw3 14840 | The "repeated symbol word"... |
| swrd2lsw 14841 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14842 | Two words of length at lea... |
| ccatw2s1ccatws2 14843 | The concatenation of a wor... |
| ccat2s1fvwALT 14844 | Alternate proof of ~ ccat2... |
| wwlktovf 14845 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14846 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14847 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14848 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14849 | There is a bijection betwe... |
| eqwrds3 14850 | A word is equal with a len... |
| wrdl3s3 14851 | A word of length 3 is a le... |
| s3sndisj 14852 | The singletons consisting ... |
| s3iunsndisj 14853 | The union of singletons co... |
| ofccat 14854 | Letterwise operations on w... |
| ofs1 14855 | Letterwise operations on a... |
| ofs2 14856 | Letterwise operations on a... |
| coss12d 14857 | Subset deduction for compo... |
| trrelssd 14858 | The composition of subclas... |
| xpcogend 14859 | The most interesting case ... |
| xpcoidgend 14860 | If two classes are not dis... |
| cotr2g 14861 | Two ways of saying that th... |
| cotr2 14862 | Two ways of saying a relat... |
| cotr3 14863 | Two ways of saying a relat... |
| coemptyd 14864 | Deduction about compositio... |
| xptrrel 14865 | The cross product is alway... |
| 0trrel 14866 | The empty class is a trans... |
| cleq1lem 14867 | Equality implies bijection... |
| cleq1 14868 | Equality of relations impl... |
| clsslem 14869 | The closure of a subclass ... |
| trcleq1 14874 | Equality of relations impl... |
| trclsslem 14875 | The transitive closure (as... |
| trcleq2lem 14876 | Equality implies bijection... |
| cvbtrcl 14877 | Change of bound variable i... |
| trcleq12lem 14878 | Equality implies bijection... |
| trclexlem 14879 | Existence of relation impl... |
| trclublem 14880 | If a relation exists then ... |
| trclubi 14881 | The Cartesian product of t... |
| trclubgi 14882 | The union with the Cartesi... |
| trclub 14883 | The Cartesian product of t... |
| trclubg 14884 | The union with the Cartesi... |
| trclfv 14885 | The transitive closure of ... |
| brintclab 14886 | Two ways to express a bina... |
| brtrclfv 14887 | Two ways of expressing the... |
| brcnvtrclfv 14888 | Two ways of expressing the... |
| brtrclfvcnv 14889 | Two ways of expressing the... |
| brcnvtrclfvcnv 14890 | Two ways of expressing the... |
| trclfvss 14891 | The transitive closure (as... |
| trclfvub 14892 | The transitive closure of ... |
| trclfvlb 14893 | The transitive closure of ... |
| trclfvcotr 14894 | The transitive closure of ... |
| trclfvlb2 14895 | The transitive closure of ... |
| trclfvlb3 14896 | The transitive closure of ... |
| cotrtrclfv 14897 | The transitive closure of ... |
| trclidm 14898 | The transitive closure of ... |
| trclun 14899 | Transitive closure of a un... |
| trclfvg 14900 | The value of the transitiv... |
| trclfvcotrg 14901 | The value of the transitiv... |
| reltrclfv 14902 | The transitive closure of ... |
| dmtrclfv 14903 | The domain of the transiti... |
| reldmrelexp 14906 | The domain of the repeated... |
| relexp0g 14907 | A relation composed zero t... |
| relexp0 14908 | A relation composed zero t... |
| relexp0d 14909 | A relation composed zero t... |
| relexpsucnnr 14910 | A reduction for relation e... |
| relexp1g 14911 | A relation composed once i... |
| dfid5 14912 | Identity relation is equal... |
| dfid6 14913 | Identity relation expresse... |
| relexp1d 14914 | A relation composed once i... |
| relexpsucnnl 14915 | A reduction for relation e... |
| relexpsucl 14916 | A reduction for relation e... |
| relexpsucr 14917 | A reduction for relation e... |
| relexpsucrd 14918 | A reduction for relation e... |
| relexpsucld 14919 | A reduction for relation e... |
| relexpcnv 14920 | Commutation of converse an... |
| relexpcnvd 14921 | Commutation of converse an... |
| relexp0rel 14922 | The exponentiation of a cl... |
| relexprelg 14923 | The exponentiation of a cl... |
| relexprel 14924 | The exponentiation of a re... |
| relexpreld 14925 | The exponentiation of a re... |
| relexpnndm 14926 | The domain of an exponenti... |
| relexpdmg 14927 | The domain of an exponenti... |
| relexpdm 14928 | The domain of an exponenti... |
| relexpdmd 14929 | The domain of an exponenti... |
| relexpnnrn 14930 | The range of an exponentia... |
| relexprng 14931 | The range of an exponentia... |
| relexprn 14932 | The range of an exponentia... |
| relexprnd 14933 | The range of an exponentia... |
| relexpfld 14934 | The field of an exponentia... |
| relexpfldd 14935 | The field of an exponentia... |
| relexpaddnn 14936 | Relation composition becom... |
| relexpuzrel 14937 | The exponentiation of a cl... |
| relexpaddg 14938 | Relation composition becom... |
| relexpaddd 14939 | Relation composition becom... |
| rtrclreclem1 14942 | The reflexive, transitive ... |
| dfrtrclrec2 14943 | If two elements are connec... |
| rtrclreclem2 14944 | The reflexive, transitive ... |
| rtrclreclem3 14945 | The reflexive, transitive ... |
| rtrclreclem4 14946 | The reflexive, transitive ... |
| dfrtrcl2 14947 | The two definitions ` t* `... |
| relexpindlem 14948 | Principle of transitive in... |
| relexpind 14949 | Principle of transitive in... |
| rtrclind 14950 | Principle of transitive in... |
| shftlem 14953 | Two ways to write a shifte... |
| shftuz 14954 | A shift of the upper integ... |
| shftfval 14955 | The value of the sequence ... |
| shftdm 14956 | Domain of a relation shift... |
| shftfib 14957 | Value of a fiber of the re... |
| shftfn 14958 | Functionality and domain o... |
| shftval 14959 | Value of a sequence shifte... |
| shftval2 14960 | Value of a sequence shifte... |
| shftval3 14961 | Value of a sequence shifte... |
| shftval4 14962 | Value of a sequence shifte... |
| shftval5 14963 | Value of a shifted sequenc... |
| shftf 14964 | Functionality of a shifted... |
| 2shfti 14965 | Composite shift operations... |
| shftidt2 14966 | Identity law for the shift... |
| shftidt 14967 | Identity law for the shift... |
| shftcan1 14968 | Cancellation law for the s... |
| shftcan2 14969 | Cancellation law for the s... |
| seqshft 14970 | Shifting the index set of ... |
| sgnval 14973 | Value of the signum functi... |
| sgn0 14974 | The signum of 0 is 0. (Co... |
| sgnp 14975 | The signum of a positive e... |
| sgnrrp 14976 | The signum of a positive r... |
| sgn1 14977 | The signum of 1 is 1. (Co... |
| sgnpnf 14978 | The signum of ` +oo ` is 1... |
| sgnn 14979 | The signum of a negative e... |
| sgnmnf 14980 | The signum of ` -oo ` is -... |
| cjval 14987 | The value of the conjugate... |
| cjth 14988 | The defining property of t... |
| cjf 14989 | Domain and codomain of the... |
| cjcl 14990 | The conjugate of a complex... |
| reval 14991 | The value of the real part... |
| imval 14992 | The value of the imaginary... |
| imre 14993 | The imaginary part of a co... |
| reim 14994 | The real part of a complex... |
| recl 14995 | The real part of a complex... |
| imcl 14996 | The imaginary part of a co... |
| ref 14997 | Domain and codomain of the... |
| imf 14998 | Domain and codomain of the... |
| crre 14999 | The real part of a complex... |
| crim 15000 | The real part of a complex... |
| replim 15001 | Reconstruct a complex numb... |
| remim 15002 | Value of the conjugate of ... |
| reim0 15003 | The imaginary part of a re... |
| reim0b 15004 | A number is real iff its i... |
| rereb 15005 | A number is real iff it eq... |
| mulre 15006 | A product with a nonzero r... |
| rere 15007 | A real number equals its r... |
| cjreb 15008 | A number is real iff it eq... |
| recj 15009 | Real part of a complex con... |
| reneg 15010 | Real part of negative. (C... |
| readd 15011 | Real part distributes over... |
| resub 15012 | Real part distributes over... |
| remullem 15013 | Lemma for ~ remul , ~ immu... |
| remul 15014 | Real part of a product. (... |
| remul2 15015 | Real part of a product. (... |
| rediv 15016 | Real part of a division. ... |
| imcj 15017 | Imaginary part of a comple... |
| imneg 15018 | The imaginary part of a ne... |
| imadd 15019 | Imaginary part distributes... |
| imsub 15020 | Imaginary part distributes... |
| immul 15021 | Imaginary part of a produc... |
| immul2 15022 | Imaginary part of a produc... |
| imdiv 15023 | Imaginary part of a divisi... |
| cjre 15024 | A real number equals its c... |
| cjcj 15025 | The conjugate of the conju... |
| cjadd 15026 | Complex conjugate distribu... |
| cjmul 15027 | Complex conjugate distribu... |
| ipcnval 15028 | Standard inner product on ... |
| cjmulrcl 15029 | A complex number times its... |
| cjmulval 15030 | A complex number times its... |
| cjmulge0 15031 | A complex number times its... |
| cjneg 15032 | Complex conjugate of negat... |
| addcj 15033 | A number plus its conjugat... |
| cjsub 15034 | Complex conjugate distribu... |
| cjexp 15035 | Complex conjugate of posit... |
| imval2 15036 | The imaginary part of a nu... |
| re0 15037 | The real part of zero. (C... |
| im0 15038 | The imaginary part of zero... |
| re1 15039 | The real part of one. (Co... |
| im1 15040 | The imaginary part of one.... |
| rei 15041 | The real part of ` _i ` . ... |
| imi 15042 | The imaginary part of ` _i... |
| cj0 15043 | The conjugate of zero. (C... |
| cji 15044 | The complex conjugate of t... |
| cjreim 15045 | The conjugate of a represe... |
| cjreim2 15046 | The conjugate of the repre... |
| cj11 15047 | Complex conjugate is a one... |
| cjne0 15048 | A number is nonzero iff it... |
| cjdiv 15049 | Complex conjugate distribu... |
| cnrecnv 15050 | The inverse to the canonic... |
| sqeqd 15051 | A deduction for showing tw... |
| recli 15052 | The real part of a complex... |
| imcli 15053 | The imaginary part of a co... |
| cjcli 15054 | Closure law for complex co... |
| replimi 15055 | Construct a complex number... |
| cjcji 15056 | The conjugate of the conju... |
| reim0bi 15057 | A number is real iff its i... |
| rerebi 15058 | A real number equals its r... |
| cjrebi 15059 | A number is real iff it eq... |
| recji 15060 | Real part of a complex con... |
| imcji 15061 | Imaginary part of a comple... |
| cjmulrcli 15062 | A complex number times its... |
| cjmulvali 15063 | A complex number times its... |
| cjmulge0i 15064 | A complex number times its... |
| renegi 15065 | Real part of negative. (C... |
| imnegi 15066 | Imaginary part of negative... |
| cjnegi 15067 | Complex conjugate of negat... |
| addcji 15068 | A number plus its conjugat... |
| readdi 15069 | Real part distributes over... |
| imaddi 15070 | Imaginary part distributes... |
| remuli 15071 | Real part of a product. (... |
| immuli 15072 | Imaginary part of a produc... |
| cjaddi 15073 | Complex conjugate distribu... |
| cjmuli 15074 | Complex conjugate distribu... |
| ipcni 15075 | Standard inner product on ... |
| cjdivi 15076 | Complex conjugate distribu... |
| crrei 15077 | The real part of a complex... |
| crimi 15078 | The imaginary part of a co... |
| recld 15079 | The real part of a complex... |
| imcld 15080 | The imaginary part of a co... |
| cjcld 15081 | Closure law for complex co... |
| replimd 15082 | Construct a complex number... |
| remimd 15083 | Value of the conjugate of ... |
| cjcjd 15084 | The conjugate of the conju... |
| reim0bd 15085 | A number is real iff its i... |
| rerebd 15086 | A real number equals its r... |
| cjrebd 15087 | A number is real iff it eq... |
| cjne0d 15088 | A number is nonzero iff it... |
| recjd 15089 | Real part of a complex con... |
| imcjd 15090 | Imaginary part of a comple... |
| cjmulrcld 15091 | A complex number times its... |
| cjmulvald 15092 | A complex number times its... |
| cjmulge0d 15093 | A complex number times its... |
| renegd 15094 | Real part of negative. (C... |
| imnegd 15095 | Imaginary part of negative... |
| cjnegd 15096 | Complex conjugate of negat... |
| addcjd 15097 | A number plus its conjugat... |
| cjexpd 15098 | Complex conjugate of posit... |
| readdd 15099 | Real part distributes over... |
| imaddd 15100 | Imaginary part distributes... |
| resubd 15101 | Real part distributes over... |
| imsubd 15102 | Imaginary part distributes... |
| remuld 15103 | Real part of a product. (... |
| immuld 15104 | Imaginary part of a produc... |
| cjaddd 15105 | Complex conjugate distribu... |
| cjmuld 15106 | Complex conjugate distribu... |
| ipcnd 15107 | Standard inner product on ... |
| cjdivd 15108 | Complex conjugate distribu... |
| rered 15109 | A real number equals its r... |
| reim0d 15110 | The imaginary part of a re... |
| cjred 15111 | A real number equals its c... |
| remul2d 15112 | Real part of a product. (... |
| immul2d 15113 | Imaginary part of a produc... |
| redivd 15114 | Real part of a division. ... |
| imdivd 15115 | Imaginary part of a divisi... |
| crred 15116 | The real part of a complex... |
| crimd 15117 | The imaginary part of a co... |
| sqrtval 15122 | Value of square root funct... |
| absval 15123 | The absolute value (modulu... |
| rennim 15124 | A real number does not lie... |
| cnpart 15125 | The specification of restr... |
| sqrt0 15126 | The square root of zero is... |
| 01sqrexlem1 15127 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem2 15128 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem3 15129 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem4 15130 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem5 15131 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem6 15132 | Lemma for ~ 01sqrex . (Co... |
| 01sqrexlem7 15133 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 15134 | Existence of a square root... |
| resqrex 15135 | Existence of a square root... |
| sqrmo 15136 | Uniqueness for the square ... |
| resqreu 15137 | Existence and uniqueness f... |
| resqrtcl 15138 | Closure of the square root... |
| resqrtthlem 15139 | Lemma for ~ resqrtth . (C... |
| resqrtth 15140 | Square root theorem over t... |
| remsqsqrt 15141 | Square of square root. (C... |
| sqrtge0 15142 | The square root function i... |
| sqrtgt0 15143 | The square root function i... |
| sqrtmul 15144 | Square root distributes ov... |
| sqrtle 15145 | Square root is monotonic. ... |
| sqrtlt 15146 | Square root is strictly mo... |
| sqrt11 15147 | The square root function i... |
| sqrt00 15148 | A square root is zero iff ... |
| rpsqrtcl 15149 | The square root of a posit... |
| sqrtdiv 15150 | Square root distributes ov... |
| sqrtneglem 15151 | The square root of a negat... |
| sqrtneg 15152 | The square root of a negat... |
| sqrtsq2 15153 | Relationship between squar... |
| sqrtsq 15154 | Square root of square. (C... |
| sqrtmsq 15155 | Square root of square. (C... |
| sqrt1 15156 | The square root of 1 is 1.... |
| sqrt4 15157 | The square root of 4 is 2.... |
| sqrt9 15158 | The square root of 9 is 3.... |
| sqrt2gt1lt2 15159 | The square root of 2 is bo... |
| sqrtm1 15160 | The imaginary unit is the ... |
| nn0sqeq1 15161 | A natural number with squa... |
| absneg 15162 | Absolute value of the oppo... |
| abscl 15163 | Real closure of absolute v... |
| abscj 15164 | The absolute value of a nu... |
| absvalsq 15165 | Square of value of absolut... |
| absvalsq2 15166 | Square of value of absolut... |
| sqabsadd 15167 | Square of absolute value o... |
| sqabssub 15168 | Square of absolute value o... |
| absval2 15169 | Value of absolute value fu... |
| abs0 15170 | The absolute value of 0. ... |
| absi 15171 | The absolute value of the ... |
| absge0 15172 | Absolute value is nonnegat... |
| absrpcl 15173 | The absolute value of a no... |
| abs00 15174 | The absolute value of a nu... |
| abs00ad 15175 | A complex number is zero i... |
| abs00bd 15176 | If a complex number is zer... |
| absreimsq 15177 | Square of the absolute val... |
| absreim 15178 | Absolute value of a number... |
| absmul 15179 | Absolute value distributes... |
| absdiv 15180 | Absolute value distributes... |
| absid 15181 | A nonnegative number is it... |
| abs1 15182 | The absolute value of one ... |
| absnid 15183 | A negative number is the n... |
| leabs 15184 | A real number is less than... |
| absor 15185 | The absolute value of a re... |
| absre 15186 | Absolute value of a real n... |
| absresq 15187 | Square of the absolute val... |
| absmod0 15188 | ` A ` is divisible by ` B ... |
| absexp 15189 | Absolute value of positive... |
| absexpz 15190 | Absolute value of integer ... |
| abssq 15191 | Square can be moved in and... |
| sqabs 15192 | The squares of two reals a... |
| absrele 15193 | The absolute value of a co... |
| absimle 15194 | The absolute value of a co... |
| max0add 15195 | The sum of the positive an... |
| absz 15196 | A real number is an intege... |
| nn0abscl 15197 | The absolute value of an i... |
| zabscl 15198 | The absolute value of an i... |
| abslt 15199 | Absolute value and 'less t... |
| absle 15200 | Absolute value and 'less t... |
| abssubne0 15201 | If the absolute value of a... |
| absdiflt 15202 | The absolute value of a di... |
| absdifle 15203 | The absolute value of a di... |
| elicc4abs 15204 | Membership in a symmetric ... |
| lenegsq 15205 | Comparison to a nonnegativ... |
| releabs 15206 | The real part of a number ... |
| recval 15207 | Reciprocal expressed with ... |
| absidm 15208 | The absolute value functio... |
| absgt0 15209 | The absolute value of a no... |
| nnabscl 15210 | The absolute value of a no... |
| abssub 15211 | Swapping order of subtract... |
| abssubge0 15212 | Absolute value of a nonneg... |
| abssuble0 15213 | Absolute value of a nonpos... |
| absmax 15214 | The maximum of two numbers... |
| abstri 15215 | Triangle inequality for ab... |
| abs3dif 15216 | Absolute value of differen... |
| abs2dif 15217 | Difference of absolute val... |
| abs2dif2 15218 | Difference of absolute val... |
| abs2difabs 15219 | Absolute value of differen... |
| abs1m 15220 | For any complex number, th... |
| recan 15221 | Cancellation law involving... |
| absf 15222 | Mapping domain and codomai... |
| abs3lem 15223 | Lemma involving absolute v... |
| abslem2 15224 | Lemma involving absolute v... |
| rddif 15225 | The difference between a r... |
| absrdbnd 15226 | Bound on the absolute valu... |
| fzomaxdiflem 15227 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 15228 | A bound on the separation ... |
| uzin2 15229 | The upper integers are clo... |
| rexanuz 15230 | Combine two different uppe... |
| rexanre 15231 | Combine two different uppe... |
| rexfiuz 15232 | Combine finitely many diff... |
| rexuz3 15233 | Restrict the base of the u... |
| rexanuz2 15234 | Combine two different uppe... |
| r19.29uz 15235 | A version of ~ 19.29 for u... |
| r19.2uz 15236 | A version of ~ r19.2z for ... |
| rexuzre 15237 | Convert an upper real quan... |
| rexico 15238 | Restrict the base of an up... |
| cau3lem 15239 | Lemma for ~ cau3 . (Contr... |
| cau3 15240 | Convert between three-quan... |
| cau4 15241 | Change the base of a Cauch... |
| caubnd2 15242 | A Cauchy sequence of compl... |
| caubnd 15243 | A Cauchy sequence of compl... |
| sqreulem 15244 | Lemma for ~ sqreu : write ... |
| sqreu 15245 | Existence and uniqueness f... |
| sqrtcl 15246 | Closure of the square root... |
| sqrtthlem 15247 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15248 | Mapping domain and codomai... |
| sqrtth 15249 | Square root theorem over t... |
| sqrtrege0 15250 | The square root function m... |
| eqsqrtor 15251 | Solve an equation containi... |
| eqsqrtd 15252 | A deduction for showing th... |
| eqsqrt2d 15253 | A deduction for showing th... |
| amgm2 15254 | Arithmetic-geometric mean ... |
| sqrtthi 15255 | Square root theorem. Theo... |
| sqrtcli 15256 | The square root of a nonne... |
| sqrtgt0i 15257 | The square root of a posit... |
| sqrtmsqi 15258 | Square root of square. (C... |
| sqrtsqi 15259 | Square root of square. (C... |
| sqsqrti 15260 | Square of square root. (C... |
| sqrtge0i 15261 | The square root of a nonne... |
| absidi 15262 | A nonnegative number is it... |
| absnidi 15263 | A negative number is the n... |
| leabsi 15264 | A real number is less than... |
| absori 15265 | The absolute value of a re... |
| absrei 15266 | Absolute value of a real n... |
| sqrtpclii 15267 | The square root of a posit... |
| sqrtgt0ii 15268 | The square root of a posit... |
| sqrt11i 15269 | The square root function i... |
| sqrtmuli 15270 | Square root distributes ov... |
| sqrtmulii 15271 | Square root distributes ov... |
| sqrtmsq2i 15272 | Relationship between squar... |
| sqrtlei 15273 | Square root is monotonic. ... |
| sqrtlti 15274 | Square root is strictly mo... |
| abslti 15275 | Absolute value and 'less t... |
| abslei 15276 | Absolute value and 'less t... |
| cnsqrt00 15277 | A square root of a complex... |
| absvalsqi 15278 | Square of value of absolut... |
| absvalsq2i 15279 | Square of value of absolut... |
| abscli 15280 | Real closure of absolute v... |
| absge0i 15281 | Absolute value is nonnegat... |
| absval2i 15282 | Value of absolute value fu... |
| abs00i 15283 | The absolute value of a nu... |
| absgt0i 15284 | The absolute value of a no... |
| absnegi 15285 | Absolute value of negative... |
| abscji 15286 | The absolute value of a nu... |
| releabsi 15287 | The real part of a number ... |
| abssubi 15288 | Swapping order of subtract... |
| absmuli 15289 | Absolute value distributes... |
| sqabsaddi 15290 | Square of absolute value o... |
| sqabssubi 15291 | Square of absolute value o... |
| absdivzi 15292 | Absolute value distributes... |
| abstrii 15293 | Triangle inequality for ab... |
| abs3difi 15294 | Absolute value of differen... |
| abs3lemi 15295 | Lemma involving absolute v... |
| rpsqrtcld 15296 | The square root of a posit... |
| sqrtgt0d 15297 | The square root of a posit... |
| absnidd 15298 | A negative number is the n... |
| leabsd 15299 | A real number is less than... |
| absord 15300 | The absolute value of a re... |
| absred 15301 | Absolute value of a real n... |
| resqrtcld 15302 | The square root of a nonne... |
| sqrtmsqd 15303 | Square root of square. (C... |
| sqrtsqd 15304 | Square root of square. (C... |
| sqrtge0d 15305 | The square root of a nonne... |
| sqrtnegd 15306 | The square root of a negat... |
| absidd 15307 | A nonnegative number is it... |
| sqrtdivd 15308 | Square root distributes ov... |
| sqrtmuld 15309 | Square root distributes ov... |
| sqrtsq2d 15310 | Relationship between squar... |
| sqrtled 15311 | Square root is monotonic. ... |
| sqrtltd 15312 | Square root is strictly mo... |
| sqr11d 15313 | The square root function i... |
| absltd 15314 | Absolute value and 'less t... |
| absled 15315 | Absolute value and 'less t... |
| abssubge0d 15316 | Absolute value of a nonneg... |
| abssuble0d 15317 | Absolute value of a nonpos... |
| absdifltd 15318 | The absolute value of a di... |
| absdifled 15319 | The absolute value of a di... |
| icodiamlt 15320 | Two elements in a half-ope... |
| abscld 15321 | Real closure of absolute v... |
| sqrtcld 15322 | Closure of the square root... |
| sqrtrege0d 15323 | The real part of the squar... |
| sqsqrtd 15324 | Square root theorem. Theo... |
| msqsqrtd 15325 | Square root theorem. Theo... |
| sqr00d 15326 | A square root is zero iff ... |
| absvalsqd 15327 | Square of value of absolut... |
| absvalsq2d 15328 | Square of value of absolut... |
| absge0d 15329 | Absolute value is nonnegat... |
| absval2d 15330 | Value of absolute value fu... |
| abs00d 15331 | The absolute value of a nu... |
| absne0d 15332 | The absolute value of a nu... |
| absrpcld 15333 | The absolute value of a no... |
| absnegd 15334 | Absolute value of negative... |
| abscjd 15335 | The absolute value of a nu... |
| releabsd 15336 | The real part of a number ... |
| absexpd 15337 | Absolute value of positive... |
| abssubd 15338 | Swapping order of subtract... |
| absmuld 15339 | Absolute value distributes... |
| absdivd 15340 | Absolute value distributes... |
| abstrid 15341 | Triangle inequality for ab... |
| abs2difd 15342 | Difference of absolute val... |
| abs2dif2d 15343 | Difference of absolute val... |
| abs2difabsd 15344 | Absolute value of differen... |
| abs3difd 15345 | Absolute value of differen... |
| abs3lemd 15346 | Lemma involving absolute v... |
| reusq0 15347 | A complex number is the sq... |
| bhmafibid1cn 15348 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15349 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15350 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15351 | The Brahmagupta-Fibonacci ... |
| limsupgord 15354 | Ordering property of the s... |
| limsupcl 15355 | Closure of the superior li... |
| limsupval 15356 | The superior limit of an i... |
| limsupgf 15357 | Closure of the superior li... |
| limsupgval 15358 | Value of the superior limi... |
| limsupgle 15359 | The defining property of t... |
| limsuple 15360 | The defining property of t... |
| limsuplt 15361 | The defining property of t... |
| limsupval2 15362 | The superior limit, relati... |
| limsupgre 15363 | If a sequence of real numb... |
| limsupbnd1 15364 | If a sequence is eventuall... |
| limsupbnd2 15365 | If a sequence is eventuall... |
| climrel 15374 | The limit relation is a re... |
| rlimrel 15375 | The limit relation is a re... |
| clim 15376 | Express the predicate: Th... |
| rlim 15377 | Express the predicate: Th... |
| rlim2 15378 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15379 | Use strictly less-than in ... |
| rlim3 15380 | Restrict the range of the ... |
| climcl 15381 | Closure of the limit of a ... |
| rlimpm 15382 | Closure of a function with... |
| rlimf 15383 | Closure of a function with... |
| rlimss 15384 | Domain closure of a functi... |
| rlimcl 15385 | Closure of the limit of a ... |
| clim2 15386 | Express the predicate: Th... |
| clim2c 15387 | Express the predicate ` F ... |
| clim0 15388 | Express the predicate ` F ... |
| clim0c 15389 | Express the predicate ` F ... |
| rlim0 15390 | Express the predicate ` B ... |
| rlim0lt 15391 | Use strictly less-than in ... |
| climi 15392 | Convergence of a sequence ... |
| climi2 15393 | Convergence of a sequence ... |
| climi0 15394 | Convergence of a sequence ... |
| rlimi 15395 | Convergence at infinity of... |
| rlimi2 15396 | Convergence at infinity of... |
| ello1 15397 | Elementhood in the set of ... |
| ello12 15398 | Elementhood in the set of ... |
| ello12r 15399 | Sufficient condition for e... |
| lo1f 15400 | An eventually upper bounde... |
| lo1dm 15401 | An eventually upper bounde... |
| lo1bdd 15402 | The defining property of a... |
| ello1mpt 15403 | Elementhood in the set of ... |
| ello1mpt2 15404 | Elementhood in the set of ... |
| ello1d 15405 | Sufficient condition for e... |
| lo1bdd2 15406 | If an eventually bounded f... |
| lo1bddrp 15407 | Refine ~ o1bdd2 to give a ... |
| elo1 15408 | Elementhood in the set of ... |
| elo12 15409 | Elementhood in the set of ... |
| elo12r 15410 | Sufficient condition for e... |
| o1f 15411 | An eventually bounded func... |
| o1dm 15412 | An eventually bounded func... |
| o1bdd 15413 | The defining property of a... |
| lo1o1 15414 | A function is eventually b... |
| lo1o12 15415 | A function is eventually b... |
| elo1mpt 15416 | Elementhood in the set of ... |
| elo1mpt2 15417 | Elementhood in the set of ... |
| elo1d 15418 | Sufficient condition for e... |
| o1lo1 15419 | A real function is eventua... |
| o1lo12 15420 | A lower bounded real funct... |
| o1lo1d 15421 | A real eventually bounded ... |
| icco1 15422 | Derive eventual boundednes... |
| o1bdd2 15423 | If an eventually bounded f... |
| o1bddrp 15424 | Refine ~ o1bdd2 to give a ... |
| climconst 15425 | An (eventually) constant s... |
| rlimconst 15426 | A constant sequence conver... |
| rlimclim1 15427 | Forward direction of ~ rli... |
| rlimclim 15428 | A sequence on an upper int... |
| climrlim2 15429 | Produce a real limit from ... |
| climconst2 15430 | A constant sequence conver... |
| climz 15431 | The zero sequence converge... |
| rlimuni 15432 | A real function whose doma... |
| rlimdm 15433 | Two ways to express that a... |
| climuni 15434 | An infinite sequence of co... |
| fclim 15435 | The limit relation is func... |
| climdm 15436 | Two ways to express that a... |
| climeu 15437 | An infinite sequence of co... |
| climreu 15438 | An infinite sequence of co... |
| climmo 15439 | An infinite sequence of co... |
| rlimres 15440 | The restriction of a funct... |
| lo1res 15441 | The restriction of an even... |
| o1res 15442 | The restriction of an even... |
| rlimres2 15443 | The restriction of a funct... |
| lo1res2 15444 | The restriction of a funct... |
| o1res2 15445 | The restriction of a funct... |
| lo1resb 15446 | The restriction of a funct... |
| rlimresb 15447 | The restriction of a funct... |
| o1resb 15448 | The restriction of a funct... |
| climeq 15449 | Two functions that are eve... |
| lo1eq 15450 | Two functions that are eve... |
| rlimeq 15451 | Two functions that are eve... |
| o1eq 15452 | Two functions that are eve... |
| climmpt 15453 | Exhibit a function ` G ` w... |
| 2clim 15454 | If two sequences converge ... |
| climmpt2 15455 | Relate an integer limit on... |
| climshftlem 15456 | A shifted function converg... |
| climres 15457 | A function restricted to u... |
| climshft 15458 | A shifted function converg... |
| serclim0 15459 | The zero series converges ... |
| rlimcld2 15460 | If ` D ` is a closed set i... |
| rlimrege0 15461 | The limit of a sequence of... |
| rlimrecl 15462 | The limit of a real sequen... |
| rlimge0 15463 | The limit of a sequence of... |
| climshft2 15464 | A shifted function converg... |
| climrecl 15465 | The limit of a convergent ... |
| climge0 15466 | A nonnegative sequence con... |
| climabs0 15467 | Convergence to zero of the... |
| o1co 15468 | Sufficient condition for t... |
| o1compt 15469 | Sufficient condition for t... |
| rlimcn1 15470 | Image of a limit under a c... |
| rlimcn1b 15471 | Image of a limit under a c... |
| rlimcn3 15472 | Image of a limit under a c... |
| rlimcn2 15473 | Image of a limit under a c... |
| climcn1 15474 | Image of a limit under a c... |
| climcn2 15475 | Image of a limit under a c... |
| addcn2 15476 | Complex number addition is... |
| subcn2 15477 | Complex number subtraction... |
| mulcn2 15478 | Complex number multiplicat... |
| reccn2 15479 | The reciprocal function is... |
| cn1lem 15480 | A sufficient condition for... |
| abscn2 15481 | The absolute value functio... |
| cjcn2 15482 | The complex conjugate func... |
| recn2 15483 | The real part function is ... |
| imcn2 15484 | The imaginary part functio... |
| climcn1lem 15485 | The limit of a continuous ... |
| climabs 15486 | Limit of the absolute valu... |
| climcj 15487 | Limit of the complex conju... |
| climre 15488 | Limit of the real part of ... |
| climim 15489 | Limit of the imaginary par... |
| rlimmptrcl 15490 | Reverse closure for a real... |
| rlimabs 15491 | Limit of the absolute valu... |
| rlimcj 15492 | Limit of the complex conju... |
| rlimre 15493 | Limit of the real part of ... |
| rlimim 15494 | Limit of the imaginary par... |
| o1of2 15495 | Show that a binary operati... |
| o1add 15496 | The sum of two eventually ... |
| o1mul 15497 | The product of two eventua... |
| o1sub 15498 | The difference of two even... |
| rlimo1 15499 | Any function with a finite... |
| rlimdmo1 15500 | A convergent function is e... |
| o1rlimmul 15501 | The product of an eventual... |
| o1const 15502 | A constant function is eve... |
| lo1const 15503 | A constant function is eve... |
| lo1mptrcl 15504 | Reverse closure for an eve... |
| o1mptrcl 15505 | Reverse closure for an eve... |
| o1add2 15506 | The sum of two eventually ... |
| o1mul2 15507 | The product of two eventua... |
| o1sub2 15508 | The product of two eventua... |
| lo1add 15509 | The sum of two eventually ... |
| lo1mul 15510 | The product of an eventual... |
| lo1mul2 15511 | The product of an eventual... |
| o1dif 15512 | If the difference of two f... |
| lo1sub 15513 | The difference of an event... |
| climadd 15514 | Limit of the sum of two co... |
| climmul 15515 | Limit of the product of tw... |
| climsub 15516 | Limit of the difference of... |
| climaddc1 15517 | Limit of a constant ` C ` ... |
| climaddc2 15518 | Limit of a constant ` C ` ... |
| climmulc2 15519 | Limit of a sequence multip... |
| climsubc1 15520 | Limit of a constant ` C ` ... |
| climsubc2 15521 | Limit of a constant ` C ` ... |
| climle 15522 | Comparison of the limits o... |
| climsqz 15523 | Convergence of a sequence ... |
| climsqz2 15524 | Convergence of a sequence ... |
| rlimadd 15525 | Limit of the sum of two co... |
| rlimaddOLD 15526 | Obsolete version of ~ rlim... |
| rlimsub 15527 | Limit of the difference of... |
| rlimmul 15528 | Limit of the product of tw... |
| rlimmulOLD 15529 | Obsolete version of ~ rlim... |
| rlimdiv 15530 | Limit of the quotient of t... |
| rlimneg 15531 | Limit of the negative of a... |
| rlimle 15532 | Comparison of the limits o... |
| rlimsqzlem 15533 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15534 | Convergence of a sequence ... |
| rlimsqz2 15535 | Convergence of a sequence ... |
| lo1le 15536 | Transfer eventual upper bo... |
| o1le 15537 | Transfer eventual boundedn... |
| rlimno1 15538 | A function whose inverse c... |
| clim2ser 15539 | The limit of an infinite s... |
| clim2ser2 15540 | The limit of an infinite s... |
| iserex 15541 | An infinite series converg... |
| isermulc2 15542 | Multiplication of an infin... |
| climlec2 15543 | Comparison of a constant t... |
| iserle 15544 | Comparison of the limits o... |
| iserge0 15545 | The limit of an infinite s... |
| climub 15546 | The limit of a monotonic s... |
| climserle 15547 | The partial sums of a conv... |
| isershft 15548 | Index shift of the limit o... |
| isercolllem1 15549 | Lemma for ~ isercoll . (C... |
| isercolllem2 15550 | Lemma for ~ isercoll . (C... |
| isercolllem3 15551 | Lemma for ~ isercoll . (C... |
| isercoll 15552 | Rearrange an infinite seri... |
| isercoll2 15553 | Generalize ~ isercoll so t... |
| climsup 15554 | A bounded monotonic sequen... |
| climcau 15555 | A converging sequence of c... |
| climbdd 15556 | A converging sequence of c... |
| caucvgrlem 15557 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15558 | A Cauchy sequence of real ... |
| caucvgrlem2 15559 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15560 | A Cauchy sequence of compl... |
| caurcvg 15561 | A Cauchy sequence of real ... |
| caurcvg2 15562 | A Cauchy sequence of real ... |
| caucvg 15563 | A Cauchy sequence of compl... |
| caucvgb 15564 | A function is convergent i... |
| serf0 15565 | If an infinite series conv... |
| iseraltlem1 15566 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15567 | Lemma for ~ iseralt . The... |
| iseraltlem3 15568 | Lemma for ~ iseralt . Fro... |
| iseralt 15569 | The alternating series tes... |
| sumex 15572 | A sum is a set. (Contribu... |
| sumeq1 15573 | Equality theorem for a sum... |
| nfsum1 15574 | Bound-variable hypothesis ... |
| nfsum 15575 | Bound-variable hypothesis ... |
| nfsumOLD 15576 | Obsolete version of ~ nfsu... |
| sumeq2w 15577 | Equality theorem for sum, ... |
| sumeq2ii 15578 | Equality theorem for sum, ... |
| sumeq2 15579 | Equality theorem for sum. ... |
| cbvsum 15580 | Change bound variable in a... |
| cbvsumv 15581 | Change bound variable in a... |
| cbvsumi 15582 | Change bound variable in a... |
| sumeq1i 15583 | Equality inference for sum... |
| sumeq2i 15584 | Equality inference for sum... |
| sumeq12i 15585 | Equality inference for sum... |
| sumeq1d 15586 | Equality deduction for sum... |
| sumeq2d 15587 | Equality deduction for sum... |
| sumeq2dv 15588 | Equality deduction for sum... |
| sumeq2sdv 15589 | Equality deduction for sum... |
| 2sumeq2dv 15590 | Equality deduction for dou... |
| sumeq12dv 15591 | Equality deduction for sum... |
| sumeq12rdv 15592 | Equality deduction for sum... |
| sum2id 15593 | The second class argument ... |
| sumfc 15594 | A lemma to facilitate conv... |
| fz1f1o 15595 | A lemma for working with f... |
| sumrblem 15596 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15597 | The sequence of partial su... |
| sumrb 15598 | Rebase the starting point ... |
| summolem3 15599 | Lemma for ~ summo . (Cont... |
| summolem2a 15600 | Lemma for ~ summo . (Cont... |
| summolem2 15601 | Lemma for ~ summo . (Cont... |
| summo 15602 | A sum has at most one limi... |
| zsum 15603 | Series sum with index set ... |
| isum 15604 | Series sum with an upper i... |
| fsum 15605 | The value of a sum over a ... |
| sum0 15606 | Any sum over the empty set... |
| sumz 15607 | Any sum of zero over a sum... |
| fsumf1o 15608 | Re-index a finite sum usin... |
| sumss 15609 | Change the index set to a ... |
| fsumss 15610 | Change the index set to a ... |
| sumss2 15611 | Change the index set of a ... |
| fsumcvg2 15612 | The sequence of partial su... |
| fsumsers 15613 | Special case of series sum... |
| fsumcvg3 15614 | A finite sum is convergent... |
| fsumser 15615 | A finite sum expressed in ... |
| fsumcl2lem 15616 | - Lemma for finite sum clo... |
| fsumcllem 15617 | - Lemma for finite sum clo... |
| fsumcl 15618 | Closure of a finite sum of... |
| fsumrecl 15619 | Closure of a finite sum of... |
| fsumzcl 15620 | Closure of a finite sum of... |
| fsumnn0cl 15621 | Closure of a finite sum of... |
| fsumrpcl 15622 | Closure of a finite sum of... |
| fsumclf 15623 | Closure of a finite sum of... |
| fsumzcl2 15624 | A finite sum with integer ... |
| fsumadd 15625 | The sum of two finite sums... |
| fsumsplit 15626 | Split a sum into two parts... |
| fsumsplitf 15627 | Split a sum into two parts... |
| sumsnf 15628 | A sum of a singleton is th... |
| fsumsplitsn 15629 | Separate out a term in a f... |
| fsumsplit1 15630 | Separate out a term in a f... |
| sumsn 15631 | A sum of a singleton is th... |
| fsum1 15632 | The finite sum of ` A ( k ... |
| sumpr 15633 | A sum over a pair is the s... |
| sumtp 15634 | A sum over a triple is the... |
| sumsns 15635 | A sum of a singleton is th... |
| fsumm1 15636 | Separate out the last term... |
| fzosump1 15637 | Separate out the last term... |
| fsum1p 15638 | Separate out the first ter... |
| fsummsnunz 15639 | A finite sum all of whose ... |
| fsumsplitsnun 15640 | Separate out a term in a f... |
| fsump1 15641 | The addition of the next t... |
| isumclim 15642 | An infinite sum equals the... |
| isumclim2 15643 | A converging series conver... |
| isumclim3 15644 | The sequence of partial fi... |
| sumnul 15645 | The sum of a non-convergen... |
| isumcl 15646 | The sum of a converging in... |
| isummulc2 15647 | An infinite sum multiplied... |
| isummulc1 15648 | An infinite sum multiplied... |
| isumdivc 15649 | An infinite sum divided by... |
| isumrecl 15650 | The sum of a converging in... |
| isumge0 15651 | An infinite sum of nonnega... |
| isumadd 15652 | Addition of infinite sums.... |
| sumsplit 15653 | Split a sum into two parts... |
| fsump1i 15654 | Optimized version of ~ fsu... |
| fsum2dlem 15655 | Lemma for ~ fsum2d - induc... |
| fsum2d 15656 | Write a double sum as a su... |
| fsumxp 15657 | Combine two sums into a si... |
| fsumcnv 15658 | Transform a region of summ... |
| fsumcom2 15659 | Interchange order of summa... |
| fsumcom 15660 | Interchange order of summa... |
| fsum0diaglem 15661 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15662 | Two ways to express "the s... |
| mptfzshft 15663 | 1-1 onto function in maps-... |
| fsumrev 15664 | Reversal of a finite sum. ... |
| fsumshft 15665 | Index shift of a finite su... |
| fsumshftm 15666 | Negative index shift of a ... |
| fsumrev2 15667 | Reversal of a finite sum. ... |
| fsum0diag2 15668 | Two ways to express "the s... |
| fsummulc2 15669 | A finite sum multiplied by... |
| fsummulc1 15670 | A finite sum multiplied by... |
| fsumdivc 15671 | A finite sum divided by a ... |
| fsumneg 15672 | Negation of a finite sum. ... |
| fsumsub 15673 | Split a finite sum over a ... |
| fsum2mul 15674 | Separate the nested sum of... |
| fsumconst 15675 | The sum of constant terms ... |
| fsumdifsnconst 15676 | The sum of constant terms ... |
| modfsummodslem1 15677 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15678 | Induction step for ~ modfs... |
| modfsummod 15679 | A finite sum modulo a posi... |
| fsumge0 15680 | If all of the terms of a f... |
| fsumless 15681 | A shorter sum of nonnegati... |
| fsumge1 15682 | A sum of nonnegative numbe... |
| fsum00 15683 | A sum of nonnegative numbe... |
| fsumle 15684 | If all of the terms of fin... |
| fsumlt 15685 | If every term in one finit... |
| fsumabs 15686 | Generalized triangle inequ... |
| telfsumo 15687 | Sum of a telescoping serie... |
| telfsumo2 15688 | Sum of a telescoping serie... |
| telfsum 15689 | Sum of a telescoping serie... |
| telfsum2 15690 | Sum of a telescoping serie... |
| fsumparts 15691 | Summation by parts. (Cont... |
| fsumrelem 15692 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15693 | The real part of a sum. (... |
| fsumim 15694 | The imaginary part of a su... |
| fsumcj 15695 | The complex conjugate of a... |
| fsumrlim 15696 | Limit of a finite sum of c... |
| fsumo1 15697 | The finite sum of eventual... |
| o1fsum 15698 | If ` A ( k ) ` is O(1), th... |
| seqabs 15699 | Generalized triangle inequ... |
| iserabs 15700 | Generalized triangle inequ... |
| cvgcmp 15701 | A comparison test for conv... |
| cvgcmpub 15702 | An upper bound for the lim... |
| cvgcmpce 15703 | A comparison test for conv... |
| abscvgcvg 15704 | An absolutely convergent s... |
| climfsum 15705 | Limit of a finite sum of c... |
| fsumiun 15706 | Sum over a disjoint indexe... |
| hashiun 15707 | The cardinality of a disjo... |
| hash2iun 15708 | The cardinality of a neste... |
| hash2iun1dif1 15709 | The cardinality of a neste... |
| hashrabrex 15710 | The number of elements in ... |
| hashuni 15711 | The cardinality of a disjo... |
| qshash 15712 | The cardinality of a set w... |
| ackbijnn 15713 | Translate the Ackermann bi... |
| binomlem 15714 | Lemma for ~ binom (binomia... |
| binom 15715 | The binomial theorem: ` ( ... |
| binom1p 15716 | Special case of the binomi... |
| binom11 15717 | Special case of the binomi... |
| binom1dif 15718 | A summation for the differ... |
| bcxmaslem1 15719 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15720 | Parallel summation (Christ... |
| incexclem 15721 | Lemma for ~ incexc . (Con... |
| incexc 15722 | The inclusion/exclusion pr... |
| incexc2 15723 | The inclusion/exclusion pr... |
| isumshft 15724 | Index shift of an infinite... |
| isumsplit 15725 | Split off the first ` N ` ... |
| isum1p 15726 | The infinite sum of a conv... |
| isumnn0nn 15727 | Sum from 0 to infinity in ... |
| isumrpcl 15728 | The infinite sum of positi... |
| isumle 15729 | Comparison of two infinite... |
| isumless 15730 | A finite sum of nonnegativ... |
| isumsup2 15731 | An infinite sum of nonnega... |
| isumsup 15732 | An infinite sum of nonnega... |
| isumltss 15733 | A partial sum of a series ... |
| climcndslem1 15734 | Lemma for ~ climcnds : bou... |
| climcndslem2 15735 | Lemma for ~ climcnds : bou... |
| climcnds 15736 | The Cauchy condensation te... |
| divrcnv 15737 | The sequence of reciprocal... |
| divcnv 15738 | The sequence of reciprocal... |
| flo1 15739 | The floor function satisfi... |
| divcnvshft 15740 | Limit of a ratio function.... |
| supcvg 15741 | Extract a sequence ` f ` i... |
| infcvgaux1i 15742 | Auxiliary theorem for appl... |
| infcvgaux2i 15743 | Auxiliary theorem for appl... |
| harmonic 15744 | The harmonic series ` H ` ... |
| arisum 15745 | Arithmetic series sum of t... |
| arisum2 15746 | Arithmetic series sum of t... |
| trireciplem 15747 | Lemma for ~ trirecip . Sh... |
| trirecip 15748 | The sum of the reciprocals... |
| expcnv 15749 | A sequence of powers of a ... |
| explecnv 15750 | A sequence of terms conver... |
| geoserg 15751 | The value of the finite ge... |
| geoser 15752 | The value of the finite ge... |
| pwdif 15753 | The difference of two numb... |
| pwm1geoser 15754 | The n-th power of a number... |
| geolim 15755 | The partial sums in the in... |
| geolim2 15756 | The partial sums in the ge... |
| georeclim 15757 | The limit of a geometric s... |
| geo2sum 15758 | The value of the finite ge... |
| geo2sum2 15759 | The value of the finite ge... |
| geo2lim 15760 | The value of the infinite ... |
| geomulcvg 15761 | The geometric series conve... |
| geoisum 15762 | The infinite sum of ` 1 + ... |
| geoisumr 15763 | The infinite sum of recipr... |
| geoisum1 15764 | The infinite sum of ` A ^ ... |
| geoisum1c 15765 | The infinite sum of ` A x.... |
| 0.999... 15766 | The recurring decimal 0.99... |
| geoihalfsum 15767 | Prove that the infinite ge... |
| cvgrat 15768 | Ratio test for convergence... |
| mertenslem1 15769 | Lemma for ~ mertens . (Co... |
| mertenslem2 15770 | Lemma for ~ mertens . (Co... |
| mertens 15771 | Mertens' theorem. If ` A ... |
| prodf 15772 | An infinite product of com... |
| clim2prod 15773 | The limit of an infinite p... |
| clim2div 15774 | The limit of an infinite p... |
| prodfmul 15775 | The product of two infinit... |
| prodf1 15776 | The value of the partial p... |
| prodf1f 15777 | A one-valued infinite prod... |
| prodfclim1 15778 | The constant one product c... |
| prodfn0 15779 | No term of a nonzero infin... |
| prodfrec 15780 | The reciprocal of an infin... |
| prodfdiv 15781 | The quotient of two infini... |
| ntrivcvg 15782 | A non-trivially converging... |
| ntrivcvgn0 15783 | A product that converges t... |
| ntrivcvgfvn0 15784 | Any value of a product seq... |
| ntrivcvgtail 15785 | A tail of a non-trivially ... |
| ntrivcvgmullem 15786 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15787 | The product of two non-tri... |
| prodex 15790 | A product is a set. (Cont... |
| prodeq1f 15791 | Equality theorem for a pro... |
| prodeq1 15792 | Equality theorem for a pro... |
| nfcprod1 15793 | Bound-variable hypothesis ... |
| nfcprod 15794 | Bound-variable hypothesis ... |
| prodeq2w 15795 | Equality theorem for produ... |
| prodeq2ii 15796 | Equality theorem for produ... |
| prodeq2 15797 | Equality theorem for produ... |
| cbvprod 15798 | Change bound variable in a... |
| cbvprodv 15799 | Change bound variable in a... |
| cbvprodi 15800 | Change bound variable in a... |
| prodeq1i 15801 | Equality inference for pro... |
| prodeq2i 15802 | Equality inference for pro... |
| prodeq12i 15803 | Equality inference for pro... |
| prodeq1d 15804 | Equality deduction for pro... |
| prodeq2d 15805 | Equality deduction for pro... |
| prodeq2dv 15806 | Equality deduction for pro... |
| prodeq2sdv 15807 | Equality deduction for pro... |
| 2cprodeq2dv 15808 | Equality deduction for dou... |
| prodeq12dv 15809 | Equality deduction for pro... |
| prodeq12rdv 15810 | Equality deduction for pro... |
| prod2id 15811 | The second class argument ... |
| prodrblem 15812 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15813 | The sequence of partial pr... |
| prodrblem2 15814 | Lemma for ~ prodrb . (Con... |
| prodrb 15815 | Rebase the starting point ... |
| prodmolem3 15816 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15817 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15818 | Lemma for ~ prodmo . (Con... |
| prodmo 15819 | A product has at most one ... |
| zprod 15820 | Series product with index ... |
| iprod 15821 | Series product with an upp... |
| zprodn0 15822 | Nonzero series product wit... |
| iprodn0 15823 | Nonzero series product wit... |
| fprod 15824 | The value of a product ove... |
| fprodntriv 15825 | A non-triviality lemma for... |
| prod0 15826 | A product over the empty s... |
| prod1 15827 | Any product of one over a ... |
| prodfc 15828 | A lemma to facilitate conv... |
| fprodf1o 15829 | Re-index a finite product ... |
| prodss 15830 | Change the index set to a ... |
| fprodss 15831 | Change the index set to a ... |
| fprodser 15832 | A finite product expressed... |
| fprodcl2lem 15833 | Finite product closure lem... |
| fprodcllem 15834 | Finite product closure lem... |
| fprodcl 15835 | Closure of a finite produc... |
| fprodrecl 15836 | Closure of a finite produc... |
| fprodzcl 15837 | Closure of a finite produc... |
| fprodnncl 15838 | Closure of a finite produc... |
| fprodrpcl 15839 | Closure of a finite produc... |
| fprodnn0cl 15840 | Closure of a finite produc... |
| fprodcllemf 15841 | Finite product closure lem... |
| fprodreclf 15842 | Closure of a finite produc... |
| fprodmul 15843 | The product of two finite ... |
| fproddiv 15844 | The quotient of two finite... |
| prodsn 15845 | A product of a singleton i... |
| fprod1 15846 | A finite product of only o... |
| prodsnf 15847 | A product of a singleton i... |
| climprod1 15848 | The limit of a product ove... |
| fprodsplit 15849 | Split a finite product int... |
| fprodm1 15850 | Separate out the last term... |
| fprod1p 15851 | Separate out the first ter... |
| fprodp1 15852 | Multiply in the last term ... |
| fprodm1s 15853 | Separate out the last term... |
| fprodp1s 15854 | Multiply in the last term ... |
| prodsns 15855 | A product of the singleton... |
| fprodfac 15856 | Factorial using product no... |
| fprodabs 15857 | The absolute value of a fi... |
| fprodeq0 15858 | Any finite product contain... |
| fprodshft 15859 | Shift the index of a finit... |
| fprodrev 15860 | Reversal of a finite produ... |
| fprodconst 15861 | The product of constant te... |
| fprodn0 15862 | A finite product of nonzer... |
| fprod2dlem 15863 | Lemma for ~ fprod2d - indu... |
| fprod2d 15864 | Write a double product as ... |
| fprodxp 15865 | Combine two products into ... |
| fprodcnv 15866 | Transform a product region... |
| fprodcom2 15867 | Interchange order of multi... |
| fprodcom 15868 | Interchange product order.... |
| fprod0diag 15869 | Two ways to express "the p... |
| fproddivf 15870 | The quotient of two finite... |
| fprodsplitf 15871 | Split a finite product int... |
| fprodsplitsn 15872 | Separate out a term in a f... |
| fprodsplit1f 15873 | Separate out a term in a f... |
| fprodn0f 15874 | A finite product of nonzer... |
| fprodclf 15875 | Closure of a finite produc... |
| fprodge0 15876 | If all the terms of a fini... |
| fprodeq0g 15877 | Any finite product contain... |
| fprodge1 15878 | If all of the terms of a f... |
| fprodle 15879 | If all the terms of two fi... |
| fprodmodd 15880 | If all factors of two fini... |
| iprodclim 15881 | An infinite product equals... |
| iprodclim2 15882 | A converging product conve... |
| iprodclim3 15883 | The sequence of partial fi... |
| iprodcl 15884 | The product of a non-trivi... |
| iprodrecl 15885 | The product of a non-trivi... |
| iprodmul 15886 | Multiplication of infinite... |
| risefacval 15891 | The value of the rising fa... |
| fallfacval 15892 | The value of the falling f... |
| risefacval2 15893 | One-based value of rising ... |
| fallfacval2 15894 | One-based value of falling... |
| fallfacval3 15895 | A product representation o... |
| risefaccllem 15896 | Lemma for rising factorial... |
| fallfaccllem 15897 | Lemma for falling factoria... |
| risefaccl 15898 | Closure law for rising fac... |
| fallfaccl 15899 | Closure law for falling fa... |
| rerisefaccl 15900 | Closure law for rising fac... |
| refallfaccl 15901 | Closure law for falling fa... |
| nnrisefaccl 15902 | Closure law for rising fac... |
| zrisefaccl 15903 | Closure law for rising fac... |
| zfallfaccl 15904 | Closure law for falling fa... |
| nn0risefaccl 15905 | Closure law for rising fac... |
| rprisefaccl 15906 | Closure law for rising fac... |
| risefallfac 15907 | A relationship between ris... |
| fallrisefac 15908 | A relationship between fal... |
| risefall0lem 15909 | Lemma for ~ risefac0 and ~... |
| risefac0 15910 | The value of the rising fa... |
| fallfac0 15911 | The value of the falling f... |
| risefacp1 15912 | The value of the rising fa... |
| fallfacp1 15913 | The value of the falling f... |
| risefacp1d 15914 | The value of the rising fa... |
| fallfacp1d 15915 | The value of the falling f... |
| risefac1 15916 | The value of rising factor... |
| fallfac1 15917 | The value of falling facto... |
| risefacfac 15918 | Relate rising factorial to... |
| fallfacfwd 15919 | The forward difference of ... |
| 0fallfac 15920 | The value of the zero fall... |
| 0risefac 15921 | The value of the zero risi... |
| binomfallfaclem1 15922 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 15923 | Lemma for ~ binomfallfac .... |
| binomfallfac 15924 | A version of the binomial ... |
| binomrisefac 15925 | A version of the binomial ... |
| fallfacval4 15926 | Represent the falling fact... |
| bcfallfac 15927 | Binomial coefficient in te... |
| fallfacfac 15928 | Relate falling factorial t... |
| bpolylem 15931 | Lemma for ~ bpolyval . (C... |
| bpolyval 15932 | The value of the Bernoulli... |
| bpoly0 15933 | The value of the Bernoulli... |
| bpoly1 15934 | The value of the Bernoulli... |
| bpolycl 15935 | Closure law for Bernoulli ... |
| bpolysum 15936 | A sum for Bernoulli polyno... |
| bpolydiflem 15937 | Lemma for ~ bpolydif . (C... |
| bpolydif 15938 | Calculate the difference b... |
| fsumkthpow 15939 | A closed-form expression f... |
| bpoly2 15940 | The Bernoulli polynomials ... |
| bpoly3 15941 | The Bernoulli polynomials ... |
| bpoly4 15942 | The Bernoulli polynomials ... |
| fsumcube 15943 | Express the sum of cubes i... |
| eftcl 15956 | Closure of a term in the s... |
| reeftcl 15957 | The terms of the series ex... |
| eftabs 15958 | The absolute value of a te... |
| eftval 15959 | The value of a term in the... |
| efcllem 15960 | Lemma for ~ efcl . The se... |
| ef0lem 15961 | The series defining the ex... |
| efval 15962 | Value of the exponential f... |
| esum 15963 | Value of Euler's constant ... |
| eff 15964 | Domain and codomain of the... |
| efcl 15965 | Closure law for the expone... |
| efval2 15966 | Value of the exponential f... |
| efcvg 15967 | The series that defines th... |
| efcvgfsum 15968 | Exponential function conve... |
| reefcl 15969 | The exponential function i... |
| reefcld 15970 | The exponential function i... |
| ere 15971 | Euler's constant ` _e ` = ... |
| ege2le3 15972 | Lemma for ~ egt2lt3 . (Co... |
| ef0 15973 | Value of the exponential f... |
| efcj 15974 | The exponential of a compl... |
| efaddlem 15975 | Lemma for ~ efadd (exponen... |
| efadd 15976 | Sum of exponents law for e... |
| fprodefsum 15977 | Move the exponential funct... |
| efcan 15978 | Cancellation law for expon... |
| efne0 15979 | The exponential of a compl... |
| efneg 15980 | The exponential of the opp... |
| eff2 15981 | The exponential function m... |
| efsub 15982 | Difference of exponents la... |
| efexp 15983 | The exponential of an inte... |
| efzval 15984 | Value of the exponential f... |
| efgt0 15985 | The exponential of a real ... |
| rpefcl 15986 | The exponential of a real ... |
| rpefcld 15987 | The exponential of a real ... |
| eftlcvg 15988 | The tail series of the exp... |
| eftlcl 15989 | Closure of the sum of an i... |
| reeftlcl 15990 | Closure of the sum of an i... |
| eftlub 15991 | An upper bound on the abso... |
| efsep 15992 | Separate out the next term... |
| effsumlt 15993 | The partial sums of the se... |
| eft0val 15994 | The value of the first ter... |
| ef4p 15995 | Separate out the first fou... |
| efgt1p2 15996 | The exponential of a posit... |
| efgt1p 15997 | The exponential of a posit... |
| efgt1 15998 | The exponential of a posit... |
| eflt 15999 | The exponential function o... |
| efle 16000 | The exponential function o... |
| reef11 16001 | The exponential function o... |
| reeff1 16002 | The exponential function m... |
| eflegeo 16003 | The exponential function o... |
| sinval 16004 | Value of the sine function... |
| cosval 16005 | Value of the cosine functi... |
| sinf 16006 | Domain and codomain of the... |
| cosf 16007 | Domain and codomain of the... |
| sincl 16008 | Closure of the sine functi... |
| coscl 16009 | Closure of the cosine func... |
| tanval 16010 | Value of the tangent funct... |
| tancl 16011 | The closure of the tangent... |
| sincld 16012 | Closure of the sine functi... |
| coscld 16013 | Closure of the cosine func... |
| tancld 16014 | Closure of the tangent fun... |
| tanval2 16015 | Express the tangent functi... |
| tanval3 16016 | Express the tangent functi... |
| resinval 16017 | The sine of a real number ... |
| recosval 16018 | The cosine of a real numbe... |
| efi4p 16019 | Separate out the first fou... |
| resin4p 16020 | Separate out the first fou... |
| recos4p 16021 | Separate out the first fou... |
| resincl 16022 | The sine of a real number ... |
| recoscl 16023 | The cosine of a real numbe... |
| retancl 16024 | The closure of the tangent... |
| resincld 16025 | Closure of the sine functi... |
| recoscld 16026 | Closure of the cosine func... |
| retancld 16027 | Closure of the tangent fun... |
| sinneg 16028 | The sine of a negative is ... |
| cosneg 16029 | The cosines of a number an... |
| tanneg 16030 | The tangent of a negative ... |
| sin0 16031 | Value of the sine function... |
| cos0 16032 | Value of the cosine functi... |
| tan0 16033 | The value of the tangent f... |
| efival 16034 | The exponential function i... |
| efmival 16035 | The exponential function i... |
| sinhval 16036 | Value of the hyperbolic si... |
| coshval 16037 | Value of the hyperbolic co... |
| resinhcl 16038 | The hyperbolic sine of a r... |
| rpcoshcl 16039 | The hyperbolic cosine of a... |
| recoshcl 16040 | The hyperbolic cosine of a... |
| retanhcl 16041 | The hyperbolic tangent of ... |
| tanhlt1 16042 | The hyperbolic tangent of ... |
| tanhbnd 16043 | The hyperbolic tangent of ... |
| efeul 16044 | Eulerian representation of... |
| efieq 16045 | The exponentials of two im... |
| sinadd 16046 | Addition formula for sine.... |
| cosadd 16047 | Addition formula for cosin... |
| tanaddlem 16048 | A useful intermediate step... |
| tanadd 16049 | Addition formula for tange... |
| sinsub 16050 | Sine of difference. (Cont... |
| cossub 16051 | Cosine of difference. (Co... |
| addsin 16052 | Sum of sines. (Contribute... |
| subsin 16053 | Difference of sines. (Con... |
| sinmul 16054 | Product of sines can be re... |
| cosmul 16055 | Product of cosines can be ... |
| addcos 16056 | Sum of cosines. (Contribu... |
| subcos 16057 | Difference of cosines. (C... |
| sincossq 16058 | Sine squared plus cosine s... |
| sin2t 16059 | Double-angle formula for s... |
| cos2t 16060 | Double-angle formula for c... |
| cos2tsin 16061 | Double-angle formula for c... |
| sinbnd 16062 | The sine of a real number ... |
| cosbnd 16063 | The cosine of a real numbe... |
| sinbnd2 16064 | The sine of a real number ... |
| cosbnd2 16065 | The cosine of a real numbe... |
| ef01bndlem 16066 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 16067 | Bounds on the sine of a po... |
| cos01bnd 16068 | Bounds on the cosine of a ... |
| cos1bnd 16069 | Bounds on the cosine of 1.... |
| cos2bnd 16070 | Bounds on the cosine of 2.... |
| sinltx 16071 | The sine of a positive rea... |
| sin01gt0 16072 | The sine of a positive rea... |
| cos01gt0 16073 | The cosine of a positive r... |
| sin02gt0 16074 | The sine of a positive rea... |
| sincos1sgn 16075 | The signs of the sine and ... |
| sincos2sgn 16076 | The signs of the sine and ... |
| sin4lt0 16077 | The sine of 4 is negative.... |
| absefi 16078 | The absolute value of the ... |
| absef 16079 | The absolute value of the ... |
| absefib 16080 | A complex number is real i... |
| efieq1re 16081 | A number whose imaginary e... |
| demoivre 16082 | De Moivre's Formula. Proo... |
| demoivreALT 16083 | Alternate proof of ~ demoi... |
| eirrlem 16086 | Lemma for ~ eirr . (Contr... |
| eirr 16087 | ` _e ` is irrational. (Co... |
| egt2lt3 16088 | Euler's constant ` _e ` = ... |
| epos 16089 | Euler's constant ` _e ` is... |
| epr 16090 | Euler's constant ` _e ` is... |
| ene0 16091 | ` _e ` is not 0. (Contrib... |
| ene1 16092 | ` _e ` is not 1. (Contrib... |
| xpnnen 16093 | The Cartesian product of t... |
| znnen 16094 | The set of integers and th... |
| qnnen 16095 | The rational numbers are c... |
| rpnnen2lem1 16096 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 16097 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 16098 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 16099 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 16100 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 16101 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 16102 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 16103 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 16104 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 16105 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 16106 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 16107 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 16108 | The other half of ~ rpnnen... |
| rpnnen 16109 | The cardinality of the con... |
| rexpen 16110 | The real numbers are equin... |
| cpnnen 16111 | The complex numbers are eq... |
| rucALT 16112 | Alternate proof of ~ ruc .... |
| ruclem1 16113 | Lemma for ~ ruc (the reals... |
| ruclem2 16114 | Lemma for ~ ruc . Orderin... |
| ruclem3 16115 | Lemma for ~ ruc . The con... |
| ruclem4 16116 | Lemma for ~ ruc . Initial... |
| ruclem6 16117 | Lemma for ~ ruc . Domain ... |
| ruclem7 16118 | Lemma for ~ ruc . Success... |
| ruclem8 16119 | Lemma for ~ ruc . The int... |
| ruclem9 16120 | Lemma for ~ ruc . The fir... |
| ruclem10 16121 | Lemma for ~ ruc . Every f... |
| ruclem11 16122 | Lemma for ~ ruc . Closure... |
| ruclem12 16123 | Lemma for ~ ruc . The sup... |
| ruclem13 16124 | Lemma for ~ ruc . There i... |
| ruc 16125 | The set of positive intege... |
| resdomq 16126 | The set of rationals is st... |
| aleph1re 16127 | There are at least aleph-o... |
| aleph1irr 16128 | There are at least aleph-o... |
| cnso 16129 | The complex numbers can be... |
| sqrt2irrlem 16130 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 16131 | The square root of 2 is ir... |
| sqrt2re 16132 | The square root of 2 exist... |
| sqrt2irr0 16133 | The square root of 2 is an... |
| nthruc 16134 | The sequence ` NN ` , ` ZZ... |
| nthruz 16135 | The sequence ` NN ` , ` NN... |
| divides 16138 | Define the divides relatio... |
| dvdsval2 16139 | One nonzero integer divide... |
| dvdsval3 16140 | One nonzero integer divide... |
| dvdszrcl 16141 | Reverse closure for the di... |
| dvdsmod0 16142 | If a positive integer divi... |
| p1modz1 16143 | If a number greater than 1... |
| dvdsmodexp 16144 | If a positive integer divi... |
| nndivdvds 16145 | Strong form of ~ dvdsval2 ... |
| nndivides 16146 | Definition of the divides ... |
| moddvds 16147 | Two ways to say ` A == B `... |
| modm1div 16148 | An integer greater than on... |
| dvds0lem 16149 | A lemma to assist theorems... |
| dvds1lem 16150 | A lemma to assist theorems... |
| dvds2lem 16151 | A lemma to assist theorems... |
| iddvds 16152 | An integer divides itself.... |
| 1dvds 16153 | 1 divides any integer. Th... |
| dvds0 16154 | Any integer divides 0. Th... |
| negdvdsb 16155 | An integer divides another... |
| dvdsnegb 16156 | An integer divides another... |
| absdvdsb 16157 | An integer divides another... |
| dvdsabsb 16158 | An integer divides another... |
| 0dvds 16159 | Only 0 is divisible by 0. ... |
| dvdsmul1 16160 | An integer divides a multi... |
| dvdsmul2 16161 | An integer divides a multi... |
| iddvdsexp 16162 | An integer divides a posit... |
| muldvds1 16163 | If a product divides an in... |
| muldvds2 16164 | If a product divides an in... |
| dvdscmul 16165 | Multiplication by a consta... |
| dvdsmulc 16166 | Multiplication by a consta... |
| dvdscmulr 16167 | Cancellation law for the d... |
| dvdsmulcr 16168 | Cancellation law for the d... |
| summodnegmod 16169 | The sum of two integers mo... |
| modmulconst 16170 | Constant multiplication in... |
| dvds2ln 16171 | If an integer divides each... |
| dvds2add 16172 | If an integer divides each... |
| dvds2sub 16173 | If an integer divides each... |
| dvds2addd 16174 | Deduction form of ~ dvds2a... |
| dvds2subd 16175 | Deduction form of ~ dvds2s... |
| dvdstr 16176 | The divides relation is tr... |
| dvdstrd 16177 | The divides relation is tr... |
| dvdsmultr1 16178 | If an integer divides anot... |
| dvdsmultr1d 16179 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 16180 | If an integer divides anot... |
| dvdsmultr2d 16181 | Deduction form of ~ dvdsmu... |
| ordvdsmul 16182 | If an integer divides eith... |
| dvdssub2 16183 | If an integer divides a di... |
| dvdsadd 16184 | An integer divides another... |
| dvdsaddr 16185 | An integer divides another... |
| dvdssub 16186 | An integer divides another... |
| dvdssubr 16187 | An integer divides another... |
| dvdsadd2b 16188 | Adding a multiple of the b... |
| dvdsaddre2b 16189 | Adding a multiple of the b... |
| fsumdvds 16190 | If every term in a sum is ... |
| dvdslelem 16191 | Lemma for ~ dvdsle . (Con... |
| dvdsle 16192 | The divisors of a positive... |
| dvdsleabs 16193 | The divisors of a nonzero ... |
| dvdsleabs2 16194 | Transfer divisibility to a... |
| dvdsabseq 16195 | If two integers divide eac... |
| dvdseq 16196 | If two nonnegative integer... |
| divconjdvds 16197 | If a nonzero integer ` M `... |
| dvdsdivcl 16198 | The complement of a diviso... |
| dvdsflip 16199 | An involution of the divis... |
| dvdsssfz1 16200 | The set of divisors of a n... |
| dvds1 16201 | The only nonnegative integ... |
| alzdvds 16202 | Only 0 is divisible by all... |
| dvdsext 16203 | Poset extensionality for d... |
| fzm1ndvds 16204 | No number between ` 1 ` an... |
| fzo0dvdseq 16205 | Zero is the only one of th... |
| fzocongeq 16206 | Two different elements of ... |
| addmodlteqALT 16207 | Two nonnegative integers l... |
| dvdsfac 16208 | A positive integer divides... |
| dvdsexp2im 16209 | If an integer divides anot... |
| dvdsexp 16210 | A power divides a power wi... |
| dvdsmod 16211 | Any number ` K ` whose mod... |
| mulmoddvds 16212 | If an integer is divisible... |
| 3dvds 16213 | A rule for divisibility by... |
| 3dvdsdec 16214 | A decimal number is divisi... |
| 3dvds2dec 16215 | A decimal number is divisi... |
| fprodfvdvdsd 16216 | A finite product of intege... |
| fproddvdsd 16217 | A finite product of intege... |
| evenelz 16218 | An even number is an integ... |
| zeo3 16219 | An integer is even or odd.... |
| zeo4 16220 | An integer is even or odd ... |
| zeneo 16221 | No even integer equals an ... |
| odd2np1lem 16222 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 16223 | An integer is odd iff it i... |
| even2n 16224 | An integer is even iff it ... |
| oddm1even 16225 | An integer is odd iff its ... |
| oddp1even 16226 | An integer is odd iff its ... |
| oexpneg 16227 | The exponential of the neg... |
| mod2eq0even 16228 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 16229 | An integer is 1 modulo 2 i... |
| oddnn02np1 16230 | A nonnegative integer is o... |
| oddge22np1 16231 | An integer greater than on... |
| evennn02n 16232 | A nonnegative integer is e... |
| evennn2n 16233 | A positive integer is even... |
| 2tp1odd 16234 | A number which is twice an... |
| mulsucdiv2z 16235 | An integer multiplied with... |
| sqoddm1div8z 16236 | A squared odd number minus... |
| 2teven 16237 | A number which is twice an... |
| zeo5 16238 | An integer is either even ... |
| evend2 16239 | An integer is even iff its... |
| oddp1d2 16240 | An integer is odd iff its ... |
| zob 16241 | Alternate characterization... |
| oddm1d2 16242 | An integer is odd iff its ... |
| ltoddhalfle 16243 | An integer is less than ha... |
| halfleoddlt 16244 | An integer is greater than... |
| opoe 16245 | The sum of two odds is eve... |
| omoe 16246 | The difference of two odds... |
| opeo 16247 | The sum of an odd and an e... |
| omeo 16248 | The difference of an odd a... |
| z0even 16249 | 2 divides 0. That means 0... |
| n2dvds1 16250 | 2 does not divide 1. That... |
| n2dvdsm1 16251 | 2 does not divide -1. Tha... |
| z2even 16252 | 2 divides 2. That means 2... |
| n2dvds3 16253 | 2 does not divide 3. That... |
| z4even 16254 | 2 divides 4. That means 4... |
| 4dvdseven 16255 | An integer which is divisi... |
| m1expe 16256 | Exponentiation of -1 by an... |
| m1expo 16257 | Exponentiation of -1 by an... |
| m1exp1 16258 | Exponentiation of negative... |
| nn0enne 16259 | A positive integer is an e... |
| nn0ehalf 16260 | The half of an even nonneg... |
| nnehalf 16261 | The half of an even positi... |
| nn0onn 16262 | An odd nonnegative integer... |
| nn0o1gt2 16263 | An odd nonnegative integer... |
| nno 16264 | An alternate characterizat... |
| nn0o 16265 | An alternate characterizat... |
| nn0ob 16266 | Alternate characterization... |
| nn0oddm1d2 16267 | A positive integer is odd ... |
| nnoddm1d2 16268 | A positive integer is odd ... |
| sumeven 16269 | If every term in a sum is ... |
| sumodd 16270 | If every term in a sum is ... |
| evensumodd 16271 | If every term in a sum wit... |
| oddsumodd 16272 | If every term in a sum wit... |
| pwp1fsum 16273 | The n-th power of a number... |
| oddpwp1fsum 16274 | An odd power of a number i... |
| divalglem0 16275 | Lemma for ~ divalg . (Con... |
| divalglem1 16276 | Lemma for ~ divalg . (Con... |
| divalglem2 16277 | Lemma for ~ divalg . (Con... |
| divalglem4 16278 | Lemma for ~ divalg . (Con... |
| divalglem5 16279 | Lemma for ~ divalg . (Con... |
| divalglem6 16280 | Lemma for ~ divalg . (Con... |
| divalglem7 16281 | Lemma for ~ divalg . (Con... |
| divalglem8 16282 | Lemma for ~ divalg . (Con... |
| divalglem9 16283 | Lemma for ~ divalg . (Con... |
| divalglem10 16284 | Lemma for ~ divalg . (Con... |
| divalg 16285 | The division algorithm (th... |
| divalgb 16286 | Express the division algor... |
| divalg2 16287 | The division algorithm (th... |
| divalgmod 16288 | The result of the ` mod ` ... |
| divalgmodcl 16289 | The result of the ` mod ` ... |
| modremain 16290 | The result of the modulo o... |
| ndvdssub 16291 | Corollary of the division ... |
| ndvdsadd 16292 | Corollary of the division ... |
| ndvdsp1 16293 | Special case of ~ ndvdsadd... |
| ndvdsi 16294 | A quick test for non-divis... |
| flodddiv4 16295 | The floor of an odd intege... |
| fldivndvdslt 16296 | The floor of an integer di... |
| flodddiv4lt 16297 | The floor of an odd number... |
| flodddiv4t2lthalf 16298 | The floor of an odd number... |
| bitsfval 16303 | Expand the definition of t... |
| bitsval 16304 | Expand the definition of t... |
| bitsval2 16305 | Expand the definition of t... |
| bitsss 16306 | The set of bits of an inte... |
| bitsf 16307 | The ` bits ` function is a... |
| bits0 16308 | Value of the zeroth bit. ... |
| bits0e 16309 | The zeroth bit of an even ... |
| bits0o 16310 | The zeroth bit of an odd n... |
| bitsp1 16311 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16312 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16313 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16314 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16315 | The bits of a number are a... |
| bitsmod 16316 | Truncating the bit sequenc... |
| bitsfi 16317 | Every number is associated... |
| bitscmp 16318 | The bit complement of ` N ... |
| 0bits 16319 | The bits of zero. (Contri... |
| m1bits 16320 | The bits of negative one. ... |
| bitsinv1lem 16321 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16322 | There is an explicit inver... |
| bitsinv2 16323 | There is an explicit inver... |
| bitsf1ocnv 16324 | The ` bits ` function rest... |
| bitsf1o 16325 | The ` bits ` function rest... |
| bitsf1 16326 | The ` bits ` function is a... |
| 2ebits 16327 | The bits of a power of two... |
| bitsinv 16328 | The inverse of the ` bits ... |
| bitsinvp1 16329 | Recursive definition of th... |
| sadadd2lem2 16330 | The core of the proof of ~... |
| sadfval 16332 | Define the addition of two... |
| sadcf 16333 | The carry sequence is a se... |
| sadc0 16334 | The initial element of the... |
| sadcp1 16335 | The carry sequence (which ... |
| sadval 16336 | The full adder sequence is... |
| sadcaddlem 16337 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16338 | Non-recursive definition o... |
| sadadd2lem 16339 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16340 | Sum of initial segments of... |
| sadadd3 16341 | Sum of initial segments of... |
| sadcl 16342 | The sum of two sequences i... |
| sadcom 16343 | The adder sequence functio... |
| saddisjlem 16344 | Lemma for ~ sadadd . (Con... |
| saddisj 16345 | The sum of disjoint sequen... |
| sadaddlem 16346 | Lemma for ~ sadadd . (Con... |
| sadadd 16347 | For sequences that corresp... |
| sadid1 16348 | The adder sequence functio... |
| sadid2 16349 | The adder sequence functio... |
| sadasslem 16350 | Lemma for ~ sadass . (Con... |
| sadass 16351 | Sequence addition is assoc... |
| sadeq 16352 | Any element of a sequence ... |
| bitsres 16353 | Restrict the bits of a num... |
| bitsuz 16354 | The bits of a number are a... |
| bitsshft 16355 | Shifting a bit sequence to... |
| smufval 16357 | The multiplication of two ... |
| smupf 16358 | The sequence of partial su... |
| smup0 16359 | The initial element of the... |
| smupp1 16360 | The initial element of the... |
| smuval 16361 | Define the addition of two... |
| smuval2 16362 | The partial sum sequence s... |
| smupvallem 16363 | If ` A ` only has elements... |
| smucl 16364 | The product of two sequenc... |
| smu01lem 16365 | Lemma for ~ smu01 and ~ sm... |
| smu01 16366 | Multiplication of a sequen... |
| smu02 16367 | Multiplication of a sequen... |
| smupval 16368 | Rewrite the elements of th... |
| smup1 16369 | Rewrite ~ smupp1 using onl... |
| smueqlem 16370 | Any element of a sequence ... |
| smueq 16371 | Any element of a sequence ... |
| smumullem 16372 | Lemma for ~ smumul . (Con... |
| smumul 16373 | For sequences that corresp... |
| gcdval 16376 | The value of the ` gcd ` o... |
| gcd0val 16377 | The value, by convention, ... |
| gcdn0val 16378 | The value of the ` gcd ` o... |
| gcdcllem1 16379 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16380 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16381 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16382 | Closure of the ` gcd ` ope... |
| gcddvds 16383 | The gcd of two integers di... |
| dvdslegcd 16384 | An integer which divides b... |
| nndvdslegcd 16385 | A positive integer which d... |
| gcdcl 16386 | Closure of the ` gcd ` ope... |
| gcdnncl 16387 | Closure of the ` gcd ` ope... |
| gcdcld 16388 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16389 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16390 | An integer is the product ... |
| divgcdz 16391 | An integer divided by the ... |
| gcdf 16392 | Domain and codomain of the... |
| gcdcom 16393 | The ` gcd ` operator is co... |
| gcdcomd 16394 | The ` gcd ` operator is co... |
| divgcdnn 16395 | A positive integer divided... |
| divgcdnnr 16396 | A positive integer divided... |
| gcdeq0 16397 | The gcd of two integers is... |
| gcdn0gt0 16398 | The gcd of two integers is... |
| gcd0id 16399 | The gcd of 0 and an intege... |
| gcdid0 16400 | The gcd of an integer and ... |
| nn0gcdid0 16401 | The gcd of a nonnegative i... |
| gcdneg 16402 | Negating one operand of th... |
| neggcd 16403 | Negating one operand of th... |
| gcdaddmlem 16404 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16405 | Adding a multiple of one o... |
| gcdadd 16406 | The GCD of two numbers is ... |
| gcdid 16407 | The gcd of a number and it... |
| gcd1 16408 | The gcd of a number with 1... |
| gcdabs1 16409 | ` gcd ` of the absolute va... |
| gcdabs2 16410 | ` gcd ` of the absolute va... |
| gcdabs 16411 | The gcd of two integers is... |
| gcdabsOLD 16412 | Obsolete version of ~ gcda... |
| modgcd 16413 | The gcd remains unchanged ... |
| 1gcd 16414 | The GCD of one and an inte... |
| gcdmultipled 16415 | The greatest common diviso... |
| gcdmultiplez 16416 | The GCD of a multiple of a... |
| gcdmultiple 16417 | The GCD of a multiple of a... |
| dvdsgcdidd 16418 | The greatest common diviso... |
| 6gcd4e2 16419 | The greatest common diviso... |
| bezoutlem1 16420 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16421 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16422 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16423 | Lemma for ~ bezout . (Con... |
| bezout 16424 | Bézout's identity: ... |
| dvdsgcd 16425 | An integer which divides e... |
| dvdsgcdb 16426 | Biconditional form of ~ dv... |
| dfgcd2 16427 | Alternate definition of th... |
| gcdass 16428 | Associative law for ` gcd ... |
| mulgcd 16429 | Distribute multiplication ... |
| absmulgcd 16430 | Distribute absolute value ... |
| mulgcdr 16431 | Reverse distribution law f... |
| gcddiv 16432 | Division law for GCD. (Con... |
| gcdzeq 16433 | A positive integer ` A ` i... |
| gcdeq 16434 | ` A ` is equal to its gcd ... |
| dvdssqim 16435 | Unidirectional form of ~ d... |
| dvdsmulgcd 16436 | A divisibility equivalent ... |
| rpmulgcd 16437 | If ` K ` and ` M ` are rel... |
| rplpwr 16438 | If ` A ` and ` B ` are rel... |
| rprpwr 16439 | If ` A ` and ` B ` are rel... |
| rppwr 16440 | If ` A ` and ` B ` are rel... |
| sqgcd 16441 | Square distributes over gc... |
| dvdssqlem 16442 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16443 | Two numbers are divisible ... |
| bezoutr 16444 | Partial converse to ~ bezo... |
| bezoutr1 16445 | Converse of ~ bezout for w... |
| nn0seqcvgd 16446 | A strictly-decreasing nonn... |
| seq1st 16447 | A sequence whose iteration... |
| algr0 16448 | The value of the algorithm... |
| algrf 16449 | An algorithm is a step fun... |
| algrp1 16450 | The value of the algorithm... |
| alginv 16451 | If ` I ` is an invariant o... |
| algcvg 16452 | One way to prove that an a... |
| algcvgblem 16453 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16454 | Two ways of expressing tha... |
| algcvga 16455 | The countdown function ` C... |
| algfx 16456 | If ` F ` reaches a fixed p... |
| eucalgval2 16457 | The value of the step func... |
| eucalgval 16458 | Euclid's Algorithm ~ eucal... |
| eucalgf 16459 | Domain and codomain of the... |
| eucalginv 16460 | The invariant of the step ... |
| eucalglt 16461 | The second member of the s... |
| eucalgcvga 16462 | Once Euclid's Algorithm ha... |
| eucalg 16463 | Euclid's Algorithm compute... |
| lcmval 16468 | Value of the ` lcm ` opera... |
| lcmcom 16469 | The ` lcm ` operator is co... |
| lcm0val 16470 | The value, by convention, ... |
| lcmn0val 16471 | The value of the ` lcm ` o... |
| lcmcllem 16472 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16473 | Closure of the ` lcm ` ope... |
| dvdslcm 16474 | The lcm of two integers is... |
| lcmledvds 16475 | A positive integer which b... |
| lcmeq0 16476 | The lcm of two integers is... |
| lcmcl 16477 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16478 | The greatest common diviso... |
| lcmneg 16479 | Negating one operand of th... |
| neglcm 16480 | Negating one operand of th... |
| lcmabs 16481 | The lcm of two integers is... |
| lcmgcdlem 16482 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16483 | The product of two numbers... |
| lcmdvds 16484 | The lcm of two integers di... |
| lcmid 16485 | The lcm of an integer and ... |
| lcm1 16486 | The lcm of an integer and ... |
| lcmgcdnn 16487 | The product of two positiv... |
| lcmgcdeq 16488 | Two integers' absolute val... |
| lcmdvdsb 16489 | Biconditional form of ~ lc... |
| lcmass 16490 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16491 | The least common multiple ... |
| 6lcm4e12 16492 | The least common multiple ... |
| absproddvds 16493 | The absolute value of the ... |
| absprodnn 16494 | The absolute value of the ... |
| fissn0dvds 16495 | For each finite subset of ... |
| fissn0dvdsn0 16496 | For each finite subset of ... |
| lcmfval 16497 | Value of the ` _lcm ` func... |
| lcmf0val 16498 | The value, by convention, ... |
| lcmfn0val 16499 | The value of the ` _lcm ` ... |
| lcmfnnval 16500 | The value of the ` _lcm ` ... |
| lcmfcllem 16501 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16502 | Closure of the ` _lcm ` fu... |
| lcmfpr 16503 | The value of the ` _lcm ` ... |
| lcmfcl 16504 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16505 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16506 | The least common multiple ... |
| dvdslcmf 16507 | The least common multiple ... |
| lcmfledvds 16508 | A positive integer which i... |
| lcmf 16509 | Characterization of the le... |
| lcmf0 16510 | The least common multiple ... |
| lcmfsn 16511 | The least common multiple ... |
| lcmftp 16512 | The least common multiple ... |
| lcmfunsnlem1 16513 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16514 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16515 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16516 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16517 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16518 | The least common multiple ... |
| lcmfdvdsb 16519 | Biconditional form of ~ lc... |
| lcmfunsn 16520 | The ` _lcm ` function for ... |
| lcmfun 16521 | The ` _lcm ` function for ... |
| lcmfass 16522 | Associative law for the ` ... |
| lcmf2a3a4e12 16523 | The least common multiple ... |
| lcmflefac 16524 | The least common multiple ... |
| coprmgcdb 16525 | Two positive integers are ... |
| ncoprmgcdne1b 16526 | Two positive integers are ... |
| ncoprmgcdgt1b 16527 | Two positive integers are ... |
| coprmdvds1 16528 | If two positive integers a... |
| coprmdvds 16529 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16530 | If an integer is divisible... |
| mulgcddvds 16531 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16532 | If ` M ` is relatively pri... |
| qredeq 16533 | Two equal reduced fraction... |
| qredeu 16534 | Every rational number has ... |
| rpmul 16535 | If ` K ` is relatively pri... |
| rpdvds 16536 | If ` K ` is relatively pri... |
| coprmprod 16537 | The product of the element... |
| coprmproddvdslem 16538 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16539 | If a positive integer is d... |
| congr 16540 | Definition of congruence b... |
| divgcdcoprm0 16541 | Integers divided by gcd ar... |
| divgcdcoprmex 16542 | Integers divided by gcd ar... |
| cncongr1 16543 | One direction of the bicon... |
| cncongr2 16544 | The other direction of the... |
| cncongr 16545 | Cancellability of Congruen... |
| cncongrcoprm 16546 | Corollary 1 of Cancellabil... |
| isprm 16549 | The predicate "is a prime ... |
| prmnn 16550 | A prime number is a positi... |
| prmz 16551 | A prime number is an integ... |
| prmssnn 16552 | The prime numbers are a su... |
| prmex 16553 | The set of prime numbers e... |
| 0nprm 16554 | 0 is not a prime number. ... |
| 1nprm 16555 | 1 is not a prime number. ... |
| 1idssfct 16556 | The positive divisors of a... |
| isprm2lem 16557 | Lemma for ~ isprm2 . (Con... |
| isprm2 16558 | The predicate "is a prime ... |
| isprm3 16559 | The predicate "is a prime ... |
| isprm4 16560 | The predicate "is a prime ... |
| prmind2 16561 | A variation on ~ prmind as... |
| prmind 16562 | Perform induction over the... |
| dvdsprime 16563 | If ` M ` divides a prime, ... |
| nprm 16564 | A product of two integers ... |
| nprmi 16565 | An inference for composite... |
| dvdsnprmd 16566 | If a number is divisible b... |
| prm2orodd 16567 | A prime number is either 2... |
| 2prm 16568 | 2 is a prime number. (Con... |
| 2mulprm 16569 | A multiple of two is prime... |
| 3prm 16570 | 3 is a prime number. (Con... |
| 4nprm 16571 | 4 is not a prime number. ... |
| prmuz2 16572 | A prime number is an integ... |
| prmgt1 16573 | A prime number is an integ... |
| prmm2nn0 16574 | Subtracting 2 from a prime... |
| oddprmgt2 16575 | An odd prime is greater th... |
| oddprmge3 16576 | An odd prime is greater th... |
| ge2nprmge4 16577 | A composite integer greate... |
| sqnprm 16578 | A square is never prime. ... |
| dvdsprm 16579 | An integer greater than or... |
| exprmfct 16580 | Every integer greater than... |
| prmdvdsfz 16581 | Each integer greater than ... |
| nprmdvds1 16582 | No prime number divides 1.... |
| isprm5 16583 | One need only check prime ... |
| isprm7 16584 | One need only check prime ... |
| maxprmfct 16585 | The set of prime factors o... |
| divgcdodd 16586 | Either ` A / ( A gcd B ) `... |
| coprm 16587 | A prime number either divi... |
| prmrp 16588 | Unequal prime numbers are ... |
| euclemma 16589 | Euclid's lemma. A prime n... |
| isprm6 16590 | A number is prime iff it s... |
| prmdvdsexp 16591 | A prime divides a positive... |
| prmdvdsexpb 16592 | A prime divides a positive... |
| prmdvdsexpr 16593 | If a prime divides a nonne... |
| prmdvdssq 16594 | Condition for a prime divi... |
| prmdvdssqOLD 16595 | Obsolete version of ~ prmd... |
| prmexpb 16596 | Two positive prime powers ... |
| prmfac1 16597 | The factorial of a number ... |
| rpexp 16598 | If two numbers ` A ` and `... |
| rpexp1i 16599 | Relative primality passes ... |
| rpexp12i 16600 | Relative primality passes ... |
| prmndvdsfaclt 16601 | A prime number does not di... |
| prmdvdsncoprmbd 16602 | Two positive integers are ... |
| ncoprmlnprm 16603 | If two positive integers a... |
| cncongrprm 16604 | Corollary 2 of Cancellabil... |
| isevengcd2 16605 | The predicate "is an even ... |
| isoddgcd1 16606 | The predicate "is an odd n... |
| 3lcm2e6 16607 | The least common multiple ... |
| qnumval 16612 | Value of the canonical num... |
| qdenval 16613 | Value of the canonical den... |
| qnumdencl 16614 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16615 | The canonical numerator of... |
| qdencl 16616 | The canonical denominator ... |
| fnum 16617 | Canonical numerator define... |
| fden 16618 | Canonical denominator defi... |
| qnumdenbi 16619 | Two numbers are the canoni... |
| qnumdencoprm 16620 | The canonical representati... |
| qeqnumdivden 16621 | Recover a rational number ... |
| qmuldeneqnum 16622 | Multiplying a rational by ... |
| divnumden 16623 | Calculate the reduced form... |
| divdenle 16624 | Reducing a quotient never ... |
| qnumgt0 16625 | A rational is positive iff... |
| qgt0numnn 16626 | A rational is positive iff... |
| nn0gcdsq 16627 | Squaring commutes with GCD... |
| zgcdsq 16628 | ~ nn0gcdsq extended to int... |
| numdensq 16629 | Squaring a rational square... |
| numsq 16630 | Square commutes with canon... |
| densq 16631 | Square commutes with canon... |
| qden1elz 16632 | A rational is an integer i... |
| zsqrtelqelz 16633 | If an integer has a ration... |
| nonsq 16634 | Any integer strictly betwe... |
| phival 16639 | Value of the Euler ` phi `... |
| phicl2 16640 | Bounds and closure for the... |
| phicl 16641 | Closure for the value of t... |
| phibndlem 16642 | Lemma for ~ phibnd . (Con... |
| phibnd 16643 | A slightly tighter bound o... |
| phicld 16644 | Closure for the value of t... |
| phi1 16645 | Value of the Euler ` phi `... |
| dfphi2 16646 | Alternate definition of th... |
| hashdvds 16647 | The number of numbers in a... |
| phiprmpw 16648 | Value of the Euler ` phi `... |
| phiprm 16649 | Value of the Euler ` phi `... |
| crth 16650 | The Chinese Remainder Theo... |
| phimullem 16651 | Lemma for ~ phimul . (Con... |
| phimul 16652 | The Euler ` phi ` function... |
| eulerthlem1 16653 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16654 | Lemma for ~ eulerth . (Co... |
| eulerth 16655 | Euler's theorem, a general... |
| fermltl 16656 | Fermat's little theorem. ... |
| prmdiv 16657 | Show an explicit expressio... |
| prmdiveq 16658 | The modular inverse of ` A... |
| prmdivdiv 16659 | The (modular) inverse of t... |
| hashgcdlem 16660 | A correspondence between e... |
| hashgcdeq 16661 | Number of initial positive... |
| phisum 16662 | The divisor sum identity o... |
| odzval 16663 | Value of the order functio... |
| odzcllem 16664 | - Lemma for ~ odzcl , show... |
| odzcl 16665 | The order of a group eleme... |
| odzid 16666 | Any element raised to the ... |
| odzdvds 16667 | The only powers of ` A ` t... |
| odzphi 16668 | The order of any group ele... |
| modprm1div 16669 | A prime number divides an ... |
| m1dvdsndvds 16670 | If an integer minus 1 is d... |
| modprminv 16671 | Show an explicit expressio... |
| modprminveq 16672 | The modular inverse of ` A... |
| vfermltl 16673 | Variant of Fermat's little... |
| vfermltlALT 16674 | Alternate proof of ~ vferm... |
| powm2modprm 16675 | If an integer minus 1 is d... |
| reumodprminv 16676 | For any prime number and f... |
| modprm0 16677 | For two positive integers ... |
| nnnn0modprm0 16678 | For a positive integer and... |
| modprmn0modprm0 16679 | For an integer not being 0... |
| coprimeprodsq 16680 | If three numbers are copri... |
| coprimeprodsq2 16681 | If three numbers are copri... |
| oddprm 16682 | A prime not equal to ` 2 `... |
| nnoddn2prm 16683 | A prime not equal to ` 2 `... |
| oddn2prm 16684 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16685 | A number is a prime number... |
| prm23lt5 16686 | A prime less than 5 is eit... |
| prm23ge5 16687 | A prime is either 2 or 3 o... |
| pythagtriplem1 16688 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16689 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16690 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16691 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16692 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16693 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16694 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16695 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16696 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16697 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16698 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16699 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16700 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16701 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16702 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16703 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16704 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16705 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16706 | Parameterize the Pythagore... |
| iserodd 16707 | Collect the odd terms in a... |
| pclem 16710 | - Lemma for the prime powe... |
| pcprecl 16711 | Closure of the prime power... |
| pcprendvds 16712 | Non-divisibility property ... |
| pcprendvds2 16713 | Non-divisibility property ... |
| pcpre1 16714 | Value of the prime power p... |
| pcpremul 16715 | Multiplicative property of... |
| pcval 16716 | The value of the prime pow... |
| pceulem 16717 | Lemma for ~ pceu . (Contr... |
| pceu 16718 | Uniqueness for the prime p... |
| pczpre 16719 | Connect the prime count pr... |
| pczcl 16720 | Closure of the prime power... |
| pccl 16721 | Closure of the prime power... |
| pccld 16722 | Closure of the prime power... |
| pcmul 16723 | Multiplication property of... |
| pcdiv 16724 | Division property of the p... |
| pcqmul 16725 | Multiplication property of... |
| pc0 16726 | The value of the prime pow... |
| pc1 16727 | Value of the prime count f... |
| pcqcl 16728 | Closure of the general pri... |
| pcqdiv 16729 | Division property of the p... |
| pcrec 16730 | Prime power of a reciproca... |
| pcexp 16731 | Prime power of an exponent... |
| pcxnn0cl 16732 | Extended nonnegative integ... |
| pcxcl 16733 | Extended real closure of t... |
| pcge0 16734 | The prime count of an inte... |
| pczdvds 16735 | Defining property of the p... |
| pcdvds 16736 | Defining property of the p... |
| pczndvds 16737 | Defining property of the p... |
| pcndvds 16738 | Defining property of the p... |
| pczndvds2 16739 | The remainder after dividi... |
| pcndvds2 16740 | The remainder after dividi... |
| pcdvdsb 16741 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16742 | There are a positive numbe... |
| pceq0 16743 | There are zero powers of a... |
| pcidlem 16744 | The prime count of a prime... |
| pcid 16745 | The prime count of a prime... |
| pcneg 16746 | The prime count of a negat... |
| pcabs 16747 | The prime count of an abso... |
| pcdvdstr 16748 | The prime count increases ... |
| pcgcd1 16749 | The prime count of a GCD i... |
| pcgcd 16750 | The prime count of a GCD i... |
| pc2dvds 16751 | A characterization of divi... |
| pc11 16752 | The prime count function, ... |
| pcz 16753 | The prime count function c... |
| pcprmpw2 16754 | Self-referential expressio... |
| pcprmpw 16755 | Self-referential expressio... |
| dvdsprmpweq 16756 | If a positive integer divi... |
| dvdsprmpweqnn 16757 | If an integer greater than... |
| dvdsprmpweqle 16758 | If a positive integer divi... |
| difsqpwdvds 16759 | If the difference of two s... |
| pcaddlem 16760 | Lemma for ~ pcadd . The o... |
| pcadd 16761 | An inequality for the prim... |
| pcadd2 16762 | The inequality of ~ pcadd ... |
| pcmptcl 16763 | Closure for the prime powe... |
| pcmpt 16764 | Construct a function with ... |
| pcmpt2 16765 | Dividing two prime count m... |
| pcmptdvds 16766 | The partial products of th... |
| pcprod 16767 | The product of the primes ... |
| sumhash 16768 | The sum of 1 over a set is... |
| fldivp1 16769 | The difference between the... |
| pcfaclem 16770 | Lemma for ~ pcfac . (Cont... |
| pcfac 16771 | Calculate the prime count ... |
| pcbc 16772 | Calculate the prime count ... |
| qexpz 16773 | If a power of a rational n... |
| expnprm 16774 | A second or higher power o... |
| oddprmdvds 16775 | Every positive integer whi... |
| prmpwdvds 16776 | A relation involving divis... |
| pockthlem 16777 | Lemma for ~ pockthg . (Co... |
| pockthg 16778 | The generalized Pocklingto... |
| pockthi 16779 | Pocklington's theorem, whi... |
| unbenlem 16780 | Lemma for ~ unben . (Cont... |
| unben 16781 | An unbounded set of positi... |
| infpnlem1 16782 | Lemma for ~ infpn . The s... |
| infpnlem2 16783 | Lemma for ~ infpn . For a... |
| infpn 16784 | There exist infinitely man... |
| infpn2 16785 | There exist infinitely man... |
| prmunb 16786 | The primes are unbounded. ... |
| prminf 16787 | There are an infinite numb... |
| prmreclem1 16788 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16789 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16790 | Lemma for ~ prmrec . The ... |
| prmreclem4 16791 | Lemma for ~ prmrec . Show... |
| prmreclem5 16792 | Lemma for ~ prmrec . Here... |
| prmreclem6 16793 | Lemma for ~ prmrec . If t... |
| prmrec 16794 | The sum of the reciprocals... |
| 1arithlem1 16795 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16796 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16797 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16798 | Lemma for ~ 1arith . (Con... |
| 1arith 16799 | Fundamental theorem of ari... |
| 1arith2 16800 | Fundamental theorem of ari... |
| elgz 16803 | Elementhood in the gaussia... |
| gzcn 16804 | A gaussian integer is a co... |
| zgz 16805 | An integer is a gaussian i... |
| igz 16806 | ` _i ` is a gaussian integ... |
| gznegcl 16807 | The gaussian integers are ... |
| gzcjcl 16808 | The gaussian integers are ... |
| gzaddcl 16809 | The gaussian integers are ... |
| gzmulcl 16810 | The gaussian integers are ... |
| gzreim 16811 | Construct a gaussian integ... |
| gzsubcl 16812 | The gaussian integers are ... |
| gzabssqcl 16813 | The squared norm of a gaus... |
| 4sqlem5 16814 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16815 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16816 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16817 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16818 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16819 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16820 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16821 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16822 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16823 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16824 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16825 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16826 | Euler's four-square identi... |
| 4sqlem11 16827 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16828 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16829 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16830 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16831 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16832 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16833 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16834 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16835 | Lemma for ~ 4sq . The pro... |
| 4sq 16836 | Lagrange's four-square the... |
| vdwapfval 16843 | Define the arithmetic prog... |
| vdwapf 16844 | The arithmetic progression... |
| vdwapval 16845 | Value of the arithmetic pr... |
| vdwapun 16846 | Remove the first element o... |
| vdwapid1 16847 | The first element of an ar... |
| vdwap0 16848 | Value of a length-1 arithm... |
| vdwap1 16849 | Value of a length-1 arithm... |
| vdwmc 16850 | The predicate " The ` <. R... |
| vdwmc2 16851 | Expand out the definition ... |
| vdwpc 16852 | The predicate " The colori... |
| vdwlem1 16853 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16854 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16855 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16856 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16857 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16858 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16859 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16860 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16861 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16862 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16863 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16864 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16865 | Lemma for ~ vdw . Main in... |
| vdw 16866 | Van der Waerden's theorem.... |
| vdwnnlem1 16867 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16868 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16869 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16870 | Van der Waerden's theorem,... |
| ramtlecl 16872 | The set ` T ` of numbers w... |
| hashbcval 16874 | Value of the "binomial set... |
| hashbccl 16875 | The binomial set is a fini... |
| hashbcss 16876 | Subset relation for the bi... |
| hashbc0 16877 | The set of subsets of size... |
| hashbc2 16878 | The size of the binomial s... |
| 0hashbc 16879 | There are no subsets of th... |
| ramval 16880 | The value of the Ramsey nu... |
| ramcl2lem 16881 | Lemma for extended real cl... |
| ramtcl 16882 | The Ramsey number has the ... |
| ramtcl2 16883 | The Ramsey number is an in... |
| ramtub 16884 | The Ramsey number is a low... |
| ramub 16885 | The Ramsey number is a low... |
| ramub2 16886 | It is sufficient to check ... |
| rami 16887 | The defining property of a... |
| ramcl2 16888 | The Ramsey number is eithe... |
| ramxrcl 16889 | The Ramsey number is an ex... |
| ramubcl 16890 | If the Ramsey number is up... |
| ramlb 16891 | Establish a lower bound on... |
| 0ram 16892 | The Ramsey number when ` M... |
| 0ram2 16893 | The Ramsey number when ` M... |
| ram0 16894 | The Ramsey number when ` R... |
| 0ramcl 16895 | Lemma for ~ ramcl : Exist... |
| ramz2 16896 | The Ramsey number when ` F... |
| ramz 16897 | The Ramsey number when ` F... |
| ramub1lem1 16898 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16899 | Lemma for ~ ramub1 . (Con... |
| ramub1 16900 | Inductive step for Ramsey'... |
| ramcl 16901 | Ramsey's theorem: the Rams... |
| ramsey 16902 | Ramsey's theorem with the ... |
| prmoval 16905 | Value of the primorial fun... |
| prmocl 16906 | Closure of the primorial f... |
| prmone0 16907 | The primorial function is ... |
| prmo0 16908 | The primorial of 0. (Cont... |
| prmo1 16909 | The primorial of 1. (Cont... |
| prmop1 16910 | The primorial of a success... |
| prmonn2 16911 | Value of the primorial fun... |
| prmo2 16912 | The primorial of 2. (Cont... |
| prmo3 16913 | The primorial of 3. (Cont... |
| prmdvdsprmo 16914 | The primorial of a number ... |
| prmdvdsprmop 16915 | The primorial of a number ... |
| fvprmselelfz 16916 | The value of the prime sel... |
| fvprmselgcd1 16917 | The greatest common diviso... |
| prmolefac 16918 | The primorial of a positiv... |
| prmodvdslcmf 16919 | The primorial of a nonnega... |
| prmolelcmf 16920 | The primorial of a positiv... |
| prmgaplem1 16921 | Lemma for ~ prmgap : The ... |
| prmgaplem2 16922 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 16923 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 16924 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 16925 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 16926 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 16927 | Lemma for ~ prmgap : for e... |
| prmgaplem6 16928 | Lemma for ~ prmgap : for e... |
| prmgaplem7 16929 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 16930 | Lemma for ~ prmgap . (Con... |
| prmgap 16931 | The prime gap theorem: for... |
| prmgaplcm 16932 | Alternate proof of ~ prmga... |
| prmgapprmolem 16933 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 16934 | Alternate proof of ~ prmga... |
| dec2dvds 16935 | Divisibility by two is obv... |
| dec5dvds 16936 | Divisibility by five is ob... |
| dec5dvds2 16937 | Divisibility by five is ob... |
| dec5nprm 16938 | Divisibility by five is ob... |
| dec2nprm 16939 | Divisibility by two is obv... |
| modxai 16940 | Add exponents in a power m... |
| mod2xi 16941 | Double exponents in a powe... |
| modxp1i 16942 | Add one to an exponent in ... |
| mod2xnegi 16943 | Version of ~ mod2xi with a... |
| modsubi 16944 | Subtract from within a mod... |
| gcdi 16945 | Calculate a GCD via Euclid... |
| gcdmodi 16946 | Calculate a GCD via Euclid... |
| decexp2 16947 | Calculate a power of two. ... |
| numexp0 16948 | Calculate an integer power... |
| numexp1 16949 | Calculate an integer power... |
| numexpp1 16950 | Calculate an integer power... |
| numexp2x 16951 | Double an integer power. ... |
| decsplit0b 16952 | Split a decimal number int... |
| decsplit0 16953 | Split a decimal number int... |
| decsplit1 16954 | Split a decimal number int... |
| decsplit 16955 | Split a decimal number int... |
| karatsuba 16956 | The Karatsuba multiplicati... |
| 2exp4 16957 | Two to the fourth power is... |
| 2exp5 16958 | Two to the fifth power is ... |
| 2exp6 16959 | Two to the sixth power is ... |
| 2exp7 16960 | Two to the seventh power i... |
| 2exp8 16961 | Two to the eighth power is... |
| 2exp11 16962 | Two to the eleventh power ... |
| 2exp16 16963 | Two to the sixteenth power... |
| 3exp3 16964 | Three to the third power i... |
| 2expltfac 16965 | The factorial grows faster... |
| cshwsidrepsw 16966 | If cyclically shifting a w... |
| cshwsidrepswmod0 16967 | If cyclically shifting a w... |
| cshwshashlem1 16968 | If cyclically shifting a w... |
| cshwshashlem2 16969 | If cyclically shifting a w... |
| cshwshashlem3 16970 | If cyclically shifting a w... |
| cshwsdisj 16971 | The singletons resulting b... |
| cshwsiun 16972 | The set of (different!) wo... |
| cshwsex 16973 | The class of (different!) ... |
| cshws0 16974 | The size of the set of (di... |
| cshwrepswhash1 16975 | The size of the set of (di... |
| cshwshashnsame 16976 | If a word (not consisting ... |
| cshwshash 16977 | If a word has a length bei... |
| prmlem0 16978 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 16979 | A quick proof skeleton to ... |
| prmlem1 16980 | A quick proof skeleton to ... |
| 5prm 16981 | 5 is a prime number. (Con... |
| 6nprm 16982 | 6 is not a prime number. ... |
| 7prm 16983 | 7 is a prime number. (Con... |
| 8nprm 16984 | 8 is not a prime number. ... |
| 9nprm 16985 | 9 is not a prime number. ... |
| 10nprm 16986 | 10 is not a prime number. ... |
| 11prm 16987 | 11 is a prime number. (Co... |
| 13prm 16988 | 13 is a prime number. (Co... |
| 17prm 16989 | 17 is a prime number. (Co... |
| 19prm 16990 | 19 is a prime number. (Co... |
| 23prm 16991 | 23 is a prime number. (Co... |
| prmlem2 16992 | Our last proving session g... |
| 37prm 16993 | 37 is a prime number. (Co... |
| 43prm 16994 | 43 is a prime number. (Co... |
| 83prm 16995 | 83 is a prime number. (Co... |
| 139prm 16996 | 139 is a prime number. (C... |
| 163prm 16997 | 163 is a prime number. (C... |
| 317prm 16998 | 317 is a prime number. (C... |
| 631prm 16999 | 631 is a prime number. (C... |
| prmo4 17000 | The primorial of 4. (Cont... |
| prmo5 17001 | The primorial of 5. (Cont... |
| prmo6 17002 | The primorial of 6. (Cont... |
| 1259lem1 17003 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 17004 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 17005 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 17006 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 17007 | Lemma for ~ 1259prm . Cal... |
| 1259prm 17008 | 1259 is a prime number. (... |
| 2503lem1 17009 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 17010 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 17011 | Lemma for ~ 2503prm . Cal... |
| 2503prm 17012 | 2503 is a prime number. (... |
| 4001lem1 17013 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 17014 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 17015 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 17016 | Lemma for ~ 4001prm . Cal... |
| 4001prm 17017 | 4001 is a prime number. (... |
| brstruct 17020 | The structure relation is ... |
| isstruct2 17021 | The property of being a st... |
| structex 17022 | A structure is a set. (Co... |
| structn0fun 17023 | A structure without the em... |
| isstruct 17024 | The property of being a st... |
| structcnvcnv 17025 | Two ways to express the re... |
| structfung 17026 | The converse of the conver... |
| structfun 17027 | Convert between two kinds ... |
| structfn 17028 | Convert between two kinds ... |
| strleun 17029 | Combine two structures int... |
| strle1 17030 | Make a structure from a si... |
| strle2 17031 | Make a structure from a pa... |
| strle3 17032 | Make a structure from a tr... |
| sbcie2s 17033 | A special version of class... |
| sbcie3s 17034 | A special version of class... |
| reldmsets 17037 | The structure override ope... |
| setsvalg 17038 | Value of the structure rep... |
| setsval 17039 | Value of the structure rep... |
| fvsetsid 17040 | The value of the structure... |
| fsets 17041 | The structure replacement ... |
| setsdm 17042 | The domain of a structure ... |
| setsfun 17043 | A structure with replaceme... |
| setsfun0 17044 | A structure with replaceme... |
| setsn0fun 17045 | The value of the structure... |
| setsstruct2 17046 | An extensible structure wi... |
| setsexstruct2 17047 | An extensible structure wi... |
| setsstruct 17048 | An extensible structure wi... |
| wunsets 17049 | Closure of structure repla... |
| setsres 17050 | The structure replacement ... |
| setsabs 17051 | Replacing the same compone... |
| setscom 17052 | Component-setting is commu... |
| sloteq 17055 | Equality theorem for the `... |
| slotfn 17056 | A slot is a function on se... |
| strfvnd 17057 | Deduction version of ~ str... |
| strfvn 17058 | Value of a structure compo... |
| strfvss 17059 | A structure component extr... |
| wunstr 17060 | Closure of a structure ind... |
| str0 17061 | All components of the empt... |
| strfvi 17062 | Structure slot extractors ... |
| fveqprc 17063 | Lemma for showing the equa... |
| oveqprc 17064 | Lemma for showing the equa... |
| wunndx 17067 | Closure of the index extra... |
| ndxarg 17068 | Get the numeric argument f... |
| ndxid 17069 | A structure component extr... |
| strndxid 17070 | The value of a structure c... |
| setsidvald 17071 | Value of the structure rep... |
| setsidvaldOLD 17072 | Obsolete version of ~ sets... |
| strfvd 17073 | Deduction version of ~ str... |
| strfv2d 17074 | Deduction version of ~ str... |
| strfv2 17075 | A variation on ~ strfv to ... |
| strfv 17076 | Extract a structure compon... |
| strfv3 17077 | Variant on ~ strfv for lar... |
| strssd 17078 | Deduction version of ~ str... |
| strss 17079 | Propagate component extrac... |
| setsid 17080 | Value of the structure rep... |
| setsnid 17081 | Value of the structure rep... |
| setsnidOLD 17082 | Obsolete proof of ~ setsni... |
| baseval 17085 | Value of the base set extr... |
| baseid 17086 | Utility theorem: index-ind... |
| basfn 17087 | The base set extractor is ... |
| base0 17088 | The base set of the empty ... |
| elbasfv 17089 | Utility theorem: reverse c... |
| elbasov 17090 | Utility theorem: reverse c... |
| strov2rcl 17091 | Partial reverse closure fo... |
| basendx 17092 | Index value of the base se... |
| basendxnn 17093 | The index value of the bas... |
| basendxnnOLD 17094 | Obsolete proof of ~ basend... |
| basndxelwund 17095 | The index of the base set ... |
| basprssdmsets 17096 | The pair of the base index... |
| opelstrbas 17097 | The base set of a structur... |
| 1strstr 17098 | A constructed one-slot str... |
| 1strstr1 17099 | A constructed one-slot str... |
| 1strbas 17100 | The base set of a construc... |
| 1strbasOLD 17101 | Obsolete proof of ~ 1strba... |
| 1strwunbndx 17102 | A constructed one-slot str... |
| 1strwun 17103 | A constructed one-slot str... |
| 1strwunOLD 17104 | Obsolete version of ~ 1str... |
| 2strstr 17105 | A constructed two-slot str... |
| 2strbas 17106 | The base set of a construc... |
| 2strop 17107 | The other slot of a constr... |
| 2strstr1 17108 | A constructed two-slot str... |
| 2strstr1OLD 17109 | Obsolete version of ~ 2str... |
| 2strbas1 17110 | The base set of a construc... |
| 2strop1 17111 | The other slot of a constr... |
| reldmress 17114 | The structure restriction ... |
| ressval 17115 | Value of structure restric... |
| ressid2 17116 | General behavior of trivia... |
| ressval2 17117 | Value of nontrivial struct... |
| ressbas 17118 | Base set of a structure re... |
| ressbasOLD 17119 | Obsolete proof of ~ ressba... |
| ressbas2 17120 | Base set of a structure re... |
| ressbasss 17121 | The base set of a restrict... |
| resseqnbas 17122 | The components of an exten... |
| resslemOLD 17123 | Obsolete version of ~ ress... |
| ress0 17124 | All restrictions of the nu... |
| ressid 17125 | Behavior of trivial restri... |
| ressinbas 17126 | Restriction only cares abo... |
| ressval3d 17127 | Value of structure restric... |
| ressval3dOLD 17128 | Obsolete version of ~ ress... |
| ressress 17129 | Restriction composition la... |
| ressabs 17130 | Restriction absorption law... |
| wunress 17131 | Closure of structure restr... |
| wunressOLD 17132 | Obsolete proof of ~ wunres... |
| plusgndx 17159 | Index value of the ~ df-pl... |
| plusgid 17160 | Utility theorem: index-ind... |
| plusgndxnn 17161 | The index of the slot for ... |
| basendxltplusgndx 17162 | The index of the slot for ... |
| basendxnplusgndx 17163 | The slot for the base set ... |
| basendxnplusgndxOLD 17164 | Obsolete version of ~ base... |
| grpstr 17165 | A constructed group is a s... |
| grpstrndx 17166 | A constructed group is a s... |
| grpbase 17167 | The base set of a construc... |
| grpbaseOLD 17168 | Obsolete version of ~ grpb... |
| grpplusg 17169 | The operation of a constru... |
| grpplusgOLD 17170 | Obsolete version of ~ grpp... |
| ressplusg 17171 | ` +g ` is unaffected by re... |
| grpbasex 17172 | The base of an explicitly ... |
| grpplusgx 17173 | The operation of an explic... |
| mulrndx 17174 | Index value of the ~ df-mu... |
| mulrid 17175 | Utility theorem: index-ind... |
| basendxnmulrndx 17176 | The slot for the base set ... |
| basendxnmulrndxOLD 17177 | Obsolete proof of ~ basend... |
| plusgndxnmulrndx 17178 | The slot for the group (ad... |
| rngstr 17179 | A constructed ring is a st... |
| rngbase 17180 | The base set of a construc... |
| rngplusg 17181 | The additive operation of ... |
| rngmulr 17182 | The multiplicative operati... |
| starvndx 17183 | Index value of the ~ df-st... |
| starvid 17184 | Utility theorem: index-ind... |
| starvndxnbasendx 17185 | The slot for the involutio... |
| starvndxnplusgndx 17186 | The slot for the involutio... |
| starvndxnmulrndx 17187 | The slot for the involutio... |
| ressmulr 17188 | ` .r ` is unaffected by re... |
| ressstarv 17189 | ` *r ` is unaffected by re... |
| srngstr 17190 | A constructed star ring is... |
| srngbase 17191 | The base set of a construc... |
| srngplusg 17192 | The addition operation of ... |
| srngmulr 17193 | The multiplication operati... |
| srnginvl 17194 | The involution function of... |
| scandx 17195 | Index value of the ~ df-sc... |
| scaid 17196 | Utility theorem: index-ind... |
| scandxnbasendx 17197 | The slot for the scalar is... |
| scandxnplusgndx 17198 | The slot for the scalar fi... |
| scandxnmulrndx 17199 | The slot for the scalar fi... |
| vscandx 17200 | Index value of the ~ df-vs... |
| vscaid 17201 | Utility theorem: index-ind... |
| vscandxnbasendx 17202 | The slot for the scalar pr... |
| vscandxnplusgndx 17203 | The slot for the scalar pr... |
| vscandxnmulrndx 17204 | The slot for the scalar pr... |
| vscandxnscandx 17205 | The slot for the scalar pr... |
| lmodstr 17206 | A constructed left module ... |
| lmodbase 17207 | The base set of a construc... |
| lmodplusg 17208 | The additive operation of ... |
| lmodsca 17209 | The set of scalars of a co... |
| lmodvsca 17210 | The scalar product operati... |
| ipndx 17211 | Index value of the ~ df-ip... |
| ipid 17212 | Utility theorem: index-ind... |
| ipndxnbasendx 17213 | The slot for the inner pro... |
| ipndxnplusgndx 17214 | The slot for the inner pro... |
| ipndxnmulrndx 17215 | The slot for the inner pro... |
| slotsdifipndx 17216 | The slot for the scalar is... |
| ipsstr 17217 | Lemma to shorten proofs of... |
| ipsbase 17218 | The base set of a construc... |
| ipsaddg 17219 | The additive operation of ... |
| ipsmulr 17220 | The multiplicative operati... |
| ipssca 17221 | The set of scalars of a co... |
| ipsvsca 17222 | The scalar product operati... |
| ipsip 17223 | The multiplicative operati... |
| resssca 17224 | ` Scalar ` is unaffected b... |
| ressvsca 17225 | ` .s ` is unaffected by re... |
| ressip 17226 | The inner product is unaff... |
| phlstr 17227 | A constructed pre-Hilbert ... |
| phlbase 17228 | The base set of a construc... |
| phlplusg 17229 | The additive operation of ... |
| phlsca 17230 | The ring of scalars of a c... |
| phlvsca 17231 | The scalar product operati... |
| phlip 17232 | The inner product (Hermiti... |
| tsetndx 17233 | Index value of the ~ df-ts... |
| tsetid 17234 | Utility theorem: index-ind... |
| tsetndxnn 17235 | The index of the slot for ... |
| basendxlttsetndx 17236 | The index of the slot for ... |
| tsetndxnbasendx 17237 | The slot for the topology ... |
| tsetndxnplusgndx 17238 | The slot for the topology ... |
| tsetndxnmulrndx 17239 | The slot for the topology ... |
| tsetndxnstarvndx 17240 | The slot for the topology ... |
| slotstnscsi 17241 | The slots ` Scalar ` , ` .... |
| topgrpstr 17242 | A constructed topological ... |
| topgrpbas 17243 | The base set of a construc... |
| topgrpplusg 17244 | The additive operation of ... |
| topgrptset 17245 | The topology of a construc... |
| resstset 17246 | ` TopSet ` is unaffected b... |
| plendx 17247 | Index value of the ~ df-pl... |
| pleid 17248 | Utility theorem: self-refe... |
| plendxnn 17249 | The index value of the ord... |
| basendxltplendx 17250 | The index value of the ` B... |
| plendxnbasendx 17251 | The slot for the order is ... |
| plendxnplusgndx 17252 | The slot for the "less tha... |
| plendxnmulrndx 17253 | The slot for the "less tha... |
| plendxnscandx 17254 | The slot for the "less tha... |
| plendxnvscandx 17255 | The slot for the "less tha... |
| slotsdifplendx 17256 | The index of the slot for ... |
| otpsstr 17257 | Functionality of a topolog... |
| otpsbas 17258 | The base set of a topologi... |
| otpstset 17259 | The open sets of a topolog... |
| otpsle 17260 | The order of a topological... |
| ressle 17261 | ` le ` is unaffected by re... |
| ocndx 17262 | Index value of the ~ df-oc... |
| ocid 17263 | Utility theorem: index-ind... |
| basendxnocndx 17264 | The slot for the orthocomp... |
| plendxnocndx 17265 | The slot for the orthocomp... |
| dsndx 17266 | Index value of the ~ df-ds... |
| dsid 17267 | Utility theorem: index-ind... |
| dsndxnn 17268 | The index of the slot for ... |
| basendxltdsndx 17269 | The index of the slot for ... |
| dsndxnbasendx 17270 | The slot for the distance ... |
| dsndxnplusgndx 17271 | The slot for the distance ... |
| dsndxnmulrndx 17272 | The slot for the distance ... |
| slotsdnscsi 17273 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17274 | The slot for the distance ... |
| slotsdifdsndx 17275 | The index of the slot for ... |
| unifndx 17276 | Index value of the ~ df-un... |
| unifid 17277 | Utility theorem: index-ind... |
| unifndxnn 17278 | The index of the slot for ... |
| basendxltunifndx 17279 | The index of the slot for ... |
| unifndxnbasendx 17280 | The slot for the uniform s... |
| unifndxntsetndx 17281 | The slot for the uniform s... |
| slotsdifunifndx 17282 | The index of the slot for ... |
| ressunif 17283 | ` UnifSet ` is unaffected ... |
| odrngstr 17284 | Functionality of an ordere... |
| odrngbas 17285 | The base set of an ordered... |
| odrngplusg 17286 | The addition operation of ... |
| odrngmulr 17287 | The multiplication operati... |
| odrngtset 17288 | The open sets of an ordere... |
| odrngle 17289 | The order of an ordered me... |
| odrngds 17290 | The metric of an ordered m... |
| ressds 17291 | ` dist ` is unaffected by ... |
| homndx 17292 | Index value of the ~ df-ho... |
| homid 17293 | Utility theorem: index-ind... |
| ccondx 17294 | Index value of the ~ df-cc... |
| ccoid 17295 | Utility theorem: index-ind... |
| slotsbhcdif 17296 | The slots ` Base ` , ` Hom... |
| slotsbhcdifOLD 17297 | Obsolete proof of ~ slotsb... |
| slotsdifplendx2 17298 | The index of the slot for ... |
| slotsdifocndx 17299 | The index of the slot for ... |
| resshom 17300 | ` Hom ` is unaffected by r... |
| ressco 17301 | ` comp ` is unaffected by ... |
| restfn 17306 | The subspace topology oper... |
| topnfn 17307 | The topology extractor fun... |
| restval 17308 | The subspace topology indu... |
| elrest 17309 | The predicate "is an open ... |
| elrestr 17310 | Sufficient condition for b... |
| 0rest 17311 | Value of the structure res... |
| restid2 17312 | The subspace topology over... |
| restsspw 17313 | The subspace topology is a... |
| firest 17314 | The finite intersections o... |
| restid 17315 | The subspace topology of t... |
| topnval 17316 | Value of the topology extr... |
| topnid 17317 | Value of the topology extr... |
| topnpropd 17318 | The topology extractor fun... |
| reldmprds 17330 | The structure product is a... |
| prdsbasex 17332 | Lemma for structure produc... |
| imasvalstr 17333 | An image structure value i... |
| prdsvalstr 17334 | Structure product value is... |
| prdsbaslem 17335 | Lemma for ~ prdsbas and si... |
| prdsvallem 17336 | Lemma for ~ prdsval . (Co... |
| prdsval 17337 | Value of the structure pro... |
| prdssca 17338 | Scalar ring of a structure... |
| prdsbas 17339 | Base set of a structure pr... |
| prdsplusg 17340 | Addition in a structure pr... |
| prdsmulr 17341 | Multiplication in a struct... |
| prdsvsca 17342 | Scalar multiplication in a... |
| prdsip 17343 | Inner product in a structu... |
| prdsle 17344 | Structure product weak ord... |
| prdsless 17345 | Closure of the order relat... |
| prdsds 17346 | Structure product distance... |
| prdsdsfn 17347 | Structure product distance... |
| prdstset 17348 | Structure product topology... |
| prdshom 17349 | Structure product hom-sets... |
| prdsco 17350 | Structure product composit... |
| prdsbas2 17351 | The base set of a structur... |
| prdsbasmpt 17352 | A constructed tuple is a p... |
| prdsbasfn 17353 | Points in the structure pr... |
| prdsbasprj 17354 | Each point in a structure ... |
| prdsplusgval 17355 | Value of a componentwise s... |
| prdsplusgfval 17356 | Value of a structure produ... |
| prdsmulrval 17357 | Value of a componentwise r... |
| prdsmulrfval 17358 | Value of a structure produ... |
| prdsleval 17359 | Value of the product order... |
| prdsdsval 17360 | Value of the metric in a s... |
| prdsvscaval 17361 | Scalar multiplication in a... |
| prdsvscafval 17362 | Scalar multiplication of a... |
| prdsbas3 17363 | The base set of an indexed... |
| prdsbasmpt2 17364 | A constructed tuple is a p... |
| prdsbascl 17365 | An element of the base has... |
| prdsdsval2 17366 | Value of the metric in a s... |
| prdsdsval3 17367 | Value of the metric in a s... |
| pwsval 17368 | Value of a structure power... |
| pwsbas 17369 | Base set of a structure po... |
| pwselbasb 17370 | Membership in the base set... |
| pwselbas 17371 | An element of a structure ... |
| pwsplusgval 17372 | Value of addition in a str... |
| pwsmulrval 17373 | Value of multiplication in... |
| pwsle 17374 | Ordering in a structure po... |
| pwsleval 17375 | Ordering in a structure po... |
| pwsvscafval 17376 | Scalar multiplication in a... |
| pwsvscaval 17377 | Scalar multiplication of a... |
| pwssca 17378 | The ring of scalars of a s... |
| pwsdiagel 17379 | Membership of diagonal ele... |
| pwssnf1o 17380 | Triviality of singleton po... |
| imasval 17393 | Value of an image structur... |
| imasbas 17394 | The base set of an image s... |
| imasds 17395 | The distance function of a... |
| imasdsfn 17396 | The distance function is a... |
| imasdsval 17397 | The distance function of a... |
| imasdsval2 17398 | The distance function of a... |
| imasplusg 17399 | The group operation in an ... |
| imasmulr 17400 | The ring multiplication in... |
| imassca 17401 | The scalar field of an ima... |
| imasvsca 17402 | The scalar multiplication ... |
| imasip 17403 | The inner product of an im... |
| imastset 17404 | The topology of an image s... |
| imasle 17405 | The ordering of an image s... |
| f1ocpbllem 17406 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17407 | An injection is compatible... |
| f1ovscpbl 17408 | An injection is compatible... |
| f1olecpbl 17409 | An injection is compatible... |
| imasaddfnlem 17410 | The image structure operat... |
| imasaddvallem 17411 | The operation of an image ... |
| imasaddflem 17412 | The image set operations a... |
| imasaddfn 17413 | The image structure's grou... |
| imasaddval 17414 | The value of an image stru... |
| imasaddf 17415 | The image structure's grou... |
| imasmulfn 17416 | The image structure's ring... |
| imasmulval 17417 | The value of an image stru... |
| imasmulf 17418 | The image structure's ring... |
| imasvscafn 17419 | The image structure's scal... |
| imasvscaval 17420 | The value of an image stru... |
| imasvscaf 17421 | The image structure's scal... |
| imasless 17422 | The order relation defined... |
| imasleval 17423 | The value of the image str... |
| qusval 17424 | Value of a quotient struct... |
| quslem 17425 | The function in ~ qusval i... |
| qusin 17426 | Restrict the equivalence r... |
| qusbas 17427 | Base set of a quotient str... |
| quss 17428 | The scalar field of a quot... |
| divsfval 17429 | Value of the function in ~... |
| ercpbllem 17430 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17431 | Translate the function com... |
| erlecpbl 17432 | Translate the relation com... |
| qusaddvallem 17433 | Value of an operation defi... |
| qusaddflem 17434 | The operation of a quotien... |
| qusaddval 17435 | The base set of an image s... |
| qusaddf 17436 | The base set of an image s... |
| qusmulval 17437 | The base set of an image s... |
| qusmulf 17438 | The base set of an image s... |
| fnpr2o 17439 | Function with a domain of ... |
| fnpr2ob 17440 | Biconditional version of ~... |
| fvpr0o 17441 | The value of a function wi... |
| fvpr1o 17442 | The value of a function wi... |
| fvprif 17443 | The value of the pair func... |
| xpsfrnel 17444 | Elementhood in the target ... |
| xpsfeq 17445 | A function on ` 2o ` is de... |
| xpsfrnel2 17446 | Elementhood in the target ... |
| xpscf 17447 | Equivalent condition for t... |
| xpsfval 17448 | The value of the function ... |
| xpsff1o 17449 | The function appearing in ... |
| xpsfrn 17450 | A short expression for the... |
| xpsff1o2 17451 | The function appearing in ... |
| xpsval 17452 | Value of the binary struct... |
| xpsrnbas 17453 | The indexed structure prod... |
| xpsbas 17454 | The base set of the binary... |
| xpsaddlem 17455 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17456 | Value of the addition oper... |
| xpsmul 17457 | Value of the multiplicatio... |
| xpssca 17458 | Value of the scalar field ... |
| xpsvsca 17459 | Value of the scalar multip... |
| xpsless 17460 | Closure of the ordering in... |
| xpsle 17461 | Value of the ordering in a... |
| ismre 17470 | Property of being a Moore ... |
| fnmre 17471 | The Moore collection gener... |
| mresspw 17472 | A Moore collection is a su... |
| mress 17473 | A Moore-closed subset is a... |
| mre1cl 17474 | In any Moore collection th... |
| mreintcl 17475 | A nonempty collection of c... |
| mreiincl 17476 | A nonempty indexed interse... |
| mrerintcl 17477 | The relative intersection ... |
| mreriincl 17478 | The relative intersection ... |
| mreincl 17479 | Two closed sets have a clo... |
| mreuni 17480 | Since the entire base set ... |
| mreunirn 17481 | Two ways to express the no... |
| ismred 17482 | Properties that determine ... |
| ismred2 17483 | Properties that determine ... |
| mremre 17484 | The Moore collections of s... |
| submre 17485 | The subcollection of a clo... |
| mrcflem 17486 | The domain and codomain of... |
| fnmrc 17487 | Moore-closure is a well-be... |
| mrcfval 17488 | Value of the function expr... |
| mrcf 17489 | The Moore closure is a fun... |
| mrcval 17490 | Evaluation of the Moore cl... |
| mrccl 17491 | The Moore closure of a set... |
| mrcsncl 17492 | The Moore closure of a sin... |
| mrcid 17493 | The closure of a closed se... |
| mrcssv 17494 | The closure of a set is a ... |
| mrcidb 17495 | A set is closed iff it is ... |
| mrcss 17496 | Closure preserves subset o... |
| mrcssid 17497 | The closure of a set is a ... |
| mrcidb2 17498 | A set is closed iff it con... |
| mrcidm 17499 | The closure operation is i... |
| mrcsscl 17500 | The closure is the minimal... |
| mrcuni 17501 | Idempotence of closure und... |
| mrcun 17502 | Idempotence of closure und... |
| mrcssvd 17503 | The Moore closure of a set... |
| mrcssd 17504 | Moore closure preserves su... |
| mrcssidd 17505 | A set is contained in its ... |
| mrcidmd 17506 | Moore closure is idempoten... |
| mressmrcd 17507 | In a Moore system, if a se... |
| submrc 17508 | In a closure system which ... |
| mrieqvlemd 17509 | In a Moore system, if ` Y ... |
| mrisval 17510 | Value of the set of indepe... |
| ismri 17511 | Criterion for a set to be ... |
| ismri2 17512 | Criterion for a subset of ... |
| ismri2d 17513 | Criterion for a subset of ... |
| ismri2dd 17514 | Definition of independence... |
| mriss 17515 | An independent set of a Mo... |
| mrissd 17516 | An independent set of a Mo... |
| ismri2dad 17517 | Consequence of a set in a ... |
| mrieqvd 17518 | In a Moore system, a set i... |
| mrieqv2d 17519 | In a Moore system, a set i... |
| mrissmrcd 17520 | In a Moore system, if an i... |
| mrissmrid 17521 | In a Moore system, subsets... |
| mreexd 17522 | In a Moore system, the clo... |
| mreexmrid 17523 | In a Moore system whose cl... |
| mreexexlemd 17524 | This lemma is used to gene... |
| mreexexlem2d 17525 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17526 | Base case of the induction... |
| mreexexlem4d 17527 | Induction step of the indu... |
| mreexexd 17528 | Exchange-type theorem. In... |
| mreexdomd 17529 | In a Moore system whose cl... |
| mreexfidimd 17530 | In a Moore system whose cl... |
| isacs 17531 | A set is an algebraic clos... |
| acsmre 17532 | Algebraic closure systems ... |
| isacs2 17533 | In the definition of an al... |
| acsfiel 17534 | A set is closed in an alge... |
| acsfiel2 17535 | A set is closed in an alge... |
| acsmred 17536 | An algebraic closure syste... |
| isacs1i 17537 | A closure system determine... |
| mreacs 17538 | Algebraicity is a composab... |
| acsfn 17539 | Algebraicity of a conditio... |
| acsfn0 17540 | Algebraicity of a point cl... |
| acsfn1 17541 | Algebraicity of a one-argu... |
| acsfn1c 17542 | Algebraicity of a one-argu... |
| acsfn2 17543 | Algebraicity of a two-argu... |
| iscat 17552 | The predicate "is a catego... |
| iscatd 17553 | Properties that determine ... |
| catidex 17554 | Each object in a category ... |
| catideu 17555 | Each object in a category ... |
| cidfval 17556 | Each object in a category ... |
| cidval 17557 | Each object in a category ... |
| cidffn 17558 | The identity arrow constru... |
| cidfn 17559 | The identity arrow operato... |
| catidd 17560 | Deduce the identity arrow ... |
| iscatd2 17561 | Version of ~ iscatd with a... |
| catidcl 17562 | Each object in a category ... |
| catlid 17563 | Left identity property of ... |
| catrid 17564 | Right identity property of... |
| catcocl 17565 | Closure of a composition a... |
| catass 17566 | Associativity of compositi... |
| catcone0 17567 | Composition of non-empty h... |
| 0catg 17568 | Any structure with an empt... |
| 0cat 17569 | The empty set is a categor... |
| homffval 17570 | Value of the functionalize... |
| fnhomeqhomf 17571 | If the Hom-set operation i... |
| homfval 17572 | Value of the functionalize... |
| homffn 17573 | The functionalized Hom-set... |
| homfeq 17574 | Condition for two categori... |
| homfeqd 17575 | If two structures have the... |
| homfeqbas 17576 | Deduce equality of base se... |
| homfeqval 17577 | Value of the functionalize... |
| comfffval 17578 | Value of the functionalize... |
| comffval 17579 | Value of the functionalize... |
| comfval 17580 | Value of the functionalize... |
| comfffval2 17581 | Value of the functionalize... |
| comffval2 17582 | Value of the functionalize... |
| comfval2 17583 | Value of the functionalize... |
| comfffn 17584 | The functionalized composi... |
| comffn 17585 | The functionalized composi... |
| comfeq 17586 | Condition for two categori... |
| comfeqd 17587 | Condition for two categori... |
| comfeqval 17588 | Equality of two compositio... |
| catpropd 17589 | Two structures with the sa... |
| cidpropd 17590 | Two structures with the sa... |
| oppcval 17593 | Value of the opposite cate... |
| oppchomfval 17594 | Hom-sets of the opposite c... |
| oppchomfvalOLD 17595 | Obsolete proof of ~ oppcho... |
| oppchom 17596 | Hom-sets of the opposite c... |
| oppccofval 17597 | Composition in the opposit... |
| oppcco 17598 | Composition in the opposit... |
| oppcbas 17599 | Base set of an opposite ca... |
| oppcbasOLD 17600 | Obsolete version of ~ oppc... |
| oppccatid 17601 | Lemma for ~ oppccat . (Co... |
| oppchomf 17602 | Hom-sets of the opposite c... |
| oppcid 17603 | Identity function of an op... |
| oppccat 17604 | An opposite category is a ... |
| 2oppcbas 17605 | The double opposite catego... |
| 2oppchomf 17606 | The double opposite catego... |
| 2oppccomf 17607 | The double opposite catego... |
| oppchomfpropd 17608 | If two categories have the... |
| oppccomfpropd 17609 | If two categories have the... |
| oppccatf 17610 | ` oppCat ` restricted to `... |
| monfval 17615 | Definition of a monomorphi... |
| ismon 17616 | Definition of a monomorphi... |
| ismon2 17617 | Write out the monomorphism... |
| monhom 17618 | A monomorphism is a morphi... |
| moni 17619 | Property of a monomorphism... |
| monpropd 17620 | If two categories have the... |
| oppcmon 17621 | A monomorphism in the oppo... |
| oppcepi 17622 | An epimorphism in the oppo... |
| isepi 17623 | Definition of an epimorphi... |
| isepi2 17624 | Write out the epimorphism ... |
| epihom 17625 | An epimorphism is a morphi... |
| epii 17626 | Property of an epimorphism... |
| sectffval 17633 | Value of the section opera... |
| sectfval 17634 | Value of the section relat... |
| sectss 17635 | The section relation is a ... |
| issect 17636 | The property " ` F ` is a ... |
| issect2 17637 | Property of being a sectio... |
| sectcan 17638 | If ` G ` is a section of `... |
| sectco 17639 | Composition of two section... |
| isofval 17640 | Function value of the func... |
| invffval 17641 | Value of the inverse relat... |
| invfval 17642 | Value of the inverse relat... |
| isinv 17643 | Value of the inverse relat... |
| invss 17644 | The inverse relation is a ... |
| invsym 17645 | The inverse relation is sy... |
| invsym2 17646 | The inverse relation is sy... |
| invfun 17647 | The inverse relation is a ... |
| isoval 17648 | The isomorphisms are the d... |
| inviso1 17649 | If ` G ` is an inverse to ... |
| inviso2 17650 | If ` G ` is an inverse to ... |
| invf 17651 | The inverse relation is a ... |
| invf1o 17652 | The inverse relation is a ... |
| invinv 17653 | The inverse of the inverse... |
| invco 17654 | The composition of two iso... |
| dfiso2 17655 | Alternate definition of an... |
| dfiso3 17656 | Alternate definition of an... |
| inveq 17657 | If there are two inverses ... |
| isofn 17658 | The function value of the ... |
| isohom 17659 | An isomorphism is a homomo... |
| isoco 17660 | The composition of two iso... |
| oppcsect 17661 | A section in the opposite ... |
| oppcsect2 17662 | A section in the opposite ... |
| oppcinv 17663 | An inverse in the opposite... |
| oppciso 17664 | An isomorphism in the oppo... |
| sectmon 17665 | If ` F ` is a section of `... |
| monsect 17666 | If ` F ` is a monomorphism... |
| sectepi 17667 | If ` F ` is a section of `... |
| episect 17668 | If ` F ` is an epimorphism... |
| sectid 17669 | The identity is a section ... |
| invid 17670 | The inverse of the identit... |
| idiso 17671 | The identity is an isomorp... |
| idinv 17672 | The inverse of the identit... |
| invisoinvl 17673 | The inverse of an isomorph... |
| invisoinvr 17674 | The inverse of an isomorph... |
| invcoisoid 17675 | The inverse of an isomorph... |
| isocoinvid 17676 | The inverse of an isomorph... |
| rcaninv 17677 | Right cancellation of an i... |
| cicfval 17680 | The set of isomorphic obje... |
| brcic 17681 | The relation "is isomorphi... |
| cic 17682 | Objects ` X ` and ` Y ` in... |
| brcici 17683 | Prove that two objects are... |
| cicref 17684 | Isomorphism is reflexive. ... |
| ciclcl 17685 | Isomorphism implies the le... |
| cicrcl 17686 | Isomorphism implies the ri... |
| cicsym 17687 | Isomorphism is symmetric. ... |
| cictr 17688 | Isomorphism is transitive.... |
| cicer 17689 | Isomorphism is an equivale... |
| sscrel 17696 | The subcategory subset rel... |
| brssc 17697 | The subcategory subset rel... |
| sscpwex 17698 | An analogue of ~ pwex for ... |
| subcrcl 17699 | Reverse closure for the su... |
| sscfn1 17700 | The subcategory subset rel... |
| sscfn2 17701 | The subcategory subset rel... |
| ssclem 17702 | Lemma for ~ ssc1 and simil... |
| isssc 17703 | Value of the subcategory s... |
| ssc1 17704 | Infer subset relation on o... |
| ssc2 17705 | Infer subset relation on m... |
| sscres 17706 | Any function restricted to... |
| sscid 17707 | The subcategory subset rel... |
| ssctr 17708 | The subcategory subset rel... |
| ssceq 17709 | The subcategory subset rel... |
| rescval 17710 | Value of the category rest... |
| rescval2 17711 | Value of the category rest... |
| rescbas 17712 | Base set of the category r... |
| rescbasOLD 17713 | Obsolete version of ~ resc... |
| reschom 17714 | Hom-sets of the category r... |
| reschomf 17715 | Hom-sets of the category r... |
| rescco 17716 | Composition in the categor... |
| resccoOLD 17717 | Obsolete proof of ~ rescco... |
| rescabs 17718 | Restriction absorption law... |
| rescabsOLD 17719 | Obsolete proof of ~ seqp1d... |
| rescabs2 17720 | Restriction absorption law... |
| issubc 17721 | Elementhood in the set of ... |
| issubc2 17722 | Elementhood in the set of ... |
| 0ssc 17723 | For any category ` C ` , t... |
| 0subcat 17724 | For any category ` C ` , t... |
| catsubcat 17725 | For any category ` C ` , `... |
| subcssc 17726 | An element in the set of s... |
| subcfn 17727 | An element in the set of s... |
| subcss1 17728 | The objects of a subcatego... |
| subcss2 17729 | The morphisms of a subcate... |
| subcidcl 17730 | The identity of the origin... |
| subccocl 17731 | A subcategory is closed un... |
| subccatid 17732 | A subcategory is a categor... |
| subcid 17733 | The identity in a subcateg... |
| subccat 17734 | A subcategory is a categor... |
| issubc3 17735 | Alternate definition of a ... |
| fullsubc 17736 | The full subcategory gener... |
| fullresc 17737 | The category formed by str... |
| resscat 17738 | A category restricted to a... |
| subsubc 17739 | A subcategory of a subcate... |
| relfunc 17748 | The set of functors is a r... |
| funcrcl 17749 | Reverse closure for a func... |
| isfunc 17750 | Value of the set of functo... |
| isfuncd 17751 | Deduce that an operation i... |
| funcf1 17752 | The object part of a funct... |
| funcixp 17753 | The morphism part of a fun... |
| funcf2 17754 | The morphism part of a fun... |
| funcfn2 17755 | The morphism part of a fun... |
| funcid 17756 | A functor maps each identi... |
| funcco 17757 | A functor maps composition... |
| funcsect 17758 | The image of a section und... |
| funcinv 17759 | The image of an inverse un... |
| funciso 17760 | The image of an isomorphis... |
| funcoppc 17761 | A functor on categories yi... |
| idfuval 17762 | Value of the identity func... |
| idfu2nd 17763 | Value of the morphism part... |
| idfu2 17764 | Value of the morphism part... |
| idfu1st 17765 | Value of the object part o... |
| idfu1 17766 | Value of the object part o... |
| idfucl 17767 | The identity functor is a ... |
| cofuval 17768 | Value of the composition o... |
| cofu1st 17769 | Value of the object part o... |
| cofu1 17770 | Value of the object part o... |
| cofu2nd 17771 | Value of the morphism part... |
| cofu2 17772 | Value of the morphism part... |
| cofuval2 17773 | Value of the composition o... |
| cofucl 17774 | The composition of two fun... |
| cofuass 17775 | Functor composition is ass... |
| cofulid 17776 | The identity functor is a ... |
| cofurid 17777 | The identity functor is a ... |
| resfval 17778 | Value of the functor restr... |
| resfval2 17779 | Value of the functor restr... |
| resf1st 17780 | Value of the functor restr... |
| resf2nd 17781 | Value of the functor restr... |
| funcres 17782 | A functor restricted to a ... |
| funcres2b 17783 | Condition for a functor to... |
| funcres2 17784 | A functor into a restricte... |
| wunfunc 17785 | A weak universe is closed ... |
| wunfuncOLD 17786 | Obsolete proof of ~ wunfun... |
| funcpropd 17787 | If two categories have the... |
| funcres2c 17788 | Condition for a functor to... |
| fullfunc 17793 | A full functor is a functo... |
| fthfunc 17794 | A faithful functor is a fu... |
| relfull 17795 | The set of full functors i... |
| relfth 17796 | The set of faithful functo... |
| isfull 17797 | Value of the set of full f... |
| isfull2 17798 | Equivalent condition for a... |
| fullfo 17799 | The morphism map of a full... |
| fulli 17800 | The morphism map of a full... |
| isfth 17801 | Value of the set of faithf... |
| isfth2 17802 | Equivalent condition for a... |
| isffth2 17803 | A fully faithful functor i... |
| fthf1 17804 | The morphism map of a fait... |
| fthi 17805 | The morphism map of a fait... |
| ffthf1o 17806 | The morphism map of a full... |
| fullpropd 17807 | If two categories have the... |
| fthpropd 17808 | If two categories have the... |
| fulloppc 17809 | The opposite functor of a ... |
| fthoppc 17810 | The opposite functor of a ... |
| ffthoppc 17811 | The opposite functor of a ... |
| fthsect 17812 | A faithful functor reflect... |
| fthinv 17813 | A faithful functor reflect... |
| fthmon 17814 | A faithful functor reflect... |
| fthepi 17815 | A faithful functor reflect... |
| ffthiso 17816 | A fully faithful functor r... |
| fthres2b 17817 | Condition for a faithful f... |
| fthres2c 17818 | Condition for a faithful f... |
| fthres2 17819 | A faithful functor into a ... |
| idffth 17820 | The identity functor is a ... |
| cofull 17821 | The composition of two ful... |
| cofth 17822 | The composition of two fai... |
| coffth 17823 | The composition of two ful... |
| rescfth 17824 | The inclusion functor from... |
| ressffth 17825 | The inclusion functor from... |
| fullres2c 17826 | Condition for a full funct... |
| ffthres2c 17827 | Condition for a fully fait... |
| fnfuc 17832 | The ` FuncCat ` operation ... |
| natfval 17833 | Value of the function givi... |
| isnat 17834 | Property of being a natura... |
| isnat2 17835 | Property of being a natura... |
| natffn 17836 | The natural transformation... |
| natrcl 17837 | Reverse closure for a natu... |
| nat1st2nd 17838 | Rewrite the natural transf... |
| natixp 17839 | A natural transformation i... |
| natcl 17840 | A component of a natural t... |
| natfn 17841 | A natural transformation i... |
| nati 17842 | Naturality property of a n... |
| wunnat 17843 | A weak universe is closed ... |
| wunnatOLD 17844 | Obsolete proof of ~ wunnat... |
| catstr 17845 | A category structure is a ... |
| fucval 17846 | Value of the functor categ... |
| fuccofval 17847 | Value of the functor categ... |
| fucbas 17848 | The objects of the functor... |
| fuchom 17849 | The morphisms in the funct... |
| fuchomOLD 17850 | Obsolete proof of ~ fuchom... |
| fucco 17851 | Value of the composition o... |
| fuccoval 17852 | Value of the functor categ... |
| fuccocl 17853 | The composition of two nat... |
| fucidcl 17854 | The identity natural trans... |
| fuclid 17855 | Left identity of natural t... |
| fucrid 17856 | Right identity of natural ... |
| fucass 17857 | Associativity of natural t... |
| fuccatid 17858 | The functor category is a ... |
| fuccat 17859 | The functor category is a ... |
| fucid 17860 | The identity morphism in t... |
| fucsect 17861 | Two natural transformation... |
| fucinv 17862 | Two natural transformation... |
| invfuc 17863 | If ` V ( x ) ` is an inver... |
| fuciso 17864 | A natural transformation i... |
| natpropd 17865 | If two categories have the... |
| fucpropd 17866 | If two categories have the... |
| initofn 17873 | ` InitO ` is a function on... |
| termofn 17874 | ` TermO ` is a function on... |
| zeroofn 17875 | ` ZeroO ` is a function on... |
| initorcl 17876 | Reverse closure for an ini... |
| termorcl 17877 | Reverse closure for a term... |
| zeroorcl 17878 | Reverse closure for a zero... |
| initoval 17879 | The value of the initial o... |
| termoval 17880 | The value of the terminal ... |
| zerooval 17881 | The value of the zero obje... |
| isinito 17882 | The predicate "is an initi... |
| istermo 17883 | The predicate "is a termin... |
| iszeroo 17884 | The predicate "is a zero o... |
| isinitoi 17885 | Implication of a class bei... |
| istermoi 17886 | Implication of a class bei... |
| initoid 17887 | For an initial object, the... |
| termoid 17888 | For a terminal object, the... |
| dfinito2 17889 | An initial object is a ter... |
| dftermo2 17890 | A terminal object is an in... |
| dfinito3 17891 | An alternate definition of... |
| dftermo3 17892 | An alternate definition of... |
| initoo 17893 | An initial object is an ob... |
| termoo 17894 | A terminal object is an ob... |
| iszeroi 17895 | Implication of a class bei... |
| 2initoinv 17896 | Morphisms between two init... |
| initoeu1 17897 | Initial objects are essent... |
| initoeu1w 17898 | Initial objects are essent... |
| initoeu2lem0 17899 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17900 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17901 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17902 | Initial objects are essent... |
| 2termoinv 17903 | Morphisms between two term... |
| termoeu1 17904 | Terminal objects are essen... |
| termoeu1w 17905 | Terminal objects are essen... |
| homarcl 17914 | Reverse closure for an arr... |
| homafval 17915 | Value of the disjointified... |
| homaf 17916 | Functionality of the disjo... |
| homaval 17917 | Value of the disjointified... |
| elhoma 17918 | Value of the disjointified... |
| elhomai 17919 | Produce an arrow from a mo... |
| elhomai2 17920 | Produce an arrow from a mo... |
| homarcl2 17921 | Reverse closure for the do... |
| homarel 17922 | An arrow is an ordered pai... |
| homa1 17923 | The first component of an ... |
| homahom2 17924 | The second component of an... |
| homahom 17925 | The second component of an... |
| homadm 17926 | The domain of an arrow wit... |
| homacd 17927 | The codomain of an arrow w... |
| homadmcd 17928 | Decompose an arrow into do... |
| arwval 17929 | The set of arrows is the u... |
| arwrcl 17930 | The first component of an ... |
| arwhoma 17931 | An arrow is contained in t... |
| homarw 17932 | A hom-set is a subset of t... |
| arwdm 17933 | The domain of an arrow is ... |
| arwcd 17934 | The codomain of an arrow i... |
| dmaf 17935 | The domain function is a f... |
| cdaf 17936 | The codomain function is a... |
| arwhom 17937 | The second component of an... |
| arwdmcd 17938 | Decompose an arrow into do... |
| idafval 17943 | Value of the identity arro... |
| idaval 17944 | Value of the identity arro... |
| ida2 17945 | Morphism part of the ident... |
| idahom 17946 | Domain and codomain of the... |
| idadm 17947 | Domain of the identity arr... |
| idacd 17948 | Codomain of the identity a... |
| idaf 17949 | The identity arrow functio... |
| coafval 17950 | The value of the compositi... |
| eldmcoa 17951 | A pair ` <. G , F >. ` is ... |
| dmcoass 17952 | The domain of composition ... |
| homdmcoa 17953 | If ` F : X --> Y ` and ` G... |
| coaval 17954 | Value of composition for c... |
| coa2 17955 | The morphism part of arrow... |
| coahom 17956 | The composition of two com... |
| coapm 17957 | Composition of arrows is a... |
| arwlid 17958 | Left identity of a categor... |
| arwrid 17959 | Right identity of a catego... |
| arwass 17960 | Associativity of compositi... |
| setcval 17963 | Value of the category of s... |
| setcbas 17964 | Set of objects of the cate... |
| setchomfval 17965 | Set of arrows of the categ... |
| setchom 17966 | Set of arrows of the categ... |
| elsetchom 17967 | A morphism of sets is a fu... |
| setccofval 17968 | Composition in the categor... |
| setcco 17969 | Composition in the categor... |
| setccatid 17970 | Lemma for ~ setccat . (Co... |
| setccat 17971 | The category of sets is a ... |
| setcid 17972 | The identity arrow in the ... |
| setcmon 17973 | A monomorphism of sets is ... |
| setcepi 17974 | An epimorphism of sets is ... |
| setcsect 17975 | A section in the category ... |
| setcinv 17976 | An inverse in the category... |
| setciso 17977 | An isomorphism in the cate... |
| resssetc 17978 | The restriction of the cat... |
| funcsetcres2 17979 | A functor into a smaller c... |
| setc2obas 17980 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 17981 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 17982 | The category of sets in a ... |
| cat1 17983 | The definition of category... |
| catcval 17986 | Value of the category of c... |
| catcbas 17987 | Set of objects of the cate... |
| catchomfval 17988 | Set of arrows of the categ... |
| catchom 17989 | Set of arrows of the categ... |
| catccofval 17990 | Composition in the categor... |
| catcco 17991 | Composition in the categor... |
| catccatid 17992 | Lemma for ~ catccat . (Co... |
| catcid 17993 | The identity arrow in the ... |
| catccat 17994 | The category of categories... |
| resscatc 17995 | The restriction of the cat... |
| catcisolem 17996 | Lemma for ~ catciso . (Co... |
| catciso 17997 | A functor is an isomorphis... |
| catcbascl 17998 | An element of the base set... |
| catcslotelcl 17999 | A slot entry of an element... |
| catcbaselcl 18000 | The base set of an element... |
| catchomcl 18001 | The Hom-set of an element ... |
| catcccocl 18002 | The composition operation ... |
| catcoppccl 18003 | The category of categories... |
| catcoppcclOLD 18004 | Obsolete proof of ~ catcop... |
| catcfuccl 18005 | The category of categories... |
| catcfucclOLD 18006 | Obsolete proof of ~ catcfu... |
| fncnvimaeqv 18007 | The inverse images of the ... |
| bascnvimaeqv 18008 | The inverse image of the u... |
| estrcval 18011 | Value of the category of e... |
| estrcbas 18012 | Set of objects of the cate... |
| estrchomfval 18013 | Set of morphisms ("arrows"... |
| estrchom 18014 | The morphisms between exte... |
| elestrchom 18015 | A morphism between extensi... |
| estrccofval 18016 | Composition in the categor... |
| estrcco 18017 | Composition in the categor... |
| estrcbasbas 18018 | An element of the base set... |
| estrccatid 18019 | Lemma for ~ estrccat . (C... |
| estrccat 18020 | The category of extensible... |
| estrcid 18021 | The identity arrow in the ... |
| estrchomfn 18022 | The Hom-set operation in t... |
| estrchomfeqhom 18023 | The functionalized Hom-set... |
| estrreslem1 18024 | Lemma 1 for ~ estrres . (... |
| estrreslem1OLD 18025 | Obsolete version of ~ estr... |
| estrreslem2 18026 | Lemma 2 for ~ estrres . (... |
| estrres 18027 | Any restriction of a categ... |
| funcestrcsetclem1 18028 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 18029 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 18030 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 18031 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 18032 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 18033 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 18034 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 18035 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 18036 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 18037 | The "natural forgetful fun... |
| fthestrcsetc 18038 | The "natural forgetful fun... |
| fullestrcsetc 18039 | The "natural forgetful fun... |
| equivestrcsetc 18040 | The "natural forgetful fun... |
| setc1strwun 18041 | A constructed one-slot str... |
| funcsetcestrclem1 18042 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 18043 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 18044 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 18045 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 18046 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 18047 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 18048 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 18049 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 18050 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 18051 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 18052 | The "embedding functor" fr... |
| fthsetcestrc 18053 | The "embedding functor" fr... |
| fullsetcestrc 18054 | The "embedding functor" fr... |
| embedsetcestrc 18055 | The "embedding functor" fr... |
| fnxpc 18064 | The binary product of cate... |
| xpcval 18065 | Value of the binary produc... |
| xpcbas 18066 | Set of objects of the bina... |
| xpchomfval 18067 | Set of morphisms of the bi... |
| xpchom 18068 | Set of morphisms of the bi... |
| relxpchom 18069 | A hom-set in the binary pr... |
| xpccofval 18070 | Value of composition in th... |
| xpcco 18071 | Value of composition in th... |
| xpcco1st 18072 | Value of composition in th... |
| xpcco2nd 18073 | Value of composition in th... |
| xpchom2 18074 | Value of the set of morphi... |
| xpcco2 18075 | Value of composition in th... |
| xpccatid 18076 | The product of two categor... |
| xpcid 18077 | The identity morphism in t... |
| xpccat 18078 | The product of two categor... |
| 1stfval 18079 | Value of the first project... |
| 1stf1 18080 | Value of the first project... |
| 1stf2 18081 | Value of the first project... |
| 2ndfval 18082 | Value of the first project... |
| 2ndf1 18083 | Value of the first project... |
| 2ndf2 18084 | Value of the first project... |
| 1stfcl 18085 | The first projection funct... |
| 2ndfcl 18086 | The second projection func... |
| prfval 18087 | Value of the pairing funct... |
| prf1 18088 | Value of the pairing funct... |
| prf2fval 18089 | Value of the pairing funct... |
| prf2 18090 | Value of the pairing funct... |
| prfcl 18091 | The pairing of functors ` ... |
| prf1st 18092 | Cancellation of pairing wi... |
| prf2nd 18093 | Cancellation of pairing wi... |
| 1st2ndprf 18094 | Break a functor into a pro... |
| catcxpccl 18095 | The category of categories... |
| catcxpcclOLD 18096 | Obsolete proof of ~ catcxp... |
| xpcpropd 18097 | If two categories have the... |
| evlfval 18106 | Value of the evaluation fu... |
| evlf2 18107 | Value of the evaluation fu... |
| evlf2val 18108 | Value of the evaluation na... |
| evlf1 18109 | Value of the evaluation fu... |
| evlfcllem 18110 | Lemma for ~ evlfcl . (Con... |
| evlfcl 18111 | The evaluation functor is ... |
| curfval 18112 | Value of the curry functor... |
| curf1fval 18113 | Value of the object part o... |
| curf1 18114 | Value of the object part o... |
| curf11 18115 | Value of the double evalua... |
| curf12 18116 | The partially evaluated cu... |
| curf1cl 18117 | The partially evaluated cu... |
| curf2 18118 | Value of the curry functor... |
| curf2val 18119 | Value of a component of th... |
| curf2cl 18120 | The curry functor at a mor... |
| curfcl 18121 | The curry functor of a fun... |
| curfpropd 18122 | If two categories have the... |
| uncfval 18123 | Value of the uncurry funct... |
| uncfcl 18124 | The uncurry operation take... |
| uncf1 18125 | Value of the uncurry funct... |
| uncf2 18126 | Value of the uncurry funct... |
| curfuncf 18127 | Cancellation of curry with... |
| uncfcurf 18128 | Cancellation of uncurry wi... |
| diagval 18129 | Define the diagonal functo... |
| diagcl 18130 | The diagonal functor is a ... |
| diag1cl 18131 | The constant functor of ` ... |
| diag11 18132 | Value of the constant func... |
| diag12 18133 | Value of the constant func... |
| diag2 18134 | Value of the diagonal func... |
| diag2cl 18135 | The diagonal functor at a ... |
| curf2ndf 18136 | As shown in ~ diagval , th... |
| hofval 18141 | Value of the Hom functor, ... |
| hof1fval 18142 | The object part of the Hom... |
| hof1 18143 | The object part of the Hom... |
| hof2fval 18144 | The morphism part of the H... |
| hof2val 18145 | The morphism part of the H... |
| hof2 18146 | The morphism part of the H... |
| hofcllem 18147 | Lemma for ~ hofcl . (Cont... |
| hofcl 18148 | Closure of the Hom functor... |
| oppchofcl 18149 | Closure of the opposite Ho... |
| yonval 18150 | Value of the Yoneda embedd... |
| yoncl 18151 | The Yoneda embedding is a ... |
| yon1cl 18152 | The Yoneda embedding at an... |
| yon11 18153 | Value of the Yoneda embedd... |
| yon12 18154 | Value of the Yoneda embedd... |
| yon2 18155 | Value of the Yoneda embedd... |
| hofpropd 18156 | If two categories have the... |
| yonpropd 18157 | If two categories have the... |
| oppcyon 18158 | Value of the opposite Yone... |
| oyoncl 18159 | The opposite Yoneda embedd... |
| oyon1cl 18160 | The opposite Yoneda embedd... |
| yonedalem1 18161 | Lemma for ~ yoneda . (Con... |
| yonedalem21 18162 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 18163 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 18164 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 18165 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 18166 | Lemma for ~ yoneda . (Con... |
| yonedalem22 18167 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 18168 | Lemma for ~ yoneda . (Con... |
| yonedalem3 18169 | Lemma for ~ yoneda . (Con... |
| yonedainv 18170 | The Yoneda Lemma with expl... |
| yonffthlem 18171 | Lemma for ~ yonffth . (Co... |
| yoneda 18172 | The Yoneda Lemma. There i... |
| yonffth 18173 | The Yoneda Lemma. The Yon... |
| yoniso 18174 | If the codomain is recover... |
| oduval 18177 | Value of an order dual str... |
| oduleval 18178 | Value of the less-equal re... |
| oduleg 18179 | Truth of the less-equal re... |
| odubas 18180 | Base set of an order dual ... |
| odubasOLD 18181 | Obsolete proof of ~ odubas... |
| isprs 18186 | Property of being a preord... |
| prslem 18187 | Lemma for ~ prsref and ~ p... |
| prsref 18188 | "Less than or equal to" is... |
| prstr 18189 | "Less than or equal to" is... |
| isdrs 18190 | Property of being a direct... |
| drsdir 18191 | Direction of a directed se... |
| drsprs 18192 | A directed set is a proset... |
| drsbn0 18193 | The base of a directed set... |
| drsdirfi 18194 | Any _finite_ number of ele... |
| isdrs2 18195 | Directed sets may be defin... |
| ispos 18203 | The predicate "is a poset"... |
| ispos2 18204 | A poset is an antisymmetri... |
| posprs 18205 | A poset is a proset. (Con... |
| posi 18206 | Lemma for poset properties... |
| posref 18207 | A poset ordering is reflex... |
| posasymb 18208 | A poset ordering is asymme... |
| postr 18209 | A poset ordering is transi... |
| 0pos 18210 | Technical lemma to simplif... |
| 0posOLD 18211 | Obsolete proof of ~ 0pos a... |
| isposd 18212 | Properties that determine ... |
| isposi 18213 | Properties that determine ... |
| isposix 18214 | Properties that determine ... |
| isposixOLD 18215 | Obsolete proof of ~ isposi... |
| pospropd 18216 | Posethood is determined on... |
| odupos 18217 | Being a poset is a self-du... |
| oduposb 18218 | Being a poset is a self-du... |
| pltfval 18220 | Value of the less-than rel... |
| pltval 18221 | Less-than relation. ( ~ d... |
| pltle 18222 | "Less than" implies "less ... |
| pltne 18223 | The "less than" relation i... |
| pltirr 18224 | The "less than" relation i... |
| pleval2i 18225 | One direction of ~ pleval2... |
| pleval2 18226 | "Less than or equal to" in... |
| pltnle 18227 | "Less than" implies not co... |
| pltval3 18228 | Alternate expression for t... |
| pltnlt 18229 | The less-than relation imp... |
| pltn2lp 18230 | The less-than relation has... |
| plttr 18231 | The less-than relation is ... |
| pltletr 18232 | Transitive law for chained... |
| plelttr 18233 | Transitive law for chained... |
| pospo 18234 | Write a poset structure in... |
| lubfval 18239 | Value of the least upper b... |
| lubdm 18240 | Domain of the least upper ... |
| lubfun 18241 | The LUB is a function. (C... |
| lubeldm 18242 | Member of the domain of th... |
| lubelss 18243 | A member of the domain of ... |
| lubeu 18244 | Unique existence proper of... |
| lubval 18245 | Value of the least upper b... |
| lubcl 18246 | The least upper bound func... |
| lubprop 18247 | Properties of greatest low... |
| luble 18248 | The greatest lower bound i... |
| lublecllem 18249 | Lemma for ~ lublecl and ~ ... |
| lublecl 18250 | The set of all elements le... |
| lubid 18251 | The LUB of elements less t... |
| glbfval 18252 | Value of the greatest lowe... |
| glbdm 18253 | Domain of the greatest low... |
| glbfun 18254 | The GLB is a function. (C... |
| glbeldm 18255 | Member of the domain of th... |
| glbelss 18256 | A member of the domain of ... |
| glbeu 18257 | Unique existence proper of... |
| glbval 18258 | Value of the greatest lowe... |
| glbcl 18259 | The least upper bound func... |
| glbprop 18260 | Properties of greatest low... |
| glble 18261 | The greatest lower bound i... |
| joinfval 18262 | Value of join function for... |
| joinfval2 18263 | Value of join function for... |
| joindm 18264 | Domain of join function fo... |
| joindef 18265 | Two ways to say that a joi... |
| joinval 18266 | Join value. Since both si... |
| joincl 18267 | Closure of join of element... |
| joindmss 18268 | Subset property of domain ... |
| joinval2lem 18269 | Lemma for ~ joinval2 and ~... |
| joinval2 18270 | Value of join for a poset ... |
| joineu 18271 | Uniqueness of join of elem... |
| joinlem 18272 | Lemma for join properties.... |
| lejoin1 18273 | A join's first argument is... |
| lejoin2 18274 | A join's second argument i... |
| joinle 18275 | A join is less than or equ... |
| meetfval 18276 | Value of meet function for... |
| meetfval2 18277 | Value of meet function for... |
| meetdm 18278 | Domain of meet function fo... |
| meetdef 18279 | Two ways to say that a mee... |
| meetval 18280 | Meet value. Since both si... |
| meetcl 18281 | Closure of meet of element... |
| meetdmss 18282 | Subset property of domain ... |
| meetval2lem 18283 | Lemma for ~ meetval2 and ~... |
| meetval2 18284 | Value of meet for a poset ... |
| meeteu 18285 | Uniqueness of meet of elem... |
| meetlem 18286 | Lemma for meet properties.... |
| lemeet1 18287 | A meet's first argument is... |
| lemeet2 18288 | A meet's second argument i... |
| meetle 18289 | A meet is less than or equ... |
| joincomALT 18290 | The join of a poset is com... |
| joincom 18291 | The join of a poset is com... |
| meetcomALT 18292 | The meet of a poset is com... |
| meetcom 18293 | The meet of a poset is com... |
| join0 18294 | Lemma for ~ odumeet . (Co... |
| meet0 18295 | Lemma for ~ odujoin . (Co... |
| odulub 18296 | Least upper bounds in a du... |
| odujoin 18297 | Joins in a dual order are ... |
| oduglb 18298 | Greatest lower bounds in a... |
| odumeet 18299 | Meets in a dual order are ... |
| poslubmo 18300 | Least upper bounds in a po... |
| posglbmo 18301 | Greatest lower bounds in a... |
| poslubd 18302 | Properties which determine... |
| poslubdg 18303 | Properties which determine... |
| posglbdg 18304 | Properties which determine... |
| istos 18307 | The predicate "is a toset"... |
| tosso 18308 | Write the totally ordered ... |
| tospos 18309 | A Toset is a Poset. (Cont... |
| tleile 18310 | In a Toset, any two elemen... |
| tltnle 18311 | In a Toset, "less than" is... |
| p0val 18316 | Value of poset zero. (Con... |
| p1val 18317 | Value of poset zero. (Con... |
| p0le 18318 | Any element is less than o... |
| ple1 18319 | Any element is less than o... |
| islat 18322 | The predicate "is a lattic... |
| odulatb 18323 | Being a lattice is self-du... |
| odulat 18324 | Being a lattice is self-du... |
| latcl2 18325 | The join and meet of any t... |
| latlem 18326 | Lemma for lattice properti... |
| latpos 18327 | A lattice is a poset. (Co... |
| latjcl 18328 | Closure of join operation ... |
| latmcl 18329 | Closure of meet operation ... |
| latref 18330 | A lattice ordering is refl... |
| latasymb 18331 | A lattice ordering is asym... |
| latasym 18332 | A lattice ordering is asym... |
| lattr 18333 | A lattice ordering is tran... |
| latasymd 18334 | Deduce equality from latti... |
| lattrd 18335 | A lattice ordering is tran... |
| latjcom 18336 | The join of a lattice comm... |
| latlej1 18337 | A join's first argument is... |
| latlej2 18338 | A join's second argument i... |
| latjle12 18339 | A join is less than or equ... |
| latleeqj1 18340 | "Less than or equal to" in... |
| latleeqj2 18341 | "Less than or equal to" in... |
| latjlej1 18342 | Add join to both sides of ... |
| latjlej2 18343 | Add join to both sides of ... |
| latjlej12 18344 | Add join to both sides of ... |
| latnlej 18345 | An idiom to express that a... |
| latnlej1l 18346 | An idiom to express that a... |
| latnlej1r 18347 | An idiom to express that a... |
| latnlej2 18348 | An idiom to express that a... |
| latnlej2l 18349 | An idiom to express that a... |
| latnlej2r 18350 | An idiom to express that a... |
| latjidm 18351 | Lattice join is idempotent... |
| latmcom 18352 | The join of a lattice comm... |
| latmle1 18353 | A meet is less than or equ... |
| latmle2 18354 | A meet is less than or equ... |
| latlem12 18355 | An element is less than or... |
| latleeqm1 18356 | "Less than or equal to" in... |
| latleeqm2 18357 | "Less than or equal to" in... |
| latmlem1 18358 | Add meet to both sides of ... |
| latmlem2 18359 | Add meet to both sides of ... |
| latmlem12 18360 | Add join to both sides of ... |
| latnlemlt 18361 | Negation of "less than or ... |
| latnle 18362 | Equivalent expressions for... |
| latmidm 18363 | Lattice meet is idempotent... |
| latabs1 18364 | Lattice absorption law. F... |
| latabs2 18365 | Lattice absorption law. F... |
| latledi 18366 | An ortholattice is distrib... |
| latmlej11 18367 | Ordering of a meet and joi... |
| latmlej12 18368 | Ordering of a meet and joi... |
| latmlej21 18369 | Ordering of a meet and joi... |
| latmlej22 18370 | Ordering of a meet and joi... |
| lubsn 18371 | The least upper bound of a... |
| latjass 18372 | Lattice join is associativ... |
| latj12 18373 | Swap 1st and 2nd members o... |
| latj32 18374 | Swap 2nd and 3rd members o... |
| latj13 18375 | Swap 1st and 3rd members o... |
| latj31 18376 | Swap 2nd and 3rd members o... |
| latjrot 18377 | Rotate lattice join of 3 c... |
| latj4 18378 | Rearrangement of lattice j... |
| latj4rot 18379 | Rotate lattice join of 4 c... |
| latjjdi 18380 | Lattice join distributes o... |
| latjjdir 18381 | Lattice join distributes o... |
| mod1ile 18382 | The weak direction of the ... |
| mod2ile 18383 | The weak direction of the ... |
| latmass 18384 | Lattice meet is associativ... |
| latdisdlem 18385 | Lemma for ~ latdisd . (Co... |
| latdisd 18386 | In a lattice, joins distri... |
| isclat 18389 | The predicate "is a comple... |
| clatpos 18390 | A complete lattice is a po... |
| clatlem 18391 | Lemma for properties of a ... |
| clatlubcl 18392 | Any subset of the base set... |
| clatlubcl2 18393 | Any subset of the base set... |
| clatglbcl 18394 | Any subset of the base set... |
| clatglbcl2 18395 | Any subset of the base set... |
| oduclatb 18396 | Being a complete lattice i... |
| clatl 18397 | A complete lattice is a la... |
| isglbd 18398 | Properties that determine ... |
| lublem 18399 | Lemma for the least upper ... |
| lubub 18400 | The LUB of a complete latt... |
| lubl 18401 | The LUB of a complete latt... |
| lubss 18402 | Subset law for least upper... |
| lubel 18403 | An element of a set is les... |
| lubun 18404 | The LUB of a union. (Cont... |
| clatglb 18405 | Properties of greatest low... |
| clatglble 18406 | The greatest lower bound i... |
| clatleglb 18407 | Two ways of expressing "le... |
| clatglbss 18408 | Subset law for greatest lo... |
| isdlat 18411 | Property of being a distri... |
| dlatmjdi 18412 | In a distributive lattice,... |
| dlatl 18413 | A distributive lattice is ... |
| odudlatb 18414 | The dual of a distributive... |
| dlatjmdi 18415 | In a distributive lattice,... |
| ipostr 18418 | The structure of ~ df-ipo ... |
| ipoval 18419 | Value of the inclusion pos... |
| ipobas 18420 | Base set of the inclusion ... |
| ipolerval 18421 | Relation of the inclusion ... |
| ipotset 18422 | Topology of the inclusion ... |
| ipole 18423 | Weak order condition of th... |
| ipolt 18424 | Strict order condition of ... |
| ipopos 18425 | The inclusion poset on a f... |
| isipodrs 18426 | Condition for a family of ... |
| ipodrscl 18427 | Direction by inclusion as ... |
| ipodrsfi 18428 | Finite upper bound propert... |
| fpwipodrs 18429 | The finite subsets of any ... |
| ipodrsima 18430 | The monotone image of a di... |
| isacs3lem 18431 | An algebraic closure syste... |
| acsdrsel 18432 | An algebraic closure syste... |
| isacs4lem 18433 | In a closure system in whi... |
| isacs5lem 18434 | If closure commutes with d... |
| acsdrscl 18435 | In an algebraic closure sy... |
| acsficl 18436 | A closure in an algebraic ... |
| isacs5 18437 | A closure system is algebr... |
| isacs4 18438 | A closure system is algebr... |
| isacs3 18439 | A closure system is algebr... |
| acsficld 18440 | In an algebraic closure sy... |
| acsficl2d 18441 | In an algebraic closure sy... |
| acsfiindd 18442 | In an algebraic closure sy... |
| acsmapd 18443 | In an algebraic closure sy... |
| acsmap2d 18444 | In an algebraic closure sy... |
| acsinfd 18445 | In an algebraic closure sy... |
| acsdomd 18446 | In an algebraic closure sy... |
| acsinfdimd 18447 | In an algebraic closure sy... |
| acsexdimd 18448 | In an algebraic closure sy... |
| mrelatglb 18449 | Greatest lower bounds in a... |
| mrelatglb0 18450 | The empty intersection in ... |
| mrelatlub 18451 | Least upper bounds in a Mo... |
| mreclatBAD 18452 | A Moore space is a complet... |
| isps 18457 | The predicate "is a poset"... |
| psrel 18458 | A poset is a relation. (C... |
| psref2 18459 | A poset is antisymmetric a... |
| pstr2 18460 | A poset is transitive. (C... |
| pslem 18461 | Lemma for ~ psref and othe... |
| psdmrn 18462 | The domain and range of a ... |
| psref 18463 | A poset is reflexive. (Co... |
| psrn 18464 | The range of a poset equal... |
| psasym 18465 | A poset is antisymmetric. ... |
| pstr 18466 | A poset is transitive. (C... |
| cnvps 18467 | The converse of a poset is... |
| cnvpsb 18468 | The converse of a poset is... |
| psss 18469 | Any subset of a partially ... |
| psssdm2 18470 | Field of a subposet. (Con... |
| psssdm 18471 | Field of a subposet. (Con... |
| istsr 18472 | The predicate is a toset. ... |
| istsr2 18473 | The predicate is a toset. ... |
| tsrlin 18474 | A toset is a linear order.... |
| tsrlemax 18475 | Two ways of saying a numbe... |
| tsrps 18476 | A toset is a poset. (Cont... |
| cnvtsr 18477 | The converse of a toset is... |
| tsrss 18478 | Any subset of a totally or... |
| ledm 18479 | The domain of ` <_ ` is ` ... |
| lern 18480 | The range of ` <_ ` is ` R... |
| lefld 18481 | The field of the 'less or ... |
| letsr 18482 | The "less than or equal to... |
| isdir 18487 | A condition for a relation... |
| reldir 18488 | A direction is a relation.... |
| dirdm 18489 | A direction's domain is eq... |
| dirref 18490 | A direction is reflexive. ... |
| dirtr 18491 | A direction is transitive.... |
| dirge 18492 | For any two elements of a ... |
| tsrdir 18493 | A totally ordered set is a... |
| ismgm 18498 | The predicate "is a magma"... |
| ismgmn0 18499 | The predicate "is a magma"... |
| mgmcl 18500 | Closure of the operation o... |
| isnmgm 18501 | A condition for a structur... |
| mgmsscl 18502 | If the base set of a magma... |
| plusffval 18503 | The group addition operati... |
| plusfval 18504 | The group addition operati... |
| plusfeq 18505 | If the addition operation ... |
| plusffn 18506 | The group addition operati... |
| mgmplusf 18507 | The group addition functio... |
| issstrmgm 18508 | Characterize a substructur... |
| intopsn 18509 | The internal operation for... |
| mgmb1mgm1 18510 | The only magma with a base... |
| mgm0 18511 | Any set with an empty base... |
| mgm0b 18512 | The structure with an empt... |
| mgm1 18513 | The structure with one ele... |
| opifismgm 18514 | A structure with a group a... |
| mgmidmo 18515 | A two-sided identity eleme... |
| grpidval 18516 | The value of the identity ... |
| grpidpropd 18517 | If two structures have the... |
| fn0g 18518 | The group zero extractor i... |
| 0g0 18519 | The identity element funct... |
| ismgmid 18520 | The identity element of a ... |
| mgmidcl 18521 | The identity element of a ... |
| mgmlrid 18522 | The identity element of a ... |
| ismgmid2 18523 | Show that a given element ... |
| lidrideqd 18524 | If there is a left and rig... |
| lidrididd 18525 | If there is a left and rig... |
| grpidd 18526 | Deduce the identity elemen... |
| mgmidsssn0 18527 | Property of the set of ide... |
| grprinvlem 18528 | Lemma for ~ grprinvd . (C... |
| grprinvd 18529 | Deduce right inverse from ... |
| grpridd 18530 | Deduce right identity from... |
| gsumvalx 18531 | Expand out the substitutio... |
| gsumval 18532 | Expand out the substitutio... |
| gsumpropd 18533 | The group sum depends only... |
| gsumpropd2lem 18534 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18535 | A stronger version of ~ gs... |
| gsummgmpropd 18536 | A stronger version of ~ gs... |
| gsumress 18537 | The group sum in a substru... |
| gsumval1 18538 | Value of the group sum ope... |
| gsum0 18539 | Value of the empty group s... |
| gsumval2a 18540 | Value of the group sum ope... |
| gsumval2 18541 | Value of the group sum ope... |
| gsumsplit1r 18542 | Splitting off the rightmos... |
| gsumprval 18543 | Value of the group sum ope... |
| gsumpr12val 18544 | Value of the group sum ope... |
| issgrp 18547 | The predicate "is a semigr... |
| issgrpv 18548 | The predicate "is a semigr... |
| issgrpn0 18549 | The predicate "is a semigr... |
| isnsgrp 18550 | A condition for a structur... |
| sgrpmgm 18551 | A semigroup is a magma. (... |
| sgrpass 18552 | A semigroup operation is a... |
| sgrp0 18553 | Any set with an empty base... |
| sgrp0b 18554 | The structure with an empt... |
| sgrp1 18555 | The structure with one ele... |
| ismnddef 18558 | The predicate "is a monoid... |
| ismnd 18559 | The predicate "is a monoid... |
| isnmnd 18560 | A condition for a structur... |
| sgrpidmnd 18561 | A semigroup with an identi... |
| mndsgrp 18562 | A monoid is a semigroup. ... |
| mndmgm 18563 | A monoid is a magma. (Con... |
| mndcl 18564 | Closure of the operation o... |
| mndass 18565 | A monoid operation is asso... |
| mndid 18566 | A monoid has a two-sided i... |
| mndideu 18567 | The two-sided identity ele... |
| mnd32g 18568 | Commutative/associative la... |
| mnd12g 18569 | Commutative/associative la... |
| mnd4g 18570 | Commutative/associative la... |
| mndidcl 18571 | The identity element of a ... |
| mndbn0 18572 | The base set of a monoid i... |
| hashfinmndnn 18573 | A finite monoid has positi... |
| mndplusf 18574 | The group addition operati... |
| mndlrid 18575 | A monoid's identity elemen... |
| mndlid 18576 | The identity element of a ... |
| mndrid 18577 | The identity element of a ... |
| ismndd 18578 | Deduce a monoid from its p... |
| mndpfo 18579 | The addition operation of ... |
| mndfo 18580 | The addition operation of ... |
| mndpropd 18581 | If two structures have the... |
| mndprop 18582 | If two structures have the... |
| issubmnd 18583 | Characterize a submonoid b... |
| ress0g 18584 | ` 0g ` is unaffected by re... |
| submnd0 18585 | The zero of a submonoid is... |
| mndinvmod 18586 | Uniqueness of an inverse e... |
| prdsplusgcl 18587 | Structure product pointwis... |
| prdsidlem 18588 | Characterization of identi... |
| prdsmndd 18589 | The product of a family of... |
| prds0g 18590 | Zero in a product of monoi... |
| pwsmnd 18591 | The structure power of a m... |
| pws0g 18592 | Zero in a structure power ... |
| imasmnd2 18593 | The image structure of a m... |
| imasmnd 18594 | The image structure of a m... |
| imasmndf1 18595 | The image of a monoid unde... |
| xpsmnd 18596 | The binary product of mono... |
| mnd1 18597 | The (smallest) structure r... |
| mnd1id 18598 | The singleton element of a... |
| ismhm 18603 | Property of a monoid homom... |
| ismhmd 18604 | Deduction version of ~ ism... |
| mhmrcl1 18605 | Reverse closure of a monoi... |
| mhmrcl2 18606 | Reverse closure of a monoi... |
| mhmf 18607 | A monoid homomorphism is a... |
| mhmpropd 18608 | Monoid homomorphism depend... |
| mhmlin 18609 | A monoid homomorphism comm... |
| mhm0 18610 | A monoid homomorphism pres... |
| idmhm 18611 | The identity homomorphism ... |
| mhmf1o 18612 | A monoid homomorphism is b... |
| submrcl 18613 | Reverse closure for submon... |
| issubm 18614 | Expand definition of a sub... |
| issubm2 18615 | Submonoids are subsets tha... |
| issubmndb 18616 | The submonoid predicate. ... |
| issubmd 18617 | Deduction for proving a su... |
| mndissubm 18618 | If the base set of a monoi... |
| resmndismnd 18619 | If the base set of a monoi... |
| submss 18620 | Submonoids are subsets of ... |
| submid 18621 | Every monoid is trivially ... |
| subm0cl 18622 | Submonoids contain zero. ... |
| submcl 18623 | Submonoids are closed unde... |
| submmnd 18624 | Submonoids are themselves ... |
| submbas 18625 | The base set of a submonoi... |
| subm0 18626 | Submonoids have the same i... |
| subsubm 18627 | A submonoid of a submonoid... |
| 0subm 18628 | The zero submonoid of an a... |
| insubm 18629 | The intersection of two su... |
| 0mhm 18630 | The constant zero linear f... |
| resmhm 18631 | Restriction of a monoid ho... |
| resmhm2 18632 | One direction of ~ resmhm2... |
| resmhm2b 18633 | Restriction of the codomai... |
| mhmco 18634 | The composition of monoid ... |
| mhmima 18635 | The homomorphic image of a... |
| mhmeql 18636 | The equalizer of two monoi... |
| submacs 18637 | Submonoids are an algebrai... |
| mndind 18638 | Induction in a monoid. In... |
| prdspjmhm 18639 | A projection from a produc... |
| pwspjmhm 18640 | A projection from a struct... |
| pwsdiagmhm 18641 | Diagonal monoid homomorphi... |
| pwsco1mhm 18642 | Right composition with a f... |
| pwsco2mhm 18643 | Left composition with a mo... |
| gsumvallem2 18644 | Lemma for properties of th... |
| gsumsubm 18645 | Evaluate a group sum in a ... |
| gsumz 18646 | Value of a group sum over ... |
| gsumwsubmcl 18647 | Closure of the composite i... |
| gsumws1 18648 | A singleton composite reco... |
| gsumwcl 18649 | Closure of the composite o... |
| gsumsgrpccat 18650 | Homomorphic property of no... |
| gsumccat 18651 | Homomorphic property of co... |
| gsumws2 18652 | Valuation of a pair in a m... |
| gsumccatsn 18653 | Homomorphic property of co... |
| gsumspl 18654 | The primary purpose of the... |
| gsumwmhm 18655 | Behavior of homomorphisms ... |
| gsumwspan 18656 | The submonoid generated by... |
| frmdval 18661 | Value of the free monoid c... |
| frmdbas 18662 | The base set of a free mon... |
| frmdelbas 18663 | An element of the base set... |
| frmdplusg 18664 | The monoid operation of a ... |
| frmdadd 18665 | Value of the monoid operat... |
| vrmdfval 18666 | The canonical injection fr... |
| vrmdval 18667 | The value of the generatin... |
| vrmdf 18668 | The mapping from the index... |
| frmdmnd 18669 | A free monoid is a monoid.... |
| frmd0 18670 | The identity of the free m... |
| frmdsssubm 18671 | The set of words taking va... |
| frmdgsum 18672 | Any word in a free monoid ... |
| frmdss2 18673 | A subset of generators is ... |
| frmdup1 18674 | Any assignment of the gene... |
| frmdup2 18675 | The evaluation map has the... |
| frmdup3lem 18676 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18677 | Universal property of the ... |
| efmnd 18680 | The monoid of endofunction... |
| efmndbas 18681 | The base set of the monoid... |
| efmndbasabf 18682 | The base set of the monoid... |
| elefmndbas 18683 | Two ways of saying a funct... |
| elefmndbas2 18684 | Two ways of saying a funct... |
| efmndbasf 18685 | Elements in the monoid of ... |
| efmndhash 18686 | The monoid of endofunction... |
| efmndbasfi 18687 | The monoid of endofunction... |
| efmndfv 18688 | The function value of an e... |
| efmndtset 18689 | The topology of the monoid... |
| efmndplusg 18690 | The group operation of a m... |
| efmndov 18691 | The value of the group ope... |
| efmndcl 18692 | The group operation of the... |
| efmndtopn 18693 | The topology of the monoid... |
| symggrplem 18694 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18695 | The monoid of endofunction... |
| efmndsgrp 18696 | The monoid of endofunction... |
| ielefmnd 18697 | The identity function rest... |
| efmndid 18698 | The identity function rest... |
| efmndmnd 18699 | The monoid of endofunction... |
| efmnd0nmnd 18700 | Even the monoid of endofun... |
| efmndbas0 18701 | The base set of the monoid... |
| efmnd1hash 18702 | The monoid of endofunction... |
| efmnd1bas 18703 | The monoid of endofunction... |
| efmnd2hash 18704 | The monoid of endofunction... |
| submefmnd 18705 | If the base set of a monoi... |
| sursubmefmnd 18706 | The set of surjective endo... |
| injsubmefmnd 18707 | The set of injective endof... |
| idressubmefmnd 18708 | The singleton containing o... |
| idresefmnd 18709 | The structure with the sin... |
| smndex1ibas 18710 | The modulo function ` I ` ... |
| smndex1iidm 18711 | The modulo function ` I ` ... |
| smndex1gbas 18712 | The constant functions ` (... |
| smndex1gid 18713 | The composition of a const... |
| smndex1igid 18714 | The composition of the mod... |
| smndex1basss 18715 | The modulo function ` I ` ... |
| smndex1bas 18716 | The base set of the monoid... |
| smndex1mgm 18717 | The monoid of endofunction... |
| smndex1sgrp 18718 | The monoid of endofunction... |
| smndex1mndlem 18719 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18720 | The monoid of endofunction... |
| smndex1id 18721 | The modulo function ` I ` ... |
| smndex1n0mnd 18722 | The identity of the monoid... |
| nsmndex1 18723 | The base set ` B ` of the ... |
| smndex2dbas 18724 | The doubling function ` D ... |
| smndex2dnrinv 18725 | The doubling function ` D ... |
| smndex2hbas 18726 | The halving functions ` H ... |
| smndex2dlinvh 18727 | The halving functions ` H ... |
| mgm2nsgrplem1 18728 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18729 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18730 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18731 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18732 | A small magma (with two el... |
| sgrp2nmndlem1 18733 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18734 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18735 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18736 | A small semigroup (with tw... |
| sgrp2rid2ex 18737 | A small semigroup (with tw... |
| sgrp2nmndlem4 18738 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18739 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18740 | A small semigroup (with tw... |
| mgmnsgrpex 18741 | There is a magma which is ... |
| sgrpnmndex 18742 | There is a semigroup which... |
| sgrpssmgm 18743 | The class of all semigroup... |
| mndsssgrp 18744 | The class of all monoids i... |
| pwmndgplus 18745 | The operation of the monoi... |
| pwmndid 18746 | The identity of the monoid... |
| pwmnd 18747 | The power set of a class `... |
| isgrp 18754 | The predicate "is a group"... |
| grpmnd 18755 | A group is a monoid. (Con... |
| grpcl 18756 | Closure of the operation o... |
| grpass 18757 | A group operation is assoc... |
| grpinvex 18758 | Every member of a group ha... |
| grpideu 18759 | The two-sided identity ele... |
| grpmndd 18760 | A group is a monoid. (Con... |
| grpcld 18761 | Closure of the operation o... |
| grpplusf 18762 | The group addition operati... |
| grpplusfo 18763 | The group addition operati... |
| resgrpplusfrn 18764 | The underlying set of a gr... |
| grppropd 18765 | If two structures have the... |
| grpprop 18766 | If two structures have the... |
| grppropstr 18767 | Generalize a specific 2-el... |
| grpss 18768 | Show that a structure exte... |
| isgrpd2e 18769 | Deduce a group from its pr... |
| isgrpd2 18770 | Deduce a group from its pr... |
| isgrpde 18771 | Deduce a group from its pr... |
| isgrpd 18772 | Deduce a group from its pr... |
| isgrpi 18773 | Properties that determine ... |
| grpsgrp 18774 | A group is a semigroup. (... |
| dfgrp2 18775 | Alternate definition of a ... |
| dfgrp2e 18776 | Alternate definition of a ... |
| isgrpix 18777 | Properties that determine ... |
| grpidcl 18778 | The identity element of a ... |
| grpbn0 18779 | The base set of a group is... |
| grplid 18780 | The identity element of a ... |
| grprid 18781 | The identity element of a ... |
| grpn0 18782 | A group is not empty. (Co... |
| hashfingrpnn 18783 | A finite group has positiv... |
| grprcan 18784 | Right cancellation law for... |
| grpinveu 18785 | The left inverse element o... |
| grpid 18786 | Two ways of saying that an... |
| isgrpid2 18787 | Properties showing that an... |
| grpidd2 18788 | Deduce the identity elemen... |
| grpinvfval 18789 | The inverse function of a ... |
| grpinvfvalALT 18790 | Shorter proof of ~ grpinvf... |
| grpinvval 18791 | The inverse of a group ele... |
| grpinvfn 18792 | Functionality of the group... |
| grpinvfvi 18793 | The group inverse function... |
| grpsubfval 18794 | Group subtraction (divisio... |
| grpsubfvalALT 18795 | Shorter proof of ~ grpsubf... |
| grpsubval 18796 | Group subtraction (divisio... |
| grpinvf 18797 | The group inversion operat... |
| grpinvcl 18798 | A group element's inverse ... |
| grpinvcld 18799 | A group element's inverse ... |
| grplinv 18800 | The left inverse of a grou... |
| grprinv 18801 | The right inverse of a gro... |
| grpinvid1 18802 | The inverse of a group ele... |
| grpinvid2 18803 | The inverse of a group ele... |
| isgrpinv 18804 | Properties showing that a ... |
| grplrinv 18805 | In a group, every member h... |
| grpidinv2 18806 | A group's properties using... |
| grpidinv 18807 | A group has a left and rig... |
| grpinvid 18808 | The inverse of the identit... |
| grplcan 18809 | Left cancellation law for ... |
| grpasscan1 18810 | An associative cancellatio... |
| grpasscan2 18811 | An associative cancellatio... |
| grpidrcan 18812 | If right adding an element... |
| grpidlcan 18813 | If left adding an element ... |
| grpinvinv 18814 | Double inverse law for gro... |
| grpinvcnv 18815 | The group inverse is its o... |
| grpinv11 18816 | The group inverse is one-t... |
| grpinvf1o 18817 | The group inverse is a one... |
| grpinvnz 18818 | The inverse of a nonzero g... |
| grpinvnzcl 18819 | The inverse of a nonzero g... |
| grpsubinv 18820 | Subtraction of an inverse.... |
| grplmulf1o 18821 | Left multiplication by a g... |
| grpinvpropd 18822 | If two structures have the... |
| grpidssd 18823 | If the base set of a group... |
| grpinvssd 18824 | If the base set of a group... |
| grpinvadd 18825 | The inverse of the group o... |
| grpsubf 18826 | Functionality of group sub... |
| grpsubcl 18827 | Closure of group subtracti... |
| grpsubrcan 18828 | Right cancellation law for... |
| grpinvsub 18829 | Inverse of a group subtrac... |
| grpinvval2 18830 | A ~ df-neg -like equation ... |
| grpsubid 18831 | Subtraction of a group ele... |
| grpsubid1 18832 | Subtraction of the identit... |
| grpsubeq0 18833 | If the difference between ... |
| grpsubadd0sub 18834 | Subtraction expressed as a... |
| grpsubadd 18835 | Relationship between group... |
| grpsubsub 18836 | Double group subtraction. ... |
| grpaddsubass 18837 | Associative-type law for g... |
| grppncan 18838 | Cancellation law for subtr... |
| grpnpcan 18839 | Cancellation law for subtr... |
| grpsubsub4 18840 | Double group subtraction (... |
| grppnpcan2 18841 | Cancellation law for mixed... |
| grpnpncan 18842 | Cancellation law for group... |
| grpnpncan0 18843 | Cancellation law for group... |
| grpnnncan2 18844 | Cancellation law for group... |
| dfgrp3lem 18845 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18846 | Alternate definition of a ... |
| dfgrp3e 18847 | Alternate definition of a ... |
| grplactfval 18848 | The left group action of e... |
| grplactval 18849 | The value of the left grou... |
| grplactcnv 18850 | The left group action of e... |
| grplactf1o 18851 | The left group action of e... |
| grpsubpropd 18852 | Weak property deduction fo... |
| grpsubpropd2 18853 | Strong property deduction ... |
| grp1 18854 | The (smallest) structure r... |
| grp1inv 18855 | The inverse function of th... |
| prdsinvlem 18856 | Characterization of invers... |
| prdsgrpd 18857 | The product of a family of... |
| prdsinvgd 18858 | Negation in a product of g... |
| pwsgrp 18859 | A structure power of a gro... |
| pwsinvg 18860 | Negation in a group power.... |
| pwssub 18861 | Subtraction in a group pow... |
| imasgrp2 18862 | The image structure of a g... |
| imasgrp 18863 | The image structure of a g... |
| imasgrpf1 18864 | The image of a group under... |
| qusgrp2 18865 | Prove that a quotient stru... |
| xpsgrp 18866 | The binary product of grou... |
| mhmlem 18867 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18868 | A surjective monoid morphi... |
| mhmmnd 18869 | The image of a monoid ` G ... |
| mhmfmhm 18870 | The function fulfilling th... |
| ghmgrp 18871 | The image of a group ` G `... |
| mulgfval 18874 | Group multiple (exponentia... |
| mulgfvalALT 18875 | Shorter proof of ~ mulgfva... |
| mulgval 18876 | Value of the group multipl... |
| mulgfn 18877 | Functionality of the group... |
| mulgfvi 18878 | The group multiple operati... |
| mulg0 18879 | Group multiple (exponentia... |
| mulgnn 18880 | Group multiple (exponentia... |
| mulgnngsum 18881 | Group multiple (exponentia... |
| mulgnn0gsum 18882 | Group multiple (exponentia... |
| mulg1 18883 | Group multiple (exponentia... |
| mulgnnp1 18884 | Group multiple (exponentia... |
| mulg2 18885 | Group multiple (exponentia... |
| mulgnegnn 18886 | Group multiple (exponentia... |
| mulgnn0p1 18887 | Group multiple (exponentia... |
| mulgnnsubcl 18888 | Closure of the group multi... |
| mulgnn0subcl 18889 | Closure of the group multi... |
| mulgsubcl 18890 | Closure of the group multi... |
| mulgnncl 18891 | Closure of the group multi... |
| mulgnn0cl 18892 | Closure of the group multi... |
| mulgcl 18893 | Closure of the group multi... |
| mulgneg 18894 | Group multiple (exponentia... |
| mulgnegneg 18895 | The inverse of a negative ... |
| mulgm1 18896 | Group multiple (exponentia... |
| mulgnn0cld 18897 | Closure of the group multi... |
| mulgcld 18898 | Deduction associated with ... |
| mulgaddcomlem 18899 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 18900 | The group multiple operato... |
| mulginvcom 18901 | The group multiple operato... |
| mulginvinv 18902 | The group multiple operato... |
| mulgnn0z 18903 | A group multiple of the id... |
| mulgz 18904 | A group multiple of the id... |
| mulgnndir 18905 | Sum of group multiples, fo... |
| mulgnn0dir 18906 | Sum of group multiples, ge... |
| mulgdirlem 18907 | Lemma for ~ mulgdir . (Co... |
| mulgdir 18908 | Sum of group multiples, ge... |
| mulgp1 18909 | Group multiple (exponentia... |
| mulgneg2 18910 | Group multiple (exponentia... |
| mulgnnass 18911 | Product of group multiples... |
| mulgnn0ass 18912 | Product of group multiples... |
| mulgass 18913 | Product of group multiples... |
| mulgassr 18914 | Reversed product of group ... |
| mulgmodid 18915 | Casting out multiples of t... |
| mulgsubdir 18916 | Distribution of group mult... |
| mhmmulg 18917 | A homomorphism of monoids ... |
| mulgpropd 18918 | Two structures with the sa... |
| submmulgcl 18919 | Closure of the group multi... |
| submmulg 18920 | A group multiple is the sa... |
| pwsmulg 18921 | Value of a group multiple ... |
| issubg 18928 | The subgroup predicate. (... |
| subgss 18929 | A subgroup is a subset. (... |
| subgid 18930 | A group is a subgroup of i... |
| subggrp 18931 | A subgroup is a group. (C... |
| subgbas 18932 | The base of the restricted... |
| subgrcl 18933 | Reverse closure for the su... |
| subg0 18934 | A subgroup of a group must... |
| subginv 18935 | The inverse of an element ... |
| subg0cl 18936 | The group identity is an e... |
| subginvcl 18937 | The inverse of an element ... |
| subgcl 18938 | A subgroup is closed under... |
| subgsubcl 18939 | A subgroup is closed under... |
| subgsub 18940 | The subtraction of element... |
| subgmulgcl 18941 | Closure of the group multi... |
| subgmulg 18942 | A group multiple is the sa... |
| issubg2 18943 | Characterize the subgroups... |
| issubgrpd2 18944 | Prove a subgroup by closur... |
| issubgrpd 18945 | Prove a subgroup by closur... |
| issubg3 18946 | A subgroup is a symmetric ... |
| issubg4 18947 | A subgroup is a nonempty s... |
| grpissubg 18948 | If the base set of a group... |
| resgrpisgrp 18949 | If the base set of a group... |
| subgsubm 18950 | A subgroup is a submonoid.... |
| subsubg 18951 | A subgroup of a subgroup i... |
| subgint 18952 | The intersection of a none... |
| 0subg 18953 | The zero subgroup of an ar... |
| 0subgOLD 18954 | Obsolete version of ~ 0sub... |
| trivsubgd 18955 | The only subgroup of a tri... |
| trivsubgsnd 18956 | The only subgroup of a tri... |
| isnsg 18957 | Property of being a normal... |
| isnsg2 18958 | Weaken the condition of ~ ... |
| nsgbi 18959 | Defining property of a nor... |
| nsgsubg 18960 | A normal subgroup is a sub... |
| nsgconj 18961 | The conjugation of an elem... |
| isnsg3 18962 | A subgroup is normal iff t... |
| subgacs 18963 | Subgroups are an algebraic... |
| nsgacs 18964 | Normal subgroups form an a... |
| elnmz 18965 | Elementhood in the normali... |
| nmzbi 18966 | Defining property of the n... |
| nmzsubg 18967 | The normalizer N_G(S) of a... |
| ssnmz 18968 | A subgroup is a subset of ... |
| isnsg4 18969 | A subgroup is normal iff i... |
| nmznsg 18970 | Any subgroup is a normal s... |
| 0nsg 18971 | The zero subgroup is norma... |
| nsgid 18972 | The whole group is a norma... |
| 0idnsgd 18973 | The whole group and the ze... |
| trivnsgd 18974 | The only normal subgroup o... |
| triv1nsgd 18975 | A trivial group has exactl... |
| 1nsgtrivd 18976 | A group with exactly one n... |
| releqg 18977 | The left coset equivalence... |
| eqgfval 18978 | Value of the subgroup left... |
| eqgval 18979 | Value of the subgroup left... |
| eqger 18980 | The subgroup coset equival... |
| eqglact 18981 | A left coset can be expres... |
| eqgid 18982 | The left coset containing ... |
| eqgen 18983 | Each coset is equipotent t... |
| eqgcpbl 18984 | The subgroup coset equival... |
| qusgrp 18985 | If ` Y ` is a normal subgr... |
| quseccl 18986 | Closure of the quotient ma... |
| qusadd 18987 | Value of the group operati... |
| qus0 18988 | Value of the group identit... |
| qusinv 18989 | Value of the group inverse... |
| qussub 18990 | Value of the group subtrac... |
| lagsubg2 18991 | Lagrange's theorem for fin... |
| lagsubg 18992 | Lagrange's theorem for Gro... |
| cycsubmel 18993 | Characterization of an ele... |
| cycsubmcl 18994 | The set of nonnegative int... |
| cycsubm 18995 | The set of nonnegative int... |
| cyccom 18996 | Condition for an operation... |
| cycsubmcom 18997 | The operation of a monoid ... |
| cycsubggend 18998 | The cyclic subgroup genera... |
| cycsubgcl 18999 | The set of integer powers ... |
| cycsubgss 19000 | The cyclic subgroup genera... |
| cycsubg 19001 | The cyclic group generated... |
| cycsubgcld 19002 | The cyclic subgroup genera... |
| cycsubg2 19003 | The subgroup generated by ... |
| cycsubg2cl 19004 | Any multiple of an element... |
| reldmghm 19007 | Lemma for group homomorphi... |
| isghm 19008 | Property of being a homomo... |
| isghm3 19009 | Property of a group homomo... |
| ghmgrp1 19010 | A group homomorphism is on... |
| ghmgrp2 19011 | A group homomorphism is on... |
| ghmf 19012 | A group homomorphism is a ... |
| ghmlin 19013 | A homomorphism of groups i... |
| ghmid 19014 | A homomorphism of groups p... |
| ghminv 19015 | A homomorphism of groups p... |
| ghmsub 19016 | Linearity of subtraction t... |
| isghmd 19017 | Deduction for a group homo... |
| ghmmhm 19018 | A group homomorphism is a ... |
| ghmmhmb 19019 | Group homomorphisms and mo... |
| ghmmulg 19020 | A homomorphism of monoids ... |
| ghmrn 19021 | The range of a homomorphis... |
| 0ghm 19022 | The constant zero linear f... |
| idghm 19023 | The identity homomorphism ... |
| resghm 19024 | Restriction of a homomorph... |
| resghm2 19025 | One direction of ~ resghm2... |
| resghm2b 19026 | Restriction of the codomai... |
| ghmghmrn 19027 | A group homomorphism from ... |
| ghmco 19028 | The composition of group h... |
| ghmima 19029 | The image of a subgroup un... |
| ghmpreima 19030 | The inverse image of a sub... |
| ghmeql 19031 | The equalizer of two group... |
| ghmnsgima 19032 | The image of a normal subg... |
| ghmnsgpreima 19033 | The inverse image of a nor... |
| ghmker 19034 | The kernel of a homomorphi... |
| ghmeqker 19035 | Two source points map to t... |
| pwsdiagghm 19036 | Diagonal homomorphism into... |
| ghmf1 19037 | Two ways of saying a group... |
| ghmf1o 19038 | A bijective group homomorp... |
| conjghm 19039 | Conjugation is an automorp... |
| conjsubg 19040 | A conjugated subgroup is a... |
| conjsubgen 19041 | A conjugated subgroup is e... |
| conjnmz 19042 | A subgroup is unchanged un... |
| conjnmzb 19043 | Alternative condition for ... |
| conjnsg 19044 | A normal subgroup is uncha... |
| qusghm 19045 | If ` Y ` is a normal subgr... |
| ghmpropd 19046 | Group homomorphism depends... |
| gimfn 19051 | The group isomorphism func... |
| isgim 19052 | An isomorphism of groups i... |
| gimf1o 19053 | An isomorphism of groups i... |
| gimghm 19054 | An isomorphism of groups i... |
| isgim2 19055 | A group isomorphism is a h... |
| subggim 19056 | Behavior of subgroups unde... |
| gimcnv 19057 | The converse of a bijectiv... |
| gimco 19058 | The composition of group i... |
| brgic 19059 | The relation "is isomorphi... |
| brgici 19060 | Prove isomorphic by an exp... |
| gicref 19061 | Isomorphism is reflexive. ... |
| giclcl 19062 | Isomorphism implies the le... |
| gicrcl 19063 | Isomorphism implies the ri... |
| gicsym 19064 | Isomorphism is symmetric. ... |
| gictr 19065 | Isomorphism is transitive.... |
| gicer 19066 | Isomorphism is an equivale... |
| gicen 19067 | Isomorphic groups have equ... |
| gicsubgen 19068 | A less trivial example of ... |
| isga 19071 | The predicate "is a (left)... |
| gagrp 19072 | The left argument of a gro... |
| gaset 19073 | The right argument of a gr... |
| gagrpid 19074 | The identity of the group ... |
| gaf 19075 | The mapping of the group a... |
| gafo 19076 | A group action is onto its... |
| gaass 19077 | An "associative" property ... |
| ga0 19078 | The action of a group on t... |
| gaid 19079 | The trivial action of a gr... |
| subgga 19080 | A subgroup acts on its par... |
| gass 19081 | A subset of a group action... |
| gasubg 19082 | The restriction of a group... |
| gaid2 19083 | A group operation is a lef... |
| galcan 19084 | The action of a particular... |
| gacan 19085 | Group inverses cancel in a... |
| gapm 19086 | The action of a particular... |
| gaorb 19087 | The orbit equivalence rela... |
| gaorber 19088 | The orbit equivalence rela... |
| gastacl 19089 | The stabilizer subgroup in... |
| gastacos 19090 | Write the coset relation f... |
| orbstafun 19091 | Existence and uniqueness f... |
| orbstaval 19092 | Value of the function at a... |
| orbsta 19093 | The Orbit-Stabilizer theor... |
| orbsta2 19094 | Relation between the size ... |
| cntrval 19099 | Substitute definition of t... |
| cntzfval 19100 | First level substitution f... |
| cntzval 19101 | Definition substitution fo... |
| elcntz 19102 | Elementhood in the central... |
| cntzel 19103 | Membership in a centralize... |
| cntzsnval 19104 | Special substitution for t... |
| elcntzsn 19105 | Value of the centralizer o... |
| sscntz 19106 | A centralizer expression f... |
| cntzrcl 19107 | Reverse closure for elemen... |
| cntzssv 19108 | The centralizer is uncondi... |
| cntzi 19109 | Membership in a centralize... |
| cntrss 19110 | The center is a subset of ... |
| cntri 19111 | Defining property of the c... |
| resscntz 19112 | Centralizer in a substruct... |
| cntz2ss 19113 | Centralizers reverse the s... |
| cntzrec 19114 | Reciprocity relationship f... |
| cntziinsn 19115 | Express any centralizer as... |
| cntzsubm 19116 | Centralizers in a monoid a... |
| cntzsubg 19117 | Centralizers in a group ar... |
| cntzidss 19118 | If the elements of ` S ` c... |
| cntzmhm 19119 | Centralizers in a monoid a... |
| cntzmhm2 19120 | Centralizers in a monoid a... |
| cntrsubgnsg 19121 | A central subgroup is norm... |
| cntrnsg 19122 | The center of a group is a... |
| oppgval 19125 | Value of the opposite grou... |
| oppgplusfval 19126 | Value of the addition oper... |
| oppgplus 19127 | Value of the addition oper... |
| setsplusg 19128 | The other components of an... |
| oppglemOLD 19129 | Obsolete version of ~ sets... |
| oppgbas 19130 | Base set of an opposite gr... |
| oppgbasOLD 19131 | Obsolete version of ~ oppg... |
| oppgtset 19132 | Topology of an opposite gr... |
| oppgtsetOLD 19133 | Obsolete version of ~ oppg... |
| oppgtopn 19134 | Topology of an opposite gr... |
| oppgmnd 19135 | The opposite of a monoid i... |
| oppgmndb 19136 | Bidirectional form of ~ op... |
| oppgid 19137 | Zero in a monoid is a symm... |
| oppggrp 19138 | The opposite of a group is... |
| oppggrpb 19139 | Bidirectional form of ~ op... |
| oppginv 19140 | Inverses in a group are a ... |
| invoppggim 19141 | The inverse is an antiauto... |
| oppggic 19142 | Every group is (naturally)... |
| oppgsubm 19143 | Being a submonoid is a sym... |
| oppgsubg 19144 | Being a subgroup is a symm... |
| oppgcntz 19145 | A centralizer in a group i... |
| oppgcntr 19146 | The center of a group is t... |
| gsumwrev 19147 | A sum in an opposite monoi... |
| symgval 19150 | The value of the symmetric... |
| permsetexOLD 19151 | Obsolete version of ~ f1os... |
| symgbas 19152 | The base set of the symmet... |
| symgbasexOLD 19153 | Obsolete as of 8-Aug-2024.... |
| elsymgbas2 19154 | Two ways of saying a funct... |
| elsymgbas 19155 | Two ways of saying a funct... |
| symgbasf1o 19156 | Elements in the symmetric ... |
| symgbasf 19157 | A permutation (element of ... |
| symgbasmap 19158 | A permutation (element of ... |
| symghash 19159 | The symmetric group on ` n... |
| symgbasfi 19160 | The symmetric group on a f... |
| symgfv 19161 | The function value of a pe... |
| symgfvne 19162 | The function values of a p... |
| symgressbas 19163 | The symmetric group on ` A... |
| symgplusg 19164 | The group operation of a s... |
| symgov 19165 | The value of the group ope... |
| symgcl 19166 | The group operation of the... |
| idresperm 19167 | The identity function rest... |
| symgmov1 19168 | For a permutation of a set... |
| symgmov2 19169 | For a permutation of a set... |
| symgbas0 19170 | The base set of the symmet... |
| symg1hash 19171 | The symmetric group on a s... |
| symg1bas 19172 | The symmetric group on a s... |
| symg2hash 19173 | The symmetric group on a (... |
| symg2bas 19174 | The symmetric group on a p... |
| 0symgefmndeq 19175 | The symmetric group on the... |
| snsymgefmndeq 19176 | The symmetric group on a s... |
| symgpssefmnd 19177 | For a set ` A ` with more ... |
| symgvalstruct 19178 | The value of the symmetric... |
| symgvalstructOLD 19179 | Obsolete proof of ~ symgva... |
| symgsubmefmnd 19180 | The symmetric group on a s... |
| symgtset 19181 | The topology of the symmet... |
| symggrp 19182 | The symmetric group on a s... |
| symgid 19183 | The group identity element... |
| symginv 19184 | The group inverse in the s... |
| symgsubmefmndALT 19185 | The symmetric group on a s... |
| galactghm 19186 | The currying of a group ac... |
| lactghmga 19187 | The converse of ~ galactgh... |
| symgtopn 19188 | The topology of the symmet... |
| symgga 19189 | The symmetric group induce... |
| pgrpsubgsymgbi 19190 | Every permutation group is... |
| pgrpsubgsymg 19191 | Every permutation group is... |
| idressubgsymg 19192 | The singleton containing o... |
| idrespermg 19193 | The structure with the sin... |
| cayleylem1 19194 | Lemma for ~ cayley . (Con... |
| cayleylem2 19195 | Lemma for ~ cayley . (Con... |
| cayley 19196 | Cayley's Theorem (construc... |
| cayleyth 19197 | Cayley's Theorem (existenc... |
| symgfix2 19198 | If a permutation does not ... |
| symgextf 19199 | The extension of a permuta... |
| symgextfv 19200 | The function value of the ... |
| symgextfve 19201 | The function value of the ... |
| symgextf1lem 19202 | Lemma for ~ symgextf1 . (... |
| symgextf1 19203 | The extension of a permuta... |
| symgextfo 19204 | The extension of a permuta... |
| symgextf1o 19205 | The extension of a permuta... |
| symgextsymg 19206 | The extension of a permuta... |
| symgextres 19207 | The restriction of the ext... |
| gsumccatsymgsn 19208 | Homomorphic property of co... |
| gsmsymgrfixlem1 19209 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 19210 | The composition of permuta... |
| fvcosymgeq 19211 | The values of two composit... |
| gsmsymgreqlem1 19212 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 19213 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 19214 | Two combination of permuta... |
| symgfixelq 19215 | A permutation of a set fix... |
| symgfixels 19216 | The restriction of a permu... |
| symgfixelsi 19217 | The restriction of a permu... |
| symgfixf 19218 | The mapping of a permutati... |
| symgfixf1 19219 | The mapping of a permutati... |
| symgfixfolem1 19220 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 19221 | The mapping of a permutati... |
| symgfixf1o 19222 | The mapping of a permutati... |
| f1omvdmvd 19225 | A permutation of any class... |
| f1omvdcnv 19226 | A permutation and its inve... |
| mvdco 19227 | Composing two permutations... |
| f1omvdconj 19228 | Conjugation of a permutati... |
| f1otrspeq 19229 | A transposition is charact... |
| f1omvdco2 19230 | If exactly one of two perm... |
| f1omvdco3 19231 | If a point is moved by exa... |
| pmtrfval 19232 | The function generating tr... |
| pmtrval 19233 | A generated transposition,... |
| pmtrfv 19234 | General value of mapping a... |
| pmtrprfv 19235 | In a transposition of two ... |
| pmtrprfv3 19236 | In a transposition of two ... |
| pmtrf 19237 | Functionality of a transpo... |
| pmtrmvd 19238 | A transposition moves prec... |
| pmtrrn 19239 | Transposing two points giv... |
| pmtrfrn 19240 | A transposition (as a kind... |
| pmtrffv 19241 | Mapping of a point under a... |
| pmtrrn2 19242 | For any transposition ther... |
| pmtrfinv 19243 | A transposition function i... |
| pmtrfmvdn0 19244 | A transposition moves at l... |
| pmtrff1o 19245 | A transposition function i... |
| pmtrfcnv 19246 | A transposition function i... |
| pmtrfb 19247 | An intrinsic characterizat... |
| pmtrfconj 19248 | Any conjugate of a transpo... |
| symgsssg 19249 | The symmetric group has su... |
| symgfisg 19250 | The symmetric group has a ... |
| symgtrf 19251 | Transpositions are element... |
| symggen 19252 | The span of the transposit... |
| symggen2 19253 | A finite permutation group... |
| symgtrinv 19254 | To invert a permutation re... |
| pmtr3ncomlem1 19255 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 19256 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 19257 | Transpositions over sets w... |
| pmtrdifellem1 19258 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19259 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19260 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19261 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19262 | A transposition of element... |
| pmtrdifwrdellem1 19263 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19264 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19265 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19266 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19267 | A sequence of transpositio... |
| pmtrdifwrdel2 19268 | A sequence of transpositio... |
| pmtrprfval 19269 | The transpositions on a pa... |
| pmtrprfvalrn 19270 | The range of the transposi... |
| psgnunilem1 19275 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19276 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19277 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19278 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19279 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19280 | Addition and subtraction o... |
| psgnuni 19281 | If the same permutation ca... |
| psgnfval 19282 | Function definition of the... |
| psgnfn 19283 | Functionality and domain o... |
| psgndmsubg 19284 | The finitary permutations ... |
| psgneldm 19285 | Property of being a finita... |
| psgneldm2 19286 | The finitary permutations ... |
| psgneldm2i 19287 | A sequence of transpositio... |
| psgneu 19288 | A finitary permutation has... |
| psgnval 19289 | Value of the permutation s... |
| psgnvali 19290 | A finitary permutation has... |
| psgnvalii 19291 | Any representation of a pe... |
| psgnpmtr 19292 | All transpositions are odd... |
| psgn0fv0 19293 | The permutation sign funct... |
| sygbasnfpfi 19294 | The class of non-fixed poi... |
| psgnfvalfi 19295 | Function definition of the... |
| psgnvalfi 19296 | Value of the permutation s... |
| psgnran 19297 | The range of the permutati... |
| gsmtrcl 19298 | The group sum of transposi... |
| psgnfitr 19299 | A permutation of a finite ... |
| psgnfieu 19300 | A permutation of a finite ... |
| pmtrsn 19301 | The value of the transposi... |
| psgnsn 19302 | The permutation sign funct... |
| psgnprfval 19303 | The permutation sign funct... |
| psgnprfval1 19304 | The permutation sign of th... |
| psgnprfval2 19305 | The permutation sign of th... |
| odfval 19314 | Value of the order functio... |
| odfvalALT 19315 | Shorter proof of ~ odfval ... |
| odval 19316 | Second substitution for th... |
| odlem1 19317 | The group element order is... |
| odcl 19318 | The order of a group eleme... |
| odf 19319 | Functionality of the group... |
| odid 19320 | Any element to the power o... |
| odlem2 19321 | Any positive annihilator o... |
| odmodnn0 19322 | Reduce the argument of a g... |
| mndodconglem 19323 | Lemma for ~ mndodcong . (... |
| mndodcong 19324 | If two multipliers are con... |
| mndodcongi 19325 | If two multipliers are con... |
| oddvdsnn0 19326 | The only multiples of ` A ... |
| odnncl 19327 | If a nonzero multiple of a... |
| odmod 19328 | Reduce the argument of a g... |
| oddvds 19329 | The only multiples of ` A ... |
| oddvdsi 19330 | Any group element is annih... |
| odcong 19331 | If two multipliers are con... |
| odeq 19332 | The ~ oddvds property uniq... |
| odval2 19333 | A non-conditional definiti... |
| odcld 19334 | The order of a group eleme... |
| odm1inv 19335 | The (order-1)th multiple o... |
| odmulgid 19336 | A relationship between the... |
| odmulg2 19337 | The order of a multiple di... |
| odmulg 19338 | Relationship between the o... |
| odmulgeq 19339 | A multiple of a point of f... |
| odbezout 19340 | If ` N ` is coprime to the... |
| od1 19341 | The order of the group ide... |
| odeq1 19342 | The group identity is the ... |
| odinv 19343 | The order of the inverse o... |
| odf1 19344 | The multiples of an elemen... |
| odinf 19345 | The multiples of an elemen... |
| dfod2 19346 | An alternative definition ... |
| odcl2 19347 | The order of an element of... |
| oddvds2 19348 | The order of an element of... |
| finodsubmsubg 19349 | A submonoid whose elements... |
| 0subgALT 19350 | A shorter proof of ~ 0subg... |
| submod 19351 | The order of an element is... |
| subgod 19352 | The order of an element is... |
| odsubdvds 19353 | The order of an element of... |
| odf1o1 19354 | An element with zero order... |
| odf1o2 19355 | An element with nonzero or... |
| odhash 19356 | An element of zero order g... |
| odhash2 19357 | If an element has nonzero ... |
| odhash3 19358 | An element which generates... |
| odngen 19359 | A cyclic subgroup of size ... |
| gexval 19360 | Value of the exponent of a... |
| gexlem1 19361 | The group element order is... |
| gexcl 19362 | The exponent of a group is... |
| gexid 19363 | Any element to the power o... |
| gexlem2 19364 | Any positive annihilator o... |
| gexdvdsi 19365 | Any group element is annih... |
| gexdvds 19366 | The only ` N ` that annihi... |
| gexdvds2 19367 | An integer divides the gro... |
| gexod 19368 | Any group element is annih... |
| gexcl3 19369 | If the order of every grou... |
| gexnnod 19370 | Every group element has fi... |
| gexcl2 19371 | The exponent of a finite g... |
| gexdvds3 19372 | The exponent of a finite g... |
| gex1 19373 | A group or monoid has expo... |
| ispgp 19374 | A group is a ` P ` -group ... |
| pgpprm 19375 | Reverse closure for the fi... |
| pgpgrp 19376 | Reverse closure for the se... |
| pgpfi1 19377 | A finite group with order ... |
| pgp0 19378 | The identity subgroup is a... |
| subgpgp 19379 | A subgroup of a p-group is... |
| sylow1lem1 19380 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19381 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19382 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19383 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19384 | Lemma for ~ sylow1 . Usin... |
| sylow1 19385 | Sylow's first theorem. If... |
| odcau 19386 | Cauchy's theorem for the o... |
| pgpfi 19387 | The converse to ~ pgpfi1 .... |
| pgpfi2 19388 | Alternate version of ~ pgp... |
| pgphash 19389 | The order of a p-group. (... |
| isslw 19390 | The property of being a Sy... |
| slwprm 19391 | Reverse closure for the fi... |
| slwsubg 19392 | A Sylow ` P ` -subgroup is... |
| slwispgp 19393 | Defining property of a Syl... |
| slwpss 19394 | A proper superset of a Syl... |
| slwpgp 19395 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19396 | Every ` P ` -subgroup is c... |
| slwn0 19397 | Every finite group contain... |
| subgslw 19398 | A Sylow subgroup that is c... |
| sylow2alem1 19399 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19400 | Lemma for ~ sylow2a . All... |
| sylow2a 19401 | A named lemma of Sylow's s... |
| sylow2blem1 19402 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19403 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19404 | Sylow's second theorem. P... |
| sylow2b 19405 | Sylow's second theorem. A... |
| slwhash 19406 | A sylow subgroup has cardi... |
| fislw 19407 | The sylow subgroups of a f... |
| sylow2 19408 | Sylow's second theorem. S... |
| sylow3lem1 19409 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19410 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19411 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19412 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19413 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19414 | Lemma for ~ sylow3 , secon... |
| sylow3 19415 | Sylow's third theorem. Th... |
| lsmfval 19420 | The subgroup sum function ... |
| lsmvalx 19421 | Subspace sum value (for a ... |
| lsmelvalx 19422 | Subspace sum membership (f... |
| lsmelvalix 19423 | Subspace sum membership (f... |
| oppglsm 19424 | The subspace sum operation... |
| lsmssv 19425 | Subgroup sum is a subset o... |
| lsmless1x 19426 | Subset implies subgroup su... |
| lsmless2x 19427 | Subset implies subgroup su... |
| lsmub1x 19428 | Subgroup sum is an upper b... |
| lsmub2x 19429 | Subgroup sum is an upper b... |
| lsmval 19430 | Subgroup sum value (for a ... |
| lsmelval 19431 | Subgroup sum membership (f... |
| lsmelvali 19432 | Subgroup sum membership (f... |
| lsmelvalm 19433 | Subgroup sum membership an... |
| lsmelvalmi 19434 | Membership of vector subtr... |
| lsmsubm 19435 | The sum of two commuting s... |
| lsmsubg 19436 | The sum of two commuting s... |
| lsmcom2 19437 | Subgroup sum commutes. (C... |
| smndlsmidm 19438 | The direct product is idem... |
| lsmub1 19439 | Subgroup sum is an upper b... |
| lsmub2 19440 | Subgroup sum is an upper b... |
| lsmunss 19441 | Union of subgroups is a su... |
| lsmless1 19442 | Subset implies subgroup su... |
| lsmless2 19443 | Subset implies subgroup su... |
| lsmless12 19444 | Subset implies subgroup su... |
| lsmidm 19445 | Subgroup sum is idempotent... |
| lsmlub 19446 | The least upper bound prop... |
| lsmss1 19447 | Subgroup sum with a subset... |
| lsmss1b 19448 | Subgroup sum with a subset... |
| lsmss2 19449 | Subgroup sum with a subset... |
| lsmss2b 19450 | Subgroup sum with a subset... |
| lsmass 19451 | Subgroup sum is associativ... |
| mndlsmidm 19452 | Subgroup sum is idempotent... |
| lsm01 19453 | Subgroup sum with the zero... |
| lsm02 19454 | Subgroup sum with the zero... |
| subglsm 19455 | The subgroup sum evaluated... |
| lssnle 19456 | Equivalent expressions for... |
| lsmmod 19457 | The modular law holds for ... |
| lsmmod2 19458 | Modular law dual for subgr... |
| lsmpropd 19459 | If two structures have the... |
| cntzrecd 19460 | Commute the "subgroups com... |
| lsmcntz 19461 | The "subgroups commute" pr... |
| lsmcntzr 19462 | The "subgroups commute" pr... |
| lsmdisj 19463 | Disjointness from a subgro... |
| lsmdisj2 19464 | Association of the disjoin... |
| lsmdisj3 19465 | Association of the disjoin... |
| lsmdisjr 19466 | Disjointness from a subgro... |
| lsmdisj2r 19467 | Association of the disjoin... |
| lsmdisj3r 19468 | Association of the disjoin... |
| lsmdisj2a 19469 | Association of the disjoin... |
| lsmdisj2b 19470 | Association of the disjoin... |
| lsmdisj3a 19471 | Association of the disjoin... |
| lsmdisj3b 19472 | Association of the disjoin... |
| subgdisj1 19473 | Vectors belonging to disjo... |
| subgdisj2 19474 | Vectors belonging to disjo... |
| subgdisjb 19475 | Vectors belonging to disjo... |
| pj1fval 19476 | The left projection functi... |
| pj1val 19477 | The left projection functi... |
| pj1eu 19478 | Uniqueness of a left proje... |
| pj1f 19479 | The left projection functi... |
| pj2f 19480 | The right projection funct... |
| pj1id 19481 | Any element of a direct su... |
| pj1eq 19482 | Any element of a direct su... |
| pj1lid 19483 | The left projection functi... |
| pj1rid 19484 | The left projection functi... |
| pj1ghm 19485 | The left projection functi... |
| pj1ghm2 19486 | The left projection functi... |
| lsmhash 19487 | The order of the direct pr... |
| efgmval 19494 | Value of the formal invers... |
| efgmf 19495 | The formal inverse operati... |
| efgmnvl 19496 | The inversion function on ... |
| efgrcl 19497 | Lemma for ~ efgval . (Con... |
| efglem 19498 | Lemma for ~ efgval . (Con... |
| efgval 19499 | Value of the free group co... |
| efger 19500 | Value of the free group co... |
| efgi 19501 | Value of the free group co... |
| efgi0 19502 | Value of the free group co... |
| efgi1 19503 | Value of the free group co... |
| efgtf 19504 | Value of the free group co... |
| efgtval 19505 | Value of the extension fun... |
| efgval2 19506 | Value of the free group co... |
| efgi2 19507 | Value of the free group co... |
| efgtlen 19508 | Value of the free group co... |
| efginvrel2 19509 | The inverse of the reverse... |
| efginvrel1 19510 | The inverse of the reverse... |
| efgsf 19511 | Value of the auxiliary fun... |
| efgsdm 19512 | Elementhood in the domain ... |
| efgsval 19513 | Value of the auxiliary fun... |
| efgsdmi 19514 | Property of the last link ... |
| efgsval2 19515 | Value of the auxiliary fun... |
| efgsrel 19516 | The start and end of any e... |
| efgs1 19517 | A singleton of an irreduci... |
| efgs1b 19518 | Every extension sequence e... |
| efgsp1 19519 | If ` F ` is an extension s... |
| efgsres 19520 | An initial segment of an e... |
| efgsfo 19521 | For any word, there is a s... |
| efgredlema 19522 | The reduced word that form... |
| efgredlemf 19523 | Lemma for ~ efgredleme . ... |
| efgredlemg 19524 | Lemma for ~ efgred . (Con... |
| efgredleme 19525 | Lemma for ~ efgred . (Con... |
| efgredlemd 19526 | The reduced word that form... |
| efgredlemc 19527 | The reduced word that form... |
| efgredlemb 19528 | The reduced word that form... |
| efgredlem 19529 | The reduced word that form... |
| efgred 19530 | The reduced word that form... |
| efgrelexlema 19531 | If two words ` A , B ` are... |
| efgrelexlemb 19532 | If two words ` A , B ` are... |
| efgrelex 19533 | If two words ` A , B ` are... |
| efgredeu 19534 | There is a unique reduced ... |
| efgred2 19535 | Two extension sequences ha... |
| efgcpbllema 19536 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19537 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19538 | Two extension sequences ha... |
| efgcpbl2 19539 | Two extension sequences ha... |
| frgpval 19540 | Value of the free group co... |
| frgpcpbl 19541 | Compatibility of the group... |
| frgp0 19542 | The free group is a group.... |
| frgpeccl 19543 | Closure of the quotient ma... |
| frgpgrp 19544 | The free group is a group.... |
| frgpadd 19545 | Addition in the free group... |
| frgpinv 19546 | The inverse of an element ... |
| frgpmhm 19547 | The "natural map" from wor... |
| vrgpfval 19548 | The canonical injection fr... |
| vrgpval 19549 | The value of the generatin... |
| vrgpf 19550 | The mapping from the index... |
| vrgpinv 19551 | The inverse of a generatin... |
| frgpuptf 19552 | Any assignment of the gene... |
| frgpuptinv 19553 | Any assignment of the gene... |
| frgpuplem 19554 | Any assignment of the gene... |
| frgpupf 19555 | Any assignment of the gene... |
| frgpupval 19556 | Any assignment of the gene... |
| frgpup1 19557 | Any assignment of the gene... |
| frgpup2 19558 | The evaluation map has the... |
| frgpup3lem 19559 | The evaluation map has the... |
| frgpup3 19560 | Universal property of the ... |
| 0frgp 19561 | The free group on zero gen... |
| isabl 19566 | The predicate "is an Abeli... |
| ablgrp 19567 | An Abelian group is a grou... |
| ablgrpd 19568 | An Abelian group is a grou... |
| ablcmn 19569 | An Abelian group is a comm... |
| ablcmnd 19570 | An Abelian group is a comm... |
| iscmn 19571 | The predicate "is a commut... |
| isabl2 19572 | The predicate "is an Abeli... |
| cmnpropd 19573 | If two structures have the... |
| ablpropd 19574 | If two structures have the... |
| ablprop 19575 | If two structures have the... |
| iscmnd 19576 | Properties that determine ... |
| isabld 19577 | Properties that determine ... |
| isabli 19578 | Properties that determine ... |
| cmnmnd 19579 | A commutative monoid is a ... |
| cmncom 19580 | A commutative monoid is co... |
| ablcom 19581 | An Abelian group operation... |
| cmn32 19582 | Commutative/associative la... |
| cmn4 19583 | Commutative/associative la... |
| cmn12 19584 | Commutative/associative la... |
| abl32 19585 | Commutative/associative la... |
| cmnmndd 19586 | A commutative monoid is a ... |
| rinvmod 19587 | Uniqueness of a right inve... |
| ablinvadd 19588 | The inverse of an Abelian ... |
| ablsub2inv 19589 | Abelian group subtraction ... |
| ablsubadd 19590 | Relationship between Abeli... |
| ablsub4 19591 | Commutative/associative su... |
| abladdsub4 19592 | Abelian group addition/sub... |
| abladdsub 19593 | Associative-type law for g... |
| ablpncan2 19594 | Cancellation law for subtr... |
| ablpncan3 19595 | A cancellation law for Abe... |
| ablsubsub 19596 | Law for double subtraction... |
| ablsubsub4 19597 | Law for double subtraction... |
| ablpnpcan 19598 | Cancellation law for mixed... |
| ablnncan 19599 | Cancellation law for group... |
| ablsub32 19600 | Swap the second and third ... |
| ablnnncan 19601 | Cancellation law for group... |
| ablnnncan1 19602 | Cancellation law for group... |
| ablsubsub23 19603 | Swap subtrahend and result... |
| mulgnn0di 19604 | Group multiple of a sum, f... |
| mulgdi 19605 | Group multiple of a sum. ... |
| mulgmhm 19606 | The map from ` x ` to ` n ... |
| mulgghm 19607 | The map from ` x ` to ` n ... |
| mulgsubdi 19608 | Group multiple of a differ... |
| ghmfghm 19609 | The function fulfilling th... |
| ghmcmn 19610 | The image of a commutative... |
| ghmabl 19611 | The image of an abelian gr... |
| invghm 19612 | The inversion map is a gro... |
| eqgabl 19613 | Value of the subgroup cose... |
| subgabl 19614 | A subgroup of an abelian g... |
| subcmn 19615 | A submonoid of a commutati... |
| submcmn 19616 | A submonoid of a commutati... |
| submcmn2 19617 | A submonoid is commutative... |
| cntzcmn 19618 | The centralizer of any sub... |
| cntzcmnss 19619 | Any subset in a commutativ... |
| cntrcmnd 19620 | The center of a monoid is ... |
| cntrabl 19621 | The center of a group is a... |
| cntzspan 19622 | If the generators commute,... |
| cntzcmnf 19623 | Discharge the centralizer ... |
| ghmplusg 19624 | The pointwise sum of two l... |
| ablnsg 19625 | Every subgroup of an abeli... |
| odadd1 19626 | The order of a product in ... |
| odadd2 19627 | The order of a product in ... |
| odadd 19628 | The order of a product is ... |
| gex2abl 19629 | A group with exponent 2 (o... |
| gexexlem 19630 | Lemma for ~ gexex . (Cont... |
| gexex 19631 | In an abelian group with f... |
| torsubg 19632 | The set of all elements of... |
| oddvdssubg 19633 | The set of all elements wh... |
| lsmcomx 19634 | Subgroup sum commutes (ext... |
| ablcntzd 19635 | All subgroups in an abelia... |
| lsmcom 19636 | Subgroup sum commutes. (C... |
| lsmsubg2 19637 | The sum of two subgroups i... |
| lsm4 19638 | Commutative/associative la... |
| prdscmnd 19639 | The product of a family of... |
| prdsabld 19640 | The product of a family of... |
| pwscmn 19641 | The structure power on a c... |
| pwsabl 19642 | The structure power on an ... |
| qusabl 19643 | If ` Y ` is a subgroup of ... |
| abl1 19644 | The (smallest) structure r... |
| abln0 19645 | Abelian groups (and theref... |
| cnaddablx 19646 | The complex numbers are an... |
| cnaddabl 19647 | The complex numbers are an... |
| cnaddid 19648 | The group identity element... |
| cnaddinv 19649 | Value of the group inverse... |
| zaddablx 19650 | The integers are an Abelia... |
| frgpnabllem1 19651 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19652 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19653 | The free group on two or m... |
| iscyg 19656 | Definition of a cyclic gro... |
| iscyggen 19657 | The property of being a cy... |
| iscyggen2 19658 | The property of being a cy... |
| iscyg2 19659 | A cyclic group is a group ... |
| cyggeninv 19660 | The inverse of a cyclic ge... |
| cyggenod 19661 | An element is the generato... |
| cyggenod2 19662 | In an infinite cyclic grou... |
| iscyg3 19663 | Definition of a cyclic gro... |
| iscygd 19664 | Definition of a cyclic gro... |
| iscygodd 19665 | Show that a group with an ... |
| cycsubmcmn 19666 | The set of nonnegative int... |
| cyggrp 19667 | A cyclic group is a group.... |
| cygabl 19668 | A cyclic group is abelian.... |
| cygctb 19669 | A cyclic group is countabl... |
| 0cyg 19670 | The trivial group is cycli... |
| prmcyg 19671 | A group with prime order i... |
| lt6abl 19672 | A group with fewer than ` ... |
| ghmcyg 19673 | The image of a cyclic grou... |
| cyggex2 19674 | The exponent of a cyclic g... |
| cyggex 19675 | The exponent of a finite c... |
| cyggexb 19676 | A finite abelian group is ... |
| giccyg 19677 | Cyclicity is a group prope... |
| cycsubgcyg 19678 | The cyclic subgroup genera... |
| cycsubgcyg2 19679 | The cyclic subgroup genera... |
| gsumval3a 19680 | Value of the group sum ope... |
| gsumval3eu 19681 | The group sum as defined i... |
| gsumval3lem1 19682 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19683 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19684 | Value of the group sum ope... |
| gsumcllem 19685 | Lemma for ~ gsumcl and rel... |
| gsumzres 19686 | Extend a finite group sum ... |
| gsumzcl2 19687 | Closure of a finite group ... |
| gsumzcl 19688 | Closure of a finite group ... |
| gsumzf1o 19689 | Re-index a finite group su... |
| gsumres 19690 | Extend a finite group sum ... |
| gsumcl2 19691 | Closure of a finite group ... |
| gsumcl 19692 | Closure of a finite group ... |
| gsumf1o 19693 | Re-index a finite group su... |
| gsumreidx 19694 | Re-index a finite group su... |
| gsumzsubmcl 19695 | Closure of a group sum in ... |
| gsumsubmcl 19696 | Closure of a group sum in ... |
| gsumsubgcl 19697 | Closure of a group sum in ... |
| gsumzaddlem 19698 | The sum of two group sums.... |
| gsumzadd 19699 | The sum of two group sums.... |
| gsumadd 19700 | The sum of two group sums.... |
| gsummptfsadd 19701 | The sum of two group sums ... |
| gsummptfidmadd 19702 | The sum of two group sums ... |
| gsummptfidmadd2 19703 | The sum of two group sums ... |
| gsumzsplit 19704 | Split a group sum into two... |
| gsumsplit 19705 | Split a group sum into two... |
| gsumsplit2 19706 | Split a group sum into two... |
| gsummptfidmsplit 19707 | Split a group sum expresse... |
| gsummptfidmsplitres 19708 | Split a group sum expresse... |
| gsummptfzsplit 19709 | Split a group sum expresse... |
| gsummptfzsplitl 19710 | Split a group sum expresse... |
| gsumconst 19711 | Sum of a constant series. ... |
| gsumconstf 19712 | Sum of a constant series. ... |
| gsummptshft 19713 | Index shift of a finite gr... |
| gsumzmhm 19714 | Apply a group homomorphism... |
| gsummhm 19715 | Apply a group homomorphism... |
| gsummhm2 19716 | Apply a group homomorphism... |
| gsummptmhm 19717 | Apply a group homomorphism... |
| gsummulglem 19718 | Lemma for ~ gsummulg and ~... |
| gsummulg 19719 | Nonnegative multiple of a ... |
| gsummulgz 19720 | Integer multiple of a grou... |
| gsumzoppg 19721 | The opposite of a group su... |
| gsumzinv 19722 | Inverse of a group sum. (... |
| gsuminv 19723 | Inverse of a group sum. (... |
| gsummptfidminv 19724 | Inverse of a group sum exp... |
| gsumsub 19725 | The difference of two grou... |
| gsummptfssub 19726 | The difference of two grou... |
| gsummptfidmsub 19727 | The difference of two grou... |
| gsumsnfd 19728 | Group sum of a singleton, ... |
| gsumsnd 19729 | Group sum of a singleton, ... |
| gsumsnf 19730 | Group sum of a singleton, ... |
| gsumsn 19731 | Group sum of a singleton. ... |
| gsumpr 19732 | Group sum of a pair. (Con... |
| gsumzunsnd 19733 | Append an element to a fin... |
| gsumunsnfd 19734 | Append an element to a fin... |
| gsumunsnd 19735 | Append an element to a fin... |
| gsumunsnf 19736 | Append an element to a fin... |
| gsumunsn 19737 | Append an element to a fin... |
| gsumdifsnd 19738 | Extract a summand from a f... |
| gsumpt 19739 | Sum of a family that is no... |
| gsummptf1o 19740 | Re-index a finite group su... |
| gsummptun 19741 | Group sum of a disjoint un... |
| gsummpt1n0 19742 | If only one summand in a f... |
| gsummptif1n0 19743 | If only one summand in a f... |
| gsummptcl 19744 | Closure of a finite group ... |
| gsummptfif1o 19745 | Re-index a finite group su... |
| gsummptfzcl 19746 | Closure of a finite group ... |
| gsum2dlem1 19747 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19748 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19749 | Write a sum over a two-dim... |
| gsum2d2lem 19750 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19751 | Write a group sum over a t... |
| gsumcom2 19752 | Two-dimensional commutatio... |
| gsumxp 19753 | Write a group sum over a c... |
| gsumcom 19754 | Commute the arguments of a... |
| gsumcom3 19755 | A commutative law for fini... |
| gsumcom3fi 19756 | A commutative law for fini... |
| gsumxp2 19757 | Write a group sum over a c... |
| prdsgsum 19758 | Finite commutative sums in... |
| pwsgsum 19759 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19760 | Replacing a finitely suppo... |
| nn0gsumfz 19761 | Replacing a finitely suppo... |
| nn0gsumfz0 19762 | Replacing a finitely suppo... |
| gsummptnn0fz 19763 | A final group sum over a f... |
| gsummptnn0fzfv 19764 | A final group sum over a f... |
| telgsumfzslem 19765 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19766 | Telescoping group sum rang... |
| telgsumfz 19767 | Telescoping group sum rang... |
| telgsumfz0s 19768 | Telescoping finite group s... |
| telgsumfz0 19769 | Telescoping finite group s... |
| telgsums 19770 | Telescoping finitely suppo... |
| telgsum 19771 | Telescoping finitely suppo... |
| reldmdprd 19776 | The domain of the internal... |
| dmdprd 19777 | The domain of definition o... |
| dmdprdd 19778 | Show that a given family i... |
| dprddomprc 19779 | A family of subgroups inde... |
| dprddomcld 19780 | If a family of subgroups i... |
| dprdval0prc 19781 | The internal direct produc... |
| dprdval 19782 | The value of the internal ... |
| eldprd 19783 | A class ` A ` is an intern... |
| dprdgrp 19784 | Reverse closure for the in... |
| dprdf 19785 | The function ` S ` is a fa... |
| dprdf2 19786 | The function ` S ` is a fa... |
| dprdcntz 19787 | The function ` S ` is a fa... |
| dprddisj 19788 | The function ` S ` is a fa... |
| dprdw 19789 | The property of being a fi... |
| dprdwd 19790 | A mapping being a finitely... |
| dprdff 19791 | A finitely supported funct... |
| dprdfcl 19792 | A finitely supported funct... |
| dprdffsupp 19793 | A finitely supported funct... |
| dprdfcntz 19794 | A function on the elements... |
| dprdssv 19795 | The internal direct produc... |
| dprdfid 19796 | A function mapping all but... |
| eldprdi 19797 | The domain of definition o... |
| dprdfinv 19798 | Take the inverse of a grou... |
| dprdfadd 19799 | Take the sum of group sums... |
| dprdfsub 19800 | Take the difference of gro... |
| dprdfeq0 19801 | The zero function is the o... |
| dprdf11 19802 | Two group sums over a dire... |
| dprdsubg 19803 | The internal direct produc... |
| dprdub 19804 | Each factor is a subset of... |
| dprdlub 19805 | The direct product is smal... |
| dprdspan 19806 | The direct product is the ... |
| dprdres 19807 | Restriction of a direct pr... |
| dprdss 19808 | Create a direct product by... |
| dprdz 19809 | A family consisting entire... |
| dprd0 19810 | The empty family is an int... |
| dprdf1o 19811 | Rearrange the index set of... |
| dprdf1 19812 | Rearrange the index set of... |
| subgdmdprd 19813 | A direct product in a subg... |
| subgdprd 19814 | A direct product in a subg... |
| dprdsn 19815 | A singleton family is an i... |
| dmdprdsplitlem 19816 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19817 | The function ` S ` is a fa... |
| dprddisj2 19818 | The function ` S ` is a fa... |
| dprd2dlem2 19819 | The direct product of a co... |
| dprd2dlem1 19820 | The direct product of a co... |
| dprd2da 19821 | The direct product of a co... |
| dprd2db 19822 | The direct product of a co... |
| dprd2d2 19823 | The direct product of a co... |
| dmdprdsplit2lem 19824 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19825 | The direct product splits ... |
| dmdprdsplit 19826 | The direct product splits ... |
| dprdsplit 19827 | The direct product is the ... |
| dmdprdpr 19828 | A singleton family is an i... |
| dprdpr 19829 | A singleton family is an i... |
| dpjlem 19830 | Lemma for theorems about d... |
| dpjcntz 19831 | The two subgroups that app... |
| dpjdisj 19832 | The two subgroups that app... |
| dpjlsm 19833 | The two subgroups that app... |
| dpjfval 19834 | Value of the direct produc... |
| dpjval 19835 | Value of the direct produc... |
| dpjf 19836 | The ` X ` -th index projec... |
| dpjidcl 19837 | The key property of projec... |
| dpjeq 19838 | Decompose a group sum into... |
| dpjid 19839 | The key property of projec... |
| dpjlid 19840 | The ` X ` -th index projec... |
| dpjrid 19841 | The ` Y ` -th index projec... |
| dpjghm 19842 | The direct product is the ... |
| dpjghm2 19843 | The direct product is the ... |
| ablfacrplem 19844 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 19845 | A finite abelian group who... |
| ablfacrp2 19846 | The factors ` K , L ` of ~... |
| ablfac1lem 19847 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 19848 | The factors of ~ ablfac1b ... |
| ablfac1b 19849 | Any abelian group is the d... |
| ablfac1c 19850 | The factors of ~ ablfac1b ... |
| ablfac1eulem 19851 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 19852 | The factorization of ~ abl... |
| pgpfac1lem1 19853 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 19854 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 19855 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 19856 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 19857 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 19858 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 19859 | Factorization of a finite ... |
| pgpfaclem1 19860 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 19861 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 19862 | Lemma for ~ pgpfac . (Con... |
| pgpfac 19863 | Full factorization of a fi... |
| ablfaclem1 19864 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 19865 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 19866 | Lemma for ~ ablfac . (Con... |
| ablfac 19867 | The Fundamental Theorem of... |
| ablfac2 19868 | Choose generators for each... |
| issimpg 19871 | The predicate "is a simple... |
| issimpgd 19872 | Deduce a simple group from... |
| simpggrp 19873 | A simple group is a group.... |
| simpggrpd 19874 | A simple group is a group.... |
| simpg2nsg 19875 | A simple group has two nor... |
| trivnsimpgd 19876 | Trivial groups are not sim... |
| simpgntrivd 19877 | Simple groups are nontrivi... |
| simpgnideld 19878 | A simple group contains a ... |
| simpgnsgd 19879 | The only normal subgroups ... |
| simpgnsgeqd 19880 | A normal subgroup of a sim... |
| 2nsgsimpgd 19881 | If any normal subgroup of ... |
| simpgnsgbid 19882 | A nontrivial group is simp... |
| ablsimpnosubgd 19883 | A subgroup of an abelian s... |
| ablsimpg1gend 19884 | An abelian simple group is... |
| ablsimpgcygd 19885 | An abelian simple group is... |
| ablsimpgfindlem1 19886 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 19887 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 19888 | Relationship between the o... |
| ablsimpgfind 19889 | An abelian simple group is... |
| fincygsubgd 19890 | The subgroup referenced in... |
| fincygsubgodd 19891 | Calculate the order of a s... |
| fincygsubgodexd 19892 | A finite cyclic group has ... |
| prmgrpsimpgd 19893 | A group of prime order is ... |
| ablsimpgprmd 19894 | An abelian simple group ha... |
| ablsimpgd 19895 | An abelian group is simple... |
| fnmgp 19898 | The multiplicative group o... |
| mgpval 19899 | Value of the multiplicatio... |
| mgpplusg 19900 | Value of the group operati... |
| mgplemOLD 19901 | Obsolete version of ~ sets... |
| mgpbas 19902 | Base set of the multiplica... |
| mgpbasOLD 19903 | Obsolete version of ~ mgpb... |
| mgpsca 19904 | The multiplication monoid ... |
| mgpscaOLD 19905 | Obsolete version of ~ mgps... |
| mgptset 19906 | Topology component of the ... |
| mgptsetOLD 19907 | Obsolete version of ~ mgpt... |
| mgptopn 19908 | Topology of the multiplica... |
| mgpds 19909 | Distance function of the m... |
| mgpdsOLD 19910 | Obsolete version of ~ mgpd... |
| mgpress 19911 | Subgroup commutes with the... |
| mgpressOLD 19912 | Obsolete version of ~ mgpr... |
| ringidval 19915 | The value of the unity ele... |
| dfur2 19916 | The multiplicative identit... |
| issrg 19919 | The predicate "is a semiri... |
| srgcmn 19920 | A semiring is a commutativ... |
| srgmnd 19921 | A semiring is a monoid. (... |
| srgmgp 19922 | A semiring is a monoid und... |
| srgdilem 19923 | Lemma for ~ srgdi and ~ sr... |
| srgcl 19924 | Closure of the multiplicat... |
| srgass 19925 | Associative law for the mu... |
| srgideu 19926 | The unity element of a sem... |
| srgfcl 19927 | Functionality of the multi... |
| srgdi 19928 | Distributive law for the m... |
| srgdir 19929 | Distributive law for the m... |
| srgidcl 19930 | The unity element of a sem... |
| srg0cl 19931 | The zero element of a semi... |
| srgidmlem 19932 | Lemma for ~ srglidm and ~ ... |
| srglidm 19933 | The unity element of a sem... |
| srgridm 19934 | The unity element of a sem... |
| issrgid 19935 | Properties showing that an... |
| srgacl 19936 | Closure of the addition op... |
| srgcom 19937 | Commutativity of the addit... |
| srgrz 19938 | The zero of a semiring is ... |
| srglz 19939 | The zero of a semiring is ... |
| srgisid 19940 | In a semiring, the only le... |
| o2timesd 19941 | An element of a ring-like ... |
| rglcom4d 19942 | Restricted commutativity o... |
| srgo2times 19943 | A semiring element plus it... |
| srgcom4lem 19944 | Lemma for ~ srgcom4 . Thi... |
| srgcom4 19945 | Restricted commutativity o... |
| srg1zr 19946 | The only semiring with a b... |
| srgen1zr 19947 | The only semiring with one... |
| srgmulgass 19948 | An associative property be... |
| srgpcomp 19949 | If two elements of a semir... |
| srgpcompp 19950 | If two elements of a semir... |
| srgpcomppsc 19951 | If two elements of a semir... |
| srglmhm 19952 | Left-multiplication in a s... |
| srgrmhm 19953 | Right-multiplication in a ... |
| srgsummulcr 19954 | A finite semiring sum mult... |
| sgsummulcl 19955 | A finite semiring sum mult... |
| srg1expzeq1 19956 | The exponentiation (by a n... |
| srgbinomlem1 19957 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 19958 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 19959 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 19960 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 19961 | Lemma for ~ srgbinom . In... |
| srgbinom 19962 | The binomial theorem for c... |
| csrgbinom 19963 | The binomial theorem for c... |
| isring 19968 | The predicate "is a (unita... |
| ringgrp 19969 | A ring is a group. (Contr... |
| ringmgp 19970 | A ring is a monoid under m... |
| iscrng 19971 | A commutative ring is a ri... |
| crngmgp 19972 | A commutative ring's multi... |
| ringgrpd 19973 | A ring is a group. (Contr... |
| ringmnd 19974 | A ring is a monoid under a... |
| ringmgm 19975 | A ring is a magma. (Contr... |
| crngring 19976 | A commutative ring is a ri... |
| crngringd 19977 | A commutative ring is a ri... |
| crnggrpd 19978 | A commutative ring is a gr... |
| mgpf 19979 | Restricted functionality o... |
| ringdilem 19980 | Properties of a unital rin... |
| ringcl 19981 | Closure of the multiplicat... |
| crngcom 19982 | A commutative ring's multi... |
| iscrng2 19983 | A commutative ring is a ri... |
| ringass 19984 | Associative law for multip... |
| ringideu 19985 | The unity element of a rin... |
| ringcld 19986 | Closure of the multiplicat... |
| ringdi 19987 | Distributive law for the m... |
| ringdir 19988 | Distributive law for the m... |
| ringidcl 19989 | The unity element of a rin... |
| ring0cl 19990 | The zero element of a ring... |
| ringidmlem 19991 | Lemma for ~ ringlidm and ~... |
| ringlidm 19992 | The unity element of a rin... |
| ringridm 19993 | The unity element of a rin... |
| isringid 19994 | Properties showing that an... |
| ringid 19995 | The multiplication operati... |
| ringo2times 19996 | A ring element plus itself... |
| ringadd2 19997 | A ring element plus itself... |
| ringidss 19998 | A subset of the multiplica... |
| ringacl 19999 | Closure of the addition op... |
| ringcomlem 20000 | Lemma for ~ ringcom . Thi... |
| ringcom 20001 | Commutativity of the addit... |
| ringabl 20002 | A ring is an Abelian group... |
| ringcmn 20003 | A ring is a commutative mo... |
| ringabld 20004 | A ring is an Abelian group... |
| ringcmnd 20005 | A ring is a commutative mo... |
| ringpropd 20006 | If two structures have the... |
| crngpropd 20007 | If two structures have the... |
| ringprop 20008 | If two structures have the... |
| isringd 20009 | Properties that determine ... |
| iscrngd 20010 | Properties that determine ... |
| ringlz 20011 | The zero of a unital ring ... |
| ringrz 20012 | The zero of a unital ring ... |
| ringsrg 20013 | Any ring is also a semirin... |
| ring1eq0 20014 | If one and zero are equal,... |
| ring1ne0 20015 | If a ring has at least two... |
| ringinvnz1ne0 20016 | In a unital ring, a left i... |
| ringinvnzdiv 20017 | In a unital ring, a left i... |
| ringnegl 20018 | Negation in a ring is the ... |
| ringnegr 20019 | Negation in a ring is the ... |
| ringmneg1 20020 | Negation of a product in a... |
| ringmneg2 20021 | Negation of a product in a... |
| ringm2neg 20022 | Double negation of a produ... |
| ringsubdi 20023 | Ring multiplication distri... |
| ringsubdir 20024 | Ring multiplication distri... |
| mulgass2 20025 | An associative property be... |
| ring1 20026 | The (smallest) structure r... |
| ringn0 20027 | Rings exist. (Contributed... |
| ringlghm 20028 | Left-multiplication in a r... |
| ringrghm 20029 | Right-multiplication in a ... |
| gsummulc1 20030 | A finite ring sum multipli... |
| gsummulc2 20031 | A finite ring sum multipli... |
| gsummgp0 20032 | If one factor in a finite ... |
| gsumdixp 20033 | Distribute a binary produc... |
| prdsmgp 20034 | The multiplicative monoid ... |
| prdsmulrcl 20035 | A structure product of rin... |
| prdsringd 20036 | A product of rings is a ri... |
| prdscrngd 20037 | A product of commutative r... |
| prds1 20038 | Value of the ring unity in... |
| pwsring 20039 | A structure power of a rin... |
| pws1 20040 | Value of the ring unity in... |
| pwscrng 20041 | A structure power of a com... |
| pwsmgp 20042 | The multiplicative group o... |
| pwspjmhmmgpd 20043 | The projection given by ~ ... |
| pwsexpg 20044 | Value of a group exponenti... |
| imasring 20045 | The image structure of a r... |
| qusring2 20046 | The quotient structure of ... |
| crngbinom 20047 | The binomial theorem for c... |
| opprval 20050 | Value of the opposite ring... |
| opprmulfval 20051 | Value of the multiplicatio... |
| opprmul 20052 | Value of the multiplicatio... |
| crngoppr 20053 | In a commutative ring, the... |
| opprlem 20054 | Lemma for ~ opprbas and ~ ... |
| opprlemOLD 20055 | Obsolete version of ~ oppr... |
| opprbas 20056 | Base set of an opposite ri... |
| opprbasOLD 20057 | Obsolete proof of ~ opprba... |
| oppradd 20058 | Addition operation of an o... |
| oppraddOLD 20059 | Obsolete proof of ~ opprba... |
| opprring 20060 | An opposite ring is a ring... |
| opprringb 20061 | Bidirectional form of ~ op... |
| oppr0 20062 | Additive identity of an op... |
| oppr1 20063 | Multiplicative identity of... |
| opprneg 20064 | The negative function in a... |
| opprsubg 20065 | Being a subgroup is a symm... |
| mulgass3 20066 | An associative property be... |
| reldvdsr 20073 | The divides relation is a ... |
| dvdsrval 20074 | Value of the divides relat... |
| dvdsr 20075 | Value of the divides relat... |
| dvdsr2 20076 | Value of the divides relat... |
| dvdsrmul 20077 | A left-multiple of ` X ` i... |
| dvdsrcl 20078 | Closure of a dividing elem... |
| dvdsrcl2 20079 | Closure of a dividing elem... |
| dvdsrid 20080 | An element in a (unital) r... |
| dvdsrtr 20081 | Divisibility is transitive... |
| dvdsrmul1 20082 | The divisibility relation ... |
| dvdsrneg 20083 | An element divides its neg... |
| dvdsr01 20084 | In a ring, zero is divisib... |
| dvdsr02 20085 | Only zero is divisible by ... |
| isunit 20086 | Property of being a unit o... |
| 1unit 20087 | The multiplicative identit... |
| unitcl 20088 | A unit is an element of th... |
| unitss 20089 | The set of units is contai... |
| opprunit 20090 | Being a unit is a symmetri... |
| crngunit 20091 | Property of being a unit i... |
| dvdsunit 20092 | A divisor of a unit is a u... |
| unitmulcl 20093 | The product of units is a ... |
| unitmulclb 20094 | Reversal of ~ unitmulcl in... |
| unitgrpbas 20095 | The base set of the group ... |
| unitgrp 20096 | The group of units is a gr... |
| unitabl 20097 | The group of units of a co... |
| unitgrpid 20098 | The identity of the group ... |
| unitsubm 20099 | The group of units is a su... |
| invrfval 20102 | Multiplicative inverse fun... |
| unitinvcl 20103 | The inverse of a unit exis... |
| unitinvinv 20104 | The inverse of the inverse... |
| ringinvcl 20105 | The inverse of a unit is a... |
| unitlinv 20106 | A unit times its inverse i... |
| unitrinv 20107 | A unit times its inverse i... |
| 1rinv 20108 | The inverse of the ring un... |
| 0unit 20109 | The additive identity is a... |
| unitnegcl 20110 | The negative of a unit is ... |
| dvrfval 20113 | Division operation in a ri... |
| dvrval 20114 | Division operation in a ri... |
| dvrcl 20115 | Closure of division operat... |
| unitdvcl 20116 | The units are closed under... |
| dvrid 20117 | A ring element divided by ... |
| dvr1 20118 | A ring element divided by ... |
| dvrass 20119 | An associative law for div... |
| dvrcan1 20120 | A cancellation law for div... |
| dvrcan3 20121 | A cancellation law for div... |
| dvreq1 20122 | Equality in terms of ratio... |
| ringinvdv 20123 | Write the inverse function... |
| rngidpropd 20124 | The ring unity depends onl... |
| dvdsrpropd 20125 | The divisibility relation ... |
| unitpropd 20126 | The set of units depends o... |
| invrpropd 20127 | The ring inverse function ... |
| isirred 20128 | An irreducible element of ... |
| isnirred 20129 | The property of being a no... |
| isirred2 20130 | Expand out the class diffe... |
| opprirred 20131 | Irreducibility is symmetri... |
| irredn0 20132 | The additive identity is n... |
| irredcl 20133 | An irreducible element is ... |
| irrednu 20134 | An irreducible element is ... |
| irredn1 20135 | The multiplicative identit... |
| irredrmul 20136 | The product of an irreduci... |
| irredlmul 20137 | The product of a unit and ... |
| irredmul 20138 | If product of two elements... |
| irredneg 20139 | The negative of an irreduc... |
| irrednegb 20140 | An element is irreducible ... |
| dfrhm2 20148 | The property of a ring hom... |
| rhmrcl1 20150 | Reverse closure of a ring ... |
| rhmrcl2 20151 | Reverse closure of a ring ... |
| isrhm 20152 | A function is a ring homom... |
| rhmmhm 20153 | A ring homomorphism is a h... |
| isrim0OLD 20154 | Obsolete version of ~ isri... |
| rimrcl 20155 | Reverse closure for an iso... |
| isrim0 20156 | A ring isomorphism is a ho... |
| rhmghm 20157 | A ring homomorphism is an ... |
| rhmf 20158 | A ring homomorphism is a f... |
| rhmmul 20159 | A homomorphism of rings pr... |
| isrhm2d 20160 | Demonstration of ring homo... |
| isrhmd 20161 | Demonstration of ring homo... |
| rhm1 20162 | Ring homomorphisms are req... |
| idrhm 20163 | The identity homomorphism ... |
| rhmf1o 20164 | A ring homomorphism is bij... |
| isrim 20165 | An isomorphism of rings is... |
| isrimOLD 20166 | Obsolete version of ~ isri... |
| rimf1o 20167 | An isomorphism of rings is... |
| rimrhmOLD 20168 | Obsolete version of ~ rimr... |
| rimrhm 20169 | A ring isomorphism is a ho... |
| rimgim 20170 | An isomorphism of rings is... |
| rhmco 20171 | The composition of ring ho... |
| pwsco1rhm 20172 | Right composition with a f... |
| pwsco2rhm 20173 | Left composition with a ri... |
| f1ghm0to0 20174 | If a group homomorphism ` ... |
| f1rhm0to0ALT 20175 | Alternate proof for ~ f1gh... |
| gim0to0 20176 | A group isomorphism maps t... |
| kerf1ghm 20177 | A group homomorphism ` F `... |
| brric 20178 | The relation "is isomorphi... |
| brric2 20179 | The relation "is isomorphi... |
| ricgic 20180 | If two rings are (ring) is... |
| rhmdvdsr 20181 | A ring homomorphism preser... |
| rhmopp 20182 | A ring homomorphism is als... |
| elrhmunit 20183 | Ring homomorphisms preserv... |
| rhmunitinv 20184 | Ring homomorphisms preserv... |
| isdrng 20189 | The predicate "is a divisi... |
| drngunit 20190 | Elementhood in the set of ... |
| drngui 20191 | The set of units of a divi... |
| drngring 20192 | A division ring is a ring.... |
| drngringd 20193 | A division ring is a ring.... |
| drnggrpd 20194 | A division ring is a group... |
| drnggrp 20195 | A division ring is a group... |
| isfld 20196 | A field is a commutative d... |
| fldcrngd 20197 | A field is a commutative r... |
| isdrng2 20198 | A division ring can equiva... |
| drngprop 20199 | If two structures have the... |
| drngmgp 20200 | A division ring contains a... |
| drngmcl 20201 | The product of two nonzero... |
| drngid 20202 | A division ring's unity is... |
| drngunz 20203 | A division ring's unity is... |
| drngid2 20204 | Properties showing that an... |
| drnginvrcl 20205 | Closure of the multiplicat... |
| drnginvrn0 20206 | The multiplicative inverse... |
| drnginvrcld 20207 | Closure of the multiplicat... |
| drnginvrl 20208 | Property of the multiplica... |
| drnginvrr 20209 | Property of the multiplica... |
| drngmul0or 20210 | A product is zero iff one ... |
| drngmulne0 20211 | A product is nonzero iff b... |
| drngmuleq0 20212 | An element is zero iff its... |
| opprdrng 20213 | The opposite of a division... |
| isdrngd 20214 | Properties that characteri... |
| isdrngrd 20215 | Properties that characteri... |
| drngpropd 20216 | If two structures have the... |
| fldpropd 20217 | If two structures have the... |
| issubrg 20222 | The subring predicate. (C... |
| subrgss 20223 | A subring is a subset. (C... |
| subrgid 20224 | Every ring is a subring of... |
| subrgring 20225 | A subring is a ring. (Con... |
| subrgcrng 20226 | A subring of a commutative... |
| subrgrcl 20227 | Reverse closure for a subr... |
| subrgsubg 20228 | A subring is a subgroup. ... |
| subrg0 20229 | A subring always has the s... |
| subrg1cl 20230 | A subring contains the mul... |
| subrgbas 20231 | Base set of a subring stru... |
| subrg1 20232 | A subring always has the s... |
| subrgacl 20233 | A subring is closed under ... |
| subrgmcl 20234 | A subgroup is closed under... |
| subrgsubm 20235 | A subring is a submonoid o... |
| subrgdvds 20236 | If an element divides anot... |
| subrguss 20237 | A unit of a subring is a u... |
| subrginv 20238 | A subring always has the s... |
| subrgdv 20239 | A subring always has the s... |
| subrgunit 20240 | An element of a ring is a ... |
| subrgugrp 20241 | The units of a subring for... |
| issubrg2 20242 | Characterize the subrings ... |
| opprsubrg 20243 | Being a subring is a symme... |
| subrgint 20244 | The intersection of a none... |
| subrgin 20245 | The intersection of two su... |
| subrgmre 20246 | The subrings of a ring are... |
| issubdrg 20247 | Characterize the subfields... |
| subsubrg 20248 | A subring of a subring is ... |
| subsubrg2 20249 | The set of subrings of a s... |
| issubrg3 20250 | A subring is an additive s... |
| resrhm 20251 | Restriction of a ring homo... |
| rhmeql 20252 | The equalizer of two ring ... |
| rhmima 20253 | The homomorphic image of a... |
| rnrhmsubrg 20254 | The range of a ring homomo... |
| cntzsubr 20255 | Centralizers in a ring are... |
| pwsdiagrhm 20256 | Diagonal homomorphism into... |
| subrgpropd 20257 | If two structures have the... |
| rhmpropd 20258 | Ring homomorphism depends ... |
| issdrg 20261 | Property of a division sub... |
| sdrgid 20262 | Every division ring is a d... |
| sdrgss 20263 | A division subring is a su... |
| issdrg2 20264 | Property of a division sub... |
| fldsdrgfld 20265 | A sub-division-ring of a f... |
| acsfn1p 20266 | Construction of a closure ... |
| subrgacs 20267 | Closure property of subrin... |
| sdrgacs 20268 | Closure property of divisi... |
| cntzsdrg 20269 | Centralizers in division r... |
| subdrgint 20270 | The intersection of a none... |
| sdrgint 20271 | The intersection of a none... |
| primefld 20272 | The smallest sub division ... |
| primefld0cl 20273 | The prime field contains t... |
| primefld1cl 20274 | The prime field contains t... |
| abvfval 20277 | Value of the set of absolu... |
| isabv 20278 | Elementhood in the set of ... |
| isabvd 20279 | Properties that determine ... |
| abvrcl 20280 | Reverse closure for the ab... |
| abvfge0 20281 | An absolute value is a fun... |
| abvf 20282 | An absolute value is a fun... |
| abvcl 20283 | An absolute value is a fun... |
| abvge0 20284 | The absolute value of a nu... |
| abveq0 20285 | The value of an absolute v... |
| abvne0 20286 | The absolute value of a no... |
| abvgt0 20287 | The absolute value of a no... |
| abvmul 20288 | An absolute value distribu... |
| abvtri 20289 | An absolute value satisfie... |
| abv0 20290 | The absolute value of zero... |
| abv1z 20291 | The absolute value of one ... |
| abv1 20292 | The absolute value of one ... |
| abvneg 20293 | The absolute value of a ne... |
| abvsubtri 20294 | An absolute value satisfie... |
| abvrec 20295 | The absolute value distrib... |
| abvdiv 20296 | The absolute value distrib... |
| abvdom 20297 | Any ring with an absolute ... |
| abvres 20298 | The restriction of an abso... |
| abvtrivd 20299 | The trivial absolute value... |
| abvtriv 20300 | The trivial absolute value... |
| abvpropd 20301 | If two structures have the... |
| staffval 20306 | The functionalization of t... |
| stafval 20307 | The functionalization of t... |
| staffn 20308 | The functionalization is e... |
| issrng 20309 | The predicate "is a star r... |
| srngrhm 20310 | The involution function in... |
| srngring 20311 | A star ring is a ring. (C... |
| srngcnv 20312 | The involution function in... |
| srngf1o 20313 | The involution function in... |
| srngcl 20314 | The involution function in... |
| srngnvl 20315 | The involution function in... |
| srngadd 20316 | The involution function in... |
| srngmul 20317 | The involution function in... |
| srng1 20318 | The conjugate of the ring ... |
| srng0 20319 | The conjugate of the ring ... |
| issrngd 20320 | Properties that determine ... |
| idsrngd 20321 | A commutative ring is a st... |
| islmod 20326 | The predicate "is a left m... |
| lmodlema 20327 | Lemma for properties of a ... |
| islmodd 20328 | Properties that determine ... |
| lmodgrp 20329 | A left module is a group. ... |
| lmodring 20330 | The scalar component of a ... |
| lmodfgrp 20331 | The scalar component of a ... |
| lmodbn0 20332 | The base set of a left mod... |
| lmodacl 20333 | Closure of ring addition f... |
| lmodmcl 20334 | Closure of ring multiplica... |
| lmodsn0 20335 | The set of scalars in a le... |
| lmodvacl 20336 | Closure of vector addition... |
| lmodass 20337 | Left module vector sum is ... |
| lmodlcan 20338 | Left cancellation law for ... |
| lmodvscl 20339 | Closure of scalar product ... |
| scaffval 20340 | The scalar multiplication ... |
| scafval 20341 | The scalar multiplication ... |
| scafeq 20342 | If the scalar multiplicati... |
| scaffn 20343 | The scalar multiplication ... |
| lmodscaf 20344 | The scalar multiplication ... |
| lmodvsdi 20345 | Distributive law for scala... |
| lmodvsdir 20346 | Distributive law for scala... |
| lmodvsass 20347 | Associative law for scalar... |
| lmod0cl 20348 | The ring zero in a left mo... |
| lmod1cl 20349 | The ring unity in a left m... |
| lmodvs1 20350 | Scalar product with the ri... |
| lmod0vcl 20351 | The zero vector is a vecto... |
| lmod0vlid 20352 | Left identity law for the ... |
| lmod0vrid 20353 | Right identity law for the... |
| lmod0vid 20354 | Identity equivalent to the... |
| lmod0vs 20355 | Zero times a vector is the... |
| lmodvs0 20356 | Anything times the zero ve... |
| lmodvsmmulgdi 20357 | Distributive law for a gro... |
| lmodfopnelem1 20358 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20359 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20360 | The (functionalized) opera... |
| lcomf 20361 | A linear-combination sum i... |
| lcomfsupp 20362 | A linear-combination sum i... |
| lmodvnegcl 20363 | Closure of vector negative... |
| lmodvnegid 20364 | Addition of a vector with ... |
| lmodvneg1 20365 | Minus 1 times a vector is ... |
| lmodvsneg 20366 | Multiplication of a vector... |
| lmodvsubcl 20367 | Closure of vector subtract... |
| lmodcom 20368 | Left module vector sum is ... |
| lmodabl 20369 | A left module is an abelia... |
| lmodcmn 20370 | A left module is a commuta... |
| lmodnegadd 20371 | Distribute negation throug... |
| lmod4 20372 | Commutative/associative la... |
| lmodvsubadd 20373 | Relationship between vecto... |
| lmodvaddsub4 20374 | Vector addition/subtractio... |
| lmodvpncan 20375 | Addition/subtraction cance... |
| lmodvnpcan 20376 | Cancellation law for vecto... |
| lmodvsubval2 20377 | Value of vector subtractio... |
| lmodsubvs 20378 | Subtraction of a scalar pr... |
| lmodsubdi 20379 | Scalar multiplication dist... |
| lmodsubdir 20380 | Scalar multiplication dist... |
| lmodsubeq0 20381 | If the difference between ... |
| lmodsubid 20382 | Subtraction of a vector fr... |
| lmodvsghm 20383 | Scalar multiplication of t... |
| lmodprop2d 20384 | If two structures have the... |
| lmodpropd 20385 | If two structures have the... |
| gsumvsmul 20386 | Pull a scalar multiplicati... |
| mptscmfsupp0 20387 | A mapping to a scalar prod... |
| mptscmfsuppd 20388 | A function mapping to a sc... |
| rmodislmodlem 20389 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20390 | The right module ` R ` ind... |
| rmodislmodOLD 20391 | Obsolete version of ~ rmod... |
| lssset 20394 | The set of all (not necess... |
| islss 20395 | The predicate "is a subspa... |
| islssd 20396 | Properties that determine ... |
| lssss 20397 | A subspace is a set of vec... |
| lssel 20398 | A subspace member is a vec... |
| lss1 20399 | The set of vectors in a le... |
| lssuni 20400 | The union of all subspaces... |
| lssn0 20401 | A subspace is not empty. ... |
| 00lss 20402 | The empty structure has no... |
| lsscl 20403 | Closure property of a subs... |
| lssvsubcl 20404 | Closure of vector subtract... |
| lssvancl1 20405 | Non-closure: if one vector... |
| lssvancl2 20406 | Non-closure: if one vector... |
| lss0cl 20407 | The zero vector belongs to... |
| lsssn0 20408 | The singleton of the zero ... |
| lss0ss 20409 | The zero subspace is inclu... |
| lssle0 20410 | No subspace is smaller tha... |
| lssne0 20411 | A nonzero subspace has a n... |
| lssvneln0 20412 | A vector ` X ` which doesn... |
| lssneln0 20413 | A vector ` X ` which doesn... |
| lssssr 20414 | Conclude subspace ordering... |
| lssvacl 20415 | Closure of vector addition... |
| lssvscl 20416 | Closure of scalar product ... |
| lssvnegcl 20417 | Closure of negative vector... |
| lsssubg 20418 | All subspaces are subgroup... |
| lsssssubg 20419 | All subspaces are subgroup... |
| islss3 20420 | A linear subspace of a mod... |
| lsslmod 20421 | A submodule is a module. ... |
| lsslss 20422 | The subspaces of a subspac... |
| islss4 20423 | A linear subspace is a sub... |
| lss1d 20424 | One-dimensional subspace (... |
| lssintcl 20425 | The intersection of a none... |
| lssincl 20426 | The intersection of two su... |
| lssmre 20427 | The subspaces of a module ... |
| lssacs 20428 | Submodules are an algebrai... |
| prdsvscacl 20429 | Pointwise scalar multiplic... |
| prdslmodd 20430 | The product of a family of... |
| pwslmod 20431 | A structure power of a lef... |
| lspfval 20434 | The span function for a le... |
| lspf 20435 | The span operator on a lef... |
| lspval 20436 | The span of a set of vecto... |
| lspcl 20437 | The span of a set of vecto... |
| lspsncl 20438 | The span of a singleton is... |
| lspprcl 20439 | The span of a pair is a su... |
| lsptpcl 20440 | The span of an unordered t... |
| lspsnsubg 20441 | The span of a singleton is... |
| 00lsp 20442 | ~ fvco4i lemma for linear ... |
| lspid 20443 | The span of a subspace is ... |
| lspssv 20444 | A span is a set of vectors... |
| lspss 20445 | Span preserves subset orde... |
| lspssid 20446 | A set of vectors is a subs... |
| lspidm 20447 | The span of a set of vecto... |
| lspun 20448 | The span of union is the s... |
| lspssp 20449 | If a set of vectors is a s... |
| mrclsp 20450 | Moore closure generalizes ... |
| lspsnss 20451 | The span of the singleton ... |
| lspsnel3 20452 | A member of the span of th... |
| lspprss 20453 | The span of a pair of vect... |
| lspsnid 20454 | A vector belongs to the sp... |
| lspsnel6 20455 | Relationship between a vec... |
| lspsnel5 20456 | Relationship between a vec... |
| lspsnel5a 20457 | Relationship between a vec... |
| lspprid1 20458 | A member of a pair of vect... |
| lspprid2 20459 | A member of a pair of vect... |
| lspprvacl 20460 | The sum of two vectors bel... |
| lssats2 20461 | A way to express atomistic... |
| lspsneli 20462 | A scalar product with a ve... |
| lspsn 20463 | Span of the singleton of a... |
| lspsnel 20464 | Member of span of the sing... |
| lspsnvsi 20465 | Span of a scalar product o... |
| lspsnss2 20466 | Comparable spans of single... |
| lspsnneg 20467 | Negation does not change t... |
| lspsnsub 20468 | Swapping subtraction order... |
| lspsn0 20469 | Span of the singleton of t... |
| lsp0 20470 | Span of the empty set. (C... |
| lspuni0 20471 | Union of the span of the e... |
| lspun0 20472 | The span of a union with t... |
| lspsneq0 20473 | Span of the singleton is t... |
| lspsneq0b 20474 | Equal singleton spans impl... |
| lmodindp1 20475 | Two independent (non-colin... |
| lsslsp 20476 | Spans in submodules corres... |
| lss0v 20477 | The zero vector in a submo... |
| lsspropd 20478 | If two structures have the... |
| lsppropd 20479 | If two structures have the... |
| reldmlmhm 20486 | Lemma for module homomorph... |
| lmimfn 20487 | Lemma for module isomorphi... |
| islmhm 20488 | Property of being a homomo... |
| islmhm3 20489 | Property of a module homom... |
| lmhmlem 20490 | Non-quantified consequence... |
| lmhmsca 20491 | A homomorphism of left mod... |
| lmghm 20492 | A homomorphism of left mod... |
| lmhmlmod2 20493 | A homomorphism of left mod... |
| lmhmlmod1 20494 | A homomorphism of left mod... |
| lmhmf 20495 | A homomorphism of left mod... |
| lmhmlin 20496 | A homomorphism of left mod... |
| lmodvsinv 20497 | Multiplication of a vector... |
| lmodvsinv2 20498 | Multiplying a negated vect... |
| islmhm2 20499 | A one-equation proof of li... |
| islmhmd 20500 | Deduction for a module hom... |
| 0lmhm 20501 | The constant zero linear f... |
| idlmhm 20502 | The identity function on a... |
| invlmhm 20503 | The negative function on a... |
| lmhmco 20504 | The composition of two mod... |
| lmhmplusg 20505 | The pointwise sum of two l... |
| lmhmvsca 20506 | The pointwise scalar produ... |
| lmhmf1o 20507 | A bijective module homomor... |
| lmhmima 20508 | The image of a subspace un... |
| lmhmpreima 20509 | The inverse image of a sub... |
| lmhmlsp 20510 | Homomorphisms preserve spa... |
| lmhmrnlss 20511 | The range of a homomorphis... |
| lmhmkerlss 20512 | The kernel of a homomorphi... |
| reslmhm 20513 | Restriction of a homomorph... |
| reslmhm2 20514 | Expansion of the codomain ... |
| reslmhm2b 20515 | Expansion of the codomain ... |
| lmhmeql 20516 | The equalizer of two modul... |
| lspextmo 20517 | A linear function is compl... |
| pwsdiaglmhm 20518 | Diagonal homomorphism into... |
| pwssplit0 20519 | Splitting for structure po... |
| pwssplit1 20520 | Splitting for structure po... |
| pwssplit2 20521 | Splitting for structure po... |
| pwssplit3 20522 | Splitting for structure po... |
| islmim 20523 | An isomorphism of left mod... |
| lmimf1o 20524 | An isomorphism of left mod... |
| lmimlmhm 20525 | An isomorphism of modules ... |
| lmimgim 20526 | An isomorphism of modules ... |
| islmim2 20527 | An isomorphism of left mod... |
| lmimcnv 20528 | The converse of a bijectiv... |
| brlmic 20529 | The relation "is isomorphi... |
| brlmici 20530 | Prove isomorphic by an exp... |
| lmiclcl 20531 | Isomorphism implies the le... |
| lmicrcl 20532 | Isomorphism implies the ri... |
| lmicsym 20533 | Module isomorphism is symm... |
| lmhmpropd 20534 | Module homomorphism depend... |
| islbs 20537 | The predicate " ` B ` is a... |
| lbsss 20538 | A basis is a set of vector... |
| lbsel 20539 | An element of a basis is a... |
| lbssp 20540 | The span of a basis is the... |
| lbsind 20541 | A basis is linearly indepe... |
| lbsind2 20542 | A basis is linearly indepe... |
| lbspss 20543 | No proper subset of a basi... |
| lsmcl 20544 | The sum of two subspaces i... |
| lsmspsn 20545 | Member of subspace sum of ... |
| lsmelval2 20546 | Subspace sum membership in... |
| lsmsp 20547 | Subspace sum in terms of s... |
| lsmsp2 20548 | Subspace sum of spans of s... |
| lsmssspx 20549 | Subspace sum (in its exten... |
| lsmpr 20550 | The span of a pair of vect... |
| lsppreli 20551 | A vector expressed as a su... |
| lsmelpr 20552 | Two ways to say that a vec... |
| lsppr0 20553 | The span of a vector paire... |
| lsppr 20554 | Span of a pair of vectors.... |
| lspprel 20555 | Member of the span of a pa... |
| lspprabs 20556 | Absorption of vector sum i... |
| lspvadd 20557 | The span of a vector sum i... |
| lspsntri 20558 | Triangle-type inequality f... |
| lspsntrim 20559 | Triangle-type inequality f... |
| lbspropd 20560 | If two structures have the... |
| pj1lmhm 20561 | The left projection functi... |
| pj1lmhm2 20562 | The left projection functi... |
| islvec 20565 | The predicate "is a left v... |
| lvecdrng 20566 | The set of scalars of a le... |
| lveclmod 20567 | A left vector space is a l... |
| lsslvec 20568 | A vector subspace is a vec... |
| lvecvs0or 20569 | If a scalar product is zer... |
| lvecvsn0 20570 | A scalar product is nonzer... |
| lssvs0or 20571 | If a scalar product belong... |
| lvecvscan 20572 | Cancellation law for scala... |
| lvecvscan2 20573 | Cancellation law for scala... |
| lvecinv 20574 | Invert coefficient of scal... |
| lspsnvs 20575 | A nonzero scalar product d... |
| lspsneleq 20576 | Membership relation that i... |
| lspsncmp 20577 | Comparable spans of nonzer... |
| lspsnne1 20578 | Two ways to express that v... |
| lspsnne2 20579 | Two ways to express that v... |
| lspsnnecom 20580 | Swap two vectors with diff... |
| lspabs2 20581 | Absorption law for span of... |
| lspabs3 20582 | Absorption law for span of... |
| lspsneq 20583 | Equal spans of singletons ... |
| lspsneu 20584 | Nonzero vectors with equal... |
| lspsnel4 20585 | A member of the span of th... |
| lspdisj 20586 | The span of a vector not i... |
| lspdisjb 20587 | A nonzero vector is not in... |
| lspdisj2 20588 | Unequal spans are disjoint... |
| lspfixed 20589 | Show membership in the spa... |
| lspexch 20590 | Exchange property for span... |
| lspexchn1 20591 | Exchange property for span... |
| lspexchn2 20592 | Exchange property for span... |
| lspindpi 20593 | Partial independence prope... |
| lspindp1 20594 | Alternate way to say 3 vec... |
| lspindp2l 20595 | Alternate way to say 3 vec... |
| lspindp2 20596 | Alternate way to say 3 vec... |
| lspindp3 20597 | Independence of 2 vectors ... |
| lspindp4 20598 | (Partial) independence of ... |
| lvecindp 20599 | Compute the ` X ` coeffici... |
| lvecindp2 20600 | Sums of independent vector... |
| lspsnsubn0 20601 | Unequal singleton spans im... |
| lsmcv 20602 | Subspace sum has the cover... |
| lspsolvlem 20603 | Lemma for ~ lspsolv . (Co... |
| lspsolv 20604 | If ` X ` is in the span of... |
| lssacsex 20605 | In a vector space, subspac... |
| lspsnat 20606 | There is no subspace stric... |
| lspsncv0 20607 | The span of a singleton co... |
| lsppratlem1 20608 | Lemma for ~ lspprat . Let... |
| lsppratlem2 20609 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 20610 | Lemma for ~ lspprat . In ... |
| lsppratlem4 20611 | Lemma for ~ lspprat . In ... |
| lsppratlem5 20612 | Lemma for ~ lspprat . Com... |
| lsppratlem6 20613 | Lemma for ~ lspprat . Neg... |
| lspprat 20614 | A proper subspace of the s... |
| islbs2 20615 | An equivalent formulation ... |
| islbs3 20616 | An equivalent formulation ... |
| lbsacsbs 20617 | Being a basis in a vector ... |
| lvecdim 20618 | The dimension theorem for ... |
| lbsextlem1 20619 | Lemma for ~ lbsext . The ... |
| lbsextlem2 20620 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 20621 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 20622 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 20623 | For any linearly independe... |
| lbsext 20624 | For any linearly independe... |
| lbsexg 20625 | Every vector space has a b... |
| lbsex 20626 | Every vector space has a b... |
| lvecprop2d 20627 | If two structures have the... |
| lvecpropd 20628 | If two structures have the... |
| sraval 20637 | Lemma for ~ srabase throug... |
| sralem 20638 | Lemma for ~ srabase and si... |
| sralemOLD 20639 | Obsolete version of ~ sral... |
| srabase 20640 | Base set of a subring alge... |
| srabaseOLD 20641 | Obsolete proof of ~ srabas... |
| sraaddg 20642 | Additive operation of a su... |
| sraaddgOLD 20643 | Obsolete proof of ~ sraadd... |
| sramulr 20644 | Multiplicative operation o... |
| sramulrOLD 20645 | Obsolete proof of ~ sramul... |
| srasca 20646 | The set of scalars of a su... |
| srascaOLD 20647 | Obsolete proof of ~ srasca... |
| sravsca 20648 | The scalar product operati... |
| sravscaOLD 20649 | Obsolete proof of ~ sravsc... |
| sraip 20650 | The inner product operatio... |
| sratset 20651 | Topology component of a su... |
| sratsetOLD 20652 | Obsolete proof of ~ sratse... |
| sratopn 20653 | Topology component of a su... |
| srads 20654 | Distance function of a sub... |
| sradsOLD 20655 | Obsolete proof of ~ srads ... |
| sralmod 20656 | The subring algebra is a l... |
| sralmod0 20657 | The subring module inherit... |
| issubrngd2 20658 | Prove a subring by closure... |
| rlmfn 20659 | ` ringLMod ` is a function... |
| rlmval 20660 | Value of the ring module. ... |
| lidlval 20661 | Value of the set of ring i... |
| rspval 20662 | Value of the ring span fun... |
| rlmval2 20663 | Value of the ring module e... |
| rlmbas 20664 | Base set of the ring modul... |
| rlmplusg 20665 | Vector addition in the rin... |
| rlm0 20666 | Zero vector in the ring mo... |
| rlmsub 20667 | Subtraction in the ring mo... |
| rlmmulr 20668 | Ring multiplication in the... |
| rlmsca 20669 | Scalars in the ring module... |
| rlmsca2 20670 | Scalars in the ring module... |
| rlmvsca 20671 | Scalar multiplication in t... |
| rlmtopn 20672 | Topology component of the ... |
| rlmds 20673 | Metric component of the ri... |
| rlmlmod 20674 | The ring module is a modul... |
| rlmlvec 20675 | The ring module over a div... |
| rlmlsm 20676 | Subgroup sum of the ring m... |
| rlmvneg 20677 | Vector negation in the rin... |
| rlmscaf 20678 | Functionalized scalar mult... |
| ixpsnbasval 20679 | The value of an infinite C... |
| lidlss 20680 | An ideal is a subset of th... |
| islidl 20681 | Predicate of being a (left... |
| lidl0cl 20682 | An ideal contains 0. (Con... |
| lidlacl 20683 | An ideal is closed under a... |
| lidlnegcl 20684 | An ideal contains negative... |
| lidlsubg 20685 | An ideal is a subgroup of ... |
| lidlsubcl 20686 | An ideal is closed under s... |
| lidlmcl 20687 | An ideal is closed under l... |
| lidl1el 20688 | An ideal contains 1 iff it... |
| lidl0 20689 | Every ring contains a zero... |
| lidl1 20690 | Every ring contains a unit... |
| lidlacs 20691 | The ideal system is an alg... |
| rspcl 20692 | The span of a set of ring ... |
| rspssid 20693 | The span of a set of ring ... |
| rsp1 20694 | The span of the identity e... |
| rsp0 20695 | The span of the zero eleme... |
| rspssp 20696 | The ideal span of a set of... |
| mrcrsp 20697 | Moore closure generalizes ... |
| lidlnz 20698 | A nonzero ideal contains a... |
| drngnidl 20699 | A division ring has only t... |
| lidlrsppropd 20700 | The left ideals and ring s... |
| 2idlval 20703 | Definition of a two-sided ... |
| 2idlcpbl 20704 | The coset equivalence rela... |
| qus1 20705 | The multiplicative identit... |
| qusring 20706 | If ` S ` is a two-sided id... |
| qusrhm 20707 | If ` S ` is a two-sided id... |
| crngridl 20708 | In a commutative ring, the... |
| crng2idl 20709 | In a commutative ring, a t... |
| quscrng 20710 | The quotient of a commutat... |
| lpival 20715 | Value of the set of princi... |
| islpidl 20716 | Property of being a princi... |
| lpi0 20717 | The zero ideal is always p... |
| lpi1 20718 | The unit ideal is always p... |
| islpir 20719 | Principal ideal rings are ... |
| lpiss 20720 | Principal ideals are a sub... |
| islpir2 20721 | Principal ideal rings are ... |
| lpirring 20722 | Principal ideal rings are ... |
| drnglpir 20723 | Division rings are princip... |
| rspsn 20724 | Membership in principal id... |
| lidldvgen 20725 | An element generates an id... |
| lpigen 20726 | An ideal is principal iff ... |
| isnzr 20729 | Property of a nonzero ring... |
| nzrnz 20730 | One and zero are different... |
| nzrring 20731 | A nonzero ring is a ring. ... |
| drngnzr 20732 | All division rings are non... |
| isnzr2 20733 | Equivalent characterizatio... |
| isnzr2hash 20734 | Equivalent characterizatio... |
| opprnzr 20735 | The opposite of a nonzero ... |
| ringelnzr 20736 | A ring is nonzero if it ha... |
| nzrunit 20737 | A unit is nonzero in any n... |
| subrgnzr 20738 | A subring of a nonzero rin... |
| 0ringnnzr 20739 | A ring is a zero ring iff ... |
| 0ring 20740 | If a ring has only one ele... |
| 0ring01eq 20741 | In a ring with only one el... |
| 01eq0ring 20742 | If the zero and the identi... |
| 0ring01eqbi 20743 | In a unital ring the zero ... |
| rng1nnzr 20744 | The (smallest) structure r... |
| ring1zr 20745 | The only (unital) ring wit... |
| rngen1zr 20746 | The only (unital) ring wit... |
| ringen1zr 20747 | The only unital ring with ... |
| rng1nfld 20748 | The zero ring is not a fie... |
| rrgval 20757 | Value of the set or left-r... |
| isrrg 20758 | Membership in the set of l... |
| rrgeq0i 20759 | Property of a left-regular... |
| rrgeq0 20760 | Left-multiplication by a l... |
| rrgsupp 20761 | Left multiplication by a l... |
| rrgss 20762 | Left-regular elements are ... |
| unitrrg 20763 | Units are regular elements... |
| isdomn 20764 | Expand definition of a dom... |
| domnnzr 20765 | A domain is a nonzero ring... |
| domnring 20766 | A domain is a ring. (Cont... |
| domneq0 20767 | In a domain, a product is ... |
| domnmuln0 20768 | In a domain, a product of ... |
| isdomn2 20769 | A ring is a domain iff all... |
| domnrrg 20770 | In a domain, any nonzero e... |
| opprdomn 20771 | The opposite of a domain i... |
| abvn0b 20772 | Another characterization o... |
| drngdomn 20773 | A division ring is a domai... |
| isidom 20774 | An integral domain is a co... |
| fldidom 20775 | A field is an integral dom... |
| fldidomOLD 20776 | Obsolete version of ~ fldi... |
| fidomndrnglem 20777 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20778 | A finite domain is a divis... |
| fiidomfld 20779 | A finite integral domain i... |
| cnfldstr 20798 | The field of complex numbe... |
| cnfldex 20799 | The field of complex numbe... |
| cnfldbas 20800 | The base set of the field ... |
| cnfldadd 20801 | The addition operation of ... |
| cnfldmul 20802 | The multiplication operati... |
| cnfldcj 20803 | The conjugation operation ... |
| cnfldtset 20804 | The topology component of ... |
| cnfldle 20805 | The ordering of the field ... |
| cnfldds 20806 | The metric of the field of... |
| cnfldunif 20807 | The uniform structure comp... |
| cnfldfun 20808 | The field of complex numbe... |
| cnfldfunALT 20809 | The field of complex numbe... |
| cnfldfunALTOLD 20810 | Obsolete proof of ~ cnfldf... |
| xrsstr 20811 | The extended real structur... |
| xrsex 20812 | The extended real structur... |
| xrsbas 20813 | The base set of the extend... |
| xrsadd 20814 | The addition operation of ... |
| xrsmul 20815 | The multiplication operati... |
| xrstset 20816 | The topology component of ... |
| xrsle 20817 | The ordering of the extend... |
| cncrng 20818 | The complex numbers form a... |
| cnring 20819 | The complex numbers form a... |
| xrsmcmn 20820 | The "multiplicative group"... |
| cnfld0 20821 | Zero is the zero element o... |
| cnfld1 20822 | One is the unity element o... |
| cnfldneg 20823 | The additive inverse in th... |
| cnfldplusf 20824 | The functionalized additio... |
| cnfldsub 20825 | The subtraction operator i... |
| cndrng 20826 | The complex numbers form a... |
| cnflddiv 20827 | The division operation in ... |
| cnfldinv 20828 | The multiplicative inverse... |
| cnfldmulg 20829 | The group multiple functio... |
| cnfldexp 20830 | The exponentiation operato... |
| cnsrng 20831 | The complex numbers form a... |
| xrsmgm 20832 | The "additive group" of th... |
| xrsnsgrp 20833 | The "additive group" of th... |
| xrsmgmdifsgrp 20834 | The "additive group" of th... |
| xrs1mnd 20835 | The extended real numbers,... |
| xrs10 20836 | The zero of the extended r... |
| xrs1cmn 20837 | The extended real numbers ... |
| xrge0subm 20838 | The nonnegative extended r... |
| xrge0cmn 20839 | The nonnegative extended r... |
| xrsds 20840 | The metric of the extended... |
| xrsdsval 20841 | The metric of the extended... |
| xrsdsreval 20842 | The metric of the extended... |
| xrsdsreclblem 20843 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 20844 | The metric of the extended... |
| cnsubmlem 20845 | Lemma for ~ nn0subm and fr... |
| cnsubglem 20846 | Lemma for ~ resubdrg and f... |
| cnsubrglem 20847 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 20848 | Lemma for ~ resubdrg and f... |
| qsubdrg 20849 | The rational numbers form ... |
| zsubrg 20850 | The integers form a subrin... |
| gzsubrg 20851 | The gaussian integers form... |
| nn0subm 20852 | The nonnegative integers f... |
| rege0subm 20853 | The nonnegative reals form... |
| absabv 20854 | The regular absolute value... |
| zsssubrg 20855 | The integers are a subset ... |
| qsssubdrg 20856 | The rational numbers are a... |
| cnsubrg 20857 | There are no subrings of t... |
| cnmgpabl 20858 | The unit group of the comp... |
| cnmgpid 20859 | The group identity element... |
| cnmsubglem 20860 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 20861 | The positive reals form a ... |
| gzrngunitlem 20862 | Lemma for ~ gzrngunit . (... |
| gzrngunit 20863 | The units on ` ZZ [ _i ] `... |
| gsumfsum 20864 | Relate a group sum on ` CC... |
| regsumfsum 20865 | Relate a group sum on ` ( ... |
| expmhm 20866 | Exponentiation is a monoid... |
| nn0srg 20867 | The nonnegative integers f... |
| rge0srg 20868 | The nonnegative real numbe... |
| zringcrng 20871 | The ring of integers is a ... |
| zringring 20872 | The ring of integers is a ... |
| zringabl 20873 | The ring of integers is an... |
| zringgrp 20874 | The ring of integers is an... |
| zringbas 20875 | The integers are the base ... |
| zringplusg 20876 | The addition operation of ... |
| zringmulg 20877 | The multiplication (group ... |
| zringmulr 20878 | The multiplication operati... |
| zring0 20879 | The zero element of the ri... |
| zring1 20880 | The unity element of the r... |
| zringnzr 20881 | The ring of integers is a ... |
| dvdsrzring 20882 | Ring divisibility in the r... |
| zringlpirlem1 20883 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 20884 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 20885 | Lemma for ~ zringlpir . A... |
| zringinvg 20886 | The additive inverse of an... |
| zringunit 20887 | The units of ` ZZ ` are th... |
| zringlpir 20888 | The integers are a princip... |
| zringndrg 20889 | The integers are not a div... |
| zringcyg 20890 | The integers are a cyclic ... |
| zringsubgval 20891 | Subtraction in the ring of... |
| zringmpg 20892 | The multiplication group o... |
| prmirredlem 20893 | A positive integer is irre... |
| dfprm2 20894 | The positive irreducible e... |
| prmirred 20895 | The irreducible elements o... |
| expghm 20896 | Exponentiation is a group ... |
| mulgghm2 20897 | The powers of a group elem... |
| mulgrhm 20898 | The powers of the element ... |
| mulgrhm2 20899 | The powers of the element ... |
| zrhval 20908 | Define the unique homomorp... |
| zrhval2 20909 | Alternate value of the ` Z... |
| zrhmulg 20910 | Value of the ` ZRHom ` hom... |
| zrhrhmb 20911 | The ` ZRHom ` homomorphism... |
| zrhrhm 20912 | The ` ZRHom ` homomorphism... |
| zrh1 20913 | Interpretation of 1 in a r... |
| zrh0 20914 | Interpretation of 0 in a r... |
| zrhpropd 20915 | The ` ZZ ` ring homomorphi... |
| zlmval 20916 | Augment an abelian group w... |
| zlmlem 20917 | Lemma for ~ zlmbas and ~ z... |
| zlmlemOLD 20918 | Obsolete version of ~ zlml... |
| zlmbas 20919 | Base set of a ` ZZ ` -modu... |
| zlmbasOLD 20920 | Obsolete version of ~ zlmb... |
| zlmplusg 20921 | Group operation of a ` ZZ ... |
| zlmplusgOLD 20922 | Obsolete version of ~ zlmb... |
| zlmmulr 20923 | Ring operation of a ` ZZ `... |
| zlmmulrOLD 20924 | Obsolete version of ~ zlmb... |
| zlmsca 20925 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 20926 | Scalar multiplication oper... |
| zlmlmod 20927 | The ` ZZ ` -module operati... |
| chrval 20928 | Definition substitution of... |
| chrcl 20929 | Closure of the characteris... |
| chrid 20930 | The canonical ` ZZ ` ring ... |
| chrdvds 20931 | The ` ZZ ` ring homomorphi... |
| chrcong 20932 | If two integers are congru... |
| chrnzr 20933 | Nonzero rings are precisel... |
| chrrhm 20934 | The characteristic restric... |
| domnchr 20935 | The characteristic of a do... |
| znlidl 20936 | The set ` n ZZ ` is an ide... |
| zncrng2 20937 | The value of the ` Z/nZ ` ... |
| znval 20938 | The value of the ` Z/nZ ` ... |
| znle 20939 | The value of the ` Z/nZ ` ... |
| znval2 20940 | Self-referential expressio... |
| znbaslem 20941 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 20942 | Obsolete version of ~ znba... |
| znbas2 20943 | The base set of ` Z/nZ ` i... |
| znbas2OLD 20944 | Obsolete version of ~ znba... |
| znadd 20945 | The additive structure of ... |
| znaddOLD 20946 | Obsolete version of ~ znad... |
| znmul 20947 | The multiplicative structu... |
| znmulOLD 20948 | Obsolete version of ~ znad... |
| znzrh 20949 | The ` ZZ ` ring homomorphi... |
| znbas 20950 | The base set of ` Z/nZ ` s... |
| zncrng 20951 | ` Z/nZ ` is a commutative ... |
| znzrh2 20952 | The ` ZZ ` ring homomorphi... |
| znzrhval 20953 | The ` ZZ ` ring homomorphi... |
| znzrhfo 20954 | The ` ZZ ` ring homomorphi... |
| zncyg 20955 | The group ` ZZ / n ZZ ` is... |
| zndvds 20956 | Express equality of equiva... |
| zndvds0 20957 | Special case of ~ zndvds w... |
| znf1o 20958 | The function ` F ` enumera... |
| zzngim 20959 | The ` ZZ ` ring homomorphi... |
| znle2 20960 | The ordering of the ` Z/nZ... |
| znleval 20961 | The ordering of the ` Z/nZ... |
| znleval2 20962 | The ordering of the ` Z/nZ... |
| zntoslem 20963 | Lemma for ~ zntos . (Cont... |
| zntos 20964 | The ` Z/nZ ` structure is ... |
| znhash 20965 | The ` Z/nZ ` structure has... |
| znfi 20966 | The ` Z/nZ ` structure is ... |
| znfld 20967 | The ` Z/nZ ` structure is ... |
| znidomb 20968 | The ` Z/nZ ` structure is ... |
| znchr 20969 | Cyclic rings are defined b... |
| znunit 20970 | The units of ` Z/nZ ` are ... |
| znunithash 20971 | The size of the unit group... |
| znrrg 20972 | The regular elements of ` ... |
| cygznlem1 20973 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 20974 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 20975 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 20976 | A cyclic group with ` n ` ... |
| cygzn 20977 | A cyclic group with ` n ` ... |
| cygth 20978 | The "fundamental theorem o... |
| cyggic 20979 | Cyclic groups are isomorph... |
| frgpcyg 20980 | A free group is cyclic iff... |
| cnmsgnsubg 20981 | The signs form a multiplic... |
| cnmsgnbas 20982 | The base set of the sign s... |
| cnmsgngrp 20983 | The group of signs under m... |
| psgnghm 20984 | The sign is a homomorphism... |
| psgnghm2 20985 | The sign is a homomorphism... |
| psgninv 20986 | The sign of a permutation ... |
| psgnco 20987 | Multiplicativity of the pe... |
| zrhpsgnmhm 20988 | Embedding of permutation s... |
| zrhpsgninv 20989 | The embedded sign of a per... |
| evpmss 20990 | Even permutations are perm... |
| psgnevpmb 20991 | A class is an even permuta... |
| psgnodpm 20992 | A permutation which is odd... |
| psgnevpm 20993 | A permutation which is eve... |
| psgnodpmr 20994 | If a permutation has sign ... |
| zrhpsgnevpm 20995 | The sign of an even permut... |
| zrhpsgnodpm 20996 | The sign of an odd permuta... |
| cofipsgn 20997 | Composition of any class `... |
| zrhpsgnelbas 20998 | Embedding of permutation s... |
| zrhcopsgnelbas 20999 | Embedding of permutation s... |
| evpmodpmf1o 21000 | The function for performin... |
| pmtrodpm 21001 | A transposition is an odd ... |
| psgnfix1 21002 | A permutation of a finite ... |
| psgnfix2 21003 | A permutation of a finite ... |
| psgndiflemB 21004 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 21005 | Lemma 2 for ~ psgndif . (... |
| psgndif 21006 | Embedding of permutation s... |
| copsgndif 21007 | Embedding of permutation s... |
| rebase 21010 | The base of the field of r... |
| remulg 21011 | The multiplication (group ... |
| resubdrg 21012 | The real numbers form a di... |
| resubgval 21013 | Subtraction in the field o... |
| replusg 21014 | The addition operation of ... |
| remulr 21015 | The multiplication operati... |
| re0g 21016 | The zero element of the fi... |
| re1r 21017 | The unity element of the f... |
| rele2 21018 | The ordering relation of t... |
| relt 21019 | The ordering relation of t... |
| reds 21020 | The distance of the field ... |
| redvr 21021 | The division operation of ... |
| retos 21022 | The real numbers are a tot... |
| refld 21023 | The real numbers form a fi... |
| refldcj 21024 | The conjugation operation ... |
| resrng 21025 | The real numbers form a st... |
| regsumsupp 21026 | The group sum over the rea... |
| rzgrp 21027 | The quotient group ` RR / ... |
| isphl 21032 | The predicate "is a genera... |
| phllvec 21033 | A pre-Hilbert space is a l... |
| phllmod 21034 | A pre-Hilbert space is a l... |
| phlsrng 21035 | The scalar ring of a pre-H... |
| phllmhm 21036 | The inner product of a pre... |
| ipcl 21037 | Closure of the inner produ... |
| ipcj 21038 | Conjugate of an inner prod... |
| iporthcom 21039 | Orthogonality (meaning inn... |
| ip0l 21040 | Inner product with a zero ... |
| ip0r 21041 | Inner product with a zero ... |
| ipeq0 21042 | The inner product of a vec... |
| ipdir 21043 | Distributive law for inner... |
| ipdi 21044 | Distributive law for inner... |
| ip2di 21045 | Distributive law for inner... |
| ipsubdir 21046 | Distributive law for inner... |
| ipsubdi 21047 | Distributive law for inner... |
| ip2subdi 21048 | Distributive law for inner... |
| ipass 21049 | Associative law for inner ... |
| ipassr 21050 | "Associative" law for seco... |
| ipassr2 21051 | "Associative" law for inne... |
| ipffval 21052 | The inner product operatio... |
| ipfval 21053 | The inner product operatio... |
| ipfeq 21054 | If the inner product opera... |
| ipffn 21055 | The inner product operatio... |
| phlipf 21056 | The inner product operatio... |
| ip2eq 21057 | Two vectors are equal iff ... |
| isphld 21058 | Properties that determine ... |
| phlpropd 21059 | If two structures have the... |
| ssipeq 21060 | The inner product on a sub... |
| phssipval 21061 | The inner product on a sub... |
| phssip 21062 | The inner product (as a fu... |
| phlssphl 21063 | A subspace of an inner pro... |
| ocvfval 21070 | The orthocomplement operat... |
| ocvval 21071 | Value of the orthocompleme... |
| elocv 21072 | Elementhood in the orthoco... |
| ocvi 21073 | Property of a member of th... |
| ocvss 21074 | The orthocomplement of a s... |
| ocvocv 21075 | A set is contained in its ... |
| ocvlss 21076 | The orthocomplement of a s... |
| ocv2ss 21077 | Orthocomplements reverse s... |
| ocvin 21078 | An orthocomplement has tri... |
| ocvsscon 21079 | Two ways to say that ` S `... |
| ocvlsp 21080 | The orthocomplement of a l... |
| ocv0 21081 | The orthocomplement of the... |
| ocvz 21082 | The orthocomplement of the... |
| ocv1 21083 | The orthocomplement of the... |
| unocv 21084 | The orthocomplement of a u... |
| iunocv 21085 | The orthocomplement of an ... |
| cssval 21086 | The set of closed subspace... |
| iscss 21087 | The predicate "is a closed... |
| cssi 21088 | Property of a closed subsp... |
| cssss 21089 | A closed subspace is a sub... |
| iscss2 21090 | It is sufficient to prove ... |
| ocvcss 21091 | The orthocomplement of any... |
| cssincl 21092 | The zero subspace is a clo... |
| css0 21093 | The zero subspace is a clo... |
| css1 21094 | The whole space is a close... |
| csslss 21095 | A closed subspace of a pre... |
| lsmcss 21096 | A subset of a pre-Hilbert ... |
| cssmre 21097 | The closed subspaces of a ... |
| mrccss 21098 | The Moore closure correspo... |
| thlval 21099 | Value of the Hilbert latti... |
| thlbas 21100 | Base set of the Hilbert la... |
| thlbasOLD 21101 | Obsolete proof of ~ thlbas... |
| thlle 21102 | Ordering on the Hilbert la... |
| thlleOLD 21103 | Obsolete proof of ~ thlle ... |
| thlleval 21104 | Ordering on the Hilbert la... |
| thloc 21105 | Orthocomplement on the Hil... |
| pjfval 21112 | The value of the projectio... |
| pjdm 21113 | A subspace is in the domai... |
| pjpm 21114 | The projection map is a pa... |
| pjfval2 21115 | Value of the projection ma... |
| pjval 21116 | Value of the projection ma... |
| pjdm2 21117 | A subspace is in the domai... |
| pjff 21118 | A projection is a linear o... |
| pjf 21119 | A projection is a function... |
| pjf2 21120 | A projection is a function... |
| pjfo 21121 | A projection is a surjecti... |
| pjcss 21122 | A projection subspace is a... |
| ocvpj 21123 | The orthocomplement of a p... |
| ishil 21124 | The predicate "is a Hilber... |
| ishil2 21125 | The predicate "is a Hilber... |
| isobs 21126 | The predicate "is an ortho... |
| obsip 21127 | The inner product of two e... |
| obsipid 21128 | A basis element has length... |
| obsrcl 21129 | Reverse closure for an ort... |
| obsss 21130 | An orthonormal basis is a ... |
| obsne0 21131 | A basis element is nonzero... |
| obsocv 21132 | An orthonormal basis has t... |
| obs2ocv 21133 | The double orthocomplement... |
| obselocv 21134 | A basis element is in the ... |
| obs2ss 21135 | A basis has no proper subs... |
| obslbs 21136 | An orthogonal basis is a l... |
| reldmdsmm 21139 | The direct sum is a well-b... |
| dsmmval 21140 | Value of the module direct... |
| dsmmbase 21141 | Base set of the module dir... |
| dsmmval2 21142 | Self-referential definitio... |
| dsmmbas2 21143 | Base set of the direct sum... |
| dsmmfi 21144 | For finite products, the d... |
| dsmmelbas 21145 | Membership in the finitely... |
| dsmm0cl 21146 | The all-zero vector is con... |
| dsmmacl 21147 | The finite hull is closed ... |
| prdsinvgd2 21148 | Negation of a single coord... |
| dsmmsubg 21149 | The finite hull of a produ... |
| dsmmlss 21150 | The finite hull of a produ... |
| dsmmlmod 21151 | The direct sum of a family... |
| frlmval 21154 | Value of the "free module"... |
| frlmlmod 21155 | The free module is a modul... |
| frlmpws 21156 | The free module as a restr... |
| frlmlss 21157 | The base set of the free m... |
| frlmpwsfi 21158 | The finite free module is ... |
| frlmsca 21159 | The ring of scalars of a f... |
| frlm0 21160 | Zero in a free module (rin... |
| frlmbas 21161 | Base set of the free modul... |
| frlmelbas 21162 | Membership in the base set... |
| frlmrcl 21163 | If a free module is inhabi... |
| frlmbasfsupp 21164 | Elements of the free modul... |
| frlmbasmap 21165 | Elements of the free modul... |
| frlmbasf 21166 | Elements of the free modul... |
| frlmlvec 21167 | The free module over a div... |
| frlmfibas 21168 | The base set of the finite... |
| elfrlmbasn0 21169 | If the dimension of a free... |
| frlmplusgval 21170 | Addition in a free module.... |
| frlmsubgval 21171 | Subtraction in a free modu... |
| frlmvscafval 21172 | Scalar multiplication in a... |
| frlmvplusgvalc 21173 | Coordinates of a sum with ... |
| frlmvscaval 21174 | Coordinates of a scalar mu... |
| frlmplusgvalb 21175 | Addition in a free module ... |
| frlmvscavalb 21176 | Scalar multiplication in a... |
| frlmvplusgscavalb 21177 | Addition combined with sca... |
| frlmgsum 21178 | Finite commutative sums in... |
| frlmsplit2 21179 | Restriction is homomorphic... |
| frlmsslss 21180 | A subset of a free module ... |
| frlmsslss2 21181 | A subset of a free module ... |
| frlmbas3 21182 | An element of the base set... |
| mpofrlmd 21183 | Elements of the free modul... |
| frlmip 21184 | The inner product of a fre... |
| frlmipval 21185 | The inner product of a fre... |
| frlmphllem 21186 | Lemma for ~ frlmphl . (Co... |
| frlmphl 21187 | Conditions for a free modu... |
| uvcfval 21190 | Value of the unit-vector g... |
| uvcval 21191 | Value of a single unit vec... |
| uvcvval 21192 | Value of a unit vector coo... |
| uvcvvcl 21193 | A coordinate of a unit vec... |
| uvcvvcl2 21194 | A unit vector coordinate i... |
| uvcvv1 21195 | The unit vector is one at ... |
| uvcvv0 21196 | The unit vector is zero at... |
| uvcff 21197 | Domain and codomain of the... |
| uvcf1 21198 | In a nonzero ring, each un... |
| uvcresum 21199 | Any element of a free modu... |
| frlmssuvc1 21200 | A scalar multiple of a uni... |
| frlmssuvc2 21201 | A nonzero scalar multiple ... |
| frlmsslsp 21202 | A subset of a free module ... |
| frlmlbs 21203 | The unit vectors comprise ... |
| frlmup1 21204 | Any assignment of unit vec... |
| frlmup2 21205 | The evaluation map has the... |
| frlmup3 21206 | The range of such an evalu... |
| frlmup4 21207 | Universal property of the ... |
| ellspd 21208 | The elements of the span o... |
| elfilspd 21209 | Simplified version of ~ el... |
| rellindf 21214 | The independent-family pre... |
| islinds 21215 | Property of an independent... |
| linds1 21216 | An independent set of vect... |
| linds2 21217 | An independent set of vect... |
| islindf 21218 | Property of an independent... |
| islinds2 21219 | Expanded property of an in... |
| islindf2 21220 | Property of an independent... |
| lindff 21221 | Functional property of a l... |
| lindfind 21222 | A linearly independent fam... |
| lindsind 21223 | A linearly independent set... |
| lindfind2 21224 | In a linearly independent ... |
| lindsind2 21225 | In a linearly independent ... |
| lindff1 21226 | A linearly independent fam... |
| lindfrn 21227 | The range of an independen... |
| f1lindf 21228 | Rearranging and deleting e... |
| lindfres 21229 | Any restriction of an inde... |
| lindsss 21230 | Any subset of an independe... |
| f1linds 21231 | A family constructed from ... |
| islindf3 21232 | In a nonzero ring, indepen... |
| lindfmm 21233 | Linear independence of a f... |
| lindsmm 21234 | Linear independence of a s... |
| lindsmm2 21235 | The monomorphic image of a... |
| lsslindf 21236 | Linear independence is unc... |
| lsslinds 21237 | Linear independence is unc... |
| islbs4 21238 | A basis is an independent ... |
| lbslinds 21239 | A basis is independent. (... |
| islinds3 21240 | A subset is linearly indep... |
| islinds4 21241 | A set is independent in a ... |
| lmimlbs 21242 | The isomorphic image of a ... |
| lmiclbs 21243 | Having a basis is an isomo... |
| islindf4 21244 | A family is independent if... |
| islindf5 21245 | A family is independent if... |
| indlcim 21246 | An independent, spanning f... |
| lbslcic 21247 | A module with a basis is i... |
| lmisfree 21248 | A module has a basis iff i... |
| lvecisfrlm 21249 | Every vector space is isom... |
| lmimco 21250 | The composition of two iso... |
| lmictra 21251 | Module isomorphism is tran... |
| uvcf1o 21252 | In a nonzero ring, the map... |
| uvcendim 21253 | In a nonzero ring, the num... |
| frlmisfrlm 21254 | A free module is isomorphi... |
| frlmiscvec 21255 | Every free module is isomo... |
| isassa 21262 | The properties of an assoc... |
| assalem 21263 | The properties of an assoc... |
| assaass 21264 | Left-associative property ... |
| assaassr 21265 | Right-associative property... |
| assalmod 21266 | An associative algebra is ... |
| assaring 21267 | An associative algebra is ... |
| assasca 21268 | An associative algebra's s... |
| assa2ass 21269 | Left- and right-associativ... |
| isassad 21270 | Sufficient condition for b... |
| issubassa3 21271 | A subring that is also a s... |
| issubassa 21272 | The subalgebras of an asso... |
| sraassa 21273 | The subring algebra over a... |
| rlmassa 21274 | The ring module over a com... |
| assapropd 21275 | If two structures have the... |
| aspval 21276 | Value of the algebraic clo... |
| asplss 21277 | The algebraic span of a se... |
| aspid 21278 | The algebraic span of a su... |
| aspsubrg 21279 | The algebraic span of a se... |
| aspss 21280 | Span preserves subset orde... |
| aspssid 21281 | A set of vectors is a subs... |
| asclfval 21282 | Function value of the alge... |
| asclval 21283 | Value of a mapped algebra ... |
| asclfn 21284 | Unconditional functionalit... |
| asclf 21285 | The algebra scalars functi... |
| asclghm 21286 | The algebra scalars functi... |
| ascl0 21287 | The scalar 0 embedded into... |
| ascl1 21288 | The scalar 1 embedded into... |
| asclmul1 21289 | Left multiplication by a l... |
| asclmul2 21290 | Right multiplication by a ... |
| ascldimul 21291 | The algebra scalars functi... |
| asclinvg 21292 | The group inverse (negatio... |
| asclrhm 21293 | The scalar injection is a ... |
| rnascl 21294 | The set of injected scalar... |
| issubassa2 21295 | A subring of a unital alge... |
| rnasclsubrg 21296 | The scalar multiples of th... |
| rnasclmulcl 21297 | (Vector) multiplication is... |
| rnasclassa 21298 | The scalar multiples of th... |
| ressascl 21299 | The injection of scalars i... |
| asclpropd 21300 | If two structures have the... |
| aspval2 21301 | The algebraic closure is t... |
| assamulgscmlem1 21302 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21303 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21304 | Exponentiation of a scalar... |
| zlmassa 21305 | The ` ZZ ` -module operati... |
| reldmpsr 21316 | The multivariate power ser... |
| psrval 21317 | Value of the multivariate ... |
| psrvalstr 21318 | The multivariate power ser... |
| psrbag 21319 | Elementhood in the set of ... |
| psrbagf 21320 | A finite bag is a function... |
| psrbagfOLD 21321 | Obsolete version of ~ psrb... |
| psrbagfsupp 21322 | Finite bags have finite su... |
| psrbagfsuppOLD 21323 | Obsolete version of ~ psrb... |
| snifpsrbag 21324 | A bag containing one eleme... |
| fczpsrbag 21325 | The constant function equa... |
| psrbaglesupp 21326 | The support of a dominated... |
| psrbaglesuppOLD 21327 | Obsolete version of ~ psrb... |
| psrbaglecl 21328 | The set of finite bags is ... |
| psrbagleclOLD 21329 | Obsolete version of ~ psrb... |
| psrbagaddcl 21330 | The sum of two finite bags... |
| psrbagaddclOLD 21331 | Obsolete version of ~ psrb... |
| psrbagcon 21332 | The analogue of the statem... |
| psrbagconOLD 21333 | Obsolete version of ~ psrb... |
| psrbaglefi 21334 | There are finitely many ba... |
| psrbaglefiOLD 21335 | Obsolete version of ~ psrb... |
| psrbagconcl 21336 | The complement of a bag is... |
| psrbagconclOLD 21337 | Obsolete version of ~ psrb... |
| psrbagconf1o 21338 | Bag complementation is a b... |
| psrbagconf1oOLD 21339 | Obsolete version of ~ psrb... |
| gsumbagdiaglemOLD 21340 | Obsolete version of ~ gsum... |
| gsumbagdiagOLD 21341 | Obsolete version of ~ gsum... |
| psrass1lemOLD 21342 | Obsolete version of ~ psra... |
| gsumbagdiaglem 21343 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21344 | Two-dimensional commutatio... |
| psrass1lem 21345 | A group sum commutation us... |
| psrbas 21346 | The base set of the multiv... |
| psrelbas 21347 | An element of the set of p... |
| psrelbasfun 21348 | An element of the set of p... |
| psrplusg 21349 | The addition operation of ... |
| psradd 21350 | The addition operation of ... |
| psraddcl 21351 | Closure of the power serie... |
| psrmulr 21352 | The multiplication operati... |
| psrmulfval 21353 | The multiplication operati... |
| psrmulval 21354 | The multiplication operati... |
| psrmulcllem 21355 | Closure of the power serie... |
| psrmulcl 21356 | Closure of the power serie... |
| psrsca 21357 | The scalar field of the mu... |
| psrvscafval 21358 | The scalar multiplication ... |
| psrvsca 21359 | The scalar multiplication ... |
| psrvscaval 21360 | The scalar multiplication ... |
| psrvscacl 21361 | Closure of the power serie... |
| psr0cl 21362 | The zero element of the ri... |
| psr0lid 21363 | The zero element of the ri... |
| psrnegcl 21364 | The negative function in t... |
| psrlinv 21365 | The negative function in t... |
| psrgrp 21366 | The ring of power series i... |
| psrgrpOLD 21367 | Obsolete proof of ~ psrgrp... |
| psr0 21368 | The zero element of the ri... |
| psrneg 21369 | The negative function of t... |
| psrlmod 21370 | The ring of power series i... |
| psr1cl 21371 | The identity element of th... |
| psrlidm 21372 | The identity element of th... |
| psrridm 21373 | The identity element of th... |
| psrass1 21374 | Associative identity for t... |
| psrdi 21375 | Distributive law for the r... |
| psrdir 21376 | Distributive law for the r... |
| psrass23l 21377 | Associative identity for t... |
| psrcom 21378 | Commutative law for the ri... |
| psrass23 21379 | Associative identities for... |
| psrring 21380 | The ring of power series i... |
| psr1 21381 | The identity element of th... |
| psrcrng 21382 | The ring of power series i... |
| psrassa 21383 | The ring of power series i... |
| resspsrbas 21384 | A restricted power series ... |
| resspsradd 21385 | A restricted power series ... |
| resspsrmul 21386 | A restricted power series ... |
| resspsrvsca 21387 | A restricted power series ... |
| subrgpsr 21388 | A subring of the base ring... |
| mvrfval 21389 | Value of the generating el... |
| mvrval 21390 | Value of the generating el... |
| mvrval2 21391 | Value of the generating el... |
| mvrid 21392 | The ` X i ` -th coefficien... |
| mvrf 21393 | The power series variable ... |
| mvrf1 21394 | The power series variable ... |
| mvrcl2 21395 | A power series variable is... |
| reldmmpl 21396 | The multivariate polynomia... |
| mplval 21397 | Value of the set of multiv... |
| mplbas 21398 | Base set of the set of mul... |
| mplelbas 21399 | Property of being a polyno... |
| mplrcl 21400 | Reverse closure for the po... |
| mplelsfi 21401 | A polynomial treated as a ... |
| mplval2 21402 | Self-referential expressio... |
| mplbasss 21403 | The set of polynomials is ... |
| mplelf 21404 | A polynomial is defined as... |
| mplsubglem 21405 | If ` A ` is an ideal of se... |
| mpllsslem 21406 | If ` A ` is an ideal of su... |
| mplsubglem2 21407 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21408 | The set of polynomials is ... |
| mpllss 21409 | The set of polynomials is ... |
| mplsubrglem 21410 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21411 | The set of polynomials is ... |
| mpl0 21412 | The zero polynomial. (Con... |
| mpladd 21413 | The addition operation on ... |
| mplneg 21414 | The negative function on m... |
| mplmul 21415 | The multiplication operati... |
| mpl1 21416 | The identity element of th... |
| mplsca 21417 | The scalar field of a mult... |
| mplvsca2 21418 | The scalar multiplication ... |
| mplvsca 21419 | The scalar multiplication ... |
| mplvscaval 21420 | The scalar multiplication ... |
| mvrcl 21421 | A power series variable is... |
| mplgrp 21422 | The polynomial ring is a g... |
| mpllmod 21423 | The polynomial ring is a l... |
| mplring 21424 | The polynomial ring is a r... |
| mpllvec 21425 | The polynomial ring is a v... |
| mplcrng 21426 | The polynomial ring is a c... |
| mplassa 21427 | The polynomial ring is an ... |
| ressmplbas2 21428 | The base set of a restrict... |
| ressmplbas 21429 | A restricted polynomial al... |
| ressmpladd 21430 | A restricted polynomial al... |
| ressmplmul 21431 | A restricted polynomial al... |
| ressmplvsca 21432 | A restricted power series ... |
| subrgmpl 21433 | A subring of the base ring... |
| subrgmvr 21434 | The variables in a subring... |
| subrgmvrf 21435 | The variables in a polynom... |
| mplmon 21436 | A monomial is a polynomial... |
| mplmonmul 21437 | The product of two monomia... |
| mplcoe1 21438 | Decompose a polynomial int... |
| mplcoe3 21439 | Decompose a monomial in on... |
| mplcoe5lem 21440 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21441 | Decompose a monomial into ... |
| mplcoe2 21442 | Decompose a monomial into ... |
| mplbas2 21443 | An alternative expression ... |
| ltbval 21444 | Value of the well-order on... |
| ltbwe 21445 | The finite bag order is a ... |
| reldmopsr 21446 | Lemma for ordered power se... |
| opsrval 21447 | The value of the "ordered ... |
| opsrle 21448 | An alternative expression ... |
| opsrval2 21449 | Self-referential expressio... |
| opsrbaslem 21450 | Get a component of the ord... |
| opsrbaslemOLD 21451 | Obsolete version of ~ opsr... |
| opsrbas 21452 | The base set of the ordere... |
| opsrbasOLD 21453 | Obsolete version of ~ opsr... |
| opsrplusg 21454 | The addition operation of ... |
| opsrplusgOLD 21455 | Obsolete version of ~ opsr... |
| opsrmulr 21456 | The multiplication operati... |
| opsrmulrOLD 21457 | Obsolete version of ~ opsr... |
| opsrvsca 21458 | The scalar product operati... |
| opsrvscaOLD 21459 | Obsolete version of ~ opsr... |
| opsrsca 21460 | The scalar ring of the ord... |
| opsrscaOLD 21461 | Obsolete version of ~ opsr... |
| opsrtoslem1 21462 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 21463 | Lemma for ~ opsrtos . (Co... |
| opsrtos 21464 | The ordered power series s... |
| opsrso 21465 | The ordered power series s... |
| opsrcrng 21466 | The ring of ordered power ... |
| opsrassa 21467 | The ring of ordered power ... |
| mvrf2 21468 | The power series/polynomia... |
| mplmon2 21469 | Express a scaled monomial.... |
| psrbag0 21470 | The empty bag is a bag. (... |
| psrbagsn 21471 | A singleton bag is a bag. ... |
| mplascl 21472 | Value of the scalar inject... |
| mplasclf 21473 | The scalar injection is a ... |
| subrgascl 21474 | The scalar injection funct... |
| subrgasclcl 21475 | The scalars in a polynomia... |
| mplmon2cl 21476 | A scaled monomial is a pol... |
| mplmon2mul 21477 | Product of scaled monomial... |
| mplind 21478 | Prove a property of polyno... |
| mplcoe4 21479 | Decompose a polynomial int... |
| evlslem4 21484 | The support of a tensor pr... |
| psrbagev1 21485 | A bag of multipliers provi... |
| psrbagev1OLD 21486 | Obsolete version of ~ psrb... |
| psrbagev2 21487 | Closure of a sum using a b... |
| psrbagev2OLD 21488 | Obsolete version of ~ psrb... |
| evlslem2 21489 | A linear function on the p... |
| evlslem3 21490 | Lemma for ~ evlseu . Poly... |
| evlslem6 21491 | Lemma for ~ evlseu . Fini... |
| evlslem1 21492 | Lemma for ~ evlseu , give ... |
| evlseu 21493 | For a given interpretation... |
| reldmevls 21494 | Well-behaved binary operat... |
| mpfrcl 21495 | Reverse closure for the se... |
| evlsval 21496 | Value of the polynomial ev... |
| evlsval2 21497 | Characterizing properties ... |
| evlsrhm 21498 | Polynomial evaluation is a... |
| evlssca 21499 | Polynomial evaluation maps... |
| evlsvar 21500 | Polynomial evaluation maps... |
| evlsgsumadd 21501 | Polynomial evaluation maps... |
| evlsgsummul 21502 | Polynomial evaluation maps... |
| evlspw 21503 | Polynomial evaluation for ... |
| evlsvarpw 21504 | Polynomial evaluation for ... |
| evlval 21505 | Value of the simple/same r... |
| evlrhm 21506 | The simple evaluation map ... |
| evlsscasrng 21507 | The evaluation of a scalar... |
| evlsca 21508 | Simple polynomial evaluati... |
| evlsvarsrng 21509 | The evaluation of the vari... |
| evlvar 21510 | Simple polynomial evaluati... |
| mpfconst 21511 | Constants are multivariate... |
| mpfproj 21512 | Projections are multivaria... |
| mpfsubrg 21513 | Polynomial functions are a... |
| mpff 21514 | Polynomial functions are f... |
| mpfaddcl 21515 | The sum of multivariate po... |
| mpfmulcl 21516 | The product of multivariat... |
| mpfind 21517 | Prove a property of polyno... |
| selvffval 21526 | Value of the "variable sel... |
| selvfval 21527 | Value of the "variable sel... |
| selvval 21528 | Value of the "variable sel... |
| mhpfval 21529 | Value of the "homogeneous ... |
| mhpval 21530 | Value of the "homogeneous ... |
| ismhp 21531 | Property of being a homoge... |
| ismhp2 21532 | Deduce a homogeneous polyn... |
| ismhp3 21533 | A polynomial is homogeneou... |
| mhpmpl 21534 | A homogeneous polynomial i... |
| mhpdeg 21535 | All nonzero terms of a hom... |
| mhp0cl 21536 | The zero polynomial is hom... |
| mhpsclcl 21537 | A scalar (or constant) pol... |
| mhpvarcl 21538 | A power series variable is... |
| mhpmulcl 21539 | A product of homogeneous p... |
| mhppwdeg 21540 | Degree of a homogeneous po... |
| mhpaddcl 21541 | Homogeneous polynomials ar... |
| mhpinvcl 21542 | Homogeneous polynomials ar... |
| mhpsubg 21543 | Homogeneous polynomials fo... |
| mhpvscacl 21544 | Homogeneous polynomials ar... |
| mhplss 21545 | Homogeneous polynomials fo... |
| psr1baslem 21556 | The set of finite bags on ... |
| psr1val 21557 | Value of the ring of univa... |
| psr1crng 21558 | The ring of univariate pow... |
| psr1assa 21559 | The ring of univariate pow... |
| psr1tos 21560 | The ordered power series s... |
| psr1bas2 21561 | The base set of the ring o... |
| psr1bas 21562 | The base set of the ring o... |
| vr1val 21563 | The value of the generator... |
| vr1cl2 21564 | The variable ` X ` is a me... |
| ply1val 21565 | The value of the set of un... |
| ply1bas 21566 | The value of the base set ... |
| ply1lss 21567 | Univariate polynomials for... |
| ply1subrg 21568 | Univariate polynomials for... |
| ply1crng 21569 | The ring of univariate pol... |
| ply1assa 21570 | The ring of univariate pol... |
| psr1bascl 21571 | A univariate power series ... |
| psr1basf 21572 | Univariate power series ba... |
| ply1basf 21573 | Univariate polynomial base... |
| ply1bascl 21574 | A univariate polynomial is... |
| ply1bascl2 21575 | A univariate polynomial is... |
| coe1fval 21576 | Value of the univariate po... |
| coe1fv 21577 | Value of an evaluated coef... |
| fvcoe1 21578 | Value of a multivariate co... |
| coe1fval3 21579 | Univariate power series co... |
| coe1f2 21580 | Functionality of univariat... |
| coe1fval2 21581 | Univariate polynomial coef... |
| coe1f 21582 | Functionality of univariat... |
| coe1fvalcl 21583 | A coefficient of a univari... |
| coe1sfi 21584 | Finite support of univaria... |
| coe1fsupp 21585 | The coefficient vector of ... |
| mptcoe1fsupp 21586 | A mapping involving coeffi... |
| coe1ae0 21587 | The coefficient vector of ... |
| vr1cl 21588 | The generator of a univari... |
| opsr0 21589 | Zero in the ordered power ... |
| opsr1 21590 | One in the ordered power s... |
| mplplusg 21591 | Value of addition in a pol... |
| mplmulr 21592 | Value of multiplication in... |
| psr1plusg 21593 | Value of addition in a uni... |
| psr1vsca 21594 | Value of scalar multiplica... |
| psr1mulr 21595 | Value of multiplication in... |
| ply1plusg 21596 | Value of addition in a uni... |
| ply1vsca 21597 | Value of scalar multiplica... |
| ply1mulr 21598 | Value of multiplication in... |
| ressply1bas2 21599 | The base set of a restrict... |
| ressply1bas 21600 | A restricted polynomial al... |
| ressply1add 21601 | A restricted polynomial al... |
| ressply1mul 21602 | A restricted polynomial al... |
| ressply1vsca 21603 | A restricted power series ... |
| subrgply1 21604 | A subring of the base ring... |
| gsumply1subr 21605 | Evaluate a group sum in a ... |
| psrbaspropd 21606 | Property deduction for pow... |
| psrplusgpropd 21607 | Property deduction for pow... |
| mplbaspropd 21608 | Property deduction for pol... |
| psropprmul 21609 | Reversing multiplication i... |
| ply1opprmul 21610 | Reversing multiplication i... |
| 00ply1bas 21611 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 21612 | Protection compatibility o... |
| ply1plusgfvi 21613 | Protection compatibility o... |
| ply1baspropd 21614 | Property deduction for uni... |
| ply1plusgpropd 21615 | Property deduction for uni... |
| opsrring 21616 | Ordered power series form ... |
| opsrlmod 21617 | Ordered power series form ... |
| psr1ring 21618 | Univariate power series fo... |
| ply1ring 21619 | Univariate polynomials for... |
| psr1lmod 21620 | Univariate power series fo... |
| psr1sca 21621 | Scalars of a univariate po... |
| psr1sca2 21622 | Scalars of a univariate po... |
| ply1lmod 21623 | Univariate polynomials for... |
| ply1sca 21624 | Scalars of a univariate po... |
| ply1sca2 21625 | Scalars of a univariate po... |
| ply1mpl0 21626 | The univariate polynomial ... |
| ply10s0 21627 | Zero times a univariate po... |
| ply1mpl1 21628 | The univariate polynomial ... |
| ply1ascl 21629 | The univariate polynomial ... |
| subrg1ascl 21630 | The scalar injection funct... |
| subrg1asclcl 21631 | The scalars in a polynomia... |
| subrgvr1 21632 | The variables in a subring... |
| subrgvr1cl 21633 | The variables in a polynom... |
| coe1z 21634 | The coefficient vector of ... |
| coe1add 21635 | The coefficient vector of ... |
| coe1addfv 21636 | A particular coefficient o... |
| coe1subfv 21637 | A particular coefficient o... |
| coe1mul2lem1 21638 | An equivalence for ~ coe1m... |
| coe1mul2lem2 21639 | An equivalence for ~ coe1m... |
| coe1mul2 21640 | The coefficient vector of ... |
| coe1mul 21641 | The coefficient vector of ... |
| ply1moncl 21642 | Closure of the expression ... |
| ply1tmcl 21643 | Closure of the expression ... |
| coe1tm 21644 | Coefficient vector of a po... |
| coe1tmfv1 21645 | Nonzero coefficient of a p... |
| coe1tmfv2 21646 | Zero coefficient of a poly... |
| coe1tmmul2 21647 | Coefficient vector of a po... |
| coe1tmmul 21648 | Coefficient vector of a po... |
| coe1tmmul2fv 21649 | Function value of a right-... |
| coe1pwmul 21650 | Coefficient vector of a po... |
| coe1pwmulfv 21651 | Function value of a right-... |
| ply1scltm 21652 | A scalar is a term with ze... |
| coe1sclmul 21653 | Coefficient vector of a po... |
| coe1sclmulfv 21654 | A single coefficient of a ... |
| coe1sclmul2 21655 | Coefficient vector of a po... |
| ply1sclf 21656 | A scalar polynomial is a p... |
| ply1sclcl 21657 | The value of the algebra s... |
| coe1scl 21658 | Coefficient vector of a sc... |
| ply1sclid 21659 | Recover the base scalar fr... |
| ply1sclf1 21660 | The polynomial scalar func... |
| ply1scl0 21661 | The zero scalar is zero. ... |
| ply1scln0 21662 | Nonzero scalars create non... |
| ply1scl1 21663 | The one scalar is the unit... |
| ply1idvr1 21664 | The identity of a polynomi... |
| cply1mul 21665 | The product of two constan... |
| ply1coefsupp 21666 | The decomposition of a uni... |
| ply1coe 21667 | Decompose a univariate pol... |
| eqcoe1ply1eq 21668 | Two polynomials over the s... |
| ply1coe1eq 21669 | Two polynomials over the s... |
| cply1coe0 21670 | All but the first coeffici... |
| cply1coe0bi 21671 | A polynomial is constant (... |
| coe1fzgsumdlem 21672 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 21673 | Value of an evaluated coef... |
| gsumsmonply1 21674 | A finite group sum of scal... |
| gsummoncoe1 21675 | A coefficient of the polyn... |
| gsumply1eq 21676 | Two univariate polynomials... |
| lply1binom 21677 | The binomial theorem for l... |
| lply1binomsc 21678 | The binomial theorem for l... |
| reldmevls1 21683 | Well-behaved binary operat... |
| ply1frcl 21684 | Reverse closure for the se... |
| evls1fval 21685 | Value of the univariate po... |
| evls1val 21686 | Value of the univariate po... |
| evls1rhmlem 21687 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 21688 | Polynomial evaluation is a... |
| evls1sca 21689 | Univariate polynomial eval... |
| evls1gsumadd 21690 | Univariate polynomial eval... |
| evls1gsummul 21691 | Univariate polynomial eval... |
| evls1pw 21692 | Univariate polynomial eval... |
| evls1varpw 21693 | Univariate polynomial eval... |
| evl1fval 21694 | Value of the simple/same r... |
| evl1val 21695 | Value of the simple/same r... |
| evl1fval1lem 21696 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 21697 | Value of the simple/same r... |
| evl1rhm 21698 | Polynomial evaluation is a... |
| fveval1fvcl 21699 | The function value of the ... |
| evl1sca 21700 | Polynomial evaluation maps... |
| evl1scad 21701 | Polynomial evaluation buil... |
| evl1var 21702 | Polynomial evaluation maps... |
| evl1vard 21703 | Polynomial evaluation buil... |
| evls1var 21704 | Univariate polynomial eval... |
| evls1scasrng 21705 | The evaluation of a scalar... |
| evls1varsrng 21706 | The evaluation of the vari... |
| evl1addd 21707 | Polynomial evaluation buil... |
| evl1subd 21708 | Polynomial evaluation buil... |
| evl1muld 21709 | Polynomial evaluation buil... |
| evl1vsd 21710 | Polynomial evaluation buil... |
| evl1expd 21711 | Polynomial evaluation buil... |
| pf1const 21712 | Constants are polynomial f... |
| pf1id 21713 | The identity is a polynomi... |
| pf1subrg 21714 | Polynomial functions are a... |
| pf1rcl 21715 | Reverse closure for the se... |
| pf1f 21716 | Polynomial functions are f... |
| mpfpf1 21717 | Convert a multivariate pol... |
| pf1mpf 21718 | Convert a univariate polyn... |
| pf1addcl 21719 | The sum of multivariate po... |
| pf1mulcl 21720 | The product of multivariat... |
| pf1ind 21721 | Prove a property of polyno... |
| evl1gsumdlem 21722 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 21723 | Polynomial evaluation buil... |
| evl1gsumadd 21724 | Univariate polynomial eval... |
| evl1gsumaddval 21725 | Value of a univariate poly... |
| evl1gsummul 21726 | Univariate polynomial eval... |
| evl1varpw 21727 | Univariate polynomial eval... |
| evl1varpwval 21728 | Value of a univariate poly... |
| evl1scvarpw 21729 | Univariate polynomial eval... |
| evl1scvarpwval 21730 | Value of a univariate poly... |
| evl1gsummon 21731 | Value of a univariate poly... |
| mamufval 21734 | Functional value of the ma... |
| mamuval 21735 | Multiplication of two matr... |
| mamufv 21736 | A cell in the multiplicati... |
| mamudm 21737 | The domain of the matrix m... |
| mamufacex 21738 | Every solution of the equa... |
| mamures 21739 | Rows in a matrix product a... |
| mndvcl 21740 | Tuple-wise additive closur... |
| mndvass 21741 | Tuple-wise associativity i... |
| mndvlid 21742 | Tuple-wise left identity i... |
| mndvrid 21743 | Tuple-wise right identity ... |
| grpvlinv 21744 | Tuple-wise left inverse in... |
| grpvrinv 21745 | Tuple-wise right inverse i... |
| mhmvlin 21746 | Tuple extension of monoid ... |
| ringvcl 21747 | Tuple-wise multiplication ... |
| mamucl 21748 | Operation closure of matri... |
| mamuass 21749 | Matrix multiplication is a... |
| mamudi 21750 | Matrix multiplication dist... |
| mamudir 21751 | Matrix multiplication dist... |
| mamuvs1 21752 | Matrix multiplication dist... |
| mamuvs2 21753 | Matrix multiplication dist... |
| matbas0pc 21756 | There is no matrix with a ... |
| matbas0 21757 | There is no matrix for a n... |
| matval 21758 | Value of the matrix algebr... |
| matrcl 21759 | Reverse closure for the ma... |
| matbas 21760 | The matrix ring has the sa... |
| matplusg 21761 | The matrix ring has the sa... |
| matsca 21762 | The matrix ring has the sa... |
| matscaOLD 21763 | Obsolete proof of ~ matsca... |
| matvsca 21764 | The matrix ring has the sa... |
| matvscaOLD 21765 | Obsolete proof of ~ matvsc... |
| mat0 21766 | The matrix ring has the sa... |
| matinvg 21767 | The matrix ring has the sa... |
| mat0op 21768 | Value of a zero matrix as ... |
| matsca2 21769 | The scalars of the matrix ... |
| matbas2 21770 | The base set of the matrix... |
| matbas2i 21771 | A matrix is a function. (... |
| matbas2d 21772 | The base set of the matrix... |
| eqmat 21773 | Two square matrices of the... |
| matecl 21774 | Each entry (according to W... |
| matecld 21775 | Each entry (according to W... |
| matplusg2 21776 | Addition in the matrix rin... |
| matvsca2 21777 | Scalar multiplication in t... |
| matlmod 21778 | The matrix ring is a linea... |
| matgrp 21779 | The matrix ring is a group... |
| matvscl 21780 | Closure of the scalar mult... |
| matsubg 21781 | The matrix ring has the sa... |
| matplusgcell 21782 | Addition in the matrix rin... |
| matsubgcell 21783 | Subtraction in the matrix ... |
| matinvgcell 21784 | Additive inversion in the ... |
| matvscacell 21785 | Scalar multiplication in t... |
| matgsum 21786 | Finite commutative sums in... |
| matmulr 21787 | Multiplication in the matr... |
| mamumat1cl 21788 | The identity matrix (as op... |
| mat1comp 21789 | The components of the iden... |
| mamulid 21790 | The identity matrix (as op... |
| mamurid 21791 | The identity matrix (as op... |
| matring 21792 | Existence of the matrix ri... |
| matassa 21793 | Existence of the matrix al... |
| matmulcell 21794 | Multiplication in the matr... |
| mpomatmul 21795 | Multiplication of two N x ... |
| mat1 21796 | Value of an identity matri... |
| mat1ov 21797 | Entries of an identity mat... |
| mat1bas 21798 | The identity matrix is a m... |
| matsc 21799 | The identity matrix multip... |
| ofco2 21800 | Distribution law for the f... |
| oftpos 21801 | The transposition of the v... |
| mattposcl 21802 | The transpose of a square ... |
| mattpostpos 21803 | The transpose of the trans... |
| mattposvs 21804 | The transposition of a mat... |
| mattpos1 21805 | The transposition of the i... |
| tposmap 21806 | The transposition of an I ... |
| mamutpos 21807 | Behavior of transposes in ... |
| mattposm 21808 | Multiplying two transposed... |
| matgsumcl 21809 | Closure of a group sum ove... |
| madetsumid 21810 | The identity summand in th... |
| matepmcl 21811 | Each entry of a matrix wit... |
| matepm2cl 21812 | Each entry of a matrix wit... |
| madetsmelbas 21813 | A summand of the determina... |
| madetsmelbas2 21814 | A summand of the determina... |
| mat0dimbas0 21815 | The empty set is the one a... |
| mat0dim0 21816 | The zero of the algebra of... |
| mat0dimid 21817 | The identity of the algebr... |
| mat0dimscm 21818 | The scalar multiplication ... |
| mat0dimcrng 21819 | The algebra of matrices wi... |
| mat1dimelbas 21820 | A matrix with dimension 1 ... |
| mat1dimbas 21821 | A matrix with dimension 1 ... |
| mat1dim0 21822 | The zero of the algebra of... |
| mat1dimid 21823 | The identity of the algebr... |
| mat1dimscm 21824 | The scalar multiplication ... |
| mat1dimmul 21825 | The ring multiplication in... |
| mat1dimcrng 21826 | The algebra of matrices wi... |
| mat1f1o 21827 | There is a 1-1 function fr... |
| mat1rhmval 21828 | The value of the ring homo... |
| mat1rhmelval 21829 | The value of the ring homo... |
| mat1rhmcl 21830 | The value of the ring homo... |
| mat1f 21831 | There is a function from a... |
| mat1ghm 21832 | There is a group homomorph... |
| mat1mhm 21833 | There is a monoid homomorp... |
| mat1rhm 21834 | There is a ring homomorphi... |
| mat1rngiso 21835 | There is a ring isomorphis... |
| mat1ric 21836 | A ring is isomorphic to th... |
| dmatval 21841 | The set of ` N ` x ` N ` d... |
| dmatel 21842 | A ` N ` x ` N ` diagonal m... |
| dmatmat 21843 | An ` N ` x ` N ` diagonal ... |
| dmatid 21844 | The identity matrix is a d... |
| dmatelnd 21845 | An extradiagonal entry of ... |
| dmatmul 21846 | The product of two diagona... |
| dmatsubcl 21847 | The difference of two diag... |
| dmatsgrp 21848 | The set of diagonal matric... |
| dmatmulcl 21849 | The product of two diagona... |
| dmatsrng 21850 | The set of diagonal matric... |
| dmatcrng 21851 | The subring of diagonal ma... |
| dmatscmcl 21852 | The multiplication of a di... |
| scmatval 21853 | The set of ` N ` x ` N ` s... |
| scmatel 21854 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 21855 | A scalar matrix can be exp... |
| scmatscmide 21856 | An entry of a scalar matri... |
| scmatscmiddistr 21857 | Distributive law for scala... |
| scmatmat 21858 | An ` N ` x ` N ` scalar ma... |
| scmate 21859 | An entry of an ` N ` x ` N... |
| scmatmats 21860 | The set of an ` N ` x ` N ... |
| scmateALT 21861 | Alternate proof of ~ scmat... |
| scmatscm 21862 | The multiplication of a ma... |
| scmatid 21863 | The identity matrix is a s... |
| scmatdmat 21864 | A scalar matrix is a diago... |
| scmataddcl 21865 | The sum of two scalar matr... |
| scmatsubcl 21866 | The difference of two scal... |
| scmatmulcl 21867 | The product of two scalar ... |
| scmatsgrp 21868 | The set of scalar matrices... |
| scmatsrng 21869 | The set of scalar matrices... |
| scmatcrng 21870 | The subring of scalar matr... |
| scmatsgrp1 21871 | The set of scalar matrices... |
| scmatsrng1 21872 | The set of scalar matrices... |
| smatvscl 21873 | Closure of the scalar mult... |
| scmatlss 21874 | The set of scalar matrices... |
| scmatstrbas 21875 | The set of scalar matrices... |
| scmatrhmval 21876 | The value of the ring homo... |
| scmatrhmcl 21877 | The value of the ring homo... |
| scmatf 21878 | There is a function from a... |
| scmatfo 21879 | There is a function from a... |
| scmatf1 21880 | There is a 1-1 function fr... |
| scmatf1o 21881 | There is a bijection betwe... |
| scmatghm 21882 | There is a group homomorph... |
| scmatmhm 21883 | There is a monoid homomorp... |
| scmatrhm 21884 | There is a ring homomorphi... |
| scmatrngiso 21885 | There is a ring isomorphis... |
| scmatric 21886 | A ring is isomorphic to ev... |
| mat0scmat 21887 | The empty matrix over a ri... |
| mat1scmat 21888 | A 1-dimensional matrix ove... |
| mvmulfval 21891 | Functional value of the ma... |
| mvmulval 21892 | Multiplication of a vector... |
| mvmulfv 21893 | A cell/element in the vect... |
| mavmulval 21894 | Multiplication of a vector... |
| mavmulfv 21895 | A cell/element in the vect... |
| mavmulcl 21896 | Multiplication of an NxN m... |
| 1mavmul 21897 | Multiplication of the iden... |
| mavmulass 21898 | Associativity of the multi... |
| mavmuldm 21899 | The domain of the matrix v... |
| mavmulsolcl 21900 | Every solution of the equa... |
| mavmul0 21901 | Multiplication of a 0-dime... |
| mavmul0g 21902 | The result of the 0-dimens... |
| mvmumamul1 21903 | The multiplication of an M... |
| mavmumamul1 21904 | The multiplication of an N... |
| marrepfval 21909 | First substitution for the... |
| marrepval0 21910 | Second substitution for th... |
| marrepval 21911 | Third substitution for the... |
| marrepeval 21912 | An entry of a matrix with ... |
| marrepcl 21913 | Closure of the row replace... |
| marepvfval 21914 | First substitution for the... |
| marepvval0 21915 | Second substitution for th... |
| marepvval 21916 | Third substitution for the... |
| marepveval 21917 | An entry of a matrix with ... |
| marepvcl 21918 | Closure of the column repl... |
| ma1repvcl 21919 | Closure of the column repl... |
| ma1repveval 21920 | An entry of an identity ma... |
| mulmarep1el 21921 | Element by element multipl... |
| mulmarep1gsum1 21922 | The sum of element by elem... |
| mulmarep1gsum2 21923 | The sum of element by elem... |
| 1marepvmarrepid 21924 | Replacing the ith row by 0... |
| submabas 21927 | Any subset of the index se... |
| submafval 21928 | First substitution for a s... |
| submaval0 21929 | Second substitution for a ... |
| submaval 21930 | Third substitution for a s... |
| submaeval 21931 | An entry of a submatrix of... |
| 1marepvsma1 21932 | The submatrix of the ident... |
| mdetfval 21935 | First substitution for the... |
| mdetleib 21936 | Full substitution of our d... |
| mdetleib2 21937 | Leibniz' formula can also ... |
| nfimdetndef 21938 | The determinant is not def... |
| mdetfval1 21939 | First substitution of an a... |
| mdetleib1 21940 | Full substitution of an al... |
| mdet0pr 21941 | The determinant function f... |
| mdet0f1o 21942 | The determinant function f... |
| mdet0fv0 21943 | The determinant of the emp... |
| mdetf 21944 | Functionality of the deter... |
| mdetcl 21945 | The determinant evaluates ... |
| m1detdiag 21946 | The determinant of a 1-dim... |
| mdetdiaglem 21947 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 21948 | The determinant of a diago... |
| mdetdiagid 21949 | The determinant of a diago... |
| mdet1 21950 | The determinant of the ide... |
| mdetrlin 21951 | The determinant function i... |
| mdetrsca 21952 | The determinant function i... |
| mdetrsca2 21953 | The determinant function i... |
| mdetr0 21954 | The determinant of a matri... |
| mdet0 21955 | The determinant of the zer... |
| mdetrlin2 21956 | The determinant function i... |
| mdetralt 21957 | The determinant function i... |
| mdetralt2 21958 | The determinant function i... |
| mdetero 21959 | The determinant function i... |
| mdettpos 21960 | Determinant is invariant u... |
| mdetunilem1 21961 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 21962 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 21963 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 21964 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 21965 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 21966 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 21967 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 21968 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 21969 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 21970 | Lemma for ~ mdetuni . (Co... |
| mdetuni 21971 | According to the definitio... |
| mdetmul 21972 | Multiplicativity of the de... |
| m2detleiblem1 21973 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 21974 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 21975 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 21976 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 21977 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 21978 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 21979 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 21980 | Leibniz' Formula for 2x2-m... |
| mndifsplit 21985 | Lemma for ~ maducoeval2 . ... |
| madufval 21986 | First substitution for the... |
| maduval 21987 | Second substitution for th... |
| maducoeval 21988 | An entry of the adjunct (c... |
| maducoeval2 21989 | An entry of the adjunct (c... |
| maduf 21990 | Creating the adjunct of ma... |
| madutpos 21991 | The adjuct of a transposed... |
| madugsum 21992 | The determinant of a matri... |
| madurid 21993 | Multiplying a matrix with ... |
| madulid 21994 | Multiplying the adjunct of... |
| minmar1fval 21995 | First substitution for the... |
| minmar1val0 21996 | Second substitution for th... |
| minmar1val 21997 | Third substitution for the... |
| minmar1eval 21998 | An entry of a matrix for a... |
| minmar1marrep 21999 | The minor matrix is a spec... |
| minmar1cl 22000 | Closure of the row replace... |
| maducoevalmin1 22001 | The coefficients of an adj... |
| symgmatr01lem 22002 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 22003 | Applying a permutation tha... |
| gsummatr01lem1 22004 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 22005 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 22006 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 22007 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 22008 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 22009 | Replacing a row of a squar... |
| smadiadetlem0 22010 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 22011 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 22012 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 22013 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 22014 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 22015 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 22016 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 22017 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 22018 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 22019 | The determinant of a subma... |
| smadiadetglem1 22020 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 22021 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 22022 | The determinant of a squar... |
| smadiadetg0 22023 | Lemma for ~ smadiadetr : v... |
| smadiadetr 22024 | The determinant of a squar... |
| invrvald 22025 | If a matrix multiplied wit... |
| matinv 22026 | The inverse of a matrix is... |
| matunit 22027 | A matrix is a unit in the ... |
| slesolvec 22028 | Every solution of a system... |
| slesolinv 22029 | The solution of a system o... |
| slesolinvbi 22030 | The solution of a system o... |
| slesolex 22031 | Every system of linear equ... |
| cramerimplem1 22032 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 22033 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 22034 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 22035 | One direction of Cramer's ... |
| cramerlem1 22036 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 22037 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 22038 | Lemma 3 for ~ cramer . (C... |
| cramer0 22039 | Special case of Cramer's r... |
| cramer 22040 | Cramer's rule. According ... |
| pmatring 22041 | The set of polynomial matr... |
| pmatlmod 22042 | The set of polynomial matr... |
| pmatassa 22043 | The set of polynomial matr... |
| pmat0op 22044 | The zero polynomial matrix... |
| pmat1op 22045 | The identity polynomial ma... |
| pmat1ovd 22046 | Entries of the identity po... |
| pmat0opsc 22047 | The zero polynomial matrix... |
| pmat1opsc 22048 | The identity polynomial ma... |
| pmat1ovscd 22049 | Entries of the identity po... |
| pmatcoe1fsupp 22050 | For a polynomial matrix th... |
| 1pmatscmul 22051 | The scalar product of the ... |
| cpmat 22058 | Value of the constructor o... |
| cpmatpmat 22059 | A constant polynomial matr... |
| cpmatel 22060 | Property of a constant pol... |
| cpmatelimp 22061 | Implication of a set being... |
| cpmatel2 22062 | Another property of a cons... |
| cpmatelimp2 22063 | Another implication of a s... |
| 1elcpmat 22064 | The identity of the ring o... |
| cpmatacl 22065 | The set of all constant po... |
| cpmatinvcl 22066 | The set of all constant po... |
| cpmatmcllem 22067 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 22068 | The set of all constant po... |
| cpmatsubgpmat 22069 | The set of all constant po... |
| cpmatsrgpmat 22070 | The set of all constant po... |
| 0elcpmat 22071 | The zero of the ring of al... |
| mat2pmatfval 22072 | Value of the matrix transf... |
| mat2pmatval 22073 | The result of a matrix tra... |
| mat2pmatvalel 22074 | A (matrix) element of the ... |
| mat2pmatbas 22075 | The result of a matrix tra... |
| mat2pmatbas0 22076 | The result of a matrix tra... |
| mat2pmatf 22077 | The matrix transformation ... |
| mat2pmatf1 22078 | The matrix transformation ... |
| mat2pmatghm 22079 | The transformation of matr... |
| mat2pmatmul 22080 | The transformation of matr... |
| mat2pmat1 22081 | The transformation of the ... |
| mat2pmatmhm 22082 | The transformation of matr... |
| mat2pmatrhm 22083 | The transformation of matr... |
| mat2pmatlin 22084 | The transformation of matr... |
| 0mat2pmat 22085 | The transformed zero matri... |
| idmatidpmat 22086 | The transformed identity m... |
| d0mat2pmat 22087 | The transformed empty set ... |
| d1mat2pmat 22088 | The transformation of a ma... |
| mat2pmatscmxcl 22089 | A transformed matrix multi... |
| m2cpm 22090 | The result of a matrix tra... |
| m2cpmf 22091 | The matrix transformation ... |
| m2cpmf1 22092 | The matrix transformation ... |
| m2cpmghm 22093 | The transformation of matr... |
| m2cpmmhm 22094 | The transformation of matr... |
| m2cpmrhm 22095 | The transformation of matr... |
| m2pmfzmap 22096 | The transformed values of ... |
| m2pmfzgsumcl 22097 | Closure of the sum of scal... |
| cpm2mfval 22098 | Value of the inverse matri... |
| cpm2mval 22099 | The result of an inverse m... |
| cpm2mvalel 22100 | A (matrix) element of the ... |
| cpm2mf 22101 | The inverse matrix transfo... |
| m2cpminvid 22102 | The inverse transformation... |
| m2cpminvid2lem 22103 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 22104 | The transformation applied... |
| m2cpmfo 22105 | The matrix transformation ... |
| m2cpmf1o 22106 | The matrix transformation ... |
| m2cpmrngiso 22107 | The transformation of matr... |
| matcpmric 22108 | The ring of matrices over ... |
| m2cpminv 22109 | The inverse matrix transfo... |
| m2cpminv0 22110 | The inverse matrix transfo... |
| decpmatval0 22113 | The matrix consisting of t... |
| decpmatval 22114 | The matrix consisting of t... |
| decpmate 22115 | An entry of the matrix con... |
| decpmatcl 22116 | Closure of the decompositi... |
| decpmataa0 22117 | The matrix consisting of t... |
| decpmatfsupp 22118 | The mapping to the matrice... |
| decpmatid 22119 | The matrix consisting of t... |
| decpmatmullem 22120 | Lemma for ~ decpmatmul . ... |
| decpmatmul 22121 | The matrix consisting of t... |
| decpmatmulsumfsupp 22122 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 22123 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 22124 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 22125 | Write a polynomial matrix ... |
| pmatcollpw2lem 22126 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 22127 | Write a polynomial matrix ... |
| monmatcollpw 22128 | The matrix consisting of t... |
| pmatcollpwlem 22129 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 22130 | Write a polynomial matrix ... |
| pmatcollpwfi 22131 | Write a polynomial matrix ... |
| pmatcollpw3lem 22132 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 22133 | Write a polynomial matrix ... |
| pmatcollpw3fi 22134 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 22135 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 22136 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 22137 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 22138 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 22139 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 22140 | Write a scalar matrix over... |
| pm2mpf1lem 22143 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 22144 | Value of the transformatio... |
| pm2mpfval 22145 | A polynomial matrix transf... |
| pm2mpcl 22146 | The transformation of poly... |
| pm2mpf 22147 | The transformation of poly... |
| pm2mpf1 22148 | The transformation of poly... |
| pm2mpcoe1 22149 | A coefficient of the polyn... |
| idpm2idmp 22150 | The transformation of the ... |
| mptcoe1matfsupp 22151 | The mapping extracting the... |
| mply1topmatcllem 22152 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 22153 | A polynomial over matrices... |
| mply1topmatcl 22154 | A polynomial over matrices... |
| mp2pm2mplem1 22155 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 22156 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 22157 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 22158 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 22159 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 22160 | A polynomial over matrices... |
| pm2mpghmlem2 22161 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 22162 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 22163 | The transformation of poly... |
| pm2mpf1o 22164 | The transformation of poly... |
| pm2mpghm 22165 | The transformation of poly... |
| pm2mpgrpiso 22166 | The transformation of poly... |
| pm2mpmhmlem1 22167 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 22168 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 22169 | The transformation of poly... |
| pm2mprhm 22170 | The transformation of poly... |
| pm2mprngiso 22171 | The transformation of poly... |
| pmmpric 22172 | The ring of polynomial mat... |
| monmat2matmon 22173 | The transformation of a po... |
| pm2mp 22174 | The transformation of a su... |
| chmatcl 22177 | Closure of the characteris... |
| chmatval 22178 | The entries of the charact... |
| chpmatfval 22179 | Value of the characteristi... |
| chpmatval 22180 | The characteristic polynom... |
| chpmatply1 22181 | The characteristic polynom... |
| chpmatval2 22182 | The characteristic polynom... |
| chpmat0d 22183 | The characteristic polynom... |
| chpmat1dlem 22184 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 22185 | The characteristic polynom... |
| chpdmatlem0 22186 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 22187 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 22188 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 22189 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 22190 | The characteristic polynom... |
| chpscmat 22191 | The characteristic polynom... |
| chpscmat0 22192 | The characteristic polynom... |
| chpscmatgsumbin 22193 | The characteristic polynom... |
| chpscmatgsummon 22194 | The characteristic polynom... |
| chp0mat 22195 | The characteristic polynom... |
| chpidmat 22196 | The characteristic polynom... |
| chmaidscmat 22197 | The characteristic polynom... |
| fvmptnn04if 22198 | The function values of a m... |
| fvmptnn04ifa 22199 | The function value of a ma... |
| fvmptnn04ifb 22200 | The function value of a ma... |
| fvmptnn04ifc 22201 | The function value of a ma... |
| fvmptnn04ifd 22202 | The function value of a ma... |
| chfacfisf 22203 | The "characteristic factor... |
| chfacfisfcpmat 22204 | The "characteristic factor... |
| chfacffsupp 22205 | The "characteristic factor... |
| chfacfscmulcl 22206 | Closure of a scaled value ... |
| chfacfscmul0 22207 | A scaled value of the "cha... |
| chfacfscmulfsupp 22208 | A mapping of scaled values... |
| chfacfscmulgsum 22209 | Breaking up a sum of value... |
| chfacfpmmulcl 22210 | Closure of the value of th... |
| chfacfpmmul0 22211 | The value of the "characte... |
| chfacfpmmulfsupp 22212 | A mapping of values of the... |
| chfacfpmmulgsum 22213 | Breaking up a sum of value... |
| chfacfpmmulgsum2 22214 | Breaking up a sum of value... |
| cayhamlem1 22215 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 22216 | The right-hand fundamental... |
| cpmidgsum 22217 | Representation of the iden... |
| cpmidgsumm2pm 22218 | Representation of the iden... |
| cpmidpmatlem1 22219 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 22220 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 22221 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 22222 | Representation of the iden... |
| cpmadugsumlemB 22223 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 22224 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 22225 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 22226 | The product of the charact... |
| cpmadugsum 22227 | The product of the charact... |
| cpmidgsum2 22228 | Representation of the iden... |
| cpmidg2sum 22229 | Equality of two sums repre... |
| cpmadumatpolylem1 22230 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 22231 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 22232 | The product of the charact... |
| cayhamlem2 22233 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 22234 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 22235 | The coefficients of the ch... |
| cayhamlem3 22236 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 22237 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 22238 | The Cayley-Hamilton theore... |
| cayleyhamilton 22239 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 22240 | Alternate proof of ~ cayle... |
| cayleyhamilton1 22241 | The Cayley-Hamilton theore... |
| istopg 22244 | Express the predicate " ` ... |
| istop2g 22245 | Express the predicate " ` ... |
| uniopn 22246 | The union of a subset of a... |
| iunopn 22247 | The indexed union of a sub... |
| inopn 22248 | The intersection of two op... |
| fitop 22249 | A topology is closed under... |
| fiinopn 22250 | The intersection of a none... |
| iinopn 22251 | The intersection of a none... |
| unopn 22252 | The union of two open sets... |
| 0opn 22253 | The empty set is an open s... |
| 0ntop 22254 | The empty set is not a top... |
| topopn 22255 | The underlying set of a to... |
| eltopss 22256 | A member of a topology is ... |
| riinopn 22257 | A finite indexed relative ... |
| rintopn 22258 | A finite relative intersec... |
| istopon 22261 | Property of being a topolo... |
| topontop 22262 | A topology on a given base... |
| toponuni 22263 | The base set of a topology... |
| topontopi 22264 | A topology on a given base... |
| toponunii 22265 | The base set of a topology... |
| toptopon 22266 | Alternative definition of ... |
| toptopon2 22267 | A topology is the same thi... |
| topontopon 22268 | A topology on a set is a t... |
| funtopon 22269 | The class ` TopOn ` is a f... |
| toponrestid 22270 | Given a topology on a set,... |
| toponsspwpw 22271 | The set of topologies on a... |
| dmtopon 22272 | The domain of ` TopOn ` is... |
| fntopon 22273 | The class ` TopOn ` is a f... |
| toprntopon 22274 | A topology is the same thi... |
| toponmax 22275 | The base set of a topology... |
| toponss 22276 | A member of a topology is ... |
| toponcom 22277 | If ` K ` is a topology on ... |
| toponcomb 22278 | Biconditional form of ~ to... |
| topgele 22279 | The topologies over the sa... |
| topsn 22280 | The only topology on a sin... |
| istps 22283 | Express the predicate "is ... |
| istps2 22284 | Express the predicate "is ... |
| tpsuni 22285 | The base set of a topologi... |
| tpstop 22286 | The topology extractor on ... |
| tpspropd 22287 | A topological space depend... |
| tpsprop2d 22288 | A topological space depend... |
| topontopn 22289 | Express the predicate "is ... |
| tsettps 22290 | If the topology component ... |
| istpsi 22291 | Properties that determine ... |
| eltpsg 22292 | Properties that determine ... |
| eltpsgOLD 22293 | Obsolete version of ~ eltp... |
| eltpsi 22294 | Properties that determine ... |
| isbasisg 22297 | Express the predicate "the... |
| isbasis2g 22298 | Express the predicate "the... |
| isbasis3g 22299 | Express the predicate "the... |
| basis1 22300 | Property of a basis. (Con... |
| basis2 22301 | Property of a basis. (Con... |
| fiinbas 22302 | If a set is closed under f... |
| basdif0 22303 | A basis is not affected by... |
| baspartn 22304 | A disjoint system of sets ... |
| tgval 22305 | The topology generated by ... |
| tgval2 22306 | Definition of a topology g... |
| eltg 22307 | Membership in a topology g... |
| eltg2 22308 | Membership in a topology g... |
| eltg2b 22309 | Membership in a topology g... |
| eltg4i 22310 | An open set in a topology ... |
| eltg3i 22311 | The union of a set of basi... |
| eltg3 22312 | Membership in a topology g... |
| tgval3 22313 | Alternate expression for t... |
| tg1 22314 | Property of a member of a ... |
| tg2 22315 | Property of a member of a ... |
| bastg 22316 | A member of a basis is a s... |
| unitg 22317 | The topology generated by ... |
| tgss 22318 | Subset relation for genera... |
| tgcl 22319 | Show that a basis generate... |
| tgclb 22320 | The property ~ tgcl can be... |
| tgtopon 22321 | A basis generates a topolo... |
| topbas 22322 | A topology is its own basi... |
| tgtop 22323 | A topology is its own basi... |
| eltop 22324 | Membership in a topology, ... |
| eltop2 22325 | Membership in a topology. ... |
| eltop3 22326 | Membership in a topology. ... |
| fibas 22327 | A collection of finite int... |
| tgdom 22328 | A space has no more open s... |
| tgiun 22329 | The indexed union of a set... |
| tgidm 22330 | The topology generator fun... |
| bastop 22331 | Two ways to express that a... |
| tgtop11 22332 | The topology generation fu... |
| 0top 22333 | The singleton of the empty... |
| en1top 22334 | ` { (/) } ` is the only to... |
| en2top 22335 | If a topology has two elem... |
| tgss3 22336 | A criterion for determinin... |
| tgss2 22337 | A criterion for determinin... |
| basgen 22338 | Given a topology ` J ` , s... |
| basgen2 22339 | Given a topology ` J ` , s... |
| 2basgen 22340 | Conditions that determine ... |
| tgfiss 22341 | If a subbase is included i... |
| tgdif0 22342 | A generated topology is no... |
| bastop1 22343 | A subset of a topology is ... |
| bastop2 22344 | A version of ~ bastop1 tha... |
| distop 22345 | The discrete topology on a... |
| topnex 22346 | The class of all topologie... |
| distopon 22347 | The discrete topology on a... |
| sn0topon 22348 | The singleton of the empty... |
| sn0top 22349 | The singleton of the empty... |
| indislem 22350 | A lemma to eliminate some ... |
| indistopon 22351 | The indiscrete topology on... |
| indistop 22352 | The indiscrete topology on... |
| indisuni 22353 | The base set of the indisc... |
| fctop 22354 | The finite complement topo... |
| fctop2 22355 | The finite complement topo... |
| cctop 22356 | The countable complement t... |
| ppttop 22357 | The particular point topol... |
| pptbas 22358 | The particular point topol... |
| epttop 22359 | The excluded point topolog... |
| indistpsx 22360 | The indiscrete topology on... |
| indistps 22361 | The indiscrete topology on... |
| indistps2 22362 | The indiscrete topology on... |
| indistpsALT 22363 | The indiscrete topology on... |
| indistpsALTOLD 22364 | Obsolete proof of ~ indist... |
| indistps2ALT 22365 | The indiscrete topology on... |
| distps 22366 | The discrete topology on a... |
| fncld 22373 | The closed-set generator i... |
| cldval 22374 | The set of closed sets of ... |
| ntrfval 22375 | The interior function on t... |
| clsfval 22376 | The closure function on th... |
| cldrcl 22377 | Reverse closure of the clo... |
| iscld 22378 | The predicate "the class `... |
| iscld2 22379 | A subset of the underlying... |
| cldss 22380 | A closed set is a subset o... |
| cldss2 22381 | The set of closed sets is ... |
| cldopn 22382 | The complement of a closed... |
| isopn2 22383 | A subset of the underlying... |
| opncld 22384 | The complement of an open ... |
| difopn 22385 | The difference of a closed... |
| topcld 22386 | The underlying set of a to... |
| ntrval 22387 | The interior of a subset o... |
| clsval 22388 | The closure of a subset of... |
| 0cld 22389 | The empty set is closed. ... |
| iincld 22390 | The indexed intersection o... |
| intcld 22391 | The intersection of a set ... |
| uncld 22392 | The union of two closed se... |
| cldcls 22393 | A closed subset equals its... |
| incld 22394 | The intersection of two cl... |
| riincld 22395 | An indexed relative inters... |
| iuncld 22396 | A finite indexed union of ... |
| unicld 22397 | A finite union of closed s... |
| clscld 22398 | The closure of a subset of... |
| clsf 22399 | The closure function is a ... |
| ntropn 22400 | The interior of a subset o... |
| clsval2 22401 | Express closure in terms o... |
| ntrval2 22402 | Interior expressed in term... |
| ntrdif 22403 | An interior of a complemen... |
| clsdif 22404 | A closure of a complement ... |
| clsss 22405 | Subset relationship for cl... |
| ntrss 22406 | Subset relationship for in... |
| sscls 22407 | A subset of a topology's u... |
| ntrss2 22408 | A subset includes its inte... |
| ssntr 22409 | An open subset of a set is... |
| clsss3 22410 | The closure of a subset of... |
| ntrss3 22411 | The interior of a subset o... |
| ntrin 22412 | A pairwise intersection of... |
| cmclsopn 22413 | The complement of a closur... |
| cmntrcld 22414 | The complement of an inter... |
| iscld3 22415 | A subset is closed iff it ... |
| iscld4 22416 | A subset is closed iff it ... |
| isopn3 22417 | A subset is open iff it eq... |
| clsidm 22418 | The closure operation is i... |
| ntridm 22419 | The interior operation is ... |
| clstop 22420 | The closure of a topology'... |
| ntrtop 22421 | The interior of a topology... |
| 0ntr 22422 | A subset with an empty int... |
| clsss2 22423 | If a subset is included in... |
| elcls 22424 | Membership in a closure. ... |
| elcls2 22425 | Membership in a closure. ... |
| clsndisj 22426 | Any open set containing a ... |
| ntrcls0 22427 | A subset whose closure has... |
| ntreq0 22428 | Two ways to say that a sub... |
| cldmre 22429 | The closed sets of a topol... |
| mrccls 22430 | Moore closure generalizes ... |
| cls0 22431 | The closure of the empty s... |
| ntr0 22432 | The interior of the empty ... |
| isopn3i 22433 | An open subset equals its ... |
| elcls3 22434 | Membership in a closure in... |
| opncldf1 22435 | A bijection useful for con... |
| opncldf2 22436 | The values of the open-clo... |
| opncldf3 22437 | The values of the converse... |
| isclo 22438 | A set ` A ` is clopen iff ... |
| isclo2 22439 | A set ` A ` is clopen iff ... |
| discld 22440 | The open sets of a discret... |
| sn0cld 22441 | The closed sets of the top... |
| indiscld 22442 | The closed sets of an indi... |
| mretopd 22443 | A Moore collection which i... |
| toponmre 22444 | The topologies over a give... |
| cldmreon 22445 | The closed sets of a topol... |
| iscldtop 22446 | A family is the closed set... |
| mreclatdemoBAD 22447 | The closed subspaces of a ... |
| neifval 22450 | Value of the neighborhood ... |
| neif 22451 | The neighborhood function ... |
| neiss2 22452 | A set with a neighborhood ... |
| neival 22453 | Value of the set of neighb... |
| isnei 22454 | The predicate "the class `... |
| neiint 22455 | An intuitive definition of... |
| isneip 22456 | The predicate "the class `... |
| neii1 22457 | A neighborhood is included... |
| neisspw 22458 | The neighborhoods of any s... |
| neii2 22459 | Property of a neighborhood... |
| neiss 22460 | Any neighborhood of a set ... |
| ssnei 22461 | A set is included in any o... |
| elnei 22462 | A point belongs to any of ... |
| 0nnei 22463 | The empty set is not a nei... |
| neips 22464 | A neighborhood of a set is... |
| opnneissb 22465 | An open set is a neighborh... |
| opnssneib 22466 | Any superset of an open se... |
| ssnei2 22467 | Any subset ` M ` of ` X ` ... |
| neindisj 22468 | Any neighborhood of an ele... |
| opnneiss 22469 | An open set is a neighborh... |
| opnneip 22470 | An open set is a neighborh... |
| opnnei 22471 | A set is open iff it is a ... |
| tpnei 22472 | The underlying set of a to... |
| neiuni 22473 | The union of the neighborh... |
| neindisj2 22474 | A point ` P ` belongs to t... |
| topssnei 22475 | A finer topology has more ... |
| innei 22476 | The intersection of two ne... |
| opnneiid 22477 | Only an open set is a neig... |
| neissex 22478 | For any neighborhood ` N `... |
| 0nei 22479 | The empty set is a neighbo... |
| neipeltop 22480 | Lemma for ~ neiptopreu . ... |
| neiptopuni 22481 | Lemma for ~ neiptopreu . ... |
| neiptoptop 22482 | Lemma for ~ neiptopreu . ... |
| neiptopnei 22483 | Lemma for ~ neiptopreu . ... |
| neiptopreu 22484 | If, to each element ` P ` ... |
| lpfval 22489 | The limit point function o... |
| lpval 22490 | The set of limit points of... |
| islp 22491 | The predicate "the class `... |
| lpsscls 22492 | The limit points of a subs... |
| lpss 22493 | The limit points of a subs... |
| lpdifsn 22494 | ` P ` is a limit point of ... |
| lpss3 22495 | Subset relationship for li... |
| islp2 22496 | The predicate " ` P ` is a... |
| islp3 22497 | The predicate " ` P ` is a... |
| maxlp 22498 | A point is a limit point o... |
| clslp 22499 | The closure of a subset of... |
| islpi 22500 | A point belonging to a set... |
| cldlp 22501 | A subset of a topological ... |
| isperf 22502 | Definition of a perfect sp... |
| isperf2 22503 | Definition of a perfect sp... |
| isperf3 22504 | A perfect space is a topol... |
| perflp 22505 | The limit points of a perf... |
| perfi 22506 | Property of a perfect spac... |
| perftop 22507 | A perfect space is a topol... |
| restrcl 22508 | Reverse closure for the su... |
| restbas 22509 | A subspace topology basis ... |
| tgrest 22510 | A subspace can be generate... |
| resttop 22511 | A subspace topology is a t... |
| resttopon 22512 | A subspace topology is a t... |
| restuni 22513 | The underlying set of a su... |
| stoig 22514 | The topological space buil... |
| restco 22515 | Composition of subspaces. ... |
| restabs 22516 | Equivalence of being a sub... |
| restin 22517 | When the subspace region i... |
| restuni2 22518 | The underlying set of a su... |
| resttopon2 22519 | The underlying set of a su... |
| rest0 22520 | The subspace topology indu... |
| restsn 22521 | The only subspace topology... |
| restsn2 22522 | The subspace topology indu... |
| restcld 22523 | A closed set of a subspace... |
| restcldi 22524 | A closed set is closed in ... |
| restcldr 22525 | A set which is closed in t... |
| restopnb 22526 | If ` B ` is an open subset... |
| ssrest 22527 | If ` K ` is a finer topolo... |
| restopn2 22528 | If ` A ` is open, then ` B... |
| restdis 22529 | A subspace of a discrete t... |
| restfpw 22530 | The restriction of the set... |
| neitr 22531 | The neighborhood of a trac... |
| restcls 22532 | A closure in a subspace to... |
| restntr 22533 | An interior in a subspace ... |
| restlp 22534 | The limit points of a subs... |
| restperf 22535 | Perfection of a subspace. ... |
| perfopn 22536 | An open subset of a perfec... |
| resstopn 22537 | The topology of a restrict... |
| resstps 22538 | A restricted topological s... |
| ordtbaslem 22539 | Lemma for ~ ordtbas . In ... |
| ordtval 22540 | Value of the order topolog... |
| ordtuni 22541 | Value of the order topolog... |
| ordtbas2 22542 | Lemma for ~ ordtbas . (Co... |
| ordtbas 22543 | In a total order, the fini... |
| ordttopon 22544 | Value of the order topolog... |
| ordtopn1 22545 | An upward ray ` ( P , +oo ... |
| ordtopn2 22546 | A downward ray ` ( -oo , P... |
| ordtopn3 22547 | An open interval ` ( A , B... |
| ordtcld1 22548 | A downward ray ` ( -oo , P... |
| ordtcld2 22549 | An upward ray ` [ P , +oo ... |
| ordtcld3 22550 | A closed interval ` [ A , ... |
| ordttop 22551 | The order topology is a to... |
| ordtcnv 22552 | The order dual generates t... |
| ordtrest 22553 | The subspace topology of a... |
| ordtrest2lem 22554 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 22555 | An interval-closed set ` A... |
| letopon 22556 | The topology of the extend... |
| letop 22557 | The topology of the extend... |
| letopuni 22558 | The topology of the extend... |
| xrstopn 22559 | The topology component of ... |
| xrstps 22560 | The extended real number s... |
| leordtvallem1 22561 | Lemma for ~ leordtval . (... |
| leordtvallem2 22562 | Lemma for ~ leordtval . (... |
| leordtval2 22563 | The topology of the extend... |
| leordtval 22564 | The topology of the extend... |
| iccordt 22565 | A closed interval is close... |
| iocpnfordt 22566 | An unbounded above open in... |
| icomnfordt 22567 | An unbounded above open in... |
| iooordt 22568 | An open interval is open i... |
| reordt 22569 | The real numbers are an op... |
| lecldbas 22570 | The set of closed interval... |
| pnfnei 22571 | A neighborhood of ` +oo ` ... |
| mnfnei 22572 | A neighborhood of ` -oo ` ... |
| ordtrestixx 22573 | The restriction of the les... |
| ordtresticc 22574 | The restriction of the les... |
| lmrel 22581 | The topological space conv... |
| lmrcl 22582 | Reverse closure for the co... |
| lmfval 22583 | The relation "sequence ` f... |
| cnfval 22584 | The set of all continuous ... |
| cnpfval 22585 | The function mapping the p... |
| iscn 22586 | The predicate "the class `... |
| cnpval 22587 | The set of all functions f... |
| iscnp 22588 | The predicate "the class `... |
| iscn2 22589 | The predicate "the class `... |
| iscnp2 22590 | The predicate "the class `... |
| cntop1 22591 | Reverse closure for a cont... |
| cntop2 22592 | Reverse closure for a cont... |
| cnptop1 22593 | Reverse closure for a func... |
| cnptop2 22594 | Reverse closure for a func... |
| iscnp3 22595 | The predicate "the class `... |
| cnprcl 22596 | Reverse closure for a func... |
| cnf 22597 | A continuous function is a... |
| cnpf 22598 | A continuous function at p... |
| cnpcl 22599 | The value of a continuous ... |
| cnf2 22600 | A continuous function is a... |
| cnpf2 22601 | A continuous function at p... |
| cnprcl2 22602 | Reverse closure for a func... |
| tgcn 22603 | The continuity predicate w... |
| tgcnp 22604 | The "continuous at a point... |
| subbascn 22605 | The continuity predicate w... |
| ssidcn 22606 | The identity function is a... |
| cnpimaex 22607 | Property of a function con... |
| idcn 22608 | A restricted identity func... |
| lmbr 22609 | Express the binary relatio... |
| lmbr2 22610 | Express the binary relatio... |
| lmbrf 22611 | Express the binary relatio... |
| lmconst 22612 | A constant sequence conver... |
| lmcvg 22613 | Convergence property of a ... |
| iscnp4 22614 | The predicate "the class `... |
| cnpnei 22615 | A condition for continuity... |
| cnima 22616 | An open subset of the codo... |
| cnco 22617 | The composition of two con... |
| cnpco 22618 | The composition of a funct... |
| cnclima 22619 | A closed subset of the cod... |
| iscncl 22620 | A characterization of a co... |
| cncls2i 22621 | Property of the preimage o... |
| cnntri 22622 | Property of the preimage o... |
| cnclsi 22623 | Property of the image of a... |
| cncls2 22624 | Continuity in terms of clo... |
| cncls 22625 | Continuity in terms of clo... |
| cnntr 22626 | Continuity in terms of int... |
| cnss1 22627 | If the topology ` K ` is f... |
| cnss2 22628 | If the topology ` K ` is f... |
| cncnpi 22629 | A continuous function is c... |
| cnsscnp 22630 | The set of continuous func... |
| cncnp 22631 | A continuous function is c... |
| cncnp2 22632 | A continuous function is c... |
| cnnei 22633 | Continuity in terms of nei... |
| cnconst2 22634 | A constant function is con... |
| cnconst 22635 | A constant function is con... |
| cnrest 22636 | Continuity of a restrictio... |
| cnrest2 22637 | Equivalence of continuity ... |
| cnrest2r 22638 | Equivalence of continuity ... |
| cnpresti 22639 | One direction of ~ cnprest... |
| cnprest 22640 | Equivalence of continuity ... |
| cnprest2 22641 | Equivalence of point-conti... |
| cndis 22642 | Every function is continuo... |
| cnindis 22643 | Every function is continuo... |
| cnpdis 22644 | If ` A ` is an isolated po... |
| paste 22645 | Pasting lemma. If ` A ` a... |
| lmfpm 22646 | If ` F ` converges, then `... |
| lmfss 22647 | Inclusion of a function ha... |
| lmcl 22648 | Closure of a limit. (Cont... |
| lmss 22649 | Limit on a subspace. (Con... |
| sslm 22650 | A finer topology has fewer... |
| lmres 22651 | A function converges iff i... |
| lmff 22652 | If ` F ` converges, there ... |
| lmcls 22653 | Any convergent sequence of... |
| lmcld 22654 | Any convergent sequence of... |
| lmcnp 22655 | The image of a convergent ... |
| lmcn 22656 | The image of a convergent ... |
| ist0 22671 | The predicate "is a T_0 sp... |
| ist1 22672 | The predicate "is a T_1 sp... |
| ishaus 22673 | The predicate "is a Hausdo... |
| iscnrm 22674 | The property of being comp... |
| t0sep 22675 | Any two topologically indi... |
| t0dist 22676 | Any two distinct points in... |
| t1sncld 22677 | In a T_1 space, singletons... |
| t1ficld 22678 | In a T_1 space, finite set... |
| hausnei 22679 | Neighborhood property of a... |
| t0top 22680 | A T_0 space is a topologic... |
| t1top 22681 | A T_1 space is a topologic... |
| haustop 22682 | A Hausdorff space is a top... |
| isreg 22683 | The predicate "is a regula... |
| regtop 22684 | A regular space is a topol... |
| regsep 22685 | In a regular space, every ... |
| isnrm 22686 | The predicate "is a normal... |
| nrmtop 22687 | A normal space is a topolo... |
| cnrmtop 22688 | A completely normal space ... |
| iscnrm2 22689 | The property of being comp... |
| ispnrm 22690 | The property of being perf... |
| pnrmnrm 22691 | A perfectly normal space i... |
| pnrmtop 22692 | A perfectly normal space i... |
| pnrmcld 22693 | A closed set in a perfectl... |
| pnrmopn 22694 | An open set in a perfectly... |
| ist0-2 22695 | The predicate "is a T_0 sp... |
| ist0-3 22696 | The predicate "is a T_0 sp... |
| cnt0 22697 | The preimage of a T_0 topo... |
| ist1-2 22698 | An alternate characterizat... |
| t1t0 22699 | A T_1 space is a T_0 space... |
| ist1-3 22700 | A space is T_1 iff every p... |
| cnt1 22701 | The preimage of a T_1 topo... |
| ishaus2 22702 | Express the predicate " ` ... |
| haust1 22703 | A Hausdorff space is a T_1... |
| hausnei2 22704 | The Hausdorff condition st... |
| cnhaus 22705 | The preimage of a Hausdorf... |
| nrmsep3 22706 | In a normal space, given a... |
| nrmsep2 22707 | In a normal space, any two... |
| nrmsep 22708 | In a normal space, disjoin... |
| isnrm2 22709 | An alternate characterizat... |
| isnrm3 22710 | A topological space is nor... |
| cnrmi 22711 | A subspace of a completely... |
| cnrmnrm 22712 | A completely normal space ... |
| restcnrm 22713 | A subspace of a completely... |
| resthauslem 22714 | Lemma for ~ resthaus and s... |
| lpcls 22715 | The limit points of the cl... |
| perfcls 22716 | A subset of a perfect spac... |
| restt0 22717 | A subspace of a T_0 topolo... |
| restt1 22718 | A subspace of a T_1 topolo... |
| resthaus 22719 | A subspace of a Hausdorff ... |
| t1sep2 22720 | Any two points in a T_1 sp... |
| t1sep 22721 | Any two distinct points in... |
| sncld 22722 | A singleton is closed in a... |
| sshauslem 22723 | Lemma for ~ sshaus and sim... |
| sst0 22724 | A topology finer than a T_... |
| sst1 22725 | A topology finer than a T_... |
| sshaus 22726 | A topology finer than a Ha... |
| regsep2 22727 | In a regular space, a clos... |
| isreg2 22728 | A topological space is reg... |
| dnsconst 22729 | If a continuous mapping to... |
| ordtt1 22730 | The order topology is T_1 ... |
| lmmo 22731 | A sequence in a Hausdorff ... |
| lmfun 22732 | The convergence relation i... |
| dishaus 22733 | A discrete topology is Hau... |
| ordthauslem 22734 | Lemma for ~ ordthaus . (C... |
| ordthaus 22735 | The order topology of a to... |
| xrhaus 22736 | The topology of the extend... |
| iscmp 22739 | The predicate "is a compac... |
| cmpcov 22740 | An open cover of a compact... |
| cmpcov2 22741 | Rewrite ~ cmpcov for the c... |
| cmpcovf 22742 | Combine ~ cmpcov with ~ ac... |
| cncmp 22743 | Compactness is respected b... |
| fincmp 22744 | A finite topology is compa... |
| 0cmp 22745 | The singleton of the empty... |
| cmptop 22746 | A compact topology is a to... |
| rncmp 22747 | The image of a compact set... |
| imacmp 22748 | The image of a compact set... |
| discmp 22749 | A discrete topology is com... |
| cmpsublem 22750 | Lemma for ~ cmpsub . (Con... |
| cmpsub 22751 | Two equivalent ways of des... |
| tgcmp 22752 | A topology generated by a ... |
| cmpcld 22753 | A closed subset of a compa... |
| uncmp 22754 | The union of two compact s... |
| fiuncmp 22755 | A finite union of compact ... |
| sscmp 22756 | A subset of a compact topo... |
| hauscmplem 22757 | Lemma for ~ hauscmp . (Co... |
| hauscmp 22758 | A compact subspace of a T2... |
| cmpfi 22759 | If a topology is compact a... |
| cmpfii 22760 | In a compact topology, a s... |
| bwth 22761 | The glorious Bolzano-Weier... |
| isconn 22764 | The predicate ` J ` is a c... |
| isconn2 22765 | The predicate ` J ` is a c... |
| connclo 22766 | The only nonempty clopen s... |
| conndisj 22767 | If a topology is connected... |
| conntop 22768 | A connected topology is a ... |
| indisconn 22769 | The indiscrete topology (o... |
| dfconn2 22770 | An alternate definition of... |
| connsuba 22771 | Connectedness for a subspa... |
| connsub 22772 | Two equivalent ways of say... |
| cnconn 22773 | Connectedness is respected... |
| nconnsubb 22774 | Disconnectedness for a sub... |
| connsubclo 22775 | If a clopen set meets a co... |
| connima 22776 | The image of a connected s... |
| conncn 22777 | A continuous function from... |
| iunconnlem 22778 | Lemma for ~ iunconn . (Co... |
| iunconn 22779 | The indexed union of conne... |
| unconn 22780 | The union of two connected... |
| clsconn 22781 | The closure of a connected... |
| conncompid 22782 | The connected component co... |
| conncompconn 22783 | The connected component co... |
| conncompss 22784 | The connected component co... |
| conncompcld 22785 | The connected component co... |
| conncompclo 22786 | The connected component co... |
| t1connperf 22787 | A connected T_1 space is p... |
| is1stc 22792 | The predicate "is a first-... |
| is1stc2 22793 | An equivalent way of sayin... |
| 1stctop 22794 | A first-countable topology... |
| 1stcclb 22795 | A property of points in a ... |
| 1stcfb 22796 | For any point ` A ` in a f... |
| is2ndc 22797 | The property of being seco... |
| 2ndctop 22798 | A second-countable topolog... |
| 2ndci 22799 | A countable basis generate... |
| 2ndcsb 22800 | Having a countable subbase... |
| 2ndcredom 22801 | A second-countable space h... |
| 2ndc1stc 22802 | A second-countable space i... |
| 1stcrestlem 22803 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 22804 | A subspace of a first-coun... |
| 2ndcrest 22805 | A subspace of a second-cou... |
| 2ndcctbss 22806 | If a topology is second-co... |
| 2ndcdisj 22807 | Any disjoint family of ope... |
| 2ndcdisj2 22808 | Any disjoint collection of... |
| 2ndcomap 22809 | A surjective continuous op... |
| 2ndcsep 22810 | A second-countable topolog... |
| dis2ndc 22811 | A discrete space is second... |
| 1stcelcls 22812 | A point belongs to the clo... |
| 1stccnp 22813 | A mapping is continuous at... |
| 1stccn 22814 | A mapping ` X --> Y ` , wh... |
| islly 22819 | The property of being a lo... |
| isnlly 22820 | The property of being an n... |
| llyeq 22821 | Equality theorem for the `... |
| nllyeq 22822 | Equality theorem for the `... |
| llytop 22823 | A locally ` A ` space is a... |
| nllytop 22824 | A locally ` A ` space is a... |
| llyi 22825 | The property of a locally ... |
| nllyi 22826 | The property of an n-local... |
| nlly2i 22827 | Eliminate the neighborhood... |
| llynlly 22828 | A locally ` A ` space is n... |
| llyssnlly 22829 | A locally ` A ` space is n... |
| llyss 22830 | The "locally" predicate re... |
| nllyss 22831 | The "n-locally" predicate ... |
| subislly 22832 | The property of a subspace... |
| restnlly 22833 | If the property ` A ` pass... |
| restlly 22834 | If the property ` A ` pass... |
| islly2 22835 | An alternative expression ... |
| llyrest 22836 | An open subspace of a loca... |
| nllyrest 22837 | An open subspace of an n-l... |
| loclly 22838 | If ` A ` is a local proper... |
| llyidm 22839 | Idempotence of the "locall... |
| nllyidm 22840 | Idempotence of the "n-loca... |
| toplly 22841 | A topology is locally a to... |
| topnlly 22842 | A topology is n-locally a ... |
| hauslly 22843 | A Hausdorff space is local... |
| hausnlly 22844 | A Hausdorff space is n-loc... |
| hausllycmp 22845 | A compact Hausdorff space ... |
| cldllycmp 22846 | A closed subspace of a loc... |
| lly1stc 22847 | First-countability is a lo... |
| dislly 22848 | The discrete space ` ~P X ... |
| disllycmp 22849 | A discrete space is locall... |
| dis1stc 22850 | A discrete space is first-... |
| hausmapdom 22851 | If ` X ` is a first-counta... |
| hauspwdom 22852 | Simplify the cardinal ` A ... |
| refrel 22859 | Refinement is a relation. ... |
| isref 22860 | The property of being a re... |
| refbas 22861 | A refinement covers the sa... |
| refssex 22862 | Every set in a refinement ... |
| ssref 22863 | A subcover is a refinement... |
| refref 22864 | Reflexivity of refinement.... |
| reftr 22865 | Refinement is transitive. ... |
| refun0 22866 | Adding the empty set prese... |
| isptfin 22867 | The statement "is a point-... |
| islocfin 22868 | The statement "is a locall... |
| finptfin 22869 | A finite cover is a point-... |
| ptfinfin 22870 | A point covered by a point... |
| finlocfin 22871 | A finite cover of a topolo... |
| locfintop 22872 | A locally finite cover cov... |
| locfinbas 22873 | A locally finite cover mus... |
| locfinnei 22874 | A point covered by a local... |
| lfinpfin 22875 | A locally finite cover is ... |
| lfinun 22876 | Adding a finite set preser... |
| locfincmp 22877 | For a compact space, the l... |
| unisngl 22878 | Taking the union of the se... |
| dissnref 22879 | The set of singletons is a... |
| dissnlocfin 22880 | The set of singletons is l... |
| locfindis 22881 | The locally finite covers ... |
| locfincf 22882 | A locally finite cover in ... |
| comppfsc 22883 | A space where every open c... |
| kgenval 22886 | Value of the compact gener... |
| elkgen 22887 | Value of the compact gener... |
| kgeni 22888 | Property of the open sets ... |
| kgentopon 22889 | The compact generator gene... |
| kgenuni 22890 | The base set of the compac... |
| kgenftop 22891 | The compact generator gene... |
| kgenf 22892 | The compact generator is a... |
| kgentop 22893 | A compactly generated spac... |
| kgenss 22894 | The compact generator gene... |
| kgenhaus 22895 | The compact generator gene... |
| kgencmp 22896 | The compact generator topo... |
| kgencmp2 22897 | The compact generator topo... |
| kgenidm 22898 | The compact generator is i... |
| iskgen2 22899 | A space is compactly gener... |
| iskgen3 22900 | Derive the usual definitio... |
| llycmpkgen2 22901 | A locally compact space is... |
| cmpkgen 22902 | A compact space is compact... |
| llycmpkgen 22903 | A locally compact space is... |
| 1stckgenlem 22904 | The one-point compactifica... |
| 1stckgen 22905 | A first-countable space is... |
| kgen2ss 22906 | The compact generator pres... |
| kgencn 22907 | A function from a compactl... |
| kgencn2 22908 | A function ` F : J --> K `... |
| kgencn3 22909 | The set of continuous func... |
| kgen2cn 22910 | A continuous function is a... |
| txval 22915 | Value of the binary topolo... |
| txuni2 22916 | The underlying set of the ... |
| txbasex 22917 | The basis for the product ... |
| txbas 22918 | The set of Cartesian produ... |
| eltx 22919 | A set in a product is open... |
| txtop 22920 | The product of two topolog... |
| ptval 22921 | The value of the product t... |
| ptpjpre1 22922 | The preimage of a projecti... |
| elpt 22923 | Elementhood in the bases o... |
| elptr 22924 | A basic open set in the pr... |
| elptr2 22925 | A basic open set in the pr... |
| ptbasid 22926 | The base set of the produc... |
| ptuni2 22927 | The base set for the produ... |
| ptbasin 22928 | The basis for a product to... |
| ptbasin2 22929 | The basis for a product to... |
| ptbas 22930 | The basis for a product to... |
| ptpjpre2 22931 | The basis for a product to... |
| ptbasfi 22932 | The basis for the product ... |
| pttop 22933 | The product topology is a ... |
| ptopn 22934 | A basic open set in the pr... |
| ptopn2 22935 | A sub-basic open set in th... |
| xkotf 22936 | Functionality of function ... |
| xkobval 22937 | Alternative expression for... |
| xkoval 22938 | Value of the compact-open ... |
| xkotop 22939 | The compact-open topology ... |
| xkoopn 22940 | A basic open set of the co... |
| txtopi 22941 | The product of two topolog... |
| txtopon 22942 | The underlying set of the ... |
| txuni 22943 | The underlying set of the ... |
| txunii 22944 | The underlying set of the ... |
| ptuni 22945 | The base set for the produ... |
| ptunimpt 22946 | Base set of a product topo... |
| pttopon 22947 | The base set for the produ... |
| pttoponconst 22948 | The base set for a product... |
| ptuniconst 22949 | The base set for a product... |
| xkouni 22950 | The base set of the compac... |
| xkotopon 22951 | The base set of the compac... |
| ptval2 22952 | The value of the product t... |
| txopn 22953 | The product of two open se... |
| txcld 22954 | The product of two closed ... |
| txcls 22955 | Closure of a rectangle in ... |
| txss12 22956 | Subset property of the top... |
| txbasval 22957 | It is sufficient to consid... |
| neitx 22958 | The Cartesian product of t... |
| txcnpi 22959 | Continuity of a two-argume... |
| tx1cn 22960 | Continuity of the first pr... |
| tx2cn 22961 | Continuity of the second p... |
| ptpjcn 22962 | Continuity of a projection... |
| ptpjopn 22963 | The projection map is an o... |
| ptcld 22964 | A closed box in the produc... |
| ptcldmpt 22965 | A closed box in the produc... |
| ptclsg 22966 | The closure of a box in th... |
| ptcls 22967 | The closure of a box in th... |
| dfac14lem 22968 | Lemma for ~ dfac14 . By e... |
| dfac14 22969 | Theorem ~ ptcls is an equi... |
| xkoccn 22970 | The "constant function" fu... |
| txcnp 22971 | If two functions are conti... |
| ptcnplem 22972 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 22973 | If every projection of a f... |
| upxp 22974 | Universal property of the ... |
| txcnmpt 22975 | A map into the product of ... |
| uptx 22976 | Universal property of the ... |
| txcn 22977 | A map into the product of ... |
| ptcn 22978 | If every projection of a f... |
| prdstopn 22979 | Topology of a structure pr... |
| prdstps 22980 | A structure product of top... |
| pwstps 22981 | A structure power of a top... |
| txrest 22982 | The subspace of a topologi... |
| txdis 22983 | The topological product of... |
| txindislem 22984 | Lemma for ~ txindis . (Co... |
| txindis 22985 | The topological product of... |
| txdis1cn 22986 | A function is jointly cont... |
| txlly 22987 | If the property ` A ` is p... |
| txnlly 22988 | If the property ` A ` is p... |
| pthaus 22989 | The product of a collectio... |
| ptrescn 22990 | Restriction is a continuou... |
| txtube 22991 | The "tube lemma". If ` X ... |
| txcmplem1 22992 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 22993 | Lemma for ~ txcmp . (Cont... |
| txcmp 22994 | The topological product of... |
| txcmpb 22995 | The topological product of... |
| hausdiag 22996 | A topology is Hausdorff if... |
| hauseqlcld 22997 | In a Hausdorff topology, t... |
| txhaus 22998 | The topological product of... |
| txlm 22999 | Two sequences converge iff... |
| lmcn2 23000 | The image of a convergent ... |
| tx1stc 23001 | The topological product of... |
| tx2ndc 23002 | The topological product of... |
| txkgen 23003 | The topological product of... |
| xkohaus 23004 | If the codomain space is H... |
| xkoptsub 23005 | The compact-open topology ... |
| xkopt 23006 | The compact-open topology ... |
| xkopjcn 23007 | Continuity of a projection... |
| xkoco1cn 23008 | If ` F ` is a continuous f... |
| xkoco2cn 23009 | If ` F ` is a continuous f... |
| xkococnlem 23010 | Continuity of the composit... |
| xkococn 23011 | Continuity of the composit... |
| cnmptid 23012 | The identity function is c... |
| cnmptc 23013 | A constant function is con... |
| cnmpt11 23014 | The composition of continu... |
| cnmpt11f 23015 | The composition of continu... |
| cnmpt1t 23016 | The composition of continu... |
| cnmpt12f 23017 | The composition of continu... |
| cnmpt12 23018 | The composition of continu... |
| cnmpt1st 23019 | The projection onto the fi... |
| cnmpt2nd 23020 | The projection onto the se... |
| cnmpt2c 23021 | A constant function is con... |
| cnmpt21 23022 | The composition of continu... |
| cnmpt21f 23023 | The composition of continu... |
| cnmpt2t 23024 | The composition of continu... |
| cnmpt22 23025 | The composition of continu... |
| cnmpt22f 23026 | The composition of continu... |
| cnmpt1res 23027 | The restriction of a conti... |
| cnmpt2res 23028 | The restriction of a conti... |
| cnmptcom 23029 | The argument converse of a... |
| cnmptkc 23030 | The curried first projecti... |
| cnmptkp 23031 | The evaluation of the inne... |
| cnmptk1 23032 | The composition of a curri... |
| cnmpt1k 23033 | The composition of a one-a... |
| cnmptkk 23034 | The composition of two cur... |
| xkofvcn 23035 | Joint continuity of the fu... |
| cnmptk1p 23036 | The evaluation of a currie... |
| cnmptk2 23037 | The uncurrying of a currie... |
| xkoinjcn 23038 | Continuity of "injection",... |
| cnmpt2k 23039 | The currying of a two-argu... |
| txconn 23040 | The topological product of... |
| imasnopn 23041 | If a relation graph is ope... |
| imasncld 23042 | If a relation graph is clo... |
| imasncls 23043 | If a relation graph is clo... |
| qtopval 23046 | Value of the quotient topo... |
| qtopval2 23047 | Value of the quotient topo... |
| elqtop 23048 | Value of the quotient topo... |
| qtopres 23049 | The quotient topology is u... |
| qtoptop2 23050 | The quotient topology is a... |
| qtoptop 23051 | The quotient topology is a... |
| elqtop2 23052 | Value of the quotient topo... |
| qtopuni 23053 | The base set of the quotie... |
| elqtop3 23054 | Value of the quotient topo... |
| qtoptopon 23055 | The base set of the quotie... |
| qtopid 23056 | A quotient map is a contin... |
| idqtop 23057 | The quotient topology indu... |
| qtopcmplem 23058 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 23059 | A quotient of a compact sp... |
| qtopconn 23060 | A quotient of a connected ... |
| qtopkgen 23061 | A quotient of a compactly ... |
| basqtop 23062 | An injection maps bases to... |
| tgqtop 23063 | An injection maps generate... |
| qtopcld 23064 | The property of being a cl... |
| qtopcn 23065 | Universal property of a qu... |
| qtopss 23066 | A surjective continuous fu... |
| qtopeu 23067 | Universal property of the ... |
| qtoprest 23068 | If ` A ` is a saturated op... |
| qtopomap 23069 | If ` F ` is a surjective c... |
| qtopcmap 23070 | If ` F ` is a surjective c... |
| imastopn 23071 | The topology of an image s... |
| imastps 23072 | The image of a topological... |
| qustps 23073 | A quotient structure is a ... |
| kqfval 23074 | Value of the function appe... |
| kqfeq 23075 | Two points in the Kolmogor... |
| kqffn 23076 | The topological indistingu... |
| kqval 23077 | Value of the quotient topo... |
| kqtopon 23078 | The Kolmogorov quotient is... |
| kqid 23079 | The topological indistingu... |
| ist0-4 23080 | The topological indistingu... |
| kqfvima 23081 | When the image set is open... |
| kqsat 23082 | Any open set is saturated ... |
| kqdisj 23083 | A version of ~ imain for t... |
| kqcldsat 23084 | Any closed set is saturate... |
| kqopn 23085 | The topological indistingu... |
| kqcld 23086 | The topological indistingu... |
| kqt0lem 23087 | Lemma for ~ kqt0 . (Contr... |
| isr0 23088 | The property " ` J ` is an... |
| r0cld 23089 | The analogue of the T_1 ax... |
| regr1lem 23090 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 23091 | A Kolmogorov quotient of a... |
| kqreglem1 23092 | A Kolmogorov quotient of a... |
| kqreglem2 23093 | If the Kolmogorov quotient... |
| kqnrmlem1 23094 | A Kolmogorov quotient of a... |
| kqnrmlem2 23095 | If the Kolmogorov quotient... |
| kqtop 23096 | The Kolmogorov quotient is... |
| kqt0 23097 | The Kolmogorov quotient is... |
| kqf 23098 | The Kolmogorov quotient is... |
| r0sep 23099 | The separation property of... |
| nrmr0reg 23100 | A normal R_0 space is also... |
| regr1 23101 | A regular space is R_1, wh... |
| kqreg 23102 | The Kolmogorov quotient of... |
| kqnrm 23103 | The Kolmogorov quotient of... |
| hmeofn 23108 | The set of homeomorphisms ... |
| hmeofval 23109 | The set of all the homeomo... |
| ishmeo 23110 | The predicate F is a homeo... |
| hmeocn 23111 | A homeomorphism is continu... |
| hmeocnvcn 23112 | The converse of a homeomor... |
| hmeocnv 23113 | The converse of a homeomor... |
| hmeof1o2 23114 | A homeomorphism is a 1-1-o... |
| hmeof1o 23115 | A homeomorphism is a 1-1-o... |
| hmeoima 23116 | The image of an open set b... |
| hmeoopn 23117 | Homeomorphisms preserve op... |
| hmeocld 23118 | Homeomorphisms preserve cl... |
| hmeocls 23119 | Homeomorphisms preserve cl... |
| hmeontr 23120 | Homeomorphisms preserve in... |
| hmeoimaf1o 23121 | The function mapping open ... |
| hmeores 23122 | The restriction of a homeo... |
| hmeoco 23123 | The composite of two homeo... |
| idhmeo 23124 | The identity function is a... |
| hmeocnvb 23125 | The converse of a homeomor... |
| hmeoqtop 23126 | A homeomorphism is a quoti... |
| hmph 23127 | Express the predicate ` J ... |
| hmphi 23128 | If there is a homeomorphis... |
| hmphtop 23129 | Reverse closure for the ho... |
| hmphtop1 23130 | The relation "being homeom... |
| hmphtop2 23131 | The relation "being homeom... |
| hmphref 23132 | "Is homeomorphic to" is re... |
| hmphsym 23133 | "Is homeomorphic to" is sy... |
| hmphtr 23134 | "Is homeomorphic to" is tr... |
| hmpher 23135 | "Is homeomorphic to" is an... |
| hmphen 23136 | Homeomorphisms preserve th... |
| hmphsymb 23137 | "Is homeomorphic to" is sy... |
| haushmphlem 23138 | Lemma for ~ haushmph and s... |
| cmphmph 23139 | Compactness is a topologic... |
| connhmph 23140 | Connectedness is a topolog... |
| t0hmph 23141 | T_0 is a topological prope... |
| t1hmph 23142 | T_1 is a topological prope... |
| haushmph 23143 | Hausdorff-ness is a topolo... |
| reghmph 23144 | Regularity is a topologica... |
| nrmhmph 23145 | Normality is a topological... |
| hmph0 23146 | A topology homeomorphic to... |
| hmphdis 23147 | Homeomorphisms preserve to... |
| hmphindis 23148 | Homeomorphisms preserve to... |
| indishmph 23149 | Equinumerous sets equipped... |
| hmphen2 23150 | Homeomorphisms preserve th... |
| cmphaushmeo 23151 | A continuous bijection fro... |
| ordthmeolem 23152 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 23153 | An order isomorphism is a ... |
| txhmeo 23154 | Lift a pair of homeomorphi... |
| txswaphmeolem 23155 | Show inverse for the "swap... |
| txswaphmeo 23156 | There is a homeomorphism f... |
| pt1hmeo 23157 | The canonical homeomorphis... |
| ptuncnv 23158 | Exhibit the converse funct... |
| ptunhmeo 23159 | Define a homeomorphism fro... |
| xpstopnlem1 23160 | The function ` F ` used in... |
| xpstps 23161 | A binary product of topolo... |
| xpstopnlem2 23162 | Lemma for ~ xpstopn . (Co... |
| xpstopn 23163 | The topology on a binary p... |
| ptcmpfi 23164 | A topological product of f... |
| xkocnv 23165 | The inverse of the "curryi... |
| xkohmeo 23166 | The Exponential Law for to... |
| qtopf1 23167 | If a quotient map is injec... |
| qtophmeo 23168 | If two functions on a base... |
| t0kq 23169 | A topological space is T_0... |
| kqhmph 23170 | A topological space is T_0... |
| ist1-5lem 23171 | Lemma for ~ ist1-5 and sim... |
| t1r0 23172 | A T_1 space is R_0. That ... |
| ist1-5 23173 | A topological space is T_1... |
| ishaus3 23174 | A topological space is Hau... |
| nrmreg 23175 | A normal T_1 space is regu... |
| reghaus 23176 | A regular T_0 space is Hau... |
| nrmhaus 23177 | A T_1 normal space is Haus... |
| elmptrab 23178 | Membership in a one-parame... |
| elmptrab2 23179 | Membership in a one-parame... |
| isfbas 23180 | The predicate " ` F ` is a... |
| fbasne0 23181 | There are no empty filter ... |
| 0nelfb 23182 | No filter base contains th... |
| fbsspw 23183 | A filter base on a set is ... |
| fbelss 23184 | An element of the filter b... |
| fbdmn0 23185 | The domain of a filter bas... |
| isfbas2 23186 | The predicate " ` F ` is a... |
| fbasssin 23187 | A filter base contains sub... |
| fbssfi 23188 | A filter base contains sub... |
| fbssint 23189 | A filter base contains sub... |
| fbncp 23190 | A filter base does not con... |
| fbun 23191 | A necessary and sufficient... |
| fbfinnfr 23192 | No filter base containing ... |
| opnfbas 23193 | The collection of open sup... |
| trfbas2 23194 | Conditions for the trace o... |
| trfbas 23195 | Conditions for the trace o... |
| isfil 23198 | The predicate "is a filter... |
| filfbas 23199 | A filter is a filter base.... |
| 0nelfil 23200 | The empty set doesn't belo... |
| fileln0 23201 | An element of a filter is ... |
| filsspw 23202 | A filter is a subset of th... |
| filelss 23203 | An element of a filter is ... |
| filss 23204 | A filter is closed under t... |
| filin 23205 | A filter is closed under t... |
| filtop 23206 | The underlying set belongs... |
| isfil2 23207 | Derive the standard axioms... |
| isfildlem 23208 | Lemma for ~ isfild . (Con... |
| isfild 23209 | Sufficient condition for a... |
| filfi 23210 | A filter is closed under t... |
| filinn0 23211 | The intersection of two el... |
| filintn0 23212 | A filter has the finite in... |
| filn0 23213 | The empty set is not a fil... |
| infil 23214 | The intersection of two fi... |
| snfil 23215 | A singleton is a filter. ... |
| fbasweak 23216 | A filter base on any set i... |
| snfbas 23217 | Condition for a singleton ... |
| fsubbas 23218 | A condition for a set to g... |
| fbasfip 23219 | A filter base has the fini... |
| fbunfip 23220 | A helpful lemma for showin... |
| fgval 23221 | The filter generating clas... |
| elfg 23222 | A condition for elements o... |
| ssfg 23223 | A filter base is a subset ... |
| fgss 23224 | A bigger base generates a ... |
| fgss2 23225 | A condition for a filter t... |
| fgfil 23226 | A filter generates itself.... |
| elfilss 23227 | An element belongs to a fi... |
| filfinnfr 23228 | No filter containing a fin... |
| fgcl 23229 | A generated filter is a fi... |
| fgabs 23230 | Absorption law for filter ... |
| neifil 23231 | The neighborhoods of a non... |
| filunibas 23232 | Recover the base set from ... |
| filunirn 23233 | Two ways to express a filt... |
| filconn 23234 | A filter gives rise to a c... |
| fbasrn 23235 | Given a filter on a domain... |
| filuni 23236 | The union of a nonempty se... |
| trfil1 23237 | Conditions for the trace o... |
| trfil2 23238 | Conditions for the trace o... |
| trfil3 23239 | Conditions for the trace o... |
| trfilss 23240 | If ` A ` is a member of th... |
| fgtr 23241 | If ` A ` is a member of th... |
| trfg 23242 | The trace operation and th... |
| trnei 23243 | The trace, over a set ` A ... |
| cfinfil 23244 | Relative complements of th... |
| csdfil 23245 | The set of all elements wh... |
| supfil 23246 | The supersets of a nonempt... |
| zfbas 23247 | The set of upper sets of i... |
| uzrest 23248 | The restriction of the set... |
| uzfbas 23249 | The set of upper sets of i... |
| isufil 23254 | The property of being an u... |
| ufilfil 23255 | An ultrafilter is a filter... |
| ufilss 23256 | For any subset of the base... |
| ufilb 23257 | The complement is in an ul... |
| ufilmax 23258 | Any filter finer than an u... |
| isufil2 23259 | The maximal property of an... |
| ufprim 23260 | An ultrafilter is a prime ... |
| trufil 23261 | Conditions for the trace o... |
| filssufilg 23262 | A filter is contained in s... |
| filssufil 23263 | A filter is contained in s... |
| isufl 23264 | Define the (strong) ultraf... |
| ufli 23265 | Property of a set that sat... |
| numufl 23266 | Consequence of ~ filssufil... |
| fiufl 23267 | A finite set satisfies the... |
| acufl 23268 | The axiom of choice implie... |
| ssufl 23269 | If ` Y ` is a subset of ` ... |
| ufileu 23270 | If the ultrafilter contain... |
| filufint 23271 | A filter is equal to the i... |
| uffix 23272 | Lemma for ~ fixufil and ~ ... |
| fixufil 23273 | The condition describing a... |
| uffixfr 23274 | An ultrafilter is either f... |
| uffix2 23275 | A classification of fixed ... |
| uffixsn 23276 | The singleton of the gener... |
| ufildom1 23277 | An ultrafilter is generate... |
| uffinfix 23278 | An ultrafilter containing ... |
| cfinufil 23279 | An ultrafilter is free iff... |
| ufinffr 23280 | An infinite subset is cont... |
| ufilen 23281 | Any infinite set has an ul... |
| ufildr 23282 | An ultrafilter gives rise ... |
| fin1aufil 23283 | There are no definable fre... |
| fmval 23294 | Introduce a function that ... |
| fmfil 23295 | A mapping filter is a filt... |
| fmf 23296 | Pushing-forward via a func... |
| fmss 23297 | A finer filter produces a ... |
| elfm 23298 | An element of a mapping fi... |
| elfm2 23299 | An element of a mapping fi... |
| fmfg 23300 | The image filter of a filt... |
| elfm3 23301 | An alternate formulation o... |
| imaelfm 23302 | An image of a filter eleme... |
| rnelfmlem 23303 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23304 | A condition for a filter t... |
| fmfnfmlem1 23305 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23306 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23307 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23308 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23309 | A filter finer than an ima... |
| fmufil 23310 | An image filter of an ultr... |
| fmid 23311 | The filter map applied to ... |
| fmco 23312 | Composition of image filte... |
| ufldom 23313 | The ultrafilter lemma prop... |
| flimval 23314 | The set of limit points of... |
| elflim2 23315 | The predicate "is a limit ... |
| flimtop 23316 | Reverse closure for the li... |
| flimneiss 23317 | A filter contains the neig... |
| flimnei 23318 | A filter contains all of t... |
| flimelbas 23319 | A limit point of a filter ... |
| flimfil 23320 | Reverse closure for the li... |
| flimtopon 23321 | Reverse closure for the li... |
| elflim 23322 | The predicate "is a limit ... |
| flimss2 23323 | A limit point of a filter ... |
| flimss1 23324 | A limit point of a filter ... |
| neiflim 23325 | A point is a limit point o... |
| flimopn 23326 | The condition for being a ... |
| fbflim 23327 | A condition for a filter t... |
| fbflim2 23328 | A condition for a filter b... |
| flimclsi 23329 | The convergent points of a... |
| hausflimlem 23330 | If ` A ` and ` B ` are bot... |
| hausflimi 23331 | One direction of ~ hausfli... |
| hausflim 23332 | A condition for a topology... |
| flimcf 23333 | Fineness is properly chara... |
| flimrest 23334 | The set of limit points in... |
| flimclslem 23335 | Lemma for ~ flimcls . (Co... |
| flimcls 23336 | Closure in terms of filter... |
| flimsncls 23337 | If ` A ` is a limit point ... |
| hauspwpwf1 23338 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23339 | If ` X ` is a Hausdorff sp... |
| flffval 23340 | Given a topology and a fil... |
| flfval 23341 | Given a function from a fi... |
| flfnei 23342 | The property of being a li... |
| flfneii 23343 | A neighborhood of a limit ... |
| isflf 23344 | The property of being a li... |
| flfelbas 23345 | A limit point of a functio... |
| flffbas 23346 | Limit points of a function... |
| flftg 23347 | Limit points of a function... |
| hausflf 23348 | If a function has its valu... |
| hausflf2 23349 | If a convergent function h... |
| cnpflfi 23350 | Forward direction of ~ cnp... |
| cnpflf2 23351 | ` F ` is continuous at poi... |
| cnpflf 23352 | Continuity of a function a... |
| cnflf 23353 | A function is continuous i... |
| cnflf2 23354 | A function is continuous i... |
| flfcnp 23355 | A continuous function pres... |
| lmflf 23356 | The topological limit rela... |
| txflf 23357 | Two sequences converge in ... |
| flfcnp2 23358 | The image of a convergent ... |
| fclsval 23359 | The set of all cluster poi... |
| isfcls 23360 | A cluster point of a filte... |
| fclsfil 23361 | Reverse closure for the cl... |
| fclstop 23362 | Reverse closure for the cl... |
| fclstopon 23363 | Reverse closure for the cl... |
| isfcls2 23364 | A cluster point of a filte... |
| fclsopn 23365 | Write the cluster point co... |
| fclsopni 23366 | An open neighborhood of a ... |
| fclselbas 23367 | A cluster point is in the ... |
| fclsneii 23368 | A neighborhood of a cluste... |
| fclssscls 23369 | The set of cluster points ... |
| fclsnei 23370 | Cluster points in terms of... |
| supnfcls 23371 | The filter of supersets of... |
| fclsbas 23372 | Cluster points in terms of... |
| fclsss1 23373 | A finer topology has fewer... |
| fclsss2 23374 | A finer filter has fewer c... |
| fclsrest 23375 | The set of cluster points ... |
| fclscf 23376 | Characterization of finene... |
| flimfcls 23377 | A limit point is a cluster... |
| fclsfnflim 23378 | A filter clusters at a poi... |
| flimfnfcls 23379 | A filter converges to a po... |
| fclscmpi 23380 | Forward direction of ~ fcl... |
| fclscmp 23381 | A space is compact iff eve... |
| uffclsflim 23382 | The cluster points of an u... |
| ufilcmp 23383 | A space is compact iff eve... |
| fcfval 23384 | The set of cluster points ... |
| isfcf 23385 | The property of being a cl... |
| fcfnei 23386 | The property of being a cl... |
| fcfelbas 23387 | A cluster point of a funct... |
| fcfneii 23388 | A neighborhood of a cluste... |
| flfssfcf 23389 | A limit point of a functio... |
| uffcfflf 23390 | If the domain filter is an... |
| cnpfcfi 23391 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23392 | A function ` F ` is contin... |
| cnfcf 23393 | Continuity of a function i... |
| flfcntr 23394 | A continuous function's va... |
| alexsublem 23395 | Lemma for ~ alexsub . (Co... |
| alexsub 23396 | The Alexander Subbase Theo... |
| alexsubb 23397 | Biconditional form of the ... |
| alexsubALTlem1 23398 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23399 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23400 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23401 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23402 | The Alexander Subbase Theo... |
| ptcmplem1 23403 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23404 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23405 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23406 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23407 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 23408 | Tychonoff's theorem: The ... |
| ptcmp 23409 | Tychonoff's theorem: The ... |
| cnextval 23412 | The function applying cont... |
| cnextfval 23413 | The continuous extension o... |
| cnextrel 23414 | In the general case, a con... |
| cnextfun 23415 | If the target space is Hau... |
| cnextfvval 23416 | The value of the continuou... |
| cnextf 23417 | Extension by continuity. ... |
| cnextcn 23418 | Extension by continuity. ... |
| cnextfres1 23419 | ` F ` and its extension by... |
| cnextfres 23420 | ` F ` and its extension by... |
| istmd 23425 | The predicate "is a topolo... |
| tmdmnd 23426 | A topological monoid is a ... |
| tmdtps 23427 | A topological monoid is a ... |
| istgp 23428 | The predicate "is a topolo... |
| tgpgrp 23429 | A topological group is a g... |
| tgptmd 23430 | A topological group is a t... |
| tgptps 23431 | A topological group is a t... |
| tmdtopon 23432 | The topology of a topologi... |
| tgptopon 23433 | The topology of a topologi... |
| tmdcn 23434 | In a topological monoid, t... |
| tgpcn 23435 | In a topological group, th... |
| tgpinv 23436 | In a topological group, th... |
| grpinvhmeo 23437 | The inverse function in a ... |
| cnmpt1plusg 23438 | Continuity of the group su... |
| cnmpt2plusg 23439 | Continuity of the group su... |
| tmdcn2 23440 | Write out the definition o... |
| tgpsubcn 23441 | In a topological group, th... |
| istgp2 23442 | A group with a topology is... |
| tmdmulg 23443 | In a topological monoid, t... |
| tgpmulg 23444 | In a topological group, th... |
| tgpmulg2 23445 | In a topological monoid, t... |
| tmdgsum 23446 | In a topological monoid, t... |
| tmdgsum2 23447 | For any neighborhood ` U `... |
| oppgtmd 23448 | The opposite of a topologi... |
| oppgtgp 23449 | The opposite of a topologi... |
| distgp 23450 | Any group equipped with th... |
| indistgp 23451 | Any group equipped with th... |
| efmndtmd 23452 | The monoid of endofunction... |
| tmdlactcn 23453 | The left group action of e... |
| tgplacthmeo 23454 | The left group action of e... |
| submtmd 23455 | A submonoid of a topologic... |
| subgtgp 23456 | A subgroup of a topologica... |
| symgtgp 23457 | The symmetric group is a t... |
| subgntr 23458 | A subgroup of a topologica... |
| opnsubg 23459 | An open subgroup of a topo... |
| clssubg 23460 | The closure of a subgroup ... |
| clsnsg 23461 | The closure of a normal su... |
| cldsubg 23462 | A subgroup of finite index... |
| tgpconncompeqg 23463 | The connected component co... |
| tgpconncomp 23464 | The identity component, th... |
| tgpconncompss 23465 | The identity component is ... |
| ghmcnp 23466 | A group homomorphism on to... |
| snclseqg 23467 | The coset of the closure o... |
| tgphaus 23468 | A topological group is Hau... |
| tgpt1 23469 | Hausdorff and T1 are equiv... |
| tgpt0 23470 | Hausdorff and T0 are equiv... |
| qustgpopn 23471 | A quotient map in a topolo... |
| qustgplem 23472 | Lemma for ~ qustgp . (Con... |
| qustgp 23473 | The quotient of a topologi... |
| qustgphaus 23474 | The quotient of a topologi... |
| prdstmdd 23475 | The product of a family of... |
| prdstgpd 23476 | The product of a family of... |
| tsmsfbas 23479 | The collection of all sets... |
| tsmslem1 23480 | The finite partial sums of... |
| tsmsval2 23481 | Definition of the topologi... |
| tsmsval 23482 | Definition of the topologi... |
| tsmspropd 23483 | The group sum depends only... |
| eltsms 23484 | The property of being a su... |
| tsmsi 23485 | The property of being a su... |
| tsmscl 23486 | A sum in a topological gro... |
| haustsms 23487 | In a Hausdorff topological... |
| haustsms2 23488 | In a Hausdorff topological... |
| tsmscls 23489 | One half of ~ tgptsmscls ,... |
| tsmsgsum 23490 | The convergent points of a... |
| tsmsid 23491 | If a sum is finite, the us... |
| haustsmsid 23492 | In a Hausdorff topological... |
| tsms0 23493 | The sum of zero is zero. ... |
| tsmssubm 23494 | Evaluate an infinite group... |
| tsmsres 23495 | Extend an infinite group s... |
| tsmsf1o 23496 | Re-index an infinite group... |
| tsmsmhm 23497 | Apply a continuous group h... |
| tsmsadd 23498 | The sum of two infinite gr... |
| tsmsinv 23499 | Inverse of an infinite gro... |
| tsmssub 23500 | The difference of two infi... |
| tgptsmscls 23501 | A sum in a topological gro... |
| tgptsmscld 23502 | The set of limit points to... |
| tsmssplit 23503 | Split a topological group ... |
| tsmsxplem1 23504 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 23505 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 23506 | Write a sum over a two-dim... |
| istrg 23515 | Express the predicate " ` ... |
| trgtmd 23516 | The multiplicative monoid ... |
| istdrg 23517 | Express the predicate " ` ... |
| tdrgunit 23518 | The unit group of a topolo... |
| trgtgp 23519 | A topological ring is a to... |
| trgtmd2 23520 | A topological ring is a to... |
| trgtps 23521 | A topological ring is a to... |
| trgring 23522 | A topological ring is a ri... |
| trggrp 23523 | A topological ring is a gr... |
| tdrgtrg 23524 | A topological division rin... |
| tdrgdrng 23525 | A topological division rin... |
| tdrgring 23526 | A topological division rin... |
| tdrgtmd 23527 | A topological division rin... |
| tdrgtps 23528 | A topological division rin... |
| istdrg2 23529 | A topological-ring divisio... |
| mulrcn 23530 | The functionalization of t... |
| invrcn2 23531 | The multiplicative inverse... |
| invrcn 23532 | The multiplicative inverse... |
| cnmpt1mulr 23533 | Continuity of ring multipl... |
| cnmpt2mulr 23534 | Continuity of ring multipl... |
| dvrcn 23535 | The division function is c... |
| istlm 23536 | The predicate " ` W ` is a... |
| vscacn 23537 | The scalar multiplication ... |
| tlmtmd 23538 | A topological module is a ... |
| tlmtps 23539 | A topological module is a ... |
| tlmlmod 23540 | A topological module is a ... |
| tlmtrg 23541 | The scalar ring of a topol... |
| tlmscatps 23542 | The scalar ring of a topol... |
| istvc 23543 | A topological vector space... |
| tvctdrg 23544 | The scalar field of a topo... |
| cnmpt1vsca 23545 | Continuity of scalar multi... |
| cnmpt2vsca 23546 | Continuity of scalar multi... |
| tlmtgp 23547 | A topological vector space... |
| tvctlm 23548 | A topological vector space... |
| tvclmod 23549 | A topological vector space... |
| tvclvec 23550 | A topological vector space... |
| ustfn 23553 | The defined uniform struct... |
| ustval 23554 | The class of all uniform s... |
| isust 23555 | The predicate " ` U ` is a... |
| ustssxp 23556 | Entourages are subsets of ... |
| ustssel 23557 | A uniform structure is upw... |
| ustbasel 23558 | The full set is always an ... |
| ustincl 23559 | A uniform structure is clo... |
| ustdiag 23560 | The diagonal set is includ... |
| ustinvel 23561 | If ` V ` is an entourage, ... |
| ustexhalf 23562 | For each entourage ` V ` t... |
| ustrel 23563 | The elements of uniform st... |
| ustfilxp 23564 | A uniform structure on a n... |
| ustne0 23565 | A uniform structure cannot... |
| ustssco 23566 | In an uniform structure, a... |
| ustexsym 23567 | In an uniform structure, f... |
| ustex2sym 23568 | In an uniform structure, f... |
| ustex3sym 23569 | In an uniform structure, f... |
| ustref 23570 | Any element of the base se... |
| ust0 23571 | The unique uniform structu... |
| ustn0 23572 | The empty set is not an un... |
| ustund 23573 | If two intersecting sets `... |
| ustelimasn 23574 | Any point ` A ` is near en... |
| ustneism 23575 | For a point ` A ` in ` X `... |
| elrnustOLD 23576 | Obsolete version of ~ elfv... |
| ustbas2 23577 | Second direction for ~ ust... |
| ustuni 23578 | The set union of a uniform... |
| ustbas 23579 | Recover the base of an uni... |
| ustimasn 23580 | Lemma for ~ ustuqtop . (C... |
| trust 23581 | The trace of a uniform str... |
| utopval 23584 | The topology induced by a ... |
| elutop 23585 | Open sets in the topology ... |
| utoptop 23586 | The topology induced by a ... |
| utopbas 23587 | The base of the topology i... |
| utoptopon 23588 | Topology induced by a unif... |
| restutop 23589 | Restriction of a topology ... |
| restutopopn 23590 | The restriction of the top... |
| ustuqtoplem 23591 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 23592 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 23593 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 23594 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 23595 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 23596 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 23597 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 23598 | For a given uniform struct... |
| utopsnneiplem 23599 | The neighborhoods of a poi... |
| utopsnneip 23600 | The neighborhoods of a poi... |
| utopsnnei 23601 | Images of singletons by en... |
| utop2nei 23602 | For any symmetrical entour... |
| utop3cls 23603 | Relation between a topolog... |
| utopreg 23604 | All Hausdorff uniform spac... |
| ussval 23611 | The uniform structure on u... |
| ussid 23612 | In case the base of the ` ... |
| isusp 23613 | The predicate ` W ` is a u... |
| ressuss 23614 | Value of the uniform struc... |
| ressust 23615 | The uniform structure of a... |
| ressusp 23616 | The restriction of a unifo... |
| tusval 23617 | The value of the uniform s... |
| tuslem 23618 | Lemma for ~ tusbas , ~ tus... |
| tuslemOLD 23619 | Obsolete proof of ~ tuslem... |
| tusbas 23620 | The base set of a construc... |
| tusunif 23621 | The uniform structure of a... |
| tususs 23622 | The uniform structure of a... |
| tustopn 23623 | The topology induced by a ... |
| tususp 23624 | A constructed uniform spac... |
| tustps 23625 | A constructed uniform spac... |
| uspreg 23626 | If a uniform space is Haus... |
| ucnval 23629 | The set of all uniformly c... |
| isucn 23630 | The predicate " ` F ` is a... |
| isucn2 23631 | The predicate " ` F ` is a... |
| ucnimalem 23632 | Reformulate the ` G ` func... |
| ucnima 23633 | An equivalent statement of... |
| ucnprima 23634 | The preimage by a uniforml... |
| iducn 23635 | The identity is uniformly ... |
| cstucnd 23636 | A constant function is uni... |
| ucncn 23637 | Uniform continuity implies... |
| iscfilu 23640 | The predicate " ` F ` is a... |
| cfilufbas 23641 | A Cauchy filter base is a ... |
| cfiluexsm 23642 | For a Cauchy filter base a... |
| fmucndlem 23643 | Lemma for ~ fmucnd . (Con... |
| fmucnd 23644 | The image of a Cauchy filt... |
| cfilufg 23645 | The filter generated by a ... |
| trcfilu 23646 | Condition for the trace of... |
| cfiluweak 23647 | A Cauchy filter base is al... |
| neipcfilu 23648 | In an uniform space, a nei... |
| iscusp 23651 | The predicate " ` W ` is a... |
| cuspusp 23652 | A complete uniform space i... |
| cuspcvg 23653 | In a complete uniform spac... |
| iscusp2 23654 | The predicate " ` W ` is a... |
| cnextucn 23655 | Extension by continuity. ... |
| ucnextcn 23656 | Extension by continuity. ... |
| ispsmet 23657 | Express the predicate " ` ... |
| psmetdmdm 23658 | Recover the base set from ... |
| psmetf 23659 | The distance function of a... |
| psmetcl 23660 | Closure of the distance fu... |
| psmet0 23661 | The distance function of a... |
| psmettri2 23662 | Triangle inequality for th... |
| psmetsym 23663 | The distance function of a... |
| psmettri 23664 | Triangle inequality for th... |
| psmetge0 23665 | The distance function of a... |
| psmetxrge0 23666 | The distance function of a... |
| psmetres2 23667 | Restriction of a pseudomet... |
| psmetlecl 23668 | Real closure of an extende... |
| distspace 23669 | A set ` X ` together with ... |
| ismet 23676 | Express the predicate " ` ... |
| isxmet 23677 | Express the predicate " ` ... |
| ismeti 23678 | Properties that determine ... |
| isxmetd 23679 | Properties that determine ... |
| isxmet2d 23680 | It is safe to only require... |
| metflem 23681 | Lemma for ~ metf and other... |
| xmetf 23682 | Mapping of the distance fu... |
| metf 23683 | Mapping of the distance fu... |
| xmetcl 23684 | Closure of the distance fu... |
| metcl 23685 | Closure of the distance fu... |
| ismet2 23686 | An extended metric is a me... |
| metxmet 23687 | A metric is an extended me... |
| xmetdmdm 23688 | Recover the base set from ... |
| metdmdm 23689 | Recover the base set from ... |
| xmetunirn 23690 | Two ways to express an ext... |
| xmeteq0 23691 | The value of an extended m... |
| meteq0 23692 | The value of a metric is z... |
| xmettri2 23693 | Triangle inequality for th... |
| mettri2 23694 | Triangle inequality for th... |
| xmet0 23695 | The distance function of a... |
| met0 23696 | The distance function of a... |
| xmetge0 23697 | The distance function of a... |
| metge0 23698 | The distance function of a... |
| xmetlecl 23699 | Real closure of an extende... |
| xmetsym 23700 | The distance function of a... |
| xmetpsmet 23701 | An extended metric is a ps... |
| xmettpos 23702 | The distance function of a... |
| metsym 23703 | The distance function of a... |
| xmettri 23704 | Triangle inequality for th... |
| mettri 23705 | Triangle inequality for th... |
| xmettri3 23706 | Triangle inequality for th... |
| mettri3 23707 | Triangle inequality for th... |
| xmetrtri 23708 | One half of the reverse tr... |
| xmetrtri2 23709 | The reverse triangle inequ... |
| metrtri 23710 | Reverse triangle inequalit... |
| xmetgt0 23711 | The distance function of a... |
| metgt0 23712 | The distance function of a... |
| metn0 23713 | A metric space is nonempty... |
| xmetres2 23714 | Restriction of an extended... |
| metreslem 23715 | Lemma for ~ metres . (Con... |
| metres2 23716 | Lemma for ~ metres . (Con... |
| xmetres 23717 | A restriction of an extend... |
| metres 23718 | A restriction of a metric ... |
| 0met 23719 | The empty metric. (Contri... |
| prdsdsf 23720 | The product metric is a fu... |
| prdsxmetlem 23721 | The product metric is an e... |
| prdsxmet 23722 | The product metric is an e... |
| prdsmet 23723 | The product metric is a me... |
| ressprdsds 23724 | Restriction of a product m... |
| resspwsds 23725 | Restriction of a power met... |
| imasdsf1olem 23726 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 23727 | The distance function is t... |
| imasf1oxmet 23728 | The image of an extended m... |
| imasf1omet 23729 | The image of a metric is a... |
| xpsdsfn 23730 | Closure of the metric in a... |
| xpsdsfn2 23731 | Closure of the metric in a... |
| xpsxmetlem 23732 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 23733 | A product metric of extend... |
| xpsdsval 23734 | Value of the metric in a b... |
| xpsmet 23735 | The direct product of two ... |
| blfvalps 23736 | The value of the ball func... |
| blfval 23737 | The value of the ball func... |
| blvalps 23738 | The ball around a point ` ... |
| blval 23739 | The ball around a point ` ... |
| elblps 23740 | Membership in a ball. (Co... |
| elbl 23741 | Membership in a ball. (Co... |
| elbl2ps 23742 | Membership in a ball. (Co... |
| elbl2 23743 | Membership in a ball. (Co... |
| elbl3ps 23744 | Membership in a ball, with... |
| elbl3 23745 | Membership in a ball, with... |
| blcomps 23746 | Commute the arguments to t... |
| blcom 23747 | Commute the arguments to t... |
| xblpnfps 23748 | The infinity ball in an ex... |
| xblpnf 23749 | The infinity ball in an ex... |
| blpnf 23750 | The infinity ball in a sta... |
| bldisj 23751 | Two balls are disjoint if ... |
| blgt0 23752 | A nonempty ball implies th... |
| bl2in 23753 | Two balls are disjoint if ... |
| xblss2ps 23754 | One ball is contained in a... |
| xblss2 23755 | One ball is contained in a... |
| blss2ps 23756 | One ball is contained in a... |
| blss2 23757 | One ball is contained in a... |
| blhalf 23758 | A ball of radius ` R / 2 `... |
| blfps 23759 | Mapping of a ball. (Contr... |
| blf 23760 | Mapping of a ball. (Contr... |
| blrnps 23761 | Membership in the range of... |
| blrn 23762 | Membership in the range of... |
| xblcntrps 23763 | A ball contains its center... |
| xblcntr 23764 | A ball contains its center... |
| blcntrps 23765 | A ball contains its center... |
| blcntr 23766 | A ball contains its center... |
| xbln0 23767 | A ball is nonempty iff the... |
| bln0 23768 | A ball is not empty. (Con... |
| blelrnps 23769 | A ball belongs to the set ... |
| blelrn 23770 | A ball belongs to the set ... |
| blssm 23771 | A ball is a subset of the ... |
| unirnblps 23772 | The union of the set of ba... |
| unirnbl 23773 | The union of the set of ba... |
| blin 23774 | The intersection of two ba... |
| ssblps 23775 | The size of a ball increas... |
| ssbl 23776 | The size of a ball increas... |
| blssps 23777 | Any point ` P ` in a ball ... |
| blss 23778 | Any point ` P ` in a ball ... |
| blssexps 23779 | Two ways to express the ex... |
| blssex 23780 | Two ways to express the ex... |
| ssblex 23781 | A nested ball exists whose... |
| blin2 23782 | Given any two balls and a ... |
| blbas 23783 | The balls of a metric spac... |
| blres 23784 | A ball in a restricted met... |
| xmeterval 23785 | Value of the "finitely sep... |
| xmeter 23786 | The "finitely separated" r... |
| xmetec 23787 | The equivalence classes un... |
| blssec 23788 | A ball centered at ` P ` i... |
| blpnfctr 23789 | The infinity ball in an ex... |
| xmetresbl 23790 | An extended metric restric... |
| mopnval 23791 | An open set is a subset of... |
| mopntopon 23792 | The set of open sets of a ... |
| mopntop 23793 | The set of open sets of a ... |
| mopnuni 23794 | The union of all open sets... |
| elmopn 23795 | The defining property of a... |
| mopnfss 23796 | The family of open sets of... |
| mopnm 23797 | The base set of a metric s... |
| elmopn2 23798 | A defining property of an ... |
| mopnss 23799 | An open set of a metric sp... |
| isxms 23800 | Express the predicate " ` ... |
| isxms2 23801 | Express the predicate " ` ... |
| isms 23802 | Express the predicate " ` ... |
| isms2 23803 | Express the predicate " ` ... |
| xmstopn 23804 | The topology component of ... |
| mstopn 23805 | The topology component of ... |
| xmstps 23806 | An extended metric space i... |
| msxms 23807 | A metric space is an exten... |
| mstps 23808 | A metric space is a topolo... |
| xmsxmet 23809 | The distance function, sui... |
| msmet 23810 | The distance function, sui... |
| msf 23811 | The distance function of a... |
| xmsxmet2 23812 | The distance function, sui... |
| msmet2 23813 | The distance function, sui... |
| mscl 23814 | Closure of the distance fu... |
| xmscl 23815 | Closure of the distance fu... |
| xmsge0 23816 | The distance function in a... |
| xmseq0 23817 | The distance between two p... |
| xmssym 23818 | The distance function in a... |
| xmstri2 23819 | Triangle inequality for th... |
| mstri2 23820 | Triangle inequality for th... |
| xmstri 23821 | Triangle inequality for th... |
| mstri 23822 | Triangle inequality for th... |
| xmstri3 23823 | Triangle inequality for th... |
| mstri3 23824 | Triangle inequality for th... |
| msrtri 23825 | Reverse triangle inequalit... |
| xmspropd 23826 | Property deduction for an ... |
| mspropd 23827 | Property deduction for a m... |
| setsmsbas 23828 | The base set of a construc... |
| setsmsbasOLD 23829 | Obsolete proof of ~ setsms... |
| setsmsds 23830 | The distance function of a... |
| setsmsdsOLD 23831 | Obsolete proof of ~ setsms... |
| setsmstset 23832 | The topology of a construc... |
| setsmstopn 23833 | The topology of a construc... |
| setsxms 23834 | The constructed metric spa... |
| setsms 23835 | The constructed metric spa... |
| tmsval 23836 | For any metric there is an... |
| tmslem 23837 | Lemma for ~ tmsbas , ~ tms... |
| tmslemOLD 23838 | Obsolete version of ~ tmsl... |
| tmsbas 23839 | The base set of a construc... |
| tmsds 23840 | The metric of a constructe... |
| tmstopn 23841 | The topology of a construc... |
| tmsxms 23842 | The constructed metric spa... |
| tmsms 23843 | The constructed metric spa... |
| imasf1obl 23844 | The image of a metric spac... |
| imasf1oxms 23845 | The image of a metric spac... |
| imasf1oms 23846 | The image of a metric spac... |
| prdsbl 23847 | A ball in the product metr... |
| mopni 23848 | An open set of a metric sp... |
| mopni2 23849 | An open set of a metric sp... |
| mopni3 23850 | An open set of a metric sp... |
| blssopn 23851 | The balls of a metric spac... |
| unimopn 23852 | The union of a collection ... |
| mopnin 23853 | The intersection of two op... |
| mopn0 23854 | The empty set is an open s... |
| rnblopn 23855 | A ball of a metric space i... |
| blopn 23856 | A ball of a metric space i... |
| neibl 23857 | The neighborhoods around a... |
| blnei 23858 | A ball around a point is a... |
| lpbl 23859 | Every ball around a limit ... |
| blsscls2 23860 | A smaller closed ball is c... |
| blcld 23861 | A "closed ball" in a metri... |
| blcls 23862 | The closure of an open bal... |
| blsscls 23863 | If two concentric balls ha... |
| metss 23864 | Two ways of saying that me... |
| metequiv 23865 | Two ways of saying that tw... |
| metequiv2 23866 | If there is a sequence of ... |
| metss2lem 23867 | Lemma for ~ metss2 . (Con... |
| metss2 23868 | If the metric ` D ` is "st... |
| comet 23869 | The composition of an exte... |
| stdbdmetval 23870 | Value of the standard boun... |
| stdbdxmet 23871 | The standard bounded metri... |
| stdbdmet 23872 | The standard bounded metri... |
| stdbdbl 23873 | The standard bounded metri... |
| stdbdmopn 23874 | The standard bounded metri... |
| mopnex 23875 | The topology generated by ... |
| methaus 23876 | The topology generated by ... |
| met1stc 23877 | The topology generated by ... |
| met2ndci 23878 | A separable metric space (... |
| met2ndc 23879 | A metric space is second-c... |
| metrest 23880 | Two alternate formulations... |
| ressxms 23881 | The restriction of a metri... |
| ressms 23882 | The restriction of a metri... |
| prdsmslem1 23883 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 23884 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 23885 | Lemma for ~ prdsxms . The... |
| prdsxms 23886 | The indexed product struct... |
| prdsms 23887 | The indexed product struct... |
| pwsxms 23888 | A power of an extended met... |
| pwsms 23889 | A power of a metric space ... |
| xpsxms 23890 | A binary product of metric... |
| xpsms 23891 | A binary product of metric... |
| tmsxps 23892 | Express the product of two... |
| tmsxpsmopn 23893 | Express the product of two... |
| tmsxpsval 23894 | Value of the product of tw... |
| tmsxpsval2 23895 | Value of the product of tw... |
| metcnp3 23896 | Two ways to express that `... |
| metcnp 23897 | Two ways to say a mapping ... |
| metcnp2 23898 | Two ways to say a mapping ... |
| metcn 23899 | Two ways to say a mapping ... |
| metcnpi 23900 | Epsilon-delta property of ... |
| metcnpi2 23901 | Epsilon-delta property of ... |
| metcnpi3 23902 | Epsilon-delta property of ... |
| txmetcnp 23903 | Continuity of a binary ope... |
| txmetcn 23904 | Continuity of a binary ope... |
| metuval 23905 | Value of the uniform struc... |
| metustel 23906 | Define a filter base ` F `... |
| metustss 23907 | Range of the elements of t... |
| metustrel 23908 | Elements of the filter bas... |
| metustto 23909 | Any two elements of the fi... |
| metustid 23910 | The identity diagonal is i... |
| metustsym 23911 | Elements of the filter bas... |
| metustexhalf 23912 | For any element ` A ` of t... |
| metustfbas 23913 | The filter base generated ... |
| metust 23914 | The uniform structure gene... |
| cfilucfil 23915 | Given a metric ` D ` and a... |
| metuust 23916 | The uniform structure gene... |
| cfilucfil2 23917 | Given a metric ` D ` and a... |
| blval2 23918 | The ball around a point ` ... |
| elbl4 23919 | Membership in a ball, alte... |
| metuel 23920 | Elementhood in the uniform... |
| metuel2 23921 | Elementhood in the uniform... |
| metustbl 23922 | The "section" image of an ... |
| psmetutop 23923 | The topology induced by a ... |
| xmetutop 23924 | The topology induced by a ... |
| xmsusp 23925 | If the uniform set of a me... |
| restmetu 23926 | The uniform structure gene... |
| metucn 23927 | Uniform continuity in metr... |
| dscmet 23928 | The discrete metric on any... |
| dscopn 23929 | The discrete metric genera... |
| nrmmetd 23930 | Show that a group norm gen... |
| abvmet 23931 | An absolute value ` F ` ge... |
| nmfval 23944 | The value of the norm func... |
| nmval 23945 | The value of the norm as t... |
| nmfval0 23946 | The value of the norm func... |
| nmfval2 23947 | The value of the norm func... |
| nmval2 23948 | The value of the norm on a... |
| nmf2 23949 | The norm on a metric group... |
| nmpropd 23950 | Weak property deduction fo... |
| nmpropd2 23951 | Strong property deduction ... |
| isngp 23952 | The property of being a no... |
| isngp2 23953 | The property of being a no... |
| isngp3 23954 | The property of being a no... |
| ngpgrp 23955 | A normed group is a group.... |
| ngpms 23956 | A normed group is a metric... |
| ngpxms 23957 | A normed group is an exten... |
| ngptps 23958 | A normed group is a topolo... |
| ngpmet 23959 | The (induced) metric of a ... |
| ngpds 23960 | Value of the distance func... |
| ngpdsr 23961 | Value of the distance func... |
| ngpds2 23962 | Write the distance between... |
| ngpds2r 23963 | Write the distance between... |
| ngpds3 23964 | Write the distance between... |
| ngpds3r 23965 | Write the distance between... |
| ngprcan 23966 | Cancel right addition insi... |
| ngplcan 23967 | Cancel left addition insid... |
| isngp4 23968 | Express the property of be... |
| ngpinvds 23969 | Two elements are the same ... |
| ngpsubcan 23970 | Cancel right subtraction i... |
| nmf 23971 | The norm on a normed group... |
| nmcl 23972 | The norm of a normed group... |
| nmge0 23973 | The norm of a normed group... |
| nmeq0 23974 | The identity is the only e... |
| nmne0 23975 | The norm of a nonzero elem... |
| nmrpcl 23976 | The norm of a nonzero elem... |
| nminv 23977 | The norm of a negated elem... |
| nmmtri 23978 | The triangle inequality fo... |
| nmsub 23979 | The norm of the difference... |
| nmrtri 23980 | Reverse triangle inequalit... |
| nm2dif 23981 | Inequality for the differe... |
| nmtri 23982 | The triangle inequality fo... |
| nmtri2 23983 | Triangle inequality for th... |
| ngpi 23984 | The properties of a normed... |
| nm0 23985 | Norm of the identity eleme... |
| nmgt0 23986 | The norm of a nonzero elem... |
| sgrim 23987 | The induced metric on a su... |
| sgrimval 23988 | The induced metric on a su... |
| subgnm 23989 | The norm in a subgroup. (... |
| subgnm2 23990 | A substructure assigns the... |
| subgngp 23991 | A normed group restricted ... |
| ngptgp 23992 | A normed abelian group is ... |
| ngppropd 23993 | Property deduction for a n... |
| reldmtng 23994 | The function ` toNrmGrp ` ... |
| tngval 23995 | Value of the function whic... |
| tnglem 23996 | Lemma for ~ tngbas and sim... |
| tnglemOLD 23997 | Obsolete version of ~ tngl... |
| tngbas 23998 | The base set of a structur... |
| tngbasOLD 23999 | Obsolete proof of ~ tngbas... |
| tngplusg 24000 | The group addition of a st... |
| tngplusgOLD 24001 | Obsolete proof of ~ tngplu... |
| tng0 24002 | The group identity of a st... |
| tngmulr 24003 | The ring multiplication of... |
| tngmulrOLD 24004 | Obsolete proof of ~ tngmul... |
| tngsca 24005 | The scalar ring of a struc... |
| tngscaOLD 24006 | Obsolete proof of ~ tngsca... |
| tngvsca 24007 | The scalar multiplication ... |
| tngvscaOLD 24008 | Obsolete proof of ~ tngvsc... |
| tngip 24009 | The inner product operatio... |
| tngipOLD 24010 | Obsolete proof of ~ tngip ... |
| tngds 24011 | The metric function of a s... |
| tngdsOLD 24012 | Obsolete proof of ~ tngds ... |
| tngtset 24013 | The topology generated by ... |
| tngtopn 24014 | The topology generated by ... |
| tngnm 24015 | The topology generated by ... |
| tngngp2 24016 | A norm turns a group into ... |
| tngngpd 24017 | Derive the axioms for a no... |
| tngngp 24018 | Derive the axioms for a no... |
| tnggrpr 24019 | If a structure equipped wi... |
| tngngp3 24020 | Alternate definition of a ... |
| nrmtngdist 24021 | The augmentation of a norm... |
| nrmtngnrm 24022 | The augmentation of a norm... |
| tngngpim 24023 | The induced metric of a no... |
| isnrg 24024 | A normed ring is a ring wi... |
| nrgabv 24025 | The norm of a normed ring ... |
| nrgngp 24026 | A normed ring is a normed ... |
| nrgring 24027 | A normed ring is a ring. ... |
| nmmul 24028 | The norm of a product in a... |
| nrgdsdi 24029 | Distribute a distance calc... |
| nrgdsdir 24030 | Distribute a distance calc... |
| nm1 24031 | The norm of one in a nonze... |
| unitnmn0 24032 | The norm of a unit is nonz... |
| nminvr 24033 | The norm of an inverse in ... |
| nmdvr 24034 | The norm of a division in ... |
| nrgdomn 24035 | A nonzero normed ring is a... |
| nrgtgp 24036 | A normed ring is a topolog... |
| subrgnrg 24037 | A normed ring restricted t... |
| tngnrg 24038 | Given any absolute value o... |
| isnlm 24039 | A normed (left) module is ... |
| nmvs 24040 | Defining property of a nor... |
| nlmngp 24041 | A normed module is a norme... |
| nlmlmod 24042 | A normed module is a left ... |
| nlmnrg 24043 | The scalar component of a ... |
| nlmngp2 24044 | The scalar component of a ... |
| nlmdsdi 24045 | Distribute a distance calc... |
| nlmdsdir 24046 | Distribute a distance calc... |
| nlmmul0or 24047 | If a scalar product is zer... |
| sranlm 24048 | The subring algebra over a... |
| nlmvscnlem2 24049 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 24050 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 24051 | The scalar multiplication ... |
| rlmnlm 24052 | The ring module over a nor... |
| rlmnm 24053 | The norm function in the r... |
| nrgtrg 24054 | A normed ring is a topolog... |
| nrginvrcnlem 24055 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 24056 | The ring inverse function ... |
| nrgtdrg 24057 | A normed division ring is ... |
| nlmtlm 24058 | A normed module is a topol... |
| isnvc 24059 | A normed vector space is j... |
| nvcnlm 24060 | A normed vector space is a... |
| nvclvec 24061 | A normed vector space is a... |
| nvclmod 24062 | A normed vector space is a... |
| isnvc2 24063 | A normed vector space is j... |
| nvctvc 24064 | A normed vector space is a... |
| lssnlm 24065 | A subspace of a normed mod... |
| lssnvc 24066 | A subspace of a normed vec... |
| rlmnvc 24067 | The ring module over a nor... |
| ngpocelbl 24068 | Membership of an off-cente... |
| nmoffn 24075 | The function producing ope... |
| reldmnghm 24076 | Lemma for normed group hom... |
| reldmnmhm 24077 | Lemma for module homomorph... |
| nmofval 24078 | Value of the operator norm... |
| nmoval 24079 | Value of the operator norm... |
| nmogelb 24080 | Property of the operator n... |
| nmolb 24081 | Any upper bound on the val... |
| nmolb2d 24082 | Any upper bound on the val... |
| nmof 24083 | The operator norm is a fun... |
| nmocl 24084 | The operator norm of an op... |
| nmoge0 24085 | The operator norm of an op... |
| nghmfval 24086 | A normed group homomorphis... |
| isnghm 24087 | A normed group homomorphis... |
| isnghm2 24088 | A normed group homomorphis... |
| isnghm3 24089 | A normed group homomorphis... |
| bddnghm 24090 | A bounded group homomorphi... |
| nghmcl 24091 | A normed group homomorphis... |
| nmoi 24092 | The operator norm achieves... |
| nmoix 24093 | The operator norm is a bou... |
| nmoi2 24094 | The operator norm is a bou... |
| nmoleub 24095 | The operator norm, defined... |
| nghmrcl1 24096 | Reverse closure for a norm... |
| nghmrcl2 24097 | Reverse closure for a norm... |
| nghmghm 24098 | A normed group homomorphis... |
| nmo0 24099 | The operator norm of the z... |
| nmoeq0 24100 | The operator norm is zero ... |
| nmoco 24101 | An upper bound on the oper... |
| nghmco 24102 | The composition of normed ... |
| nmotri 24103 | Triangle inequality for th... |
| nghmplusg 24104 | The sum of two bounded lin... |
| 0nghm 24105 | The zero operator is a nor... |
| nmoid 24106 | The operator norm of the i... |
| idnghm 24107 | The identity operator is a... |
| nmods 24108 | Upper bound for the distan... |
| nghmcn 24109 | A normed group homomorphis... |
| isnmhm 24110 | A normed module homomorphi... |
| nmhmrcl1 24111 | Reverse closure for a norm... |
| nmhmrcl2 24112 | Reverse closure for a norm... |
| nmhmlmhm 24113 | A normed module homomorphi... |
| nmhmnghm 24114 | A normed module homomorphi... |
| nmhmghm 24115 | A normed module homomorphi... |
| isnmhm2 24116 | A normed module homomorphi... |
| nmhmcl 24117 | A normed module homomorphi... |
| idnmhm 24118 | The identity operator is a... |
| 0nmhm 24119 | The zero operator is a bou... |
| nmhmco 24120 | The composition of bounded... |
| nmhmplusg 24121 | The sum of two bounded lin... |
| qtopbaslem 24122 | The set of open intervals ... |
| qtopbas 24123 | The set of open intervals ... |
| retopbas 24124 | A basis for the standard t... |
| retop 24125 | The standard topology on t... |
| uniretop 24126 | The underlying set of the ... |
| retopon 24127 | The standard topology on t... |
| retps 24128 | The standard topological s... |
| iooretop 24129 | Open intervals are open se... |
| icccld 24130 | Closed intervals are close... |
| icopnfcld 24131 | Right-unbounded closed int... |
| iocmnfcld 24132 | Left-unbounded closed inte... |
| qdensere 24133 | ` QQ ` is dense in the sta... |
| cnmetdval 24134 | Value of the distance func... |
| cnmet 24135 | The absolute value metric ... |
| cnxmet 24136 | The absolute value metric ... |
| cnbl0 24137 | Two ways to write the open... |
| cnblcld 24138 | Two ways to write the clos... |
| cnfldms 24139 | The complex number field i... |
| cnfldxms 24140 | The complex number field i... |
| cnfldtps 24141 | The complex number field i... |
| cnfldnm 24142 | The norm of the field of c... |
| cnngp 24143 | The complex numbers form a... |
| cnnrg 24144 | The complex numbers form a... |
| cnfldtopn 24145 | The topology of the comple... |
| cnfldtopon 24146 | The topology of the comple... |
| cnfldtop 24147 | The topology of the comple... |
| cnfldhaus 24148 | The topology of the comple... |
| unicntop 24149 | The underlying set of the ... |
| cnopn 24150 | The set of complex numbers... |
| zringnrg 24151 | The ring of integers is a ... |
| remetdval 24152 | Value of the distance func... |
| remet 24153 | The absolute value metric ... |
| rexmet 24154 | The absolute value metric ... |
| bl2ioo 24155 | A ball in terms of an open... |
| ioo2bl 24156 | An open interval of reals ... |
| ioo2blex 24157 | An open interval of reals ... |
| blssioo 24158 | The balls of the standard ... |
| tgioo 24159 | The topology generated by ... |
| qdensere2 24160 | ` QQ ` is dense in ` RR ` ... |
| blcvx 24161 | An open ball in the comple... |
| rehaus 24162 | The standard topology on t... |
| tgqioo 24163 | The topology generated by ... |
| re2ndc 24164 | The standard topology on t... |
| resubmet 24165 | The subspace topology indu... |
| tgioo2 24166 | The standard topology on t... |
| rerest 24167 | The subspace topology indu... |
| tgioo3 24168 | The standard topology on t... |
| xrtgioo 24169 | The topology on the extend... |
| xrrest 24170 | The subspace topology indu... |
| xrrest2 24171 | The subspace topology indu... |
| xrsxmet 24172 | The metric on the extended... |
| xrsdsre 24173 | The metric on the extended... |
| xrsblre 24174 | Any ball of the metric of ... |
| xrsmopn 24175 | The metric on the extended... |
| zcld 24176 | The integers are a closed ... |
| recld2 24177 | The real numbers are a clo... |
| zcld2 24178 | The integers are a closed ... |
| zdis 24179 | The integers are a discret... |
| sszcld 24180 | Every subset of the intege... |
| reperflem 24181 | A subset of the real numbe... |
| reperf 24182 | The real numbers are a per... |
| cnperf 24183 | The complex numbers are a ... |
| iccntr 24184 | The interior of a closed i... |
| icccmplem1 24185 | Lemma for ~ icccmp . (Con... |
| icccmplem2 24186 | Lemma for ~ icccmp . (Con... |
| icccmplem3 24187 | Lemma for ~ icccmp . (Con... |
| icccmp 24188 | A closed interval in ` RR ... |
| reconnlem1 24189 | Lemma for ~ reconn . Conn... |
| reconnlem2 24190 | Lemma for ~ reconn . (Con... |
| reconn 24191 | A subset of the reals is c... |
| retopconn 24192 | Corollary of ~ reconn . T... |
| iccconn 24193 | A closed interval is conne... |
| opnreen 24194 | Every nonempty open set is... |
| rectbntr0 24195 | A countable subset of the ... |
| xrge0gsumle 24196 | A finite sum in the nonneg... |
| xrge0tsms 24197 | Any finite or infinite sum... |
| xrge0tsms2 24198 | Any finite or infinite sum... |
| metdcnlem 24199 | The metric function of a m... |
| xmetdcn2 24200 | The metric function of an ... |
| xmetdcn 24201 | The metric function of an ... |
| metdcn2 24202 | The metric function of a m... |
| metdcn 24203 | The metric function of a m... |
| msdcn 24204 | The metric function of a m... |
| cnmpt1ds 24205 | Continuity of the metric f... |
| cnmpt2ds 24206 | Continuity of the metric f... |
| nmcn 24207 | The norm of a normed group... |
| ngnmcncn 24208 | The norm of a normed group... |
| abscn 24209 | The absolute value functio... |
| metdsval 24210 | Value of the "distance to ... |
| metdsf 24211 | The distance from a point ... |
| metdsge 24212 | The distance from the poin... |
| metds0 24213 | If a point is in a set, it... |
| metdstri 24214 | A generalization of the tr... |
| metdsle 24215 | The distance from a point ... |
| metdsre 24216 | The distance from a point ... |
| metdseq0 24217 | The distance from a point ... |
| metdscnlem 24218 | Lemma for ~ metdscn . (Co... |
| metdscn 24219 | The function ` F ` which g... |
| metdscn2 24220 | The function ` F ` which g... |
| metnrmlem1a 24221 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 24222 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 24223 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 24224 | Lemma for ~ metnrm . (Con... |
| metnrm 24225 | A metric space is normal. ... |
| metreg 24226 | A metric space is regular.... |
| addcnlem 24227 | Lemma for ~ addcn , ~ subc... |
| addcn 24228 | Complex number addition is... |
| subcn 24229 | Complex number subtraction... |
| mulcn 24230 | Complex number multiplicat... |
| divcn 24231 | Complex number division is... |
| cnfldtgp 24232 | The complex numbers form a... |
| fsumcn 24233 | A finite sum of functions ... |
| fsum2cn 24234 | Version of ~ fsumcn for tw... |
| expcn 24235 | The power function on comp... |
| divccn 24236 | Division by a nonzero cons... |
| sqcn 24237 | The square function on com... |
| iitopon 24242 | The unit interval is a top... |
| iitop 24243 | The unit interval is a top... |
| iiuni 24244 | The base set of the unit i... |
| dfii2 24245 | Alternate definition of th... |
| dfii3 24246 | Alternate definition of th... |
| dfii4 24247 | Alternate definition of th... |
| dfii5 24248 | The unit interval expresse... |
| iicmp 24249 | The unit interval is compa... |
| iiconn 24250 | The unit interval is conne... |
| cncfval 24251 | The value of the continuou... |
| elcncf 24252 | Membership in the set of c... |
| elcncf2 24253 | Version of ~ elcncf with a... |
| cncfrss 24254 | Reverse closure of the con... |
| cncfrss2 24255 | Reverse closure of the con... |
| cncff 24256 | A continuous complex funct... |
| cncfi 24257 | Defining property of a con... |
| elcncf1di 24258 | Membership in the set of c... |
| elcncf1ii 24259 | Membership in the set of c... |
| rescncf 24260 | A continuous complex funct... |
| cncfcdm 24261 | Change the codomain of a c... |
| cncfss 24262 | The set of continuous func... |
| climcncf 24263 | Image of a limit under a c... |
| abscncf 24264 | Absolute value is continuo... |
| recncf 24265 | Real part is continuous. ... |
| imcncf 24266 | Imaginary part is continuo... |
| cjcncf 24267 | Complex conjugate is conti... |
| mulc1cncf 24268 | Multiplication by a consta... |
| divccncf 24269 | Division by a constant is ... |
| cncfco 24270 | The composition of two con... |
| cncfcompt2 24271 | Composition of continuous ... |
| cncfmet 24272 | Relate complex function co... |
| cncfcn 24273 | Relate complex function co... |
| cncfcn1 24274 | Relate complex function co... |
| cncfmptc 24275 | A constant function is a c... |
| cncfmptid 24276 | The identity function is a... |
| cncfmpt1f 24277 | Composition of continuous ... |
| cncfmpt2f 24278 | Composition of continuous ... |
| cncfmpt2ss 24279 | Composition of continuous ... |
| addccncf 24280 | Adding a constant is a con... |
| idcncf 24281 | The identity function is a... |
| sub1cncf 24282 | Subtracting a constant is ... |
| sub2cncf 24283 | Subtraction from a constan... |
| cdivcncf 24284 | Division with a constant n... |
| negcncf 24285 | The negative function is c... |
| negfcncf 24286 | The negative of a continuo... |
| abscncfALT 24287 | Absolute value is continuo... |
| cncfcnvcn 24288 | Rewrite ~ cmphaushmeo for ... |
| expcncf 24289 | The power function on comp... |
| cnmptre 24290 | Lemma for ~ iirevcn and re... |
| cnmpopc 24291 | Piecewise definition of a ... |
| iirev 24292 | Reverse the unit interval.... |
| iirevcn 24293 | The reversion function is ... |
| iihalf1 24294 | Map the first half of ` II... |
| iihalf1cn 24295 | The first half function is... |
| iihalf2 24296 | Map the second half of ` I... |
| iihalf2cn 24297 | The second half function i... |
| elii1 24298 | Divide the unit interval i... |
| elii2 24299 | Divide the unit interval i... |
| iimulcl 24300 | The unit interval is close... |
| iimulcn 24301 | Multiplication is a contin... |
| icoopnst 24302 | A half-open interval start... |
| iocopnst 24303 | A half-open interval endin... |
| icchmeo 24304 | The natural bijection from... |
| icopnfcnv 24305 | Define a bijection from ` ... |
| icopnfhmeo 24306 | The defined bijection from... |
| iccpnfcnv 24307 | Define a bijection from ` ... |
| iccpnfhmeo 24308 | The defined bijection from... |
| xrhmeo 24309 | The bijection from ` [ -u ... |
| xrhmph 24310 | The extended reals are hom... |
| xrcmp 24311 | The topology of the extend... |
| xrconn 24312 | The topology of the extend... |
| icccvx 24313 | A linear combination of tw... |
| oprpiece1res1 24314 | Restriction to the first p... |
| oprpiece1res2 24315 | Restriction to the second ... |
| cnrehmeo 24316 | The canonical bijection fr... |
| cnheiborlem 24317 | Lemma for ~ cnheibor . (C... |
| cnheibor 24318 | Heine-Borel theorem for co... |
| cnllycmp 24319 | The topology on the comple... |
| rellycmp 24320 | The topology on the reals ... |
| bndth 24321 | The Boundedness Theorem. ... |
| evth 24322 | The Extreme Value Theorem.... |
| evth2 24323 | The Extreme Value Theorem,... |
| lebnumlem1 24324 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24325 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24326 | Lemma for ~ lebnum . By t... |
| lebnum 24327 | The Lebesgue number lemma,... |
| xlebnum 24328 | Generalize ~ lebnum to ext... |
| lebnumii 24329 | Specialize the Lebesgue nu... |
| ishtpy 24335 | Membership in the class of... |
| htpycn 24336 | A homotopy is a continuous... |
| htpyi 24337 | A homotopy evaluated at it... |
| ishtpyd 24338 | Deduction for membership i... |
| htpycom 24339 | Given a homotopy from ` F ... |
| htpyid 24340 | A homotopy from a function... |
| htpyco1 24341 | Compose a homotopy with a ... |
| htpyco2 24342 | Compose a homotopy with a ... |
| htpycc 24343 | Concatenate two homotopies... |
| isphtpy 24344 | Membership in the class of... |
| phtpyhtpy 24345 | A path homotopy is a homot... |
| phtpycn 24346 | A path homotopy is a conti... |
| phtpyi 24347 | Membership in the class of... |
| phtpy01 24348 | Two path-homotopic paths h... |
| isphtpyd 24349 | Deduction for membership i... |
| isphtpy2d 24350 | Deduction for membership i... |
| phtpycom 24351 | Given a homotopy from ` F ... |
| phtpyid 24352 | A homotopy from a path to ... |
| phtpyco2 24353 | Compose a path homotopy wi... |
| phtpycc 24354 | Concatenate two path homot... |
| phtpcrel 24356 | The path homotopy relation... |
| isphtpc 24357 | The relation "is path homo... |
| phtpcer 24358 | Path homotopy is an equiva... |
| phtpc01 24359 | Path homotopic paths have ... |
| reparphti 24360 | Lemma for ~ reparpht . (C... |
| reparpht 24361 | Reparametrization lemma. ... |
| phtpcco2 24362 | Compose a path homotopy wi... |
| pcofval 24373 | The value of the path conc... |
| pcoval 24374 | The concatenation of two p... |
| pcovalg 24375 | Evaluate the concatenation... |
| pcoval1 24376 | Evaluate the concatenation... |
| pco0 24377 | The starting point of a pa... |
| pco1 24378 | The ending point of a path... |
| pcoval2 24379 | Evaluate the concatenation... |
| pcocn 24380 | The concatenation of two p... |
| copco 24381 | The composition of a conca... |
| pcohtpylem 24382 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24383 | Homotopy invariance of pat... |
| pcoptcl 24384 | A constant function is a p... |
| pcopt 24385 | Concatenation with a point... |
| pcopt2 24386 | Concatenation with a point... |
| pcoass 24387 | Order of concatenation doe... |
| pcorevcl 24388 | Closure for a reversed pat... |
| pcorevlem 24389 | Lemma for ~ pcorev . Prov... |
| pcorev 24390 | Concatenation with the rev... |
| pcorev2 24391 | Concatenation with the rev... |
| pcophtb 24392 | The path homotopy equivale... |
| om1val 24393 | The definition of the loop... |
| om1bas 24394 | The base set of the loop s... |
| om1elbas 24395 | Elementhood in the base se... |
| om1addcl 24396 | Closure of the group opera... |
| om1plusg 24397 | The group operation (which... |
| om1tset 24398 | The topology of the loop s... |
| om1opn 24399 | The topology of the loop s... |
| pi1val 24400 | The definition of the fund... |
| pi1bas 24401 | The base set of the fundam... |
| pi1blem 24402 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24403 | Another way to write the l... |
| pi1bas2 24404 | The base set of the fundam... |
| pi1eluni 24405 | Elementhood in the base se... |
| pi1bas3 24406 | The base set of the fundam... |
| pi1cpbl 24407 | The group operation, loop ... |
| elpi1 24408 | The elements of the fundam... |
| elpi1i 24409 | The elements of the fundam... |
| pi1addf 24410 | The group operation of ` p... |
| pi1addval 24411 | The concatenation of two p... |
| pi1grplem 24412 | Lemma for ~ pi1grp . (Con... |
| pi1grp 24413 | The fundamental group is a... |
| pi1id 24414 | The identity element of th... |
| pi1inv 24415 | An inverse in the fundamen... |
| pi1xfrf 24416 | Functionality of the loop ... |
| pi1xfrval 24417 | The value of the loop tran... |
| pi1xfr 24418 | Given a path ` F ` and its... |
| pi1xfrcnvlem 24419 | Given a path ` F ` between... |
| pi1xfrcnv 24420 | Given a path ` F ` between... |
| pi1xfrgim 24421 | The mapping ` G ` between ... |
| pi1cof 24422 | Functionality of the loop ... |
| pi1coval 24423 | The value of the loop tran... |
| pi1coghm 24424 | The mapping ` G ` between ... |
| isclm 24427 | A subcomplex module is a l... |
| clmsca 24428 | The ring of scalars ` F ` ... |
| clmsubrg 24429 | The base set of the ring o... |
| clmlmod 24430 | A subcomplex module is a l... |
| clmgrp 24431 | A subcomplex module is an ... |
| clmabl 24432 | A subcomplex module is an ... |
| clmring 24433 | The scalar ring of a subco... |
| clmfgrp 24434 | The scalar ring of a subco... |
| clm0 24435 | The zero of the scalar rin... |
| clm1 24436 | The identity of the scalar... |
| clmadd 24437 | The addition of the scalar... |
| clmmul 24438 | The multiplication of the ... |
| clmcj 24439 | The conjugation of the sca... |
| isclmi 24440 | Reverse direction of ~ isc... |
| clmzss 24441 | The scalar ring of a subco... |
| clmsscn 24442 | The scalar ring of a subco... |
| clmsub 24443 | Subtraction in the scalar ... |
| clmneg 24444 | Negation in the scalar rin... |
| clmneg1 24445 | Minus one is in the scalar... |
| clmabs 24446 | Norm in the scalar ring of... |
| clmacl 24447 | Closure of ring addition f... |
| clmmcl 24448 | Closure of ring multiplica... |
| clmsubcl 24449 | Closure of ring subtractio... |
| lmhmclm 24450 | The domain of a linear ope... |
| clmvscl 24451 | Closure of scalar product ... |
| clmvsass 24452 | Associative law for scalar... |
| clmvscom 24453 | Commutative law for the sc... |
| clmvsdir 24454 | Distributive law for scala... |
| clmvsdi 24455 | Distributive law for scala... |
| clmvs1 24456 | Scalar product with ring u... |
| clmvs2 24457 | A vector plus itself is tw... |
| clm0vs 24458 | Zero times a vector is the... |
| clmopfne 24459 | The (functionalized) opera... |
| isclmp 24460 | The predicate "is a subcom... |
| isclmi0 24461 | Properties that determine ... |
| clmvneg1 24462 | Minus 1 times a vector is ... |
| clmvsneg 24463 | Multiplication of a vector... |
| clmmulg 24464 | The group multiple functio... |
| clmsubdir 24465 | Scalar multiplication dist... |
| clmpm1dir 24466 | Subtractive distributive l... |
| clmnegneg 24467 | Double negative of a vecto... |
| clmnegsubdi2 24468 | Distribution of negative o... |
| clmsub4 24469 | Rearrangement of 4 terms i... |
| clmvsrinv 24470 | A vector minus itself. (C... |
| clmvslinv 24471 | Minus a vector plus itself... |
| clmvsubval 24472 | Value of vector subtractio... |
| clmvsubval2 24473 | Value of vector subtractio... |
| clmvz 24474 | Two ways to express the ne... |
| zlmclm 24475 | The ` ZZ ` -module operati... |
| clmzlmvsca 24476 | The scalar product of a su... |
| nmoleub2lem 24477 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 24478 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 24479 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 24480 | The operator norm is the s... |
| nmoleub2b 24481 | The operator norm is the s... |
| nmoleub3 24482 | The operator norm is the s... |
| nmhmcn 24483 | A linear operator over a n... |
| cmodscexp 24484 | The powers of ` _i ` belon... |
| cmodscmulexp 24485 | The scalar product of a ve... |
| cvslvec 24488 | A subcomplex vector space ... |
| cvsclm 24489 | A subcomplex vector space ... |
| iscvs 24490 | A subcomplex vector space ... |
| iscvsp 24491 | The predicate "is a subcom... |
| iscvsi 24492 | Properties that determine ... |
| cvsi 24493 | The properties of a subcom... |
| cvsunit 24494 | Unit group of the scalar r... |
| cvsdiv 24495 | Division of the scalar rin... |
| cvsdivcl 24496 | The scalar field of a subc... |
| cvsmuleqdivd 24497 | An equality involving rati... |
| cvsdiveqd 24498 | An equality involving rati... |
| cnlmodlem1 24499 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 24500 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 24501 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 24502 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 24503 | The set of complex numbers... |
| cnstrcvs 24504 | The set of complex numbers... |
| cnrbas 24505 | The set of complex numbers... |
| cnrlmod 24506 | The complex left module of... |
| cnrlvec 24507 | The complex left module of... |
| cncvs 24508 | The complex left module of... |
| recvs 24509 | The field of the real numb... |
| recvsOLD 24510 | Obsolete version of ~ recv... |
| qcvs 24511 | The field of rational numb... |
| zclmncvs 24512 | The ring of integers as le... |
| isncvsngp 24513 | A normed subcomplex vector... |
| isncvsngpd 24514 | Properties that determine ... |
| ncvsi 24515 | The properties of a normed... |
| ncvsprp 24516 | Proportionality property o... |
| ncvsge0 24517 | The norm of a scalar produ... |
| ncvsm1 24518 | The norm of the opposite o... |
| ncvsdif 24519 | The norm of the difference... |
| ncvspi 24520 | The norm of a vector plus ... |
| ncvs1 24521 | From any nonzero vector of... |
| cnrnvc 24522 | The module of complex numb... |
| cnncvs 24523 | The module of complex numb... |
| cnnm 24524 | The norm of the normed sub... |
| ncvspds 24525 | Value of the distance func... |
| cnindmet 24526 | The metric induced on the ... |
| cnncvsaddassdemo 24527 | Derive the associative law... |
| cnncvsmulassdemo 24528 | Derive the associative law... |
| cnncvsabsnegdemo 24529 | Derive the absolute value ... |
| iscph 24534 | A subcomplex pre-Hilbert s... |
| cphphl 24535 | A subcomplex pre-Hilbert s... |
| cphnlm 24536 | A subcomplex pre-Hilbert s... |
| cphngp 24537 | A subcomplex pre-Hilbert s... |
| cphlmod 24538 | A subcomplex pre-Hilbert s... |
| cphlvec 24539 | A subcomplex pre-Hilbert s... |
| cphnvc 24540 | A subcomplex pre-Hilbert s... |
| cphsubrglem 24541 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 24542 | Lemma for ~ cphreccl . (C... |
| cphsca 24543 | A subcomplex pre-Hilbert s... |
| cphsubrg 24544 | The scalar field of a subc... |
| cphreccl 24545 | The scalar field of a subc... |
| cphdivcl 24546 | The scalar field of a subc... |
| cphcjcl 24547 | The scalar field of a subc... |
| cphsqrtcl 24548 | The scalar field of a subc... |
| cphabscl 24549 | The scalar field of a subc... |
| cphsqrtcl2 24550 | The scalar field of a subc... |
| cphsqrtcl3 24551 | If the scalar field of a s... |
| cphqss 24552 | The scalar field of a subc... |
| cphclm 24553 | A subcomplex pre-Hilbert s... |
| cphnmvs 24554 | Norm of a scalar product. ... |
| cphipcl 24555 | An inner product is a memb... |
| cphnmfval 24556 | The value of the norm in a... |
| cphnm 24557 | The square of the norm is ... |
| nmsq 24558 | The square of the norm is ... |
| cphnmf 24559 | The norm of a vector is a ... |
| cphnmcl 24560 | The norm of a vector is a ... |
| reipcl 24561 | An inner product of an ele... |
| ipge0 24562 | The inner product in a sub... |
| cphipcj 24563 | Conjugate of an inner prod... |
| cphipipcj 24564 | An inner product times its... |
| cphorthcom 24565 | Orthogonality (meaning inn... |
| cphip0l 24566 | Inner product with a zero ... |
| cphip0r 24567 | Inner product with a zero ... |
| cphipeq0 24568 | The inner product of a vec... |
| cphdir 24569 | Distributive law for inner... |
| cphdi 24570 | Distributive law for inner... |
| cph2di 24571 | Distributive law for inner... |
| cphsubdir 24572 | Distributive law for inner... |
| cphsubdi 24573 | Distributive law for inner... |
| cph2subdi 24574 | Distributive law for inner... |
| cphass 24575 | Associative law for inner ... |
| cphassr 24576 | "Associative" law for seco... |
| cph2ass 24577 | Move scalar multiplication... |
| cphassi 24578 | Associative law for the fi... |
| cphassir 24579 | "Associative" law for the ... |
| cphpyth 24580 | The pythagorean theorem fo... |
| tcphex 24581 | Lemma for ~ tcphbas and si... |
| tcphval 24582 | Define a function to augme... |
| tcphbas 24583 | The base set of a subcompl... |
| tchplusg 24584 | The addition operation of ... |
| tcphsub 24585 | The subtraction operation ... |
| tcphmulr 24586 | The ring operation of a su... |
| tcphsca 24587 | The scalar field of a subc... |
| tcphvsca 24588 | The scalar multiplication ... |
| tcphip 24589 | The inner product of a sub... |
| tcphtopn 24590 | The topology of a subcompl... |
| tcphphl 24591 | Augmentation of a subcompl... |
| tchnmfval 24592 | The norm of a subcomplex p... |
| tcphnmval 24593 | The norm of a subcomplex p... |
| cphtcphnm 24594 | The norm of a norm-augment... |
| tcphds 24595 | The distance of a pre-Hilb... |
| phclm 24596 | A pre-Hilbert space whose ... |
| tcphcphlem3 24597 | Lemma for ~ tcphcph : real... |
| ipcau2 24598 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 24599 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 24600 | Lemma for ~ tcphcph : homo... |
| tcphcph 24601 | The standard definition of... |
| ipcau 24602 | The Cauchy-Schwarz inequal... |
| nmparlem 24603 | Lemma for ~ nmpar . (Cont... |
| nmpar 24604 | A subcomplex pre-Hilbert s... |
| cphipval2 24605 | Value of the inner product... |
| 4cphipval2 24606 | Four times the inner produ... |
| cphipval 24607 | Value of the inner product... |
| ipcnlem2 24608 | The inner product operatio... |
| ipcnlem1 24609 | The inner product operatio... |
| ipcn 24610 | The inner product operatio... |
| cnmpt1ip 24611 | Continuity of inner produc... |
| cnmpt2ip 24612 | Continuity of inner produc... |
| csscld 24613 | A "closed subspace" in a s... |
| clsocv 24614 | The orthogonal complement ... |
| cphsscph 24615 | A subspace of a subcomplex... |
| lmmbr 24622 | Express the binary relatio... |
| lmmbr2 24623 | Express the binary relatio... |
| lmmbr3 24624 | Express the binary relatio... |
| lmmcvg 24625 | Convergence property of a ... |
| lmmbrf 24626 | Express the binary relatio... |
| lmnn 24627 | A condition that implies c... |
| cfilfval 24628 | The set of Cauchy filters ... |
| iscfil 24629 | The property of being a Ca... |
| iscfil2 24630 | The property of being a Ca... |
| cfilfil 24631 | A Cauchy filter is a filte... |
| cfili 24632 | Property of a Cauchy filte... |
| cfil3i 24633 | A Cauchy filter contains b... |
| cfilss 24634 | A filter finer than a Cauc... |
| fgcfil 24635 | The Cauchy filter conditio... |
| fmcfil 24636 | The Cauchy filter conditio... |
| iscfil3 24637 | A filter is Cauchy iff it ... |
| cfilfcls 24638 | Similar to ultrafilters ( ... |
| caufval 24639 | The set of Cauchy sequence... |
| iscau 24640 | Express the property " ` F... |
| iscau2 24641 | Express the property " ` F... |
| iscau3 24642 | Express the Cauchy sequenc... |
| iscau4 24643 | Express the property " ` F... |
| iscauf 24644 | Express the property " ` F... |
| caun0 24645 | A metric with a Cauchy seq... |
| caufpm 24646 | Inclusion of a Cauchy sequ... |
| caucfil 24647 | A Cauchy sequence predicat... |
| iscmet 24648 | The property " ` D ` is a ... |
| cmetcvg 24649 | The convergence of a Cauch... |
| cmetmet 24650 | A complete metric space is... |
| cmetmeti 24651 | A complete metric space is... |
| cmetcaulem 24652 | Lemma for ~ cmetcau . (Co... |
| cmetcau 24653 | The convergence of a Cauch... |
| iscmet3lem3 24654 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 24655 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 24656 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 24657 | The property " ` D ` is a ... |
| iscmet2 24658 | A metric ` D ` is complete... |
| cfilresi 24659 | A Cauchy filter on a metri... |
| cfilres 24660 | Cauchy filter on a metric ... |
| caussi 24661 | Cauchy sequence on a metri... |
| causs 24662 | Cauchy sequence on a metri... |
| equivcfil 24663 | If the metric ` D ` is "st... |
| equivcau 24664 | If the metric ` D ` is "st... |
| lmle 24665 | If the distance from each ... |
| nglmle 24666 | If the norm of each member... |
| lmclim 24667 | Relate a limit on the metr... |
| lmclimf 24668 | Relate a limit on the metr... |
| metelcls 24669 | A point belongs to the clo... |
| metcld 24670 | A subset of a metric space... |
| metcld2 24671 | A subset of a metric space... |
| caubl 24672 | Sufficient condition to en... |
| caublcls 24673 | The convergent point of a ... |
| metcnp4 24674 | Two ways to say a mapping ... |
| metcn4 24675 | Two ways to say a mapping ... |
| iscmet3i 24676 | Properties that determine ... |
| lmcau 24677 | Every convergent sequence ... |
| flimcfil 24678 | Every convergent filter in... |
| metsscmetcld 24679 | A complete subspace of a m... |
| cmetss 24680 | A subspace of a complete m... |
| equivcmet 24681 | If two metrics are strongl... |
| relcmpcmet 24682 | If ` D ` is a metric space... |
| cmpcmet 24683 | A compact metric space is ... |
| cfilucfil3 24684 | Given a metric ` D ` and a... |
| cfilucfil4 24685 | Given a metric ` D ` and a... |
| cncmet 24686 | The set of complex numbers... |
| recmet 24687 | The real numbers are a com... |
| bcthlem1 24688 | Lemma for ~ bcth . Substi... |
| bcthlem2 24689 | Lemma for ~ bcth . The ba... |
| bcthlem3 24690 | Lemma for ~ bcth . The li... |
| bcthlem4 24691 | Lemma for ~ bcth . Given ... |
| bcthlem5 24692 | Lemma for ~ bcth . The pr... |
| bcth 24693 | Baire's Category Theorem. ... |
| bcth2 24694 | Baire's Category Theorem, ... |
| bcth3 24695 | Baire's Category Theorem, ... |
| isbn 24702 | A Banach space is a normed... |
| bnsca 24703 | The scalar field of a Bana... |
| bnnvc 24704 | A Banach space is a normed... |
| bnnlm 24705 | A Banach space is a normed... |
| bnngp 24706 | A Banach space is a normed... |
| bnlmod 24707 | A Banach space is a left m... |
| bncms 24708 | A Banach space is a comple... |
| iscms 24709 | A complete metric space is... |
| cmscmet 24710 | The induced metric on a co... |
| bncmet 24711 | The induced metric on Bana... |
| cmsms 24712 | A complete metric space is... |
| cmspropd 24713 | Property deduction for a c... |
| cmssmscld 24714 | The restriction of a metri... |
| cmsss 24715 | The restriction of a compl... |
| lssbn 24716 | A subspace of a Banach spa... |
| cmetcusp1 24717 | If the uniform set of a co... |
| cmetcusp 24718 | The uniform space generate... |
| cncms 24719 | The field of complex numbe... |
| cnflduss 24720 | The uniform structure of t... |
| cnfldcusp 24721 | The field of complex numbe... |
| resscdrg 24722 | The real numbers are a sub... |
| cncdrg 24723 | The only complete subfield... |
| srabn 24724 | The subring algebra over a... |
| rlmbn 24725 | The ring module over a com... |
| ishl 24726 | The predicate "is a subcom... |
| hlbn 24727 | Every subcomplex Hilbert s... |
| hlcph 24728 | Every subcomplex Hilbert s... |
| hlphl 24729 | Every subcomplex Hilbert s... |
| hlcms 24730 | Every subcomplex Hilbert s... |
| hlprlem 24731 | Lemma for ~ hlpr . (Contr... |
| hlress 24732 | The scalar field of a subc... |
| hlpr 24733 | The scalar field of a subc... |
| ishl2 24734 | A Hilbert space is a compl... |
| cphssphl 24735 | A Banach subspace of a sub... |
| cmslssbn 24736 | A complete linear subspace... |
| cmscsscms 24737 | A closed subspace of a com... |
| bncssbn 24738 | A closed subspace of a Ban... |
| cssbn 24739 | A complete subspace of a n... |
| csschl 24740 | A complete subspace of a c... |
| cmslsschl 24741 | A complete linear subspace... |
| chlcsschl 24742 | A closed subspace of a sub... |
| retopn 24743 | The topology of the real n... |
| recms 24744 | The real numbers form a co... |
| reust 24745 | The Uniform structure of t... |
| recusp 24746 | The real numbers form a co... |
| rrxval 24751 | Value of the generalized E... |
| rrxbase 24752 | The base of the generalize... |
| rrxprds 24753 | Expand the definition of t... |
| rrxip 24754 | The inner product of the g... |
| rrxnm 24755 | The norm of the generalize... |
| rrxcph 24756 | Generalized Euclidean real... |
| rrxds 24757 | The distance over generali... |
| rrxvsca 24758 | The scalar product over ge... |
| rrxplusgvscavalb 24759 | The result of the addition... |
| rrxsca 24760 | The field of real numbers ... |
| rrx0 24761 | The zero ("origin") in a g... |
| rrx0el 24762 | The zero ("origin") in a g... |
| csbren 24763 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 24764 | Triangle inequality in R^n... |
| rrxf 24765 | Euclidean vectors as funct... |
| rrxfsupp 24766 | Euclidean vectors are of f... |
| rrxsuppss 24767 | Support of Euclidean vecto... |
| rrxmvallem 24768 | Support of the function us... |
| rrxmval 24769 | The value of the Euclidean... |
| rrxmfval 24770 | The value of the Euclidean... |
| rrxmetlem 24771 | Lemma for ~ rrxmet . (Con... |
| rrxmet 24772 | Euclidean space is a metri... |
| rrxdstprj1 24773 | The distance between two p... |
| rrxbasefi 24774 | The base of the generalize... |
| rrxdsfi 24775 | The distance over generali... |
| rrxmetfi 24776 | Euclidean space is a metri... |
| rrxdsfival 24777 | The value of the Euclidean... |
| ehlval 24778 | Value of the Euclidean spa... |
| ehlbase 24779 | The base of the Euclidean ... |
| ehl0base 24780 | The base of the Euclidean ... |
| ehl0 24781 | The Euclidean space of dim... |
| ehleudis 24782 | The Euclidean distance fun... |
| ehleudisval 24783 | The value of the Euclidean... |
| ehl1eudis 24784 | The Euclidean distance fun... |
| ehl1eudisval 24785 | The value of the Euclidean... |
| ehl2eudis 24786 | The Euclidean distance fun... |
| ehl2eudisval 24787 | The value of the Euclidean... |
| minveclem1 24788 | Lemma for ~ minvec . The ... |
| minveclem4c 24789 | Lemma for ~ minvec . The ... |
| minveclem2 24790 | Lemma for ~ minvec . Any ... |
| minveclem3a 24791 | Lemma for ~ minvec . ` D `... |
| minveclem3b 24792 | Lemma for ~ minvec . The ... |
| minveclem3 24793 | Lemma for ~ minvec . The ... |
| minveclem4a 24794 | Lemma for ~ minvec . ` F `... |
| minveclem4b 24795 | Lemma for ~ minvec . The ... |
| minveclem4 24796 | Lemma for ~ minvec . The ... |
| minveclem5 24797 | Lemma for ~ minvec . Disc... |
| minveclem6 24798 | Lemma for ~ minvec . Any ... |
| minveclem7 24799 | Lemma for ~ minvec . Sinc... |
| minvec 24800 | Minimizing vector theorem,... |
| pjthlem1 24801 | Lemma for ~ pjth . (Contr... |
| pjthlem2 24802 | Lemma for ~ pjth . (Contr... |
| pjth 24803 | Projection Theorem: Any H... |
| pjth2 24804 | Projection Theorem with ab... |
| cldcss 24805 | Corollary of the Projectio... |
| cldcss2 24806 | Corollary of the Projectio... |
| hlhil 24807 | Corollary of the Projectio... |
| addcncf 24808 | The addition of two contin... |
| subcncf 24809 | The addition of two contin... |
| mulcncf 24810 | The multiplication of two ... |
| divcncf 24811 | The quotient of two contin... |
| pmltpclem1 24812 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 24813 | Lemma for ~ pmltpc . (Con... |
| pmltpc 24814 | Any function on the reals ... |
| ivthlem1 24815 | Lemma for ~ ivth . The se... |
| ivthlem2 24816 | Lemma for ~ ivth . Show t... |
| ivthlem3 24817 | Lemma for ~ ivth , the int... |
| ivth 24818 | The intermediate value the... |
| ivth2 24819 | The intermediate value the... |
| ivthle 24820 | The intermediate value the... |
| ivthle2 24821 | The intermediate value the... |
| ivthicc 24822 | The interval between any t... |
| evthicc 24823 | Specialization of the Extr... |
| evthicc2 24824 | Combine ~ ivthicc with ~ e... |
| cniccbdd 24825 | A continuous function on a... |
| ovolfcl 24830 | Closure for the interval e... |
| ovolfioo 24831 | Unpack the interval coveri... |
| ovolficc 24832 | Unpack the interval coveri... |
| ovolficcss 24833 | Any (closed) interval cove... |
| ovolfsval 24834 | The value of the interval ... |
| ovolfsf 24835 | Closure for the interval l... |
| ovolsf 24836 | Closure for the partial su... |
| ovolval 24837 | The value of the outer mea... |
| elovolmlem 24838 | Lemma for ~ elovolm and re... |
| elovolm 24839 | Elementhood in the set ` M... |
| elovolmr 24840 | Sufficient condition for e... |
| ovolmge0 24841 | The set ` M ` is composed ... |
| ovolcl 24842 | The volume of a set is an ... |
| ovollb 24843 | The outer volume is a lowe... |
| ovolgelb 24844 | The outer volume is the gr... |
| ovolge0 24845 | The volume of a set is alw... |
| ovolf 24846 | The domain and codomain of... |
| ovollecl 24847 | If an outer volume is boun... |
| ovolsslem 24848 | Lemma for ~ ovolss . (Con... |
| ovolss 24849 | The volume of a set is mon... |
| ovolsscl 24850 | If a set is contained in a... |
| ovolssnul 24851 | A subset of a nullset is n... |
| ovollb2lem 24852 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 24853 | It is often more convenien... |
| ovolctb 24854 | The volume of a denumerabl... |
| ovolq 24855 | The rational numbers have ... |
| ovolctb2 24856 | The volume of a countable ... |
| ovol0 24857 | The empty set has 0 outer ... |
| ovolfi 24858 | A finite set has 0 outer L... |
| ovolsn 24859 | A singleton has 0 outer Le... |
| ovolunlem1a 24860 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 24861 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 24862 | Lemma for ~ ovolun . (Con... |
| ovolun 24863 | The Lebesgue outer measure... |
| ovolunnul 24864 | Adding a nullset does not ... |
| ovolfiniun 24865 | The Lebesgue outer measure... |
| ovoliunlem1 24866 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 24867 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 24868 | Lemma for ~ ovoliun . (Co... |
| ovoliun 24869 | The Lebesgue outer measure... |
| ovoliun2 24870 | The Lebesgue outer measure... |
| ovoliunnul 24871 | A countable union of nulls... |
| shft2rab 24872 | If ` B ` is a shift of ` A... |
| ovolshftlem1 24873 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 24874 | Lemma for ~ ovolshft . (C... |
| ovolshft 24875 | The Lebesgue outer measure... |
| sca2rab 24876 | If ` B ` is a scale of ` A... |
| ovolscalem1 24877 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 24878 | Lemma for ~ ovolshft . (C... |
| ovolsca 24879 | The Lebesgue outer measure... |
| ovolicc1 24880 | The measure of a closed in... |
| ovolicc2lem1 24881 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 24882 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 24883 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 24884 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 24885 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 24886 | The measure of a closed in... |
| ovolicc 24887 | The measure of a closed in... |
| ovolicopnf 24888 | The measure of a right-unb... |
| ovolre 24889 | The measure of the real nu... |
| ismbl 24890 | The predicate " ` A ` is L... |
| ismbl2 24891 | From ~ ovolun , it suffice... |
| volres 24892 | A self-referencing abbrevi... |
| volf 24893 | The domain and codomain of... |
| mblvol 24894 | The volume of a measurable... |
| mblss 24895 | A measurable set is a subs... |
| mblsplit 24896 | The defining property of m... |
| volss 24897 | The Lebesgue measure is mo... |
| cmmbl 24898 | The complement of a measur... |
| nulmbl 24899 | A nullset is measurable. ... |
| nulmbl2 24900 | A set of outer measure zer... |
| unmbl 24901 | A union of measurable sets... |
| shftmbl 24902 | A shift of a measurable se... |
| 0mbl 24903 | The empty set is measurabl... |
| rembl 24904 | The set of all real number... |
| unidmvol 24905 | The union of the Lebesgue ... |
| inmbl 24906 | An intersection of measura... |
| difmbl 24907 | A difference of measurable... |
| finiunmbl 24908 | A finite union of measurab... |
| volun 24909 | The Lebesgue measure funct... |
| volinun 24910 | Addition of non-disjoint s... |
| volfiniun 24911 | The volume of a disjoint f... |
| iundisj 24912 | Rewrite a countable union ... |
| iundisj2 24913 | A disjoint union is disjoi... |
| voliunlem1 24914 | Lemma for ~ voliun . (Con... |
| voliunlem2 24915 | Lemma for ~ voliun . (Con... |
| voliunlem3 24916 | Lemma for ~ voliun . (Con... |
| iunmbl 24917 | The measurable sets are cl... |
| voliun 24918 | The Lebesgue measure funct... |
| volsuplem 24919 | Lemma for ~ volsup . (Con... |
| volsup 24920 | The volume of the limit of... |
| iunmbl2 24921 | The measurable sets are cl... |
| ioombl1lem1 24922 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 24923 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 24924 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 24925 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 24926 | An open right-unbounded in... |
| icombl1 24927 | A closed unbounded-above i... |
| icombl 24928 | A closed-below, open-above... |
| ioombl 24929 | An open real interval is m... |
| iccmbl 24930 | A closed real interval is ... |
| iccvolcl 24931 | A closed real interval has... |
| ovolioo 24932 | The measure of an open int... |
| volioo 24933 | The measure of an open int... |
| ioovolcl 24934 | An open real interval has ... |
| ovolfs2 24935 | Alternative expression for... |
| ioorcl2 24936 | An open interval with fini... |
| ioorf 24937 | Define a function from ope... |
| ioorval 24938 | Define a function from ope... |
| ioorinv2 24939 | The function ` F ` is an "... |
| ioorinv 24940 | The function ` F ` is an "... |
| ioorcl 24941 | The function ` F ` does no... |
| uniiccdif 24942 | A union of closed interval... |
| uniioovol 24943 | A disjoint union of open i... |
| uniiccvol 24944 | An almost-disjoint union o... |
| uniioombllem1 24945 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 24946 | Lemma for ~ uniioombl . (... |
| uniioombllem2 24947 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 24948 | Lemma for ~ uniioombl . (... |
| uniioombllem3 24949 | Lemma for ~ uniioombl . (... |
| uniioombllem4 24950 | Lemma for ~ uniioombl . (... |
| uniioombllem5 24951 | Lemma for ~ uniioombl . (... |
| uniioombllem6 24952 | Lemma for ~ uniioombl . (... |
| uniioombl 24953 | A disjoint union of open i... |
| uniiccmbl 24954 | An almost-disjoint union o... |
| dyadf 24955 | The function ` F ` returns... |
| dyadval 24956 | Value of the dyadic ration... |
| dyadovol 24957 | Volume of a dyadic rationa... |
| dyadss 24958 | Two closed dyadic rational... |
| dyaddisjlem 24959 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 24960 | Two closed dyadic rational... |
| dyadmaxlem 24961 | Lemma for ~ dyadmax . (Co... |
| dyadmax 24962 | Any nonempty set of dyadic... |
| dyadmbllem 24963 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 24964 | Any union of dyadic ration... |
| opnmbllem 24965 | Lemma for ~ opnmbl . (Con... |
| opnmbl 24966 | All open sets are measurab... |
| opnmblALT 24967 | All open sets are measurab... |
| subopnmbl 24968 | Sets which are open in a m... |
| volsup2 24969 | The volume of ` A ` is the... |
| volcn 24970 | The function formed by res... |
| volivth 24971 | The Intermediate Value The... |
| vitalilem1 24972 | Lemma for ~ vitali . (Con... |
| vitalilem2 24973 | Lemma for ~ vitali . (Con... |
| vitalilem3 24974 | Lemma for ~ vitali . (Con... |
| vitalilem4 24975 | Lemma for ~ vitali . (Con... |
| vitalilem5 24976 | Lemma for ~ vitali . (Con... |
| vitali 24977 | If the reals can be well-o... |
| ismbf1 24988 | The predicate " ` F ` is a... |
| mbff 24989 | A measurable function is a... |
| mbfdm 24990 | The domain of a measurable... |
| mbfconstlem 24991 | Lemma for ~ mbfconst and r... |
| ismbf 24992 | The predicate " ` F ` is a... |
| ismbfcn 24993 | A complex function is meas... |
| mbfima 24994 | Definitional property of a... |
| mbfimaicc 24995 | The preimage of any closed... |
| mbfimasn 24996 | The preimage of a point un... |
| mbfconst 24997 | A constant function is mea... |
| mbf0 24998 | The empty function is meas... |
| mbfid 24999 | The identity function is m... |
| mbfmptcl 25000 | Lemma for the ` MblFn ` pr... |
| mbfdm2 25001 | The domain of a measurable... |
| ismbfcn2 25002 | A complex function is meas... |
| ismbfd 25003 | Deduction to prove measura... |
| ismbf2d 25004 | Deduction to prove measura... |
| mbfeqalem1 25005 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 25006 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 25007 | If two functions are equal... |
| mbfres 25008 | The restriction of a measu... |
| mbfres2 25009 | Measurability of a piecewi... |
| mbfss 25010 | Change the domain of a mea... |
| mbfmulc2lem 25011 | Multiplication by a consta... |
| mbfmulc2re 25012 | Multiplication by a consta... |
| mbfmax 25013 | The maximum of two functio... |
| mbfneg 25014 | The negative of a measurab... |
| mbfpos 25015 | The positive part of a mea... |
| mbfposr 25016 | Converse to ~ mbfpos . (C... |
| mbfposb 25017 | A function is measurable i... |
| ismbf3d 25018 | Simplified form of ~ ismbf... |
| mbfimaopnlem 25019 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 25020 | The preimage of any open s... |
| mbfimaopn2 25021 | The preimage of any set op... |
| cncombf 25022 | The composition of a conti... |
| cnmbf 25023 | A continuous function is m... |
| mbfaddlem 25024 | The sum of two measurable ... |
| mbfadd 25025 | The sum of two measurable ... |
| mbfsub 25026 | The difference of two meas... |
| mbfmulc2 25027 | A complex constant times a... |
| mbfsup 25028 | The supremum of a sequence... |
| mbfinf 25029 | The infimum of a sequence ... |
| mbflimsup 25030 | The limit supremum of a se... |
| mbflimlem 25031 | The pointwise limit of a s... |
| mbflim 25032 | The pointwise limit of a s... |
| 0pval 25035 | The zero function evaluate... |
| 0plef 25036 | Two ways to say that the f... |
| 0pledm 25037 | Adjust the domain of the l... |
| isi1f 25038 | The predicate " ` F ` is a... |
| i1fmbf 25039 | Simple functions are measu... |
| i1ff 25040 | A simple function is a fun... |
| i1frn 25041 | A simple function has fini... |
| i1fima 25042 | Any preimage of a simple f... |
| i1fima2 25043 | Any preimage of a simple f... |
| i1fima2sn 25044 | Preimage of a singleton. ... |
| i1fd 25045 | A simplified set of assump... |
| i1f0rn 25046 | Any simple function takes ... |
| itg1val 25047 | The value of the integral ... |
| itg1val2 25048 | The value of the integral ... |
| itg1cl 25049 | Closure of the integral on... |
| itg1ge0 25050 | Closure of the integral on... |
| i1f0 25051 | The zero function is simpl... |
| itg10 25052 | The zero function has zero... |
| i1f1lem 25053 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 25054 | Base case simple functions... |
| itg11 25055 | The integral of an indicat... |
| itg1addlem1 25056 | Decompose a preimage, whic... |
| i1faddlem 25057 | Decompose the preimage of ... |
| i1fmullem 25058 | Decompose the preimage of ... |
| i1fadd 25059 | The sum of two simple func... |
| i1fmul 25060 | The pointwise product of t... |
| itg1addlem2 25061 | Lemma for ~ itg1add . The... |
| itg1addlem3 25062 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 25063 | Lemma for ~ itg1add . (Co... |
| itg1addlem4OLD 25064 | Obsolete version of ~ itg1... |
| itg1addlem5 25065 | Lemma for ~ itg1add . (Co... |
| itg1add 25066 | The integral of a sum of s... |
| i1fmulclem 25067 | Decompose the preimage of ... |
| i1fmulc 25068 | A nonnegative constant tim... |
| itg1mulc 25069 | The integral of a constant... |
| i1fres 25070 | The "restriction" of a sim... |
| i1fpos 25071 | The positive part of a sim... |
| i1fposd 25072 | Deduction form of ~ i1fpos... |
| i1fsub 25073 | The difference of two simp... |
| itg1sub 25074 | The integral of a differen... |
| itg10a 25075 | The integral of a simple f... |
| itg1ge0a 25076 | The integral of an almost ... |
| itg1lea 25077 | Approximate version of ~ i... |
| itg1le 25078 | If one simple function dom... |
| itg1climres 25079 | Restricting the simple fun... |
| mbfi1fseqlem1 25080 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 25081 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 25082 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 25083 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 25084 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 25085 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 25086 | A characterization of meas... |
| mbfi1flimlem 25087 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 25088 | Any real measurable functi... |
| mbfmullem2 25089 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 25090 | Lemma for ~ mbfmul . (Con... |
| mbfmul 25091 | The product of two measura... |
| itg2lcl 25092 | The set of lower sums is a... |
| itg2val 25093 | Value of the integral on n... |
| itg2l 25094 | Elementhood in the set ` L... |
| itg2lr 25095 | Sufficient condition for e... |
| xrge0f 25096 | A real function is a nonne... |
| itg2cl 25097 | The integral of a nonnegat... |
| itg2ub 25098 | The integral of a nonnegat... |
| itg2leub 25099 | Any upper bound on the int... |
| itg2ge0 25100 | The integral of a nonnegat... |
| itg2itg1 25101 | The integral of a nonnegat... |
| itg20 25102 | The integral of the zero f... |
| itg2lecl 25103 | If an ` S.2 ` integral is ... |
| itg2le 25104 | If one function dominates ... |
| itg2const 25105 | Integral of a constant fun... |
| itg2const2 25106 | When the base set of a con... |
| itg2seq 25107 | Definitional property of t... |
| itg2uba 25108 | Approximate version of ~ i... |
| itg2lea 25109 | Approximate version of ~ i... |
| itg2eqa 25110 | Approximate equality of in... |
| itg2mulclem 25111 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 25112 | The integral of a nonnegat... |
| itg2splitlem 25113 | Lemma for ~ itg2split . (... |
| itg2split 25114 | The ` S.2 ` integral split... |
| itg2monolem1 25115 | Lemma for ~ itg2mono . We... |
| itg2monolem2 25116 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 25117 | Lemma for ~ itg2mono . (C... |
| itg2mono 25118 | The Monotone Convergence T... |
| itg2i1fseqle 25119 | Subject to the conditions ... |
| itg2i1fseq 25120 | Subject to the conditions ... |
| itg2i1fseq2 25121 | In an extension to the res... |
| itg2i1fseq3 25122 | Special case of ~ itg2i1fs... |
| itg2addlem 25123 | Lemma for ~ itg2add . (Co... |
| itg2add 25124 | The ` S.2 ` integral is li... |
| itg2gt0 25125 | If the function ` F ` is s... |
| itg2cnlem1 25126 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 25127 | Lemma for ~ itgcn . (Cont... |
| itg2cn 25128 | A sort of absolute continu... |
| ibllem 25129 | Conditioned equality theor... |
| isibl 25130 | The predicate " ` F ` is i... |
| isibl2 25131 | The predicate " ` F ` is i... |
| iblmbf 25132 | An integrable function is ... |
| iblitg 25133 | If a function is integrabl... |
| dfitg 25134 | Evaluate the class substit... |
| itgex 25135 | An integral is a set. (Co... |
| itgeq1f 25136 | Equality theorem for an in... |
| itgeq1 25137 | Equality theorem for an in... |
| nfitg1 25138 | Bound-variable hypothesis ... |
| nfitg 25139 | Bound-variable hypothesis ... |
| cbvitg 25140 | Change bound variable in a... |
| cbvitgv 25141 | Change bound variable in a... |
| itgeq2 25142 | Equality theorem for an in... |
| itgresr 25143 | The domain of an integral ... |
| itg0 25144 | The integral of anything o... |
| itgz 25145 | The integral of zero on an... |
| itgeq2dv 25146 | Equality theorem for an in... |
| itgmpt 25147 | Change bound variable in a... |
| itgcl 25148 | The integral of an integra... |
| itgvallem 25149 | Substitution lemma. (Cont... |
| itgvallem3 25150 | Lemma for ~ itgposval and ... |
| ibl0 25151 | The zero function is integ... |
| iblcnlem1 25152 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 25153 | Expand out the universal q... |
| itgcnlem 25154 | Expand out the sum in ~ df... |
| iblrelem 25155 | Integrability of a real fu... |
| iblposlem 25156 | Lemma for ~ iblpos . (Con... |
| iblpos 25157 | Integrability of a nonnega... |
| iblre 25158 | Integrability of a real fu... |
| itgrevallem1 25159 | Lemma for ~ itgposval and ... |
| itgposval 25160 | The integral of a nonnegat... |
| itgreval 25161 | Decompose the integral of ... |
| itgrecl 25162 | Real closure of an integra... |
| iblcn 25163 | Integrability of a complex... |
| itgcnval 25164 | Decompose the integral of ... |
| itgre 25165 | Real part of an integral. ... |
| itgim 25166 | Imaginary part of an integ... |
| iblneg 25167 | The negative of an integra... |
| itgneg 25168 | Negation of an integral. ... |
| iblss 25169 | A subset of an integrable ... |
| iblss2 25170 | Change the domain of an in... |
| itgitg2 25171 | Transfer an integral using... |
| i1fibl 25172 | A simple function is integ... |
| itgitg1 25173 | Transfer an integral using... |
| itgle 25174 | Monotonicity of an integra... |
| itgge0 25175 | The integral of a positive... |
| itgss 25176 | Expand the set of an integ... |
| itgss2 25177 | Expand the set of an integ... |
| itgeqa 25178 | Approximate equality of in... |
| itgss3 25179 | Expand the set of an integ... |
| itgioo 25180 | Equality of integrals on o... |
| itgless 25181 | Expand the integral of a n... |
| iblconst 25182 | A constant function is int... |
| itgconst 25183 | Integral of a constant fun... |
| ibladdlem 25184 | Lemma for ~ ibladd . (Con... |
| ibladd 25185 | Add two integrals over the... |
| iblsub 25186 | Subtract two integrals ove... |
| itgaddlem1 25187 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 25188 | Lemma for ~ itgadd . (Con... |
| itgadd 25189 | Add two integrals over the... |
| itgsub 25190 | Subtract two integrals ove... |
| itgfsum 25191 | Take a finite sum of integ... |
| iblabslem 25192 | Lemma for ~ iblabs . (Con... |
| iblabs 25193 | The absolute value of an i... |
| iblabsr 25194 | A measurable function is i... |
| iblmulc2 25195 | Multiply an integral by a ... |
| itgmulc2lem1 25196 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 25197 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 25198 | Multiply an integral by a ... |
| itgabs 25199 | The triangle inequality fo... |
| itgsplit 25200 | The ` S. ` integral splits... |
| itgspliticc 25201 | The ` S. ` integral splits... |
| itgsplitioo 25202 | The ` S. ` integral splits... |
| bddmulibl 25203 | A bounded function times a... |
| bddibl 25204 | A bounded function is inte... |
| cniccibl 25205 | A continuous function on a... |
| bddiblnc 25206 | Choice-free proof of ~ bdd... |
| cnicciblnc 25207 | Choice-free proof of ~ cni... |
| itggt0 25208 | The integral of a strictly... |
| itgcn 25209 | Transfer ~ itg2cn to the f... |
| ditgeq1 25212 | Equality theorem for the d... |
| ditgeq2 25213 | Equality theorem for the d... |
| ditgeq3 25214 | Equality theorem for the d... |
| ditgeq3dv 25215 | Equality theorem for the d... |
| ditgex 25216 | A directed integral is a s... |
| ditg0 25217 | Value of the directed inte... |
| cbvditg 25218 | Change bound variable in a... |
| cbvditgv 25219 | Change bound variable in a... |
| ditgpos 25220 | Value of the directed inte... |
| ditgneg 25221 | Value of the directed inte... |
| ditgcl 25222 | Closure of a directed inte... |
| ditgswap 25223 | Reverse a directed integra... |
| ditgsplitlem 25224 | Lemma for ~ ditgsplit . (... |
| ditgsplit 25225 | This theorem is the raison... |
| reldv 25234 | The derivative function is... |
| limcvallem 25235 | Lemma for ~ ellimc . (Con... |
| limcfval 25236 | Value and set bounds on th... |
| ellimc 25237 | Value of the limit predica... |
| limcrcl 25238 | Reverse closure for the li... |
| limccl 25239 | Closure of the limit opera... |
| limcdif 25240 | It suffices to consider fu... |
| ellimc2 25241 | Write the definition of a ... |
| limcnlp 25242 | If ` B ` is not a limit po... |
| ellimc3 25243 | Write the epsilon-delta de... |
| limcflflem 25244 | Lemma for ~ limcflf . (Co... |
| limcflf 25245 | The limit operator can be ... |
| limcmo 25246 | If ` B ` is a limit point ... |
| limcmpt 25247 | Express the limit operator... |
| limcmpt2 25248 | Express the limit operator... |
| limcresi 25249 | Any limit of ` F ` is also... |
| limcres 25250 | If ` B ` is an interior po... |
| cnplimc 25251 | A function is continuous a... |
| cnlimc 25252 | ` F ` is a continuous func... |
| cnlimci 25253 | If ` F ` is a continuous f... |
| cnmptlimc 25254 | If ` F ` is a continuous f... |
| limccnp 25255 | If the limit of ` F ` at `... |
| limccnp2 25256 | The image of a convergent ... |
| limcco 25257 | Composition of two limits.... |
| limciun 25258 | A point is a limit of ` F ... |
| limcun 25259 | A point is a limit of ` F ... |
| dvlem 25260 | Closure for a difference q... |
| dvfval 25261 | Value and set bounds on th... |
| eldv 25262 | The differentiable predica... |
| dvcl 25263 | The derivative function ta... |
| dvbssntr 25264 | The set of differentiable ... |
| dvbss 25265 | The set of differentiable ... |
| dvbsss 25266 | The set of differentiable ... |
| perfdvf 25267 | The derivative is a functi... |
| recnprss 25268 | Both ` RR ` and ` CC ` are... |
| recnperf 25269 | Both ` RR ` and ` CC ` are... |
| dvfg 25270 | Explicitly write out the f... |
| dvf 25271 | The derivative is a functi... |
| dvfcn 25272 | The derivative is a functi... |
| dvreslem 25273 | Lemma for ~ dvres . (Cont... |
| dvres2lem 25274 | Lemma for ~ dvres2 . (Con... |
| dvres 25275 | Restriction of a derivativ... |
| dvres2 25276 | Restriction of the base se... |
| dvres3 25277 | Restriction of a complex d... |
| dvres3a 25278 | Restriction of a complex d... |
| dvidlem 25279 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 25280 | Derivative of a function r... |
| dvconst 25281 | Derivative of a constant f... |
| dvid 25282 | Derivative of the identity... |
| dvcnp 25283 | The difference quotient is... |
| dvcnp2 25284 | A function is continuous a... |
| dvcn 25285 | A differentiable function ... |
| dvnfval 25286 | Value of the iterated deri... |
| dvnff 25287 | The iterated derivative is... |
| dvn0 25288 | Zero times iterated deriva... |
| dvnp1 25289 | Successor iterated derivat... |
| dvn1 25290 | One times iterated derivat... |
| dvnf 25291 | The N-times derivative is ... |
| dvnbss 25292 | The set of N-times differe... |
| dvnadd 25293 | The ` N ` -th derivative o... |
| dvn2bss 25294 | An N-times differentiable ... |
| dvnres 25295 | Multiple derivative versio... |
| cpnfval 25296 | Condition for n-times cont... |
| fncpn 25297 | The ` C^n ` object is a fu... |
| elcpn 25298 | Condition for n-times cont... |
| cpnord 25299 | ` C^n ` conditions are ord... |
| cpncn 25300 | A ` C^n ` function is cont... |
| cpnres 25301 | The restriction of a ` C^n... |
| dvaddbr 25302 | The sum rule for derivativ... |
| dvmulbr 25303 | The product rule for deriv... |
| dvadd 25304 | The sum rule for derivativ... |
| dvmul 25305 | The product rule for deriv... |
| dvaddf 25306 | The sum rule for everywher... |
| dvmulf 25307 | The product rule for every... |
| dvcmul 25308 | The product rule when one ... |
| dvcmulf 25309 | The product rule when one ... |
| dvcobr 25310 | The chain rule for derivat... |
| dvco 25311 | The chain rule for derivat... |
| dvcof 25312 | The chain rule for everywh... |
| dvcjbr 25313 | The derivative of the conj... |
| dvcj 25314 | The derivative of the conj... |
| dvfre 25315 | The derivative of a real f... |
| dvnfre 25316 | The ` N ` -th derivative o... |
| dvexp 25317 | Derivative of a power func... |
| dvexp2 25318 | Derivative of an exponenti... |
| dvrec 25319 | Derivative of the reciproc... |
| dvmptres3 25320 | Function-builder for deriv... |
| dvmptid 25321 | Function-builder for deriv... |
| dvmptc 25322 | Function-builder for deriv... |
| dvmptcl 25323 | Closure lemma for ~ dvmptc... |
| dvmptadd 25324 | Function-builder for deriv... |
| dvmptmul 25325 | Function-builder for deriv... |
| dvmptres2 25326 | Function-builder for deriv... |
| dvmptres 25327 | Function-builder for deriv... |
| dvmptcmul 25328 | Function-builder for deriv... |
| dvmptdivc 25329 | Function-builder for deriv... |
| dvmptneg 25330 | Function-builder for deriv... |
| dvmptsub 25331 | Function-builder for deriv... |
| dvmptcj 25332 | Function-builder for deriv... |
| dvmptre 25333 | Function-builder for deriv... |
| dvmptim 25334 | Function-builder for deriv... |
| dvmptntr 25335 | Function-builder for deriv... |
| dvmptco 25336 | Function-builder for deriv... |
| dvrecg 25337 | Derivative of the reciproc... |
| dvmptdiv 25338 | Function-builder for deriv... |
| dvmptfsum 25339 | Function-builder for deriv... |
| dvcnvlem 25340 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25341 | A weak version of ~ dvcnvr... |
| dvexp3 25342 | Derivative of an exponenti... |
| dveflem 25343 | Derivative of the exponent... |
| dvef 25344 | Derivative of the exponent... |
| dvsincos 25345 | Derivative of the sine and... |
| dvsin 25346 | Derivative of the sine fun... |
| dvcos 25347 | Derivative of the cosine f... |
| dvferm1lem 25348 | Lemma for ~ dvferm . (Con... |
| dvferm1 25349 | One-sided version of ~ dvf... |
| dvferm2lem 25350 | Lemma for ~ dvferm . (Con... |
| dvferm2 25351 | One-sided version of ~ dvf... |
| dvferm 25352 | Fermat's theorem on statio... |
| rollelem 25353 | Lemma for ~ rolle . (Cont... |
| rolle 25354 | Rolle's theorem. If ` F `... |
| cmvth 25355 | Cauchy's Mean Value Theore... |
| mvth 25356 | The Mean Value Theorem. I... |
| dvlip 25357 | A function with derivative... |
| dvlipcn 25358 | A complex function with de... |
| dvlip2 25359 | Combine the results of ~ d... |
| c1liplem1 25360 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25361 | C^1 functions are Lipschit... |
| c1lip2 25362 | C^1 functions are Lipschit... |
| c1lip3 25363 | C^1 functions are Lipschit... |
| dveq0 25364 | If a continuous function h... |
| dv11cn 25365 | Two functions defined on a... |
| dvgt0lem1 25366 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25367 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25368 | A function on a closed int... |
| dvlt0 25369 | A function on a closed int... |
| dvge0 25370 | A function on a closed int... |
| dvle 25371 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25372 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25373 | Lemma for ~ dvivth . (Con... |
| dvivth 25374 | Darboux' theorem, or the i... |
| dvne0 25375 | A function on a closed int... |
| dvne0f1 25376 | A function on a closed int... |
| lhop1lem 25377 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25378 | L'Hôpital's Rule for... |
| lhop2 25379 | L'Hôpital's Rule for... |
| lhop 25380 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25381 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25382 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25383 | The derivative rule for in... |
| dvcvx 25384 | A real function with stric... |
| dvfsumle 25385 | Compare a finite sum to an... |
| dvfsumge 25386 | Compare a finite sum to an... |
| dvfsumabs 25387 | Compare a finite sum to an... |
| dvmptrecl 25388 | Real closure of a derivati... |
| dvfsumrlimf 25389 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25390 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25391 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 25392 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25393 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25394 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25395 | Compare a finite sum to an... |
| dvfsumrlim2 25396 | Compare a finite sum to an... |
| dvfsumrlim3 25397 | Conjoin the statements of ... |
| dvfsum2 25398 | The reverse of ~ dvfsumrli... |
| ftc1lem1 25399 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 25400 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 25401 | The Fundamental Theorem of... |
| ftc1lem3 25402 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 25403 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 25404 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 25405 | Lemma for ~ ftc1 . (Contr... |
| ftc1 25406 | The Fundamental Theorem of... |
| ftc1cn 25407 | Strengthen the assumptions... |
| ftc2 25408 | The Fundamental Theorem of... |
| ftc2ditglem 25409 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 25410 | Directed integral analogue... |
| itgparts 25411 | Integration by parts. If ... |
| itgsubstlem 25412 | Lemma for ~ itgsubst . (C... |
| itgsubst 25413 | Integration by ` u ` -subs... |
| itgpowd 25414 | The integral of a monomial... |
| reldmmdeg 25419 | Multivariate degree is a b... |
| tdeglem1 25420 | Functionality of the total... |
| tdeglem1OLD 25421 | Obsolete version of ~ tdeg... |
| tdeglem3 25422 | Additivity of the total de... |
| tdeglem3OLD 25423 | Obsolete version of ~ tdeg... |
| tdeglem4 25424 | There is only one multi-in... |
| tdeglem4OLD 25425 | Obsolete version of ~ tdeg... |
| tdeglem2 25426 | Simplification of total de... |
| mdegfval 25427 | Value of the multivariate ... |
| mdegval 25428 | Value of the multivariate ... |
| mdegleb 25429 | Property of being of limit... |
| mdeglt 25430 | If there is an upper limit... |
| mdegldg 25431 | A nonzero polynomial has s... |
| mdegxrcl 25432 | Closure of polynomial degr... |
| mdegxrf 25433 | Functionality of polynomia... |
| mdegcl 25434 | Sharp closure for multivar... |
| mdeg0 25435 | Degree of the zero polynom... |
| mdegnn0cl 25436 | Degree of a nonzero polyno... |
| degltlem1 25437 | Theorem on arithmetic of e... |
| degltp1le 25438 | Theorem on arithmetic of e... |
| mdegaddle 25439 | The degree of a sum is at ... |
| mdegvscale 25440 | The degree of a scalar mul... |
| mdegvsca 25441 | The degree of a scalar mul... |
| mdegle0 25442 | A polynomial has nonpositi... |
| mdegmullem 25443 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 25444 | The multivariate degree of... |
| deg1fval 25445 | Relate univariate polynomi... |
| deg1xrf 25446 | Functionality of univariat... |
| deg1xrcl 25447 | Closure of univariate poly... |
| deg1cl 25448 | Sharp closure of univariat... |
| mdegpropd 25449 | Property deduction for pol... |
| deg1fvi 25450 | Univariate polynomial degr... |
| deg1propd 25451 | Property deduction for pol... |
| deg1z 25452 | Degree of the zero univari... |
| deg1nn0cl 25453 | Degree of a nonzero univar... |
| deg1n0ima 25454 | Degree image of a set of p... |
| deg1nn0clb 25455 | A polynomial is nonzero if... |
| deg1lt0 25456 | A polynomial is zero iff i... |
| deg1ldg 25457 | A nonzero univariate polyn... |
| deg1ldgn 25458 | An index at which a polyno... |
| deg1ldgdomn 25459 | A nonzero univariate polyn... |
| deg1leb 25460 | Property of being of limit... |
| deg1val 25461 | Value of the univariate de... |
| deg1lt 25462 | If the degree of a univari... |
| deg1ge 25463 | Conversely, a nonzero coef... |
| coe1mul3 25464 | The coefficient vector of ... |
| coe1mul4 25465 | Value of the "leading" coe... |
| deg1addle 25466 | The degree of a sum is at ... |
| deg1addle2 25467 | If both factors have degre... |
| deg1add 25468 | Exact degree of a sum of t... |
| deg1vscale 25469 | The degree of a scalar tim... |
| deg1vsca 25470 | The degree of a scalar tim... |
| deg1invg 25471 | The degree of the negated ... |
| deg1suble 25472 | The degree of a difference... |
| deg1sub 25473 | Exact degree of a differen... |
| deg1mulle2 25474 | Produce a bound on the pro... |
| deg1sublt 25475 | Subtraction of two polynom... |
| deg1le0 25476 | A polynomial has nonpositi... |
| deg1sclle 25477 | A scalar polynomial has no... |
| deg1scl 25478 | A nonzero scalar polynomia... |
| deg1mul2 25479 | Degree of multiplication o... |
| deg1mul3 25480 | Degree of multiplication o... |
| deg1mul3le 25481 | Degree of multiplication o... |
| deg1tmle 25482 | Limiting degree of a polyn... |
| deg1tm 25483 | Exact degree of a polynomi... |
| deg1pwle 25484 | Limiting degree of a varia... |
| deg1pw 25485 | Exact degree of a variable... |
| ply1nz 25486 | Univariate polynomials ove... |
| ply1nzb 25487 | Univariate polynomials are... |
| ply1domn 25488 | Corollary of ~ deg1mul2 : ... |
| ply1idom 25489 | The ring of univariate pol... |
| ply1divmo 25500 | Uniqueness of a quotient i... |
| ply1divex 25501 | Lemma for ~ ply1divalg : e... |
| ply1divalg 25502 | The division algorithm for... |
| ply1divalg2 25503 | Reverse the order of multi... |
| uc1pval 25504 | Value of the set of unitic... |
| isuc1p 25505 | Being a unitic polynomial.... |
| mon1pval 25506 | Value of the set of monic ... |
| ismon1p 25507 | Being a monic polynomial. ... |
| uc1pcl 25508 | Unitic polynomials are pol... |
| mon1pcl 25509 | Monic polynomials are poly... |
| uc1pn0 25510 | Unitic polynomials are not... |
| mon1pn0 25511 | Monic polynomials are not ... |
| uc1pdeg 25512 | Unitic polynomials have no... |
| uc1pldg 25513 | Unitic polynomials have un... |
| mon1pldg 25514 | Unitic polynomials have on... |
| mon1puc1p 25515 | Monic polynomials are unit... |
| uc1pmon1p 25516 | Make a unitic polynomial m... |
| deg1submon1p 25517 | The difference of two moni... |
| q1pval 25518 | Value of the univariate po... |
| q1peqb 25519 | Characterizing property of... |
| q1pcl 25520 | Closure of the quotient by... |
| r1pval 25521 | Value of the polynomial re... |
| r1pcl 25522 | Closure of remainder follo... |
| r1pdeglt 25523 | The remainder has a degree... |
| r1pid 25524 | Express the original polyn... |
| dvdsq1p 25525 | Divisibility in a polynomi... |
| dvdsr1p 25526 | Divisibility in a polynomi... |
| ply1remlem 25527 | A term of the form ` x - N... |
| ply1rem 25528 | The polynomial remainder t... |
| facth1 25529 | The factor theorem and its... |
| fta1glem1 25530 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 25531 | Lemma for ~ fta1g . (Cont... |
| fta1g 25532 | The one-sided fundamental ... |
| fta1blem 25533 | Lemma for ~ fta1b . (Cont... |
| fta1b 25534 | The assumption that ` R ` ... |
| drnguc1p 25535 | Over a division ring, all ... |
| ig1peu 25536 | There is a unique monic po... |
| ig1pval 25537 | Substitutions for the poly... |
| ig1pval2 25538 | Generator of the zero idea... |
| ig1pval3 25539 | Characterizing properties ... |
| ig1pcl 25540 | The monic generator of an ... |
| ig1pdvds 25541 | The monic generator of an ... |
| ig1prsp 25542 | Any ideal of polynomials o... |
| ply1lpir 25543 | The ring of polynomials ov... |
| ply1pid 25544 | The polynomials over a fie... |
| plyco0 25553 | Two ways to say that a fun... |
| plyval 25554 | Value of the polynomial se... |
| plybss 25555 | Reverse closure of the par... |
| elply 25556 | Definition of a polynomial... |
| elply2 25557 | The coefficient function c... |
| plyun0 25558 | The set of polynomials is ... |
| plyf 25559 | The polynomial is a functi... |
| plyss 25560 | The polynomial set functio... |
| plyssc 25561 | Every polynomial ring is c... |
| elplyr 25562 | Sufficient condition for e... |
| elplyd 25563 | Sufficient condition for e... |
| ply1termlem 25564 | Lemma for ~ ply1term . (C... |
| ply1term 25565 | A one-term polynomial. (C... |
| plypow 25566 | A power is a polynomial. ... |
| plyconst 25567 | A constant function is a p... |
| ne0p 25568 | A test to show that a poly... |
| ply0 25569 | The zero function is a pol... |
| plyid 25570 | The identity function is a... |
| plyeq0lem 25571 | Lemma for ~ plyeq0 . If `... |
| plyeq0 25572 | If a polynomial is zero at... |
| plypf1 25573 | Write the set of complex p... |
| plyaddlem1 25574 | Derive the coefficient fun... |
| plymullem1 25575 | Derive the coefficient fun... |
| plyaddlem 25576 | Lemma for ~ plyadd . (Con... |
| plymullem 25577 | Lemma for ~ plymul . (Con... |
| plyadd 25578 | The sum of two polynomials... |
| plymul 25579 | The product of two polynom... |
| plysub 25580 | The difference of two poly... |
| plyaddcl 25581 | The sum of two polynomials... |
| plymulcl 25582 | The product of two polynom... |
| plysubcl 25583 | The difference of two poly... |
| coeval 25584 | Value of the coefficient f... |
| coeeulem 25585 | Lemma for ~ coeeu . (Cont... |
| coeeu 25586 | Uniqueness of the coeffici... |
| coelem 25587 | Lemma for properties of th... |
| coeeq 25588 | If ` A ` satisfies the pro... |
| dgrval 25589 | Value of the degree functi... |
| dgrlem 25590 | Lemma for ~ dgrcl and simi... |
| coef 25591 | The domain and codomain of... |
| coef2 25592 | The domain and codomain of... |
| coef3 25593 | The domain and codomain of... |
| dgrcl 25594 | The degree of any polynomi... |
| dgrub 25595 | If the ` M ` -th coefficie... |
| dgrub2 25596 | All the coefficients above... |
| dgrlb 25597 | If all the coefficients ab... |
| coeidlem 25598 | Lemma for ~ coeid . (Cont... |
| coeid 25599 | Reconstruct a polynomial a... |
| coeid2 25600 | Reconstruct a polynomial a... |
| coeid3 25601 | Reconstruct a polynomial a... |
| plyco 25602 | The composition of two pol... |
| coeeq2 25603 | Compute the coefficient fu... |
| dgrle 25604 | Given an explicit expressi... |
| dgreq 25605 | If the highest term in a p... |
| 0dgr 25606 | A constant function has de... |
| 0dgrb 25607 | A function has degree zero... |
| dgrnznn 25608 | A nonzero polynomial with ... |
| coefv0 25609 | The result of evaluating a... |
| coeaddlem 25610 | Lemma for ~ coeadd and ~ d... |
| coemullem 25611 | Lemma for ~ coemul and ~ d... |
| coeadd 25612 | The coefficient function o... |
| coemul 25613 | A coefficient of a product... |
| coe11 25614 | The coefficient function i... |
| coemulhi 25615 | The leading coefficient of... |
| coemulc 25616 | The coefficient function i... |
| coe0 25617 | The coefficients of the ze... |
| coesub 25618 | The coefficient function o... |
| coe1termlem 25619 | The coefficient function o... |
| coe1term 25620 | The coefficient function o... |
| dgr1term 25621 | The degree of a monomial. ... |
| plycn 25622 | A polynomial is a continuo... |
| dgr0 25623 | The degree of the zero pol... |
| coeidp 25624 | The coefficients of the id... |
| dgrid 25625 | The degree of the identity... |
| dgreq0 25626 | The leading coefficient of... |
| dgrlt 25627 | Two ways to say that the d... |
| dgradd 25628 | The degree of a sum of pol... |
| dgradd2 25629 | The degree of a sum of pol... |
| dgrmul2 25630 | The degree of a product of... |
| dgrmul 25631 | The degree of a product of... |
| dgrmulc 25632 | Scalar multiplication by a... |
| dgrsub 25633 | The degree of a difference... |
| dgrcolem1 25634 | The degree of a compositio... |
| dgrcolem2 25635 | Lemma for ~ dgrco . (Cont... |
| dgrco 25636 | The degree of a compositio... |
| plycjlem 25637 | Lemma for ~ plycj and ~ co... |
| plycj 25638 | The double conjugation of ... |
| coecj 25639 | Double conjugation of a po... |
| plyrecj 25640 | A polynomial with real coe... |
| plymul0or 25641 | Polynomial multiplication ... |
| ofmulrt 25642 | The set of roots of a prod... |
| plyreres 25643 | Real-coefficient polynomia... |
| dvply1 25644 | Derivative of a polynomial... |
| dvply2g 25645 | The derivative of a polyno... |
| dvply2 25646 | The derivative of a polyno... |
| dvnply2 25647 | Polynomials have polynomia... |
| dvnply 25648 | Polynomials have polynomia... |
| plycpn 25649 | Polynomials are smooth. (... |
| quotval 25652 | Value of the quotient func... |
| plydivlem1 25653 | Lemma for ~ plydivalg . (... |
| plydivlem2 25654 | Lemma for ~ plydivalg . (... |
| plydivlem3 25655 | Lemma for ~ plydivex . Ba... |
| plydivlem4 25656 | Lemma for ~ plydivex . In... |
| plydivex 25657 | Lemma for ~ plydivalg . (... |
| plydiveu 25658 | Lemma for ~ plydivalg . (... |
| plydivalg 25659 | The division algorithm on ... |
| quotlem 25660 | Lemma for properties of th... |
| quotcl 25661 | The quotient of two polyno... |
| quotcl2 25662 | Closure of the quotient fu... |
| quotdgr 25663 | Remainder property of the ... |
| plyremlem 25664 | Closure of a linear factor... |
| plyrem 25665 | The polynomial remainder t... |
| facth 25666 | The factor theorem. If a ... |
| fta1lem 25667 | Lemma for ~ fta1 . (Contr... |
| fta1 25668 | The easy direction of the ... |
| quotcan 25669 | Exact division with a mult... |
| vieta1lem1 25670 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 25671 | Lemma for ~ vieta1 : induc... |
| vieta1 25672 | The first-order Vieta's fo... |
| plyexmo 25673 | An infinite set of values ... |
| elaa 25676 | Elementhood in the set of ... |
| aacn 25677 | An algebraic number is a c... |
| aasscn 25678 | The algebraic numbers are ... |
| elqaalem1 25679 | Lemma for ~ elqaa . The f... |
| elqaalem2 25680 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 25681 | Lemma for ~ elqaa . (Cont... |
| elqaa 25682 | The set of numbers generat... |
| qaa 25683 | Every rational number is a... |
| qssaa 25684 | The rational numbers are c... |
| iaa 25685 | The imaginary unit is alge... |
| aareccl 25686 | The reciprocal of an algeb... |
| aacjcl 25687 | The conjugate of an algebr... |
| aannenlem1 25688 | Lemma for ~ aannen . (Con... |
| aannenlem2 25689 | Lemma for ~ aannen . (Con... |
| aannenlem3 25690 | The algebraic numbers are ... |
| aannen 25691 | The algebraic numbers are ... |
| aalioulem1 25692 | Lemma for ~ aaliou . An i... |
| aalioulem2 25693 | Lemma for ~ aaliou . (Con... |
| aalioulem3 25694 | Lemma for ~ aaliou . (Con... |
| aalioulem4 25695 | Lemma for ~ aaliou . (Con... |
| aalioulem5 25696 | Lemma for ~ aaliou . (Con... |
| aalioulem6 25697 | Lemma for ~ aaliou . (Con... |
| aaliou 25698 | Liouville's theorem on dio... |
| geolim3 25699 | Geometric series convergen... |
| aaliou2 25700 | Liouville's approximation ... |
| aaliou2b 25701 | Liouville's approximation ... |
| aaliou3lem1 25702 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 25703 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 25704 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 25705 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 25706 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 25707 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 25708 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 25709 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 25710 | Example of a "Liouville nu... |
| aaliou3 25711 | Example of a "Liouville nu... |
| taylfvallem1 25716 | Lemma for ~ taylfval . (C... |
| taylfvallem 25717 | Lemma for ~ taylfval . (C... |
| taylfval 25718 | Define the Taylor polynomi... |
| eltayl 25719 | Value of the Taylor series... |
| taylf 25720 | The Taylor series defines ... |
| tayl0 25721 | The Taylor series is alway... |
| taylplem1 25722 | Lemma for ~ taylpfval and ... |
| taylplem2 25723 | Lemma for ~ taylpfval and ... |
| taylpfval 25724 | Define the Taylor polynomi... |
| taylpf 25725 | The Taylor polynomial is a... |
| taylpval 25726 | Value of the Taylor polyno... |
| taylply2 25727 | The Taylor polynomial is a... |
| taylply 25728 | The Taylor polynomial is a... |
| dvtaylp 25729 | The derivative of the Tayl... |
| dvntaylp 25730 | The ` M ` -th derivative o... |
| dvntaylp0 25731 | The first ` N ` derivative... |
| taylthlem1 25732 | Lemma for ~ taylth . This... |
| taylthlem2 25733 | Lemma for ~ taylth . (Con... |
| taylth 25734 | Taylor's theorem. The Tay... |
| ulmrel 25737 | The uniform limit relation... |
| ulmscl 25738 | Closure of the base set in... |
| ulmval 25739 | Express the predicate: Th... |
| ulmcl 25740 | Closure of a uniform limit... |
| ulmf 25741 | Closure of a uniform limit... |
| ulmpm 25742 | Closure of a uniform limit... |
| ulmf2 25743 | Closure of a uniform limit... |
| ulm2 25744 | Simplify ~ ulmval when ` F... |
| ulmi 25745 | The uniform limit property... |
| ulmclm 25746 | A uniform limit of functio... |
| ulmres 25747 | A sequence of functions co... |
| ulmshftlem 25748 | Lemma for ~ ulmshft . (Co... |
| ulmshft 25749 | A sequence of functions co... |
| ulm0 25750 | Every function converges u... |
| ulmuni 25751 | A sequence of functions un... |
| ulmdm 25752 | Two ways to express that a... |
| ulmcaulem 25753 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 25754 | A sequence of functions co... |
| ulmcau2 25755 | A sequence of functions co... |
| ulmss 25756 | A uniform limit of functio... |
| ulmbdd 25757 | A uniform limit of bounded... |
| ulmcn 25758 | A uniform limit of continu... |
| ulmdvlem1 25759 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 25760 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 25761 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 25762 | If ` F ` is a sequence of ... |
| mtest 25763 | The Weierstrass M-test. I... |
| mtestbdd 25764 | Given the hypotheses of th... |
| mbfulm 25765 | A uniform limit of measura... |
| iblulm 25766 | A uniform limit of integra... |
| itgulm 25767 | A uniform limit of integra... |
| itgulm2 25768 | A uniform limit of integra... |
| pserval 25769 | Value of the function ` G ... |
| pserval2 25770 | Value of the function ` G ... |
| psergf 25771 | The sequence of terms in t... |
| radcnvlem1 25772 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 25773 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 25774 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 25775 | Zero is always a convergen... |
| radcnvcl 25776 | The radius of convergence ... |
| radcnvlt1 25777 | If ` X ` is within the ope... |
| radcnvlt2 25778 | If ` X ` is within the ope... |
| radcnvle 25779 | If ` X ` is a convergent p... |
| dvradcnv 25780 | The radius of convergence ... |
| pserulm 25781 | If ` S ` is a region conta... |
| psercn2 25782 | Since by ~ pserulm the ser... |
| psercnlem2 25783 | Lemma for ~ psercn . (Con... |
| psercnlem1 25784 | Lemma for ~ psercn . (Con... |
| psercn 25785 | An infinite series converg... |
| pserdvlem1 25786 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 25787 | Lemma for ~ pserdv . (Con... |
| pserdv 25788 | The derivative of a power ... |
| pserdv2 25789 | The derivative of a power ... |
| abelthlem1 25790 | Lemma for ~ abelth . (Con... |
| abelthlem2 25791 | Lemma for ~ abelth . The ... |
| abelthlem3 25792 | Lemma for ~ abelth . (Con... |
| abelthlem4 25793 | Lemma for ~ abelth . (Con... |
| abelthlem5 25794 | Lemma for ~ abelth . (Con... |
| abelthlem6 25795 | Lemma for ~ abelth . (Con... |
| abelthlem7a 25796 | Lemma for ~ abelth . (Con... |
| abelthlem7 25797 | Lemma for ~ abelth . (Con... |
| abelthlem8 25798 | Lemma for ~ abelth . (Con... |
| abelthlem9 25799 | Lemma for ~ abelth . By a... |
| abelth 25800 | Abel's theorem. If the po... |
| abelth2 25801 | Abel's theorem, restricted... |
| efcn 25802 | The exponential function i... |
| sincn 25803 | Sine is continuous. (Cont... |
| coscn 25804 | Cosine is continuous. (Co... |
| reeff1olem 25805 | Lemma for ~ reeff1o . (Co... |
| reeff1o 25806 | The real exponential funct... |
| reefiso 25807 | The exponential function o... |
| efcvx 25808 | The exponential function o... |
| reefgim 25809 | The exponential function i... |
| pilem1 25810 | Lemma for ~ pire , ~ pigt2... |
| pilem2 25811 | Lemma for ~ pire , ~ pigt2... |
| pilem3 25812 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 25813 | ` _pi ` is between 2 and 4... |
| sinpi 25814 | The sine of ` _pi ` is 0. ... |
| pire 25815 | ` _pi ` is a real number. ... |
| picn 25816 | ` _pi ` is a complex numbe... |
| pipos 25817 | ` _pi ` is positive. (Con... |
| pirp 25818 | ` _pi ` is a positive real... |
| negpicn 25819 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 25820 | Lemma for ~ sinhalfpi and ... |
| halfpire 25821 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 25822 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 25823 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 25824 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 25825 | The sine of ` _pi / 2 ` is... |
| coshalfpi 25826 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 25827 | The cosine of ` -u _pi / 2... |
| efhalfpi 25828 | The exponential of ` _i _p... |
| cospi 25829 | The cosine of ` _pi ` is `... |
| efipi 25830 | The exponential of ` _i x.... |
| eulerid 25831 | Euler's identity. (Contri... |
| sin2pi 25832 | The sine of ` 2 _pi ` is 0... |
| cos2pi 25833 | The cosine of ` 2 _pi ` is... |
| ef2pi 25834 | The exponential of ` 2 _pi... |
| ef2kpi 25835 | If ` K ` is an integer, th... |
| efper 25836 | The exponential function i... |
| sinperlem 25837 | Lemma for ~ sinper and ~ c... |
| sinper 25838 | The sine function is perio... |
| cosper 25839 | The cosine function is per... |
| sin2kpi 25840 | If ` K ` is an integer, th... |
| cos2kpi 25841 | If ` K ` is an integer, th... |
| sin2pim 25842 | Sine of a number subtracte... |
| cos2pim 25843 | Cosine of a number subtrac... |
| sinmpi 25844 | Sine of a number less ` _p... |
| cosmpi 25845 | Cosine of a number less ` ... |
| sinppi 25846 | Sine of a number plus ` _p... |
| cosppi 25847 | Cosine of a number plus ` ... |
| efimpi 25848 | The exponential function a... |
| sinhalfpip 25849 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 25850 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 25851 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 25852 | The cosine of ` _pi / 2 ` ... |
| ptolemy 25853 | Ptolemy's Theorem. This t... |
| sincosq1lem 25854 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 25855 | The signs of the sine and ... |
| sincosq2sgn 25856 | The signs of the sine and ... |
| sincosq3sgn 25857 | The signs of the sine and ... |
| sincosq4sgn 25858 | The signs of the sine and ... |
| coseq00topi 25859 | Location of the zeroes of ... |
| coseq0negpitopi 25860 | Location of the zeroes of ... |
| tanrpcl 25861 | Positive real closure of t... |
| tangtx 25862 | The tangent function is gr... |
| tanabsge 25863 | The tangent function is gr... |
| sinq12gt0 25864 | The sine of a number stric... |
| sinq12ge0 25865 | The sine of a number betwe... |
| sinq34lt0t 25866 | The sine of a number stric... |
| cosq14gt0 25867 | The cosine of a number str... |
| cosq14ge0 25868 | The cosine of a number bet... |
| sincosq1eq 25869 | Complementarity of the sin... |
| sincos4thpi 25870 | The sine and cosine of ` _... |
| tan4thpi 25871 | The tangent of ` _pi / 4 `... |
| sincos6thpi 25872 | The sine and cosine of ` _... |
| sincos3rdpi 25873 | The sine and cosine of ` _... |
| pigt3 25874 | ` _pi ` is greater than 3.... |
| pige3 25875 | ` _pi ` is greater than or... |
| pige3ALT 25876 | Alternate proof of ~ pige3... |
| abssinper 25877 | The absolute value of sine... |
| sinkpi 25878 | The sine of an integer mul... |
| coskpi 25879 | The absolute value of the ... |
| sineq0 25880 | A complex number whose sin... |
| coseq1 25881 | A complex number whose cos... |
| cos02pilt1 25882 | Cosine is less than one be... |
| cosq34lt1 25883 | Cosine is less than one in... |
| efeq1 25884 | A complex number whose exp... |
| cosne0 25885 | The cosine function has no... |
| cosordlem 25886 | Lemma for ~ cosord . (Con... |
| cosord 25887 | Cosine is decreasing over ... |
| cos0pilt1 25888 | Cosine is between minus on... |
| cos11 25889 | Cosine is one-to-one over ... |
| sinord 25890 | Sine is increasing over th... |
| recosf1o 25891 | The cosine function is a b... |
| resinf1o 25892 | The sine function is a bij... |
| tanord1 25893 | The tangent function is st... |
| tanord 25894 | The tangent function is st... |
| tanregt0 25895 | The real part of the tange... |
| negpitopissre 25896 | The interval ` ( -u _pi (,... |
| efgh 25897 | The exponential function o... |
| efif1olem1 25898 | Lemma for ~ efif1o . (Con... |
| efif1olem2 25899 | Lemma for ~ efif1o . (Con... |
| efif1olem3 25900 | Lemma for ~ efif1o . (Con... |
| efif1olem4 25901 | The exponential function o... |
| efif1o 25902 | The exponential function o... |
| efifo 25903 | The exponential function o... |
| eff1olem 25904 | The exponential function m... |
| eff1o 25905 | The exponential function m... |
| efabl 25906 | The image of a subgroup of... |
| efsubm 25907 | The image of a subgroup of... |
| circgrp 25908 | The circle group ` T ` is ... |
| circsubm 25909 | The circle group ` T ` is ... |
| logrn 25914 | The range of the natural l... |
| ellogrn 25915 | Write out the property ` A... |
| dflog2 25916 | The natural logarithm func... |
| relogrn 25917 | The range of the natural l... |
| logrncn 25918 | The range of the natural l... |
| eff1o2 25919 | The exponential function r... |
| logf1o 25920 | The natural logarithm func... |
| dfrelog 25921 | The natural logarithm func... |
| relogf1o 25922 | The natural logarithm func... |
| logrncl 25923 | Closure of the natural log... |
| logcl 25924 | Closure of the natural log... |
| logimcl 25925 | Closure of the imaginary p... |
| logcld 25926 | The logarithm of a nonzero... |
| logimcld 25927 | The imaginary part of the ... |
| logimclad 25928 | The imaginary part of the ... |
| abslogimle 25929 | The imaginary part of the ... |
| logrnaddcl 25930 | The range of the natural l... |
| relogcl 25931 | Closure of the natural log... |
| eflog 25932 | Relationship between the n... |
| logeq0im1 25933 | If the logarithm of a numb... |
| logccne0 25934 | The logarithm isn't 0 if i... |
| logne0 25935 | Logarithm of a non-1 posit... |
| reeflog 25936 | Relationship between the n... |
| logef 25937 | Relationship between the n... |
| relogef 25938 | Relationship between the n... |
| logeftb 25939 | Relationship between the n... |
| relogeftb 25940 | Relationship between the n... |
| log1 25941 | The natural logarithm of `... |
| loge 25942 | The natural logarithm of `... |
| logneg 25943 | The natural logarithm of a... |
| logm1 25944 | The natural logarithm of n... |
| lognegb 25945 | If a number has imaginary ... |
| relogoprlem 25946 | Lemma for ~ relogmul and ~... |
| relogmul 25947 | The natural logarithm of t... |
| relogdiv 25948 | The natural logarithm of t... |
| explog 25949 | Exponentiation of a nonzer... |
| reexplog 25950 | Exponentiation of a positi... |
| relogexp 25951 | The natural logarithm of p... |
| relog 25952 | Real part of a logarithm. ... |
| relogiso 25953 | The natural logarithm func... |
| reloggim 25954 | The natural logarithm is a... |
| logltb 25955 | The natural logarithm func... |
| logfac 25956 | The logarithm of a factori... |
| eflogeq 25957 | Solve an equation involvin... |
| logleb 25958 | Natural logarithm preserve... |
| rplogcl 25959 | Closure of the logarithm f... |
| logge0 25960 | The logarithm of a number ... |
| logcj 25961 | The natural logarithm dist... |
| efiarg 25962 | The exponential of the "ar... |
| cosargd 25963 | The cosine of the argument... |
| cosarg0d 25964 | The cosine of the argument... |
| argregt0 25965 | Closure of the argument of... |
| argrege0 25966 | Closure of the argument of... |
| argimgt0 25967 | Closure of the argument of... |
| argimlt0 25968 | Closure of the argument of... |
| logimul 25969 | Multiplying a number by ` ... |
| logneg2 25970 | The logarithm of the negat... |
| logmul2 25971 | Generalization of ~ relogm... |
| logdiv2 25972 | Generalization of ~ relogd... |
| abslogle 25973 | Bound on the magnitude of ... |
| tanarg 25974 | The basic relation between... |
| logdivlti 25975 | The ` log x / x ` function... |
| logdivlt 25976 | The ` log x / x ` function... |
| logdivle 25977 | The ` log x / x ` function... |
| relogcld 25978 | Closure of the natural log... |
| reeflogd 25979 | Relationship between the n... |
| relogmuld 25980 | The natural logarithm of t... |
| relogdivd 25981 | The natural logarithm of t... |
| logled 25982 | Natural logarithm preserve... |
| relogefd 25983 | Relationship between the n... |
| rplogcld 25984 | Closure of the logarithm f... |
| logge0d 25985 | The logarithm of a number ... |
| logge0b 25986 | The logarithm of a number ... |
| loggt0b 25987 | The logarithm of a number ... |
| logle1b 25988 | The logarithm of a number ... |
| loglt1b 25989 | The logarithm of a number ... |
| divlogrlim 25990 | The inverse logarithm func... |
| logno1 25991 | The logarithm function is ... |
| dvrelog 25992 | The derivative of the real... |
| relogcn 25993 | The real logarithm functio... |
| ellogdm 25994 | Elementhood in the "contin... |
| logdmn0 25995 | A number in the continuous... |
| logdmnrp 25996 | A number in the continuous... |
| logdmss 25997 | The continuity domain of `... |
| logcnlem2 25998 | Lemma for ~ logcn . (Cont... |
| logcnlem3 25999 | Lemma for ~ logcn . (Cont... |
| logcnlem4 26000 | Lemma for ~ logcn . (Cont... |
| logcnlem5 26001 | Lemma for ~ logcn . (Cont... |
| logcn 26002 | The logarithm function is ... |
| dvloglem 26003 | Lemma for ~ dvlog . (Cont... |
| logdmopn 26004 | The "continuous domain" of... |
| logf1o2 26005 | The logarithm maps its con... |
| dvlog 26006 | The derivative of the comp... |
| dvlog2lem 26007 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 26008 | The derivative of the comp... |
| advlog 26009 | The antiderivative of the ... |
| advlogexp 26010 | The antiderivative of a po... |
| efopnlem1 26011 | Lemma for ~ efopn . (Cont... |
| efopnlem2 26012 | Lemma for ~ efopn . (Cont... |
| efopn 26013 | The exponential map is an ... |
| logtayllem 26014 | Lemma for ~ logtayl . (Co... |
| logtayl 26015 | The Taylor series for ` -u... |
| logtaylsum 26016 | The Taylor series for ` -u... |
| logtayl2 26017 | Power series expression fo... |
| logccv 26018 | The natural logarithm func... |
| cxpval 26019 | Value of the complex power... |
| cxpef 26020 | Value of the complex power... |
| 0cxp 26021 | Value of the complex power... |
| cxpexpz 26022 | Relate the complex power f... |
| cxpexp 26023 | Relate the complex power f... |
| logcxp 26024 | Logarithm of a complex pow... |
| cxp0 26025 | Value of the complex power... |
| cxp1 26026 | Value of the complex power... |
| 1cxp 26027 | Value of the complex power... |
| ecxp 26028 | Write the exponential func... |
| cxpcl 26029 | Closure of the complex pow... |
| recxpcl 26030 | Real closure of the comple... |
| rpcxpcl 26031 | Positive real closure of t... |
| cxpne0 26032 | Complex exponentiation is ... |
| cxpeq0 26033 | Complex exponentiation is ... |
| cxpadd 26034 | Sum of exponents law for c... |
| cxpp1 26035 | Value of a nonzero complex... |
| cxpneg 26036 | Value of a complex number ... |
| cxpsub 26037 | Exponent subtraction law f... |
| cxpge0 26038 | Nonnegative exponentiation... |
| mulcxplem 26039 | Lemma for ~ mulcxp . (Con... |
| mulcxp 26040 | Complex exponentiation of ... |
| cxprec 26041 | Complex exponentiation of ... |
| divcxp 26042 | Complex exponentiation of ... |
| cxpmul 26043 | Product of exponents law f... |
| cxpmul2 26044 | Product of exponents law f... |
| cxproot 26045 | The complex power function... |
| cxpmul2z 26046 | Generalize ~ cxpmul2 to ne... |
| abscxp 26047 | Absolute value of a power,... |
| abscxp2 26048 | Absolute value of a power,... |
| cxplt 26049 | Ordering property for comp... |
| cxple 26050 | Ordering property for comp... |
| cxplea 26051 | Ordering property for comp... |
| cxple2 26052 | Ordering property for comp... |
| cxplt2 26053 | Ordering property for comp... |
| cxple2a 26054 | Ordering property for comp... |
| cxplt3 26055 | Ordering property for comp... |
| cxple3 26056 | Ordering property for comp... |
| cxpsqrtlem 26057 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 26058 | The complex exponential fu... |
| logsqrt 26059 | Logarithm of a square root... |
| cxp0d 26060 | Value of the complex power... |
| cxp1d 26061 | Value of the complex power... |
| 1cxpd 26062 | Value of the complex power... |
| cxpcld 26063 | Closure of the complex pow... |
| cxpmul2d 26064 | Product of exponents law f... |
| 0cxpd 26065 | Value of the complex power... |
| cxpexpzd 26066 | Relate the complex power f... |
| cxpefd 26067 | Value of the complex power... |
| cxpne0d 26068 | Complex exponentiation is ... |
| cxpp1d 26069 | Value of a nonzero complex... |
| cxpnegd 26070 | Value of a complex number ... |
| cxpmul2zd 26071 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 26072 | Sum of exponents law for c... |
| cxpsubd 26073 | Exponent subtraction law f... |
| cxpltd 26074 | Ordering property for comp... |
| cxpled 26075 | Ordering property for comp... |
| cxplead 26076 | Ordering property for comp... |
| divcxpd 26077 | Complex exponentiation of ... |
| recxpcld 26078 | Positive real closure of t... |
| cxpge0d 26079 | Nonnegative exponentiation... |
| cxple2ad 26080 | Ordering property for comp... |
| cxplt2d 26081 | Ordering property for comp... |
| cxple2d 26082 | Ordering property for comp... |
| mulcxpd 26083 | Complex exponentiation of ... |
| cxpsqrtth 26084 | Square root theorem over t... |
| 2irrexpq 26085 | There exist irrational num... |
| cxprecd 26086 | Complex exponentiation of ... |
| rpcxpcld 26087 | Positive real closure of t... |
| logcxpd 26088 | Logarithm of a complex pow... |
| cxplt3d 26089 | Ordering property for comp... |
| cxple3d 26090 | Ordering property for comp... |
| cxpmuld 26091 | Product of exponents law f... |
| cxpcom 26092 | Commutative law for real e... |
| dvcxp1 26093 | The derivative of a comple... |
| dvcxp2 26094 | The derivative of a comple... |
| dvsqrt 26095 | The derivative of the real... |
| dvcncxp1 26096 | Derivative of complex powe... |
| dvcnsqrt 26097 | Derivative of square root ... |
| cxpcn 26098 | Domain of continuity of th... |
| cxpcn2 26099 | Continuity of the complex ... |
| cxpcn3lem 26100 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 26101 | Extend continuity of the c... |
| resqrtcn 26102 | Continuity of the real squ... |
| sqrtcn 26103 | Continuity of the square r... |
| cxpaddlelem 26104 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 26105 | Ordering property for comp... |
| abscxpbnd 26106 | Bound on the absolute valu... |
| root1id 26107 | Property of an ` N ` -th r... |
| root1eq1 26108 | The only powers of an ` N ... |
| root1cj 26109 | Within the ` N ` -th roots... |
| cxpeq 26110 | Solve an equation involvin... |
| loglesqrt 26111 | An upper bound on the loga... |
| logreclem 26112 | Symmetry of the natural lo... |
| logrec 26113 | Logarithm of a reciprocal ... |
| logbval 26116 | Define the value of the ` ... |
| logbcl 26117 | General logarithm closure.... |
| logbid1 26118 | General logarithm is 1 whe... |
| logb1 26119 | The logarithm of ` 1 ` to ... |
| elogb 26120 | The general logarithm of a... |
| logbchbase 26121 | Change of base for logarit... |
| relogbval 26122 | Value of the general logar... |
| relogbcl 26123 | Closure of the general log... |
| relogbzcl 26124 | Closure of the general log... |
| relogbreexp 26125 | Power law for the general ... |
| relogbzexp 26126 | Power law for the general ... |
| relogbmul 26127 | The logarithm of the produ... |
| relogbmulexp 26128 | The logarithm of the produ... |
| relogbdiv 26129 | The logarithm of the quoti... |
| relogbexp 26130 | Identity law for general l... |
| nnlogbexp 26131 | Identity law for general l... |
| logbrec 26132 | Logarithm of a reciprocal ... |
| logbleb 26133 | The general logarithm func... |
| logblt 26134 | The general logarithm func... |
| relogbcxp 26135 | Identity law for the gener... |
| cxplogb 26136 | Identity law for the gener... |
| relogbcxpb 26137 | The logarithm is the inver... |
| logbmpt 26138 | The general logarithm to a... |
| logbf 26139 | The general logarithm to a... |
| logbfval 26140 | The general logarithm of a... |
| relogbf 26141 | The general logarithm to a... |
| logblog 26142 | The general logarithm to t... |
| logbgt0b 26143 | The logarithm of a positiv... |
| logbgcd1irr 26144 | The logarithm of an intege... |
| 2logb9irr 26145 | Example for ~ logbgcd1irr ... |
| logbprmirr 26146 | The logarithm of a prime t... |
| 2logb3irr 26147 | Example for ~ logbprmirr .... |
| 2logb9irrALT 26148 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 26149 | The square root of two to ... |
| 2irrexpqALT 26150 | Alternate proof of ~ 2irre... |
| angval 26151 | Define the angle function,... |
| angcan 26152 | Cancel a constant multipli... |
| angneg 26153 | Cancel a negative sign in ... |
| angvald 26154 | The (signed) angle between... |
| angcld 26155 | The (signed) angle between... |
| angrteqvd 26156 | Two vectors are at a right... |
| cosangneg2d 26157 | The cosine of the angle be... |
| angrtmuld 26158 | Perpendicularity of two ve... |
| ang180lem1 26159 | Lemma for ~ ang180 . Show... |
| ang180lem2 26160 | Lemma for ~ ang180 . Show... |
| ang180lem3 26161 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 26162 | Lemma for ~ ang180 . Redu... |
| ang180lem5 26163 | Lemma for ~ ang180 : Redu... |
| ang180 26164 | The sum of angles ` m A B ... |
| lawcoslem1 26165 | Lemma for ~ lawcos . Here... |
| lawcos 26166 | Law of cosines (also known... |
| pythag 26167 | Pythagorean theorem. Give... |
| isosctrlem1 26168 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 26169 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 26170 | Lemma for ~ isosctr . Cor... |
| isosctr 26171 | Isosceles triangle theorem... |
| ssscongptld 26172 | If two triangles have equa... |
| affineequiv 26173 | Equivalence between two wa... |
| affineequiv2 26174 | Equivalence between two wa... |
| affineequiv3 26175 | Equivalence between two wa... |
| affineequiv4 26176 | Equivalence between two wa... |
| affineequivne 26177 | Equivalence between two wa... |
| angpieqvdlem 26178 | Equivalence used in the pr... |
| angpieqvdlem2 26179 | Equivalence used in ~ angp... |
| angpined 26180 | If the angle at ABC is ` _... |
| angpieqvd 26181 | The angle ABC is ` _pi ` i... |
| chordthmlem 26182 | If ` M ` is the midpoint o... |
| chordthmlem2 26183 | If M is the midpoint of AB... |
| chordthmlem3 26184 | If M is the midpoint of AB... |
| chordthmlem4 26185 | If P is on the segment AB ... |
| chordthmlem5 26186 | If P is on the segment AB ... |
| chordthm 26187 | The intersecting chords th... |
| heron 26188 | Heron's formula gives the ... |
| quad2 26189 | The quadratic equation, wi... |
| quad 26190 | The quadratic equation. (... |
| 1cubrlem 26191 | The cube roots of unity. ... |
| 1cubr 26192 | The cube roots of unity. ... |
| dcubic1lem 26193 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 26194 | Reverse direction of ~ dcu... |
| dcubic1 26195 | Forward direction of ~ dcu... |
| dcubic 26196 | Solutions to the depressed... |
| mcubic 26197 | Solutions to a monic cubic... |
| cubic2 26198 | The solution to the genera... |
| cubic 26199 | The cubic equation, which ... |
| binom4 26200 | Work out a quartic binomia... |
| dquartlem1 26201 | Lemma for ~ dquart . (Con... |
| dquartlem2 26202 | Lemma for ~ dquart . (Con... |
| dquart 26203 | Solve a depressed quartic ... |
| quart1cl 26204 | Closure lemmas for ~ quart... |
| quart1lem 26205 | Lemma for ~ quart1 . (Con... |
| quart1 26206 | Depress a quartic equation... |
| quartlem1 26207 | Lemma for ~ quart . (Cont... |
| quartlem2 26208 | Closure lemmas for ~ quart... |
| quartlem3 26209 | Closure lemmas for ~ quart... |
| quartlem4 26210 | Closure lemmas for ~ quart... |
| quart 26211 | The quartic equation, writ... |
| asinlem 26218 | The argument to the logari... |
| asinlem2 26219 | The argument to the logari... |
| asinlem3a 26220 | Lemma for ~ asinlem3 . (C... |
| asinlem3 26221 | The argument to the logari... |
| asinf 26222 | Domain and codomain of the... |
| asincl 26223 | Closure for the arcsin fun... |
| acosf 26224 | Domain and codoamin of the... |
| acoscl 26225 | Closure for the arccos fun... |
| atandm 26226 | Since the property is a li... |
| atandm2 26227 | This form of ~ atandm is a... |
| atandm3 26228 | A compact form of ~ atandm... |
| atandm4 26229 | A compact form of ~ atandm... |
| atanf 26230 | Domain and codoamin of the... |
| atancl 26231 | Closure for the arctan fun... |
| asinval 26232 | Value of the arcsin functi... |
| acosval 26233 | Value of the arccos functi... |
| atanval 26234 | Value of the arctan functi... |
| atanre 26235 | A real number is in the do... |
| asinneg 26236 | The arcsine function is od... |
| acosneg 26237 | The negative symmetry rela... |
| efiasin 26238 | The exponential of the arc... |
| sinasin 26239 | The arcsine function is an... |
| cosacos 26240 | The arccosine function is ... |
| asinsinlem 26241 | Lemma for ~ asinsin . (Co... |
| asinsin 26242 | The arcsine function compo... |
| acoscos 26243 | The arccosine function is ... |
| asin1 26244 | The arcsine of ` 1 ` is ` ... |
| acos1 26245 | The arccosine of ` 1 ` is ... |
| reasinsin 26246 | The arcsine function compo... |
| asinsinb 26247 | Relationship between sine ... |
| acoscosb 26248 | Relationship between cosin... |
| asinbnd 26249 | The arcsine function has r... |
| acosbnd 26250 | The arccosine function has... |
| asinrebnd 26251 | Bounds on the arcsine func... |
| asinrecl 26252 | The arcsine function is re... |
| acosrecl 26253 | The arccosine function is ... |
| cosasin 26254 | The cosine of the arcsine ... |
| sinacos 26255 | The sine of the arccosine ... |
| atandmneg 26256 | The domain of the arctange... |
| atanneg 26257 | The arctangent function is... |
| atan0 26258 | The arctangent of zero is ... |
| atandmcj 26259 | The arctangent function di... |
| atancj 26260 | The arctangent function di... |
| atanrecl 26261 | The arctangent function is... |
| efiatan 26262 | Value of the exponential o... |
| atanlogaddlem 26263 | Lemma for ~ atanlogadd . ... |
| atanlogadd 26264 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 26265 | Lemma for ~ atanlogsub . ... |
| atanlogsub 26266 | A variation on ~ atanlogad... |
| efiatan2 26267 | Value of the exponential o... |
| 2efiatan 26268 | Value of the exponential o... |
| tanatan 26269 | The arctangent function is... |
| atandmtan 26270 | The tangent function has r... |
| cosatan 26271 | The cosine of an arctangen... |
| cosatanne0 26272 | The arctangent function ha... |
| atantan 26273 | The arctangent function is... |
| atantanb 26274 | Relationship between tange... |
| atanbndlem 26275 | Lemma for ~ atanbnd . (Co... |
| atanbnd 26276 | The arctangent function is... |
| atanord 26277 | The arctangent function is... |
| atan1 26278 | The arctangent of ` 1 ` is... |
| bndatandm 26279 | A point in the open unit d... |
| atans 26280 | The "domain of continuity"... |
| atans2 26281 | It suffices to show that `... |
| atansopn 26282 | The domain of continuity o... |
| atansssdm 26283 | The domain of continuity o... |
| ressatans 26284 | The real number line is a ... |
| dvatan 26285 | The derivative of the arct... |
| atancn 26286 | The arctangent is a contin... |
| atantayl 26287 | The Taylor series for ` ar... |
| atantayl2 26288 | The Taylor series for ` ar... |
| atantayl3 26289 | The Taylor series for ` ar... |
| leibpilem1 26290 | Lemma for ~ leibpi . (Con... |
| leibpilem2 26291 | The Leibniz formula for ` ... |
| leibpi 26292 | The Leibniz formula for ` ... |
| leibpisum 26293 | The Leibniz formula for ` ... |
| log2cnv 26294 | Using the Taylor series fo... |
| log2tlbnd 26295 | Bound the error term in th... |
| log2ublem1 26296 | Lemma for ~ log2ub . The ... |
| log2ublem2 26297 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26298 | Lemma for ~ log2ub . In d... |
| log2ub 26299 | ` log 2 ` is less than ` 2... |
| log2le1 26300 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26301 | Lemma for ~ birthday . (C... |
| birthdaylem2 26302 | For general ` N ` and ` K ... |
| birthdaylem3 26303 | For general ` N ` and ` K ... |
| birthday 26304 | The Birthday Problem. The... |
| dmarea 26307 | The domain of the area fun... |
| areambl 26308 | The fibers of a measurable... |
| areass 26309 | A measurable region is a s... |
| dfarea 26310 | Rewrite ~ df-area self-ref... |
| areaf 26311 | Area measurement is a func... |
| areacl 26312 | The area of a measurable r... |
| areage0 26313 | The area of a measurable r... |
| areaval 26314 | The area of a measurable r... |
| rlimcnp 26315 | Relate a limit of a real-v... |
| rlimcnp2 26316 | Relate a limit of a real-v... |
| rlimcnp3 26317 | Relate a limit of a real-v... |
| xrlimcnp 26318 | Relate a limit of a real-v... |
| efrlim 26319 | The limit of the sequence ... |
| dfef2 26320 | The limit of the sequence ... |
| cxplim 26321 | A power to a negative expo... |
| sqrtlim 26322 | The inverse square root fu... |
| rlimcxp 26323 | Any power to a positive ex... |
| o1cxp 26324 | An eventually bounded func... |
| cxp2limlem 26325 | A linear factor grows slow... |
| cxp2lim 26326 | Any power grows slower tha... |
| cxploglim 26327 | The logarithm grows slower... |
| cxploglim2 26328 | Every power of the logarit... |
| divsqrtsumlem 26329 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26330 | The function ` F ` used in... |
| divsqrsum 26331 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26332 | A bound on the distance of... |
| divsqrtsumo1 26333 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26334 | Closure of a 0-1 linear co... |
| scvxcvx 26335 | A strictly convex function... |
| jensenlem1 26336 | Lemma for ~ jensen . (Con... |
| jensenlem2 26337 | Lemma for ~ jensen . (Con... |
| jensen 26338 | Jensen's inequality, a fin... |
| amgmlem 26339 | Lemma for ~ amgm . (Contr... |
| amgm 26340 | Inequality of arithmetic a... |
| logdifbnd 26343 | Bound on the difference of... |
| logdiflbnd 26344 | Lower bound on the differe... |
| emcllem1 26345 | Lemma for ~ emcl . The se... |
| emcllem2 26346 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26347 | Lemma for ~ emcl . The fu... |
| emcllem4 26348 | Lemma for ~ emcl . The di... |
| emcllem5 26349 | Lemma for ~ emcl . The pa... |
| emcllem6 26350 | Lemma for ~ emcl . By the... |
| emcllem7 26351 | Lemma for ~ emcl and ~ har... |
| emcl 26352 | Closure and bounds for the... |
| harmonicbnd 26353 | A bound on the harmonic se... |
| harmonicbnd2 26354 | A bound on the harmonic se... |
| emre 26355 | The Euler-Mascheroni const... |
| emgt0 26356 | The Euler-Mascheroni const... |
| harmonicbnd3 26357 | A bound on the harmonic se... |
| harmoniclbnd 26358 | A bound on the harmonic se... |
| harmonicubnd 26359 | A bound on the harmonic se... |
| harmonicbnd4 26360 | The asymptotic behavior of... |
| fsumharmonic 26361 | Bound a finite sum based o... |
| zetacvg 26364 | The zeta series is converg... |
| eldmgm 26371 | Elementhood in the set of ... |
| dmgmaddn0 26372 | If ` A ` is not a nonposit... |
| dmlogdmgm 26373 | If ` A ` is in the continu... |
| rpdmgm 26374 | A positive real number is ... |
| dmgmn0 26375 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26376 | If ` A ` is not a nonposit... |
| dmgmdivn0 26377 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26378 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26379 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26380 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26381 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26382 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26383 | The series ` G ` is unifor... |
| lgamgulm 26384 | The series ` G ` is unifor... |
| lgamgulm2 26385 | Rewrite the limit of the s... |
| lgambdd 26386 | The log-Gamma function is ... |
| lgamucov 26387 | The ` U ` regions used in ... |
| lgamucov2 26388 | The ` U ` regions used in ... |
| lgamcvglem 26389 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 26390 | The log-Gamma function is ... |
| lgamf 26391 | The log-Gamma function is ... |
| gamf 26392 | The Gamma function is a co... |
| gamcl 26393 | The exponential of the log... |
| eflgam 26394 | The exponential of the log... |
| gamne0 26395 | The Gamma function is neve... |
| igamval 26396 | Value of the inverse Gamma... |
| igamz 26397 | Value of the inverse Gamma... |
| igamgam 26398 | Value of the inverse Gamma... |
| igamlgam 26399 | Value of the inverse Gamma... |
| igamf 26400 | Closure of the inverse Gam... |
| igamcl 26401 | Closure of the inverse Gam... |
| gamigam 26402 | The Gamma function is the ... |
| lgamcvg 26403 | The series ` G ` converges... |
| lgamcvg2 26404 | The series ` G ` converges... |
| gamcvg 26405 | The pointwise exponential ... |
| lgamp1 26406 | The functional equation of... |
| gamp1 26407 | The functional equation of... |
| gamcvg2lem 26408 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 26409 | An infinite product expres... |
| regamcl 26410 | The Gamma function is real... |
| relgamcl 26411 | The log-Gamma function is ... |
| rpgamcl 26412 | The log-Gamma function is ... |
| lgam1 26413 | The log-Gamma function at ... |
| gam1 26414 | The log-Gamma function at ... |
| facgam 26415 | The Gamma function general... |
| gamfac 26416 | The Gamma function general... |
| wilthlem1 26417 | The only elements that are... |
| wilthlem2 26418 | Lemma for ~ wilth : induct... |
| wilthlem3 26419 | Lemma for ~ wilth . Here ... |
| wilth 26420 | Wilson's theorem. A numbe... |
| wilthimp 26421 | The forward implication of... |
| ftalem1 26422 | Lemma for ~ fta : "growth... |
| ftalem2 26423 | Lemma for ~ fta . There e... |
| ftalem3 26424 | Lemma for ~ fta . There e... |
| ftalem4 26425 | Lemma for ~ fta : Closure... |
| ftalem5 26426 | Lemma for ~ fta : Main pr... |
| ftalem6 26427 | Lemma for ~ fta : Dischar... |
| ftalem7 26428 | Lemma for ~ fta . Shift t... |
| fta 26429 | The Fundamental Theorem of... |
| basellem1 26430 | Lemma for ~ basel . Closu... |
| basellem2 26431 | Lemma for ~ basel . Show ... |
| basellem3 26432 | Lemma for ~ basel . Using... |
| basellem4 26433 | Lemma for ~ basel . By ~ ... |
| basellem5 26434 | Lemma for ~ basel . Using... |
| basellem6 26435 | Lemma for ~ basel . The f... |
| basellem7 26436 | Lemma for ~ basel . The f... |
| basellem8 26437 | Lemma for ~ basel . The f... |
| basellem9 26438 | Lemma for ~ basel . Since... |
| basel 26439 | The sum of the inverse squ... |
| efnnfsumcl 26452 | Finite sum closure in the ... |
| ppisval 26453 | The set of primes less tha... |
| ppisval2 26454 | The set of primes less tha... |
| ppifi 26455 | The set of primes less tha... |
| prmdvdsfi 26456 | The set of prime divisors ... |
| chtf 26457 | Domain and codoamin of the... |
| chtcl 26458 | Real closure of the Chebys... |
| chtval 26459 | Value of the Chebyshev fun... |
| efchtcl 26460 | The Chebyshev function is ... |
| chtge0 26461 | The Chebyshev function is ... |
| vmaval 26462 | Value of the von Mangoldt ... |
| isppw 26463 | Two ways to say that ` A `... |
| isppw2 26464 | Two ways to say that ` A `... |
| vmappw 26465 | Value of the von Mangoldt ... |
| vmaprm 26466 | Value of the von Mangoldt ... |
| vmacl 26467 | Closure for the von Mangol... |
| vmaf 26468 | Functionality of the von M... |
| efvmacl 26469 | The von Mangoldt is closed... |
| vmage0 26470 | The von Mangoldt function ... |
| chpval 26471 | Value of the second Chebys... |
| chpf 26472 | Functionality of the secon... |
| chpcl 26473 | Closure for the second Che... |
| efchpcl 26474 | The second Chebyshev funct... |
| chpge0 26475 | The second Chebyshev funct... |
| ppival 26476 | Value of the prime-countin... |
| ppival2 26477 | Value of the prime-countin... |
| ppival2g 26478 | Value of the prime-countin... |
| ppif 26479 | Domain and codomain of the... |
| ppicl 26480 | Real closure of the prime-... |
| muval 26481 | The value of the Möbi... |
| muval1 26482 | The value of the Möbi... |
| muval2 26483 | The value of the Möbi... |
| isnsqf 26484 | Two ways to say that a num... |
| issqf 26485 | Two ways to say that a num... |
| sqfpc 26486 | The prime count of a squar... |
| dvdssqf 26487 | A divisor of a squarefree ... |
| sqf11 26488 | A squarefree number is com... |
| muf 26489 | The Möbius function i... |
| mucl 26490 | Closure of the Möbius... |
| sgmval 26491 | The value of the divisor f... |
| sgmval2 26492 | The value of the divisor f... |
| 0sgm 26493 | The value of the sum-of-di... |
| sgmf 26494 | The divisor function is a ... |
| sgmcl 26495 | Closure of the divisor fun... |
| sgmnncl 26496 | Closure of the divisor fun... |
| mule1 26497 | The Möbius function t... |
| chtfl 26498 | The Chebyshev function doe... |
| chpfl 26499 | The second Chebyshev funct... |
| ppiprm 26500 | The prime-counting functio... |
| ppinprm 26501 | The prime-counting functio... |
| chtprm 26502 | The Chebyshev function at ... |
| chtnprm 26503 | The Chebyshev function at ... |
| chpp1 26504 | The second Chebyshev funct... |
| chtwordi 26505 | The Chebyshev function is ... |
| chpwordi 26506 | The second Chebyshev funct... |
| chtdif 26507 | The difference of the Cheb... |
| efchtdvds 26508 | The exponentiated Chebyshe... |
| ppifl 26509 | The prime-counting functio... |
| ppip1le 26510 | The prime-counting functio... |
| ppiwordi 26511 | The prime-counting functio... |
| ppidif 26512 | The difference of the prim... |
| ppi1 26513 | The prime-counting functio... |
| cht1 26514 | The Chebyshev function at ... |
| vma1 26515 | The von Mangoldt function ... |
| chp1 26516 | The second Chebyshev funct... |
| ppi1i 26517 | Inference form of ~ ppiprm... |
| ppi2i 26518 | Inference form of ~ ppinpr... |
| ppi2 26519 | The prime-counting functio... |
| ppi3 26520 | The prime-counting functio... |
| cht2 26521 | The Chebyshev function at ... |
| cht3 26522 | The Chebyshev function at ... |
| ppinncl 26523 | Closure of the prime-count... |
| chtrpcl 26524 | Closure of the Chebyshev f... |
| ppieq0 26525 | The prime-counting functio... |
| ppiltx 26526 | The prime-counting functio... |
| prmorcht 26527 | Relate the primorial (prod... |
| mumullem1 26528 | Lemma for ~ mumul . A mul... |
| mumullem2 26529 | Lemma for ~ mumul . The p... |
| mumul 26530 | The Möbius function i... |
| sqff1o 26531 | There is a bijection from ... |
| fsumdvdsdiaglem 26532 | A "diagonal commutation" o... |
| fsumdvdsdiag 26533 | A "diagonal commutation" o... |
| fsumdvdscom 26534 | A double commutation of di... |
| dvdsppwf1o 26535 | A bijection from the divis... |
| dvdsflf1o 26536 | A bijection from the numbe... |
| dvdsflsumcom 26537 | A sum commutation from ` s... |
| fsumfldivdiaglem 26538 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 26539 | The right-hand side of ~ d... |
| musum 26540 | The sum of the Möbius... |
| musumsum 26541 | Evaluate a collapsing sum ... |
| muinv 26542 | The Möbius inversion ... |
| dvdsmulf1o 26543 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 26544 | Product of two divisor sum... |
| sgmppw 26545 | The value of the divisor f... |
| 0sgmppw 26546 | A prime power ` P ^ K ` ha... |
| 1sgmprm 26547 | The sum of divisors for a ... |
| 1sgm2ppw 26548 | The sum of the divisors of... |
| sgmmul 26549 | The divisor function for f... |
| ppiublem1 26550 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 26551 | A prime greater than ` 3 `... |
| ppiub 26552 | An upper bound on the prim... |
| vmalelog 26553 | The von Mangoldt function ... |
| chtlepsi 26554 | The first Chebyshev functi... |
| chprpcl 26555 | Closure of the second Cheb... |
| chpeq0 26556 | The second Chebyshev funct... |
| chteq0 26557 | The first Chebyshev functi... |
| chtleppi 26558 | Upper bound on the ` theta... |
| chtublem 26559 | Lemma for ~ chtub . (Cont... |
| chtub 26560 | An upper bound on the Cheb... |
| fsumvma 26561 | Rewrite a sum over the von... |
| fsumvma2 26562 | Apply ~ fsumvma for the co... |
| pclogsum 26563 | The logarithmic analogue o... |
| vmasum 26564 | The sum of the von Mangold... |
| logfac2 26565 | Another expression for the... |
| chpval2 26566 | Express the second Chebysh... |
| chpchtsum 26567 | The second Chebyshev funct... |
| chpub 26568 | An upper bound on the seco... |
| logfacubnd 26569 | A simple upper bound on th... |
| logfaclbnd 26570 | A lower bound on the logar... |
| logfacbnd3 26571 | Show the stronger statemen... |
| logfacrlim 26572 | Combine the estimates ~ lo... |
| logexprlim 26573 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 26574 | Write out ~ logfacrlim as ... |
| mersenne 26575 | A Mersenne prime is a prim... |
| perfect1 26576 | Euclid's contribution to t... |
| perfectlem1 26577 | Lemma for ~ perfect . (Co... |
| perfectlem2 26578 | Lemma for ~ perfect . (Co... |
| perfect 26579 | The Euclid-Euler theorem, ... |
| dchrval 26582 | Value of the group of Diri... |
| dchrbas 26583 | Base set of the group of D... |
| dchrelbas 26584 | A Dirichlet character is a... |
| dchrelbas2 26585 | A Dirichlet character is a... |
| dchrelbas3 26586 | A Dirichlet character is a... |
| dchrelbasd 26587 | A Dirichlet character is a... |
| dchrrcl 26588 | Reverse closure for a Diri... |
| dchrmhm 26589 | A Dirichlet character is a... |
| dchrf 26590 | A Dirichlet character is a... |
| dchrelbas4 26591 | A Dirichlet character is a... |
| dchrzrh1 26592 | Value of a Dirichlet chara... |
| dchrzrhcl 26593 | A Dirichlet character take... |
| dchrzrhmul 26594 | A Dirichlet character is c... |
| dchrplusg 26595 | Group operation on the gro... |
| dchrmul 26596 | Group operation on the gro... |
| dchrmulcl 26597 | Closure of the group opera... |
| dchrn0 26598 | A Dirichlet character is n... |
| dchr1cl 26599 | Closure of the principal D... |
| dchrmulid2 26600 | Left identity for the prin... |
| dchrinvcl 26601 | Closure of the group inver... |
| dchrabl 26602 | The set of Dirichlet chara... |
| dchrfi 26603 | The group of Dirichlet cha... |
| dchrghm 26604 | A Dirichlet character rest... |
| dchr1 26605 | Value of the principal Dir... |
| dchreq 26606 | A Dirichlet character is d... |
| dchrresb 26607 | A Dirichlet character is d... |
| dchrabs 26608 | A Dirichlet character take... |
| dchrinv 26609 | The inverse of a Dirichlet... |
| dchrabs2 26610 | A Dirichlet character take... |
| dchr1re 26611 | The principal Dirichlet ch... |
| dchrptlem1 26612 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 26613 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 26614 | Lemma for ~ dchrpt . (Con... |
| dchrpt 26615 | For any element other than... |
| dchrsum2 26616 | An orthogonality relation ... |
| dchrsum 26617 | An orthogonality relation ... |
| sumdchr2 26618 | Lemma for ~ sumdchr . (Co... |
| dchrhash 26619 | There are exactly ` phi ( ... |
| sumdchr 26620 | An orthogonality relation ... |
| dchr2sum 26621 | An orthogonality relation ... |
| sum2dchr 26622 | An orthogonality relation ... |
| bcctr 26623 | Value of the central binom... |
| pcbcctr 26624 | Prime count of a central b... |
| bcmono 26625 | The binomial coefficient i... |
| bcmax 26626 | The binomial coefficient t... |
| bcp1ctr 26627 | Ratio of two central binom... |
| bclbnd 26628 | A bound on the binomial co... |
| efexple 26629 | Convert a bound on a power... |
| bpos1lem 26630 | Lemma for ~ bpos1 . (Cont... |
| bpos1 26631 | Bertrand's postulate, chec... |
| bposlem1 26632 | An upper bound on the prim... |
| bposlem2 26633 | There are no odd primes in... |
| bposlem3 26634 | Lemma for ~ bpos . Since ... |
| bposlem4 26635 | Lemma for ~ bpos . (Contr... |
| bposlem5 26636 | Lemma for ~ bpos . Bound ... |
| bposlem6 26637 | Lemma for ~ bpos . By usi... |
| bposlem7 26638 | Lemma for ~ bpos . The fu... |
| bposlem8 26639 | Lemma for ~ bpos . Evalua... |
| bposlem9 26640 | Lemma for ~ bpos . Derive... |
| bpos 26641 | Bertrand's postulate: ther... |
| zabsle1 26644 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 26645 | When ` a ` is coprime to t... |
| lgslem2 26646 | The set ` Z ` of all integ... |
| lgslem3 26647 | The set ` Z ` of all integ... |
| lgslem4 26648 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 26649 | Value of the Legendre symb... |
| lgsfval 26650 | Value of the function ` F ... |
| lgsfcl2 26651 | The function ` F ` is clos... |
| lgscllem 26652 | The Legendre symbol is an ... |
| lgsfcl 26653 | Closure of the function ` ... |
| lgsfle1 26654 | The function ` F ` has mag... |
| lgsval2lem 26655 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 26656 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 26657 | The Legendre symbol is an ... |
| lgs0 26658 | The Legendre symbol when t... |
| lgscl 26659 | The Legendre symbol is an ... |
| lgsle1 26660 | The Legendre symbol has ab... |
| lgsval2 26661 | The Legendre symbol at a p... |
| lgs2 26662 | The Legendre symbol at ` 2... |
| lgsval3 26663 | The Legendre symbol at an ... |
| lgsvalmod 26664 | The Legendre symbol is equ... |
| lgsval4 26665 | Restate ~ lgsval for nonze... |
| lgsfcl3 26666 | Closure of the function ` ... |
| lgsval4a 26667 | Same as ~ lgsval4 for posi... |
| lgscl1 26668 | The value of the Legendre ... |
| lgsneg 26669 | The Legendre symbol is eit... |
| lgsneg1 26670 | The Legendre symbol for no... |
| lgsmod 26671 | The Legendre (Jacobi) symb... |
| lgsdilem 26672 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 26673 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 26674 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 26675 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 26676 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 26677 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 26678 | The Legendre symbol is com... |
| lgsdirprm 26679 | The Legendre symbol is com... |
| lgsdir 26680 | The Legendre symbol is com... |
| lgsdilem2 26681 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 26682 | The Legendre symbol is com... |
| lgsne0 26683 | The Legendre symbol is non... |
| lgsabs1 26684 | The Legendre symbol is non... |
| lgssq 26685 | The Legendre symbol at a s... |
| lgssq2 26686 | The Legendre symbol at a s... |
| lgsprme0 26687 | The Legendre symbol at any... |
| 1lgs 26688 | The Legendre symbol at ` 1... |
| lgs1 26689 | The Legendre symbol at ` 1... |
| lgsmodeq 26690 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 26691 | The Legendre (Jacobi) symb... |
| lgsdirnn0 26692 | Variation on ~ lgsdir vali... |
| lgsdinn0 26693 | Variation on ~ lgsdi valid... |
| lgsqrlem1 26694 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 26695 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 26696 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 26697 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 26698 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 26699 | The Legendre symbol for od... |
| lgsqrmod 26700 | If the Legendre symbol of ... |
| lgsqrmodndvds 26701 | If the Legendre symbol of ... |
| lgsdchrval 26702 | The Legendre symbol functi... |
| lgsdchr 26703 | The Legendre symbol functi... |
| gausslemma2dlem0a 26704 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 26705 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 26706 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 26707 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 26708 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 26709 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 26710 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 26711 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 26712 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 26713 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 26714 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 26715 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 26716 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 26717 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 26718 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 26719 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 26720 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 26721 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 26722 | Gauss' Lemma (see also the... |
| lgseisenlem1 26723 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 26724 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 26725 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 26726 | Lemma for ~ lgseisen . Th... |
| lgseisen 26727 | Eisenstein's lemma, an exp... |
| lgsquadlem1 26728 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 26729 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 26730 | Lemma for ~ lgsquad . (Co... |
| lgsquad 26731 | The Law of Quadratic Recip... |
| lgsquad2lem1 26732 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 26733 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 26734 | Extend ~ lgsquad to coprim... |
| lgsquad3 26735 | Extend ~ lgsquad2 to integ... |
| m1lgs 26736 | The first supplement to th... |
| 2lgslem1a1 26737 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 26738 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 26739 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 26740 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 26741 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 26742 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 26743 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 26744 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 26745 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 26746 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 26747 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 26748 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 26749 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 26750 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 26751 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 26752 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 26753 | The Legendre symbol for ` ... |
| 2lgslem4 26754 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 26755 | The second supplement to t... |
| 2lgsoddprmlem1 26756 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 26757 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 26758 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 26759 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 26760 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 26761 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 26762 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 26763 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 26764 | The second supplement to t... |
| 2sqlem1 26765 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 26766 | Lemma for ~ 2sq . (Contri... |
| mul2sq 26767 | Fibonacci's identity (actu... |
| 2sqlem3 26768 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 26769 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 26770 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 26771 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 26772 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 26773 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 26774 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 26775 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 26776 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 26777 | Lemma for ~ 2sq . (Contri... |
| 2sq 26778 | All primes of the form ` 4... |
| 2sqblem 26779 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 26780 | The converse to ~ 2sq . (... |
| 2sq2 26781 | ` 2 ` is the sum of square... |
| 2sqn0 26782 | If the sum of two squares ... |
| 2sqcoprm 26783 | If the sum of two squares ... |
| 2sqmod 26784 | Given two decompositions o... |
| 2sqmo 26785 | There exists at most one d... |
| 2sqnn0 26786 | All primes of the form ` 4... |
| 2sqnn 26787 | All primes of the form ` 4... |
| addsq2reu 26788 | For each complex number ` ... |
| addsqn2reu 26789 | For each complex number ` ... |
| addsqrexnreu 26790 | For each complex number, t... |
| addsqnreup 26791 | There is no unique decompo... |
| addsq2nreurex 26792 | For each complex number ` ... |
| addsqn2reurex2 26793 | For each complex number ` ... |
| 2sqreulem1 26794 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 26795 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 26796 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 26797 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 26798 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 26799 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 26800 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 26801 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 26802 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 26803 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 26804 | There exists a unique deco... |
| 2sqreunn 26805 | There exists a unique deco... |
| 2sqreult 26806 | There exists a unique deco... |
| 2sqreultb 26807 | There exists a unique deco... |
| 2sqreunnlt 26808 | There exists a unique deco... |
| 2sqreunnltb 26809 | There exists a unique deco... |
| 2sqreuop 26810 | There exists a unique deco... |
| 2sqreuopnn 26811 | There exists a unique deco... |
| 2sqreuoplt 26812 | There exists a unique deco... |
| 2sqreuopltb 26813 | There exists a unique deco... |
| 2sqreuopnnlt 26814 | There exists a unique deco... |
| 2sqreuopnnltb 26815 | There exists a unique deco... |
| 2sqreuopb 26816 | There exists a unique deco... |
| chebbnd1lem1 26817 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 26818 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 26819 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 26820 | The Chebyshev bound: The ... |
| chtppilimlem1 26821 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 26822 | Lemma for ~ chtppilim . (... |
| chtppilim 26823 | The ` theta ` function is ... |
| chto1ub 26824 | The ` theta ` function is ... |
| chebbnd2 26825 | The Chebyshev bound, part ... |
| chto1lb 26826 | The ` theta ` function is ... |
| chpchtlim 26827 | The ` psi ` and ` theta ` ... |
| chpo1ub 26828 | The ` psi ` function is up... |
| chpo1ubb 26829 | The ` psi ` function is up... |
| vmadivsum 26830 | The sum of the von Mangold... |
| vmadivsumb 26831 | Give a total bound on the ... |
| rplogsumlem1 26832 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 26833 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 26834 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 26835 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 26836 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 26837 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 26838 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 26839 | Lemma for ~ dchrisum . Le... |
| dchrisum 26840 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 26841 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 26842 | The sum of the Möbius... |
| dchrvmasumlem1 26843 | An alternative expression ... |
| dchrvmasum2lem 26844 | Give an expression for ` l... |
| dchrvmasum2if 26845 | Combine the results of ~ d... |
| dchrvmasumlem2 26846 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 26847 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 26848 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 26849 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 26850 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 26851 | An asymptotic approximatio... |
| dchrvmaeq0 26852 | The set ` W ` is the colle... |
| dchrisum0fval 26853 | Value of the function ` F ... |
| dchrisum0fmul 26854 | The function ` F ` , the d... |
| dchrisum0ff 26855 | The function ` F ` is a re... |
| dchrisum0flblem1 26856 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 26857 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 26858 | The divisor sum of a real ... |
| dchrisum0fno1 26859 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 26860 | A partial result along the... |
| dchrisum0re 26861 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 26862 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 26863 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 26864 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 26865 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 26866 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 26867 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 26868 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 26869 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 26870 | The sum of the Möbius... |
| dchrvmasumlem 26871 | The sum of the Möbius... |
| dchrmusum 26872 | The sum of the Möbius... |
| dchrvmasum 26873 | The sum of the von Mangold... |
| rpvmasum 26874 | The sum of the von Mangold... |
| rplogsum 26875 | The sum of ` log p / p ` o... |
| dirith2 26876 | Dirichlet's theorem: there... |
| dirith 26877 | Dirichlet's theorem: there... |
| mudivsum 26878 | Asymptotic formula for ` s... |
| mulogsumlem 26879 | Lemma for ~ mulogsum . (C... |
| mulogsum 26880 | Asymptotic formula for ... |
| logdivsum 26881 | Asymptotic analysis of ... |
| mulog2sumlem1 26882 | Asymptotic formula for ... |
| mulog2sumlem2 26883 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 26884 | Lemma for ~ mulog2sum . (... |
| mulog2sum 26885 | Asymptotic formula for ... |
| vmalogdivsum2 26886 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 26887 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 26888 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 26889 | The sum ` sum_ m n <_ x , ... |
| logsqvma 26890 | A formula for ` log ^ 2 ( ... |
| logsqvma2 26891 | The Möbius inverse of... |
| log2sumbnd 26892 | Bound on the difference be... |
| selberglem1 26893 | Lemma for ~ selberg . Est... |
| selberglem2 26894 | Lemma for ~ selberg . (Co... |
| selberglem3 26895 | Lemma for ~ selberg . Est... |
| selberg 26896 | Selberg's symmetry formula... |
| selbergb 26897 | Convert eventual boundedne... |
| selberg2lem 26898 | Lemma for ~ selberg2 . Eq... |
| selberg2 26899 | Selberg's symmetry formula... |
| selberg2b 26900 | Convert eventual boundedne... |
| chpdifbndlem1 26901 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 26902 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 26903 | A bound on the difference ... |
| logdivbnd 26904 | A bound on a sum of logs, ... |
| selberg3lem1 26905 | Introduce a log weighting ... |
| selberg3lem2 26906 | Lemma for ~ selberg3 . Eq... |
| selberg3 26907 | Introduce a log weighting ... |
| selberg4lem1 26908 | Lemma for ~ selberg4 . Eq... |
| selberg4 26909 | The Selberg symmetry formu... |
| pntrval 26910 | Define the residual of the... |
| pntrf 26911 | Functionality of the resid... |
| pntrmax 26912 | There is a bound on the re... |
| pntrsumo1 26913 | A bound on a sum over ` R ... |
| pntrsumbnd 26914 | A bound on a sum over ` R ... |
| pntrsumbnd2 26915 | A bound on a sum over ` R ... |
| selbergr 26916 | Selberg's symmetry formula... |
| selberg3r 26917 | Selberg's symmetry formula... |
| selberg4r 26918 | Selberg's symmetry formula... |
| selberg34r 26919 | The sum of ~ selberg3r and... |
| pntsval 26920 | Define the "Selberg functi... |
| pntsf 26921 | Functionality of the Selbe... |
| selbergs 26922 | Selberg's symmetry formula... |
| selbergsb 26923 | Selberg's symmetry formula... |
| pntsval2 26924 | The Selberg function can b... |
| pntrlog2bndlem1 26925 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 26926 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 26927 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 26928 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 26929 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 26930 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 26931 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 26932 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 26933 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 26934 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 26935 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 26936 | Lemma for ~ pnt . Establi... |
| pntibndlem1 26937 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 26938 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 26939 | Lemma for ~ pntibnd . The... |
| pntibndlem3 26940 | Lemma for ~ pntibnd . Pac... |
| pntibnd 26941 | Lemma for ~ pnt . Establi... |
| pntlemd 26942 | Lemma for ~ pnt . Closure... |
| pntlemc 26943 | Lemma for ~ pnt . Closure... |
| pntlema 26944 | Lemma for ~ pnt . Closure... |
| pntlemb 26945 | Lemma for ~ pnt . Unpack ... |
| pntlemg 26946 | Lemma for ~ pnt . Closure... |
| pntlemh 26947 | Lemma for ~ pnt . Bounds ... |
| pntlemn 26948 | Lemma for ~ pnt . The "na... |
| pntlemq 26949 | Lemma for ~ pntlemj . (Co... |
| pntlemr 26950 | Lemma for ~ pntlemj . (Co... |
| pntlemj 26951 | Lemma for ~ pnt . The ind... |
| pntlemi 26952 | Lemma for ~ pnt . Elimina... |
| pntlemf 26953 | Lemma for ~ pnt . Add up ... |
| pntlemk 26954 | Lemma for ~ pnt . Evaluat... |
| pntlemo 26955 | Lemma for ~ pnt . Combine... |
| pntleme 26956 | Lemma for ~ pnt . Package... |
| pntlem3 26957 | Lemma for ~ pnt . Equatio... |
| pntlemp 26958 | Lemma for ~ pnt . Wrappin... |
| pntleml 26959 | Lemma for ~ pnt . Equatio... |
| pnt3 26960 | The Prime Number Theorem, ... |
| pnt2 26961 | The Prime Number Theorem, ... |
| pnt 26962 | The Prime Number Theorem: ... |
| abvcxp 26963 | Raising an absolute value ... |
| padicfval 26964 | Value of the p-adic absolu... |
| padicval 26965 | Value of the p-adic absolu... |
| ostth2lem1 26966 | Lemma for ~ ostth2 , altho... |
| qrngbas 26967 | The base set of the field ... |
| qdrng 26968 | The rationals form a divis... |
| qrng0 26969 | The zero element of the fi... |
| qrng1 26970 | The unity element of the f... |
| qrngneg 26971 | The additive inverse in th... |
| qrngdiv 26972 | The division operation in ... |
| qabvle 26973 | By using induction on ` N ... |
| qabvexp 26974 | Induct the product rule ~ ... |
| ostthlem1 26975 | Lemma for ~ ostth . If tw... |
| ostthlem2 26976 | Lemma for ~ ostth . Refin... |
| qabsabv 26977 | The regular absolute value... |
| padicabv 26978 | The p-adic absolute value ... |
| padicabvf 26979 | The p-adic absolute value ... |
| padicabvcxp 26980 | All positive powers of the... |
| ostth1 26981 | - Lemma for ~ ostth : triv... |
| ostth2lem2 26982 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 26983 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 26984 | Lemma for ~ ostth2 . (Con... |
| ostth2 26985 | - Lemma for ~ ostth : regu... |
| ostth3 26986 | - Lemma for ~ ostth : p-ad... |
| ostth 26987 | Ostrowski's theorem, which... |
| elno 26994 | Membership in the surreals... |
| sltval 26995 | The value of the surreal l... |
| bdayval 26996 | The value of the birthday ... |
| nofun 26997 | A surreal is a function. ... |
| nodmon 26998 | The domain of a surreal is... |
| norn 26999 | The range of a surreal is ... |
| nofnbday 27000 | A surreal is a function ov... |
| nodmord 27001 | The domain of a surreal ha... |
| elno2 27002 | An alternative condition f... |
| elno3 27003 | Another condition for memb... |
| sltval2 27004 | Alternate expression for s... |
| nofv 27005 | The function value of a su... |
| nosgnn0 27006 | ` (/) ` is not a surreal s... |
| nosgnn0i 27007 | If ` X ` is a surreal sign... |
| noreson 27008 | The restriction of a surre... |
| sltintdifex 27009 |
If ` A |
| sltres 27010 | If the restrictions of two... |
| noxp1o 27011 | The Cartesian product of a... |
| noseponlem 27012 | Lemma for ~ nosepon . Con... |
| nosepon 27013 | Given two unequal surreals... |
| noextend 27014 | Extending a surreal by one... |
| noextendseq 27015 | Extend a surreal by a sequ... |
| noextenddif 27016 | Calculate the place where ... |
| noextendlt 27017 | Extending a surreal with a... |
| noextendgt 27018 | Extending a surreal with a... |
| nolesgn2o 27019 | Given ` A ` less-than or e... |
| nolesgn2ores 27020 | Given ` A ` less-than or e... |
| nogesgn1o 27021 | Given ` A ` greater than o... |
| nogesgn1ores 27022 | Given ` A ` greater than o... |
| sltsolem1 27023 | Lemma for ~ sltso . The "... |
| sltso 27024 | Less-than totally orders t... |
| bdayfo 27025 | The birthday function maps... |
| fvnobday 27026 | The value of a surreal at ... |
| nosepnelem 27027 | Lemma for ~ nosepne . (Co... |
| nosepne 27028 | The value of two non-equal... |
| nosep1o 27029 | If the value of a surreal ... |
| nosep2o 27030 | If the value of a surreal ... |
| nosepdmlem 27031 | Lemma for ~ nosepdm . (Co... |
| nosepdm 27032 | The first place two surrea... |
| nosepeq 27033 | The values of two surreals... |
| nosepssdm 27034 | Given two non-equal surrea... |
| nodenselem4 27035 | Lemma for ~ nodense . Sho... |
| nodenselem5 27036 | Lemma for ~ nodense . If ... |
| nodenselem6 27037 | The restriction of a surre... |
| nodenselem7 27038 | Lemma for ~ nodense . ` A ... |
| nodenselem8 27039 | Lemma for ~ nodense . Giv... |
| nodense 27040 | Given two distinct surreal... |
| bdayimaon 27041 | Lemma for full-eta propert... |
| nolt02olem 27042 | Lemma for ~ nolt02o . If ... |
| nolt02o 27043 | Given ` A ` less-than ` B ... |
| nogt01o 27044 | Given ` A ` greater than `... |
| noresle 27045 | Restriction law for surrea... |
| nomaxmo 27046 | A class of surreals has at... |
| nominmo 27047 | A class of surreals has at... |
| nosupprefixmo 27048 | In any class of surreals, ... |
| noinfprefixmo 27049 | In any class of surreals, ... |
| nosupcbv 27050 | Lemma to change bound vari... |
| nosupno 27051 | The next several theorems ... |
| nosupdm 27052 | The domain of the surreal ... |
| nosupbday 27053 | Birthday bounding law for ... |
| nosupfv 27054 | The value of surreal supre... |
| nosupres 27055 | A restriction law for surr... |
| nosupbnd1lem1 27056 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 27057 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 27058 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 27059 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 27060 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 27061 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 27062 | Bounding law from below fo... |
| nosupbnd2lem1 27063 | Bounding law from above wh... |
| nosupbnd2 27064 | Bounding law from above fo... |
| noinfcbv 27065 | Change bound variables for... |
| noinfno 27066 | The next several theorems ... |
| noinfdm 27067 | Next, we calculate the dom... |
| noinfbday 27068 | Birthday bounding law for ... |
| noinffv 27069 | The value of surreal infim... |
| noinfres 27070 | The restriction of surreal... |
| noinfbnd1lem1 27071 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 27072 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 27073 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 27074 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 27075 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 27076 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 27077 | Bounding law from above fo... |
| noinfbnd2lem1 27078 | Bounding law from below wh... |
| noinfbnd2 27079 | Bounding law from below fo... |
| nosupinfsep 27080 | Given two sets of surreals... |
| noetasuplem1 27081 | Lemma for ~ noeta . Estab... |
| noetasuplem2 27082 | Lemma for ~ noeta . The r... |
| noetasuplem3 27083 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 27084 | Lemma for ~ noeta . When ... |
| noetainflem1 27085 | Lemma for ~ noeta . Estab... |
| noetainflem2 27086 | Lemma for ~ noeta . The r... |
| noetainflem3 27087 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 27088 | Lemma for ~ noeta . If ` ... |
| noetalem1 27089 | Lemma for ~ noeta . Eithe... |
| noetalem2 27090 | Lemma for ~ noeta . The f... |
| noeta 27091 | The full-eta axiom for the... |
| sltirr 27094 | Surreal less-than is irref... |
| slttr 27095 | Surreal less-than is trans... |
| sltasym 27096 | Surreal less-than is asymm... |
| sltlin 27097 | Surreal less-than obeys tr... |
| slttrieq2 27098 | Trichotomy law for surreal... |
| slttrine 27099 | Trichotomy law for surreal... |
| slenlt 27100 | Surreal less-than or equal... |
| sltnle 27101 | Surreal less-than in terms... |
| sleloe 27102 | Surreal less-than or equal... |
| sletri3 27103 | Trichotomy law for surreal... |
| sltletr 27104 | Surreal transitive law. (... |
| slelttr 27105 | Surreal transitive law. (... |
| sletr 27106 | Surreal transitive law. (... |
| slttrd 27107 | Surreal less-than is trans... |
| sltletrd 27108 | Surreal less-than is trans... |
| slelttrd 27109 | Surreal less-than is trans... |
| sletrd 27110 | Surreal less-than or equal... |
| slerflex 27111 | Surreal less-than or equal... |
| bdayfun 27112 | The birthday function is a... |
| bdayfn 27113 | The birthday function is a... |
| bdaydm 27114 | The birthday function's do... |
| bdayrn 27115 | The birthday function's ra... |
| bdayelon 27116 | The value of the birthday ... |
| nocvxminlem 27117 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 27118 | Given a nonempty convex cl... |
| noprc 27119 | The surreal numbers are a ... |
| noeta2 27124 | A version of ~ noeta with ... |
| brsslt 27125 | Binary relation form of th... |
| ssltex1 27126 | The first argument of surr... |
| ssltex2 27127 | The second argument of sur... |
| ssltss1 27128 | The first argument of surr... |
| ssltss2 27129 | The second argument of sur... |
| ssltsep 27130 | The separation property of... |
| ssltd 27131 | Deduce surreal set less-th... |
| ssltsepc 27132 | Two elements of separated ... |
| ssltsepcd 27133 | Two elements of separated ... |
| sssslt1 27134 | Relation between surreal s... |
| sssslt2 27135 | Relation between surreal s... |
| nulsslt 27136 | The empty set is less-than... |
| nulssgt 27137 | The empty set is greater t... |
| conway 27138 | Conway's Simplicity Theore... |
| scutval 27139 | The value of the surreal c... |
| scutcut 27140 | Cut properties of the surr... |
| scutcl 27141 | Closure law for surreal cu... |
| scutcld 27142 | Closure law for surreal cu... |
| scutbday 27143 | The birthday of the surrea... |
| eqscut 27144 | Condition for equality to ... |
| eqscut2 27145 | Condition for equality to ... |
| sslttr 27146 | Transitive law for surreal... |
| ssltun1 27147 | Union law for surreal set ... |
| ssltun2 27148 | Union law for surreal set ... |
| scutun12 27149 | Union law for surreal cuts... |
| dmscut 27150 | The domain of the surreal ... |
| scutf 27151 | Functionality statement fo... |
| etasslt 27152 | A restatement of ~ noeta u... |
| etasslt2 27153 | A version of ~ etasslt wit... |
| scutbdaybnd 27154 | An upper bound on the birt... |
| scutbdaybnd2 27155 | An upper bound on the birt... |
| scutbdaybnd2lim 27156 | An upper bound on the birt... |
| scutbdaylt 27157 | If a surreal lies in a gap... |
| slerec 27158 | A comparison law for surre... |
| sltrec 27159 | A comparison law for surre... |
| ssltdisj 27160 | If ` A ` preceeds ` B ` , ... |
| 0sno 27165 | Surreal zero is a surreal.... |
| 1sno 27166 | Surreal one is a surreal. ... |
| bday0s 27167 | Calculate the birthday of ... |
| 0slt1s 27168 | Surreal zero is less than ... |
| bday0b 27169 | The only surreal with birt... |
| bday1s 27170 | The birthday of surreal on... |
| cuteq0 27171 | Condition for a surreal cu... |
| madeval 27182 | The value of the made by f... |
| madeval2 27183 | Alternative characterizati... |
| oldval 27184 | The value of the old optio... |
| newval 27185 | The value of the new optio... |
| madef 27186 | The made function is a fun... |
| oldf 27187 | The older function is a fu... |
| newf 27188 | The new function is a func... |
| old0 27189 | No surreal is older than `... |
| madessno 27190 | Made sets are surreals. (... |
| oldssno 27191 | Old sets are surreals. (C... |
| newssno 27192 | New sets are surreals. (C... |
| leftval 27193 | The value of the left opti... |
| rightval 27194 | The value of the right opt... |
| leftf 27195 | The functionality of the l... |
| rightf 27196 | The functionality of the r... |
| elmade 27197 | Membership in the made fun... |
| elmade2 27198 | Membership in the made fun... |
| elold 27199 | Membership in an old set. ... |
| ssltleft 27200 | A surreal is greater than ... |
| ssltright 27201 | A surreal is less than its... |
| lltropt 27202 | The left options of a surr... |
| made0 27203 | The only surreal made on d... |
| new0 27204 | The only surreal new on da... |
| old1 27205 | The only surreal older tha... |
| madess 27206 | If ` A ` is less than or e... |
| oldssmade 27207 | The older-than set is a su... |
| leftssold 27208 | The left options are a sub... |
| rightssold 27209 | The right options are a su... |
| leftssno 27210 | The left set of a surreal ... |
| rightssno 27211 | The right set of a surreal... |
| madecut 27212 | Given a section that is a ... |
| madeun 27213 | The made set is the union ... |
| madeoldsuc 27214 | The made set is the old se... |
| oldsuc 27215 | The value of the old set a... |
| oldlim 27216 | The value of the old set a... |
| madebdayim 27217 | If a surreal is a member o... |
| oldbdayim 27218 | If ` X ` is in the old set... |
| oldirr 27219 | No surreal is a member of ... |
| leftirr 27220 | No surreal is a member of ... |
| rightirr 27221 | No surreal is a member of ... |
| left0s 27222 | The left set of ` 0s ` is ... |
| right0s 27223 | The right set of ` 0s ` is... |
| left1s 27224 | The left set of ` 1s ` is ... |
| right1s 27225 | The right set of ` 1s ` is... |
| lrold 27226 | The union of the left and ... |
| madebdaylemold 27227 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 27228 | Lemma for ~ madebday . If... |
| madebday 27229 | A surreal is part of the s... |
| oldbday 27230 | A surreal is part of the s... |
| newbday 27231 | A surreal is an element of... |
| lrcut 27232 | A surreal is equal to the ... |
| scutfo 27233 | The surreal cut function i... |
| sltn0 27234 | If ` X ` is less than ` Y ... |
| lruneq 27235 | If two surreals share a bi... |
| sltlpss 27236 | If two surreals share a bi... |
| cofsslt 27237 | If every element of ` A ` ... |
| coinitsslt 27238 | If ` B ` is coinitial with... |
| cofcut1 27239 | If ` C ` is cofinal with `... |
| cofcut1d 27240 | If ` C ` is cofinal with `... |
| cofcut2 27241 | If ` A ` and ` C ` are mut... |
| cofcut2d 27242 | If ` A ` and ` C ` are mut... |
| cofcutr 27243 | If ` X ` is the cut of ` A... |
| cofcutr1d 27244 | If ` X ` is the cut of ` A... |
| cofcutr2d 27245 | If ` X ` is the cut of ` A... |
| cofcutrtime 27246 | If ` X ` is the cut of ` A... |
| cofcutrtime1d 27247 | If ` X ` is a timely cut o... |
| cofcutrtime2d 27248 | If ` X ` is a timely cut o... |
| lrrecval 27251 | The next step in the devel... |
| lrrecval2 27252 | Next, we establish an alte... |
| lrrecpo 27253 | Now, we establish that ` R... |
| lrrecse 27254 | Next, we show that ` R ` i... |
| lrrecfr 27255 | Now we show that ` R ` is ... |
| lrrecpred 27256 | Finally, we calculate the ... |
| noinds 27257 | Induction principle for a ... |
| norecfn 27258 | Surreal recursion over one... |
| norecov 27259 | Calculate the value of the... |
| noxpordpo 27262 | To get through most of the... |
| noxpordfr 27263 | Next we establish the foun... |
| noxpordse 27264 | Next we establish the set-... |
| noxpordpred 27265 | Next we calculate the pred... |
| no2indslem 27266 | Double induction on surrea... |
| no2inds 27267 | Double induction on surrea... |
| norec2fn 27268 | The double-recursion opera... |
| norec2ov 27269 | The value of the double-re... |
| no3inds 27270 | Triple induction over surr... |
| addsfn 27273 | Surreal addition is a func... |
| addsval 27274 | The value of surreal addit... |
| addsval2 27275 | The value of surreal addit... |
| addsid1 27276 | Surreal addition to zero i... |
| addsid1d 27277 | Surreal addition to zero i... |
| addscom 27278 | Surreal addition commutes.... |
| addscomd 27279 | Surreal addition commutes.... |
| addsid2 27280 | Surreal addition to zero i... |
| addsproplem1 27281 | Lemma for surreal addition... |
| addsproplem2 27282 | Lemma for surreal addition... |
| addsproplem3 27283 | Lemma for surreal addition... |
| addsproplem4 27284 | Lemma for surreal addition... |
| addsproplem5 27285 | Lemma for surreal addition... |
| addsproplem6 27286 | Lemma for surreal addition... |
| addsproplem7 27287 | Lemma for surreal addition... |
| addsprop 27288 | Inductively show that surr... |
| addscut 27289 | Demonstrate the cut proper... |
| addscld 27290 | Surreal numbers are closed... |
| addscl 27291 | Surreal numbers are closed... |
| addsf 27292 | Function statement for sur... |
| addsfo 27293 | Surreal addition is onto. ... |
| sltadd1im 27294 | Surreal less-than is prese... |
| sltadd2im 27295 | Surreal less-than is prese... |
| sleadd1im 27296 | Surreal less-than or equal... |
| sleadd2im 27297 | Surreal less-than or equal... |
| sleadd1 27298 | Addition to both sides of ... |
| sleadd2 27299 | Addition to both sides of ... |
| sltadd2 27300 | Addition to both sides of ... |
| sltadd1 27301 | Addition to both sides of ... |
| addscan2 27302 | Cancellation law for surre... |
| addscan1 27303 | Cancellation law for surre... |
| sleadd1d 27304 | Addition to both sides of ... |
| sleadd2d 27305 | Addition to both sides of ... |
| sltadd2d 27306 | Addition to both sides of ... |
| sltadd1d 27307 | Addition to both sides of ... |
| addscan2d 27308 | Cancellation law for surre... |
| addscan1d 27309 | Cancellation law for surre... |
| addsunif 27310 | Uniformity theorem for sur... |
| addsasslem1 27311 | Lemma for addition associa... |
| addsasslem2 27312 | Lemma for addition associa... |
| addsass 27313 | Surreal addition is associ... |
| addsassd 27314 | Surreal addition is associ... |
| adds32d 27315 | Commutative/associative la... |
| adds4d 27316 | Rearrangement of four term... |
| adds42d 27317 | Rearrangement of four term... |
| negsfn 27322 | Surreal negation is a func... |
| subsfn 27323 | Surreal subtraction is a f... |
| negsval 27324 | The value of the surreal n... |
| negs0s 27325 | Negative surreal zero is s... |
| negsproplem1 27326 | Lemma for surreal negation... |
| negsproplem2 27327 | Lemma for surreal negation... |
| negsproplem3 27328 | Lemma for surreal negation... |
| negsproplem4 27329 | Lemma for surreal negation... |
| negsproplem5 27330 | Lemma for surreal negation... |
| negsproplem6 27331 | Lemma for surreal negation... |
| negsproplem7 27332 | Lemma for surreal negation... |
| negsprop 27333 | Show closure and ordering ... |
| negscl 27334 | The surreals are closed un... |
| negscld 27335 | The surreals are closed un... |
| sltnegim 27336 | The forward direction of t... |
| negscut 27337 | The cut properties of surr... |
| negscut2 27338 | The cut that defines surre... |
| negsid 27339 | Surreal addition of a numb... |
| negsidd 27340 | Surreal addition of a numb... |
| negsex 27341 | Every surreal has a negati... |
| negnegs 27342 | A surreal is equal to the ... |
| sltneg 27343 | Negative of both sides of ... |
| sleneg 27344 | Negative of both sides of ... |
| negs11 27345 | Surreal negation is one-to... |
| negsdi 27346 | Distribution of surreal ne... |
| negsf 27347 | Function statement for sur... |
| negsfo 27348 | Function statement for sur... |
| negsf1o 27349 | Surreal negation is a bije... |
| negsunif 27350 | Uniformity property for su... |
| subsval 27351 | The value of surreal subtr... |
| subsvald 27352 | The value of surreal subtr... |
| subscl 27353 | Closure law for surreal su... |
| subscld 27354 | Closure law for surreal su... |
| subsid1 27355 | Identity law for subtracti... |
| subsid 27356 | Subtraction of a surreal f... |
| subadds 27357 | Relationship between addit... |
| subaddsd 27358 | Relationship between addit... |
| pncans 27359 | Cancellation law for surre... |
| pncan3s 27360 | Subtraction and addition o... |
| npcans 27361 | Cancellation law for surre... |
| sltsub1 27362 | Subtraction from both side... |
| sltsub2 27363 | Subtraction from both side... |
| sltsub1d 27364 | Subtraction from both side... |
| sltsub2d 27365 | Subtraction from both side... |
| negsubsdi2d 27366 | Distribution of negative o... |
| addsubsassd 27367 | Associative-type law for s... |
| sltsubsubbd 27368 | Equivalence for the surrea... |
| itvndx 27379 | Index value of the Interva... |
| lngndx 27380 | Index value of the "line" ... |
| itvid 27381 | Utility theorem: index-ind... |
| lngid 27382 | Utility theorem: index-ind... |
| slotsinbpsd 27383 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 27384 | The slots ` Base ` , ` +g ... |
| lngndxnitvndx 27385 | The slot for the line is n... |
| trkgstr 27386 | Functionality of a Tarski ... |
| trkgbas 27387 | The base set of a Tarski g... |
| trkgdist 27388 | The measure of a distance ... |
| trkgitv 27389 | The congruence relation in... |
| istrkgc 27396 | Property of being a Tarski... |
| istrkgb 27397 | Property of being a Tarski... |
| istrkgcb 27398 | Property of being a Tarski... |
| istrkge 27399 | Property of fulfilling Euc... |
| istrkgl 27400 | Building lines from the se... |
| istrkgld 27401 | Property of fulfilling the... |
| istrkg2ld 27402 | Property of fulfilling the... |
| istrkg3ld 27403 | Property of fulfilling the... |
| axtgcgrrflx 27404 | Axiom of reflexivity of co... |
| axtgcgrid 27405 | Axiom of identity of congr... |
| axtgsegcon 27406 | Axiom of segment construct... |
| axtg5seg 27407 | Five segments axiom, Axiom... |
| axtgbtwnid 27408 | Identity of Betweenness. ... |
| axtgpasch 27409 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 27410 | Axiom of Continuity. Axio... |
| axtgcont 27411 | Axiom of Continuity. Axio... |
| axtglowdim2 27412 | Lower dimension axiom for ... |
| axtgupdim2 27413 | Upper dimension axiom for ... |
| axtgeucl 27414 | Euclid's Axiom. Axiom A10... |
| tgjustf 27415 | Given any function ` F ` ,... |
| tgjustr 27416 | Given any equivalence rela... |
| tgjustc1 27417 | A justification for using ... |
| tgjustc2 27418 | A justification for using ... |
| tgcgrcomimp 27419 | Congruence commutes on the... |
| tgcgrcomr 27420 | Congruence commutes on the... |
| tgcgrcoml 27421 | Congruence commutes on the... |
| tgcgrcomlr 27422 | Congruence commutes on bot... |
| tgcgreqb 27423 | Congruence and equality. ... |
| tgcgreq 27424 | Congruence and equality. ... |
| tgcgrneq 27425 | Congruence and equality. ... |
| tgcgrtriv 27426 | Degenerate segments are co... |
| tgcgrextend 27427 | Link congruence over a pai... |
| tgsegconeq 27428 | Two points that satisfy th... |
| tgbtwntriv2 27429 | Betweenness always holds f... |
| tgbtwncom 27430 | Betweenness commutes. The... |
| tgbtwncomb 27431 | Betweenness commutes, bico... |
| tgbtwnne 27432 | Betweenness and inequality... |
| tgbtwntriv1 27433 | Betweenness always holds f... |
| tgbtwnswapid 27434 | If you can swap the first ... |
| tgbtwnintr 27435 | Inner transitivity law for... |
| tgbtwnexch3 27436 | Exchange the first endpoin... |
| tgbtwnouttr2 27437 | Outer transitivity law for... |
| tgbtwnexch2 27438 | Exchange the outer point o... |
| tgbtwnouttr 27439 | Outer transitivity law for... |
| tgbtwnexch 27440 | Outer transitivity law for... |
| tgtrisegint 27441 | A line segment between two... |
| tglowdim1 27442 | Lower dimension axiom for ... |
| tglowdim1i 27443 | Lower dimension axiom for ... |
| tgldimor 27444 | Excluded-middle like state... |
| tgldim0eq 27445 | In dimension zero, any two... |
| tgldim0itv 27446 | In dimension zero, any two... |
| tgldim0cgr 27447 | In dimension zero, any two... |
| tgbtwndiff 27448 | There is always a ` c ` di... |
| tgdim01 27449 | In geometries of dimension... |
| tgifscgr 27450 | Inner five segment congrue... |
| tgcgrsub 27451 | Removing identical parts f... |
| iscgrg 27454 | The congruence property fo... |
| iscgrgd 27455 | The property for two seque... |
| iscgrglt 27456 | The property for two seque... |
| trgcgrg 27457 | The property for two trian... |
| trgcgr 27458 | Triangle congruence. (Con... |
| ercgrg 27459 | The shape congruence relat... |
| tgcgrxfr 27460 | A line segment can be divi... |
| cgr3id 27461 | Reflexivity law for three-... |
| cgr3simp1 27462 | Deduce segment congruence ... |
| cgr3simp2 27463 | Deduce segment congruence ... |
| cgr3simp3 27464 | Deduce segment congruence ... |
| cgr3swap12 27465 | Permutation law for three-... |
| cgr3swap23 27466 | Permutation law for three-... |
| cgr3swap13 27467 | Permutation law for three-... |
| cgr3rotr 27468 | Permutation law for three-... |
| cgr3rotl 27469 | Permutation law for three-... |
| trgcgrcom 27470 | Commutative law for three-... |
| cgr3tr 27471 | Transitivity law for three... |
| tgbtwnxfr 27472 | A condition for extending ... |
| tgcgr4 27473 | Two quadrilaterals to be c... |
| isismt 27476 | Property of being an isome... |
| ismot 27477 | Property of being an isome... |
| motcgr 27478 | Property of a motion: dist... |
| idmot 27479 | The identity is a motion. ... |
| motf1o 27480 | Motions are bijections. (... |
| motcl 27481 | Closure of motions. (Cont... |
| motco 27482 | The composition of two mot... |
| cnvmot 27483 | The converse of a motion i... |
| motplusg 27484 | The operation for motions ... |
| motgrp 27485 | The motions of a geometry ... |
| motcgrg 27486 | Property of a motion: dist... |
| motcgr3 27487 | Property of a motion: dist... |
| tglng 27488 | Lines of a Tarski Geometry... |
| tglnfn 27489 | Lines as functions. (Cont... |
| tglnunirn 27490 | Lines are sets of points. ... |
| tglnpt 27491 | Lines are sets of points. ... |
| tglngne 27492 | It takes two different poi... |
| tglngval 27493 | The line going through poi... |
| tglnssp 27494 | Lines are subset of the ge... |
| tgellng 27495 | Property of lying on the l... |
| tgcolg 27496 | We choose the notation ` (... |
| btwncolg1 27497 | Betweenness implies coline... |
| btwncolg2 27498 | Betweenness implies coline... |
| btwncolg3 27499 | Betweenness implies coline... |
| colcom 27500 | Swapping the points defini... |
| colrot1 27501 | Rotating the points defini... |
| colrot2 27502 | Rotating the points defini... |
| ncolcom 27503 | Swapping non-colinear poin... |
| ncolrot1 27504 | Rotating non-colinear poin... |
| ncolrot2 27505 | Rotating non-colinear poin... |
| tgdim01ln 27506 | In geometries of dimension... |
| ncoltgdim2 27507 | If there are three non-col... |
| lnxfr 27508 | Transfer law for colineari... |
| lnext 27509 | Extend a line with a missi... |
| tgfscgr 27510 | Congruence law for the gen... |
| lncgr 27511 | Congruence rule for lines.... |
| lnid 27512 | Identity law for points on... |
| tgidinside 27513 | Law for finding a point in... |
| tgbtwnconn1lem1 27514 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 27515 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 27516 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 27517 | Connectivity law for betwe... |
| tgbtwnconn2 27518 | Another connectivity law f... |
| tgbtwnconn3 27519 | Inner connectivity law for... |
| tgbtwnconnln3 27520 | Derive colinearity from be... |
| tgbtwnconn22 27521 | Double connectivity law fo... |
| tgbtwnconnln1 27522 | Derive colinearity from be... |
| tgbtwnconnln2 27523 | Derive colinearity from be... |
| legval 27526 | Value of the less-than rel... |
| legov 27527 | Value of the less-than rel... |
| legov2 27528 | An equivalent definition o... |
| legid 27529 | Reflexivity of the less-th... |
| btwnleg 27530 | Betweenness implies less-t... |
| legtrd 27531 | Transitivity of the less-t... |
| legtri3 27532 | Equality from the less-tha... |
| legtrid 27533 | Trichotomy law for the les... |
| leg0 27534 | Degenerated (zero-length) ... |
| legeq 27535 | Deduce equality from "less... |
| legbtwn 27536 | Deduce betweenness from "l... |
| tgcgrsub2 27537 | Removing identical parts f... |
| ltgseg 27538 | The set ` E ` denotes the ... |
| ltgov 27539 | Strict "shorter than" geom... |
| legov3 27540 | An equivalent definition o... |
| legso 27541 | The "shorter than" relatio... |
| ishlg 27544 | Rays : Definition 6.1 of ... |
| hlcomb 27545 | The half-line relation com... |
| hlcomd 27546 | The half-line relation com... |
| hlne1 27547 | The half-line relation imp... |
| hlne2 27548 | The half-line relation imp... |
| hlln 27549 | The half-line relation imp... |
| hleqnid 27550 | The endpoint does not belo... |
| hlid 27551 | The half-line relation is ... |
| hltr 27552 | The half-line relation is ... |
| hlbtwn 27553 | Betweenness is a sufficien... |
| btwnhl1 27554 | Deduce half-line from betw... |
| btwnhl2 27555 | Deduce half-line from betw... |
| btwnhl 27556 | Swap betweenness for a hal... |
| lnhl 27557 | Either a point ` C ` on th... |
| hlcgrex 27558 | Construct a point on a hal... |
| hlcgreulem 27559 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 27560 | The point constructed in ~... |
| btwnlng1 27561 | Betweenness implies coline... |
| btwnlng2 27562 | Betweenness implies coline... |
| btwnlng3 27563 | Betweenness implies coline... |
| lncom 27564 | Swapping the points defini... |
| lnrot1 27565 | Rotating the points defini... |
| lnrot2 27566 | Rotating the points defini... |
| ncolne1 27567 | Non-colinear points are di... |
| ncolne2 27568 | Non-colinear points are di... |
| tgisline 27569 | The property of being a pr... |
| tglnne 27570 | It takes two different poi... |
| tglndim0 27571 | There are no lines in dime... |
| tgelrnln 27572 | The property of being a pr... |
| tglineeltr 27573 | Transitivity law for lines... |
| tglineelsb2 27574 | If ` S ` lies on PQ , then... |
| tglinerflx1 27575 | Reflexivity law for line m... |
| tglinerflx2 27576 | Reflexivity law for line m... |
| tglinecom 27577 | Commutativity law for line... |
| tglinethru 27578 | If ` A ` is a line contain... |
| tghilberti1 27579 | There is a line through an... |
| tghilberti2 27580 | There is at most one line ... |
| tglinethrueu 27581 | There is a unique line goi... |
| tglnne0 27582 | A line ` A ` has at least ... |
| tglnpt2 27583 | Find a second point on a l... |
| tglineintmo 27584 | Two distinct lines interse... |
| tglineineq 27585 | Two distinct lines interse... |
| tglineneq 27586 | Given three non-colinear p... |
| tglineinteq 27587 | Two distinct lines interse... |
| ncolncol 27588 | Deduce non-colinearity fro... |
| coltr 27589 | A transitivity law for col... |
| coltr3 27590 | A transitivity law for col... |
| colline 27591 | Three points are colinear ... |
| tglowdim2l 27592 | Reformulation of the lower... |
| tglowdim2ln 27593 | There is always one point ... |
| mirreu3 27596 | Existential uniqueness of ... |
| mirval 27597 | Value of the point inversi... |
| mirfv 27598 | Value of the point inversi... |
| mircgr 27599 | Property of the image by t... |
| mirbtwn 27600 | Property of the image by t... |
| ismir 27601 | Property of the image by t... |
| mirf 27602 | Point inversion as functio... |
| mircl 27603 | Closure of the point inver... |
| mirmir 27604 | The point inversion functi... |
| mircom 27605 | Variation on ~ mirmir . (... |
| mirreu 27606 | Any point has a unique ant... |
| mireq 27607 | Equality deduction for poi... |
| mirinv 27608 | The only invariant point o... |
| mirne 27609 | Mirror of non-center point... |
| mircinv 27610 | The center point is invari... |
| mirf1o 27611 | The point inversion functi... |
| miriso 27612 | The point inversion functi... |
| mirbtwni 27613 | Point inversion preserves ... |
| mirbtwnb 27614 | Point inversion preserves ... |
| mircgrs 27615 | Point inversion preserves ... |
| mirmir2 27616 | Point inversion of a point... |
| mirmot 27617 | Point investion is a motio... |
| mirln 27618 | If two points are on the s... |
| mirln2 27619 | If a point and its mirror ... |
| mirconn 27620 | Point inversion of connect... |
| mirhl 27621 | If two points ` X ` and ` ... |
| mirbtwnhl 27622 | If the center of the point... |
| mirhl2 27623 | Deduce half-line relation ... |
| mircgrextend 27624 | Link congruence over a pai... |
| mirtrcgr 27625 | Point inversion of one poi... |
| mirauto 27626 | Point inversion preserves ... |
| miduniq 27627 | Uniqueness of the middle p... |
| miduniq1 27628 | Uniqueness of the middle p... |
| miduniq2 27629 | If two point inversions co... |
| colmid 27630 | Colinearity and equidistan... |
| symquadlem 27631 | Lemma of the symetrial qua... |
| krippenlem 27632 | Lemma for ~ krippen . We ... |
| krippen 27633 | Krippenlemma (German for c... |
| midexlem 27634 | Lemma for the existence of... |
| israg 27639 | Property for 3 points A, B... |
| ragcom 27640 | Commutative rule for right... |
| ragcol 27641 | The right angle property i... |
| ragmir 27642 | Right angle property is pr... |
| mirrag 27643 | Right angle is conserved b... |
| ragtrivb 27644 | Trivial right angle. Theo... |
| ragflat2 27645 | Deduce equality from two r... |
| ragflat 27646 | Deduce equality from two r... |
| ragtriva 27647 | Trivial right angle. Theo... |
| ragflat3 27648 | Right angle and colinearit... |
| ragcgr 27649 | Right angle and colinearit... |
| motrag 27650 | Right angles are preserved... |
| ragncol 27651 | Right angle implies non-co... |
| perpln1 27652 | Derive a line from perpend... |
| perpln2 27653 | Derive a line from perpend... |
| isperp 27654 | Property for 2 lines A, B ... |
| perpcom 27655 | The "perpendicular" relati... |
| perpneq 27656 | Two perpendicular lines ar... |
| isperp2 27657 | Property for 2 lines A, B,... |
| isperp2d 27658 | One direction of ~ isperp2... |
| ragperp 27659 | Deduce that two lines are ... |
| footexALT 27660 | Alternative version of ~ f... |
| footexlem1 27661 | Lemma for ~ footex . (Con... |
| footexlem2 27662 | Lemma for ~ footex . (Con... |
| footex 27663 | From a point ` C ` outside... |
| foot 27664 | From a point ` C ` outside... |
| footne 27665 | Uniqueness of the foot poi... |
| footeq 27666 | Uniqueness of the foot poi... |
| hlperpnel 27667 | A point on a half-line whi... |
| perprag 27668 | Deduce a right angle from ... |
| perpdragALT 27669 | Deduce a right angle from ... |
| perpdrag 27670 | Deduce a right angle from ... |
| colperp 27671 | Deduce a perpendicularity ... |
| colperpexlem1 27672 | Lemma for ~ colperp . Fir... |
| colperpexlem2 27673 | Lemma for ~ colperpex . S... |
| colperpexlem3 27674 | Lemma for ~ colperpex . C... |
| colperpex 27675 | In dimension 2 and above, ... |
| mideulem2 27676 | Lemma for ~ opphllem , whi... |
| opphllem 27677 | Lemma 8.24 of [Schwabhause... |
| mideulem 27678 | Lemma for ~ mideu . We ca... |
| midex 27679 | Existence of the midpoint,... |
| mideu 27680 | Existence and uniqueness o... |
| islnopp 27681 | The property for two point... |
| islnoppd 27682 | Deduce that ` A ` and ` B ... |
| oppne1 27683 | Points lying on opposite s... |
| oppne2 27684 | Points lying on opposite s... |
| oppne3 27685 | Points lying on opposite s... |
| oppcom 27686 | Commutativity rule for "op... |
| opptgdim2 27687 | If two points opposite to ... |
| oppnid 27688 | The "opposite to a line" r... |
| opphllem1 27689 | Lemma for ~ opphl . (Cont... |
| opphllem2 27690 | Lemma for ~ opphl . Lemma... |
| opphllem3 27691 | Lemma for ~ opphl : We as... |
| opphllem4 27692 | Lemma for ~ opphl . (Cont... |
| opphllem5 27693 | Second part of Lemma 9.4 o... |
| opphllem6 27694 | First part of Lemma 9.4 of... |
| oppperpex 27695 | Restating ~ colperpex usin... |
| opphl 27696 | If two points ` A ` and ` ... |
| outpasch 27697 | Axiom of Pasch, outer form... |
| hlpasch 27698 | An application of the axio... |
| ishpg 27701 | Value of the half-plane re... |
| hpgbr 27702 | Half-planes : property for... |
| hpgne1 27703 | Points on the open half pl... |
| hpgne2 27704 | Points on the open half pl... |
| lnopp2hpgb 27705 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 27706 | If two points lie on the o... |
| hpgerlem 27707 | Lemma for the proof that t... |
| hpgid 27708 | The half-plane relation is... |
| hpgcom 27709 | The half-plane relation co... |
| hpgtr 27710 | The half-plane relation is... |
| colopp 27711 | Opposite sides of a line f... |
| colhp 27712 | Half-plane relation for co... |
| hphl 27713 | If two points are on the s... |
| midf 27718 | Midpoint as a function. (... |
| midcl 27719 | Closure of the midpoint. ... |
| ismidb 27720 | Property of the midpoint. ... |
| midbtwn 27721 | Betweenness of midpoint. ... |
| midcgr 27722 | Congruence of midpoint. (... |
| midid 27723 | Midpoint of a null segment... |
| midcom 27724 | Commutativity rule for the... |
| mirmid 27725 | Point inversion preserves ... |
| lmieu 27726 | Uniqueness of the line mir... |
| lmif 27727 | Line mirror as a function.... |
| lmicl 27728 | Closure of the line mirror... |
| islmib 27729 | Property of the line mirro... |
| lmicom 27730 | The line mirroring functio... |
| lmilmi 27731 | Line mirroring is an invol... |
| lmireu 27732 | Any point has a unique ant... |
| lmieq 27733 | Equality deduction for lin... |
| lmiinv 27734 | The invariants of the line... |
| lmicinv 27735 | The mirroring line is an i... |
| lmimid 27736 | If we have a right angle, ... |
| lmif1o 27737 | The line mirroring functio... |
| lmiisolem 27738 | Lemma for ~ lmiiso . (Con... |
| lmiiso 27739 | The line mirroring functio... |
| lmimot 27740 | Line mirroring is a motion... |
| hypcgrlem1 27741 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 27742 | Lemma for ~ hypcgr , case ... |
| hypcgr 27743 | If the catheti of two righ... |
| lmiopp 27744 | Line mirroring produces po... |
| lnperpex 27745 | Existence of a perpendicul... |
| trgcopy 27746 | Triangle construction: a c... |
| trgcopyeulem 27747 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 27748 | Triangle construction: a c... |
| iscgra 27751 | Property for two angles AB... |
| iscgra1 27752 | A special version of ~ isc... |
| iscgrad 27753 | Sufficient conditions for ... |
| cgrane1 27754 | Angles imply inequality. ... |
| cgrane2 27755 | Angles imply inequality. ... |
| cgrane3 27756 | Angles imply inequality. ... |
| cgrane4 27757 | Angles imply inequality. ... |
| cgrahl1 27758 | Angle congruence is indepe... |
| cgrahl2 27759 | Angle congruence is indepe... |
| cgracgr 27760 | First direction of proposi... |
| cgraid 27761 | Angle congruence is reflex... |
| cgraswap 27762 | Swap rays in a congruence ... |
| cgrcgra 27763 | Triangle congruence implie... |
| cgracom 27764 | Angle congruence commutes.... |
| cgratr 27765 | Angle congruence is transi... |
| flatcgra 27766 | Flat angles are congruent.... |
| cgraswaplr 27767 | Swap both side of angle co... |
| cgrabtwn 27768 | Angle congruence preserves... |
| cgrahl 27769 | Angle congruence preserves... |
| cgracol 27770 | Angle congruence preserves... |
| cgrancol 27771 | Angle congruence preserves... |
| dfcgra2 27772 | This is the full statement... |
| sacgr 27773 | Supplementary angles of co... |
| oacgr 27774 | Vertical angle theorem. V... |
| acopy 27775 | Angle construction. Theor... |
| acopyeu 27776 | Angle construction. Theor... |
| isinag 27780 | Property for point ` X ` t... |
| isinagd 27781 | Sufficient conditions for ... |
| inagflat 27782 | Any point lies in a flat a... |
| inagswap 27783 | Swap the order of the half... |
| inagne1 27784 | Deduce inequality from the... |
| inagne2 27785 | Deduce inequality from the... |
| inagne3 27786 | Deduce inequality from the... |
| inaghl 27787 | The "point lie in angle" r... |
| isleag 27789 | Geometrical "less than" pr... |
| isleagd 27790 | Sufficient condition for "... |
| leagne1 27791 | Deduce inequality from the... |
| leagne2 27792 | Deduce inequality from the... |
| leagne3 27793 | Deduce inequality from the... |
| leagne4 27794 | Deduce inequality from the... |
| cgrg3col4 27795 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 27796 | First congruence theorem: ... |
| tgsas 27797 | First congruence theorem: ... |
| tgsas2 27798 | First congruence theorem: ... |
| tgsas3 27799 | First congruence theorem: ... |
| tgasa1 27800 | Second congruence theorem:... |
| tgasa 27801 | Second congruence theorem:... |
| tgsss1 27802 | Third congruence theorem: ... |
| tgsss2 27803 | Third congruence theorem: ... |
| tgsss3 27804 | Third congruence theorem: ... |
| dfcgrg2 27805 | Congruence for two triangl... |
| isoas 27806 | Congruence theorem for iso... |
| iseqlg 27809 | Property of a triangle bei... |
| iseqlgd 27810 | Condition for a triangle t... |
| f1otrgds 27811 | Convenient lemma for ~ f1o... |
| f1otrgitv 27812 | Convenient lemma for ~ f1o... |
| f1otrg 27813 | A bijection between bases ... |
| f1otrge 27814 | A bijection between bases ... |
| ttgval 27817 | Define a function to augme... |
| ttgvalOLD 27818 | Obsolete proof of ~ ttgval... |
| ttglem 27819 | Lemma for ~ ttgbas , ~ ttg... |
| ttglemOLD 27820 | Obsolete version of ~ ttgl... |
| ttgbas 27821 | The base set of a subcompl... |
| ttgbasOLD 27822 | Obsolete proof of ~ ttgbas... |
| ttgplusg 27823 | The addition operation of ... |
| ttgplusgOLD 27824 | Obsolete proof of ~ ttgplu... |
| ttgsub 27825 | The subtraction operation ... |
| ttgvsca 27826 | The scalar product of a su... |
| ttgvscaOLD 27827 | Obsolete proof of ~ ttgvsc... |
| ttgds 27828 | The metric of a subcomplex... |
| ttgdsOLD 27829 | Obsolete proof of ~ ttgds ... |
| ttgitvval 27830 | Betweenness for a subcompl... |
| ttgelitv 27831 | Betweenness for a subcompl... |
| ttgbtwnid 27832 | Any subcomplex module equi... |
| ttgcontlem1 27833 | Lemma for % ttgcont . (Co... |
| xmstrkgc 27834 | Any metric space fulfills ... |
| cchhllem 27835 | Lemma for chlbas and chlvs... |
| cchhllemOLD 27836 | Obsolete version of ~ cchh... |
| elee 27843 | Membership in a Euclidean ... |
| mptelee 27844 | A condition for a mapping ... |
| eleenn 27845 | If ` A ` is in ` ( EE `` N... |
| eleei 27846 | The forward direction of ~... |
| eedimeq 27847 | A point belongs to at most... |
| brbtwn 27848 | The binary relation form o... |
| brcgr 27849 | The binary relation form o... |
| fveere 27850 | The function value of a po... |
| fveecn 27851 | The function value of a po... |
| eqeefv 27852 | Two points are equal iff t... |
| eqeelen 27853 | Two points are equal iff t... |
| brbtwn2 27854 | Alternate characterization... |
| colinearalglem1 27855 | Lemma for ~ colinearalg . ... |
| colinearalglem2 27856 | Lemma for ~ colinearalg . ... |
| colinearalglem3 27857 | Lemma for ~ colinearalg . ... |
| colinearalglem4 27858 | Lemma for ~ colinearalg . ... |
| colinearalg 27859 | An algebraic characterizat... |
| eleesub 27860 | Membership of a subtractio... |
| eleesubd 27861 | Membership of a subtractio... |
| axdimuniq 27862 | The unique dimension axiom... |
| axcgrrflx 27863 | ` A ` is as far from ` B `... |
| axcgrtr 27864 | Congruence is transitive. ... |
| axcgrid 27865 | If there is no distance be... |
| axsegconlem1 27866 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 27867 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 27868 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 27869 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 27870 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 27871 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 27872 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 27873 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 27874 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 27875 | Lemma for ~ axsegcon . Sh... |
| axsegcon 27876 | Any segment ` A B ` can be... |
| ax5seglem1 27877 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 27878 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 27879 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 27880 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 27881 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 27882 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 27883 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 27884 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 27885 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 27886 | Lemma for ~ ax5seg . Take... |
| ax5seg 27887 | The five segment axiom. T... |
| axbtwnid 27888 | Points are indivisible. T... |
| axpaschlem 27889 | Lemma for ~ axpasch . Set... |
| axpasch 27890 | The inner Pasch axiom. Ta... |
| axlowdimlem1 27891 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 27892 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 27893 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 27894 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 27895 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 27896 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 27897 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 27898 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 27899 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 27900 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 27901 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 27902 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 27903 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 27904 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 27905 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 27906 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 27907 | Lemma for ~ axlowdim . Es... |
| axlowdim1 27908 | The lower dimension axiom ... |
| axlowdim2 27909 | The lower two-dimensional ... |
| axlowdim 27910 | The general lower dimensio... |
| axeuclidlem 27911 | Lemma for ~ axeuclid . Ha... |
| axeuclid 27912 | Euclid's axiom. Take an a... |
| axcontlem1 27913 | Lemma for ~ axcont . Chan... |
| axcontlem2 27914 | Lemma for ~ axcont . The ... |
| axcontlem3 27915 | Lemma for ~ axcont . Give... |
| axcontlem4 27916 | Lemma for ~ axcont . Give... |
| axcontlem5 27917 | Lemma for ~ axcont . Comp... |
| axcontlem6 27918 | Lemma for ~ axcont . Stat... |
| axcontlem7 27919 | Lemma for ~ axcont . Give... |
| axcontlem8 27920 | Lemma for ~ axcont . A po... |
| axcontlem9 27921 | Lemma for ~ axcont . Give... |
| axcontlem10 27922 | Lemma for ~ axcont . Give... |
| axcontlem11 27923 | Lemma for ~ axcont . Elim... |
| axcontlem12 27924 | Lemma for ~ axcont . Elim... |
| axcont 27925 | The axiom of continuity. ... |
| eengv 27928 | The value of the Euclidean... |
| eengstr 27929 | The Euclidean geometry as ... |
| eengbas 27930 | The Base of the Euclidean ... |
| ebtwntg 27931 | The betweenness relation u... |
| ecgrtg 27932 | The congruence relation us... |
| elntg 27933 | The line definition in the... |
| elntg2 27934 | The line definition in the... |
| eengtrkg 27935 | The geometry structure for... |
| eengtrkge 27936 | The geometry structure for... |
| edgfid 27939 | Utility theorem: index-ind... |
| edgfndx 27940 | Index value of the ~ df-ed... |
| edgfndxnn 27941 | The index value of the edg... |
| edgfndxid 27942 | The value of the edge func... |
| edgfndxidOLD 27943 | Obsolete version of ~ edgf... |
| basendxltedgfndx 27944 | The index value of the ` B... |
| baseltedgfOLD 27945 | Obsolete proof of ~ basend... |
| basendxnedgfndx 27946 | The slots ` Base ` and ` .... |
| vtxval 27951 | The set of vertices of a g... |
| iedgval 27952 | The set of indexed edges o... |
| 1vgrex 27953 | A graph with at least one ... |
| opvtxval 27954 | The set of vertices of a g... |
| opvtxfv 27955 | The set of vertices of a g... |
| opvtxov 27956 | The set of vertices of a g... |
| opiedgval 27957 | The set of indexed edges o... |
| opiedgfv 27958 | The set of indexed edges o... |
| opiedgov 27959 | The set of indexed edges o... |
| opvtxfvi 27960 | The set of vertices of a g... |
| opiedgfvi 27961 | The set of indexed edges o... |
| funvtxdmge2val 27962 | The set of vertices of an ... |
| funiedgdmge2val 27963 | The set of indexed edges o... |
| funvtxdm2val 27964 | The set of vertices of an ... |
| funiedgdm2val 27965 | The set of indexed edges o... |
| funvtxval0 27966 | The set of vertices of an ... |
| basvtxval 27967 | The set of vertices of a g... |
| edgfiedgval 27968 | The set of indexed edges o... |
| funvtxval 27969 | The set of vertices of a g... |
| funiedgval 27970 | The set of indexed edges o... |
| structvtxvallem 27971 | Lemma for ~ structvtxval a... |
| structvtxval 27972 | The set of vertices of an ... |
| structiedg0val 27973 | The set of indexed edges o... |
| structgrssvtxlem 27974 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 27975 | The set of vertices of a g... |
| structgrssiedg 27976 | The set of indexed edges o... |
| struct2grstr 27977 | A graph represented as an ... |
| struct2grvtx 27978 | The set of vertices of a g... |
| struct2griedg 27979 | The set of indexed edges o... |
| graop 27980 | Any representation of a gr... |
| grastruct 27981 | Any representation of a gr... |
| gropd 27982 | If any representation of a... |
| grstructd 27983 | If any representation of a... |
| gropeld 27984 | If any representation of a... |
| grstructeld 27985 | If any representation of a... |
| setsvtx 27986 | The vertices of a structur... |
| setsiedg 27987 | The (indexed) edges of a s... |
| snstrvtxval 27988 | The set of vertices of a g... |
| snstriedgval 27989 | The set of indexed edges o... |
| vtxval0 27990 | Degenerated case 1 for ver... |
| iedgval0 27991 | Degenerated case 1 for edg... |
| vtxvalsnop 27992 | Degenerated case 2 for ver... |
| iedgvalsnop 27993 | Degenerated case 2 for edg... |
| vtxval3sn 27994 | Degenerated case 3 for ver... |
| iedgval3sn 27995 | Degenerated case 3 for edg... |
| vtxvalprc 27996 | Degenerated case 4 for ver... |
| iedgvalprc 27997 | Degenerated case 4 for edg... |
| edgval 28000 | The edges of a graph. (Co... |
| iedgedg 28001 | An indexed edge is an edge... |
| edgopval 28002 | The edges of a graph repre... |
| edgov 28003 | The edges of a graph repre... |
| edgstruct 28004 | The edges of a graph repre... |
| edgiedgb 28005 | A set is an edge iff it is... |
| edg0iedg0 28006 | There is no edge in a grap... |
| isuhgr 28011 | The predicate "is an undir... |
| isushgr 28012 | The predicate "is an undir... |
| uhgrf 28013 | The edge function of an un... |
| ushgrf 28014 | The edge function of an un... |
| uhgrss 28015 | An edge is a subset of ver... |
| uhgreq12g 28016 | If two sets have the same ... |
| uhgrfun 28017 | The edge function of an un... |
| uhgrn0 28018 | An edge is a nonempty subs... |
| lpvtx 28019 | The endpoints of a loop (w... |
| ushgruhgr 28020 | An undirected simple hyper... |
| isuhgrop 28021 | The property of being an u... |
| uhgr0e 28022 | The empty graph, with vert... |
| uhgr0vb 28023 | The null graph, with no ve... |
| uhgr0 28024 | The null graph represented... |
| uhgrun 28025 | The union ` U ` of two (un... |
| uhgrunop 28026 | The union of two (undirect... |
| ushgrun 28027 | The union ` U ` of two (un... |
| ushgrunop 28028 | The union of two (undirect... |
| uhgrstrrepe 28029 | Replacing (or adding) the ... |
| incistruhgr 28030 | An _incidence structure_ `... |
| isupgr 28035 | The property of being an u... |
| wrdupgr 28036 | The property of being an u... |
| upgrf 28037 | The edge function of an un... |
| upgrfn 28038 | The edge function of an un... |
| upgrss 28039 | An edge is a subset of ver... |
| upgrn0 28040 | An edge is a nonempty subs... |
| upgrle 28041 | An edge of an undirected p... |
| upgrfi 28042 | An edge is a finite subset... |
| upgrex 28043 | An edge is an unordered pa... |
| upgrbi 28044 | Show that an unordered pai... |
| upgrop 28045 | A pseudograph represented ... |
| isumgr 28046 | The property of being an u... |
| isumgrs 28047 | The simplified property of... |
| wrdumgr 28048 | The property of being an u... |
| umgrf 28049 | The edge function of an un... |
| umgrfn 28050 | The edge function of an un... |
| umgredg2 28051 | An edge of a multigraph ha... |
| umgrbi 28052 | Show that an unordered pai... |
| upgruhgr 28053 | An undirected pseudograph ... |
| umgrupgr 28054 | An undirected multigraph i... |
| umgruhgr 28055 | An undirected multigraph i... |
| upgrle2 28056 | An edge of an undirected p... |
| umgrnloopv 28057 | In a multigraph, there is ... |
| umgredgprv 28058 | In a multigraph, an edge i... |
| umgrnloop 28059 | In a multigraph, there is ... |
| umgrnloop0 28060 | A multigraph has no loops.... |
| umgr0e 28061 | The empty graph, with vert... |
| upgr0e 28062 | The empty graph, with vert... |
| upgr1elem 28063 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 28064 | A pseudograph with one edg... |
| upgr0eop 28065 | The empty graph, with vert... |
| upgr1eop 28066 | A pseudograph with one edg... |
| upgr0eopALT 28067 | Alternate proof of ~ upgr0... |
| upgr1eopALT 28068 | Alternate proof of ~ upgr1... |
| upgrun 28069 | The union ` U ` of two pse... |
| upgrunop 28070 | The union of two pseudogra... |
| umgrun 28071 | The union ` U ` of two mul... |
| umgrunop 28072 | The union of two multigrap... |
| umgrislfupgrlem 28073 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 28074 | A multigraph is a loop-fre... |
| lfgredgge2 28075 | An edge of a loop-free gra... |
| lfgrnloop 28076 | A loop-free graph has no l... |
| uhgredgiedgb 28077 | In a hypergraph, a set is ... |
| uhgriedg0edg0 28078 | A hypergraph has no edges ... |
| uhgredgn0 28079 | An edge of a hypergraph is... |
| edguhgr 28080 | An edge of a hypergraph is... |
| uhgredgrnv 28081 | An edge of a hypergraph co... |
| uhgredgss 28082 | The set of edges of a hype... |
| upgredgss 28083 | The set of edges of a pseu... |
| umgredgss 28084 | The set of edges of a mult... |
| edgupgr 28085 | Properties of an edge of a... |
| edgumgr 28086 | Properties of an edge of a... |
| uhgrvtxedgiedgb 28087 | In a hypergraph, a vertex ... |
| upgredg 28088 | For each edge in a pseudog... |
| umgredg 28089 | For each edge in a multigr... |
| upgrpredgv 28090 | An edge of a pseudograph a... |
| umgrpredgv 28091 | An edge of a multigraph al... |
| upgredg2vtx 28092 | For a vertex incident to a... |
| upgredgpr 28093 | If a proper pair (of verti... |
| edglnl 28094 | The edges incident with a ... |
| numedglnl 28095 | The number of edges incide... |
| umgredgne 28096 | An edge of a multigraph al... |
| umgrnloop2 28097 | A multigraph has no loops.... |
| umgredgnlp 28098 | An edge of a multigraph is... |
| isuspgr 28103 | The property of being a si... |
| isusgr 28104 | The property of being a si... |
| uspgrf 28105 | The edge function of a sim... |
| usgrf 28106 | The edge function of a sim... |
| isusgrs 28107 | The property of being a si... |
| usgrfs 28108 | The edge function of a sim... |
| usgrfun 28109 | The edge function of a sim... |
| usgredgss 28110 | The set of edges of a simp... |
| edgusgr 28111 | An edge of a simple graph ... |
| isuspgrop 28112 | The property of being an u... |
| isusgrop 28113 | The property of being an u... |
| usgrop 28114 | A simple graph represented... |
| isausgr 28115 | The property of an unorder... |
| ausgrusgrb 28116 | The equivalence of the def... |
| usgrausgri 28117 | A simple graph represented... |
| ausgrumgri 28118 | If an alternatively define... |
| ausgrusgri 28119 | The equivalence of the def... |
| usgrausgrb 28120 | The equivalence of the def... |
| usgredgop 28121 | An edge of a simple graph ... |
| usgrf1o 28122 | The edge function of a sim... |
| usgrf1 28123 | The edge function of a sim... |
| uspgrf1oedg 28124 | The edge function of a sim... |
| usgrss 28125 | An edge is a subset of ver... |
| uspgrushgr 28126 | A simple pseudograph is an... |
| uspgrupgr 28127 | A simple pseudograph is an... |
| uspgrupgrushgr 28128 | A graph is a simple pseudo... |
| usgruspgr 28129 | A simple graph is a simple... |
| usgrumgr 28130 | A simple graph is an undir... |
| usgrumgruspgr 28131 | A graph is a simple graph ... |
| usgruspgrb 28132 | A class is a simple graph ... |
| usgrupgr 28133 | A simple graph is an undir... |
| usgruhgr 28134 | A simple graph is an undir... |
| usgrislfuspgr 28135 | A simple graph is a loop-f... |
| uspgrun 28136 | The union ` U ` of two sim... |
| uspgrunop 28137 | The union of two simple ps... |
| usgrun 28138 | The union ` U ` of two sim... |
| usgrunop 28139 | The union of two simple gr... |
| usgredg2 28140 | The value of the "edge fun... |
| usgredg2ALT 28141 | Alternate proof of ~ usgre... |
| usgredgprv 28142 | In a simple graph, an edge... |
| usgredgprvALT 28143 | Alternate proof of ~ usgre... |
| usgredgppr 28144 | An edge of a simple graph ... |
| usgrpredgv 28145 | An edge of a simple graph ... |
| edgssv2 28146 | An edge of a simple graph ... |
| usgredg 28147 | For each edge in a simple ... |
| usgrnloopv 28148 | In a simple graph, there i... |
| usgrnloopvALT 28149 | Alternate proof of ~ usgrn... |
| usgrnloop 28150 | In a simple graph, there i... |
| usgrnloopALT 28151 | Alternate proof of ~ usgrn... |
| usgrnloop0 28152 | A simple graph has no loop... |
| usgrnloop0ALT 28153 | Alternate proof of ~ usgrn... |
| usgredgne 28154 | An edge of a simple graph ... |
| usgrf1oedg 28155 | The edge function of a sim... |
| uhgr2edg 28156 | If a vertex is adjacent to... |
| umgr2edg 28157 | If a vertex is adjacent to... |
| usgr2edg 28158 | If a vertex is adjacent to... |
| umgr2edg1 28159 | If a vertex is adjacent to... |
| usgr2edg1 28160 | If a vertex is adjacent to... |
| umgrvad2edg 28161 | If a vertex is adjacent to... |
| umgr2edgneu 28162 | If a vertex is adjacent to... |
| usgrsizedg 28163 | In a simple graph, the siz... |
| usgredg3 28164 | The value of the "edge fun... |
| usgredg4 28165 | For a vertex incident to a... |
| usgredgreu 28166 | For a vertex incident to a... |
| usgredg2vtx 28167 | For a vertex incident to a... |
| uspgredg2vtxeu 28168 | For a vertex incident to a... |
| usgredg2vtxeu 28169 | For a vertex incident to a... |
| usgredg2vtxeuALT 28170 | Alternate proof of ~ usgre... |
| uspgredg2vlem 28171 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 28172 | In a simple pseudograph, t... |
| usgredg2vlem1 28173 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 28174 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 28175 | In a simple graph, the map... |
| usgriedgleord 28176 | Alternate version of ~ usg... |
| ushgredgedg 28177 | In a simple hypergraph the... |
| usgredgedg 28178 | In a simple graph there is... |
| ushgredgedgloop 28179 | In a simple hypergraph the... |
| uspgredgleord 28180 | In a simple pseudograph th... |
| usgredgleord 28181 | In a simple graph the numb... |
| usgredgleordALT 28182 | Alternate proof for ~ usgr... |
| usgrstrrepe 28183 | Replacing (or adding) the ... |
| usgr0e 28184 | The empty graph, with vert... |
| usgr0vb 28185 | The null graph, with no ve... |
| uhgr0v0e 28186 | The null graph, with no ve... |
| uhgr0vsize0 28187 | The size of a hypergraph w... |
| uhgr0edgfi 28188 | A graph of order 0 (i.e. w... |
| usgr0v 28189 | The null graph, with no ve... |
| uhgr0vusgr 28190 | The null graph, with no ve... |
| usgr0 28191 | The null graph represented... |
| uspgr1e 28192 | A simple pseudograph with ... |
| usgr1e 28193 | A simple graph with one ed... |
| usgr0eop 28194 | The empty graph, with vert... |
| uspgr1eop 28195 | A simple pseudograph with ... |
| uspgr1ewop 28196 | A simple pseudograph with ... |
| uspgr1v1eop 28197 | A simple pseudograph with ... |
| usgr1eop 28198 | A simple graph with (at le... |
| uspgr2v1e2w 28199 | A simple pseudograph with ... |
| usgr2v1e2w 28200 | A simple graph with two ve... |
| edg0usgr 28201 | A class without edges is a... |
| lfuhgr1v0e 28202 | A loop-free hypergraph wit... |
| usgr1vr 28203 | A simple graph with one ve... |
| usgr1v 28204 | A class with one (or no) v... |
| usgr1v0edg 28205 | A class with one (or no) v... |
| usgrexmpldifpr 28206 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 28207 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 28208 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 28209 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 28210 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 28211 | ` G ` is a simple graph of... |
| griedg0prc 28212 | The class of empty graphs ... |
| griedg0ssusgr 28213 | The class of all simple gr... |
| usgrprc 28214 | The class of simple graphs... |
| relsubgr 28217 | The class of the subgraph ... |
| subgrv 28218 | If a class is a subgraph o... |
| issubgr 28219 | The property of a set to b... |
| issubgr2 28220 | The property of a set to b... |
| subgrprop 28221 | The properties of a subgra... |
| subgrprop2 28222 | The properties of a subgra... |
| uhgrissubgr 28223 | The property of a hypergra... |
| subgrprop3 28224 | The properties of a subgra... |
| egrsubgr 28225 | An empty graph consisting ... |
| 0grsubgr 28226 | The null graph (represente... |
| 0uhgrsubgr 28227 | The null graph (as hypergr... |
| uhgrsubgrself 28228 | A hypergraph is a subgraph... |
| subgrfun 28229 | The edge function of a sub... |
| subgruhgrfun 28230 | The edge function of a sub... |
| subgreldmiedg 28231 | An element of the domain o... |
| subgruhgredgd 28232 | An edge of a subgraph of a... |
| subumgredg2 28233 | An edge of a subgraph of a... |
| subuhgr 28234 | A subgraph of a hypergraph... |
| subupgr 28235 | A subgraph of a pseudograp... |
| subumgr 28236 | A subgraph of a multigraph... |
| subusgr 28237 | A subgraph of a simple gra... |
| uhgrspansubgrlem 28238 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 28239 | A spanning subgraph ` S ` ... |
| uhgrspan 28240 | A spanning subgraph ` S ` ... |
| upgrspan 28241 | A spanning subgraph ` S ` ... |
| umgrspan 28242 | A spanning subgraph ` S ` ... |
| usgrspan 28243 | A spanning subgraph ` S ` ... |
| uhgrspanop 28244 | A spanning subgraph of a h... |
| upgrspanop 28245 | A spanning subgraph of a p... |
| umgrspanop 28246 | A spanning subgraph of a m... |
| usgrspanop 28247 | A spanning subgraph of a s... |
| uhgrspan1lem1 28248 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 28249 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 28250 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 28251 | The induced subgraph ` S `... |
| upgrreslem 28252 | Lemma for ~ upgrres . (Co... |
| umgrreslem 28253 | Lemma for ~ umgrres and ~ ... |
| upgrres 28254 | A subgraph obtained by rem... |
| umgrres 28255 | A subgraph obtained by rem... |
| usgrres 28256 | A subgraph obtained by rem... |
| upgrres1lem1 28257 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 28258 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 28259 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 28260 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 28261 | A pseudograph obtained by ... |
| umgrres1 28262 | A multigraph obtained by r... |
| usgrres1 28263 | Restricting a simple graph... |
| isfusgr 28266 | The property of being a fi... |
| fusgrvtxfi 28267 | A finite simple graph has ... |
| isfusgrf1 28268 | The property of being a fi... |
| isfusgrcl 28269 | The property of being a fi... |
| fusgrusgr 28270 | A finite simple graph is a... |
| opfusgr 28271 | A finite simple graph repr... |
| usgredgffibi 28272 | The number of edges in a s... |
| fusgredgfi 28273 | In a finite simple graph t... |
| usgr1v0e 28274 | The size of a (finite) sim... |
| usgrfilem 28275 | In a finite simple graph, ... |
| fusgrfisbase 28276 | Induction base for ~ fusgr... |
| fusgrfisstep 28277 | Induction step in ~ fusgrf... |
| fusgrfis 28278 | A finite simple graph is o... |
| fusgrfupgrfs 28279 | A finite simple graph is a... |
| nbgrprc0 28282 | The set of neighbors is em... |
| nbgrcl 28283 | If a class ` X ` has at le... |
| nbgrval 28284 | The set of neighbors of a ... |
| dfnbgr2 28285 | Alternate definition of th... |
| dfnbgr3 28286 | Alternate definition of th... |
| nbgrnvtx0 28287 | If a class ` X ` is not a ... |
| nbgrel 28288 | Characterization of a neig... |
| nbgrisvtx 28289 | Every neighbor ` N ` of a ... |
| nbgrssvtx 28290 | The neighbors of a vertex ... |
| nbuhgr 28291 | The set of neighbors of a ... |
| nbupgr 28292 | The set of neighbors of a ... |
| nbupgrel 28293 | A neighbor of a vertex in ... |
| nbumgrvtx 28294 | The set of neighbors of a ... |
| nbumgr 28295 | The set of neighbors of an... |
| nbusgrvtx 28296 | The set of neighbors of a ... |
| nbusgr 28297 | The set of neighbors of an... |
| nbgr2vtx1edg 28298 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 28299 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 28300 | If a hypergraph has two ve... |
| nbusgreledg 28301 | A class/vertex is a neighb... |
| uhgrnbgr0nb 28302 | A vertex which is not endp... |
| nbgr0vtxlem 28303 | Lemma for ~ nbgr0vtx and ~... |
| nbgr0vtx 28304 | In a null graph (with no v... |
| nbgr0edg 28305 | In an empty graph (with no... |
| nbgr1vtx 28306 | In a graph with one vertex... |
| nbgrnself 28307 | A vertex in a graph is not... |
| nbgrnself2 28308 | A class ` X ` is not a nei... |
| nbgrssovtx 28309 | The neighbors of a vertex ... |
| nbgrssvwo2 28310 | The neighbors of a vertex ... |
| nbgrsym 28311 | In a graph, the neighborho... |
| nbupgrres 28312 | The neighborhood of a vert... |
| usgrnbcnvfv 28313 | Applying the edge function... |
| nbusgredgeu 28314 | For each neighbor of a ver... |
| edgnbusgreu 28315 | For each edge incident to ... |
| nbusgredgeu0 28316 | For each neighbor of a ver... |
| nbusgrf1o0 28317 | The mapping of neighbors o... |
| nbusgrf1o1 28318 | The set of neighbors of a ... |
| nbusgrf1o 28319 | The set of neighbors of a ... |
| nbedgusgr 28320 | The number of neighbors of... |
| edgusgrnbfin 28321 | The number of neighbors of... |
| nbusgrfi 28322 | The class of neighbors of ... |
| nbfiusgrfi 28323 | The class of neighbors of ... |
| hashnbusgrnn0 28324 | The number of neighbors of... |
| nbfusgrlevtxm1 28325 | The number of neighbors of... |
| nbfusgrlevtxm2 28326 | If there is a vertex which... |
| nbusgrvtxm1 28327 | If the number of neighbors... |
| nb3grprlem1 28328 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 28329 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 28330 | The neighbors of a vertex ... |
| nb3grpr2 28331 | The neighbors of a vertex ... |
| nb3gr2nb 28332 | If the neighbors of two ve... |
| uvtxval 28335 | The set of all universal v... |
| uvtxel 28336 | A universal vertex, i.e. a... |
| uvtxisvtx 28337 | A universal vertex is a ve... |
| uvtxssvtx 28338 | The set of the universal v... |
| vtxnbuvtx 28339 | A universal vertex has all... |
| uvtxnbgrss 28340 | A universal vertex has all... |
| uvtxnbgrvtx 28341 | A universal vertex is neig... |
| uvtx0 28342 | There is no universal vert... |
| isuvtx 28343 | The set of all universal v... |
| uvtxel1 28344 | Characterization of a univ... |
| uvtx01vtx 28345 | If a graph/class has no ed... |
| uvtx2vtx1edg 28346 | If a graph has two vertice... |
| uvtx2vtx1edgb 28347 | If a hypergraph has two ve... |
| uvtxnbgr 28348 | A universal vertex has all... |
| uvtxnbgrb 28349 | A vertex is universal iff ... |
| uvtxusgr 28350 | The set of all universal v... |
| uvtxusgrel 28351 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 28352 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 28353 | If the number of neighbors... |
| uvtxnbvtxm1 28354 | A universal vertex has ` n... |
| nbupgruvtxres 28355 | The neighborhood of a univ... |
| uvtxupgrres 28356 | A universal vertex is univ... |
| cplgruvtxb 28361 | A graph ` G ` is complete ... |
| prcliscplgr 28362 | A proper class (representi... |
| iscplgr 28363 | The property of being a co... |
| iscplgrnb 28364 | A graph is complete iff al... |
| iscplgredg 28365 | A graph ` G ` is complete ... |
| iscusgr 28366 | The property of being a co... |
| cusgrusgr 28367 | A complete simple graph is... |
| cusgrcplgr 28368 | A complete simple graph is... |
| iscusgrvtx 28369 | A simple graph is complete... |
| cusgruvtxb 28370 | A simple graph is complete... |
| iscusgredg 28371 | A simple graph is complete... |
| cusgredg 28372 | In a complete simple graph... |
| cplgr0 28373 | The null graph (with no ve... |
| cusgr0 28374 | The null graph (with no ve... |
| cplgr0v 28375 | A null graph (with no vert... |
| cusgr0v 28376 | A graph with no vertices a... |
| cplgr1vlem 28377 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 28378 | A graph with one vertex is... |
| cusgr1v 28379 | A graph with one vertex an... |
| cplgr2v 28380 | An undirected hypergraph w... |
| cplgr2vpr 28381 | An undirected hypergraph w... |
| nbcplgr 28382 | In a complete graph, each ... |
| cplgr3v 28383 | A pseudograph with three (... |
| cusgr3vnbpr 28384 | The neighbors of a vertex ... |
| cplgrop 28385 | A complete graph represent... |
| cusgrop 28386 | A complete simple graph re... |
| cusgrexilem1 28387 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 28388 | Lemma for ~ usgrexi . (Co... |
| usgrexi 28389 | An arbitrary set regarded ... |
| cusgrexilem2 28390 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 28391 | An arbitrary set ` V ` reg... |
| cusgrexg 28392 | For each set there is a se... |
| structtousgr 28393 | Any (extensible) structure... |
| structtocusgr 28394 | Any (extensible) structure... |
| cffldtocusgr 28395 | The field of complex numbe... |
| cusgrres 28396 | Restricting a complete sim... |
| cusgrsizeindb0 28397 | Base case of the induction... |
| cusgrsizeindb1 28398 | Base case of the induction... |
| cusgrsizeindslem 28399 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 28400 | Part 1 of induction step i... |
| cusgrsize2inds 28401 | Induction step in ~ cusgrs... |
| cusgrsize 28402 | The size of a finite compl... |
| cusgrfilem1 28403 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 28404 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 28405 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 28406 | If the size of a complete ... |
| usgredgsscusgredg 28407 | A simple graph is a subgra... |
| usgrsscusgr 28408 | A simple graph is a subgra... |
| sizusglecusglem1 28409 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 28410 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 28411 | The size of a simple graph... |
| fusgrmaxsize 28412 | The maximum size of a fini... |
| vtxdgfval 28415 | The value of the vertex de... |
| vtxdgval 28416 | The degree of a vertex. (... |
| vtxdgfival 28417 | The degree of a vertex for... |
| vtxdgop 28418 | The vertex degree expresse... |
| vtxdgf 28419 | The vertex degree function... |
| vtxdgelxnn0 28420 | The degree of a vertex is ... |
| vtxdg0v 28421 | The degree of a vertex in ... |
| vtxdg0e 28422 | The degree of a vertex in ... |
| vtxdgfisnn0 28423 | The degree of a vertex in ... |
| vtxdgfisf 28424 | The vertex degree function... |
| vtxdeqd 28425 | Equality theorem for the v... |
| vtxduhgr0e 28426 | The degree of a vertex in ... |
| vtxdlfuhgr1v 28427 | The degree of the vertex i... |
| vdumgr0 28428 | A vertex in a multigraph h... |
| vtxdun 28429 | The degree of a vertex in ... |
| vtxdfiun 28430 | The degree of a vertex in ... |
| vtxduhgrun 28431 | The degree of a vertex in ... |
| vtxduhgrfiun 28432 | The degree of a vertex in ... |
| vtxdlfgrval 28433 | The value of the vertex de... |
| vtxdumgrval 28434 | The value of the vertex de... |
| vtxdusgrval 28435 | The value of the vertex de... |
| vtxd0nedgb 28436 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 28437 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 28438 | The value of the vertex de... |
| vtxdusgrfvedg 28439 | The value of the vertex de... |
| vtxduhgr0nedg 28440 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 28441 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 28442 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 28443 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 28444 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 28445 | The vertex degree function... |
| vtxdgfusgr 28446 | In a finite simple graph, ... |
| fusgrn0degnn0 28447 | In a nonempty, finite grap... |
| 1loopgruspgr 28448 | A graph with one edge whic... |
| 1loopgredg 28449 | The set of edges in a grap... |
| 1loopgrnb0 28450 | In a graph (simple pseudog... |
| 1loopgrvd2 28451 | The vertex degree of a one... |
| 1loopgrvd0 28452 | The vertex degree of a one... |
| 1hevtxdg0 28453 | The vertex degree of verte... |
| 1hevtxdg1 28454 | The vertex degree of verte... |
| 1hegrvtxdg1 28455 | The vertex degree of a gra... |
| 1hegrvtxdg1r 28456 | The vertex degree of a gra... |
| 1egrvtxdg1 28457 | The vertex degree of a one... |
| 1egrvtxdg1r 28458 | The vertex degree of a one... |
| 1egrvtxdg0 28459 | The vertex degree of a one... |
| p1evtxdeqlem 28460 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 28461 | If an edge ` E ` which doe... |
| p1evtxdp1 28462 | If an edge ` E ` (not bein... |
| uspgrloopvtx 28463 | The set of vertices in a g... |
| uspgrloopvtxel 28464 | A vertex in a graph (simpl... |
| uspgrloopiedg 28465 | The set of edges in a grap... |
| uspgrloopedg 28466 | The set of edges in a grap... |
| uspgrloopnb0 28467 | In a graph (simple pseudog... |
| uspgrloopvd2 28468 | The vertex degree of a one... |
| umgr2v2evtx 28469 | The set of vertices in a m... |
| umgr2v2evtxel 28470 | A vertex in a multigraph w... |
| umgr2v2eiedg 28471 | The edge function in a mul... |
| umgr2v2eedg 28472 | The set of edges in a mult... |
| umgr2v2e 28473 | A multigraph with two edge... |
| umgr2v2enb1 28474 | In a multigraph with two e... |
| umgr2v2evd2 28475 | In a multigraph with two e... |
| hashnbusgrvd 28476 | In a simple graph, the num... |
| usgruvtxvdb 28477 | In a finite simple graph w... |
| vdiscusgrb 28478 | A finite simple graph with... |
| vdiscusgr 28479 | In a finite complete simpl... |
| vtxdusgradjvtx 28480 | The degree of a vertex in ... |
| usgrvd0nedg 28481 | If a vertex in a simple gr... |
| uhgrvd00 28482 | If every vertex in a hyper... |
| usgrvd00 28483 | If every vertex in a simpl... |
| vdegp1ai 28484 | The induction step for a v... |
| vdegp1bi 28485 | The induction step for a v... |
| vdegp1ci 28486 | The induction step for a v... |
| vtxdginducedm1lem1 28487 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 28488 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 28489 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 28490 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 28491 | The degree of a vertex ` v... |
| vtxdginducedm1fi 28492 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 28493 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 28494 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 28495 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 28496 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 28497 | Induction step of ~ finsum... |
| finsumvtxdg2size 28498 | The sum of the degrees of ... |
| fusgr1th 28499 | The sum of the degrees of ... |
| finsumvtxdgeven 28500 | The sum of the degrees of ... |
| vtxdgoddnumeven 28501 | The number of vertices of ... |
| fusgrvtxdgonume 28502 | The number of vertices of ... |
| isrgr 28507 | The property of a class be... |
| rgrprop 28508 | The properties of a k-regu... |
| isrusgr 28509 | The property of being a k-... |
| rusgrprop 28510 | The properties of a k-regu... |
| rusgrrgr 28511 | A k-regular simple graph i... |
| rusgrusgr 28512 | A k-regular simple graph i... |
| finrusgrfusgr 28513 | A finite regular simple gr... |
| isrusgr0 28514 | The property of being a k-... |
| rusgrprop0 28515 | The properties of a k-regu... |
| usgreqdrusgr 28516 | If all vertices in a simpl... |
| fusgrregdegfi 28517 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 28518 | If all vertices in a nonem... |
| frusgrnn0 28519 | In a nonempty finite k-reg... |
| 0edg0rgr 28520 | A graph is 0-regular if it... |
| uhgr0edg0rgr 28521 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 28522 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 28523 | A simple graph is 0-regula... |
| 0vtxrgr 28524 | A null graph (with no vert... |
| 0vtxrusgr 28525 | A graph with no vertices a... |
| 0uhgrrusgr 28526 | The null graph as hypergra... |
| 0grrusgr 28527 | The null graph represented... |
| 0grrgr 28528 | The null graph represented... |
| cusgrrusgr 28529 | A complete simple graph wi... |
| cusgrm1rusgr 28530 | A finite simple graph with... |
| rusgrpropnb 28531 | The properties of a k-regu... |
| rusgrpropedg 28532 | The properties of a k-regu... |
| rusgrpropadjvtx 28533 | The properties of a k-regu... |
| rusgrnumwrdl2 28534 | In a k-regular simple grap... |
| rusgr1vtxlem 28535 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 28536 | If a k-regular simple grap... |
| rgrusgrprc 28537 | The class of 0-regular sim... |
| rusgrprc 28538 | The class of 0-regular sim... |
| rgrprc 28539 | The class of 0-regular gra... |
| rgrprcx 28540 | The class of 0-regular gra... |
| rgrx0ndm 28541 | 0 is not in the domain of ... |
| rgrx0nd 28542 | The potentially alternativ... |
| ewlksfval 28549 | The set of s-walks of edge... |
| isewlk 28550 | Conditions for a function ... |
| ewlkprop 28551 | Properties of an s-walk of... |
| ewlkinedg 28552 | The intersection (common v... |
| ewlkle 28553 | An s-walk of edges is also... |
| upgrewlkle2 28554 | In a pseudograph, there is... |
| wkslem1 28555 | Lemma 1 for walks to subst... |
| wkslem2 28556 | Lemma 2 for walks to subst... |
| wksfval 28557 | The set of walks (in an un... |
| iswlk 28558 | Properties of a pair of fu... |
| wlkprop 28559 | Properties of a walk. (Co... |
| wlkv 28560 | The classes involved in a ... |
| iswlkg 28561 | Generalization of ~ iswlk ... |
| wlkf 28562 | The mapping enumerating th... |
| wlkcl 28563 | A walk has length ` # ( F ... |
| wlkp 28564 | The mapping enumerating th... |
| wlkpwrd 28565 | The sequence of vertices o... |
| wlklenvp1 28566 | The number of vertices of ... |
| wksv 28567 | The class of walks is a se... |
| wksvOLD 28568 | Obsolete version of ~ wksv... |
| wlkn0 28569 | The sequence of vertices o... |
| wlklenvm1 28570 | The number of edges of a w... |
| ifpsnprss 28571 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 28572 | Each pair of adjacent vert... |
| wlkvtxiedg 28573 | The vertices of a walk are... |
| relwlk 28574 | The set ` ( Walks `` G ) `... |
| wlkvv 28575 | If there is at least one w... |
| wlkop 28576 | A walk is an ordered pair.... |
| wlkcpr 28577 | A walk as class with two c... |
| wlk2f 28578 | If there is a walk ` W ` t... |
| wlkcomp 28579 | A walk expressed by proper... |
| wlkcompim 28580 | Implications for the prope... |
| wlkelwrd 28581 | The components of a walk a... |
| wlkeq 28582 | Conditions for two walks (... |
| edginwlk 28583 | The value of the edge func... |
| upgredginwlk 28584 | The value of the edge func... |
| iedginwlk 28585 | The value of the edge func... |
| wlkl1loop 28586 | A walk of length 1 from a ... |
| wlk1walk 28587 | A walk is a 1-walk "on the... |
| wlk1ewlk 28588 | A walk is an s-walk "on th... |
| upgriswlk 28589 | Properties of a pair of fu... |
| upgrwlkedg 28590 | The edges of a walk in a p... |
| upgrwlkcompim 28591 | Implications for the prope... |
| wlkvtxedg 28592 | The vertices of a walk are... |
| upgrwlkvtxedg 28593 | The pairs of connected ver... |
| uspgr2wlkeq 28594 | Conditions for two walks w... |
| uspgr2wlkeq2 28595 | Conditions for two walks w... |
| uspgr2wlkeqi 28596 | Conditions for two walks w... |
| umgrwlknloop 28597 | In a multigraph, each walk... |
| wlkResOLD 28598 | Obsolete version of ~ opab... |
| wlkv0 28599 | If there is a walk in the ... |
| g0wlk0 28600 | There is no walk in a null... |
| 0wlk0 28601 | There is no walk for the e... |
| wlk0prc 28602 | There is no walk in a null... |
| wlklenvclwlk 28603 | The number of vertices in ... |
| wlkson 28604 | The set of walks between t... |
| iswlkon 28605 | Properties of a pair of fu... |
| wlkonprop 28606 | Properties of a walk betwe... |
| wlkpvtx 28607 | A walk connects vertices. ... |
| wlkepvtx 28608 | The endpoints of a walk ar... |
| wlkoniswlk 28609 | A walk between two vertice... |
| wlkonwlk 28610 | A walk is a walk between i... |
| wlkonwlk1l 28611 | A walk is a walk from its ... |
| wlksoneq1eq2 28612 | Two walks with identical s... |
| wlkonl1iedg 28613 | If there is a walk between... |
| wlkon2n0 28614 | The length of a walk betwe... |
| 2wlklem 28615 | Lemma for theorems for wal... |
| upgr2wlk 28616 | Properties of a pair of fu... |
| wlkreslem 28617 | Lemma for ~ wlkres . (Con... |
| wlkres 28618 | The restriction ` <. H , Q... |
| redwlklem 28619 | Lemma for ~ redwlk . (Con... |
| redwlk 28620 | A walk ending at the last ... |
| wlkp1lem1 28621 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 28622 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 28623 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 28624 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 28625 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 28626 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 28627 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 28628 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 28629 | Append one path segment (e... |
| wlkdlem1 28630 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 28631 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 28632 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 28633 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 28634 | Two words representing a w... |
| lfgrwlkprop 28635 | Two adjacent vertices in a... |
| lfgriswlk 28636 | Conditions for a pair of f... |
| lfgrwlknloop 28637 | In a loop-free graph, each... |
| reltrls 28642 | The set ` ( Trails `` G ) ... |
| trlsfval 28643 | The set of trails (in an u... |
| istrl 28644 | Conditions for a pair of c... |
| trliswlk 28645 | A trail is a walk. (Contr... |
| trlf1 28646 | The enumeration ` F ` of a... |
| trlreslem 28647 | Lemma for ~ trlres . Form... |
| trlres 28648 | The restriction ` <. H , Q... |
| upgrtrls 28649 | The set of trails in a pse... |
| upgristrl 28650 | Properties of a pair of fu... |
| upgrf1istrl 28651 | Properties of a pair of a ... |
| wksonproplem 28652 | Lemma for theorems for pro... |
| wksonproplemOLD 28653 | Obsolete version of ~ wkso... |
| trlsonfval 28654 | The set of trails between ... |
| istrlson 28655 | Properties of a pair of fu... |
| trlsonprop 28656 | Properties of a trail betw... |
| trlsonistrl 28657 | A trail between two vertic... |
| trlsonwlkon 28658 | A trail between two vertic... |
| trlontrl 28659 | A trail is a trail between... |
| relpths 28668 | The set ` ( Paths `` G ) `... |
| pthsfval 28669 | The set of paths (in an un... |
| spthsfval 28670 | The set of simple paths (i... |
| ispth 28671 | Conditions for a pair of c... |
| isspth 28672 | Conditions for a pair of c... |
| pthistrl 28673 | A path is a trail (in an u... |
| spthispth 28674 | A simple path is a path (i... |
| pthiswlk 28675 | A path is a walk (in an un... |
| spthiswlk 28676 | A simple path is a walk (i... |
| pthdivtx 28677 | The inner vertices of a pa... |
| pthdadjvtx 28678 | The adjacent vertices of a... |
| 2pthnloop 28679 | A path of length at least ... |
| upgr2pthnlp 28680 | A path of length at least ... |
| spthdifv 28681 | The vertices of a simple p... |
| spthdep 28682 | A simple path (at least of... |
| pthdepisspth 28683 | A path with different star... |
| upgrwlkdvdelem 28684 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 28685 | In a pseudograph, all edge... |
| upgrspthswlk 28686 | The set of simple paths in... |
| upgrwlkdvspth 28687 | A walk consisting of diffe... |
| pthsonfval 28688 | The set of paths between t... |
| spthson 28689 | The set of simple paths be... |
| ispthson 28690 | Properties of a pair of fu... |
| isspthson 28691 | Properties of a pair of fu... |
| pthsonprop 28692 | Properties of a path betwe... |
| spthonprop 28693 | Properties of a simple pat... |
| pthonispth 28694 | A path between two vertice... |
| pthontrlon 28695 | A path between two vertice... |
| pthonpth 28696 | A path is a path between i... |
| isspthonpth 28697 | A pair of functions is a s... |
| spthonisspth 28698 | A simple path between to v... |
| spthonpthon 28699 | A simple path between two ... |
| spthonepeq 28700 | The endpoints of a simple ... |
| uhgrwkspthlem1 28701 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 28702 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 28703 | Any walk of length 1 betwe... |
| usgr2wlkneq 28704 | The vertices and edges are... |
| usgr2wlkspthlem1 28705 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 28706 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 28707 | In a simple graph, any wal... |
| usgr2trlncl 28708 | In a simple graph, any tra... |
| usgr2trlspth 28709 | In a simple graph, any tra... |
| usgr2pthspth 28710 | In a simple graph, any pat... |
| usgr2pthlem 28711 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 28712 | In a simple graph, there i... |
| usgr2pth0 28713 | In a simply graph, there i... |
| pthdlem1 28714 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 28715 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 28716 | Lemma 2 for ~ pthd . (Con... |
| pthd 28717 | Two words representing a t... |
| clwlks 28720 | The set of closed walks (i... |
| isclwlk 28721 | A pair of functions repres... |
| clwlkiswlk 28722 | A closed walk is a walk (i... |
| clwlkwlk 28723 | Closed walks are walks (in... |
| clwlkswks 28724 | Closed walks are walks (in... |
| isclwlke 28725 | Properties of a pair of fu... |
| isclwlkupgr 28726 | Properties of a pair of fu... |
| clwlkcomp 28727 | A closed walk expressed by... |
| clwlkcompim 28728 | Implications for the prope... |
| upgrclwlkcompim 28729 | Implications for the prope... |
| clwlkcompbp 28730 | Basic properties of the co... |
| clwlkl1loop 28731 | A closed walk of length 1 ... |
| crcts 28736 | The set of circuits (in an... |
| cycls 28737 | The set of cycles (in an u... |
| iscrct 28738 | Sufficient and necessary c... |
| iscycl 28739 | Sufficient and necessary c... |
| crctprop 28740 | The properties of a circui... |
| cyclprop 28741 | The properties of a cycle:... |
| crctisclwlk 28742 | A circuit is a closed walk... |
| crctistrl 28743 | A circuit is a trail. (Co... |
| crctiswlk 28744 | A circuit is a walk. (Con... |
| cyclispth 28745 | A cycle is a path. (Contr... |
| cycliswlk 28746 | A cycle is a walk. (Contr... |
| cycliscrct 28747 | A cycle is a circuit. (Co... |
| cyclnspth 28748 | A (non-trivial) cycle is n... |
| cyclispthon 28749 | A cycle is a path starting... |
| lfgrn1cycl 28750 | In a loop-free graph there... |
| usgr2trlncrct 28751 | In a simple graph, any tra... |
| umgrn1cycl 28752 | In a multigraph graph (wit... |
| uspgrn2crct 28753 | In a simple pseudograph th... |
| usgrn2cycl 28754 | In a simple graph there ar... |
| crctcshwlkn0lem1 28755 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 28756 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 28757 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 28758 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 28759 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 28760 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 28761 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 28762 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 28763 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 28764 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 28765 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 28766 | Cyclically shifting the in... |
| crctcshwlk 28767 | Cyclically shifting the in... |
| crctcshtrl 28768 | Cyclically shifting the in... |
| crctcsh 28769 | Cyclically shifting the in... |
| wwlks 28780 | The set of walks (in an un... |
| iswwlks 28781 | A word over the set of ver... |
| wwlksn 28782 | The set of walks (in an un... |
| iswwlksn 28783 | A word over the set of ver... |
| wwlksnprcl 28784 | Derivation of the length o... |
| iswwlksnx 28785 | Properties of a word to re... |
| wwlkbp 28786 | Basic properties of a walk... |
| wwlknbp 28787 | Basic properties of a walk... |
| wwlknp 28788 | Properties of a set being ... |
| wwlknbp1 28789 | Other basic properties of ... |
| wwlknvtx 28790 | The symbols of a word ` W ... |
| wwlknllvtx 28791 | If a word ` W ` represents... |
| wwlknlsw 28792 | If a word represents a wal... |
| wspthsn 28793 | The set of simple paths of... |
| iswspthn 28794 | An element of the set of s... |
| wspthnp 28795 | Properties of a set being ... |
| wwlksnon 28796 | The set of walks of a fixe... |
| wspthsnon 28797 | The set of simple paths of... |
| iswwlksnon 28798 | The set of walks of a fixe... |
| wwlksnon0 28799 | Sufficient conditions for ... |
| wwlksonvtx 28800 | If a word ` W ` represents... |
| iswspthsnon 28801 | The set of simple paths of... |
| wwlknon 28802 | An element of the set of w... |
| wspthnon 28803 | An element of the set of s... |
| wspthnonp 28804 | Properties of a set being ... |
| wspthneq1eq2 28805 | Two simple paths with iden... |
| wwlksn0s 28806 | The set of all walks as wo... |
| wwlkssswrd 28807 | Walks (represented by word... |
| wwlksn0 28808 | A walk of length 0 is repr... |
| 0enwwlksnge1 28809 | In graphs without edges, t... |
| wwlkswwlksn 28810 | A walk of a fixed length a... |
| wwlkssswwlksn 28811 | The walks of a fixed lengt... |
| wlkiswwlks1 28812 | The sequence of vertices i... |
| wlklnwwlkln1 28813 | The sequence of vertices i... |
| wlkiswwlks2lem1 28814 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 28815 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 28816 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 28817 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 28818 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 28819 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 28820 | A walk as word corresponds... |
| wlkiswwlks 28821 | A walk as word corresponds... |
| wlkiswwlksupgr2 28822 | A walk as word corresponds... |
| wlkiswwlkupgr 28823 | A walk as word corresponds... |
| wlkswwlksf1o 28824 | The mapping of (ordinary) ... |
| wlkswwlksen 28825 | The set of walks as words ... |
| wwlksm1edg 28826 | Removing the trailing edge... |
| wlklnwwlkln2lem 28827 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 28828 | A walk of length ` N ` as ... |
| wlklnwwlkn 28829 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 28830 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 28831 | A walk of length ` N ` as ... |
| wlknewwlksn 28832 | If a walk in a pseudograph... |
| wlknwwlksnbij 28833 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 28834 | In a simple pseudograph, t... |
| wlknwwlksneqs 28835 | The set of walks of a fixe... |
| wwlkseq 28836 | Equality of two walks (as ... |
| wwlksnred 28837 | Reduction of a walk (as wo... |
| wwlksnext 28838 | Extension of a walk (as wo... |
| wwlksnextbi 28839 | Extension of a walk (as wo... |
| wwlksnredwwlkn 28840 | For each walk (as word) of... |
| wwlksnredwwlkn0 28841 | For each walk (as word) of... |
| wwlksnextwrd 28842 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 28843 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 28844 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 28845 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 28846 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 28847 | There is a bijection betwe... |
| wwlksnexthasheq 28848 | The number of the extensio... |
| disjxwwlksn 28849 | Sets of walks (as words) e... |
| wwlksnndef 28850 | Conditions for ` WWalksN `... |
| wwlksnfi 28851 | The number of walks repres... |
| wlksnfi 28852 | The number of walks of fix... |
| wlksnwwlknvbij 28853 | There is a bijection betwe... |
| wwlksnextproplem1 28854 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 28855 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 28856 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 28857 | Adding additional properti... |
| disjxwwlkn 28858 | Sets of walks (as words) e... |
| hashwwlksnext 28859 | Number of walks (as words)... |
| wwlksnwwlksnon 28860 | A walk of fixed length is ... |
| wspthsnwspthsnon 28861 | A simple path of fixed len... |
| wspthsnonn0vne 28862 | If the set of simple paths... |
| wspthsswwlkn 28863 | The set of simple paths of... |
| wspthnfi 28864 | In a finite graph, the set... |
| wwlksnonfi 28865 | In a finite graph, the set... |
| wspthsswwlknon 28866 | The set of simple paths of... |
| wspthnonfi 28867 | In a finite graph, the set... |
| wspniunwspnon 28868 | The set of nonempty simple... |
| wspn0 28869 | If there are no vertices, ... |
| 2wlkdlem1 28870 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 28871 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 28872 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 28873 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 28874 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 28875 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 28876 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 28877 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 28878 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 28879 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 28880 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 28881 | Construction of a walk fro... |
| 2wlkond 28882 | A walk of length 2 from on... |
| 2trld 28883 | Construction of a trail fr... |
| 2trlond 28884 | A trail of length 2 from o... |
| 2pthd 28885 | A path of length 2 from on... |
| 2spthd 28886 | A simple path of length 2 ... |
| 2pthond 28887 | A simple path of length 2 ... |
| 2pthon3v 28888 | For a vertex adjacent to t... |
| umgr2adedgwlklem 28889 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 28890 | In a multigraph, two adjac... |
| umgr2adedgwlkon 28891 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 28892 | Alternate proof for ~ umgr... |
| umgr2adedgspth 28893 | In a multigraph, two adjac... |
| umgr2wlk 28894 | In a multigraph, there is ... |
| umgr2wlkon 28895 | For each pair of adjacent ... |
| elwwlks2s3 28896 | A walk of length 2 as word... |
| midwwlks2s3 28897 | There is a vertex between ... |
| wwlks2onv 28898 | If a length 3 string repre... |
| elwwlks2ons3im 28899 | A walk as word of length 2... |
| elwwlks2ons3 28900 | For each walk of length 2 ... |
| s3wwlks2on 28901 | A length 3 string which re... |
| umgrwwlks2on 28902 | A walk of length 2 between... |
| wwlks2onsym 28903 | There is a walk of length ... |
| elwwlks2on 28904 | A walk of length 2 between... |
| elwspths2on 28905 | A simple path of length 2 ... |
| wpthswwlks2on 28906 | For two different vertices... |
| 2wspdisj 28907 | All simple paths of length... |
| 2wspiundisj 28908 | All simple paths of length... |
| usgr2wspthons3 28909 | A simple path of length 2 ... |
| usgr2wspthon 28910 | A simple path of length 2 ... |
| elwwlks2 28911 | A walk of length 2 between... |
| elwspths2spth 28912 | A simple path of length 2 ... |
| rusgrnumwwlkl1 28913 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 28914 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 28915 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 28916 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 28917 | Induction base 1 for ~ rus... |
| rusgr0edg 28918 | Special case for graphs wi... |
| rusgrnumwwlks 28919 | Induction step for ~ rusgr... |
| rusgrnumwwlk 28920 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 28921 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 28922 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 28923 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 28924 | In a ` K `-regular graph, ... |
| clwwlk 28927 | The set of closed walks (i... |
| isclwwlk 28928 | Properties of a word to re... |
| clwwlkbp 28929 | Basic properties of a clos... |
| clwwlkgt0 28930 | There is no empty closed w... |
| clwwlksswrd 28931 | Closed walks (represented ... |
| clwwlk1loop 28932 | A closed walk of length 1 ... |
| clwwlkccatlem 28933 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 28934 | The concatenation of two w... |
| umgrclwwlkge2 28935 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 28936 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 28937 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 28938 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 28939 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 28940 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 28941 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 28942 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 28943 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 28944 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 28945 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 28946 | A closed walk as word of l... |
| clwlkclwwlk2 28947 | A closed walk corresponds ... |
| clwlkclwwlkflem 28948 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 28949 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 28950 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 28951 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 28952 | ` F ` is a function from t... |
| clwlkclwwlkfo 28953 | ` F ` is a function from t... |
| clwlkclwwlkf1 28954 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 28955 | ` F ` is a bijection betwe... |
| clwlkclwwlken 28956 | The set of the nonempty cl... |
| clwwisshclwwslemlem 28957 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 28958 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 28959 | Cyclically shifting a clos... |
| clwwisshclwwsn 28960 | Cyclically shifting a clos... |
| erclwwlkrel 28961 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 28962 | Two classes are equivalent... |
| erclwwlkeqlen 28963 | If two classes are equival... |
| erclwwlkref 28964 | ` .~ ` is a reflexive rela... |
| erclwwlksym 28965 | ` .~ ` is a symmetric rela... |
| erclwwlktr 28966 | ` .~ ` is a transitive rel... |
| erclwwlk 28967 | ` .~ ` is an equivalence r... |
| clwwlkn 28970 | The set of closed walks of... |
| isclwwlkn 28971 | A word over the set of ver... |
| clwwlkn0 28972 | There is no closed walk of... |
| clwwlkneq0 28973 | Sufficient conditions for ... |
| clwwlkclwwlkn 28974 | A closed walk of a fixed l... |
| clwwlksclwwlkn 28975 | The closed walks of a fixe... |
| clwwlknlen 28976 | The length of a word repre... |
| clwwlknnn 28977 | The length of a closed wal... |
| clwwlknwrd 28978 | A closed walk of a fixed l... |
| clwwlknbp 28979 | Basic properties of a clos... |
| isclwwlknx 28980 | Characterization of a word... |
| clwwlknp 28981 | Properties of a set being ... |
| clwwlknwwlksn 28982 | A word representing a clos... |
| clwwlknlbonbgr1 28983 | The last but one vertex in... |
| clwwlkinwwlk 28984 | If the initial vertex of a... |
| clwwlkn1 28985 | A closed walk of length 1 ... |
| loopclwwlkn1b 28986 | The singleton word consist... |
| clwwlkn1loopb 28987 | A word represents a closed... |
| clwwlkn2 28988 | A closed walk of length 2 ... |
| clwwlknfi 28989 | If there is only a finite ... |
| clwwlkel 28990 | Obtaining a closed walk (a... |
| clwwlkf 28991 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 28992 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 28993 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 28994 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 28995 | F is a 1-1 onto function, ... |
| clwwlken 28996 | The set of closed walks of... |
| clwwlknwwlkncl 28997 | Obtaining a closed walk (a... |
| clwwlkwwlksb 28998 | A nonempty word over verti... |
| clwwlknwwlksnb 28999 | A word over vertices repre... |
| clwwlkext2edg 29000 | If a word concatenated wit... |
| wwlksext2clwwlk 29001 | If a word represents a wal... |
| wwlksubclwwlk 29002 | Any prefix of a word repre... |
| clwwnisshclwwsn 29003 | Cyclically shifting a clos... |
| eleclclwwlknlem1 29004 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 29005 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 29006 | The set of cyclical shifts... |
| clwwlknccat 29007 | The concatenation of two w... |
| umgr2cwwk2dif 29008 | If a word represents a clo... |
| umgr2cwwkdifex 29009 | If a word represents a clo... |
| erclwwlknrel 29010 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 29011 | Two classes are equivalent... |
| erclwwlkneqlen 29012 | If two classes are equival... |
| erclwwlknref 29013 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 29014 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 29015 | ` .~ ` is a transitive rel... |
| erclwwlkn 29016 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 29017 | The quotient set of the se... |
| hashclwwlkn0 29018 | The number of closed walks... |
| eclclwwlkn1 29019 | An equivalence class accor... |
| eleclclwwlkn 29020 | A member of an equivalence... |
| hashecclwwlkn1 29021 | The size of every equivale... |
| umgrhashecclwwlk 29022 | The size of every equivale... |
| fusgrhashclwwlkn 29023 | The size of the set of clo... |
| clwwlkndivn 29024 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 29025 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 29026 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 29027 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 29028 | There is a one-to-one onto... |
| clwlkssizeeq 29029 | The size of the set of clo... |
| clwlksndivn 29030 | The size of the set of clo... |
| clwwlknonmpo 29033 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 29034 | The set of closed walks on... |
| isclwwlknon 29035 | A word over the set of ver... |
| clwwlk0on0 29036 | There is no word over the ... |
| clwwlknon0 29037 | Sufficient conditions for ... |
| clwwlknonfin 29038 | In a finite graph ` G ` , ... |
| clwwlknonel 29039 | Characterization of a word... |
| clwwlknonccat 29040 | The concatenation of two w... |
| clwwlknon1 29041 | The set of closed walks on... |
| clwwlknon1loop 29042 | If there is a loop at vert... |
| clwwlknon1nloop 29043 | If there is no loop at ver... |
| clwwlknon1sn 29044 | The set of (closed) walks ... |
| clwwlknon1le1 29045 | There is at most one (clos... |
| clwwlknon2 29046 | The set of closed walks on... |
| clwwlknon2x 29047 | The set of closed walks on... |
| s2elclwwlknon2 29048 | Sufficient conditions of a... |
| clwwlknon2num 29049 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 29050 | A word over vertices repre... |
| clwwlknonex2lem1 29051 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 29052 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 29053 | Extending a closed walk ` ... |
| clwwlknonex2e 29054 | Extending a closed walk ` ... |
| clwwlknondisj 29055 | The sets of closed walks o... |
| clwwlknun 29056 | The set of closed walks of... |
| clwwlkvbij 29057 | There is a bijection betwe... |
| 0ewlk 29058 | The empty set (empty seque... |
| 1ewlk 29059 | A sequence of 1 edge is an... |
| 0wlk 29060 | A pair of an empty set (of... |
| is0wlk 29061 | A pair of an empty set (of... |
| 0wlkonlem1 29062 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 29063 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 29064 | A walk of length 0 from a ... |
| 0wlkons1 29065 | A walk of length 0 from a ... |
| 0trl 29066 | A pair of an empty set (of... |
| is0trl 29067 | A pair of an empty set (of... |
| 0trlon 29068 | A trail of length 0 from a... |
| 0pth 29069 | A pair of an empty set (of... |
| 0spth 29070 | A pair of an empty set (of... |
| 0pthon 29071 | A path of length 0 from a ... |
| 0pthon1 29072 | A path of length 0 from a ... |
| 0pthonv 29073 | For each vertex there is a... |
| 0clwlk 29074 | A pair of an empty set (of... |
| 0clwlkv 29075 | Any vertex (more precisely... |
| 0clwlk0 29076 | There is no closed walk in... |
| 0crct 29077 | A pair of an empty set (of... |
| 0cycl 29078 | A pair of an empty set (of... |
| 1pthdlem1 29079 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 29080 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 29081 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 29082 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 29083 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 29084 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 29085 | In a graph with two vertic... |
| 1trld 29086 | In a graph with two vertic... |
| 1pthd 29087 | In a graph with two vertic... |
| 1pthond 29088 | In a graph with two vertic... |
| upgr1wlkdlem1 29089 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 29090 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 29091 | In a pseudograph with two ... |
| upgr1trld 29092 | In a pseudograph with two ... |
| upgr1pthd 29093 | In a pseudograph with two ... |
| upgr1pthond 29094 | In a pseudograph with two ... |
| lppthon 29095 | A loop (which is an edge a... |
| lp1cycl 29096 | A loop (which is an edge a... |
| 1pthon2v 29097 | For each pair of adjacent ... |
| 1pthon2ve 29098 | For each pair of adjacent ... |
| wlk2v2elem1 29099 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 29100 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 29101 | In a graph with two vertic... |
| ntrl2v2e 29102 | A walk which is not a trai... |
| 3wlkdlem1 29103 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 29104 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 29105 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 29106 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 29107 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 29108 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 29109 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 29110 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 29111 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 29112 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 29113 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 29114 | Construction of a walk fro... |
| 3wlkond 29115 | A walk of length 3 from on... |
| 3trld 29116 | Construction of a trail fr... |
| 3trlond 29117 | A trail of length 3 from o... |
| 3pthd 29118 | A path of length 3 from on... |
| 3pthond 29119 | A path of length 3 from on... |
| 3spthd 29120 | A simple path of length 3 ... |
| 3spthond 29121 | A simple path of length 3 ... |
| 3cycld 29122 | Construction of a 3-cycle ... |
| 3cyclpd 29123 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 29124 | If there is a cycle of len... |
| uhgr3cyclexlem 29125 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 29126 | If there are three differe... |
| umgr3cyclex 29127 | If there are three (differ... |
| umgr3v3e3cycl 29128 | If and only if there is a ... |
| upgr4cycl4dv4e 29129 | If there is a cycle of len... |
| dfconngr1 29132 | Alternative definition of ... |
| isconngr 29133 | The property of being a co... |
| isconngr1 29134 | The property of being a co... |
| cusconngr 29135 | A complete hypergraph is c... |
| 0conngr 29136 | A graph without vertices i... |
| 0vconngr 29137 | A graph without vertices i... |
| 1conngr 29138 | A graph with (at most) one... |
| conngrv2edg 29139 | A vertex in a connected gr... |
| vdn0conngrumgrv2 29140 | A vertex in a connected mu... |
| releupth 29143 | The set ` ( EulerPaths `` ... |
| eupths 29144 | The Eulerian paths on the ... |
| iseupth 29145 | The property " ` <. F , P ... |
| iseupthf1o 29146 | The property " ` <. F , P ... |
| eupthi 29147 | Properties of an Eulerian ... |
| eupthf1o 29148 | The ` F ` function in an E... |
| eupthfi 29149 | Any graph with an Eulerian... |
| eupthseg 29150 | The ` N ` -th edge in an e... |
| upgriseupth 29151 | The property " ` <. F , P ... |
| upgreupthi 29152 | Properties of an Eulerian ... |
| upgreupthseg 29153 | The ` N ` -th edge in an e... |
| eupthcl 29154 | An Eulerian path has lengt... |
| eupthistrl 29155 | An Eulerian path is a trai... |
| eupthiswlk 29156 | An Eulerian path is a walk... |
| eupthpf 29157 | The ` P ` function in an E... |
| eupth0 29158 | There is an Eulerian path ... |
| eupthres 29159 | The restriction ` <. H , Q... |
| eupthp1 29160 | Append one path segment to... |
| eupth2eucrct 29161 | Append one path segment to... |
| eupth2lem1 29162 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 29163 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 29164 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 29165 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 29166 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 29167 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 29168 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 29169 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 29170 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 29171 | Formerly part of proof of ... |
| eupth2lem3lem1 29172 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 29173 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 29174 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 29175 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 29176 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 29177 | Formerly part of proof of ... |
| eupth2lem3lem7 29178 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 29179 | Formerly part of proof of ... |
| eupth2lem3 29180 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 29181 | Lemma for ~ eupth2 (induct... |
| eupth2lems 29182 | Lemma for ~ eupth2 (induct... |
| eupth2 29183 | The only vertices of odd d... |
| eulerpathpr 29184 | A graph with an Eulerian p... |
| eulerpath 29185 | A pseudograph with an Eule... |
| eulercrct 29186 | A pseudograph with an Eule... |
| eucrctshift 29187 | Cyclically shifting the in... |
| eucrct2eupth1 29188 | Removing one edge ` ( I ``... |
| eucrct2eupth 29189 | Removing one edge ` ( I ``... |
| konigsbergvtx 29190 | The set of vertices of the... |
| konigsbergiedg 29191 | The indexed edges of the K... |
| konigsbergiedgw 29192 | The indexed edges of the K... |
| konigsbergssiedgwpr 29193 | Each subset of the indexed... |
| konigsbergssiedgw 29194 | Each subset of the indexed... |
| konigsbergumgr 29195 | The Königsberg graph ... |
| konigsberglem1 29196 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 29197 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 29198 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 29199 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 29200 | Lemma 5 for ~ konigsberg :... |
| konigsberg 29201 | The Königsberg Bridge... |
| isfrgr 29204 | The property of being a fr... |
| frgrusgr 29205 | A friendship graph is a si... |
| frgr0v 29206 | Any null graph (set with n... |
| frgr0vb 29207 | Any null graph (without ve... |
| frgruhgr0v 29208 | Any null graph (without ve... |
| frgr0 29209 | The null graph (graph with... |
| frcond1 29210 | The friendship condition: ... |
| frcond2 29211 | The friendship condition: ... |
| frgreu 29212 | Variant of ~ frcond2 : An... |
| frcond3 29213 | The friendship condition, ... |
| frcond4 29214 | The friendship condition, ... |
| frgr1v 29215 | Any graph with (at most) o... |
| nfrgr2v 29216 | Any graph with two (differ... |
| frgr3vlem1 29217 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 29218 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 29219 | Any graph with three verti... |
| 1vwmgr 29220 | Every graph with one verte... |
| 3vfriswmgrlem 29221 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 29222 | Every friendship graph wit... |
| 1to2vfriswmgr 29223 | Every friendship graph wit... |
| 1to3vfriswmgr 29224 | Every friendship graph wit... |
| 1to3vfriendship 29225 | The friendship theorem for... |
| 2pthfrgrrn 29226 | Between any two (different... |
| 2pthfrgrrn2 29227 | Between any two (different... |
| 2pthfrgr 29228 | Between any two (different... |
| 3cyclfrgrrn1 29229 | Every vertex in a friendsh... |
| 3cyclfrgrrn 29230 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 29231 | Every vertex in a friendsh... |
| 3cyclfrgr 29232 | Every vertex in a friendsh... |
| 4cycl2v2nb 29233 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 29234 | In a 4-cycle, two distinct... |
| n4cyclfrgr 29235 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 29236 | A graph with a 4-cycle is ... |
| frgrnbnb 29237 | If two neighbors ` U ` and... |
| frgrconngr 29238 | A friendship graph is conn... |
| vdgn0frgrv2 29239 | A vertex in a friendship g... |
| vdgn1frgrv2 29240 | Any vertex in a friendship... |
| vdgn1frgrv3 29241 | Any vertex in a friendship... |
| vdgfrgrgt2 29242 | Any vertex in a friendship... |
| frgrncvvdeqlem1 29243 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 29244 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 29245 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 29246 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 29247 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 29248 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 29249 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 29250 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 29251 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 29252 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 29253 | In a friendship graph, two... |
| frgrwopreglem4a 29254 | In a friendship graph any ... |
| frgrwopreglem5a 29255 | If a friendship graph has ... |
| frgrwopreglem1 29256 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 29257 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 29258 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 29259 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 29260 | According to statement 5 i... |
| frgrwopregbsn 29261 | According to statement 5 i... |
| frgrwopreg1 29262 | According to statement 5 i... |
| frgrwopreg2 29263 | According to statement 5 i... |
| frgrwopreglem5lem 29264 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 29265 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 29266 | Alternate direct proof of ... |
| frgrwopreg 29267 | In a friendship graph ther... |
| frgrregorufr0 29268 | In a friendship graph ther... |
| frgrregorufr 29269 | If there is a vertex havin... |
| frgrregorufrg 29270 | If there is a vertex havin... |
| frgr2wwlkeu 29271 | For two different vertices... |
| frgr2wwlkn0 29272 | In a friendship graph, the... |
| frgr2wwlk1 29273 | In a friendship graph, the... |
| frgr2wsp1 29274 | In a friendship graph, the... |
| frgr2wwlkeqm 29275 | If there is a (simple) pat... |
| frgrhash2wsp 29276 | The number of simple paths... |
| fusgreg2wsplem 29277 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 29278 | The set of paths of length... |
| fusgreghash2wspv 29279 | According to statement 7 i... |
| fusgreg2wsp 29280 | In a finite simple graph, ... |
| 2wspmdisj 29281 | The sets of paths of lengt... |
| fusgreghash2wsp 29282 | In a finite k-regular grap... |
| frrusgrord0lem 29283 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 29284 | If a nonempty finite frien... |
| frrusgrord 29285 | If a nonempty finite frien... |
| numclwwlk2lem1lem 29286 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 29287 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 29288 | If the initial vertex of a... |
| clwwnonrepclwwnon 29289 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 29290 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 29291 | Value of operation ` C ` ,... |
| 2clwwlk2 29292 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 29293 | Characterization of an ele... |
| 2clwwlk2clwwlk 29294 | An element of the value of... |
| numclwwlk1lem2foalem 29295 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 29296 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 29297 | Characterization of an ele... |
| numclwwlk1lem2foa 29298 | Going forth and back from ... |
| numclwwlk1lem2f 29299 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 29300 | Value of the function ` T ... |
| numclwwlk1lem2f1 29301 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 29302 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 29303 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 29304 | The set of double loops of... |
| numclwwlk1 29305 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 29306 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 29307 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 29308 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 29309 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 29310 | The sets of the two repres... |
| wlkl0 29311 | There is exactly one walk ... |
| clwlknon2num 29312 | There are k walks of lengt... |
| numclwlk1lem1 29313 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 29314 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 29315 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 29316 | Value of operation ` H ` ,... |
| numclwwlkovh 29317 | Value of operation ` H ` ,... |
| numclwwlkovq 29318 | Value of operation ` Q ` ,... |
| numclwwlkqhash 29319 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 29320 | In a friendship graph, for... |
| numclwlk2lem2f 29321 | ` R ` is a function mappin... |
| numclwlk2lem2fv 29322 | Value of the function ` R ... |
| numclwlk2lem2f1o 29323 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 29324 | In a friendship graph, the... |
| numclwwlk2 29325 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 29326 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 29327 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 29328 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 29329 | Statement 12 in [Huneke] p... |
| numclwwlk4 29330 | The total number of closed... |
| numclwwlk5lem 29331 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 29332 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 29333 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 29334 | For a prime divisor ` P ` ... |
| numclwwlk7 29335 | Statement 14 in [Huneke] p... |
| numclwwlk8 29336 | The size of the set of clo... |
| frgrreggt1 29337 | If a finite nonempty frien... |
| frgrreg 29338 | If a finite nonempty frien... |
| frgrregord013 29339 | If a finite friendship gra... |
| frgrregord13 29340 | If a nonempty finite frien... |
| frgrogt3nreg 29341 | If a finite friendship gra... |
| friendshipgt3 29342 | The friendship theorem for... |
| friendship 29343 | The friendship theorem: I... |
| conventions 29344 |
H... |
| conventions-labels 29345 |
... |
| conventions-comments 29346 |
... |
| natded 29347 | Here are typical n... |
| ex-natded5.2 29348 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 29349 | A more efficient proof of ... |
| ex-natded5.2i 29350 | The same as ~ ex-natded5.2... |
| ex-natded5.3 29351 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 29352 | A more efficient proof of ... |
| ex-natded5.3i 29353 | The same as ~ ex-natded5.3... |
| ex-natded5.5 29354 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 29355 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 29356 | A more efficient proof of ... |
| ex-natded5.8 29357 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 29358 | A more efficient proof of ... |
| ex-natded5.13 29359 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 29360 | A more efficient proof of ... |
| ex-natded9.20 29361 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 29362 | A more efficient proof of ... |
| ex-natded9.26 29363 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 29364 | A more efficient proof of ... |
| ex-or 29365 | Example for ~ df-or . Exa... |
| ex-an 29366 | Example for ~ df-an . Exa... |
| ex-dif 29367 | Example for ~ df-dif . Ex... |
| ex-un 29368 | Example for ~ df-un . Exa... |
| ex-in 29369 | Example for ~ df-in . Exa... |
| ex-uni 29370 | Example for ~ df-uni . Ex... |
| ex-ss 29371 | Example for ~ df-ss . Exa... |
| ex-pss 29372 | Example for ~ df-pss . Ex... |
| ex-pw 29373 | Example for ~ df-pw . Exa... |
| ex-pr 29374 | Example for ~ df-pr . (Co... |
| ex-br 29375 | Example for ~ df-br . Exa... |
| ex-opab 29376 | Example for ~ df-opab . E... |
| ex-eprel 29377 | Example for ~ df-eprel . ... |
| ex-id 29378 | Example for ~ df-id . Exa... |
| ex-po 29379 | Example for ~ df-po . Exa... |
| ex-xp 29380 | Example for ~ df-xp . Exa... |
| ex-cnv 29381 | Example for ~ df-cnv . Ex... |
| ex-co 29382 | Example for ~ df-co . Exa... |
| ex-dm 29383 | Example for ~ df-dm . Exa... |
| ex-rn 29384 | Example for ~ df-rn . Exa... |
| ex-res 29385 | Example for ~ df-res . Ex... |
| ex-ima 29386 | Example for ~ df-ima . Ex... |
| ex-fv 29387 | Example for ~ df-fv . Exa... |
| ex-1st 29388 | Example for ~ df-1st . Ex... |
| ex-2nd 29389 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 29390 | Example for ~ df-dec , 100... |
| ex-fl 29391 | Example for ~ df-fl . Exa... |
| ex-ceil 29392 | Example for ~ df-ceil . (... |
| ex-mod 29393 | Example for ~ df-mod . (C... |
| ex-exp 29394 | Example for ~ df-exp . (C... |
| ex-fac 29395 | Example for ~ df-fac . (C... |
| ex-bc 29396 | Example for ~ df-bc . (Co... |
| ex-hash 29397 | Example for ~ df-hash . (... |
| ex-sqrt 29398 | Example for ~ df-sqrt . (... |
| ex-abs 29399 | Example for ~ df-abs . (C... |
| ex-dvds 29400 | Example for ~ df-dvds : 3 ... |
| ex-gcd 29401 | Example for ~ df-gcd . (C... |
| ex-lcm 29402 | Example for ~ df-lcm . (C... |
| ex-prmo 29403 | Example for ~ df-prmo : ` ... |
| aevdemo 29404 | Proof illustrating the com... |
| ex-ind-dvds 29405 | Example of a proof by indu... |
| ex-fpar 29406 | Formalized example provide... |
| avril1 29407 | Poisson d'Avril's Theorem.... |
| 2bornot2b 29408 | The law of excluded middle... |
| helloworld 29409 | The classic "Hello world" ... |
| 1p1e2apr1 29410 | One plus one equals two. ... |
| eqid1 29411 | Law of identity (reflexivi... |
| 1div0apr 29412 | Division by zero is forbid... |
| topnfbey 29413 | Nothing seems to be imposs... |
| 9p10ne21 29414 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 29415 | 9 + 10 equals 21. This as... |
| isplig 29418 | The predicate "is a planar... |
| ispligb 29419 | The predicate "is a planar... |
| tncp 29420 | In any planar incidence ge... |
| l2p 29421 | For any line in a planar i... |
| lpni 29422 | For any line in a planar i... |
| nsnlplig 29423 | There is no "one-point lin... |
| nsnlpligALT 29424 | Alternate version of ~ nsn... |
| n0lplig 29425 | There is no "empty line" i... |
| n0lpligALT 29426 | Alternate version of ~ n0l... |
| eulplig 29427 | Through two distinct point... |
| pliguhgr 29428 | Any planar incidence geome... |
| dummylink 29429 | Alias for ~ a1ii that may ... |
| id1 29430 | Alias for ~ idALT that may... |
| isgrpo 29439 | The predicate "is a group ... |
| isgrpoi 29440 | Properties that determine ... |
| grpofo 29441 | A group operation maps ont... |
| grpocl 29442 | Closure law for a group op... |
| grpolidinv 29443 | A group has a left identit... |
| grpon0 29444 | The base set of a group is... |
| grpoass 29445 | A group operation is assoc... |
| grpoidinvlem1 29446 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 29447 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 29448 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 29449 | Lemma for ~ grpoidinv . (... |
| grpoidinv 29450 | A group has a left and rig... |
| grpoideu 29451 | The left identity element ... |
| grporndm 29452 | A group's range in terms o... |
| 0ngrp 29453 | The empty set is not a gro... |
| gidval 29454 | The value of the identity ... |
| grpoidval 29455 | Lemma for ~ grpoidcl and o... |
| grpoidcl 29456 | The identity element of a ... |
| grpoidinv2 29457 | A group's properties using... |
| grpolid 29458 | The identity element of a ... |
| grporid 29459 | The identity element of a ... |
| grporcan 29460 | Right cancellation law for... |
| grpoinveu 29461 | The left inverse element o... |
| grpoid 29462 | Two ways of saying that an... |
| grporn 29463 | The range of a group opera... |
| grpoinvfval 29464 | The inverse function of a ... |
| grpoinvval 29465 | The inverse of a group ele... |
| grpoinvcl 29466 | A group element's inverse ... |
| grpoinv 29467 | The properties of a group ... |
| grpolinv 29468 | The left inverse of a grou... |
| grporinv 29469 | The right inverse of a gro... |
| grpoinvid1 29470 | The inverse of a group ele... |
| grpoinvid2 29471 | The inverse of a group ele... |
| grpolcan 29472 | Left cancellation law for ... |
| grpo2inv 29473 | Double inverse law for gro... |
| grpoinvf 29474 | Mapping of the inverse fun... |
| grpoinvop 29475 | The inverse of the group o... |
| grpodivfval 29476 | Group division (or subtrac... |
| grpodivval 29477 | Group division (or subtrac... |
| grpodivinv 29478 | Group division by an inver... |
| grpoinvdiv 29479 | Inverse of a group divisio... |
| grpodivf 29480 | Mapping for group division... |
| grpodivcl 29481 | Closure of group division ... |
| grpodivdiv 29482 | Double group division. (C... |
| grpomuldivass 29483 | Associative-type law for m... |
| grpodivid 29484 | Division of a group member... |
| grponpcan 29485 | Cancellation law for group... |
| isablo 29488 | The predicate "is an Abeli... |
| ablogrpo 29489 | An Abelian group operation... |
| ablocom 29490 | An Abelian group operation... |
| ablo32 29491 | Commutative/associative la... |
| ablo4 29492 | Commutative/associative la... |
| isabloi 29493 | Properties that determine ... |
| ablomuldiv 29494 | Law for group multiplicati... |
| ablodivdiv 29495 | Law for double group divis... |
| ablodivdiv4 29496 | Law for double group divis... |
| ablodiv32 29497 | Swap the second and third ... |
| ablonncan 29498 | Cancellation law for group... |
| ablonnncan1 29499 | Cancellation law for group... |
| vcrel 29502 | The class of all complex v... |
| vciOLD 29503 | Obsolete version of ~ cvsi... |
| vcsm 29504 | Functionality of th scalar... |
| vccl 29505 | Closure of the scalar prod... |
| vcidOLD 29506 | Identity element for the s... |
| vcdi 29507 | Distributive law for the s... |
| vcdir 29508 | Distributive law for the s... |
| vcass 29509 | Associative law for the sc... |
| vc2OLD 29510 | A vector plus itself is tw... |
| vcablo 29511 | Vector addition is an Abel... |
| vcgrp 29512 | Vector addition is a group... |
| vclcan 29513 | Left cancellation law for ... |
| vczcl 29514 | The zero vector is a vecto... |
| vc0rid 29515 | The zero vector is a right... |
| vc0 29516 | Zero times a vector is the... |
| vcz 29517 | Anything times the zero ve... |
| vcm 29518 | Minus 1 times a vector is ... |
| isvclem 29519 | Lemma for ~ isvcOLD . (Co... |
| vcex 29520 | The components of a comple... |
| isvcOLD 29521 | The predicate "is a comple... |
| isvciOLD 29522 | Properties that determine ... |
| cnaddabloOLD 29523 | Obsolete version of ~ cnad... |
| cnidOLD 29524 | Obsolete version of ~ cnad... |
| cncvcOLD 29525 | Obsolete version of ~ cncv... |
| nvss 29535 | Structure of the class of ... |
| nvvcop 29536 | A normed complex vector sp... |
| nvrel 29544 | The class of all normed co... |
| vafval 29545 | Value of the function for ... |
| bafval 29546 | Value of the function for ... |
| smfval 29547 | Value of the function for ... |
| 0vfval 29548 | Value of the function for ... |
| nmcvfval 29549 | Value of the norm function... |
| nvop2 29550 | A normed complex vector sp... |
| nvvop 29551 | The vector space component... |
| isnvlem 29552 | Lemma for ~ isnv . (Contr... |
| nvex 29553 | The components of a normed... |
| isnv 29554 | The predicate "is a normed... |
| isnvi 29555 | Properties that determine ... |
| nvi 29556 | The properties of a normed... |
| nvvc 29557 | The vector space component... |
| nvablo 29558 | The vector addition operat... |
| nvgrp 29559 | The vector addition operat... |
| nvgf 29560 | Mapping for the vector add... |
| nvsf 29561 | Mapping for the scalar mul... |
| nvgcl 29562 | Closure law for the vector... |
| nvcom 29563 | The vector addition (group... |
| nvass 29564 | The vector addition (group... |
| nvadd32 29565 | Commutative/associative la... |
| nvrcan 29566 | Right cancellation law for... |
| nvadd4 29567 | Rearrangement of 4 terms i... |
| nvscl 29568 | Closure law for the scalar... |
| nvsid 29569 | Identity element for the s... |
| nvsass 29570 | Associative law for the sc... |
| nvscom 29571 | Commutative law for the sc... |
| nvdi 29572 | Distributive law for the s... |
| nvdir 29573 | Distributive law for the s... |
| nv2 29574 | A vector plus itself is tw... |
| vsfval 29575 | Value of the function for ... |
| nvzcl 29576 | Closure law for the zero v... |
| nv0rid 29577 | The zero vector is a right... |
| nv0lid 29578 | The zero vector is a left ... |
| nv0 29579 | Zero times a vector is the... |
| nvsz 29580 | Anything times the zero ve... |
| nvinv 29581 | Minus 1 times a vector is ... |
| nvinvfval 29582 | Function for the negative ... |
| nvm 29583 | Vector subtraction in term... |
| nvmval 29584 | Value of vector subtractio... |
| nvmval2 29585 | Value of vector subtractio... |
| nvmfval 29586 | Value of the function for ... |
| nvmf 29587 | Mapping for the vector sub... |
| nvmcl 29588 | Closure law for the vector... |
| nvnnncan1 29589 | Cancellation law for vecto... |
| nvmdi 29590 | Distributive law for scala... |
| nvnegneg 29591 | Double negative of a vecto... |
| nvmul0or 29592 | If a scalar product is zer... |
| nvrinv 29593 | A vector minus itself. (C... |
| nvlinv 29594 | Minus a vector plus itself... |
| nvpncan2 29595 | Cancellation law for vecto... |
| nvpncan 29596 | Cancellation law for vecto... |
| nvaddsub 29597 | Commutative/associative la... |
| nvnpcan 29598 | Cancellation law for a nor... |
| nvaddsub4 29599 | Rearrangement of 4 terms i... |
| nvmeq0 29600 | The difference between two... |
| nvmid 29601 | A vector minus itself is t... |
| nvf 29602 | Mapping for the norm funct... |
| nvcl 29603 | The norm of a normed compl... |
| nvcli 29604 | The norm of a normed compl... |
| nvs 29605 | Proportionality property o... |
| nvsge0 29606 | The norm of a scalar produ... |
| nvm1 29607 | The norm of the negative o... |
| nvdif 29608 | The norm of the difference... |
| nvpi 29609 | The norm of a vector plus ... |
| nvz0 29610 | The norm of a zero vector ... |
| nvz 29611 | The norm of a vector is ze... |
| nvtri 29612 | Triangle inequality for th... |
| nvmtri 29613 | Triangle inequality for th... |
| nvabs 29614 | Norm difference property o... |
| nvge0 29615 | The norm of a normed compl... |
| nvgt0 29616 | A nonzero norm is positive... |
| nv1 29617 | From any nonzero vector, c... |
| nvop 29618 | A complex inner product sp... |
| cnnv 29619 | The set of complex numbers... |
| cnnvg 29620 | The vector addition (group... |
| cnnvba 29621 | The base set of the normed... |
| cnnvs 29622 | The scalar product operati... |
| cnnvnm 29623 | The norm operation of the ... |
| cnnvm 29624 | The vector subtraction ope... |
| elimnv 29625 | Hypothesis elimination lem... |
| elimnvu 29626 | Hypothesis elimination lem... |
| imsval 29627 | Value of the induced metri... |
| imsdval 29628 | Value of the induced metri... |
| imsdval2 29629 | Value of the distance func... |
| nvnd 29630 | The norm of a normed compl... |
| imsdf 29631 | Mapping for the induced me... |
| imsmetlem 29632 | Lemma for ~ imsmet . (Con... |
| imsmet 29633 | The induced metric of a no... |
| imsxmet 29634 | The induced metric of a no... |
| cnims 29635 | The metric induced on the ... |
| vacn 29636 | Vector addition is jointly... |
| nmcvcn 29637 | The norm of a normed compl... |
| nmcnc 29638 | The norm of a normed compl... |
| smcnlem 29639 | Lemma for ~ smcn . (Contr... |
| smcn 29640 | Scalar multiplication is j... |
| vmcn 29641 | Vector subtraction is join... |
| dipfval 29644 | The inner product function... |
| ipval 29645 | Value of the inner product... |
| ipval2lem2 29646 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 29647 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 29648 | Lemma for ~ ipval3 . (Con... |
| ipval2 29649 | Expansion of the inner pro... |
| 4ipval2 29650 | Four times the inner produ... |
| ipval3 29651 | Expansion of the inner pro... |
| ipidsq 29652 | The inner product of a vec... |
| ipnm 29653 | Norm expressed in terms of... |
| dipcl 29654 | An inner product is a comp... |
| ipf 29655 | Mapping for the inner prod... |
| dipcj 29656 | The complex conjugate of a... |
| ipipcj 29657 | An inner product times its... |
| diporthcom 29658 | Orthogonality (meaning inn... |
| dip0r 29659 | Inner product with a zero ... |
| dip0l 29660 | Inner product with a zero ... |
| ipz 29661 | The inner product of a vec... |
| dipcn 29662 | Inner product is jointly c... |
| sspval 29665 | The set of all subspaces o... |
| isssp 29666 | The predicate "is a subspa... |
| sspid 29667 | A normed complex vector sp... |
| sspnv 29668 | A subspace is a normed com... |
| sspba 29669 | The base set of a subspace... |
| sspg 29670 | Vector addition on a subsp... |
| sspgval 29671 | Vector addition on a subsp... |
| ssps 29672 | Scalar multiplication on a... |
| sspsval 29673 | Scalar multiplication on a... |
| sspmlem 29674 | Lemma for ~ sspm and other... |
| sspmval 29675 | Vector addition on a subsp... |
| sspm 29676 | Vector subtraction on a su... |
| sspz 29677 | The zero vector of a subsp... |
| sspn 29678 | The norm on a subspace is ... |
| sspnval 29679 | The norm on a subspace in ... |
| sspimsval 29680 | The induced metric on a su... |
| sspims 29681 | The induced metric on a su... |
| lnoval 29694 | The set of linear operator... |
| islno 29695 | The predicate "is a linear... |
| lnolin 29696 | Basic linearity property o... |
| lnof 29697 | A linear operator is a map... |
| lno0 29698 | The value of a linear oper... |
| lnocoi 29699 | The composition of two lin... |
| lnoadd 29700 | Addition property of a lin... |
| lnosub 29701 | Subtraction property of a ... |
| lnomul 29702 | Scalar multiplication prop... |
| nvo00 29703 | Two ways to express a zero... |
| nmoofval 29704 | The operator norm function... |
| nmooval 29705 | The operator norm function... |
| nmosetre 29706 | The set in the supremum of... |
| nmosetn0 29707 | The set in the supremum of... |
| nmoxr 29708 | The norm of an operator is... |
| nmooge0 29709 | The norm of an operator is... |
| nmorepnf 29710 | The norm of an operator is... |
| nmoreltpnf 29711 | The norm of any operator i... |
| nmogtmnf 29712 | The norm of an operator is... |
| nmoolb 29713 | A lower bound for an opera... |
| nmoubi 29714 | An upper bound for an oper... |
| nmoub3i 29715 | An upper bound for an oper... |
| nmoub2i 29716 | An upper bound for an oper... |
| nmobndi 29717 | Two ways to express that a... |
| nmounbi 29718 | Two ways two express that ... |
| nmounbseqi 29719 | An unbounded operator dete... |
| nmounbseqiALT 29720 | Alternate shorter proof of... |
| nmobndseqi 29721 | A bounded sequence determi... |
| nmobndseqiALT 29722 | Alternate shorter proof of... |
| bloval 29723 | The class of bounded linea... |
| isblo 29724 | The predicate "is a bounde... |
| isblo2 29725 | The predicate "is a bounde... |
| bloln 29726 | A bounded operator is a li... |
| blof 29727 | A bounded operator is an o... |
| nmblore 29728 | The norm of a bounded oper... |
| 0ofval 29729 | The zero operator between ... |
| 0oval 29730 | Value of the zero operator... |
| 0oo 29731 | The zero operator is an op... |
| 0lno 29732 | The zero operator is linea... |
| nmoo0 29733 | The operator norm of the z... |
| 0blo 29734 | The zero operator is a bou... |
| nmlno0lem 29735 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 29736 | The norm of a linear opera... |
| nmlno0 29737 | The norm of a linear opera... |
| nmlnoubi 29738 | An upper bound for the ope... |
| nmlnogt0 29739 | The norm of a nonzero line... |
| lnon0 29740 | The domain of a nonzero li... |
| nmblolbii 29741 | A lower bound for the norm... |
| nmblolbi 29742 | A lower bound for the norm... |
| isblo3i 29743 | The predicate "is a bounde... |
| blo3i 29744 | Properties that determine ... |
| blometi 29745 | Upper bound for the distan... |
| blocnilem 29746 | Lemma for ~ blocni and ~ l... |
| blocni 29747 | A linear operator is conti... |
| lnocni 29748 | If a linear operator is co... |
| blocn 29749 | A linear operator is conti... |
| blocn2 29750 | A bounded linear operator ... |
| ajfval 29751 | The adjoint function. (Co... |
| hmoval 29752 | The set of Hermitian (self... |
| ishmo 29753 | The predicate "is a hermit... |
| phnv 29756 | Every complex inner produc... |
| phrel 29757 | The class of all complex i... |
| phnvi 29758 | Every complex inner produc... |
| isphg 29759 | The predicate "is a comple... |
| phop 29760 | A complex inner product sp... |
| cncph 29761 | The set of complex numbers... |
| elimph 29762 | Hypothesis elimination lem... |
| elimphu 29763 | Hypothesis elimination lem... |
| isph 29764 | The predicate "is an inner... |
| phpar2 29765 | The parallelogram law for ... |
| phpar 29766 | The parallelogram law for ... |
| ip0i 29767 | A slight variant of Equati... |
| ip1ilem 29768 | Lemma for ~ ip1i . (Contr... |
| ip1i 29769 | Equation 6.47 of [Ponnusam... |
| ip2i 29770 | Equation 6.48 of [Ponnusam... |
| ipdirilem 29771 | Lemma for ~ ipdiri . (Con... |
| ipdiri 29772 | Distributive law for inner... |
| ipasslem1 29773 | Lemma for ~ ipassi . Show... |
| ipasslem2 29774 | Lemma for ~ ipassi . Show... |
| ipasslem3 29775 | Lemma for ~ ipassi . Show... |
| ipasslem4 29776 | Lemma for ~ ipassi . Show... |
| ipasslem5 29777 | Lemma for ~ ipassi . Show... |
| ipasslem7 29778 | Lemma for ~ ipassi . Show... |
| ipasslem8 29779 | Lemma for ~ ipassi . By ~... |
| ipasslem9 29780 | Lemma for ~ ipassi . Conc... |
| ipasslem10 29781 | Lemma for ~ ipassi . Show... |
| ipasslem11 29782 | Lemma for ~ ipassi . Show... |
| ipassi 29783 | Associative law for inner ... |
| dipdir 29784 | Distributive law for inner... |
| dipdi 29785 | Distributive law for inner... |
| ip2dii 29786 | Inner product of two sums.... |
| dipass 29787 | Associative law for inner ... |
| dipassr 29788 | "Associative" law for seco... |
| dipassr2 29789 | "Associative" law for inne... |
| dipsubdir 29790 | Distributive law for inner... |
| dipsubdi 29791 | Distributive law for inner... |
| pythi 29792 | The Pythagorean theorem fo... |
| siilem1 29793 | Lemma for ~ sii . (Contri... |
| siilem2 29794 | Lemma for ~ sii . (Contri... |
| siii 29795 | Inference from ~ sii . (C... |
| sii 29796 | Obsolete version of ~ ipca... |
| ipblnfi 29797 | A function ` F ` generated... |
| ip2eqi 29798 | Two vectors are equal iff ... |
| phoeqi 29799 | A condition implying that ... |
| ajmoi 29800 | Every operator has at most... |
| ajfuni 29801 | The adjoint function is a ... |
| ajfun 29802 | The adjoint function is a ... |
| ajval 29803 | Value of the adjoint funct... |
| iscbn 29806 | A complex Banach space is ... |
| cbncms 29807 | The induced metric on comp... |
| bnnv 29808 | Every complex Banach space... |
| bnrel 29809 | The class of all complex B... |
| bnsscmcl 29810 | A subspace of a Banach spa... |
| cnbn 29811 | The set of complex numbers... |
| ubthlem1 29812 | Lemma for ~ ubth . The fu... |
| ubthlem2 29813 | Lemma for ~ ubth . Given ... |
| ubthlem3 29814 | Lemma for ~ ubth . Prove ... |
| ubth 29815 | Uniform Boundedness Theore... |
| minvecolem1 29816 | Lemma for ~ minveco . The... |
| minvecolem2 29817 | Lemma for ~ minveco . Any... |
| minvecolem3 29818 | Lemma for ~ minveco . The... |
| minvecolem4a 29819 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 29820 | Lemma for ~ minveco . The... |
| minvecolem4c 29821 | Lemma for ~ minveco . The... |
| minvecolem4 29822 | Lemma for ~ minveco . The... |
| minvecolem5 29823 | Lemma for ~ minveco . Dis... |
| minvecolem6 29824 | Lemma for ~ minveco . Any... |
| minvecolem7 29825 | Lemma for ~ minveco . Sin... |
| minveco 29826 | Minimizing vector theorem,... |
| ishlo 29829 | The predicate "is a comple... |
| hlobn 29830 | Every complex Hilbert spac... |
| hlph 29831 | Every complex Hilbert spac... |
| hlrel 29832 | The class of all complex H... |
| hlnv 29833 | Every complex Hilbert spac... |
| hlnvi 29834 | Every complex Hilbert spac... |
| hlvc 29835 | Every complex Hilbert spac... |
| hlcmet 29836 | The induced metric on a co... |
| hlmet 29837 | The induced metric on a co... |
| hlpar2 29838 | The parallelogram law sati... |
| hlpar 29839 | The parallelogram law sati... |
| hlex 29840 | The base set of a Hilbert ... |
| hladdf 29841 | Mapping for Hilbert space ... |
| hlcom 29842 | Hilbert space vector addit... |
| hlass 29843 | Hilbert space vector addit... |
| hl0cl 29844 | The Hilbert space zero vec... |
| hladdid 29845 | Hilbert space addition wit... |
| hlmulf 29846 | Mapping for Hilbert space ... |
| hlmulid 29847 | Hilbert space scalar multi... |
| hlmulass 29848 | Hilbert space scalar multi... |
| hldi 29849 | Hilbert space scalar multi... |
| hldir 29850 | Hilbert space scalar multi... |
| hlmul0 29851 | Hilbert space scalar multi... |
| hlipf 29852 | Mapping for Hilbert space ... |
| hlipcj 29853 | Conjugate law for Hilbert ... |
| hlipdir 29854 | Distributive law for Hilbe... |
| hlipass 29855 | Associative law for Hilber... |
| hlipgt0 29856 | The inner product of a Hil... |
| hlcompl 29857 | Completeness of a Hilbert ... |
| cnchl 29858 | The set of complex numbers... |
| htthlem 29859 | Lemma for ~ htth . The co... |
| htth 29860 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 29916 | The group (addition) opera... |
| h2hsm 29917 | The scalar product operati... |
| h2hnm 29918 | The norm function of Hilbe... |
| h2hvs 29919 | The vector subtraction ope... |
| h2hmetdval 29920 | Value of the distance func... |
| h2hcau 29921 | The Cauchy sequences of Hi... |
| h2hlm 29922 | The limit sequences of Hil... |
| axhilex-zf 29923 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 29924 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 29925 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 29926 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 29927 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 29928 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 29929 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 29930 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 29931 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 29932 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 29933 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 29934 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 29935 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 29936 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 29937 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 29938 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 29939 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 29940 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 29953 | The Hilbert space scalar p... |
| hvaddcl 29954 | Closure of vector addition... |
| hvmulcl 29955 | Closure of scalar multipli... |
| hvmulcli 29956 | Closure inference for scal... |
| hvsubf 29957 | Mapping domain and codomai... |
| hvsubval 29958 | Value of vector subtractio... |
| hvsubcl 29959 | Closure of vector subtract... |
| hvaddcli 29960 | Closure of vector addition... |
| hvcomi 29961 | Commutation of vector addi... |
| hvsubvali 29962 | Value of vector subtractio... |
| hvsubcli 29963 | Closure of vector subtract... |
| ifhvhv0 29964 | Prove ` if ( A e. ~H , A ,... |
| hvaddid2 29965 | Addition with the zero vec... |
| hvmul0 29966 | Scalar multiplication with... |
| hvmul0or 29967 | If a scalar product is zer... |
| hvsubid 29968 | Subtraction of a vector fr... |
| hvnegid 29969 | Addition of negative of a ... |
| hv2neg 29970 | Two ways to express the ne... |
| hvaddid2i 29971 | Addition with the zero vec... |
| hvnegidi 29972 | Addition of negative of a ... |
| hv2negi 29973 | Two ways to express the ne... |
| hvm1neg 29974 | Convert minus one times a ... |
| hvaddsubval 29975 | Value of vector addition i... |
| hvadd32 29976 | Commutative/associative la... |
| hvadd12 29977 | Commutative/associative la... |
| hvadd4 29978 | Hilbert vector space addit... |
| hvsub4 29979 | Hilbert vector space addit... |
| hvaddsub12 29980 | Commutative/associative la... |
| hvpncan 29981 | Addition/subtraction cance... |
| hvpncan2 29982 | Addition/subtraction cance... |
| hvaddsubass 29983 | Associativity of sum and d... |
| hvpncan3 29984 | Subtraction and addition o... |
| hvmulcom 29985 | Scalar multiplication comm... |
| hvsubass 29986 | Hilbert vector space assoc... |
| hvsub32 29987 | Hilbert vector space commu... |
| hvmulassi 29988 | Scalar multiplication asso... |
| hvmulcomi 29989 | Scalar multiplication comm... |
| hvmul2negi 29990 | Double negative in scalar ... |
| hvsubdistr1 29991 | Scalar multiplication dist... |
| hvsubdistr2 29992 | Scalar multiplication dist... |
| hvdistr1i 29993 | Scalar multiplication dist... |
| hvsubdistr1i 29994 | Scalar multiplication dist... |
| hvassi 29995 | Hilbert vector space assoc... |
| hvadd32i 29996 | Hilbert vector space commu... |
| hvsubassi 29997 | Hilbert vector space assoc... |
| hvsub32i 29998 | Hilbert vector space commu... |
| hvadd12i 29999 | Hilbert vector space commu... |
| hvadd4i 30000 | Hilbert vector space addit... |
| hvsubsub4i 30001 | Hilbert vector space addit... |
| hvsubsub4 30002 | Hilbert vector space addit... |
| hv2times 30003 | Two times a vector. (Cont... |
| hvnegdii 30004 | Distribution of negative o... |
| hvsubeq0i 30005 | If the difference between ... |
| hvsubcan2i 30006 | Vector cancellation law. ... |
| hvaddcani 30007 | Cancellation law for vecto... |
| hvsubaddi 30008 | Relationship between vecto... |
| hvnegdi 30009 | Distribution of negative o... |
| hvsubeq0 30010 | If the difference between ... |
| hvaddeq0 30011 | If the sum of two vectors ... |
| hvaddcan 30012 | Cancellation law for vecto... |
| hvaddcan2 30013 | Cancellation law for vecto... |
| hvmulcan 30014 | Cancellation law for scala... |
| hvmulcan2 30015 | Cancellation law for scala... |
| hvsubcan 30016 | Cancellation law for vecto... |
| hvsubcan2 30017 | Cancellation law for vecto... |
| hvsub0 30018 | Subtraction of a zero vect... |
| hvsubadd 30019 | Relationship between vecto... |
| hvaddsub4 30020 | Hilbert vector space addit... |
| hicl 30022 | Closure of inner product. ... |
| hicli 30023 | Closure inference for inne... |
| his5 30028 | Associative law for inner ... |
| his52 30029 | Associative law for inner ... |
| his35 30030 | Move scalar multiplication... |
| his35i 30031 | Move scalar multiplication... |
| his7 30032 | Distributive law for inner... |
| hiassdi 30033 | Distributive/associative l... |
| his2sub 30034 | Distributive law for inner... |
| his2sub2 30035 | Distributive law for inner... |
| hire 30036 | A necessary and sufficient... |
| hiidrcl 30037 | Real closure of inner prod... |
| hi01 30038 | Inner product with the 0 v... |
| hi02 30039 | Inner product with the 0 v... |
| hiidge0 30040 | Inner product with self is... |
| his6 30041 | Zero inner product with se... |
| his1i 30042 | Conjugate law for inner pr... |
| abshicom 30043 | Commuted inner products ha... |
| hial0 30044 | A vector whose inner produ... |
| hial02 30045 | A vector whose inner produ... |
| hisubcomi 30046 | Two vector subtractions si... |
| hi2eq 30047 | Lemma used to prove equali... |
| hial2eq 30048 | Two vectors whose inner pr... |
| hial2eq2 30049 | Two vectors whose inner pr... |
| orthcom 30050 | Orthogonality commutes. (... |
| normlem0 30051 | Lemma used to derive prope... |
| normlem1 30052 | Lemma used to derive prope... |
| normlem2 30053 | Lemma used to derive prope... |
| normlem3 30054 | Lemma used to derive prope... |
| normlem4 30055 | Lemma used to derive prope... |
| normlem5 30056 | Lemma used to derive prope... |
| normlem6 30057 | Lemma used to derive prope... |
| normlem7 30058 | Lemma used to derive prope... |
| normlem8 30059 | Lemma used to derive prope... |
| normlem9 30060 | Lemma used to derive prope... |
| normlem7tALT 30061 | Lemma used to derive prope... |
| bcseqi 30062 | Equality case of Bunjakova... |
| normlem9at 30063 | Lemma used to derive prope... |
| dfhnorm2 30064 | Alternate definition of th... |
| normf 30065 | The norm function maps fro... |
| normval 30066 | The value of the norm of a... |
| normcl 30067 | Real closure of the norm o... |
| normge0 30068 | The norm of a vector is no... |
| normgt0 30069 | The norm of nonzero vector... |
| norm0 30070 | The norm of a zero vector.... |
| norm-i 30071 | Theorem 3.3(i) of [Beran] ... |
| normne0 30072 | A norm is nonzero iff its ... |
| normcli 30073 | Real closure of the norm o... |
| normsqi 30074 | The square of a norm. (Co... |
| norm-i-i 30075 | Theorem 3.3(i) of [Beran] ... |
| normsq 30076 | The square of a norm. (Co... |
| normsub0i 30077 | Two vectors are equal iff ... |
| normsub0 30078 | Two vectors are equal iff ... |
| norm-ii-i 30079 | Triangle inequality for no... |
| norm-ii 30080 | Triangle inequality for no... |
| norm-iii-i 30081 | Theorem 3.3(iii) of [Beran... |
| norm-iii 30082 | Theorem 3.3(iii) of [Beran... |
| normsubi 30083 | Negative doesn't change th... |
| normpythi 30084 | Analogy to Pythagorean the... |
| normsub 30085 | Swapping order of subtract... |
| normneg 30086 | The norm of a vector equal... |
| normpyth 30087 | Analogy to Pythagorean the... |
| normpyc 30088 | Corollary to Pythagorean t... |
| norm3difi 30089 | Norm of differences around... |
| norm3adifii 30090 | Norm of differences around... |
| norm3lem 30091 | Lemma involving norm of di... |
| norm3dif 30092 | Norm of differences around... |
| norm3dif2 30093 | Norm of differences around... |
| norm3lemt 30094 | Lemma involving norm of di... |
| norm3adifi 30095 | Norm of differences around... |
| normpari 30096 | Parallelogram law for norm... |
| normpar 30097 | Parallelogram law for norm... |
| normpar2i 30098 | Corollary of parallelogram... |
| polid2i 30099 | Generalized polarization i... |
| polidi 30100 | Polarization identity. Re... |
| polid 30101 | Polarization identity. Re... |
| hilablo 30102 | Hilbert space vector addit... |
| hilid 30103 | The group identity element... |
| hilvc 30104 | Hilbert space is a complex... |
| hilnormi 30105 | Hilbert space norm in term... |
| hilhhi 30106 | Deduce the structure of Hi... |
| hhnv 30107 | Hilbert space is a normed ... |
| hhva 30108 | The group (addition) opera... |
| hhba 30109 | The base set of Hilbert sp... |
| hh0v 30110 | The zero vector of Hilbert... |
| hhsm 30111 | The scalar product operati... |
| hhvs 30112 | The vector subtraction ope... |
| hhnm 30113 | The norm function of Hilbe... |
| hhims 30114 | The induced metric of Hilb... |
| hhims2 30115 | Hilbert space distance met... |
| hhmet 30116 | The induced metric of Hilb... |
| hhxmet 30117 | The induced metric of Hilb... |
| hhmetdval 30118 | Value of the distance func... |
| hhip 30119 | The inner product operatio... |
| hhph 30120 | The Hilbert space of the H... |
| bcsiALT 30121 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 30122 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 30123 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 30124 | Corollary of the Bunjakova... |
| bcs3 30125 | Corollary of the Bunjakova... |
| hcau 30126 | Member of the set of Cauch... |
| hcauseq 30127 | A Cauchy sequences on a Hi... |
| hcaucvg 30128 | A Cauchy sequence on a Hil... |
| seq1hcau 30129 | A sequence on a Hilbert sp... |
| hlimi 30130 | Express the predicate: Th... |
| hlimseqi 30131 | A sequence with a limit on... |
| hlimveci 30132 | Closure of the limit of a ... |
| hlimconvi 30133 | Convergence of a sequence ... |
| hlim2 30134 | The limit of a sequence on... |
| hlimadd 30135 | Limit of the sum of two se... |
| hilmet 30136 | The Hilbert space norm det... |
| hilxmet 30137 | The Hilbert space norm det... |
| hilmetdval 30138 | Value of the distance func... |
| hilims 30139 | Hilbert space distance met... |
| hhcau 30140 | The Cauchy sequences of Hi... |
| hhlm 30141 | The limit sequences of Hil... |
| hhcmpl 30142 | Lemma used for derivation ... |
| hilcompl 30143 | Lemma used for derivation ... |
| hhcms 30145 | The Hilbert space induced ... |
| hhhl 30146 | The Hilbert space structur... |
| hilcms 30147 | The Hilbert space norm det... |
| hilhl 30148 | The Hilbert space of the H... |
| issh 30150 | Subspace ` H ` of a Hilber... |
| issh2 30151 | Subspace ` H ` of a Hilber... |
| shss 30152 | A subspace is a subset of ... |
| shel 30153 | A member of a subspace of ... |
| shex 30154 | The set of subspaces of a ... |
| shssii 30155 | A closed subspace of a Hil... |
| sheli 30156 | A member of a subspace of ... |
| shelii 30157 | A member of a subspace of ... |
| sh0 30158 | The zero vector belongs to... |
| shaddcl 30159 | Closure of vector addition... |
| shmulcl 30160 | Closure of vector scalar m... |
| issh3 30161 | Subspace ` H ` of a Hilber... |
| shsubcl 30162 | Closure of vector subtract... |
| isch 30164 | Closed subspace ` H ` of a... |
| isch2 30165 | Closed subspace ` H ` of a... |
| chsh 30166 | A closed subspace is a sub... |
| chsssh 30167 | Closed subspaces are subsp... |
| chex 30168 | The set of closed subspace... |
| chshii 30169 | A closed subspace is a sub... |
| ch0 30170 | The zero vector belongs to... |
| chss 30171 | A closed subspace of a Hil... |
| chel 30172 | A member of a closed subsp... |
| chssii 30173 | A closed subspace of a Hil... |
| cheli 30174 | A member of a closed subsp... |
| chelii 30175 | A member of a closed subsp... |
| chlimi 30176 | The limit property of a cl... |
| hlim0 30177 | The zero sequence in Hilbe... |
| hlimcaui 30178 | If a sequence in Hilbert s... |
| hlimf 30179 | Function-like behavior of ... |
| hlimuni 30180 | A Hilbert space sequence c... |
| hlimreui 30181 | The limit of a Hilbert spa... |
| hlimeui 30182 | The limit of a Hilbert spa... |
| isch3 30183 | A Hilbert subspace is clos... |
| chcompl 30184 | Completeness of a closed s... |
| helch 30185 | The Hilbert lattice one (w... |
| ifchhv 30186 | Prove ` if ( A e. CH , A ,... |
| helsh 30187 | Hilbert space is a subspac... |
| shsspwh 30188 | Subspaces are subsets of H... |
| chsspwh 30189 | Closed subspaces are subse... |
| hsn0elch 30190 | The zero subspace belongs ... |
| norm1 30191 | From any nonzero Hilbert s... |
| norm1exi 30192 | A normalized vector exists... |
| norm1hex 30193 | A normalized vector can ex... |
| elch0 30196 | Membership in zero for clo... |
| h0elch 30197 | The zero subspace is a clo... |
| h0elsh 30198 | The zero subspace is a sub... |
| hhssva 30199 | The vector addition operat... |
| hhsssm 30200 | The scalar multiplication ... |
| hhssnm 30201 | The norm operation on a su... |
| issubgoilem 30202 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 30203 | Lemma for ~ hhssabloi . F... |
| hhssabloi 30204 | Abelian group property of ... |
| hhssablo 30205 | Abelian group property of ... |
| hhssnv 30206 | Normed complex vector spac... |
| hhssnvt 30207 | Normed complex vector spac... |
| hhsst 30208 | A member of ` SH ` is a su... |
| hhshsslem1 30209 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 30210 | Lemma for ~ hhsssh . (Con... |
| hhsssh 30211 | The predicate " ` H ` is a... |
| hhsssh2 30212 | The predicate " ` H ` is a... |
| hhssba 30213 | The base set of a subspace... |
| hhssvs 30214 | The vector subtraction ope... |
| hhssvsf 30215 | Mapping of the vector subt... |
| hhssims 30216 | Induced metric of a subspa... |
| hhssims2 30217 | Induced metric of a subspa... |
| hhssmet 30218 | Induced metric of a subspa... |
| hhssmetdval 30219 | Value of the distance func... |
| hhsscms 30220 | The induced metric of a cl... |
| hhssbnOLD 30221 | Obsolete version of ~ cssb... |
| ocval 30222 | Value of orthogonal comple... |
| ocel 30223 | Membership in orthogonal c... |
| shocel 30224 | Membership in orthogonal c... |
| ocsh 30225 | The orthogonal complement ... |
| shocsh 30226 | The orthogonal complement ... |
| ocss 30227 | An orthogonal complement i... |
| shocss 30228 | An orthogonal complement i... |
| occon 30229 | Contraposition law for ort... |
| occon2 30230 | Double contraposition for ... |
| occon2i 30231 | Double contraposition for ... |
| oc0 30232 | The zero vector belongs to... |
| ocorth 30233 | Members of a subset and it... |
| shocorth 30234 | Members of a subspace and ... |
| ococss 30235 | Inclusion in complement of... |
| shococss 30236 | Inclusion in complement of... |
| shorth 30237 | Members of orthogonal subs... |
| ocin 30238 | Intersection of a Hilbert ... |
| occon3 30239 | Hilbert lattice contraposi... |
| ocnel 30240 | A nonzero vector in the co... |
| chocvali 30241 | Value of the orthogonal co... |
| shuni 30242 | Two subspaces with trivial... |
| chocunii 30243 | Lemma for uniqueness part ... |
| pjhthmo 30244 | Projection Theorem, unique... |
| occllem 30245 | Lemma for ~ occl . (Contr... |
| occl 30246 | Closure of complement of H... |
| shoccl 30247 | Closure of complement of H... |
| choccl 30248 | Closure of complement of H... |
| choccli 30249 | Closure of ` CH ` orthocom... |
| shsval 30254 | Value of subspace sum of t... |
| shsss 30255 | The subspace sum is a subs... |
| shsel 30256 | Membership in the subspace... |
| shsel3 30257 | Membership in the subspace... |
| shseli 30258 | Membership in subspace sum... |
| shscli 30259 | Closure of subspace sum. ... |
| shscl 30260 | Closure of subspace sum. ... |
| shscom 30261 | Commutative law for subspa... |
| shsva 30262 | Vector sum belongs to subs... |
| shsel1 30263 | A subspace sum contains a ... |
| shsel2 30264 | A subspace sum contains a ... |
| shsvs 30265 | Vector subtraction belongs... |
| shsub1 30266 | Subspace sum is an upper b... |
| shsub2 30267 | Subspace sum is an upper b... |
| choc0 30268 | The orthocomplement of the... |
| choc1 30269 | The orthocomplement of the... |
| chocnul 30270 | Orthogonal complement of t... |
| shintcli 30271 | Closure of intersection of... |
| shintcl 30272 | The intersection of a none... |
| chintcli 30273 | The intersection of a none... |
| chintcl 30274 | The intersection (infimum)... |
| spanval 30275 | Value of the linear span o... |
| hsupval 30276 | Value of supremum of set o... |
| chsupval 30277 | The value of the supremum ... |
| spancl 30278 | The span of a subset of Hi... |
| elspancl 30279 | A member of a span is a ve... |
| shsupcl 30280 | Closure of the subspace su... |
| hsupcl 30281 | Closure of supremum of set... |
| chsupcl 30282 | Closure of supremum of sub... |
| hsupss 30283 | Subset relation for suprem... |
| chsupss 30284 | Subset relation for suprem... |
| hsupunss 30285 | The union of a set of Hilb... |
| chsupunss 30286 | The union of a set of clos... |
| spanss2 30287 | A subset of Hilbert space ... |
| shsupunss 30288 | The union of a set of subs... |
| spanid 30289 | A subspace of Hilbert spac... |
| spanss 30290 | Ordering relationship for ... |
| spanssoc 30291 | The span of a subset of Hi... |
| sshjval 30292 | Value of join for subsets ... |
| shjval 30293 | Value of join in ` SH ` . ... |
| chjval 30294 | Value of join in ` CH ` . ... |
| chjvali 30295 | Value of join in ` CH ` . ... |
| sshjval3 30296 | Value of join for subsets ... |
| sshjcl 30297 | Closure of join for subset... |
| shjcl 30298 | Closure of join in ` SH ` ... |
| chjcl 30299 | Closure of join in ` CH ` ... |
| shjcom 30300 | Commutative law for Hilber... |
| shless 30301 | Subset implies subset of s... |
| shlej1 30302 | Add disjunct to both sides... |
| shlej2 30303 | Add disjunct to both sides... |
| shincli 30304 | Closure of intersection of... |
| shscomi 30305 | Commutative law for subspa... |
| shsvai 30306 | Vector sum belongs to subs... |
| shsel1i 30307 | A subspace sum contains a ... |
| shsel2i 30308 | A subspace sum contains a ... |
| shsvsi 30309 | Vector subtraction belongs... |
| shunssi 30310 | Union is smaller than subs... |
| shunssji 30311 | Union is smaller than Hilb... |
| shsleji 30312 | Subspace sum is smaller th... |
| shjcomi 30313 | Commutative law for join i... |
| shsub1i 30314 | Subspace sum is an upper b... |
| shsub2i 30315 | Subspace sum is an upper b... |
| shub1i 30316 | Hilbert lattice join is an... |
| shjcli 30317 | Closure of ` CH ` join. (... |
| shjshcli 30318 | ` SH ` closure of join. (... |
| shlessi 30319 | Subset implies subset of s... |
| shlej1i 30320 | Add disjunct to both sides... |
| shlej2i 30321 | Add disjunct to both sides... |
| shslej 30322 | Subspace sum is smaller th... |
| shincl 30323 | Closure of intersection of... |
| shub1 30324 | Hilbert lattice join is an... |
| shub2 30325 | A subspace is a subset of ... |
| shsidmi 30326 | Idempotent law for Hilbert... |
| shslubi 30327 | The least upper bound law ... |
| shlesb1i 30328 | Hilbert lattice ordering i... |
| shsval2i 30329 | An alternate way to expres... |
| shsval3i 30330 | An alternate way to expres... |
| shmodsi 30331 | The modular law holds for ... |
| shmodi 30332 | The modular law is implied... |
| pjhthlem1 30333 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 30334 | Lemma for ~ pjhth . (Cont... |
| pjhth 30335 | Projection Theorem: Any H... |
| pjhtheu 30336 | Projection Theorem: Any H... |
| pjhfval 30338 | The value of the projectio... |
| pjhval 30339 | Value of a projection. (C... |
| pjpreeq 30340 | Equality with a projection... |
| pjeq 30341 | Equality with a projection... |
| axpjcl 30342 | Closure of a projection in... |
| pjhcl 30343 | Closure of a projection in... |
| omlsilem 30344 | Lemma for orthomodular law... |
| omlsii 30345 | Subspace inference form of... |
| omlsi 30346 | Subspace form of orthomodu... |
| ococi 30347 | Complement of complement o... |
| ococ 30348 | Complement of complement o... |
| dfch2 30349 | Alternate definition of th... |
| ococin 30350 | The double complement is t... |
| hsupval2 30351 | Alternate definition of su... |
| chsupval2 30352 | The value of the supremum ... |
| sshjval2 30353 | Value of join in the set o... |
| chsupid 30354 | A subspace is the supremum... |
| chsupsn 30355 | Value of supremum of subse... |
| shlub 30356 | Hilbert lattice join is th... |
| shlubi 30357 | Hilbert lattice join is th... |
| pjhtheu2 30358 | Uniqueness of ` y ` for th... |
| pjcli 30359 | Closure of a projection in... |
| pjhcli 30360 | Closure of a projection in... |
| pjpjpre 30361 | Decomposition of a vector ... |
| axpjpj 30362 | Decomposition of a vector ... |
| pjclii 30363 | Closure of a projection in... |
| pjhclii 30364 | Closure of a projection in... |
| pjpj0i 30365 | Decomposition of a vector ... |
| pjpji 30366 | Decomposition of a vector ... |
| pjpjhth 30367 | Projection Theorem: Any H... |
| pjpjhthi 30368 | Projection Theorem: Any H... |
| pjop 30369 | Orthocomplement projection... |
| pjpo 30370 | Projection in terms of ort... |
| pjopi 30371 | Orthocomplement projection... |
| pjpoi 30372 | Projection in terms of ort... |
| pjoc1i 30373 | Projection of a vector in ... |
| pjchi 30374 | Projection of a vector in ... |
| pjoccl 30375 | The part of a vector that ... |
| pjoc1 30376 | Projection of a vector in ... |
| pjomli 30377 | Subspace form of orthomodu... |
| pjoml 30378 | Subspace form of orthomodu... |
| pjococi 30379 | Proof of orthocomplement t... |
| pjoc2i 30380 | Projection of a vector in ... |
| pjoc2 30381 | Projection of a vector in ... |
| sh0le 30382 | The zero subspace is the s... |
| ch0le 30383 | The zero subspace is the s... |
| shle0 30384 | No subspace is smaller tha... |
| chle0 30385 | No Hilbert lattice element... |
| chnlen0 30386 | A Hilbert lattice element ... |
| ch0pss 30387 | The zero subspace is a pro... |
| orthin 30388 | The intersection of orthog... |
| ssjo 30389 | The lattice join of a subs... |
| shne0i 30390 | A nonzero subspace has a n... |
| shs0i 30391 | Hilbert subspace sum with ... |
| shs00i 30392 | Two subspaces are zero iff... |
| ch0lei 30393 | The closed subspace zero i... |
| chle0i 30394 | No Hilbert closed subspace... |
| chne0i 30395 | A nonzero closed subspace ... |
| chocini 30396 | Intersection of a closed s... |
| chj0i 30397 | Join with lattice zero in ... |
| chm1i 30398 | Meet with lattice one in `... |
| chjcli 30399 | Closure of ` CH ` join. (... |
| chsleji 30400 | Subspace sum is smaller th... |
| chseli 30401 | Membership in subspace sum... |
| chincli 30402 | Closure of Hilbert lattice... |
| chsscon3i 30403 | Hilbert lattice contraposi... |
| chsscon1i 30404 | Hilbert lattice contraposi... |
| chsscon2i 30405 | Hilbert lattice contraposi... |
| chcon2i 30406 | Hilbert lattice contraposi... |
| chcon1i 30407 | Hilbert lattice contraposi... |
| chcon3i 30408 | Hilbert lattice contraposi... |
| chunssji 30409 | Union is smaller than ` CH... |
| chjcomi 30410 | Commutative law for join i... |
| chub1i 30411 | ` CH ` join is an upper bo... |
| chub2i 30412 | ` CH ` join is an upper bo... |
| chlubi 30413 | Hilbert lattice join is th... |
| chlubii 30414 | Hilbert lattice join is th... |
| chlej1i 30415 | Add join to both sides of ... |
| chlej2i 30416 | Add join to both sides of ... |
| chlej12i 30417 | Add join to both sides of ... |
| chlejb1i 30418 | Hilbert lattice ordering i... |
| chdmm1i 30419 | De Morgan's law for meet i... |
| chdmm2i 30420 | De Morgan's law for meet i... |
| chdmm3i 30421 | De Morgan's law for meet i... |
| chdmm4i 30422 | De Morgan's law for meet i... |
| chdmj1i 30423 | De Morgan's law for join i... |
| chdmj2i 30424 | De Morgan's law for join i... |
| chdmj3i 30425 | De Morgan's law for join i... |
| chdmj4i 30426 | De Morgan's law for join i... |
| chnlei 30427 | Equivalent expressions for... |
| chjassi 30428 | Associative law for Hilber... |
| chj00i 30429 | Two Hilbert lattice elemen... |
| chjoi 30430 | The join of a closed subsp... |
| chj1i 30431 | Join with Hilbert lattice ... |
| chm0i 30432 | Meet with Hilbert lattice ... |
| chm0 30433 | Meet with Hilbert lattice ... |
| shjshsi 30434 | Hilbert lattice join equal... |
| shjshseli 30435 | A closed subspace sum equa... |
| chne0 30436 | A nonzero closed subspace ... |
| chocin 30437 | Intersection of a closed s... |
| chssoc 30438 | A closed subspace less tha... |
| chj0 30439 | Join with Hilbert lattice ... |
| chslej 30440 | Subspace sum is smaller th... |
| chincl 30441 | Closure of Hilbert lattice... |
| chsscon3 30442 | Hilbert lattice contraposi... |
| chsscon1 30443 | Hilbert lattice contraposi... |
| chsscon2 30444 | Hilbert lattice contraposi... |
| chpsscon3 30445 | Hilbert lattice contraposi... |
| chpsscon1 30446 | Hilbert lattice contraposi... |
| chpsscon2 30447 | Hilbert lattice contraposi... |
| chjcom 30448 | Commutative law for Hilber... |
| chub1 30449 | Hilbert lattice join is gr... |
| chub2 30450 | Hilbert lattice join is gr... |
| chlub 30451 | Hilbert lattice join is th... |
| chlej1 30452 | Add join to both sides of ... |
| chlej2 30453 | Add join to both sides of ... |
| chlejb1 30454 | Hilbert lattice ordering i... |
| chlejb2 30455 | Hilbert lattice ordering i... |
| chnle 30456 | Equivalent expressions for... |
| chjo 30457 | The join of a closed subsp... |
| chabs1 30458 | Hilbert lattice absorption... |
| chabs2 30459 | Hilbert lattice absorption... |
| chabs1i 30460 | Hilbert lattice absorption... |
| chabs2i 30461 | Hilbert lattice absorption... |
| chjidm 30462 | Idempotent law for Hilbert... |
| chjidmi 30463 | Idempotent law for Hilbert... |
| chj12i 30464 | A rearrangement of Hilbert... |
| chj4i 30465 | Rearrangement of the join ... |
| chjjdiri 30466 | Hilbert lattice join distr... |
| chdmm1 30467 | De Morgan's law for meet i... |
| chdmm2 30468 | De Morgan's law for meet i... |
| chdmm3 30469 | De Morgan's law for meet i... |
| chdmm4 30470 | De Morgan's law for meet i... |
| chdmj1 30471 | De Morgan's law for join i... |
| chdmj2 30472 | De Morgan's law for join i... |
| chdmj3 30473 | De Morgan's law for join i... |
| chdmj4 30474 | De Morgan's law for join i... |
| chjass 30475 | Associative law for Hilber... |
| chj12 30476 | A rearrangement of Hilbert... |
| chj4 30477 | Rearrangement of the join ... |
| ledii 30478 | An ortholattice is distrib... |
| lediri 30479 | An ortholattice is distrib... |
| lejdii 30480 | An ortholattice is distrib... |
| lejdiri 30481 | An ortholattice is distrib... |
| ledi 30482 | An ortholattice is distrib... |
| spansn0 30483 | The span of the singleton ... |
| span0 30484 | The span of the empty set ... |
| elspani 30485 | Membership in the span of ... |
| spanuni 30486 | The span of a union is the... |
| spanun 30487 | The span of a union is the... |
| sshhococi 30488 | The join of two Hilbert sp... |
| hne0 30489 | Hilbert space has a nonzer... |
| chsup0 30490 | The supremum of the empty ... |
| h1deoi 30491 | Membership in orthocomplem... |
| h1dei 30492 | Membership in 1-dimensiona... |
| h1did 30493 | A generating vector belong... |
| h1dn0 30494 | A nonzero vector generates... |
| h1de2i 30495 | Membership in 1-dimensiona... |
| h1de2bi 30496 | Membership in 1-dimensiona... |
| h1de2ctlem 30497 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 30498 | Membership in 1-dimensiona... |
| spansni 30499 | The span of a singleton in... |
| elspansni 30500 | Membership in the span of ... |
| spansn 30501 | The span of a singleton in... |
| spansnch 30502 | The span of a Hilbert spac... |
| spansnsh 30503 | The span of a Hilbert spac... |
| spansnchi 30504 | The span of a singleton in... |
| spansnid 30505 | A vector belongs to the sp... |
| spansnmul 30506 | A scalar product with a ve... |
| elspansncl 30507 | A member of a span of a si... |
| elspansn 30508 | Membership in the span of ... |
| elspansn2 30509 | Membership in the span of ... |
| spansncol 30510 | The singletons of collinea... |
| spansneleqi 30511 | Membership relation implie... |
| spansneleq 30512 | Membership relation that i... |
| spansnss 30513 | The span of the singleton ... |
| elspansn3 30514 | A member of the span of th... |
| elspansn4 30515 | A span membership conditio... |
| elspansn5 30516 | A vector belonging to both... |
| spansnss2 30517 | The span of the singleton ... |
| normcan 30518 | Cancellation-type law that... |
| pjspansn 30519 | A projection on the span o... |
| spansnpji 30520 | A subset of Hilbert space ... |
| spanunsni 30521 | The span of the union of a... |
| spanpr 30522 | The span of a pair of vect... |
| h1datomi 30523 | A 1-dimensional subspace i... |
| h1datom 30524 | A 1-dimensional subspace i... |
| cmbr 30526 | Binary relation expressing... |
| pjoml2i 30527 | Variation of orthomodular ... |
| pjoml3i 30528 | Variation of orthomodular ... |
| pjoml4i 30529 | Variation of orthomodular ... |
| pjoml5i 30530 | The orthomodular law. Rem... |
| pjoml6i 30531 | An equivalent of the ortho... |
| cmbri 30532 | Binary relation expressing... |
| cmcmlem 30533 | Commutation is symmetric. ... |
| cmcmi 30534 | Commutation is symmetric. ... |
| cmcm2i 30535 | Commutation with orthocomp... |
| cmcm3i 30536 | Commutation with orthocomp... |
| cmcm4i 30537 | Commutation with orthocomp... |
| cmbr2i 30538 | Alternate definition of th... |
| cmcmii 30539 | Commutation is symmetric. ... |
| cmcm2ii 30540 | Commutation with orthocomp... |
| cmcm3ii 30541 | Commutation with orthocomp... |
| cmbr3i 30542 | Alternate definition for t... |
| cmbr4i 30543 | Alternate definition for t... |
| lecmi 30544 | Comparable Hilbert lattice... |
| lecmii 30545 | Comparable Hilbert lattice... |
| cmj1i 30546 | A Hilbert lattice element ... |
| cmj2i 30547 | A Hilbert lattice element ... |
| cmm1i 30548 | A Hilbert lattice element ... |
| cmm2i 30549 | A Hilbert lattice element ... |
| cmbr3 30550 | Alternate definition for t... |
| cm0 30551 | The zero Hilbert lattice e... |
| cmidi 30552 | The commutes relation is r... |
| pjoml2 30553 | Variation of orthomodular ... |
| pjoml3 30554 | Variation of orthomodular ... |
| pjoml5 30555 | The orthomodular law. Rem... |
| cmcm 30556 | Commutation is symmetric. ... |
| cmcm3 30557 | Commutation with orthocomp... |
| cmcm2 30558 | Commutation with orthocomp... |
| lecm 30559 | Comparable Hilbert lattice... |
| fh1 30560 | Foulis-Holland Theorem. I... |
| fh2 30561 | Foulis-Holland Theorem. I... |
| cm2j 30562 | A lattice element that com... |
| fh1i 30563 | Foulis-Holland Theorem. I... |
| fh2i 30564 | Foulis-Holland Theorem. I... |
| fh3i 30565 | Variation of the Foulis-Ho... |
| fh4i 30566 | Variation of the Foulis-Ho... |
| cm2ji 30567 | A lattice element that com... |
| cm2mi 30568 | A lattice element that com... |
| qlax1i 30569 | One of the equations showi... |
| qlax2i 30570 | One of the equations showi... |
| qlax3i 30571 | One of the equations showi... |
| qlax4i 30572 | One of the equations showi... |
| qlax5i 30573 | One of the equations showi... |
| qlaxr1i 30574 | One of the conditions show... |
| qlaxr2i 30575 | One of the conditions show... |
| qlaxr4i 30576 | One of the conditions show... |
| qlaxr5i 30577 | One of the conditions show... |
| qlaxr3i 30578 | A variation of the orthomo... |
| chscllem1 30579 | Lemma for ~ chscl . (Cont... |
| chscllem2 30580 | Lemma for ~ chscl . (Cont... |
| chscllem3 30581 | Lemma for ~ chscl . (Cont... |
| chscllem4 30582 | Lemma for ~ chscl . (Cont... |
| chscl 30583 | The subspace sum of two cl... |
| osumi 30584 | If two closed subspaces of... |
| osumcori 30585 | Corollary of ~ osumi . (C... |
| osumcor2i 30586 | Corollary of ~ osumi , sho... |
| osum 30587 | If two closed subspaces of... |
| spansnji 30588 | The subspace sum of a clos... |
| spansnj 30589 | The subspace sum of a clos... |
| spansnscl 30590 | The subspace sum of a clos... |
| sumspansn 30591 | The sum of two vectors bel... |
| spansnm0i 30592 | The meet of different one-... |
| nonbooli 30593 | A Hilbert lattice with two... |
| spansncvi 30594 | Hilbert space has the cove... |
| spansncv 30595 | Hilbert space has the cove... |
| 5oalem1 30596 | Lemma for orthoarguesian l... |
| 5oalem2 30597 | Lemma for orthoarguesian l... |
| 5oalem3 30598 | Lemma for orthoarguesian l... |
| 5oalem4 30599 | Lemma for orthoarguesian l... |
| 5oalem5 30600 | Lemma for orthoarguesian l... |
| 5oalem6 30601 | Lemma for orthoarguesian l... |
| 5oalem7 30602 | Lemma for orthoarguesian l... |
| 5oai 30603 | Orthoarguesian law 5OA. Th... |
| 3oalem1 30604 | Lemma for 3OA (weak) ortho... |
| 3oalem2 30605 | Lemma for 3OA (weak) ortho... |
| 3oalem3 30606 | Lemma for 3OA (weak) ortho... |
| 3oalem4 30607 | Lemma for 3OA (weak) ortho... |
| 3oalem5 30608 | Lemma for 3OA (weak) ortho... |
| 3oalem6 30609 | Lemma for 3OA (weak) ortho... |
| 3oai 30610 | 3OA (weak) orthoarguesian ... |
| pjorthi 30611 | Projection components on o... |
| pjch1 30612 | Property of identity proje... |
| pjo 30613 | The orthogonal projection.... |
| pjcompi 30614 | Component of a projection.... |
| pjidmi 30615 | A projection is idempotent... |
| pjadjii 30616 | A projection is self-adjoi... |
| pjaddii 30617 | Projection of vector sum i... |
| pjinormii 30618 | The inner product of a pro... |
| pjmulii 30619 | Projection of (scalar) pro... |
| pjsubii 30620 | Projection of vector diffe... |
| pjsslem 30621 | Lemma for subset relations... |
| pjss2i 30622 | Subset relationship for pr... |
| pjssmii 30623 | Projection meet property. ... |
| pjssge0ii 30624 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 30625 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 30626 | The projection on a subspa... |
| pjadji 30627 | A projection is self-adjoi... |
| pjaddi 30628 | Projection of vector sum i... |
| pjinormi 30629 | The inner product of a pro... |
| pjsubi 30630 | Projection of vector diffe... |
| pjmuli 30631 | Projection of scalar produ... |
| pjige0i 30632 | The inner product of a pro... |
| pjige0 30633 | The inner product of a pro... |
| pjcjt2 30634 | The projection on a subspa... |
| pj0i 30635 | The projection of the zero... |
| pjch 30636 | Projection of a vector in ... |
| pjid 30637 | The projection of a vector... |
| pjvec 30638 | The set of vectors belongi... |
| pjocvec 30639 | The set of vectors belongi... |
| pjocini 30640 | Membership of projection i... |
| pjini 30641 | Membership of projection i... |
| pjjsi 30642 | A sufficient condition for... |
| pjfni 30643 | Functionality of a project... |
| pjrni 30644 | The range of a projection.... |
| pjfoi 30645 | A projection maps onto its... |
| pjfi 30646 | The mapping of a projectio... |
| pjvi 30647 | The value of a projection ... |
| pjhfo 30648 | A projection maps onto its... |
| pjrn 30649 | The range of a projection.... |
| pjhf 30650 | The mapping of a projectio... |
| pjfn 30651 | Functionality of a project... |
| pjsumi 30652 | The projection on a subspa... |
| pj11i 30653 | One-to-one correspondence ... |
| pjdsi 30654 | Vector decomposition into ... |
| pjds3i 30655 | Vector decomposition into ... |
| pj11 30656 | One-to-one correspondence ... |
| pjmfn 30657 | Functionality of the proje... |
| pjmf1 30658 | The projector function map... |
| pjoi0 30659 | The inner product of proje... |
| pjoi0i 30660 | The inner product of proje... |
| pjopythi 30661 | Pythagorean theorem for pr... |
| pjopyth 30662 | Pythagorean theorem for pr... |
| pjnormi 30663 | The norm of the projection... |
| pjpythi 30664 | Pythagorean theorem for pr... |
| pjneli 30665 | If a vector does not belon... |
| pjnorm 30666 | The norm of the projection... |
| pjpyth 30667 | Pythagorean theorem for pr... |
| pjnel 30668 | If a vector does not belon... |
| pjnorm2 30669 | A vector belongs to the su... |
| mayete3i 30670 | Mayet's equation E_3. Par... |
| mayetes3i 30671 | Mayet's equation E^*_3, de... |
| hosmval 30677 | Value of the sum of two Hi... |
| hommval 30678 | Value of the scalar produc... |
| hodmval 30679 | Value of the difference of... |
| hfsmval 30680 | Value of the sum of two Hi... |
| hfmmval 30681 | Value of the scalar produc... |
| hosval 30682 | Value of the sum of two Hi... |
| homval 30683 | Value of the scalar produc... |
| hodval 30684 | Value of the difference of... |
| hfsval 30685 | Value of the sum of two Hi... |
| hfmval 30686 | Value of the scalar produc... |
| hoscl 30687 | Closure of the sum of two ... |
| homcl 30688 | Closure of the scalar prod... |
| hodcl 30689 | Closure of the difference ... |
| ho0val 30692 | Value of the zero Hilbert ... |
| ho0f 30693 | Functionality of the zero ... |
| df0op2 30694 | Alternate definition of Hi... |
| dfiop2 30695 | Alternate definition of Hi... |
| hoif 30696 | Functionality of the Hilbe... |
| hoival 30697 | The value of the Hilbert s... |
| hoico1 30698 | Composition with the Hilbe... |
| hoico2 30699 | Composition with the Hilbe... |
| hoaddcl 30700 | The sum of Hilbert space o... |
| homulcl 30701 | The scalar product of a Hi... |
| hoeq 30702 | Equality of Hilbert space ... |
| hoeqi 30703 | Equality of Hilbert space ... |
| hoscli 30704 | Closure of Hilbert space o... |
| hodcli 30705 | Closure of Hilbert space o... |
| hocoi 30706 | Composition of Hilbert spa... |
| hococli 30707 | Closure of composition of ... |
| hocofi 30708 | Mapping of composition of ... |
| hocofni 30709 | Functionality of compositi... |
| hoaddcli 30710 | Mapping of sum of Hilbert ... |
| hosubcli 30711 | Mapping of difference of H... |
| hoaddfni 30712 | Functionality of sum of Hi... |
| hosubfni 30713 | Functionality of differenc... |
| hoaddcomi 30714 | Commutativity of sum of Hi... |
| hosubcl 30715 | Mapping of difference of H... |
| hoaddcom 30716 | Commutativity of sum of Hi... |
| hodsi 30717 | Relationship between Hilbe... |
| hoaddassi 30718 | Associativity of sum of Hi... |
| hoadd12i 30719 | Commutative/associative la... |
| hoadd32i 30720 | Commutative/associative la... |
| hocadddiri 30721 | Distributive law for Hilbe... |
| hocsubdiri 30722 | Distributive law for Hilbe... |
| ho2coi 30723 | Double composition of Hilb... |
| hoaddass 30724 | Associativity of sum of Hi... |
| hoadd32 30725 | Commutative/associative la... |
| hoadd4 30726 | Rearrangement of 4 terms i... |
| hocsubdir 30727 | Distributive law for Hilbe... |
| hoaddid1i 30728 | Sum of a Hilbert space ope... |
| hodidi 30729 | Difference of a Hilbert sp... |
| ho0coi 30730 | Composition of the zero op... |
| hoid1i 30731 | Composition of Hilbert spa... |
| hoid1ri 30732 | Composition of Hilbert spa... |
| hoaddid1 30733 | Sum of a Hilbert space ope... |
| hodid 30734 | Difference of a Hilbert sp... |
| hon0 30735 | A Hilbert space operator i... |
| hodseqi 30736 | Subtraction and addition o... |
| ho0subi 30737 | Subtraction of Hilbert spa... |
| honegsubi 30738 | Relationship between Hilbe... |
| ho0sub 30739 | Subtraction of Hilbert spa... |
| hosubid1 30740 | The zero operator subtract... |
| honegsub 30741 | Relationship between Hilbe... |
| homulid2 30742 | An operator equals its sca... |
| homco1 30743 | Associative law for scalar... |
| homulass 30744 | Scalar product associative... |
| hoadddi 30745 | Scalar product distributiv... |
| hoadddir 30746 | Scalar product reverse dis... |
| homul12 30747 | Swap first and second fact... |
| honegneg 30748 | Double negative of a Hilbe... |
| hosubneg 30749 | Relationship between opera... |
| hosubdi 30750 | Scalar product distributiv... |
| honegdi 30751 | Distribution of negative o... |
| honegsubdi 30752 | Distribution of negative o... |
| honegsubdi2 30753 | Distribution of negative o... |
| hosubsub2 30754 | Law for double subtraction... |
| hosub4 30755 | Rearrangement of 4 terms i... |
| hosubadd4 30756 | Rearrangement of 4 terms i... |
| hoaddsubass 30757 | Associative-type law for a... |
| hoaddsub 30758 | Law for operator addition ... |
| hosubsub 30759 | Law for double subtraction... |
| hosubsub4 30760 | Law for double subtraction... |
| ho2times 30761 | Two times a Hilbert space ... |
| hoaddsubassi 30762 | Associativity of sum and d... |
| hoaddsubi 30763 | Law for sum and difference... |
| hosd1i 30764 | Hilbert space operator sum... |
| hosd2i 30765 | Hilbert space operator sum... |
| hopncani 30766 | Hilbert space operator can... |
| honpcani 30767 | Hilbert space operator can... |
| hosubeq0i 30768 | If the difference between ... |
| honpncani 30769 | Hilbert space operator can... |
| ho01i 30770 | A condition implying that ... |
| ho02i 30771 | A condition implying that ... |
| hoeq1 30772 | A condition implying that ... |
| hoeq2 30773 | A condition implying that ... |
| adjmo 30774 | Every Hilbert space operat... |
| adjsym 30775 | Symmetry property of an ad... |
| eigrei 30776 | A necessary and sufficient... |
| eigre 30777 | A necessary and sufficient... |
| eigposi 30778 | A sufficient condition (fi... |
| eigorthi 30779 | A necessary and sufficient... |
| eigorth 30780 | A necessary and sufficient... |
| nmopval 30798 | Value of the norm of a Hil... |
| elcnop 30799 | Property defining a contin... |
| ellnop 30800 | Property defining a linear... |
| lnopf 30801 | A linear Hilbert space ope... |
| elbdop 30802 | Property defining a bounde... |
| bdopln 30803 | A bounded linear Hilbert s... |
| bdopf 30804 | A bounded linear Hilbert s... |
| nmopsetretALT 30805 | The set in the supremum of... |
| nmopsetretHIL 30806 | The set in the supremum of... |
| nmopsetn0 30807 | The set in the supremum of... |
| nmopxr 30808 | The norm of a Hilbert spac... |
| nmoprepnf 30809 | The norm of a Hilbert spac... |
| nmopgtmnf 30810 | The norm of a Hilbert spac... |
| nmopreltpnf 30811 | The norm of a Hilbert spac... |
| nmopre 30812 | The norm of a bounded oper... |
| elbdop2 30813 | Property defining a bounde... |
| elunop 30814 | Property defining a unitar... |
| elhmop 30815 | Property defining a Hermit... |
| hmopf 30816 | A Hermitian operator is a ... |
| hmopex 30817 | The class of Hermitian ope... |
| nmfnval 30818 | Value of the norm of a Hil... |
| nmfnsetre 30819 | The set in the supremum of... |
| nmfnsetn0 30820 | The set in the supremum of... |
| nmfnxr 30821 | The norm of any Hilbert sp... |
| nmfnrepnf 30822 | The norm of a Hilbert spac... |
| nlfnval 30823 | Value of the null space of... |
| elcnfn 30824 | Property defining a contin... |
| ellnfn 30825 | Property defining a linear... |
| lnfnf 30826 | A linear Hilbert space fun... |
| dfadj2 30827 | Alternate definition of th... |
| funadj 30828 | Functionality of the adjoi... |
| dmadjss 30829 | The domain of the adjoint ... |
| dmadjop 30830 | A member of the domain of ... |
| adjeu 30831 | Elementhood in the domain ... |
| adjval 30832 | Value of the adjoint funct... |
| adjval2 30833 | Value of the adjoint funct... |
| cnvadj 30834 | The adjoint function equal... |
| funcnvadj 30835 | The converse of the adjoin... |
| adj1o 30836 | The adjoint function maps ... |
| dmadjrn 30837 | The adjoint of an operator... |
| eigvecval 30838 | The set of eigenvectors of... |
| eigvalfval 30839 | The eigenvalues of eigenve... |
| specval 30840 | The value of the spectrum ... |
| speccl 30841 | The spectrum of an operato... |
| hhlnoi 30842 | The linear operators of Hi... |
| hhnmoi 30843 | The norm of an operator in... |
| hhbloi 30844 | A bounded linear operator ... |
| hh0oi 30845 | The zero operator in Hilbe... |
| hhcno 30846 | The continuous operators o... |
| hhcnf 30847 | The continuous functionals... |
| dmadjrnb 30848 | The adjoint of an operator... |
| nmoplb 30849 | A lower bound for an opera... |
| nmopub 30850 | An upper bound for an oper... |
| nmopub2tALT 30851 | An upper bound for an oper... |
| nmopub2tHIL 30852 | An upper bound for an oper... |
| nmopge0 30853 | The norm of any Hilbert sp... |
| nmopgt0 30854 | A linear Hilbert space ope... |
| cnopc 30855 | Basic continuity property ... |
| lnopl 30856 | Basic linearity property o... |
| unop 30857 | Basic inner product proper... |
| unopf1o 30858 | A unitary operator in Hilb... |
| unopnorm 30859 | A unitary operator is idem... |
| cnvunop 30860 | The inverse (converse) of ... |
| unopadj 30861 | The inverse (converse) of ... |
| unoplin 30862 | A unitary operator is line... |
| counop 30863 | The composition of two uni... |
| hmop 30864 | Basic inner product proper... |
| hmopre 30865 | The inner product of the v... |
| nmfnlb 30866 | A lower bound for a functi... |
| nmfnleub 30867 | An upper bound for the nor... |
| nmfnleub2 30868 | An upper bound for the nor... |
| nmfnge0 30869 | The norm of any Hilbert sp... |
| elnlfn 30870 | Membership in the null spa... |
| elnlfn2 30871 | Membership in the null spa... |
| cnfnc 30872 | Basic continuity property ... |
| lnfnl 30873 | Basic linearity property o... |
| adjcl 30874 | Closure of the adjoint of ... |
| adj1 30875 | Property of an adjoint Hil... |
| adj2 30876 | Property of an adjoint Hil... |
| adjeq 30877 | A property that determines... |
| adjadj 30878 | Double adjoint. Theorem 3... |
| adjvalval 30879 | Value of the value of the ... |
| unopadj2 30880 | The adjoint of a unitary o... |
| hmopadj 30881 | A Hermitian operator is se... |
| hmdmadj 30882 | Every Hermitian operator h... |
| hmopadj2 30883 | An operator is Hermitian i... |
| hmoplin 30884 | A Hermitian operator is li... |
| brafval 30885 | The bra of a vector, expre... |
| braval 30886 | A bra-ket juxtaposition, e... |
| braadd 30887 | Linearity property of bra ... |
| bramul 30888 | Linearity property of bra ... |
| brafn 30889 | The bra function is a func... |
| bralnfn 30890 | The Dirac bra function is ... |
| bracl 30891 | Closure of the bra functio... |
| bra0 30892 | The Dirac bra of the zero ... |
| brafnmul 30893 | Anti-linearity property of... |
| kbfval 30894 | The outer product of two v... |
| kbop 30895 | The outer product of two v... |
| kbval 30896 | The value of the operator ... |
| kbmul 30897 | Multiplication property of... |
| kbpj 30898 | If a vector ` A ` has norm... |
| eleigvec 30899 | Membership in the set of e... |
| eleigvec2 30900 | Membership in the set of e... |
| eleigveccl 30901 | Closure of an eigenvector ... |
| eigvalval 30902 | The eigenvalue of an eigen... |
| eigvalcl 30903 | An eigenvalue is a complex... |
| eigvec1 30904 | Property of an eigenvector... |
| eighmre 30905 | The eigenvalues of a Hermi... |
| eighmorth 30906 | Eigenvectors of a Hermitia... |
| nmopnegi 30907 | Value of the norm of the n... |
| lnop0 30908 | The value of a linear Hilb... |
| lnopmul 30909 | Multiplicative property of... |
| lnopli 30910 | Basic scalar product prope... |
| lnopfi 30911 | A linear Hilbert space ope... |
| lnop0i 30912 | The value of a linear Hilb... |
| lnopaddi 30913 | Additive property of a lin... |
| lnopmuli 30914 | Multiplicative property of... |
| lnopaddmuli 30915 | Sum/product property of a ... |
| lnopsubi 30916 | Subtraction property for a... |
| lnopsubmuli 30917 | Subtraction/product proper... |
| lnopmulsubi 30918 | Product/subtraction proper... |
| homco2 30919 | Move a scalar product out ... |
| idunop 30920 | The identity function (res... |
| 0cnop 30921 | The identically zero funct... |
| 0cnfn 30922 | The identically zero funct... |
| idcnop 30923 | The identity function (res... |
| idhmop 30924 | The Hilbert space identity... |
| 0hmop 30925 | The identically zero funct... |
| 0lnop 30926 | The identically zero funct... |
| 0lnfn 30927 | The identically zero funct... |
| nmop0 30928 | The norm of the zero opera... |
| nmfn0 30929 | The norm of the identicall... |
| hmopbdoptHIL 30930 | A Hermitian operator is a ... |
| hoddii 30931 | Distributive law for Hilbe... |
| hoddi 30932 | Distributive law for Hilbe... |
| nmop0h 30933 | The norm of any operator o... |
| idlnop 30934 | The identity function (res... |
| 0bdop 30935 | The identically zero opera... |
| adj0 30936 | Adjoint of the zero operat... |
| nmlnop0iALT 30937 | A linear operator with a z... |
| nmlnop0iHIL 30938 | A linear operator with a z... |
| nmlnopgt0i 30939 | A linear Hilbert space ope... |
| nmlnop0 30940 | A linear operator with a z... |
| nmlnopne0 30941 | A linear operator with a n... |
| lnopmi 30942 | The scalar product of a li... |
| lnophsi 30943 | The sum of two linear oper... |
| lnophdi 30944 | The difference of two line... |
| lnopcoi 30945 | The composition of two lin... |
| lnopco0i 30946 | The composition of a linea... |
| lnopeq0lem1 30947 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 30948 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 30949 | A condition implying that ... |
| lnopeqi 30950 | Two linear Hilbert space o... |
| lnopeq 30951 | Two linear Hilbert space o... |
| lnopunilem1 30952 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 30953 | Lemma for ~ lnopunii . (C... |
| lnopunii 30954 | If a linear operator (whos... |
| elunop2 30955 | An operator is unitary iff... |
| nmopun 30956 | Norm of a unitary Hilbert ... |
| unopbd 30957 | A unitary operator is a bo... |
| lnophmlem1 30958 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 30959 | Lemma for ~ lnophmi . (Co... |
| lnophmi 30960 | A linear operator is Hermi... |
| lnophm 30961 | A linear operator is Hermi... |
| hmops 30962 | The sum of two Hermitian o... |
| hmopm 30963 | The scalar product of a He... |
| hmopd 30964 | The difference of two Herm... |
| hmopco 30965 | The composition of two com... |
| nmbdoplbi 30966 | A lower bound for the norm... |
| nmbdoplb 30967 | A lower bound for the norm... |
| nmcexi 30968 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 30969 | The norm of a continuous l... |
| nmcoplbi 30970 | A lower bound for the norm... |
| nmcopex 30971 | The norm of a continuous l... |
| nmcoplb 30972 | A lower bound for the norm... |
| nmophmi 30973 | The norm of the scalar pro... |
| bdophmi 30974 | The scalar product of a bo... |
| lnconi 30975 | Lemma for ~ lnopconi and ~... |
| lnopconi 30976 | A condition equivalent to ... |
| lnopcon 30977 | A condition equivalent to ... |
| lnopcnbd 30978 | A linear operator is conti... |
| lncnopbd 30979 | A continuous linear operat... |
| lncnbd 30980 | A continuous linear operat... |
| lnopcnre 30981 | A linear operator is conti... |
| lnfnli 30982 | Basic property of a linear... |
| lnfnfi 30983 | A linear Hilbert space fun... |
| lnfn0i 30984 | The value of a linear Hilb... |
| lnfnaddi 30985 | Additive property of a lin... |
| lnfnmuli 30986 | Multiplicative property of... |
| lnfnaddmuli 30987 | Sum/product property of a ... |
| lnfnsubi 30988 | Subtraction property for a... |
| lnfn0 30989 | The value of a linear Hilb... |
| lnfnmul 30990 | Multiplicative property of... |
| nmbdfnlbi 30991 | A lower bound for the norm... |
| nmbdfnlb 30992 | A lower bound for the norm... |
| nmcfnexi 30993 | The norm of a continuous l... |
| nmcfnlbi 30994 | A lower bound for the norm... |
| nmcfnex 30995 | The norm of a continuous l... |
| nmcfnlb 30996 | A lower bound of the norm ... |
| lnfnconi 30997 | A condition equivalent to ... |
| lnfncon 30998 | A condition equivalent to ... |
| lnfncnbd 30999 | A linear functional is con... |
| imaelshi 31000 | The image of a subspace un... |
| rnelshi 31001 | The range of a linear oper... |
| nlelshi 31002 | The null space of a linear... |
| nlelchi 31003 | The null space of a contin... |
| riesz3i 31004 | A continuous linear functi... |
| riesz4i 31005 | A continuous linear functi... |
| riesz4 31006 | A continuous linear functi... |
| riesz1 31007 | Part 1 of the Riesz repres... |
| riesz2 31008 | Part 2 of the Riesz repres... |
| cnlnadjlem1 31009 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 31010 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 31011 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 31012 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 31013 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 31014 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 31015 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 31016 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 31017 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 31018 | Every continuous linear op... |
| cnlnadjeui 31019 | Every continuous linear op... |
| cnlnadjeu 31020 | Every continuous linear op... |
| cnlnadj 31021 | Every continuous linear op... |
| cnlnssadj 31022 | Every continuous linear Hi... |
| bdopssadj 31023 | Every bounded linear Hilbe... |
| bdopadj 31024 | Every bounded linear Hilbe... |
| adjbdln 31025 | The adjoint of a bounded l... |
| adjbdlnb 31026 | An operator is bounded and... |
| adjbd1o 31027 | The mapping of adjoints of... |
| adjlnop 31028 | The adjoint of an operator... |
| adjsslnop 31029 | Every operator with an adj... |
| nmopadjlei 31030 | Property of the norm of an... |
| nmopadjlem 31031 | Lemma for ~ nmopadji . (C... |
| nmopadji 31032 | Property of the norm of an... |
| adjeq0 31033 | An operator is zero iff it... |
| adjmul 31034 | The adjoint of the scalar ... |
| adjadd 31035 | The adjoint of the sum of ... |
| nmoptrii 31036 | Triangle inequality for th... |
| nmopcoi 31037 | Upper bound for the norm o... |
| bdophsi 31038 | The sum of two bounded lin... |
| bdophdi 31039 | The difference between two... |
| bdopcoi 31040 | The composition of two bou... |
| nmoptri2i 31041 | Triangle-type inequality f... |
| adjcoi 31042 | The adjoint of a compositi... |
| nmopcoadji 31043 | The norm of an operator co... |
| nmopcoadj2i 31044 | The norm of an operator co... |
| nmopcoadj0i 31045 | An operator composed with ... |
| unierri 31046 | If we approximate a chain ... |
| branmfn 31047 | The norm of the bra functi... |
| brabn 31048 | The bra of a vector is a b... |
| rnbra 31049 | The set of bras equals the... |
| bra11 31050 | The bra function maps vect... |
| bracnln 31051 | A bra is a continuous line... |
| cnvbraval 31052 | Value of the converse of t... |
| cnvbracl 31053 | Closure of the converse of... |
| cnvbrabra 31054 | The converse bra of the br... |
| bracnvbra 31055 | The bra of the converse br... |
| bracnlnval 31056 | The vector that a continuo... |
| cnvbramul 31057 | Multiplication property of... |
| kbass1 31058 | Dirac bra-ket associative ... |
| kbass2 31059 | Dirac bra-ket associative ... |
| kbass3 31060 | Dirac bra-ket associative ... |
| kbass4 31061 | Dirac bra-ket associative ... |
| kbass5 31062 | Dirac bra-ket associative ... |
| kbass6 31063 | Dirac bra-ket associative ... |
| leopg 31064 | Ordering relation for posi... |
| leop 31065 | Ordering relation for oper... |
| leop2 31066 | Ordering relation for oper... |
| leop3 31067 | Operator ordering in terms... |
| leoppos 31068 | Binary relation defining a... |
| leoprf2 31069 | The ordering relation for ... |
| leoprf 31070 | The ordering relation for ... |
| leopsq 31071 | The square of a Hermitian ... |
| 0leop 31072 | The zero operator is a pos... |
| idleop 31073 | The identity operator is a... |
| leopadd 31074 | The sum of two positive op... |
| leopmuli 31075 | The scalar product of a no... |
| leopmul 31076 | The scalar product of a po... |
| leopmul2i 31077 | Scalar product applied to ... |
| leoptri 31078 | The positive operator orde... |
| leoptr 31079 | The positive operator orde... |
| leopnmid 31080 | A bounded Hermitian operat... |
| nmopleid 31081 | A nonzero, bounded Hermiti... |
| opsqrlem1 31082 | Lemma for opsqri . (Contr... |
| opsqrlem2 31083 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 31084 | Lemma for opsqri . (Contr... |
| opsqrlem4 31085 | Lemma for opsqri . (Contr... |
| opsqrlem5 31086 | Lemma for opsqri . (Contr... |
| opsqrlem6 31087 | Lemma for opsqri . (Contr... |
| pjhmopi 31088 | A projector is a Hermitian... |
| pjlnopi 31089 | A projector is a linear op... |
| pjnmopi 31090 | The operator norm of a pro... |
| pjbdlni 31091 | A projector is a bounded l... |
| pjhmop 31092 | A projection is a Hermitia... |
| hmopidmchi 31093 | An idempotent Hermitian op... |
| hmopidmpji 31094 | An idempotent Hermitian op... |
| hmopidmch 31095 | An idempotent Hermitian op... |
| hmopidmpj 31096 | An idempotent Hermitian op... |
| pjsdii 31097 | Distributive law for Hilbe... |
| pjddii 31098 | Distributive law for Hilbe... |
| pjsdi2i 31099 | Chained distributive law f... |
| pjcoi 31100 | Composition of projections... |
| pjcocli 31101 | Closure of composition of ... |
| pjcohcli 31102 | Closure of composition of ... |
| pjadjcoi 31103 | Adjoint of composition of ... |
| pjcofni 31104 | Functionality of compositi... |
| pjss1coi 31105 | Subset relationship for pr... |
| pjss2coi 31106 | Subset relationship for pr... |
| pjssmi 31107 | Projection meet property. ... |
| pjssge0i 31108 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 31109 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 31110 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 31111 | Composition of projections... |
| pjscji 31112 | The projection of orthogon... |
| pjssumi 31113 | The projection on a subspa... |
| pjssposi 31114 | Projector ordering can be ... |
| pjordi 31115 | The definition of projecto... |
| pjssdif2i 31116 | The projection subspace of... |
| pjssdif1i 31117 | A necessary and sufficient... |
| pjimai 31118 | The image of a projection.... |
| pjidmcoi 31119 | A projection is idempotent... |
| pjoccoi 31120 | Composition of projections... |
| pjtoi 31121 | Subspace sum of projection... |
| pjoci 31122 | Projection of orthocomplem... |
| pjidmco 31123 | A projection operator is i... |
| dfpjop 31124 | Definition of projection o... |
| pjhmopidm 31125 | Two ways to express the se... |
| elpjidm 31126 | A projection operator is i... |
| elpjhmop 31127 | A projection operator is H... |
| 0leopj 31128 | A projector is a positive ... |
| pjadj2 31129 | A projector is self-adjoin... |
| pjadj3 31130 | A projector is self-adjoin... |
| elpjch 31131 | Reconstruction of the subs... |
| elpjrn 31132 | Reconstruction of the subs... |
| pjinvari 31133 | A closed subspace ` H ` wi... |
| pjin1i 31134 | Lemma for Theorem 1.22 of ... |
| pjin2i 31135 | Lemma for Theorem 1.22 of ... |
| pjin3i 31136 | Lemma for Theorem 1.22 of ... |
| pjclem1 31137 | Lemma for projection commu... |
| pjclem2 31138 | Lemma for projection commu... |
| pjclem3 31139 | Lemma for projection commu... |
| pjclem4a 31140 | Lemma for projection commu... |
| pjclem4 31141 | Lemma for projection commu... |
| pjci 31142 | Two subspaces commute iff ... |
| pjcmul1i 31143 | A necessary and sufficient... |
| pjcmul2i 31144 | The projection subspace of... |
| pjcohocli 31145 | Closure of composition of ... |
| pjadj2coi 31146 | Adjoint of double composit... |
| pj2cocli 31147 | Closure of double composit... |
| pj3lem1 31148 | Lemma for projection tripl... |
| pj3si 31149 | Stronger projection triple... |
| pj3i 31150 | Projection triplet theorem... |
| pj3cor1i 31151 | Projection triplet corolla... |
| pjs14i 31152 | Theorem S-14 of Watanabe, ... |
| isst 31155 | Property of a state. (Con... |
| ishst 31156 | Property of a complex Hilb... |
| sticl 31157 | ` [ 0 , 1 ] ` closure of t... |
| stcl 31158 | Real closure of the value ... |
| hstcl 31159 | Closure of the value of a ... |
| hst1a 31160 | Unit value of a Hilbert-sp... |
| hstel2 31161 | Properties of a Hilbert-sp... |
| hstorth 31162 | Orthogonality property of ... |
| hstosum 31163 | Orthogonal sum property of... |
| hstoc 31164 | Sum of a Hilbert-space-val... |
| hstnmoc 31165 | Sum of norms of a Hilbert-... |
| stge0 31166 | The value of a state is no... |
| stle1 31167 | The value of a state is le... |
| hstle1 31168 | The norm of the value of a... |
| hst1h 31169 | The norm of a Hilbert-spac... |
| hst0h 31170 | The norm of a Hilbert-spac... |
| hstpyth 31171 | Pythagorean property of a ... |
| hstle 31172 | Ordering property of a Hil... |
| hstles 31173 | Ordering property of a Hil... |
| hstoh 31174 | A Hilbert-space-valued sta... |
| hst0 31175 | A Hilbert-space-valued sta... |
| sthil 31176 | The value of a state at th... |
| stj 31177 | The value of a state on a ... |
| sto1i 31178 | The state of a subspace pl... |
| sto2i 31179 | The state of the orthocomp... |
| stge1i 31180 | If a state is greater than... |
| stle0i 31181 | If a state is less than or... |
| stlei 31182 | Ordering law for states. ... |
| stlesi 31183 | Ordering law for states. ... |
| stji1i 31184 | Join of components of Sasa... |
| stm1i 31185 | State of component of unit... |
| stm1ri 31186 | State of component of unit... |
| stm1addi 31187 | Sum of states whose meet i... |
| staddi 31188 | If the sum of 2 states is ... |
| stm1add3i 31189 | Sum of states whose meet i... |
| stadd3i 31190 | If the sum of 3 states is ... |
| st0 31191 | The state of the zero subs... |
| strlem1 31192 | Lemma for strong state the... |
| strlem2 31193 | Lemma for strong state the... |
| strlem3a 31194 | Lemma for strong state the... |
| strlem3 31195 | Lemma for strong state the... |
| strlem4 31196 | Lemma for strong state the... |
| strlem5 31197 | Lemma for strong state the... |
| strlem6 31198 | Lemma for strong state the... |
| stri 31199 | Strong state theorem. The... |
| strb 31200 | Strong state theorem (bidi... |
| hstrlem2 31201 | Lemma for strong set of CH... |
| hstrlem3a 31202 | Lemma for strong set of CH... |
| hstrlem3 31203 | Lemma for strong set of CH... |
| hstrlem4 31204 | Lemma for strong set of CH... |
| hstrlem5 31205 | Lemma for strong set of CH... |
| hstrlem6 31206 | Lemma for strong set of CH... |
| hstri 31207 | Hilbert space admits a str... |
| hstrbi 31208 | Strong CH-state theorem (b... |
| largei 31209 | A Hilbert lattice admits a... |
| jplem1 31210 | Lemma for Jauch-Piron theo... |
| jplem2 31211 | Lemma for Jauch-Piron theo... |
| jpi 31212 | The function ` S ` , that ... |
| golem1 31213 | Lemma for Godowski's equat... |
| golem2 31214 | Lemma for Godowski's equat... |
| goeqi 31215 | Godowski's equation, shown... |
| stcltr1i 31216 | Property of a strong class... |
| stcltr2i 31217 | Property of a strong class... |
| stcltrlem1 31218 | Lemma for strong classical... |
| stcltrlem2 31219 | Lemma for strong classical... |
| stcltrthi 31220 | Theorem for classically st... |
| cvbr 31224 | Binary relation expressing... |
| cvbr2 31225 | Binary relation expressing... |
| cvcon3 31226 | Contraposition law for the... |
| cvpss 31227 | The covers relation implie... |
| cvnbtwn 31228 | The covers relation implie... |
| cvnbtwn2 31229 | The covers relation implie... |
| cvnbtwn3 31230 | The covers relation implie... |
| cvnbtwn4 31231 | The covers relation implie... |
| cvnsym 31232 | The covers relation is not... |
| cvnref 31233 | The covers relation is not... |
| cvntr 31234 | The covers relation is not... |
| spansncv2 31235 | Hilbert space has the cove... |
| mdbr 31236 | Binary relation expressing... |
| mdi 31237 | Consequence of the modular... |
| mdbr2 31238 | Binary relation expressing... |
| mdbr3 31239 | Binary relation expressing... |
| mdbr4 31240 | Binary relation expressing... |
| dmdbr 31241 | Binary relation expressing... |
| dmdmd 31242 | The dual modular pair prop... |
| mddmd 31243 | The modular pair property ... |
| dmdi 31244 | Consequence of the dual mo... |
| dmdbr2 31245 | Binary relation expressing... |
| dmdi2 31246 | Consequence of the dual mo... |
| dmdbr3 31247 | Binary relation expressing... |
| dmdbr4 31248 | Binary relation expressing... |
| dmdi4 31249 | Consequence of the dual mo... |
| dmdbr5 31250 | Binary relation expressing... |
| mddmd2 31251 | Relationship between modul... |
| mdsl0 31252 | A sublattice condition tha... |
| ssmd1 31253 | Ordering implies the modul... |
| ssmd2 31254 | Ordering implies the modul... |
| ssdmd1 31255 | Ordering implies the dual ... |
| ssdmd2 31256 | Ordering implies the dual ... |
| dmdsl3 31257 | Sublattice mapping for a d... |
| mdsl3 31258 | Sublattice mapping for a m... |
| mdslle1i 31259 | Order preservation of the ... |
| mdslle2i 31260 | Order preservation of the ... |
| mdslj1i 31261 | Join preservation of the o... |
| mdslj2i 31262 | Meet preservation of the r... |
| mdsl1i 31263 | If the modular pair proper... |
| mdsl2i 31264 | If the modular pair proper... |
| mdsl2bi 31265 | If the modular pair proper... |
| cvmdi 31266 | The covering property impl... |
| mdslmd1lem1 31267 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 31268 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 31269 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 31270 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 31271 | Preservation of the modula... |
| mdslmd2i 31272 | Preservation of the modula... |
| mdsldmd1i 31273 | Preservation of the dual m... |
| mdslmd3i 31274 | Modular pair conditions th... |
| mdslmd4i 31275 | Modular pair condition tha... |
| csmdsymi 31276 | Cross-symmetry implies M-s... |
| mdexchi 31277 | An exchange lemma for modu... |
| cvmd 31278 | The covering property impl... |
| cvdmd 31279 | The covering property impl... |
| ela 31281 | Atoms in a Hilbert lattice... |
| elat2 31282 | Expanded membership relati... |
| elatcv0 31283 | A Hilbert lattice element ... |
| atcv0 31284 | An atom covers the zero su... |
| atssch 31285 | Atoms are a subset of the ... |
| atelch 31286 | An atom is a Hilbert latti... |
| atne0 31287 | An atom is not the Hilbert... |
| atss 31288 | A lattice element smaller ... |
| atsseq 31289 | Two atoms in a subset rela... |
| atcveq0 31290 | A Hilbert lattice element ... |
| h1da 31291 | A 1-dimensional subspace i... |
| spansna 31292 | The span of the singleton ... |
| sh1dle 31293 | A 1-dimensional subspace i... |
| ch1dle 31294 | A 1-dimensional subspace i... |
| atom1d 31295 | The 1-dimensional subspace... |
| superpos 31296 | Superposition Principle. ... |
| chcv1 31297 | The Hilbert lattice has th... |
| chcv2 31298 | The Hilbert lattice has th... |
| chjatom 31299 | The join of a closed subsp... |
| shatomici 31300 | The lattice of Hilbert sub... |
| hatomici 31301 | The Hilbert lattice is ato... |
| hatomic 31302 | A Hilbert lattice is atomi... |
| shatomistici 31303 | The lattice of Hilbert sub... |
| hatomistici 31304 | ` CH ` is atomistic, i.e. ... |
| chpssati 31305 | Two Hilbert lattice elemen... |
| chrelati 31306 | The Hilbert lattice is rel... |
| chrelat2i 31307 | A consequence of relative ... |
| cvati 31308 | If a Hilbert lattice eleme... |
| cvbr4i 31309 | An alternate way to expres... |
| cvexchlem 31310 | Lemma for ~ cvexchi . (Co... |
| cvexchi 31311 | The Hilbert lattice satisf... |
| chrelat2 31312 | A consequence of relative ... |
| chrelat3 31313 | A consequence of relative ... |
| chrelat3i 31314 | A consequence of the relat... |
| chrelat4i 31315 | A consequence of relative ... |
| cvexch 31316 | The Hilbert lattice satisf... |
| cvp 31317 | The Hilbert lattice satisf... |
| atnssm0 31318 | The meet of a Hilbert latt... |
| atnemeq0 31319 | The meet of distinct atoms... |
| atssma 31320 | The meet with an atom's su... |
| atcv0eq 31321 | Two atoms covering the zer... |
| atcv1 31322 | Two atoms covering the zer... |
| atexch 31323 | The Hilbert lattice satisf... |
| atomli 31324 | An assertion holding in at... |
| atoml2i 31325 | An assertion holding in at... |
| atordi 31326 | An ordering law for a Hilb... |
| atcvatlem 31327 | Lemma for ~ atcvati . (Co... |
| atcvati 31328 | A nonzero Hilbert lattice ... |
| atcvat2i 31329 | A Hilbert lattice element ... |
| atord 31330 | An ordering law for a Hilb... |
| atcvat2 31331 | A Hilbert lattice element ... |
| chirredlem1 31332 | Lemma for ~ chirredi . (C... |
| chirredlem2 31333 | Lemma for ~ chirredi . (C... |
| chirredlem3 31334 | Lemma for ~ chirredi . (C... |
| chirredlem4 31335 | Lemma for ~ chirredi . (C... |
| chirredi 31336 | The Hilbert lattice is irr... |
| chirred 31337 | The Hilbert lattice is irr... |
| atcvat3i 31338 | A condition implying that ... |
| atcvat4i 31339 | A condition implying exist... |
| atdmd 31340 | Two Hilbert lattice elemen... |
| atmd 31341 | Two Hilbert lattice elemen... |
| atmd2 31342 | Two Hilbert lattice elemen... |
| atabsi 31343 | Absorption of an incompara... |
| atabs2i 31344 | Absorption of an incompara... |
| mdsymlem1 31345 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 31346 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 31347 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 31348 | Lemma for ~ mdsymi . This... |
| mdsymlem5 31349 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 31350 | Lemma for ~ mdsymi . This... |
| mdsymlem7 31351 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 31352 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 31353 | M-symmetry of the Hilbert ... |
| mdsym 31354 | M-symmetry of the Hilbert ... |
| dmdsym 31355 | Dual M-symmetry of the Hil... |
| atdmd2 31356 | Two Hilbert lattice elemen... |
| sumdmdii 31357 | If the subspace sum of two... |
| cmmdi 31358 | Commuting subspaces form a... |
| cmdmdi 31359 | Commuting subspaces form a... |
| sumdmdlem 31360 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 31361 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 31362 | The subspace sum of two Hi... |
| dmdbr4ati 31363 | Dual modular pair property... |
| dmdbr5ati 31364 | Dual modular pair property... |
| dmdbr6ati 31365 | Dual modular pair property... |
| dmdbr7ati 31366 | Dual modular pair property... |
| mdoc1i 31367 | Orthocomplements form a mo... |
| mdoc2i 31368 | Orthocomplements form a mo... |
| dmdoc1i 31369 | Orthocomplements form a du... |
| dmdoc2i 31370 | Orthocomplements form a du... |
| mdcompli 31371 | A condition equivalent to ... |
| dmdcompli 31372 | A condition equivalent to ... |
| mddmdin0i 31373 | If dual modular implies mo... |
| cdjreui 31374 | A member of the sum of dis... |
| cdj1i 31375 | Two ways to express " ` A ... |
| cdj3lem1 31376 | A property of " ` A ` and ... |
| cdj3lem2 31377 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 31378 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 31379 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 31380 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 31381 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 31382 | Lemma for ~ cdj3i . The s... |
| cdj3i 31383 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 31384 | (_This theorem is a dummy ... |
| sa-abvi 31385 | A theorem about the univer... |
| xfree 31386 | A partial converse to ~ 19... |
| xfree2 31387 | A partial converse to ~ 19... |
| addltmulALT 31388 | A proof readability experi... |
| bian1d 31389 | Adding a superfluous conju... |
| or3di 31390 | Distributive law for disju... |
| or3dir 31391 | Distributive law for disju... |
| 3o1cs 31392 | Deduction eliminating disj... |
| 3o2cs 31393 | Deduction eliminating disj... |
| 3o3cs 31394 | Deduction eliminating disj... |
| 13an22anass 31395 | Associative law for four c... |
| sbc2iedf 31396 | Conversion of implicit sub... |
| rspc2daf 31397 | Double restricted speciali... |
| ralcom4f 31398 | Commutation of restricted ... |
| rexcom4f 31399 | Commutation of restricted ... |
| 19.9d2rf 31400 | A deduction version of one... |
| 19.9d2r 31401 | A deduction version of one... |
| r19.29ffa 31402 | A commonly used pattern ba... |
| eqtrb 31403 | A transposition of equalit... |
| opsbc2ie 31404 | Conversion of implicit sub... |
| opreu2reuALT 31405 | Correspondence between uni... |
| 2reucom 31408 | Double restricted existent... |
| 2reu2rex1 31409 | Double restricted existent... |
| 2reureurex 31410 | Double restricted existent... |
| 2reu2reu2 31411 | Double restricted existent... |
| opreu2reu1 31412 | Equivalent definition of t... |
| sq2reunnltb 31413 | There exists a unique deco... |
| addsqnot2reu 31414 | For each complex number ` ... |
| sbceqbidf 31415 | Equality theorem for class... |
| sbcies 31416 | A special version of class... |
| mo5f 31417 | Alternate definition of "a... |
| nmo 31418 | Negation of "at most one".... |
| reuxfrdf 31419 | Transfer existential uniqu... |
| rexunirn 31420 | Restricted existential qua... |
| rmoxfrd 31421 | Transfer "at most one" res... |
| rmoun 31422 | "At most one" restricted e... |
| rmounid 31423 | A case where an "at most o... |
| riotaeqbidva 31424 | Equivalent wff's yield equ... |
| dmrab 31425 | Domain of a restricted cla... |
| difrab2 31426 | Difference of two restrict... |
| rabexgfGS 31427 | Separation Scheme in terms... |
| rabsnel 31428 | Truth implied by equality ... |
| rabeqsnd 31429 | Conditions for a restricte... |
| eqrrabd 31430 | Deduce equality with a res... |
| foresf1o 31431 | From a surjective function... |
| rabfodom 31432 | Domination relation for re... |
| abrexdomjm 31433 | An indexed set is dominate... |
| abrexdom2jm 31434 | An indexed set is dominate... |
| abrexexd 31435 | Existence of a class abstr... |
| elabreximd 31436 | Class substitution in an i... |
| elabreximdv 31437 | Class substitution in an i... |
| abrexss 31438 | A necessary condition for ... |
| elunsn 31439 | Elementhood to a union wit... |
| nelun 31440 | Negated membership for a u... |
| snsssng 31441 | If a singleton is a subset... |
| inin 31442 | Intersection with an inter... |
| inindif 31443 | See ~ inundif . (Contribu... |
| difininv 31444 | Condition for the intersec... |
| difeq 31445 | Rewriting an equation with... |
| eqdif 31446 | If both set differences of... |
| indifbi 31447 | Two ways to express equali... |
| diffib 31448 | Case where ~ diffi is a bi... |
| difxp1ss 31449 | Difference law for Cartesi... |
| difxp2ss 31450 | Difference law for Cartesi... |
| undifr 31451 | Union of complementary par... |
| indifundif 31452 | A remarkable equation with... |
| elpwincl1 31453 | Closure of intersection wi... |
| elpwdifcl 31454 | Closure of class differenc... |
| elpwiuncl 31455 | Closure of indexed union w... |
| eqsnd 31456 | Deduce that a set is a sin... |
| elpreq 31457 | Equality wihin a pair. (C... |
| nelpr 31458 | A set ` A ` not in a pair ... |
| inpr0 31459 | Rewrite an empty intersect... |
| neldifpr1 31460 | The first element of a pai... |
| neldifpr2 31461 | The second element of a pa... |
| unidifsnel 31462 | The other element of a pai... |
| unidifsnne 31463 | The other element of a pai... |
| ifeqeqx 31464 | An equality theorem tailor... |
| elimifd 31465 | Elimination of a condition... |
| elim2if 31466 | Elimination of two conditi... |
| elim2ifim 31467 | Elimination of two conditi... |
| ifeq3da 31468 | Given an expression ` C ` ... |
| uniinn0 31469 | Sufficient and necessary c... |
| uniin1 31470 | Union of intersection. Ge... |
| uniin2 31471 | Union of intersection. Ge... |
| difuncomp 31472 | Express a class difference... |
| elpwunicl 31473 | Closure of a set union wit... |
| cbviunf 31474 | Rule used to change the bo... |
| iuneq12daf 31475 | Equality deduction for ind... |
| iunin1f 31476 | Indexed union of intersect... |
| ssiun3 31477 | Subset equivalence for an ... |
| ssiun2sf 31478 | Subset relationship for an... |
| iuninc 31479 | The union of an increasing... |
| iundifdifd 31480 | The intersection of a set ... |
| iundifdif 31481 | The intersection of a set ... |
| iunrdx 31482 | Re-index an indexed union.... |
| iunpreima 31483 | Preimage of an indexed uni... |
| iunrnmptss 31484 | A subset relation for an i... |
| iunxunsn 31485 | Appending a set to an inde... |
| iunxunpr 31486 | Appending two sets to an i... |
| iinabrex 31487 | Rewriting an indexed inter... |
| disjnf 31488 | In case ` x ` is not free ... |
| cbvdisjf 31489 | Change bound variables in ... |
| disjss1f 31490 | A subset of a disjoint col... |
| disjeq1f 31491 | Equality theorem for disjo... |
| disjxun0 31492 | Simplify a disjoint union.... |
| disjdifprg 31493 | A trivial partition into a... |
| disjdifprg2 31494 | A trivial partition of a s... |
| disji2f 31495 | Property of a disjoint col... |
| disjif 31496 | Property of a disjoint col... |
| disjorf 31497 | Two ways to say that a col... |
| disjorsf 31498 | Two ways to say that a col... |
| disjif2 31499 | Property of a disjoint col... |
| disjabrex 31500 | Rewriting a disjoint colle... |
| disjabrexf 31501 | Rewriting a disjoint colle... |
| disjpreima 31502 | A preimage of a disjoint s... |
| disjrnmpt 31503 | Rewriting a disjoint colle... |
| disjin 31504 | If a collection is disjoin... |
| disjin2 31505 | If a collection is disjoin... |
| disjxpin 31506 | Derive a disjunction over ... |
| iundisjf 31507 | Rewrite a countable union ... |
| iundisj2f 31508 | A disjoint union is disjoi... |
| disjrdx 31509 | Re-index a disjunct collec... |
| disjex 31510 | Two ways to say that two c... |
| disjexc 31511 | A variant of ~ disjex , ap... |
| disjunsn 31512 | Append an element to a dis... |
| disjun0 31513 | Adding the empty element p... |
| disjiunel 31514 | A set of elements B of a d... |
| disjuniel 31515 | A set of elements B of a d... |
| xpdisjres 31516 | Restriction of a constant ... |
| opeldifid 31517 | Ordered pair elementhood o... |
| difres 31518 | Case when class difference... |
| imadifxp 31519 | Image of the difference wi... |
| relfi 31520 | A relation (set) is finite... |
| reldisjun 31521 | Split a relation into two ... |
| 0res 31522 | Restriction of the empty f... |
| funresdm1 31523 | Restriction of a disjoint ... |
| fnunres1 31524 | Restriction of a disjoint ... |
| fcoinver 31525 | Build an equivalence relat... |
| fcoinvbr 31526 | Binary relation for the eq... |
| brabgaf 31527 | The law of concretion for ... |
| brelg 31528 | Two things in a binary rel... |
| br8d 31529 | Substitution for an eight-... |
| opabdm 31530 | Domain of an ordered-pair ... |
| opabrn 31531 | Range of an ordered-pair c... |
| opabssi 31532 | Sufficient condition for a... |
| opabid2ss 31533 | One direction of ~ opabid2... |
| ssrelf 31534 | A subclass relationship de... |
| eqrelrd2 31535 | A version of ~ eqrelrdv2 w... |
| erbr3b 31536 | Biconditional for equivale... |
| iunsnima 31537 | Image of a singleton by an... |
| iunsnima2 31538 | Version of ~ iunsnima with... |
| ac6sf2 31539 | Alternate version of ~ ac6... |
| fnresin 31540 | Restriction of a function ... |
| f1o3d 31541 | Describe an implicit one-t... |
| eldmne0 31542 | A function of nonempty dom... |
| f1rnen 31543 | Equinumerosity of the rang... |
| rinvf1o 31544 | Sufficient conditions for ... |
| fresf1o 31545 | Conditions for a restricti... |
| nfpconfp 31546 | The set of fixed points of... |
| fmptco1f1o 31547 | The action of composing (t... |
| cofmpt2 31548 | Express composition of a m... |
| f1mptrn 31549 | Express injection for a ma... |
| dfimafnf 31550 | Alternate definition of th... |
| funimass4f 31551 | Membership relation for th... |
| elimampt 31552 | Membership in the image of... |
| suppss2f 31553 | Show that the support of a... |
| fovcld 31554 | Closure law for an operati... |
| ofrn 31555 | The range of the function ... |
| ofrn2 31556 | The range of the function ... |
| off2 31557 | The function operation pro... |
| ofresid 31558 | Applying an operation rest... |
| fimarab 31559 | Expressing the image of a ... |
| unipreima 31560 | Preimage of a class union.... |
| opfv 31561 | Value of a function produc... |
| xppreima 31562 | The preimage of a Cartesia... |
| 2ndimaxp 31563 | Image of a cartesian produ... |
| djussxp2 31564 | Stronger version of ~ djus... |
| 2ndresdju 31565 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 31566 | The ` 2nd ` function restr... |
| xppreima2 31567 | The preimage of a Cartesia... |
| abfmpunirn 31568 | Membership in a union of a... |
| rabfmpunirn 31569 | Membership in a union of a... |
| abfmpeld 31570 | Membership in an element o... |
| abfmpel 31571 | Membership in an element o... |
| fmptdF 31572 | Domain and codomain of the... |
| fmptcof2 31573 | Composition of two functio... |
| fcomptf 31574 | Express composition of two... |
| acunirnmpt 31575 | Axiom of choice for the un... |
| acunirnmpt2 31576 | Axiom of choice for the un... |
| acunirnmpt2f 31577 | Axiom of choice for the un... |
| aciunf1lem 31578 | Choice in an index union. ... |
| aciunf1 31579 | Choice in an index union. ... |
| ofoprabco 31580 | Function operation as a co... |
| ofpreima 31581 | Express the preimage of a ... |
| ofpreima2 31582 | Express the preimage of a ... |
| funcnvmpt 31583 | Condition for a function i... |
| funcnv5mpt 31584 | Two ways to say that a fun... |
| funcnv4mpt 31585 | Two ways to say that a fun... |
| preimane 31586 | Different elements have di... |
| fnpreimac 31587 | Choose a set ` x ` contain... |
| fgreu 31588 | Exactly one point of a fun... |
| fcnvgreu 31589 | If the converse of a relat... |
| rnmposs 31590 | The range of an operation ... |
| mptssALT 31591 | Deduce subset relation of ... |
| dfcnv2 31592 | Alternative definition of ... |
| fnimatp 31593 | The image of an unordered ... |
| fnunres2 31594 | Restriction of a disjoint ... |
| rnexd 31595 | The range of a set is a se... |
| imaexd 31596 | The image of a set is a se... |
| mpomptxf 31597 | Express a two-argument fun... |
| suppovss 31598 | A bound for the support of... |
| fvdifsupp 31599 | Function value is zero out... |
| fmptssfisupp 31600 | The restriction of a mappi... |
| suppiniseg 31601 | Relation between the suppo... |
| fsuppinisegfi 31602 | The initial segment ` ( ``... |
| fressupp 31603 | The restriction of a funct... |
| fdifsuppconst 31604 | A function is a zero const... |
| ressupprn 31605 | The range of a function re... |
| supppreima 31606 | Express the support of a f... |
| fsupprnfi 31607 | Finite support implies fin... |
| cosnopne 31608 | Composition of two ordered... |
| cosnop 31609 | Composition of two ordered... |
| cnvprop 31610 | Converse of a pair of orde... |
| brprop 31611 | Binary relation for a pair... |
| mptprop 31612 | Rewrite pairs of ordered p... |
| coprprop 31613 | Composition of two pairs o... |
| gtiso 31614 | Two ways to write a strict... |
| isoun 31615 | Infer an isomorphism from ... |
| disjdsct 31616 | A disjoint collection is d... |
| df1stres 31617 | Definition for a restricti... |
| df2ndres 31618 | Definition for a restricti... |
| 1stpreimas 31619 | The preimage of a singleto... |
| 1stpreima 31620 | The preimage by ` 1st ` is... |
| 2ndpreima 31621 | The preimage by ` 2nd ` is... |
| curry2ima 31622 | The image of a curried fun... |
| preiman0 31623 | The preimage of a nonempty... |
| intimafv 31624 | The intersection of an ima... |
| ecref 31625 | All elements are in their ... |
| supssd 31626 | Inequality deduction for s... |
| infssd 31627 | Inequality deduction for i... |
| imafi2 31628 | The image by a finite set ... |
| unifi3 31629 | If a union is finite, then... |
| snct 31630 | A singleton is countable. ... |
| prct 31631 | An unordered pair is count... |
| mpocti 31632 | An operation is countable ... |
| abrexct 31633 | An image set of a countabl... |
| mptctf 31634 | A countable mapping set is... |
| abrexctf 31635 | An image set of a countabl... |
| padct 31636 | Index a countable set with... |
| cnvoprabOLD 31637 | The converse of a class ab... |
| f1od2 31638 | Sufficient condition for a... |
| fcobij 31639 | Composing functions with a... |
| fcobijfs 31640 | Composing finitely support... |
| suppss3 31641 | Deduce a function's suppor... |
| fsuppcurry1 31642 | Finite support of a currie... |
| fsuppcurry2 31643 | Finite support of a currie... |
| offinsupp1 31644 | Finite support for a funct... |
| ffs2 31645 | Rewrite a function's suppo... |
| ffsrn 31646 | The range of a finitely su... |
| resf1o 31647 | Restriction of functions t... |
| maprnin 31648 | Restricting the range of t... |
| fpwrelmapffslem 31649 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 31650 | Define a canonical mapping... |
| fpwrelmapffs 31651 | Define a canonical mapping... |
| creq0 31652 | The real representation of... |
| 1nei 31653 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 31654 | An integer unit times itse... |
| nnmulge 31655 | Multiplying by a positive ... |
| lt2addrd 31656 | If the right-hand side of ... |
| xrlelttric 31657 | Trichotomy law for extende... |
| xaddeq0 31658 | Two extended reals which a... |
| xrinfm 31659 | The extended real numbers ... |
| le2halvesd 31660 | A sum is less than the who... |
| xraddge02 31661 | A number is less than or e... |
| xrge0addge 31662 | A number is less than or e... |
| xlt2addrd 31663 | If the right-hand side of ... |
| xrsupssd 31664 | Inequality deduction for s... |
| xrge0infss 31665 | Any subset of nonnegative ... |
| xrge0infssd 31666 | Inequality deduction for i... |
| xrge0addcld 31667 | Nonnegative extended reals... |
| xrge0subcld 31668 | Condition for closure of n... |
| infxrge0lb 31669 | A member of a set of nonne... |
| infxrge0glb 31670 | The infimum of a set of no... |
| infxrge0gelb 31671 | The infimum of a set of no... |
| xrofsup 31672 | The supremum is preserved ... |
| supxrnemnf 31673 | The supremum of a nonempty... |
| xnn0gt0 31674 | Nonzero extended nonnegati... |
| xnn01gt 31675 | An extended nonnegative in... |
| nn0xmulclb 31676 | Finite multiplication in t... |
| joiniooico 31677 | Disjoint joining an open i... |
| ubico 31678 | A right-open interval does... |
| xeqlelt 31679 | Equality in terms of 'less... |
| eliccelico 31680 | Relate elementhood to a cl... |
| elicoelioo 31681 | Relate elementhood to a cl... |
| iocinioc2 31682 | Intersection between two o... |
| xrdifh 31683 | Class difference of a half... |
| iocinif 31684 | Relate intersection of two... |
| difioo 31685 | The difference between two... |
| difico 31686 | The difference between two... |
| uzssico 31687 | Upper integer sets are a s... |
| fz2ssnn0 31688 | A finite set of sequential... |
| nndiffz1 31689 | Upper set of the positive ... |
| ssnnssfz 31690 | For any finite subset of `... |
| fzne1 31691 | Elementhood in a finite se... |
| fzm1ne1 31692 | Elementhood of an integer ... |
| fzspl 31693 | Split the last element of ... |
| fzdif2 31694 | Split the last element of ... |
| fzodif2 31695 | Split the last element of ... |
| fzodif1 31696 | Set difference of two half... |
| fzsplit3 31697 | Split a finite interval of... |
| bcm1n 31698 | The proportion of one bino... |
| iundisjfi 31699 | Rewrite a countable union ... |
| iundisj2fi 31700 | A disjoint union is disjoi... |
| iundisjcnt 31701 | Rewrite a countable union ... |
| iundisj2cnt 31702 | A countable disjoint union... |
| fzone1 31703 | Elementhood in a half-open... |
| fzom1ne1 31704 | Elementhood in a half-open... |
| f1ocnt 31705 | Given a countable set ` A ... |
| fz1nnct 31706 | NN and integer ranges star... |
| fz1nntr 31707 | NN and integer ranges star... |
| hashunif 31708 | The cardinality of a disjo... |
| hashxpe 31709 | The size of the Cartesian ... |
| hashgt1 31710 | Restate "set contains at l... |
| dvdszzq 31711 | Divisibility for an intege... |
| prmdvdsbc 31712 | Condition for a prime numb... |
| numdenneg 31713 | Numerator and denominator ... |
| divnumden2 31714 | Calculate the reduced form... |
| nnindf 31715 | Principle of Mathematical ... |
| nn0min 31716 | Extracting the minimum pos... |
| subne0nn 31717 | A nonnegative difference i... |
| ltesubnnd 31718 | Subtracting an integer num... |
| fprodeq02 31719 | If one of the factors is z... |
| pr01ssre 31720 | The range of the indicator... |
| fprodex01 31721 | A product of factors equal... |
| prodpr 31722 | A product over a pair is t... |
| prodtp 31723 | A product over a triple is... |
| fsumub 31724 | An upper bound for a term ... |
| fsumiunle 31725 | Upper bound for a sum of n... |
| dfdec100 31726 | Split the hundreds from a ... |
| dp2eq1 31729 | Equality theorem for the d... |
| dp2eq2 31730 | Equality theorem for the d... |
| dp2eq1i 31731 | Equality theorem for the d... |
| dp2eq2i 31732 | Equality theorem for the d... |
| dp2eq12i 31733 | Equality theorem for the d... |
| dp20u 31734 | Add a zero in the tenths (... |
| dp20h 31735 | Add a zero in the unit pla... |
| dp2cl 31736 | Closure for the decimal fr... |
| dp2clq 31737 | Closure for a decimal frac... |
| rpdp2cl 31738 | Closure for a decimal frac... |
| rpdp2cl2 31739 | Closure for a decimal frac... |
| dp2lt10 31740 | Decimal fraction builds re... |
| dp2lt 31741 | Comparing two decimal frac... |
| dp2ltsuc 31742 | Comparing a decimal fracti... |
| dp2ltc 31743 | Comparing two decimal expa... |
| dpval 31746 | Define the value of the de... |
| dpcl 31747 | Prove that the closure of ... |
| dpfrac1 31748 | Prove a simple equivalence... |
| dpval2 31749 | Value of the decimal point... |
| dpval3 31750 | Value of the decimal point... |
| dpmul10 31751 | Multiply by 10 a decimal e... |
| decdiv10 31752 | Divide a decimal number by... |
| dpmul100 31753 | Multiply by 100 a decimal ... |
| dp3mul10 31754 | Multiply by 10 a decimal e... |
| dpmul1000 31755 | Multiply by 1000 a decimal... |
| dpval3rp 31756 | Value of the decimal point... |
| dp0u 31757 | Add a zero in the tenths p... |
| dp0h 31758 | Remove a zero in the units... |
| rpdpcl 31759 | Closure of the decimal poi... |
| dplt 31760 | Comparing two decimal expa... |
| dplti 31761 | Comparing a decimal expans... |
| dpgti 31762 | Comparing a decimal expans... |
| dpltc 31763 | Comparing two decimal inte... |
| dpexpp1 31764 | Add one zero to the mantis... |
| 0dp2dp 31765 | Multiply by 10 a decimal e... |
| dpadd2 31766 | Addition with one decimal,... |
| dpadd 31767 | Addition with one decimal.... |
| dpadd3 31768 | Addition with two decimals... |
| dpmul 31769 | Multiplication with one de... |
| dpmul4 31770 | An upper bound to multipli... |
| threehalves 31771 | Example theorem demonstrat... |
| 1mhdrd 31772 | Example theorem demonstrat... |
| xdivval 31775 | Value of division: the (un... |
| xrecex 31776 | Existence of reciprocal of... |
| xmulcand 31777 | Cancellation law for exten... |
| xreceu 31778 | Existential uniqueness of ... |
| xdivcld 31779 | Closure law for the extend... |
| xdivcl 31780 | Closure law for the extend... |
| xdivmul 31781 | Relationship between divis... |
| rexdiv 31782 | The extended real division... |
| xdivrec 31783 | Relationship between divis... |
| xdivid 31784 | A number divided by itself... |
| xdiv0 31785 | Division into zero is zero... |
| xdiv0rp 31786 | Division into zero is zero... |
| eliccioo 31787 | Membership in a closed int... |
| elxrge02 31788 | Elementhood in the set of ... |
| xdivpnfrp 31789 | Plus infinity divided by a... |
| rpxdivcld 31790 | Closure law for extended d... |
| xrpxdivcld 31791 | Closure law for extended d... |
| wrdfd 31792 | A word is a zero-based seq... |
| wrdres 31793 | Condition for the restrict... |
| wrdsplex 31794 | Existence of a split of a ... |
| pfx1s2 31795 | The prefix of length 1 of ... |
| pfxrn2 31796 | The range of a prefix of a... |
| pfxrn3 31797 | Express the range of a pre... |
| pfxf1 31798 | Condition for a prefix to ... |
| s1f1 31799 | Conditions for a length 1 ... |
| s2rn 31800 | Range of a length 2 string... |
| s2f1 31801 | Conditions for a length 2 ... |
| s3rn 31802 | Range of a length 3 string... |
| s3f1 31803 | Conditions for a length 3 ... |
| s3clhash 31804 | Closure of the words of le... |
| ccatf1 31805 | Conditions for a concatena... |
| pfxlsw2ccat 31806 | Reconstruct a word from it... |
| wrdt2ind 31807 | Perform an induction over ... |
| swrdrn2 31808 | The range of a subword is ... |
| swrdrn3 31809 | Express the range of a sub... |
| swrdf1 31810 | Condition for a subword to... |
| swrdrndisj 31811 | Condition for the range of... |
| splfv3 31812 | Symbols to the right of a ... |
| 1cshid 31813 | Cyclically shifting a sing... |
| cshw1s2 31814 | Cyclically shifting a leng... |
| cshwrnid 31815 | Cyclically shifting a word... |
| cshf1o 31816 | Condition for the cyclic s... |
| ressplusf 31817 | The group operation functi... |
| ressnm 31818 | The norm in a restricted s... |
| abvpropd2 31819 | Weaker version of ~ abvpro... |
| oppgle 31820 | less-than relation of an o... |
| oppgleOLD 31821 | Obsolete version of ~ oppg... |
| oppglt 31822 | less-than relation of an o... |
| ressprs 31823 | The restriction of a prose... |
| oduprs 31824 | Being a proset is a self-d... |
| posrasymb 31825 | A poset ordering is asymet... |
| resspos 31826 | The restriction of a Poset... |
| resstos 31827 | The restriction of a Toset... |
| odutos 31828 | Being a toset is a self-du... |
| tlt2 31829 | In a Toset, two elements m... |
| tlt3 31830 | In a Toset, two elements m... |
| trleile 31831 | In a Toset, two elements m... |
| toslublem 31832 | Lemma for ~ toslub and ~ x... |
| toslub 31833 | In a toset, the lowest upp... |
| tosglblem 31834 | Lemma for ~ tosglb and ~ x... |
| tosglb 31835 | Same theorem as ~ toslub ,... |
| clatp0cl 31836 | The poset zero of a comple... |
| clatp1cl 31837 | The poset one of a complet... |
| mntoval 31842 | Operation value of the mon... |
| ismnt 31843 | Express the statement " ` ... |
| ismntd 31844 | Property of being a monoto... |
| mntf 31845 | A monotone function is a f... |
| mgcoval 31846 | Operation value of the mon... |
| mgcval 31847 | Monotone Galois connection... |
| mgcf1 31848 | The lower adjoint ` F ` of... |
| mgcf2 31849 | The upper adjoint ` G ` of... |
| mgccole1 31850 | An inequality for the kern... |
| mgccole2 31851 | Inequality for the closure... |
| mgcmnt1 31852 | The lower adjoint ` F ` of... |
| mgcmnt2 31853 | The upper adjoint ` G ` of... |
| mgcmntco 31854 | A Galois connection like s... |
| dfmgc2lem 31855 | Lemma for dfmgc2, backward... |
| dfmgc2 31856 | Alternate definition of th... |
| mgcmnt1d 31857 | Galois connection implies ... |
| mgcmnt2d 31858 | Galois connection implies ... |
| mgccnv 31859 | The inverse Galois connect... |
| pwrssmgc 31860 | Given a function ` F ` , e... |
| mgcf1olem1 31861 | Property of a Galois conne... |
| mgcf1olem2 31862 | Property of a Galois conne... |
| mgcf1o 31863 | Given a Galois connection,... |
| xrs0 31866 | The zero of the extended r... |
| xrslt 31867 | The "strictly less than" r... |
| xrsinvgval 31868 | The inversion operation in... |
| xrsmulgzz 31869 | The "multiple" function in... |
| xrstos 31870 | The extended real numbers ... |
| xrsclat 31871 | The extended real numbers ... |
| xrsp0 31872 | The poset 0 of the extende... |
| xrsp1 31873 | The poset 1 of the extende... |
| ressmulgnn 31874 | Values for the group multi... |
| ressmulgnn0 31875 | Values for the group multi... |
| xrge0base 31876 | The base of the extended n... |
| xrge00 31877 | The zero of the extended n... |
| xrge0plusg 31878 | The additive law of the ex... |
| xrge0le 31879 | The "less than or equal to... |
| xrge0mulgnn0 31880 | The group multiple functio... |
| xrge0addass 31881 | Associativity of extended ... |
| xrge0addgt0 31882 | The sum of nonnegative and... |
| xrge0adddir 31883 | Right-distributivity of ex... |
| xrge0adddi 31884 | Left-distributivity of ext... |
| xrge0npcan 31885 | Extended nonnegative real ... |
| fsumrp0cl 31886 | Closure of a finite sum of... |
| abliso 31887 | The image of an Abelian gr... |
| gsumsubg 31888 | The group sum in a subgrou... |
| gsumsra 31889 | The group sum in a subring... |
| gsummpt2co 31890 | Split a finite sum into a ... |
| gsummpt2d 31891 | Express a finite sum over ... |
| lmodvslmhm 31892 | Scalar multiplication in a... |
| gsumvsmul1 31893 | Pull a scalar multiplicati... |
| gsummptres 31894 | Extend a finite group sum ... |
| gsummptres2 31895 | Extend a finite group sum ... |
| gsumzresunsn 31896 | Append an element to a fin... |
| gsumpart 31897 | Express a group sum as a d... |
| gsumhashmul 31898 | Express a group sum by gro... |
| xrge0tsmsd 31899 | Any finite or infinite sum... |
| xrge0tsmsbi 31900 | Any limit of a finite or i... |
| xrge0tsmseq 31901 | Any limit of a finite or i... |
| cntzun 31902 | The centralizer of a union... |
| cntzsnid 31903 | The centralizer of the ide... |
| cntrcrng 31904 | The center of a ring is a ... |
| isomnd 31909 | A (left) ordered monoid is... |
| isogrp 31910 | A (left-)ordered group is ... |
| ogrpgrp 31911 | A left-ordered group is a ... |
| omndmnd 31912 | A left-ordered monoid is a... |
| omndtos 31913 | A left-ordered monoid is a... |
| omndadd 31914 | In an ordered monoid, the ... |
| omndaddr 31915 | In a right ordered monoid,... |
| omndadd2d 31916 | In a commutative left orde... |
| omndadd2rd 31917 | In a left- and right- orde... |
| submomnd 31918 | A submonoid of an ordered ... |
| xrge0omnd 31919 | The nonnegative extended r... |
| omndmul2 31920 | In an ordered monoid, the ... |
| omndmul3 31921 | In an ordered monoid, the ... |
| omndmul 31922 | In a commutative ordered m... |
| ogrpinv0le 31923 | In an ordered group, the o... |
| ogrpsub 31924 | In an ordered group, the o... |
| ogrpaddlt 31925 | In an ordered group, stric... |
| ogrpaddltbi 31926 | In a right ordered group, ... |
| ogrpaddltrd 31927 | In a right ordered group, ... |
| ogrpaddltrbid 31928 | In a right ordered group, ... |
| ogrpsublt 31929 | In an ordered group, stric... |
| ogrpinv0lt 31930 | In an ordered group, the o... |
| ogrpinvlt 31931 | In an ordered group, the o... |
| gsumle 31932 | A finite sum in an ordered... |
| symgfcoeu 31933 | Uniqueness property of per... |
| symgcom 31934 | Two permutations ` X ` and... |
| symgcom2 31935 | Two permutations ` X ` and... |
| symgcntz 31936 | All elements of a (finite)... |
| odpmco 31937 | The composition of two odd... |
| symgsubg 31938 | The value of the group sub... |
| pmtrprfv2 31939 | In a transposition of two ... |
| pmtrcnel 31940 | Composing a permutation ` ... |
| pmtrcnel2 31941 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 31942 | Composing a permutation ` ... |
| pmtridf1o 31943 | Transpositions of ` X ` an... |
| pmtridfv1 31944 | Value at X of the transpos... |
| pmtridfv2 31945 | Value at Y of the transpos... |
| psgnid 31946 | Permutation sign of the id... |
| psgndmfi 31947 | For a finite base set, the... |
| pmtrto1cl 31948 | Useful lemma for the follo... |
| psgnfzto1stlem 31949 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 31950 | Value of our permutation `... |
| fzto1st1 31951 | Special case where the per... |
| fzto1st 31952 | The function moving one el... |
| fzto1stinvn 31953 | Value of the inverse of ou... |
| psgnfzto1st 31954 | The permutation sign for m... |
| tocycval 31957 | Value of the cycle builder... |
| tocycfv 31958 | Function value of a permut... |
| tocycfvres1 31959 | A cyclic permutation is a ... |
| tocycfvres2 31960 | A cyclic permutation is th... |
| cycpmfvlem 31961 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 31962 | Value of a cycle function ... |
| cycpmfv2 31963 | Value of a cycle function ... |
| cycpmfv3 31964 | Values outside of the orbi... |
| cycpmcl 31965 | Cyclic permutations are pe... |
| tocycf 31966 | The permutation cycle buil... |
| tocyc01 31967 | Permutation cycles built f... |
| cycpm2tr 31968 | A cyclic permutation of 2 ... |
| cycpm2cl 31969 | Closure for the 2-cycles. ... |
| cyc2fv1 31970 | Function value of a 2-cycl... |
| cyc2fv2 31971 | Function value of a 2-cycl... |
| trsp2cyc 31972 | Exhibit the word a transpo... |
| cycpmco2f1 31973 | The word U used in ~ cycpm... |
| cycpmco2rn 31974 | The orbit of the compositi... |
| cycpmco2lem1 31975 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 31976 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 31977 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 31978 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 31979 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 31980 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 31981 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 31982 | The composition of a cycli... |
| cyc2fvx 31983 | Function value of a 2-cycl... |
| cycpm3cl 31984 | Closure of the 3-cycles in... |
| cycpm3cl2 31985 | Closure of the 3-cycles in... |
| cyc3fv1 31986 | Function value of a 3-cycl... |
| cyc3fv2 31987 | Function value of a 3-cycl... |
| cyc3fv3 31988 | Function value of a 3-cycl... |
| cyc3co2 31989 | Represent a 3-cycle as a c... |
| cycpmconjvlem 31990 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 31991 | A formula for computing co... |
| cycpmrn 31992 | The range of the word used... |
| tocyccntz 31993 | All elements of a (finite)... |
| evpmval 31994 | Value of the set of even p... |
| cnmsgn0g 31995 | The neutral element of the... |
| evpmsubg 31996 | The alternating group is a... |
| evpmid 31997 | The identity is an even pe... |
| altgnsg 31998 | The alternating group ` ( ... |
| cyc3evpm 31999 | 3-Cycles are even permutat... |
| cyc3genpmlem 32000 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 32001 | The alternating group ` A ... |
| cycpmgcl 32002 | Cyclic permutations are pe... |
| cycpmconjslem1 32003 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 32004 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 32005 | All cycles of the same len... |
| cyc3conja 32006 | All 3-cycles are conjugate... |
| sgnsv 32009 | The sign mapping. (Contri... |
| sgnsval 32010 | The sign value. (Contribu... |
| sgnsf 32011 | The sign function. (Contr... |
| inftmrel 32016 | The infinitesimal relation... |
| isinftm 32017 | Express ` x ` is infinites... |
| isarchi 32018 | Express the predicate " ` ... |
| pnfinf 32019 | Plus infinity is an infini... |
| xrnarchi 32020 | The completed real line is... |
| isarchi2 32021 | Alternative way to express... |
| submarchi 32022 | A submonoid is archimedean... |
| isarchi3 32023 | This is the usual definiti... |
| archirng 32024 | Property of Archimedean or... |
| archirngz 32025 | Property of Archimedean le... |
| archiexdiv 32026 | In an Archimedean group, g... |
| archiabllem1a 32027 | Lemma for ~ archiabl : In... |
| archiabllem1b 32028 | Lemma for ~ archiabl . (C... |
| archiabllem1 32029 | Archimedean ordered groups... |
| archiabllem2a 32030 | Lemma for ~ archiabl , whi... |
| archiabllem2c 32031 | Lemma for ~ archiabl . (C... |
| archiabllem2b 32032 | Lemma for ~ archiabl . (C... |
| archiabllem2 32033 | Archimedean ordered groups... |
| archiabl 32034 | Archimedean left- and righ... |
| isslmd 32037 | The predicate "is a semimo... |
| slmdlema 32038 | Lemma for properties of a ... |
| lmodslmd 32039 | Left semimodules generaliz... |
| slmdcmn 32040 | A semimodule is a commutat... |
| slmdmnd 32041 | A semimodule is a monoid. ... |
| slmdsrg 32042 | The scalar component of a ... |
| slmdbn0 32043 | The base set of a semimodu... |
| slmdacl 32044 | Closure of ring addition f... |
| slmdmcl 32045 | Closure of ring multiplica... |
| slmdsn0 32046 | The set of scalars in a se... |
| slmdvacl 32047 | Closure of vector addition... |
| slmdass 32048 | Semiring left module vecto... |
| slmdvscl 32049 | Closure of scalar product ... |
| slmdvsdi 32050 | Distributive law for scala... |
| slmdvsdir 32051 | Distributive law for scala... |
| slmdvsass 32052 | Associative law for scalar... |
| slmd0cl 32053 | The ring zero in a semimod... |
| slmd1cl 32054 | The ring unity in a semiri... |
| slmdvs1 32055 | Scalar product with ring u... |
| slmd0vcl 32056 | The zero vector is a vecto... |
| slmd0vlid 32057 | Left identity law for the ... |
| slmd0vrid 32058 | Right identity law for the... |
| slmd0vs 32059 | Zero times a vector is the... |
| slmdvs0 32060 | Anything times the zero ve... |
| gsumvsca1 32061 | Scalar product of a finite... |
| gsumvsca2 32062 | Scalar product of a finite... |
| prmsimpcyc 32063 | A group of prime order is ... |
| 0ringsubrg 32064 | A subring of a zero ring i... |
| rngurd 32065 | Deduce the unity element o... |
| dvdschrmulg 32066 | In a ring, any multiple of... |
| freshmansdream 32067 | For a prime number ` P ` ,... |
| frobrhm 32068 | In a commutative ring with... |
| ress1r 32069 | ` 1r ` is unaffected by re... |
| dvrdir 32070 | Distributive law for the d... |
| rdivmuldivd 32071 | Multiplication of two rati... |
| ringinvval 32072 | The ring inverse expressed... |
| dvrcan5 32073 | Cancellation law for commo... |
| subrgchr 32074 | If ` A ` is a subring of `... |
| rmfsupp2 32075 | A mapping of a multiplicat... |
| sdrgdvcl 32076 | A sub-division-ring is clo... |
| sdrginvcl 32077 | A sub-division-ring is clo... |
| primefldchr 32078 | The characteristic of a pr... |
| fldgenval 32081 | Value of the field generat... |
| fldgenssid 32082 | The field generated by a s... |
| fldgensdrg 32083 | A generated subfield is a ... |
| fldgenss 32084 | Generated subfields preser... |
| fldgenidfld 32085 | The subfield generated by ... |
| fldgenid 32086 | The subfield of a field ` ... |
| fldgenfld 32087 | A generated subfield is a ... |
| primefldgen1 32088 | The prime field of a divis... |
| 1fldgenq 32089 | The field of rational numb... |
| isorng 32094 | An ordered ring is a ring ... |
| orngring 32095 | An ordered ring is a ring.... |
| orngogrp 32096 | An ordered ring is an orde... |
| isofld 32097 | An ordered field is a fiel... |
| orngmul 32098 | In an ordered ring, the or... |
| orngsqr 32099 | In an ordered ring, all sq... |
| ornglmulle 32100 | In an ordered ring, multip... |
| orngrmulle 32101 | In an ordered ring, multip... |
| ornglmullt 32102 | In an ordered ring, multip... |
| orngrmullt 32103 | In an ordered ring, multip... |
| orngmullt 32104 | In an ordered ring, the st... |
| ofldfld 32105 | An ordered field is a fiel... |
| ofldtos 32106 | An ordered field is a tota... |
| orng0le1 32107 | In an ordered ring, the ri... |
| ofldlt1 32108 | In an ordered field, the r... |
| ofldchr 32109 | The characteristic of an o... |
| suborng 32110 | Every subring of an ordere... |
| subofld 32111 | Every subfield of an order... |
| isarchiofld 32112 | Axiom of Archimedes : a ch... |
| rhmdvd 32113 | A ring homomorphism preser... |
| kerunit 32114 | If a unit element lies in ... |
| reldmresv 32117 | The scalar restriction is ... |
| resvval 32118 | Value of structure restric... |
| resvid2 32119 | General behavior of trivia... |
| resvval2 32120 | Value of nontrivial struct... |
| resvsca 32121 | Base set of a structure re... |
| resvlem 32122 | Other elements of a scalar... |
| resvlemOLD 32123 | Obsolete version of ~ resv... |
| resvbas 32124 | ` Base ` is unaffected by ... |
| resvbasOLD 32125 | Obsolete proof of ~ resvba... |
| resvplusg 32126 | ` +g ` is unaffected by sc... |
| resvplusgOLD 32127 | Obsolete proof of ~ resvpl... |
| resvvsca 32128 | ` .s ` is unaffected by sc... |
| resvvscaOLD 32129 | Obsolete proof of ~ resvvs... |
| resvmulr 32130 | ` .r ` is unaffected by sc... |
| resvmulrOLD 32131 | Obsolete proof of ~ resvmu... |
| resv0g 32132 | ` 0g ` is unaffected by sc... |
| resv1r 32133 | ` 1r ` is unaffected by sc... |
| resvcmn 32134 | Scalar restriction preserv... |
| gzcrng 32135 | The gaussian integers form... |
| reofld 32136 | The real numbers form an o... |
| nn0omnd 32137 | The nonnegative integers f... |
| rearchi 32138 | The field of the real numb... |
| nn0archi 32139 | The monoid of the nonnegat... |
| xrge0slmod 32140 | The extended nonnegative r... |
| qusker 32141 | The kernel of a quotient m... |
| eqgvscpbl 32142 | The left coset equivalence... |
| qusvscpbl 32143 | The quotient map distribut... |
| qusscaval 32144 | Value of the scalar multip... |
| imaslmod 32145 | The image structure of a l... |
| quslmod 32146 | If ` G ` is a submodule in... |
| quslmhm 32147 | If ` G ` is a submodule of... |
| ecxpid 32148 | The equivalence class of a... |
| eqg0el 32149 | Equivalence class of a quo... |
| qsxpid 32150 | The quotient set of a cart... |
| qusxpid 32151 | The Group quotient equival... |
| qustriv 32152 | The quotient of a group ` ... |
| qustrivr 32153 | Converse of ~ qustriv . (... |
| fermltlchr 32154 | A generalization of Fermat... |
| znfermltl 32155 | Fermat's little theorem in... |
| islinds5 32156 | A set is linearly independ... |
| ellspds 32157 | Variation on ~ ellspd . (... |
| 0ellsp 32158 | Zero is in all spans. (Co... |
| 0nellinds 32159 | The group identity cannot ... |
| rspsnel 32160 | Membership in a principal ... |
| rspsnid 32161 | A principal ideal contains... |
| elrsp 32162 | Write the elements of a ri... |
| rspidlid 32163 | The ideal span of an ideal... |
| pidlnz 32164 | A principal ideal generate... |
| lbslsp 32165 | Any element of a left modu... |
| lindssn 32166 | Any singleton of a nonzero... |
| lindflbs 32167 | Conditions for an independ... |
| linds2eq 32168 | Deduce equality of element... |
| lindfpropd 32169 | Property deduction for lin... |
| lindspropd 32170 | Property deduction for lin... |
| elgrplsmsn 32171 | Membership in a sumset wit... |
| lsmsnorb 32172 | The sumset of a group with... |
| lsmsnorb2 32173 | The sumset of a single ele... |
| elringlsm 32174 | Membership in a product of... |
| elringlsmd 32175 | Membership in a product of... |
| ringlsmss 32176 | Closure of the product of ... |
| ringlsmss1 32177 | The product of an ideal ` ... |
| ringlsmss2 32178 | The product with an ideal ... |
| lsmsnpridl 32179 | The product of the ring wi... |
| lsmsnidl 32180 | The product of the ring wi... |
| lsmidllsp 32181 | The sum of two ideals is t... |
| lsmidl 32182 | The sum of two ideals is a... |
| lsmssass 32183 | Group sum is associative, ... |
| grplsm0l 32184 | Sumset with the identity s... |
| grplsmid 32185 | The direct sum of an eleme... |
| quslsm 32186 | Express the image by the q... |
| qusima 32187 | The image of a subgroup by... |
| nsgqus0 32188 | A normal subgroup ` N ` is... |
| nsgmgclem 32189 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 32190 | There is a monotone Galois... |
| nsgqusf1olem1 32191 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 32192 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 32193 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 32194 | The canonical projection h... |
| ghmquskerlem1 32195 | Lemma for ~ ghmqusker (Con... |
| ghmquskerco 32196 | In the case of theorem ~ g... |
| ghmquskerlem2 32197 | Lemma for ~ ghmqusker . (... |
| ghmqusker 32198 | A surjective group homomor... |
| intlidl 32199 | The intersection of a none... |
| rhmpreimaidl 32200 | The preimage of an ideal b... |
| kerlidl 32201 | The kernel of a ring homom... |
| 0ringidl 32202 | The zero ideal is the only... |
| elrspunidl 32203 | Elementhood to the span of... |
| lidlincl 32204 | Ideals are closed under in... |
| idlinsubrg 32205 | The intersection between a... |
| rhmimaidl 32206 | The image of an ideal ` I ... |
| prmidlval 32209 | The class of prime ideals ... |
| isprmidl 32210 | The predicate "is a prime ... |
| prmidlnr 32211 | A prime ideal is a proper ... |
| prmidl 32212 | The main property of a pri... |
| prmidl2 32213 | A condition that shows an ... |
| idlmulssprm 32214 | Let ` P ` be a prime ideal... |
| pridln1 32215 | A proper ideal cannot cont... |
| prmidlidl 32216 | A prime ideal is an ideal.... |
| prmidlssidl 32217 | Prime ideals as a subset o... |
| lidlnsg 32218 | An ideal is a normal subgr... |
| cringm4 32219 | Commutative/associative la... |
| isprmidlc 32220 | The predicate "is prime id... |
| prmidlc 32221 | Property of a prime ideal ... |
| 0ringprmidl 32222 | The trivial ring does not ... |
| prmidl0 32223 | The zero ideal of a commut... |
| rhmpreimaprmidl 32224 | The preimage of a prime id... |
| qsidomlem1 32225 | If the quotient ring of a ... |
| qsidomlem2 32226 | A quotient by a prime idea... |
| qsidom 32227 | An ideal ` I ` in the comm... |
| mxidlval 32230 | The set of maximal ideals ... |
| ismxidl 32231 | The predicate "is a maxima... |
| mxidlidl 32232 | A maximal ideal is an idea... |
| mxidlnr 32233 | A maximal ideal is proper.... |
| mxidlmax 32234 | A maximal ideal is a maxim... |
| mxidln1 32235 | One is not contained in an... |
| mxidlnzr 32236 | A ring with a maximal idea... |
| mxidlprm 32237 | Every maximal ideal is pri... |
| ssmxidllem 32238 | The set ` P ` used in the ... |
| ssmxidl 32239 | Let ` R ` be a ring, and l... |
| krull 32240 | Krull's theorem: Any nonz... |
| mxidlnzrb 32241 | A ring is nonzero if and o... |
| idlsrgstr 32244 | A constructed semiring of ... |
| idlsrgval 32245 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 32246 | Baae of the ideals of a ri... |
| idlsrgplusg 32247 | Additive operation of the ... |
| idlsrg0g 32248 | The zero ideal is the addi... |
| idlsrgmulr 32249 | Multiplicative operation o... |
| idlsrgtset 32250 | Topology component of the ... |
| idlsrgmulrval 32251 | Value of the ring multipli... |
| idlsrgmulrcl 32252 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 32253 | In a commutative ring, the... |
| idlsrgmulrss2 32254 | The product of two ideals ... |
| idlsrgmulrssin 32255 | In a commutative ring, the... |
| idlsrgmnd 32256 | The ideals of a ring form ... |
| idlsrgcmnd 32257 | The ideals of a ring form ... |
| isufd 32260 | The property of being a Un... |
| rprmval 32261 | The prime elements of a ri... |
| isrprm 32262 | Property for ` P ` to be a... |
| asclmulg 32263 | Apply group multiplication... |
| 0ringmon1p 32264 | There are no monic polynom... |
| fply1 32265 | Conditions for a function ... |
| ply1scleq 32266 | Equality of a constant pol... |
| evls1fn 32267 | Functionality of the subri... |
| evls1scafv 32268 | Value of the univariate po... |
| evls1expd 32269 | Univariate polynomial eval... |
| evls1varpwval 32270 | Univariate polynomial eval... |
| evls1fpws 32271 | Evaluation of a univariate... |
| ressply1evl 32272 | Evaluation of a univariate... |
| evls1addd 32273 | Univariate polynomial eval... |
| evls1muld 32274 | Univariate polynomial eval... |
| ressdeg1 32275 | The degree of a univariate... |
| ply1ascl0 32276 | The zero scalar as a polyn... |
| ressply10g 32277 | A restricted polynomial al... |
| ressply1mon1p 32278 | The monic polynomials of a... |
| ressply1invg 32279 | An element of a restricted... |
| ressply1sub 32280 | A restricted polynomial al... |
| asclply1subcl 32281 | Closure of the algebra sca... |
| ply1chr 32282 | The characteristic of a po... |
| ply1fermltlchr 32283 | Fermat's little theorem fo... |
| ply1fermltl 32284 | Fermat's little theorem fo... |
| sra1r 32285 | The unity element of a sub... |
| sraring 32286 | Condition for a subring al... |
| sradrng 32287 | Condition for a subring al... |
| srasubrg 32288 | A subring of the original ... |
| sralvec 32289 | Given a sub division ring ... |
| srafldlvec 32290 | Given a subfield ` F ` of ... |
| drgext0g 32291 | The additive neutral eleme... |
| drgextvsca 32292 | The scalar multiplication ... |
| drgext0gsca 32293 | The additive neutral eleme... |
| drgextsubrg 32294 | The scalar field is a subr... |
| drgextlsp 32295 | The scalar field is a subs... |
| drgextgsum 32296 | Group sum in a division ri... |
| lvecdimfi 32297 | Finite version of ~ lvecdi... |
| dimval 32300 | The dimension of a vector ... |
| dimvalfi 32301 | The dimension of a vector ... |
| dimcl 32302 | Closure of the vector spac... |
| lvecdim0i 32303 | A vector space of dimensio... |
| lvecdim0 32304 | A vector space of dimensio... |
| lssdimle 32305 | The dimension of a linear ... |
| dimpropd 32306 | If two structures have the... |
| rgmoddim 32307 | The left vector space indu... |
| frlmdim 32308 | Dimension of a free left m... |
| tnglvec 32309 | Augmenting a structure wit... |
| tngdim 32310 | Dimension of a left vector... |
| rrxdim 32311 | Dimension of the generaliz... |
| matdim 32312 | Dimension of the space of ... |
| lbslsat 32313 | A nonzero vector ` X ` is ... |
| lsatdim 32314 | A line, spanned by a nonze... |
| drngdimgt0 32315 | The dimension of a vector ... |
| lmhmlvec2 32316 | A homomorphism of left vec... |
| kerlmhm 32317 | The kernel of a vector spa... |
| imlmhm 32318 | The image of a vector spac... |
| lindsunlem 32319 | Lemma for ~ lindsun . (Co... |
| lindsun 32320 | Condition for the union of... |
| lbsdiflsp0 32321 | The linear spans of two di... |
| dimkerim 32322 | Given a linear map ` F ` b... |
| qusdimsum 32323 | Let ` W ` be a vector spac... |
| fedgmullem1 32324 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 32325 | Lemma for ~ fedgmul . (Co... |
| fedgmul 32326 | The multiplicativity formu... |
| relfldext 32335 | The field extension is a r... |
| brfldext 32336 | The field extension relati... |
| ccfldextrr 32337 | The field of the complex n... |
| fldextfld1 32338 | A field extension is only ... |
| fldextfld2 32339 | A field extension is only ... |
| fldextsubrg 32340 | Field extension implies a ... |
| fldextress 32341 | Field extension implies a ... |
| brfinext 32342 | The finite field extension... |
| extdgval 32343 | Value of the field extensi... |
| fldextsralvec 32344 | The subring algebra associ... |
| extdgcl 32345 | Closure of the field exten... |
| extdggt0 32346 | Degrees of field extension... |
| fldexttr 32347 | Field extension is a trans... |
| fldextid 32348 | The field extension relati... |
| extdgid 32349 | A trivial field extension ... |
| extdgmul 32350 | The multiplicativity formu... |
| finexttrb 32351 | The extension ` E ` of ` K... |
| extdg1id 32352 | If the degree of the exten... |
| extdg1b 32353 | The degree of the extensio... |
| fldextchr 32354 | The characteristic of a su... |
| ccfldsrarelvec 32355 | The subring algebra of the... |
| ccfldextdgrr 32356 | The degree of the field ex... |
| irngval 32359 | The elements of a field ` ... |
| elirng 32360 | Property for an element ` ... |
| irngss 32361 | All elements of a subring ... |
| irngssv 32362 | An integral element is an ... |
| 0ringirng 32363 | A zero ring ` R ` has no i... |
| irngnzply1lem 32364 | In the case of a field ` E... |
| irngnzply1 32365 | In the case of a field ` E... |
| evls1maprhm 32368 | The function ` F ` mapping... |
| ply1annidllem 32369 | Write the set ` Q ` of pol... |
| ply1annidl 32370 | The set ` Q ` of polynomia... |
| ply1annig1p 32371 | The ideal ` Q ` of polynom... |
| minplyval 32372 | Expand the value of the mi... |
| ply1annprmidl 32373 | The set ` Q ` of polynomia... |
| smatfval 32376 | Value of the submatrix. (... |
| smatrcl 32377 | Closure of the rectangular... |
| smatlem 32378 | Lemma for the next theorem... |
| smattl 32379 | Entries of a submatrix, to... |
| smattr 32380 | Entries of a submatrix, to... |
| smatbl 32381 | Entries of a submatrix, bo... |
| smatbr 32382 | Entries of a submatrix, bo... |
| smatcl 32383 | Closure of the square subm... |
| matmpo 32384 | Write a square matrix as a... |
| 1smat1 32385 | The submatrix of the ident... |
| submat1n 32386 | One case where the submatr... |
| submatres 32387 | Special case where the sub... |
| submateqlem1 32388 | Lemma for ~ submateq . (C... |
| submateqlem2 32389 | Lemma for ~ submateq . (C... |
| submateq 32390 | Sufficient condition for t... |
| submatminr1 32391 | If we take a submatrix by ... |
| lmatval 32394 | Value of the literal matri... |
| lmatfval 32395 | Entries of a literal matri... |
| lmatfvlem 32396 | Useful lemma to extract li... |
| lmatcl 32397 | Closure of the literal mat... |
| lmat22lem 32398 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 32399 | Entry of a 2x2 literal mat... |
| lmat22e12 32400 | Entry of a 2x2 literal mat... |
| lmat22e21 32401 | Entry of a 2x2 literal mat... |
| lmat22e22 32402 | Entry of a 2x2 literal mat... |
| lmat22det 32403 | The determinant of a liter... |
| mdetpmtr1 32404 | The determinant of a matri... |
| mdetpmtr2 32405 | The determinant of a matri... |
| mdetpmtr12 32406 | The determinant of a matri... |
| mdetlap1 32407 | A Laplace expansion of the... |
| madjusmdetlem1 32408 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 32409 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 32410 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 32411 | Lemma for ~ madjusmdet . ... |
| madjusmdet 32412 | Express the cofactor of th... |
| mdetlap 32413 | Laplace expansion of the d... |
| ist0cld 32414 | The predicate "is a T_0 sp... |
| txomap 32415 | Given two open maps ` F ` ... |
| qtopt1 32416 | If every equivalence class... |
| qtophaus 32417 | If an open map's graph in ... |
| circtopn 32418 | The topology of the unit c... |
| circcn 32419 | The function gluing the re... |
| reff 32420 | For any cover refinement, ... |
| locfinreflem 32421 | A locally finite refinemen... |
| locfinref 32422 | A locally finite refinemen... |
| iscref 32425 | The property that every op... |
| crefeq 32426 | Equality theorem for the "... |
| creftop 32427 | A space where every open c... |
| crefi 32428 | The property that every op... |
| crefdf 32429 | A formulation of ~ crefi e... |
| crefss 32430 | The "every open cover has ... |
| cmpcref 32431 | Equivalent definition of c... |
| cmpfiref 32432 | Every open cover of a Comp... |
| ldlfcntref 32435 | Every open cover of a Lind... |
| ispcmp 32438 | The predicate "is a paraco... |
| cmppcmp 32439 | Every compact space is par... |
| dispcmp 32440 | Every discrete space is pa... |
| pcmplfin 32441 | Given a paracompact topolo... |
| pcmplfinf 32442 | Given a paracompact topolo... |
| rspecval 32445 | Value of the spectrum of t... |
| rspecbas 32446 | The prime ideals form the ... |
| rspectset 32447 | Topology component of the ... |
| rspectopn 32448 | The topology component of ... |
| zarcls0 32449 | The closure of the identit... |
| zarcls1 32450 | The unit ideal ` B ` is th... |
| zarclsun 32451 | The union of two closed se... |
| zarclsiin 32452 | In a Zariski topology, the... |
| zarclsint 32453 | The intersection of a fami... |
| zarclssn 32454 | The closed points of Zaris... |
| zarcls 32455 | The open sets of the Zaris... |
| zartopn 32456 | The Zariski topology is a ... |
| zartop 32457 | The Zariski topology is a ... |
| zartopon 32458 | The points of the Zariski ... |
| zar0ring 32459 | The Zariski Topology of th... |
| zart0 32460 | The Zariski topology is T_... |
| zarmxt1 32461 | The Zariski topology restr... |
| zarcmplem 32462 | Lemma for ~ zarcmp . (Con... |
| zarcmp 32463 | The Zariski topology is co... |
| rspectps 32464 | The spectrum of a ring ` R... |
| rhmpreimacnlem 32465 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 32466 | The function mapping a pri... |
| metidval 32471 | Value of the metric identi... |
| metidss 32472 | As a relation, the metric ... |
| metidv 32473 | ` A ` and ` B ` identify b... |
| metideq 32474 | Basic property of the metr... |
| metider 32475 | The metric identification ... |
| pstmval 32476 | Value of the metric induce... |
| pstmfval 32477 | Function value of the metr... |
| pstmxmet 32478 | The metric induced by a ps... |
| hauseqcn 32479 | In a Hausdorff topology, t... |
| elunitge0 32480 | An element of the closed u... |
| unitssxrge0 32481 | The closed unit interval i... |
| unitdivcld 32482 | Necessary conditions for a... |
| iistmd 32483 | The closed unit interval f... |
| unicls 32484 | The union of the closed se... |
| tpr2tp 32485 | The usual topology on ` ( ... |
| tpr2uni 32486 | The usual topology on ` ( ... |
| xpinpreima 32487 | Rewrite the cartesian prod... |
| xpinpreima2 32488 | Rewrite the cartesian prod... |
| sqsscirc1 32489 | The complex square of side... |
| sqsscirc2 32490 | The complex square of side... |
| cnre2csqlem 32491 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 32492 | Image of a centered square... |
| tpr2rico 32493 | For any point of an open s... |
| cnvordtrestixx 32494 | The restriction of the 'gr... |
| prsdm 32495 | Domain of the relation of ... |
| prsrn 32496 | Range of the relation of a... |
| prsss 32497 | Relation of a subproset. ... |
| prsssdm 32498 | Domain of a subproset rela... |
| ordtprsval 32499 | Value of the order topolog... |
| ordtprsuni 32500 | Value of the order topolog... |
| ordtcnvNEW 32501 | The order dual generates t... |
| ordtrestNEW 32502 | The subspace topology of a... |
| ordtrest2NEWlem 32503 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 32504 | An interval-closed set ` A... |
| ordtconnlem1 32505 | Connectedness in the order... |
| ordtconn 32506 | Connectedness in the order... |
| mndpluscn 32507 | A mapping that is both a h... |
| mhmhmeotmd 32508 | Deduce a Topological Monoi... |
| rmulccn 32509 | Multiplication by a real c... |
| raddcn 32510 | Addition in the real numbe... |
| xrmulc1cn 32511 | The operation multiplying ... |
| fmcncfil 32512 | The image of a Cauchy filt... |
| xrge0hmph 32513 | The extended nonnegative r... |
| xrge0iifcnv 32514 | Define a bijection from ` ... |
| xrge0iifcv 32515 | The defined function's val... |
| xrge0iifiso 32516 | The defined bijection from... |
| xrge0iifhmeo 32517 | Expose a homeomorphism fro... |
| xrge0iifhom 32518 | The defined function from ... |
| xrge0iif1 32519 | Condition for the defined ... |
| xrge0iifmhm 32520 | The defined function from ... |
| xrge0pluscn 32521 | The addition operation of ... |
| xrge0mulc1cn 32522 | The operation multiplying ... |
| xrge0tps 32523 | The extended nonnegative r... |
| xrge0topn 32524 | The topology of the extend... |
| xrge0haus 32525 | The topology of the extend... |
| xrge0tmd 32526 | The extended nonnegative r... |
| xrge0tmdALT 32527 | Alternate proof of ~ xrge0... |
| lmlim 32528 | Relate a limit in a given ... |
| lmlimxrge0 32529 | Relate a limit in the nonn... |
| rge0scvg 32530 | Implication of convergence... |
| fsumcvg4 32531 | A serie with finite suppor... |
| pnfneige0 32532 | A neighborhood of ` +oo ` ... |
| lmxrge0 32533 | Express "sequence ` F ` co... |
| lmdvg 32534 | If a monotonic sequence of... |
| lmdvglim 32535 | If a monotonic real number... |
| pl1cn 32536 | A univariate polynomial is... |
| zringnm 32539 | The norm (function) for a ... |
| zzsnm 32540 | The norm of the ring of th... |
| zlm0 32541 | Zero of a ` ZZ ` -module. ... |
| zlm1 32542 | Unity element of a ` ZZ ` ... |
| zlmds 32543 | Distance in a ` ZZ ` -modu... |
| zlmdsOLD 32544 | Obsolete proof of ~ zlmds ... |
| zlmtset 32545 | Topology in a ` ZZ ` -modu... |
| zlmtsetOLD 32546 | Obsolete proof of ~ zlmtse... |
| zlmnm 32547 | Norm of a ` ZZ ` -module (... |
| zhmnrg 32548 | The ` ZZ ` -module built f... |
| nmmulg 32549 | The norm of a group produc... |
| zrhnm 32550 | The norm of the image by `... |
| cnzh 32551 | The ` ZZ ` -module of ` CC... |
| rezh 32552 | The ` ZZ ` -module of ` RR... |
| qqhval 32555 | Value of the canonical hom... |
| zrhf1ker 32556 | The kernel of the homomorp... |
| zrhchr 32557 | The kernel of the homomorp... |
| zrhker 32558 | The kernel of the homomorp... |
| zrhunitpreima 32559 | The preimage by ` ZRHom ` ... |
| elzrhunit 32560 | Condition for the image by... |
| elzdif0 32561 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 32562 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 32563 | Value of the canonical hom... |
| qqhvval 32564 | Value of the canonical hom... |
| qqh0 32565 | The image of ` 0 ` by the ... |
| qqh1 32566 | The image of ` 1 ` by the ... |
| qqhf 32567 | ` QQHom ` as a function. ... |
| qqhvq 32568 | The image of a quotient by... |
| qqhghm 32569 | The ` QQHom ` homomorphism... |
| qqhrhm 32570 | The ` QQHom ` homomorphism... |
| qqhnm 32571 | The norm of the image by `... |
| qqhcn 32572 | The ` QQHom ` homomorphism... |
| qqhucn 32573 | The ` QQHom ` homomorphism... |
| rrhval 32577 | Value of the canonical hom... |
| rrhcn 32578 | If the topology of ` R ` i... |
| rrhf 32579 | If the topology of ` R ` i... |
| isrrext 32581 | Express the property " ` R... |
| rrextnrg 32582 | An extension of ` RR ` is ... |
| rrextdrg 32583 | An extension of ` RR ` is ... |
| rrextnlm 32584 | The norm of an extension o... |
| rrextchr 32585 | The ring characteristic of... |
| rrextcusp 32586 | An extension of ` RR ` is ... |
| rrexttps 32587 | An extension of ` RR ` is ... |
| rrexthaus 32588 | The topology of an extensi... |
| rrextust 32589 | The uniformity of an exten... |
| rerrext 32590 | The field of the real numb... |
| cnrrext 32591 | The field of the complex n... |
| qqtopn 32592 | The topology of the field ... |
| rrhfe 32593 | If ` R ` is an extension o... |
| rrhcne 32594 | If ` R ` is an extension o... |
| rrhqima 32595 | The ` RRHom ` homomorphism... |
| rrh0 32596 | The image of ` 0 ` by the ... |
| xrhval 32599 | The value of the embedding... |
| zrhre 32600 | The ` ZRHom ` homomorphism... |
| qqhre 32601 | The ` QQHom ` homomorphism... |
| rrhre 32602 | The ` RRHom ` homomorphism... |
| relmntop 32605 | Manifold is a relation. (... |
| ismntoplly 32606 | Property of being a manifo... |
| ismntop 32607 | Property of being a manifo... |
| nexple 32608 | A lower bound for an expon... |
| indv 32611 | Value of the indicator fun... |
| indval 32612 | Value of the indicator fun... |
| indval2 32613 | Alternate value of the ind... |
| indf 32614 | An indicator function as a... |
| indfval 32615 | Value of the indicator fun... |
| ind1 32616 | Value of the indicator fun... |
| ind0 32617 | Value of the indicator fun... |
| ind1a 32618 | Value of the indicator fun... |
| indpi1 32619 | Preimage of the singleton ... |
| indsum 32620 | Finite sum of a product wi... |
| indsumin 32621 | Finite sum of a product wi... |
| prodindf 32622 | The product of indicators ... |
| indf1o 32623 | The bijection between a po... |
| indpreima 32624 | A function with range ` { ... |
| indf1ofs 32625 | The bijection between fini... |
| esumex 32628 | An extended sum is a set b... |
| esumcl 32629 | Closure for extended sum i... |
| esumeq12dvaf 32630 | Equality deduction for ext... |
| esumeq12dva 32631 | Equality deduction for ext... |
| esumeq12d 32632 | Equality deduction for ext... |
| esumeq1 32633 | Equality theorem for an ex... |
| esumeq1d 32634 | Equality theorem for an ex... |
| esumeq2 32635 | Equality theorem for exten... |
| esumeq2d 32636 | Equality deduction for ext... |
| esumeq2dv 32637 | Equality deduction for ext... |
| esumeq2sdv 32638 | Equality deduction for ext... |
| nfesum1 32639 | Bound-variable hypothesis ... |
| nfesum2 32640 | Bound-variable hypothesis ... |
| cbvesum 32641 | Change bound variable in a... |
| cbvesumv 32642 | Change bound variable in a... |
| esumid 32643 | Identify the extended sum ... |
| esumgsum 32644 | A finite extended sum is t... |
| esumval 32645 | Develop the value of the e... |
| esumel 32646 | The extended sum is a limi... |
| esumnul 32647 | Extended sum over the empt... |
| esum0 32648 | Extended sum of zero. (Co... |
| esumf1o 32649 | Re-index an extended sum u... |
| esumc 32650 | Convert from the collectio... |
| esumrnmpt 32651 | Rewrite an extended sum in... |
| esumsplit 32652 | Split an extended sum into... |
| esummono 32653 | Extended sum is monotonic.... |
| esumpad 32654 | Extend an extended sum by ... |
| esumpad2 32655 | Remove zeroes from an exte... |
| esumadd 32656 | Addition of infinite sums.... |
| esumle 32657 | If all of the terms of an ... |
| gsumesum 32658 | Relate a group sum on ` ( ... |
| esumlub 32659 | The extended sum is the lo... |
| esumaddf 32660 | Addition of infinite sums.... |
| esumlef 32661 | If all of the terms of an ... |
| esumcst 32662 | The extended sum of a cons... |
| esumsnf 32663 | The extended sum of a sing... |
| esumsn 32664 | The extended sum of a sing... |
| esumpr 32665 | Extended sum over a pair. ... |
| esumpr2 32666 | Extended sum over a pair, ... |
| esumrnmpt2 32667 | Rewrite an extended sum in... |
| esumfzf 32668 | Formulating a partial exte... |
| esumfsup 32669 | Formulating an extended su... |
| esumfsupre 32670 | Formulating an extended su... |
| esumss 32671 | Change the index set to a ... |
| esumpinfval 32672 | The value of the extended ... |
| esumpfinvallem 32673 | Lemma for ~ esumpfinval . ... |
| esumpfinval 32674 | The value of the extended ... |
| esumpfinvalf 32675 | Same as ~ esumpfinval , mi... |
| esumpinfsum 32676 | The value of the extended ... |
| esumpcvgval 32677 | The value of the extended ... |
| esumpmono 32678 | The partial sums in an ext... |
| esumcocn 32679 | Lemma for ~ esummulc2 and ... |
| esummulc1 32680 | An extended sum multiplied... |
| esummulc2 32681 | An extended sum multiplied... |
| esumdivc 32682 | An extended sum divided by... |
| hashf2 32683 | Lemma for ~ hasheuni . (C... |
| hasheuni 32684 | The cardinality of a disjo... |
| esumcvg 32685 | The sequence of partial su... |
| esumcvg2 32686 | Simpler version of ~ esumc... |
| esumcvgsum 32687 | The value of the extended ... |
| esumsup 32688 | Express an extended sum as... |
| esumgect 32689 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 32690 | All terms of a converging ... |
| esum2dlem 32691 | Lemma for ~ esum2d (finite... |
| esum2d 32692 | Write a double extended su... |
| esumiun 32693 | Sum over a nonnecessarily ... |
| ofceq 32696 | Equality theorem for funct... |
| ofcfval 32697 | Value of an operation appl... |
| ofcval 32698 | Evaluate a function/consta... |
| ofcfn 32699 | The function operation pro... |
| ofcfeqd2 32700 | Equality theorem for funct... |
| ofcfval3 32701 | General value of ` ( F oFC... |
| ofcf 32702 | The function/constant oper... |
| ofcfval2 32703 | The function operation exp... |
| ofcfval4 32704 | The function/constant oper... |
| ofcc 32705 | Left operation by a consta... |
| ofcof 32706 | Relate function operation ... |
| sigaex 32709 | Lemma for ~ issiga and ~ i... |
| sigaval 32710 | The set of sigma-algebra w... |
| issiga 32711 | An alternative definition ... |
| isrnsiga 32712 | The property of being a si... |
| 0elsiga 32713 | A sigma-algebra contains t... |
| baselsiga 32714 | A sigma-algebra contains i... |
| sigasspw 32715 | A sigma-algebra is a set o... |
| sigaclcu 32716 | A sigma-algebra is closed ... |
| sigaclcuni 32717 | A sigma-algebra is closed ... |
| sigaclfu 32718 | A sigma-algebra is closed ... |
| sigaclcu2 32719 | A sigma-algebra is closed ... |
| sigaclfu2 32720 | A sigma-algebra is closed ... |
| sigaclcu3 32721 | A sigma-algebra is closed ... |
| issgon 32722 | Property of being a sigma-... |
| sgon 32723 | A sigma-algebra is a sigma... |
| elsigass 32724 | An element of a sigma-alge... |
| elrnsiga 32725 | Dropping the base informat... |
| isrnsigau 32726 | The property of being a si... |
| unielsiga 32727 | A sigma-algebra contains i... |
| dmvlsiga 32728 | Lebesgue-measurable subset... |
| pwsiga 32729 | Any power set forms a sigm... |
| prsiga 32730 | The smallest possible sigm... |
| sigaclci 32731 | A sigma-algebra is closed ... |
| difelsiga 32732 | A sigma-algebra is closed ... |
| unelsiga 32733 | A sigma-algebra is closed ... |
| inelsiga 32734 | A sigma-algebra is closed ... |
| sigainb 32735 | Building a sigma-algebra f... |
| insiga 32736 | The intersection of a coll... |
| sigagenval 32739 | Value of the generated sig... |
| sigagensiga 32740 | A generated sigma-algebra ... |
| sgsiga 32741 | A generated sigma-algebra ... |
| unisg 32742 | The sigma-algebra generate... |
| dmsigagen 32743 | A sigma-algebra can be gen... |
| sssigagen 32744 | A set is a subset of the s... |
| sssigagen2 32745 | A subset of the generating... |
| elsigagen 32746 | Any element of a set is al... |
| elsigagen2 32747 | Any countable union of ele... |
| sigagenss 32748 | The generated sigma-algebr... |
| sigagenss2 32749 | Sufficient condition for i... |
| sigagenid 32750 | The sigma-algebra generate... |
| ispisys 32751 | The property of being a pi... |
| ispisys2 32752 | The property of being a pi... |
| inelpisys 32753 | Pi-systems are closed unde... |
| sigapisys 32754 | All sigma-algebras are pi-... |
| isldsys 32755 | The property of being a la... |
| pwldsys 32756 | The power set of the unive... |
| unelldsys 32757 | Lambda-systems are closed ... |
| sigaldsys 32758 | All sigma-algebras are lam... |
| ldsysgenld 32759 | The intersection of all la... |
| sigapildsyslem 32760 | Lemma for ~ sigapildsys . ... |
| sigapildsys 32761 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 32762 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 32763 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 32764 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 32765 | The lambda system ` E ` ge... |
| dynkin 32766 | Dynkin's lambda-pi theorem... |
| isros 32767 | The property of being a ri... |
| rossspw 32768 | A ring of sets is a collec... |
| 0elros 32769 | A ring of sets contains th... |
| unelros 32770 | A ring of sets is closed u... |
| difelros 32771 | A ring of sets is closed u... |
| inelros 32772 | A ring of sets is closed u... |
| fiunelros 32773 | A ring of sets is closed u... |
| issros 32774 | The property of being a se... |
| srossspw 32775 | A semiring of sets is a co... |
| 0elsros 32776 | A semiring of sets contain... |
| inelsros 32777 | A semiring of sets is clos... |
| diffiunisros 32778 | In semiring of sets, compl... |
| rossros 32779 | Rings of sets are semiring... |
| brsiga 32782 | The Borel Algebra on real ... |
| brsigarn 32783 | The Borel Algebra is a sig... |
| brsigasspwrn 32784 | The Borel Algebra is a set... |
| unibrsiga 32785 | The union of the Borel Alg... |
| cldssbrsiga 32786 | A Borel Algebra contains a... |
| sxval 32789 | Value of the product sigma... |
| sxsiga 32790 | A product sigma-algebra is... |
| sxsigon 32791 | A product sigma-algebra is... |
| sxuni 32792 | The base set of a product ... |
| elsx 32793 | The cartesian product of t... |
| measbase 32796 | The base set of a measure ... |
| measval 32797 | The value of the ` measure... |
| ismeas 32798 | The property of being a me... |
| isrnmeas 32799 | The property of being a me... |
| dmmeas 32800 | The domain of a measure is... |
| measbasedom 32801 | The base set of a measure ... |
| measfrge0 32802 | A measure is a function ov... |
| measfn 32803 | A measure is a function on... |
| measvxrge0 32804 | The values of a measure ar... |
| measvnul 32805 | The measure of the empty s... |
| measge0 32806 | A measure is nonnegative. ... |
| measle0 32807 | If the measure of a given ... |
| measvun 32808 | The measure of a countable... |
| measxun2 32809 | The measure the union of t... |
| measun 32810 | The measure the union of t... |
| measvunilem 32811 | Lemma for ~ measvuni . (C... |
| measvunilem0 32812 | Lemma for ~ measvuni . (C... |
| measvuni 32813 | The measure of a countable... |
| measssd 32814 | A measure is monotone with... |
| measunl 32815 | A measure is sub-additive ... |
| measiuns 32816 | The measure of the union o... |
| measiun 32817 | A measure is sub-additive.... |
| meascnbl 32818 | A measure is continuous fr... |
| measinblem 32819 | Lemma for ~ measinb . (Co... |
| measinb 32820 | Building a measure restric... |
| measres 32821 | Building a measure restric... |
| measinb2 32822 | Building a measure restric... |
| measdivcst 32823 | Division of a measure by a... |
| measdivcstALTV 32824 | Alternate version of ~ mea... |
| cntmeas 32825 | The Counting measure is a ... |
| pwcntmeas 32826 | The counting measure is a ... |
| cntnevol 32827 | Counting and Lebesgue meas... |
| voliune 32828 | The Lebesgue measure funct... |
| volfiniune 32829 | The Lebesgue measure funct... |
| volmeas 32830 | The Lebesgue measure is a ... |
| ddeval1 32833 | Value of the delta measure... |
| ddeval0 32834 | Value of the delta measure... |
| ddemeas 32835 | The Dirac delta measure is... |
| relae 32839 | 'almost everywhere' is a r... |
| brae 32840 | 'almost everywhere' relati... |
| braew 32841 | 'almost everywhere' relati... |
| truae 32842 | A truth holds almost every... |
| aean 32843 | A conjunction holds almost... |
| faeval 32845 | Value of the 'almost every... |
| relfae 32846 | The 'almost everywhere' bu... |
| brfae 32847 | 'almost everywhere' relati... |
| ismbfm 32850 | The predicate " ` F ` is a... |
| elunirnmbfm 32851 | The property of being a me... |
| mbfmfun 32852 | A measurable function is a... |
| mbfmf 32853 | A measurable function as a... |
| isanmbfmOLD 32854 | Obsolete version of ~ isan... |
| mbfmcnvima 32855 | The preimage by a measurab... |
| isanmbfm 32856 | The predicate to be a meas... |
| mbfmbfmOLD 32857 | A measurable function to a... |
| mbfmbfm 32858 | A measurable function to a... |
| mbfmcst 32859 | A constant function is mea... |
| 1stmbfm 32860 | The first projection map i... |
| 2ndmbfm 32861 | The second projection map ... |
| imambfm 32862 | If the sigma-algebra in th... |
| cnmbfm 32863 | A continuous function is m... |
| mbfmco 32864 | The composition of two mea... |
| mbfmco2 32865 | The pair building of two m... |
| mbfmvolf 32866 | Measurable functions with ... |
| elmbfmvol2 32867 | Measurable functions with ... |
| mbfmcnt 32868 | All functions are measurab... |
| br2base 32869 | The base set for the gener... |
| dya2ub 32870 | An upper bound for a dyadi... |
| sxbrsigalem0 32871 | The closed half-spaces of ... |
| sxbrsigalem3 32872 | The sigma-algebra generate... |
| dya2iocival 32873 | The function ` I ` returns... |
| dya2iocress 32874 | Dyadic intervals are subse... |
| dya2iocbrsiga 32875 | Dyadic intervals are Borel... |
| dya2icobrsiga 32876 | Dyadic intervals are Borel... |
| dya2icoseg 32877 | For any point and any clos... |
| dya2icoseg2 32878 | For any point and any open... |
| dya2iocrfn 32879 | The function returning dya... |
| dya2iocct 32880 | The dyadic rectangle set i... |
| dya2iocnrect 32881 | For any point of an open r... |
| dya2iocnei 32882 | For any point of an open s... |
| dya2iocuni 32883 | Every open set of ` ( RR X... |
| dya2iocucvr 32884 | The dyadic rectangular set... |
| sxbrsigalem1 32885 | The Borel algebra on ` ( R... |
| sxbrsigalem2 32886 | The sigma-algebra generate... |
| sxbrsigalem4 32887 | The Borel algebra on ` ( R... |
| sxbrsigalem5 32888 | First direction for ~ sxbr... |
| sxbrsigalem6 32889 | First direction for ~ sxbr... |
| sxbrsiga 32890 | The product sigma-algebra ... |
| omsval 32893 | Value of the function mapp... |
| omsfval 32894 | Value of the outer measure... |
| omscl 32895 | A closure lemma for the co... |
| omsf 32896 | A constructed outer measur... |
| oms0 32897 | A constructed outer measur... |
| omsmon 32898 | A constructed outer measur... |
| omssubaddlem 32899 | For any small margin ` E `... |
| omssubadd 32900 | A constructed outer measur... |
| carsgval 32903 | Value of the Caratheodory ... |
| carsgcl 32904 | Closure of the Caratheodor... |
| elcarsg 32905 | Property of being a Carath... |
| baselcarsg 32906 | The universe set, ` O ` , ... |
| 0elcarsg 32907 | The empty set is Caratheod... |
| carsguni 32908 | The union of all Caratheod... |
| elcarsgss 32909 | Caratheodory measurable se... |
| difelcarsg 32910 | The Caratheodory measurabl... |
| inelcarsg 32911 | The Caratheodory measurabl... |
| unelcarsg 32912 | The Caratheodory-measurabl... |
| difelcarsg2 32913 | The Caratheodory-measurabl... |
| carsgmon 32914 | Utility lemma: Apply mono... |
| carsgsigalem 32915 | Lemma for the following th... |
| fiunelcarsg 32916 | The Caratheodory measurabl... |
| carsgclctunlem1 32917 | Lemma for ~ carsgclctun . ... |
| carsggect 32918 | The outer measure is count... |
| carsgclctunlem2 32919 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 32920 | Lemma for ~ carsgclctun . ... |
| carsgclctun 32921 | The Caratheodory measurabl... |
| carsgsiga 32922 | The Caratheodory measurabl... |
| omsmeas 32923 | The restriction of a const... |
| pmeasmono 32924 | This theorem's hypotheses ... |
| pmeasadd 32925 | A premeasure on a ring of ... |
| itgeq12dv 32926 | Equality theorem for an in... |
| sitgval 32932 | Value of the simple functi... |
| issibf 32933 | The predicate " ` F ` is a... |
| sibf0 32934 | The constant zero function... |
| sibfmbl 32935 | A simple function is measu... |
| sibff 32936 | A simple function is a fun... |
| sibfrn 32937 | A simple function has fini... |
| sibfima 32938 | Any preimage of a singleto... |
| sibfinima 32939 | The measure of the interse... |
| sibfof 32940 | Applying function operatio... |
| sitgfval 32941 | Value of the Bochner integ... |
| sitgclg 32942 | Closure of the Bochner int... |
| sitgclbn 32943 | Closure of the Bochner int... |
| sitgclcn 32944 | Closure of the Bochner int... |
| sitgclre 32945 | Closure of the Bochner int... |
| sitg0 32946 | The integral of the consta... |
| sitgf 32947 | The integral for simple fu... |
| sitgaddlemb 32948 | Lemma for * sitgadd . (Co... |
| sitmval 32949 | Value of the simple functi... |
| sitmfval 32950 | Value of the integral dist... |
| sitmcl 32951 | Closure of the integral di... |
| sitmf 32952 | The integral metric as a f... |
| oddpwdc 32954 | Lemma for ~ eulerpart . T... |
| oddpwdcv 32955 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 32956 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 32957 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 32958 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 32959 | Lemma for ~ eulerpart . (... |
| eulerpartlems 32960 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 32961 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 32962 | Lemma for ~ eulerpart . (... |
| eulerpartleme 32963 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 32964 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 32965 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 32966 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 32967 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 32968 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 32969 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 32970 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 32971 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 32972 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 32973 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 32974 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 32975 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 32976 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 32977 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 32978 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 32979 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 32980 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 32981 | Lemma for ~ eulerpart . (... |
| eulerpart 32982 | Euler's theorem on partiti... |
| subiwrd 32985 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 32986 | Length of a subword of an ... |
| iwrdsplit 32987 | Lemma for ~ sseqp1 . (Con... |
| sseqval 32988 | Value of the strong sequen... |
| sseqfv1 32989 | Value of the strong sequen... |
| sseqfn 32990 | A strong recursive sequenc... |
| sseqmw 32991 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 32992 | A strong recursive sequenc... |
| sseqfres 32993 | The first elements in the ... |
| sseqfv2 32994 | Value of the strong sequen... |
| sseqp1 32995 | Value of the strong sequen... |
| fiblem 32998 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 32999 | Value of the Fibonacci seq... |
| fib1 33000 | Value of the Fibonacci seq... |
| fibp1 33001 | Value of the Fibonacci seq... |
| fib2 33002 | Value of the Fibonacci seq... |
| fib3 33003 | Value of the Fibonacci seq... |
| fib4 33004 | Value of the Fibonacci seq... |
| fib5 33005 | Value of the Fibonacci seq... |
| fib6 33006 | Value of the Fibonacci seq... |
| elprob 33009 | The property of being a pr... |
| domprobmeas 33010 | A probability measure is a... |
| domprobsiga 33011 | The domain of a probabilit... |
| probtot 33012 | The probability of the uni... |
| prob01 33013 | A probability is an elemen... |
| probnul 33014 | The probability of the emp... |
| unveldomd 33015 | The universe is an element... |
| unveldom 33016 | The universe is an element... |
| nuleldmp 33017 | The empty set is an elemen... |
| probcun 33018 | The probability of the uni... |
| probun 33019 | The probability of the uni... |
| probdif 33020 | The probability of the dif... |
| probinc 33021 | A probability law is incre... |
| probdsb 33022 | The probability of the com... |
| probmeasd 33023 | A probability measure is a... |
| probvalrnd 33024 | The value of a probability... |
| probtotrnd 33025 | The probability of the uni... |
| totprobd 33026 | Law of total probability, ... |
| totprob 33027 | Law of total probability. ... |
| probfinmeasb 33028 | Build a probability measur... |
| probfinmeasbALTV 33029 | Alternate version of ~ pro... |
| probmeasb 33030 | Build a probability from a... |
| cndprobval 33033 | The value of the condition... |
| cndprobin 33034 | An identity linking condit... |
| cndprob01 33035 | The conditional probabilit... |
| cndprobtot 33036 | The conditional probabilit... |
| cndprobnul 33037 | The conditional probabilit... |
| cndprobprob 33038 | The conditional probabilit... |
| bayesth 33039 | Bayes Theorem. (Contribut... |
| rrvmbfm 33042 | A real-valued random varia... |
| isrrvv 33043 | Elementhood to the set of ... |
| rrvvf 33044 | A real-valued random varia... |
| rrvfn 33045 | A real-valued random varia... |
| rrvdm 33046 | The domain of a random var... |
| rrvrnss 33047 | The range of a random vari... |
| rrvf2 33048 | A real-valued random varia... |
| rrvdmss 33049 | The domain of a random var... |
| rrvfinvima 33050 | For a real-value random va... |
| 0rrv 33051 | The constant function equa... |
| rrvadd 33052 | The sum of two random vari... |
| rrvmulc 33053 | A random variable multipli... |
| rrvsum 33054 | An indexed sum of random v... |
| orvcval 33057 | Value of the preimage mapp... |
| orvcval2 33058 | Another way to express the... |
| elorvc 33059 | Elementhood of a preimage.... |
| orvcval4 33060 | The value of the preimage ... |
| orvcoel 33061 | If the relation produces o... |
| orvccel 33062 | If the relation produces c... |
| elorrvc 33063 | Elementhood of a preimage ... |
| orrvcval4 33064 | The value of the preimage ... |
| orrvcoel 33065 | If the relation produces o... |
| orrvccel 33066 | If the relation produces c... |
| orvcgteel 33067 | Preimage maps produced by ... |
| orvcelval 33068 | Preimage maps produced by ... |
| orvcelel 33069 | Preimage maps produced by ... |
| dstrvval 33070 | The value of the distribut... |
| dstrvprob 33071 | The distribution of a rand... |
| orvclteel 33072 | Preimage maps produced by ... |
| dstfrvel 33073 | Elementhood of preimage ma... |
| dstfrvunirn 33074 | The limit of all preimage ... |
| orvclteinc 33075 | Preimage maps produced by ... |
| dstfrvinc 33076 | A cumulative distribution ... |
| dstfrvclim1 33077 | The limit of the cumulativ... |
| coinfliplem 33078 | Division in the extended r... |
| coinflipprob 33079 | The ` P ` we defined for c... |
| coinflipspace 33080 | The space of our coin-flip... |
| coinflipuniv 33081 | The universe of our coin-f... |
| coinfliprv 33082 | The ` X ` we defined for c... |
| coinflippv 33083 | The probability of heads i... |
| coinflippvt 33084 | The probability of tails i... |
| ballotlemoex 33085 | ` O ` is a set. (Contribu... |
| ballotlem1 33086 | The size of the universe i... |
| ballotlemelo 33087 | Elementhood in ` O ` . (C... |
| ballotlem2 33088 | The probability that the f... |
| ballotlemfval 33089 | The value of ` F ` . (Con... |
| ballotlemfelz 33090 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 33091 | If the ` J ` th ballot is ... |
| ballotlemfc0 33092 | ` F ` takes value 0 betwee... |
| ballotlemfcc 33093 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 33094 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 33095 | ` ( F `` C ) ` always star... |
| ballotleme 33096 | Elements of ` E ` . (Cont... |
| ballotlemodife 33097 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 33098 | If the first pick is a vot... |
| ballotlem5 33099 | If A is not ahead througho... |
| ballotlemi 33100 | Value of ` I ` for a given... |
| ballotlemiex 33101 | Properties of ` ( I `` C )... |
| ballotlemi1 33102 | The first tie cannot be re... |
| ballotlemii 33103 | The first tie cannot be re... |
| ballotlemsup 33104 | The set of zeroes of ` F `... |
| ballotlemimin 33105 | ` ( I `` C ) ` is the firs... |
| ballotlemic 33106 | If the first vote is for B... |
| ballotlem1c 33107 | If the first vote is for A... |
| ballotlemsval 33108 | Value of ` S ` . (Contrib... |
| ballotlemsv 33109 | Value of ` S ` evaluated a... |
| ballotlemsgt1 33110 | ` S ` maps values less tha... |
| ballotlemsdom 33111 | Domain of ` S ` for a give... |
| ballotlemsel1i 33112 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 33113 | The defined ` S ` is a bij... |
| ballotlemsi 33114 | The image by ` S ` of the ... |
| ballotlemsima 33115 | The image by ` S ` of an i... |
| ballotlemieq 33116 | If two countings share the... |
| ballotlemrval 33117 | Value of ` R ` . (Contrib... |
| ballotlemscr 33118 | The image of ` ( R `` C ) ... |
| ballotlemrv 33119 | Value of ` R ` evaluated a... |
| ballotlemrv1 33120 | Value of ` R ` before the ... |
| ballotlemrv2 33121 | Value of ` R ` after the t... |
| ballotlemro 33122 | Range of ` R ` is included... |
| ballotlemgval 33123 | Expand the value of ` .^ `... |
| ballotlemgun 33124 | A property of the defined ... |
| ballotlemfg 33125 | Express the value of ` ( F... |
| ballotlemfrc 33126 | Express the value of ` ( F... |
| ballotlemfrci 33127 | Reverse counting preserves... |
| ballotlemfrceq 33128 | Value of ` F ` for a rever... |
| ballotlemfrcn0 33129 | Value of ` F ` for a rever... |
| ballotlemrc 33130 | Range of ` R ` . (Contrib... |
| ballotlemirc 33131 | Applying ` R ` does not ch... |
| ballotlemrinv0 33132 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 33133 | ` R ` is its own inverse :... |
| ballotlem1ri 33134 | When the vote on the first... |
| ballotlem7 33135 | ` R ` is a bijection betwe... |
| ballotlem8 33136 | There are as many counting... |
| ballotth 33137 | Bertrand's ballot problem ... |
| sgncl 33138 | Closure of the signum. (C... |
| sgnclre 33139 | Closure of the signum. (C... |
| sgnneg 33140 | Negation of the signum. (... |
| sgn3da 33141 | A conditional containing a... |
| sgnmul 33142 | Signum of a product. (Con... |
| sgnmulrp2 33143 | Multiplication by a positi... |
| sgnsub 33144 | Subtraction of a number of... |
| sgnnbi 33145 | Negative signum. (Contrib... |
| sgnpbi 33146 | Positive signum. (Contrib... |
| sgn0bi 33147 | Zero signum. (Contributed... |
| sgnsgn 33148 | Signum is idempotent. (Co... |
| sgnmulsgn 33149 | If two real numbers are of... |
| sgnmulsgp 33150 | If two real numbers are of... |
| fzssfzo 33151 | Condition for an integer i... |
| gsumncl 33152 | Closure of a group sum in ... |
| gsumnunsn 33153 | Closure of a group sum in ... |
| ccatmulgnn0dir 33154 | Concatenation of words fol... |
| ofcccat 33155 | Letterwise operations on w... |
| ofcs1 33156 | Letterwise operations on a... |
| ofcs2 33157 | Letterwise operations on a... |
| plymul02 33158 | Product of a polynomial wi... |
| plymulx0 33159 | Coefficients of a polynomi... |
| plymulx 33160 | Coefficients of a polynomi... |
| plyrecld 33161 | Closure of a polynomial wi... |
| signsplypnf 33162 | The quotient of a polynomi... |
| signsply0 33163 | Lemma for the rule of sign... |
| signspval 33164 | The value of the skipping ... |
| signsw0glem 33165 | Neutral element property o... |
| signswbase 33166 | The base of ` W ` is the u... |
| signswplusg 33167 | The operation of ` W ` . ... |
| signsw0g 33168 | The neutral element of ` W... |
| signswmnd 33169 | ` W ` is a monoid structur... |
| signswrid 33170 | The zero-skipping operatio... |
| signswlid 33171 | The zero-skipping operatio... |
| signswn0 33172 | The zero-skipping operatio... |
| signswch 33173 | The zero-skipping operatio... |
| signslema 33174 | Computational part of ~~? ... |
| signstfv 33175 | Value of the zero-skipping... |
| signstfval 33176 | Value of the zero-skipping... |
| signstcl 33177 | Closure of the zero skippi... |
| signstf 33178 | The zero skipping sign wor... |
| signstlen 33179 | Length of the zero skippin... |
| signstf0 33180 | Sign of a single letter wo... |
| signstfvn 33181 | Zero-skipping sign in a wo... |
| signsvtn0 33182 | If the last letter is nonz... |
| signstfvp 33183 | Zero-skipping sign in a wo... |
| signstfvneq0 33184 | In case the first letter i... |
| signstfvcl 33185 | Closure of the zero skippi... |
| signstfvc 33186 | Zero-skipping sign in a wo... |
| signstres 33187 | Restriction of a zero skip... |
| signstfveq0a 33188 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 33189 | In case the last letter is... |
| signsvvfval 33190 | The value of ` V ` , which... |
| signsvvf 33191 | ` V ` is a function. (Con... |
| signsvf0 33192 | There is no change of sign... |
| signsvf1 33193 | In a single-letter word, w... |
| signsvfn 33194 | Number of changes in a wor... |
| signsvtp 33195 | Adding a letter of the sam... |
| signsvtn 33196 | Adding a letter of a diffe... |
| signsvfpn 33197 | Adding a letter of the sam... |
| signsvfnn 33198 | Adding a letter of a diffe... |
| signlem0 33199 | Adding a zero as the highe... |
| signshf 33200 | ` H ` , corresponding to t... |
| signshwrd 33201 | ` H ` , corresponding to t... |
| signshlen 33202 | Length of ` H ` , correspo... |
| signshnz 33203 | ` H ` is not the empty wor... |
| efcld 33204 | Closure law for the expone... |
| iblidicc 33205 | The identity function is i... |
| rpsqrtcn 33206 | Continuity of the real pos... |
| divsqrtid 33207 | A real number divided by i... |
| cxpcncf1 33208 | The power function on comp... |
| efmul2picn 33209 | Multiplying by ` ( _i x. (... |
| fct2relem 33210 | Lemma for ~ ftc2re . (Con... |
| ftc2re 33211 | The Fundamental Theorem of... |
| fdvposlt 33212 | Functions with a positive ... |
| fdvneggt 33213 | Functions with a negative ... |
| fdvposle 33214 | Functions with a nonnegati... |
| fdvnegge 33215 | Functions with a nonpositi... |
| prodfzo03 33216 | A product of three factors... |
| actfunsnf1o 33217 | The action ` F ` of extend... |
| actfunsnrndisj 33218 | The action ` F ` of extend... |
| itgexpif 33219 | The basis for the circle m... |
| fsum2dsub 33220 | Lemma for ~ breprexp - Re-... |
| reprval 33223 | Value of the representatio... |
| repr0 33224 | There is exactly one repre... |
| reprf 33225 | Members of the representat... |
| reprsum 33226 | Sums of values of the memb... |
| reprle 33227 | Upper bound to the terms i... |
| reprsuc 33228 | Express the representation... |
| reprfi 33229 | Bounded representations ar... |
| reprss 33230 | Representations with terms... |
| reprinrn 33231 | Representations with term ... |
| reprlt 33232 | There are no representatio... |
| hashreprin 33233 | Express a sum of represent... |
| reprgt 33234 | There are no representatio... |
| reprinfz1 33235 | For the representation of ... |
| reprfi2 33236 | Corollary of ~ reprinfz1 .... |
| reprfz1 33237 | Corollary of ~ reprinfz1 .... |
| hashrepr 33238 | Develop the number of repr... |
| reprpmtf1o 33239 | Transposing ` 0 ` and ` X ... |
| reprdifc 33240 | Express the representation... |
| chpvalz 33241 | Value of the second Chebys... |
| chtvalz 33242 | Value of the Chebyshev fun... |
| breprexplema 33243 | Lemma for ~ breprexp (indu... |
| breprexplemb 33244 | Lemma for ~ breprexp (clos... |
| breprexplemc 33245 | Lemma for ~ breprexp (indu... |
| breprexp 33246 | Express the ` S ` th power... |
| breprexpnat 33247 | Express the ` S ` th power... |
| vtsval 33250 | Value of the Vinogradov tr... |
| vtscl 33251 | Closure of the Vinogradov ... |
| vtsprod 33252 | Express the Vinogradov tri... |
| circlemeth 33253 | The Hardy, Littlewood and ... |
| circlemethnat 33254 | The Hardy, Littlewood and ... |
| circlevma 33255 | The Circle Method, where t... |
| circlemethhgt 33256 | The circle method, where t... |
| hgt750lemc 33260 | An upper bound to the summ... |
| hgt750lemd 33261 | An upper bound to the summ... |
| hgt749d 33262 | A deduction version of ~ a... |
| logdivsqrle 33263 | Conditions for ` ( ( log `... |
| hgt750lem 33264 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 33265 | Decimal multiplication gal... |
| hgt750lemf 33266 | Lemma for the statement 7.... |
| hgt750lemg 33267 | Lemma for the statement 7.... |
| oddprm2 33268 | Two ways to write the set ... |
| hgt750lemb 33269 | An upper bound on the cont... |
| hgt750lema 33270 | An upper bound on the cont... |
| hgt750leme 33271 | An upper bound on the cont... |
| tgoldbachgnn 33272 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 33273 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 33274 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 33275 | Odd integers greater than ... |
| tgoldbachgt 33276 | Odd integers greater than ... |
| istrkg2d 33279 | Property of fulfilling dim... |
| axtglowdim2ALTV 33280 | Alternate version of ~ axt... |
| axtgupdim2ALTV 33281 | Alternate version of ~ axt... |
| afsval 33284 | Value of the AFS relation ... |
| brafs 33285 | Binary relation form of th... |
| tg5segofs 33286 | Rephrase ~ axtg5seg using ... |
| lpadval 33289 | Value of the ` leftpad ` f... |
| lpadlem1 33290 | Lemma for the ` leftpad ` ... |
| lpadlem3 33291 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 33292 | Length of a left-padded wo... |
| lpadlem2 33293 | Lemma for the ` leftpad ` ... |
| lpadlen2 33294 | Length of a left-padded wo... |
| lpadmax 33295 | Length of a left-padded wo... |
| lpadleft 33296 | The contents of prefix of ... |
| lpadright 33297 | The suffix of a left-padde... |
| bnj170 33310 | ` /\ ` -manipulation. (Co... |
| bnj240 33311 | ` /\ ` -manipulation. (Co... |
| bnj248 33312 | ` /\ ` -manipulation. (Co... |
| bnj250 33313 | ` /\ ` -manipulation. (Co... |
| bnj251 33314 | ` /\ ` -manipulation. (Co... |
| bnj252 33315 | ` /\ ` -manipulation. (Co... |
| bnj253 33316 | ` /\ ` -manipulation. (Co... |
| bnj255 33317 | ` /\ ` -manipulation. (Co... |
| bnj256 33318 | ` /\ ` -manipulation. (Co... |
| bnj257 33319 | ` /\ ` -manipulation. (Co... |
| bnj258 33320 | ` /\ ` -manipulation. (Co... |
| bnj268 33321 | ` /\ ` -manipulation. (Co... |
| bnj290 33322 | ` /\ ` -manipulation. (Co... |
| bnj291 33323 | ` /\ ` -manipulation. (Co... |
| bnj312 33324 | ` /\ ` -manipulation. (Co... |
| bnj334 33325 | ` /\ ` -manipulation. (Co... |
| bnj345 33326 | ` /\ ` -manipulation. (Co... |
| bnj422 33327 | ` /\ ` -manipulation. (Co... |
| bnj432 33328 | ` /\ ` -manipulation. (Co... |
| bnj446 33329 | ` /\ ` -manipulation. (Co... |
| bnj23 33330 | First-order logic and set ... |
| bnj31 33331 | First-order logic and set ... |
| bnj62 33332 | First-order logic and set ... |
| bnj89 33333 | First-order logic and set ... |
| bnj90 33334 | First-order logic and set ... |
| bnj101 33335 | First-order logic and set ... |
| bnj105 33336 | First-order logic and set ... |
| bnj115 33337 | First-order logic and set ... |
| bnj132 33338 | First-order logic and set ... |
| bnj133 33339 | First-order logic and set ... |
| bnj156 33340 | First-order logic and set ... |
| bnj158 33341 | First-order logic and set ... |
| bnj168 33342 | First-order logic and set ... |
| bnj206 33343 | First-order logic and set ... |
| bnj216 33344 | First-order logic and set ... |
| bnj219 33345 | First-order logic and set ... |
| bnj226 33346 | First-order logic and set ... |
| bnj228 33347 | First-order logic and set ... |
| bnj519 33348 | First-order logic and set ... |
| bnj524 33349 | First-order logic and set ... |
| bnj525 33350 | First-order logic and set ... |
| bnj534 33351 | First-order logic and set ... |
| bnj538 33352 | First-order logic and set ... |
| bnj529 33353 | First-order logic and set ... |
| bnj551 33354 | First-order logic and set ... |
| bnj563 33355 | First-order logic and set ... |
| bnj564 33356 | First-order logic and set ... |
| bnj593 33357 | First-order logic and set ... |
| bnj596 33358 | First-order logic and set ... |
| bnj610 33359 | Pass from equality ( ` x =... |
| bnj642 33360 | ` /\ ` -manipulation. (Co... |
| bnj643 33361 | ` /\ ` -manipulation. (Co... |
| bnj645 33362 | ` /\ ` -manipulation. (Co... |
| bnj658 33363 | ` /\ ` -manipulation. (Co... |
| bnj667 33364 | ` /\ ` -manipulation. (Co... |
| bnj705 33365 | ` /\ ` -manipulation. (Co... |
| bnj706 33366 | ` /\ ` -manipulation. (Co... |
| bnj707 33367 | ` /\ ` -manipulation. (Co... |
| bnj708 33368 | ` /\ ` -manipulation. (Co... |
| bnj721 33369 | ` /\ ` -manipulation. (Co... |
| bnj832 33370 | ` /\ ` -manipulation. (Co... |
| bnj835 33371 | ` /\ ` -manipulation. (Co... |
| bnj836 33372 | ` /\ ` -manipulation. (Co... |
| bnj837 33373 | ` /\ ` -manipulation. (Co... |
| bnj769 33374 | ` /\ ` -manipulation. (Co... |
| bnj770 33375 | ` /\ ` -manipulation. (Co... |
| bnj771 33376 | ` /\ ` -manipulation. (Co... |
| bnj887 33377 | ` /\ ` -manipulation. (Co... |
| bnj918 33378 | First-order logic and set ... |
| bnj919 33379 | First-order logic and set ... |
| bnj923 33380 | First-order logic and set ... |
| bnj927 33381 | First-order logic and set ... |
| bnj931 33382 | First-order logic and set ... |
| bnj937 33383 | First-order logic and set ... |
| bnj941 33384 | First-order logic and set ... |
| bnj945 33385 | Technical lemma for ~ bnj6... |
| bnj946 33386 | First-order logic and set ... |
| bnj951 33387 | ` /\ ` -manipulation. (Co... |
| bnj956 33388 | First-order logic and set ... |
| bnj976 33389 | First-order logic and set ... |
| bnj982 33390 | First-order logic and set ... |
| bnj1019 33391 | First-order logic and set ... |
| bnj1023 33392 | First-order logic and set ... |
| bnj1095 33393 | First-order logic and set ... |
| bnj1096 33394 | First-order logic and set ... |
| bnj1098 33395 | First-order logic and set ... |
| bnj1101 33396 | First-order logic and set ... |
| bnj1113 33397 | First-order logic and set ... |
| bnj1109 33398 | First-order logic and set ... |
| bnj1131 33399 | First-order logic and set ... |
| bnj1138 33400 | First-order logic and set ... |
| bnj1142 33401 | First-order logic and set ... |
| bnj1143 33402 | First-order logic and set ... |
| bnj1146 33403 | First-order logic and set ... |
| bnj1149 33404 | First-order logic and set ... |
| bnj1185 33405 | First-order logic and set ... |
| bnj1196 33406 | First-order logic and set ... |
| bnj1198 33407 | First-order logic and set ... |
| bnj1209 33408 | First-order logic and set ... |
| bnj1211 33409 | First-order logic and set ... |
| bnj1213 33410 | First-order logic and set ... |
| bnj1212 33411 | First-order logic and set ... |
| bnj1219 33412 | First-order logic and set ... |
| bnj1224 33413 | First-order logic and set ... |
| bnj1230 33414 | First-order logic and set ... |
| bnj1232 33415 | First-order logic and set ... |
| bnj1235 33416 | First-order logic and set ... |
| bnj1239 33417 | First-order logic and set ... |
| bnj1238 33418 | First-order logic and set ... |
| bnj1241 33419 | First-order logic and set ... |
| bnj1247 33420 | First-order logic and set ... |
| bnj1254 33421 | First-order logic and set ... |
| bnj1262 33422 | First-order logic and set ... |
| bnj1266 33423 | First-order logic and set ... |
| bnj1265 33424 | First-order logic and set ... |
| bnj1275 33425 | First-order logic and set ... |
| bnj1276 33426 | First-order logic and set ... |
| bnj1292 33427 | First-order logic and set ... |
| bnj1293 33428 | First-order logic and set ... |
| bnj1294 33429 | First-order logic and set ... |
| bnj1299 33430 | First-order logic and set ... |
| bnj1304 33431 | First-order logic and set ... |
| bnj1316 33432 | First-order logic and set ... |
| bnj1317 33433 | First-order logic and set ... |
| bnj1322 33434 | First-order logic and set ... |
| bnj1340 33435 | First-order logic and set ... |
| bnj1345 33436 | First-order logic and set ... |
| bnj1350 33437 | First-order logic and set ... |
| bnj1351 33438 | First-order logic and set ... |
| bnj1352 33439 | First-order logic and set ... |
| bnj1361 33440 | First-order logic and set ... |
| bnj1366 33441 | First-order logic and set ... |
| bnj1379 33442 | First-order logic and set ... |
| bnj1383 33443 | First-order logic and set ... |
| bnj1385 33444 | First-order logic and set ... |
| bnj1386 33445 | First-order logic and set ... |
| bnj1397 33446 | First-order logic and set ... |
| bnj1400 33447 | First-order logic and set ... |
| bnj1405 33448 | First-order logic and set ... |
| bnj1422 33449 | First-order logic and set ... |
| bnj1424 33450 | First-order logic and set ... |
| bnj1436 33451 | First-order logic and set ... |
| bnj1441 33452 | First-order logic and set ... |
| bnj1441g 33453 | First-order logic and set ... |
| bnj1454 33454 | First-order logic and set ... |
| bnj1459 33455 | First-order logic and set ... |
| bnj1464 33456 | Conversion of implicit sub... |
| bnj1465 33457 | First-order logic and set ... |
| bnj1468 33458 | Conversion of implicit sub... |
| bnj1476 33459 | First-order logic and set ... |
| bnj1502 33460 | First-order logic and set ... |
| bnj1503 33461 | First-order logic and set ... |
| bnj1517 33462 | First-order logic and set ... |
| bnj1521 33463 | First-order logic and set ... |
| bnj1533 33464 | First-order logic and set ... |
| bnj1534 33465 | First-order logic and set ... |
| bnj1536 33466 | First-order logic and set ... |
| bnj1538 33467 | First-order logic and set ... |
| bnj1541 33468 | First-order logic and set ... |
| bnj1542 33469 | First-order logic and set ... |
| bnj110 33470 | Well-founded induction res... |
| bnj157 33471 | Well-founded induction res... |
| bnj66 33472 | Technical lemma for ~ bnj6... |
| bnj91 33473 | First-order logic and set ... |
| bnj92 33474 | First-order logic and set ... |
| bnj93 33475 | Technical lemma for ~ bnj9... |
| bnj95 33476 | Technical lemma for ~ bnj1... |
| bnj96 33477 | Technical lemma for ~ bnj1... |
| bnj97 33478 | Technical lemma for ~ bnj1... |
| bnj98 33479 | Technical lemma for ~ bnj1... |
| bnj106 33480 | First-order logic and set ... |
| bnj118 33481 | First-order logic and set ... |
| bnj121 33482 | First-order logic and set ... |
| bnj124 33483 | Technical lemma for ~ bnj1... |
| bnj125 33484 | Technical lemma for ~ bnj1... |
| bnj126 33485 | Technical lemma for ~ bnj1... |
| bnj130 33486 | Technical lemma for ~ bnj1... |
| bnj149 33487 | Technical lemma for ~ bnj1... |
| bnj150 33488 | Technical lemma for ~ bnj1... |
| bnj151 33489 | Technical lemma for ~ bnj1... |
| bnj154 33490 | Technical lemma for ~ bnj1... |
| bnj155 33491 | Technical lemma for ~ bnj1... |
| bnj153 33492 | Technical lemma for ~ bnj8... |
| bnj207 33493 | Technical lemma for ~ bnj8... |
| bnj213 33494 | First-order logic and set ... |
| bnj222 33495 | Technical lemma for ~ bnj2... |
| bnj229 33496 | Technical lemma for ~ bnj5... |
| bnj517 33497 | Technical lemma for ~ bnj5... |
| bnj518 33498 | Technical lemma for ~ bnj8... |
| bnj523 33499 | Technical lemma for ~ bnj8... |
| bnj526 33500 | Technical lemma for ~ bnj8... |
| bnj528 33501 | Technical lemma for ~ bnj8... |
| bnj535 33502 | Technical lemma for ~ bnj8... |
| bnj539 33503 | Technical lemma for ~ bnj8... |
| bnj540 33504 | Technical lemma for ~ bnj8... |
| bnj543 33505 | Technical lemma for ~ bnj8... |
| bnj544 33506 | Technical lemma for ~ bnj8... |
| bnj545 33507 | Technical lemma for ~ bnj8... |
| bnj546 33508 | Technical lemma for ~ bnj8... |
| bnj548 33509 | Technical lemma for ~ bnj8... |
| bnj553 33510 | Technical lemma for ~ bnj8... |
| bnj554 33511 | Technical lemma for ~ bnj8... |
| bnj556 33512 | Technical lemma for ~ bnj8... |
| bnj557 33513 | Technical lemma for ~ bnj8... |
| bnj558 33514 | Technical lemma for ~ bnj8... |
| bnj561 33515 | Technical lemma for ~ bnj8... |
| bnj562 33516 | Technical lemma for ~ bnj8... |
| bnj570 33517 | Technical lemma for ~ bnj8... |
| bnj571 33518 | Technical lemma for ~ bnj8... |
| bnj605 33519 | Technical lemma. This lem... |
| bnj581 33520 | Technical lemma for ~ bnj5... |
| bnj589 33521 | Technical lemma for ~ bnj8... |
| bnj590 33522 | Technical lemma for ~ bnj8... |
| bnj591 33523 | Technical lemma for ~ bnj8... |
| bnj594 33524 | Technical lemma for ~ bnj8... |
| bnj580 33525 | Technical lemma for ~ bnj5... |
| bnj579 33526 | Technical lemma for ~ bnj8... |
| bnj602 33527 | Equality theorem for the `... |
| bnj607 33528 | Technical lemma for ~ bnj8... |
| bnj609 33529 | Technical lemma for ~ bnj8... |
| bnj611 33530 | Technical lemma for ~ bnj8... |
| bnj600 33531 | Technical lemma for ~ bnj8... |
| bnj601 33532 | Technical lemma for ~ bnj8... |
| bnj852 33533 | Technical lemma for ~ bnj6... |
| bnj864 33534 | Technical lemma for ~ bnj6... |
| bnj865 33535 | Technical lemma for ~ bnj6... |
| bnj873 33536 | Technical lemma for ~ bnj6... |
| bnj849 33537 | Technical lemma for ~ bnj6... |
| bnj882 33538 | Definition (using hypothes... |
| bnj18eq1 33539 | Equality theorem for trans... |
| bnj893 33540 | Property of ` _trCl ` . U... |
| bnj900 33541 | Technical lemma for ~ bnj6... |
| bnj906 33542 | Property of ` _trCl ` . (... |
| bnj908 33543 | Technical lemma for ~ bnj6... |
| bnj911 33544 | Technical lemma for ~ bnj6... |
| bnj916 33545 | Technical lemma for ~ bnj6... |
| bnj917 33546 | Technical lemma for ~ bnj6... |
| bnj934 33547 | Technical lemma for ~ bnj6... |
| bnj929 33548 | Technical lemma for ~ bnj6... |
| bnj938 33549 | Technical lemma for ~ bnj6... |
| bnj944 33550 | Technical lemma for ~ bnj6... |
| bnj953 33551 | Technical lemma for ~ bnj6... |
| bnj958 33552 | Technical lemma for ~ bnj6... |
| bnj1000 33553 | Technical lemma for ~ bnj8... |
| bnj965 33554 | Technical lemma for ~ bnj8... |
| bnj964 33555 | Technical lemma for ~ bnj6... |
| bnj966 33556 | Technical lemma for ~ bnj6... |
| bnj967 33557 | Technical lemma for ~ bnj6... |
| bnj969 33558 | Technical lemma for ~ bnj6... |
| bnj970 33559 | Technical lemma for ~ bnj6... |
| bnj910 33560 | Technical lemma for ~ bnj6... |
| bnj978 33561 | Technical lemma for ~ bnj6... |
| bnj981 33562 | Technical lemma for ~ bnj6... |
| bnj983 33563 | Technical lemma for ~ bnj6... |
| bnj984 33564 | Technical lemma for ~ bnj6... |
| bnj985v 33565 | Version of ~ bnj985 with a... |
| bnj985 33566 | Technical lemma for ~ bnj6... |
| bnj986 33567 | Technical lemma for ~ bnj6... |
| bnj996 33568 | Technical lemma for ~ bnj6... |
| bnj998 33569 | Technical lemma for ~ bnj6... |
| bnj999 33570 | Technical lemma for ~ bnj6... |
| bnj1001 33571 | Technical lemma for ~ bnj6... |
| bnj1006 33572 | Technical lemma for ~ bnj6... |
| bnj1014 33573 | Technical lemma for ~ bnj6... |
| bnj1015 33574 | Technical lemma for ~ bnj6... |
| bnj1018g 33575 | Version of ~ bnj1018 with ... |
| bnj1018 33576 | Technical lemma for ~ bnj6... |
| bnj1020 33577 | Technical lemma for ~ bnj6... |
| bnj1021 33578 | Technical lemma for ~ bnj6... |
| bnj907 33579 | Technical lemma for ~ bnj6... |
| bnj1029 33580 | Property of ` _trCl ` . (... |
| bnj1033 33581 | Technical lemma for ~ bnj6... |
| bnj1034 33582 | Technical lemma for ~ bnj6... |
| bnj1039 33583 | Technical lemma for ~ bnj6... |
| bnj1040 33584 | Technical lemma for ~ bnj6... |
| bnj1047 33585 | Technical lemma for ~ bnj6... |
| bnj1049 33586 | Technical lemma for ~ bnj6... |
| bnj1052 33587 | Technical lemma for ~ bnj6... |
| bnj1053 33588 | Technical lemma for ~ bnj6... |
| bnj1071 33589 | Technical lemma for ~ bnj6... |
| bnj1083 33590 | Technical lemma for ~ bnj6... |
| bnj1090 33591 | Technical lemma for ~ bnj6... |
| bnj1093 33592 | Technical lemma for ~ bnj6... |
| bnj1097 33593 | Technical lemma for ~ bnj6... |
| bnj1110 33594 | Technical lemma for ~ bnj6... |
| bnj1112 33595 | Technical lemma for ~ bnj6... |
| bnj1118 33596 | Technical lemma for ~ bnj6... |
| bnj1121 33597 | Technical lemma for ~ bnj6... |
| bnj1123 33598 | Technical lemma for ~ bnj6... |
| bnj1030 33599 | Technical lemma for ~ bnj6... |
| bnj1124 33600 | Property of ` _trCl ` . (... |
| bnj1133 33601 | Technical lemma for ~ bnj6... |
| bnj1128 33602 | Technical lemma for ~ bnj6... |
| bnj1127 33603 | Property of ` _trCl ` . (... |
| bnj1125 33604 | Property of ` _trCl ` . (... |
| bnj1145 33605 | Technical lemma for ~ bnj6... |
| bnj1147 33606 | Property of ` _trCl ` . (... |
| bnj1137 33607 | Property of ` _trCl ` . (... |
| bnj1148 33608 | Property of ` _pred ` . (... |
| bnj1136 33609 | Technical lemma for ~ bnj6... |
| bnj1152 33610 | Technical lemma for ~ bnj6... |
| bnj1154 33611 | Property of ` Fr ` . (Con... |
| bnj1171 33612 | Technical lemma for ~ bnj6... |
| bnj1172 33613 | Technical lemma for ~ bnj6... |
| bnj1173 33614 | Technical lemma for ~ bnj6... |
| bnj1174 33615 | Technical lemma for ~ bnj6... |
| bnj1175 33616 | Technical lemma for ~ bnj6... |
| bnj1176 33617 | Technical lemma for ~ bnj6... |
| bnj1177 33618 | Technical lemma for ~ bnj6... |
| bnj1186 33619 | Technical lemma for ~ bnj6... |
| bnj1190 33620 | Technical lemma for ~ bnj6... |
| bnj1189 33621 | Technical lemma for ~ bnj6... |
| bnj69 33622 | Existence of a minimal ele... |
| bnj1228 33623 | Existence of a minimal ele... |
| bnj1204 33624 | Well-founded induction. T... |
| bnj1234 33625 | Technical lemma for ~ bnj6... |
| bnj1245 33626 | Technical lemma for ~ bnj6... |
| bnj1256 33627 | Technical lemma for ~ bnj6... |
| bnj1259 33628 | Technical lemma for ~ bnj6... |
| bnj1253 33629 | Technical lemma for ~ bnj6... |
| bnj1279 33630 | Technical lemma for ~ bnj6... |
| bnj1286 33631 | Technical lemma for ~ bnj6... |
| bnj1280 33632 | Technical lemma for ~ bnj6... |
| bnj1296 33633 | Technical lemma for ~ bnj6... |
| bnj1309 33634 | Technical lemma for ~ bnj6... |
| bnj1307 33635 | Technical lemma for ~ bnj6... |
| bnj1311 33636 | Technical lemma for ~ bnj6... |
| bnj1318 33637 | Technical lemma for ~ bnj6... |
| bnj1326 33638 | Technical lemma for ~ bnj6... |
| bnj1321 33639 | Technical lemma for ~ bnj6... |
| bnj1364 33640 | Property of ` _FrSe ` . (... |
| bnj1371 33641 | Technical lemma for ~ bnj6... |
| bnj1373 33642 | Technical lemma for ~ bnj6... |
| bnj1374 33643 | Technical lemma for ~ bnj6... |
| bnj1384 33644 | Technical lemma for ~ bnj6... |
| bnj1388 33645 | Technical lemma for ~ bnj6... |
| bnj1398 33646 | Technical lemma for ~ bnj6... |
| bnj1413 33647 | Property of ` _trCl ` . (... |
| bnj1408 33648 | Technical lemma for ~ bnj1... |
| bnj1414 33649 | Property of ` _trCl ` . (... |
| bnj1415 33650 | Technical lemma for ~ bnj6... |
| bnj1416 33651 | Technical lemma for ~ bnj6... |
| bnj1418 33652 | Property of ` _pred ` . (... |
| bnj1417 33653 | Technical lemma for ~ bnj6... |
| bnj1421 33654 | Technical lemma for ~ bnj6... |
| bnj1444 33655 | Technical lemma for ~ bnj6... |
| bnj1445 33656 | Technical lemma for ~ bnj6... |
| bnj1446 33657 | Technical lemma for ~ bnj6... |
| bnj1447 33658 | Technical lemma for ~ bnj6... |
| bnj1448 33659 | Technical lemma for ~ bnj6... |
| bnj1449 33660 | Technical lemma for ~ bnj6... |
| bnj1442 33661 | Technical lemma for ~ bnj6... |
| bnj1450 33662 | Technical lemma for ~ bnj6... |
| bnj1423 33663 | Technical lemma for ~ bnj6... |
| bnj1452 33664 | Technical lemma for ~ bnj6... |
| bnj1466 33665 | Technical lemma for ~ bnj6... |
| bnj1467 33666 | Technical lemma for ~ bnj6... |
| bnj1463 33667 | Technical lemma for ~ bnj6... |
| bnj1489 33668 | Technical lemma for ~ bnj6... |
| bnj1491 33669 | Technical lemma for ~ bnj6... |
| bnj1312 33670 | Technical lemma for ~ bnj6... |
| bnj1493 33671 | Technical lemma for ~ bnj6... |
| bnj1497 33672 | Technical lemma for ~ bnj6... |
| bnj1498 33673 | Technical lemma for ~ bnj6... |
| bnj60 33674 | Well-founded recursion, pa... |
| bnj1514 33675 | Technical lemma for ~ bnj1... |
| bnj1518 33676 | Technical lemma for ~ bnj1... |
| bnj1519 33677 | Technical lemma for ~ bnj1... |
| bnj1520 33678 | Technical lemma for ~ bnj1... |
| bnj1501 33679 | Technical lemma for ~ bnj1... |
| bnj1500 33680 | Well-founded recursion, pa... |
| bnj1525 33681 | Technical lemma for ~ bnj1... |
| bnj1529 33682 | Technical lemma for ~ bnj1... |
| bnj1523 33683 | Technical lemma for ~ bnj1... |
| bnj1522 33684 | Well-founded recursion, pa... |
| exdifsn 33685 | There exists an element in... |
| srcmpltd 33686 | If a statement is true for... |
| prsrcmpltd 33687 | If a statement is true for... |
| dff15 33688 | A one-to-one function in t... |
| f1resveqaeq 33689 | If a function restricted t... |
| f1resrcmplf1dlem 33690 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 33691 | If a function's restrictio... |
| funen1cnv 33692 | If a function is equinumer... |
| fnrelpredd 33693 | A function that preserves ... |
| cardpred 33694 | The cardinality function p... |
| nummin 33695 | Every nonempty class of nu... |
| fineqvrep 33696 | If the Axiom of Infinity i... |
| fineqvpow 33697 | If the Axiom of Infinity i... |
| fineqvac 33698 | If the Axiom of Infinity i... |
| fineqvacALT 33699 | Shorter proof of ~ fineqva... |
| zltp1ne 33700 | Integer ordering relation.... |
| nnltp1ne 33701 | Positive integer ordering ... |
| nn0ltp1ne 33702 | Nonnegative integer orderi... |
| 0nn0m1nnn0 33703 | A number is zero if and on... |
| f1resfz0f1d 33704 | If a function with a seque... |
| fisshasheq 33705 | A finite set is equal to i... |
| hashfundm 33706 | The size of a set function... |
| hashf1dmrn 33707 | The size of the domain of ... |
| hashf1dmcdm 33708 | The size of the domain of ... |
| revpfxsfxrev 33709 | The reverse of a prefix of... |
| swrdrevpfx 33710 | A subword expressed in ter... |
| lfuhgr 33711 | A hypergraph is loop-free ... |
| lfuhgr2 33712 | A hypergraph is loop-free ... |
| lfuhgr3 33713 | A hypergraph is loop-free ... |
| cplgredgex 33714 | Any two (distinct) vertice... |
| cusgredgex 33715 | Any two (distinct) vertice... |
| cusgredgex2 33716 | Any two distinct vertices ... |
| pfxwlk 33717 | A prefix of a walk is a wa... |
| revwlk 33718 | The reverse of a walk is a... |
| revwlkb 33719 | Two words represent a walk... |
| swrdwlk 33720 | Two matching subwords of a... |
| pthhashvtx 33721 | A graph containing a path ... |
| pthisspthorcycl 33722 | A path is either a simple ... |
| spthcycl 33723 | A walk is a trivial path i... |
| usgrgt2cycl 33724 | A non-trivial cycle in a s... |
| usgrcyclgt2v 33725 | A simple graph with a non-... |
| subgrwlk 33726 | If a walk exists in a subg... |
| subgrtrl 33727 | If a trail exists in a sub... |
| subgrpth 33728 | If a path exists in a subg... |
| subgrcycl 33729 | If a cycle exists in a sub... |
| cusgr3cyclex 33730 | Every complete simple grap... |
| loop1cycl 33731 | A hypergraph has a cycle o... |
| 2cycld 33732 | Construction of a 2-cycle ... |
| 2cycl2d 33733 | Construction of a 2-cycle ... |
| umgr2cycllem 33734 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 33735 | A multigraph with two dist... |
| dfacycgr1 33738 | An alternate definition of... |
| isacycgr 33739 | The property of being an a... |
| isacycgr1 33740 | The property of being an a... |
| acycgrcycl 33741 | Any cycle in an acyclic gr... |
| acycgr0v 33742 | A null graph (with no vert... |
| acycgr1v 33743 | A multigraph with one vert... |
| acycgr2v 33744 | A simple graph with two ve... |
| prclisacycgr 33745 | A proper class (representi... |
| acycgrislfgr 33746 | An acyclic hypergraph is a... |
| upgracycumgr 33747 | An acyclic pseudograph is ... |
| umgracycusgr 33748 | An acyclic multigraph is a... |
| upgracycusgr 33749 | An acyclic pseudograph is ... |
| cusgracyclt3v 33750 | A complete simple graph is... |
| pthacycspth 33751 | A path in an acyclic graph... |
| acycgrsubgr 33752 | The subgraph of an acyclic... |
| quartfull 33759 | The quartic equation, writ... |
| deranglem 33760 | Lemma for derangements. (... |
| derangval 33761 | Define the derangement fun... |
| derangf 33762 | The derangement number is ... |
| derang0 33763 | The derangement number of ... |
| derangsn 33764 | The derangement number of ... |
| derangenlem 33765 | One half of ~ derangen . ... |
| derangen 33766 | The derangement number is ... |
| subfacval 33767 | The subfactorial is define... |
| derangen2 33768 | Write the derangement numb... |
| subfacf 33769 | The subfactorial is a func... |
| subfaclefac 33770 | The subfactorial is less t... |
| subfac0 33771 | The subfactorial at zero. ... |
| subfac1 33772 | The subfactorial at one. ... |
| subfacp1lem1 33773 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 33774 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 33775 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 33776 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 33777 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 33778 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 33779 | Lemma for ~ subfacp1 . By... |
| subfacp1 33780 | A two-term recurrence for ... |
| subfacval2 33781 | A closed-form expression f... |
| subfaclim 33782 | The subfactorial converges... |
| subfacval3 33783 | Another closed form expres... |
| derangfmla 33784 | The derangements formula, ... |
| erdszelem1 33785 | Lemma for ~ erdsze . (Con... |
| erdszelem2 33786 | Lemma for ~ erdsze . (Con... |
| erdszelem3 33787 | Lemma for ~ erdsze . (Con... |
| erdszelem4 33788 | Lemma for ~ erdsze . (Con... |
| erdszelem5 33789 | Lemma for ~ erdsze . (Con... |
| erdszelem6 33790 | Lemma for ~ erdsze . (Con... |
| erdszelem7 33791 | Lemma for ~ erdsze . (Con... |
| erdszelem8 33792 | Lemma for ~ erdsze . (Con... |
| erdszelem9 33793 | Lemma for ~ erdsze . (Con... |
| erdszelem10 33794 | Lemma for ~ erdsze . (Con... |
| erdszelem11 33795 | Lemma for ~ erdsze . (Con... |
| erdsze 33796 | The Erdős-Szekeres th... |
| erdsze2lem1 33797 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 33798 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 33799 | Generalize the statement o... |
| kur14lem1 33800 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 33801 | Lemma for ~ kur14 . Write... |
| kur14lem3 33802 | Lemma for ~ kur14 . A clo... |
| kur14lem4 33803 | Lemma for ~ kur14 . Compl... |
| kur14lem5 33804 | Lemma for ~ kur14 . Closu... |
| kur14lem6 33805 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 33806 | Lemma for ~ kur14 : main p... |
| kur14lem8 33807 | Lemma for ~ kur14 . Show ... |
| kur14lem9 33808 | Lemma for ~ kur14 . Since... |
| kur14lem10 33809 | Lemma for ~ kur14 . Disch... |
| kur14 33810 | Kuratowski's closure-compl... |
| ispconn 33817 | The property of being a pa... |
| pconncn 33818 | The property of being a pa... |
| pconntop 33819 | A simply connected space i... |
| issconn 33820 | The property of being a si... |
| sconnpconn 33821 | A simply connected space i... |
| sconntop 33822 | A simply connected space i... |
| sconnpht 33823 | A closed path in a simply ... |
| cnpconn 33824 | An image of a path-connect... |
| pconnconn 33825 | A path-connected space is ... |
| txpconn 33826 | The topological product of... |
| ptpconn 33827 | The topological product of... |
| indispconn 33828 | The indiscrete topology (o... |
| connpconn 33829 | A connected and locally pa... |
| qtoppconn 33830 | A quotient of a path-conne... |
| pconnpi1 33831 | All fundamental groups in ... |
| sconnpht2 33832 | Any two paths in a simply ... |
| sconnpi1 33833 | A path-connected topologic... |
| txsconnlem 33834 | Lemma for ~ txsconn . (Co... |
| txsconn 33835 | The topological product of... |
| cvxpconn 33836 | A convex subset of the com... |
| cvxsconn 33837 | A convex subset of the com... |
| blsconn 33838 | An open ball in the comple... |
| cnllysconn 33839 | The topology of the comple... |
| resconn 33840 | A subset of ` RR ` is simp... |
| ioosconn 33841 | An open interval is simply... |
| iccsconn 33842 | A closed interval is simpl... |
| retopsconn 33843 | The real numbers are simpl... |
| iccllysconn 33844 | A closed interval is local... |
| rellysconn 33845 | The real numbers are local... |
| iisconn 33846 | The unit interval is simpl... |
| iillysconn 33847 | The unit interval is local... |
| iinllyconn 33848 | The unit interval is local... |
| fncvm 33851 | Lemma for covering maps. ... |
| cvmscbv 33852 | Change bound variables in ... |
| iscvm 33853 | The property of being a co... |
| cvmtop1 33854 | Reverse closure for a cove... |
| cvmtop2 33855 | Reverse closure for a cove... |
| cvmcn 33856 | A covering map is a contin... |
| cvmcov 33857 | Property of a covering map... |
| cvmsrcl 33858 | Reverse closure for an eve... |
| cvmsi 33859 | One direction of ~ cvmsval... |
| cvmsval 33860 | Elementhood in the set ` S... |
| cvmsss 33861 | An even covering is a subs... |
| cvmsn0 33862 | An even covering is nonemp... |
| cvmsuni 33863 | An even covering of ` U ` ... |
| cvmsdisj 33864 | An even covering of ` U ` ... |
| cvmshmeo 33865 | Every element of an even c... |
| cvmsf1o 33866 | ` F ` , localized to an el... |
| cvmscld 33867 | The sets of an even coveri... |
| cvmsss2 33868 | An open subset of an evenl... |
| cvmcov2 33869 | The covering map property ... |
| cvmseu 33870 | Every element in ` U. T ` ... |
| cvmsiota 33871 | Identify the unique elemen... |
| cvmopnlem 33872 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 33873 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 33874 | A covering map is an open ... |
| cvmliftmolem1 33875 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 33876 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 33877 | A lift of a continuous fun... |
| cvmliftmo 33878 | A lift of a continuous fun... |
| cvmliftlem1 33879 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 33880 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 33881 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 33882 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 33883 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 33884 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 33885 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 33886 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 33887 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 33888 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 33889 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 33890 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 33891 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 33892 | Lemma for ~ cvmlift . Dis... |
| cvmlift 33893 | One of the important prope... |
| cvmfo 33894 | A covering map is an onto ... |
| cvmliftiota 33895 | Write out a function ` H `... |
| cvmlift2lem1 33896 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 33897 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 33898 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 33899 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 33900 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 33901 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 33902 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 33903 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 33904 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 33905 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 33906 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 33907 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 33908 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 33909 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 33910 | A two-dimensional version ... |
| cvmliftphtlem 33911 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 33912 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 33913 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 33914 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 33915 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 33916 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 33917 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 33918 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 33919 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 33920 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 33921 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 33922 | A general version of ~ cvm... |
| snmlff 33923 | The function ` F ` from ~ ... |
| snmlfval 33924 | The function ` F ` from ~ ... |
| snmlval 33925 | The property " ` A ` is si... |
| snmlflim 33926 | If ` A ` is simply normal,... |
| goel 33941 | A "Godel-set of membership... |
| goelel3xp 33942 | A "Godel-set of membership... |
| goeleq12bg 33943 | Two "Godel-set of membersh... |
| gonafv 33944 | The "Godel-set for the She... |
| goaleq12d 33945 | Equality of the "Godel-set... |
| gonanegoal 33946 | The Godel-set for the Shef... |
| satf 33947 | The satisfaction predicate... |
| satfsucom 33948 | The satisfaction predicate... |
| satfn 33949 | The satisfaction predicate... |
| satom 33950 | The satisfaction predicate... |
| satfvsucom 33951 | The satisfaction predicate... |
| satfv0 33952 | The value of the satisfact... |
| satfvsuclem1 33953 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 33954 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 33955 | The value of the satisfact... |
| satfv1lem 33956 | Lemma for ~ satfv1 . (Con... |
| satfv1 33957 | The value of the satisfact... |
| satfsschain 33958 | The binary relation of a s... |
| satfvsucsuc 33959 | The satisfaction predicate... |
| satfbrsuc 33960 | The binary relation of a s... |
| satfrel 33961 | The value of the satisfact... |
| satfdmlem 33962 | Lemma for ~ satfdm . (Con... |
| satfdm 33963 | The domain of the satisfac... |
| satfrnmapom 33964 | The range of the satisfact... |
| satfv0fun 33965 | The value of the satisfact... |
| satf0 33966 | The satisfaction predicate... |
| satf0sucom 33967 | The satisfaction predicate... |
| satf00 33968 | The value of the satisfact... |
| satf0suclem 33969 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 33970 | The value of the satisfact... |
| satf0op 33971 | An element of a value of t... |
| satf0n0 33972 | The value of the satisfact... |
| sat1el2xp 33973 | The first component of an ... |
| fmlafv 33974 | The valid Godel formulas o... |
| fmla 33975 | The set of all valid Godel... |
| fmla0 33976 | The valid Godel formulas o... |
| fmla0xp 33977 | The valid Godel formulas o... |
| fmlasuc0 33978 | The valid Godel formulas o... |
| fmlafvel 33979 | A class is a valid Godel f... |
| fmlasuc 33980 | The valid Godel formulas o... |
| fmla1 33981 | The valid Godel formulas o... |
| isfmlasuc 33982 | The characterization of a ... |
| fmlasssuc 33983 | The Godel formulas of heig... |
| fmlaomn0 33984 | The empty set is not a God... |
| fmlan0 33985 | The empty set is not a God... |
| gonan0 33986 | The "Godel-set of NAND" is... |
| goaln0 33987 | The "Godel-set of universa... |
| gonarlem 33988 | Lemma for ~ gonar (inducti... |
| gonar 33989 | If the "Godel-set of NAND"... |
| goalrlem 33990 | Lemma for ~ goalr (inducti... |
| goalr 33991 | If the "Godel-set of unive... |
| fmla0disjsuc 33992 | The set of valid Godel for... |
| fmlasucdisj 33993 | The valid Godel formulas o... |
| satfdmfmla 33994 | The domain of the satisfac... |
| satffunlem 33995 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 33996 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 33997 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 33998 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 33999 | The domain of an ordered p... |
| satffunlem2lem2 34000 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 34001 | Lemma 1 for ~ satffun : in... |
| satffunlem2 34002 | Lemma 2 for ~ satffun : in... |
| satffun 34003 | The value of the satisfact... |
| satff 34004 | The satisfaction predicate... |
| satfun 34005 | The satisfaction predicate... |
| satfvel 34006 | An element of the value of... |
| satfv0fvfmla0 34007 | The value of the satisfact... |
| satefv 34008 | The simplified satisfactio... |
| sate0 34009 | The simplified satisfactio... |
| satef 34010 | The simplified satisfactio... |
| sate0fv0 34011 | A simplified satisfaction ... |
| satefvfmla0 34012 | The simplified satisfactio... |
| sategoelfvb 34013 | Characterization of a valu... |
| sategoelfv 34014 | Condition of a valuation `... |
| ex-sategoelel 34015 | Example of a valuation of ... |
| ex-sategoel 34016 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 34017 | The value of the satisfact... |
| 2goelgoanfmla1 34018 | Two Godel-sets of membersh... |
| satefvfmla1 34019 | The simplified satisfactio... |
| ex-sategoelelomsuc 34020 | Example of a valuation of ... |
| ex-sategoelel12 34021 | Example of a valuation of ... |
| prv 34022 | The "proves" relation on a... |
| elnanelprv 34023 | The wff ` ( A e. B -/\ B e... |
| prv0 34024 | Every wff encoded as ` U `... |
| prv1n 34025 | No wff encoded as a Godel-... |
| mvtval 34094 | The set of variable typeco... |
| mrexval 34095 | The set of "raw expression... |
| mexval 34096 | The set of expressions, wh... |
| mexval2 34097 | The set of expressions, wh... |
| mdvval 34098 | The set of disjoint variab... |
| mvrsval 34099 | The set of variables in an... |
| mvrsfpw 34100 | The set of variables in an... |
| mrsubffval 34101 | The substitution of some v... |
| mrsubfval 34102 | The substitution of some v... |
| mrsubval 34103 | The substitution of some v... |
| mrsubcv 34104 | The value of a substituted... |
| mrsubvr 34105 | The value of a substituted... |
| mrsubff 34106 | A substitution is a functi... |
| mrsubrn 34107 | Although it is defined for... |
| mrsubff1 34108 | When restricted to complet... |
| mrsubff1o 34109 | When restricted to complet... |
| mrsub0 34110 | The value of the substitut... |
| mrsubf 34111 | A substitution is a functi... |
| mrsubccat 34112 | Substitution distributes o... |
| mrsubcn 34113 | A substitution does not ch... |
| elmrsubrn 34114 | Characterization of the su... |
| mrsubco 34115 | The composition of two sub... |
| mrsubvrs 34116 | The set of variables in a ... |
| msubffval 34117 | A substitution applied to ... |
| msubfval 34118 | A substitution applied to ... |
| msubval 34119 | A substitution applied to ... |
| msubrsub 34120 | A substitution applied to ... |
| msubty 34121 | The type of a substituted ... |
| elmsubrn 34122 | Characterization of substi... |
| msubrn 34123 | Although it is defined for... |
| msubff 34124 | A substitution is a functi... |
| msubco 34125 | The composition of two sub... |
| msubf 34126 | A substitution is a functi... |
| mvhfval 34127 | Value of the function mapp... |
| mvhval 34128 | Value of the function mapp... |
| mpstval 34129 | A pre-statement is an orde... |
| elmpst 34130 | Property of being a pre-st... |
| msrfval 34131 | Value of the reduct of a p... |
| msrval 34132 | Value of the reduct of a p... |
| mpstssv 34133 | A pre-statement is an orde... |
| mpst123 34134 | Decompose a pre-statement ... |
| mpstrcl 34135 | The elements of a pre-stat... |
| msrf 34136 | The reduct of a pre-statem... |
| msrrcl 34137 | If ` X ` and ` Y ` have th... |
| mstaval 34138 | Value of the set of statem... |
| msrid 34139 | The reduct of a statement ... |
| msrfo 34140 | The reduct of a pre-statem... |
| mstapst 34141 | A statement is a pre-state... |
| elmsta 34142 | Property of being a statem... |
| ismfs 34143 | A formal system is a tuple... |
| mfsdisj 34144 | The constants and variable... |
| mtyf2 34145 | The type function maps var... |
| mtyf 34146 | The type function maps var... |
| mvtss 34147 | The set of variable typeco... |
| maxsta 34148 | An axiom is a statement. ... |
| mvtinf 34149 | Each variable typecode has... |
| msubff1 34150 | When restricted to complet... |
| msubff1o 34151 | When restricted to complet... |
| mvhf 34152 | The function mapping varia... |
| mvhf1 34153 | The function mapping varia... |
| msubvrs 34154 | The set of variables in a ... |
| mclsrcl 34155 | Reverse closure for the cl... |
| mclsssvlem 34156 | Lemma for ~ mclsssv . (Co... |
| mclsval 34157 | The function mapping varia... |
| mclsssv 34158 | The closure of a set of ex... |
| ssmclslem 34159 | Lemma for ~ ssmcls . (Con... |
| vhmcls 34160 | All variable hypotheses ar... |
| ssmcls 34161 | The original expressions a... |
| ss2mcls 34162 | The closure is monotonic u... |
| mclsax 34163 | The closure is closed unde... |
| mclsind 34164 | Induction theorem for clos... |
| mppspstlem 34165 | Lemma for ~ mppspst . (Co... |
| mppsval 34166 | Definition of a provable p... |
| elmpps 34167 | Definition of a provable p... |
| mppspst 34168 | A provable pre-statement i... |
| mthmval 34169 | A theorem is a pre-stateme... |
| elmthm 34170 | A theorem is a pre-stateme... |
| mthmi 34171 | A statement whose reduct i... |
| mthmsta 34172 | A theorem is a pre-stateme... |
| mppsthm 34173 | A provable pre-statement i... |
| mthmblem 34174 | Lemma for ~ mthmb . (Cont... |
| mthmb 34175 | If two statements have the... |
| mthmpps 34176 | Given a theorem, there is ... |
| mclsppslem 34177 | The closure is closed unde... |
| mclspps 34178 | The closure is closed unde... |
| problem1 34253 | Practice problem 1. Clues... |
| problem2 34254 | Practice problem 2. Clues... |
| problem3 34255 | Practice problem 3. Clues... |
| problem4 34256 | Practice problem 4. Clues... |
| problem5 34257 | Practice problem 5. Clues... |
| quad3 34258 | Variant of quadratic equat... |
| climuzcnv 34259 | Utility lemma to convert b... |
| sinccvglem 34260 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 34261 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 34262 | The circumference of a cir... |
| elfzm12 34263 | Membership in a curtailed ... |
| nn0seqcvg 34264 | A strictly-decreasing nonn... |
| lediv2aALT 34265 | Division of both sides of ... |
| abs2sqlei 34266 | The absolute values of two... |
| abs2sqlti 34267 | The absolute values of two... |
| abs2sqle 34268 | The absolute values of two... |
| abs2sqlt 34269 | The absolute values of two... |
| abs2difi 34270 | Difference of absolute val... |
| abs2difabsi 34271 | Absolute value of differen... |
| axextprim 34272 | ~ ax-ext without distinct ... |
| axrepprim 34273 | ~ ax-rep without distinct ... |
| axunprim 34274 | ~ ax-un without distinct v... |
| axpowprim 34275 | ~ ax-pow without distinct ... |
| axregprim 34276 | ~ ax-reg without distinct ... |
| axinfprim 34277 | ~ ax-inf without distinct ... |
| axacprim 34278 | ~ ax-ac without distinct v... |
| untelirr 34279 | We call a class "untanged"... |
| untuni 34280 | The union of a class is un... |
| untsucf 34281 | If a class is untangled, t... |
| unt0 34282 | The null set is untangled.... |
| untint 34283 | If there is an untangled e... |
| efrunt 34284 | If ` A ` is well-founded b... |
| untangtr 34285 | A transitive class is unta... |
| 3jaodd 34286 | Double deduction form of ~... |
| 3orit 34287 | Closed form of ~ 3ori . (... |
| biimpexp 34288 | A biconditional in the ant... |
| nepss 34289 | Two classes are unequal if... |
| 3ccased 34290 | Triple disjunction form of... |
| dfso3 34291 | Expansion of the definitio... |
| brtpid1 34292 | A binary relation involvin... |
| brtpid2 34293 | A binary relation involvin... |
| brtpid3 34294 | A binary relation involvin... |
| iota5f 34295 | A method for computing iot... |
| jath 34296 | Closed form of ~ ja . Pro... |
| xpab 34297 | Cartesian product of two c... |
| nnuni 34298 | The union of a finite ordi... |
| rspc4v 34299 | 4-variable restricted spec... |
| sqdivzi 34300 | Distribution of square ove... |
| supfz 34301 | The supremum of a finite s... |
| inffz 34302 | The infimum of a finite se... |
| fz0n 34303 | The sequence ` ( 0 ... ( N... |
| shftvalg 34304 | Value of a sequence shifte... |
| divcnvlin 34305 | Limit of the ratio of two ... |
| climlec3 34306 | Comparison of a constant t... |
| logi 34307 | Calculate the logarithm of... |
| iexpire 34308 | ` _i ` raised to itself is... |
| bcneg1 34309 | The binomial coefficent ov... |
| bcm1nt 34310 | The proportion of one bion... |
| bcprod 34311 | A product identity for bin... |
| bccolsum 34312 | A column-sum rule for bino... |
| iprodefisumlem 34313 | Lemma for ~ iprodefisum . ... |
| iprodefisum 34314 | Applying the exponential f... |
| iprodgam 34315 | An infinite product versio... |
| faclimlem1 34316 | Lemma for ~ faclim . Clos... |
| faclimlem2 34317 | Lemma for ~ faclim . Show... |
| faclimlem3 34318 | Lemma for ~ faclim . Alge... |
| faclim 34319 | An infinite product expres... |
| iprodfac 34320 | An infinite product expres... |
| faclim2 34321 | Another factorial limit du... |
| gcd32 34322 | Swap the second and third ... |
| gcdabsorb 34323 | Absorption law for gcd. (... |
| dftr6 34324 | A potential definition of ... |
| coep 34325 | Composition with the membe... |
| coepr 34326 | Composition with the conve... |
| dffr5 34327 | A quantifier-free definiti... |
| dfso2 34328 | Quantifier-free definition... |
| br8 34329 | Substitution for an eight-... |
| br6 34330 | Substitution for a six-pla... |
| br4 34331 | Substitution for a four-pl... |
| cnvco1 34332 | Another distributive law o... |
| cnvco2 34333 | Another distributive law o... |
| eldm3 34334 | Quantifier-free definition... |
| elrn3 34335 | Quantifier-free definition... |
| pocnv 34336 | The converse of a partial ... |
| socnv 34337 | The converse of a strict o... |
| sotrd 34338 | Transitivity law for stric... |
| elintfv 34339 | Membership in an intersect... |
| funpsstri 34340 | A condition for subset tri... |
| fundmpss 34341 | If a class ` F ` is a prop... |
| funsseq 34342 | Given two functions with e... |
| fununiq 34343 | The uniqueness condition o... |
| funbreq 34344 | An equality condition for ... |
| br1steq 34345 | Uniqueness condition for t... |
| br2ndeq 34346 | Uniqueness condition for t... |
| dfdm5 34347 | Definition of domain in te... |
| dfrn5 34348 | Definition of range in ter... |
| opelco3 34349 | Alternate way of saying th... |
| elima4 34350 | Quantifier-free expression... |
| fv1stcnv 34351 | The value of the converse ... |
| fv2ndcnv 34352 | The value of the converse ... |
| setinds 34353 | Principle of set induction... |
| setinds2f 34354 | ` _E ` induction schema, u... |
| setinds2 34355 | ` _E ` induction schema, u... |
| elpotr 34356 | A class of transitive sets... |
| dford5reg 34357 | Given ~ ax-reg , an ordina... |
| dfon2lem1 34358 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 34359 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 34360 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 34361 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 34362 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 34363 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 34364 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 34365 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 34366 | Lemma for ~ dfon2 . A cla... |
| dfon2 34367 | ` On ` consists of all set... |
| rdgprc0 34368 | The value of the recursive... |
| rdgprc 34369 | The value of the recursive... |
| dfrdg2 34370 | Alternate definition of th... |
| dfrdg3 34371 | Generalization of ~ dfrdg2... |
| axextdfeq 34372 | A version of ~ ax-ext for ... |
| ax8dfeq 34373 | A version of ~ ax-8 for us... |
| axextdist 34374 | ~ ax-ext with distinctors ... |
| axextbdist 34375 | ~ axextb with distinctors ... |
| 19.12b 34376 | Version of ~ 19.12vv with ... |
| exnel 34377 | There is always a set not ... |
| distel 34378 | Distinctors in terms of me... |
| axextndbi 34379 | ~ axextnd as a bicondition... |
| hbntg 34380 | A more general form of ~ h... |
| hbimtg 34381 | A more general and closed ... |
| hbaltg 34382 | A more general and closed ... |
| hbng 34383 | A more general form of ~ h... |
| hbimg 34384 | A more general form of ~ h... |
| wsuceq123 34389 | Equality theorem for well-... |
| wsuceq1 34390 | Equality theorem for well-... |
| wsuceq2 34391 | Equality theorem for well-... |
| wsuceq3 34392 | Equality theorem for well-... |
| nfwsuc 34393 | Bound-variable hypothesis ... |
| wlimeq12 34394 | Equality theorem for the l... |
| wlimeq1 34395 | Equality theorem for the l... |
| wlimeq2 34396 | Equality theorem for the l... |
| nfwlim 34397 | Bound-variable hypothesis ... |
| elwlim 34398 | Membership in the limit cl... |
| wzel 34399 | The zero of a well-founded... |
| wsuclem 34400 | Lemma for the supremum pro... |
| wsucex 34401 | Existence theorem for well... |
| wsuccl 34402 | If ` X ` is a set with an ... |
| wsuclb 34403 | A well-founded successor i... |
| wlimss 34404 | The class of limit points ... |
| mulsfn 34407 | Surreal multiplication is ... |
| mulsval 34408 | The value of surreal multi... |
| muls01 34409 | Surreal multiplication by ... |
| muls02 34410 | Surreal multiplication by ... |
| mulsid1 34411 | Surreal one is an identity... |
| mulsid2 34412 | Surreal one is an identity... |
| mulsproplem1 34413 | Lemma for surreal multipli... |
| mulsproplem2 34414 | Lemma for surreal multipli... |
| txpss3v 34463 | A tail Cartesian product i... |
| txprel 34464 | A tail Cartesian product i... |
| brtxp 34465 | Characterize a ternary rel... |
| brtxp2 34466 | The binary relation over a... |
| dfpprod2 34467 | Expanded definition of par... |
| pprodcnveq 34468 | A converse law for paralle... |
| pprodss4v 34469 | The parallel product is a ... |
| brpprod 34470 | Characterize a quaternary ... |
| brpprod3a 34471 | Condition for parallel pro... |
| brpprod3b 34472 | Condition for parallel pro... |
| relsset 34473 | The subset class is a bina... |
| brsset 34474 | For sets, the ` SSet ` bin... |
| idsset 34475 | ` _I ` is equal to the int... |
| eltrans 34476 | Membership in the class of... |
| dfon3 34477 | A quantifier-free definiti... |
| dfon4 34478 | Another quantifier-free de... |
| brtxpsd 34479 | Expansion of a common form... |
| brtxpsd2 34480 | Another common abbreviatio... |
| brtxpsd3 34481 | A third common abbreviatio... |
| relbigcup 34482 | The ` Bigcup ` relationshi... |
| brbigcup 34483 | Binary relation over ` Big... |
| dfbigcup2 34484 | ` Bigcup ` using maps-to n... |
| fobigcup 34485 | ` Bigcup ` maps the univer... |
| fnbigcup 34486 | ` Bigcup ` is a function o... |
| fvbigcup 34487 | For sets, ` Bigcup ` yield... |
| elfix 34488 | Membership in the fixpoint... |
| elfix2 34489 | Alternative membership in ... |
| dffix2 34490 | The fixpoints of a class i... |
| fixssdm 34491 | The fixpoints of a class a... |
| fixssrn 34492 | The fixpoints of a class a... |
| fixcnv 34493 | The fixpoints of a class a... |
| fixun 34494 | The fixpoint operator dist... |
| ellimits 34495 | Membership in the class of... |
| limitssson 34496 | The class of all limit ord... |
| dfom5b 34497 | A quantifier-free definiti... |
| sscoid 34498 | A condition for subset and... |
| dffun10 34499 | Another potential definiti... |
| elfuns 34500 | Membership in the class of... |
| elfunsg 34501 | Closed form of ~ elfuns . ... |
| brsingle 34502 | The binary relation form o... |
| elsingles 34503 | Membership in the class of... |
| fnsingle 34504 | The singleton relationship... |
| fvsingle 34505 | The value of the singleton... |
| dfsingles2 34506 | Alternate definition of th... |
| snelsingles 34507 | A singleton is a member of... |
| dfiota3 34508 | A definition of iota using... |
| dffv5 34509 | Another quantifier-free de... |
| unisnif 34510 | Express union of singleton... |
| brimage 34511 | Binary relation form of th... |
| brimageg 34512 | Closed form of ~ brimage .... |
| funimage 34513 | ` Image A ` is a function.... |
| fnimage 34514 | ` Image R ` is a function ... |
| imageval 34515 | The image functor in maps-... |
| fvimage 34516 | Value of the image functor... |
| brcart 34517 | Binary relation form of th... |
| brdomain 34518 | Binary relation form of th... |
| brrange 34519 | Binary relation form of th... |
| brdomaing 34520 | Closed form of ~ brdomain ... |
| brrangeg 34521 | Closed form of ~ brrange .... |
| brimg 34522 | Binary relation form of th... |
| brapply 34523 | Binary relation form of th... |
| brcup 34524 | Binary relation form of th... |
| brcap 34525 | Binary relation form of th... |
| brsuccf 34526 | Binary relation form of th... |
| funpartlem 34527 | Lemma for ~ funpartfun . ... |
| funpartfun 34528 | The functional part of ` F... |
| funpartss 34529 | The functional part of ` F... |
| funpartfv 34530 | The function value of the ... |
| fullfunfnv 34531 | The full functional part o... |
| fullfunfv 34532 | The function value of the ... |
| brfullfun 34533 | A binary relation form con... |
| brrestrict 34534 | Binary relation form of th... |
| dfrecs2 34535 | A quantifier-free definiti... |
| dfrdg4 34536 | A quantifier-free definiti... |
| dfint3 34537 | Quantifier-free definition... |
| imagesset 34538 | The Image functor applied ... |
| brub 34539 | Binary relation form of th... |
| brlb 34540 | Binary relation form of th... |
| altopex 34545 | Alternative ordered pairs ... |
| altopthsn 34546 | Two alternate ordered pair... |
| altopeq12 34547 | Equality for alternate ord... |
| altopeq1 34548 | Equality for alternate ord... |
| altopeq2 34549 | Equality for alternate ord... |
| altopth1 34550 | Equality of the first memb... |
| altopth2 34551 | Equality of the second mem... |
| altopthg 34552 | Alternate ordered pair the... |
| altopthbg 34553 | Alternate ordered pair the... |
| altopth 34554 | The alternate ordered pair... |
| altopthb 34555 | Alternate ordered pair the... |
| altopthc 34556 | Alternate ordered pair the... |
| altopthd 34557 | Alternate ordered pair the... |
| altxpeq1 34558 | Equality for alternate Car... |
| altxpeq2 34559 | Equality for alternate Car... |
| elaltxp 34560 | Membership in alternate Ca... |
| altopelaltxp 34561 | Alternate ordered pair mem... |
| altxpsspw 34562 | An inclusion rule for alte... |
| altxpexg 34563 | The alternate Cartesian pr... |
| rankaltopb 34564 | Compute the rank of an alt... |
| nfaltop 34565 | Bound-variable hypothesis ... |
| sbcaltop 34566 | Distribution of class subs... |
| cgrrflx2d 34569 | Deduction form of ~ axcgrr... |
| cgrtr4d 34570 | Deduction form of ~ axcgrt... |
| cgrtr4and 34571 | Deduction form of ~ axcgrt... |
| cgrrflx 34572 | Reflexivity law for congru... |
| cgrrflxd 34573 | Deduction form of ~ cgrrfl... |
| cgrcomim 34574 | Congruence commutes on the... |
| cgrcom 34575 | Congruence commutes betwee... |
| cgrcomand 34576 | Deduction form of ~ cgrcom... |
| cgrtr 34577 | Transitivity law for congr... |
| cgrtrand 34578 | Deduction form of ~ cgrtr ... |
| cgrtr3 34579 | Transitivity law for congr... |
| cgrtr3and 34580 | Deduction form of ~ cgrtr3... |
| cgrcoml 34581 | Congruence commutes on the... |
| cgrcomr 34582 | Congruence commutes on the... |
| cgrcomlr 34583 | Congruence commutes on bot... |
| cgrcomland 34584 | Deduction form of ~ cgrcom... |
| cgrcomrand 34585 | Deduction form of ~ cgrcom... |
| cgrcomlrand 34586 | Deduction form of ~ cgrcom... |
| cgrtriv 34587 | Degenerate segments are co... |
| cgrid2 34588 | Identity law for congruenc... |
| cgrdegen 34589 | Two congruent segments are... |
| brofs 34590 | Binary relation form of th... |
| 5segofs 34591 | Rephrase ~ ax5seg using th... |
| ofscom 34592 | The outer five segment pre... |
| cgrextend 34593 | Link congruence over a pai... |
| cgrextendand 34594 | Deduction form of ~ cgrext... |
| segconeq 34595 | Two points that satisfy th... |
| segconeu 34596 | Existential uniqueness ver... |
| btwntriv2 34597 | Betweenness always holds f... |
| btwncomim 34598 | Betweenness commutes. Imp... |
| btwncom 34599 | Betweenness commutes. (Co... |
| btwncomand 34600 | Deduction form of ~ btwnco... |
| btwntriv1 34601 | Betweenness always holds f... |
| btwnswapid 34602 | If you can swap the first ... |
| btwnswapid2 34603 | If you can swap arguments ... |
| btwnintr 34604 | Inner transitivity law for... |
| btwnexch3 34605 | Exchange the first endpoin... |
| btwnexch3and 34606 | Deduction form of ~ btwnex... |
| btwnouttr2 34607 | Outer transitivity law for... |
| btwnexch2 34608 | Exchange the outer point o... |
| btwnouttr 34609 | Outer transitivity law for... |
| btwnexch 34610 | Outer transitivity law for... |
| btwnexchand 34611 | Deduction form of ~ btwnex... |
| btwndiff 34612 | There is always a ` c ` di... |
| trisegint 34613 | A line segment between two... |
| funtransport 34616 | The ` TransportTo ` relati... |
| fvtransport 34617 | Calculate the value of the... |
| transportcl 34618 | Closure law for segment tr... |
| transportprops 34619 | Calculate the defining pro... |
| brifs 34628 | Binary relation form of th... |
| ifscgr 34629 | Inner five segment congrue... |
| cgrsub 34630 | Removing identical parts f... |
| brcgr3 34631 | Binary relation form of th... |
| cgr3permute3 34632 | Permutation law for three-... |
| cgr3permute1 34633 | Permutation law for three-... |
| cgr3permute2 34634 | Permutation law for three-... |
| cgr3permute4 34635 | Permutation law for three-... |
| cgr3permute5 34636 | Permutation law for three-... |
| cgr3tr4 34637 | Transitivity law for three... |
| cgr3com 34638 | Commutativity law for thre... |
| cgr3rflx 34639 | Identity law for three-pla... |
| cgrxfr 34640 | A line segment can be divi... |
| btwnxfr 34641 | A condition for extending ... |
| colinrel 34642 | Colinearity is a relations... |
| brcolinear2 34643 | Alternate colinearity bina... |
| brcolinear 34644 | The binary relation form o... |
| colinearex 34645 | The colinear predicate exi... |
| colineardim1 34646 | If ` A ` is colinear with ... |
| colinearperm1 34647 | Permutation law for coline... |
| colinearperm3 34648 | Permutation law for coline... |
| colinearperm2 34649 | Permutation law for coline... |
| colinearperm4 34650 | Permutation law for coline... |
| colinearperm5 34651 | Permutation law for coline... |
| colineartriv1 34652 | Trivial case of colinearit... |
| colineartriv2 34653 | Trivial case of colinearit... |
| btwncolinear1 34654 | Betweenness implies coline... |
| btwncolinear2 34655 | Betweenness implies coline... |
| btwncolinear3 34656 | Betweenness implies coline... |
| btwncolinear4 34657 | Betweenness implies coline... |
| btwncolinear5 34658 | Betweenness implies coline... |
| btwncolinear6 34659 | Betweenness implies coline... |
| colinearxfr 34660 | Transfer law for colineari... |
| lineext 34661 | Extend a line with a missi... |
| brofs2 34662 | Change some conditions for... |
| brifs2 34663 | Change some conditions for... |
| brfs 34664 | Binary relation form of th... |
| fscgr 34665 | Congruence law for the gen... |
| linecgr 34666 | Congruence rule for lines.... |
| linecgrand 34667 | Deduction form of ~ linecg... |
| lineid 34668 | Identity law for points on... |
| idinside 34669 | Law for finding a point in... |
| endofsegid 34670 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 34671 | Deduction form of ~ endofs... |
| btwnconn1lem1 34672 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 34673 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 34674 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 34675 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 34676 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 34677 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 34678 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 34679 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 34680 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 34681 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 34682 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 34683 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 34684 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 34685 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 34686 | Connectitivy law for betwe... |
| btwnconn2 34687 | Another connectivity law f... |
| btwnconn3 34688 | Inner connectivity law for... |
| midofsegid 34689 | If two points fall in the ... |
| segcon2 34690 | Generalization of ~ axsegc... |
| brsegle 34693 | Binary relation form of th... |
| brsegle2 34694 | Alternate characterization... |
| seglecgr12im 34695 | Substitution law for segme... |
| seglecgr12 34696 | Substitution law for segme... |
| seglerflx 34697 | Segment comparison is refl... |
| seglemin 34698 | Any segment is at least as... |
| segletr 34699 | Segment less than is trans... |
| segleantisym 34700 | Antisymmetry law for segme... |
| seglelin 34701 | Linearity law for segment ... |
| btwnsegle 34702 | If ` B ` falls between ` A... |
| colinbtwnle 34703 | Given three colinear point... |
| broutsideof 34706 | Binary relation form of ` ... |
| broutsideof2 34707 | Alternate form of ` Outsid... |
| outsidene1 34708 | Outsideness implies inequa... |
| outsidene2 34709 | Outsideness implies inequa... |
| btwnoutside 34710 | A principle linking outsid... |
| broutsideof3 34711 | Characterization of outsid... |
| outsideofrflx 34712 | Reflexivity of outsideness... |
| outsideofcom 34713 | Commutativity law for outs... |
| outsideoftr 34714 | Transitivity law for outsi... |
| outsideofeq 34715 | Uniqueness law for ` Outsi... |
| outsideofeu 34716 | Given a nondegenerate ray,... |
| outsidele 34717 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 34718 | Outside of implies colinea... |
| funray 34725 | Show that the ` Ray ` rela... |
| fvray 34726 | Calculate the value of the... |
| funline 34727 | Show that the ` Line ` rel... |
| linedegen 34728 | When ` Line ` is applied w... |
| fvline 34729 | Calculate the value of the... |
| liness 34730 | A line is a subset of the ... |
| fvline2 34731 | Alternate definition of a ... |
| lineunray 34732 | A line is composed of a po... |
| lineelsb2 34733 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 34734 | Reflexivity law for line m... |
| linecom 34735 | Commutativity law for line... |
| linerflx2 34736 | Reflexivity law for line m... |
| ellines 34737 | Membership in the set of a... |
| linethru 34738 | If ` A ` is a line contain... |
| hilbert1.1 34739 | There is a line through an... |
| hilbert1.2 34740 | There is at most one line ... |
| linethrueu 34741 | There is a unique line goi... |
| lineintmo 34742 | Two distinct lines interse... |
| fwddifval 34747 | Calculate the value of the... |
| fwddifnval 34748 | The value of the forward d... |
| fwddifn0 34749 | The value of the n-iterate... |
| fwddifnp1 34750 | The value of the n-iterate... |
| rankung 34751 | The rank of the union of t... |
| ranksng 34752 | The rank of a singleton. ... |
| rankelg 34753 | The membership relation is... |
| rankpwg 34754 | The rank of a power set. ... |
| rank0 34755 | The rank of the empty set ... |
| rankeq1o 34756 | The only set with rank ` 1... |
| elhf 34759 | Membership in the heredita... |
| elhf2 34760 | Alternate form of membersh... |
| elhf2g 34761 | Hereditarily finiteness vi... |
| 0hf 34762 | The empty set is a heredit... |
| hfun 34763 | The union of two HF sets i... |
| hfsn 34764 | The singleton of an HF set... |
| hfadj 34765 | Adjoining one HF element t... |
| hfelhf 34766 | Any member of an HF set is... |
| hftr 34767 | The class of all hereditar... |
| hfext 34768 | Extensionality for HF sets... |
| hfuni 34769 | The union of an HF set is ... |
| hfpw 34770 | The power class of an HF s... |
| hfninf 34771 | ` _om ` is not hereditaril... |
| a1i14 34772 | Add two antecedents to a w... |
| a1i24 34773 | Add two antecedents to a w... |
| exp5d 34774 | An exportation inference. ... |
| exp5g 34775 | An exportation inference. ... |
| exp5k 34776 | An exportation inference. ... |
| exp56 34777 | An exportation inference. ... |
| exp58 34778 | An exportation inference. ... |
| exp510 34779 | An exportation inference. ... |
| exp511 34780 | An exportation inference. ... |
| exp512 34781 | An exportation inference. ... |
| 3com12d 34782 | Commutation in consequent.... |
| imp5p 34783 | A triple importation infer... |
| imp5q 34784 | A triple importation infer... |
| ecase13d 34785 | Deduction for elimination ... |
| subtr 34786 | Transitivity of implicit s... |
| subtr2 34787 | Transitivity of implicit s... |
| trer 34788 | A relation intersected wit... |
| elicc3 34789 | An equivalent membership c... |
| finminlem 34790 | A useful lemma about finit... |
| gtinf 34791 | Any number greater than an... |
| opnrebl 34792 | A set is open in the stand... |
| opnrebl2 34793 | A set is open in the stand... |
| nn0prpwlem 34794 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 34795 | Two nonnegative integers a... |
| topbnd 34796 | Two equivalent expressions... |
| opnbnd 34797 | A set is open iff it is di... |
| cldbnd 34798 | A set is closed iff it con... |
| ntruni 34799 | A union of interiors is a ... |
| clsun 34800 | A pairwise union of closur... |
| clsint2 34801 | The closure of an intersec... |
| opnregcld 34802 | A set is regularly closed ... |
| cldregopn 34803 | A set if regularly open if... |
| neiin 34804 | Two neighborhoods intersec... |
| hmeoclda 34805 | Homeomorphisms preserve cl... |
| hmeocldb 34806 | Homeomorphisms preserve cl... |
| ivthALT 34807 | An alternate proof of the ... |
| fnerel 34810 | Fineness is a relation. (... |
| isfne 34811 | The predicate " ` B ` is f... |
| isfne4 34812 | The predicate " ` B ` is f... |
| isfne4b 34813 | A condition for a topology... |
| isfne2 34814 | The predicate " ` B ` is f... |
| isfne3 34815 | The predicate " ` B ` is f... |
| fnebas 34816 | A finer cover covers the s... |
| fnetg 34817 | A finer cover generates a ... |
| fnessex 34818 | If ` B ` is finer than ` A... |
| fneuni 34819 | If ` B ` is finer than ` A... |
| fneint 34820 | If a cover is finer than a... |
| fness 34821 | A cover is finer than its ... |
| fneref 34822 | Reflexivity of the finenes... |
| fnetr 34823 | Transitivity of the finene... |
| fneval 34824 | Two covers are finer than ... |
| fneer 34825 | Fineness intersected with ... |
| topfne 34826 | Fineness for covers corres... |
| topfneec 34827 | A cover is equivalent to a... |
| topfneec2 34828 | A topology is precisely id... |
| fnessref 34829 | A cover is finer iff it ha... |
| refssfne 34830 | A cover is a refinement if... |
| neibastop1 34831 | A collection of neighborho... |
| neibastop2lem 34832 | Lemma for ~ neibastop2 . ... |
| neibastop2 34833 | In the topology generated ... |
| neibastop3 34834 | The topology generated by ... |
| topmtcl 34835 | The meet of a collection o... |
| topmeet 34836 | Two equivalent formulation... |
| topjoin 34837 | Two equivalent formulation... |
| fnemeet1 34838 | The meet of a collection o... |
| fnemeet2 34839 | The meet of equivalence cl... |
| fnejoin1 34840 | Join of equivalence classe... |
| fnejoin2 34841 | Join of equivalence classe... |
| fgmin 34842 | Minimality property of a g... |
| neifg 34843 | The neighborhood filter of... |
| tailfval 34844 | The tail function for a di... |
| tailval 34845 | The tail of an element in ... |
| eltail 34846 | An element of a tail. (Co... |
| tailf 34847 | The tail function of a dir... |
| tailini 34848 | A tail contains its initia... |
| tailfb 34849 | The collection of tails of... |
| filnetlem1 34850 | Lemma for ~ filnet . Chan... |
| filnetlem2 34851 | Lemma for ~ filnet . The ... |
| filnetlem3 34852 | Lemma for ~ filnet . (Con... |
| filnetlem4 34853 | Lemma for ~ filnet . (Con... |
| filnet 34854 | A filter has the same conv... |
| tb-ax1 34855 | The first of three axioms ... |
| tb-ax2 34856 | The second of three axioms... |
| tb-ax3 34857 | The third of three axioms ... |
| tbsyl 34858 | The weak syllogism from Ta... |
| re1ax2lem 34859 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 34860 | ~ ax-2 rederived from the ... |
| naim1 34861 | Constructor theorem for ` ... |
| naim2 34862 | Constructor theorem for ` ... |
| naim1i 34863 | Constructor rule for ` -/\... |
| naim2i 34864 | Constructor rule for ` -/\... |
| naim12i 34865 | Constructor rule for ` -/\... |
| nabi1i 34866 | Constructor rule for ` -/\... |
| nabi2i 34867 | Constructor rule for ` -/\... |
| nabi12i 34868 | Constructor rule for ` -/\... |
| df3nandALT1 34871 | The double nand expressed ... |
| df3nandALT2 34872 | The double nand expressed ... |
| andnand1 34873 | Double and in terms of dou... |
| imnand2 34874 | An ` -> ` nand relation. ... |
| nalfal 34875 | Not all sets hold ` F. ` a... |
| nexntru 34876 | There does not exist a set... |
| nexfal 34877 | There does not exist a set... |
| neufal 34878 | There does not exist exact... |
| neutru 34879 | There does not exist exact... |
| nmotru 34880 | There does not exist at mo... |
| mofal 34881 | There exist at most one se... |
| nrmo 34882 | "At most one" restricted e... |
| meran1 34883 | A single axiom for proposi... |
| meran2 34884 | A single axiom for proposi... |
| meran3 34885 | A single axiom for proposi... |
| waj-ax 34886 | A single axiom for proposi... |
| lukshef-ax2 34887 | A single axiom for proposi... |
| arg-ax 34888 | A single axiom for proposi... |
| negsym1 34889 | In the paper "On Variable ... |
| imsym1 34890 | A symmetry with ` -> ` . ... |
| bisym1 34891 | A symmetry with ` <-> ` . ... |
| consym1 34892 | A symmetry with ` /\ ` . ... |
| dissym1 34893 | A symmetry with ` \/ ` . ... |
| nandsym1 34894 | A symmetry with ` -/\ ` . ... |
| unisym1 34895 | A symmetry with ` A. ` . ... |
| exisym1 34896 | A symmetry with ` E. ` . ... |
| unqsym1 34897 | A symmetry with ` E! ` . ... |
| amosym1 34898 | A symmetry with ` E* ` . ... |
| subsym1 34899 | A symmetry with ` [ x / y ... |
| ontopbas 34900 | An ordinal number is a top... |
| onsstopbas 34901 | The class of ordinal numbe... |
| onpsstopbas 34902 | The class of ordinal numbe... |
| ontgval 34903 | The topology generated fro... |
| ontgsucval 34904 | The topology generated fro... |
| onsuctop 34905 | A successor ordinal number... |
| onsuctopon 34906 | One of the topologies on a... |
| ordtoplem 34907 | Membership of the class of... |
| ordtop 34908 | An ordinal is a topology i... |
| onsucconni 34909 | A successor ordinal number... |
| onsucconn 34910 | A successor ordinal number... |
| ordtopconn 34911 | An ordinal topology is con... |
| onintopssconn 34912 | An ordinal topology is con... |
| onsuct0 34913 | A successor ordinal number... |
| ordtopt0 34914 | An ordinal topology is T_0... |
| onsucsuccmpi 34915 | The successor of a success... |
| onsucsuccmp 34916 | The successor of a success... |
| limsucncmpi 34917 | The successor of a limit o... |
| limsucncmp 34918 | The successor of a limit o... |
| ordcmp 34919 | An ordinal topology is com... |
| ssoninhaus 34920 | The ordinal topologies ` 1... |
| onint1 34921 | The ordinal T_1 spaces are... |
| oninhaus 34922 | The ordinal Hausdorff spac... |
| fveleq 34923 | Please add description her... |
| findfvcl 34924 | Please add description her... |
| findreccl 34925 | Please add description her... |
| findabrcl 34926 | Please add description her... |
| nnssi2 34927 | Convert a theorem for real... |
| nnssi3 34928 | Convert a theorem for real... |
| nndivsub 34929 | Please add description her... |
| nndivlub 34930 | A factor of a positive int... |
| ee7.2aOLD 34933 | Lemma for Euclid's Element... |
| dnival 34934 | Value of the "distance to ... |
| dnicld1 34935 | Closure theorem for the "d... |
| dnicld2 34936 | Closure theorem for the "d... |
| dnif 34937 | The "distance to nearest i... |
| dnizeq0 34938 | The distance to nearest in... |
| dnizphlfeqhlf 34939 | The distance to nearest in... |
| rddif2 34940 | Variant of ~ rddif . (Con... |
| dnibndlem1 34941 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 34942 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 34943 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 34944 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 34945 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 34946 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 34947 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 34948 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 34949 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 34950 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 34951 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 34952 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 34953 | Lemma for ~ dnibnd . (Con... |
| dnibnd 34954 | The "distance to nearest i... |
| dnicn 34955 | The "distance to nearest i... |
| knoppcnlem1 34956 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 34957 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 34958 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 34959 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 34960 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 34961 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 34962 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 34963 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 34964 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 34965 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 34966 | Lemma for ~ knoppcn . (Co... |
| knoppcn 34967 | The continuous nowhere dif... |
| knoppcld 34968 | Closure theorem for Knopp'... |
| unblimceq0lem 34969 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 34970 | If ` F ` is unbounded near... |
| unbdqndv1 34971 | If the difference quotient... |
| unbdqndv2lem1 34972 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 34973 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 34974 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 34975 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 34976 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 34977 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 34978 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 34979 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 34980 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 34981 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 34982 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 34983 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 34984 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 34985 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 34986 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 34987 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 34988 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 34989 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 34990 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 34991 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 34992 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 34993 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 34994 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 34995 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 34996 | Lemma for ~ knoppndv . (C... |
| knoppndv 34997 | The continuous nowhere dif... |
| knoppf 34998 | Knopp's function is a func... |
| knoppcn2 34999 | Variant of ~ knoppcn with ... |
| cnndvlem1 35000 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 35001 | Lemma for ~ cnndv . (Cont... |
| cnndv 35002 | There exists a continuous ... |
| bj-mp2c 35003 | A double modus ponens infe... |
| bj-mp2d 35004 | A double modus ponens infe... |
| bj-0 35005 | A syntactic theorem. See ... |
| bj-1 35006 | In this proof, the use of ... |
| bj-a1k 35007 | Weakening of ~ ax-1 . As ... |
| bj-poni 35008 | Inference associated with ... |
| bj-nnclav 35009 | When ` F. ` is substituted... |
| bj-nnclavi 35010 | Inference associated with ... |
| bj-nnclavc 35011 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 35012 | Inference associated with ... |
| bj-jarrii 35013 | Inference associated with ... |
| bj-imim21 35014 | The propositional function... |
| bj-imim21i 35015 | Inference associated with ... |
| bj-peircestab 35016 | Over minimal implicational... |
| bj-stabpeirce 35017 | This minimal implicational... |
| bj-syl66ib 35018 | A mixed syllogism inferenc... |
| bj-orim2 35019 | Proof of ~ orim2 from the ... |
| bj-currypeirce 35020 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 35021 | Peirce's axiom ~ peirce im... |
| bj-animbi 35022 | Conjunction in terms of im... |
| bj-currypara 35023 | Curry's paradox. Note tha... |
| bj-con2com 35024 | A commuted form of the con... |
| bj-con2comi 35025 | Inference associated with ... |
| bj-pm2.01i 35026 | Inference associated with ... |
| bj-nimn 35027 | If a formula is true, then... |
| bj-nimni 35028 | Inference associated with ... |
| bj-peircei 35029 | Inference associated with ... |
| bj-looinvi 35030 | Inference associated with ... |
| bj-looinvii 35031 | Inference associated with ... |
| bj-mt2bi 35032 | Version of ~ mt2 where the... |
| bj-ntrufal 35033 | The negation of a theorem ... |
| bj-fal 35034 | Shortening of ~ fal using ... |
| bj-jaoi1 35035 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 35036 | Shortens ~ consensus (110>... |
| bj-dfbi4 35037 | Alternate definition of th... |
| bj-dfbi5 35038 | Alternate definition of th... |
| bj-dfbi6 35039 | Alternate definition of th... |
| bj-bijust0ALT 35040 | Alternate proof of ~ bijus... |
| bj-bijust00 35041 | A self-implication does no... |
| bj-consensus 35042 | Version of ~ consensus exp... |
| bj-consensusALT 35043 | Alternate proof of ~ bj-co... |
| bj-df-ifc 35044 | Candidate definition for t... |
| bj-dfif 35045 | Alternate definition of th... |
| bj-ififc 35046 | A biconditional connecting... |
| bj-imbi12 35047 | Uncurried (imported) form ... |
| bj-biorfi 35048 | This should be labeled "bi... |
| bj-falor 35049 | Dual of ~ truan (which has... |
| bj-falor2 35050 | Dual of ~ truan . (Contri... |
| bj-bibibi 35051 | A property of the bicondit... |
| bj-imn3ani 35052 | Duplication of ~ bnj1224 .... |
| bj-andnotim 35053 | Two ways of expressing a c... |
| bj-bi3ant 35054 | This used to be in the mai... |
| bj-bisym 35055 | This used to be in the mai... |
| bj-bixor 35056 | Equivalence of two ternary... |
| bj-axdd2 35057 | This implication, proved u... |
| bj-axd2d 35058 | This implication, proved u... |
| bj-axtd 35059 | This implication, proved f... |
| bj-gl4 35060 | In a normal modal logic, t... |
| bj-axc4 35061 | Over minimal calculus, the... |
| prvlem1 35066 | An elementary property of ... |
| prvlem2 35067 | An elementary property of ... |
| bj-babygodel 35068 | See the section header com... |
| bj-babylob 35069 | See the section header com... |
| bj-godellob 35070 | Proof of Gödel's theo... |
| bj-genr 35071 | Generalization rule on the... |
| bj-genl 35072 | Generalization rule on the... |
| bj-genan 35073 | Generalization rule on a c... |
| bj-mpgs 35074 | From a closed form theorem... |
| bj-2alim 35075 | Closed form of ~ 2alimi . ... |
| bj-2exim 35076 | Closed form of ~ 2eximi . ... |
| bj-alanim 35077 | Closed form of ~ alanimi .... |
| bj-2albi 35078 | Closed form of ~ 2albii . ... |
| bj-notalbii 35079 | Equivalence of universal q... |
| bj-2exbi 35080 | Closed form of ~ 2exbii . ... |
| bj-3exbi 35081 | Closed form of ~ 3exbii . ... |
| bj-sylgt2 35082 | Uncurried (imported) form ... |
| bj-alrimg 35083 | The general form of the *a... |
| bj-alrimd 35084 | A slightly more general ~ ... |
| bj-sylget 35085 | Dual statement of ~ sylgt ... |
| bj-sylget2 35086 | Uncurried (imported) form ... |
| bj-exlimg 35087 | The general form of the *e... |
| bj-sylge 35088 | Dual statement of ~ sylg (... |
| bj-exlimd 35089 | A slightly more general ~ ... |
| bj-nfimexal 35090 | A weak from of nonfreeness... |
| bj-alexim 35091 | Closed form of ~ aleximi .... |
| bj-nexdh 35092 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 35093 | Uncurried (imported) form ... |
| bj-hbxfrbi 35094 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 35095 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 35096 | Distribute quantifiers ove... |
| bj-exalimi 35097 | An inference for distribut... |
| bj-exalims 35098 | Distributing quantifiers o... |
| bj-exalimsi 35099 | An inference for distribut... |
| bj-ax12ig 35100 | A lemma used to prove a we... |
| bj-ax12i 35101 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 35102 | Closed form of ~ nfim and ... |
| bj-cbvalimt 35103 | A lemma in closed form use... |
| bj-cbveximt 35104 | A lemma in closed form use... |
| bj-eximALT 35105 | Alternate proof of ~ exim ... |
| bj-aleximiALT 35106 | Alternate proof of ~ alexi... |
| bj-eximcom 35107 | A commuted form of ~ exim ... |
| bj-ax12wlem 35108 | A lemma used to prove a we... |
| bj-cbvalim 35109 | A lemma used to prove ~ bj... |
| bj-cbvexim 35110 | A lemma used to prove ~ bj... |
| bj-cbvalimi 35111 | An equality-free general i... |
| bj-cbveximi 35112 | An equality-free general i... |
| bj-cbval 35113 | Changing a bound variable ... |
| bj-cbvex 35114 | Changing a bound variable ... |
| bj-ssbeq 35117 | Substitution in an equalit... |
| bj-ssblem1 35118 | A lemma for the definiens ... |
| bj-ssblem2 35119 | An instance of ~ ax-11 pro... |
| bj-ax12v 35120 | A weaker form of ~ ax-12 a... |
| bj-ax12 35121 | Remove a DV condition from... |
| bj-ax12ssb 35122 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 35123 | Special case of ~ 19.41 pr... |
| bj-equsexval 35124 | Special case of ~ equsexv ... |
| bj-subst 35125 | Proof of ~ sbalex from cor... |
| bj-ssbid2 35126 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 35127 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 35128 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 35129 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 35130 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 35131 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 35132 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 35133 | Closed form of ~ spimvw . ... |
| bj-spnfw 35134 | Theorem close to a closed ... |
| bj-cbvexiw 35135 | Change bound variable. Th... |
| bj-cbvexivw 35136 | Change bound variable. Th... |
| bj-modald 35137 | A short form of the axiom ... |
| bj-denot 35138 | A weakening of ~ ax-6 and ... |
| bj-eqs 35139 | A lemma for substitutions,... |
| bj-cbvexw 35140 | Change bound variable. Th... |
| bj-ax12w 35141 | The general statement that... |
| bj-ax89 35142 | A theorem which could be u... |
| bj-elequ12 35143 | An identity law for the no... |
| bj-cleljusti 35144 | One direction of ~ cleljus... |
| bj-alcomexcom 35145 | Commutation of two existen... |
| bj-hbalt 35146 | Closed form of ~ hbal . W... |
| axc11n11 35147 | Proof of ~ axc11n from { ~... |
| axc11n11r 35148 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 35149 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 35150 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 35151 | Alternate proof of ~ bj-ax... |
| bj-sb 35152 | A weak variant of ~ sbid2 ... |
| bj-modalbe 35153 | The predicate-calculus ver... |
| bj-spst 35154 | Closed form of ~ sps . On... |
| bj-19.21bit 35155 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 35156 | Closed form of ~ 19.23bi .... |
| bj-nexrt 35157 | Closed form of ~ nexr . C... |
| bj-alrim 35158 | Closed form of ~ alrimi . ... |
| bj-alrim2 35159 | Uncurried (imported) form ... |
| bj-nfdt0 35160 | A theorem close to a close... |
| bj-nfdt 35161 | Closed form of ~ nf5d and ... |
| bj-nexdt 35162 | Closed form of ~ nexd . (... |
| bj-nexdvt 35163 | Closed form of ~ nexdv . ... |
| bj-alexbiex 35164 | Adding a second quantifier... |
| bj-exexbiex 35165 | Adding a second quantifier... |
| bj-alalbial 35166 | Adding a second quantifier... |
| bj-exalbial 35167 | Adding a second quantifier... |
| bj-19.9htbi 35168 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 35169 | Strengthening ~ hbnt by re... |
| bj-biexal1 35170 | A general FOL biconditiona... |
| bj-biexal2 35171 | When ` ph ` is substituted... |
| bj-biexal3 35172 | When ` ph ` is substituted... |
| bj-bialal 35173 | When ` ph ` is substituted... |
| bj-biexex 35174 | When ` ph ` is substituted... |
| bj-hbext 35175 | Closed form of ~ hbex . (... |
| bj-nfalt 35176 | Closed form of ~ nfal . (... |
| bj-nfext 35177 | Closed form of ~ nfex . (... |
| bj-eeanvw 35178 | Version of ~ exdistrv with... |
| bj-modal4 35179 | First-order logic form of ... |
| bj-modal4e 35180 | First-order logic form of ... |
| bj-modalb 35181 | A short form of the axiom ... |
| bj-wnf1 35182 | When ` ph ` is substituted... |
| bj-wnf2 35183 | When ` ph ` is substituted... |
| bj-wnfanf 35184 | When ` ph ` is substituted... |
| bj-wnfenf 35185 | When ` ph ` is substituted... |
| bj-substax12 35186 | Equivalent form of the axi... |
| bj-substw 35187 | Weak form of the LHS of ~ ... |
| bj-nnfbi 35190 | If two formulas are equiva... |
| bj-nnfbd 35191 | If two formulas are equiva... |
| bj-nnfbii 35192 | If two formulas are equiva... |
| bj-nnfa 35193 | Nonfreeness implies the eq... |
| bj-nnfad 35194 | Nonfreeness implies the eq... |
| bj-nnfai 35195 | Nonfreeness implies the eq... |
| bj-nnfe 35196 | Nonfreeness implies the eq... |
| bj-nnfed 35197 | Nonfreeness implies the eq... |
| bj-nnfei 35198 | Nonfreeness implies the eq... |
| bj-nnfea 35199 | Nonfreeness implies the eq... |
| bj-nnfead 35200 | Nonfreeness implies the eq... |
| bj-nnfeai 35201 | Nonfreeness implies the eq... |
| bj-dfnnf2 35202 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 35203 | New nonfreeness implies ol... |
| bj-wnfnf 35204 | When ` ph ` is substituted... |
| bj-nnfnt 35205 | A variable is nonfree in a... |
| bj-nnftht 35206 | A variable is nonfree in a... |
| bj-nnfth 35207 | A variable is nonfree in a... |
| bj-nnfnth 35208 | A variable is nonfree in t... |
| bj-nnfim1 35209 | A consequence of nonfreene... |
| bj-nnfim2 35210 | A consequence of nonfreene... |
| bj-nnfim 35211 | Nonfreeness in the anteced... |
| bj-nnfimd 35212 | Nonfreeness in the anteced... |
| bj-nnfan 35213 | Nonfreeness in both conjun... |
| bj-nnfand 35214 | Nonfreeness in both conjun... |
| bj-nnfor 35215 | Nonfreeness in both disjun... |
| bj-nnford 35216 | Nonfreeness in both disjun... |
| bj-nnfbit 35217 | Nonfreeness in both sides ... |
| bj-nnfbid 35218 | Nonfreeness in both sides ... |
| bj-nnfv 35219 | A non-occurring variable i... |
| bj-nnf-alrim 35220 | Proof of the closed form o... |
| bj-nnf-exlim 35221 | Proof of the closed form o... |
| bj-dfnnf3 35222 | Alternate definition of no... |
| bj-nfnnfTEMP 35223 | New nonfreeness is equival... |
| bj-nnfa1 35224 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 35225 | See ~ nfe1 . (Contributed... |
| bj-19.12 35226 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 35227 | One of four lemmas for non... |
| bj-nnflemee 35228 | One of four lemmas for non... |
| bj-nnflemae 35229 | One of four lemmas for non... |
| bj-nnflemea 35230 | One of four lemmas for non... |
| bj-nnfalt 35231 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 35232 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 35233 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 35234 | Statement ~ 19.21t proved ... |
| bj-19.23t 35235 | Statement ~ 19.23t proved ... |
| bj-19.36im 35236 | One direction of ~ 19.36 f... |
| bj-19.37im 35237 | One direction of ~ 19.37 f... |
| bj-19.42t 35238 | Closed form of ~ 19.42 fro... |
| bj-19.41t 35239 | Closed form of ~ 19.41 fro... |
| bj-sbft 35240 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 35241 | Version of ~ pm11.53v with... |
| bj-pm11.53v 35242 | Version of ~ pm11.53v with... |
| bj-pm11.53a 35243 | A variant of ~ pm11.53v . ... |
| bj-equsvt 35244 | A variant of ~ equsv . (C... |
| bj-equsalvwd 35245 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 35246 | Variant of ~ equsexvw . (... |
| bj-sbievwd 35247 | Variant of ~ sbievw . (Co... |
| bj-axc10 35248 | Alternate proof of ~ axc10... |
| bj-alequex 35249 | A fol lemma. See ~ aleque... |
| bj-spimt2 35250 | A step in the proof of ~ s... |
| bj-cbv3ta 35251 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 35252 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 35253 | A theorem close to a close... |
| bj-hbsb3 35254 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 35255 | A theorem close to a close... |
| bj-nfs1t2 35256 | A theorem close to a close... |
| bj-nfs1 35257 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 35258 | Version of ~ axc10 with a ... |
| bj-spimtv 35259 | Version of ~ spimt with a ... |
| bj-cbv3hv2 35260 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 35261 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 35262 | Version of ~ cbv2h with a ... |
| bj-cbv2v 35263 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 35264 | Version of ~ cbvald with a... |
| bj-cbvexdv 35265 | Version of ~ cbvexd with a... |
| bj-cbval2vv 35266 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 35267 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 35268 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 35269 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 35270 | Version of ~ cbvex4v with ... |
| bj-equsalhv 35271 | Version of ~ equsalh with ... |
| bj-axc11nv 35272 | Version of ~ axc11n with a... |
| bj-aecomsv 35273 | Version of ~ aecoms with a... |
| bj-axc11v 35274 | Version of ~ axc11 with a ... |
| bj-drnf2v 35275 | Version of ~ drnf2 with a ... |
| bj-equs45fv 35276 | Version of ~ equs45f with ... |
| bj-hbs1 35277 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 35278 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 35279 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 35280 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 35281 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 35282 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 35283 | Biconditional version of a... |
| bj-hbaeb 35284 | Biconditional version of ~... |
| bj-hbnaeb 35285 | Biconditional version of ~... |
| bj-dvv 35286 | A special instance of ~ bj... |
| bj-equsal1t 35287 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 35288 | Inference associated with ... |
| bj-equsal1 35289 | One direction of ~ equsal ... |
| bj-equsal2 35290 | One direction of ~ equsal ... |
| bj-equsal 35291 | Shorter proof of ~ equsal ... |
| stdpc5t 35292 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 35293 | More direct proof of ~ std... |
| 2stdpc5 35294 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 35295 | Proof of ~ 19.21t from ~ s... |
| exlimii 35296 | Inference associated with ... |
| ax11-pm 35297 | Proof of ~ ax-11 similar t... |
| ax6er 35298 | Commuted form of ~ ax6e . ... |
| exlimiieq1 35299 | Inferring a theorem when i... |
| exlimiieq2 35300 | Inferring a theorem when i... |
| ax11-pm2 35301 | Proof of ~ ax-11 from the ... |
| bj-sbsb 35302 | Biconditional showing two ... |
| bj-dfsb2 35303 | Alternate (dual) definitio... |
| bj-sbf3 35304 | Substitution has no effect... |
| bj-sbf4 35305 | Substitution has no effect... |
| bj-sbnf 35306 | Move nonfree predicate in ... |
| bj-eu3f 35307 | Version of ~ eu3v where th... |
| bj-sblem1 35308 | Lemma for substitution. (... |
| bj-sblem2 35309 | Lemma for substitution. (... |
| bj-sblem 35310 | Lemma for substitution. (... |
| bj-sbievw1 35311 | Lemma for substitution. (... |
| bj-sbievw2 35312 | Lemma for substitution. (... |
| bj-sbievw 35313 | Lemma for substitution. C... |
| bj-sbievv 35314 | Version of ~ sbie with a s... |
| bj-moeub 35315 | Uniqueness is equivalent t... |
| bj-sbidmOLD 35316 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 35317 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 35318 | Curried (exported) form of... |
| bj-dvelimv 35319 | A version of ~ dvelim usin... |
| bj-nfeel2 35320 | Nonfreeness in a membershi... |
| bj-axc14nf 35321 | Proof of a version of ~ ax... |
| bj-axc14 35322 | Alternate proof of ~ axc14... |
| mobidvALT 35323 | Alternate proof of ~ mobid... |
| sbn1ALT 35324 | Alternate proof of ~ sbn1 ... |
| eliminable1 35325 | A theorem used to prove th... |
| eliminable2a 35326 | A theorem used to prove th... |
| eliminable2b 35327 | A theorem used to prove th... |
| eliminable2c 35328 | A theorem used to prove th... |
| eliminable3a 35329 | A theorem used to prove th... |
| eliminable3b 35330 | A theorem used to prove th... |
| eliminable-velab 35331 | A theorem used to prove th... |
| eliminable-veqab 35332 | A theorem used to prove th... |
| eliminable-abeqv 35333 | A theorem used to prove th... |
| eliminable-abeqab 35334 | A theorem used to prove th... |
| eliminable-abelv 35335 | A theorem used to prove th... |
| eliminable-abelab 35336 | A theorem used to prove th... |
| bj-denoteslem 35337 | Lemma for ~ bj-denotes . ... |
| bj-denotes 35338 | This would be the justific... |
| bj-issettru 35339 | Weak version of ~ isset wi... |
| bj-elabtru 35340 | This is as close as we can... |
| bj-issetwt 35341 | Closed form of ~ bj-issetw... |
| bj-issetw 35342 | The closest one can get to... |
| bj-elissetALT 35343 | Alternate proof of ~ eliss... |
| bj-issetiv 35344 | Version of ~ bj-isseti wit... |
| bj-isseti 35345 | Version of ~ isseti with a... |
| bj-ralvw 35346 | A weak version of ~ ralv n... |
| bj-rexvw 35347 | A weak version of ~ rexv n... |
| bj-rababw 35348 | A weak version of ~ rabab ... |
| bj-rexcom4bv 35349 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 35350 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 35351 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 35352 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 35353 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 35354 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 35355 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 35356 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 35357 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 35358 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 35359 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 35360 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 35361 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 35362 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 35363 | Remove from ~ spcimdv depe... |
| elelb 35364 | Equivalence between two co... |
| bj-pwvrelb 35365 | Characterization of the el... |
| bj-nfcsym 35366 | The nonfreeness quantifier... |
| bj-sbeqALT 35367 | Substitution in an equalit... |
| bj-sbeq 35368 | Distribute proper substitu... |
| bj-sbceqgALT 35369 | Distribute proper substitu... |
| bj-csbsnlem 35370 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 35371 | Substitution in a singleto... |
| bj-sbel1 35372 | Version of ~ sbcel1g when ... |
| bj-abv 35373 | The class of sets verifyin... |
| bj-abvALT 35374 | Alternate version of ~ bj-... |
| bj-ab0 35375 | The class of sets verifyin... |
| bj-abf 35376 | Shorter proof of ~ abf (wh... |
| bj-csbprc 35377 | More direct proof of ~ csb... |
| bj-exlimvmpi 35378 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 35379 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 35380 | Lemma for theorems of the ... |
| bj-exlimmpbir 35381 | Lemma for theorems of the ... |
| bj-vtoclf 35382 | Remove dependency on ~ ax-... |
| bj-vtocl 35383 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 35384 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 35385 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 35386 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 35387 | A version of ~ vtoclg with... |
| bj-rabbida2 35388 | Version of ~ rabbidva2 wit... |
| bj-rabeqd 35389 | Deduction form of ~ rabeq ... |
| bj-rabeqbid 35390 | Version of ~ rabeqbidv wit... |
| bj-rabeqbida 35391 | Version of ~ rabeqbidva wi... |
| bj-seex 35392 | Version of ~ seex with a d... |
| bj-nfcf 35393 | Version of ~ df-nfc with a... |
| bj-zfauscl 35394 | General version of ~ zfaus... |
| bj-elabd2ALT 35395 | Alternate proof of ~ elabd... |
| bj-unrab 35396 | Generalization of ~ unrab ... |
| bj-inrab 35397 | Generalization of ~ inrab ... |
| bj-inrab2 35398 | Shorter proof of ~ inrab .... |
| bj-inrab3 35399 | Generalization of ~ dfrab3... |
| bj-rabtr 35400 | Restricted class abstracti... |
| bj-rabtrALT 35401 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 35402 | Proof of ~ bj-rabtr found ... |
| bj-gabss 35405 | Inclusion of generalized c... |
| bj-gabssd 35406 | Inclusion of generalized c... |
| bj-gabeqd 35407 | Equality of generalized cl... |
| bj-gabeqis 35408 | Equality of generalized cl... |
| bj-elgab 35409 | Elements of a generalized ... |
| bj-gabima 35410 | Generalized class abstract... |
| bj-ru0 35413 | The FOL part of Russell's ... |
| bj-ru1 35414 | A version of Russell's par... |
| bj-ru 35415 | Remove dependency on ~ ax-... |
| currysetlem 35416 | Lemma for ~ currysetlem , ... |
| curryset 35417 | Curry's paradox in set the... |
| currysetlem1 35418 | Lemma for ~ currysetALT . ... |
| currysetlem2 35419 | Lemma for ~ currysetALT . ... |
| currysetlem3 35420 | Lemma for ~ currysetALT . ... |
| currysetALT 35421 | Alternate proof of ~ curry... |
| bj-n0i 35422 | Inference associated with ... |
| bj-disjsn01 35423 | Disjointness of the single... |
| bj-0nel1 35424 | The empty set does not bel... |
| bj-1nel0 35425 | ` 1o ` does not belong to ... |
| bj-xpimasn 35426 | The image of a singleton, ... |
| bj-xpima1sn 35427 | The image of a singleton b... |
| bj-xpima1snALT 35428 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 35429 | The image of a singleton b... |
| bj-xpnzex 35430 | If the first factor of a p... |
| bj-xpexg2 35431 | Curried (exported) form of... |
| bj-xpnzexb 35432 | If the first factor of a p... |
| bj-cleq 35433 | Substitution property for ... |
| bj-snsetex 35434 | The class of sets "whose s... |
| bj-clexab 35435 | Sethood of certain classes... |
| bj-sngleq 35438 | Substitution property for ... |
| bj-elsngl 35439 | Characterization of the el... |
| bj-snglc 35440 | Characterization of the el... |
| bj-snglss 35441 | The singletonization of a ... |
| bj-0nelsngl 35442 | The empty set is not a mem... |
| bj-snglinv 35443 | Inverse of singletonizatio... |
| bj-snglex 35444 | A class is a set if and on... |
| bj-tageq 35447 | Substitution property for ... |
| bj-eltag 35448 | Characterization of the el... |
| bj-0eltag 35449 | The empty set belongs to t... |
| bj-tagn0 35450 | The tagging of a class is ... |
| bj-tagss 35451 | The tagging of a class is ... |
| bj-snglsstag 35452 | The singletonization is in... |
| bj-sngltagi 35453 | The singletonization is in... |
| bj-sngltag 35454 | The singletonization and t... |
| bj-tagci 35455 | Characterization of the el... |
| bj-tagcg 35456 | Characterization of the el... |
| bj-taginv 35457 | Inverse of tagging. (Cont... |
| bj-tagex 35458 | A class is a set if and on... |
| bj-xtageq 35459 | The products of a given cl... |
| bj-xtagex 35460 | The product of a set and t... |
| bj-projeq 35463 | Substitution property for ... |
| bj-projeq2 35464 | Substitution property for ... |
| bj-projun 35465 | The class projection on a ... |
| bj-projex 35466 | Sethood of the class proje... |
| bj-projval 35467 | Value of the class project... |
| bj-1upleq 35470 | Substitution property for ... |
| bj-pr1eq 35473 | Substitution property for ... |
| bj-pr1un 35474 | The first projection prese... |
| bj-pr1val 35475 | Value of the first project... |
| bj-pr11val 35476 | Value of the first project... |
| bj-pr1ex 35477 | Sethood of the first proje... |
| bj-1uplth 35478 | The characteristic propert... |
| bj-1uplex 35479 | A monuple is a set if and ... |
| bj-1upln0 35480 | A monuple is nonempty. (C... |
| bj-2upleq 35483 | Substitution property for ... |
| bj-pr21val 35484 | Value of the first project... |
| bj-pr2eq 35487 | Substitution property for ... |
| bj-pr2un 35488 | The second projection pres... |
| bj-pr2val 35489 | Value of the second projec... |
| bj-pr22val 35490 | Value of the second projec... |
| bj-pr2ex 35491 | Sethood of the second proj... |
| bj-2uplth 35492 | The characteristic propert... |
| bj-2uplex 35493 | A couple is a set if and o... |
| bj-2upln0 35494 | A couple is nonempty. (Co... |
| bj-2upln1upl 35495 | A couple is never equal to... |
| bj-rcleqf 35496 | Relative version of ~ cleq... |
| bj-rcleq 35497 | Relative version of ~ dfcl... |
| bj-reabeq 35498 | Relative form of ~ eqab . ... |
| bj-disj2r 35499 | Relative version of ~ ssdi... |
| bj-sscon 35500 | Contraposition law for rel... |
| bj-abex 35501 | Two ways of stating that t... |
| bj-clex 35502 | Two ways of stating that a... |
| bj-axsn 35503 | Two ways of stating the ax... |
| bj-snexg 35505 | A singleton built on a set... |
| bj-snex 35506 | A singleton is a set. See... |
| bj-axbun 35507 | Two ways of stating the ax... |
| bj-unexg 35509 | Existence of binary unions... |
| bj-prexg 35510 | Existence of unordered pai... |
| bj-prex 35511 | Existence of unordered pai... |
| bj-axadj 35512 | Two ways of stating the ax... |
| bj-adjg1 35514 | Existence of the result of... |
| bj-snfromadj 35515 | Singleton from adjunction ... |
| bj-prfromadj 35516 | Unordered pair from adjunc... |
| bj-adjfrombun 35517 | Adjunction from singleton ... |
| eleq2w2ALT 35518 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 35519 | Alternate proof of ~ clel3... |
| bj-pw0ALT 35520 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 35521 | Quantitative version of ~ ... |
| bj-unirel 35522 | Quantitative version of ~ ... |
| bj-elpwg 35523 | If the intersection of two... |
| bj-velpwALT 35524 | This theorem ~ bj-velpwALT... |
| bj-elpwgALT 35525 | Alternate proof of ~ elpwg... |
| bj-vjust 35526 | Justification theorem for ... |
| bj-nul 35527 | Two formulations of the ax... |
| bj-nuliota 35528 | Definition of the empty se... |
| bj-nuliotaALT 35529 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 35530 | Alternate proof of ~ vtocl... |
| bj-elsn12g 35531 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 35532 | Biconditional version of ~... |
| bj-pwcfsdom 35533 | Remove hypothesis from ~ p... |
| bj-grur1 35534 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 35535 | The extension of a predica... |
| bj-dfid2ALT 35536 | Alternate version of ~ dfi... |
| bj-0nelopab 35537 | The empty set is never an ... |
| bj-brrelex12ALT 35538 | Two classes related by a b... |
| bj-epelg 35539 | The membership relation an... |
| bj-epelb 35540 | Two classes are related by... |
| bj-nsnid 35541 | A set does not contain the... |
| bj-rdg0gALT 35542 | Alternate proof of ~ rdg0g... |
| bj-evaleq 35543 | Equality theorem for the `... |
| bj-evalfun 35544 | The evaluation at a class ... |
| bj-evalfn 35545 | The evaluation at a class ... |
| bj-evalval 35546 | Value of the evaluation at... |
| bj-evalid 35547 | The evaluation at a set of... |
| bj-ndxarg 35548 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 35549 | Closed general form of ~ s... |
| bj-rest00 35552 | An elementwise intersectio... |
| bj-restsn 35553 | An elementwise intersectio... |
| bj-restsnss 35554 | Special case of ~ bj-rests... |
| bj-restsnss2 35555 | Special case of ~ bj-rests... |
| bj-restsn0 35556 | An elementwise intersectio... |
| bj-restsn10 35557 | Special case of ~ bj-rests... |
| bj-restsnid 35558 | The elementwise intersecti... |
| bj-rest10 35559 | An elementwise intersectio... |
| bj-rest10b 35560 | Alternate version of ~ bj-... |
| bj-restn0 35561 | An elementwise intersectio... |
| bj-restn0b 35562 | Alternate version of ~ bj-... |
| bj-restpw 35563 | The elementwise intersecti... |
| bj-rest0 35564 | An elementwise intersectio... |
| bj-restb 35565 | An elementwise intersectio... |
| bj-restv 35566 | An elementwise intersectio... |
| bj-resta 35567 | An elementwise intersectio... |
| bj-restuni 35568 | The union of an elementwis... |
| bj-restuni2 35569 | The union of an elementwis... |
| bj-restreg 35570 | A reformulation of the axi... |
| bj-raldifsn 35571 | All elements in a set sati... |
| bj-0int 35572 | If ` A ` is a collection o... |
| bj-mooreset 35573 | A Moore collection is a se... |
| bj-ismoore 35576 | Characterization of Moore ... |
| bj-ismoored0 35577 | Necessary condition to be ... |
| bj-ismoored 35578 | Necessary condition to be ... |
| bj-ismoored2 35579 | Necessary condition to be ... |
| bj-ismooredr 35580 | Sufficient condition to be... |
| bj-ismooredr2 35581 | Sufficient condition to be... |
| bj-discrmoore 35582 | The powerclass ` ~P A ` is... |
| bj-0nmoore 35583 | The empty set is not a Moo... |
| bj-snmoore 35584 | A singleton is a Moore col... |
| bj-snmooreb 35585 | A singleton is a Moore col... |
| bj-prmoore 35586 | A pair formed of two neste... |
| bj-0nelmpt 35587 | The empty set is not an el... |
| bj-mptval 35588 | Value of a function given ... |
| bj-dfmpoa 35589 | An equivalent definition o... |
| bj-mpomptALT 35590 | Alternate proof of ~ mpomp... |
| setsstrset 35607 | Relation between ~ df-sets... |
| bj-nfald 35608 | Variant of ~ nfald . (Con... |
| bj-nfexd 35609 | Variant of ~ nfexd . (Con... |
| copsex2d 35610 | Implicit substitution dedu... |
| copsex2b 35611 | Biconditional form of ~ co... |
| opelopabd 35612 | Membership of an ordere pa... |
| opelopabb 35613 | Membership of an ordered p... |
| opelopabbv 35614 | Membership of an ordered p... |
| bj-opelrelex 35615 | The coordinates of an orde... |
| bj-opelresdm 35616 | If an ordered pair is in a... |
| bj-brresdm 35617 | If two classes are related... |
| brabd0 35618 | Expressing that two sets a... |
| brabd 35619 | Expressing that two sets a... |
| bj-brab2a1 35620 | "Unbounded" version of ~ b... |
| bj-opabssvv 35621 | A variant of ~ relopabiv (... |
| bj-funidres 35622 | The restricted identity re... |
| bj-opelidb 35623 | Characterization of the or... |
| bj-opelidb1 35624 | Characterization of the or... |
| bj-inexeqex 35625 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 35626 | If the intersection of two... |
| bj-opelid 35627 | Characterization of the or... |
| bj-ideqg 35628 | Characterization of the cl... |
| bj-ideqgALT 35629 | Alternate proof of ~ bj-id... |
| bj-ideqb 35630 | Characterization of classe... |
| bj-idres 35631 | Alternate expression for t... |
| bj-opelidres 35632 | Characterization of the or... |
| bj-idreseq 35633 | Sufficient condition for t... |
| bj-idreseqb 35634 | Characterization for two c... |
| bj-ideqg1 35635 | For sets, the identity rel... |
| bj-ideqg1ALT 35636 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 35637 | Characterization of the co... |
| bj-elid3 35638 | Characterization of the co... |
| bj-elid4 35639 | Characterization of the el... |
| bj-elid5 35640 | Characterization of the el... |
| bj-elid6 35641 | Characterization of the el... |
| bj-elid7 35642 | Characterization of the el... |
| bj-diagval 35645 | Value of the functionalize... |
| bj-diagval2 35646 | Value of the functionalize... |
| bj-eldiag 35647 | Characterization of the el... |
| bj-eldiag2 35648 | Characterization of the el... |
| bj-imdirvallem 35651 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 35652 | Value of the functionalize... |
| bj-imdirval2lem 35653 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 35654 | Value of the functionalize... |
| bj-imdirval3 35655 | Value of the functionalize... |
| bj-imdiridlem 35656 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 35657 | Functorial property of the... |
| bj-opelopabid 35658 | Membership in an ordered-p... |
| bj-opabco 35659 | Composition of ordered-pai... |
| bj-xpcossxp 35660 | The composition of two Car... |
| bj-imdirco 35661 | Functorial property of the... |
| bj-iminvval 35664 | Value of the functionalize... |
| bj-iminvval2 35665 | Value of the functionalize... |
| bj-iminvid 35666 | Functorial property of the... |
| bj-inftyexpitaufo 35673 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 35676 | An element of the circle a... |
| bj-inftyexpiinv 35679 | Utility theorem for the in... |
| bj-inftyexpiinj 35680 | Injectivity of the paramet... |
| bj-inftyexpidisj 35681 | An element of the circle a... |
| bj-ccinftydisj 35684 | The circle at infinity is ... |
| bj-elccinfty 35685 | A lemma for infinite exten... |
| bj-ccssccbar 35688 | Complex numbers are extend... |
| bj-ccinftyssccbar 35689 | Infinite extended complex ... |
| bj-pinftyccb 35692 | The class ` pinfty ` is an... |
| bj-pinftynrr 35693 | The extended complex numbe... |
| bj-minftyccb 35696 | The class ` minfty ` is an... |
| bj-minftynrr 35697 | The extended complex numbe... |
| bj-pinftynminfty 35698 | The extended complex numbe... |
| bj-rrhatsscchat 35707 | The real projective line i... |
| bj-imafv 35722 | If the direct image of a s... |
| bj-funun 35723 | Value of a function expres... |
| bj-fununsn1 35724 | Value of a function expres... |
| bj-fununsn2 35725 | Value of a function expres... |
| bj-fvsnun1 35726 | The value of a function wi... |
| bj-fvsnun2 35727 | The value of a function wi... |
| bj-fvmptunsn1 35728 | Value of a function expres... |
| bj-fvmptunsn2 35729 | Value of a function expres... |
| bj-iomnnom 35730 | The canonical bijection fr... |
| bj-smgrpssmgm 35739 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 35740 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 35741 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 35742 | Monoids are semigroups (el... |
| bj-cmnssmnd 35743 | Commutative monoids are mo... |
| bj-cmnssmndel 35744 | Commutative monoids are mo... |
| bj-grpssmnd 35745 | Groups are monoids. (Cont... |
| bj-grpssmndel 35746 | Groups are monoids (elemen... |
| bj-ablssgrp 35747 | Abelian groups are groups.... |
| bj-ablssgrpel 35748 | Abelian groups are groups ... |
| bj-ablsscmn 35749 | Abelian groups are commuta... |
| bj-ablsscmnel 35750 | Abelian groups are commuta... |
| bj-modssabl 35751 | (The additive groups of) m... |
| bj-vecssmod 35752 | Vector spaces are modules.... |
| bj-vecssmodel 35753 | Vector spaces are modules ... |
| bj-finsumval0 35756 | Value of a finite sum. (C... |
| bj-fvimacnv0 35757 | Variant of ~ fvimacnv wher... |
| bj-isvec 35758 | The predicate "is a vector... |
| bj-fldssdrng 35759 | Fields are division rings.... |
| bj-flddrng 35760 | Fields are division rings ... |
| bj-rrdrg 35761 | The field of real numbers ... |
| bj-isclm 35762 | The predicate "is a subcom... |
| bj-isrvec 35765 | The predicate "is a real v... |
| bj-rvecmod 35766 | Real vector spaces are mod... |
| bj-rvecssmod 35767 | Real vector spaces are mod... |
| bj-rvecrr 35768 | The field of scalars of a ... |
| bj-isrvecd 35769 | The predicate "is a real v... |
| bj-rvecvec 35770 | Real vector spaces are vec... |
| bj-isrvec2 35771 | The predicate "is a real v... |
| bj-rvecssvec 35772 | Real vector spaces are vec... |
| bj-rveccmod 35773 | Real vector spaces are sub... |
| bj-rvecsscmod 35774 | Real vector spaces are sub... |
| bj-rvecsscvec 35775 | Real vector spaces are sub... |
| bj-rveccvec 35776 | Real vector spaces are sub... |
| bj-rvecssabl 35777 | (The additive groups of) r... |
| bj-rvecabl 35778 | (The additive groups of) r... |
| bj-subcom 35779 | A consequence of commutati... |
| bj-lineqi 35780 | Solution of a (scalar) lin... |
| bj-bary1lem 35781 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 35782 | Lemma for bj-bary1: comput... |
| bj-bary1 35783 | Barycentric coordinates in... |
| bj-endval 35786 | Value of the monoid of end... |
| bj-endbase 35787 | Base set of the monoid of ... |
| bj-endcomp 35788 | Composition law of the mon... |
| bj-endmnd 35789 | The monoid of endomorphism... |
| taupilem3 35790 | Lemma for tau-related theo... |
| taupilemrplb 35791 | A set of positive reals ha... |
| taupilem1 35792 | Lemma for ~ taupi . A pos... |
| taupilem2 35793 | Lemma for ~ taupi . The s... |
| taupi 35794 | Relationship between ` _ta... |
| dfgcd3 35795 | Alternate definition of th... |
| irrdifflemf 35796 | Lemma for ~ irrdiff . The... |
| irrdiff 35797 | The irrationals are exactl... |
| iccioo01 35798 | The closed unit interval i... |
| csbrecsg 35799 | Move class substitution in... |
| csbrdgg 35800 | Move class substitution in... |
| csboprabg 35801 | Move class substitution in... |
| csbmpo123 35802 | Move class substitution in... |
| con1bii2 35803 | A contraposition inference... |
| con2bii2 35804 | A contraposition inference... |
| vtoclefex 35805 | Implicit substitution of a... |
| rnmptsn 35806 | The range of a function ma... |
| f1omptsnlem 35807 | This is the core of the pr... |
| f1omptsn 35808 | A function mapping to sing... |
| mptsnunlem 35809 | This is the core of the pr... |
| mptsnun 35810 | A class ` B ` is equal to ... |
| dissneqlem 35811 | This is the core of the pr... |
| dissneq 35812 | Any topology that contains... |
| exlimim 35813 | Closed form of ~ exlimimd ... |
| exlimimd 35814 | Existential elimination ru... |
| exellim 35815 | Closed form of ~ exellimdd... |
| exellimddv 35816 | Eliminate an antecedent wh... |
| topdifinfindis 35817 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 35818 | This is the core of the pr... |
| topdifinffin 35819 | Part of Exercise 3 of [Mun... |
| topdifinf 35820 | Part of Exercise 3 of [Mun... |
| topdifinfeq 35821 | Two different ways of defi... |
| icorempo 35822 | Closed-below, open-above i... |
| icoreresf 35823 | Closed-below, open-above i... |
| icoreval 35824 | Value of the closed-below,... |
| icoreelrnab 35825 | Elementhood in the set of ... |
| isbasisrelowllem1 35826 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 35827 | Lemma for ~ isbasisrelowl ... |
| icoreclin 35828 | The set of closed-below, o... |
| isbasisrelowl 35829 | The set of all closed-belo... |
| icoreunrn 35830 | The union of all closed-be... |
| istoprelowl 35831 | The set of all closed-belo... |
| icoreelrn 35832 | A class abstraction which ... |
| iooelexlt 35833 | An element of an open inte... |
| relowlssretop 35834 | The lower limit topology o... |
| relowlpssretop 35835 | The lower limit topology o... |
| sucneqond 35836 | Inequality of an ordinal s... |
| sucneqoni 35837 | Inequality of an ordinal s... |
| onsucuni3 35838 | If an ordinal number has a... |
| 1oequni2o 35839 | The ordinal number ` 1o ` ... |
| rdgsucuni 35840 | If an ordinal number has a... |
| rdgeqoa 35841 | If a recursive function wi... |
| elxp8 35842 | Membership in a Cartesian ... |
| cbveud 35843 | Deduction used to change b... |
| cbvreud 35844 | Deduction used to change b... |
| difunieq 35845 | The difference of unions i... |
| inunissunidif 35846 | Theorem about subsets of t... |
| rdgellim 35847 | Elementhood in a recursive... |
| rdglimss 35848 | A recursive definition at ... |
| rdgssun 35849 | In a recursive definition ... |
| exrecfnlem 35850 | Lemma for ~ exrecfn . (Co... |
| exrecfn 35851 | Theorem about the existenc... |
| exrecfnpw 35852 | For any base set, a set wh... |
| finorwe 35853 | If the Axiom of Infinity i... |
| dffinxpf 35856 | This theorem is the same a... |
| finxpeq1 35857 | Equality theorem for Carte... |
| finxpeq2 35858 | Equality theorem for Carte... |
| csbfinxpg 35859 | Distribute proper substitu... |
| finxpreclem1 35860 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 35861 | Lemma for ` ^^ ` recursion... |
| finxp0 35862 | The value of Cartesian exp... |
| finxp1o 35863 | The value of Cartesian exp... |
| finxpreclem3 35864 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 35865 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 35866 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 35867 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 35868 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 35869 | The value of Cartesian exp... |
| finxp2o 35870 | The value of Cartesian exp... |
| finxp3o 35871 | The value of Cartesian exp... |
| finxpnom 35872 | Cartesian exponentiation w... |
| finxp00 35873 | Cartesian exponentiation o... |
| iunctb2 35874 | Using the axiom of countab... |
| domalom 35875 | A class which dominates ev... |
| isinf2 35876 | The converse of ~ isinf . ... |
| ctbssinf 35877 | Using the axiom of choice,... |
| ralssiun 35878 | The index set of an indexe... |
| nlpineqsn 35879 | For every point ` p ` of a... |
| nlpfvineqsn 35880 | Given a subset ` A ` of ` ... |
| fvineqsnf1 35881 | A theorem about functions ... |
| fvineqsneu 35882 | A theorem about functions ... |
| fvineqsneq 35883 | A theorem about functions ... |
| pibp16 35884 | Property P000016 of pi-bas... |
| pibp19 35885 | Property P000019 of pi-bas... |
| pibp21 35886 | Property P000021 of pi-bas... |
| pibt1 35887 | Theorem T000001 of pi-base... |
| pibt2 35888 | Theorem T000002 of pi-base... |
| wl-section-prop 35889 | Intuitionistic logic is no... |
| wl-section-boot 35893 | In this section, I provide... |
| wl-luk-imim1i 35894 | Inference adding common co... |
| wl-luk-syl 35895 | An inference version of th... |
| wl-luk-imtrid 35896 | A syllogism rule of infere... |
| wl-luk-pm2.18d 35897 | Deduction based on reducti... |
| wl-luk-con4i 35898 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 35899 | Inference rule. Copy of ~... |
| wl-luk-a1i 35900 | Inference rule. Copy of ~... |
| wl-luk-mpi 35901 | A nested modus ponens infe... |
| wl-luk-imim2i 35902 | Inference adding common an... |
| wl-luk-imtrdi 35903 | A syllogism rule of infere... |
| wl-luk-ax3 35904 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 35905 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 35906 | This theorem, called "Asse... |
| wl-luk-com12 35907 | Inference that swaps (comm... |
| wl-luk-pm2.21 35908 | From a wff and its negatio... |
| wl-luk-con1i 35909 | A contraposition inference... |
| wl-luk-ja 35910 | Inference joining the ante... |
| wl-luk-imim2 35911 | A closed form of syllogism... |
| wl-luk-a1d 35912 | Deduction introducing an e... |
| wl-luk-ax2 35913 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 35914 | Principle of identity. Th... |
| wl-luk-notnotr 35915 | Converse of double negatio... |
| wl-luk-pm2.04 35916 | Swap antecedents. Theorem... |
| wl-section-impchain 35917 | An implication like ` ( ps... |
| wl-impchain-mp-x 35918 | This series of theorems pr... |
| wl-impchain-mp-0 35919 | This theorem is the start ... |
| wl-impchain-mp-1 35920 | This theorem is in fact a ... |
| wl-impchain-mp-2 35921 | This theorem is in fact a ... |
| wl-impchain-com-1.x 35922 | It is often convenient to ... |
| wl-impchain-com-1.1 35923 | A degenerate form of antec... |
| wl-impchain-com-1.2 35924 | This theorem is in fact a ... |
| wl-impchain-com-1.3 35925 | This theorem is in fact a ... |
| wl-impchain-com-1.4 35926 | This theorem is in fact a ... |
| wl-impchain-com-n.m 35927 | This series of theorems al... |
| wl-impchain-com-2.3 35928 | This theorem is in fact a ... |
| wl-impchain-com-2.4 35929 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 35930 | This theorem is in fact a ... |
| wl-impchain-a1-x 35931 | If an implication chain is... |
| wl-impchain-a1-1 35932 | Inference rule, a copy of ... |
| wl-impchain-a1-2 35933 | Inference rule, a copy of ... |
| wl-impchain-a1-3 35934 | Inference rule, a copy of ... |
| wl-ifp-ncond1 35935 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 35936 | If one case of an ` if- ` ... |
| wl-ifpimpr 35937 | If one case of an ` if- ` ... |
| wl-ifp4impr 35938 | If one case of an ` if- ` ... |
| wl-df-3xor 35939 | Alternative definition of ... |
| wl-df3xor2 35940 | Alternative definition of ... |
| wl-df3xor3 35941 | Alternative form of ~ wl-d... |
| wl-3xortru 35942 | If the first input is true... |
| wl-3xorfal 35943 | If the first input is fals... |
| wl-3xorbi 35944 | Triple xor can be replaced... |
| wl-3xorbi2 35945 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 35946 | Equivalence theorem for tr... |
| wl-3xorbi123i 35947 | Equivalence theorem for tr... |
| wl-3xorrot 35948 | Rotation law for triple xo... |
| wl-3xorcoma 35949 | Commutative law for triple... |
| wl-3xorcomb 35950 | Commutative law for triple... |
| wl-3xornot1 35951 | Flipping the first input f... |
| wl-3xornot 35952 | Triple xor distributes ove... |
| wl-1xor 35953 | In the recursive scheme ... |
| wl-2xor 35954 | In the recursive scheme ... |
| wl-df-3mintru2 35955 | Alternative definition of ... |
| wl-df2-3mintru2 35956 | The adder carry in disjunc... |
| wl-df3-3mintru2 35957 | The adder carry in conjunc... |
| wl-df4-3mintru2 35958 | An alternative definition ... |
| wl-1mintru1 35959 | Using the recursion formul... |
| wl-1mintru2 35960 | Using the recursion formul... |
| wl-2mintru1 35961 | Using the recursion formul... |
| wl-2mintru2 35962 | Using the recursion formul... |
| wl-df3maxtru1 35963 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 35965 | A version of ~ ax-wl-13v w... |
| wl-mps 35966 | Replacing a nested consequ... |
| wl-syls1 35967 | Replacing a nested consequ... |
| wl-syls2 35968 | Replacing a nested anteced... |
| wl-embant 35969 | A true wff can always be a... |
| wl-orel12 35970 | In a conjunctive normal fo... |
| wl-cases2-dnf 35971 | A particular instance of ~... |
| wl-cbvmotv 35972 | Change bound variable. Us... |
| wl-moteq 35973 | Change bound variable. Us... |
| wl-motae 35974 | Change bound variable. Us... |
| wl-moae 35975 | Two ways to express "at mo... |
| wl-euae 35976 | Two ways to express "exact... |
| wl-nax6im 35977 | The following series of th... |
| wl-hbae1 35978 | This specialization of ~ h... |
| wl-naevhba1v 35979 | An instance of ~ hbn1w app... |
| wl-spae 35980 | Prove an instance of ~ sp ... |
| wl-speqv 35981 | Under the assumption ` -. ... |
| wl-19.8eqv 35982 | Under the assumption ` -. ... |
| wl-19.2reqv 35983 | Under the assumption ` -. ... |
| wl-nfalv 35984 | If ` x ` is not present in... |
| wl-nfimf1 35985 | An antecedent is irrelevan... |
| wl-nfae1 35986 | Unlike ~ nfae , this speci... |
| wl-nfnae1 35987 | Unlike ~ nfnae , this spec... |
| wl-aetr 35988 | A transitive law for varia... |
| wl-axc11r 35989 | Same as ~ axc11r , but usi... |
| wl-dral1d 35990 | A version of ~ dral1 with ... |
| wl-cbvalnaed 35991 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 35992 | A more general version of ... |
| wl-exeq 35993 | The semantics of ` E. x y ... |
| wl-aleq 35994 | The semantics of ` A. x y ... |
| wl-nfeqfb 35995 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 35996 | If ` y ` is not free in ` ... |
| wl-equsalvw 35997 | Version of ~ equsalv with ... |
| wl-equsald 35998 | Deduction version of ~ equ... |
| wl-equsal 35999 | A useful equivalence relat... |
| wl-equsal1t 36000 | The expression ` x = y ` i... |
| wl-equsalcom 36001 | This simple equivalence ea... |
| wl-equsal1i 36002 | The antecedent ` x = y ` i... |
| wl-sb6rft 36003 | A specialization of ~ wl-e... |
| wl-cbvalsbi 36004 | Change bounded variables i... |
| wl-sbrimt 36005 | Substitution with a variab... |
| wl-sblimt 36006 | Substitution with a variab... |
| wl-sb8t 36007 | Substitution of variable i... |
| wl-sb8et 36008 | Substitution of variable i... |
| wl-sbhbt 36009 | Closed form of ~ sbhb . C... |
| wl-sbnf1 36010 | Two ways expressing that `... |
| wl-equsb3 36011 | ~ equsb3 with a distinctor... |
| wl-equsb4 36012 | Substitution applied to an... |
| wl-2sb6d 36013 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 36014 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 36015 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 36016 | Version of ~ sbcom2 with a... |
| wl-sbalnae 36017 | A theorem used in eliminat... |
| wl-sbal1 36018 | A theorem used in eliminat... |
| wl-sbal2 36019 | Move quantifier in and out... |
| wl-2spsbbi 36020 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 36021 | This theorem provides a ba... |
| wl-lem-nexmo 36022 | This theorem provides a ba... |
| wl-lem-moexsb 36023 | The antecedent ` A. x ( ph... |
| wl-alanbii 36024 | This theorem extends ~ ala... |
| wl-mo2df 36025 | Version of ~ mof with a co... |
| wl-mo2tf 36026 | Closed form of ~ mof with ... |
| wl-eudf 36027 | Version of ~ eu6 with a co... |
| wl-eutf 36028 | Closed form of ~ eu6 with ... |
| wl-euequf 36029 | ~ euequ proved with a dist... |
| wl-mo2t 36030 | Closed form of ~ mof . (C... |
| wl-mo3t 36031 | Closed form of ~ mo3 . (C... |
| wl-sb8eut 36032 | Substitution of variable i... |
| wl-sb8mot 36033 | Substitution of variable i... |
| wl-issetft 36034 | A closed form of ~ issetf ... |
| wl-axc11rc11 36035 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 36037 | A transitive law for varia... |
| wl-ax11-lem2 36038 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 36039 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 36040 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 36041 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 36042 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 36043 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 36044 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 36045 | The easy part when ` x ` c... |
| wl-ax11-lem10 36046 | We now have prepared every... |
| wl-clabv 36047 | Variant of ~ df-clab , whe... |
| wl-dfclab 36048 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 36049 | Using class abstraction in... |
| wl-clabt 36050 | Using class abstraction in... |
| rabiun 36051 | Abstraction restricted to ... |
| iundif1 36052 | Indexed union of class dif... |
| imadifss 36053 | The difference of images i... |
| cureq 36054 | Equality theorem for curry... |
| unceq 36055 | Equality theorem for uncur... |
| curf 36056 | Functional property of cur... |
| uncf 36057 | Functional property of unc... |
| curfv 36058 | Value of currying. (Contr... |
| uncov 36059 | Value of uncurrying. (Con... |
| curunc 36060 | Currying of uncurrying. (... |
| unccur 36061 | Uncurrying of currying. (... |
| phpreu 36062 | Theorem related to pigeonh... |
| finixpnum 36063 | A finite Cartesian product... |
| fin2solem 36064 | Lemma for ~ fin2so . (Con... |
| fin2so 36065 | Any totally ordered Tarski... |
| ltflcei 36066 | Theorem to move the floor ... |
| leceifl 36067 | Theorem to move the floor ... |
| sin2h 36068 | Half-angle rule for sine. ... |
| cos2h 36069 | Half-angle rule for cosine... |
| tan2h 36070 | Half-angle rule for tangen... |
| lindsadd 36071 | In a vector space, the uni... |
| lindsdom 36072 | A linearly independent set... |
| lindsenlbs 36073 | A maximal linearly indepen... |
| matunitlindflem1 36074 | One direction of ~ matunit... |
| matunitlindflem2 36075 | One direction of ~ matunit... |
| matunitlindf 36076 | A matrix over a field is i... |
| ptrest 36077 | Expressing a restriction o... |
| ptrecube 36078 | Any point in an open set o... |
| poimirlem1 36079 | Lemma for ~ poimir - the v... |
| poimirlem2 36080 | Lemma for ~ poimir - conse... |
| poimirlem3 36081 | Lemma for ~ poimir to add ... |
| poimirlem4 36082 | Lemma for ~ poimir connect... |
| poimirlem5 36083 | Lemma for ~ poimir to esta... |
| poimirlem6 36084 | Lemma for ~ poimir establi... |
| poimirlem7 36085 | Lemma for ~ poimir , simil... |
| poimirlem8 36086 | Lemma for ~ poimir , estab... |
| poimirlem9 36087 | Lemma for ~ poimir , estab... |
| poimirlem10 36088 | Lemma for ~ poimir establi... |
| poimirlem11 36089 | Lemma for ~ poimir connect... |
| poimirlem12 36090 | Lemma for ~ poimir connect... |
| poimirlem13 36091 | Lemma for ~ poimir - for a... |
| poimirlem14 36092 | Lemma for ~ poimir - for a... |
| poimirlem15 36093 | Lemma for ~ poimir , that ... |
| poimirlem16 36094 | Lemma for ~ poimir establi... |
| poimirlem17 36095 | Lemma for ~ poimir establi... |
| poimirlem18 36096 | Lemma for ~ poimir stating... |
| poimirlem19 36097 | Lemma for ~ poimir establi... |
| poimirlem20 36098 | Lemma for ~ poimir establi... |
| poimirlem21 36099 | Lemma for ~ poimir stating... |
| poimirlem22 36100 | Lemma for ~ poimir , that ... |
| poimirlem23 36101 | Lemma for ~ poimir , two w... |
| poimirlem24 36102 | Lemma for ~ poimir , two w... |
| poimirlem25 36103 | Lemma for ~ poimir stating... |
| poimirlem26 36104 | Lemma for ~ poimir showing... |
| poimirlem27 36105 | Lemma for ~ poimir showing... |
| poimirlem28 36106 | Lemma for ~ poimir , a var... |
| poimirlem29 36107 | Lemma for ~ poimir connect... |
| poimirlem30 36108 | Lemma for ~ poimir combini... |
| poimirlem31 36109 | Lemma for ~ poimir , assig... |
| poimirlem32 36110 | Lemma for ~ poimir , combi... |
| poimir 36111 | Poincare-Miranda theorem. ... |
| broucube 36112 | Brouwer - or as Kulpa call... |
| heicant 36113 | Heine-Cantor theorem: a co... |
| opnmbllem0 36114 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 36115 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 36116 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 36117 | The difference between two... |
| mblfinlem4 36118 | Backward direction of ~ is... |
| ismblfin 36119 | Measurability in terms of ... |
| ovoliunnfl 36120 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 36121 | Demonstration of ~ ovoliun... |
| voliunnfl 36122 | ~ voliun is incompatible w... |
| volsupnfl 36123 | ~ volsup is incompatible w... |
| mbfresfi 36124 | Measurability of a piecewi... |
| mbfposadd 36125 | If the sum of two measurab... |
| cnambfre 36126 | A real-valued, a.e. contin... |
| dvtanlem 36127 | Lemma for ~ dvtan - the do... |
| dvtan 36128 | Derivative of tangent. (C... |
| itg2addnclem 36129 | An alternate expression fo... |
| itg2addnclem2 36130 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 36131 | Lemma incomprehensible in ... |
| itg2addnc 36132 | Alternate proof of ~ itg2a... |
| itg2gt0cn 36133 | ~ itg2gt0 holds on functio... |
| ibladdnclem 36134 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 36135 | Choice-free analogue of ~ ... |
| itgaddnclem1 36136 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 36137 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 36138 | Choice-free analogue of ~ ... |
| iblsubnc 36139 | Choice-free analogue of ~ ... |
| itgsubnc 36140 | Choice-free analogue of ~ ... |
| iblabsnclem 36141 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 36142 | Choice-free analogue of ~ ... |
| iblmulc2nc 36143 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 36144 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 36145 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 36146 | Choice-free analogue of ~ ... |
| itgabsnc 36147 | Choice-free analogue of ~ ... |
| itggt0cn 36148 | ~ itggt0 holds for continu... |
| ftc1cnnclem 36149 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 36150 | Choice-free proof of ~ ftc... |
| ftc1anclem1 36151 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 36152 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 36153 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 36154 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 36155 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 36156 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 36157 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 36158 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 36159 | ~ ftc1a holds for function... |
| ftc2nc 36160 | Choice-free proof of ~ ftc... |
| asindmre 36161 | Real part of domain of dif... |
| dvasin 36162 | Derivative of arcsine. (C... |
| dvacos 36163 | Derivative of arccosine. ... |
| dvreasin 36164 | Real derivative of arcsine... |
| dvreacos 36165 | Real derivative of arccosi... |
| areacirclem1 36166 | Antiderivative of cross-se... |
| areacirclem2 36167 | Endpoint-inclusive continu... |
| areacirclem3 36168 | Integrability of cross-sec... |
| areacirclem4 36169 | Endpoint-inclusive continu... |
| areacirclem5 36170 | Finding the cross-section ... |
| areacirc 36171 | The area of a circle of ra... |
| unirep 36172 | Define a quantity whose de... |
| cover2 36173 | Two ways of expressing the... |
| cover2g 36174 | Two ways of expressing the... |
| brabg2 36175 | Relation by a binary relat... |
| opelopab3 36176 | Ordered pair membership in... |
| cocanfo 36177 | Cancellation of a surjecti... |
| brresi2 36178 | Restriction of a binary re... |
| fnopabeqd 36179 | Equality deduction for fun... |
| fvopabf4g 36180 | Function value of an opera... |
| eqfnun 36181 | Two functions on ` A u. B ... |
| fnopabco 36182 | Composition of a function ... |
| opropabco 36183 | Composition of an operator... |
| cocnv 36184 | Composition with a functio... |
| f1ocan1fv 36185 | Cancel a composition by a ... |
| f1ocan2fv 36186 | Cancel a composition by th... |
| inixp 36187 | Intersection of Cartesian ... |
| upixp 36188 | Universal property of the ... |
| abrexdom 36189 | An indexed set is dominate... |
| abrexdom2 36190 | An indexed set is dominate... |
| ac6gf 36191 | Axiom of Choice. (Contrib... |
| indexa 36192 | If for every element of an... |
| indexdom 36193 | If for every element of an... |
| frinfm 36194 | A subset of a well-founded... |
| welb 36195 | A nonempty subset of a wel... |
| supex2g 36196 | Existence of supremum. (C... |
| supclt 36197 | Closure of supremum. (Con... |
| supubt 36198 | Upper bound property of su... |
| filbcmb 36199 | Combine a finite set of lo... |
| fzmul 36200 | Membership of a product in... |
| sdclem2 36201 | Lemma for ~ sdc . (Contri... |
| sdclem1 36202 | Lemma for ~ sdc . (Contri... |
| sdc 36203 | Strong dependent choice. ... |
| fdc 36204 | Finite version of dependen... |
| fdc1 36205 | Variant of ~ fdc with no s... |
| seqpo 36206 | Two ways to say that a seq... |
| incsequz 36207 | An increasing sequence of ... |
| incsequz2 36208 | An increasing sequence of ... |
| nnubfi 36209 | A bounded above set of pos... |
| nninfnub 36210 | An infinite set of positiv... |
| subspopn 36211 | An open set is open in the... |
| neificl 36212 | Neighborhoods are closed u... |
| lpss2 36213 | Limit points of a subset a... |
| metf1o 36214 | Use a bijection with a met... |
| blssp 36215 | A ball in the subspace met... |
| mettrifi 36216 | Generalized triangle inequ... |
| lmclim2 36217 | A sequence in a metric spa... |
| geomcau 36218 | If the distance between co... |
| caures 36219 | The restriction of a Cauch... |
| caushft 36220 | A shifted Cauchy sequence ... |
| constcncf 36221 | A constant function is a c... |
| cnres2 36222 | The restriction of a conti... |
| cnresima 36223 | A continuous function is c... |
| cncfres 36224 | A continuous function on c... |
| istotbnd 36228 | The predicate "is a totall... |
| istotbnd2 36229 | The predicate "is a totall... |
| istotbnd3 36230 | A metric space is totally ... |
| totbndmet 36231 | The predicate "totally bou... |
| 0totbnd 36232 | The metric (there is only ... |
| sstotbnd2 36233 | Condition for a subset of ... |
| sstotbnd 36234 | Condition for a subset of ... |
| sstotbnd3 36235 | Use a net that is not nece... |
| totbndss 36236 | A subset of a totally boun... |
| equivtotbnd 36237 | If the metric ` M ` is "st... |
| isbnd 36239 | The predicate "is a bounde... |
| bndmet 36240 | A bounded metric space is ... |
| isbndx 36241 | A "bounded extended metric... |
| isbnd2 36242 | The predicate "is a bounde... |
| isbnd3 36243 | A metric space is bounded ... |
| isbnd3b 36244 | A metric space is bounded ... |
| bndss 36245 | A subset of a bounded metr... |
| blbnd 36246 | A ball is bounded. (Contr... |
| ssbnd 36247 | A subset of a metric space... |
| totbndbnd 36248 | A totally bounded metric s... |
| equivbnd 36249 | If the metric ` M ` is "st... |
| bnd2lem 36250 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 36251 | If balls are totally bound... |
| prdsbnd 36252 | The product metric over fi... |
| prdstotbnd 36253 | The product metric over fi... |
| prdsbnd2 36254 | If balls are totally bound... |
| cntotbnd 36255 | A subset of the complex nu... |
| cnpwstotbnd 36256 | A subset of ` A ^ I ` , wh... |
| ismtyval 36259 | The set of isometries betw... |
| isismty 36260 | The condition "is an isome... |
| ismtycnv 36261 | The inverse of an isometry... |
| ismtyima 36262 | The image of a ball under ... |
| ismtyhmeolem 36263 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 36264 | An isometry is a homeomorp... |
| ismtybndlem 36265 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 36266 | Isometries preserve bounde... |
| ismtyres 36267 | A restriction of an isomet... |
| heibor1lem 36268 | Lemma for ~ heibor1 . A c... |
| heibor1 36269 | One half of ~ heibor , tha... |
| heiborlem1 36270 | Lemma for ~ heibor . We w... |
| heiborlem2 36271 | Lemma for ~ heibor . Subs... |
| heiborlem3 36272 | Lemma for ~ heibor . Usin... |
| heiborlem4 36273 | Lemma for ~ heibor . Usin... |
| heiborlem5 36274 | Lemma for ~ heibor . The ... |
| heiborlem6 36275 | Lemma for ~ heibor . Sinc... |
| heiborlem7 36276 | Lemma for ~ heibor . Sinc... |
| heiborlem8 36277 | Lemma for ~ heibor . The ... |
| heiborlem9 36278 | Lemma for ~ heibor . Disc... |
| heiborlem10 36279 | Lemma for ~ heibor . The ... |
| heibor 36280 | Generalized Heine-Borel Th... |
| bfplem1 36281 | Lemma for ~ bfp . The seq... |
| bfplem2 36282 | Lemma for ~ bfp . Using t... |
| bfp 36283 | Banach fixed point theorem... |
| rrnval 36286 | The n-dimensional Euclidea... |
| rrnmval 36287 | The value of the Euclidean... |
| rrnmet 36288 | Euclidean space is a metri... |
| rrndstprj1 36289 | The distance between two p... |
| rrndstprj2 36290 | Bound on the distance betw... |
| rrncmslem 36291 | Lemma for ~ rrncms . (Con... |
| rrncms 36292 | Euclidean space is complet... |
| repwsmet 36293 | The supremum metric on ` R... |
| rrnequiv 36294 | The supremum metric on ` R... |
| rrntotbnd 36295 | A set in Euclidean space i... |
| rrnheibor 36296 | Heine-Borel theorem for Eu... |
| ismrer1 36297 | An isometry between ` RR `... |
| reheibor 36298 | Heine-Borel theorem for re... |
| iccbnd 36299 | A closed interval in ` RR ... |
| icccmpALT 36300 | A closed interval in ` RR ... |
| isass 36305 | The predicate "is an assoc... |
| isexid 36306 | The predicate ` G ` has a ... |
| ismgmOLD 36309 | Obsolete version of ~ ismg... |
| clmgmOLD 36310 | Obsolete version of ~ mgmc... |
| opidonOLD 36311 | Obsolete version of ~ mndp... |
| rngopidOLD 36312 | Obsolete version of ~ mndp... |
| opidon2OLD 36313 | Obsolete version of ~ mndp... |
| isexid2 36314 | If ` G e. ( Magma i^i ExId... |
| exidu1 36315 | Uniqueness of the left and... |
| idrval 36316 | The value of the identity ... |
| iorlid 36317 | A magma right and left ide... |
| cmpidelt 36318 | A magma right and left ide... |
| smgrpismgmOLD 36321 | Obsolete version of ~ sgrp... |
| issmgrpOLD 36322 | Obsolete version of ~ issg... |
| smgrpmgm 36323 | A semigroup is a magma. (... |
| smgrpassOLD 36324 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 36327 | Obsolete version of ~ mnds... |
| mndoisexid 36328 | A monoid has an identity e... |
| mndoismgmOLD 36329 | Obsolete version of ~ mndm... |
| mndomgmid 36330 | A monoid is a magma with a... |
| ismndo 36331 | The predicate "is a monoid... |
| ismndo1 36332 | The predicate "is a monoid... |
| ismndo2 36333 | The predicate "is a monoid... |
| grpomndo 36334 | A group is a monoid. (Con... |
| exidcl 36335 | Closure of the binary oper... |
| exidreslem 36336 | Lemma for ~ exidres and ~ ... |
| exidres 36337 | The restriction of a binar... |
| exidresid 36338 | The restriction of a binar... |
| ablo4pnp 36339 | A commutative/associative ... |
| grpoeqdivid 36340 | Two group elements are equ... |
| grposnOLD 36341 | The group operation for th... |
| elghomlem1OLD 36344 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 36345 | Obsolete as of 15-Mar-2020... |
| elghomOLD 36346 | Obsolete version of ~ isgh... |
| ghomlinOLD 36347 | Obsolete version of ~ ghml... |
| ghomidOLD 36348 | Obsolete version of ~ ghmi... |
| ghomf 36349 | Mapping property of a grou... |
| ghomco 36350 | The composition of two gro... |
| ghomdiv 36351 | Group homomorphisms preser... |
| grpokerinj 36352 | A group homomorphism is in... |
| relrngo 36355 | The class of all unital ri... |
| isrngo 36356 | The predicate "is a (unita... |
| isrngod 36357 | Conditions that determine ... |
| rngoi 36358 | The properties of a unital... |
| rngosm 36359 | Functionality of the multi... |
| rngocl 36360 | Closure of the multiplicat... |
| rngoid 36361 | The multiplication operati... |
| rngoideu 36362 | The unity element of a rin... |
| rngodi 36363 | Distributive law for the m... |
| rngodir 36364 | Distributive law for the m... |
| rngoass 36365 | Associative law for the mu... |
| rngo2 36366 | A ring element plus itself... |
| rngoablo 36367 | A ring's addition operatio... |
| rngoablo2 36368 | In a unital ring the addit... |
| rngogrpo 36369 | A ring's addition operatio... |
| rngone0 36370 | The base set of a ring is ... |
| rngogcl 36371 | Closure law for the additi... |
| rngocom 36372 | The addition operation of ... |
| rngoaass 36373 | The addition operation of ... |
| rngoa32 36374 | The addition operation of ... |
| rngoa4 36375 | Rearrangement of 4 terms i... |
| rngorcan 36376 | Right cancellation law for... |
| rngolcan 36377 | Left cancellation law for ... |
| rngo0cl 36378 | A ring has an additive ide... |
| rngo0rid 36379 | The additive identity of a... |
| rngo0lid 36380 | The additive identity of a... |
| rngolz 36381 | The zero of a unital ring ... |
| rngorz 36382 | The zero of a unital ring ... |
| rngosn3 36383 | Obsolete as of 25-Jan-2020... |
| rngosn4 36384 | Obsolete as of 25-Jan-2020... |
| rngosn6 36385 | Obsolete as of 25-Jan-2020... |
| rngonegcl 36386 | A ring is closed under neg... |
| rngoaddneg1 36387 | Adding the negative in a r... |
| rngoaddneg2 36388 | Adding the negative in a r... |
| rngosub 36389 | Subtraction in a ring, in ... |
| rngmgmbs4 36390 | The range of an internal o... |
| rngodm1dm2 36391 | In a unital ring the domai... |
| rngorn1 36392 | In a unital ring the range... |
| rngorn1eq 36393 | In a unital ring the range... |
| rngomndo 36394 | In a unital ring the multi... |
| rngoidmlem 36395 | The unity element of a rin... |
| rngolidm 36396 | The unity element of a rin... |
| rngoridm 36397 | The unity element of a rin... |
| rngo1cl 36398 | The unity element of a rin... |
| rngoueqz 36399 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 36400 | Negation in a ring is the ... |
| rngonegmn1r 36401 | Negation in a ring is the ... |
| rngoneglmul 36402 | Negation of a product in a... |
| rngonegrmul 36403 | Negation of a product in a... |
| rngosubdi 36404 | Ring multiplication distri... |
| rngosubdir 36405 | Ring multiplication distri... |
| zerdivemp1x 36406 | In a unital ring a left in... |
| isdivrngo 36409 | The predicate "is a divisi... |
| drngoi 36410 | The properties of a divisi... |
| gidsn 36411 | Obsolete as of 23-Jan-2020... |
| zrdivrng 36412 | The zero ring is not a div... |
| dvrunz 36413 | In a division ring the rin... |
| isgrpda 36414 | Properties that determine ... |
| isdrngo1 36415 | The predicate "is a divisi... |
| divrngcl 36416 | The product of two nonzero... |
| isdrngo2 36417 | A division ring is a ring ... |
| isdrngo3 36418 | A division ring is a ring ... |
| rngohomval 36423 | The set of ring homomorphi... |
| isrngohom 36424 | The predicate "is a ring h... |
| rngohomf 36425 | A ring homomorphism is a f... |
| rngohomcl 36426 | Closure law for a ring hom... |
| rngohom1 36427 | A ring homomorphism preser... |
| rngohomadd 36428 | Ring homomorphisms preserv... |
| rngohommul 36429 | Ring homomorphisms preserv... |
| rngogrphom 36430 | A ring homomorphism is a g... |
| rngohom0 36431 | A ring homomorphism preser... |
| rngohomsub 36432 | Ring homomorphisms preserv... |
| rngohomco 36433 | The composition of two rin... |
| rngokerinj 36434 | A ring homomorphism is inj... |
| rngoisoval 36436 | The set of ring isomorphis... |
| isrngoiso 36437 | The predicate "is a ring i... |
| rngoiso1o 36438 | A ring isomorphism is a bi... |
| rngoisohom 36439 | A ring isomorphism is a ri... |
| rngoisocnv 36440 | The inverse of a ring isom... |
| rngoisoco 36441 | The composition of two rin... |
| isriscg 36443 | The ring isomorphism relat... |
| isrisc 36444 | The ring isomorphism relat... |
| risc 36445 | The ring isomorphism relat... |
| risci 36446 | Determine that two rings a... |
| riscer 36447 | Ring isomorphism is an equ... |
| iscom2 36454 | A device to add commutativ... |
| iscrngo 36455 | The predicate "is a commut... |
| iscrngo2 36456 | The predicate "is a commut... |
| iscringd 36457 | Conditions that determine ... |
| flddivrng 36458 | A field is a division ring... |
| crngorngo 36459 | A commutative ring is a ri... |
| crngocom 36460 | The multiplication operati... |
| crngm23 36461 | Commutative/associative la... |
| crngm4 36462 | Commutative/associative la... |
| fldcrngo 36463 | A field is a commutative r... |
| isfld2 36464 | The predicate "is a field"... |
| crngohomfo 36465 | The image of a homomorphis... |
| idlval 36472 | The class of ideals of a r... |
| isidl 36473 | The predicate "is an ideal... |
| isidlc 36474 | The predicate "is an ideal... |
| idlss 36475 | An ideal of ` R ` is a sub... |
| idlcl 36476 | An element of an ideal is ... |
| idl0cl 36477 | An ideal contains ` 0 ` . ... |
| idladdcl 36478 | An ideal is closed under a... |
| idllmulcl 36479 | An ideal is closed under m... |
| idlrmulcl 36480 | An ideal is closed under m... |
| idlnegcl 36481 | An ideal is closed under n... |
| idlsubcl 36482 | An ideal is closed under s... |
| rngoidl 36483 | A ring ` R ` is an ` R ` i... |
| 0idl 36484 | The set containing only ` ... |
| 1idl 36485 | Two ways of expressing the... |
| 0rngo 36486 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 36487 | The only ideals in a divis... |
| intidl 36488 | The intersection of a none... |
| inidl 36489 | The intersection of two id... |
| unichnidl 36490 | The union of a nonempty ch... |
| keridl 36491 | The kernel of a ring homom... |
| pridlval 36492 | The class of prime ideals ... |
| ispridl 36493 | The predicate "is a prime ... |
| pridlidl 36494 | A prime ideal is an ideal.... |
| pridlnr 36495 | A prime ideal is a proper ... |
| pridl 36496 | The main property of a pri... |
| ispridl2 36497 | A condition that shows an ... |
| maxidlval 36498 | The set of maximal ideals ... |
| ismaxidl 36499 | The predicate "is a maxima... |
| maxidlidl 36500 | A maximal ideal is an idea... |
| maxidlnr 36501 | A maximal ideal is proper.... |
| maxidlmax 36502 | A maximal ideal is a maxim... |
| maxidln1 36503 | One is not contained in an... |
| maxidln0 36504 | A ring with a maximal idea... |
| isprrngo 36509 | The predicate "is a prime ... |
| prrngorngo 36510 | A prime ring is a ring. (... |
| smprngopr 36511 | A simple ring (one whose o... |
| divrngpr 36512 | A division ring is a prime... |
| isdmn 36513 | The predicate "is a domain... |
| isdmn2 36514 | The predicate "is a domain... |
| dmncrng 36515 | A domain is a commutative ... |
| dmnrngo 36516 | A domain is a ring. (Cont... |
| flddmn 36517 | A field is a domain. (Con... |
| igenval 36520 | The ideal generated by a s... |
| igenss 36521 | A set is a subset of the i... |
| igenidl 36522 | The ideal generated by a s... |
| igenmin 36523 | The ideal generated by a s... |
| igenidl2 36524 | The ideal generated by an ... |
| igenval2 36525 | The ideal generated by a s... |
| prnc 36526 | A principal ideal (an idea... |
| isfldidl 36527 | Determine if a ring is a f... |
| isfldidl2 36528 | Determine if a ring is a f... |
| ispridlc 36529 | The predicate "is a prime ... |
| pridlc 36530 | Property of a prime ideal ... |
| pridlc2 36531 | Property of a prime ideal ... |
| pridlc3 36532 | Property of a prime ideal ... |
| isdmn3 36533 | The predicate "is a domain... |
| dmnnzd 36534 | A domain has no zero-divis... |
| dmncan1 36535 | Cancellation law for domai... |
| dmncan2 36536 | Cancellation law for domai... |
| efald2 36537 | A proof by contradiction. ... |
| notbinot1 36538 | Simplification rule of neg... |
| bicontr 36539 | Biconditional of its own n... |
| impor 36540 | An equivalent formula for ... |
| orfa 36541 | The falsum ` F. ` can be r... |
| notbinot2 36542 | Commutation rule between n... |
| biimpor 36543 | A rewriting rule for bicon... |
| orfa1 36544 | Add a contradicting disjun... |
| orfa2 36545 | Remove a contradicting dis... |
| bifald 36546 | Infer the equivalence to a... |
| orsild 36547 | A lemma for not-or-not eli... |
| orsird 36548 | A lemma for not-or-not eli... |
| cnf1dd 36549 | A lemma for Conjunctive No... |
| cnf2dd 36550 | A lemma for Conjunctive No... |
| cnfn1dd 36551 | A lemma for Conjunctive No... |
| cnfn2dd 36552 | A lemma for Conjunctive No... |
| or32dd 36553 | A rearrangement of disjunc... |
| notornotel1 36554 | A lemma for not-or-not eli... |
| notornotel2 36555 | A lemma for not-or-not eli... |
| contrd 36556 | A proof by contradiction, ... |
| an12i 36557 | An inference from commutin... |
| exmid2 36558 | An excluded middle law. (... |
| selconj 36559 | An inference for selecting... |
| truconj 36560 | Add true as a conjunct. (... |
| orel 36561 | An inference for disjuncti... |
| negel 36562 | An inference for negation ... |
| botel 36563 | An inference for bottom el... |
| tradd 36564 | Add top ad a conjunct. (C... |
| gm-sbtru 36565 | Substitution does not chan... |
| sbfal 36566 | Substitution does not chan... |
| sbcani 36567 | Distribution of class subs... |
| sbcori 36568 | Distribution of class subs... |
| sbcimi 36569 | Distribution of class subs... |
| sbcni 36570 | Move class substitution in... |
| sbali 36571 | Discard class substitution... |
| sbexi 36572 | Discard class substitution... |
| sbcalf 36573 | Move universal quantifier ... |
| sbcexf 36574 | Move existential quantifie... |
| sbcalfi 36575 | Move universal quantifier ... |
| sbcexfi 36576 | Move existential quantifie... |
| spsbcdi 36577 | A lemma for eliminating a ... |
| alrimii 36578 | A lemma for introducing a ... |
| spesbcdi 36579 | A lemma for introducing an... |
| exlimddvf 36580 | A lemma for eliminating an... |
| exlimddvfi 36581 | A lemma for eliminating an... |
| sbceq1ddi 36582 | A lemma for eliminating in... |
| sbccom2lem 36583 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 36584 | Commutative law for double... |
| sbccom2f 36585 | Commutative law for double... |
| sbccom2fi 36586 | Commutative law for double... |
| csbcom2fi 36587 | Commutative law for double... |
| fald 36588 | Refutation of falsity, in ... |
| tsim1 36589 | A Tseitin axiom for logica... |
| tsim2 36590 | A Tseitin axiom for logica... |
| tsim3 36591 | A Tseitin axiom for logica... |
| tsbi1 36592 | A Tseitin axiom for logica... |
| tsbi2 36593 | A Tseitin axiom for logica... |
| tsbi3 36594 | A Tseitin axiom for logica... |
| tsbi4 36595 | A Tseitin axiom for logica... |
| tsxo1 36596 | A Tseitin axiom for logica... |
| tsxo2 36597 | A Tseitin axiom for logica... |
| tsxo3 36598 | A Tseitin axiom for logica... |
| tsxo4 36599 | A Tseitin axiom for logica... |
| tsan1 36600 | A Tseitin axiom for logica... |
| tsan2 36601 | A Tseitin axiom for logica... |
| tsan3 36602 | A Tseitin axiom for logica... |
| tsna1 36603 | A Tseitin axiom for logica... |
| tsna2 36604 | A Tseitin axiom for logica... |
| tsna3 36605 | A Tseitin axiom for logica... |
| tsor1 36606 | A Tseitin axiom for logica... |
| tsor2 36607 | A Tseitin axiom for logica... |
| tsor3 36608 | A Tseitin axiom for logica... |
| ts3an1 36609 | A Tseitin axiom for triple... |
| ts3an2 36610 | A Tseitin axiom for triple... |
| ts3an3 36611 | A Tseitin axiom for triple... |
| ts3or1 36612 | A Tseitin axiom for triple... |
| ts3or2 36613 | A Tseitin axiom for triple... |
| ts3or3 36614 | A Tseitin axiom for triple... |
| iuneq2f 36615 | Equality deduction for ind... |
| rabeq12f 36616 | Equality deduction for res... |
| csbeq12 36617 | Equality deduction for sub... |
| sbeqi 36618 | Equality deduction for sub... |
| ralbi12f 36619 | Equality deduction for res... |
| oprabbi 36620 | Equality deduction for cla... |
| mpobi123f 36621 | Equality deduction for map... |
| iuneq12f 36622 | Equality deduction for ind... |
| iineq12f 36623 | Equality deduction for ind... |
| opabbi 36624 | Equality deduction for cla... |
| mptbi12f 36625 | Equality deduction for map... |
| orcomdd 36626 | Commutativity of logic dis... |
| scottexf 36627 | A version of ~ scottex wit... |
| scott0f 36628 | A version of ~ scott0 with... |
| scottn0f 36629 | A version of ~ scott0f wit... |
| ac6s3f 36630 | Generalization of the Axio... |
| ac6s6 36631 | Generalization of the Axio... |
| ac6s6f 36632 | Generalization of the Axio... |
| el2v1 36676 | New way ( ~ elv , and the ... |
| el3v 36677 | New way ( ~ elv , and the ... |
| el3v1 36678 | New way ( ~ elv , and the ... |
| el3v2 36679 | New way ( ~ elv , and the ... |
| el3v3 36680 | New way ( ~ elv , and the ... |
| el3v12 36681 | New way ( ~ elv , and the ... |
| el3v13 36682 | New way ( ~ elv , and the ... |
| el3v23 36683 | New way ( ~ elv , and the ... |
| anan 36684 | Multiple commutations in c... |
| triantru3 36685 | A wff is equivalent to its... |
| bianbi 36686 | Exchanging conjunction in ... |
| bianim 36687 | Exchanging conjunction in ... |
| biorfd 36688 | A wff is equivalent to its... |
| eqbrtr 36689 | Substitution of equal clas... |
| eqbrb 36690 | Substitution of equal clas... |
| eqeltr 36691 | Substitution of equal clas... |
| eqelb 36692 | Substitution of equal clas... |
| eqeqan2d 36693 | Implication of introducing... |
| suceqsneq 36694 | One-to-one relationship be... |
| sucdifsn2 36695 | Absorption of union with a... |
| sucdifsn 36696 | The difference between the... |
| disjresin 36697 | The restriction to a disjo... |
| disjresdisj 36698 | The intersection of restri... |
| disjresdif 36699 | The difference between res... |
| disjresundif 36700 | Lemma for ~ ressucdifsn2 .... |
| ressucdifsn2 36701 | The difference between res... |
| ressucdifsn 36702 | The difference between res... |
| inres2 36703 | Two ways of expressing the... |
| coideq 36704 | Equality theorem for compo... |
| nexmo1 36705 | If there is no case where ... |
| ralin 36706 | Restricted universal quant... |
| r2alan 36707 | Double restricted universa... |
| 3ralbii 36708 | Inference adding three res... |
| ssrabi 36709 | Inference of restricted ab... |
| rabbieq 36710 | Equivalent wff's correspon... |
| rabimbieq 36711 | Restricted equivalent wff'... |
| abeqin 36712 | Intersection with class ab... |
| abeqinbi 36713 | Intersection with class ab... |
| rabeqel 36714 | Class element of a restric... |
| eqrelf 36715 | The equality connective be... |
| br1cnvinxp 36716 | Binary relation on the con... |
| releleccnv 36717 | Elementhood in a converse ... |
| releccnveq 36718 | Equality of converse ` R `... |
| opelvvdif 36719 | Negated elementhood of ord... |
| vvdifopab 36720 | Ordered-pair class abstrac... |
| brvdif 36721 | Binary relation with unive... |
| brvdif2 36722 | Binary relation with unive... |
| brvvdif 36723 | Binary relation with the c... |
| brvbrvvdif 36724 | Binary relation with the c... |
| brcnvep 36725 | The converse of the binary... |
| elecALTV 36726 | Elementhood in the ` R ` -... |
| brcnvepres 36727 | Restricted converse epsilo... |
| brres2 36728 | Binary relation on a restr... |
| br1cnvres 36729 | Binary relation on the con... |
| eldmres 36730 | Elementhood in the domain ... |
| elrnres 36731 | Element of the range of a ... |
| eldmressnALTV 36732 | Element of the domain of a... |
| elrnressn 36733 | Element of the range of a ... |
| eldm4 36734 | Elementhood in a domain. ... |
| eldmres2 36735 | Elementhood in the domain ... |
| eceq1i 36736 | Equality theorem for ` C `... |
| elecres 36737 | Elementhood in the restric... |
| ecres 36738 | Restricted coset of ` B ` ... |
| ecres2 36739 | The restricted coset of ` ... |
| eccnvepres 36740 | Restricted converse epsilo... |
| eleccnvep 36741 | Elementhood in the convers... |
| eccnvep 36742 | The converse epsilon coset... |
| extep 36743 | Property of epsilon relati... |
| disjeccnvep 36744 | Property of the epsilon re... |
| eccnvepres2 36745 | The restricted converse ep... |
| eccnvepres3 36746 | Condition for a restricted... |
| eldmqsres 36747 | Elementhood in a restricte... |
| eldmqsres2 36748 | Elementhood in a restricte... |
| qsss1 36749 | Subclass theorem for quoti... |
| qseq1i 36750 | Equality theorem for quoti... |
| qseq1d 36751 | Equality theorem for quoti... |
| brinxprnres 36752 | Binary relation on a restr... |
| inxprnres 36753 | Restriction of a class as ... |
| dfres4 36754 | Alternate definition of th... |
| exan3 36755 | Equivalent expressions wit... |
| exanres 36756 | Equivalent expressions wit... |
| exanres3 36757 | Equivalent expressions wit... |
| exanres2 36758 | Equivalent expressions wit... |
| cnvepres 36759 | Restricted converse epsilo... |
| eqrel2 36760 | Equality of relations. (C... |
| rncnv 36761 | Range of converse is the d... |
| dfdm6 36762 | Alternate definition of do... |
| dfrn6 36763 | Alternate definition of ra... |
| rncnvepres 36764 | The range of the restricte... |
| dmecd 36765 | Equality of the coset of `... |
| dmec2d 36766 | Equality of the coset of `... |
| brid 36767 | Property of the identity b... |
| ideq2 36768 | For sets, the identity bin... |
| idresssidinxp 36769 | Condition for the identity... |
| idreseqidinxp 36770 | Condition for the identity... |
| extid 36771 | Property of identity relat... |
| inxpss 36772 | Two ways to say that an in... |
| idinxpss 36773 | Two ways to say that an in... |
| ref5 36774 | Two ways to say that an in... |
| inxpss3 36775 | Two ways to say that an in... |
| inxpss2 36776 | Two ways to say that inter... |
| inxpssidinxp 36777 | Two ways to say that inter... |
| idinxpssinxp 36778 | Two ways to say that inter... |
| idinxpssinxp2 36779 | Identity intersection with... |
| idinxpssinxp3 36780 | Identity intersection with... |
| idinxpssinxp4 36781 | Identity intersection with... |
| relcnveq3 36782 | Two ways of saying a relat... |
| relcnveq 36783 | Two ways of saying a relat... |
| relcnveq2 36784 | Two ways of saying a relat... |
| relcnveq4 36785 | Two ways of saying a relat... |
| qsresid 36786 | Simplification of a specia... |
| n0elqs 36787 | Two ways of expressing tha... |
| n0elqs2 36788 | Two ways of expressing tha... |
| ecex2 36789 | Condition for a coset to b... |
| uniqsALTV 36790 | The union of a quotient se... |
| imaexALTV 36791 | Existence of an image of a... |
| ecexALTV 36792 | Existence of a coset, like... |
| rnresequniqs 36793 | The range of a restriction... |
| n0el2 36794 | Two ways of expressing tha... |
| cnvepresex 36795 | Sethood condition for the ... |
| eccnvepex 36796 | The converse epsilon coset... |
| cnvepimaex 36797 | The image of converse epsi... |
| cnvepima 36798 | The image of converse epsi... |
| inex3 36799 | Sufficient condition for t... |
| inxpex 36800 | Sufficient condition for a... |
| eqres 36801 | Converting a class constan... |
| brrabga 36802 | The law of concretion for ... |
| brcnvrabga 36803 | The law of concretion for ... |
| opideq 36804 | Equality conditions for or... |
| iss2 36805 | A subclass of the identity... |
| eldmcnv 36806 | Elementhood in a domain of... |
| dfrel5 36807 | Alternate definition of th... |
| dfrel6 36808 | Alternate definition of th... |
| cnvresrn 36809 | Converse restricted to ran... |
| relssinxpdmrn 36810 | Subset of restriction, spe... |
| cnvref4 36811 | Two ways to say that a rel... |
| cnvref5 36812 | Two ways to say that a rel... |
| ecin0 36813 | Two ways of saying that th... |
| ecinn0 36814 | Two ways of saying that th... |
| ineleq 36815 | Equivalence of restricted ... |
| inecmo 36816 | Equivalence of a double re... |
| inecmo2 36817 | Equivalence of a double re... |
| ineccnvmo 36818 | Equivalence of a double re... |
| alrmomorn 36819 | Equivalence of an "at most... |
| alrmomodm 36820 | Equivalence of an "at most... |
| ineccnvmo2 36821 | Equivalence of a double un... |
| inecmo3 36822 | Equivalence of a double un... |
| moeu2 36823 | Uniqueness is equivalent t... |
| mopickr 36824 | "At most one" picks a vari... |
| moantr 36825 | Sufficient condition for t... |
| brabidgaw 36826 | The law of concretion for ... |
| brabidga 36827 | The law of concretion for ... |
| inxp2 36828 | Intersection with a Cartes... |
| opabf 36829 | A class abstraction of a c... |
| ec0 36830 | The empty-coset of a class... |
| 0qs 36831 | Quotient set with the empt... |
| brcnvin 36832 | Intersection with a conver... |
| xrnss3v 36834 | A range Cartesian product ... |
| xrnrel 36835 | A range Cartesian product ... |
| brxrn 36836 | Characterize a ternary rel... |
| brxrn2 36837 | A characterization of the ... |
| dfxrn2 36838 | Alternate definition of th... |
| xrneq1 36839 | Equality theorem for the r... |
| xrneq1i 36840 | Equality theorem for the r... |
| xrneq1d 36841 | Equality theorem for the r... |
| xrneq2 36842 | Equality theorem for the r... |
| xrneq2i 36843 | Equality theorem for the r... |
| xrneq2d 36844 | Equality theorem for the r... |
| xrneq12 36845 | Equality theorem for the r... |
| xrneq12i 36846 | Equality theorem for the r... |
| xrneq12d 36847 | Equality theorem for the r... |
| elecxrn 36848 | Elementhood in the ` ( R |... |
| ecxrn 36849 | The ` ( R |X. S ) ` -coset... |
| disjressuc2 36850 | Double restricted quantifi... |
| disjecxrn 36851 | Two ways of saying that ` ... |
| disjecxrncnvep 36852 | Two ways of saying that co... |
| disjsuc2 36853 | Double restricted quantifi... |
| xrninxp 36854 | Intersection of a range Ca... |
| xrninxp2 36855 | Intersection of a range Ca... |
| xrninxpex 36856 | Sufficient condition for t... |
| inxpxrn 36857 | Two ways to express the in... |
| br1cnvxrn2 36858 | The converse of a binary r... |
| elec1cnvxrn2 36859 | Elementhood in the convers... |
| rnxrn 36860 | Range of the range Cartesi... |
| rnxrnres 36861 | Range of a range Cartesian... |
| rnxrncnvepres 36862 | Range of a range Cartesian... |
| rnxrnidres 36863 | Range of a range Cartesian... |
| xrnres 36864 | Two ways to express restri... |
| xrnres2 36865 | Two ways to express restri... |
| xrnres3 36866 | Two ways to express restri... |
| xrnres4 36867 | Two ways to express restri... |
| xrnresex 36868 | Sufficient condition for a... |
| xrnidresex 36869 | Sufficient condition for a... |
| xrncnvepresex 36870 | Sufficient condition for a... |
| brin2 36871 | Binary relation on an inte... |
| brin3 36872 | Binary relation on an inte... |
| dfcoss2 36875 | Alternate definition of th... |
| dfcoss3 36876 | Alternate definition of th... |
| dfcoss4 36877 | Alternate definition of th... |
| cosscnv 36878 | Class of cosets by the con... |
| coss1cnvres 36879 | Class of cosets by the con... |
| coss2cnvepres 36880 | Special case of ~ coss1cnv... |
| cossex 36881 | If ` A ` is a set then the... |
| cosscnvex 36882 | If ` A ` is a set then the... |
| 1cosscnvepresex 36883 | Sufficient condition for a... |
| 1cossxrncnvepresex 36884 | Sufficient condition for a... |
| relcoss 36885 | Cosets by ` R ` is a relat... |
| relcoels 36886 | Coelements on ` A ` is a r... |
| cossss 36887 | Subclass theorem for the c... |
| cosseq 36888 | Equality theorem for the c... |
| cosseqi 36889 | Equality theorem for the c... |
| cosseqd 36890 | Equality theorem for the c... |
| 1cossres 36891 | The class of cosets by a r... |
| dfcoels 36892 | Alternate definition of th... |
| brcoss 36893 | ` A ` and ` B ` are cosets... |
| brcoss2 36894 | Alternate form of the ` A ... |
| brcoss3 36895 | Alternate form of the ` A ... |
| brcosscnvcoss 36896 | For sets, the ` A ` and ` ... |
| brcoels 36897 | ` B ` and ` C ` are coelem... |
| cocossss 36898 | Two ways of saying that co... |
| cnvcosseq 36899 | The converse of cosets by ... |
| br2coss 36900 | Cosets by ` ,~ R ` binary ... |
| br1cossres 36901 | ` B ` and ` C ` are cosets... |
| br1cossres2 36902 | ` B ` and ` C ` are cosets... |
| brressn 36903 | Binary relation on a restr... |
| ressn2 36904 | A class ' R ' restricted t... |
| refressn 36905 | Any class ' R ' restricted... |
| antisymressn 36906 | Every class ' R ' restrict... |
| trressn 36907 | Any class ' R ' restricted... |
| relbrcoss 36908 | ` A ` and ` B ` are cosets... |
| br1cossinres 36909 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 36910 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 36911 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 36912 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 36913 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 36914 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 36915 | The domain of cosets is th... |
| dmcoss2 36916 | The domain of cosets is th... |
| rncossdmcoss 36917 | The range of cosets is the... |
| dm1cosscnvepres 36918 | The domain of cosets of th... |
| dmcoels 36919 | The domain of coelements i... |
| eldmcoss 36920 | Elementhood in the domain ... |
| eldmcoss2 36921 | Elementhood in the domain ... |
| eldm1cossres 36922 | Elementhood in the domain ... |
| eldm1cossres2 36923 | Elementhood in the domain ... |
| refrelcosslem 36924 | Lemma for the left side of... |
| refrelcoss3 36925 | The class of cosets by ` R... |
| refrelcoss2 36926 | The class of cosets by ` R... |
| symrelcoss3 36927 | The class of cosets by ` R... |
| symrelcoss2 36928 | The class of cosets by ` R... |
| cossssid 36929 | Equivalent expressions for... |
| cossssid2 36930 | Equivalent expressions for... |
| cossssid3 36931 | Equivalent expressions for... |
| cossssid4 36932 | Equivalent expressions for... |
| cossssid5 36933 | Equivalent expressions for... |
| brcosscnv 36934 | ` A ` and ` B ` are cosets... |
| brcosscnv2 36935 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 36936 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 36937 | Cosets by the converse ran... |
| cosscnvssid3 36938 | Equivalent expressions for... |
| cosscnvssid4 36939 | Equivalent expressions for... |
| cosscnvssid5 36940 | Equivalent expressions for... |
| coss0 36941 | Cosets by the empty set ar... |
| cossid 36942 | Cosets by the identity rel... |
| cosscnvid 36943 | Cosets by the converse ide... |
| trcoss 36944 | Sufficient condition for t... |
| eleccossin 36945 | Two ways of saying that th... |
| trcoss2 36946 | Equivalent expressions for... |
| elrels2 36948 | The element of the relatio... |
| elrelsrel 36949 | The element of the relatio... |
| elrelsrelim 36950 | The element of the relatio... |
| elrels5 36951 | Equivalent expressions for... |
| elrels6 36952 | Equivalent expressions for... |
| elrelscnveq3 36953 | Two ways of saying a relat... |
| elrelscnveq 36954 | Two ways of saying a relat... |
| elrelscnveq2 36955 | Two ways of saying a relat... |
| elrelscnveq4 36956 | Two ways of saying a relat... |
| cnvelrels 36957 | The converse of a set is a... |
| cosselrels 36958 | Cosets of sets are element... |
| cosscnvelrels 36959 | Cosets of converse sets ar... |
| dfssr2 36961 | Alternate definition of th... |
| relssr 36962 | The subset relation is a r... |
| brssr 36963 | The subset relation and su... |
| brssrid 36964 | Any set is a subset of its... |
| issetssr 36965 | Two ways of expressing set... |
| brssrres 36966 | Restricted subset binary r... |
| br1cnvssrres 36967 | Restricted converse subset... |
| brcnvssr 36968 | The converse of a subset r... |
| brcnvssrid 36969 | Any set is a converse subs... |
| br1cossxrncnvssrres 36970 | ` <. B , C >. ` and ` <. D... |
| extssr 36971 | Property of subset relatio... |
| dfrefrels2 36975 | Alternate definition of th... |
| dfrefrels3 36976 | Alternate definition of th... |
| dfrefrel2 36977 | Alternate definition of th... |
| dfrefrel3 36978 | Alternate definition of th... |
| dfrefrel5 36979 | Alternate definition of th... |
| elrefrels2 36980 | Element of the class of re... |
| elrefrels3 36981 | Element of the class of re... |
| elrefrelsrel 36982 | For sets, being an element... |
| refreleq 36983 | Equality theorem for refle... |
| refrelid 36984 | Identity relation is refle... |
| refrelcoss 36985 | The class of cosets by ` R... |
| refrelressn 36986 | Any class ' R ' restricted... |
| dfcnvrefrels2 36990 | Alternate definition of th... |
| dfcnvrefrels3 36991 | Alternate definition of th... |
| dfcnvrefrel2 36992 | Alternate definition of th... |
| dfcnvrefrel3 36993 | Alternate definition of th... |
| dfcnvrefrel4 36994 | Alternate definition of th... |
| dfcnvrefrel5 36995 | Alternate definition of th... |
| elcnvrefrels2 36996 | Element of the class of co... |
| elcnvrefrels3 36997 | Element of the class of co... |
| elcnvrefrelsrel 36998 | For sets, being an element... |
| cnvrefrelcoss2 36999 | Necessary and sufficient c... |
| cosselcnvrefrels2 37000 | Necessary and sufficient c... |
| cosselcnvrefrels3 37001 | Necessary and sufficient c... |
| cosselcnvrefrels4 37002 | Necessary and sufficient c... |
| cosselcnvrefrels5 37003 | Necessary and sufficient c... |
| dfsymrels2 37007 | Alternate definition of th... |
| dfsymrels3 37008 | Alternate definition of th... |
| dfsymrels4 37009 | Alternate definition of th... |
| dfsymrels5 37010 | Alternate definition of th... |
| dfsymrel2 37011 | Alternate definition of th... |
| dfsymrel3 37012 | Alternate definition of th... |
| dfsymrel4 37013 | Alternate definition of th... |
| dfsymrel5 37014 | Alternate definition of th... |
| elsymrels2 37015 | Element of the class of sy... |
| elsymrels3 37016 | Element of the class of sy... |
| elsymrels4 37017 | Element of the class of sy... |
| elsymrels5 37018 | Element of the class of sy... |
| elsymrelsrel 37019 | For sets, being an element... |
| symreleq 37020 | Equality theorem for symme... |
| symrelim 37021 | Symmetric relation implies... |
| symrelcoss 37022 | The class of cosets by ` R... |
| idsymrel 37023 | The identity relation is s... |
| epnsymrel 37024 | The membership (epsilon) r... |
| symrefref2 37025 | Symmetry is a sufficient c... |
| symrefref3 37026 | Symmetry is a sufficient c... |
| refsymrels2 37027 | Elements of the class of r... |
| refsymrels3 37028 | Elements of the class of r... |
| refsymrel2 37029 | A relation which is reflex... |
| refsymrel3 37030 | A relation which is reflex... |
| elrefsymrels2 37031 | Elements of the class of r... |
| elrefsymrels3 37032 | Elements of the class of r... |
| elrefsymrelsrel 37033 | For sets, being an element... |
| dftrrels2 37037 | Alternate definition of th... |
| dftrrels3 37038 | Alternate definition of th... |
| dftrrel2 37039 | Alternate definition of th... |
| dftrrel3 37040 | Alternate definition of th... |
| eltrrels2 37041 | Element of the class of tr... |
| eltrrels3 37042 | Element of the class of tr... |
| eltrrelsrel 37043 | For sets, being an element... |
| trreleq 37044 | Equality theorem for the t... |
| trrelressn 37045 | Any class ' R ' restricted... |
| dfeqvrels2 37050 | Alternate definition of th... |
| dfeqvrels3 37051 | Alternate definition of th... |
| dfeqvrel2 37052 | Alternate definition of th... |
| dfeqvrel3 37053 | Alternate definition of th... |
| eleqvrels2 37054 | Element of the class of eq... |
| eleqvrels3 37055 | Element of the class of eq... |
| eleqvrelsrel 37056 | For sets, being an element... |
| elcoeleqvrels 37057 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 37058 | For sets, being an element... |
| eqvrelrel 37059 | An equivalence relation is... |
| eqvrelrefrel 37060 | An equivalence relation is... |
| eqvrelsymrel 37061 | An equivalence relation is... |
| eqvreltrrel 37062 | An equivalence relation is... |
| eqvrelim 37063 | Equivalence relation impli... |
| eqvreleq 37064 | Equality theorem for equiv... |
| eqvreleqi 37065 | Equality theorem for equiv... |
| eqvreleqd 37066 | Equality theorem for equiv... |
| eqvrelsym 37067 | An equivalence relation is... |
| eqvrelsymb 37068 | An equivalence relation is... |
| eqvreltr 37069 | An equivalence relation is... |
| eqvreltrd 37070 | A transitivity relation fo... |
| eqvreltr4d 37071 | A transitivity relation fo... |
| eqvrelref 37072 | An equivalence relation is... |
| eqvrelth 37073 | Basic property of equivale... |
| eqvrelcl 37074 | Elementhood in the field o... |
| eqvrelthi 37075 | Basic property of equivale... |
| eqvreldisj 37076 | Equivalence classes do not... |
| qsdisjALTV 37077 | Elements of a quotient set... |
| eqvrelqsel 37078 | If an element of a quotien... |
| eqvrelcoss 37079 | Two ways to express equiva... |
| eqvrelcoss3 37080 | Two ways to express equiva... |
| eqvrelcoss2 37081 | Two ways to express equiva... |
| eqvrelcoss4 37082 | Two ways to express equiva... |
| dfcoeleqvrels 37083 | Alternate definition of th... |
| dfcoeleqvrel 37084 | Alternate definition of th... |
| brredunds 37088 | Binary relation on the cla... |
| brredundsredund 37089 | For sets, binary relation ... |
| redundss3 37090 | Implication of redundancy ... |
| redundeq1 37091 | Equivalence of redundancy ... |
| redundpim3 37092 | Implication of redundancy ... |
| redundpbi1 37093 | Equivalence of redundancy ... |
| refrelsredund4 37094 | The naive version of the c... |
| refrelsredund2 37095 | The naive version of the c... |
| refrelsredund3 37096 | The naive version of the c... |
| refrelredund4 37097 | The naive version of the d... |
| refrelredund2 37098 | The naive version of the d... |
| refrelredund3 37099 | The naive version of the d... |
| dmqseq 37102 | Equality theorem for domai... |
| dmqseqi 37103 | Equality theorem for domai... |
| dmqseqd 37104 | Equality theorem for domai... |
| dmqseqeq1 37105 | Equality theorem for domai... |
| dmqseqeq1i 37106 | Equality theorem for domai... |
| dmqseqeq1d 37107 | Equality theorem for domai... |
| brdmqss 37108 | The domain quotient binary... |
| brdmqssqs 37109 | If ` A ` and ` R ` are set... |
| n0eldmqs 37110 | The empty set is not an el... |
| n0eldmqseq 37111 | The empty set is not an el... |
| n0elim 37112 | Implication of that the em... |
| n0el3 37113 | Two ways of expressing tha... |
| cnvepresdmqss 37114 | The domain quotient binary... |
| cnvepresdmqs 37115 | The domain quotient predic... |
| unidmqs 37116 | The range of a relation is... |
| unidmqseq 37117 | The union of the domain qu... |
| dmqseqim 37118 | If the domain quotient of ... |
| dmqseqim2 37119 | Lemma for ~ erimeq2 . (Co... |
| releldmqs 37120 | Elementhood in the domain ... |
| eldmqs1cossres 37121 | Elementhood in the domain ... |
| releldmqscoss 37122 | Elementhood in the domain ... |
| dmqscoelseq 37123 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 37124 | Two ways to express the eq... |
| brers 37129 | Binary equivalence relatio... |
| dferALTV2 37130 | Equivalence relation with ... |
| erALTVeq1 37131 | Equality theorem for equiv... |
| erALTVeq1i 37132 | Equality theorem for equiv... |
| erALTVeq1d 37133 | Equality theorem for equiv... |
| dfcomember 37134 | Alternate definition of th... |
| dfcomember2 37135 | Alternate definition of th... |
| dfcomember3 37136 | Alternate definition of th... |
| eqvreldmqs 37137 | Two ways to express comemb... |
| eqvreldmqs2 37138 | Two ways to express comemb... |
| brerser 37139 | Binary equivalence relatio... |
| erimeq2 37140 | Equivalence relation on it... |
| erimeq 37141 | Equivalence relation on it... |
| dffunsALTV 37145 | Alternate definition of th... |
| dffunsALTV2 37146 | Alternate definition of th... |
| dffunsALTV3 37147 | Alternate definition of th... |
| dffunsALTV4 37148 | Alternate definition of th... |
| dffunsALTV5 37149 | Alternate definition of th... |
| dffunALTV2 37150 | Alternate definition of th... |
| dffunALTV3 37151 | Alternate definition of th... |
| dffunALTV4 37152 | Alternate definition of th... |
| dffunALTV5 37153 | Alternate definition of th... |
| elfunsALTV 37154 | Elementhood in the class o... |
| elfunsALTV2 37155 | Elementhood in the class o... |
| elfunsALTV3 37156 | Elementhood in the class o... |
| elfunsALTV4 37157 | Elementhood in the class o... |
| elfunsALTV5 37158 | Elementhood in the class o... |
| elfunsALTVfunALTV 37159 | The element of the class o... |
| funALTVfun 37160 | Our definition of the func... |
| funALTVss 37161 | Subclass theorem for funct... |
| funALTVeq 37162 | Equality theorem for funct... |
| funALTVeqi 37163 | Equality inference for the... |
| funALTVeqd 37164 | Equality deduction for the... |
| dfdisjs 37170 | Alternate definition of th... |
| dfdisjs2 37171 | Alternate definition of th... |
| dfdisjs3 37172 | Alternate definition of th... |
| dfdisjs4 37173 | Alternate definition of th... |
| dfdisjs5 37174 | Alternate definition of th... |
| dfdisjALTV 37175 | Alternate definition of th... |
| dfdisjALTV2 37176 | Alternate definition of th... |
| dfdisjALTV3 37177 | Alternate definition of th... |
| dfdisjALTV4 37178 | Alternate definition of th... |
| dfdisjALTV5 37179 | Alternate definition of th... |
| dfeldisj2 37180 | Alternate definition of th... |
| dfeldisj3 37181 | Alternate definition of th... |
| dfeldisj4 37182 | Alternate definition of th... |
| dfeldisj5 37183 | Alternate definition of th... |
| eldisjs 37184 | Elementhood in the class o... |
| eldisjs2 37185 | Elementhood in the class o... |
| eldisjs3 37186 | Elementhood in the class o... |
| eldisjs4 37187 | Elementhood in the class o... |
| eldisjs5 37188 | Elementhood in the class o... |
| eldisjsdisj 37189 | The element of the class o... |
| eleldisjs 37190 | Elementhood in the disjoin... |
| eleldisjseldisj 37191 | The element of the disjoin... |
| disjrel 37192 | Disjoint relation is a rel... |
| disjss 37193 | Subclass theorem for disjo... |
| disjssi 37194 | Subclass theorem for disjo... |
| disjssd 37195 | Subclass theorem for disjo... |
| disjeq 37196 | Equality theorem for disjo... |
| disjeqi 37197 | Equality theorem for disjo... |
| disjeqd 37198 | Equality theorem for disjo... |
| disjdmqseqeq1 37199 | Lemma for the equality the... |
| eldisjss 37200 | Subclass theorem for disjo... |
| eldisjssi 37201 | Subclass theorem for disjo... |
| eldisjssd 37202 | Subclass theorem for disjo... |
| eldisjeq 37203 | Equality theorem for disjo... |
| eldisjeqi 37204 | Equality theorem for disjo... |
| eldisjeqd 37205 | Equality theorem for disjo... |
| disjres 37206 | Disjoint restriction. (Co... |
| eldisjn0elb 37207 | Two forms of disjoint elem... |
| disjxrn 37208 | Two ways of saying that a ... |
| disjxrnres5 37209 | Disjoint range Cartesian p... |
| disjorimxrn 37210 | Disjointness condition for... |
| disjimxrn 37211 | Disjointness condition for... |
| disjimres 37212 | Disjointness condition for... |
| disjimin 37213 | Disjointness condition for... |
| disjiminres 37214 | Disjointness condition for... |
| disjimxrnres 37215 | Disjointness condition for... |
| disjALTV0 37216 | The null class is disjoint... |
| disjALTVid 37217 | The class of identity rela... |
| disjALTVidres 37218 | The class of identity rela... |
| disjALTVinidres 37219 | The intersection with rest... |
| disjALTVxrnidres 37220 | The class of range Cartesi... |
| disjsuc 37221 | Disjoint range Cartesian p... |
| dfantisymrel4 37223 | Alternate definition of th... |
| dfantisymrel5 37224 | Alternate definition of th... |
| antisymrelres 37225 | (Contributed by Peter Mazs... |
| antisymrelressn 37226 | (Contributed by Peter Mazs... |
| dfpart2 37231 | Alternate definition of th... |
| dfmembpart2 37232 | Alternate definition of th... |
| brparts 37233 | Binary partitions relation... |
| brparts2 37234 | Binary partitions relation... |
| brpartspart 37235 | Binary partition and the p... |
| parteq1 37236 | Equality theorem for parti... |
| parteq2 37237 | Equality theorem for parti... |
| parteq12 37238 | Equality theorem for parti... |
| parteq1i 37239 | Equality theorem for parti... |
| parteq1d 37240 | Equality theorem for parti... |
| partsuc2 37241 | Property of the partition.... |
| partsuc 37242 | Property of the partition.... |
| disjim 37243 | The "Divide et Aequivalere... |
| disjimi 37244 | Every disjoint relation ge... |
| detlem 37245 | If a relation is disjoint,... |
| eldisjim 37246 | If the elements of ` A ` a... |
| eldisjim2 37247 | Alternate form of ~ eldisj... |
| eqvrel0 37248 | The null class is an equiv... |
| det0 37249 | The cosets by the null cla... |
| eqvrelcoss0 37250 | The cosets by the null cla... |
| eqvrelid 37251 | The identity relation is a... |
| eqvrel1cossidres 37252 | The cosets by a restricted... |
| eqvrel1cossinidres 37253 | The cosets by an intersect... |
| eqvrel1cossxrnidres 37254 | The cosets by a range Cart... |
| detid 37255 | The cosets by the identity... |
| eqvrelcossid 37256 | The cosets by the identity... |
| detidres 37257 | The cosets by the restrict... |
| detinidres 37258 | The cosets by the intersec... |
| detxrnidres 37259 | The cosets by the range Ca... |
| disjlem14 37260 | Lemma for ~ disjdmqseq , ~... |
| disjlem17 37261 | Lemma for ~ disjdmqseq , ~... |
| disjlem18 37262 | Lemma for ~ disjdmqseq , ~... |
| disjlem19 37263 | Lemma for ~ disjdmqseq , ~... |
| disjdmqsss 37264 | Lemma for ~ disjdmqseq via... |
| disjdmqscossss 37265 | Lemma for ~ disjdmqseq via... |
| disjdmqs 37266 | If a relation is disjoint,... |
| disjdmqseq 37267 | If a relation is disjoint,... |
| eldisjn0el 37268 | Special case of ~ disjdmqs... |
| partim2 37269 | Disjoint relation on its n... |
| partim 37270 | Partition implies equivale... |
| partimeq 37271 | Partition implies that the... |
| eldisjlem19 37272 | Special case of ~ disjlem1... |
| membpartlem19 37273 | Together with ~ disjlem19 ... |
| petlem 37274 | If you can prove that the ... |
| petlemi 37275 | If you can prove disjointn... |
| pet02 37276 | Class ` A ` is a partition... |
| pet0 37277 | Class ` A ` is a partition... |
| petid2 37278 | Class ` A ` is a partition... |
| petid 37279 | A class is a partition by ... |
| petidres2 37280 | Class ` A ` is a partition... |
| petidres 37281 | A class is a partition by ... |
| petinidres2 37282 | Class ` A ` is a partition... |
| petinidres 37283 | A class is a partition by ... |
| petxrnidres2 37284 | Class ` A ` is a partition... |
| petxrnidres 37285 | A class is a partition by ... |
| eqvreldisj1 37286 | The elements of the quotie... |
| eqvreldisj2 37287 | The elements of the quotie... |
| eqvreldisj3 37288 | The elements of the quotie... |
| eqvreldisj4 37289 | Intersection with the conv... |
| eqvreldisj5 37290 | Range Cartesian product wi... |
| eqvrelqseqdisj2 37291 | Implication of ~ eqvreldis... |
| fences3 37292 | Implication of ~ eqvrelqse... |
| eqvrelqseqdisj3 37293 | Implication of ~ eqvreldis... |
| eqvrelqseqdisj4 37294 | Lemma for ~ petincnvepres2... |
| eqvrelqseqdisj5 37295 | Lemma for the Partition-Eq... |
| mainer 37296 | The Main Theorem of Equiva... |
| partimcomember 37297 | Partition with general ` R... |
| mpet3 37298 | Member Partition-Equivalen... |
| cpet2 37299 | The conventional form of t... |
| cpet 37300 | The conventional form of M... |
| mpet 37301 | Member Partition-Equivalen... |
| mpet2 37302 | Member Partition-Equivalen... |
| mpets2 37303 | Member Partition-Equivalen... |
| mpets 37304 | Member Partition-Equivalen... |
| mainpart 37305 | Partition with general ` R... |
| fences 37306 | The Theorem of Fences by E... |
| fences2 37307 | The Theorem of Fences by E... |
| mainer2 37308 | The Main Theorem of Equiva... |
| mainerim 37309 | Every equivalence relation... |
| petincnvepres2 37310 | A partition-equivalence th... |
| petincnvepres 37311 | The shortest form of a par... |
| pet2 37312 | Partition-Equivalence Theo... |
| pet 37313 | Partition-Equivalence Theo... |
| pets 37314 | Partition-Equivalence Theo... |
| prtlem60 37315 | Lemma for ~ prter3 . (Con... |
| bicomdd 37316 | Commute two sides of a bic... |
| jca2r 37317 | Inference conjoining the c... |
| jca3 37318 | Inference conjoining the c... |
| prtlem70 37319 | Lemma for ~ prter3 : a rea... |
| ibdr 37320 | Reverse of ~ ibd . (Contr... |
| prtlem100 37321 | Lemma for ~ prter3 . (Con... |
| prtlem5 37322 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 37323 | Lemma for ~ prter2 . (Con... |
| brabsb2 37324 | A closed form of ~ brabsb ... |
| eqbrrdv2 37325 | Other version of ~ eqbrrdi... |
| prtlem9 37326 | Lemma for ~ prter3 . (Con... |
| prtlem10 37327 | Lemma for ~ prter3 . (Con... |
| prtlem11 37328 | Lemma for ~ prter2 . (Con... |
| prtlem12 37329 | Lemma for ~ prtex and ~ pr... |
| prtlem13 37330 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 37331 | Lemma for ~ prtex , ~ prte... |
| prtlem400 37332 | Lemma for ~ prter2 and als... |
| erprt 37335 | The quotient set of an equ... |
| prtlem14 37336 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 37337 | Lemma for ~ prter1 and ~ p... |
| prtlem17 37338 | Lemma for ~ prter2 . (Con... |
| prtlem18 37339 | Lemma for ~ prter2 . (Con... |
| prtlem19 37340 | Lemma for ~ prter2 . (Con... |
| prter1 37341 | Every partition generates ... |
| prtex 37342 | The equivalence relation g... |
| prter2 37343 | The quotient set of the eq... |
| prter3 37344 | For every partition there ... |
| axc5 37355 | This theorem repeats ~ sp ... |
| ax4fromc4 37356 | Rederivation of Axiom ~ ax... |
| ax10fromc7 37357 | Rederivation of Axiom ~ ax... |
| ax6fromc10 37358 | Rederivation of Axiom ~ ax... |
| hba1-o 37359 | The setvar ` x ` is not fr... |
| axc4i-o 37360 | Inference version of ~ ax-... |
| equid1 37361 | Proof of ~ equid from our ... |
| equcomi1 37362 | Proof of ~ equcomi from ~ ... |
| aecom-o 37363 | Commutation law for identi... |
| aecoms-o 37364 | A commutation rule for ide... |
| hbae-o 37365 | All variables are effectiv... |
| dral1-o 37366 | Formula-building lemma for... |
| ax12fromc15 37367 | Rederivation of Axiom ~ ax... |
| ax13fromc9 37368 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 37369 | Axiom to quantify a variab... |
| sps-o 37370 | Generalization of antecede... |
| hbequid 37371 | Bound-variable hypothesis ... |
| nfequid-o 37372 | Bound-variable hypothesis ... |
| axc5c7 37373 | Proof of a single axiom th... |
| axc5c7toc5 37374 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 37375 | Rederivation of ~ ax-c7 fr... |
| axc711 37376 | Proof of a single axiom th... |
| nfa1-o 37377 | ` x ` is not free in ` A. ... |
| axc711toc7 37378 | Rederivation of ~ ax-c7 fr... |
| axc711to11 37379 | Rederivation of ~ ax-11 fr... |
| axc5c711 37380 | Proof of a single axiom th... |
| axc5c711toc5 37381 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 37382 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 37383 | Rederivation of ~ ax-11 fr... |
| equidqe 37384 | ~ equid with existential q... |
| axc5sp1 37385 | A special case of ~ ax-c5 ... |
| equidq 37386 | ~ equid with universal qua... |
| equid1ALT 37387 | Alternate proof of ~ equid... |
| axc11nfromc11 37388 | Rederivation of ~ ax-c11n ... |
| naecoms-o 37389 | A commutation rule for dis... |
| hbnae-o 37390 | All variables are effectiv... |
| dvelimf-o 37391 | Proof of ~ dvelimh that us... |
| dral2-o 37392 | Formula-building lemma for... |
| aev-o 37393 | A "distinctor elimination"... |
| ax5eq 37394 | Theorem to add distinct qu... |
| dveeq2-o 37395 | Quantifier introduction wh... |
| axc16g-o 37396 | A generalization of Axiom ... |
| dveeq1-o 37397 | Quantifier introduction wh... |
| dveeq1-o16 37398 | Version of ~ dveeq1 using ... |
| ax5el 37399 | Theorem to add distinct qu... |
| axc11n-16 37400 | This theorem shows that, g... |
| dveel2ALT 37401 | Alternate proof of ~ dveel... |
| ax12f 37402 | Basis step for constructin... |
| ax12eq 37403 | Basis step for constructin... |
| ax12el 37404 | Basis step for constructin... |
| ax12indn 37405 | Induction step for constru... |
| ax12indi 37406 | Induction step for constru... |
| ax12indalem 37407 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 37408 | Alternate proof of ~ ax12i... |
| ax12inda2 37409 | Induction step for constru... |
| ax12inda 37410 | Induction step for constru... |
| ax12v2-o 37411 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 37412 | Derive ~ ax-c15 from a hyp... |
| axc11-o 37413 | Show that ~ ax-c11 can be ... |
| fsumshftd 37414 | Index shift of a finite su... |
| riotaclbgBAD 37416 | Closure of restricted iota... |
| riotaclbBAD 37417 | Closure of restricted iota... |
| riotasvd 37418 | Deduction version of ~ rio... |
| riotasv2d 37419 | Value of description binde... |
| riotasv2s 37420 | The value of description b... |
| riotasv 37421 | Value of description binde... |
| riotasv3d 37422 | A property ` ch ` holding ... |
| elimhyps 37423 | A version of ~ elimhyp usi... |
| dedths 37424 | A version of weak deductio... |
| renegclALT 37425 | Closure law for negative o... |
| elimhyps2 37426 | Generalization of ~ elimhy... |
| dedths2 37427 | Generalization of ~ dedths... |
| nfcxfrdf 37428 | A utility lemma to transfe... |
| nfded 37429 | A deduction theorem that c... |
| nfded2 37430 | A deduction theorem that c... |
| nfunidALT2 37431 | Deduction version of ~ nfu... |
| nfunidALT 37432 | Deduction version of ~ nfu... |
| nfopdALT 37433 | Deduction version of bound... |
| cnaddcom 37434 | Recover the commutative la... |
| toycom 37435 | Show the commutative law f... |
| lshpset 37440 | The set of all hyperplanes... |
| islshp 37441 | The predicate "is a hyperp... |
| islshpsm 37442 | Hyperplane properties expr... |
| lshplss 37443 | A hyperplane is a subspace... |
| lshpne 37444 | A hyperplane is not equal ... |
| lshpnel 37445 | A hyperplane's generating ... |
| lshpnelb 37446 | The subspace sum of a hype... |
| lshpnel2N 37447 | Condition that determines ... |
| lshpne0 37448 | The member of the span in ... |
| lshpdisj 37449 | A hyperplane and the span ... |
| lshpcmp 37450 | If two hyperplanes are com... |
| lshpinN 37451 | The intersection of two di... |
| lsatset 37452 | The set of all 1-dim subsp... |
| islsat 37453 | The predicate "is a 1-dim ... |
| lsatlspsn2 37454 | The span of a nonzero sing... |
| lsatlspsn 37455 | The span of a nonzero sing... |
| islsati 37456 | A 1-dim subspace (atom) (o... |
| lsateln0 37457 | A 1-dim subspace (atom) (o... |
| lsatlss 37458 | The set of 1-dim subspaces... |
| lsatlssel 37459 | An atom is a subspace. (C... |
| lsatssv 37460 | An atom is a set of vector... |
| lsatn0 37461 | A 1-dim subspace (atom) of... |
| lsatspn0 37462 | The span of a vector is an... |
| lsator0sp 37463 | The span of a vector is ei... |
| lsatssn0 37464 | A subspace (or any class) ... |
| lsatcmp 37465 | If two atoms are comparabl... |
| lsatcmp2 37466 | If an atom is included in ... |
| lsatel 37467 | A nonzero vector in an ato... |
| lsatelbN 37468 | A nonzero vector in an ato... |
| lsat2el 37469 | Two atoms sharing a nonzer... |
| lsmsat 37470 | Convert comparison of atom... |
| lsatfixedN 37471 | Show equality with the spa... |
| lsmsatcv 37472 | Subspace sum has the cover... |
| lssatomic 37473 | The lattice of subspaces i... |
| lssats 37474 | The lattice of subspaces i... |
| lpssat 37475 | Two subspaces in a proper ... |
| lrelat 37476 | Subspaces are relatively a... |
| lssatle 37477 | The ordering of two subspa... |
| lssat 37478 | Two subspaces in a proper ... |
| islshpat 37479 | Hyperplane properties expr... |
| lcvfbr 37482 | The covers relation for a ... |
| lcvbr 37483 | The covers relation for a ... |
| lcvbr2 37484 | The covers relation for a ... |
| lcvbr3 37485 | The covers relation for a ... |
| lcvpss 37486 | The covers relation implie... |
| lcvnbtwn 37487 | The covers relation implie... |
| lcvntr 37488 | The covers relation is not... |
| lcvnbtwn2 37489 | The covers relation implie... |
| lcvnbtwn3 37490 | The covers relation implie... |
| lsmcv2 37491 | Subspace sum has the cover... |
| lcvat 37492 | If a subspace covers anoth... |
| lsatcv0 37493 | An atom covers the zero su... |
| lsatcveq0 37494 | A subspace covered by an a... |
| lsat0cv 37495 | A subspace is an atom iff ... |
| lcvexchlem1 37496 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 37497 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 37498 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 37499 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 37500 | Lemma for ~ lcvexch . (Co... |
| lcvexch 37501 | Subspaces satisfy the exch... |
| lcvp 37502 | Covering property of Defin... |
| lcv1 37503 | Covering property of a sub... |
| lcv2 37504 | Covering property of a sub... |
| lsatexch 37505 | The atom exchange property... |
| lsatnle 37506 | The meet of a subspace and... |
| lsatnem0 37507 | The meet of distinct atoms... |
| lsatexch1 37508 | The atom exch1ange propert... |
| lsatcv0eq 37509 | If the sum of two atoms co... |
| lsatcv1 37510 | Two atoms covering the zer... |
| lsatcvatlem 37511 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 37512 | A nonzero subspace less th... |
| lsatcvat2 37513 | A subspace covered by the ... |
| lsatcvat3 37514 | A condition implying that ... |
| islshpcv 37515 | Hyperplane properties expr... |
| l1cvpat 37516 | A subspace covered by the ... |
| l1cvat 37517 | Create an atom under an el... |
| lshpat 37518 | Create an atom under a hyp... |
| lflset 37521 | The set of linear function... |
| islfl 37522 | The predicate "is a linear... |
| lfli 37523 | Property of a linear funct... |
| islfld 37524 | Properties that determine ... |
| lflf 37525 | A linear functional is a f... |
| lflcl 37526 | A linear functional value ... |
| lfl0 37527 | A linear functional is zer... |
| lfladd 37528 | Property of a linear funct... |
| lflsub 37529 | Property of a linear funct... |
| lflmul 37530 | Property of a linear funct... |
| lfl0f 37531 | The zero function is a fun... |
| lfl1 37532 | A nonzero functional has a... |
| lfladdcl 37533 | Closure of addition of two... |
| lfladdcom 37534 | Commutativity of functiona... |
| lfladdass 37535 | Associativity of functiona... |
| lfladd0l 37536 | Functional addition with t... |
| lflnegcl 37537 | Closure of the negative of... |
| lflnegl 37538 | A functional plus its nega... |
| lflvscl 37539 | Closure of a scalar produc... |
| lflvsdi1 37540 | Distributive law for (righ... |
| lflvsdi2 37541 | Reverse distributive law f... |
| lflvsdi2a 37542 | Reverse distributive law f... |
| lflvsass 37543 | Associative law for (right... |
| lfl0sc 37544 | The (right vector space) s... |
| lflsc0N 37545 | The scalar product with th... |
| lfl1sc 37546 | The (right vector space) s... |
| lkrfval 37549 | The kernel of a functional... |
| lkrval 37550 | Value of the kernel of a f... |
| ellkr 37551 | Membership in the kernel o... |
| lkrval2 37552 | Value of the kernel of a f... |
| ellkr2 37553 | Membership in the kernel o... |
| lkrcl 37554 | A member of the kernel of ... |
| lkrf0 37555 | The value of a functional ... |
| lkr0f 37556 | The kernel of the zero fun... |
| lkrlss 37557 | The kernel of a linear fun... |
| lkrssv 37558 | The kernel of a linear fun... |
| lkrsc 37559 | The kernel of a nonzero sc... |
| lkrscss 37560 | The kernel of a scalar pro... |
| eqlkr 37561 | Two functionals with the s... |
| eqlkr2 37562 | Two functionals with the s... |
| eqlkr3 37563 | Two functionals with the s... |
| lkrlsp 37564 | The subspace sum of a kern... |
| lkrlsp2 37565 | The subspace sum of a kern... |
| lkrlsp3 37566 | The subspace sum of a kern... |
| lkrshp 37567 | The kernel of a nonzero fu... |
| lkrshp3 37568 | The kernels of nonzero fun... |
| lkrshpor 37569 | The kernel of a functional... |
| lkrshp4 37570 | A kernel is a hyperplane i... |
| lshpsmreu 37571 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 37572 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 37573 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 37574 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 37575 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 37576 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 37577 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 37578 | The set ` G ` defined by h... |
| lshpkr 37579 | The kernel of functional `... |
| lshpkrex 37580 | There exists a functional ... |
| lshpset2N 37581 | The set of all hyperplanes... |
| islshpkrN 37582 | The predicate "is a hyperp... |
| lfl1dim 37583 | Equivalent expressions for... |
| lfl1dim2N 37584 | Equivalent expressions for... |
| ldualset 37587 | Define the (left) dual of ... |
| ldualvbase 37588 | The vectors of a dual spac... |
| ldualelvbase 37589 | Utility theorem for conver... |
| ldualfvadd 37590 | Vector addition in the dua... |
| ldualvadd 37591 | Vector addition in the dua... |
| ldualvaddcl 37592 | The value of vector additi... |
| ldualvaddval 37593 | The value of the value of ... |
| ldualsca 37594 | The ring of scalars of the... |
| ldualsbase 37595 | Base set of scalar ring fo... |
| ldualsaddN 37596 | Scalar addition for the du... |
| ldualsmul 37597 | Scalar multiplication for ... |
| ldualfvs 37598 | Scalar product operation f... |
| ldualvs 37599 | Scalar product operation v... |
| ldualvsval 37600 | Value of scalar product op... |
| ldualvscl 37601 | The scalar product operati... |
| ldualvaddcom 37602 | Commutative law for vector... |
| ldualvsass 37603 | Associative law for scalar... |
| ldualvsass2 37604 | Associative law for scalar... |
| ldualvsdi1 37605 | Distributive law for scala... |
| ldualvsdi2 37606 | Reverse distributive law f... |
| ldualgrplem 37607 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 37608 | The dual of a vector space... |
| ldual0 37609 | The zero scalar of the dua... |
| ldual1 37610 | The unit scalar of the dua... |
| ldualneg 37611 | The negative of a scalar o... |
| ldual0v 37612 | The zero vector of the dua... |
| ldual0vcl 37613 | The dual zero vector is a ... |
| lduallmodlem 37614 | Lemma for ~ lduallmod . (... |
| lduallmod 37615 | The dual of a left module ... |
| lduallvec 37616 | The dual of a left vector ... |
| ldualvsub 37617 | The value of vector subtra... |
| ldualvsubcl 37618 | Closure of vector subtract... |
| ldualvsubval 37619 | The value of the value of ... |
| ldualssvscl 37620 | Closure of scalar product ... |
| ldualssvsubcl 37621 | Closure of vector subtract... |
| ldual0vs 37622 | Scalar zero times a functi... |
| lkr0f2 37623 | The kernel of the zero fun... |
| lduallkr3 37624 | The kernels of nonzero fun... |
| lkrpssN 37625 | Proper subset relation bet... |
| lkrin 37626 | Intersection of the kernel... |
| eqlkr4 37627 | Two functionals with the s... |
| ldual1dim 37628 | Equivalent expressions for... |
| ldualkrsc 37629 | The kernel of a nonzero sc... |
| lkrss 37630 | The kernel of a scalar pro... |
| lkrss2N 37631 | Two functionals with kerne... |
| lkreqN 37632 | Proportional functionals h... |
| lkrlspeqN 37633 | Condition for colinear fun... |
| isopos 37642 | The predicate "is an ortho... |
| opposet 37643 | Every orthoposet is a pose... |
| oposlem 37644 | Lemma for orthoposet prope... |
| op01dm 37645 | Conditions necessary for z... |
| op0cl 37646 | An orthoposet has a zero e... |
| op1cl 37647 | An orthoposet has a unity ... |
| op0le 37648 | Orthoposet zero is less th... |
| ople0 37649 | An element less than or eq... |
| opnlen0 37650 | An element not less than a... |
| lub0N 37651 | The least upper bound of t... |
| opltn0 37652 | A lattice element greater ... |
| ople1 37653 | Any element is less than t... |
| op1le 37654 | If the orthoposet unity is... |
| glb0N 37655 | The greatest lower bound o... |
| opoccl 37656 | Closure of orthocomplement... |
| opococ 37657 | Double negative law for or... |
| opcon3b 37658 | Contraposition law for ort... |
| opcon2b 37659 | Orthocomplement contraposi... |
| opcon1b 37660 | Orthocomplement contraposi... |
| oplecon3 37661 | Contraposition law for ort... |
| oplecon3b 37662 | Contraposition law for ort... |
| oplecon1b 37663 | Contraposition law for str... |
| opoc1 37664 | Orthocomplement of orthopo... |
| opoc0 37665 | Orthocomplement of orthopo... |
| opltcon3b 37666 | Contraposition law for str... |
| opltcon1b 37667 | Contraposition law for str... |
| opltcon2b 37668 | Contraposition law for str... |
| opexmid 37669 | Law of excluded middle for... |
| opnoncon 37670 | Law of contradiction for o... |
| riotaocN 37671 | The orthocomplement of the... |
| cmtfvalN 37672 | Value of commutes relation... |
| cmtvalN 37673 | Equivalence for commutes r... |
| isolat 37674 | The predicate "is an ortho... |
| ollat 37675 | An ortholattice is a latti... |
| olop 37676 | An ortholattice is an orth... |
| olposN 37677 | An ortholattice is a poset... |
| isolatiN 37678 | Properties that determine ... |
| oldmm1 37679 | De Morgan's law for meet i... |
| oldmm2 37680 | De Morgan's law for meet i... |
| oldmm3N 37681 | De Morgan's law for meet i... |
| oldmm4 37682 | De Morgan's law for meet i... |
| oldmj1 37683 | De Morgan's law for join i... |
| oldmj2 37684 | De Morgan's law for join i... |
| oldmj3 37685 | De Morgan's law for join i... |
| oldmj4 37686 | De Morgan's law for join i... |
| olj01 37687 | An ortholattice element jo... |
| olj02 37688 | An ortholattice element jo... |
| olm11 37689 | The meet of an ortholattic... |
| olm12 37690 | The meet of an ortholattic... |
| latmassOLD 37691 | Ortholattice meet is assoc... |
| latm12 37692 | A rearrangement of lattice... |
| latm32 37693 | A rearrangement of lattice... |
| latmrot 37694 | Rotate lattice meet of 3 c... |
| latm4 37695 | Rearrangement of lattice m... |
| latmmdiN 37696 | Lattice meet distributes o... |
| latmmdir 37697 | Lattice meet distributes o... |
| olm01 37698 | Meet with lattice zero is ... |
| olm02 37699 | Meet with lattice zero is ... |
| isoml 37700 | The predicate "is an ortho... |
| isomliN 37701 | Properties that determine ... |
| omlol 37702 | An orthomodular lattice is... |
| omlop 37703 | An orthomodular lattice is... |
| omllat 37704 | An orthomodular lattice is... |
| omllaw 37705 | The orthomodular law. (Co... |
| omllaw2N 37706 | Variation of orthomodular ... |
| omllaw3 37707 | Orthomodular law equivalen... |
| omllaw4 37708 | Orthomodular law equivalen... |
| omllaw5N 37709 | The orthomodular law. Rem... |
| cmtcomlemN 37710 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 37711 | Commutation is symmetric. ... |
| cmt2N 37712 | Commutation with orthocomp... |
| cmt3N 37713 | Commutation with orthocomp... |
| cmt4N 37714 | Commutation with orthocomp... |
| cmtbr2N 37715 | Alternate definition of th... |
| cmtbr3N 37716 | Alternate definition for t... |
| cmtbr4N 37717 | Alternate definition for t... |
| lecmtN 37718 | Ordered elements commute. ... |
| cmtidN 37719 | Any element commutes with ... |
| omlfh1N 37720 | Foulis-Holland Theorem, pa... |
| omlfh3N 37721 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 37722 | Analogue of modular law ~ ... |
| omlspjN 37723 | Contraction of a Sasaki pr... |
| cvrfval 37730 | Value of covers relation "... |
| cvrval 37731 | Binary relation expressing... |
| cvrlt 37732 | The covers relation implie... |
| cvrnbtwn 37733 | There is no element betwee... |
| ncvr1 37734 | No element covers the latt... |
| cvrletrN 37735 | Property of an element abo... |
| cvrval2 37736 | Binary relation expressing... |
| cvrnbtwn2 37737 | The covers relation implie... |
| cvrnbtwn3 37738 | The covers relation implie... |
| cvrcon3b 37739 | Contraposition law for the... |
| cvrle 37740 | The covers relation implie... |
| cvrnbtwn4 37741 | The covers relation implie... |
| cvrnle 37742 | The covers relation implie... |
| cvrne 37743 | The covers relation implie... |
| cvrnrefN 37744 | The covers relation is not... |
| cvrcmp 37745 | If two lattice elements th... |
| cvrcmp2 37746 | If two lattice elements co... |
| pats 37747 | The set of atoms in a pose... |
| isat 37748 | The predicate "is an atom"... |
| isat2 37749 | The predicate "is an atom"... |
| atcvr0 37750 | An atom covers zero. ( ~ ... |
| atbase 37751 | An atom is a member of the... |
| atssbase 37752 | The set of atoms is a subs... |
| 0ltat 37753 | An atom is greater than ze... |
| leatb 37754 | A poset element less than ... |
| leat 37755 | A poset element less than ... |
| leat2 37756 | A nonzero poset element le... |
| leat3 37757 | A poset element less than ... |
| meetat 37758 | The meet of any element wi... |
| meetat2 37759 | The meet of any element wi... |
| isatl 37761 | The predicate "is an atomi... |
| atllat 37762 | An atomic lattice is a lat... |
| atlpos 37763 | An atomic lattice is a pos... |
| atl0dm 37764 | Condition necessary for ze... |
| atl0cl 37765 | An atomic lattice has a ze... |
| atl0le 37766 | Orthoposet zero is less th... |
| atlle0 37767 | An element less than or eq... |
| atlltn0 37768 | A lattice element greater ... |
| isat3 37769 | The predicate "is an atom"... |
| atn0 37770 | An atom is not zero. ( ~ ... |
| atnle0 37771 | An atom is not less than o... |
| atlen0 37772 | A lattice element is nonze... |
| atcmp 37773 | If two atoms are comparabl... |
| atncmp 37774 | Frequently-used variation ... |
| atnlt 37775 | Two atoms cannot satisfy t... |
| atcvreq0 37776 | An element covered by an a... |
| atncvrN 37777 | Two atoms cannot satisfy t... |
| atlex 37778 | Every nonzero element of a... |
| atnle 37779 | Two ways of expressing "an... |
| atnem0 37780 | The meet of distinct atoms... |
| atlatmstc 37781 | An atomic, complete, ortho... |
| atlatle 37782 | The ordering of two Hilber... |
| atlrelat1 37783 | An atomistic lattice with ... |
| iscvlat 37785 | The predicate "is an atomi... |
| iscvlat2N 37786 | The predicate "is an atomi... |
| cvlatl 37787 | An atomic lattice with the... |
| cvllat 37788 | An atomic lattice with the... |
| cvlposN 37789 | An atomic lattice with the... |
| cvlexch1 37790 | An atomic covering lattice... |
| cvlexch2 37791 | An atomic covering lattice... |
| cvlexchb1 37792 | An atomic covering lattice... |
| cvlexchb2 37793 | An atomic covering lattice... |
| cvlexch3 37794 | An atomic covering lattice... |
| cvlexch4N 37795 | An atomic covering lattice... |
| cvlatexchb1 37796 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 37797 | A version of ~ cvlexchb2 f... |
| cvlatexch1 37798 | Atom exchange property. (... |
| cvlatexch2 37799 | Atom exchange property. (... |
| cvlatexch3 37800 | Atom exchange property. (... |
| cvlcvr1 37801 | The covering property. Pr... |
| cvlcvrp 37802 | A Hilbert lattice satisfie... |
| cvlatcvr1 37803 | An atom is covered by its ... |
| cvlatcvr2 37804 | An atom is covered by its ... |
| cvlsupr2 37805 | Two equivalent ways of exp... |
| cvlsupr3 37806 | Two equivalent ways of exp... |
| cvlsupr4 37807 | Consequence of superpositi... |
| cvlsupr5 37808 | Consequence of superpositi... |
| cvlsupr6 37809 | Consequence of superpositi... |
| cvlsupr7 37810 | Consequence of superpositi... |
| cvlsupr8 37811 | Consequence of superpositi... |
| ishlat1 37814 | The predicate "is a Hilber... |
| ishlat2 37815 | The predicate "is a Hilber... |
| ishlat3N 37816 | The predicate "is a Hilber... |
| ishlatiN 37817 | Properties that determine ... |
| hlomcmcv 37818 | A Hilbert lattice is ortho... |
| hloml 37819 | A Hilbert lattice is ortho... |
| hlclat 37820 | A Hilbert lattice is compl... |
| hlcvl 37821 | A Hilbert lattice is an at... |
| hlatl 37822 | A Hilbert lattice is atomi... |
| hlol 37823 | A Hilbert lattice is an or... |
| hlop 37824 | A Hilbert lattice is an or... |
| hllat 37825 | A Hilbert lattice is a lat... |
| hllatd 37826 | Deduction form of ~ hllat ... |
| hlomcmat 37827 | A Hilbert lattice is ortho... |
| hlpos 37828 | A Hilbert lattice is a pos... |
| hlatjcl 37829 | Closure of join operation.... |
| hlatjcom 37830 | Commutatitivity of join op... |
| hlatjidm 37831 | Idempotence of join operat... |
| hlatjass 37832 | Lattice join is associativ... |
| hlatj12 37833 | Swap 1st and 2nd members o... |
| hlatj32 37834 | Swap 2nd and 3rd members o... |
| hlatjrot 37835 | Rotate lattice join of 3 c... |
| hlatj4 37836 | Rearrangement of lattice j... |
| hlatlej1 37837 | A join's first argument is... |
| hlatlej2 37838 | A join's second argument i... |
| glbconN 37839 | De Morgan's law for GLB an... |
| glbconNOLD 37840 | Obsolete version of ~ glbc... |
| glbconxN 37841 | De Morgan's law for GLB an... |
| atnlej1 37842 | If an atom is not less tha... |
| atnlej2 37843 | If an atom is not less tha... |
| hlsuprexch 37844 | A Hilbert lattice has the ... |
| hlexch1 37845 | A Hilbert lattice has the ... |
| hlexch2 37846 | A Hilbert lattice has the ... |
| hlexchb1 37847 | A Hilbert lattice has the ... |
| hlexchb2 37848 | A Hilbert lattice has the ... |
| hlsupr 37849 | A Hilbert lattice has the ... |
| hlsupr2 37850 | A Hilbert lattice has the ... |
| hlhgt4 37851 | A Hilbert lattice has a he... |
| hlhgt2 37852 | A Hilbert lattice has a he... |
| hl0lt1N 37853 | Lattice 0 is less than lat... |
| hlexch3 37854 | A Hilbert lattice has the ... |
| hlexch4N 37855 | A Hilbert lattice has the ... |
| hlatexchb1 37856 | A version of ~ hlexchb1 fo... |
| hlatexchb2 37857 | A version of ~ hlexchb2 fo... |
| hlatexch1 37858 | Atom exchange property. (... |
| hlatexch2 37859 | Atom exchange property. (... |
| hlatmstcOLDN 37860 | An atomic, complete, ortho... |
| hlatle 37861 | The ordering of two Hilber... |
| hlateq 37862 | The equality of two Hilber... |
| hlrelat1 37863 | An atomistic lattice with ... |
| hlrelat5N 37864 | An atomistic lattice with ... |
| hlrelat 37865 | A Hilbert lattice is relat... |
| hlrelat2 37866 | A consequence of relative ... |
| exatleN 37867 | A condition for an atom to... |
| hl2at 37868 | A Hilbert lattice has at l... |
| atex 37869 | At least one atom exists. ... |
| intnatN 37870 | If the intersection with a... |
| 2llnne2N 37871 | Condition implying that tw... |
| 2llnneN 37872 | Condition implying that tw... |
| cvr1 37873 | A Hilbert lattice has the ... |
| cvr2N 37874 | Less-than and covers equiv... |
| hlrelat3 37875 | The Hilbert lattice is rel... |
| cvrval3 37876 | Binary relation expressing... |
| cvrval4N 37877 | Binary relation expressing... |
| cvrval5 37878 | Binary relation expressing... |
| cvrp 37879 | A Hilbert lattice satisfie... |
| atcvr1 37880 | An atom is covered by its ... |
| atcvr2 37881 | An atom is covered by its ... |
| cvrexchlem 37882 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 37883 | A Hilbert lattice satisfie... |
| cvratlem 37884 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 37885 | A nonzero Hilbert lattice ... |
| ltltncvr 37886 | A chained strong ordering ... |
| ltcvrntr 37887 | Non-transitive condition f... |
| cvrntr 37888 | The covers relation is not... |
| atcvr0eq 37889 | The covers relation is not... |
| lnnat 37890 | A line (the join of two di... |
| atcvrj0 37891 | Two atoms covering the zer... |
| cvrat2 37892 | A Hilbert lattice element ... |
| atcvrneN 37893 | Inequality derived from at... |
| atcvrj1 37894 | Condition for an atom to b... |
| atcvrj2b 37895 | Condition for an atom to b... |
| atcvrj2 37896 | Condition for an atom to b... |
| atleneN 37897 | Inequality derived from at... |
| atltcvr 37898 | An equivalence of less-tha... |
| atle 37899 | Any nonzero element has an... |
| atlt 37900 | Two atoms are unequal iff ... |
| atlelt 37901 | Transfer less-than relatio... |
| 2atlt 37902 | Given an atom less than an... |
| atexchcvrN 37903 | Atom exchange property. V... |
| atexchltN 37904 | Atom exchange property. V... |
| cvrat3 37905 | A condition implying that ... |
| cvrat4 37906 | A condition implying exist... |
| cvrat42 37907 | Commuted version of ~ cvra... |
| 2atjm 37908 | The meet of a line (expres... |
| atbtwn 37909 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 37910 | There exists a 3rd atom ` ... |
| atbtwnex 37911 | Given atoms ` P ` in ` X `... |
| 3noncolr2 37912 | Two ways to express 3 non-... |
| 3noncolr1N 37913 | Two ways to express 3 non-... |
| hlatcon3 37914 | Atom exchange combined wit... |
| hlatcon2 37915 | Atom exchange combined wit... |
| 4noncolr3 37916 | A way to express 4 non-col... |
| 4noncolr2 37917 | A way to express 4 non-col... |
| 4noncolr1 37918 | A way to express 4 non-col... |
| athgt 37919 | A Hilbert lattice, whose h... |
| 3dim0 37920 | There exists a 3-dimension... |
| 3dimlem1 37921 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 37922 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 37923 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 37924 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 37925 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 37926 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 37927 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 37928 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 37929 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 37930 | Construct a 3-dimensional ... |
| 3dim2 37931 | Construct 2 new layers on ... |
| 3dim3 37932 | Construct a new layer on t... |
| 2dim 37933 | Generate a height-3 elemen... |
| 1dimN 37934 | An atom is covered by a he... |
| 1cvrco 37935 | The orthocomplement of an ... |
| 1cvratex 37936 | There exists an atom less ... |
| 1cvratlt 37937 | An atom less than or equal... |
| 1cvrjat 37938 | An element covered by the ... |
| 1cvrat 37939 | Create an atom under an el... |
| ps-1 37940 | The join of two atoms ` R ... |
| ps-2 37941 | Lattice analogue for the p... |
| 2atjlej 37942 | Two atoms are different if... |
| hlatexch3N 37943 | Rearrange join of atoms in... |
| hlatexch4 37944 | Exchange 2 atoms. (Contri... |
| ps-2b 37945 | Variation of projective ge... |
| 3atlem1 37946 | Lemma for ~ 3at . (Contri... |
| 3atlem2 37947 | Lemma for ~ 3at . (Contri... |
| 3atlem3 37948 | Lemma for ~ 3at . (Contri... |
| 3atlem4 37949 | Lemma for ~ 3at . (Contri... |
| 3atlem5 37950 | Lemma for ~ 3at . (Contri... |
| 3atlem6 37951 | Lemma for ~ 3at . (Contri... |
| 3atlem7 37952 | Lemma for ~ 3at . (Contri... |
| 3at 37953 | Any three non-colinear ato... |
| llnset 37968 | The set of lattice lines i... |
| islln 37969 | The predicate "is a lattic... |
| islln4 37970 | The predicate "is a lattic... |
| llni 37971 | Condition implying a latti... |
| llnbase 37972 | A lattice line is a lattic... |
| islln3 37973 | The predicate "is a lattic... |
| islln2 37974 | The predicate "is a lattic... |
| llni2 37975 | The join of two different ... |
| llnnleat 37976 | An atom cannot majorize a ... |
| llnneat 37977 | A lattice line is not an a... |
| 2atneat 37978 | The join of two distinct a... |
| llnn0 37979 | A lattice line is nonzero.... |
| islln2a 37980 | The predicate "is a lattic... |
| llnle 37981 | Any element greater than 0... |
| atcvrlln2 37982 | An atom under a line is co... |
| atcvrlln 37983 | An element covering an ato... |
| llnexatN 37984 | Given an atom on a line, t... |
| llncmp 37985 | If two lattice lines are c... |
| llnnlt 37986 | Two lattice lines cannot s... |
| 2llnmat 37987 | Two intersecting lines int... |
| 2at0mat0 37988 | Special case of ~ 2atmat0 ... |
| 2atmat0 37989 | The meet of two unequal li... |
| 2atm 37990 | An atom majorized by two d... |
| ps-2c 37991 | Variation of projective ge... |
| lplnset 37992 | The set of lattice planes ... |
| islpln 37993 | The predicate "is a lattic... |
| islpln4 37994 | The predicate "is a lattic... |
| lplni 37995 | Condition implying a latti... |
| islpln3 37996 | The predicate "is a lattic... |
| lplnbase 37997 | A lattice plane is a latti... |
| islpln5 37998 | The predicate "is a lattic... |
| islpln2 37999 | The predicate "is a lattic... |
| lplni2 38000 | The join of 3 different at... |
| lvolex3N 38001 | There is an atom outside o... |
| llnmlplnN 38002 | The intersection of a line... |
| lplnle 38003 | Any element greater than 0... |
| lplnnle2at 38004 | A lattice line (or atom) c... |
| lplnnleat 38005 | A lattice plane cannot maj... |
| lplnnlelln 38006 | A lattice plane is not les... |
| 2atnelpln 38007 | The join of two atoms is n... |
| lplnneat 38008 | No lattice plane is an ato... |
| lplnnelln 38009 | No lattice plane is a latt... |
| lplnn0N 38010 | A lattice plane is nonzero... |
| islpln2a 38011 | The predicate "is a lattic... |
| islpln2ah 38012 | The predicate "is a lattic... |
| lplnriaN 38013 | Property of a lattice plan... |
| lplnribN 38014 | Property of a lattice plan... |
| lplnric 38015 | Property of a lattice plan... |
| lplnri1 38016 | Property of a lattice plan... |
| lplnri2N 38017 | Property of a lattice plan... |
| lplnri3N 38018 | Property of a lattice plan... |
| lplnllnneN 38019 | Two lattice lines defined ... |
| llncvrlpln2 38020 | A lattice line under a lat... |
| llncvrlpln 38021 | An element covering a latt... |
| 2lplnmN 38022 | If the join of two lattice... |
| 2llnmj 38023 | The meet of two lattice li... |
| 2atmat 38024 | The meet of two intersecti... |
| lplncmp 38025 | If two lattice planes are ... |
| lplnexatN 38026 | Given a lattice line on a ... |
| lplnexllnN 38027 | Given an atom on a lattice... |
| lplnnlt 38028 | Two lattice planes cannot ... |
| 2llnjaN 38029 | The join of two different ... |
| 2llnjN 38030 | The join of two different ... |
| 2llnm2N 38031 | The meet of two different ... |
| 2llnm3N 38032 | Two lattice lines in a lat... |
| 2llnm4 38033 | Two lattice lines that maj... |
| 2llnmeqat 38034 | An atom equals the interse... |
| lvolset 38035 | The set of 3-dim lattice v... |
| islvol 38036 | The predicate "is a 3-dim ... |
| islvol4 38037 | The predicate "is a 3-dim ... |
| lvoli 38038 | Condition implying a 3-dim... |
| islvol3 38039 | The predicate "is a 3-dim ... |
| lvoli3 38040 | Condition implying a 3-dim... |
| lvolbase 38041 | A 3-dim lattice volume is ... |
| islvol5 38042 | The predicate "is a 3-dim ... |
| islvol2 38043 | The predicate "is a 3-dim ... |
| lvoli2 38044 | The join of 4 different at... |
| lvolnle3at 38045 | A lattice plane (or lattic... |
| lvolnleat 38046 | An atom cannot majorize a ... |
| lvolnlelln 38047 | A lattice line cannot majo... |
| lvolnlelpln 38048 | A lattice plane cannot maj... |
| 3atnelvolN 38049 | The join of 3 atoms is not... |
| 2atnelvolN 38050 | The join of two atoms is n... |
| lvolneatN 38051 | No lattice volume is an at... |
| lvolnelln 38052 | No lattice volume is a lat... |
| lvolnelpln 38053 | No lattice volume is a lat... |
| lvoln0N 38054 | A lattice volume is nonzer... |
| islvol2aN 38055 | The predicate "is a lattic... |
| 4atlem0a 38056 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 38057 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 38058 | Lemma for ~ 4at . (Contri... |
| 4atlem3 38059 | Lemma for ~ 4at . Break i... |
| 4atlem3a 38060 | Lemma for ~ 4at . Break i... |
| 4atlem3b 38061 | Lemma for ~ 4at . Break i... |
| 4atlem4a 38062 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 38063 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 38064 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 38065 | Lemma for ~ 4at . Frequen... |
| 4atlem9 38066 | Lemma for ~ 4at . Substit... |
| 4atlem10a 38067 | Lemma for ~ 4at . Substit... |
| 4atlem10b 38068 | Lemma for ~ 4at . Substit... |
| 4atlem10 38069 | Lemma for ~ 4at . Combine... |
| 4atlem11a 38070 | Lemma for ~ 4at . Substit... |
| 4atlem11b 38071 | Lemma for ~ 4at . Substit... |
| 4atlem11 38072 | Lemma for ~ 4at . Combine... |
| 4atlem12a 38073 | Lemma for ~ 4at . Substit... |
| 4atlem12b 38074 | Lemma for ~ 4at . Substit... |
| 4atlem12 38075 | Lemma for ~ 4at . Combine... |
| 4at 38076 | Four atoms determine a lat... |
| 4at2 38077 | Four atoms determine a lat... |
| lplncvrlvol2 38078 | A lattice line under a lat... |
| lplncvrlvol 38079 | An element covering a latt... |
| lvolcmp 38080 | If two lattice planes are ... |
| lvolnltN 38081 | Two lattice volumes cannot... |
| 2lplnja 38082 | The join of two different ... |
| 2lplnj 38083 | The join of two different ... |
| 2lplnm2N 38084 | The meet of two different ... |
| 2lplnmj 38085 | The meet of two lattice pl... |
| dalemkehl 38086 | Lemma for ~ dath . Freque... |
| dalemkelat 38087 | Lemma for ~ dath . Freque... |
| dalemkeop 38088 | Lemma for ~ dath . Freque... |
| dalempea 38089 | Lemma for ~ dath . Freque... |
| dalemqea 38090 | Lemma for ~ dath . Freque... |
| dalemrea 38091 | Lemma for ~ dath . Freque... |
| dalemsea 38092 | Lemma for ~ dath . Freque... |
| dalemtea 38093 | Lemma for ~ dath . Freque... |
| dalemuea 38094 | Lemma for ~ dath . Freque... |
| dalemyeo 38095 | Lemma for ~ dath . Freque... |
| dalemzeo 38096 | Lemma for ~ dath . Freque... |
| dalemclpjs 38097 | Lemma for ~ dath . Freque... |
| dalemclqjt 38098 | Lemma for ~ dath . Freque... |
| dalemclrju 38099 | Lemma for ~ dath . Freque... |
| dalem-clpjq 38100 | Lemma for ~ dath . Freque... |
| dalemceb 38101 | Lemma for ~ dath . Freque... |
| dalempeb 38102 | Lemma for ~ dath . Freque... |
| dalemqeb 38103 | Lemma for ~ dath . Freque... |
| dalemreb 38104 | Lemma for ~ dath . Freque... |
| dalemseb 38105 | Lemma for ~ dath . Freque... |
| dalemteb 38106 | Lemma for ~ dath . Freque... |
| dalemueb 38107 | Lemma for ~ dath . Freque... |
| dalempjqeb 38108 | Lemma for ~ dath . Freque... |
| dalemsjteb 38109 | Lemma for ~ dath . Freque... |
| dalemtjueb 38110 | Lemma for ~ dath . Freque... |
| dalemqrprot 38111 | Lemma for ~ dath . Freque... |
| dalemyeb 38112 | Lemma for ~ dath . Freque... |
| dalemcnes 38113 | Lemma for ~ dath . Freque... |
| dalempnes 38114 | Lemma for ~ dath . Freque... |
| dalemqnet 38115 | Lemma for ~ dath . Freque... |
| dalempjsen 38116 | Lemma for ~ dath . Freque... |
| dalemply 38117 | Lemma for ~ dath . Freque... |
| dalemsly 38118 | Lemma for ~ dath . Freque... |
| dalemswapyz 38119 | Lemma for ~ dath . Swap t... |
| dalemrot 38120 | Lemma for ~ dath . Rotate... |
| dalemrotyz 38121 | Lemma for ~ dath . Rotate... |
| dalem1 38122 | Lemma for ~ dath . Show t... |
| dalemcea 38123 | Lemma for ~ dath . Freque... |
| dalem2 38124 | Lemma for ~ dath . Show t... |
| dalemdea 38125 | Lemma for ~ dath . Freque... |
| dalemeea 38126 | Lemma for ~ dath . Freque... |
| dalem3 38127 | Lemma for ~ dalemdnee . (... |
| dalem4 38128 | Lemma for ~ dalemdnee . (... |
| dalemdnee 38129 | Lemma for ~ dath . Axis o... |
| dalem5 38130 | Lemma for ~ dath . Atom `... |
| dalem6 38131 | Lemma for ~ dath . Analog... |
| dalem7 38132 | Lemma for ~ dath . Analog... |
| dalem8 38133 | Lemma for ~ dath . Plane ... |
| dalem-cly 38134 | Lemma for ~ dalem9 . Cent... |
| dalem9 38135 | Lemma for ~ dath . Since ... |
| dalem10 38136 | Lemma for ~ dath . Atom `... |
| dalem11 38137 | Lemma for ~ dath . Analog... |
| dalem12 38138 | Lemma for ~ dath . Analog... |
| dalem13 38139 | Lemma for ~ dalem14 . (Co... |
| dalem14 38140 | Lemma for ~ dath . Planes... |
| dalem15 38141 | Lemma for ~ dath . The ax... |
| dalem16 38142 | Lemma for ~ dath . The at... |
| dalem17 38143 | Lemma for ~ dath . When p... |
| dalem18 38144 | Lemma for ~ dath . Show t... |
| dalem19 38145 | Lemma for ~ dath . Show t... |
| dalemccea 38146 | Lemma for ~ dath . Freque... |
| dalemddea 38147 | Lemma for ~ dath . Freque... |
| dalem-ccly 38148 | Lemma for ~ dath . Freque... |
| dalem-ddly 38149 | Lemma for ~ dath . Freque... |
| dalemccnedd 38150 | Lemma for ~ dath . Freque... |
| dalemclccjdd 38151 | Lemma for ~ dath . Freque... |
| dalemcceb 38152 | Lemma for ~ dath . Freque... |
| dalemswapyzps 38153 | Lemma for ~ dath . Swap t... |
| dalemrotps 38154 | Lemma for ~ dath . Rotate... |
| dalemcjden 38155 | Lemma for ~ dath . Show t... |
| dalem20 38156 | Lemma for ~ dath . Show t... |
| dalem21 38157 | Lemma for ~ dath . Show t... |
| dalem22 38158 | Lemma for ~ dath . Show t... |
| dalem23 38159 | Lemma for ~ dath . Show t... |
| dalem24 38160 | Lemma for ~ dath . Show t... |
| dalem25 38161 | Lemma for ~ dath . Show t... |
| dalem27 38162 | Lemma for ~ dath . Show t... |
| dalem28 38163 | Lemma for ~ dath . Lemma ... |
| dalem29 38164 | Lemma for ~ dath . Analog... |
| dalem30 38165 | Lemma for ~ dath . Analog... |
| dalem31N 38166 | Lemma for ~ dath . Analog... |
| dalem32 38167 | Lemma for ~ dath . Analog... |
| dalem33 38168 | Lemma for ~ dath . Analog... |
| dalem34 38169 | Lemma for ~ dath . Analog... |
| dalem35 38170 | Lemma for ~ dath . Analog... |
| dalem36 38171 | Lemma for ~ dath . Analog... |
| dalem37 38172 | Lemma for ~ dath . Analog... |
| dalem38 38173 | Lemma for ~ dath . Plane ... |
| dalem39 38174 | Lemma for ~ dath . Auxili... |
| dalem40 38175 | Lemma for ~ dath . Analog... |
| dalem41 38176 | Lemma for ~ dath . (Contr... |
| dalem42 38177 | Lemma for ~ dath . Auxili... |
| dalem43 38178 | Lemma for ~ dath . Planes... |
| dalem44 38179 | Lemma for ~ dath . Dummy ... |
| dalem45 38180 | Lemma for ~ dath . Dummy ... |
| dalem46 38181 | Lemma for ~ dath . Analog... |
| dalem47 38182 | Lemma for ~ dath . Analog... |
| dalem48 38183 | Lemma for ~ dath . Analog... |
| dalem49 38184 | Lemma for ~ dath . Analog... |
| dalem50 38185 | Lemma for ~ dath . Analog... |
| dalem51 38186 | Lemma for ~ dath . Constr... |
| dalem52 38187 | Lemma for ~ dath . Lines ... |
| dalem53 38188 | Lemma for ~ dath . The au... |
| dalem54 38189 | Lemma for ~ dath . Line `... |
| dalem55 38190 | Lemma for ~ dath . Lines ... |
| dalem56 38191 | Lemma for ~ dath . Analog... |
| dalem57 38192 | Lemma for ~ dath . Axis o... |
| dalem58 38193 | Lemma for ~ dath . Analog... |
| dalem59 38194 | Lemma for ~ dath . Analog... |
| dalem60 38195 | Lemma for ~ dath . ` B ` i... |
| dalem61 38196 | Lemma for ~ dath . Show t... |
| dalem62 38197 | Lemma for ~ dath . Elimin... |
| dalem63 38198 | Lemma for ~ dath . Combin... |
| dath 38199 | Desargues's theorem of pro... |
| dath2 38200 | Version of Desargues's the... |
| lineset 38201 | The set of lines in a Hilb... |
| isline 38202 | The predicate "is a line".... |
| islinei 38203 | Condition implying "is a l... |
| pointsetN 38204 | The set of points in a Hil... |
| ispointN 38205 | The predicate "is a point"... |
| atpointN 38206 | The singleton of an atom i... |
| psubspset 38207 | The set of projective subs... |
| ispsubsp 38208 | The predicate "is a projec... |
| ispsubsp2 38209 | The predicate "is a projec... |
| psubspi 38210 | Property of a projective s... |
| psubspi2N 38211 | Property of a projective s... |
| 0psubN 38212 | The empty set is a project... |
| snatpsubN 38213 | The singleton of an atom i... |
| pointpsubN 38214 | A point (singleton of an a... |
| linepsubN 38215 | A line is a projective sub... |
| atpsubN 38216 | The set of all atoms is a ... |
| psubssat 38217 | A projective subspace cons... |
| psubatN 38218 | A member of a projective s... |
| pmapfval 38219 | The projective map of a Hi... |
| pmapval 38220 | Value of the projective ma... |
| elpmap 38221 | Member of a projective map... |
| pmapssat 38222 | The projective map of a Hi... |
| pmapssbaN 38223 | A weakening of ~ pmapssat ... |
| pmaple 38224 | The projective map of a Hi... |
| pmap11 38225 | The projective map of a Hi... |
| pmapat 38226 | The projective map of an a... |
| elpmapat 38227 | Member of the projective m... |
| pmap0 38228 | Value of the projective ma... |
| pmapeq0 38229 | A projective map value is ... |
| pmap1N 38230 | Value of the projective ma... |
| pmapsub 38231 | The projective map of a Hi... |
| pmapglbx 38232 | The projective map of the ... |
| pmapglb 38233 | The projective map of the ... |
| pmapglb2N 38234 | The projective map of the ... |
| pmapglb2xN 38235 | The projective map of the ... |
| pmapmeet 38236 | The projective map of a me... |
| isline2 38237 | Definition of line in term... |
| linepmap 38238 | A line described with a pr... |
| isline3 38239 | Definition of line in term... |
| isline4N 38240 | Definition of line in term... |
| lneq2at 38241 | A line equals the join of ... |
| lnatexN 38242 | There is an atom in a line... |
| lnjatN 38243 | Given an atom in a line, t... |
| lncvrelatN 38244 | A lattice element covered ... |
| lncvrat 38245 | A line covers the atoms it... |
| lncmp 38246 | If two lines are comparabl... |
| 2lnat 38247 | Two intersecting lines int... |
| 2atm2atN 38248 | Two joins with a common at... |
| 2llnma1b 38249 | Generalization of ~ 2llnma... |
| 2llnma1 38250 | Two different intersecting... |
| 2llnma3r 38251 | Two different intersecting... |
| 2llnma2 38252 | Two different intersecting... |
| 2llnma2rN 38253 | Two different intersecting... |
| cdlema1N 38254 | A condition for required f... |
| cdlema2N 38255 | A condition for required f... |
| cdlemblem 38256 | Lemma for ~ cdlemb . (Con... |
| cdlemb 38257 | Given two atoms not less t... |
| paddfval 38260 | Projective subspace sum op... |
| paddval 38261 | Projective subspace sum op... |
| elpadd 38262 | Member of a projective sub... |
| elpaddn0 38263 | Member of projective subsp... |
| paddvaln0N 38264 | Projective subspace sum op... |
| elpaddri 38265 | Condition implying members... |
| elpaddatriN 38266 | Condition implying members... |
| elpaddat 38267 | Membership in a projective... |
| elpaddatiN 38268 | Consequence of membership ... |
| elpadd2at 38269 | Membership in a projective... |
| elpadd2at2 38270 | Membership in a projective... |
| paddunssN 38271 | Projective subspace sum in... |
| elpadd0 38272 | Member of projective subsp... |
| paddval0 38273 | Projective subspace sum wi... |
| padd01 38274 | Projective subspace sum wi... |
| padd02 38275 | Projective subspace sum wi... |
| paddcom 38276 | Projective subspace sum co... |
| paddssat 38277 | A projective subspace sum ... |
| sspadd1 38278 | A projective subspace sum ... |
| sspadd2 38279 | A projective subspace sum ... |
| paddss1 38280 | Subset law for projective ... |
| paddss2 38281 | Subset law for projective ... |
| paddss12 38282 | Subset law for projective ... |
| paddasslem1 38283 | Lemma for ~ paddass . (Co... |
| paddasslem2 38284 | Lemma for ~ paddass . (Co... |
| paddasslem3 38285 | Lemma for ~ paddass . Res... |
| paddasslem4 38286 | Lemma for ~ paddass . Com... |
| paddasslem5 38287 | Lemma for ~ paddass . Sho... |
| paddasslem6 38288 | Lemma for ~ paddass . (Co... |
| paddasslem7 38289 | Lemma for ~ paddass . Com... |
| paddasslem8 38290 | Lemma for ~ paddass . (Co... |
| paddasslem9 38291 | Lemma for ~ paddass . Com... |
| paddasslem10 38292 | Lemma for ~ paddass . Use... |
| paddasslem11 38293 | Lemma for ~ paddass . The... |
| paddasslem12 38294 | Lemma for ~ paddass . The... |
| paddasslem13 38295 | Lemma for ~ paddass . The... |
| paddasslem14 38296 | Lemma for ~ paddass . Rem... |
| paddasslem15 38297 | Lemma for ~ paddass . Use... |
| paddasslem16 38298 | Lemma for ~ paddass . Use... |
| paddasslem17 38299 | Lemma for ~ paddass . The... |
| paddasslem18 38300 | Lemma for ~ paddass . Com... |
| paddass 38301 | Projective subspace sum is... |
| padd12N 38302 | Commutative/associative la... |
| padd4N 38303 | Rearrangement of 4 terms i... |
| paddidm 38304 | Projective subspace sum is... |
| paddclN 38305 | The projective sum of two ... |
| paddssw1 38306 | Subset law for projective ... |
| paddssw2 38307 | Subset law for projective ... |
| paddss 38308 | Subset law for projective ... |
| pmodlem1 38309 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 38310 | Lemma for ~ pmod1i . (Con... |
| pmod1i 38311 | The modular law holds in a... |
| pmod2iN 38312 | Dual of the modular law. ... |
| pmodN 38313 | The modular law for projec... |
| pmodl42N 38314 | Lemma derived from modular... |
| pmapjoin 38315 | The projective map of the ... |
| pmapjat1 38316 | The projective map of the ... |
| pmapjat2 38317 | The projective map of the ... |
| pmapjlln1 38318 | The projective map of the ... |
| hlmod1i 38319 | A version of the modular l... |
| atmod1i1 38320 | Version of modular law ~ p... |
| atmod1i1m 38321 | Version of modular law ~ p... |
| atmod1i2 38322 | Version of modular law ~ p... |
| llnmod1i2 38323 | Version of modular law ~ p... |
| atmod2i1 38324 | Version of modular law ~ p... |
| atmod2i2 38325 | Version of modular law ~ p... |
| llnmod2i2 38326 | Version of modular law ~ p... |
| atmod3i1 38327 | Version of modular law tha... |
| atmod3i2 38328 | Version of modular law tha... |
| atmod4i1 38329 | Version of modular law tha... |
| atmod4i2 38330 | Version of modular law tha... |
| llnexchb2lem 38331 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 38332 | Line exchange property (co... |
| llnexch2N 38333 | Line exchange property (co... |
| dalawlem1 38334 | Lemma for ~ dalaw . Speci... |
| dalawlem2 38335 | Lemma for ~ dalaw . Utili... |
| dalawlem3 38336 | Lemma for ~ dalaw . First... |
| dalawlem4 38337 | Lemma for ~ dalaw . Secon... |
| dalawlem5 38338 | Lemma for ~ dalaw . Speci... |
| dalawlem6 38339 | Lemma for ~ dalaw . First... |
| dalawlem7 38340 | Lemma for ~ dalaw . Secon... |
| dalawlem8 38341 | Lemma for ~ dalaw . Speci... |
| dalawlem9 38342 | Lemma for ~ dalaw . Speci... |
| dalawlem10 38343 | Lemma for ~ dalaw . Combi... |
| dalawlem11 38344 | Lemma for ~ dalaw . First... |
| dalawlem12 38345 | Lemma for ~ dalaw . Secon... |
| dalawlem13 38346 | Lemma for ~ dalaw . Speci... |
| dalawlem14 38347 | Lemma for ~ dalaw . Combi... |
| dalawlem15 38348 | Lemma for ~ dalaw . Swap ... |
| dalaw 38349 | Desargues's law, derived f... |
| pclfvalN 38352 | The projective subspace cl... |
| pclvalN 38353 | Value of the projective su... |
| pclclN 38354 | Closure of the projective ... |
| elpclN 38355 | Membership in the projecti... |
| elpcliN 38356 | Implication of membership ... |
| pclssN 38357 | Ordering is preserved by s... |
| pclssidN 38358 | A set of atoms is included... |
| pclidN 38359 | The projective subspace cl... |
| pclbtwnN 38360 | A projective subspace sand... |
| pclunN 38361 | The projective subspace cl... |
| pclun2N 38362 | The projective subspace cl... |
| pclfinN 38363 | The projective subspace cl... |
| pclcmpatN 38364 | The set of projective subs... |
| polfvalN 38367 | The projective subspace po... |
| polvalN 38368 | Value of the projective su... |
| polval2N 38369 | Alternate expression for v... |
| polsubN 38370 | The polarity of a set of a... |
| polssatN 38371 | The polarity of a set of a... |
| pol0N 38372 | The polarity of the empty ... |
| pol1N 38373 | The polarity of the whole ... |
| 2pol0N 38374 | The closed subspace closur... |
| polpmapN 38375 | The polarity of a projecti... |
| 2polpmapN 38376 | Double polarity of a proje... |
| 2polvalN 38377 | Value of double polarity. ... |
| 2polssN 38378 | A set of atoms is a subset... |
| 3polN 38379 | Triple polarity cancels to... |
| polcon3N 38380 | Contraposition law for pol... |
| 2polcon4bN 38381 | Contraposition law for pol... |
| polcon2N 38382 | Contraposition law for pol... |
| polcon2bN 38383 | Contraposition law for pol... |
| pclss2polN 38384 | The projective subspace cl... |
| pcl0N 38385 | The projective subspace cl... |
| pcl0bN 38386 | The projective subspace cl... |
| pmaplubN 38387 | The LUB of a projective ma... |
| sspmaplubN 38388 | A set of atoms is a subset... |
| 2pmaplubN 38389 | Double projective map of a... |
| paddunN 38390 | The closure of the project... |
| poldmj1N 38391 | De Morgan's law for polari... |
| pmapj2N 38392 | The projective map of the ... |
| pmapocjN 38393 | The projective map of the ... |
| polatN 38394 | The polarity of the single... |
| 2polatN 38395 | Double polarity of the sin... |
| pnonsingN 38396 | The intersection of a set ... |
| psubclsetN 38399 | The set of closed projecti... |
| ispsubclN 38400 | The predicate "is a closed... |
| psubcliN 38401 | Property of a closed proje... |
| psubcli2N 38402 | Property of a closed proje... |
| psubclsubN 38403 | A closed projective subspa... |
| psubclssatN 38404 | A closed projective subspa... |
| pmapidclN 38405 | Projective map of the LUB ... |
| 0psubclN 38406 | The empty set is a closed ... |
| 1psubclN 38407 | The set of all atoms is a ... |
| atpsubclN 38408 | A point (singleton of an a... |
| pmapsubclN 38409 | A projective map value is ... |
| ispsubcl2N 38410 | Alternate predicate for "i... |
| psubclinN 38411 | The intersection of two cl... |
| paddatclN 38412 | The projective sum of a cl... |
| pclfinclN 38413 | The projective subspace cl... |
| linepsubclN 38414 | A line is a closed project... |
| polsubclN 38415 | A polarity is a closed pro... |
| poml4N 38416 | Orthomodular law for proje... |
| poml5N 38417 | Orthomodular law for proje... |
| poml6N 38418 | Orthomodular law for proje... |
| osumcllem1N 38419 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 38420 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 38421 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 38422 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 38423 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 38424 | Lemma for ~ osumclN . Use... |
| osumcllem7N 38425 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 38426 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 38427 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 38428 | Lemma for ~ osumclN . Con... |
| osumcllem11N 38429 | Lemma for ~ osumclN . (Co... |
| osumclN 38430 | Closure of orthogonal sum.... |
| pmapojoinN 38431 | For orthogonal elements, p... |
| pexmidN 38432 | Excluded middle law for cl... |
| pexmidlem1N 38433 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 38434 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 38435 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 38436 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 38437 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 38438 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 38439 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 38440 | Lemma for ~ pexmidN . The... |
| pexmidALTN 38441 | Excluded middle law for cl... |
| pl42lem1N 38442 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 38443 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 38444 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 38445 | Lemma for ~ pl42N . (Cont... |
| pl42N 38446 | Law holding in a Hilbert l... |
| watfvalN 38455 | The W atoms function. (Co... |
| watvalN 38456 | Value of the W atoms funct... |
| iswatN 38457 | The predicate "is a W atom... |
| lhpset 38458 | The set of co-atoms (latti... |
| islhp 38459 | The predicate "is a co-ato... |
| islhp2 38460 | The predicate "is a co-ato... |
| lhpbase 38461 | A co-atom is a member of t... |
| lhp1cvr 38462 | The lattice unity covers a... |
| lhplt 38463 | An atom under a co-atom is... |
| lhp2lt 38464 | The join of two atoms unde... |
| lhpexlt 38465 | There exists an atom less ... |
| lhp0lt 38466 | A co-atom is greater than ... |
| lhpn0 38467 | A co-atom is nonzero. TOD... |
| lhpexle 38468 | There exists an atom under... |
| lhpexnle 38469 | There exists an atom not u... |
| lhpexle1lem 38470 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 38471 | There exists an atom under... |
| lhpexle2lem 38472 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 38473 | There exists atom under a ... |
| lhpexle3lem 38474 | There exists atom under a ... |
| lhpexle3 38475 | There exists atom under a ... |
| lhpex2leN 38476 | There exist at least two d... |
| lhpoc 38477 | The orthocomplement of a c... |
| lhpoc2N 38478 | The orthocomplement of an ... |
| lhpocnle 38479 | The orthocomplement of a c... |
| lhpocat 38480 | The orthocomplement of a c... |
| lhpocnel 38481 | The orthocomplement of a c... |
| lhpocnel2 38482 | The orthocomplement of a c... |
| lhpjat1 38483 | The join of a co-atom (hyp... |
| lhpjat2 38484 | The join of a co-atom (hyp... |
| lhpj1 38485 | The join of a co-atom (hyp... |
| lhpmcvr 38486 | The meet of a lattice hype... |
| lhpmcvr2 38487 | Alternate way to express t... |
| lhpmcvr3 38488 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 38489 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 38490 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 38491 | Specialization of ~ lhpmcv... |
| lhpm0atN 38492 | If the meet of a lattice h... |
| lhpmat 38493 | An element covered by the ... |
| lhpmatb 38494 | An element covered by the ... |
| lhp2at0 38495 | Join and meet with differe... |
| lhp2atnle 38496 | Inequality for 2 different... |
| lhp2atne 38497 | Inequality for joins with ... |
| lhp2at0nle 38498 | Inequality for 2 different... |
| lhp2at0ne 38499 | Inequality for joins with ... |
| lhpelim 38500 | Eliminate an atom not unde... |
| lhpmod2i2 38501 | Modular law for hyperplane... |
| lhpmod6i1 38502 | Modular law for hyperplane... |
| lhprelat3N 38503 | The Hilbert lattice is rel... |
| cdlemb2 38504 | Given two atoms not under ... |
| lhple 38505 | Property of a lattice elem... |
| lhpat 38506 | Create an atom under a co-... |
| lhpat4N 38507 | Property of an atom under ... |
| lhpat2 38508 | Create an atom under a co-... |
| lhpat3 38509 | There is only one atom und... |
| 4atexlemk 38510 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 38511 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 38512 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 38513 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 38514 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 38515 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 38516 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 38517 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 38518 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 38519 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 38520 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 38521 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 38522 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 38523 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 38524 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 38525 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 38526 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 38527 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 38528 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 38529 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 38530 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 38531 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 38532 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 38533 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 38534 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 38535 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 38536 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 38537 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 38538 | Whenever there are at leas... |
| 4atex 38539 | Whenever there are at leas... |
| 4atex2 38540 | More general version of ~ ... |
| 4atex2-0aOLDN 38541 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 38542 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 38543 | Same as ~ 4atex2 except th... |
| 4atex3 38544 | More general version of ~ ... |
| lautset 38545 | The set of lattice automor... |
| islaut 38546 | The predicate "is a lattic... |
| lautle 38547 | Less-than or equal propert... |
| laut1o 38548 | A lattice automorphism is ... |
| laut11 38549 | One-to-one property of a l... |
| lautcl 38550 | A lattice automorphism val... |
| lautcnvclN 38551 | Reverse closure of a latti... |
| lautcnvle 38552 | Less-than or equal propert... |
| lautcnv 38553 | The converse of a lattice ... |
| lautlt 38554 | Less-than property of a la... |
| lautcvr 38555 | Covering property of a lat... |
| lautj 38556 | Meet property of a lattice... |
| lautm 38557 | Meet property of a lattice... |
| lauteq 38558 | A lattice automorphism arg... |
| idlaut 38559 | The identity function is a... |
| lautco 38560 | The composition of two lat... |
| pautsetN 38561 | The set of projective auto... |
| ispautN 38562 | The predicate "is a projec... |
| ldilfset 38571 | The mapping from fiducial ... |
| ldilset 38572 | The set of lattice dilatio... |
| isldil 38573 | The predicate "is a lattic... |
| ldillaut 38574 | A lattice dilation is an a... |
| ldil1o 38575 | A lattice dilation is a on... |
| ldilval 38576 | Value of a lattice dilatio... |
| idldil 38577 | The identity function is a... |
| ldilcnv 38578 | The converse of a lattice ... |
| ldilco 38579 | The composition of two lat... |
| ltrnfset 38580 | The set of all lattice tra... |
| ltrnset 38581 | The set of lattice transla... |
| isltrn 38582 | The predicate "is a lattic... |
| isltrn2N 38583 | The predicate "is a lattic... |
| ltrnu 38584 | Uniqueness property of a l... |
| ltrnldil 38585 | A lattice translation is a... |
| ltrnlaut 38586 | A lattice translation is a... |
| ltrn1o 38587 | A lattice translation is a... |
| ltrncl 38588 | Closure of a lattice trans... |
| ltrn11 38589 | One-to-one property of a l... |
| ltrncnvnid 38590 | If a translation is differ... |
| ltrncoidN 38591 | Two translations are equal... |
| ltrnle 38592 | Less-than or equal propert... |
| ltrncnvleN 38593 | Less-than or equal propert... |
| ltrnm 38594 | Lattice translation of a m... |
| ltrnj 38595 | Lattice translation of a m... |
| ltrncvr 38596 | Covering property of a lat... |
| ltrnval1 38597 | Value of a lattice transla... |
| ltrnid 38598 | A lattice translation is t... |
| ltrnnid 38599 | If a lattice translation i... |
| ltrnatb 38600 | The lattice translation of... |
| ltrncnvatb 38601 | The converse of the lattic... |
| ltrnel 38602 | The lattice translation of... |
| ltrnat 38603 | The lattice translation of... |
| ltrncnvat 38604 | The converse of the lattic... |
| ltrncnvel 38605 | The converse of the lattic... |
| ltrncoelN 38606 | Composition of lattice tra... |
| ltrncoat 38607 | Composition of lattice tra... |
| ltrncoval 38608 | Two ways to express value ... |
| ltrncnv 38609 | The converse of a lattice ... |
| ltrn11at 38610 | Frequently used one-to-one... |
| ltrneq2 38611 | The equality of two transl... |
| ltrneq 38612 | The equality of two transl... |
| idltrn 38613 | The identity function is a... |
| ltrnmw 38614 | Property of lattice transl... |
| dilfsetN 38615 | The mapping from fiducial ... |
| dilsetN 38616 | The set of dilations for a... |
| isdilN 38617 | The predicate "is a dilati... |
| trnfsetN 38618 | The mapping from fiducial ... |
| trnsetN 38619 | The set of translations fo... |
| istrnN 38620 | The predicate "is a transl... |
| trlfset 38623 | The set of all traces of l... |
| trlset 38624 | The set of traces of latti... |
| trlval 38625 | The value of the trace of ... |
| trlval2 38626 | The value of the trace of ... |
| trlcl 38627 | Closure of the trace of a ... |
| trlcnv 38628 | The trace of the converse ... |
| trljat1 38629 | The value of a translation... |
| trljat2 38630 | The value of a translation... |
| trljat3 38631 | The value of a translation... |
| trlat 38632 | If an atom differs from it... |
| trl0 38633 | If an atom not under the f... |
| trlator0 38634 | The trace of a lattice tra... |
| trlatn0 38635 | The trace of a lattice tra... |
| trlnidat 38636 | The trace of a lattice tra... |
| ltrnnidn 38637 | If a lattice translation i... |
| ltrnideq 38638 | Property of the identity l... |
| trlid0 38639 | The trace of the identity ... |
| trlnidatb 38640 | A lattice translation is n... |
| trlid0b 38641 | A lattice translation is t... |
| trlnid 38642 | Different translations wit... |
| ltrn2ateq 38643 | Property of the equality o... |
| ltrnateq 38644 | If any atom (under ` W ` )... |
| ltrnatneq 38645 | If any atom (under ` W ` )... |
| ltrnatlw 38646 | If the value of an atom eq... |
| trlle 38647 | The trace of a lattice tra... |
| trlne 38648 | The trace of a lattice tra... |
| trlnle 38649 | The atom not under the fid... |
| trlval3 38650 | The value of the trace of ... |
| trlval4 38651 | The value of the trace of ... |
| trlval5 38652 | The value of the trace of ... |
| arglem1N 38653 | Lemma for Desargues's law.... |
| cdlemc1 38654 | Part of proof of Lemma C i... |
| cdlemc2 38655 | Part of proof of Lemma C i... |
| cdlemc3 38656 | Part of proof of Lemma C i... |
| cdlemc4 38657 | Part of proof of Lemma C i... |
| cdlemc5 38658 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 38659 | Lemma for ~ cdlemc . (Con... |
| cdlemc 38660 | Lemma C in [Crawley] p. 11... |
| cdlemd1 38661 | Part of proof of Lemma D i... |
| cdlemd2 38662 | Part of proof of Lemma D i... |
| cdlemd3 38663 | Part of proof of Lemma D i... |
| cdlemd4 38664 | Part of proof of Lemma D i... |
| cdlemd5 38665 | Part of proof of Lemma D i... |
| cdlemd6 38666 | Part of proof of Lemma D i... |
| cdlemd7 38667 | Part of proof of Lemma D i... |
| cdlemd8 38668 | Part of proof of Lemma D i... |
| cdlemd9 38669 | Part of proof of Lemma D i... |
| cdlemd 38670 | If two translations agree ... |
| ltrneq3 38671 | Two translations agree at ... |
| cdleme00a 38672 | Part of proof of Lemma E i... |
| cdleme0aa 38673 | Part of proof of Lemma E i... |
| cdleme0a 38674 | Part of proof of Lemma E i... |
| cdleme0b 38675 | Part of proof of Lemma E i... |
| cdleme0c 38676 | Part of proof of Lemma E i... |
| cdleme0cp 38677 | Part of proof of Lemma E i... |
| cdleme0cq 38678 | Part of proof of Lemma E i... |
| cdleme0dN 38679 | Part of proof of Lemma E i... |
| cdleme0e 38680 | Part of proof of Lemma E i... |
| cdleme0fN 38681 | Part of proof of Lemma E i... |
| cdleme0gN 38682 | Part of proof of Lemma E i... |
| cdlemeulpq 38683 | Part of proof of Lemma E i... |
| cdleme01N 38684 | Part of proof of Lemma E i... |
| cdleme02N 38685 | Part of proof of Lemma E i... |
| cdleme0ex1N 38686 | Part of proof of Lemma E i... |
| cdleme0ex2N 38687 | Part of proof of Lemma E i... |
| cdleme0moN 38688 | Part of proof of Lemma E i... |
| cdleme1b 38689 | Part of proof of Lemma E i... |
| cdleme1 38690 | Part of proof of Lemma E i... |
| cdleme2 38691 | Part of proof of Lemma E i... |
| cdleme3b 38692 | Part of proof of Lemma E i... |
| cdleme3c 38693 | Part of proof of Lemma E i... |
| cdleme3d 38694 | Part of proof of Lemma E i... |
| cdleme3e 38695 | Part of proof of Lemma E i... |
| cdleme3fN 38696 | Part of proof of Lemma E i... |
| cdleme3g 38697 | Part of proof of Lemma E i... |
| cdleme3h 38698 | Part of proof of Lemma E i... |
| cdleme3fa 38699 | Part of proof of Lemma E i... |
| cdleme3 38700 | Part of proof of Lemma E i... |
| cdleme4 38701 | Part of proof of Lemma E i... |
| cdleme4a 38702 | Part of proof of Lemma E i... |
| cdleme5 38703 | Part of proof of Lemma E i... |
| cdleme6 38704 | Part of proof of Lemma E i... |
| cdleme7aa 38705 | Part of proof of Lemma E i... |
| cdleme7a 38706 | Part of proof of Lemma E i... |
| cdleme7b 38707 | Part of proof of Lemma E i... |
| cdleme7c 38708 | Part of proof of Lemma E i... |
| cdleme7d 38709 | Part of proof of Lemma E i... |
| cdleme7e 38710 | Part of proof of Lemma E i... |
| cdleme7ga 38711 | Part of proof of Lemma E i... |
| cdleme7 38712 | Part of proof of Lemma E i... |
| cdleme8 38713 | Part of proof of Lemma E i... |
| cdleme9a 38714 | Part of proof of Lemma E i... |
| cdleme9b 38715 | Utility lemma for Lemma E ... |
| cdleme9 38716 | Part of proof of Lemma E i... |
| cdleme10 38717 | Part of proof of Lemma E i... |
| cdleme8tN 38718 | Part of proof of Lemma E i... |
| cdleme9taN 38719 | Part of proof of Lemma E i... |
| cdleme9tN 38720 | Part of proof of Lemma E i... |
| cdleme10tN 38721 | Part of proof of Lemma E i... |
| cdleme16aN 38722 | Part of proof of Lemma E i... |
| cdleme11a 38723 | Part of proof of Lemma E i... |
| cdleme11c 38724 | Part of proof of Lemma E i... |
| cdleme11dN 38725 | Part of proof of Lemma E i... |
| cdleme11e 38726 | Part of proof of Lemma E i... |
| cdleme11fN 38727 | Part of proof of Lemma E i... |
| cdleme11g 38728 | Part of proof of Lemma E i... |
| cdleme11h 38729 | Part of proof of Lemma E i... |
| cdleme11j 38730 | Part of proof of Lemma E i... |
| cdleme11k 38731 | Part of proof of Lemma E i... |
| cdleme11l 38732 | Part of proof of Lemma E i... |
| cdleme11 38733 | Part of proof of Lemma E i... |
| cdleme12 38734 | Part of proof of Lemma E i... |
| cdleme13 38735 | Part of proof of Lemma E i... |
| cdleme14 38736 | Part of proof of Lemma E i... |
| cdleme15a 38737 | Part of proof of Lemma E i... |
| cdleme15b 38738 | Part of proof of Lemma E i... |
| cdleme15c 38739 | Part of proof of Lemma E i... |
| cdleme15d 38740 | Part of proof of Lemma E i... |
| cdleme15 38741 | Part of proof of Lemma E i... |
| cdleme16b 38742 | Part of proof of Lemma E i... |
| cdleme16c 38743 | Part of proof of Lemma E i... |
| cdleme16d 38744 | Part of proof of Lemma E i... |
| cdleme16e 38745 | Part of proof of Lemma E i... |
| cdleme16f 38746 | Part of proof of Lemma E i... |
| cdleme16g 38747 | Part of proof of Lemma E i... |
| cdleme16 38748 | Part of proof of Lemma E i... |
| cdleme17a 38749 | Part of proof of Lemma E i... |
| cdleme17b 38750 | Lemma leading to ~ cdleme1... |
| cdleme17c 38751 | Part of proof of Lemma E i... |
| cdleme17d1 38752 | Part of proof of Lemma E i... |
| cdleme0nex 38753 | Part of proof of Lemma E i... |
| cdleme18a 38754 | Part of proof of Lemma E i... |
| cdleme18b 38755 | Part of proof of Lemma E i... |
| cdleme18c 38756 | Part of proof of Lemma E i... |
| cdleme22gb 38757 | Utility lemma for Lemma E ... |
| cdleme18d 38758 | Part of proof of Lemma E i... |
| cdlemesner 38759 | Part of proof of Lemma E i... |
| cdlemedb 38760 | Part of proof of Lemma E i... |
| cdlemeda 38761 | Part of proof of Lemma E i... |
| cdlemednpq 38762 | Part of proof of Lemma E i... |
| cdlemednuN 38763 | Part of proof of Lemma E i... |
| cdleme20zN 38764 | Part of proof of Lemma E i... |
| cdleme20y 38765 | Part of proof of Lemma E i... |
| cdleme19a 38766 | Part of proof of Lemma E i... |
| cdleme19b 38767 | Part of proof of Lemma E i... |
| cdleme19c 38768 | Part of proof of Lemma E i... |
| cdleme19d 38769 | Part of proof of Lemma E i... |
| cdleme19e 38770 | Part of proof of Lemma E i... |
| cdleme19f 38771 | Part of proof of Lemma E i... |
| cdleme20aN 38772 | Part of proof of Lemma E i... |
| cdleme20bN 38773 | Part of proof of Lemma E i... |
| cdleme20c 38774 | Part of proof of Lemma E i... |
| cdleme20d 38775 | Part of proof of Lemma E i... |
| cdleme20e 38776 | Part of proof of Lemma E i... |
| cdleme20f 38777 | Part of proof of Lemma E i... |
| cdleme20g 38778 | Part of proof of Lemma E i... |
| cdleme20h 38779 | Part of proof of Lemma E i... |
| cdleme20i 38780 | Part of proof of Lemma E i... |
| cdleme20j 38781 | Part of proof of Lemma E i... |
| cdleme20k 38782 | Part of proof of Lemma E i... |
| cdleme20l1 38783 | Part of proof of Lemma E i... |
| cdleme20l2 38784 | Part of proof of Lemma E i... |
| cdleme20l 38785 | Part of proof of Lemma E i... |
| cdleme20m 38786 | Part of proof of Lemma E i... |
| cdleme20 38787 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 38788 | Part of proof of Lemma E i... |
| cdleme21b 38789 | Part of proof of Lemma E i... |
| cdleme21c 38790 | Part of proof of Lemma E i... |
| cdleme21at 38791 | Part of proof of Lemma E i... |
| cdleme21ct 38792 | Part of proof of Lemma E i... |
| cdleme21d 38793 | Part of proof of Lemma E i... |
| cdleme21e 38794 | Part of proof of Lemma E i... |
| cdleme21f 38795 | Part of proof of Lemma E i... |
| cdleme21g 38796 | Part of proof of Lemma E i... |
| cdleme21h 38797 | Part of proof of Lemma E i... |
| cdleme21i 38798 | Part of proof of Lemma E i... |
| cdleme21j 38799 | Combine ~ cdleme20 and ~ c... |
| cdleme21 38800 | Part of proof of Lemma E i... |
| cdleme21k 38801 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 38802 | Part of proof of Lemma E i... |
| cdleme22a 38803 | Part of proof of Lemma E i... |
| cdleme22b 38804 | Part of proof of Lemma E i... |
| cdleme22cN 38805 | Part of proof of Lemma E i... |
| cdleme22d 38806 | Part of proof of Lemma E i... |
| cdleme22e 38807 | Part of proof of Lemma E i... |
| cdleme22eALTN 38808 | Part of proof of Lemma E i... |
| cdleme22f 38809 | Part of proof of Lemma E i... |
| cdleme22f2 38810 | Part of proof of Lemma E i... |
| cdleme22g 38811 | Part of proof of Lemma E i... |
| cdleme23a 38812 | Part of proof of Lemma E i... |
| cdleme23b 38813 | Part of proof of Lemma E i... |
| cdleme23c 38814 | Part of proof of Lemma E i... |
| cdleme24 38815 | Quantified version of ~ cd... |
| cdleme25a 38816 | Lemma for ~ cdleme25b . (... |
| cdleme25b 38817 | Transform ~ cdleme24 . TO... |
| cdleme25c 38818 | Transform ~ cdleme25b . (... |
| cdleme25dN 38819 | Transform ~ cdleme25c . (... |
| cdleme25cl 38820 | Show closure of the unique... |
| cdleme25cv 38821 | Change bound variables in ... |
| cdleme26e 38822 | Part of proof of Lemma E i... |
| cdleme26ee 38823 | Part of proof of Lemma E i... |
| cdleme26eALTN 38824 | Part of proof of Lemma E i... |
| cdleme26fALTN 38825 | Part of proof of Lemma E i... |
| cdleme26f 38826 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 38827 | Part of proof of Lemma E i... |
| cdleme26f2 38828 | Part of proof of Lemma E i... |
| cdleme27cl 38829 | Part of proof of Lemma E i... |
| cdleme27a 38830 | Part of proof of Lemma E i... |
| cdleme27b 38831 | Lemma for ~ cdleme27N . (... |
| cdleme27N 38832 | Part of proof of Lemma E i... |
| cdleme28a 38833 | Lemma for ~ cdleme25b . T... |
| cdleme28b 38834 | Lemma for ~ cdleme25b . T... |
| cdleme28c 38835 | Part of proof of Lemma E i... |
| cdleme28 38836 | Quantified version of ~ cd... |
| cdleme29ex 38837 | Lemma for ~ cdleme29b . (... |
| cdleme29b 38838 | Transform ~ cdleme28 . (C... |
| cdleme29c 38839 | Transform ~ cdleme28b . (... |
| cdleme29cl 38840 | Show closure of the unique... |
| cdleme30a 38841 | Part of proof of Lemma E i... |
| cdleme31so 38842 | Part of proof of Lemma E i... |
| cdleme31sn 38843 | Part of proof of Lemma E i... |
| cdleme31sn1 38844 | Part of proof of Lemma E i... |
| cdleme31se 38845 | Part of proof of Lemma D i... |
| cdleme31se2 38846 | Part of proof of Lemma D i... |
| cdleme31sc 38847 | Part of proof of Lemma E i... |
| cdleme31sde 38848 | Part of proof of Lemma D i... |
| cdleme31snd 38849 | Part of proof of Lemma D i... |
| cdleme31sdnN 38850 | Part of proof of Lemma E i... |
| cdleme31sn1c 38851 | Part of proof of Lemma E i... |
| cdleme31sn2 38852 | Part of proof of Lemma E i... |
| cdleme31fv 38853 | Part of proof of Lemma E i... |
| cdleme31fv1 38854 | Part of proof of Lemma E i... |
| cdleme31fv1s 38855 | Part of proof of Lemma E i... |
| cdleme31fv2 38856 | Part of proof of Lemma E i... |
| cdleme31id 38857 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 38858 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 38859 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 38860 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 38861 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 38862 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 38863 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 38864 | Part of proof of Lemma E i... |
| cdlemefr29exN 38865 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 38866 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 38867 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 38868 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 38869 | TODO fix comment. (Contri... |
| cdlemefr29clN 38870 | Show closure of the unique... |
| cdleme43frv1snN 38871 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 38872 | Part of proof of Lemma E i... |
| cdlemefr32fva1 38873 | Part of proof of Lemma E i... |
| cdlemefr31fv1 38874 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 38875 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 38876 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 38877 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 38878 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 38879 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 38880 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 38881 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 38882 | Show closure of the unique... |
| cdleme43fsv1snlem 38883 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 38884 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 38885 | Part of proof of Lemma E i... |
| cdlemefs32fva1 38886 | Part of proof of Lemma E i... |
| cdlemefs31fv1 38887 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 38888 | Value of f(r) when r is an... |
| cdlemefs44 38889 | Value of f_s(r) when r is ... |
| cdlemefr45 38890 | Value of f(r) when r is an... |
| cdlemefr45e 38891 | Explicit expansion of ~ cd... |
| cdlemefs45 38892 | Value of f_s(r) when r is ... |
| cdlemefs45ee 38893 | Explicit expansion of ~ cd... |
| cdlemefs45eN 38894 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 38895 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 38896 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 38897 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 38898 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 38899 | Show closure of ` [_ R / s... |
| cdleme32fva 38900 | Part of proof of Lemma D i... |
| cdleme32fva1 38901 | Part of proof of Lemma D i... |
| cdleme32fvaw 38902 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 38903 | Part of proof of Lemma D i... |
| cdleme32a 38904 | Part of proof of Lemma D i... |
| cdleme32b 38905 | Part of proof of Lemma D i... |
| cdleme32c 38906 | Part of proof of Lemma D i... |
| cdleme32d 38907 | Part of proof of Lemma D i... |
| cdleme32e 38908 | Part of proof of Lemma D i... |
| cdleme32f 38909 | Part of proof of Lemma D i... |
| cdleme32le 38910 | Part of proof of Lemma D i... |
| cdleme35a 38911 | Part of proof of Lemma E i... |
| cdleme35fnpq 38912 | Part of proof of Lemma E i... |
| cdleme35b 38913 | Part of proof of Lemma E i... |
| cdleme35c 38914 | Part of proof of Lemma E i... |
| cdleme35d 38915 | Part of proof of Lemma E i... |
| cdleme35e 38916 | Part of proof of Lemma E i... |
| cdleme35f 38917 | Part of proof of Lemma E i... |
| cdleme35g 38918 | Part of proof of Lemma E i... |
| cdleme35h 38919 | Part of proof of Lemma E i... |
| cdleme35h2 38920 | Part of proof of Lemma E i... |
| cdleme35sn2aw 38921 | Part of proof of Lemma E i... |
| cdleme35sn3a 38922 | Part of proof of Lemma E i... |
| cdleme36a 38923 | Part of proof of Lemma E i... |
| cdleme36m 38924 | Part of proof of Lemma E i... |
| cdleme37m 38925 | Part of proof of Lemma E i... |
| cdleme38m 38926 | Part of proof of Lemma E i... |
| cdleme38n 38927 | Part of proof of Lemma E i... |
| cdleme39a 38928 | Part of proof of Lemma E i... |
| cdleme39n 38929 | Part of proof of Lemma E i... |
| cdleme40m 38930 | Part of proof of Lemma E i... |
| cdleme40n 38931 | Part of proof of Lemma E i... |
| cdleme40v 38932 | Part of proof of Lemma E i... |
| cdleme40w 38933 | Part of proof of Lemma E i... |
| cdleme42a 38934 | Part of proof of Lemma E i... |
| cdleme42c 38935 | Part of proof of Lemma E i... |
| cdleme42d 38936 | Part of proof of Lemma E i... |
| cdleme41sn3aw 38937 | Part of proof of Lemma E i... |
| cdleme41sn4aw 38938 | Part of proof of Lemma E i... |
| cdleme41snaw 38939 | Part of proof of Lemma E i... |
| cdleme41fva11 38940 | Part of proof of Lemma E i... |
| cdleme42b 38941 | Part of proof of Lemma E i... |
| cdleme42e 38942 | Part of proof of Lemma E i... |
| cdleme42f 38943 | Part of proof of Lemma E i... |
| cdleme42g 38944 | Part of proof of Lemma E i... |
| cdleme42h 38945 | Part of proof of Lemma E i... |
| cdleme42i 38946 | Part of proof of Lemma E i... |
| cdleme42k 38947 | Part of proof of Lemma E i... |
| cdleme42ke 38948 | Part of proof of Lemma E i... |
| cdleme42keg 38949 | Part of proof of Lemma E i... |
| cdleme42mN 38950 | Part of proof of Lemma E i... |
| cdleme42mgN 38951 | Part of proof of Lemma E i... |
| cdleme43aN 38952 | Part of proof of Lemma E i... |
| cdleme43bN 38953 | Lemma for Lemma E in [Craw... |
| cdleme43cN 38954 | Part of proof of Lemma E i... |
| cdleme43dN 38955 | Part of proof of Lemma E i... |
| cdleme46f2g2 38956 | Conversion for ` G ` to re... |
| cdleme46f2g1 38957 | Conversion for ` G ` to re... |
| cdleme17d2 38958 | Part of proof of Lemma E i... |
| cdleme17d3 38959 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 38960 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 38961 | Part of proof of Lemma E i... |
| cdleme48fv 38962 | Part of proof of Lemma D i... |
| cdleme48fvg 38963 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 38964 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 38965 | TODO: fix comment. TODO: ... |
| cdleme48b 38966 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 38967 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 38968 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 38969 | TODO: fix comment. (Contr... |
| cdleme4gfv 38970 | Part of proof of Lemma D i... |
| cdlemeg47b 38971 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 38972 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 38973 | Value of g_s(r) when r is ... |
| cdlemeg49le 38974 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 38975 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 38976 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 38977 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 38978 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 38979 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 38980 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 38981 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 38982 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 38983 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 38984 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 38985 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 38986 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 38987 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 38988 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 38989 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 38990 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 38991 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 38992 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 38993 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 38994 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 38995 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 38996 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 38997 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 38998 | TODO: fix comment. (Contr... |
| cdleme48gfv1 38999 | TODO: fix comment. (Contr... |
| cdleme48gfv 39000 | TODO: fix comment. (Contr... |
| cdleme48fgv 39001 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 39002 | Part of proof of Lemma D i... |
| cdleme50lebi 39003 | Part of proof of Lemma D i... |
| cdleme50eq 39004 | Part of proof of Lemma D i... |
| cdleme50f 39005 | Part of proof of Lemma D i... |
| cdleme50f1 39006 | Part of proof of Lemma D i... |
| cdleme50rnlem 39007 | Part of proof of Lemma D i... |
| cdleme50rn 39008 | Part of proof of Lemma D i... |
| cdleme50f1o 39009 | Part of proof of Lemma D i... |
| cdleme50laut 39010 | Part of proof of Lemma D i... |
| cdleme50ldil 39011 | Part of proof of Lemma D i... |
| cdleme50trn1 39012 | Part of proof that ` F ` i... |
| cdleme50trn2a 39013 | Part of proof that ` F ` i... |
| cdleme50trn2 39014 | Part of proof that ` F ` i... |
| cdleme50trn12 39015 | Part of proof that ` F ` i... |
| cdleme50trn3 39016 | Part of proof that ` F ` i... |
| cdleme50trn123 39017 | Part of proof that ` F ` i... |
| cdleme51finvfvN 39018 | Part of proof of Lemma E i... |
| cdleme51finvN 39019 | Part of proof of Lemma E i... |
| cdleme50ltrn 39020 | Part of proof of Lemma E i... |
| cdleme51finvtrN 39021 | Part of proof of Lemma E i... |
| cdleme50ex 39022 | Part of Lemma E in [Crawle... |
| cdleme 39023 | Lemma E in [Crawley] p. 11... |
| cdlemf1 39024 | Part of Lemma F in [Crawle... |
| cdlemf2 39025 | Part of Lemma F in [Crawle... |
| cdlemf 39026 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 39027 | ~ cdlemf with additional c... |
| cdlemftr3 39028 | Special case of ~ cdlemf s... |
| cdlemftr2 39029 | Special case of ~ cdlemf s... |
| cdlemftr1 39030 | Part of proof of Lemma G o... |
| cdlemftr0 39031 | Special case of ~ cdlemf s... |
| trlord 39032 | The ordering of two Hilber... |
| cdlemg1a 39033 | Shorter expression for ` G... |
| cdlemg1b2 39034 | This theorem can be used t... |
| cdlemg1idlemN 39035 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 39036 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 39037 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 39038 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 39039 | This theorem can be used t... |
| cdlemg1idN 39040 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 39041 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 39042 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 39043 | Version of ~ cdleme51finvt... |
| ltrniotaval 39044 | Value of the unique transl... |
| ltrniotacnvval 39045 | Converse value of the uniq... |
| ltrniotaidvalN 39046 | Value of the unique transl... |
| ltrniotavalbN 39047 | Value of the unique transl... |
| cdlemeiota 39048 | A translation is uniquely ... |
| cdlemg1ci2 39049 | Any function of the form o... |
| cdlemg1cN 39050 | Any translation belongs to... |
| cdlemg1cex 39051 | Any translation is one of ... |
| cdlemg2cN 39052 | Any translation belongs to... |
| cdlemg2dN 39053 | This theorem can be used t... |
| cdlemg2cex 39054 | Any translation is one of ... |
| cdlemg2ce 39055 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 39056 | Part of proof of Lemma E i... |
| cdlemg2fvlem 39057 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 39058 | ~ cdleme42keg with simpler... |
| cdlemg2idN 39059 | Version of ~ cdleme31id wi... |
| cdlemg3a 39060 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 39061 | TODO: Replace this with ~... |
| cdlemg2fv 39062 | Value of a translation in ... |
| cdlemg2fv2 39063 | Value of a translation in ... |
| cdlemg2k 39064 | ~ cdleme42keg with simpler... |
| cdlemg2kq 39065 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 39066 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 39067 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 39068 | TODO: Is there a simpler ... |
| cdlemb3 39069 | Given two atoms not under ... |
| cdlemg7fvbwN 39070 | Properties of a translatio... |
| cdlemg4a 39071 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 39072 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 39073 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 39074 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 39075 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 39076 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 39077 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 39078 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 39079 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 39080 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 39081 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 39082 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 39083 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 39084 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 39085 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 39086 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 39087 | Value of a translation com... |
| cdlemg7aN 39088 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 39089 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 39090 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 39091 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 39092 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 39093 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 39094 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 39095 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 39096 | The triples ` <. P , ( F `... |
| cdlemg9 39097 | The triples ` <. P , ( F `... |
| cdlemg10b 39098 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 39099 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 39100 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 39101 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 39102 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 39103 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 39104 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 39105 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 39106 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 39107 | The triples ` <. P , ( F `... |
| cdlemg12c 39108 | The triples ` <. P , ( F `... |
| cdlemg12d 39109 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 39110 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 39111 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 39112 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 39113 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 39114 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 39115 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 39116 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 39117 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 39118 | Eliminate the ` ( F `` P )... |
| cdlemg15 39119 | Eliminate the ` ( (... |
| cdlemg16 39120 | Part of proof of Lemma G o... |
| cdlemg16ALTN 39121 | This version of ~ cdlemg16... |
| cdlemg16z 39122 | Eliminate ` ( ( F `... |
| cdlemg16zz 39123 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 39124 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 39125 | Part of proof of Lemma G i... |
| cdlemg17dN 39126 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 39127 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 39128 | TODO: fix comment. (Contr... |
| cdlemg17f 39129 | TODO: fix comment. (Contr... |
| cdlemg17g 39130 | TODO: fix comment. (Contr... |
| cdlemg17h 39131 | TODO: fix comment. (Contr... |
| cdlemg17i 39132 | TODO: fix comment. (Contr... |
| cdlemg17ir 39133 | TODO: fix comment. (Contr... |
| cdlemg17j 39134 | TODO: fix comment. (Contr... |
| cdlemg17pq 39135 | Utility theorem for swappi... |
| cdlemg17bq 39136 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 39137 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 39138 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 39139 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 39140 | Part of Lemma G of [Crawle... |
| cdlemg18a 39141 | Show two lines are differe... |
| cdlemg18b 39142 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 39143 | Show two lines intersect a... |
| cdlemg18d 39144 | Show two lines intersect a... |
| cdlemg18 39145 | Show two lines intersect a... |
| cdlemg19a 39146 | Show two lines intersect a... |
| cdlemg19 39147 | Show two lines intersect a... |
| cdlemg20 39148 | Show two lines intersect a... |
| cdlemg21 39149 | Version of cdlemg19 with `... |
| cdlemg22 39150 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 39151 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 39152 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 39153 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 39154 | ~ cdlemg16zz restated for ... |
| cdlemg27a 39155 | For use with case when ` (... |
| cdlemg28a 39156 | Part of proof of Lemma G o... |
| cdlemg31b0N 39157 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 39158 | TODO: Fix comment. (Cont... |
| cdlemg27b 39159 | TODO: Fix comment. (Cont... |
| cdlemg31a 39160 | TODO: fix comment. (Contr... |
| cdlemg31b 39161 | TODO: fix comment. (Contr... |
| cdlemg31c 39162 | Show that when ` N ` is an... |
| cdlemg31d 39163 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 39164 | TODO: Fix comment. (Cont... |
| cdlemg33c0 39165 | TODO: Fix comment. (Cont... |
| cdlemg28b 39166 | Part of proof of Lemma G o... |
| cdlemg28 39167 | Part of proof of Lemma G o... |
| cdlemg29 39168 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 39169 | TODO: Fix comment. (Cont... |
| cdlemg33b 39170 | TODO: Fix comment. (Cont... |
| cdlemg33c 39171 | TODO: Fix comment. (Cont... |
| cdlemg33d 39172 | TODO: Fix comment. (Cont... |
| cdlemg33e 39173 | TODO: Fix comment. (Cont... |
| cdlemg33 39174 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 39175 | Use cdlemg33 to eliminate ... |
| cdlemg35 39176 | TODO: Fix comment. TODO:... |
| cdlemg36 39177 | Use cdlemg35 to eliminate ... |
| cdlemg38 39178 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 39179 | Eliminate ` =/= ` conditio... |
| cdlemg40 39180 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 39181 | Convert ~ cdlemg40 to func... |
| ltrnco 39182 | The composition of two tra... |
| trlcocnv 39183 | Swap the arguments of the ... |
| trlcoabs 39184 | Absorption into a composit... |
| trlcoabs2N 39185 | Absorption of the trace of... |
| trlcoat 39186 | The trace of a composition... |
| trlcocnvat 39187 | Commonly used special case... |
| trlconid 39188 | The composition of two dif... |
| trlcolem 39189 | Lemma for ~ trlco . (Cont... |
| trlco 39190 | The trace of a composition... |
| trlcone 39191 | If two translations have d... |
| cdlemg42 39192 | Part of proof of Lemma G o... |
| cdlemg43 39193 | Part of proof of Lemma G o... |
| cdlemg44a 39194 | Part of proof of Lemma G o... |
| cdlemg44b 39195 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 39196 | Part of proof of Lemma G o... |
| cdlemg47a 39197 | TODO: fix comment. TODO: ... |
| cdlemg46 39198 | Part of proof of Lemma G o... |
| cdlemg47 39199 | Part of proof of Lemma G o... |
| cdlemg48 39200 | Eliminate ` h ` from ~ cdl... |
| ltrncom 39201 | Composition is commutative... |
| ltrnco4 39202 | Rearrange a composition of... |
| trljco 39203 | Trace joined with trace of... |
| trljco2 39204 | Trace joined with trace of... |
| tgrpfset 39207 | The translation group maps... |
| tgrpset 39208 | The translation group for ... |
| tgrpbase 39209 | The base set of the transl... |
| tgrpopr 39210 | The group operation of the... |
| tgrpov 39211 | The group operation value ... |
| tgrpgrplem 39212 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 39213 | The translation group is a... |
| tgrpabl 39214 | The translation group is a... |
| tendofset 39221 | The set of all trace-prese... |
| tendoset 39222 | The set of trace-preservin... |
| istendo 39223 | The predicate "is a trace-... |
| tendotp 39224 | Trace-preserving property ... |
| istendod 39225 | Deduce the predicate "is a... |
| tendof 39226 | Functionality of a trace-p... |
| tendoeq1 39227 | Condition determining equa... |
| tendovalco 39228 | Value of composition of tr... |
| tendocoval 39229 | Value of composition of en... |
| tendocl 39230 | Closure of a trace-preserv... |
| tendoco2 39231 | Distribution of compositio... |
| tendoidcl 39232 | The identity is a trace-pr... |
| tendo1mul 39233 | Multiplicative identity mu... |
| tendo1mulr 39234 | Multiplicative identity mu... |
| tendococl 39235 | The composition of two tra... |
| tendoid 39236 | The identity value of a tr... |
| tendoeq2 39237 | Condition determining equa... |
| tendoplcbv 39238 | Define sum operation for t... |
| tendopl 39239 | Value of endomorphism sum ... |
| tendopl2 39240 | Value of result of endomor... |
| tendoplcl2 39241 | Value of result of endomor... |
| tendoplco2 39242 | Value of result of endomor... |
| tendopltp 39243 | Trace-preserving property ... |
| tendoplcl 39244 | Endomorphism sum is a trac... |
| tendoplcom 39245 | The endomorphism sum opera... |
| tendoplass 39246 | The endomorphism sum opera... |
| tendodi1 39247 | Endomorphism composition d... |
| tendodi2 39248 | Endomorphism composition d... |
| tendo0cbv 39249 | Define additive identity f... |
| tendo02 39250 | Value of additive identity... |
| tendo0co2 39251 | The additive identity trac... |
| tendo0tp 39252 | Trace-preserving property ... |
| tendo0cl 39253 | The additive identity is a... |
| tendo0pl 39254 | Property of the additive i... |
| tendo0plr 39255 | Property of the additive i... |
| tendoicbv 39256 | Define inverse function fo... |
| tendoi 39257 | Value of inverse endomorph... |
| tendoi2 39258 | Value of additive inverse ... |
| tendoicl 39259 | Closure of the additive in... |
| tendoipl 39260 | Property of the additive i... |
| tendoipl2 39261 | Property of the additive i... |
| erngfset 39262 | The division rings on trac... |
| erngset 39263 | The division ring on trace... |
| erngbase 39264 | The base set of the divisi... |
| erngfplus 39265 | Ring addition operation. ... |
| erngplus 39266 | Ring addition operation. ... |
| erngplus2 39267 | Ring addition operation. ... |
| erngfmul 39268 | Ring multiplication operat... |
| erngmul 39269 | Ring addition operation. ... |
| erngfset-rN 39270 | The division rings on trac... |
| erngset-rN 39271 | The division ring on trace... |
| erngbase-rN 39272 | The base set of the divisi... |
| erngfplus-rN 39273 | Ring addition operation. ... |
| erngplus-rN 39274 | Ring addition operation. ... |
| erngplus2-rN 39275 | Ring addition operation. ... |
| erngfmul-rN 39276 | Ring multiplication operat... |
| erngmul-rN 39277 | Ring addition operation. ... |
| cdlemh1 39278 | Part of proof of Lemma H o... |
| cdlemh2 39279 | Part of proof of Lemma H o... |
| cdlemh 39280 | Lemma H of [Crawley] p. 11... |
| cdlemi1 39281 | Part of proof of Lemma I o... |
| cdlemi2 39282 | Part of proof of Lemma I o... |
| cdlemi 39283 | Lemma I of [Crawley] p. 11... |
| cdlemj1 39284 | Part of proof of Lemma J o... |
| cdlemj2 39285 | Part of proof of Lemma J o... |
| cdlemj3 39286 | Part of proof of Lemma J o... |
| tendocan 39287 | Cancellation law: if the v... |
| tendoid0 39288 | A trace-preserving endomor... |
| tendo0mul 39289 | Additive identity multipli... |
| tendo0mulr 39290 | Additive identity multipli... |
| tendo1ne0 39291 | The identity (unity) is no... |
| tendoconid 39292 | The composition (product) ... |
| tendotr 39293 | The trace of the value of ... |
| cdlemk1 39294 | Part of proof of Lemma K o... |
| cdlemk2 39295 | Part of proof of Lemma K o... |
| cdlemk3 39296 | Part of proof of Lemma K o... |
| cdlemk4 39297 | Part of proof of Lemma K o... |
| cdlemk5a 39298 | Part of proof of Lemma K o... |
| cdlemk5 39299 | Part of proof of Lemma K o... |
| cdlemk6 39300 | Part of proof of Lemma K o... |
| cdlemk8 39301 | Part of proof of Lemma K o... |
| cdlemk9 39302 | Part of proof of Lemma K o... |
| cdlemk9bN 39303 | Part of proof of Lemma K o... |
| cdlemki 39304 | Part of proof of Lemma K o... |
| cdlemkvcl 39305 | Part of proof of Lemma K o... |
| cdlemk10 39306 | Part of proof of Lemma K o... |
| cdlemksv 39307 | Part of proof of Lemma K o... |
| cdlemksel 39308 | Part of proof of Lemma K o... |
| cdlemksat 39309 | Part of proof of Lemma K o... |
| cdlemksv2 39310 | Part of proof of Lemma K o... |
| cdlemk7 39311 | Part of proof of Lemma K o... |
| cdlemk11 39312 | Part of proof of Lemma K o... |
| cdlemk12 39313 | Part of proof of Lemma K o... |
| cdlemkoatnle 39314 | Utility lemma. (Contribut... |
| cdlemk13 39315 | Part of proof of Lemma K o... |
| cdlemkole 39316 | Utility lemma. (Contribut... |
| cdlemk14 39317 | Part of proof of Lemma K o... |
| cdlemk15 39318 | Part of proof of Lemma K o... |
| cdlemk16a 39319 | Part of proof of Lemma K o... |
| cdlemk16 39320 | Part of proof of Lemma K o... |
| cdlemk17 39321 | Part of proof of Lemma K o... |
| cdlemk1u 39322 | Part of proof of Lemma K o... |
| cdlemk5auN 39323 | Part of proof of Lemma K o... |
| cdlemk5u 39324 | Part of proof of Lemma K o... |
| cdlemk6u 39325 | Part of proof of Lemma K o... |
| cdlemkj 39326 | Part of proof of Lemma K o... |
| cdlemkuvN 39327 | Part of proof of Lemma K o... |
| cdlemkuel 39328 | Part of proof of Lemma K o... |
| cdlemkuat 39329 | Part of proof of Lemma K o... |
| cdlemkuv2 39330 | Part of proof of Lemma K o... |
| cdlemk18 39331 | Part of proof of Lemma K o... |
| cdlemk19 39332 | Part of proof of Lemma K o... |
| cdlemk7u 39333 | Part of proof of Lemma K o... |
| cdlemk11u 39334 | Part of proof of Lemma K o... |
| cdlemk12u 39335 | Part of proof of Lemma K o... |
| cdlemk21N 39336 | Part of proof of Lemma K o... |
| cdlemk20 39337 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 39338 | Utility lemma. (Contribut... |
| cdlemk13-2N 39339 | Part of proof of Lemma K o... |
| cdlemkole-2N 39340 | Utility lemma. (Contribut... |
| cdlemk14-2N 39341 | Part of proof of Lemma K o... |
| cdlemk15-2N 39342 | Part of proof of Lemma K o... |
| cdlemk16-2N 39343 | Part of proof of Lemma K o... |
| cdlemk17-2N 39344 | Part of proof of Lemma K o... |
| cdlemkj-2N 39345 | Part of proof of Lemma K o... |
| cdlemkuv-2N 39346 | Part of proof of Lemma K o... |
| cdlemkuel-2N 39347 | Part of proof of Lemma K o... |
| cdlemkuv2-2 39348 | Part of proof of Lemma K o... |
| cdlemk18-2N 39349 | Part of proof of Lemma K o... |
| cdlemk19-2N 39350 | Part of proof of Lemma K o... |
| cdlemk7u-2N 39351 | Part of proof of Lemma K o... |
| cdlemk11u-2N 39352 | Part of proof of Lemma K o... |
| cdlemk12u-2N 39353 | Part of proof of Lemma K o... |
| cdlemk21-2N 39354 | Part of proof of Lemma K o... |
| cdlemk20-2N 39355 | Part of proof of Lemma K o... |
| cdlemk22 39356 | Part of proof of Lemma K o... |
| cdlemk30 39357 | Part of proof of Lemma K o... |
| cdlemkuu 39358 | Convert between function a... |
| cdlemk31 39359 | Part of proof of Lemma K o... |
| cdlemk32 39360 | Part of proof of Lemma K o... |
| cdlemkuel-3 39361 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 39362 | Part of proof of Lemma K o... |
| cdlemk18-3N 39363 | Part of proof of Lemma K o... |
| cdlemk22-3 39364 | Part of proof of Lemma K o... |
| cdlemk23-3 39365 | Part of proof of Lemma K o... |
| cdlemk24-3 39366 | Part of proof of Lemma K o... |
| cdlemk25-3 39367 | Part of proof of Lemma K o... |
| cdlemk26b-3 39368 | Part of proof of Lemma K o... |
| cdlemk26-3 39369 | Part of proof of Lemma K o... |
| cdlemk27-3 39370 | Part of proof of Lemma K o... |
| cdlemk28-3 39371 | Part of proof of Lemma K o... |
| cdlemk33N 39372 | Part of proof of Lemma K o... |
| cdlemk34 39373 | Part of proof of Lemma K o... |
| cdlemk29-3 39374 | Part of proof of Lemma K o... |
| cdlemk35 39375 | Part of proof of Lemma K o... |
| cdlemk36 39376 | Part of proof of Lemma K o... |
| cdlemk37 39377 | Part of proof of Lemma K o... |
| cdlemk38 39378 | Part of proof of Lemma K o... |
| cdlemk39 39379 | Part of proof of Lemma K o... |
| cdlemk40 39380 | TODO: fix comment. (Contr... |
| cdlemk40t 39381 | TODO: fix comment. (Contr... |
| cdlemk40f 39382 | TODO: fix comment. (Contr... |
| cdlemk41 39383 | Part of proof of Lemma K o... |
| cdlemkfid1N 39384 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 39385 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 39386 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 39387 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 39388 | TODO: is this useful or sh... |
| cdlemky 39389 | Part of proof of Lemma K o... |
| cdlemkyu 39390 | Convert between function a... |
| cdlemkyuu 39391 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 39392 | Part of proof of Lemma K o... |
| cdlemk19ylem 39393 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 39394 | Part of proof of Lemma K o... |
| cdlemk19y 39395 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 39396 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 39397 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 39398 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 39399 | The value of the tau funct... |
| cdlemk35s 39400 | Substitution version of ~ ... |
| cdlemk35s-id 39401 | Substitution version of ~ ... |
| cdlemk39s 39402 | Substitution version of ~ ... |
| cdlemk39s-id 39403 | Substitution version of ~ ... |
| cdlemk42 39404 | Part of proof of Lemma K o... |
| cdlemk19xlem 39405 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 39406 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 39407 | Part of proof of Lemma K o... |
| cdlemk11tc 39408 | Part of proof of Lemma K o... |
| cdlemk11t 39409 | Part of proof of Lemma K o... |
| cdlemk45 39410 | Part of proof of Lemma K o... |
| cdlemk46 39411 | Part of proof of Lemma K o... |
| cdlemk47 39412 | Part of proof of Lemma K o... |
| cdlemk48 39413 | Part of proof of Lemma K o... |
| cdlemk49 39414 | Part of proof of Lemma K o... |
| cdlemk50 39415 | Part of proof of Lemma K o... |
| cdlemk51 39416 | Part of proof of Lemma K o... |
| cdlemk52 39417 | Part of proof of Lemma K o... |
| cdlemk53a 39418 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 39419 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 39420 | Part of proof of Lemma K o... |
| cdlemk54 39421 | Part of proof of Lemma K o... |
| cdlemk55a 39422 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 39423 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 39424 | Part of proof of Lemma K o... |
| cdlemkyyN 39425 | Part of proof of Lemma K o... |
| cdlemk43N 39426 | Part of proof of Lemma K o... |
| cdlemk35u 39427 | Substitution version of ~ ... |
| cdlemk55u1 39428 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 39429 | Part of proof of Lemma K o... |
| cdlemk39u1 39430 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 39431 | Part of proof of Lemma K o... |
| cdlemk19u1 39432 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 39433 | Part of Lemma K of [Crawle... |
| cdlemk56 39434 | Part of Lemma K of [Crawle... |
| cdlemk19w 39435 | Use a fixed element to eli... |
| cdlemk56w 39436 | Use a fixed element to eli... |
| cdlemk 39437 | Lemma K of [Crawley] p. 11... |
| tendoex 39438 | Generalization of Lemma K ... |
| cdleml1N 39439 | Part of proof of Lemma L o... |
| cdleml2N 39440 | Part of proof of Lemma L o... |
| cdleml3N 39441 | Part of proof of Lemma L o... |
| cdleml4N 39442 | Part of proof of Lemma L o... |
| cdleml5N 39443 | Part of proof of Lemma L o... |
| cdleml6 39444 | Part of proof of Lemma L o... |
| cdleml7 39445 | Part of proof of Lemma L o... |
| cdleml8 39446 | Part of proof of Lemma L o... |
| cdleml9 39447 | Part of proof of Lemma L o... |
| dva1dim 39448 | Two expressions for the 1-... |
| dvhb1dimN 39449 | Two expressions for the 1-... |
| erng1lem 39450 | Value of the endomorphism ... |
| erngdvlem1 39451 | Lemma for ~ eringring . (... |
| erngdvlem2N 39452 | Lemma for ~ eringring . (... |
| erngdvlem3 39453 | Lemma for ~ eringring . (... |
| erngdvlem4 39454 | Lemma for ~ erngdv . (Con... |
| eringring 39455 | An endomorphism ring is a ... |
| erngdv 39456 | An endomorphism ring is a ... |
| erng0g 39457 | The division ring zero of ... |
| erng1r 39458 | The division ring unity of... |
| erngdvlem1-rN 39459 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 39460 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 39461 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 39462 | Lemma for ~ erngdv . (Con... |
| erngring-rN 39463 | An endomorphism ring is a ... |
| erngdv-rN 39464 | An endomorphism ring is a ... |
| dvafset 39467 | The constructed partial ve... |
| dvaset 39468 | The constructed partial ve... |
| dvasca 39469 | The ring base set of the c... |
| dvabase 39470 | The ring base set of the c... |
| dvafplusg 39471 | Ring addition operation fo... |
| dvaplusg 39472 | Ring addition operation fo... |
| dvaplusgv 39473 | Ring addition operation fo... |
| dvafmulr 39474 | Ring multiplication operat... |
| dvamulr 39475 | Ring multiplication operat... |
| dvavbase 39476 | The vectors (vector base s... |
| dvafvadd 39477 | The vector sum operation f... |
| dvavadd 39478 | Ring addition operation fo... |
| dvafvsca 39479 | Ring addition operation fo... |
| dvavsca 39480 | Ring addition operation fo... |
| tendospcl 39481 | Closure of endomorphism sc... |
| tendospass 39482 | Associative law for endomo... |
| tendospdi1 39483 | Forward distributive law f... |
| tendocnv 39484 | Converse of a trace-preser... |
| tendospdi2 39485 | Reverse distributive law f... |
| tendospcanN 39486 | Cancellation law for trace... |
| dvaabl 39487 | The constructed partial ve... |
| dvalveclem 39488 | Lemma for ~ dvalvec . (Co... |
| dvalvec 39489 | The constructed partial ve... |
| dva0g 39490 | The zero vector of partial... |
| diaffval 39493 | The partial isomorphism A ... |
| diafval 39494 | The partial isomorphism A ... |
| diaval 39495 | The partial isomorphism A ... |
| diaelval 39496 | Member of the partial isom... |
| diafn 39497 | Functionality and domain o... |
| diadm 39498 | Domain of the partial isom... |
| diaeldm 39499 | Member of domain of the pa... |
| diadmclN 39500 | A member of domain of the ... |
| diadmleN 39501 | A member of domain of the ... |
| dian0 39502 | The value of the partial i... |
| dia0eldmN 39503 | The lattice zero belongs t... |
| dia1eldmN 39504 | The fiducial hyperplane (t... |
| diass 39505 | The value of the partial i... |
| diael 39506 | A member of the value of t... |
| diatrl 39507 | Trace of a member of the p... |
| diaelrnN 39508 | Any value of the partial i... |
| dialss 39509 | The value of partial isomo... |
| diaord 39510 | The partial isomorphism A ... |
| dia11N 39511 | The partial isomorphism A ... |
| diaf11N 39512 | The partial isomorphism A ... |
| diaclN 39513 | Closure of partial isomorp... |
| diacnvclN 39514 | Closure of partial isomorp... |
| dia0 39515 | The value of the partial i... |
| dia1N 39516 | The value of the partial i... |
| dia1elN 39517 | The largest subspace in th... |
| diaglbN 39518 | Partial isomorphism A of a... |
| diameetN 39519 | Partial isomorphism A of a... |
| diainN 39520 | Inverse partial isomorphis... |
| diaintclN 39521 | The intersection of partia... |
| diasslssN 39522 | The partial isomorphism A ... |
| diassdvaN 39523 | The partial isomorphism A ... |
| dia1dim 39524 | Two expressions for the 1-... |
| dia1dim2 39525 | Two expressions for a 1-di... |
| dia1dimid 39526 | A vector (translation) bel... |
| dia2dimlem1 39527 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 39528 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 39529 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 39530 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 39531 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 39532 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 39533 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 39534 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 39535 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 39536 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 39537 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 39538 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 39539 | Lemma for ~ dia2dim . Eli... |
| dia2dim 39540 | A two-dimensional subspace... |
| dvhfset 39543 | The constructed full vecto... |
| dvhset 39544 | The constructed full vecto... |
| dvhsca 39545 | The ring of scalars of the... |
| dvhbase 39546 | The ring base set of the c... |
| dvhfplusr 39547 | Ring addition operation fo... |
| dvhfmulr 39548 | Ring multiplication operat... |
| dvhmulr 39549 | Ring multiplication operat... |
| dvhvbase 39550 | The vectors (vector base s... |
| dvhelvbasei 39551 | Vector membership in the c... |
| dvhvaddcbv 39552 | Change bound variables to ... |
| dvhvaddval 39553 | The vector sum operation f... |
| dvhfvadd 39554 | The vector sum operation f... |
| dvhvadd 39555 | The vector sum operation f... |
| dvhopvadd 39556 | The vector sum operation f... |
| dvhopvadd2 39557 | The vector sum operation f... |
| dvhvaddcl 39558 | Closure of the vector sum ... |
| dvhvaddcomN 39559 | Commutativity of vector su... |
| dvhvaddass 39560 | Associativity of vector su... |
| dvhvscacbv 39561 | Change bound variables to ... |
| dvhvscaval 39562 | The scalar product operati... |
| dvhfvsca 39563 | Scalar product operation f... |
| dvhvsca 39564 | Scalar product operation f... |
| dvhopvsca 39565 | Scalar product operation f... |
| dvhvscacl 39566 | Closure of the scalar prod... |
| tendoinvcl 39567 | Closure of multiplicative ... |
| tendolinv 39568 | Left multiplicative invers... |
| tendorinv 39569 | Right multiplicative inver... |
| dvhgrp 39570 | The full vector space ` U ... |
| dvhlveclem 39571 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 39572 | The full vector space ` U ... |
| dvhlmod 39573 | The full vector space ` U ... |
| dvh0g 39574 | The zero vector of vector ... |
| dvheveccl 39575 | Properties of a unit vecto... |
| dvhopclN 39576 | Closure of a ` DVecH ` vec... |
| dvhopaddN 39577 | Sum of ` DVecH ` vectors e... |
| dvhopspN 39578 | Scalar product of ` DVecH ... |
| dvhopN 39579 | Decompose a ` DVecH ` vect... |
| dvhopellsm 39580 | Ordered pair membership in... |
| cdlemm10N 39581 | The image of the map ` G `... |
| docaffvalN 39584 | Subspace orthocomplement f... |
| docafvalN 39585 | Subspace orthocomplement f... |
| docavalN 39586 | Subspace orthocomplement f... |
| docaclN 39587 | Closure of subspace orthoc... |
| diaocN 39588 | Value of partial isomorphi... |
| doca2N 39589 | Double orthocomplement of ... |
| doca3N 39590 | Double orthocomplement of ... |
| dvadiaN 39591 | Any closed subspace is a m... |
| diarnN 39592 | Partial isomorphism A maps... |
| diaf1oN 39593 | The partial isomorphism A ... |
| djaffvalN 39596 | Subspace join for ` DVecA ... |
| djafvalN 39597 | Subspace join for ` DVecA ... |
| djavalN 39598 | Subspace join for ` DVecA ... |
| djaclN 39599 | Closure of subspace join f... |
| djajN 39600 | Transfer lattice join to `... |
| dibffval 39603 | The partial isomorphism B ... |
| dibfval 39604 | The partial isomorphism B ... |
| dibval 39605 | The partial isomorphism B ... |
| dibopelvalN 39606 | Member of the partial isom... |
| dibval2 39607 | Value of the partial isomo... |
| dibopelval2 39608 | Member of the partial isom... |
| dibval3N 39609 | Value of the partial isomo... |
| dibelval3 39610 | Member of the partial isom... |
| dibopelval3 39611 | Member of the partial isom... |
| dibelval1st 39612 | Membership in value of the... |
| dibelval1st1 39613 | Membership in value of the... |
| dibelval1st2N 39614 | Membership in value of the... |
| dibelval2nd 39615 | Membership in value of the... |
| dibn0 39616 | The value of the partial i... |
| dibfna 39617 | Functionality and domain o... |
| dibdiadm 39618 | Domain of the partial isom... |
| dibfnN 39619 | Functionality and domain o... |
| dibdmN 39620 | Domain of the partial isom... |
| dibeldmN 39621 | Member of domain of the pa... |
| dibord 39622 | The isomorphism B for a la... |
| dib11N 39623 | The isomorphism B for a la... |
| dibf11N 39624 | The partial isomorphism A ... |
| dibclN 39625 | Closure of partial isomorp... |
| dibvalrel 39626 | The value of partial isomo... |
| dib0 39627 | The value of partial isomo... |
| dib1dim 39628 | Two expressions for the 1-... |
| dibglbN 39629 | Partial isomorphism B of a... |
| dibintclN 39630 | The intersection of partia... |
| dib1dim2 39631 | Two expressions for a 1-di... |
| dibss 39632 | The partial isomorphism B ... |
| diblss 39633 | The value of partial isomo... |
| diblsmopel 39634 | Membership in subspace sum... |
| dicffval 39637 | The partial isomorphism C ... |
| dicfval 39638 | The partial isomorphism C ... |
| dicval 39639 | The partial isomorphism C ... |
| dicopelval 39640 | Membership in value of the... |
| dicelvalN 39641 | Membership in value of the... |
| dicval2 39642 | The partial isomorphism C ... |
| dicelval3 39643 | Member of the partial isom... |
| dicopelval2 39644 | Membership in value of the... |
| dicelval2N 39645 | Membership in value of the... |
| dicfnN 39646 | Functionality and domain o... |
| dicdmN 39647 | Domain of the partial isom... |
| dicvalrelN 39648 | The value of partial isomo... |
| dicssdvh 39649 | The partial isomorphism C ... |
| dicelval1sta 39650 | Membership in value of the... |
| dicelval1stN 39651 | Membership in value of the... |
| dicelval2nd 39652 | Membership in value of the... |
| dicvaddcl 39653 | Membership in value of the... |
| dicvscacl 39654 | Membership in value of the... |
| dicn0 39655 | The value of the partial i... |
| diclss 39656 | The value of partial isomo... |
| diclspsn 39657 | The value of isomorphism C... |
| cdlemn2 39658 | Part of proof of Lemma N o... |
| cdlemn2a 39659 | Part of proof of Lemma N o... |
| cdlemn3 39660 | Part of proof of Lemma N o... |
| cdlemn4 39661 | Part of proof of Lemma N o... |
| cdlemn4a 39662 | Part of proof of Lemma N o... |
| cdlemn5pre 39663 | Part of proof of Lemma N o... |
| cdlemn5 39664 | Part of proof of Lemma N o... |
| cdlemn6 39665 | Part of proof of Lemma N o... |
| cdlemn7 39666 | Part of proof of Lemma N o... |
| cdlemn8 39667 | Part of proof of Lemma N o... |
| cdlemn9 39668 | Part of proof of Lemma N o... |
| cdlemn10 39669 | Part of proof of Lemma N o... |
| cdlemn11a 39670 | Part of proof of Lemma N o... |
| cdlemn11b 39671 | Part of proof of Lemma N o... |
| cdlemn11c 39672 | Part of proof of Lemma N o... |
| cdlemn11pre 39673 | Part of proof of Lemma N o... |
| cdlemn11 39674 | Part of proof of Lemma N o... |
| cdlemn 39675 | Lemma N of [Crawley] p. 12... |
| dihordlem6 39676 | Part of proof of Lemma N o... |
| dihordlem7 39677 | Part of proof of Lemma N o... |
| dihordlem7b 39678 | Part of proof of Lemma N o... |
| dihjustlem 39679 | Part of proof after Lemma ... |
| dihjust 39680 | Part of proof after Lemma ... |
| dihord1 39681 | Part of proof after Lemma ... |
| dihord2a 39682 | Part of proof after Lemma ... |
| dihord2b 39683 | Part of proof after Lemma ... |
| dihord2cN 39684 | Part of proof after Lemma ... |
| dihord11b 39685 | Part of proof after Lemma ... |
| dihord10 39686 | Part of proof after Lemma ... |
| dihord11c 39687 | Part of proof after Lemma ... |
| dihord2pre 39688 | Part of proof after Lemma ... |
| dihord2pre2 39689 | Part of proof after Lemma ... |
| dihord2 39690 | Part of proof after Lemma ... |
| dihffval 39693 | The isomorphism H for a la... |
| dihfval 39694 | Isomorphism H for a lattic... |
| dihval 39695 | Value of isomorphism H for... |
| dihvalc 39696 | Value of isomorphism H for... |
| dihlsscpre 39697 | Closure of isomorphism H f... |
| dihvalcqpre 39698 | Value of isomorphism H for... |
| dihvalcq 39699 | Value of isomorphism H for... |
| dihvalb 39700 | Value of isomorphism H for... |
| dihopelvalbN 39701 | Ordered pair member of the... |
| dihvalcqat 39702 | Value of isomorphism H for... |
| dih1dimb 39703 | Two expressions for a 1-di... |
| dih1dimb2 39704 | Isomorphism H at an atom u... |
| dih1dimc 39705 | Isomorphism H at an atom n... |
| dib2dim 39706 | Extend ~ dia2dim to partia... |
| dih2dimb 39707 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 39708 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 39709 | Ordered pair member of the... |
| dihvalcq2 39710 | Value of isomorphism H for... |
| dihopelvalcpre 39711 | Member of value of isomorp... |
| dihopelvalc 39712 | Member of value of isomorp... |
| dihlss 39713 | The value of isomorphism H... |
| dihss 39714 | The value of isomorphism H... |
| dihssxp 39715 | An isomorphism H value is ... |
| dihopcl 39716 | Closure of an ordered pair... |
| xihopellsmN 39717 | Ordered pair membership in... |
| dihopellsm 39718 | Ordered pair membership in... |
| dihord6apre 39719 | Part of proof that isomorp... |
| dihord3 39720 | The isomorphism H for a la... |
| dihord4 39721 | The isomorphism H for a la... |
| dihord5b 39722 | Part of proof that isomorp... |
| dihord6b 39723 | Part of proof that isomorp... |
| dihord6a 39724 | Part of proof that isomorp... |
| dihord5apre 39725 | Part of proof that isomorp... |
| dihord5a 39726 | Part of proof that isomorp... |
| dihord 39727 | The isomorphism H is order... |
| dih11 39728 | The isomorphism H is one-t... |
| dihf11lem 39729 | Functionality of the isomo... |
| dihf11 39730 | The isomorphism H for a la... |
| dihfn 39731 | Functionality and domain o... |
| dihdm 39732 | Domain of isomorphism H. (... |
| dihcl 39733 | Closure of isomorphism H. ... |
| dihcnvcl 39734 | Closure of isomorphism H c... |
| dihcnvid1 39735 | The converse isomorphism o... |
| dihcnvid2 39736 | The isomorphism of a conve... |
| dihcnvord 39737 | Ordering property for conv... |
| dihcnv11 39738 | The converse of isomorphis... |
| dihsslss 39739 | The isomorphism H maps to ... |
| dihrnlss 39740 | The isomorphism H maps to ... |
| dihrnss 39741 | The isomorphism H maps to ... |
| dihvalrel 39742 | The value of isomorphism H... |
| dih0 39743 | The value of isomorphism H... |
| dih0bN 39744 | A lattice element is zero ... |
| dih0vbN 39745 | A vector is zero iff its s... |
| dih0cnv 39746 | The isomorphism H converse... |
| dih0rn 39747 | The zero subspace belongs ... |
| dih0sb 39748 | A subspace is zero iff the... |
| dih1 39749 | The value of isomorphism H... |
| dih1rn 39750 | The full vector space belo... |
| dih1cnv 39751 | The isomorphism H converse... |
| dihwN 39752 | Value of isomorphism H at ... |
| dihmeetlem1N 39753 | Isomorphism H of a conjunc... |
| dihglblem5apreN 39754 | A conjunction property of ... |
| dihglblem5aN 39755 | A conjunction property of ... |
| dihglblem2aN 39756 | Lemma for isomorphism H of... |
| dihglblem2N 39757 | The GLB of a set of lattic... |
| dihglblem3N 39758 | Isomorphism H of a lattice... |
| dihglblem3aN 39759 | Isomorphism H of a lattice... |
| dihglblem4 39760 | Isomorphism H of a lattice... |
| dihglblem5 39761 | Isomorphism H of a lattice... |
| dihmeetlem2N 39762 | Isomorphism H of a conjunc... |
| dihglbcpreN 39763 | Isomorphism H of a lattice... |
| dihglbcN 39764 | Isomorphism H of a lattice... |
| dihmeetcN 39765 | Isomorphism H of a lattice... |
| dihmeetbN 39766 | Isomorphism H of a lattice... |
| dihmeetbclemN 39767 | Lemma for isomorphism H of... |
| dihmeetlem3N 39768 | Lemma for isomorphism H of... |
| dihmeetlem4preN 39769 | Lemma for isomorphism H of... |
| dihmeetlem4N 39770 | Lemma for isomorphism H of... |
| dihmeetlem5 39771 | Part of proof that isomorp... |
| dihmeetlem6 39772 | Lemma for isomorphism H of... |
| dihmeetlem7N 39773 | Lemma for isomorphism H of... |
| dihjatc1 39774 | Lemma for isomorphism H of... |
| dihjatc2N 39775 | Isomorphism H of join with... |
| dihjatc3 39776 | Isomorphism H of join with... |
| dihmeetlem8N 39777 | Lemma for isomorphism H of... |
| dihmeetlem9N 39778 | Lemma for isomorphism H of... |
| dihmeetlem10N 39779 | Lemma for isomorphism H of... |
| dihmeetlem11N 39780 | Lemma for isomorphism H of... |
| dihmeetlem12N 39781 | Lemma for isomorphism H of... |
| dihmeetlem13N 39782 | Lemma for isomorphism H of... |
| dihmeetlem14N 39783 | Lemma for isomorphism H of... |
| dihmeetlem15N 39784 | Lemma for isomorphism H of... |
| dihmeetlem16N 39785 | Lemma for isomorphism H of... |
| dihmeetlem17N 39786 | Lemma for isomorphism H of... |
| dihmeetlem18N 39787 | Lemma for isomorphism H of... |
| dihmeetlem19N 39788 | Lemma for isomorphism H of... |
| dihmeetlem20N 39789 | Lemma for isomorphism H of... |
| dihmeetALTN 39790 | Isomorphism H of a lattice... |
| dih1dimatlem0 39791 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 39792 | Lemma for ~ dih1dimat . (... |
| dih1dimat 39793 | Any 1-dimensional subspace... |
| dihlsprn 39794 | The span of a vector belon... |
| dihlspsnssN 39795 | A subspace included in a 1... |
| dihlspsnat 39796 | The inverse isomorphism H ... |
| dihatlat 39797 | The isomorphism H of an at... |
| dihat 39798 | There exists at least one ... |
| dihpN 39799 | The value of isomorphism H... |
| dihlatat 39800 | The reverse isomorphism H ... |
| dihatexv 39801 | There is a nonzero vector ... |
| dihatexv2 39802 | There is a nonzero vector ... |
| dihglblem6 39803 | Isomorphism H of a lattice... |
| dihglb 39804 | Isomorphism H of a lattice... |
| dihglb2 39805 | Isomorphism H of a lattice... |
| dihmeet 39806 | Isomorphism H of a lattice... |
| dihintcl 39807 | The intersection of closed... |
| dihmeetcl 39808 | Closure of closed subspace... |
| dihmeet2 39809 | Reverse isomorphism H of a... |
| dochffval 39812 | Subspace orthocomplement f... |
| dochfval 39813 | Subspace orthocomplement f... |
| dochval 39814 | Subspace orthocomplement f... |
| dochval2 39815 | Subspace orthocomplement f... |
| dochcl 39816 | Closure of subspace orthoc... |
| dochlss 39817 | A subspace orthocomplement... |
| dochssv 39818 | A subspace orthocomplement... |
| dochfN 39819 | Domain and codomain of the... |
| dochvalr 39820 | Orthocomplement of a close... |
| doch0 39821 | Orthocomplement of the zer... |
| doch1 39822 | Orthocomplement of the uni... |
| dochoc0 39823 | The zero subspace is close... |
| dochoc1 39824 | The unit subspace (all vec... |
| dochvalr2 39825 | Orthocomplement of a close... |
| dochvalr3 39826 | Orthocomplement of a close... |
| doch2val2 39827 | Double orthocomplement for... |
| dochss 39828 | Subset law for orthocomple... |
| dochocss 39829 | Double negative law for or... |
| dochoc 39830 | Double negative law for or... |
| dochsscl 39831 | If a set of vectors is inc... |
| dochoccl 39832 | A set of vectors is closed... |
| dochord 39833 | Ordering law for orthocomp... |
| dochord2N 39834 | Ordering law for orthocomp... |
| dochord3 39835 | Ordering law for orthocomp... |
| doch11 39836 | Orthocomplement is one-to-... |
| dochsordN 39837 | Strict ordering law for or... |
| dochn0nv 39838 | An orthocomplement is nonz... |
| dihoml4c 39839 | Version of ~ dihoml4 with ... |
| dihoml4 39840 | Orthomodular law for const... |
| dochspss 39841 | The span of a set of vecto... |
| dochocsp 39842 | The span of an orthocomple... |
| dochspocN 39843 | The span of an orthocomple... |
| dochocsn 39844 | The double orthocomplement... |
| dochsncom 39845 | Swap vectors in an orthoco... |
| dochsat 39846 | The double orthocomplement... |
| dochshpncl 39847 | If a hyperplane is not clo... |
| dochlkr 39848 | Equivalent conditions for ... |
| dochkrshp 39849 | The closure of a kernel is... |
| dochkrshp2 39850 | Properties of the closure ... |
| dochkrshp3 39851 | Properties of the closure ... |
| dochkrshp4 39852 | Properties of the closure ... |
| dochdmj1 39853 | De Morgan-like law for sub... |
| dochnoncon 39854 | Law of noncontradiction. ... |
| dochnel2 39855 | A nonzero member of a subs... |
| dochnel 39856 | A nonzero vector doesn't b... |
| djhffval 39859 | Subspace join for ` DVecH ... |
| djhfval 39860 | Subspace join for ` DVecH ... |
| djhval 39861 | Subspace join for ` DVecH ... |
| djhval2 39862 | Value of subspace join for... |
| djhcl 39863 | Closure of subspace join f... |
| djhlj 39864 | Transfer lattice join to `... |
| djhljjN 39865 | Lattice join in terms of `... |
| djhjlj 39866 | ` DVecH ` vector space clo... |
| djhj 39867 | ` DVecH ` vector space clo... |
| djhcom 39868 | Subspace join commutes. (... |
| djhspss 39869 | Subspace span of union is ... |
| djhsumss 39870 | Subspace sum is a subset o... |
| dihsumssj 39871 | The subspace sum of two is... |
| djhunssN 39872 | Subspace union is a subset... |
| dochdmm1 39873 | De Morgan-like law for clo... |
| djhexmid 39874 | Excluded middle property o... |
| djh01 39875 | Closed subspace join with ... |
| djh02 39876 | Closed subspace join with ... |
| djhlsmcl 39877 | A closed subspace sum equa... |
| djhcvat42 39878 | A covering property. ( ~ ... |
| dihjatb 39879 | Isomorphism H of lattice j... |
| dihjatc 39880 | Isomorphism H of lattice j... |
| dihjatcclem1 39881 | Lemma for isomorphism H of... |
| dihjatcclem2 39882 | Lemma for isomorphism H of... |
| dihjatcclem3 39883 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 39884 | Lemma for isomorphism H of... |
| dihjatcc 39885 | Isomorphism H of lattice j... |
| dihjat 39886 | Isomorphism H of lattice j... |
| dihprrnlem1N 39887 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 39888 | Lemma for ~ dihprrn . (Co... |
| dihprrn 39889 | The span of a vector pair ... |
| djhlsmat 39890 | The sum of two subspace at... |
| dihjat1lem 39891 | Subspace sum of a closed s... |
| dihjat1 39892 | Subspace sum of a closed s... |
| dihsmsprn 39893 | Subspace sum of a closed s... |
| dihjat2 39894 | The subspace sum of a clos... |
| dihjat3 39895 | Isomorphism H of lattice j... |
| dihjat4 39896 | Transfer the subspace sum ... |
| dihjat6 39897 | Transfer the subspace sum ... |
| dihsmsnrn 39898 | The subspace sum of two si... |
| dihsmatrn 39899 | The subspace sum of a clos... |
| dihjat5N 39900 | Transfer lattice join with... |
| dvh4dimat 39901 | There is an atom that is o... |
| dvh3dimatN 39902 | There is an atom that is o... |
| dvh2dimatN 39903 | Given an atom, there exist... |
| dvh1dimat 39904 | There exists an atom. (Co... |
| dvh1dim 39905 | There exists a nonzero vec... |
| dvh4dimlem 39906 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 39907 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 39908 | There is a vector that is ... |
| dvh3dim 39909 | There is a vector that is ... |
| dvh4dimN 39910 | There is a vector that is ... |
| dvh3dim2 39911 | There is a vector that is ... |
| dvh3dim3N 39912 | There is a vector that is ... |
| dochsnnz 39913 | The orthocomplement of a s... |
| dochsatshp 39914 | The orthocomplement of a s... |
| dochsatshpb 39915 | The orthocomplement of a s... |
| dochsnshp 39916 | The orthocomplement of a n... |
| dochshpsat 39917 | A hyperplane is closed iff... |
| dochkrsat 39918 | The orthocomplement of a k... |
| dochkrsat2 39919 | The orthocomplement of a k... |
| dochsat0 39920 | The orthocomplement of a k... |
| dochkrsm 39921 | The subspace sum of a clos... |
| dochexmidat 39922 | Special case of excluded m... |
| dochexmidlem1 39923 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 39924 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 39925 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 39926 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 39927 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 39928 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 39929 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 39930 | Lemma for ~ dochexmid . T... |
| dochexmid 39931 | Excluded middle law for cl... |
| dochsnkrlem1 39932 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 39933 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 39934 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 39935 | A (closed) kernel expresse... |
| dochsnkr2 39936 | Kernel of the explicit fun... |
| dochsnkr2cl 39937 | The ` X ` determining func... |
| dochflcl 39938 | Closure of the explicit fu... |
| dochfl1 39939 | The value of the explicit ... |
| dochfln0 39940 | The value of a functional ... |
| dochkr1 39941 | A nonzero functional has a... |
| dochkr1OLDN 39942 | A nonzero functional has a... |
| lpolsetN 39945 | The set of polarities of a... |
| islpolN 39946 | The predicate "is a polari... |
| islpoldN 39947 | Properties that determine ... |
| lpolfN 39948 | Functionality of a polarit... |
| lpolvN 39949 | The polarity of the whole ... |
| lpolconN 39950 | Contraposition property of... |
| lpolsatN 39951 | The polarity of an atomic ... |
| lpolpolsatN 39952 | Property of a polarity. (... |
| dochpolN 39953 | The subspace orthocompleme... |
| lcfl1lem 39954 | Property of a functional w... |
| lcfl1 39955 | Property of a functional w... |
| lcfl2 39956 | Property of a functional w... |
| lcfl3 39957 | Property of a functional w... |
| lcfl4N 39958 | Property of a functional w... |
| lcfl5 39959 | Property of a functional w... |
| lcfl5a 39960 | Property of a functional w... |
| lcfl6lem 39961 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 39962 | Lemma for ~ lcfl7N . If t... |
| lcfl6 39963 | Property of a functional w... |
| lcfl7N 39964 | Property of a functional w... |
| lcfl8 39965 | Property of a functional w... |
| lcfl8a 39966 | Property of a functional w... |
| lcfl8b 39967 | Property of a nonzero func... |
| lcfl9a 39968 | Property implying that a f... |
| lclkrlem1 39969 | The set of functionals hav... |
| lclkrlem2a 39970 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 39971 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 39972 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 39973 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 39974 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 39975 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 39976 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 39977 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 39978 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 39979 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 39980 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 39981 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 39982 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 39983 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 39984 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 39985 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 39986 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 39987 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 39988 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 39989 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 39990 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 39991 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 39992 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 39993 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 39994 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 39995 | The set of functionals hav... |
| lclkr 39996 | The set of functionals wit... |
| lcfls1lem 39997 | Property of a functional w... |
| lcfls1N 39998 | Property of a functional w... |
| lcfls1c 39999 | Property of a functional w... |
| lclkrslem1 40000 | The set of functionals hav... |
| lclkrslem2 40001 | The set of functionals hav... |
| lclkrs 40002 | The set of functionals hav... |
| lclkrs2 40003 | The set of functionals wit... |
| lcfrvalsnN 40004 | Reconstruction from the du... |
| lcfrlem1 40005 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 40006 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 40007 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 40008 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 40009 | Lemma for ~ lcfr . The se... |
| lcfrlem6 40010 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 40011 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 40012 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 40013 | Lemma for ~ lcf1o . (This... |
| lcf1o 40014 | Define a function ` J ` th... |
| lcfrlem10 40015 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 40016 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 40017 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 40018 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 40019 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 40020 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 40021 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 40022 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 40023 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 40024 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 40025 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 40026 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 40027 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 40028 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 40029 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 40030 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 40031 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 40032 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 40033 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 40034 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 40035 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 40036 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 40037 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 40038 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 40039 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 40040 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 40041 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 40042 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 40043 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 40044 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 40045 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 40046 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 40047 | Lemma for ~ lcfr . Elimin... |
| lcfr 40048 | Reconstruction of a subspa... |
| lcdfval 40051 | Dual vector space of funct... |
| lcdval 40052 | Dual vector space of funct... |
| lcdval2 40053 | Dual vector space of funct... |
| lcdlvec 40054 | The dual vector space of f... |
| lcdlmod 40055 | The dual vector space of f... |
| lcdvbase 40056 | Vector base set of a dual ... |
| lcdvbasess 40057 | The vector base set of the... |
| lcdvbaselfl 40058 | A vector in the base set o... |
| lcdvbasecl 40059 | Closure of the value of a ... |
| lcdvadd 40060 | Vector addition for the cl... |
| lcdvaddval 40061 | The value of the value of ... |
| lcdsca 40062 | The ring of scalars of the... |
| lcdsbase 40063 | Base set of scalar ring fo... |
| lcdsadd 40064 | Scalar addition for the cl... |
| lcdsmul 40065 | Scalar multiplication for ... |
| lcdvs 40066 | Scalar product for the clo... |
| lcdvsval 40067 | Value of scalar product op... |
| lcdvscl 40068 | The scalar product operati... |
| lcdlssvscl 40069 | Closure of scalar product ... |
| lcdvsass 40070 | Associative law for scalar... |
| lcd0 40071 | The zero scalar of the clo... |
| lcd1 40072 | The unit scalar of the clo... |
| lcdneg 40073 | The unit scalar of the clo... |
| lcd0v 40074 | The zero functional in the... |
| lcd0v2 40075 | The zero functional in the... |
| lcd0vvalN 40076 | Value of the zero function... |
| lcd0vcl 40077 | Closure of the zero functi... |
| lcd0vs 40078 | A scalar zero times a func... |
| lcdvs0N 40079 | A scalar times the zero fu... |
| lcdvsub 40080 | The value of vector subtra... |
| lcdvsubval 40081 | The value of the value of ... |
| lcdlss 40082 | Subspaces of a dual vector... |
| lcdlss2N 40083 | Subspaces of a dual vector... |
| lcdlsp 40084 | Span in the set of functio... |
| lcdlkreqN 40085 | Colinear functionals have ... |
| lcdlkreq2N 40086 | Colinear functionals have ... |
| mapdffval 40089 | Projectivity from vector s... |
| mapdfval 40090 | Projectivity from vector s... |
| mapdval 40091 | Value of projectivity from... |
| mapdvalc 40092 | Value of projectivity from... |
| mapdval2N 40093 | Value of projectivity from... |
| mapdval3N 40094 | Value of projectivity from... |
| mapdval4N 40095 | Value of projectivity from... |
| mapdval5N 40096 | Value of projectivity from... |
| mapdordlem1a 40097 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 40098 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 40099 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 40100 | Lemma for ~ mapdord . Ord... |
| mapdord 40101 | Ordering property of the m... |
| mapd11 40102 | The map defined by ~ df-ma... |
| mapddlssN 40103 | The mapping of a subspace ... |
| mapdsn 40104 | Value of the map defined b... |
| mapdsn2 40105 | Value of the map defined b... |
| mapdsn3 40106 | Value of the map defined b... |
| mapd1dim2lem1N 40107 | Value of the map defined b... |
| mapdrvallem2 40108 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 40109 | Lemma for ~ mapdrval . (C... |
| mapdrval 40110 | Given a dual subspace ` R ... |
| mapd1o 40111 | The map defined by ~ df-ma... |
| mapdrn 40112 | Range of the map defined b... |
| mapdunirnN 40113 | Union of the range of the ... |
| mapdrn2 40114 | Range of the map defined b... |
| mapdcnvcl 40115 | Closure of the converse of... |
| mapdcl 40116 | Closure the value of the m... |
| mapdcnvid1N 40117 | Converse of the value of t... |
| mapdsord 40118 | Strong ordering property o... |
| mapdcl2 40119 | The mapping of a subspace ... |
| mapdcnvid2 40120 | Value of the converse of t... |
| mapdcnvordN 40121 | Ordering property of the c... |
| mapdcnv11N 40122 | The converse of the map de... |
| mapdcv 40123 | Covering property of the c... |
| mapdincl 40124 | Closure of dual subspace i... |
| mapdin 40125 | Subspace intersection is p... |
| mapdlsmcl 40126 | Closure of dual subspace s... |
| mapdlsm 40127 | Subspace sum is preserved ... |
| mapd0 40128 | Projectivity map of the ze... |
| mapdcnvatN 40129 | Atoms are preserved by the... |
| mapdat 40130 | Atoms are preserved by the... |
| mapdspex 40131 | The map of a span equals t... |
| mapdn0 40132 | Transfer nonzero property ... |
| mapdncol 40133 | Transfer non-colinearity f... |
| mapdindp 40134 | Transfer (part of) vector ... |
| mapdpglem1 40135 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 40136 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 40137 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 40138 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 40139 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 40140 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 40141 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 40142 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 40143 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 40144 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 40145 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 40146 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 40147 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 40148 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 40149 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 40150 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 40151 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 40152 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 40153 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 40154 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 40155 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 40156 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 40157 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 40158 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 40159 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 40160 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 40161 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 40162 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 40163 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 40164 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 40165 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 40166 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 40167 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 40168 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 40169 | Part 1 of proof of the fir... |
| baerlem3lem1 40170 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 40171 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 40172 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 40173 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 40174 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 40175 | Lemma for ~ baerlem5b . (... |
| baerlem3 40176 | An equality that holds whe... |
| baerlem5a 40177 | An equality that holds whe... |
| baerlem5b 40178 | An equality that holds whe... |
| baerlem5amN 40179 | An equality that holds whe... |
| baerlem5bmN 40180 | An equality that holds whe... |
| baerlem5abmN 40181 | An equality that holds whe... |
| mapdindp0 40182 | Vector independence lemma.... |
| mapdindp1 40183 | Vector independence lemma.... |
| mapdindp2 40184 | Vector independence lemma.... |
| mapdindp3 40185 | Vector independence lemma.... |
| mapdindp4 40186 | Vector independence lemma.... |
| mapdhval 40187 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 40188 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 40189 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 40190 | Lemmma for ~~? mapdh . (C... |
| mapdheq 40191 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 40192 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 40193 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 40194 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 40195 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 40196 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 40197 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 40198 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 40199 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 40200 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 40201 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 40202 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 40203 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 40204 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 40205 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 40206 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 40207 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 40208 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 40209 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 40210 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 40211 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 40212 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 40213 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 40214 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 40215 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 40216 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 40217 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 40218 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 40221 | Map from nonzero vectors t... |
| hvmapfval 40222 | Map from nonzero vectors t... |
| hvmapval 40223 | Value of map from nonzero ... |
| hvmapvalvalN 40224 | Value of value of map (i.e... |
| hvmapidN 40225 | The value of the vector to... |
| hvmap1o 40226 | The vector to functional m... |
| hvmapclN 40227 | Closure of the vector to f... |
| hvmap1o2 40228 | The vector to functional m... |
| hvmapcl2 40229 | Closure of the vector to f... |
| hvmaplfl 40230 | The vector to functional m... |
| hvmaplkr 40231 | Kernel of the vector to fu... |
| mapdhvmap 40232 | Relationship between ` map... |
| lspindp5 40233 | Obtain an independent vect... |
| hdmaplem1 40234 | Lemma to convert a frequen... |
| hdmaplem2N 40235 | Lemma to convert a frequen... |
| hdmaplem3 40236 | Lemma to convert a frequen... |
| hdmaplem4 40237 | Lemma to convert a frequen... |
| mapdh8a 40238 | Part of Part (8) in [Baer]... |
| mapdh8aa 40239 | Part of Part (8) in [Baer]... |
| mapdh8ab 40240 | Part of Part (8) in [Baer]... |
| mapdh8ac 40241 | Part of Part (8) in [Baer]... |
| mapdh8ad 40242 | Part of Part (8) in [Baer]... |
| mapdh8b 40243 | Part of Part (8) in [Baer]... |
| mapdh8c 40244 | Part of Part (8) in [Baer]... |
| mapdh8d0N 40245 | Part of Part (8) in [Baer]... |
| mapdh8d 40246 | Part of Part (8) in [Baer]... |
| mapdh8e 40247 | Part of Part (8) in [Baer]... |
| mapdh8g 40248 | Part of Part (8) in [Baer]... |
| mapdh8i 40249 | Part of Part (8) in [Baer]... |
| mapdh8j 40250 | Part of Part (8) in [Baer]... |
| mapdh8 40251 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 40252 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 40253 | Lemma for part (9) in [Bae... |
| hdmap1ffval 40258 | Preliminary map from vecto... |
| hdmap1fval 40259 | Preliminary map from vecto... |
| hdmap1vallem 40260 | Value of preliminary map f... |
| hdmap1val 40261 | Value of preliminary map f... |
| hdmap1val0 40262 | Value of preliminary map f... |
| hdmap1val2 40263 | Value of preliminary map f... |
| hdmap1eq 40264 | The defining equation for ... |
| hdmap1cbv 40265 | Frequently used lemma to c... |
| hdmap1valc 40266 | Connect the value of the p... |
| hdmap1cl 40267 | Convert closure theorem ~ ... |
| hdmap1eq2 40268 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 40269 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 40270 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 40271 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 40272 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 40273 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 40274 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 40275 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 40276 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 40277 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 40278 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 40279 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 40280 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 40281 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 40282 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 40283 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 40284 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 40285 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 40286 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 40287 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 40288 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 40289 | Map from vectors to functi... |
| hdmapfval 40290 | Map from vectors to functi... |
| hdmapval 40291 | Value of map from vectors ... |
| hdmapfnN 40292 | Functionality of map from ... |
| hdmapcl 40293 | Closure of map from vector... |
| hdmapval2lem 40294 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 40295 | Value of map from vectors ... |
| hdmapval0 40296 | Value of map from vectors ... |
| hdmapeveclem 40297 | Lemma for ~ hdmapevec . T... |
| hdmapevec 40298 | Value of map from vectors ... |
| hdmapevec2 40299 | The inner product of the r... |
| hdmapval3lemN 40300 | Value of map from vectors ... |
| hdmapval3N 40301 | Value of map from vectors ... |
| hdmap10lem 40302 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 40303 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 40304 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 40305 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 40306 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 40307 | Part of proof of part 12 i... |
| hdmapnzcl 40308 | Nonzero vector closure of ... |
| hdmapneg 40309 | Part of proof of part 12 i... |
| hdmapsub 40310 | Part of proof of part 12 i... |
| hdmap11 40311 | Part of proof of part 12 i... |
| hdmaprnlem1N 40312 | Part of proof of part 12 i... |
| hdmaprnlem3N 40313 | Part of proof of part 12 i... |
| hdmaprnlem3uN 40314 | Part of proof of part 12 i... |
| hdmaprnlem4tN 40315 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 40316 | Part of proof of part 12 i... |
| hdmaprnlem6N 40317 | Part of proof of part 12 i... |
| hdmaprnlem7N 40318 | Part of proof of part 12 i... |
| hdmaprnlem8N 40319 | Part of proof of part 12 i... |
| hdmaprnlem9N 40320 | Part of proof of part 12 i... |
| hdmaprnlem3eN 40321 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 40322 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 40323 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 40324 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 40325 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 40326 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 40327 | Part of proof of part 12 i... |
| hdmapf1oN 40328 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 40329 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 40330 | Prior to part 14 in [Baer]... |
| hdmap14lem1 40331 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 40332 | Prior to part 14 in [Baer]... |
| hdmap14lem3 40333 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 40334 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 40335 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 40336 | Case where ` F ` is zero. ... |
| hdmap14lem7 40337 | Combine cases of ` F ` . ... |
| hdmap14lem8 40338 | Part of proof of part 14 i... |
| hdmap14lem9 40339 | Part of proof of part 14 i... |
| hdmap14lem10 40340 | Part of proof of part 14 i... |
| hdmap14lem11 40341 | Part of proof of part 14 i... |
| hdmap14lem12 40342 | Lemma for proof of part 14... |
| hdmap14lem13 40343 | Lemma for proof of part 14... |
| hdmap14lem14 40344 | Part of proof of part 14 i... |
| hdmap14lem15 40345 | Part of proof of part 14 i... |
| hgmapffval 40348 | Map from the scalar divisi... |
| hgmapfval 40349 | Map from the scalar divisi... |
| hgmapval 40350 | Value of map from the scal... |
| hgmapfnN 40351 | Functionality of scalar si... |
| hgmapcl 40352 | Closure of scalar sigma ma... |
| hgmapdcl 40353 | Closure of the vector spac... |
| hgmapvs 40354 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 40355 | Value of the scalar sigma ... |
| hgmapval1 40356 | Value of the scalar sigma ... |
| hgmapadd 40357 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 40358 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 40359 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 40360 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 40361 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 40362 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 40363 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 40364 | Part of proof of part 16 i... |
| hgmap11 40365 | The scalar sigma map is on... |
| hgmapf1oN 40366 | The scalar sigma map is a ... |
| hgmapeq0 40367 | The scalar sigma map is ze... |
| hdmapipcl 40368 | The inner product (Hermiti... |
| hdmapln1 40369 | Linearity property that wi... |
| hdmaplna1 40370 | Additive property of first... |
| hdmaplns1 40371 | Subtraction property of fi... |
| hdmaplnm1 40372 | Multiplicative property of... |
| hdmaplna2 40373 | Additive property of secon... |
| hdmapglnm2 40374 | g-linear property of secon... |
| hdmapgln2 40375 | g-linear property that wil... |
| hdmaplkr 40376 | Kernel of the vector to du... |
| hdmapellkr 40377 | Membership in the kernel (... |
| hdmapip0 40378 | Zero property that will be... |
| hdmapip1 40379 | Construct a proportional v... |
| hdmapip0com 40380 | Commutation property of Ba... |
| hdmapinvlem1 40381 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 40382 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 40383 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 40384 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 40385 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 40386 | Involution property of sca... |
| hgmapvvlem2 40387 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 40388 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 40389 | Value of a double involuti... |
| hdmapglem7a 40390 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 40391 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 40392 | Lemma for ~ hdmapg . Line... |
| hdmapg 40393 | Apply the scalar sigma fun... |
| hdmapoc 40394 | Express our constructed or... |
| hlhilset 40397 | The final Hilbert space co... |
| hlhilsca 40398 | The scalar of the final co... |
| hlhilbase 40399 | The base set of the final ... |
| hlhilplus 40400 | The vector addition for th... |
| hlhilslem 40401 | Lemma for ~ hlhilsbase etc... |
| hlhilslemOLD 40402 | Obsolete version of ~ hlhi... |
| hlhilsbase 40403 | The scalar base set of the... |
| hlhilsbaseOLD 40404 | Obsolete version of ~ hlhi... |
| hlhilsplus 40405 | Scalar addition for the fi... |
| hlhilsplusOLD 40406 | Obsolete version of ~ hlhi... |
| hlhilsmul 40407 | Scalar multiplication for ... |
| hlhilsmulOLD 40408 | Obsolete version of ~ hlhi... |
| hlhilsbase2 40409 | The scalar base set of the... |
| hlhilsplus2 40410 | Scalar addition for the fi... |
| hlhilsmul2 40411 | Scalar multiplication for ... |
| hlhils0 40412 | The scalar ring zero for t... |
| hlhils1N 40413 | The scalar ring unity for ... |
| hlhilvsca 40414 | The scalar product for the... |
| hlhilip 40415 | Inner product operation fo... |
| hlhilipval 40416 | Value of inner product ope... |
| hlhilnvl 40417 | The involution operation o... |
| hlhillvec 40418 | The final constructed Hilb... |
| hlhildrng 40419 | The star division ring for... |
| hlhilsrnglem 40420 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 40421 | The star division ring for... |
| hlhil0 40422 | The zero vector for the fi... |
| hlhillsm 40423 | The vector sum operation f... |
| hlhilocv 40424 | The orthocomplement for th... |
| hlhillcs 40425 | The closed subspaces of th... |
| hlhilphllem 40426 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 40427 | Lemma for ~ hlhil . (Cont... |
| hlathil 40428 | Construction of a Hilbert ... |
| leexp1ad 40429 | Weak base ordering relatio... |
| relogbcld 40430 | Closure of the general log... |
| relogbexpd 40431 | Identity law for general l... |
| relogbzexpd 40432 | Power law for the general ... |
| logblebd 40433 | The general logarithm is m... |
| uzindd 40434 | Induction on the upper int... |
| fzadd2d 40435 | Membership of a sum in a f... |
| zltlem1d 40436 | Integer ordering relation,... |
| zltp1led 40437 | Integer ordering relation,... |
| fzne2d 40438 | Elementhood in a finite se... |
| eqfnfv2d2 40439 | Equality of functions is d... |
| fzsplitnd 40440 | Split a finite interval of... |
| fzsplitnr 40441 | Split a finite interval of... |
| addassnni 40442 | Associative law for additi... |
| addcomnni 40443 | Commutative law for additi... |
| mulassnni 40444 | Associative law for multip... |
| mulcomnni 40445 | Commutative law for multip... |
| gcdcomnni 40446 | Commutative law for gcd. ... |
| gcdnegnni 40447 | Negation invariance for gc... |
| neggcdnni 40448 | Negation invariance for gc... |
| bccl2d 40449 | Closure of the binomial co... |
| recbothd 40450 | Take reciprocal on both si... |
| gcdmultiplei 40451 | The GCD of a multiple of a... |
| gcdaddmzz2nni 40452 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 40453 | Adding a multiple of one o... |
| gcdnncli 40454 | Closure of the gcd operato... |
| muldvds1d 40455 | If a product divides an in... |
| muldvds2d 40456 | If a product divides an in... |
| nndivdvdsd 40457 | A positive integer divides... |
| nnproddivdvdsd 40458 | A product of natural numbe... |
| coprmdvds2d 40459 | If an integer is divisible... |
| 12gcd5e1 40460 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 40461 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 40462 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 40463 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 40464 | Calculate the least common... |
| 12lcm5e60 40465 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 40466 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 40467 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 40468 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 40469 | Useful equation to calcula... |
| lcm1un 40470 | Least common multiple of n... |
| lcm2un 40471 | Least common multiple of n... |
| lcm3un 40472 | Least common multiple of n... |
| lcm4un 40473 | Least common multiple of n... |
| lcm5un 40474 | Least common multiple of n... |
| lcm6un 40475 | Least common multiple of n... |
| lcm7un 40476 | Least common multiple of n... |
| lcm8un 40477 | Least common multiple of n... |
| 3factsumint1 40478 | Move constants out of inte... |
| 3factsumint2 40479 | Move constants out of inte... |
| 3factsumint3 40480 | Move constants out of inte... |
| 3factsumint4 40481 | Move constants out of inte... |
| 3factsumint 40482 | Helpful equation for lcm i... |
| resopunitintvd 40483 | Restrict continuous functi... |
| resclunitintvd 40484 | Restrict continuous functi... |
| resdvopclptsd 40485 | Restrict derivative on uni... |
| lcmineqlem1 40486 | Part of lcm inequality lem... |
| lcmineqlem2 40487 | Part of lcm inequality lem... |
| lcmineqlem3 40488 | Part of lcm inequality lem... |
| lcmineqlem4 40489 | Part of lcm inequality lem... |
| lcmineqlem5 40490 | Technical lemma for recipr... |
| lcmineqlem6 40491 | Part of lcm inequality lem... |
| lcmineqlem7 40492 | Derivative of 1-x for chai... |
| lcmineqlem8 40493 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 40494 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 40495 | Induction step of ~ lcmine... |
| lcmineqlem11 40496 | Induction step, continuati... |
| lcmineqlem12 40497 | Base case for induction. ... |
| lcmineqlem13 40498 | Induction proof for lcm in... |
| lcmineqlem14 40499 | Technical lemma for inequa... |
| lcmineqlem15 40500 | F times the least common m... |
| lcmineqlem16 40501 | Technical divisibility lem... |
| lcmineqlem17 40502 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 40503 | Technical lemma to shift f... |
| lcmineqlem19 40504 | Dividing implies inequalit... |
| lcmineqlem20 40505 | Inequality for lcm lemma. ... |
| lcmineqlem21 40506 | The lcm inequality lemma w... |
| lcmineqlem22 40507 | The lcm inequality lemma w... |
| lcmineqlem23 40508 | Penultimate step to the lc... |
| lcmineqlem 40509 | The least common multiple ... |
| 3exp7 40510 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 40511 | First inequality in inequa... |
| 3lexlogpow5ineq2 40512 | Second inequality in inequ... |
| 3lexlogpow5ineq4 40513 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 40514 | Combined inequality chain ... |
| 3lexlogpow2ineq1 40515 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 40516 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 40517 | Result for bound in AKS in... |
| intlewftc 40518 | Inequality inference by in... |
| aks4d1lem1 40519 | Technical lemma to reduce ... |
| aks4d1p1p1 40520 | Exponential law for finite... |
| dvrelog2 40521 | The derivative of the loga... |
| dvrelog3 40522 | The derivative of the loga... |
| dvrelog2b 40523 | Derivative of the binary l... |
| 0nonelalab 40524 | Technical lemma for open i... |
| dvrelogpow2b 40525 | Derivative of the power of... |
| aks4d1p1p3 40526 | Bound of a ceiling of the ... |
| aks4d1p1p2 40527 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 40528 | Technical step for inequal... |
| dvle2 40529 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 40530 | Inequality lift to differe... |
| aks4d1p1p7 40531 | Bound of intermediary of i... |
| aks4d1p1p5 40532 | Show inequality for existe... |
| aks4d1p1 40533 | Show inequality for existe... |
| aks4d1p2 40534 | Technical lemma for existe... |
| aks4d1p3 40535 | There exists a small enoug... |
| aks4d1p4 40536 | There exists a small enoug... |
| aks4d1p5 40537 | Show that ` N ` and ` R ` ... |
| aks4d1p6 40538 | The maximal prime power ex... |
| aks4d1p7d1 40539 | Technical step in AKS lemm... |
| aks4d1p7 40540 | Technical step in AKS lemm... |
| aks4d1p8d1 40541 | If a prime divides one num... |
| aks4d1p8d2 40542 | Any prime power dividing a... |
| aks4d1p8d3 40543 | The remainder of a divisio... |
| aks4d1p8 40544 | Show that ` N ` and ` R ` ... |
| aks4d1p9 40545 | Show that the order is bou... |
| aks4d1 40546 | Lemma 4.1 from ~ https://w... |
| fldhmf1 40547 | A field homomorphism is in... |
| aks6d1c2p1 40548 | In the AKS-theorem the sub... |
| aks6d1c2p2 40549 | Injective condition for co... |
| 5bc2eq10 40550 | The value of 5 choose 2. ... |
| facp2 40551 | The factorial of a success... |
| 2np3bcnp1 40552 | Part of induction step for... |
| 2ap1caineq 40553 | Inequality for Theorem 6.6... |
| sticksstones1 40554 | Different strictly monoton... |
| sticksstones2 40555 | The range function on stri... |
| sticksstones3 40556 | The range function on stri... |
| sticksstones4 40557 | Equinumerosity lemma for s... |
| sticksstones5 40558 | Count the number of strict... |
| sticksstones6 40559 | Function induces an order ... |
| sticksstones7 40560 | Closure property of sticks... |
| sticksstones8 40561 | Establish mapping between ... |
| sticksstones9 40562 | Establish mapping between ... |
| sticksstones10 40563 | Establish mapping between ... |
| sticksstones11 40564 | Establish bijective mappin... |
| sticksstones12a 40565 | Establish bijective mappin... |
| sticksstones12 40566 | Establish bijective mappin... |
| sticksstones13 40567 | Establish bijective mappin... |
| sticksstones14 40568 | Sticks and stones with def... |
| sticksstones15 40569 | Sticks and stones with alm... |
| sticksstones16 40570 | Sticks and stones with col... |
| sticksstones17 40571 | Extend sticks and stones t... |
| sticksstones18 40572 | Extend sticks and stones t... |
| sticksstones19 40573 | Extend sticks and stones t... |
| sticksstones20 40574 | Lift sticks and stones to ... |
| sticksstones21 40575 | Lift sticks and stones to ... |
| sticksstones22 40576 | Non-exhaustive sticks and ... |
| metakunt1 40577 | A is an endomapping. (Con... |
| metakunt2 40578 | A is an endomapping. (Con... |
| metakunt3 40579 | Value of A. (Contributed b... |
| metakunt4 40580 | Value of A. (Contributed b... |
| metakunt5 40581 | C is the left inverse for ... |
| metakunt6 40582 | C is the left inverse for ... |
| metakunt7 40583 | C is the left inverse for ... |
| metakunt8 40584 | C is the left inverse for ... |
| metakunt9 40585 | C is the left inverse for ... |
| metakunt10 40586 | C is the right inverse for... |
| metakunt11 40587 | C is the right inverse for... |
| metakunt12 40588 | C is the right inverse for... |
| metakunt13 40589 | C is the right inverse for... |
| metakunt14 40590 | A is a primitive permutati... |
| metakunt15 40591 | Construction of another pe... |
| metakunt16 40592 | Construction of another pe... |
| metakunt17 40593 | The union of three disjoin... |
| metakunt18 40594 | Disjoint domains and codom... |
| metakunt19 40595 | Domains on restrictions of... |
| metakunt20 40596 | Show that B coincides on t... |
| metakunt21 40597 | Show that B coincides on t... |
| metakunt22 40598 | Show that B coincides on t... |
| metakunt23 40599 | B coincides on the union o... |
| metakunt24 40600 | Technical condition such t... |
| metakunt25 40601 | B is a permutation. (Cont... |
| metakunt26 40602 | Construction of one soluti... |
| metakunt27 40603 | Construction of one soluti... |
| metakunt28 40604 | Construction of one soluti... |
| metakunt29 40605 | Construction of one soluti... |
| metakunt30 40606 | Construction of one soluti... |
| metakunt31 40607 | Construction of one soluti... |
| metakunt32 40608 | Construction of one soluti... |
| metakunt33 40609 | Construction of one soluti... |
| metakunt34 40610 | ` D ` is a permutation. (... |
| andiff 40611 | Adding biconditional when ... |
| fac2xp3 40612 | Factorial of 2x+3, sublemm... |
| prodsplit 40613 | Product split into two fac... |
| 2xp3dxp2ge1d 40614 | 2x+3 is greater than or eq... |
| factwoffsmonot 40615 | A factorial with offset is... |
| bicomdALT 40616 | Alternate proof of ~ bicom... |
| elabgw 40617 | Membership in a class abst... |
| elab2gw 40618 | Membership in a class abst... |
| elrab2w 40619 | Membership in a restricted... |
| ruvALT 40620 | Alternate proof of ~ ruv w... |
| sn-wcdeq 40621 | Alternative to ~ wcdeq and... |
| acos1half 40622 | The arccosine of ` 1 / 2 `... |
| isdomn5 40623 | The right conjunct in the ... |
| isdomn4 40624 | A ring is a domain iff it ... |
| ioin9i8 40625 | Miscellaneous inference cr... |
| jaodd 40626 | Double deduction form of ~... |
| syl3an12 40627 | A double syllogism inferen... |
| sbtd 40628 | A true statement is true u... |
| sbor2 40629 | One direction of ~ sbor , ... |
| 19.9dev 40630 | ~ 19.9d in the case of an ... |
| rspcedvdw 40631 | Version of ~ rspcedvd wher... |
| 2rspcedvdw 40632 | Double application of ~ rs... |
| 3rspcedvdw 40633 | Triple application of ~ rs... |
| 3rspcedvd 40634 | Triple application of ~ rs... |
| eqimssd 40635 | Equality implies inclusion... |
| rabdif 40636 | Move difference in and out... |
| sn-axrep5v 40637 | A condensed form of ~ axre... |
| sn-axprlem3 40638 | ~ axprlem3 using only Tars... |
| sn-exelALT 40639 | Alternate proof of ~ exel ... |
| ss2ab1 40640 | Class abstractions in a su... |
| ssabdv 40641 | Deduction of abstraction s... |
| sn-iotalem 40642 | An unused lemma showing th... |
| sn-iotalemcor 40643 | Corollary of ~ sn-iotalem ... |
| abbi1sn 40644 | Originally part of ~ uniab... |
| brif1 40645 | Move a relation inside and... |
| brif2 40646 | Move a relation inside and... |
| brif12 40647 | Move a relation inside and... |
| pssexg 40648 | The proper subset of a set... |
| pssn0 40649 | A proper superset is nonem... |
| psspwb 40650 | Classes are proper subclas... |
| xppss12 40651 | Proper subset theorem for ... |
| coexd 40652 | The composition of two set... |
| elpwbi 40653 | Membership in a power set,... |
| opelxpii 40654 | Ordered pair membership in... |
| imaopab 40655 | The image of a class of or... |
| fnsnbt 40656 | A function's domain is a s... |
| fnimasnd 40657 | The image of a function by... |
| fvmptd4 40658 | Deduction version of ~ fvm... |
| ofun 40659 | A function operation of un... |
| dfqs2 40660 | Alternate definition of qu... |
| dfqs3 40661 | Alternate definition of qu... |
| qseq12d 40662 | Equality theorem for quoti... |
| qsalrel 40663 | The quotient set is equal ... |
| elmapdd 40664 | Deduction associated with ... |
| isfsuppd 40665 | Deduction form of ~ isfsup... |
| fzosumm1 40666 | Separate out the last term... |
| ccatcan2d 40667 | Cancellation law for conca... |
| ressbasssg 40668 | The base set of a restrict... |
| ressbasss2 40669 | The base set of a restrict... |
| nelsubginvcld 40670 | The inverse of a non-subgr... |
| nelsubgcld 40671 | A non-subgroup-member plus... |
| nelsubgsubcld 40672 | A non-subgroup-member minu... |
| rnasclg 40673 | The set of injected scalar... |
| frlmfielbas 40674 | The vectors of a finite fr... |
| frlmfzwrd 40675 | A vector of a module with ... |
| frlmfzowrd 40676 | A vector of a module with ... |
| frlmfzolen 40677 | The dimension of a vector ... |
| frlmfzowrdb 40678 | The vectors of a module wi... |
| frlmfzoccat 40679 | The concatenation of two v... |
| frlmvscadiccat 40680 | Scalar multiplication dist... |
| sn-grplidd 40681 | The identity element of a ... |
| sn-grpridd 40682 | The identity element of a ... |
| grpassd 40683 | A group operation is assoc... |
| grplinvd 40684 | The left inverse of a grou... |
| sn-grprinvd 40685 | The right inverse of a gro... |
| grpasscan2d 40686 | An associative cancellatio... |
| grpcominv1 40687 | If two elements commute, t... |
| grpcominv2 40688 | If two elements commute, t... |
| finsubmsubg 40689 | A submonoid of a finite gr... |
| ringassd 40690 | Associative law for multip... |
| ringlidmd 40691 | The unity element of a rin... |
| ringridmd 40692 | The unity element of a rin... |
| resrhm2b 40693 | Restriction of the codomai... |
| rncrhmcl 40694 | The range of a commutative... |
| rimcnv 40695 | The converse of a ring iso... |
| rimco 40696 | The composition of ring is... |
| brrici 40697 | Prove isomorphic by an exp... |
| ricsym 40698 | Ring isomorphism is symmet... |
| rictr 40699 | Ring isomorphism is transi... |
| riccrng1 40700 | Ring isomorphism preserves... |
| riccrng 40701 | A ring is commutative if a... |
| drnginvrn0d 40702 | A multiplicative inverse i... |
| drnginvrld 40703 | Property of the multiplica... |
| drnginvrrd 40704 | Property of the multiplica... |
| drngmulcanad 40705 | Cancellation of a nonzero ... |
| drngmulcan2ad 40706 | Cancellation of a nonzero ... |
| drnginvmuld 40707 | Inverse of a nonzero produ... |
| flddrngd 40708 | A field is a division ring... |
| lmodgrpd 40709 | A left module is a group. ... |
| lvecgrp 40710 | A vector space is a group.... |
| lveclmodd 40711 | A vector space is a left m... |
| lvecgrpd 40712 | A vector space is a group.... |
| lvecring 40713 | The scalar component of a ... |
| lmhmlvec 40714 | The property for modules t... |
| frlm0vald 40715 | All coordinates of the zer... |
| frlmsnic 40716 | Given a free module with a... |
| uvccl 40717 | A unit vector is a vector.... |
| uvcn0 40718 | A unit vector is nonzero. ... |
| pwselbasr 40719 | The reverse direction of ~... |
| pwsgprod 40720 | Finite products in a power... |
| mplringd 40721 | The polynomial ring is a r... |
| mplcrngd 40722 | The polynomial ring is a c... |
| mplsubrgcl 40723 | An element of a polynomial... |
| mhmcompl 40724 | The composition of a monoi... |
| rhmmpllem1 40725 | Lemma for ~ rhmmpl . A su... |
| rhmmpllem2 40726 | Lemma for ~ rhmmpl . A su... |
| mhmcoaddmpl 40727 | Show that the ring homomor... |
| rhmcomulmpl 40728 | Show that the ring homomor... |
| rhmmpl 40729 | Provide a ring homomorphis... |
| mplascl0 40730 | The zero scalar as a polyn... |
| evl0 40731 | The zero polynomial evalua... |
| evlsval3 40732 | Give a formula for the pol... |
| evlsvval 40733 | Give a formula for the eva... |
| evlsscaval 40734 | Polynomial evaluation buil... |
| evlsvarval 40735 | Polynomial evaluation buil... |
| evlsbagval 40736 | Polynomial evaluation buil... |
| evlsexpval 40737 | Polynomial evaluation buil... |
| evlsaddval 40738 | Polynomial evaluation buil... |
| evlsmulval 40739 | Polynomial evaluation buil... |
| evlsevl 40740 | Evaluation in a subring is... |
| evladdval 40741 | Polynomial evaluation buil... |
| selvcllem1 40742 | ` T ` is an associative al... |
| selvcllem2 40743 | ` D ` is a ring homomorphi... |
| selvcllem3 40744 | The third argument passed ... |
| selvcllemh 40745 | Apply the third argument (... |
| selvcllem4 40746 | The fourth argument passed... |
| selvcllem5 40747 | The fifth argument passed ... |
| selvcl 40748 | Closure of the "variable s... |
| selvval2 40749 | Value of the "variable sel... |
| selvadd 40750 | The "variable selection" f... |
| fsuppind 40751 | Induction on functions ` F... |
| fsuppssindlem1 40752 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 40753 | Lemma for ~ fsuppssind . ... |
| fsuppssind 40754 | Induction on functions ` F... |
| mhpind 40755 | The homogeneous polynomial... |
| mhphflem 40756 | Lemma for ~ mhphf . Add s... |
| mhphf 40757 | A homogeneous polynomial d... |
| mhphf2 40758 | A homogeneous polynomial d... |
| mhphf3 40759 | A homogeneous polynomial d... |
| mhphf4 40760 | A homogeneous polynomial d... |
| c0exALT 40761 | Alternate proof of ~ c0ex ... |
| 0cnALT3 40762 | Alternate proof of ~ 0cn u... |
| elre0re 40763 | Specialized version of ~ 0... |
| 1t1e1ALT 40764 | Alternate proof of ~ 1t1e1... |
| remulcan2d 40765 | ~ mulcan2d for real number... |
| readdid1addid2d 40766 | Given some real number ` B... |
| sn-1ne2 40767 | A proof of ~ 1ne2 without ... |
| nnn1suc 40768 | A positive integer that is... |
| nnadd1com 40769 | Addition with 1 is commuta... |
| nnaddcom 40770 | Addition is commutative fo... |
| nnaddcomli 40771 | Version of ~ addcomli for ... |
| nnadddir 40772 | Right-distributivity for n... |
| nnmul1com 40773 | Multiplication with 1 is c... |
| nnmulcom 40774 | Multiplication is commutat... |
| mvrrsubd 40775 | Move a subtraction in the ... |
| laddrotrd 40776 | Rotate the variables right... |
| raddcom12d 40777 | Swap the first two variabl... |
| lsubrotld 40778 | Rotate the variables left ... |
| lsubcom23d 40779 | Swap the second and third ... |
| addsubeq4com 40780 | Relation between sums and ... |
| sqsumi 40781 | A sum squared. (Contribut... |
| negn0nposznnd 40782 | Lemma for ~ dffltz . (Con... |
| sqmid3api 40783 | Value of the square of the... |
| decaddcom 40784 | Commute ones place in addi... |
| sqn5i 40785 | The square of a number end... |
| sqn5ii 40786 | The square of a number end... |
| decpmulnc 40787 | Partial products algorithm... |
| decpmul 40788 | Partial products algorithm... |
| sqdeccom12 40789 | The square of a number in ... |
| sq3deccom12 40790 | Variant of ~ sqdeccom12 wi... |
| 235t711 40791 | Calculate a product by lon... |
| ex-decpmul 40792 | Example usage of ~ decpmul... |
| oexpreposd 40793 | Lemma for ~ dffltz . TODO... |
| ltexp1d 40794 | ~ ltmul1d for exponentiati... |
| ltexp1dd 40795 | Raising both sides of 'les... |
| exp11nnd 40796 | ~ sq11d for positive real ... |
| exp11d 40797 | ~ exp11nnd for nonzero int... |
| 0dvds0 40798 | 0 divides 0. (Contributed... |
| absdvdsabsb 40799 | Divisibility is invariant ... |
| dvdsexpim 40800 | ~ dvdssqim generalized to ... |
| gcdnn0id 40801 | The ` gcd ` of a nonnegati... |
| gcdle1d 40802 | The greatest common diviso... |
| gcdle2d 40803 | The greatest common diviso... |
| dvdsexpad 40804 | Deduction associated with ... |
| nn0rppwr 40805 | If ` A ` and ` B ` are rel... |
| expgcd 40806 | Exponentiation distributes... |
| nn0expgcd 40807 | Exponentiation distributes... |
| zexpgcd 40808 | Exponentiation distributes... |
| numdenexp 40809 | ~ numdensq extended to non... |
| numexp 40810 | ~ numsq extended to nonneg... |
| denexp 40811 | ~ densq extended to nonneg... |
| dvdsexpnn 40812 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 40813 | ~ dvdsexpnn generalized to... |
| dvdsexpb 40814 | ~ dvdssq generalized to po... |
| posqsqznn 40815 | When a positive rational s... |
| cxpgt0d 40816 | A positive real raised to ... |
| zrtelqelz 40817 | ~ zsqrtelqelz generalized ... |
| zrtdvds 40818 | A positive integer root di... |
| rtprmirr 40819 | The root of a prime number... |
| resubval 40822 | Value of real subtraction,... |
| renegeulemv 40823 | Lemma for ~ renegeu and si... |
| renegeulem 40824 | Lemma for ~ renegeu and si... |
| renegeu 40825 | Existential uniqueness of ... |
| rernegcl 40826 | Closure law for negative r... |
| renegadd 40827 | Relationship between real ... |
| renegid 40828 | Addition of a real number ... |
| reneg0addid2 40829 | Negative zero is a left ad... |
| resubeulem1 40830 | Lemma for ~ resubeu . A v... |
| resubeulem2 40831 | Lemma for ~ resubeu . A v... |
| resubeu 40832 | Existential uniqueness of ... |
| rersubcl 40833 | Closure for real subtracti... |
| resubadd 40834 | Relation between real subt... |
| resubaddd 40835 | Relationship between subtr... |
| resubf 40836 | Real subtraction is an ope... |
| repncan2 40837 | Addition and subtraction o... |
| repncan3 40838 | Addition and subtraction o... |
| readdsub 40839 | Law for addition and subtr... |
| reladdrsub 40840 | Move LHS of a sum into RHS... |
| reltsub1 40841 | Subtraction from both side... |
| reltsubadd2 40842 | 'Less than' relationship b... |
| resubcan2 40843 | Cancellation law for real ... |
| resubsub4 40844 | Law for double subtraction... |
| rennncan2 40845 | Cancellation law for real ... |
| renpncan3 40846 | Cancellation law for real ... |
| repnpcan 40847 | Cancellation law for addit... |
| reppncan 40848 | Cancellation law for mixed... |
| resubidaddid1lem 40849 | Lemma for ~ resubidaddid1 ... |
| resubidaddid1 40850 | Any real number subtracted... |
| resubdi 40851 | Distribution of multiplica... |
| re1m1e0m0 40852 | Equality of two left-addit... |
| sn-00idlem1 40853 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 40854 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 40855 | Lemma for ~ sn-00id . (Co... |
| sn-00id 40856 | ~ 00id proven without ~ ax... |
| re0m0e0 40857 | Real number version of ~ 0... |
| readdid2 40858 | Real number version of ~ a... |
| sn-addid2 40859 | ~ addid2 without ~ ax-mulc... |
| remul02 40860 | Real number version of ~ m... |
| sn-0ne2 40861 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 40862 | Real number version of ~ m... |
| resubid 40863 | Subtraction of a real numb... |
| readdid1 40864 | Real number version of ~ a... |
| resubid1 40865 | Real number version of ~ s... |
| renegneg 40866 | A real number is equal to ... |
| readdcan2 40867 | Commuted version of ~ read... |
| renegid2 40868 | Commuted version of ~ rene... |
| remulneg2d 40869 | Product with negative is n... |
| sn-it0e0 40870 | Proof of ~ it0e0 without ~... |
| sn-negex12 40871 | A combination of ~ cnegex ... |
| sn-negex 40872 | Proof of ~ cnegex without ... |
| sn-negex2 40873 | Proof of ~ cnegex2 without... |
| sn-addcand 40874 | ~ addcand without ~ ax-mul... |
| sn-addid1 40875 | ~ addid1 without ~ ax-mulc... |
| sn-addcan2d 40876 | ~ addcan2d without ~ ax-mu... |
| reixi 40877 | ~ ixi without ~ ax-mulcom ... |
| rei4 40878 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 40879 | A number that sums to itse... |
| sn-mul01 40880 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 40881 | ~ negeu without ~ ax-mulco... |
| sn-subcl 40882 | ~ subcl without ~ ax-mulco... |
| sn-subf 40883 | ~ subf without ~ ax-mulcom... |
| resubeqsub 40884 | Equivalence between real s... |
| subresre 40885 | Subtraction restricted to ... |
| addinvcom 40886 | A number commutes with its... |
| remulinvcom 40887 | A left multiplicative inve... |
| remulid2 40888 | Commuted version of ~ ax-1... |
| sn-1ticom 40889 | Lemma for ~ sn-mulid2 and ... |
| sn-mulid2 40890 | ~ mulid2 without ~ ax-mulc... |
| it1ei 40891 | ` 1 ` is a multiplicative ... |
| ipiiie0 40892 | The multiplicative inverse... |
| remulcand 40893 | Commuted version of ~ remu... |
| sn-0tie0 40894 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 40895 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 40896 | ~ ltaddpos without ~ ax-mu... |
| sn-ltaddneg 40897 | ~ ltaddneg without ~ ax-mu... |
| reposdif 40898 | Comparison of two numbers ... |
| relt0neg1 40899 | Comparison of a real and i... |
| relt0neg2 40900 | Comparison of a real and i... |
| sn-addlt0d 40901 | The sum of negative number... |
| sn-addgt0d 40902 | The sum of positive number... |
| sn-nnne0 40903 | ~ nnne0 without ~ ax-mulco... |
| reelznn0nn 40904 | ~ elznn0nn restated using ... |
| nn0addcom 40905 | Addition is commutative fo... |
| zaddcomlem 40906 | Lemma for ~ zaddcom . (Co... |
| zaddcom 40907 | Addition is commutative fo... |
| renegmulnnass 40908 | Move multiplication by a n... |
| nn0mulcom 40909 | Multiplication is commutat... |
| zmulcomlem 40910 | Lemma for ~ zmulcom . (Co... |
| zmulcom 40911 | Multiplication is commutat... |
| mulgt0con1dlem 40912 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 40913 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 40914 | Lemma for ~ mulgt0b2d and ... |
| mulgt0b2d 40915 | Biconditional, deductive f... |
| sn-ltmul2d 40916 | ~ ltmul2d without ~ ax-mul... |
| sn-0lt1 40917 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 40918 | ~ ltp1 without ~ ax-mulcom... |
| reneg1lt0 40919 | Lemma for ~ sn-inelr . (C... |
| sn-inelr 40920 | ~ inelr without ~ ax-mulco... |
| itrere 40921 | ` _i ` times a real is rea... |
| retire 40922 | Commuted version of ~ itre... |
| cnreeu 40923 | The reals in the expressio... |
| sn-sup2 40924 | ~ sup2 with exactly the sa... |
| prjspval 40927 | Value of the projective sp... |
| prjsprel 40928 | Utility theorem regarding ... |
| prjspertr 40929 | The relation in ` PrjSp ` ... |
| prjsperref 40930 | The relation in ` PrjSp ` ... |
| prjspersym 40931 | The relation in ` PrjSp ` ... |
| prjsper 40932 | The relation used to defin... |
| prjspreln0 40933 | Two nonzero vectors are eq... |
| prjspvs 40934 | A nonzero multiple of a ve... |
| prjsprellsp 40935 | Two vectors are equivalent... |
| prjspeclsp 40936 | The vectors equivalent to ... |
| prjspval2 40937 | Alternate definition of pr... |
| prjspnval 40940 | Value of the n-dimensional... |
| prjspnerlem 40941 | A lemma showing that the e... |
| prjspnval2 40942 | Value of the n-dimensional... |
| prjspner 40943 | The relation used to defin... |
| prjspnvs 40944 | A nonzero multiple of a ve... |
| prjspnssbas 40945 | A projective point spans a... |
| prjspnn0 40946 | A projective point is none... |
| 0prjspnlem 40947 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 40948 | Any vector is equivalent t... |
| prjspner01 40949 | Any vector is equivalent t... |
| prjspner1 40950 | Two vectors whose zeroth c... |
| 0prjspnrel 40951 | In the zero-dimensional pr... |
| 0prjspn 40952 | A zero-dimensional project... |
| prjcrvfval 40955 | Value of the projective cu... |
| prjcrvval 40956 | Value of the projective cu... |
| prjcrv0 40957 | The "curve" (zero set) cor... |
| dffltz 40958 | Fermat's Last Theorem (FLT... |
| fltmul 40959 | A counterexample to FLT st... |
| fltdiv 40960 | A counterexample to FLT st... |
| flt0 40961 | A counterexample for FLT d... |
| fltdvdsabdvdsc 40962 | Any factor of both ` A ` a... |
| fltabcoprmex 40963 | A counterexample to FLT im... |
| fltaccoprm 40964 | A counterexample to FLT wi... |
| fltbccoprm 40965 | A counterexample to FLT wi... |
| fltabcoprm 40966 | A counterexample to FLT wi... |
| infdesc 40967 | Infinite descent. The hyp... |
| fltne 40968 | If a counterexample to FLT... |
| flt4lem 40969 | Raising a number to the fo... |
| flt4lem1 40970 | Satisfy the antecedent use... |
| flt4lem2 40971 | If ` A ` is even, ` B ` is... |
| flt4lem3 40972 | Equivalent to ~ pythagtrip... |
| flt4lem4 40973 | If the product of two copr... |
| flt4lem5 40974 | In the context of the lemm... |
| flt4lem5elem 40975 | Version of ~ fltaccoprm an... |
| flt4lem5a 40976 | Part 1 of Equation 1 of ... |
| flt4lem5b 40977 | Part 2 of Equation 1 of ... |
| flt4lem5c 40978 | Part 2 of Equation 2 of ... |
| flt4lem5d 40979 | Part 3 of Equation 2 of ... |
| flt4lem5e 40980 | Satisfy the hypotheses of ... |
| flt4lem5f 40981 | Final equation of ~... |
| flt4lem6 40982 | Remove shared factors in a... |
| flt4lem7 40983 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 40984 | Strengthening of Fermat's ... |
| fltltc 40985 | ` ( C ^ N ) ` is the large... |
| fltnltalem 40986 | Lemma for ~ fltnlta . A l... |
| fltnlta 40987 | In a Fermat counterexample... |
| binom2d 40988 | Deduction form of binom2. ... |
| cu3addd 40989 | Cube of sum of three numbe... |
| sqnegd 40990 | The square of the negative... |
| negexpidd 40991 | The sum of a real number t... |
| rexlimdv3d 40992 | An extended version of ~ r... |
| 3cubeslem1 40993 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 40994 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 40995 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 40996 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 40997 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 40998 | Lemma for ~ 3cubes . This... |
| 3cubes 40999 | Every rational number is a... |
| rntrclfvOAI 41000 | The range of the transitiv... |
| moxfr 41001 | Transfer at-most-one betwe... |
| imaiinfv 41002 | Indexed intersection of an... |
| elrfi 41003 | Elementhood in a set of re... |
| elrfirn 41004 | Elementhood in a set of re... |
| elrfirn2 41005 | Elementhood in a set of re... |
| cmpfiiin 41006 | In a compact topology, a s... |
| ismrcd1 41007 | Any function from the subs... |
| ismrcd2 41008 | Second half of ~ ismrcd1 .... |
| istopclsd 41009 | A closure function which s... |
| ismrc 41010 | A function is a Moore clos... |
| isnacs 41013 | Expand definition of Noeth... |
| nacsfg 41014 | In a Noetherian-type closu... |
| isnacs2 41015 | Express Noetherian-type cl... |
| mrefg2 41016 | Slight variation on finite... |
| mrefg3 41017 | Slight variation on finite... |
| nacsacs 41018 | A closure system of Noethe... |
| isnacs3 41019 | A choice-free order equiva... |
| incssnn0 41020 | Transitivity induction of ... |
| nacsfix 41021 | An increasing sequence of ... |
| constmap 41022 | A constant (represented wi... |
| mapco2g 41023 | Renaming indices in a tupl... |
| mapco2 41024 | Post-composition (renaming... |
| mapfzcons 41025 | Extending a one-based mapp... |
| mapfzcons1 41026 | Recover prefix mapping fro... |
| mapfzcons1cl 41027 | A nonempty mapping has a p... |
| mapfzcons2 41028 | Recover added element from... |
| mptfcl 41029 | Interpret range of a maps-... |
| mzpclval 41034 | Substitution lemma for ` m... |
| elmzpcl 41035 | Double substitution lemma ... |
| mzpclall 41036 | The set of all functions w... |
| mzpcln0 41037 | Corollary of ~ mzpclall : ... |
| mzpcl1 41038 | Defining property 1 of a p... |
| mzpcl2 41039 | Defining property 2 of a p... |
| mzpcl34 41040 | Defining properties 3 and ... |
| mzpval 41041 | Value of the ` mzPoly ` fu... |
| dmmzp 41042 | ` mzPoly ` is defined for ... |
| mzpincl 41043 | Polynomial closedness is a... |
| mzpconst 41044 | Constant functions are pol... |
| mzpf 41045 | A polynomial function is a... |
| mzpproj 41046 | A projection function is p... |
| mzpadd 41047 | The pointwise sum of two p... |
| mzpmul 41048 | The pointwise product of t... |
| mzpconstmpt 41049 | A constant function expres... |
| mzpaddmpt 41050 | Sum of polynomial function... |
| mzpmulmpt 41051 | Product of polynomial func... |
| mzpsubmpt 41052 | The difference of two poly... |
| mzpnegmpt 41053 | Negation of a polynomial f... |
| mzpexpmpt 41054 | Raise a polynomial functio... |
| mzpindd 41055 | "Structural" induction to ... |
| mzpmfp 41056 | Relationship between multi... |
| mzpsubst 41057 | Substituting polynomials f... |
| mzprename 41058 | Simplified version of ~ mz... |
| mzpresrename 41059 | A polynomial is a polynomi... |
| mzpcompact2lem 41060 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 41061 | Polynomials are finitary o... |
| coeq0i 41062 | ~ coeq0 but without explic... |
| fzsplit1nn0 41063 | Split a finite 1-based set... |
| eldiophb 41066 | Initial expression of Diop... |
| eldioph 41067 | Condition for a set to be ... |
| diophrw 41068 | Renaming and adding unused... |
| eldioph2lem1 41069 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 41070 | Lemma for ~ eldioph2 . Co... |
| eldioph2 41071 | Construct a Diophantine se... |
| eldioph2b 41072 | While Diophantine sets wer... |
| eldiophelnn0 41073 | Remove antecedent on ` B `... |
| eldioph3b 41074 | Define Diophantine sets in... |
| eldioph3 41075 | Inference version of ~ eld... |
| ellz1 41076 | Membership in a lower set ... |
| lzunuz 41077 | The union of a lower set o... |
| fz1eqin 41078 | Express a one-based finite... |
| lzenom 41079 | Lower integers are countab... |
| elmapresaunres2 41080 | ~ fresaunres2 transposed t... |
| diophin 41081 | If two sets are Diophantin... |
| diophun 41082 | If two sets are Diophantin... |
| eldiophss 41083 | Diophantine sets are sets ... |
| diophrex 41084 | Projecting a Diophantine s... |
| eq0rabdioph 41085 | This is the first of a num... |
| eqrabdioph 41086 | Diophantine set builder fo... |
| 0dioph 41087 | The null set is Diophantin... |
| vdioph 41088 | The "universal" set (as la... |
| anrabdioph 41089 | Diophantine set builder fo... |
| orrabdioph 41090 | Diophantine set builder fo... |
| 3anrabdioph 41091 | Diophantine set builder fo... |
| 3orrabdioph 41092 | Diophantine set builder fo... |
| 2sbcrex 41093 | Exchange an existential qu... |
| sbcrexgOLD 41094 | Interchange class substitu... |
| 2sbcrexOLD 41095 | Exchange an existential qu... |
| sbc2rex 41096 | Exchange a substitution wi... |
| sbc2rexgOLD 41097 | Exchange a substitution wi... |
| sbc4rex 41098 | Exchange a substitution wi... |
| sbc4rexgOLD 41099 | Exchange a substitution wi... |
| sbcrot3 41100 | Rotate a sequence of three... |
| sbcrot5 41101 | Rotate a sequence of five ... |
| sbccomieg 41102 | Commute two explicit subst... |
| rexrabdioph 41103 | Diophantine set builder fo... |
| rexfrabdioph 41104 | Diophantine set builder fo... |
| 2rexfrabdioph 41105 | Diophantine set builder fo... |
| 3rexfrabdioph 41106 | Diophantine set builder fo... |
| 4rexfrabdioph 41107 | Diophantine set builder fo... |
| 6rexfrabdioph 41108 | Diophantine set builder fo... |
| 7rexfrabdioph 41109 | Diophantine set builder fo... |
| rabdiophlem1 41110 | Lemma for arithmetic dioph... |
| rabdiophlem2 41111 | Lemma for arithmetic dioph... |
| elnn0rabdioph 41112 | Diophantine set builder fo... |
| rexzrexnn0 41113 | Rewrite an existential qua... |
| lerabdioph 41114 | Diophantine set builder fo... |
| eluzrabdioph 41115 | Diophantine set builder fo... |
| elnnrabdioph 41116 | Diophantine set builder fo... |
| ltrabdioph 41117 | Diophantine set builder fo... |
| nerabdioph 41118 | Diophantine set builder fo... |
| dvdsrabdioph 41119 | Divisibility is a Diophant... |
| eldioph4b 41120 | Membership in ` Dioph ` ex... |
| eldioph4i 41121 | Forward-only version of ~ ... |
| diophren 41122 | Change variables in a Diop... |
| rabrenfdioph 41123 | Change variable numbers in... |
| rabren3dioph 41124 | Change variable numbers in... |
| fphpd 41125 | Pigeonhole principle expre... |
| fphpdo 41126 | Pigeonhole principle for s... |
| ctbnfien 41127 | An infinite subset of a co... |
| fiphp3d 41128 | Infinite pigeonhole princi... |
| rencldnfilem 41129 | Lemma for ~ rencldnfi . (... |
| rencldnfi 41130 | A set of real numbers whic... |
| irrapxlem1 41131 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 41132 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 41133 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 41134 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 41135 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 41136 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 41137 | Dirichlet's approximation ... |
| pellexlem1 41138 | Lemma for ~ pellex . Arit... |
| pellexlem2 41139 | Lemma for ~ pellex . Arit... |
| pellexlem3 41140 | Lemma for ~ pellex . To e... |
| pellexlem4 41141 | Lemma for ~ pellex . Invo... |
| pellexlem5 41142 | Lemma for ~ pellex . Invo... |
| pellexlem6 41143 | Lemma for ~ pellex . Doin... |
| pellex 41144 | Every Pell equation has a ... |
| pell1qrval 41155 | Value of the set of first-... |
| elpell1qr 41156 | Membership in a first-quad... |
| pell14qrval 41157 | Value of the set of positi... |
| elpell14qr 41158 | Membership in the set of p... |
| pell1234qrval 41159 | Value of the set of genera... |
| elpell1234qr 41160 | Membership in the set of g... |
| pell1234qrre 41161 | General Pell solutions are... |
| pell1234qrne0 41162 | No solution to a Pell equa... |
| pell1234qrreccl 41163 | General solutions of the P... |
| pell1234qrmulcl 41164 | General solutions of the P... |
| pell14qrss1234 41165 | A positive Pell solution i... |
| pell14qrre 41166 | A positive Pell solution i... |
| pell14qrne0 41167 | A positive Pell solution i... |
| pell14qrgt0 41168 | A positive Pell solution i... |
| pell14qrrp 41169 | A positive Pell solution i... |
| pell1234qrdich 41170 | A general Pell solution is... |
| elpell14qr2 41171 | A number is a positive Pel... |
| pell14qrmulcl 41172 | Positive Pell solutions ar... |
| pell14qrreccl 41173 | Positive Pell solutions ar... |
| pell14qrdivcl 41174 | Positive Pell solutions ar... |
| pell14qrexpclnn0 41175 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 41176 | Positive Pell solutions ar... |
| pell1qrss14 41177 | First-quadrant Pell soluti... |
| pell14qrdich 41178 | A positive Pell solution i... |
| pell1qrge1 41179 | A Pell solution in the fir... |
| pell1qr1 41180 | 1 is a Pell solution and i... |
| elpell1qr2 41181 | The first quadrant solutio... |
| pell1qrgaplem 41182 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 41183 | First-quadrant Pell soluti... |
| pell14qrgap 41184 | Positive Pell solutions ar... |
| pell14qrgapw 41185 | Positive Pell solutions ar... |
| pellqrexplicit 41186 | Condition for a calculated... |
| infmrgelbi 41187 | Any lower bound of a nonem... |
| pellqrex 41188 | There is a nontrivial solu... |
| pellfundval 41189 | Value of the fundamental s... |
| pellfundre 41190 | The fundamental solution o... |
| pellfundge 41191 | Lower bound on the fundame... |
| pellfundgt1 41192 | Weak lower bound on the Pe... |
| pellfundlb 41193 | A nontrivial first quadran... |
| pellfundglb 41194 | If a real is larger than t... |
| pellfundex 41195 | The fundamental solution a... |
| pellfund14gap 41196 | There are no solutions bet... |
| pellfundrp 41197 | The fundamental Pell solut... |
| pellfundne1 41198 | The fundamental Pell solut... |
| reglogcl 41199 | General logarithm is a rea... |
| reglogltb 41200 | General logarithm preserve... |
| reglogleb 41201 | General logarithm preserve... |
| reglogmul 41202 | Multiplication law for gen... |
| reglogexp 41203 | Power law for general log.... |
| reglogbas 41204 | General log of the base is... |
| reglog1 41205 | General log of 1 is 0. (C... |
| reglogexpbas 41206 | General log of a power of ... |
| pellfund14 41207 | Every positive Pell soluti... |
| pellfund14b 41208 | The positive Pell solution... |
| rmxfval 41213 | Value of the X sequence. ... |
| rmyfval 41214 | Value of the Y sequence. ... |
| rmspecsqrtnq 41215 | The discriminant used to d... |
| rmspecnonsq 41216 | The discriminant used to d... |
| qirropth 41217 | This lemma implements the ... |
| rmspecfund 41218 | The base of exponent used ... |
| rmxyelqirr 41219 | The solutions used to cons... |
| rmxyelqirrOLD 41220 | Obsolete version of ~ rmxy... |
| rmxypairf1o 41221 | The function used to extra... |
| rmxyelxp 41222 | Lemma for ~ frmx and ~ frm... |
| frmx 41223 | The X sequence is a nonneg... |
| frmy 41224 | The Y sequence is an integ... |
| rmxyval 41225 | Main definition of the X a... |
| rmspecpos 41226 | The discriminant used to d... |
| rmxycomplete 41227 | The X and Y sequences take... |
| rmxynorm 41228 | The X and Y sequences defi... |
| rmbaserp 41229 | The base of exponentiation... |
| rmxyneg 41230 | Negation law for X and Y s... |
| rmxyadd 41231 | Addition formula for X and... |
| rmxy1 41232 | Value of the X and Y seque... |
| rmxy0 41233 | Value of the X and Y seque... |
| rmxneg 41234 | Negation law (even functio... |
| rmx0 41235 | Value of X sequence at 0. ... |
| rmx1 41236 | Value of X sequence at 1. ... |
| rmxadd 41237 | Addition formula for X seq... |
| rmyneg 41238 | Negation formula for Y seq... |
| rmy0 41239 | Value of Y sequence at 0. ... |
| rmy1 41240 | Value of Y sequence at 1. ... |
| rmyadd 41241 | Addition formula for Y seq... |
| rmxp1 41242 | Special addition-of-1 form... |
| rmyp1 41243 | Special addition of 1 form... |
| rmxm1 41244 | Subtraction of 1 formula f... |
| rmym1 41245 | Subtraction of 1 formula f... |
| rmxluc 41246 | The X sequence is a Lucas ... |
| rmyluc 41247 | The Y sequence is a Lucas ... |
| rmyluc2 41248 | Lucas sequence property of... |
| rmxdbl 41249 | "Double-angle formula" for... |
| rmydbl 41250 | "Double-angle formula" for... |
| monotuz 41251 | A function defined on an u... |
| monotoddzzfi 41252 | A function which is odd an... |
| monotoddzz 41253 | A function (given implicit... |
| oddcomabszz 41254 | An odd function which take... |
| 2nn0ind 41255 | Induction on nonnegative i... |
| zindbi 41256 | Inductively transfer a pro... |
| rmxypos 41257 | For all nonnegative indice... |
| ltrmynn0 41258 | The Y-sequence is strictly... |
| ltrmxnn0 41259 | The X-sequence is strictly... |
| lermxnn0 41260 | The X-sequence is monotoni... |
| rmxnn 41261 | The X-sequence is defined ... |
| ltrmy 41262 | The Y-sequence is strictly... |
| rmyeq0 41263 | Y is zero only at zero. (... |
| rmyeq 41264 | Y is one-to-one. (Contrib... |
| lermy 41265 | Y is monotonic (non-strict... |
| rmynn 41266 | ` rmY ` is positive for po... |
| rmynn0 41267 | ` rmY ` is nonnegative for... |
| rmyabs 41268 | ` rmY ` commutes with ` ab... |
| jm2.24nn 41269 | X(n) is strictly greater t... |
| jm2.17a 41270 | First half of lemma 2.17 o... |
| jm2.17b 41271 | Weak form of the second ha... |
| jm2.17c 41272 | Second half of lemma 2.17 ... |
| jm2.24 41273 | Lemma 2.24 of [JonesMatija... |
| rmygeid 41274 | Y(n) increases faster than... |
| congtr 41275 | A wff of the form ` A || (... |
| congadd 41276 | If two pairs of numbers ar... |
| congmul 41277 | If two pairs of numbers ar... |
| congsym 41278 | Congruence mod ` A ` is a ... |
| congneg 41279 | If two integers are congru... |
| congsub 41280 | If two pairs of numbers ar... |
| congid 41281 | Every integer is congruent... |
| mzpcong 41282 | Polynomials commute with c... |
| congrep 41283 | Every integer is congruent... |
| congabseq 41284 | If two integers are congru... |
| acongid 41285 | A wff like that in this th... |
| acongsym 41286 | Symmetry of alternating co... |
| acongneg2 41287 | Negate right side of alter... |
| acongtr 41288 | Transitivity of alternatin... |
| acongeq12d 41289 | Substitution deduction for... |
| acongrep 41290 | Every integer is alternati... |
| fzmaxdif 41291 | Bound on the difference be... |
| fzneg 41292 | Reflection of a finite ran... |
| acongeq 41293 | Two numbers in the fundame... |
| dvdsacongtr 41294 | Alternating congruence pas... |
| coprmdvdsb 41295 | Multiplication by a coprim... |
| modabsdifz 41296 | Divisibility in terms of m... |
| dvdsabsmod0 41297 | Divisibility in terms of m... |
| jm2.18 41298 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 41299 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 41300 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 41301 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 41302 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 41303 | Lemma 2.19 of [JonesMatija... |
| jm2.21 41304 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 41305 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 41306 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 41307 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 41308 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 41309 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 41310 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 41311 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 41312 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 41313 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 41314 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 41315 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 41316 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 41317 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 41318 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 41319 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 41320 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 41321 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 41322 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 41323 | Lemma for ~ rmydioph . Us... |
| rmydioph 41324 | ~ jm2.27 restated in terms... |
| rmxdiophlem 41325 | X can be expressed in term... |
| rmxdioph 41326 | X is a Diophantine functio... |
| jm3.1lem1 41327 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 41328 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 41329 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 41330 | Diophantine expression for... |
| expdiophlem1 41331 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 41332 | Lemma for ~ expdioph . Ex... |
| expdioph 41333 | The exponential function i... |
| setindtr 41334 | Set induction for sets con... |
| setindtrs 41335 | Set induction scheme witho... |
| dford3lem1 41336 | Lemma for ~ dford3 . (Con... |
| dford3lem2 41337 | Lemma for ~ dford3 . (Con... |
| dford3 41338 | Ordinals are precisely the... |
| dford4 41339 | ~ dford3 expressed in prim... |
| wopprc 41340 | Unrelated: Wiener pairs t... |
| rpnnen3lem 41341 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 41342 | Dedekind cut injection of ... |
| axac10 41343 | Characterization of choice... |
| harinf 41344 | The Hartogs number of an i... |
| wdom2d2 41345 | Deduction for weak dominan... |
| ttac 41346 | Tarski's theorem about cho... |
| pw2f1ocnv 41347 | Define a bijection between... |
| pw2f1o2 41348 | Define a bijection between... |
| pw2f1o2val 41349 | Function value of the ~ pw... |
| pw2f1o2val2 41350 | Membership in a mapped set... |
| soeq12d 41351 | Equality deduction for tot... |
| freq12d 41352 | Equality deduction for fou... |
| weeq12d 41353 | Equality deduction for wel... |
| limsuc2 41354 | Limit ordinals in the sens... |
| wepwsolem 41355 | Transfer an ordering on ch... |
| wepwso 41356 | A well-ordering induces a ... |
| dnnumch1 41357 | Define an enumeration of a... |
| dnnumch2 41358 | Define an enumeration (wea... |
| dnnumch3lem 41359 | Value of the ordinal injec... |
| dnnumch3 41360 | Define an injection from a... |
| dnwech 41361 | Define a well-ordering fro... |
| fnwe2val 41362 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 41363 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 41364 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 41365 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 41366 | A well-ordering can be con... |
| aomclem1 41367 | Lemma for ~ dfac11 . This... |
| aomclem2 41368 | Lemma for ~ dfac11 . Succ... |
| aomclem3 41369 | Lemma for ~ dfac11 . Succ... |
| aomclem4 41370 | Lemma for ~ dfac11 . Limi... |
| aomclem5 41371 | Lemma for ~ dfac11 . Comb... |
| aomclem6 41372 | Lemma for ~ dfac11 . Tran... |
| aomclem7 41373 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 41374 | Lemma for ~ dfac11 . Perf... |
| dfac11 41375 | The right-hand side of thi... |
| kelac1 41376 | Kelley's choice, basic for... |
| kelac2lem 41377 | Lemma for ~ kelac2 and ~ d... |
| kelac2 41378 | Kelley's choice, most comm... |
| dfac21 41379 | Tychonoff's theorem is a c... |
| islmodfg 41382 | Property of a finitely gen... |
| islssfg 41383 | Property of a finitely gen... |
| islssfg2 41384 | Property of a finitely gen... |
| islssfgi 41385 | Finitely spanned subspaces... |
| fglmod 41386 | Finitely generated left mo... |
| lsmfgcl 41387 | The sum of two finitely ge... |
| islnm 41390 | Property of being a Noethe... |
| islnm2 41391 | Property of being a Noethe... |
| lnmlmod 41392 | A Noetherian left module i... |
| lnmlssfg 41393 | A submodule of Noetherian ... |
| lnmlsslnm 41394 | All submodules of a Noethe... |
| lnmfg 41395 | A Noetherian left module i... |
| kercvrlsm 41396 | The domain of a linear fun... |
| lmhmfgima 41397 | A homomorphism maps finite... |
| lnmepi 41398 | Epimorphic images of Noeth... |
| lmhmfgsplit 41399 | If the kernel and range of... |
| lmhmlnmsplit 41400 | If the kernel and range of... |
| lnmlmic 41401 | Noetherian is an invariant... |
| pwssplit4 41402 | Splitting for structure po... |
| filnm 41403 | Finite left modules are No... |
| pwslnmlem0 41404 | Zeroeth powers are Noether... |
| pwslnmlem1 41405 | First powers are Noetheria... |
| pwslnmlem2 41406 | A sum of powers is Noether... |
| pwslnm 41407 | Finite powers of Noetheria... |
| unxpwdom3 41408 | Weaker version of ~ unxpwd... |
| pwfi2f1o 41409 | The ~ pw2f1o bijection rel... |
| pwfi2en 41410 | Finitely supported indicat... |
| frlmpwfi 41411 | Formal linear combinations... |
| gicabl 41412 | Being Abelian is a group i... |
| imasgim 41413 | A relabeling of the elemen... |
| isnumbasgrplem1 41414 | A set which is equipollent... |
| harn0 41415 | The Hartogs number of a se... |
| numinfctb 41416 | A numerable infinite set c... |
| isnumbasgrplem2 41417 | If the (to be thought of a... |
| isnumbasgrplem3 41418 | Every nonempty numerable s... |
| isnumbasabl 41419 | A set is numerable iff it ... |
| isnumbasgrp 41420 | A set is numerable iff it ... |
| dfacbasgrp 41421 | A choice equivalent in abs... |
| islnr 41424 | Property of a left-Noether... |
| lnrring 41425 | Left-Noetherian rings are ... |
| lnrlnm 41426 | Left-Noetherian rings have... |
| islnr2 41427 | Property of being a left-N... |
| islnr3 41428 | Relate left-Noetherian rin... |
| lnr2i 41429 | Given an ideal in a left-N... |
| lpirlnr 41430 | Left principal ideal rings... |
| lnrfrlm 41431 | Finite-dimensional free mo... |
| lnrfg 41432 | Finitely-generated modules... |
| lnrfgtr 41433 | A submodule of a finitely ... |
| hbtlem1 41436 | Value of the leading coeff... |
| hbtlem2 41437 | Leading coefficient ideals... |
| hbtlem7 41438 | Functionality of leading c... |
| hbtlem4 41439 | The leading ideal function... |
| hbtlem3 41440 | The leading ideal function... |
| hbtlem5 41441 | The leading ideal function... |
| hbtlem6 41442 | There is a finite set of p... |
| hbt 41443 | The Hilbert Basis Theorem ... |
| dgrsub2 41448 | Subtracting two polynomial... |
| elmnc 41449 | Property of a monic polyno... |
| mncply 41450 | A monic polynomial is a po... |
| mnccoe 41451 | A monic polynomial has lea... |
| mncn0 41452 | A monic polynomial is not ... |
| dgraaval 41457 | Value of the degree functi... |
| dgraalem 41458 | Properties of the degree o... |
| dgraacl 41459 | Closure of the degree func... |
| dgraaf 41460 | Degree function on algebra... |
| dgraaub 41461 | Upper bound on degree of a... |
| dgraa0p 41462 | A rational polynomial of d... |
| mpaaeu 41463 | An algebraic number has ex... |
| mpaaval 41464 | Value of the minimal polyn... |
| mpaalem 41465 | Properties of the minimal ... |
| mpaacl 41466 | Minimal polynomial is a po... |
| mpaadgr 41467 | Minimal polynomial has deg... |
| mpaaroot 41468 | The minimal polynomial of ... |
| mpaamn 41469 | Minimal polynomial is moni... |
| itgoval 41474 | Value of the integral-over... |
| aaitgo 41475 | The standard algebraic num... |
| itgoss 41476 | An integral element is int... |
| itgocn 41477 | All integral elements are ... |
| cnsrexpcl 41478 | Exponentiation is closed i... |
| fsumcnsrcl 41479 | Finite sums are closed in ... |
| cnsrplycl 41480 | Polynomials are closed in ... |
| rgspnval 41481 | Value of the ring-span of ... |
| rgspncl 41482 | The ring-span of a set is ... |
| rgspnssid 41483 | The ring-span of a set con... |
| rgspnmin 41484 | The ring-span is contained... |
| rgspnid 41485 | The span of a subring is i... |
| rngunsnply 41486 | Adjoining one element to a... |
| flcidc 41487 | Finite linear combinations... |
| algstr 41490 | Lemma to shorten proofs of... |
| algbase 41491 | The base set of a construc... |
| algaddg 41492 | The additive operation of ... |
| algmulr 41493 | The multiplicative operati... |
| algsca 41494 | The set of scalars of a co... |
| algvsca 41495 | The scalar product operati... |
| mendval 41496 | Value of the module endomo... |
| mendbas 41497 | Base set of the module end... |
| mendplusgfval 41498 | Addition in the module end... |
| mendplusg 41499 | A specific addition in the... |
| mendmulrfval 41500 | Multiplication in the modu... |
| mendmulr 41501 | A specific multiplication ... |
| mendsca 41502 | The module endomorphism al... |
| mendvscafval 41503 | Scalar multiplication in t... |
| mendvsca 41504 | A specific scalar multipli... |
| mendring 41505 | The module endomorphism al... |
| mendlmod 41506 | The module endomorphism al... |
| mendassa 41507 | The module endomorphism al... |
| idomrootle 41508 | No element of an integral ... |
| idomodle 41509 | Limit on the number of ` N... |
| fiuneneq 41510 | Two finite sets of equal s... |
| idomsubgmo 41511 | The units of an integral d... |
| proot1mul 41512 | Any primitive ` N ` -th ro... |
| proot1hash 41513 | If an integral domain has ... |
| proot1ex 41514 | The complex field has prim... |
| isdomn3 41517 | Nonzero elements form a mu... |
| mon1pid 41518 | Monicity and degree of the... |
| mon1psubm 41519 | Monic polynomials are a mu... |
| deg1mhm 41520 | Homomorphic property of th... |
| cytpfn 41521 | Functionality of the cyclo... |
| cytpval 41522 | Substitutions for the Nth ... |
| fgraphopab 41523 | Express a function as a su... |
| fgraphxp 41524 | Express a function as a su... |
| hausgraph 41525 | The graph of a continuous ... |
| r1sssucd 41530 | Deductive form of ~ r1sssu... |
| iocunico 41531 | Split an open interval int... |
| iocinico 41532 | The intersection of two se... |
| iocmbl 41533 | An open-below, closed-abov... |
| cnioobibld 41534 | A bounded, continuous func... |
| arearect 41535 | The area of a rectangle wh... |
| areaquad 41536 | The area of a quadrilatera... |
| uniel 41537 | Two ways to say a union is... |
| unielss 41538 | Two ways to say the union ... |
| unielid 41539 | Two ways to say the union ... |
| ssunib 41540 | Two ways to say a class is... |
| rp-intrabeq 41541 | Equality theorem for supre... |
| rp-unirabeq 41542 | Equality theorem for infim... |
| onmaxnelsup 41543 | Two ways to say the maximu... |
| onsupneqmaxlim0 41544 | If the supremum of a class... |
| onsupcl2 41545 | The supremum of a set of o... |
| onuniintrab 41546 | The union of a set of ordi... |
| onintunirab 41547 | The intersection of a non-... |
| onsupnmax 41548 | If the union of a class of... |
| onsupuni 41549 | The supremum of a set of o... |
| onsupuni2 41550 | The supremum of a set of o... |
| onsupintrab 41551 | The supremum of a set of o... |
| onsupintrab2 41552 | The supremum of a set of o... |
| onsupcl3 41553 | The supremum of a set of o... |
| onsupex3 41554 | The supremum of a set of o... |
| onuniintrab2 41555 | The union of a set of ordi... |
| oninfint 41556 | The infimum of a non-empty... |
| oninfunirab 41557 | The infimum of a non-empty... |
| oninfcl2 41558 | The infimum of a non-empty... |
| onsupmaxb 41559 | The union of a class of or... |
| onexgt 41560 | For any ordinal, there is ... |
| onexomgt 41561 | For any ordinal, there is ... |
| omlimcl2 41562 | The product of a limit ord... |
| onexlimgt 41563 | For any ordinal, there is ... |
| onexoegt 41564 | For any ordinal, there is ... |
| oninfex2 41565 | The infimum of a non-empty... |
| onsupeqmax 41566 | Condition when the supremu... |
| onsupeqnmax 41567 | Condition when the supremu... |
| onsuplub 41568 | The supremum of a set of o... |
| onsupnub 41569 | An upper bound of a set of... |
| onfisupcl 41570 | Sufficient condition when ... |
| onelord 41571 | Every element of a ordinal... |
| onepsuc 41572 | Every ordinal is less than... |
| epsoon 41573 | The ordinals are strictly ... |
| epirron 41574 | The strict order on the or... |
| oneptr 41575 | The strict order on the or... |
| oneltr 41576 | The elementhood relation o... |
| oneptri 41577 | The strict, complete (line... |
| oneltri 41578 | The elementhood relation o... |
| ordeldif 41579 | Membership in the differen... |
| ordeldifsucon 41580 | Membership in the differen... |
| ordeldif1o 41581 | Membership in the differen... |
| ordne0gt0 41582 | Ordinal zero is less than ... |
| ondif1i 41583 | Ordinal zero is less than ... |
| onsucelab 41584 | The successor of every ord... |
| dflim6 41585 | A limit ordinal is a non-z... |
| limnsuc 41586 | A limit ordinal is not an ... |
| onsucss 41587 | If one ordinal is less tha... |
| ordnexbtwnsuc 41588 | For any distinct pair of o... |
| orddif0suc 41589 | For any distinct pair of o... |
| onsucf1lem 41590 | For ordinals, the successo... |
| onsucf1olem 41591 | The successor operation is... |
| onsucrn 41592 | The successor operation is... |
| onsucf1o 41593 | The successor operation is... |
| dflim7 41594 | A limit ordinal is a non-z... |
| onov0suclim 41595 | Compactly express rules fo... |
| oa0suclim 41596 | Closed form expression of ... |
| om0suclim 41597 | Closed form expression of ... |
| oe0suclim 41598 | Closed form expression of ... |
| oaomoecl 41599 | The operations of addition... |
| onsupsucismax 41600 | If the union of a set of o... |
| onsssupeqcond 41601 | If for every element of a ... |
| limexissup 41602 | An ordinal which is a limi... |
| limiun 41603 | A limit ordinal is the uni... |
| limexissupab 41604 | An ordinal which is a limi... |
| om1om1r 41605 | Ordinal one is both a left... |
| oe0rif 41606 | Ordinal zero raised to any... |
| oasubex 41607 | While subtraction can't be... |
| nnamecl 41608 | Natural numbers are closed... |
| onsucwordi 41609 | The successor operation pr... |
| oalim2cl 41610 | The ordinal sum of any ord... |
| oaltublim 41611 | Given ` C ` is a limit ord... |
| oaordi3 41612 | Ordinal addition of the sa... |
| oaord3 41613 | When the same ordinal is a... |
| 1oaomeqom 41614 | Ordinal one plus omega is ... |
| oaordnrex 41615 | When omega is added on the... |
| oaordnr 41616 | When the same ordinal is a... |
| omge1 41617 | Any non-zero ordinal produ... |
| omge2 41618 | Any non-zero ordinal produ... |
| omlim2 41619 | The non-zero product with ... |
| omord2lim 41620 | Given a limit ordinal, the... |
| omord2i 41621 | Ordinal multiplication of ... |
| omord2com 41622 | When the same non-zero ord... |
| 2omomeqom 41623 | Ordinal two times omega is... |
| omnord1ex 41624 | When omega is multiplied o... |
| omnord1 41625 | When the same non-zero ord... |
| oege1 41626 | Any non-zero ordinal power... |
| oege2 41627 | Any power of an ordinal at... |
| rp-oelim2 41628 | The power of an ordinal at... |
| oeord2lim 41629 | Given a limit ordinal, the... |
| oeord2i 41630 | Ordinal exponentiation of ... |
| oeord2com 41631 | When the same base at leas... |
| nnoeomeqom 41632 | Any natural number at leas... |
| df3o2 41633 | Ordinal 3 is the unordered... |
| df3o3 41634 | Ordinal 3, fully expanded.... |
| oenord1ex 41635 | When ordinals two and thre... |
| oenord1 41636 | When two ordinals (both at... |
| oaomoencom 41637 | Ordinal addition, multipli... |
| oenassex 41638 | Ordinal two raised to two ... |
| oenass 41639 | Ordinal exponentiation is ... |
| cantnftermord 41640 | For terms of the form of a... |
| cantnfub 41641 | Given a finite number of t... |
| cantnfub2 41642 | Given a finite number of t... |
| bropabg 41643 | Equivalence for two classe... |
| cantnfresb 41644 | A Cantor normal form which... |
| cantnf2 41645 | For every ordinal, ` A ` ,... |
| oawordex2 41646 | If ` C ` is between ` A ` ... |
| nnawordexg 41647 | If an ordinal, ` B ` , is ... |
| succlg 41648 | Closure law for ordinal su... |
| dflim5 41649 | A limit ordinal is either ... |
| oacl2g 41650 | Closure law for ordinal ad... |
| omabs2 41651 | Ordinal multiplication by ... |
| omcl2 41652 | Closure law for ordinal mu... |
| omcl3g 41653 | Closure law for ordinal mu... |
| ofoafg 41654 | Addition operator for func... |
| ofoaf 41655 | Addition operator for func... |
| ofoafo 41656 | Addition operator for func... |
| ofoacl 41657 | Closure law for component ... |
| ofoaid1 41658 | Identity law for component... |
| ofoaid2 41659 | Identity law for component... |
| ofoaass 41660 | Component-wise addition of... |
| ofoacom 41661 | Component-wise addition of... |
| naddcnff 41662 | Addition operator for Cant... |
| naddcnffn 41663 | Addition operator for Cant... |
| naddcnffo 41664 | Addition of Cantor normal ... |
| naddcnfcl 41665 | Closure law for component-... |
| naddcnfcom 41666 | Component-wise ordinal add... |
| naddcnfid1 41667 | Identity law for component... |
| naddcnfid2 41668 | Identity law for component... |
| naddcnfass 41669 | Component-wise addition of... |
| abeqabi 41670 | Generalized condition for ... |
| abpr 41671 | Condition for a class abst... |
| abtp 41672 | Condition for a class abst... |
| ralopabb 41673 | Restricted universal quant... |
| fpwfvss 41674 | Functions into a powerset ... |
| sdomne0 41675 | A class that strictly domi... |
| sdomne0d 41676 | A class that strictly domi... |
| safesnsupfiss 41677 | If ` B ` is a finite subse... |
| safesnsupfiub 41678 | If ` B ` is a finite subse... |
| safesnsupfidom1o 41679 | If ` B ` is a finite subse... |
| safesnsupfilb 41680 | If ` B ` is a finite subse... |
| isoeq145d 41681 | Equality deduction for iso... |
| resisoeq45d 41682 | Equality deduction for equ... |
| negslem1 41683 | An equivalence between ide... |
| nvocnvb 41684 | Equivalence to saying the ... |
| rp-brsslt 41685 | Binary relation form of a ... |
| nla0002 41686 | Extending a linear order t... |
| nla0003 41687 | Extending a linear order t... |
| nla0001 41688 | Extending a linear order t... |
| faosnf0.11b 41689 | ` B ` is called a non-limi... |
| dfno2 41690 | A surreal number, in the f... |
| onnog 41691 | Every ordinal maps to a su... |
| onnobdayg 41692 | Every ordinal maps to a su... |
| bdaybndex 41693 | Bounds formed from the bir... |
| bdaybndbday 41694 | Bounds formed from the bir... |
| onno 41695 | Every ordinal maps to a su... |
| onnoi 41696 | Every ordinal maps to a su... |
| 0no 41697 | Ordinal zero maps to a sur... |
| 1no 41698 | Ordinal one maps to a surr... |
| 2no 41699 | Ordinal two maps to a surr... |
| 3no 41700 | Ordinal three maps to a su... |
| 4no 41701 | Ordinal four maps to a sur... |
| fnimafnex 41702 | The functional image of a ... |
| nlimsuc 41703 | A successor is not a limit... |
| nlim1NEW 41704 | 1 is not a limit ordinal. ... |
| nlim2NEW 41705 | 2 is not a limit ordinal. ... |
| nlim3 41706 | 3 is not a limit ordinal. ... |
| nlim4 41707 | 4 is not a limit ordinal. ... |
| oa1un 41708 | Given ` A e. On ` , let ` ... |
| oa1cl 41709 | ` A +o 1o ` is in ` On ` .... |
| 0finon 41710 | 0 is a finite ordinal. Se... |
| 1finon 41711 | 1 is a finite ordinal. Se... |
| 2finon 41712 | 2 is a finite ordinal. Se... |
| 3finon 41713 | 3 is a finite ordinal. Se... |
| 4finon 41714 | 4 is a finite ordinal. Se... |
| finona1cl 41715 | The finite ordinals are cl... |
| finonex 41716 | The finite ordinals are a ... |
| fzunt 41717 | Union of two adjacent fini... |
| fzuntd 41718 | Union of two adjacent fini... |
| fzunt1d 41719 | Union of two overlapping f... |
| fzuntgd 41720 | Union of two adjacent or o... |
| ifpan123g 41721 | Conjunction of conditional... |
| ifpan23 41722 | Conjunction of conditional... |
| ifpdfor2 41723 | Define or in terms of cond... |
| ifporcor 41724 | Corollary of commutation o... |
| ifpdfan2 41725 | Define and with conditiona... |
| ifpancor 41726 | Corollary of commutation o... |
| ifpdfor 41727 | Define or in terms of cond... |
| ifpdfan 41728 | Define and with conditiona... |
| ifpbi2 41729 | Equivalence theorem for co... |
| ifpbi3 41730 | Equivalence theorem for co... |
| ifpim1 41731 | Restate implication as con... |
| ifpnot 41732 | Restate negated wff as con... |
| ifpid2 41733 | Restate wff as conditional... |
| ifpim2 41734 | Restate implication as con... |
| ifpbi23 41735 | Equivalence theorem for co... |
| ifpbiidcor 41736 | Restatement of ~ biid . (... |
| ifpbicor 41737 | Corollary of commutation o... |
| ifpxorcor 41738 | Corollary of commutation o... |
| ifpbi1 41739 | Equivalence theorem for co... |
| ifpnot23 41740 | Negation of conditional lo... |
| ifpnotnotb 41741 | Factor conditional logic o... |
| ifpnorcor 41742 | Corollary of commutation o... |
| ifpnancor 41743 | Corollary of commutation o... |
| ifpnot23b 41744 | Negation of conditional lo... |
| ifpbiidcor2 41745 | Restatement of ~ biid . (... |
| ifpnot23c 41746 | Negation of conditional lo... |
| ifpnot23d 41747 | Negation of conditional lo... |
| ifpdfnan 41748 | Define nand as conditional... |
| ifpdfxor 41749 | Define xor as conditional ... |
| ifpbi12 41750 | Equivalence theorem for co... |
| ifpbi13 41751 | Equivalence theorem for co... |
| ifpbi123 41752 | Equivalence theorem for co... |
| ifpidg 41753 | Restate wff as conditional... |
| ifpid3g 41754 | Restate wff as conditional... |
| ifpid2g 41755 | Restate wff as conditional... |
| ifpid1g 41756 | Restate wff as conditional... |
| ifpim23g 41757 | Restate implication as con... |
| ifpim3 41758 | Restate implication as con... |
| ifpnim1 41759 | Restate negated implicatio... |
| ifpim4 41760 | Restate implication as con... |
| ifpnim2 41761 | Restate negated implicatio... |
| ifpim123g 41762 | Implication of conditional... |
| ifpim1g 41763 | Implication of conditional... |
| ifp1bi 41764 | Substitute the first eleme... |
| ifpbi1b 41765 | When the first variable is... |
| ifpimimb 41766 | Factor conditional logic o... |
| ifpororb 41767 | Factor conditional logic o... |
| ifpananb 41768 | Factor conditional logic o... |
| ifpnannanb 41769 | Factor conditional logic o... |
| ifpor123g 41770 | Disjunction of conditional... |
| ifpimim 41771 | Consequnce of implication.... |
| ifpbibib 41772 | Factor conditional logic o... |
| ifpxorxorb 41773 | Factor conditional logic o... |
| rp-fakeimass 41774 | A special case where impli... |
| rp-fakeanorass 41775 | A special case where a mix... |
| rp-fakeoranass 41776 | A special case where a mix... |
| rp-fakeinunass 41777 | A special case where a mix... |
| rp-fakeuninass 41778 | A special case where a mix... |
| rp-isfinite5 41779 | A set is said to be finite... |
| rp-isfinite6 41780 | A set is said to be finite... |
| intabssd 41781 | When for each element ` y ... |
| eu0 41782 | There is only one empty se... |
| epelon2 41783 | Over the ordinal numbers, ... |
| ontric3g 41784 | For all ` x , y e. On ` , ... |
| dfsucon 41785 | ` A ` is called a successo... |
| snen1g 41786 | A singleton is equinumerou... |
| snen1el 41787 | A singleton is equinumerou... |
| sn1dom 41788 | A singleton is dominated b... |
| pr2dom 41789 | An unordered pair is domin... |
| tr3dom 41790 | An unordered triple is dom... |
| ensucne0 41791 | A class equinumerous to a ... |
| ensucne0OLD 41792 | A class equinumerous to a ... |
| dfom6 41793 | Let ` _om ` be defined to ... |
| infordmin 41794 | ` _om ` is the smallest in... |
| iscard4 41795 | Two ways to express the pr... |
| minregex 41796 | Given any cardinal number ... |
| minregex2 41797 | Given any cardinal number ... |
| iscard5 41798 | Two ways to express the pr... |
| elrncard 41799 | Let us define a cardinal n... |
| harval3 41800 | ` ( har `` A ) ` is the le... |
| harval3on 41801 | For any ordinal number ` A... |
| omssrncard 41802 | All natural numbers are ca... |
| 0iscard 41803 | 0 is a cardinal number. (... |
| 1iscard 41804 | 1 is a cardinal number. (... |
| omiscard 41805 | ` _om ` is a cardinal numb... |
| sucomisnotcard 41806 | ` _om +o 1o ` is not a car... |
| nna1iscard 41807 | For any natural number, th... |
| har2o 41808 | The least cardinal greater... |
| en2pr 41809 | A class is equinumerous to... |
| pr2cv 41810 | If an unordered pair is eq... |
| pr2el1 41811 | If an unordered pair is eq... |
| pr2cv1 41812 | If an unordered pair is eq... |
| pr2el2 41813 | If an unordered pair is eq... |
| pr2cv2 41814 | If an unordered pair is eq... |
| pren2 41815 | An unordered pair is equin... |
| pr2eldif1 41816 | If an unordered pair is eq... |
| pr2eldif2 41817 | If an unordered pair is eq... |
| pren2d 41818 | A pair of two distinct set... |
| aleph1min 41819 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 41820 | ` aleph ` is a strictly or... |
| alephiso3 41821 | ` aleph ` is a strictly or... |
| pwelg 41822 | The powerclass is an eleme... |
| pwinfig 41823 | The powerclass of an infin... |
| pwinfi2 41824 | The powerclass of an infin... |
| pwinfi3 41825 | The powerclass of an infin... |
| pwinfi 41826 | The powerclass of an infin... |
| fipjust 41827 | A definition of the finite... |
| cllem0 41828 | The class of all sets with... |
| superficl 41829 | The class of all supersets... |
| superuncl 41830 | The class of all supersets... |
| ssficl 41831 | The class of all subsets o... |
| ssuncl 41832 | The class of all subsets o... |
| ssdifcl 41833 | The class of all subsets o... |
| sssymdifcl 41834 | The class of all subsets o... |
| fiinfi 41835 | If two classes have the fi... |
| rababg 41836 | Condition when restricted ... |
| elinintab 41837 | Two ways of saying a set i... |
| elmapintrab 41838 | Two ways to say a set is a... |
| elinintrab 41839 | Two ways of saying a set i... |
| inintabss 41840 | Upper bound on intersectio... |
| inintabd 41841 | Value of the intersection ... |
| xpinintabd 41842 | Value of the intersection ... |
| relintabex 41843 | If the intersection of a c... |
| elcnvcnvintab 41844 | Two ways of saying a set i... |
| relintab 41845 | Value of the intersection ... |
| nonrel 41846 | A non-relation is equal to... |
| elnonrel 41847 | Only an ordered pair where... |
| cnvssb 41848 | Subclass theorem for conve... |
| relnonrel 41849 | The non-relation part of a... |
| cnvnonrel 41850 | The converse of the non-re... |
| brnonrel 41851 | A non-relation cannot rela... |
| dmnonrel 41852 | The domain of the non-rela... |
| rnnonrel 41853 | The range of the non-relat... |
| resnonrel 41854 | A restriction of the non-r... |
| imanonrel 41855 | An image under the non-rel... |
| cononrel1 41856 | Composition with the non-r... |
| cononrel2 41857 | Composition with the non-r... |
| elmapintab 41858 | Two ways to say a set is a... |
| fvnonrel 41859 | The function value of any ... |
| elinlem 41860 | Two ways to say a set is a... |
| elcnvcnvlem 41861 | Two ways to say a set is a... |
| cnvcnvintabd 41862 | Value of the relationship ... |
| elcnvlem 41863 | Two ways to say a set is a... |
| elcnvintab 41864 | Two ways of saying a set i... |
| cnvintabd 41865 | Value of the converse of t... |
| undmrnresiss 41866 | Two ways of saying the ide... |
| reflexg 41867 | Two ways of saying a relat... |
| cnvssco 41868 | A condition weaker than re... |
| refimssco 41869 | Reflexive relations are su... |
| cleq2lem 41870 | Equality implies bijection... |
| cbvcllem 41871 | Change of bound variable i... |
| clublem 41872 | If a superset ` Y ` of ` X... |
| clss2lem 41873 | The closure of a property ... |
| dfid7 41874 | Definition of identity rel... |
| mptrcllem 41875 | Show two versions of a clo... |
| cotrintab 41876 | The intersection of a clas... |
| rclexi 41877 | The reflexive closure of a... |
| rtrclexlem 41878 | Existence of relation impl... |
| rtrclex 41879 | The reflexive-transitive c... |
| trclubgNEW 41880 | If a relation exists then ... |
| trclubNEW 41881 | If a relation exists then ... |
| trclexi 41882 | The transitive closure of ... |
| rtrclexi 41883 | The reflexive-transitive c... |
| clrellem 41884 | When the property ` ps ` h... |
| clcnvlem 41885 | When ` A ` , an upper boun... |
| cnvtrucl0 41886 | The converse of the trivia... |
| cnvrcl0 41887 | The converse of the reflex... |
| cnvtrcl0 41888 | The converse of the transi... |
| dmtrcl 41889 | The domain of the transiti... |
| rntrcl 41890 | The range of the transitiv... |
| dfrtrcl5 41891 | Definition of reflexive-tr... |
| trcleq2lemRP 41892 | Equality implies bijection... |
| sqrtcvallem1 41893 | Two ways of saying a compl... |
| reabsifneg 41894 | Alternate expression for t... |
| reabsifnpos 41895 | Alternate expression for t... |
| reabsifpos 41896 | Alternate expression for t... |
| reabsifnneg 41897 | Alternate expression for t... |
| reabssgn 41898 | Alternate expression for t... |
| sqrtcvallem2 41899 | Equivalent to saying that ... |
| sqrtcvallem3 41900 | Equivalent to saying that ... |
| sqrtcvallem4 41901 | Equivalent to saying that ... |
| sqrtcvallem5 41902 | Equivalent to saying that ... |
| sqrtcval 41903 | Explicit formula for the c... |
| sqrtcval2 41904 | Explicit formula for the c... |
| resqrtval 41905 | Real part of the complex s... |
| imsqrtval 41906 | Imaginary part of the comp... |
| resqrtvalex 41907 | Example for ~ resqrtval . ... |
| imsqrtvalex 41908 | Example for ~ imsqrtval . ... |
| al3im 41909 | Version of ~ ax-4 for a ne... |
| intima0 41910 | Two ways of expressing the... |
| elimaint 41911 | Element of image of inters... |
| cnviun 41912 | Converse of indexed union.... |
| imaiun1 41913 | The image of an indexed un... |
| coiun1 41914 | Composition with an indexe... |
| elintima 41915 | Element of intersection of... |
| intimass 41916 | The image under the inters... |
| intimass2 41917 | The image under the inters... |
| intimag 41918 | Requirement for the image ... |
| intimasn 41919 | Two ways to express the im... |
| intimasn2 41920 | Two ways to express the im... |
| ss2iundf 41921 | Subclass theorem for index... |
| ss2iundv 41922 | Subclass theorem for index... |
| cbviuneq12df 41923 | Rule used to change the bo... |
| cbviuneq12dv 41924 | Rule used to change the bo... |
| conrel1d 41925 | Deduction about compositio... |
| conrel2d 41926 | Deduction about compositio... |
| trrelind 41927 | The intersection of transi... |
| xpintrreld 41928 | The intersection of a tran... |
| restrreld 41929 | The restriction of a trans... |
| trrelsuperreldg 41930 | Concrete construction of a... |
| trficl 41931 | The class of all transitiv... |
| cnvtrrel 41932 | The converse of a transiti... |
| trrelsuperrel2dg 41933 | Concrete construction of a... |
| dfrcl2 41936 | Reflexive closure of a rel... |
| dfrcl3 41937 | Reflexive closure of a rel... |
| dfrcl4 41938 | Reflexive closure of a rel... |
| relexp2 41939 | A set operated on by the r... |
| relexpnul 41940 | If the domain and range of... |
| eliunov2 41941 | Membership in the indexed ... |
| eltrclrec 41942 | Membership in the indexed ... |
| elrtrclrec 41943 | Membership in the indexed ... |
| briunov2 41944 | Two classes related by the... |
| brmptiunrelexpd 41945 | If two elements are connec... |
| fvmptiunrelexplb0d 41946 | If the indexed union range... |
| fvmptiunrelexplb0da 41947 | If the indexed union range... |
| fvmptiunrelexplb1d 41948 | If the indexed union range... |
| brfvid 41949 | If two elements are connec... |
| brfvidRP 41950 | If two elements are connec... |
| fvilbd 41951 | A set is a subset of its i... |
| fvilbdRP 41952 | A set is a subset of its i... |
| brfvrcld 41953 | If two elements are connec... |
| brfvrcld2 41954 | If two elements are connec... |
| fvrcllb0d 41955 | A restriction of the ident... |
| fvrcllb0da 41956 | A restriction of the ident... |
| fvrcllb1d 41957 | A set is a subset of its i... |
| brtrclrec 41958 | Two classes related by the... |
| brrtrclrec 41959 | Two classes related by the... |
| briunov2uz 41960 | Two classes related by the... |
| eliunov2uz 41961 | Membership in the indexed ... |
| ov2ssiunov2 41962 | Any particular operator va... |
| relexp0eq 41963 | The zeroth power of relati... |
| iunrelexp0 41964 | Simplification of zeroth p... |
| relexpxpnnidm 41965 | Any positive power of a Ca... |
| relexpiidm 41966 | Any power of any restricti... |
| relexpss1d 41967 | The relational power of a ... |
| comptiunov2i 41968 | The composition two indexe... |
| corclrcl 41969 | The reflexive closure is i... |
| iunrelexpmin1 41970 | The indexed union of relat... |
| relexpmulnn 41971 | With exponents limited to ... |
| relexpmulg 41972 | With ordered exponents, th... |
| trclrelexplem 41973 | The union of relational po... |
| iunrelexpmin2 41974 | The indexed union of relat... |
| relexp01min 41975 | With exponents limited to ... |
| relexp1idm 41976 | Repeated raising a relatio... |
| relexp0idm 41977 | Repeated raising a relatio... |
| relexp0a 41978 | Absorption law for zeroth ... |
| relexpxpmin 41979 | The composition of powers ... |
| relexpaddss 41980 | The composition of two pow... |
| iunrelexpuztr 41981 | The indexed union of relat... |
| dftrcl3 41982 | Transitive closure of a re... |
| brfvtrcld 41983 | If two elements are connec... |
| fvtrcllb1d 41984 | A set is a subset of its i... |
| trclfvcom 41985 | The transitive closure of ... |
| cnvtrclfv 41986 | The converse of the transi... |
| cotrcltrcl 41987 | The transitive closure is ... |
| trclimalb2 41988 | Lower bound for image unde... |
| brtrclfv2 41989 | Two ways to indicate two e... |
| trclfvdecomr 41990 | The transitive closure of ... |
| trclfvdecoml 41991 | The transitive closure of ... |
| dmtrclfvRP 41992 | The domain of the transiti... |
| rntrclfvRP 41993 | The range of the transitiv... |
| rntrclfv 41994 | The range of the transitiv... |
| dfrtrcl3 41995 | Reflexive-transitive closu... |
| brfvrtrcld 41996 | If two elements are connec... |
| fvrtrcllb0d 41997 | A restriction of the ident... |
| fvrtrcllb0da 41998 | A restriction of the ident... |
| fvrtrcllb1d 41999 | A set is a subset of its i... |
| dfrtrcl4 42000 | Reflexive-transitive closu... |
| corcltrcl 42001 | The composition of the ref... |
| cortrcltrcl 42002 | Composition with the refle... |
| corclrtrcl 42003 | Composition with the refle... |
| cotrclrcl 42004 | The composition of the ref... |
| cortrclrcl 42005 | Composition with the refle... |
| cotrclrtrcl 42006 | Composition with the refle... |
| cortrclrtrcl 42007 | The reflexive-transitive c... |
| frege77d 42008 | If the images of both ` { ... |
| frege81d 42009 | If the image of ` U ` is a... |
| frege83d 42010 | If the image of the union ... |
| frege96d 42011 | If ` C ` follows ` A ` in ... |
| frege87d 42012 | If the images of both ` { ... |
| frege91d 42013 | If ` B ` follows ` A ` in ... |
| frege97d 42014 | If ` A ` contains all elem... |
| frege98d 42015 | If ` C ` follows ` A ` and... |
| frege102d 42016 | If either ` A ` and ` C ` ... |
| frege106d 42017 | If ` B ` follows ` A ` in ... |
| frege108d 42018 | If either ` A ` and ` C ` ... |
| frege109d 42019 | If ` A ` contains all elem... |
| frege114d 42020 | If either ` R ` relates ` ... |
| frege111d 42021 | If either ` A ` and ` C ` ... |
| frege122d 42022 | If ` F ` is a function, ` ... |
| frege124d 42023 | If ` F ` is a function, ` ... |
| frege126d 42024 | If ` F ` is a function, ` ... |
| frege129d 42025 | If ` F ` is a function and... |
| frege131d 42026 | If ` F ` is a function and... |
| frege133d 42027 | If ` F ` is a function and... |
| dfxor4 42028 | Express exclusive-or in te... |
| dfxor5 42029 | Express exclusive-or in te... |
| df3or2 42030 | Express triple-or in terms... |
| df3an2 42031 | Express triple-and in term... |
| nev 42032 | Express that not every set... |
| 0pssin 42033 | Express that an intersecti... |
| dfhe2 42036 | The property of relation `... |
| dfhe3 42037 | The property of relation `... |
| heeq12 42038 | Equality law for relations... |
| heeq1 42039 | Equality law for relations... |
| heeq2 42040 | Equality law for relations... |
| sbcheg 42041 | Distribute proper substitu... |
| hess 42042 | Subclass law for relations... |
| xphe 42043 | Any Cartesian product is h... |
| 0he 42044 | The empty relation is here... |
| 0heALT 42045 | The empty relation is here... |
| he0 42046 | Any relation is hereditary... |
| unhe1 42047 | The union of two relations... |
| snhesn 42048 | Any singleton is hereditar... |
| idhe 42049 | The identity relation is h... |
| psshepw 42050 | The relation between sets ... |
| sshepw 42051 | The relation between sets ... |
| rp-simp2-frege 42054 | Simplification of triple c... |
| rp-simp2 42055 | Simplification of triple c... |
| rp-frege3g 42056 | Add antecedent to ~ ax-fre... |
| frege3 42057 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 42058 | Double-use of ~ ax-frege2 ... |
| rp-frege24 42059 | Introducing an embedded an... |
| rp-frege4g 42060 | Deduction related to distr... |
| frege4 42061 | Special case of closed for... |
| frege5 42062 | A closed form of ~ syl . ... |
| rp-7frege 42063 | Distribute antecedent and ... |
| rp-4frege 42064 | Elimination of a nested an... |
| rp-6frege 42065 | Elimination of a nested an... |
| rp-8frege 42066 | Eliminate antecedent when ... |
| rp-frege25 42067 | Closed form for ~ a1dd . ... |
| frege6 42068 | A closed form of ~ imim2d ... |
| axfrege8 42069 | Swap antecedents. Identic... |
| frege7 42070 | A closed form of ~ syl6 . ... |
| frege26 42072 | Identical to ~ idd . Prop... |
| frege27 42073 | We cannot (at the same tim... |
| frege9 42074 | Closed form of ~ syl with ... |
| frege12 42075 | A closed form of ~ com23 .... |
| frege11 42076 | Elimination of a nested an... |
| frege24 42077 | Closed form for ~ a1d . D... |
| frege16 42078 | A closed form of ~ com34 .... |
| frege25 42079 | Closed form for ~ a1dd . ... |
| frege18 42080 | Closed form of a syllogism... |
| frege22 42081 | A closed form of ~ com45 .... |
| frege10 42082 | Result commuting anteceden... |
| frege17 42083 | A closed form of ~ com3l .... |
| frege13 42084 | A closed form of ~ com3r .... |
| frege14 42085 | Closed form of a deduction... |
| frege19 42086 | A closed form of ~ syl6 . ... |
| frege23 42087 | Syllogism followed by rota... |
| frege15 42088 | A closed form of ~ com4r .... |
| frege21 42089 | Replace antecedent in ante... |
| frege20 42090 | A closed form of ~ syl8 . ... |
| axfrege28 42091 | Contraposition. Identical... |
| frege29 42093 | Closed form of ~ con3d . ... |
| frege30 42094 | Commuted, closed form of ~... |
| axfrege31 42095 | Identical to ~ notnotr . ... |
| frege32 42097 | Deduce ~ con1 from ~ con3 ... |
| frege33 42098 | If ` ph ` or ` ps ` takes ... |
| frege34 42099 | If as a consequence of the... |
| frege35 42100 | Commuted, closed form of ~... |
| frege36 42101 | The case in which ` ps ` i... |
| frege37 42102 | If ` ch ` is a necessary c... |
| frege38 42103 | Identical to ~ pm2.21 . P... |
| frege39 42104 | Syllogism between ~ pm2.18... |
| frege40 42105 | Anything implies ~ pm2.18 ... |
| axfrege41 42106 | Identical to ~ notnot . A... |
| frege42 42108 | Not not ~ id . Propositio... |
| frege43 42109 | If there is a choice only ... |
| frege44 42110 | Similar to a commuted ~ pm... |
| frege45 42111 | Deduce ~ pm2.6 from ~ con1... |
| frege46 42112 | If ` ps ` holds when ` ph ... |
| frege47 42113 | Deduce consequence follows... |
| frege48 42114 | Closed form of syllogism w... |
| frege49 42115 | Closed form of deduction w... |
| frege50 42116 | Closed form of ~ jaoi . P... |
| frege51 42117 | Compare with ~ jaod . Pro... |
| axfrege52a 42118 | Justification for ~ ax-fre... |
| frege52aid 42120 | The case when the content ... |
| frege53aid 42121 | Specialization of ~ frege5... |
| frege53a 42122 | Lemma for ~ frege55a . Pr... |
| axfrege54a 42123 | Justification for ~ ax-fre... |
| frege54cor0a 42125 | Synonym for logical equiva... |
| frege54cor1a 42126 | Reflexive equality. (Cont... |
| frege55aid 42127 | Lemma for ~ frege57aid . ... |
| frege55lem1a 42128 | Necessary deduction regard... |
| frege55lem2a 42129 | Core proof of Proposition ... |
| frege55a 42130 | Proposition 55 of [Frege18... |
| frege55cor1a 42131 | Proposition 55 of [Frege18... |
| frege56aid 42132 | Lemma for ~ frege57aid . ... |
| frege56a 42133 | Proposition 56 of [Frege18... |
| frege57aid 42134 | This is the all imporant f... |
| frege57a 42135 | Analogue of ~ frege57aid .... |
| axfrege58a 42136 | Identical to ~ anifp . Ju... |
| frege58acor 42138 | Lemma for ~ frege59a . (C... |
| frege59a 42139 | A kind of Aristotelian inf... |
| frege60a 42140 | Swap antecedents of ~ ax-f... |
| frege61a 42141 | Lemma for ~ frege65a . Pr... |
| frege62a 42142 | A kind of Aristotelian inf... |
| frege63a 42143 | Proposition 63 of [Frege18... |
| frege64a 42144 | Lemma for ~ frege65a . Pr... |
| frege65a 42145 | A kind of Aristotelian inf... |
| frege66a 42146 | Swap antecedents of ~ freg... |
| frege67a 42147 | Lemma for ~ frege68a . Pr... |
| frege68a 42148 | Combination of applying a ... |
| axfrege52c 42149 | Justification for ~ ax-fre... |
| frege52b 42151 | The case when the content ... |
| frege53b 42152 | Lemma for frege102 (via ~ ... |
| axfrege54c 42153 | Reflexive equality of clas... |
| frege54b 42155 | Reflexive equality of sets... |
| frege54cor1b 42156 | Reflexive equality. (Cont... |
| frege55lem1b 42157 | Necessary deduction regard... |
| frege55lem2b 42158 | Lemma for ~ frege55b . Co... |
| frege55b 42159 | Lemma for ~ frege57b . Pr... |
| frege56b 42160 | Lemma for ~ frege57b . Pr... |
| frege57b 42161 | Analogue of ~ frege57aid .... |
| axfrege58b 42162 | If ` A. x ph ` is affirmed... |
| frege58bid 42164 | If ` A. x ph ` is affirmed... |
| frege58bcor 42165 | Lemma for ~ frege59b . (C... |
| frege59b 42166 | A kind of Aristotelian inf... |
| frege60b 42167 | Swap antecedents of ~ ax-f... |
| frege61b 42168 | Lemma for ~ frege65b . Pr... |
| frege62b 42169 | A kind of Aristotelian inf... |
| frege63b 42170 | Lemma for ~ frege91 . Pro... |
| frege64b 42171 | Lemma for ~ frege65b . Pr... |
| frege65b 42172 | A kind of Aristotelian inf... |
| frege66b 42173 | Swap antecedents of ~ freg... |
| frege67b 42174 | Lemma for ~ frege68b . Pr... |
| frege68b 42175 | Combination of applying a ... |
| frege53c 42176 | Proposition 53 of [Frege18... |
| frege54cor1c 42177 | Reflexive equality. (Cont... |
| frege55lem1c 42178 | Necessary deduction regard... |
| frege55lem2c 42179 | Core proof of Proposition ... |
| frege55c 42180 | Proposition 55 of [Frege18... |
| frege56c 42181 | Lemma for ~ frege57c . Pr... |
| frege57c 42182 | Swap order of implication ... |
| frege58c 42183 | Principle related to ~ sp ... |
| frege59c 42184 | A kind of Aristotelian inf... |
| frege60c 42185 | Swap antecedents of ~ freg... |
| frege61c 42186 | Lemma for ~ frege65c . Pr... |
| frege62c 42187 | A kind of Aristotelian inf... |
| frege63c 42188 | Analogue of ~ frege63b . ... |
| frege64c 42189 | Lemma for ~ frege65c . Pr... |
| frege65c 42190 | A kind of Aristotelian inf... |
| frege66c 42191 | Swap antecedents of ~ freg... |
| frege67c 42192 | Lemma for ~ frege68c . Pr... |
| frege68c 42193 | Combination of applying a ... |
| dffrege69 42194 | If from the proposition th... |
| frege70 42195 | Lemma for ~ frege72 . Pro... |
| frege71 42196 | Lemma for ~ frege72 . Pro... |
| frege72 42197 | If property ` A ` is hered... |
| frege73 42198 | Lemma for ~ frege87 . Pro... |
| frege74 42199 | If ` X ` has a property ` ... |
| frege75 42200 | If from the proposition th... |
| dffrege76 42201 | If from the two propositio... |
| frege77 42202 | If ` Y ` follows ` X ` in ... |
| frege78 42203 | Commuted form of of ~ freg... |
| frege79 42204 | Distributed form of ~ freg... |
| frege80 42205 | Add additional condition t... |
| frege81 42206 | If ` X ` has a property ` ... |
| frege82 42207 | Closed-form deduction base... |
| frege83 42208 | Apply commuted form of ~ f... |
| frege84 42209 | Commuted form of ~ frege81... |
| frege85 42210 | Commuted form of ~ frege77... |
| frege86 42211 | Conclusion about element o... |
| frege87 42212 | If ` Z ` is a result of an... |
| frege88 42213 | Commuted form of ~ frege87... |
| frege89 42214 | One direction of ~ dffrege... |
| frege90 42215 | Add antecedent to ~ frege8... |
| frege91 42216 | Every result of an applica... |
| frege92 42217 | Inference from ~ frege91 .... |
| frege93 42218 | Necessary condition for tw... |
| frege94 42219 | Looking one past a pair re... |
| frege95 42220 | Looking one past a pair re... |
| frege96 42221 | Every result of an applica... |
| frege97 42222 | The property of following ... |
| frege98 42223 | If ` Y ` follows ` X ` and... |
| dffrege99 42224 | If ` Z ` is identical with... |
| frege100 42225 | One direction of ~ dffrege... |
| frege101 42226 | Lemma for ~ frege102 . Pr... |
| frege102 42227 | If ` Z ` belongs to the ` ... |
| frege103 42228 | Proposition 103 of [Frege1... |
| frege104 42229 | Proposition 104 of [Frege1... |
| frege105 42230 | Proposition 105 of [Frege1... |
| frege106 42231 | Whatever follows ` X ` in ... |
| frege107 42232 | Proposition 107 of [Frege1... |
| frege108 42233 | If ` Y ` belongs to the ` ... |
| frege109 42234 | The property of belonging ... |
| frege110 42235 | Proposition 110 of [Frege1... |
| frege111 42236 | If ` Y ` belongs to the ` ... |
| frege112 42237 | Identity implies belonging... |
| frege113 42238 | Proposition 113 of [Frege1... |
| frege114 42239 | If ` X ` belongs to the ` ... |
| dffrege115 42240 | If from the circumstance t... |
| frege116 42241 | One direction of ~ dffrege... |
| frege117 42242 | Lemma for ~ frege118 . Pr... |
| frege118 42243 | Simplified application of ... |
| frege119 42244 | Lemma for ~ frege120 . Pr... |
| frege120 42245 | Simplified application of ... |
| frege121 42246 | Lemma for ~ frege122 . Pr... |
| frege122 42247 | If ` X ` is a result of an... |
| frege123 42248 | Lemma for ~ frege124 . Pr... |
| frege124 42249 | If ` X ` is a result of an... |
| frege125 42250 | Lemma for ~ frege126 . Pr... |
| frege126 42251 | If ` M ` follows ` Y ` in ... |
| frege127 42252 | Communte antecedents of ~ ... |
| frege128 42253 | Lemma for ~ frege129 . Pr... |
| frege129 42254 | If the procedure ` R ` is ... |
| frege130 42255 | Lemma for ~ frege131 . Pr... |
| frege131 42256 | If the procedure ` R ` is ... |
| frege132 42257 | Lemma for ~ frege133 . Pr... |
| frege133 42258 | If the procedure ` R ` is ... |
| enrelmap 42259 | The set of all possible re... |
| enrelmapr 42260 | The set of all possible re... |
| enmappw 42261 | The set of all mappings fr... |
| enmappwid 42262 | The set of all mappings fr... |
| rfovd 42263 | Value of the operator, ` (... |
| rfovfvd 42264 | Value of the operator, ` (... |
| rfovfvfvd 42265 | Value of the operator, ` (... |
| rfovcnvf1od 42266 | Properties of the operator... |
| rfovcnvd 42267 | Value of the converse of t... |
| rfovf1od 42268 | The value of the operator,... |
| rfovcnvfvd 42269 | Value of the converse of t... |
| fsovd 42270 | Value of the operator, ` (... |
| fsovrfovd 42271 | The operator which gives a... |
| fsovfvd 42272 | Value of the operator, ` (... |
| fsovfvfvd 42273 | Value of the operator, ` (... |
| fsovfd 42274 | The operator, ` ( A O B ) ... |
| fsovcnvlem 42275 | The ` O ` operator, which ... |
| fsovcnvd 42276 | The value of the converse ... |
| fsovcnvfvd 42277 | The value of the converse ... |
| fsovf1od 42278 | The value of ` ( A O B ) `... |
| dssmapfvd 42279 | Value of the duality opera... |
| dssmapfv2d 42280 | Value of the duality opera... |
| dssmapfv3d 42281 | Value of the duality opera... |
| dssmapnvod 42282 | For any base set ` B ` the... |
| dssmapf1od 42283 | For any base set ` B ` the... |
| dssmap2d 42284 | For any base set ` B ` the... |
| or3or 42285 | Decompose disjunction into... |
| andi3or 42286 | Distribute over triple dis... |
| uneqsn 42287 | If a union of classes is e... |
| brfvimex 42288 | If a binary relation holds... |
| brovmptimex 42289 | If a binary relation holds... |
| brovmptimex1 42290 | If a binary relation holds... |
| brovmptimex2 42291 | If a binary relation holds... |
| brcoffn 42292 | Conditions allowing the de... |
| brcofffn 42293 | Conditions allowing the de... |
| brco2f1o 42294 | Conditions allowing the de... |
| brco3f1o 42295 | Conditions allowing the de... |
| ntrclsbex 42296 | If (pseudo-)interior and (... |
| ntrclsrcomplex 42297 | The relative complement of... |
| neik0imk0p 42298 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 42299 | If an interior function is... |
| ntrkbimka 42300 | If the interiors of disjoi... |
| ntrk0kbimka 42301 | If the interiors of disjoi... |
| clsk3nimkb 42302 | If the base set is not emp... |
| clsk1indlem0 42303 | The ansatz closure functio... |
| clsk1indlem2 42304 | The ansatz closure functio... |
| clsk1indlem3 42305 | The ansatz closure functio... |
| clsk1indlem4 42306 | The ansatz closure functio... |
| clsk1indlem1 42307 | The ansatz closure functio... |
| clsk1independent 42308 | For generalized closure fu... |
| neik0pk1imk0 42309 | Kuratowski's K0' and K1 ax... |
| isotone1 42310 | Two different ways to say ... |
| isotone2 42311 | Two different ways to say ... |
| ntrk1k3eqk13 42312 | An interior function is bo... |
| ntrclsf1o 42313 | If (pseudo-)interior and (... |
| ntrclsnvobr 42314 | If (pseudo-)interior and (... |
| ntrclsiex 42315 | If (pseudo-)interior and (... |
| ntrclskex 42316 | If (pseudo-)interior and (... |
| ntrclsfv1 42317 | If (pseudo-)interior and (... |
| ntrclsfv2 42318 | If (pseudo-)interior and (... |
| ntrclselnel1 42319 | If (pseudo-)interior and (... |
| ntrclselnel2 42320 | If (pseudo-)interior and (... |
| ntrclsfv 42321 | The value of the interior ... |
| ntrclsfveq1 42322 | If interior and closure fu... |
| ntrclsfveq2 42323 | If interior and closure fu... |
| ntrclsfveq 42324 | If interior and closure fu... |
| ntrclsss 42325 | If interior and closure fu... |
| ntrclsneine0lem 42326 | If (pseudo-)interior and (... |
| ntrclsneine0 42327 | If (pseudo-)interior and (... |
| ntrclscls00 42328 | If (pseudo-)interior and (... |
| ntrclsiso 42329 | If (pseudo-)interior and (... |
| ntrclsk2 42330 | An interior function is co... |
| ntrclskb 42331 | The interiors of disjoint ... |
| ntrclsk3 42332 | The intersection of interi... |
| ntrclsk13 42333 | The interior of the inters... |
| ntrclsk4 42334 | Idempotence of the interio... |
| ntrneibex 42335 | If (pseudo-)interior and (... |
| ntrneircomplex 42336 | The relative complement of... |
| ntrneif1o 42337 | If (pseudo-)interior and (... |
| ntrneiiex 42338 | If (pseudo-)interior and (... |
| ntrneinex 42339 | If (pseudo-)interior and (... |
| ntrneicnv 42340 | If (pseudo-)interior and (... |
| ntrneifv1 42341 | If (pseudo-)interior and (... |
| ntrneifv2 42342 | If (pseudo-)interior and (... |
| ntrneiel 42343 | If (pseudo-)interior and (... |
| ntrneifv3 42344 | The value of the neighbors... |
| ntrneineine0lem 42345 | If (pseudo-)interior and (... |
| ntrneineine1lem 42346 | If (pseudo-)interior and (... |
| ntrneifv4 42347 | The value of the interior ... |
| ntrneiel2 42348 | Membership in iterated int... |
| ntrneineine0 42349 | If (pseudo-)interior and (... |
| ntrneineine1 42350 | If (pseudo-)interior and (... |
| ntrneicls00 42351 | If (pseudo-)interior and (... |
| ntrneicls11 42352 | If (pseudo-)interior and (... |
| ntrneiiso 42353 | If (pseudo-)interior and (... |
| ntrneik2 42354 | An interior function is co... |
| ntrneix2 42355 | An interior (closure) func... |
| ntrneikb 42356 | The interiors of disjoint ... |
| ntrneixb 42357 | The interiors (closures) o... |
| ntrneik3 42358 | The intersection of interi... |
| ntrneix3 42359 | The closure of the union o... |
| ntrneik13 42360 | The interior of the inters... |
| ntrneix13 42361 | The closure of the union o... |
| ntrneik4w 42362 | Idempotence of the interio... |
| ntrneik4 42363 | Idempotence of the interio... |
| clsneibex 42364 | If (pseudo-)closure and (p... |
| clsneircomplex 42365 | The relative complement of... |
| clsneif1o 42366 | If a (pseudo-)closure func... |
| clsneicnv 42367 | If a (pseudo-)closure func... |
| clsneikex 42368 | If closure and neighborhoo... |
| clsneinex 42369 | If closure and neighborhoo... |
| clsneiel1 42370 | If a (pseudo-)closure func... |
| clsneiel2 42371 | If a (pseudo-)closure func... |
| clsneifv3 42372 | Value of the neighborhoods... |
| clsneifv4 42373 | Value of the closure (inte... |
| neicvgbex 42374 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 42375 | The relative complement of... |
| neicvgf1o 42376 | If neighborhood and conver... |
| neicvgnvo 42377 | If neighborhood and conver... |
| neicvgnvor 42378 | If neighborhood and conver... |
| neicvgmex 42379 | If the neighborhoods and c... |
| neicvgnex 42380 | If the neighborhoods and c... |
| neicvgel1 42381 | A subset being an element ... |
| neicvgel2 42382 | The complement of a subset... |
| neicvgfv 42383 | The value of the neighborh... |
| ntrrn 42384 | The range of the interior ... |
| ntrf 42385 | The interior function of a... |
| ntrf2 42386 | The interior function is a... |
| ntrelmap 42387 | The interior function is a... |
| clsf2 42388 | The closure function is a ... |
| clselmap 42389 | The closure function is a ... |
| dssmapntrcls 42390 | The interior and closure o... |
| dssmapclsntr 42391 | The closure and interior o... |
| gneispa 42392 | Each point ` p ` of the ne... |
| gneispb 42393 | Given a neighborhood ` N `... |
| gneispace2 42394 | The predicate that ` F ` i... |
| gneispace3 42395 | The predicate that ` F ` i... |
| gneispace 42396 | The predicate that ` F ` i... |
| gneispacef 42397 | A generic neighborhood spa... |
| gneispacef2 42398 | A generic neighborhood spa... |
| gneispacefun 42399 | A generic neighborhood spa... |
| gneispacern 42400 | A generic neighborhood spa... |
| gneispacern2 42401 | A generic neighborhood spa... |
| gneispace0nelrn 42402 | A generic neighborhood spa... |
| gneispace0nelrn2 42403 | A generic neighborhood spa... |
| gneispace0nelrn3 42404 | A generic neighborhood spa... |
| gneispaceel 42405 | Every neighborhood of a po... |
| gneispaceel2 42406 | Every neighborhood of a po... |
| gneispacess 42407 | All supersets of a neighbo... |
| gneispacess2 42408 | All supersets of a neighbo... |
| k0004lem1 42409 | Application of ~ ssin to r... |
| k0004lem2 42410 | A mapping with a particula... |
| k0004lem3 42411 | When the value of a mappin... |
| k0004val 42412 | The topological simplex of... |
| k0004ss1 42413 | The topological simplex of... |
| k0004ss2 42414 | The topological simplex of... |
| k0004ss3 42415 | The topological simplex of... |
| k0004val0 42416 | The topological simplex of... |
| inductionexd 42417 | Simple induction example. ... |
| wwlemuld 42418 | Natural deduction form of ... |
| leeq1d 42419 | Specialization of ~ breq1d... |
| leeq2d 42420 | Specialization of ~ breq2d... |
| absmulrposd 42421 | Specialization of absmuld ... |
| imadisjld 42422 | Natural dduction form of o... |
| imadisjlnd 42423 | Natural deduction form of ... |
| wnefimgd 42424 | The image of a mapping fro... |
| fco2d 42425 | Natural deduction form of ... |
| wfximgfd 42426 | The value of a function on... |
| extoimad 42427 | If |f(x)| <= C for all x t... |
| imo72b2lem0 42428 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 42429 | Natural deduction form of ... |
| imo72b2lem2 42430 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 42431 | Natural deduction form of ... |
| imo72b2lem1 42432 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 42433 | 'Less than or equal to' re... |
| lemuldiv4d 42434 | 'Less than or equal to' re... |
| imo72b2 42435 | IMO 1972 B2. (14th Intern... |
| int-addcomd 42436 | AdditionCommutativity gene... |
| int-addassocd 42437 | AdditionAssociativity gene... |
| int-addsimpd 42438 | AdditionSimplification gen... |
| int-mulcomd 42439 | MultiplicationCommutativit... |
| int-mulassocd 42440 | MultiplicationAssociativit... |
| int-mulsimpd 42441 | MultiplicationSimplificati... |
| int-leftdistd 42442 | AdditionMultiplicationLeft... |
| int-rightdistd 42443 | AdditionMultiplicationRigh... |
| int-sqdefd 42444 | SquareDefinition generator... |
| int-mul11d 42445 | First MultiplicationOne ge... |
| int-mul12d 42446 | Second MultiplicationOne g... |
| int-add01d 42447 | First AdditionZero generat... |
| int-add02d 42448 | Second AdditionZero genera... |
| int-sqgeq0d 42449 | SquareGEQZero generator ru... |
| int-eqprincd 42450 | PrincipleOfEquality genera... |
| int-eqtransd 42451 | EqualityTransitivity gener... |
| int-eqmvtd 42452 | EquMoveTerm generator rule... |
| int-eqineqd 42453 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 42454 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 42455 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 42456 | SecondPrincipleOfInequalit... |
| int-ineqtransd 42457 | InequalityTransitivity gen... |
| unitadd 42458 | Theorem used in conjunctio... |
| gsumws3 42459 | Valuation of a length 3 wo... |
| gsumws4 42460 | Valuation of a length 4 wo... |
| amgm2d 42461 | Arithmetic-geometric mean ... |
| amgm3d 42462 | Arithmetic-geometric mean ... |
| amgm4d 42463 | Arithmetic-geometric mean ... |
| spALT 42464 | ~ sp can be proven from th... |
| elnelneqd 42465 | Two classes are not equal ... |
| elnelneq2d 42466 | Two classes are not equal ... |
| rr-spce 42467 | Prove an existential. (Co... |
| rexlimdvaacbv 42468 | Unpack a restricted existe... |
| rexlimddvcbvw 42469 | Unpack a restricted existe... |
| rexlimddvcbv 42470 | Unpack a restricted existe... |
| rr-elrnmpt3d 42471 | Elementhood in an image se... |
| finnzfsuppd 42472 | If a function is zero outs... |
| rr-phpd 42473 | Equivalent of ~ php withou... |
| suceqd 42474 | Deduction associated with ... |
| tfindsd 42475 | Deduction associated with ... |
| mnringvald 42478 | Value of the monoid ring f... |
| mnringnmulrd 42479 | Components of a monoid rin... |
| mnringnmulrdOLD 42480 | Obsolete version of ~ mnri... |
| mnringbased 42481 | The base set of a monoid r... |
| mnringbasedOLD 42482 | Obsolete version of ~ mnri... |
| mnringbaserd 42483 | The base set of a monoid r... |
| mnringelbased 42484 | Membership in the base set... |
| mnringbasefd 42485 | Elements of a monoid ring ... |
| mnringbasefsuppd 42486 | Elements of a monoid ring ... |
| mnringaddgd 42487 | The additive operation of ... |
| mnringaddgdOLD 42488 | Obsolete version of ~ mnri... |
| mnring0gd 42489 | The additive identity of a... |
| mnring0g2d 42490 | The additive identity of a... |
| mnringmulrd 42491 | The ring product of a mono... |
| mnringscad 42492 | The scalar ring of a monoi... |
| mnringscadOLD 42493 | Obsolete version of ~ mnri... |
| mnringvscad 42494 | The scalar product of a mo... |
| mnringvscadOLD 42495 | Obsolete version of ~ mnri... |
| mnringlmodd 42496 | Monoid rings are left modu... |
| mnringmulrvald 42497 | Value of multiplication in... |
| mnringmulrcld 42498 | Monoid rings are closed un... |
| gru0eld 42499 | A nonempty Grothendieck un... |
| grusucd 42500 | Grothendieck universes are... |
| r1rankcld 42501 | Any rank of the cumulative... |
| grur1cld 42502 | Grothendieck universes are... |
| grurankcld 42503 | Grothendieck universes are... |
| grurankrcld 42504 | If a Grothendieck universe... |
| scotteqd 42507 | Equality theorem for the S... |
| scotteq 42508 | Closed form of ~ scotteqd ... |
| nfscott 42509 | Bound-variable hypothesis ... |
| scottabf 42510 | Value of the Scott operati... |
| scottab 42511 | Value of the Scott operati... |
| scottabes 42512 | Value of the Scott operati... |
| scottss 42513 | Scott's trick produces a s... |
| elscottab 42514 | An element of the output o... |
| scottex2 42515 | ~ scottex expressed using ... |
| scotteld 42516 | The Scott operation sends ... |
| scottelrankd 42517 | Property of a Scott's tric... |
| scottrankd 42518 | Rank of a nonempty Scott's... |
| gruscottcld 42519 | If a Grothendieck universe... |
| dfcoll2 42522 | Alternate definition of th... |
| colleq12d 42523 | Equality theorem for the c... |
| colleq1 42524 | Equality theorem for the c... |
| colleq2 42525 | Equality theorem for the c... |
| nfcoll 42526 | Bound-variable hypothesis ... |
| collexd 42527 | The output of the collecti... |
| cpcolld 42528 | Property of the collection... |
| cpcoll2d 42529 | ~ cpcolld with an extra ex... |
| grucollcld 42530 | A Grothendieck universe co... |
| ismnu 42531 | The hypothesis of this the... |
| mnuop123d 42532 | Operations of a minimal un... |
| mnussd 42533 | Minimal universes are clos... |
| mnuss2d 42534 | ~ mnussd with arguments pr... |
| mnu0eld 42535 | A nonempty minimal univers... |
| mnuop23d 42536 | Second and third operation... |
| mnupwd 42537 | Minimal universes are clos... |
| mnusnd 42538 | Minimal universes are clos... |
| mnuprssd 42539 | A minimal universe contain... |
| mnuprss2d 42540 | Special case of ~ mnuprssd... |
| mnuop3d 42541 | Third operation of a minim... |
| mnuprdlem1 42542 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 42543 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 42544 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 42545 | Lemma for ~ mnuprd . Gene... |
| mnuprd 42546 | Minimal universes are clos... |
| mnuunid 42547 | Minimal universes are clos... |
| mnuund 42548 | Minimal universes are clos... |
| mnutrcld 42549 | Minimal universes contain ... |
| mnutrd 42550 | Minimal universes are tran... |
| mnurndlem1 42551 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 42552 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 42553 | Minimal universes contain ... |
| mnugrud 42554 | Minimal universes are Grot... |
| grumnudlem 42555 | Lemma for ~ grumnud . (Co... |
| grumnud 42556 | Grothendieck universes are... |
| grumnueq 42557 | The class of Grothendieck ... |
| expandan 42558 | Expand conjunction to prim... |
| expandexn 42559 | Expand an existential quan... |
| expandral 42560 | Expand a restricted univer... |
| expandrexn 42561 | Expand a restricted existe... |
| expandrex 42562 | Expand a restricted existe... |
| expanduniss 42563 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 42564 | Express the predicate on `... |
| rr-grothprimbi 42565 | Express "every set is cont... |
| inagrud 42566 | Inaccessible levels of the... |
| inaex 42567 | Assuming the Tarski-Grothe... |
| gruex 42568 | Assuming the Tarski-Grothe... |
| rr-groth 42569 | An equivalent of ~ ax-grot... |
| rr-grothprim 42570 | An equivalent of ~ ax-grot... |
| ismnushort 42571 | Express the predicate on `... |
| dfuniv2 42572 | Alternative definition of ... |
| rr-grothshortbi 42573 | Express "every set is cont... |
| rr-grothshort 42574 | A shorter equivalent of ~ ... |
| nanorxor 42575 | 'nand' is equivalent to th... |
| undisjrab 42576 | Union of two disjoint rest... |
| iso0 42577 | The empty set is an ` R , ... |
| ssrecnpr 42578 | ` RR ` is a subset of both... |
| seff 42579 | Let set ` S ` be the real ... |
| sblpnf 42580 | The infinity ball in the a... |
| prmunb2 42581 | The primes are unbounded. ... |
| dvgrat 42582 | Ratio test for divergence ... |
| cvgdvgrat 42583 | Ratio test for convergence... |
| radcnvrat 42584 | Let ` L ` be the limit, if... |
| reldvds 42585 | The divides relation is in... |
| nznngen 42586 | All positive integers in t... |
| nzss 42587 | The set of multiples of _m... |
| nzin 42588 | The intersection of the se... |
| nzprmdif 42589 | Subtract one prime's multi... |
| hashnzfz 42590 | Special case of ~ hashdvds... |
| hashnzfz2 42591 | Special case of ~ hashnzfz... |
| hashnzfzclim 42592 | As the upper bound ` K ` o... |
| caofcan 42593 | Transfer a cancellation la... |
| ofsubid 42594 | Function analogue of ~ sub... |
| ofmul12 42595 | Function analogue of ~ mul... |
| ofdivrec 42596 | Function analogue of ~ div... |
| ofdivcan4 42597 | Function analogue of ~ div... |
| ofdivdiv2 42598 | Function analogue of ~ div... |
| lhe4.4ex1a 42599 | Example of the Fundamental... |
| dvsconst 42600 | Derivative of a constant f... |
| dvsid 42601 | Derivative of the identity... |
| dvsef 42602 | Derivative of the exponent... |
| expgrowthi 42603 | Exponential growth and dec... |
| dvconstbi 42604 | The derivative of a functi... |
| expgrowth 42605 | Exponential growth and dec... |
| bccval 42608 | Value of the generalized b... |
| bcccl 42609 | Closure of the generalized... |
| bcc0 42610 | The generalized binomial c... |
| bccp1k 42611 | Generalized binomial coeff... |
| bccm1k 42612 | Generalized binomial coeff... |
| bccn0 42613 | Generalized binomial coeff... |
| bccn1 42614 | Generalized binomial coeff... |
| bccbc 42615 | The binomial coefficient a... |
| uzmptshftfval 42616 | When ` F ` is a maps-to fu... |
| dvradcnv2 42617 | The radius of convergence ... |
| binomcxplemwb 42618 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 42619 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 42620 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 42621 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 42622 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 42623 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 42624 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 42625 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 42626 | Lemma for ~ binomcxp . Wh... |
| binomcxp 42627 | Generalize the binomial th... |
| pm10.12 42628 | Theorem *10.12 in [Whitehe... |
| pm10.14 42629 | Theorem *10.14 in [Whitehe... |
| pm10.251 42630 | Theorem *10.251 in [Whiteh... |
| pm10.252 42631 | Theorem *10.252 in [Whiteh... |
| pm10.253 42632 | Theorem *10.253 in [Whiteh... |
| albitr 42633 | Theorem *10.301 in [Whiteh... |
| pm10.42 42634 | Theorem *10.42 in [Whitehe... |
| pm10.52 42635 | Theorem *10.52 in [Whitehe... |
| pm10.53 42636 | Theorem *10.53 in [Whitehe... |
| pm10.541 42637 | Theorem *10.541 in [Whiteh... |
| pm10.542 42638 | Theorem *10.542 in [Whiteh... |
| pm10.55 42639 | Theorem *10.55 in [Whitehe... |
| pm10.56 42640 | Theorem *10.56 in [Whitehe... |
| pm10.57 42641 | Theorem *10.57 in [Whitehe... |
| 2alanimi 42642 | Removes two universal quan... |
| 2al2imi 42643 | Removes two universal quan... |
| pm11.11 42644 | Theorem *11.11 in [Whitehe... |
| pm11.12 42645 | Theorem *11.12 in [Whitehe... |
| 19.21vv 42646 | Compare Theorem *11.3 in [... |
| 2alim 42647 | Theorem *11.32 in [Whitehe... |
| 2albi 42648 | Theorem *11.33 in [Whitehe... |
| 2exim 42649 | Theorem *11.34 in [Whitehe... |
| 2exbi 42650 | Theorem *11.341 in [Whiteh... |
| spsbce-2 42651 | Theorem *11.36 in [Whitehe... |
| 19.33-2 42652 | Theorem *11.421 in [Whiteh... |
| 19.36vv 42653 | Theorem *11.43 in [Whitehe... |
| 19.31vv 42654 | Theorem *11.44 in [Whitehe... |
| 19.37vv 42655 | Theorem *11.46 in [Whitehe... |
| 19.28vv 42656 | Theorem *11.47 in [Whitehe... |
| pm11.52 42657 | Theorem *11.52 in [Whitehe... |
| aaanv 42658 | Theorem *11.56 in [Whitehe... |
| pm11.57 42659 | Theorem *11.57 in [Whitehe... |
| pm11.58 42660 | Theorem *11.58 in [Whitehe... |
| pm11.59 42661 | Theorem *11.59 in [Whitehe... |
| pm11.6 42662 | Theorem *11.6 in [Whitehea... |
| pm11.61 42663 | Theorem *11.61 in [Whitehe... |
| pm11.62 42664 | Theorem *11.62 in [Whitehe... |
| pm11.63 42665 | Theorem *11.63 in [Whitehe... |
| pm11.7 42666 | Theorem *11.7 in [Whitehea... |
| pm11.71 42667 | Theorem *11.71 in [Whitehe... |
| sbeqal1 42668 | If ` x = y ` always implie... |
| sbeqal1i 42669 | Suppose you know ` x = y `... |
| sbeqal2i 42670 | If ` x = y ` implies ` x =... |
| axc5c4c711 42671 | Proof of a theorem that ca... |
| axc5c4c711toc5 42672 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 42673 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 42674 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 42675 | Rederivation of ~ ax-11 fr... |
| axc11next 42676 | This theorem shows that, g... |
| pm13.13a 42677 | One result of theorem *13.... |
| pm13.13b 42678 | Theorem *13.13 in [Whitehe... |
| pm13.14 42679 | Theorem *13.14 in [Whitehe... |
| pm13.192 42680 | Theorem *13.192 in [Whiteh... |
| pm13.193 42681 | Theorem *13.193 in [Whiteh... |
| pm13.194 42682 | Theorem *13.194 in [Whiteh... |
| pm13.195 42683 | Theorem *13.195 in [Whiteh... |
| pm13.196a 42684 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 42685 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 42686 | Theorem *13.22 in [Whitehe... |
| iotain 42687 | Equivalence between two di... |
| iotaexeu 42688 | The iota class exists. Th... |
| iotasbc 42689 | Definition *14.01 in [Whit... |
| iotasbc2 42690 | Theorem *14.111 in [Whiteh... |
| pm14.12 42691 | Theorem *14.12 in [Whitehe... |
| pm14.122a 42692 | Theorem *14.122 in [Whiteh... |
| pm14.122b 42693 | Theorem *14.122 in [Whiteh... |
| pm14.122c 42694 | Theorem *14.122 in [Whiteh... |
| pm14.123a 42695 | Theorem *14.123 in [Whiteh... |
| pm14.123b 42696 | Theorem *14.123 in [Whiteh... |
| pm14.123c 42697 | Theorem *14.123 in [Whiteh... |
| pm14.18 42698 | Theorem *14.18 in [Whitehe... |
| iotaequ 42699 | Theorem *14.2 in [Whitehea... |
| iotavalb 42700 | Theorem *14.202 in [Whiteh... |
| iotasbc5 42701 | Theorem *14.205 in [Whiteh... |
| pm14.24 42702 | Theorem *14.24 in [Whitehe... |
| iotavalsb 42703 | Theorem *14.242 in [Whiteh... |
| sbiota1 42704 | Theorem *14.25 in [Whitehe... |
| sbaniota 42705 | Theorem *14.26 in [Whitehe... |
| eubiOLD 42706 | Obsolete proof of ~ eubi a... |
| iotasbcq 42707 | Theorem *14.272 in [Whiteh... |
| elnev 42708 | Any set that contains one ... |
| rusbcALT 42709 | A version of Russell's par... |
| compeq 42710 | Equality between two ways ... |
| compne 42711 | The complement of ` A ` is... |
| compab 42712 | Two ways of saying "the co... |
| conss2 42713 | Contrapositive law for sub... |
| conss1 42714 | Contrapositive law for sub... |
| ralbidar 42715 | More general form of ~ ral... |
| rexbidar 42716 | More general form of ~ rex... |
| dropab1 42717 | Theorem to aid use of the ... |
| dropab2 42718 | Theorem to aid use of the ... |
| ipo0 42719 | If the identity relation p... |
| ifr0 42720 | A class that is founded by... |
| ordpss 42721 | ~ ordelpss with an anteced... |
| fvsb 42722 | Explicit substitution of a... |
| fveqsb 42723 | Implicit substitution of a... |
| xpexb 42724 | A Cartesian product exists... |
| trelpss 42725 | An element of a transitive... |
| addcomgi 42726 | Generalization of commutat... |
| addrval 42736 | Value of the operation of ... |
| subrval 42737 | Value of the operation of ... |
| mulvval 42738 | Value of the operation of ... |
| addrfv 42739 | Vector addition at a value... |
| subrfv 42740 | Vector subtraction at a va... |
| mulvfv 42741 | Scalar multiplication at a... |
| addrfn 42742 | Vector addition produces a... |
| subrfn 42743 | Vector subtraction produce... |
| mulvfn 42744 | Scalar multiplication prod... |
| addrcom 42745 | Vector addition is commuta... |
| idiALT 42749 | Placeholder for ~ idi . T... |
| exbir 42750 | Exportation implication al... |
| 3impexpbicom 42751 | Version of ~ 3impexp where... |
| 3impexpbicomi 42752 | Inference associated with ... |
| bi1imp 42753 | Importation inference simi... |
| bi2imp 42754 | Importation inference simi... |
| bi3impb 42755 | Similar to ~ 3impb with im... |
| bi3impa 42756 | Similar to ~ 3impa with im... |
| bi23impib 42757 | ~ 3impib with the inner im... |
| bi13impib 42758 | ~ 3impib with the outer im... |
| bi123impib 42759 | ~ 3impib with the implicat... |
| bi13impia 42760 | ~ 3impia with the outer im... |
| bi123impia 42761 | ~ 3impia with the implicat... |
| bi33imp12 42762 | ~ 3imp with innermost impl... |
| bi23imp13 42763 | ~ 3imp with middle implica... |
| bi13imp23 42764 | ~ 3imp with outermost impl... |
| bi13imp2 42765 | Similar to ~ 3imp except t... |
| bi12imp3 42766 | Similar to ~ 3imp except a... |
| bi23imp1 42767 | Similar to ~ 3imp except a... |
| bi123imp0 42768 | Similar to ~ 3imp except a... |
| 4animp1 42769 | A single hypothesis unific... |
| 4an31 42770 | A rearrangement of conjunc... |
| 4an4132 42771 | A rearrangement of conjunc... |
| expcomdg 42772 | Biconditional form of ~ ex... |
| iidn3 42773 | ~ idn3 without virtual ded... |
| ee222 42774 | ~ e222 without virtual ded... |
| ee3bir 42775 | Right-biconditional form o... |
| ee13 42776 | ~ e13 without virtual dedu... |
| ee121 42777 | ~ e121 without virtual ded... |
| ee122 42778 | ~ e122 without virtual ded... |
| ee333 42779 | ~ e333 without virtual ded... |
| ee323 42780 | ~ e323 without virtual ded... |
| 3ornot23 42781 | If the second and third di... |
| orbi1r 42782 | ~ orbi1 with order of disj... |
| 3orbi123 42783 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 42784 | Closed form of ~ syl5 . D... |
| impexpd 42785 | The following User's Proof... |
| com3rgbi 42786 | The following User's Proof... |
| impexpdcom 42787 | The following User's Proof... |
| ee1111 42788 | Non-virtual deduction form... |
| pm2.43bgbi 42789 | Logical equivalence of a 2... |
| pm2.43cbi 42790 | Logical equivalence of a 3... |
| ee233 42791 | Non-virtual deduction form... |
| imbi13 42792 | Join three logical equival... |
| ee33 42793 | Non-virtual deduction form... |
| con5 42794 | Biconditional contrapositi... |
| con5i 42795 | Inference form of ~ con5 .... |
| exlimexi 42796 | Inference similar to Theor... |
| sb5ALT 42797 | Equivalence for substituti... |
| eexinst01 42798 | ~ exinst01 without virtual... |
| eexinst11 42799 | ~ exinst11 without virtual... |
| vk15.4j 42800 | Excercise 4j of Unit 15 of... |
| notnotrALT 42801 | Converse of double negatio... |
| con3ALT2 42802 | Contraposition. Alternate... |
| ssralv2 42803 | Quantification restricted ... |
| sbc3or 42804 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 42805 | Closed form of ~ alrimi wi... |
| rspsbc2 42806 | ~ rspsbc with two quantify... |
| sbcoreleleq 42807 | Substitution of a setvar v... |
| tratrb 42808 | If a class is transitive a... |
| ordelordALT 42809 | An element of an ordinal c... |
| sbcim2g 42810 | Distribution of class subs... |
| sbcbi 42811 | Implication form of ~ sbcb... |
| trsbc 42812 | Formula-building inference... |
| truniALT 42813 | The union of a class of tr... |
| onfrALTlem5 42814 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 42815 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 42816 | Lemma for ~ onfrALT . (Co... |
| ggen31 42817 | ~ gen31 without virtual de... |
| onfrALTlem2 42818 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 42819 | A theorem pertaining to th... |
| onfrALTlem1 42820 | Lemma for ~ onfrALT . (Co... |
| onfrALT 42821 | The membership relation is... |
| 19.41rg 42822 | Closed form of right-to-le... |
| opelopab4 42823 | Ordered pair membership in... |
| 2pm13.193 42824 | ~ pm13.193 for two variabl... |
| hbntal 42825 | A closed form of ~ hbn . ~... |
| hbimpg 42826 | A closed form of ~ hbim . ... |
| hbalg 42827 | Closed form of ~ hbal . D... |
| hbexg 42828 | Closed form of ~ nfex . D... |
| ax6e2eq 42829 | Alternate form of ~ ax6e f... |
| ax6e2nd 42830 | If at least two sets exist... |
| ax6e2ndeq 42831 | "At least two sets exist" ... |
| 2sb5nd 42832 | Equivalence for double sub... |
| 2uasbanh 42833 | Distribute the unabbreviat... |
| 2uasban 42834 | Distribute the unabbreviat... |
| e2ebind 42835 | Absorption of an existenti... |
| elpwgded 42836 | ~ elpwgdedVD in convention... |
| trelded 42837 | Deduction form of ~ trel .... |
| jaoded 42838 | Deduction form of ~ jao . ... |
| sbtT 42839 | A substitution into a theo... |
| not12an2impnot1 42840 | If a double conjunction is... |
| in1 42843 | Inference form of ~ df-vd1... |
| iin1 42844 | ~ in1 without virtual dedu... |
| dfvd1ir 42845 | Inference form of ~ df-vd1... |
| idn1 42846 | Virtual deduction identity... |
| dfvd1imp 42847 | Left-to-right part of defi... |
| dfvd1impr 42848 | Right-to-left part of defi... |
| dfvd2 42851 | Definition of a 2-hypothes... |
| dfvd2an 42854 | Definition of a 2-hypothes... |
| dfvd2ani 42855 | Inference form of ~ dfvd2a... |
| dfvd2anir 42856 | Right-to-left inference fo... |
| dfvd2i 42857 | Inference form of ~ dfvd2 ... |
| dfvd2ir 42858 | Right-to-left inference fo... |
| dfvd3 42863 | Definition of a 3-hypothes... |
| dfvd3i 42864 | Inference form of ~ dfvd3 ... |
| dfvd3ir 42865 | Right-to-left inference fo... |
| dfvd3an 42866 | Definition of a 3-hypothes... |
| dfvd3ani 42867 | Inference form of ~ dfvd3a... |
| dfvd3anir 42868 | Right-to-left inference fo... |
| vd01 42869 | A virtual hypothesis virtu... |
| vd02 42870 | Two virtual hypotheses vir... |
| vd03 42871 | A theorem is virtually inf... |
| vd12 42872 | A virtual deduction with 1... |
| vd13 42873 | A virtual deduction with 1... |
| vd23 42874 | A virtual deduction with 2... |
| dfvd2imp 42875 | The virtual deduction form... |
| dfvd2impr 42876 | A 2-antecedent nested impl... |
| in2 42877 | The virtual deduction intr... |
| int2 42878 | The virtual deduction intr... |
| iin2 42879 | ~ in2 without virtual dedu... |
| in2an 42880 | The virtual deduction intr... |
| in3 42881 | The virtual deduction intr... |
| iin3 42882 | ~ in3 without virtual dedu... |
| in3an 42883 | The virtual deduction intr... |
| int3 42884 | The virtual deduction intr... |
| idn2 42885 | Virtual deduction identity... |
| iden2 42886 | Virtual deduction identity... |
| idn3 42887 | Virtual deduction identity... |
| gen11 42888 | Virtual deduction generali... |
| gen11nv 42889 | Virtual deduction generali... |
| gen12 42890 | Virtual deduction generali... |
| gen21 42891 | Virtual deduction generali... |
| gen21nv 42892 | Virtual deduction form of ... |
| gen31 42893 | Virtual deduction generali... |
| gen22 42894 | Virtual deduction generali... |
| ggen22 42895 | ~ gen22 without virtual de... |
| exinst 42896 | Existential Instantiation.... |
| exinst01 42897 | Existential Instantiation.... |
| exinst11 42898 | Existential Instantiation.... |
| e1a 42899 | A Virtual deduction elimin... |
| el1 42900 | A Virtual deduction elimin... |
| e1bi 42901 | Biconditional form of ~ e1... |
| e1bir 42902 | Right biconditional form o... |
| e2 42903 | A virtual deduction elimin... |
| e2bi 42904 | Biconditional form of ~ e2... |
| e2bir 42905 | Right biconditional form o... |
| ee223 42906 | ~ e223 without virtual ded... |
| e223 42907 | A virtual deduction elimin... |
| e222 42908 | A virtual deduction elimin... |
| e220 42909 | A virtual deduction elimin... |
| ee220 42910 | ~ e220 without virtual ded... |
| e202 42911 | A virtual deduction elimin... |
| ee202 42912 | ~ e202 without virtual ded... |
| e022 42913 | A virtual deduction elimin... |
| ee022 42914 | ~ e022 without virtual ded... |
| e002 42915 | A virtual deduction elimin... |
| ee002 42916 | ~ e002 without virtual ded... |
| e020 42917 | A virtual deduction elimin... |
| ee020 42918 | ~ e020 without virtual ded... |
| e200 42919 | A virtual deduction elimin... |
| ee200 42920 | ~ e200 without virtual ded... |
| e221 42921 | A virtual deduction elimin... |
| ee221 42922 | ~ e221 without virtual ded... |
| e212 42923 | A virtual deduction elimin... |
| ee212 42924 | ~ e212 without virtual ded... |
| e122 42925 | A virtual deduction elimin... |
| e112 42926 | A virtual deduction elimin... |
| ee112 42927 | ~ e112 without virtual ded... |
| e121 42928 | A virtual deduction elimin... |
| e211 42929 | A virtual deduction elimin... |
| ee211 42930 | ~ e211 without virtual ded... |
| e210 42931 | A virtual deduction elimin... |
| ee210 42932 | ~ e210 without virtual ded... |
| e201 42933 | A virtual deduction elimin... |
| ee201 42934 | ~ e201 without virtual ded... |
| e120 42935 | A virtual deduction elimin... |
| ee120 42936 | Virtual deduction rule ~ e... |
| e021 42937 | A virtual deduction elimin... |
| ee021 42938 | ~ e021 without virtual ded... |
| e012 42939 | A virtual deduction elimin... |
| ee012 42940 | ~ e012 without virtual ded... |
| e102 42941 | A virtual deduction elimin... |
| ee102 42942 | ~ e102 without virtual ded... |
| e22 42943 | A virtual deduction elimin... |
| e22an 42944 | Conjunction form of ~ e22 ... |
| ee22an 42945 | ~ e22an without virtual de... |
| e111 42946 | A virtual deduction elimin... |
| e1111 42947 | A virtual deduction elimin... |
| e110 42948 | A virtual deduction elimin... |
| ee110 42949 | ~ e110 without virtual ded... |
| e101 42950 | A virtual deduction elimin... |
| ee101 42951 | ~ e101 without virtual ded... |
| e011 42952 | A virtual deduction elimin... |
| ee011 42953 | ~ e011 without virtual ded... |
| e100 42954 | A virtual deduction elimin... |
| ee100 42955 | ~ e100 without virtual ded... |
| e010 42956 | A virtual deduction elimin... |
| ee010 42957 | ~ e010 without virtual ded... |
| e001 42958 | A virtual deduction elimin... |
| ee001 42959 | ~ e001 without virtual ded... |
| e11 42960 | A virtual deduction elimin... |
| e11an 42961 | Conjunction form of ~ e11 ... |
| ee11an 42962 | ~ e11an without virtual de... |
| e01 42963 | A virtual deduction elimin... |
| e01an 42964 | Conjunction form of ~ e01 ... |
| ee01an 42965 | ~ e01an without virtual de... |
| e10 42966 | A virtual deduction elimin... |
| e10an 42967 | Conjunction form of ~ e10 ... |
| ee10an 42968 | ~ e10an without virtual de... |
| e02 42969 | A virtual deduction elimin... |
| e02an 42970 | Conjunction form of ~ e02 ... |
| ee02an 42971 | ~ e02an without virtual de... |
| eel021old 42972 | ~ el021old without virtual... |
| el021old 42973 | A virtual deduction elimin... |
| eel132 42974 | ~ syl2an with antecedents ... |
| eel000cT 42975 | An elimination deduction. ... |
| eel0TT 42976 | An elimination deduction. ... |
| eelT00 42977 | An elimination deduction. ... |
| eelTTT 42978 | An elimination deduction. ... |
| eelT11 42979 | An elimination deduction. ... |
| eelT1 42980 | Syllogism inference combin... |
| eelT12 42981 | An elimination deduction. ... |
| eelTT1 42982 | An elimination deduction. ... |
| eelT01 42983 | An elimination deduction. ... |
| eel0T1 42984 | An elimination deduction. ... |
| eel12131 42985 | An elimination deduction. ... |
| eel2131 42986 | ~ syl2an with antecedents ... |
| eel3132 42987 | ~ syl2an with antecedents ... |
| eel0321old 42988 | ~ el0321old without virtua... |
| el0321old 42989 | A virtual deduction elimin... |
| eel2122old 42990 | ~ el2122old without virtua... |
| el2122old 42991 | A virtual deduction elimin... |
| eel0000 42992 | Elimination rule similar t... |
| eel00001 42993 | An elimination deduction. ... |
| eel00000 42994 | Elimination rule similar ~... |
| eel11111 42995 | Five-hypothesis eliminatio... |
| e12 42996 | A virtual deduction elimin... |
| e12an 42997 | Conjunction form of ~ e12 ... |
| el12 42998 | Virtual deduction form of ... |
| e20 42999 | A virtual deduction elimin... |
| e20an 43000 | Conjunction form of ~ e20 ... |
| ee20an 43001 | ~ e20an without virtual de... |
| e21 43002 | A virtual deduction elimin... |
| e21an 43003 | Conjunction form of ~ e21 ... |
| ee21an 43004 | ~ e21an without virtual de... |
| e333 43005 | A virtual deduction elimin... |
| e33 43006 | A virtual deduction elimin... |
| e33an 43007 | Conjunction form of ~ e33 ... |
| ee33an 43008 | ~ e33an without virtual de... |
| e3 43009 | Meta-connective form of ~ ... |
| e3bi 43010 | Biconditional form of ~ e3... |
| e3bir 43011 | Right biconditional form o... |
| e03 43012 | A virtual deduction elimin... |
| ee03 43013 | ~ e03 without virtual dedu... |
| e03an 43014 | Conjunction form of ~ e03 ... |
| ee03an 43015 | Conjunction form of ~ ee03... |
| e30 43016 | A virtual deduction elimin... |
| ee30 43017 | ~ e30 without virtual dedu... |
| e30an 43018 | A virtual deduction elimin... |
| ee30an 43019 | Conjunction form of ~ ee30... |
| e13 43020 | A virtual deduction elimin... |
| e13an 43021 | A virtual deduction elimin... |
| ee13an 43022 | ~ e13an without virtual de... |
| e31 43023 | A virtual deduction elimin... |
| ee31 43024 | ~ e31 without virtual dedu... |
| e31an 43025 | A virtual deduction elimin... |
| ee31an 43026 | ~ e31an without virtual de... |
| e23 43027 | A virtual deduction elimin... |
| e23an 43028 | A virtual deduction elimin... |
| ee23an 43029 | ~ e23an without virtual de... |
| e32 43030 | A virtual deduction elimin... |
| ee32 43031 | ~ e32 without virtual dedu... |
| e32an 43032 | A virtual deduction elimin... |
| ee32an 43033 | ~ e33an without virtual de... |
| e123 43034 | A virtual deduction elimin... |
| ee123 43035 | ~ e123 without virtual ded... |
| el123 43036 | A virtual deduction elimin... |
| e233 43037 | A virtual deduction elimin... |
| e323 43038 | A virtual deduction elimin... |
| e000 43039 | A virtual deduction elimin... |
| e00 43040 | Elimination rule identical... |
| e00an 43041 | Elimination rule identical... |
| eel00cT 43042 | An elimination deduction. ... |
| eelTT 43043 | An elimination deduction. ... |
| e0a 43044 | Elimination rule identical... |
| eelT 43045 | An elimination deduction. ... |
| eel0cT 43046 | An elimination deduction. ... |
| eelT0 43047 | An elimination deduction. ... |
| e0bi 43048 | Elimination rule identical... |
| e0bir 43049 | Elimination rule identical... |
| uun0.1 43050 | Convention notation form o... |
| un0.1 43051 | ` T. ` is the constant tru... |
| uunT1 43052 | A deduction unionizing a n... |
| uunT1p1 43053 | A deduction unionizing a n... |
| uunT21 43054 | A deduction unionizing a n... |
| uun121 43055 | A deduction unionizing a n... |
| uun121p1 43056 | A deduction unionizing a n... |
| uun132 43057 | A deduction unionizing a n... |
| uun132p1 43058 | A deduction unionizing a n... |
| anabss7p1 43059 | A deduction unionizing a n... |
| un10 43060 | A unionizing deduction. (... |
| un01 43061 | A unionizing deduction. (... |
| un2122 43062 | A deduction unionizing a n... |
| uun2131 43063 | A deduction unionizing a n... |
| uun2131p1 43064 | A deduction unionizing a n... |
| uunTT1 43065 | A deduction unionizing a n... |
| uunTT1p1 43066 | A deduction unionizing a n... |
| uunTT1p2 43067 | A deduction unionizing a n... |
| uunT11 43068 | A deduction unionizing a n... |
| uunT11p1 43069 | A deduction unionizing a n... |
| uunT11p2 43070 | A deduction unionizing a n... |
| uunT12 43071 | A deduction unionizing a n... |
| uunT12p1 43072 | A deduction unionizing a n... |
| uunT12p2 43073 | A deduction unionizing a n... |
| uunT12p3 43074 | A deduction unionizing a n... |
| uunT12p4 43075 | A deduction unionizing a n... |
| uunT12p5 43076 | A deduction unionizing a n... |
| uun111 43077 | A deduction unionizing a n... |
| 3anidm12p1 43078 | A deduction unionizing a n... |
| 3anidm12p2 43079 | A deduction unionizing a n... |
| uun123 43080 | A deduction unionizing a n... |
| uun123p1 43081 | A deduction unionizing a n... |
| uun123p2 43082 | A deduction unionizing a n... |
| uun123p3 43083 | A deduction unionizing a n... |
| uun123p4 43084 | A deduction unionizing a n... |
| uun2221 43085 | A deduction unionizing a n... |
| uun2221p1 43086 | A deduction unionizing a n... |
| uun2221p2 43087 | A deduction unionizing a n... |
| 3impdirp1 43088 | A deduction unionizing a n... |
| 3impcombi 43089 | A 1-hypothesis proposition... |
| trsspwALT 43090 | Virtual deduction proof of... |
| trsspwALT2 43091 | Virtual deduction proof of... |
| trsspwALT3 43092 | Short predicate calculus p... |
| sspwtr 43093 | Virtual deduction proof of... |
| sspwtrALT 43094 | Virtual deduction proof of... |
| sspwtrALT2 43095 | Short predicate calculus p... |
| pwtrVD 43096 | Virtual deduction proof of... |
| pwtrrVD 43097 | Virtual deduction proof of... |
| suctrALT 43098 | The successor of a transit... |
| snssiALTVD 43099 | Virtual deduction proof of... |
| snssiALT 43100 | If a class is an element o... |
| snsslVD 43101 | Virtual deduction proof of... |
| snssl 43102 | If a singleton is a subcla... |
| snelpwrVD 43103 | Virtual deduction proof of... |
| unipwrVD 43104 | Virtual deduction proof of... |
| unipwr 43105 | A class is a subclass of t... |
| sstrALT2VD 43106 | Virtual deduction proof of... |
| sstrALT2 43107 | Virtual deduction proof of... |
| suctrALT2VD 43108 | Virtual deduction proof of... |
| suctrALT2 43109 | Virtual deduction proof of... |
| elex2VD 43110 | Virtual deduction proof of... |
| elex22VD 43111 | Virtual deduction proof of... |
| eqsbc2VD 43112 | Virtual deduction proof of... |
| zfregs2VD 43113 | Virtual deduction proof of... |
| tpid3gVD 43114 | Virtual deduction proof of... |
| en3lplem1VD 43115 | Virtual deduction proof of... |
| en3lplem2VD 43116 | Virtual deduction proof of... |
| en3lpVD 43117 | Virtual deduction proof of... |
| simplbi2VD 43118 | Virtual deduction proof of... |
| 3ornot23VD 43119 | Virtual deduction proof of... |
| orbi1rVD 43120 | Virtual deduction proof of... |
| bitr3VD 43121 | Virtual deduction proof of... |
| 3orbi123VD 43122 | Virtual deduction proof of... |
| sbc3orgVD 43123 | Virtual deduction proof of... |
| 19.21a3con13vVD 43124 | Virtual deduction proof of... |
| exbirVD 43125 | Virtual deduction proof of... |
| exbiriVD 43126 | Virtual deduction proof of... |
| rspsbc2VD 43127 | Virtual deduction proof of... |
| 3impexpVD 43128 | Virtual deduction proof of... |
| 3impexpbicomVD 43129 | Virtual deduction proof of... |
| 3impexpbicomiVD 43130 | Virtual deduction proof of... |
| sbcoreleleqVD 43131 | Virtual deduction proof of... |
| hbra2VD 43132 | Virtual deduction proof of... |
| tratrbVD 43133 | Virtual deduction proof of... |
| al2imVD 43134 | Virtual deduction proof of... |
| syl5impVD 43135 | Virtual deduction proof of... |
| idiVD 43136 | Virtual deduction proof of... |
| ancomstVD 43137 | Closed form of ~ ancoms . ... |
| ssralv2VD 43138 | Quantification restricted ... |
| ordelordALTVD 43139 | An element of an ordinal c... |
| equncomVD 43140 | If a class equals the unio... |
| equncomiVD 43141 | Inference form of ~ equnco... |
| sucidALTVD 43142 | A set belongs to its succe... |
| sucidALT 43143 | A set belongs to its succe... |
| sucidVD 43144 | A set belongs to its succe... |
| imbi12VD 43145 | Implication form of ~ imbi... |
| imbi13VD 43146 | Join three logical equival... |
| sbcim2gVD 43147 | Distribution of class subs... |
| sbcbiVD 43148 | Implication form of ~ sbcb... |
| trsbcVD 43149 | Formula-building inference... |
| truniALTVD 43150 | The union of a class of tr... |
| ee33VD 43151 | Non-virtual deduction form... |
| trintALTVD 43152 | The intersection of a clas... |
| trintALT 43153 | The intersection of a clas... |
| undif3VD 43154 | The first equality of Exer... |
| sbcssgVD 43155 | Virtual deduction proof of... |
| csbingVD 43156 | Virtual deduction proof of... |
| onfrALTlem5VD 43157 | Virtual deduction proof of... |
| onfrALTlem4VD 43158 | Virtual deduction proof of... |
| onfrALTlem3VD 43159 | Virtual deduction proof of... |
| simplbi2comtVD 43160 | Virtual deduction proof of... |
| onfrALTlem2VD 43161 | Virtual deduction proof of... |
| onfrALTlem1VD 43162 | Virtual deduction proof of... |
| onfrALTVD 43163 | Virtual deduction proof of... |
| csbeq2gVD 43164 | Virtual deduction proof of... |
| csbsngVD 43165 | Virtual deduction proof of... |
| csbxpgVD 43166 | Virtual deduction proof of... |
| csbresgVD 43167 | Virtual deduction proof of... |
| csbrngVD 43168 | Virtual deduction proof of... |
| csbima12gALTVD 43169 | Virtual deduction proof of... |
| csbunigVD 43170 | Virtual deduction proof of... |
| csbfv12gALTVD 43171 | Virtual deduction proof of... |
| con5VD 43172 | Virtual deduction proof of... |
| relopabVD 43173 | Virtual deduction proof of... |
| 19.41rgVD 43174 | Virtual deduction proof of... |
| 2pm13.193VD 43175 | Virtual deduction proof of... |
| hbimpgVD 43176 | Virtual deduction proof of... |
| hbalgVD 43177 | Virtual deduction proof of... |
| hbexgVD 43178 | Virtual deduction proof of... |
| ax6e2eqVD 43179 | The following User's Proof... |
| ax6e2ndVD 43180 | The following User's Proof... |
| ax6e2ndeqVD 43181 | The following User's Proof... |
| 2sb5ndVD 43182 | The following User's Proof... |
| 2uasbanhVD 43183 | The following User's Proof... |
| e2ebindVD 43184 | The following User's Proof... |
| sb5ALTVD 43185 | The following User's Proof... |
| vk15.4jVD 43186 | The following User's Proof... |
| notnotrALTVD 43187 | The following User's Proof... |
| con3ALTVD 43188 | The following User's Proof... |
| elpwgdedVD 43189 | Membership in a power clas... |
| sspwimp 43190 | If a class is a subclass o... |
| sspwimpVD 43191 | The following User's Proof... |
| sspwimpcf 43192 | If a class is a subclass o... |
| sspwimpcfVD 43193 | The following User's Proof... |
| suctrALTcf 43194 | The sucessor of a transiti... |
| suctrALTcfVD 43195 | The following User's Proof... |
| suctrALT3 43196 | The successor of a transit... |
| sspwimpALT 43197 | If a class is a subclass o... |
| unisnALT 43198 | A set equals the union of ... |
| notnotrALT2 43199 | Converse of double negatio... |
| sspwimpALT2 43200 | If a class is a subclass o... |
| e2ebindALT 43201 | Absorption of an existenti... |
| ax6e2ndALT 43202 | If at least two sets exist... |
| ax6e2ndeqALT 43203 | "At least two sets exist" ... |
| 2sb5ndALT 43204 | Equivalence for double sub... |
| chordthmALT 43205 | The intersecting chords th... |
| isosctrlem1ALT 43206 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 43207 | The indexed union of conne... |
| iunconnALT 43208 | The indexed union of conne... |
| sineq0ALT 43209 | A complex number whose sin... |
| evth2f 43210 | A version of ~ evth2 using... |
| elunif 43211 | A version of ~ eluni using... |
| rzalf 43212 | A version of ~ rzal using ... |
| fvelrnbf 43213 | A version of ~ fvelrnb usi... |
| rfcnpre1 43214 | If F is a continuous funct... |
| ubelsupr 43215 | If U belongs to A and U is... |
| fsumcnf 43216 | A finite sum of functions ... |
| mulltgt0 43217 | The product of a negative ... |
| rspcegf 43218 | A version of ~ rspcev usin... |
| rabexgf 43219 | A version of ~ rabexg usin... |
| fcnre 43220 | A function continuous with... |
| sumsnd 43221 | A sum of a singleton is th... |
| evthf 43222 | A version of ~ evth using ... |
| cnfex 43223 | The class of continuous fu... |
| fnchoice 43224 | For a finite set, a choice... |
| refsumcn 43225 | A finite sum of continuous... |
| rfcnpre2 43226 | If ` F ` is a continuous f... |
| cncmpmax 43227 | When the hypothesis for th... |
| rfcnpre3 43228 | If F is a continuous funct... |
| rfcnpre4 43229 | If F is a continuous funct... |
| sumpair 43230 | Sum of two distinct comple... |
| rfcnnnub 43231 | Given a real continuous fu... |
| refsum2cnlem1 43232 | This is the core Lemma for... |
| refsum2cn 43233 | The sum of two continuus r... |
| adantlllr 43234 | Deduction adding a conjunc... |
| 3adantlr3 43235 | Deduction adding a conjunc... |
| 3adantll2 43236 | Deduction adding a conjunc... |
| 3adantll3 43237 | Deduction adding a conjunc... |
| ssnel 43238 | If not element of a set, t... |
| elabrexg 43239 | Elementhood in an image se... |
| sncldre 43240 | A singleton is closed w.r.... |
| n0p 43241 | A polynomial with a nonzer... |
| pm2.65ni 43242 | Inference rule for proof b... |
| pwssfi 43243 | Every element of the power... |
| iuneq2df 43244 | Equality deduction for ind... |
| nnfoctb 43245 | There exists a mapping fro... |
| ssinss1d 43246 | Intersection preserves sub... |
| elpwinss 43247 | An element of the powerset... |
| unidmex 43248 | If ` F ` is a set, then ` ... |
| ndisj2 43249 | A non-disjointness conditi... |
| zenom 43250 | The set of integer numbers... |
| uzwo4 43251 | Well-ordering principle: a... |
| unisn0 43252 | The union of the singleton... |
| ssin0 43253 | If two classes are disjoin... |
| inabs3 43254 | Absorption law for interse... |
| pwpwuni 43255 | Relationship between power... |
| disjiun2 43256 | In a disjoint collection, ... |
| 0pwfi 43257 | The empty set is in any po... |
| ssinss2d 43258 | Intersection preserves sub... |
| zct 43259 | The set of integer numbers... |
| pwfin0 43260 | A finite set always belong... |
| uzct 43261 | An upper integer set is co... |
| iunxsnf 43262 | A singleton index picks ou... |
| fiiuncl 43263 | If a set is closed under t... |
| iunp1 43264 | The addition of the next s... |
| fiunicl 43265 | If a set is closed under t... |
| ixpeq2d 43266 | Equality theorem for infin... |
| disjxp1 43267 | The sets of a cartesian pr... |
| disjsnxp 43268 | The sets in the cartesian ... |
| eliind 43269 | Membership in indexed inte... |
| rspcef 43270 | Restricted existential spe... |
| inn0f 43271 | A nonempty intersection. ... |
| ixpssmapc 43272 | An infinite Cartesian prod... |
| inn0 43273 | A nonempty intersection. ... |
| elintd 43274 | Membership in class inters... |
| ssdf 43275 | A sufficient condition for... |
| brneqtrd 43276 | Substitution of equal clas... |
| ssnct 43277 | A set containing an uncoun... |
| ssuniint 43278 | Sufficient condition for b... |
| elintdv 43279 | Membership in class inters... |
| ssd 43280 | A sufficient condition for... |
| ralimralim 43281 | Introducing any antecedent... |
| snelmap 43282 | Membership of the element ... |
| xrnmnfpnf 43283 | An extended real that is n... |
| nelrnmpt 43284 | Non-membership in the rang... |
| iuneq1i 43285 | Equality theorem for index... |
| nssrex 43286 | Negation of subclass relat... |
| ssinc 43287 | Inclusion relation for a m... |
| ssdec 43288 | Inclusion relation for a m... |
| elixpconstg 43289 | Membership in an infinite ... |
| iineq1d 43290 | Equality theorem for index... |
| metpsmet 43291 | A metric is a pseudometric... |
| ixpssixp 43292 | Subclass theorem for infin... |
| ballss3 43293 | A sufficient condition for... |
| iunincfi 43294 | Given a sequence of increa... |
| nsstr 43295 | If it's not a subclass, it... |
| rexanuz3 43296 | Combine two different uppe... |
| cbvmpo2 43297 | Rule to change the second ... |
| cbvmpo1 43298 | Rule to change the first b... |
| eliuniin 43299 | Indexed union of indexed i... |
| ssabf 43300 | Subclass of a class abstra... |
| pssnssi 43301 | A proper subclass does not... |
| rabidim2 43302 | Membership in a restricted... |
| eluni2f 43303 | Membership in class union.... |
| eliin2f 43304 | Membership in indexed inte... |
| nssd 43305 | Negation of subclass relat... |
| iineq12dv 43306 | Equality deduction for ind... |
| supxrcld 43307 | The supremum of an arbitra... |
| elrestd 43308 | A sufficient condition for... |
| eliuniincex 43309 | Counterexample to show tha... |
| eliincex 43310 | Counterexample to show tha... |
| eliinid 43311 | Membership in an indexed i... |
| abssf 43312 | Class abstraction in a sub... |
| supxrubd 43313 | A member of a set of exten... |
| ssrabf 43314 | Subclass of a restricted c... |
| ssrabdf 43315 | Subclass of a restricted c... |
| eliin2 43316 | Membership in indexed inte... |
| ssrab2f 43317 | Subclass relation for a re... |
| restuni3 43318 | The underlying set of a su... |
| rabssf 43319 | Restricted class abstracti... |
| eliuniin2 43320 | Indexed union of indexed i... |
| restuni4 43321 | The underlying set of a su... |
| restuni6 43322 | The underlying set of a su... |
| restuni5 43323 | The underlying set of a su... |
| unirestss 43324 | The union of an elementwis... |
| iniin1 43325 | Indexed intersection of in... |
| iniin2 43326 | Indexed intersection of in... |
| cbvrabv2 43327 | A more general version of ... |
| cbvrabv2w 43328 | A more general version of ... |
| iinssiin 43329 | Subset implication for an ... |
| eliind2 43330 | Membership in indexed inte... |
| iinssd 43331 | Subset implication for an ... |
| rabbida2 43332 | Equivalent wff's yield equ... |
| iinexd 43333 | The existence of an indexe... |
| rabexf 43334 | Separation Scheme in terms... |
| rabbida3 43335 | Equivalent wff's yield equ... |
| r19.36vf 43336 | Restricted quantifier vers... |
| raleqd 43337 | Equality deduction for res... |
| iinssf 43338 | Subset implication for an ... |
| iinssdf 43339 | Subset implication for an ... |
| resabs2i 43340 | Absorption law for restric... |
| ssdf2 43341 | A sufficient condition for... |
| rabssd 43342 | Restricted class abstracti... |
| rexnegd 43343 | Minus a real number. (Con... |
| rexlimd3 43344 | * Inference from Theorem 1... |
| resabs1i 43345 | Absorption law for restric... |
| nel1nelin 43346 | Membership in an intersect... |
| nel2nelin 43347 | Membership in an intersect... |
| nel1nelini 43348 | Membership in an intersect... |
| nel2nelini 43349 | Membership in an intersect... |
| eliunid 43350 | Membership in indexed unio... |
| reximddv3 43351 | Deduction from Theorem 19.... |
| reximdd 43352 | Deduction from Theorem 19.... |
| unfid 43353 | The union of two finite se... |
| inopnd 43354 | The intersection of two op... |
| ss2rabdf 43355 | Deduction of restricted ab... |
| restopn3 43356 | If ` A ` is open, then ` A... |
| restopnssd 43357 | A topology restricted to a... |
| restsubel 43358 | A subset belongs in the sp... |
| toprestsubel 43359 | A subset is open in the to... |
| rabidd 43360 | An "identity" law of concr... |
| iunssdf 43361 | Subset theorem for an inde... |
| iinss2d 43362 | Subset implication for an ... |
| r19.3rzf 43363 | Restricted quantification ... |
| r19.28zf 43364 | Restricted quantifier vers... |
| iindif2f 43365 | Indexed intersection of cl... |
| ralfal 43366 | Two ways of expressing emp... |
| archd 43367 | Archimedean property of re... |
| eliund 43368 | Membership in indexed unio... |
| nimnbi 43369 | If an implication is false... |
| nimnbi2 43370 | If an implication is false... |
| notbicom 43371 | Commutative law for the ne... |
| rexeqif 43372 | Equality inference for res... |
| rspced 43373 | Restricted existential spe... |
| feq1dd 43374 | Equality deduction for fun... |
| fnresdmss 43375 | A function does not change... |
| fmptsnxp 43376 | Maps-to notation and Carte... |
| fvmpt2bd 43377 | Value of a function given ... |
| rnmptfi 43378 | The range of a function wi... |
| fresin2 43379 | Restriction of a function ... |
| ffi 43380 | A function with finite dom... |
| suprnmpt 43381 | An explicit bound for the ... |
| rnffi 43382 | The range of a function wi... |
| mptelpm 43383 | A function in maps-to nota... |
| rnmptpr 43384 | Range of a function define... |
| resmpti 43385 | Restriction of the mapping... |
| founiiun 43386 | Union expressed as an inde... |
| rnresun 43387 | Distribution law for range... |
| dffo3f 43388 | An onto mapping expressed ... |
| elrnmptf 43389 | The range of a function in... |
| rnmptssrn 43390 | Inclusion relation for two... |
| disjf1 43391 | A 1 to 1 mapping built fro... |
| rnsnf 43392 | The range of a function wh... |
| wessf1ornlem 43393 | Given a function ` F ` on ... |
| wessf1orn 43394 | Given a function ` F ` on ... |
| foelrnf 43395 | Property of a surjective f... |
| nelrnres 43396 | If ` A ` is not in the ran... |
| disjrnmpt2 43397 | Disjointness of the range ... |
| elrnmpt1sf 43398 | Elementhood in an image se... |
| founiiun0 43399 | Union expressed as an inde... |
| disjf1o 43400 | A bijection built from dis... |
| fompt 43401 | Express being onto for a m... |
| disjinfi 43402 | Only a finite number of di... |
| fvovco 43403 | Value of the composition o... |
| ssnnf1octb 43404 | There exists a bijection b... |
| nnf1oxpnn 43405 | There is a bijection betwe... |
| rnmptssd 43406 | The range of a function gi... |
| projf1o 43407 | A biijection from a set to... |
| fvmap 43408 | Function value for a membe... |
| fvixp2 43409 | Projection of a factor of ... |
| fidmfisupp 43410 | A function with a finite d... |
| choicefi 43411 | For a finite set, a choice... |
| mpct 43412 | The exponentiation of a co... |
| cnmetcoval 43413 | Value of the distance func... |
| fcomptss 43414 | Express composition of two... |
| elmapsnd 43415 | Membership in a set expone... |
| mapss2 43416 | Subset inheritance for set... |
| fsneq 43417 | Equality condition for two... |
| difmap 43418 | Difference of two sets exp... |
| unirnmap 43419 | Given a subset of a set ex... |
| inmap 43420 | Intersection of two sets e... |
| fcoss 43421 | Composition of two mapping... |
| fsneqrn 43422 | Equality condition for two... |
| difmapsn 43423 | Difference of two sets exp... |
| mapssbi 43424 | Subset inheritance for set... |
| unirnmapsn 43425 | Equality theorem for a sub... |
| iunmapss 43426 | The indexed union of set e... |
| ssmapsn 43427 | A subset ` C ` of a set ex... |
| iunmapsn 43428 | The indexed union of set e... |
| absfico 43429 | Mapping domain and codomai... |
| icof 43430 | The set of left-closed rig... |
| elpmrn 43431 | The range of a partial fun... |
| imaexi 43432 | The image of a set is a se... |
| axccdom 43433 | Relax the constraint on ax... |
| dmmptdff 43434 | The domain of the mapping ... |
| dmmptdf 43435 | The domain of the mapping ... |
| elpmi2 43436 | The domain of a partial fu... |
| dmrelrnrel 43437 | A relation preserving func... |
| fvcod 43438 | Value of a function compos... |
| elrnmpoid 43439 | Membership in the range of... |
| axccd 43440 | An alternative version of ... |
| axccd2 43441 | An alternative version of ... |
| funimassd 43442 | Sufficient condition for t... |
| fimassd 43443 | The image of a class is a ... |
| feqresmptf 43444 | Express a restricted funct... |
| elrnmpt1d 43445 | Elementhood in an image se... |
| dmresss 43446 | The domain of a restrictio... |
| dmmptssf 43447 | The domain of a mapping is... |
| dmmptdf2 43448 | The domain of the mapping ... |
| dmuz 43449 | Domain of the upper intege... |
| fmptd2f 43450 | Domain and codomain of the... |
| mpteq1df 43451 | An equality theorem for th... |
| mpteq1dfOLD 43452 | Obsolete version of ~ mpte... |
| mptexf 43453 | If the domain of a functio... |
| fvmpt4 43454 | Value of a function given ... |
| fmptf 43455 | Functionality of the mappi... |
| resimass 43456 | The image of a restriction... |
| mptssid 43457 | The mapping operation expr... |
| mptfnd 43458 | The maps-to notation defin... |
| mpteq12daOLD 43459 | Obsolete version of ~ mpte... |
| rnmptlb 43460 | Boundness below of the ran... |
| rnmptbddlem 43461 | Boundness of the range of ... |
| rnmptbdd 43462 | Boundness of the range of ... |
| mptima2 43463 | Image of a function in map... |
| funimaeq 43464 | Membership relation for th... |
| rnmptssf 43465 | The range of a function gi... |
| rnmptbd2lem 43466 | Boundness below of the ran... |
| rnmptbd2 43467 | Boundness below of the ran... |
| infnsuprnmpt 43468 | The indexed infimum of rea... |
| suprclrnmpt 43469 | Closure of the indexed sup... |
| suprubrnmpt2 43470 | A member of a nonempty ind... |
| suprubrnmpt 43471 | A member of a nonempty ind... |
| rnmptssdf 43472 | The range of a function gi... |
| rnmptbdlem 43473 | Boundness above of the ran... |
| rnmptbd 43474 | Boundness above of the ran... |
| rnmptss2 43475 | The range of a function gi... |
| elmptima 43476 | The image of a function in... |
| ralrnmpt3 43477 | A restricted quantifier ov... |
| fvelima2 43478 | Function value in an image... |
| rnmptssbi 43479 | The range of a function gi... |
| imass2d 43480 | Subset theorem for image. ... |
| imassmpt 43481 | Membership relation for th... |
| fpmd 43482 | A total function is a part... |
| fconst7 43483 | An alternative way to expr... |
| fnmptif 43484 | Functionality and domain o... |
| dmmptif 43485 | Domain of the mapping oper... |
| mpteq2dfa 43486 | Slightly more general equa... |
| dmmpt1 43487 | The domain of the mapping ... |
| fmptff 43488 | Functionality of the mappi... |
| fvmptelcdmf 43489 | The value of a function at... |
| fmptdff 43490 | A version of ~ fmptd using... |
| fvmpt2df 43491 | Deduction version of ~ fvm... |
| rn1st 43492 | The range of a function wi... |
| rnmptssff 43493 | The range of a function gi... |
| rnmptssdff 43494 | The range of a function gi... |
| fvmpt4d 43495 | Value of a function given ... |
| sub2times 43496 | Subtracting from a number,... |
| nnxrd 43497 | A natural number is an ext... |
| nnxr 43498 | A natural number is an ext... |
| abssubrp 43499 | The distance of two distin... |
| elfzfzo 43500 | Relationship between membe... |
| oddfl 43501 | Odd number representation ... |
| abscosbd 43502 | Bound for the absolute val... |
| mul13d 43503 | Commutative/associative la... |
| negpilt0 43504 | Negative ` _pi ` is negati... |
| dstregt0 43505 | A complex number ` A ` tha... |
| subadd4b 43506 | Rearrangement of 4 terms i... |
| xrlttri5d 43507 | Not equal and not larger i... |
| neglt 43508 | The negative of a positive... |
| zltlesub 43509 | If an integer ` N ` is les... |
| divlt0gt0d 43510 | The ratio of a negative nu... |
| subsub23d 43511 | Swap subtrahend and result... |
| 2timesgt 43512 | Double of a positive real ... |
| reopn 43513 | The reals are open with re... |
| sub31 43514 | Swap the first and third t... |
| nnne1ge2 43515 | A positive integer which i... |
| lefldiveq 43516 | A closed enough, smaller r... |
| negsubdi3d 43517 | Distribution of negative o... |
| ltdiv2dd 43518 | Division of a positive num... |
| abssinbd 43519 | Bound for the absolute val... |
| halffl 43520 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 43521 | Ordering relation for a st... |
| hashssle 43522 | The size of a subset of a ... |
| lttri5d 43523 | Not equal and not larger i... |
| fzisoeu 43524 | A finite ordered set has a... |
| lt3addmuld 43525 | If three real numbers are ... |
| absnpncan2d 43526 | Triangular inequality, com... |
| fperiodmullem 43527 | A function with period ` T... |
| fperiodmul 43528 | A function with period T i... |
| upbdrech 43529 | Choice of an upper bound f... |
| lt4addmuld 43530 | If four real numbers are l... |
| absnpncan3d 43531 | Triangular inequality, com... |
| upbdrech2 43532 | Choice of an upper bound f... |
| ssfiunibd 43533 | A finite union of bounded ... |
| fzdifsuc2 43534 | Remove a successor from th... |
| fzsscn 43535 | A finite sequence of integ... |
| divcan8d 43536 | A cancellation law for div... |
| dmmcand 43537 | Cancellation law for divis... |
| fzssre 43538 | A finite sequence of integ... |
| bccld 43539 | A binomial coefficient, in... |
| leadd12dd 43540 | Addition to both sides of ... |
| fzssnn0 43541 | A finite set of sequential... |
| xreqle 43542 | Equality implies 'less tha... |
| xaddid2d 43543 | ` 0 ` is a left identity f... |
| xadd0ge 43544 | A number is less than or e... |
| elfzolem1 43545 | A member in a half-open in... |
| xrgtned 43546 | 'Greater than' implies not... |
| xrleneltd 43547 | 'Less than or equal to' an... |
| xaddcomd 43548 | The extended real addition... |
| supxrre3 43549 | The supremum of a nonempty... |
| uzfissfz 43550 | For any finite subset of t... |
| xleadd2d 43551 | Addition of extended reals... |
| suprltrp 43552 | The supremum of a nonempty... |
| xleadd1d 43553 | Addition of extended reals... |
| xreqled 43554 | Equality implies 'less tha... |
| xrgepnfd 43555 | An extended real greater t... |
| xrge0nemnfd 43556 | A nonnegative extended rea... |
| supxrgere 43557 | If a real number can be ap... |
| iuneqfzuzlem 43558 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 43559 | If two unions indexed by u... |
| xle2addd 43560 | Adding both side of two in... |
| supxrgelem 43561 | If an extended real number... |
| supxrge 43562 | If an extended real number... |
| suplesup 43563 | If any element of ` A ` ca... |
| infxrglb 43564 | The infimum of a set of ex... |
| xadd0ge2 43565 | A number is less than or e... |
| nepnfltpnf 43566 | An extended real that is n... |
| ltadd12dd 43567 | Addition to both sides of ... |
| nemnftgtmnft 43568 | An extended real that is n... |
| xrgtso 43569 | 'Greater than' is a strict... |
| rpex 43570 | The positive reals form a ... |
| xrge0ge0 43571 | A nonnegative extended rea... |
| xrssre 43572 | A subset of extended reals... |
| ssuzfz 43573 | A finite subset of the upp... |
| absfun 43574 | The absolute value is a fu... |
| infrpge 43575 | The infimum of a nonempty,... |
| xrlexaddrp 43576 | If an extended real number... |
| supsubc 43577 | The supremum function dist... |
| xralrple2 43578 | Show that ` A ` is less th... |
| nnuzdisj 43579 | The first ` N ` elements o... |
| ltdivgt1 43580 | Divsion by a number greate... |
| xrltned 43581 | 'Less than' implies not eq... |
| nnsplit 43582 | Express the set of positiv... |
| divdiv3d 43583 | Division into a fraction. ... |
| abslt2sqd 43584 | Comparison of the square o... |
| qenom 43585 | The set of rational number... |
| qct 43586 | The set of rational number... |
| xrltnled 43587 | 'Less than' in terms of 'l... |
| lenlteq 43588 | 'less than or equal to' bu... |
| xrred 43589 | An extended real that is n... |
| rr2sscn2 43590 | The cartesian square of ` ... |
| infxr 43591 | The infimum of a set of ex... |
| infxrunb2 43592 | The infimum of an unbounde... |
| infxrbnd2 43593 | The infimum of a bounded-b... |
| infleinflem1 43594 | Lemma for ~ infleinf , cas... |
| infleinflem2 43595 | Lemma for ~ infleinf , whe... |
| infleinf 43596 | If any element of ` B ` ca... |
| xralrple4 43597 | Show that ` A ` is less th... |
| xralrple3 43598 | Show that ` A ` is less th... |
| eluzelzd 43599 | A member of an upper set o... |
| suplesup2 43600 | If any element of ` A ` is... |
| recnnltrp 43601 | ` N ` is a natural number ... |
| nnn0 43602 | The set of positive intege... |
| fzct 43603 | A finite set of sequential... |
| rpgtrecnn 43604 | Any positive real number i... |
| fzossuz 43605 | A half-open integer interv... |
| infxrrefi 43606 | The real and extended real... |
| xrralrecnnle 43607 | Show that ` A ` is less th... |
| fzoct 43608 | A finite set of sequential... |
| frexr 43609 | A function taking real val... |
| nnrecrp 43610 | The reciprocal of a positi... |
| reclt0d 43611 | The reciprocal of a negati... |
| lt0neg1dd 43612 | If a number is negative, i... |
| mnfled 43613 | Minus infinity is less tha... |
| infxrcld 43614 | The infimum of an arbitrar... |
| xrralrecnnge 43615 | Show that ` A ` is less th... |
| reclt0 43616 | The reciprocal of a negati... |
| ltmulneg 43617 | Multiplying by a negative ... |
| allbutfi 43618 | For all but finitely many.... |
| ltdiv23neg 43619 | Swap denominator with othe... |
| xreqnltd 43620 | A consequence of trichotom... |
| mnfnre2 43621 | Minus infinity is not a re... |
| zssxr 43622 | The integers are a subset ... |
| fisupclrnmpt 43623 | A nonempty finite indexed ... |
| supxrunb3 43624 | The supremum of an unbound... |
| elfzod 43625 | Membership in a half-open ... |
| fimaxre4 43626 | A nonempty finite set of r... |
| ren0 43627 | The set of reals is nonemp... |
| eluzelz2 43628 | A member of an upper set o... |
| resabs2d 43629 | Absorption law for restric... |
| uzid2 43630 | Membership of the least me... |
| supxrleubrnmpt 43631 | The supremum of a nonempty... |
| uzssre2 43632 | An upper set of integers i... |
| uzssd 43633 | Subset relationship for tw... |
| eluzd 43634 | Membership in an upper set... |
| infxrlbrnmpt2 43635 | A member of a nonempty ind... |
| xrre4 43636 | An extended real is real i... |
| uz0 43637 | The upper integers functio... |
| eluzelz2d 43638 | A member of an upper set o... |
| infleinf2 43639 | If any element in ` B ` is... |
| unb2ltle 43640 | "Unbounded below" expresse... |
| uzidd2 43641 | Membership of the least me... |
| uzssd2 43642 | Subset relationship for tw... |
| rexabslelem 43643 | An indexed set of absolute... |
| rexabsle 43644 | An indexed set of absolute... |
| allbutfiinf 43645 | Given a "for all but finit... |
| supxrrernmpt 43646 | The real and extended real... |
| suprleubrnmpt 43647 | The supremum of a nonempty... |
| infrnmptle 43648 | An indexed infimum of exte... |
| infxrunb3 43649 | The infimum of an unbounde... |
| uzn0d 43650 | The upper integers are all... |
| uzssd3 43651 | Subset relationship for tw... |
| rexabsle2 43652 | An indexed set of absolute... |
| infxrunb3rnmpt 43653 | The infimum of an unbounde... |
| supxrre3rnmpt 43654 | The indexed supremum of a ... |
| uzublem 43655 | A set of reals, indexed by... |
| uzub 43656 | A set of reals, indexed by... |
| ssrexr 43657 | A subset of the reals is a... |
| supxrmnf2 43658 | Removing minus infinity fr... |
| supxrcli 43659 | The supremum of an arbitra... |
| uzid3 43660 | Membership of the least me... |
| infxrlesupxr 43661 | The supremum of a nonempty... |
| xnegeqd 43662 | Equality of two extended n... |
| xnegrecl 43663 | The extended real negative... |
| xnegnegi 43664 | Extended real version of ~... |
| xnegeqi 43665 | Equality of two extended n... |
| nfxnegd 43666 | Deduction version of ~ nfx... |
| xnegnegd 43667 | Extended real version of ~... |
| uzred 43668 | An upper integer is a real... |
| xnegcli 43669 | Closure of extended real n... |
| supminfrnmpt 43670 | The indexed supremum of a ... |
| infxrpnf 43671 | Adding plus infinity to a ... |
| infxrrnmptcl 43672 | The infimum of an arbitrar... |
| leneg2d 43673 | Negative of one side of 'l... |
| supxrltinfxr 43674 | The supremum of the empty ... |
| max1d 43675 | A number is less than or e... |
| supxrleubrnmptf 43676 | The supremum of a nonempty... |
| nleltd 43677 | 'Not less than or equal to... |
| zxrd 43678 | An integer is an extended ... |
| infxrgelbrnmpt 43679 | The infimum of an indexed ... |
| rphalfltd 43680 | Half of a positive real is... |
| uzssz2 43681 | An upper set of integers i... |
| leneg3d 43682 | Negative of one side of 'l... |
| max2d 43683 | A number is less than or e... |
| uzn0bi 43684 | The upper integers functio... |
| xnegrecl2 43685 | If the extended real negat... |
| nfxneg 43686 | Bound-variable hypothesis ... |
| uzxrd 43687 | An upper integer is an ext... |
| infxrpnf2 43688 | Removing plus infinity fro... |
| supminfxr 43689 | The extended real suprema ... |
| infrpgernmpt 43690 | The infimum of a nonempty,... |
| xnegre 43691 | An extended real is real i... |
| xnegrecl2d 43692 | If the extended real negat... |
| uzxr 43693 | An upper integer is an ext... |
| supminfxr2 43694 | The extended real suprema ... |
| xnegred 43695 | An extended real is real i... |
| supminfxrrnmpt 43696 | The indexed supremum of a ... |
| min1d 43697 | The minimum of two numbers... |
| min2d 43698 | The minimum of two numbers... |
| pnfged 43699 | Plus infinity is an upper ... |
| xrnpnfmnf 43700 | An extended real that is n... |
| uzsscn 43701 | An upper set of integers i... |
| absimnre 43702 | The absolute value of the ... |
| uzsscn2 43703 | An upper set of integers i... |
| xrtgcntopre 43704 | The standard topologies on... |
| absimlere 43705 | The absolute value of the ... |
| rpssxr 43706 | The positive reals are a s... |
| monoordxrv 43707 | Ordering relation for a mo... |
| monoordxr 43708 | Ordering relation for a mo... |
| monoord2xrv 43709 | Ordering relation for a mo... |
| monoord2xr 43710 | Ordering relation for a mo... |
| xrpnf 43711 | An extended real is plus i... |
| xlenegcon1 43712 | Extended real version of ~... |
| xlenegcon2 43713 | Extended real version of ~... |
| pimxrneun 43714 | The preimage of a set of e... |
| caucvgbf 43715 | A function is convergent i... |
| cvgcau 43716 | A convergent function is C... |
| cvgcaule 43717 | A convergent function is C... |
| rexanuz2nf 43718 | A simple counterexample re... |
| gtnelioc 43719 | A real number larger than ... |
| ioossioc 43720 | An open interval is a subs... |
| ioondisj2 43721 | A condition for two open i... |
| ioondisj1 43722 | A condition for two open i... |
| ioogtlb 43723 | An element of a closed int... |
| evthiccabs 43724 | Extreme Value Theorem on y... |
| ltnelicc 43725 | A real number smaller than... |
| eliood 43726 | Membership in an open real... |
| iooabslt 43727 | An upper bound for the dis... |
| gtnelicc 43728 | A real number greater than... |
| iooinlbub 43729 | An open interval has empty... |
| iocgtlb 43730 | An element of a left-open ... |
| iocleub 43731 | An element of a left-open ... |
| eliccd 43732 | Membership in a closed rea... |
| eliccre 43733 | A member of a closed inter... |
| eliooshift 43734 | Element of an open interva... |
| eliocd 43735 | Membership in a left-open ... |
| icoltub 43736 | An element of a left-close... |
| eliocre 43737 | A member of a left-open ri... |
| iooltub 43738 | An element of an open inte... |
| ioontr 43739 | The interior of an interva... |
| snunioo1 43740 | The closure of one end of ... |
| lbioc 43741 | A left-open right-closed i... |
| ioomidp 43742 | The midpoint is an element... |
| iccdifioo 43743 | If the open inverval is re... |
| iccdifprioo 43744 | An open interval is the cl... |
| ioossioobi 43745 | Biconditional form of ~ io... |
| iccshift 43746 | A closed interval shifted ... |
| iccsuble 43747 | An upper bound to the dist... |
| iocopn 43748 | A left-open right-closed i... |
| eliccelioc 43749 | Membership in a closed int... |
| iooshift 43750 | An open interval shifted b... |
| iccintsng 43751 | Intersection of two adiace... |
| icoiccdif 43752 | Left-closed right-open int... |
| icoopn 43753 | A left-closed right-open i... |
| icoub 43754 | A left-closed, right-open ... |
| eliccxrd 43755 | Membership in a closed rea... |
| pnfel0pnf 43756 | ` +oo ` is a nonnegative e... |
| eliccnelico 43757 | An element of a closed int... |
| eliccelicod 43758 | A member of a closed inter... |
| ge0xrre 43759 | A nonnegative extended rea... |
| ge0lere 43760 | A nonnegative extended Rea... |
| elicores 43761 | Membership in a left-close... |
| inficc 43762 | The infimum of a nonempty ... |
| qinioo 43763 | The rational numbers are d... |
| lenelioc 43764 | A real number smaller than... |
| ioonct 43765 | A nonempty open interval i... |
| xrgtnelicc 43766 | A real number greater than... |
| iccdificc 43767 | The difference of two clos... |
| iocnct 43768 | A nonempty left-open, righ... |
| iccnct 43769 | A closed interval, with mo... |
| iooiinicc 43770 | A closed interval expresse... |
| iccgelbd 43771 | An element of a closed int... |
| iooltubd 43772 | An element of an open inte... |
| icoltubd 43773 | An element of a left-close... |
| qelioo 43774 | The rational numbers are d... |
| tgqioo2 43775 | Every open set of reals is... |
| iccleubd 43776 | An element of a closed int... |
| elioored 43777 | A member of an open interv... |
| ioogtlbd 43778 | An element of a closed int... |
| ioofun 43779 | ` (,) ` is a function. (C... |
| icomnfinre 43780 | A left-closed, right-open,... |
| sqrlearg 43781 | The square compared with i... |
| ressiocsup 43782 | If the supremum belongs to... |
| ressioosup 43783 | If the supremum does not b... |
| iooiinioc 43784 | A left-open, right-closed ... |
| ressiooinf 43785 | If the infimum does not be... |
| icogelbd 43786 | An element of a left-close... |
| iocleubd 43787 | An element of a left-open ... |
| uzinico 43788 | An upper interval of integ... |
| preimaiocmnf 43789 | Preimage of a right-closed... |
| uzinico2 43790 | An upper interval of integ... |
| uzinico3 43791 | An upper interval of integ... |
| icossico2 43792 | Condition for a closed-bel... |
| dmico 43793 | The domain of the closed-b... |
| ndmico 43794 | The closed-below, open-abo... |
| uzubioo 43795 | The upper integers are unb... |
| uzubico 43796 | The upper integers are unb... |
| uzubioo2 43797 | The upper integers are unb... |
| uzubico2 43798 | The upper integers are unb... |
| iocgtlbd 43799 | An element of a left-open ... |
| xrtgioo2 43800 | The topology on the extend... |
| tgioo4 43801 | The standard topology on t... |
| fsummulc1f 43802 | Closure of a finite sum of... |
| fsumnncl 43803 | Closure of a nonempty, fin... |
| fsumge0cl 43804 | The finite sum of nonnegat... |
| fsumf1of 43805 | Re-index a finite sum usin... |
| fsumiunss 43806 | Sum over a disjoint indexe... |
| fsumreclf 43807 | Closure of a finite sum of... |
| fsumlessf 43808 | A shorter sum of nonnegati... |
| fsumsupp0 43809 | Finite sum of function val... |
| fsumsermpt 43810 | A finite sum expressed in ... |
| fmul01 43811 | Multiplying a finite numbe... |
| fmulcl 43812 | If ' Y ' is closed under t... |
| fmuldfeqlem1 43813 | induction step for the pro... |
| fmuldfeq 43814 | X and Z are two equivalent... |
| fmul01lt1lem1 43815 | Given a finite multiplicat... |
| fmul01lt1lem2 43816 | Given a finite multiplicat... |
| fmul01lt1 43817 | Given a finite multiplicat... |
| cncfmptss 43818 | A continuous complex funct... |
| rrpsscn 43819 | The positive reals are a s... |
| mulc1cncfg 43820 | A version of ~ mulc1cncf u... |
| infrglb 43821 | The infimum of a nonempty ... |
| expcnfg 43822 | If ` F ` is a complex cont... |
| prodeq2ad 43823 | Equality deduction for pro... |
| fprodsplit1 43824 | Separate out a term in a f... |
| fprodexp 43825 | Positive integer exponenti... |
| fprodabs2 43826 | The absolute value of a fi... |
| fprod0 43827 | A finite product with a ze... |
| mccllem 43828 | * Induction step for ~ mcc... |
| mccl 43829 | A multinomial coefficient,... |
| fprodcnlem 43830 | A finite product of functi... |
| fprodcn 43831 | A finite product of functi... |
| clim1fr1 43832 | A class of sequences of fr... |
| isumneg 43833 | Negation of a converging s... |
| climrec 43834 | Limit of the reciprocal of... |
| climmulf 43835 | A version of ~ climmul usi... |
| climexp 43836 | The limit of natural power... |
| climinf 43837 | A bounded monotonic noninc... |
| climsuselem1 43838 | The subsequence index ` I ... |
| climsuse 43839 | A subsequence ` G ` of a c... |
| climrecf 43840 | A version of ~ climrec usi... |
| climneg 43841 | Complex limit of the negat... |
| climinff 43842 | A version of ~ climinf usi... |
| climdivf 43843 | Limit of the ratio of two ... |
| climreeq 43844 | If ` F ` is a real functio... |
| ellimciota 43845 | An explicit value for the ... |
| climaddf 43846 | A version of ~ climadd usi... |
| mullimc 43847 | Limit of the product of tw... |
| ellimcabssub0 43848 | An equivalent condition fo... |
| limcdm0 43849 | If a function has empty do... |
| islptre 43850 | An equivalence condition f... |
| limccog 43851 | Limit of the composition o... |
| limciccioolb 43852 | The limit of a function at... |
| climf 43853 | Express the predicate: Th... |
| mullimcf 43854 | Limit of the multiplicatio... |
| constlimc 43855 | Limit of constant function... |
| rexlim2d 43856 | Inference removing two res... |
| idlimc 43857 | Limit of the identity func... |
| divcnvg 43858 | The sequence of reciprocal... |
| limcperiod 43859 | If ` F ` is a periodic fun... |
| limcrecl 43860 | If ` F ` is a real-valued ... |
| sumnnodd 43861 | A series indexed by ` NN `... |
| lptioo2 43862 | The upper bound of an open... |
| lptioo1 43863 | The lower bound of an open... |
| elprn1 43864 | A member of an unordered p... |
| elprn2 43865 | A member of an unordered p... |
| limcmptdm 43866 | The domain of a maps-to fu... |
| clim2f 43867 | Express the predicate: Th... |
| limcicciooub 43868 | The limit of a function at... |
| ltmod 43869 | A sufficient condition for... |
| islpcn 43870 | A characterization for a l... |
| lptre2pt 43871 | If a set in the real line ... |
| limsupre 43872 | If a sequence is bounded, ... |
| limcresiooub 43873 | The left limit doesn't cha... |
| limcresioolb 43874 | The right limit doesn't ch... |
| limcleqr 43875 | If the left and the right ... |
| lptioo2cn 43876 | The upper bound of an open... |
| lptioo1cn 43877 | The lower bound of an open... |
| neglimc 43878 | Limit of the negative func... |
| addlimc 43879 | Sum of two limits. (Contr... |
| 0ellimcdiv 43880 | If the numerator converges... |
| clim2cf 43881 | Express the predicate ` F ... |
| limclner 43882 | For a limit point, both fr... |
| sublimc 43883 | Subtraction of two limits.... |
| reclimc 43884 | Limit of the reciprocal of... |
| clim0cf 43885 | Express the predicate ` F ... |
| limclr 43886 | For a limit point, both fr... |
| divlimc 43887 | Limit of the quotient of t... |
| expfac 43888 | Factorial grows faster tha... |
| climconstmpt 43889 | A constant sequence conver... |
| climresmpt 43890 | A function restricted to u... |
| climsubmpt 43891 | Limit of the difference of... |
| climsubc2mpt 43892 | Limit of the difference of... |
| climsubc1mpt 43893 | Limit of the difference of... |
| fnlimfv 43894 | The value of the limit fun... |
| climreclf 43895 | The limit of a convergent ... |
| climeldmeq 43896 | Two functions that are eve... |
| climf2 43897 | Express the predicate: Th... |
| fnlimcnv 43898 | The sequence of function v... |
| climeldmeqmpt 43899 | Two functions that are eve... |
| climfveq 43900 | Two functions that are eve... |
| clim2f2 43901 | Express the predicate: Th... |
| climfveqmpt 43902 | Two functions that are eve... |
| climd 43903 | Express the predicate: Th... |
| clim2d 43904 | The limit of complex numbe... |
| fnlimfvre 43905 | The limit function of real... |
| allbutfifvre 43906 | Given a sequence of real-v... |
| climleltrp 43907 | The limit of complex numbe... |
| fnlimfvre2 43908 | The limit function of real... |
| fnlimf 43909 | The limit function of real... |
| fnlimabslt 43910 | A sequence of function val... |
| climfveqf 43911 | Two functions that are eve... |
| climmptf 43912 | Exhibit a function ` G ` w... |
| climfveqmpt3 43913 | Two functions that are eve... |
| climeldmeqf 43914 | Two functions that are eve... |
| climreclmpt 43915 | The limit of B convergent ... |
| limsupref 43916 | If a sequence is bounded, ... |
| limsupbnd1f 43917 | If a sequence is eventuall... |
| climbddf 43918 | A converging sequence of c... |
| climeqf 43919 | Two functions that are eve... |
| climeldmeqmpt3 43920 | Two functions that are eve... |
| limsupcld 43921 | Closure of the superior li... |
| climfv 43922 | The limit of a convergent ... |
| limsupval3 43923 | The superior limit of an i... |
| climfveqmpt2 43924 | Two functions that are eve... |
| limsup0 43925 | The superior limit of the ... |
| climeldmeqmpt2 43926 | Two functions that are eve... |
| limsupresre 43927 | The supremum limit of a fu... |
| climeqmpt 43928 | Two functions that are eve... |
| climfvd 43929 | The limit of a convergent ... |
| limsuplesup 43930 | An upper bound for the sup... |
| limsupresico 43931 | The superior limit doesn't... |
| limsuppnfdlem 43932 | If the restriction of a fu... |
| limsuppnfd 43933 | If the restriction of a fu... |
| limsupresuz 43934 | If the real part of the do... |
| limsupub 43935 | If the limsup is not ` +oo... |
| limsupres 43936 | The superior limit of a re... |
| climinf2lem 43937 | A convergent, nonincreasin... |
| climinf2 43938 | A convergent, nonincreasin... |
| limsupvaluz 43939 | The superior limit, when t... |
| limsupresuz2 43940 | If the domain of a functio... |
| limsuppnflem 43941 | If the restriction of a fu... |
| limsuppnf 43942 | If the restriction of a fu... |
| limsupubuzlem 43943 | If the limsup is not ` +oo... |
| limsupubuz 43944 | For a real-valued function... |
| climinf2mpt 43945 | A bounded below, monotonic... |
| climinfmpt 43946 | A bounded below, monotonic... |
| climinf3 43947 | A convergent, nonincreasin... |
| limsupvaluzmpt 43948 | The superior limit, when t... |
| limsupequzmpt2 43949 | Two functions that are eve... |
| limsupubuzmpt 43950 | If the limsup is not ` +oo... |
| limsupmnflem 43951 | The superior limit of a fu... |
| limsupmnf 43952 | The superior limit of a fu... |
| limsupequzlem 43953 | Two functions that are eve... |
| limsupequz 43954 | Two functions that are eve... |
| limsupre2lem 43955 | Given a function on the ex... |
| limsupre2 43956 | Given a function on the ex... |
| limsupmnfuzlem 43957 | The superior limit of a fu... |
| limsupmnfuz 43958 | The superior limit of a fu... |
| limsupequzmptlem 43959 | Two functions that are eve... |
| limsupequzmpt 43960 | Two functions that are eve... |
| limsupre2mpt 43961 | Given a function on the ex... |
| limsupequzmptf 43962 | Two functions that are eve... |
| limsupre3lem 43963 | Given a function on the ex... |
| limsupre3 43964 | Given a function on the ex... |
| limsupre3mpt 43965 | Given a function on the ex... |
| limsupre3uzlem 43966 | Given a function on the ex... |
| limsupre3uz 43967 | Given a function on the ex... |
| limsupreuz 43968 | Given a function on the re... |
| limsupvaluz2 43969 | The superior limit, when t... |
| limsupreuzmpt 43970 | Given a function on the re... |
| supcnvlimsup 43971 | If a function on a set of ... |
| supcnvlimsupmpt 43972 | If a function on a set of ... |
| 0cnv 43973 | If ` (/) ` is a complex nu... |
| climuzlem 43974 | Express the predicate: Th... |
| climuz 43975 | Express the predicate: Th... |
| lmbr3v 43976 | Express the binary relatio... |
| climisp 43977 | If a sequence converges to... |
| lmbr3 43978 | Express the binary relatio... |
| climrescn 43979 | A sequence converging w.r.... |
| climxrrelem 43980 | If a sequence ranging over... |
| climxrre 43981 | If a sequence ranging over... |
| limsuplt2 43984 | The defining property of t... |
| liminfgord 43985 | Ordering property of the i... |
| limsupvald 43986 | The superior limit of a se... |
| limsupresicompt 43987 | The superior limit doesn't... |
| limsupcli 43988 | Closure of the superior li... |
| liminfgf 43989 | Closure of the inferior li... |
| liminfval 43990 | The inferior limit of a se... |
| climlimsup 43991 | A sequence of real numbers... |
| limsupge 43992 | The defining property of t... |
| liminfgval 43993 | Value of the inferior limi... |
| liminfcl 43994 | Closure of the inferior li... |
| liminfvald 43995 | The inferior limit of a se... |
| liminfval5 43996 | The inferior limit of an i... |
| limsupresxr 43997 | The superior limit of a fu... |
| liminfresxr 43998 | The inferior limit of a fu... |
| liminfval2 43999 | The superior limit, relati... |
| climlimsupcex 44000 | Counterexample for ~ climl... |
| liminfcld 44001 | Closure of the inferior li... |
| liminfresico 44002 | The inferior limit doesn't... |
| limsup10exlem 44003 | The range of the given fun... |
| limsup10ex 44004 | The superior limit of a fu... |
| liminf10ex 44005 | The inferior limit of a fu... |
| liminflelimsuplem 44006 | The superior limit is grea... |
| liminflelimsup 44007 | The superior limit is grea... |
| limsupgtlem 44008 | For any positive real, the... |
| limsupgt 44009 | Given a sequence of real n... |
| liminfresre 44010 | The inferior limit of a fu... |
| liminfresicompt 44011 | The inferior limit doesn't... |
| liminfltlimsupex 44012 | An example where the ` lim... |
| liminfgelimsup 44013 | The inferior limit is grea... |
| liminfvalxr 44014 | Alternate definition of ` ... |
| liminfresuz 44015 | If the real part of the do... |
| liminflelimsupuz 44016 | The superior limit is grea... |
| liminfvalxrmpt 44017 | Alternate definition of ` ... |
| liminfresuz2 44018 | If the domain of a functio... |
| liminfgelimsupuz 44019 | The inferior limit is grea... |
| liminfval4 44020 | Alternate definition of ` ... |
| liminfval3 44021 | Alternate definition of ` ... |
| liminfequzmpt2 44022 | Two functions that are eve... |
| liminfvaluz 44023 | Alternate definition of ` ... |
| liminf0 44024 | The inferior limit of the ... |
| limsupval4 44025 | Alternate definition of ` ... |
| liminfvaluz2 44026 | Alternate definition of ` ... |
| liminfvaluz3 44027 | Alternate definition of ` ... |
| liminflelimsupcex 44028 | A counterexample for ~ lim... |
| limsupvaluz3 44029 | Alternate definition of ` ... |
| liminfvaluz4 44030 | Alternate definition of ` ... |
| limsupvaluz4 44031 | Alternate definition of ` ... |
| climliminflimsupd 44032 | If a sequence of real numb... |
| liminfreuzlem 44033 | Given a function on the re... |
| liminfreuz 44034 | Given a function on the re... |
| liminfltlem 44035 | Given a sequence of real n... |
| liminflt 44036 | Given a sequence of real n... |
| climliminf 44037 | A sequence of real numbers... |
| liminflimsupclim 44038 | A sequence of real numbers... |
| climliminflimsup 44039 | A sequence of real numbers... |
| climliminflimsup2 44040 | A sequence of real numbers... |
| climliminflimsup3 44041 | A sequence of real numbers... |
| climliminflimsup4 44042 | A sequence of real numbers... |
| limsupub2 44043 | A extended real valued fun... |
| limsupubuz2 44044 | A sequence with values in ... |
| xlimpnfxnegmnf 44045 | A sequence converges to ` ... |
| liminflbuz2 44046 | A sequence with values in ... |
| liminfpnfuz 44047 | The inferior limit of a fu... |
| liminflimsupxrre 44048 | A sequence with values in ... |
| xlimrel 44051 | The limit on extended real... |
| xlimres 44052 | A function converges iff i... |
| xlimcl 44053 | The limit of a sequence of... |
| rexlimddv2 44054 | Restricted existential eli... |
| xlimclim 44055 | Given a sequence of reals,... |
| xlimconst 44056 | A constant sequence conver... |
| climxlim 44057 | A converging sequence in t... |
| xlimbr 44058 | Express the binary relatio... |
| fuzxrpmcn 44059 | A function mapping from an... |
| cnrefiisplem 44060 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 44061 | A non-real, complex number... |
| xlimxrre 44062 | If a sequence ranging over... |
| xlimmnfvlem1 44063 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 44064 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 44065 | A function converges to mi... |
| xlimconst2 44066 | A sequence that eventually... |
| xlimpnfvlem1 44067 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 44068 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 44069 | A function converges to pl... |
| xlimclim2lem 44070 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 44071 | Given a sequence of extend... |
| xlimmnf 44072 | A function converges to mi... |
| xlimpnf 44073 | A function converges to pl... |
| xlimmnfmpt 44074 | A function converges to pl... |
| xlimpnfmpt 44075 | A function converges to pl... |
| climxlim2lem 44076 | In this lemma for ~ climxl... |
| climxlim2 44077 | A sequence of extended rea... |
| dfxlim2v 44078 | An alternative definition ... |
| dfxlim2 44079 | An alternative definition ... |
| climresd 44080 | A function restricted to u... |
| climresdm 44081 | A real function converges ... |
| dmclimxlim 44082 | A real valued sequence tha... |
| xlimmnflimsup2 44083 | A sequence of extended rea... |
| xlimuni 44084 | An infinite sequence conve... |
| xlimclimdm 44085 | A sequence of extended rea... |
| xlimfun 44086 | The convergence relation o... |
| xlimmnflimsup 44087 | If a sequence of extended ... |
| xlimdm 44088 | Two ways to express that a... |
| xlimpnfxnegmnf2 44089 | A sequence converges to ` ... |
| xlimresdm 44090 | A function converges in th... |
| xlimpnfliminf 44091 | If a sequence of extended ... |
| xlimpnfliminf2 44092 | A sequence of extended rea... |
| xlimliminflimsup 44093 | A sequence of extended rea... |
| xlimlimsupleliminf 44094 | A sequence of extended rea... |
| coseq0 44095 | A complex number whose cos... |
| sinmulcos 44096 | Multiplication formula for... |
| coskpi2 44097 | The cosine of an integer m... |
| cosnegpi 44098 | The cosine of negative ` _... |
| sinaover2ne0 44099 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 44100 | The cosine of an integer m... |
| mulcncff 44101 | The multiplication of two ... |
| cncfmptssg 44102 | A continuous complex funct... |
| constcncfg 44103 | A constant function is a c... |
| idcncfg 44104 | The identity function is a... |
| cncfshift 44105 | A periodic continuous func... |
| resincncf 44106 | ` sin ` restricted to real... |
| addccncf2 44107 | Adding a constant is a con... |
| 0cnf 44108 | The empty set is a continu... |
| fsumcncf 44109 | The finite sum of continuo... |
| cncfperiod 44110 | A periodic continuous func... |
| subcncff 44111 | The subtraction of two con... |
| negcncfg 44112 | The opposite of a continuo... |
| cnfdmsn 44113 | A function with a singleto... |
| cncfcompt 44114 | Composition of continuous ... |
| addcncff 44115 | The sum of two continuous ... |
| ioccncflimc 44116 | Limit at the upper bound o... |
| cncfuni 44117 | A complex function on a su... |
| icccncfext 44118 | A continuous function on a... |
| cncficcgt0 44119 | A the absolute value of a ... |
| icocncflimc 44120 | Limit at the lower bound, ... |
| cncfdmsn 44121 | A complex function with a ... |
| divcncff 44122 | The quotient of two contin... |
| cncfshiftioo 44123 | A periodic continuous func... |
| cncfiooicclem1 44124 | A continuous function ` F ... |
| cncfiooicc 44125 | A continuous function ` F ... |
| cncfiooiccre 44126 | A continuous function ` F ... |
| cncfioobdlem 44127 | ` G ` actually extends ` F... |
| cncfioobd 44128 | A continuous function ` F ... |
| jumpncnp 44129 | Jump discontinuity or disc... |
| cxpcncf2 44130 | The complex power function... |
| fprodcncf 44131 | The finite product of cont... |
| add1cncf 44132 | Addition to a constant is ... |
| add2cncf 44133 | Addition to a constant is ... |
| sub1cncfd 44134 | Subtracting a constant is ... |
| sub2cncfd 44135 | Subtraction from a constan... |
| fprodsub2cncf 44136 | ` F ` is continuous. (Con... |
| fprodadd2cncf 44137 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 44138 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 44139 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 44140 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 44141 | The sequence ` S ` of fini... |
| dvsinexp 44142 | The derivative of sin^N . ... |
| dvcosre 44143 | The real derivative of the... |
| dvsinax 44144 | Derivative exercise: the d... |
| dvsubf 44145 | The subtraction rule for e... |
| dvmptconst 44146 | Function-builder for deriv... |
| dvcnre 44147 | From complex differentiati... |
| dvmptidg 44148 | Function-builder for deriv... |
| dvresntr 44149 | Function-builder for deriv... |
| fperdvper 44150 | The derivative of a period... |
| dvasinbx 44151 | Derivative exercise: the d... |
| dvresioo 44152 | Restriction of a derivativ... |
| dvdivf 44153 | The quotient rule for ever... |
| dvdivbd 44154 | A sufficient condition for... |
| dvsubcncf 44155 | A sufficient condition for... |
| dvmulcncf 44156 | A sufficient condition for... |
| dvcosax 44157 | Derivative exercise: the d... |
| dvdivcncf 44158 | A sufficient condition for... |
| dvbdfbdioolem1 44159 | Given a function with boun... |
| dvbdfbdioolem2 44160 | A function on an open inte... |
| dvbdfbdioo 44161 | A function on an open inte... |
| ioodvbdlimc1lem1 44162 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 44163 | Limit at the lower bound o... |
| ioodvbdlimc1 44164 | A real function with bound... |
| ioodvbdlimc2lem 44165 | Limit at the upper bound o... |
| ioodvbdlimc2 44166 | A real function with bound... |
| dvdmsscn 44167 | ` X ` is a subset of ` CC ... |
| dvmptmulf 44168 | Function-builder for deriv... |
| dvnmptdivc 44169 | Function-builder for itera... |
| dvdsn1add 44170 | If ` K ` divides ` N ` but... |
| dvxpaek 44171 | Derivative of the polynomi... |
| dvnmptconst 44172 | The ` N ` -th derivative o... |
| dvnxpaek 44173 | The ` n ` -th derivative o... |
| dvnmul 44174 | Function-builder for the `... |
| dvmptfprodlem 44175 | Induction step for ~ dvmpt... |
| dvmptfprod 44176 | Function-builder for deriv... |
| dvnprodlem1 44177 | ` D ` is bijective. (Cont... |
| dvnprodlem2 44178 | Induction step for ~ dvnpr... |
| dvnprodlem3 44179 | The multinomial formula fo... |
| dvnprod 44180 | The multinomial formula fo... |
| itgsin0pilem1 44181 | Calculation of the integra... |
| ibliccsinexp 44182 | sin^n on a closed interval... |
| itgsin0pi 44183 | Calculation of the integra... |
| iblioosinexp 44184 | sin^n on an open integral ... |
| itgsinexplem1 44185 | Integration by parts is ap... |
| itgsinexp 44186 | A recursive formula for th... |
| iblconstmpt 44187 | A constant function is int... |
| itgeq1d 44188 | Equality theorem for an in... |
| mbfres2cn 44189 | Measurability of a piecewi... |
| vol0 44190 | The measure of the empty s... |
| ditgeqiooicc 44191 | A function ` F ` on an ope... |
| volge0 44192 | The volume of a set is alw... |
| cnbdibl 44193 | A continuous bounded funct... |
| snmbl 44194 | A singleton is measurable.... |
| ditgeq3d 44195 | Equality theorem for the d... |
| iblempty 44196 | The empty function is inte... |
| iblsplit 44197 | The union of two integrabl... |
| volsn 44198 | A singleton has 0 Lebesgue... |
| itgvol0 44199 | If the domani is negligibl... |
| itgcoscmulx 44200 | Exercise: the integral of ... |
| iblsplitf 44201 | A version of ~ iblsplit us... |
| ibliooicc 44202 | If a function is integrabl... |
| volioc 44203 | The measure of a left-open... |
| iblspltprt 44204 | If a function is integrabl... |
| itgsincmulx 44205 | Exercise: the integral of ... |
| itgsubsticclem 44206 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 44207 | Integration by u-substitut... |
| itgioocnicc 44208 | The integral of a piecewis... |
| iblcncfioo 44209 | A continuous function ` F ... |
| itgspltprt 44210 | The ` S. ` integral splits... |
| itgiccshift 44211 | The integral of a function... |
| itgperiod 44212 | The integral of a periodic... |
| itgsbtaddcnst 44213 | Integral substitution, add... |
| volico 44214 | The measure of left-closed... |
| sublevolico 44215 | The Lebesgue measure of a ... |
| dmvolss 44216 | Lebesgue measurable sets a... |
| ismbl3 44217 | The predicate " ` A ` is L... |
| volioof 44218 | The function that assigns ... |
| ovolsplit 44219 | The Lebesgue outer measure... |
| fvvolioof 44220 | The function value of the ... |
| volioore 44221 | The measure of an open int... |
| fvvolicof 44222 | The function value of the ... |
| voliooico 44223 | An open interval and a lef... |
| ismbl4 44224 | The predicate " ` A ` is L... |
| volioofmpt 44225 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 44226 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 44227 | The Lebesgue measure of op... |
| volicofmpt 44228 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 44229 | The Lebesgue measure of a ... |
| voliccico 44230 | A closed interval and a le... |
| mbfdmssre 44231 | The domain of a measurable... |
| stoweidlem1 44232 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 44233 | lemma for ~ stoweid : here... |
| stoweidlem3 44234 | Lemma for ~ stoweid : if `... |
| stoweidlem4 44235 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 44236 | There exists a δ as ... |
| stoweidlem6 44237 | Lemma for ~ stoweid : two ... |
| stoweidlem7 44238 | This lemma is used to prov... |
| stoweidlem8 44239 | Lemma for ~ stoweid : two ... |
| stoweidlem9 44240 | Lemma for ~ stoweid : here... |
| stoweidlem10 44241 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 44242 | This lemma is used to prov... |
| stoweidlem12 44243 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 44244 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 44245 | There exists a ` k ` as in... |
| stoweidlem15 44246 | This lemma is used to prov... |
| stoweidlem16 44247 | Lemma for ~ stoweid . The... |
| stoweidlem17 44248 | This lemma proves that the... |
| stoweidlem18 44249 | This theorem proves Lemma ... |
| stoweidlem19 44250 | If a set of real functions... |
| stoweidlem20 44251 | If a set A of real functio... |
| stoweidlem21 44252 | Once the Stone Weierstrass... |
| stoweidlem22 44253 | If a set of real functions... |
| stoweidlem23 44254 | This lemma is used to prov... |
| stoweidlem24 44255 | This lemma proves that for... |
| stoweidlem25 44256 | This lemma proves that for... |
| stoweidlem26 44257 | This lemma is used to prov... |
| stoweidlem27 44258 | This lemma is used to prov... |
| stoweidlem28 44259 | There exists a δ as ... |
| stoweidlem29 44260 | When the hypothesis for th... |
| stoweidlem30 44261 | This lemma is used to prov... |
| stoweidlem31 44262 | This lemma is used to prov... |
| stoweidlem32 44263 | If a set A of real functio... |
| stoweidlem33 44264 | If a set of real functions... |
| stoweidlem34 44265 | This lemma proves that for... |
| stoweidlem35 44266 | This lemma is used to prov... |
| stoweidlem36 44267 | This lemma is used to prov... |
| stoweidlem37 44268 | This lemma is used to prov... |
| stoweidlem38 44269 | This lemma is used to prov... |
| stoweidlem39 44270 | This lemma is used to prov... |
| stoweidlem40 44271 | This lemma proves that q_n... |
| stoweidlem41 44272 | This lemma is used to prov... |
| stoweidlem42 44273 | This lemma is used to prov... |
| stoweidlem43 44274 | This lemma is used to prov... |
| stoweidlem44 44275 | This lemma is used to prov... |
| stoweidlem45 44276 | This lemma proves that, gi... |
| stoweidlem46 44277 | This lemma proves that set... |
| stoweidlem47 44278 | Subtracting a constant fro... |
| stoweidlem48 44279 | This lemma is used to prov... |
| stoweidlem49 44280 | There exists a function q_... |
| stoweidlem50 44281 | This lemma proves that set... |
| stoweidlem51 44282 | There exists a function x ... |
| stoweidlem52 44283 | There exists a neighborhoo... |
| stoweidlem53 44284 | This lemma is used to prov... |
| stoweidlem54 44285 | There exists a function ` ... |
| stoweidlem55 44286 | This lemma proves the exis... |
| stoweidlem56 44287 | This theorem proves Lemma ... |
| stoweidlem57 44288 | There exists a function x ... |
| stoweidlem58 44289 | This theorem proves Lemma ... |
| stoweidlem59 44290 | This lemma proves that the... |
| stoweidlem60 44291 | This lemma proves that the... |
| stoweidlem61 44292 | This lemma proves that the... |
| stoweidlem62 44293 | This theorem proves the St... |
| stoweid 44294 | This theorem proves the St... |
| stowei 44295 | This theorem proves the St... |
| wallispilem1 44296 | ` I ` is monotone: increas... |
| wallispilem2 44297 | A first set of properties ... |
| wallispilem3 44298 | I maps to real values. (C... |
| wallispilem4 44299 | ` F ` maps to explicit exp... |
| wallispilem5 44300 | The sequence ` H ` converg... |
| wallispi 44301 | Wallis' formula for π :... |
| wallispi2lem1 44302 | An intermediate step betwe... |
| wallispi2lem2 44303 | Two expressions are proven... |
| wallispi2 44304 | An alternative version of ... |
| stirlinglem1 44305 | A simple limit of fraction... |
| stirlinglem2 44306 | ` A ` maps to positive rea... |
| stirlinglem3 44307 | Long but simple algebraic ... |
| stirlinglem4 44308 | Algebraic manipulation of ... |
| stirlinglem5 44309 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 44310 | A series that converges to... |
| stirlinglem7 44311 | Algebraic manipulation of ... |
| stirlinglem8 44312 | If ` A ` converges to ` C ... |
| stirlinglem9 44313 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 44314 | A bound for any B(N)-B(N +... |
| stirlinglem11 44315 | ` B ` is decreasing. (Con... |
| stirlinglem12 44316 | The sequence ` B ` is boun... |
| stirlinglem13 44317 | ` B ` is decreasing and ha... |
| stirlinglem14 44318 | The sequence ` A ` converg... |
| stirlinglem15 44319 | The Stirling's formula is ... |
| stirling 44320 | Stirling's approximation f... |
| stirlingr 44321 | Stirling's approximation f... |
| dirkerval 44322 | The N_th Dirichlet Kernel.... |
| dirker2re 44323 | The Dirichlet Kernel value... |
| dirkerdenne0 44324 | The Dirichlet Kernel denom... |
| dirkerval2 44325 | The N_th Dirichlet Kernel ... |
| dirkerre 44326 | The Dirichlet Kernel at an... |
| dirkerper 44327 | the Dirichlet Kernel has p... |
| dirkerf 44328 | For any natural number ` N... |
| dirkertrigeqlem1 44329 | Sum of an even number of a... |
| dirkertrigeqlem2 44330 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 44331 | Trigonometric equality lem... |
| dirkertrigeq 44332 | Trigonometric equality for... |
| dirkeritg 44333 | The definite integral of t... |
| dirkercncflem1 44334 | If ` Y ` is a multiple of ... |
| dirkercncflem2 44335 | Lemma used to prove that t... |
| dirkercncflem3 44336 | The Dirichlet Kernel is co... |
| dirkercncflem4 44337 | The Dirichlet Kernel is co... |
| dirkercncf 44338 | For any natural number ` N... |
| fourierdlem1 44339 | A partition interval is a ... |
| fourierdlem2 44340 | Membership in a partition.... |
| fourierdlem3 44341 | Membership in a partition.... |
| fourierdlem4 44342 | ` E ` is a function that m... |
| fourierdlem5 44343 | ` S ` is a function. (Con... |
| fourierdlem6 44344 | ` X ` is in the periodic p... |
| fourierdlem7 44345 | The difference between the... |
| fourierdlem8 44346 | A partition interval is a ... |
| fourierdlem9 44347 | ` H ` is a complex functio... |
| fourierdlem10 44348 | Condition on the bounds of... |
| fourierdlem11 44349 | If there is a partition, t... |
| fourierdlem12 44350 | A point of a partition is ... |
| fourierdlem13 44351 | Value of ` V ` in terms of... |
| fourierdlem14 44352 | Given the partition ` V ` ... |
| fourierdlem15 44353 | The range of the partition... |
| fourierdlem16 44354 | The coefficients of the fo... |
| fourierdlem17 44355 | The defined ` L ` is actua... |
| fourierdlem18 44356 | The function ` S ` is cont... |
| fourierdlem19 44357 | If two elements of ` D ` h... |
| fourierdlem20 44358 | Every interval in the part... |
| fourierdlem21 44359 | The coefficients of the fo... |
| fourierdlem22 44360 | The coefficients of the fo... |
| fourierdlem23 44361 | If ` F ` is continuous and... |
| fourierdlem24 44362 | A sufficient condition for... |
| fourierdlem25 44363 | If ` C ` is not in the ran... |
| fourierdlem26 44364 | Periodic image of a point ... |
| fourierdlem27 44365 | A partition open interval ... |
| fourierdlem28 44366 | Derivative of ` ( F `` ( X... |
| fourierdlem29 44367 | Explicit function value fo... |
| fourierdlem30 44368 | Sum of three small pieces ... |
| fourierdlem31 44369 | If ` A ` is finite and for... |
| fourierdlem32 44370 | Limit of a continuous func... |
| fourierdlem33 44371 | Limit of a continuous func... |
| fourierdlem34 44372 | A partition is one to one.... |
| fourierdlem35 44373 | There is a single point in... |
| fourierdlem36 44374 | ` F ` is an isomorphism. ... |
| fourierdlem37 44375 | ` I ` is a function that m... |
| fourierdlem38 44376 | The function ` F ` is cont... |
| fourierdlem39 44377 | Integration by parts of ... |
| fourierdlem40 44378 | ` H ` is a continuous func... |
| fourierdlem41 44379 | Lemma used to prove that e... |
| fourierdlem42 44380 | The set of points in a mov... |
| fourierdlem43 44381 | ` K ` is a real function. ... |
| fourierdlem44 44382 | A condition for having ` (... |
| fourierdlem46 44383 | The function ` F ` has a l... |
| fourierdlem47 44384 | For ` r ` large enough, th... |
| fourierdlem48 44385 | The given periodic functio... |
| fourierdlem49 44386 | The given periodic functio... |
| fourierdlem50 44387 | Continuity of ` O ` and it... |
| fourierdlem51 44388 | ` X ` is in the periodic p... |
| fourierdlem52 44389 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 44390 | The limit of ` F ( s ) ` a... |
| fourierdlem54 44391 | Given a partition ` Q ` an... |
| fourierdlem55 44392 | ` U ` is a real function. ... |
| fourierdlem56 44393 | Derivative of the ` K ` fu... |
| fourierdlem57 44394 | The derivative of ` O ` . ... |
| fourierdlem58 44395 | The derivative of ` K ` is... |
| fourierdlem59 44396 | The derivative of ` H ` is... |
| fourierdlem60 44397 | Given a differentiable fun... |
| fourierdlem61 44398 | Given a differentiable fun... |
| fourierdlem62 44399 | The function ` K ` is cont... |
| fourierdlem63 44400 | The upper bound of interva... |
| fourierdlem64 44401 | The partition ` V ` is fin... |
| fourierdlem65 44402 | The distance of two adjace... |
| fourierdlem66 44403 | Value of the ` G ` functio... |
| fourierdlem67 44404 | ` G ` is a function. (Con... |
| fourierdlem68 44405 | The derivative of ` O ` is... |
| fourierdlem69 44406 | A piecewise continuous fun... |
| fourierdlem70 44407 | A piecewise continuous fun... |
| fourierdlem71 44408 | A periodic piecewise conti... |
| fourierdlem72 44409 | The derivative of ` O ` is... |
| fourierdlem73 44410 | A version of the Riemann L... |
| fourierdlem74 44411 | Given a piecewise smooth f... |
| fourierdlem75 44412 | Given a piecewise smooth f... |
| fourierdlem76 44413 | Continuity of ` O ` and it... |
| fourierdlem77 44414 | If ` H ` is bounded, then ... |
| fourierdlem78 44415 | ` G ` is continuous when r... |
| fourierdlem79 44416 | ` E ` projects every inter... |
| fourierdlem80 44417 | The derivative of ` O ` is... |
| fourierdlem81 44418 | The integral of a piecewis... |
| fourierdlem82 44419 | Integral by substitution, ... |
| fourierdlem83 44420 | The fourier partial sum fo... |
| fourierdlem84 44421 | If ` F ` is piecewise coni... |
| fourierdlem85 44422 | Limit of the function ` G ... |
| fourierdlem86 44423 | Continuity of ` O ` and it... |
| fourierdlem87 44424 | The integral of ` G ` goes... |
| fourierdlem88 44425 | Given a piecewise continuo... |
| fourierdlem89 44426 | Given a piecewise continuo... |
| fourierdlem90 44427 | Given a piecewise continuo... |
| fourierdlem91 44428 | Given a piecewise continuo... |
| fourierdlem92 44429 | The integral of a piecewis... |
| fourierdlem93 44430 | Integral by substitution (... |
| fourierdlem94 44431 | For a piecewise smooth fun... |
| fourierdlem95 44432 | Algebraic manipulation of ... |
| fourierdlem96 44433 | limit for ` F ` at the low... |
| fourierdlem97 44434 | ` F ` is continuous on the... |
| fourierdlem98 44435 | ` F ` is continuous on the... |
| fourierdlem99 44436 | limit for ` F ` at the upp... |
| fourierdlem100 44437 | A piecewise continuous fun... |
| fourierdlem101 44438 | Integral by substitution f... |
| fourierdlem102 44439 | For a piecewise smooth fun... |
| fourierdlem103 44440 | The half lower part of the... |
| fourierdlem104 44441 | The half upper part of the... |
| fourierdlem105 44442 | A piecewise continuous fun... |
| fourierdlem106 44443 | For a piecewise smooth fun... |
| fourierdlem107 44444 | The integral of a piecewis... |
| fourierdlem108 44445 | The integral of a piecewis... |
| fourierdlem109 44446 | The integral of a piecewis... |
| fourierdlem110 44447 | The integral of a piecewis... |
| fourierdlem111 44448 | The fourier partial sum fo... |
| fourierdlem112 44449 | Here abbreviations (local ... |
| fourierdlem113 44450 | Fourier series convergence... |
| fourierdlem114 44451 | Fourier series convergence... |
| fourierdlem115 44452 | Fourier serier convergence... |
| fourierd 44453 | Fourier series convergence... |
| fourierclimd 44454 | Fourier series convergence... |
| fourierclim 44455 | Fourier series convergence... |
| fourier 44456 | Fourier series convergence... |
| fouriercnp 44457 | If ` F ` is continuous at ... |
| fourier2 44458 | Fourier series convergence... |
| sqwvfoura 44459 | Fourier coefficients for t... |
| sqwvfourb 44460 | Fourier series ` B ` coeff... |
| fourierswlem 44461 | The Fourier series for the... |
| fouriersw 44462 | Fourier series convergence... |
| fouriercn 44463 | If the derivative of ` F `... |
| elaa2lem 44464 | Elementhood in the set of ... |
| elaa2 44465 | Elementhood in the set of ... |
| etransclem1 44466 | ` H ` is a function. (Con... |
| etransclem2 44467 | Derivative of ` G ` . (Co... |
| etransclem3 44468 | The given ` if ` term is a... |
| etransclem4 44469 | ` F ` expressed as a finit... |
| etransclem5 44470 | A change of bound variable... |
| etransclem6 44471 | A change of bound variable... |
| etransclem7 44472 | The given product is an in... |
| etransclem8 44473 | ` F ` is a function. (Con... |
| etransclem9 44474 | If ` K ` divides ` N ` but... |
| etransclem10 44475 | The given ` if ` term is a... |
| etransclem11 44476 | A change of bound variable... |
| etransclem12 44477 | ` C ` applied to ` N ` . ... |
| etransclem13 44478 | ` F ` applied to ` Y ` . ... |
| etransclem14 44479 | Value of the term ` T ` , ... |
| etransclem15 44480 | Value of the term ` T ` , ... |
| etransclem16 44481 | Every element in the range... |
| etransclem17 44482 | The ` N ` -th derivative o... |
| etransclem18 44483 | The given function is inte... |
| etransclem19 44484 | The ` N ` -th derivative o... |
| etransclem20 44485 | ` H ` is smooth. (Contrib... |
| etransclem21 44486 | The ` N ` -th derivative o... |
| etransclem22 44487 | The ` N ` -th derivative o... |
| etransclem23 44488 | This is the claim proof in... |
| etransclem24 44489 | ` P ` divides the I -th de... |
| etransclem25 44490 | ` P ` factorial divides th... |
| etransclem26 44491 | Every term in the sum of t... |
| etransclem27 44492 | The ` N ` -th derivative o... |
| etransclem28 44493 | ` ( P - 1 ) ` factorial di... |
| etransclem29 44494 | The ` N ` -th derivative o... |
| etransclem30 44495 | The ` N ` -th derivative o... |
| etransclem31 44496 | The ` N ` -th derivative o... |
| etransclem32 44497 | This is the proof for the ... |
| etransclem33 44498 | ` F ` is smooth. (Contrib... |
| etransclem34 44499 | The ` N ` -th derivative o... |
| etransclem35 44500 | ` P ` does not divide the ... |
| etransclem36 44501 | The ` N ` -th derivative o... |
| etransclem37 44502 | ` ( P - 1 ) ` factorial di... |
| etransclem38 44503 | ` P ` divides the I -th de... |
| etransclem39 44504 | ` G ` is a function. (Con... |
| etransclem40 44505 | The ` N ` -th derivative o... |
| etransclem41 44506 | ` P ` does not divide the ... |
| etransclem42 44507 | The ` N ` -th derivative o... |
| etransclem43 44508 | ` G ` is a continuous func... |
| etransclem44 44509 | The given finite sum is no... |
| etransclem45 44510 | ` K ` is an integer. (Con... |
| etransclem46 44511 | This is the proof for equa... |
| etransclem47 44512 | ` _e ` is transcendental. ... |
| etransclem48 44513 | ` _e ` is transcendental. ... |
| etransc 44514 | ` _e ` is transcendental. ... |
| rrxtopn 44515 | The topology of the genera... |
| rrxngp 44516 | Generalized Euclidean real... |
| rrxtps 44517 | Generalized Euclidean real... |
| rrxtopnfi 44518 | The topology of the n-dime... |
| rrxtopon 44519 | The topology on generalize... |
| rrxtop 44520 | The topology on generalize... |
| rrndistlt 44521 | Given two points in the sp... |
| rrxtoponfi 44522 | The topology on n-dimensio... |
| rrxunitopnfi 44523 | The base set of the standa... |
| rrxtopn0 44524 | The topology of the zero-d... |
| qndenserrnbllem 44525 | n-dimensional rational num... |
| qndenserrnbl 44526 | n-dimensional rational num... |
| rrxtopn0b 44527 | The topology of the zero-d... |
| qndenserrnopnlem 44528 | n-dimensional rational num... |
| qndenserrnopn 44529 | n-dimensional rational num... |
| qndenserrn 44530 | n-dimensional rational num... |
| rrxsnicc 44531 | A multidimensional singlet... |
| rrnprjdstle 44532 | The distance between two p... |
| rrndsmet 44533 | ` D ` is a metric for the ... |
| rrndsxmet 44534 | ` D ` is an extended metri... |
| ioorrnopnlem 44535 | The a point in an indexed ... |
| ioorrnopn 44536 | The indexed product of ope... |
| ioorrnopnxrlem 44537 | Given a point ` F ` that b... |
| ioorrnopnxr 44538 | The indexed product of ope... |
| issal 44545 | Express the predicate " ` ... |
| pwsal 44546 | The power set of a given s... |
| salunicl 44547 | SAlg sigma-algebra is clos... |
| saluncl 44548 | The union of two sets in a... |
| prsal 44549 | The pair of the empty set ... |
| saldifcl 44550 | The complement of an eleme... |
| 0sal 44551 | The empty set belongs to e... |
| salgenval 44552 | The sigma-algebra generate... |
| saliunclf 44553 | SAlg sigma-algebra is clos... |
| saliuncl 44554 | SAlg sigma-algebra is clos... |
| salincl 44555 | The intersection of two se... |
| saluni 44556 | A set is an element of any... |
| saliinclf 44557 | SAlg sigma-algebra is clos... |
| saliincl 44558 | SAlg sigma-algebra is clos... |
| saldifcl2 44559 | The difference of two elem... |
| intsaluni 44560 | The union of an arbitrary ... |
| intsal 44561 | The arbitrary intersection... |
| salgenn0 44562 | The set used in the defini... |
| salgencl 44563 | ` SalGen ` actually genera... |
| issald 44564 | Sufficient condition to pr... |
| salexct 44565 | An example of nontrivial s... |
| sssalgen 44566 | A set is a subset of the s... |
| salgenss 44567 | The sigma-algebra generate... |
| salgenuni 44568 | The base set of the sigma-... |
| issalgend 44569 | One side of ~ dfsalgen2 . ... |
| salexct2 44570 | An example of a subset tha... |
| unisalgen 44571 | The union of a set belongs... |
| dfsalgen2 44572 | Alternate characterization... |
| salexct3 44573 | An example of a sigma-alge... |
| salgencntex 44574 | This counterexample shows ... |
| salgensscntex 44575 | This counterexample shows ... |
| issalnnd 44576 | Sufficient condition to pr... |
| dmvolsal 44577 | Lebesgue measurable sets f... |
| saldifcld 44578 | The complement of an eleme... |
| saluncld 44579 | The union of two sets in a... |
| salgencld 44580 | ` SalGen ` actually genera... |
| 0sald 44581 | The empty set belongs to e... |
| iooborel 44582 | An open interval is a Bore... |
| salincld 44583 | The intersection of two se... |
| salunid 44584 | A set is an element of any... |
| unisalgen2 44585 | The union of a set belongs... |
| bor1sal 44586 | The Borel sigma-algebra on... |
| iocborel 44587 | A left-open, right-closed ... |
| subsaliuncllem 44588 | A subspace sigma-algebra i... |
| subsaliuncl 44589 | A subspace sigma-algebra i... |
| subsalsal 44590 | A subspace sigma-algebra i... |
| subsaluni 44591 | A set belongs to the subsp... |
| salrestss 44592 | A sigma-algebra restricted... |
| sge0rnre 44595 | When ` sum^ ` is applied t... |
| fge0icoicc 44596 | If ` F ` maps to nonnegati... |
| sge0val 44597 | The value of the sum of no... |
| fge0npnf 44598 | If ` F ` maps to nonnegati... |
| sge0rnn0 44599 | The range used in the defi... |
| sge0vald 44600 | The value of the sum of no... |
| fge0iccico 44601 | A range of nonnegative ext... |
| gsumge0cl 44602 | Closure of group sum, for ... |
| sge0reval 44603 | Value of the sum of nonneg... |
| sge0pnfval 44604 | If a term in the sum of no... |
| fge0iccre 44605 | A range of nonnegative ext... |
| sge0z 44606 | Any nonnegative extended s... |
| sge00 44607 | The sum of nonnegative ext... |
| fsumlesge0 44608 | Every finite subsum of non... |
| sge0revalmpt 44609 | Value of the sum of nonneg... |
| sge0sn 44610 | A sum of a nonnegative ext... |
| sge0tsms 44611 | ` sum^ ` applied to a nonn... |
| sge0cl 44612 | The arbitrary sum of nonne... |
| sge0f1o 44613 | Re-index a nonnegative ext... |
| sge0snmpt 44614 | A sum of a nonnegative ext... |
| sge0ge0 44615 | The sum of nonnegative ext... |
| sge0xrcl 44616 | The arbitrary sum of nonne... |
| sge0repnf 44617 | The of nonnegative extende... |
| sge0fsum 44618 | The arbitrary sum of a fin... |
| sge0rern 44619 | If the sum of nonnegative ... |
| sge0supre 44620 | If the arbitrary sum of no... |
| sge0fsummpt 44621 | The arbitrary sum of a fin... |
| sge0sup 44622 | The arbitrary sum of nonne... |
| sge0less 44623 | A shorter sum of nonnegati... |
| sge0rnbnd 44624 | The range used in the defi... |
| sge0pr 44625 | Sum of a pair of nonnegati... |
| sge0gerp 44626 | The arbitrary sum of nonne... |
| sge0pnffigt 44627 | If the sum of nonnegative ... |
| sge0ssre 44628 | If a sum of nonnegative ex... |
| sge0lefi 44629 | A sum of nonnegative exten... |
| sge0lessmpt 44630 | A shorter sum of nonnegati... |
| sge0ltfirp 44631 | If the sum of nonnegative ... |
| sge0prle 44632 | The sum of a pair of nonne... |
| sge0gerpmpt 44633 | The arbitrary sum of nonne... |
| sge0resrnlem 44634 | The sum of nonnegative ext... |
| sge0resrn 44635 | The sum of nonnegative ext... |
| sge0ssrempt 44636 | If a sum of nonnegative ex... |
| sge0resplit 44637 | ` sum^ ` splits into two p... |
| sge0le 44638 | If all of the terms of sum... |
| sge0ltfirpmpt 44639 | If the extended sum of non... |
| sge0split 44640 | Split a sum of nonnegative... |
| sge0lempt 44641 | If all of the terms of sum... |
| sge0splitmpt 44642 | Split a sum of nonnegative... |
| sge0ss 44643 | Change the index set to a ... |
| sge0iunmptlemfi 44644 | Sum of nonnegative extende... |
| sge0p1 44645 | The addition of the next t... |
| sge0iunmptlemre 44646 | Sum of nonnegative extende... |
| sge0fodjrnlem 44647 | Re-index a nonnegative ext... |
| sge0fodjrn 44648 | Re-index a nonnegative ext... |
| sge0iunmpt 44649 | Sum of nonnegative extende... |
| sge0iun 44650 | Sum of nonnegative extende... |
| sge0nemnf 44651 | The generalized sum of non... |
| sge0rpcpnf 44652 | The sum of an infinite num... |
| sge0rernmpt 44653 | If the sum of nonnegative ... |
| sge0lefimpt 44654 | A sum of nonnegative exten... |
| nn0ssge0 44655 | Nonnegative integers are n... |
| sge0clmpt 44656 | The generalized sum of non... |
| sge0ltfirpmpt2 44657 | If the extended sum of non... |
| sge0isum 44658 | If a series of nonnegative... |
| sge0xrclmpt 44659 | The generalized sum of non... |
| sge0xp 44660 | Combine two generalized su... |
| sge0isummpt 44661 | If a series of nonnegative... |
| sge0ad2en 44662 | The value of the infinite ... |
| sge0isummpt2 44663 | If a series of nonnegative... |
| sge0xaddlem1 44664 | The extended addition of t... |
| sge0xaddlem2 44665 | The extended addition of t... |
| sge0xadd 44666 | The extended addition of t... |
| sge0fsummptf 44667 | The generalized sum of a f... |
| sge0snmptf 44668 | A sum of a nonnegative ext... |
| sge0ge0mpt 44669 | The sum of nonnegative ext... |
| sge0repnfmpt 44670 | The of nonnegative extende... |
| sge0pnffigtmpt 44671 | If the generalized sum of ... |
| sge0splitsn 44672 | Separate out a term in a g... |
| sge0pnffsumgt 44673 | If the sum of nonnegative ... |
| sge0gtfsumgt 44674 | If the generalized sum of ... |
| sge0uzfsumgt 44675 | If a real number is smalle... |
| sge0pnfmpt 44676 | If a term in the sum of no... |
| sge0seq 44677 | A series of nonnegative re... |
| sge0reuz 44678 | Value of the generalized s... |
| sge0reuzb 44679 | Value of the generalized s... |
| ismea 44682 | Express the predicate " ` ... |
| dmmeasal 44683 | The domain of a measure is... |
| meaf 44684 | A measure is a function th... |
| mea0 44685 | The measure of the empty s... |
| nnfoctbdjlem 44686 | There exists a mapping fro... |
| nnfoctbdj 44687 | There exists a mapping fro... |
| meadjuni 44688 | The measure of the disjoin... |
| meacl 44689 | The measure of a set is a ... |
| iundjiunlem 44690 | The sets in the sequence `... |
| iundjiun 44691 | Given a sequence ` E ` of ... |
| meaxrcl 44692 | The measure of a set is an... |
| meadjun 44693 | The measure of the union o... |
| meassle 44694 | The measure of a set is gr... |
| meaunle 44695 | The measure of the union o... |
| meadjiunlem 44696 | The sum of nonnegative ext... |
| meadjiun 44697 | The measure of the disjoin... |
| ismeannd 44698 | Sufficient condition to pr... |
| meaiunlelem 44699 | The measure of the union o... |
| meaiunle 44700 | The measure of the union o... |
| psmeasurelem 44701 | ` M ` applied to a disjoin... |
| psmeasure 44702 | Point supported measure, R... |
| voliunsge0lem 44703 | The Lebesgue measure funct... |
| voliunsge0 44704 | The Lebesgue measure funct... |
| volmea 44705 | The Lebeasgue measure on t... |
| meage0 44706 | If the measure of a measur... |
| meadjunre 44707 | The measure of the union o... |
| meassre 44708 | If the measure of a measur... |
| meale0eq0 44709 | A measure that is less tha... |
| meadif 44710 | The measure of the differe... |
| meaiuninclem 44711 | Measures are continuous fr... |
| meaiuninc 44712 | Measures are continuous fr... |
| meaiuninc2 44713 | Measures are continuous fr... |
| meaiunincf 44714 | Measures are continuous fr... |
| meaiuninc3v 44715 | Measures are continuous fr... |
| meaiuninc3 44716 | Measures are continuous fr... |
| meaiininclem 44717 | Measures are continuous fr... |
| meaiininc 44718 | Measures are continuous fr... |
| meaiininc2 44719 | Measures are continuous fr... |
| caragenval 44724 | The sigma-algebra generate... |
| isome 44725 | Express the predicate " ` ... |
| caragenel 44726 | Membership in the Caratheo... |
| omef 44727 | An outer measure is a func... |
| ome0 44728 | The outer measure of the e... |
| omessle 44729 | The outer measure of a set... |
| omedm 44730 | The domain of an outer mea... |
| caragensplit 44731 | If ` E ` is in the set gen... |
| caragenelss 44732 | An element of the Caratheo... |
| carageneld 44733 | Membership in the Caratheo... |
| omecl 44734 | The outer measure of a set... |
| caragenss 44735 | The sigma-algebra generate... |
| omeunile 44736 | The outer measure of the u... |
| caragen0 44737 | The empty set belongs to a... |
| omexrcl 44738 | The outer measure of a set... |
| caragenunidm 44739 | The base set of an outer m... |
| caragensspw 44740 | The sigma-algebra generate... |
| omessre 44741 | If the outer measure of a ... |
| caragenuni 44742 | The base set of the sigma-... |
| caragenuncllem 44743 | The Caratheodory's constru... |
| caragenuncl 44744 | The Caratheodory's constru... |
| caragendifcl 44745 | The Caratheodory's constru... |
| caragenfiiuncl 44746 | The Caratheodory's constru... |
| omeunle 44747 | The outer measure of the u... |
| omeiunle 44748 | The outer measure of the i... |
| omelesplit 44749 | The outer measure of a set... |
| omeiunltfirp 44750 | If the outer measure of a ... |
| omeiunlempt 44751 | The outer measure of the i... |
| carageniuncllem1 44752 | The outer measure of ` A i... |
| carageniuncllem2 44753 | The Caratheodory's constru... |
| carageniuncl 44754 | The Caratheodory's constru... |
| caragenunicl 44755 | The Caratheodory's constru... |
| caragensal 44756 | Caratheodory's method gene... |
| caratheodorylem1 44757 | Lemma used to prove that C... |
| caratheodorylem2 44758 | Caratheodory's constructio... |
| caratheodory 44759 | Caratheodory's constructio... |
| 0ome 44760 | The map that assigns 0 to ... |
| isomenndlem 44761 | ` O ` is sub-additive w.r.... |
| isomennd 44762 | Sufficient condition to pr... |
| caragenel2d 44763 | Membership in the Caratheo... |
| omege0 44764 | If the outer measure of a ... |
| omess0 44765 | If the outer measure of a ... |
| caragencmpl 44766 | A measure built with the C... |
| vonval 44771 | Value of the Lebesgue meas... |
| ovnval 44772 | Value of the Lebesgue oute... |
| elhoi 44773 | Membership in a multidimen... |
| icoresmbl 44774 | A closed-below, open-above... |
| hoissre 44775 | The projection of a half-o... |
| ovnval2 44776 | Value of the Lebesgue oute... |
| volicorecl 44777 | The Lebesgue measure of a ... |
| hoiprodcl 44778 | The pre-measure of half-op... |
| hoicvr 44779 | ` I ` is a countable set o... |
| hoissrrn 44780 | A half-open interval is a ... |
| ovn0val 44781 | The Lebesgue outer measure... |
| ovnn0val 44782 | The value of a (multidimen... |
| ovnval2b 44783 | Value of the Lebesgue oute... |
| volicorescl 44784 | The Lebesgue measure of a ... |
| ovnprodcl 44785 | The product used in the de... |
| hoiprodcl2 44786 | The pre-measure of half-op... |
| hoicvrrex 44787 | Any subset of the multidim... |
| ovnsupge0 44788 | The set used in the defini... |
| ovnlecvr 44789 | Given a subset of multidim... |
| ovnpnfelsup 44790 | ` +oo ` is an element of t... |
| ovnsslelem 44791 | The (multidimensional, non... |
| ovnssle 44792 | The (multidimensional) Leb... |
| ovnlerp 44793 | The Lebesgue outer measure... |
| ovnf 44794 | The Lebesgue outer measure... |
| ovncvrrp 44795 | The Lebesgue outer measure... |
| ovn0lem 44796 | For any finite dimension, ... |
| ovn0 44797 | For any finite dimension, ... |
| ovncl 44798 | The Lebesgue outer measure... |
| ovn02 44799 | For the zero-dimensional s... |
| ovnxrcl 44800 | The Lebesgue outer measure... |
| ovnsubaddlem1 44801 | The Lebesgue outer measure... |
| ovnsubaddlem2 44802 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 44803 | ` ( voln* `` X ) ` is suba... |
| ovnome 44804 | ` ( voln* `` X ) ` is an o... |
| vonmea 44805 | ` ( voln `` X ) ` is a mea... |
| volicon0 44806 | The measure of a nonempty ... |
| hsphoif 44807 | ` H ` is a function (that ... |
| hoidmvval 44808 | The dimensional volume of ... |
| hoissrrn2 44809 | A half-open interval is a ... |
| hsphoival 44810 | ` H ` is a function (that ... |
| hoiprodcl3 44811 | The pre-measure of half-op... |
| volicore 44812 | The Lebesgue measure of a ... |
| hoidmvcl 44813 | The dimensional volume of ... |
| hoidmv0val 44814 | The dimensional volume of ... |
| hoidmvn0val 44815 | The dimensional volume of ... |
| hsphoidmvle2 44816 | The dimensional volume of ... |
| hsphoidmvle 44817 | The dimensional volume of ... |
| hoidmvval0 44818 | The dimensional volume of ... |
| hoiprodp1 44819 | The dimensional volume of ... |
| sge0hsphoire 44820 | If the generalized sum of ... |
| hoidmvval0b 44821 | The dimensional volume of ... |
| hoidmv1lelem1 44822 | The supremum of ` U ` belo... |
| hoidmv1lelem2 44823 | This is the contradiction ... |
| hoidmv1lelem3 44824 | The dimensional volume of ... |
| hoidmv1le 44825 | The dimensional volume of ... |
| hoidmvlelem1 44826 | The supremum of ` U ` belo... |
| hoidmvlelem2 44827 | This is the contradiction ... |
| hoidmvlelem3 44828 | This is the contradiction ... |
| hoidmvlelem4 44829 | The dimensional volume of ... |
| hoidmvlelem5 44830 | The dimensional volume of ... |
| hoidmvle 44831 | The dimensional volume of ... |
| ovnhoilem1 44832 | The Lebesgue outer measure... |
| ovnhoilem2 44833 | The Lebesgue outer measure... |
| ovnhoi 44834 | The Lebesgue outer measure... |
| dmovn 44835 | The domain of the Lebesgue... |
| hoicoto2 44836 | The half-open interval exp... |
| dmvon 44837 | Lebesgue measurable n-dime... |
| hoi2toco 44838 | The half-open interval exp... |
| hoidifhspval 44839 | ` D ` is a function that r... |
| hspval 44840 | The value of the half-spac... |
| ovnlecvr2 44841 | Given a subset of multidim... |
| ovncvr2 44842 | ` B ` and ` T ` are the le... |
| dmovnsal 44843 | The domain of the Lebesgue... |
| unidmovn 44844 | Base set of the n-dimensio... |
| rrnmbl 44845 | The set of n-dimensional R... |
| hoidifhspval2 44846 | ` D ` is a function that r... |
| hspdifhsp 44847 | A n-dimensional half-open ... |
| unidmvon 44848 | Base set of the n-dimensio... |
| hoidifhspf 44849 | ` D ` is a function that r... |
| hoidifhspval3 44850 | ` D ` is a function that r... |
| hoidifhspdmvle 44851 | The dimensional volume of ... |
| voncmpl 44852 | The Lebesgue measure is co... |
| hoiqssbllem1 44853 | The center of the n-dimens... |
| hoiqssbllem2 44854 | The center of the n-dimens... |
| hoiqssbllem3 44855 | A n-dimensional ball conta... |
| hoiqssbl 44856 | A n-dimensional ball conta... |
| hspmbllem1 44857 | Any half-space of the n-di... |
| hspmbllem2 44858 | Any half-space of the n-di... |
| hspmbllem3 44859 | Any half-space of the n-di... |
| hspmbl 44860 | Any half-space of the n-di... |
| hoimbllem 44861 | Any n-dimensional half-ope... |
| hoimbl 44862 | Any n-dimensional half-ope... |
| opnvonmbllem1 44863 | The half-open interval exp... |
| opnvonmbllem2 44864 | An open subset of the n-di... |
| opnvonmbl 44865 | An open subset of the n-di... |
| opnssborel 44866 | Open sets of a generalized... |
| borelmbl 44867 | All Borel subsets of the n... |
| volicorege0 44868 | The Lebesgue measure of a ... |
| isvonmbl 44869 | The predicate " ` A ` is m... |
| mblvon 44870 | The n-dimensional Lebesgue... |
| vonmblss 44871 | n-dimensional Lebesgue mea... |
| volico2 44872 | The measure of left-closed... |
| vonmblss2 44873 | n-dimensional Lebesgue mea... |
| ovolval2lem 44874 | The value of the Lebesgue ... |
| ovolval2 44875 | The value of the Lebesgue ... |
| ovnsubadd2lem 44876 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 44877 | ` ( voln* `` X ) ` is suba... |
| ovolval3 44878 | The value of the Lebesgue ... |
| ovnsplit 44879 | The n-dimensional Lebesgue... |
| ovolval4lem1 44880 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 44881 | The value of the Lebesgue ... |
| ovolval4 44882 | The value of the Lebesgue ... |
| ovolval5lem1 44883 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 44884 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 44885 | The value of the Lebesgue ... |
| ovolval5 44886 | The value of the Lebesgue ... |
| ovnovollem1 44887 | if ` F ` is a cover of ` B... |
| ovnovollem2 44888 | if ` I ` is a cover of ` (... |
| ovnovollem3 44889 | The 1-dimensional Lebesgue... |
| ovnovol 44890 | The 1-dimensional Lebesgue... |
| vonvolmbllem 44891 | If a subset ` B ` of real ... |
| vonvolmbl 44892 | A subset of Real numbers i... |
| vonvol 44893 | The 1-dimensional Lebesgue... |
| vonvolmbl2 44894 | A subset ` X ` of the spac... |
| vonvol2 44895 | The 1-dimensional Lebesgue... |
| hoimbl2 44896 | Any n-dimensional half-ope... |
| voncl 44897 | The Lebesgue measure of a ... |
| vonhoi 44898 | The Lebesgue outer measure... |
| vonxrcl 44899 | The Lebesgue measure of a ... |
| ioosshoi 44900 | A n-dimensional open inter... |
| vonn0hoi 44901 | The Lebesgue outer measure... |
| von0val 44902 | The Lebesgue measure (for ... |
| vonhoire 44903 | The Lebesgue measure of a ... |
| iinhoiicclem 44904 | A n-dimensional closed int... |
| iinhoiicc 44905 | A n-dimensional closed int... |
| iunhoiioolem 44906 | A n-dimensional open inter... |
| iunhoiioo 44907 | A n-dimensional open inter... |
| ioovonmbl 44908 | Any n-dimensional open int... |
| iccvonmbllem 44909 | Any n-dimensional closed i... |
| iccvonmbl 44910 | Any n-dimensional closed i... |
| vonioolem1 44911 | The sequence of the measur... |
| vonioolem2 44912 | The n-dimensional Lebesgue... |
| vonioo 44913 | The n-dimensional Lebesgue... |
| vonicclem1 44914 | The sequence of the measur... |
| vonicclem2 44915 | The n-dimensional Lebesgue... |
| vonicc 44916 | The n-dimensional Lebesgue... |
| snvonmbl 44917 | A n-dimensional singleton ... |
| vonn0ioo 44918 | The n-dimensional Lebesgue... |
| vonn0icc 44919 | The n-dimensional Lebesgue... |
| ctvonmbl 44920 | Any n-dimensional countabl... |
| vonn0ioo2 44921 | The n-dimensional Lebesgue... |
| vonsn 44922 | The n-dimensional Lebesgue... |
| vonn0icc2 44923 | The n-dimensional Lebesgue... |
| vonct 44924 | The n-dimensional Lebesgue... |
| vitali2 44925 | There are non-measurable s... |
| pimltmnf2f 44928 | Given a real-valued functi... |
| pimltmnf2 44929 | Given a real-valued functi... |
| preimagelt 44930 | The preimage of a right-op... |
| preimalegt 44931 | The preimage of a left-ope... |
| pimconstlt0 44932 | Given a constant function,... |
| pimconstlt1 44933 | Given a constant function,... |
| pimltpnff 44934 | Given a real-valued functi... |
| pimltpnf 44935 | Given a real-valued functi... |
| pimgtpnf2f 44936 | Given a real-valued functi... |
| pimgtpnf2 44937 | Given a real-valued functi... |
| salpreimagelt 44938 | If all the preimages of le... |
| pimrecltpos 44939 | The preimage of an unbound... |
| salpreimalegt 44940 | If all the preimages of ri... |
| pimiooltgt 44941 | The preimage of an open in... |
| preimaicomnf 44942 | Preimage of an open interv... |
| pimltpnf2f 44943 | Given a real-valued functi... |
| pimltpnf2 44944 | Given a real-valued functi... |
| pimgtmnf2 44945 | Given a real-valued functi... |
| pimdecfgtioc 44946 | Given a nonincreasing func... |
| pimincfltioc 44947 | Given a nondecreasing func... |
| pimdecfgtioo 44948 | Given a nondecreasing func... |
| pimincfltioo 44949 | Given a nondecreasing func... |
| preimaioomnf 44950 | Preimage of an open interv... |
| preimageiingt 44951 | A preimage of a left-close... |
| preimaleiinlt 44952 | A preimage of a left-open,... |
| pimgtmnff 44953 | Given a real-valued functi... |
| pimgtmnf 44954 | Given a real-valued functi... |
| pimrecltneg 44955 | The preimage of an unbound... |
| salpreimagtge 44956 | If all the preimages of le... |
| salpreimaltle 44957 | If all the preimages of ri... |
| issmflem 44958 | The predicate " ` F ` is a... |
| issmf 44959 | The predicate " ` F ` is a... |
| salpreimalelt 44960 | If all the preimages of ri... |
| salpreimagtlt 44961 | If all the preimages of le... |
| smfpreimalt 44962 | Given a function measurabl... |
| smff 44963 | A function measurable w.r.... |
| smfdmss 44964 | The domain of a function m... |
| issmff 44965 | The predicate " ` F ` is a... |
| issmfd 44966 | A sufficient condition for... |
| smfpreimaltf 44967 | Given a function measurabl... |
| issmfdf 44968 | A sufficient condition for... |
| sssmf 44969 | The restriction of a sigma... |
| mbfresmf 44970 | A real-valued measurable f... |
| cnfsmf 44971 | A continuous function is m... |
| incsmflem 44972 | A nondecreasing function i... |
| incsmf 44973 | A real-valued, nondecreasi... |
| smfsssmf 44974 | If a function is measurabl... |
| issmflelem 44975 | The predicate " ` F ` is a... |
| issmfle 44976 | The predicate " ` F ` is a... |
| smfpimltmpt 44977 | Given a function measurabl... |
| smfpimltxr 44978 | Given a function measurabl... |
| issmfdmpt 44979 | A sufficient condition for... |
| smfconst 44980 | Given a sigma-algebra over... |
| sssmfmpt 44981 | The restriction of a sigma... |
| cnfrrnsmf 44982 | A function, continuous fro... |
| smfid 44983 | The identity function is B... |
| bormflebmf 44984 | A Borel measurable functio... |
| smfpreimale 44985 | Given a function measurabl... |
| issmfgtlem 44986 | The predicate " ` F ` is a... |
| issmfgt 44987 | The predicate " ` F ` is a... |
| issmfled 44988 | A sufficient condition for... |
| smfpimltxrmptf 44989 | Given a function measurabl... |
| smfpimltxrmpt 44990 | Given a function measurabl... |
| smfmbfcex 44991 | A constant function, with ... |
| issmfgtd 44992 | A sufficient condition for... |
| smfpreimagt 44993 | Given a function measurabl... |
| smfaddlem1 44994 | Given the sum of two funct... |
| smfaddlem2 44995 | The sum of two sigma-measu... |
| smfadd 44996 | The sum of two sigma-measu... |
| decsmflem 44997 | A nonincreasing function i... |
| decsmf 44998 | A real-valued, nonincreasi... |
| smfpreimagtf 44999 | Given a function measurabl... |
| issmfgelem 45000 | The predicate " ` F ` is a... |
| issmfge 45001 | The predicate " ` F ` is a... |
| smflimlem1 45002 | Lemma for the proof that t... |
| smflimlem2 45003 | Lemma for the proof that t... |
| smflimlem3 45004 | The limit of sigma-measura... |
| smflimlem4 45005 | Lemma for the proof that t... |
| smflimlem5 45006 | Lemma for the proof that t... |
| smflimlem6 45007 | Lemma for the proof that t... |
| smflim 45008 | The limit of sigma-measura... |
| nsssmfmbflem 45009 | The sigma-measurable funct... |
| nsssmfmbf 45010 | The sigma-measurable funct... |
| smfpimgtxr 45011 | Given a function measurabl... |
| smfpimgtmpt 45012 | Given a function measurabl... |
| smfpreimage 45013 | Given a function measurabl... |
| mbfpsssmf 45014 | Real-valued measurable fun... |
| smfpimgtxrmptf 45015 | Given a function measurabl... |
| smfpimgtxrmpt 45016 | Given a function measurabl... |
| smfpimioompt 45017 | Given a function measurabl... |
| smfpimioo 45018 | Given a function measurabl... |
| smfresal 45019 | Given a sigma-measurable f... |
| smfrec 45020 | The reciprocal of a sigma-... |
| smfres 45021 | The restriction of sigma-m... |
| smfmullem1 45022 | The multiplication of two ... |
| smfmullem2 45023 | The multiplication of two ... |
| smfmullem3 45024 | The multiplication of two ... |
| smfmullem4 45025 | The multiplication of two ... |
| smfmul 45026 | The multiplication of two ... |
| smfmulc1 45027 | A sigma-measurable functio... |
| smfdiv 45028 | The fraction of two sigma-... |
| smfpimbor1lem1 45029 | Every open set belongs to ... |
| smfpimbor1lem2 45030 | Given a sigma-measurable f... |
| smfpimbor1 45031 | Given a sigma-measurable f... |
| smf2id 45032 | Twice the identity functio... |
| smfco 45033 | The composition of a Borel... |
| smfneg 45034 | The negative of a sigma-me... |
| smffmptf 45035 | A function measurable w.r.... |
| smffmpt 45036 | A function measurable w.r.... |
| smflim2 45037 | The limit of a sequence of... |
| smfpimcclem 45038 | Lemma for ~ smfpimcc given... |
| smfpimcc 45039 | Given a countable set of s... |
| issmfle2d 45040 | A sufficient condition for... |
| smflimmpt 45041 | The limit of a sequence of... |
| smfsuplem1 45042 | The supremum of a countabl... |
| smfsuplem2 45043 | The supremum of a countabl... |
| smfsuplem3 45044 | The supremum of a countabl... |
| smfsup 45045 | The supremum of a countabl... |
| smfsupmpt 45046 | The supremum of a countabl... |
| smfsupxr 45047 | The supremum of a countabl... |
| smfinflem 45048 | The infimum of a countable... |
| smfinf 45049 | The infimum of a countable... |
| smfinfmpt 45050 | The infimum of a countable... |
| smflimsuplem1 45051 | If ` H ` converges, the ` ... |
| smflimsuplem2 45052 | The superior limit of a se... |
| smflimsuplem3 45053 | The limit of the ` ( H `` ... |
| smflimsuplem4 45054 | If ` H ` converges, the ` ... |
| smflimsuplem5 45055 | ` H ` converges to the sup... |
| smflimsuplem6 45056 | The superior limit of a se... |
| smflimsuplem7 45057 | The superior limit of a se... |
| smflimsuplem8 45058 | The superior limit of a se... |
| smflimsup 45059 | The superior limit of a se... |
| smflimsupmpt 45060 | The superior limit of a se... |
| smfliminflem 45061 | The inferior limit of a co... |
| smfliminf 45062 | The inferior limit of a co... |
| smfliminfmpt 45063 | The inferior limit of a co... |
| adddmmbl 45064 | If two functions have doma... |
| adddmmbl2 45065 | If two functions have doma... |
| muldmmbl 45066 | If two functions have doma... |
| muldmmbl2 45067 | If two functions have doma... |
| smfdmmblpimne 45068 | If a measurable function w... |
| smfdivdmmbl 45069 | If a functions and a sigma... |
| smfpimne 45070 | Given a function measurabl... |
| smfpimne2 45071 | Given a function measurabl... |
| smfdivdmmbl2 45072 | If a functions and a sigma... |
| fsupdm 45073 | The domain of the sup func... |
| fsupdm2 45074 | The domain of the sup func... |
| smfsupdmmbllem 45075 | If a countable set of sigm... |
| smfsupdmmbl 45076 | If a countable set of sigm... |
| finfdm 45077 | The domain of the inf func... |
| finfdm2 45078 | The domain of the inf func... |
| smfinfdmmbllem 45079 | If a countable set of sigm... |
| smfinfdmmbl 45080 | If a countable set of sigm... |
| sigarval 45081 | Define the signed area by ... |
| sigarim 45082 | Signed area takes value in... |
| sigarac 45083 | Signed area is anticommuta... |
| sigaraf 45084 | Signed area is additive by... |
| sigarmf 45085 | Signed area is additive (w... |
| sigaras 45086 | Signed area is additive by... |
| sigarms 45087 | Signed area is additive (w... |
| sigarls 45088 | Signed area is linear by t... |
| sigarid 45089 | Signed area of a flat para... |
| sigarexp 45090 | Expand the signed area for... |
| sigarperm 45091 | Signed area ` ( A - C ) G ... |
| sigardiv 45092 | If signed area between vec... |
| sigarimcd 45093 | Signed area takes value in... |
| sigariz 45094 | If signed area is zero, th... |
| sigarcol 45095 | Given three points ` A ` ,... |
| sharhght 45096 | Let ` A B C ` be a triangl... |
| sigaradd 45097 | Subtracting (double) area ... |
| cevathlem1 45098 | Ceva's theorem first lemma... |
| cevathlem2 45099 | Ceva's theorem second lemm... |
| cevath 45100 | Ceva's theorem. Let ` A B... |
| simpcntrab 45101 | The center of a simple gro... |
| et-ltneverrefl 45102 | Less-than class is never r... |
| et-equeucl 45103 | Alternative proof that equ... |
| et-sqrtnegnre 45104 | The square root of a negat... |
| natlocalincr 45105 | Global monotonicity on hal... |
| natglobalincr 45106 | Local monotonicity on half... |
| upwordnul 45109 | Empty set is an increasing... |
| upwordisword 45110 | Any increasing sequence is... |
| singoutnword 45111 | Singleton with character o... |
| singoutnupword 45112 | Singleton with character o... |
| upwordsing 45113 | Singleton is an increasing... |
| upwordsseti 45114 | Strictly increasing sequen... |
| tworepnotupword 45115 | Concatenation of identical... |
| upwrdfi 45116 | There is a finite number o... |
| hirstL-ax3 45117 | The third axiom of a syste... |
| ax3h 45118 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 45119 | A closed form showing (a i... |
| aibandbiaiaiffb 45120 | A closed form showing (a i... |
| notatnand 45121 | Do not use. Use intnanr i... |
| aistia 45122 | Given a is equivalent to `... |
| aisfina 45123 | Given a is equivalent to `... |
| bothtbothsame 45124 | Given both a, b are equiva... |
| bothfbothsame 45125 | Given both a, b are equiva... |
| aiffbbtat 45126 | Given a is equivalent to b... |
| aisbbisfaisf 45127 | Given a is equivalent to b... |
| axorbtnotaiffb 45128 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 45129 | Given a is equivalent to (... |
| axorbciffatcxorb 45130 | Given a is equivalent to (... |
| aibnbna 45131 | Given a implies b, (not b)... |
| aibnbaif 45132 | Given a implies b, not b, ... |
| aiffbtbat 45133 | Given a is equivalent to b... |
| astbstanbst 45134 | Given a is equivalent to T... |
| aistbistaandb 45135 | Given a is equivalent to T... |
| aisbnaxb 45136 | Given a is equivalent to b... |
| atbiffatnnb 45137 | If a implies b, then a imp... |
| bisaiaisb 45138 | Application of bicom1 with... |
| atbiffatnnbalt 45139 | If a implies b, then a imp... |
| abnotbtaxb 45140 | Assuming a, not b, there e... |
| abnotataxb 45141 | Assuming not a, b, there e... |
| conimpf 45142 | Assuming a, not b, and a i... |
| conimpfalt 45143 | Assuming a, not b, and a i... |
| aistbisfiaxb 45144 | Given a is equivalent to T... |
| aisfbistiaxb 45145 | Given a is equivalent to F... |
| aifftbifffaibif 45146 | Given a is equivalent to T... |
| aifftbifffaibifff 45147 | Given a is equivalent to T... |
| atnaiana 45148 | Given a, it is not the cas... |
| ainaiaandna 45149 | Given a, a implies it is n... |
| abcdta 45150 | Given (((a and b) and c) a... |
| abcdtb 45151 | Given (((a and b) and c) a... |
| abcdtc 45152 | Given (((a and b) and c) a... |
| abcdtd 45153 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 45154 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 45155 | Operands in a biconditiona... |
| nabctnabc 45156 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 45157 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 45158 | From one negated implicati... |
| twonotinotbothi 45159 | From these two negated imp... |
| clifte 45160 | show d is the same as an i... |
| cliftet 45161 | show d is the same as an i... |
| clifteta 45162 | show d is the same as an i... |
| cliftetb 45163 | show d is the same as an i... |
| confun 45164 | Given the hypotheses there... |
| confun2 45165 | Confun simplified to two p... |
| confun3 45166 | Confun's more complex form... |
| confun4 45167 | An attempt at derivative. ... |
| confun5 45168 | An attempt at derivative. ... |
| plcofph 45169 | Given, a,b and a "definiti... |
| pldofph 45170 | Given, a,b c, d, "definiti... |
| plvcofph 45171 | Given, a,b,d, and "definit... |
| plvcofphax 45172 | Given, a,b,d, and "definit... |
| plvofpos 45173 | rh is derivable because ON... |
| mdandyv0 45174 | Given the equivalences set... |
| mdandyv1 45175 | Given the equivalences set... |
| mdandyv2 45176 | Given the equivalences set... |
| mdandyv3 45177 | Given the equivalences set... |
| mdandyv4 45178 | Given the equivalences set... |
| mdandyv5 45179 | Given the equivalences set... |
| mdandyv6 45180 | Given the equivalences set... |
| mdandyv7 45181 | Given the equivalences set... |
| mdandyv8 45182 | Given the equivalences set... |
| mdandyv9 45183 | Given the equivalences set... |
| mdandyv10 45184 | Given the equivalences set... |
| mdandyv11 45185 | Given the equivalences set... |
| mdandyv12 45186 | Given the equivalences set... |
| mdandyv13 45187 | Given the equivalences set... |
| mdandyv14 45188 | Given the equivalences set... |
| mdandyv15 45189 | Given the equivalences set... |
| mdandyvr0 45190 | Given the equivalences set... |
| mdandyvr1 45191 | Given the equivalences set... |
| mdandyvr2 45192 | Given the equivalences set... |
| mdandyvr3 45193 | Given the equivalences set... |
| mdandyvr4 45194 | Given the equivalences set... |
| mdandyvr5 45195 | Given the equivalences set... |
| mdandyvr6 45196 | Given the equivalences set... |
| mdandyvr7 45197 | Given the equivalences set... |
| mdandyvr8 45198 | Given the equivalences set... |
| mdandyvr9 45199 | Given the equivalences set... |
| mdandyvr10 45200 | Given the equivalences set... |
| mdandyvr11 45201 | Given the equivalences set... |
| mdandyvr12 45202 | Given the equivalences set... |
| mdandyvr13 45203 | Given the equivalences set... |
| mdandyvr14 45204 | Given the equivalences set... |
| mdandyvr15 45205 | Given the equivalences set... |
| mdandyvrx0 45206 | Given the exclusivities se... |
| mdandyvrx1 45207 | Given the exclusivities se... |
| mdandyvrx2 45208 | Given the exclusivities se... |
| mdandyvrx3 45209 | Given the exclusivities se... |
| mdandyvrx4 45210 | Given the exclusivities se... |
| mdandyvrx5 45211 | Given the exclusivities se... |
| mdandyvrx6 45212 | Given the exclusivities se... |
| mdandyvrx7 45213 | Given the exclusivities se... |
| mdandyvrx8 45214 | Given the exclusivities se... |
| mdandyvrx9 45215 | Given the exclusivities se... |
| mdandyvrx10 45216 | Given the exclusivities se... |
| mdandyvrx11 45217 | Given the exclusivities se... |
| mdandyvrx12 45218 | Given the exclusivities se... |
| mdandyvrx13 45219 | Given the exclusivities se... |
| mdandyvrx14 45220 | Given the exclusivities se... |
| mdandyvrx15 45221 | Given the exclusivities se... |
| H15NH16TH15IH16 45222 | Given 15 hypotheses and a ... |
| dandysum2p2e4 45223 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 45224 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 45225 | Replacement of a nested an... |
| adh-minim 45226 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 45227 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 45228 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 45229 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 45230 | Fourth lemma for the deriv... |
| adh-minim-ax1 45231 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 45232 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 45233 | Sixth lemma for the deriva... |
| adh-minim-ax2c 45234 | Derivation of a commuted f... |
| adh-minim-ax2 45235 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 45236 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 45237 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 45238 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 45239 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 45240 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 45241 | Third lemma for the deriva... |
| adh-minimp-sylsimp 45242 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 45243 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 45244 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 45245 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 45246 | Fourth lemma for the deriv... |
| adh-minimp-ax2 45247 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 45248 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 45249 | Derivation of ~ pm2.43 Whi... |
| eusnsn 45250 | There is a unique element ... |
| absnsb 45251 | If the class abstraction `... |
| euabsneu 45252 | Another way to express exi... |
| elprneb 45253 | An element of a proper uno... |
| oppr 45254 | Equality for ordered pairs... |
| opprb 45255 | Equality for unordered pai... |
| or2expropbilem1 45256 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 45257 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 45258 | If two classes are strictl... |
| eubrv 45259 | If there is a unique set w... |
| eubrdm 45260 | If there is a unique set w... |
| eldmressn 45261 | Element of the domain of a... |
| iota0def 45262 | Example for a defined iota... |
| iota0ndef 45263 | Example for an undefined i... |
| fveqvfvv 45264 | If a function's value at a... |
| fnresfnco 45265 | Composition of two functio... |
| funcoressn 45266 | A composition restricted t... |
| funressnfv 45267 | A restriction to a singlet... |
| funressndmfvrn 45268 | The value of a function ` ... |
| funressnvmo 45269 | A function restricted to a... |
| funressnmo 45270 | A function restricted to a... |
| funressneu 45271 | There is exactly one value... |
| fresfo 45272 | Conditions for a restricti... |
| fsetsniunop 45273 | The class of all functions... |
| fsetabsnop 45274 | The class of all functions... |
| fsetsnf 45275 | The mapping of an element ... |
| fsetsnf1 45276 | The mapping of an element ... |
| fsetsnfo 45277 | The mapping of an element ... |
| fsetsnf1o 45278 | The mapping of an element ... |
| fsetsnprcnex 45279 | The class of all functions... |
| cfsetssfset 45280 | The class of constant func... |
| cfsetsnfsetfv 45281 | The function value of the ... |
| cfsetsnfsetf 45282 | The mapping of the class o... |
| cfsetsnfsetf1 45283 | The mapping of the class o... |
| cfsetsnfsetfo 45284 | The mapping of the class o... |
| cfsetsnfsetf1o 45285 | The mapping of the class o... |
| fsetprcnexALT 45286 | First version of proof for... |
| fcoreslem1 45287 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 45288 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 45289 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 45290 | Lemma 4 for ~ fcores . (C... |
| fcores 45291 | Every composite function `... |
| fcoresf1lem 45292 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 45293 | If a composition is inject... |
| fcoresf1b 45294 | A composition is injective... |
| fcoresfo 45295 | If a composition is surjec... |
| fcoresfob 45296 | A composition is surjectiv... |
| fcoresf1ob 45297 | A composition is bijective... |
| f1cof1blem 45298 | Lemma for ~ f1cof1b and ~ ... |
| f1cof1b 45299 | If the range of ` F ` equa... |
| funfocofob 45300 | If the domain of a functio... |
| fnfocofob 45301 | If the domain of a functio... |
| focofob 45302 | If the domain of a functio... |
| f1ocof1ob 45303 | If the range of ` F ` equa... |
| f1ocof1ob2 45304 | If the range of ` F ` equa... |
| aiotajust 45306 | Soundness justification th... |
| dfaiota2 45308 | Alternate definition of th... |
| reuabaiotaiota 45309 | The iota and the alternate... |
| reuaiotaiota 45310 | The iota and the alternate... |
| aiotaexb 45311 | The alternate iota over a ... |
| aiotavb 45312 | The alternate iota over a ... |
| aiotaint 45313 | This is to ~ df-aiota what... |
| dfaiota3 45314 | Alternate definition of ` ... |
| iotan0aiotaex 45315 | If the iota over a wff ` p... |
| aiotaexaiotaiota 45316 | The alternate iota over a ... |
| aiotaval 45317 | Theorem 8.19 in [Quine] p.... |
| aiota0def 45318 | Example for a defined alte... |
| aiota0ndef 45319 | Example for an undefined a... |
| r19.32 45320 | Theorem 19.32 of [Margaris... |
| rexsb 45321 | An equivalent expression f... |
| rexrsb 45322 | An equivalent expression f... |
| 2rexsb 45323 | An equivalent expression f... |
| 2rexrsb 45324 | An equivalent expression f... |
| cbvral2 45325 | Change bound variables of ... |
| cbvrex2 45326 | Change bound variables of ... |
| ralndv1 45327 | Example for a theorem abou... |
| ralndv2 45328 | Second example for a theor... |
| reuf1odnf 45329 | There is exactly one eleme... |
| reuf1od 45330 | There is exactly one eleme... |
| euoreqb 45331 | There is a set which is eq... |
| 2reu3 45332 | Double restricted existent... |
| 2reu7 45333 | Two equivalent expressions... |
| 2reu8 45334 | Two equivalent expressions... |
| 2reu8i 45335 | Implication of a double re... |
| 2reuimp0 45336 | Implication of a double re... |
| 2reuimp 45337 | Implication of a double re... |
| ralbinrald 45344 | Elemination of a restricte... |
| nvelim 45345 | If a class is the universa... |
| alneu 45346 | If a statement holds for a... |
| eu2ndop1stv 45347 | If there is a unique secon... |
| dfateq12d 45348 | Equality deduction for "de... |
| nfdfat 45349 | Bound-variable hypothesis ... |
| dfdfat2 45350 | Alternate definition of th... |
| fundmdfat 45351 | A function is defined at a... |
| dfatprc 45352 | A function is not defined ... |
| dfatelrn 45353 | The value of a function ` ... |
| dfafv2 45354 | Alternative definition of ... |
| afveq12d 45355 | Equality deduction for fun... |
| afveq1 45356 | Equality theorem for funct... |
| afveq2 45357 | Equality theorem for funct... |
| nfafv 45358 | Bound-variable hypothesis ... |
| csbafv12g 45359 | Move class substitution in... |
| afvfundmfveq 45360 | If a class is a function r... |
| afvnfundmuv 45361 | If a set is not in the dom... |
| ndmafv 45362 | The value of a class outsi... |
| afvvdm 45363 | If the function value of a... |
| nfunsnafv 45364 | If the restriction of a cl... |
| afvvfunressn 45365 | If the function value of a... |
| afvprc 45366 | A function's value at a pr... |
| afvvv 45367 | If a function's value at a... |
| afvpcfv0 45368 | If the value of the altern... |
| afvnufveq 45369 | The value of the alternati... |
| afvvfveq 45370 | The value of the alternati... |
| afv0fv0 45371 | If the value of the altern... |
| afvfvn0fveq 45372 | If the function's value at... |
| afv0nbfvbi 45373 | The function's value at an... |
| afvfv0bi 45374 | The function's value at an... |
| afveu 45375 | The value of a function at... |
| fnbrafvb 45376 | Equivalence of function va... |
| fnopafvb 45377 | Equivalence of function va... |
| funbrafvb 45378 | Equivalence of function va... |
| funopafvb 45379 | Equivalence of function va... |
| funbrafv 45380 | The second argument of a b... |
| funbrafv2b 45381 | Function value in terms of... |
| dfafn5a 45382 | Representation of a functi... |
| dfafn5b 45383 | Representation of a functi... |
| fnrnafv 45384 | The range of a function ex... |
| afvelrnb 45385 | A member of a function's r... |
| afvelrnb0 45386 | A member of a function's r... |
| dfaimafn 45387 | Alternate definition of th... |
| dfaimafn2 45388 | Alternate definition of th... |
| afvelima 45389 | Function value in an image... |
| afvelrn 45390 | A function's value belongs... |
| fnafvelrn 45391 | A function's value belongs... |
| fafvelcdm 45392 | A function's value belongs... |
| ffnafv 45393 | A function maps to a class... |
| afvres 45394 | The value of a restricted ... |
| tz6.12-afv 45395 | Function value. Theorem 6... |
| tz6.12-1-afv 45396 | Function value (Theorem 6.... |
| dmfcoafv 45397 | Domains of a function comp... |
| afvco2 45398 | Value of a function compos... |
| rlimdmafv 45399 | Two ways to express that a... |
| aoveq123d 45400 | Equality deduction for ope... |
| nfaov 45401 | Bound-variable hypothesis ... |
| csbaovg 45402 | Move class substitution in... |
| aovfundmoveq 45403 | If a class is a function r... |
| aovnfundmuv 45404 | If an ordered pair is not ... |
| ndmaov 45405 | The value of an operation ... |
| ndmaovg 45406 | The value of an operation ... |
| aovvdm 45407 | If the operation value of ... |
| nfunsnaov 45408 | If the restriction of a cl... |
| aovvfunressn 45409 | If the operation value of ... |
| aovprc 45410 | The value of an operation ... |
| aovrcl 45411 | Reverse closure for an ope... |
| aovpcov0 45412 | If the alternative value o... |
| aovnuoveq 45413 | The alternative value of t... |
| aovvoveq 45414 | The alternative value of t... |
| aov0ov0 45415 | If the alternative value o... |
| aovovn0oveq 45416 | If the operation's value a... |
| aov0nbovbi 45417 | The operation's value on a... |
| aovov0bi 45418 | The operation's value on a... |
| rspceaov 45419 | A frequently used special ... |
| fnotaovb 45420 | Equivalence of operation v... |
| ffnaov 45421 | An operation maps to a cla... |
| faovcl 45422 | Closure law for an operati... |
| aovmpt4g 45423 | Value of a function given ... |
| aoprssdm 45424 | Domain of closure of an op... |
| ndmaovcl 45425 | The "closure" of an operat... |
| ndmaovrcl 45426 | Reverse closure law, in co... |
| ndmaovcom 45427 | Any operation is commutati... |
| ndmaovass 45428 | Any operation is associati... |
| ndmaovdistr 45429 | Any operation is distribut... |
| dfatafv2iota 45432 | If a function is defined a... |
| ndfatafv2 45433 | The alternate function val... |
| ndfatafv2undef 45434 | The alternate function val... |
| dfatafv2ex 45435 | The alternate function val... |
| afv2ex 45436 | The alternate function val... |
| afv2eq12d 45437 | Equality deduction for fun... |
| afv2eq1 45438 | Equality theorem for funct... |
| afv2eq2 45439 | Equality theorem for funct... |
| nfafv2 45440 | Bound-variable hypothesis ... |
| csbafv212g 45441 | Move class substitution in... |
| fexafv2ex 45442 | The alternate function val... |
| ndfatafv2nrn 45443 | The alternate function val... |
| ndmafv2nrn 45444 | The value of a class outsi... |
| funressndmafv2rn 45445 | The alternate function val... |
| afv2ndefb 45446 | Two ways to say that an al... |
| nfunsnafv2 45447 | If the restriction of a cl... |
| afv2prc 45448 | A function's value at a pr... |
| dfatafv2rnb 45449 | The alternate function val... |
| afv2orxorb 45450 | If a set is in the range o... |
| dmafv2rnb 45451 | The alternate function val... |
| fundmafv2rnb 45452 | The alternate function val... |
| afv2elrn 45453 | An alternate function valu... |
| afv20defat 45454 | If the alternate function ... |
| fnafv2elrn 45455 | An alternate function valu... |
| fafv2elcdm 45456 | An alternate function valu... |
| fafv2elrnb 45457 | An alternate function valu... |
| fcdmvafv2v 45458 | If the codomain of a funct... |
| tz6.12-2-afv2 45459 | Function value when ` F ` ... |
| afv2eu 45460 | The value of a function at... |
| afv2res 45461 | The value of a restricted ... |
| tz6.12-afv2 45462 | Function value (Theorem 6.... |
| tz6.12-1-afv2 45463 | Function value (Theorem 6.... |
| tz6.12c-afv2 45464 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 45465 | Corollary of Theorem 6.12(... |
| funressnbrafv2 45466 | The second argument of a b... |
| dfatbrafv2b 45467 | Equivalence of function va... |
| dfatopafv2b 45468 | Equivalence of function va... |
| funbrafv2 45469 | The second argument of a b... |
| fnbrafv2b 45470 | Equivalence of function va... |
| fnopafv2b 45471 | Equivalence of function va... |
| funbrafv22b 45472 | Equivalence of function va... |
| funopafv2b 45473 | Equivalence of function va... |
| dfatsnafv2 45474 | Singleton of function valu... |
| dfafv23 45475 | A definition of function v... |
| dfatdmfcoafv2 45476 | Domain of a function compo... |
| dfatcolem 45477 | Lemma for ~ dfatco . (Con... |
| dfatco 45478 | The predicate "defined at"... |
| afv2co2 45479 | Value of a function compos... |
| rlimdmafv2 45480 | Two ways to express that a... |
| dfafv22 45481 | Alternate definition of ` ... |
| afv2ndeffv0 45482 | If the alternate function ... |
| dfatafv2eqfv 45483 | If a function is defined a... |
| afv2rnfveq 45484 | If the alternate function ... |
| afv20fv0 45485 | If the alternate function ... |
| afv2fvn0fveq 45486 | If the function's value at... |
| afv2fv0 45487 | If the function's value at... |
| afv2fv0b 45488 | The function's value at an... |
| afv2fv0xorb 45489 | If a set is in the range o... |
| an4com24 45490 | Rearrangement of 4 conjunc... |
| 3an4ancom24 45491 | Commutative law for a conj... |
| 4an21 45492 | Rearrangement of 4 conjunc... |
| dfnelbr2 45495 | Alternate definition of th... |
| nelbr 45496 | The binary relation of a s... |
| nelbrim 45497 | If a set is related to ano... |
| nelbrnel 45498 | A set is related to anothe... |
| nelbrnelim 45499 | If a set is related to ano... |
| ralralimp 45500 | Selecting one of two alter... |
| otiunsndisjX 45501 | The union of singletons co... |
| fvifeq 45502 | Equality of function value... |
| rnfdmpr 45503 | The range of a one-to-one ... |
| imarnf1pr 45504 | The image of the range of ... |
| funop1 45505 | A function is an ordered p... |
| fun2dmnopgexmpl 45506 | A function with a domain c... |
| opabresex0d 45507 | A collection of ordered pa... |
| opabbrfex0d 45508 | A collection of ordered pa... |
| opabresexd 45509 | A collection of ordered pa... |
| opabbrfexd 45510 | A collection of ordered pa... |
| f1oresf1orab 45511 | Build a bijection by restr... |
| f1oresf1o 45512 | Build a bijection by restr... |
| f1oresf1o2 45513 | Build a bijection by restr... |
| fvmptrab 45514 | Value of a function mappin... |
| fvmptrabdm 45515 | Value of a function mappin... |
| cnambpcma 45516 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 45517 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 45518 | The sum of two complex num... |
| leaddsuble 45519 | Addition and subtraction o... |
| 2leaddle2 45520 | If two real numbers are le... |
| ltnltne 45521 | Variant of trichotomy law ... |
| p1lep2 45522 | A real number increasd by ... |
| ltsubsubaddltsub 45523 | If the result of subtracti... |
| zm1nn 45524 | An integer minus 1 is posi... |
| readdcnnred 45525 | The sum of a real number a... |
| resubcnnred 45526 | The difference of a real n... |
| recnmulnred 45527 | The product of a real numb... |
| cndivrenred 45528 | The quotient of an imagina... |
| sqrtnegnre 45529 | The square root of a negat... |
| nn0resubcl 45530 | Closure law for subtractio... |
| zgeltp1eq 45531 | If an integer is between a... |
| 1t10e1p1e11 45532 | 11 is 1 times 10 to the po... |
| deccarry 45533 | Add 1 to a 2 digit number ... |
| eluzge0nn0 45534 | If an integer is greater t... |
| nltle2tri 45535 | Negated extended trichotom... |
| ssfz12 45536 | Subset relationship for fi... |
| elfz2z 45537 | Membership of an integer i... |
| 2elfz3nn0 45538 | If there are two elements ... |
| fz0addcom 45539 | The addition of two member... |
| 2elfz2melfz 45540 | If the sum of two integers... |
| fz0addge0 45541 | The sum of two integers in... |
| elfzlble 45542 | Membership of an integer i... |
| elfzelfzlble 45543 | Membership of an element o... |
| fzopred 45544 | Join a predecessor to the ... |
| fzopredsuc 45545 | Join a predecessor and a s... |
| 1fzopredsuc 45546 | Join 0 and a successor to ... |
| el1fzopredsuc 45547 | An element of an open inte... |
| subsubelfzo0 45548 | Subtracting a difference f... |
| fzoopth 45549 | A half-open integer range ... |
| 2ffzoeq 45550 | Two functions over a half-... |
| m1mod0mod1 45551 | An integer decreased by 1 ... |
| elmod2 45552 | An integer modulo 2 is eit... |
| smonoord 45553 | Ordering relation for a st... |
| fsummsndifre 45554 | A finite sum with one of i... |
| fsumsplitsndif 45555 | Separate out a term in a f... |
| fsummmodsndifre 45556 | A finite sum of summands m... |
| fsummmodsnunz 45557 | A finite sum of summands m... |
| setsidel 45558 | The injected slot is an el... |
| setsnidel 45559 | The injected slot is an el... |
| setsv 45560 | The value of the structure... |
| preimafvsnel 45561 | The preimage of a function... |
| preimafvn0 45562 | The preimage of a function... |
| uniimafveqt 45563 | The union of the image of ... |
| uniimaprimaeqfv 45564 | The union of the image of ... |
| setpreimafvex 45565 | The class ` P ` of all pre... |
| elsetpreimafvb 45566 | The characterization of an... |
| elsetpreimafv 45567 | An element of the class ` ... |
| elsetpreimafvssdm 45568 | An element of the class ` ... |
| fvelsetpreimafv 45569 | There is an element in a p... |
| preimafvelsetpreimafv 45570 | The preimage of a function... |
| preimafvsspwdm 45571 | The class ` P ` of all pre... |
| 0nelsetpreimafv 45572 | The empty set is not an el... |
| elsetpreimafvbi 45573 | An element of the preimage... |
| elsetpreimafveqfv 45574 | The elements of the preima... |
| eqfvelsetpreimafv 45575 | If an element of the domai... |
| elsetpreimafvrab 45576 | An element of the preimage... |
| imaelsetpreimafv 45577 | The image of an element of... |
| uniimaelsetpreimafv 45578 | The union of the image of ... |
| elsetpreimafveq 45579 | If two preimages of functi... |
| fundcmpsurinjlem1 45580 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 45581 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 45582 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 45583 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 45584 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 45585 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 45586 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 45587 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 45588 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 45589 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 45590 | Every function ` F : A -->... |
| fundcmpsurinj 45591 | Every function ` F : A -->... |
| fundcmpsurbijinj 45592 | Every function ` F : A -->... |
| fundcmpsurinjimaid 45593 | Every function ` F : A -->... |
| fundcmpsurinjALT 45594 | Alternate proof of ~ fundc... |
| iccpval 45597 | Partition consisting of a ... |
| iccpart 45598 | A special partition. Corr... |
| iccpartimp 45599 | Implications for a class b... |
| iccpartres 45600 | The restriction of a parti... |
| iccpartxr 45601 | If there is a partition, t... |
| iccpartgtprec 45602 | If there is a partition, t... |
| iccpartipre 45603 | If there is a partition, t... |
| iccpartiltu 45604 | If there is a partition, t... |
| iccpartigtl 45605 | If there is a partition, t... |
| iccpartlt 45606 | If there is a partition, t... |
| iccpartltu 45607 | If there is a partition, t... |
| iccpartgtl 45608 | If there is a partition, t... |
| iccpartgt 45609 | If there is a partition, t... |
| iccpartleu 45610 | If there is a partition, t... |
| iccpartgel 45611 | If there is a partition, t... |
| iccpartrn 45612 | If there is a partition, t... |
| iccpartf 45613 | The range of the partition... |
| iccpartel 45614 | If there is a partition, t... |
| iccelpart 45615 | An element of any partitio... |
| iccpartiun 45616 | A half-open interval of ex... |
| icceuelpartlem 45617 | Lemma for ~ icceuelpart . ... |
| icceuelpart 45618 | An element of a partitione... |
| iccpartdisj 45619 | The segments of a partitio... |
| iccpartnel 45620 | A point of a partition is ... |
| fargshiftfv 45621 | If a class is a function, ... |
| fargshiftf 45622 | If a class is a function, ... |
| fargshiftf1 45623 | If a function is 1-1, then... |
| fargshiftfo 45624 | If a function is onto, the... |
| fargshiftfva 45625 | The values of a shifted fu... |
| lswn0 45626 | The last symbol of a not e... |
| nfich1 45629 | The first interchangeable ... |
| nfich2 45630 | The second interchangeable... |
| ichv 45631 | Setvar variables are inter... |
| ichf 45632 | Setvar variables are inter... |
| ichid 45633 | A setvar variable is alway... |
| icht 45634 | A theorem is interchangeab... |
| ichbidv 45635 | Formula building rule for ... |
| ichcircshi 45636 | The setvar variables are i... |
| ichan 45637 | If two setvar variables ar... |
| ichn 45638 | Negation does not affect i... |
| ichim 45639 | Formula building rule for ... |
| dfich2 45640 | Alternate definition of th... |
| ichcom 45641 | The interchangeability of ... |
| ichbi12i 45642 | Equivalence for interchang... |
| icheqid 45643 | In an equality for the sam... |
| icheq 45644 | In an equality of setvar v... |
| ichnfimlem 45645 | Lemma for ~ ichnfim : A s... |
| ichnfim 45646 | If in an interchangeabilit... |
| ichnfb 45647 | If ` x ` and ` y ` are int... |
| ichal 45648 | Move a universal quantifie... |
| ich2al 45649 | Two setvar variables are a... |
| ich2ex 45650 | Two setvar variables are a... |
| ichexmpl1 45651 | Example for interchangeabl... |
| ichexmpl2 45652 | Example for interchangeabl... |
| ich2exprop 45653 | If the setvar variables ar... |
| ichnreuop 45654 | If the setvar variables ar... |
| ichreuopeq 45655 | If the setvar variables ar... |
| sprid 45656 | Two identical representati... |
| elsprel 45657 | An unordered pair is an el... |
| spr0nelg 45658 | The empty set is not an el... |
| sprval 45661 | The set of all unordered p... |
| sprvalpw 45662 | The set of all unordered p... |
| sprssspr 45663 | The set of all unordered p... |
| spr0el 45664 | The empty set is not an un... |
| sprvalpwn0 45665 | The set of all unordered p... |
| sprel 45666 | An element of the set of a... |
| prssspr 45667 | An element of a subset of ... |
| prelspr 45668 | An unordered pair of eleme... |
| prsprel 45669 | The elements of a pair fro... |
| prsssprel 45670 | The elements of a pair fro... |
| sprvalpwle2 45671 | The set of all unordered p... |
| sprsymrelfvlem 45672 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 45673 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 45674 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 45675 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 45676 | The value of the function ... |
| sprsymrelf 45677 | The mapping ` F ` is a fun... |
| sprsymrelf1 45678 | The mapping ` F ` is a one... |
| sprsymrelfo 45679 | The mapping ` F ` is a fun... |
| sprsymrelf1o 45680 | The mapping ` F ` is a bij... |
| sprbisymrel 45681 | There is a bijection betwe... |
| sprsymrelen 45682 | The class ` P ` of subsets... |
| prpair 45683 | Characterization of a prop... |
| prproropf1olem0 45684 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 45685 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 45686 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 45687 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 45688 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 45689 | There is a bijection betwe... |
| prproropen 45690 | The set of proper pairs an... |
| prproropreud 45691 | There is exactly one order... |
| pairreueq 45692 | Two equivalent representat... |
| paireqne 45693 | Two sets are not equal iff... |
| prprval 45696 | The set of all proper unor... |
| prprvalpw 45697 | The set of all proper unor... |
| prprelb 45698 | An element of the set of a... |
| prprelprb 45699 | A set is an element of the... |
| prprspr2 45700 | The set of all proper unor... |
| prprsprreu 45701 | There is a unique proper u... |
| prprreueq 45702 | There is a unique proper u... |
| sbcpr 45703 | The proper substitution of... |
| reupr 45704 | There is a unique unordere... |
| reuprpr 45705 | There is a unique proper u... |
| poprelb 45706 | Equality for unordered pai... |
| 2exopprim 45707 | The existence of an ordere... |
| reuopreuprim 45708 | There is a unique unordere... |
| fmtno 45711 | The ` N ` th Fermat number... |
| fmtnoge3 45712 | Each Fermat number is grea... |
| fmtnonn 45713 | Each Fermat number is a po... |
| fmtnom1nn 45714 | A Fermat number minus one ... |
| fmtnoodd 45715 | Each Fermat number is odd.... |
| fmtnorn 45716 | A Fermat number is a funct... |
| fmtnof1 45717 | The enumeration of the Fer... |
| fmtnoinf 45718 | The set of Fermat numbers ... |
| fmtnorec1 45719 | The first recurrence relat... |
| sqrtpwpw2p 45720 | The floor of the square ro... |
| fmtnosqrt 45721 | The floor of the square ro... |
| fmtno0 45722 | The ` 0 ` th Fermat number... |
| fmtno1 45723 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 45724 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 45725 | The second recurrence rela... |
| fmtnodvds 45726 | Any Fermat number divides ... |
| goldbachthlem1 45727 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 45728 | Lemma 2 for ~ goldbachth .... |
| goldbachth 45729 | Goldbach's theorem: Two d... |
| fmtnorec3 45730 | The third recurrence relat... |
| fmtnorec4 45731 | The fourth recurrence rela... |
| fmtno2 45732 | The ` 2 ` nd Fermat number... |
| fmtno3 45733 | The ` 3 ` rd Fermat number... |
| fmtno4 45734 | The ` 4 ` th Fermat number... |
| fmtno5lem1 45735 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 45736 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 45737 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 45738 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 45739 | The ` 5 ` th Fermat number... |
| fmtno0prm 45740 | The ` 0 ` th Fermat number... |
| fmtno1prm 45741 | The ` 1 ` st Fermat number... |
| fmtno2prm 45742 | The ` 2 ` nd Fermat number... |
| 257prm 45743 | 257 is a prime number (the... |
| fmtno3prm 45744 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 45745 | Any power of two is coprim... |
| fmtnoprmfac1lem 45746 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 45747 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 45748 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 45749 | Divisor of Fermat number (... |
| fmtnofac2lem 45750 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 45751 | Divisor of Fermat number (... |
| fmtnofac1 45752 | Divisor of Fermat number (... |
| fmtno4sqrt 45753 | The floor of the square ro... |
| fmtno4prmfac 45754 | If P was a (prime) factor ... |
| fmtno4prmfac193 45755 | If P was a (prime) factor ... |
| fmtno4nprmfac193 45756 | 193 is not a (prime) facto... |
| fmtno4prm 45757 | The ` 4 `-th Fermat number... |
| 65537prm 45758 | 65537 is a prime number (t... |
| fmtnofz04prm 45759 | The first five Fermat numb... |
| fmtnole4prm 45760 | The first five Fermat numb... |
| fmtno5faclem1 45761 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 45762 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 45763 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 45764 | The factorisation of the `... |
| fmtno5nprm 45765 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 45766 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 45767 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 45768 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 45769 | The mapping of a Fermat nu... |
| prminf2 45770 | The set of prime numbers i... |
| 2pwp1prm 45771 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 45772 | Every prime number of the ... |
| m2prm 45773 | The second Mersenne number... |
| m3prm 45774 | The third Mersenne number ... |
| flsqrt 45775 | A condition equivalent to ... |
| flsqrt5 45776 | The floor of the square ro... |
| 3ndvds4 45777 | 3 does not divide 4. (Con... |
| 139prmALT 45778 | 139 is a prime number. In... |
| 31prm 45779 | 31 is a prime number. In ... |
| m5prm 45780 | The fifth Mersenne number ... |
| 127prm 45781 | 127 is a prime number. (C... |
| m7prm 45782 | The seventh Mersenne numbe... |
| m11nprm 45783 | The eleventh Mersenne numb... |
| mod42tp1mod8 45784 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 45785 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 45786 | If ` P ` is a Sophie Germa... |
| lighneallem1 45787 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 45788 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 45789 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 45790 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 45791 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 45792 | Lemma 3 for ~ lighneal . ... |
| lighneal 45793 | If a power of a prime ` P ... |
| modexp2m1d 45794 | The square of an integer w... |
| proththdlem 45795 | Lemma for ~ proththd . (C... |
| proththd 45796 | Proth's theorem (1878). I... |
| 5tcu2e40 45797 | 5 times the cube of 2 is 4... |
| 3exp4mod41 45798 | 3 to the fourth power is -... |
| 41prothprmlem1 45799 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 45800 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 45801 | 41 is a _Proth prime_. (C... |
| quad1 45802 | A condition for a quadrati... |
| requad01 45803 | A condition for a quadrati... |
| requad1 45804 | A condition for a quadrati... |
| requad2 45805 | A condition for a quadrati... |
| iseven 45810 | The predicate "is an even ... |
| isodd 45811 | The predicate "is an odd n... |
| evenz 45812 | An even number is an integ... |
| oddz 45813 | An odd number is an intege... |
| evendiv2z 45814 | The result of dividing an ... |
| oddp1div2z 45815 | The result of dividing an ... |
| oddm1div2z 45816 | The result of dividing an ... |
| isodd2 45817 | The predicate "is an odd n... |
| dfodd2 45818 | Alternate definition for o... |
| dfodd6 45819 | Alternate definition for o... |
| dfeven4 45820 | Alternate definition for e... |
| evenm1odd 45821 | The predecessor of an even... |
| evenp1odd 45822 | The successor of an even n... |
| oddp1eveni 45823 | The successor of an odd nu... |
| oddm1eveni 45824 | The predecessor of an odd ... |
| evennodd 45825 | An even number is not an o... |
| oddneven 45826 | An odd number is not an ev... |
| enege 45827 | The negative of an even nu... |
| onego 45828 | The negative of an odd num... |
| m1expevenALTV 45829 | Exponentiation of -1 by an... |
| m1expoddALTV 45830 | Exponentiation of -1 by an... |
| dfeven2 45831 | Alternate definition for e... |
| dfodd3 45832 | Alternate definition for o... |
| iseven2 45833 | The predicate "is an even ... |
| isodd3 45834 | The predicate "is an odd n... |
| 2dvdseven 45835 | 2 divides an even number. ... |
| m2even 45836 | A multiple of 2 is an even... |
| 2ndvdsodd 45837 | 2 does not divide an odd n... |
| 2dvdsoddp1 45838 | 2 divides an odd number in... |
| 2dvdsoddm1 45839 | 2 divides an odd number de... |
| dfeven3 45840 | Alternate definition for e... |
| dfodd4 45841 | Alternate definition for o... |
| dfodd5 45842 | Alternate definition for o... |
| zefldiv2ALTV 45843 | The floor of an even numbe... |
| zofldiv2ALTV 45844 | The floor of an odd numer ... |
| oddflALTV 45845 | Odd number representation ... |
| iseven5 45846 | The predicate "is an even ... |
| isodd7 45847 | The predicate "is an odd n... |
| dfeven5 45848 | Alternate definition for e... |
| dfodd7 45849 | Alternate definition for o... |
| gcd2odd1 45850 | The greatest common diviso... |
| zneoALTV 45851 | No even integer equals an ... |
| zeoALTV 45852 | An integer is even or odd.... |
| zeo2ALTV 45853 | An integer is even or odd ... |
| nneoALTV 45854 | A positive integer is even... |
| nneoiALTV 45855 | A positive integer is even... |
| odd2np1ALTV 45856 | An integer is odd iff it i... |
| oddm1evenALTV 45857 | An integer is odd iff its ... |
| oddp1evenALTV 45858 | An integer is odd iff its ... |
| oexpnegALTV 45859 | The exponential of the neg... |
| oexpnegnz 45860 | The exponential of the neg... |
| bits0ALTV 45861 | Value of the zeroth bit. ... |
| bits0eALTV 45862 | The zeroth bit of an even ... |
| bits0oALTV 45863 | The zeroth bit of an odd n... |
| divgcdoddALTV 45864 | Either ` A / ( A gcd B ) `... |
| opoeALTV 45865 | The sum of two odds is eve... |
| opeoALTV 45866 | The sum of an odd and an e... |
| omoeALTV 45867 | The difference of two odds... |
| omeoALTV 45868 | The difference of an odd a... |
| oddprmALTV 45869 | A prime not equal to ` 2 `... |
| 0evenALTV 45870 | 0 is an even number. (Con... |
| 0noddALTV 45871 | 0 is not an odd number. (... |
| 1oddALTV 45872 | 1 is an odd number. (Cont... |
| 1nevenALTV 45873 | 1 is not an even number. ... |
| 2evenALTV 45874 | 2 is an even number. (Con... |
| 2noddALTV 45875 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 45876 | An odd nonnegative integer... |
| nnoALTV 45877 | An alternate characterizat... |
| nn0oALTV 45878 | An alternate characterizat... |
| nn0e 45879 | An alternate characterizat... |
| nneven 45880 | An alternate characterizat... |
| nn0onn0exALTV 45881 | For each odd nonnegative i... |
| nn0enn0exALTV 45882 | For each even nonnegative ... |
| nnennexALTV 45883 | For each even positive int... |
| nnpw2evenALTV 45884 | 2 to the power of a positi... |
| epoo 45885 | The sum of an even and an ... |
| emoo 45886 | The difference of an even ... |
| epee 45887 | The sum of two even number... |
| emee 45888 | The difference of two even... |
| evensumeven 45889 | If a summand is even, the ... |
| 3odd 45890 | 3 is an odd number. (Cont... |
| 4even 45891 | 4 is an even number. (Con... |
| 5odd 45892 | 5 is an odd number. (Cont... |
| 6even 45893 | 6 is an even number. (Con... |
| 7odd 45894 | 7 is an odd number. (Cont... |
| 8even 45895 | 8 is an even number. (Con... |
| evenprm2 45896 | A prime number is even iff... |
| oddprmne2 45897 | Every prime number not bei... |
| oddprmuzge3 45898 | A prime number which is od... |
| evenltle 45899 | If an even number is great... |
| odd2prm2 45900 | If an odd number is the su... |
| even3prm2 45901 | If an even number is the s... |
| mogoldbblem 45902 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 45903 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 45904 | Lemma for ~ perfectALTV . ... |
| perfectALTV 45905 | The Euclid-Euler theorem, ... |
| fppr 45908 | The set of Fermat pseudopr... |
| fpprmod 45909 | The set of Fermat pseudopr... |
| fpprel 45910 | A Fermat pseudoprime to th... |
| fpprbasnn 45911 | The base of a Fermat pseud... |
| fpprnn 45912 | A Fermat pseudoprime to th... |
| fppr2odd 45913 | A Fermat pseudoprime to th... |
| 11t31e341 45914 | 341 is the product of 11 a... |
| 2exp340mod341 45915 | Eight to the eighth power ... |
| 341fppr2 45916 | 341 is the (smallest) _Pou... |
| 4fppr1 45917 | 4 is the (smallest) Fermat... |
| 8exp8mod9 45918 | Eight to the eighth power ... |
| 9fppr8 45919 | 9 is the (smallest) Fermat... |
| dfwppr 45920 | Alternate definition of a ... |
| fpprwppr 45921 | A Fermat pseudoprime to th... |
| fpprwpprb 45922 | An integer ` X ` which is ... |
| fpprel2 45923 | An alternate definition fo... |
| nfermltl8rev 45924 | Fermat's little theorem wi... |
| nfermltl2rev 45925 | Fermat's little theorem wi... |
| nfermltlrev 45926 | Fermat's little theorem re... |
| isgbe 45933 | The predicate "is an even ... |
| isgbow 45934 | The predicate "is a weak o... |
| isgbo 45935 | The predicate "is an odd G... |
| gbeeven 45936 | An even Goldbach number is... |
| gbowodd 45937 | A weak odd Goldbach number... |
| gbogbow 45938 | A (strong) odd Goldbach nu... |
| gboodd 45939 | An odd Goldbach number is ... |
| gbepos 45940 | Any even Goldbach number i... |
| gbowpos 45941 | Any weak odd Goldbach numb... |
| gbopos 45942 | Any odd Goldbach number is... |
| gbegt5 45943 | Any even Goldbach number i... |
| gbowgt5 45944 | Any weak odd Goldbach numb... |
| gbowge7 45945 | Any weak odd Goldbach numb... |
| gboge9 45946 | Any odd Goldbach number is... |
| gbege6 45947 | Any even Goldbach number i... |
| gbpart6 45948 | The Goldbach partition of ... |
| gbpart7 45949 | The (weak) Goldbach partit... |
| gbpart8 45950 | The Goldbach partition of ... |
| gbpart9 45951 | The (strong) Goldbach part... |
| gbpart11 45952 | The (strong) Goldbach part... |
| 6gbe 45953 | 6 is an even Goldbach numb... |
| 7gbow 45954 | 7 is a weak odd Goldbach n... |
| 8gbe 45955 | 8 is an even Goldbach numb... |
| 9gbo 45956 | 9 is an odd Goldbach numbe... |
| 11gbo 45957 | 11 is an odd Goldbach numb... |
| stgoldbwt 45958 | If the strong ternary Gold... |
| sbgoldbwt 45959 | If the strong binary Goldb... |
| sbgoldbst 45960 | If the strong binary Goldb... |
| sbgoldbaltlem1 45961 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 45962 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 45963 | An alternate (related to t... |
| sbgoldbb 45964 | If the strong binary Goldb... |
| sgoldbeven3prm 45965 | If the binary Goldbach con... |
| sbgoldbm 45966 | If the strong binary Goldb... |
| mogoldbb 45967 | If the modern version of t... |
| sbgoldbmb 45968 | The strong binary Goldbach... |
| sbgoldbo 45969 | If the strong binary Goldb... |
| nnsum3primes4 45970 | 4 is the sum of at most 3 ... |
| nnsum4primes4 45971 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 45972 | Every prime is "the sum of... |
| nnsum4primesprm 45973 | Every prime is "the sum of... |
| nnsum3primesgbe 45974 | Any even Goldbach number i... |
| nnsum4primesgbe 45975 | Any even Goldbach number i... |
| nnsum3primesle9 45976 | Every integer greater than... |
| nnsum4primesle9 45977 | Every integer greater than... |
| nnsum4primesodd 45978 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 45979 | If the (strong) ternary Go... |
| evengpop3 45980 | If the (weak) ternary Gold... |
| evengpoap3 45981 | If the (strong) ternary Go... |
| nnsum4primeseven 45982 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 45983 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 45984 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 45985 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 45986 | If the binary Goldbach con... |
| bgoldbtbndlem1 45987 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 45988 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 45989 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 45990 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 45991 | If the binary Goldbach con... |
| tgoldbachgtALTV 45994 | Variant of Thierry Arnoux'... |
| bgoldbachlt 45995 | The binary Goldbach conjec... |
| tgblthelfgott 45997 | The ternary Goldbach conje... |
| tgoldbachlt 45998 | The ternary Goldbach conje... |
| tgoldbach 45999 | The ternary Goldbach conje... |
| isomgrrel 46004 | The isomorphy relation for... |
| isomgr 46005 | The isomorphy relation for... |
| isisomgr 46006 | Implications of two graphs... |
| isomgreqve 46007 | A set is isomorphic to a h... |
| isomushgr 46008 | The isomorphy relation for... |
| isomuspgrlem1 46009 | Lemma 1 for ~ isomuspgr . ... |
| isomuspgrlem2a 46010 | Lemma 1 for ~ isomuspgrlem... |
| isomuspgrlem2b 46011 | Lemma 2 for ~ isomuspgrlem... |
| isomuspgrlem2c 46012 | Lemma 3 for ~ isomuspgrlem... |
| isomuspgrlem2d 46013 | Lemma 4 for ~ isomuspgrlem... |
| isomuspgrlem2e 46014 | Lemma 5 for ~ isomuspgrlem... |
| isomuspgrlem2 46015 | Lemma 2 for ~ isomuspgr . ... |
| isomuspgr 46016 | The isomorphy relation for... |
| isomgrref 46017 | The isomorphy relation is ... |
| isomgrsym 46018 | The isomorphy relation is ... |
| isomgrsymb 46019 | The isomorphy relation is ... |
| isomgrtrlem 46020 | Lemma for ~ isomgrtr . (C... |
| isomgrtr 46021 | The isomorphy relation is ... |
| strisomgrop 46022 | A graph represented as an ... |
| ushrisomgr 46023 | A simple hypergraph (with ... |
| 1hegrlfgr 46024 | A graph ` G ` with one hyp... |
| upwlksfval 46027 | The set of simple walks (i... |
| isupwlk 46028 | Properties of a pair of fu... |
| isupwlkg 46029 | Generalization of ~ isupwl... |
| upwlkbprop 46030 | Basic properties of a simp... |
| upwlkwlk 46031 | A simple walk is a walk. ... |
| upgrwlkupwlk 46032 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 46033 | In a pseudograph, the defi... |
| upgrisupwlkALT 46034 | Alternate proof of ~ upgri... |
| upgredgssspr 46035 | The set of edges of a pseu... |
| uspgropssxp 46036 | The set ` G ` of "simple p... |
| uspgrsprfv 46037 | The value of the function ... |
| uspgrsprf 46038 | The mapping ` F ` is a fun... |
| uspgrsprf1 46039 | The mapping ` F ` is a one... |
| uspgrsprfo 46040 | The mapping ` F ` is a fun... |
| uspgrsprf1o 46041 | The mapping ` F ` is a bij... |
| uspgrex 46042 | The class ` G ` of all "si... |
| uspgrbispr 46043 | There is a bijection betwe... |
| uspgrspren 46044 | The set ` G ` of the "simp... |
| uspgrymrelen 46045 | The set ` G ` of the "simp... |
| uspgrbisymrel 46046 | There is a bijection betwe... |
| uspgrbisymrelALT 46047 | Alternate proof of ~ uspgr... |
| ovn0dmfun 46048 | If a class operation value... |
| xpsnopab 46049 | A Cartesian product with a... |
| xpiun 46050 | A Cartesian product expres... |
| ovn0ssdmfun 46051 | If a class' operation valu... |
| fnxpdmdm 46052 | The domain of the domain o... |
| cnfldsrngbas 46053 | The base set of a subring ... |
| cnfldsrngadd 46054 | The group addition operati... |
| cnfldsrngmul 46055 | The ring multiplication op... |
| plusfreseq 46056 | If the empty set is not co... |
| mgmplusfreseq 46057 | If the empty set is not co... |
| 0mgm 46058 | A set with an empty base s... |
| mgmpropd 46059 | If two structures have the... |
| ismgmd 46060 | Deduce a magma from its pr... |
| mgmhmrcl 46065 | Reverse closure of a magma... |
| submgmrcl 46066 | Reverse closure for submag... |
| ismgmhm 46067 | Property of a magma homomo... |
| mgmhmf 46068 | A magma homomorphism is a ... |
| mgmhmpropd 46069 | Magma homomorphism depends... |
| mgmhmlin 46070 | A magma homomorphism prese... |
| mgmhmf1o 46071 | A magma homomorphism is bi... |
| idmgmhm 46072 | The identity homomorphism ... |
| issubmgm 46073 | Expand definition of a sub... |
| issubmgm2 46074 | Submagmas are subsets that... |
| rabsubmgmd 46075 | Deduction for proving that... |
| submgmss 46076 | Submagmas are subsets of t... |
| submgmid 46077 | Every magma is trivially a... |
| submgmcl 46078 | Submagmas are closed under... |
| submgmmgm 46079 | Submagmas are themselves m... |
| submgmbas 46080 | The base set of a submagma... |
| subsubmgm 46081 | A submagma of a submagma i... |
| resmgmhm 46082 | Restriction of a magma hom... |
| resmgmhm2 46083 | One direction of ~ resmgmh... |
| resmgmhm2b 46084 | Restriction of the codomai... |
| mgmhmco 46085 | The composition of magma h... |
| mgmhmima 46086 | The homomorphic image of a... |
| mgmhmeql 46087 | The equalizer of two magma... |
| submgmacs 46088 | Submagmas are an algebraic... |
| ismhm0 46089 | Property of a monoid homom... |
| mhmismgmhm 46090 | Each monoid homomorphism i... |
| opmpoismgm 46091 | A structure with a group a... |
| copissgrp 46092 | A structure with a constan... |
| copisnmnd 46093 | A structure with a constan... |
| 0nodd 46094 | 0 is not an odd integer. ... |
| 1odd 46095 | 1 is an odd integer. (Con... |
| 2nodd 46096 | 2 is not an odd integer. ... |
| oddibas 46097 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 46098 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 46099 | The structure of all odd i... |
| nnsgrpmgm 46100 | The structure of positive ... |
| nnsgrp 46101 | The structure of positive ... |
| nnsgrpnmnd 46102 | The structure of positive ... |
| nn0mnd 46103 | The set of nonnegative int... |
| gsumsplit2f 46104 | Split a group sum into two... |
| gsumdifsndf 46105 | Extract a summand from a f... |
| gsumfsupp 46106 | A group sum of a family ca... |
| iscllaw 46113 | The predicate "is a closed... |
| iscomlaw 46114 | The predicate "is a commut... |
| clcllaw 46115 | Closure of a closed operat... |
| isasslaw 46116 | The predicate "is an assoc... |
| asslawass 46117 | Associativity of an associ... |
| mgmplusgiopALT 46118 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 46119 | Slot 2 (group operation) o... |
| intopval 46126 | The internal (binary) oper... |
| intop 46127 | An internal (binary) opera... |
| clintopval 46128 | The closed (internal binar... |
| assintopval 46129 | The associative (closed in... |
| assintopmap 46130 | The associative (closed in... |
| isclintop 46131 | The predicate "is a closed... |
| clintop 46132 | A closed (internal binary)... |
| assintop 46133 | An associative (closed int... |
| isassintop 46134 | The predicate "is an assoc... |
| clintopcllaw 46135 | The closure law holds for ... |
| assintopcllaw 46136 | The closure low holds for ... |
| assintopasslaw 46137 | The associative low holds ... |
| assintopass 46138 | An associative (closed int... |
| ismgmALT 46147 | The predicate "is a magma"... |
| iscmgmALT 46148 | The predicate "is a commut... |
| issgrpALT 46149 | The predicate "is a semigr... |
| iscsgrpALT 46150 | The predicate "is a commut... |
| mgm2mgm 46151 | Equivalence of the two def... |
| sgrp2sgrp 46152 | Equivalence of the two def... |
| idfusubc0 46153 | The identity functor for a... |
| idfusubc 46154 | The identity functor for a... |
| inclfusubc 46155 | The "inclusion functor" fr... |
| lmod0rng 46156 | If the scalar ring of a mo... |
| nzrneg1ne0 46157 | The additive inverse of th... |
| 0ringdif 46158 | A zero ring is a ring whic... |
| 0ringbas 46159 | The base set of a zero rin... |
| 0ring1eq0 46160 | In a zero ring, a ring whi... |
| nrhmzr 46161 | There is no ring homomorph... |
| isrng 46164 | The predicate "is a non-un... |
| rngabl 46165 | A non-unital ring is an (a... |
| rngmgp 46166 | A non-unital ring is a sem... |
| ringrng 46167 | A unital ring is a non-uni... |
| ringssrng 46168 | The unital rings are non-u... |
| isringrng 46169 | The predicate "is a unital... |
| rngdir 46170 | Distributive law for the m... |
| rngcl 46171 | Closure of the multiplicat... |
| rnglz 46172 | The zero of a non-unital r... |
| rnghmrcl 46177 | Reverse closure of a non-u... |
| rnghmfn 46178 | The mapping of two non-uni... |
| rnghmval 46179 | The set of the non-unital ... |
| isrnghm 46180 | A function is a non-unital... |
| isrnghmmul 46181 | A function is a non-unital... |
| rnghmmgmhm 46182 | A non-unital ring homomorp... |
| rnghmval2 46183 | The non-unital ring homomo... |
| isrngisom 46184 | An isomorphism of non-unit... |
| rngimrcl 46185 | Reverse closure for an iso... |
| rnghmghm 46186 | A non-unital ring homomorp... |
| rnghmf 46187 | A ring homomorphism is a f... |
| rnghmmul 46188 | A homomorphism of non-unit... |
| isrnghm2d 46189 | Demonstration of non-unita... |
| isrnghmd 46190 | Demonstration of non-unita... |
| rnghmf1o 46191 | A non-unital ring homomorp... |
| isrngim 46192 | An isomorphism of non-unit... |
| rngimf1o 46193 | An isomorphism of non-unit... |
| rngimrnghm 46194 | An isomorphism of non-unit... |
| rnghmco 46195 | The composition of non-uni... |
| idrnghm 46196 | The identity homomorphism ... |
| c0mgm 46197 | The constant mapping to ze... |
| c0mhm 46198 | The constant mapping to ze... |
| c0ghm 46199 | The constant mapping to ze... |
| c0rhm 46200 | The constant mapping to ze... |
| c0rnghm 46201 | The constant mapping to ze... |
| c0snmgmhm 46202 | The constant mapping to ze... |
| c0snmhm 46203 | The constant mapping to ze... |
| c0snghm 46204 | The constant mapping to ze... |
| zrrnghm 46205 | The constant mapping to ze... |
| rhmfn 46206 | The mapping of two rings t... |
| rhmval 46207 | The ring homomorphisms bet... |
| rhmisrnghm 46208 | Each unital ring homomorph... |
| lidldomn1 46209 | If a (left) ideal (which i... |
| lidlssbas 46210 | The base set of the restri... |
| lidlbas 46211 | A (left) ideal of a ring i... |
| lidlabl 46212 | A (left) ideal of a ring i... |
| lidlmmgm 46213 | The multiplicative group o... |
| lidlmsgrp 46214 | The multiplicative group o... |
| lidlrng 46215 | A (left) ideal of a ring i... |
| zlidlring 46216 | The zero (left) ideal of a... |
| uzlidlring 46217 | Only the zero (left) ideal... |
| lidldomnnring 46218 | A (left) ideal of a domain... |
| 0even 46219 | 0 is an even integer. (Co... |
| 1neven 46220 | 1 is not an even integer. ... |
| 2even 46221 | 2 is an even integer. (Co... |
| 2zlidl 46222 | The even integers are a (l... |
| 2zrng 46223 | The ring of integers restr... |
| 2zrngbas 46224 | The base set of R is the s... |
| 2zrngadd 46225 | The group addition operati... |
| 2zrng0 46226 | The additive identity of R... |
| 2zrngamgm 46227 | R is an (additive) magma. ... |
| 2zrngasgrp 46228 | R is an (additive) semigro... |
| 2zrngamnd 46229 | R is an (additive) monoid.... |
| 2zrngacmnd 46230 | R is a commutative (additi... |
| 2zrngagrp 46231 | R is an (additive) group. ... |
| 2zrngaabl 46232 | R is an (additive) abelian... |
| 2zrngmul 46233 | The ring multiplication op... |
| 2zrngmmgm 46234 | R is a (multiplicative) ma... |
| 2zrngmsgrp 46235 | R is a (multiplicative) se... |
| 2zrngALT 46236 | The ring of integers restr... |
| 2zrngnmlid 46237 | R has no multiplicative (l... |
| 2zrngnmrid 46238 | R has no multiplicative (r... |
| 2zrngnmlid2 46239 | R has no multiplicative (l... |
| 2zrngnring 46240 | R is not a unital ring. (... |
| cznrnglem 46241 | Lemma for ~ cznrng : The ... |
| cznabel 46242 | The ring constructed from ... |
| cznrng 46243 | The ring constructed from ... |
| cznnring 46244 | The ring constructed from ... |
| rngcvalALTV 46249 | Value of the category of n... |
| rngcval 46250 | Value of the category of n... |
| rnghmresfn 46251 | The class of non-unital ri... |
| rnghmresel 46252 | An element of the non-unit... |
| rngcbas 46253 | Set of objects of the cate... |
| rngchomfval 46254 | Set of arrows of the categ... |
| rngchom 46255 | Set of arrows of the categ... |
| elrngchom 46256 | A morphism of non-unital r... |
| rngchomfeqhom 46257 | The functionalized Hom-set... |
| rngccofval 46258 | Composition in the categor... |
| rngcco 46259 | Composition in the categor... |
| dfrngc2 46260 | Alternate definition of th... |
| rnghmsscmap2 46261 | The non-unital ring homomo... |
| rnghmsscmap 46262 | The non-unital ring homomo... |
| rnghmsubcsetclem1 46263 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 46264 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 46265 | The non-unital ring homomo... |
| rngccat 46266 | The category of non-unital... |
| rngcid 46267 | The identity arrow in the ... |
| rngcsect 46268 | A section in the category ... |
| rngcinv 46269 | An inverse in the category... |
| rngciso 46270 | An isomorphism in the cate... |
| rngcbasALTV 46271 | Set of objects of the cate... |
| rngchomfvalALTV 46272 | Set of arrows of the categ... |
| rngchomALTV 46273 | Set of arrows of the categ... |
| elrngchomALTV 46274 | A morphism of non-unital r... |
| rngccofvalALTV 46275 | Composition in the categor... |
| rngccoALTV 46276 | Composition in the categor... |
| rngccatidALTV 46277 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 46278 | The category of non-unital... |
| rngcidALTV 46279 | The identity arrow in the ... |
| rngcsectALTV 46280 | A section in the category ... |
| rngcinvALTV 46281 | An inverse in the category... |
| rngcisoALTV 46282 | An isomorphism in the cate... |
| rngchomffvalALTV 46283 | The value of the functiona... |
| rngchomrnghmresALTV 46284 | The value of the functiona... |
| rngcifuestrc 46285 | The "inclusion functor" fr... |
| funcrngcsetc 46286 | The "natural forgetful fun... |
| funcrngcsetcALT 46287 | Alternate proof of ~ funcr... |
| zrinitorngc 46288 | The zero ring is an initia... |
| zrtermorngc 46289 | The zero ring is a termina... |
| zrzeroorngc 46290 | The zero ring is a zero ob... |
| ringcvalALTV 46295 | Value of the category of r... |
| ringcval 46296 | Value of the category of u... |
| rhmresfn 46297 | The class of unital ring h... |
| rhmresel 46298 | An element of the unital r... |
| ringcbas 46299 | Set of objects of the cate... |
| ringchomfval 46300 | Set of arrows of the categ... |
| ringchom 46301 | Set of arrows of the categ... |
| elringchom 46302 | A morphism of unital rings... |
| ringchomfeqhom 46303 | The functionalized Hom-set... |
| ringccofval 46304 | Composition in the categor... |
| ringcco 46305 | Composition in the categor... |
| dfringc2 46306 | Alternate definition of th... |
| rhmsscmap2 46307 | The unital ring homomorphi... |
| rhmsscmap 46308 | The unital ring homomorphi... |
| rhmsubcsetclem1 46309 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 46310 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 46311 | The unital ring homomorphi... |
| ringccat 46312 | The category of unital rin... |
| ringcid 46313 | The identity arrow in the ... |
| rhmsscrnghm 46314 | The unital ring homomorphi... |
| rhmsubcrngclem1 46315 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 46316 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 46317 | The unital ring homomorphi... |
| rngcresringcat 46318 | The restriction of the cat... |
| ringcsect 46319 | A section in the category ... |
| ringcinv 46320 | An inverse in the category... |
| ringciso 46321 | An isomorphism in the cate... |
| ringcbasbas 46322 | An element of the base set... |
| funcringcsetc 46323 | The "natural forgetful fun... |
| funcringcsetcALTV2lem1 46324 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 46325 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 46326 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 46327 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 46328 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 46329 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 46330 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 46331 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 46332 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 46333 | The "natural forgetful fun... |
| ringcbasALTV 46334 | Set of objects of the cate... |
| ringchomfvalALTV 46335 | Set of arrows of the categ... |
| ringchomALTV 46336 | Set of arrows of the categ... |
| elringchomALTV 46337 | A morphism of rings is a f... |
| ringccofvalALTV 46338 | Composition in the categor... |
| ringccoALTV 46339 | Composition in the categor... |
| ringccatidALTV 46340 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 46341 | The category of rings is a... |
| ringcidALTV 46342 | The identity arrow in the ... |
| ringcsectALTV 46343 | A section in the category ... |
| ringcinvALTV 46344 | An inverse in the category... |
| ringcisoALTV 46345 | An isomorphism in the cate... |
| ringcbasbasALTV 46346 | An element of the base set... |
| funcringcsetclem1ALTV 46347 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 46348 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 46349 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 46350 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 46351 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 46352 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 46353 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 46354 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 46355 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 46356 | The "natural forgetful fun... |
| irinitoringc 46357 | The ring of integers is an... |
| zrtermoringc 46358 | The zero ring is a termina... |
| zrninitoringc 46359 | The zero ring is not an in... |
| nzerooringczr 46360 | There is no zero object in... |
| srhmsubclem1 46361 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 46362 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 46363 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 46364 | According to ~ df-subc , t... |
| sringcat 46365 | The restriction of the cat... |
| crhmsubc 46366 | According to ~ df-subc , t... |
| cringcat 46367 | The restriction of the cat... |
| drhmsubc 46368 | According to ~ df-subc , t... |
| drngcat 46369 | The restriction of the cat... |
| fldcat 46370 | The restriction of the cat... |
| fldc 46371 | The restriction of the cat... |
| fldhmsubc 46372 | According to ~ df-subc , t... |
| rngcrescrhm 46373 | The category of non-unital... |
| rhmsubclem1 46374 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 46375 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 46376 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 46377 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 46378 | According to ~ df-subc , t... |
| rhmsubccat 46379 | The restriction of the cat... |
| srhmsubcALTVlem1 46380 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 46381 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 46382 | According to ~ df-subc , t... |
| sringcatALTV 46383 | The restriction of the cat... |
| crhmsubcALTV 46384 | According to ~ df-subc , t... |
| cringcatALTV 46385 | The restriction of the cat... |
| drhmsubcALTV 46386 | According to ~ df-subc , t... |
| drngcatALTV 46387 | The restriction of the cat... |
| fldcatALTV 46388 | The restriction of the cat... |
| fldcALTV 46389 | The restriction of the cat... |
| fldhmsubcALTV 46390 | According to ~ df-subc , t... |
| rngcrescrhmALTV 46391 | The category of non-unital... |
| rhmsubcALTVlem1 46392 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 46393 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 46394 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 46395 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 46396 | According to ~ df-subc , t... |
| rhmsubcALTVcat 46397 | The restriction of the cat... |
| opeliun2xp 46398 | Membership of an ordered p... |
| eliunxp2 46399 | Membership in a union of C... |
| mpomptx2 46400 | Express a two-argument fun... |
| cbvmpox2 46401 | Rule to change the bound v... |
| dmmpossx2 46402 | The domain of a mapping is... |
| mpoexxg2 46403 | Existence of an operation ... |
| ovmpordxf 46404 | Value of an operation give... |
| ovmpordx 46405 | Value of an operation give... |
| ovmpox2 46406 | The value of an operation ... |
| fdmdifeqresdif 46407 | The restriction of a condi... |
| offvalfv 46408 | The function operation exp... |
| ofaddmndmap 46409 | The function operation app... |
| mapsnop 46410 | A singleton of an ordered ... |
| fprmappr 46411 | A function with a domain o... |
| mapprop 46412 | An unordered pair containi... |
| ztprmneprm 46413 | A prime is not an integer ... |
| 2t6m3t4e0 46414 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 46415 | For any finite subset of `... |
| nn0sumltlt 46416 | If the sum of two nonnegat... |
| bcpascm1 46417 | Pascal's rule for the bino... |
| altgsumbc 46418 | The sum of binomial coeffi... |
| altgsumbcALT 46419 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 46420 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 46421 | An element of the (base se... |
| zlmodzxz0 46422 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 46423 | The scalar multiplication ... |
| zlmodzxzadd 46424 | The addition of the ` ZZ `... |
| zlmodzxzsubm 46425 | The subtraction of the ` Z... |
| zlmodzxzsub 46426 | The subtraction of the ` Z... |
| mgpsumunsn 46427 | Extract a summand/factor f... |
| mgpsumz 46428 | If the group sum for the m... |
| mgpsumn 46429 | If the group sum for the m... |
| exple2lt6 46430 | A nonnegative integer to t... |
| pgrple2abl 46431 | Every symmetric group on a... |
| pgrpgt2nabl 46432 | Every symmetric group on a... |
| invginvrid 46433 | Identity for a multiplicat... |
| rmsupp0 46434 | The support of a mapping o... |
| domnmsuppn0 46435 | The support of a mapping o... |
| rmsuppss 46436 | The support of a mapping o... |
| mndpsuppss 46437 | The support of a mapping o... |
| scmsuppss 46438 | The support of a mapping o... |
| rmsuppfi 46439 | The support of a mapping o... |
| rmfsupp 46440 | A mapping of a multiplicat... |
| mndpsuppfi 46441 | The support of a mapping o... |
| mndpfsupp 46442 | A mapping of a scalar mult... |
| scmsuppfi 46443 | The support of a mapping o... |
| scmfsupp 46444 | A mapping of a scalar mult... |
| suppmptcfin 46445 | The support of a mapping w... |
| mptcfsupp 46446 | A mapping with value 0 exc... |
| fsuppmptdmf 46447 | A mapping with a finite do... |
| lmodvsmdi 46448 | Multiple distributive law ... |
| gsumlsscl 46449 | Closure of a group sum in ... |
| assaascl0 46450 | The scalar 0 embedded into... |
| assaascl1 46451 | The scalar 1 embedded into... |
| ply1vr1smo 46452 | The variable in a polynomi... |
| ply1ass23l 46453 | Associative identity with ... |
| ply1sclrmsm 46454 | The ring multiplication of... |
| coe1id 46455 | Coefficient vector of the ... |
| coe1sclmulval 46456 | The value of the coefficie... |
| ply1mulgsumlem1 46457 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 46458 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 46459 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 46460 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 46461 | The product of two polynom... |
| evl1at0 46462 | Polynomial evaluation for ... |
| evl1at1 46463 | Polynomial evaluation for ... |
| linply1 46464 | A term of the form ` x - C... |
| lineval 46465 | A term of the form ` x - C... |
| linevalexample 46466 | The polynomial ` x - 3 ` o... |
| dmatALTval 46471 | The algebra of ` N ` x ` N... |
| dmatALTbas 46472 | The base set of the algebr... |
| dmatALTbasel 46473 | An element of the base set... |
| dmatbas 46474 | The set of all ` N ` x ` N... |
| lincop 46479 | A linear combination as op... |
| lincval 46480 | The value of a linear comb... |
| dflinc2 46481 | Alternative definition of ... |
| lcoop 46482 | A linear combination as op... |
| lcoval 46483 | The value of a linear comb... |
| lincfsuppcl 46484 | A linear combination of ve... |
| linccl 46485 | A linear combination of ve... |
| lincval0 46486 | The value of an empty line... |
| lincvalsng 46487 | The linear combination ove... |
| lincvalsn 46488 | The linear combination ove... |
| lincvalpr 46489 | The linear combination ove... |
| lincval1 46490 | The linear combination ove... |
| lcosn0 46491 | Properties of a linear com... |
| lincvalsc0 46492 | The linear combination whe... |
| lcoc0 46493 | Properties of a linear com... |
| linc0scn0 46494 | If a set contains the zero... |
| lincdifsn 46495 | A vector is a linear combi... |
| linc1 46496 | A vector is a linear combi... |
| lincellss 46497 | A linear combination of a ... |
| lco0 46498 | The set of empty linear co... |
| lcoel0 46499 | The zero vector is always ... |
| lincsum 46500 | The sum of two linear comb... |
| lincscm 46501 | A linear combinations mult... |
| lincsumcl 46502 | The sum of two linear comb... |
| lincscmcl 46503 | The multiplication of a li... |
| lincsumscmcl 46504 | The sum of a linear combin... |
| lincolss 46505 | According to the statement... |
| ellcoellss 46506 | Every linear combination o... |
| lcoss 46507 | A set of vectors of a modu... |
| lspsslco 46508 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 46509 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 46510 | Equivalence of a _span_ of... |
| rellininds 46514 | The class defining the rel... |
| linindsv 46516 | The classes of the module ... |
| islininds 46517 | The property of being a li... |
| linindsi 46518 | The implications of being ... |
| linindslinci 46519 | The implications of being ... |
| islinindfis 46520 | The property of being a li... |
| islinindfiss 46521 | The property of being a li... |
| linindscl 46522 | A linearly independent set... |
| lindepsnlininds 46523 | A linearly dependent subse... |
| islindeps 46524 | The property of being a li... |
| lincext1 46525 | Property 1 of an extension... |
| lincext2 46526 | Property 2 of an extension... |
| lincext3 46527 | Property 3 of an extension... |
| lindslinindsimp1 46528 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 46529 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 46530 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 46531 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 46532 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 46533 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 46534 | Implication 2 for ~ lindsl... |
| lindslininds 46535 | Equivalence of definitions... |
| linds0 46536 | The empty set is always a ... |
| el0ldep 46537 | A set containing the zero ... |
| el0ldepsnzr 46538 | A set containing the zero ... |
| lindsrng01 46539 | Any subset of a module is ... |
| lindszr 46540 | Any subset of a module ove... |
| snlindsntorlem 46541 | Lemma for ~ snlindsntor . ... |
| snlindsntor 46542 | A singleton is linearly in... |
| ldepsprlem 46543 | Lemma for ~ ldepspr . (Co... |
| ldepspr 46544 | If a vector is a scalar mu... |
| lincresunit3lem3 46545 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 46546 | Lemma 1 for properties of ... |
| lincresunitlem2 46547 | Lemma for properties of a ... |
| lincresunit1 46548 | Property 1 of a specially ... |
| lincresunit2 46549 | Property 2 of a specially ... |
| lincresunit3lem1 46550 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 46551 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 46552 | Property 3 of a specially ... |
| lincreslvec3 46553 | Property 3 of a specially ... |
| islindeps2 46554 | Conditions for being a lin... |
| islininds2 46555 | Implication of being a lin... |
| isldepslvec2 46556 | Alternative definition of ... |
| lindssnlvec 46557 | A singleton not containing... |
| lmod1lem1 46558 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 46559 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 46560 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 46561 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 46562 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 46563 | The (smallest) structure r... |
| lmod1zr 46564 | The (smallest) structure r... |
| lmod1zrnlvec 46565 | There is a (left) module (... |
| lmodn0 46566 | Left modules exist. (Cont... |
| zlmodzxzequa 46567 | Example of an equation wit... |
| zlmodzxznm 46568 | Example of a linearly depe... |
| zlmodzxzldeplem 46569 | A and B are not equal. (C... |
| zlmodzxzequap 46570 | Example of an equation wit... |
| zlmodzxzldeplem1 46571 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 46572 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 46573 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 46574 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 46575 | { A , B } is a linearly de... |
| ldepsnlinclem1 46576 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 46577 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 46578 | The class of all (left) ve... |
| ldepsnlinc 46579 | The reverse implication of... |
| ldepslinc 46580 | For (left) vector spaces, ... |
| suppdm 46581 | If the range of a function... |
| eluz2cnn0n1 46582 | An integer greater than 1 ... |
| divge1b 46583 | The ratio of a real number... |
| divgt1b 46584 | The ratio of a real number... |
| ltsubaddb 46585 | Equivalence for the "less ... |
| ltsubsubb 46586 | Equivalence for the "less ... |
| ltsubadd2b 46587 | Equivalence for the "less ... |
| divsub1dir 46588 | Distribution of division o... |
| expnegico01 46589 | An integer greater than 1 ... |
| elfzolborelfzop1 46590 | An element of a half-open ... |
| pw2m1lepw2m1 46591 | 2 to the power of a positi... |
| zgtp1leeq 46592 | If an integer is between a... |
| flsubz 46593 | An integer can be moved in... |
| fldivmod 46594 | Expressing the floor of a ... |
| mod0mul 46595 | If an integer is 0 modulo ... |
| modn0mul 46596 | If an integer is not 0 mod... |
| m1modmmod 46597 | An integer decreased by 1 ... |
| difmodm1lt 46598 | The difference between an ... |
| nn0onn0ex 46599 | For each odd nonnegative i... |
| nn0enn0ex 46600 | For each even nonnegative ... |
| nnennex 46601 | For each even positive int... |
| nneop 46602 | A positive integer is even... |
| nneom 46603 | A positive integer is even... |
| nn0eo 46604 | A nonnegative integer is e... |
| nnpw2even 46605 | 2 to the power of a positi... |
| zefldiv2 46606 | The floor of an even integ... |
| zofldiv2 46607 | The floor of an odd intege... |
| nn0ofldiv2 46608 | The floor of an odd nonneg... |
| flnn0div2ge 46609 | The floor of a positive in... |
| flnn0ohalf 46610 | The floor of the half of a... |
| logcxp0 46611 | Logarithm of a complex pow... |
| regt1loggt0 46612 | The natural logarithm for ... |
| fdivval 46615 | The quotient of two functi... |
| fdivmpt 46616 | The quotient of two functi... |
| fdivmptf 46617 | The quotient of two functi... |
| refdivmptf 46618 | The quotient of two functi... |
| fdivpm 46619 | The quotient of two functi... |
| refdivpm 46620 | The quotient of two functi... |
| fdivmptfv 46621 | The function value of a qu... |
| refdivmptfv 46622 | The function value of a qu... |
| bigoval 46625 | Set of functions of order ... |
| elbigofrcl 46626 | Reverse closure of the "bi... |
| elbigo 46627 | Properties of a function o... |
| elbigo2 46628 | Properties of a function o... |
| elbigo2r 46629 | Sufficient condition for a... |
| elbigof 46630 | A function of order G(x) i... |
| elbigodm 46631 | The domain of a function o... |
| elbigoimp 46632 | The defining property of a... |
| elbigolo1 46633 | A function (into the posit... |
| rege1logbrege0 46634 | The general logarithm, wit... |
| rege1logbzge0 46635 | The general logarithm, wit... |
| fllogbd 46636 | A real number is between t... |
| relogbmulbexp 46637 | The logarithm of the produ... |
| relogbdivb 46638 | The logarithm of the quoti... |
| logbge0b 46639 | The logarithm of a number ... |
| logblt1b 46640 | The logarithm of a number ... |
| fldivexpfllog2 46641 | The floor of a positive re... |
| nnlog2ge0lt1 46642 | A positive integer is 1 if... |
| logbpw2m1 46643 | The floor of the binary lo... |
| fllog2 46644 | The floor of the binary lo... |
| blenval 46647 | The binary length of an in... |
| blen0 46648 | The binary length of 0. (... |
| blenn0 46649 | The binary length of a "nu... |
| blenre 46650 | The binary length of a pos... |
| blennn 46651 | The binary length of a pos... |
| blennnelnn 46652 | The binary length of a pos... |
| blennn0elnn 46653 | The binary length of a non... |
| blenpw2 46654 | The binary length of a pow... |
| blenpw2m1 46655 | The binary length of a pow... |
| nnpw2blen 46656 | A positive integer is betw... |
| nnpw2blenfzo 46657 | A positive integer is betw... |
| nnpw2blenfzo2 46658 | A positive integer is eith... |
| nnpw2pmod 46659 | Every positive integer can... |
| blen1 46660 | The binary length of 1. (... |
| blen2 46661 | The binary length of 2. (... |
| nnpw2p 46662 | Every positive integer can... |
| nnpw2pb 46663 | A number is a positive int... |
| blen1b 46664 | The binary length of a non... |
| blennnt2 46665 | The binary length of a pos... |
| nnolog2flm1 46666 | The floor of the binary lo... |
| blennn0em1 46667 | The binary length of the h... |
| blennngt2o2 46668 | The binary length of an od... |
| blengt1fldiv2p1 46669 | The binary length of an in... |
| blennn0e2 46670 | The binary length of an ev... |
| digfval 46673 | Operation to obtain the ` ... |
| digval 46674 | The ` K ` th digit of a no... |
| digvalnn0 46675 | The ` K ` th digit of a no... |
| nn0digval 46676 | The ` K ` th digit of a no... |
| dignn0fr 46677 | The digits of the fraction... |
| dignn0ldlem 46678 | Lemma for ~ dignnld . (Co... |
| dignnld 46679 | The leading digits of a po... |
| dig2nn0ld 46680 | The leading digits of a po... |
| dig2nn1st 46681 | The first (relevant) digit... |
| dig0 46682 | All digits of 0 are 0. (C... |
| digexp 46683 | The ` K ` th digit of a po... |
| dig1 46684 | All but one digits of 1 ar... |
| 0dig1 46685 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 46686 | The integers 0 and 1 corre... |
| dig2nn0 46687 | A digit of a nonnegative i... |
| 0dig2nn0e 46688 | The last bit of an even in... |
| 0dig2nn0o 46689 | The last bit of an odd int... |
| dig2bits 46690 | The ` K ` th digit of a no... |
| dignn0flhalflem1 46691 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 46692 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 46693 | The digits of the half of ... |
| dignn0flhalf 46694 | The digits of the rounded ... |
| nn0sumshdiglemA 46695 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 46696 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 46697 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 46698 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 46699 | A nonnegative integer can ... |
| nn0mulfsum 46700 | Trivial algorithm to calcu... |
| nn0mullong 46701 | Standard algorithm (also k... |
| naryfval 46704 | The set of the n-ary (endo... |
| naryfvalixp 46705 | The set of the n-ary (endo... |
| naryfvalel 46706 | An n-ary (endo)function on... |
| naryrcl 46707 | Reverse closure for n-ary ... |
| naryfvalelfv 46708 | The value of an n-ary (end... |
| naryfvalelwrdf 46709 | An n-ary (endo)function on... |
| 0aryfvalel 46710 | A nullary (endo)function o... |
| 0aryfvalelfv 46711 | The value of a nullary (en... |
| 1aryfvalel 46712 | A unary (endo)function on ... |
| fv1arycl 46713 | Closure of a unary (endo)f... |
| 1arympt1 46714 | A unary (endo)function in ... |
| 1arympt1fv 46715 | The value of a unary (endo... |
| 1arymaptfv 46716 | The value of the mapping o... |
| 1arymaptf 46717 | The mapping of unary (endo... |
| 1arymaptf1 46718 | The mapping of unary (endo... |
| 1arymaptfo 46719 | The mapping of unary (endo... |
| 1arymaptf1o 46720 | The mapping of unary (endo... |
| 1aryenef 46721 | The set of unary (endo)fun... |
| 1aryenefmnd 46722 | The set of unary (endo)fun... |
| 2aryfvalel 46723 | A binary (endo)function on... |
| fv2arycl 46724 | Closure of a binary (endo)... |
| 2arympt 46725 | A binary (endo)function in... |
| 2arymptfv 46726 | The value of a binary (end... |
| 2arymaptfv 46727 | The value of the mapping o... |
| 2arymaptf 46728 | The mapping of binary (end... |
| 2arymaptf1 46729 | The mapping of binary (end... |
| 2arymaptfo 46730 | The mapping of binary (end... |
| 2arymaptf1o 46731 | The mapping of binary (end... |
| 2aryenef 46732 | The set of binary (endo)fu... |
| itcoval 46737 | The value of the function ... |
| itcoval0 46738 | A function iterated zero t... |
| itcoval1 46739 | A function iterated once. ... |
| itcoval2 46740 | A function iterated twice.... |
| itcoval3 46741 | A function iterated three ... |
| itcoval0mpt 46742 | A mapping iterated zero ti... |
| itcovalsuc 46743 | The value of the function ... |
| itcovalsucov 46744 | The value of the function ... |
| itcovalendof 46745 | The n-th iterate of an end... |
| itcovalpclem1 46746 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 46747 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 46748 | The value of the function ... |
| itcovalt2lem2lem1 46749 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 46750 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 46751 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 46752 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 46753 | The value of the function ... |
| ackvalsuc1mpt 46754 | The Ackermann function at ... |
| ackvalsuc1 46755 | The Ackermann function at ... |
| ackval0 46756 | The Ackermann function at ... |
| ackval1 46757 | The Ackermann function at ... |
| ackval2 46758 | The Ackermann function at ... |
| ackval3 46759 | The Ackermann function at ... |
| ackendofnn0 46760 | The Ackermann function at ... |
| ackfnnn0 46761 | The Ackermann function at ... |
| ackval0val 46762 | The Ackermann function at ... |
| ackvalsuc0val 46763 | The Ackermann function at ... |
| ackvalsucsucval 46764 | The Ackermann function at ... |
| ackval0012 46765 | The Ackermann function at ... |
| ackval1012 46766 | The Ackermann function at ... |
| ackval2012 46767 | The Ackermann function at ... |
| ackval3012 46768 | The Ackermann function at ... |
| ackval40 46769 | The Ackermann function at ... |
| ackval41a 46770 | The Ackermann function at ... |
| ackval41 46771 | The Ackermann function at ... |
| ackval42 46772 | The Ackermann function at ... |
| ackval42a 46773 | The Ackermann function at ... |
| ackval50 46774 | The Ackermann function at ... |
| fv1prop 46775 | The function value of unor... |
| fv2prop 46776 | The function value of unor... |
| submuladdmuld 46777 | Transformation of a sum of... |
| affinecomb1 46778 | Combination of two real af... |
| affinecomb2 46779 | Combination of two real af... |
| affineid 46780 | Identity of an affine comb... |
| 1subrec1sub 46781 | Subtract the reciprocal of... |
| resum2sqcl 46782 | The sum of two squares of ... |
| resum2sqgt0 46783 | The sum of the square of a... |
| resum2sqrp 46784 | The sum of the square of a... |
| resum2sqorgt0 46785 | The sum of the square of t... |
| reorelicc 46786 | Membership in and outside ... |
| rrx2pxel 46787 | The x-coordinate of a poin... |
| rrx2pyel 46788 | The y-coordinate of a poin... |
| prelrrx2 46789 | An unordered pair of order... |
| prelrrx2b 46790 | An unordered pair of order... |
| rrx2pnecoorneor 46791 | If two different points ` ... |
| rrx2pnedifcoorneor 46792 | If two different points ` ... |
| rrx2pnedifcoorneorr 46793 | If two different points ` ... |
| rrx2xpref1o 46794 | There is a bijection betwe... |
| rrx2xpreen 46795 | The set of points in the t... |
| rrx2plord 46796 | The lexicographical orderi... |
| rrx2plord1 46797 | The lexicographical orderi... |
| rrx2plord2 46798 | The lexicographical orderi... |
| rrx2plordisom 46799 | The set of points in the t... |
| rrx2plordso 46800 | The lexicographical orderi... |
| ehl2eudisval0 46801 | The Euclidean distance of ... |
| ehl2eudis0lt 46802 | An upper bound of the Eucl... |
| lines 46807 | The lines passing through ... |
| line 46808 | The line passing through t... |
| rrxlines 46809 | Definition of lines passin... |
| rrxline 46810 | The line passing through t... |
| rrxlinesc 46811 | Definition of lines passin... |
| rrxlinec 46812 | The line passing through t... |
| eenglngeehlnmlem1 46813 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 46814 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 46815 | The line definition in the... |
| rrx2line 46816 | The line passing through t... |
| rrx2vlinest 46817 | The vertical line passing ... |
| rrx2linest 46818 | The line passing through t... |
| rrx2linesl 46819 | The line passing through t... |
| rrx2linest2 46820 | The line passing through t... |
| elrrx2linest2 46821 | The line passing through t... |
| spheres 46822 | The spheres for given cent... |
| sphere 46823 | A sphere with center ` X `... |
| rrxsphere 46824 | The sphere with center ` M... |
| 2sphere 46825 | The sphere with center ` M... |
| 2sphere0 46826 | The sphere around the orig... |
| line2ylem 46827 | Lemma for ~ line2y . This... |
| line2 46828 | Example for a line ` G ` p... |
| line2xlem 46829 | Lemma for ~ line2x . This... |
| line2x 46830 | Example for a horizontal l... |
| line2y 46831 | Example for a vertical lin... |
| itsclc0lem1 46832 | Lemma for theorems about i... |
| itsclc0lem2 46833 | Lemma for theorems about i... |
| itsclc0lem3 46834 | Lemma for theorems about i... |
| itscnhlc0yqe 46835 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 46836 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 46837 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 46838 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 46839 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 46840 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 46841 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 46842 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 46843 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 46844 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 46845 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 46846 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 46847 | The intersection points of... |
| itsclc0b 46848 | The intersection points of... |
| itsclinecirc0 46849 | The intersection points of... |
| itsclinecirc0b 46850 | The intersection points of... |
| itsclinecirc0in 46851 | The intersection points of... |
| itsclquadb 46852 | Quadratic equation for the... |
| itsclquadeu 46853 | Quadratic equation for the... |
| 2itscplem1 46854 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 46855 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 46856 | Lemma D for ~ 2itscp . (C... |
| 2itscp 46857 | A condition for a quadrati... |
| itscnhlinecirc02plem1 46858 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 46859 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 46860 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 46861 | Intersection of a nonhoriz... |
| inlinecirc02plem 46862 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 46863 | Intersection of a line wit... |
| inlinecirc02preu 46864 | Intersection of a line wit... |
| pm4.71da 46865 | Deduction converting a bic... |
| logic1 46866 | Distribution of implicatio... |
| logic1a 46867 | Variant of ~ logic1 . (Co... |
| logic2 46868 | Variant of ~ logic1 . (Co... |
| pm5.32dav 46869 | Distribution of implicatio... |
| pm5.32dra 46870 | Reverse distribution of im... |
| exp12bd 46871 | The import-export theorem ... |
| mpbiran3d 46872 | Equivalence with a conjunc... |
| mpbiran4d 46873 | Equivalence with a conjunc... |
| dtrucor3 46874 | An example of how ~ ax-5 w... |
| ralbidb 46875 | Formula-building rule for ... |
| ralbidc 46876 | Formula-building rule for ... |
| r19.41dv 46877 | A complex deduction form o... |
| rspceb2dv 46878 | Restricted existential spe... |
| rmotru 46879 | Two ways of expressing "at... |
| reutru 46880 | Two ways of expressing "ex... |
| reutruALT 46881 | Alternate proof for ~ reut... |
| ssdisjd 46882 | Subset preserves disjointn... |
| ssdisjdr 46883 | Subset preserves disjointn... |
| disjdifb 46884 | Relative complement is ant... |
| predisj 46885 | Preimages of disjoint sets... |
| vsn 46886 | The singleton of the unive... |
| mosn 46887 | "At most one" element in a... |
| mo0 46888 | "At most one" element in a... |
| mosssn 46889 | "At most one" element in a... |
| mo0sn 46890 | Two ways of expressing "at... |
| mosssn2 46891 | Two ways of expressing "at... |
| unilbss 46892 | Superclass of the greatest... |
| inpw 46893 | Two ways of expressing a c... |
| mof0 46894 | There is at most one funct... |
| mof02 46895 | A variant of ~ mof0 . (Co... |
| mof0ALT 46896 | Alternate proof for ~ mof0... |
| eufsnlem 46897 | There is exactly one funct... |
| eufsn 46898 | There is exactly one funct... |
| eufsn2 46899 | There is exactly one funct... |
| mofsn 46900 | There is at most one funct... |
| mofsn2 46901 | There is at most one funct... |
| mofsssn 46902 | There is at most one funct... |
| mofmo 46903 | There is at most one funct... |
| mofeu 46904 | The uniqueness of a functi... |
| elfvne0 46905 | If a function value has a ... |
| fdomne0 46906 | A function with non-empty ... |
| f1sn2g 46907 | A function that maps a sin... |
| f102g 46908 | A function that maps the e... |
| f1mo 46909 | A function that maps a set... |
| f002 46910 | A function with an empty c... |
| map0cor 46911 | A function exists iff an e... |
| fvconstr 46912 | Two ways of expressing ` A... |
| fvconstrn0 46913 | Two ways of expressing ` A... |
| fvconstr2 46914 | Two ways of expressing ` A... |
| fvconst0ci 46915 | A constant function's valu... |
| fvconstdomi 46916 | A constant function's valu... |
| f1omo 46917 | There is at most one eleme... |
| f1omoALT 46918 | There is at most one eleme... |
| iccin 46919 | Intersection of two closed... |
| iccdisj2 46920 | If the upper bound of one ... |
| iccdisj 46921 | If the upper bound of one ... |
| mreuniss 46922 | The union of a collection ... |
| clduni 46923 | The union of closed sets i... |
| opncldeqv 46924 | Conditions on open sets ar... |
| opndisj 46925 | Two ways of saying that tw... |
| clddisj 46926 | Two ways of saying that tw... |
| neircl 46927 | Reverse closure of the nei... |
| opnneilem 46928 | Lemma factoring out common... |
| opnneir 46929 | If something is true for a... |
| opnneirv 46930 | A variant of ~ opnneir wit... |
| opnneilv 46931 | The converse of ~ opnneir ... |
| opnneil 46932 | A variant of ~ opnneilv . ... |
| opnneieqv 46933 | The equivalence between ne... |
| opnneieqvv 46934 | The equivalence between ne... |
| restcls2lem 46935 | A closed set in a subspace... |
| restcls2 46936 | A closed set in a subspace... |
| restclsseplem 46937 | Lemma for ~ restclssep . ... |
| restclssep 46938 | Two disjoint closed sets i... |
| cnneiima 46939 | Given a continuous functio... |
| iooii 46940 | Open intervals are open se... |
| icccldii 46941 | Closed intervals are close... |
| i0oii 46942 | ` ( 0 [,) A ) ` is open in... |
| io1ii 46943 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 46944 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 46945 | Open neighborhood and neig... |
| sepnsepo 46946 | Open neighborhood and neig... |
| sepdisj 46947 | Separated sets are disjoin... |
| seposep 46948 | If two sets are separated ... |
| sepcsepo 46949 | If two sets are separated ... |
| sepfsepc 46950 | If two sets are separated ... |
| seppsepf 46951 | If two sets are precisely ... |
| seppcld 46952 | If two sets are precisely ... |
| isnrm4 46953 | A topological space is nor... |
| dfnrm2 46954 | A topological space is nor... |
| dfnrm3 46955 | A topological space is nor... |
| iscnrm3lem1 46956 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 46957 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem3 46958 | Lemma for ~ iscnrm3lem4 . ... |
| iscnrm3lem4 46959 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 46960 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 46961 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 46962 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 46963 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 46964 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 46965 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 46966 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 46967 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 46968 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 46969 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 46970 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 46971 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 46972 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 46973 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 46974 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 46975 | A completely normal topolo... |
| iscnrm3v 46976 | A topology is completely n... |
| iscnrm4 46977 | A completely normal topolo... |
| isprsd 46978 | Property of being a preord... |
| lubeldm2 46979 | Member of the domain of th... |
| glbeldm2 46980 | Member of the domain of th... |
| lubeldm2d 46981 | Member of the domain of th... |
| glbeldm2d 46982 | Member of the domain of th... |
| lubsscl 46983 | If a subset of ` S ` conta... |
| glbsscl 46984 | If a subset of ` S ` conta... |
| lubprlem 46985 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 46986 | The set of two comparable ... |
| lubpr 46987 | The LUB of the set of two ... |
| glbprlem 46988 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 46989 | The set of two comparable ... |
| glbpr 46990 | The GLB of the set of two ... |
| joindm2 46991 | The join of any two elemen... |
| joindm3 46992 | The join of any two elemen... |
| meetdm2 46993 | The meet of any two elemen... |
| meetdm3 46994 | The meet of any two elemen... |
| posjidm 46995 | Poset join is idempotent. ... |
| posmidm 46996 | Poset meet is idempotent. ... |
| toslat 46997 | A toset is a lattice. (Co... |
| isclatd 46998 | The predicate "is a comple... |
| intubeu 46999 | Existential uniqueness of ... |
| unilbeu 47000 | Existential uniqueness of ... |
| ipolublem 47001 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 47002 | The domain of the LUB of t... |
| ipolub 47003 | The LUB of the inclusion p... |
| ipoglblem 47004 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 47005 | The domain of the GLB of t... |
| ipoglb 47006 | The GLB of the inclusion p... |
| ipolub0 47007 | The LUB of the empty set i... |
| ipolub00 47008 | The LUB of the empty set i... |
| ipoglb0 47009 | The GLB of the empty set i... |
| mrelatlubALT 47010 | Least upper bounds in a Mo... |
| mrelatglbALT 47011 | Greatest lower bounds in a... |
| mreclat 47012 | A Moore space is a complet... |
| topclat 47013 | A topology is a complete l... |
| toplatglb0 47014 | The empty intersection in ... |
| toplatlub 47015 | Least upper bounds in a to... |
| toplatglb 47016 | Greatest lower bounds in a... |
| toplatjoin 47017 | Joins in a topology are re... |
| toplatmeet 47018 | Meets in a topology are re... |
| topdlat 47019 | A topology is a distributi... |
| catprslem 47020 | Lemma for ~ catprs . (Con... |
| catprs 47021 | A preorder can be extracte... |
| catprs2 47022 | A category equipped with t... |
| catprsc 47023 | A construction of the preo... |
| catprsc2 47024 | An alternate construction ... |
| endmndlem 47025 | A diagonal hom-set in a ca... |
| idmon 47026 | An identity arrow, or an i... |
| idepi 47027 | An identity arrow, or an i... |
| funcf2lem 47028 | A utility theorem for prov... |
| isthinc 47031 | The predicate "is a thin c... |
| isthinc2 47032 | A thin category is a categ... |
| isthinc3 47033 | A thin category is a categ... |
| thincc 47034 | A thin category is a categ... |
| thinccd 47035 | A thin category is a categ... |
| thincssc 47036 | A thin category is a categ... |
| isthincd2lem1 47037 | Lemma for ~ isthincd2 and ... |
| thincmo2 47038 | Morphisms in the same hom-... |
| thincmo 47039 | There is at most one morph... |
| thincmoALT 47040 | Alternate proof for ~ thin... |
| thincmod 47041 | At most one morphism in ea... |
| thincn0eu 47042 | In a thin category, a hom-... |
| thincid 47043 | In a thin category, a morp... |
| thincmon 47044 | In a thin category, all mo... |
| thincepi 47045 | In a thin category, all mo... |
| isthincd2lem2 47046 | Lemma for ~ isthincd2 . (... |
| isthincd 47047 | The predicate "is a thin c... |
| isthincd2 47048 | The predicate " ` C ` is a... |
| oppcthin 47049 | The opposite category of a... |
| subthinc 47050 | A subcategory of a thin ca... |
| functhinclem1 47051 | Lemma for ~ functhinc . G... |
| functhinclem2 47052 | Lemma for ~ functhinc . (... |
| functhinclem3 47053 | Lemma for ~ functhinc . T... |
| functhinclem4 47054 | Lemma for ~ functhinc . O... |
| functhinc 47055 | A functor to a thin catego... |
| fullthinc 47056 | A functor to a thin catego... |
| fullthinc2 47057 | A full functor to a thin c... |
| thincfth 47058 | A functor from a thin cate... |
| thincciso 47059 | Two thin categories are is... |
| 0thincg 47060 | Any structure with an empt... |
| 0thinc 47061 | The empty category (see ~ ... |
| indthinc 47062 | An indiscrete category in ... |
| indthincALT 47063 | An alternate proof for ~ i... |
| prsthinc 47064 | Preordered sets as categor... |
| setcthin 47065 | A category of sets all of ... |
| setc2othin 47066 | The category ` ( SetCat ``... |
| thincsect 47067 | In a thin category, one mo... |
| thincsect2 47068 | In a thin category, ` F ` ... |
| thincinv 47069 | In a thin category, ` F ` ... |
| thinciso 47070 | In a thin category, ` F : ... |
| thinccic 47071 | In a thin category, two ob... |
| prstcval 47074 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 47075 | Lemma for ~ prstcnid and ~... |
| prstcnid 47076 | Components other than ` Ho... |
| prstcbas 47077 | The base set is unchanged.... |
| prstcleval 47078 | Value of the less-than-or-... |
| prstclevalOLD 47079 | Obsolete proof of ~ prstcl... |
| prstcle 47080 | Value of the less-than-or-... |
| prstcocval 47081 | Orthocomplementation is un... |
| prstcocvalOLD 47082 | Obsolete proof of ~ prstco... |
| prstcoc 47083 | Orthocomplementation is un... |
| prstchomval 47084 | Hom-sets of the constructe... |
| prstcprs 47085 | The category is a preorder... |
| prstcthin 47086 | The preordered set is equi... |
| prstchom 47087 | Hom-sets of the constructe... |
| prstchom2 47088 | Hom-sets of the constructe... |
| prstchom2ALT 47089 | Hom-sets of the constructe... |
| postcpos 47090 | The converted category is ... |
| postcposALT 47091 | Alternate proof for ~ post... |
| postc 47092 | The converted category is ... |
| mndtcval 47095 | Value of the category buil... |
| mndtcbasval 47096 | The base set of the catego... |
| mndtcbas 47097 | The category built from a ... |
| mndtcob 47098 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 47099 | Two objects in a category ... |
| mndtchom 47100 | The only hom-set of the ca... |
| mndtcco 47101 | The composition of the cat... |
| mndtcco2 47102 | The composition of the cat... |
| mndtccatid 47103 | Lemma for ~ mndtccat and ~... |
| mndtccat 47104 | The function value is a ca... |
| mndtcid 47105 | The identity morphism, or ... |
| grptcmon 47106 | All morphisms in a categor... |
| grptcepi 47107 | All morphisms in a categor... |
| nfintd 47108 | Bound-variable hypothesis ... |
| nfiund 47109 | Bound-variable hypothesis ... |
| nfiundg 47110 | Bound-variable hypothesis ... |
| iunord 47111 | The indexed union of a col... |
| iunordi 47112 | The indexed union of a col... |
| spd 47113 | Specialization deduction, ... |
| spcdvw 47114 | A version of ~ spcdv where... |
| tfis2d 47115 | Transfinite Induction Sche... |
| bnd2d 47116 | Deduction form of ~ bnd2 .... |
| dffun3f 47117 | Alternate definition of fu... |
| setrecseq 47120 | Equality theorem for set r... |
| nfsetrecs 47121 | Bound-variable hypothesis ... |
| setrec1lem1 47122 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 47123 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 47124 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 47125 | Lemma for ~ setrec1 . If ... |
| setrec1 47126 | This is the first of two f... |
| setrec2fun 47127 | This is the second of two ... |
| setrec2lem1 47128 | Lemma for ~ setrec2 . The... |
| setrec2lem2 47129 | Lemma for ~ setrec2 . The... |
| setrec2 47130 | This is the second of two ... |
| setrec2v 47131 | Version of ~ setrec2 with ... |
| setrec2mpt 47132 | Version of ~ setrec2 where... |
| setis 47133 | Version of ~ setrec2 expre... |
| elsetrecslem 47134 | Lemma for ~ elsetrecs . A... |
| elsetrecs 47135 | A set ` A ` is an element ... |
| setrecsss 47136 | The ` setrecs ` operator r... |
| setrecsres 47137 | A recursively generated cl... |
| vsetrec 47138 | Construct ` _V ` using set... |
| 0setrec 47139 | If a function sends the em... |
| onsetreclem1 47140 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 47141 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 47142 | Lemma for ~ onsetrec . (C... |
| onsetrec 47143 | Construct ` On ` using set... |
| elpglem1 47146 | Lemma for ~ elpg . (Contr... |
| elpglem2 47147 | Lemma for ~ elpg . (Contr... |
| elpglem3 47148 | Lemma for ~ elpg . (Contr... |
| elpg 47149 | Membership in the class of... |
| pgindlem 47150 | Lemma for ~ pgind . (Cont... |
| pgindnf 47151 | Version of ~ pgind with ex... |
| pgind 47152 | Induction on partizan game... |
| sbidd 47153 | An identity theorem for su... |
| sbidd-misc 47154 | An identity theorem for su... |
| gte-lte 47159 | Simple relationship betwee... |
| gt-lt 47160 | Simple relationship betwee... |
| gte-lteh 47161 | Relationship between ` <_ ... |
| gt-lth 47162 | Relationship between ` < `... |
| ex-gt 47163 | Simple example of ` > ` , ... |
| ex-gte 47164 | Simple example of ` >_ ` ,... |
| sinhval-named 47171 | Value of the named sinh fu... |
| coshval-named 47172 | Value of the named cosh fu... |
| tanhval-named 47173 | Value of the named tanh fu... |
| sinh-conventional 47174 | Conventional definition of... |
| sinhpcosh 47175 | Prove that ` ( sinh `` A )... |
| secval 47182 | Value of the secant functi... |
| cscval 47183 | Value of the cosecant func... |
| cotval 47184 | Value of the cotangent fun... |
| seccl 47185 | The closure of the secant ... |
| csccl 47186 | The closure of the cosecan... |
| cotcl 47187 | The closure of the cotange... |
| reseccl 47188 | The closure of the secant ... |
| recsccl 47189 | The closure of the cosecan... |
| recotcl 47190 | The closure of the cotange... |
| recsec 47191 | The reciprocal of secant i... |
| reccsc 47192 | The reciprocal of cosecant... |
| reccot 47193 | The reciprocal of cotangen... |
| rectan 47194 | The reciprocal of tangent ... |
| sec0 47195 | The value of the secant fu... |
| onetansqsecsq 47196 | Prove the tangent squared ... |
| cotsqcscsq 47197 | Prove the tangent squared ... |
| ifnmfalse 47198 | If A is not a member of B,... |
| logb2aval 47199 | Define the value of the ` ... |
| comraddi 47206 | Commute RHS addition. See... |
| mvlraddi 47207 | Move the right term in a s... |
| mvrladdi 47208 | Move the left term in a su... |
| assraddsubi 47209 | Associate RHS addition-sub... |
| joinlmuladdmuli 47210 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 47211 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 47212 | Join AB-CB into (A-C) on L... |
| mvlrmuld 47213 | Move the right term in a p... |
| mvlrmuli 47214 | Move the right term in a p... |
| i2linesi 47215 | Solve for the intersection... |
| i2linesd 47216 | Solve for the intersection... |
| alimp-surprise 47217 | Demonstrate that when usin... |
| alimp-no-surprise 47218 | There is no "surprise" in ... |
| empty-surprise 47219 | Demonstrate that when usin... |
| empty-surprise2 47220 | "Prove" that false is true... |
| eximp-surprise 47221 | Show what implication insi... |
| eximp-surprise2 47222 | Show that "there exists" w... |
| alsconv 47227 | There is an equivalence be... |
| alsi1d 47228 | Deduction rule: Given "al... |
| alsi2d 47229 | Deduction rule: Given "al... |
| alsc1d 47230 | Deduction rule: Given "al... |
| alsc2d 47231 | Deduction rule: Given "al... |
| alscn0d 47232 | Deduction rule: Given "al... |
| alsi-no-surprise 47233 | Demonstrate that there is ... |
| 5m4e1 47234 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 47235 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 47236 | Resolution rule. This is ... |
| testable 47237 | In classical logic all wff... |
| aacllem 47238 | Lemma for other theorems a... |
| amgmwlem 47239 | Weighted version of ~ amgm... |
| amgmlemALT 47240 | Alternate proof of ~ amgml... |
| amgmw2d 47241 | Weighted arithmetic-geomet... |
| young2d 47242 | Young's inequality for ` n... |
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